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# Gröbner-Shirshov basis for the finitely presented algebras defined by
permutation relations of symmetric type111Supported by the NNSF of China
(11171118), the Research Fund for the Doctoral Program of Higher Education of
China (20114407110007), the NSF of Guangdong Province
(S2011010003374,S2012040007369), the Program on International Cooperation and
Innovation, Department of Education, Guangdong Province (2012gjhz0007), and
the NSF of Zhanjiang Normal University (QL0902).
Jianjun Qiu
Mathematics and Computational Science School, Zhanjiang Normal University
Zhanjiang 524048, China
[email protected]
Yuqun Chen222Corresponding author.
School of Mathematical Sciences, South China Normal University
Guangzhou 510631, China
[email protected]
###### Abstract
In this paper, we give a Gröbner-Shirshov basis for the finitely presented
semigroup algebra $\mathbf{k}[S_{n}(Sym_{n})]$ defined by permutation
relations of symmetric type. As an application, by the Composition-Diamond
Lemma, we obtain normal forms of elements of momoid $S_{n}(Sym_{n})$, which
gives an answer to an open problem posted by F. Cedó, E. Jespers and J.
Okniński [7] for the symmetric group case.
AMS Mathematics Subject Classification (2000): 16S15, 16S35, 20M25.
Keywords: Gröbner-Shirshov basis, finitely presented, normal form, semigroup
algebra.
## 1 Introduction
Let $Sym_{n}$ be the symmetric group of degree $n$ and $H$ a subset of
$Sym_{n}$. Recently, F. Cedó, E. Jespers and J. Okniński [7] introduced a new
class of finitely presented semigroup algebra $\mathbf{k}[S_{n}(H)]$ over a
field $\mathbf{k}$, where the monoid $S_{n}(H)$ is defined by a set of
generators $x_{1},x_{2},\ldots,x_{n}$ and homogenous permutation relations,
i.e.
$S_{n}(H)=\langle x_{1},x_{2},\ldots,x_{n}|x_{\sigma(1)}x_{\sigma(2)}\cdots
x_{\sigma(n)}=x_{1}x_{2}\cdots x_{n},\sigma\in H\rangle.$
There are some results on this new algebraic structure, for example, the
alternating type [6, 8], the abelian type [9], and the $n$-cyclic type [7].
Let $\varsigma$ be the cyclic permutation
$\displaystyle\varsigma=\left(\begin{array}[]{ccccc}1&2&\cdots&n-1&n\\\
2&3&\cdots&n&1\\\ \end{array}\right)$ (3)
By using the rewriting system method, Cedó, Jespers and Okniński [7] obtained
normal forms of elements of $S_{n}(H)$ for the case when $H$ is the cyclic
subgroup of $Sym_{n}$ generated by the cyclic permutation $\varsigma$. They
also proposed some open problems at the end of the same paper [7]. One of the
open problems is: “For an arbitrary subgroup $H$ of symmetric group $Sym_{n}$,
what does every element of $S_{n}(H)$ have a unique canonical form, as is the
case of the monoid defined by permutation relations of cyclic subgroup type.”
In this paper, we use the Gröbner-Shirshov bases method to study the finitely
presented algebra defined by permutation relations of symmetric type
$\mathbf{k}[S_{n}(Sym_{n})]$. We find a Gröbner-Shirshov basis for the algebra
$\mathbf{k}[S_{n}(Sym_{n})]$. As an application, we get normal forms of
elements of monoid $S_{n}(Sym_{n})$, which gives an answer to the above
problem for the case when $H$ is the symmetric group $Sym_{n}$.
## 2 Composition-Diamond Lemma for associative algebra
We first cite some concepts and results from the literature [2, 10] which are
related to Gröbner-Shirshov bases for associative algebras.
Let $\mathbf{k}$ be a field, $\mathbf{k}\langle X\rangle$ the free associative
algebra over $\mathbf{k}$ generated by $X$. Denote $X^{*}$ the free monoid
generated by $X$, where the empty word is the identity which is denoted by 1.
For a word $w\in X^{*}$, we denote the length of $w$ by $|w|$. Let $X^{*}$ be
a well ordered set. Then every nonzero polynomial $f\in\mathbf{k}\langle
X\rangle$ has the leading word $\bar{f}$. If the coefficient of $\bar{f}$ in
$f$ is equal to 1, then $f$ is called monic.
Let $f$ and $g$ be two monic polynomials in $\mathbf{k}\langle X\rangle$.
Then, there are two kinds of compositions:
$(i)$ If $w$ is a word such that $w=\bar{f}b=a\bar{g}$ for some $a,b\in X^{*}$
with $|\bar{f}|+|\bar{g}|>|w|$, then the polynomial $(f,g)_{w}=fb-ag$ is
called the intersection composition of $f$ and $g$ with respect to $w$.
$(ii)$ If $w=\bar{f}=a\bar{g}b$ for some $a,b\in X^{*}$, then the polynomial
$(f,g)_{w}=f-agb$ is called the inclusion composition of $f$ and $g$ with
respect to $w$.
In (i) and (ii), the word $w$ is called an ambiguity.
Let $S\subseteq$ $\mathbf{k}\langle X\rangle$ with each $s\in S$ monic. Then
the composition $(f,g)_{w}$ is called trivial modulo $(S,\ w)$ if
$(f,g)_{w}=\sum\alpha_{i}a_{i}s_{i}b_{i}$, where each
$\alpha_{i}\in\mathbf{k}$, $a_{i},b_{i}\in X^{*},\ s_{i}\in S$ and
$a_{i}\overline{s_{i}}b_{i}<w$. If this is the case, then we write
$(f,g)_{w}\equiv 0\quad mod(S,w).$
In general, for $p,q\in\mathbf{k}\langle X\rangle$, we write
$p\equiv q\quad mod(S,w)$
which means that $p-q=\sum\alpha_{i}a_{i}s_{i}b_{i}$, where each
$\alpha_{i}\in\mathbf{k},a_{i},b_{i}\in X^{*},\ s_{i}\in S$ and
$a_{i}\overline{s_{i}}b_{i}<w$.
We call the set $S$ endowed with the well order $<$ a Gröbner-Shirshov basis
in $\mathbf{k}\langle X\rangle$ if any composition of polynomials in $S$ is
trivial modulo $S$ and corresponding $w$.
A well order $<$ on $X^{*}$ is monomial if for $u,v\in X^{*}$, we have
$u<v\Rightarrow w_{1}uw_{2}<w_{1}vw_{2},\ for\ all\ w_{1},\ w_{2}\in X^{*}.$
The following lemma was proved by Shirshov [10] for free Lie algebras (with
deg-lex order) in 1962 (see also Bokut [2]). In 1976, Bokut [3] specialized
the approach of Shirshov to associative algebras (see also Bergman [1]). For
commutative polynomials, this lemma is known as the Buchberger’s Theorem (see
[4] and [5]).
Composition-Diamond Lemma. Let $\mathbf{k}$ be a field, $\mathbf{k}\langle
X|S\rangle=\mathbf{k}\langle X\rangle/Id(S)$ and $>$ a monomial order on
$X^{*}$, where $Id(S)$ is the ideal of $\mathbf{k}\langle X\rangle$ generated
by $S$. Then the following statements are equivalent:
1. (i)
$S$ is a Gröbner-Shirshov basis in $\mathbf{k}\langle X\rangle$.
2. (ii)
$f\in Id(S)\Rightarrow\bar{f}=a\bar{s}b$ for some $s\in S$ and $a,b\in X^{*}$.
3. (iii)
$Irr(S)=\\{u\in X^{*}|u\neq a\bar{s}b,s\in S,a,b\in X^{*}\\}$ is a
$\mathbf{k}$-linear basis of the algebra $\mathbf{k}\langle X|S\rangle$.
If a subset $S$ of $\mathbf{k}\langle X\rangle$ is not a Gröbner-Shirshov
basis, then we can add to $S$ all nontrivial compositions of polynomials of
$S$, and by continuing this process (may be infinitely) many times, we
eventually obtain a Gröbner-Shirshov basis $S^{comp}$. Such a process is
called the Shirshov algorithm.
Let $M=\langle X|S\rangle$ be a monoid presentation. Then $S$ is a subset of
$\mathbf{k}\langle X\rangle$ and hence one can find a Gröbner-Shirshov basis
$S^{comp}$. We also call $S^{comp}$ a Gröbner-Shirshov basis of monoid $M$.
The set $Irr(S^{comp})=\\{u\in X^{*}|u\neq a\overline{s}b,\ a,b\in X^{*},\
s\in S^{comp}\\}$ is a $\mathbf{k}$-linear basis of $\mathbf{k}\langle
X|S\rangle$ which is also normal forms of elements of monoid $M$.
## 3 A Gröbner-Shirshov basis for $\mathbf{k}[S_{n}(Sym_{n})]$
Let $S_{n}(Sym_{n})$ be the finitely presented momoid defined by permutation
relations of symmetric type, i.e.
$S_{n}(Sym_{n})=\langle
x_{1},x_{2},\ldots,x_{n}|x_{\sigma(1)}x_{\sigma(2)}\cdots
x_{\sigma(n)}=x_{1}x_{2}\cdots x_{n},\sigma\in Sym_{n}\rangle,$
where $Sym_{n}$ is the symmetric group of degree $n$.
We give some notations which will be used in this section. Let $\varepsilon\in
Sym_{n}$ be the identity map of $Sym_{n}$ and
$Sym_{n}^{0}=Sym_{n}\backslash\\{\varepsilon\\}$. Let $\mathbb{N}$ be the set
of positive integers. Denote $\mathbf{n}=\\{1,2,\ldots,n\\}$, and
$[n_{1},n_{2}]=\\{n_{1},n_{1}+1,\ldots,n_{2}\\}$ for any
$n_{1},n_{2}\in\mathbf{n}$ and $n_{1}\leq n_{2}$. For any $\sigma\in Sym_{n}$,
denote
$\mathbf{x}_{\sigma}:=x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)},$
in particular,
$\mathbf{x}_{\varepsilon}:=x_{1}x_{2}\cdots x_{n}.$
For any $x_{i_{1}},\ x_{i_{2}},\ \cdots,x_{i_{m}}\in X,\ m\geq 2$, define
$\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}}:=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{m}},$
where $j_{1},j_{2},\cdots,j_{m}$ is the permutation of
$i_{1},i_{2},\cdots,i_{m}$ such that $j_{1}\leq j_{2}\leq\cdots\leq j_{m}$.
For example,
$\underline{x_{2}x_{5}x_{4}x_{3}x_{2}x_{3}}=x_{2}x_{2}x_{3}x_{3}x_{4}x_{5}$.
Let $X=\\{x_{1},x_{2},\ldots,x_{n}\\}$, $x_{1}<x_{2}<\cdots<x_{n}$ and $``<"$
the degree-lexicographic order on $X^{*}$. Denote
$S=\\{\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}|\sigma\in Sym_{n}^{0}\\}$
and $\widetilde{S}$ the subset of $\mathbf{k}\langle X\rangle$ consisting of
the following polynomials:
1. 1
$\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon},$
2. 2
$x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
3. 3
$x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
4. 4
$\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$, $x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$,
5. 5
$\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$,
where $\sigma\in Sym_{n}^{0}$, $m\geq 1,\ 2\leq
i,i_{1},i_{2},\cdots,i_{m+1}\leq n$.
###### Lemma 3.1
$\mathbf{k}[S_{n}(Sym_{n})]=\mathbf{k}\langle X|S\rangle=\mathbf{k}\langle
X|\widetilde{S}\rangle$.
Proof. For any $s_{1},s_{2}\in\mathbf{k}\langle X\rangle$, we write
$s_{1}\equiv_{I}s_{2}$ if $s_{1}-s_{2}\in Id(S)$. Since
$S\subseteq\widetilde{S}$, we just have to prove that $\widetilde{S}\subseteq
Id(S)$. It suffices to prove that $s\equiv_{I}0$ for any $s\in\widetilde{S}$.
For $2\leq i\leq n$, there exist $\sigma_{1},\sigma_{2}\in Sym_{n}^{0}$ such
that $\mathbf{x}_{\sigma_{1}}x_{i}=x_{i}\mathbf{x}_{\sigma_{2}}$. Therefore
$x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}=(\mathbf{x}_{\sigma_{1}}-\mathbf{x}_{\varepsilon})x_{i}-x_{i}(\mathbf{x}_{\sigma_{2}}-\mathbf{x}_{\varepsilon})\equiv_{I}0.$
Now we use induction on $m$ to prove that all the polynomials of type 3, 4, 5
are in $Id(S)$.
(a) For $m=1$ and $2\leq i\leq n$, there exist $\sigma_{1},\sigma_{2}\in
Sym_{n}^{0}$ such that
$\mathbf{x}_{\sigma_{1}}x_{1}x_{i}=x_{i}x_{1}\mathbf{x}_{\sigma_{2}}$.
Therefore
$\displaystyle
x_{i}x_{1}\mathbf{x}_{\varepsilon}-x_{1}\mathbf{x}_{\varepsilon}x_{i}$
$\displaystyle=$
$\displaystyle(\mathbf{x}_{\sigma_{1}}-\mathbf{x}_{\varepsilon})x_{1}x_{i}-x_{i}x_{1}(\mathbf{x}_{\sigma_{2}}-\mathbf{x}_{\varepsilon})+x_{1}(\mathbf{x}_{\varsigma}-\mathbf{x}_{\varepsilon})x_{i}$
$\displaystyle\equiv_{I}$ $\displaystyle 0,$
where $\varsigma$ is the cyclic permutation defined by (3).
(b) For $m=1$ and $2\leq i_{2}<i_{1}\leq n$, there exist
$\sigma_{1},\sigma_{2}\in Sym_{n}^{0}$ such that
$\mathbf{x}_{\sigma_{1}}x_{i_{2}}x_{i_{1}}=x_{i_{1}}x_{i_{2}}\mathbf{x}_{\sigma_{2}}.$
Therefore,
$\displaystyle\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}}$
$\displaystyle=$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}-\mathbf{x}_{\varepsilon}x_{i_{2}}x_{i_{1}}$
$\displaystyle\equiv_{I}$ $\displaystyle
x_{i_{1}}x_{i_{2}}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{2}}x_{i_{1}}\
(\mbox{by type 2})$ $\displaystyle\equiv_{I}$
$\displaystyle(\mathbf{x}_{\sigma_{1}}-\mathbf{x}_{\varepsilon})x_{i_{2}}x_{i_{1}}-x_{i_{1}}x_{i_{2}}(\mathbf{x}_{\sigma_{2}}-\mathbf{x}_{\varepsilon})$
$\displaystyle\equiv_{I}$ $\displaystyle 0.$
(c) For $m=1$ and $2\leq i_{1}\leq n$, since
$\mathbf{x}_{\varepsilon}x_{1}\equiv_{I}x_{1}\mathbf{x}_{\varepsilon}$, we
have
$\displaystyle\mathbf{x}_{\varepsilon}x_{i_{1}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}$
$\displaystyle=$
$\displaystyle(\mathbf{x}_{\varepsilon}x_{i_{1}}-x_{i_{1}}\mathbf{x}_{\varepsilon})x_{1}+x_{i_{1}}\mathbf{x}_{\varepsilon}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}$
$\displaystyle\equiv_{I}$
$\displaystyle(\mathbf{x}_{\varepsilon}x_{i_{1}}-x_{i_{1}}\mathbf{x}_{\varepsilon})x_{1}+(x_{i_{1}}x_{1}\mathbf{x}_{\varepsilon}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}})$
$\displaystyle\equiv_{I}$ $\displaystyle 0\ (\mbox{by (a) and type 2}).$
Now we assume that all the polynomials of type 3, 4, 5 are in $Id(S)$ for
$m^{\prime},\ 1\leq m^{\prime}<m$.
(i) For $2\leq i\leq n$, since
$x_{1}\mathbf{x}_{\varsigma}=\mathbf{x}_{\varepsilon}x_{1}$, we have
$\displaystyle
x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$
$\displaystyle=$ $\displaystyle
x_{i}x_{1}^{m}(-\mathbf{x}_{\varsigma}+\mathbf{x}_{\varepsilon})+x_{i}x_{1}^{m}\mathbf{x}_{\varsigma}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$
$\displaystyle\equiv_{I}$ $\displaystyle
x_{i}x_{1}^{m-1}\mathbf{x}_{\varepsilon}x_{1}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$
$\displaystyle\equiv_{I}$ $\displaystyle
x_{1}^{m-1}\mathbf{x}_{\varepsilon}x_{i}x_{1}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}\
(\mbox{by induction})$ $\displaystyle\equiv_{I}$ $\displaystyle
x_{1}^{m-1}x_{1}\mathbf{x}_{\varepsilon}x_{i}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$
$\displaystyle\equiv_{I}$ $\displaystyle 0.$
This shows that all polynomials of type 3 are in $Id(S)$.
(ii) For $2\leq i_{1},i_{2},\cdots,i_{m+1}\leq n$, let
$x_{i_{t}}=\max\\{x_{i_{1}},x_{i_{2}},\cdots,x_{i_{m+1}}\\}.$
There are two cases to consider.
Case 1: If $x_{i_{t}}=x_{i_{m+1}}$, then
$\displaystyle\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{i_{m+1}}}$ $\displaystyle\equiv_{I}$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}}x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{i_{m+1}}}\ (\mbox{by induction})$ $\displaystyle\equiv_{I}$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{i_{m+1}}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{i_{m+1}}}$ $\displaystyle\equiv_{I}$ $\displaystyle 0.$
Case 2: If $x_{it}\neq x_{i_{m+1}}$, then by induction, we have
$\displaystyle\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{t}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots x_{i_{t}}\cdots
x_{i_{m+1}}}$ $\displaystyle\equiv_{I}$ $\displaystyle x_{i_{1}}\cdots
x_{i_{t-1}}\mathbf{x}_{\varepsilon}x_{i_{t}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots x_{i_{t}}\cdots
x_{i_{m+1}}}\ (\mbox{by\ type\ 2 })$ $\displaystyle\equiv_{I}$ $\displaystyle
x_{i_{1}}\cdots x_{i_{t-1}}\mathbf{x}_{\varepsilon}\underline{x_{i_{t}}\cdots
x_{i_{m+1}}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{t}}\cdots x_{i_{m+1}}}\ (\mbox{by induction})$ $\displaystyle\equiv_{I}$
$\displaystyle x_{i_{1}}\cdots
x_{i_{t-1}}\mathbf{x}_{\varepsilon}\underline{x_{i_{t+1}}\cdots
x_{i_{m+1}}}x_{i_{t}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{t}}\cdots x_{i_{m+1}}}$ $\displaystyle\equiv_{I}$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{t-1}}\underline{x_{i_{t+1}}\cdots
x_{i_{m+1}}}x_{i_{t}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{t}}\cdots x_{i_{m+1}}}$ $\displaystyle\equiv_{I}$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{t-1}}\underline{x_{i_{t+1}}\cdots
x_{i_{m+1}}}}x_{i_{t}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{t}}\cdots x_{i_{m+1}}}$ $\displaystyle\equiv_{I}$ $\displaystyle 0.$
This shows that all polynomials of type 4 are in $Id(S)$.
(iii) For $2\leq i_{1},i_{2},\cdots,i_{m}\leq n$, we have,
$\displaystyle\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$ $\displaystyle\equiv_{I}$ $\displaystyle
x_{i_{1}}\mathbf{x}_{\varepsilon}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}\ (\mbox{by type 2 })$ $\displaystyle\equiv_{I}$ $\displaystyle
x_{i_{1}}x_{1}\mathbf{x}_{\varepsilon}x_{i_{2}}\cdots
x_{i_{m}}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}\
(\mbox{by induction})$ $\displaystyle\equiv_{I}$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}\
(\mbox{by type 3 })$ $\displaystyle\equiv_{I}$ $\displaystyle 0.$
This shows that all polynomials of type 5 are in $Id(S)$.
The proof is complete. $\blacksquare$
The following theorem is the main result in this paper.
###### Theorem 3.2
With the degree-lexicographic order on $X^{*}$, $\widetilde{S}$ is a Gröbner-
Shirshov basis in $\mathbf{k}\langle X\rangle$.
Proof. Let $f_{i}$ or $f_{i}^{\prime}$ be the polynomial of type $i$ in
$\widetilde{S}$, $i=1,2,\ldots,5$ and $\sigma,\sigma^{\prime}\in Sym_{n}^{0}$.
Denote $i\wedge j$ the composition of the polynomials of type $i$ and type
$j$.
All possible compositions of the polynomials in $\widetilde{S}$ are only as
below:
$1\wedge 1$,
$f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon},f_{1}^{\prime}=\mathbf{x}_{\sigma^{\prime}}-\mathbf{x}_{\varepsilon},$
$w=x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}\Delta x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}},$ $\mathbf{x}_{\sigma}=x_{i_{1}}x_{i_{2}}\cdots
x_{i_{r}}\Delta$, $\mathbf{x}_{\sigma^{\prime}}=\Delta
x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots x_{i_{\pi(r)}}$, $\pi\in Sym_{r}$, $1\leq
r<n$.
$1\wedge 2$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
$w=x_{i_{1}}\cdots x_{i_{t}}x_{i}\mathbf{x}_{\varepsilon}$,
$\mathbf{x}_{\sigma}=x_{i_{1}}\cdots x_{i_{t}}x_{i}\\\ x_{1}x_{2}\ldots
x_{n-t-1}$, $\\{i_{1},i_{2},\ldots,i_{t}\\}=[n-t,n]\backslash\\{i\\}$, $0\leq
n-t-1<i$, $2\leq i\leq n$.
$1\wedge 3$, there are two cases. Let
$f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$ and
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$.
$w_{1}=\mathbf{x}_{\sigma}x_{1}^{m-1}\mathbf{x}_{\varepsilon}$,
$\mathbf{x}_{\sigma}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{n-2}}x_{i}x_{1}$,
$\\{i_{1},i_{2},\ldots,i_{n-2}\\}=\mathbf{n}\backslash\\{i,1\\}$, $m\geq 1$,
$2\leq i\leq n$.
$w_{2}=\mathbf{x}_{\sigma}x_{1}^{m}\mathbf{x}_{\varepsilon}$,
$\mathbf{x}_{\sigma}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{n-1}}x_{i}$,
$\\{i_{1},i_{2},\ldots,i_{n-1}\\}=\mathbf{n}\backslash\\{i\\}$, $m\geq 1$,
$2\leq i\leq n$.
$1\wedge 4$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$, $w=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}$,
$\mathbf{x}_{\sigma}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}x_{1}x_{2}\cdots
x_{n-t}$, $\\{j_{1},j_{2},\ldots,j_{t}\\}=[n-t+1,n]$, $2\leq
i_{1},i_{2},\cdots,i_{m+1}\leq n,$ $m\geq 1$, $1\leq t\leq n-1$,
$x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$.
$1\wedge 5$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$, $w=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{1}$,
$\mathbf{x}_{\sigma}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}x_{1}x_{2}\cdots
x_{n-t}$, $\\{j_{1},j_{2},\ldots,j_{t}\\}=[n-t+1,n]$, $2\leq
i_{1},i_{2},\cdots,i_{m}\leq n$, $m\geq 1$, $1\leq t\leq n-1$.
$2\wedge 1$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i},$
$f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$w=x_{i}\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}$,
$\mathbf{x}_{\sigma}=x_{t+1}\cdots x_{n}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}$,
$\\{j_{1},j_{2},\ldots,j_{t}\\}=\mathbf{n}\backslash[t+1,n]$, $2\leq i\leq n$,
$1\leq t\leq n-1$.
$2\wedge 2$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i},$,
$f_{2}^{\prime}=x_{n}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{n}$,
$w=x_{i}\mathbf{x}_{\varepsilon}\mathbf{x}_{\varepsilon}$, $2\leq i\leq n$.
$2\wedge 3$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
$f_{3}=x_{n}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{n}$,
$w=x_{i}\mathbf{x}_{\varepsilon}x_{1}^{m}\mathbf{x}_{\varepsilon}$, $2\leq
i\leq n$, $m\geq 1$.
$2\wedge 4$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
$f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$, $w=x_{i}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}$, $m\geq 1$, $2\leq i,i_{1},i_{2},\ldots,i_{m+1}\leq n$,
$x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$.
$2\wedge 5$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
$f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$, $w=x_{i}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}$, $m\geq 1$, $2\leq i,i_{1},i_{2},\ldots,i_{m}\leq n$.
$3\wedge 1$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{t}}$,
$\mathbf{x}_{\sigma}=x_{t+1}x_{t+2}\cdots x_{n}x_{i_{1}}\cdots x_{i_{t}}$,
$\\{i_{1},i_{2},\ldots,i_{t}\\}=[1,t]$, $1\leq t\leq n-1$, $2\leq i\leq n$,
$m\geq 1$.
$3\wedge 2$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{2}=x_{n}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{n}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}\mathbf{x}_{\varepsilon}$, $2\leq
i\leq n$, $m\geq 1$.
$3\wedge 3$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{3}^{\prime}=x_{n}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}-x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}x_{n}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}$,
$2\leq i\leq n$, $m,m_{1}\geq 1$.
$3\wedge 4$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m_{1}+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m_{1}+1}}}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}+1}}$, $2\leq i,i_{1},i_{2},\ldots,i_{m_{1}+1}\leq n$, $m,m_{1}\geq
1$, $x_{i_{1}}\cdots x_{i_{m_{1}+1}}>\underline{x_{i_{1}}\cdots
x_{i_{m_{1}+1}}}$.
$3\wedge 5$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}}}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}}}x_{1}$, $2\leq i,i_{1},i_{2},\ldots,i_{m_{1}}\leq n$,
$m,m_{1}\geq 1$.
$4\wedge 1$, there are two cases. Let
$f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$, $m\geq
1$, $2\leq i_{1},i_{2},\cdots,i_{m+1}\leq n$, $x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$.
$w_{1}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m+1}}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}$,
$\mathbf{x}_{\sigma}=x_{i_{m+2-n+t}}\cdots x_{i_{m+1}}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}$,
$\\{j_{1},j_{2},\ldots,j_{t}\\}=\mathbf{n}\backslash\\{i_{m+2-n+t},\ldots,i_{m+1}\\}$,
$w_{2}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m+1}}x_{j_{1}}\cdots
x_{j_{t-m-1}}$, $\mathbf{x}_{\sigma}=x_{t+1}\cdots x_{n}x_{i_{1}}\cdots
x_{i_{m+1}}x_{j_{1}}\cdots x_{j_{t-m-1}}$, $2\leq
i_{1},i_{2},\cdots,i_{m+1}\leq n$,
$\\{j_{1},j_{2},\ldots,j_{t-m-1}\\}=\mathbf{n}\backslash([t+1,n]\cup\\{i_{1},i_{2},\ldots,i_{m+1}\\})$,
$t-m-1\geq 1$.
$4\wedge 2$, $f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$,
$f_{2}=x_{i_{m+1}}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{m+1}}$,
$w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}\mathbf{x}_{\varepsilon}$, $m\geq 1$, $2\leq
i_{1},\cdots,i_{m+1}\leq n$, $x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$.
$4\wedge 3$, $f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$,
$f_{3}=x_{i_{m+1}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}-x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}x_{i_{m+1}}$,
$w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}$, $2\leq
i_{1},\cdots,i_{m+1}\leq n$, $m_{1},m\geq 1$, $x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$.
$5\wedge 1$, there are two cases. Let
$f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$, $2\leq
i_{1},i_{2},\cdots,i_{m}\leq n$, $m\geq 1$.
$w_{1}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{1}x_{j_{1}}\cdots
x_{j_{n+t-m-2}}$, $\mathbf{x}_{\sigma}=x_{i_{t}}x_{i_{t+1}}\cdots
x_{i_{m}}x_{1}x_{j_{1}}\cdots x_{j_{n+t-m-2}}$, $2\leq
i_{1},i_{2},\cdots,i_{m}\leq n$,
$\\{j_{1},j_{2},\ldots,j_{n+t-m-2}\\}=\mathbf{n}\backslash(\\{i_{t},i_{t+1},\ldots,i_{m}\\}\cup\\{1\\})$.
$w_{2}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{1}$,
$\mathbf{x}_{\sigma}=x_{t+1}\cdots x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}x_{j_{1}}\cdots x_{j_{t-m}}$,
$2\leq i_{1},i_{2},\cdots,i_{m}\leq n$,
$\\{j_{1},j_{2},\ldots,j_{t-m}\\}=\mathbf{n}\backslash(\\{i_{1},i_{2},\ldots,i_{m}\\}\cup[t+1,n]\cup\\{1\\})$,
$1\leq m\leq n-2$, $t-m\geq 0$.
$5\wedge 2$, $f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$,
$f_{2}=x_{i_{m}}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{m}}$
$w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}\mathbf{x}_{\varepsilon}$, $2\leq i_{1},i_{2},\cdots,i_{m}\leq n$,
$m\geq 1$.
$5\wedge 3$, $f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$,
$f_{3}=x_{i_{m}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}-x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}x_{i_{m}}$,
$w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}$, $2\leq
i_{1},i_{2},\cdots,i_{m}\leq n$, $m,m_{1}\geq 1$.
$5\wedge 4$, $f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$, $f_{4}=\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{m_{1}+1}}$
$-\mathbf{x}_{\varepsilon}\underline{x_{j_{1}}x_{j_{2}}\cdots
x_{j_{m_{1}+1}}}$, $w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots x_{j_{m_{1}+1}}$,
$2\leq i_{1},i_{2},\cdots,i_{m},\\\ j_{1},j_{2},\cdots,j_{m_{1}+1}\leq n,$
$m,m_{1}\geq 1$, $x_{j_{1}}x_{j_{2}}\cdots
x_{j_{m_{1}+1}}>\underline{x_{j_{1}}x_{j_{2}}\cdots x_{j_{m_{1}+1}}}$.
$5\wedge 5$, $f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$,
$f_{5}^{\prime}=\mathbf{x}_{\varepsilon}x_{i_{1}^{\prime}}x_{i_{2}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}x_{1}$
$-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}^{\prime}}x_{i_{2}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}$, $w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}\mathbf{x}_{\varepsilon}x_{i_{1}^{\prime}}x_{i_{2}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}x_{1}$, $2\leq i_{1},i_{2},\cdots,i_{m},\\\
i_{1}^{\prime},i_{2}^{\prime},\cdots,i_{m_{1}}^{\prime}\leq n$.
We prove that all the above compositions are trivial. Here, we just check
$1\wedge 1$, $1\wedge 4$, $2\wedge 5$. Others are similarly proved.
For $1\wedge 1$, there are two cases to consider.
Case 1: If $1\notin\\{i_{1},i_{2},\ldots,i_{r}\\}$, then
$\displaystyle 1\wedge 1$ $\displaystyle=$ $\displaystyle
f_{1}x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}-x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}f_{1}^{\prime}$
$\displaystyle=$
$\displaystyle-\mathbf{x}_{\varepsilon}x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}+x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}\mathbf{x}_{\varepsilon}$
$\displaystyle\equiv$
$\displaystyle-\mathbf{x}_{\varepsilon}x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}+\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}\
(\mbox{by type 2 })$ $\displaystyle\equiv$
$\displaystyle-\mathbf{x}_{\varepsilon}\underline{x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}}+\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{r}}}\ (\mbox{by type 4 })$ $\displaystyle\equiv$ $\displaystyle 0\
mod(\widetilde{S},w),$
Case 2: If $1\in\\{i_{1},i_{2},\ldots,i_{r}\\}$, say,
$x_{1}=x_{i_{t}}=x_{i_{\pi(s)}},1\leq s,t\leq r$, then by type 5 and 4, we
have
$\displaystyle 1\wedge 1$ $\displaystyle=$ $\displaystyle
f_{1}x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}-x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}f_{1}^{\prime}$
$\displaystyle=$ $\displaystyle-\mathbf{x}_{\varepsilon}x_{i_{\pi(1)}}\cdots
x_{i_{\pi(s)}}\cdots x_{i_{\pi(r)}}+x_{i_{1}}\cdots x_{i_{t}}\cdots
x_{i_{r}}\mathbf{x}_{\varepsilon}$ $\displaystyle=$ $\displaystyle-
x_{1}\mathbf{x}_{\varepsilon}x_{i_{\pi(1)}}\cdots
x_{i_{\pi(s-1)}}x_{i_{\pi(s+1)}}\cdots
x_{i_{\pi(r)}}+x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{t-1}}x_{i_{t+1}}\cdots x_{i_{r}}$ $\displaystyle=$ $\displaystyle-
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{\pi(1)}}\cdots
x_{i_{\pi(s-1)}}x_{i_{\pi(s+1)}}\cdots
x_{i_{\pi(r)}}}+x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{t-1}}x_{i_{t+1}}\cdots x_{i_{r}}}$ $\displaystyle\equiv$ $\displaystyle
0\ mod(\widetilde{S},w).$ $\displaystyle 1\wedge 4$ $\displaystyle=$
$\displaystyle f_{1}x_{n-t+1}x_{n-t+2}\cdots x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}f_{4}$ $\displaystyle=$
$\displaystyle-\mathbf{x}_{\varepsilon}x_{n-t+1}x_{n-t+2}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}+x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$ $\displaystyle\equiv$
$\displaystyle-\mathbf{x}_{\varepsilon}x_{n-t+1}x_{n-t+2}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}+\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$
$\displaystyle\equiv$
$\displaystyle-\mathbf{x}_{\varepsilon}\underline{x_{n-t+1}x_{n-t+2}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}+\mathbf{x}_{\varepsilon}\underline{x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w).$ $\displaystyle
2\wedge 5$ $\displaystyle=$ $\displaystyle f_{2}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{i}f_{5}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{i}x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}-x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$ $\displaystyle\equiv$ $\displaystyle mod(\widetilde{S},w).$
The proof is complete. $\blacksquare$
By the Composition-Diamond Lemma and Theorem 3.2, we have the following
corollary.
###### Corollary 3.3
The set
$Irr(\widetilde{S})=(X^{*}\backslash\bigcup_{\sigma\in
Sym_{n}}X^{*}\\{\mathbf{x}_{\sigma}\\}X^{*})\bigcup\\{x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}x_{2}^{m_{2}}\cdots
x_{n}^{m_{n}}|m_{i}\geq 0,i=1,2,\ldots,n\\}$
is a $\mathbf{k}$-linear basis of algebra $\mathbf{k}[S_{n}(Sym_{n})]$.
Moreover, $Irr(\widetilde{S})$ is normal forms of elements of monoid
$S_{n}(Sym_{n})$.
## 4 Appendix
In this section, we will check that all the compositions are trivial.
$1\wedge 1$,
$f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon},f_{1}^{\prime}=\mathbf{x}_{\sigma^{\prime}}-\mathbf{x}_{\varepsilon},$
$w=x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}\Delta x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}},$ $\mathbf{x}_{\sigma}=x_{i_{1}}x_{i_{2}}\cdots
x_{i_{r}}\Delta$, $\mathbf{x}_{\sigma^{\prime}}=\Delta
x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots x_{i_{\pi(r)}}$, $\pi\in Sym_{r}$, $1\leq
r<n$.
For $1\wedge 1$, there are two cases to consider.
Case 1: If $1\notin\\{i_{1},i_{2},\ldots,i_{r}\\}$, then
$\displaystyle 1\wedge 1$ $\displaystyle=$ $\displaystyle
f_{1}x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}-x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}f_{1}^{\prime}$
$\displaystyle=$
$\displaystyle-\mathbf{x}_{\varepsilon}x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}+x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}\mathbf{x}_{\varepsilon}$
$\displaystyle\equiv$
$\displaystyle-\mathbf{x}_{\varepsilon}x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}+\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}\
(\mbox{by type 2 })$ $\displaystyle\equiv$
$\displaystyle-\mathbf{x}_{\varepsilon}\underline{x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}}+\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{r}}}\ (\mbox{by type 4 })$ $\displaystyle\equiv$ $\displaystyle 0\
mod(\widetilde{S},w),$
Case 2: If $1\in\\{i_{1},i_{2},\ldots,i_{r}\\}$, say,
$x_{1}=x_{i_{t}}=x_{i_{\pi(s)}},1\leq s,t\leq r$, then by type 5 and 4, we
have
$\displaystyle 1\wedge 1$ $\displaystyle=$ $\displaystyle
f_{1}x_{i_{\pi(1)}}x_{i_{\pi(2)}}\cdots
x_{i_{\pi(r)}}-x_{i_{1}}x_{i_{2}}\cdots x_{i_{r}}f_{1}^{\prime}$
$\displaystyle=$ $\displaystyle-\mathbf{x}_{\varepsilon}x_{i_{\pi(1)}}\cdots
x_{i_{\pi(s)}}\cdots x_{i_{\pi(r)}}+x_{i_{1}}\cdots x_{i_{t}}\cdots
x_{i_{r}}\mathbf{x}_{\varepsilon}$ $\displaystyle=$ $\displaystyle-
x_{1}\mathbf{x}_{\varepsilon}x_{i_{\pi(1)}}\cdots
x_{i_{\pi(s-1)}}x_{i_{\pi(s+1)}}\cdots
x_{i_{\pi(r)}}+x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{t-1}}x_{i_{t+1}}\cdots x_{i_{r}}$ $\displaystyle=$ $\displaystyle-
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{\pi(1)}}\cdots
x_{i_{\pi(s-1)}}x_{i_{\pi(s+1)}}\cdots
x_{i_{\pi(r)}}}+x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{t-1}}x_{i_{t+1}}\cdots x_{i_{r}}}$ $\displaystyle\equiv$ $\displaystyle
0\ mod(\widetilde{S},w).$
$1\wedge 2$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
$w=x_{i_{1}}\cdots x_{i_{t}}x_{i}\mathbf{x}_{\varepsilon}$,
$\mathbf{x}_{\sigma}=x_{i_{1}}\cdots x_{i_{t}}x_{i}\\\ x_{1}x_{2}\ldots
x_{n-t-1}$, $\\{i_{1},i_{2},\ldots,i_{t}\\}=[n-t,n]\backslash\\{i\\}$, $0\leq
n-t-1<i$, $2\leq i\leq n$.
$\displaystyle 1\wedge 2$ $\displaystyle=$ $\displaystyle f_{1}x_{n-t}\cdots
x_{n}-x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}f_{2}$ $\displaystyle=$
$\displaystyle\mathbf{x}_{\varepsilon}x_{n-t}\cdots
x_{n}-x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}\mathbf{x}_{\varepsilon}x_{i}$
$\displaystyle=$ $\displaystyle\mathbf{x}_{\varepsilon}x_{n-t}\cdots
x_{n}-\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{t}}x_{i}$
$\displaystyle\equiv$ $\displaystyle\mathbf{x}_{\varepsilon}x_{n-t}\cdots
x_{n}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{t}}x_{i}}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{n-t}\cdots
x_{n}-\mathbf{x}_{\varepsilon}x_{n-t}\cdots x_{n}$ $\displaystyle\equiv$
$\displaystyle 0\ mod(\widetilde{S},w)$
$1\wedge 3$, there are two cases. Let
$f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$ and
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$.
$w_{1}=\mathbf{x}_{\sigma}x_{1}^{m-1}\mathbf{x}_{\varepsilon}$,
$\mathbf{x}_{\sigma}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{n-2}}x_{i}x_{1}$,
$\\{i_{1},i_{2},\ldots,i_{n-2}\\}=\mathbf{n}\backslash\\{i,1\\}$, $m\geq 1$,
$2\leq i\leq n$.
$w_{2}=\mathbf{x}_{\sigma}x_{1}^{m}\mathbf{x}_{\varepsilon}$,
$\mathbf{x}_{\sigma}=x_{i_{1}}x_{i_{2}}\cdots x_{i_{n-1}}x_{i}$,
$\\{i_{1},i_{2},\ldots,i_{n-1}\\}=\mathbf{n}\backslash\\{i\\}$, $m\geq 1$,
$2\leq i\leq n$.
$\displaystyle 1\wedge 3$ $\displaystyle=$ $\displaystyle
f_{1}x_{1}^{m-1}\mathbf{x}_{\varepsilon}-x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n-2}}f_{3}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{1}^{m-1}\mathbf{x}_{\varepsilon}-x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n-2}}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{1}^{m-1}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n-2}}x_{i}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m}\mathbf{x}_{\varepsilon}x_{2}\cdots
x_{n}-x_{1}^{m}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n-2}}x_{i}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m}\mathbf{x}_{\varepsilon}x_{2}\cdots
x_{n}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{2}\cdots x_{n}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w_{1})$
$\displaystyle 1\wedge 3$ $\displaystyle=$ $\displaystyle
f_{1}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n-1}}f_{3}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n-1}}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{n-1}}x_{i}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{2}\cdots
x_{n}-x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{i_{1}^{\prime}}x_{i_{2}^{\prime}}\cdots
x_{i_{n-2}^{\prime}}x_{i}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{2}\cdots
x_{n}-x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{2}\cdots x_{n}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w_{2}),$
where
$\\{i_{1}^{\prime},i_{2}^{\prime},\ldots,i_{n-2}^{\prime}\\}=[2,n]\backslash\\{i\\}$.
$1\wedge 4$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$, $w=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}$,
$\mathbf{x}_{\sigma}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}x_{1}x_{2}\cdots
x_{n-t}$, $\\{j_{1},j_{2},\ldots,j_{t}\\}=[n-t+1,n]$, $2\leq
i_{1},i_{2},\cdots,i_{m+1}\leq n,$ $m\geq 1$, $1\leq t\leq n-1$,
$x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$.
$\displaystyle 1\wedge 4$ $\displaystyle=$ $\displaystyle f_{1}x_{n-t+1}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}-x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}f_{4}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{n-t+1}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}-x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{n-t+1}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}-\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$
$\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{n-t+1}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}-\mathbf{x}_{\varepsilon}\underline{x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w)$
$1\wedge 5$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$, $w=x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{1}$,
$\mathbf{x}_{\sigma}=x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}x_{1}x_{2}\cdots
x_{n-t}$, $\\{j_{1},j_{2},\ldots,j_{t}\\}=[n-t+1,n]$, $2\leq
i_{1},i_{2},\cdots,i_{m}\leq n$, $m\geq 1$, $1\leq t\leq n-1$.
$\displaystyle 1\wedge 5$ $\displaystyle=$ $\displaystyle f_{1}x_{n-t+1}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{1}-x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}f_{4}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{n-t+1}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{1}-x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{n-t+1}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}}-x_{1}\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}}$ $\displaystyle\equiv$
$\displaystyle x_{1}\mathbf{x}_{\varepsilon}\underline{x_{n-t+1}\cdots
x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}}-x_{1}\mathbf{x}_{\varepsilon}\underline{x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}}}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w)$
$2\wedge 1$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i},$
$f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$w=x_{i}\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}$,
$\mathbf{x}_{\sigma}=x_{t+1}\cdots x_{n}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}$,
$\\{j_{1},j_{2},\ldots,j_{t}\\}=\mathbf{n}\backslash[t+1,n]$, $2\leq i\leq n$,
$1\leq t\leq n-1$.
$\displaystyle 2\wedge 1$ $\displaystyle=$ $\displaystyle
f_{2}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}-x_{i}x_{1}x_{2}\cdots x_{t}f_{1}$
$\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}-x_{i}x_{1}x_{2}\cdots x_{t}\mathbf{x}_{\varepsilon}$
$\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{j_{2}^{\prime}}x_{j_{3}^{\prime}}\cdots
x_{j_{t}^{\prime}}-x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{1}x_{2}\cdots x_{t}$
$\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i}x_{j_{2}^{\prime}}x_{j_{3}^{\prime}}\cdots
x_{j_{t}^{\prime}}}-x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i}x_{1}x_{2}\cdots
x_{t}}$ $\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w)$
where $\\{j_{2}^{\prime},j_{3}^{\prime},\ldots,j_{t}^{\prime}\\}=[2,t]$.
$2\wedge 2$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i},$,
$f_{2}^{\prime}=x_{n}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{n}$,
$w=x_{i}\mathbf{x}_{\varepsilon}\mathbf{x}_{\varepsilon}$, $2\leq i\leq n$.
$\displaystyle 2\wedge 2$ $\displaystyle=$ $\displaystyle
f_{2}\mathbf{x}_{\varepsilon}-x_{i}x_{1}\cdots x_{n-1}f_{2}^{\prime}$
$\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i}\mathbf{x}_{\varepsilon}-x_{i}x_{1}\cdots
x_{n-1}\mathbf{x}_{\varepsilon}x_{n}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{2}\cdots
x_{n}-x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{2}\cdots x_{n-1}x_{n}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w)$
$2\wedge 3$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
$f_{3}=x_{n}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{n}$,
$w=x_{i}\mathbf{x}_{\varepsilon}x_{1}^{m}\mathbf{x}_{\varepsilon}$, $2\leq
i\leq n$, $m\geq 1$.
$\displaystyle 2\wedge 3$ $\displaystyle=$ $\displaystyle
f_{2}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{i}x_{1}\cdots x_{n-1}f_{3}$
$\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{i}x_{1}\cdots
x_{n-1}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{n}$ $\displaystyle\equiv$
$\displaystyle x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{i}x_{2}\cdots
x_{n}-x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{i}x_{2}\cdots x_{n}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w)$
$2\wedge 4$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
$f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$, $w=x_{i}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}$, $m\geq 1$, $2\leq i,i_{1},i_{2},\ldots,i_{m+1}\leq n$,
$x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$.
$\displaystyle 2\wedge 4$ $\displaystyle=$ $\displaystyle
f_{2}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}-x_{i}f_{4}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-x_{i}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}-\mathbf{x}_{\varepsilon}x_{i}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}-\mathbf{x}_{\varepsilon}\underline{x_{i}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}}$ $\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w)$
$2\wedge 5$,
$f_{2}=x_{i}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i}$,
$f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$, $w=x_{i}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}$, $m\geq 1$, $2\leq i,i_{1},i_{2},\ldots,i_{m}\leq n$.
$\displaystyle 2\wedge 5$ $\displaystyle=$ $\displaystyle
f_{2}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{1}-x_{i}f_{5}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{i}x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}-x_{1}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$ $\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w).$
$3\wedge 1$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{t}}$,
$\mathbf{x}_{\sigma}=x_{t+1}x_{t+2}\cdots x_{n}x_{i_{1}}\cdots x_{i_{t}}$,
$\\{i_{1},i_{2},\ldots,i_{t}\\}=[1,t]$, $1\leq t\leq n-1$, $2\leq i\leq n$,
$m\geq 1$.
$\displaystyle 3\wedge 1$ $\displaystyle=$ $\displaystyle f_{3}x_{i_{1}}\cdots
x_{i_{t}}-x_{i}x_{1}^{m}x_{1}\cdots x_{t}f_{1}$ $\displaystyle\equiv$
$\displaystyle x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}\cdots
x_{i_{t}}-x_{i}x_{1}^{m}x_{1}\cdots x_{t}\mathbf{x}_{\varepsilon}$
$\displaystyle\equiv$ $\displaystyle
x_{1}^{m+1}\mathbf{x}_{\varepsilon}\underline{x_{i}x_{i_{2}^{\prime}}\cdots
x_{i_{t}^{\prime}}}-x_{1}^{m+1}\mathbf{x}_{\varepsilon}\underline{x_{i}x_{2}\cdots
x_{t}}$ $\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w).$
where $\\{i_{2}^{\prime},i_{3}^{\prime},\cdots,i_{t}^{\prime}\\}=[2,t]$.
$3\wedge 2$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{2}=x_{n}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{n}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}\mathbf{x}_{\varepsilon}$, $2\leq
i\leq n$, $m\geq 1$.
$\displaystyle 3\wedge 2$ $\displaystyle=$ $\displaystyle
f_{3}\mathbf{x}_{\varepsilon}-x_{i}x_{1}^{m}x_{1}\cdots x_{n-1}f_{2}$
$\displaystyle\equiv$ $\displaystyle
x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}\mathbf{x}_{\varepsilon}-x_{i}x_{1}^{m}x_{1}\cdots
x_{n-1}\mathbf{x}_{\varepsilon}x_{n}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{i}x_{2}\cdots
x_{n}-x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{i}x_{2}\cdots x_{n}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w).$
$3\wedge 3$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{3}^{\prime}=x_{n}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}-x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}x_{n}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}$,
$2\leq i\leq n$, $m,m_{1}\geq 1$.
$\displaystyle 3\wedge 3$ $\displaystyle=$ $\displaystyle
f_{3}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}-x_{i}x_{1}^{m}x_{1}\cdots
x_{n-1}f_{3}^{\prime}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}-x_{i}x_{1}^{m}x_{1}\cdots
x_{n-1}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}x_{n}$ $\displaystyle\equiv$
$\displaystyle x_{1}^{m+m_{1}+1}\mathbf{x}_{\varepsilon}x_{i}x_{2}\cdots
x_{n}-x_{1}^{m+m_{1}+1}\mathbf{x}_{\varepsilon}x_{i}x_{2}\cdots x_{n}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w).$
$3\wedge 4$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m_{1}+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m_{1}+1}}}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}+1}}$, $2\leq i,i_{1},i_{2},\ldots,i_{m_{1}+1}\leq n$, $m,m_{1}\geq
1$, $x_{i_{1}}\cdots x_{i_{m_{1}+1}}>\underline{x_{i_{1}}\cdots
x_{i_{m_{1}+1}}}$.
$\displaystyle 3\wedge 4$ $\displaystyle=$ $\displaystyle f_{3}x_{i_{1}}\cdots
x_{i_{m_{1}+1}}-x_{i}x_{1}^{m}f_{4}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}\cdots
x_{i_{m_{1}+1}}-x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m_{1}+1}}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m}\mathbf{x}_{\varepsilon}\underline{x_{i}x_{i_{1}}\cdots
x_{i_{m_{1}+1}}}-x_{1}^{m}\mathbf{x}_{\varepsilon}\underline{x_{i}x_{i_{1}}\cdots
x_{i_{m_{1}+1}}}$ $\displaystyle\equiv$ $\displaystyle 0\
mod(\widetilde{S},w).$
$3\wedge 5$,
$f_{3}=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}-x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}$,
$f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}}}$,
$w=x_{i}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}}}x_{1}$, $2\leq i,i_{1},i_{2},\ldots,i_{m_{1}}\leq n$,
$m,m_{1}\geq 1$.
$\displaystyle 3\wedge 5$ $\displaystyle=$ $\displaystyle
f_{3}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m_{1}}}x_{1}-x_{i}x_{1}^{m}f_{5}$
$\displaystyle\equiv$ $\displaystyle
x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m_{1}}}x_{1}-x_{i}x_{1}^{m}x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m_{1}}}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m+1}\mathbf{x}_{\varepsilon}\underline{x_{i}x_{i_{1}}\cdots
x_{i_{m_{1}}}}-x_{1}^{m+1}\mathbf{x}_{\varepsilon}\underline{x_{i}x_{i_{1}}\cdots
x_{i_{m_{1}}}}$ $\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w).$
$4\wedge 1$, there are two cases. Let
$f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$, $m\geq
1$, $2\leq i_{1},i_{2},\cdots,i_{m+1}\leq n$, $x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$.
$w_{1}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m+1}}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t}}$,
$\mathbf{x}_{\sigma}=x_{i_{m+2-n+t}}\cdots x_{i_{m+1}}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}$,
$\\{j_{1},j_{2},\ldots,j_{t}\\}=\mathbf{n}\backslash\\{i_{m+2-n+t},\ldots,i_{m+1}\\}$,
$w_{2}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m+1}}x_{j_{1}}\cdots
x_{j_{t-m-1}}$, $\mathbf{x}_{\sigma}=x_{t+1}\cdots x_{n}x_{i_{1}}\cdots
x_{i_{m+1}}x_{j_{1}}\cdots x_{j_{t-m-1}}$, $2\leq
i_{1},i_{2},\cdots,i_{m+1}\leq n$,
$\\{j_{1},j_{2},\ldots,j_{t-m-1}\\}=\mathbf{n}\backslash([t+1,n]\cup\\{i_{1},i_{2},\ldots,i_{m+1}\\})$,
$t-m-1\geq 1$.
$\displaystyle 4\wedge 1$ $\displaystyle=$ $\displaystyle
f_{4}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m+1-n+t}}f_{1}$
$\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{t}}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m+1-n+t}}\mathbf{x}_{\varepsilon}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}x_{j_{2}^{\prime}}x_{j_{3}^{\prime}}\cdots
x_{j_{t^{\prime}}}}-x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m+1-n+t}}x_{2}\cdots x_{n}}$ $\displaystyle\equiv$ $\displaystyle 0\
mod(\widetilde{S},w_{1}).$
where
$\\{j_{2}^{\prime},j_{3}^{\prime},\ldots,j_{t^{\prime}}\\}=\mathbf{n}\backslash\\{i_{m+2-n+t},\ldots,i_{m+1},1\\}$.
$\displaystyle 4\wedge 1$ $\displaystyle=$ $\displaystyle
f_{4}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t-m-1}}-x_{1}\cdots x_{t}f_{1}$
$\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}x_{j_{1}}x_{j_{2}}\cdots x_{j_{t-m-1}}-x_{1}\cdots
x_{t}\mathbf{x}_{\varepsilon}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}x_{j_{2}^{\prime}}x_{j_{3}^{\prime}}\cdots
x_{j_{t-m-1}^{\prime}}}-x_{1}\mathbf{x}_{\varepsilon}x_{2}\cdots x_{t}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w_{1}).$
where
$\\{j_{2}^{\prime},j_{3}^{\prime},\ldots,j_{t-m-1}^{\prime}\\}=[2,t]\backslash\\{i_{1},i_{2},\ldots,i_{m+1}\\}$.
$4\wedge 2$, $f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$,
$f_{2}=x_{i_{m+1}}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{m+1}}$,
$w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}\mathbf{x}_{\varepsilon}$, $m\geq 1$, $2\leq
i_{1},\cdots,i_{m+1}\leq n$, $x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$.
$\displaystyle 4\wedge 2$ $\displaystyle=$ $\displaystyle
f_{4}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}f_{2}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}\mathbf{x}_{\varepsilon}x_{i_{m+1}}$ $\displaystyle\equiv$
$\displaystyle
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}x_{2}\cdots
x_{n}}-x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{2}\cdots x_{n}x_{i_{m+1}}}$ $\displaystyle\equiv$ $\displaystyle
0\ mod(\widetilde{S},w).$
$4\wedge 3$, $f_{4}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}-\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}$,
$f_{3}=x_{i_{m+1}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}-x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}x_{i_{m+1}}$,
$w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}$, $2\leq
i_{1},\cdots,i_{m+1}\leq n$, $m_{1},m\geq 1$, $x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}>\underline{x_{i_{1}}x_{i_{2}}\cdots x_{i_{m+1}}}$.
$\displaystyle 4\wedge 3$ $\displaystyle=$ $\displaystyle
f_{4}x_{1}^{m}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}f_{3}$ $\displaystyle\equiv$
$\displaystyle\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m+1}}}x_{1}^{m}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}^{m}\mathbf{x}_{\varepsilon}x_{i_{m+1}}$ $\displaystyle\equiv$
$\displaystyle x_{1}^{m+1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m+1}}}x_{2}\cdots
x_{n}-x_{1}^{m+1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{2}\cdots
x_{n}x_{i_{m+1}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{m+1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m+1}}x_{2}\cdots
x_{n}}-x_{1}^{m+1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m}}x_{2}\cdots x_{n}x_{i_{m+1}}}$ $\displaystyle\equiv$ $\displaystyle
0\ mod(\widetilde{S},w).$
$5\wedge 1$, there are two cases. Let
$f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$, $f_{1}=\mathbf{x}_{\sigma}-\mathbf{x}_{\varepsilon}$, $2\leq
i_{1},i_{2},\cdots,i_{m}\leq n$, $m\geq 1$.
$w_{1}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{1}x_{j_{1}}\cdots
x_{j_{n+t-m-2}}$, $\mathbf{x}_{\sigma}=x_{i_{t}}x_{i_{t+1}}\cdots
x_{i_{m}}x_{1}x_{j_{1}}\cdots x_{j_{n+t-m-2}}$, $2\leq
i_{1},i_{2},\cdots,i_{m}\leq n$,
$\\{j_{1},j_{2},\ldots,j_{n+t-m-2}\\}=\mathbf{n}\backslash(\\{i_{t},i_{t+1},\ldots,i_{m}\\}\cup\\{1\\})$.
$w_{2}=\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{1}$,
$\mathbf{x}_{\sigma}=x_{t+1}\cdots x_{n}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}x_{j_{1}}\cdots x_{j_{t-m}}$,
$2\leq i_{1},i_{2},\cdots,i_{m}\leq n$,
$\\{j_{1},j_{2},\ldots,j_{t-m}\\}=\mathbf{n}\backslash(\\{i_{1},i_{2},\ldots,i_{m}\\}\cup[t+1,n]\cup\\{1\\})$,
$1\leq m\leq n-2$, $t-m\geq 0$.
$\displaystyle 5\wedge 1$ $\displaystyle=$ $\displaystyle f_{5}x_{j_{1}}\cdots
x_{j_{n+t-m-2}}-\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{t-1}}f_{1}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{j_{1}}\cdots
x_{j_{n+t-m-2}}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{t-1}}\mathbf{x}_{\varepsilon}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{j_{1}}\cdots
x_{j_{n+t-m-2}}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{t-1}}x_{2}\cdots x_{n}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{j_{1}}\cdots
x_{j_{n+t-m-2}}}-x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{t-1}}x_{2}\cdots x_{n}}$ $\displaystyle\equiv$ $\displaystyle 0\
mod(\widetilde{S},w_{1}).$ $\displaystyle 5\wedge 1$ $\displaystyle=$
$\displaystyle f_{5}x_{j_{1}}\cdots x_{j_{t-m}}-x_{1}x_{2}\cdots x_{t}f_{1}$
$\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{j_{1}}\cdots
x_{j_{t-m}}-x_{1}x_{2}\cdots x_{t}\mathbf{x}_{\varepsilon}$
$\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{j_{1}}\cdots
x_{j_{t-m}}-x_{1}\mathbf{x}_{\varepsilon}x_{2}\cdots x_{t}$
$\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{j_{1}}\cdots
x_{j_{t-m}}}-x_{1}\mathbf{x}_{\varepsilon}\underline{x_{2}\cdots x_{t}}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w_{2}).$
$5\wedge 2$, $f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$,
$f_{2}=x_{i_{m}}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{m}}$
$w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}\mathbf{x}_{\varepsilon}$, $2\leq i_{1},i_{2},\cdots,i_{m}\leq n$,
$m\geq 1$.
$\displaystyle 5\wedge 2$ $\displaystyle=$ $\displaystyle f_{5}x_{2}\cdots
x_{n}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m-1}}f_{2}$
$\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots x_{i_{m}}x_{2}\cdots
x_{n}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m-1}}\mathbf{x}_{\varepsilon}x_{i_{m}}$ $\displaystyle\equiv$
$\displaystyle x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{2}\cdots x_{n}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m-1}}x_{2}\cdots x_{n}x_{i_{m}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{2}\cdots
x_{n}}-x_{1}\mathbf{x}_{\varepsilon}\underline{x_{i_{1}}\cdots
x_{i_{m-1}}x_{2}\cdots x_{n}x_{i_{m}}}$ $\displaystyle\equiv$ $\displaystyle
0\ mod(\widetilde{S},w).$
$5\wedge 3$, $f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$,
$f_{3}=x_{i_{m}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}-x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}x_{i_{m}}$,
$w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}$, $2\leq
i_{1},i_{2},\cdots,i_{m}\leq n$, $m,m_{1}\geq 1$.
$\displaystyle 5\wedge 3$ $\displaystyle=$ $\displaystyle
f_{5}x_{1}^{m_{1}-1}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m-1}}f_{3}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}^{m_{1}-1}\mathbf{x}_{\varepsilon}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m-1}}x_{1}^{m_{1}}\mathbf{x}_{\varepsilon}$ $\displaystyle\equiv$
$\displaystyle x_{1}^{m_{1}+1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m}}x_{2}\cdots
x_{n}-x_{1}^{m_{1}+1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m}}x_{2}\cdots x_{n}$ $\displaystyle\equiv$ $\displaystyle 0\
mod(\widetilde{S},w).$
$5\wedge 4$, $f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$, $f_{4}=\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{m_{1}+1}}$
$-\mathbf{x}_{\varepsilon}\underline{x_{j_{1}}x_{j_{2}}\cdots
x_{j_{m_{1}+1}}}$, $w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}\mathbf{x}_{\varepsilon}x_{j_{1}}x_{j_{2}}\cdots x_{j_{m_{1}+1}}$,
$2\leq i_{1},i_{2},\cdots,i_{m},\\\ j_{1},j_{2},\cdots,j_{m_{1}+1}\leq n,$
$m,m_{1}\geq 1$, $x_{j_{1}}x_{j_{2}}\cdots
x_{j_{m_{1}+1}}>\underline{x_{j_{1}}x_{j_{2}}\cdots x_{j_{m_{1}+1}}}$.
$\displaystyle 5\wedge 4$ $\displaystyle=$ $\displaystyle f_{5}x_{2}\cdots
x_{n}x_{j_{1}}x_{j_{2}}\cdots
x_{j_{m_{1}+1}}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}f_{4}$
$\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{2}\cdots
x_{n}x_{j_{1}}\cdots x_{j_{m_{1}+1}}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m}}\mathbf{x}_{\varepsilon}x_{j_{1}}\cdots x_{j_{m_{1}+1}}$
$\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{2}\cdots
x_{n}x_{j_{1}}\cdots
x_{j_{m_{1}+1}}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m}}x_{2}\cdots x_{n}x_{j_{1}}\cdots x_{j_{m_{1}+1}}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w).$
$5\wedge 5$, $f_{5}=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}x_{1}-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}$,
$f_{5}^{\prime}=\mathbf{x}_{\varepsilon}x_{i_{1}^{\prime}}x_{i_{2}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}x_{1}$
$-x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}^{\prime}}x_{i_{2}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}$, $w=\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}\mathbf{x}_{\varepsilon}x_{i_{1}^{\prime}}x_{i_{2}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}x_{1}$, $2\leq i_{1},i_{2},\cdots,i_{m},\\\
i_{1}^{\prime},i_{2}^{\prime},\cdots,i_{m_{1}}^{\prime}\leq n$.
$\displaystyle 5\wedge 5$ $\displaystyle=$ $\displaystyle f_{5}x_{2}\cdots
x_{n}x_{i_{1}^{\prime}}x_{i_{2}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}x_{1}-\mathbf{x}_{\varepsilon}x_{i_{1}}x_{i_{2}}\cdots
x_{i_{m}}f_{5}^{\prime}$ $\displaystyle\equiv$ $\displaystyle
x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{2}\cdots
x_{n}x_{i_{1}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}x_{1}-\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m}}x_{1}\mathbf{x}_{\varepsilon}x_{i_{1}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}$ $\displaystyle\equiv$ $\displaystyle
x_{1}^{2}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots x_{i_{m}}x_{2}\cdots
x_{n}x_{i_{1}^{\prime}}\cdots
x_{i_{m_{1}}^{\prime}}-x_{1}^{2}\mathbf{x}_{\varepsilon}x_{i_{1}}\cdots
x_{i_{m}}x_{2}\cdots x_{n}x_{i_{1}^{\prime}}\cdots x_{i_{m_{1}}^{\prime}}$
$\displaystyle\equiv$ $\displaystyle 0\ mod(\widetilde{S},w).$
## References
* [1] G.M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29 (1978), 178-218.
* [2] L.A. Bokut, Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras, Izv. Akad. Nauk. SSSR Ser. Mat., 36 (1972), 1173-1219.
* [3] L.A. Bokut, Imbeddings into simple associative algebras, Algebra i Logika, 15 (1976), 117-142.
* [4] B. Buchberger, An algorithm for finding a basis for the residue class ring of a zero-dimensional polynomial ideal [in German], Ph.D. thesis, University of Innsbruck, Austria, 1965.
* [5] B. Buchberger, An algorithmical criteria for the solvability of algebraic systems of equations [in German], Aequationes Math., 4 (1970), 374-383.
* [6] F. Cedó, E. Jespers, J. Okniński, The radical of the four generated algebra of alternating type, Contemporary Mathematics, 499 (2009), 1-26.
* [7] F. Cedó, E. Jespers, J. Okniński, Finitely presented algebras and groups defined by permutation relations, J. Pure App. Algebra, 214(7) (2010), 1095-1102.
* [8] F. Cedó, E. Jespers, J. Okniński, Algebras and groups defined by permutation relations of alternating type, J. Algebra, 324(6) (2010), 1290-1313.
* [9] F. Cedó, E. Jespers, G. Kleinb, Finitely presented monoids and algebras defined by permutation relations of abelian type, J. Pure App. Algebra, 216(5) (2012), 1033-1039.
* [10] A.I. Shirshov, Some algorithmic problem for Lie algebras, Sibirsk. Mat. Z., 3 (1962), 292-296 (in Russian); English translation in SIGSAM Bull., 33(2) (1999), 3-6.
|
arxiv-papers
| 2014-03-25T15:16:39 |
2024-09-04T02:50:00.445970
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jianjun Qiu, Yuqun Chen",
"submitter": "Jianjun Qiu",
"url": "https://arxiv.org/abs/1403.8076"
}
|
1403.8109
|
# A Study on Sprout Graphs
Johan Kok Tshwane Metropolitan Police Department
City of Tshwane, Republic of South Africa
[email protected] Naduvath Sudev Department of Mathematics
Vidya Academy of Science & Technology
Thalakkottukara, Thrissur - 680501, India.
[email protected]
###### Abstract
Sprout graphs are finite directed graphs matured over a finite subset of the
non-negative time line. A simple undirected connected graph on at least two
vertices is required to construct an infant graph to mature from. The maxi-max
arc-weight principle and the maxi-min arc-weight principle are introduced in
this paper to determine the maximum and minimum maturity weight of a sprout
graph. These principles demand more mathematical debates for logical closure.
Since complete graphs, paths, stars and possibly cycles form part of the
skeleton of all graphs, the introduction of results for these family of sprout
graphs is expected to lay a good research foundation.
Keywords: Sprouting, sprout graph, infant graph, directed graph, index
pattern, arc weight, maturity weight.
AMS Classification Numbers: 05C05, 05C20, 05C38, 05C62.
## 1 Introduction
For general notation and concepts in graph and digraph theory, we refer to [1,
2, 3, 5]. Generally all graphs mentioned in this paper other than sprout
graphs, are non-trivial, simple, connected, finite and undirected graphs. The
trivial graph $K_{1}$ will be addressed as a special case wherever applicable.
## 2 Sprout Graphs
The idea of sprouting resembles neurological growth of good or malicious
networks or virus infection through Information and Communication Technology
Networks. A sprout graph can generally be considered as a simple and finite
directed graph matured on a time line from graphs on at least two vertices.
The idea of sprouting and the notion of sprout graphs can be described as
follows.
###### Definition 2.1.
Consider a graph $G$ on $n$ vertices, where $n\geq 2$, with a fixed default
vertex labeling
$D=\\{d_{1},d_{2},d_{3},\ldots,d_{n}\\}\subseteq\mathbb{D}=\\{d_{i}:i\in\mathbb{N}\\}$
and let $V(G)=\\{v_{i}:1\leq i\leq
n\\}\subseteq\mathbb{V}=\\{v_{i}:i\in\mathbb{N}\\}$. Define a random bijective
function $f:D\to V$ so that the vertices of $G$ are labeled according to the
range of $f(D)$. The range of $f$ is called the index pattern of $G$ and is
denoted by $\mathscr{I}$.
In other words, we have $\mathscr{I}=\\{f(d_{i})=v_{j}:1\leq i,j\leq n\\}$. It
can be noted that a graph $G$ on $n$ vertices can have $n!$ possible index
patterns.
We denote the time line corresponding to the index pattern $\mathscr{I}$ by
$\mathsf{T}_{\mathscr{I}}$. By the term sprout, we mean an ordered pair
$(i,j)$ of positive integers and an edge $v_{i}v_{j}$ of $G$ can be reduced to
a sprout $(i,j)$ if and only if $i<j$. If all edges of $G$ are reduced to
sprouts the resultant graph is called an infant graph. When the context is
clear we shall refer to either the graph $G$ or the infant graph $G$. Invoking
these definitions, the notion of a sprout graph matured from a given graph can
be described as follows.
###### Definition 2.2.
Let $\mathbb{G}_{\mathscr{I}}$ be a directed graph formed from the infant
graph $G$ such that every arc $(v_{i},v_{j})$ of $\mathbb{G}_{\mathscr{I}}$ is
formed from the sprout $(i,j)$ at time $t=|i-j|$ with,
$t\in\mathsf{T}_{\mathscr{I}}=\\{0,1,2,\ldots,m_{\mathscr{I}}\\}\subseteq\\{0,1,2,3,\ldots,n-1\\}$
and $m_{\mathscr{I}}=max|i-j|,~{}\forall~{}(i,j)$. At $t=m_{\mathscr{I}}$ the
sprouting has matured and the resultant directed graph
$\mathbb{G}_{\mathscr{I}}$ is called the sprout graph matured from the given
graph $G$.
###### Definition 2.3.
A sprouting graph, denoted by $\mathbb{G}_{t=k}$, is the directed graph
maturing from the given infant graph $G$ which has maturity level,
$t=k<m_{\mathscr{I}}$.
In real applications the sprout $(i,j)$ can evolve (or grow) over the time
interval $[0,j-i)$ with arcing at $t=j-i$. Note that at $t=0$ the graph $G$ is
reduced to an infant graph with sprouts $(i,j),\forall~{}v_{i}v_{j}\in E(G)$
and $i<j$, attached to vertex $v_{i}$ and the number of sprouts attached to
$v_{i}$ at $t=0$ is equal to $d^{+}_{\mathbb{G}}(v_{i})$ in the sprout graph
$\mathbb{G}_{\mathscr{I}}$. In view of this fact let us define the following
notions.
###### Definition 2.4.
A vertex $v$ in a sprout graph $\mathbb{G}_{\mathscr{I}}$, having
$d_{\mathbb{G}}(v_{i})=d^{-}_{\mathbb{G}}(v_{i})$ is called an adult vertex
and a vertex $u$ in $\mathbb{G}_{\mathscr{I}}$ having
$d_{\mathbb{G}}(v_{i})=d^{+}_{\mathbb{G}}(v_{i})$ is called an initiator
vertex.
In view of the above notions, the existence of initiator and adult vertices
for the sprout graphs matured from the infant graphs in respect of a given
graph is established in the following proposition.
###### Proposition 2.5.
All sprout graphs $\mathbb{G}_{\mathscr{I}}$, matured from the infant graphs
in respect of a graph $G$ on $n\geq 2$ vertices, have at least one adult
vertex and at least one initiator vertex.
###### Proof.
Since a sprout $(n,i),i\leq n-1$ can never exist, we have
$d_{\mathbb{G}}(v_{n})=d^{-}_{\mathbb{G}}(v_{n})$ in
$\mathbb{G}_{\mathscr{I}}$. Similarly, since a sprout $(i,1)$, $i\geq 2$ can
never exist, we have $d_{\mathbb{G}}(v_{1})=d^{+}_{\mathbb{G}}(v_{1})$ in
$\mathbb{G}_{\mathscr{I}}$. Therefore, every sprout graph
$\mathbb{G}_{\mathscr{I}}$ has at least one initiator and an adult vertex. ∎
In terms of the index patterns of two or more graphs, an index pattern for
different operations of these graphs can be formed as follows.
###### Definition 2.6.
Let the two given graphs $G_{1}$ and $G_{2}$ have the initial default index
patterns $D_{1}=\\{d_{1},d_{2},d_{3},\ldots,d_{n}\\}$ and
$D_{2}=\\{d^{\prime}_{1},d^{\prime}_{2},d^{\prime}_{3},\ldots,d^{\prime}_{m}\\}$,
where $d^{\prime}_{j}\neq d_{j}$, in $\mathbb{D}$. We define a new labeling
set $D_{1}\uplus D_{2}$ for the extended graph $G_{1}\ast G_{2}$ by
$D_{1}\uplus
D_{2}=\\{d_{1},d_{2},d_{3},\ldots,d_{n},d^{\prime}_{1+n},d^{\prime}_{2+n},d^{\prime}_{3+n},\ldots,d^{\prime}_{m+n}\\}$,
where $\ast$ is some binary operation (either union or join of $G_{1}$ and
$G_{2}$) between $G_{1}$ and $G_{2}$.
Note that the sets $D_{1}\uplus D_{2}$ and $D_{2}\uplus D_{1}$ need not be
equal. Also, note that this notion can be applied to index patterns
$\mathscr{I}_{1}$ and $\mathscr{I}_{2}$ as well. Hence, we propose the
following result.
###### Corollary 2.7.
If $G=\bigcup\limits^{k}_{i=1}G_{i}$, then
$\mathbb{G}_{\mathscr{I}}=\bigcup\limits^{k}_{i=1}\mathbb{G}_{i,\mathscr{I}}$
has at least $k$ adult and initiator vertices.
###### Proof.
The result is an immediate consequence of Proposition 2.5. ∎
Recall that the pendant vertices of a tree are called leafs of that tree. If
the given graph $G$ has a pendant vertex, say $v_{j}$, then we say that
$G-v_{j}$ is the graph obtained by lobbing off $v_{j}$.
###### Lemma 2.8.
For a tree $T$ on $n$ vertices there exists at least two index patterns
$\mathscr{I}_{1}$, $\mathscr{I}_{2}$ such that $\mathbb{T}_{\mathscr{I}_{1}}$
has exactly one adult vertex and $\mathbb{T}_{\mathscr{I}_{2}}$ has exactly
one initiator vertex.
###### Proof.
Consider a tree $T$ on $n$ vertices with $t$ leafs. Label the leafs randomly
by $v_{1},v_{2},v_{3},\ldots,v_{t}$ in an injective manner. Now, lob off the
leafs to obtain the subtree $T^{\prime}$ on $n-t$ vertices having $t^{\prime}$
leafs. Label these leafs by $v_{t+1},v_{t+2},v_{t+3},\ldots,v_{t+t^{\prime}}$
injectively in a random manner. Lob off these $t^{\prime}$ leafs. Repeat the
procedure iteratively until we get a single vertex which can be labeled by
$v_{n}$ or we get a $K_{2}$ whose end vertices can be labeled by $v_{n-1}$ and
$v_{n}$. Then, by Definition 2.2, $v_{n}$ will be the unique adult vertex in
the corresponding sprout graph $\mathbb{T}_{\mathscr{I}_{1}}$.
In the similar way, we can find out another sprout graph
$\mathbb{T}_{\mathscr{I}_{2}}$ whose vertices can be labeled in the reverse
order so that $v_{1}$ is the unique initiator vertex of
$\mathbb{T}_{\mathscr{I}_{2}}$. ∎
###### Corollary 2.9.
Every graph $G$ has at least two index patterns $\mathscr{I}_{1}$,
$\mathscr{I}_{2}$ such that the corresponding sprout graph
$\mathbb{G}_{\mathscr{I}_{1}}$ has exactly one adult vertex and
$\mathbb{G}_{\mathscr{I}_{2}}$ has exactly one initiator vertex.
###### Proof.
The result follows immediately from Lemma 2.8 and from the fact that every
connected graph has a spanning tree. ∎
###### Definition 2.10.
The arc-weight of an arc $(v_{i},v_{j})$ of a sprout graph, denoted by
$w(v_{i},v_{j})$, is defined as $w(v_{i},v_{j})=j-i$. If all arcs are labeled
by $a_{i},i=1,2,3,\ldots,\epsilon(\mathbb{G}_{\mathscr{I}})$ in an injective
manner, then the maturity weight of the sprout graph
$\mathbb{G}_{\mathscr{I}}$, denoted by $mw(\mathbb{G}_{\mathscr{I}})$, is
defined to be
$mw(\mathbb{G}_{\mathscr{I}})=\sum\limits_{i=1}^{\epsilon(\mathbb{G}_{\mathscr{I}})}w(a_{i})$.
It can be observed that the sum of arc-weights in a sprout graph
$\mathbb{G}_{\mathscr{I}},$ $\forall\mathscr{I}$ need not be a constant, and
this value depends on the random labeling of its vertices. Hence, for some
index pattern $\mathscr{I}^{\ast}$ within the possible $n!$ index patterns we
obtain,
$\sum\limits_{i=1}^{\epsilon(\mathbb{G}_{\mathscr{I}^{\ast}})}w(a_{i})=\min\\{\sum\limits_{i=1}^{\epsilon(\mathbb{G}_{\mathscr{I}})}w(a_{i})\\}$.
Similarly, for some index pattern $\mathscr{I}^{\prime}$ within the possible
$n!$ index patterns, we have
$\sum\limits_{i=1}^{\epsilon(\mathbb{G}_{\mathscr{I}^{\prime}})}w(a_{i})=\max\\{\sum\limits_{i=1}^{\epsilon(\mathbb{G}_{\mathscr{I}})}w(a_{i})\\}$.
Note that the index patterns $\mathscr{I}^{\ast}$ and $\mathscr{I}^{\prime}$
need not be necessarily unique. Henceforth, $\mathscr{I}^{*}_{i}$ and
$\mathscr{I}^{\prime}_{i}$ will denote index patterns corresponding to
$\min\\{\sum\limits_{i=1}^{\epsilon(\mathbb{G}_{\mathscr{I}})}w(a_{i})\\}$ and
$\max\\{\sum\limits_{i=1}^{\epsilon(\mathbb{G}_{\mathscr{I}})}w(a_{i})\\},$
respectively.
The following is a straight forward result which is important in our further
studies.
###### Lemma 2.11.
Consider a graph $G$ on $n$ vertices and the index patterns
$\mathscr{I}_{1}=\\{v_{1},v_{2},v_{3},\ldots,v_{n}\\}$ and
$\mathscr{I}_{2}=\\{v_{1+k},v_{2+k},v_{3+k},\ldots,v_{n+k}\\}$, where
$k\in\mathbb{N}_{0}$. Then,
$mw(\mathbb{G}_{\mathscr{I}_{1}})=mw(\mathbb{G}_{\mathscr{I}_{2}})$,
$\min(mw(\mathbb{G}_{\mathscr{I}^{\ast}_{1}}))=\min(mw(\mathbb{G}_{\mathscr{I}^{\ast}_{2}}))$
and
$\max(mw(\mathbb{G}_{\mathscr{I}^{\prime}_{1}}))=\max(mw(\mathbb{G}_{\mathscr{I}^{\prime}_{2}}))$.
###### Proof.
The results follow from the fact that $|(i+k)-(j+k)|=|i-j|$. ∎
For graphs $G_{i}$, $1\leq i\leq t$ and corresponding index patterns
$\mathscr{I}_{1},\mathscr{I}_{2},\mathscr{I}_{3},\ldots,\mathscr{I}_{t}$,
consider $H=\bigcup\limits_{i=1}^{t}G_{i}$. Then, by Definition 2.6 and
Corollary 2.7 it can be followed that
$mw(\mathbb{H}_{\mathscr{I}})=\sum\limits_{i=1}^{t}mw(\mathbb{G}_{\mathscr{I}_{i}})$.
We note that the underlying graph of $\mathbb{G}_{t=i}$ is a subgraph of $G$.
Hence, for some $j$ and $j\leq i\leq m_{\mathscr{I}}$, the underlying graph of
$\mathbb{G}_{t=i}$ is a spanning subgraph of $G$.
###### Theorem 2.12.
For any graph $G$, there exists an index pattern $\mathscr{I}$ such that
$\mathbb{G}_{t=1}$ is a directed Hamilton path of the sprout graph
$\mathbb{G}_{\mathscr{I}}$ if and only if $G$ contains a Hamilton path.
###### Proof.
Assume that the given graph $G$ on $n$ vertices has a Hamilton path, say
$P_{n}$. Label the vertices from any end vertex of $P_{n}$ through the
consecutive adjacent vertices by $v_{1},v_{2},v_{3},\ldots,v_{n}$ in an
injective manner. Clearly, $(i+1)-i=1$, for all $1\leq i\leq n-1$. Hence,
$\mathbb{G}_{t=1}=P^{\rightarrow}_{n}$.
Conversely, assume that for a graph $G$ on $n$ vertices,
$\mathbb{G}_{t=1}=P^{\rightarrow}_{n}$. As time proceeds, $2\leq t\leq
m_{\mathscr{I}}$, only arcs between some vertex pairs are added. Hence,
$P_{n}$ is contained completely in the underlying graph of the sprout graph
$\mathbb{G}_{\mathscr{I}}$. Therefore, graph $G$ contains a Hamilton path. ∎
###### Corollary 2.13.
The underlying graph of the sprouting graph $\mathbb{G}_{t=1}$ of a given
graph $G$ is acyclic.
###### Proof.
By Theorem 2.12, we have $\mathbb{G}_{t=1}=P^{\rightarrow}_{n}$, a directed
Hamilton path and hence the underlying graph of $\mathbb{G}_{t=1}$ is a
Hamilton path in $G$. Therefore, the underlying graph of $\mathbb{G}_{t=1}$ is
acyclic. ∎
## 3 Two Fundamental Arc-Weight Principles
For any three positive integers $x,y,z\in\mathbb{N}$ such that $x>z$ and
$y>z$, we have $x-z>y-z\iff x>y$ and $x-y\leq y-z\iff x+z\leq 2y$. Invoking
these inequalities, we introduce two fundamental arc-weight principles. It is
to be noted that the application of the principles may be a complex problem by
itself.
### 3.1 The Maxi-Max Arc-Weight Principle
The maximum maturity weight of a sprout graph $\mathbb{G}_{\mathscr{I}}$ is
obtained by indexing the vertices of $G$ such that, the maximal adjacent
vertex pairs exist such that the $|$arc-weights$|$ are a maximum over all
index patterns.
The maxi-max arc-weight principle (MMAW-Principle) describes an index pattern
$\mathscr{I}^{\prime}$ of the vertices of the given graph $G$ that ensures the
maximum maturity weight of the sprout graph
$\mathbb{G}_{\mathscr{I}^{\prime}}$.
#### Fundamental MMAW-Principle Algorithm
Consider the set of consecutive integers $I=\\{1,2,3,\ldots,n\\},n\geq 2$ and
let $x_{i_{j}}\in I$. First arrange the integers
$x_{i_{1}},x_{i_{2}},x_{i_{3}},\ldots,x_{i_{n}}$ such that $x_{i_{k}}\neq
x_{i_{t}}$ and
$|x_{i_{1}}-x_{i_{2}}|+|x_{i_{2}}-x_{i_{3}}|+|x_{i_{3}}-x_{i_{4}}|+\ldots+|x_{i_{n-1}}-x_{i_{n}}|$
is a maximum. Thereafter arrange the integers such that $x_{i_{k}}\neq
x_{i_{t}}$ and
$|x_{i_{1}}-x_{i_{2}}|+|x_{i_{2}}-x_{i_{3}}|+|x_{i_{3}}-x_{i_{4}}|+\ldots+|x_{i_{n-1}}-x_{i_{n}}|$
is a minimum.
1. S-1:
Begin with: $...,n,1,n-1,..$. $\rightarrow$
2. S-2:
Extend to: $...,2,n,1,n-1,3,..$. $\rightarrow$
3. S-3:
Extend to: $...,n-2,2,n,1,n-1,3,n-3,..$. and so on $\rightarrow$
4. S-4:
Exhaust the procedure to obtain $\ell_{1},\ldots
n-2,2,n,1,n-1,3,n-3,...,\ell_{2}$ with
$(\ell_{1},\ell_{2})=(\lceil\frac{n}{2}\rceil,\lceil\frac{n}{2}\rceil+1)$ or
$(\lceil\frac{n}{2}\rceil+1,\lceil\frac{n}{2}\rceil)$ or
$(\lceil\frac{n-1}{2}\rceil,\lceil\frac{n-1}{2}\rceil+1)$ $\rightarrow$
5. S-5:
Exit.
### 3.2 The Maxi-Min Arc-Weight Principle
The maxi-min arc-weight principle (MmAW-Principle) describes an index pattern
$\mathscr{I}^{\ast}$ of the vertices of the given graph $G$ that ensures the
minimum maturity weight of the sprout graph $\mathbb{G}_{\mathscr{I}^{\ast}}$.
The maxi-min arc-weight principle states that the minimum maturity weight of a
graph $G$ is obtained by indexing the vertices in such a way that the maximal
adjacent vertex pairs exist such that the the absolute values of the arc-
weights are a minimum over all index patterns.
#### MmAW-Principle Algorithm
1. S-1:
Extend to: $1,2,3\ldots,n-1,n$ $\rightarrow$
2. S-2:
Exit.
It is easy to see that both of these informal algorithms are well-defined and
converges. These algorithms find immediate application for paths $P_{n}$,
$n\geq 2$.
Since a graph $G$ on $n$ vertices is a spanning subgraph of $K_{n}$, the
vertices of $K_{n}$ can be labeled randomly
$\mathscr{I}=\\{v_{1},v_{2},v_{3},...,v_{n}\\}$. Certainly, the graph $G$ can
be obtained by removing $\frac{n(n-1)}{2}-\epsilon(G)$ carefully selected
edges from $K_{n}$. Let $\overline{G}$ denote the complement of $G$.
It implies that if the edges are carefully selected for removal so as to
ensure maxi-min arc-weights remaining in
$\mathbb{G}_{\mathscr{I}^{\ast}_{1}}$, then
$\max(mw(\overline{\mathbb{G}_{\mathscr{I}^{\ast}_{1}}})$, corresponding to
the index pattern $\mathscr{I}^{\ast}_{1}$ concerned, is obtained.
Similarly, if the edges are carefully selected for removal so as to ensure the
maxi-max arc-weights remaining in $\mathbb{G}_{\mathscr{I}^{\prime}_{2}}$,
then $\min(mw(\overline{\mathbb{G}_{\mathscr{I}^{\prime}_{2}}})$,
corresponding to the index pattern $\mathscr{I}^{\prime}_{2}$ concerned, is
obtained.
From the observations above the next useful result follows.
###### Theorem 3.1.
Consider a graph $G$ on $n$ vertices. Let $\mathscr{I}^{\ast}_{1}$ and
$\mathscr{I}^{\prime}_{2}$ be two index patterns such that
$mw(\mathbb{G}_{\mathscr{I}^{\ast}_{1}})=\min(mw(\mathbb{G}_{\mathscr{I}}))$
and
$mw(\mathbb{G}_{\mathscr{I}^{\prime}_{2}})=\max(mw(\mathbb{G}_{\mathscr{I}}))$.
Then
$mw(\overline{\mathbb{G}_{\mathscr{I}^{\ast}_{1}}})=\max(mw(\overline{\mathbb{G}_{\mathscr{I}}}))$
and
$mw(\overline{\mathbb{G}_{\mathscr{I}^{\prime}_{2}}})=\min(mw(\overline{\mathbb{G}_{\mathscr{I}}}))$.
###### Proof.
(i) Let $\mathscr{I}^{\ast}_{1}$ be such that
$mw(\mathbb{G}_{\mathscr{I}^{\ast}_{1}})=\min(mw(\mathbb{G}_{\mathscr{I}}))$.
It implies that maximal number of edges with maximum index differences have
been removed from $K_{n}$ to obtain $G$. Hence, the maxi-max index differences
edges are the edges of $\overline{G}$. Therefore,
$mw(\overline{\mathbb{G}_{\mathscr{I}^{\ast}_{1}}})=\max(mw(\overline{\mathbb{G}_{\mathscr{I}}}))$.
(ii) A similar reasoning as in (i) can be applied to prove part (ii). ∎
###### Lemma 3.2.
If for a graph $G$ an index pattern $\mathscr{I}$ exists such that
$\mathsf{T}_{\mathscr{I}}=\\{0,1\\}$ then,
$mw(\mathbb{G}_{\mathscr{I}})=\min(mw(\mathbb{G}_{\mathscr{I}}))=\epsilon(G)$.
###### Proof.
For any index pattern $\mathscr{I}$ we have $|w(v_{i},v_{j})|\geq 1$, for any
arc $(v_{i},v_{j})$ in $\mathbb{G}_{\mathscr{I}}$. Hence, the proof is
obvious. ∎
Lemma 3.2 implies that $mw(\mathbb{G}_{\mathscr{I}})=\epsilon(G)$ if and only
if $G=P_{n}$, where $n\geq 1$.
## 4 Sprout Graphs of Certain Classes of Graphs
Since complete graphs, paths and possibly cycles and stars amongst others form
part of the skeleton of all graphs the introduction of sprouting to these
graph classes will nourish further studies.
### 4.1 Sprouting of Complete Graphs
###### Proposition 4.1.
For all indexing sets $\mathscr{I}$, the maturity weight of the sprout graph
of a complete graph $K_{n}$ is
$mw(\mathbb{K}_{n,{\mathscr{I}}})=\sum\limits_{j=1}^{n-1}\sum\limits_{i=1}^{n-j}(n-i)=\sum\limits_{j=1}^{n-1}\sum\limits_{i=1}^{n-j}i$.
###### Proof.
Randomly label the vertices of the complete graph $K_{n}$ by
$v_{1},v_{2},v_{3},\ldots,v_{n}$. Now, consider the matured sprout graph
$\mathbb{K}_{n,{\mathscr{I}}}$. Regardless of the random indexing of the
vertices we have the following arc-weights,
$w(v_{1},v_{i})=i-1~{}\forall\,2\leq i\leq n$,
$w(v_{2},v_{i})=i-2~{}\forall\,3\leq i\leq n$, ……, $w(v_{n-1},v_{n})=1$.
Hence, by summing all columns carrying equal arc-weights, across all the above
mentioned rows, we have
$mw(\mathbb{K}_{n,{\mathscr{I}}})=\sum\limits_{i=1}^{n-1}(n-i)+\sum\limits_{i=1}^{n-2}(n-i)+\sum\limits_{i=1}^{n-3}(n-i)+\ldots+\sum\limits_{i=1}^{n-(n-1)}(n-i)=\sum\limits_{j=1}^{n-1}\sum\limits_{i=1}^{n-j}(n-i)=\sum\limits_{j=1}^{n-1}\sum\limits_{i=1}^{n-j}i$
and hence the result follows. ∎
###### Corollary 4.2.
For every index pattern $\mathscr{I}$, the complete sprout graph
$\mathbb{K}_{n,{\mathscr{I}}}$ has one (unique) adult vertex, $v_{n}$ and one
(unique) initiator vertex, $v_{1}$.
###### Proof.
Write $\mathbb{K}_{n,{\mathscr{I}}}$ as $\mathbb{K}_{n}$ for brevity. Since
any vertex $v_{i},i<n$ is always a tail to $v_{n}$, the vertex $v_{i}$ will
always have $d^{+}_{\mathbb{K}_{n}}(v_{i})\geq 1$ in
$\mathbb{K}_{n,{\mathscr{I}}}\implies
d^{-}_{\mathbb{K}_{n}}(v_{i})<d_{\mathbb{K}_{n}}(v_{i})$. Since it contradicts
Definition 2.2, the result follows from Proposition 2.5. By similar arguments,
we can establish the result for the unique initiator vertex also. ∎
###### Lemma 4.3.
For a graph $G$ on $n$ vertices, we have
$\max(mw(\mathbb{G}_{\mathscr{I}}))\leq\min(mw(\mathbb{K}_{n,{\mathscr{I}}}))=\max(mw(\mathbb{K}_{n,{\mathscr{I}}}))$.
###### Proof.
It follows from Corollary 4.2 that
$\min(mw(\mathbb{K}_{n,{\mathscr{I}}}))=\max(mw(\mathbb{K}_{n,{\mathscr{I}}}))$.
Since $|\epsilon(K_{n})|\geq|\epsilon(G)|$, where $G$ is a graph on $n$
vertices, the removal of edges from $K_{n}$ to obtain $G$ results in reducing
the corresponding terms in the summation
$mw(\mathbb{K}_{n,{\mathscr{I}}})=\sum\limits_{i=1}^{n-1}(n-i)+\sum\limits_{i=1}^{n-2}(n-i)+\sum\limits_{i=1}^{n-3}(n-i)+\ldots+\sum\limits_{i=1}^{n-(n-1)}(n-i)=\sum\limits_{j=1}^{n-1}\sum\limits_{i=1}^{n-j}(n-i)=\sum\limits_{j=1}^{n-1}\sum\limits_{i=1}^{n-j}i$,
to zero. Therefore, $max(mw(\mathbb{G}_{\mathscr{I}}))\leq
min(mw(\mathbb{K}_{n,{\mathscr{I}}}))=\max(mw(\mathbb{K}_{n,{\mathscr{I}}}))$.
∎
### 4.2 Sprouting of Paths
###### Proposition 4.4.
For the path $P_{n}$, for $n\geq 2$, we have
$\min(mw(\mathbb{P}_{n,{\mathscr{I}^{\ast}_{1}}}))=n-1$, and
$\max(mw(\mathbb{P}_{n,{\mathscr{I}^{\prime}_{2}}}))=\sum\limits_{i=0}^{\lceil\frac{n}{2}\rceil-2}(2n-3-4i)$.
###### Proof.
Consider the path $P_{n},n\geq 2$ as a graph with its $n$ vertices seated on a
horizontal line. The labeling of vertices of $G$ can be done as explained
below.
(i) Label the vertices consecutively by $v_{1},v_{2},v_{3},...,v_{n-1},v_{n}$
, from the leftmost vertex onwards. Let this be the index pattern
$\mathscr{I}^{\ast}_{1}$. Clearly, we have
$m_{\mathscr{I}^{\ast}_{1}}=\max\\{|i-j|\\}=1$, for all sprouts $(i,j)$ and
hence $\mathbb{T}_{\mathscr{I}^{\ast}_{1}}=\\{0,1\\}$. Therefore, arcs having
arc-weight $1$, will arc at $t=1$. Since there are exactly $(n-1)$ such arcs
in $\mathbb{P}_{n,{\mathscr{I}^{\ast}_{1}}}$, by Lemma 3.2, the first part of
the result follows.
(ii) Label the vertices from left to right consecutively with the default
labelling $\\{d_{i}:1\leq i\leq n\\}$. Then we have to consider the following
cases.
Case 1: Let $n$ be odd. Label the central vertex
$d_{\lceil\frac{n}{2}\rceil+1}$ to be $v_{1}$. Label the vertices adjacent to
$v_{1}$ respectively $v_{n}$ and $v_{n-1}$ in accordance with Step-2 of MMAW-
Principle algorithm. Now, label the other vertices exhaustively in accordance
with the MMAW-Principle algorithm to get the index pattern
$\mathscr{I}^{\prime}_{2}=\\{v_{\ell_{1}},\ldots,v_{n-2},v_{2},v_{n},\\\
v_{1},v_{3},v_{n-3},\ldots,v_{\ell_{2}}\\}$,
$(\ell_{1},\ell_{2})=(\lceil\frac{n}{2}\rceil,\lceil\frac{n}{2}\rceil+1)$ or
$(\lceil\frac{n}{2}\rceil+1,\lceil\frac{n}{2}\rceil)$ or
$(\lceil\frac{n-1}{2}\rceil,\lceil\frac{n-1}{2}\rceil+1)$. Then, by the MMAW-
Principle and invoking Definition 2.10, we have the required condition
$\max(mw(\mathbb{P}_{n,{\mathscr{I}^{\prime}_{2}}}))=\sum\limits_{i=0}^{\lceil\frac{n}{2}\rceil-2}(2n-3-4i)$.
Case 2: Let $n$ be even. Now, the path does not have a central vertex, instead
a pair of central vertices exists. Without loss of generality, label the
rightmost central vertex (that is, the $\frac{n+1}{2}$-th vertex) by $v_{1}$
and label the vertex to the left adjacent to $v_{1}$ by $v_{n}$ and the vertex
to the right adjacent to $v_{1}$ by $v_{n-1}$. Proceed with this labeling
exhaustively as explained in Case-1. Therefore, the required result follows as
explained in Case-1. ∎
###### Corollary 4.5.
For the path $P_{n}$ we have
$\min(mw(\mathbb{P}_{2,t=1}))=\max(mw(\mathbb{P}_{2,t=1}))$ and
$\min(mw(\mathbb{P}_{n,{\mathscr{I}^{\ast}_{1}}}))=mw(\mathbb{P}_{n,t=1})<\max(mw(\mathbb{P}_{n,{\mathscr{I}^{\prime}_{2}}}))$,
for $n\geq 3$.
###### Proof.
The result is an immediate consequence of Proposition 4.4. ∎
From Proposition 4.1 and Proposition 4.4, we have for a graph $G$, $n-1\leq
mw(\mathbb{G}_{\mathscr{I}})\leq\sum\limits_{j=1}^{n-1}\sum\limits_{i=1}^{n-j}i$.
Hence, we get a result which states that $\forall n\in\mathbb{N}$, there exist
a graph $G$ and an index pattern $\mathscr{I}$ for which
$\min(mw(\mathbb{G}_{\mathscr{I}}))=n$. The graph $G$ is the path $P_{n+1}$
with index pattern found in the first part of the proof of Proposition 4.4. A
similar result cannot be found for $\max(mw(\mathbb{G}_{\mathscr{I}}))$.
###### Theorem 4.6.
[Zané’s111The first author wishes to dedicate this theorem to Zané van der
Merwe who it is hoped, will grow up to be a great mathematician.] Consider the
set of graphs $\mathcal{G}=\\{G:\epsilon(G)=q\\}$. For a graph
$H\in\mathcal{G}$, we have
$\min(mw(\mathbb{H}_{\mathscr{I}}))=\min(\min(mw(\mathbb{G}_{\mathscr{I}^{\prime}})))$
if and only if $H\cong P_{q+1}$.
###### Proof.
Clearly the result holds for $q=1,2,3$. Assume the result holds for all
$G\in\mathcal{G}$ with $4\leq\epsilon(G)\leq k$. Hence,
$\min(mw(\mathbb{P}_{k+1,t=1}))=\min(\min(mw(\mathbb{G}_{\mathscr{I}^{\prime}})))$,
$\epsilon(G)=k$. Now, consider consider a graph $G$ with $\epsilon(G)=k+1$ and
let $H\cong P_{k+2}$. Clearly,
$\min(mw(\mathbb{P}_{k+2,t=1}))=\min(mw(\mathbb{P}_{k+1,t=1}))+1$. It is the
minimum increase in maturity weight possible and hence,
$\min(\min(mw(\mathbb{G}_{\mathscr{I}^{\prime}})))=\min(mw(\mathbb{H}_{\mathscr{I}}))=\min(mw(\mathbb{P}_{k+2,t=1}))$.
Conversely, assume there exists a graph $H\ncong P_{k+2}$ such that
$\min(mw(\mathbb{H}_{\mathscr{I}}))=\min(\min(mw(\mathbb{G}_{\mathscr{I}^{\prime}})))$,
with $\epsilon(H)=k+1$. Then, it follows that $P_{k+1}$ is a subgraph of $H$.
Hence, to add the additional edge an additional pendant vertex (leaf) was
added to $P_{k+1}$ to obtain $H$. This, however, implies that the increase in
minimum maturity weight by
$\min(mw(\mathbb{H}_{\mathscr{I}}))-\min(mw(\mathbb{P}_{t=1}))\geq 2$. It is a
contradiction, since
$\min(mw(\mathbb{P}_{k+2,t=1}))-\min(mw(\mathbb{P}_{k+1,t=1}))=1$. Therefore,
we must have $H\cong P_{k+2}$. ∎
Note that we can not find a complete graph $K_{q}$ such that
$\epsilon(K_{q})=n$ for all integral values of $n$. Hence, a result analogous
to Theorem 4.6 to determine $\max(\max(mw(\mathbb{G}_{\mathscr{I}})))$ does
not exist.
### 4.3 Sprouting of Cycles
###### Proposition 4.7.
For the cycle $C_{n}$, where $n\geq 4$, we have
$\min(mw(\mathbb{C}_{n,{\mathscr{I}^{\ast}_{1}}}))=2(n-1)$ and
$\max(mw(\mathbb{C}_{n,{\mathscr{I}^{\prime}_{2}}}))=\max(mw(\mathbb{P}_{k+1,{\mathscr{I}^{\prime}_{2}}}))+1=\sum\limits_{i=0}^{\lceil\frac{n}{2}\rceil-2}(2n-3-4i)+1$.
###### Proof.
Identify the cycle $C_{n},n\geq 4$ as the graph with the $n$ vertices seated
on the circumference of a circle with a vertex seated centre at the top. Then,
the labeling of vertices of $C_{n}$ can be done as explained below.
(i) Label the top vertex by $v_{1}$ and label the other vertices clockwise
$v_{2},v_{3},...,v_{n}$. Call the index pattern $\mathscr{I}^{\ast}_{1}$.
Clearly, the arc-weights, $w(v_{i},v_{j})=1$ except for the arc
$w(v_{1},v_{n})=n-1$. Hence,
$\mathbb{T}_{\mathscr{I}^{\ast}_{1}}=\\{0,1,n-1\\}$. Therefore, all arcs
having arc-weight $1$, will arc at $t=1$. There are exactly $(n-1)$ such arcs
in $\mathbb{C}_{n,t=1}$ and the last arc $(v_{1},v_{n})$ arcs at $t=(n-1)$, so
$mw(\mathbb{C}_{n,t=n-1})=2(n-1)$. Without loss of generality, interchange the
vertex labeling $v_{n}$ and $v_{i}$, $i<n$ to obtain $\mathscr{I}^{\prime}$.
The only possible decrease in the maturity weight is on condition that
$n-2<i<n$, $i\in\mathbb{R}$. For $i=n-1$ we have
$mw(\mathbb{P}_{n,{\mathscr{I}^{\prime}}})=mw(\mathbb{P}_{n,t=n-1})$. Hence,
$\min(mw(\mathbb{P}_{n,{\mathscr{I}^{\ast}_{1}}}))=mw(\mathbb{P}_{n,t=n-1})$.
(ii) Consider the path $P_{3}$ and label the vertices $v_{3},v_{1},v_{2}$ in
accordance with the MMAW-Principle. Then, here the end vertices have index
difference $1$ hence
$\max(mw(\mathbb{C}_{3,t=2}))=4=\max(mw(\mathbb{P}_{3,t=2}))+1$, as the end
vertices has the indexes $\lceil\frac{3}{2}\rceil$,
$\lceil\frac{3}{2}\rceil+1$ respectively. Next, assume that the result holds
for $C_{k}$, $k\geq 4$. Hence,
$\max(mw(\mathbb{C}_{k,{\mathscr{I}^{\prime}_{2}}}))=\max(mw(\mathbb{P}_{k,1}))+1$
and the end vertices of the path $P_{k}$ have indexes either
$\lceil\frac{k}{2}\rceil$, $\lceil\frac{k}{2}\rceil+1$ or
$\lceil\frac{k-1}{2}\rceil$, $\lceil\frac{k-1}{2}\rceil+1$ respectively. Now
consider the path $P_{k+1}$. Clearly, after labeling the vertices in
accordance with the MMAW-Principle, the end vertices have indexes either
$\lceil\frac{k+1}{2}\rceil$, $\lceil\frac{k+1}{2}\rceil+1$ or
$\lceil\frac{k+1}{2}\rceil$, $\lceil\frac{k}{2}\rceil+1$. In both cases the
index difference between the end vertices is $1$ and hence the result
$\max(mw(\mathbb{C}_{k,{\mathscr{I}^{\prime}_{2}}}))=\max(mw(\mathbb{P}_{k+1,{\mathscr{I}^{\prime}_{2}}}))+1$
holds. Hence, the result follows by induction. ∎
###### Corollary 4.8.
For the cycle $C_{n}$ we have
$\min(mw(\mathbb{C}_{3,t=1}))=\max(mw(\mathbb{C}_{3,t=1}^{s}))$ and
$\min(mw(\mathbb{C}_{n,{\mathscr{I}^{\ast}_{1}}}))=mw(\mathbb{C}_{n,t=(n-1)})<\max(mw(\mathbb{C}_{n,{\mathscr{I}^{\prime}_{2}}}))$,
where $n\geq 4$.
###### Proof.
The proof follows immediately from Proposition 4.7. ∎
From Proposition 4.7, it follows that for every positive integer $n\geq 3$,
there exist a graph and an index pattern $\mathscr{I}$ for which
$\min(mw(\mathbb{G}_{\mathscr{I}}))=2n$. The graph is the cycle $C_{n+1}$ with
an index pattern found in the first part of the proof of Proposition 4.7. An
analogous result cannot be found for $\max(mw(\mathbb{G}_{\mathscr{I}}))$.
It has been established that if two different random index patterns of a graph
$G$ say $\mathscr{I}_{1}$ and $\mathscr{I}_{2}$ result in
$\mathbb{T}_{\mathscr{I}_{1}}$ and $\mathbb{T}_{\mathscr{I}_{2}}$
respectively, such that
$\mathbb{T}_{\mathscr{I}_{1}}=\mathbb{T}_{\mathscr{I}_{2}}$ then,
$\mathbb{T}_{\mathscr{I}_{1}}=\mathbb{T}_{\mathscr{I}_{2}}\centernot\implies
mw(\mathbb{G}_{\mathscr{I}_{1}})=mw(\mathbb{G}_{\mathscr{I}_{2}})$.
### 4.4 Sprouting of Stars
###### Theorem 4.9.
The sprout graph of star $K_{(1,n)}$ has
1. (i)
$\min(mw(\mathbb{K}_{(1,n),{\mathscr{I}^{\ast}_{1}}}))=\begin{cases}2\sum\limits_{i=1}^{\lceil\frac{(n+1)}{2}\rceil-1}i+\lceil\frac{(n+1)}{2}\rceil,&\text{if
$n\geq 3$ and odd},\\\
2\sum\limits_{i=1}^{\lceil\frac{(n+1)}{2}\rceil-1}i,&\text{if $n\geq 2$ and
even}.\end{cases}$
2. (ii)
$\max(mw(\mathbb{K}_{(1,n),{\mathscr{I}^{\prime}_{2}}}))=\sum\limits_{i=1}^{n}i,~{}\forall
n\in\mathbb{N}.$
###### Proof.
(i) First consider the star graph $K_{(1,3)}$. Note that in the table that
follows; $i,j,k,l\in\\{1,2,3,4\\}$. The possible $4!$ index patterns with the
corresponding values of $mw(\mathbb{K}_{(1,3),\mathscr{I}})$ are given in the
following table.
central vertex $v_{i}$ | leaf $v_{j}$ | leaf $v_{k}$ | leaf $v_{l}$ | $mw(\mathbb{K}_{(1,3),\mathscr{I}})$
---|---|---|---|---
1 | 2 | 3 | 4 | 6
1 | 2 | 4 | 3 | 6
1 | 3 | 2 | 4 | 6
1 | 3 | 4 | 2 | 6
1 | 4 | 2 | 3 | 6
1 | 4 | 3 | 2 | 6
2 | 1 | 3 | 4 | 4
2 | 1 | 4 | 3 | 4
2 | 3 | 1 | 4 | 4
2 | 3 | 4 | 1 | 4
2 | 4 | 1 | 3 | 4
2 | 4 | 3 | 1 | 4
3 | 1 | 2 | 4 | 4
3 | 1 | 4 | 2 | 4
3 | 2 | 1 | 4 | 4
3 | 2 | 4 | 1 | 4
3 | 4 | 1 | 2 | 4
3 | 4 | 2 | 1 | 4
4 | 1 | 2 | 3 | 6
4 | 1 | 3 | 2 | 6
4 | 2 | 1 | 3 | 6
4 | 2 | 3 | 1 | 6
4 | 3 | 1 | 2 | 6
4 | 3 | 2 | 1 | 6
Clearly,
$\min(mw(\mathbb{K}_{(1,3),t=2}))=4=2\sum\limits_{i=1}^{\lceil\frac{(3+1)}{2}\rceil-1}i+\lceil\frac{(3+1)}{2}\rceil$.
Therefore, the results holds for $K_{(1,3)}$. Next, assume it holds for
$K_{(1,q)},~{}q>3$ and $q$ is odd. Hence, it is assumed that
$\min(mw(\mathbb{K}_{(1,q),\mathscr{I}}))=2\sum\limits_{i=1}^{\lceil\frac{(q+1)}{2}\rceil-1}i+\lceil\frac{(q+1)}{2}\rceil$.
Now, consider the graph $K_{(1,q+2)}$. We have
$\lceil\frac{q+3}{2}\rceil=\lceil\frac{(q+1)}{2}\rceil+1$. Hence, the central
vertex index increases to $\lceil\frac{(q+1)}{2}\rceil+1$. This results in the
central vertex, $v_{\lceil\frac{(q+1)}{2}\rceil}$ in $K_{(1,q)}$ to become a
leaf in $K_{(1,q+2)}$, and vertex $v_{(\lceil\frac{(q+1)}{2}\rceil+1)}$
becomes the central vertex in $K_{(1,q+2)}$. Thus, for all
$v_{i},i<\lceil\frac{q+1}{2}\rceil$ in $K_{(1,q)}$, the difference
$|\lceil\frac{(q+1)}{2}\rceil-i|$ increases by $1$ in $K_{(1,q+2)}$. The value
$|(\lceil\frac{(q+1)}{2}\rceil+1)-\lceil\frac{(q+1)}{2}\rceil|$ repeats twice
due to the index interchanging of the central vertex. Then, exact mirror
values follow with the value
$\lceil\frac{q+1}{2}\rceil+1=\lceil\frac{q+3}{2}\rceil$, added as well. Hence,
the result (i) holds for $K_{(1,q)},q>3$ and $q$ is odd. Hence, in general it
follows that
$\min(mw(\mathbb{K}_{(1,n),{\mathscr{I}^{\ast}_{1}}}))=2\sum\limits_{i=1}^{\lceil\frac{(n+1)}{2}\rceil-1}i)+\lceil\frac{(n+1)}{2}\rceil$,
if $n\geq 3$ and is odd.
Now, consider the graph $K_{(1,2)}$. Note that in the table that follows;
$i,j,k\in\\{1,2,3\\}$. The possible $3!$ index patterns with the corresponding
values $mw(\mathbb{K}_{(1,2),\mathscr{I}})$ are given in the following table.
Central vertex $v_{i}$ | leaf $v_{j}$ | leaf $v_{k}$ | $mw(\mathbb{K}_{(1,2),\mathscr{I}})$
---|---|---|---
1 | 2 | 3 | 3
1 | 3 | 2 | 3
2 | 1 | 3 | 2
2 | 3 | 1 | 2
3 | 1 | 2 | 3
3 | 2 | 1 | 3
Clearly,
$\min(mw(\mathbb{K}_{(1,2),\mathscr{I}}))=2=2\sum\limits_{i=1}^{\lceil\frac{(2+1)}{2}\rceil-1}i$.
Hence, the results holds for $K_{(1,2)}$. Assume it holds for $K_{(1,q)},q>2$
and $q$ is even. Hence, it is assumed that
$\min(mw(K_{(1,q),\mathscr{I}}))=2\sum\limits_{i=1}^{\lceil\frac{(q+1)}{2}\rceil-1}i$.
Now, consider the graph $K_{(1,q+2)}$. As explained in the first part of (i),
we have that $\lceil\frac{(q+3)}{2}\rceil=\lceil\frac{(q+1)}{2}\rceil+1$.
Hence, the central vertex index increases to $\lceil\frac{(q+1)}{2}\rceil+1$.
This results in the central vertex $v_{\lceil\frac{(q+1)}{2}\rceil}$ in
$K_{(1,q)}$ to become a leaf in $K_{(1,q+2)}$ and vertex
$v_{(\lceil\frac{(q+1)}{2}\rceil+1)}$ becomes central vertex in $K_{(1,q+2)}$.
Thus, for all $v_{i},i<\lceil\frac{(q+1)}{2}\rceil$ in $K_{(1,q)}$, the
difference $|\lceil\frac{(q+1)}{2}\rceil-i|$ increases by $1$ in
$K_{(1,q+2)}$. Therefore, as explained in the previous case, the result
follows.
(ii) By labeling the central vertex by $v_{1}$ and the leafs by
$v_{2},v_{3},\ldots,v_{n+1}$ the result can be proved similarly explained in
$(i)$. ∎
###### Corollary 4.10.
For the sprout star $\mathbb{K}_{(1,n),\mathscr{I}}$, the central vertex is
indexed $\ell$ where
$\ell\in\begin{cases}\\{\lceil\frac{(n+1)}{2}\rceil,\lceil\frac{(n+1)}{2}\rceil+1\\},&\text{for
$\min(mw(\mathbb{K}_{(1,n),\mathscr{I}}))$ if $n\geq 3$ is odd},\\\
\\{\lceil\frac{(n+1)}{2}\rceil\\},&\text{for
$\min(mw(\mathbb{K}_{(1,n),\mathscr{I}}))$ if $n\geq 3$ is even},\\\
\\{1,n+1\\},&\text{for
$\max(mw(\mathbb{K}_{(1,n),\mathscr{I}}))$}.\end{cases}$
###### Proof.
If $n$ is odd, $n+1$ is even and we have two central indexes namely,
$\lceil\frac{(n+1)}{2}\rceil$ and $\lceil\frac{(n+1)}{2}\rceil+1$ allowing
minimal mirror image vertex index differences. Hence,
$\min(mw(\mathbb{K}_{(1,n),\mathscr{I}}))=2\sum\limits_{i=1}^{\lceil\frac{(n+1)}{2}\rceil-1}i+\lceil\frac{(n+1)}{2}\rceil=2\sum\limits_{i=\lceil\frac{(n+1)}{2}\rceil-1}^{n+1}{|(\lceil\frac{(n+1)}{2}\rceil+1)-i|}+\lceil\frac{(n+1)}{2}\rceil$.
If $n$ is even, $n+1$ is odd and hence we have a unique central vertex index
allowing minimal mirror image vertex index differences to be exactly,
$\lceil\frac{(n+1)}{2}\rceil$. Hence,
$\min(mw(\mathbb{K}_{(1,n),\mathscr{I}}))=2\sum\limits_{i=1}^{\lceil\frac{(n+1)}{2}\rceil-1}i)=2\sum\limits_{i=\lceil\frac{(n+1)}{2}\rceil-1}^{n+1}{|\lceil\frac{(n+1)}{2}\rceil-i|)}$.
Hence we have the result as
$\ell\in\begin{cases}\\{\lceil\frac{(n+1)}{2}\rceil,\lceil\frac{(n+1)}{2}\rceil+1\\},&\text{for
$\min(mw(\mathbb{K}_{(1,n),\mathscr{I}}))$ if $n\geq 3$ and is odd},\\\
\\{\lceil\frac{(n+1)}{2}\rceil\\},&\text{for
$\min(mw(\mathbb{K}_{(1,n),\mathscr{I}}))$ if $n\geq 3$ and is
even},\end{cases}$
follows.
Since $|1-(i+1)|$, for all $i\in\\{2,3,\ldots,n\\}$, is equal to $|(n+1)-i|$,
for all $i\in\\{1,2,\ldots,n-1\\}$ and assures maximal vertex index
differences, the result $\ell\in\\{1,n+1\\}$ for determining
$\max(mw(\mathbb{K}_{(1,n),\mathscr{I}}))$ follows. ∎
### 4.5 Sprout Complete Bi-partite Graphs
A complete bi-partite graph $K_{(n,m)}$ has $V(K_{(n,m)})=\\{d_{i}:1\leq i\leq
n\\}\bigcup\\{d^{\prime}_{i}:1\leq i\leq m\\}$ and
$E(K_{(n,m)})=\\{d_{i}d^{\prime}_{j}:1\leq i\leq n$, $1\leq j\leq m\\}$.
###### Proposition 4.11.
For a complete bi-partite graph $K_{(n,m)}$, $n,m\geq 2$ we have
(i)
$\min(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\ast}_{1}}}))=\begin{cases}\frac{nm}{2}(n+m-1)+\frac{m}{2}(m+1),&\text{if
$n+m$ is odd},\\\ \frac{nm}{2}(n+m),&\text{if $n+m$ is even}.\end{cases}$
(ii)
$\max(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\prime}_{2}}}))=\begin{cases}(n(n-1)+\frac{nm}{2}(n+m),&\text{if
$n+m$ is even},\\\
(n(\lfloor\frac{n+m}{2}\rfloor-1)+\frac{n}{2}(n+1))(n+m),&\text{if $n+m$ is
odd}.\end{cases}$
###### Proof.
(i) Consider a complete bi-partite graph $K_{(n,m)}$, $n,m\geq 2$ and $n\geq
m$. Without loss of generality let the left column have $n$ vertices and the
right column have $m$ vertices. Label the vertices according to
$\mathscr{I}^{\ast}_{1}$ as follows; the left column from top down,
$v_{1},v_{m+2},v_{m+3},\ldots,v_{m+n}$ and the right column from top down,
$v_{2},v_{3},v_{4},\ldots,v_{m+1}$. Clearly in terms of the MMAW-Principle,
the maximum number of edges have been removed from $K_{(n+m)}$, all with
maximum index difference, to construct $K_{(n,m)}$.
Subcase (i)(a): Assume that $n+m$ is odd. The index differences $((m+2)+i)-j$,
where $1\leq i\leq n-2$ and $2\leq j\leq m+1$, can be written in a
$(n-1)\times(n+m)-2$ matrix form as
$\mathbb{A}=\begin{pmatrix}1&2&3&\ldots&m&0&0&\ldots&0\\\
0&2&3&4&\ldots&m+1&0&\ldots&0\\\ {9}\\\
0&0&\ldots&0&n-1&n&n+1&\ldots&(n+m)-2\end{pmatrix}.$
Hence, we have
$\min(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\ast}_{1}}}))=\sum\limits^{n}_{i=1}\sum\limits^{m}_{j=1}a_{ij}$,
$a_{ij}\in\mathbb{A}$.
Alternatively, let $t=n+m$. It follows from $\mathbb{A}$ that the matrix can
rather be written as a $m$ x $(n+m)-2$ triangular array $\mathbb{A}^{\ast}$,
with each row having odd number of entries namely
$1$, $2$, $3$, $\ldots$, $m$, $m+1$, $\ldots$, $\frac{(n+m)-1}{2}$, $\ldots$,
$n-2$, $n-1$, $\ldots$, $t-4$, $t-3$, $t-2$
$2$, $3$, $\ldots$, $m$, $m+1$, $\ldots$$,\frac{(n+m)-1}{2}$, $\ldots$, $n-2$,
$n-1$$,\ldots$, $t-4$, $t-3$
$3$, $\ldots$, $m$, $m+1$, $\ldots$$,\frac{(n+m)-1}{2}$, $\ldots$, $n-2$,
$n-1$, $\ldots$, $t-4$
$\vdots$
$m$, $m+1$, $\ldots$$,\frac{(n+m)-1}{2}$, $\ldots$, $n-2$, $n-1$.
Clearly,
$\min(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\ast}_{1}}}))>\sum\limits_{a_{ij}\in\mathbb{A}^{\ast}}a_{ij}$.
Then, the above expressions can be written as
$\min(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\ast}_{1}}}))>\sum\limits^{(n-1)}_{i=1}\sum\limits^{i+(m-1)}_{j=i}j$.
Equality is obtained by adding the index difference $k-1$, where $2\leq k\leq
m+1$. Hence,
$\min(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\ast}_{1}}}))=\sum\limits^{(n-1)}_{i=1}\sum\limits^{i+(m-1)}_{j=i}j+\sum\limits^{m}_{i=1}i$.
Therefore, the subcase (i)(a) is settled.
Subcase (i)(b): Assume $n+m$ is even. Similar reasoning can be applied as in
subcase (i)(a) except the fact that each row has even number of entries.
(ii) Consider a complete bi-partite graph $K_{(n,m)}$, $n,m\geq 2$ and $n\leq
m$. Without loss of generality let the left column have $n$ vertices and the
right column have $m$ vertices. Label the vertices according to
$\mathscr{I}^{\prime}_{2}$ as follows; the left column from top down,
$v_{1},v_{2},v_{3},\ldots,v_{n}$ and the right column from top down,
$v_{n+1},v_{n+2},v_{n+3},\ldots,v_{n+m}$. Clearly, by the MmAW-Principle, the
maximum number of edges have been removed from $K_{(n+m)}$, all with minimum
index difference, to construct $K_{(n,m)}$.
Subcase (ii)(a): Assume $n+m$ is even. The index differences $(n+i)-j$, $1\leq
i\leq m$ and $1\leq j\leq n$ can be written in a $n$ x $m$ matrix form as
$\mathbb{A}=\begin{pmatrix}1&2&3&\ldots&m\\\ 2&3&4&\ldots&m+1\\\ {5}\\\
n&n+1&n+2&\ldots&(n+m)-1\end{pmatrix}.$
Hence, we have
$\max(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\prime}_{2}}}))=\sum\limits^{n}_{i=1}\sum\limits^{m}_{j=1}a_{ij}$,
$a_{ij}\in\mathbb{A}$.
Alternatively, let $t=n+m$. It follows from $\mathbb{A}$ that the matrix can
rather be written as a $(n+m)-1$ x $n$ triangular array $\mathbb{A}^{*}$, with
each row having odd number of entries namely
$1$, $2$, $3$, $\ldots$, $n$, $n+1$, $\ldots$, $\frac{n+m}{2}$, $\ldots$,
$m-1$, $m$, $\ldots$, $t-3$, $t-2$, $t-1$
$2$, $3$, $\ldots$, $n$, $n+1$, $\ldots$$,\frac{n+m}{2}$, $\ldots$, $m-1$,
$m$$,\ldots$, $t-3$, $t-2$
$3$, $\ldots$, $n$, $n+1$, $\ldots$$,\frac{n+m}{2}$, $\ldots$, $m-1$, $m$,
$\ldots$, $t-3$
$\vdots$
$n$, $n+1$, $\ldots$$,\frac{n+m}{2}$, $\ldots$, $m-1$, $m$.
Clearly,
$\max(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\prime}_{2}}}))=\sum\limits_{a_{ij}\in\mathbb{A}^{\ast}}a_{ij}$.
The above expressions can be written as
$\max(mw(\mathbb{K}_{(n,m),{\mathscr{I}^{\prime}_{2}}}))=(n(n-1)+\frac{1}{2}n(m-1))(n+m)+\frac{n}{2}(n+m)$.
Hence, the subcase (ii)(a) is as also settled.
Subcase (ii)(b) Assume $n+m$ is odd. Similar reasoning as in subcase (ii)(a)
except each row has even number of entries. ∎
### 4.6 Sprouting of an Edge-joint Graph
Let us first recall the definition of the edge-joint graph of two given
graphs.
###### Definition 4.12.
[4] The edge-joint of two simple undirected graphs $G$ and $H$ is the graph
obtained by adding the edge $vu_{\arrowvert_{{}_{v\in V(G),u\in V(H)}}}$, and
is denoted by $G\rightsquigarrow_{vu}H$.
Consider the graphs $G$ and $H$ on $n$ and $m$ vertices respectively, with
$m\leq n$. Let the vertices of $G$ be labeled according to the index pattern
$\mathscr{I}_{1}=\\{v_{1},v_{2},v_{3},\ldots,v_{n}\\}$ and the vertices of $H$
be labeled according to the index pattern
$\mathscr{I}_{2}=\\{u_{1},u_{2},u_{3},\ldots,u_{m}\\}$. In the edge-joint
graph $G\rightsquigarrow_{v_{k}u_{l}}H$, relabel the vertices of graph $H$ to
$v_{n+1},v_{n+2},\ldots,v_{n+m}$. Also, let the new index pattern be
$\mathscr{I}=\\{v_{1},v_{2},v_{3},\ldots,v_{n},v_{n+1},v_{n+2},\ldots,v_{n+m}\\}$.
Invoking this concept, we have the next result.
###### Theorem 4.13.
For the graphs $G$ and $H$ on $n$ and $m$ vertices respectively, with $m\leq
n$, we have
1. (i)
$mw((\mathbb{G}\rightsquigarrow_{v_{k}u_{l}}\mathbb{H})_{\mathscr{I}})=mw(\mathbb{G}_{\mathscr{I}_{1}})+mw(\mathbb{H}_{\mathscr{I}^{\prime}_{2}})+|k-(l+n)|$.
2. (ii)
$mw((\mathbb{H}\rightsquigarrow_{u_{l}v_{k}}\mathbb{G})_{\mathscr{I}})=mw(\mathbb{G}_{\mathscr{I}^{\prime}_{1}})+mw(\mathbb{H}_{\mathscr{I}_{2}})+|(k+m)-l|$.
###### Proof.
(i) In graph $G$ indexing did not change and hence
$mw(\mathbb{G}_{\mathscr{I}_{1}})$ remains the same. In graph $H$ the indexing
changed consistently with $+n$ and hence for each pair of adjacent vertices
say, $v_{i+n},v_{j+n}$ we have $|(i+n)-(j+n)|=|i-j|$. Thus,
$mw(\mathbb{H}_{\mathscr{I}^{\prime}_{2}})$ remains the same. Finally, the
arc-weight of the new arc $(v_{k},v_{l+n})=|k-(l+n)|$ is evident and hence the
result follows.
(ii) Similar reasoning as in (i). ∎
## 5 Application to Certain Small Graphs
### 5.1 Sprout Wheels
The next result follows from Proposition 4.7 and Theorem 4.9. A wheel is
defined as $W_{n+1}=C_{n}+K_{1}$.
###### Proposition 5.1.
For a wheel $W_{n+1}$, $n\geq 4$ we have
1. (i)
$\min(mw(\mathbb{W}_{n+1,{\mathscr{I}^{\ast}_{1}}}))=\min(mw(\mathbb{C}_{n,{\mathscr{I}^{\ast}_{1}}}))+\min(mw(\mathbb{K}_{(1,n),{\mathscr{I}^{\ast}_{1}}}))+2$.
2. (ii)
$\max(mw(\mathbb{W}_{n+1,{\mathscr{I}^{\prime}_{2}}}))=\max(mw(\mathbb{C}_{n,{\mathscr{I}^{\prime}_{2}}}))+\max(mw(\mathbb{K}_{(1,n),{\mathscr{I}^{\prime}_{2}}}))$.
###### Proof.
(i) Consider the index pattern of $K_{(1,n)}$ in accordance to Theorem 4.9 and
without loss of generality, let the central vertex be $v_{t}$, where
$t=\lceil\frac{(n+1)}{2}\rceil$. Adding the cycle $C_{n}$ changes the index of
only arcs. We now have arcs $(v_{t_{1}},v_{t_{2}})$ and either
$t_{1}=\lceil\frac{n}{2}\rceil-1$, $t_{2}=\lceil\frac{n}{2}\rceil+1$ and
$(v_{1},v_{n+1})$ or $t_{1}=\lceil\frac{(n+1)}{2}\rceil-1$,
$t_{2}=\lceil\frac{(n+1)}{2}\rceil+1$ and $(v_{1},v_{n+1})$. If we consider
the cycle $C_{n}$ only we have an increase in
$\min(mw(\mathbb{C}_{n,{\mathscr{I}^{\ast}_{1}}}))$ of either
$(\lceil\frac{n}{2}\rceil+1)-(\lceil\frac{n}{2}\rceil-1)=2$ or
$(\lceil\frac{(n+1)}{2}\rceil+1)-(\lceil\frac{(n+1)}{2}\rceil-1)=2$. Hence, in
both cases, we have an increase of $1$ and therefore, a total increase of $2$
is effected and hence the first part of the result is settled.
(ii) Consider the cycle $C_{n}$ and label the vertices according to
$\mathscr{I}^{\prime}_{2}$ as determined in Proposition 4.7. Without loss of
generality add the central vertex $v_{1}$ (See Corollary 4.10) and add $1$ to
each index of the cycle vertices, and denote this index pattern of $C_{n}$ to
be $\mathscr{I}^{\prime}_{2}$. Denote the index pattern of $K_{(1,n)}$ to be
$\mathscr{I}^{\prime\prime}_{2}$. For every vertex $v_{i}$ in the cycle of
$W_{n+1}$ we now have the pattern index $\mathscr{I}^{\prime}_{2}$ plus $1$.
Considering the cycle only and invoking Lemma 1.4, we have
$\max(mw(\mathbb{C}_{n,{\mathscr{I}^{\prime}_{2}}}))=\max(mw(\mathbb{C}_{n,{\mathscr{I}^{\prime}_{2}}}))$.
Considering the star $K_{(1,n)}$ only we have
$\max(mw(\mathbb{K}_{(1,n),{\mathscr{I}^{\prime\prime}_{2}}}))=\max(mw(K_{(1,n),{\mathscr{I}^{\prime}_{2}}}))$.
Therefore part (ii) of the result also follows. ∎
### 5.2 Sprout Ladder Graphs
A ladder $L_{n}$, where $n\geq 3$, is defined to be $L_{n}=(P_{n}\cup
P_{n})+\\{d_{i}d^{\prime}_{i}:2\leq i\leq n-1\\}$. The edges
$\\{d_{i}d^{\prime}_{i}:2\leq i\leq n-1\\}$ are called steps.
###### Proposition 5.2.
For a ladder $L_{n}$, we have
$\min(mw(\mathbb{L}_{n,{\mathscr{I}^{\ast}_{1}}}))=2\min(mw(\mathbb{P}_{n,{\mathscr{I}^{\ast}_{1}}}))+n(n-2)$.
###### Proof.
Let the ladder be constructed upright, that is, left and right pillars are
both vertical $P_{n}$ with horizontal steps. Label the left pillar from top
down, $v_{1},v_{2},\ldots,v_{n}$ and label the right pillar from top down,
$v_{n+1},v_{n+2},v_{n+3},\ldots,v_{2n}$. Clearly, by MMAW-Principle, the
maximum number of edges have been removed from $K_{2n}$, all with maximum
index difference, to construct $L_{n}$. For a step $v_{i}v_{i+n}$ we have
index difference $n$. Since $n-2$ steps exist, the result,
$\min(mw(\mathbb{L}_{n,{\mathscr{I}^{\ast}_{1}}}))=2\min(mw(\mathbb{P}_{n,{\mathscr{I}^{\ast}_{1}}}))+n(n-2)$
follows. ∎
Determining $\max(mw(\mathbb{L}_{n,{\mathscr{I}^{\prime}_{2}}}))$ is an open
problem.
## 6 Conclusion
The main focus of this study is that a matured sprout graph is a directed
clone of an initial graph. In real world application, it means that the
underlying graph of the resultant sprout graph must be known structurally or
genetically in advance. If sprouts may re-direct through a probability
function to arc elsewhere, the matured sprout graph may not resemble the
initial graph. The latter calls for further research and could assist in
understanding less predictable neurological growth of good or malicious
networks or cell growth in biological structures.
The maxi-max arc-weight principle and the maxi-min arc-weight principle have
been introduced. The authors suggest that these principles require further
mathematical discussions for logical closure. Determining the minimum and
maximum maturity weight of a wide rage of graphs classes and small graphs such
as, the sun graph, the crown graph, the armed crown graph, the house graph,
the rasta graph, the helm graph etc. might lead us to interesting results as
well as methodologies of applications of these principles.
Most of the results in this paper can be derived by simply labeling the
vertices of a graph without the notion of sprouting. However, further in-depth
research is required to explore the application of the process of sprouting as
a dynamical concept. The graph structure can be conceptualised as a cancer
type and $\mathsf{T}_{\mathscr{I}}$ can be conceptualised as the
aggressiveness index of cancerous growth (grade) whilst
$t\in\mathsf{T}_{\mathscr{I}}$ can be conceptualised as the stage or phase of
growth.
Some of the open problems we have identified during our present study are the
following.
###### Problem 6.1.
Prove or disprove the pattern conjecture which states that if two different
random index patterns of a graph $G$ say $\mathscr{I}_{1}$ and
$\mathscr{I}_{2}$ result in $\mathsf{T}_{\mathscr{I}_{1}}$ and
$\mathsf{T}_{\mathscr{I}_{2}}$ respectively, then
$\mathsf{T}_{\mathscr{I}_{1}}\subset\mathsf{T}_{\mathscr{I}_{2}}\implies
mw(\mathbb{G}_{\mathscr{I}_{1}})<mw(\mathbb{G}_{\mathscr{I}_{2}})$.
###### Problem 6.2.
Determine $\min(mw(\mathbb{G}_{\mathscr{I}}))$ and
$\max(mw(\mathbb{G}_{\mathscr{I}}))$ with $G\cong
K_{(r_{1},r_{2},r_{3},...,r_{n})}$, where $2\leq r_{1}\leq r_{2}\leq
r_{3}\leq...\leq r_{n},\forall r_{i}\in\mathbb{N}$, being a complete
$n$-partite graph.
###### Problem 6.3.
Describe an algorithm to determine $max(mw(\mathbb{T}_{\mathscr{I}})),T$ a
tree.
###### Problem 6.4.
Describe a formal MMAW-Principle algorithm.
## References
* [1] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Macmillan Press, London, 1976.
* [2] G. Chartrand and L. Lesniak, Graphs and Digraphs, CRC Press, 2000.
* [3] J. T. Gross and J. Yellen, Graph Theory and its Applications, CRC Press, 2006.
* [4] J. Kok and N. Sudev , Certain Types of Total Irregularities of Graphs and Digraphs, arXiv: 1406.6863v3 [math; CO].
* [5] D. B. West, Introduction to Graph Theory, Pearson Education Incorporated, 2001.
|
arxiv-papers
| 2014-03-27T07:48:59 |
2024-09-04T02:50:00.456426
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Johan Kok, Naduvath Sudev",
"submitter": "Johan Kok",
"url": "https://arxiv.org/abs/1403.8109"
}
|
1404.0019
|
# Environmental correlations and Markovian to non-Markovian transitions in
collisional models
N. K. Bernardes1 [email protected] A. R. R. Carvalho2 C. H. Monken1 M. F.
Santos1 [email protected] 1Departamento de Física, Universidade Federal
de Minas Gerais, Belo Horizonte, Caixa Postal 702, 30161-970, Brazil 2Centre
for Quantum Computation and Communication Technology, Department of Quantum
Sciences, Research School of Physics and Engineering, The Australian National
University, Canberra, ACT 0200 Australia
###### Abstract
We investigate the smallest set of requirements for inducing non-Markovian
dynamics in a collisional model of open quantum systems. This is done by
introducing correlations in the state of the environment and analyzing the
divisibility of the quantum maps from consecutive time steps. Our model and
results serve as a platform for the microscopic study of non-Markovian
behavior as well as an example of a simple scenario of non-Markovianity with
purely contractive maps, i.e. with no backflow of information between system
and environment.
###### pacs:
03.65.Yz 03.67.-a
††preprint: PRE/003
Introduction. The dynamics of open quantum systems is characterized for
continuous Markov processes in terms of master equations in the so-called
Lindblad form breuer . However, the required assumption of a memoryless
environment is in reality, in most of the cases, an approximation. In general,
the environment presents memory effects that may lead to non-Markovian
dynamics breuer ; gardiner ; lai ; aharonov . Understanding these effects is
an important fundamental question with potential applications in the
engineering of reservoirs for quantum computation verstraete and as a
resource for quantum information processing such as quantum key distribution
vasile , quantum metrology matsuzaki ; chin , quantum teleportation elsi , and
quantum communication Bogna .
There have been different approaches to study the time evolution of quantum
systems subjected to the action of external environments. Some focus on the
macroscopic characteristics of the environment, such as its spectral
decomposition gardiner , others analyze the general mathematical properties of
the quantum maps they produce gorini ; lindblad ; wolf ; wolf2 . In a line of
possible approaches, one could argue that these two form the borders, each of
which presenting powerful but yet incomplete pictures. The mathematical
approach, based on the infinite divisibility of the time evolution into trace-
preserving completely positive quantum maps (from now on CP maps), is formally
absolute but difficult to connect to practical examples. On the other hand,
the macroscopic approach relates to many experiments realized (or realizable)
in labs but can only draw generic pictures, failing to address the detailed
microscopic origins for the behavior of the system.
There is a third approach, based on collisional models, which initially was
proposed to study the relaxation phenomena of simple systems as coupled spins
and coupled harmonic oscillators rau and has been used in Cavity QED
experiments for decades as an effective way to simulate Markovian reservoirs
sargent ; brune ; haroche1 ; haroche2 ; haroche3 . In most of these models,
the system $\rho$ is made to interact (collide), one at a time and
sequentially, with particles $\omega_{1},\omega_{2}...\omega_{n}$ that form a
controlled environment. Each of these interactions of strength $\eta_{i}$ and
duration $\tau_{i}$ is described by a unitary transformation $U_{i}$, as
illustrated in Fig. 1. It is assumed that each environmental particle
interacts only once with the system and that initially system $\rho$ and
environment $\omega_{env}$ are decoupled, $\rho\otimes\omega_{env}$. Under
these assumptions the evolution of the system after the j$th$ collision is
given by
$\rho\rightarrow\rho^{\prime}=\text{Tr}_{env}\left[U_{j}...U_{1}(\rho\otimes\omega_{env})U_{1}^{\dagger}...U_{j}^{\dagger}\right],$
(1)
where $\text{Tr}_{env}$ denotes the partial trace over the environmental
degrees of freedom. In standard collisional models the environmental particles
are in a factorized state $\omega_{env}=\omega^{\otimes n}$ and in the limit
of vanishing product $\eta\tau$, the effective evolution of the system can be
approximated by a standard Lindblad form ziman1 ; ziman2 .
Figure 1: (Color online) Schematic picture of the collisional model.
More recently, collisional models have also drawn theoretical attention for
their potential advantages in the microscopic study of environmental memory
effects. In particular, many recent theoretical works have connected
variations of collisional models to the study of non-Markovianity giovannetti
; tomas ; ciccarello ; vacchini ; budini ; Paternostro exploring the specific
roles played by the reservoir in the dynamics of the system. There are many
different ways in which a reservoir may induce a non-Markovian time evolution
on a given quantum system, correlations in its quantum state, time dependent
interactions with the system, and the internal dynamics of the reservoir
itself being the most common breuer ; gardiner . In most of the papers
studying these models, the authors either focus on building up and
understanding a memory kernel function (MKF) giovannetti ; ciccarello ;
vacchini ; budini or they study non-Markovian effects that arrive from the
accumulation over many collisions and that are due to strong long range
correlations in the environment tomas ; Paternostro . The first one involves
studying some integro-differential type of Master equation and deriving
general conditions on the MKF present in this equation so that the evolution
of the system is either Markovian or non-Markovian. This approach can be
connected to standard quantum statistical general properties such as the
effects of sub or super-ohmic reservoirs and so on. The second approach, on
the other hand, produces an effect equivalent to the one obtained when the
system interacts with a low dimensional environment (a qubit in the limiting
case) which is as far from a thermodynamical reservoir as possible.
However, and more important for us, it happens that in both cases the three
main ingredients that may lead to a non-Markovian evolution, namely the
correlations in the environment, its internal dynamics and the detailed
interaction between the system and the environment, they all end up bunched
together as a sole effect. For example, these effects combine to form the
kernel function of the MKF based results, and from then on the analysis focus
solely on its general properties. On the other hand, in the long range
correlation works, they also combine, this time to produce an overall periodic
system-environment dynamics that cannot be described by a Markovian evolution
of the system alone (see, for example, Ref. tomas ).
In this paper, we would like to address a different set of questions: keeping
the system-reservoir interaction constant and assuming no internal dynamics of
the reservoir between collisions, which would be the minimal amount of
correlation in the environmental state in order to generate a non-Markovian
evolution of the system? Would any correlation be sufficient? Which type of
correlation? How would it depend on the specific details of the interaction
between system and environment? Which would be the order of magnitude of these
non-Markovian effects?
In order to answer these questions, we divide our paper as follows: first, we
design a new collisional model that is able to generate, in principle, any
unital open system dynamics on a target qubit and, in particular, a master
equation in the Lindblad form. Then, we introduce small modifications to the
model, specifically concerning correlations in the environmental state to
study the influence of these modifications in the emergence of memory effects
and on the eventual induction of non-Markovian evolutions in the system. In
order to do so, we adopt the general criterion of divisibility of quantum
maps, defined in wolf , used as a measure in rivas and extensively reviewed
in section 3 of Ref. Plenio3 , and relate it to the properties of the Choi
matrix (also known as the dynamical matrix) choi ; bengtsson ; simon .
Finally, we analyze the evolution of the correlation between the system and
the environment and whether there is backflow of information to the system in
the non-Markovian regime.
Note that the relation between the system-environment correlation and non-
Markovianity has been investigated in Refs. modi ; rodriguez-rosario .
However, these works do not focus on how the details of the internal
properties of the environment and its interaction with the system will
influence the dynamics of the latter which are exactly the answers we seek in
this work.
Model. The first goal is to show that our model reproduces the master equation
in its most standard form and that all the three ingredients are easily and
independently modifiable. For simplicity and experimental reproducibility, we
take one qubit ($\\{|0\rangle,|1\rangle\\}$) as our system and consider its
sequential interaction, one at a time, with an environment made of a string of
qudits ($\\{|0\rangle,|1\rangle,|2\rangle,...,|d-1\rangle\\}^{\otimes n}$,
$n\rightarrow\infty$). The system is initially decoupled from the environment
whose state before the interactions is given by $\omega_{env}=\omega^{\otimes
n}$ where $\omega=(1-\sum_{i}\epsilon_{i})|0\rangle\langle
0|+\sum_{i}\epsilon_{i}|i\rangle\langle i|$, $0\leq\epsilon_{i}\leq 1$,
$0\leq\sum_{i}\epsilon_{i}\leq 1$ and $i=1..d-1$. The interaction Hamiltonian
for each collision is given by $H=\eta(\mathds{1}\otimes|0\rangle_{R}\langle
0|+\sum_{i}\sigma_{i}\otimes|i\rangle_{R}\langle i|)$, in units of $\hbar$.
The $\sigma_{i}$ matrices are unitary operators that act on the system and
“point”, for now, into arbitrary directions given by unitary vectors
$\hat{s}_{i}$, that is, $\sigma_{i}=\vec{\sigma}.\hat{s}_{i}$, where
$\vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$. Note that
$H^{2}\propto\mathds{1}_{s}\otimes\mathds{1}_{R}$, therefore, the overall time
evolution operator $U$ for any given collision can be expanded as
$U(\tau)=\cos(\eta\tau)-i\frac{H}{\eta}\sin(\eta\tau).$ (2)
Each collision is then set to take a time $\tau=\pi/(2\eta)$ so that the state
of the system after a collision is given by
$\displaystyle\rho(t+\textrm{collision})=\text{Tr}_{env}\left[U(\tau)(\rho(t)\otimes\omega)U(\tau)^{\dagger}\right]$
$\displaystyle=(1-\sum_{i}\epsilon_{i})\rho(t)+\sum_{i}\epsilon_{i}\sigma_{i}\rho(t)\sigma_{i}.$
(3)
The differential standard master equation in the Lindblad form for different
unital channels can be written as $\rho(t+\Delta
t)=\rho(t)-\sum_{i}\gamma_{i}\Delta t\rho(t)+\sum_{i}\gamma_{i}\Delta
t\sigma_{i}\rho(t)\sigma_{i}$ where $\gamma_{i}\Delta t$ is a dimensionless
quantity related to the probability of channel “i” changing the state of the
system in time $\Delta t$, the relative rate of change between different
channels being given by $\gamma_{i}/\gamma_{j}$ or, equivalently by
$\gamma_{i}\Delta t/(\gamma_{j}\Delta t)$. A quick look in the equation
produced by our model shows that a simple identification of
$\epsilon_{i}\equiv\gamma_{i}\Delta t$ provides exactly the same evolution for
the system after any given collision so that the state of the system after the
$j$th collision reads,
$\displaystyle\frac{\rho(T+\Delta t)-\rho(T)}{\Delta
t}=-(\sum_{i}\gamma_{i})\rho(T)+\sum_{i}\gamma_{i}\sigma_{i}\rho(T)\sigma_{i},$
(4)
where $T=(j-1)\Delta t$ and $\epsilon_{i}\equiv\gamma_{i}\Delta t$. This is
physically reasonable: $\epsilon_{i}$ is encoded in the environmental state
and gives exactly the probability of channel “i” acting on the system.
Besides, in the limit of $\Delta t\rightarrow 0$ ($\epsilon_{i}\rightarrow
0$), this is the differential version of a standard master equation
corresponding, therefore, to a single time step physical implementation of a
CP map. Note that because $\epsilon$ can be as small as needed, this
environment, albeit highly organized, still truthfully reproduces the effects
and the physics of $d$ independent unital
($\sigma_{i}^{\dagger}\sigma_{i}=\textit{I}$) Markovian reservoirs where the
overall time interval $T$ is determined by the number of collisions. Also note
that, contrary to the standard collisional models, as in Refs. giovannetti ;
tomas ; ciccarello ; vacchini ; budini ; Paternostro , we rely not on the
duration of the collision itself (which is always the same) but on the state
of the reservoir in order to map the evolution of the system onto the standard
Lindblad form111 The main effect of different choices of $\tau$ would be to
add terms of the form $\epsilon_{i}[\sigma_{i},\rho(t)]$ to Eq. (3). These
terms, that could be seen as Hamiltonian terms generated by the collision,
would not change the results presented in this work but explicit expressions
for state $\rho(T+2\Delta t)$ would become much more complicated to analyze
and deviations from the standard master equation due to non-Markovian effects
much more difficult to understand.. Finally, note that taking the same state
$\omega$ for each environmental particle is equivalent to choosing constant
decoherence rates for the evolution of the system.
There are some clear advantages for using this model to simulate reservoirs
and specially to study Markovian to non-Markovian transitions. First of all,
it connects directly even to the more restrictive mathematical description of
Markovianity (as described above) and is simple enough to implement in
basically any quantum optical setup (same concepts used in Liu ; Chiuri ;
Steve , for example). Second, it is flexible enough to allow for the
generation of basically any unital time evolution of the system: through a
suitable choice of the interaction Hamiltonian and the specific set of
$\epsilon_{i}$ encoded in $\omega$ for each environmental particle, one can
adjust the number and type ($\sigma_{i}$) of independent channels acting on
the system as well as their effective $\gamma_{i}$ rates as a function of
time. And, because time intervals can be chosen as small as necessary ($\Delta
t\propto\epsilon$) and the effective number, type and strength of the channels
can be controlled, step by step, by choosing the respective $\epsilon_{i}$ to
be either finite or zero, one can use our model to design any unital map from
one collision to the following which translates into any unital overall open
system dynamics for the target qubit. Third, note that we can also take the
reservoir particles to be in a pure state, making it easy to calculate
correlations among them and also between the system and the reservoir after
each collision. For example, state $\omega$ or its pure state version
$\omega^{\prime}=|R\rangle\langle R|$,
where$|R\rangle=\sqrt{1-\sum_{i}\epsilon_{i}}\ |0\rangle+\sqrt{\epsilon_{i}}\
|i\rangle$, produce exactly the same $\rho(\Delta t)$ after the first
collision. Finally, also note that, in its mixed state version, the
environmental state $\omega^{\otimes n}$ does not change in time, another
desirable characteristic of a true reservoir.
Non-Markovianity. In its standard version, presented above, our model renders
the Lindblad form. However, and more importantly, it can be easily modified to
generate memory effects that may ultimately lead to non-Markovian evolutions
of the system. One possibility is to increase $\epsilon_{i}$ (change state
$|R\rangle$) of at least one environmental particle in such a way as to not
obtain the differential form of the master equation after the corresponding
collision. This, however, creates a somewhat trivial non-Markovianity that
arises even for a “single particle” environment which is as far as conceivable
from a true reservoir. This is the origin of the non-Markovian effect in Steve
; Paternostro for example. On the contrary, we want to understand memory
effects due to correlations in the environment while preserving its reservoir-
like characteristics. In particular, the one that guarantees small changes to
the system in each time step which translates to $\epsilon_{i}\ll 1$ in our
model.
Even within these restrictions, there are still many different ways to
correlate the state of the environment, each leading to different dynamics of
the system. For instance, one possibility similar to the case studied in tomas
, is to collide the system with infinitely correlated particles which, in our
model, would translate into an environmental state
$\omega_{env}=\sum_{i}p_{i}|ii...iiii..\rangle\langle ii...iiii..|$ where
$i=0,1,2,...d-1$. Note, however, that the effect of such environment after $n$
collisions is to generate state $\rho(t+n\Delta
t)=p_{0}\rho(t)+\sum_{i=1}^{d-1}p_{i}\sigma_{i}^{n}\rho(t)\sigma_{i}^{n}$ for
the system which is either $\rho(t)$ for $n$ even (because $\sigma_{i}^{2}=1$)
or the single collision state $\rho(t+n\Delta
t)=p_{0}\rho(t)+\sum_{i=1}^{d-1}p_{i}\sigma_{i}\rho(t)\sigma_{i}$ if $n$ is
odd. This dynamics would be naturally non-Markovian but in the same trivial
way of the “single particle, large $\epsilon$” scenario analyzed in the
previous paragraph.
Here, we are interested in identifying the smallest set of conditions for non-
Markovian behavior in our model, therefore we will ask the simplest question:
when can correlations between two consecutive collisions generate memory
effects? In order to answer this question, we will analyze, from now on, the
two-channel environment made of a sequence of qutrits, for reasons that become
clear later. In this scenario, two correlated consecutive collisions are
obtained if one replaces any two neighboring $\omega\otimes\omega$ portion of
the environment by a two-particle state $\Omega=|R_{2}\rangle\langle R_{2}|$
where
$\displaystyle|R_{2}\rangle=(1-\epsilon_{1}-\epsilon_{2})|00\rangle+\sqrt{1-\epsilon_{1}-\epsilon_{2}}(\sqrt{\epsilon_{1}}|01\rangle+\sqrt{\epsilon_{2}}|02\rangle)$
$\displaystyle+\sqrt{\epsilon_{1}}\sqrt{1-2[q\epsilon_{1}+(1-q)\epsilon_{2}]}|10\rangle$
(5)
$\displaystyle+\sqrt{\epsilon_{2}}\sqrt{1-2[q\epsilon_{2}+(1-q)\epsilon_{1}]}|20\rangle$
$\displaystyle+\sqrt{2(1-q)\epsilon_{1}\epsilon_{2}}(|12\rangle+|21\rangle)+\sqrt{2q}(\epsilon_{1}|11\rangle+\epsilon_{2}|22\rangle),$
and $0\leq q\leq 1$, as shown in Fig. (2). A quick inspection of this state
shows that when $q=1/2$ it reduces to the uncorrelated product
$|R\rangle|R\rangle$ and when $q>1/2$ ($q<1/2$) the collisions are correlated
(anti-correlated). Also note that for any $q$, state $\rho(T+\Delta t)$ still
follows Eq. (4) for the chosen $\sigma_{1,2}$ channels and, as expected,
correlation will affect the system only after the second collision. Finally,
note that $\Omega$ or its mixed version made of just its diagonal terms in the
same $\\{|00\rangle,|01\rangle,...\\}$ basis produce, once again, exactly the
same dynamics on the qubit system.
Figure 2: (Color online) Schematic picture of our collisional model with an
environment that is correlated between $T$ and $T+2\Delta t$
On the other hand, when $q\neq 1/2$, there are three independent sets of
parameters that can influence the dynamics of the system:
$\\{\epsilon_{1},\epsilon_{2}\\}$, $\\{\sigma_{1},\sigma_{2}\\}$ and $q$ each
translating respectively into different decoherence rates, different unitaries
induced on the system and the quantity of correlation in the environmental
state. Similarly to Eq. (3) and provided $|R_{2}\rangle$ is used, the state of
the system after two time steps reads:
$\displaystyle\rho(T+2\Delta
t)=\textrm{Tr}_{env}\left[U(\tau)U(\tau)(\rho(T)\otimes|R_{2}\rangle\langle
R_{2}|)U^{\dagger}(\tau)U^{\dagger}(\tau)\right]$
$\displaystyle=\left(1-\epsilon_{1}-\epsilon_{2}\right)\left[(1-\epsilon_{1}-\epsilon_{2})\rho(T)+\epsilon_{1}\sigma_{1}\rho(T)\sigma_{1}\right.$
$\displaystyle\left.+\epsilon_{2}\sigma_{2}\rho(T)\sigma_{2}\right]+2q(\epsilon_{1}^{2}+\epsilon_{2}^{2})\rho(T)+$
(6)
$\displaystyle+\epsilon_{1}\left\\{1-2[q\epsilon_{1}+(1-q)\epsilon_{2}]\right\\}\sigma_{1}\rho(T)\sigma_{1}$
$\displaystyle+\epsilon_{2}\left\\{1-2[q\epsilon_{2}+(1-q)\epsilon_{1}]\right\\}\sigma_{2}\rho(T)\sigma_{2}$
$\displaystyle+2(1-q)\epsilon_{1}\epsilon_{2}(\sigma_{1}\sigma_{2}\rho(T)\sigma_{2}\sigma_{1}+\sigma_{2}\sigma_{1}\rho(T)\sigma_{1}\sigma_{2}),$
where $\rho(T)$ is the state of the system immediately before interacting with
the correlated part of the environment. We can rewrite this state in terms of
$\rho(T+\Delta t)$ as follows
$\displaystyle\rho(T+2\Delta t)=\rho(T+\Delta
t)+\epsilon_{1}\mathcal{L}_{1}\rho(T+\Delta
t)+\epsilon_{2}\mathcal{L}_{2}\rho(T+\Delta t)$
$\displaystyle+(2q-1)\\{(\epsilon_{2}-\epsilon_{1})[(\epsilon_{1}\mathcal{L}_{1}\rho(T)-\epsilon_{2}\mathcal{L}_{2}\rho(T)]$
$\displaystyle+\epsilon_{1}\epsilon_{2}[2\rho(T)-\sigma_{1}\sigma_{2}\rho(T)\sigma_{2}\sigma_{1}-\sigma_{2}\sigma_{1}\rho(T)\sigma_{1}\sigma_{2}]\\},$
(7)
where $\mathcal{L}_{i}\rho=-\rho+\sigma_{i}\rho\sigma_{i}$. First thing to
notice is that, as expected, when $q=1/2$ we recover the original Lindblad
form for the second time step, i.e. any non-Markovian behavior must come from
the terms proportional to the correlation factor $Q\equiv(2q-1)$. Also note
that these terms are of the order of $\epsilon^{2}$ and, therefore, are much
smaller than the terms of the Markovian time evolution obtained for $q=1/2$
($Q=0$). As we analyze later on, that means that there is no backflow of
information from the reservoir to the system in consecutive time steps and any
non-Markovianity will not be witnessed by measurements such as the
distinguishability of two arbitrary states breuer2 . This type of non-
Markovianity has been analyzed recently in sabrina and our model provides a
simple framework to study its consequences and characteristics.
One can use Eq. (Environmental correlations and Markovian to non-Markovian
transitions in collisional models) to calculate $\rho(T+2\Delta t)$ for an
arbitrary $\rho(T)$ and, with the corresponding $\rho(T+\Delta t)$, extract
the overall map $\Phi_{21}$ that implements $\rho(T+2\Delta
t)=\Phi_{21}[\rho(T+\Delta t)]$. Testing Markovianity of these two collisions
translates into checking the complete positivity of $\Phi_{21}$. We discuss
how to do this and the most general maps later on. But first, we will simplify
the problem to two different scenarios that already illustrate all the
important effects brought by the correlation terms in the dynamics of the
system. And, because $\rho(T)$ is arbitrary, we will set $T=0$ from now on
with no loss of generality.
Same channels. The first case is obtained when the two standard channels are
the same, i.e. $\sigma_{1}=\sigma_{2}=\sigma$. In this case, correlation can
only arise for different parameters $\epsilon_{1}\neq\epsilon_{2}$ such that
if the system changes by a certain amount in the first collision it is more
(or less) likely to change by that same amount in the second collision. Under
these conditions, Eq. (Environmental correlations and Markovian to non-
Markovian transitions in collisional models) can be rewritten as
$\displaystyle\frac{\rho(2\Delta t)-\rho(\Delta t)}{\Delta
t}=\gamma_{q}[-\rho(\Delta t)+\sigma\rho(\Delta t)\sigma],$ (8)
where
$\gamma_{q}=\gamma_{1}+\gamma_{2}-Q(\gamma_{1}-\gamma_{2})(\epsilon_{1}-\epsilon_{2})/[1-2(\epsilon_{1}+\epsilon_{2})]$.
This equation is still in the Lindblad form but with a modified rate which
means that, for channels of the same type, the effect of $Q\neq 0$ ($q\neq
1/2$) in consecutive collisions is just to modulate the decoherence rate while
still preserving the overall Markovianity of the evolution. Also note that
this correction is of the order of $\gamma\epsilon$ ($\epsilon\ll 1$), i.e. it
is much smaller than the effect of the standard, uncorrelated, reservoir.
Same coefficients. The other simple and meaningful scenario happens when the
two decoherence rates of the standard channels are the same,
$\epsilon_{1}=\epsilon_{2}=\epsilon$ in which case correlation necessarily
means different channels acting on the system ($[\sigma_{1},\sigma_{2}]\neq
0$). At this point, just for the sake of simplicity but with no loss of
generality, we will set $\sigma_{1}=\sqrt{a}\ \sigma_{x}+\sqrt{1-a}\
\sigma_{z}$ and $\sigma_{2}=\sigma_{z}$. Note that these two channels
correspond to axes in the Bloch sphere and any arbitrary non-commuting set
$\left\\{\sigma_{1},\sigma_{2}\right\\}$ can always be rotated into our
particular choice. The general solution in this case is presented in the
Appendix but, more importantly, under our primary condition of $\epsilon\ll
1$, it reads
$\displaystyle\frac{\rho(2\Delta t)-\rho(\Delta t)}{\Delta
t}=\gamma(\mathcal{L}_{1}+\mathcal{L}_{2})\rho(\Delta t)$
$\displaystyle-2\gamma a\epsilon Q\left[-\rho(\Delta t)+\sigma_{y}\rho(\Delta
t)\sigma_{y}\right],$ (9)
where we keep terms up to $O(\epsilon^{2})$. Some things pop out immediately
when looking at Eq. (9): first, and as expected, when $Q=0$ (or $a=0$), it
reduces back to the Markovian evolution of two independent channels (or one).
Second, the term proportional to the correlation factor $Q$ is, again, of the
order of $\gamma\epsilon$ and, hence, much smaller than the ones generated by
the uncorrelated evolution. Third, and more important, this term looks like an
effective reservoir created by the correlation but it is only in the Lindblad
form when $Q<0$ ($q<1/2$), i.e. when the environment is anti-correlated. When
$Q>0$ ($q>1/2$), the effective “decoherence” rate
$\gamma_{\mbox{\scriptsize{eff}}}=-2\gamma a\epsilon Q$ is negative and the
map $\Phi_{21}$ is not CP, i.e. the evolution is non-Markovian. Note that the
non-Markovian transition exists for any value of $a\neq 0$ (i.e. for any non-
commuting set ${\sigma_{1},\sigma_{2}}$), whose effect is just to modulate the
$\gamma_{\mbox{\scriptsize{eff}}}$ created by the correlation. Also note that
the addition of an extra Markovian channel in the $y$ direction,
$\mathcal{L}_{y}$, of rate $\gamma_{y}$, would require larger values of $Q$
for the non-Markovian evolution to happen. In this case, the evolution would
be non-Markovian only for $Q>\gamma_{y}/(2a\gamma\epsilon)$ and would be
strictly Markovian as soon as $\gamma_{y}=2a\gamma\epsilon$. In our model,
that would correspond to adding an extra level to the environmental particles
(ququarts instead of qutrits), making $\sigma_{3}=\sigma_{y}$,
$\epsilon_{3}=\gamma_{y}\Delta t$ and adjusting state $|R_{2}\rangle$ to
include this extra channel. Finally, note that the fact that it is correlation
and not anti-correlation that generates non-Markovianity is a consequence of
the particular choices we took in our model, such as the specific interaction
Hamiltonian $H$. What is important is the transition from one regime to the
other depending on the parameters that dictate the time evolution of the
system and that are contained not only in $H$ but specially in $\omega_{env}$.
These two scenarios contain all the effects that correlations in two
consecutive collisions may generate in the evolution of the system:
modulations of the standard decoherence rates and/or the creation of effective
channels in the directions orthogonal to the standard ones. Both effects
($O(\epsilon^{2})$) are much smaller than the standard changes to the state of
the system (of the order of $\epsilon$), which means that only the second one
can be associated to a non-Markovian evolution – the first effect is
equivalent to just introducing a small change in the already existing
$\epsilon_{i}$ associated to the second collision. Here, we considered only
two standard channels, $\sigma_{1,2}$, but the analysis in this paragraph is
general and it holds for an arbitrary number of channels in the model:
correlations between any two of them will be equivalent to redefining their
standard decoherence rates relative to the second collision and to creating an
effective channel in an orthogonal direction. This effective channel may not
represent a CP map, i.e. it may feature a negative decoherence rate and that
will translate into the non-Markovian evolution of the system as long as its
modulus is larger than an eventual standard rate in the same direction. The
example that leads to Eq. (9) features the most favorable case for non-
Markovianity in our model, where the standard rate of the orthogonal channel
is actually equal to zero, i.e., there is no standard $\sigma_{y}$ channel to
compete with the non-Markovian effect brought by correlations in the
environmental state.
These simple cases already encompass the important properties of the model.
However, one can still solve the most complete scenario by extracting the
general map $\Phi_{21}$ from Eq. (Environmental correlations and Markovian to
non-Markovian transitions in collisional models) and $\rho(\Delta t)$. By
construction, our model produces a CP map $\Phi_{20}$ that takes state
$\rho(0)$ into Eq. (Environmental correlations and Markovian to non-Markovian
transitions in collisional models) and based on the divisibility criterion
wolf ; Plenio3 , this map is Markovian if it is divisible into two consecutive
CP maps $\Phi_{20}=\Phi_{21}\Phi_{10}$. $\Phi_{10}$ is CP by construction and
the question reduces to the characteristics of $\Phi_{21}$. The density matrix
of a qubit state can be represented by
$\rho=\left(\mathbb{1}+\vec{r}\cdot\vec{\sigma}\right)/2$, where $\vec{r}$ is
the Bloch vector ($r_{i}=\textrm{Tr}(\rho\sigma_{i})$). The action of a map
$\Phi$ on a qubit quantum state will be given, in general, by
$\Phi:\vec{r}\mapsto\vec{r^{\prime}}=\Lambda\vec{r}+\vec{t}$, where $\Lambda$
is responsible to change the norm and rotate the Bloch vector and
$\vec{t}=(t_{x},t_{y},t_{z})$ to translate it. For unital maps ($\vec{t}=0$),
we can associate to this map an Hermitian matrix
$\mathcal{H}=\left(\mathds{1}+\sum_{\mu,\nu=x,y,z}\Lambda_{\mu\nu}\sigma_{\mu}\otimes\sigma^{*}_{\nu}\right)/2$.
The map $\Phi$ is completely positive iff $\mathcal{H}\geq 0$ choi ; simon .
We solved this problem in our model by extracting the matrix $\mathcal{H}$
from our map and diagonalizing it just to find that only one of its
eigenvalues can be smaller than zero. In Fig. (3) we plot this eigenvalue
$\lambda$ as a function of $Q$ and $a$ and also as a function of $Q$ for one
particular value of $a$, for different $\epsilon_{1},\epsilon_{2}$. As already
anticipated, $Q=0$ ($q=1/2$) draws the line of non-Markovian to Markovian
transition and any $a\neq 0$, corresponding to non-commuting uncorrelated
channels, only defines the intensity of the effective reservoir.
Figure 3: (Color online) Eigenvalue of $\mathcal{H}$ in terms of the
correlation factor $Q$ and parameter $a$ for $\epsilon_{1}=0.01$ and
$\epsilon_{2}=0.02$ (top). Eigenvalue of $\mathcal{H}$ in terms of the
correlation factor $Q$ for $a=0.05$, $\epsilon_{1}=0.01$, and
$\epsilon_{2}=0.02$ (bottom).
Finally note that even though we are particularly interested in the small
$\epsilon$ situation, in order to filter out eventual non-Markovian effects
that can be linked to the finite size of the reservoir, the analysis here
presented based on divisibility is more general and apply for any physical
value of $\epsilon_{i}$ (still bounded by $0\leq\epsilon_{i}\leq 1$ and
$0\leq\sum_{i}\epsilon_{i}\leq 1$). The difference is that for large values of
$\epsilon$ we can no longer talk about a time differential equation for $\rho$
but rather about the divisibility of a discrete set of maps. Furthermore, it
is possible to show that the dominating term of the eigenvalue $\lambda$ that
determines the non-Markovian behavior is $-8aQ\epsilon_{1}\epsilon_{2}$. This
means that the non-Markovian effect will always increase with $\epsilon$. Note
that in our model, $\epsilon_{i}=\gamma_{i}\Delta t$ which means that larger
values of $\epsilon_{i}$ can be understood either as larger decoherence rates
and/or larger time intervals, both situations in which non-Markovian effects
are known to exist.
Non-Markovianity and Backflow of Information. A question that often arises
when studying non-Markovianity is that of the backflow of information from the
environment to the system. Quantities like the distinguishability of two
arbitrary states, among others, can witness such behavior and, therefore, are
constantly linked to these problems. Our model is a simple and easily
computable example of non-Markovian evolution that cannot be witnessed by
these quantities because, in fact, there is no backflow of information in it.
A quick look at Eq. (Environmental correlations and Markovian to non-Markovian
transitions in collisional models) shows that the effect of the uncorrelated
channels produces a contractive map of the order of $\epsilon$ in each time
step, i.e. it typically deforms the Bloch Sphere that represents the possible
initial states of the system into an ellipsoid whose axes shrink
proportionally to $\epsilon$ after each collision. Since the effects of
correlation that generate non-Markovianity are of the order of $\epsilon^{2}$,
any quantity based on distances in the parameter space (Bloch sphere) will not
witness any non-Markovian behavior in our model. Another way of saying this is
that the bona-fide measure of information for a qubit is its von Neumann
entropy $S=-\textrm{Tr}(\rho\log\rho)$. In our results, $S$ always increases
after each collision, i.e. the information is always “flowing” from the system
to the reservoir. This can be easily explained by the fact that the eventual
non-Markovian effect, related to a particular negative $\gamma_{i}$ rate is
always of the order of $\epsilon^{2}$ while the standard Markovian channels
are of the order of $\epsilon$. Therefore, even though the evolution may not
be given by a CP map (may not be Markovian), the map is always contractive
leading to guaranteed loss of information on the qubit after each collision.
Furthermore, as we have shown, correlations in the environment change the rate
of flow of the information but these changes happen both for the Markovian and
non-Markovian situations.
Recent works have also related non-Markovianity and backflow of information to
the entanglement (or discord) created between the system and the environment
as, for example, in rivas ; alipour . This can also be easily analyzed in our
model because we can assume, with no loss of generality, initially
uncorrelated pure states for system and environment. In this case the
correlation between them due to the time evolution is simply the entanglement
$E(t)$ created by the collisions which is given by the von Neumann entropy of
the system $S=-\textrm{Tr}(\rho\log\rho)$. Under these conditions, the
$\epsilon$ contraction described in the last paragraph also means that, in
general, the system is getting more entangled with the environment in each
time step, the exceptional cases being those in which the state of the system
simply does not evolve in time like, for example, when there is only one
channel $\sigma$ and the initial state is an eigenstate of this operator.
These exceptions are trivial Markovian examples, though, and not interesting
for us. First of all, note that this ever increasing entanglement means that
our model provides another counter-example, one that can be easily tested in
the labs by the way, to the results suggested in alipour and shabani and
already disproved in brodutch relating Markovian evolution to vanishing
discord between system and reservoir. Because our initial state is pure and
the overall evolution unitary, the discord between both systems is equal to
the entanglement which always increase in each time step.
Figure 4: (Color online) The difference of entanglement $\Delta E$ versus the
correlation factor $Q$ for $a=1$ and $\epsilon_{1}=\epsilon_{2}=0.01$ with
initial states belonging to the $xz$-plane of Bloch sphere (blue solid line)
and for $a=0$ and $\epsilon_{1}=0.01$ and $\epsilon_{2}=0.02$ with initial
states with some component in the $z$-direction (red dashed line).
There are, however, corrections of the contraction rate, proportional to
$\epsilon^{2}$, that are due to the correlation factor of the environmental
state. These corrections can be made explicit by calculating the difference
$\Delta E=E(2\Delta t)-E_{Q=0}(2\Delta t)$ between the real entanglement of
the system and environment for the second time step and that of an
uncorrelated evolution ($q=1/2$). In Fig. (4) we display this difference for
the two meaningful examples analyzed in the text:
$\\{a=0,\epsilon_{1}\neq\epsilon_{2}\\}$ and
$\\{a=1,\epsilon_{1}=\epsilon_{2}\\}$. In each case we choose the initial
state that accentuate this difference the most and the figure for all possible
initial states is in the Appendix. Note that in both cases, the correlation
factor defines the sign of the correction to the entanglement rate for the
second time step, but because the evolution is still Markovian for $a=0$, this
cannot be used to establish any direct relation between this rate and the
character of the dynamics of the system. In this sense, our model also
provides an example of non-Markovian behavior that cannot be detected by the
entanglement between the system and the environment.
Figure 5: (Color online) Simple quantum optical experimental setup for the
implementation of the collisional model. The blue lines represent the possible
paths for the polarized photon. The photon passes through the setup where beam
splitters, represented by squares, establish the probability of the action of
the half-wave plate (black rectangles). The beam splitters have reflectivity
of $2\epsilon$ (green color), of 0.5 (black color), or $q$ (red color). The
half-wave plates can have their optical axis at $(\arcsin{\sqrt{a}})/2$ with
respect to the $x$ direction ($\sigma_{1}$) or at $0^{\circ}$ ($\sigma_{2}$).
The dashed black line is just for illustrative reasons and it delimits the
first collision. After passing through this setup, the system’s state should
be tomographically determined, which is not represented in this figure.
A simple quantum optical example. Before concluding, we would like to present
a simple quantum optical setup where this kind of analysis can be implemented.
Assume a source of single photons, either on demand or heralded (such as
parametric down conversion). The setup shown in Fig. 5 implements exactly the
desired time evolution for the polarization qubit of the incoming photon. For
simplicity, we show just the case where $\epsilon_{1}=\epsilon_{2}=\epsilon$,
but the most general case can be realized with a similar scheme. The
environment state is encoded in the different possible paths that the photon
may follow. The operation $\sigma_{1}$ is given by a half-wave plate with the
fast axis at an angle of $(\arcsin{\sqrt{a}})/2$ with respect to the $x$
direction and the operation $\sigma_{2}$ is given by a half-wave plate with
the fast axis at $0^{\circ}$. The reflectivities of different beam splitters
(indicated at their lower left corners) establish the probabilities that the
system will suffer a change. The first collision is represented in the setup
with the operations before the dashed (black) line: with a probability of
$1-2\epsilon$ the photon will not suffer any change, but with probability
$\epsilon$, its polarization will suffer one of the possible rotations
($\sigma_{1}$ or $\sigma_{2}$). For the second collision, the scheme follows
the same logic. In order to measure the system’s state, the outputs of the
setup should be combined in a detector and quantum state tomography should be
realized. By reconstructing the state after one and two collisions, we can
verify how the dynamics of evolution is.
Conclusions. In this work, we have designed a new collisional model that
provides an intuitive way to approximate the mathematical definition of
Markovian evolution into a feasible quantum optical experiment. For this
model, the smallest set of requirements to simulate non-Markovian dynamics has
been established. We have also studied the effects of correlations in the
environmental states in the dynamics of the system and, in particular, we have
shown that correlations alone are not sufficient to generate non-Markovianity,
which will also depend on the particularities of the interaction between
system and reservoir, and we have analyzed under which conditions this happens
in our model. In order to do so, we have derived the map that describes the
evolution of the system and checked its complete positivity as a function of
the parameters of the model. We have also shown an example of non-Markovian
evolution that does not violate the criterion of Ref. breuer2 . Our case also
exemplified that Markovian dynamics appear in system-environment states that
are not only correlated (with non-vanishing discord) but, in fact, whose
correlation increases with time, contradicting Ref. alipour ; shabani .
Furthermore, we have showed an example where the correlation in the
environment modulates the rate at which it entangles with the system but this
modulation is detached from the particular character of the dynamics and,
therefore, cannot be used as a witness of non-Markovianity. Finally, we would
like to stress that the model we have designed is very simple and yet
physically meaningful: for example, with some adaptations it could be used to
study situations like diluted gases where a single particle collides
consecutively with others, one at a time. This can be a possible line of
extension of the current work.
## Acknowledgments
The authors would like to thank P. Haikka for useful discussions on non-
Markovianity. N.K.B, C.H.M. and M.F.S. would like to thank the support from
the Brazilian agencies CNPq and CAPES. M.F.S. would like to thank the support
of FAPEMIG, project PPM IV. This work is part of the INCT-IQ from CNPq and
also of the Australian Research Council Centre of Excellence for Quantum
Computation and Communication Technology (project number CE110001027).
## Appendix A
The action of a quantum map on a state can be characterized by
$\Phi(\rho)=\sum_{i}\lambda_{i}T_{i}\rho T_{i}^{\dagger}$ where
$\\{\lambda_{i}\\}$ are the eigenvalues of the matrix $\mathcal{H}$ aiello .
The operators $\\{T_{i}\\}$ are formed by the eigenvectors $\\{u_{i}\\}$ of
$\mathcal{H}$ as follows
$T_{i}=\begin{pmatrix}\left[\vec{u}_{i}\right]_{0}&\left[\vec{u}_{i}\right]_{1}\\\
\left[\vec{u}_{i}\right]_{2}&\left[\vec{u}_{i}\right]_{3}\end{pmatrix},$ (10)
where $\left[\vec{u}_{i}\right]_{j}$ is the $j$th element of the vector
$\vec{u}_{i}$. For a CP map $\\{\sqrt{\lambda_{i}}T_{i}\\}$ are the Kraus
operators.
We would like to calculate the map that describes the evolution of the system
from the first to the second collision for
$\epsilon_{1}=\epsilon_{2}=\epsilon$. It is possible to show that
$\displaystyle\rho(2\Delta t)=(1-2\epsilon)\rho(\Delta
t)+C_{1}\left[\rho(\Delta t)-\sigma_{y}\rho(\Delta t)\sigma_{y}\right]+$
$\displaystyle C_{2}\sigma_{z}\rho(\Delta
t)\sigma_{z}+C_{3}\sigma_{x}\rho(\Delta t)\sigma_{x}+$ (11) $\displaystyle
C_{4}\left[\sigma_{x}\rho(\Delta t)\sigma_{z}+\sigma_{z}\rho(\Delta
t)\sigma_{x}\right],$
where $\\{C_{i}\\}$ are functions of the parameters $a$, $\epsilon$ and $q$
and are given by
$\displaystyle
C_{1}=\frac{2a(2q-1)\epsilon^{2}(1-2\epsilon)}{1+4\epsilon(a\epsilon-1)},$
$\displaystyle C_{2}=\frac{\epsilon\\{2+8\epsilon
a-32a\epsilon^{2}[\epsilon[(2a-1)(q-1)-q+2]+1]\\}}{1+4\epsilon(a\epsilon-1)}$
$\displaystyle
C_{3}=\frac{a\epsilon\\{1-4\epsilon-4\epsilon^{2}[(2a-1)(q-1)-q]\\}}{1+4\epsilon(a\epsilon-1)},$
$\displaystyle
C_{4}=\frac{\sqrt{(1-a)a}\epsilon\\{1-4\epsilon[2a(q-1)\epsilon+1]\\}}{1+4\epsilon(a\epsilon-1)}.$
Figure 6: (Color online) The difference $\Delta E$ versus the correlation
factor $Q$ for (a) $a=1$ and $\epsilon_{1}=\epsilon_{2}=0.01$ with initial
states belonging to the $xz$-plane of Bloch sphere and (b) for $a=0$ and
$\epsilon_{1}=0.01$ and $\epsilon_{2}=0.02$ with initial states with some
component in the $z$-direction. The Bloch vector is defined as
$\vec{r}=(\sin{\theta}\cos{\phi},\sin{\theta}\sin{\phi},\cos{\theta})$.
Notice here that because $\epsilon\ll 1$ and $0\leq a\leq 1$, the only
coefficient whose sign depends on the correlation factor $Q$ is $C_{1}$. All
the other coefficients will be either zero or positive numbers. When $Q<0$
($q<1/2$), $C_{1}$ is negative and the $\sigma_{y}$ term in Eq. (9) will be in
the Lindblad form. However, when $Q>0$ ($q>1/2$), $C_{1}$ is positive and the
evolution of the system cannot be described by a master equation in the
Lindblad form anymore. This becomes clearer if we expand the coefficients
until terms of the second order of $\epsilon$, assuming $\epsilon$
sufficiently small, as follows
$\displaystyle\rho(2\Delta t)\approx(1-2\epsilon)\rho(\Delta
t)+(2-a)\epsilon\sigma_{z}\rho(\Delta
t)\sigma_{z}+a\epsilon\sigma_{x}\rho(\Delta
t)\sigma_{x}+\sqrt{a(1-a)}\epsilon\left[\sigma_{x}\rho(\Delta
t)\sigma_{z}+\sigma_{z}\rho(\Delta t)\sigma_{x}\right]+$ (12) $\displaystyle
2aQ\epsilon^{2}\left[\rho(\Delta t)-\sigma_{y}\rho(\Delta
t)\sigma_{y}\right]=$ $\displaystyle(1-2\epsilon)\rho(\Delta
t)+\epsilon\sigma_{1}\rho(\Delta t)\sigma_{1}+\epsilon\sigma_{2}\rho(\Delta
t)\sigma_{2}+2aQ\epsilon^{2}\left[\rho(\Delta t)-\sigma_{y}\rho(\Delta
t)\sigma_{y}\right].$
## Appendix B
Fig. 6 shows the difference $\Delta E=E(2\Delta t)-E_{Q=0}(2\Delta t)$ between
the entanglement of the system and the environment in the actual evolution (as
a function of $Q$) and that obtained when $Q=0$ and as a function of different
initial states of the system, defined by Bloch vectors given by
$\vec{r}=(\sin{\theta}\cos{\phi},\sin{\theta}\sin{\phi},\cos{\theta})$. Either
in the case $\epsilon_{1}=\epsilon_{2}$ and $a=1$ or in the case
$\epsilon_{1}\neq\epsilon_{2}$ and $a=0$, the environment where its qutrits
are more correlated ($Q>0$) will result in a system less correlated to its
environment than in the case of the equivalent uncorrelated evolution ($Q=0$).
On the other hand, for $Q<0$, the system will become even more correlated to
its environment than for $Q=0$ after the second collision. The entanglement
between system and environment shows a similar behavior in both cases, but
only in the first case ($\epsilon_{1}=\epsilon_{2}$, $a=1$) there is a
Markovian to non-Markovian transition in the evolution of the system,
therefore, the analysis of $\Delta E$ alone is not enough to characterize the
dynamics of the system.
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|
arxiv-papers
| 2014-03-31T20:03:32 |
2024-09-04T02:50:00.474150
|
{
"license": "Public Domain",
"authors": "N. K. Bernardes, A. R. R. Carvalho, C. H. Monken, and M. F. Santos",
"submitter": "Nadja Kolb Bernardes",
"url": "https://arxiv.org/abs/1404.0019"
}
|
1404.0126
|
pdefnPrototypical Definition
dthrmDual Theorem
esaxEilenberg–Steenrod Axiom
|
arxiv-papers
| 2014-04-01T05:03:17 |
2024-09-04T02:50:00.485841
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anastasis Kratsios",
"submitter": "Anastasis Kratsios",
"url": "https://arxiv.org/abs/1404.0126"
}
|
1404.0140
|
# BCS-BEC crossover and quantum phase transition in an ultracold Fermi gas
under spin-orbit coupling
Fan Wu Key Laboratory of Quantum Information, University of Science and
Technology of China, CAS, Hefei, Anhui, 230026, People’s Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, People’s
Republic of China Ren Zhang Department of Physics, Renmin University of
China, Beijing 100872, People’s Republic of China Tian-Shu Deng Key
Laboratory of Quantum Information, University of Science and Technology of
China, CAS, Hefei, Anhui, 230026, People’s Republic of China Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, People’s Republic of
China Wei Zhang [email protected] Department of Physics, Renmin University
of China, Beijing 100872, People’s Republic of China Beijing Key Laboratory
of Opto-electronic Functional Materials and Micro-nano Devices, Beijing
100872, People’s Republic of China Wei Yi [email protected] Key Laboratory of
Quantum Information, University of Science and Technology of China, CAS,
Hefei, Anhui, 230026, People’s Republic of China Synergetic Innovation Center
of Quantum Information and Quantum Physics, University of Science and
Technology of China, Hefei, Anhui 230026, People’s Republic of China Guang-
Can Guo Key Laboratory of Quantum Information, University of Science and
Technology of China, CAS, Hefei, Anhui, 230026, People’s Republic of China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, People’s
Republic of China
###### Abstract
In this work, we study the BCS-BEC crossover and quantum phase transition in a
Fermi gas under Rashba spin-orbit coupling close to a Feshbach resonance. By
adopting a two-channel model, we take into account of the closed channel
molecules, and show that combined with spin-orbit coupling, a finite
background scattering in the open channel can lead to two branches of solution
for both the two-body and the many-body ground states. The branching of the
two-body bound state solution originates from the avoided crossing between
bound states in the open and the closed channels, respectively. For the many-
body states, we identify a quantum phase transition in the upper branch
regardless of the sign of the background scattering length, which is in clear
contrast to the case without spin-orbit coupling. For systems with negative
background scattering length in particular, we show that the bound state in
the open channel, and hence the quantum phase transition in the upper branch,
are induced by spin-orbit coupling. We then characterize the critical detuning
of the quantum phase transition for both positive and negative background
scattering lengths, and demonstrate the optimal parameters for the critical
point to be probed experimentally.
###### pacs:
67.85.Lm, 03.75.Ss, 05.30.Fk
## I Introduction
Synthetic spin-orbit coupling (SOC), a recent addition to the toolbox
available for quantum simulation in ultracold atomic gases, can give rise to
interesting two-body and many-body properties by modifying the single-particle
dispersion spectrum of the underlying system gauge2exp ; fermisocexp1 ;
fermisocexp2 ; fermisocfeshnist ; fermisocexpfeshbach . In ultracold Fermi
gases, it has been shown that the implementation of synthetic SOC can lead to
an unconventional superfluid with non-trivial topological features, or a
superfluid with non-zero center-of-mass (CoM) momentum, or a combination of
both, depending on the spatial dimensions of the gas, and on the form of
synthetic SOC implemented soc3 ; TFZheng ; TFWei ; Ren_W ; chenggang ;
huhuitwo-c ; WF_A ; zhang ; sato ; chuanwei ; soc4 ; soc6 ; iskin ; thermo ;
2d2 ; 2d1 ; melo ; wmliu ; helianyi ; wy2d ; xiaosen ; wypolaron ;
iskinnistsoc ; puhantwobody ; shenoy ; wyfflo ; melosupp2 ; puhan3dsoc ; XF ;
xiongjun ; hufflo ; chuanweifflo ; bdg1 ; bdg2 ; HUHUI_2 ; zhangpeng ;
zhangpeng2 ; shenoy2 . Most of these studies have assumed the system to be
close to a Feshbach resonance, so that the interaction is tunable via an
external magnetic field. However, to characterize the Feshbach resonance, most
of the previous studies on spin-orbit coupled Fermi systems have adopted a
single-channel model TFZheng ; TFWei ; WF_A ; zhang ; sato ; chuanwei ; soc4 ;
soc6 ; iskin ; thermo ; 2d2 ; 2d1 ; melo ; wmliu ; helianyi ; wy2d ; xiaosen ;
wypolaron ; iskinnistsoc ; puhantwobody ; wyfflo ; melosupp2 ; puhan3dsoc ; XF
; xiongjun ; hufflo ; chuanweifflo ; bdg1 ; bdg2 ; HUHUI_2 ; zhangpeng ;
zhangpeng2 ; shenoy2 . While on a phenomenological level, a two-channel model
is more appropriate, where the Feshbach resonance is described as a multi-
channel resonant scattering process when the bound state in a closed channel
is tuned close to the continuum threshold of an open channel FBR . The two-
channel model reduces to a single-channel model only when the population of
the closed channel molecule becomes negligible, which is not always the case,
particularly under SOC huhuitwo-c ; Ren_W .
Recently, there have been several studies using two-channel models for the
characterization of spin-orbit coupled Fermi gases near a Feshbach resonance
huhuitwo-c ; chenggang ; Ren_W ; shenoy . A particularly interesting finding
is that the SOC can induce a new branch of two-body bound state. While it has
been reported before that in the absence of SOC, two branches of bound state
can be found near a Feshbach resonance for a Fermi gas with positive
background scattering length, the extra bound state under negative background
scattering length is purely induced by SOC. The existence of this new two-body
bound state should leave signatures on the many-body level. Indeed, for a
Fermi gas without SOC, a quantum phase transition exists for a positive
background scattering length, which is intimately connected with the
corresponding two-body bound state WY_D . We expect that similar phase
transitions may appear in a spin-orbit coupled Fermi gas when the proper two-
channel resonant scattering process is considered.
In this work, we study an ultracold Fermi gas under Rashba SOC close to a
Feshbach resonance using a two-channel model. We first confirm the two-body
calculations in Refs. shenoy ; chenggang , and study the branching of the two-
body bound state in the presence of a finite background scattering length. For
a positive background scattering length, the open channel supports a bound
state even in the absence of SOC, and the two branches of two-body bound state
originate from the avoided level crossing between the bound states in the open
and the closed channels. For a negative background scattering length, an SOC-
induced bound state emerges in the open channel for any finite SOC. The SOC-
induced bound state then couples with the bound state in the closed channel,
also leading to two branches of bound state.
With these understandings, we characterize many-body properties of the system
using a Bardeen-Cooper-Schrieffer (BCS) mean field approach. As expected, we
find two-branches of many-body solutions, the upper and the lower branch, for
any finite background scattering length. With a positive background scattering
length, we find that the lower branch is always bosonic with negative chemical
potential, essentially a condensate of tightly bound molecules. A quantum
phase transition exists in the upper branch, across which the ground state of
the Fermi gas changes from a superfluid state to a normal state. The position
of the phase transition can be controlled by tuning the SOC strength. We also
notice that by tuning the interaction or the SOC strength, a Bardeen-Cooper-
Schrieffer to Bose-Einstein condensate (BCS-BEC) crossover occurs in the upper
branch. With a negative background scattering length, the upper branch emerges
from the scattering threshold on the low-field-side of the Feshbach resonance
via a quantum phase transition for any finite SOC.
We discuss in detail the many-body properties of different branches under
various parameters, and characterize the critical detuning for the onset of
the quantum phase transition. We show that the results in this work, the
quantum phase transitions in particular, should best be observed in narrow
Feshbach resonances under appropriate SOC. While experimentally, only an equal
mixture of Rashba and Dresselhaus SOC has been realized in cold atomic gases
gauge2exp ; fermisocexp1 ; fermisocexp2 , there have been various proposals
for realizing the Rashba-type SOC xiongjun ; rashbagen1 ; rashbagen2 ;
rashbagen3 . With the recent experimental implementation of Feshbach resonance
in spin-orbit coupled degenerate Fermi gases fermisocfeshnist ;
fermisocexpfeshbach , we expect that the SOC-induced quantum phase transition
reported here can be experimentally probed in the future.
The paper is organized as the following: in Sec. II, we introduce the two-
channel model for a Fermi gas under Rashba SOC and close to a Feshbach
resonance. In Sec. III, we study the two-body bound state solutions under a
finite background scattering length. In Sec. IV, we discuss in detail the
many-body ground state of the two-channel model using the standard BCS mean-
field theory. For a finite background scattering length and a finite SOC,
there are typically two branches of ground state, where a quantum phase
transition can be identified in the upper branch. We then characterize the
critical point of the quantum phase transition in Sec. V, and finally
summarize in Sec. VI.
## II Two-channel model
We consider a three-dimensional two-component Fermi gas close to a Feshbach
resonance under Rashba SOC. This system can be described by a two-channel
model
$\displaystyle H=H_{0}+H_{\rm SOC}+H_{\rm bf}+H_{\rm int},$ (1)
where the terms take the following forms
$\displaystyle H_{0}=$
$\displaystyle\sum_{\mathbf{k},\sigma=\uparrow,\downarrow}\epsilon_{\mathbf{k}}a^{{\dagger}}_{\mathbf{k}\sigma}a_{\mathbf{k}\sigma}+\sum_{\mathbf{q}}({\gamma}+\frac{\epsilon_{\mathbf{q}}}{2})b^{{\dagger}}_{\mathbf{q}}b_{\mathbf{q}},$
$\displaystyle H_{\rm SOC}=$
$\displaystyle\sum_{\mathbf{k}}{\alpha}[(k_{x}-ik_{y})a^{{\dagger}}_{\mathbf{k},\uparrow}a_{\mathbf{k},\downarrow}+h.c.],$
$\displaystyle H_{\rm bf}=$
$\displaystyle\frac{{g}}{\sqrt{V}}\sum_{\mathbf{k},\mathbf{q}}(a^{{\dagger}}_{\mathbf{k}+\mathbf{q}\uparrow}a^{{\dagger}}_{-\mathbf{k}+\mathbf{q}\downarrow}b_{\mathbf{q}}+h.c.),$
$\displaystyle H_{\rm int}=$
$\displaystyle\frac{{U}}{V}\sum_{\mathbf{k},\mathbf{k}^{\prime},\mathbf{q}}a^{{\dagger}}_{\mathbf{k}+\mathbf{q}\uparrow}a^{{\dagger}}_{-\mathbf{k}+\mathbf{q}\downarrow}a_{-\mathbf{k}^{\prime}+\mathbf{q}\downarrow}a_{\mathbf{k}^{\prime}+\mathbf{q}\uparrow}.$
Here, $a_{\mathbf{k},\sigma}$($a^{{\dagger}}_{\mathbf{k},\sigma}$) is the
annihilation (creation) operator for atoms with pseudo-spin $\sigma$ and
momentum ${\bf k}$, $\epsilon_{\mathbf{k}}=\hbar^{2}k^{2}/2m$ is the single
fermion dispersion, $b_{\mathbf{q}}$($b^{{\dagger}}_{\mathbf{q}}$) is the
annihilation (creation) operator for the closed channel molecules, ${\alpha}$
is the Rashba SOC strength, $V$ is the quantization volume, and $h.c.$ stands
for Hermitian conjugate. The bare atom-molecule coupling rate ${g}$, the bare
background interaction rate ${U}$, and the bare detuning ${\gamma}$ are
connected with the physical ones $\\{{g}_{p},{U}_{p},{\gamma}_{p}\\}$ through
the standard renormalization relations:
${U}=\Gamma{U}_{p},{g}=\Gamma{g}_{p},{\gamma}={\gamma}_{p}-\Gamma{g}^{2}_{p}/U_{c}$,
where
$\Gamma=(1+{U}_{p}/U_{c})^{-1},U^{-1}_{c}=-\sum_{\mathbf{k}}1/2\epsilon_{\mathbf{k}}$,
and ${U}_{p}=4\pi\hbar^{2}a_{\rm bg}/m,{g}^{2}_{p}=4\pi\hbar^{2}a_{\rm
bg}W\mu_{\rm co}/m$, and ${\gamma}_{p}=\mu_{\rm co}(B-B_{0})$ crossoverreview
. Here, $a_{\rm bg}$ is the background scattering length in the open channel,
$W$ is the Feshbach resonance width, $\mu_{\rm co}$ is the magnetic moment
difference between the closed and open channels, and $B-B_{0}$ is the magnetic
field detuning with $B_{0}$ the Feshbach resonance point.
To be consistent with the experimental parameters, we adopt the unit of energy
as $E_{0}$, and define the unit of the momentum $k_{0}$ and the unit of
density $n_{0}$ as
$k_{0}=\sqrt{\frac{2mE_{0}}{\hbar^{2}}},\quad\quad
n_{0}=\frac{k^{3}_{0}}{3\pi^{2}}.$ (2)
We then obtain a dimensionless version of the parameters in the Hamiltonian,
which will be used in the following discussion.
## III Two-body bound states
In this section, we investigate the two-body bound-state solution under the
Hamiltonian Eq. (1). Due to the presence of SOC, the relative motion of the
fermions is dependent on the CoM motion. As a result, the bound-state energy
also acquires dependence on the CoM momentum. For the lowest energy case of
zero CoM momentum, the two-body bound state wave function can be written as
$\displaystyle|\Psi\rangle$ $\displaystyle=$ $\displaystyle\bigg{\\{}\beta
b_{0}^{\dagger}+\sum_{{\bf k}}{}^{\prime}\Big{[}\eta^{\uparrow\downarrow}({\bf
k})a_{{\bf k},\uparrow}^{\dagger}a_{-{\bf
k},\downarrow}^{\dagger}+\eta^{\downarrow\uparrow}({\bf k})a_{{\bf
k},\downarrow}^{\dagger}a_{-{\bf k},\uparrow}^{\dagger}$ (3)
$\displaystyle+\eta^{\uparrow\uparrow}({\bf k})a_{{\bf
k},\uparrow}^{\dagger}a_{-{\bf
k},\uparrow}^{\dagger}+\eta^{\downarrow\downarrow}({\bf k})a_{{\bf
k},\downarrow}^{\dagger}a_{-{\bf
k},\downarrow}^{\dagger}\Big{]}\bigg{\\}}|0\rangle,$
where $\beta$ and $\eta^{\sigma\sigma^{\prime}}$ denote the closed channel and
open channel coefficients, respectively, and the summation over momentum space
$\sum_{\bf k}^{\prime}$ runs over half of the momentum space with $k_{y}>0$.
By solving the Shrödinger’s equation $H|\Psi\rangle=E|\Psi\rangle$ and
matching coefficients, we obtain the equation for the two-body binding energy
$E$
$\displaystyle\left[{U}_{p}-\frac{{g}_{p}^{2}}{{\gamma}_{p}-E}\right]^{-1}={\cal
S}({\alpha},E),$ (4)
where ${\cal S}({\alpha},E)$ is defined as,
$\displaystyle{\cal
S}({\alpha},E)\equiv\frac{3\pi}{8\sqrt{2}}\left[\sqrt{-E}-\frac{\alpha}{\sqrt{2}}{\rm
arctanh}\left(\frac{{\alpha}}{\sqrt{-2E}}\right)\right].$ (5)
Figure 1: (Color online) Two-body binding energy as a function of detuning for
(a-b) positive background scattering length, and (c-d) negative background
scattering length. The results for a two-channel model (solid, red) are
compared with those from a single-channel model (dashed, blue). Using the unit
system defined within the text, the dimensionless atom-molecule coupling
constant $g_{p}=7$ for all panels. Other parameters are (a)
$U_{p}=0.17,\alpha=0$; (b) $U_{p}=0.17,\alpha=5$; (c) $U_{p}=-0.17,\alpha=0$;
(d) $U_{p}=-0.17,\alpha=5$.
In Fig. 1, we plot typical results for the two-body binding energy
$E_{b}\equiv E-E_{\rm th}$ for both cases of positive and negative background
scattering lengths, where $E_{\rm th}=-\alpha^{2}/2$ is the threshold. In the
absence of SOC [see Fig. 1(a) and (c)], there are two branches of bound state
solution for positive background scattering length, while there is only one
branch for negative background scattering length. This is consistent with the
calculations of Ref. WY_D . For finite SOC, an additional branch of bound
state emerges for the case with negative background scattering length [see
Fig. 1(d)], which is consistent with the results of Ref. shenoy ; chenggang .
As a comparison, results obtained from a single-channel model with
corresponding parameters are also shown.
The position of the bound state threshold in the upper branch can be
determined analytically for both cases of positive and negative background
scattering length, leading to
$\displaystyle\gamma_{c}=-\frac{\alpha^{2}}{2}+\frac{g_{p}^{2}}{U_{p}}.$ (6)
It is clear that the positions of the bound state threshold in both cases are
pushed towards the BEC-side of the Feshbach resonance as the SOC strength
increases. This is the direct result of decreasing threshold energy with
increasing SOC.
In the large-detuning limit, the binding energy in either branch
asymptotically approaches a common value $E_{\inf}$, which is determined by
the following equation,
$\displaystyle{U}_{p}=\frac{16}{3\pi}\frac{1}{\sqrt{-2E_{\inf}}-{\alpha}{\rm
arctanh}\left({\alpha}/\sqrt{-2E_{\inf}}\right)}.$ (7)
One can easily read from this result that $E_{\rm inf}$ becomes more negative
with increasing SOC strength.
Figure 2: (Color online) The superfluid order parameter $\Delta$, the shifted
chemical potential $\mu_{b}\equiv\mu+\alpha^{2}/4$, and the molecular fraction
$n_{b}$ associated with (a-c) the upper branch solution, and (d-f) the lower
branch solution for the case of a positive background scattering length. For
all panels, dimensionless parameters are chose as
$U_{p}=0.17,g_{p}=7,\alpha=1$ (solid, red), $U_{p}=0.14,g_{p}=5,\alpha=1$
(dashed, black), and $U_{p}=0.1,g_{p}=2.5,\alpha=1$ (dash-dotted, blue).
## IV Manby-body pairing states
In this section, we characterize the many-body properties of the system at
zero temperature. Following the standard BCS mean-field theory, the effective
Hamiltonian can be written in a matrix form in the pseudo-spin basis
$\\{a_{\mathbf{k},\uparrow},a_{-\mathbf{k},\uparrow}^{{\dagger}},a_{\mathbf{k},\downarrow}a_{-\mathbf{k},\downarrow}^{{\dagger}}\\}^{T}$:
$\displaystyle H_{\text{eff}}-{\mu}N$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\mathbf{k}}\left[\begin{array}[]{cccc}\lambda_{\mathbf{k}}&{\Delta}&0&\kappa_{\mathbf{k}}^{-}\\\
{\Delta}&-\lambda_{\mathbf{k}}&\kappa_{\mathbf{k}}^{-}&0\\\
0&\kappa_{\mathbf{k}}^{+}&-\lambda_{\mathbf{k}}&-{\Delta}\\\
\kappa_{\mathbf{k}}^{+}&0&-{\Delta}&\lambda_{\mathbf{k}}\end{array}\right]$
(13)
$\displaystyle+\sum_{\mathbf{k}}\left(\epsilon_{\mathbf{k}}+{U}f\right)+({\gamma}-2{\mu})|\psi_{m}|^{2}-{U}(|p|^{2}+f^{2}).$
Here,
$\lambda_{\mathbf{k}}=\epsilon_{\mathbf{k}}-{\mu}+Uf,\kappa_{\mathbf{k}}^{\pm}={\alpha}(k_{x}\pm
ik_{y})$, the order parameter ${\Delta}={g}\psi_{m}+{U}p$, and the mean-field
parameters are defined as:
$\displaystyle\psi_{m}=$ $\displaystyle<b_{0}>$ $\displaystyle p=$
$\displaystyle\sum_{\mathbf{k}}<a_{-\mathbf{k},\downarrow}a_{\mathbf{k},\uparrow}>$
$\displaystyle f=$
$\displaystyle\sum_{\mathbf{k}}<a^{{\dagger}}_{\mathbf{k},\uparrow}a_{\mathbf{k},\uparrow}>=\sum_{\mathbf{k}}<a^{{\dagger}}_{\mathbf{k},\downarrow}a_{\mathbf{k},\downarrow}>.$
(14)
Notice that the dimensionless total particle number $N=1$ in the unit system
defined in Sec. II.
By diagonalizing the effective Hamiltonian Eq. (13) and by imposing the
renormalization condition with physical parameters, we obtain the expression
for the ground state thermodynamic potential at zero temperature
$\displaystyle\Omega=$
$\displaystyle\sum_{\mathbf{k}}\left[\epsilon_{\mathbf{k}}-\frac{1}{2}(E_{\mathbf{k},+}+E_{\mathbf{k},-})+{U}f\right]$
$\displaystyle+({\gamma}-2{\mu})|\psi_{m}|^{2}-{U}(|p|^{2}+f^{2}),$ (15)
where the quasi-particle dispersion
$E_{\mathbf{k},\pm}=\sqrt{A^{2}_{\mathbf{k},\pm}+{\Delta}^{2}}$, and
$A_{\mathbf{k},\pm}=\epsilon_{\mathbf{k}}-{\mu}+{U}f\pm{\alpha}k_{\perp}$ with
$k_{\perp}=\sqrt{k_{x}^{2}+k_{z}^{2}}$ is defined to simplify notation.
From the extrema conditions $\partial\Omega/\partial f=0$,
$\partial\Omega/\partial p=0$, $\partial\Omega/\partial\psi_{m}=0$, and the
number equation $N=-\partial\Omega/\partial{\mu}$, we have a set of self-
consistent equations huhuitwo-c :
$\displaystyle\psi_{m}=$
$\displaystyle-\frac{{g}_{p}p}{({\gamma}_{p}-2{\mu})},$ $\displaystyle 2f=$
$\displaystyle\sum_{\mathbf{k}}\left(1-\frac{A_{+}}{2E_{+}}-\frac{A_{-}}{2E_{-}}\right),$
$\displaystyle 1=$
$\displaystyle\left({U}_{p}-\frac{{g}^{2}_{p}}{{\gamma}_{p}-2{\mu}}\right)\sum_{\mathbf{k}}\left(\frac{1}{2\epsilon_{\mathbf{k}}}-\frac{1}{4E_{\mathbf{k},+}}-\frac{1}{4E_{\mathbf{k},-}}\right),$
$\displaystyle 1=$ $\displaystyle
2f+2|{\Delta}|^{2}\left[{g}_{p}-\frac{({\gamma}_{p}-2{\mu}){U}_{p}}{{g}_{p}}\right]^{-2},$
(16)
from which the ground state parameters can be determined. Note that for the
parameter regime discussed in this work, we find the influence on the chemical
potential induced by the Hartree term $f$ remains negligible. Thus, the quasi-
particle dispersion can be approximated as
$E_{\mathbf{k},\pm}\approx\sqrt{(\epsilon_{\mathbf{k}}-{\mu}\pm{\alpha}k_{\perp})^{2}+{\Delta}^{2}}.$
(17)
Note that the self-consistent Eqs. (16) can be reduced to the more familiar
forms of the gap and number equations under SOC in a single-channel model by
setting ${g}_{p}=0$ wy2d . We also define the closed channel fraction as
$n_{b}=2|{\Delta}|^{2}\left[({g}_{p}-\frac{({\gamma}_{p}-2{\mu}){U}_{p}}{{g}_{p}})^{2}\right]^{-1},$
(18)
which will be used to describe the properties of the underlying system.
Figure 3: (Color online) The superfluid order parameter $\Delta$, the shifted
chemical potential $\mu_{b}\equiv\mu+\alpha^{2}/4$, and the molecular fraction
$n_{b}$ associated with (a-c) the upper branch solution, and (d-f) the lower
branch solution for the case of a negative background scattering length.
Dimensionless parameters used in this figure are
$U_{p}=-0.17,g_{p}=7,\alpha=5$ (solid, red), $U_{p}=-0.14,g_{p}=5,\alpha=5$
(dashed, black), and $U_{p}=-0.1,g_{p}=2.5,\alpha=5$ (dash-dotted, blue).
### IV.1 Positive background scattering length
For the case of a positive background scattering length $a_{\rm bg}>0$, there
exists a weakly-bound state in the open channel away from the Fesbach
resonance. As the magnetic field is tuned close to the resonance point, the
coupling between the bound states in the open and the closed channels gives
rise to the two branches of many-body solution in the absence of SOC. Previous
studies have shown the existence of a quantum phase transition in the upper
branch where the many-body ground state changes from a superfluid state to a
normal state WY_D . The qualitative picture remains valid in the presence of
SOC, where quantitative modifications can be induced by the SOC. In Fig. 2, we
map out various mean-field quantities as functions of the detuning for several
scattering parameters.
In Fig. 2(a-c) we show the results of order parameter, the chemical potential,
and the molecular fraction as functions of detuning for the upper branch,
which corresponds to the weakly-bound state in the two-body case. Here, the
chemical potential is plotted after subtracting the single-particle threshold
with SOC, i.e., $\mu_{b}\equiv\mu+\alpha^{2}/4$. An interesting feature here
is the existence of a quantum phase transition, whose location can be
identified as the detuning where the order parameter approaches zero [see Fig.
2 (a)]. As the resonance width narrows, the location of the phase transition
point moves towards the BEC-side of the Feshbach resonance. Importantly, with
appropriate resonance width and SOC strength, the location of the quantum
phase transition point may cross the Feshbach resonance and reach the BEC
side, as we will show in Sec. V. We also note that the shifted chemical
potential $\mu_{b}$ in the upper branch crosses zero on the BEC side of the
resonance, demonstrating the existence of a BCS-BEC crossover [see the inset
of Fig. 2(b)]. We show in Fig. 2(d-f) the properties of the lower branch,
which corresponds to the deeply-bound state in the two-body case. It is
apparent that the shifted chemical potential stays negative with an order
parameter approaching finite values in both the weak and strong coupling
limit. Physically, the solution in the lower branch corresponds to a
condensate of tightly bound molecules, which become Rashbons in the large SOC
limit soc3 .
### IV.2 Negative scattering length
We now turn to the case with negative background scattering length $a_{\rm
bg}<0$. In the absence of SOC, there is only one branch of many-body solution,
which features a BCS-BEC crossover as the interaction strength is tuned WY_D .
When SOC is turned on, however, this picture is drastically modified. Similar
to the two-body case, a new branch (upper branch) of many-body solution
emerges. Interestingly, a quantum phase transition can also be identified in
this upper branch.
In Fig. 3 (a-c) we show the results of the order parameter, the shifted
chemical potential, and the molecular fraction as functions of detuning for
the upper branch. Here, the location of the quantum phase transition is pushed
towards the BEC-limit with increasing SOC strength or Feshbach resonance
width, and the shifted chemical potential $\mu_{b}$ remains positive for
arbitrary detuning. In Fig. 3 (d-f), we show the same quantities for the lower
branch. Notice that there is no quantum phase transition in this branch, as
the order parameter is always finite. For small SOC, the shifted chemical
potential is positive in the BCS-limit, indicating the existence of a Fermi
surface. Hence, the system in the lower branch undergoes a BCS-BEC crossover
as the interaction becomes stronger or as the SOC strength increases.
## V Quantum Phase Transition
In previous sections, we see that the quantum phase transition in the upper
branch is intimately connected with the SOC. For positive background
scattering length, SOC can modify the location of the phase transition point,
while for negative background scattering length, SOC can induce a new quantum
phase transition. In this section, we discuss in detail the dependence of the
phase transition point on various parameters.
The condition for the onset of this quantum phase transition can in fact be
obtained analytically by examining the gap and the number equations (16). At
the critical detuning where the quantum phase transition occurs, we have
$\Delta=0$. For the upper branch, regardless of the sign of the background
scattering length, this is only possible when the right-hand-side of the gap
equation also tends to infinity as the quantum critical point is approached.
Therefore, at the critical point, the denominator of the left-hand-side of the
gap equation must vanish, leading to
$\displaystyle U_{p}=\frac{g^{2}_{p}}{\gamma_{p}^{c}-2\mu}.$ (19)
Here, $\gamma_{p}^{c}$ is the critical detuning of the quantum phase
transition point in the upper branch. Similarly, the number equation at the
critical point takes the form
$\displaystyle
1=\sum_{\mathbf{k}}\left(1-\frac{A^{\prime}_{k,+}}{2|A^{\prime}_{k,+}|}-\frac{A^{\prime}_{k,-}}{2|A^{\prime}_{k,-}|}\right),$
(20)
where $A^{\prime}_{\mathbf{k},\pm}=\epsilon_{\mathbf{k}}-{\mu}\pm\alpha
k_{\perp}$. From these equations, we see that SOC only affects the quantum
critical point through the chemical potential $\mu$.
Figure 4: Critical detuning $\gamma_{p}^{c}$ for the quantum phase transition
in the upper branch as functions of $g_{p}$ and $\alpha$. Dimensionless
parameters used in these plots are (a) $\alpha=2,U_{p}=0.06$; (b)
$\alpha=5,U_{p}=-0.06$; (c) $g_{p}=0.5,U_{p}=0.06$; and (d)
$g_{p}=0.5,U_{p}=-0.06$.
We show in Fig. 4 the phase transition point in the upper branch as functions
of the scattering parameter $g_{p}$ and the SOC strength. For a fixed
background interaction rate $U_{p}$, the width of the Feshbach resonance
typically narrows with decreasing $g_{p}$. For a positive background
scattering length, the critical detuning can be made to cross the resonance
point when either the resonance width or the SOC strength is tuned [see Fig.
4(a) and 4(c)]. For systems with negative background scattering length,
however, the critical point is always lying on the BEC side of the Feshbach
resonance in any realistic situation. By choosing a narrow resonance with
small SOC strength, this quantum phase transition can be tuned closer to the
resonance points, as can be seen in Fig. 4(b) and 4(d). This observation
suggests that for the experimental observation of this quantum phase
transition, a system with narrow Feshbach resonance under moderate SOC
strength should be preferred.
## VI SUMMARY
We have studied a spin-orbit coupled ultracold Fermi gas near a Feshbach
resonance by using a two-channel model. We find that under a finite SOC and
with a finite background scattering length, there are in general two branches
of solution for both the two-body and the many-body ground states. This is in
contrast to the conventional BCS-BEC crossover picture, where the background
scattering length is typically neglected; and is different from the case
without SOC, where the upper-branch solution only exists for a positive
background scattering length. As a result, for a negative background
scattering length, the bound state in the open channel is purely SOC induced.
These lead to the interesting situation that a quantum phase transition exists
in the upper branch of the many-body solution. The location of the quantum
phase transition can be tuned by the SOC strength, or by choosing Feshbach
resonances with different resonance widths. In particular, the critical point
of the quantum phase transition can be tuned close to or across the resonance
point, where the Fermi gas is most stable against three-body losses. It is
therefore hopeful that such phase transitions can be observed in experiments.
###### Acknowledgements.
This work is supported by NFRP (2011CB921200, 2011CBA00200), NKBRP
(2013CB922000), NNSF (60921091), NSFC (11105134, 11274009, 11374283), SRFDP
(20113402120022), the Fundamental Research Funds for the Central Universities
(WK2470000006), and the Research Funds of Renmin University of China
(10XNL016, 13XNH123).
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|
arxiv-papers
| 2014-04-01T06:30:43 |
2024-09-04T02:50:00.491366
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fan Wu, Ren Zhang, Tian-Shu Deng, Wei Zhang, Wei Yi and Guang-Can Guo",
"submitter": "Fan Wu",
"url": "https://arxiv.org/abs/1404.0140"
}
|
1404.0177
|
# Multiple values and finiteness problem of meromorphic mappings sharing
different families of moving hyperplanes
Ha Huong Giang Faculty of Fundamental Sciences, Electric Power University
235-Hoang Quoc Viet, Tu Liem , Ha Noi, Vietnam. [email protected]
###### Abstract.
In this article, we show some uniqueness theorems for meromorphic mappings of
${\mathbf{C}}^{n}$ into the complex projective space
${\mathbf{P}^{n}{(\mathbf{C})}}$ sharing different families of moving
hyperplanes regardless of multiplicites, where all intersecting points between
these mappings and moving hyperplanes with multiplicities more than a certain
number do not need to be counted.
††footnotetext:
## 1\. Introduction
In 1926, Nevanlinna [5] showed that for two nonconstant meromorphic functions
$f$ and $g$ on the complex plane ${\mathbf{C}}$, if they have the same inverse
images for five distinct values, then $f\equiv g$. After that, many
mathematicians have generalized the Nevanlinna’s result to the case of
meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}^{n}{(\mathbf{C})}}$. Specially, in 1975, Fujimoto [3] proved that
for two linearly nondegenerate meromorphic mappings $f$ and $g$ of
${\mathbf{C}}^{m}$ into ${\mathbf{P}^{n}{(\mathbf{C})}}$, if they have the
same inverse images counted with multiplicities for $3n+2$ hyperplanes in
general position in ${\mathbf{P}^{n}{(\mathbf{C})}}$, then $f\equiv g$.
In 1983, L.Smiley [9] considered meromorphic mappings with share $3n+2$
hyperplanes of ${\mathbf{P}^{n}{(\mathbf{C})}}$ without counting
multiplicities and he proved the following. Theorem A (see [9]). Let
$f,g:{\mathbf{C}}^{m}\rightarrow{\mathbf{P}^{n}{(\mathbf{C})}}$ be linearly
nondegenerate meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}^{n}{(\mathbf{C})}}$ . Let $\\{H_{i}\\}_{i=1}^{q}$ $(q\geq 3n+2)$
be hyperlanes in ${\mathbf{P}^{n}{(\mathbf{C})}}$ in general position. Assume
that
$\displaystyle(i)f^{-1}(H_{i})=g^{-1}(H_{i}),\quad for\quad 1\leq i\leq q$
$\displaystyle(ii)\dim(f^{-1}(H_{i})\cap f^{-1}(H_{j}))\leq m-2,\quad for\quad
all\quad 1\leq i<j\leq q$ $\displaystyle(iii)f=g\quad
on\quad\bigcup_{i=1}^{q}f^{-1}(H_{i})$
then $f=g$.
In 2010, Gerd Dethloff , Si Duc Quang and Tran Van Tan [2] considered the case
where the mappings sharing different families of hyperplanes. They showed that
Theorem B (see [2]). Let
$f,g:{\mathbf{C}}^{m}\rightarrow{\mathbf{P}^{n}{(\mathbf{C})}}$ be a
meromorphic mapping. Let $\\{H_{i}\\}_{i=1}^{q}$ and $\\{L_{i}\\}_{i=1}^{q}$,
$(q\geq 3n+2)$ be families of hyperplanes in ${\mathbf{P}^{n}{(\mathbf{C})}}$
in general position. Assume that
$\displaystyle(i)f^{-1}(H_{i})=g^{-1}(L_{i}),\quad for\quad 1\leq i\leq q$
$\displaystyle(ii)\dim(f^{-1}(H_{i})\cap f^{-1}(H_{j}))\leq m-2,\quad for\quad
all\quad 1\leq i<j\leq q$
$\displaystyle(iii)\frac{(f,H_{i})}{(g,L_{i})}=\frac{(f,H_{j})}{(g,L_{j})}\quad
on\quad\bigcup_{k=1}^{q}f^{-1}(H_{k})\setminus(f^{-1}(H_{i})\cap
f^{-1}(H_{j})),\quad for\quad all\quad 1\leq i<j\leq q.$
Then the following assertions hold:
$dim\langle Imf\rangle=dim\langle Img\rangle:=p$
where for a subset $X\subset{\mathbf{P}^{n}{(\mathbf{C})}}$, we denote by
$\langle X\rangle$ the smallest projective subspace of
${\mathbf{P}^{n}{(\mathbf{C})}}$ containing $X$.
$If\quad q>\dfrac{2n+3-p+\sqrt{(2n+3-p)^{2}+8(p-1)(2n-p+1)}}{2}(\geq 2n+2)$
then
$\displaystyle\frac{(f,H_{1})}{(g,L_{1})}\equiv...\equiv\frac{(f,H_{q})}{(g,L_{q})}$
Furthermore, there exists a linear projective transformation $\mathcal{L}$ of
${\mathbf{P}^{n}{(\mathbf{C})}}$ into itself such that $\mathcal{L}(f)\equiv
g$ and $\mathcal{L}(H_{i}\cap\langle
Imf\rangle)=\mathcal{L}_{i}\cap\mathcal{L}(\langle Imf\rangle)$ for all
$i\in\\{1,...,q\\}$.
In 2011, Ting-Bin Cao and Hong-Xun Yi [1] showed the following result
Theorem C (see [1]). Let $f$ and $g$ be two linearly non-degenerate
meromorphic mappings of ${\mathbf{C}}^{m}$ into
${\mathbf{P}^{n}{(\mathbf{C})}}$, and let $H_{1},H_{2},...,H_{q}$ be $q$
$(q\geq 2n)$ hyperplanes in general position such that $dimf^{-1}(H_{i}\cap
H_{j})\leq m-2$ for $i\neq j$. Take $m_{j}$ $(j=1,2,...,q)$ be positive
integers or $\infty$ such that $m_{1}\geq m_{2}\geq...\geq m_{n}\geq n$,
$\nu_{(f,H_{j}),\leq m_{j}}^{1}=\nu_{(g,H_{j}),\leq
m_{j}}^{1}\quad(j=1,2,...,q)$
and $f(z)=g(z)$ on
$\bigcup_{j=1}^{q}\\{z\in{\mathbf{C}}^{m}:0<\nu_{(f,H_{j})}\leq m_{j}\\}$. If
$\sum_{j=3}^{q}\dfrac{m_{j}}{1+m_{j}}>\dfrac{nq-q+n+1}{n}-\dfrac{4n-4}{q+2n-2}+\biggl{(}\dfrac{1}{1+m_{1}}+\dfrac{1}{1+m_{2}}\biggl{)}$
then $f(z)\equiv g(z)$.
Recently, Zhonghua Wang and Zhenhan Tu proved a uniqueness theorem for
meromorphic mappings in several complex variables into the complex projective
space ${\mathbf{P}^{n}{(\mathbf{C})}}$ with two families of moving targets as
follows. Theorem D (see [11]). Let
$f,g,a_{i},b_{i}:\mathbf{C}^{m}\rightarrow{\mathbf{P}^{n}{(\mathbf{C})}}$ be
meromorphic mappings $(i=1,2,...,q)$. Suppose that $\\{a_{i}\\}_{i=1}^{q}$ are
“small” (with respect to f) and located in the general position, and that
$\\{b_{i}\\}_{i=1}^{q}$ are “small” (with respect to g) and located in the
general position such that f and g are linearly nondegenerate over
$\mathcal{R}(\\{a_{i},b_{i}\\}_{i=1}^{q})$. For any reduced representations
$a_{i}=(a_{i0},...,a_{in})$ and $b_{i}=(b_{i0},...,b_{in})$ $(i=1,2,...,q)$,
we may assume $a_{i0}\not\equiv 0$ and $b_{i0}\not\equiv 0$ $(i=1,2,...,q)$ by
changing the homogeneous coodinate system of ${\mathbf{P}^{n}{(\mathbf{C})}}$.
Let $\widetilde{a}_{i}=\frac{a_{i}}{a_{i0}}$ and
$\widetilde{b}_{i}=\frac{b_{i}}{b_{i0}}$ $(i=1,2,...,q)$. Assume that
$\displaystyle(i)\nu_{(f,\widetilde{a}_{i})}^{1}(z)=\nu_{(g,\widetilde{b}_{i})}^{1}(z),for1\leq
i\leq q$
$\displaystyle(ii)\dim\\{z\in{\mathbf{C}}^{m}:(f(z),a_{i}(z))=(f(z),a_{j}(z))=0\\}\leq
m-2,for1\leq i<j\leq q$
$\displaystyle(iii)\frac{(f,\widetilde{a}_{i})}{(g,\widetilde{b}_{i})}=\frac{(f,\widetilde{a}_{j})}{(g,\widetilde{b}_{j})}on\bigcup_{k=1;k\neq
i,j}^{q}\\{z\in C^{n}:(f(z),a_{k}(z))=0\\},for1\leq i<j\leq q.$
Then
If $q=2n^{2}+2n+3$ then there exist
$\\{i_{1},...,i_{n+1}\\}\subset\\{1,...,q\\}$such that
$\displaystyle\frac{(f,\widetilde{a}_{i_{1}})}{(g,\widetilde{b}_{i_{1}})}\equiv...\equiv\frac{(f,\widetilde{a}_{i_{n+1}})}{(g,\widetilde{b}_{i_{n+1}})}$
which immediately means that there exists a matrix L with its elements
$L_{ij}$ in $\mathcal{R}(\\{a_{i},b_{i}\\})_{i=1}^{q}$ such that $L(f)=g$
We note that, in the above theorem the mappings are assumed to be linearly
nondegenerate. Our purpose in this paper is to study the case where the
mappings may be degenerate. We will show some uniqueness theorems for mappings
sharing different families of moving hyperplanes regardless of multiplicities,
which are improvements and extensions of some recent results in this direction
when reduced to the case of mappings sharing the same family of moving
hyperplanes. Our main results of this work are stated as follows.
Let $f^{t}:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be meromorphic
mapping. Let $\\{a_{i}^{t}\\}_{i=1}^{q}$ be family of moving hyperplanes in
${\mathbf{P}}^{n}({\mathbf{C}})$ in general position such that $a_{i}^{t}$ be
“slowly” with respect to $f^{t}$. By changing the homogeneous coodinate system
of ${\mathbf{P}^{n}{(\mathbf{C})}}$ if necessary, we may assume that
$a^{t}_{i0}\not\equiv 0$ $(1\leq i\leq q)$ any given meromorphic mapping
$a_{i}^{t}=(a^{t}_{i0},...,a^{t}_{in})$. Let
$\widetilde{a}^{t}_{i}=\frac{a^{t}_{i}}{a^{t}_{i0}}$, $1\leq i\leq q$.
We will prove the following.
###### Theorem 1.1.
Let $f^{1},f^{2}:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be two
meromorphic mappings. Let $k_{i}$ $(1\leq i\leq q)$ be positive integers or
$\infty$. Let $\\{a_{i}^{t}\\}_{i=1}^{q}$ $(t=1,2)$ be two families of moving
hyperplanes in ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position such that
$a_{i}^{t}$ is ”slowly” with respect to $f^{t}$ and $\dim\
\\{z\in{\mathbf{C}}^{m}:\nu_{(f^{t},a^{t}_{i}),\leq
k_{i}}.\nu_{(f^{t},a^{t}_{j}),\leq k_{j}}>0\\}\leq m-2$ $(1\leq i<j\leq
q,t=1,2)$. We assum that:
(a) $\min\\{\nu_{(f^{2},\widetilde{a}_{i}^{2}),\leq
k_{i}}(z),1\\}=\min\\{\nu_{(f^{1},\widetilde{a}_{i}^{1}),\leq k_{i}}(z),1\\}$
$(1\leq i\leq q),$ for all $z\in{\mathbf{C}}^{m}$,
(b)
$\frac{(f^{1},\widetilde{a}_{i}^{1})}{(f^{2},\widetilde{a}_{i}^{2})}=\frac{(f^{1},\widetilde{a}_{j}^{1})}{(f^{2},\widetilde{a}_{j}^{2})}$
on $\bigcup_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,j}}}}}^{q}\mathrm{Supp}\,\\{z\in{\mathbf{C}}^{m}:\nu_{(f^{1},a^{1}_{v}),\leq
k_{v}}(z)\\}$, for $1\leq i<j\leq q.$
If $q>3n^{2}+n+2$ and
$\sum_{i=1}^{q}\frac{1}{k_{i}+1}<\biggl{(}\frac{2q}{3n(n+1)}-\frac{2q}{q+2n-2}\biggl{)}$,
then there exist $n+1$ indices $1\leq i_{1}<i_{2}<\cdots<i_{n+1}\leq q$ such
that
(1.2)
$\displaystyle\dfrac{(f^{1},\widetilde{a}_{i_{1}}^{1})}{(f^{2},\widetilde{a}_{i_{1}}^{2})}=\cdots=\dfrac{(f^{1},\widetilde{a}_{i_{n+1}}^{1})}{(f^{2},\widetilde{a}_{i_{n+1}}^{2})}.$
###### Theorem 1.3.
Let $f^{1},f^{2},f^{3}:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be
three meromorphic mappings. Let $k_{i}$ $(1\leq i\leq q)$ be positive integers
or $\infty$. Let $\\{a_{i}^{t}\\}_{i=1}^{q}$ $(t=1,2,3)$ be three families of
moving hyperplanes in ${\mathbf{P}}^{n}({\mathbf{C}})$ in general position
such that $a_{i}^{t}$ be “slowly” with respect to $f^{t}$ and $\dim\
\\{z\in{\mathbf{C}}^{m}:\nu_{(f^{t},a^{t}_{i}),\leq
k_{i}}.\nu_{(f^{t},a^{t}_{j}),\leq k_{j}}>0\\}\leq m-2\quad(1\leq i<j\leq
q,1\leq t\leq 3)$. We assume that:
1. (a)
$\min\\{\nu_{(f^{t},\widetilde{a}_{i}^{t}),\leq
k_{i}}(z),1\\}=\min\\{\nu_{(f^{1},\widetilde{a}_{i}^{1}),\leq k_{i}}(z),1\\}$
$(1\leq i\leq q,t=2,3),\ \forall z\in{\mathbf{C}}^{m}$,
2. (b)
$\frac{(f^{1},\widetilde{a}_{i}^{1})}{(f^{t},\widetilde{a}_{i}^{t})}=\frac{(f^{1},\widetilde{a}_{j}^{1})}{(f^{t},\widetilde{a}_{j}^{t})}$
on $\bigcup_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,j}}}}}^{q}\mathrm{Supp}\,\\{z\in{\mathbf{C}}^{m}:\nu_{(f^{1},a^{1}_{v}),\leq
k_{v}}(z)\\}$, $1\leq i<j\leq q,$ $t=2,3.$
If $q>\dfrac{9n^{2}+7n+6}{4}$ and
$\sum_{i=1}^{q}\dfrac{1}{k_{i}+1}<\dfrac{q-3n+2}{2q-5n+10}\biggl{(}\dfrac{2(2q+n-3)}{3n(n+1)}-3\biggl{)}$,
then there are two maps $f^{s},f^{t}\ (1\leq s<t\leq 3)$ and $n+1$ indices
$1\leq i_{1}<i_{2}<\cdots<i_{n+1}\leq q$ such that
$\dfrac{(f^{s},\widetilde{a}_{i_{1}}^{1})}{(f^{t},\widetilde{a}_{i_{1}}^{2})}=\cdots=\dfrac{(f^{s},\widetilde{a}_{i_{n+1}}^{1})}{(f^{t},\widetilde{a}_{i_{n+1}}^{2})}.$
## 2\. Basic notions and auxiliary results from Nevanlinna theory
(a) Counting function of divisor.
For $z=(z_{1},\dots,z_{m})\in{\mathbf{C}}^{m}$, we set
$\|z\|=\Big{(}\sum\limits_{j=1}^{m}|z_{j}|^{2}\Big{)}^{1/2}$ and define
$\displaystyle B(r)$ $\displaystyle=\\{z\in{\mathbf{C}}^{m};\|z\|<r\\},\quad
S(r)=\\{z\in{\mathbf{C}}^{m};\|z\|=r\\},$ $\displaystyle d^{c}$
$\displaystyle=\dfrac{\sqrt{-1}}{4\pi}(\overline{\partial}-\partial),\quad\sigma=\big{(}dd^{c}\|z\|^{2}\big{)}^{m-1},$
$\displaystyle\eta$
$\displaystyle=d^{c}\text{log}\|z\|^{2}\land\big{(}dd^{c}\text{log}\|z\|\big{)}^{m-1}.$
Thoughout this paper, we denote by $\mathcal{M}$ the set of all meromorphic
functions on ${\mathbf{C}}^{m}$. A divisor $E$ on ${\mathbf{C}}^{m}$ is given
by a formal sum $E=\sum\mu_{\nu}X_{\nu}$, where $\\{X_{\nu}\\}$ is a locally
family of distinct irreducible analytic hypersurfaces in ${\mathbf{C}}^{m}$
and $\mu_{\nu}\in\mathbf{Z}$. We define the support of the divisor $E$ by
setting $\mathrm{Supp}\,(E)=\cup_{\nu\neq 0}X_{\nu}$. Sometimes, we identify
the divisor $E$ with a function $E(z)$ from ${\mathbf{C}}^{m}$ into
$\mathbf{Z}$ defined by $E(z):=\sum_{X_{\nu}\ni z}\mu_{\nu}$.
Let $M,k$ be a positive integer or $+\infty$. We define the truncated divisor
$E^{[M]}$ and $E_{\leq k}^{[M]}$ by
$\displaystyle E^{[M]}:=\sum_{\nu}\min\\{\mu_{\nu},M\\}X_{\nu},$
$\displaystyle E_{\leq k}^{M}:=\begin{cases}0,&\text{ if }E(z)>k,\\\
E^{M},&\text{ if }E(z)\leq k.\end{cases}$
and the truncated counting function to level $M$ of $E$ by
$\displaystyle
N^{[M]}(r,E):=\int\limits_{1}^{r}\frac{n^{[M]}(t,E)}{t^{2m-1}}dt\quad(1<r<+\infty),$
Similarly, we define $N(r,E_{>k}^{[M]})$ and $N(r,E_{\leq k}^{[M]})$ and
denote them by $N_{>k}^{[M]}(r,E)$ and $N_{\leq k}^{[M]}(r,E)$ respectively.
where
$\displaystyle n^{[M]}(t,E):=\begin{cases}\int\limits_{\mathrm{Supp}\,(E)\cap
B(t)}E^{[M]}\sigma&\text{ if }m\geq 2,\\\ \sum_{|z|\leq t}E^{[M]}(z)&\text{ if
}m=1.\end{cases}$
Similarly, we define $n_{>k}^{[M]}(t,E)$ and $n_{\leq k}^{[M]}(t,E)$.
We omit the character [M] if $M=+\infty$.
For an analytic hypersurface $E$ of ${\mathbf{C}}^{m}$, we may consider it as
a reduced divisor and denote by $N(r,E)$ its counting function.
Let $\varphi$ be a nonzero meromorphic function on ${\mathbf{C}}^{m}$. We
denote by $\nu^{0}_{\varphi}$ (resp. $\nu^{\infty}_{\varphi}$) the divisor of
zeros (resp. divisor of poles) of $\varphi$. The divisor of $\varphi$ is
defined by
$\nu_{\varphi}=\nu^{0}_{\varphi}-\nu^{\infty}_{\varphi}.$
We have the following Jensen’s formula:
$\displaystyle
N(r,\nu^{0}_{\varphi})-N(r,\nu^{\infty}_{\varphi})=\int\limits_{S(r)}\text{log}|\varphi|\eta-\int\limits_{S(1)}\text{log}|\varphi|\eta.$
For convenience, we will write $N_{\varphi}(r)$ and $N^{[M]}_{\varphi}(r)$ for
$N(r,\nu^{0}_{\varphi})$ and $N^{[M]}(r,\nu^{0}_{\varphi})$, respectively.
(b) The first main theorem.
Let $f$ be a meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})$. For arbitrary fixed homogeneous coordinates
$(w_{0}:\cdots:w_{n})$ of ${\mathbf{P}}^{n}({\mathbf{C}})$, we take a reduced
representation $f=(f_{0}:\cdots:f_{n})$, which means that each $f_{i}$ is
holomorphic function on ${\mathbf{C}}^{m}$ and
$f(z)=(f_{0}(z):\cdots:f_{n}(z))$ outside the analytic set
$I(f):=\\{z;f_{0}(z)=\cdots=f_{n}(z)=0\\}$ of codimension at least $2$.
Denote by $\Omega$ the Fubini Study form of ${\mathbf{P}}^{n}({\mathbf{C}})$.
The characteristic function of $f$ (with respect to $\Omega$) is defined by
$\displaystyle
T_{f}(r):=\int_{1}^{r}\dfrac{dt}{t^{2m-1}}\int_{B(t)}f^{*}\Omega\wedge\sigma,\quad\quad
1<r<+\infty.$
By Jensen’s formula we have
$\displaystyle T_{f}(r)=\int_{S(r)}\log||f||\eta+O(1),$
where $\|f\|=\max\\{|f_{0}|,\dots,|f_{n}|\\}$.
Let $a$ be a meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ with reduced representation
$a=(a_{0}:\dots:a_{n})$. We define
$m_{f,a}(r)=\int\limits_{S(r)}\text{log}\dfrac{||f||\cdot||a||}{|(f,a)|}\eta-\int\limits_{S(1)}\text{log}\dfrac{||f||\cdot||a||}{|(f,a)|}\eta,$
where $\|a\|=\big{(}|a_{0}|^{2}+\dots+|a_{n}|^{2}\big{)}^{1/2}$ and
$(f,a)=\sum_{i=0}^{n}f_{i}\cdot a_{i}.$
Let $f$ and $a$ be as above. If $(f,a)\not\equiv 0$, then the first main
theorem for moving hyperplaness in value distribution theory states
$T_{f}(r)+T_{a}(r)=m_{f,a}(r)+N_{(f,a)}(r)+O(1)\ (r>1).$
For a meromorphic function $\varphi$ on ${\mathbf{C}}^{m}$, the proximity
function $m(r,\varphi)$ is defined by
$m(r,\varphi)=\int\limits_{S(r)}\log^{+}|\varphi|\eta,$
where $\log^{+}x=\max\big{\\{}\log x,0\big{\\}}$ for $x\geqslant 0$. The
Nevanlinna’s characteristic function is defined by
$T(r,\varphi)=N(r,\nu^{\infty}_{\varphi})+m(r,\varphi).$
We regard $\varphi$ as a meromorphic mapping of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{1}({\mathbf{C}})^{*}$, there is a fact that
$T_{\varphi}(r)=T(r,\varphi)+O(1).$
(c) Lemma on logarithmic derivative.
As usual, by the notation $``||\ P"$ we mean the assertion $P$ holds for all
$r\in[0,\infty)$ excluding a Borel subset $E$ of the interval $[0,\infty)$
with $\int_{E}dr<\infty$. Denote by $\mathbf{Z}_{+}$ the set of all
nonnegative integers. The lemma on logarithmic derivative in Nevanlinna theory
is stated as follows.
###### Lemma 2.1 (see [8, Lemma 3.11]).
Let $f$ be a nonzero meromorphic function on ${\mathbf{C}}^{m}.$ Then
$\biggl{|}\biggl{|}\quad
m\biggl{(}r,\dfrac{\mathcal{D}^{\alpha}(f)}{f}\biggl{)}=O(\log^{+}T_{f}(r))\
(\alpha\in\mathbf{Z}^{m}_{+}).$
(d) Family of moving hyperplanes.
We assume that thoughout this paper, the homogeneous coordinates of
${\mathbf{P}}^{n}({\mathbf{C}})$ is chosen so that for each given meromorphic
mapping $a=(a_{0}:\cdots:a_{n})$ of ${\mathbf{C}}^{m}$ into
${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ then $a_{0}\not\equiv 0$. We set
$\tilde{a}_{i}=\dfrac{a_{i}}{a_{0}}\text{ and
}\tilde{a}=(\tilde{a}_{0}:\tilde{a}_{1}:\cdots:\tilde{a}_{n}).$
Let $f:{\mathbf{C}}^{m}\rightarrow{\mathbf{P}}^{n}({\mathbf{C}})$ be a
meromorphic mapping with the reduced representation $f=(f_{0}:\cdots:f_{n}).$
We put $(f,a):=\sum_{i=0}^{n}f_{i}a_{i}$ and
$(f,\tilde{a}):=\sum_{i=0}^{n}f_{i}\tilde{a}_{i}.$
Let $\\{a_{i}\\}_{i=1}^{q}$ be $q$ meromorphic mappings of ${\mathbf{C}}^{m}$
into ${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ with reduced representations
$a_{i}=(a_{i0}:\cdots:a_{in})\ (1\leq i\leq q).$ We denote by
$\mathcal{R}(\\{a_{i}\\})$ (for brevity we will write $\mathcal{R}$ if there
is no confusion) the smallest subfield of $\mathcal{M}$ which contains
${\mathbf{C}}$ and all ${a_{i_{j}}}/{a_{i_{k}}}$ with $a_{i_{k}}\not\equiv 0.$
###### Definition 2.2.
The family $\\{a_{i}\\}_{i=1}^{q}$ is said to be in general position if
$\dim(\\{a_{i_{0}},\ldots,a_{i_{n}}\\})_{\mathcal{M}}=n+1$ for any $1\leq
i_{0}\leq\cdots\leq i_{n}\leq q$, where
$(\\{a_{i_{0}},\ldots,a_{i_{n}}\\})_{\mathcal{M}}$ is the linear span of
$\\{a_{i_{0}},\ldots,a_{i_{N}}\\}$ over the field $\mathcal{M}.$
###### Theorem 2.3 (The second main theorem [7, Corollary 1.2]).
Let $f:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$ be a meromorphic
mapping. Let $\\{a_{i}\\}_{i=1}^{q}\ (q\geq 2n+1)$ be meromorphic mappings of
${\mathbf{C}}^{m}$ into ${\mathbf{P}}^{n}({\mathbf{C}})^{*}$ in general
position such that $(f,a_{i})\not\equiv 0\ (1\leq i\leq q).$
$\mathrm{(a)}$ If $q\geq 3n+3$ then
$||\dfrac{2q}{3(n+1)}T_{f}(r)\leq\sum_{i=1}^{q}N_{(f,a_{i})}^{[n]}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
$\mathrm{(b)}$ If $q<3n+3$ then
$||\dfrac{q-n+1}{n+2}T_{f}(r)\leq\sum_{i=1}^{q}N_{(f,a_{i})}^{[n]}(r)+o(T_{f}(r))+O(\max_{1\leq
i\leq q}T_{a_{i}}(r)).$
## 3\. Proof of Theorem 1.1
Assume that
$\sum_{v=1}^{q}\dfrac{1}{k_{v}+1}<\biggl{(}\dfrac{2q}{3n(n+1)}-\dfrac{2q}{q+2n-2}\biggl{)}.$
Suppose that the conclussion 1.2 does not hold. By changing indices if
necessary, we may assume that
$\underbrace{\dfrac{(f^{1},\widetilde{a}_{1}^{1})}{(f^{2},\widetilde{a}_{1}^{2})}\equiv\dfrac{(f^{1},\widetilde{a}_{2}^{1})}{(f^{2},\widetilde{a}_{2}^{2})}\equiv\cdots\equiv\dfrac{(f^{1},\widetilde{a}_{v_{1}}^{1})}{(f^{2},\widetilde{a}_{v_{1}}^{2})}}_{\text{
group
}1}\not\equiv\underbrace{\dfrac{(f^{1},\widetilde{a}_{v_{1}+1}^{1})}{(f^{2},\widetilde{a}_{v_{1}+1}^{2})}\equiv\cdots\equiv\dfrac{(f^{1},\widetilde{a}_{v_{2}}^{1})}{(f^{2},\widetilde{a}_{v_{2}}^{2})}}_{\text{
group }2}$
$\not\equiv\underbrace{\dfrac{(f^{1},\widetilde{a}_{v_{2}+1}^{1})}{(f^{2},\widetilde{a}_{v_{2}+1}^{2})}\equiv\cdots\equiv\dfrac{(f^{1},\widetilde{a}_{v_{3}}^{1})}{(f^{2},\widetilde{a}_{v_{3}}^{2})}}_{\text{
group
}3}\not\equiv\cdots\not\equiv\underbrace{\dfrac{(f^{1},\widetilde{a}_{v_{s-1}+1}^{1})}{(f^{2},\widetilde{a}_{v_{s-1}+1}^{2})}\equiv\cdots\equiv\dfrac{(f^{1},\widetilde{a}_{v_{s}}^{1})}{(f^{2},\widetilde{a}_{v_{s}}^{2})}}_{\text{
group }s},$
where $v_{s}=q.$
For each $1\leq i\leq q,$ we set
$\sigma(i)=\begin{cases}i+n&\text{ if $i+n\leq q$},\\\ i+n-q&\text{ if
$i+n>q$}\end{cases}$
and
$P_{i}=(f^{1},\widetilde{a}_{i}^{1})(f^{2},\widetilde{a}_{\sigma(i)}^{2})-(f^{2},\widetilde{a}_{i}^{2})(f^{1},\widetilde{a}_{\sigma(i)}^{1}).$
By supposition, the number of elements of each group is at most $n$. Hence
$\dfrac{(f^{1},\widetilde{a}_{i}^{1})}{(f^{2},\widetilde{a}_{i}^{2})}$ and
$\dfrac{(f^{1},\widetilde{a}_{\sigma(i)}^{1})}{(f^{2},\widetilde{a}_{\sigma(i)}^{2})}$
belong to distinct groups. This means that $P_{i}\not\equiv 0\ (1\leq i\leq
q)$.
Fix an index $i$ with $1\leq i\leq q.$ It is easy to see that
$\displaystyle\nu_{P_{i}}(z)\geq\min\\{\nu_{(f^{1},\widetilde{a}_{i}^{1})},\nu_{(f^{2},\widetilde{a}_{i}^{2})}\\}+\min\\{\nu_{(f^{1},\widetilde{a}_{\sigma(i)}^{1})},\nu_{(f^{2},\widetilde{a}_{\sigma(i)}^{2})}\\}+\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,\sigma(i)}}}}}^{q}\nu_{(f^{1},\widetilde{a}_{v}^{1})}^{[1]}(z)$
outside a finite union of analytic sets of dimension $\leq m-2.$ Since
$\min\\{a,b\\}+n\geq\min\\{a,n\\}+\min\\{b,n\\}$ for all positive integers $a$
and $b$, the above inequality implies that
$\displaystyle
N_{P_{i}}(r)\geq\sum_{v=i,\sigma(i)}\left(N^{[n]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq
k_{v}}(r)+N^{[n]}_{(f^{2},\widetilde{a}_{v}^{2}),\leq
k_{v}}(r)-nN^{[1]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq
k_{v}}(r)\right)+\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,\sigma(i)}}}}}^{q}N^{[1]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq k_{v}}(r).$
On the other hand, by the Jensen formula, we have
$\displaystyle N_{P_{i}}(r)=$ $\displaystyle\int_{S(r)}\log|P_{i}|\eta+O(1)$
$\displaystyle\leq$
$\displaystyle\int_{S(r)}\log(|(f^{1},\widetilde{a}_{i}^{1})|^{2}+|(f^{1},\widetilde{a}_{\sigma(i)}^{1}|^{2})^{\frac{1}{2}}\eta+\int_{S(r)}\log(|(f^{2},\widetilde{a}_{i}^{2})|^{2}+|(f^{2},\widetilde{a}_{\sigma(i)}^{2}|^{2})^{\frac{1}{2}}\eta+O(1)$
$\displaystyle\leq$ $\displaystyle
T_{f^{1}}(r)+T_{f^{2}}(r)+o(T_{f^{1}}(r)+T_{f^{2}}(r)).$
This implies that
$\displaystyle T_{f^{1}}(r)+T_{f^{2}}(r)\geq$
$\displaystyle\sum_{v=i,\sigma(i)}\left(N^{[n]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq
k_{v}}(r)+N^{[n]}_{(f^{2},\widetilde{a}_{v}^{2}),\leq
k_{v}}(r)-nN^{[1]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq k_{v}}(r)\right)$
$\displaystyle+$ $\displaystyle\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,\sigma(i)}}}}}^{q}N^{[1]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq
k_{v}}(r)+o(T_{f^{1}}(r)+T_{f^{2}}(r)).$
Summing-up both sides of the above inequality over $i=1,\ldots,q$, we have
$\displaystyle q(T_{f^{1}}(r)+T_{f^{2}}(r))\geq$ $\displaystyle
2\sum_{v=1}^{q}\left(N^{[n]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq
k_{v}}(r)+N^{[n]}_{(f^{2},\widetilde{a}_{v}^{2}),\leq k_{v}}(r)\right)$
$\displaystyle+(q-2n-2)\sum_{v=1}^{q}N^{[1]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq
k_{v}}(r)+o(T_{f^{1}}(r)+T_{f^{2}}(r))$ $\displaystyle\geq$
$\displaystyle(2+\frac{q-2n-2}{2n})\sum_{v=1}^{q}\left(N^{[n]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq
k_{v}}(r)+N^{[n]}_{(f^{2},\widetilde{a}_{v}^{2}),\leq k_{v}}(r)\right)$
$\displaystyle+o(T_{f^{1}}(r)+T_{f^{2}}(r))$
We get
$\displaystyle\dfrac{2qn}{q+2n-2}(T_{f^{1}}(r)+T_{f^{2}}(r))$
$\displaystyle\geq\sum_{v=1}^{q}\left(N^{[n]}_{(f^{1},\widetilde{a}_{v}^{1}),\leq
k_{v}}(r)+N^{[n]}_{(f^{2},\widetilde{a}_{v}^{2}),\leq
k_{v}}(r)\right)+o(T_{f^{1}}(r)+T_{f^{2}}(r))$
$\displaystyle=\sum_{v=1}^{q}(N^{[n]}_{(f^{1},\widetilde{a}_{v}^{1})}(r)+N_{(f^{2},\widetilde{a}_{v}^{2})}^{[n]}(r)-N_{(f^{1},\widetilde{a}_{v}^{1}),>k_{v}}^{[n]}(r)-N_{(f^{2},\widetilde{a}_{v}^{2}),>k_{v}}^{[n]}(r))$
$\displaystyle+o(T_{f^{1}}(r)+T_{f^{2}}(r))$
$\displaystyle\geq\sum_{v=1}^{q}(N^{[n]}_{(f^{1},a_{v}^{1})}(r)+N_{(f^{2},a_{v}^{2})}^{[n]}(r)-N_{(f^{1},\widetilde{a}_{v}^{1}),>k_{v}}^{[n]}(r)-N_{(f^{2},\widetilde{a}_{v}^{2}),>k_{v}}^{[n]}(r))$
$\displaystyle+o(T_{f^{1}}(r)+T_{f^{2}}(r))$
By theorem 2.3, we have
$\displaystyle\biggl{|}\biggl{|}\dfrac{2q}{3(n+1)}(T_{f^{1}}(r)+T_{f^{2}}(r))$
$\displaystyle\leq\sum_{v=1}^{q}(N^{[n]}_{(f^{1},a_{v}^{1})}(r)+N_{(f^{2},a_{v}^{2})}^{[n]}(r))+o(T_{f^{1}}(r)+T_{f^{2}}(r))$
From the above inequalities, we have
$\displaystyle\biggl{(}\frac{2q}{3(n+1)}$
$\displaystyle-\frac{2qn}{q+2n-2}\biggl{)}(T_{f^{1}}(r)+T_{f^{2}}(r))$
$\displaystyle\leq\sum_{v=1}^{q}\biggl{(}N_{(f^{1},\widetilde{a}_{v}^{1}),>k_{v}}^{[n]}(r)+N_{(f^{2},\widetilde{a}_{v}^{2}),>k_{v}}^{[n]}(r)\biggl{)}$
$\displaystyle\ +o(T_{f^{1}}(r)+T_{f^{2}}(r))$
$\displaystyle\leq\sum_{v=1}^{q}\dfrac{n}{k_{v}+1}(N_{(f^{1},\widetilde{a}_{v}^{1})}(r)+N_{(f^{2},\widetilde{a}_{v}^{2})}(r))$
$\displaystyle\ +o(T_{f^{1}}(r)+T_{f^{2}}(r))$ $\displaystyle\leq
n\sum_{v=1}^{q}\dfrac{1}{k_{v}+1}(T_{f^{1}}(r)+T_{f^{2}}(r))+o(T_{f^{1}}(r)+T_{f^{2}}(r))$
Letting $r\to\infty$, we get
$\biggl{(}\dfrac{2q}{3n(n+1)}-\dfrac{2q}{q+2n-2}\biggl{)}\leq\sum_{v=1}^{q}\dfrac{1}{k_{v}+1}.$
This is a contradiction.
Then the supposition is impossible. Hence the theorem is proved. $\square$
## 4\. Proof of Theorem 1.3
In order to prove Theorem 1.3, we need the following.
3.1. Let $f^{1},f^{2},f^{3}:{\mathbf{C}}^{m}\to{\mathbf{P}}^{n}({\mathbf{C}})$
be three meromorphic mappings. Let $k_{i}$ $(1\leq i\leq q)$ be positive
integers or $\infty$. Let $\\{a_{i}^{t}\\}_{i=1}^{q}$ $(t=1,2,3)$ be 3
families of moving hyperplanes in ${\mathbf{P}}^{n}({\mathbf{C}})$ in general
position such that $a_{i}^{t}$ be ”slowly” with respect to $f^{t}$ and $\dim\
\\{z\in{\mathbf{C}}^{m}:\nu_{(f^{t},a^{t}_{i}),\leq
k_{i}}.\nu_{(f^{t},a^{t}_{j}),\leq k_{j}}>0\\}\leq m-2\quad(1\leq i<j\leq
q,1\leq t\leq 3)$, we put
$T(r)=\sum_{t=1}^{3}T_{f^{t}}(r).$
Assume that $a_{i}^{t}$ has a reduced representation
$a_{i}^{t}=(a_{i0}^{t}:\cdots:a_{in}^{t}).$ By changing the homogeneous
coordinate system of ${\mathbf{P}}^{n}({\mathbf{C}}),$ we may assume that
$a_{i0}^{t}\not\equiv 0\ (1\leq i\leq q,1\leq t\leq 3).$ For each
$c=(c_{1},...,c_{q})\in{\mathbf{C}}^{q}\setminus\\{0\\}$, we set
$\displaystyle
a^{t}_{c}:=(\sum_{i=1}^{q}c_{i}\widetilde{a}^{t}_{i0},...,\sum_{i=1}^{q}c_{i}\widetilde{a}^{t}_{in}),\text{
}||a^{t}_{c}||:=(\sum_{j=0}^{n}|\sum_{i=1}^{q}c_{i}\widetilde{a}^{t}_{ij}|^{2})^{\frac{1}{2}}$
$\displaystyle(f^{t},a^{t}_{c}):=\sum_{j=0}^{n}\sum_{i=1}^{q}c_{i}\widetilde{a}_{ij}f^{t}_{j}=\sum_{i=1}^{q}c_{i}(f^{t},\widetilde{a}_{i})\quad(1\leq
t\leq 3)$
We denote by $\beta$ the union of all irreducible components with dimension
$m-1$ of the analytic set $\bigcap_{i=1}^{q}Zero(f^{t},a^{t}_{i})$ $(1\leq
t\leq 3)$. Then $\beta$ is either an analytic set of pure dimension $m-1$ or
empty set. With $c\in{\mathbf{C}}^{q},$ we denote by $S_{c}^{jt}$ the closure
of set $(Zero(f^{t},a^{t}_{j})\cap Zero(f^{t},a^{t}_{c}))\setminus\beta$. Then
$S^{jt}_{c}$ is an analytic set. We also denote by $\mathcal{C}$ the set of
all $c\in{\mathbf{C}}^{q}\setminus\\{0\\}$ such that
$\dim\mathcal{S}_{c}^{jk}\leq m-2$
###### Lemma 4.1.
$\mathcal{C}$ is dense in ${\mathbf{C}}^{q}.$
Proof. For $1\leq i\leq q,1\leq t\leq 3$ and for each irreducible component
$\nu$ of the analytic set $\mathrm{Zero}(f^{t},a^{t}_{i})$ with
$\nu\not\subset\beta$, we set
$V^{it}_{\nu}=\\{c=(c_{1},\ldots,c_{q})\in{\mathbf{C}}^{q}\ :\
(f^{t},a^{t}_{c})(z)=0,\ \forall z\in\nu\\}.$
Then, $V^{it}_{\nu}$ is an complex vector subspace of ${\mathbf{C}}^{q}$.
Since $\nu\not\subset\beta$, there exists an index $j$ such that
$\nu\not\subset\mathrm{Zero}(f^{t},a^{t}_{j})$. Therefore the element
$c=(0,\ldots,0,{\mathrel{\mathop{{1}}\limits_{{j-th}}}},0,\ldots,0)$ does not
belong to $V^{it}_{\nu}$. Hence $\dim V^{it}_{\nu}\leq q-1$. Let
$K=\bigcup_{i=1}^{q}\bigcup_{t=1}^{3}\bigcup_{\nu}V^{it}_{\nu}$. Then $K$ is a
union of at most a countable number of $(q-1)$-dimensional complex vector
subspaces in ${\mathbf{C}}^{q}$. It is easy to see that
$\mathcal{C}\supset{\mathbf{C}}^{q}\setminus K$. Therefore $\mathcal{C}$ is
dense in ${\mathbf{C}}^{q}$. The lemma is proved. $\square$
###### Lemma 4.2.
For every $c\in\mathcal{C}$, we put
$F^{jt}_{c}:=\dfrac{(f^{t},\widetilde{a}_{j}^{t})}{(f^{t},a_{c}^{t})}\quad(1\leq
j\leq q,\ 1\leq t\leq 3).$ Then
$||T(r,F_{c}^{jt})\leq T_{f^{t}}(r)+o(T(r))$
Proof. Let $h$ be a meromorphic function on ${\mathbf{C}}^{m}$ such that
$\big{(}h(f^{t},\tilde{a}^{t}_{j}):h(f^{t},a^{t}_{c})\big{)}$ is a reduced
representation of a meromorphic mapping into ${\mathbf{P}}^{1}({\mathbf{C}})$.
It is easy to see that
$\nu^{0}_{h}\leq\sum_{i=1}^{q}\nu_{a_{j0}}.$
This implies that
$||\
N_{h}(r)\leq\sum_{j=1}^{q}N_{a^{t}_{j0}}(r)\leq\sum_{j=1}^{q}T_{a^{t}_{j}}(r)=o(T(r)).$
By the definition of the characteristic function and by Jensen formula, we
have
$\displaystyle||\ T(r,F^{jt}_{c})$
$\displaystyle=\int\limits_{S(r)}\log\left(|h(f^{t},\tilde{a}^{t}_{j})|^{2}+|h(f^{t},a^{t}_{c})|^{2}\right)^{\frac{1}{2}}\eta$
$\displaystyle\leq\int\limits_{S(r)}\log||f^{t}||\eta+\int\limits_{S(r)}\log|h|\eta+\int\limits_{S(r)}\log(||\tilde{a}^{t}_{j}||^{2}+||a^{t}_{c}||^{2})^{\frac{1}{2}}\eta+O(1)$
$\displaystyle\leq
T_{f^{t}}(r)+N_{h}(r)+\int\limits_{S(r)}\log^{+}||\tilde{a}^{t}_{j}||\eta+\int\limits_{S(r)}\log^{+}||\tilde{a}^{t}_{c}||\eta+O(1)$
$\displaystyle\leq
T_{f^{t}}(r)+\sum_{i=1}^{n}\int\limits_{S(r)}\log^{+}|\dfrac{a^{t}_{ji}}{a_{j0}}|\eta+\sum_{v=1}^{q}\sum_{i=1}^{n}\int\limits_{S(r)}\log^{+}|\dfrac{a^{t}_{vi}}{a_{v0}}|\eta+o(T(r))$
$\displaystyle=T_{f^{t}}(r)+\sum_{i=1}^{n}m(r,\dfrac{a^{t}_{ji}}{a^{t}_{j0}})+\sum_{v=1}^{q}\sum_{i=1}^{n}m(r,\dfrac{a^{t}_{vi}}{a_{v0}})+o(T(r))$
$\displaystyle\leq
T_{f^{t}}(r)+nT_{a^{t}_{j}}(r)+n\sum_{v=1}^{q}T_{a^{t}_{v}}(r)+o(T(r))=T_{f^{t}}(r)+o(T(r)).$
The lemma is proved. $\square$
###### Definition 4.3 (see [4, p. 138]).
Let $F_{1},F_{2},F_{3}$ be nonzero meromorphic functions on
${\mathbf{C}}^{m}$. Take a set
$\alpha=(\alpha_{1},...,\alpha_{m})\in(\mathbf{Z}^{+})^{m}$ with
$|\alpha|=\sum_{i=1}^{m}\alpha_{i}=1$. We define Cartan’s auxiliary function
by
$\Phi^{\alpha}\equiv\Phi^{\alpha}(F_{1},F_{2},F_{3}):=F_{1}F_{2}F_{3}\left|\begin{array}[]{cccc}1&1&1\\\
\frac{1}{F_{1}}&\frac{1}{F_{2}}&\frac{1}{F_{3}}\\\
\mathcal{D}^{\alpha}(\frac{1}{F_{1}})&\mathcal{D}^{\alpha}(\frac{1}{F_{2}})&\mathcal{D}^{\alpha}(\frac{1}{F_{3}})\\\
\end{array}\right|$
By simple computation, we have
(4.4)
$\displaystyle\Phi^{\alpha}(F_{1},F_{2},F_{3})=F_{1}\biggl{(}\dfrac{\mathcal{D}^{\alpha}F_{2}}{F_{2}}-\dfrac{\mathcal{D}^{\alpha}F_{3}}{F_{3}}\biggl{)}+F_{2}\biggl{(}\dfrac{\mathcal{D}^{\alpha}F_{3}}{F_{3}}-\dfrac{\mathcal{D}^{\alpha}F_{1}}{F_{1}}\biggl{)}+F_{3}\biggl{(}\dfrac{\mathcal{D}^{\alpha}F_{1}}{F_{1}}-\dfrac{\mathcal{D}^{\alpha}F_{2}}{F_{2}}\biggl{)}.$
###### Lemma 4.5 (see [4, Proposition 3.4]).
If $\Phi^{\alpha}(F,G,H)=0$ and
$\Phi^{\alpha}(\frac{1}{F},\frac{1}{G},\frac{1}{H})=0$ for all $\alpha$ with
$|\alpha|=1$, then one of the following assertions holds :
(i) $F=G,G=H$ or $H=F$
(ii) $\frac{F}{G},\frac{G}{H}$ and $\frac{H}{F}$ are all constant.
###### Lemma 4.6 (see [10, Lemma 4.7]).
Suppose that there exists
$\Phi^{\alpha}=\Phi^{\alpha}(F_{c}^{j_{0}1},F_{c}^{j_{0}2},F_{c}^{j_{3}})\not\equiv
0$ for some $c\in\mathcal{C},$ $|\alpha|=1$. Then, for each $1\leq t\leq 3$,
the following holds:
$\displaystyle 2\sum_{j=1}^{3}N^{[1]}_{(f^{1},\widetilde{a}^{1}_{j}),\leq
k_{j}}(r)$
$\displaystyle+\sum_{t=1}^{3}N^{[n]}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}(r)-(2n+3)N^{[1]}_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(r)-2\sum_{t=1}^{3}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(r)$
$\displaystyle\leq N_{\Phi^{\alpha}}(r)+o(T(r))\leq
T(r)+\sum_{t=1}^{3}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(r)+o(T(r)).$
Proof. (a)Firstly, we will prove the first inequality. We set
$\displaystyle\mathcal{A}=\\{z\in{\mathbf{C}}^{m}:\nu^{0}_{(f^{t},\widetilde{a}^{t}_{j_{0}})}>0\\}$
$\displaystyle\mathcal{V}=\\{z\in{\mathbf{C}}^{m}:\nu^{0}_{(f^{t},\widetilde{a}^{t}_{i}),\leq
k_{i}}.\nu^{0}_{(f^{t},\widetilde{a}^{t}_{j}),\leq k_{j}}>0\\}\quad(1\leq
i<j\leq q)$
Then $\mathcal{V}$ is an analytic set of codimension at least 2. We also set
$\mathcal{D}=\bigcup_{i=1}^{q}\\{z\in{\mathbf{C}}^{m}:\nu^{0}_{(f^{t},\widetilde{a}^{t}_{i}),\leq
k_{i}}>0\\}\text{ and
}\mathcal{S}=(\bigcup_{i=1}^{q}\bigcup_{t=1}^{3}S_{c}^{it})\cup\mathcal{A}\cup\beta$
Let $z_{0}$ be a regular point of the analytic set $\mathcal{D}$ such that
$z_{0}\not\in\mathcal{V}\cup\mathcal{S}$. There are three cases:
Case 1. $z_{0}\not\in\mathcal{A}$. Let $\nu$ be the irreducible component of
$\mathcal{D}$ which contains $z_{0}$. Then, there exist a neighborhood $U$ of
$z_{0}$ and a holomorphic function $h$ on $U$ such that $dh$ has nonzero point
and $U\cap\mathrm{Zero}h=\nu$. Moreover, we may assume that
$U\cap(\mathcal{V}\cup\mathcal{S}\cup\mathcal{A})=\emptyset$. Since
$\dfrac{(f^{t},\widetilde{a}^{t}_{i})}{(f^{s},\widetilde{a}^{s}_{i})}=\dfrac{(f^{t},\widetilde{a}^{t}_{j})}{(f^{s},\widetilde{a}^{s}_{j})}$
for all $z\in\nu$, $1\leq i\neq j\leq q$, $1\leq t\neq s\leq 3$, there exist
holomorphic functions $\varphi_{v}$ defined on $U$ such that
$F^{cv}_{j_{0}}=h\varphi_{v}$ on U $(1\leq v\leq 3)$
Then, we rewrite the function $\Phi^{\alpha}$ on $U$ as follows
$\displaystyle\Phi^{\alpha}(F_{c}^{j_{0}1},F_{c}^{j_{0}2},F_{c}^{j_{0}3}):$
$\displaystyle=F_{c}^{j_{0}1}F_{c}^{j_{0}2}F_{c}^{j_{0}3}\left|\begin{array}[]{cccc}1&1&1\\\
F_{j_{0}}^{c1}&F_{j_{0}}^{c2}&F_{j_{0}}^{3}\\\
\mathcal{D}^{\alpha}(F_{j_{0}}^{c1})&\mathcal{D}^{\alpha}(F_{j_{0}}^{c2})&\mathcal{D}^{\alpha}(F_{j_{0}}^{c3})\\\
\end{array}\right|$
$\displaystyle=F_{c}^{j_{0}1}F_{c}^{j_{0}2}F_{c}^{j_{0}3}\left|\begin{array}[]{cccc}F^{c2}_{j_{0}}-F^{c1}_{j_{0}}&F^{c3}_{j_{0}}-F^{c1}_{j_{0}}\\\
\mathcal{D}^{\alpha}(F^{c2}_{j_{0}}-F^{c1}_{j_{0}})&\mathcal{D}^{\alpha}(F^{c3}_{j_{0}}-F^{c1}_{j_{0}})\\\
\end{array}\right|$
$\displaystyle=F_{c}^{j_{0}1}F_{c}^{j_{0}2}F_{c}^{j_{0}3}h^{2}\left|\begin{array}[]{cccc}\varphi_{2}-\varphi_{1}&\varphi_{3}-\varphi_{1}\\\
\mathcal{D}^{\alpha}(\varphi_{2}-\varphi_{1})&\mathcal{D}^{\alpha}(\varphi_{3}-\varphi_{1})\\\
\end{array}\right|$ $\displaystyle\nu^{0}_{\Phi^{\alpha}}(z_{0})\geq 2$
$\displaystyle\geq
2\sum_{j=1}^{q}\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j}),\leq
k_{j}}(z_{0})\\}+\sum_{t=1}^{3}\min\\{n,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}$ $\displaystyle\
-(2n+3)\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}-2\sum_{t=1}^{3}\min\\{1,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(z_{0})\\}$
Case 2.
$z_{0}\in\\{z\in{\mathbf{C}}^{m}:\nu^{0}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}>0\\}$. Without loss of generality, we may assume that
$0<\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\leq\nu_{(f^{2},\widetilde{a}^{2}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\leq\nu_{(f^{3},\widetilde{a}^{3}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})$.
$\displaystyle\Phi^{\alpha}$
$\displaystyle=F_{c}^{j_{0}1}\left|\begin{array}[]{cccc}F^{c2}_{j_{0}}(F^{c2}_{j_{0}}-F^{c1}_{j_{0}})&F^{c3}_{j_{0}}(F^{c3}_{j_{0}}-F^{c1}_{j_{0}})\\\
F^{c2}_{j_{0}}\mathcal{D}^{\alpha}(F^{c2}_{j_{0}}-F^{c1}_{j_{0}})&F^{c3}_{j_{0}}\mathcal{D}^{\alpha}(F^{c3}_{j_{0}}-F^{c1}_{j_{0}})\\\
\end{array}\right|$
$\displaystyle=F_{c}^{j_{0}1}\biggl{(}F^{c2}_{j_{0}}(F^{c2}_{j_{0}}-F^{c1}_{j_{0}}).F^{c3}_{j_{0}}\mathcal{D}^{\alpha}(F^{c3}_{j_{0}}-F^{c1}_{j_{0}})-F^{c3}_{j_{0}}(F^{c3}_{j_{0}}-F^{c1}_{j_{0}}).F^{c2}_{j_{0}}\mathcal{D}^{\alpha}(F^{c2}_{j_{0}}-F^{c1}_{j_{0}})\biggl{)}$
Because of the assumption, we see that
$F^{c2}_{j_{0}}(F^{c2}_{j_{0}}-F^{c1}_{j_{0}})$ and
$F^{c3}_{j_{0}}(F^{c3}_{j_{0}}-F^{c1}_{j_{0}})$ are holomorphic on a
neighborhood of $z_{0}$. Moreover, we have
$\displaystyle\nu^{\infty}_{F^{c2}_{j_{0}}\mathcal{D}^{\alpha}(F^{c2}_{j_{0}}-F^{c1}_{j_{0}})}(z_{0})\leq|\alpha|=1$
$\displaystyle\nu^{\infty}_{F^{c3}_{j_{0}}\mathcal{D}^{\alpha}(F^{c3}_{j_{0}}-F^{c1}_{j_{0}})}(z_{0})\leq|\alpha|=1$
Therefore
$\displaystyle\nu^{0}_{\Phi^{\alpha}}(z_{0})$
$\displaystyle\geq\nu_{F_{c}^{j_{0}1}}(z_{0})-1=\min_{1\leq t\leq
3}\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})-\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}$
$\displaystyle\geq\sum_{t=1}^{3}\min\\{n,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}-2n\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}-\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}$
$\displaystyle=\sum_{t=1}^{3}\min\\{n,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}-(2n+1)\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}$ $\displaystyle\geq
2\sum_{j=1}^{q}\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j}),\leq
k_{j}}(z_{0})\\}+\sum_{t=1}^{3}\min\\{n,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}$ $\displaystyle\
-(2n+3)\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}-2\sum_{t=1}^{3}\min\\{1,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(z_{0})\\}$
Case 3.
$z_{0}\in\\{z\in{\mathbf{C}}^{m}:\nu^{0}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}>0\\}$.
$\displaystyle\nu^{0}_{\Phi^{\alpha}}(z_{0})$ $\displaystyle\geq 0\geq
2\sum_{j=1}^{q}\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j}),\leq
k_{j}}(z_{0})\\}+\sum_{v=1}^{3}\min\\{n,\nu_{(f^{v},\widetilde{a}^{v}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}$ $\displaystyle\
-(2n+3)\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}-2\sum_{t=1}^{3}\min\\{1,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(z_{0})\\}$
Then, from the above three cases it follows that
$\displaystyle\nu^{0}_{\Phi^{\alpha}}(z)$ $\displaystyle\geq
2\sum_{j=1}^{q}\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j}),\leq
k_{j}}(z_{0})\\}+\sum_{t=1}^{3}\min\\{n,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}$
$\displaystyle-(2n+3)\min\\{1,\nu_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})\\}-2\sum_{t=1}^{3}\min\\{1,\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(z_{0})\\}$
for every z outside the analytic set of codimension 2. Integrating both sides
of this inequality, we get
$\displaystyle||N_{\Phi^{\alpha}}(r)$ $\displaystyle\geq
2\sum_{j=1}^{q}N^{[1]}_{(f^{1},\widetilde{a}^{1}_{j}),\leq
k_{j}}(z_{0})+\sum_{t=1}^{3}N^{[n]}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})$
$\displaystyle-(2n+3)N^{[1]}_{(f^{1},\widetilde{a}^{1}_{j_{0}}),\leq
k_{j_{0}}}(z_{0})-2\sum_{t=1}^{3}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(z_{0})+o(T(r))$
for each $1\leq t\leq 3$. Hence, the first inequality of lemma is proved.
(b) We now prove the second inequality. By the definition of the Nevanlinna
characteristic function, we have
$N_{\Phi^{\alpha}}(r)\leq
T(r,\Phi^{\alpha})+O(1)=N_{\frac{1}{\Phi^{\alpha}}}(r)+m(r,\Phi^{\alpha})+O(1)$
We see that a pole of $\Phi^{\alpha}$ must be zero or pole of $F^{j_{0}t}_{c}\
(1\leq t\leq 3)$. Let $z_{0}\not\in\mathcal{V}\cup\mathcal{S}$. There are
three cases:
Case 1. If
$z_{0}\in\\{z\in{\mathbf{C}}^{m}:\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),\leq
k_{j_{0}}}^{0}(z_{0})>0\\}$, then by (4.4) we easily see that
$\nu^{\infty}_{\Phi^{\alpha}}(z_{0})\leq\max_{1\leq t\leq
3}\nu^{\infty}_{(f^{t},\widetilde{a}^{t}_{j_{0}})}+1.$
Case 2. If
$z_{0}\in\\{z\in{\mathbf{C}}^{m}:\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}^{0}(z_{0})>0\\}$,
we rewrite the function $\Phi^{\alpha}$ as follows
$\displaystyle\Phi^{\alpha}(F_{c}^{j_{0}1},F_{c}^{j_{0}2},F_{c}^{j_{0}3}):$
$\displaystyle=F_{c}^{j_{0}1}F_{c}^{j_{0}2}F_{c}^{j_{0}3}\left|\begin{array}[]{cccc}1&1&1\\\
F_{j_{0}}^{c1}&F_{j_{0}}^{c2}&F_{j_{0}}^{c3})\\\
\mathcal{D}^{\alpha}(F_{j_{0}}^{c1})&\mathcal{D}^{\alpha}(F_{j_{0}}^{c2})&\mathcal{D}^{\alpha}(F_{j_{0}}^{c3})\\\
\end{array}\right|$
$\displaystyle=F^{j_{0}1}_{c}(F^{j_{0}2}_{c}-F^{j_{0}3}_{c})D^{\alpha}(F^{c1}_{j_{0}})+F^{j_{0}2}_{c}(F^{j_{0}3}_{c}-F^{j_{0}1}_{c})D^{\alpha}(F^{c2}_{j_{0}})$
$\displaystyle+F^{j_{0}3}_{c}(F^{j_{0}1}_{c}-F^{j_{0}2}_{c})D^{\alpha}(F^{c3}_{j_{0}})$
It is easy to see that $\nu_{\Phi^{\alpha}}^{\infty}(z_{0})\leq\max_{1\leq
t\leq
3}\\{\nu^{\infty}_{F^{j_{0}t}_{c}D^{\alpha}(F^{ct}_{j_{0}})}(z_{0})\\}\leq|\alpha|=1$
Case 3. If $z_{0}\in\\{z\in{\mathbf{C}}^{m}:\nu_{(f^{t},a^{t}_{c})}>0\\}$ then
$\nu_{\Phi^{\alpha}}^{\infty}(z_{0})\leq\sum_{t=1}^{3}\nu_{F^{j_{0}t}_{c}}^{\infty}(z_{0})$.
Thus, every $z\not\in\mathcal{V}\cup\mathcal{S}$, we have
$\nu_{\Phi^{\alpha}}^{\infty}(z_{0})\leq\sum_{t=1}^{3}\nu_{F^{j_{0}t}_{c}}^{\infty}(z_{0})+\sum_{t=1}^{3}\min\\{\nu_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}},1\\}$.
Therefore, we have
$\displaystyle||N_{\frac{1}{\Phi^{\alpha}}}(r)$
$\displaystyle\leq\sum_{t=1}^{3}N_{\frac{1}{F_{c}^{j_{0}t}}}(r)+\sum_{t=1}^{3}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(r)+o(T(r))$
By the logarithmic derivative lemma (Lemma 2.1), we have
$\displaystyle||m(r,\Phi^{\alpha})$
$\displaystyle\leq\sum_{t=1}^{3}m(r,F_{c}^{j_{0}t})+O\biggl{(}\sum
m\biggl{(}r,\dfrac{{\mathcal{D}}^{\alpha^{i}}(F_{j_{0}}^{ct})}{F_{j_{0}}^{ct}}\biggl{)}\biggl{)}+O(1)$
$\displaystyle\leq\sum_{t=1}^{3}m(r,F_{c}^{j_{0}t})+\sum_{t=1}^{3}o(T(r,F_{j_{0}}^{ct}))+O(1)$
$\displaystyle=\sum_{t=1}^{3}m(r,F_{c}^{j_{0}t})+o(T(r)).$
This implies that
$\displaystyle\biggl{|}\biggl{|}N_{\Phi^{\alpha}}(r)$
$\displaystyle\leq\sum_{t=1}^{3}T(r,F_{c}^{j_{0}t})+o(T(r))\leq\sum_{t=1}^{3}T_{f^{t}}(r)+\sum_{t=1}^{3}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(r)+o(T(r))$
$\displaystyle\leq
T(r)+\sum_{t=1}^{3}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{j_{0}}),>k_{j_{0}}}(r)+o(T(r))\quad\hfill\square$
3.2. Proof of Theorem 1.3. Assume that
$\sum_{i=1}^{q}\dfrac{1}{k_{i}+1}<\dfrac{q-3n+2}{2q-5n+10}\biggl{(}\dfrac{2(2q+n-3)}{3n(n+1)}-3\biggl{)}.$
Denote by $\mathcal{Q}$ be the set of all indices $j\in\\{1,..,q\\}$
satisfying the following: there exist $c\in\mathcal{C}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{m})\in\mathbf{Z}_{+}^{m}$ with $|\alpha|=1$
such that $\Phi^{\alpha}(F_{c}^{j1},F_{c}^{j2},F_{c}^{j3})\not\equiv 0$. We
put $p=\sharp\mathcal{Q}$.
Suppose that $p\geq q-3n+2$. Without loss of generality, we may assume that
$1,...,q-3n+2\in\mathcal{Q}$. Then by Lemma 4.6, for $j\in\mathcal{Q}$, $1\leq
t\leq 3$, we have
$\displaystyle T(r)$ $\displaystyle\geq
2\sum_{j=1}^{q}N^{[1]}_{(f^{v},\widetilde{a}^{v}_{j}),\leq
k_{j}}(r)+\sum_{t=1}^{3}N^{[n]}_{(f^{t},\widetilde{a}^{t}_{i}),\leq k_{i}}(r)$
$\displaystyle-(2n+3)N^{[1]}_{(f^{v},\widetilde{a}^{v}_{i}),\leq
k_{i}}(r)-3\sum_{t=1}^{3}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{i}),>k_{i}}(r)+o(T(r)).$
By summing up both side of above inequality over $1\leq j\leq q-3n+2$ and
$1\leq t\leq 3$, we have
$\displaystyle\parallel 3(q-3n+2)T(r)$ $\displaystyle\geq
2(q-3n+2)\sum_{t=1}^{3}\sum_{i=1}^{q}N_{(f^{t},\widetilde{a}_{i}^{t}),\leq
k_{i}}^{[1]}(r)+3\sum_{t=1}^{3}\sum_{i=1}^{q-3n+2}N^{[n]}_{(f^{t},\widetilde{a}^{t}_{i}),\leq
k_{i}}(r)$
$\displaystyle-(2n+3)\sum_{t=1}^{3}\sum_{i=1}^{q-3n+2}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{i}),\leq
k_{i}}(r)-9\sum_{t=1}^{3}\sum_{i=1}^{q-3n+2}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{i}),>k_{i}}(r)$
$\displaystyle=\sum_{t=1}^{3}\biggl{(}(2q-8n+1)\sum_{i=1}^{q-3n+2}N_{(f^{t},\widetilde{a}_{i}^{t}),\leq
k_{i}}^{[1]}(r)+3\sum_{i=1}^{q-3n+2}N^{[n]}_{(f^{t},\widetilde{a}^{t}_{i}),\leq
k_{i}}(r)$
$\displaystyle+(2q-6n+4)\sum_{i=q-3n+3}^{q}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{i}),\leq
k_{i}}(r)-9\sum_{i=1}^{q-3n+2}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{i}),>k_{i}}(r)\biggl{)}$
$\displaystyle=\sum_{t=1}^{3}\biggl{(}(2q-8n+1)\sum_{i=1}^{q-3n+2}N_{(f^{t},\widetilde{a}_{i}^{t})}^{[1]}(r)+3\sum_{i=1}^{q-3n+2}(N^{[n]}_{(f^{t},a^{t}_{i})}(r)-N^{[n]}_{(f^{t},\widetilde{a}^{t}_{i}),>k_{i}}(r))$
$\displaystyle+(2q-6n+4)\sum_{i=q-3n+3}^{q}(N^{[1]}_{(f^{t},\widetilde{a}^{t}_{i})}(r)-N^{[1]}_{(f^{t},\widetilde{a}^{t}_{i}),>k_{i}}(r))$
$\displaystyle\
-(2q-8n+10)\sum_{i=1}^{q-3n+2}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{i}),>k_{i}}(r)\biggl{)}$
$\displaystyle\geq\sum_{t=1}^{3}\biggl{(}\dfrac{2q-5n+1}{n}\sum_{i=1}^{q-3n+2}N_{(f^{t},\widetilde{a}_{i}^{t})}^{[n]}(r)+\dfrac{2q-6n+4}{n}\sum_{i=q-3n+3}^{q}N^{[n]}_{(f^{t},\widetilde{a}^{t}_{i})}(r)$
$\displaystyle-\sum_{i=1}^{q-3n+2}\dfrac{2q-5n+10}{k_{i}+1}N_{(f^{t},\widetilde{a}^{t}_{i})}(r)-\sum_{i=q-3n+3}^{q}\dfrac{2q-6n+4}{k_{i}+1}N_{(f^{t},\widetilde{a}^{t}_{i})}(r)\biggl{)}$
$\displaystyle\geq\sum_{t=1}^{3}\biggl{(}\dfrac{2q-5n+1}{n}\sum_{i=1}^{q-3n+2}N_{(f^{t},a_{i}^{t})}^{[n]}(r)+\dfrac{2q-6n+4}{n}\sum_{i=q-3n+3}^{q}N^{[n]}_{(f^{t},a^{t}_{i})}(r)$
$\displaystyle-\sum_{i=1}^{q-3n+2}\dfrac{2q-5n+10}{k_{i}+1}N_{(f^{t},a^{t}_{i})}(r)-\sum_{i=q-3n+3}^{q}\dfrac{2q-6n+4}{k_{i}+1}N_{(f^{t},a^{t}_{i})}(r)\biggl{)}+o(T(r))$
On the other hand, by theorem 2.3, we have
$\displaystyle||3(q-3n+2)T(r)$
$\displaystyle\geq\biggl{(}\dfrac{2q-5n+1}{n}.\dfrac{2(q-3n+2)}{3(n+1)}+\dfrac{2q-6n+4}{n}.\dfrac{2(3n-2)}{3(n+1)}$
$\displaystyle\quad-\sum_{i=1}^{q-3n+2}\dfrac{2q-5n+10}{k_{i}+1}-\sum_{i=q-3n+3}^{q}\dfrac{2q-6n+4}{k_{i}+1}\biggl{)}T(r)+o(T(r))$
$\displaystyle\geq\biggl{(}\dfrac{(2q-6n+4)(2q+n-3)}{3n(n+1)}-(2q-5n+10)\sum_{i=1}^{q}\dfrac{1}{k_{i}+1}\biggl{)}T(r)+o(T(r))$
Letting $r\longrightarrow+\infty$, we get
$\displaystyle
3(q-3n+2)\geq\dfrac{(2q-6n+4)(2q+n-3)}{3n(n+1)}-(2q-5n+10)\sum_{i=1}^{q}\dfrac{1}{k_{i}+1}$
$\displaystyle\text{ i.e.,
}\sum_{i=1}^{q}\dfrac{1}{k_{i}+1}\geq\dfrac{q-3n+2}{2q-5n+10}\biggl{(}\dfrac{2(2q+n-3)}{3n(n+1)}-3\biggl{)}$
This is a contradiction.
Then $\sharp\mathcal{Q}\leq q-3n+1$. Without loss of generality, we may assume
that $1,2,...,3n-1\not\in\mathcal{Q}$. This mean that
$\Phi^{\alpha}(F_{c}^{j1},F_{c}^{j2},F_{c}^{j3})\equiv 0,$
for all $c\in\mathcal{C}$, $\alpha=(\alpha_{1},\ldots,\alpha_{m})\ \text{ with
}|\alpha|=1$. By the density of $\mathcal{C}$ in ${\mathbf{C}}^{q}$, the above
equality holds for all $c\in{\mathbf{C}}^{q}$, and $|\alpha|=1$. For each
$i\in\\{1,...,3n-1\\}$, chosing
$c_{i}=(0,...,0,{\mathrel{\mathop{{1}}\limits_{{i-th}}}},0,...,0)$ we have
$\Phi^{\alpha}(F_{c_{i}}^{j1},F_{c_{i}}^{j2},F_{c_{i}}^{j3})\equiv 0\
\forall|\alpha|=1.$
Then by Lemma 4.5, there exists a constant $\lambda$ such that
$F_{c_{i}}^{j1}=\lambda F_{c_{i}}^{j2},F_{c_{i}}^{j2}=\lambda
F_{c_{i}}^{j3},\text{ or }F_{c_{i}}^{j3}=\lambda F_{c_{i}}^{j1}.$
For instance, we assume that $F_{c_{i}}^{j1}=\lambda F_{c_{i}}^{j2}$. We will
show that $\lambda=1.$
Indeed, suppose that $\lambda\neq 1$, we have
$\displaystyle 0=T(r,\dfrac{F^{j1}_{c_{i}}}{F^{j2}_{c_{i}}})$
$\displaystyle\geq
N^{[1]}_{F^{j1}_{c_{i}}-F^{j2}_{c_{i}}}(r)\geq\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,j}}}}}^{q}N^{[1]}_{(f^{1},\widetilde{a}^{1}_{v}),\leq
k_{v}}-\sum_{t=1}^{2}\sum_{v=i,j}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{v}),>k_{v}}(r)$
$\displaystyle\geq\dfrac{1}{2}\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,j}}}}}^{q}\biggl{(}N^{[1]}_{(f^{1},\widetilde{a}^{1}_{v}),\leq
k_{v}}+N^{[1]}_{(f^{2},\widetilde{a}^{2}_{v}),\leq
k_{v}}\biggl{)}-\sum_{t=1}^{2}\sum_{v=i,j}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{v}),>k_{v}}(r)$
$\displaystyle\geq\dfrac{1}{2}\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,j}}}}}^{q}\biggl{(}N^{[1]}_{(f^{1},\widetilde{a}^{1}_{v})}+N^{[1]}_{(f^{2},\widetilde{a}^{2}_{v})}-N^{[1]}_{(f^{1},\widetilde{a}^{1}_{v}),>k_{v}}-N^{[1]}_{(f^{2},\widetilde{a}^{2}_{v}),>k_{v}}\biggl{)}$
$\displaystyle\
-\sum_{t=1}^{2}\sum_{v=i,j}N^{[1]}_{(f^{t},\widetilde{a}^{t}_{v}),>k_{v}}(r)$
$\displaystyle\geq\dfrac{1}{2}\sum_{v\neq
i}\biggl{(}\dfrac{1}{n}(N^{[n]}_{(f^{1},a^{1}_{v})}+N^{[n]}_{(f^{2},a^{2}_{v})})-\dfrac{1}{k_{v}+1}(N_{(f^{1},a^{1}_{v})}+N_{(f^{2},a^{2}_{v})})\biggl{)}$
$\displaystyle\
-\sum_{t=1}^{2}\sum_{v=i,j}\dfrac{1}{k_{v}+1}N_{(f^{t},a^{t}_{v})}(r)+o(T(r))$
$\displaystyle\geq\biggl{(}\dfrac{q-2}{3n(n+1)}-\dfrac{1}{2}\sum_{{\mathrel{\mathop{{v=1}}\limits_{{v\neq
i,j}}}}}^{q}\dfrac{1}{k_{v}+1}-\sum_{v=i,j}\dfrac{1}{k_{v}+1}\biggl{)}T(r)+o(T(r))$
$\displaystyle\geq\biggl{(}\dfrac{q-2}{3n(n+1)}-\sum_{v=1}^{q}\dfrac{1}{k_{v}+1}\biggl{)}T(r)+o(T(r))$
Thus, $\sum_{v=1}^{q}\dfrac{1}{k_{v}+1}\geq\dfrac{q-2}{3n(n+1)}$. This is a
contradiction. Thus $\lambda=1\ (1\leq i<j\leq q).$
Define
$I_{1}=\\{j\in\\{2,\ldots,3n-1\\}:F_{1}^{j1}=F_{1}^{j2}\\},$
$I_{2}=\\{j\in\\{2,\ldots,3n-1\\}:F_{1}^{j2}=F_{1}^{j3}\\},$
$I_{3}=\\{j\in\\{2,\ldots,3n-1\\}:F_{1}^{j3}=F_{1}^{j1}\\}.$
Since $\sharp(I_{1}\cup I_{2}\cup I_{3})=\sharp\\{2,\ldots,3n-1\\}=3n-2$,
there exists $1\leq v\leq 3$ such that $\sharp\ I_{v}\geq n$. Without loss of
generality, we may assume that $\sharp\ I_{1}\geq n$. This implies that
$\dfrac{(f^{1},\widetilde{a}^{1}_{1})}{(f^{2},\widetilde{a}^{2}_{1})}=\dfrac{(f^{1},\widetilde{a}^{1}_{j})}{(f^{2},\widetilde{a}^{2}_{j})}\
\forall j\in I_{1}.$
The theorem is proved. $\square$
## References
* [1] T.B Cao and H.X. Yi, Uniqueness theorems for meromorphic mappings sharing hyperplanes in general position, arXiv:1011.5828v4 [math.CV] 10 Dec. 2010.
* [2] G. Dethloff, S.D Quang and T.V Tan, A uniqueness theorem for meromorphic mappings with two families of hyperplanes, Proc. Amer. Math. Soc. 140 No. 1 (2012), 189-197
* [3] H. Fujimoto, Non-integrated defect relation for meromorphic maps of complete Kähler manifolds into ${\mathbf{P}}^{N_{1}}({\mathbf{C}})\times\ldots\times{\mathbf{P}}^{N_{k}}({\mathbf{C}}),$ Japanese J. Math. 11 (1985), 233-264.
* [4] H. Fujimoto, Uniqueness problem with truncated multiplicities in value distribution theory, Nagoya Math. J. 152 (1998), 131-152.
* [5] R. Nevanlinna, Einige Eideutigkeitssätze in der Theorie der meromorphen Funktionen, Acta. Math., 48 (1926), 367-391.
* [6] J. Noguchi and T. Ochiai, Introduction to Geometric Function Theory in Several Complex Variables, Trans. Math. Monogr. 80, Amer. Math. Soc., Providence, Rhode Island, 1990.
* [7] S. D. Quang, Second main theorem for meromorphic mappings and moving hyperplanes with truncated counting function, Preprint.
* [8] B. Shiffman, Introduction to the Carlson - Griffiths equidistribution theory, Lecture Notes in Math. 981 (1983), 44-89.
* [9] L. Smiley, Geometric conditions for unicity of holomorphic curves, Contemp. Math. 25 (1983), 149-154.
* [10] D. D. Thai and S. D. Quang, Uniqueness problem with truncated multiplicities of meromorphic mappings in several complex variables for moving targets, Internat. J. Math., 16 (2005), 903-939.
* [11] Z. Wang and Z. Tu, Uniqueness theorems for meromorphic mappings in several complex variables into ${\mathbf{P}}^{N}({\mathbf{C}})$ with two families of moving targets, Chin. Ann. Math, 33B No. 1 (2013), 719-732.
|
arxiv-papers
| 2014-04-01T09:32:03 |
2024-09-04T02:50:00.499665
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Giang Ha Huong",
"submitter": "Giang Ha Huong",
"url": "https://arxiv.org/abs/1404.0177"
}
|
1404.0188
|
_In silico_ prediction of mutant HIV-1 proteases cleaving a target sequence
Jan H. Jensen,1 Martin Willemoës,2 Jakob R. Winther,2 Luca De Vico1,∗
1 Department of Chemistry, University of Copenhagen, Universitetsparken 5,
DK-2100 Copenhagen, Denmark
2 Department of Biology, University of Copenhagen, Ole Maaløes Vej 5, DK-2200
Copenhagen, Denmark
$\ast$ Corresponding Author, Email: [email protected]
## Abstract
HIV-1 protease represents an appealing system for directed enzyme re-design,
since it has various different endogenous targets, a relatively simple
structure and it is well studied. Recently Chaudhury and Gray (Structure
(2009) 17: 1636 – 1648) published a computational algorithm to discern the
specificity determining residues of HIV-1 protease. In this paper we present
two computational tools aimed at re-designing HIV-1 protease, derived from the
algorithm of Chaudhuri and Gray. First, we present an energy-only based
methodology to discriminate cleavable and non cleavable peptides for HIV-1
proteases, both wild type and mutant. Secondly, we show an algorithm we
developed to predict mutant HIV-1 proteases capable of cleaving a new target
substrate peptide, different from the natural targets of HIV-1 protease. The
obtained _in silico_ mutant enzymes were analyzed in terms of cleavability and
specificity towards the target peptide using the energy-only methodology. We
found two mutant proteases as best candidates for specificity and cleavability
towards the target sequence.
## List of Abbreviations
PR
HIV-1 protease.
WT-PR
Wild type HIV-1 protease.
mutant PR
Mutant HIV-1 protease.
Pr3 set
Set of mutant proteases derived from Pr3, a mutant protease developed by
Alvizo _et al_. These were heterodimer proteases.
DR set
Set of mutant proteases derived as a subset of HIV-1 proteases that have been
found to be drug resistant. These were homodimer proteases.
## Introduction
Proteases represent a class of enzymes ubiquitous in all living organisms,
with multiple applications in industry and biotechnology research [1, 2, 3].
There is thus interest in designing new proteases capable of cleaving specific
peptide sequences [4]. HIV-1 protease (PR) represents an attractive starting
structure for directed enzyme re-design, since it is known to cleave a variety
of sequences. PR is the enzyme responsible for processing the gag – pol fusion
polyproteins of the HIV virus [5]. PR is an aspartic protease [6, 7, 8] and is
a homodimer where each chain is composed of 99 residues. Wild type PR (WT-PR)
is very specific for the endogenous cleavage sequences of the polyprotein
(endogenous substrate peptides, Table Supporting Material for), even if the
source of this specificity is still not completely clear. A series of other
non-endogenous peptides have also been found to be cleaved by PR. The latest
hypothesis on the origin of this specificity, called dynamic substrate
envelope [9, 10], states that peptides fitting into the protease cavity
through a certain number of hydrogen bonds will be bound and possibly cleaved
nearly regardless of their amino acid composition. In fact, there is no clear
trend in amino acid sequence (e.g. a negatively charged amino acid in position
P1 or a hydrophobic one in position P2’). This suggests that with few
mutations PR could be made to cleave other target peptide sequences in a
specific manner.
Many computational studies on PR, both wild type (WT) and drug resistant
mutant enzymes, are aimed at elucidating the affinity of the enzymes towards
endogenous substrates and inhibitors to be used as drug candidates [11, 12,
13, 14]. Recently Chaudhury and Gray [15] published a computational algorithm
specifically tailored for PR and aimed at the identification of the
specificity determining residues. The algorithm is based on PyRosetta [16], a
python script-based interface to Rosetta [17]. Thanks to the algorithm the
authors were able to predict accurate protease – substrate complex structures
(within 1.1 Å rms of the corresponding crystal structure) and introduced an
energetic discrimination of cleavable peptides. More recently Alvizo _et al._
[18] employed computational methods to re-engineer a mutant PR (Pr3) more
specific for one of the endogenous peptide sequences over two others.
The first aim of this study is to develop an energy-only based methodology to
discern cleavable and non cleavable peptides for PRs, WT and mutant. This
methodology is based on the qualitative evaluation of PR:peptide complexes
binding energies and is derived from the algorithm developed by Chaudhury and
Gray. The second aim is to search and define an algorithm to predict mutant
PRs capable of cleaving a specific target peptide sequence different from any
endogenous substrate. We use our cleavability discerning methodology on the
suggested mutant proteases, in order to define the best guess in terms of
specificity towards the peptide sequence. In other words, the sought after
mutant structure has to show better and worse binding towards the target and
endogenous peptides, respectively, than WT-PR. To the best of our knowledge
ours is the first study aimed at predicting a mutant PR capable of cleaving
specifically a non endogenous peptide sequence.
The paper is organized as follows: first, we present our computed binding
energies for known cleavable and non-cleavable peptides bound to WT-PR,
selected peptides bound to a set of single, double and triple mutants (Pr3
set) derived from Pr3 as developed by Alvizo _et al._ , and a set of known
mutant PRs and peptide derived from drug resistance (DR set) studies [19, 20,
21]. Secondly, we present two different versions of our algorithm to determine
mutant PRs that will cleave the sequence HFLSF*MAIP, where the * symbol
indicates the desired cleaving site. A discussion about the best strategy to
suggest mutant enzymes follows. The conclusions summarize the main findings of
the paper, followed by a detailed description of the employed computational
methods.
## Results and Discussion
### Development of a Cleavability Test
In general, the activity of an enzyme towards two similar substrates is
regulated by (i) the strength of the enzyme-substrate binding and (ii) the
efficiency of the enzymatic reaction. The two processes are regulated by two
constants, usually indicated as $k_{m}$ and $k_{cat}$, respectively. The
overall enzymatic efficiency is given by the ratio of these two constants. The
dynamic substrate envelope hypothesis [10] suggests that if a peptide is bound
to PR it will be cleaved. Thus, we decided to evaluate the binding energy of
different peptides to PR, which can be correlated to $k_{m}$. We then compared
the computed binding energies to PR of known cleavable and non cleavable
peptides, to be correlated to corresponding ranges of binding energies. By so
doing we disregarded $k_{cat}$, that is we did not consider possible effects
from the enzymatic reaction.
The cleavability test was developed by considering binding energies of WT-PR
with its endogenous and known cleavable substrates and known non-cleavable
peptides. Afterwards we investigated the reliability of the test with mutant
PRs (the Pr3 set) when binding PR endogenous substrates. Finally, we assessed
the test on mutant PRs (the DR set) when binding mutant substrates.
The complete methodology for evaluating binding energies is described in the
computational methods section. In brief, it is composed by a structure
optimization algorithm, followed by an energetic re-evaluation of the obtained
structures. In the following paragraphs we evaluate our methodology in terms
of binding energies versus cleavability for: (1) WT-PR and its endogenous
substrates and known cleavable and non-cleavable peptides, (2) the Pr3 set of
mutant PRs and endogenous substrates and (3) the DR set with wild type and
mutated endogenous peptides. Binding energies were computed also for WT-PR and
all mutant PRs in complex with octa-alanine (poly-Ala) and octa-arginine
(poly-Arg) peptides to test for aspecific binding.
(1) Table 1 reports the computed binding energies of the set of known
cleavable endogenous peptides of WT-PR. The sequence of the tested endogenous
cleavable peptides is reported in Table Supporting Material for. Alongside the
endogenous peptides, a set of 59 known cleavable peptides was also tested. The
sequence of the 59 tested non-endogenous cleavable peptides was obtained as
previously described [15, 22, 23, 24, 25, 26]. Table Supporting Material for
reports the computed binding energies to WT-PR and Table Supporting Material
for the sequences of these non-endogenous peptides. Table Supporting Material
for reports the computed binding energies of a set of peptides supposedly non-
cleavable by WT-PR. The sequence of the 43 tested non-cleavable peptides was
obtained as previously described [15, 26, 27] and is reported in Table
Supporting Material for.
We performed a Mann-Whitney’s U test [28] to compare the computed binding
energies, and found a significant difference between the cleavable and non-
cleavable sets (p $\approx 10^{-7}$), as reported in Table 2. Thus, we deemed
the binding energy criterion sufficient to achieve discrimination. We further
analyzed the computed binding energies through an ROC plot [29] relative to
different cutoff values, so as to differentiate between cleavable and non-
clevable peptides. The plot is reported in Figure 1, and the relative data in
Table Supporting Material for. The computed area under ROC [30] is 0.79 and
0.80 for FMO and RosettaDock energies, respectively, being the values of 0.50
and 1.00 typical correspondingly of a useless and a perfect test. Through the
ROC plot, we found the best cutoff values discerning cleavable and non-
cleavable peptides as those closest to (0, 1), which represents the
theoretical perfect test. We found that cutoff values of -25 kcal/mol and -3
kT are best at discerning FMO and RosettaDock computed binding energies,
respectively. Both FMO and RosettaDock perform well in computing binding
energies capable of discerning cleavable and non-cleavable peptides. However,
Figure Supporting Material for shows that there is no apparent correlation
between FMO and RosettaDock computed binding energies. Thus, we repeated the
Mann-Whitney’s U test and ROC analysis excluding the set of non-endogenous
known cleavable peptides binding energies. The rationale behind this analysis
is that we expect WT-PR to bind the endogenous peptides with higher affinity,
as opposed to the broader range of the complete cleavable set, characterized
only by cleavability and not specificity. Consequently, we assume that the
endogenous peptides set have better binding energies, than the complete set of
cleavable peptides. The Mann-Whitney’s U test (Table 2) shows that the
RosettaDock based binding energies are in this case two orders of magnitude
worse than FMO at discerning cleavable and non-cleavable peptides. The
relative ROC plot (Figure Supporting Material for) shows as well that the FMO
data performs better than RosettaDock, in terms of more strict best cutoff
value and larger area under the ROC. Thus, we concluded that FMO computed
binding energies are better than RosettaDock ones since are capable of
discerning expected effects, such as the usage of a better performing subset
of peptides. In the rest of this paper we will discuss only binding energies
computed through FMO energy re-evaluation.
From Table 1 it is expected that WT-PR exhibits qualitatively different
binding to the poly-protein substrates, given their computed binding energies
ranging from -41 for the binding of p6pol-PR to -72 kcal/mol for p2-NC, with
an average value of -60 kcal/mol. However, available experimental $K_{m}$
values[22] do not show any trend similar to the computed data. Still, one has
to remember that these computed binding energies should be considered only
qualitatively and only compared to others obtained in the same manner. See the
Computational Methods section for further details. Furthermore, the span of
both computed energies for which experimental data are available (20 kcal/mol)
and the $K_{m}$ values (2 orders of magnitude) is too small to allow a clear
trend. The computed binding energies for the set of cleavable non-endogenous
peptides span a wide range of values, from -2 to -86 kcal/mol, with average
-40 kcal/mol. These peptides not being the natural target of WT-PR may account
for this large span. The average computed binding energy for all cleavable
peptides is -43 kcal/mol. The computed binding energies for the non-cleavable
set of peptides (Table Supporting Material for) span an even wider range of
values than those of the cleavable ones. Some PR – peptides complexes show
positive energies. The majority (56%) of the computed binding energies are in
the range -35 – 0 kcal/mol. However, a few peptides show a binding energy to
WT-PR similar to those of the cleavable peptides.
(2) Recently Alvizo _et al._ [18] suggested through computational means a
triple mutant (Pr3) with increased binding capability towards the endogenous
RTp51-RTp66 cleavage sequence peptide compared to that towards other two
cleavage sequences CA-p2 and p2-NC. The efficiency of Pr3 in cleaving
preferentially RTp51-RTp66 was later experimentally verified. Pr3 was made by
tethering a mutated chain of protease (A28S, D30F, G48R) to a wild type one.
For comparison with our predicted mutant PRs, Table 3 reports our computed
binding energies for the Pr3 three-fold mutant, as well for simpler one- and
two-fold mutant PRs derived from Pr3 (Pr3 set), as compared to WT. Note,
however, that experimental data are available only for the three-fold mutant
PR. In our calculations, Pr3 set carried mutations only on chain A, while
still being formed by two separate chains. We expected to find that Pr3
computed binding would be stronger towards RTp51-RTp66, while weaker towards
CA-p2 and p2-NC, compared to WT-PR. The computed binding energies of the Pr3
set show that the mutant enzymes often have higher affinity for the desired
RTp51-RTp66 peptide compared to CA-p2 and p2-NC. Most notably the double
mutant A28S/G48R has a stronger computed binding energy towards the target
peptide than WT-PR, while lower for the other two endogenous substrates. The
binding energy test indicates that A28S/G48R (for which there is no
experimental data available) would have been a more successful mutation than
Pr3. Nevertheless, the possibility of using the binding energy test with
mutant PRs was found viable.
(3) Finally we decided to apply the binding energy test to series of mutant
PRs binding mutant endogenous substrates. Thus, we evaluated the binding
energies of drug resistant HIV-1 proteases towards wild type and mutant
substrate peptides. It has been found that mutations of the cleavage sites are
correlated to mutations of the protease, often leading to drug resistance. We
analyzed the K436R and A431V mutations of the NC-p1 Gag substrate peptide
cleavage sequence in relation to a series of single mutations and one double
mutation of HIV-1 protease (DR set). It has been reported [19] that a K436R
mutation increases resistance to protease inhibitor drugs when combined with
I50V, I84V and I84V/L90M PR mutations, while the A431V mutation results in a
more efficient PR regardless of other mutations. We expected that the more
efficient mutant PR – mutant peptide combinations were also characterized by
stronger binding energies. Table 4 reports the results of our binding energy
test for the DR set. Our methodology indicates cleavability for all
combinations of mutant PRs and mutated NC-p1 substrate peptides. While there
are some fluctuations in the binding energies, no clear pattern arises that
can be related to the experimental findings. Possibly, the increased
efficiency of drug resistant mutant proteases towards mutated peptides is
related to $k_{cat}$. As previously stated, the effects of this constant are
not considered by the present approach. Nevertheless, the binding energy test
was found suitable also for combinations of mutant PRs with any peptide.
### Prediction and Analysis of Mutant PRs
The second aim of this study was to develop a computational methodology for
the design of a mutant PR. The sought after enzyme had to be capable of
cleaving a new target substrate different from the endogenous ones. The
obtained mutant PR should also be specific for the target peptide sequence
compared to the endogenous peptides. The chosen sequence for the target
peptide was HLSF*MAIP, where the * symbol indicates the desired cleaving site.
The sequence was extracted from that of $\kappa$-casein. Once candidate mutant
PRs were obtained, we employed the binding energy test to asses the enzymes
cleaving capabilities. The possibility of an increase in cleaving capability
towards the target substrate was asserted by differences in binding energy
between WT-PR and mutant PRs. We evaluated the binding energies of mutant PRs
in complex with the TF-PR peptide, used as a starting template (see the
Computational Methods section), and the CA-p2 and p2-NC peptides (for selected
mutant PRs) in order to test the specificity of our mutant PRs.
The mutant-generating algorithm is described in details in the Computational
Methods section. Two main strategies (Strategy1 and Strategy2) were employed
for generating mutant PRs. In Strategy1, the side chains of only the 6
residues previously indicated as specificity determining[15] were allowed to
change. The analysis of the binding energies of the mutant PRs generated by
Strategy1 found the enzymes insufficient to perform the desired scope. This
prompted us to further develop the algorithm. In Strategy2, the side chains of
26 residues were allowed to change. See the Computational Methods section for
further details on the residues choice. The analysis of the binding energies
of these mutant PRs found some of the predicted enzymes to be adequate to
cleave the desired target sequence.
Tables Supporting Material for and Supporting Material for in the Supporting
material reports the Strategy1 mutant PRs (M1 – M16) and their computed
binding energies towards the targetpeptide and TF-PR, CA-p2 and p2-NC
endogenous peptides. Among these mutant PRs, M5 shows the strongest binding
energy towards the target peptide. However it has to be noted that the
computed binding energy of M5 towards the TF-PR peptide (used as a starting
template for all mutant enzymes) is also stronger with respect to WT. Possibly
M5 is simply a better generic binder. To verify this hypothesis we tested M5
as a binder also for other two endogenous peptide sequences, CA-p2 and p2-NC.
Compared to WT-PR, M5 has weaker binding energy for the former peptide, but
equal for the latter. In conclusion, M5 is not predicted to be more specific
for the target sequence than for the endogenous peptides. Moreover, M5 was not
directly predicted through Strategy1, but as a homodimeric derivative of M2,
which shows only a small improvement in binding of the target peptide. All
other mutant PRs suggested by Strategy1, M1 – M4 and M6 – M16, were found
having a weak binding energy towards the target peptide, with some of them
showing prominently positive binding energies. It can be concluded that
Strategy1 is unsatisfactory at predicting a mutant PR with an increased and
specific affinity towards the target peptide. This is possibly due to the fact
that allowing only six residues to change is too strict a condition to achieve
a suitable mutant PR.
Thus, we decided to further improve the mutation algorithm by including more
residues among those that can be changed. The six generations of mutant PRs
computed through our Strategy2 mutant algorithm are presented in Table 5. We
refer to them as generations since at each macro step of the algorithm the
lowest in energy (as computed with the standard RosettaDock energy function)
structure was used as starting point for the next step. The sixth generation
(M23) did not produce any new change with respect to the fifth (M22), and the
algorithm was consequently terminated. For each generation the structure with
the lowest absolute energy was further optimized. After generation 1 two
mutant structures were chosen (M17 and M18) since they are very close in
energy (as evaluated with the RosettaDock energy function, data not shown) but
relatively different as mutation sites. In addition, an extra mutant PR (M24)
was generated as homodimer of M22. The computed binding energies of the
Strategy2 mutant PRs (M17 – M24) are shown in Table 6. All Strategy2 mutant
PRs show a binding energy towards the target sequence two to four fold
stronger than WT-PR, with M17 displaying the strongest binding energy.
However, as for M5, binding energies towards the template peptide TF-PR as
well as CA-p2 and p2-NC are also stronger than WT. Possibly M17 is also a good
but generic binder. Through the subsequent generations of mutant proteases, at
last M22 shows a binding energy towards the target peptide more than three
fold stronger than WT, while the computed binding energy towards the natural
endogenous substrates is weaker than WT. Similar results were obtained for its
homodimer M24. M22 and M24 show binding energies below the cutoff value of -25
kcal/mol, and thus represent the best candidates to be further studied
experimentally.
We compared the structures of WT-PR and M24 as optimized while binding the
target peptide. Figure 2 reports the superimposed backbones of the two enzymes
after structure alignment. The two computed structures are quite coincident.
Hence, it is expected that M24 should retain the main structural features of
the wild type enzyme. We also tried to analyze the choice of changed residues.
Figure 3 shows that the residues that were changed from WT-PR to M24 are
disposed all around the bound peptide. Figures Supporting Material for –
Supporting Material for given as supporting material compare each residue that
differs between WT-PR and M24, while bound to the target peptide. Although it
is evident that the A28S substitution on chain A introduces a hydrogen bond
between the residue and the side chain of the serine in the peptide (Figure
Supporting Material for), the other substitutions are less easily
rationalized. On going study aims at elucidating the role of the other
residues substitutions.
It is interesting to note that Strategy2 mutated only 7 out of the 26 residues
that were set as mutable in the method. It is also worth noting that of the 7
residues (A28, D30, K45, I50, P81, V82, I84) suggested by Strategy2 in the
various mutant generations, A28, K45, P81 are not included in the set of major
mutations site of HIV-1 protease responsible for drug resistance [31], that
is: D30, V32, M46, I47, G48, I50, I54, Q58, T74, L76 V82, N83, I84, N88, L90.
A28, K45, and P81 together with I50 are also not included in the specificity
determining residues set [15]. However, A28 was located by Alvizo _et al_. for
the Pr3 mutant [18]. We envision Strategy2 also as a tool to locate those
residues most involved in binding a given substrate peptide.
From the analysis of the different PRs, mutant and wild type, and their
binding energies, it is worth to note that WT-PR has a certain affinity with
the octa-arginine peptide. Its computed binding energy is at the limit to
consider the octa-arginine peptide as cleavable by WT-PR. Possibly this
relatively strong binding is given by very few interactions. Accordingly, the
single D30F change on chain A, that is changing one negatively charged residue
into an aromatic hydrophobic one, is able to drop the computed binding energy
to 0, as shown in Table 4. The currently going analysis of the residue by
residue interactions for the modified side chains will give further
information also on this aspect of the binding of PR.
Finally, it is interesting to note that the algorithm is not always preserving
amino acid side chain changes through the generations. For example, I84V on
chain A is introduced in M18 and kept in M19, M20 and M21, but later reverted.
Possibly, an isoleucin in position 84 is energetically more favorable, given
the other side chain changes.
## Conclusions
In the first part of this study we developed a methodology to test the
cleavability of a peptide by HIV-1 protease (Tables 1 and Supporting Material
for), solely based on the binding energy between the enzyme and the substrate.
The methodology can also be applied to mutant PRs, Table 3. The technique is
based on a PyRosetta algorithm generating, iteratively, optimized structures,
coupled with an energy re-evaluation at a higher level of theory (FMO/PCM
MP2/6-31G(d)).
In the second part of this study, the optimization algorithm was extended to
permit the stochastic change of the side chain of selected residues, in order
to better bind a given target peptide sequence. The selected target peptide
was required to be different from the endogenous peptides. The desired outcome
was a mutant PR with stronger and weaker predicted binding energy for the
target and endogenous peptides, respectively, compared to WT-PR. The mutant
PRs M22 and M24 generated through Strategy2 exhibit such desired
characteristics (Table 6). We analyzed the backbone structure of WT-PR and M24
and found no major differences, thus indicating that M24 should retain the
general structure features of wild type HIV-1 protease. Strategy2 algorithm is
able to predict mutations outside the usual set of residues involved in drug
resistance, possibly giving an ulterior insight into the binding process of
HIV-1 protease.
Ongoing experimental studies will show if and how well M22 and/or M24 bind and
cleave the target sequence. Our current experimental and computational studies
are also aimed at analyzing M24 mutations, residue by residue and in
combination, and their possible role in binding the target sequence. It is our
hope that the experimental tests will provide enough information to be used to
further improve the mutant generating algorithm. If the combination of
computational algorithm and experimental verification is successful it will
maybe permit the design of mutant PRs specific for any given substrate
peptide.
## Computational Methods
In general, the activity of an enzyme towards two similar substrates is
regulated by (i) how good the enzyme-substrate binding is and (ii) how
efficient the enzymatic reaction is. Following the dynamic substrate envelope
hypothesis [9, 10], we assume a correlation between the binding of different
peptides to PR and cleavability of the former. Thus we compute qualitative
binding energies, on the premise that lower binding energy equals better
cleavability.
### Binding Energies
#### PyRosetta Algorithm
The structure of wild type (WT) HIV-1 protease in complex with different octa-
peptides was optimized using PyRosetta 1.1,[16] a python script-based
interface to Rosetta,[17] and the algorithm depicted in Figure 4. The
algorithm is based on the flexible peptide-docking algorithm used by Chaudhury
and Gray[15] to identify in WT HIV-1 protease the active-site residues mostly
involved in the discrimination of cleavable and non-cleavable peptides.
Following their algorithm, the HIV-1 protease – peptide complexes are
represented in atomic resolution, as opposed to a coarse-grain representation.
With respect to the algorithm described in [15], our algorithm (Figure 4) has
a larger number of cycles (8x4x6=192 compared to 8x12=96), and more ’small’
and ’shear’ moves for the perturbation of both the side chain and the backbone
atoms. The side chain conformations are further optimized through a repacking
algorithm[32] and using the extended Dunbrack library[33, 34]. The moves are
applied to all residues of the substrate peptide plus a selected number of
residues of the protease, with the following criterion: all residues inside a
5 Å distance from any atom of the substrate peptide, plus all the residues
reported as active by Chaudhury and Gray[15], plus their $\pm$1 neighbours,
plus if one residue is included on only one chain it is made to be included in
both. After the moves, an energy minimization step is performed, based on the
Davidon-Fletcher-Powell method [35, 36]. Each structure is then accepted or
rejected based on a Monte Carlo (MC) criterion depending on the standard
RosettaDock energy function [37, 33, 32, 38, 39]. Along the optimization a
temperature gradient was applied, from an initial value of kT = 3.0 to 1.0,
unless differently stated. 500 decoy structures were generated using 5
parallel algorithm runs, each producing 100 structures.
The main difference with the algorithm of [15] is that after the algorithm
produced 500 decoy structures, the lowest in energy is chosen and used as a
starting structure for another cycle of optimization. This process is repeated
$K$ times, until convergence. It was found that, after at least 5 cycles, the
computed RosettaDock energy did not change between subsequent cycles as soon
as all 5 parallel runs of a single cycle produced structures with the same
energy. Consequently, in order to render as automatic as possible the
algorithm, the fact that $K>5$ and that each parallel run produced, as best
structure, a decoy with the same energy was taken as a mark for convergence.
It was found that, on average, a value of $K=20$ was sufficient. As an
example, Figure 5 reports the energy of WT-PR bound to TF-PR along the
optimization. The points at each step corresponds to the RosettaDock energy of
the lowest in energy decoy out of the 500 computed at that particular step.
Such structure would then be used as starting point for the next cycle. At the
end of the $K$ cycles the lowest in energy decoy is chosen as the PyRosetta
optimized structure.
The same algorithm was also used for the optimization of mutant HIV-1
proteases (_vide infra_), the octa-peptides alone, and the protease alone as
apo-protein.
The starting structures were prepared from that of HIV-1 protease in complex
with an inhibitor (PDB accession code 1HXB [40]), considered as apo-protein.
In order to place the substrate peptide, the structure of a D25N deactivated
protease in complex with the natural substrate peptide p2-NC (PDB accession
code 1KJ7 [9]) was aligned with respect to the backbone atoms of the protease
(RMS = 0.436 Å). The starting structure was then composed using the apo-
protein from 1HXB and the substrate peptide from 1KJ7. All subsequent
protease-peptide complexes were created starting from this structure and
mutating the peptide accordingly. See Table Supporting Material for, Table
Supporting Material for and Table Supporting Material for for a complete list
of the considered substrate peptides. Hydrogen atoms were added through the
program Pymol [41].
#### Further Structures Optimization and Energetic Re-evaluation
The position of the hydrogen atoms of each PyRosetta generated structure was
optimized using Open Babel[42] with the MMFF94[43, 44, 45, 46, 47] force
field. The energy of each structure was finally re-evaluated at the higher
level of theory ‘FMO2-MP2/6-31G(d)/PCM[1]’. Single point energy evaluations
were carried out using the fragment molecular orbital (FMO) approximation [48,
49], as implemented in GAMESS [50]. Each FMO calculation was carried out at
the MP2 level of theory [51] with the 6-31G(d) basis set [52, 53] and the
Polarazible Continuum Model (PCM) approximation [54, 55]. Pairs of fragments
separated by more than two van der Waals radii were calculated using a Coulomb
expression for the interaction energy and ignoring correlation effects
(RESDIM=2.0 RCORSD=2.0 in $FMO). The input files for the FMO calculations were
prepared using the program FRAGIT [56].
#### Binding Energies Evaluation
The re-evaluated energy of every optimized structure was used to compute the
binding energy of PR with different substrate peptides. The binding energy
($E_{Bind}$) of HIV-1 protease (wild type or mutated) and a peptide was
evaluated with equation (1), where $E_{Complex}$ is the energy of the complex,
$E_{APO}$ the energy of the protease optimized as apo-protein, $E_{Pep}$ the
energy of the optimized peptide.
$E_{Bind}=E_{Complex}-\left(E_{APO}+E_{Pep}\right)$ (1)
These binding energies can not be directly compared to experimental values,
for which a much more complex and accurate methodology is required [57]. These
energies were used only to qualitatively compare different PR – peptide
combinations.
### Mutation Algorithm
A similar procedure as that described in Figure 4 was used to produce mutant
HIV-1 proteases, possibly capable of cleaving a given peptide different from
the endogenous substrate peptides. The general idea was to ’expose’ the
protease to a different peptide and allow some residues to change in order to
accommodate it better. A target octa-peptide was chosen: HLSF*MAIP, where the
* symbol indicates the desired cleaving site. The peptide sequence was
extracted from that of $\kappa$-casein.
The assumption behind the algorithm is that lowering the energy of the PR –
peptide complex by changing the side chains of selected residues would
decrease also the binding energy, thus increasing the cleavability.
Two different methodologies were designed to predict mutant PRs, Strategy1 and
Strategy2. The Strategy1 mutation algorithm is depicted in Figure 6. Each
optimization step corresponds to the algorithm of Figure 4. In the mutation
steps (also based on the previous algorithm), the Dunbrack library of rotamers
includes all rotamers of all amino acids, but only for a selected number of
residues. The six specificity determining residues, as found by [15], are
chosen to be altered. In other words, during the mutation step, whenever one
of the alterable residues is being optimized, the random choice of a test
rotamer is among all possible amino acids. In Scheme A alterations are allowed
on all 6 residues on both chains, for a total of 12 alterable residues. Thus,
side-chain perturbation and repacking rotamer choice is performed randomly
selecting among 12 x 20 = 240 possible amino acids. In Scheme B only
alterations on L76 and V82 of Chain A and D30, I47, G48, and I84 of Chain B
are allowed, for a total of 6 alterable residues. In this case, side-chain
perturbation and repacking rotamer choice is performed with a random selection
among 6 x 20 = 120 possible amino acids. Each mutation step took ca. 40 hours
on 5 cpus to produce 500 decoys. The lowest energy decoy is then chosen as
starting structure for the next step. The energy of the structure is evaluated
with the standard RosettaDock energy function. The residue reference energy
part of the energy function [32] takes into account also the differences
between different amino acids. In other words, energy differences between two
mutant structures originates solely from different side chain interactions
rather than from a different number of atoms.
Both the mutation and the optimization steps were repeated $K^{\prime}$ and
$K$ times, respectively. The mutation cycles are considered converged once two
following cycles do not introduce new mutations. Different values of $K$ and
$K^{\prime}$ were found necessary to reach convergence. After a series of
mutation cycles ($K^{\prime}\geq 8$), a series of optimization cycles was
performed ($K\geq 8$), followed by another usually shorter mutation cycle
($K^{\prime}\leq 3$) and finally a short optimization cycle ($K\leq 3$).
Among the naturally cleaved peptides, TF-PR (sequence SFNF*PQIT) was chosen as
a starting substrate peptide, since it is the most similar, in terms of
conserved residues, to the target peptide (sequence HLSF*MAIP). Consequently,
the optimized structure of WT protease in complex with the TF-PR peptide was
chosen as starting template. The substrate peptide sequence was altered one
amino acid at the time, as reported in Table Supporting Material for. After
each peptide alteration, a series of protease mutation and optimization cycles
were performed. Once convergence was reached, a new peptide single amino acid
change was introduced and the procedure repeated. Different mutant PRs were
obtained from different runs by changing a few parameters, e.g. the initial
temperature of the simulation. These parameters are specified in Table
Supporting Material for. Some mutant PRs were also produced by ’exposing’ the
protease directly to the target peptide without prior intermediates (mutation
Scheme F). This last process required a higher number of $K^{\prime}$ cycles
($K^{\prime}\geq 15$), but without having to cycle through one substrate
peptide residue at the time.
All mutant PRs obtained through Strategy1 were heterodimers. By simply
equalizing alterations on both chains a number of extra homodimer mutant PRs
were also obtained. These structures were subsequently optimized as previously
described.
In Strategy2 the number of residues allowed to change was increased in order
to include all amino acids residing inside a 3 Å radius from the TF-PR
peptide. In other words, we chose those residues with at least one atom that
is distant at most 3 Å from any atom of the substrate peptide. The specificity
determining residues were also included in the set of alterable amino acids,
if not already present. The residues Asp25, Thr26 and Gly27 of both chains
were excluded from the set, since they represent the catalytic triad [5]. The
full set of 26 residues is reported in Table Supporting Material for. Thus,
side-chain perturbation and repacking rotamer choice is performed randomly
selecting among 26 x 20 = 520 possible amino acids. The mutant PRs were
generated using the target peptide directly (Scheme F). Each mutation step
took a bit more than 3 days on 5 cpus to produce 500 decoy structures. An
initial temperature of 9 kT was usued. $K^{\prime}$ = 6 mutation cycles were
performed. The lowest in energy decoy after each mutation step was
subsequently optimized (two after the first step). The sixth mutation step did
not introduce any new mutation in PR and the mutation cycle was stopped.
Also the mutant PRs obtained through Strategy2 were heterodimers. Only the
homodimer of the final mutant PR was considered, see Table 5.
## Acknowledgments
Computational resources were provided by the Danish Center for Scientific
Computing (DCSC). LDV acknowledges S. Chaudhury and J. J. Gray for fruitful
discussions about PyRosetta, for providing the set of non cleavable peptides
and a copy of their algorithm script. C. Steinmann is acknowledged for help
with the program FRAGIT and the FMO based calculations.
## Supporting material
Supporting material available: Tables Supporting Material for – Supporting
Material for, Figures Supporting Material for – Supporting Material for, a
movie showing the three dimensional structure of WT-PR bound to the target
peptide, with highlighted the residues that are changed in M24.
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## Figure Legends
Figure 1: ROC plot comparing different cutoff values for binding energies
computed through FMO energy re-evaluation or RosettaDock energy function. The
values for each method closest to the theoretical optimum (0,1) are
highlighted. The computed area under the ROC curve is 0.79 and 0.80 for FMO
and Rosetta, respectively. The raw data is reported in Table Supporting
Material for. Figure 2: Backbone difference between PyRosetta computed
structures of WT-PR and M24. The optimized structures of WT-PR and M24 binding
the target peptide were aligned with respect to their $\alpha$-carbon atoms
using PyMol. The backbone of M24 (red) is almost coincident with that of WT-PR
(green) with a RMS of 0.227 Å.
---
Figure 3: Spatial disposition of the residues changed by Strategy2. The six
residues of chain A (top) and 6 residues of chain B (bottom) are highlighted
in ball-and-sticks. The reported structure (as semi-transparent cartoon) is
that of WT-PR optimized when binding the target peptide (only the backbone is
shown in sticks). Figures Supporting Material for \- Supporting Material for
report the full residue by residue changes. A movie showing the three
dimensional structure is included as Supporting Material. Figure 4:
PyRosetta based optimization algorithm. $\phi$, $\psi$, $\chi$ represent
perturbations applied to both backbone and side chain dihedral angles. MC
criterion stands for a Monte Carlo based check of decoy structures. Figure 5:
Optimization algorithm convergence. Example of energy convergence during the
various macro cycles of the optimization algorithm for WT-PR in complex with
TF-PR peptide. Each point along the graph corresponds to the energy (computed
with the RosettaDock energy function) of the lowest in energy decoy out of 500
produced during each of the _K_ steps. Figure 6: PyRosetta based mutation
algorithm. The optimization step is the algorithm presented in Figure 4. In
the mutation step the side chain perturbation for the six specificity
determining residues is among all possible rotamer of all 20 amino acids.
## Tables
Table 1: Computed binding energies of WT-PR and cleavable endogenous peptides. Substrate Peptide | FMO (kcal/mol) | RosettaDock (kT) | exp $K_{m}$ (mM)[22]
---|---|---|---
MA-CA | -57 | -9 | 0.15
CA-p2 | -52 | -6 | 0.01
p2-NC | -72 | -4 | 0.05
NC-p1 | -68 | -3 |
p1-p6 | -47 | 1 |
p6pol-PR | -41 | -6 |
TF-PR | -62 | -5 | $<$0.01
PR-RTp51 | -64 | -7 | 0.07
RTp51-RTp66 | -68 | -12 | 0.04
RTp66-INT | -62 | -6 |
RH-IN | -63 | -10 | 0.006
Table 2: Comparison of WT-PR computed binding energies.
| RosettaDock Energy Function | FMO Energy Re-evaluation
---|---|---
Average endogenousa | -6 (kT) | -60 (kcal/mol)
(Standard deviation) | (3) (kT) | (10) (kcal/mol)
Average all cleavableb | -5 (kT) | -43 (kcal/mol)
(Standard deviation) | (3) (kT) | (22) (kcal/mol)
Average non cleavablec | -1 (kT) | -15 (kcal/mol)
(Standard deviation) | (4) (kT) | (28) (kcal/mol)
U test probability (all cleavable VS non-cleavable) | $1.46\cdot 10^{-7}$ | $2.21\cdot 10^{-7}$
U test probability (only endogenous VS non-cleavable) | $3.49\cdot 10^{-4}$ | $6.16\cdot 10^{-6}$
a Table 1.
b Table 1 plus Table Supporting Material for.
c Table Supporting Material for.
Energies were computed with the standard RosettaDock energy function, as
described in [15] and with the FMO re-evaluation. A Mann-Whitney’s U test
probability was evaluated by comparing the binding energies of the set of
endogenous peptide against the non-cleavable and the entire set of cleavable
peptides against the non-cleavable. The FMO based binding energies are more
clear in discriminating cleavable and non cleavable peptides than the Rosetta
based ones.
Table 3: FMO computed binding energies of HIV-1 protease WT and Pr3 set of
mutant PRs.
PR | Peptides | | | | |
---|---|---|---|---|---|---
| RTp51-RTp66 | poly-Ala | poly-Arg | TF-PR | CA-p2 | p2-NC
WT-PR | -68 | -15 | -41 | -62 | -52 | -72
_Single mutant_ | | | | | |
A28S | -65 | -12 | -35 | -41 | -13 | -67
D30F | -48 | -1 | 0 | -7 | -4 | -43
G48R | -66 | -44 | -27 | -55 | -15 | -43
_Double mutant_ | | | | | |
A28SD30F | -44 | -16 | 30 | -35 | -22 | -55
A28SG48R | -96 | -16 | -54 | -35 | -14 | -60
D30FG48R | -76 | 3 | 18 | -19 | -1 | -54
_Triple mutant_ | | | | | |
A28SD30FG48R | -42 | -21 | 12 | -32 | -12 | -63
Computed binding energies (kcal/mol) of WT and mutant HIV-1 proteases in
complex with RTp51-RTp66, poly-alanine, poly-arginine, TF-PR, CA-p2 and p2-NC
peptides.
Table 4: FMO computed binding energies of HIV-1 protease WT and selected drug
resistance mutant PRs (DR set).
PR | Peptides | | | |
---|---|---|---|---|---
| NC-p1WT | NC-p1K436R | NC-p1A431V | poly-Ala | poly-Arg
WT-PR | -49 | -70 | -56 | -15 | -41
D30N | -29 | -44 | -36 | -15 | -23
I50L | -64 | -49 | -54 | -16 | -49
I50V | -54 | -46 | -52 | -18 | -34
V82A | -45 | -54 | -52 | -16 | -38
I84V | -46 | -75 | -35 | -23 | -38
I84V L90M | -55 | -67 | -55 | -20 | -46
Computed binding energies (kcal/mol) of WT-PR and selected drug resistance
mutant proteases in complex with NC-p1 as wild type, K436R and A431V drug
resistance associated mutant peptides, poly-alanine and poly-arginine
peptides.
Table 5: Strategy2 suggested mutant PRs.
Mutant ID | Chain A | Chain B | Mutation Scheme | Notes
---|---|---|---|---
M17 | A28S D30T | A28S D30T K45M I50L V82F | F | After one mutation step
M18 | A28S D30T I50L P81D V82R I84V | A28S D30T K45M I50L V82Y | F | After one mutation step
M19 | A28S D30T I50L P81D V82R I84V | A28S D30T K45A I50L V82Y I84L | F | After two mutation steps
M20 | A28S D30T I50L P81D V82R I84V | A28S D30T K45D I50L V82Y I84L | F | After three mutation steps
M21 | A28S D30T I50L P81L V82Y I84V | A28S D30T K45D I50L V82Y | F | After four mutation steps
M22 | A28S D30T I50L P81L V82Y | A28S D30T K45A I50L V82Y | F | After five mutation steps
M23 | A28S D30T I50L P81L V82Y | A28S D30T K45A I50L V82Y | F | After six mutation steps
M24 | A28S D30T K45A I50L P81L V82Y | A28S D30T K45A I50L P81L V82Y | – | Homodimer of M22
M17 – M23 represent the subsequent generations of mutant PRs suggested by
Strategy2. All mutant enzymes were generated following Scheme F.
Table 6: FMO computed binding energies of HIV-1 protease WT and Strategy2
mutant PRs.
PR | Peptides | | | | |
---|---|---|---|---|---|---
| Target | poly-Ala | poly-Arg | TF-PR | CA-p2 | p2-NC
WT-PR | -9 | -15 | -41 | -62 | -52 | -72
_Gen 1_ | | | | | |
M17 | -34 | -13 | -47 | -68 | -82 | -74
M18 | -24 | -19 | -45 | -82 | -62 | -63
_Gen 2_ | | | | | |
M19 | -17 | 2 | 1 | -67 | -46 | -81
_Gen 3_ | | | | | |
M20 | -23 | -2 | -19 | -67 | -33 | -84
_Gen 4_ | | | | | |
M21 | -20 | 2 | 7 | -37 | -32 | -18
_Gen 5_ | | | | | |
M22 | -29 | -6 | 10 | -42 | -25 | -30
M24 | -29 | -11 | 7 | -44 | -33 | -33
Computed binding energies (kcal/mol) of WT-PR and Strategy2 mutant proteases
in complex with Target, poly-alanine, poly-arginine, TF-PR, CA-p2 and p2-NC
peptides.
## Supporting Material for
_In silico_ prediction of mutant HIV-1 proteases cleaving a target sequence
Jan H. Jensen, Martin Willemoës, Jakob R. Winther, Luca De Vico
The video animation of the optimized structure of WT-PR binding the target
peptide with highlighted residues can be found at this link:
http://youtu.be/NEXKojTw2Bc .
[h] Cleavable peptides. P4 P3 P2 P1 * P1’ P2’ P3’ P4’ MA-CA S Q N Y * P I V Q
CA-p2 A R V L * A E A M p2-NC A T I M * M Q R G NC-p1 R Q A N * F L G K p1-p6
P G N F * L Q S R p6pol-PR S F N F * P Q V T TF-PR S F N F * P Q I T PR-RTp51
T L N F * P I S P RTp51-RTp66 A E T F * Y V D G RTp66-INT R K V L * F L D G
RH-IN R K I L * F L D G
List of the cleavable endogenous peptides considered in this work
Extra cleavable peptides. P4 P3 P2 P1 * P1’ P2’ P3’ P4’ K001 T Q I M * F E T F
K002 G Q V N * Y E E F K003 P F I F * E E E P K005 D T V L * E E M S K007 A E
E L * A E I F K008 S L N L * R E T Q K010 A E C F * R I F D K011 D Q I L * I E
I C K012 D D L F * F E A D K013 Y E E F * V Q M M K014 P I V G * A E T F K016
R E A F * R V F D K018 A Q T F * Y V N L K019 P T L L * T E A P K020 S F I G *
M E F K K021 D A I N * T E F K K022 Q I T L * W Q R P K023 E L E F * P E G G
K029 K E L Y * P L T S K031 S R S L * Y A S S K032 A E A M * S Q V T K034 G S
H L * V E A L K035 G G V Y * A T R S K036 F R S G * V E T T K037 V E V A * E E
E E K038 L P V N * G E F S K039 E T T A * L V C D K040 H L V E * A L Y L K041
H Y G F * P T Y G K042 D S A D * A E E D K043 G W I L * G E H G K045 Q A I Y *
L A L Q K046 E K V Y * L A W V K047 V E I C * T E M E K048 T Q D F * W E V Q
K049 L W M G * Y E L H K050 G D A Y * F S V P K051 E L E L * A E N R K052 S K
D L * I A E I K053 L E V N * I V T D K054 I I V A * C E G N K056 G G N Y * P V
Q H K057 A R L M * A E A L K058 P F A A * A Q Q R K059 P R N F * P V A Q K060
G L A A * P Q F S K061 S L N L * P V A K K063 R Q V L * F L E K K064 Q M I F *
E E H G SUB3 Q I T L * W K R P T035 V E I C * T E M E T084 T Q D F * W E V Q
T112 G D A Y * F S V P T228 L W M G * Y E L H T300 E L E L * A E N R T322 S K
D L * I A E I T480 Q A I Y * L A L Q T491 L E V N * I V T D T529 E K V Y * L A
W V
List of the cleavable non-endogenous peptides considered in this work
Other peptides P4 P3 P2 P1 * P1’ P2’ P3’ P4’ Target H L S F * M A I P
NC-p1A431V R Q V N * F L G K NC-p1K436R R Q A N * F L G R poli-Ala A A A A * A
A A A poli-Arg R R R R * R R R R
List of other peptides considered in this work
Non-cleavable peptides P4 P3 P2 P1 * P1’ P2’ P3’ P4’ NBP1 V N C A * K K I V
NBP2 W R N R * C K G T NBP3 M M K S * R N L T NBP4 L A A A * M K R H NBP5 T T
Q A * N K H I T015 G M D G * P K V K T031 I K A L * V E I C T033 A L V E * I C
T E T037 I C T E * M E K E T039 T E M E * K E G K T080 L N K R * T Q D F T082
K R T Q * D F W E T086 D F W E * V Q L G T088 W E V Q * L G I P T108 V L D V *
G D A Y T110 D V G D * A Y F S T114 A Y F S * V P L D T116 F S V P * L D E D
T224 E P P F * L W M G T226 P F L W * M G Y E T230 M G Y E * L H P D T232 Y E
L H * P D K W T296 T E E A * E L E L T298 E A E L * E L A E T302 E L A E * N R
E I T304 A E N R * E I L K T318 Y Y D P * S K D L T320 D P S K * D L I A T324
D L I A * E I Q K T326 I A E I * Q K Q G T441 Y V D G * A A N R T476 K T E L *
Q A I Y T478 E L Q A * I Y L A T482 I Y L A * L Q D S T484 L A L Q * D S G L
T487 Q D S G * L E V N T489 S G L E * V N I V T493 V N I V * T D S Q T495 I V
T D * S Q Y A T525 L I K K * E K L A T527 K K E K * V Y L A T531 V Y L A * W V
P A T533 L A W V * P A H K
List of non-cleavable peptides considered in this work
[htp!] Computed binding energies of WT-PR and non-endogenous cleavable
peptides. Substrate Peptide FMO (kcal/mol) RosettaDock (kT) Substrate Peptide
FMO (kcal/mol) RosettaDock (kT) K001 -22 -2 K043 -67 -5 K002 -33 -4 K045 -55
-5 K003 -16 -8 K046 -39 -4 K005 -24 -4 K047 -79 -5 K007 -27 -1 K048 -86 -9
K008 -40 -2 K049 -40 -5 K010 -55 -4 K050 -36 -7 K011 -42 -3 K051 -31 -2 K012
-22 -7 K052 -4 1 K013 -36 -2 K053 -51 -4 K014 -29 -3 K054 -56 -1 K016 -73 -3
K056 -70 -5 K018 -72 -4 K057 -7 -6 K019 -61 -9 K058 -35 -2 K020 -8 -3 K059 -58
-6 K021 -64 -8 K060 -36 -5 K022 -45 -5 K061 -34 -5 K023 -63 -11 K063 -61 -3
K029 -48 -4 K064 -2 -7 K031 -32 -5 SUB3 -30 -6 K032 -51 -6 T035 -67 -4 K034 -8
-8 T084 -81 -7 K035 -30 -8 T112 -69 -11 K036 -3 -3 T228 -29 -5 K037 -2 -4 T300
-29 -1 K038 -39 -5 T322 -21 1 K039 -10 0 T480 -62 -5 K040 -30 -6 T491 -56 -1
K041 -53 -6 T529 -27 1 K042 -29 -6
[htp!] Computed binding energies of WT-PR and non-cleavable peptides.
Substrate Peptide FMO (kcal/mol) RosettaDock (kT) Substrate Peptide FMO
(kcal/mol) RosettaDock (kT) NBP1 -18 3 T296 43 2 NBP2 -21 3 T298 -54 -2 NBP3
-63 0 T302 -44 4 NBP4 -18 4 T304 1 -2 NBP5 -68 7 T318 23 0 T015 2 -3 T320 23
-2 T031 -29 -6 T324 -18 0 T033 -10 -2 T326 -10 5 T037 -28 -5 T441 -30 -2 T039
-36 -3 T476 -10 1 T080 -45 4 T478 -33 -2 T082 9 -2 T482 -45 -1 T086 -23 0 T484
-24 -2 T088 -42 3 T487 31 -3 T108 -12 -4 T489 -24 -6 T110 -21 -3 T493 -19 -1
T114 -16 -8 T495 -15 1 T116 -10 -2 T525 61 7 T224 48 -1 T527 -9 1 T226 -49 -9
T531 -5 -5 T230 19 -2 T533 -23 -7 T232 -13 -1
[htp!] Strategy1 suggested mutant PRs. Mutant ID Chain A Chain B Mutation
Scheme Notes M1 V82R I84V D30Y V82I A M2 V82Y D30V B M3 D30T D30V V82I B F M4
D30Y V82R D30Y V82R – Homodimer of M1 M5 D30V V82Y D30V V82Y – Homodimer of M2
M6 D30V D30V – Homodimer of M3 M7 D30T I84V D30V V82F A F Initial temperature
= 9 kT M8 D30V B F Initial temperature = 9 kT M9 D30T I47L L76F V82R I84T D30E
V82Y A Initial temperature = 6 kT M10 V82Y D30T I84L B Initial temperature = 6
kT M11 D30V V82Y I84V D30H I47L L76F V82Y A Initial temperature = 12 kT M12
V82Y D30T B Initial temperature = 12 kT M13 D30E L76F V82R D30E L76F V82R –
Homodimer of M9 M14 D30T V82Y I84L D30T V82Y I84L – Homodimer of M10 M15 D30H
I47L V82Y D30H I47L V82Y – Homodimer of M11 M16 D30T V82Y D30T V82Y –
Homodimer of M12
Different mutant PRs were obtained by small modifications of the mutation
algorithm. Inside Strategy1 two different schemes were used when choosing
which residues could mutate. In Scheme A all six specificity determining
residues were allowed to mutate on both chains. In Scheme B only residues 76
and 82 were set as mutable on Chain A and 30, 47, 48, and 84 on Chain B. In
addition, a straight forward variant of the algorithm was tested, as opposed
to the step-wise one presented in Table Supporting Material for. In this
variant (Scheme F) the protease was directly ’exposed’ to the final target
peptide sequence. A $K$ value in the order of 20 was necessary. Other
parameters that differ from those specified in the Computational Methods
section are also highlighted.
[htp!] FMO computed binding energies of HIV-1 protease WT and Strategy1
mutant PRs. PR Peptides Target poly-Ala poly-Arg TF-PR CA-p2 p2-NC WT-PR -9
-15 -41 -62 -52 -72 M1 -7 3 28 -41 M2 -13 -9 12 -41 M3 -8 -4 13 -43 M4 -18 -24
17 -52 M5 -30 -14 4 -54 -27 -73 M6 -14 2 -1 -14 M7 -7 -2 3 -49 M8 -10 -9 -5
-55 M9 20 17 14 -36 M10 40 11 -3 -43 M11 5 10 45 -44 M12 2 2 16 -43 M13 3 2
-24 -52 M14 3 8 17 -60 M15 -9 2 71 -51 M16 -8 -6 26 -44
Computed binding energies (kcal/mol) of WT-PR and Strategy1 mutant proteases
in complex with target, poly-alanine, poly-arginine, TF-PR, CA-p2 and p2-NC
peptides.
[htp!] Residues set as mutable in Strategy 2. Chain A Chain B Arg 8 Arg 8 Ala
28 Leu 23 Asp 29 Ala 28 Asp 30 Asp 29 Val 32 Asp 30 Gly 48 Lys 45 Gly 49 Ile
47 Ile 50 Gly 48 Leu 76 Gly 49 Thr 80 Ile 50 Pro 81 Pro 81 Val 82 Val 82 Ile
84 Ile 84
The residues were selected as those inside a 3 Å radius from the substrate
peptide plus the specificity determining residues, if not included, minus the
catalytic triad Asp25, Thr26 and Gly27 on both chains. The optimized structure
of WT protease in complex with the TF-PR peptide was used as template.
[!ht] Substrate peptide mutation sequence P4 P3 P2 P1 * P1’ P2’ P3’ P4’ Start
Ser Phe Asn Phe * Pro Gln Ile Thr His Phe Asn Phe * Pro Gln Ile Thr His Phe
Asn Phe * Pro Gln Ile Pro His Leu Asn Phe * Pro Gln Ile Pro His Leu Asn Phe *
Pro Ala Ile Pro His Leu Ser Phe * Pro Ala Ile Pro Target His Leu Ser Phe * Met
Ala Ile Pro
Step wise sequence of substrate peptides employed in the mutation algorithm.
The starting sequence corresponds to the natural substrate TF-PR. This
sequence is altered one amino acid at the time towards that of the desired
target sequence. The P1 and P3’ position were not changed during the sequence.
[htp!] ROC data. _Total_ FMO Rosetta cutoff True positive False positive
Distance to (0, 1) cutoff True positive False positive Distance to (0, 1) -inf
1.00 1.00 1.00 -inf 1.00 1.00 1.00 -10 0.90 0.67 0.68 4 1.00 0.88 0.88 -15
0.89 0.56 0.57 3 1.00 0.84 0.84 -20 0.87 0.44 0.46 2 1.00 0.77 0.77 -25 0.81
0.30 0.35 1 0.99 0.74 0.74 -30 0.71 0.26 0.38 0 0.94 0.63 0.63 -35 0.61 0.21
0.44 -1 0.93 0.58 0.59 -40 0.53 0.19 0.51 -2 0.84 0.37 0.40 -45 0.47 0.09 0.54
-3 0.74 0.23 0.35 -50 0.44 0.07 0.56 -4 0.61 0.19 0.43 -55 0.39 0.05 0.62 -5
0.50 0.14 0.52 -60 0.30 0.05 0.70 -6 0.27 0.09 0.73 -65 0.17 0.02 0.83 -7 0.20
0.05 0.80 -70 0.09 0.00 0.91 -8 0.13 0.05 0.87 -75 0.04 0.00 0.96 -9 0.06 0.00
0.94 -80 0.03 0.00 0.97 -10 0.04 0.00 0.96 -85 0.01 0.00 0.99 -11 0.01 0.00
0.99 +inf 0.00 0.00 1.00 +inf 0.00 0.00 1.00 _Only endogenous_ FMO Rosetta
cutoff True positive False positive Distance to (0, 1) cutoff True positive
False positive Distance to (0, 1) -inf 1.00 1.00 1.00 -inf 1.00 1.00 1.00 -10
1.00 0.67 0.67 4 1.00 0.88 0.88 -15 1.00 0.56 0.56 3 1.00 0.84 0.84 -20 1.00
0.44 0.44 2 1.00 0.77 0.77 -25 1.00 0.30 0.30 1 0.91 0.74 0.75 -30 1.00 0.26
0.26 0 0.91 0.63 0.63 -35 1.00 0.21 0.21 -1 0.91 0.58 0.59 -40 1.00 0.19 0.19
-2 0.91 0.37 0.38 -45 0.91 0.09 0.13 -3 0.91 0.23 0.25 -50 0.82 0.07 0.19 -4
0.73 0.19 0.33 -55 0.73 0.05 0.28 -5 0.73 0.14 0.31 -60 0.64 0.05 0.37 -6 0.36
0.09 0.64 -65 0.27 0.02 0.73 -7 0.36 0.05 0.64 -70 0.09 0.00 0.91 -8 0.27 0.05
0.73 -75 0.00 0.00 1.00 -9 0.18 0.00 0.82 -80 0.00 0.00 1.00 -10 0.09 0.00
0.91 -85 0.00 0.00 1.00 -11 0.09 0.00 0.91 +inf 0.00 0.00 1.00 +inf 0.00 0.00
1.00
Comparison of ROC data for FMO energy re-evaluation and RosettaDock energy
function generated binding energies, while considering different cutoff
values. The upper part of the table reports the full comparison between known
cleavable and non cleavable peptides. In the lower part, data for only the
endogenous peptides was used for the cleavable part. True positive data is
reported in the graphs as sensitivity, false positive as 1 - specificity.
Theoretical values for $\pm$ infinite cutoff have been added.
[!ht]
Correlation plot between FMO and RosettaDock computed binding energies. The
linear trend line shows no correlation between the data (R2 = 0.15435).
[!ht]
ROC plot comparing different cutoff values for binding energies computed
through FMO energy re-evaluation or RosettaDock energy function. The values
for each method closest to the theoretical optimum (0,1) are highlighted. The
comparison was done using the data of only the endogenous peptides for the
cleavable peptides part. The computed area under the ROC curve is 0.96 and
0.84 for FMO and Rosetta, respectively. The raw data is reported in Table
Supporting Material for.
Figures Supporting Material for \- Supporting Material for compare the changes
between WT-PR and M24 , residue by residue. In each figure the enzyme is
represented as semi-transparent ribbon, the peptide as sticks and the changing
residue as ball-and-sticks. The peptide residues numbering is from 2 to 9.
Each residue changed by Strategy2 is indicated by a label containing the
chain, the residue name and its number.
A28S
---
WT |
M24 |
Chain A, residue 28.
D30T
---
WT |
M24 |
Chain A, residue 30.
K45A
---
WT |
M24 |
Chain A, residue 45.
I50L
---
WT |
M24 |
Chain A, residue 50.
P81L
---
WT |
M24 |
Chain A, residue 81.
V82Y
---
WT |
M24 |
Chain A, residue 82.
A28S
---
WT |
M24 |
Chain B, residue 28.
D30T
---
WT |
M24 |
Chain B, residue 30.
K45A
---
WT |
M24 |
Chain B, residue 45.
I50L
---
WT |
M24 |
Chain B, residue 50.
P81L
---
WT |
M24 |
Chain B, residue 81.
V82Y
---
WT |
M24 |
Chain B, residue 82.
|
arxiv-papers
| 2014-04-01T10:37:50 |
2024-09-04T02:50:00.509433
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan H. Jensen, Martin Willemo\\\"es, Jakob R. Winther, Luca De Vico",
"submitter": "Luca De Vico",
"url": "https://arxiv.org/abs/1404.0188"
}
|
1404.0260
|
# Horizon Thermodynamics and Gravitational Field Equations in Quasi-
Topological Gravity
A. Sheykhi 1,2, M. H. Dehghani1,2 and R. Dehghani 1 [email protected]
[email protected] 1 Physics Department and Biruni Observatory, College of
Sciences, Shiraz University, Shiraz 71454, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran
###### Abstract
In this paper we show that the gravitational field equations of
$(n+1)$-dimensional topological black holes with constant horizon curvature,
in cubic and quartic quasi-topological gravity, can be recast in the form of
the first law of thermodynamics, $dE=TdS-PdV$, at the black hole horizon. This
procedure leads to extract an expression for the horizon entropy as well as
the energy (mass) in terms of the horizon radius, which coincide exactly with
those obtained in quasi-topological gravity by solving the field equations and
using the Wald’s method. We also argue that this approach is powerful and can
be extended to all higher order quasi-topological gravity for extracting the
corresponding entropy and energy in terms of horizon radius.
Keywords: quasi-topological; thermodynamics; gravity. reconstruction.
## I Introduction
In recent years, most theoretical physicists as well as cosmologists, have
been convinced that there should be a deep relation between the gravitational
field equations and the laws of thermodynamics. It was pointed out by Jacobson
for the first time in $1995$, that the hyperbolic second order partial
differential Einstein equation has a predisposition to the first law of
thermodynamics Jac . Indeed, Jacobson derived the Einstein field equations of
general relativity, in its tensorial form, by applying the Clausius relation,
$\delta Q=T\delta S$, on the horizon of spacetime, here $\delta S$ is the
change in the entropy and $\delta Q$ and $T$ are the energy flux across the
horizon and the Unruh temperature seen by an accelerating observer just inside
the horizon. Also, by applying the Clauius relation to the apparent horizon of
the Friedmann-Robertson-Walker universe, the corresponding Friedmann equations
can be derived in Einstein, Gauss-Bonnet and more general Lovelock gravity
CaiKim . Following these investigations, a lot of attempts have been done to
reveal the connection between thermodynamics and gravity in different setups
Par ; Pad ; Cai0 ; Cai1 ; Cai2 ; Cai3 ; Cai4 ; Shey1 ; Shey2 ; ShHL . For
example, in Ref. CaiHL the relationship between the first law of
thermodynamics and the gravitational field equation of a static, spherically
symmetric black hole in Horava-Lifshitz gravity has been explored. It was
shown that, the gravitational field equations of static, spherically symmetric
black holes in Horava-Lifshitz theory can be expressed as the first law of
thermodynamics on the event horizon CaiHL . This approach can lead to extract
expressions for the entropy and mass of Horava-Lifshitz black holes which are
consistent with those obtained from other approaches CaiHL . These results
further support the idea that gravitation on a macroscopic scale is a
manifestation of thermodynamics.
It is well known that the natural generalization of the Einstein-Hilbert
action to higher dimensional spacetime, and higher order gravity with second
order equation of motion, is the Lovelock action Lov . However, because of the
topological origin of the Lovelock terms, the second term of the Lovelock
action (the Gauss-Bonnet term) does not have any dynamical effect in four
dimensions. Similarly, the cubic interaction only contributes to the equations
of motion when the bulk dimension is seven or greater. Recently, a
modification of third order Lovelock gravity was proposed by Olive1 ; Myer1
which contains cubic terms of Riemann tensor and contribute to the equation of
motions in five dimensions. This new theory, which is called “quasi-
topological” gravity, was also extended to include the quartic terms of
Riemann tensor MHD2 . Quasi-topological gravity provides a useful toy model to
study a broader class of four (and higher) dimensional CFT’s, involving three
or more independent parameters Myer2 . Black hole spacetimes in higher order
quasi-topological gravity which have at most second order derivatives of the
metric in the field equations, have been explored and their thermodynamics
have been investigated MHD1 ; MHD3 ; MHD4 ; QT .
In this paper we turn to investigate the connection between the gravitational
field equations and the first law of thermodynamics in quasi-topological
gravity. For a static topological black hole spacetime with constant horizon
curvature, we will show that the gravitational field equations can be
transformed to the first law of thermodynamics, $TdS-dE=PdV$, on the black
hole horizon. This allows us to extract the entropy expression in terms of the
horizon radius, which is useful in studying thermodynamics of these kind of
black holes.
The structure of this paper is as follow. In the next section, we show that
the gravitational field equations in Einstein and Gauss-Bonnet gravity can be
recast as the first law of thermodynamics on the black hole horizon. In
section III, we briefly review the action of the quasi-topological gravity. In
section IV, we apply the method to cubic quasi-topological theory. In section
V, we will generalize our approach to the quartic and higher order quasi-
topological gravity and extract the entropy and energy expressions of these
theories. We finish our paper with conclusions in section VI.
## II Horizon thermodynamics in Einstein and Gauss-Bonnet gravity
Let us start with Einstein-Hilbert and Gauss-Bonnet cases to set the stage and
to see how the method works Par . One can derive the equations of motion for
gravitational theory by either, varying the action of the theory with respect
to $g_{\mu\nu}$, without specifying the form of the spacetime metric, or by
specifying the spacetime metric, then inserting the metric in the action, and
finally varying the resulting action with respect to the metric functions. In
both cases one arrives at the field equations. To see how the two approaches
leads to the same result, in this section, we use both the field equations as
well as variation method, for transforming the equations of motion to the
first law of thermodynamics at the black hole horizon.
### II.1 The Einstein-Hilbert Case
The Einstein field equations is ($c=1$)
$R_{ab}-\frac{1}{2}Rg_{ab}=8\pi GT_{ab}.$ (1)
We consider a four-dimensional static, spherically symmetric spacetime with a
horizon, which is described by the metric
$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}.$ (2)
Inserting metric (2) in Eq. (1), the $(rr)$ component of the Einstein
equations can be written
$\frac{1}{r^{2}}\left[rf^{\prime}(r)-1+f(r)\right]=8\pi GP,$ (3)
where $P=T^{r}_{r}$ is the radial pressure of matter at the horizon CaiHL .
Here prime denotes derivative with respect to $r$. We assume that the
spacetime has a horizon at $r=r_{+}$ which is the simple root of $f(r_{+})=0$.
We also propose at $r=r_{+}$ the surface gravity $\kappa=f^{\prime}(r_{+})/2$
has non zero value which implies a finite non zero temperature
$T={f^{\prime}(r_{+})}/{4\pi}$ at the horizon. Evaluating the field equation
(3) at the horizon where $f(r_{+})=0$, we find
$\frac{1}{G}\left(\frac{r_{+}f^{\prime}(r_{+})}{2}-\frac{1}{2}\right)=4\pi
r_{+}^{2}P.$ (4)
Multiplying both sides of Eq. (4) by $dr_{+}$, we arrive at
$\underbrace{\frac{f^{\prime}(r_{+})}{4\pi}}_{\text{T}}\underbrace{\frac{1}{G}d\left(\frac{4\pi
r_{+}^{2}}{4}\right)}_{\text{dS}}\underbrace{-\frac{1}{2}\left(\frac{dr_{+}}{G}\right)}_{\text{-dE}}=\underbrace{Pd\left(\frac{4\pi
r_{+}^{3}}{3}\right)}_{\text{P d V}}.$ (5)
If we invoke the expressions for the entropy and energy (mass) of the black
hole,
$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{A}{4G},$ (6)
$\displaystyle E$ $\displaystyle=$ $\displaystyle\frac{r_{+}}{2G},$ (7)
where $A=4\pi r_{+}^{2}$ is the area of the horizon, we find that Eq. (5) is
nothing but the first law of thermodynamics,
$\displaystyle dE=TdS-PdV.$ (8)
### II.2 The Gauss-Bonnet Gravity
We shall now turn our attention to the more general case, namely the Gauss-
Bonnet gravity. In an effective action approach to the string theory, the
Gauss-Bonnet term corresponds to the leading order quantum corrections to
gravity, and its presence guarantees a ghost-free actionzwiebach . This theory
contains a special combination of curvature-squared term, added to the
Einstein-Hilbert action. The Gauss-Bonnet term does not have any dynamical
effect in four dimensions since it is just a topological term in four
dimensions. Static black hole solutions of Gauss-Bonnet gravity have been
found and their thermodynamics have been investigated in ample details Caigb .
The action of the Einstein- Hilbert in the presence of the Gauss-Bonnet
correction term, in $(n+1)$-dimensions, is given by
$I=\frac{1}{16\pi
G_{n+1}}\int{d^{n+1}x\sqrt{-g}\left(-2\Lambda+R+\alpha\mathcal{L}_{GB}\right)}+\int
d^{n+1}x\mathcal{L}_{M},$ (9)
where $\alpha$ is the Gauss-Bonnet coefficient with dimension
$($length$)^{2}$, and $\mathcal{L}_{GB}$ is the Gauss-Bonnet Lagrangian which
has the form,
$\mathcal{L}_{GB}=R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^{2}.$ (10)
The field equations can be obtained by varying the above action with respect
to the metric $g_{ab}$. We find
$G_{ab}+\Lambda g_{ab}+2\alpha H_{ab}=8\pi G_{n+1}T_{ab},$ (11)
where,
$\displaystyle G_{ab}=R_{ab}-\frac{1}{2}Rg_{ab},$ (12) $\displaystyle
H_{ab}=RR_{ab}-2R_{a}{}^{c}R_{bc}-2R^{cd}R_{acbd}+R_{a}{}^{cde}R_{bcde}-\textstyle{1\over
4}g_{ab}{\cal L}_{GB},$ (13)
are the Einstein and the second-order Lovelock tensor, respectively. Consider
again a static spherically symmetric solution of the form
$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{n-1}^{2}.$ (14)
Substituting metric (14) in Eq. (11), the $(rr)$ component of the field
equations reduces to
$\frac{1}{2}\frac{(n-1)}{r^{2}}\Big{\\{}rf^{\prime}-\left(n-2\right)\left(1-f\right)+\frac{\tilde{\alpha}}{r^{2}}\left(1-f\right)\left[2rf^{\prime}-\left(n-4\right)\left(1-f\right)\right]-\frac{nr^{2}}{l^{2}}\Big{\\}}=8\pi
G_{n+1}P.$ (15)
where again $P=T^{r}_{r}$ is the radial pressure of matter at the horizon, and
we have defined $\tilde{\alpha}=(n-2)(n-3)\alpha$.
On the other hand, by substituting the metric (14) in action (10), and varying
the action (10) with respect to $g^{rr}$, after multiplying both sides in
$(-g)^{-1/2}g^{rr}$, we get
$\frac{(n-1)}{16\pi
G_{n+1}}\frac{1}{r^{2}}\Bigg{\\{}\left(r+\frac{2\tilde{\alpha}}{r}-\frac{2\tilde{\alpha}f}{r}\right)f^{\prime}-\left(n-2\right)\left(1-f\right)-\frac{\tilde{\alpha}}{r^{2}}\left(n-4\right)(1-f)^{2}-\frac{nr^{2}}{l^{2}}\Bigg{\\}}=P.$
(16)
where $P$ is defined as,
$P=T_{r}^{r}=g^{rr}T_{rr}=g^{rr}\left\\{\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{L}_{M}}{\delta
g^{rr}}\right\\}.$ (17)
As one can see, the resulting equation derived by variational principle in
(16) is precisely the same as one obtained in (15) directly from the field
equations (11). As we mentioned already, we have two approaches for deriving
the components of the field equations; varying the action with respect to
$g_{\mu\nu}$, and then inserting metric in the field equations, or by
substituting the metric in the action, and varying the resulting action with
respect to the metric functions.
The variational method leads to (16), can also be applied to the more general
metric
$ds^{2}=-N^{2}(r)f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Sigma_{k,n-1}^{2},$
(18)
where $d\Sigma_{k,n-1}^{2}$ represents the line elements of an
$(n-1)$-dimensional constant curvature hypersurface with unit radius. Without
loss of the generality, one may take $k=1,-1,$ and $0$, corresponding to
spherical, hyperbolic and planar hypersurface. Inserting metric (18) in action
(10), after variation with respect to $g^{rr}$, we arrive at
$\displaystyle\frac{(n-1)}{16\pi
G_{n+1}}\frac{1}{Nr^{3}}\Bigg{\\{}\left(r^{2}+2k\tilde{\alpha}-2\tilde{\alpha}f\right)\left(N^{2}f\right)^{\prime}+N^{2}\left[r\left(n-2\right)\left(f-k\right)-\frac{\tilde{\alpha}}{r}\left(n-4\right)(k^{2}-2kf+f^{2})-\frac{nr^{3}}{l^{2}}\right]\Bigg{\\}}$
$\displaystyle=T_{r}^{r}=P,$ (19)
where again the radial pressure is given by (17). We also assume the function
$f(r)$ has a simple zero at $r=r_{+}$ with $f(r=r_{+})=0$ and non-vanishing
surface gravity $\kappa=Nf^{\prime}(r_{+})/2$. The temperature associated with
the horizon is now defined as $T={\kappa}/{2\pi}={Nf^{\prime}(r_{+})}/{4\pi}$.
Evaluating Eq. (II.2) at $r=r_{+}$, we obtain
$\frac{N(n-1)}{16\pi
G_{n+1}r_{+}^{2}}\left[f^{\prime}(r_{+})\left(r_{+}+\tilde{\alpha}\frac{2k}{r_{+}}\right)-k\left(n-2\right)-\frac{\tilde{\alpha}}{r_{+}^{2}}k^{2}\left(n-4\right)-\frac{nr_{+}^{2}}{l^{2}}\right]=P.$
(20)
Consider two equilibrium states of the system with an infinitesimal different
in the extensive variables entropy, energy, and volume $dS$, $dE$ and $dV$,
respectively, while the values of the intensive quantities of the system are
the temperature $T$ and pressure $P$. Our aim is to introduce in Eq. (20) a
factor $dV$ and see whether we can rewrite it in the form $TdS-dE=PdV$.
Multiplying both sides of Eq. (20) by the factor
$\Sigma_{k}r_{+}^{n-3}dr_{+}$, where $\Sigma_{k}$ is the area of a unit
$(n-1)$-dimensional constant hypersurface with volume
$V=\Sigma_{k}r_{+}^{n}/n$, we arrive at
$\frac{\kappa}{2\pi}d\left[\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\left(1+\frac{(n-1)}{(n-3)}\frac{2k\tilde{\alpha}}{r_{+}^{2}}\right)\right]-d\left[\frac{(n-1)\Sigma_{k}r_{+}^{n-2}}{16\pi
G_{n+1}}\left(k+\frac{\tilde{\alpha}k^{2}}{r_{+}^{2}}+\frac{r_{+}^{2}}{l^{2}}\right)\right]=P\Sigma_{k}r_{+}^{n-1}dr_{+}=PdV.$
(21)
The first term in the left hand side is in the form $TdS$ and our analysis
allows us to read off the expression of entropy $S$ for the horizon as,
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\left(1+\frac{(n-1)}{(n-3)}\frac{2\tilde{\alpha}k}{r_{+}^{2}}\right).$
(22)
In addition, the second term in (21) can be interpreted as $dE$, where the
energy of the system is given by
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{(n-1)\Sigma_{k}r_{+}^{n-2}}{16\pi
G_{n+1}}\left(k+\frac{\tilde{\alpha}k^{2}}{r_{+}^{2}}+\frac{r_{+}^{2}}{l^{2}}\right).$
(23)
Thus, we have transformed the field equations in Guass-Bonnet gravity to the
first law of thermodynamics, $TdS-dE=PdV$, on the black hole horizon. The
obtained expressions for the entropy and energy in (22) and (23), coincide
with the expressions of entropy and energy for Gauss-Bonnet black holes in AdS
spaces derived by solving the field equations Caigb .
In our analysis, we have supposed that both $l^{2}=-n(n-1)/{\Lambda}$ and
$\tilde{\alpha}$ are fixed. Recently, there were a lot of interest in denoting
$\Lambda$ as a thermodynamical variable proportional to the pressure PVcrit .
In this case the last term in the left-hand side of Eq. (21), $n(n-1)/(16\pi
G_{n+1}l^{2})dV$, may be moved to the right-hand side as $P_{\Lambda}dV$. One
should note that the authors of Ref. PVcrit have considered the solutions of
Einstein field equations, and therefore only the term $P_{\Lambda}dV$ will be
appeared in the right-hand side of Eq. (21) for the solutions of Einstein
equation. Also, one may note that $P_{\Lambda}$ is the $T^{r}_{r}$-component
of energy momentum tensor of cosmological constant term, if one denotes the
$\Lambda$-term as energy term in the right-hand side of Einstein equation.
Also, if one denotes $\tilde{\alpha}$ as a thermodynamical variable, then a
term $\tilde{\alpha}dA$ will appear in the right hand side of Eq. (21), where
$A$ is the conjugate quantity to the Gauss-Bonnet coefficient CaiPV . In this
case, since the Gauss-Bonnet coefficient has not the dimensions of pressure in
geometric units, its conjugate quantity has not the dimensions of volume.
Here, before going to the case of quasi-topological gravity, we pause to give
a comment on the expressions for energy density $\rho$ and pressure $P$. In
general, these quantities are not the same for a general perfect fluid. Indeed
$P$ is given in Eq. (II.2), while $\rho$ can be calculated by varying the
action (10) with respect to $g_{tt}$ and multiplying both sides in
$(-g)^{-1/2}g_{tt}$ as
$\frac{(n-1)}{16\pi
G_{n+1}}\frac{N}{r^{3}}\Bigg{\\{}\left(r^{2}+2k\tilde{\alpha}-2\tilde{\alpha}f\right)f^{\prime}+r\left(n-2\right)\left(f-k\right)-\frac{\tilde{\alpha}}{r}\left(n-4\right)\left[k^{2}-2kf+f^{2}\right]-\frac{nr^{3}}{l^{2}}\Bigg{\\}}=-T_{t}^{t}=\rho.$
(24)
Of course, one may note that the pressure and energy density are the same on
the horizon as one may see by calculating $\rho$ on the horizon.
## III Quasi-Topological Gravity
The action of the quasi-topological theory in $(n+1)$-dimensions is given by
$I=\int d^{n+1}x\left(\mathcal{L}_{G}+\mathcal{L}_{M}\right),$ (25)
where $\mathcal{L}_{M}$ is the Lagrangian of the matter and
$\mathcal{L}_{G}=\frac{\sqrt{-g}}{16\pi
G_{n+1}}\left(-2\Lambda+{\mu}_{1}\mathcal{L}_{1}+{\mu}_{2}\mathcal{L}_{2}+{\mu}_{3}\mathcal{X}_{3}+{\mu}_{4}\mathcal{X}_{4}+...\right).$
(26)
In the above equation $\Lambda=-(n-2)(n-3)/2l^{2}$ is the cosmological
constant, $\mathcal{L}_{1}=R$ is the Einstein-Hilbert Lagrangian,
$\mathcal{L}_{2}=R_{abcd}{R}^{abcd}-4{R}_{ab}{R}^{ab}+{R}^{2}$ is the second
order Lovelock (Gauss-Bonnet) Lagrangian, $\mathcal{X}_{3}$ is the curvature-
cubed Lagrangian given by Myer1
$\displaystyle\mathcal{X}_{3}$ $\displaystyle=$ $\displaystyle
R_{ab}^{cd}R_{cd}^{\,\,e\,\,\,f}R_{e\,\,f}^{\,\,a\,\,\,b}+\frac{1}{(2n-1)(n-3)}\left(\frac{3(3n-5)}{8}R_{abcd}R^{abcd}R\right.$
(27)
$\displaystyle-3(n-1)R_{abcd}R^{abc}{}_{e}R^{de}+3(n+1)R_{abcd}R^{ac}R^{bd}$
$\displaystyle\left.+\,6(n-1)R_{a}{}^{b}R_{b}{}^{c}R_{c}{}^{a}-\frac{3(3n-1)}{2}R_{a}^{\,\,b}R_{b}^{\,\,a}R+\frac{3(n+1)}{8}R^{3}\right),$
and $\mathcal{X}_{4}$ is the fourth order term of quasi-topological gravity
MHD2
$\displaystyle\mathcal{X}_{4}$ $\displaystyle=$ $\displaystyle
c_{1}R_{abcd}R^{cdef}R_{\phantom{hg}{ef}}^{hg}R_{hg}{}^{ab}+c_{2}R_{abcd}R^{abcd}R_{ef}R^{ef}+c_{3}RR_{ab}R^{ac}R_{c}{}^{b}+c_{4}(R_{abcd}R^{abcd})^{2}$
(28)
$\displaystyle+c_{5}R_{ab}R^{ac}R_{cd}R^{db}+c_{6}RR_{abcd}R^{ac}R^{db}+c_{7}R_{abcd}R^{ac}R^{be}R_{\phantom{d}{e}}^{d}+c_{8}R_{abcd}R^{acef}R_{\phantom{b}{e}}^{b}R_{\phantom{d}{f}}^{d}$
$\displaystyle+c_{9}R_{abcd}R^{ac}R_{ef}R^{bedf}+c_{10}R^{4}+c_{11}R^{2}R_{abcd}R^{abcd}+c_{12}R^{2}R_{ab}R^{ab}$
$\displaystyle+c_{13}R_{abcd}R^{abef}R_{ef}{}_{g}^{c}R^{dg}+c_{14}R_{abcd}R^{aecf}R_{gehf}R^{gbhd},$
where the coefficients $c_{i}$ are given by MHD2
$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle-\left(n-1\right)\left({n}^{7}-3\,{n}^{6}-29\,{n}^{5}+170\,{n}^{4}-349\,{n}^{3}+348\,{n}^{2}-180\,n+36\right),$
$\displaystyle c_{2}$ $\displaystyle=$
$\displaystyle-4\,\left(n-3\right)\left(2\,{n}^{6}-20\,{n}^{5}+65\,{n}^{4}-81\,{n}^{3}+13\,{n}^{2}+45\,n-18\right),$
$\displaystyle c_{3}$ $\displaystyle=$
$\displaystyle-64\,\left(n-1\right)\left(3\,{n}^{2}-8\,n+3\right)\left({n}^{2}-3\,n+3\right),$
$\displaystyle c_{4}$ $\displaystyle=$
$\displaystyle-{(n}^{8}-6\,{n}^{7}+12\,{n}^{6}-22\,{n}^{5}+114\,{n}^{4}-345\,{n}^{3}+468\,{n}^{2}-270\,n+54),$
$\displaystyle c_{5}$ $\displaystyle=$ $\displaystyle
16\,\left(n-1\right)\left(10\,{n}^{4}-51\,{n}^{3}+93\,{n}^{2}-72\,n+18\right),$
$\displaystyle c_{6}$ $\displaystyle=$ $\displaystyle--
32\,\left(n-1\right)^{2}\left(n-3\right)^{2}\left(3\,{n}^{2}-8\,n+3\right),$
$\displaystyle c_{7}$ $\displaystyle=$ $\displaystyle
64\,\left(n-2\right)\left(n-1\right)^{2}\left(4\,{n}^{3}-18\,{n}^{2}+27\,n-9\right),$
$\displaystyle c_{8}$ $\displaystyle=$
$\displaystyle-96\,\left(n-1\right)\left(n-2\right)\left(2\,{n}^{4}-7\,{n}^{3}+4\,{n}^{2}+6\,n-3\right),$
$\displaystyle c_{9}$ $\displaystyle=$ $\displaystyle
16\left(n-1\right)^{3}\left(2\,{n}^{4}-26\,{n}^{3}+93\,{n}^{2}-117\,n+36\right),$
$\displaystyle c_{10}$ $\displaystyle=$
$\displaystyle{n}^{5}-31\,{n}^{4}+168\,{n}^{3}-360\,{n}^{2}+330\,n-90,$
$\displaystyle c_{11}$ $\displaystyle=$ $\displaystyle
2\,(6\,{n}^{6}-67\,{n}^{5}+311\,{n}^{4}-742\,{n}^{3}+936\,{n}^{2}-576\,n+126),$
$\displaystyle c_{12}$ $\displaystyle=$ $\displaystyle
8\,{(}7\,{n}^{5}-47\,{n}^{4}+121\,{n}^{3}-141\,{n}^{2}+63\,n-9),$
$\displaystyle c_{13}$ $\displaystyle=$ $\displaystyle
16\,n\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(3\,{n}^{2}-8\,n+3\right),$
$\displaystyle c_{14}$ $\displaystyle=$ $\displaystyle
8\,\left(n-1\right)\left({n}^{7}-4\,{n}^{6}-15\,{n}^{5}+122\,{n}^{4}-287\,{n}^{3}+297\,{n}^{2}-126\,n+18\right).$
For spherically symmetric metric, the action (25) yields second-order
equations of motion in $(n+1)$-dimensional spacetimes except in $n=2m-1$,
where $m$ is the order of quasi-topological theory MHD2 . In the remaining
part of this paper, we will show that the gravitational field equations
describing by the spacetime metric (18) can be recast in the form of the first
law of thermodynamics at the black hole horizon.
## IV Horizon Thermodynamics in cubic quasi-topological gravity
In section II, we showed that the field equations of static black hole
spacetimes, with constant horizon curvature, in Einstein and Gauss-Bonnet
gravities can be reexpressed as the first law of thermodynamics, $dE=TdS-PdV$,
on the horizon. Here, we want to see whether the above procedure works or not
in other gravity theories such as quasi-topological gravity. As we discussed,
and explicitly showed, for the Gauss-Bonnet case, one can use both the field
equations as well as the variational principle, instead of using the field
equations, and transform the equations of motion to the first law at the
spacetime horizon. However, since the field equations for cubic Myer1 and
quartic DV quasi-topological gravity are very long, in this section and also
the next one, we present the resulting equations from variational principle
for economic reason, which are clearly the components of the field equations.
We shall now continue the previous procedure for the cubic term of quasi-
topological gravity. The total action of the cubic quasi-topological gravity
in $(n+1)$ dimensions can be written as
$I=\frac{1}{16\pi G_{n+1}}\int
d^{n+1}x\sqrt{-g}[-2\Lambda+{\mu}_{1}\mathcal{L}_{1}+{\mu}_{2}\mathcal{L}_{2}+{\mu}_{3}\mathcal{X}_{3}]+\int
d^{n+1}x\mathcal{L}_{M}.$ (29)
Varying the action (29) with respect to $g^{rr}$ and multiplying both sides in
$(-g)^{-1/2}g^{rr}$, one obtains
$\displaystyle\frac{(n-1)}{16\pi
G_{n+1}}\frac{1}{Nr^{5}}\Bigg{\\{}\frac{d}{dr}\left(N^{2}f\right)\left(r^{4}+2kr^{2}\hat{\mu}_{2}l^{2}-2r^{2}\hat{\mu}_{2}l^{2}f+3k^{2}\hat{\mu}_{3}l^{4}-6k\hat{\mu}_{3}l^{4}f+3k\hat{\mu}_{3}l^{4}f^{2}\right)$
$\displaystyle+N^{2}\left[r^{3}\left(n-2\right)\left(f-k\right)-\hat{\mu}_{2}l^{2}r\left(n-4\right)\left(k^{2}-2kf+f^{2}\right)-\frac{\hat{\mu}_{3}}{r}l^{4}\left(n-6\right)\left(-k+3k^{2}f-3kf^{2}+f^{3}\right)-\frac{nr^{5}}{l^{2}}\right]\Bigg{\\}}=T^{r}_{r},$
where $T^{r}_{r}$ is given by (17), and
$\hat{\mu}_{1}=1,\text{ \ \
}\hat{\mu}_{2}=\frac{(n-2)(n-3)}{l^{2}}\mu_{2},\text{ \ \
}\hat{\mu}_{3}=\frac{(n-2)(n-5)(3n^{2}-9n+4)}{8(2n-1)l^{4}}\mu_{3}.$
Next, we evaluate Eq. (IV) at $r=r_{+}$ and using the fact that $f(r_{+})=0$,
to obtain
$\displaystyle\frac{N(n-1)}{16\pi
G_{n+1}}\Bigg{\\{}f^{\prime}(r_{+})\left(r_{+}+\frac{2k}{r_{+}}\hat{\mu}_{2}l^{2}+\frac{3k^{2}}{r_{+}^{3}}\hat{\mu}_{3}l^{4}\right)-k\left(n-2\right)$
$\displaystyle-\frac{\hat{\mu}_{2}l^{2}}{r_{+}^{2}}k^{2}\left(n-4\right)-\frac{\hat{\mu}_{3}l^{4}k}{r_{+}^{4}}\left(n-6\right)-\frac{nr_{+}^{2}}{l^{2}}\Bigg{\\}}=r_{+}^{2}P.$
(31)
Multiplying both sides of the above equation by the factor
$\Sigma_{k}r_{+}^{n-3}dr_{+}$, and setting $f^{\prime}(r_{+})=2\kappa$, we
have
$\displaystyle\frac{\kappa}{2\pi}d\left[\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\left(1+2k\hat{\mu}_{2}\frac{(n-1)}{(n-3)}\frac{l^{2}}{r_{+}^{2}}+3k^{2}\hat{\mu}_{3}\frac{(n-1)}{(n-5)}\frac{l^{4}}{r_{+}^{4}}\right)\right]$
$\displaystyle-d\left[\frac{(n-1)\Sigma_{k}r_{+}^{n-2}}{16\pi
G_{n+1}}\left(k+\frac{k^{2}\hat{\mu}_{2}l^{2}}{r_{+}^{2}}+\frac{k^{3}\hat{\mu}_{3}l^{4}}{r_{+}^{4}}+\frac{r_{+}^{2}}{l^{2}}\right)\right]=P\Sigma_{k}r{+}^{n-1}dr_{+}.$
(32)
We can rewrite this equation in the form,
$\displaystyle
Td\left[\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\left(1+2k\hat{\mu}_{2}\frac{(n-1)}{(n-3)}\frac{l^{2}}{r_{+}^{2}}+3k^{2}\hat{\mu}_{3}\frac{(n-1)}{(n-5)}\frac{l^{4}}{r_{+}^{4}}\right)\right]$
$\displaystyle-d\left[\frac{(n-1)\Sigma_{k}r_{+}^{n}}{16\pi
G_{n+1}l^{2}}\left(1+k\frac{l^{2}}{r_{+}^{2}}+k^{2}\hat{\mu}_{2}\frac{l^{4}}{r_{+}^{4}}+k^{3}\hat{\mu}_{3}\frac{l^{6}}{r_{+}^{6}}\right)\right]=PdV.$
(33)
The first term in the left hand side of the above equation is in the form
$TdS$, and so one may recognize the entropy expression for the horizon in
cubic quasi-topological gravity as,
$S=\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\left(1+2k\hat{\mu}_{2}\frac{(n-1)}{(n-3)}\frac{l^{2}}{r_{+}^{2}}+3k^{2}\hat{\mu}_{3}\frac{(n-1)}{(n-5)}\frac{l^{4}}{r_{+}^{4}}\right).$
(34)
According to the first low of thermodynamics, we can interpret the second term
in the left hand side of (IV) as the energy of the system,
$E=\frac{(n-1)\Sigma_{k}r_{+}^{n}}{16\pi
G_{n+1}l^{2}}\left(1+k\frac{l^{2}}{r_{+}^{2}}+k^{2}\hat{\mu}_{2}\frac{l^{4}}{r_{+}^{4}}+k^{3}\hat{\mu}_{3}\frac{l^{6}}{r_{+}^{6}}\right).$
(35)
These expressions for entropy and energy of the black holes coincide exactly
with those obtained in quasi-topological gravity by solving the field
equations and using the Wald’s method Myer1 . Here we arrived at the same
result by transforming the field equations to the form of the first law on the
black hole horizon. This indicates that the approach presented here is enough
powerful and further reveals the deep connection between the gravitational
field equations and the first law of thermodynamics on the horizon of the
black hole. Again, one can show that although $P$ and $\rho$ are different in
general but they are the same at the horizon.
## V Horizon Thermodynamics of Quasi-topological Gravity
In this section, we would like to extend the above study to the case of $m$-th
order quasi-topological gravity. First, we consider the quartic quasi-
topological gravity. Varying the action (25) with respect to $g^{rr}$ and
multiplying both sides in $(-g)^{-1/2}g^{rr}$, we obtain
$\displaystyle\frac{(n-1)}{16\pi
G_{n+1}}\frac{1}{Nr^{7}}\Bigg{\\{}\frac{d}{dr}\left(N^{2}f\right)\left(r^{6}+2kr^{4}\hat{\mu}_{2}l^{2}-2r^{4}\hat{\mu}_{2}l^{2}f+3k^{2}r^{2}\hat{\mu}_{3}l^{4}-6kr^{2}\hat{\mu}_{3}l^{4}f+3r^{2}\hat{\mu}_{3}l^{4}f^{2}\right)$
$\displaystyle+N^{2}[f^{\prime}\left(4k\hat{\mu}_{4}l^{6}-12k^{2}\hat{\mu}_{4}l^{6}f+12k\hat{\mu}_{4}l^{6}f^{2}-4\hat{\mu}_{4}l^{6}f^{3}\right)+r^{5}\left(n-2\right)\left(f-k\right)+\hat{\mu}_{2}l^{2}r^{3}\left(n-4\right)\left(2kf-k^{2}-f^{2}\right)$
$\displaystyle-\hat{\mu}_{3}l^{4}r\left(n-6\right)\left(k-3k^{2}f+3kf^{2}-f^{3}\right)+\frac{\hat{\mu}_{4}}{r}l^{6}\left(n-8\right)\left(-k^{2}+4kf-6k^{2}f^{2}+4kf^{3}-f^{4}\right)-\frac{nr^{7}}{l^{2}}]\Bigg{\\}}=T_{r}^{r},$
(36)
where
$\hat{\mu}_{4}=\frac{n(n-1)(n-2)^{2}(n-3)(n-7)(n^{5}-15n^{4}+72n^{3}-156n^{2}+150n-42)}{l^{6}}\mu_{4}.$
Evaluating Eq. (V) at $r=r_{+}$ and setting $f(r_{+})=0$, we arrive at
$\displaystyle\frac{N(n-1)}{16\pi
G_{n+1}}\Bigg{\\{}f^{\prime}\left(r_{+}+\frac{2k}{r_{+}}\hat{\mu}_{2}l^{2}+\frac{3k^{2}}{r_{+}^{3}}\hat{\mu}_{3}l^{4}+\frac{4k}{r_{+}^{5}}\hat{\mu}_{4}l^{6}\right)-k\left(n-2\right)$
$\displaystyle-\hat{\mu}_{2}\frac{k^{2}l^{2}}{r_{+}^{2}}\left(n-4\right)-\hat{\mu}_{3}\frac{kl^{4}}{r_{+}^{4}}\left(n-6\right)-\frac{\hat{\mu}_{4}l^{6}k^{2}}{r_{+}^{6}}\left(n-8\right)-\frac{nr_{+}^{2}}{l^{2}}\Bigg{\\}}=Pr_{+}^{2},$
(37)
Multiplying both sides of the above equation by the factor
$\Sigma_{k}r_{+}^{n-3}dr_{+}$ and setting $f^{\prime}(r_{+})=2\kappa$, we get
$\displaystyle\frac{\kappa}{2\pi}d\left[\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\left(1+2k\hat{\mu}_{2}\frac{(n-1)}{(n-3)}\frac{l^{2}}{r_{+}^{2}}+3k^{2}\hat{\mu}_{3}\frac{(n-1)}{(n-5)}\frac{l^{4}}{r_{+}^{4}}+4k^{3}\hat{\mu}_{4}\frac{(n-1)}{(n-7)}\frac{l^{6}}{r_{+}^{6}}\right)\right]$
$\displaystyle-d\left[\frac{(n-1)\Sigma_{k}r_{+}^{n-2}}{16\pi
G_{n+1}}\left(k+\frac{\hat{\mu}_{2}l^{2}k^{2}}{r_{+}^{2}}+\frac{\hat{\mu}_{3}l^{4}k^{3}}{r_{+}^{4}}+\frac{\hat{\mu}_{4}l^{6}k^{4}}{r_{+}^{6}}+\frac{r_{+}^{2}}{l^{2}}\right)\right]=P\Sigma_{k}r{+}^{n-1}dr_{+}$
(38)
Using the definition $T=\kappa/2\pi$, the above equation can be rewritten in
the form
$\displaystyle
Td\left[\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\left(1+2k\hat{\mu}_{2}\frac{(n-1)}{(n-3)}\frac{l^{2}}{r_{+}^{2}}+3k^{2}\hat{\mu}_{3}\frac{(n-1)}{(n-5)}\frac{l^{4}}{r_{+}^{4}}+4k^{3}\hat{\mu}_{4}\frac{(n-1)}{(n-7)}\frac{l^{6}}{r_{+}^{6}}\right)\right]$
$\displaystyle-d\left[\frac{(n-1)\Sigma_{k}r_{+}^{n}}{16\pi
G_{n+1}l^{2}}\left(1+k\frac{l^{2}}{r_{+}^{2}}+\hat{\mu}_{2}k^{2}\frac{l^{4}}{r_{+}^{4}}+\hat{\mu}_{3}k^{3}\frac{l^{6}}{r_{+}^{6}}+\hat{\mu}_{4}k^{4}\frac{l^{8}}{r_{+}^{8}}\right)\right]=PdV.$
(39)
Equation (V) is nothing, but the first law of thermodynamics on the horizon,
$TdS-dE=PdV$. We can define the entropy expression as
$S=\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\left(1+2k\hat{\mu}_{2}\frac{(n-1)}{(n-3)}\frac{l^{2}}{r_{+}^{2}}+3k^{2}\hat{\mu}_{3}\frac{(n-1)}{(n-5)}\frac{l^{4}}{r_{+}^{4}}+4k^{3}\hat{\mu}_{4}\frac{(n-1)}{(n-7)}\frac{l^{6}}{r_{+}^{6}}\right),$
(40)
and the total energy (mass) of the black hole as
$E=\frac{(n-1)\Sigma_{k}r_{+}^{n}}{16\pi
G_{n+1}l^{2}}\left(1+k\frac{l^{2}}{r_{+}^{2}}+\hat{\mu}_{2}k^{2}\frac{l^{4}}{r_{+}^{4}}+\hat{\mu}_{3}k^{3}\frac{l^{6}}{r_{+}^{6}}+\hat{\mu}_{4}k^{4}\frac{l^{8}}{r_{+}^{8}}\right).$
(41)
The obtained expressions for entropy and energy are precisely the expressions
calculated by other authors for topological black holes in quartic quasi-
topological gravity MHD2 . In this way we show that the field equations of
quartic quasi-topological gravity can be transformed to the form of the first
law of thermodynamics on the event horizon of $(n+1)$-dimensional topological
black holes.
Having the results for the cubic and quartic cases at hand, one may conjecture
that there exists similar connection for $m$-th order quasi-topological
gravity in $n\neq 2m-1$ dimensions. For the topological black holes with
metric (18), one may conjecture that the gravity part of the action (25) is
given by MHD2
$I_{G}=\int
dtdrN(r)\left(r^{n}\sum_{i=0}^{m}\hat{\mu}_{i}\left[l^{2}r^{-2}(k-f)\right]^{i}\right)^{\prime},$
(42)
where $\hat{\mu}_{i}$ are coefficients of the $i$-th powered curvature term
with $\hat{\mu}_{1}=1$. The corresponding field equations evaluated on the
horizon may be rewritten as
$Td\left[\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\sum_{i=1}^{m}i\frac{(n-1)}{(n+1-2i)}\frac{\hat{\mu}_{i}k^{i-1}l^{2i-2}}{r_{+}^{2i-2}}\right]-d\left[\frac{(n-1)\Sigma_{k}r_{+}^{n}}{16\pi
G_{n+1}l^{2}}\left(1+\sum_{i=1}^{m}\hat{\mu}_{i}k^{i}\frac{l^{2i}}{r_{+}^{2i}}\right)\right]=PdV.$
(43)
Equation (43) is the first law of thermodynamics provided we define
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{k}r_{+}^{n-1}}{4G_{n+1}}\sum_{i=1}^{m}i\frac{(n-1)}{(n+1-2i)}\frac{\hat{\mu}_{i}k^{i-1}l^{2i-2}}{r_{+}^{2i-2}},$
(44) $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{(n-1)\Sigma_{k}r_{+}^{n}}{16\pi
G_{n+1}l^{2}}\left(1+\sum_{i=1}^{m}\hat{\mu}_{i}k^{i}\frac{l^{2i}}{r_{+}^{2i}}\right).$
(45)
These are the most general expressions for entropy and energy of static black
hole spacetimes with spherical, hyperbolic or planar horizon topology in the
most general quasi-topological theory of gravity. We expect to confirm our
general results (44) and (44) in the future by solving explicitly the field
equations. Again, one can show that although $P$ and $\rho$ are different in
general but they are the same at the horizon.
## VI CONCLUSIONS
According to the black hole thermodynamics, a black hole can be regarded as a
thermodynamic system which has entropy and temperature associated with its
horizon. Since the discovery of black hole thermodynamics in $1970$’s
physicists have been speculating that there should be some deep connection
between thermodynamics and gravity. This is due to the fact that thermodynamic
quantities of black holes such as temperature and entropy are closely related
to their geometrical quantities such as surface gravity and horizon area. In
this paper, we have investigated the thermodynamics of topological black
holes, with spherical, hyperbolic or planar horizon topology, in quasi-
topological theory of gravity. We showed that one can always rewrite the field
equations of quasi-topological gravity in the form of the first law of
thermodynamics, $dE=TdS-PdV$, at the black hole horizon. This procedure allows
us to obtain the entropy and the mass expressions in terms of the radius of
black hole horizon, which are exactly the same as those resulting from the
Wald’s method for black hole entropy and the Hamiltonian approach for black
hole mass. The novelty and advantages of the present study is that in the
process of deriving the entropy and the mass of black holes, we have not
solved the field equation of the quasi-topological theory and we had no
difficulties of Wald’s method for calculating entropy. This completely differs
from the previous works in the literature Myer1 ; MHD2 .
The thermodynamic interpretation of the gravitational field equations in the
most general quasi-topological gravity indicates that the connection between
thermodynamics and gravity is not just an accident, but something with deep
physical meaning. On the other hand the disclosed relation on the field
equations and the first law of thermodynamics on the black hole horizon also
sheds the light on holography, since the gravitational field equations
persists the information in the bulk and the first law of thermodynamics on
the event horizon contains the information on the boundary. Our study shows
that the approach here is powerful to find an expression of entropy in term of
the horizon radius. It could help to extract an expression of entropy
associated with the event horizon in quasi-topological gravity, which is
useful in studying the thermodynamical properties of black holes in this
theory.
###### Acknowledgements.
We thank from the Research Council of Shiraz University. This work has been
supported financially by Research Institute for Astronomy & Astrophysics of
Maragha (RIAAM), Iran.
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|
arxiv-papers
| 2014-03-31T16:37:58 |
2024-09-04T02:50:00.527163
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "A. Sheykhi, M. H. Dehghani and R. Dehghani",
"submitter": "Ahmad Sheykhi",
"url": "https://arxiv.org/abs/1404.0260"
}
|
1404.0275
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-050 LHCb-PAPER-2014-008 April, 1, 2014
Evidence for the decay
$\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}$
The LHCb collaboration†††Authors are listed on the following pages.
Evidence for the decay mode
$\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}$ in
${{{\mathrm{B}}^{+}}}\rightarrow\mathrm{X}(3872){{\mathrm{K}}^{+}}$ decays is
found with a significance of 4.4 standard deviations. The analysis is based on
a data sample of proton-proton collisions, corresponding to an integrated
luminosity of 3$\mbox{\,fb}^{-1}$, collected with the LHCb detector, at
centre-of-mass energies of 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$. The ratio
of the branching fraction of the
$\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}$ decay to that of
the
$\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}$ decay is measured to be
$\dfrac{{\cal
B}(\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma})}{{\cal
B}(\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma})}=2.46\pm 0.64\pm 0.29,$
where the first uncertainty is statistical and the second is systematic. The
measured value agrees with expectations for a pure charmonium interpretation
of the $\mathrm{X}(3872)$ state and a mixture of charmonium and molecular
interpretations. However, it does not support a pure
${\mathrm{D}}{{\bar{}\mathrm{D}}^{*}}$ molecular interpretation of the
$\mathrm{X}(3872)$ state.
Submitted to Nucl. Phys. B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L. An3, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30,38,
W. Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,q, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,o, A. Camboni36, P. Campana18,38, D. Campora
Perez38, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,38,j, A. Cardini15, H.
Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20,
L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph.
Charpentier38, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38,
X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47,
J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A.
Comerma-Montells11, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G.
Corti38, M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C.
Craik48, M. Cruz Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J.
Dalseno46, P. David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De
Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De
Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D.
Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto38, H. Dijkstra38, S.
Donleavy52, F. Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A.
Dovbnya43, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A.
Dzyuba30, S. Easo49, U. Egede53, V. Egorychev31, S. Eidelman34, S.
Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5, Ch.
Elsasser40, S. Esen11, T. Evans55, A. Falabella16,f, C. Färber11, C.
Farinelli41, S. Farry52, D. Ferguson50, V. Fernandez Albor37, F. Ferreira
Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M. Fiorini16,f, M.
Firlej27, C. Fitzpatrick38, T. Fiutowski27, M. Fontana10, F. Fontanelli19,j,
R. Forty38, O. Francisco2, M. Frank38, C. Frei38, M. Frosini17,38,g, J.
Fu21,38, E. Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, S. Gallorini22,
S. Gambetta19,j, M. Gandelman2, P. Gandini59, Y. Gao3, J. Garofoli59, J. Garra
Tico47, L. Garrido36, C. Gaspar38, R. Gauld55, L. Gavardi9, E. Gersabeck11, M.
Gersabeck54, T. Gershon48, Ph. Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47,
L. Giubega29, V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38,
A. Gomes1,a, H. Gordon38, C. Gotti20, M. Grabalosa Gándara5, R. Graciani
Diaz36, L.A. Granado Cardoso38, E. Graugés36, G. Graziani17, A. Grecu29, E.
Greening55, S. Gregson47, P. Griffith45, L. Grillo11, O. Grünberg62, B. Gui59,
E. Gushchin33, Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C.
Haen38, S.C. Haines47, S. Hall53, B. Hamilton58, T. Hampson46, X. Han11, S.
Hansmann-Menzemer11, N. Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann62,
J. He38, T. Head38, V. Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A.
Hernando Morata37, E. van Herwijnen38, M. Heß62, A. Hicheur1, D. Hill55, M.
Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55, N. Hussain55, D.
Hutchcroft52, D. Hynds51, M. Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11,
J. Jalocha55, E. Jans41, P. Jaton39, A. Jawahery58, M. Jezabek26, F. Jing3, M.
John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N. Jurik59, M.
Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38, M.
Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C. Khurewathanakul39, S.
Klaver54, O. Kochebina7, M. Kolpin11, I. Komarov39, R.F. Koopman42, P.
Koppenburg41,38, M. Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11,
M. Kreps48, G. Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k,
V. Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, B. Langhans38, T. Latham48, C.
Lazzeroni45, R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A.
Leflat32, J. Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11,
Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G.
Liu38, S. Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40,
H. Lu3, D. Lucchesi22,q, H. Luo50, A. Lupato22, E. Luppi16,f, O. Lupton55, F.
Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O. Maev30, S. Malde55, G.
Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5, J.F. Marchand4, U.
Marconi14, C. Marin Benito36, P. Marino23,s, R. Märki39, J. Marks11, G.
Martellotti25, A. Martens8, A. Martín Sánchez7, M. Martinelli41, D. Martinez
Santos42, F. Martinez Vidal64, D. Martins Tostes2, A. Massafferri1, R.
Matev38, Z. Mathe38, C. Matteuzzi20, A. Mazurov16,f, M. McCann53, J.
McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B. Meadows57,55, F.
Meier9, M. Meissner11, M. Merk41, D.A. Milanes8, M.-N. Minard4, J. Molina
Rodriguez60, S. Monteil5, D. Moran54, M. Morandin22, P. Morawski26, A. Mordà6,
M.J. Morello23,s, J. Moron27, R. Mountain59, F. Muheim50, K. Müller40, R.
Muresan29, B. Muster39, P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva2,
M. Needham50, N. Neri21, S. Neubert38, N. Neufeld38, M. Neuner11, A.D.
Nguyen39, T.D. Nguyen39, C. Nguyen-Mau39,p, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-Mucha27, V.
Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R. Oldeman15,e, G.
Onderwater65, M. Orlandea29, J.M. Otalora Goicochea2, P. Owen53, A.
Oyanguren64, B.K. Pal59, A. Palano13,c, F. Palombo21,t, M. Palutan18, J.
Panman38, A. Papanestis49,38, M. Pappagallo51, C. Parkes54, C.J. Parkinson9,
G. Passaleva17, G.D. Patel52, M. Patel53, C. Patrignani19,j, A. Pazos
Alvarez37, A. Pearce54, A. Pellegrino41, M. Pepe Altarelli38, S.
Perazzini14,d, E. Perez Trigo37, P. Perret5, M. Perrin-Terrin6, L.
Pescatore45, E. Pesen66, K. Petridis53, A. Petrolini19,j, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, F. Polci8, A. Poluektov48,34, I. Polyakov31, E. Polycarpo2,
A. Popov35, D. Popov10, B. Popovici29, C. Potterat2, A. Powell55, J.
Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39,
G. Punzi23,r, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S. Ricciardi49, A.
Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5, P. Robbe7, A.B.
Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V.
Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F.
Ruffini23, H. Ruiz36, P. Ruiz Valls64, G. Sabatino25,l, J.J. Saborido Silva37,
N. Sagidova30, P. Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez
Mayordomo64, B. Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37,
E. Santovetti24,l, M. Sapunov6, A. Sarti18,m, C. Satriano25,n, A. Satta24, M.
Savrie16,f, D. Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M.
Schmelling10, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25, M. Seco37, A. Semennikov31, K.
Senderowska27, I. Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11,
M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34,
V. Shevchenko63, A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N.
Skidmore46, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39,
D. Souza46, B. Souza De Paula2, B. Spaan9, A. Sparkes50, F. Spinella23, P.
Spradlin51, F. Stagni38, S. Stahl11, O. Steinkamp40, O. Stenyakin35, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, E. Teodorescu29, F. Teubert38, C. Thomas55, E. Thomas38, J. van
Tilburg41, V. Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D.
Tonelli38, S. Topp-Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39,
M.T. Tran39, M. Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M.
Ubeda Garcia38, A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez
Sierra37, S. Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M.
Vesterinen11, B. Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-
Cardona36,o, A. Vollhardt40, D. Volyanskyy10, D. Voong46, A. Vorobyev30, V.
Vorobyev34, C. Voß62, H. Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R.
Wallace12, J. Walsh23, S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45,
A.D. Webber54, D. Websdale53, M. Whitehead48, J. Wicht38, D. Wiedner11, G.
Wilkinson55, M.P. Williams45, M. Williams56, F.F. Wilson49, J. Wimberley58, J.
Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A. Wotton47, S. Wright47,
S. Wu3, K. Wyllie38, Y. Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O.
Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L. Zhang59, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A. Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
## 1 Introduction
The $\mathrm{X}(3872)$ state was discovered in 2003 by the Belle collaboration
[1]. Subsequently, it has been studied by several other experiments [2, 3, 4,
5, 6]. Several properties of the $\mathrm{X}(3872)$ state have been
determined, including the precise value of its mass [7, 5] and the dipion mass
spectrum in the decay
$\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}{{\uppi}^{-}}$ [1, 8, 6]. Recently, its quantum numbers
were determined to be $J^{PC}=1^{++}$ by combination of the measurements
performed by the CDF [9] and the LHCb [10] collaborations.
Despite a large amount of experimental information, the nature of
$\mathrm{X}(3872)$ state and other similar states is still uncertain [11, 12].
In particular for the $\mathrm{X}(3872)$ state, interpretation as a
${\mathrm{D}}{{\bar{}\mathrm{D}}^{*}}$ molecule [13], tetraquark [14],
${{\mathrm{c}}{\overline{\mathrm{c}}}}\mathrm{g}$ hybrid meson [15], vector
glueball [16] or mixed state [17, 18] are proposed. Radiative decays of the
$\mathrm{X}(3872)$ provide a valuable opportunity to understand its nature.
Studies of the decay modes
$\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}$ resulted in the determination of its ${C\text{-parity}}$
[19, 20]. Evidence for the
$\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}$ decay and the
branching fraction ratio,
$R_{\uppsi{\upgamma}}\equiv\frac{{\cal
B}(\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma})}{{\cal
B}(\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma})}=3.4\pm 1.4,$
were reported by the BaBar collaboration [21]. In contrast, no significant
signal was found for the
${\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}}$ decay by the
Belle collaboration, therefore only an upper limit for
${R_{\uppsi{\upgamma}}<2.1~{}(\text{at 90\% confidence level})}$ was reported
[20]. The ratio $R_{\uppsi{\upgamma}}$ is predicted to be in the range
$(3-4)\times 10^{-3}$ for a ${\mathrm{D}}{{\bar{}\mathrm{D}}^{*}}$ molecule
[22, 23, 24], $1.2-15$ for a pure charmonium state [25, 26, 27, 28, 29, 30,
31] and $0.5-5$ for a molecule-charmonium mixture [29, 32].
In this paper, evidence for the decay
$\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}$ and a measurement
of the ratio $R_{\uppsi{\upgamma}}$ using
${{{\mathrm{B}}^{+}}}\rightarrow\mathrm{X}(3872){{\mathrm{K}}^{+}}$ decays are
presented.111The inclusion of charged conjugate processes is implied
throughout. The analysis is based on a data sample of proton-proton
(${\mathrm{p}}{\mathrm{p}}$) collisions, corresponding to an integrated
luminosity of 1$\mbox{\,fb}^{-1}$ at a centre-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$ and 2$\mbox{\,fb}^{-1}$ at
$8\mathrm{\,Te\kern-1.00006ptV}$, collected with the LHCb detector.
## 2 Detector and software
The LHCb detector [33] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\upeta<5$, designed for the study of particles
containing $\mathrm{b}$ or $\mathrm{c}$ quarks. The detector includes a high-
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $\mathrm{pp}$ interaction region, a large-area silicon-strip
detector located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes placed downstream. The combined tracking system provides a momentum
measurement with relative uncertainty that varies from 0.4 % at
5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6 % at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20${\,\upmu\rm m}$ for tracks with high transverse momentum. Charged hadrons
are identified using two ring-imaging Cherenkov detectors [34]. The
calorimeter system consists of a scintillating pad detector (SPD) and a pre-
shower system (PS), followed by electromagnetic (ECAL) and hadron
calorimeters. The SPD and PS are designed to distinguish between signals from
photons and electrons. Muons are identified by a system composed of
alternating layers of iron and multiwire proportional chambers [35].
The trigger [36] consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage where a full event
reconstruction is applied. Events are first required to pass the hardware
trigger, which selects muons with a transverse momentum, $p_{\rm T}$, greater
than 1.48${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the subsequent software
trigger, at least one of the final state particles is required to have both
$\mbox{$p_{\rm T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact
parameter in excess of $100{\,\upmu\rm m}$ with respect to all of the primary
${\mathrm{p}}{\mathrm{p}}$ interaction vertexes (PVs) in the event. Finally,
the tracks of two or more of the final state particles are required to form a
vertex that is significantly displaced from the PVs.
The analysis technique reported below has been validated using simulated
events. The ${\mathrm{p}}{\mathrm{p}}$ collisions are generated using Pythia
[37, *Sjostrand:2007gs] with a specific LHCb configuration described in Ref.
[39]. Decays of hadronic particles are described by EvtGen [40] in which final
state radiation is generated using Photos package [41]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [42, 43] as described in Ref. [44].
## 3 Event selection
Candidate ${{{\mathrm{B}}^{+}}}\rightarrow\mathrm{X}(3872){{\mathrm{K}}^{+}}$
decays, followed by $\mathrm{X}(3872)\rightarrow\uppsi{\upgamma}$, where
$\uppsi$ denotes a ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
or $\uppsi{\mathrm{(2S)}}$ meson, are reconstructed using the
$\uppsi\rightarrow{\upmu^{+}\upmu^{-}}$ channel. The
$\uppsi{\mathrm{(2S)}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}{{\uppi}^{-}}$ decay mode is not used due to low
reconstruction efficiency. Most selection criteria are common for the two
channels, except where directly related to the photon kinematics, due to the
difference in the energy release in these two channels. The selection criteria
follow those used in Refs. [45, 46, 47].
The track quality of reconstructed charged particles is ensured by requiring
that the $\chi^{2}$ per degree of freedom, $\chi^{2}/\mathrm{ndf}$, is less
than 3. Well-identified muons are selected by requiring that the difference in
the logarithms of the muon hypothesis likelihood with respect to the pion
hypothesis likelihood, $\Delta\log\mathcal{L}_{\mu/{\uppi}}$ [48], is larger
than zero. To select kaons, the corresponding difference in the logarithms of
likelihoods of the kaon and pion hypotheses [34] is required to satisfy
$\Delta\log\mathcal{L}_{{\mathrm{K}}/{\uppi}}>0$.
To ensure that the muons and kaons do not originate from a $\mathrm{pp}$
interaction vertex, the impact parameter $\chi^{2}$, defined as the difference
between the $\chi^{2}$ of a given PV formed with and without the considered
track, is required to be $\chi^{2}_{\mathrm{IP}}>4$. When more than one PV is
reconstructed, the smallest value of $\chi^{2}_{\mathrm{IP}}$ is chosen.
Pairs of oppositely charged tracks identified as muons, each having
$\mbox{$p_{\rm T}$}>0.55{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, are combined to
form $\uppsi\rightarrow{\upmu^{+}\upmu^{-}}$ candidates. The fit of the common
two-prong vertex is required to satisfy $\chi^{2}<20$. The vertex is required
to be well separated from the reconstructed PV by selecting candidates with
decay length significance greater than 3. The invariant mass of the dimuon
combination is required to be between 3.020 and
3.135${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ candidates and
between 3.597 and 3.730${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ for the
$\uppsi{\mathrm{(2S)}}$ candidates.
Photons are reconstructed using the electromagnetic calorimeter and identified
using a likelihood-based estimator, constructed from variables that rely on
calorimeter and tracking information [49]. Candidate photon clusters must not
be matched to the trajectory of a track extrapolated from the tracking system
to the cluster position in the calorimeter. Further photon quality refinement
is done using information from the PS and SPD detectors. The photon transverse
momentum is required to be greater than
1${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ or
0.6${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ or
$\uppsi{\mathrm{(2S)}}$ in the final state, respectively. To suppress the
large combinatorial background from
${{{\uppi}^{0}}\rightarrow{\upgamma}{\upgamma}}$ decays, a pion veto is
applied [46]. The photons that, when combined with another photon, form a
${{{\uppi}^{0}}\rightarrow{\upgamma}{\upgamma}}$ candidate with invariant mass
within $25{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the ${{\uppi}^{0}}$
mass, corresponding to $\pm 3$ times the mass resolution [46, 50], are not
used in the reconstruction.
To form $\mathrm{X}(3872)$ candidates, the selected $\uppsi$ candidates are
combined with a reconstructed photon. To be considered as a $\mathrm{X}(3872)$
candidate, the ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}$ or $\uppsi{\mathrm{(2S)}}{\upgamma}$ combination must have
an invariant mass in the range ${3.7\text{ --
}4.1}$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ or ${3.75\text{ --
}4.05}$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, respectively, to account
for the different available phase space.
The $\mathrm{X}(3872)$ candidates are combined with selected kaons to create
${{\mathrm{B}}^{+}}$ meson candidates. The kaons are required to have
transverse momentum larger than 0.8${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
quality of the ${{\mathrm{B}}^{+}}$ vertex is ensured by requiring the
$\chi^{2}$ of the vertex fit to be less than 25. In addition, the decay time
of the ${{\mathrm{B}}^{+}}$ is required to be larger than 150${\,\upmu\rm
m}$/$c$ to reduce the large combinatorial background from particles produced
at the PV.
To improve the invariant mass resolution of the $\mathrm{X}(3872)$ candidate,
a kinematic fit [51] is performed. In this fit, the invariant mass of the
$\uppsi$ candidate is constrained to its nominal value [52], the decay
products of the ${{\mathrm{B}}^{+}}$ candidate are required to originate from
a common vertex, and the momentum vector of the ${{\mathrm{B}}^{+}}$ candidate
is required to point back to the PV. The $\chi^{2}$/ndf for this fit is
required to be less than 5. To improve the resolution on the
${{\mathrm{B}}^{+}}$ candidate invariant mass, and minimize its correlation
with the reconstructed $\mathrm{X}(3872)$ candidate mass, the
${{\mathrm{B}}^{+}}$ mass is determined from a similar kinematic fit with an
additional constraint applied to the mass of the $\mathrm{X}(3872)$ resonance
[52]. The ${{\mathrm{B}}^{+}}$ candidates are required to have invariant mass
in the range ${5.0-5.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$. To reject
possible contributions from
${{{\mathrm{B}}^{+}}}\rightarrow\uppsi{{\mathrm{K}}^{+}}$ decays with an
additional random soft photon, the invariant mass of the
$\uppsi{{\mathrm{K}}^{+}}$ combination is required to be outside a $\pm
40{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass window around the known
${{\mathrm{B}}^{+}}$ mass [52].
## 4 Signal yields
To determine the signal yield of the
${{{{\mathrm{B}}^{+}}}\rightarrow\mathrm{X}(3872){{\mathrm{K}}^{+}}}$ decays
followed by ${\mathrm{X}(3872)\rightarrow\uppsi{\upgamma}}$, an unbinned
extended maximum likelihood two-dimensional fit in
${\uppsi{\upgamma}{{\mathrm{K}}^{+}}}$ and $\uppsi{\upgamma}$ invariant masses
is performed. The probability density function used in the fit consists of
three components to describe the mass spectrum: signal, background from other
$\mathrm{B}$ decays that peaks in the ${\uppsi{\upgamma}{{\mathrm{K}}^{+}}}$
and $\uppsi{\upgamma}$ invariant mass distributions (henceforth called
“peaking background”) and combinatorial background. The signal component is
modelled as a product of a Gaussian function in the
${\uppsi{\upgamma}{{\mathrm{K}}^{+}}}$ invariant mass and a Crystal Ball
function [53] in the $\uppsi{\upgamma}$ invariant mass. The mass resolution
and tail parameters of the Crystal Ball function are fixed to those determined
from simulated signal events.
The peaking background is studied using simulation. The sources of the peaking
background are different in the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\uppsi{\mathrm{(2S)}}$ channels due to differences in the photon spectra and
in the photon selection requirements in these two channels. The main source of
the peaking background in the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ channel is the
partially reconstructed
${{{{\mathrm{B}}^{+}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{*+}}}$ decays followed by
${{{\mathrm{K}}^{*+}}\rightarrow{{\mathrm{K}}^{+}}{{\uppi}^{0}}}$ where one
photon from the ${\uppi}^{0}$ decay is not detected. In the
$\uppsi{\mathrm{(2S)}}$ channel the peaking background arises from partially
reconstructed
${{\mathrm{B}}\rightarrow\uppsi{\mathrm{(2S)}}{{\mathrm{K}}^{+}}\mathrm{Y}}$
decays combined with a random photon, where $\mathrm{B}$ denotes a
$\mathrm{b}$ hadron and $\mathrm{Y}$ denotes additional particles of the
$\mathrm{B}$ decay. These background contributions are modelled in the fit
using non-parametric kernel probability density functions [54], obtained from
simulation of $\mathrm{B}$ decays to final states containing a
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ or
$\uppsi{\mathrm{(2S)}}$ meson.
Combinatorial background is modelled as the product of an exponential function
of the $\uppsi{\upgamma}{{\mathrm{K}}^{+}}$ invariant mass and a second-order
polynomial function of the ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}$ invariant mass or a third-order polynomial function of the
$\uppsi{\mathrm{(2S)}}{\upgamma}$ invariant mass. For the latter case, the
polynomial function is constrained to account for the small available phase
space, allowing only two polynomial degrees of freedom to vary in the fit.
The fit results for the position of the ${{\mathrm{B}}^{+}}$ and
$\mathrm{X}(3872)$ mass peaks, $m_{{{{\mathrm{B}}^{+}}}}$ and
$m_{\mathrm{X}(3872)}$, respectively, and the signal yields $N_{\uppsi}$ are
listed in Table 1. Projections of the fit on
$\uppsi{\upgamma}{{\mathrm{K}}^{+}}$ and $\uppsi{\upgamma}$ invariant masses
are shown in Figs. 1 and 2 for the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\uppsi{\mathrm{(2S)}}$ channels, respectively.
Table 1: Parameters of the signal functions of the fits to the two-dimensional mass distributions of the ${{{{\mathrm{B}}^{+}}}\rightarrow\mathrm{X}(3872){{\mathrm{K}}^{+}}}$ decays followed by ${\mathrm{X}(3872)\rightarrow\uppsi{\upgamma}}$. Uncertainties are statistical only. Parameter | Decay mode
---|---
$\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}{\upgamma}$ | $\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}$
$m_{{{{\mathrm{B}}^{+}}}}~{}~{}~{}~{}\,~{}\left[\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\right]$ | $5277.7\pm 0.8$ | $5281.9\pm 2.4$
$m_{\mathrm{X}(3872)}~{}\left[\\!{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\right]$ | $3873.4\pm 3.4$ | $3869.5\pm 3.4$
$N_{\uppsi}$ | $\phantom{0.0}591\pm 48\phantom{.}$ | $\phantom{00}36.4\pm 9.0$
Candidates/(10${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$)
Candidates/(10${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$)
$m_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}{{\mathrm{K}}^{+}}}$$\left[\mathrm{GeV}/c^{2}\right]$$m_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}}$$\left[\mathrm{GeV}/c^{2}\right]$LHCbLHCba)b) Figure 1: a)
Distribution of the ${{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}{{\mathrm{K}}^{+}}}$ invariant mass with fit projection
overlaid, restricted to those candidates with
${{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}{\upgamma}}$
invariant mass within $\pm 3\sigma$ from the $\mathrm{X}(3872)$ peak position.
b) Distribution of the ${{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}}$ invariant mass with fit projection overlaid, restricted to
those candidates with ${{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}{{\mathrm{K}}^{+}}}$ invariant mass within $\pm 3\sigma$ from
the ${{\mathrm{B}}^{+}}$ peak position. The total fit (thick solid blue)
together with the signal (thin solid green) and background components (dash-
dotted orange for the combinatorial, dashed magenta for the peaking component
and long dashed blue for their sum) are shown.
Candidates/(10${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$)
Candidates/(15${\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$)
$m_{\uppsi{\mathrm{(2S)}}{\upgamma}{{\mathrm{K}}^{+}}}$$\left[\mathrm{GeV}/c^{2}\right]$$m_{\uppsi{\mathrm{(2S)}}{\upgamma}}$$\left[\mathrm{GeV}/c^{2}\right]$LHCbLHCba)b)
Figure 2: a) Distribution of the
${\uppsi{\mathrm{(2S)}}{\upgamma}{{\mathrm{K}}^{+}}}$ invariant mass with fit
projection overlaid, restricted to those candidates with
${\uppsi{\mathrm{(2S)}}{\upgamma}}$ invariant mass within $\pm 3\sigma$ from
the $\mathrm{X}(3872)$ peak position. b) Distribution of the
${\uppsi{\mathrm{(2S)}}{\upgamma}}$ invariant mass with fit projection
overlaid, restricted to those candidates with
${\uppsi{\mathrm{(2S)}}{\upgamma}{{\mathrm{K}}^{+}}}$ invariant mass within
$\pm 3\sigma$ from the ${{\mathrm{B}}^{+}}$ peak position. The total fit
(thick solid blue) together with the signal (thin solid green) and background
components (dash-dotted orange for the combinatorial, dashed magenta for the
peaking component and long dashed blue for their sum) are shown.
The significance of the observed signal in the $\uppsi{\mathrm{(2S)}}$ channel
is determined by simulating a large number of background-only experiments,
taking into account all uncertainties in the shape of the background
distribution. The probability for the background to fluctuate to at least the
number of observed events is found to be $1.2\times 10^{-5}$, corresponding to
a significance of 4.4 standard deviations for the
${{{{\mathrm{B}}^{+}}}\rightarrow\mathrm{X}(3872){{\mathrm{K}}^{+}}}$ decay
followed by ${\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}}$.
## 5 Efficiencies and systematic uncertainties
The ratio of the $\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}$
and
$\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}$ branching fractions is calculated using the formula
$R_{\uppsi{\upgamma}}=\dfrac{N_{\uppsi{\mathrm{(2S)}}}}{N_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}}}\times\dfrac{\upvarepsilon_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}}}{\upvarepsilon_{\uppsi{\mathrm{(2S)}}}}\times\dfrac{{\cal
B}({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\rightarrow{\upmu^{+}\upmu^{-}})}{{\cal
B}(\uppsi{\mathrm{(2S)}}\rightarrow{\upmu^{+}\upmu^{-}})},$ (1)
where $N_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}}$ and
$N_{\uppsi{\mathrm{(2S)}}}$ are the measured yields listed in Table 1, and
$\upvarepsilon_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}}$
and $\upvarepsilon_{\uppsi{\mathrm{(2S)}}}$ are the total efficiencies. For
the ratio of the $\uppsi\rightarrow{\upmu^{+}\upmu^{-}}$ branching fractions,
lepton universality is assumed and a ratio of dielectron branching fractions
equal to $7.60\pm 0.18$ [52] is used. The uncertainty is treated as a
systematic uncertainty.
The total efficiency is the product of the geometrical acceptance, the
detection, reconstruction, selection and trigger efficiencies. The
efficiencies are estimated using simulated events that have been corrected to
reproduce the observed kinematics of ${{\mathrm{B}}^{+}}$ mesons using the
high-yield decay
${{{{\mathrm{B}}^{+}}}\rightarrow{\upchi_{{\mathrm{c}}1}}{{\mathrm{K}}^{+}}}$
with
${{\upchi_{{\mathrm{c}}1}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}}$, which has a topology and kinematics similar to those of
the decays under study. The ratio of the efficiencies is found to be
${\upvarepsilon_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}}}/{\upvarepsilon_{\uppsi{\mathrm{(2S)}}}}=5.25\pm 0.04$, where the
uncertainty is due to finite size of the simulated samples. The ratio of
efficiencies is different from unity mainly because of the different photon
spectra in the decays with ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ and $\uppsi{\mathrm{(2S)}}$ in the final state.
Most sources of systematic uncertainty cancel in the ratio, in particular
those related to the kaon, muon and $\uppsi$ reconstruction and
identification. The remaining systematic uncertainties are summarized in Table
2 and discussed in turn in the following.
Table 2: Relative systematic uncertainties on the ratio of branching fractions ($R_{\uppsi{\upgamma}}$). Source | Uncertainty [%]
---|---
$\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}\gamma$ yield determination | $6$
$\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}\gamma$ yield determination | $7$
Photon reconstruction | $6$
${{\mathrm{B}}^{+}}$ kinematics | $3$
Selection criteria | $2$
Trigger | $1$
${\cal B}({{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}\rightarrow{\mathrm{e}^{+}\mathrm{e}^{-}})/{\cal B}(\uppsi{\mathrm{(2S)}}\rightarrow{\mathrm{e}^{+}\mathrm{e}^{-}})$ | $2$
Simulation sample size | $1$
Sum in quadrature | $12$
Systematic uncertainties related to the signal yield determination are
considered in four categories: signal, peaking background, combinatorial
background and intervals used in the fit. For each category individual
uncertainties are estimated using a number of alternative fit models. The
maximum deviations from the baseline values of the yields are taken as
individual systematic uncertainties, which are then added in quadrature. The
systematic uncertainties on the event yields are dominated by uncertainties in
the description of backgrounds and are 6 % and 7 % in the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\uppsi{\mathrm{(2S)}}$ channels, respectively.
Another important source of systematic uncertainty arises from the potential
disagreement between data and simulation in the estimation of efficiencies.
This includes the photon reconstruction efficiency, the imperfect knowledge of
${{\mathrm{B}}^{+}}$ kinematics and the description of the selection criteria
efficiencies. The photon reconstruction efficiency is studied using a large
sample of
${{{\mathrm{B}}^{+}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{*+}}$ decays, followed by
${{\mathrm{K}}^{*+}}\rightarrow{{\mathrm{K}}^{+}}{{\uppi}^{0}}$ and
${{\uppi}^{0}}\rightarrow{\upgamma}{\upgamma}$ decays. The relative yields of
${{{\mathrm{B}}^{+}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{*+}}$ and
${{{\mathrm{B}}^{+}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{+}}$ decays are compared in data and simulation. For
photons with transverse momentum greater than
0.6${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, the agreement between data and
simulation is within 6 %, which is assigned as the systematic uncertainty due
to the photon reconstruction.
The systematic uncertainty related to the knowledge of the
${{\mathrm{B}}^{+}}$ production properties is estimated by comparing the ratio
of efficiencies determined without making corrections to the
${{\mathrm{B}}^{+}}$ transverse momentum and rapidity spectra to the default
ratio of efficiencies determined after the corrections. The relative
difference between the two methods is found to be 3 % and is conservatively
assigned as the systematic uncertainty from this source.
To study the uncertainty due to selection criteria, the high-yield decay
${{{{\mathrm{B}}^{+}}}\rightarrow{\upchi_{{\mathrm{c}}1}}{{\mathrm{K}}^{+}}}$,
followed by
${{\upchi_{{\mathrm{c}}1}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}}$, which has a similar topology to the decays studied in
this analysis, is used. The selection criteria for the photon and kaon
transverse momentum, the ${{\uppi}^{0}}\rightarrow{\upgamma}{\upgamma}$ veto
and the $\chi^{2}$/ndf of the kinematic fit are studied. The selection
criteria are varied in ranges corresponding to as much as a $30\,\%$ change in
the signal yields and the ratios of the selection and reconstruction
efficiencies are compared between data and simulation. The largest difference
of 2 % is assigned as the corresponding systematic uncertainty.
The systematic uncertainty related to the trigger efficiency is obtained by
comparing the trigger efficiency ratios in data and simulation for the high
yield decay modes
${{{\mathrm{B}}^{+}}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{+}}$ and
${{{\mathrm{B}}^{+}}}\rightarrow\uppsi{\mathrm{(2S)}}{{\mathrm{K}}^{+}}$,
which have similar kinematics and the same trigger requirements as the
channels under study in this analysis [55]. An agreement within 1 % is found,
which is assigned as the corresponding systematic uncertainty.
## 6 Results and summary
Using a sample of ${\mathrm{p}}{\mathrm{p}}$ collisions at centre-of-mass
energies of 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$, corresponding to an
integrated luminosity of 3$\mbox{\,fb}^{-1}$, evidence for the decay
$\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma}$ in
${{{\mathrm{B}}^{+}}}\rightarrow\mathrm{X}(3872){{\mathrm{K}}^{+}}$ decays is
found with a significance of 4.4 standard deviations. Its branching fraction,
normalized to that of the
$\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma}$ decay mode is measured to be
$R_{\uppsi{\upgamma}}=\dfrac{{\cal
B}(\mathrm{X}(3872)\rightarrow\uppsi{\mathrm{(2S)}}{\upgamma})}{{\cal
B}(\mathrm{X}(3872)\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\upgamma})}=2.46\pm 0.64\pm 0.29,$
where the first uncertainty is statistical and the second is systematic. This
result is compatible with, but more precise than, previous measurements [21,
20]. The measured value of $R_{\uppsi{\upgamma}}$ agrees with expectations for
a pure charmonium interpretation of the $\mathrm{X}(3872)$ state [25, 26, 27,
28, 29, 30, 31] and a molecular-charmonium mixture interpretations [29, 32].
However, it does not support a pure ${\mathrm{D}}{{\bar{}\mathrm{D}}^{*}}$
molecular interpretation [22, 23, 24] of the $\mathrm{X}(3872)$ state.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC and the Royal Society (United Kingdom);
NSF (USA). We also acknowledge the support received from EPLANET, Marie Curie
Actions and the ERC under FP7. The Tier1 computing centres are supported by
IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The
Netherlands), PIC (Spain), GridPP (United Kingdom). We are indebted to the
communities behind the multiple open source software packages on which we
depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia).
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|
arxiv-papers
| 2014-04-01T15:27:17 |
2024-09-04T02:50:00.535512
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. An, L.\n Anderlini, J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V.\n Batozskaya, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J.\n Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S.\n Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw,\n S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia,\n M. Borsato, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den\n Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown,\n A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M.\n Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez, A. Carbone,\n G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K.\n Carvalho Akiba, G. Casse, L. Cassina, L. Castillo Garcia, M. Cattaneo, Ch.\n Cauet, R. Cenci, M. Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic,\n H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti,\n M. Corvo, I. Counts, B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S.\n Cunliffe, R. Currie, C. D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A.\n Davis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W.\n De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n S. Donleavy, F. Dordei, M. Dorigo, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof,\n L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, T. Evans, A. Falabella, C.\n F\\\"arber, C. Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F.\n Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, M.\n Firlej, C. Fitzpatrick, T. Fiutowski, M. Fontana, F. Fontanelli, R. Forty, O.\n Francisco, M. Frank, C. Frei, M. Frosini, J. Fu, E. Furfaro, A. Gallas\n Torreira, D. Galli, S. Gallorini, S. Gambetta, M. Gandelman, P. Gandini, Y.\n Gao, J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, L. Gavardi,\n E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, A. Gianelle, S. Giani', V.\n Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, C. Gotti, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, P. Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz,\n T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B.\n Hamilton, T. Hampson, X. Han, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew,\n J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n L. Henry, J.A. Hernando Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, D.\n Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, N. Hussain, D.\n Hutchcroft, D. Hynds, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, J.\n Jalocha, E. Jans, P. Jaton, A. Jawahery, M. Jezabek, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, N. Jurik, M. Kaballo, S. Kandybei, W.\n Kanso, M. Karacson, T.M. Karbach, M. Kelsey, I.R. Kenyon, T. Ketel, B.\n Khanji, C. Khurewathanakul, S. Klaver, O. Kochebina, M. Kolpin, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, B. Langhans, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam,\n J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T.\n Lesiak, B. Leverington, Y. Li, M. Liles, R. Lindner, C. Linn, F. Lionetto, B.\n Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, P. Lowdon, H.\n Lu, D. Lucchesi, H. Luo, A. Lupato, E. Luppi, O. Lupton, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, M.\n Manzali, J. Maratas, J.F. Marchand, U. Marconi, C. Marin Benito, P. Marino,\n R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, F. Martinez Vidal, D. Martins Tostes, A.\n Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M. McCann, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, J.\n Moron, R. Mountain, F. Muheim, K. M\\\"uller, R. Muresan, B. Muster, P. Naik,\n T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N.\n Neufeld, M. Neuner, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M.\n Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano,\n F. Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes,\n C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, C. Patrignani, A. Pazos\n Alvarez, A. Pearce, A. Pellegrino, M. Pepe Altarelli, S. Perazzini, E. Perez\n Trigo, P. Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, K. Petridis, A.\n Petrolini, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A.\n Pistone, S. Playfer, M. Plo Casasus, F. Polci, A. Poluektov, I. Polyakov, E.\n Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J.\n Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi,\n W. Qian, B. Rachwal, J.H. Rademacker, B. Rakotomiaramanana, M. Rama, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Reichert, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S.\n Roiser, V. Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F.\n Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, C. Sanchez Mayordomo,\n B. Sanmartin Sedes, R. Santacesaria, C. Santamarina Rios, E. Santovetti, M.\n Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, M. Schiller,\n H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A.\n Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, L. Sestini, P.\n Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, V.\n Shevchenko, A. Shires, R. Silva Coutinho, G. Simi, M. Sirendi, N. Skidmore,\n T. Skwarnicki, N.A. Smith, E. Smith, E. Smith, J. Smith, M. Smith, H. Snoek,\n M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B.\n Spaan, A. Sparkes, F. Spinella, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, O. Stenyakin, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S.\n Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah, L. Sun, W.\n Sutcliffe, K. Swientek, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini,\n E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V.\n Tisserand, M. Tobin, S. Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen,\n N. Torr, E. Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev,\n P. Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer,\n V. Vagnoni, G. Valenti, A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C.\n V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M.\n Vesterinen, B. Viaud, D. Vieira, M. Vieites Diaz, X. Vilasis-Cardona, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, J.A. de Vries, R. Waldi, C. Wallace, R. Wallace, J. Walsh, S.\n Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, D. Wiedner, G. Wilkinson, M.P. Williams, M. Williams,\n F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G. Wormser,\n S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Xu, Z. Yang, X.\n Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Ivan Belyaev",
"url": "https://arxiv.org/abs/1404.0275"
}
|
1404.0287
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-054 LHCb-PAPER-2014-009 April 1, 2014
Evidence for the decay
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$
The LHCb collaboration†††Authors are listed on the following pages.
Evidence is presented for the decay
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ using proton-proton collision data,
corresponding to an integrated luminosity of 3$\mbox{\,fb}^{-1}$, collected
with the LHCb detector. A signal yield of $32\pm 8$ decays is found with a
significance of 4.5 standard deviations. The ratio of the branching fraction
of the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ decay to that of the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ decay is measured to be
$\dfrac{{\cal
B}\left({{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}\right)}{{\cal
B}\left({{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}\right)}=1.74\pm 0.44\pm 0.24,$
where the first uncertainty is statistical and the second is systematic.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L. An3, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,m, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, J. Beddow51, F. Bedeschi23, I. Bediaga1, S. Belogurov31, K.
Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S. Benson50, J.
Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van Beuzekom41, A.
Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M. Bjørnstad54, T. Blake48,
F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A. Bondar34, N. Bondar30,38,
W. Bonivento15,38, S. Borghi54, A. Borgia59, M. Borsato7, T.J.V. Bowcock52, E.
Bowen40, C. Bozzi16, T. Brambach9, J. van den Brand42, J. Bressieux39, D.
Brett54, M. Britsch10, T. Britton59, N.H. Brook46, H. Brown52, A. Bursche40,
G. Busetto22,p, J. Buytaert38, S. Cadeddu15, R. Calabrese16,f, O. Callot7, M.
Calvi20,k, M. Calvo Gomez36,n, A. Camboni36, P. Campana18,38, D. Campora
Perez38, F. Caponio21,t, A. Carbone14,d, G. Carboni24,l, R. Cardinale19,38,j,
A. Cardini15, H. Carranza-Mejia50, L. Carson50, K. Carvalho Akiba2, G.
Casse52, L. Cassina20, L. Castillo Garcia38, M. Cattaneo38, Ch. Cauet9, R.
Cenci58, M. Charles8, Ph. Charpentier38, S.-F. Cheung55, N. Chiapolini40, M.
Chrzaszcz40,26, K. Ciba38, X. Cid Vidal38, G. Ciezarek53, P.E.L. Clarke50, M.
Clemencic38, H.V. Cliff47, J. Closier38, C. Coca29, V. Coco38, J. Cogan6, E.
Cogneras5, P. Collins38, A. Comerma-Montells11, A. Contu15,38, A. Cook46, M.
Coombes46, S. Coquereau8, G. Corti38, I. Counts56, B. Couturier38, G.A.
Cowan50, D.C. Craik48, M. Cruz Torres60, S. Cunliffe53, R. Currie50, C.
D’Ambrosio38, J. Dalseno46, P. David8, P.N.Y. David41, A. Davis57, K. De
Bruyn41, S. De Capua54, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De
Silva57, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N.
Déléage4, D. Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto38, H.
Dijkstra38, S. Donleavy52, F. Dordei11, M. Dorigo39, A. Dosil Suárez37, D.
Dossett48, A. Dovbnya43, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49, U. Egede53, V. Egorychev31, S. Eidelman34,
S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5,
Ch. Elsasser40, S. Esen11, A. Falabella16,f, C. Färber11, C. Farinelli41, N.
Farley45, S. Farry52, D. Ferguson50, V. Fernandez Albor37, F. Ferreira
Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M. Fiorini16,f, M.
Firlej27, C. Fitzpatrick38, T. Fiutowski27, M. Fontana10, F. Fontanelli19,j,
R. Forty38, O. Francisco2, M. Frank38, C. Frei38, M. Frosini17,38,g, J. Fu21,
E. Furfaro24,l, A. Gallas Torreira37, D. Galli14,d, S. Gallorini22, S.
Gambetta19,j, M. Gandelman2, P. Gandini59, Y. Gao3, J. Garofoli59, J. Garra
Tico47, L. Garrido36, C. Gaspar38, R. Gauld55, L. Gavardi9, E. Gersabeck11, M.
Gersabeck54, T. Gershon48, Ph. Ghez4, A. Gianelle22, S. Giani’39, V. Gibson47,
L. Giubega29, V.V. Gligorov38, C. Göbel60, D. Golubkov31, A. Golutvin53,31,38,
A. Gomes1,a, H. Gordon38, C. Gotti20, M. Grabalosa Gándara5, R. Graciani
Diaz36, L.A. Granado Cardoso38, E. Graugés36, G. Graziani17, A. Grecu29, E.
Greening55, S. Gregson47, P. Griffith45, L. Grillo11, O. Grünberg62, B. Gui59,
E. Gushchin33, Yu. Guz35,38, T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C.
Haen38, T.W. Hafkenscheid65, S.C. Haines47, S. Hall53, B. Hamilton58, T.
Hampson46, X. Han11, S. Hansmann-Menzemer11, N. Harnew55, S.T. Harnew46, J.
Harrison54, T. Hartmann62, J. He38, T. Head38, V. Heijne41, K. Hennessy52, P.
Henrard5, L. Henry8, J.A. Hernando Morata37, E. van Herwijnen38, M. Heß62, A.
Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54, W. Hulsbergen41, P. Hunt55,
N. Hussain55, D. Hutchcroft52, D. Hynds51, M. Idzik27, P. Ilten56, R.
Jacobsson38, A. Jaeger11, E. Jans41, P. Jaton39, A. Jawahery58, M. Jezabek26,
F. Jing3, M. John55, D. Johnson55, C.R. Jones47, C. Joram38, B. Jost38, N.
Jurik59, M. Kaballo9, S. Kandybei43, W. Kanso6, M. Karacson38, T.M. Karbach38,
M. Kelsey59, I.R. Kenyon45, T. Ketel42, B. Khanji20, C. Khurewathanakul39, S.
Klaver54, O. Kochebina7, M. Kolpin11, I. Komarov39, R.F. Koopman42, P.
Koppenburg41,38, M. Korolev32, A. Kozlinskiy41, L. Kravchuk33, K. Kreplin11,
M. Kreps48, G. Krocker11, P. Krokovny34, F. Kruse9, M. Kucharczyk20,26,38,k,
V. Kudryavtsev34, K. Kurek28, T. Kvaratskheliya31, V.N. La Thi39, D.
Lacarrere38, G. Lafferty54, A. Lai15, D. Lambert50, R.W. Lambert42, E.
Lanciotti38, G. Lanfranchi18, C. Langenbruch38, B. Langhans38, T. Latham48, C.
Lazzeroni45, R. Le Gac6, J. van Leerdam41, J.-P. Lees4, R. Lefèvre5, A.
Leflat32, J. Lefrançois7, S. Leo23, O. Leroy6, T. Lesiak26, B. Leverington11,
Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F. Lionetto40, B. Liu15, G.
Liu38, S. Lohn38, I. Longstaff51, J.H. Lopes2, N. Lopez-March39, P. Lowdon40,
H. Lu3, D. Lucchesi22,p, H. Luo50, E. Luppi16,f, O. Lupton55, F. Machefert7,
I.V. Machikhiliyan31, F. Maciuc29, O. Maev30, S. Malde55, G. Manca15,e, G.
Mancinelli6, M. Manzali16,f, J. Maratas5, J.F. Marchand4, U. Marconi14, C.
Marin Benito36, P. Marino23,r, R. Märki39, J. Marks11, G. Martellotti25, A.
Martens8, A. Martín Sánchez7, M. Martinelli41, D. Martinez Santos42, F.
Martinez Vidal64, D. Martins Tostes2, A. Massafferri1, R. Matev38, Z. Mathe38,
C. Matteuzzi20, A. Mazurov16,f, M. McCann53, J. McCarthy45, A. McNab54, R.
McNulty12, B. McSkelly52, B. Meadows57,55, F. Meier9, M. Meissner11, M.
Merk41, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez60, S. Monteil5, D.
Moran54, M. Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,r, J.
Moron27, R. Mountain59, F. Muheim50, K. Müller40, R. Muresan29, B. Muster39,
P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21,
S. Neubert38, N. Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C.
Nguyen-Mau39,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A.
Novoselov35, A. Oblakowska-Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51,
O. Okhrimenko44, R. Oldeman15,e, G. Onderwater65, M. Orlandea29, J.M. Otalora
Goicochea2, P. Owen53, A. Oyanguren36, B.K. Pal59, A. Palano13,c, F.
Palombo21,s, M. Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51,
L. Pappalardo16, C. Parkes54, C.J. Parkinson9, G. Passaleva17, G.D. Patel52,
M. Patel53, C. Patrignani19,j, A. Pazos Alvarez37, A. Pearce54, A.
Pellegrino41, M. Pepe Altarelli38, S. Perazzini14,d, E. Perez Trigo37, P.
Perret5, M. Perrin-Terrin6, L. Pescatore45, E. Pesen66, K. Petridis53, A.
Petrolini19,j, E. Picatoste Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25,
A. Pistone19, S. Playfer50, M. Plo Casasus37, F. Polci8, A. Poluektov48,34, E.
Polycarpo2, A. Popov35, D. Popov10, B. Popovici29, C. Potterat2, A. Powell55,
J. Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig
Navarro39, G. Punzi23,q, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S. Ricciardi49, A.
Richards53, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5, P. Robbe7, D.A.
Roberts58, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S. Roiser38,
V. Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F.
Ruffini23, H. Ruiz36, P. Ruiz Valls36, G. Sabatino25,l, J.J. Saborido Silva37,
N. Sagidova30, P. Sail51, B. Saitta15,e, V. Salustino Guimaraes2, B. Sanmartin
Sedes37, R. Santacesaria25, C. Santamarina Rios37, E. Santovetti24,l, M.
Sapunov6, A. Sarti18, C. Satriano25,m, A. Satta24, M. Savrie16,f, D.
Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M. Schmelling10, B.
Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R. Schwemmer38, B.
Sciascia18, A. Sciubba25, M. Seco37, A. Semennikov31, K. Senderowska27, I.
Sepp53, N. Serra40, J. Serrano6, P. Seyfert11, M. Shapkin35, I.
Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34, O. Shevchenko43,
V. Shevchenko63, A. Shires9, F. Sidorov31, R. Silva Coutinho48, G. Simi22, M.
Sirendi47, N. Skidmore46, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E.
Smith53, J. Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51,
F. Soomro39, D. Souza46, B. Souza De Paula2, B. Spaan9, A. Sparkes50, F.
Spinella23, P. Spradlin51, F. Stagni38, S. Stahl11, O. Steinkamp40, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38,
A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B.
Viaud7, D. Vieira2, X. Vilasis-Cardona36,n, A. Vollhardt40, D. Volyanskyy10,
D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H. Voss10, J.A. de
Vries41, R. Waldi62, C. Wallace48, R. Wallace12, S. Wandernoth11, J. Wang59,
D.R. Ward47, N.K. Watson45, D. Websdale53, M. Whitehead48, J. Wicht38, D.
Wiedner11, G. Wilkinson55, M.P. Williams48,49, M. Williams56, F.F. Wilson49,
J. Wimberley58, J. Wishahi9, W. Wislicki28, M. Witek26, G. Wormser7, S.A.
Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y. Xie61, Z. Xing59, Z. Yang3, X.
Yuan3, O. Yushchenko35, M. Zangoli14, M. Zavertyaev10,b, F. Zhang3, L.
Zhang59, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov31, L. Zhong3, A.
Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pUniversità di Padova, Padova, Italy
qUniversità di Pisa, Pisa, Italy
rScuola Normale Superiore, Pisa, Italy
sUniversità degli Studi di Milano, Milano, Italy
tPolitecnico di Milano, Milano, Italy
## 1 Introduction
The ${\mathrm{B}}_{\mathrm{c}}^{+}$ meson is the only meson consisting of two
heavy quarks of different flavours. It was discovered by the CDF collaboration
through the semileptonic decay
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\ell^{+}\upnu_{\ell}\mathrm{X}$ [1], where $\mathrm{X}$ denotes
possible unobserved particles.111The inclusion of charge conjugate modes is
implicit throughout this paper. The CDF collaboration also observed the
hadronic decay mode
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ [2]. Recently, the LHCb experiment has observed several
new channels including
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}{{\uppi}^{+}}{{\uppi}^{-}}$ [3],
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{\uppsi{\mathrm{(2S)}}}{{\uppi}^{+}}$
[4],
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{D}}^{+}_{\mathrm{s}}}$ and
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\mathrm{D}}_{{\mathrm{s}}}^{*+}$ [5],
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{+}}$ [6],
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{+}}{{\mathrm{K}}^{-}}{{\uppi}^{+}}$ [7] and
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{B}}^{0}_{\mathrm{s}}}{{\uppi}^{+}}$
[8]. The lifetime of the ${\mathrm{B}}_{\mathrm{c}}^{+}$ meson [9, 10] is
about three times shorter than that of the ${{\mathrm{B}}^{0}}$ and
${{\mathrm{B}}^{+}}$ mesons, confirming the important role played by the
$\mathrm{c}$ quark in ${\mathrm{B}}_{\mathrm{c}}^{+}$ decays. The decays of
${\mathrm{B}}_{\mathrm{c}}^{+}$ mesons into charmonia and light hadrons are
expected to be well described by the factorization approximation [11,
*Wirbel:1988ft]. In this scheme, the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ decay is characterized by the form
factors of the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{\mathrm{W}}^{+}$ transition and the spectral functions for the virtual
${\mathrm{W}}^{+}$ boson into light hadrons [13]. The predictions for the
ratio of branching fractions
$R_{5\uppi}\equiv\dfrac{{\cal
B}\left({{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}\right)}{{\cal
B}\left({{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}\right)}$ (1)
are 0.95 and 1.1 [14], using form factor calculations from Refs. [15] and
[16], respectively.
In this article, the first evidence for the decay
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ and a measurement of $R_{5\uppi}$ are
reported. The analysis is based on a data sample of proton-proton
(${\mathrm{p}}{\mathrm{p}}$) collisions, corresponding to an integrated
luminosity of 1$\mbox{\,fb}^{-1}$ at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ and 2$\mbox{\,fb}^{-1}$ at
8$\mathrm{\,Te\kern-1.00006ptV}$, collected with the LHCb detector.
## 2 Detector
The LHCb detector [17] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\upeta<5$, designed for the study of particles
containing $\mathrm{b}$ or $\mathrm{c}$ quarks. The detector includes a high-
precision tracking system consisting of a silicon-strip vertex detector
surrounding the ${\mathrm{p}}{\mathrm{p}}$ interaction region, a large-area
silicon-strip detector located upstream of a dipole magnet with a bending
power of about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and
straw drift tubes [18] placed downstream. The combined tracking system
provides a momentum measurement with relative uncertainty that varies from
0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at
100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter resolution of
20${\,\upmu\rm m}$ for tracks with large transverse momentum. Different types
of charged hadrons are distinguished using information from two ring-imaging
Cherenkov detectors [19]. Photon, electron and hadron candidates are
identified by a calorimeter system consisting of scintillating-pad and
preshower detectors, an electromagnetic calorimeter and a hadronic
calorimeter. Muons are identified by a system composed of alternating layers
of iron and multiwire proportional chambers [20]. The trigger [21] consists of
a hardware stage, based on information from the calorimeter and muon systems,
followed by a software stage, which applies a full event reconstruction.
This analysis uses events collected by triggers that select the
$\upmu^{+}\upmu^{-}$ pair from the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ decay with high
efficiency. At the hardware stage either one or two muon candidates are
required to trigger the event. In the case of single muon triggers, the
transverse momentum, $p_{\rm T}$, of the muon candidate is required to be
greater than $1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. For dimuon candidates,
the product of the $p_{\rm T}$ of muon candidates is required to satisfy
$\sqrt{\mbox{$p_{\rm T}$}_{1}\mbox{$p_{\rm
T}$}_{2}}>1.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. At the subsequent software
trigger stage, two muons are selected with an invariant mass in the range
$2.97<m_{{\upmu^{+}\upmu^{-}}}<3.21{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$
and consistent with originating from a common vertex. The common vertex is
required to be significantly displaced from the ${\mathrm{p}}{\mathrm{p}}$
collision vertices.
Simulated ${\mathrm{p}}{\mathrm{p}}$ collisions are generated using Pythia 6.4
[22] with the configuration described in Ref. [23]. Final-state QED radiative
corrections are included using the Photos package [24]. The
${\mathrm{B}}_{\mathrm{c}}^{+}$ mesons are produced by a dedicated generator,
Bcvegpy [25]. The decays of all hadrons are performed by EvtGen [26], and a
specific model is implemented to generate the decays
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$, assuming factorization [14]. The model
allows the implementation of different form factors for this decay, calculated
using QCD sum rules [15] or a relativistic quark model [16]. These predictions
lead to very similar values and those based on the relativistic quark model
are used in the simulation. The coupling of the five pion
($3{{\uppi}^{+}}2{{\uppi}^{-}}$) system to the virtual $\mathrm{W}^{+}$ is
taken from $\uptau^{+}$ lepton decays [27]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [28, *Agostinelli:2002hh] as described in Ref. [30].
## 3 Candidate selection
The decays
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ and
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ are reconstructed using the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\\!\rightarrow{\upmu^{+}\upmu^{-}}$ decay mode. The selection criteria
chosen are similar for both channels.
All tracks are required to be in the pseudorapidity range $2<\upeta<4.9$. Good
track quality of charged particles is ensured by requiring the $\chi^{2}$ per
number of degrees of freedom, $\chi^{2}/\mathrm{ndf}$, provided by the track
fit, to be less than 3. Suppression of fake tracks created by the
reconstruction is achieved by a neural network trained with simulated samples
to discriminate between fake tracks and tracks associated with real particles
[31], ensuring the rate of fake tracks below 0.3 %.
Two dedicated neural networks are used for muon and pion identification. These
networks use the information from the Cherenkov detectors [19], muon chambers
[32] and the calorimeter system [33], together with the tracking information.
The momentum of the pion candidates is required to be between
3.2${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and
150${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ in order to ensure good quality
particle identification in Cherenkov detectors. The requirements on the neural
network output are chosen to ensure good agreement between data and simulation
and significant reduction of the background due to misidentification.
Pairs of oppositely charged muons, originating from a common vertex, are
combined to form ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\\!\rightarrow{\upmu^{+}\upmu^{-}}$ candidates. The $p_{\rm T}$ of each
muon is required to be greater than 550${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$.
Good vertex reconstruction is ensured by requiring the $\chi^{2}$ of the
vertex fit, $\chi^{2}_{\rm vtx}$, to be less than 20. To select dimuon
vertices that are well-separated from the reconstructed
${\mathrm{p}}{\mathrm{p}}$ interaction vertices, the decay length is required
to be at least three times its uncertainty. The invariant mass of the dimuon
combination is required to be between 3.020 and
3.135${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The asymmetric mass range
with respect to the known ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ meson mass [9] is chosen to include the QED radiative tail.
The selected ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
candidates are combined with pions to form
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ and
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ candidates. The transverse momentum of each pion is
required to be greater than 400${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. To
ensure that the pions are inconsistent with being directly produced in a
$\mathrm{pp}$ interaction, the impact parameter $\chi^{2}$, defined as the
difference between the $\chi^{2}$ values of the fits of the
${\mathrm{p}}{\mathrm{p}}$ collision vertex formed with and without the
considered pion track, is required to satisfy $\chi^{2}_{\mathrm{IP}}>4$. When
more than one primary vertex is reconstructed, the vertex with the smallest
value of $\chi^{2}_{\mathrm{IP}}$ is chosen. Good vertex reconstruction for
the ${\mathrm{B}}_{\mathrm{c}}^{+}$ candidate vertex is ensured by requiring
the $\chi^{2}_{\rm vtx}/\mathrm{ndf}$ to be less than 12. To suppress the
large combinatorial background in the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ sample, the $\chi^{2}$ of the vertex fit
for all ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}\uppi^{\pm}$
combinations, as well as for all dipion combinations, is required to be less
than 20. To improve the invariant mass resolution, a kinematic fit [34] is
performed that constrains the ${\upmu^{+}\upmu^{-}}$ pair to the known mass of
the ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ meson. It is
also required that the ${\mathrm{B}}_{\mathrm{c}}^{+}$ candidate’s momentum
vector points back to from the associated ${\mathrm{p}}{\mathrm{p}}$
interaction vertex. When more than one ${\mathrm{p}}{\mathrm{p}}$ collision
vertex is found, that with the smallest value of $\chi^{2}_{\mathrm{IP}}$ is
chosen. The $\chi^{2}$ per number of degrees of freedom of the fit,
$\chi^{2}_{\mathrm{fit}}/\mathrm{ndf}$, is required to be less than 5. The
measured decay time of the ${\mathrm{B}}_{\mathrm{c}}^{+}$ candidate,
calculated with respect to the associated primary vertex, is required to be
between $150{\,\upmu\rm m}/c$ and 1 mm$/c$.
## 4 Signal and normalization yields
The mass distribution for the selected
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ candidates is shown in Fig. 1. To
estimate the signal yield, an extended maximum likelihood fit to the unbinned
mass distribution is made. The
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ signal is modelled by a Gaussian
distribution and the background by a constant function. The fit results for
the fitted mass and mass resolution of ${\mathrm{B}}_{\mathrm{c}}^{+}$ signal,
$m_{{{\mathrm{B}}_{\mathrm{c}}^{+}}}$ and
$\sigma_{{{\mathrm{B}}_{\mathrm{c}}^{+}}}$, and signal yield
$N_{{{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}}$, are listed in Table 1,
$m_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}}$$\left[\mathrm{GeV}/c^{2}\right]$
Candidates/(10${\mathrm{\,Me\kern-1.20007ptV\\!/}c^{2}}$)
LHCb Figure 1: Mass distribution for selected
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ candidates. The result of a fit using the
model described in the text (red solid line) is shown together with the
background component (blue dashed line).
The statistical significance for the observed signal is determined as
$\mathcal{S}_{\upsigma}=\sqrt{-2\log{\frac{\mathcal{L}_{\mathrm{B}}}{\mathcal{L}_{\mathrm{S+B}}}}}$
where ${\mathcal{L}_{\mathrm{S+B}}}$ and ${\mathcal{L}_{\mathrm{B}}}$ denote
the likelihood associated with the signal-plus-background and background-only
hypothesis, respectively. The likelihoods are calculated with the peak
position fixed to the known mass of ${\mathrm{B}}_{\mathrm{c}}^{+}$ meson [9,
5] and the mass resolution fixed to
10.1${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ as expected from simulation.
The statistical significance of the
$\mathrm{B_{c}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ signal is 4.5 standard deviations.
For the selected ${\mathrm{B}}_{\mathrm{c}}^{+}$ candidates, the existence of
resonant structures is searched for in the ${{\uppi}^{+}}{{\uppi}^{-}}$,
${{\uppi}^{+}}{{\uppi}^{+}}{{\uppi}^{-}}$,
${{\uppi}^{+}}{{\uppi}^{-}}{{\uppi}^{-}}$, $2{{\uppi}^{+}}2{{\uppi}^{-}}$,
$3{{\uppi}^{+}}2{{\uppi}^{-}}$ and
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}{{\uppi}^{-}}$ combinations of final state particles using
the sPlot technique [35], with the reconstructed
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ mass as discriminating variable, to
subtract the background. No significant narrow structures are observed; in
particular, no indication of a contribution from
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{\uppsi{\mathrm{(2S)}}}{{\uppi}^{+}}{{\uppi}^{+}}{{\uppi}^{-}}$,
followed by the
${\uppsi{\mathrm{(2S)}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}{{\uppi}^{-}}$ decay, is seen. The background-subtracted
five-pion mass distribution is shown in Fig. 2, along with the theoretical
prediction in Ref. [14], which describes the data well. The consistency
between data and the model prediction is estimated using a $\chi^{2}$-test and
gives a $p$-value of 14 %. The corresponding $p$-value for the phase space
decay model is 4 %.
Table 1: Signal parameters of the unbinned extended maximum likelihood fit to the ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ mass distribution. Uncertainties are statistical only. Parameter | Value
---|---
$m_{{{\mathrm{B}}_{\mathrm{c}}^{+}}}$ | $\left[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\right]$ | $6273\pm 3\phantom{000}$
$\sigma_{{{\mathrm{B}}_{\mathrm{c}}^{+}}}$ | $\left[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}\right]$ | $11.4\pm 3.4\phantom{0}$
$N_{{{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}}$ | | $32\pm 8\phantom{0}$
$m_{3{{\uppi}^{+}}2{{\uppi}^{-}}}$$\left[\mathrm{GeV}/c^{2}\right]$
Yield/(400${\mathrm{\,Me\kern-1.20007ptV\\!/}c^{2}}$)
LHCb Figure 2: Background-subtracted distribution of five-pion mass from
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ events (points with error bars). The
model prediction from Ref. [14] is shown by a red solid line, and the
expectation from the phase space model is shown by a blue dashed line.
The mass distribution of the selected
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ candidates is shown in Fig. 3, together with the result
of an extended unbinned maximum likelihood fit. The
${\mathrm{B}}_{\mathrm{c}}^{+}$ signal is modelled by a Gaussian distribution
and the background by an exponential function. The fit gives a yield of
$2271\pm 63$ events.
$m_{{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}}$$\left[\mathrm{GeV}/c^{2}\right]$
Candidates/(10${\mathrm{\,Me\kern-1.20007ptV\\!/}c^{2}}$)
LHCb Figure 3: Mass distribution for selected
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ candidates. The result of a fit using the model
described in the text (red solid line) is shown together with the background
component (blue dashed line).
## 5 Efficiency and systematic uncertainties
The overall efficiency for each decay is the product of the geometrical
acceptance of the detector, reconstruction, selection and trigger
efficiencies. These are estimated using simulation and the ratio of the
efficiencies is found to be
$\dfrac{\upvarepsilon({{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}})}{\upvarepsilon({{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}})}=123.8\pm 5.6,$ (2)
where the uncertainty reflects the size of the simulated sample. The large
difference in efficiencies is due to the reconstruction of four additional
low-$p_{\rm T}$ pions in the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ mode. The efficiencies for the data
samples collected at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$ and 8$\mathrm{\,Te\kern-1.00006ptV}$ are
found to be similar and a luminosity-weighted average is used, with the
corresponding systematic uncertainty discussed below.
Many sources of systematic uncertainty cancel in the ratio, in particular
those related to the muon and
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}$ reconstruction and
identification. Those that do not cancel are discussed below and summarized in
Table 2.
Table 2: Relative systematic uncertainties for the ratio $R_{5\uppi}$. The total uncertainty is the quadratic sum of the individual components. Source | Uncertainty $\left[\%\right]$
---|---
Fit model | 6.6
Decay model |
$m_{3{{\uppi}^{+}}2{{\uppi}^{-}}}$ reweighting | 7.7
${\uppsi{\mathrm{(2S)}}}$ mass veto | 3.1
Data-simulation agreement |
Hadron interactions | $4\times 2.0$
Track quality selection | $4\times 0.6$
Trigger | 1.1
Pion identification | 0.7
Selection variables | 1.0
${{\mathrm{B}}_{\mathrm{c}}^{+}}$ lifetime | 0.9
Stability for various data taking conditions | 2.5
Acceptance | 0.8
Total | 13.90
A systematic uncertainty arises from the imperfect knowledge of the shape of
the signal and background in the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ and
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}{{\uppi}^{+}}$ mass
distributions. The dependence of the signal yields on the fit model is studied
by varying the signal and background parameterizations. This is assessed by
using Crystal Ball [36] and double-sided Crystal Ball [37] functions for the
parameterization of the ${\mathrm{B}}_{\mathrm{c}}^{+}$ signals. The
background parametrization is performed using both exponential and polynomial
functions. The maximum observed change of 6.6 % in the ratio of
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ and
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ yields is assigned as a systematic uncertainty.
To assess the systematic uncertainty related to the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ decay model used in the simulation [14],
the reconstructed mass distribution of the five-pion system in simulated
events is reweighted to reproduce the distribution observed in data. As a
cross-check the efficiency is also recalculated using a phase space model for
the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ decays. There is a maximal change in
efficiency of 7.7 %, which is taken as the systematic uncertainty for the
decay model. In addition, the analysis is repeated with the removal of all
${\mathrm{B}}_{\mathrm{c}}^{+}$ candidates where the
${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}{{\uppi}^{-}}$ mass is compatible with originating from
${\uppsi{\mathrm{(2S)}}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}{{\uppi}^{-}}$ decays. The observed difference of 3.1 % is
assigned as an additional systematic uncertainty.
A large class of uncertainties arises from the differences between data and
simulation, in particular those affecting the efficiency for reconstruction of
charged-particle tracks. The largest of these arises from the simulation of
hadronic interactions in the detector, which has an uncertainty of $2\,\%$ per
track [31, 38, 39]. An additional uncertainty associated with the track
quality requirements for the additional four pions in the signal decay is
estimated to be $0.6\,\%$ per track [5, 7]. The trigger efficiency for events
with ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}\\!\rightarrow{\upmu^{+}\upmu^{-}}$ produced in beauty hadron decays is
studied on data in high-yield modes [40, 5] and a systematic uncertainty of
1.1 % is assigned based on the comparison of the ratio of trigger efficiencies
for high-yield samples of
${{\mathrm{B}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{+}}$ and
${{\mathrm{B}}^{+}}\rightarrow{\uppsi{\mathrm{(2S)}}}{{\mathrm{K}}^{+}}$
decays on data and simulation [40].
The systematic uncertainty associated with pion identification is studied
using a sample of
${{\mathrm{B}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{+}}{{\uppi}^{+}}{{\uppi}^{-}}$ decays. The efficiency to
identify a ${{\uppi}^{+}}{{\uppi}^{-}}$ pair is compared for data and
simulation. This comparison shows a 0.35% difference between the data and
simulation in the efficiency to identify a pion pair. As a result of this
study an uncertainty of 0.7 % is assigned for the four additional pions in the
analysis.
The transverse momentum and rapidity spectra for the selected
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ candidates, as well their daughter
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons and pions,
are found to be in good agreement with the predictions from the Bcvegpy
generator. Good agreement in efficiencies determined from the data and
simulation has been observed for all variables used in the selection of
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ candidates. The differences do not exceed 1 %, which is
used as a conservative estimate for the systematic uncertainty from the
selection variables. The agreement between data and simulation has also been
cross-checked using the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ signal by varying the selection criteria
to the values that correspond to a $20\,\%$ change in the signal yield in
simulation. No unexpectedly large deviation is found.
The different acceptance as a function of decay time for the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ and
${{\mathrm{B}}_{\mathrm{c}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ decay modes results in an additional systematic
uncertainty related to the imprecise knowledge of the
${\mathrm{B}}_{\mathrm{c}}^{+}$ lifetime. To assess the related uncertainty,
the decay time distributions for simulated events are reweighted after
changing the ${\mathrm{B}}_{\mathrm{c}}^{+}$ lifetime by one standard
deviation around the value of $509\pm 8\pm 12\rm\,fs$ [10] measured by LHCb
and the efficiencies are recomputed. The observed 0.9 % variation in the ratio
of efficiencies is used as the systematic uncertainty.
The uncertainty related to the stability of the analysis results against
variations of the detector and trigger configurations occuring in different
data-taking periods are tested by studying the ratio of the yields of
${{\mathrm{B}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{+}}{{\uppi}^{+}}{{\uppi}^{-}}$ and
${{\mathrm{B}}^{+}}\\!\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\mathrm{K}}^{+}}$ decays as a function of the data-taking period.
According to this study an additional systematic uncertainty of 2.5 % is
assigned [5].
The last systematic uncertainty originates from the dependence of the
geometrical acceptance on both the beam crossing angle and the position of the
luminosity region. The resulting 0.8 % difference in the efficiency ratios is
taken as an estimate of the systematic uncertainty.
A summary of systematic uncertainties is presented in Table 2. The total
systematic uncertainty on the ratio of the branching fractions $R_{5\uppi}$ is
$13.9\,\%$.
## 6 Results and summary
The first evidence for the decay
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}$ is found using ${\mathrm{p}}{\mathrm{p}}$
collisions, corresponding to an integrated luminosity of 3$\mbox{\,fb}^{-1}$,
collected with the LHCb detector A signal yield of $32\pm 8$ events is found.
The significance, taking into account the systematic uncertainties due to the
fit function, peak position and mass resolution in the fit, is estimated to be
4.5 standard deviations.
Using the
${{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}$ mode as a normalization channel, the ratio of branching
fractions is calculated as
$R_{5\uppi}=\dfrac{N\left({{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}\right)}{N\left({{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}\right)}\times\dfrac{\upvarepsilon({{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}})}{\upvarepsilon({{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}})},$ (3)
where $N$ is the number of reconstructed decays obtained from the fit
described in Sect. 4 and the efficiency ratio is taken from Eq. (2). The ratio
of branching fractions is measured to be
$\dfrac{{\cal
B}\left({{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}3{{\uppi}^{+}}2{{\uppi}^{-}}\right)}{{\cal
B}\left({{\mathrm{B}}_{\mathrm{c}}^{+}}\rightarrow{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}{{\uppi}^{+}}\right)}=1.74\pm 0.44\pm 0.24,$
where the first uncertainty is statistical and the second is systematic. The
result is in agreement with theoretical predictions [14] of 0.95 and 1.1 using
the form factors from Refs. [15] and [16], respectively. This result is also
consistent with analogous measurements in ${\mathrm{B}}^{0}$ and
${\mathrm{B}}^{+}$ meson decays [9]
$\displaystyle\dfrac{{\cal
B}\left({{\mathrm{B}}^{0}}\\!\rightarrow{{\mathrm{D}}^{*-}}3{{\uppi}^{+}}2{{\uppi}^{-}}\right)}{{\cal
B}\left({{\mathrm{B}}^{0}}\\!\rightarrow{{\mathrm{D}}^{*-}}{{\uppi}^{+}}\right)}$
$\displaystyle=$ $\displaystyle 1.70\pm 0.34,$ $\displaystyle\dfrac{{\cal
B}\left({{\mathrm{B}}^{+}}\\!\rightarrow{\bar{\mathrm{D}}^{*0}}3{{\uppi}^{+}}2{{\uppi}^{-}}\right)}{{\cal
B}\left({{\mathrm{B}}^{+}}\\!\rightarrow{\bar{\mathrm{D}}^{*0}}{{\uppi}^{+}}\right)}$
$\displaystyle=$ $\displaystyle 1.10\pm 0.24,$
as expected from factorization.
## Acknowledgements
We thank A.K. Likhoded and A.V. Luchinky for fruitful discussions about the
dynamics of ${\mathrm{B}}_{\mathrm{c}}^{+}$ decays. We express our gratitude
to our colleagues in the CERN accelerator departments for the excellent
performance of the LHC. We thank the technical and administrative staff at the
LHCb institutes. We acknowledge support from CERN and from the national
agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and
Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland);
INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania);
MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal
and GENCAT (Spain); SNSF and SER (Switzerland); NASU (Ukraine); STFC and the
Royal Society (United Kingdom); NSF (USA). We also acknowledge the support
received from EPLANET, Marie Curie Actions and the ERC under FP7. The Tier1
computing centres are supported by IN2P3 (France), KIT and BMBF (Germany),
INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United
Kingdom). We are indebted to the communities behind the multiple open source
software packages on which we depend. We are also thankful for the computing
resources and the access to software R&D tools provided by Yandex LLC
(Russia).
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arxiv-papers
| 2014-04-01T15:56:36 |
2024-09-04T02:50:00.546693
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. An, L.\n Anderlini, J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V.\n Batozskaya, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga, S.\n Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson, J.\n Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien, S.\n Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw,\n S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A. Borgia,\n M. Borsato, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den\n Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown,\n A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M.\n Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez, F. Caponio,\n A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L.\n Carson, K. Carvalho Akiba, G. Casse, L. Cassina, L. Castillo Garcia, M.\n Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, S.-F. Cheung, N.\n Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke,\n M. Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E.\n Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S.\n Coquereau, G. Corti, I. Counts, B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz\n Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, J. Dalseno, P. David, P.N.Y.\n David, A. Davis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De\n Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n S. Donleavy, F. Dordei, M. Dorigo, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof,\n L. Eklund, I. El Rifai, Ch. Elsasser, S. Esen, A. Falabella, C. F\\\"arber, C.\n Farinelli, N. Farley, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira\n Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, M. Firlej, C.\n Fitzpatrick, T. Fiutowski, M. Fontana, F. Fontanelli, R. Forty, O. Francisco,\n M. Frank, C. Frei, M. Frosini, J. Fu, E. Furfaro, A. Gallas Torreira, D.\n Galli, S. Gallorini, S. Gambetta, M. Gandelman, P. Gandini, Y. Gao, J.\n Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, L. Gavardi, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, A. Gianelle, S. Giani', V.\n Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A.\n Gomes, H. Gordon, C. Gotti, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, P. Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz,\n T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, T.W. Hafkenscheid, S.C.\n Haines, S. Hall, B. Hamilton, T. Hampson, X. Han, S. Hansmann-Menzemer, N.\n Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K.\n Hennessy, P. Henrard, L. Henry, J.A. Hernando Morata, E. van Herwijnen, M.\n He\\ss, A. Hicheur, D. Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt,\n N. Hussain, D. Hutchcroft, D. Hynds, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, M. Jezabek, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, N. Jurik, M. Kaballo, S. Kandybei, W.\n Kanso, M. Karacson, T.M. Karbach, M. Kelsey, I.R. Kenyon, T. Ketel, B.\n Khanji, C. Khurewathanakul, S. Klaver, O. Kochebina, M. Kolpin, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, B. Langhans, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam,\n J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T.\n Lesiak, B. Leverington, Y. Li, M. Liles, R. Lindner, C. Linn, F. Lionetto, B.\n Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, P. Lowdon, H.\n Lu, D. Lucchesi, H. Luo, E. Luppi, O. Lupton, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, M.\n Manzali, J. Maratas, J.F. Marchand, U. Marconi, C. Marin Benito, P. Marino,\n R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, F. Martinez Vidal, D. Martins Tostes, A.\n Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M. McCann, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, J.\n Moron, R. Mountain, F. Muheim, K. M\\\"uller, R. Muresan, B. Muster, P. Naik,\n T. Nakada, R. Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N.\n Neufeld, M. Neuner, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V.\n Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M.\n Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano,\n F. Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, L.\n Pappalardo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, C.\n Patrignani, A. Pazos Alvarez, A. Pearce, A. Pellegrino, M. Pepe Altarelli, S.\n Perazzini, E. Perez Trigo, P. Perret, M. Perrin-Terrin, L. Pescatore, E.\n Pesen, K. Petridis, A. Petrolini, E. Picatoste Olloqui, B. Pietrzyk, T.\n Pila\\v{r}, D. Pinci, A. Pistone, S. Playfer, M. Plo Casasus, F. Polci, A.\n Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici, C. Potterat, A.\n Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A. Puig\n Navarro, G. Punzi, W. Qian, B. Rachwal, J.H. Rademacker, B.\n Rakotomiaramanana, M. Rama, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven,\n S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert,\n V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, A.B. Rodrigues, E.\n Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, M.\n Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G.\n Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V. Salustino\n Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O.\n Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba,\n M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, O.\n Shevchenko, V. Shevchenko, A. Shires, F. Sidorov, R. Silva Coutinho, G. Simi,\n M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E. Smith, J.\n Smith, M. Smith, H. Snoek, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, F. Spinella, P. Spradlin, F. Stagni,\n S. Stahl, O. Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S.\n Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah, L. Sun, W.\n Sutcliffe, K. Swientek, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini,\n F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier,\n S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning,\n M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti,\n A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S.\n Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D.\n Vieira, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A.\n Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de Vries, R. Waldi, C.\n Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, D.\n Websdale, M. Whitehead, J. Wicht, D. Wiedner, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G.\n Wormser, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang,\n X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Ivan Belyaev",
"url": "https://arxiv.org/abs/1404.0287"
}
|
1404.0289
|
††thanks: This work was partially supported by DARPA’s QuASAR program.
Publication of the U.S. government, not subject to U.S. copyright.
# Sub-Wavelength Imaging and Field Mapping
via EIT and Autler-Townes Splitting In Rydberg Atoms
Christopher L. Holloway [email protected] Joshua A. Gordon National
Institute of Standards and Technology (NIST), Electromagnetics Division, U.S.
Department of Commerce, Boulder Laboratories, Boulder, CO 80305 Andrew
Schwarzkopf David A. Anderson Stephanie A. Miller Nithiwadee Thaicharoen
Georg Raithel Department of Physics, University of Michigan, Ann Arbor, MI
48109
###### Abstract
We present a technique for measuring radio-frequency (RF) electric field
strengths with sub-wavelength resolution. We use Rydberg states of rubidium
atoms to probe the RF field. The RF field causes an energy splitting of the
Rydberg states via the Autler-Townes effect, and we detect the splitting via
electromagnetically induced transparency (EIT). We use this technique to
measure the electric field distribution inside a glass cylinder with applied
RF fields at 17.04 GHz and 104.77 GHz. We achieve a spatial resolution of
$\bf{\approx}$100 $\bf{\mu}$m, limited by the widths of the laser beams
utilized for the EIT spectroscopy. We numerically simulate the fields in the
glass cylinder and find good agreement with the measured fields. Our results
suggest that this technique could be applied to image fields on a small
spatial scale over a large range of frequencies, up into the sub-THz regime.
atom based metrology, Autler-Townes Splitting, broadband probe, electrical
field measurements and sensors, EIT, sub-wavelength imaging, Rydberg atoms
## I Introduction
The typical probe (sensor) to measure an electric (E) field has a size on the
order of $\lambda/2$ or $\lambda/4$ (where $\lambda$ is the free-space
wavelength of the radiation one intends to measure). One example is a dipole
loaded field probe dipole . These conventional probes can only measure an
E-field strength averaged over the length of the probe. If one is interested
in measuring a field distribution in the neighborhood of a structure with
spatial features smaller than $\lambda$, the conventional probe would be
problematic. For example, consider metasurface structures apm . These
metasurfaces are typically composed of periodic arrays of inclusions or
scatterers. These scatterers are typically sub-wavelength in size (on the
order of $\lambda/10$ or smaller). Furthermore, these scatterers have even
smaller sub-structures (gaps, holes, apertures) that can be as small as
$\sim\lambda/100$. With current methods, it is virtually impossible to measure
E-field distributions on these sub-wavelength scales. The technique we
demonstrate to perform high spatial resolution mapping of RF fields could be
used in design verification and characterization of the metasurfaces and other
sub-wavelength devices and structures.
Figure 1: A four-level system.
Figure 2: Vapor cell setup for measuring EIT, with counter-propagating probe
(red) and coupling (blue) beams. The RF is applied transverse to the optical
beam propagation in the vapor cell.
To measure E-fields with sub-wavelength resolution, we take advantage of a
recently-demonstrated technique that uses atoms as field probes jim -gordon .
Here we demonstrate, for the first time, its spatial resolution capability.
The technique uses room-temperature rubidium atoms as probes, exploiting the
sensitivity of their high-lying Rydberg states to radio frequency (RF)
radiation. (The term RF is used here to cover the conventional RF, microwave,
millimeter wave, and sub-terahertz spectra.) This sensitivity reflects the
large transition matrix elements ($\wp$, on the order of $10^{3}$ to
$10^{4}ea_{0}$) for RF transitions between Rydberg states. We measure an
Autler-Townes splitting autler of Rydberg energy levels in these atoms to
obtain the RF field strength. The energy levels are measured using
electromagnetically induced transparency (EIT) EIT ; EIT2 .
Here we describe the physical principles underlying the technique. Consider a
sample of stationary four-level atoms illuminated by a single weak (“probe”)
light field, as depicted in Fig 1. When the frequency of the light matches the
$|1\rangle$ to $|2\rangle$ atomic resonance, the atoms scatter light from the
incident beam and reduce the transmitted light intensity. If a second strong
(“coupling”) light field is applied resonant with the $|2\rangle$ to
$|3\rangle$ transition, the $|2\rangle$ and $|3\rangle$ states are mixed to
form dressed states which are close in energy. The excitation amplitudes from
$|1\rangle$ to each of these two dressed states then have opposite signs,
leading to destructive quantum interference of these excitation pathways. As
such, a transparency window is opened for the probe light: probe light
transmission is increased. This is the phenomenon known as EIT EIT . If one
applies an RF field which couples states $|3\rangle$ and $|4\rangle$, a third
dressed state is introduced between the two involved in EIT which leads to
constructive interference in the probe absorption. This splits the EIT
resonance in two, and for resonant driving fields the new transmission maxima
are split by the Rabi frequency $\Omega_{RF}$ of the $|3\rangle$ – $|4\rangle$
transition Lukin1999 ; Dutta2007 . This is known as Autler-Townes splitting of
the EIT signal tony . Therefore the frequency splitting $\Delta f_{0}$ between
the transmission maxima allows a measurement of the E-field amplitude of the
RF field via
$|E|=\frac{\hbar}{\wp}{\Omega_{RF}}=2\pi\frac{\hbar}{\wp}{\Delta f_{0}}$ (1)
where $\hbar$ is Planck’s constant and $\wp$ is the electric dipole moment of
the RF transition $|3\rangle$ to $|4\rangle$.
In order to measure the field amplitude for different RF frequencies,
different states $|3\rangle$ and $|4\rangle$ can be chosen. State $|3\rangle$
is selected by tuning the wavelength of the coupling laser. A large range of
atomic transitions can be selected, allowing measurements of microwave fields
over a correspondingly wide selection of frequencies. In essence, the atoms
act as highly-tunable, resonant RF detectors. This is a significant benefit of
using Rydberg atoms as field probes. In broadband , we use this fact to show
how the technique can be used for broadband measurements of RF E-fields,
ranging from 1 GHz to 500 GHz.
## II Experimental Setup
The experimental setup is shown in Fig. 2. We use a cylindrical glass vapor
cell of length 75 mm and diameter 25 mm containing rubidium-85 (85Rb) atoms.
The levels $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$ correspond
respectively to the 85Rb $5S_{1/2}$ ground state, $5P_{3/2}$ excited state,
and two Rydberg states. The probe is a 780 nm laser (“red”) which is scanned
across the $5S_{1/2}$ – $5P_{3/2}$ transition. The probe beam is focused to a
full-width at half maximum (FWHM) of 80 $\mu$m, with a power of 120 nW to keep
the intensity below the saturation intensity of the transition. Figure 3 shows
a typical transmission signal as a function of relative probe detuning
$\Delta_{p}$. The global shape of the curve is the Doppler absorption spectrum
of 85Rb at room temperature. To produce an EIT signal, we apply a counter-
propagating coupling laser (wavelength $\lambda_{c}\approx 480$ nm, “blue”)
with a power of 22 mW, focused to a FWHM of 100 $\mu$m. As an example, tuning
the coupling laser near the $5P_{3/2}$ – $50D$ Rydberg transition results in
distinct EIT transmission peaks corresponding to the transitions from
$5S_{1/2}$ to the allowed $5P_{3/2}$ hyperfine sublevels ($F=4,3,2$), which
are strongly coupled to the fine-structure-split $50D_{5/2}$ and $50D_{3/2}$
Rydberg states. The peaks for the strongest of these cascades are visible atop
the Doppler profile in Fig 3. Differential Doppler shifts between the probe
and coupling beams alter the frequency separations between EIT peaks in the
probe transmission spectrum. Splittings of $5P_{3/2}$ hyperfine states are
scaled by $1-\lambda_{c}/\lambda_{p}$, while splittings of Rydberg states are
scaled by $\lambda_{c}/\lambda_{p}$ EIT_Adams . The latter factor is relevant
to measurements of RF-induced splittings of EIT peaks.
Hereafter, for each EIT system we investigate, we focus on the strongest peak,
corresponding to the $5P_{3/2}(F=4)$ – $nD_{5/2}$ transition. We take this
peak to be at $\Delta_{p}=0$. In order to improve the signal-to-noise ratio,
we use heterodyne detection. We modulate the blue laser amplitude with a 30
kHz square wave and detect any resulting modulation of the probe transmission
with a lock-in amplifier. This removes the Doppler background and isolates the
EIT signal as shown in the black curve of Fig. 4. Here we tune the coupling
laser near the $5P_{3/2}$ – $28D_{5/2}$ transition. Application of RF to
couple states $28D_{5/2}$ and $29P_{3/2}$ splits the EIT peak as shown in the
gray curve. We measure the frequency splitting of the EIT peaks in the probe
spectrum, $\Delta f$, and determine the E-field amplitude by
$|E|=2\pi\frac{\hbar}{\wp}\frac{\lambda_{p}}{\lambda_{c}}\Delta f\quad.$ (2)
Note here the use of the Doppler scaling factor, not present in Eq. 1 for
stationary atoms.
The E-field sensing volume is determined by the overlap of the RF, probe beam,
and coupling beam within the vapor cell. Based on the geometries given above,
this volume is a cylinder of length 75 mm and diameter 80 $\mu$m. The small
optical beam diameter gives the measurement high spatial resolution in the
dimensions transverse to the optical beams, which is crucial in the
experiments presented next.
Figure 3: Probe transmission as a function of $\Delta_{p}$ for the three-level
$5S_{1/2}-5P_{3/2}-50D$ EIT system. EIT peaks are visible for transitions
corresponding to (right to left) $5P_{3/2}(F=4)$ – $50D_{5/2}$,
$5P_{3/2}(F=3)$ – $50D_{5/2}$, and $5P_{3/2}(F=4)$ – $50D_{3/2}$.
Figure 4: Black curve: EIT-signal as a function of $\Delta_{p}$ for the EIT
system $5S_{1/2}-5P_{3/2}-28D_{5/2}$. Gray curve: The $28D_{5/2}$ level is
coupled to the $29P_{3/2}$ level by a 104.77 GHz RF field.
## III Sub-Wavelength Field Mapping
When an electromagnetic (EM) wave is incident onto a hollow dielectric
cylinder, standing waves typically develop inside the cylinder due to internal
reflections from the cylinder walls. The resulting field distribution will
vary depending on the EM frequency. Using the technique explained in the
previous section, we image the field inside our glass vapor cell (which is,
electromagnetically, a hollow dielectric cylinder) for two different RF
frequencies: 104.77 GHz and 17.04 GHz. For the 104.77 GHz measurements, we
deliver the RF with an open-ended waveguide (see Fig. 5); for 17.04 GHz we use
a horn antenna. In each of these measurements, the vapor cell is placed on a
translation stage with 12 mm of travel, see Fig. 5. The cell is then
translated in a direction perpendicular to the propagation directions of the
optical beams. This allows the imaging of RF fields inside the cell as a
function of the spatial coordinate parallel to the translation axis. The
spatial resolution is limited by the optical beam diameter (80 $\mu$m).
Figure 5: Experimental setup for field-mapping measurements with EIT. The
vapor cell is on a translation stage and is scanned with respect to the
optical beams. The waveguide in the photo is closer than that used in the
measurement.
We first perform measurements at 104.77 GHz. The blue laser is tuned to
$\approx 482.23$ nm to couple states $5{\rm P}_{3/2}$ and $28{\rm D}_{5/2}$,
and the 104.77 GHz field couples $28{\rm D}_{5/2}-29{\rm P}_{3/2}$. The open-
ended waveguide is spaced 0.14 m from the focal axis of the lasers, and is
supplied with 0.58 mW of RF power (measured with a power meter attached to the
end of the waveguide).
The cell is translated away from the source in discrete steps. At each step
position we measure the splitting of the EIT signal. We convert to an electric
field using Eq. 2, and the dipole matrix element for this transition,
$\wp=473.14ea_{0}$. Here, $e$ is the electron charge and $a_{0}$ is the Bohr
radius. The dashed line in Fig. 6 shows the measured splitting (right axis)
and the E-field amplitude (left axis) as a function of position for a step
size of 0.25 mm. The crosses show a second scan at higher resolution with a
step size of 0.10 mm, which corresponds to the larger of the two laser beam
widths. The origin in Fig. 6 corresponds to a distance of approximately 8.4 mm
between the laser beams and the inside edge of the cell that is furthest from
the source.
Figure 6: Measured EIT splitting, $\Delta f$ (right axis) and corresponding
electric field amplitude $|E|$ (left axis) as a function of position inside
the cell at 104.77 GHz.
These two sets of measurements lie on top of one another, showing the
measurement is repeatable. The results further demonstrate significant
periodic field variation inside the cell. We see up to approximately $\pm
20~{}\%$ variation in the field amplitude over the 12 mm scan. The average
period of the observed pattern is approximately half the wavelength of the RF
($\lambda_{RF}/2=1.43$ mm). The two sets of measurements yield field-mapping
resolutions of $\approx\lambda/10$ and $\approx\lambda/30$, respectively. We
have thus shown that this method enables sub-wavelength mapping of 104.77 GHz
radiation fields, and, importantly, that the achieved spatial resolution is
comparable to the laser beam spot size.
Next, we compare the measured field distribution inside the cell to the
results of a three-dimensional numerical full wave simulation performed using
a commercial finite element code. It is challenging to perform such a
numerical simulation of the actual experimental setup at high frequencies
because of computer memory requirements. This is partially due to the small RF
wavelength at 104.77 GHz, and the relatively large cell size (several RF
wavelengths at 104.77 GHz). To overcome these issues we did the following.
Instead of modeling the actual open-ended waveguide placed 0.14 m from the
cell, we performed a numerical simulation for a plane-wave impinging on the
cell with a field strength of 2.8 V/m. In order to determine this field-
strength value for the plane-wave, the open-ended waveguide source antenna was
modeled independently and the numerical simulated far-field radiation pattern
and field strength was determined. Using the measured working distance and
power mentioned above, this yields a field amplitude of 2.8 V/m at the
location of the laser beams in the experiment. The cell in the simulation has
dimensions as mentioned above for the experiment, with glass having
$\epsilon_{r}=5.5$ at RF frequencies. We assumed $\epsilon_{r}=1$ for the
region inside the cell. Figure 7 shows the numerical results for the field
inside the cell with the incident plane wave source. This contour plot shows
the expected field distribution, in which the field variation is primarily
along the propagation direction of the incident RF wave. This indicates that
our measurement method gives good spatial resolution in the only dimension
along which there is significant field strength variation. To quantitatively
compare simulation with measurement, we average the numerical results along 92
$\%$ of the length of the cell. The comparison is shown in Fig. 8, which shows
that the simulation and measurements give similar spatial variation. Both show
field distributions with a period roughly equal to half the RF wavelength. We
see good qualitative agreement between the numerical results and the data.
(a) (b)
Figure 7: Simulation of electric-field amplitude $|E|$ for a plane wave
incident onto the vapor cell from the right: (a) incident + scattered, and (b)
scattered. The region shown in the figure is a horizontal planar cut through
the center of the cell, with half of the cell shown. The E-field is on a
linear colorscale ranging from 0.7 V/m (blue) to 5 V/m (red).
Figure 8: Comparison of experimental and simulated $|E|$ as a function of
position inside the cell at 104.77 GHz.
To study field mapping in the cm-wave regime we repeat the experiment with a
17.04 GHz source. Here, the blue laser is tuned to $\approx 480.13$ nm to
couple states $5P_{3/2}$ and $50D_{5/2}$. The RF couples states $50D_{5/2}$
and $51P_{3/2}$. We use a horn antenna at a distance of 0.880 m from the laser
beams. Figure 9 shows the measured E-field as a function of position inside
the cell, where we have used $\wp=1574.83ea_{0}$. For this measurement the
origin of the position axis corresponds to the laser beams being approximately
10 mm from the inner surface of the cell that is furthest from the RF source.
The variation of the measured field is approximately $\pm 50$ $\%$ of its
average, and the observed separation between the maximum and minimum is
$\approx\lambda_{RF}/4$.
We perform numerical simulations for this case as well, using a plane wave
source with a field amplitude at the vapor cell determined from a far-field
calculation. Based on source power, cable losses, and known antenna
characteristics, the field amplitude at the location of the laser beams is
0.76 V/m. In Fig. 9, we compare the data with results obtained from the
numerical simulation. Here, agreement is good.
Figure 9: Experimental and simulated $|E|$ as a function of position inside
the cell at 17.04 GHz.
## IV Discussion and Conclusion
In this paper we have demonstrated a technique based on Rydberg atoms, EIT,
and Autler-Townes splitting which can perform sub-wavelength imaging and field
mapping of RF radiation. We have validated the approach by comparing the
measured field inside a hollow glass cylinder to results obtained from a full-
wave numerical simulation.
While the spatial resolution of our measurements is determined by the $\approx
100\mu$m beam widths of our lasers, the spatial resolution of the method is in
principle limited by the optical diffraction limit. This is a significant
improvement over the measurement resolution achievable by conventional probes.
There are many possible applications of this technique. For example, the
sensing volume could be scanned over a printed-circuit-board (PCB) or a
metasurface in order to map their fields, as well as other applications where
E-field measurements on a small spatial resolution are desired. We aim to
demonstrate these applications in future work.
## References
* (1) M. Kanda and L. Driver, “An isotropic electric-field probe with tapered resistive dipoles for broad-band use, 100 kHz to 18 GHz,” IEEE Trans. on Microwave Theory and Techniques, vol. 35, no. 7, pp. 124-130, Frb., 1987.
* (2) C.L. Holloway, E.F. Kuester, J.A. Gordon, J. O’Hara, J. Booth and D.R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials”, IEEE Antennas and Propagation Magazine, vol. 54, no. 2, pp. 10-35, April 2012.
* (3) J.A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau and J. P. Shaffer, “Microwave electrometry with Rydberg atoms in a vapor cell using bright atomic resonances”, Nature Physics 8, 819, 2012.
* (4) C.L. Holloway, J.A. Gordon, A. Schwarzkopf, D. A. Anderson, S. A. Miller, N. Thaicharoen, G. Raithel, S. Jefferts, and T. Heavner, “Broadband Rydberg Atom-Based Electric-Field Probe/Sensor: From Self-Calibrating Measurements to Sub-Wavelength Imaging,” IEEE Trans. on Antenna and Propag., 2014\.
* (5) J.A. Sedlacek, A. Schwettmann, H. Kübler, and J. P. Shaffer, “Atom-Based Vector Microwave Electrometry Using Rubidium Rydberg Atoms in a Vapor Cell”, Phys. Rev. Lett., 111, 6, 2013
* (6) J.A. Gordon, C.L., Holloway, S., Jefferts, and T. Heavner, “Quantum-Based SI Traceable Electric-Field Probe,”, Proc 2010 Int. Symp. on Electromagnetic Compatibility, Fort Lauderdale, FL, pp. 321-324, 25-30 July, 2010.
* (7) J.A. Gordon, C.L. Holloway, A. Schwarzkopf, D. A. Anderson, S.A. Miller, N. Thaicharoen, and G. Raithel, “Millimeter-Wave Detection via Autler-Townes Splitting In Rubidium Rydberg Atoms,” IEEE Trans. on Antenna and Propag., 2014.
* (8) S.H. Autler and C.H. Townes, “Stark Effect in Rapidly Varying Fields,” Physical Review, vol. 100, no. 2, pp. 703–722, 1955.
* (9) M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in Coherent Media,” Reviews Modern Physics, vol. 77, pp. 633-673, April, 2005.
* (10) K.J. Boller, A. Imamolu, and S.Ee Harris, “Observation of electromagnetically induced transparency,” Physical Review Letters, vol. 66, no. 20, pp. 2593-2596, May, 1991.
* (11) M. D. Lukin, S. F. Yelin, M. Fleischhauer, and M. O. Scully. “Quantum interference effects induced by interacting dark resonances,” Phys. Rev. A 60, 3225 (1999)
* (12) B. K. Dutta and P. K. Mahapatra. Phys. Scr. 75 345, (2007)
* (13) T.Y. Abi-Salloum, “Electromagnetically induced transparency and Autler-Townes splitting: Two similar but distinct phenomena in two categories of three-level atomic systems,” it Phys. Rev. A, 81, 053836, 2010.
* (14) A.K. Mohapatra, T.R. Jackson, C.S Adams, “Coherent optical detection of highly excited Rydberg states using electromagnetically induced transparency,” Phys. Rev. Lett. 98, 113003 (2007).
|
arxiv-papers
| 2014-04-01T15:58:04 |
2024-09-04T02:50:00.556820
|
{
"license": "Public Domain",
"authors": "Christopher L. Holloway, Joshua A. Gordon, Andrew Schwarzkopf, David\n A. Anderson, Stephanie A. Miller, Nithiwadee Thaicharoen, and Georg Raithel",
"submitter": "Christopher Holloway",
"url": "https://arxiv.org/abs/1404.0289"
}
|
1404.0318
|
# Rheology of weakly wetted granular materials \- a comparison of experimental
and numerical data
Rüdiger Schwarze Institute of Mechanics and Fluid Dynamics, TU Bergakademie
Freiberg. Lampadiusstr. 4, 09596 Freiberg, Germany Anton Gladkyy Institute
of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg. Lampadiusstr. 4,
09596 Freiberg, Germany Fabian Uhlig Institute of Mechanics and Fluid
Dynamics, TU Bergakademie Freiberg. Lampadiusstr. 4, 09596 Freiberg, Germany
Stefan Luding Multi Scale Mechanics (MSM), Engineering Technology (CTW) and
MESA+, University of Twente. P.O.Box 217, 7500 AE Enschede, The Netherlands
###### Abstract
Shear cell simulations and experiments of weakly wetted particles (a few
volume percent liquid binders) are compared, with the goal to understand their
flow rheology. Application examples are cores for metal casting by core
shooting made of sand and liquid binding materials.
The experiments are carried out with a Couette-like rotating viscometer. The
weakly wetted granular materials are made of quartz sand and small amounts of
Newtonian liquids. For comparison, experiments on dry sand are also performed
with a modified configuration of the viscometer. The numerical model involves
spherical, monodisperse particles with contact forces and a simple liquid
bridge model for individual capillary bridges between two particles. Different
liquid content and properties lead to different flow rheology when measuring
the shear stress-strain relations. In the experiments of the weakly wetted
granular material, the apparent shear viscosity $\eta_{g}$ scales inversely
proportional to the inertial number $I$, for all shear rates. On the contrary,
in the dry case, an intermediate scaling regime inversely quadratic in $I$ is
observed for moderate shear rates. In the simulations, both scaling regimes
are found for dry and wet granular material as well.
## 1 Introduction
Dry granular matter and its flow rheology have been the subject of detailed
studies during the last years and slowly their interesting behavior becomes
more and more clear [1, 2]. The other extreme case are particles suspended in
fluids – a field of wide relevance in industrial processes – which nowadays
are mostly understood and can be modeled reasonably well [3, 4, 5, 6, 7].
However, weakly wetted granular materials have recently attracted new
attention, see e.g. Ref. [8, 9, 10, 11, 12], even though they were studied
earlier [13, 14, 15, 16]. Wet as well as dry granular rheology [17, 18] plays
an important role in geotechnical and geophysical context [19, 20], as well as
in several technical processes e.g. in growth agglomeration [21], for die-
filling [22, 23], or in the production of sand cores for casting by core
shooting [24]. In the latter example, weakly wetted granular materials are
mixtures of a granular matter and few volume-percent of liquid binder. Like
dry granular materials, they exhibit non-Newtonian flow behavior, where the
relations between shear stresses and shear rates, for example, can be
expressed by nonlinear functions. Since the presence of small amounts of
liquid change the rheological behavior of the granular material markedly [25,
26], detailed knowledge of the constitutive equations of these materials is of
fundamental importance for the control of the corresponding processes. As an
example, the filling flow – even in complex core boxes – can be analyzed by
CFD simulations [27] when the rheological model of the material is known.
Figure 1: Characteristics of core shooting material with quartz sand F35
[28]: Rheological data from measurements with dry F35 and with core shooting
material with a mass ratio $m_{SR}/m_{F}=0.02$ between the mass $m_{SR}$ of
the synthetic resin binder and the mass $m_{F}$ of F35, see section 2 and Ref.
[25] for more details. Inset: SEM picture of a sample of F35.
As an example, Fig. 1 shows some rheometer experiment [25] flow curves of a
typical core shooting material made of quartz sand F35 (with mean particle
diameter $d_{P}=0.18$ mm) [28] and synthetic resin binder and the pure, dry
F35. The scaling $\eta_{g}\left(\dot{\gamma}_{g}\right)$ of the apparent
viscosity $\eta_{g}$ with the shear rates $\dot{\gamma}_{g}$ differs markedly
between the core shooting material and the dry granular F35.
Heuristically, this difference can be explained by the capillary bridges of
the liquid binder between individual sand particles. A more detailed review of
the rheological measurements is given in the next section.
Unfortunately, the realization of rheometer experiments involving weakly
wetted granular materials is complicated and time consuming. Therefore,
alternative and more efficient methods for rheological investigations are
highly desirable. In this paper, we use the split-bottom ring shear setup of a
rheometer [29] for discrete-element method (DEM) simulations of these partly
wet granular materials. Similar DEM simulations of wet granular materials have
been performed in order to study the micro-mechanics in cohesive mixing
processes [30], the discharge from hoppers [31] or the mixing in a blade mixer
[32]. In these papers, explicit capillary forces are added to the contact
forces in order to properly describe the interactions between two particles in
the wet granular material. As an alternative to the most simple approach
pursued below, Grima and Wypych [33] employ an implicit model of the capillary
force in their simulations, while Mani et al. [8] explicitly allow for liquid
migration between the particles and across the bridges.
Starting from the DEM results, we apply a local micro-macro transition [34,
35, 36] in order to obtain rheological flow rules for weakly wetted granular
materials. First results indicate, that the numerically determined flow rules
exhibit similar differences between dry and wet granular materials as in the
experiments. The device used to measure the stress-strain relations is the
split-bottom ring-shear cell as invented by Fenistein et al. [29] and used by
others for dry, frictional and (van der Waals) elasto-plastic adhesive
particle systems [34, 35, 36, 37, 38, 39, 40, 41, 42].
## 2 Previous rheological measurements
Rheological data of weakly wetted granular materials (core shooting materials)
have been recently measured in a Searl-type rotating viscometer [25] with a
fixed bottom and outer cylinder wall and a rotating inner cylinder wall. Sand
particles are glued to the fixed and rotating cylinder walls in order to
define proper wall shear stress conditions and reduce wall-slip. In these
shear cell experiments, a shear band width $w_{sb}$ of about 10 - 20 particle
diameters $d_{P}$ was expected.
For measurements on dry sand, the gap width $b_{dry}=2$ mm $\simeq
10\,d_{P}\simeq w_{sb}$ between the inner and outer cylinder was fitted to the
assumed width of the shear zone in the granular material. With this setup, all
the material in the gap should be sheared. Here, the upper annulus of the
viscometer gap was open.
For measurements on core shooting materials, the upper annulus was closed by a
movable circular ring in order to superimpose an external pressure $p_{e}$ to
the weakly wetted granular material. Without $p_{e}$, fissures arose in the
shear zone, which induced the disruption of the measurements. In all
measurements, $p_{e}$ exceeded the pure hydrostatic pressure level
$p_{h}=\rho\,g\,h\simeq 1$ kPa of the core shooting material, with bulk
density $\rho\simeq 1400$ kg/m3 and fill level $h\simeq 70$ mm in the
viscometer gap. The gap width $b_{wet}=10$ mm $\simeq 50\,d_{P}\simeq
5\,w_{sb}$ between the inner and outer cylinder was increased for these
experiments. We assume, that parts of the material in the gap were not sheared
in this configuration. Therefore, the exact width of the shear zone remains
unknown, which induces an uncertainty in the absolute values of the shear
rate. However, the scaling $\eta_{g}\left(\dot{\gamma}_{g}\right)$ should have
been correctly resolved in these measurements, too, only amplitudes of
$\eta_{g}$ are maybe somewhat shifted.
Figure 2: Apparent shear viscosity $\eta_{g}\left(\dot{\gamma}_{g}\right)$ for
core shooting material made of F35 and synthetic resin binder with
$m_{SR}/m_{F}=0.01$ (top), and made of F35 and sodium silicate binder with
$m_{SS}/m_{F}=0.02$ (bottom); error bars indicate the standard deviation of
the experimental data. Results of dry F35 are also given for comparison,
performed with smaller gap $b$. Measurements are made with different external
pressure levels $p_{e}$. Continuous and dashed lines indicate the scaling
between $\eta_{g}$ and $\dot{\gamma}_{g}$, obtained from least square fits of
Eq. (2) to the wet data, for $0.4$ s${}^{-1}<\dot{\gamma}_{g}<200$ s-1.
In Fig. 2, some results of the rheometer measurements on two core shooting
materials with F35 as basic sand but different liquid binders are displayed.
For both materials, the apparent shear viscosity
$\eta_{g}=\frac{\left|\tau_{g}\right|}{\dot{\gamma}_{g}}$ (1)
of the core shooting material exceeds the values of the dry granular F35
markedly for shear rates $\dot{\gamma}_{g}>10$ s-1. Here, the quantities
$\tau_{g}$ and $\dot{\gamma}_{g}$ are _globally_ defined, i.e. they describe
the dynamics of the bulk granular material and set-up: The mean shear stress
$\tau_{g}=F_{t,o}/A_{o}$ is the quotient of the tangential force $F_{t,o}$ at
the outer cylinder wall, which is measured by a force transducer, and the
outer cylinder wall area $A_{o}$. The mean shear rate
$\dot{\gamma}_{g}=U_{i}/b$ is approximated by the quotient of the velocity
$U_{i}$ of the rotating inner cylinder wall and the gap widths $b=b_{dry}$ or
$b=b_{wet}$.
Problems in the realization of the measurements are indicated by the large
uncertainty in the results for the core shooting material with synthetic resin
binder and $p_{e}=8$ kPa.
| F35, synthetic resin | F35, sodium silicate
---|---|---
| $m_{SR}/m_{F}=0.01$ | $m_{SS}/m_{F}=0.02$
$p_{e}$ [kPa] | $4$ kPa | $8$ kPa | $4$ kPa | $8$ kPa
$K$ | 2084 | 3291 | 1807 | 2603
$\alpha$ | 0.90 | 1.07 | 1.11 | 1.10
Table 1: Parameters of the flow rules for core shooting materials of Fig. 2 in
Eq. (2). Values of $K$ and $\alpha$ are obtained from a least square fit of
Eq. (2) to the wet data, for $0.4$ s${}^{-1}<\dot{\gamma}_{g}<200$ s-1. Here,
values of $K$ are mere fit-parameters since their units are non-linearly
dependent on the values of $\alpha$ and are thus not given.
The flow rules of the core shooting materials in the measured shear rate
intervals can be described as a first approximation by a power law
$\displaystyle\eta_{g}\left(\dot{\gamma}_{g}\right)$
$\displaystyle=K\,\left|\dot{\gamma}_{g}\right|^{-\alpha}$ (2)
with the consistency factor $K$ and a power law index $\alpha$. (Note that the
nonlinear power-law spoils the units and has to be phrased better using
dimensionless quantities as discussed below.) Table 1 summarizes the values of
$K$ and $\alpha$ for the different core shooting materials for different
levels of $p_{e}$.
All measurements indicate shear-thinning behavior of the core-shooting
materials with power laws close to $\left(\dot{\gamma}_{g}\right)^{-1}$.
However, some interesting differences are also found: At the same levels of
$p_{e}$, the parameters $K$ and $\alpha$ differ markedly between the two core
shooting materials with the same basic sand but different liquid binders.
Obviously, the presence of small amounts of different liquids in the same
basic granular material affects the flow behavior significantly. For
increasing $p_{e}$, the power law index $\alpha$ in case of core shooting
material made of F35 with synthetic resin binder decreases, whereas $\alpha$
rises for core shooting material made of F35 with sodium silicate binder.
The proposed scaling $\eta_{g}\propto p_{e}/\dot{\gamma}_{g}$ of Jop et al.
[17] is only found in the core shooting material with sodium silicate binder,
where the continuous and dashed curves are nearly parallel. In case of the
core shooting material with synthetic resin binder, where the continuous and
dashed curves intersect, the scaling seems to be more complex.
The rheological behavior of dry sand in the same rheometer (but with smaller
gap $b$) seems to be more complex; we fail to fit the experimental data by a
simple power law ansatz, Eq. (2), with constant $K$ and $\alpha$. Instead, we
found three distinct regimes for low, medium and high shear rates. The scaling
is $\eta_{g}\propto\dot{\gamma}_{g}^{-1.1}$ for low,
$\eta_{g}\propto\dot{\gamma}_{g}^{-1.7}$ for medium, and
$\eta_{g}\propto\dot{\gamma}_{g}^{-1.2}$ for high shear rates, see Fig.
2(top). These different regimes are not found for the weakly wetted granular
materials.
On the way to a better theoretical description of this complex rheology, it is
advisable/necessary to use a dimensionless shear rate:
$\displaystyle I_{g}$
$\displaystyle=\frac{\dot{\gamma}_{g}\,d_{P}}{\sqrt{{p_{g}}/{\rho}}}$ (3)
as defined in [1]. This number describes the ratio of inertial and confining
stress, i.e. pressure, of the sheared material. For the evaluation of the
different measurements, the global pressure level $p_{g}$ is estimated by
$p_{g}\simeq p_{e}$, because $p_{e}\gg p_{h}$ as explained above. The
quantities $\dot{\gamma}_{g}$, $p_{g}$, and $I_{g}$ are _global_ parameters,
i.e. are obtained from an external measurement that implies an average over
the whole shear cell. In case of the dry sand, where no external pressure
$p_{e}$ was applied, $I_{g}$ cannot be properly evaluated for these
measurements.
Figure 3: Apparent shear viscosity as a function of dimensionless shear rate
for the two core shooting materials and the two external pressure levels
$p_{e}$ of Fig. 2. Continuous and dashed lines indicate the scaling
$\eta_{g}\sim I_{g}^{-\alpha}$, with $\alpha$ from table 1.
Fig. 3 summarizes the observed apparent shear viscosities
$\eta_{g}\left(I_{g}\right)$ for the different core shooting materials and the
different levels of $p_{e}$. In this semi-dimensionless representation, all
measurements follow roughly the same shear-thinning behavior of the core-
shooting materials with scaling as $\eta_{g}\propto I_{g}^{-\alpha}$ with
$\alpha\simeq 1$, which are indicated by continuous and dashed lines as in
Fig. 2.
Interestingly, the synthetic binder is sensitive to the confining stress,
whereas the sodium silicate binder is not. The former displays considerably
higher apparent viscosity for smaller $p_{e}$. The more systematic
investigation of these subtle differences are ongoing and will be published
elsewhere.
## 3 Model fundamentals
The flow of a sheared, weakly wetted granular material is now investigated by
means of DEM simulations.
### 3.1 Equations of motion
The DEM model is based on Newton’s equations of motion for the translational
and rotational degrees of freedom of a spherical particle
$\displaystyle m_{i}\dfrac{d^{2}\underline{r}_{i}}{dt^{2}}$
$\displaystyle=\sum\limits_{\left\\{i,j\right\\}=c}\,\underline{f}_{ij}+m_{i}\,\underline{g}$
(4) $\displaystyle J_{i}\dfrac{d\underline{\omega}_{i}}{dt}$
$\displaystyle=\sum\limits_{\left\\{i,j\right\\}=c}\,\underline{l}_{ij}\times\underline{f}_{ij}$
(5)
with mass $m_{i}$, position $\underline{r}_{i}$, moment of inertia $J_{i}$,
and angular velocity $\underline{\omega}_{i}$ of particle $i$. The right hand
side terms in Eq. (4) are the sum of the inter-particle forces
$\underline{f}_{ij}$ due to contacts $c$ with particles $j$ and volume/body
forces, here from gravity $\underline{g}$. The right hand side in Eq. (5) is
the torque arising from the contacts $c$ with the branch vector
$\underline{l}_{ij}$, i.e., the distance vector from the particle center to
the contact point of the two particles $i$ and $j$.
The inter-particle forces $\underline{f}_{ij}$ are modeled by well-known
force-overlap relations in combination with capillary forces, which are
induced by liquid bridges between the interacting particles. Details of the
contact force modeling are given in the next subsection.
### 3.2 Contact force law
Since the realistic modeling of the deformation of two interacting particles,
e.g. in a core shooting material, is much too complicated, the inter-particle
force is described by a force-overlap relation. The modeling of the contact
force is based on the quantities/symbols given in Figs. 4 and 5.
The force $\underline{f}_{ij}$ on particle $i$, from particle $j$, at contact
$c$, can be decomposed into a normal and a tangential part as
$\displaystyle\underline{f}_{ij}$
$\displaystyle=f_{ij}^{n}\,\hat{n}_{ij}+f_{ij}^{t}\,\hat{t}_{ij}+f_{ij,c}\,\hat{n}_{ij}\,.$
(6)
with normal $\hat{n}_{ij}$ and tangential unit vector $\hat{t}_{ij}$ at the
contact point $c$. The normal force $f_{ij}^{n}$, the tangential force
$f_{ij}^{t}$ and the capillary force $f_{ij,c}$ due to a liquid bridge between
particle $i$ and $j$ are specified below.
Figure 4: Sketch of the contact between two particles $i$, $j$.
Two dry particles $i$, $j$ with radius $R$, which are moving with velocities
$\underline{v}_{i}$ and $\underline{v}_{j}$ and rotating with angular
velocities $\underline{\omega}_{i}$ and $\underline{\omega}_{j}$, interact, if
the normal overlap $\delta$ is positive
$\displaystyle\delta$
$\displaystyle=2R-\left(\underline{r}_{i}-\underline{r}_{j}\right)\cdot\hat{n}_{ij}>0\,.$
(7)
The normal contact force involves a linear repulsive and a linear dissipative
force,
$\displaystyle f_{ij}^{n}$
$\displaystyle=k_{n}\delta+\gamma_{n}\,v_{ij}^{n}\,,$ (8)
with normal spring stiffness $k_{n}$, normal viscous damping $\gamma_{n}$ and
normal velocity $v_{ij}^{n}$.
Our model is able to capture friction forces and torques, as well as rolling
and torsion torques, as described in Ref. [43]. For the sake of brevity we did
set the latter interactions to zero and focus on friction only, i.e. the
tangential friction force is
$\displaystyle f_{ij}^{t}$ $\displaystyle=k_{t}\chi+\gamma_{t}\,v_{ij}^{t}\,,$
(9)
with tangential spring stiffness $k_{t}$, tangential viscous damping
$\gamma_{t}$ and tangential velocity $v_{ij}^{t}$, where $\chi$ is the
integral of $v_{ij}^{t}$ over time, adapted such that the tangential force is
limited by Coulomb sliding friction
$\displaystyle f_{ij}^{t}$ $\displaystyle\leq\mu_{C}\,f_{ij}^{n}\,,$ (10)
with Coulomb’s coefficient of friction $\mu_{C}$.
Note, that ${\displaystyle f_{ij}^{n}}$ and $f_{ij}^{t}$ give only non-zero
contributions to $\underline{f}_{ij}$, when the two particles are in contact,
$\delta>0$. The capillary force $f_{ij,c}$ is also active when two particles
separate after a contact. Details of the modeling of $f_{ij,c}$ are given in
the next subsection.
### 3.3 Capillary forces
The shapes of liquid bridges between individual particles of a granular medium
depend strongly on the amount of the added liquid, see e.g. Refs. [14, 16].
For core shooting materials with low mass ratios between binder and dry sand
or sand-like materials we expect, based on the findings in [16], that the
grains are connected by individual capillary bridges. The relevant parameters
of such a bridge are indicated in Fig. 5.
Figure 5: Sketch of a liquid bridge between particles $i$ and $j$ with the
bridge length $s$, the bridge volume $V_{b}$ and the contact angle $\theta$.
The liquid volume of the bridge is $V_{b}$. The length $s=-\delta$ of the
bridge is given by the surface distance of the two particles, which are
connected by the bridge. Finally, the equilibrium contact angle
$\theta<90^{o}$ is found at the bridge-particle contact line.
With these parameters, we approximate the inter-particle force $f_{ij,c}$ of
the capillary bridge according to the proposal of Willett et al. [15], see
also [16, 32], who calculated $f_{ij,c}$ with the gorge method at the bridge
neck
$\displaystyle f_{ij,c}$
$\displaystyle=\frac{2\,\pi\,\gamma\,R\,\cos\left(\theta\right)}{1+1.05\hat{s}+2.5\hat{s}^{2}}\,.$
(11)
with surface tension, $\gamma$ and dimensionless $\hat{s}=s\,\sqrt{R/V_{b}}$.
Note, that $f_{ij,c}$ exists only during and past a contact between particle
$i$ and $j$, providing a non-zero contribution to $\underline{f}_{ij}$ until
the total distance $s$ between $i$ and $j$ rises above the critical bridge
length $s>s_{crit}$. Then, the bridge ruptures and $f_{ij,c}$ becomes zero.
Several authors have proposed correlations between $s_{crit}$ and other
parameters of the capillary bridge [13, 14]. We use the approximation of
Willett et al. [15]
$\displaystyle s_{crit}$
$\displaystyle=R\,\left(1+\frac{1}{2}\,\theta\right)\,\left[\left(\frac{V_{b}}{R^{3}}\right)^{\frac{1}{3}}+0.1\,\left(\frac{V_{b}}{R^{3}}\right)^{\frac{2}{3}}\right]\,.$
(12)
The model equations presented above were implemented into the open-source DEM
software package LIGGGHTS, version 2.3.2. The simulations were carried out at
the HPC cluster CVC at the University Computer Center of the TU Bergakademie
Freiberg.
### 3.4 Capillary force model validation
Figure 6: Forces in a capillary bridge with contact angle $\theta=0^{\circ}$
between two equal-sized particles; lines indicate results of the DEM
simulation, points give the experimental data of Willett et al. [15]. Liquid
bridge volumes $V_{b}$ are given in nano-liters ($10^{-9}$l$=$nl).
For the validation of the capillary force calculation, we perform DEM
simulations according to the experiments described in Willett et al. [15].
There, the separation process of two equal-sized particles with $R=2.4$ mm
connected by a capillary bridge has been investigated. The liquid surface
tension was $\gamma=20.6$ mN/m. The particles were perfectly wetted by the
liquid, i.e. $\theta=0^{\circ}$. Experiments with different liquid bridge
volumes $V_{b}=13.6,\,31.3$ and $74.2$ nl have been carried out.
In the DEM simulations, we track the capillary force $f_{12,c}$ due to Eq.
(11) when the particles separate with velocity $v_{12}^{n}=0.001\,\mbox{m/s}$
from the distance $s=0$ (at the end of the mechanical contact between the two
particles) until $s=s_{crit}$ (the rupture distance, Eq. (12)).
Fig. 6 gives a comparison of the DEM results and the experimental data of
Willett et al. [15], showing the good agreement between our data and the
measurements.
### 3.5 Setup of the numerical rheometer
The flow and rheology of dry and wet granular materials in a three-dimensional
split bottom shear cell [29, 34, 35, 36] are investigated with the DEM model.
Basic parameters of the shear cell geometry are sketched in Fig. 7, with the
values: $r_{i}=14.7$ mm, $r_{s}=85$ mm, $r_{o}=110$ mm, and $U_{o}=6.9$ mm/s.
Figure 7: Setup of the numerical rheometer, light gray: sheared granular
material, medium gray: static, inner part of the shear cell, dark gray:
rotating, outer part of the shear cell (the medium and dark gray particles are
part of the rough walls of the shear cell and thus displayed as particles).
In contrast to previous studies, where only a quarter of the rheometer was
modeled, see Refs. [34, 35, 36], here we study the full ring-geometry. The
rheometer is filled with $n_{P}\simeq 210000$ particles, which all have the
same diameter $d_{P}=2$ mm, with density $\rho_{P}=2000$ kg/m3. We assume,
that polydispersity has in a first approximation little influence on the
macroscopic flow behavior of the granular material, similar to the findings in
Refs. [44, 45]. The expected small effects of polydispersity should be
investigated in more detail in the future, but are not subject of the present
study.
The mechanical parameters of the particles are chosen as $k_{n}=110$ N/m,
$\gamma_{n}=2\cdot 10^{-3}$ kg/s, $k_{t}=12$ N/m, $\gamma_{t}=0.5\cdot
10^{-3}$ kg/s, and $\mu_{C}=0.01$ (if not mentioned otherwise) in order to
match those of the numerical simulations by Luding [34, 36] for
straightforward validation purposes.
An adaption of the contact model to real sand F35 is presently not possible
and must be postponed since the material parameters are actually unknown.
However, this is no essential constriction, because details of the contact
model, including stiffness, have been found to cause only small differences
for granular flows in the collisional and dense regimes, as long as the
particles are not too soft, see e.g. Refs. [46, 47], where the effect of
softness of the material was studied in more detail.
The results of our calculations with dry granular material ($f_{c}=0$) are
used to verify and validate the DEM code and force model implementation, in
comparison with previous results [34]. The simulations with wet granular
materials should reveal if the influence of the liquid bridges (the model of
which was validated in subsection 3.4) on the apparent shear viscosity is the
same as in the experiments.
For the simulations of wet granular materials, the equilibrium contact angle
$\theta$ was varied between $0^{\circ}$ and $20^{\circ}$, whereas the surface
tension $\gamma=20.6\,$mN/m was kept constant. The simulations were conducted
for two different bridge volumes $V_{b}$ ($4.2$ nl, $42$ nl) for each
capillary bridge. Assuming an average number $n_{c,i}=6$ of capillary bridges
per particle, this corresponds approximately to a mass ratio $m_{l}/m_{ap}$ =
0.15%, 1,5% between the liquid and the dry granular material, respectively.
Here, the total mass $m_{l}=\sum_{i=1}^{n_{P}}\rho_{l}\,V_{b}\,n_{c,i}/2$ of
the liquid is calculated with an arbitrary fixed value $\rho_{l}$ = 1000
kg/m3, because $\rho_{l}$ is not applied in the DEM model. The total mass of
all dry particles is $m_{ap}=\sum_{i=1}^{n_{P}}m_{i}$. Note, that
$m_{l}/m_{ap}$ was found to change during the simulation, because $n_{c,i}$
varies in the interval $6\lesssim n_{c,i}\lesssim 7$, however this small
effect was ignored and $n_{c,i}=6$ was kept constant for the calculation of
$V_{b}$.
The influence of other, more complex liquid distributions is analyzed in
ongoing simulations and will be reported elsewhere.
## 4 Results
### 4.1 Micro-macro transition
As in Refs. [34, 35, 36], continuum quantities like the solid-fraction $\phi$,
the velocity-field $\underline{u}$ or the stress-field
$\underline{\underline{\sigma}}$ are computed by a micro-macro transition
method from the DEM results, e.g.,
$\displaystyle\phi(\underline{r})$ $\displaystyle=\frac{1}{\Delta t\,\Delta
V}\int\limits_{\Delta t}\ \sum_{i\in\Delta V}V_{i}dt\,,$ (13)
$\displaystyle\underline{u}\left(\underline{r}\right)$
$\displaystyle=\frac{1}{\Delta t\,\Delta V}\int\limits_{\Delta
t}\left(\sum_{i\in\Delta
V}V_{i}\,\underline{v}_{i}\right)dt\times\frac{1}{\phi(\underline{r})}\,,$
(14) $\displaystyle\underline{\underline{\sigma}}\left(\underline{r}\right)$
$\displaystyle=\frac{1}{\Delta t\,\Delta V}\int\limits_{\Delta
t}\left(\sum_{i\in\Delta
V}m_{i}\underline{v}^{\prime}_{i}\otimes\underline{v}^{\prime}_{i}+\sum_{c\in\Delta
V}^{\left\\{i,j\right\\}=c}\underline{f}_{ij}\otimes\underline{l}_{ij}\right)dt\,,$
(15)
with fluctuation velocity
$\underline{v}^{\prime}_{i}=\underline{v}_{i}-\underline{u}(\underline{r})$,
averaging time intervals of typically $\Delta t=5$ s, a discrete averaging
time-step $dt=0.05$ s, and particle volume $V_{i}$, together with the
(ring/torus-shaped) averaging volume $\Delta V$ at various positions
$\underline{r}=(r,z)$ in the system (with cylindrical coordinates). Further
parameter-fields like strain rate magnitude, shear stress magnitude,
hydrostatic pressure, apparent viscosity and inertial number can be calculated
from these variables as:
$\displaystyle\dot{\gamma}$
$\displaystyle=\frac{1}{2}\sqrt{\left(\frac{\partial u_{\varphi}}{\partial
r}-\frac{u_{\varphi}}{r}\right)^{2}+\left(\frac{\partial u_{\varphi}}{\partial
z}\right)^{2}}\,,$ (16) $\displaystyle\left|\tau\right|$
$\displaystyle=\sqrt{\sigma_{r\varphi}^{2}+\sigma_{z\varphi}^{2}}\,,$ (17)
$\displaystyle p$
$\displaystyle=\frac{1}{3}\left(\sigma_{rr}+\sigma_{zz}+\sigma_{\varphi\varphi}\right)\,,$
(18) $\displaystyle\eta$
$\displaystyle=\frac{\left|\tau\right|}{\dot{\gamma}}\,,$ (19) $\displaystyle
I$ $\displaystyle=\frac{\dot{\gamma}\,d_{P}}{\sqrt{{p}/{\rho}}}\,.$ (20)
In contrast to the experimental setup, these parameters can be investigated
_locally_ , i.e., at arbitrary positions $\underline{r}$ anywhere in the
filled measurement volume (gap) of the rheometer.
All simulations which are discussed afterwards run for 20 s. For the average,
only the period between 15 s and 20 s are is into account. Therefore, the
system is examined in quasi-steady state flow conditions (the transient
behavior at the onset of shear is disregarded). However, it cannot be
excluded, that long-time relaxation effects may have an impact on our
findings, which is not adequately resolved by our relatively short
simulations.
### 4.2 Dry granular material
Fig. 8 compares the shear stress intensity $|\tau|/p=\mu_{m}$ in our DEM
simulations to the previous findings of Luding [34, 36], with the same
macroscopic friction value $\mu_{m}=0.14$ for $\mu_{C}=0.01$ as reported
earlier.
Figure 8: Local shear stress $\left|\tau\right|$ plotted against local
pressure $p$. Different symbol sizes indicate the magnitude of the strain rate
$\dot{\gamma}$ at different locations within the rheometer gap, with
$\dot{\gamma}$ given in the inset in units of $\mbox{s}^{-1}$; larger symbols
correspond also to larger $\dot{\gamma}$. The solid line represents the
function $\left|\tau\right|=\mu_{m}\,p$, with the macroscopic friction
coefficient $\mu_{m}=0.14$. Figure 9: Strain rate $\dot{\gamma}$ as function
of radial and vertical position within the rheometer gap. As in Fig. 8,
different symbol sizes indicate the magnitude of $\dot{\gamma}$. The lines
indicate the center location $R_{c}$ (middle line) and the width $W$ of the
shear bands (outer lines), as obtained from the fit function Eq. (19).
In order to allow for a more quantitative analysis of the dynamic behavior of
the dry granular material, its shear band structure is analyzed. The shear
bands in dry granular matter have been intensively studied in experiments and
DEM simulations, see e.g., Refs. [29, 34]. Among other details, as will be
discussed elsewhere, the profiles of the velocity field are well approximated
by error functions:
$\displaystyle\omega\left(r\right)=A+B\,\mathrm{erf}\,\left(\frac{r-R_{c}}{W}\right)$
(21)
where the dimensionless amplitudes are $A\simeq B\simeq 0.50$, $R_{c}$ is the
center of the shear band, and $W$ is its width. The strain rate as function of
$r$ and $z$ and the shear band structure in the rheometer gap are shown in
Fig. 9.
Obviously, our results for both the shear stress intensity and the shear band
structure agree well with the findings in [34], which were obtained with other
software for DEM simulation and analysis. Therefore we conclude, that there
are no implementation errors in the basic LIGGGHTS code and our dry contact
model implementation, for the parameters used here.
### 4.3 Wet granular material
#### 4.3.1 Macroscopic friction and cohesion
Figure 10: Shear stress $\left|\tau\right|$ plotted against pressure $p$.
Different symbols indicate results from dry material ($\bullet$), and from two
wetted materials, which contain liquid bridges with (i) $\theta=0^{\circ}$,
$V_{b}=4.2$ nl ($\Box$) and (ii) $\theta=20^{\circ}$, $V_{b}=42$ nl
($\triangle$). The magnitude of the strain rate $\dot{\gamma}$ is indicated by
the size of the symbols similar as in Fig. 8. The solid and the dashed lines
represent the function $\left|\tau\right|=\mu_{m}\,p+c$, with the macroscopic
friction coefficients $\mu_{m}$ for the dry ($c=0$) and the wet ($c=5$ Pa)
materials, respectively.
In Fig. 10, the dry results are compared to two simulations with liquid
bridges. The addition of the liquid bridge forces leads to larger shear stress
magnitudes so that the macroscopic yield stress at critical state (the
termination locus, reached after large shear strain) is shifted upwards. The
offset on the vertical axis is referred to as the macroscopic cohesion $c$.
While the dry data are fitted well by the macroscopic line fit with $c=0$, the
wet data for small $p<100$ Pa considerably drop below the fit result with
constant $c>0$.
#### 4.3.2 Local shear viscosities
Figure 11: Local shear viscosity $\eta\left(I\right)$ from the DEM
simulations. Different symbols indicate results from dry ($\bullet$) and wet
material (open symbols). The wet material contains liquid bridges with (top)
$V_{b}=4.2$ nl ($\Box$) and (bottom) $V_{b}=42$ nl ($\circ$), whereas
$\theta=0^{\circ}$ is the same for both cases. The magnitude of the local
strain rate $\dot{\gamma}$ is indicated by the size of the symbols similar as
in Fig. 8.
Figure 12: Apparent shear viscosity $\eta\left(I\right)$ from the DEM
simulations. Different symbols indicate results from dry ($\bullet$) and wet
material (open symbols). The wet material contains liquid bridges with (top)
$\theta=10^{\circ}$ ($\triangle$) and (bottom) $\theta=20^{\circ}$
($\triangledown$), whereas $V_{b}=42$ nl is constant in all cases. The
magnitude of the strain rate $\dot{\gamma}$ is indicated by the size of the
symbols similar as in Fig. 8.
Next, the local shear viscosities $\eta$ are compared for dry and different
wet granular materials. Figs. 11 and 12 give the correlations
$\eta\left(I\right)$, which are found from several DEM simulations with
different liquid bridge volumes $V_{b}=4.2$ nl, $42$ nl and contact angles
$\theta=0^{\circ}$, $10^{\circ}$ and $20^{\circ}$. The DEM simulations are
evaluated locally in the granular medium. Therefore, in contrast to the
experiments, a single DEM simulation provides many data-points at various
bulk-densities, shear-stresses, pressures and shear-rates that are plotted as
different symbols for different $V_{b}$ and $\theta$.
As in the experiments, the inverse proportional dependence of the apparent
viscosity on $I$ is evidenced for the dry and wet material, and, as to be
expected, the addition of small amounts of liquid increases the viscosity by
about 10 to 20%. (The change appears not that large only due to the
logarithmic vertical axis in the plots). The scaling $\eta\sim I^{-\alpha}$
changes from $\alpha=0.9$ to $\alpha=2$ in all materials, in different
regimes. The influence of the contact angle and the liquid bridge volume
$V_{b}$ on the rheology of the wet granular materials is presently
investigated in more detail and results will be published elsewhere.
#### 4.3.3 Shear bands
Figure 13: Shear bands in dry and wet material, see Fig. 9. Different line
styles indicate results from dry (continuous) and wet material (dashed). Here,
points serve only for distinction but don’t represent flow data. In the upper
subfigure, the wet material contains liquid bridges with (i) $V_{b}=4.2$ nl
and (ii) $V_{b}=42$ nl, whereas $\theta=0^{\circ}$ is constant in both cases.
In the lower subfigure, the wet material contains liquid bridges with (i)
$\theta=0^{\circ}$, (ii) $\theta=10^{\circ}$ and (iii) $\theta=20^{\circ}$,
whereas $V_{b}=42$ nl is constant here.
Next, the influence of the liquid on the dynamic behavior of the granular
material is investigated. The shear bands for dry and wet materials are given
in Fig. 13. Variations in $V_{b}$ and $\theta$ induce noticeable changes in
the shear band structure. With increasing liquid content, its center position
moves inwards to smaller radial distances. For small liquid content, the shear
band width decreases, whereas it increases for the larger liquid content
simulations. The correlation for changing contact angle $\theta$ is more
complex. Due to the high liquid content, the shear band moves inwards. Not
surprisingly, the shift is largest for lowest contact angle
$\theta=0^{\circ}$. But against expectation, the lowest shift is found for
$\theta=10^{\circ}$ and not for $\theta=20^{\circ}$. Additionally, the
qualitative change of the shear band as reported for very strong van der Waals
type adhesion [36] is not reproduced here, possibly since the liquid bridge
forces never become strong enough to have a similar effect. These interesting
findings are presently investigated in more detail, results will be published
elsewhere.
## 5 Conclusions
Shear experiments are complemented by a numerical rheometer study with core
shooting materials as application in mind, but with a much more general
perspective concerning concepts and methods. The simple DEM contact and liquid
bridge model is validated using previous results from a three-dimensional
split bottom ring shear cell for dry materials, and with two-particle
collision data from more advanced numerical and experimental studies for
weakly wet materials.
The DEM simulations of wet granular material show that the internal structures
of the sheared material, i.e., the shear bands, are qualitatively the same as
for dry materials. However, they move inwards with increasing liquid content
and while getting a little narrower for small liquid content, then become
wider for very high liquid content. Finally, changing contact angles influence
shear bands in non-trivial manner.
The apparent local shear viscosity of the granular material significantly
increases when only small amounts of liquid are added to the material,
representing well the trend as seen from the experiments.
Future studies will involve a more quantitative study of the constitutive
relations that describe the rheology of the material and their implementation
into CFD or FEM codes to predict large scale core shooting flow behavior. The
simulations and the experiments have to be performed in a more comparable way
between wet and dry configurations, i.e. using the same gap width; the
relation between the local and global viscosities (due to local and global
shear rates) has to be better understood. An open question is how much a
pendular liquid bridge contact model could be simplified, i.e. which details
and non-linearities are important and which are not. Nevertheless, more
complex capillary bridge models have to be used, which allow the description
of other than the pendular bridge regimes, too.
## Acknowledgments
We acknowledge the support of the ERASMUS program which allowed us to host FU
during his Master study at the University of Twente. Helpful discussions with
T. Weinhart, A. Singh, V. Magnanimo are appreciated. RS acknowledges the
German Science Foundation (DFG) for funding parts of the work under project
no. SCHW 1168/6-1, and SL acknowledges the NWO/STW, VICI grant 10828, and the
DFG, project SPP1482 B12, for partial financial support. Finally, we
acknowledge the constructive criticism of the referees of the paper.
The final publication is available at Springer via
http://dx.doi.org/10.1007/s10035-013-0430-z
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|
arxiv-papers
| 2014-03-31T11:56:17 |
2024-09-04T02:50:00.564304
|
{
"license": "Public Domain",
"authors": "Ruediger Schwarze and Anton Gladkyy and Fabian Uhlig and Stefan Luding",
"submitter": "Anton Gladkyy",
"url": "https://arxiv.org/abs/1404.0318"
}
|
1404.0484
|
Characteristics of Finite Jaco Graphs, $J_{n}(1),n\in\mathbb{N}$
(Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian
Mukungunugwa)111Affiliation of authors:
Johan Kok (Tshwane Metropolitan Police Department), City of Tshwane, Republic
of South Africa
e-mail: [email protected]
Paul Fisher (Department of Mathematics, University of Botswana), City of
Gaborone, Republic of Botswana
e-mail: [email protected]
Bettina Wilkens (Department of Mathematics, University of Botswana), City of
Gaborone, Republic of Botswana
e-mail: [email protected]
Mokhwetha Mabula (Department of Mathematics and Applied Mathematics,
University of Pretoria), City of Tshwane, Republic of South Africa
e-mail: [email protected]
Vivian Mukungunugwa (Department of Mathematics, University of Zimbabwe), City
of Harare, Republic of Zimbabwe
e-mail: [email protected]
###### Abstract
We introduce the concept of a family of finite directed graphs (_order 1_)
which are directed graphs derived from a infinite directed graph (_order 1_),
called the _1_ -root digraph. The _1_ -root digraph has four fundamental
properties which are; $V(J_{\infty}(1))=\\{v_{i}|i\in\mathbb{N}\\}$ and, if
$v_{j}$ is the head of an edge (arc) then the tail is always a vertex
$v_{i},i<j$ and, if $v_{k},$ for smallest $k\in\mathbb{N}$ is a tail vertex
then all vertices $v_{\ell},k<\ell<j$ are tails of arcs to $v_{j}$ and
finally, the degree of vertex $k$ is $d(v_{k})=k.$ The family of finite
directed graphs are those limited to $n\in\mathbb{N}$ vertices by lobbing off
all vertices (and edges arcing to vertices) $v_{t},t>n.$ Hence, trivially we
have $d(v_{i})\leq i$ for $i\in\mathbb{N}.$ We present an interesting
Fibonaccian-Zeckendorf result and present the Fisher Algorithm to table
particular values of interest. It is meant to be an _introductory paper_ to
encourage exploratory research.
Keywords: Jaco graph, Directed graph, Jaconian vertex, Jaconian set, Number of
edges, Shortest path, Fisher Algorithm, Zeckendorf representation
AMS Classification Numbers: 05C07, 05C12, 05C20, 11B39
## 1 Introduction
We introduce the concept of a family of finite Jaco Graphs (_order 1_) which
are directed graphs derived from the infinite Jaco Graph (_order 1_), called
the _1_ -root digraph. The _1_ -root digraph has four fundamental properties
which are; $V(J_{\infty}(1))=\\{v_{i}|i\in\mathbb{N}\\}$ and, if $v_{j}$ is
the head of an edge (arc) then the tail is always a vertex $v_{i},i<j$ and, if
$v_{k},$ for smallest $k\in\mathbb{N}$ is a tail vertex then all vertices
$v_{\ell},k<\ell<j$ are tails of arcs to $v_{j}$ and finally, the degree of
vertex $k$ is $d(v_{k})=k.$
###### Definition 1.1.
The infinite Jaco Graph $J_{\infty}(1)$ is defined by
$V(J_{\infty}(1))=\\{v_{i}|i\in\mathbb{N}\\}$,
$E(J_{\infty}(1))\subseteq\\{(v_{i},v_{j})|i,j\in\mathbb{N},i<j\\}$ and
$(v_{i},v_{j})\in E(J_{\infty}(1))$ if and only if $2i-d^{-}(v_{i})\geq j.$
###### Definition 1.2.
The family of finite Jaco Graphs are defined by $\\{J_{n}(1)\subseteq
J_{\infty}(1)|n\in\mathbb{N}\\}.$ A member of the family is referred to as the
Jaco Graph, $J_{n}(1).$
###### Definition 1.3.
The set of vertices attaining degree $\Delta(J_{n}(1))$ is called the Jaconian
vertices of the Jaco Graph $J_{n}(1),$ and denoted, $\mathbb{J}(J_{n}(1))$ or,
$\mathbb{J}_{n}(1)$ for brevity.
###### Definition 1.4.
The lowest numbered (indiced) Jaconian vertex is called the prime Jaconian
vertex of a Jaco Graph.
###### Definition 1.5.
If $v_{i}$ is the prime Jaconian vertex of a Jaco Graph $J_{n}(1)$, the
complete subgraph on vertices $v_{i+1},v_{i+2},\cdots,v_{n}$ is called the
Hope subgraph of a Jaco Graph and denoted, $\mathbb{H}(J_{n}(1))$ or,
$\mathbb{H}_{n}(1)$ for brevity.
###### Definition 1.6.
If, in applying definition 1.1 to vertex $v_{i}$ (not necessarily
exhaustively), or for logical method of proof we have the edge $(v_{i},v_{k})$
linked in a Jaco Graph $J_{n}(1),$ then the degree vertex $v_{i}$ attains at
$v_{k}$ is called the, ”at degree of $v_{i}$ at $v_{k}$”, and is denoted,
$d^{*}(v_{i})@v_{k}.$
###### Definition 1.7.
In $J_{\infty}(1)$ we have $n=d^{+}(v_{n})+d^{-}(v_{n})$ whilst in $J_{n}(1)$
we have $d(v_{i})=\lceil d^{+}(v_{i})\rceil+d^{-}(v_{i}),i\leq n.$
Property 1: From the definition of a Jaco Graph $J_{n}(1),$ it follows that
for the prime Jaconian vertex $v_{i},$ we have $d(v_{m})=m$ for all
$m\in\\{1,2,3,\cdots,i\\}.$
Property 2: From the definition of a Jaco Graph $J_{n}(1),$ it follows that
$\Delta(J_{k}(1))\leq\Delta(J_{n}(1))$ for all $k\leq n.$
Property 3: The $d^{-}(v_{k})$ for any vertex $v_{k}$ of a Jaco Graph
$J_{n}(1),~{}n\geq k$ is equal to $d(v_{k})$ in the underlying Jaco Graph
$J_{k}(1).$
###### Lemma 1.1.
If in a Jaco Graph $J_{n}(1),$ and for smallest $i$ with $d(v_{i})=i,$ the
edge $(v_{i},v_{n})$ is defined, then $v_{i}$ is the prime Jaconian vertex of
$J_{n}(1).$
###### Proof.
If by definition 1.1 and for smallest $i$ with $d(v_{i})=i$ the edge
$(v_{i},v_{n})$ is defined, we have in the underlying graph of $J_{n}(1)$ that
$d(v_{j})\leq d(v_{i})$ for all $j>i$. We also have that
$d(v_{s})<d(v_{i}),s<i.$ So it follows that $d(v_{i})=\Delta(J_{n}(1))$ hence
by definition 1.4 the vertex $v_{i}$ is the prime Jaconian vertex of
$J_{n}(1).$ ∎
###### Lemma 1.2.
For all Jaco Graphs $J_{n}(1),~{}n\geq 2$ and, $v_{i},v_{i-1}\in V(J_{n}(1))$
we have that in the underlying graph $|(d(v_{i})-d(v_{i-1})|\leq 1.$
###### Proof.
Consider the Jaco Graph $J_{n}(1),~{}n\geq 2.$ The result is trivially true
for all vertices $v_{1},v_{2},v_{3},\cdots,v_{k}$ if $v_{k}$ is the prime
Jaconian vertex of $J_{n}(1).$ Now consider the Hope subraph
$\mathbb{H}(J_{n}(1)).$ All vertices of $\mathbb{H}(J_{n}(1))$ have equal
degree so the result holds for the Hope subraph _per se_. Furthermore if a
vertex $v_{j},~{}(k+1)\leq j\leq n$ is linked to a vertex $v_{t},~{}1\leq
t\leq k$ then all vertices $v_{l},~{}(k+1)\leq l<j$ are linked to $v_{t}$
which implies $|d(v_{j})-d(v_{l})|=0<1$ hence $|(d(v_{j+1})-d(v_{j})|\leq 1.$
∎
###### Corollary 1.3.
For a Jaco Graph $J_{n}(1)$ the maximum degree $\Delta(J_{n}(1))$ might repeat
itself as $n$ increases to $n+1,$ (i.e. $\Delta J_{n}(1)=\Delta J_{n+1}(1)$)
but on an increase of we always obtain $\Delta J_{n+1}(1)=\Delta(J_{n}(1))+1.$
###### Proof.
The result follows from Lemma 1.2. ∎
## 2 The Fisher Algorithm for
$\\{J_{i}(1),i\in\\{4,5,6,\cdots,s\in\mathbb{N}\\}$
The family of finite Jaco Graphs are those limited to $n\in\mathbb{N}$
vertices by lobbing off all vertices (and edges arcing to vertices)
$v_{t},t>n.$ Hence, trivially we have $d(v_{i})\leq i$ for $i\in\mathbb{N}.$
Column 1 is the map: $\phi(v_{i})\rightarrow i,\forall i.$
Column 2 is the in-degree of vertex $v_{i}.$
Column 3 is the out-degree of vertex $v_{i}$ in $J_{\infty}(1).$
Column 4 is the set $\mathbb{J}(J_{i}(1)).$
Column 5 is $\Delta(J_{i}(1)).$
Column 6 is the distance $d_{J_{i}(1)}(v_{1},v_{i}).$
We generally refer to the entries in a row $i$ as: $ent_{1i}=i,$
$ent_{2i}=d^{-}(v_{i}),~{}ent_{3i}=d^{+}(v_{i}),~{}ent_{4i}=\mathbb{J}(J_{i}(1)),~{}ent_{5i}=\Delta(J_{i}(1)),~{}ent_{6i}=d_{J_{i}(1)}(v_{1},v_{i})$
as interchangeable.
Note that rows 1, 2 and 3 follow easily from definition 1.1.
Step 0: Set $j=4$, then set $i=j$ and $s\geq 4.$
Step 1: Set $ent_{1i}=i.$
Step 2: Set $ent_{2i}=ent_{1(i-1)}-ent_{5(i-1)}.$ (Note that
$d^{-}(v_{i})=v(\mathbb{H}_{i-1}(1))=(i-1)-\Delta(J_{i-1}(1))).$
Step 3: Set $ent_{3i}=ent_{1i}-ent_{2i}.$ (Note that
$d^{+}(v_{i})=i-d^{-}(v_{i})).$
Step 4: Consider $ent_{4(i-1)}.$ If $ent_{4(i-1)}=\\{v_{k}\\},$ set $t=k,$
else set $t=k+1.$
Step 5: Set the prime Jaconian vertex as $v_{t}$ so
$\mathbb{J}(J_{i}(1))=\\{v_{t}\\}$ to begin with. Let
$l=t+1,~{}t+2,\cdots,i-1$ and recursively calculate $i-ent_{1l}+ent_{2l}$. If
$i-ent_{1l}+ent_{2l}=t,$ add $v_{l}$ to the set of Jaconian vertices, else go
to Step 6.
Step 6: Set $ent_{5i}=t.$ (Note that if
$\mathbb{J}_{i}(1)=\\{v_{t},v_{t+1},..v_{\ell}\\},$ then,
$\Delta(J_{i}(1))=t).$
Step 7: Select smallest $k$ such that, $k+ent_{3k}\geq i$ then set
$ent_{6i}=ent_{6k}+1.$
Step 8: Set $j=i+1$, then set $i=j.$ If $i\leq s$, go to Step 1, else go to
Step 9.
Step 9: Exit.
###### Proposition 2.1.
Consider the Jaco Graph $J_{i}(1),~{}i\geq 4.$ If the Jaconian vertex of
$J_{i-1}(1)$ is unique say, $v_{k}$ then $k+d^{+}(v_{k})<i$ and
$(k+1)+d^{+}(v_{k+1})>i.$
###### Proof.
Because the Jaconian vertex $v_{k}$ is unique to $J_{i-1}(1)$ it implies that
edge $(v_{k},v_{i-1})$ exists (see Theorem 2.11), so $k+d^{+}(v_{k})=i-1<i.$
And since edge $(v_{k+1},v_{i})$ does not exist in $J_{i-1}(1)$ we have
$d(v_{k+1})=k-1.$
By extending to $J_{i}(1)$ the edge $(v_{k+1},v_{i})$ is linked. So degree of
$v_{k+1}$ increases to $d(v_{k+1})=k$ implying $d^{+}(v_{k+1})$ increased by
$1.$ Thus, $(k+1)+d^{+}(v_{k+1})=(k+1)+(d^{+}(v_{k})+1)=(i-1)+2=i+1>i.$ ∎
###### Lemma 2.2.
(Conjectured): If for $n\in\mathbb{N}$ we have that $d^{+}(v_{n})=\ell$ is
non-repetitive (meaning $d^{+}(v_{n-1})<d^{+}(v_{n})<d^{+}(v_{n+1})$) then,
$\mathbb{J}(J_{n}(1))=\\{v_{\ell}\\}.$
###### Theorem 2.3.
(Morrie’s Theorem): If a Jaco Graph $J_{n}(1),n\geq 2$ has a prime Jaconian
vertex $v_{k}$ then:
(a) $d^{-}(v_{k})=d^{-}(v_{k+1})$ and $d^{-}(v_{k+2})=d^{-}(v_{k+1})+1$ if and
only if $\mathbb{J}(J_{n})=\\{v_{k}\\}$ and
$\mathbb{J}(J_{n+1}(1))=\\{v_{k},v_{k+1},v_{k+2}\\}$,
(b) $d^{-}(v_{k})=d^{-}(v_{k+1})=d^{-}(v_{k+2})$ if and only if
$\mathbb{J}(J_{n})=\\{v_{k}\\}$ and
$\mathbb{J}(J_{n+1}(1))=\\{v_{k},v_{k+1}\\}.$
###### Proof.
Let $d^{-}(v_{k})=d^{-}(v_{k+1})$ and $d^{-}(v_{k+2})=d^{-}(v_{k+1})+1$ for
the Jaco Graph $J_{n}(1),n\geq 2.$ and let $\mathbb{J}(J_{n}(1))=\\{v_{k}\\}.$
From definition 1.7 and Steps 1, 2 and 3 of the Fisher Algorithm it follow
that we have associated entries:
$ent_{1k}=k,ent_{2k}=d^{-}(v_{k}),ent_{3k}=d^{+}(v_{k})=k-d^{-}(v_{k})$ and,
$ent_{1(k+1)}=k+1,ent_{2(k+1)}=d^{-}(v_{k+1})=d^{-}(v_{k}),ent_{3(k+1)}=d^{+}(v_{k+1})=(k+1)-d^{-}(v_{k})$
and,
$ent_{1(k+2)}=k+2,ent_{2(k+2)}=d^{-}(v_{k+2})=d^{-}(v_{k})+1,ent_{3(k+1)}=d^{+}(v_{k+2})=(k+2)-d^{-}(v_{k})-1$
and,
$ent_{1(k+3)}=k+3,ent_{2(k+3)}=d^{-}(v_{k+3})=d^{-}(v_{k})+1,ent_{3(k+3)}=d^{+}(v_{k+3})=(k+3)-d^{-}(v_{k})-1.$
Let $n=2k-d^{-}(v_{k})$ and it easily follows from Step 5 that
$\mathbb{J}(J_{n}(1))=\\{v_{k}\\}.$ Now let $n=2k-d^{-}(v_{k})+1$ and
initialise $\mathbb{J}(J_{n+1}(1))=\\{v_{k}\\}$ and set $t=k.$ Also let
$l=k+1,k+2,...,2k-d^{-}(v_{k}).$
For $l=k+1$ we have that $(2k-d^{-}(v_{k})+1)-(k+1)+d^{-}(v_{k})=k=t,$ so
$v_{k+1}\in\mathbb{J}(J_{n+1}(1)).$
For $l=k+2$ we have that $(2k-d^{-}(v_{k})+1)-(k+2)+d^{-}(v_{k})+1=k=t,$ so
$v_{k+2}\in\mathbb{J}(J_{n+1}(1)).$
For $l=k+3$ we have that $(2k-d^{-}(v_{k})+1)-(k+3)+d^{-}(v_{k})+1=k-1\neq t,$
so $v_{k+3}\notin\mathbb{J}(J_{n+1}(1)).$
So it follows that if $d^{-}(v_{k})=d^{-}(v_{k+1})$ and
$d^{-}(v_{k+2})=d^{-}(v_{k+1})+1$ then $\mathbb{J}(J_{n}(1))=\\{v_{k}\\}$ and
$\mathbb{J}(J_{n+1}(1))=\\{v_{k},v_{k+1},v_{k+2}\\}$ with
$n\in\\{2k-d^{-}(v_{k}),2k-d^{-}(v_{k})+1\\}.$
Conversely, if $\mathbb{J}(J_{n}(1))=\\{v_{k}\\}$ and
$\mathbb{J}(J_{n+1}(1))=\\{v_{k},v_{k+1},v_{k+2}\\}$ we have from the inverse
of definition 1.7 and Steps 1, 2, 3, 4, 5 and 6 of the Fisher Algorithm the
associated entries:
$ent_{1n}=n,ent_{2n}=(n-k),ent_{3n}=k,ent_{4n}=\\{v_{k}\\},ent_{5n}=k\Rightarrow$
$ent_{1k}=k,ent_{2k}=2k-n,ent_{3k}=n-k,ent_{5k}=k\Rightarrow$
$ent_{1(k+1)}=k+1,ent_{2(k+1)}=2k-n,ent_{3(k+1)}=n-k+1,ent_{5(k+1)}=k\Rightarrow$
$ent_{1(k+2)}=k+2,ent_{2(k+2)}=2k-n+1,ent_{3(k+2)}=n-k+1,ent_{5(k+2)}=k+1.$
$\therefore d^{-}(v_{k})=d^{-}(v_{k+1})$ and
$d^{-}(v_{k+2})=d^{-}(v_{k+1})+1.$
Result $(b)$ follows similarly to $(a)$. ∎
###### Proposition 2.4.
For all Jaco Graphs $J_{n}(1),$ we have Card $\mathbb{J}(J_{n}(1))\leq 3.$
###### Proof.
It is evident that for some $m\in\mathbb{N},$ Card $\mathbb{J}(J_{m}(1))=3.$
Let $\mathbb{J}(J_{m}(1))=\\{v_{k},v_{k+1},v_{k+2}\\}.$ So in Step 4 of the
Fisher Algorithm we initially set $i=m$ and $t=k.$ We also have that
$i-(k+2)+d^{-}(v_{k+2})=t.$
From Morrie’s theorem it follows that $d^{-}(v_{k})=d^{-}(v_{k+1})$ and
$d^{-}(v_{k+2})=d^{-}(v_{k+1})+1.$ It follows that,
$d^{-}(v_{k+3})=d^{-}(v_{k+1})+1.$ However, in Step 5 we have
$i-(k+3)+d^{-}(v_{k+3})=i-(k+3)+d^{-}(v_{k+1})+1=(i-(k+2)+d^{-}(v_{k+2}))-1<t.$
So vertex $v_{k+3}$ cannot be added to $\mathbb{J}(J_{m}(1)).$ ∎
###### Corollary 2.5.
From Proposition 2.4 it follows that if and only if the Jaconian vertex of
$J_{i-1}(1),~{}i\geq 2$ is unique say, $v_{k}$ then $\mathbb{J}(J_{i}(1))=$
either $\\{v_{k},v_{k+1}\\}$ or $\\{v_{k},v_{k+1},v_{k+2}\\}.$
###### Proof.
By extending from to $J_{i-1}(1)$ to $J_{i}(1)$ the edge $(v_{k},v_{i})$ is
not linked. Because $d(v_{k+1})=d(v_{k})-1$ in $J_{i-1}(1)$ and increases by
$1$ in $J_{i}(1)$ it follows that $d(v_{k+1})=d(v_{k})$ in $J_{i}(1).$ Hence,
at least $\mathbb{J}_{i}(1)=\\{v_{k},v_{k+1}\\}.$ If
$d^{-}(v_{k+2})=d^{-}(v_{k+1})+1,$ then
$\mathbb{J}_{i}(1)=\\{v_{k},v_{k+1},v_{k+2}\\}.$ So it follows that
$\mathbb{J}_{i}(1)=$ either $\\{v_{k},v_{k+1}\\}$ or
$\\{v_{k},~{}v_{k+1},~{}v_{k+2}\\}.$
Conversely, assume that $\mathbb{J}_{i}(1)=$ either $\\{v_{k},~{}v_{k+1}\\}$
or $\\{v_{k},~{}v_{k+1},~{}v_{k+2}\\}.$
Case 1: Let $\mathbb{J}(J_{i}(1))=\\{v_{k},~{}v_{k+1}\\}.$ So
$i-(k+1)+d^{-}(v_{k+1})=k$ in $J_{i}(1).$ Hence in $J_{i-1}(1)$ we have
$(i-1)-(k+1)+d^{-}(v_{k+1})=(i-(k+1))+d^{-}(v_{k+1})-1=k-1.$ So from Step 5 of
the Fisher Algorithm it follows that $v_{k+1}\notin\mathbb{J}(J_{i-1}(1)).$
However, $v_{k}\in\mathbb{J}(J_{i-1}(1))=\\{v_{k}\\}.$
Case 2: Let $\mathbb{J}(J_{i}(1))=\\{v_{k},~{}v_{k+1},~{}v_{k+2}\\}.$ Same
reasoning as in case 1, follows. ∎
###### Corollary 2.6.
If $k+d^{+}(v_{k})=i$ and $(k+1)+d^{+}(v_{k+1})>i+1$ then $v_{k}$ is the
unique Jaconian vertex of $J_{i}(1).$
###### Proof.
The result follows directly from Step 5 of the Fisher Algorithm. ∎
###### Proposition 2.7.
If we have $d^{-}(v_{k-1})=d^{-}(v_{k})=d^{-}(v_{k+1})$ then $v_{k}$ is the
unique Jaconian vertex of $J_{l}(1),~{}l=2k-d^{-}(v_{k}).$
###### Proof.
For $d^{+}(v_{k})=k-d^{-}(v_{k})$ and
$l=k+d^{+}(v_{k})=k+k-d^{-}(v_{k})=2k-d^{-}(v_{k})$ it follows that $v_{k}$ is
a Jaconian vertex of $J_{l}(1).$ Furthermore, $v_{k}$ is the unique Jaconian
vertex of $J_{l}(1)$ because:
Case 1: For $v_{k-1}$ and because $d^{-}(v_{k-1})=d^{-}(v_{k}),$ we have
$l-(k-1)+d^{-}(v_{k})=2k-d^{-}(v_{k})-(k-1)+d^{-}(v_{k})=2k-d^{-}(v_{k})-k+1+d^{-}(v_{k})=2k-k+1=k+1>k.$
Hence, $v_{k-1}$ is not a Jaconian vertex of $J_{l}(1).$
Case 2: For $v_{k+1}$ and because $d^{-}v_{k+1})=d^{-}(v_{k}),$ we have
$l-(k+1)+d^{-}(v_{k})=2k-d^{-}(v_{k})-(k+1)+d^{-}(v_{k})=2k-d^{-}(v_{k})-k-1+d^{-}(v_{k})=2k-k-1=k-1<k.$
Hence, $v_{k+1}$ is not a Jaconian vertex of $J_{l}(1).$ ∎
###### Proposition 2.8.
$\mathbb{J}(J_{k-1}(1))=\\{v_{l-1}\\}$ if and only if
$d^{+}(v_{k})=d^{+}(v_{k+1})=l.$
###### Proof.
If $\mathbb{J}(J_{k-1}(1)=\\{v_{l-1}\\},$ implying $\Delta(J_{k-1}(1))=l-1,$
it follows from Step 2 of the Fisher Algorithm that
$d^{-}(v_{k})=(k-1)-\Delta(J_{k-1})=(k-1)-d^{+}(v_{k-1}).$ So because
$k=(k-1)+1,$ it follows that $d^{+}(v_{k})=l$ because
$d(v_{i})=d^{+}(v_{i})+d^{-}(v_{i}),\forall i.$ By similar reasoning
$d^{+}(v_{k+1})=l,$ so it holds that $d^{+}(v_{k})=d^{+}(v_{k+1})=l.$
Conversely, if $d^{+}(v_{k})=d^{+}(v_{k+1})=l,$ it follows by inversing the
convergence properties of the Fisher Algorithm that
$\mathbb{J}(J_{k-1}(1))=\\{v_{l-1}\\}.$ ∎
###### Theorem 2.9.
Let $m=n+\Delta(J_{n}(1)),$ then $\Delta(J_{m}(1))=$ either $n$ or $n-1.$
###### Proof.
Let $m=n+\Delta(J_{n}(1)).$
Case 1: Assume $\mathbb{J}(J_{n}(1))=\\{v_{k}\\}$ so $\Delta(J_{n}(1))=k.$ We
have that $d^{+}(v_{n})=k$ so in $J_{m}(1)$ the edge $(v_{n},v_{m})$ is
defined to attain $d(v_{n})=n,$ and $v_{n}$ is the prime Jaconian vertex of
$J_{m}(1),~{}m=n+\Delta(J_{n}(1)).$ Hence, $\Delta(J_{m}(1))=n.$
Case 2: Assume $\mathbb{J}(J_{n}(1))=\\{v_{k},v_{k+1}\\}$ or
$\\{v_{k},v_{k+1},v_{k+2}\\}.$ We have that $d^{+}(v_{n})=k$ or $k+1.$ So by
the same reasoning as in Case 1 it follows that $\Delta(J_{m}(1))=n$ or $n-1.$
∎
###### Theorem 2.10 (Conjectured).
For the Jaco Graphs $J_{n}(1),$ $J_{m}(1)$ with $n\geq 3,~{}m\geq 3,n\neq m$
we have
$\Delta(J_{n+m}(1))=\begin{cases}\Delta(J_{n}(1))+\Delta(J_{m}(1)),&\text{if
$J_{n}(1)$ or $J_{m}(1)$ has a unique Jaconian vertex}\\\
\Delta(J_{n}(1))+\Delta(J_{m}(1))+1,&\text{otherwise.}\end{cases}$
###### Theorem 2.11.
If the Jaco Graph $J_{n}(1)$ has a unique Jaconian vertex (prime Jaconian
vertex only) at $v_{i},$ then:
(a) Edge $(v_{i},v_{n})$ exists and,
(b) $\Delta(J_{n}(1))+d(v_{n})=n.$
###### Proof.
The proof follows through contra absurdum. Assume the Jaco Graph $J_{n}(1)$
has a unique Jaconian vertex $v_{i}$. If the edge $(v_{i},v_{n})$ is undefined
then at most, the edge $(v_{i},v_{n-1})$ is defined. When considering vertex
$v_{i+1}$ and proceeding with construction per definition, at least the vertex
$v_{i+1},$ can at most, be linked to $v_{n}$ to have the edge
$(v_{i+1},v_{n})$ defined. So, $d(v_{i+1})\geq d(v_{i})=\Delta(J_{n}(1)),$
renders a contradiction on the uniqueness of the Jaconian vertex $v_{i}.$
Through contra absurdum we conclude that $(v_{i},v_{n})$ is defined. Hence,
result $(a)$ follows.
The Hope subgraph on the vertices $v_{i+1},v_{i+2},\cdots,v_{n}$ allows for
$d(v_{n})=(n-i)-1.$ But with the edge $(v_{i},v_{n})$ added we have
$d(v_{n})=(n-i)-1+1=n-i.$
$\therefore$ $\Delta(J_{n}(1))+d(v_{n})=i+(n-i)=n.$ Hence, result $(b)$
follows. ∎
Note that $\Delta(J_{n}(1))+d(v_{n})=n\nRightarrow$ uniqueness of the Jaconian
vertex.
###### Theorem 2.12.
Consider the Jaco Graph $J_{n}(1).$ For $m<i<k\leq n,$ the edge
$(v_{m},v_{i})$ can only exist if the edge $(v_{m},v_{i-1})$ exists.
Furthermore, if the edge $(v_{i},v_{k})$ exists then the edges
$(v_{i+1},v_{k}),\cdots,(v_{k-1},v_{k})$ exist.
###### Proof.
(Part 1): After applying definition 1.1 exhaustively to the vertex $v_{m-1}$
the vertex $v_{m}$ has attained $d^{-}(v_{m}).$ Applying definition 1.1
exhaustively to vertex $v_{m}$ proceeds by linking the edges
$(v_{m},v_{m+1}),(v_{i},v_{m+2}),\cdots$ in such a way as to attain
$d(v_{m})=\max(abs(\min(d(v_{m})))\leq m.$ So after linking the edge
$(v_{m},v_{i-1})$ and, if and only if $d^{*}(v_{m})@v_{i-1}<m,$ can the edge
$(v_{m},v_{i})$ be linked.
(Part 2): If the edge $(v_{i},v_{k})$ exists then $d^{*}(v_{i})@v_{k}\leq i.$
So because $i+1>i$ it follows that $d^{*}(v_{i+1})@v_{k}<i+1.$ Hence, by
definition 1.1 the edge $(v_{i+1},v_{k})$ exists. ∎
###### Lemma 2.13.
The vertex $v_{i}$ is the prime Jaconian vertex of a Jaco Graph $J_{n}(1),$ if
and only $d(v_{l})\leq d(v_{i})=i$ for $l=i+1,i+2,\cdots,n.$
###### Proof.
Let the vertex $v_{i}$ be the prime Jaconian vertex of the Jaco Graph
$J_{n}(1),~{}n\in\mathbb{N}.$ If for any $v_{l},l=i+1,i+2,\cdots,n.$ we have
$d(v_{l})>d(v_{i})$ then $v_{i}$ cannot be the prime Jaconian vertex of
$J_{n}(1)$ as it then, contradicts definition 1.3.
Conversely: If $d(v_{l})\leq d(v_{i})=i$ for $l=i+1,i+2,\cdots,n,$ then if
follows from definition 1.2 and 1.3 as well as from property 1,
$(d(v_{i})>d(v_{m})=m$ for all $m\in\\{1,2,3,\cdots,i-1\\}),$ that vertex
$v_{i}$ is the prime Jaconian vertex. ∎
###### Theorem 2.14.
If for the Jaco Graph $J_{n}(1),$ we have $\Delta(J_{n}(1))=k,$ then the out-
degrees of the vertices $v_{k+1},v_{k+2},v_{k+3},\cdots,v_{n}$ are
respectively, $\lceil d^{+}(v_{k+1})\rceil=(n-k-1),\lceil
d^{+}(v_{k+2})\rceil=(n-k-2),\cdots,\lceil d^{+}(v_{n-1})\rceil=1,\lceil
d^{+}(v_{n})\rceil=0.$
###### Proof.
From definition 1.4 we have that with $v_{k}$ the prime Jaconian vertex, the
Hope subgraph $\mathbb{H}(J_{n}(1)),$ on vertices
$v_{k+1},v_{k+2},v_{j+3},\cdots,v_{n}$ is a complete graph. On applying
definition 1.1 exhaustively to vertex $v_{k+1}$ it is clearly possible to link
$n-(k+1)$ edges hence, $\lceil d^{+}(v_{k+1})\rceil=n-(k+1)=n-k-1.$
Furthermore, it follows from definition 1.4 that the subgraph on vertices
$v_{k+2},v_{k+3},v_{j+4},\cdots,v_{n}$ is a complete graph as well. On
applying definition 1.1 exhaustively to vertex $v_{k+2}$ it is clearly
possible to link $n-(k+2)$ edges hence, $\lceil
d^{+}(v_{k+2})\rceil=n-(k+2)=n-k-2.$
By repeating the immediate above to vertices $v_{k+3},\cdots,v_{n}$ and noting
that $\lceil d^{+}(v_{n})\rceil=n-n=0,$ the result follows. ∎
###### Theorem 2.15.
If for the Jaco Graph $J_{n}(1),$ we can express $n=7+3k,$
$k\in\\{0,1,2,\cdots\\},$ then: $\Delta(J_{n}(1))\leq n-(3+k)$ and, edge
$(v_{\Delta(J_{n}(1))},v_{n})$ exists.
###### Proof.
For $k=0$ it follows from the Fisher Algorithm that $\Delta(J_{7}(1))\leq
7–(3+0).$ Furthermore, since $4+d^{+}(v_{4})=7$ the edge $(v_{4},v_{7})$
exists.
Assume the result holds for $m=7+3k,~{}k>0,~{}k\in\\{1,2,\cdots\\}$ so for the
Jaco Graph $J_{m}(1)$ we have $\Delta(J_{m}(1))\leq m-(3+k)$ and, edge
$(v_{\Delta(J)},v_{m})$ exists.
Now consider the Jaco Graph on $n$ vertices with, $n=7+3(k+1).$ Noting that
$\Delta(J_{\ell}(1))\leq\ell-1,\forall\ell\in\mathbb{N},$ it follows that
$\Delta(J_{n}(1))-\Delta(J_{m}(1))<7+3(k+1)-(7+3k)=3$ hence,
$\Delta(J_{n}(1))<3+\Delta(J_{m}(1))<3+m-(3+k)=n-(3+k).$ So it follows that
$\Delta(J_{n}(1))\leq n-(3+k)-1=n-(3+(k+1)).$ ∎
## 3 Fibonaccian-Zeckendorf Result
###### Theorem 3.1.
(Bettina’s Theorem): Let $\mathbb{F}=\\{f_{0},f_{1},f_{2},f_{3},...\\}$ be the
set of Fibonacci numbers and let
$n=f_{i_{1}}+f_{i_{2}}+...+f_{i_{r}},n\in\mathbb{N}$ be the Zeckendorf
representation of $n.$ Then
$d^{+}(v_{n})=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r}-1}.$
###### Proof.
Through induction we have that first of all, $1=f_{2}$ and
$d^{+}(v_{1})=1=f_{1}.$ Let $2\leq n=f_{i_{1}}+f_{i_{2}}+...+f_{i_{r}}$ and
let $k=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r}-1}.$ If $i_{r}\geq 3,$ then
$k=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r}-1}$ is the Zeckendorf representation
of $k$, such that induction yields
$d^{+}(v_{k})=k=f_{i_{1}-2}+f_{i_{2}-2}+...+f_{i_{r}-2}.$ Since
$k+d^{+}(v_{k})=f_{i_{1}-1}+f_{i_{1}-2}+f_{i_{2}-1}+f{i_{2}-2}+...f_{i_{r}-1}+f_{i_{r}-2}=f_{i_{1}}+f_{i_{2}}+...f_{i_{r}}=n,$
we have $d^{+}(v_{n})=k.$
Finally consider $n=f_{i_{1}}+f_{i_{2}}+...+f_{i_{r}},i_{r}=2.$ Note that
$n>1$ implies that $i_{r-1}\geq 4$ and that the Zeckendorf representation of
$n-1$ given by $n-1=f_{i_{1}}+f_{i_{2}}+...+f_{i_{r-1}}.$ Let
$k=d^{+}(v_{n-1}).$ Through induction we have that,
$k=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r-1}-1},$ and since $i_{r-1}\leq 4,$ this
is the Zeckendorf representation of $k.$ Accordingly,
$d(v_{k})=f_{i_{1}-2}+f_{i_{2}-2}+...+f_{i_{r-1}-2},$ and
$k+d^{+}(v_{k})=f_{i_{1}-1}+f_{i_{1}-2}+f_{i_{2}-1}+f{i_{2}-2}+...f_{i_{r-1}-1}+f_{i_{r-1}-2}=n-1.$
It follows that $d^{+}(v_{n})>k=d^{+}(v_{n-1}).$ Hence, it follows that
$d^{+}(v_{n})=k+1=(f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r-1}-1})+f_{1}=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r}-1}.$
∎
###### Proposition 3.2.
For Fibonacci numbers $t=f_{n-1},~{}h=f_{n}$ and $l=f_{m},~{}t\geq 3,~{}h\geq
3,~{}l\geq 3$ we have:
(a) $\Delta(J_{h}(1))=t=f_{n-1},$
(b) $\mathbb{J}(J_{h}(1))=\\{v_{t}\\},$
(c) $\Delta(J_{h+l}(1))=\Delta(J_{h}(1))+\Delta(J_{l}(1)),$
(d) $\mathbb{J}(J_{h+l}(1))=\\{v_{\Delta(J_{h}(1))+\Delta(J_{l}(1))}\\}.$
###### Proof.
From the Fisher Algorithm it follows that for $s=f_{2},~{}u=f_{3},w=f_{4}$ we
have:
$f_{2}=1,~{}d^{+}(v_{s})=1,~{}\Delta(J_{s}(1))=0$ and
$\mathbb{J}(J_{1}(1))=\\{v_{1}\\};$
$f_{3}=2,~{}d^{+}(v_{u})=1,~{}\Delta(J_{u}(1))=1$ and
$\mathbb{J}(J_{2}(1))=\\{v_{1},v_{2}\\};$
$f_{4}=3,~{}d^{+}(v_{w})=2,~{}\Delta(J_{w}(1))=2$ and
$\mathbb{J}(J_{3}(1))=\\{v_{2}\\}.$
So assume for $i=f_{k-1}$ and $r=f_{k}$ we have
$d^{+}(v_{r})=f_{k-1},\Delta(J_{r}(1))=f_{k-1}$ and
$\mathbb{J}(J_{r}(1))=\\{v_{i}\\}.$ So for $q=f_{k+1},$ we have that:
$f_{k+1}=f_{k}+f_{k-1}=f_{k}+d^{+}(v_{r})\Rightarrow v_{r}$
is the prime Jaconian vertex of $J_{q}(1)$ so $\Delta(J_{q}(1))=r=f_{k}.$ So
result $(a)$ namely, $\Delta(J_{h}(1))=t=f_{n-1}$ holds in general.
Bettina’s Theorem yields $d^{+}(v_{r+1})=d^{+}(v_{r})+1.$ So we have
$(f_{k}+1)+d^{+}(v_{r+1})=(f_{k}+1)+d^{+}(v_{r})+1=(f_{k}+d^{+}(v_{r}))+2>f_{k+1}+1.$
From Corollary 2.6 it follows that $\mathbb{J}(J_{q}(1))=\\{v_{r}\\}.$ So
result $(b)$ namely, $\mathbb{J}(J_{h}(1))=\\{v_{t}\\}$ holds in general.
The result $\Delta(J_{h+l}(1))=\Delta(J_{h}(1))+\Delta(J_{l}(1))$ follows from
$(b)$ read together with Theorem 2.10.
The result
$\mathbb{J}(J_{h+l}(1))=\\{v_{\Delta(J_{h}(1))+\Delta(J_{l}(1))}\\}$ follows
from $(c)$ read together with Theorem 2.10. ∎
[Open problem: Note that for $P_{0}=\\{n\in\mathbb{N}|1\leq n\leq 44\\}$ we
have, $\Delta(J_{n}(1))=3k$ if $n=5k.$ Then for
$P_{1}=\\{n\in\mathbb{N}|45\leq n\leq 99\\}$ we have, $\Delta(J_{n}(1))=3k+1$
if $n=5k.$ Then for $P_{2}=\\{n\in\mathbb{N}|100\leq n\leq????\\}$ we have,
$\Delta(J_{n}(1))=3k+2$ if $n=5k$ and seemingly so on $\cdots.$ Find a
partitioning $\mathbb{P}=\cup_{i\in\mathbb{N}}P_{i}$ to close the result.]
[Open problem: Prove Lemma (conjectured) 2.2.]
[Open problem: Prove Theorem (conjectured) 2.10.]
[Open problem: Determine the number of spanning trees of $J_{n}(1)$.]
_Open access:_ This paper is distributed under the terms of the Creative
Commons Attribution License which permits any use, distribution and
reproduction in any medium, provided the original author(s) and the source are
credited.
References
$[1]$ Bondy, J.A., Murty, U.S.R., _Graph Theory with Applications,_ Macmillan
Press, London, 1976.
$[2]$ Zeckendorf, E., _Représentation des nombres naturels par une somme de
nombres de Fibonacci ou de nombres de Lucas,_ Bulletin de la Société Royale
des Sciences de Liège, Vol 41, (1972): pp 179-182.
Acknowledgement will be given to colleagues for preliminary peer review and
other contributions on the content of this paper during the preprint arXiv
publication term:222
[Remark: The concept of Jaco Graphs followed from a dream during the night of
10/11 January 2013 which was the first dream Kokkie could recall about his
daddy after his daddy passed away in the peaceful morning hours of 24 May
2012, shortly before the awakening of Bob Dylan, celebrating Dylan’s 71st
birthday]
|
arxiv-papers
| 2014-04-02T08:37:41 |
2024-09-04T02:50:00.580267
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian\n Mukungunugwa",
"submitter": "Johan Kok",
"url": "https://arxiv.org/abs/1404.0484"
}
|
1404.0488
|
Sufficient Criteria for Existence of Pullback Attractors for Stochastic
Lattice Dynamical Systems with Deterministic Non-autonomous Terms 111This work
has been partially supported by NSFC Grants 11071199, NSF of Guangxi Grants
2013GXNSFBA019008 and Guangxi Provincial Department of Research Project Grants
2013YB102.
Anhui Gu, Yangrong Li
School of Mathematics and Statistics, Southwest University, Chongqing 400715,
China
Abstract: We consider the pullback attractors for non-autonomous dynamical
systems generated by stochastic lattice differential equations with non-
autonomous deterministic terms. We first establish a sufficient condition for
existence of pullback attractors of lattice dynamical systems with both non-
autonomous deterministic and random forcing terms. As an application of the
abstract theory, we prove the existence of a unique pullback attractor for the
first-order lattice dynamical systems with both deterministic non-autonomous
forcing terms and multiplicative white noise. Our results recover many
existing ones on the existences of pullback attractors for lattice dynamical
systems with autonomous terms or white noises.
Keywords: Random attractor, stochastic lattice dynamical system,
multiplicative white noise.
## 1 Introduction
The study of non-autonomous evolution equations has attracted several
interests from both mathematicians and physicists due to the effects of time-
dependent linear/non-linear forces from natural phenomena are represented by
non-autonomous terms in the associated models. One of the important concepts
for describing the asymptotic behavior of non-autonomous evolution equations
is the pullback attractor, which generalized the notation of global attractor
for non-autonomous dynamical systems [2, 10, 11]. The pullback attractors are
different from the uniform attractor (see e.g. [7, 8]) in that they employ
techniques of non-autonomous equations more straightly.
Global attractors, uniform attractors and pullback attractors all play
important roles in the fields of asymptotic behavior of autonomous and non-
autonomous infinite dynamical systems [7, 8, 15, 17, 19, 20]. Sometimes, the
forwards dynamics may be hard to describe, in this case, there is not even an
attracting trajectory (which would in general be a moving object) that
describes the dynamics. Especially, in the stochastic cases, the pullback
process produces a fixed subset of the phrase space. Pullback attractors
attract all bounded set, then become appropriate alternatives to study the
asymptotic behavior of dynamical systems.
Lattice dynamical systems, which are coupled systems with ODEs on infinite
lattices, have drawn much attention from mathematicians and physicists
recently, due to the wide range of applications in various areas (e.g. [9]).
For autonomous deterministic lattice dynamical systems, we can see e.g. [3,
26, 27, 28, 31] for the existence and approximations of attractors. For non-
autonomous deterministic cases, we can see e.g. [25, 30, 32, 33] for the
existence and continuity of kernel section, uniform attractors and pullback
exponential attractors. As in the stochastic cases, stochastic lattice
dynamical systems (SLDS) arise naturally while random influences or
uncertainties are taken into account in lattice dynamical systems, these
noises may play an important role as intrinsic phenomena rather than just
compensation of defects in deterministic models. Since Bates et al. [4]
initiated the study of SLDS, lots of work have been done regarding the
existence of global random attractors for SLDS with white
additive/multiplicative noises in regular or weight spaces of infinite
sequences, see e.g. [5, 6, 16, 29]. For lattice dynamical systems perturbed by
other “rough” noises, we can refer to e.g. [13, 14] for more details. As we
can seen that all the systems above are considered with the autonomous
deterministic external forcing terms (if indeed exist!). There is no results
on pullback attractors for general non-autonomous SLDS (with time-dependent
deterministic coefficients and external forcing terms) as far as we know.
Motivated by [22] and [30], we consider the existence of non-autonomous
dynamical systems generated by lattice differential equations with both non-
autonomous deterministic and stochastic forcing terms. By borrowing the main
framework of [22] on two parametric space, we first set up the abstract
structure for the continuous cocycle. As a typical example, we investigate the
following stochastic lattice dynamical systems (SLDS) with time-dependent
external forcing terms:
$\frac{du_{i}(t)}{dt}=\nu_{i}(t)(u_{i-1}-2u_{i}+u_{i+1})-\lambda_{i}(t)u_{i}-f_{i}(u_{i},t)+g_{i}(t)+u_{i}\circ\frac{dw(t)}{dt},$
(1.1)
where $i\in\mathbb{Z}$, $\mathbb{Z}$ denotes the integer set;
$u_{i}\in\mathbb{R}$, $\nu_{i}(t)$ and $\lambda_{i}(t)$ are locally integrable
in $t$; $g_{i}\in C(\mathbb{R},\mathbb{R})$ and $f_{i}\in
C(\mathbb{R}\times\mathbb{R},\mathbb{R})$ satisfies proper dissipative
conditions; $w(t)$ is a Brownian motion (Wiener process) and $\circ$ denotes
the Stratonovich sense of the stochastic term.
Stochastic systems similar to (1.1) are discrete of the Reaction-Diffusion
equation which used to model the phenomena of stochastic resonance in biology
and physics, where $f$ is a time-dependent input signal and $w$ is a Wiener
process used to test the impact of stochastic fluctuations on $f$. For this
topic, we can see e.g. [12, 21, 23, 24] and the references therein. The main
difference between system (1.1) and the model considered in [4] is the
coefficients and deterministic external forcing terms are time-dependent. In
this case, the existing results of one parametric space cannot be applied
directly. We first need to introduce two parametric space to describe the
dynamics of the SLDS: one is responsible for deterministic forcing and the
other is responsible for stochastic perturbations. Then we applied the
skeleton to (1.1).
The outline of the paper is as follows. In the next section, we recall some
results regarding pullback attractor for non-autonomous dynamical systems over
two parametric spaces in [22]. In section 3, we establish the conditions on
the existence of pullback attractors for cocycles over two parametric spaces.
In section 4, a sufficient condition for the existence of pullback attractors
for lattice differential equations with both non-autonomous deterministic and
random forcing terms is given. As a example of the result in previous
sections, the existence of pullback attractor for the first-order SLDS with
time-dependent deterministic force and multiplicative white noise is studied
in the last section.
## 2 Preliminaries
For the reader’s convenience, we recall the theory of pullback random
dynamical systems over two parametric spaces in [22].
Let $\Omega_{1}$ be a nonempty set and $\\{\theta_{1,t}\\}_{t\in\mathbb{R}}$
be a family of mappings from $\Omega_{1}$ into itself such that $\theta_{1,0}$
is the identity on $\Omega_{1}$ and $\theta_{1,s+t}=\theta_{1,t}\theta_{1,s}$
for all $t,s\in\mathbb{R}$. Let $(\Omega_{2},\mathcal{F}_{2},P)$ be a
probability space and $\theta_{2}:\mathbb{R}\times\Omega_{2}\to\Omega_{2}$ be
a $(\mathcal{B}(\mathbb{R})\times\mathcal{F}_{2},\mathcal{F}_{2})$ -measurable
mapping such that $\theta_{2}(0,\cdot)$ is the identity on $\Omega_{2}$,
$\theta_{2}(s+t,\cdot)=\theta_{2}(t,\cdot)\theta_{2}(s,\cdot)$ for all
$t,s\in\mathbb{R}$ and $P\theta_{2}(t,\cdot)=P$ for all $t\in\mathbb{R}$. We
usually write $\theta_{2}(t,\cdot)$ as $\theta_{2,t}$ and call both
$(\Omega_{1},\\{\theta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},P,\\{\theta_{2,t}\\}_{t\in\mathbb{R}})$ a
parametric dynamical system.
Let $(X,d)$ be a complete separable metric space with Borel $\sigma$-algebra
$\mathcal{B}(X)$. Denote by $2^{X}$ the collection of all subsets of $X$. A
set-valued mapping $K:\Omega_{1}\times\Omega_{2}\to 2^{X}$ is called
measurable with respect to $\mathcal{F}_{2}$ in $\Omega_{2}$ if the value
$K(\omega_{1},\omega_{2})$ is a closed nonempty subset of $X$ for all
$\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$, and the mapping
$\omega_{2}\in\Omega_{2}\to d(x,K(\omega_{1},\omega_{2}))$ is
$(\mathcal{F}_{2},\ \mathcal{B}(\mathbb{R}))$-measurable for every fixed $x\in
X$ and $\omega_{1}\in\Omega_{1}$. If $K$ is measurable with respect to
$\mathcal{F}_{2}$ in $\Omega_{2}$, then we say that the family
$\\{K(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}$
is measurable with respect to $\mathcal{F}_{2}$ in $\Omega_{2}$. We now define
a cocycle on $X$ over two parametric spaces.
###### Definition 2.1.
Let $(\Omega_{1},\\{\theta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},P,\\{\theta_{2,t}\\}_{t\in\mathbb{R}})$ be
parametric dynamical systems. A mapping $\Phi$:
$\mathbb{R}^{+}\times\Omega_{1}\times\Omega_{2}\times X\to X$ is called a
continuous cocycle on $X$ over
$(\Omega_{1},\\{\theta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},P,\\{\theta_{2,t}\\}_{t\in\mathbb{R}})$ if for
all $\omega_{1}\in\Omega_{1}$, $\omega_{2}\in\Omega_{2}$ and
$t,\tau\in\mathbb{R}^{+}$, the following conditions (i)-(iv) are satisfied:
* (i)
$\Phi(\cdot,\omega_{1},\cdot,\cdot):\mathbb{R}^{+}\times\Omega_{2}\times X\to
X$ is $(\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}_{2}\times\mathcal{B}(X),\
\mathcal{B}(X))$-measurable;
* (ii)
$\Phi(0,\omega_{1},\omega_{2},\cdot)$ is the identity on $X$;
* (iii)
$\Phi(t+\tau,\omega_{1},\omega_{2},\cdot)=\Phi(t,\theta_{1,\tau}\omega_{1},\theta_{2,\tau}\omega_{2},\cdot)\Phi(\tau,\omega_{1},\omega_{2},\cdot)$;
* (iv)
$\Phi(t,\omega_{1},\omega_{2},\cdot):X\to X$ is continuous.
In the sequel, we use $\mathcal{D}(X)$ to denote a collection of some families
of nonempty subsets of $X$:
${\mathcal{D}(X)}=\\{D=\\{D(\omega_{1},\omega_{2})\subseteq X:\
D(\omega_{1},\omega_{2})\neq\emptyset,\ \omega_{1}\in\Omega_{1},\
\omega_{2}\in\Omega_{2}\\}\\}.$
###### Definition 2.2.
Let $\mathcal{D}(X)$ be a collection of some families of nonempty subsets of
$X$ and $K=\\{K(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\
\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}(X)$. Then $K$ is called a
$\mathcal{D}(X)$-pullback absorbing set for $\Phi$ if for all
$\omega_{1}\in\Omega_{1}$, $\omega_{2}\in\Omega_{2}$ and for every
$B\in\mathcal{D}(X)$, there exists $T=T(B,\omega_{1},\omega_{2})>0$ such that
$\Phi(t,\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2},B(\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2}))\subseteq
K(\omega_{1},\omega_{2})\quad\text{for all}\ t\geq T.$
If, in addition, for all $\omega_{1}\in\Omega_{1}$ and
$\omega_{2}\in\Omega_{2}$, $K(\omega_{1},\omega_{2})$ is a closed nonempty
subset of $X$ and $K$ is measurable with respect to the $P$-completion of
$\mathcal{F}_{2}$ in $\Omega_{2}$, then we say $K$ is a closed measurable
$\mathcal{D}(X)$-pullback absorbing set for $\Phi$.
###### Definition 2.3.
Let $\mathcal{D}(X)$ be a collection of some families of nonempty subsets of
$X$. Then $\Phi$ is said to be $\mathcal{D}(X)$-pullback asymptotically
compact in $X$ if for all $\omega_{1}\in\Omega_{1}$ and
$\omega_{2}\in\Omega_{2}$, the sequence
$\\{\Phi(t_{n},\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2},x_{n})\\}_{n=1}^{\infty}\text{
has a convergent subsequence in }X$
whenever $t_{n}\to\infty$, and $x_{n}\in
B(\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2})$ with
$\\{B(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\
\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}(X)$.
###### Definition 2.4.
Let $\mathcal{D}(X)$ be a collection of some families of nonempty subsets of
$X$ and
$\mathcal{A}=\\{\mathcal{A}(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}(X)$.
Then $\mathcal{A}$ is called a $\mathcal{D}(X)$-pullback attractor for $\Phi$
if the following conditions (i)-(iii) are fulfilled:
* (i)
$\mathcal{A}$ is measurable with respect to the $P$-completion of
$\mathcal{F}_{2}$ in $\Omega_{2}$ and $\mathcal{A}(\omega_{1},\omega_{2})$ is
compact for all $\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$.
* (ii)
$\mathcal{A}$ is invariant, that is, for every $\omega_{1}\in\Omega_{1}$ and
$\omega_{2}\in\Omega_{2}$,
$\Phi(t,\omega_{1},\omega_{2},\mathcal{A}(\omega_{1},\omega_{2}))=\mathcal{A}(\theta_{1,t}\omega_{1},\theta_{2,t}\omega_{2}),\
\ \forall\ t\geq 0.$
* (iii)
$\mathcal{A}$ attracts every member of $\mathcal{D}(X)$, that is, for every
$B=\\{B(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}(X)$
and for every $\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$,
$\lim_{t\to\infty}d(\Phi(t,\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2},B(\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2})),\mathcal{A}(\omega_{1},\omega_{2}))=0.$
The following result on the existence and uniqueness of
$\mathcal{D}(X)$-pullback attractors for $\Phi$ can be found in [22].
###### Proposition 2.5.
Let $\Phi$ be a continuous cocycle on $X$ over
$(\Omega_{1},\\{\theta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},P,\\{\theta_{2,t}\\}_{t\in\mathbb{R}})$. Suppose
that
$K=\\{K(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}(X)$
is a closed measurable (w.r.t. the $P$-completion of $\mathcal{F}_{2}$)
$\mathcal{D}(X)$-pullback absorbing set for $\Phi$ in $\mathcal{D}(X)$ and
$\Phi$ is $\mathcal{D}(X)$-pullback asymptotically compact in $X$. Then $\Phi$
has a unique $\mathcal{D}(X)$-pullback attractor
$\mathcal{A}=\\{\mathcal{A}(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}\in\mathcal{D}(X)$
which is given by
$\displaystyle\mathcal{A}(\omega_{1},\omega_{2})$
$\displaystyle=\bigcap_{\tau\geq
0}\overline{\bigcup_{t\geq\tau}\Phi(t,\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2},K(\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2}))}.$
To describe the size of subsets in a Banach space $X$, we introduce the
concept of Kolmogorov’s $\varepsilon$-entropy. Let $Y$ be a subset of $X$.
Given $\varepsilon>0$, we define
$n_{\varepsilon}(Y):=\min\\{n\geq
1:Y\subset\bigcup_{i=1}^{n}\mathcal{N}(x_{i},\varepsilon)\quad\mbox{for
some}\quad x_{1},\ldots,x_{n}\in X\\},$
where $\mathcal{N}(x_{i},\varepsilon)=\\{y\in
X:\|y-x_{i}\|_{X}<\varepsilon\\}$. The Kolmogorov $\varepsilon$-entropy of the
subset $Y$ of $X$ is the number
$\mathbb{K}_{\varepsilon}(Y):=\ln n_{\varepsilon}(Y)\in[0,+\infty].$ (2.1)
## 3 Pullback attractors for cocycles in $\ell^{2}$
In this section, we provide some sufficient conditions for the existence of
pullback attractors for cocycles in $\ell^{2}$.
Let $D$ be a bounded nonempty subset of $\ell^{2}$, denote by
$\|D\|=\sup_{u\in D}\|u\|$. Suppose
$D=\\{D(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}$
is a family of bounded nonempty subsets of $\ell^{2}$ satisfying, for every
$\gamma>0$,
$\lim_{s\rightarrow+\infty}e^{-\gamma
s}\|D(\theta_{1,-s}\omega_{1},\theta_{2,-s}\omega_{2})\|^{2}=0.$ (3.1)
Denote by $\mathcal{D}(\ell^{2})$ the collection of all family of bounded
nonempty subsets of $\ell^{2}$,
$\mathcal{D}(\ell^{2})=\\{D=\\{D(\omega_{1},\omega_{2}):\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}\\}:D\
\mbox{satisfies}\ \eqref{s1}\\}.$
###### Definition 3.1.
A mapping $\Phi$:
$\mathbb{R}^{+}\times\Omega_{1}\times\Omega_{2}\times\ell^{2}\to\ell^{2}$ is
said to be asymptotically null in $\mathcal{D}(\ell^{2})$ if for a.e.
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$, any
$B(\omega_{1},\omega_{2})\in\mathcal{D}(\ell^{2})$, and any $\varepsilon>0$,
there exist $T(\varepsilon,\omega_{1},\omega_{2},B(\omega_{1},\omega_{2}))>0$
and
$I(\varepsilon,\omega_{1},\omega_{2},B(\omega_{1},\omega_{2}))\in\mathbb{N}$
such that
$\sum_{|i|>I(\varepsilon,\omega_{1},\omega_{2},B(\omega_{1},\omega_{2}))}|(\Phi(t,\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2},u(\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2})))_{i}|^{2}\leq\varepsilon^{2},$
for all $t\geq T(\varepsilon,\omega_{1},\omega_{2},B(\omega_{1},\omega_{2}))$
and $u(\omega_{1},\omega_{2})\in B(\omega_{1},\omega_{2})$.
###### Theorem 3.2.
Suppose that
* $(\mathbf{a})$
there exists a closed measurable (w.r.t. the $P$-completion of
$\mathcal{F}_{2}$) $\mathcal{D}(\ell^{2})$-pullback absorbing set $K$ in
$\mathcal{D}(\ell^{2})$ such that for a.e.
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$, any
$B(\omega_{1},\omega_{2})\in\mathcal{D}(\ell^{2})$, there exists
$T_{B}(\omega_{1},\omega_{2})>0$ yielding
$\Phi(t,\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2})B(\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2})\subset
K$ for all $t\geq T_{B}(\omega_{1},\omega_{2})$;
* $(\mathbf{b})$
$\Phi$:
$\mathbb{R}^{+}\times\Omega_{1}\times\Omega_{2}\times\ell^{2}\to\ell^{2}$ is
asymptotically null on $K$, i.e., for a.e.
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$, any
$B(\omega_{1},\omega_{2})\in\mathcal{D}(\ell^{2})$, and any $\varepsilon>0$,
there exist $T(\varepsilon,\omega_{1},\omega_{2},K)>0$ and
$I_{0}(\varepsilon,\omega_{1},\omega_{2},K)\in\mathbb{N}$ such that
$\displaystyle\sup_{u\in
K}\sum_{|i|>I_{0}(\varepsilon,\omega_{1},\omega_{2},K)}|(\Phi(t,\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2},u(\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2})))_{i}|^{2}\leq\varepsilon^{2},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\forall t\geq
T(\varepsilon,\omega_{1},\omega_{2},B(\omega_{1},\omega_{2})).$
Then
* (i)
$\Phi$ possesses a unique $\mathcal{D}(\ell^{2})$-pullback attractor is given
by, for each $\omega_{1}\in\Omega_{1}$ and $\omega_{2}\in\Omega_{2}$,
$\displaystyle\mathcal{A}(\omega_{1},\omega_{2})$
$\displaystyle=\bigcap_{\tau\geq
T_{K}(\omega_{1},\omega_{2})}\overline{\bigcup_{t\geq\tau}\Phi(t,\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2},K(\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2}))};$
* (ii)
the Kolmogorov $\epsilon$-entropy of $\mathcal{A}(\omega_{1},\omega_{2})$
satisfies
$\displaystyle\mathbb{K}_{\varepsilon}\leq(2I_{0}(\varepsilon,\omega_{1},\omega_{2},K)+1)\ln(\lfloor\frac{2r_{0}(\omega_{1},\omega_{2})\sqrt{2I_{0}(\varepsilon,\omega_{1},\omega_{2},K)+1}}{\varepsilon}\rfloor+1),$
where $r_{0}(\omega_{1},\omega_{2})=\sup_{u(\omega_{1},\omega_{2})\in
K}\|u(\omega_{1},\omega_{2})\|$.
###### Proof.
The proof is based on Theorem 3.1 in [16] and Proposition 2.5 under slightly
modifications.
(i) For a.e. $\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$ and
$t_{n}\rightarrow\infty$ as $n\rightarrow\infty$, let
$p_{n}(\omega_{1},\omega_{2})\in
K(\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2})\in\mathcal{D}(\ell^{2})\
(n=1,2,\cdots)$ and
$u^{(n)}(\omega_{1},\omega_{2})=\Phi(t_{n},\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2}),$
where
$u_{i}^{(n)}(\omega_{1},\omega_{2})=(\Phi(t_{n},\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2}))_{i},i\in\mathbb{Z}$.
By $(\mathbf{a})$, there exists $N_{1}(\omega_{1},\omega_{2},K)\in\mathbb{N}$
such that $t_{n}\geq T_{K}(\omega_{1},\omega_{2})$ if $n\geq
N_{1}(\omega_{1},\omega_{2},K)$. Hence
$u^{(n)}(\omega_{1},\omega_{2})=\Phi(t_{n},\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2})p_{n}(\omega_{1},\omega_{2})\in
K,\ \forall n\geq N_{1}(\omega_{1},\omega_{2},K).$
Now let us prove that the set
$\Lambda=\\{u^{(n)}(\omega_{1},\omega_{2})=\Phi(t_{n},\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2})p_{n}(\omega_{1},\omega_{2})\\}_{n\geq
N_{1}(\omega_{1},\omega_{2},K)}$
is pre-compact, that is, for any given $\varepsilon>0$, $\Lambda$ has a finite
covering of balls of radius $\varepsilon$. By condition $(\mathbf{b})$, there
exists $T_{1}(\varepsilon,\omega_{1},\omega_{2},K)>0$ and
$I_{0}(\varepsilon,\omega_{1},\omega_{2},K)\in\mathbb{N}$ such that for $n\geq
N_{2}(\varepsilon,\omega_{1},\omega_{2},K)$, we have that $t_{n}\geq
T_{1}(\varepsilon,\omega_{1},\omega_{2},K)$ and
$\displaystyle\sup_{n\geq
N_{1}(\omega_{1},\omega_{2},K)}(\sum_{|i|>I_{0}(\varepsilon,\omega_{1},\omega_{2},K)}|\Phi(t_{n},\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2})p_{n}(\omega_{1},\omega_{2}))_{i}|^{2})^{\frac{1}{2}}\leq\frac{\varepsilon}{2}.$
Let
$N_{3}(\varepsilon,\omega_{1},\omega_{2},K)=\max\\{N_{1}(\omega_{1},\omega_{2},K),N_{2}(\varepsilon,\omega_{1},\omega_{2},K)\\}$.
Thus for any $n\geq N_{3}(\varepsilon,\omega_{1},\omega_{2},K)$,
$u^{(n)}(\omega_{1},\omega_{2})=(u_{i}^{(n)}(\omega_{1},\omega_{2}))_{i\in\mathbb{Z}}$
can be decomposed into
$\displaystyle
u^{(n)}(\omega_{1},\omega_{2})=(u_{i}^{(n)}(\omega_{1},\omega_{2}))_{i\in\mathbb{Z}}$
$\displaystyle~{}~{}~{}~{}=(v_{i}^{(n)}(\omega_{1},\omega_{2}))_{i\in\mathbb{Z}}+(o_{i}^{(n)}(\omega_{1},\omega_{2}))_{i\in\mathbb{Z}}$
$\displaystyle~{}~{}~{}~{}~{}~{}=v^{(n)}(\omega_{1},\omega_{2})+o^{(n)}(\omega_{1},\omega_{2}),$
(3.2)
where
$v_{i}^{(n)}(\omega_{1},\omega_{2})=\left\\{\begin{array}[]{lll}u_{i}^{(n)}(\omega_{1},\omega_{2}),&|i|\leq
I_{0}(\varepsilon,\omega_{1},\omega_{2},K),\\\
0,&|i|>I_{0}(\varepsilon,\omega_{1},\omega_{2},K),\end{array}\right.$
and
$o_{i}^{(n)}(\omega_{1},\omega_{2})=\left\\{\begin{array}[]{lll}0,&|i|\leq
I_{0}(\varepsilon,\omega_{1},\omega_{2},K),\\\
u_{i}^{(n)}(\omega_{1},\omega_{2}),&|i|>I_{0}(\varepsilon,\omega_{1},\omega_{2},K).\end{array}\right.$
Then for $n\geq N_{3}(\varepsilon,\omega_{1},\omega_{2},K)$, we obtain
$\displaystyle\|v^{(n)}(\omega_{1},\omega_{2})\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{|i|\leq
I_{0}(\varepsilon,\omega_{1},\omega_{2},K)}|u_{i}^{(n)}(\omega_{1},\omega_{2})|^{2}$
$\displaystyle\leq$ $\displaystyle\|u^{(n)}(\omega_{1},\omega_{2})\|^{2}\leq
r^{2}_{0}(\omega_{1},\omega_{2}),$
$\displaystyle\|o^{(n)}(\omega_{1},\omega_{2})\|^{2}=\sum_{|i|>I_{0}(\varepsilon,\omega_{1},\omega_{2},K)}|u_{i}^{(n)}(\omega_{1},\omega_{2})|^{2}\leq\frac{\varepsilon^{2}}{4},$
and
$|v_{i}^{(n)}(\omega_{1},\omega_{2})|\leq r_{0}(\omega_{1},\omega_{2}),$
for all $|i|\leq I_{0}(\varepsilon,\omega_{1},\omega_{2},K)$, where
$r_{0}(\omega_{1},\omega_{2})$ defined in (ii). Now let
$\displaystyle\Gamma(\omega_{1},\omega_{2})=\\{v=(v_{i})_{|i|\leq
I_{0}(\varepsilon,\omega_{1},\omega_{2},K)}\in\mathbb{R}^{2I_{0}(\varepsilon,\omega_{1},\omega_{2},K)+1}:$
$\displaystyle~{}~{}~{}~{}~{}~{}v_{i}\in\mathbb{R},|v_{i}|\leq
r_{0}(\omega_{1},\omega_{2})\\},$
and
$\displaystyle
n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))$
$\displaystyle~{}~{}~{}~{}~{}~{}=(\lfloor\frac{2r_{0}(\omega_{1},\omega_{2})\sqrt{2I_{0}(\varepsilon,\omega_{1},\omega_{2},K)+1}}{\varepsilon}\rfloor+1)^{2I_{0}(\varepsilon,\omega_{1},\omega_{2},K)+1}.$
Then
$\Gamma(\omega_{1},\omega_{2})\subset\mathbb{R}^{2I_{0}(\varepsilon,\omega_{1},\omega_{2},K)+1}$
is a $(2I_{0}(\varepsilon,\omega_{1},\omega_{2},K)+1)$-dimensional regular
polyhedron which is covered by
$n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))$ open
balls of radius $\frac{\varepsilon}{2}$ centered at
$u^{*}_{m}=(u^{*}_{m,i})_{|i|\leq
I_{0}(\varepsilon,\omega_{1},\omega_{2},K)}$, $u^{*}_{m,i}\in\mathbb{R},1\leq
m\leq n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))$,
in the norm of $\mathbb{R}^{2I_{0}(\varepsilon,\omega_{1},\omega_{2},K)+1}$.
For each $1\leq m\leq
n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))$, we set
$v_{i}=(v_{m,i})_{i\in\mathbb{Z}}\in\ell^{2}$ such that
$v_{m,i}=\left\\{\begin{array}[]{lll}u^{*}_{m,i},&|i|\leq
I_{0}(\varepsilon,\omega_{1},\omega_{2},K),\\\
0,&|i|>I_{0}(\varepsilon,\omega_{1},\omega_{2},K).\end{array}\right.$
Then for
$v^{(n)}(\omega_{1},\omega_{2})=(v_{i}^{(n)}(\omega_{1},\omega_{2}))_{i\in\mathbb{Z}}\
(n\geq N_{3}(\varepsilon,\omega_{1},\omega_{2},K))$ in the decomposition (3),
there exists
$m_{0}\in\\{1,2,\cdots,n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))\\}$
such that
$\displaystyle\|v^{(n)}(\omega_{1},\omega_{2})-v_{m_{0}}\|^{2}=\sum_{|i|\leq
I_{0}(\varepsilon,\omega_{1},\omega_{2},K)}|u_{i}^{(n)}(\omega_{1},\omega_{2})-u_{m_{0},i}|^{2}\leq\frac{\varepsilon^{2}}{4},$
and hence, we get
$\displaystyle\|u^{(n)}(\omega_{1},\omega_{2})-v_{m_{0}}\|^{2}=\|v^{(n)}(\omega_{1},\omega_{2})-v_{m_{0}}+o^{(n)}(\omega_{1},\omega_{2})\|^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\leq
2\|v^{(n)}(\omega_{1},\omega_{2})-v_{m_{0}}\|^{2}+2\|o^{(n)}(\omega_{1},\omega_{2})\|^{2}\leq\varepsilon^{2}.$
Therefore,
$\\{u_{i}^{(n)}(\omega_{1},\omega_{2})=\Phi(t_{n},\theta_{1,-t_{n}}\omega_{1},\theta_{2,-t_{n}}\omega_{2})_{n\geq
N_{3}(\varepsilon,\omega_{1},\omega_{2},K)}\\}\subset\ell^{2}$ can be covered
by
$n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))$ open
balls of radius $\varepsilon$ centered at
$v_{m}=(v_{m,i})_{i\in\mathbb{Z}},1\leq m\leq
n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))$.
(ii) By the invariant property of $\mathcal{D}(X)$-pullback attractors, we
have
$\mathcal{A}(\omega_{1},\omega_{2})=\Phi(t,\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2})\mathcal{A}(\theta_{1,-t}\omega_{1},\theta_{2,-t}\omega_{2})\subset
K$
for $t\geq T(\omega_{1},\omega_{2},K)$ and a.e.
$\omega_{1}\in\Omega_{1},\omega_{2}\in\Omega_{2}$. For any $\varepsilon>0$, we
can see that $\mathcal{A}(\omega_{1},\omega_{2})$ can be covered under the
norm of $\ell^{2}$, by
$n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))$-balls
in $\ell^{2}$ with center $v_{m}=(v_{m,i})_{i\in\mathbb{Z}},1\leq m\leq
n_{\varepsilon,\omega_{1},\omega_{2}}(\Gamma(\omega_{1},\omega_{2}))$ and
radius $\varepsilon$. Thus, by the definition of (2.1), the proof is
completed. ∎
## 4 Pullback attractors for lattice differential equations in $\ell^{2}$
In this section, we discuss the proper choice of parametric spaces
$\Omega_{1}$ and $\Omega_{2}$ to consider pullback attractors for lattice
differential equations with both non-autonomous deterministic and random
forcing terms by using the abstract theory presented in the previous section.
Suppose now $\Omega_{1}=\mathbb{R}$. Define a family
$\\{\theta_{1,t}\\}_{t\in\mathbb{R}}$ of shift operators by
$\theta_{1,t}(\tau)=\tau+t,\ \ \forall t,\tau\in\mathbb{R}.$ (4.1)
Let $\Phi$:
$\mathbb{R}^{+}\times\mathbb{R}\times\Omega_{2}\times\ell^{2}\to\ell^{2}$ be a
continuous cocycle on $\ell^{2}$ over
$(\mathbb{R},\\{\theta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},P,\\{\theta_{2,t}\\}_{t\in\mathbb{R}})$ where
$\\{\theta_{1,t}\\}_{t\in\mathbb{R}}$ is defined in (4.1). Due to Theorem 3.2,
we obtain the following result:
###### Theorem 4.1.
Suppose that
* $(\mathbf{a})$
there exists a closed measurable (w.r.t. the $P$-completion of
$\mathcal{F}_{2}$) $\mathcal{D}(\ell^{2})$-pullback absorbing set $K$ in
$\mathcal{D}(\ell^{2})$ such that for a.e.
$\tau\in\mathbb{R},\omega\in\Omega_{2}$, any
$B(\tau,\omega)\in\mathcal{D}(\ell^{2})$, there exists $T_{B}(\tau,\omega)>0$
yielding
$\Phi(t,\tau-t,\theta_{2,-t}\omega)B(\tau-t,\theta_{2,-t}\omega)\subset K$ for
all $t\geq T_{B}(\tau,\omega)$;
* $(\mathbf{b})$
$\Phi$:
$\mathbb{R}^{+}\times\mathbb{R}\times\Omega_{2}\times\ell^{2}\to\ell^{2}$ is
asymptotically null on $K$, i.e., for a.e.
$\tau\in\mathbb{R},\omega\in\Omega_{2}$, any
$B(\tau,\omega)\in\mathcal{D}(\ell^{2})$, and any $\varepsilon>0$, there exist
$T(\varepsilon,\tau,\omega,K)>0$ and
$I_{0}(\varepsilon,\tau,\omega,K)\in\mathbb{N}$ such that
$\displaystyle\sup_{u\in
K}\sum_{|i|>I_{0}(\varepsilon,\tau,\omega,K)}|(\Phi(t,\tau-t,\theta_{2,-t}\omega,u(\tau-t,\theta_{2,-t}\omega)))_{i}|^{2}\leq\varepsilon^{2},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}\forall t\geq
T(\varepsilon,\tau,\omega,B(\tau,\omega)).$ (4.2)
Then
* (i)
$\Phi$ possesses a unique $\mathcal{D}(\ell^{2})$-pullback attractor is given
by, for each $\tau\in\mathbb{R}$ and $\omega\in\Omega_{2}$,
$\displaystyle\mathcal{A}(\tau,\omega)$ $\displaystyle=\bigcap_{s\geq
T_{K}(\tau,\omega)}\overline{\bigcup_{t\geq
s}\Phi(t,\tau-t,\theta_{2,-t}\omega,K(\tau-t,\theta_{2,-t}\omega))};$ (4.3)
* (ii)
the Kolmogorov $\varepsilon$-entropy of $\mathcal{A}(\tau,\omega)$ satisyies
$\displaystyle\mathbb{K}_{\varepsilon}\leq(2I_{0}(\varepsilon,\tau,\omega,K)+1)\ln(\lfloor\frac{2r_{0}(\tau,\omega)\sqrt{2I_{0}(\varepsilon,\tau,\omega,K)+1}}{\varepsilon}\rfloor+1),$
where $r_{0}(\tau,\omega)=\sup_{u(\tau,\omega)\in K}\|u(\tau,\omega)\|,\
\forall\tau\in\mathbb{R},\omega\in\Omega_{2}$.
## 5 Pullback attractors for SLDS in $\ell^{2}$
In this section, we will apply Theorem 4.1 to prove the existence of a
pullback attractor for non-autonomous first order stochastic lattice dynamical
system.
### 5.1 Mathematical Settings
Denote $C_{b}(\mathbb{R},\ell^{2})$ be the space of all continuous bounded
functions from $\mathbb{R}$ into $\ell^{2}$. Consider the following non-
autonomous first order lattice differential equations with time-dependent
external forcing terms and multiplicative white noise
$\frac{du_{i}(t)}{dt}=\nu_{i}(t)(u_{i-1}-2u_{i}+u_{i+1})-\lambda_{i}(t)u_{i}-f_{i}(u_{i},t)+g_{i}(t)+u_{i}\circ\frac{dw(t)}{dt},\
\ i\in\mathbb{Z},$ (5.1)
with initial data
$u_{i}(\tau)=u_{i,\tau},\ \ i\in\mathbb{Z},\tau\in\mathbb{R},$ (5.2)
where $u_{i}\in\mathbb{R}$, $\mathbb{Z}$ denotes the integer set; $\nu_{i}(t)$
and $\lambda_{i}(t)$ are locally integrable in $t$; $g_{i}\in
C(\mathbb{R},\mathbb{R})$ and $f_{i}\in
C(\mathbb{R}\times\mathbb{R},\mathbb{R})$ for $i\in\mathbb{Z}$; $w$ is an
independent Brownian motion. Note that system (5.1)-(5.2) can be written as
for $t\geq\tau\in\mathbb{R}$,
$\frac{du}{dt}=-\nu(t)Au-\lambda(t)u-f(u,t)+g(t)+u\circ\frac{dw(t)}{dt},\
u(\tau)=u_{\tau}=(u_{i,\tau})_{i\in\mathbb{Z}},$ (5.3)
where $u=(u_{i})_{i\in\mathbb{Z}}$,
$f(u,t)=(f_{i}(u_{i},t))_{i\in\mathbb{Z}},g(t)=(g_{i}(t))_{i\in\mathbb{Z}}$,
$Au=(-u_{i-1}+2u_{i}-u_{i+1})_{i\in\mathbb{Z}}$ and $w(t)$ is the white noise
with values in $\ell^{2}$ defined on the probability space
$(\Omega,\mathcal{F},P)$ and
$\Omega=\\{\omega\in C(\mathbb{R},\ell^{2}):\omega(0)=0\\},$
the Borel sigma-algebra $\mathcal{F}$ is generated by the compact open
topology, and $P$ is the corresponding Wiener measure on $\mathcal{F}$. Define
a group $\\{\theta_{2,t}\\}_{t\in\mathbb{R}}$ acting on
$(\Omega,\mathcal{F},P)$ by
$\theta_{2,t}\omega(\cdot)=\omega(\cdot+t)-\omega(t),\ \
\omega\in\Omega,t\in\mathbb{R}.$ (5.4)
Then $(\Omega,\mathcal{F},P,\\{\theta_{2,t}\\}_{t\in\mathbb{R}})$ is a
parametric dynamical system. We make the following assumptions:
* $(\mathbf{A1})$
$\lambda_{i}(t),\nu_{i}(t)\in L_{loc}^{1}(\mathbb{R})$ in $t$ and there exist
positive constants $\lambda^{0},\lambda_{0}$ and $\nu^{0},\nu_{0}$ such that
for $\forall i\in\mathbb{Z}$, $t\in\mathbb{R}$,
$0<\lambda_{0}\leq\lambda_{i}(t)\leq\lambda^{0}<+\infty,$
$0<\nu_{0}\leq\nu_{i}(t)\leq\nu^{0}<+\infty;$
* $(\mathbf{A2})$
$f_{i}(x,t)$ is differentiable in $x$ and continuous in $t$; $f_{i}(0,t)=0$;
$xf_{i}(x,t)\geq-\alpha_{i}^{2}(t)$, where
$\alpha(t)=(\alpha_{i}(t))_{i\in\mathbb{Z}}\in C_{b}(\mathbb{R},\ell^{2})$,
and there exists a constant $\beta\geq 0$ such that
$\partial_{x}f_{i}(x,t)\geq-\beta$, $\forall x,t\in\mathbb{R},i\in\mathbb{Z}$;
* $(\mathbf{A3})$
There exists a positive-valued continuous function $\zeta(\iota,t)\in
C(\mathbb{R}^{+}\times\mathbb{R},\mathbb{R}^{+})$ such that
$\sup_{i\in\mathbb{Z}}\max_{x\in[-\iota,\iota]}|\partial_{x}f_{i}(x,t)|\leq\zeta(\iota,t),\
\forall\iota\in\mathbb{R}^{+},t\in\mathbb{R};$
* $(\mathbf{A4})$
$g(t)=(g_{i}(t))_{i\in\mathbb{Z}}\in C_{b}(\mathbb{R},\ell^{2})$.
Now, let $\\{\theta_{1,t}\\}_{t\in\mathbb{R}}$ be the group acting on
$\mathbb{R}$ given by (4.1). We next define a continuous cocycle for system
(5.3) over $(\mathbb{R},\\{\theta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega_{2},\mathcal{F}_{2},P,\\{\theta_{2,t}\\}_{t\in\mathbb{R}})$. This can
be done by first transferring the stochastic system into a corresponding non-
autonomous deterministic one. Given $\omega\in\Omega$, denote by
$z(\omega)=-\int^{0}_{-\infty}e^{r}\omega(r)dr.$ (5.5)
Then the random variable $z$ given in (5.5) is a stationary solution of the
one-dimensional Ornstein-Uhlenbeck equation
$dz+zdt=dw(t).$
In other words, we get
$dz(\theta_{2,t}\omega)+z(\theta_{2,t}\omega)dt=dw(t).$ (5.6)
By [4, 6], we know that there exists a $\theta_{2,t}$-variant set
$\Omega^{\prime}\subseteq\Omega$ of full $P$ measure such that
$z(\theta_{2,t}\omega)$ is continuous in $t$ for every
$\omega\in\Omega^{\prime}$, and the random variable $|z(\omega)|$ is tempered.
In addition, for every $\omega\in\Omega^{\prime}$, we have the following
limits:
$\lim_{t\rightarrow\pm\infty}\frac{|\omega(t)|}{|t|}=0,\quad\lim_{t\rightarrow\pm\infty}\frac{|z(\theta_{2,t}\omega)|}{|t|}=0\quad\mbox{and}\quad\lim_{t\rightarrow\pm\infty}\frac{1}{t}\int_{0}^{t}z(\theta_{2,s}\omega)ds=0.$
(5.7)
Hereafter, we will write $\Omega$ as $\Omega^{\prime}$ and $\theta_{2,t}$ as
$\theta_{t}$ instead.
### 5.2 Existence and Uniqueness of a Mild Solution
Let $u(t)$ be the solution of system (5.3), then
$v(t)=u(t)e^{-z(\theta_{t}\omega)}$ satisfies
$\frac{dv}{dt}=-\nu(t)Av-\lambda(t)v-e^{-z(\theta_{t}\omega)}f(e^{z(\theta_{t}\omega)}v,t)+e^{-z(\theta_{t}\omega)}g(t)+z(\theta_{t}\omega)v,$
(5.8)
with initial condition
$v_{\tau}=v(\tau,\omega)=u_{\tau}e^{-z(\theta_{\tau}\omega)},\
t>\tau,\tau\in\mathbb{R},\omega\in\Omega$. We recall
$v:[\tau,\tau+T)\rightarrow\ell^{2}\ (T>0)$ a mild solution of the following
random differential equation
$\frac{dv(t)}{dt}=G(v,t,\theta_{t}\omega),\ \
v=(v_{i})_{i\in\mathbb{Z}},G=(G_{i})_{i\in\mathbb{Z}},\ \
t\geq\tau\in\mathbb{R},$
where $\omega\in\Omega$, if $v\in C([\tau,\tau+T),\ell^{2})$ and
$v_{i}(t,\tau)=v_{i}(\tau)+\int_{\tau}^{t}G_{i}(v(s),s,\theta_{s}\omega)ds\ \
\text{for}\ i\in\mathbb{Z}\ \text{and}\ t\in[\tau,\tau+T).$
In this subsection, we will prove the existence and uniqueness of the mild
solution of system (5.8).
###### Proposition 5.1.
Let $T>0$ and assumptions $(\mathbf{A1}$-$\mathbf{A4})$ hold. Then for
$\tau\in\mathbb{R},\omega\in\Omega$ and any initial data
$v_{\tau}\in\ell^{2}$, system (5.8) has a unique
$(\mathcal{F},\mathcal{B}(\ell^{2}))$-measurable mild solution
$v(\cdot,\tau;\omega,v_{\tau}$, $g)\in C([\tau,\tau+T),\ell^{2})$ with
$v(\tau,\tau;\omega,v_{\tau},g)=v_{\tau}$,
$v(t,\tau;\omega,v_{\tau},g)\in\ell^{2}$ being continuous in
$v_{\tau}\in\ell^{2}$ and $g\in C_{b}(\mathbb{R},\ell^{2})$. Moreover, the
solution $v(t,\tau;\omega,v_{\tau},g)$ exists globally on $[\tau,+\infty)$ for
any $\tau\in\mathbb{R}$. Moreover, for given
$t\in\mathbb{R}^{+},\tau\in\mathbb{R},\omega\in\Omega$ and
$u_{\tau}\in\ell^{2}$, the mapping
$\Phi(t,\tau,\omega,v_{\tau},g)=v(t+\tau,\tau;\theta_{-\tau}\omega,v_{\tau},g)=u(t+\tau,\tau;\theta_{-\tau}\omega,u_{\tau},g)e^{-z(\theta_{t}\omega)},$
generates a continuous cocycle from
$\mathbb{R}^{+}\times\mathbb{R}\times\Omega\times\ell^{2}$ to $\ell^{2}$ over
$(\mathbb{R},\\{\theta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},P,\\{\theta_{t}\\}_{t\in\mathbb{R}})$, where
$v_{\tau}=u_{\tau}e^{-z(\theta_{\tau}\omega)}$.
###### Proof.
We first show that if $v_{\tau}\in\ell^{2}$, system (5.8) has a unique
measurable mild solution $v(t,\tau;\omega,v_{\tau},g)\in\ell^{2}$ on
$[\tau,\tau+T)$ with $v(\tau,\tau;\omega,v_{\tau},g)=v_{\tau}$ for $T>0$ and
$\omega\in\Omega$.
Given $\omega\in\Omega,v_{\tau}\in\ell^{2}$ and $g\in
C_{b}(\mathbb{R},\ell^{2})$, let
$F(v,t,\omega)=-\nu(t)Av-\lambda(t)v-e^{-z(\omega)}f(ve^{z(\omega)},t)+e^{-z(\omega)}g(t)+vz(\omega).$
Note that $F(v,t,\omega)$ is continuous in $v$ and locally integrable in $t$
and measurable in $\omega$ from $\ell^{2}\times\mathbb{R}\times\Omega$ into
$\ell^{2}$. Denote $|\|\cdot|\|=\sup_{t\in\mathbb{R}}\|\cdot(t)\|$, then by
$(\mathbf{A1}$-$\mathbf{A4})$,
$\displaystyle\begin{split}\|F(v,t,\omega)\|&\leq(\lambda^{0}+4\nu^{0}+\max\\{\zeta(\|v\||e^{z(\omega)}|,t),\beta\\}+|z(\omega)|)\|v\|\\\
&\quad+|e^{-z(\omega)}||\|g|\|.\end{split}$
Hence for any
$v^{(1)}=(v_{i}^{(1)})_{i\in\mathbb{Z}},v^{(2)}=(v_{i}^{(2)})_{i\in\mathbb{Z}}\in\ell^{2}$,
$\displaystyle\|F(v^{(1)},t,\omega)-F(v^{(2)},t,\omega)\|$
$\displaystyle~{}~{}\leq(\lambda^{0}+4\nu^{0}+\max\\{\zeta((\|v^{(1)}\|+\|v^{(2)}\|)|e^{z(\omega)}|,t),\beta\\}+|z(\omega)|)\|v^{(1)}-v^{(2)}\|.$
For any bounded set $B\subset\ell^{2}$ with $\sup_{u\in B}\|v\|\leq\iota$, and
define
$\kappa_{B}(t,\omega)=(\lambda^{0}+4\nu^{0}+\max\\{\zeta(\iota|e^{z(\omega)}|,t),\beta\\}+|z(\omega)|)\iota+|e^{-z(\omega)}||\|g|\|\geq
0,$
then for any $v,v^{(1)},v^{(2)}\in B$,
$F(v,t,\omega)\leq\kappa_{B}(t,\omega),\ \
\|F(v^{(1)},t,\omega)-F(v^{(2)},t,\omega)\|\leq\kappa_{B}(t,\omega)\|v^{(1)}-v^{(2)}\|$
and
$\int_{\tau}^{\tau+1}\kappa_{B}(s,\theta_{s}\omega)ds<\infty,\
\forall\tau\in\mathbb{R}.$
By [10, Proposition 2.1.1], problem (5.8) possesses a unique local mild
solution $v(\cdot,\tau,\omega;v_{\tau},g)\in C([\tau,\tau+T_{\max}),\ell^{2})$
satisfying the integral equation
$\displaystyle\begin{split}v(t)&=v_{\tau}+\int_{\tau}^{t}(-\nu(s)Av-\lambda(s)v-e^{-z(\theta_{s}\omega)}f(ve^{z(\theta_{s}\omega)},s)\\\
&\quad\quad+e^{-z(\theta_{s}\omega)}g(s)+vz(\theta_{s}\omega))ds,\
t\in[\tau,\tau+T_{\max})\ (0<T_{\max}\leq T),\end{split}$ (5.9)
where $[\tau,\tau+T_{\max})$ is the maximal interval of existence of the
solution of (5.8).
We next show that $T_{\max}=T$. Since $\lambda_{i}(t),\nu_{i}(t)\in
L_{loc}^{1}(\mathbb{R})$ in $t$, by [18], there exist sequences of continuous
functions in $t\in\mathbb{R}$,
$\lambda_{i}^{(m)}(t),\nu_{i}^{(m)}(t),m\in\mathbb{N}$, such that
$\lim_{m\rightarrow\infty}\int_{\tau}^{t}|\lambda_{i}^{(m)}(s)-\lambda_{i}(s)|ds=0\
\mbox{and}\
\lambda_{0}\leq\lambda_{i}^{(m)}(t)\leq\lambda^{0},\forall\tau,t\in\mathbb{R},$
(5.10)
$\lim_{m\rightarrow\infty}\int_{\tau}^{t}|\nu_{i}^{(m)}(s)-\nu_{i}(s)|ds=0\
\mbox{and}\
\nu_{0}\leq\nu_{i}^{(m)}(t)\leq\nu^{0},\forall\tau,t\in\mathbb{R}.$ (5.11)
Consider the following differential equations with initial data
$v_{\tau}\in\ell^{2}$,
$\frac{dv^{(m)}}{dt}=F^{(m)}(v^{(m)},t,\omega),$ (5.12)
where
$F^{(m)}(v^{(m)},t,\omega)=(F^{(m)}_{i}(v^{(m)},t,\omega))_{i\in\mathbb{Z}}$
and
$\begin{split}F^{(m)}_{i}(v^{(m)},t,\omega)&=-\nu_{i}^{(m)}(t)Av_{i}^{(m)}-\lambda_{i}^{(m)}(t)v_{i}^{(m)}\\\
&\quad-e^{-z(\omega)}f_{i}(v_{i}^{(m)}e^{z(\omega)},t)+e^{-z(\omega)}g_{i}(t)+v_{i}^{(m)}z(\omega).\end{split}$
(5.13)
For $\omega\in\Omega$, by the continuity of $F^{(m)}_{i}(v^{(m)},t,\omega)$ in
$t$, (5.12) has a unique solution $v(\cdot,\tau;\omega,v_{\tau},g)\in
C([\tau,\tau+T_{\max}^{(m)}),\ell^{2})\cap
C^{1}((\tau,\tau+T_{\max}^{(m)}),\ell^{2})$ such that
$\begin{split}\frac{dv_{i}^{(m)}}{dt}=F^{(m)}_{i}(v^{(m)},t,\omega)\end{split}$
(5.14)
and
$\begin{split}v_{i}^{(m)}=v_{\tau}+\int_{\tau}^{t}F^{(m)}_{i}(v^{(m)}(s),s,\omega)ds.\end{split}$
(5.15)
Taking the inner product in $\ell^{2}$ in (5.14) yields
$\displaystyle\frac{d\|v^{(m)}\|^{2}}{dt}=2(-\nu^{(m)}(t)Av^{(m)}-\lambda^{(m)}(t)v^{(m)}+z(\theta_{t}\omega)v^{(m)},v^{(m)})$
$\displaystyle\quad-2(e^{-z(\theta_{t}\omega)}f^{(m)}(v^{(m)}e^{z(\theta_{t}\omega)},t),v^{(m)})+2(e^{-z(\theta_{t}\omega)}g^{(m)}(t),v^{(m)}).$
(5.16)
Note that
$\begin{split}-\beta
e^{2z(\theta_{t}\omega)}\|v^{(m)}\|^{2}&\leq(f(v^{(m)}e^{z(\theta_{t}\omega)},t),v^{(m)}e^{z(\theta_{t}\omega)})\\\
&\quad\quad\leq\iota(e^{z(\theta_{t}\omega)}\|v^{(m)}\|,s)e^{2z(\theta_{t}\omega)}\|v^{(m)}\|^{2}.\end{split}$
It follows from (5.2) that
$\displaystyle\frac{d\|v^{(m)}\|^{2}}{dt}\leq(-\lambda_{0}+2\beta+2z(\theta_{s}\omega))\|v^{(m)}\|^{2}+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})e^{-2z(\theta_{s}\omega)}.$
(5.17)
Applying Gronwall’s inequality to (5.17), we obtain that
$\displaystyle\begin{split}&\|v^{(m)}(t)\|^{2}\leq\|v_{\tau}\|^{2}e^{(2\beta-\lambda_{0})(t-\tau)+2\int_{\tau}^{t}z(\theta_{r}\omega)dr}\\\
&\quad+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})e^{(2\beta-\lambda_{0})t+2\int_{0}^{t}z(\theta_{r}\omega)dr}\int_{\tau}^{t}e^{(\lambda_{0}-2\beta)s-2z(\theta_{s}\omega)-2\int_{0}^{s}z(\theta_{r}\omega)dr}ds\\\
&\quad\quad:=\eta^{2}(t,\tau,\omega),\quad
t\in[\tau,\tau+T_{\max}^{(m)}),\end{split}$
where $\eta^{2}(t,\tau,\omega)\in C([\tau,\tau+T),\mathbb{R}^{+})$ is
independent of $m$, which implies that
$\displaystyle\begin{split}|v_{i}^{(m)}(t)|\leq\eta(t,\tau,\omega),\ \mbox{for
all}\ m\in\mathbb{N},t\in[\tau,\tau+T),\omega\in\Omega.\end{split}$ (5.18)
It then follows that for some $\tilde{\eta}(T,\tau,\omega)>0$, which is
independent on $m$ such that
$|F_{i}^{(m)}(v^{(m)}(t),t)|\leq\tilde{\eta}(T,\tau,\omega)$ and
$\displaystyle\begin{split}|v_{i}^{(m)}(t)-v_{i}^{(m)}(s)|&=\int_{s}^{t}|F_{i}^{(m)}(v^{(m)}(r),r)|dr\\\
&\leq\tilde{\eta}(T,\tau,\omega)|t-s|,\forall
t,s\in[\tau,\tau+T),m\in\mathbb{N},\omega\in\Omega.\end{split}$
By the Arzela-Ascoli Theorem, there exists a convergent subsequence
$\\{v_{i}^{(m_{k})}(t)$, $t\in[\tau,\tau+T)\\}$ of
$\\{v_{i}^{(m)}(t),t\in[\tau,\tau+T)\\}$ such that
$\displaystyle\begin{split}v_{i}^{(m_{k})}(t)\rightarrow\bar{v}_{i}(t)\
\mbox{as}\ k\rightarrow\infty\ \mbox{for}\
t\in[\tau,\tau+T),i\in\mathbb{Z}\end{split}$
and $\bar{v}_{i}(t)$ is continuous in $t\in[\tau,\tau+T)$. Moreover,
$|\bar{v}_{i}(t)|\leq\eta(t,\tau,\omega)$ for
$t\in[\tau,\tau+T),\omega\in\Omega$.
By (5.10), (5.11), (5.18) and the Lebesgue Dominated Convergence Theorem, we
have
$\lim_{k\rightarrow\infty}\int_{\tau}^{t}|\lambda_{i}^{(m_{k})}(s)v_{i}^{(m_{k})}(s)-\lambda_{i}(s)\bar{v}_{i}(s)|ds=0,$
(5.19)
$\lim_{k\rightarrow\infty}\int_{\tau}^{t}|\nu_{i}^{(m_{k})}(s)v_{i}^{(m_{k})}(s)-\nu_{i}(s)\bar{v}_{i}(s)|ds=0.$
(5.20)
Thus by replacing $m$ by $m_{k}$ in (5.15) and letting $k\rightarrow\infty$,
we obtain
$\begin{split}\bar{v}_{i}(t)=v_{\tau}+\int_{\tau}^{t}F_{i}(\bar{v}(s),s,\omega)ds\
\mbox{for all}\ t\in[\tau,\tau+T),\omega\in\Omega,\end{split}$
which implies that $\bar{u}(t)=(\bar{u}_{i}(t))_{i\in\mathbb{Z}}$ is a mild
solution of (5.8). Then by the uniqueness of the mild solutions of (5.8),
$T_{\max}=T$. Moreover, this means that $v(t,\tau;\omega,v_{\tau},g)$ exists
globally on $[\tau,+\infty)$ for any $\tau\in\mathbb{R}$. Here we remain to
show for given $t\in\mathbb{R}^{+},\tau\in\mathbb{R},\omega\in\Omega$ and
$u_{\tau}\in\ell^{2}$, the mapping
$\Phi(t,\tau,\omega,v_{\tau},g)=v(t+\tau,\tau;\theta_{-\tau}\omega,v_{\tau},g)=u(t+\tau,\tau;\theta_{-\tau}\omega,u_{\tau},g)e^{-z(\theta_{t}\omega)},$
generates a continuous cocycle from
$\mathbb{R}^{+}\times\mathbb{R}\times\Omega\times\ell^{2}$ to $\ell^{2}$ over
$(\mathbb{R},\\{\theta_{1,t}\\}_{t\in\mathbb{R}})$ and
$(\Omega,\mathcal{F},P,\\{\theta_{t}\\}_{t\in\mathbb{R}})$ in the sense of
Definition 2.1. In fact, the function $F(v,t,\omega)$ is continuous in $v,g$
and measurable in $t,\omega$, which implies that
$v:(\mathbb{R}^{+})\times\mathbb{R}\times\Omega\times\ell^{2}\rightarrow\ell^{2}$,
$(t,\cdot;\omega,v_{\tau},g)\mapsto v(t,\cdot;\omega,v_{\tau},g)$ is
$(\mathcal{B}(\mathbb{R}^{+})\times\mathcal{F}\times\mathcal{B}(\ell^{2}),\mathcal{B}(\ell^{2}))$-measurable
(see [2]). The proof is complete.
∎
### 5.3 Existence of a Pullback Absorbing Set
In this subsection, we will get the existence of a
$\mathcal{D}(\ell^{2})$-pullback absorbing set for the continuous cocycle
$\Phi$.
###### Lemma 5.2.
Let $\tilde{\lambda}=\lambda_{0}-\beta-\frac{2}{\sqrt{\pi}}>0$. Assume that
$(\mathbf{A1}$-$\mathbf{A4})$ hold, then there exists a closed measurable
$\mathcal{D}(\ell^{2})$-pullback absorbing set
$\mathcal{K}=\\{\mathcal{K}(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\\}$
for $\Phi$ in $\mathcal{D}(\ell^{2})$ such that for any
$B(\tau,\omega)\in\mathcal{D}(\ell^{2})$, there exists
$T_{B}=T_{B}(\tau,\omega)>0$ yielding
$\Phi(t,\tau-t,\theta_{-t}\omega)B(\tau-t,\theta_{-t}\omega)\subseteq\mathcal{K}(\tau,\omega)$
for all $t\geq T_{B}$ and $v_{\tau-t}\in B(\tau-t,\theta_{-t}\omega)$.
###### Proof.
Let $\Phi^{(m)}$ be a solution of system (5.12), then $\Phi^{(m)}\in\ell^{2}$
for all $t\geq\tau$. From Proposition 5.1, we know that
$v^{(m)}(\tau,\tau-t,\theta_{-\tau}\omega)=\Phi^{(m)}(t,\tau-t,\theta_{-t}\omega)$.
Denote $\hat{\lambda}=\lambda_{0}-\beta$ and apply Gronwall’s inequality over
$(\tau-t,\tau)$ to (5.17), it follows that
$\displaystyle\begin{split}&\|v^{(m)}(\tau,\tau-t,\omega,v_{\tau-t})\|^{2}+\beta\int_{\tau-t}^{\tau}e^{-\hat{\lambda}(\tau-s)+2\int_{s}^{\tau}z(\theta_{r}\omega)}\|v^{(m)}(s,\tau-t,\omega,v_{\tau-t})\|^{2}ds\\\
&\quad\leq
e^{-\hat{\lambda}t-2\int_{\tau}^{\tau-t}z(\theta_{r}\omega)dr}\|v_{\tau-t}\|^{2}\\\
&\quad\quad+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})e^{-\hat{\lambda}\tau+2\int_{0}^{\tau}z(\theta_{r}\omega)dr}\int_{\tau-t}^{\tau}e^{\hat{\lambda}s-2z(\theta_{s}\omega)-2\int_{0}^{s}z(\theta_{r}\omega)dr}ds.\end{split}$
Since $v^{(m_{k})}\rightarrow v$ for some $m_{k}\rightarrow\infty$, where $v$
is the mild solution of (5.8), then the estimation above still holds with
$v^{(m_{k})}$ being replaced by $v$. Now, by replacing $\omega$ with
$\theta_{-\tau}\omega$ in the expression $v$, we obtain
$\begin{split}&\|v(\tau,\tau-t,\theta_{-\tau}\omega,v_{\tau-t}\|^{2}\\\
&\quad\quad+\beta\int_{\tau-t}^{\tau}e^{-\hat{\lambda}(\tau-s)+2\int_{s}^{\tau}z(\theta_{r-\tau}\omega)dr}\|v(s,\tau-t,\theta_{-\tau}\omega,v_{\tau-t})\|^{2}ds\\\
&=\|v(\tau,\tau-t,\theta_{-\tau}\omega,v(\tau-t,\theta_{-\tau}\omega)\|^{2}\\\
&\quad\quad+\beta\int_{-t}^{0}e^{-\hat{\lambda}s+2\int_{-t}^{0}z(\theta_{r}\omega)dr}\|v(s+\tau,\tau-t,\theta_{-\tau}\omega,v_{\tau-t})\|^{2}ds\\\
&\leq
e^{-\hat{\lambda}t-2\int_{\tau}^{\tau-t}z(\theta_{r-\tau}\omega)dr}\|v_{\tau-t}\|^{2}\\\
&\quad\quad+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})\int_{\tau-t}^{\tau}e^{\hat{\lambda}(s-\tau)-2z(\theta_{s-\tau}\omega)+2\int_{s}^{\tau}z(\theta_{r-\tau}\omega)dr}ds\\\
&\leq
e^{-\hat{\lambda}t-2\int_{-t}^{0}z(\theta_{r}\omega)dr}\|v_{\tau-t}\|^{2}\\\
&\quad\quad+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})\int_{-t}^{0}e^{\hat{\lambda}s-2z(\theta_{s}\omega)+2\int_{s}^{0}z(\theta_{r}\omega)dr}ds\\\
&\leq
e^{-\hat{\lambda}t-2\int_{-t}^{0}z(\theta_{r}\omega)dr}\|v_{\tau-t}\|^{2}\\\
&\quad\quad+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})\int_{-\infty}^{0}e^{\hat{\lambda}s-2z(\theta_{s}\omega)+2\int_{s}^{0}z(\theta_{r}\omega)dr}ds.\end{split}$
(5.21)
Due to (5.7), we know that
$\int_{-\infty}^{0}e^{\hat{\lambda}s-2z(\theta_{s}\omega)+2\int_{s}^{0}z(\theta_{r}\omega)dr}ds<+\infty,$
and
$\lim_{t\rightarrow\pm\infty}\frac{1}{t}\int_{0}^{t}|z(\theta_{s}\omega)|ds=\frac{1}{\sqrt{\pi}}.$
Let $\tilde{\lambda}=\lambda_{0}-\beta-\frac{2}{\sqrt{\pi}}$ and consider for
any $v_{\tau-t}\in B(\tau-t,\theta_{-t}\omega)$, we have for
$\tilde{\lambda}>0$, $\tau\in\mathbb{R}$ from (3.1) that
$\displaystyle
e^{-\hat{\lambda}t-2\int_{-t}^{0}z(\theta_{r}\omega)dr}\|v_{\tau-t}\|^{2}$
$\displaystyle\quad\leq
e^{-\hat{\lambda}t-2\int_{-t}^{0}z(\theta_{s}\omega)ds}\|B(\tau-t,\theta_{-t}\omega)\|^{2}\rightarrow
0\ \mbox{as}\ t\rightarrow+\infty.$ (5.22)
By (5.21) and (5.3), it follows that
$\displaystyle\|v(\tau,\tau-t,\theta_{-\tau}\omega,v_{\tau-t})\|^{2}$
$\displaystyle\quad\quad\quad\quad\leq
1+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})\int_{-\infty}^{0}e^{\hat{\lambda}s-2z(\theta_{s}\omega)+2\int_{s}^{0}z(\theta_{r}\omega)dr}ds.$
Now denoting
$\displaystyle
R^{2}(\omega)=1+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})\int_{-\infty}^{0}e^{\hat{\lambda}s-2z(\theta_{s}\omega)+2\int_{s}^{0}z(\theta_{r}\omega)dr}ds,$
(5.23)
we conclude that
$\displaystyle\mathcal{K}(\tau,\omega)=\overline{B_{\ell^{2}}(0,R(\omega))}$
(5.24)
is a closed measurable $\mathcal{D}(\ell^{2})$-pullback absorbing set. In
fact, for all $\gamma>0$,
$\displaystyle\begin{split}e^{-\gamma t}R^{2}(\theta_{-t}\omega)&=e^{-\gamma
t}+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})e^{-\gamma
t}\int_{-\infty}^{0}e^{\hat{\lambda}s-2z(\theta_{s-t}\omega)+2\int_{s}^{0}z(\theta_{r-t}\omega)dr}ds\\\
&=e^{-\gamma t}+(2|\|\alpha|\|^{2}+\frac{|\|g|\|^{2}}{\lambda_{0}})e^{-\gamma
t}\int_{-\infty}^{-t}e^{\hat{\lambda}(s+t)-2z(\theta_{s}\omega)+2\int_{s}^{0}z(\theta_{r}\omega)dr}ds\\\
&\quad\quad\rightarrow 0\ \ \mbox{as}\ \ t\rightarrow+\infty.\end{split}$
∎
### 5.4 Asymptotically Null of the Solutions
In this subsection, the property of asymptotically null for the solution
$\Phi$ of system (5.8) will be established.
###### Lemma 5.3.
Let $\mathcal{K}(\tau,\omega)$ be the absorbing set given by (5.24). Then for
every $\epsilon>0$, there exist
$\tilde{T}(\epsilon,\tau,\omega,\mathcal{K}(\tau,\omega))>0$ and
$\tilde{N}(\epsilon,\tau,\omega,\mathcal{K}(\tau,\omega))\geq 1$, such that
the solution
$\Phi(t,\tau-t,\theta_{-t}\omega)=v(\tau,\tau-t,\theta_{-\tau}\omega)$ of
problem (5.8) is asymptotically null, that is, for all
$t\geq\tilde{T}(\epsilon,\tau,\omega,\mathcal{K}(\tau,\omega))$,
$v_{\tau-t}\in B(\tau-t,\theta_{-t}\omega)$,
$\displaystyle\sum_{|i|>\tilde{N}(\epsilon,\tau,\omega,\mathcal{K}(\tau,\omega))}|(v(\tau,\tau-t,\theta_{-\tau}\omega,v_{\tau-t},g)_{i}|^{2}\leq\epsilon^{2}.$
###### Proof.
Choose a smooth cut-off function satisfying $0\leq\rho(s)\leq 1$ for
$s\in\mathbb{R^{+}}$ and $\rho(s)=0$ for $0\leq s\leq 1$, $\rho(s)=1$ for
$s\geq 2$. Suppose there exists a constant $c_{0}$ such that
$|\rho^{\prime}(s)|\leq c_{0}$ for $s\in\mathbb{R}^{+}$. For any $n\geq 1$,
let
$v^{(m)}_{n}=v^{(m)}(\tau,\tau-t,\omega,v_{\tau-t,n},g_{n})=(v^{(m)}_{n,i})_{i\in\mathbb{Z}}$
be a mild solution of (5.12). Let $N$ be a fixed integer which will be
specified later, and set $x^{(m)}_{n}=(x^{(m)}_{n,i})_{i\in\mathbb{Z}}$ where
$x^{(m)}_{n,i}=\rho(\frac{|i|}{N})v^{(m)}_{n,i}$ for any $i\in\mathbb{Z}$.
Then taking the inner product of (5.12) with $x$ in $\ell^{2}$, we obtain
$\begin{split}&\frac{d}{dt}\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|v^{(m)}_{n,i}|^{2}\\\
&\quad=-2\nu^{(m)}(t)(A_{m}v^{(m)}_{n},x^{(m)}_{n})-2(\lambda^{(m)}(t)-z(\theta_{t}\omega))\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|v^{(m)}_{n,i}|^{2}\\\
&\quad\quad-2e^{-z(\theta_{t}\omega)}\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})f(e^{z(\theta_{t}\omega)}v^{(m)}_{n,i},t)v^{(m)}_{n,i}\\\
&\quad\quad+2e^{-z(\theta_{t}\omega)}(g_{n}(t),x^{(m)}_{n}).\end{split}$
(5.25)
We now estimate terms in (5.25) one by one. First, we have
$\displaystyle(A_{m}v^{(m)}_{n},x^{(m)}_{n})=(\tilde{B}_{m}v^{(m)},\tilde{B}_{m}x^{(m)}_{n})\geq-\frac{2c_{0}}{N}\|v^{(m)}\|^{2}.$
(5.26)
For the second term in (5.25), it follows from the assumption
$(\mathbf{A}_{2})$ that
$\displaystyle-\infty<-2e^{-z(\theta_{t}\omega)}\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})f_{i}(e^{z(\theta_{t}\omega)}v^{(m)}_{n,i},t)v^{(m)}_{n,i}\leq
2e^{-2z(\theta_{t}\omega)}\sum_{|i|\geq N}\alpha^{2}_{i}(t).$
For the last term in (5.25),
$\displaystyle
2e^{-z(\theta_{t}\omega)}(g_{n}(t),x^{(m)}_{n})\leq\lambda^{(m)}(t)\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|v_{i}|^{2}+\frac{1}{\lambda_{0}}e^{-2z(\theta_{t}\omega)}\sum_{|i|\geq
N}g^{2}_{i}(t).$ (5.27)
Combining (5.25)-(5.27), it yields
$\begin{split}&\frac{d}{dt}\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|v^{(m)}_{n,i}|^{2}+(\lambda_{0}-2z(\theta_{t}\omega))\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|v^{(m)}_{n,i}|^{2}\\\
&\quad\leq\frac{4\nu^{0}c_{0}}{N}\|v^{(m)}\|^{2}+(2+\frac{1}{\lambda_{0}})e^{-2z(\theta_{t}\omega)}\sum_{|i|\geq
N}(\alpha^{2}_{i}(t)+g^{2}_{i}(t)).\end{split}$ (5.28)
Apply Gronwall’s inequality to (5.28) over $(\tau-t,\tau)$, we obtain that
$\begin{split}&\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|v^{(m)}_{n,i}(\tau,\tau-t,\omega,v^{(m)}_{\tau-t,n},g_{n})|^{2}\\\
&\leq
e^{-\lambda_{0}t-2\int_{\tau}^{\tau-t}z(\theta_{r}\omega)dr}\|v^{(m)}_{\tau-t,n}\|^{2}\\\
&\quad+\frac{4\nu^{0}c_{0}}{N}\int_{\tau-t}^{\tau}e^{-\lambda_{0}(\tau-s)+2\int_{s}^{\tau}z(\theta_{r}\omega)dr}\|v^{(m)}(s,\tau-t,\omega,v^{(m)}_{\tau-t,n},g_{n})\|^{2}ds\\\
&\quad+(2+\frac{1}{\lambda_{0}})\sum_{|i|\geq
N}(\alpha^{2}_{i}(t)+g^{2}_{i}(t))\int_{\tau-t}^{\tau}e^{-\lambda_{0}(\tau-s)+2\int_{s}^{\tau}z(\theta_{r}\omega)dr-2z(\theta_{s}\omega)}ds.\end{split}$
(5.29)
Now, for $\tau\in\mathbb{R}$, substitute $\theta_{-\tau}\omega$ for $\omega$
and estimate each term in (5.29)
$\begin{split}&\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|v^{(m)}_{n,i}(\tau,\tau-t,\theta_{-\tau}\omega,v^{(m)}_{\tau-t,n},g_{n})|^{2}\\\
&\leq
e^{-\lambda_{0}t-2\int_{\tau}^{\tau-t}z(\theta_{r-\tau}\omega)dr}\|v^{(m)}_{\tau-t,n}\|^{2}\\\
&\quad+\frac{4\nu^{0}c_{0}}{N}\int_{\tau-t}^{\tau}e^{-\lambda_{0}(\tau-s)+2\int_{s}^{\tau}z(\theta_{r-\tau}\omega)dr}\|v^{(m)}(s,\tau-t,\theta_{-\tau}\omega,v^{(m)}_{\tau-t,n},g_{n})\|^{2}ds\\\
&\quad+(2+\frac{1}{\lambda_{0}})\sum_{|i|\geq
N}(\alpha^{2}_{i}(t)+g^{2}_{i}(t))\int_{\tau-t}^{\tau}e^{-\lambda_{0}(\tau-s)+2\int_{s}^{\tau}z(\theta_{r-\tau}\omega)dr-2z(\theta_{s-\tau}\omega)}ds\\\
&\leq
e^{-\lambda_{0}t-2\int_{-t}^{0}z(\theta_{r}\omega)dr}\|v^{(m)}_{\tau-t,n}\|^{2}\\\
&\quad+\frac{4\nu^{0}c_{0}}{N}\int_{-t}^{0}e^{-\lambda_{0}s+2\int_{-t}^{0}z(\theta_{r}\omega)dr}\|v^{(m)}(s+\tau,\tau-t,\theta_{-\tau}\omega,v^{(m)}_{\tau-t,n},g_{n})\|^{2}ds\\\
&\quad+(2+\frac{1}{\lambda_{0}})\sum_{|i|\geq
N}(\alpha^{2}_{i}(t)+g^{2}_{i}(t))\int_{-t}^{0}e^{\lambda_{0}s+2\int_{s}^{0}z(\theta_{r}\omega)dr-2z(\theta_{s}\omega)}ds.\end{split}$
(5.30)
By Lemma 5.2, there exists $T_{1}(\epsilon,\tau,\omega,\mathcal{K}(\omega))>0$
such that for all $t\geq T_{1}(\epsilon,\tau,\omega,\mathcal{K}(\omega))$,
$\begin{split}&\frac{4\nu^{0}c_{0}}{N}\int_{-t}^{0}e^{-\lambda_{0}s+2\int_{-t}^{0}z(\theta_{r}\omega)dr}\|v^{(m)}(s+\tau,\tau-t,\theta_{-\tau}\omega,v^{(m)}_{\tau-t,n},g_{n})\|^{2}ds\\\
&\quad\quad\leq\frac{4\nu^{0}c_{0}}{\beta N}R^{2}(\omega),\end{split}$ (5.31)
where $R^{2}(\omega)$ is given by (5.23). Since $g(t),\alpha(t)\in
C_{b}(\mathbb{R},\ell^{2})$, by using (5.7) again, we know
$\begin{split}(2+\frac{1}{\lambda_{0}})\sum_{|i|\geq
N}(\alpha^{2}_{i}(t)+g^{2}_{i}(t))\int_{-\infty}^{0}e^{\lambda_{0}s+2\int_{s}^{0}z(\theta_{r}\omega)dr-2z(\theta_{s}\omega)}ds<\infty,\end{split}$
and hence
$\begin{split}\lim_{N\rightarrow\infty}(2+\frac{1}{\lambda_{0}})\sum_{|i|\geq
N}(\alpha^{2}_{i}(t)+g^{2}_{i}(t))\int_{-\infty}^{0}e^{\lambda_{0}s+2\int_{s}^{0}z(\theta_{r}\omega)dr-2z(\theta_{s}\omega)}ds=0.\end{split}$
(5.32)
Now, by means of (5.3) and (5.30)-(5.32), there exist
$\tilde{T}(\epsilon,\tau,\omega,\mathcal{K}(\omega))\geq
T_{1}(\epsilon,\tau,\omega,\mathcal{K}(\omega))$ and
$\tilde{N}(\epsilon,\tau,\omega,\mathcal{K}(\omega))\geq 1$ such that
$\displaystyle\sum_{|i|\geq\tilde{N}(\epsilon,\tau,\omega,\mathcal{K}(\omega))}|v^{(m)}_{n,i}(\tau,\tau-t,\theta_{-\tau}\omega,v^{(m)}_{\tau-t,n},g_{n})|^{2}$
(5.33) $\displaystyle\leq$
$\displaystyle\sum_{i\in\mathbb{Z}}\rho(\frac{|i|}{N})|v^{(m)}_{n,i}(\tau,\tau-t,\theta_{-\tau}\omega,v^{(m)}_{\tau-t,n},g_{n})|^{2}\leq\epsilon^{2}.$
Since there is $m_{k}$ such that
$v^{(m_{k})}_{n,i}(\tau,\tau-t,\theta_{-\tau}\omega,v^{(m)}_{\tau-t,n},g_{n})\rightarrow(v(\tau,\tau-t,\theta_{-\tau}\omega,v_{\tau-t,n},g_{n}))_{i}$
as $m_{k}\rightarrow\infty$, by (5.33)
$\displaystyle\sum_{|i|\geq\tilde{N}(\epsilon,\tau,\omega,\mathcal{K}(\omega))}|(v(\tau,\tau-t,\theta_{-\tau}\omega,v_{\tau-t,n},g_{n}))_{i}|^{2}\leq\epsilon^{2}$
for any $n\geq 1$. Now, letting $n\rightarrow\infty$ we can obtain the
conclusion. ∎
### 5.5 Existence of Pullback Attractors
We are now in a position to give our main result in this section.
###### Theorem 5.4.
Suppose that $(\mathbf{A1}$-$\mathbf{A4})$ hold. The lattice dynamical system
$\Phi$ with both non-autonomous deterministic and random forcing terms
generated by system (5.8) has a unique pullback attractor.
###### Proof.
The result follows directly from Lemmas 5.2, 5.3 and Theorem 4.1. ∎
## Acknowledgments
The first author thanks Prof. Bixiang Wang for emailing him the file of
reference [5].
## References
* [1] A. Adams and J. Fournier, “Sobolev Spaces,” 2nd edition, Elsevier Ltd., Amsterdam, 2003.
* [2] L. Arnold, “Random Dynamical Systems,” Springer-Verlag, Berlin, 1998.
* [3] P.W. Bates, K. Lu and B. Wang, _Attractors for lattice dynamical systems_ , Internat. J. Bifur. Chaos 11 (2001), 143–153.
* [4] P. W. Bates, H. Lisei and K. Lu, _Attractors for stochastic lattice dynamical systems_ , Stoch. Dyn., 6 (2006), 1–21.
* [5] P. W. Bates, K. Lu and B. Wang, _Attractors of non-autonomous stochastic lattice systems in weighted spaces_ , (2013), preprint.
* [6] T. Caraballo and K. Lu, _Attractors for stochastic lattice dynamical systems with a multiplicative noise_ , Front. Math. China, 3 (2008), 317–335.
* [7] A. Carvalho, J. Langa and J. Robinson, “Attractors for infinite-dimensional nonautonomous dynamical systems,” Springer, New York, 2012.
* [8] V. Chepyzhov and M. Vishik, “Attractors for equations of mathematical physics,” American Mathematical Society, Providence, 2002\.
* [9] S. Chow, “Lattice dynamical systems,” in “Dynamical systems”,1–102, Springer, Berlin, 2003.
* [10] I. Chueshov, “Monotone Random Systems Theory and Applications,” Springer-Verlag, New York, 2002.
* [11] H. Crauel, A. Debussche and F. Flandoli, _Random attractors_ , J. Dyn. Differ. Equat., 9 (1997), 307–341.
* [12] L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni, _Stochastic resonance_ , Reviews of Modern Physics, 70 (1998), 223–287.
* [13] A. Gu, _Random attractors of stochastic lattice dynamical systems driven by fractional Brownian motions_ , Internat. J. Bifur. Chaos, 23 (2013), 1–9.
* [14] A. Gu and W. Ai, _Random attractor for stochastic lattice dynamical systems with $\alpha$-stable Lévy noises_, Commun. Nonlinear Sci. Numer. Simulat., 19 (2014), 1433–1441.
* [15] J. Hale, “Aymptotic behavior of dissipative systems,” American Mathematical Society, Providence, 1988.
* [16] X. Han, W. Shen and S. Zhou, _Random attractors for stochastic lattice dynamical system in weighted spaces_ , J. Differ. Eqns., 250 (2011), 1235–1266.
* [17] P. Kloeden and M. Rasmussen, “Nonautonomous Dynamical Systems,” American Mathematical Society, Providence, 2011.
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* [20] R. Temam, “Infinite-dimensional dynamical systems in mechanics and physics,” Springer-Verlag, New York, 1988.
* [21] H. Tuckwell, “Introduction to Theoretical Neurobiology: Vol. 2, Nonlinear and Stochastic Theories,” Cambridge University Press, Cambridge, 1998.
* [22] B. Wang, _Sufficient and necessary criteria for existence of pullback attactors for non-compact random dynamical systems_ , J. Differ. Eqns., 253 (2012), 1544–1583.
* [23] K. Wiesenfeld and F. Moss, _Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs_ , Nature, 373 (1995), 33–35.
* [24] K. Wiesenfeld, D. Pierson, E. Pantazelou, C. Dames and F. Moss, _Stochastic resonance on a circle_ , Phys. Rev. Lett., 72 (1994), 2125–2129.
* [25] X. Zhao and S. Zhou, _Kernel sections for processes and nonautonomous lattice systems_ , Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 763–785.
* [26] S. Zhou, _Attractors for second order lattice dynamical systems_ , J. Differ. Eqns., 179 (2002), 605–624.
* [27] S. Zhou, _Attractors for lattice systems corresponding to evolution equations_ , Nonlinearity, 15 (2002), 1079–1095.
* [28] S. Zhou, _Attractors and approximations for lattice dynamical systems_ , J. Differ. Eqns., 200 (2004), 342–368.
* [29] S. Zhou, _Upper-semicontinuity of attractors for random lattice systems perturbed by small white noises_ , Nonlinear Analysis, 75 (2012), 2793–2805.
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|
arxiv-papers
| 2014-04-02T08:52:10 |
2024-09-04T02:50:00.587617
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Anhui Gu and Yangrong Li",
"submitter": "Anhui Gu Dr.",
"url": "https://arxiv.org/abs/1404.0488"
}
|
1404.0619
|
# On sinh-Gordon Thermodynamic Bethe Ansatz and fermionic basis.
S. Negro SN Department of mathematical sciences, University of Durham,
Science Laboratories, South Rd, Durham DH1 3LE, United Kingdom. Dip. di Fisica
and INFN, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy
[email protected] [email protected]
###### Abstract.
We review the construction of the fermionic basis for sinh-Gordon model and
investigate numerically the ultra-violet limit of the one-point functions. We
then compare the predictions obtained from this formalism against previously
established results.
## 1\. Introduction
In the study of a Quantum Field Theory (QFT), the one-point functions play a
fundamental rôle; indeed, when using the Operator Product Expansion (OPE) to
calculate the ultraviolet asymptotics of a correlation function, one needs to
know both the coefficients of the said expansion and the one-point functions
of the local operators in the theory. While the former are purely ultra-violet
objects and can, in principle, be extracted via perturbation theory of the
corresponding ultra-violet Conformal Field Theory (CFT), the one-point
functions depend essentially on the infra-red structure of the theory, where
perturbative techniques are of no help at all. Thus the development of new
methods to explore the infra-red region is of primary importance.
Integrable models are the perfect playground where one can experiment with new
analytical methods aimed at extracting data; in particular the sinh-Gordon
model is the simplest example of massive integrable QFT and, at the same time,
is complicated enough to display interesting structures. Moreover this model,
along with its twin, the sine-Gordon model, has received plenty of attention
in the last 30 years and nowadays most of its features are known.
Computing one-point functions in an integrable deformation of a CFT is
anything but an easy task and we wish to explain the reasons for this fact
clearly. Although in our deformed theory the conformal invariance is broken,
the local fields retain a one-to-one correspondence with those of the original
CFT, which are organized according to the corresponding Virasoro algebra in
the usual way. This means that in the perturbed theory there exist fields
$\Phi_{a}(z,\bar{z})=e^{a\eta(z,\bar{z})}$ which can be deemed as _primary_ ,
whose space of descendants can be identified with the tensor product of Verma
modules $\mathcal{V}_{a}\otimes\bar{\mathcal{V}}_{a}$ of the unperturbed CFT.
The operators acting in the space of states of the perturbed CFT can thus be
interpreted as operators acting on the corresponding Verma modules and,
consequently, one-point functions appear to be functionals on the tensor
product $\mathcal{V}_{a}\otimes\bar{\mathcal{V}}_{a}$. However, we still have
not taken in account the integrable structure of the model; in fact _all one-
point functions of descendants built out of integrals of motion identically
vanish_. This means that the correct space on which the one-point function
should be defined as a linear functional is the tensor product
$\mathcal{V}^{\textrm{quo}}_{a}\otimes\bar{\mathcal{V}}^{\textrm{quo}}_{a}$ of
the two quotient spaces
(1.1)
$\mathcal{V}^{\textrm{quo}}_{a}\doteq\mathcal{V}_{a}\Bigg{/}\sum_{k=1}^{\infty}\mathbf{i}_{2k-1}\mathcal{V}_{a}\
,\qquad\bar{\mathcal{V}}^{\textrm{quo}}_{a}\doteq\bar{\mathcal{V}}_{a}\Bigg{/}\sum_{k=1}^{\infty}\bar{\mathbf{i}}_{2k-1}\bar{\mathcal{V}}_{a}\;,$
where with $\mathbf{i}_{2k-1}$ (respectively $\bar{\mathbf{i}}_{2k-1}$) we
denote the action of the chiral (antichiral) integrals of motion on the Verma
module.
It’s now becoming clear what is the main issue: the basis we introduced above,
composed of the primary fields $\Phi_{a}(z,\bar{z})$ and their “conformal”
descendants, is a basis for the full Verma module! In order to reduce this
last to a basis of the quotient space, one has to factor out by hand all the
null vectors which arise from the action of the integrals of motion and their
form quickly becomes rather involved. One would rather work directly in the
quotient space, where the factoring of null vectors is automatically taken in
account, and fix uniquely a basis by means of some physical requirement. A
basis of this kind was actually discovered some years ago for the six-vertex
model [1, 2, 3] and immediately extended to CFT [4], sine-Gordon [5, 6, 7] and
sinh-Gordon models [8].
The building blocks of this basis are the primary fields $\Phi_{a}(z,\bar{z})$
and creation operators which, acting on the former, produce the descendants,
much like what happens for the usual conformal basis; the peculiar fact is
that these creation operators are _fermions_. There are two of them for each
chirality : $\boldsymbol{\beta}^{\ast}_{2j-1}$,
$\boldsymbol{\gamma}^{\ast}_{2j-1}$, $\bar{\boldsymbol{\beta}}^{\ast}_{2j-1}$
and $\bar{\boldsymbol{\gamma}}^{\ast}_{2j-1}$. In the above-cited articles,
these fermions were defined, in a mathematically rigorous fashion for six-
vertex, CFT and sine-Gordon models and as an educated conjecture for the sinh-
Gordon model, and their properties were thoroughly analysed; in particular for
sin(h)-Gordon model111Here and in the following, the shorthand sin(h)-Gordon
is used to denote both sine-Gordon and sinh-Gordon., the quotient space
$\mathcal{V}^{\textrm{quo}}_{a}\otimes\bar{\mathcal{V}}^{\textrm{quo}}_{a}$
was shown to allow the following basis:
(1.2)
$\boldsymbol{\beta}^{\ast}_{I^{+}}\bar{\boldsymbol{\beta}}^{\ast}_{\bar{I}^{+}}\bar{\boldsymbol{\gamma}}^{\ast}_{\bar{I}^{-}}\boldsymbol{\gamma}^{\ast}_{I^{-}}\Phi_{a}(0)\
,\qquad\mathfrak{C}(I^{+})=\mathfrak{C}(I^{-})\ ,\
\mathfrak{C}(\bar{I}^{+})=\mathfrak{C}(\bar{I}^{-})\;,$
where $I^{\pm}=\\{2i_{1}^{\pm}-1,\ldots,2i_{n}^{\pm}-1\\}$ and similarly for
$\bar{I}^{\pm}$. The symbol $\mathfrak{C}(I)$ stands for the _cardinality_ of
the set $I$ and the following multi-index notation is introduced:
(1.3) $A_{I}=A_{i_{1}}A_{i_{2}}\ldots A_{i_{n}}\
;\quad|I|\doteq\sum_{p=1}^{\mathfrak{C}(I)}i_{p}\ ;\quad
aI+b=\\{ai_{1}+b,\ldots,ai_{n}+b\\}\;,$
where $a,b\in\mathbb{Z}$ and $I=\\{i_{1},\ldots,i_{n}\\}$.
While the rigorous construction of this basis, presented in Refs. [4]-[6],
might appear somewhat cumbersome and hard to understand, when the dust raised
by their construction has fallen, the fermions reveal their true strength in
the simple and beautiful determinant formula for the one-point functions:
(1.4)
$\frac{\langle\boldsymbol{\beta}^{\ast}_{I^{+}}\bar{\boldsymbol{\beta}}^{\ast}_{\bar{I}^{+}}\bar{\boldsymbol{\gamma}}^{\ast}_{\bar{I}^{-}}\boldsymbol{\gamma}^{\ast}_{I^{-}}\Phi_{a}(0)\rangle_{R}}{\langle\Phi_{a}(0)\rangle_{R}}=\mathcal{D}\Big{(}I^{+}\cup(-\bar{I}^{+})\Big{|}I^{-}\cup(-\bar{I}^{-})\Big{|}\alpha\Big{)}\;,$
where, for two sets $A=\\{a_{j}\\}_{j=1}^{n}$ and $B=\\{b_{j}\\}_{j=1}^{n}$,
the function $\mathcal{D}$ is defined as follows
$\displaystyle\mathcal{D}(A|B|\alpha)$
$\displaystyle\doteq\left(\prod_{\ell=1}^{n}\frac{\textrm{sgn}(a_{\ell})\textrm{sgn}(b_{\ell})}{\pi}\right)\times$
(1.5)
$\displaystyle\times\det\Big{[}\Theta(ia_{j},ib_{k}|\alpha)-\pi\textrm{sgn}(a_{j})t_{a_{j}}(\alpha)\;\delta_{a_{j},-b_{k}}\Big{]}_{j,k=1}^{n}$
and the functions $\Theta(a,b|\alpha)$ and $t_{a}(\alpha)$ will be defined
below. The parameter $\alpha$ is related to the conformal dimension of the
primary field by
(1.6) $\alpha=\frac{2}{b+b^{-1}}a\;.$
A very important property of the fermions is that, aside from allowing the
construction of the descendants, they can be used in order to _shift_ the
primary and descendant fields in their conformal dimension $a$. As it is shown
in Ref. [6], if we give up the conditions
$\mathfrak{C}(I^{+})=\mathfrak{C}(I^{-}),\
\mathfrak{C}(\bar{I}^{+})=\mathfrak{C}(\bar{I}^{-})$ in favour of the less
restraining
$\mathfrak{C}(I^{+})-\mathfrak{C}(I^{-})=\mathfrak{C}(\bar{I}^{-})-\mathfrak{C}(\bar{I}^{+})=m$,
then the following relation holds
$\displaystyle\boldsymbol{\beta}^{\ast}_{I^{+}}\bar{\boldsymbol{\beta}}^{\ast}_{\bar{I}^{+}}\bar{\boldsymbol{\gamma}}^{\ast}_{\bar{I}^{-}}\boldsymbol{\gamma}^{\ast}_{I^{-}}\Phi_{a-mb}(0)\cong$
(1.7)
$\displaystyle\cong\frac{C_{m}(a)}{\prod_{j=1}^{m}t_{2j-1}(a)}\boldsymbol{\beta}^{\ast}_{I^{+}+2m}\bar{\boldsymbol{\beta}}^{\ast}_{\bar{I}^{+}-2m}\bar{\boldsymbol{\gamma}}^{\ast}_{\bar{I}^{-}+2m}\boldsymbol{\gamma}^{\ast}_{I^{-}-2m}\boldsymbol{\beta}^{\ast}_{I_{\textrm{odd}}(m)}\bar{\boldsymbol{\gamma}}^{\ast}_{I_{\textrm{odd}}(m)}\Phi_{a}(0)$
where $I_{\textrm{odd}}(m)=\\{1,3,\ldots,2m-1\\}$ and we use the symbol
$\cong$ to denote identification in weak sense (that is, under expectation
value).
As was mentioned above, for the sine-Gordon model the fermionic basis can be
build in a mathematically rigorous fashion; the authors of Ref. [6] performed
this task by relying on the fact that sine-Gordon model allows for a lattice
regularization in the form of the eight-vertex model, which is well studied
and relatively easy to manage. Conversely, for its twin, the sinh-Gordon
model, the situation is not so simple: the lattice regularization, in this
case, takes the form of a much more complicated model, where the Boltzmann
weights are defined in terms of the R-matrix of the tensor product of two
infinite-dimensional representations of $U_{q}(\widehat{\mathfrak{sl}}_{2})$
[9, 10]. So far the status of the phase transition for this model has not been
clarified and, thus, relying on the lattice regularization is not a viable
strategy for sinh-Gordon model.
An alternative approach is to start directly from the Thermodynamic Bethe
Ansatz (TBA) equations which, for the sinh-Gordon model, exhibit a very simple
structure, given the fact that the spectrum of the theory consists of a single
particle. This fact let the authors of Ref. [8] straightforwardly define all
the functions involved in the formula (1.4). However that same formula, along
with the very existence of the fermionic basis, had to be introduced as a
conjecture based on two facts:
* •
From purely algebraic point of view, the ultra-violet (UV) limit of the sinh-
Gordon model corresponds to the CFT considered in Ref. [3]; this last theory
is, at the same time, the UV limit of the sine-Gordon model.
* •
There are two possible interpretations of the sin(h)-Gordon action, as a
perturbation of the free boson CFT222Here and in the following the notation
$\partial=\frac{\partial}{\partial z}$ and
$\bar{\partial}=\frac{\partial}{\partial\bar{z}}$ will be used.:
$\mathcal{A}=\int\left\\{\left[\frac{1}{4\pi}\partial\eta(z,\bar{z})\bar{\partial}\eta(z,\bar{z})\right]+\frac{2\boldsymbol{\mu}^{2}}{\sin(\pi
b^{2})}\cosh[b\,\eta(z,\bar{z})]\right\\}\frac{dz\wedge d\bar{z}}{2}\;,$
or as a perturbation of the Liouville model, conventionally identified as the
minimal CFT with central charge $c=1+6Q^{2}$, where $Q=b+b^{-1}$:
$\mathcal{A}=\int\left\\{\left[\frac{1}{4\pi}\partial\eta(z,\bar{z})\bar{\partial}\eta(z,\bar{z})+\frac{\boldsymbol{\mu}^{2}}{\sin(\pi
b^{2})}e^{b\,\eta(z,\bar{z})}\right]+\frac{\boldsymbol{\mu}^{2}}{\sin(\pi
b^{2})}e^{-b\,\eta(z,\bar{z})}\right\\}\frac{dz\wedge d\bar{z}}{2}\;.$
This twofold interpretation of the action led the authors of Ref. [11] to some
functional relations for the one-point functions of sine-Gordon model, which
were named _reflection relations_. In Ref. [12] it was shown how the fermionic
basis can be interpreted as a basis of the space of states for which these
reflection relations are trivially satisfied.
A remark about the choice for the normalization of the dimensional constant is
necessary. As discussed in Ref. [8], this choice, aside from being extremely
convenient for the calculations, encloses serious physical reasons. Firstly it
takes automatically into account the change of sign in the potential energy
when passing from sinh- to sine-Gordon and encodes also the pole at $b=i$ of
this last333Due to the fact that the perturbing operator becomes irrelevant
for $b^{2}<-1$. Note that there are poles also for $b\in\mathbb{Z}$ which look
natural once one consider the physical scale of the model, namely the mass of
the particle [13].; more importantly, this normalisation allows the mass m of
both the sinh-Gordon particle and that of the sine-Gordon lowest breather to
be expressed by a universal formula:
(1.8)
$\boldsymbol{\mu}\Gamma(1+b^{2})=\left[\frac{m}{4\sqrt{\pi}}\Gamma\left(\frac{1}{2(1+b^{2})}\right)\Gamma\left(1+\frac{b^{2}}{2(1+b^{2})}\right)\right]^{1+b^{2}}\;.$
Since, for the sinh-Gordon model, the formula (1.4) and the existence of the
fermionic basis still retain the status of conjectures, it is of utmost
importance to obtain a posteriori confirmations of their validity, by checking
the predictions against known results. Analytic comparison with results of
Refs. [14] and [15] were already performed in Ref. [8].
The purpose of this paper is to obtain further confirmations of the validity
of (1.4), by means of numerical simulations. In particular the one-point
functions of the sinh-Gordon model, defined on a cylinder of radius $2\pi R$,
were numerically evaluated for very small values of the radius $R\sim 0$,
limit in which the model approaches its UV limit; these numerical results were
then compared against the theoretical behaviours obtained in Refs. [4] and
[15]. As in the last-cited article a rescaling of the model to a circumference
of fixed radius $2\pi$ is to be performed; this amounts to a renormalisation
of the physical mass $m\rightarrow mR$, so that $\boldsymbol{\mu}\propto
R^{1+b^{2}}$.
It has to be noted that, since the goal is to compare the results obtained
from (1.4) with the known “CFT behaviour”, it is wise to avoid the possible
complications arising in the regions of the parameter space where $a-b<0$. Let
us clarify this point.
Looking at the formula for the conformal dimension shift of the fields (1.7)
we see that the ratio of expectation values of the two primary fields
$\Phi_{a-b}(0)$ and $\Phi_{a}(0)$ can be expressed in terms of the ratio of
the one-point function of the descendant
$\boldsymbol{\beta}_{1}^{\ast}\bar{\boldsymbol{\gamma}}_{1}^{\ast}\Phi_{a}(0)$
with that of the primary field $\Phi_{a}(0)$; in formulae
(1.9)
$\frac{\langle\Phi_{a-b}(0)\rangle}{\langle\Phi_{a}(0)\rangle}=\frac{C_{1}(a)}{t_{1}(a)}\frac{\langle\boldsymbol{\beta}_{1}^{\ast}\bar{\boldsymbol{\gamma}}_{1}^{\ast}\Phi_{a}(0)\rangle}{\langle\Phi_{a}(0)\rangle}\;.$
As one approaches the UV limit $R\sim 0$, the one-point functions of primary
fields are believed to behave as three-point functions of the Liouville CFT
[15] with two additional fields, of dimensions
$\Delta_{\pm}=\frac{Q^{2}}{4}-P(R)^{2}$ where $P(R)$ is the quantized momentum
of Liouville CFT, placed at $\pm\infty$. This, however, holds true only if the
dimensions of the fields are positive, which means
(1.10) $\left\\{\begin{array}[]{l}0<a<Q\\\
0<a-b<Q\end{array}\right.\quad\Rightarrow\quad b<a<Q\;.$
This fact becomes evident, for example, sending $a\rightarrow 0$; in this
case, the expectation value of the field $\Phi_{-b}(0)=e^{-b\eta(0)}$ can be
calculated directly in terms of the ground-state energy
$E(R)\underset{R\rightarrow 0}{\sim}-\frac{\pi}{6R}c_{\textrm{eff}}(R)$ where
[15]
(1.11) $c_{\textrm{eff}}(R)\underset{R\rightarrow
0}{\sim}1-\frac{24\pi}{\Big{(}\delta_{1}-4Q\log\frac{R}{2\pi}\Big{)}^{2}}\;,$
and $\delta_{1}$ is a constant. With (1.4), (1.9) and the relation between the
function $\Theta(i,-i|0)$ and $E(R)$ shown in Ref. [6], one obtains:
(1.12) $\langle e^{-b\eta(0)}\rangle=-\frac{C_{1}(0)}{\pi
m^{2}t_{1}(0)}\left(\frac{1}{R}+\frac{d}{dR}\right)E(R)\underset{R\rightarrow
0}{\sim}\frac{\pi c_{1}(0,b)}{2m^{2}Q^{2}}\frac{R^{-2(b^{2}+1)}}{(-\log
R)^{3}}\;,$
where the definition (LABEL:eq:C-constants) was applied, setting
$C_{1}(a)/t_{1}(a)\underset{R\rightarrow 0}{\sim}c_{1}(a,b)R^{2b(2a-b)}$,
$c_{1}(a,b)$ being a function of $a$ and $b$ only. On the other hand, using
the formula for the Liouville three-point amplitude (4.63) found in Refs. [16]
and [17], the result is radically different
(1.13) $\displaystyle\langle
e^{-b\eta(0)}\rangle=\frac{\langle\Phi_{\frac{Q}{2}-P}(-\infty)|\Phi_{-b}(0)|\Phi_{\frac{Q}{2}+P}(\infty)\rangle}{\langle\Phi_{\frac{Q}{2}-P}(-\infty)|\Phi_{\frac{Q}{2}+P}(\infty)\rangle}\underset{R\rightarrow
0}{\sim}k(a,b)R^{2(1+b^{2})}\;.$
It is clear that outside the natural region $b<a<Q$, the sinh-Gordon model do
no more approaches naïvely the Liouville CFT: there are contributions not
taken in account which become important. However, as said above, rather than
exploring the UV limit of the sinh-Gordon model per se, the goal of this paper
is to use it in order to obtain evidence of the agreement between the
predictions obtained from the fermionic basis and the results known in the
literature: for this reason from now on the parameter space will be restricted
to the region $0<b<a<Q$.
## 2\. The fermionic basis
Let us review briefly the properties of the fermionic basis.
The two-fold interpretation of the sinh-Gordon action that we mentioned above
has an interesting and important consequence. If we look at sinh-Gordon as a
deformation of the free boson CFT, then the natural choice for the descendants
of the primary field $\Phi_{a}(0)=e^{a\eta(0)}$ are normal ordered products of
$e^{a\eta(0)}$ with polynomials of even degree444We limit ourselves to even
degree polynomials, since $\mathcal{V}_{a}^{\textrm{quo}}$ non-trivial
subspaces are of even dimension only. in the derivatives of $\eta(0)$. In this
_Heisenberg basis_ the one-point functions inherit the natural free boson
symmetry
(2.14) $\sigma_{1}\;:a\rightarrow-a\;.$
On the other hand, if we consider the sinh-Gordon model as a deformation of
the Liouville CFT, the descendants of $\Phi_{a}(0)$ are more naturally defined
as normal-ordered products of $e^{a\eta(0)}$ with polynomials in even-degree
derivatives of $T(z,\overline{z})$ and $\bar{T}(z,\overline{z})$, where
$\displaystyle T(z,\overline{z})\doteq
T_{z,z}(z,\overline{z})=-\frac{1}{4}\Big{[}\partial\eta(z,\overline{z})\Big{]}^{2}+\frac{Q}{2}\partial^{2}\eta(z,\overline{z})\>,$
$\displaystyle\bar{T}(z,\overline{z})\doteq
T_{\overline{z},\overline{z}}(z,\overline{z})=-\frac{1}{4}\Big{[}\bar{\partial}\eta(z,\overline{z})\Big{]}^{2}+\frac{Q}{2}\bar{\partial}^{2}\eta(z,\overline{z})\>,$
are the components of Liouville energy-momentum tensor. It is natural to
assume that in this basis, that we call the _Virasoro basis_ , the one-point
functions retain the symmetry of the Liouville model
(2.16) $\sigma_{2}\;:a\rightarrow Q-a\;.$
Since both the Heisenberg and the Virasoro basis are, when the action of the
integrals of motion has been factored out, fully fledged bases of sinh-Gordon
space of states, the one-point functions in any possible basis have to
transform in some definite way under the symmetries $\sigma_{1}$ and
$\sigma_{2}$. This fact gives rise to the above-mentioned reflection relations
and suggests that there must exist a basis in which both these symmetries act
in a simple, multiplicative way: this particular basis is the _fermionic
basis_ ; we consider then the fermions as defined by their behaviour under the
symmetries $\sigma_{1}$ and $\sigma_{2}$. Starting from the Liouville CFT,
where the fermions $\boldsymbol{\beta}^{\textrm{CFT}\,\ast}$ and
$\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}$ can be defined as an intrinsic
property of the model [8], we see that the reflections act on the fermionic
basis as follows:
$\displaystyle\phantom{\sigma_{1}\;:\;}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow
u(a)\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\phantom{\qquad,\qquad\sigma_{2}\;:\qquad\;}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}$
(2.17)
$\displaystyle\sigma_{1}\;:\;\phantom{\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow
u^{-1}(-a)\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}}\qquad,\qquad\sigma_{2}\;:\;$
$\displaystyle\phantom{\sigma_{1}\;:\;}\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow
u^{-1}(-a)\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\phantom{\qquad,\qquad\sigma_{2}\;:\;\;}\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}$
where
(2.18)
$u(a)\doteq\frac{-2a+b(2m-1)}{2a+b^{-1}(2m-1)}=\frac{-Q\alpha+b(2m-1)}{Q\alpha+b^{-1}(2m-1)}$
and for the second chirality we only have to change $a$ in $-a$ in the above
function. There is an additional symmetry which was considered in Ref. [8],
that is the duality $b\rightarrow b^{-1}$, under which our fermions simply
exchange
$\displaystyle\phantom{\textrm{duality}\quad\;}\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}$
(2.19) $\displaystyle\textrm{duality}\;:$
$\displaystyle\phantom{\textrm{duality}\quad\;}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\rightarrow\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}$
The normalization of the fermions is such that when expressing the descendants
in fermionic basis in terms of Virasoro descendants we have
(2.20)
$\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{I^{+}}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{I^{-}}\Phi_{a}=C_{I^{+},I^{-}}\left\\{\mathbf{l}^{n}_{-2}+\cdots\right\\}\Phi_{a}\;,\qquad\mathfrak{C}(I^{+})=\mathfrak{C}(I^{-})=n\;,$
with $\mathbf{l}_{n}$ being the coefficients of the Laurent expansion of the
Liouville energy-momentum tensor component $T(z,\overline{z})$ while
$C_{I^{+},I^{-}}$ is the determinant of the Cauchy matrix
$\\{1/(i^{+}_{j}+i^{-}_{k}-1)\\}_{j,k=1}^{n}$.
The fermions for the sinh-Gordon model are obtained from the CFT ones simply
by multiplication by a constant:
$\displaystyle\boldsymbol{\beta}^{\ast}_{2m-1}=D_{2m-1}(a)\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2m-1}\qquad,\qquad\boldsymbol{\gamma}^{\ast}_{2m-1}=D_{2m-1}(Q-a)\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2m-1}\;,$
$\displaystyle\overline{\boldsymbol{\gamma}}^{\ast}_{2m-1}=D_{2m-1}(a)\overline{\boldsymbol{\gamma}}^{\textrm{CFT}\,\ast}_{2m-1}\qquad,\qquad\overline{\boldsymbol{\beta}}^{\ast}_{2m-1}=D_{2m-1}(Q-a)\overline{\boldsymbol{\beta}}^{\textrm{CFT}\,\ast}_{2m-1}\;,$
where
(2.22) $D_{2m-1}(a)=\frac{1}{2\pi
i}\left(\frac{\boldsymbol{\mu}\Gamma(1+b^{2})}{b^{1+b^{2}}}\right)^{-\frac{2m-1}{1+b^{2}}}\frac{\Gamma\left(\frac{a}{Q}+\frac{2m-1}{2bQ}\right)\Gamma\left(\frac{Q-a}{Q}+b\frac{2m-1}{2Q}\right)}{(m-1)!}\;.$
Note that this definition for the constants $D_{2m-1}$ differs from the one
used in Refs. [4]-[6] by the factor
(2.23)
$(-1)^{m}\sqrt{\frac{1+b^{2}}{i}}\frac{\boldsymbol{\mu}^{-\frac{2m-1}{1+b^{2}}}}{2\sin\left[\pi\left(\frac{a}{Q}-b\frac{2m-1}{2Q}\right)\right]}\;.$
The reason for this choice is twofold: on one side, the presence of
$\boldsymbol{\mu}^{-\frac{2m-1}{1+b^{2}}}$ makes the fermions dimensionless
while, on the other, the Q-periodic
$\sin\left[\pi\left(\frac{a}{Q}-b\frac{2m-1}{2Q}\right)\right]$ lets the non-
CFT fermions inherit the duality (2.19). Of course this last holds iff the
following term is “self-dual”
(2.24) $\frac{[\boldsymbol{\mu}\Gamma(1+b^{2})]^{\frac{1}{1+b^{2}}}}{b}\;,$
but this follows automatically when expressing $\boldsymbol{\mu}$ in terms of
the sinh-Gordon particle mass, which is explicitly self-dual:
(2.25)
$\boldsymbol{\mu}\Gamma(1+b^{2})=\left[\frac{m}{4\sqrt{\pi}}\Gamma\left(\frac{1}{2(1+b^{2})}\right)\Gamma\left(1+\frac{b^{2}}{2(1+b^{2})}\right)\right]^{1+b^{2}}\;.$
The constants $t_{\ell}(a)$ and $C_{m}(a)$ introduced in (1.7) are defined as
follows
(2.26)
$t_{\ell}(a)\doteq-\frac{1}{2}\sin^{-1}\left[\frac{\pi}{Q}\left(2a+\frac{\ell}{b}\right)\right]$
$\displaystyle C_{m}(a)\doteq\prod_{j=0}^{m-1}C_{1}(a-2bj)\;,$ $\displaystyle
C_{1}(a)\doteq[\boldsymbol{\mu}\Gamma(1+b^{2})]^{4x}\frac{\gamma(x)\gamma\left(\frac{1}{2}-x\right)}{2bQ\gamma(2bxQ)}\;,$
where $2Qx=2a-b$ and we denote $\gamma(y)\doteq\Gamma(y)/\Gamma(1-y)\;,\
\forall y\in\mathbb{C}$.
## 3\. TBA and one-point functions
As said above, since the TBA equations for the sinh-Gordon model are extremely
simple, it is quite straightforward to chose them as a starting point and
proceed to the construction of the function $\Theta(l,m|\alpha)$ relying on
the consistency equations which derive from the symmetries of the fermions.
Let us consider the sinh-Gordon model defined on an infinite cylinder of
circumference $2\pi R$; we call the infinite direction the _space direction_
and the compact one the _Matsubara direction_. The TBA for this model consists
of a single integral equation:
(3.28) $\epsilon(\theta)=2\pi
mR\cosh\theta-\int\limits_{-\infty}^{\infty}\Phi(\theta-\theta^{\prime})\log\left(1+e^{-\epsilon(\theta^{\prime})}\right)d\theta^{\prime}\;,$
with
$\displaystyle\Phi(\theta)$ $\displaystyle=\frac{1}{2\pi\cosh\left(\theta+\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)}+\frac{1}{2\pi\cosh\left(\theta-\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)}=\int\limits_{-\infty}^{\infty}e^{ik\theta}\widehat{\Phi}(k)\frac{dk}{2\pi}\;,$
$\displaystyle\widehat{\Phi}(k)$
$\displaystyle=\frac{\cosh\left(\pi\frac{b^{2}-1}{2(b^{2}+1)}k\right)}{\cosh\left(\pi\frac{k}{2}\right)}\;.$
Starting from this basic equation, one can build Baxter $Q$-functions in the
Matsubara direction; namely define
(3.30) $\log Q(\theta)=-\frac{\pi
mR\cosh\theta}{\sin\frac{\pi}{b^{2}+1}}+\int\limits_{-\infty}^{\infty}\frac{\log\left(1+e^{-\epsilon(\theta^{\prime})}\right)}{\cosh(\theta-\theta^{\prime})}\frac{d\theta^{\prime}}{2\pi}\;,$
where we have chosen the first term on the right-hand side for consistency. It
is straightforward to check that
(3.31) $e^{-\epsilon(\theta)}=Q\left(\theta+\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)Q\left(\theta-\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)\;,$
from which, recalling the Dirac delta representation
$\cosh(\theta+i\frac{\pi}{2})+\cosh(\theta-i\frac{\pi}{2})=2\pi\delta(\theta)$,
one can derive the bilinear equation555Actually, one should be careful and
define correctly the analyticity conditions for the function $Q(\theta)$; a
discussion can be found in Ref. [18].
(3.32) $Q\left(\theta+\frac{\pi i}{2}\right)Q\left(\theta-\frac{\pi
i}{2}\right)-Q\left(\theta+\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)Q\left(\theta-\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)=1\;.$
Introducing $\zeta=e^{(b^{2}+1)\theta}$, it’s not difficult to see how (3.32)
implies that the function $T(\zeta)$, defined from the equation
(3.33) $T(\zeta)Q(\theta)=Q\left(\theta+\pi
i\frac{b^{2}}{b^{2}+1}\right)+Q\left(\theta-\pi
i\frac{b^{2}}{b^{2}+1}\right)\;,$
is a single-valued function of $\zeta^{2}$, with essential singularities at
$\zeta=0$ and $\zeta=\infty$. This equation is a second order finite
difference equation for the function $Q(\theta)$ and thus admits two different
solutions: $Q(\theta)$ and $Q(\theta+i\frac{\pi}{b^{2}+1})$, the equation
(3.32) being their quantum Wronskian.
It’s important to stress that the equations for the functions $Q(\theta)$ and
$T(\theta)$ given here are to be considered as _definitions_ , so one should
check that they are reasonable. A verification of the correctness of these
definition was carried out in Ref. [15], where the behaviour of $T(\zeta)$ in
the ultraviolet region $R\rightarrow 0$ is investigated numerically, showing
how the asymptotics of $T(\zeta)$ for $\zeta\rightarrow 0$ and for
$\zeta\rightarrow\infty$ correcly reproduce the eigenvalues of CFT integrals
of motion and, moreover, that their normalisation is the same as in the sine-
Gordon case [19, 20, 21]; this is an extremely convincing argument.
Now, having the TBA equation (3.28) at our disposal, we introduce a deformed
kernel $\Phi_{\alpha}(\theta)$ requiring that its Fourier image
$\widehat{\Phi}(k,\alpha)$ satisfy $\widehat{\Phi}(k,0)=\widehat{\Phi}(k)$ and
the following symmetries
$\displaystyle\widehat{\Phi}(k,\alpha+$ $\displaystyle
2)=\widehat{\Phi}(k,\alpha)\;,\qquad\widehat{\Phi}(k,-\alpha)=\widehat{\Phi}(-k,\alpha)\;,$
$\displaystyle\widehat{\Phi}(k,\alpha-2\frac{b^{2}}{b^{2}+1})=\widehat{\Phi}(k+2i,\alpha)\;.$
The first two relations directly derive from the request that the fermions
transform in the correct way under the transformations $\sigma_{1}$ and
$\sigma_{2}$; the third one, on the other hand, is necessary in order to grant
the validity of the shift relation (1.7), as was shown in Ref. [8].
It’s not hard to find that the kernel we’re looking for has the following
form:
$\displaystyle\Phi_{\alpha}(\theta)=\frac{e^{i\pi\alpha}}{2\pi\cosh\left(\theta+\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)}+\frac{e^{-i\pi\alpha}}{2\pi\cosh\left(\theta-\pi
i\frac{b^{2}-1}{2(b^{2}+1)}\right)}=\int\limits_{-\infty}^{\infty}e^{ik\theta}\widehat{\Phi}(k,\alpha)\frac{dk}{2\pi}\;,$
$\displaystyle\widehat{\Phi}(k,\alpha)=\frac{\cosh\left(\pi\frac{b^{2}-1}{2(b^{2}+1)}k-i\pi\alpha\right)}{\cosh\left(\pi\frac{k}{2}\right)}\;.$
It is interesting to notice that, contrary to the function
$\widehat{R}(k,\alpha)$ of the sine-Gordon model [6], the deformed kernel
$\widehat{\Phi}$ does not have poles in the $k$-plane whose positions depend
on $\alpha$. This simplification in the kernel structure is directly
correlated to the fact that sinh-Gordon one-point functions, as functions of
$\alpha$, have much simpler analytical properties than those of sine-Gordon.
Let us proceed by defining the dressed resolvent, which satisfies to the
equation
(3.36)
$R_{\textrm{dress}}(\theta,\theta^{\prime}|\alpha)-\left[\Phi\ast\,R_{\textrm{dress}}\right](\theta,\theta^{\prime}|\alpha)=\Phi(\theta,\theta^{\prime}|\alpha)\;,$
where
$\Phi(\theta,\theta^{\prime}|\alpha)\equiv\Phi_{\alpha}(\theta-\theta^{\prime})$
and the $\ast$ denotes a deformed convolution
(3.37)
$[f\ast\,g](\theta,\theta^{\prime})\doteq\int\limits_{-\infty}^{\infty}f(\theta,\phi)g(\phi,\theta^{\prime})dm(\phi)\;,\qquad
dm(\phi)\doteq\frac{d\phi}{1+e^{\epsilon(\phi)}}\;.$
Now, using the dressed resolvent, we build the function
$\Theta^{\textrm{shG}}_{R}$:
(3.38)
$R_{\textrm{dress}}(\theta,\theta^{\prime}|\alpha)-\Phi_{\alpha}(\theta-\theta^{\prime})=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{dl}{2\pi}\frac{dm}{2\pi}\widehat{\Phi}(l,\alpha)\Theta^{\textrm{shG}}_{R}(l,m|\alpha)\widehat{\Phi}(m,-\alpha)e^{il\theta+im\theta^{\prime}}\;.$
Straightforward calculations show that the function
$\Theta^{\textrm{shG}}_{R}$ satisfies the following equation
(3.39)
$\Theta^{\textrm{shG}}_{R}(l,m|\alpha)-G(l+m)-\int\limits_{-\infty}^{\infty}G(l-k)\widehat{\Phi}(k,\alpha)\Theta^{\textrm{shG}}_{R}(k,m|\alpha)\frac{dk}{2\pi}=0\;,$
with the function $G(k)$ being the $k$-moment of the measure $dm(\theta)$
(3.40)
$G(k)\doteq\int\limits_{-\infty}^{\infty}e^{-ik\theta}\frac{d\theta}{1+e^{\epsilon(\theta)}}\;.$
A useful way to express the function $\Theta^{\textrm{shG}}_{R}$ is the
following
(3.41)
$\Theta^{\textrm{shG}}_{R}(il,im|\alpha)=e_{l}\ast\,e_{m}+e_{l}\ast\,R_{\textrm{dress}}^{(\alpha)}\ast\,e_{m}\;,$
where we have introduced the shorthand notation $e_{l}(\theta)\doteq
e^{l\,\theta}$.
Since, for the ground state, the function $\epsilon(\theta)$ is even, from the
symmetries of $\widehat{\Phi}(k,\alpha)$ one easily derives the following
relations:
(3.42)
$\Theta^{\textrm{shG}}_{R}(l,m|-\alpha)=\Theta^{\textrm{shG}}_{R}(m,l|\alpha)\;,\qquad\Theta^{\textrm{shG}}_{R}(l,m|\alpha+2)=\Theta^{\textrm{shG}}_{R}(l,m|\alpha)\;,$
$\displaystyle\Theta^{\textrm{shG}}_{R}(l,m|\alpha-2\frac{b^{2}}{b^{2}+1})$
$\displaystyle-\Theta^{\textrm{shG}}_{R}(l+2i,m-2i|\alpha)=$ (3.43)
$\displaystyle=\frac{\Theta^{\textrm{shG}}_{R}(l+2i,-i|\alpha)\Theta^{\textrm{shG}}_{R}(i,m-2i|\alpha)}{\pi
t_{1}(\frac{Q}{2}\alpha)-\Theta^{\textrm{shG}}_{R}(i,-i|\alpha)}\;.$
As has been said in the introduction, the function $\Theta_{R}^{\textrm{shG}}$
can be used in order to calculate the expectation values of descendants in the
fermionic basis:
###### Conjecture 1.
We conjecture that, as in the sine-Gordon model, the one-point functions of
the sinh-Gordon model in the fermionic basis are expressed in terms of a
determinant
(3.44)
$\frac{\langle\boldsymbol{\beta}^{\ast}_{I^{+}}\overline{\boldsymbol{\beta}}^{\ast}_{\overline{I}^{+}}\overline{\boldsymbol{\gamma}}^{\ast}_{\overline{I}^{-}}\boldsymbol{\gamma}^{\ast}_{I^{-}}\Phi_{\alpha}(0)\rangle_{R}}{\langle\Phi_{\alpha}(0)\rangle_{R}}=\mathcal{D}\left(I^{+}\cup(-\overline{I}^{+})|I^{-}\cup(-\overline{I}^{-})|\alpha\right)\;,$
where, for two sets $A=\\{a_{j}\\}_{j=1}^{n}$ and $B=\\{b_{j}\\}_{j=1}^{n}$,
we have
(3.45)
$\mathcal{D}(A|B|\alpha)\doteq\left(\prod_{\ell=1}^{n}\frac{\textrm{sgn}(a_{\ell})\textrm{sgn}(b_{\ell})}{\pi}\right)\det\left[\Theta^{\textrm{shG}}_{R}(ia_{j},ib_{k}|\alpha)-\pi\delta_{a_{j},-b_{k}}\textrm{sgn}(a_{j})t_{a_{j}}(\alpha)\right]_{j,k=1}^{n}$
Notice how, since
$\Theta^{\textrm{shG}}_{R}(l,m|\alpha)\underset{R\rightarrow\infty}{\rightarrow}0$,
in the infinite volume limit $R\rightarrow\infty$ the formulae for the one-
point functions in sinh-Gordon coincide with the analytic continuation with
respect to $b$ of the corresponding ones in sine-Gordon model [6].
## 4\. Numerical analysis in the $R\rightarrow 0$ limit
We now turn to the numerical evaluation of the one-point functions of the
sinh-Gordon model in the UV limit $R\rightarrow 0$. We will begin by studying
the behaviour of the descendant fields and then move to the primary ones. As
mentioned in the introduction, we rescale the model on a cylinder of radius
$2\pi$ and take $r=2\pi mR$ as the parameter to be sent to zero.
### 4.1. Descendant fields
We are interested in the UV behaviour of the following class of one-point
functions
(4.46)
$F_{2j-1,2k-1}(\alpha,r)\doteq\frac{\langle\boldsymbol{\beta}^{\ast}_{2j-1}\boldsymbol{\gamma}^{\ast}_{2k-1}\Phi_{\alpha}\rangle_{r}}{\langle\Phi_{\alpha}\rangle_{r}}\;,\qquad
j,k\in\mathbb{N}\;,$
which can be rewritten using (LABEL:eq:cfttosinhfermions) as
$F_{2j-1,2k-1}(\alpha,r)=D_{2j-1}(\alpha)D_{2k-1}(2-\alpha)\frac{\langle\boldsymbol{\beta}^{\textrm{CFT}\,\ast}_{2j-1}\boldsymbol{\gamma}^{\textrm{CFT}\,\ast}_{2k-1}\Phi_{\alpha}(0)\rangle_{r}}{\langle\Phi_{\alpha}(0)\rangle_{r}}\;,$
In the $r\rightarrow 0$ limit, these functions should behave like ratios of
CFT one-point functions. In particular, using the formulae found in the
appendix of Ref. [6], we see that
(4.47) $F_{2j-1,2k-1}\underset{r\rightarrow 0}{\sim}-\left(\frac{2\pi
m}{r}\right)^{2j+2k-2}\frac{D_{2j-1}(\alpha)D_{2k-1}(2-\alpha)}{j+k-1}\Omega_{2j-1,2k-1}\;,$
where $\Omega_{2j-1,2k-1}$ are functions of the vacuum eigenvalues $I_{2n-1}$
of the integrals of motion, which can be found, for example, in Ref. [20]. For
the cases we are interested in we have
$\displaystyle\Omega_{1,1}(\alpha,r)=I_{1}(r)-\frac{\Delta_{\alpha}}{12}\;,$
$\displaystyle\Omega_{1,3}(\alpha,r)=I_{3}(r)-\frac{\Delta_{\alpha}}{6}I_{1}(r)+\frac{\Delta_{\alpha}^{2}}{144}+\frac{c+5}{1080}\Delta_{\alpha}-\frac{\Delta_{\alpha}}{360}d_{\alpha}\;,$
where
(4.49) $\Delta_{\alpha}=\frac{Q^{2}}{4}\alpha(2-\alpha)\;,\qquad
d_{\alpha}=\frac{1}{6}\sqrt{(25-c)(24\Delta_{\alpha}+1-c)}$
The vacuum eigenvalues of the integrals of motion do not depend directly on
the radius $r$, but rather on the momentum $P(r)$, which is itself a function
of $r$:
(4.50) $I_{1}(r)=P(r)^{2}-\frac{1}{24}\;,\qquad
I_{3}(r)=I_{1}(r)^{2}+\frac{1}{6}I_{1}(r)+\frac{c}{1440}\;.$
As explained neatly in Ref. [17], in the limit $r\rightarrow 0$ the main
contribution to the one-point functions $\langle e^{a\eta}\rangle$, with
$a>0$, comes from the following region in the configuration space
(4.51) $|b\eta_{0}|<-\log\frac{\mu^{2}}{\sin\pi b^{2}}\;,$
where $\eta_{0}$ is the zero mode of the field $\eta(z,\overline{z})$; here
the interaction term in sinh-Gordon action can be neglected. This means that
in this region we can consider $\eta$ as a free field and that the ground
state wave functional $\boldsymbol{\Psi}_{0}[\eta]$ can be approximated by the
superposition of two zero-mode plane waves
(4.52)
$\boldsymbol{\Psi}_{0}[\eta]\underset{r\rightarrow\infty}{\sim}\left(c_{1}e^{iP(r)\eta_{0}}+c_{2}e^{-iP(r)\eta_{0}}\right)\;,$
where the momentum $P(r)$ is quantised thanks to the presence of the potential
walls $b\eta_{0}\sim\pm\log\frac{\mu^{2}}{\sin\pi b^{2}}$. The quantisation
condition reads
(4.53) $S(P)^{2}=1\;\Rightarrow\;\delta(P)=\pi\;,\ S(P)\doteq
e^{-i\delta(P)}\;,$
where $S(P)$ is the Liouville reflection amplitude
(4.54)
$S(P)=-\left(\boldsymbol{\mu}\frac{\Gamma(1+b^{2})}{b^{2}}\right)^{-4i\frac{P(r)}{b}}\frac{\Gamma\big{(}1+2iP(r)b\big{)}\Gamma\big{(}1+2iP(r)b^{-1}\big{)}}{\Gamma\big{(}1-2iP(r)b\big{)}\Gamma\big{(}1-2iP(r)b^{-1}\big{)}}\;.$
Using (1.8) and remembering that we rescaled the mass $m\rightarrow mR$, we
easily obtain the quantisation condition for the momentum
$\displaystyle
2P(r)Q\log\left[\frac{r}{8\pi^{\frac{3}{2}}\big{(}b^{2}\big{)}^{\frac{1}{1+b^{2}}}}\Gamma\Big{(}\frac{1}{2(1+b^{2})}\Big{)}\Gamma\Big{(}1+\frac{b^{2}}{2(1+b^{2})}\Big{)}\right]=$
$\displaystyle=-\frac{\pi}{2}+\frac{1}{2i}\log\left[\frac{\Gamma\big{(}1+2iP(r)b\big{)}\Gamma\big{(}1+2iP(r)b^{-1}\big{)}}{\Gamma\big{(}1-2iP(r)b\big{)}\Gamma\big{(}1-2iP(r)b^{-1}\big{)}}\right]\;.$
We have considered the two following ratios of expectation values
(4.56)
$F_{1,1}(\alpha,r)\doteq\frac{\langle\boldsymbol{\beta}^{\ast}_{1}\boldsymbol{\gamma}^{\ast}_{1}\Phi_{\alpha}(0)\rangle_{r}}{\langle\Phi_{\alpha}(0)\rangle_{r}}\;,\qquad
F_{1,3}(\alpha,r)\doteq\frac{\langle\boldsymbol{\beta}^{\ast}_{1}\boldsymbol{\gamma}^{\ast}_{3}\Phi_{\alpha}(0)\rangle_{r}}{\langle\Phi_{\alpha}(0)\rangle_{r}}\;,$
and evaluated numerically the corresponding functions
$\Theta_{r}^{\textrm{shG}}(i,i|\alpha)$ and
$\Theta_{r}^{\textrm{shG}}(i,3i|\alpha)$ for values of $\alpha$ ranging from
$0.75$ up to $1.5$, with $b\in[0.4,1.0]$ and $r\in[0.005,0.95]$. Figures 1-8
show some of these numerical estimates plotted against the curve (4.47); the
agreement of the data with the theoretical prevision is very good for the
whole range of $r$ considered.
Figure 1. Plot of $F_{1,1}(\alpha,r)$ against its theoretical behaviour for
$\alpha=0.75$ and $b=0.4$
Figure 2. Plot of $F_{1,1}(\alpha,r)$ against its theoretical behaviour for
$\alpha=0.75$ and $b=0.8$
Figure 3. Plot of $F_{1,1}(\alpha,r)$ against its theoretical behaviour for
$\alpha=1.1$ and $b=0.4$
Figure 4. Plot of $F_{1,1}(\alpha,r)$ against its theoretical behaviour for
$\alpha=1.1$ and $b=0.8$
Figure 5. Plot of $F_{1,3}(\alpha,r)$ against its theoretical behaviour for
$\alpha=0.75$ and $b=0.4$
Figure 6. Plot of $F_{1,3}(\alpha,r)$ against its theoretical behaviour for
$\alpha=0.75$ and $b=0.8$
Figure 7. Plot of $F_{1,3}(\alpha,r)$ against its theoretical behaviour for
$\alpha=1$ and $b=0.4$
Figure 8. Plot of $F_{1,3}(\alpha,r)$ against its theoretical behaviour for
$\alpha=1$ and $b=0.8$
The tables 1 and Tab.2, displaying the values of the relative error $\sigma$
(4.57)
$\sigma_{2j-1,2k-1}\doteq\left|1-\frac{F_{2j-1,2k-1}(\alpha,r)}{F_{2j-1,2k-1}^{\textrm{CFT}}(\alpha,r)}\right|$
with
(4.58) $F_{2j-1,2k-1}^{\textrm{CFT}}(\alpha,r)=-\left(\frac{2\pi
m}{r}\right)^{2j+2k-2}\frac{D_{2j-1}(\alpha)D_{2k-1}(2-\alpha)}{j+k-1}\Omega_{2j-1,2k-1}\;,$
are a remarkable evidence in support of the conjecture 3.45.
Table 1. Values of the relative error for $F_{1,1}(\alpha,r)$. | $\sigma_{1,1}$
---|---
| $\alpha=0.75$ | | $\alpha=1.1$
$r$ | $b=0.4$ | $b=0.8$ | | $b=0.4$ | $b=0.8$
0.005 | $1.5\times 10^{-4}$ | $2.0\times 10^{-5}$ | | $2.4\times 10^{-4}$ | $6.0\times 10^{-5}$
0.01 | $5.5\times 10^{-5}$ | $3.3\times 10^{-6}$ | | $1.1\times 10^{-4}$ | $1.5\times 10^{-5}$
0.015 | $2.3\times 10^{-5}$ | $1.2\times 10^{-6}$ | | $6.1\times 10^{-5}$ | $8.1\times 10^{-6}$
0.02 | $1.3\times 10^{-5}$ | $1.8\times 10^{-6}$ | | $3.7\times 10^{-5}$ | $4.6\times 10^{-6}$
0.025 | $7.5\times 10^{-6}$ | $3.0\times 10^{-7}$ | | $2.2\times 10^{-5}$ | $2.3\times 10^{-6}$
0.03 | $5.1\times 10^{-6}$ | $7.3\times 10^{-7}$ | | $1.6\times 10^{-5}$ | $2.8\times 10^{-6}$
0.035 | $1.7\times 10^{-6}$ | $1.1\times 10^{-6}$ | | $1.1\times 10^{-5}$ | $1.1\times 10^{-6}$
0.04 | $1.4\times 10^{-6}$ | $1.1\times 10^{-6}$ | | $7.0\times 10^{-6}$ | $3.1\times 10^{-7}$
0.045 | $1.4\times 10^{-6}$ | $1.1\times 10^{-6}$ | | $6.7\times 10^{-6}$ | $2.2\times 10^{-6}$
0.05 | $1.3\times 10^{-6}$ | $1.3\times 10^{-6}$ | | $2.4\times 10^{-6}$ | $2.5\times 10^{-6}$
0.055 | $4.2\times 10^{-6}$ | $3.3\times 10^{-6}$ | | $7.1\times 10^{-6}$ | $8.0\times 10^{-7}$
0.06 | $1.1\times 10^{-6}$ | $2.5\times 10^{-6}$ | | $2.2\times 10^{-6}$ | $3.2\times 10^{-7}$
0.065 | $4.0\times 10^{-7}$ | $2.4\times 10^{-7}$ | | $2.4\times 10^{-6}$ | $4.4\times 10^{-7}$
0.07 | $2.7\times 10^{-7}$ | $2.9\times 10^{-7}$ | | $2.2\times 10^{-6}$ | $1.7\times 10^{-7}$
0.075 | $3.1\times 10^{-7}$ | $2.8\times 10^{-7}$ | | $1.3\times 10^{-6}$ | $1.2\times 10^{-6}$
0.08 | $1.3\times 10^{-7}$ | $1.0\times 10^{-7}$ | | $1.0\times 10^{-6}$ | $4.3\times 10^{-8}$
0.085 | $5.4\times 10^{-7}$ | $1.3\times 10^{-7}$ | | $3.1\times 10^{-8}$ | $5.4\times 10^{-7}$
0.09 | $2.8\times 10^{-8}$ | $2.2\times 10^{-7}$ | | $1.4\times 10^{-6}$ | $2.8\times 10^{-6}$
0.095 | $1.2\times 10^{-6}$ | $2.3\times 10^{-7}$ | | $2.6\times 10^{-6}$ | $6.8\times 10^{-11}$
Table 2. Values of the relative error for $F_{1,3}(\alpha,r)$. | $\sigma_{1,3}$
---|---
| $\alpha=0.75$ | | $\alpha=1$
$r$ | $b=0.4$ | $b=0.8$ | | $b=0.4$ | $b=0.8$
0.005 | $2.4\times 10^{-4}$ | $6.2\times 10^{-5}$ | | $1.4\times 10^{-4}$ | $1.8\times 10^{-5}$
0.01 | $1.0\times 10^{-4}$ | $1.6\times 10^{-5}$ | | $5.2\times 10^{-5}$ | $5.0\times 10^{-6}$
0.015 | $6.2\times 10^{-5}$ | $9.3\times 10^{-6}$ | | $2.5\times 10^{-5}$ | $3.1\times 10^{-6}$
0.02 | $3.7\times 10^{-5}$ | $4.1\times 10^{-6}$ | | $1.2\times 10^{-5}$ | $1.1\times 10^{-6}$
0.025 | $2.1\times 10^{-5}$ | $8.9\times 10^{-7}$ | | $7.0\times 10^{-6}$ | $5.0\times 10^{-7}$
0.03 | $1.8\times 10^{-5}$ | $8.1\times 10^{-7}$ | | $4.2\times 10^{-6}$ | $8.8\times 10^{-7}$
0.035 | $1.0\times 10^{-5}$ | $8.5\times 10^{-7}$ | | $1.7\times 10^{-6}$ | $9.9\times 10^{-7}$
0.04 | $7.4\times 10^{-6}$ | $1.4\times 10^{-6}$ | | $1.0\times 10^{-6}$ | $6.7\times 10^{-7}$
0.045 | $5.9\times 10^{-6}$ | $1.3\times 10^{-6}$ | | $4.1\times 10^{-7}$ | $2.1\times 10^{-6}$
0.05 | $3.3\times 10^{-6}$ | $2.2\times 10^{-6}$ | | $3.4\times 10^{-7}$ | $1.0\times 10^{-6}$
0.055 | $2.4\times 10^{-6}$ | $1.2\times 10^{-6}$ | | $1.3\times 10^{-6}$ | $2.2\times 10^{-6}$
0.06 | $2.2\times 10^{-6}$ | $4.8\times 10^{-7}$ | | $7.1\times 10^{-7}$ | $8.0\times 10^{-7}$
0.065 | $1.6\times 10^{-6}$ | $7.0\times 10^{-7}$ | | $3.9\times 10^{-7}$ | $4.2\times 10^{-8}$
0.07 | $1.3\times 10^{-8}$ | $4.9\times 10^{-7}$ | | $3.0\times 10^{-7}$ | $4.7\times 10^{-7}$
0.075 | $3.8\times 10^{-7}$ | $1.1\times 10^{-6}$ | | $7.6\times 10^{-9}$ | $5.8\times 10^{-7}$
0.08 | $8.0\times 10^{-7}$ | $1.2\times 10^{-6}$ | | $9.2\times 10^{-8}$ | $3.6\times 10^{-7}$
0.085 | $2.8\times 10^{-6}$ | $4.2\times 10^{-7}$ | | $4.5\times 10^{-7}$ | $6.1\times 10^{-7}$
0.09 | $1.6\times 10^{-6}$ | $2.8\times 10^{-6}$ | | $7.0\times 10^{-7}$ | $4.0\times 10^{-7}$
0.095 | $3.0\times 10^{-6}$ | $1.0\times 10^{-6}$ | | $7.1\times 10^{-7}$ | $6.1\times 10^{-7}$
### 4.2. Primary fields
Let us now consider the following ratio of primary fields’ expectation values
(4.59)
$\mathcal{F}(\alpha,r)\doteq\frac{\langle\Phi_{\alpha-2\frac{b^{2}}{b^{2}+1}}\rangle^{\textrm{shG}}_{r}}{\langle\Phi_{\alpha}\rangle^{\textrm{shG}}_{r}}\;.$
Using the shift formula (1.7) and the determinant one (3.44) we can write
(4.60)
$\mathcal{F}(\alpha,r)=\frac{C_{1}(\alpha)}{t_{1}(\alpha)}\frac{\langle\boldsymbol{\beta}^{\ast}_{1}\bar{\boldsymbol{\gamma}}^{\ast}_{1}\Phi_{\alpha}\rangle^{\textrm{shG}}_{r}}{\langle\Phi_{\alpha}\rangle^{\textrm{shG}}_{r}}=-\frac{C_{1}(\alpha)}{\pi
t_{1}(\alpha)}\Big{[}\Theta(i,-i|\alpha)-\pi t_{1}(\alpha)\Big{]}\;.$
On the other hand, from Ref. [15] we know that we can approximate the
behaviour of the expectation value of a primary field $\Phi_{\alpha}$ in the
region (4.51) with that of a three-point function of Liouville CFT:
(4.61) $\langle\Phi_{\alpha}\rangle^{\textrm{shG}}_{r}\underset{r\rightarrow
0}{\sim}\mathcal{N}(r,b)\langle
0|e^{a(-P)\eta(-\infty)}\Phi_{\alpha}e^{a(P)\eta(\infty)}|0\rangle_{r}^{\textrm{Liou}}$
where the function $\mathcal{N}(r,b)$ is a normalization constant and
(4.62)
$a(P)=\frac{Q}{2}+iP(r)\;\Rightarrow\qquad\Delta_{a(P)}=\frac{Q^{2}}{4}-P(r)^{2}$
with $P(r)$ satisfying the quantization condition (LABEL:eq:quantcond).
The form of Liouville three-point function was found in Refs. [16] and [17]
and reads
(4.63) $\langle
0|e^{a(-P)\eta(-\infty)}\Phi_{\alpha}e^{a(P)\eta(\infty)}|0\rangle_{r}^{\textrm{Liou}}=\left(\boldsymbol{\mu}\frac{\Gamma(1+b^{2})}{b^{1+b^{2}}}\right)^{-Q\frac{\alpha}{b}}\Upsilon_{0}\frac{\Upsilon(2a)\Upsilon(Q-2iP)\Upsilon(Q+2iP)}{\Upsilon(a)^{2}\Upsilon(a-2iP)\Upsilon(a+2iP)}\;,$
where the function $\Upsilon(x)$ is defined by the equations
$\frac{\Upsilon(x+b)}{\Upsilon(x)}=\gamma(b\,x)b^{1-2bx}\;,\quad\frac{\Upsilon(x+b^{-1})}{\Upsilon(x)}=\gamma\left(\frac{x}{b}\right)b^{-1+2\frac{x}{b}}\;,\quad\Upsilon_{0}\doteq\frac{d\Upsilon}{dx}\Big{|}_{x=0}\;.$
The general form of the normalization $\mathcal{N}(r,b)$ is not known, but
this is irrelevant to our needs, since we are considering the ratio of two
one-point functions.
With some simple calculations one finds
$\displaystyle\mathcal{F}(\alpha,r)$ $\displaystyle\underset{r\rightarrow
0}{\sim}\mathcal{F}^{\textrm{CFT}}(\alpha,r)=\left[\frac{r}{8\pi^{\frac{3}{2}}}\Gamma\left(\frac{1}{2(1+b^{2})}\right)\Gamma\left(1+\frac{b^{2}}{2(1+b^{2})}\right)\right]^{2}\times$
(4.64)
$\displaystyle\times\frac{\gamma\big{(}b(a-b)\big{)}^{2}}{\gamma\big{(}b(2a-b)\big{)}\gamma\big{(}2b(a-b)\big{)}}\gamma\big{(}b(a-b+2iP)\big{)}\gamma\big{(}b(a-b-2iP)\big{)}\;.$
We have evaluated numerically the function $\Theta(i,-i|\alpha)$ and used it
to extract the value of $\mathcal{F}(\alpha,r)$ by means of the formula
(4.60). We then compared the data we obtained with the theoretical CFT
behaviour (4.64). Figures 9-13 show the results for various values of $\alpha$
and $b$.
Figure 9. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=0.75$ and $b=0.4$
Figure 10. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=0.75$ and $b=0.7$
Figure 11. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=0.75$ and $b=0.8$
Figure 12. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=1.5$ and $b=0.4$
Figure 13. Plot of $\mathcal{F}(\alpha,r)$ against its theoretical behaviour
for $\alpha=1.5$ and $b=0.8$
In table 3 are collected the values of the relative error $\varsigma$
(4.65)
$\varsigma\doteq\left|1-\frac{\mathcal{F}(\alpha,r)}{\mathcal{F}^{\textrm{CFT}}(\alpha,r)}\right|\;.$
Table 3. Values of the relative error for $\mathcal{F}(\alpha,r)$. | $\varsigma$
---|---
| $\alpha=0.75$ | | $\alpha=1.5$
$r$ | $b=0.4$ | $b=0.7$ | $b=0.8$ | | $b=0.4$ | $b=0.8$
0.005 | $6.2\times 10^{-3}$ | $1.1\times 10^{-3}$ | $1.2$ | | $1.1\times 10^{-2}$ | $8.2\times 10^{-4}$
0.01 | $2.7\times 10^{-3}$ | $1.7\times 10^{-3}$ | $1.2$ | | $3.6\times 10^{-3}$ | $2.5\times 10^{-4}$
0.015 | $1.4\times 10^{-3}$ | $5.9\times 10^{-3}$ | $1.3$ | | $1.7\times 10^{-3}$ | $1.1\times 10^{-4}$
0.02 | $7.8\times 10^{-4}$ | $7.3\times 10^{-3}$ | $1.3$ | | $9.6\times 10^{-4}$ | $5.4\times 10^{-5}$
0.025 | $5.7\times 10^{-4}$ | $9.4\times 10^{-3}$ | $1.3$ | | $5.8\times 10^{-4}$ | $2.7\times 10^{-5}$
0.03 | $3.1\times 10^{-4}$ | $1.1\times 10^{-2}$ | $1.3$ | | $3.7\times 10^{-4}$ | $1.6\times 10^{-5}$
0.035 | $2.2\times 10^{-4}$ | $1.2\times 10^{-2}$ | $1.3$ | | $2.4\times 10^{-4}$ | $1.0\times 10^{-5}$
0.04 | $1.7\times 10^{-4}$ | $1.4\times 10^{-2}$ | $1.4$ | | $1.7\times 10^{-4}$ | $5.3\times 10^{-6}$
0.045 | $1.6\times 10^{-4}$ | $1.6\times 10^{-2}$ | $1.4$ | | $1.2\times 10^{-4}$ | $1.2\times 10^{-6}$
0.05 | $3.6\times 10^{-5}$ | $1.7\times 10^{-2}$ | $1.4$ | | $8.5\times 10^{-5}$ | $5.9\times 10^{-6}$
0.055 | $9.5\times 10^{-5}$ | $1.9\times 10^{-2}$ | $1.4$ | | $5.9\times 10^{-5}$ | $1.1\times 10^{-6}$
0.06 | $3.6\times 10^{-5}$ | $2.0\times 10^{-2}$ | $1.4$ | | $4.4\times 10^{-5}$ | $6.6\times 10^{-7}$
0.065 | $7.2\times 10^{-5}$ | $2.1\times 10^{-2}$ | $1.4$ | | $3.5\times 10^{-5}$ | $5.6\times 10^{-7}$
0.07 | $5.5\times 10^{-5}$ | $2.3\times 10^{-2}$ | $1.4$ | | $2.6\times 10^{-5}$ | $9.8\times 10^{-7}$
0.075 | $2.8\times 10^{-5}$ | $2.4\times 10^{-2}$ | $1.4$ | | $1.8\times 10^{-5}$ | $3.3\times 10^{-7}$
0.08 | $3.1\times 10^{-5}$ | $2.5\times 10^{-2}$ | $1.5$ | | $1.3\times 10^{-5}$ | $9.2\times 10^{-8}$
0.085 | $3.1\times 10^{-5}$ | $2.7\times 10^{-2}$ | $1.5$ | | $9.3\times 10^{-6}$ | $8.2\times 10^{-8}$
0.09 | $7.5\times 10^{-6}$ | $2.8\times 10^{-2}$ | $1.5$ | | $9.1\times 10^{-6}$ | $2.0\times 10^{-7}$
0.095 | $3.7\times 10^{-6}$ | $2.9\times 10^{-2}$ | $1.5$ | | $4.8\times 10^{-6}$ | $3.9\times 10^{-7}$
The agreement between the data and the CFT behaviour is incredibly good until
$b\gtrsim 0.7$, when $\alpha=0.75$, as is clearly visible from figures 10 and
11. The reason for this discrepancy is that, as we explained in the
introduction, the supposition that sinh-Gordon approaches naïvely the
Liouville CFT in its UV limit is no longer valid when
$b\geq\sqrt{\frac{\alpha}{2-\alpha}}$. When $\alpha=0.75$, the _critical_
value is $b^{\textrm{crit}}=\sqrt{3/5}\sim 0.774$, which explains why figure
10 still shows a good agreement for very small values of $r$, while in figure
11 we see that the data and the CFT curve behave in radically different ways.
## 5\. Conclusion
We investigated numerically the behaviour of the conjecture 3.45 in the UV
limit $R\rightarrow 0$ of the sinh-Gordon model defined on an infinite
cylinder of radius $2\pi R$. We found an extremely good agreement with the
theoretical predictions in Ref. [15], up to the $4^{\textrm{th}}$ decimal
place, in the cases of both primary and descendant fields. In figures 1-13 and
tables 1-3 part of these results are collected. We consider these, along with
the analytical results of Ref. [8], as a very strong confirmation of the
correctness of the fermionic basis description for the sinh-Gordon model.
We have also verified that the limiting behaviour of the primary fields’
expectation values is very well described by that of a particular three point
function in Liouville CFT only if the parameters are such that the scaling
dimensions of the involved fields are all positive, meaning that $0<b<a<Q$. It
would be interesting to study the behaviour of sinh-Gordon model’s UV limit
outside this region.
## Acknowledgments
I am grateful to F. Smirnov whose valuable advices helped considerably to
direct my analysis and organize this work.
Research of SN is supported by the People Programme (Marie Curie Actions) of
the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA
Grant Agreement No 317089 (GATIS).
This project was partially supported by INFN grant IS FTECP and the UniTo-
SanPaolo research grant Nr TO-Call3-2012-0088.
## References
* [1] H. Boos, M. Jimbo, T. Miwa, F. Smirnov and Y. Takeyama, _Hidden Grassmann structure in the XXZ model_ , Commun. Math. Phys. 272 (2007) 263-281.
* [2] H. Boos, M. Jimbo, T. Miwa, F. Smirnov and Y. Takeyama, _Hidden Grassmann structure in the XXZ model II: creation operators_ , Commun. Math. Phys. 286 (2009) 875-932.
* [3] M. Jimbo, T. Miwa and F. Smirnov, _Hidden Grassmann structure in the XXZ model III: introducing Matsubara direction_ , J. Phys. A42 (2009) 304018.
* [4] H. Boos, M. Jimbo, T. Miwa and F. Smirnov, _Hidden Grassmann structure in the XXZ model IV: CFT limit_ , Commun. Math. Phys. 299 (2010) 825-866.
* [5] M. Jimbo, T. Miwa and F. Smirnov, _On one point functions of descendents in Sine-Gordon model_ , arXiv: 0912.0934
* [6] M. Jimbo, T. Miwa and F. Smirnov, _Hidden Grassmann structure in the XXZ model V: sine-Gordon model_ , Lett. Math. Phys. 96 (2011) 325-365.
* [7] M. Jimbo, T. Miwa and F. Smirnov, _Fermionic structure in the sine-Gordon model: form factors and null-vectors_ , Nucl. Phys. B852 (2011) 390-440.
* [8] S. Negro and F. Smirnov, _On one-point functions for sinh-Gordon model at finite temperature_ , Nucl. Phys. B875 (2013) 166-185, arXiv: 1306.1476.
* [9] A.G. Bytsko and J. Teschner, _Quantization of models with non-compact quantum group symmetry. Modular XXZ magnet and lattice sinh-Gordon model_ , J. Phys. A39 (2006) 12927-12981.
* [10] J. Teschner, _On the spectrum of the Sinh-Gordon model in finite volume_ , Nucl. Phys. B799 (2008) 403-429, [arXiv:hep-th/0702214].
* [11] V. Fateev, D. Fradkin, S. Lukyanov, A. Zamolodchikov and Al. Zamolodchikov, _Expectation values of descendent fields in the sine-Gordon model_ , Nucl. Phys. B540 (1999) 587–609.
* [12] S. Negro and F. Smirnov, _Reflection relations and fermionic basis_ , Lett. Math. Phys. 103 (2013) 1293-1311, arXiv: 1304.1860.
* [13] Al. Zamolodchikov, _On the thermodynamic Bethe ansatz equation in the sinh-Gordon model_ , J. Phys. A39 (2006) 12863-12887.
* [14] A. LeClair and G. Mussardo, _Finite Temperature Correlation Functions in Integrable QFT_ , Nucl. Phys. B552 (1999) 624-642, arXiv: hep-th/9902075.
* [15] S. Lukyanov, _Finite temperature expectation values of local fields in the sinh-Gordon model_ , Nucl. Phys. B612 (2001) 391-412, arXiv: hep-th/0005027.
* [16] H. Dorn and H.-J. Otto, _Two and three point functions in Liouville Theory_ , Nucl. Phys. B429, 375-388 (1994)
* [17] A. Zamolodchikov and Al. Zamolodchikov, _Structure constants and conformal bootstrap in Liouville field theory_ , Nucl. Phys. B477 (1996) 577–605, arXiv: hep-th/9506136.
* [18] Al. Zamolodchikov, _On the thermodynamic Bethe ansatz equation in sinh-Gordon model_ , J. Phys. A39 (2006) 12863-12887, arXiv: hep-th/0005181.
* [19] V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, _Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz_ , Commun. Math. Phys. 177 (1996) 381, [arXiv:hep-th/9412229].
* [20] V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, _Integrable structure of conformal field theory II. Q-operator and DDV equation_ , Commun. Math. Phys. 190 (1997) 247, [arXiv:hep-th/9604044]
* [21] V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, \- _Integrable structure of conformal field theory III. The Yang-Baxter relation_ , Commun. Math. Phys. 200 (1999) 297, [arXiv:hep-th/9805008].
|
arxiv-papers
| 2014-04-02T17:14:38 |
2024-09-04T02:50:00.613591
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stefano Negro",
"submitter": "Stefano Negro",
"url": "https://arxiv.org/abs/1404.0619"
}
|
1404.0743
|
# Pentago is a First Player Win: Strongly Solving a Game Using Parallel In-
Core Retrograde Analysis
Geoffrey Irving Otherlab
San Francisco, CA
[email protected]
###### Abstract
We present a strong solution of the board game pentago, computed using
exhaustive parallel retrograde analysis in 4 hours on 98304 ($3\times 2^{15}$)
threads of NERSC’s Cray Edison. At $3.0\times 10^{15}$ states, pentago is the
largest divergent game solved to date by two orders of magnitude, and the only
example of a nontrivial divergent game solved using retrograde analysis.
Unlike previous retrograde analyses, our computation was performed entirely
in-core, writing only a small portion of the results to disk; an out-of-core
implementation would have been much slower. Symmetry was used to reduce
branching factor and exploit instruction level parallelism. Despite a
theoretically embarrassingly parallel structure, asynchronous message passing
was required to fit the computation into available RAM, causing latency
problems on an older Cray machine. All code and data for the project are open
source, together with a website which combines database lookup and on-the-fly
computation to interactively explore the strong solution.
## I Introduction
Computer play of combinatorial games such as chess, checkers, and go has been
an active area of research since the early days of computer science [1]. The
limit of computer play is a solved game, when a computer can play perfectly
either from the start position (weakly solved) or from any position (strongly
solved). The first nontrivial weakly solved game was Connect-Four in 1988 by
both Allen and Allis [2], later strongly solved by Tromp [3]. Many games have
been solved since, the most challenging being the weak solution of checkers
[4]. The checkers solution involved 18 years of parallel out-of-core
retrograde analysis culminating in a $3.9\times 10^{13}$ position endgame
database together with a $10^{14}$ operation forward search.
To date, all solved games have been either convergent (fewer positions near
the end of the game) or amenable to knowledge-based strategies. Checkers is an
example of a convergent game: while the entire $10^{20}$ state space is too
large to explore fully, the set of positions with 10 or fewer pieces has a
more manageable $3.9\times 10^{13}$ positions. Pieces are removed but never
added, so a database of $\leq 10$ piece positions can be computed via
retrograde (backward) analysis starting with 1 piece, then 2 pieces, and so on
up to 10 pieces. The computed database is then used to prune a forward search
starting from the beginning of the game.
wintieloss
Figure 1: (Left) With perfect play, the first player wins with any opening
move except the corners, which tie. (Right) A more delicate position with
black to play. The full strong solution can be explored at http://perfect-
pentago.net.
Figure 2: Counts of pentago positions vs. stones on the board, with symmetries
removed. Run ``web/counts+ in the source repository to reproduce.
In contrast to convergent games, the number of positions in a divergent game
increases with time (typically as more stones are added to the board), making
traditional retrograde analysis plus forward search impractical. Thus, all
nontrivial divergent games solved to date have game-specific knowledge based
strategies which can be used to avoid brute force: zuzgwang control in
Connect-Four [2], threat-space search for gomoku and renju [5, 6], and
H-search in hex [7]. For discussion on the characteristic of various solved
games, see [8].
Pentago is a divergent game designed by Tomas Flodén and sold by Mindtwister
[9]. We reproduce the rules here for completeness. Pentago is played on a
$6\times 6$ board, divided into four $3\times 3$ quadrants. There are two
players, black and white, who alternate turns. The goal of each player is to
get five stones of their color in a row, either horizontally, vertically, or
diagonally. Each turn, a player places a stone in an empty space in some
quadrant, then chooses a possibly different quadrant to rotate 90 degrees left
or right. If both players get five in a row at the same time, or the last move
is played with no five in a row, the game is a tie. If a player makes five a
row by placing a stone, there is no need to rotate a quadrant: the player wins
immediately.
Unlike divergent games solved to date, no strong knowledge based strategies
are known for pentago, and existing programs are capable of searching only to
fairly low depths [10, 11]. This is primarily a consequence of the high
branching factor of pentago: there are $36\cdot 8=288$ possible first moves
including rotation and an average branching factor of $97.3$ over all
states.111To reproduce the branching factor average, run ``bin/analyze branch+
in https://github.com/girving/pentago. For the rest of the paper, only the
command will be given. To reduce the branching factor to a manageable level,
our solver performs all computations in terms of _rotation abstracted
positions_ consisting of all 256 ways to rotate the quadrants of a given
board; this eliminates the factor of 8 due to rotation for an average
branching factor of only $12.2$. Operating on more than one board at a time
lets us take full advantage of SSE acceleration.
Unfortunately, symmetry techniques alone are insufficient to solve pentago on
commodity hardware. The game has 3,009,081,623,421,558 ($3\times 10^{15}$)
states with symmetries removed, all but $0.3\%$ of which are reachable with
valid play;222Run ``analyze counts+ and ``analyze reachable+, respectively.
the number of states over time is shown in Figure 2. To solve the game using
retrograde analysis, we traverse all positions in reverse order of the number
of stones, starting from the 35 stone _slice_ (the 36th is computed on demand)
and iteratively computing the $n$-stone slice from the $(n+1)$-stone slice up
to the beginning of the game. This requires storing two adjacent slices at a
time, requiring $213$ TB at peak before compression. Our initial target was to
fit into half of the NERSC Cray Hopper’s $217$ TB, which was plausible using
fast but weak compression only if minimal memory was wasted communication
buffers and working storage.
In order to minimize working memory, our parallel solver grabs inputs from
other processes immediately before they are used, overlapping a small number
of work chunks to hide latency. Since computing each chunk takes a variable
amount of time (see below), we opted for a fully asynchronous communication
pattern: when a process needs an input block, it sends a message to the owner
of that block, and the owner replies asynchronously with the data.
The solver was run exactly once at full scale, generating a $3.7$ TB database
of perfect results with $0$ through $18$ stones and establishing that pentago
is a win for the first player to move (Figure 1). The full strong solution can
be explored online at http://perfect-pentago.net.
At $3.0\times 10^{15}$ states, pentago is the largest divergent game
computation by a factor of 150 (vs. $2\times 10^{13}$ for $9\times 6$ Connect-
Four), and the largest strongly solved game by a factor of 660 (vs. $4.5\times
10^{12}$ for $7\times 6$ Connect-Four). Among retrograde analyses used to
solve games, it is the largest by state space by a factor of 77 (vs.
$3.9\times 10^{13}$ in the solution of Checkers). However, it is not the
largest endgame database over any game: the 7-piece Lomonosov Endgame
Tablebases for chess are 140 TB in size, and were computed over six months at
Moscow State’s Lomonosov supercomputer [12]. Unfortunately, the technical
details of the Lomonosov computation are unpublished, so a detailed comparison
is difficult.
## II Problem definition
Let $S$ be the set of arrangements of black and white stones on a $6\times 6$
board. Only some of these are valid pentago positions: if we let black play
first, we have equal numbers of black and white stones on black’s turn and one
extra black stone on white’s turn. Define predicates
$f_{b},f_{w}:S\to\\{0,1\\}$ by $f_{c}(s)=1$ if color $c$ has a five in a row.
Given color $c$, let $\bar{c}$ be the other color. For $s\in S$, let
$p_{c}(s)\subset S$ be the positions reached by placing a stone of color $c$,
$r(s)$ the positions reached by rotating exactly one quadrant $90^{\circ}$
left or right. Let $v_{c}(s)$ be the value of position $s$ with $c$ to play:
$v_{c}(s)=-1,0,1$ if $c$ loses, ties, or wins, respectively. If $f_{b}(s)$,
$f_{w}(s)$, or $s$ has 36 stones, the game is over and
$v_{c}(s)=f_{c}(s)-f_{\bar{c}}(s)$. Otherwise
$\displaystyle v_{c}(s)=\max_{a\in p_{c}(s)}\begin{cases}1&\mbox{if
}f_{c}(a)\\\ \max_{b\in r(a)}h_{c}(b)&\mbox{otherwise}\end{cases}$
where
$\displaystyle h_{c}(s)$
$\displaystyle=\begin{cases}f_{c}(s)-f_{\bar{c}}(s)&\mbox{if }f_{c}(s)\wedge
f_{\bar{c}}(s)\\\ -v_{\bar{c}}(s)&\mbox{otherwise}\end{cases}$
Slice $n+1$Slice
$n$ComputeScatter$\vdots$$\vdots$$\vdots$$\curvearrowright$rotateGatherInputInputInputCombine$\vdots$
Figure 3: We decompose the set of pentago position into sections, each a 4D
array of blocks (shown here as 2D). The results for a given block are the
combination of results from each block line that contains it, with each such
block line depending on exactly one block line from a different section. Each
computation from input line to output line can be performed on a different
processor, first gathering the input blocks together into a complete line, and
finally scattering the output blocks to their owners. Each pink rounded
rectangle lies in a possibly different process. Since we compute only those
sections which are unique with symmetries removed, some input lines must be
rotated before computation.
## III Abstracting over rotations
The exact symmetry group of pentago is the 8 element dihedral group $D_{4}$
with 4 global reflections and 4 global rotations. Computing only one element
from each $D_{4}$ equivalence class saves a factor of $8$, but does nothing
for the large branching factor of the game. Thus, we consider the _local_
group of all 256 ways to rotate the four quadrants, which has the abelian
group structure
$L=\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}\times\mathbb{Z}_{4}$.
Combined with the group of global symmetries, the full group of _approximate
symmetries_ is a semidirect product $G=\mathbb{Z}_{4}^{4}\rtimes D_{4}$ with
$2048$ elements.
Computing one board $b$ from each equivalence class w. r. t. $G$ is not
enough; we must compute a function
$f_{b}:L\to\\{-1,0,1\\}$
mapping $g\in L$ to the result of the quadrant rotated board $gb$. Each board
is a win, loss, or tie, so there are $3^{256}$ such functions. To avoid
ternary arithmetic we use 2 bits per value for uncompressed data: one bit for
win vs. loss/tie and one for win/tie vs. loss. Thus, for each board we have
two functions $\mathbb{Z}_{4}^{4}\to\\{0,1\\}$, each a $4\times 4\times
4\times 4$ array of bits. Each such function $L\to\\{0,1\\}$ is packed into a
256 bit table.
Since quadrant rotations do not change the equivalence class w. r. t. $G$,
operating on these functions $f_{b}$ removes the branching factor due to
rotations. In its place, we have the mixing operation
$\displaystyle\operatorname{rmax}$
$\displaystyle:\left(L\to\\{0,1\\}\right)\to\left(L\to\\{0,1\\}\right)$
$\displaystyle\operatorname{rmax}$ $\displaystyle(f)(g)=\max_{r\in R}f(g+r)$
where $R\subset L$ is the set of $90^{\circ}$ degree rotations left or right,
and we use $+$ because the group $L$ is abelian. In addition to
$\operatorname{rmax}$, two other rotation abstracted routines are needed.
First, given the position of stones of one color, we must be able to compute
the set of rotations $g\in L$ which produce five in a row. Second, our
equivalence class representative w. r. t. $G$ can change when we add a stone,
so we must be able to transform $f_{b}$ into $f_{gb}$ for any $g\in G$; this
involves cyclic shifts, dimension transpositions, and reflections of $4\times
4\times 4\times 4$ bit tables.
Although the code required for these operations is complex, verifying their
correctness was a straightforward process of checking group theoretic
definitions against the much simpler routines operating on one board at a
time. The ease of verification frees us to make the routines as complicated as
required for speed without reducing confidence in the code.
## IV Data layout and distribution
Given a board $b$, we must choose a unique representative out of the
equivalence class $Gb$. This choice should be made such that adding a stone
changes the representative choice in as few ways as possible, so that the
effective branching factor will be smaller once we take data layout into
account. Concretely, since we have eliminated branching factor due to
rotation, an average board has $12.2$ child boards which are needed as input;
if representatives were chosen arbitrarily, the representatives of the child
equivalent classes w. r. t. $G$ might be located in up to $12.2$ processes
depending on how data is distributed.
$0$$1$$2$$3$$4$$\cdot$$\cdot$$\cdot$$0$$1$$\cdot$$\cdot$$\cdot$$f_{00cd}$$f_{01cd}$$\cdot$$\cdot$$\cdot$$f_{10cd}$$f_{11cd}$$\cdot$$\cdot$$\cdot$$f_{20cd}$$f_{21cd}$$\cdot$$\cdot$$\cdot$$f_{30cd}$$f_{31cd}$$\cdot$$\cdot$$\cdot$$f_{40cd}$$f_{41cd}$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$$\cdot$
Figure 4: Each section is a 4D array of functions $f_{abcd}:L\to\\{-1,0,1\\}$.
Each dimension corresponds to all patterns of stones in one of the four
quadrants with fixed counts of black and white stones, including only patterns
lexicographically minimal under rotation. For each $(a,b,c,d)$ describing the
four quadrants of a board, $f_{abcd}$ gives the loss/tie/win values for all
256 ways to rotate the four quadrants. The order is chosen so that reflected
pairs are adjacent so that reflection preserves the block structure. The
figure shows 2D slices of the full 4D array.
Therefore, we partition all boards in a given slice (fixed number of stones)
into _sections_ defined by the numbers of stones of each color in the four
quadrants, computing only sections whose counts are lexicographically minimal
under $D_{4}$ symmetry. Within a given section, we consider only boards whose
quadrants are lexicographically minimal under per-quadrant rotations; each
quadrant is independent under this requirement, so the section becomes a four
dimensional rectangular array where each dimension defines the stones in one
quadrant. We precompute an ordering of these rotation minimal quadrant states
so that we can convert from a position in the four dimensional section array
to a board state using table lookup. The structure of one such section is
shown in Figure 4. There are at most four quadrants to chose from when placing
a stone (some may be full near the end of the game), so at most four child
sections contribute to the results for a given parent. In other words, we have
reduced the effective branching factor from $12.2$ to $4$.
Sections alone provide insufficiently fine parallelism (the largest is $1.8$
TB uncompressed), so we divide each 4D section array into $8\times 8\times
8\times 8$ blocks and partition the blocks for all sections among the
different processes. When a stone is added in a quadrant, we move to a child
section with index layout different from the parent only for the quadrant
where the stone was added since the other quadrants have the same pattern of
stones. Therefore, a single block in a parent section depends on inputs from
one _line_ of blocks in up to four child sections. Since the different input
lines for a block correspond to moves in different quadrants, we can compute
each line contribution separately on different processes and combine them with
$\max$ on whichever process owns the output block.
The structure of the computation is illustrated in Figure 3. Say we want to
compute an output block line in section $n$, which is a $k\times 1\times
1\times 1$ grid of blocks (possibly transposed) corresponding to an $8k\times
8\times 8\times 5$ grid of boards (block sizes may differ from $8$ at section
boundaries). Our output block line depends on a single input block line in
section $n+1$, which (possibly after rotation) is an $8k^{\prime}\times
8\times 8\times 5$ grid of nodes. The input and output block lines differ in
size only along the long dimension, since the long dimension corresponds to
the quadrant where we will place a stone. When we compute index $(a,b,c,d)$ of
the output line, we mix together several indices $(a^{\prime},b,c,d)$ with
different $a^{\prime}$ corresponding to the different places to put a stone in
the lines’ quadrant. Since the map from $a$ to $a^{\prime}$ is many to many,
computing the entire block line on a single processor gives an effective
branching factor of $4$ for communication cost even though the underlying
branching factor is $12.2$. Once the block line is computed, its component
blocks are scattered to their owners to be merged together via $\max$ with
other contributions (each block needs up to four such block line
contributions).
Since our block structure is symmetric w. r. t. dimension and our quadrant
positions are always minimal with respect to rotation, the block structure of
sections is preserved when the board is rotated. However, the ordering of
quadrant states does change when a quadrant is reflected, since a
lexicographically least quadrant may no longer be lexicographically least
after reflection. To maintain the block structure, we require even sized
blocks (8 in our case) and adjust our precomputed quadrant state ordering so
that reflected pairs occur next to each other in the same block. With this
trick, the block structure is invariant to all symmetries.
The relatively simple structure of sections, blocks, and rotation-abstracted
values within blocks does have a cost: if a position or section is preserved
by a symmetry it will be double counted in the data layout. Abstracting over
rotations increases the number of effective positions by 5.4% and removing
symmetries only at the section level costs an additional 9.3%, for a total
overcounting of 15.2% relative to storing each symmetry-unique position
once.333Run ``analyze ratio+.
### IV-A Deterministic pseudorandom partitioning
In parallel, we must partition the set of blocks across processes to balance
memory usage and the set of block lines across processes to balance compute.
Ideally, the process computing a given block line would also own many of the
input and output blocks in order to minimize communication. Unfortunately,
these desires couple together the partition for all slices. Over all slices,
there are 3,654,002,393 blocks and 996,084,744 block lines.444Run ``analyze
approx+. Thus, we have a graph partitioning problem with 4,650,087,137 nodes
divided into 72 clusters, each cluster defining a load balancing constraint.
Although existing graph partitioning codes such as ParMETIS[13] might be
sufficient for our problem, we have sufficient computation to hide
communication latency and opt for a simple randomized partitioning scheme
instead.
We partition each slice independently. Since there are at most 8239 sections
to a slice, and each section is a regular 4D grid of blocks, we can define an
ordering of all block lines by arranging the sections back to back. We choose
a pseudorandom permutation of the ordered block lines and give each process a
contiguous chunk of the scrambled ordering. Each block is then randomly
assigned to one of its four lines, and given to the process which owns that
line. In both cases, these choices can be made consistently with only an
$O(1)$ size random seed shared between processes: we use the arbitrary size
cipher technique of [14] for random access random permutations and the
Threefry generator of [15] for conventional random numbers. Since the cipher
permutations are invertible, we can find the process owning a given block or
block line in $O(1)$ time.
Figure 5: Load balance ratio ($\max/\min$) for various quantities as a
function of stones on the board using deterministic pseudorandom partitioning.
89% of the computation occurs from slice $20$ to $28$, where all quantities
balance to within 20%. Run ``paper/numbers load+ to reproduce.
At scale, a pseudorandom partitioning scheme automatically balances any
quantity where the central limit theorem applies. In particular, though our
scheme does not explicitly account for the different amounts of work required
to compute different block lines, or the different sizes of blocks at the
boundary of sections, there are enough blocks and block lines to keep the
$\max/\min$ ratio to within 10-20% for all large slices and all relevant
quantities (Figure 5).
### IV-B Compression
Since our uncompressed memory usage would be at least $213$ TB, we compress
all data in memory until needed using the fast but weak compression library
Snappy [16]. Most blocks are $256$ KB ($64\cdot 8^{4}$ bytes) uncompressed,
large enough to compress each block separately without harming compression
ratio. Despite its speed relative to stronger compression such as ZLIB or
LZMA[17, 18], Snappy still consumed about 29% of our compute time ignoring
I/O.555See ``snappy fraction+ in ``paper/numbers+. Stronger compression is
thus out of reach for in memory purposes, although we do use LZMA when writing
out the smaller final data set.
With compression the memory usage varies unpredictably, with two consequences.
First, repeatedly allocating and deallocating irregular block sizes results in
significant fragmentation. During early testing on BlueGene, which has no
virtual memory system, fragmentation caused the code to run out of memory much
earlier than necessary. We solved this with a manual compacting garbage
collector for bulk data storage, which is straightforward in our case due to
the lack of pointers. Second, estimates from slices near the end of the game
gave a compression ratio of roughly $1/3$. Since we were uncertain whether
this ratio should grow or shrink at the peak of the computation, and wanted a
high probability of solving the game in a single run, we used a conservative
estimate of $0.4$ when determining how many nodes to use. However, the actual
average compression ratio was $0.26$.666See ``total data+ in ``paper/numbers+.
Taking advantage of the unexpectedly good compression would have required
dynamic partitioning, or even (ideally) a dynamic number of MPI nodes.
## V Asynchronous control flow
0123456789101112131415Process rank020406080100120140160180200220240260Time
(s)computesnappywaitunsnappycompactcountaccumulateother230.1230.2230.3230.4230.5Time
(s)
Figure 6: Images from a trace visualization tool used to diagnose performance
problems in asynchronous code. (Left) The history of a 16 process, 96 thread
run computing the section with four stones of each color in each quadrant.
Each process has one communication thread (mostly red for waiting) and five
worker threads performing computation, with colors showing the type of
computation performed. (Right) A zoom showing the information flow related to
part of a block line computation. At the leftmost point in the graph shown,
the process decides to compute a given block line, and sends out requests for
input data to other processes. Once all responses arrive, the computation
begins. When the computation finishes, the results are scattered to other
processes and compressed for storage. To reproduce, run ``paper/history+.
Within slice $n$, each process can compute its allocation of block lines in
any order, since all inputs are from slice $n+1$ which has already been
computed. However, most of these inputs are stored on other processes, and due
to memory limitations only a small fraction of them can be stored locally at
any given time. Moreover, the time to compute a given block line varies with
size, ruling out a lockstep communication/compute cycle. Instead, we use an
asynchronous control flow where each process sends requests for input data for
at most five block lines at time, begins computing as soon as all inputs for a
block line are in place, sends out output data when ready, and listens for
incoming output data from other processes to be merged.
We emphasize that asynchrony is needed only because of the memory constraint:
if we had 4 times as much memory in order to store all inputs locally, we
could split the computation into communication / compute epochs and use an
embarrassingly parallel control flow during compute.
We use a hybrid MPI/Pthread model where each 6 thread process has 1
communication thread and 5 worker threads (with 8 processes per 48-hyperthread
Edison node). A hybrid structure reduces the memory usage by allowing several
threads to share the same temporary storage required when computing a block
line. The communication thread must simultaneously listen for incoming remote
messages and completed tasks from the worker threads; this can be done with
self-to-self MPI messages in environments which support `MPI_THREAD_MULTIPLE`
but requires alternatively polling between `MPI_Testsome` and
`pthread_spin_trylock` if only `MPI_THREAD_FUNNELED` is available.
At the time the code was written, the MPI 3 standard was not yet available on
the target machine, and the one sided communication primitives in MPI 2 were
not sufficient for our communication pattern.777For more discussion, see
http://scicomp.stackexchange.com/questions/2846/simulating-the-mpi-isend-
irecv-wait-model-with-one-sided-communication. Specifically, the MPI 2 one
sided primitives provide no asynchronous way to know when a request completes;
and our only synchronization points are between entire slices. MPI 3 solves
this problem: after an initial communication phase exchanging pointers to the
required compressed slice $n+1$ blocks, all input requests during slice $n$
computation could be handled with `MPI_Rget`[19]. Unfortunately, MPI 3 does
not solve the reverse problem of output messages: when an output block
arrives, any previous data for that block must be uncompressed, combined with
the new data, and recompressed for storage. `MPI_Accumulate` has no support
for user defined operations, so output messages would still be limited to two
sided communication. Finally, MPI 3 provides the useful `MPI_Ibarrier`
primitive which is exactly what we need to know when all processes have
finished computing and thus when the previous slice can be deallocated; since
we use MPI 2 we must simulate `MPI_Ibarrier` using a manual tree reduction.
The asynchronous control flow was tricky to write but straightforward to
debug, since most bugs manifested as deadlocks. Each message and response is
labeled with a unique global id, so deadlocks were easy to eliminate by
reading traces of events. However, the performance characteristics of the code
were harder to understand, since high latency might be a result of unrelated
communication at the same time. Existing profiling tools such as TAU [20] were
insufficient for tracing the dependencies between asynchronous messages
combined with control flow across threads. Thus, we wrote a custom trace
visualizer with knowledge of the information flow between inputs through
compute to output; an example visualization is shown in Figure 6. On NERSC’s
Cray XE6 Hopper, this tool confirmed that long idle periods were due to high
latency, but was not sufficient to diagnose the underlying cause of the
problem. Unfortunately, we still do not know the cause of this latency.
Testing on the Argonne’s BlueGene/Q Vesta was inconclusive since the code
easily saturated BlueGene’s poor integer performance. On the newer Cray Edison
used for the final production run, the problem went away: worker threads were
idle only 16.4% of the time with I/O excluded.888See ``Idle vs. total time+ in
``paper/numbers+.
## VI Performance
Our final production job ran on NERSC’s Cray XC30 Edison, using 98304
($3\times 2^{15}$) threads including hyperthreading (49152 cores, 2048 nodes).
The bulk of the computation from slice $35$ down to $19$ took $2.7$ hours.
Starting at slice $18$ we began writing output results to disk, though our
first computation finished writing only slices $17$ and $18$ before hitting an
unfortunately chosen wall clock limit of $4$ hours. Two smaller jobs on 192
and 128 nodes were used to finish the computation down to slice $0$, the start
of the game.
Figure 7: Time profile of the main production run, showing total worker thread
time usage over all processes. All game logic is contained in the _compute_
(green) section; the other sections are overhead due to load imbalance, non-
hidden communication latency, compression/decompression, and I/O. Slices 17
and 18 include LZMA compression and final I/O. During I/O, all workers are
idle and the communication thread on each process is inside MPI/IO.
The time profile of the computation is shown in Figure 7. Excluding I/O, only
16.4% of the total worker thread time is spent idle,8 confirming that our
random load balancing scheme is sufficient for near peak performance. Since
only 5 out of 6 threads per rank are workers, we could theoretically speed up
the computation by up to 6/5 if the communication thread performed useful
work. Unfortunately, existing MPI implementations do not implement performant
asynchronous progress, even ignoring our need for active responses to messages
(see [21] for a good discussion).
Including I/O, our performance is further from optimal: 51.0% of worker thread
time is idle, with 34.6% due entirely to I/O. This is due to both very high
latency when writing small files during every slice (around 200 seconds
independent of file size) and low bandwidth when writing file results in
slices 17 and 18. The high latency was a consequence of using
`MPI_File_write_ordered` when writing small files, since MPICH and thus Cray
MPI implement this routine using shared files for synchronization rather than
fast network collectives. Unfortunately, the low bandwidth is likely user
error: we accidentally wrote to NERSC’s global scratch filesystem rather than
the special filesystem optimized for Edison.
Since much of the complexity of our implementation derives from the memory
constrained in core structure, it is important to estimate how much slower the
computation would have been if run out of core. Edison’s peak I/O bandwidth is
168 GB/s, or 66 GB/s on our 2048 out of 5192 nodes if bandwidth is shared
proportionally. An out of core version of our algorithm would write each block
once and read each block four times, for a total of 3.6 PB of I/O uncompressed
or 0.94 PB with Snappy compression. Thus, at peak I/O bandwidth, a Snappy
compressed out of core version of our code would take 4.0 hours for I/O.999See
``Total I/O time estimates+ in ``paper/numbers+. In contrast, the non-I/O
portion of our main run took 1.8 hours, for a speedup of 2.25. If peak I/O
performance could not be achieved, or I/O and compute could not be fully
overlapped, the speedup would be larger.
The rotation abstracted compute kernel uses SSE for instruction level
parallelism, packing each 256-bit $L\to\\{0,1\\}$ function into two 128 bit
SSE registers. $\operatorname{rmax}$ can be computed for one such table in
$152$ SSE instructions, or $3/5$ths of an instruction per position since each
function encodes $256$ positions. Computing which of the 256 quadrant
rotations of a board give five in a row takes $190$ instructions and $640$
bytes of cache coherent table lookup, and transforming a function $f_{b}$ into
$f_{gb}$ takes between $60$ and $200$ instructions dependent on the particular
$g\in G$. Although bit-level representations of board state are standard in
computational games, we believe this is the first instance where values of
many distinct positions are evaluated in parallel using bit twiddling.
Measuring over the entire compute kernel (which excludes idle time,
communication, and (de)compression), our SSE routines achieve a $1.81\times$
speedup on Edison over 64-bit versions and a $2.25\times$ speedup on an Intel
Core i7, compared to a naive speedup of $2$ for twice as many bits per
instruction.101010See ``SSE vs. non-SSE speedup+ in ``paper/numbers+. We are
not sure what caused the superlinear speedup in the $2.25$ case; one
possibility is reduced register pressure.
Figure 8: Latency for 8-byte request-for-input messages between different
nodes in a 96-thread test run on Edison. Around 20% of the messages complete
in under $10$ ms, but the tail is quite long. To reproduce, run
``paper/numbers messages 0,2+.
Even on Edison’s faster network, our typical message latency is still quite
high as shown in Figure 8. Here latency is measured from immediately before
`MPI_Isend` to the time we start responding to the finished message when
`MPI_Testsome` succeeds. The large latencies may be due to interference
between small messages and larger messages, as the communication thread
processes different types of messages asynchronously. Since the latency on
Edison is low enough for our purposes, we have not investigated in detail.
Within each node, we used the PAPI hardware counter library [22] to measure
instruction issue, cache misses, and branch mispredications. The results show
that our workload has minimal memory bandwidth requirements (under $15$ MB/s
per core in performance critical sections), mispredicts branches mostly during
Snappy (de)compression, and makes significant use of dual instruction issue
only for the one out of five worker threads that share a core with the less
active communication thread; if hyperthreading is turned off, the issue rate
jumps from $1.21$ to $1.89$ instructions per cycle.111111Run ``paper/numbers
papi+.
## VII Correctness and fault detection
Since our goal is a database of perfect play, it is important to consider the
possible sources of error in the computation. We are interested only in
undetected errors, since empirically the code ran without crashing or failing
an assertion.
On the software side, we make heavy use of unit tests throughout the code,
including simple tests for correctness of simple routines, bootstrap tests
comparing simple routines to more complicated variants (such as when
abstracting over rotations), and comparison tests between different
algorithms. In particular, we compare our parallel backward code against the
results of forward search, both at the beginning and end of the game. Near the
end of the game this is easy, as forward search can quickly compute perfect
play. Near the beginning, our tests replace all values at slice 4 or 5 with
random values, compute optimal play up to the random slice with both backward
and forward algorithms, and compare.
On the hardware side, the main failure points are DRAM, CPU, network, and
disk. Undetected disk errors are unlikely since the primary output files are
checksummed as part of LZMA compression. Of DRAM, CPU, and network, DRAM
errors dominate according to [23] (Tables 6.11 and 6.12), at least on
BlueGene. Edison memory and network are SECDED (single error correct, double
error detect), so an undetectable error would require three simultaneous
failures. Unfortunately, we do not know a reliable method to estimate this
probability conditional on an apparently successful run, since DRAM errors are
far from uncorrelated events [24]. However, we believe the probability is
quite small, and in particular that undetected hardware errors are less likely
than software errors.
As a test on software errors when running the code at scale, we write out
small sample files with the results of randomly chosen boards during each
slice. Large numbers of samples generated by the main run were validated
against forward search for slices 20 and up, so any remaining software errors
in the parallel code must manifest only on a small set of positions. We also
write out win/loss/tie counts for each slice. Both sample and count files
would be useful for cross-validation should someone reproduce the calculation
in the future.
Unfortunately, the sample files are insufficient to detect rare software bugs
or hardware failures, and indeed we know of no cheap method for detecting this
kind of unlikely error without rerunning or rewriting the code. Solving
pentago falls most naturally into the complexity class PSPACE (polynomial
space), and indeed the similar five-in-a-row game gomoku has been proven
PSPACE-complete [25]. Unless $\textrm{NP}=\textrm{PSPACE}$, it is unlikely
that a short certificate exists proving that pentago is a first player win,
especially if we require a strong solution with perfect play known from all
positions.
## VIII Open source and data
All code for this project is open source, and is available online at
https://github.com/girving/pentago. The repository includes the paper source
and all log files used to generate timing and other numbers. To regenerate any
reported number from the data, run either `bin/analyze <command>` or
`paper/numbers`; see the footnotes and figure captions.
The 3.7 TB strong solution is hosted on Rackspace Cloud Files; see the
download instructions at https://github.com/girving/pentago/#data. We store
small sparse sample and count files in Numpy’s `.npy` format [26], and the
main solution files in a custom `.pentago` format using the described block
structure with LZMA compression per block. The format is described at
https://github.com/girving/pentago/blob/master/pentago/data/supertensor.h.
Both `.npy` and `.pentago` formats are easy to write in parallel using MPI
I/O.
The strong solution is useless without a convenient method for exploring the
data, so we have built a website showing which moves win, lose, or tie from
any position: http://perfect-pentago.net. The frontend Javascript uses a
backend server at http://backend.perfect-pentago.net:2048 to look up the value
of positions. Any position with 18 or fewer stones is fetched from the
database using an HTTP range request to download the surrounding compressed
block. As in the parallel algorithm, the children of a position fall into at
most four blocks; we cache the uncompressed blocks to take advantage of this
locality.
Positions with more than 18 stones fall outside the database and are
recomputed from scratch using a specialized serial retrograde solver. Since
there are at least 18 stones already on the board, usually in an asymmetric
configuration, this solver rotates the board only through the
$\operatorname{rmax}$ function, avoiding the complexity of standardizing
positions into rotation minimal configurations. In addition, we use the fact
that $\operatorname{rmax}$ flips the parity of the $\mathbb{Z}_{4}^{4}$
symmetry group to store half the required bits, reducing the storage per
rotation abstracted position from 64 bytes to 32. With these optimizations,
evaluating all child values of an 18 stone position takes 16 seconds on a
single 2.6 GHz Intel Xeon thread, fast enough for interactive use.
Both remote lookups and from-scratch computation have significant latency, so
the backend server is written in Javascript using Node.js [27] for
asynchronous use by multiple clients. The Javascript handles asynchronous
logic and I/O, but calls down to C++ for performance intensive computation.
The backend server has a simple JSON API, and anyone wishing to develop their
own frontend is welcome to query it directly.
## IX Conclusion
We have strongly solved the board game pentago using retrograde analysis on
98304 threads of Edison, a Cray XC30 machine at NERSC. Symmetry techniques
were used to improve the branching factor of the game and take advantage of
SSE instruction level parallelism. Unlike previous retrograde analysis, the
computation was almost entirely in-core, writing results to disk only near the
end. To fit safely into memory, we use a fully asynchronous communication
structure where each process requests data from other processes as needed,
performs computation, and scatters results to their destinations.
The asynchronous control flow was a primary complicating factor during
development and optimization of the code, and runs against several limitations
of MPI including difficulties in synchronizing between MPI and threads, lack
of support for asynchronous progress in existing implementations, poor one-
sided communication in MPI 2 (fixed in MPI 3 too late for use in this
project), and lack of user defined operations in one-sided `MPI_Accumulate`.
Unfortunately, the latter would require both strong asynchronous progress and
careful consideration of threading semantics. Profiling tools were also a
significant limitation, leading us to implement our own tracing and
visualization tool to understand the flow of information across processes and
between threads without one process. Although our custom tool helped localize
the problem to high latency, we were unable to diagnose the underlying cause;
further analysis would likely require network profiling and visualization
tools incorporating knowledge of network topology.
Compression was a requirement to fit into memory, but we were limited to the
fast and weak Snappy library to prevent compression from becoming a compute
bottleneck. Compression also makes memory usage difficult to predict in
advance, causing us to overestimate memory requirements and use more Edison
nodes than required. Avoiding such overestimate without the I/O cost of
checkpointing would require a dynamic number of MPI nodes.
Our computation shares many characteristics with other irregularly structured,
data intensive HPC workloads. These characteristics include multiple levels of
structure (slices, sections, block lines, blocks, boards, bits), memory
restrictions, asynchronous control flow, reliance on integer performance
(compression and game logic), and reliance on both fast compute and fast
communication. Multiple levels of structure are important in many applications
(e.g., domains, pages, paragraphs, sentences, words for web search) and often
warrant different parallelism strategies at different levels. In addition to
allowing larger problem sizes either in RAM or on-package RAM, the ability to
operate near a memory limit improves performance for codes with imperfect
parallel scaling and eases the scheduling problem for shared clusters,
increasing both latency and bandwidth for users. Asynchronous control flow
adds flexibility which can be spent on memory constraints or irregular work
chunk sizes (common with multiple levels of structure). Poor integer
performance ruled out BlueGene for our purposes, which is problematic even for
floating point codes if compression is required. Finally, traditional Big Data
applications often have less tightly coupled communication patterns such as
MapReduce [28]; our application is sufficiently latency-critical to obtain
clear benefit from the faster network on Edison compared to Hopper, and serves
as an intermediate example between traditional HPC and Big Data (see the Graph
500 benchmark suite for other examples [29]).
## Acknowledgments
I am grateful to Jeff Hammond for valuable advice throughout the project, and
to Jed Brown for numerous helpful suggestions including the initial suggestion
that supercomputer time might be a possibility. Hosting for the 3.7 TB data
set and compute servers for http://perfect-pentago.net were generously donated
by Rackspace; thanks especially to Jesse Noller at Rackspace for offering to
host this open source project. Since a primary goal of this project is open
data, accessible hosting for the final results is essential. This research
used resources of the National Energy Research Scientific Computing Center,
which is supported by the Office of Science of the U.S. Department of Energy
under Contract No. DE-AC02-05CH11231. This research also used resources of the
Argonne Leadership Computing Facility at Argonne National Laboratory, which is
supported by the Office of Science of the U.S. Department of Energy under
contract DE-AC02-06CH11357.
## References
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* [5] L. V. Allis, H. J. Herik, and M. Huntjens, _Go-moku and threat-space search_. Univ. of Limburg, Department of Computer Science, 1993.
* [6] J. Wágner and I. Virág, “Solving renju.” _ICGA Journal_ , vol. 24, no. 1, pp. 30–35, 2001.
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* [9] Mindtwister. (2013) Pentago ce. [Online]. http://mindtwisterusa.com/products/games/pentago-ce
* [10] N. Buescher, “On solving pentago,” Bachlor’s thesis, Technische Universität Darmstadt, 2011.
* [11] T. Ewalds, “Playing and solving havannah,” Ph.D. dissertation, University of Alberta, 2012.
* [12] V. Makhnychev and V. Zakharov. (2012, Jul.) Lomonosov endgame tablebases. [Online]. http://chessok.com/?page_id=27966
* [13] K. Schloegel, G. Karypis, and V. Kumar, “Parallel static and dynamic multi-constraint graph partitioning,” _Concurrency and Computation: Practice and Experience_ , vol. 14, no. 3, pp. 219–240, 2002.
* [14] J. Black and P. Rogaway, “Ciphers with arbitrary finite domains,” in _Topics in Cryptology – CT-RSA 2002_. Springer, 2002, pp. 114–130.
* [15] J. K. Salmon, M. A. Moraes, R. O. Dror, and D. E. Shaw, “Parallel random numbers: as easy as 1, 2, 3,” in _High Performance Computing, Networking, Storage and Analysis (SC), 2011 International Conference for_. IEEE, 2011, pp. 1–12.
* [16] Z. Tarantov and S. Gunderson. (2014) Snappy: A fast compressor/decompressor. [Online]. http://code.google.com/p/snappy
* [17] P. Deutsch and J.-L. Gailly, “Zlib compressed data format specification version 3.3,” 1996.
* [18] T. Tukaani Project. (2014) XZ Utils. [Online]. http://tukaani.org/xz
* [19] MPI Forum, “MPI: A message-passing interface standard. Version 3.0.” Nov. 2012.
* [20] S. S. Shende and A. D. Malony, “The TAU parallel performance system,” _International Journal of High Performance Computing Applications_ , vol. 20, no. 2, pp. 287–311, 2006.
* [21] J. Squyres. (2012) MPI progress. [Online]. http://blogs.cisco.com/performance/mpi-progress
* [22] P. J. Mucci, S. Browne, C. Deane, and G. Ho, “PAPI: A portable interface to hardware performance counters,” in _Proc. of the Department of Defense HPCMP Users Group Conference_ , 1999, pp. 7–10.
* [23] K. Bergman, S. Borkar, D. Campbell, W. Carlson, W. Dally, M. Denneau, P. Franzon, W. Harrod, K. Hill, J. Hiller _et al._ , “Exascale computing study: Technology challenges in achieving exascale systems,” _Defense Advanced Research Projects Agency Information Processing Techniques Office (DARPA IPTO), Tech. Rep_ , vol. 15, 2008.
* [24] B. Schroeder, E. Pinheiro, and W.-D. Weber, “DRAM errors in the wild: a large-scale field study,” in _ACM SIGMETRICS Performance Evaluation Review_ , vol. 37, no. 1. ACM, 2009, pp. 193–204.
* [25] S. Reisch, “Gobang ist PSPACE-vollständig,” _Acta Informatica_ , vol. 13, no. 1, pp. 59–66, 1980.
* [26] R. Kern. (2007, Dec.) A simple file format for Numpy arrays. [Online]. https://github.com/numpy/numpy/blob/master/doc/neps/npy-format.txt
* [27] R. L. Dahl. (2014) Node.js: evented I/O for v8 javascript. [Online]. http://nodejs.org
* [28] J. Dean and S. Ghemawat, “MapReduce: simplified data processing on large clusters,” _Comm. of the ACM_ , vol. 51, no. 1, pp. 107–113, 2008.
* [29] R. C. Murphy, K. B. Wheeler, B. W. Barrett, and J. A. Ang, “Introducing the Graph 500,” _Cray User’s Group (CUG)_ , 2010.
|
arxiv-papers
| 2014-04-03T01:01:21 |
2024-09-04T02:50:00.626184
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Geoffrey Irving",
"submitter": "Geoffrey Irving",
"url": "https://arxiv.org/abs/1404.0743"
}
|
1404.0791
|
# Phase transition, Quasinormal modes and Hawking radiation of Schwarzschild
black hole in Quintessence field
R. [email protected] Nijo [email protected]
and V C [email protected] Department of Physics, Cochin University
of Science and Technology, Kochi 682022, India
###### Abstract
Black hole thermodynamic stability can be determined by studying the nature of
heat capacity of the system. For Schwarzschild black hole the heat capacity is
negative, but in the quintessence field this system shows a second order phase
transition, implying the existence of a stable phase. We further discuss the
equation of state of the present system. While analyzing the quasinormal
modes, we find that the massive scalar quasinormal mode frequencies in the
complex $\omega$ plane shows a dramatic change when we plot it as a
progressive function of quintessence state parameter. We also find the Hawking
temperature of the system via the method of tunneling.
###### keywords:
Phase transition, Quintessence, Quasinormal modes, Hawking radiation.
Received (Day Month Year)Revised (Day Month Year)
PACS Nos.: 04.70.Dy, 04.70.-s
## 1 Introduction
The thermodynamic properties of black holes have received considerable
attraction in recent times, as it is hoped that these studies can establish a
connection among thermodynamics, gravitation and quantum statistical mechanics
and eventually leading to quantum gravity. Since the seminal works of
Hawking[1] and Bekenstein[2, 3], it is understood that black holes behave as
thermodynamic objects, with characteristic temperature and entropy. Hawking
radiation has not been yet directly observed but the thermodynamic properties
are thoroughly understood. The realization that black hole laws are
thermodynamic in nature implies that there should be an underlying statistical
description of them in terms of some microscopic states. Black hole
thermodynamics is now widely studied. It is well known that, for Schwarzschild
black hole the heat capacity is negative and it is thermodynamically unstable.
Accelerating expansion of the universe is a most recent fascinating result of
observational cosmology. To explain the accelerated expansion of the universe,
it is proposed that the universe is regarded as being dominated by an exotic
scalar field with a large negative pressure called “dark energy” which
constitutes about 70 percent of the total energy of the universe. There are
several candidates for dark energy. “Quintessence”[4, 5] is one among them. It
is characterized by a parameter $\epsilon$, the ratio of the pressure to
energy density of the dark energy, and the value of $\epsilon$ falls in the
range $-1\leq\epsilon\leq-\frac{1}{3}$. In our previous study[6] of
Schwarzschild black hole surrounded by quintessence, we observe a second order
thermodynamic phase transition for the black hole, thus it possesses a
positive heat capacity regime and thermodynamic stability.
The study of quasinormal modes gained great attention since the existence of
QNMs was first pointed out by Vishveshwara[7] in the calculation of the
scattering of gravitational waves by a black hole. They are considered to be
the characteristic sound of the black holes. The quasi normal modes of
different black holes surrounded by quintessence have been studied earlier[8,
9]. Connection between black hole quasinormal modes and their phase transition
was studied in[11] and later it was found that the relation is not so
trivial[12]. Here we are investigating the massive scalar QNMs of the
Schwarzschild-Quintessence black hole.
It will be very interesting to study the thermal emission from the
Schwarzschild-Quintessence black hole. Several derivations of Hawking
radiation exist in literature. Parikh and Wilczek[13] put forward a semi-
classical quantum tunneling model that implemented Hawking radiation as a
tunneling process. More specifically they considered the effects of a positive
energy matter shell propagating outward through the horizon of the
Schwarzschild and Reissener-Nordstrom black holes. The back reaction[14, 15]
and noncommutative effects[16] have also been discussed by tunneling
mechanicm. We find the Blotzman factor via the method of tunneling and study
its variation with respect to the quintessence state parameter.
The paper is organized as follows. In section 2 we discuss the second order
phase transition and equation of state of the black hole. In section 3 we
calculate the QNMs of a massive scalar field and we observe a connection
between QNMs and phase transition and in section 4 we calculate the Hawking
radiation through tunneling mechanism. This paper ends with conclusion in
section 5.
## 2 Thermodynamics
### 2.1 Second order phase transition
Phase transition is an important phenomenon in thermodynamics, so it is
natural to probe the same in black hole thermodynamics. The work of Hawking
and Page[17] proved that there is a phase transition between thermal AdS state
and AdS black hole in 4 dimensions as the temperature changes. And later the
black hole phase transition has been extended and indicates that there exist
different phase transitions under various circumstances [18, 19, 20, 21, 22,
23, 24, 25]. The phase transition is always identified with the sign change of
heat capacity. Davies[26] argued that the point at which the specific heat
travels from positive to negative values through an infinite discontinuity
marks a phase transition.
The discovery of thermal emission of elementary particles by Schwarzschild
black holes has initiated deeper investigations of thermodynamic properties of
stationary, rotating and charged black holes. Those investigations studied
stable equilibrium of non-rotating black holes with a thermal radiation
bath[27, 28, 29], the fluctuation-dissipation theorem in irreversible
thermodynamics[30], the black-hole version for the third law of thermodynamics
and specific heats of black holes in thermal equilibrium[26, 27]. It has been
found that Black hole thermodynamics differs from the normal theory of
thermodynamics in a number of ways: apart from the unsolved problem of a
proper definition of stable equilibrium for Kerr black holes, Hawking[29] has
shown that black holes cannot be described by means of a canonical ensemble
(this is closely related to the fact that the black-hole entropy is a global
property, since it cannot be divided up into a number of weakly interacting
parts).
Figure 1: Variation of heat capacity with entropy and with quintessence state
parameter $\epsilon$ keeping $a=0.1$.
We are considering the Schwarzschild black hole surrounded by quintessence. In
the present study, the black hole is regarded as a thermal system and it is
then natural to apply the laws of thermodynamics. However, a crucial
difference from other thermal systems is that it is a gravitational object
whose entropy is identified with the area of the black hole(here we are using
$c=G=\hbar=1$). The metric of a Schwarzschild black hole surrounded by
quintessence[31] is given by,
$ds^{2}=f(r)dt^{2}-\frac{1}{f(r)}dr^{2}-r^{2}(d\theta^{2}+\sin\theta^{2}d\phi^{2}),$
(1)
where
$f(r)=1-\frac{2M}{r}-\frac{a}{r^{3\epsilon+1}}.$ (2)
Here M is the black hole mass and $a$ is the normalization factor, which is
positive, depending on the energy density of quintessence. Quintessence is a
scalar field whose equation of state parameter $\epsilon$ is defined as the
ratio of its pressure $p$ and its energy density $\rho$, which is given by a
kinetic term and a potential term as[4],
$\epsilon\equiv\frac{p}{\rho}=\frac{\frac{1}{2}\dot{Q}^{2}-V(Q)}{\frac{1}{2}\dot{Q}^{2}+V(Q)}$.
Following Kiselev[31], the energy density can be written as
$\rho_{\epsilon}=-\frac{a}{2}\frac{3\epsilon}{r^{3(1+\epsilon)}}$.
We can establish the relation between mass of a black hole and its horizon
radius directly from (2) as,
$M=\frac{r}{2}-\frac{a}{2r^{3\epsilon}},$ (3)
and we know that entropy can be written as
$S=\frac{A}{4}=\pi r^{2},$ (4)
so that $r$ can be written in terms of $S$ as
$r=\sqrt{\frac{S}{\pi}}.$ (5)
Let us rewrite (3) using (5) as
$M=\frac{1}{2}\left[\sqrt{\frac{S}{\pi}}-a(\frac{\pi}{S})^{\frac{3\epsilon}{2}}\right].$
(6)
Now we can deduce the heat capacity from the above expression for mass in
terms of entropy. Heat capacity in terms of entropy and quintessence parameter
is given by
$C=T\frac{\partial S}{\partial
T}=-\frac{16S^{3\epsilon+5}+96a\epsilon\pi^{\frac{3\epsilon+1}{2}}S^{\frac{3\epsilon+9}{2}}}{8S^{3\epsilon+4}+144a\epsilon^{2}\pi^{\frac{3\epsilon+1}{2}}S^{3\epsilon+2}+96a\epsilon\pi^{\frac{3\epsilon+1}{2}}S^{\frac{3\epsilon+7}{2}}}.$
(7)
In Fig.1 we have drawn the heat capacity as a three dimensional plot by
introducing the quintessence state parameter along the third axis and it is
clear that there is a second order phase transition. From the plot we can find
the critical point of phase transition for each value of quintessence state
parameter. The quintessence effect in fact makes the thermodynamically
unstable Schwarzschild system stable and changes the transition point with
respect to the state parameter. The 3 dimensional plot actually enables us to
find the dependence of heat capacity on the quintessence parameter. It is
obvious that the infinite discontinuity has not been shown for all values of
quintessence state parameters. From a certain value of quintessence parameter
onwards the phase transition behaviour begins. We could see that for the
Schwarzschild like case in the quintessence field, i.e., for
$\epsilon=-\frac{1}{3}$, the heat capacity does not show any kind of phase
transition. Thus it agrees with the existing results of Schwarzschild case.
### 2.2 Equation of state of the Black hole
Figure 2: P-V isotherms with quintessence state parameter as the third axis
The cosmological constant related term in the metric, will act as a pressure
term[33, 34]. Thus we could write
$P=-\frac{a}{8\pi},$ (8)
and the mass of the black hole, $M$ is most naturally associated with the
enthalpy $H$ of the black hole, hence
$H=E+PV.$ (9)
In black hole thermodynamics also, volume has been considered as a
thermodynamic variable[33, 35]. So we find the volume of the black hole
thermodynamically and find the equation of state. The natural variables for
enthalpy are entropy and pressure, so we could write $H$, in turn $M$, as a
function of $S$ and $P$,
$M=H(S,P).$ (10)
Now using (3) and (8), enthalpy can be written as
$H(S,P)=\frac{1}{2}\left(\frac{S}{\pi}\right)^{\frac{1}{2}}\left[1+\frac{8\pi^{\frac{3\epsilon+3}{2}}P}{S^{\frac{3\epsilon+1}{2}}}\right].$
(11)
We can find the volume of the Black hole using Legendre transformation,
$V=\left(\frac{\partial H}{\partial P}\right)_{S}=\frac{4\pi}{r^{3\epsilon}}.$
(12)
The equation of state of black hole can be written as,
$T=\frac{1}{4\pi}\left[\left(\frac{V}{4\pi}\right)^{\frac{1}{3\epsilon}}-\frac{6\epsilon
P}{(4\pi)^{\frac{2}{3\epsilon}}}V^{(1+\frac{2}{3\epsilon})}\right].$ (13)
We have plotted the P-V isotherms with the quintessence state parameter
$\epsilon$ in Fig.2.
## 3 Quasinormal modes and phase transition
The massive scalar field in a curved background is governed by the Klein-
Gordon equation:
$\Box{\Phi}-m^{2}\Phi=\frac{1}{\sqrt{-g}}(g^{\mu\nu}\sqrt{-g}\Phi_{,\mu})_{,\nu}-m^{2}\Phi=0,$
(14)
where $\Phi$ is the scalar field.
Using(1) in (14) and separating angular and time variables, we obtain the
radial equation:
$\frac{d^{2}}{dr_{*}^{2}}+[\omega^{2}-V(r)]\Phi(r)=0,$ (15)
where,
$V(r)=(1-\frac{2M}{r}-\frac{a}{r^{3\epsilon+1}})(\frac{l(l+1)}{r^{2}}+\frac{2M}{r^{3}}+\frac{a(3\epsilon+1)}{r^{3\epsilon+3}}+m^{2}),$
(16) $dr_{*}=\frac{dr}{1-\frac{2M}{r}-\frac{a}{r^{3\epsilon+1}}},$ (17)
and $l=0,1,2,3...$ parameterizes the field angular harmonic index. The
effective potential $V(r)$ approaches to a constant both at the event horizon
and at spatial infinity. It is clear that the effective potential relates to
the value of $r$, angular harmonic index $l$, the state parameter $\epsilon$,
the scalar field mass $m$, the normalization factor $a$ and the mass of the
black hole $M$. However, in this paper, we only want to investigate the
relationship between the state parameter $\epsilon$ and the scalar field mass
$m$ with the quasinormal modes. Therefore, taking $M=1$ and $a=0.1$, we
compute the quasinormal frequencies stipulated by the above potential using
the third-order WKB method developed by Schutz, Will and Iyer[36, 37, 38].
Fig.3 represents the quasinormal mode frequencies for different values of
quintessence parameter, including the Schwarzschild case for which $a=0$. The
values of QNMs for each quintessence state parameter and for different values
of ‘$m$’, is given in Tab.3. It is clear that the quintessence effect is to
shift the frequencies away from the original Schwarzschild case. Now we are
probing the QNM frequencies to get some notion about the phase transition,
which can be understood by plotting the complex frequencies with progressing
values of quintessence parameter. Tab.3 gives the QNM frequencies for
different values of quintessence parameter.
Fig.4 represents the QNM spectrum with respect to the varying quintessence
state parameter keeping mass ‘$m$’ fixed. We can now see that the value of
$\epsilon$ at which the heat capacity shows a phase transition(Fig.1)
coincides with the value of $\epsilon$ at which the QNM spectrum showing a
change in its slope. In the previous studies also, such a numerical
coincidence has been found[11]. So we conclude that there may be a connection
between thermodynamic and perturbative stabilities in the case of
Schwarzschild black hole surrounded by quintessence.
The importance of this study lies on the fact that the phase transition is
driven by the quintessence field. In the more generic case of quintessence
field, such as the Reissener-Nordström-Quintessence black hole, the phase
transition[10] is mainly driven by the charge $Q$ and the quintessence state
parameter $a$ has got least significance. Of course it can be effective, when
we use heavy quintessence field, but for any realistic case the quintessence
densities will be much lower than this. Thus, the similar study of connecting
the phase transition and QNM spectra in the Reissener-Nordström-Quintessence
black hole will not make much difference from the work[11]. Whereas in the
Schwarzschild-Quintessence black hole the system achieves the stable phase by
the presence of quintessence only. So this study is quite important to check
the influence of the quintessence field rather than that of charge.
Figure 3: Figure represents massive-scalar QNMs of Schwarzschild black hole
surrounded by Quintessence, with $l=4$, $n=0$, $a=0.1$ and we plot it for
different values of mass( $m=0.1,0.2,0.3$ etc), ($\epsilon=0$ is the
Schwarzschild case with $a=0$).
Values of the quasinormal frequencies for low overtones($n=0$) in the
Schwarzschild black hole($a=0$) and in the Schwarzschild black hole surrounded
by quintessence($a=0.1$) for fixed $l=4$. $a$ $\epsilon$ $\omega(m=0.1)$
$\omega(m=0.2)$ $\omega(m=0.3)$ $\omega(m=0.4)$ 000 00 00.869210-0.096046i
0.874830-0.094995i 0.884224-0.093232i 0.897437-0.090742i 0.100 0-0.3
00.756040-0.079890i 0.762003-0.078819i 0.771979-0.077021i 0.786025-0.074475i
0.100 0-0.4 00.707477-0.072560i 0.713355-0.071551i 0.723185-0.069858i
0.737026-0.067465i 0.100 0-0.5 00.632884-0.062241i 0.638489-0.061371i
0.647857-0.059917i 0.661032-0.057876i 0.100 0-0.6 00.508557-0.047248i
0.513416-0.046673i 0.521512-0.045727i 0.532845-0.044432i
Figure 4: Figure represents massive-scalar QNMs of Schwarzschild black hole
surrounded by Quintessence, with $l=4$, $n=0$, $a=0.1$,$m=0.4$ and we plot it
for different values of quintessence state parameter $\epsilon$. Here
$\epsilon=-0.33$ at the right extreme of the curve, $\epsilon=-0.66$ at the
turning point and it terminates at $\epsilon=-1$.
Values of the quasinormal frequencies for low overtones($n=0$) in the
Schwarzschild black hole surrounded by quintessence($a=0.1$) for fixed $l=4$
and fixed mass ($m=0.4$).
$\epsilon\hphantom{0000000000000000000000000000000}$ $\omega_{R}+i\omega_{I}$
-0.3 00000000000000000000000000000000 00.786025-0.074475i -0.4
00000000000000000000000000000000 00.737026-0.067465i -0.5
00000000000000000000000000000000 00.661032-0.057876i -0.6
00000000000000000000000000000000 00.532845-0.044432i -0.7
00000000000000000000000000000000 00.246904-0.020032i -0.8
00000000000000000000000000000000 00.043126-0.511825i -0.9
00000000000000000000000000000000 00.078842-0.841099i
## 4 Hawking radiation via tunneling
We present a short and direct derivation of Hawking radiation, considering it
as a tunneling process based on particles in a dynamical geometry for a
Schwarzschild black hole surrounded by quintessence. To describe tunneling as
an across horizon phenomena, it is necessary to choose coordinates which,
unlike Schwarzschild coordinates, are not singular at the event horizon. Thus
we rescale the time coordinate into Eddington-Finkelstein coordinates as
$t=T\pm r_{*}$, where the $+$ and $-$ represent ingoing and outgoing particles
respectively[39, 40]. The tortoise coordinate $r_{*}$ is defined as,
$\frac{dr_{*}}{dr}=f(r)^{-1}.$ (18)
In the following we study the outgoing particle only which is radiated from
the black hole. The background metric thus can be transformed to
$ds^{2}=-f(r)dT^{2}+2dTdr+r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}).$ (19)
The apparent horizon of the metric is given by the equation
$f(r)=1-\frac{2M}{r}-\frac{a}{r^{3\epsilon+1}}=0$ (20)
In the absence of quintessence $(a=0)$, this equation arrives at the solution
$r=2M$; now we consider the quintessence field strength as small and thus we
could treat the whole quintessence field as a perturbation to the original
background metric. Eventually the new horizon radius will be slightly modified
from the original horizon radius as
$R=r+\delta.$ (21)
Substituting this in (20), we obtain
$1-\frac{2M}{r}\left(1-\frac{\delta}{r}\right)-\frac{a}{r^{(3\epsilon+1)}}\left(1-(1+3\epsilon)\frac{\delta}{r}\right)=0,$
(22)
which gives us
$\delta\backsimeq\frac{a}{r^{3\epsilon}},$ (23)
in the first approximation.
The radial null geodesic is given by
$\dot{r}=\frac{dr}{dT}=\frac{1}{2}\left(1-\frac{2M}{r}-\frac{a}{r^{(3\epsilon+1)}}\right).$
(24)
When a particle of energy $E$ is radiated away from the black hole, $\dot{r}$
becomes
$\dot{r}=\frac{1}{2}\left(1-\frac{2(M-E)}{r}-\frac{a}{r^{(3\epsilon+1)}}\right).$
(25)
The imaginary part of the action is
$Im\mathcal{S}=Im\int P_{r}dr=Im\int\int
dP_{r}dr=Im\int\int\frac{dH}{\dot{r}}dr.$ (26)
where we have used the Hamilton’s equation $\frac{dH}{dp_{r}}=\dot{r}$ and
$H=M-E^{\prime}\Rightarrow dH=-dE^{\prime}$. Thus, the imaginary part of
action takes the form
$Im\mathcal{S}=Im\int_{M}^{M-E}\int\frac{2dr}{1-\frac{2M}{r}-\frac{a}{r^{3\epsilon+1}}}(-dE^{\prime}),$
(27)
$=Im\int\frac{2Erdr}{(r-R)}.$ (28)
We use the method of tunneling to evaluate the integral over $r$ and obtain
$Im\mathcal{S}=(4\pi R)E.$ (29)
Now using the WKB approximation, the rate of radiation is expressed as
$\Gamma\varpropto e^{-2Im\mathcal{S}}=e^{-\beta E}.$ (30)
where $\beta$ is,
$\beta=\frac{1}{T}=8\pi\left(r+\frac{a}{r^{3\epsilon}}\right).$ (31)
Fig.5 represents the variation of $\beta$ with respect to $r$. We can see that
for different quintessence parameter values, $\beta$ diverges as $r$
increases.
But the variation of $\beta$ with respect to the quintessence parameter
$\epsilon$ is plotted in Fig.6, in which $\beta$ increases sharply below a
particular value of $\epsilon$. This can be read along with the phase
transition behaviour, which has been obtained in both the thermodynamic and
perturbative approaches of sections 2.1 and 3.
Figure 5: Figure represents the variation of Boltzman factor with radius for
different $\epsilon$
Now we are going to find the change in entropy of the Black hole after
emitting a particle out. For that we need to take the energy conservation into
account. Then the radial null geodesic after emitting a particle of energy $E$
is given by (25).
The imaginary part of the action of the massive particle is[41]
$Im\mathcal{S}=Im\int_{t_{i}}^{t_{f}}Ldt=Im\int_{r_{ie}}^{r_{fe}}(P_{r}\dot{r})\frac{dr}{\dot{r}}=Im\int_{r_{ie}}^{r_{fe}}\left[\int_{0}^{P_{r}}\dot{r}dP^{\prime}_{r}\right]\frac{dr}{\dot{r}}.$
(32)
where $r_{ie}$ and $r_{fe}$ represent the localization of the event horizon
before and after the emission of a particle with energy $E$. $\dot{r}$ is
given from the Hamilton’s canonical equation of motion,
$\dot{r}=\frac{dH}{dP_{r}}|_{r},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}dH|_{r}=d(M-E).$
(33)
Now substituting (25) and (33) in (32) we find,
$Im\mathcal{S}=Im\int_{r_{ie}}^{r_{fe}}\int_{M}^{M-E}\frac{2\left[d(M-E^{\prime})\right]dr}{\left(1-\frac{2(M-E^{\prime})}{r}-\frac{a}{r^{(3\epsilon+1)}}\right)},$
(34)
which can be written as
$Im\mathcal{S}=Im\int_{r_{ie}}^{r_{fe}}\int_{M}^{M-E}\frac{2rdrd(M-E^{\prime})}{(r-R)},$
(35)
where $R=2(M-E)+\delta$.
Figure 6: Figure represents the variation of Boltzman factor with $\epsilon$
for different horizon radius
.
Now the second integral can be deformed as a contour, so as to ensure that
positive energy solutions decay in time. That is, we are taking the contour in
the lower $E$ plane.
Using the method of Parikh and Wilczek[13] we obtain
$Im\mathcal{S}=-Im\int_{r_{ie}}^{r_{fe}}rdr(\pi
i)=\frac{\pi}{2}(r_{ie}^{2}-r_{fe}^{2}).$ (36)
Using WKB approximation, we can get the tunneling rate of radiation as
$\Gamma\propto e^{-2Im\mathcal{S}}=e^{\pi(r_{ie}^{2}-r_{fe}^{2})}=e^{\Delta
S_{BH}},$ (37)
where $\Delta S_{BH}$ denotes the change in the Bekenstein-Hawking entropy at
the event horizon before and after the particle tunnels out. It is obvious
that the energy carried away by the tunneled particle will change the energy
of the black hole and thus the entropy of the black hole should be decreased.
In the perspective of area theorem, the tunneling of particle results in
decrease in the area as a few number of area quanta. The change in entropy
found here can be quantized and in the semi-classical approach we can see that
tunneling phenomenon and area quantization give the same results.
The calculation of Hawking temperature via the tunneling method is also
described in a more general way by Banerji et al[42], in which a general
discussion of temperature for a general static, spherically symmetric black
hole has been presented. The present expression can also be obtained from the
general expression.
## 5 Summary and conclusion
In an earlier work[6], we found that Schwarzschild black hole can have a
stable phase when it is immersed in quintessence field. Here we analyze the
second order thermodynamic phase transition in detail. We first plot the heat
capacity in 3 dimensions taking the quintessence state parameter as one of the
axes(Fig.1). From which we could find the critical point changes as $\epsilon$
changes. It is evident from the plot that for certain values of
$\epsilon$(lower values of $\epsilon$, between $-\frac{1}{3}$ to
$-\frac{2}{3}$), there is no phase transition. It is in general agreement with
the result of Schwarzschild case( i. e.,$\epsilon=-\frac{1}{3}$) that the
system does not show any phase transition.
Then we analyzed QNMs for the massive-scalar field of the same system(here we
have used the same value of $a$, which we used to find its thermodynamic phase
transition). The complex frequency plot for different values of quintessence
parameter does not give any striking evidence of the phase transition we
observed, but when we plot the imaginary frequencies as a progressing function
of quintessence parameter, we could see a turning point in the plot(Fig.4).
The value of quintessence state parameter $\epsilon$, at which the plot shows
a change in slope coincides with the value of $\epsilon$ at which the heat
capacity started showing phase transition(Fig.1). We have made a thorough
investigation on the pressure and volume of the same system, and derived the
equation of state.
We have also found the Hawking radiation via the method of tunneling for the
same system. We have plotted the Boltzman factor as a function of both horizon
radius and quintessence state parameter. The plot of $\beta$ verses $\epsilon$
also implies an indication of phase transition(Fig.6).
In summary, the present study shows that the value of quintessence state
parameter $\epsilon$ at which the heat capacity shows a phase transition
coincides with the value of $\epsilon$ at which the QNM spectrum showing a
change in its slope. In the case of Hawking radiation, the plot of $\beta$
verses $\epsilon$ also shows a significant change at the same value of
$\epsilon$. According to Berti[12], the connection between QNMs and phase
transition is not so trivial. But we could see a coincidence in the values of
quintessence state parameter in thermodynamic phase transition, complex QNM
spectrum and Hawking radiation.
## 6 Acknowledgments
TR wishes to thank UGC, New Delhi for financial support under RFSMS scheme. NV
Wishes to thank UGC, for the financial support under Kothari fellowship
scheme. VCK is thankful to UGC, New Delhi for financial support through a
Major Research Project and wishes to acknowledge Associateship of IUCAA, Pune,
India.
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|
arxiv-papers
| 2014-04-03T07:46:13 |
2024-09-04T02:50:00.637018
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Tharanath, Nijo Varghese and V C Kuriakose",
"submitter": "Tharanath R",
"url": "https://arxiv.org/abs/1404.0791"
}
|
1404.0886
|
# $(\alpha,\beta,\lambda,\delta,m,\Omega)_{p}-$Neighborhood for some families
of analytic and multivalent functions
Halit ORHAN Department of Mathematics, Faculty of Science, Ataturk
University, 25240 Erzurum, Turkey. [email protected]
###### Abstract.
In the present investigation, we give some interesting results related with
neighborhoods of $\ p-$valent functions. Relevant connections with some other
recent works are also pointed out.
###### Key words and phrases:
$p-$valent functions, inclusion relations, neighborhood properties, Salagean
differential operator, Miller and Mocanu’s lemma.
###### 1991 Mathematics Subject Classification:
30C45
## 1\. INTRODUCTION AND DEFINITIONS
Let $A$ demonstrate the family of functions $f(z)$ of the form
$f(z)=z+\sum\limits_{n=2}^{\infty}{a_{n}z^{n}}\text{ \ \ \ }$
which are analytic in the open unit disk
$\mathcal{U}=\left\\{{z\in\mathbb{C}:\left|z\right|<1}\right\\}.$
We denote by $\mathcal{A}_{p}(n)$ the class of functions $f(z)$ normalized by
(1.1) $f(z)=z^{p}+\sum\limits_{k=n}^{\infty}{a_{k+p}z^{k+p}}\text{ \ \ \
}(n,p\in\mathbb{N}:=\left\\{1,2,3,...\right\\})$
which are analytic and $p$-valent in $\mathcal{U}$.
Upon differentiating both sides of (1.1) $m$ times with respect to $z,$ we
have
(1.2)
$f^{(m)}(z)=\frac{p!}{(p-m)!}z^{p-m}+\sum\limits_{k=n}^{\infty}\frac{(k+p)!}{(k+p-m)!}{a_{k+p}z^{k+p-m}}$
$(n,p\in\mathbb{N};m\in\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\};p>m).$
We show by $\mathcal{A}_{p}(n,m)$ the class of functions of the form (1.2)
which are analytic and $p$-valent in $\mathcal{U}$.
The concept of neighborhood for $f(z)\in\mathcal{A}$ was first given by
Goodman [7]. The concept of $\delta$-neighborhoods $N_{\delta}(f)$ of analytic
functions $f(z)\in\mathcal{A}$ was first studied by Ruscheweyh [8]. Walker
[12], defined a neighborhood of analytic functions having positive real part.
Later, Owa et al.[13] generalized the results given by Walker. In 1996,
Altıntaş and Owa [14] gave $(n,\delta)$-neighborhoods for functions
$f(z)\in\mathcal{A}$ with negative coefficients. In 2007,
$(n,\delta)$-neighborhoods for $p$-valent functions with negative coefficients
were considered by Srivastava et al. [4], and Orhan [5]. Very recently, Orhan
et al.[1], introduced a new definition of $(n,\delta)$-neighborhood for
analytic functions $f(z)\in\mathcal{A}.$ Orhan et al.’s [1] results were
generalized for the functions $f(z)\in\mathcal{A}$ and
$f(z)\in\mathcal{A}_{p}(n)$ by many author (see, [6, 9, 10, 15]).
In this paper, we introduce the neighborhoods
$(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g)$ and
$(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-M(g)$ of a function $f^{(m)}(z)$
when $f(z)\in\mathcal{A}_{p}(n).$
Using the Salagean derivative operator [3]; we can write the following
equalities for the function $f^{(m)}(z)$ given by
$D^{0}f^{(m)}(z)=f^{(m)}(z)$
$D^{1}f^{(m)}(z)=\frac{z}{(p-m)}\left(f^{(m)}(z)\right)^{\prime}=\frac{p!}{(p-m)!}z^{p-m}+\sum\limits_{k=n}^{\infty}\frac{(k+p-m)(k+p)!}{(p-m)(k+p-m)!}{a_{k+p}z^{k+p-m}}$
$D^{2}f^{(m)}(z)=D(Df^{(m)}(z))=\frac{p!}{(p-m)!}z^{p-m}+\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{2}(k+p)!}{(p-m)^{2}(k+p-m)!}{a_{k+p}z^{k+p-m}}$
$.........................................................$
$D^{\Omega}f^{(m)}(z)=D(D^{\Omega-1}f^{(m)}(z))=\frac{p!}{(p-m)!}z^{p-m}+\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega}(k+p)!}{(p-m)^{\Omega}(k+p-m)!}{a_{k+p}z^{k+p-m}.}$
We define $\tciFourier:\mathcal{A}_{p}(n,m)\rightarrow\mathcal{A}_{p}(n,m)$
such that
$\tciFourier(f^{(m)}(z))=(1-\lambda)\left(D^{\Omega}f^{(m)}(z)\right)+\frac{\lambda
z}{(p-m)}\left(D^{\Omega}f^{(m)}(z)\right)^{\prime}\text{ }$
(1.3)
$=\frac{p!}{(p-m)!}z^{p-m}+\sum\limits_{k=n}^{\infty}\frac{(k+p)!(k+p-m)^{\Omega}(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}a_{k+p}z^{k+p-m}$
$\text{\ }(0\leq\lambda\leq 1;\text{ }\Omega,m\in\mathbb{N}_{0};\text{ }p>m).$
Let $\tciFourier(\lambda,m,\Omega)$ denote class of functions of the form
(1.3) which are analytic in $\mathcal{U}$.
For $f,g\in\tciFourier(\lambda,m,\Omega)$, $f$ said to be
$(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-$neighborhood for $g$ if it
satisfies
$\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f^{\left(m\right)}(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m-1}}\right|<\delta\text{
\ \ }(z\in\mathcal{U)}$
for some $-\pi\leq\alpha-\beta\leq\pi$ and
${\delta>}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}.$ We show this
neighborhood by $(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g).$
Also, we say that $f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-M(g)$ if
it satisfies
$\left|\frac{e^{i\alpha}\tciFourier(f^{\left(m\right)}(z))}{z^{p-m}}-\frac{e^{i\beta}\tciFourier(g^{(m)}(z))}{z^{p-m}}\right|<\delta\text{
\ \ }(z\in\mathcal{U)}$
for some $-\pi\leq\alpha-\beta\leq\pi$ and
${\delta>}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}.$
We give some results for functions belonging to
$(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g)$ and
$(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-M(g).$
## 2\. Main Results
Now we can establish our main results.
###### Theorem 2.1.
If $f\in\tciFourier(\lambda,m,\Omega)$ satisfies
$\ \ \ \sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}{a_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|$
(2.1) ${\leq}{\delta-}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}$
for some $-\pi\leq\alpha-\beta\leq\pi;$ $p>m$ and
${\delta>}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]},$ then $\
f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g).$
###### Proof.
By virtue of (1.3), we can write
$\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m-1}}\right|=$
$\displaystyle\left|\frac{p!(p-m)}{(p-m)!}e^{i\alpha}+e^{i\alpha}\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}{a_{k+p}z^{k}-}\frac{p!(p-m)}{(p-m)!}e^{i\beta}\right.$
$\displaystyle\left.-e^{i\beta}\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}b{{}_{k+p}z^{k}}\right|$ $\displaystyle<$
$\displaystyle\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}$
$\displaystyle+\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}{a_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|.$
If
$\ \ \sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}{a_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|$
${\leq}{\delta-}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]},$
then we observe that
$\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m-1}}\right|<\delta\text{
\ \ }(z\in\mathcal{U)}\mathit{.}$
Thus, $f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g).$ This evidently
completes the proof of Theorem 2.1.
###### Remark 2.2.
In its special case when
(2.2) $m=\Omega=\lambda=\alpha=0\text{ and }\beta=\alpha,$
in Theorem 2.1 yields a result given earlier by Altuntaş et al. ([9] p.3,
Theorem 1).
We give an example for Theorem 2.1.
###### Example 2.1.
For given
$g(z)=\frac{p!}{(p-m)!}z^{p-m}+\sum\limits_{k=n}^{\infty}B_{k+p}(\alpha,\beta,\lambda,m,{\delta,\Omega)}z^{k+p-m}\in\tciFourier(\lambda,m,\Omega)$
$\text{\ }(n,p\in\mathbb{N}=\left\\{1,2,3,...\right\\};\text{ }p>m;\text{
}\Omega,m\in\mathbb{N}_{0})$
we consider
$f(z)=\frac{p!}{(p-m)!}z^{p-m}+\sum\limits_{k=n}^{\infty}A_{k+p}(\alpha,\beta,\lambda,m,{\delta,\Omega)}z^{k+p-m}\in\tciFourier(\lambda,m,\Omega)$
$(n,p\in\mathbb{N}=\left\\{1,2,3,...\right\\};\text{ }p>m;\text{
}\Omega,m\in\mathbb{N}_{0})$
with
$A_{k+p}=\frac{(p-m)^{\Omega}\\{{\delta-}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}\\}(k+p-m)!(n+p-1)}{(1+\lambda
k(p-m)^{-1})(k+p-m)^{\Omega+1}(k+p-1)!(k+p)^{2}(k+p-1)}e^{-i\alpha}+{e}^{i(\beta-\alpha)}B{{}_{k+p}.}$
Then we have that
$\ \sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}A{{}_{k+p}-e}^{i\beta}B{{}_{k+p}}\right|{=}$
(2.3)
$(n+p-1)\left({\delta-}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}\right)\sum\limits_{k=n}^{\infty}\frac{1}{(k+p-1)(k+p)}.$
Finally, in view of the telescopic series, we obtain
$\displaystyle\sum\limits_{k=n}^{\infty}\frac{1}{(k+p-1)(k+p)}$
$\displaystyle=$
$\displaystyle\underset{\zeta\longrightarrow\infty}{\lim}\sum\limits_{k=n}^{\zeta}\left[\frac{1}{k+p-1}-\frac{1}{k+p}\right]$
$\displaystyle=$
$\displaystyle\underset{\zeta\longrightarrow\infty}{\lim}\left[\frac{1}{n+p-1}-\frac{1}{\zeta+p}\right]$
$\displaystyle=$ $\displaystyle\frac{1}{n+p-1}.$
Using (2.4) in (2.3), we get
$\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}{A_{k+p}-e}^{i\beta}B{{}_{k+p}}\right|$
${=}{\delta-}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}.$
Therefore, we say that
$f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g).$
Also, Theorem 1 gives us the following corollary.
###### Corollary 2.3.
If $f\in\tciFourier(\lambda,m,\Omega)$ satisfies
$\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|\left|a{{}_{k+p}}\right|{-}\left|b{{}_{k+p}}\right|\right|$
${\leq\delta-}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}$
for some $-\pi\leq\alpha-\beta\leq\pi$ and
${\delta>}\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]},$ and
$\arg(a_{k+p})-\arg(b_{k+p})=\beta-\alpha$
$(n,p\in\mathbb{N}=\left\\{1,2,3,...\right\\};$ $m\in\mathbb{N}_{0},$ $p>m),$
then $\ f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g).$
###### Proof.
By theorem (2.1), we see the inequality (2.1) which implies that
$f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g).$
Since $\arg(a_{k+p})-\arg(b_{k+p})=\beta-\alpha,$ if
$\arg(a_{k+p})=\alpha_{k+p},$ we see
$\arg(b_{k+p})=\alpha_{k+p}-\beta+\alpha.$ Therefore,
$e^{i\alpha}a_{k+p}-e^{i\beta}b_{k+p}=e^{i\alpha}\left|a_{k+p}\right|e^{i\alpha_{k+p}}-e^{i\beta}\left|b_{k+p}\right|e^{i(\alpha_{k+p}-\beta+\alpha)}=(\left|a_{k+p}\right|-\left|b_{k+p}\right|)e^{i(\alpha_{k+p}+\alpha)}$
implies that
(2.5)
$\left|e^{i\alpha}a_{k+p}-e^{i\beta}b_{k+p}\right|=\left|\left|a_{k+p}\right|-\left|b_{k+p}\right|\right|.$
Using (2.5) in (2.1) the proof of the corollary is complete.
Next, we can prove the following theorem.
###### Theorem 2.4.
If $f\in\tciFourier(\lambda,m,\Omega)$ satisfies
$\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|\leq\delta-\frac{p!}{(p-m)!}\sqrt{2[1-\cos(\alpha-\beta)]}\text{
\ \ }\left(z\in\mathcal{U}\right).$
for some $-\pi\leq\alpha-\beta\leq\pi;$ $p>m$ and
${\delta>}\frac{p!}{(p-m)!}\sqrt{2[1-\cos(\alpha-\beta)]}$ then
$f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-M(g).$
The proof of this teorem is similar with Theorem 2.1.
###### Corollary 2.5.
If $f\in\tciFourier(\lambda,m,\Omega)$ satisfies
$\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|\left|a{{}_{k+p}}\right|{-}\left|b{{}_{k+p}}\right|\right|\leq\delta-\frac{p!}{(p-m)!}\sqrt{2[1-\cos(\alpha-\beta)]}\text{
\ \ }\left(z\in\mathcal{U}\right).$
for some $-\pi\leq\alpha-\beta\leq\pi;$ $p>m$ and
${\delta>}\frac{p!}{(p-m)!}\sqrt{2[1-\cos(\alpha-\beta)]}$ and
$\arg(a_{k+p})-\arg(b_{k+p})=\beta-\alpha,$ then
$f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-M(g).$
Our next result as follows.
###### Theorem 2.6.
If
$f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g),0\leq\alpha<\beta\leq\pi;$
$p>m$ and $\arg(e^{i\alpha}a_{k+p}-e^{i\beta}b_{k+p})=k\phi,$ then
$\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|{\leq\delta-}\frac{p!}{(p-m-1)!}(\cos\alpha-\cos\beta).$
###### Proof.
For $f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-N(g),$ we have
$\displaystyle\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f^{\left(m\right)}(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m-1}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{p!(e^{i\alpha}-e^{i\beta})}{(p-m-1)!}+\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}(e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p})}z^{k}\right|$
$=\left|\frac{p!(e^{i\alpha}-e^{i\beta})}{(p-m-1)!}+\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}(e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p})}e^{ik\phi}z^{k}\right|<\delta.$
Let us consider $z$ such that $\arg z=-\phi.$ Then
$z^{k}=\left|z\right|^{k}e^{-ik\phi}.$ For such a point $z\in\mathcal{U},$ we
see that
$\displaystyle\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g(z))}{z^{p-m-1}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{p!(e^{i\alpha}-e^{i\beta})}{(p-m-1)!}+\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|\left|z\right|^{k}\right|$
$=\left[\left(\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|\left|z\right|^{k}+\frac{p!(\cos\alpha-\cos\beta)}{(p-m-1)!}\right)^{2}\right.$
$+\left.\left(\frac{p!(\sin\alpha-\sin\beta)}{(p-m-1)!}\right)^{2}\right]^{\frac{1}{2}}<\delta.$
This implies that
$\left(\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|\left|z\right|^{k}+\frac{p!(\cos\alpha-\cos\beta)}{(p-m-1)!}\right)^{2}\text{
}<\delta^{2},$
or
$\frac{p!}{(p-m-1)!}(\cos\alpha-\cos\beta)+\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|\left|z\right|^{k}<\delta$
for $z\in\mathcal{U}$. Letting $\left|z\right|\longrightarrow 1^{-},$ we have
that
$\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega+1}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|\leq\delta-\frac{p!}{(p-m-1)!}(\cos\alpha-\cos\beta).$
###### Remark 2.7.
Applying the parametric substitutions listed in (2.2), Theorem 2.4 and 2.6
would yield a set of known results due to Altuntaş et al. ([9] p.5, Theorem 4;
p.6, Theorem 7).
###### Theorem 2.8.
If
$f\in(\alpha,\beta,\lambda,m,{\delta,\Omega)}_{p}-M(g),0\leq\alpha<\beta\leq\pi$
and $\arg(e^{i\alpha}a_{k+p}-e^{i\beta}b_{k+p})=k\phi,$ then
$\sum\limits_{k=n}^{\infty}\frac{(k+p-m)^{\Omega}(k+p)!(1+\lambda
k(p-m)^{-1})}{(p-m)^{\Omega}(k+p-m)!}\left|e^{i\alpha}a{{}_{k+p}-e}^{i\beta}b{{}_{k+p}}\right|\leq\delta+\frac{p!}{(p-m-1)!}(\cos\beta-\cos\alpha).$
The proof of this theorem is similar with Theorem 2.6.
###### Remark 2.9.
Taking $\lambda=\alpha=\Omega=m=0,$ $\beta=\alpha$ and $p=1,$in Theorem 2.8,
we arrive at the following Theorem due to Orhan et al.[1].
###### Theorem 2.10.
If $f\in(\alpha,\delta)-N(g)$ and $\arg(a_{n}-e^{i\alpha}b_{n})=(n-1)\varphi$
$\ (n=2,3,4,...),$ then
$\sum\limits_{n=2}^{\infty}n\left|a_{n}{-e}^{i\alpha}b{{}_{n}}\right|\leq\delta+\cos\alpha-1.$
We give an application of following lemma due to Miller and Mocanu [2] (see
also, [11]).
###### Lemma 2.1.
Let the function
$w(z)=b_{n}z^{n}+b_{n+1}z^{n+1}+b_{n+2}z^{n+2}+...\text{ \ \ \
}(n\in\mathcal{U)}$
be regular in $\mathcal{U}$ with $w(z)\neq 0,$ $(n\in\mathcal{U)}$. If
$z_{0}=r_{0}e^{i\theta_{0}}$ $(r_{0}<1)$ and
$\left|w(z_{0})\right|=\max_{\left|z\right|\leq r_{0}}\left|w(z)\right|,$ then
$z_{0}w^{\prime}(z_{0})=qw(z_{0})$ where $q$ is real and $q\geq n\geq 1.$
Applying the above lemma, we derive
###### Theorem 2.11.
If $f\in\tciFourier(\lambda,m,\Omega)$ satisfies
$\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m-1}}\right|<\delta(p+n-m)-\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}$
for some $-\pi\leq\alpha-\beta\leq\pi;$ $p>m$ and
${\delta>}\left(\frac{p!}{(p+n-m)(p-m-1)!}\right)\sqrt{2[1-\cos(\alpha-\beta)]},$
then
$\left|\frac{e^{i\alpha}\tciFourier(f^{(m)}(z))}{z^{p-m}}-\frac{e^{i\beta}\tciFourier(g^{(m)}(z))}{z^{p-m}}\right|<\delta+\frac{p!}{(p-m)!}\sqrt{2[1-\cos(\alpha-\beta)]}\
\ \left(z\in\mathcal{U}\right).$
###### Proof.
Let us define $w(z)$ by
(2.6)
$\frac{e^{i\alpha}\tciFourier(f^{(m)}(z))}{z^{p-m}}-\frac{e^{i\beta}\tciFourier(g^{(m)}(z))}{z^{p-m}}=\frac{p!}{(p-m)!}(e^{i\alpha}{-e}^{i\beta})+\delta
w(z).$
Then $w(z)$ is analytic in $\mathcal{U}$ and $w(0)=0.$ By logarithmic
differentiation, we get from (2.6) that
$\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))-e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{e^{i\alpha}\tciFourier(f^{(m)}(z))-e^{i\beta}\tciFourier(g^{(m)}(z))}-\frac{p-m}{z}=\frac{\delta
w\prime(z)}{\frac{p!}{(p-m)!}(e^{i\alpha}{-e}^{i\beta})+\delta w(z)}.$
Since
$\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))-e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m}\left(\frac{p!}{(p-m)!}(e^{i\alpha}{-e}^{i\beta})+\delta
w(z)\right)}=\frac{p-m}{z}+\frac{\delta
w\prime(z)}{\frac{p!}{(p-m)!}(e^{i\alpha}{-e}^{i\beta})+\delta w(z)},$
we see that
$\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m-1}}=\frac{p!}{(p-m-1)!}(e^{i\alpha}{-e}^{i\beta})+\delta
w(z)\left(p-m+\frac{zw\prime(z)}{w(z)}\right).$
This implies that
$\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m-1}}\right|=\left|\frac{p!}{(p-m-1)!}(e^{i\alpha}{-e}^{i\beta})+\delta
w(z)\left(p-m+\frac{zw\prime(z)}{w(z)}\right)\right|.$
We claim that
$\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))}{z^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z^{p-m-1}}\right|<\delta(p-m+n)-\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}$
in $\mathcal{U}$.
Otherwise, there exists a point $z_{0}\in\mathcal{U}$ such that
$z_{0}w^{\prime}(z_{0})=qw(z_{0})$ (by Miller and Mocanu’s Lemma) where
$w(z_{0})=e^{i\theta}$ and $q\geq n\geq 1.$
Therefore, we obtain that
$\displaystyle\left|\frac{e^{i\alpha}\tciFourier^{\prime}(f^{(m)}(z))}{z_{0}^{p-m-1}}-\frac{e^{i\beta}\tciFourier^{\prime}(g^{(m)}(z))}{z_{0}^{p-m-1}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{p!}{(p-m-1)!}(e^{i\alpha}{-e}^{i\beta})+\delta
e^{i\theta}\left(p-m+q\right)\right|$ $\displaystyle\geq$
$\displaystyle\delta\left(p+q-m\right)-\left|\frac{p!}{(p-m-1)!}(e^{i\alpha}{-e}^{i\beta})\right|$
$\displaystyle\geq$
$\displaystyle\delta\left(p+n-m\right)-\frac{p!}{(p-m-1)!}\sqrt{2[1-\cos(\alpha-\beta)]}.$
This contradicts our condition in Theorem 2.11.
Hence, there is no $z_{0}\in\mathcal{U}$ such that $\left|w(z_{0})\right|=1.$
This means that $\left|w(z)\right|<1$ for all$\ z\in\mathcal{U}.$
Thus, have that
$\displaystyle\left|\frac{e^{i\alpha}\tciFourier(f^{(m)}(z))}{z^{p-m}}-\frac{e^{i\beta}\tciFourier(g^{(m)}(z))}{z^{p-m}}\right|$
$\displaystyle=$
$\displaystyle\left|\frac{p!}{(p-m)!}(e^{i\alpha}{-e}^{i\beta})+\delta
w(z)\right|$ $\displaystyle\leq$
$\displaystyle\frac{p!}{(p-m)!}\left|e^{i\alpha}{-e}^{i\beta}\right|+\delta\left|w(z)\right|$
$\displaystyle<$
$\displaystyle\delta+\frac{p!}{(p-m)!}\sqrt{2[1-\cos(\alpha-\beta)]}.$
Upon setting $m=0,$ $\alpha=\varphi,\wp=\tciFourier$ and $\beta=\alpha$ in
Theorem 2.11, we have the following corollary given by Sağsöz et al.[6].
###### Corollary 2.12.
If $f\in\wp(\Omega,\lambda)$ satisfies
$\left|\frac{e^{i\alpha}\wp^{\prime}(f(z))}{z^{p-1}}-\frac{e^{i\beta}\wp^{\prime}(g(z))}{z^{p-1}}\right|<\delta(p+n)-p\sqrt{2[1-\cos(\varphi-\alpha)]}$
for some $-\pi\leq\alpha-\beta\leq\pi;$and
${\delta>}\left(\frac{p}{(p+n)}\right)\sqrt{2[1-\cos(\alpha-\beta)]},$ then
$\left|\frac{e^{i\alpha}\wp(f(z))}{z^{p}}-\frac{e^{i\beta}\wp(g(z))}{z^{p}}\right|<\delta+\sqrt{2[1-\cos(\varphi-\alpha)]}\text{
\ \ }\left(z\in\mathcal{U}\right).$
## References
* [1] H. Orhan, E. Kadıoğlu and S. Owa, ($\alpha,\delta)-$Neighborhood for certain analytic functions, International Symposium on Geometric Function Theory and Applications (Edited by S. Owa and Y. Polatoğlu), T. C. Istanbul Kultur University Publ., 2008, 207-213.
* [2] Miller, S.S and Mocanu, P.T, Second order differential inequalities in the complex plane, J. Math. Anal.Appl., 65, 289-305, (1978).
* [3] G. S. Sălăgean, Subclasses of univalent functions, Complex analysis –Proc. 5th Rom.-Finn. Semin., Bucharest 1981, Part 1, Lect. Notes Math. 1013 (1983) 362-372.
* [4] H. M. Srivastava and H. Orhan, Coefficient inequalities and inclusion relations for some families of analytic and multivalent functions, Appl. Math. Lett., 20 (2007), 686-691.
* [5] H. Orhan, Neighborhoods of a certain class of $p$-valent functions with negative coefficients defined by using a differential operator, Math. Ineq. Appl. Vol. 12, Number 2, (2009), 335-349.
* [6] F. Sağsöz, and M. Kamali, $(\varphi,\alpha,\delta,\lambda,\Omega)_{p}$-Neighborhood for some classes of multivalent functions, J. Ineq. Appl., (2013), 2013:152.
* [7] Goodman, A.W, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc.,8, 598-601, (1957).
* [8] Ruscheweyh, S., Neighborhoods of univalent functions,Proc. Amer. Math. Soc., 81, 521-527, (1981).
* [9] F. Altuntaş, S. Owa and M. Kamali, $(\alpha,\delta)_{p}$-Neighborhood for certain class of multivalent functions, PanAmer. Math. J., Vol. 19, (2009), No. 235-46.
* [10] Frasin, B. A., $(\alpha,\beta,\delta)$-Neighborhood for certain analytic functions with negative coefficients, Eur. J. Pure Appl.Math., 4(1), 14-19 (2011).
* [11] I. S. Jack, Functions starlikenes and convex of order $\alpha,$ J. London Math. Soc. 2(3) (1971), 469-474.
* [12] Walker, J. B., A note on neighborhoods of analytic functions having positive real part, Int. J. Math. Math. Sci., 13, 425–430, (1990).
* [13] Owa, S., Saitoh, H. and Nunokawa, M., Neighborhoods of certain analytic functions, Appl. Math. Lett., 6, 73-77, (1993).
* [14] Altıntaş, O. and Owa, S., Neighborhood for certain analytic functions with negative coefficients, Int. J. Math. Math. Sci. 19, 797-800, (1996).
* [15] Kugita, K., Kuroki, K. and Owa, S., $(\alpha,\beta)$-Neighborhood for functions associated with Salagean differential operator and Alexander integral operator, Int. J. Math. Anal. Vol. 4, 2010, No. 5, 211-220.
|
arxiv-papers
| 2014-04-03T12:49:35 |
2024-09-04T02:50:00.646112
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Halit Orhan",
"submitter": "Halit Orhan",
"url": "https://arxiv.org/abs/1404.0886"
}
|
1404.0931
|
# The Minimal Total Irregularity of Graphs111Research supported by the
Zhujiang Technology New Star Foundation of Guangzhou (No.2011J2200090), and
Program on International Cooperation and Innovation, Department of Education,
Guangdong Province (No.2012gjhz0007).
Yingxue [email protected] Lihua You333Corresponding author:
[email protected] Jieshan [email protected]
(School of Mathematical Sciences, South China Normal University, Guangzhou,
510631, P.R. China
)
###### Abstract
In [2], Abdo and Dimitov defined the total irregularity of a graph $G=(V,E)$
as
$\rm irr_{t}$$(G)=\frac{1}{2}\sum_{u,v\in V}|d_{G}(u)-d_{G}(v)|,$
where $d_{G}(u)$ denotes the vertex degree of a vertex $u\in V$. In this
paper, we investigate the minimal total irregularity of the connected graphs,
determine the minimal, the second minimal, the third minimal total
irregularity of trees, unicyclic graphs, bicyclic graphs on $n$ vertices, and
propose an open problem for further research. Keywords: total irregularity;
minimal; tree; unicyclic graph; bicyclic graph.
## 1 Introduction
Let $G=(V,E)$ be a simple undirected graph with vertex set $V$ and edge set
$E$. For any vertices $v\in V$, the degree of a vertex $v$ in $G$, denoted by
$d_{G}(v)$, is the number of edges of $G$ incident with $v$. If
$V=\\{v_{1},v_{2},\ldots,v_{n}\\}$, then the sequence
$(d_{G}(v_{1}),d_{G}(v_{2}),\ldots,d_{G}(v_{n}))$ is called a degree sequence
of $G$ ([1]). Without loss of generality, we assume $d_{G}(v_{1})\geq
d_{G}(v_{2})\geq\ldots\geq d_{G}(v_{n})$.
A graph is regular if all its vertices have the same degree, otherwise it is
irregular. Several approaches that characterize how irregular a graph is have
been proposed. In [3], Alberson defined the imbalance of an edge $e=uv\in E$
as $|d_{G}(u)-d_{G}(v)|$ and the irregularity of $G$ as
$\rm irr$$(G)=\sum\limits_{uv\in E}|d_{G}(u)-d_{G}(v)|.$ $(1)$
More results on the imbalance, the irregularity of a graph $G$ can be found in
[3]-[6].
Inspired by the structure and meaning of the equation (1), Abdo and Dimitov
[2] introduced a new irregularity measure, called the total irregularity. For
a graph $G$, it is defined as
$\rm irr_{t}$$(G)=\frac{1}{2}\sum\limits_{u,v\in V}|d_{G}(u)-d_{G}(v)|.$ $(2)$
Although the two irregularity measures capture the irregularity only by a
single parameter, namely the degree of a vertex, the new measure is more
superior than the old one in some aspects. For example, (2) has an expected
property of an irregularity measure that graphs with the same degree sequences
have the same total irregularity, while (1) does not have. Both measures also
have common properties, including that they are zero if and only if $G$ is
regular.
Obviously, $\rm irr_{t}$$(G)$ is an upper bound of $\rm irr$$(G)$. In [7], the
authors derived relation between $\rm irr_{t}$$(G)$ and $\rm irr$$(G)$ for a
connected graph $G$ with $n$ vertices, that is, $\rm irr_{t}$$(G)\leq
n^{2}$$\rm irr$$(G)/4.$ Furthermore, they showed that $\rm
irr_{t}$$(T)\leq(n-2)$$\rm irr$$(T)$ for any tree $T$.
Let $P_{n}$, $C_{n}$ and $S_{n}$ be the path, cycle and star on $n$ vertices,
respectively. In [2], the authors obtained the upper bound of the total
irregularity among all graphs on $n$ vertices, and they showed the star graph
$S_{n}$ is the tree with the maximal total irregularity among all trees on $n$
vertices.
###### Theorem 1.
([2]) Let $G$ be a simple, undirected graph on $n$ vertices. Then
(1) $\rm irr_{t}$$(G)\leq\frac{1}{12}(2n^{3}-3n^{2}-2n+3).$
(2) If $G$ is a tree, then $\rm irr_{t}$$(G)\leq(n-1)(n-2),$ with equality
holds if and only if $G\cong S_{n}$.
In [8], the authors investigated the total irregularity of unicyclic graphs,
and determined the graph with the maximal total irregularity $n^{2}-n-6$ among
all unicyclic graphs on $n$ vertices. In [9], the authors investigated the
total irregularity of bicyclic graphs, and determined the graph with the
maximal total irregularity $n^{2}+n-16$ among all bicyclic graphs on $n$
vertices.
Recently, Abdo and Dimitrov ([10]) also obtained the upper bounds on the total
irregularity of graphs under several graph operations including join,
lexicographic product, Cartesian product, strong product, direct product,
corona product, disjunction and symmetric difference and so on.
In this paper, we introduce an important transformation to investigate the
minimal total irregularity of graphs in Section 2, determine the minimal, the
second minimal, the third minimal total irregularity of trees, unicyclic
graphs, bicyclic graphs on $n$ vertices in Sections 3-5, and propose an open
problem for further research.
## 2 Branch-transformation
In this section, we introduce an important transformation to investigate the
minimal total irregularity of graphs on $n$ vertices.
Let $G$ be a graph on $n$ vertices, $T$ be an induced subtree of $G$. We call
$T$ is a hanging tree of $G$ if $G$ can be formed by connecting a vertex of
$T$ and a vertex of $G-T$.
Branch-transformation: Let $G$ be a simple graph with at least two pendent
vertices. Without loss of generality, let $u$ be a vertex of $G$ with
$d_{G}(u)\geq 3$, $T$ be a hanging tree of $G$ connecting to $u$ with
$|V(T)|\geq 1$, and $v$ be a pendent vertex of $G$ with $v\notin T$. Let
$G^{\prime}$ be the graph obtained from $G$ by deleting $T$ from vertex $u$
and attaching it to vertex $v$. We call the transformation from $G$ to
$G^{\prime}$ is a branch-transformation on $G$ from vertex $u$ to vertex $v$
(see Figure 1).
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\put(150.0,50.0){\circle*{2.0}} \put(110.0,50.0){\circle{40.0}}
\put(166.0,50.0){\circle{30.0}} \put(70.0,50.0){\line(1,0){20.0}}
\put(130.0,50.0){\line(1,0){20.0}} \put(105.0,45.0){$G_{0}$}
\put(65.0,53.0){$v$} \put(131.0,53.0){$u$} \put(160.0,45.0){$T$}
\par\put(195.0,50.0){$\Longrightarrow$} \par\put(260.0,50.0){\circle*{2.0}}
\put(280.0,50.0){\circle*{2.0}} \put(300.0,50.0){\circle*{2.0}}
\put(340.0,50.0){\circle*{2.0}} \put(320.0,50.0){\circle{40.0}}
\put(244.0,50.0){\circle{30.0}} \put(260.0,50.0){\line(1,0){20.0}}
\put(280.0,50.0){\line(1,0){20.0}} \put(315.0,45.0){$G_{0}$}
\put(280.0,53.0){$v$} \put(341.0,53.0){$u$} \put(240.0,45.0){$T$}
\put(110.0,15.0){$G$} \put(280.0,15.0){$G^{\prime}$} \par\put(70.0,0.0){Figure
1. branch-transformation on $G$ from $u$ to $v$} \par\end{picture}
$G_{0}$$v$$u$$T$
$\Longrightarrow$
$G_{0}$$v$$u$$T$$G$$G^{\prime}$
Figure 1. branch-transformation on $G$ from $u$ to $v$
###### Lemma 2.
Let $G^{\prime}$ be the graph obtained from $G$ by branch-transformation from
$u$ to $v$. Then $\rm irr_{t}$$(G)>\rm irr_{t}$$(G^{\prime}).$
###### Proof.
Let $G=(V,E)$, $V_{1}=${$w|d_{G}(w)\geq d_{G}(u),w\in V$},
$V_{2}=${$w|d_{G}(w)=1,w\in V$}, $V_{3}=\\{w|2\leq d_{G}(w)<d_{G}(u),w\in
V\\}.$ Clearly, $u\in V_{1}$, $v\in V_{2}$, and $V_{1}\cup V_{2}\cup V_{3}=V$.
Let $|V_{1}|=s$, $|V_{2}|=h$, $|V_{3}|=r$, then $s\geq 1$, $h\geq 2$ and
$s+h+r=n$.
Note that after banch-transformation, only the degrees of $u$ and $v$ have
been changed, namely, $d_{G^{\prime}}(u)=d_{G}(u)-1$,
$d_{G^{\prime}}(v)=d_{G}(v)+1=2$ and $d_{G^{\prime}}(x)=d_{G}(x)$ for any
$x\in V\backslash\\{u,v\\}$. Let $U=V\backslash\\{u,v\\}$. Then
$|d_{G^{\prime}}(u)-d_{G^{\prime}}(v)|-|d_{G}(u)-d_{G}(v)|=-2,$
$\sum\limits_{x\in
U}(|d_{G^{\prime}}(u)-d_{G^{\prime}}(x)|-|d_{G}(u)-d_{G}(x)|)=(s-1)-(r+h-1)=s-r-h,$
$\sum\limits_{x\in
U}(|d_{G^{\prime}}(v)-d_{G^{\prime}}(x)|-|d_{G}(v)-d_{G}(x)|)=-(s-1)-r+(h-1)=-s-r+h.$
Thus, we have
$\rm irr_{t}$$(G^{\prime})-$$\rm irr_{t}$$(G)$
$=|d_{G^{\prime}}(u)-d_{G^{\prime}}(v)|+\sum\limits_{x\in
U}|d_{G^{\prime}}(u)-d_{G^{\prime}}(x)|+\sum\limits_{x\in
U}|d_{G^{\prime}}(v)-d_{G^{\prime}}(x)|$
$-(|d_{G}(u)-d_{G}(v)|+\sum\limits_{x\in
U}|d_{G}(u)-d_{G}(x)|+\sum\limits_{x\in U}|d_{G}(v)-d_{G}(x)|)$
$=(|d_{G^{\prime}}(u)-d_{G^{\prime}}(v)|-|d_{G}(u)-d_{G}(v)|)$
$+\sum\limits_{x\in
U}(|d_{G^{\prime}}(u)-d_{G^{\prime}}(x)|-|d_{G}(u)-d_{G}(x)|)$
$+\sum\limits_{x\in
U}(|d_{G^{\prime}}(v)-d_{G^{\prime}}(x)|-|d_{G}(v)-d_{G}(x)|)$
$=-2+(s-r-h)+(-s-r+h)$
$=-2r-2$
$<0$. ∎
###### Remark 3.
Let $G^{\prime}$ be the graph obtained from $G$ by branch-transformation from
$u$ to $v$. Then by branch-transformation and Lemma 2, we have
$d_{G^{\prime}}(u)=d_{G}(u)-1\geq 2$ and $d_{G^{\prime}}(v)=d_{G}(v)+1=2$,
namely, $|\\{w|d_{G^{\prime}}(w)=1,w\in V\\}|=|\\{w|d_{G}(w)=1,w\in V\\}|-1$.
If $d_{G^{\prime}}(u)\geq 3$, $G^{\prime}$ has at least two pendent vertices,
and there exists a hanging tree of $G^{\prime}$ connecting to the vertex $u$,
we can repeat branch-transformation on $G^{\prime}$ from the vertex $u$, till
the degree of $u$ is equal to 2, or there is only one pendent vertex in the
resulting graph, or there is not any hanging tree connecting to the vertex
$u$.
From the above arguments, we see that we can do branch-transformation on $G$
if and only if the following three conditions hold:
(1) there exists a vertex $u$ with $d_{G}(u)\geq 3$;
(2) there exists a hanging tree of $G$ connecting to the vertex $u$;
(3) $G$ has at least two pendent vertices.
## 3 The minimal total irregularity of trees
In this section, we determine the minimal, the second minimal, the third
minimal total irregularity of trees on $n$ vertices and characterize the
extremal graphs.
###### Lemma 4.
([1]) Let $G=(V,E)$ be a graph and $|E|=m$. Then $\sum\limits_{v\in
V}d_{G}(v)=2m$.
Let $G=(V,E)$ be a tree. Then for any vertex $u\in V$, $d_{G}(u)\geq 2$
implies there must exist a hanging tree of $G$ connecting to the vertex $u$,
thus we can obtain the following results by branch-transformation.
###### Theorem 5.
Let $G=(V,E)$ be a tree on $n$ vertices. Then $\rm irr_{t}$$(G)\geq 2n-4$, and
the equality holds if and only if $G\cong P_{n}$.
###### Proof.
Clearly, $2(n-1)=\sum\limits_{v\in V}d_{G}(v)$ by Lemma 4. Let
$s=|\\{w|d_{G}(w)\geq 3,w\in V\\}|$, and $h=|\\{w|d_{G}(w)=1,w\in V\\}|$. Then
$s\geq 0$ and $h\geq 2$. Let $\triangle{(G)}$ be the maximum degree of the
vertices of $G$. Now we complete the proof by the following two cases.
Case 1: $s=0$.
Then $h=2$ by $2(n-1)=\sum\limits_{v\in V}d_{G}(v)=2(n-h)+h$, and the degree
sequence of $G$ is $(2,\ldots,2,1,1)$. Thus $G\cong P_{n}$ and $\rm
irr_{t}$$(G)=2n-4$.
Case 2: $s\geq 1$.
Then $\triangle(G)\geq 3$ by $s\geq 1$, and $h\geq\triangle(G)+s-1\geq 3$ by
$2(n-1)=\sum\limits_{v\in V}d_{G}(v)\geq\triangle(G)+3(s-1)+2(n-s-h)+h$. So we
can do branch-transformation $h-2$ times on $G$ till the degree sequence of
the resulting graph is $(2,\ldots,2,1,1)$, denoted by $H_{1}$, and thus $\rm
irr_{t}$$(G)>$$\rm irr_{t}$$(H_{1})=2n-4$ by Lemma 2. ∎
###### Theorem 6.
Let $n\geq 5$, $G=(V,E)$ be a tree on $n$ vertices and $G\ncong P_{n}$. Then
$\rm irr_{t}$$(G)\geq 4n-10$, and the equality holds if and only if the degree
sequence of $G$ is $(3,2,\ldots,2,1,1,1)$.
###### Proof.
It is obvious that $2(n-1)=\sum\limits_{v\in V}d_{G}(v)$ by Lemma 4. Let
$s=|\\{w|d_{G}(w)\geq 3,w\in V\\}|$, and $h=|\\{w|d_{G}(w)=1,w\in V\\}|$. Then
$\triangle{(G)}\geq 3$ and $s\geq 1$ since $G\ncong P_{n}$. Now we complete
the proof by the following two cases.
Case 1: $s+\triangle(G)=4$.
Clearly, $s=1,\triangle(G)=3$. Then $h=3$ by $2(n-1)=\sum\limits_{v\in
V}d_{G}(v)=3+2(n-1-h)+h$, and the degree sequence of $G$ is
$(3,2,\ldots,2,1,1,1)$. Thus $\rm irr_{t}$$(G)=4n-10$.
Case 2: $s+\triangle(G)\geq 5$.
Then $h\geq\triangle(G)+s-1\geq 4$ by $2(n-1)=\sum\limits_{v\in
V}d_{G}(v)\geq\triangle(G)+3(s-1)+2(n-s-h)+h$. Now we can do branch-
transformation $h-3$ times on $G$ till the degree sequence of the resulting
graph is $(3,2,\ldots,2,1,1,1)$, denoted by $H_{2}$, and thus $\rm
irr_{t}$$(G)>$$\rm irr_{t}$$(H_{2})=4n-10$ by Lemma 2. ∎
###### Theorem 7.
Let $n\geq 6$, $G=(V,E)$ be a tree on $n$ vertices, $G\ncong P_{n}.$ If the
sequence $(3,2,\ldots,2,1,1,1)$ is not the degree sequence of $G,$ then $\rm
irr_{t}$$(G)\geq 6n-20$, and the equality holds if and only if the degree
sequence of $G$ is $(3,3,2,\ldots,2,1,1,1,1)$.
###### Proof.
Clearly, $2(n-1)=\sum\limits_{v\in V}d_{G}(v)$ by Lemma 4. Let
$s=|\\{w|d_{G}(w)\geq 3,w\in V\\}|$, and $h=|\\{w|d_{G}(w)=1,w\in V\\}|$. Then
$s\geq 1$ and $\triangle{(G)}\geq 3$ since $G\ncong P_{n}$. Now we complete
the proof by the following two cases.
Case 1: $s=1$.
Then $\triangle(G)\geq 4$ because sequence $(3,2,\ldots,2,1,1,1)$ is not the
degree sequence of $G$.
Subcase 1.1: $\triangle(G)=4$.
Then $h=4$ by $2(n-1)=\sum\limits_{v\in V}d_{G}(v)=4+2(n-1-h)+h$, and the
degree sequence of $G$ is $(4,2,\ldots,2,1,1,1,1)$. Thus $\rm
irr_{t}$$(G)=6n-18>6n-20$.
Subcase 1.2: $\triangle(G)\geq 5$.
Then $h=\triangle(G)\geq 5$ by $2(n-1)=\sum\limits_{v\in
V}d_{G}(v)=\triangle(G)+2(n-1-h)+h$. Now we can do branch-transformation $h-4$
times on $G$ till the degree sequence of the resulting graph is
$(4,2,\ldots,2,1,1,1,1)$, denoted by $H_{3}$, and thus $\rm irr_{t}$$(G)>$$\rm
irr_{t}$$(H_{3})=6n-18>6n-20$ by Lemma 2.
Case 2: $s\geq 2$.
Case 2.1: $s+\triangle(G)=5$.
Then the degree sequence of $G$ is $(3,3,2,\ldots,2,1,1,1,1)$, and thus $\rm
irr_{t}$$(G)=6n-20$.
Case 2.2: $s+\triangle(G)\geq 6$.
Then $h\geq\triangle(G)+s-1\geq 5$ by $2(n-1)=\sum\limits_{v\in
V}d_{G}(v)\geq\triangle(G)+3(s-1)+2(n-s-h)+h$. Now we can do branch-
transformation $h-4$ times on $G$ till the degree sequence of the resulting
graph is $(3,3,2,\ldots,2,1,1,1,1)$, denoted by $H_{4}$, and thus $\rm
irr_{t}$$(G)>$$\rm irr_{t}$$(H_{4})=6n-20$ by Lemma 2. ∎
###### Remark 8.
Let $n\geq 6$, by Theorems 5-7, we know the minimal, the second minimal, the
third minimal total irregularity of trees on $n$ vertices are $2n-4$, $4n-10$,
$6n-20$, respectively, and the degree sequences of the corresponding extremal
graphs are $(2,\ldots,2,1,1)$, $(3,2,\ldots,2,1,1,1)$,
$(3,3,2,\ldots,2,1,1,1,1)$, respectively.
## 4 The minimal total irregularity of unicyclic graphs
In this section, we determine the minimal, the second minimal, the third
minimal total irregularity of unicyclic graphs on $n$ vertices and
characterize the extremal graphs.
An unicyclic graph is a simple connected graph in which the number of edges
equals the number of vertices. Let $G=(V,E)$ be an unicyclic graph. Then for
any vertex $u\in V$, $d_{G}(u)\geq 3$ implies there must exist a hanging tree
of $G$ connecting to the vertex $u$, thus we can obtain the following results
by branch-transformation.
###### Theorem 9.
Let $n\geq 3$ and $G=(V,E)$ be an unicyclic graph on $n$ vertices.
(1) $\rm irr_{t}$$(G)\geq 0$, and the equality holds if and only if $G\cong
C_{n}$.
(2) Let $n\geq 4$, and $G\ncong C_{n}$. Then $\rm irr_{t}$$(G)\geq 2n-2$, and
the equality holds if and only if the degree sequence of $G$ is
$(3,2,\ldots,2,1)$.
###### Proof.
(1) is obvious. Now we show (2) holds.
It is obvious that $2n=\sum\limits_{v\in V}d_{G}(v)$ by Lemma 4. Let
$s=|\\{w|d_{G}(w)\geq 3,w\in V\\}|$, and $h=|\\{w|d_{G}(w)=1,w\in V\\}|$. Then
$s\geq 1$, $h\geq 1$, $\triangle(G)\geq 3$ by $G\ncong C_{n}$ and
$2n=\sum\limits_{v\in V}d_{G}(v)$. Now we complete the proof by the following
two cases.
Case 1: $s+\triangle(G)=4$.
Then $(3,2,\ldots,2,1)$ is the degree sequence of $G$ by $2n=\sum\limits_{v\in
V}d_{G}(v)=3+2(n-1-h)+h$, and thus $\rm irr_{t}$$(G)=2n-2$.
Case 2: $s+\triangle(G)\geq 5$.
Then $h\geq\triangle(G)+s-3\geq 2$ by $2n=\sum\limits_{v\in
V}d_{G}(v)\geq\triangle(G)+3(s-1)+2(n-s-h)+h$, and we can do branch-
transformation $h-1$ times on $G$ till the degree sequence of the resulting
graph is $(3,2,\ldots,2,1)$, denoted by $H_{5}$, and thus $\rm
irr_{t}$$(G)>$$\rm irr_{t}$$(H_{5})=2n-2$ by Lemma 2. ∎
###### Theorem 10.
Let $n\geq 5$, $G=(V,E)$ be an unicyclic graph on $n$ vertices with $G\ncong
C_{n}.$ If the sequence $(3,2,\ldots,2,1)$ is not the degree sequence of $G,$
then $\rm irr_{t}$$(G)\geq 4n-8$, and the equality holds if and only if the
degree sequence of $G$ is $(3,3,2,\ldots,2,1,1)$.
###### Proof.
Clearly, $2n=\sum\limits_{v\in V}d_{G}(v)$ by Lemma 4. Let
$s=|\\{w|d_{G}(w)\geq 3,w\in V\\}|$, and $h=|\\{w|d_{G}(w)=1,w\in V\\}|$. Then
$s\geq 1$, $h\geq 1$ by $G\ncong C_{n}$ and $2n=\sum\limits_{v\in V}d_{G}(v)$.
Now we complete the proof by the following two cases.
Case 1: $s=1$.
Then $\triangle(G)\geq 4$ because sequence $(3,2,\ldots,2,1)$ is not the
degree sequence of $G$.
Subcase 1.1: $\triangle(G)=4$.
Then $h=2$ and the degree sequence is $(4,2,\ldots,2,1,1)$ by
$2n=\sum\limits_{v\in V}d_{G}(v)=4+2(n-1-h)+h$, and thus $\rm
irr_{t}$$(G)=4n-6>4n-8$.
Subcase 1.2: $\triangle(G)\geq 5$.
Then $h=\triangle(G)-2\geq 3$ by $2n=\sum\limits_{v\in
V}d_{G}(v)=\triangle(G)+2(n-1-h)+h$, and we can do branch-transformation $h-2$
times on $G$ till the degree sequence of the resulting graph is
$(4,2,\ldots,2,1,1)$, denoted by $H_{6}$, and thus $\rm irr_{t}$$(G)>$$\rm
irr_{t}$$(H_{6})=4n-6$ by Lemma 2.
Case 2: $s\geq 2$.
Subcase 2.1: $s+\triangle(G)=5$.
Then $h=2$ and the degree sequence is $(3,3,2,\ldots,2,1,1)$ by
$2n=\sum\limits_{v\in V}d(v)=6+2(n-2-h)+h$, and thus $\rm irr_{t}$$(G)=4n-8$.
Subcase 2.2: $s+\triangle(G)\geq 6$.
Then $h\geq\triangle(G)+s-3\geq 3$ by $2n=\sum\limits_{v\in
V}d_{G}(v)\geq\triangle(G)+3(s-1)+2(n-s-h)+h$, and we can do branch-
transformation $h-2$ times on $G$ till the degree sequence of the resulting
graph is $(3,3,2,\ldots,2,1,1)$, denoted by $H_{7}$, and thus $\rm
irr_{t}$$(G)>$$\rm irr_{t}$$(H_{7})=4n-8$ by Lemma 2. ∎
###### Remark 11.
Let $n\geq 5$, by Theorems 9-10, we know the minimal, the second minimal, the
third minimal total irregularity of unicyclic graphs on $n$ vertices are $0$,
$2n-2$, $4n-8$, respectively, and the degree sequences of the corresponding
extremal graphs are $(2,\ldots,2)$, $(3,2,\ldots,2,1)$,
$(3,3,2,\ldots,2,1,1)$, respectively.
## 5 The minimal total irregularity of bicyclic graphs
In this section, we determine the minimal, the second minimal, the third
minimal total irregularity of bicyclic graphs on $n$ vertices and characterize
the extremal graphs.
A bicyclic graph is a simple connected graph in which the number of edges
equals the number of vertices plus one. There are two basic bicyclic graphs:
$\infty$-graph and $\Theta$-graph. An $\infty$-graph, denoted by
$\infty(p,q,l)$ (see Figure 2), is obtained from two vertex-disjoint cycles
$C_{p}$ and $C_{q}$ by connecting one vertex of $C_{p}$ and one of $C_{q}$
with a path $P_{l}$ of length $l-1$ (in the case of $l=1$, identifying the
above two vertices, see Figure 3) where $p,q\geq 3$ and $l\geq 1$; and a
$\Theta$-graph, denoted by $\theta(p,q,l)$ (see Figure 4), is a graph on
$p+q-l$ vertices with the two cycles $C_{p}$ and $C_{q}$ have $l$ common
vertices, where $p,q\geq 3$ and $l\geq 2$.
\begin{picture}(400.0,50.0)\par\put(180.0,25.0){\circle{40.0}}
\put(200.0,25.0){\circle*{2.0}} \put(200.0,25.0){\line(1,0){20.0}}
\put(220.0,25.0){\circle*{2.0}} \put(223.0,22.0){$\cdots$}
\put(240.0,25.0){\circle*{2.0}} \put(240.0,25.0){\line(1,0){20.0}}
\put(260.0,25.0){\circle*{2.0}} \put(280.0,25.0){\circle{40.0}}
\par\par\put(175.0,23.0){\small$C_{p}$} \put(275.0,23.0){\small$C_{q}$}
\put(130.0,-20.0){\small Figure 2. The graph $\infty(p,q,l)$ with $l\geq 2$}
\end{picture}
$\cdots$
$C_{p}$$C_{q}$Figure 2. The graph $\infty(p,q,l)$ with $l\geq 2$
$C_{p}$$C_{q}$Figure 3. The graph $\infty(p,q,1)$
\begin{picture}(500.0,110.0)\par\par\put(200.0,50.0){\circle*{2.0}}
\put(220.0,50.0){\circle*{2.0}} \put(240.0,50.0){\circle*{2.0}}
\put(260.0,50.0){\circle*{2.0}} \put(200.0,90.0){\circle*{2.0}}
\put(220.0,90.0){\circle*{2.0}} \put(240.0,90.0){\circle*{2.0}}
\put(260.0,90.0){\circle*{2.0}} \put(200.0,10.0){\circle*{2.0}}
\put(220.0,10.0){\circle*{2.0}} \put(240.0,10.0){\circle*{2.0}}
\put(260.0,10.0){\circle*{2.0}} \par\put(225.0,10.0){\circle*{1.0}}
\put(230.0,10.0){\circle*{1.0}} \put(235.0,10.0){\circle*{1.0}}
\par\put(225.0,50.0){\circle*{1.0}} \put(230.0,50.0){\circle*{1.0}}
\put(235.0,50.0){\circle*{1.0}} \par\put(225.0,90.0){\circle*{1.0}}
\put(230.0,90.0){\circle*{1.0}} \put(235.0,90.0){\circle*{1.0}}
\par\put(200.0,50.0){\line(1,0){20.0}} \put(200.0,90.0){\line(1,0){20.0}}
\put(200.0,10.0){\line(1,0){20.0}} \par\put(240.0,50.0){\line(1,0){20.0}}
\put(240.0,90.0){\line(1,0){20.0}} \put(240.0,10.0){\line(1,0){20.0}}
\par\put(200.0,50.0){\line(0,-1){40.0}} \put(200.0,90.0){\line(0,-1){40.0}}
\par\put(260.0,50.0){\line(0,-1){40.0}} \put(260.0,90.0){\line(0,-1){40.0}}
\par\put(190.0,53.0){\footnotesize$z_{1}$}
\put(215.0,53.0){\footnotesize$z_{2}$} \put(265.0,53.0){\footnotesize$z_{l}$}
\par\put(190.0,94.0){\footnotesize$x_{1}$}
\put(215.0,94.0){\footnotesize$x_{2}$}
\put(265.0,94.0){\footnotesize$x_{p-l}$}
\par\put(190.0,2.0){\footnotesize$y_{1}$}
\put(215.0,2.0){\footnotesize$y_{2}$} \put(265.0,2.0){\footnotesize$y_{q-l}$}
\par\put(225.0,70.0){$C_{p}$} \put(225.0,25.0){$C_{q}$}
\par\par\put(165.0,-25.0){\small Figure 4. The graph $\theta(p,q,l)$}
\end{picture}
$z_{1}$$z_{2}$$z_{l}$
$x_{1}$$x_{2}$$x_{p-l}$
$y_{1}$$y_{2}$$y_{q-l}$
$C_{p}$$C_{q}$
Figure 4. The graph $\theta(p,q,l)$
Denoted by $\mathcal{B}_{n}$ is the set of all bicyclic graphs on $n$
vertices. Obviously, $\mathcal{B}_{n}$ consists of three types of graphs:
first type denoted by $B^{+}_{n}$, is the set of those graphs each of which is
an $\infty$-graph, $\infty(p,q,l)$, with trees attached when $l=1$; second
type denoted by $B^{++}_{n}$, is the set of those graphs each of which is an
$\infty$-graph, $\infty(p,q,l)$, with trees attached when $l\geq 2$; third
type denoted by $\Theta_{n}$, is the set of those graphs each of which is a
$\Theta$-graph, $\theta(p,q,l)$, with trees attached. Then
$\mathcal{B}_{n}=B_{n}^{+}\cup B_{n}^{++}\cup\Theta_{n}.$
### 5.1 The graph with minimal total irregularity in $B^{+}_{n}$
In this subsection, the minimal, the second minimal total irregularity of the
bicyclic graphs in $B^{+}_{n}$ are determined.
###### Theorem 12.
Let $n\geq 6$, $G=(V,E)\in B^{+}_{n}$.
(1) $\rm irr_{t}$$(G)\geq 2n-2$, and the equality holds if and only if the
degree sequence of $G$ is $(4,2,\ldots,2)$.
(2) If $(4,2,\ldots,2)$ is not the degree sequence of $G$, then $\rm
irr_{t}$$(G)\geq 4n-6$, and the equality holds if and only if the degree
sequence of $G$ is $(4,3,2,\ldots,2,1)$.
###### Proof.
Clearly, $\sum\limits_{v\in V}d_{G}(v)=2(n+1)$ by Lemma 4. Let
$s=|\\{w|d_{G}(w)\geq 3,w\in V\\}|$, $h=|\\{w|d_{G}(w)=1,w\in V\\}|$ and
$t=|\\{w|d_{G}(w)=\triangle(G),w\in V\\}|$. Then $s\geq 1$, $h\geq 0$, $1\leq
t\leq s$ and $\triangle(G)\geq 4$ by $G\in B^{+}_{n}$.
Note that $G\in B^{+}_{n}$, if $s=1$, $\triangle(G)\geq 5$ or $s\geq 2$, there
must exist a vertex $u$ with $d_{G}(u)\geq 3$ and there exists a hanging tree
of $G$ connecting to $u$. Then we complete the proof by the following two
cases.
Case 1: $s=1$.
Subcase 1.1: $\triangle(G)=4$.
Then $h=0$ and the degree sequence of $G$ is $(4,2,\ldots,2)$ by the fact
$2(n+1)=\sum\limits_{v\in V}d_{G}(v)=4+2(n-1-h)+h$, and thus $\rm
irr_{t}$$(G)=2n-2$.
Subcase 1.2: $\triangle(G)=5$.
Then $h=1$ and the degree sequence of $G$ is $(5,2,\ldots,2,1)$ by the fact
$2(n+1)=\sum\limits_{v\in V}d_{G}(v)=5+2(n-1-h)+h$, and thus $\rm
irr_{t}$$(G)=4n-4>4n-6$.
Subcase 1.3: $\triangle(G)\geq 6$.
Then $h=\triangle(G)-4\geq 2$ by the fact $2(n+1)=\sum\limits_{v\in
V}d_{G}(v)=\triangle(G)+2(n-1-h)+h$, and we can do branch-transformation $h-1$
times on $G$ till the degree sequence of the resulting graph is
$(5,2,\ldots,2,1)$, denoted by $H_{8}$, and thus $\rm irr_{t}$$(G)>$$\rm
irr_{t}$$(H_{8})=4n-4$ by Lemma 2.
Case 2: $s\geq 2$.
Subcase 2.1: $s+\triangle(G)=6$.
Then $s=2$, $\triangle(G)=4$ and $1\leq t\leq 2$.
If $t=1$, then $h=1$ and the degree sequence of $G$ is $(4,3,2,\ldots,2,1)$ by
the fact $2(n+1)=\sum\limits_{v\in V}d_{G}(v)=4+3+2(n-2-h)+h$, and thus $\rm
irr_{t}$$(G)=4n-6$.
If $t=2$, then $h=2$ by the fact $2(n+1)=\sum\limits_{v\in
V}d_{G}(v)=4+4+2(n-2-h)+h$, and we can do branch-transformation once on $G$
such that the degree sequence of the resulting graph is $(4,3,2\ldots,2,1)$,
denoted by $H_{9}$, and thus $\rm irr_{t}$$(G)>$$\rm irr_{t}$$(H_{9})=4n-6$ by
Lemma 2.
Subcase 2.2: $s+\triangle(G)\geq 7$.
Then $h\geq\triangle(G)+s-5\geq 2$ by the fact $2(n+1)=\sum\limits_{v\in
V}d_{G}(v)\geq\triangle(G)+3(s-1)+2(n-s-h)+h$, and we can do branch-
transformation $h-1$ times on $G$ such that the degree sequence of the
resulting graph is $(4,3,2\ldots,2,1)$, denoted by $H_{9}$, and thus $\rm
irr_{t}$$(G)>$$\rm irr_{t}$$(H_{9})=4n-6$ by Lemma 2. ∎
### 5.2 The graph with minimal total irregularity in $B^{++}_{n}$
In this subsection, the minimal, the second minimal total irregularity of the
bicyclic graphs in $B^{++}_{n}$ are determined.
###### Theorem 13.
Let $n\geq 7$, $G=(V,E)\in B^{++}_{n}$.
(1) $\rm irr_{t}$$(G)\geq 2n-4$, and the equality holds if and only if the
degree sequence of $G$ is $(3,3,2,\ldots,2)$.
(2) If $(3,3,2,\ldots,2)$ is not the degree sequence of $G$, then $\rm
irr_{t}$$(G)\geq 4n-10$, and the equality holds if and only if the degree
sequence of $G$ is $(3,3,3,2,\ldots,2,1)$.
###### Proof.
Clearly, $\sum\limits_{v\in V}d_{G}(v)=2(n+1)$ by Lemma 4. Let
$s=|\\{w|d_{G}(w)\geq 3,w\in V\\}|$, $h=|\\{w|d_{G}(w)=1,w\in V\\}|$ and
$t=|\\{w|d_{G}(w)=\triangle(G),w\in V\\}|$. Then $s\geq 2$, $h\geq 0$, $1\leq
t\leq s$ and $\triangle(G)\geq 3$ by $G\in B^{++}_{n}$.
Note that $G\in B^{++}_{n}$, if $s=2$, $\triangle(G)\geq 4$ or $s\geq 3$,
there must exist a vertex $u$ with $d_{G}(u)\geq 3$ and there exists a hanging
tree of $G$ connecting to $u$. Then we complete the proof by the following two
cases.
Case 1: $s=2$.
Subcase 1.1: $\triangle(G)=3$.
Then $h=0$ and the degree sequence of $G$ is $(3,3,2,\ldots,2)$ by the fact
$2(n+1)=\sum\limits_{v\in V}d_{G}(v)=3+3+2(n-2-h)+h$, and thus $\rm
irr_{t}$$(G)=2n-4$.
Subcase 1.2: $\triangle(G)=4$.
Then $t=1$ or $t=2$ by $1\leq t\leq s$.
If $t=1$, then $h=1$ and the degree sequence of $G$ is $(4,3,2,\ldots,2,1)$ by
the fact $2(n+1)=\sum\limits_{v\in V}d_{G}(v)=4+3+2(n-2-h)+h$, and thus $\rm
irr_{t}$$(G)=4n-6>4n-10$.
If $t=2$, then $h=2$ by the fact $2(n+1)=\sum\limits_{v\in
V}d_{G}(v)=4+4+2(n-2-h)+h$, and we can do branch-transformation once on $G$
such that the degree sequence of the resulting graph is $(4,3,2\ldots,2,1)$,
denoted by $H_{10}$, and thus $\rm irr_{t}$$(G)>$$\rm irr_{t}$$(H_{10})=4n-6$
by Lemma 2.
Subcase 1.3: $\triangle(G)\geq 5$.
Then $h\geq\triangle(G)-3\geq 2$ by the fact $2(n+1)=\sum\limits_{v\in
V}d_{G}(v)\geq\triangle(G)+3+2(n-2-h)+h$, and we can do branch-transformation
$h-1$ times on $G$ such that the degree sequence of the resulting graph is
$(4,3,2\ldots,2,1)$, denoted by $H_{10}$, and thus $\rm irr_{t}$$(G)>$$\rm
irr_{t}$$(H_{10})=4n-6$ by Lemma 2.
Case 2: $s\geq 3$.
Subcase 2.1: $s+\triangle(G)=6$.
Then $h=1$ and the degree sequence of $G$ is $(3,3,3,2,\ldots,2,1)$ by the
fact $2(n+1)=\sum\limits_{v\in V}d_{G}(v)=3+3+3+2(n-3-h)+h$, and thus $\rm
irr_{t}$$(G)=4n-10$.
Subcase 2.2: $s+\triangle(G)\geq 7$.
Then $h\geq\triangle(G)+s-5\geq 2$ by the fact $2(n+1)=\sum\limits_{v\in
V}d_{G}(v)\geq\triangle(G)+3(s-1)+2(n-s-h)+h$, and we can do branch-
transformation $h-1$ times on $G$ such that the degree sequence of the
resulting graph is $(3,3,3,2\ldots,2,1)$, denoted by $H_{11}$, and thus $\rm
irr_{t}$$(G)>$$\rm irr_{t}$$(H_{11})=4n-10$ by Lemma 2. ∎
### 5.3 The graph with minimal total irregularity in $\Theta_{n}$
By the same proof of Theorem 13, we can determine the minimal, the second
minimal total irregularity of the bicyclic graphs in $\Theta_{n}$ immediately.
###### Theorem 14.
Let $n\geq 5$, $G=(V,E)\in\Theta_{n}$.
(1) $\rm irr_{t}$$(G)\geq 2n-4$, and the equality holds if and only if the
degree sequence of $G$ is $(3,3,2,\ldots,2)$.
(2) If $(3,3,2,\ldots,2)$ is not the degree sequence of $G$, then $\rm
irr_{t}$$(G)\geq 4n-10$, and the equality holds if and only if the degree
sequence of $G$ is $(3,3,3,2,\ldots,2,1)$.
### 5.4 The graph with minimal total irregularity in $\mathcal{B}_{n}$
By Theorems 12-14, we can determine the minimal, the second minimal, the third
minimal total irregularity of the bicyclic graphs on $n$ vertices immediately.
###### Theorem 15.
Let $n\geq 7$, $G\in\mathcal{B}_{n}$.
(1) $\rm irr_{t}$$(G)\geq 2n-4$, and the equality holds if and only if the
degree sequence of $G$ is $(3,3,2,\ldots,2)$.
(2) If $(3,3,2,\ldots,2)$ is not the degree sequence of $G$, then $\rm
irr_{t}$$(G)\geq 2n-2$, and the equality holds if and only if the degree
sequence of $G$ is $(4,2,\ldots,2)$.
(3) If $(3,3,2,\ldots,2)$ and $(4,2,\ldots,2)$ are not the degree sequence of
$G$, then $\rm irr_{t}$$(G)\geq 4n-10$, and the equality holds if and only if
the degree sequence of $G$ is $(3,3,3,2,\ldots,2,1)$.
## 6 Open problem for further research
By the results of Sections 3-5, we know the minimal irregularity of simple
connected graphs on $n$ vertices is zero, and the corresponding extremal
graphs are regular graphs. Furthermore, we suppose $2n-4$ is the second
minimal and $2n-2$ is the third minimal.
###### Conjecture 16.
Let $G$ be a simple connected graph with $n$ vertices. If $G$ is not a regular
graph, then $\rm irr_{t}$$(G)\geq 2n-4$.
## References
* [1] J.A. Bondy, U.S. R. Murty, Graph theory with applications, MacMillan, London, 1976.
* [2] H. Abdo, D. Dimitrov, The total irregularity of a graph, arXiv:1207.5267v1 24 July 2012.
* [3] M.O. Albertson, The irregularity of a graph, Ars Comb., 46 (1997) 219–225.
* [4] H. Abdo, N. Cohen, D. Dimitrov, Bounds and computation of irregularity of a graph, Filomat, in press, 2014.
* [5] M.A. Henning, D. Rautenbach, On the irregularity of bipartite graphs, Discrete Math., 307 (2007) 1467–1472.
* [6] P. Hansen, H. Mélot, Variable neighborhood search for extremal graphs. 9. Bounding the irregularity of a graph, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 69 (2005) 253–264.
* [7] D. Dimitrov, R. Škrekovski, Comparing the irregularity and the total irregularity of graphs, Ars. Math. Contemp., in press, 2014.
* [8] L.H. You, J.S. Yang and Z.F. You, The maximal total irregularity of unicyclic graphs, Ars Comb., 114 (2014), in press.
* [9] L.H. You, J.S. Yang, Y.X. Zhu and Z.F. You, The maximal total irregularity of bicyclic graphs, Journal of Applied Mathematics, 2014, accepted.
* [10] H. Abdo, D. Dimitrov, The total irregularity of graphs under graph operations, arXiv:1304.0185v1 31 Mar 2013 .
|
arxiv-papers
| 2014-04-03T14:26:20 |
2024-09-04T02:50:00.663524
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yingxue Zhu, Lihua You, Jieshan Yang",
"submitter": "Lihua You",
"url": "https://arxiv.org/abs/1404.0931"
}
|
1404.1017
|
# Stability and response of polygenic traits to stabilizing selection and
mutation
Harold P. de Vladar, Nick Barton [email protected]@ist.ac.at
###### Abstract
When polygenic traits are under stabilizing selection, many different
combinations of alleles allow close adaptation to the optimum. If alleles have
equal effects, all combinations that result in the same deviation from the
optimum are equivalent. Furthermore, the genetic variance that is maintained
by mutation-selection balance is $2\mu/S$ per locus, where $\mu$ is the
mutation rate and $S$ the strength of stabilizing selection. In reality,
alleles vary in their effects, making the fitness landscape asymmetric, and
complicating analysis of the equilibria. We show that that the resulting
genetic variance depends on the fraction of alleles near fixation, which
contribute by $2\mu/S$, and on the total mutational effects of alleles that
are at intermediate frequency. The interplay between stabilizing selection and
mutation leads to a sharp transition: alleles with effects smaller than a
threshold value of $2\sqrt{\mu/S}$ remain polymorphic, whereas those with
larger effects are fixed. The genetic load in equilibrium is less than for
traits of equal effects, and the fitness equilibria are more similar. We find
that if the optimum is displaced, alleles with effects close to the threshold
value sweep first, and their rate of increase is bounded by $\sqrt{\mu S}$.
Long term response leads in general to well-adapted traits, unlike the case of
equal effects that often end up at a sub-optimal fitness peak. However, the
particular peaks to which the populations converge are extremely sensitive to
the initial states, and to the speed of the shift of the optimum trait value.
## Introduction
Understanding quantitative genetics in terms of population genetics is crucial
for both scientific and practical reasons. However, the development of a
consistent theory for long term evolution has had limited success, because the
polygenic basis of quantitative traits makes the prediction of their response
to selection immensely intricate, even under the simplest assumptions (e.g.
additivity, equal effects of an allele on the trait, and linkage equilibrium)
[Barton and Turelli 1989, Keightley and Hill 1990, Turelli and Barton 1994].
Most traits seem to be under some form of stabilizing selection, either by the
direct action of selection on a trait whose extreme values are unfit, or
indirectly by compromising individual fitness due to pleiotropic detrimental
effects [Keightley and Hill 1990, Mackay 2001, Hill and Zhang 2012, Mackay
2010]. The joint effects of stabilizing selection and mutation lead to very
complicated allele frequency equilibria and evolution, and it is not obvious
how much genetic variation they can maintain [Turelli 1984, Turelli 1988,
Bürger 2000].
An exact analysis in terms of allele frequencies is lacking for polygenic
traits with loci of unequal effects. This is desirable, as data from genome
wide association studies (GWAS) yield information about the distribution of
single nucleotide polymorphisms (SNPs) relevant to several traits based on
population sequence data [Hindorff et al. 2009, Visscher et al. 2012]. This
makes it urgent to understand how variation at the molecular level explains
phenotypic variation. How quantitative variation depends on the number of loci
and on the distribution of allelic effects is not clear, and is the central
question of this article.
Figure 1: Response to selection of traits determined by 50 loci of equal (red
and dash-dotted, color online) and unequal effects (black). (A) Deviation of
the trait mean from the optimum value. (B) Genetic variance. The dotted black
line show the HoC variance. Allele frequencies under (C) equal and (D) unequal
effects. The equal effects have $\gamma=1/10$, and the unequal effects are
distributed as an exponential (mean=1/10). Mutation rate $\mu=10^{-4}$,
selection intensity $S=10^{-1}$. The dynamics are numerical solutions to
ordinary differential equations (Eq. 6 in the text).
?) showed that when a population is at equilibrium with the trait mean is at
the optimum and the alleles are very close to fixation, then the genetic
variance, $\nu$, is $2n\mu/S$, where $n$ is the number of contributing loci,
$\mu$ is the per locus mutation rate and $S$ is the strength of stabilizing
selection. In general, the genetic load $L$ is due to both deviation from the
optimum and genetic variance, $L\propto\Delta z^{2}+\nu$, where $\Delta z$ is
the deviation of the trait mean from the optimum. The contribution of the
genetic variance is much more significant because compared to it, the
deviations from the optimum are very small. Moreover, if the trait is not at
the optimum, higher variance can be maintained. These calculations assumed a
trait with diallelic loci of equal effects. However, we will show that under
unequal effects, deviations from the optimum can also maintain less variance.
The response to a shift in the optimum trait value will be radically different
under equal and unequal effects. For example, Fig. 1 shows the response of two
equivalent populations that differ only in their distribution of allelic
effects. Notice that although the traits match the optimum almost perfectly in
both cases (Fig. 1A), under equal effects much more variation is maintained
than under unequal effects (Fig. 1B), which implies a greater mutation load.
We will see that under unequal effects, the equilibria depend on the magnitude
of allelic effects. With equal effects, there is a high degree of symmetry in
the sense that many allelic combinations match a given optimum value, making
it easier to characterize the possible equilibria [Barton 1986]. However, this
analysis fails under unequal effects because the symmetry is absent. For
example, Figs. 1C,D show the response of the allele frequencies; in C and D
the alleles have equal and unequal effects, respectively. Initially, the
alleles rest at a stable equilibrium that have comparable mean and variance.
We see that the response under unequal effects is more heterogeneous in Fig.
1D (unequal effects), whereas the alleles respond homogeneously under equal
effects (Fig. 1C). This difference in the response accounts for the eventual
mal-adaptation of traits with loci of equal effects. Moreover, alleles that
have very large effects are at very low frequency and might take substantial
time to achieve a higher representation in the population. Thus, anticipating
when they will reach intermediate frequencies and make a notable contribution
to the genetic variance is difficult. Also, we don’t know which alleles
contribute preferentially to the response and eventual adaptation of the
trait. Under equal effects, all alleles have the same contribution, but the
symmetry of the solutions effectively reduces the genetic degrees of freedom,
which in turn limits the possible paths to find a global fitness optimum.
The House of Cards (HoC) is a mutation-selection balance model that assumes
that each allele is new and arises independently from the previous allele from
which it mutated, so that the effects of new mutations are uncorrelated from
the previous ones. In equilibrium, the variance of allelic effects is larger
than the genetic variance, and is predicted to be $2n\mu/S$, where $n$ is the
number of loci, $\mu$ is the per-locus mutation rate, and $S$ is the strength
of stabilizing selection [Turelli 1984, Kingman 1978, Bürger 2000, , Ch. IV].
Exactly this amount of genetic variance is maintained for traits with several
diallelic loci of equal effects, when they are adapted to the optimum [Barton
1986]. However, numerical experiments such as the one shown in Fig. 1B reveals
that with unequal effects, the genetic variance is decreased even further
below this bound. It is not immediately clear why this difference between
traits with equal and unequal effects occurs.
Although the HoC assumes a continuous production of new alleles with varying
effects [Turelli 1984], this model can be interpreted as a limit of a trait
constituted of many loci [Barton 1986], in which case each locus composing a
trait $z$ under stabilizing selection evolves according (as will be explained
in detail below) to:
$\frac{dp}{dt}=-S\gamma p(1-p)(2\Delta z+\gamma(1-2p))+\mu(1-2p)~{},$ (1)
where $p$ is the frequency of the ‘+’ allele, $\gamma$ is its allelic effect,
and $\Delta z$ the deviation of the trait mean from the optimum. Detailed
analyses of this system under equal effects were performed by ?). At
equilibrium, the equation above can have one or three solutions for each
locus, given by the cubic polynomial on $p$ that results from equating
$\frac{dp}{dt}=0$. If we plot the equilibrium value of $p$ against $\Delta
z$/$\gamma$ we find that there are two types of curves, depending on the
mutation rate, $\mu$ (Fig. 2A). The first type (Fig. 2A, thin curve) occurs at
high mutation rates; in this case and at small deviations from the trait
optimum, the equilibrium is maintained at intermediate frequencies,
maintaining substantial variability. The other type of equilibrium (Fig. 2A,
thick curve) occurs when mutation rates are low compared to the mutational
effects: for well-adapted traits either of the two alleles can be near
fixation, each one contributing to the genetic variance by 2$\mu$/$S$, as the
HoC predicts.
Figure 2: (A) Equilibria of allele frequencies as a function of scaled
deviation from the optimum. Thin curve: equilibria for alleles of small
effects $\left(\gamma^{2}<4\mu/S\right)$; for each value of $\Delta z/\gamma$
there is one stable allele frequency. Thick curve: equilibria for alleles of
large effects, $\left(\gamma^{2}>4\mu/S\right)$; for small deviations from the
optimum, there are two possible equilibria near fixation. the dashed segment
are unstable equilibria. (B) Equilibria of allele frequencies as a function of
the scaled parameter $m=\mu\left/\gamma^{2}\right.S$. Thick gray curve: no
deviations from the optimum, $\Delta z$=0. Thin black curve: small deviations
from the optimum. Black thick curve: large deviations from the optimum.
Circles: end point of Fig. 1D.
Under equal effects the equilibrium value of the trait only depends on the
number of ‘+’ and ‘-’ alleles, thus allowing many equivalent genetic
combinations; there are many other stable but suboptimal combinations [Barton
1986]. The particular state to which the population converges is thus strongly
determined by its previous history. All these suboptimal combinations trap the
population in local fitness peaks that deviate considerably from the optimum
trait value. We can see in Fig. 2A (thick line) that if the effect of each
locus on the trait is fairly large, then deviations from the trait that are at
most equally large as the effect can maintain any ‘+’ or ‘-’ alleles at
equilibrium. Thus, many of the suboptimal combinations are realizable. Also,
if the population is resting at an initial equilibrium and the optimum is
shifted (either slowly or abruptly), the allele frequencies respond in a
coordinated way. Thus, the trait is resilient to perturbations in the sense
that all allele frequencies are always equidistant from the bifurcation point
where their stability changes, and thus resist large deviations from the
optimum. Once the bifurcation point is reached, all loci become unstable at
once and suddenly jump to a suboptimal state. Therefore, it is unlikely that
populations reach an optimal peak.
In this paper we show that if the loci that constitute the trait have
different effects, there is a more heterogeneous distribution of equilibria,
with no symmetry amongst peaks. There are still many sub-optimal states where
the population could get stuck, but we will see that under unequal effects,
these suboptimal equilibria are much more similar (and closer) to the optimum.
However, the trait is also less resilient to deviations from the optimum, and
smaller perturbations render the configurations unstable. In fact, we see in
Fig. 2A that alleles of very small effects will make $\Delta z/\gamma$ large,
implying that the allelic configurations become unstable. Naturally, the
occurrence of small effects is contingent on the distribution of allelic
effects, which is unknown in detail; we will explore this aspect in this
article.
Summarizing, under equal effects precise adaptation to the optimum is harder
because the population might get stuck at suboptimal peaks that have large
variation and larger mutation load. At equilibrium, selection purges the new
mutations, and irrespectively of their allelic effects, each locus contributes
$2\mu/S$ to the genetic variance. In fact, this is an upper-bound achieved
when the trait is perfectly adapted to the optimum, irrespective of the
distribution of genetic effects (as long as these are larger than their
contribution to the genetic variance in equilibrium). However, if the trait
mean deviates from the optimum, the genetic variance can differ from that of
the HoC [Bürger and Hofbauer 1994] (Fig. 1).
A different situation is realized if the allelic effects are smaller than the
equilibrium variance, for which the HoC model does not apply. Another classic
approximation, which supposes multiple alleles and is often referred as the
Gaussian Model (GM) [Kimura 1965, Lande 1976], makes the opposite assumption
about the allelic effects, i.e. that these are small compared to their
contribution to the genetic variance in equilibrium. The GM assumes that there
is a continuous production of new alleles that follows a Gaussian distribution
of effects at each locus, that is centered at the parental genotypic value. ?)
showed that in polygenic diallelic traits under equal effects, changes in the
optimum can lead the population towards stable albeit maladapted equilibria
which can have much larger variation than that of the HoC, and fall into a
limit that is better approximated by the GM.
The analyses for polygenic systems with unequal effects that we perform here
are more challenging than for equal effects. Our current understanding of
unequal effects derives from models that deal with a few loci, from which
general results are hard to extrapolate [Bürger 2000, Turelli 1984, Chevin and
Hospital 2008, Pavlidis et al. 2012]. In this article we aim to understand how
a trait determined by arbitrarily many loci of unequal effects responds to
stabilizing selection and mutation. Putting aside the technical complexities,
we regard this problem as fundamental to understanding the bigger picture of
the evolution of polygenic traits, namely that of finite populations subject
to drift, and how these are constrained by pleiotropic effects caused by
selection on multiple characters. But first, we need to understand in detail
the nature of the equilibria and the response of allele frequencies to factors
such as stabilizing selection and mutation. Thus, we address the simplest case
of deterministic selection on a single trait.
We start by studying the equilibria, and find that there can be multiple loci
with high polymorphism, provided that they have small effects. For these
alleles of small effect, deviations from the optimum trait value are tolerated
without affecting their equilibrium. However, we also find that there is a
threshold $\hat{\gamma}=2\sqrt{\mu/S}$ that objectively defines which alleles
are of “small” effect ($\gamma<\hat{\gamma}$), and which are of “large” effect
($\gamma>\hat{\gamma}$). The former remain at intermediate frequencies and
latter near fixation most of the time. Alleles of large effect can be
sensitive even to small deviations from the optimum. In particular, if the
optimum is suddenly shifted, we find that the alleles that respond first are
those with effects closer to the threshold value, $\gamma\sim\hat{\gamma}$. In
the long-term, however, the dynamics are intricate. Different initial
equilibrium configurations that are equally well adapted may lead the
population to totally different regions of the fitness landscape. However,
these different genetic states have very similar phenotypic values.
## Model of stabilizing selection and mutation on additive traits
We consider the simplest diploid genotype-phenotype map, which assumes an
additive trait for diallelic loci, without dominance or epistasis:
$z=\sum_{i=1}^{n}\gamma_{i}(X_{i}+X_{i}^{\prime}-1)$ (2)
where $\gamma_{i}$ is the allelic effect at locus $i$, and $n$ is the number
of loci composing the trait, and $X$ and $X^{\prime}$ are indicators of the
‘-’ allele ($X,X^{\prime}=0$) or of the ‘+’ allele ($X,X^{\prime}=1$). We will
allow each $\gamma_{i}$ to vary across loci. Specific values will be drawn
from a given distribution (we explore mainly $\Gamma$-distributed effects),
although in every run they will be kept constant. Assuming linkage
equilibrium, the trait mean and the genetic variance are given by:
$\displaystyle\bar{z}=\sum_{i=1}^{n}\gamma_{i}\left(2p_{i}-1\right)$ (3)
$\displaystyle\nu=2\sum_{i=1}^{n}\gamma_{i}^{2}p_{i}\left(1-p_{i}\right)$ (4)
where $p_{i}=\mathcal{E}[X_{i}]$, the allele frequency of the ‘+’ allele,
given by the expectation of $X_{i}$ in the population. (Unless otherwise
stated, the expectations are on the population, not on the distribution of
effects).
We assume a Gaussian fitness, $W_{z}=\exp[-\frac{S}{2}(z-z_{\circ})^{2}]$ so
the mean fitness of the population is:
$\bar{W}=\exp\left[-\frac{S}{2}\Delta z^{2}-\frac{S}{2}\nu\right]$ (5)
which assumes weak selection. The genetic load is due to both terms: the
deviations from the optimum $\Delta z=\bar{z}-z_{\circ}$ and the genetic
variance $\nu$. The maximum mean fitness is 1, which occurs if an optimal
genotype is fixed, with no genetic variance.
In an infinite, random mating population, the change in allele frequencies is
given by the selection-mutation equation:
$\frac{dp_{i}}{dt}=-S\gamma_{i}p_{i}\left(1-p_{i}\right)\left(2\Delta
z+\gamma_{i}\left(1-2p_{i}\right)\right)+\mu\left(1-2p_{i}\right)~{},$ (6)
for $i=1,\ldots,n$, and $\mu,S<<1$ (see for example [Barton 1986]). This
equation for the dynamics of allele frequencies assumes linkage equilibrium
and weak selection.
To understand the complexities of the fitness landscape and how the trait
evolves, we first study the properties of the equilibria. An exact solution of
the equilibria of the system defined by Eq. 6 for the $n$ loci is possible,
but the formulae are complicated, being solutions to coupled cubic equations.
Therefore, instead of taking this exact but intricate approach, we will first
study the qualitative aspects of the equilibria of Eq. 6, by assuming that
$\Delta z$ is given, which uncouples the equations. This will give a solid
intuition to understand detailed equilibrium analyses, and how much genetic
variance can be maintained at the different sub-optimal peaks.
Afterwards, we will explore the dynamics and understand the irregular
behaviours by using the intuition derived from the equilibrium analyses. We
numerically solve the system for all the allele frequencies at each locus.
Since we assume diallelic loci, there are $n$ equations to track. We calculate
the genetic variance and trait means from the definitions given by Eqns. 3 and
4. We typically randomize the initial conditions and the realization of
allelic effects, unless otherwise stated.
## Allelic equilibria have two defined regimes
Equation 6 shows that the equilibrium condition for every locus is given by a
set of cubic equations coupled through $\Delta z$. When there are many loci,
we can assume a particular value for $\Delta z$ and treat each of the $n$
equations independently. Therefore, for each locus the number of valid roots
of the cubic equation depends on four quantities: the deviation from the
optimum $\Delta z$, the allelic effect $\gamma$ of the focal locus, the
mutation rate $\mu$, and the strength of stabilizing selection $S$. However,
these four variables can be combined into just two scaled parameters,
$\delta=\Delta z/\gamma$ and $m=\mu/S\gamma^{2}$ and the equilibrium solution
at each locus is given by the scaled equation:
$p^{3}-p^{2}\left(\frac{3}{2}+\delta\right)+p\left(\frac{1}{2}+\delta+m\right)-\frac{m}{2}=0.$
(7)
Figure 2A shows how the equilibrium frequencies depend on the scaled deviation
from the optimum, $\delta$. These diagrams also hold for unequal effects,
except that the equilibria for each locus are represented by a specific
diagram. In Appendix A we give the precise expression of the critical points
of Eq. 7. Figure 2A shows that there may be two types of equilibria: either
near fixation of one or other allele, or a single equilibrium at intermediate
frequency 1/2. The factor that determines which equilibrium is attained at a
given locus is the scaled variable $m=\mu/S\gamma^{2}$.
Figure 2B shows the equilibrium allele frequencies as a function of $m$.
Consider $\delta$=0: we see a partitioning of two qualitative regions with
stable states that are near fixation ($m<1/4$, to the left) and intermediate
equilibrium ($m>1/4$, to the right). Notice that since $m$ in inversely
proportional to $\gamma^{2}$, the smaller the effects, the more to the right
the alleles are represented in Fig. 2B, and vice versa. Thus
$\hat{\gamma}=2\sqrt{\mu/S}$ is a threshold that objectively defines alleles
of large and small effect: if $\gamma>\hat{\gamma}$, these fall into the
category of “large” effects, and if $\gamma<\hat{\gamma}$, these fall into the
category of “small” effects.
Figure 2B shows how this diagram is modified for $\delta\neq 0$: the
bifurcation is shifted, and the intermediate equilibria close to $m\geq 1/4$
are displaced from 1/2. This has two main implications. Firstly, assume a
small deviation of $\delta>0$ ($\delta<0$); some of the alleles of large
effect that would have been close to fixation, at the “+” (“-”) state, are
forced to sweep to the alternative state. Secondly, some of the alleles of
small effect that would be at the intermediate state, $p$=1/2, will show
reduced (increased) frequencies. Most notably, the alleles that are displaced
are those that are close to $m=\hat{m}$, which are those that are close to the
threshold $\hat{\gamma}=2\sqrt{\mu/S}$.
A bigger picture emerges when we consider how the equilibria depend on
combinations of both $\delta$ and $m$ (see Appendix A for the exact
calculations) which is shown in Fig. 3. On a logarithmic scale, the allelic
effects fall on a straight line, with the distribution of effects determining
their spread along this line: smaller effects fall towards the right and
larger effects falling towards the left of the diagram; different values of
$\delta$ are represented as parallel lines of slope 1/2.
When effects are large enough that $m<\hat{m}=1/4$ the alleles can be in a bi-
stable regime: there are two stable points close to fixation and one unstable
at intermediate frequency (the thick line in Fig. 2A). This is provided that
deviations from the optimum are small ($\delta\sim 0$). In this case, the
stability is not affected. However (and as we will explain below), deviations
that are of the order $\delta~{}1/2$ or larger disturb this equilibrium (see
Fig. 2B).
Figure 3: Diagram showing the two regions of qualitatively different
equilibria of allele frequencies. For $m<1/4$ (the shaded region) the allele
frequencies are near fixation points. When $m>1/4$ only polymorphisms can be
maintained. On a log scale the effects are distributed along parallel lines
whose height is determined by $\log(\Delta
z)-\log(\gamma),\log(\mu/S)-2\log(\gamma)$, and therefore have a slope of 1/2.
Effects that fall at the right hand side of the point $m=1/4$ can never fall
into the bistable regime, and correspond to the alleles with the smallest
effect. Depicted in the figure are effects distributed (bullet)
$\gamma\sim\exp(10)$; (upwards triangle) trait with equal but large effects
($\gamma=3$); (downwards triangle) trait with equal but small effects
($\gamma=0.001$).
The situation is very different for small effects, when $m>1/4$, since there
is only one valid root of the cubic above (the thin line in Fig. 2A). When the
trait is close to the optimum ($\delta\sim$ 0) intermediate frequencies can be
maintained, as explained above. Small deviations from the optimum will
readjust frequencies slightly, but the stability of the equilibrium is not
modified (there is no qualitative change in the stability).
As a consequence, the frequency of alleles of small effect varies smoothly
with the deviations from the optimum, whereas those with larger effect
experience discontinuous transitions when the magnitude of the deviation
approaches half of their respective effects.
The scaling properties indicate that the parameters that really matter are
$m=\mu/\gamma^{2}S$ and $\delta=\Delta z/\gamma$. Thus, the specific numerical
choices of $\mu$ and $S$ are not in themselves decisive.
## Stability and variation at equilibrium
### Equilibria under mutation-selection balance.
As above, when mutation is present there might be one, or three solutions for
each locus, with the stability depending on the particular multidimensional
adaptive peak. Consequently, there might be up to $3^{n}$ possible equilibria,
although only a fraction of these can be stable. Assuming equal effects, it
would be enough to count the number of loci that are fixed or intermediate,
since the symmetry of the landscape makes it feasible to understand the
stability of any of these peaks [Barton 1986]. With unequal effects, we need
to consider each of these configurations separately. These are tractable as
long as we assume that $\Delta z=0$, in which case the equilibria at each
locus are given by
$0=\left(\mu-S\gamma^{2}p(1-p)\right)(1-2p).$ (8)
In this case there are three possible states for each locus, namely
$\displaystyle p=1/2$ (9) $\displaystyle
p=\frac{1}{2}\left(1\pm\sqrt{1-4\frac{\mu}{S\gamma^{2}}}\right)~{}.$ (10)
In Supplementary Information 1 we detail the stability analyses that we now
summarize. In the absence of mutation, at most one allele can be maintained
polymorphic, irrespective of the magnitudes of the allelic effects and of the
deviation from the optimum (this result was anticipated by ?)). If mutation
rates are small compared to selection, the trait mean is exactly at the
optimum, and all alleles are of large effects, then the configurations where
all loci are close to fixation will be stable (all eigenvalues are negative).
Furthermore, configurations where alleles of large effect are at intermediate
frequency will be unstable. We also find that configurations where alleles of
small effect are near fixation, are unstable.
Why is this? Alleles with large effects increase the genetic variance
substantially. We saw that each allele near fixation contributes $2\mu/S$,
whereas if it has intermediate frequency, it contributes $\gamma^{2}/2$ to the
load. Since the alleles with large effects fulfill that $\gamma^{2}/2>2\mu/S$,
the genetic load would be much larger if the alleles of large effect were
maintained polymorphic. A similar argument applies for allele of small effect.
Because $\gamma^{2}/2<2\mu/S$, then the load would be significantly higher if
these alleles were near fixation. We can interpret this by thinking that the
amount of selection required to fix an alleles of small effect would need to
be considerably high, as to make them fall into a“large effect” class.
### Distribution of allelic equilibria.
We saw that alleles of large effect will be in near fixation. However, whether
they are more likely to be at the ‘+’ or ‘-’ state depends on details such as
the position of the optimum and the deviation from it. For example, in
Supplementary Information 2 we show that optima positioned towards the largest
(smallest) trait value $z_{x}$ bias alleles to the ‘+’ (‘-’) states. Can we
estimate how likely are alleles to be close to a particular state?
We assume that the trait mean is at the optimum, and focus on the state of one
particular allele. We will study how the probability $\rho$ that the focal
allele is at the ‘+’ state depends on its effect $\gamma$. In this case, we
take $\rho$ to be a probability calculated over all possible states (peaks),
where we assume that the rest of the (background) loci contribute in a way
that keep the trait at the optimum. We assume that for all alleles of large
effects the initial conditions are such that
$\Pr_{0}(`-^{\prime})=\Pr_{0}(`+^{\prime})=1/2$. Numerically, we perform many
runs that start close to uniformly randomly selected peaks, and let the system
reach equilibrium. Then we count how often alleles of effect $\gamma$ are in
the ‘+’ state. In Appendices B and C we show that that
$\rho_{j}=1\left/\left(1+\exp\left[-2\frac{z_{\circ}}{V}\sqrt{\gamma_{j}^{2}-4\frac{\mu}{S}}\right]\right)\right..$
(11)
where $V=\sum^{\prime}_{i\neq j}(\gamma_{i}^{2}-\frac{4\mu}{S})$, where
$\sum^{\prime}$ indicates summation on the set of alleles of large effect.
Figure 4 shows that the predictions of Eq. 11 are consistent with the
simulations. The distribution of effects does not affect the probability of an
allele being in the ‘+’ or ‘-’ state, only the effect of the focal allele
matters.
The larger the effect of the focal allele is, the larger the probability it is
in the ‘+’ state (this assumes positive positioning of the optimum; for
negative positioning, the converse would be true). The reason is that alleles
that are closer to the threshold value are more prone to the instabilities
resulting from small deterministic fluctuations around the optimum. Large
alleles, on the other hand, are more often at the ‘+’ state since they are
more resilient to perturbations from the optimum value. Thus, once a
population attains equilibrium, large alleles with effects close to
$\hat{\gamma}$ are much more likely to be stuck in alternative equilibria than
larger alleles.
We also find, in Fig. 4B that the larger the value of $z_{\circ}$, the larger
the probability for all loci to be in the ‘+’ peak. This is expected, because
lager trait values require more ‘+’ alleles. This obvious observation,
although supported by the model, is quantitatively underestimated by it. In
principle, deviations from the optimum trait value can be accommodated in Eq.
11 (Appendix C). But this correction, at least to first order on $\Delta z$,
does not fully account for the underestimation of the model at large optimum
values (data not shown). What actually happens is that as the optimum is
positioned closer to the range of response of the trait, the distribution of
traits is considerable skewed and the Gaussian assumption fails.
Figure 4: The probability of ‘+’ alleles increases as the magnitude of their
effect gets larger. Lines: following Eq. 11; symbols: average of occurrences
of the ‘+’ state from 100 simulations. (A) The optimum is fixed and the
distribution of allelic effects is varied. $z_{\circ}=10$ (roughly half-way
from the maximum trait value); solid line and bullets: exponential
distribution (mean=1/5); dashed line and gray squares: Gamma distribution
(shape=20, scale=1/100). Dotted line and crosses: equal effects, $\gamma=1/5$.
(B) The distribution of allelic effects is fixed and the position of the
optimum is varied. Effects distributed as an exponential (mean=1/5); thick
solid line and bullets $z_{\circ}=0$, dashed line and squares $z_{\circ}=5$,
dot-dashed line and crosses $z_{\circ}=10$, dotted line and rings
$z_{\circ}=15$, thin solid line an stars $z_{\circ}=20$. In all cases the
trait is determined by 100 loci. $\mu=10^{-4},S=0.1$. The initial conditions
were uniformly and independently drawn for each locus.
Above we saw that alleles with effects that are close to the critical point
are more susceptible to small perturbations to the optimum. Thus how stable
the trait is to small shifts of the optimum, depends on how far the
frequencies are from the critical point, which in turn depends on the
particular distribution of allelic effects. Under equal effects, all alleles
are equally far from the critical point, and thus remain stable for a long
period until the deviation is large enough. But when the deviation reaches the
critical value all alleles are perturbed at the same instant.
Under unequal effects the picture is more complex. The individual equilibria
of each allele are perturbed differently by deviations from the optimum.
Moreover, once a given allele is perturbed and placed at an alternative state,
the newly attained equilibrium is characterized by a different deviation from
the optimum, potentially perturbing yet another allele. The interplay amongst
the complex equilibria are hard to characterize in detail.
Now we will determine the size of the deviations from the optimum. By
employing perturbation analysis (Appendix B) we find that positive deviations
from the optimum push allele frequencies closer to fixation. We also prove
that the maximum deviation close to a given peak is of the order
$\tilde{\Delta}z\simeq\min_{i\in\mathbb{L}}\gamma_{i}/2$, where $\mathbb{L}$
is the set of large effects (in Appendix B we give an exact expression to the
maximum deviation). Clearly, this is bounded below by $\hat{\gamma}/2$, and
$\tilde{\Delta}z$ depends on the particular draw of effects. This limit for
the deviation is suggested by the diagram Fig. 2A: we see that the shoulders
of the black lines actually occur relatively near $\delta=1/2$ (as long as
effects are large enough). Consequently, we expect that most of the time the
traits will be fairly well adapted, and most of the load is given by the
genetic variance, rather than by large deviations of the mean from the
optimum.
### Genetic variance.
By direct substitution of Eqns. 9-10 into Eq. 4 (see Table 1) we see that the
genetic variance that is maintained by mutation-selection balance is
$\gamma^{2}/2$ per locus at the intermediate state, and $2\mu/S$ per locus
near fixation. Contrast this to the genetic variance predicted by the HoC,
which is same as for traits controlled by equal but large effects, i.e.
$\nu=2n\mu/S$. Under unequal effects , if $\Delta z=0$, the genetic variance
is
$\nu=2n_{f}\frac{\mu}{S}+\frac{1}{2}\sum_{k\in\mathbb{S}}\gamma_{k}^{2}$ (12)
where $n_{f}$ is the number of alleles of large effect, and the set
$\mathbb{S}$ contains the $n_{s}$ alleles with small effects,
$\gamma^{2}/2<2\mu/S$; clearly, $n=n_{f}+n_{s}$. Notice that the first term is
due to alleles that are close to fixation, and their contribution to the
genetic variance is independent of their effect, and the second term is due to
alleles of small effect, which are at intermediate frequency. Equation 12 is
one of our central results.
With this result we come back to Fig. 1: in panel B the genetic variance of
the trait with unequal effects is lower than the HoC. That is because 24
alleles are of small effect. Equation 12 correctly predicts the equilibrium
variance, $\nu=0.064$. However, note that if we use the HoC with $n_{f}$
(instead of $n$) then $\nu\simeq 0.052$, more than 80% of the total variance.
Thus, the HoC variance bounds the genetic variance under unequal effects.
Specifically, Eq. 12 implies that $\nu_{HoC}(n_{f})\leq\nu\leq\nu_{Hoc}(n)$,
where $\nu_{HoC}(m)$ is the HoC variance with $m$ loci. With no deviations
from the optimum, the load is proportional to the genetic variance. Under the
HoC the load is always $L=S\nu/2=n\mu$. However because under unequal effects
the genetic variance is smaller, the mutational load will also be smaller, and
dependent on the distribution of alleles.
The equilibrium genetic variance depends on the distribution
$\mathcal{P}(\gamma)$ of allelic effects. Even though alleles near fixation
contribute to $\nu$ independently of $\gamma$, the proportion of alleles of
large effects will change with $\mathcal{P}$. For example, fixing the expected
value of $\gamma$ at a value larger than $\hat{\gamma}$, but allowing the
shape of the distribution to change, will result in different proportions
$P=n_{f}/n$ of alleles of large effect (Fig. 5). In this way, we keep the
whole range of response of the trait comparable across different distributions
of effects. Distributions peaked around the mean will correspond to traits
with alleles of large effect, all of which will be near fixation (Fig. 5A,
yellow stars), thus 100% of the variance is due to alleles of large effects
and will match the HoC variance (Fig. 5B, yellow curve). For distributions
that are more spread, the traits will have mixed effects (Fig. 5 red
squares/line and green circles/line): $n_{s}$ increases to 96, with about 7%
of the variance due to alleles of small effects, and $n_{s}=342$, about 17% of
the variance is due to alleles of small effects, red and green respectively.
The extreme case will be for positively skewed distributions, such as the
exponential, where the proportion of alleles of large effects is much smaller
(Fig. 5 black diamonds), and the genetic variance will be considerably lower
than that of the the HoC (Fig. 5B, black curve): roughly half of the alleles
($n_{s}=478$) are of small effect, but contribute by 20% to the total
variance.
Figure 5: (A) Equilibria under different distributions of allelic effects.
Symbols: data from simulations. Initial frequencies were drawn uniformly in
$(0,1)$ and the system numerically evolved to equilibrium. Gray lines:
equilibria of allele frequency; symbols: numerical equilibria. (B) Cumulative
contribution to the genetic variance under different distributions of allelic
effects $\mathcal{P}(\gamma)$. Solid curves: data corresponding to the
simulations in panel A. Solid horizontal lines: equilibrium genetic variance
(Eq. 12). Dotted line: genetic variance of the HoC. $\mathcal{P}(\gamma)$ is a
Gamma distribution of mean=0.1 with shape parameters $\pi$ as: yellow, stars:
$\pi=10^{-3}$, red, squares $\pi=10^{-2}$, green, circles $\pi=5\times
10^{-2}$, black, diamonds $\pi=10^{-1}$.
$\mu=10^{-4},S=0.1,n=1000,z_{\circ}=0$.
### Distribution of phenotypic equilibria
For many loci, the number of allelic equilibria can be astronomical.
Nevertheless, under equal effects it can be calculated explicitly [Barton
1986]. When mapped to trait mean and genetic variance, the number of distinct
equilibria is smaller, since many combinations of allelic effects have
equivalent, or at least very similar, trait mean and variance. Fig. 6 shows
how the phenotypic states change when we keep the mean of the allelic effects
constant, but increase its variance: the genetic variance decreases (see also
Fig. 5), and deviations from the optimum have less effect on the genetic
variance, making Eq. 12 a good approximation. Notably, we find that the number
of values of trait mean and genetic variance increase when the distribution of
effects spreads. However, these equilibria become more similar and closer to
each other.
Figure 6: Phenotypic equilibria under different distribution of effects. Gray
circles: equal effects; dots: effects tightly clustered around the mean
(gamma-distributed with variance=1/1000); crosses: exponentially-distributed
effects (variance=1/100). In all cases the mean effect is 0.1.Points are
results from numerical calculations for 11 equidistant optima
$z_{\circ}\in[-z_{x}/2,z_{x}/2],z_{x}=\bar{\gamma}=5$, at each point employing
200 runs with uniform random initial conditions. $\mu=10^{-4},S=0.1,n=50$.
Figure 7: (A) Connectedness of macroscopic fitness peaks as the number of
different effects of a polygenic trait are systematically increased from 1 to
20. The black thick lines originate at sub-optimal equilibria, and the thin
blue lines (color online) originate at the optimal equilibrium ($\delta z=0$).
(B) Number of new equilibria derived from the sub-optimal equilibria under
equal effects (black circles and green diamonds) and at the optimum equilibria
(blue stars) as the number of different effects is increased.
$\mu=10^{-4},S=0.1,n=50,z_{\circ}=0$.
Although it is hard to count the states precisely, we can study how a given
phenotypic equilibrium is affected as the asymmetry of the unequal effects is
increased. For instance, suppose under equal effects we track a set of initial
conditions $\Pi$ that lead to a particular point in the “phenotypic”
$(\bar{z},\nu)$-space. The states to which these trajectories converge (basins
of attraction) are symmetric in the sense that exchanging ‘+’ alleles at one
locus with ‘-’ alleles at another does not change the phenotypic states. If we
keep constant the set $\Pi$, but now change one effect by a small amount, how
is the distribution of phenotypic states affected? Many of the trajectories
starting at $\Pi$ that under equal effects converged to equilibria
characterized by the same trait mean and variance will now converge to
different points in $(\bar{z},\nu)$-space. The symmetries are broken and
exchanging alleles at that locus with another one affect the trait mean and
variance. Hence, the phenotypic equilibria show bifurcations when we perturb
the effects. We can repeat this procedure by then perturbing a second locus,
and so on. Thus, if we represent the phenotypic states as nodes, and we
connect these nodes according to the initial condition that led to their
corresponding phenotypic states, we will have a graph that represents the
increase in complexity of the adaptive peaks. Unless we have a way to cover
the initial space uniformly, this does not ensure complete counting of the
number of phenotypic equilibria. However, even if the subsampling of the
basins of attraction is poor, the method quantifies how complex the space
becomes as we increase the variance of the allelic effects.
If we carry out this procedure for different phenotypic states under equal
effects, then we will have several of these graphs. If these graphs share
nodes, then it means that the adaptive landscape is more accessible to better-
adapted equilibria (because under unequal effects the equilibria have less
genetic load, Fig. 6). In fact, in Fig. 7 we see an example for three graphs
derived from the optimal peak and two sub-optimal peaks.
Altogether, this exposes that under unequal effects the fitness landscape is
more complex or “rough”, but the solutions are generally closer to the
optimum. Surprisingly, perturbing slightly the effect of only one or two loci
is enough to overlap different graphs with common nodes, indicating that
unequal effects act like a funnel to guide the trajectories to nearly optimal
states.
## Initial response to selection
We saw that there are two well-defined regimes which clearly separate alleles
of large effect from alleles of small effect. If the optimum shifts, which
alleles respond first? A related question is: is the initial rate of change of
the trait driven mainly by alleles of large or small effect? Although these
two questions are related, they are not the same; even if, for example alleles
of small effect sweep first, they might not drive a substantial displacement
of the trait. Conversely, even if an allele of large effect sweeps first, its
overall effect might be negligible when compared to a background of very many
loci of small effect.
To calculate the rate of response of an allele, assume that the population is
at equilibrium at a local peak with no deviation from the optimum that is at
$z_{\circ}$. Suddenly, the optimum is placed at another value $z_{f}$.
Equation 6 implies that at each locus
$\frac{dp}{dt}=2S\gamma pq\Delta\Omega~{},$ (13)
where $\Delta\Omega=z_{f}-z_{\circ}$. For alleles of small effect, the right
hand side is $\frac{S\gamma}{2}\Delta\Omega$. Therefore, alleles with
infinitesimally small effects will be nearly neutral and will have a
vanishingly small rate of response to selection. As the effects become closer
to (but still smaller than) $\hat{\gamma}$, the rate of response is larger.
Consider now alleles that have infinitely large effects. The right hand side
of Eq. 13 is $\frac{2\mu}{\gamma}\Delta\Omega$, and implies that since these
alleles will be almost fixed, there is little variation to select on.
Consequently, their rate of response is also vanishingly small. As the effect
become closer to (but still larger than) $\hat{\gamma}$, the rate of response
become larger. Therefore, those alleles with effects close to $\hat{\gamma}$
will have the earliest response to selection because they are the most
sensitive to deviations from the optimum. Thus, the maximum response for each
limit is given by the effect that is exactly at the critical
value$\hat{\gamma}$. Evaluating Eq. 13 at $\hat{\gamma}$ we find that the rate
of response is at most
$\left(\frac{dp}{dt}\right)_{\max}=\sqrt{\mu S}\Delta\Omega~{},$ (14)
indicating that alleles with effects close to $\hat{\gamma}$ drive the initial
response of the trait to selection.
## Long-term response to selection
The long-term response of a polygenic character to the displacement of the
optimum trait value can be driven by alleles other than those of intermediate
effect. Although the alleles of large effect evolve slowly in the beginning,
they can eventually gain representation and evolve much faster. A general
closed solution for the dynamics is neither possible nor useful, as the
behaviour of the allele frequencies is rather complicated. The question is
whether the theory developed above can be useful to gain insight into the
long-term response of the trait.
### Abrupt displacement of the optimum.
We will assume that re-positioning the optimum happens always within the range
of response of the trait, and far from the extreme values given by
$z_{x}=\sum\gamma_{i}$; i.e. $-z_{x}<<z_{\circ}<<z_{x}$. The equilibrium
analyses revealed that the particular position of the optimum is not decisive
for equilibria or stability. Instead the deviation from the optimum is the
important factor. Thus, if the population eventually adapts to the new optimum
value, the genetic variance that is maintained at the newly established
equilibrium will be more or less the same as in the beginning. In the
transient time, the dynamics will be complicated and depend on the specific
initial conditions (the adaptive peaks where the population initially stands),
and on the distance to the optimum.
If the optimum changes abruptly and is larger in magnitude than the largest of
the effects, all the equilibria will be perturbed, favouring an increase in
the frequency of the those alleles that diminish the deviation from the
optimum. That is, if the new optimum value is smaller (larger) than that of
the original optimum value, ‘-’ (‘+’) alleles will increase in frequency. This
displacement is seen in the diagram of Fig. 3 towards the top (where only one
stable border is initially beneficial), and then a gradual return of the line
to low values of $\Delta z$. Therefore, we expect to observe a transient
increase in the genetic variance. Fig. 8 provides an example where these
patterns are in fact found.
In the example of Fig. 8, of 50 loci, 26 are of large effect and contribute
more than 80$\%$ of the initial variance, whereas 24 are of small effect,
contributing the remaining 20$\%$ of the variation. In Fig. 8C we see that
many of the large alleles shift in frequency and some sweep, transiently
raising the genetic variance. In this case, since the optimum shifters from
$z_{\circ}=2$ to $z_{\circ}=-2$, the ‘+’ (‘-’) alleles decrease (increase) in
frequency. Alleles of small effect are displaced, but not substantially. Most
of the transient variation that is generated is due to sweeps of alleles of
large effects.
As hypothesized above, even if transient dynamics are very complex (Fig. 8),
when a new equilibrium is attained, the final deviations from the optimum are
small, and the genetic variance is close to that of Eq. 12 (see also Fig. 1).
As we saw in the introduction, under equal effects the population will evolve
to a sub-optimal state where plenty of genetic variation is maintained. Why do
populations end up better adapted under unequal effects?
At first we might think that a bulk of alleles of small effects could provide
enough background variation, allowing the population to explore the genetic
space more efficiently. However, the response of a trait constituted only by
alleles of large effect is virtually the same as that of a trait that contains
also alleles of small effect, in as much as the initial genetic variation
contributed by the latter alleles is small, and the optimum is not too close
to the maximum trait value (Supplementary Information 3). Thus, concerning the
response, alleles of small effect can be regarded as nearly neutral.
Another plausible explanation lies in the rate of “beneficial mutations” (in
the sense that these are mutations that approach the trait mean to the new
optimum). Because under equal effects the response of allele frequencies is
synchronized, most mutations are initially beneficial. However, due to the
epistatic nature of the fitness landscape, those initially beneficial
mutations are not necessarily beneficial once the rarer alleles have increased
their representation in the population, and might even become detrimental.
Furthermore, these alleles arise and increase their frequency on the same time
scale. Under unequal effects different mutations arise at different times and
can compensate the load contributed by previous mutations. In fact, because of
epistasis, we expect and in fact we find (Fig. 8C), that some alleles that are
initially beneficial increase in frequency, but afterwards, become detrimental
and decrease in frequency again. This “prevention of sweeps” has been observed
in polygenic traits with up to 8 loci [Pavlidis et al. 2012]. However, under
equal effects allele frequencies remain synchronized along evolution, and it
is unlikely that the initial conditions and the shift in the optimum are in
general finely tuned in such a way to allow the population to reach a local
peak that is close to the global optimum.
Figure 8: Response to an abrupt displacement of the optimum of a polygenic
trait. (A) Deviation of the trait mean from the newly positioned optimum. (B)
Genetic variance. Black: exact numerical results. Dashed black line: house of
cards prediction, ($\nu=2n\mu/S$). Dotted black line: exact value from Eq. 12.
(C) Response of the allele frequencies; black lines: alleles of large effect;
thin red lines (color online): alleles of small effect. The trait is
constituted by $n=50$ loci; 26 of large effects, and 24 of small effects,
distributed as an exponential of mean $=1/10$.
$\mu=10^{-4},S=10^{-1},z_{\circ}\simeq-2$.
Since the previous examples suggest that adaptation is driven mostly by
alleles of large effect, an interesting question that follows is: what happens
when traits are controlled principally by alleles of small effect? First of
all, Eq. 12 indicates that the genetic variance will be much lower than that
of the HoC. We find that the population eventually adapts (although somewhat
slower), but with virtually no change in the genetic variance. In this
example, the trait has only three alleles of large effect, which contribute by
11% of the genetic variance, and by 397 loci with alleles of small effect,
which contribute the remaining 89 % of the variation. The three alleles of
large effect sweep, but ultimately don’t affect the genetic variation
substantially (assuming their HoC contribution), and some alleles of small
effect are strongly shifted.
Figure 9 shows the response of a system of alleles of principally small
effects. Although 400 loci determine the trait, we estimate that the effective
number of alleles $n_{e}\simeq 28$ (see Supplementary Information 4). Assuming
constant genetic variance given by the HoC (but using $n_{e}$) , we find that
directional selection towards the optimum explains the response of the trait
(although it fails to predict a minor final deviation from the optimum). This
experiment highlights why we might find stasis of the genetic variance and a
sustained response to selection, which is caused by innumerable alleles of
small effect. Under these circumstances, although experimental essays would be
able to detect only a few major loci [Hindorff et al. 2009, Visscher et al.
2012], these turn out to be the least relevant to explain the quantitative
genetic variation.
Figure 9: Response to an abrupt displacement of the optimum of a polygenic
trait constituted mainly by alleles of small effect. (A) Deviation of the
trait mean from the newly positioned optimum. Black line: exact numerical
results; Dashed gray line: approximation assuming an effective number of loci,
$n_{eff}=28$ and constant genetic variance, $\nu\simeq 2n_{e}\mu/S$ (see
text). (B) Genetic variance. Black: exact numerical results. Dashed black
line: house of cards prediction, ($\nu=2n\mu/S$). Dotted black line: exact
value from Eq. 12. (C) Response of the allele frequencies; black lines:
alleles of large effect; thin red lines (color online): alleles of small
effect. The trait has $n=400$ loci; 3 of large effects, and 378 of small
effects, distributed as an exponential of mean $=1/80$.
$\mu=10^{-4},S=10^{-1},z_{\circ}\simeq-2$.
### Slowly moving optimum.
If the optimum is shifted slowly enough, the deviation $\Delta z$ remains very
small. We also see that the genetic variance then hardly changes (for example,
Fig. 10). After some time the population keeps evolving but reaches a
stationary state. If the optimum suddenly stops, the population settles at a
state characterized by a lower genetic variance, but larger deviation from the
optimum as in the case when it adapts to a rapid shift of the optimum, as in
the previous section. How can we explain these patterns?
Figure 10: Response to a gradually shifting optimum of a polygenic trait. (A)
Deviation of the trait mean from the newly positioned optimum. (B) Black:
exact numerical results. Dashed black line: house of cards prediction,
($\nu=2n\mu/S$). Dotted black line: exact value from Eq. 12. (C) Response of
the allele frequencies; black lines: alleles of large effect; thin red lines
(color online): alleles of small effect. The optimum linearly moves from
$-z_{\circ}$ to $z_{\circ}$ between $t=0$ and $t=10^{4}$, and afterwards stays
constant at $z_{\circ}$. Other parameters as in Fig. 8.
Under the infinitesimal model the population achieves a stationary lag from
the optimum given by $\Delta z^{*}=-\kappa/2S\nu$, where $\kappa$ is the speed
of the moving optimum [Lynch and Lande 1993, Jones et al. 2004]. However, we
see in Fig. 11 that this approximation fails for finite number of loci of
unequal effects.
Suppose that a moving optimum changes linearly in time:
$z_{\circ}(t):=\Omega_{0}+(\Omega_{f}-\Omega_{0})t/T$. For simplicity we will
consider optima starting at $-\Omega$ and ending at $\Omega$. Hence the speed
of the moving optimum is $\kappa=2\Delta\Omega/T$.
By summing Eq. 6 over loci and using Eqns. 3-4, we find that during transient
evolution the deviation from the optimum is given by
$\frac{d\Delta z}{dt}=-2\nu\Delta z+Sm_{3}-2\mu\bar{z}-\kappa$ (15)
where $m_{3}=\sum_{i}\gamma_{i}^{3}p_{i}q_{i}(2p_{i}-1)$ is the third moment
of the allelic effects. If the deviation from the optimum reaches a stationary
state $\Delta z^{*}$ where $\frac{d\Delta z^{*}}{dt}=0$, then:
$\Delta z^{*}=\frac{\kappa+2\mu\bar{z}-Sm_{3}}{2S\nu}.$ (16)
Under the infinitesimal model, the breeding values are normally distributed
which implies that $m_{3}=0$. The genetic variance due to mutational effects
is finite, but the mutation rate decreases with the inverse of $n$ and the
term $\mu\bar{z}$ can be neglected. Consequently, Eq. 16 reduces to the
approximation of the infinitesimal model [Lynch and Lande 1993, Jones et al.
2004]. What limits are then necessary from the point of view of our model with
a finite number of loci?
The stationary lag is not a constant; it represents a quasi-equilibrium state,
and so we need to know how $\bar{z},\nu$ and $m_{3}$ change in time. This is
not feasible in an exact way, except under restrictive limits such as the
infinitesimal model. Even under other simple assumptions, such as the HoC,
predicting higher moments is hard [Barton 1986, Barton and Turelli 1987,
Bürger 1991].
Figure 11 shows that the third moment of allelic effects, $m_{3}$, is relevant
for an accurate prediction of the lag. In fact, if all terms of Eq. 16 are
considered, there is virtually no distinction between the stationary lag
approximation and the actual lag. However, neglecting the third moment does
affect the prediction substantially. But the extent to which $m_{3}$ is
relevant, depends on the distribution of effects.
Fig. 11B shows an example for a trait constituted only by alleles of small
effects. The third moment is small, and neglecting it leads to a good
approximation of the stationary lag. This is consistent with the infinitesimal
model as a limit of many loci of small effects.
When traits are determined by alleles of large but equal effects the
distribution of allelic effects is also asymmetric. As the optimum advances,
traits with unequal effects allow many small adjustments. These gradual
changes allow fine tuning of the deviation from the optimum. Under unequal
effects, this results in high frequency but low amplitude fluctuations of the
lag. However, under equal effects the allele frequencies change in a
coordinated fashion and the equilibria are more robust to deviations from the
optimum. Thus, we observe fewer but larger fluctuations.
Figure 11: Stationary lag approximation for the response of polygenic traits
to a steadily moving optimum. In all cases the trait is is constituted by
$n=1000$ loci. The optimum moves steadily shifting from -$\Omega$ to $\Omega$
in $T=300,000$ time units. The value of $\Omega$ was chosen to match the
random initial condition. Black: lag $\bar{z}-z_{\circ}$; dotted red (color
online) stationary lag $\Delta z^{*}$ (from Eq. 16); dashed purple (color
online) stationary lag neglecting the third moment of the allelic effects. The
insets compare $\bar{z}$ (black) and $z_{\circ}$ (dashed pink, color online)
along all times. The trait is determined by (A) exponentially distributed
effects (mean=0.1), $n_{s}=435,\Omega=48$; (B) exponentially distributed
effects (mean=0.01) $n_{s}=1000,\Omega=3.72$, and (C) Equal effects with mean
$\simeq 0.1,n_{s}=0,\Omega=-46$. In all cases $S=0.1,\mu=10^{-4}$.
Comparing Figs. 11 A and B we find that traits with mixed allelic effects
initially respond smoothly, but eventually enter a highly fluctuating phase.
This does not happen when all the effects are small. We must point out that
strong fluctuations can occur when the optimum is very close to the range of
response of the trait. In the Supplementary Information 5 we show that
different initial conditions converge to the same erratic trajectories, and
show that these are not chaotic, but quasi-periodic (deterministic but
unpredictable fluctuations of many different frequencies, characterized by a
zero Lyapunov exponent).
The existence of the quasi-periodic phase explains why, if the optimum
suddenly halts, the population remains stuck at a local optimum. This is
because the populations are not able to wander freely in the fitness space.
Being driven by the moving optimum, they are forced to stay in states that
keep a certain deviation from it. In turn, in equilibrium, this poses some
directional selective pressure, which biases even further the allele
frequencies, resulting in a loss of genetic variation.
In the Supplementary Information 5 we also study a moving optimum that
oscillates smoothly with different frequencies and amplitudes. The lag enters
a periodic phase of many frequencies (i.e. it is not smooth), and the
fluctuations increase as the frequency and the amplitude increase. We also
study damped oscillations. Surprisingly, once the oscillations stop the
population end up better adapted when compared with linearly moving optima.
## Discussion
Our analyses help to understand the relative contributions of alleles of large
and of small effect in the maintenance of genetic variation and in the
response to stabilizing selection. Interestingly, our analysis questions
whether the specific distribution of allelic effects is relevant. Naturally, a
more robust interpretation of these results requires understanding how genetic
drift affects the distribution of allele frequencies. In this model, alleles
of small effect are at intermediate frequencies. However these will be fixed
by genetic drift. This could induce major changes to our results when there
are many alleles of small effect. Thus, selection on many traits and genetic
drift might change the picture substantially by maintaining larger variation,
even though the details of the distribution of allelic effects might not be
relevant.
Under genetic drift, the eventual fate of any allele is fixation or loss.
Although drift may seem an additional complication, it also has an interesting
effect, namely to allow access to parts of the fitness landscape that were
inaccessible from a given state of a deterministic population [de Vladar and
Barton 2010]. In this sense, genetic drift smooths the landscape, and although
stochastic effects are introduced, the expected trajectories are somewhat
regularized, because the populations can easily escape suboptimal peaks
[Wright 1935, Barton 1989], converging to fitter states. In this sense, drift
aids adaptation, allowing alleles to jump across peaks by mutation and genetic
drift [Wright 1931, Wright 1932, Coyne et al. 1997].
The stochastic HoC [Bürger 2000, P. 270] shows that, on average, each locus at
equilibrium contributes to the genetic variance by
$2\gamma^{2}N\mu\left/\left(1+\gamma^{2}NS\right)\right.$. Thus, the expected
genetic variation, $\langle\nu\rangle$, is smaller than the $\nu_{{HoC}}$.
Alleles of small effect will be fixed by drift, but their contribution is
still small compared to alleles of large effect. Consequently, on average, the
response of the trait is slower in finite populations than in infinite
populations, which was already observed by us for the case of equal effects
[de Vladar and Barton 2010]. However, in a strong selection regime, i.e.
$\gamma^{2}NS>>1$, even alleles of small effect ($gamma^{2}NS<\mu$) will be
near fixation. Thus for sufficiently large populations, the HoC approximation
should hold. However, the problem is far from trivial because these fixed
alleles of small effect can further induce deviations of the trait mean from
the optimum.
These analyses assume that at equilibrium the trait is well adapted. Other
factors such as asymmetric mutation rates can maintain a deviation from the
optimum [Charlesworth 2013]. In this case, a stationary population effectively
experiences directional selection, and maintains even more genetic variance
than when the trait matches the optimum. In our analyses we find in some cases
that when traits deviate from the optimum, there is more genetic variance.
Under asymmetric mutation rates, as in Charlesworth’s model, the deviation
from the optimum is maintained by two opposing forces: an asymmetric flux of
mutations, and directional selection towards the optimum. In our model the
populations simply stand at a suboptimal peak.
### Selection on many traits and pleiotropy.
It is often argued that stabilizing selection acts on multiple traits. Under a
common polygenic basis, two traits that are subject to antagonistic selective
pressures remain at an intermediate value that is a compromise amongst the
optimal solutions. Alleles of large effect that are not subject to these
pleiotropic effects can contribute significantly for the response to
selection, even though when their contribution to the genetic variance is
negligible due to the large number of loci [Kelly and Rausher 2009, provide
many examples]. In this section, we show that our previous result are relevant
in this larger context.
A more general calculation for many traits under selection shows that if
several traits are all adapted to their optima, there are still two classes of
alleles: near fixation and at intermediate frequency, but the criterion for
locus $i$ to be near fixation is
$\Gamma_{i}\equiv\sum_{k}\gamma_{ki}^{2}S_{k}>4\mu$, where $S_{k}$ is the
selection strength on trait $k$ and $\gamma_{ki}$ is the allelic effect of
locus $i$ on trait $k$ (Appendix D). However, if the optimum for one trait
favors alleles at the ‘+’ state, and the optimum of the other trait favor
alleles at the ‘-’ state, then the net effect of selection on an allele might
partly neutralize. In this case, deviations from the optima will exist and
both alleles will be maintained at intermediate frequency, and the genetic
variance in the population will be high, as they will contribute by
$\gamma_{ki}^{2}/2$ even if $\gamma_{ki}>\hat{\gamma}$.
For multiple traits that share a polygenic basis, only a few principal
components will experience strong stabilizing selection; all the other
components will be subject to only weak selection, otherwise the genetic load
would be prohibitively large [Barton 1990]. Hence, it remains unclear when
(and unlikely that) a particular focal trait is the main component of fitness
[Barton 1990]. Therefore, if the stabilizing nature of selection is attributed
to pleotropic factors, the equilibrium genetic variance will be decreased
[Barton 1990, Turelli 1985, Slatkin and Frank 1990]: if the strength of
selection on the $M$ traits are the same we get that $\nu=\nu_{HoC}/M$.
The observed differences in fitness can be due to other correlated traits, as
explained above, or to pleiotropic effects directly affecting fitness. For
morphological traits, the distribution of allelic effects is positively
correlated with fitness effect [Keightley and Hill 1990]. However it remains
difficult to disentangle whether pleiotropy or multivariate selection is the
acting mode of fitness reduction [Barton 1990, Zhang and Hill 2003]
Under antagonistic selection the picture is different. The HoC model for many
traits [Turelli 1985] shows that the genetic variance of one trait depends on
the strength of selection of the other traits (even if the traits are
uncorrelated). In this case each locus near fixation contributes by
$2\gamma_{1i}\gamma_{2i}\mu/\Gamma_{i}$. However, we must consider that the
condition $\Gamma_{i}>4\mu$ is dependent not only on the distribution of
allelic effects, but also on the distribution of selective coefficients $S$.
If the latter has a mean of zero and small variance (weak selection), the
fixation condition would be hard to fulfill, and most alleles will be at
intermediate frequency leading to high genetic variance (consistent with ?)).
### Admixed populations and genetic incompatibilities.
Suppose that two populations that are genetically differentiated come into
contact. Will a subsequent admixture result on maladapted offspring? In SI 6
we show that the admixed population necessarily has larger genetic variance
than the source populations, even if the latter have the same trait mean and
variance. This is because the populations might be at different adaptive peaks
that have the same or very similar phenotypic distribution. However, there are
$\tilde{n}$ loci with distinct alleles in each population, which cause the
excess variance relative to the parental mean. This will be caused by alleles
of large effects, each one contributing by
$\gamma_{i}^{2}-2\tilde{n}\frac{\mu}{S}$. This can be interpreted as the
expression of genetic incompatibilities between the two divergent populations,
and emphasizes the role of stabilizing selection and epistasis in the process
of speciation [Barton 1989, Barton 2001]. (However, this mechanism is of a
different nature than the paradigm of Dobzhansky-Müller incompatibilities).
After secondary contact the population might develop isolation and re-adapt to
its original state, retaining the incompatible alleles, or it might hybridize
and re-adapt to a new state. This will depend on the initial degree of
admixture, but also on the otherwise negligible deviations from the optimum as
well as on genetic drift, factors that we have not considered.
### SNPs as genomic signatures of stabilizing selection.
Under the assumptions of our analyses, most loci with high heterozygosity will
have small effects, whereas alleles of large effect will have much lower
heterozygosity, a result consistent with early results of the neutral theory
[Kimura 1969]. In turn, our results support the well-known idea that there can
be substantial measurement bias in the estimation of allelic effects from QTL
or GWAS: alleles of large effect will be harder to detect than polymorphisms
of alleles with small effect. For instance, most effects that we can map are
expected to be small. This is consistent with the knowledge that most alleles
have small effects. Furthermore, if alleles of large effect are common, our
results indicate that they will be close to fixation, and thus rare in the
population, and consequently less likely to be detected.
Ignoring drift and equating $n_{s}$ to the SNPs on a genome of size $n$, and
the proportion of fixed alleles to $P=n_{f}/n$, and assuming that this
proportion is homogeneous not only across the genome, but also across the set
of loci that affect different traits (questionable suppositions of course),
this implies that traits are approximately at a fraction $P$ of the total
genetic variation, as we saw above. The ratio $n_{s}/n$ is on the order of
1:100 or 1:1000, thus $P\simeq.99$. The evolution of these traits are
mutation-limited rather than by standing genetic variability. These
estimations assume linkage equilibrium, as are the SNPs identified by GWAS,
which are often spread across the genome. (Clearly, this does not apply within
genes, coding or regulatory sequences, as linkage is tight).
Different populations that show similar trait distribution and genetic
variation may still differ at individual SNPs, especially if these have large
effects. Thus, a particular allele might not be uniquely associated with a
particular trait, even if they are causally related. This justifies and is
consistent with GWAS findings that several variants can be associated with
different alleles. What we have shown is that these causal variants are
expected to contribute equally to the genetic variance, irrespective of the
specific genetic makeup of the quantitative trait(s).
### Acknowledgements.
The authors would like to thank Tiago Paixao and Daniel Weissman for the
discussions. This project was funded by the ERC-2009-AdG Grant for project
250152 SELECTIONINFORMATION.
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## Appendix A: Bifurcation points
The bifurcation points occur when there is a change in the stability of the
equilibrium allele frequencies by varying a parameter. This means that, in
addition to the equilibrium condition $dp/dt=0$, we also require that the
eigenvalue vanishes at the point of equilibrium. If we rescale the equations
in terms of $\delta$ and $m$, we have to solve the following system:
$\begin{cases}m(1-2p)-(1-p)p(2\delta-2p+1)=0&\\\
-2\delta-2m+2p(2\delta-3p+3)-1=0&\end{cases}$ (17)
By eliminating $p$ from the two equations we get that
$8\delta^{2}\left(2\delta^{2}+2(m-5)m-1\right)=(4m-1)^{3}$ (18)
This formula defines the boundary in the diagram of Fig. 3 in the main text.
Clearly. if there are no deviations from the trait, $\delta=0$, the right hand
side gives the critical value $\hat{m}=1/4$. Also, as $m$ vanishes
$\delta\rightarrow 1/2$. In general, the last equation gives the boundary for
any arbitrary deviation.
We can also eliminate $m$ and get the allele frequency $p$ at which the
bifurcation occurs, given by the solutions to the cubic:
$-8p^{3}+4(\delta+3)p^{2}+(-4\delta-6)p+1+2\delta=0$ (19)
Notice that in order to keep allele frequencies $0\leq p\leq 1$, it must be
fulfilled that $-1/2\leq\delta\leq 1/2$.
## Appendix B: Perturbation analysis for small deviations from the optimum
Consider a approximate solution to Eq. 6 expressed as $p=P_{0}+(\Delta
z)P_{1}+(\Delta z)^{2}P_{2}+O[(\Delta z)^{3}]$. The time derivative of $p$
neglecting terms of order $(\Delta z)^{3}$ is
$\begin{array}[]{lcl}\dot{p}&\simeq&\dot{P}_{0}+(\Delta z)\dot{P}_{1}+(\Delta
z)^{2}\dot{P}_{2}\\\
&=&(1-2P_{0})\left[\mu-S\gamma^{2}P_{0}(1-P_{0})\right]+\\\ &&+(\Delta
z)\left[-S\gamma\left(2P_{0}(1-P_{0})(1-\gamma P_{1})+\gamma
P_{1}(1-2P_{0})^{2}\right)-2\mu P_{1}\right]+\\\ &&+(\Delta
z)^{2}\left[S\gamma\left(\left(2P_{0}-1\right)P_{1}\left(3P_{1}+2\right)+6\left(P_{0}-1\right)P_{0}P_{2}+P_{2}\right)-2\mu
P_{2}\right]~{}.\end{array}$ (20)
The unperturbed solutions for $P_{0}$ are those given in the main text, i.e.
Eq. 9. In equilibrium, we require the terms proportional to $\Delta z^{m}$ in
the last equation to vanish, which gives the following two solutions for
alleles of small and of large effects respectively:
$P_{1}=\left\\{\begin{array}[]{lc}\frac{S\gamma}{S\gamma^{2}-4\mu}&\left(\gamma^{2}<4\mu/S\right)\\\
\frac{2\mu/\gamma}{4\mu-S\gamma^{2}}&\left(\gamma^{2}>4\mu/S\right)\end{array}\right.$
(21)
Notice that both quantities are negative.
The second order perturbations give $P_{2}=0$ for alleles of small effect, and
for alleles of large effect
$P_{2}=\pm\frac{4\mu\sqrt{\gamma S}(\mu+\gamma S)}{(4\mu+\gamma S)^{5/2}}$
(22)
where the sign indicates whether the allele is in the ‘+’ or ‘-’ state.
#### Deviations from the optimum.
We can estimate the deviation from the optimum by summing over all alleles.
This leads to a quadratic equation for $\Delta z$ with the following solution
$\Delta
z=\frac{1-\zeta_{1}}{2\zeta_{2}}\pm\left[\left(\frac{1-\zeta_{1}}{2\zeta_{2}}\right)^{2}-\frac{\zeta_{0}-z_{\circ}}{\zeta_{2}}\right]^{1/2}~{},$
(23)
where the factors $\zeta_{k}$ are
$\zeta_{k}=2\sum_{i=1}^{n}\gamma_{i}{P_{k}}_{i}$ (24)
(${P_{k}}_{i}$ is the $k$’th perturbation at the locus $i$).
#### Maximum trait deviation.
Taking the solution for alleles of large effect, we can calculate what is the
maximum allowed deviation from the trait, $\tilde{\Delta}z$. For this, we
equate the allele frequency to zero (for positive deviations) or equivalently
to one (for negative deviations). Assuming positive deviations:
$\min{p}=0=\frac{1}{2}\left[1-\sqrt{1-\frac{4\mu}{s\gamma^{2}}}\right]+(\tilde{\Delta}z)\frac{2\mu/\gamma}{4\mu-S\gamma^{2}}$
(25)
Which gives a deviation of
$\tilde{\Delta}z=\gamma\left(\frac{S\gamma^{2}}{4\mu}-1\right)\left[1-\sqrt{1-\frac{4\mu}{S\gamma^{2}}}\right]$
(26)
Notice that the larger effects tolerate larger deviations. Hence we need to
take the minimum of all these deviations, to ensure stability for all alleles.
This is thus given by the smallest allele of large effect. Assuming that
$S\gamma^{2}>>4\mu$ the expression above simplifies to
$\tilde{\Delta}z=\frac{\gamma}{2}\left(1-\frac{3\mu}{S\gamma^{2}}\right)~{},$
(27)
and for alleles of extremely large effect, the deviations can be at most of
order $\tilde{\Delta}z\simeq\frac{\gamma}{2}$, as reported in the main text.
## Appendix C: Probability of allelic states
In this appendix we derive the probability $\rho$ of finding an allele at the
‘+’ state, that is, Eq. 11 on the main text.
As mentioned in the main text, we assume that the trait mean has a value
$Z=\bar{z}=z_{\circ}$, and calculate the probability $\rho$ that given allele
$X$ is at the ‘+’ state. That is $\rho\equiv\Pr[X=1|Z=\bar{z}]$. We first
decompose this probability using Bayes’ theorem:
$\Pr[X=1|Z=\bar{z}]=\Pr[X=1]\frac{\Pr[Z=\bar{z}|X=1]}{\sum_{y}\Pr[Z=\bar{z}|X=y]Pr[X=y]}.$
Then, express the trait mean as the sum over loci
$\bar{z}=\sum_{i}(2x_{i}-1)\sqrt{\gamma_{i}^{2}-4\mu/S}$, where $x_{i}$
indicates whether the allele is close to $x=1$ or $x=0$. Here we assumed that
the background alleles are near fixation. Summarizing:
$\begin{array}[]{ccl}\Pr[\bar{z}|x_{j}=1]&=&\Pr[\sum_{i}(2x_{i}-1)\sqrt{\gamma_{i}^{2}-4\mu/S}|x_{j}=1]\\\
&=&\Pr[\bar{z}-\sqrt{\gamma_{j}^{2}-4\mu/S}]\end{array}$ (28)
Since the trait is a sum over independent loci, we can approximation that the
trait distribution is normal (central limit theorem). Its variance $V$, is
given by summing over the background loci of large effect:
$V=\sum_{i\neq j}\left(\gamma_{i}^{2}-\frac{4\mu}{S}\right)~{}.$ (29)
Now assume that the initial configurations between ‘+’ and ‘-’ alleles are
chosen uniformly, i.e. $\Pr[X=1]=\Pr[X=0]=1/2$. The sum in the denominator of
Bayes’ theorem involves only two gaussian terms. Putting the pieces together
this leads to Eq. 11:
$\rho_{j}=1\left/\left(1+\exp\left[-2\frac{z_{\circ}}{V}\sqrt{\gamma_{j}^{2}-4\frac{\mu}{S}}\right]\right)\right.~{}.$
(30)
To accommodate deviations from the optimum trait value, we proceed in a
similar way, but using the first order perturbation on the allele frequencies.
Notably, the variance V does not change, since all alleles are displaced
proportionally to $\Delta z$. We arrive at the expression
$\rho_{j}=1\left/\left(1+\exp\left[-2\frac{1}{V}\sqrt{\gamma_{j}^{2}-4\frac{\mu}{S}}\left(z_{\circ}+\frac{\Delta
zS\gamma^{2}_{j}}{S\gamma^{2}_{j}-4\mu}\right)\right]\right)\right.~{}.$ (31)
Notice that the term proportional to $\Delta z$ denotes the strength of the
directional selection component on allele $j$. For alleles of very large
effect ($S\gamma^{2}_{j}>>4\mu$, and the term is approximately $\Delta z$,
which as we saw before is very small. Thus the deviations from the optimum
affect mainly alleles of large effect that are closer to the critical value
$\hat{\gamma}$.
## Appendix D: Stabilizing selection on multiple traits
We consider a simple extension of our model, where stabilizing selection acts
independently on many traits. Call $\boldsymbol{z}=(z_{1},\ldots,z_{m})$, an
array of $m$ traits that are under selection
$\boldsymbol{S}=(S_{1},\ldots,S_{m})$, and $\boldsymbol{z}_{\circ}$ their
corresponding optima. Calling $\mathcal{S}=\text{diag}(S)$, we define fitness
as
$W_{\boldsymbol{z}}=\exp\left[-(\boldsymbol{z}-\boldsymbol{z}_{\circ})\cdot\mathcal{S}\cdot(\boldsymbol{z}-\boldsymbol{z}_{\circ})^{T}/2\right)$,
where “$\cdot$” represents inner product. In principle we could accommodate
correlation selection in the model by allowing the matrix $\mathcal{S}$ to
have non-zero off-diagonal elements, but we leave out that possibility at the
moments. Under weak selection, mean fitness becomes
$\bar{W}\simeq\exp\left[-\frac{1}{2}(\Delta\boldsymbol{z}\cdot\mathcal{S}\cdot\Delta\boldsymbol{z}^{T}+\boldsymbol{S}\cdot\boldsymbol{\nu})\right]$
(32)
where $\Delta\boldsymbol{z}=(\bar{\boldsymbol{z}}-\boldsymbol{z}_{\circ})$ is
the vector of deviations from the optima and $\boldsymbol{\nu}$ is the vector
of genetic variances.
The trait means and genetic variances are
$\displaystyle\bar{z}_{k}=\sum_{i=1}^{n}\gamma_{ki}(2p_{i}-1)$ (33)
$\displaystyle\nu_{k}=2\sum_{i=1}^{n}\gamma_{ki}^{2}p_{i}q_{i}$ (34)
where $\gamma_{ki}$ is the allelic effect of locus $i$ on trait $k$. We are
assuming that all $n$ alleles contribute to $m$ traits. This can be relaxed by
simply assuming that $\gamma_{ki}=0$ for some $i,k$.
The equilibria for this system is given by
$0=-p_{i}q_{i}\left[2\beta_{i}-\Gamma_{i}(1-2p_{i})\right]+\mu(1-2p_{i})$ (35)
where
$\displaystyle\beta_{i}=\sum_{k}^{m}S_{k}\Delta z_{k}\gamma_{ki}$ (36)
$\displaystyle\Gamma_{i}=\sum_{k}^{m}S_{k}\gamma_{ki}^{2}$ (37)
We find that the two quantities above take the role of the deviation from the
optimum and allelic effects on the single trait model. In fact, Eq. 7 in the
main text holds, when we define $\delta=\beta/\Gamma$ and $m=\mu/\Gamma$.
Consequently, the critical points, the scaled equilibria, and their stability
are the same.
|
arxiv-papers
| 2014-04-03T17:13:29 |
2024-09-04T02:50:00.691330
|
{
"license": "Public Domain",
"authors": "Harold P. de Vladar and Nick Barton",
"submitter": "Harold P. de Vladar",
"url": "https://arxiv.org/abs/1404.1017"
}
|
1404.1116
|
# Resolving Multi-path Interference in Time-of-Flight Imaging via Modulation
Frequency Diversity and Sparse Regularization
Ayush Bhandari
Massachusetts Institute of Technology &Achuta Kadambi
Massachusetts Institute of Technology Refael Whyte
University of Waikato and
Massachusetts Institute of Technology &Christopher Barsi
Massachusetts Institute of Technology Micha Feigin
Massachusetts Institute of Technology &Adrian Dorrington
University of Waikato Ramesh Raskar
Massachusetts Institute of Technology [email protected]
[email protected] [email protected] [email protected] [email protected]
[email protected] [email protected] [email protected]
Ayush Bhandari Resolving multi-path interference in time-of-flight imaging
Resolving Multi-path Interference in Time-of-Flight Imaging Time-of-flight
(ToF) cameras calculate depth maps by reconstructing phase shifts of
amplitude-modulated signals. For broad illumination or transparent objects,
reflections from multiple scene points can illuminate a given pixel, giving
rise to an erroneous depth map. We report here a sparsity regularized solution
that separates $K$ interfering components using multiple modulation frequency
measurements. The method maps ToF imaging to the general framework of spectral
estimation theory and has applications in improving depth profiles and
exploiting multiple scattering. Depth imaging, multi–path interference, sparse
regularization, time–of–flight imaging 39 06 April 2014 20XX-XX-XX 20XX-XX-XX
Ayush Bhandari
Media Laboratory, 75 Amherst St.
Massachusetts Institute of Technology
02139, Cambridge. USA
E-mail: ,
URL: http://mit.edu/~ayush
Achuta Kadambi
Media Laboratory, 75 Amherst St.
Massachusetts Institute of Technology
02139, Cambridge. USA
E-mail:
Refael Whyte
School of Engineering, University of Waikato
Private Bag 3105
Hamilton 3240. New Zealand
E-mail:
Christopher Barsi
Media Laboratory, 75 Amherst St.
Massachusetts Institute of Technology
02139, Cambridge. USA
E-mail:
Micha Feigin
Media Laboratory, 75 Amherst St.
Massachusetts Institute of Technology
02139, Cambridge. USA
E-mail:
Adrian Dorrington
School of Engineering, University of Waikato
Private Bag 3105
Hamilton 3240. New Zealand
E-mail:
Ramesh Raskar
Media Laboratory, 75 Amherst St.
Massachusetts Institute of Technology
02139, Cambridge. USA
E-mail:
Optical ranging and surface profiling have widespread applications in image-
guided surgery [5], gesture recognition [4], remote sensing [1], shape
measurement [7], and novel phase imaging [17]. Generally, the characteristic
wavelength of the probe determines the resolution of the image, making time-
of-flight (ToF) methods suitable for macroscopic scenes[10, 22, 18]. Although
ToF sensors can be implemented with impulsive sources, commercial ToF cameras
rely on the continuous wave approach: the source intensity is modulated at
radio frequencies ($\sim$10s of MHz), and the sensor reconstructs the phase
shift between the reflected and emitted signals. Distance is calculated by
scaling the phase by the modulation frequency (Fig. 1 (a)). This method,
amplitude modulated continuous wave (AMCW) ToF, offers high SNR in real time.
However, AMCW ToF suffers from multipath interference (MPI) [3, 21, 15, 12,
20, 9, 8, 16, 14]. Consider, for example, the scenes in Figs. 1 (b,c). Light
rays from multiple reflectors scatter to the observation point. Each path
acquires a different phase shift, and the measurement consists of the sum of
these components. The recovered phase, therefore, will be incorrect. Such
“mixed” pixels contain depth errors and arise whenever global lighting effects
exist. In some cases (Fig. 1 (d)), the measurement comprises a continuum of
scattering paths. This can be improved with structured light or mechanical
scanning [6, 11], but these are limited by the source resolution.
Computational optimization [13, 19] schemes rely on radiometric assumptions
and have limited applicability.
Figure 1: (a) ToF principle: the phase delay of an emitted AMCW wave
proportionally encodes the distance of the reflecting object. (b) Mirror-like
and (c) semi-transparent reflections produce MPI at a given camera pixel and
yields an incorrect phase. (c) A complicated scene with severe MPI.
Here, we resolve MPI via sparse regularization of multiple modulation
frequency measurements. The formulation allows us to recast this problem into
the general framework of spectral estimation theory [23]. This contribution
generalizes the two-component, dual-frequency approach [15, 8, 16], beyond
which the two-component optimization methods fail. Thus, our method here has
two significant benefits. First, we separate MPI from direct illumination to
produce improved depth maps. Second, we resolve MPI into its components, so
that we can characterize and exploit multiple scattering phenomena. The
procedure has two steps: (1) record a scene with multiple modulation
frequencies and (2) reconstruct the MPI components using a sparsity
constraint.
Consider first the single-component case. Mathematically, the camera emits the
normalized time-modulated intensity $s(t)$111Here, we consider continuous wave
imaging and hence the sinusoidal model, but the discussion is generally
applicable to any periodic function. and detects a signal $r(t)$:
$\displaystyle s\left(t\right)=1+s_{0}\cos\left({\omega
t}\right),t\in\mathbb{R}$ (1a) $\displaystyle
r\left(t\right)=\Gamma(1+s_{0}\cos\left({\omega t-\phi}\right)).$ (1b)
Here, $s_{0}$ and $\Gamma\in[0,1]$ are the signal modulation depth and the
reflection amplitude, respectively, $\omega$ is the modulation frequency, and
$\phi$ is the phase delay between the reference waveform $s\left(t\right)$ and
the delayed version $r\left(t\right)$. For a co-located source and detector,
the distance to the object from the camera is given by the relation
$d=c\phi/2\omega$, where $c$ is the speed of light.
Electronically, each pixel acts as a homodyne detector, measuring the cross-
correlation between the reflected signal and the reference. Denoting the
complex conjugate of $f\in\mathbb{C}$ by $f^{*}$, the cross-correlation of two
functions $f$ and $g$ is
$\mathsf{C}_{f,g}\left(\tau\right)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathop{\lim}\limits_{T\to\infty}\frac{1}{{2T}}\int_{-T}^{+T}{f^{*}\left({t+\tau}\right)g\left(t\right)dt}.$
(2)
Note that infinite limits are approximately valid when the integration window
$2T$ is such that $T\gg\omega^{-1}$. A shorter time window produces residual
errors, but this is easily avoidable in practice. The pixel samples the cross-
correlation at discrete times $\tau_{q}$:
$m_{\omega}\left[q\right]\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\mathsf{C}_{s,r}\left(\tau_{q}\right)\stackrel{{\scriptstyle(\ref{CC})}}{{=}}\Gamma\left(1+\frac{{s_{0}^{2}}}{2}\cos({\omega\tau_{q}+\phi})\right).$
(3)
Using the “$4$ Bucket Sampling” technique [10], we calculate the estimated
reflection amplitude and the phase, $\widetilde{\Gamma},\widetilde{\phi}$,
using four samples ${\tau_{q}}=\pi q/2\omega$ with $q=0,...,3$:
$\displaystyle\widetilde{\Gamma}=\sqrt{{{\left({{m_{\omega}}\left[3\right]-{m_{\omega}}\left[1\right]}\right)}^{2}}+\left({{m_{\omega}}\left[0\right]-{m_{\omega}}\left[2\right]}\right)^{2}}/s_{0}^{2},$
(4a)
$\displaystyle\tan\widetilde{\phi}=\left({\frac{{{m_{\omega}}\left[3\right]-{m_{\omega}}\left[1\right]}}{{{m_{\omega}}\left[0\right]-{m_{\omega}}\left[2\right]}}}\right).$
(4b)
Therefore, we associate a complex value, $z_{\omega}$, with a pixel
measurement:
$z_{\omega}=\widetilde{\Gamma}e^{\jmath\widetilde{\phi}(\omega)}.$ (5)
Note that these results are formally equivalent to wavefront reconstruction
via phase-shifting digital holography [25].
When multiple reflections contribute to a single measurement, the return
signal comprises a sum. In phasor notation, for $K$ components,
$r\left(t\right)=C_{0}+\sum\nolimits_{k=0}^{K-1}{{\Gamma_{k}}{e^{\jmath\left({\omega
t-{\phi_{k}}\left(\omega\right)}\right)}}},$ (6)
where $C_{0}$ is a constant, ${\phi_{k}}\left(\omega\right)=2d_{k}\omega/c$,
and $\left\\{{{d_{k}}}\right\\}_{k=0}^{K-1}$ are $K$ depths at which the
corresponding reflection takes place. The reflection amplitude of the
$k^{\text{th}}$ surface is $\Gamma_{k}$. Each pixel records
${m_{\omega}^{K}}[q]=C_{0}+\frac{s_{0}^{2}}{2}e^{\jmath\omega\tau_{q}}\sum\nolimits_{k=0}^{K-1}{{\Gamma_{k}}{e^{\jmath{{\phi_{k}}\left(\omega\right)}}}}.$
(7)
Importantly, for a given modulation frequency $\omega_{0}$ (ignoring a
constant DC term),
$m_{{\omega_{0}}}^{K}[\tau_{q}]\propto{\exp{\jmath{\omega_{0}}\tau_{q}}}$,
i.e., there is no variation with respect to individual depth components
$\left\\{{{\Gamma_{k}(\omega)},{\phi_{k}}}\right\\}_{k=0}^{K-1}$ [3],
regardless of the sampling density. Equivalently, the camera measurement,
$z_{\omega}^{(K)}=\widetilde{\Gamma}(\omega)e^{\jmath\widetilde{\phi}(\omega)}=\sum\nolimits_{k=0}^{K-1}{{\Gamma_{k}(\omega)}{e^{\jmath{{\phi_{k}}\left(\omega\right)}}}}$
(8)
is now a complex sum of $K$ reflections, which cannot be separated without
independent measurements. Thus, at a given frequency, the measured phase, and
hence the depth, is a nonlinear mixture of all interefering components.
Our method separates these components by recording the scene with equi-spaced
frequencies $\omega=n\omega_{0}$ ($n\in\mathbb{N}$) and acquiring a set of
measurements $\mathbf{z}$:
${\mathbf{z}}={\left[{z_{\omega_{0}}^{(K)},z_{{2\omega_{0}}}^{(K)},\ldots,z_{N{\omega_{0}}}^{(K)}}\right]^{\top}}.$
(9)
The forward model can be written compactly in vector-matrix form as
${\mathbf{z}}=\boldsymbol{\Phi g}+\boldsymbol{\sigma}$, where
$\boldsymbol{\Phi}\in\mathbb{C}^{N\times K}$ is identified as a Vandermonde
matrix,
$\boldsymbol{\Phi}=\begin{pmatrix}e^{\jmath{\omega_{0}}{\phi_{0}}}&e^{\jmath{\omega_{0}}{\phi_{1}}}&\cdots&e^{\jmath{\omega_{0}}{\phi_{K-1}}}\\\
e^{\jmath 2{\omega_{0}}{\phi_{0}}}&e^{\jmath
2{\omega_{0}}{\phi_{1}}}&\cdots&e^{\jmath 2{\omega_{0}}{\phi_{K-1}}}\\\
\vdots&\vdots&\ddots&\vdots\\\ e^{\jmath N{\omega_{0}}{\phi_{0}}}&e^{\jmath
N{\omega_{0}}{\phi_{1}}}&\cdots&e^{\jmath N{\omega_{0}}{\phi_{K-1}}}\\\
\end{pmatrix},$ (10)
$\boldsymbol{g}={\left[{{\Gamma_{0}},\ldots,{\Gamma_{K-1}}}\right]^{\top}}\in\mathbb{R}^{K\times
1}$, and $\boldsymbol{\sigma}$ represents zero-mean Gaussian i.i.d. noise,
which controls the error $\varepsilon_{0}$ in our reconstruction algorithm.
Our goal is to estimate the phases
$\boldsymbol{\phi}={\left[{{\phi_{0}},\ldots,{\phi_{K-1}}}\right]^{\top}}\in\mathbb{R}^{K\times
1}$ and the reflection amplitude vector $\boldsymbol{g}$.
To recover these quantities, first note the similarity between
$\boldsymbol{\Phi}$ and an oversampled $N\times L$ discrete Fourier transform
(DFT) matrix $\boldsymbol{\Psi}$, with elements $\Psi_{nl}=\exp(\jmath nl/L)$.
If $L\gg K$, the discretization of $\boldsymbol{\Psi}$ is small enough to
assume that the columns of $\boldsymbol{\Phi}$ are contained in
$\boldsymbol{\Psi}$. We can also define a vector
$\boldsymbol{g}^{\prime}\in\mathbb{R}^{L\times 1}$, whose elements are zero
except for $K$ reflection amplitudes
$\left\\{{{\Gamma_{k}}}\right\\}_{k=0}^{K-1}$, such that
$\mathbf{z}=\boldsymbol{\Psi}\boldsymbol{g}^{\prime}$. We use the
($K$-)sparsity of $\boldsymbol{g}^{\prime}$ to regularize the problem:
$\qquad\underbrace{\left\|{{\mathbf{z}}-\boldsymbol{\Psi
g^{\prime}}}\right\|_{{\ell_{2}}}^{2}}_{{\textsf{Data-
Fidelity}}}<{\varepsilon_{0}}\quad\textrm{ such
that}\quad\underbrace{{{\left\|{\boldsymbol{g}^{\prime}}\right\|}_{{\ell_{0}}}}=K}_{{\textsf{Sparsity}}},$
(11)
where the $\ell_{p}$–norm as
$\left\|{\mathbf{x}}\right\|_{{\ell_{p}}}^{p}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\sum\nolimits_{n}{{{\left|{{x_{n}}}\right|}^{p}}}$.
The case of $p\rightarrow 0$ is used to define
${{\left\|{\boldsymbol{g}^{\prime}}\right\|}_{{\ell_{0}}}}$ as the number of
nonzero elements of $\boldsymbol{g}^{\prime}$. Eq. 11 demands a least-squares
solution to the data-fidelity problem $\left\|{{\mathbf{z}}-\boldsymbol{\Psi
g^{\prime}}}\right\|_{{\ell_{2}}}^{2}$ up to some error tolerance
$\varepsilon_{0}$, with the constraint that we accommodate up to $K$ nonzero
values of $\boldsymbol{g}^{\prime}$.
The sparsity of $\boldsymbol{g}^{\prime}$ arises from two underlying
assumptions. First, we do not consider the case of volumetric scattering,
which would preclude discrete reflections and require a different
parametrization (e.g., through the diffusion coefficient). Second, we ignore
the contributions of inter-reflections between scattering layers, as their
amplitudes fall off quickly. They could be incorporated, into our formulation,
with the result of changing the sparsity of $\boldsymbol{g}^{\prime}$ from $K$
to $K^{\prime}$, where $K^{\prime}-K$ is the number of inter-reflections
considered.
We solve Eq. 11 via orthogonal matching pursuit (OMP), which is an iterative
algorithm that searches for the best-fit projections (in the least-squares
sense) of the coefficients onto an over-complete dictionary. We input
$\boldsymbol{\Psi}$ and measurements $\mathbf{z}$ into the algorithm. The
outputs are the set of reflection coefficients $\Gamma_{k}$ and their
positions in $\boldsymbol{g^{\prime}}$. With the position of each $\Gamma_{k}$
reconstructed, the corresponding phases $\phi_{k}$ are recovered through the
elements of $\boldsymbol{\Psi}$: $\phi_{k}=(\jmath
n)^{-1}\log(\Psi_{nl_{k}})=l_{k}/L$, where $l_{k}$ is the location of
$\Gamma_{k}$ in $\boldsymbol{g}^{\prime}$.
Figure 2: Left: experimental setup. Two transparencies block the left side of
the camera (for a three-component measurement), and one transparency blocks
the right (two-component measurement). Right: measured amplitude and depth at
$\omega=3\omega_{0}$. Dashed line indicates edge of second transparency.
We verify this theory with the experimental setup shown in Fig. 2. A PMD19k-2
$160\times 120$ sensor array is controlled by a Stratix III FPGA. Analog pixel
values are converted to 16-bit unsigned values by an ADC during the pixel
readout process. Eight 100 mW Sony SLD 1239JL-54 laser diodes illuminate the
scene. The lasers are placed symmetrically around the detector for a coaxial
configuration. The base frequency modulation is
$f_{0}=\omega_{0}/(2\pi)=0.7937\textrm{ MHz}$, and the integration time is 47
ms. The scene consists of three layers. Farthest, at 8.1 m, is an opaque wall
with gray-scale text (“MIT”) printed on it. Closest, at 0.3 m is a semi-
transparent sheet. Between the two layers is another semi-transparent sheet
that covers only the left half of the field of view. Therefore, the left-hand
side records three bounces and the right only two. All three layers are within
the depth of field of the camera to avoid mixed pixels from blurring.
Depth and amplitude maps acquired at a specific frequency are shown in Fig. 2.
Due to MPI, the measured depths do not correspond to any physical layer in the
scene. All depth and amplitude information from the three scene layers is
mixed nonlinearly into a set of composite measurements (pixels) and cannot be
recovered.
Figure 3: Reconstructed amplitudes and depths via sparse regularization.
Dashed lines indicate edge of second transparency.
We repeat the acquisition $77$ times, with modulation frequencies spaced
$0.7937\textrm{ MHz}$ apart and input these data into the OMP algorithm with
$K=3$. The reconstruction, shown in Fig. 3, shows each depth correctly
recovered. The closest depth map (Fig. 3 (a), first transparency) is constant.
The second map (Fig. 3 (b)) contains two depths: the second transparency on
the LHS and the wall on the RHS. The third depth map contains the wall depth
on the LHS (Fig. 3 (c)). The third-bounce amplitude (Fig. 3 (f)) is zero where
there are only two layers (RHS). The depth here is therefore undefined, though
we set the distance to be 10 m to avoid random fluctuations. Further, the text
is recovered properly in the amplitude maps corresponding to the correct
depths (Figs. 3 (e,f)). Note that accurate depths are recovered even in the
presence of strong specularity (Fig. 3 (e)).
A phase histogram is shown in Fig. 4. The histogram from the single frequency
measurement in Fig. 1 varies from 0.6 to 1.8 rad. Recovered phases are
centered around the ground truth values. The third-phase variance is wider
because OMP computes the first two components, leaving little residual energy,
so that several columns in $\boldsymbol{\Psi}$ can minimize the least-squares
error.
Figure 4: Phase histogram for reconstructed and measured depth maps.
Reconstructed phases cluster around the correct depths, whereas the measured
depth map has a wide variance across the entire range.
In principle, the technique can be extended to any number of bounces, provided
enough modulation frequencies are used (though a first-principles derivation
is beyond the scope of this contribution). In practice, however, the reflected
amplitudes decrease with increasing component number, so that higher-order
components diminish in importance. Furthermore, OMP need not assume a number
of components that is the same as that of the physical implementation. If the
assumed number is greater than the physical number, OMP will reconstruct all
the physical components, with higher-order ones having an amplitude on order
of the system noise. Conversely, if the assumed number is less than the
physical number, OMP will recover the strongest reflections.
Therefore, the method is a generalization of global/direct illumination
separation and can decompose different elements of global lighting. This is
useful not only for improved depth accuracy, but also imaging in the presence
of multiple scatterers such as diffuse layers, sediment, turbulence, and
turbid media, as well as in places where third-component scattering must be
extracted [24]. Furthermore, because it is based on phase measurements, this
technique can be mapped to multiple scattering in holography [2] by
substituting optical frequency for the modulation frequency.
In conclusion, we implemented a multi-frequency approach for decomposing
multiple depths for a ToF camera. The result is general and holds for any
number of bounces, and it can be extended to non-harmonic signals [21]. Future
work includes calculating bounds on measurements and resolution. The method
can be incorporated with structured illumination and pixel correlations and
for edge detection, and refocusing. The result holds promise for mitigating
and exploiting multipath for a wide variety of scenes.
## References
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* [10] Sergi Foix, Guillem Alenya, and Carme Torras. Lock-in time-of-flight (tof) cameras: a survey. IEEE Sensors Journal, 11(9):1917–1926, 2011.
* [11] Active Sensor Plnanning for Multiview Vision Tasks. Chen, S. Y. Springer, 2008.
* [12] Mario Frank, Matthias Plaue, Holger Rapp, Ullrich Köthe, Bernd Jähne, and Fred A Hamprecht. Theoretical and experimental error analysis of continuous-wave Time-of-Flight range cameras. Proc. SPIE Conf. on Vis. Commun. and Image Proc., 48(1):013602—013602, 2009.
* [13] S. Fuchs. Multipath interference compensation in time-of-flight camera images. In Proc. Computer Vision and Pattern Recognition, 2010.
* [14] John P Godbaz, Michael J Cree, and Adrian A Dorrington. Understanding and ameliorating non-linear phase and amplitude responses in amcw lidar. Remote Sensing, 4(1):21–42, 2011.
* [15] John P Godbaz, Adrian A Dorrington, and Michael J Cree. Understanding and ameliorating mixed pixels and multipath interference in amcw lidar. In TOF Range-Imaging Cameras, pages 91–116. Springer, 2013.
* [16] J.P. Godbaz, M.J. Cree, and A.A. Dorrington. Closed-form inverses for the mixed pixel/multipath interference problem in amcw lidar. In IS&T/SPIE Electronic Imaging, pages 829618–829618, 2012.
* [17] J. C. Halimeh and M. Wegener. Time-of-flight imaging of invisibility cloaks. Opt. Express, 20:63–74, 2012.
* [18] Miles Hansard, Seungkyu Lee, Ouk Choi, and Radu Horaud. Time-of-flight cameras: principles, methods and applications. Springer, 2013.
* [19] D. Jimenez, D. Pizarro, M. Mazo, and S. Palazuelos. Modelling and correction of multipath interference in time of flight cameras. In Proc. Computer Vision and Pattern Recognition, 2012.
* [20] A. P P Jongenelen, D. G. Bailey, A. D. Payne, A. A. Dorrington, and D. A. Carnegie. Analysis of errors in tof range imaging with dual-frequency modulation. IEEE Trans. on Instrumentation and Measurement, 60(5):1861–1868, 2011.
* [21] A. Kadambi, R. Whyte, A. Bhandari, L. Streeter, C. Barsi, A. A. Dorrington, and R. Raskar. Coded time of flight cameras: sparse deconvolution to address multipath interference and recover time profiles. ACM Trans. Graph., to appear.
* [22] Andreas Kolb, Erhardt Barth, Reinhard Koch, and Rasmus Larsen. Time-of-Flight sensors in computer graphics. In Proc. Eurographics, pages 119–134, 2009.
* [23] P. Stoica and R. L. Moses. Introduction to Spectral Analysis. Prentice Hall, 1997.
* [24] Andreas Velten, Thomas Willwacher, Otkrist Gupta, Ashok Veeraraghavan, M. G. Bawendi, and Ramesh Raskar. Recovering three-dimensional shape around a corner using ultrafast time-of-flight imaging. Nat. Commun., 3:745, 2012.
* [25] T. Yamaguchi, I.and Zhang. Phase shifting digital holography. Opt. Lett., 22:1268, 1997.
|
arxiv-papers
| 2014-04-03T23:22:48 |
2024-09-04T02:50:00.707808
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ayush Bhandari, Achuta Kadambi, Refael Whyte, Christopher Barsi, Micha\n Feigin, Adrian Dorrington, and Ramesh Raskar",
"submitter": "Ayush Bhandari",
"url": "https://arxiv.org/abs/1404.1116"
}
|
1404.1201
|
11institutetext: GMD Berlin,
[email protected]
http://www.first.gmd.de/persons/Burghardt.Jochen.html
# Regular Substitution Sets: A Means of Controlling E-Unification
Jochen Burghardt
###### Abstract
A method for selecting solution constructors in narrowing is presented. The
method is based on a sort discipline that describes regular sets of ground
constructor terms as sorts. It is extended to cope with regular sets of ground
substitutions, thus allowing different sorts to be computed for terms with
different variable bindings. An algorithm for computing signatures of
equationally defined functions is given that allows potentially infinite
overloading. Applications to formal program development are sketched.
Technical Report
---
Arbeitspapiere der GMD 926
July 1995
ISSN 0723–0508
GMD – Forschungszentrum
---
Informationstechnik GmbH
D–53754 Sankt Augustin
Tel. | *49–2241–14–0
Fax | *49–2241–14–2618
Telex | 889469 gmd d
http://www.gmd.de
## 1 Motivation
Solving equations by narrowing has important applications, e.g. in the area of
formal software development. However, the usual narrowing strategies are only
able to restrict the set of application positions111Cf. e.g. the mathematical
definition of the notion of strategy in [6].. Ordered paramodulation [2] is
able to provide a succession in which the defining equations have to be
selected, but it cannot guarantee that an appropriate one is selected first.
Bockmayr [3] has shown that, under certain general conditions, narrowing
strategies essentially enumerate the whole term universe rather than
specifically selecting the appropriate equations of a defined function to
narrow with or the appropriate constructor to insert into the solution. In
this paper, we present an approach for restricting the set of applicable
defining equations in a narrowing step that is based on the dynamic
computation of function signatures, rather than their declaration by a user.
The main idea is as follows222Notations and naming conventions are consistent
with Def. 2 below. : As e.g. in [7], we distinguish between constructors and
equationally defined functions; each well-defined ground term can be reduced
to a ground constructor term, viz. its unique normal form. For a term $v$, let
$V$ be the set of all possible values of $v$, i.e., the set of all normal
forms of admitted ground constructor instances of $v$. Then, a goal equation
$v_{1}=v_{2}$ cannot be solved if $V_{1}\cap V_{2}=\\{\\}$; in this case, it
can be pruned from the search space of narrowing. Unfortunately, $V_{1}$ and
$V_{2}$ are undecidable in general; to overcome this problem, we will define
computable upper approximations $\overline{V}_{1}\supset V_{1}$ and
$\overline{V}_{2}\supset V_{2}$, respectively, and base the pruning decision
on the consideration of $\overline{V}_{1}\cap\overline{V}_{2}$.
To this end, we provide a framework of “extended sorts” to describe infinite
sets of ground constructor terms like $\overline{V}$ in a closed form, which
is based on regular tree grammars (e.g. [13]). It is essential that extended
sorts are closed wrt. intersection and that their inhabitance can be decided
in order to conduct the above disjointness test. Moreover, set equality and
subsort property can be decided, and $\overline{V}=V$ always holds if $v$ is a
constructor term.
An algorithm for computing the extended sort $\overline{V}$ from a term $v$ is
presented. In terms of conventional order-sorted rewriting, we thereby achieve
potentially infinite overloading, since for an arbitrary input sort $S$ we can
compute a signature $f:S\rightarrow\overline{f[S]}$ rather than being
restricted to a few user-defined signatures which are generally too coarse for
the disjointness test to be successfully applied. It is clear that the impact
of this test on search-space reduction depends on the expressiveness of the
sort framework and on the quality of signature approximation.
Consider, for example, the theory comprising equations a. to i. in Fig. 15.
When trying to solve a goal equation like $val(x)=s^{5}(0)$ wrt. this theory,
conventional strategies are unable to decide which of the equations g., h., i.
is to be used for a first narrowing step. Narrowing (at root position) with
equation g., h., and i. results in the new goal equations $0=s^{5}(0)$,
$dup(val(x^{\prime}))=s^{5}(0)$, and $s(dup(val(x^{\prime})))=s^{5}(0)$,
respectively. While the first one is obviously false, the unsatisfiability of
the second one can be detected as our algorithm computes the sort of its left-
hand side as $Even$ and recognizes that this is disjoint from its right-hand
side’s sort, $\\{s^{5}(0)\\}$; similarly, the third one is considered to be
“possibly satisfiable” by the disjointness test. Hence, narrowing only makes
sense with equation i., and any solution to the above goal equation must take
the form $x=x^{\prime}\\!::\\!i$. In Sect. 7 and App. 0.B, examples of the
pruning of infinite search-tree branches are given. Note that if a user were
to declare the signatures $+\\!:\\!Nat\\!\times\\!Nat\\!\rightarrow\\!Nat$,
$dup\\!:\\!Nat\\!\rightarrow\\!Nat$, and $val\\!:\\!Bin\\!\rightarrow\\!Nat$,
the disjointness test would allow narrowing with equations h. and i. In more
complicated applications, a user cannot know in advance which signatures might
become essential to disjointness tests in the course of the narrowing proof.
This example also shows that it is important to consider variable bindings
during the computation of a term’s sort in order to get good approximations.
For example, when computing a signature for $dup$, the term $x+x$ should be
assigned the sort $Even$, whereas $x+y$ can only be assigned $Nat$, assuming
that $x$ and $y$ range over $Nat$. In conventional order-sorted approaches,
the mapping from a term to its sort is usually a homomorphic extension of the
sort assignment of variables, thus necessarily ignoring variable bindings,
e.g.:
| $sortof(x+x)$
---|---
$=$ | $get\\_range\\_from\\_signatures(+,sortof(x),sortof(x))$
$=$ | $get\\_range\\_from\\_signatures(+,sortof(x),sortof(y))$
$=$ | $sortof(x+y)$.
Instead, we use infinite sets of ground substitutions to denote sorts of
variables, e.g. $\\{[x\\!:=\\!s^{i}(0),y\\!:=\\!s^{j}(0)]\mid i,j\in
I\\!\\!N\\}$ to indicate that $x$ and $y$ range over $Nat$. The mapping from a
term to its set of possible values can then be achieved by applying each
element of the substitution set, e.g.:
| $\\{[x\\!:=\\!s^{i}(0)]\mid i\in I\\!\\!N\\}\;\;\;(x+x)$
---|---
$=$ | $\\{[x\\!:=\\!s^{i}(0)]\;(x+x)\;\mid\;i\in I\\!\\!N\\}$
$=$ | $\\{s^{i}(0)+s^{i}(0)\mid i\in I\\!\\!N\\}$.
Similarly, $\\{[x\\!:=\\!s^{i}(0),y\\!:=\\!s^{j}(0)]\mid i,j\in
I\\!\\!N\\}\;(x+y)=\\{s^{i}(0)+s^{j}(0)\mid i,j\in I\\!\\!N\\}$. Both sets are
different, hence the chance of finding different approximations for them
within our extended sort framework is not forfeited333Schmidt-Schauß [12]
admits “term declarations”, allowing the user to declare different sorts for
terms with different bindings. In our approach, however, the sorts are to be
computed automatically. . In Fig. 13, we show that $Even$ can in fact be
obtained as the sort of $x+x$; obtaining $Nat$ for $x+y$ is similar.
In order to have finite descriptions of such ground substitution sets, we
express ground substitutions as ground constructor terms (“t-substitutions”)
in a lifted algebra, allowing sets of them to be treated as tree languages
(“t-sets”), and, in particular, to be described by regular tree grammars
(“regular t-sets”). Regular t-sets can also express simple relations between
distinct variables, allowing e.g. the representation of certain conditional
equations by unconditional ones.
We provide a new class of tree languages, called “extended sorts”, which can
be described by applying substitutions from a regular t-set $\sigma$ to an
arbitrary constructor term $u$ with $vars(u)\subset dom(\sigma)$. In this way,
the set of ground-constructor instances of an arbitrary constructor term can
be expressed as an extended sort.
Regular string languages have been used e.g. by Mishra [11] as a basis for
sort inference on Horn clauses. Owing to the restriction to string languages
describing admissible paths in term trees, he is only able to express infinite
sets that are closed wrt. all constructors; e.g. the set of all lists of
naturals containing at least one $0$ cannot be modeled. Comon [5] uses regular
tree languages to describe sets of ground constructor terms as sorts, and the
corresponding automaton constructions to implement sort operations. He
provides a transformation system to decide first-order formulas with equality
and sort membership as the only predicates. He shows the decision of inductive
reducibility as an application. However, he does not consider equationally
defined functions, e.g. $(\forall x,y\;\;x\\!+\\!y=y\\!+\\!x)\rightarrow
0\\!+\\!1=1\\!+\\!0$ reduces to $(\forall
x,y\;\;x\\!+\\!y=y\\!+\\!x)\rightarrow false$ in his calculus.
Uribe [15] provides a unification algorithm for order-sorted terms in the
presence of semilinear term declarations. The set of all ground constructor
instances of a constructor term can then be described by a regular tree
automaton with equality tests for direct subterms; allowing equality tests for
arbitrary subterms makes the disjointness of two tree languages undecidable
[14]. In our approach, arbitrary equality constraints may be imposed on
subterms up to a fixed finite depth, whereas below that depth no equality
constraints are allowed at all. Antimirov [1] suggested allowing regular
t-sets with equality tests in extended sorts, thus extending the class of
describable tree languages. This approach still remains to be investigated.
This paper is organized as follows. After a short introduction on regular
sorts in Sect. 2, regular substitution sets and extended sorts are presented
in Sect. 3 – 5. In Sect. 6, the algorithm for computing signatures of
equationally defined functions is given. It is shown that an unsorted root-
narrowing calculus from [9] remains complete if extended by appropriate sort
restrictions. Section 7 sketches the application of narrowing to synthesize
programs from formal specifications. Appendices 0.A and 0.B contain two case
studies in program synthesis. For a full version including all proofs, see
[4].
## 2 Regular Sorts
Definition 1.
Let ${\cal V}$ be a countable set of variables, ${\cal CR}$ a finite set of
term constructor symbols, each with fixed arity, ${\cal F}$ a finite set of
symbols for non-constructor functions, and ${\cal S}$ a countable set of sort
names. Let $ar(g)$ denote the arity of a function symbol $g$. For a
set444“$\subset$” denotes subset or equality, “$\subsetneqq$” denotes proper
subset. of symbols $X\subset{\cal V}\cup{\cal CR}\cup{\cal F}\cup{\cal S}$,
let ${\cal T}_{X}$ be the set of terms formed of symbols from $X$; we
abbreviate ${\cal T}_{X\cup Y}$ to ${\cal T}_{X,Y}$. For example, the elements
of ${\cal T}_{\cal CR}$, ${\cal T}_{{\cal CR},{\cal V}}$, and ${\cal T}_{{\cal
CR},{\cal F},{\cal V}}$ are called ground constructor terms, constructor
terms, and terms, respectively; the set ${\cal T}_{{\cal CR},{\cal F},{\cal
V},{\cal S}}$ is introduced in Sect. 6 for technical reasons. Let identifiers
like $u,u^{\prime},u_{i},\ldots$ always denote members of ${\cal T}_{{\cal
CR},{\cal V}}$; similarly, $v\in{\cal T}_{{\cal CR},{\cal F},{\cal V}}$,
$w\in{\cal T}_{{\cal CR},{\cal F},{\cal V},{\cal S}}$, $x,y,z\in{\cal V}$,
$f\in{\cal F}$, $g\in{\cal F}\cup{\cal CR}$, $cr\in{\cal CR}$, and $S\in{\cal
S}$.
Definition 2. $\langle v_{1},\ldots,v_{n}\rangle$ denotes an $n$-tuple,
$\langle v_{i}\mid p(v_{i}),\;i=1,\ldots,n\rangle$ denotes a tuple containing
each $v_{i}$ such that $p(v_{i})$ holds. We assume the existence of at least
one nullary (e.g. $nil$) and one binary constructor (e.g. $cons$), so we can
model arbitrary tuples as constructor terms. To improve readability, we
sometimes write the application of a unary function $f$ to its argument $x$ as
$f\\!\cdot\\!x$; $x\raisebox{5.69054pt}{\scriptsize:S}$ stands, in the
following, for the variable or constant $x$ of sort $S$. We define the
elementwise extension of a function $f:A\rightarrow B$ to a set
$A^{\prime}\subset A$ by $f[A^{\prime}]:=\\{f(a)\mid a\in A^{\prime}\\}$.
$A\times B$ denotes the Cartesian product of sets $A$ and $B$. For a finite
set $A$, we denote its cardinality by $\\#A$. We tacitly extend notations like
$\bigcup_{i=1}^{n}A_{i}$ to several binary operators defined in this paper,
e.g. $\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip
2.84544pt_{i=1}^{n}S_{i}:=S_{1}\mid\ldots\mid S_{n}$.
Definition 3. Let $vars(v_{1},\ldots,v_{n})$ denote the set of variables
occurring in any of the terms $v_{i}$. A term is called linear if it contains
no multiple occurrences of the same variable; it is called pseudolinear, if
any two occurrences of the same variable are at the same depth; it is called
semilinear if, for any two occurrences of the same variable, the lists of
function symbols on each path from the root to an occurrence are equal. We
write $v_{1}\mathrel{\mathop{\sphericalangle}\limits_{\neq}}v_{2}$ to express
that $v_{1}$ is a proper subterm of $v_{2}$; we write
$v_{1}\mathrel{\sphericalangle}v_{2}$ for
$v_{1}\mathrel{\mathop{\sphericalangle}\limits_{\neq}}v_{2}\;\vee\;v_{1}=v_{2}$.
The depth of a position in a term is its distance from the root. We
distinguish between “ordinary” substitutions, defined as usual (denoted by
$\beta,\gamma,\ldots$), and “t-substitutions”, defined as constructor terms in
Sect. 3, and denoted by $\sigma^{\prime},\tau^{\prime},\ldots$. Application of
a substitution $\beta$ to a term $v$ is written in prefix form, i.e. $\beta
v$. For an ordinary substitution $\beta$, let $dom(\beta):=\\{x\in{\cal
V}\mid\beta x\neq x\\}$, and $ran(\beta):=\bigcup_{x\in dom(\beta)}vars(\beta
x)$. $[x_{1}\\!:=\\!v_{1},\ldots,x_{n}\\!:=\\!v_{n}]$ denotes the substitution
that maps each $x_{i}$ to $v_{i}$.
$\beta\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$ denotes the domain
restriction of $\beta$ to a set $V$ of variables. We assume all substitutions
to be idempotent. If $\beta_{1}$ and $\beta_{2}$ agree on the intersection of
their domains, $\beta_{1}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}\beta_{2}$ denotes a “parallel
composition” of them, i.e.
$(\beta_{1}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}\beta_{2})\;(x):=\left\\{\begin{tabular}[]{@{}l@{\hspace*{0.5cm}}l@{}}$\beta_{1}x$\hfil\hskip
14.22636pt&if $x\in dom(\beta_{1})$\\\ $\beta_{2}x$\hfil\hskip 14.22636pt&if
$x\in dom(\beta_{2})$\\\ \end{tabular}\right.$
$\beta_{1}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}\beta_{2}$ is undefined if
$\beta_{1}$ and $\beta_{2}$ do not agree on $dom(\beta_{1})\cap
dom(\beta_{2})$. A substitution $\beta$ is called linear if the term
$\langle\beta x_{1},\ldots,\beta x_{n}\rangle$ is linear, where
$\\{x_{1},\ldots,x_{n}\\}=dom(\beta)$; similarly, $\beta$ is called
pseudolinear if $\langle\beta x_{1},\ldots,\beta x_{n}\rangle$ is
pseudolinear. We use the common notions of renaming substitution and most
general unifier $\beta=mgu(v_{1},v_{2})$, however, we will additionally assume
that $v_{1}$ and $v_{2}$ have disjoint variables and write $\beta$ as
$\beta_{1}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}\beta_{2}$ with
$dom(\beta_{1})=vars(v_{1})$ and $dom(\beta_{2})=vars(v_{2})$. $mgu$ is
tacitly extended to finite sets of terms.
We follow the approach of [4] in describing regular sets of ground constructor
terms as fixed points of sort equations, which is equivalent to the approach
using finite tree automata [5], but provides a unique methodology for
algorithms and proofs.
Definition 4. We allow sort definitions of the following syntax:
$\mathrm{SortName\doteq SortName\mid\ldots\mid SortName}$,
---
$\mathrm{SortName\doteq Constructor(SortName,\ldots,SortName)}$
Let $\stackrel{{\scriptstyle\mbox{\Large\bf.}}}{{<}}$ be the transitive
closure of the relation
$S_{i}\stackrel{{\scriptstyle\mbox{\Large\bf..}}}{{<}}S:\Leftrightarrow
S\doteq S_{1}\mid\ldots\mid S_{n}$. We admit finite systems of sort
definitions such that $\stackrel{{\scriptstyle\mbox{\Large\bf.}}}{{<}}$ is an
irreflexive partial, hence well-founded, order. For example, the sort system
consisting of $A\doteq B$ and $B\doteq A$ is forbidden. Each occurring sort
name has to be defined. In examples, we generally use arbitrary sort
expressions built from sort names, constructors, and “$\mid$” on the right-
hand side of a sort definition. Any such sort system can be transformed to
meet the above requirements while maintaining the least-fixed-point semantics
given below.
For example, consider the sort definition $Bin\doteq nil\mid Bin\\!::\\!o\mid
Bin\\!::\\!i$ from Fig. 15 on page 15, which denotes the lists of binary
digits, where “$o$” denotes zero, “$i$” denotes one, and $::$ is an
infix-$snoc$, i.e. reversed cons. The sort definition can be transformed into
the corresponding definitions shown in Fig. 1, which obey Def. 2, by
introducing new auxiliary sort names $Nil$, $Bino$, $Bini$, $O$, and $I$.
Let $X$ be an arbitrary mapping from sort names $S$ to subsets $S^{X}$ of
${\cal T}_{\cal CR}$. $X$ is extended to sort expressions as follows:
$(S_{1}\mid S_{1})^{X}$ | $=S_{1}^{X}\cup S_{2}^{X}$
---|---
$cr(S_{1},\ldots,S_{n})^{X}$ | $=cr[S_{1}^{X}\times\ldots\times S_{n}^{X}]$
$cr^{X}$ | $=\\{cr\\}$
We say that $X_{1}\subset X_{2}$ if $S^{X_{1}}\subset S^{X_{2}}$ for all sort
names $S$. According to Thm. 2 below, for each admitted system of sort
definition there exists exactly one mapping $M$, such that $S^{M}=S^{\prime
M}$ for each sort definition $S\doteq S^{\prime}$. The semantics of a sort
expression $S$ is then defined as $S^{M}$.
Theorem 5. Each admitted system of sort definitions has exactly one fixed
point.
Proof. If $M$ and $M^{\prime}$ are fixed points of the sort definitions, use
induction on the the lexicographic combination of $\mathrel{\sphericalangle}$
and $\stackrel{{\scriptstyle\mbox{\Large\bf.}}}{{<}}$ to show $\forall
u\;\forall S\;\;\;u\in S^{M}\Rightarrow u\in S^{M^{\prime}}$.
Definition 6. For a sort name $S$, let $use(S)$ denote the set of all sort
names that occur directly or indirectly in the definition of $S$. For example,
$use(Bin)=\\{O,I,Bin,Bino,Bini\\}$, cf. Fig. 1 and 2.
A subset $T\subset{\cal T}_{\cal CR}$ is called regular if a system of sort
definitions exists, such that $T=S^{M}$ for some sort expression $S$. Note
that $u^{M}=\\{u\\}$ for all $u\in{\cal T}_{\cal CR}$, e.g.,
$s(0)^{M}=\\{s(0)\\}$. The empty sort is denoted by $\bot$; it can be defined
e.g. by $\bot\doteq s(\bot)$. The uniqueness of fixed points validates the
following induction principle, which is used in almost all correctness proofs
of sort algorithms, cf. Alg. 3, 4, 7, 7, 7, 7, and 4.
Theorem 7. Let $p$ be a family of unary predicates, indexed over the set of
all defined sort names. Show for each defined sort name $S$:
$\forall u\\!\in\\!{\cal T}_{\cal CR}\;\;p_{S}(u)\leftrightarrow$ | $p_{S_{1}}(u)\vee\ldots\vee p_{S_{n}}(u)$ | if $S\doteq S_{1}\mid\ldots\mid S_{n}$
---|---|---
$\forall u\\!\in\\!{\cal T}_{\cal CR}\;\;p_{S}(u)\leftrightarrow$ | $\exists u_{1},\ldots,u_{n}\in{\cal T}_{\cal CR}\;\;u\\!=\\!cr(u_{1},\ldots,u_{n})$ |
| $\wedge p_{S_{1}}(u_{1})\wedge\ldots\wedge p_{S_{n}}(u_{n})$ | if $S\doteq cr(S_{1},\ldots,S_{n})$
$\forall u\\!\in\\!{\cal T}_{\cal CR}\;\;p_{S}(u)\leftrightarrow$ | $u\\!=\\!cr$ | if $S\doteq cr$
Then, $\forall u\\!\in\\!{\cal T}_{\cal CR}\;\;u\in S^{M}\leftrightarrow
p_{S}(u)$ holds for each defined sort name $S$.
Proof. The mapping $S\mapsto\\{u\in{\cal T}_{\cal CR}\mid p_{S}(u)\\}$ is a
fixed point of the sort definitions, hence the only one by Thm. 2.
Theorem 8. Let $p$ be a family of unary predicates, indexed over the set of
all defined sort names. Show for each defined sort name $S$:
$\forall u\\!\in\\!{\cal T}_{\cal CR}$ | $p_{S}(u)$ | $\leftarrow p_{S_{1}}(u)\vee\ldots\vee p_{S_{n}}(u)$ | if $S\doteq S_{1}\mid\ldots\mid S_{n}$
---|---|---|---
$\forall u_{1},\ldots,u_{n}\\!\in\\!{\cal T}_{\cal CR}\;\;$ | $p_{S}(cr(u_{1},\ldots,u_{n}))$ | $\leftarrow p_{S_{1}}(u_{1})\wedge\ldots\wedge p_{S_{n}}(u_{n})$ | if $S\doteq cr(S_{1},\ldots,S_{n})$
$\forall u\\!\in\\!{\cal T}_{\cal CR}$ | $p_{S}(cr)$ | | if $S\doteq cr$
Then, $\forall u\in S^{M}\;\;\;p_{S}(u)$ holds for each defined sort name $S$.
Proof. Use Scott’s fixed-point induction. The Thm. remains valid even if
$\stackrel{{\scriptstyle\mbox{\Large\bf.}}}{{<}}$ is not irreflexive.
Corollary 9. For each sort name $S$, we provide the following structural
induction principle: show for each sort definition $S^{\prime}\doteq
cr(S^{\prime}_{1},\ldots,S^{\prime}_{n})$ such that $S^{\prime}\in use(S)$,
and $S^{\prime M}\subset S^{M}$:
$\displaystyle\forall u_{1},\ldots,u_{n}\in{\cal T}_{\cal
CR}\;\;\;(\bigwedge_{\shortstack{$\scriptstyle i=1\ldots n$ \\\ $\scriptstyle
S^{\prime M}_{i}\subset S^{M}$}}u_{i}\in S^{\prime M}_{i}\wedge
p(u_{i}))\longrightarrow p(cr(u_{1},\ldots,u_{n}))$
Then, $\forall u\in S^{M}\;\;p(u)$ holds. $S^{\prime M}\subset S^{M}$ can be
decided using Alg. 4 below.
Proof. Use Thm. 2 with
$p_{S^{\prime}}(u):\Leftrightarrow\left\\{\begin{tabular}[]{l@{\hspace{0.5cm}}l}$p(u)$\hfil\hskip
14.22636pt&if $S^{\prime}\in use(S)$ and $S^{\prime M}\subset S^{M}$\\\
$true$\hfil\hskip 14.22636pt&else\\\ \end{tabular}\right.$ .
$Bin$ | $\doteq Nil\mid Bino\mid Bini$ | | binary numbers, i.e. lists of binary digits
---|---|---|---
$Bin^{0}$ | $\doteq Nil\mid Bino^{0}$ | | binary numbers that contain no ones
$Bin^{n+1}$ | $\doteq Nil\mid Bino^{n+1}\mid Bini^{n}$ | for $n\geqslant 0$ | binary numbers that contain at most $n+1$ ones
$Bin_{0}$ | $\doteq Bin$ | | binary numbers that contain at least $0$ ones
$Bin_{n+1}$ | $\doteq Bino_{n+1}\mid Bini_{n}$ | for $n\geqslant 0$ | binary numbers that contain at least $n+1$ ones
$Bino$ | $\doteq Bin\\!::\\!O$ | | binary numbers with a trailing zero
$Bini$ | $\doteq Bin\\!::\\!I$ | | binary numbers with a trailing one
$Bino^{n}$ | $\doteq Bin^{n}\\!::\\!O$ | |
$Bini^{n}$ | $\doteq Bin^{n}\\!::\\!I$ | |
$Bino_{n}$ | $\doteq Bin_{n}\\!::\\!O$ | |
$Bini_{n}$ | $\doteq Bin_{n}\\!::\\!I$ | |
$Nil$ | $\doteq nil$ | | empty list
$O$ | $\doteq o$ | | zero-digit
$I$ | $\doteq i$ | | one-digit
Figure 1: Examples of sort definitions | $S^{\prime M}\\!\subset\\!Bin^{M}$
---|---
| $S^{\prime}\\!\doteq\\!cr(\ldots)$ |
| $S^{\prime}\\!\in\\!use(Bin)$ | $\downarrow$ | $\downarrow$
| $O$ | $+$ | $-$
| $I$ | $+$ | $-$
| $Bin$ | $-$ | $+$
$p(nil)$ | $Nil$ | $+$ | $+$
$\forall u_{1}\;\;u_{1}\in Bin^{M}\wedge p(u_{1})\rightarrow p(u_{1}\\!::\\!o)$ | $Bino$ | $+$ | $+$
$\forall u_{1}\;\;u_{1}\in Bin^{M}\wedge p(u_{1})\rightarrow p(u_{1}\\!::\\!i)$ | $Bini$ | $+$ | $+$
$\forall u\\!\in\\!Bin^{M}\;\;p(u)$ | | |
Figure 2: Induction principle for sort $Bin$
Figure 2 shows an induction principle for sort $Bin$ from Fig. 1, using Cor.
2. Algorithms for computing the intersection and the relative complement of
two regular sorts, as well as for deciding the inhabitance of a sort, and thus
of the subsort and sort equivalence property, are given below. They consist
essentially of distributivity rules, constructor-matching rules, and loop-
checking rules. The latter stop the algorithm, when it calls itself
recursively with the same arguments, and generate a corresponding new
recursive sort definition.
$inf(Bin^{2},Bin_{1})$ | $=Sort_{1}$ | $\doteq inf(Nil,Bin_{1})\mid inf(Bino^{2},Bin_{1})\mid inf(Bini^{1},Bin_{1})$ | by 2.
---|---|---|---
$inf(Nil,Bin_{1})$ | $=Sort_{2}$ | $\doteq inf(Nil,Bino_{1})\mid inf(Nil,Bini_{0})$ | by 3.
$inf(Bino^{2},Bin_{1})$ | $=Sort_{3}$ | $\doteq inf(Bino^{2},Bino_{1})\mid inf(Bino^{2},Bini_{0})$ | by 3.
$inf(Bini^{1},Bin_{1})$ | $=Sort_{4}$ | $\doteq inf(Bini^{1},Bino_{1})\mid inf(Bini^{1},Bini_{0})$ | by 3.
$inf(Nil,Bino_{1})$ | $=Sort_{5}$ | $\doteq\bot$ | by 5.
$inf(Nil,Bini_{0})$ | $=Sort_{6}$ | $\doteq\bot$ | by 5.
$inf(Bino^{2},Bino_{1})$ | $=Sort_{7}$ | $\doteq inf(Bin^{2},Bin_{1})\\!::\\!inf(O,O)$ | by 4.
$inf(Bino^{2},Bini_{0})$ | $=Sort_{8}$ | $\doteq inf(Bin^{2},Bin_{0})\\!::\\!inf(O,I)$ | by 4.
$inf(Bini^{1},Bino_{1})$ | $=Sort_{9}$ | $\doteq inf(Bin^{1},Bin_{1})\\!::\\!inf(I,O)$ | by 4.
$inf(Bini^{1},Bini_{0})$ | $=Sort_{10}$ | $\doteq inf(Bin^{1},Bin_{0})\\!::\\!inf(I,I)$ | by 4.
$inf(Bin^{2},Bin_{1})$ | $=Sort_{1}$ | | by 1.
$inf(O,O)$ | $=Sort_{11}$ | $\doteq O$ | by 4.
$inf(Bin^{2},Bin_{0})$ | $=Sort_{12}$ | $\doteq Bin^{2}$ by similar computations |
$inf(O,I)$ | $=Sort_{13}$ | $\doteq\bot$ | by 5.
$inf(Bin^{1},Bin_{1})$ | $=Sort_{14}$ | $\doteq\ldots$ by similar computations |
$inf(I,O)$ | $=Sort_{15}$ | $\doteq\bot$ | by 5.
$inf(Bin^{1},Bin_{0})$ | $=Sort_{16}$ | $\doteq inf(Bin^{1},Bin)$ | by 3.
$inf(I,I)$ | $=Sort_{17}$ | $\doteq I$ | by 4.
$inf(Bin^{1},Bin)$ | $=Bin^{1}$ | by similar computations |
Hence, | | |
$Sort_{1}$ | $\doteq Sort_{2}\mid Sort_{3}\mid Sort_{4}$, |
$Sort_{2}$ | $\doteq Sort_{5}\mid Sort_{6}$, |
$Sort_{3}$ | $\doteq Sort_{7}\mid Sort_{8}$, |
$Sort_{4}$ | $\doteq Sort_{9}\mid Sort_{10}$, |
$Sort_{5}$ | $\doteq\bot$ |
$Sort_{6}$ | $\doteq\bot$ |
$Sort_{7}$ | $\doteq Sort_{1}\\!::\\!Sort_{11}$, |
$Sort_{8}$ | $\doteq Sort_{12}\\!::\\!Sort_{13}$, |
$Sort_{9}$ | $\doteq Sort_{14}\\!::\\!Sort_{15}$, |
$Sort_{10}$ | $\doteq Sort_{16}\\!::\\!Sort_{17}$, |
$Sort_{16}$ | $\doteq Bin^{1}$ |
$Sort_{11}$ | $\doteq O$ |
$Sort_{12}$ | $=Bin^{2}$ |
$Sort_{13}$ | $\doteq\bot$ |
$Sort_{14}$ | $\doteq\ldots$ |
$Sort_{15}$ | $\doteq\bot$ |
$Sort_{17}$ | $\doteq I$ |
Figure 3: Example computation of sort infimum
Algorithm 10. The following algorithm computes the intersection of two
regular sorts. Let $S_{1}$ and $S_{2}$ be sort names, let $S$ be a new sort
name. Define $inf(S_{1},S_{2})=S$, where a new sort definition is introduced
for $S$:
1. 1.
If $inf(S_{1},S_{2})$ has already been called earlier, $S$ is already defined
(loop check).
2. 2.
Else, if $S_{1}\doteq S_{11}\mid\ldots\mid S_{1n}$, define $S\doteq
inf(S_{11},S_{2})\mid\ldots\mid inf(S_{1n},S_{2})$
3. 3.
Else, if $S_{2}\doteq S_{21}\mid\ldots\mid S_{2n}$, define $S\doteq
inf(S_{1},S_{21})\mid\ldots\mid inf(S_{1},S_{2n})$
4. 4.
Else, if $S_{1}\doteq cr(S_{11},\ldots,S_{1n})$ and $S_{2}\doteq
cr(S_{21},\ldots,S_{2n})$,
define $S\doteq cr(inf(S_{11},S_{21}),\ldots,inf(S_{1n},S_{2n}))$
5. 5.
Else, define $S\doteq\bot$
Using Thm. 2 with
$p_{S}(u):\Leftrightarrow\left\\{\begin{tabular}[]{@{}l@{\hspace*{0.3cm}}l@{}}$u\in
S_{1}^{M}\cap S_{2}^{M}$\hfil\hskip 8.5359pt&if $S=inf(S_{1},S_{2})$\\\ $u\in
S^{M}$\hfil\hskip 8.5359pt&else\\\ \end{tabular}\right.$
it can be shown that $inf(S_{1},S_{2})^{M}=S_{1}^{M}\cap S_{2}^{M}$. The
$else$-case in the definition of $p_{S}(u)$ causes only trivial proof
obligations; in later applications of Thm. 2, it will be tacitly omitted for
the sake of brevity. The algorithm obviously needs at most
$\\#use(S_{1})*\\#use(S_{2})$ recursive calls to compute $inf(S_{1},S_{2})$.
$diff(Bin,Bin_{1})$ | $=$ | $Sort_{18}$ | $\doteq$ | $diff(Nil,Bin_{1})\mid diff(Bino,Bin_{1})$ | by 2.
---|---|---|---|---|---
| | | $\mid$ | $diff(Bini,Bin_{1})$ |
$diff(Nil,Bin_{1})$ | $=$ | $Sort_{19}$ | $\doteq$ | $diff(Nil,Bino_{1}\mid Bini_{0})$ | by 3.
$diff(Bino,Bin_{1})$ | $=$ | $Sort_{20}$ | $\doteq$ | $diff(Bino,Bino_{1}\mid Bini_{0})$ | by 3.
$diff(Bini,Bin_{1})$ | $=$ | $Sort_{21}$ | $\doteq$ | $diff(Bini,Bino_{1}\mid Bini_{0})$ | by 3.
$diff(Nil,Bino_{1}\mid Bini_{1})$ | $=$ | $Sort_{22}$ | $\doteq$ | $diff(Nil,Bino_{1})$ | by 6.
$diff(Nil,Bino_{1})$ | $=$ | $Sort_{23}$ | $\doteq$ | $Nil$ | by 7.
$diff(Bino,Bino_{1}\mid Bini_{0})$ | $=$ | $Sort_{24}$ | $\doteq$ | $Sort_{25}\mid Sort_{26}\mid Sort_{27}\mid Sort_{28}$ | by 4.
| | | $Sort_{25}$
---
$Sort_{26}$
$Sort_{27}$
$Sort_{28}$
| $\doteq$
---
$\doteq$
$\doteq$
$\doteq$
| $diff(Bin,$ | $Bin_{1}$ | $\mid$ | $Bin_{0}$ | $)\\!::\\!diff(O,$ | $\bot$ | $)$
---|---|---|---|---|---|---
$diff(Bin,$ | $Bin_{1}$ | | | $)\\!::\\!diff(O,$ | | | $I$ | $)$
$diff(Bin,$ | | | $Bin_{0}$ | $)\\!::\\!diff(O,$ | $O$ | | | $)$
$diff(Bin,$ | $\bot$ | $)\\!::\\!diff(O,$ | $O$ | $\mid$ | $I$ | $)$
| 1, | 1
---|---
1, | 2
2, | 1
2, | 2
$diff(Bini,Bino_{1}\mid Bini_{0})$ | $=$ | $Sort_{29}$ | $\doteq$ | $Sort_{30}\mid Sort_{31}\mid Sort_{32}\mid Sort_{33}$ | by 4.
| | | $Sort_{30}$
---
$Sort_{31}$
$Sort_{32}$
$Sort_{33}$
| $\doteq$
---
$\doteq$
$\doteq$
$\doteq$
| $diff(Bin,$ | $Bin_{1}$ | $\mid$ | $Bin_{0}$ | $)\\!::\\!diff(I,$ | $\bot$ | $)$
---|---|---|---|---|---|---
$diff(Bin,$ | $Bin_{1}$ | | | $)\\!::\\!diff(I,$ | | | $I$ | $)$
$diff(Bin,$ | | | $Bin_{0}$ | $)\\!::\\!diff(I,$ | $O$ | | | $)$
$diff(Bin,$ | $\bot$ | $)\\!::\\!diff(I,$ | $O$ | $\mid$ | $I$ | $)$
| 1, | 1
---|---
1, | 2
2, | 1
2, | 2
$diff(Bin,Bin_{1}\mid Bin_{0})$ | $=$ | $\bot$ | | by similar computations |
$diff(Bin,Bin_{1})$ | $=$ | $Sort_{18}$ | | | by 1.
$diff(Bin,Bin_{0})$ | $=$ | $\bot$ | | by similar computations |
$diff(Bin,\bot)$ | $=$ | $Bin$ | | | by 8.
$diff(O,\bot)$ | $=$ | $O$ | | | by 8.
$diff(O,I)$ | $=$ | $O$ | | | by 7.
$diff(O,O)$ | $=$ | $\bot$ | | | by 5.
$diff(O,O\mid I)$ | $=$ | $\bot$ | | | by 6.,5.
$diff(I,\bot)$ | $=$ | $I$ | | | by 8.
$diff(I,I)$ | $=$ | $\bot$ | | | by 5.
$diff(I,O)$ | $=$ | $I$ | | | by 7.
$diff(I,O\mid I)$ | $=$ | $\bot$ | | | by 6.,5.
Hence, | | | | |
$Sort_{18}$ | $\doteq$ | $Sort_{19}\mid Sort_{20}\mid Sort_{21}$ |
$Sort_{19}$ | $\doteq$ | $Sort_{22}$ |
$Sort_{20}$ | $\doteq$ | $Sort_{24}$ |
$Sort_{21}$ | $\doteq$ | $Sort_{29}$ |
$Sort_{22}$ | $\doteq$ | $Sort_{23}$ |
$Sort_{23}$ | $\doteq$ | $Nil$ |
$Sort_{24}$ | $\doteq$ | $Sort_{25}\mid Sort_{26}\mid Sort_{27}\mid Sort_{28}$ |
$Sort_{25}$ | $\doteq$ | $\bot\\!::\\!O$ |
$Sort_{26}$ | $\doteq$ | $Sort_{18}\\!::\\!O$ |
$Sort_{27}$ | $\doteq$ | $\bot\\!::\\!\bot$ |
$Sort_{28}$ | $\doteq$ | $Bin\\!::\\!\bot$ |
$Sort_{29}$ | $\doteq$ | $Sort_{30}\mid Sort_{31}\mid Sort_{32}\mid Sort_{33}$ |
$Sort_{30}$ | $\doteq$ | $\bot\\!::\\!I$ |
$Sort_{31}$ | $\doteq$ | $Sort_{18}\\!::\\!\bot$ |
$Sort_{32}$ | $\doteq$ | $\bot\\!::\\!I$ |
$Sort_{33}$ | $\doteq$ | $Bin\\!::\\!\bot$ |
Figure 4: Example computation of sort difference
Algorithm 11. The following algorithm computes the relative complement of two
regular sorts. For technical reasons, the second argument may be an arbitrary
union of sort names. Let $S_{1},\ldots,S_{m}$ be sort names, let $S$ be a new
sort name. Define $diff(S_{1},S_{2}\mid\ldots\mid S_{m})=S$, where a new sort
definition is introduced for $S$:
1. 1.
If $diff(S_{1},S_{2}\\!\mid\\!\ldots\\!\mid\\!S_{m})$ has already been called
earlier, $S$ is already defined (loop check).
2. 2.
If $S_{1}\doteq S_{11}\mid\ldots\mid S_{1n}$, define $S\doteq
diff(S_{11},S_{2}\\!\mid\\!\ldots\\!\mid\\!S_{m})\mid\ldots\mid
diff(S_{1n},S_{2}\\!\mid\\!\ldots\\!\mid\\!S_{m})$
3. 3.
If $S_{i}\doteq S_{i1}\mid\ldots\mid S_{in}$ for $2\leqslant i\leqslant m$,
define $S\doteq
diff(S_{1},S_{2}\\!\mid\\!\ldots\\!\mid\\!S_{i\\!-\\!1}\\!\mid\\!S_{i\\!+\\!1}\\!\mid\\!\ldots\\!\mid\\!S_{m}\;\mid\;S_{i1}\\!\mid\\!\ldots\\!\mid\\!S_{in})$
4. 4.
If $S_{1}\doteq cr(S_{11},\ldots,S_{1n})$ …, $S_{m}\doteq
cr(S_{m1},\ldots,S_{mn})$, with $n>0$,
let $S_{l_{1},\ldots,l_{m}}$ be a new sort name for each
$l_{1},\ldots,l_{m}\in\\{1,\ldots,n\\}$,
define $\displaystyle S\doteq\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip
2.84544pt_{l_{1}\\!=\\!1}^{n}\ldots\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip
2.84544pt_{l_{m}\\!=\\!1}^{n}\;S_{l_{1},\ldots,l_{m}}$
and $\displaystyle S_{l_{1}\ldots l_{m}}\doteq
cr(diff(S_{11},\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip
2.84544pt_{j\geqslant
2,\;l_{j}\\!=\\!1}S_{j1}),\ldots,diff(S_{1n},\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip
2.84544pt_{j\geqslant 2,\;l_{j}\\!=\\!n}S_{jn}))$.
5. 5.
If $S_{1}\doteq cr$, $S_{2}\doteq cr$ and $m=2$, define $S\doteq\bot$
6. 6.
If $S_{1}\doteq cr(\ldots)$ and $S_{m}\doteq cr^{\prime}(\ldots)$ with $cr\neq
cr^{\prime}$ and $m>2$,
define $S\doteq diff(S_{1},S_{2}\\!\mid\\!\ldots\\!\mid\\!S_{m-1})$
7. 7.
If $S_{1}\doteq cr(\ldots)$ and $S_{m}\doteq cr^{\prime}(\ldots)$ with $cr\neq
cr^{\prime}$ and $m=2$,
define $S\doteq S_{1}$
8. 8.
If $S_{2}\doteq\bot$ and $m=2$, define $S\doteq S_{1}$.
Using Thm. 2 with
$p_{S}(u)$ | $:\Leftrightarrow$ | $u\in S_{1}^{M}\setminus(S_{2}^{M}\cup\ldots\cup S_{m}^{M})$ | if $S=diff(S_{1},S_{2})$ and
---|---|---|---
$p_{S_{l_{1},\ldots,l_{m}}}(u)$ | $:\Leftrightarrow$ | $\exists u_{1},\ldots,u_{n}\;\;u=cr(u_{1},\ldots,u_{n})$ |
| $\wedge$ | $\bigwedge_{i=1}^{n}u_{i}\in(S_{1i}^{M}\setminus\bigcup_{j\geqslant 2,\;l_{j}=i}S_{ji}^{M})$ | if $S_{l_{1},\ldots,l_{m}}$ was defined in rule 4.,
it can be shown that $diff(S_{1},S_{2}\mid\ldots\mid
S_{m})^{M}=S_{1}^{M}\setminus(S_{2}^{M}\cup\ldots\cup S_{m}^{M})$.
The algorithm needs at most $\\#use(S_{1})*2^{\\#use(S_{2})}$ recursive calls
to compute $diff(S_{1},S_{2})$.
Algorithm 12. Let $S$ be a sort name, define $inh(S,Occ)=\langle
A,B,C,D\rangle$ where $A$ is a finite set of ground constructor terms,
$B\in\\{true,false\\}$, $C$, $D$, and $Occ$ are finite sets of sort names, as
follows:
1. 1.
If $S\in Occ$, define $inh(S,Occ)=\langle\\{\\},false,\\{S\\},\\{S\\}\rangle$
2. 2.
Else, if $S\doteq S_{1}\mid\ldots\mid S_{n}$, define $inh(S,Occ)=\langle
A_{1}\cup\ldots\cup A_{n},B,C,D\rangle$
3. 3.
Else, if $S\doteq cr(S_{1},\ldots,S_{n})$, define $inh(S,Occ)=\langle
cr[A_{1}\times\ldots\times A_{n}],B,C,D\rangle$
4. 4.
Else, if $S\doteq cr$, define
$inh(S,Occ)=\langle\\{cr\\},false,\\{S\\},\\{\\}\rangle$
where $\langle A_{i},B_{i},C_{i},D_{i}\rangle:=inh(S_{i},Occ\cup\\{S\\})$ for
$i=1,\ldots,n$, $B:=B_{1}\vee\ldots\vee B_{n}\vee S\in
C_{1}\\!\cup\\!\ldots\\!\cup\\!C_{n}$, $C:=C_{1}\cup\ldots\cup
C_{n}\cup\\{S\\}$, and $D:=(D_{1}\cup\ldots\cup D_{n})\setminus\\{S\\}$. $A$
is used to decide $S^{M}\neq\\{\\}$, $B$ is $true$ if a loop occurs in the
definition of $S$, $C$ is used to compute $B$, and $D$ is used only for proof
technical reasons and need not be computed in a practical implementation. Let
$inh(S,\\{\\})=\langle A,B,C,D\rangle$. Then, $S^{M}\neq\\{\\}$ iff
$A\neq\\{\\}$; $S^{M}$ finite iff $B\Leftrightarrow false$, and in this case
$A=S^{M}$. The algorithm needs at most $\\#use(S)*2^{\\#use(S)}$ recursive
calls to compute $inh(S,\\{\\})$. Define $single(S):\Leftrightarrow
inh(S,\\{\\})=\langle\\{u\\},false,C,D\rangle$ for some $u$, $C$, $D$.
Proof.
1. 1.
Use $Occ_{1}<Occ_{2}:\Leftrightarrow Occ_{2}\subsetneqq Occ_{1}$ to show
termination and complexity;
note that $Occ_{1},Occ_{2}$ is bounded from above by the finite set $use(S)$.
2. 2.
Let $inh(S,Occ)=\langle A,B,C,D\rangle$, and $E:=\\{u\mid\exists S^{\prime}\in
D,u^{\prime}\in S^{\prime M}\;\;\;u^{\prime}\mathrel{\sphericalangle}u\\}$;
all following statements are proven by induction on the computation tree of
$inh(S,Occ)$.
3. 3.
Show $D\subset Occ\cap C$, hence $E=\\{\\}$ if $Occ=\\{\\}$.
4. 4.
If $B=false$, show $S^{M}\subset A\cup E$.
5. 5.
Show $A\subset S^{M}$.
6. 6.
If $A=\\{\\}$, show $S^{M}\subset E$ by induction on the computation tree, and
(nested) induction on $u\in S^{M}$.
7. 7.
If $Occ=\\{\\}$, from 5. (“$\Leftarrow$”), 3. and 6. (“$\Rightarrow$”) follows
$S^{M}\neq\\{\\}$ iff $A\neq\\{\\}$.
8. 8.
Show $B\Leftrightarrow false$ iff $S$ contains no loops (iff $S^{M}$ is finite
by the pumping lemma).
9. 9.
If $Occ=\\{\\}$, from 5. (“$\subset$”), 3. and 4. (“$\supset$”) follows
$A=S^{M}$ if $B\Leftrightarrow false$.
Using the sort definitions from Fig. 1, Fig. 3 shows the computation of the
intersection of $Bin^{2}$ and $Bin_{1}$ by Alg. 3; the result may be
simplified to $Sort_{1}\doteq Sort_{1}\\!::\\!o\mid Bin^{1}\\!::\\!i$, which
uses the sloppy notation for sort definitions mentioned in Def. 2, and
intuitively denotes all binary lists with one or two $i$-digits. Figure 4
shows the computation of the complement of $Bin_{1}$ relative to $Bin$ by Alg.
4; the result may be simplified to $Sort_{18}\doteq nil\mid
Sort_{18}\\!::\\!o$ which is equivalent to $Bin^{0}$.
In [4], sort definitions may include “constraint formulas” which are not to be
considered by the sort algorithms, but rather collected and passed to an
external prover in which the sort algorithms are meant to be embedded. A sort
definition (cf. Def. 2) may also have the form
$SortName\doteq Constructor(Id:SortName,\ldots,Id:SortName)\lhd
Constraint(Id,\ldots,Id)$,
with the semantics
$(cr(i_{1}:S_{1},\ldots,i_{n}:S_{n})\lhd
p(i_{1},\ldots,i_{n}))^{M}=\\{cr(u_{1},\ldots,u_{n})\mid u_{1}\in
S_{1}^{M},\ldots,u_{n}\in S_{n}^{M},p(u_{1},\ldots,u_{n})\\}$;
and e.g. rule 4. of Alg. 3 has then the following form
If $S_{1}\doteq cr(S_{11},\ldots,S_{1n})\lhd ct_{1}$, and $S_{2}\doteq
cr(S_{21},\ldots,S_{2n})\lhd ct_{2}$,
define $S\doteq cr(inf(S_{11},S_{21}),\ldots,inf(S_{1n},S_{2n}))\lhd
ct_{1}\wedge ct_{2}$.
The same applies to Alg. 4. Algorithm 4 may yield a proper predicate
$B\not\in\\{true,false\\}$ if a nontrivial constraint formula occurs above a
loop in the sort definition. Constraint formulas are mentioned here only
because they appear in App. 0.B.
## 3 T-Substitutions
In this section, we apply the formalism from Sect. 2 to define possibly
infinite regular sets of ground substitutions. We define suitable free
constructors from which ground substitutions can be built as terms of a lifted
algebra ${\cal T}^{*}_{\mathchoice{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal
V\\!\hookrightarrow\\!\cal CR)}{(\cal V\hookrightarrow\cal CR)}{(\cal
V\hookrightarrow\cal CR)}}$. We call such terms t-substitutions. Note that the
classical approach, constructing substitutions by functional composition from
simple substitutions, cannot be used, since functional composition is not free
but obeys e.g. the associativity law. We provide the necessary notions and
properties of t-substitutions and of sets of them, called t-sets. All results
in this section hold for arbitrary t-sets.
We first define suitable free constructors from which ground substitutions can
be built as terms of a lifted algebra. Expressed informally, to build a
substitution term corresponding to
$[x_{1}\\!:=\\!u_{1},\ldots,x_{n}\\!:=\\!u_{n}]$ with $u_{i}$ ground, we
“overlay” the $u_{i}$ to obtain the substitution term; on the right, an
example is shown for $[x\\!:=\\!cons(0,nil),y\\!:=\\!s(s(0))]$.
$\begin{array}[c]{@{}l*{8}{c@{}}c@{}}x:=&cons&(&0&&&&,&nil&)\\\
y:=&s&(&s&(&0&)&&&)\\\
\cline{2-10}\cr&cons_{x}s_{y}&(&0_{x}s_{y}&(&0_{y}&)&,&nil_{x}&)\\\
\end{array}$
Definition 13. Given a set $V\subset{\cal V}$ of variables, define the
constructors for t-substitutions with domain $V$ as the set
$\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal
CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}$ of all total mappings from
$V$ to $\cal CR$. T-substitution constructors are denoted by
$\vec{cr},\vec{cr}^{\prime},\ldots$ , the empty mapping by $\varepsilon$.
Function application is written as $\vec{cr}_{x}$, the arity is defined as
$ar(\vec{cr}):=\max_{x\in dom(\vec{cr})}ar(\vec{cr}_{x})$. For
${V^{\prime}}\subset{\cal V}$, let
$\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V^{\prime}$}}$ denote
the restriction of $\vec{cr}$ to the variables in ${V^{\prime}}$, i.e.
$dom(\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V^{\prime}$}})={V^{\prime}}$ and
$(\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V^{\prime}$}})_{x}=\vec{cr}_{x}$ for all $x\in{V^{\prime}}$. For $\vec{cr}$
and $\vec{cr}^{\prime}$ such that $\vec{cr}_{x}=\vec{cr}^{\prime}_{x}$ for all
$x\in dom(\vec{cr})\cap dom(\vec{cr}^{\prime})$, let
$\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\vec{cr}^{\prime}$ denote the
“parallel composition” of $\vec{cr}$ and $\vec{cr}^{\prime}$, i.e.
$dom(\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\vec{cr}^{\prime})=dom(\vec{cr})\cup dom(\vec{cr}^{\prime})$, and
$(\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\vec{cr}^{\prime})_{x}=\left\\{\begin{tabular}[]{l@{\hspace*{0.5cm}}l}$\vec{cr}_{x}$\hfil\hskip
14.22636pt&if $x\in dom(\vec{cr})$\\\ $\vec{cr}^{\prime}_{x}$\hfil\hskip
14.22636pt&if $x\in dom(\vec{cr}^{\prime})$\\\ \end{tabular}\right.$ .
Note that $\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\vec{cr}^{\prime}$ is undefined
if $\vec{cr}$ and $\vec{cr}^{\prime}$ do not agree on their domain
intersection.
Example 14. In examples, we write e.g. $0_{x}s_{y}$ to denote the mapping
$(x\\!\mapsto\\!0,y\\!\mapsto\\!s)$. We have $0_{x}s_{y}$
$\in\mathchoice{(\\{x,y\\}\\!\rightarrow\\!\cal
CR)}{(\\{x,y\\}\\!\rightarrow\\!\cal CR)}{(\\{x,y\\}\rightarrow\cal
CR)}{(\\{x,y\\}\rightarrow\cal CR)}$, $ar(0_{x}s_{y})$ $=max(0,1)=1$,
$(0_{x}s_{y})\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle\\{y\\}$}}$
$=s_{y}$, and
$(0_{x}s_{y})\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}(s_{y}cons_{z})$
$=0_{x}s_{y}cons_{z}$.
Definition 15. Once we have defined t-substitution constructors, we inherit
the initial term algebra ${\cal T}_{\mathchoice{(V\\!\rightarrow\\!\cal
CR)}{(V\\!\rightarrow\\!\cal CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal
CR)}}$ over them. However, we have to exclude some nonsense terms. Define the
subset ${\cal T}^{*}_{\mathchoice{(V\\!\rightarrow\\!\cal
CR)}{(V\\!\rightarrow\\!\cal CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal
CR)}}\subset{\cal T}_{\mathchoice{(V\\!\rightarrow\\!\cal
CR)}{(V\\!\rightarrow\\!\cal CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal
CR)}}$ of admissible t-substitutions with domain $V$ as the least set such
that
$\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{ar(\vec{cr})})\in{\cal
T}^{*}_{\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal
CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}}$
---
if $\vec{cr}\in\mathchoice{(V\\!\rightarrow\\!\cal
CR)}{(V\\!\rightarrow\\!\cal CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal
CR)}$ and $\sigma^{\prime}_{i}\in{\cal T}^{*}_{\mathchoice{(\\{x\in V\mid
ar(\vec{cr}_{x})\geqslant i\\}\\!\rightarrow\\!\cal CR)}{(\\{x\in V\mid
ar(\vec{cr}_{x})\geqslant i\\}\\!\rightarrow\\!\cal CR)}{(\\{x\in V\mid
ar(\vec{cr}_{x})\geqslant i\\}\rightarrow\cal CR)}{(\\{x\in V\mid
ar(\vec{cr}_{x})\geqslant i\\}\rightarrow\cal CR)}}$ for
$i=1,\ldots,ar(\vec{cr})$.
We denote t-substitutions by
$\sigma^{\prime},\tau^{\prime},\mu^{\prime},\ldots$ . Sets of t-substitutions
are called t-sets and are denoted by $\sigma,\tau,\mu,\ldots$ .
T-substitutions, built as constructor terms:
---
$s_{x}(s_{x}(0_{x}))$ | $\mbox{\^{=}}[x\\!:=\\!s(s(0))]$
$s_{x}s_{y}(0_{x}0_{y})$ | $\mbox{\^{=}}[x\\!:=\\!s(0),y\\!:=\\!s(0)]$
$s_{x}0_{y}(0_{x})$ | $\mbox{\^{=}}[x\\!:=\\!s(0),y\\!:=\\!0]$
Figure 5: Examples of t-substitutions
Example 16. Definition 3 implies that $\vec{cr}\in{\cal
T}^{*}_{\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal
CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}}$ if
$\vec{cr}\in\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal
CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}$ is a nullary
t-substitution constructor. For example, we have $0_{y}\in{\cal
T}^{*}_{\mathchoice{(\\{y\\}\\!\rightarrow\\!\cal
CR)}{(\\{y\\}\\!\rightarrow\\!\cal CR)}{(\\{y\\}\rightarrow\cal
CR)}{(\\{y\\}\rightarrow\cal CR)}}$, and hence $0_{x}s_{y}(0_{y})\in{\cal
T}^{*}_{\mathchoice{(\\{x,y\\}\\!\rightarrow\\!\cal
CR)}{(\\{x,y\\}\\!\rightarrow\\!\cal CR)}{(\\{x,y\\}\rightarrow\cal
CR)}{(\\{x,y\\}\rightarrow\cal CR)}}$, but neither $0_{y}\in{\cal
T}^{*}_{\mathchoice{(\\{x,y\\}\\!\rightarrow\\!\cal
CR)}{(\\{x,y\\}\\!\rightarrow\\!\cal CR)}{(\\{x,y\\}\rightarrow\cal
CR)}{(\\{x,y\\}\rightarrow\cal CR)}}$, nor $0_{x}s_{y}(0_{x}0_{y})\in{\cal
T}^{*}_{\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal
CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}}$ for any $V$. Figure 5
shows some more t-substitutions together with their intended semantics.
Definition 17. Since ${\cal T}^{*}_{\mathchoice{(V_{1}\\!\rightarrow\\!\cal
CR)}{(V_{1}\\!\rightarrow\\!\cal CR)}{(V_{1}\rightarrow\cal
CR)}{(V_{1}\rightarrow\cal CR)}}\cap{\cal
T}^{*}_{\mathchoice{(V_{2}\\!\rightarrow\\!\cal
CR)}{(V_{2}\\!\rightarrow\\!\cal CR)}{(V_{2}\rightarrow\cal
CR)}{(V_{2}\rightarrow\cal CR)}}=\\{\\}$ for $V_{1}\neq V_{2}$, we may define
$dom(\sigma^{\prime}):=V$ iff $\sigma^{\prime}\in{\cal
T}^{*}_{\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal
CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}}$. Let
$\mathchoice{(V\\!\hookrightarrow\\!\cal CR)}{(V\\!\hookrightarrow\\!\cal
CR)}{(V\hookrightarrow\cal CR)}{(V\hookrightarrow\cal CR)}$ be the set of all
partial mappings from $V$ to ${\cal CR}$; define the set of admissible
t-substitutions with a subset of $V$ as domain by ${\cal
T}^{*}_{\mathchoice{(V\\!\hookrightarrow\\!\cal
CR)}{(V\\!\hookrightarrow\\!\cal CR)}{(V\hookrightarrow\cal
CR)}{(V\hookrightarrow\cal CR)}}:=\bigcup_{V^{\prime}\subset
V,V^{\prime}\mbox{\scriptsize\ finite}}{\cal
T}^{*}_{\mathchoice{(V^{\prime}\\!\rightarrow\\!\cal
CR)}{(V^{\prime}\\!\rightarrow\\!\cal CR)}{(V^{\prime}\rightarrow\cal
CR)}{(V^{\prime}\rightarrow\cal CR)}}$.
Definition 18. Define the t-substitution application $\sigma^{\prime}u$ by
$\sigma^{\prime}(cr(u_{1},\ldots,u_{k}))$
$:=cr[\sigma^{\prime}u_{1}\times\ldots\times\sigma^{\prime}u_{k}]$
$\sigma^{\prime}(cr)$ $:=\\{cr\\}$
$(\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{n}))(x)$
$:=\vec{cr}_{x}[\sigma^{\prime}_{1}x\times\ldots\times\sigma^{\prime}_{ar(\vec{cr}_{x})}x]$
if $x\in dom(\vec{cr})$, $n>0$ $(\vec{cr})(x)$ $:=\\{\vec{cr}_{x}\\}$ if $x\in
dom(\vec{cr})$, $n=0$
$(\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{n}))(x)$ $:=\\{\\}$ if
$x\not\in dom(\vec{cr})$
$\sigma^{\prime}u$ yields a set with at most one ground constructor term.
Application is extended elementwise to t-sets by $\sigma
u:=\bigcup_{\sigma^{\prime}\in\sigma}\sigma^{\prime}u$.
Note that, in contrast to an ordinary substitution $\beta$, a t-substitution
$\sigma^{\prime}$ is undefined outside its domain, i.e. it returns the empty
set. We have $\varepsilon u=\\{\\}$ if $u$ contains variables, $\varepsilon
u=u$ if $u$ is ground, and always $\bot u=\\{\\}$.
Lemma 19. $\sigma^{\prime}=\tau^{\prime}$ iff
$\sigma^{\prime}x=\tau^{\prime}x$ for all $x\in{\cal V}$,
where “$=$” on the left-hand side denotes the syntactic equality in ${\cal
T}^{*}_{\mathchoice{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal
V\\!\hookrightarrow\\!\cal CR)}{(\cal V\hookrightarrow\cal CR)}{(\cal
V\hookrightarrow\cal CR)}}$.
Although constructors may be written in different ways, e.g.
$0_{x}s_{y}=s_{y}0_{x}$, the initiality condition
$\vec{cr}(u_{1},\ldots,u_{n})=\vec{cr}^{\prime}(u^{\prime}_{1},\ldots,u^{\prime}_{n^{\prime}})\Rightarrow\vec{cr}=\vec{cr}^{\prime}\wedge
n=n^{\prime}\wedge u_{1}=u^{\prime}_{1}\wedge\ldots\wedge
u_{n}=u^{\prime}_{n}$ is satisfied in ${\cal T}^{*}_{\mathchoice{(\cal
V\\!\hookrightarrow\\!\cal CR)}{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal
V\hookrightarrow\cal CR)}{(\cal V\hookrightarrow\cal CR)}}$. The desired
equivalence of term equality and function equality from Lemma 5 is the reason
for restricting t-substitutions to a subset ${\cal T}^{*}_{\mathchoice{(\cal
V\\!\hookrightarrow\\!\cal CR)}{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal
V\hookrightarrow\cal CR)}{(\cal V\hookrightarrow\cal CR)}}$ of the initial
algebra ${\cal T}_{\mathchoice{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal
V\\!\hookrightarrow\\!\cal CR)}{(\cal V\hookrightarrow\cal CR)}{(\cal
V\hookrightarrow\cal CR)}}$, excluding nonsense terms like e.g.
$0_{x}s_{y}(0_{x}0_{y})$ and $0_{x}s_{y}(0_{x}1_{y})$ which would contradict
the initiality requirement.
Lemma 20. ${\cal T}^{*}_{\mathchoice{(\cal V\\!\hookrightarrow\\!\cal
CR)}{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal V\hookrightarrow\cal
CR)}{(\cal V\hookrightarrow\cal CR)}}$ corresponds to the set of all ordinary
ground substitutions in the following sense: For each $\sigma^{\prime}$ there
exists a $\beta$, such that $\sigma^{\prime}u=\\{\beta u\\}$ whenever
$vars(u)\subset dom(\sigma^{\prime})$. Conversely, for each $\beta$ there
exists a $\sigma^{\prime}$ with the respective property; cf. Fig. 5 which
shows some example correspondences.
Proof. Induction on $\sigma^{\prime}$ with
$\beta_{\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{n})}(x):=\vec{cr}_{x}(\beta_{\sigma^{\prime}_{1}}x,\ldots,\beta_{\sigma^{\prime}_{n}}x)$.
Conversely: define
$\\{\sigma^{\prime}_{\beta}\\}:=\Diamond\raisebox{1.1pt}{$\cdot$}\hskip
2.5pt_{x\in dom(\beta)}[x\\!:=\\!\beta x]$. Then,
$\sigma^{\prime}_{\beta}u=\\{\beta u\\}$ whenever $vars(u)\subset dom(\beta)$.
Definition 21. Define the t-substitution restriction
$\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$ by
$\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$ as defined in
Def. 3 if $ar(\vec{cr})=0$
$(\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{n}))\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}$ $:=(\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}})\;(\sigma^{\prime}_{1}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}},\ldots,\sigma^{\prime}_{m}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}})$ if $ar(\vec{cr})>0$ and
$ar(\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}})=m$
Restriction is extended elementwise to t-sets by
$\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}:=\\{\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}\;\mid\;\sigma^{\prime}\in\sigma\\}$.
Definition 22. Define the parallel composition of t-substitutions
$\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau^{\prime}$ by
$\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{n})\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\vec{cr}^{\prime}(\tau^{\prime}_{1},\ldots,\tau^{\prime}_{m}):=$
$\left\\{\begin{tabular}[]{@{}l@{\hspace*{1em}}l@{}}$(\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\vec{cr}^{\prime})\;[(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\tau^{\prime}_{1})\times\ldots\times(\sigma^{\prime}_{n}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\tau^{\prime}_{n})\times\\{\tau^{\prime}_{n+1}\\}\times\ldots\times\\{\tau^{\prime}_{m}\\}]$\hfil\hskip
10.00002pt&if $\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\vec{cr}^{\prime}$ is defined,
and $n\leqslant m$\\\
$(\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\vec{cr}^{\prime})\;[(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\tau^{\prime}_{1})\times\ldots\times(\sigma^{\prime}_{m}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\tau^{\prime}_{m})\times\\{\sigma^{\prime}_{m+1}\\}\times\ldots\times\\{\sigma^{\prime}_{n}\\}]$\hfil\hskip
10.00002pt&if $\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\vec{cr}^{\prime}$ is defined,
and $m\leqslant n$\\\ $\\{\\}$\hfil\hskip 10.00002pt&if
$\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\vec{cr}^{\prime}$ is
undefined\\\ \end{tabular}\right.$
$\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau^{\prime}$ yields a set with
at most one t-substitution. Parallel composition is extended elementwise to
t-sets by $\sigma\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\tau:=\bigcup_{\sigma^{\prime}\in\sigma,\tau^{\prime}\in\tau}\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau^{\prime}$. Note that
$\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau^{\prime}=\\{\\}$ if
$\sigma^{\prime}$ and $\tau^{\prime}$ do not agree on
$dom(\sigma^{\prime})\cap dom(\tau^{\prime})$.
Definition 23. Define the lifting of a ground constructor term $u$ to a
t-substitution $[x\\!:=\\!u]$, using the notation from Def. 3, by
$[x\\!:=\\!cr]$ $:=(x\\!\mapsto\\!cr)$ if $ar(cr)=0$
$[x\\!:=\\!cr(u_{1},\ldots,u_{n})]$
$:=(x\\!\mapsto\\!cr)\;([x\\!:=\\!u_{1}],\ldots,[x\\!:=\\!u_{n}])$ if
$ar(cr)=n>0$
Lifting is extended elementwise to sets of ground constructor terms by
$[x\\!:=\\!S]:=\\{[x\\!:=\\!u]\;\mid\;u\in S\\}$.
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\;(x)$ | $=\\{0\\}$
---|---
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\;(y)$ | $=\\{s(0)\\}$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\;(z)$ | $=\\{cons(0,nil)\\}$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\;(cons(x,cons(y,nil)))$ | $=\\{cons(0,cons(s(0),nil))\\}$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\;(x^{\prime})$ | $=\\{\\}$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle\\{x\\}$}}$ | $=0_{x}$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle\\{y\\}$}}$ | $=s_{y}(0_{y})$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle\\{z\\}$}}$ | $=cons_{z}(0_{z},nil_{z})$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle\\{x,y\\}$}}$ | $=0_{x}s_{y}(0_{y})$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle\\{x,z\\}$}}$ | $=0_{x}cons_{z}(0_{z},nil_{z})$
$(0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z}))\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle\\{y,z\\}$}}$ | $=s_{y}cons_{z}(0_{y}0_{z},nil_{z})$
$(0_{x})$ | $\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}(s_{y}(0_{y}))$ | $=\\{0_{x}s_{y}(0_{y})\\}$
$(0_{x}s_{y}(0_{y}))$ | $\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}(0_{x}cons_{z}(0_{z},nil_{z}))$ | $=\\{0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z})\\}$
$(0_{x}cons_{z}(0_{z},nil_{z}))$ | $\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}(s_{y}cons_{z}(0_{y}0_{z},nil_{z}))$ | $=\\{0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z})\\}$
$(0_{x}s_{y}(0_{y}))$ | $\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}(s_{y}cons_{z}(0_{y}0_{z},nil_{z}))$ | $=\\{0_{x}s_{y}cons_{z}(0_{y}0_{z},nil_{z})\\}$
$(0_{x}s_{y}(0_{y}))$ | $\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}(0_{y}cons_{z}(0_{z},nil_{z}))$ | $=\\{\\}$
$[x\\!:=\\!0]$ | $=0_{x}$
$[y\\!:=\\!s(0)]$ | $=s_{y}(0_{y})$
$[z\\!:=\\!cons(0,nil)]$ | $=cons_{z}(0_{z},nil_{z})$
Figure 6: Some example computations according to Defs. 5 – 5
Definition 24. Let $\beta$ be an ordinary idempotent substitution with
$n\geqslant 1$, let $\sigma^{\prime}$ be a t-substitution with
$ran(\beta)\subset dom(\sigma^{\prime})$ and $dom(\beta)\cap
dom(\sigma^{\prime})=\\{\\}$, define
$\sigma^{\prime}\circ\beta:=\Diamond\raisebox{1.1pt}{$\cdot$}\hskip
2.5pt_{x\in dom(\beta)}[x\\!:=\\!\sigma^{\prime}\beta x]$. We always have
$dom(\sigma^{\prime}\circ\beta)=dom(\beta)$,
$(\sigma^{\prime}\circ\beta)v=\sigma^{\prime}(\beta v)$ for all $v$ with
$vars(v)\subset dom(\beta)$, and
$(\sigma^{\prime}\circ\beta)\\!/\\!_{\beta}=\sigma^{\prime}$.
For a t-set $\sigma$ with the same domain as $\sigma^{\prime}$, define
$\sigma\circ\beta:=\bigcup_{\sigma^{\prime}\in\sigma}\sigma^{\prime}\circ\beta$.
We have $dom(\sigma\circ\beta)=dom(\beta)$, $(\sigma\circ\beta)v=\sigma(\beta
v)$ for all $v$ with $vars(v)\subset dom(\beta)$, and
$(\sigma\circ\beta)\\!/\\!_{\beta}=\sigma$.
Lemma 25. Some properties of application, restriction, parallel composition,
and abstraction are:
* •
$\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}u=\sigma^{\prime}u$ if $vars(u)\subset V$;
$\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}u=\\{\\}$,
else
* •
$dom(\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}})=dom(\sigma^{\prime})\cap V$
* •
$(\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V_{1}$}})\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V_{2}$}}=\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V_{1}\cap V_{2}$}}$
* •
$\sigma\subset\tau\Rightarrow\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}\subset\tau\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$
* •
$\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}u=\sigma u$ if
$vars(u)\subset V$
* •
$(\sigma\cap\tau)\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}=\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}\cap\tau\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$
* •
$\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}$ is associative
* •
$\sigma\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\tau=(\sigma\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\tau\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle T\setminus
S$}})\cap(\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle S\setminus
T$}}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau)$ where $S=dom(\sigma)$,
$T=dom(\tau)$
* •
$\sigma^{\prime}u=\tau^{\prime}u\Leftrightarrow\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
vars(u)$}}=\tau^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
vars(u)$}}$.
* •
$\sigma^{\prime}\langle
x_{1},\ldots,x_{n}\rangle=\langle\sigma^{\prime}x_{1},\ldots,\sigma^{\prime}x_{n}\rangle$
* •
$\sigma\langle x_{1},\ldots,x_{n}\rangle\subset\langle\sigma
x_{1},\ldots,\sigma x_{n}\rangle$.
Definition 26. Define the factorization $\sigma^{\prime}\\!/\\!_{\beta}$ of a
t-substitution $\sigma^{\prime}$ wrt. to an ordinary substitution $\beta$ with
$dom(\beta)\subset dom(\sigma^{\prime})$ as follows, let
$k:=ar(\vec{cr}_{x})$:
1. $\sigma^{\prime}\\!/\\!_{[x_{1}{:=}u_{1},\ldots,x_{n}{:=}u_{n}]}$ $:=\sigma^{\prime}\\!/\\!_{[x_{1}{:=}u_{1}]}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\ldots\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\sigma^{\prime}\\!/\\!_{[x_{n}{:=}u_{n}]}$ if $n>1$ 2. $\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{n})\\!/\\!_{[x{:=}\vec{cr}_{x}(u_{1},\ldots,u_{k})]}$ $:=\sigma^{\prime}_{1}\\!/\\!_{[x{:=}u_{1}]}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\ldots\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\sigma^{\prime}_{k}\\!/\\!_{[x{:=}u_{k}]}$ if $k>0$ 3. $\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{n})\\!/\\!_{[x{:=}\vec{cr}_{x}]}$ $:=\\{\varepsilon\\}$ if $k=0$ 4. $\vec{cr}(\sigma^{\prime}_{1},\ldots,\sigma^{\prime}_{n})\\!/\\!_{[x{:=}cr^{\prime}(u_{1},\ldots,u_{k})]}$ $:=\\{\\}$ if $\vec{cr}_{x}\neq cr^{\prime}$ 5. $\sigma^{\prime}\\!/\\!_{[x{:=}y]}$ $:=[y\\!:=\\!\sigma^{\prime}x]$ if $x\\!\neq\\!y\\!\in\\!{\cal V}$
$\sigma^{\prime}\\!/\\!_{\beta}$ yields a set with at most one t-substitution;
it is extended elementwise to t-sets by
$\sigma\\!/\\!_{\beta}:=\bigcup_{\sigma^{\prime}\in\sigma}\sigma^{\prime}\\!/\\!_{\beta}$.
Note that $[y\\!:=\\!\sigma^{\prime}x]$ is a singleton or empty set by Defs. 5
and 5. Factorization by the identity substitution is undefined. We have
$dom(\sigma^{\prime}\\!/\\!_{\beta})=ran(\beta)$ if
$\sigma^{\prime}\\!/\\!_{\beta}\neq\\{\\}$.
Lemma 27. (Pattern-Matching Properties)
a. $\sigma^{\prime}\\!/\\!_{\beta}\beta u=\sigma^{\prime}u$, if
$\sigma^{\prime}\\!/\\!_{\beta}\neq\\{\\}$ b. $\sigma\\!/\\!_{\beta}\beta
u=\sigma u\cap\top\beta u$
Proof.
a. Induction on $n=\\#dom(\beta)$: show $n=1$ by induction on $u$, show
$n\leadsto n+1$ by induction on $u$.
b. follows from a.
Example 28. We have
$(0_{x}s_{y}(0_{y}))\\!/\\!_{[y\\!:=\\!s(z)]}$ $=$
$0_{y}\\!/\\!_{[y\\!:=\\!z]}$ by Def. 6.2 $=$ $[z\\!:=\\!(0_{y})(y)]$ by Def.
6.5 $=$ $[z\\!:=\\!\\{0\\}]$ by Def. 5 $=$ $\\{0_{z}\\}$ by Def. 5 but
$(0_{x}s_{y}(0_{y}))\\!/\\!_{[y\\!:=\\!s(s(z))]}$ $=$
$0_{y}\\!/\\!_{[y\\!:=\\!s(z)]}$ by Def. 6.2 $=$ $\\{\\}$ by Def. 6.4
and
$\\{0_{z}\\}([y\\!:=\\!s(z)](y))=\\{0_{z}\\}(s(z))=\\{s(0)\\}=(0_{x}s_{y}(0_{y}))(y)$
by Def. 5.
Lemma 29. The following propositions are equivalent:
a. $\sigma^{\prime}\\!/\\!_{\beta}\neq\\{\\}$ b.
$\sigma^{\prime}u\cap\top\beta u\neq\\{\\}$ for all $u$ with $vars(u)\subset
dom(\beta)$ c. $\sigma^{\prime}u\cap\top\beta u\neq\\{\\}$ for some $u$ with
$vars(u)=dom(\beta)$
Proof.
a. $\Rightarrow$ b. by induction on $u$;
b. $\Rightarrow$ c. trivial;
c. $\Rightarrow$ a. Show $\sigma^{\prime}u=\tau^{\prime}\beta
u\Rightarrow\sigma^{\prime}x=\tau^{\prime}\beta x$ for all
$\tau^{\prime}\in\top$ and $x\in vars(u)$ by induction on $u$.
Show $\sigma^{\prime}u\cap\tau^{\prime}\beta
u\neq\\{\\}\Rightarrow\sigma^{\prime}\\!/\\!_{[x_{i}\\!:=\\!u_{i}]}y=\sigma^{\prime}\\!/\\!_{[x_{j}\\!:=\\!u_{j}]}y$
for all $\tau^{\prime}\in\top$ and $y\in vars(u_{i})\cap vars(u_{j})$,
$x_{i},x_{j}\in vars(u)$.
Show by induction on $\\#dom(\beta)$ that both conditions together imply a.
Lemma 30. Let $u_{1},\ldots,u_{n}$ have pairwise disjoint variables, and let
$vars(u_{i})\subset dom(\sigma^{\prime}_{i})$. Then,
$\sigma^{\prime}_{1}u_{1}=\ldots=\sigma^{\prime}_{n}u_{n}$ iff
$u_{1},\ldots,u_{n}$ are simultaneously unifiable by
$\beta_{1}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}\ldots\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}\beta_{n}$ with
$dom(\beta_{i})=vars(u_{i})$ and
$\sigma^{\prime}_{1}\\!/\\!_{\beta_{1}}=\ldots=\sigma^{\prime}_{n}\\!/\\!_{\beta_{n}}\neq\\{\\}$.
Proof.
“$\Rightarrow$”: Unifiability is obvious, minimality of $\beta_{i}$ implies
$\sigma^{\prime}_{i}u_{i}\cap\top\beta_{i}u_{i}\neq\\{\\}$, hence
$\sigma^{\prime}_{i}\\!/\\!_{\beta_{i}}\neq\\{\\}$ by 6. According to 6,
$\sigma^{\prime}_{i}\\!/\\!_{\beta_{i}}\beta_{i}u_{i}=\sigma^{\prime}_{i}u_{i}=\sigma^{\prime}_{j}u_{j}=\sigma^{\prime}_{j}\\!/\\!_{\beta_{j}}\beta_{j}u_{j}=\sigma^{\prime}_{j}\\!/\\!_{\beta_{j}}\beta_{i}u_{i}$,
hence
$\sigma^{\prime}_{i}\\!/\\!_{\beta_{i}}=\sigma^{\prime}_{j}\\!/\\!_{\beta_{j}}$.
“$\Leftarrow$”: According to 6, we have
$\sigma^{\prime}_{i}u_{i}=\sigma^{\prime}_{i}\\!/\\!_{\beta_{i}}\beta_{i}u_{i}=\sigma^{\prime}_{j}\\!/\\!_{\beta_{j}}\beta_{j}u_{j}=\sigma^{\prime}_{j}u_{j}$.
Domain conditions as in Lemma 3 can always be satisfied by bounded renaming,
factorizing by a renaming substitution, cf. Alg. 4 below. If $u_{1}$ and
$u_{2}$ cannot be unified, $\sigma_{1}u_{1}$ and $\sigma_{2}u_{2}$ are always
disjoint.
Theorem 31. Let $\beta=mgu(u_{1},u_{2})$, $dom(\beta)=vars(u_{1},u_{2})$,
$vars(u_{i})\subset dom(\sigma_{i})$,
$dom(\sigma_{1})\cap dom(\sigma_{2})=\\{\\}$, and $u=\beta u_{1}$; then
$\sigma_{1}u_{1}\cap\sigma_{2}u_{2}=(\sigma_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\sigma_{2})\\!/\\!_{\beta}\;u$.
Proof. 555 Remember that e.g. $\sigma^{\prime}_{1}u_{1}$ yields a set of
ground constructor terms with at most one element.
“$\subset$”: Suppose
$\sigma^{\prime}_{1}u_{1}=\sigma^{\prime}_{2}u_{2}\subset\sigma_{1}u_{1}\cap\sigma_{2}u_{2}$,
let $\gamma$ be a renaming substitution; then,
$(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\sigma^{\prime}_{2})\langle
u_{1},u_{2}\rangle=(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\sigma^{\prime}_{2})\langle
u_{2},u_{1}\rangle\stackrel{{\scriptstyle*}}{{=}}(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}_{2})\\!/\\!_{\gamma}\gamma\langle u_{2},u_{1}\rangle$.
Applying Lemma 3 yields
$(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}_{2})\\!/\\!_{\beta}\neq\\{\\}$, since
$\beta\circ\gamma^{-1}=mgu(\langle u_{1},u_{2}\rangle,\langle
u_{2},u_{1}\rangle)$, hence
$\sigma^{\prime}_{1}u_{1}=(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}_{2})\;u_{1}\stackrel{{\scriptstyle*}}{{=}}(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}_{2})\\!/\\!_{\beta}\beta
u_{1}=(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}_{2})\\!/\\!_{\beta}u\subset(\sigma_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\sigma_{2})\\!/\\!_{\beta}u$;
similarly,
$\sigma^{\prime}_{2}u_{2}\subset(\sigma_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\sigma_{2})\\!/\\!_{\beta}u$.
“$\supset$”: Suppose
$(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}_{2})\\!/\\!_{\beta}\subset(\sigma_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\sigma_{2})\\!/\\!_{\beta}$,
i.e. $(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}_{2})\\!/\\!_{\beta}\neq\\{\\}$ hence
$\sigma^{\prime}_{1}\\!/\\!_{\beta}\neq\\{\\}$, and
$(\sigma^{\prime}_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}_{2})\\!/\\!_{\beta}u=\sigma^{\prime}_{1}\\!/\\!_{\beta}\beta
u_{1}\stackrel{{\scriptstyle*}}{{=}}\sigma^{\prime}_{1}u_{1}\subset\sigma_{1}u_{1}$;
similarly, $(\sigma_{1}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma_{2})\\!/\\!_{\beta}u\subset\sigma_{2}u_{2}$.
The equations marked “$\stackrel{{\scriptstyle*}}{{=}}$” hold by Lemma 6.
Theorem 32. $\sigma u\neq\\{\\}$ iff
$\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle vars(u)$}}\cap{\cal
T}^{*}_{\mathchoice{(vars(u)\\!\rightarrow\\!\cal
CR)}{(vars(u)\\!\rightarrow\\!\cal CR)}{(vars(u)\rightarrow\cal
CR)}{(vars(u)\rightarrow\cal CR)}}\neq\\{\\}$.
Note that ${\cal
T}^{*}_{\mathchoice{(\\{x_{1},\ldots,x_{n}\\}\\!\rightarrow\\!\cal
CR)}{(\\{x_{1},\ldots,x_{n}\\}\\!\rightarrow\\!\cal
CR)}{(\\{x_{1},\ldots,x_{n}\\}\rightarrow\cal
CR)}{(\\{x_{1},\ldots,x_{n}\\}\rightarrow\cal
CR)}}=compose(\\{abstract(x_{1},{\cal T}_{\cal
CR}),\ldots,abstract(x_{n},{\cal T}_{\cal CR})\\})$ is regular since ${\cal
T}_{\cal CR}$ is regular.
Theorem 33. Let $u,u_{1},\ldots,u_{n}$ have pairwise disjoint variables, let
$\beta_{i}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}\gamma_{i}=mgu(u,u_{i})$ exist for
all $i$. Then, $\sigma u\subset\tau_{1}u_{1}\cup\ldots\cup\tau_{n}u_{n}$ iff
$\forall
i\;\;(\sigma\\!/\\!_{\beta_{i}}\;\setminus\;\tau_{i}\\!/\\!_{\gamma_{i}})\;\beta_{i}u\subset\bigcup_{j=1,\;j\neq
i}^{n}\tau_{j}u_{j}$ and $\sigma\subset\bigcup_{i=1}^{n}\top\circ\beta_{i}$.
Note that we provide no algorithm to decide the latter condition.
Proof. “$\Rightarrow$”:
Let $\sigma^{\prime}\in\sigma$ such that
$\sigma^{\prime}\\!/\\!_{\beta_{i}}\neq\\{\\}$ and
$\sigma^{\prime}\\!/\\!_{\beta_{i}}\not\subset\tau_{i}\\!/\\!_{\gamma_{i}}$
for all $i$.
By assumption, $j\in\\{1,\ldots,n\\}$ and $\tau^{\prime}_{j}\in\tau_{j}$ exist
such that $\sigma^{\prime}u=\tau^{\prime}_{j}u_{j}$.
By 3,
$\sigma^{\prime}\\!/\\!_{\beta_{j}}=\tau^{\prime}_{j}\\!/\\!_{\gamma_{i}}\neq\\{\\}$,
hence $j\neq i$, and
$\sigma^{\prime}\\!/\\!_{\beta_{i}}\beta_{i}u=\sigma^{\prime}u=\tau^{\prime}_{j}u_{j}$
by 6.
Next, consider an arbitrary $\sigma^{\prime}\in\sigma$. By assumption, $i$ and
$\tau^{\prime}_{i}\in\tau_{i}$ exist such that
$\sigma^{\prime}u=\tau^{\prime}_{i}u_{i}$.
By 3 and 6, $\sigma^{\prime}\\!/\\!_{\beta_{i}}\beta_{i}u=\sigma^{\prime}u$;
hence,
$\\{\sigma^{\prime}\\}=\sigma^{\prime}\\!/\\!_{\beta_{i}}\circ\beta_{i}\subset\top\circ\beta_{i}$
“$\Leftarrow$”:
Let $\sigma^{\prime}\in\sigma$; by assumption, an $i$ exists such that
$\sigma^{\prime}=\tau^{\prime}\circ\beta_{i}$ for some $\tau^{\prime}\in\top$.
By 3, $\sigma^{\prime}\\!/\\!_{\beta_{i}}\neq\\{\\}$. Case distinction:
* •
$\sigma^{\prime}\\!/\\!_{\beta_{i}}\not\subset\tau_{i}\\!/\\!_{\gamma_{i}}$,
then by assumption
$\sigma^{\prime}u=\sigma^{\prime}\\!/\\!_{\beta_{i}}\beta_{i}u=\tau^{\prime}_{j}u_{j}$
for some $j\neq i$ and $\tau^{\prime}_{j}\in\tau_{j}$.
* •
$\sigma^{\prime}\\!/\\!_{\beta_{i}}=\tau^{\prime}_{i}\\!/\\!_{\gamma_{i}}\neq\\{\\}$
for some $\tau^{\prime}_{i}\in\tau_{i}$, then
$\sigma^{\prime}u=\sigma^{\prime}\\!/\\!_{\beta_{i}}\beta_{i}u=\tau^{\prime}_{i}\\!/\\!_{\gamma_{i}}\gamma_{i}u_{i}=\tau^{\prime}_{i}u_{i}$.
Theorem 34. Let $u,u_{1},\ldots,u_{n}$ have pairwise disjoint variables, let
$u$ be unifiable with each $u_{i}$.
For $I\subset\\{1,\ldots,n\\}$ let
$\beta_{I}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}\bigcirc\raisebox{1.1pt}{$\cdot$}\hskip 2.5pt_{i\in
I}\beta_{I,i}=mgu(\\{u\\}\cup\\{u_{i}\mid i\in I\\})$ if it exists,
$dom(\beta_{I})=vars(u)$, $dom(\beta_{I,i})=vars(u_{i})$ 666I.e.
$\beta_{\\{\\}}$ is a renaming substitution on $u$,
let $J$ be the set of all $I$ with existing mgu.
Let
$\sigma_{I}:=\\{\sigma^{\prime}\in\sigma\mid\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\neq\\{\\}\leftrightarrow
i\in I\\}$.
Then, $\sigma u\subset\tau_{1}u_{1}\cup\ldots\cup\tau_{n}u_{n}$ iff
$\sigma_{I}\\!/\\!_{\beta_{I}}\subset\bigcup_{i\in
I}\tau_{i}\\!/\\!_{\beta_{I,i}}$ for all $I\in J$. The latter condition reads
$\sigma_{\\{\\}}\subset\\{\\}$ for $I=\\{\\}$. Note that the $\sigma_{I}$ are
not regular, in general.
Proof. First observe $(*)$: for $\sigma^{\prime}\in\sigma_{I}$,
$\\{u\\}\cup\\{u_{i}\mid i\in I\\}$ is simultaneously unifiable
since
$\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\beta_{\\{i\\}}u=\sigma^{\prime}u=\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\beta_{\\{i\\},i}u_{i}$
for all $i\in I$.
“$\Rightarrow$”:
Let $\sigma^{\prime}\in\sigma_{I}$ for some $I\in J$; by assumption,
$\sigma^{\prime}u=\tau^{\prime}_{i}u_{i}$ for some $i$ and
$\tau^{\prime}_{i}\in\tau_{i}$.
Applying 3 to $\sigma^{\prime}u=\tau^{\prime}_{i}u_{i}$ yields
$\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\neq\\{\\}$; hence $i\in I$.
Applying 3 to $(*)$ and $\sigma^{\prime}u=\tau^{\prime}_{i}u_{i}$ yields
$\sigma^{\prime}\\!/\\!_{\beta_{I}}=\tau^{\prime}_{i}\\!/\\!_{\beta_{I,i}}\neq\\{\\}$.
“$\Leftarrow$”:
Let $\sigma^{\prime}\in\sigma$, and let
$I:=\\{i\mid\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\neq\\{\\}\\}$. Then,
$\sigma^{\prime}\in\sigma_{I}$, and $I\in J$ by $(*)$.
By 6.1 below, it follows that $\sigma^{\prime}\\!/\\!_{\beta_{I}}\neq\\{\\}$,
hence
$\sigma^{\prime}\\!/\\!_{\beta_{I}}=\tau^{\prime}_{i}\\!/\\!_{\beta_{I,i}}$
for some $i\in I$ and $\tau^{\prime}_{i}\in\tau_{i}$ by assumption.
Hence,
$\sigma^{\prime}u=\sigma^{\prime}\\!/\\!_{\beta_{I}}\beta_{I}u=\tau^{\prime}_{i}\\!/\\!_{\beta_{I,i}}\beta_{I,i}u_{i}=\tau^{\prime}_{i}u_{i}$.
Lemma 35. Using the notions of 6,
let $I,I_{1},I_{2},I_{3}\in J$ with $I_{1}\subset I_{2}\cap I_{3}$ and
$I_{2}\neq I_{3}$, $\sigma^{\prime}\in\sigma$,
$\sigma^{\prime}_{2}\in\sigma_{I_{2}}$,
$\sigma^{\prime}_{3}\in\sigma_{I_{3}}$, then:
1. 1.
$\sigma^{\prime}\\!/\\!_{\beta_{I}}\neq\\{\\}$ iff $\forall i\in
I\;\;\;\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\neq\\{\\}$;
2. 2.
$\sigma^{\prime}_{2}\\!/\\!_{\beta_{I_{1}}}\neq\sigma^{\prime}_{3}\\!/\\!_{\beta_{I_{1}}}$;
3. 3.
$\sigma_{I_{2}}\\!/\\!_{\beta_{I_{1}}}=\sigma\\!/\\!_{\beta_{I_{1}}}\setminus\bigcup_{I_{3}\in
J,I_{2}\neq I_{3}\supset I_{1}}\sigma_{I_{3}}\\!/\\!_{\beta_{I_{1}}}$.
Proof.
1. 1\. “$\Rightarrow$”:
Let $i\in I$, then
$\sigma^{\prime}u=\sigma^{\prime}\\!/\\!_{\beta_{I}}\beta_{I}u=\sigma^{\prime}\\!/\\!_{\beta_{I}}\beta_{I,i}u_{i}$;
by 3, $\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\neq\\{\\}$.
2. 1\. “$\Leftarrow$”:
Let $I=\\{i_{1},\ldots,i_{m}\\}$, then
$\sigma^{\prime}u\\!=\\!\sigma^{\prime}\\!/\\!_{\beta_{\\{i_{1}\\}}}\beta_{\\{i_{1}\\}}u\\!=\\!\sigma^{\prime}\\!/\\!_{\beta_{\\{i_{1}\\}}}\beta_{\\{i_{1}\\},i_{1}}u_{i_{1}}\\!=\\!\ldots\\!=\\!\sigma^{\prime}\\!/\\!_{\beta_{\\{i_{m}\\}}}\beta_{\\{i_{m}\\}}u\\!=\\!\sigma^{\prime}\\!/\\!_{\beta_{\\{i_{m}\\}}}\beta_{\\{i_{m}\\},i_{m}}u_{i_{m}}$,
hence $\sigma^{\prime}\\!/\\!_{\beta_{I}}\neq\\{\\}$ by 3.
3. 2.:
By 1., we have
$\sigma^{\prime}_{2}\\!/\\!_{\beta_{I_{1}}}\neq\\{\\}\neq\sigma^{\prime}_{3}\\!/\\!_{\beta_{I_{1}}}$;
assume
$\sigma^{\prime}_{2}\\!/\\!_{\beta_{I_{1}}}=\sigma^{\prime}_{3}\\!/\\!_{\beta_{I_{1}}}$.
W.l.o.g., let $i\in I_{3}\setminus I_{2}$,
then
$\sigma^{\prime}_{2}u=\sigma^{\prime}_{2}\\!/\\!_{\beta_{I_{1}}}\beta_{I_{1}}u=\sigma^{\prime}_{3}\\!/\\!_{\beta_{I_{1}}}\beta_{I_{1}}u=\sigma^{\prime}_{3}u=\sigma^{\prime}_{3}\\!/\\!_{\beta_{\\{i\\}}}\beta_{\\{i\\}}u=\sigma^{\prime}_{3}\\!/\\!_{\beta_{\\{i\\}}}\beta_{\\{i\\},i}u_{i}$,
hence $\sigma^{\prime}_{2}\\!/\\!_{\beta_{\\{i\\}}}\neq\\{\\}$ contradicting
$i\not\in I_{2}$.
4. 3\. “$\subset$”:
Let $\sigma^{\prime}\in\sigma_{I_{2}}$, then $\sigma^{\prime}\in\sigma$ and
$\sigma^{\prime}\\!/\\!_{\beta_{I_{1}}}\neq\sigma^{\prime\prime}\\!/\\!_{\beta_{I_{1}}}$
for all $\sigma^{\prime\prime}\in\sigma_{I_{3}}$, $I_{2}\neq I_{3}\supset
I_{1}$ by 2.
5. 3\. “$\supset$:
Let $\sigma^{\prime}\in\sigma$ with
$\sigma^{\prime}\\!/\\!_{\beta_{I_{1}}}\neq\\{\\}$; define
$I:=\\{i\in\\{1,\ldots,n\\}\mid\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\neq\\{\\}\\}$,
then $I\in J$, since
$\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\Diamond\raisebox{1.1pt}{$\cdot$}\hskip 2.5pt_{i\in
I}\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\beta_{\\{i\\}}$ unifies
$\\{u\\}\cup\\{u_{i}\mid i\in I\\}$,
and $I_{1}\subset I$, since
$\sigma^{\prime}\\!/\\!_{\beta_{\\{i\\}}}\neq\\{\\}$ for all $i\in I_{1}$ by
1.
Case distinction:
1. (a)
$I\neq I_{2}$; then
$\sigma^{\prime}\\!/\\!_{\beta_{I_{1}}}\subset\sigma_{I}\\!/\\!_{\beta_{I_{1}}}$,
with $I_{2}\neq I\supset I_{1}$, hence $I$ is one of the $I_{3}$,
i.e., $\sigma^{\prime}\\!/\\!_{\beta_{I_{1}}}$ is not contained in the right
hand side, and we have nothing to show.
2. (b)
$I=I_{2}$; then, $\sigma^{\prime}\\!/\\!_{\beta_{I_{1}}}$ is contained in the
left hand side.
Example 36. Let $J=\\{\\{\\},\\{1\\},\\{2\\},\\{3\\},\\{1,2\\},\\{1,3\\}\\}$,
then 6.3 yields following equations, where e.g. $\sigma_{\\{i,j\\}}$ is
written as $\sigma_{ij}$; similarly for $\beta$:
$\sigma_{12}\\!/\\!_{\beta_{12}}$ $=\sigma\\!/\\!_{\beta_{12}}$
$\sigma_{13}\\!/\\!_{\beta_{13}}$ $=\sigma\\!/\\!_{\beta_{13}}$
$\sigma_{12}\\!/\\!_{\beta_{1}}$ $=\sigma\\!/\\!_{\beta_{1}}$
$\setminus(\sigma_{1}\\!/\\!_{\beta_{1}}\cup\sigma_{13}\\!/\\!_{\beta_{1}})$
$\sigma_{1}\\!/\\!_{\beta_{1}}$ $=\sigma\\!/\\!_{\beta_{1}}$
$\setminus(\sigma_{12}\\!/\\!_{\beta_{1}}\cup\sigma_{13}\\!/\\!_{\beta_{1}})$
$\sigma_{12}\\!/\\!_{\beta_{2}}$ $=\sigma\\!/\\!_{\beta_{2}}$
$\setminus\sigma_{2}\\!/\\!_{\beta_{2}}$ $\sigma_{2}\\!/\\!_{\beta_{2}}$
$=\sigma\\!/\\!_{\beta_{2}}$ $\setminus\sigma_{12}\\!/\\!_{\beta_{2}}$
$\sigma_{13}\\!/\\!_{\beta_{3}}$ $=\sigma\\!/\\!_{\beta_{3}}$
$\setminus\sigma_{3}\\!/\\!_{\beta_{3}}$ $\sigma_{3}\\!/\\!_{\beta_{3}}$
$=\sigma\\!/\\!_{\beta_{3}}$ $\setminus\sigma_{13}\\!/\\!_{\beta_{3}}$
## 4 Regular T-Sets and Algorithms
In this section, we introduce the notion of a regular t-set and provide
algorithms to compute with them. We obtain a decidability result for a class
of Horn clauses that is isomorphic to regular t-sets (Cor. 9). We present some
simple relations like $x<y$ that can be expressed by regular t-sets (Figs. 7
and 8), and operations on relations that can be computed (Fig. 9).
Using the result from Sect. 3, we can describe regular sets of ground
substitutions as subsets of the initial term algebra ${\cal
T}^{*}_{\mathchoice{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal
V\\!\hookrightarrow\\!\cal CR)}{(\cal V\hookrightarrow\cal CR)}{(\cal
V\hookrightarrow\cal CR)}}$. We will only consider t-sets with a unique domain
$dom(\sigma^{\prime})=V_{\sigma}$ for all $\sigma^{\prime}\in\sigma$; define
$dom(\sigma):=V_{\sigma}$. The empty t-set is again denoted by $\bot$; it will
be clear from the context whether $\bot$ denotes the empty sort or the empty
t-set. For each finite $V$, $\top_{V}:={\cal
T}^{*}_{\mathchoice{(V\\!\hookrightarrow\\!\cal
CR)}{(V\\!\hookrightarrow\\!\cal CR)}{(V\hookrightarrow\cal
CR)}{(V\hookrightarrow\cal CR)}}$ is expressible as a regular set. We write
$\top$ for $\top_{V}$ when $V$ is clear from the context; note that ${\cal
T}^{*}_{\mathchoice{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal
V\\!\hookrightarrow\\!\cal CR)}{(\cal V\hookrightarrow\cal CR)}{(\cal
V\hookrightarrow\cal CR)}}$ is not expressible since infinitely many
t-substitution constructors exist.
We immediately inherit the mechanisms and algorithms given in Sect. 2, i.e.
for intersection, relative complement, and inhabitance. In addition, the
operations defined in 5, 5, 5, and 5 can be computed for regular t-sets.
T-sets, described as regular sets:
---
$Nat_{x}$ | $\doteq 0_{x}\mid s_{x}(Nat_{x})$ | $\mbox{\^{=}}\\{[x\\!:=\\!s^{i}(0)]\mid i\in I\\!\\!N\\}$
$Nat_{y}$ | $\doteq 0_{y}\mid s_{y}(Nat_{y})$ | $\mbox{\^{=}}\\{[y\\!:=\\!s^{i}(0)]\mid i\in I\\!\\!N\\}$
$Nat_{x\\!=\\!y}$ | $\doteq 0_{x}0_{y}\mid s_{x}s_{y}(Nat_{x\\!=\\!y})$ | $\mbox{\^{=}}\\{[x\\!:=\\!s^{i}(0),y\\!:=\\!s^{i}(0)]\mid i\in I\\!\\!N\\}$
$Nat_{x,y}$ | $\doteq 0_{x}0_{y}\mid s_{x}s_{y}(Nat_{x,y})\mid$ |
| $\;\;\;\;0_{x}s_{y}(Nat_{y})\mid s_{x}0_{y}(Nat_{x})$ | $\mbox{\^{=}}\\{[x\\!:=\\!s^{i}(0),y\\!:=\\!s^{j}(0)]\mid i,j\in I\\!\\!N\\}$
$Nat_{x\\!<\\!y}$ | $\doteq 0_{x}s_{y}(Nat_{y})\mid s_{x}s_{y}(Nat_{x\\!<\\!y})$ | $\mbox{\^{=}}\\{[x\\!:=\\!s^{i}(0),y\\!:=\\!s^{j}(0)]\mid i,j\in I\\!\\!N,i<j\\}$
Figure 7: Examples of regular t-sets
Algorithm 37. The following algorithm computes the elementwise application of
a regular t-set to a variable. Let $\sigma$ be the name of a regular t-set,
and let $S$ be a new sort name. Define $apply(\sigma,x)=S$, where the
algorithm introduces a new sort definition for $S$:
1. 1.
If $apply(\sigma,x)$ has already been called earlier, $S$ is already defined
(loop check).
2. 2.
Else, if $\sigma\doteq\sigma_{1}\mid\ldots\mid\sigma_{n}$, define $S\doteq
apply(\sigma_{1},x)\mid\ldots\mid apply(\sigma_{n},x)$
3. 3.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$ with $x\in
dom(\vec{cr})$,
define
$S\doteq\vec{cr}_{x}(apply(\sigma_{1},x),\ldots,apply(\sigma_{ar(\vec{cr}_{x})},x))$,
4. 4.
Else, define $S\doteq\bot$.
Using the t-set version of Thm. 2 with $p_{S}(u):\Leftrightarrow
u\in\sigma^{M}x$ if $S=apply(\sigma,x)$, it can be shown that
$apply(\sigma,x)^{M}=\sigma^{M}x$. The algorithm needs at most
$\\#use(\sigma)$ recursive calls to compute $apply(\sigma,x)$.
Although t-substitutions are homomorphic wrt. all constructors in ${\cal CR}$,
t-sets are generally not; e.g., using the definitions from Fig. 7,
$Nat_{x<y}^{M}(\langle x,y\rangle)\subsetneqq\langle
Nat_{x<y}^{M}(x),Nat_{x<y}^{M}(y)\rangle$, cf. Lemma 6. Such t-sets can
express certain relations between distinct variables, e.g. $Nat_{x\\!<\\!y}$
always assigns a value to $x$ that is less than the value assigned to $y$.
Figure 8 shows some more nontrivial relations that are expressible by regular
t-sets. Figure 9 shows operations on relations that can be computed for
t-sets.
Definition 38. We call a t-set $\sigma$ independent if it is homomorphic on
linear terms, i.e. if $\sigma\langle x_{1},\ldots,x_{n}\rangle=\langle\sigma
x_{1},\ldots,\sigma x_{n}\rangle$, otherwise we call it “dependent”. An
independent t-set assigns the value of one variable independently of the value
of the others, e.g. $Nat_{x,y}$ in Fig. 7. A finite union
$\sigma_{1}\cup\ldots\cup\sigma_{n}$ of independent t-sets $\sigma_{i}$ is
called semi-independent. The intersection of two (semi-)independent t-sets is
again (semi-)independent; the union of two semi-independent t-sets is
trivially semi-independent.
Algorithm 39. The following algorithm computes the elementwise application of
a regular t-set to a linear constructor term. Let $u$ be a linear constructor
term, let $\sigma$ be the name of a regular t-set such that
$\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle vars(u)$}}$ is
independent. Define $apply(\sigma,cr(u_{1},\ldots,u_{n}))\doteq
cr(apply(\sigma,u_{1}),\ldots,apply(\sigma,u_{n}))$; if $u\in{\cal V}$,
compute $apply(\sigma,u)$ by Alg. 7. Then, $apply(\sigma,u)^{M}=\sigma^{M}u$.
Algorithm 40. The following algorithm computes the elementwise restriction of
a regular t-set to a set of variables. Let $\sigma$ be the name of a regular
t-set, $V\subset{\cal V}$, and let $\tau$ be a new name for a regular t-set.
Define $restrict(\sigma,V)=\tau$, where the algorithm introduces a new t-set
definition for $\tau$:
1. 1.
If $restrict(\sigma,V)$ has already been called earlier, $\tau$ is already
defined (loop check).
2. 2.
Else, if $\sigma\doteq\sigma_{1}\mid\ldots\mid\sigma_{n}$, define $\tau\doteq
restrict(\sigma_{1},V)\mid\ldots\mid restrict(\sigma_{n},V)$
3. 3.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$, define
$\tau\doteq(\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}})\;(restrict(\sigma_{1},V),\ldots,restrict(\sigma_{m},V))$,
where $m=ar(\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}})$.
Using Thm. 2 with
$p_{\tau}(\tau^{\prime}):\Leftrightarrow\tau^{\prime}\in\sigma^{M}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}$ if $\tau=restrict(\sigma,V)$,
it can be shown that
$restrict(\sigma,V)^{M}=\sigma^{M}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
V$}}$. The algorithm needs at most $\\#use(\sigma)$ recursive calls to compute
$restrict(\sigma,V)$. If $\sigma$ is (semi-)independent, then so is
$restrict(\sigma,V)$.
Algorithm 41. The following algorithm computes the elementwise parallel
composition of two regular t-sets $\sigma$ and $\tau$. Let $\mu$ be a new name
for a regular t-set. Define $compose(\sigma,\tau)=\mu$, where the algorithm
introduces a new t-set definition for $\mu$:
1. 1.
If $compose(\sigma,\tau)$ has already been called earlier, $\mu$ is already
defined (loop check).
2. 2.
Else, if $\sigma\doteq\sigma_{1}\mid\ldots\mid\sigma_{n}$, define $\mu\doteq
compose(\sigma_{1},\tau)\mid\ldots\mid compose(\sigma_{n},\tau)$.
3. 3.
Else, if $\tau\doteq\tau_{1}\mid\ldots\mid\tau_{n}$, define $\mu\doteq
compose(\sigma,\tau_{1})\mid\ldots\mid compose(\sigma,\tau_{n})$.
4. 4.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$,
$\tau\doteq\vec{cr}^{\prime}(\tau_{1},\ldots,\tau_{m})$, $\vec{cr}$ and
$\vec{cr}^{\prime}$ agree on their domain intersection,
and w.l.o.g. $n\leqslant m$, define
$\mu\doteq(\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\vec{cr}^{\prime})\;(compose(\sigma_{1},\tau_{1}),\ldots,compose(\sigma_{n},\tau_{n}),\tau_{n+1},\ldots,\tau_{m})$.
5. 5.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$,
$\tau\doteq\vec{cr}^{\prime}(\tau_{1},\ldots,\tau_{m})$, and $\vec{cr}$ and
$\vec{cr}^{\prime}$ do not agree on their domain intersection, define
$\mu\doteq\bot$.
Using Thm. 2 with
$p_{\mu}(\mu^{\prime}):\Leftrightarrow\mu^{\prime}\in\sigma^{M}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau^{M}$ if
$\mu=compose(\sigma,\tau)$, it can be shown that
$compose(\sigma,\tau)^{M}=\sigma^{M}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau^{M}$. The algorithm needs
at most $\\#use(\sigma)*\\#use(\tau)$ recursive calls to compute
$compose(\sigma,\tau)$. If $\sigma$ and $\tau$ are both (semi-)independent,
then so is $compose(\sigma,\tau)$.
We write $compose(\\{\sigma_{1},\ldots,\sigma_{n}\\})$ for
$compose(\sigma_{1},compose(\ldots,compose(\sigma_{n-1},\sigma_{n})\ldots))$.
Algorithm 42. The following algorithm computes the elementwise lifting of a
regular sort to a regular t-set. Let $S$ be the name of a regular sort,
$x\in{\cal V}$, and let $\sigma$ be a new name for a regular t-set. Define
$abstract(S,x)=\sigma$, where the algorithm introduces a new t-set definition
for $\sigma$:
1. 1.
If $abstract(S,x)$ has already been called earlier, $\sigma$ is already
defined (loop check).
2. 2.
Else, if $S\doteq S_{1}\mid\ldots\mid S_{n}$, define $\sigma\doteq
abstract(S_{1},x)\mid\ldots\mid abstract(S_{n},x)$.
3. 3.
Else, if $S\doteq cr(S_{1},\ldots,S_{n})$, define
$\sigma\doteq(x\\!\mapsto\\!cr)\;(abstract(S_{1},x),\ldots,abstract(S_{n},x))$.
Using Thm. 2 with
$p_{\sigma}(\sigma^{\prime}):\Leftrightarrow\sigma^{\prime}\in[x\\!:=\\!S^{M}]$
if $\sigma=abstract(S,x)$,
it can be shown that $abstract(S,x)^{M}=[x\\!:=\\!S^{M}]$. The algorithm needs
at most $\\#use(S)$ recursive calls to compute $abstract(x,S)$.
$abstract(x,S)$ always yields an independent t-set.
Expressible relations e.g.:
---
|
prefix $x$ of length $y$ of a $snoc$-list $z$ with regular element sort $Elem$
$Pref_{x,y,z}$ | $\doteq nil_{x}0_{y}nil_{z}\mid snoc_{x}s_{y}snoc_{z}(Pref_{x,y,z},Elem_{x=z})\mid nil_{x}0_{y}snoc_{z}(List_{x},Elem_{x})$
$List_{x}$ | $\doteq nil_{x}\mid snoc_{x}(List_{x},Elem_{x})$
$Elem_{x}$ | $=abstract(x,Elem)$
$Elem_{x=z}$ | $=dup(Elem_{x},[z\\!:=\\!x])$
lexicographical order on $cons$-lists wrt. regular element ordering
$Elem_{x<y}$
$Lex_{x<y}$ | $\doteq nil_{x}cons_{y}(Elem_{y},List_{y})\mid cons_{x}cons_{y}(Elem_{x=y},Lex_{x<y})$
| $\mid cons_{x}cons_{y}(Elem_{x<y},List_{x,y})$
$Elem_{x}$ | $=abstract(x,Elem)$
$Elem_{y}$ | $=abstract(y,Elem)$
$Elem_{x=y}$ | $=dup(Elem_{x},[y\\!:=\\!x])$
$List_{x}$ | $\doteq nil_{x}\mid cons_{x}(Elem_{x},List_{x})$
$List_{y}$ | $\doteq nil_{y}\mid cons_{y}(Elem_{y},List_{y})$
$List_{x,y}$ | $=compose(List_{x},List_{y})$
matching of tree $x$ at the root of tree $y$ (variable bindings not
considered)
set of functions symbols $F$, only one variable symbol $v$
$Mtch_{x,y}$ | $\doteq\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip 2.84544pt_{f\in F}\;f_{x}f_{y}(Mtch_{x,y},\ldots,Mtch_{x,y})\mid v_{x}f_{y}(Term_{y},\ldots,Term_{y})$
$Term$ | $\doteq v\mid\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip 2.84544pt_{f\in F}\;f(Term,\ldots,Term)$
$Term_{y}$ | $=abstract(y,Term)$
sum $z$ of binary strings $x$ and $y$ ($cons$-lists, least bit first)
$Sum_{x,y,z}$ | $\doteq Sum_{0,x,y,z}$
$Sum_{0,x,y,z}$ | $\doteq nil_{x}nil_{y}nil_{z}$
| $\mid cons_{x}nil_{y}cons_{z}(i_{x}i_{z}\\!\mid\\!o_{x}o_{z},Bin_{0,x=z})$
| $\mid
cons_{x}cons_{y}cons_{z}(o_{x}o_{y}o_{z}\\!\mid\\!o_{x}i_{y}i_{z}\\!\mid\\!i_{x}o_{y}i_{z},Sum_{0,x,y,z})$
| $\mid cons_{x}cons_{y}cons_{z}(i_{x}i_{y}o_{z},Sum_{1,x,y,z})$
$Sum_{1,x,y,z}$ | $\doteq nil_{x}nil_{y}cons_{z}(i_{z},nil_{z})$
| $\mid cons_{x}nil_{y}cons_{z}(i_{x}o_{z},Bin_{1,x=z})$
| $\mid cons_{x}cons_{y}cons_{z}(o_{x}o_{y}i_{z},Sum_{0,x,y,z})$
| $\mid
cons_{x}cons_{y}cons_{z}(o_{x}i_{y}o_{z}\\!\mid\\!i_{x}o_{y}o_{z}\\!\mid\\!i_{x}i_{y}i_{z},Sum_{1,x,y,z})$
$Bin^{\prime}$ | $\doteq nil\mid cons(o,Bin^{\prime})\mid cons(i,Bin^{\prime})$
$Bin_{x}$ | $=abstract(x,Bin^{\prime})$
$Bin_{0,x=z}$ | $=dup(Bin_{x},[z\\!:=\\!x])$
$Bin_{1,x=z}$ | $\doteq nil_{x}cons_{z}(i_{z},nil_{z})\mid cons_{x}cons_{z}(o_{x}i_{z},Bin_{0,x=z})$
| $\mid cons_{x}cons_{z}(i_{x}o_{z},Bin_{1,x=z})$
Figure 8: Some relations expressible by regular t-sets
Operations on relations e.g.:
---
relation join | $\sigma\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau$
relational image |
$R[x_{0}]=\\{y\mid x_{0}\mathrel{R}y\\}$ | $R[x_{0}]=apply(compose(R,abstract(x,x_{0})),y)$
factorization wrt. equivalence |
$x\mathrel{R_{E}}y\Leftrightarrow\exists x^{\prime},y^{\prime}\;\;x\mathrel{E}x^{\prime}\mathrel{R}y^{\prime}\mathrel{E}y$ | $R_{E}=compose(\\{E,R,E\\})$
equivalence from mapping |
$x\mathrel{E_{M}}y\Leftrightarrow M(x)=M(y)$ | $E_{M}=compose(fact(M,[x\\!:=\\!x^{\prime},y\\!:=\\!y^{\prime}]),fact(M,[x\\!:=\\!y^{\prime},y\\!:=\\!x^{\prime}]))$
restriction | $\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$
bounded renaming | $\sigma\\!/\\!_{\beta}$
conjunction | $\sigma\cap\tau$
disjunction | $\sigma\mid\tau$
negation | $\top\setminus\sigma$
For example, using the definitions from Fig. 8,
$restrict(Pref_{x,y,z},\\{x,y\\})$ yields the length function on $snoc$-lists,
and $apply(compose(Pref_{x,y,z},abstract(y,s^{3}(0))),x)$ yields the regular
sort of all $snoc$-lists of length 3.
Figure 9: Computable operations on relations in t-set form
Regular sorts from Sect. 2 can be shown to correspond to Horn clauses with
unary predicates and thus decide this theory class by extending the form of
sort expressions allowed on the right-hand side of a sort definition to
include intersections, too.
$\scriptstyle\vec{y}_{1}$$\scriptstyle\vec{y}_{j}$$\scriptstyle\vec{y}_{n}$$\scriptstyle\vec{x}_{1}$$\scriptstyle\vec{x}_{m}$$\scriptstyle
x_{11}$$\scriptstyle x_{1a}$$\scriptstyle x_{m1}$$\scriptstyle x_{ma}$
Similarly, regular t-sets correspond to Horn clauses of the following form:
$p(cr_{1}(\vec{x}_{1}),\ldots,cr_{m}(\vec{x}_{m}))\leftarrow
p_{1}(\vec{y}_{1})\wedge\ldots\wedge p_{1}(\vec{y}_{n})$ where
$\vec{x}_{i}:=\langle x_{i1},\ldots,x_{ia_{i}}\rangle$ for $i=1,\ldots,m$, and
$\vec{y}_{j}:=\langle x_{ij}\mid i=1,\ldots,m,j\leqslant ar(cr_{i})\rangle$
for $j=1,\ldots,n$ with $n:=max_{i=1,\ldots,m}ar(cr_{i})$. The relation
between $\vec{x}_{i}$ and $\vec{y}_{j}$ is shown in the above diagram. If all
term constructors $cr_{i}$ have the same arity $n$, the $\vec{y}_{j}$ are the
column vectors of an $m\times n$ matrix built from the $\vec{x}_{i}$ as line
vectors. For example, the definition $Lgth_{x,y}\doteq 0_{x}nil_{y}\mid
s_{x}snoc_{y}(Lgth_{x,y},Nat_{y})$ corresponds to the Horn clauses
$lgth(0,nil)$ and $lgth(s(x),snoc(y_{1},y_{2}))\leftarrow lgth(x,y_{1})\wedge
nat(y_{2})$. We thus have the following
Corollary 43. The satisfiability of any predicate defined by Horn clauses of
the above form can be decided. The set of such predicates is closed wrt.
conjunction, disjunction, and negation.
Algorithm 44. The following algorithm “duplicates” each t-substitution
$\sigma^{\prime}$ in $\sigma$, i.e., it composes $\sigma^{\prime}$ with a
renamed copy of itself. Let $\sigma$ be the name of a regular t-set, let
$\beta$ be an ordinary idempotent substitution with $\beta x\in{\cal V}$ for
all $x\in dom(\beta)$, $\beta$ need not be linear. Let $\tau$ be a new name
for a regular t-set. Define $dup(\sigma,\beta)=\tau$, where the algorithm
introduces a new t-set definition for $\tau$:
1. 1.
If $dup(\sigma,\beta)$ has already been called earlier, $\tau$ is already
defined (loop check).
2. 2.
Else, if $\sigma\doteq\sigma_{1}\mid\ldots\mid\sigma_{n}$, define $\tau\doteq
dup(\sigma_{1},\beta)\mid\ldots\mid dup(\sigma_{n},\beta)$.
3. 3.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$ and
$cr_{x_{i}}=cr_{x_{j}}$ whenever $\beta x_{i}=\beta x_{j}$,
let $\vec{cr}^{\prime}$ be a t-substitution constructor such that
$\vec{cr}^{\prime}_{\beta x}=\vec{cr}_{x}$ for all $x\in dom(\beta)$,
define
$\tau\doteq(\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\vec{cr}^{\prime})\;(dup(\sigma_{1},\beta),\ldots,dup(\sigma_{n},\beta))$.
4. 4.
Else, define $\theta\doteq\bot$.
Using Thm. 2 with
$p_{\tau}(\tau^{\prime}):\Leftrightarrow\tau^{\prime}\in\bigcup_{\sigma^{\prime}\in\sigma}\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}(\sigma^{\prime}\\!/\\!_{\beta})$, it can be shown that
$dup(\sigma,\beta)^{M}=\bigcup_{\sigma^{\prime}\in\sigma}\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}(\sigma^{\prime}\\!/\\!_{\beta})$. The algorithm needs at most
$\\#use(\sigma)$ recursive calls to compute $dup(\sigma,\beta)$.
Example 45. Using the definitions in Fig. 7, we get
$dup(Nat_{x},[y\\!:=\\!x])=Nat_{x\\!=\\!y}$.
Algorithm 46. If $\sigma$ is regular and $\beta x\in{\cal V}$ for all $x\in
dom(\beta)$, $\sigma\circ\beta$ is again regular; in general, it is not. In
the former case, the following algorithm computes a regular t-set definition
for $\sigma\circ\beta$:
1. 1.
If $\beta=\beta_{1}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}\beta_{2}$ such that $\beta_{1}$
and $\beta_{2}$ are each injective, i.e. renamings, let $\gamma$ be a renaming
on $ran(\beta_{2})$, then
$\sigma\circ\beta=fact(dup(\sigma,\gamma),\beta_{1}^{-1}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}(\beta_{2}^{-1}\circ\gamma^{-1}))$.
2. 2.
Any other $\beta$ can be represented as $\beta_{1}\circ\ldots\circ\beta_{n}$
such that $1\leqslant\\#\\{x\mid\beta_{i}x=y\\}\leqslant 2$ for all $y$ and
for all $i$, i.e. each $\beta_{i}$ has the form required by 1.; then
$\sigma\circ\beta=(\ldots(\sigma\circ\beta_{1})\circ\ldots)\circ\beta_{n}$.
Algorithm 47. The following algorithm computes $fact(\sigma,\beta)$ if $\beta
x\in{\cal V}$ for all $x$. Let $\sigma$ be a regular t-set, let $\mu$ be a new
name for a regular t-set; define $fact(\sigma,\beta)=\mu$, where a new t-set
definition is introduced for $\mu$:
1. 1.
If $fact(\sigma,\beta)$ has been called earlier, $\mu$ is already defined
(loop checking).
2. 2.
Else, if $\sigma\doteq\sigma_{1}\mid\ldots\mid\sigma_{n}$, define $\mu\doteq
fact(\sigma_{1},\beta)\mid\ldots\mid fact(\sigma_{n},\beta)$.
3. 3.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$ with
$dom(\beta)\subset dom(\vec{cr})$,
and $\vec{cr}_{x}=\vec{cr}_{y}$ whenever $\beta x=\beta y$, define
$\vec{cr}^{\prime}:ran(\beta)\rightarrow{\cal CR}$ by
$\vec{cr}^{\prime}_{\beta x}:=\vec{cr}_{x}$,
define
$\mu\doteq\vec{cr}^{\prime}(fact(\sigma_{1},\beta),\ldots,fact(\sigma_{ar(\vec{cr}^{\prime})},\beta))$.
4. 4.
Else, define $\mu\doteq\bot$.
Using the induction principle from Thm. 2, lifted to t-sets, with
$p_{\mu}(\sigma^{\prime}):\Leftrightarrow\sigma^{\prime}\in\sigma^{M}\\!/\\!_{\beta}$
if $\mu=fact(\sigma,\beta)$, it can be shown that
$fact(\sigma,\beta)^{M}=\sigma^{M}\\!/\\!_{\beta}$. The algorithm needs at
most $\\#use(\sigma)$ recursive calls to compute $fact(\sigma,\beta)$.
Definition 48. $\beta$ is called homogeneous if all variables in the range of
$\beta$ occur at the same depth, i.e., if $\beta x\in{\cal V}$ for each $x\in
dom(\beta)$, or if $\beta x=cr_{x}(u_{x1},\ldots,u_{x\;ar(cr_{x})})$ for all
$x\in dom(\beta)$ and appropriate $u_{xi}$, and $[x\\!:=\\!u_{xi}\mid x\in
dom(\beta),ar(cr_{x})\geqslant i]$ is again homogeneous for each $i$.
Algorithm 49. The following algorithm computes $fact(\sigma,\beta)$ if
$\beta$ is homogeneous. Let $\sigma$ be a regular t-set, define
$fact(\sigma,\beta)$ by:
1. 1.
If $\beta=[\;]$, define $fact(\sigma,\beta):=\\{\varepsilon\\}$.
2. 2.
Else, if $\beta x\in{\cal V}$ for all $x$, compute $fact(\sigma,\beta)$ by
Alg. 4.
3. 3.
Else, if $\sigma\doteq\sigma_{1}\mid\ldots\mid\sigma_{n}$, define
$fact(\sigma,\beta):=fact(\sigma_{1},\beta)\mid\ldots\mid
fact(\sigma_{n},\beta)$.
4. 4.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$,
$dom(\beta)\subset dom(\vec{cr})$, and $\beta
x=\vec{cr}_{x}(u_{x,1},\ldots,u_{x,ar(\vec{cr}_{x})})$ for all $x\in
dom(\beta)$,
define $fact(\sigma,\beta):=compose(\\{$
$fact(\sigma_{1},[x\\!:=\\!u_{x,1}\mid x\in
dom(\beta),ar(\vec{cr}_{x})\geqslant 1]),\ldots,$
$fact(\sigma_{n},[x\\!:=\\!u_{x,n}\mid x\in
dom(\beta),ar(\vec{cr}_{x})\geqslant n])\\})$.
5. 5.
Else, define $fact(\sigma,\beta):=\bot$.
Using the lexicographic combination of the size of range terms of $\beta$ and
$\stackrel{{\scriptstyle\mbox{\Large\bf.}}}{{<}}$, it can be shown that the
algorithm always terminates and yields
$fact(\sigma,\beta)^{M}=\sigma^{M}\\!/\\!_{\beta}$ for $\beta\neq[\;]$. The
algorithm needs at most $depth(\beta)$ recursive calls to compute
$fact(\sigma,\beta)$. If $\sigma$ is semi-independent, then so is
$fact(\sigma,\beta)$.
Algorithm 50. Let $\beta$ be pseudolinear, $dom(\beta)\subset dom(\sigma)$,
$V:=\\{x\in dom(\beta)\mid\beta x\in{\cal V}\\}$. Define the finite set
$hom(\sigma,\beta)$ of ordinary homogenizing substitutions for $\beta$ wrt.
$\sigma$:
1. 1.
If $\sigma\doteq\sigma_{1}\mid\ldots\mid\sigma_{n}$, define
$hom(\sigma,\beta):=hom(\sigma_{1},\beta)\cup\ldots\cup
hom(\sigma_{n},\beta)$.
2. 2.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$, let
$\gamma_{0}:=[\beta
x\\!:=\\!\vec{cr}_{x}(y_{x,1},\ldots,y_{x,ar(\vec{cr}_{x})})\mid x\in V]$
where the $y_{x,i}$ are new variables, define
$hom(\sigma,\beta):=(hom(\sigma,\gamma_{0}\circ\beta)\circ\gamma_{0})\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
ran(\beta)$}}$.
3. 3.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$, $\beta$ not
homogeneous, $V=\\{\\}$,
and $\beta x=\vec{cr}_{x}(u_{x1},\ldots,u_{x\;ar(\vec{cr}_{x})})$ for all
$x\in dom(\beta)$,
define $hom(\sigma,\beta):=\bigcirc\raisebox{1.1pt}{$\cdot$}\hskip
2.5pt_{i=1}^{n}hom(\sigma_{i},[x\\!:=\\!u_{xi}\mid x\in
dom(\beta),ar(\vec{cr}_{x})\geqslant i])$.
4. 4.
Else, if $\beta$ homogeneous, define
$hom(\sigma,\beta):=\\{[x\\!:=\\!y_{x}\mid x\in ran(\beta)]\\}$,
where each $y_{x}$ is a new variable.
5. 5.
Else, define $hom(\sigma,\beta):=\\{\\}$.
Then, for each $\gamma\in hom(\sigma,\beta)$:
1. 1.
$\gamma\circ\beta$ is homogeneous,
2. 2.
$dom(\gamma)=ran(\beta)$,
3. 3.
for each $\sigma^{\prime}\in\sigma$ with
$\sigma^{\prime}\\!/\\!_{\beta}\neq\\{\\}$ there exists a $\gamma\in
hom(\sigma,\beta)$ such that
$\sigma^{\prime}\\!/\\!_{\gamma\circ\beta}\neq\\{\\}$, and
4. 4.
for each $\sigma^{\prime}\in\sigma$ with
$\sigma^{\prime}\\!/\\!_{\beta}\neq\\{\\}$ there exists at most one $\gamma\in
hom(\sigma,\beta)$ such that
$\sigma^{\prime}\\!/\\!_{\gamma\circ\beta}\neq\\{\\}$.
Proof. Use the lexicographic combination of the size of range terms of
$\beta$ and $\stackrel{{\scriptstyle\mbox{\Large\bf.}}}{{<}}$ as termination
ordering and as induction ordering for 1. to 4. The algorithm needs at most
$\\#use(\sigma)*depth(\beta)$ recursive calls to compute $hom(\sigma,\beta)$.
Theorem 51. If $\sigma$ regular and $\beta$ pseudolinear, then
$\sigma\\!/\\!_{\beta}u=\bigcup_{\gamma\in
hom(\sigma,\beta)}\sigma\\!/\\!_{\gamma\circ\beta}\gamma u$ for all $u$ with
$vars(u)\subset ran(\beta)$, where the $\sigma\\!/\\!_{\gamma\circ\beta}$ are
regular.
Proof. Regularity follows from 4 and 4. Since for each
$\\{\\}\neq\sigma^{\prime}\\!/\\!_{\beta}\in\sigma\\!/\\!_{\beta}$ there
exists exactly one $\gamma$ with
$\sigma^{\prime}\\!/\\!_{\gamma\circ\beta}\neq\\{\\}$ by 4, the claimed
equality follows from 6.
Example 52. Consider the definition of $nat_{x\\!<\\!y}$ in Fig. 7; let
$\beta:=[x\\!:=\\!x^{\prime},y\\!:=\\!s(y^{\prime})]$. We first homogenize
$\beta$ wrt. $nat_{x\\!<\\!y}$ by 4, yielding
$hom(nat_{x\\!<\\!y},\beta)=\\{[x^{\prime}\\!:=\\!0,y^{\prime}\\!:=\\!y^{\prime\prime}],[x^{\prime}\\!:=\\!s(x^{\prime\prime}),y^{\prime}\\!:=\\!y^{\prime\prime}]\\}$.
Then, using 4, we factorize $nat_{x\\!<\\!y}$ wrt. the homogeneous
substitutions
$[x^{\prime}\\!:=\\!0,y^{\prime}\\!:=\\!y^{\prime\prime}]\circ\beta$ and
$[x^{\prime}\\!:=\\!s(x^{\prime\prime}),y^{\prime}\\!:=\\!y^{\prime\prime}]\circ\beta$,
yielding
$fact(nat_{x\\!<\\!y},[x\\!:=\\!0,y\\!:=\\!s(y^{\prime\prime})])=nat_{y^{\prime\prime}}$
and
$fact(nat_{x\\!<\\!y},[x\\!:=\\!s(x^{\prime\prime}),y\\!:=\\!s(y^{\prime\prime})])=nat_{x^{\prime\prime}\\!<\\!y^{\prime\prime}}$,
respectively. Using 4, we can thus compute
$nat_{x\\!<\\!y}\\!/\\!_{\beta}\langle x^{\prime},y^{\prime}\rangle$ as
$nat_{y^{\prime\prime}}\langle 0,y^{\prime\prime}\rangle\mid
nat_{x^{\prime\prime}\\!<\\!y^{\prime\prime}}\langle
s(x^{\prime\prime}),y^{\prime\prime}\rangle$.
Example 53. Let $\sigma\doteq 0_{x}cr_{y}(0_{y},0_{y})\mid
cr_{x}cr_{y}(\sigma,\sigma)$, and
$\beta=[x\\!:=\\!x^{\prime},y\\!:=\\!cr(y^{\prime},x^{\prime})]$. Then,
$\sigma^{M}\\!/\\!_{\beta}$ is the infinite set of complete binary trees $A$
that is minimal with $0_{x^{\prime}}0_{y^{\prime}}\in A$ and
$cr_{x^{\prime}}cr_{y^{\prime}}(\sigma^{\prime},\sigma^{\prime})\in A$ for
$\sigma^{\prime}\in A$. $\sigma\\!/\\!_{\beta}x^{\prime}$ is a similar set of
complete binary trees which cannot be written as
$\tau_{1}u_{1}\cup\ldots\cup\tau_{n}u_{n}$ with regular $\tau_{i}$.
Proof.
1. 2. 1.
Show $\sigma^{\prime}\in
A\Rightarrow\sigma^{\prime}\in\sigma^{M}\\!/\\!_{\beta}$ by induction on
$\sigma^{\prime}$.
3. 2.
Show
$\sigma^{M}\\!/\\!_{\beta}=\\{0_{x^{\prime}}0_{y^{\prime}}\\}\cup(\sigma^{M}\\!/\\!_{[x\\!:=\\!x_{1},y\\!:=\\!y^{\prime}]}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{M}\\!/\\!_{\beta}\\!/\\!_{[x^{\prime}\\!:=\\!x_{2},y^{\prime}\\!:=\\!x_{1}]})\circ[x^{\prime}\\!:=\\!cr(x_{1},x_{2})]$
by direct computation.
4. 3.
Show
$\sigma^{M}\\!/\\!_{[x\\!:=\\!x_{1},y\\!:=\\!y^{\prime}]}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\sigma^{\prime}\\!/\\!_{[x^{\prime}\\!:=\\!x_{2},y^{\prime}\\!:=\\!x_{1}]}=\sigma^{\prime}\\!/\\!_{[x^{\prime}\\!:=\\!x_{2},y^{\prime}\\!:=\\!x_{1}]}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}cr_{y^{\prime}}(\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
y^{\prime}$}},\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
y^{\prime}$}})$ by induction on $\sigma^{\prime}$ using 2.
5. 4.
Show $\sigma^{M}\\!/\\!_{\beta}\subset A$ by induction on the order
$\sigma^{\prime}_{1}<\sigma^{\prime}_{2}:\Leftrightarrow\sigma^{\prime}_{1}x^{\prime}\mathrel{\mathop{\sphericalangle}\limits_{\neq}}\sigma^{\prime}_{2}x^{\prime}$
using 3. and 4.
6. 5.
Show that no infinite set of complete binary trees can be represented as $\tau
u$ with regular t-set $\tau$ and constructor term $u$ by induction on $u$,
using in the base case 7 and a pumping lemma.
Theorem 54. If $\sigma$ is independent, then
$\sigma\\!/\\!_{\beta}=\Diamond\raisebox{1.1pt}{$\cdot$}\hskip 2.5pt_{x\in
dom(\beta)}\sigma\\!/\\!_{[x{:=}\beta x]}$.
The right-hand side can be algorithmically computed using 4, since
$[x\\!:=\\!\beta x]$ is always homogeneous.
Algorithm 55. Let $\sigma$ be the name of a regular t-set. The following
algorithm decides whether $\sigma\subset\top\circ\beta$,
i.e. whether $\sigma^{\prime}\\!/\\!_{\beta}\neq\\{\\}$ for all
$\sigma^{\prime}\in\sigma$.
Define $div(\sigma,\beta)$ $:\Leftrightarrow$ $\bigwedge_{x\in
dom(\beta)}div(\sigma,[x\\!:=\\!\beta x])$ $\wedge$
$\bigwedge_{x,x^{\prime}\in dom(\beta),x\neq x^{\prime}}\bigwedge_{y\in
vars(\beta x)\cap vars(\beta x^{\prime})}$
$single(apply(fact(\sigma,[x\\!:=\\!\beta x]),y)\mid
apply(fact(\sigma,[x^{\prime}\\!:=\\!\beta x^{\prime}]),y))$;
where $div(\sigma,[x\\!:=\\!u])$ is computed as follows:
1. 1.
If $\sigma\doteq\sigma_{1}\mid\ldots\mid\sigma_{n}$, define
$div(\sigma,[x\\!:=\\!u]):\Leftrightarrow
div(\sigma_{1},[x\\!:=\\!u])\wedge\ldots\wedge div(\sigma_{n},[x\\!:=\\!u])$.
2. 2.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$ and
$u=\vec{cr}_{x}(u_{1},\ldots,u_{k})$,
define $div(\sigma,[x\\!:=\\!u])$ $:\Leftrightarrow$
$\bigwedge_{i=1}^{k}div(\sigma_{i},[x\\!:=\\!u_{i}])$ $\wedge$
$\bigwedge_{1\leqslant i<j\leqslant k}\bigwedge_{y\in vars(u_{i})\cap
vars(u_{j})}$ $single(apply(fact(\sigma_{i},[x\\!:=\\!u_{i}]),y)\mid
apply(fact(\sigma_{j},[x\\!:=\\!u_{j}]),y))$.
3. 3.
Else, if $\sigma\doteq\vec{cr}(\sigma_{1},\ldots,\sigma_{n})$ and
$u=cr^{\prime}(u_{1},\ldots,u_{k})$ with $cr^{\prime}\neq\vec{cr}_{x}$, define
$div(\sigma,[x\\!:=\\!u]):\Leftrightarrow false$.
4. 4.
Else, if $u\in{\cal V}$ (also $u=x$), define
$div(\sigma,[x\\!:=\\!u]):\Leftrightarrow x\in dom(\sigma)$.
Using the lexicographical combination of $\mathrel{\sphericalangle}$ and
$\stackrel{{\scriptstyle\mbox{\Large\bf.}}}{{<}}$, the correctness and
termination of the computation of $div(\sigma,[x\\!:=\\!u])$ can be shown by
induction on $\langle\sigma,u\rangle$. The proof, as well as the correctness
proof for $div(\sigma,\beta)$, uses the fact that $\sigma y\cup\tau y$ is a
singleton set for all $y\in dom(\sigma)\cap dom(\tau)$ iff
$\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau^{\prime}\neq\\{\\}$ for all
$\sigma^{\prime}\in\sigma$, $\tau^{\prime}\in\tau$. The algorithm needs at
most $\\#use(\sigma)$ recursive calls to decide $div(\sigma,[x\\!:=\\!u])$.
## 5 Extended Sorts
In this section, we discuss several possible ways of defining a class of sorts
that can express more subsets of ${\cal T}_{\cal CR}$ than regular tree
languages. In particular, we define a class called “extended sorts” that can
express for arbitrary $u\in{\cal T}_{{\cal CR},{\cal V}}$ the set of all
possible values $U$ of $u$, mentioned in Sect. 1, as an extended sort. We
define the notion of an “annotated term” ${{}^{\sigma}\\!v}$ where the t-set
$\sigma$ indicates the set of admitted ground constructor instances of $v$’s
variables, i.e. their sort.
The results obtained in Sect. 4 allow us to define three different language
classes which are all proper extensions of regular tree languages. In each
class, a language of ground constructor terms is described by applying regular
t-sets $\sigma$ to constructor terms $u$, the classes differing in the form
that is allowed for $\sigma$ and $u$:
1. 1.
$\sigma u$ with $\sigma$ semi-independent, $u$ arbitrary.
The intersection can be computed using Thms. 6 and 9. The subset property
$\sigma u\subset\tau_{1}u_{1}$ (hence equivalence and inhabitance) can be
decided using Thm. 6 and Lemma 6, since we have $J=\\{\\{\\},\\{1\\}\\}$,
$\sigma_{\\{\\}}\subset\\{\\}\Leftrightarrow div(\sigma,\beta_{\\{1\\}})$, and
$\sigma_{\\{1\\}}\\!/\\!_{\beta_{\\{1\\}}}=\sigma\\!/\\!_{\beta_{\\{1\\}}}$.
However, this class is not closed wrt. union. Any regular sort $S^{M}$ from
Sect. 2 can be expressed as $[x\\!:=\\!S^{M}]\;x$, but the converse is false,
e.g. $Nat_{x}^{M}\;\langle x,x\rangle$ is not a regular sort, as can be shown
using a pumping lemma [4].
2. 2.
$\sigma_{1}u_{1}\cup\ldots\cup\sigma_{n}u_{n}$ with $\sigma_{i}$ independent,
$u_{i}$ arbitrary.
This is a proper superclass of the class given in 1. The intersection can be
computed using Thms. 6 and 9; union is trivial; inhabitance can be decided
using Thm. 6. However, we do not provide an algorithm to decide the subset
property in general. Again, any regular sort $S^{M}$ can be expressed as
$[x\\!:=\\!S^{M}]\;x$.
3. 3.
$\sigma_{1}u_{1}\cup\ldots\cup\sigma_{n}u_{n}$ with $\sigma_{i}$ arbitrary,
$u_{i}\in T$ for some set $T$ such that for any $u,u^{\prime}\in T$,
$\beta=mgu(u,u^{\prime})$ is always pseudolinear if it exists, and again
$\beta u\in T$.
The intersection can be computed using Thms. 6, and 4. Inhabitance can be
decided as in class 2., but again we do not provide an algorithm to decide
subsort in general. If we take $T$ to be the set of all constructor terms in
which variables occur only at a fixed unique depth $n$, the requirements to
$T$ are fulfilled, and each regular sort can be expressed.
As shown in section 4, dependent regular t-sets can express certain relations
between distinct variables, e.g. the conditional equation
$x\raisebox{5.69054pt}{\scriptsize:Nat}<y\raisebox{5.69054pt}{\scriptsize:Nat}\rightarrow
f(x,y)=g(x,y)$ can be expressed unconditionally by $f(x,y)=g(x,y)$, where the
value combinations of $x$ and $y$ are restricted by $Nat_{x<y}$ from Fig. 7.
Since we have the problem that Thms. 6 and 6 use factorization
$\sigma\\!/\\!_{\beta}$ which is not always a regular t-set, cf. Ex. 4, we
have to restrict $T$ as above. It is an unsolved problem whether a superclass
of 2. exists that allows dependent t-sets but is still closed wrt. the
required operations, especially intersection.
All classes follow the philosophy of allowing arbitrary nonlinearities up to a
finite depth and forbidding any below. Since class 1. is sufficient to
represent the set of all possible values $U$ of an arbitrary constructor term
$u$, we will use this class in the rest of this paper. In classical order-
sorted approaches, each variable in a term is assigned a sort, e.g.
$x\raisebox{5.69054pt}{\scriptsize:Nat}$ \+ x:Nat. We will, instead, use a
semi-independent t-set to specify the set of possible ground instances,
written e.g. ${{}^{(Nat_{x})}\\!\;}(x+x)$, with the informal meaning that each
(“admissible”) substitution instantiating $x+x$ must be extendable to a ground
substitution contained in $Nat_{x}^{M}$. This approach still allows variable
bindings in a term to be reflected by its sort, as sketched in Sect. 1; what
we lose is the possibility of expressing nontrivial relations between
variables.
Definition 56. We define an annotated term as a pair of a semi-independent
regular t-set $\sigma$ and an (unsorted) term $v$; it is written as
${{}^{\sigma}\\!v}$. The t-set $\sigma$ denotes the admitted instances of $v$,
cf. the use of ${{}^{\sigma}\\!v}$ in Defs. 6 and 14 below.
Definition 57. We call an expression of the form $\sigma u$ with semi-
independent regular $\sigma$ and $u\in{\cal T}_{{\cal CR},{\cal V}}$ an
extended sort. The set of all possible values $U$ of an annotated term
${{}^{\sigma}\\!u}$ is always an extended sort, viz. $U=\sigma u$. Sets of the
form $\sigma v$, where $v$ contains non-constructor functions, will be
approximated by extended sorts later, cf. Sect. 6.
Corollary 58. A sorted equation ${{}^{\sigma}\\!u}_{1}={{}^{\sigma}\\!u}_{2}$
between annotated constructor terms is solvable iff an (unsorted) mgu $\beta$
of $u_{1}$ and $u_{2}$ exists and $\sigma\\!/\\!_{\beta}\neq\\{\\}$. The
latter set contains the admissible ground instances of variables in
$ran(\beta)$. An equation system
${{}^{\sigma_{1}}\\!\,}u_{1}={{}^{\tau_{1}}\\!\,}u^{\prime}_{1}\wedge\ldots\wedge{{}^{\sigma_{n}}\\!\,}u_{n}={{}^{\tau_{n}}\\!\,}u^{\prime}_{n}$
can be reduced to a single equation ${{}^{\sigma}\\!\langle
u_{1},\ldots,u_{n}\rangle}={{}^{\sigma}\\!\langle
u^{\prime}_{1},\ldots,u^{\prime}_{n}\rangle}$ by defining
$\sigma:=\Diamond\raisebox{1.1pt}{$\cdot$}\hskip
2.5pt_{i=1}^{n}\sigma_{i}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau_{i}$.
Example 59. Using the definitions from 4, $\sigma\langle
x,y\rangle\cap\top_{\\{x^{\prime\prime},y^{\prime\prime}\\}}\langle
x^{\prime\prime},cr(y^{\prime\prime},x^{\prime\prime})\rangle$ cannot be
represented as an extended sort.
Proof. Observe that
$\beta^{\prime}=\beta\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}{\circ\mkern-7.0mu\cdot\mkern
2.0mu}}[x^{\prime\prime}\\!:=\\!x^{\prime},y^{\prime\prime}\\!:=\\!y^{\prime}]=mgu(\langle
x,y\rangle,\langle
x^{\prime\prime},cr(y^{\prime\prime},x^{\prime\prime})\rangle)$,
and $(\sigma\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\top_{\\{x^{\prime\prime},y^{\prime\prime}\\}})\\!/\\!_{\beta^{\prime}}=\sigma\\!/\\!_{\beta}$;
hence, by 6, $\sigma\langle
x,y\rangle\cap\top_{\\{x^{\prime\prime},y^{\prime\prime}\\}}\langle
x^{\prime\prime},cr(y^{\prime\prime},x^{\prime\prime})\rangle=(\sigma\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\top_{\\{x^{\prime\prime},y^{\prime\prime}\\}})\\!/\\!_{\beta^{\prime}}\langle
x^{\prime},cr(y^{\prime},x^{\prime})\rangle=\sigma\\!/\\!_{\beta}\langle
x^{\prime},cr(y^{\prime},x^{\prime})\rangle$,
the latter cannot be written as $\tau_{1}u_{1}\cup\ldots\cup\tau_{n}u_{n}$
with $\tau_{i}$ regular, by an argument similar to 4.5.
## 6 Equational Theories
In this section, we extend the previous formalism to allow equationally
defined functions $f$. We allow defining equations of the form given in Def.
6, thus ensuring the “executability” of $f$. Signatures of such a function are
computed from its defining equations by the $rg$ algorithm presented below in
Alg. 10, which will play a central role in pruning the search space of
narrowing. The algorithm takes a regular t-set and a term with non-constructor
functions and computes an upper approximation by an extended sort, e.g.
$rg([x,y\\!:=\\!Nat],x+y)=[z\\!:=\\!Nat]\;z$. In terms of Sect. 1, we have
$rg(\sigma,v)=\overline{V}$ where $\sigma$ denotes the values over which the
variables in $v$ may range. The $rg$ algorithm consists of local
transformations like rewriting and some simplification rules (cf. Def. 10),
global transformations looking at a sequence of local transformation steps and
recognizing certain kinds of self-references (cf. Lemma 10), and an
approximation rule. Only the main rules can be discussed here; the complete
algorithm is given in [4].
In Theorem 14, a narrowing calculus from [9] is equipped with sorts. In [4],
the calculus is shown to remain complete if the applicability of its main rule
is restricted by the disjointness test from Sect. 1.
Definition 60. In the rest of this section, we assume that $f$ has the
following defining equations:
${{{}^{\mu_{1}}\\!f}(u_{11},\ldots,u_{1n})}$ | $={{}^{\mu_{1}}\\!\,}v_{1},$
---|---
| $\ldots$,
${{}^{\mu_{m}}\\!f}(u_{m1},\ldots,u_{mn})$ | $={{}^{\mu_{m}}\\!\,}v_{m}$,
where $vars(v_{i})\subset vars(u_{i1},\ldots,u_{in})$. We assume that the
variables of different defining equations are disjoint. Define
$dom(f,I):=\bigcup_{i\in I}\mu_{i}\langle u_{i1},\ldots,u_{in}\rangle$, and
$dom(f):=dom(f,\\{1,\ldots,m\\})$.
Definition 61. Define the rewrite relation induced by the defining equations
by:
${{}^{\sigma}\\!v}_{1}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}}{{}^{\sigma}\\!v}_{2}$ iff
1. 1.
a defining equation ${{}^{\mu}\\!f}(u_{1},\ldots,u_{n})={{}^{\mu}\\!v}$, a
substitution $\beta$, and a term $v^{\prime}(x)$ linear in $x$ exist such that
$v_{1}=v^{\prime}(\beta f(u_{1},\ldots,u_{n}))$, $v_{2}=v^{\prime}(\beta v)$,
2. 2.
and for all $\sigma^{\prime}\in\sigma$ there exists $\mu^{\prime}\in\mu$ such
that for all $x\in vars(u_{1},\ldots,u_{n})$
${{}^{\varepsilon}\\!\sigma}^{\prime}\beta
x\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}{{}^{\varepsilon}\\!\mu}^{\prime}x$ if
${{}^{\varepsilon}\\!\sigma}^{\prime}\beta x$ is well-defined.
While the former condition is merely rewriting by pattern matching, the latter
is an analogue to the classical well-sortedness requirement for $\beta$,
requiring any well-defined variable instance to be admitted by the defining
equation’s sort. A ground term is called well-defined if it is reducible to a
ground constructor term.
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}$
and
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}$
are defined as usual; the definition of
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$ is
recursive, but well-founded. We require confluence and termination of
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$,
ensuring ${\cal T}_{\cal CR}\subset{\cal T}_{{\cal CR},{\cal
F}}/{\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}}$, where ${\cal T}_{{\cal CR},{\cal
F}}/{\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}}$
denotes the set of equivalence classes of terms in ${\cal T}_{{\cal CR},{\cal
F}}$ modulo
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}$.
In other words,
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}$
does not identify terms in ${\cal T}_{\cal CR}$, but new irreducible terms
like $nil+nil$ may arise which we will regard as “junk terms” and exclude from
equation solutions. For a well-defined ground term $v$, let $nf(v)\in{\cal
T}_{\cal CR}$ denote its unique normal form; for $A\subset{\cal T}_{{\cal
CR},{\cal F}}$, let $nf[A]:=\\{nf(v)\mid v\in A,v\mbox{ well-defined}\\}$.
| Classical order-sorted terms | Annotated terms
---|---|---
term | $w$ where $x_{1}\raisebox{5.69054pt}{\scriptsize:$s_{1}$},\ldots,x_{m}\raisebox{5.69054pt}{\scriptsize:$s_{m}$}$ | ${{}^{\sigma}\\!w}$ where $dom(\sigma)=\\{x_{1},\ldots,x_{m}\\}$
sort | $sortof(w)$ | $\sigma w$
def. eq. | $f(l_{1},\ldots,l_{n})=r$ where | ${{}^{\mu}\\!f}(l_{1},\ldots,l_{n})={{}^{\mu}\\!r}$ where
| $y_{1}\raisebox{5.69054pt}{\scriptsize:$t_{1}$},\ldots,y_{m}\raisebox{5.69054pt}{\scriptsize:$t_{m}$}$ | $dom(\mu)=\\{y_{1},\ldots,y_{m}\\}$
rewriting | $v(\beta f(l_{1},\ldots,l_{n}))\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}v(\beta r)$ where | ${{}^{\sigma}\\!v}(\beta f(l_{1},\ldots,l_{n}))\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}{{}^{\sigma}\\!v}(\beta r)$ where
| $\forall y\in V\;\;sortof(\beta y)\subset sortof(y)$ | $\forall\sigma^{\prime}\\!\in\\!\sigma\;\exists\mu^{\prime}\\!\in\\!\mu\;\forall y\\!\in\\!V\;\;{{}^{\varepsilon}\\!\sigma}^{\prime}\beta y\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}{{}^{\varepsilon}\\!\mu}^{\prime}y$
equation | $w_{1}=w_{2}$ where $x_{1}\raisebox{5.69054pt}{\scriptsize:$s_{1}$},\ldots,x_{m}\raisebox{5.69054pt}{\scriptsize:$s_{m}$}$ | ${{}^{\sigma}\\!w}_{1}={{}^{\sigma}\\!w}_{2}$
solution | $\gamma w_{1}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}\gamma w_{2}$ where | $\gamma w_{1}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}\gamma w_{2}$ and $\tau$ where
| $\forall x\in W\;\;sortof(\gamma x)\subset sortof(x)$ | $nf[\tau\gamma\langle x_{1},\ldots,x_{m}\rangle]\subset\sigma\langle x_{1},\ldots,x_{m}\rangle$
| | and $nf[\tau\beta\langle u_{1},\ldots,u_{n}\rangle]\neq\\{\\}$
$w,w_{1},w_{2}\in{\cal T}_{{\cal CR},{\cal F},{\cal V}}$,
$l_{1},\ldots,l_{n}\in{\cal T}_{{\cal CR},{\cal V}}$,
$\sigma,\tau,\mu\subset{\cal T}^{*}_{\mathchoice{(\cal
V\\!\hookrightarrow\\!\cal CR)}{(\cal V\\!\hookrightarrow\\!\cal CR)}{(\cal
V\hookrightarrow\cal CR)}{(\cal V\hookrightarrow\cal CR)}}$,
$W=vars(w)=vars(w_{1},w_{2})=\\{x_{1},\ldots,x_{m}\\}=dom(\sigma)$,
$V=vars(l_{1},\ldots,l_{n})=\\{y_{1},\ldots,y_{m}\\}$
Examples:
---
| Classical order-sorted terms | Annotated terms
term | $x\raisebox{5.69054pt}{\scriptsize:Nat}+x\raisebox{5.69054pt}{\scriptsize:Nat}$ | ${{}^{[x\\!:=\\!Nat]\;}\\!x}+x$
sort | $Nat+Nat=Nat$ | $[x\\!:=\\!Nat]\;(x+x)=Even$
def. eq. | | $a\raisebox{5.69054pt}{\scriptsize:Nat}+0$ | $=a\raisebox{5.69054pt}{\scriptsize:Nat}$
---|---
$a\raisebox{5.69054pt}{\scriptsize:Nat}+s(b\raisebox{5.69054pt}{\scriptsize:Nat})$ | $=s(a\raisebox{5.69054pt}{\scriptsize:Nat}+b\raisebox{5.69054pt}{\scriptsize:Nat})$
$s(a\raisebox{5.69054pt}{\scriptsize:Nat})+b\raisebox{5.69054pt}{\scriptsize:Nat}$ | $=s(a\raisebox{5.69054pt}{\scriptsize:Nat}+b\raisebox{5.69054pt}{\scriptsize:Nat})$
| ${{}^{[a\\!:=\\!Nat]\;}\\!a}+0$ | $={{}^{[a\\!:=\\!Nat]\;}\\!a}$
---|---
${{}^{[a,b\\!:=\\!Nat]\;}\\!a}\\!+\\!s(b)$ | $={{}^{[a,b\\!:=\\!Nat]\;}\\!s}(a\\!+\\!b)$
${{}^{[a,b\\!:=\\!Nat]\;}\\!s}(a)\\!+\\!b$ | $={{}^{[a,b\\!:=\\!Nat]\;}\\!s}(a\\!+\\!b)$
rewriting | | | $s(x\raisebox{5.69054pt}{\scriptsize:Nat}+x\raisebox{5.69054pt}{\scriptsize:Nat}+s(y\raisebox{5.69054pt}{\scriptsize:Nat}))$
---|---
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$ | $s(s(x\raisebox{5.69054pt}{\scriptsize:Nat}+x\raisebox{5.69054pt}{\scriptsize:Nat}+y\raisebox{5.69054pt}{\scriptsize:Nat}))$
where $\beta=[a\\!:=\\!x+x,b\\!:=\\!y]$
| | ${{}^{[x,y\\!:=\\!Nat]\;}\\!s}(x+x+s(y))$
---|---
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$ | ${{}^{[x,y\\!:=\\!Nat]\;}\\!s}(s(x+x+y))$
where $\beta=[a\\!:=\\!x+x,b\\!:=\\!y]$
| $sortof(\beta a)=Nat\\!+\\!Nat=sortof(a)$ | for $[x\\!:=\\!s^{i}(0),y\\!:=\\!s^{j}(0)]$
| $sortof(\beta b)=Nat=sortof(b)$ | choose $[x\\!:=\\!s^{2\\!\cdot\\!i}(0),y\\!:=\\!s^{j}(0)]$
equation | $s(s(x\raisebox{5.69054pt}{\scriptsize:Nat}))=y\raisebox{5.69054pt}{\scriptsize:Nat}+y\raisebox{5.69054pt}{\scriptsize:Nat}$ | ${{}^{[x,y\\!:=\\!Nat]\;}\\!s}(s(x))=y+y$
solution | | $\gamma=[x\\!:=\\!z\raisebox{5.69054pt}{\scriptsize:Nat}+z\raisebox{5.69054pt}{\scriptsize:Nat},y\\!:=\\!s(z\raisebox{5.69054pt}{\scriptsize:Nat})]$
---
$sortof(\gamma x)=Nat\\!+\\!Nat=sortof(x)$
$sortof(\gamma y)=s(Nat)\subset sortof(y)$
| $\gamma=[x\\!:=\\!z+z,y\\!:=\\!s(z)]$
---
and $\tau=[z\\!:=\\!Nat]$, $nf[\tau\gamma\langle x,y\rangle]$
$=\\{\langle s^{2\\!\cdot\\!i}(0),s^{i\\!+\\!1}(0)\rangle\mid i\in
I\\!\\!N\\}$
$\subset\langle Nat,Nat\rangle$
Figure 10: Comparison of classical order-sorted terms and annotated terms
Figure 10 shows a comparison of classical order-sorted terms and annotated
terms. The applicability of
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$ is
not decidable in general owing to the well-sortedness condition 6.2. It is
possible to compute sufficiently large t-sets $\mu_{i}$ for the defining
equations such that 6.2 becomes trivial, cf. Alg. 10; however, if the
$\mu_{i}$ are too large, well-sorted terms arise that are not well-defined. As
in any order-sorted term rewriting approach, we cannot overcome both problems
simultaneously.
Range sorts are computed using expressions of the form
$(w_{1}\\!:\\!u_{1})\ldots(w_{n}\\!:\\!u_{n})$, which can intuitively be
thought of as generalized equation systems; the semantic is the set of all
t-substitutions making each $u_{i}$ equal
($\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}$)
to an element of $w_{i}$. For example, $(Nat:x)$ denotes
$\\{[x\\!:=\\!s^{i}(0)]\mid i\in I\\!\\!N\\}$, and $(x+x:z)$ can be evaluated
to $\\{[x\\!:=\\!s^{i}(0),z\\!:=\\!s^{2*i}(0)]\mid i\in I\\!\\!N\\}$.
Definition 62. Define $w^{M}\subset{\cal T}_{{\cal CR},{\cal F},{\cal V}}$
by:
$S^{M}$ $:=S^{M}$ as in Def. 2 for $S\in{\cal S}$ $g(v_{1},\ldots,v_{n})^{M}$
$:=\\{g(v^{\prime}_{1},\ldots,v^{\prime}_{n})\mid v^{\prime}_{1}\in
v_{1}^{M},\ldots,v^{\prime}_{n}\in v_{n}^{M}\\}$ for $g\in{\cal CR}\cup{\cal
F}$ $x^{M}$ $:=\\{x\\}$ for $x\in{\cal V}$
For $w\in{\cal T}_{{\cal CR},{\cal S}}$, $w^{M}$ agrees with Def. 2. For
$w\in{\cal T}_{{\cal CR},{\cal F},{\cal V}}$ we always have $w^{M}=\\{w\\}$.
We tacitly extend the operations of Sect. 3 to ${\cal T}_{{\cal CR},{\cal
F},{\cal V}}$ by treating function symbols from ${\cal F}$ like constructors
from ${\cal CR}$, e.g. $(0_{x})\;(x+x)=\\{0+0\\}$.
Let $(w:u)^{M}:=\\{\sigma^{\prime}\mid dom(\sigma^{\prime})=vars(w,u),\exists
w^{\prime}\in w^{M},u^{\prime}\in
u^{M}\;\;\sigma^{\prime}w^{\prime}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}\sigma^{\prime}u^{\prime}\\}$ and
$((w_{1}:u_{1})\;(w_{2}:u_{2}))^{M}:=(w_{1}:u_{1})^{M}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}(w_{2}:u_{2})^{M}$. We write
$(\sigma)$ to denote an expression
$(w_{1}\\!:\\!u_{1})\ldots(w_{n}\\!:\\!u_{n})$ such that
$((w_{1}\\!:\\!u_{1})\ldots(w_{n}\\!:\\!u_{n}))^{M}=\sigma$, e.g.
$(Nat_{x,y})$ denotes $(Nat:x)(Nat:y)$, but note that $\sigma$ need neither be
independent nor even regular. The terms are unsorted in order to deal with
t-sets explicitly; $({{}^{\sigma}\\!w}:{{}^{\tau}\\!u})$ can be written as
$(\sigma)\;(\tau)\;(w:u)$. Note that the t-substitutions in $(w:u)^{M}$ always
yield ground constructor terms.
Lemma 63. Let
${{}^{\varepsilon}\\!f}(w_{1},\ldots,w_{n})\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}{{}^{\varepsilon}\\!u}$, for $w_{1},\ldots,w_{n}\in{\cal
T}_{{\cal CR},{\cal F}}$ and $u\in{\cal T}_{\cal CR}$; let
$I\subset\\{1,\ldots,m\\}$ such that $nf[\top\langle
w_{1},\ldots,w_{n}\rangle]\cap dom(f,I)=nf[\top\langle
w_{1},\ldots,w_{n}\rangle]\cap dom(f)$;
then, $i\in I$ and $\mu^{\prime}_{i}\in\mu_{i}$ exists such that
$w_{j}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}\mu^{\prime}_{i}u_{ij}$ for $j=1,\ldots,n$ and
$\mu^{\prime}_{i}v_{i}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}u$.
Proof. Consider the first reduction at root position within the chain
${{}^{\varepsilon}\\!f}(w_{1},\ldots,w_{n})\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}{{}^{\varepsilon}\\!u}$.
Definition 64. The following local transformation rules for $rg$ are defined
(excerpt):
1. 1.
$(f(v^{\prime}_{1},\ldots,v^{\prime}_{n}):u)=\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip
2.84544pt_{i\in
I}(v_{i}:u)\;(v^{\prime}_{1}:u_{i1})\;\ldots\;(v^{\prime}_{n}:u_{in})\;(\mu_{i})$
if $I\subset\\{1,\ldots,m\\}$ arbitrary such that $nf[\top\langle
w_{1},\ldots,w_{n}\rangle]\cap dom(f,I)=nf[\top\langle
w_{1},\ldots,w_{n}\rangle]\cap dom(f)$, cf. the remarks on page 12.
2. 2.
$(u^{\prime}:x)\;(v:u)=(u^{\prime}:x)\;([x\\!:=\\!u^{\prime}]\;v:[x\\!:=\\!u^{\prime}]\;u)$
if $x\not\in vars(u^{\prime})$,
3. 3.
$(S:x)\;(S^{\prime}:x)=(S\cap S^{\prime}:x)$,
Rules 2. and 3. also show “$(\cdot:\cdot)$” as a generalization of term
equality and sort membership, respectively. All local rules satisfy the
correctness criterion $lhs^{M}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
vars(lhs)$}}=rhs^{M}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle
vars(rhs)$}}$.
Proof. Use 10 for correctness of rule 1.; correctness of 2. and 3. follows by
simple computations.
Only one proper approximation rule is needed, viz.
$(w_{1}:u_{1})\;\ldots\;(w_{n}:u_{n})^{M}\subset(w_{2}:u_{2})\;\ldots\;(w_{n}:u_{n})^{M}$;
all other rules can be made exact by including the left-hand side in the
right-hand side.
Applying local transformations creates a computation tree with alternatives
(separated by “$\mid$”) as nodes, each alternative having a unique computation
path from the root, cf. Fig. 13. Global transformations operate on such
computation trees. A proof methodology (“rank induction”) is provided in Def.
10 and Lemma 10 for their verification that also allows the introduction of
new global rules, if necessary, for some class of applications.
Definition 65. Let
$\sigma^{\prime}\in((w_{1}\\!:\\!u_{1})\;\ldots\;(w_{n}\\!:\\!u_{n}))^{M}$,
then e.g.
$\sigma^{\prime}w_{1}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}\sigma^{\prime}u_{1}$, where $\sigma^{\prime}u_{1}\in{\cal
T}_{\cal CR}$ is in normal form. Owing to confluence and termination, each
“$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$”
chain starting from $\sigma^{\prime}w_{1}$ ends after finitely many steps at
$\sigma^{\prime}u_{1}$.
Define $rank(\sigma^{\prime},(w_{1}:u_{1}))$ as the length of the longest such
chain, which always exists. Define
$rank(\sigma^{\prime},(w_{1}:u_{1})\;\ldots\;(w_{n}:u_{n})):=\sum_{i=1}^{n}rank(\sigma^{\prime},(w_{i}:u_{i}))$.
We always have $rank(\sigma^{\prime},(\sigma))\in I\\!\\!N$ and
$rank(\sigma^{\prime},(\tau))=0$ for $\tau$ regular t-set.
Lemma 66. Let $(\sigma)=(\sigma_{1})\mid\ldots\mid(\sigma_{m})$ be the result
of repeated application of the rules from Def. 10, let $z\in dom(\sigma)\cap
dom(\sigma_{1})\cap\ldots\cap dom(\sigma_{m})$. Then for each
$\sigma^{\prime}\in(\sigma)^{M}$, an $i\in\\{1,\ldots,m\\}$ and a
$\sigma^{\prime}_{i}\in(\sigma_{i})^{M}$ exists such that
$\sigma^{\prime}z=\sigma^{\prime}_{i}z$ and
$rank(\sigma^{\prime},(\sigma))\geqslant
rank(\sigma^{\prime}_{i},(\sigma_{i}))+n_{i}$, where $n_{i}$ denotes the
number of applications of Def. 10.1 in the path from $(\sigma)$ to
$(\sigma_{i})$.
Proof. Induction on the number of applications of rules from Def. 10.
Lemma 67. (Global Transformation: Loop-Checking Rule)
Assume $z\not\in dom(\sigma)\supset vars(v)\not\ni x$ and a computation tree
of the form
$(\sigma)\;(v:z)\;\;\;=\ldots$ $=$
$(\sigma)\;(v:x)\;(u_{1}(x):z)\mid\ldots\mid(\sigma)\;(v:x)\;(u_{n}(x):z)\mid(u_{n+1}:z)\mid\ldots\mid(u_{m}:z)$
where in each alternative’s path at least one application of rule 10.1
occurred. Then, $((\sigma)\;(v:z))^{M}\subset(S:z)^{M}$, where $S$ is a new
sort name defined by $S\doteq u_{1}(S)\mid\ldots\mid u_{n}(S)\mid
u_{n+1}\mid\ldots\mid u_{m}$. If all $u_{i}(x)$ are linear in $x$, we have
equality.
Proof. Show
$\sigma^{\prime}\in((\sigma)\;(v:z))^{M}\;\Rightarrow\;\sigma^{\prime}z\in
s^{M}$ by induction on $rank(\sigma^{\prime},(\sigma)\;(v:z))$, using Lemma
10.
Lemma 68. Let $1\leqslant k\leqslant n\leqslant m$,
assume $(\sigma)$ $(v:z)$ $=$ $(\sigma)$ $(v:u_{1})$ $(u^{\prime}_{1}:z)$
$\mid\ldots\mid$ $(\sigma)$ $(v:u_{n})$ $(u^{\prime}_{n}:z)$ $\mid$
$(\sigma_{n\\!+\\!1})$ $(v_{n\\!+\\!1}\\!:\\!u_{n\\!+\\!1})$
$(u^{\prime}_{n\\!+\\!1}\\!:\\!z)$ $\mid\ldots\mid$ $(\sigma_{m})$
$(v_{m}:u_{m})$ $(u^{\prime}_{m}:z)$ where in each alternative’s path at least
one application of Def. 10.1 occurred. Then, $(\sigma)$ $(v:z)$ $=$ $(\sigma)$
$(v:u_{k\\!+\\!1})$ $(u^{\prime}_{k\\!+\\!1}:z)$ $\mid\ldots\mid$ $(\sigma)$
$(v:u_{n})$ $(u^{\prime}_{n}:z)$ $\mid$ $(\sigma_{n\\!+\\!1})$
$(v_{n\\!+\\!1}\\!:\\!u_{n\\!+\\!1})$ $(u^{\prime}_{n\\!+\\!1}\\!:\\!z)$
$\mid\ldots\mid$ $(\sigma_{m})$ $(v_{m}:u_{m})$ $(u^{\prime}_{m}:z)$
provided $(u^{\prime}_{i}:u_{j})^{M}=\\{\\}$ for all
$i\in\\{k+1,\ldots,m\\},j\in\\{1,\ldots,k\\}$.
Intuitively, constructor terms $u_{1},\ldots,u_{k}$ may be produced as the
value of $z$ or $v$ only in alternatives $1,\ldots,k$, but this in turn
requires a constructor term $u_{1},\ldots,u_{k}$. Hence, there is no recursion
basis, i.e. $v$ may not have a $u_{i}$ as its value, i.e. the first $k$
alternatives are superfluous.
Proof. Show
$\sigma^{\prime}\in((\sigma)\;(v:z))^{M}\;\;\Longrightarrow\;\;\bigwedge_{j=1}^{k}\forall\tau^{\prime}\\!\in\\!\top\;\;\sigma^{\prime}z\neq\tau^{\prime}u_{j}$
by induction on $rank(\sigma^{\prime},(\sigma)\;(v:z))$, using 10.
Lemma 69. Let $y,x_{1},\ldots,x_{k}\in{\cal V}$, $w\in{\cal T}_{{\cal
CR},{\cal F},{\cal V}}$, $u(x_{1},\ldots,x_{k})\in{\cal T}_{{\cal
CR},\\{x_{1},\ldots,x_{k}\\}}$, $u_{ij},v_{ij},v_{i}\in{\cal T}_{{\cal
CR},{\cal V}}$,
$v_{i}(y)\in{\cal T}_{{\cal CR},\\{y\\}}$ for
$1\\!\leqslant\\!i\\!\leqslant\\!n_{1}$, $v_{i}\in{\cal T}_{\cal CR}$ for
$n_{1}+1\\!\leqslant\\!i\\!\leqslant\\!n_{2}$,
We abbreviate $u(y,u_{12},\ldots,u_{1k})$ to $u(y,\vec{u}_{1})$.
Assume
$(\sigma)$ $(\tau)$ $(w:z)$ $=$ $(\beta_{1}\sigma)$ $(\tau_{1})$
$(\beta_{1}w:u(y,\vec{u}_{1}))$ $(u(v_{1}(y),\vec{v}_{1}):z)$ $\mid$ $\ldots$
$\mid$ $(\beta_{n_{1}}\sigma)$ $(\tau_{n_{1}})$
$(\beta_{n_{1}}w:u(y,\vec{u}_{n_{1}}))$ $(u(v_{n_{1}}(y),\vec{v}_{n_{1}}):z)$
$\mid$ $(\beta_{n_{1}+1}\sigma)$ $(\tau_{n_{1}+1})$
$(\beta_{n_{1}+1}w:u_{n_{1}+1})$ $(u(v_{n_{1}+1},\vec{v}_{n_{1}+1}):z)$ $\mid$
$\ldots$ $\mid$ $(\beta_{n_{2}}\sigma)$ $(\tau_{n_{2}})$
$(\beta_{n_{2}}w:u_{n_{2}})$ $(u(v_{n_{2}},\vec{v}_{n_{2}}):z)$ $\mid$
$(\tau_{n_{2}+1})$ $(\beta_{n_{2}+1}w:u_{n_{2}+1})$ $(v_{n_{2}+1}:z)$ $\mid$
$\ldots$ $\mid$ $(\tau_{n_{3}})$ $(\beta_{n_{3}}w:u_{n_{3}})$ $(v_{n_{3}}:z)$
and $(u_{i}:u(x_{1},\vec{x}))=\bot_{s}=(v_{j}:u(x_{1},\vec{x}))$ for
$n_{1}+1\leqslant i\leqslant n_{2}$ and $n_{2}+1\leqslant j\leqslant n_{3}$.
Define $S\doteq v_{1}(S)\mid\ldots\mid v_{n_{1}}(S)\mid
v_{n_{1}+1}\mid\ldots\mid v_{n_{2}}$.
Then,
$(\sigma)$ $(\tau)$ $(w:z)$ $=$ $(\beta_{1}\sigma)$ $(\tau_{1})$
$(\beta_{1}w:u(y,\vec{u}_{1}))$ $(u(v_{1}(y),\vec{v}_{1}):z)$ $(S:y)$ $\mid$
$\ldots$ $\mid$ $(\beta_{n_{1}}\sigma)$ $(\tau_{n_{1}})$
$(\beta_{n_{1}}w:u(y,\vec{u}_{n_{1}}))$ $(u(v_{n_{1}}(y),\vec{v}_{n_{1}}):z)$
$(S:y)$ $\mid$ $(\beta_{n_{1}+1}\sigma)$ $(\tau_{n_{1}+1})$
$(\beta_{n_{1}+1}w:u_{n_{1}+1})$ $(u(v_{n_{1}+1},\vec{v}_{n_{1}+1}):z)$ $\mid$
$\ldots$ $\mid$ $(\beta_{n_{2}}\sigma)$ $(\tau_{n_{2}})$
$(\beta_{n_{2}}w:u_{n_{2}})$ $(u(v_{n_{2}},\vec{v}_{n_{2}}):z)$ $\mid$ $\mid$
$(\tau_{n_{2}+1})$ $(\beta_{n_{2}+1}w:u_{n_{2}+1})$ $(v_{n_{2}+1}:z)$ $\mid$
$\ldots$ $\mid$ $(\tau_{n_{3}})$ $(\beta_{n_{3}}w:u_{n_{3}})$ $(v_{n_{3}}:z)$
Intuitively, a constructor term of the form $u(v,\ldots)$ can occur only in
two places:
* •
in one of the alternatives $1,\ldots,n_{1}$, $v$ having the form
$v_{i}(v^{\prime})$ where $u(v^{\prime},\ldots)$ occurred earlier; or
* •
in one of the alternatives $n_{1}+1,\ldots,n_{2}$, $v$ having the form
$v_{i}$.
Thus, it is always true that $v\in S^{M}$.
Proof. Show
$\exists\sigma^{\prime}\in(\sigma)(w:z)\;^{M}\;\;\;u(u^{\prime}_{1},\vec{u}^{\prime})=\sigma^{\prime}z\Rightarrow
u^{\prime}_{1}\in S^{M}$ by induction along the order
$u(u^{\prime}_{1},\vec{u}^{\prime})<u(u^{\prime\prime}_{1},\vec{u}^{\prime\prime}):\Leftrightarrow
u^{\prime}_{1}\mathrel{\sphericalangle}u^{\prime\prime}_{1}$.
Algorithm 70. The following algorithm provides an initial, coarse
approximation $max_{f}$ of the range sorts for an equationally defined
function $f$.
$max_{f}$ $\doteq max_{\mu_{1},v_{1}}\mid\ldots\mid max_{\mu_{m},v_{m}}$ where
$max_{\mu,w}$ for $w\in{\cal T}_{{\cal CR},{\cal F},{\cal V}}$ is defined by:
$max_{\mu,g(w_{1},\ldots,w_{n})}$ $:=max_{g}$ if $g\in{\cal F}$
$max_{\mu,cr(w_{1},\ldots,w_{n})}$ $\doteq
cr(max_{\mu,w_{1}},\ldots,max_{\mu,w_{n}})$ if $cr\in{\cal CR}$ $max_{\mu,x}$
$\doteq apply(\mu,x)$ if $x\in{\cal V}$
If
$f(w_{1},\ldots,w_{n})\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}u\in{\cal T}_{\cal CR}$, then $u\in max_{f}^{M}$, as can be
shown by induction on the length of the
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$
chain.
Algorithm 71. Assume $f$ is defined by the (yet unsorted) equations
$f(u_{11},\ldots,u_{1n_{1}})$ $=v_{1}$ … $f(u_{m1},\ldots,u_{mn_{m}})$
$=v_{m}$
Assume that for each $f\in{\cal F}$ a set $F_{f}\subset{\cal F}$ of admitted
function symbols for arguments of $f$ is given. The following algorithm finds
minimal independent t-sets $\mu_{i}$ such that the applicability of
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$
becomes trivial if only subterms starting with a $g\in F_{f}$ appear at the
argument positions of $f$.
$max^{\prime}_{f}$ $\doteq max^{\prime}_{f,v_{1}}\mid\ldots\mid
max^{\prime}_{f,v_{m}}$ where $max^{\prime}_{f,w}$ for $w\in{\cal T}_{{\cal
CR},{\cal F},{\cal V}}$ is defined by:
$max^{\prime}_{f,g(w_{1},\ldots,w_{n})}$ $:=max^{\prime}_{g}$ if $g\in{\cal
F}$ $max^{\prime}_{f,cr(w_{1},\ldots,w_{n})}$ $\doteq
cr(max^{\prime}_{f,w_{1}},\ldots,max^{\prime}_{f,w_{n}})$ if $cr\in{\cal CR}$
$max^{\prime}_{f,x}$ $\doteq\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip
2.84544pt_{g\in F_{f}}max^{\prime}_{g}$ if $x\in{\cal V}$
Define $\mu_{i}=compose(\\{abstract(x,max^{\prime}_{f,x})\;\mid\;x\in
vars(u_{i1},\ldots,u_{in_{i}})\\})$.
Then,
$f(v_{1},\ldots,v_{n})\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}v$ iff
6.1 is satisfied and ($v_{i}\in{\cal V}$ or $v_{i}=g(\vec{v}_{i})$ with
$g\in{\cal CR}\cup{\cal F}_{f}$).
Example 72. For the functions defined by the unstarred equations of Fig. 15,
allowing arbitrary argument terms for $+$ and $dup$, but only constructor
terms from $Bin$ as arguments for $val$, one gets
$F_{+}=F_{dup}=\\{+,dup,val\\}$, $F_{val}=\\{\\}$, and
$max^{\prime}_{+}$ $\doteq max^{\prime}_{+,x}\mid s(max^{\prime}_{+})$
$max^{\prime}_{+,x}$ $\doteq max^{\prime}_{+}\mid max^{\prime}_{dup}\mid
max^{\prime}_{val}$ $max^{\prime}_{dup}$ $\doteq max^{\prime}_{+}$
$max^{\prime}_{val}$ $\doteq 0\mid max^{\prime}_{dup}\mid
s(max^{\prime}_{dup})$ or, simplified: $max^{\prime}_{+}$ $\doteq Nat$
$max^{\prime}_{+,x}$ $\doteq Nat$ $max^{\prime}_{dup}$ $\doteq Nat$
$max^{\prime}_{val}$ $\doteq Nat$
which corresponds to the implicit t-set shown in Fig. 15.
Algorithm 73. To compute $rg(\sigma,v)$, start with the expression
$(\sigma)\;(v:z)$, where $z$ is new, and repeatedly apply rules in the
following order: global rules, approximation rule, simplifying local rules
(like Defs. 10.2 and 10.3), and rewriting (Def. 10.1). Apply approximation
only if certain conditions make it necessary; apply all other rules wherever
possible. By setting certain parameters in the termination criterion, the
trade-off between computation time and precision of the result can be
controlled. On termination, an expression
$(\sigma_{1})\;(u_{1}:z)\mid\ldots\mid(\sigma_{n})\;(u_{n}:z)$ with regular
t-sets $\sigma_{i}$ is obtained. The final result is then
$rg(\sigma,v):=\sigma_{1}u_{1}\mid\ldots\mid\sigma_{n}u_{n}$, satisfying
$nf[\sigma^{M}v]\subset rg(\sigma,v)^{M}$.
$x+0$ | $=x$
---|---
$x+s(y)$ | $=s(x)+y$
| $(Nat:x)\;(Nat:y)\;(x+y:z)$ |
---|---|---
$=$ | $(Nat:x)\;(Nat:y)\;(x:x_{1})\;(y:0)\;(x_{1}:z)$ | 10.1
$\mid$ | $(Nat:x)\;(Nat:y)\;(x:x_{1})\;(y:s(y_{1}))\;(s(x_{1})+y_{1}:z)$ |
$=$ | $(Nat:x)\;(x:z)$ | simplification
$\mid$ | $(Nat:x_{1})\;(Nat:y_{1})\;(s(x_{1})+y_{1}:z)$ |
$=$ | $(Nat:x)\;(x:z)$ | 10.1
$\mid$ | $(Nat:x_{2})\;(x_{2}:z)$ | \+ simplification
$\mid$ | $(Nat:x_{1})\;(Nat:y_{2})\;(s(s(x_{1}))+y_{2}:z)$ |
$=$ | $\ldots$ |
$=$ | $(Nat:x)\;(x:z)$ | 10.1
$\mid$ | $(Nat:x)\;(Nat:y)\;(s^{i}(x)+y:z)$ | \+ simplification
$=$ | $\ldots$ |
Figure 11: Nonterminating sort rewriting computation
The termination of Alg. 10 has to be artificially enforced. Certainly, the
rewrite relation
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}$ is
Noetherian, i.e. each computation chain starting from a term will terminate.
However, Alg. 10 computes with sorts that represent infinitely many terms in
general, and the length of their computation chains may increase unboundedly.
Hence, sort rewriting need not terminate even though term rewriting
terminates. As an example, consider the equational theory and the computation
shown in Fig. 11.
In principle, Alg. 10 can be stopped after every step, using the approximation
by $max_{f}$; in other words, there is a trade-off between computation time
and the precision of the result. We suggest the following termination
criterion: applying Rule 10.1 to an expression $(f(\ldots):\ldots)$ is allowed
only if less than $\\#use(dom(f))$ rewrite steps wrt. $f$ have occurred in the
current path777 For an extended sort $S=\sigma u$, we define
$use(S)=use(\sigma)$. . The – heuristic – justification considers that $f$ is
defined recursively over the structure of $dom(f)$ and that no more than
$\\#use(dom(f))$ rewrite steps are necessary to “get back to the starting
expression”, thus making e.g. Rule 10 applicable. Since all other local and
global transformations except 10.1 can be applied only a finite number of
times, this criterion ensures termination.
Index sets:
---
$f_{1}:$ | $\\{1,2\\}$ | $\\{1,3\\}$
$f_{2}:$ | $\\{1,2\\}$ | $\\{2,3\\}$
$f_{3}:$ | $\\{1,2\\}$ |
$f_{4}:$ | $\\{1,2\\}$ | $\\{2,3\\}$
$e$$\bf f$$\bf g$$\bf b$$\bf h$$i$$c$$j$$\bf k$$\bf l$$\bf d$$\bf
a$$f_{1}$$f_{2}$$f_{3}$$f_{4}$
Figure 12: Applying a global transformation in an example computation tree
The defining equations of a function $f$ need not be independent in a logical
/ axiomatic sense; arbitrarily many “derived” equations may be added, cf. Fig.
13. Accordingly, it suffices to use only a subset of the equations for a
rewrite step by 10.1, provided $dom(f)$ is still completely covered (cf. the
role of the index set $I$ in 10.1). Since it is not possible to select a
suitable $I$ at the time the rewrite step is conducted we proceed the other
way round: we use all equations for $f$ in each rewrite step, making a global
transformation applicable not only if all alternatives have the required form
but even when a subset of alternatives has the required form and confinement
to these alternatives still leads to index sets completely covering $dom(f)$
in all relevant rewrite steps.
In this way, supplying additional derived function equations may result in
making “better” global transformations applicable, and hence in enhancing the
precision of the computed sort. Thus, we may get an effect similar to that
obtained by term declarations in [12].
The test for applicability of a global transformation works as follows: for
each alternative $(\sigma)$ that does not meet the applicability criterion,
delete all complete index sets in the last rewrite step leading to $(\sigma)$.
If no index set remains, omit this rewrite step, and delete in turn all
complete index sets in the previous rewrite step in the computation tree. If
no previous rewrite step exists, the global transformation cannot be made
applicable. If a complete index set still exists in the first, highest-level
rewrite step after all deletions are done, the transformation has been made
applicable.
As an example, consider the computation tree shown in Fig. 12. Rewrite steps
have been conducted for functions $f_{1},f_{2},f_{3},f_{4}$, transforming
alternative $a$ into $b\mid c\mid d$, and in turn to $e\mid f\mid g\mid h\mid
i\mid j\mid k\mid l$ (only applications of 10.1 are shown). The complete index
sets for each function are listed in the table. Suppose a global
transformation were applicable if we could restrict our attention to $f\mid
g\mid h\mid k\mid l$ (shown in bold face). The procedure described above
results in selection of the index set $\\{2,3\\}$ for both, $f_{2}$ and
$f_{4}$, and $\\{1,3\\}$ for $f_{1}$, deleting alternative $c$, as well.
Hence, the transformation has been made applicable.
Taking the definitions in Fig. 15, we can compute $rg(Nat_{x},x+x)$:
| $(Nat:x)\;(x+x:z)$
---|---
$=$ | $(Nat:x)\;(x_{1}:z)\;(x:x_{1})\;(x:0)\;\mid\;(Nat:x)\;(x_{1}:z)\;(x:0)\;(x:x_{1})\;\mid$
|
$(Nat\\!:\\!x)\;(s(x_{1}+y_{1})\\!:\\!z)\;(x\\!:\\!x_{1})\;(x\\!:\\!sy_{1})\mid(Nat\\!:\\!x)\;(s(x_{1}+y_{1})\\!:\\!z)\;(x\\!:\\!sx_{1})\;(x\\!:\\!y_{1})$
$=$ | $(0:z)$ | $\mid(Nat:y_{1})\;(sy_{1}+y_{1}:z_{1})\;(sz_{1}:z)$ | $\mid(Nat:x_{1})\;(x_{1}+sx_{1}:z_{1})\;(sz_{1}:z)$
$=$ | $(0:z)$ | $\mid(Nat:y_{2})\;(ssy_{2}\\!+\\!y_{2}:z_{2})\;(ssz_{2}\\!:\\!z)$ | $\mid(Nat:y_{2})\;(y_{2}+y_{2}:z_{2})\;(ssz_{2}:z)$
| | $\mid(Nat:y_{2})\;(y_{2}+y_{2}:z_{2})\;(ssz_{2}:z)$ | $\mid(Nat:x_{2})\;(x_{2}\\!+\\!ssx_{2}:z_{2})\;(ssz_{2}\\!:\\!z)$
$=$ | $(0:z)$ | $\mid(Nat:y_{2})\;(y_{2}+y_{2}:z_{2})\;(ssz_{2}:z)$ |
$=$ | $(Even:z)$
where the new sort definition $Even\doteq 0\mid s(s(Even))$ is generated. The
performed steps are: Rule 10.1 with equations a.-d.; simplification; Rule 10.1
with c.-d. twice in parallel, including simplification; deletion of the 2nd,
4th, and 5th alternative, since they are covered by the 3rd one; this makes
Lemma 10 applicable as the final step.
Figure 13: Range sort computation for $x+x$
| $(Bin:x)\;(val\\!\cdot\\!x:z)$ |
---|---|---
$=$ | $(Bin:x)\;(x:nil)\;(0:z)$ | Def. $val$
$\mid$ | $(Bin:x)\;(x:x_{1}\\!::\\!o)\;(dup\\!\cdot\\!val\\!\cdot\\!x_{1}:z)$ |
$\mid$ | $(Bin:x)\;(x:x_{1}\\!::\\!i)\;(s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!x_{1}:z)$ |
$=$ | $(0:z)$ |
$\mid$ | $(Bin:x_{1})\;(dup\\!\cdot\\!val\\!\cdot\\!x_{1}:z)$ |
$\mid$ | $(Bin:x_{1})\;(dup\\!\cdot\\!val\\!\cdot\\!x_{1}:z_{1})\;(s\\!\cdot\\!z_{1}:z)$ |
$=$ | $(0:z)$ | Def. $dup$
$\mid$ | $(Bin:x_{1})\;(val\\!\cdot\\!x_{1}:0)\;(0:z)$ |
$\mid$ | $(Bin:x_{1})\;(val\\!\cdot\\!x_{1}:s\\!\cdot\\!x_{2})\;(s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!x_{2}:z)$ |
$\mid$ | $(Bin:x_{1})\;(val\\!\cdot\\!x_{1}:0)\;(0:z_{1})\;(s\\!\cdot\\!z_{1}:z)$ |
$\mid$ | $(Bin:x_{1})\;(val\\!\cdot\\!x_{1}:s\\!\cdot\\!x_{2})\;(s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!x_{2}:z_{1})\;(s\\!\cdot\\!z_{1}:z)$ |
$=$ | $(0:z)$ |
$\mid$ | $(Bin:x_{1})\;(val\\!\cdot\\!x_{1}:s\\!\cdot\\!x_{2})\;(dup\\!\cdot\\!x_{2}:z_{2})\;(s\\!\cdot\\!s\\!\cdot\\!z_{2}:z)$ |
$\mid$ | $(Bin:x_{1})\;(val\\!\cdot\\!x_{1}:0)\;(s\\!\cdot\\!0:z)$ |
$\mid$ | $(Bin:x_{1})\;(val\\!\cdot\\!x_{1}:s\\!\cdot\\!x_{2})\;(dup\\!\cdot\\!x_{2}:z_{2})\;(s\\!\cdot\\!s\\!\cdot\\!s\\!\cdot\\!z_{2}:z)$ |
$\stackrel{{\scriptstyle(\subset)}}{{=}}$ | $(0:z)$ | $(*)$
$\mid$ | $(Bin:x_{1})\;(max_{val}:s\\!\cdot\\!x_{2})\;(dup\\!\cdot\\!x_{2}:z_{2})\;(s\\!\cdot\\!s\\!\cdot\\!z_{2}:z)$ |
$\mid$ | $(Bin:x_{1})\;(max_{val}:0)\;(s\\!\cdot\\!0:z)$ |
$\mid$ | $(Bin:x_{1})\;(max_{val}:s\\!\cdot\\!x_{2})\;(dup\\!\cdot\\!x_{2}:z_{2})\;(s\\!\cdot\\!s\\!\cdot\\!s\\!\cdot\\!z_{2}:z)$ |
$=$ | $(0:z)$ | $(**)$
$\mid$ | $(Nat:s\\!\cdot\\!x_{2})\;(dup\\!\cdot\\!x_{2}:z_{2})\;(s\\!\cdot\\!s\\!\cdot\\!z_{2}:z)$ |
$\mid$ | $(Nat:0)\;(s\\!\cdot\\!0:z)$ |
$\mid$ | $(Nat:s\\!\cdot\\!x_{2})\;(dup\\!\cdot\\!x_{2}:z_{2})\;(s\\!\cdot\\!s\\!\cdot\\!s\\!\cdot\\!z_{2}:z)$ |
$=$ | $(0:z)$ | $dup$, see above
$\mid$ | $(s\\!\cdot\\!s\\!\cdot\\!Even:z)\mid(s\\!\cdot\\!0:z)\mid(s\\!\cdot\\!s\\!\cdot\\!s\\!\cdot\\!Even:z)$ |
$=$ | $(Nat:z)$ |
$(*)$: | Here, $val\\!\cdot\\!x_{1}$ is estimated upwards, since the original expression $(val\\!\cdot\\!x_{1}:\ldots)$ occurs as part of the actual expression, but the introduction of a new recursive sort definition is prohibited by the presence of the non-constructor function $dup$. We write “$\stackrel{{\scriptstyle(\subset)}}{{=}}$” to indicate that $rg$ here differs from the real, semantic range sort.
---|---
$(**)$: | One can trivially transform the defining equations of $val$ and $dup$ into sort definitions by replacing function applications with corresponding sort names. This yields an upper bound for the range sorts:
| $max_{val}$ $\doteq 0\mid s\\!\cdot\\!max_{dup}\mid max_{dup}$ and
$max_{dup}$ $\doteq 0\mid s\\!\cdot\\!s\\!\cdot\\!max_{dup}$ i.e.
$max_{dup}^{M}=Even$ and $max_{val}^{M}=Nat^{M}$
Figure 14: Range sort computation for $val$
Definition 74. A substitution $\beta$ is called a solution of an equation
${{}^{\sigma}\\!v}_{1}={{}^{\sigma}\\!v}_{2}$ iff a t-set $\tau$ exists that
denotes the sorts of variables in the $ran(\beta)$ such that
1. 1.
${{}^{\tau}\\!\beta}v_{1}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}{{}^{\tau}\\!\beta}v_{2}$,
2. 2.
$\forall\tau^{\prime}\in\tau\;\;\exists\sigma^{\prime}\in\sigma\;\;\forall
x\in vars(v_{1},v_{2})\;\;\;\tau^{\prime}\beta x\mbox{ well-defined
}\Rightarrow\tau^{\prime}\beta
x\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}}
\end{picture}^{*}}\sigma^{\prime}x$,
or equivalently: $nf[\tau\beta\langle
x_{1},\ldots,x_{n}\rangle]\subset\sigma\langle x_{1},\ldots,x_{n}\rangle$,
where $\\{x_{1},\ldots,x_{n}\\}=vars(v_{1},v_{2})$,
similar to the classical well-sortedness requirement for $\beta$, and
3. 3.
$nf[\tau\beta v_{1}]\neq\\{\\}$, i.e. the solution has at least one well-
defined ground instance.
Theorem 75. An arbitrary narrowing calculus preserving solution sets remains
complete if restricted appropriately by sorts. For example, for lazy narrowing
[9], abbreviating
$\tau:=\sigma\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}\top_{vars(u_{1},\ldots,u_{n})}$, we get for the main rules:
(ln)
${{}^{\tau}\\!v}_{1}={{}^{\tau}\\!u}_{1}\wedge\ldots\wedge{{}^{\tau}\\!v}_{n}={{}^{\tau}\\!u}_{n}\wedge{{}^{\tau}\\!v}={{}^{\tau}\\!v^{\prime}}$
${{}^{\sigma}\\!f}(v_{1},\ldots,v_{n})={{}^{\sigma}\\!v}$
${{}^{\top}\\!f}(u_{1},\ldots,u_{n})={{}^{\top}\\!v^{\prime}}$ defining
equation $inh(inf(rg(\tau,v),rg(\tau,v^{\prime})))$,
$inh(inf(rg(\tau,v_{1}),rg(\tau,u_{1})))$, …
$inh(inf(rg(\tau,v_{n}),rg(\tau,u_{n})))$,
(d)
${{}^{\tau}\\!u}_{1}={{}^{\tau}\\!v}_{1}\;\wedge\;\ldots\;\wedge\;{{}^{\tau}\\!u}_{n}={{}^{\tau}\\!v}_{n}$
${{}^{\sigma}\\!f}(u_{1},\ldots,u_{n})={{}^{\sigma}\\!f}(v_{1},\ldots,v_{n})$
$inh(inf(rg(\tau,u_{1}),rg(\tau,v_{1})))$,…,
$inh(inf(rg(\tau,u_{n}),rg(\tau,v_{n})))$
In rule (ln), the remaining equations
${{}^{\tau}\\!\;}v_{1}={{}^{\tau}\\!\;}u_{1}$, …,
${{}^{\tau}\\!\;}v_{n}={{}^{\tau}\\!\;}u_{n}$ can often be solved by purely
syntactic unification. In this case, the non-disjointness criteria
$inh(inf(rg(\tau,v_{1}),rg(\tau,u_{1})))$,…,
$inh(inf(rg(\tau,v_{n}),rg(\tau,u_{n})))$ are trivially satisfied and may be
omitted in practical implementations. Note that the variables in defining
equations have to be assigned the sort $\top$. Starting from a conditional
narrowing calculus, nontrivially sorted defining equations become possible.
Proof. Rules may be restricted using the fact that the solution set of
${{}^{\sigma_{1}}\\!v}_{1}={{}^{\sigma_{2}}\\!v}_{2}$ is inhabited only if
$rg(\sigma_{1},v_{1})^{M}\cap rg(\sigma_{2},v_{2})^{M}\supset
nf[\sigma_{1}^{M}v_{1}]\cap nf[\sigma_{2}^{M}v_{2}]\neq\\{\\}$.
To prove the completeness of assigning sorts to variables in goal equations,
observe that each definition of a regular t-set $\sigma$ can be transformed
into a definition of a function $f_{\sigma}$ admitted by Def. 6 such that
$\sigma^{\prime}\in\sigma^{M}$ iff
$f_{\sigma}(\sigma^{\prime}x_{1},\ldots,\sigma^{\prime}x_{n})=true$, where
$dom(\sigma)=\\{x_{1},\ldots,x_{n}\\}$ and $true\in{\cal CR}$. Hence, a sorted
equation ${{}^{\sigma}\\!v}_{1}={{}^{\sigma}\\!v}_{2}$ can be simulated
without sorts by $v_{1}=v_{2}\wedge f_{\sigma}(x_{1},\ldots,x_{n})=true$.
Assigning non-trivial sorts to variables in defining equations is possible for
conditional calculi in a similar way.
Lemma 76. Let $\sigma$ be independent, and let $x$ be new; then, the equation
${{}^{\sigma}\\!v}_{1}={{}^{\sigma}\\!v}_{2}$ has a solution iff
$rg(\sigma,\langle v_{1},v_{2}\rangle)^{M}\cap\top\langle
x,x\rangle\neq\\{\\}$, provided the approximation rule was not used in $rg$
computation.
Lemma 14 shows that the amount of search space reduction by the sorts depends
only on the quality of approximations by $rg$ and the expressiveness of our
sort language. Without the reflection of variable bindings in sorts, such a
result is impossible, even if no “occur check” and no non-constructor
functions are involved, e.g. $\langle x,y\rangle=\langle 0,s(0)\rangle$ is
solvable, but $\langle x,x\rangle=\langle 0,s(0)\rangle$ is not.
It is possible to extend the presented framework to cope with unfree
constructors, too. This allows us, for example, to define a sort $Set$ of sets
of natural numbers, cf. App. 0.B. As we show below, it is sufficient to be
able to compute the closure of a sort wrt. the congruence relation induced by
the equations between constructors.
Definition 77. Assume we are given certain equations between constructors in
addition to the equations for defined functions. As in Def. 6, we define the
rewrite relation
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c$}} \end{picture}}$ to
be induced by the equations between constructors. We do not require
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c$}} \end{picture}}$ to
be confluent, nor to be Noetherian. For the union of both equation sets, we
similarly define
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c\\!d$}}
\end{picture}}$. We require the defining equations to be compatible with the
constructor equations, i.e.
$\bigwedge_{i=1}^{n}v_{i}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c$}}
\end{picture}^{*}}v^{\prime}_{i}\Longrightarrow
nf(f(v_{1},\ldots,v_{n}))\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c$}}
\end{picture}^{*}}nf(f(v^{\prime}_{1},\ldots,v^{\prime}_{n}))$
whenever at least one of the two normal forms exists. Note that the sort
algorithms work only on free sorts and hence ignore the relation
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c$}}
\end{picture}^{*}}$.
Lemma 78. If $v,w\in{\cal T}_{{\cal CR},{\cal F}}$ are well-defined, we have
$v\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c\\!d$}}
\end{picture}^{*}}w\;\;\Leftrightarrow\;\;nf(v)\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c$}}
\end{picture}^{*}}nf(w)$.
Proof. “$\Leftarrow$” trivial; “$\Rightarrow$” by induction on the length of
the
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c\\!d$}}
\end{picture}^{*}}$ chain.
In each equivalence class wrt.
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c$}}
\end{picture}^{*}}$, we may select an arbitrary element and declare it to be
the normal form, thus defining $nf_{c}$. We adapt the notion of solution of an
equation system from Def. 14 by replacing
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}$
with
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}}
\put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c\\!d$}}
\end{picture}^{*}}$, leaving condition 14.3 unchanged. Then, using Lemma 14,
we can show that an equation ${{}^{\sigma}\\!v}={{}^{\sigma}\\!v}^{\prime}$
has a solution only if $nf_{c}[rg(\sigma,v)^{M}]\cap
nf_{c}[rg(\sigma,v^{\prime})^{M}]\neq\\{\\}$. If we have an algorithm $rg_{c}$
to compute upper approximations for $nf_{c}[\cdot]$, similar to $rg$ for
$nf[\cdot]$, we can extend the sorted narrowing rules from Def. 14 to cope
with unfree constructors by replacing $rg(\tau,v)$ with $rg_{c}(rg(\tau,v))$,
etc. However, such an algorithm is not provided here.
## 7 Application in Formal Program Development
To support formal program development, we employ the paradigm of
implementation proof, starting from an “abstract” operation $ao$ on abstract
data of sort $as_{1}$ or $as_{2}$ which are to be implemented by a
corresponding “concrete” operation $co$ on concrete data of sorts $cs_{1}$ or
$cs_{2}$, respectively. The connection between abstract and concrete data is
established by representation func-
$cs_{1}$$cs_{2}$$as_{1}$$as_{2}$$co$$ao$$r_{1}$$r_{2}$
tions $r_{1}:cs_{1}\rightarrow as_{1}$ and $r_{2}:cs_{2}\rightarrow as_{2}$,
representing each concrete data term as an abstract one. Different concrete
terms may represent the same abstract term. Thus, it is possible to perform
the computation on the concrete level, and interpret the result on the
abstract level. The correspondence between the concrete and abstract operation
imposes correctness requirements on the concrete operation.
We wish to synthesize the concrete operation $co$ as the Skolem function for
$y$ in the formula $\forall\vec{x}\;\exists y\;\;ao(r_{1}(\vec{x}))=r_{2}(y)$.
A suitable method for the constructive correctness proof is induction on the
form of a data term $\vec{x}\in cs_{1}$, leading to a case distinction
according to (one of) the head constructor(s) of $\vec{x}$. In each case, we
have to solve an equation $ao(r_{1}(\vec{x}_{i}))=r_{2}(y)$ wrt. $y$. The
synthesized function $co$ is then given by equations
$co(\vec{x}_{i})=\beta_{i}y$, where $\vec{x}_{i}$ is a data term starting with
the $i^{th}$ constructor, and $\beta_{i}$ is the solving substitution for this
case. After having solved an equation, one still has to check whether the
solution $\beta_{i}y$ is of the required sort $cs_{2}$, if not, a different
solution must be found.
The sort discipline presented here supports specifically this method. Besides
allowing recursive sort definitions of $cs_{1}$, $cs_{2}$, $as_{1}$, and
$as_{2}$ as well as recursive function definitions of $ao$, $r_{1}$, and
$r_{2}$, the induction principle from Thm. 2 provides the case distinction and
proof goals for an induction on $x\in cs_{1}^{M}$. The sort discipline is able
to cope with the additional problems of synthesis as compared with
verification, i.e., to direct the construction of the solution term, to the
extent that disjoint subsorts of a concrete sort are assigned with disjoint
subsorts of the corresponding abstract sort. In this manner, the sort of an
“abstract” term indicates which “concrete” terms are representing it.
As an example, consider the formal development of algorithms for binary
numbers, Consider the sort and function definitions in Fig. 15. All terms are
sorted by the t-set
$[x\\!:=\\!Nat]\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}}[y\\!:=\\!Nat]\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern
2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}[z\\!:=\\!Bin]$, which is
omitted in the equations for the sake of brevity. Equations marked by “∗” are
redundant and can be proven by structural induction. The rightmost column
contains the sort computed by $rg$ for each equation; all sorts happen to be
regular. We have axioms defining the “representation function”
$val:Bin\longrightarrow Nat$, and an auxiliary function $dup$ to duplicate
natural numbers which uses the addition $+$ on natural numbers.
$Nat$ | $\doteq 0\mid s(Nat)$
---|---
$Bin$ | $\doteq nil\mid Bin\\!::\\!o\mid Bin\\!::\\!i$
E.g.
$val(nil\\!::\\!i\\!::\\!o\\!::\\!i)=s^{5}(0)$
Prove
$\forall c\;\exists z\;\;s(val(c))=val(z)$
$Bin$$Bin$$Nat$$Nat$$incr$$s$$val$$val$
a. | $x+0$ | $=x$ | $Nat$
---|---|---|---
b.∗ | $0+x$ | $=x$ | $Nat$
c. | $x+s(y)$ | $=s(x+y)$ | $s(Nat)$
d.∗ | $s(x)+y$ | $=s(x+y)$ | $s(Nat)$
e. | $dup(x)$ | $=x+x$ | $Even$
f.∗ | $dup(x+y)$ | $=dup(x)+dup(y)$ | $Even$
g. | $val(nil)$ | $=0$ | $0$
h. | $val(z\\!::\\!o)$ | $=dup(val(z))$ | $Even$
i. | $val(z\\!::\\!i)$ | $=s(dup(val(z)))$ | $s(Even)$
Figure 15: Sort and function definitions for synthesis of binary arithmetic
algorithms
The main contribution of the sorts is the computation of $rg(Nat_{x},dup(x))$
$=[z\\!:=\\!Even]\;z=Even$, where the sort definition $Even\doteq 0\mid
s(s(Even))$ is automatically introduced. Although only independent t-sets are
involved in the example, the variable bindings in $x+x$ are reflected by its
sort, viz. $Even$. For this range-sort computation, the redundant equation d.
is necessary, cf. Fig. 13.
Taking the easiest example, let us synthesize an algorithm $incr$ for
incrementing a binary number; the synthesis of algorithms for addition and
multiplication is shown in App. 0.A. The goal $\forall c\;\exists
z\;\;s(val(c))=val(z)$ is proved by structural induction on $c$, the
appropriate induction scheme being provided by Thm. 2, cf. Fig. 2. For
example, in case $c=c^{\prime}\\!::\\!o$, we have to solve the equation
$s(val(c^{\prime}\\!::\\!o))=val(z)$ wrt. $z$, and the sorted narrowing rule
from Thm. 14 is only applicable to equation i. since the left-hand side’s sort
is computed as $s(Even)$. Note that, in order to get the full benefit of the
sort calculus, narrowing should be applied only at the root of a term, since
then additional sort information is supplied from the other side of the
equation; this is the reason for using lazy narrowing. The employed calculus’
drawback of admitting only trivially sorted defining equations is overcome by
subsequently checking the solutions obtained for well-sortedness.
Narrowing with equation h. instead of i. would lead into an infinite branch888
Cf. Figs. 19 and 21 in App. 0.A, where the search space for this example is
shown for both unsorted and sorted narrowing. , trying to solve an equation
$s(dup(\ldots))=dup(\ldots)$. Such infinite branches are cut off by the sorts,
especially by the global transformation rules which detect certain kinds of
recursion loops. This seems to justify the computational overhead of sort
computation. Thanks to the provided proof methodology based on regular t-sets,
new global rules for detecting new recursion patterns can easily be added if
required.
The control information provided by the sort calculus acquires particular
importance in “proper” narrowing steps, i.e., the ones actually contributing
to the solution term. While conventional narrowing procedures essentially
enumerate each element of the constructor term algebra and test whether it is
a solution, the presented sort calculus approaches the solutions directly,
depending on the precision of computed range sorts.
The sort algorithms, especially $rg$, perform, in fact, simple induction
proofs. For example, it is easy to prove by induction that $x+x$ always has
sort $Even$, and that sorts $Even$ and $s(Even)$ are disjoint, once these
claims have been guessed or intuitively recognized. However, while a
conventional induction prover would not propose these claims as auxiliary
lemmas during the proof of $\forall c\;\exists z\;\;s(val(c))=val(z)$, they
are implicitly generated by the sort algorithms. The sort calculus allows the
“recognition of new concepts”, so to speak, although only within the rather
limited framework given by the sort language. In [8], an approach to the
automatic generation of more complex auxiliary lemmas is presented based on
E-generalization using regular sorts, too.
A prototype support system written in Quintus-Prolog takes a total of 41
seconds user time on a Sparc 1 to automatically conduct the 9 induction
proofs, with 135 narrowing subgoals necessary for the development of
incrementation, addition, and multiplication algorithms on binary numbers, cf.
App. 0.A. In the form of a paper case study from the area of compiler
construction, an implementation of sets of lists of natural numbers by ordered
son-brother trees has been proved, cf. App. 0.B. The algorithm for inserting a
new list into a tree is used to construct comb vectors for parse table
compression; it is specified as an implementation of $(\\{\cdot\\}\cup\cdot)$.
The use of sorts reduces the search space of the synthesis proof to that of a
verification proof, i.e. it uniquely determines all proper narrowing steps or
solution constructors. The computed signatures are too complex for there to be
much likelihood of their being declared by a user who does not know the proof
in advance.
## 8 References
## References
* [1] V. Antimirov. Personal communication, Apr 1995.
* [2] Leo Bachmair and Harald Ganzinger. On restrictions of ordered paramodulation with simplification. In Proc. 10th CADE, volume 449 of LNAI, pages 427–441, Jul 1990.
* [3] Alexander Bockmayr. Beitr ge zur Theorie des logisch-funktionalen Programmierens. PhD thesis, University Karlsruhe, 1991.
* [4] Jochen Burghardt. Eine feink rnige Sortendisziplin und ihre Anwendung in der Programmkonstruktion. PhD thesis, Univ. Karlsruhe, 1993.
* [5] Hubert Comon. Equational formulas in order-sorted algebras. In Proc. ICALP, 1990.
* [6] R. Echahed. On Completeness of Narrowing Strategies, volume 298 of LNCS. Springer, 1988.
* [7] L. Fribourg. A narrowing procedure for theories with constructors. In Proc. 7. CADE, volume 170 of LNCS, pages 259–279, 1984\.
* [8] Birgit Heinz. Lemma discovery by anti-unification of regular sorts. Technical Report 94–21, TU Berlin, 1994.
* [9] S. H lldobler. Foundations of Equational Programming, volume 353 of LNAI. Springer, 1989.
* [10] Eduard Klein and M. Martin. The parser generating system PGS. Software Practice and Experience, 19(11):1015–1028, 1989.
* [11] P. Mishra. Towards a theory of types in Prolog. In Proc. 1984 International Symposium on Logic Programming, pages 289–298. IEEE, 1984.
* [12] Manfred Schmidt-Schau . Computational Aspects of an Order-Sorted Logic with Term Declarations. PhD thesis, Univ. Kaiserslautern, Apr 1988.
* [13] J.W. Thatcher and J.B. Wright. Generalized finite automata theory with an application to a decision problem of second-order logic. Mathematical Systems Theory, 2(1), 1968.
* [14] M. Tommasi. Automates d’Arbres avec Tests d’ galit s entre Cousins Germains. Technical report, LIFL-IT, 1991.
* [15] T.E. Uribe. Sorted unification using set constraints. In Proc. CADE–11, volume 607 of LNCS, pages 163–177, 1992\.
Appendix
## Appendix 0.A Case Study “Binary Arithmetic”
In this appendix, the synthesis of algorithms for incrementation, addition,
and multiplication of binary numbers is shown. Figure 16 gives an overview of
the induction proofs conducted, together with the induction variable, the
computation time (in seconds on a Sparc 1 under Quintus Prolog), and the
number of subgoals. $a$ and $b$ denote (Skolem) constants, while $x$ denotes a
variable with respect to which the equation is to be solved. Note that in our
paradigm, universally quantified variables are skolemized into constant
symbols; induction is performed over just some of these constants. Figure 17
lists the synthesized algorithms.
On page 0.A, a protocol of the synthesis session is given. Pure induction
proofs, i.e. ones that do not solve an equation wrt. some variable, are
omitted. The notation in the Prolog implementation differs slightly from the
one used in this paper. The predicate “$init\\_sort\\_system$” computes a
range sort for every defining equation. The predicate “$solve$” tries to solve
the given equation; every time the actual narrowing step is not uniquely
determined by the sorts, the user is shown a menu and prompted to make a
decision. During the session, the user always had to decide to start an
induction, indicated by “ind”, optionally followed by a list of induction
variables. Note that none of the proofs required any further user interaction.
The execution trace shows the actual subgoal on entry into “$solve$”, and the
solved subgoal together with the solving substitution on exit. At the end of
the session, an example computation ($5*6=30$) is performed (predicate
“$eval\\_term$”) using the newly synthesized algorithms, and the sort
definitions (incomplete), proved laws, and function-defining equations are
listed. Functions $f_{24}$, $f_{188}$, and $f_{429}$ compute the successor of
a binary number, the sum of two binary numbers, and the product of two binary
numbers, respectively.
Starting on page 18, the search space is shown in particular for the synthesis
of the $incr$ algorithm. Figures 18 to 20 show the search space in cases where
no sorts are used to control narrowing; Fig. 21 shows the search space where
sorts are used.
formula | ind. | time | sub
---|---|---|---
| | var. | (sec.) | goals
initialization | | 3 |
$s(val(a))$ | $=val(x)$ | $a$ | 5 | 10
$0+a$ | $=a$ | $a$ | 1 | 5
$s(a)+b$ | $=s(a+b)$ | $b$ | 1 | 7
$dup(a)+dup(b)$ | $=dup(a+b)$ | $b$ | 2 | 11
$val(a)+val(b)$ | $=val(x)$ | $a,b$ | 19 | 47
$a+b+b$ | $=a+dup(b)$ | $b$ | 2 | 11
$a*dup(b)$ | $=dup(a*b)$ | $b$ | 2 | 13
$dup(val(a))+val(b)$ | $=val(add(a\\!::\\!o,b))$ | $b$ | 3 | 18
$val(a)*val(b)$ | $=val(x)$ | $b$ | 6 | 13
total | | 41 | 135
Figure 16: Implementation proofs for binary arithmetic $incr(nil)$ | $=nil\\!::\\!i$
---|---
$incr(x\\!::\\!o)$ | $=x\\!::\\!i$
$incr(x\\!::\\!i)$ | $=incr(x)::o$
$add(nil,nil)$ | $=nil$
$add(nil,y\\!::\\!o)$ | $=y\\!::\\!o$
$add(nil,y\\!::\\!i)$ | $=y\\!::\\!i$
$add(x\\!::\\!o,nil)$ | $=x\\!::\\!o$
$add(x\\!::\\!o,y\\!::\\!o)$ | $=add(x,y)\\!::\\!o$
$add(x\\!::\\!o,y\\!::\\!i)$ | $=add(x,y)\\!::\\!i$
$add(x\\!::\\!i,nil)$ | $=x\\!::\\!i$
$add(x\\!::\\!i,y\\!::\\!o)$ | $=add(x,y)\\!::\\!i$
$add(x\\!::\\!i,y\\!::\\!i)$ | $=incr(add(x,y))\\!::\\!o$
$mult(x,nil)$ | $=nil$
$mult(x,y\\!::\\!o)$ | $=mult(x,y)\\!::\\!o$
$mult(x,y\\!::\\!i)$ | $=add(mult(x,y)\\!::\\!o,x)$
Figure 17: Synthesized algorithms for binary arithmetic
Synthesis session protocol
---
?- $init\\_sort\\_system.$
$val\;nil=0$ | $0$
$val(x:o)=dup\;val\;x$ | $sort_{1}$
$val(x:i)=s\;dup\;val\;x$ | $s\;sort_{1}$
$dup\;0=0$ | $0$
$dup\;s\;u=s\;s\;dup\;u$ | $s\;s\;sort_{1}$
$u+0=u$ | $nat$
$u+s\;v=s(u+v)$ | $s\;sort_{4}$
$u*0=0$ | $0$
$u*(s\;v)=u*v+u$ | $0\mid nat\mid s\;nat$
?- $solve(s(val(c))=val(x),S).$
$s\;val\;c=val\;x$
$1$ $[dup\;val\;x_{19}=s\;val\;c]\leftarrow[x:=x_{19}:o]$
$2$ $[dup\;val\;x_{23}=val\;c]\leftarrow[x:=x_{23}:i]$ | ind
$s\;val\;nil=val\;x$
$s\;0=val\;x$
$dup\;val\;x_{27}=0$
$0=val\;x_{27}$
$0=val\;x_{27}$ | $\leftarrow[x_{27}:=nil]$
$dup\;val\;x_{27}=0$ | $\leftarrow[x_{27}:=nil]$
$s\;0=val\;x$ | $\leftarrow[x:=nil:i]$
$s\;val\;nil=val\;x$ | $\leftarrow[x:=nil:i]$
$s\;val(c_{25}:o)=val\;x$
$s\;dup\;val\;c_{25}=val\;x$
$s\;dup\;val\;c_{25}=val\;x$ | $\leftarrow[x:=c_{25}:i]$
$s\;val(c_{25}:o)=val\;x$ | $\leftarrow[x:=c_{25}:i]$
$s\;val(c_{26}:i)=val\;x$
$s\;s\;dup\;val\;c_{26}=val\;x$
$dup\;val\;x_{38}=s\;s\;dup\;val\;c_{26}$
$val\;f_{24}(c_{26})=val\;x_{38}$
$val\;f_{24}(c_{26})=val\;x_{38}$ | $\leftarrow[x_{38}:=f_{24}(c_{26})]$
$dup\;val\;x_{38}=s\;s\;dup\;val\;c_{26}$ | $\leftarrow[x_{38}:=f_{24}(c_{26})]$
$s\;s\;dup\;val\;c_{26}=val\;x$ | $\leftarrow[x:=f_{24}(c_{26}):o]$
$s\;val(c_{26}:i)=val\;x$ | $\leftarrow[x:=f_{24}(c_{26}):o]$
$s\;val\;c=val\;x$ | $\leftarrow[x:=f_{24}(c)]$
$S=[x:=f_{24}(c)]$
?- $solve(0+a=a,S).$
$S=[\;]$
?- $solve(s(a)+b=s(a+b),S).$
$S=[\;]$
?- $solve(dup(a)+dup(b)=dup(a+b),S).$
$S=[\;]$
?- $solve(val(c)+val(d)=val(x),S).$
$val\;c+val\;d=val\;x$
$1$ $[0=val\;c+val\;d]\leftarrow[x:=nil]$
$2$ $[dup\;val\;x_{141}=val\;c+val\;d]\leftarrow[x:=x_{141}:o]$
$3$ $[s\;dup\;val\;x_{153}=val\;c+val\;d]\leftarrow[x:=x_{153}:i]$
$4$ $[0=val\;d]\leftarrow[u_{162}:=val\;x,x:=c]$
$5$ $[s(u_{186}+v_{187})=val\;x,s\;v_{187}=val\;d]\leftarrow[u_{186}:=val\;c]$ | ind
$val\;nil+val\;nil=val\;x$
$0+val\;nil=val\;x$
$0+0=val\;x$
$0=val\;x$
$0=val\;x$ | $\leftarrow[x:=nil]$
$0+0=val\;x$ | $\leftarrow[x:=nil]$
$0+val\;nil=val\;x$ | $\leftarrow[x:=nil]$
$val\;nil+val\;nil=val\;x$ | $\leftarrow[x:=nil]$
$val\;nil+val(d_{191}:o)=val\;x$
$0+val(d_{191}:o)=val\;x$
$0+dup\;val\;d_{191}=val\;x$
$dup\;val\;d_{191}=val\;x$
$dup\;val\;d_{191}=val\;x$ | $\leftarrow[x:=d_{191}:o]$
$0+dup\;val\;d_{191}=val\;x$ | $\leftarrow[x:=d_{191}:o]$
$0+val(d_{191}:o)=val\;x$ | $\leftarrow[x:=d_{191}:o]$
$val\;nil+val(d_{191}:o)=val\;x$ | $\leftarrow[x:=d_{191}:o]$
$val\;nil+val(d_{192}:i)=val\;x$
$0+val(d_{192}:i)=val\;x$
$0+s\;dup\;val\;d_{192}=val\;x$
$s(0+dup\;val\;d_{192})=val\;x$
$s\;dup\;val\;d_{192}=val\;x$
$s\;dup\;val\;d_{192}=val\;x$ | $\leftarrow[x:=d_{192}:i]$
$s(0+dup\;val\;d_{192})=val\;x$ | $\leftarrow[x:=d_{192}:i]$
$0+s\;dup\;val\;d_{192}=val\;x$ | $\leftarrow[x:=d_{192}:i]$
$0+val(d_{192}:i)=val\;x$ | $\leftarrow[x:=d_{192}:i]$
$val\;nil+val(d_{192}:i)=val\;x$ | $\leftarrow[x:=d_{192}:i]$
$val(c_{189}:o)+val\;nil=val\;x$
$val(c_{189}:o)+0=val\;x$
$dup\;val\;c_{189}+0=val\;x$
$dup\;val\;c_{189}=val\;x$
$dup\;val\;c_{189}=val\;x$ | $\leftarrow[x:=c_{189}:o]$
$dup\;val\;c_{189}+0=val\;x$ | $\leftarrow[x:=c_{189}:o]$
$val(c_{189}:o)+0=val\;x$ | $\leftarrow[x:=c_{189}:o]$
$val(c_{189}:o)+val\;nil=val\;x$ | $\leftarrow[x:=c_{189}:o]$
$val(c_{189}:o)+val(d_{191}:o)=val\;x$
$dup\;val\;c_{189}+val(d_{191}:o)=val\;x$
$dup\;val\;c_{189}+dup\;val\;d_{191}=val\;x$
$dup(val\;c_{189}+val\;d_{191})=val\;x$
$dup\;val\;f_{188}(c_{189},d_{191})=val\;x$
$dup\;val\;f_{188}(c_{189},d_{191})=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{191}):o]$
$dup(val\;c_{189}+val\;d_{191})=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{191}):o]$
$dup\;val\;c_{189}+dup\;val\;d_{191}=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{191}):o]$
$dup\;val\;c_{189}+val(d_{191}:o)=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{191}):o]$
$val(c_{189}:o)+val(d_{191}:o)=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{191}):o]$
$val(c_{189}:o)+val(d_{192}:i)=val\;x$
$dup\;val\;c_{189}+val(d_{192}:i)=val\;x$
$dup\;val\;c_{189}+s\;dup\;val\;d_{192}=val\;x$
$s(dup\;val\;c_{189}+dup\;val\;d_{192})=val\;x$
$s\;dup(val\;c_{189}+val\;d_{192})=val\;x$
$s\;dup\;val\;f_{188}(c_{189},d_{192})=val\;x$
$s\;dup\;val\;f_{188}(c_{189},d_{192})=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{192}):i]$
$s\;dup(val\;c_{189}+val\;d_{192})=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{192}):i]$
$s(dup\;val\;c_{189}+dup\;val\;d_{192})=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{192}):i]$
$dup\;val\;c_{189}+s\;dup\;val\;d_{192}=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{192}):i]$
$dup\;val\;c_{189}+val(d_{192}:i)=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{192}):i]$
$val(c_{189}:o)+val(d_{192}:i)=val\;x$ | $\leftarrow[x:=f_{188}(c_{189},d_{192}):i]$
$val(c_{190}:i)+val\;nil=val\;x$
$val(c_{190}:i)+0=val\;x$
$s\;dup\;val\;c_{190}+0=val\;x$
$s\;dup\;val\;c_{190}=val\;x$
$s\;dup\;val\;c_{190}=val\;x$ | $\leftarrow[x:=c_{190}:i]$
$s\;dup\;val\;c_{190}+0=val\;x$ | $\leftarrow[x:=c_{190}:i]$
$val(c_{190}:i)+0=val\;x$ | $\leftarrow[x:=c_{190}:i]$
$val(c_{190}:i)+val\;nil=val\;x$ | $\leftarrow[x:=c_{190}:i]$
$val(c_{190}:i)+val(d_{191}:o)=val\;x$
$val(c_{190}:i)+dup\;val\;d_{191}=val\;x$
$s\;dup\;val\;c_{190}+dup\;val\;d_{191}=val\;x$
$s(dup\;val\;c_{190}+dup\;val\;d_{191})=val\;x$
$s\;dup(val\;c_{190}+val\;d_{191})=val\;x$
$s\;dup\;val\;f_{188}(c_{190},d_{191})=val\;x$
$s\;dup\;val\;f_{188}(c_{190},d_{191})=val\;x$ | $\leftarrow[x:=f_{188}(c_{190},d_{191}):i]$
$s\;dup(val\;c_{190}+val\;d_{191})=val\;x$ | $\leftarrow[x:=f_{188}(c_{190},d_{191}):i]$
$s(dup\;val\;c_{190}+dup\;val\;d_{191})=val\;x$ | $\leftarrow[x:=f_{188}(c_{190},d_{191}):i]$
$s\;dup\;val\;c_{190}+dup\;val\;d_{191}=val\;x$ | $\leftarrow[x:=f_{188}(c_{190},d_{191}):i]$
$val(c_{190}:i)+dup\;val\;d_{191}=val\;x$ | $\leftarrow[x:=f_{188}(c_{190},d_{191}):i]$
$val(c_{190}:i)+val(d_{191}:o)=val\;x$ | $\leftarrow[x:=f_{188}(c_{190},d_{191}):i]$
$val(c_{190}:i)+val(d_{192}:i)=val\;x$
$s\;dup\;val\;c_{190}+val(d_{192}:i)=val\;x$
$s\;dup\;val\;c_{190}+s\;dup\;val\;d_{192}=val\;x$
$s(s\;dup\;val\;c_{190}+dup\;val\;d_{192})=val\;x$
$s\;s(dup\;val\;c_{190}+dup\;val\;d_{192})=val\;x$
$s\;s\;dup(val\;c_{190}+val\;d_{192})=val\;x$
$s\;s\;dup\;val\;f_{188}(c_{190},d_{192})=val\;x$
$dup\;val\;x_{248}=s\;s\;dup\;val\;f_{188}(c_{190},d_{192})$
$val\;f_{24}(f_{188}(c_{190},d_{192}))=val\;x_{248}$
$val\;f_{24}(f_{188}(c_{190},d_{192}))=val\;x_{248}$ | $\leftarrow[x_{248}:=f_{24}(f_{188}(c_{190},d_{192}))]$
$dup\;val\;x_{248}=s\;s\;dup\;val\;f_{188}(c_{190},d_{192})$ | $\leftarrow[x_{248}:=f_{24}(f_{188}(c_{190},d_{192}))]$
$s\;s\;dup\;val\;f_{188}(c_{190},d_{192})=val\;x$ | $\leftarrow[x:=f_{24}(f_{188}(c_{190},d_{192})):o]$
$s\;s\;dup(val\;c_{190}+val\;d_{192})=val\;x$ | $\leftarrow[x:=f_{24}(f_{188}(c_{190},d_{192})):o]$
$s\;s(dup\;val\;c_{190}+dup\;val\;d_{192})=val\;x$ | $\leftarrow[x:=f_{24}(f_{188}(c_{190},d_{192})):o]$
$s(s\;dup\;val\;c_{190}+dup\;val\;d_{192})=val\;x$ | $\leftarrow[x:=f_{24}(f_{188}(c_{190},d_{192})):o]$
$s\;dup\;val\;c_{190}+s\;dup\;val\;d_{192}=val\;x$ | $\leftarrow[x:=f_{24}(f_{188}(c_{190},d_{192})):o]$
$s\;dup\;val\;c_{190}+val(d_{192}:i)=val\;x$ | $\leftarrow[x:=f_{24}(f_{188}(c_{190},d_{192})):o]$
$val(c_{190}:i)+val(d_{192}:i)=val\;x$ | $\leftarrow[x:=f_{24}(f_{188}(c_{190},d_{192})):o]$
$val\;c+val\;d=val\;x$ | $\leftarrow[x:=f_{188}(c,d)]$
$S=[x:=f_{188}(c,d)]$
?- $solve(a+b+b=a+dup(b),S).$
$S=[\;]$
?- $solve(a*dup(b)=dup(a*b),S).$
$S=[\;]$
?- $solve(dup(val(c))+val(d)=val(f_{188}(c:o,d)),S).$
$S=[\;]$
?- $solve(val(c)*val(d)=val(x),S).$
$(val\;c)*(val\;d)=val\;x$
$1$ $[0=(val\;c)*(val\;d)]\leftarrow[x:=nil]$
$2$ $[dup\;val\;x_{412}=(val\;c)*(val\;d)]\leftarrow[x:=x_{412}:o]$
$3$ $[s\;dup\;val\;x_{419}=(val\;c)*(val\;d)]\leftarrow[x:=x_{419}:i]$
$4$ $[0=val\;x,0=val\;d]\leftarrow[u_{420}:=val\;c]$
$5$ $[u_{427}*v_{428}+u_{427}=val\;x,s\;v_{428}=val\;d]\leftarrow[u_{427}:=val\;c]$ | ind[d].
$(val\;c)*(val\;nil)=val\;x$
$(val\;c)*0=val\;x$
$0=val\;x$
$0=val\;x$ | $\leftarrow[x:=nil]$
$(val\;c)*0=val\;x$ | $\leftarrow[x:=nil]$
$(val\;c)*(val\;nil)=val\;x$ | $\leftarrow[x:=nil]$
$(val\;c)*(val(d_{430}:o))=val\;x$
$(val\;c)*(dup\;val\;d_{430})=val\;x$
$dup(val\;c)*(val\;d_{430})=val\;x$
$dup\;val\;f_{429}(c,d_{430})=val\;x$
$dup\;val\;f_{429}(c,d_{430})=val\;x$ | $\leftarrow[x:=f_{429}(c,d_{430}):o]$
$dup(val\;c)*(val\;d_{430})=val\;x$ | $\leftarrow[x:=f_{429}(c,d_{430}):o]$
$(val\;c)*(dup\;val\;d_{430})=val\;x$ | $\leftarrow[x:=f_{429}(c,d_{430}):o]$
$(val\;c)*(val(d_{430}:o))=val\;x$ | $\leftarrow[x:=f_{429}(c,d_{430}):o]$
$(val\;c)*(val(d_{431}:i))=val\;x$
$(val\;c)*(s\;dup\;val\;d_{431})=val\;x$
$(val\;c)*(dup\;val\;d_{431})+val\;c=val\;x$
$dup(val\;c)*(val\;d_{431})+val\;c=val\;x$
$dup\;val\;f_{429}(c,d_{431})+val\;c=val\;x$
$val\;f_{188}(f_{429}(c,d_{431}):o,c)=val\;x$
$val\;f_{188}(f_{429}(c,d_{431}):o,c)=val\;x$ | $\leftarrow[x:=f_{188}(f_{429}(c,d_{431}):o,c)]$
$dup\;val\;f_{429}(c,d_{431})+val\;c=val\;x$ | $\leftarrow[x:=f_{188}(f_{429}(c,d_{431}):o,c)]$
$dup(val\;c)*(val\;d_{431})+val\;c=val\;x$ | $\leftarrow[x:=f_{188}(f_{429}(c,d_{431}):o,c)]$
$(val\;c)*(dup\;val\;d_{431})+val\;c=val\;x$ | $\leftarrow[x:=f_{188}(f_{429}(c,d_{431}):o,c)]$
$(val\;c)*(s\;dup\;val\;d_{431})=val\;x$ | $\leftarrow[x:=f_{188}(f_{429}(c,d_{431}):o,c)]$
$(val\;c)*(val(d_{431}:i))=val\;x$ | $\leftarrow[x:=f_{188}(f_{429}(c,d_{431}):o,c)]$
$(val\;c)*(val\;d)=val\;x$ | $\leftarrow[x:=f_{429}(c,d)]$
$S=[x:=f_{429}(c,d)]$
?- $eval\\_term(f_{429}(nil:i:o:i,nil:i:i:o),T).$
$T=nil:i:i:i:i:o$
?- $listing(\doteq),\;listing({\bf law}),\;listing({\bf def}).$
$nat\doteq 0\mid s\;nat.$
$bin\doteq nil\mid bin:o\mid bin:i.$
$sort_{1}\doteq 0\mid s\;s\;sort_{1}.$
law $s\;val\;v_{43}=val\;f_{24}(v_{43}).$
law $0+v_{50}=v_{50}.$
law $s\;v_{87}+v_{88}=s(v_{87}+v_{88}).$
law $dup\;v_{117}+dup\;v_{118}=dup(v_{117}+v_{118}).$
law $val\;v_{254}+val\;v_{255}=val\;f_{188}(v_{254},v_{255}).$
law $v_{340}+v_{341}+v_{341}=v_{340}+dup\;v_{341}.$
law $v_{356}*(dup\;v_{357})=dup\;v_{356}*v_{357}.$
law $dup\;val\;v_{397}+val\;v_{398}=val\;f_{188}(v_{397}:o,v_{398}).$
law $(val\;v_{446})*(val\;v_{447})=val\;f_{429}(v_{446},v_{447}).$
def $val\;nil=0.$
def $val(x:o)=dup\;val\;x.$
def $val(x:i)=s\;dup\;val\;x.$
def $dup\;0=0.$
def $dup\;s\;u=s\;s\;dup\;u.$
def $u+0=u.$
def $u+s\;v=s(u+v).$
def $u*0=0.$
def $u*(s\;v)=u*v+u.$
def $f_{24}(nil)=nil:i.$
def $f_{24}(v_{35}:o)=v_{35}:i.$
def $f_{24}(v_{42}:i)=f_{24}(v_{42}):o.$
def $f_{188}(nil,nil)=nil.$
def $f_{188}(nil,v_{201}:o)=v_{201}:o.$
def $f_{188}(nil,v_{207}:i)=v_{207}:i.$
def $f_{188}(v_{215}:o,nil)=v_{215}:o.$
def $f_{188}(v_{224}:o,v_{225}:o)=f_{188}(v_{224},v_{225}):o.$
def $f_{188}(v_{231}:o,v_{232}:i)=f_{188}(v_{231},v_{232}):i.$
def $f_{188}(v_{238}:i,nil)=v_{238}:i.$
def $f_{188}(v_{244}:i,v_{245}:o)=f_{188}(v_{244},v_{245}):i.$
def $f_{188}(v_{252}:i,v_{253}:i)=f_{24}(f_{188}(v_{252},v_{253})):o.$
def $f_{429}(v_{433},nil)=nil.$
def $f_{429}(v_{442},v_{443}:o)=f_{429}(v_{442},v_{443}):o.$
def $f_{429}(v_{444},v_{445}:i)=f_{188}(f_{429}(v_{444},v_{445}):o,v_{444}).$
$val\\!\cdot\\!z=s\\!\cdot\\!val\\!\cdot\\!nil$$val\\!\cdot\\!z=s\\!\cdot\\!0$$[z:=nil]$$0=s\\!\cdot\\!0$Fail$[z:=z_{1}\\!::\\!o]$$dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!0$$val\\!\cdot\\!z_{1}=0$$0=s\\!\cdot\\!0$Fail$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{2}=s\\!\cdot\\!0$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!dup\\!\cdot\\!z_{2}=0$Fail$[z:=z_{1}\\!::\\!i]$$s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!0$$dup\\!\cdot\\!val\\!\cdot\\!z_{1}=0$$val\\!\cdot\\!z_{1}=0$$0=0$$[z_{1}:=nil]$Success$[z:=nil\\!::\\!i]$$[z_{1}:=z_{2}\\!::\\!o]$$dup\\!\cdot\\!val\\!\cdot\\!z_{2}=0$$val\\!\cdot\\!z_{2}=0$$0=0$$[z_{2}:=nil]$Success$[z:=nil\\!::\\!o\\!::\\!i]$$[z_{2}:=z_{3}\\!::\\!o]$$dup\\!\cdot\\!val\\!\cdot\\!z_{3}=0$…
$[z_{2}:=z_{3}\\!::\\!i]$$s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!z_{3}=0$Fail$val\\!\cdot\\!z_{2}=s\\!\cdot\\!z_{3}$$s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{3}=0$Fail$[z_{1}:=z_{2}\\!::\\!i]$$s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!z_{2}=0$Fail$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{2}=0$Fail
Figure 18: Brute-force search space for equation
$val\\!\cdot\\!z=s\\!\cdot\\!val\\!\cdot\\!nil$
$val\\!\cdot\\!z=s\\!\cdot\\!val(c_{1}\\!::\\!o)$$val\\!\cdot\\!z=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{1}$$[z:=nil]$$0=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{1}$Fail$[z:=z_{1}\\!::\\!o]$$dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{1}$$val\\!\cdot\\!z_{1}=0$$0=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{1}$Fail$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{2}=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{1}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!dup\\!\cdot\\!z_{2}=dup\\!\cdot\\!val\\!\cdot\\!c_{1}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$val\\!\cdot\\!c_{1}=0$$s\\!\cdot\\!dup\\!\cdot\\!z_{2}=0$Fail$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$val\\!\cdot\\!c_{1}=s\\!\cdot\\!z_{3}$$s\\!\cdot\\!dup\\!\cdot\\!z_{2}=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{3}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$val\\!\cdot\\!c_{1}=s\\!\cdot\\!z_{3}$$dup\\!\cdot\\!z_{2}=s\\!\cdot\\!dup\\!\cdot\\!z_{3}$…
$[z:=z_{1}\\!::\\!i]$$s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{1}$$[z_{1}:=c_{1}]$Success$[z:=c_{1}\\!::\\!i]$
Figure 19: Brute-force search space for equation
$val\\!\cdot\\!z=s\\!\cdot\\!val(c_{1}::o)$
$val\\!\cdot\\!z=s\\!\cdot\\!val(c_{2}\\!::\\!i)$$val\\!\cdot\\!z=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$[z:=nil]$$0=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$Fail$[z:=z_{1}\\!::\\!o]$$dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$val\\!\cdot\\!z_{1}=0$$0=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$Fail$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{2}=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$[z_{2}:=val\\!\cdot\\!c_{2}]$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!val\\!\cdot\\!c_{2}$Induction$[z:=incr(c_{2})::o]$$[z:=z_{1}\\!::\\!i]$$s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$val\\!\cdot\\!z_{1}=0$$0=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$Fail$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{2}=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!dup\\!\cdot\\!z_{2}=dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$val\\!\cdot\\!c_{2}=0$$s\\!\cdot\\!dup\\!\cdot\\!z_{2}=0$Fail$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$val\\!\cdot\\!c_{2}=s\\!\cdot\\!z_{3}$$s\\!\cdot\\!dup\\!\cdot\\!z_{2}=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{3}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$val\\!\cdot\\!c_{2}=s\\!\cdot\\!z_{3}$$dup\\!\cdot\\!z_{2}=s\\!\cdot\\!dup\\!\cdot\\!z_{3}$…
Figure 20: Brute-force search space for equation
$val\\!\cdot\\!z=s\\!\cdot\\!val(c_{2}::i)$
$val\\!\cdot\\!z=s\\!\cdot\\!val\\!\cdot\\!nil$$val\\!\cdot\\!z=s\\!\cdot\\!0$$[z:=z_{1}\\!::\\!i]$$s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!0$$dup\\!\cdot\\!val\\!\cdot\\!z_{1}=0$$val\\!\cdot\\!z_{1}=0$$[z_{1}:=nil]$Success$[z:=nil\\!::\\!i]$$[z_{1}:=z_{2}\\!::\\!o]$$dup\\!\cdot\\!val\\!\cdot\\!z_{2}=0$$val\\!\cdot\\!z_{2}=0$$[z_{2}:=nil]$Success$[z:=nil\\!::\\!o\\!::\\!i]$$[z_{2}:=z_{3}\\!::\\!o]$$dup\\!\cdot\\!val\\!\cdot\\!z_{3}=0$…
$val\\!\cdot\\!z=s\\!\cdot\\!val(c_{1}\\!::\\!o)$$val\\!\cdot\\!z=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{1}$$[z:=z_{1}\\!::\\!i]$$s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{1}$$[z_{1}:=c_{1}]$Success$[z:=c_{1}\\!::\\!i]$
$val\\!\cdot\\!z=s\\!\cdot\\!val(c_{2}\\!::\\!i)$$val\\!\cdot\\!z=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$[z:=z_{1}\\!::\\!o]$$dup\\!\cdot\\!val\\!\cdot\\!z_{1}=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!z_{2}=s\\!\cdot\\!s\\!\cdot\\!dup\\!\cdot\\!val\\!\cdot\\!c_{2}$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!z_{2}$$[z_{2}:=val\\!\cdot\\!c_{2}]$$val\\!\cdot\\!z_{1}=s\\!\cdot\\!val\\!\cdot\\!c_{2}$Induction$[z:=incr(c_{2})::o]$
Figure 21: Search space of $incr$ synthesis using sorts
## Appendix 0.B Case Study “Comb Vector Construction”
In this section, we demonstrate the use of the sort discipline by applying it
in a paper case study from the area of compiler construction. The parser
generating system PGS is a tool for generating a syntax analyzer for a
programming language or, in general, any structured input [10]. The user of
PGS has to specify the language to be analyzed by a grammar. The main
applications of PGS are in the area of compiler construction, e.g. parsing,
syntax analysis or syntax-directed translation.
PGS uses a comb vector technique to compress the two-dimensional array
representation of parse tables. A parse table can be merged into one array,
called $cont$, where the beginning of each original row is indicated by an
entry in an additional array called $base$. In order to be able to distinguish
between error and non-error entries, an array called $row$ is introduced in
parallel to $cont$, containing the row number from which the associated entry
in the $cont$ array originated.
Given a two-dimensional array, one way of constructing a comb vector is to
enter each row into a search tree, lexicographically sorted by the list of its
distances. The tree is a son-brother tree, a vertical link pointing to the
first son of a node, a horizontal link to the next brother. There are two
kinds of nodes, depending on whether a vertical link is necessary or not. A
node which has a vertical link corresponds to a distance; brother nodes of
this kind are in ascending order with respect to it. A node without a vertical
link corresponds to a row number (shown in italics in Fig. 22).
The tree is then traversed in post order, and the corresponding rows are
entered into the comb vector. Figure 22 shows an example two-dimensional array
together with the constructed search tree. For example, the path $down$,
$right$, $down$, $down$ corresponds to the distance list $1,3,0$ of row 4.
Figure 23 shows the constructed comb vector and its access function.
Our aim is to define the data structure of a search tree and to construct an
algorithm for inserting a list of distances into a search tree. For the sake
of simplicity, we do not distinguish between distances and row numbers,
representing both by natural numbers.
Assume the constructors
for search trees: | $nil_{t}$ | empty search tree,
---|---|---
| $node1(\cdot,\cdot,\cdot)$ | node with vertical and horizontal link,
| $node2(\cdot,\cdot)$ | node with horizontal link only,
for distance lists: | $nil_{l}$ | empty list,
| $(\cdot){\scriptstyle+}(\cdot)$ | list “cons”,
for sets of distance lists: | $mt$ | empty set, and
| $add(\cdot,\cdot)$ | add an element
The search tree in Fig. 22 is represented by the term
$\begin{array}[]{@{}l@{}l@{}l@{}l@{}l@{}}node1(1,&node1(1,&node2(3,&nil_{t}),\\\
&&node1(3,&node1(0,&node2(4,nil_{t}),\\\ &&&&node2(2,node2(5,nil_{t}))),\\\
&&&nil_{t})),\\\ &node1(2,&node2(1,&nil_{t}),\\\ &&nil_{t}))\;;\\\
\end{array}$
its set of distance lists can be represented by
$add(1{\scriptstyle+}1{\scriptstyle+}3{\scriptstyle+}nil_{l},add(1{\scriptstyle+}3{\scriptstyle+}0{\scriptstyle+}4{\scriptstyle+}nil_{l},add(1{\scriptstyle+}3{\scriptstyle+}2{\scriptstyle+}nil_{l},add(1{\scriptstyle+}3{\scriptstyle+}5{\scriptstyle+}nil_{l},add(2{\scriptstyle+}1{\scriptstyle+}nil_{l},mt)))))$.
Matrix | | Distances | | Search tree
---|---|---|---|---
1 | $\cdot$ | A | $\cdot$ | $\cdot$ | B | $\cdot$ | $\cdot$ | $\cdot$ | | 2 | | 1213103254
2 | C | $\cdot$ | D | $\cdot$ | $\cdot$ | $\cdot$ | E | $\cdot$ | | 1,3 | |
3 | $\cdot$ | $\cdot$ | $\cdot$ | F | $\cdot$ | G | $\cdot$ | H | | 1,1 | |
4 | I | $\cdot$ | J | $\cdot$ | $\cdot$ | $\cdot$ | K | L | | 1,3,0 | |
5 | M | $\cdot$ | N | $\cdot$ | $\cdot$ | $\cdot$ | O | $\cdot$ | | 1,3 | |
Figure 22: Search-tree construction for comb vectors
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | …
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
row | 3 | 4 | 3 | 4 | 3 | . | . | 4 | 4 | 2 | 5 | 2 | 5 | . | . | 2 | 5 |
cont | F | I | G | J | H | . | . | K | L | C | M | D | N | . | . | E | O |
| 1 | 2 | 3 | 4 | 5
---|---|---|---|---|---
base | $+16$ | $+9$ | $-3$ | $+1$ | $+10$
$get(i,j)=$ | if | $row[base[i]+j]==i$
---|---|---
| then | $cont[base[i]+j]$
| else | $0$
| fi |
Figure 23: Comb vector and access function
To form a valid search tree, a term has to satisfy the following conditions:
* •
a vertical link may not be $nil_{t}$ (38,39),
* •
the horizontal link of a $node2$ never points to a $node1$ (39),
* •
each horizontal chain of $node1$s is in ascending order (first line of 38).
This leads to the sort definitions:
$Tree$ | $\doteq$ | $nil_{t}\mid Tree1\mid Tree2$ | (37)
---|---|---|---
$Tree1$ | $\doteq$ | $node1(n\\!:\\!Nat,t_{1}\\!:\\!Tree1\\!\mid\\!Tree2,t_{2}\\!:\\!Tree1)\lhd n<t_{2}$ | (38)
| $\mid$ | $node1(n\\!:\\!Nat,t_{1}\\!:\\!Tree1\\!\mid\\!Tree2,t_{2}\\!:\\!Tree2\\!\mid\\!nil_{t})$ |
$Tree2$ | $\doteq$ | $node2(Nat,Tree2\\!\mid\\!nil_{t})$ | (39)
$List$ | $\doteq$ | $nil_{l}\mid Nat{\scriptstyle+}List$ | (40)
$Set$ | $\doteq$ | $mt\mid add(List,Set)$ | (41)
The definition of $Tree1$ makes use of a constraint predicate, cf. the remarks
at the end of Sect. 2. $Tree1$ and $Tree2$ denote the sort of all search trees
starting with a $node1$ and a $node2$, respectively. The constraint predicate
is defined by the axiom $n_{1}<node1(n_{2},t_{3},t_{4})\leftrightarrow
n_{1}<n_{2}$.
$Tree\\!\times\\!(Nat+List)$$Set\\!\times\\!List$$Tree$$Set$$insert$$rep$$id$$rep$$\cdot\cup\\{\cdot\\}$
Figure 24: Specification of the insert algorithm $rep:Tree\rightarrow Set$
gives the set of distance lists represented by a (sub)tree:
---
$rep(nil_{t})$ | $=mt$ | (42)
$rep(node1(n,t_{1},t_{2}))$ | $=n{\scriptstyle\oplus}rep\\!\cdot\\!t_{1}\cup rep\\!\cdot\\!t_{2}$ | (43)
$rep(node2(n,t))$ | $=add(n{\scriptstyle+}nil_{l},rep\\!\cdot\\!t)$ | (44)
${\scriptstyle\oplus}:Nat\times Set\rightarrow Set$ pointwise prefixes a set
of distance lists by a new distance:
$n{\scriptstyle\oplus}mt$ | $=mt$ | (45)
$n{\scriptstyle\oplus}add(l,s)$ | $=add(n{\scriptstyle+}l,n{\scriptstyle\oplus}s)$ | (46)
$\cup:Set\times Set\rightarrow Set$ is the ordinary set union:
$mt\cup s$ | $=s$ | (47)
$add(l,s_{1})\cup s_{2}$ | $=add(l,s_{1}\cup s_{2})$ | (48)
We have the following equations between constructors:
$add(l_{1},add(l_{2},s))$ | $=add(l_{2},add(l_{1},s))$ | (49)
$add(l,add(l,s))$ | $=add(l,s)$ | (50)
Finally, we need the following derived lemma:
$s_{1}\cup s_{2}$ | $=s_{2}\cup s_{1}$ | (51)
Figure 25: Auxiliary function definitions for search-tree specification
Using the terminology introduced in Sect. 7, we have
$as_{1}=Set\times(Nat+List)$, $as_{2}=Set$, $ao(s,l)=add(l,s)$,
$cs_{1}=Tree\times List$, $cs_{2}=Tree$, $co(t,l)=insert(t,l)$ is to be
synthesized, $r_{1}(t,l)=\langle rep(t),l\rangle$, and $r_{2}(t)=rep(t)$.
The specification uses several auxiliary functions defined in Fig. 25.
Expressed in informal terms, it says: “Given a tree $t$ and a non-empty
distance list $l$, find a tree $T$ that contains the same distance lists as
$t$ and additional $l$”; and in formal terms: $\forall t\in
Tree^{M},l\in(Nat{\scriptstyle+}List)^{M}\;\exists T\in
Tree^{M}\;\;\;rep\\!\cdot\\!T=add(l,rep\\!\cdot\\!t)$. The $insert$ function
will be synthesized as Skolem function for $T$.
Using Alg. 10, we obtain the following range sorts of $rep$, cf. Fig. 26:
$rg([x\\!:=\\!nil_{t}],rep\\!\cdot\\!x)$ | $=mt$
---|---
$rg([x\\!:=\\!Tree1],rep\\!\cdot\\!x)$ | $=Sort_{{52}}$
$rg([x\\!:=\\!Tree2],rep\\!\cdot\\!x)$ | $=Sort_{{53}}$
Since the data type $Set$ is built up from unfree constructors (cf. Eqns. (49)
and (50)), we have to somehow compute the normal form sorts, cf. the remarks
at the end of Sect. 6. Setting $Sort_{{54}}=nf_{c}(Sort_{{52}})$ and
$Sort_{{55}}=nf_{c}(Sort_{{53}})$, we may get:
$Sort_{52}$ | $\doteq add(Nat{\scriptstyle+}Nat{\scriptstyle+}List,Set)$
---|---
$Sort_{53}$ | $\doteq add(Nat{\scriptstyle+}nil_{l},mt)\;\;\mid\;\;add(Nat{\scriptstyle+}nil_{l},Sort_{53})$
$Sort_{54}$ | $\doteq add(Nat{\scriptstyle+}Nat{\scriptstyle+}List,Set)\mid add(List,Sort_{54})$
$Sort_{55}$ | $\doteq Sort_{53}$
Intuitively, a term of sort $Sort_{54}$ denotes a set of distance sequences of
which at least one has a length $\geqslant 2$, while a term of sort
$Sort_{55}$ denotes a set of distance sequences of length 1. $Sort_{52}$,
$Sort_{53}$, and $mt$ are pairwise disjoint, as are $Sort_{54}$, $Sort_{55}$,
and $mt$. These signatures are too complex for there to be much likelihood of
their being declared by a user who does not know the proof in advance. The
estimation of range sorts, especially of $rep$, with such precision that
inputs starting with different constructors result in disjoint output sorts is
the main contribution of the sort discipline to search-space reduction in this
example.
When verifying by hand, without use of the sort discipline, some intuition is
needed to find out which values $rep(node1(n,t_{1},t_{2}))$ can have:
> First, we always have
> $rep(node2(n,t))=add(n{\scriptstyle+}nil_{l},rep(t))\neq mt$.
>
> Then, $rep(node1(n,t_{1},t_{2}))=n{\scriptstyle\oplus}rep(t_{1})\cup
> rep(t_{2})$, where $t_{2}$ may be $nil_{t}$ and thus $rep(t_{2})=mt$, but
> $t_{1}$ has again the form $node1(n^{\prime},t^{\prime}_{1},t^{\prime}_{2})$
> 999Or $node2(n^{\prime},t^{\prime})$, see above. and thus (by I.H.)
> $rep(t_{1})\neq mt$, hence also $n{\scriptstyle\oplus}rep(t_{1})\neq mt$.
> Thus, we always have $rep(node1(n,t_{1},t_{2}))\neq mt$.
>
> Finally, $rep(t_{1})$ contains at least one distance sequence of length
> $\geqslant 1$ (for $t_{1}=node2(n^{\prime},t^{\prime})$ trivial, for
> $t_{1}=node1(n^{\prime},t^{\prime}_{1},t^{\prime}_{2})$ by I.H.); that is
> why $n{\scriptstyle\oplus}rep(t_{1})\subset rep(node1(n,t_{1},t_{2}))$ has
> to contain at least one distance sequence of length $\geqslant 2$.
The “intuition” in this argumentation consists in recognizing two induction
hypotheses and verifying them as valid. The main difficulty here consists in
recognizing suitable hypotheses; checking of their validity could probably be
carried out by an arbitrary induction prover. It is precisely this task of
recognition that is performed by the sort discipline. The two implicitly made
inductions in the intuitive argumentation correspond to applications of the
global transformation rules from Lemmas 6 and 10, cf. Fig. 26.
| $(Tree1\\!:\\!x)\;(rep\\!\cdot\\!x\\!:\\!z)$ |
---|---|---
$=$ | $(Nat\\!:\\!n_{1})\;(Tree1\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{1}\cup rep\\!\cdot\\!t_{2}\\!:\\!z)$ |
$\mid$ | $(Nat\\!:\\!n_{1})\;(Tree2\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{1}\cup rep\\!\cdot\\!t_{2}\\!:\\!z)$ |
$=$ | … |
$=$ | $(Nat\\!:\\!n_{1})\;(Tree1\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(rep\\!\cdot\\!t_{1}\\!:\\!mt)\;(mt\cup rep\\!\cdot\\!t_{2}\\!:\\!z)$ |
$\mid$ | $(Nat\\!:\\!n_{1})\;(Tree1\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(rep\\!\cdot\\!t_{1}\\!:\\!add(l_{1},s_{1}))(add(n_{1}{\scriptstyle+}l_{1},n_{1}{\scriptstyle\oplus}s_{1}\cup rep\\!\cdot\\!t_{2})\\!:\\!z)$
$\mid$ | $(Nat\\!:\\!n_{1})\;(Tree2\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(add(Nat{\scriptstyle+}Nat{\scriptstyle+}nil_{l},Set)\\!:\\!z)$ |
$=$ | $(Nat\\!:\\!n_{1})\;(Tree1\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(rep\\!\cdot\\!t_{1}\\!:\\!add(l_{1},s_{1}))(add(n_{1}{\scriptstyle+}l_{1},n_{1}{\scriptstyle\oplus}s_{1}\cup rep\\!\cdot\\!t_{2})\\!:\\!z)$
$\mid$ | $(Nat\\!:\\!n_{1})\;(Tree2\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(add(Nat{\scriptstyle+}Nat{\scriptstyle+}nil_{l},Set)\\!:\\!z)$ | $(*)$
$\stackrel{{\scriptstyle(\subset)}}{{=}}$ | $(Nat\\!:\\!n_{1})\;(Tree1\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(rep\\!\cdot\\!t_{1}\\!:\\!add(l_{1},s_{1}))\;(add(n_{1}{\scriptstyle+}l_{1},max_{\cup})\\!:\\!z)$ | $(**)$
$\mid$ | $(Nat\\!:\\!n_{1})\;(Tree2\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(add(Nat{\scriptstyle+}Nat{\scriptstyle+}nil_{l},Set)\\!:\\!z)$ |
$=$ | $(Nat\\!:\\!n_{1})\;(Tree1\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(rep\\!\cdot\\!t_{1}\\!:\\!add(l_{1},s_{1}))\;(add(n_{1}{\scriptstyle+}l_{1},max_{\cup})\\!:\\!z)\;(l_{1}\\!:\\!Sort_{{56}})$
$\mid$ | $(Nat\\!:\\!n_{1})\;(Tree2\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(add(Nat{\scriptstyle+}Nat{\scriptstyle+}nil_{l},Set)\\!:\\!z)$ | $(***)$
$\stackrel{{\scriptstyle(\subset)}}{{=}}$ | $(Nat\\!:\\!n_{1})\;(Tree1\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(add(n_{1}{\scriptstyle+}l_{1},max_{\cup})\\!:\\!z)\;(l_{1}\\!:\\!Sort_{{56}})$ |
$\mid$ | $(Nat\\!:\\!n_{1})\;(Tree2\\!:\\!t_{1})\;(Tree\\!:\\!t_{2})\;(add(Nat{\scriptstyle+}Nat{\scriptstyle+}nil_{l},Set)\\!:\\!z)$ |
$(*)$: | Constructor term deletion (cf. Lemma 6): the first alternative can be removed since $rep\\!\cdot\\!x$ does not produce $mt$ outside of it, but it requires, in turn, the production of $mt$ by $rep\\!\cdot\\!t_{1}$. Note that the constructors $mt$ and $add(\cdot,\cdot)$ are regarded as free such that $add(x,y)\neq mt$ always holds. It has been shown that it is sufficient to consider Eqns. (49) and (50) only outside the $rg$ computation.
---|---
$(**)$: | Estimation by trivial upper bound, cf. Alg. 10, and $(**)$ in Fig. 14; $max_{\cup}=Set$.
$(***)$: | Constructor argument estimation (cf. Lemma 10): if $rep\\!\cdot\\!x$ yields $add(l_{1},s_{1})$, $l_{1}$ has the form $Nat{\scriptstyle+}Nat{\scriptstyle+}nil_{l}$ from the second alternative or $add(n^{\prime}_{1}{\scriptstyle+}l^{\prime}_{1},\ldots)$ from the first one, where the same holds, in turn, for $l^{\prime}_{1}$.
| Hence, $l_{1}$ belongs to $Sort_{{56}}$, where $Sort_{56}\doteq
Nat{\scriptstyle+}Nat{\scriptstyle+}nil_{l}\mid Nat{\scriptstyle+}Sort_{56}$.
The result we get is
$(Tree1\\!:\\!x)\;(rep\\!\cdot\\!x\\!:\\!z)=(Sort_{52}\\!:\\!z)$
---
where $Sort_{52}\doteq add(Nat{\scriptstyle+}Sort_{56},Set)\mid
add(Nat{\scriptstyle+}Nat{\scriptstyle+}nil_{l},Set)$,
i.e. $Sort_{52}=add(Nat{\scriptstyle+}Nat{\scriptstyle+}List,Set)$.
Figure 26: Range sort computation for $rep$
Figures 28 to 35 show the synthesis proof. Variables are denoted by upper-case
letters, constants by lower-case letters. A number in the right-hand column
refers to the equation that has been used for narrowing (rule (ln) in Thm.
14), an exponent “-” denoting the reversed equation; “dec” and “I.H.” mean the
application of the decomposition rule (rule (d) in Thm. 14), and the induction
hypothesis, respectively. Narrowing steps that are not uniquely determined by
the sort discipline are marked with “$*$”. They all occur as a series of
backward applications of laws for ${\scriptstyle\oplus}$ or $\cup$ in order to
get the right-hand side close to the syntactic structure of the left-hand side
and then perform a decomposition. Application of the induction hypothesis is
marked with “$(*)$” since it need only be taken into account if the actual
equation’s sort is too large to determine a narrowing step uniquely.
In cases that do not use the induction hypothesis, the sort restrictions
enable us to find the solution automatically. For example, in case 2.1
($t=node2(n_{1},t_{2}),l=n_{4}{\scriptstyle+}nil_{l}$, cf. Fig. 30), owing to
sort restrictions, only Eqn. (44) can be used in the narrowing step, since the
right-hand side has the sort $add(Nat{\scriptstyle+}nil_{l},Sort_{53})\subset
Sort_{53}$. In cases that actually use the induction hypothesis, the sort
restrictions prune the search space to the size of a verification proof,
solving the additional problems of synthesis. Moreover, the use of the sort
discipline allows us to perform the crucial proper narrowing step as the very
first one, providing syntactic information at the equations’ left-hand side,
which can be used by subsequent steps concerned with E-unification wrt. the
$Set$ equations.
The breaking-down of case 3.2 ($t=node1(n_{1},t_{2},t_{3})$,
$l=n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6}$) into three subcases 3.2.1 –
3.2.3 can be done automatically. Each solution has to fulfill the additional
requirement that $insert(\ldots)\in Tree^{M}$, since this is not ensured by
the narrowing process itself. After having found the solution
$T=node1(n_{4},insert(nil_{t},n_{5}{\scriptstyle+}l_{6}),t)$ shown in Fig. 33,
it can be determined that $T\in Tree^{M}$ only if
$insert(nil_{t},n_{5}{\scriptstyle+}l_{6})\in Tree1^{M}\cup Tree2^{M}$ and
$n_{4}<n_{1}$. The former condition is delayed until the synthesis of the
algorithm is complete and can then be verified by an easy induction. The
latter condition is intended to be passed to a prover in which the sort
algorithms are embedded; it must be able to detect that
$n_{4}<n_{1}\not\Leftrightarrow true$ and to initiate the search for further
solutions of case 3.2. In this way, the solutions shown in Figs. 34 and 35 are
found. Finally, the prover must be able to detect that all subcases have been
covered, i.e. $n_{4}\\!<\\!n_{1}\vee n_{1}\\!<\\!n_{4}\vee
n_{4}\\!=\\!n_{1}\Leftrightarrow true$.
The synthesized algorithm is shown in Fig. 27.
$insert(nil_{t},n_{4}{\scriptstyle+}nil_{l})$ | $=node2(n_{4},nil_{t})$
---|---
$insert(nil_{t},n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6})$ | $=node1(n_{4},insert(nil_{t},n_{5}{\scriptstyle+}l_{6}),nil_{t})$
$insert(node2(n_{1},t_{2}),n_{4}{\scriptstyle+}nil_{l})$ | $=node2(n_{4},node2(n_{1},t_{2}))$
$insert(node2(n_{1},t_{2}),n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6})$ | $=node1(n_{4},insert(nil_{t},n_{5}{\scriptstyle+}l_{6}),node2(n_{1},t_{2}))$
$insert(node1(n_{1},t_{2},t_{3}),n_{4}{\scriptstyle+}nil_{l})$ | $=node1(n_{1},t_{2},insert(t_{3},n_{4}{\scriptstyle+}nil_{l}))$
$insert(node1(n_{1},t_{2},t_{3}),n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6})$ | $=node1(n_{4},insert(nil_{t},n_{5}{\scriptstyle+}l_{6}),node1(n_{1},t_{2},t_{3}))$
| | $\longleftarrow n_{4}<n_{1}$
$insert(node1(n_{1},t_{2},t_{3}),n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6})$ | $=node1(n_{1},t_{2},insert(t_{3},n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6}))$ | $\longleftarrow n_{1}<n_{4}$
$insert(node1(n_{1},t_{2},t_{3}),n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6})$ | $=node1(n_{1},insert(t_{2},n_{5}{\scriptstyle+}l_{6}),t_{3})$ | $\longleftarrow n_{4}=n_{1}$
Figure 27: Synthesized comb vector insertion algorithm
$rep\\!\cdot\\!T$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!nil_{t})$ |
---|---|---
$add(N_{7}{\scriptstyle+}nil_{l},rep\\!\cdot\\!T_{8})$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!nil_{t})\;\;\wedge$ | (44)
$T$ | $=node2(N_{7},T_{8})$ |
$add(N_{7}{\scriptstyle+}nil_{l},rep\\!\cdot\\!T_{8})$ | $=add(n_{4}{\scriptstyle+}nil_{l},mt)$ | (42)
$N_{7}$ | $=n_{4}\;\;\wedge$ | dec.
$rep\\!\cdot\\!T_{8}$ | $=mt$ |
$T_{8}$ | $=nil_{t}$ | (42)
Answer substitution: $[N_{7}\\!:=\\!n_{4},T_{8}\\!:=\\!nil_{t}]$ $\circ$
$[T\\!:=\\!node2(N_{7},T_{8})]$
Figure 28: Case 1.1 — $t=nil_{t}$, $l=n_{4}{\scriptstyle+}nil_{l}$
$rep\\!\cdot\\!T$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!nil_{t})$ | | |
---|---|---|---|---
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!nil_{t})\;\;\wedge$ | (43) | |
$T$ | $=node1(N_{7},T_{8},T_{9})$ | | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},mt\cup rep\\!\cdot\\!nil_{t})$ | (47)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},mt)\cup rep\\!\cdot\\!nil_{t}$ | (48)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},n_{4}{\scriptstyle\oplus}mt)\cup rep\\!\cdot\\!nil_{t}$ | (45)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=n_{4}{\scriptstyle\oplus}add(n_{5}{\scriptstyle+}l_{6},mt)\cup rep\\!\cdot\\!nil_{t}$ | (46)- | | $*$
$N_{7}$ | $=n_{4}\;\;\wedge$ | dec. | | $*$
$rep\\!\cdot\\!T_{8}$ | $=add(n_{5}{\scriptstyle+}l_{6},mt)\;\;\wedge$ | | |
$T_{9}$ | $=nil_{t}$ | | |
$add(L_{11},rep\\!\cdot\\!T_{10})$ | $=add(n_{5}{\scriptstyle+}l_{6},mt)\;\;\wedge$ | I.H. | | $(*)$
$T_{8}$ | $=insert(T_{10},L_{11})$ | | |
$L_{11}$ | $=n_{5}{\scriptstyle+}l_{6}\;\;\wedge$ | dec. | |
$rep\\!\cdot\\!T_{10}$ | $=mt$ | | |
$T_{10}$ | $=nil_{t}$ | (42) | |
Answer substitution:
$[L_{11}\\!:=\\!n_{5}{\scriptstyle+}l_{6},T_{10}\\!:=\\!nil_{t}]$ $\circ$
$[N_{7}\\!:=\\!n_{4},T_{9}\\!:=\\!nil_{t},T_{8}\\!:=\\!insert(T_{10},L_{11})]$
$\circ$ $[T\\!:=\\!node1(N_{7},T_{8},T_{9})]$
Figure 29: Case 1.2 — $t=nil_{t}$,
$l=n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6}$
$rep\\!\cdot\\!T$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!node2(n_{1},t_{2}))$ |
---|---|---
$add(N_{7}{\scriptstyle+}nil_{l},rep\\!\cdot\\!T_{8})$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!node2(n_{1},t_{2}))\;\;\wedge$ | (44)
$T$ | $=node2(N_{7},T_{8})$ |
$N_{7}$ | $=n_{4}\;\;\wedge$ | dec.
$T_{8}$ | $=node2(n_{1},t_{2})$ |
Answer substitution: $[N_{7}\\!:=\\!n_{4},T_{8}\\!:=\\!node2(n_{1},t_{2})]$
$\circ$ $[T\\!:=\\!node2(N_{7},T_{8})]$
Figure 30: Case 2.1 — $t=node2(n_{1},t_{2})$, $l=n_{4}{\scriptstyle+}nil_{l}$
$rep\\!\cdot\\!T$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!node2(n_{1},t_{2}))$ | | |
---|---|---|---|---
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!node2(n_{1},t_{2}))\;\;\wedge$ | (43) | |
$T$ | $=node1(N_{7},T_{8},T_{9})$ | | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},mt\cup rep\\!\cdot\\!node2(n_{1},t_{2}))$ | (47)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},mt)\cup rep\\!\cdot\\!node2(n_{1},t_{2})$ | (48)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},n_{4}{\scriptstyle\oplus}mt)\cup rep\\!\cdot\\!node2(n_{1},t_{2})$ | (45)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=n_{4}{\scriptstyle\oplus}add(n_{5}{\scriptstyle+}l_{6},mt)\cup rep\\!\cdot\\!node2(n_{1},t_{2})$ | (46)- | | $*$
$N_{7}$ | $=n_{4}\;\;\wedge$ | dec. | | $*$
$rep\\!\cdot\\!T_{8}$ | $=add(n_{5}{\scriptstyle+}l_{6},mt)\;\;\wedge$ | | |
$T_{9}$ | $=node2(n_{1},t_{2})$ | | |
$add(L_{11},rep\\!\cdot\\!T_{10})$ | $=add(n_{5}{\scriptstyle+}l_{6},mt)\;\;\wedge$ | I.H. | | $(*)$
$T_{8}$ | $=insert(T_{10},L_{11})$ | | |
$L_{11}$ | $=n_{5}{\scriptstyle+}l_{6}\;\;\wedge$ | dec. | |
$rep\\!\cdot\\!T_{10}$ | $=mt$ | | |
$T_{10}$ | $=nil_{t}$ | (42) | |
Answer substitution:
$[L_{11}\\!:=\\!n_{5}{\scriptstyle+}l_{6},T_{10}\\!:=\\!nil_{t}]$ $\circ$
$[N_{7}\\!:=\\!n_{4},T_{9}\\!:=\\!node2(n_{1},t_{2}),T_{8}\\!:=\\!insert(T_{10},L_{11})]$
$\circ$ $[T\\!:=\\!node1(N_{7},T_{8},T_{9})]$
Figure 31: Case 2.2 — $t=node2(n_{1},t_{2})$,
$l=n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6}$
$rep\\!\cdot\\!T$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))$ | | |
---|---|---|---|---
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))\;\;\wedge$ | (43) | |
$T$ | $=node1(N_{7},T_{8},T_{9})$ | | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}nil_{l},n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2}\cup rep\\!\cdot\\!t_{3})$ | (43) | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!t_{3}\cup n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2})$ | (51) | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!t_{3})\cup n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2}$ | (48)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2}\cup add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!t_{3})$ | (51) | | $*$
$N_{7}$ | $=n_{1}\;\;\wedge$ | dec. | | $*$
$T_{8}$ | $=t_{2}\;\;\wedge$ | | |
$rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!t_{3})$ | | |
$add(L_{11},rep\\!\cdot\\!T_{10})$ | $=add(n_{4}{\scriptstyle+}nil_{l},rep\\!\cdot\\!t_{3})\;\;\wedge$ | I.H. | | $(*)$
$T_{9}$ | $=insert(T_{10},L_{11})$ | | |
$L_{11}$ | $=n_{4}{\scriptstyle+}nil_{l}\;\;\wedge$ | dec. | |
$T_{10}$ | $=t_{3}$ | | |
Answer substitution:
$[L_{11}\\!:=\\!n_{4}{\scriptstyle+}nil_{l},T_{10}\\!:=\\!t_{3}]$ $\circ$
$[N_{7}\\!:=\\!n_{1},T_{8}\\!:=\\!t_{2},T_{9}\\!:=\\!insert(T_{10},L_{11})]$
$\circ$ $[T\\!:=\\!node1(N_{7},T_{8},T_{9})]$
Figure 32: Case 3.1 — $t=node1(n_{1},t_{2},t_{3})$,
$l=n_{4}{\scriptstyle+}nil_{l}$
$rep\\!\cdot\\!T$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))$ | | |
---|---|---|---|---
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))\;\;\wedge$ | (43) | |
$T$ | $=node1(N_{7},T_{8},T_{9})$ | | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},mt\cup rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))$ | (47)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},mt)\cup rep\\!\cdot\\!node1(n_{1},t_{2},t_{3})$ | (48)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}\\!{\scriptstyle+}\\!n_{5}\\!{\scriptstyle+}\\!l_{6},n_{4}{\scriptstyle\oplus}mt)\cup rep\\!\cdot\\!node1(n_{1},t_{2},t_{3})$ | (45)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=n_{4}{\scriptstyle\oplus}add(n_{5}{\scriptstyle+}l_{6},mt)\cup rep\\!\cdot\\!node1(n_{1},t_{2},t_{3})$ | (46)- | | $*$
$N_{7}$ | $=n_{4}\;\;\wedge$ | dec. | | $*$
$rep\\!\cdot\\!T_{8}$ | $=add(n_{5}{\scriptstyle+}l_{6},mt)\;\;\wedge$ | | |
$T_{9}$ | $=node1(n_{1},t_{2},t_{3})$ | | |
$add(L_{11},rep\\!\cdot\\!T_{10})$ | $=add(n_{5}{\scriptstyle+}l_{6},mt)\;\;\wedge$ | I.H. | | $(*)$
$T_{8}$ | $=insert(T_{10},L_{11})$ | | |
$L_{11}$ | $=n_{5}{\scriptstyle+}l_{6}\;\;\wedge$ | dec. | |
$rep\\!\cdot\\!T_{10}$ | $=mt$ | | |
$T_{10}$ | $=nil_{t}$ | (42) | |
Answer substitution:
$[L_{11}\\!:=\\!n_{5}{\scriptstyle+}l_{6},T_{10}\\!:=\\!nil_{t}]$ $\circ$
$[N_{7}\\!:=\\!n_{4},T_{9}\\!:=\\!node1(n_{1},t_{2},t_{3}),T_{8}\\!:=\\!insert(T_{10},L_{11})]$
$\circ$ $[T\\!:=\\!node1(N_{7},T_{8},T_{9})]$
Figure 33: Case 3.2.1 — $t=node1(n_{1},t_{2},t_{3})$,
$l=n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6}$, $n_{4}<n_{1}$
$rep\\!\cdot\\!T$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))$ | | |
---|---|---|---|---
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))\;\;\wedge$ | (43) | |
$T$ | $=node1(N_{7},T_{8},T_{9})$ | | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2}\cup rep\\!\cdot\\!t_{3})$ | (43) | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!t_{3}\cup n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2})$ | (51) | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!t_{3})\cup n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2}$ | (48)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2}\cup add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!t_{3})$ | (51) | | $*$
$N_{7}$ | $=n_{1}\;\;\wedge$ | dec. | | $*$
$T_{8}$ | $=t_{2}\;\;\wedge$ | | |
$rep\\!\cdot\\!T_{9}$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!t_{3})$ | | |
$add(L_{11},rep\\!\cdot\\!T_{10})$ | $=add(n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!t_{3})\;\;\wedge$ | I.H. | | $(*)$
$T_{9}$ | $=insert(T_{10},L_{11})$ | | |
$L_{11}$ | $=n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6}\;\;\wedge$ | dec. | |
$T_{10}$ | $=t_{3}$ | | |
Answer substitution:
$[L_{11}\\!:=\\!n_{r}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},T_{10}\\!:=\\!t_{3}]$
$\circ$
$[N_{7}\\!:=\\!n_{1},T_{8}\\!:=\\!t_{2},T_{9}\\!:=\\!insert(T_{10},L_{11})]$
$\circ$ $[T\\!:=\\!node1(N_{7},T_{8},T_{9})]$
Figure 34: Case 3.2.2 — $t=node1(n_{1},t_{2},t_{3})$,
$l=n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6}$, $n_{1}<n_{4}$
$rep\\!\cdot\\!T$ | $=add(n_{1}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))$ | | |
---|---|---|---|---
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{1}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!node1(n_{1},t_{2},t_{3}))\;\;\wedge$ | (43) | |
$T$ | $=node1(N_{7},T_{8},T_{9})$ | | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{1}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2}\cup rep\\!\cdot\\!t_{3})$ | (43) | |
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=add(n_{1}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6},n_{1}{\scriptstyle\oplus}rep\\!\cdot\\!t_{2})\cup rep\\!\cdot\\!t_{3}$ | (48)- | | $*$
$N_{7}{\scriptstyle\oplus}rep\\!\cdot\\!T_{8}\cup rep\\!\cdot\\!T_{9}$ | $=n_{1}{\scriptstyle\oplus}add(n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!t_{2})\cup rep\\!\cdot\\!t_{3}$ | (46)- | | $*$
$N_{7}$ | $=n_{1}\;\;\wedge$ | dec. | | $*$
$rep\\!\cdot\\!T_{8}$ | $=add(n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!t_{2})\;\;\wedge$ | | |
$T_{9}$ | $=t_{3}$ | | |
$add(L_{11},rep\\!\cdot\\!T_{10})$ | $=add(n_{5}{\scriptstyle+}l_{6},rep\\!\cdot\\!t_{2})\;\;\wedge$ | I.H. | | $(*)$
$T_{8}$ | $=insert(T_{10},L_{11})$ | | |
$L_{11}$ | $=n_{5}{\scriptstyle+}l_{6}\;\;\wedge$ | dec. | |
$T_{10}$ | $=t_{2}$ | | |
Answer substitution:
$[L_{11}\\!:=\\!n_{5}{\scriptstyle+}l_{6},T_{10}\\!:=\\!t_{2}]$ $\circ$
$[N_{7}\\!:=\\!n_{1},T_{9}\\!:=\\!t_{3},T_{8}\\!:=\\!insert(T_{10},L_{11})]$
$\circ$ $[T\\!:=\\!node1(N_{7},T_{8},T_{9})]$
Figure 35: Case 3.2.3 — $t=node1(n_{1},t_{2},t_{3})$,
$l=n_{4}{\scriptstyle+}n_{5}{\scriptstyle+}l_{6}$, $n_{4}=n_{1}$
## Appendix 0.C Index
NOTION | Nr. | Page
---|---|---
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c\\!d$}} \end{picture}}$ | 14 | 14
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle c$}} \end{picture}}$ | 14 | 14
$\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}$ | 6 | 6
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${{}^{\sigma}\\!v}_{1}\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\rightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}}{{}^{\sigma}\\!v}_{2}$ | 6 | 6
${{}^{\mu_{i}}\\!f}(u_{i1},\ldots,u_{in})$ | 6 | 6
$\subsetneqq$ | 2 | 2
$\subset$ | 2 | 2
$A\times B$ | 2 | 2
$\doteq$ | 2 | 2
$\stackrel{{\scriptstyle\mbox{\Large\bf..}}}{{<}}$ | 2 | 2
$\stackrel{{\scriptstyle\mbox{\Large\bf.}}}{{<}}$ | 2 | 2
$\rule[-1.9919pt]{1.13791pt}{11.38092pt}\hskip 2.84544pt_{i=1}^{n}S_{i}$ | 2 | 2
$\mid$ | 2 | 2
$\bot$ | 6 | 6
$\top$ | 6 | 6
$\top_{V}$ | 6 | 6
$\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}$ | 3 | 3
$\mathchoice{(V\\!\hookrightarrow\\!\cal CR)}{(V\\!\hookrightarrow\\!\cal CR)}{(V\hookrightarrow\cal CR)}{(V\hookrightarrow\cal CR)}$ | 5 | 5
$\varepsilon$ | 3 | 3
$\beta$ | 2 | 2
$\beta\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$ | 2 | 2
$\beta v$ | 2 | 2
$\beta_{1}\mathbin{\mathchoice{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}{\circ\mkern-7.0mu\cdot\mkern 2.0mu}}\beta_{2}$ | 2 | 2
$\gamma$ | 2 | 2
$\mu$ | 3 | 3
$\mu^{\prime}$ | 3 | 3
$\sigma$ | 3 | 3
$\sigma^{\prime}$ | 3 | 3
$\sigma\\!/\\!_{\beta}$ | 6 | 6
$\sigma^{\prime}\\!/\\!_{\beta}$ | 6 | 6
$\sigma\circ\beta$ | 6 | 6
$\sigma^{\prime}\circ\beta$ | 6 | 6
$\sigma\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau$ | 5 | 5
$\sigma^{\prime}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\tau^{\prime}$ | 5 | 5
$\sigma\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$ | 5 | 5
$\sigma^{\prime}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V$}}$ | 5 | 5
$\sigma u$ | 5 | 5
$\sigma^{\prime}u$ | 5 | 5
$\tau$ | 3 | 3
$\tau^{\prime}$ | 3 | 3
${{}^{\sigma}\\!v}$ | 5 | 5
$::$ | 2 | 2
$0_{x}s_{y}$ | 3 | 3
$abstract(S,x)$ | 7 | 7
admissible t-substitutions | 3 | 3
alternatives | 10 | 10
annotated term | 5 | 5
application | 3 | 3
application | 5 | 5
$apply(\sigma,x)$ | 7 | 7
approximation rule | 10 | 10
$ar(g)$ | 2 | 2
arity | 2 | 2
$ar(\vec{cr})$ | 3 | 3
$Bin$ | 2 | 2
$Bin$ | 14 | 14
$compose(\sigma,\tau)$ | 7 | 7
computation path | 10 | 10
computation tree | 10 | 10
constraint | 4 | 4
constructor symbols | 2 | 2
constructor terms | 2 | 2
constructor-matching rules | 2 | 2
constructors for t-substitutions | 3 | 3
$cr$ | 2 | 2
$\vec{cr}$ | 3 | 3
$\vec{cr}\mathbin{\mathchoice{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}{\diamond\mkern-7.0mu\cdot\mkern 2.0mu}}\vec{cr}^{\prime}$ | 3 | 3
$\vec{cr}\mid_{\raisebox{-0.70004pt}{$\scriptscriptstyle V^{\prime}$}}$ | 3 | 3
$\vec{cr}_{x}$ | 3 | 3
${\cal CR}$ | 2 | 2
(d) | 14 | 14
defining equations | 5 | 5
defining equations | 6 | 6
dependent | 7 | 7
depth | 2 | 2
$diff(S_{1},S_{2}\mid\ldots\mid S_{m})$ | 4 | 4
distributivity rules | 2 | 2
$div(\sigma,\beta)$ | 9 | 9
$dom(\beta)$ | 2 | 2
$dom(f)$ | 6 | 6
$dom(f,I)$ | 6 | 6
$dom(\sigma^{\prime})$ | 5 | 5
$dom(\sigma)$ | 6 | 6
$dup(\sigma,\beta)$ | 9 | 9
$dup(x)$ | 14 | 14
elementwise extension | 2 | 2
equations between constructors | 14 | 14
$Even$ | 14 | 14
extended sort | 5 | 5
$f$ | 2 | 2
${\cal F}$ | 2 | 2
$f\\!\cdot\\!x$ | 2 | 2
$f[A^{\prime}]$ | 2 | 2
factorization | 6 | 6
$fact(\sigma,\beta)$ | 4 | 4
$fact(\sigma,\beta)$ | 4 | 4
$g$ | 2 | 2
ground constructor terms | 2 | 2
homogeneous | 4 | 4
$i$ | 2 | 2
independent | 7 | 7
Induction Principle | 2 | 2
$inf(S_{1},S_{2})$ | 3 | 3
inhabitance | 2 | 2
$inh(S,Occ)$ | 4 | 4
intersection | 2 | 2
junk terms | 6 | 6
lazy narrowing | 14 | 14
$Lex_{x<y}$ | 7 | 7
lifting | 5 | 5
linear | 2 | 2
linear | 2 | 2
(ln) | 14 | 14
local transformation rules | 10 | 10
loop-checking rules | 2 | 2
$M$ | 2 | 2
$max_{f}$ | 10 | 10
$max^{\prime}_{f}$ | 10 | 10
$mgu(v_{1},v_{2})$ | 2 | 2
most general unifier | 2 | 2
$Mtch_{x,y}$ | 7 | 7
$Nat$ | 14 | 14
$Nat_{x}$ | 6 | 6
$Nat_{x,y}$ | 6 | 6
$Nat_{x\\!<\\!y}$ | 6 | 6
$Nat_{x\\!=\\!y}$ | 6 | 6
$Nat_{y}$ | 6 | 6
$nf[A]$ | 6 | 6
$nf_{c}$ | 14 | 14
$nf(v)$ | 6 | 6
non-constructor functions | 2 | 2
$o$ | 2 | 2
ordinary substitution | 2 | 2
parallel composition | 3 | 3
parallel composition | 5 | 5
parallel composition | 2 | 2
partial mappings | 5 | 5
$Pref_{x,y,z}$ | 7 | 7
pseudolinear | 2 | 2
pseudolinear | 2 | 2
$ran(\beta)$ | 2 | 2
Rank of a T-Substitution | 10 | 10
$rank(\sigma^{\prime},(w_{1}:u_{1}))$ | 10 | 10
$rank(\sigma^{\prime},(w_{1}:u_{1})\;\ldots\;(w_{n}:u_{n}))$ | 10 | 10
regular | 2 | 2
relative complement | 2 | 2
renaming substitution | 2 | 2
restriction | 3 | 3
restriction | 5 | 5
restriction | 2 | 2
$restrict(\sigma,V)$ | 7 | 7
rewrite relation | 6 | 6
$rg$ | 5 | 5
$rg_{c}$ | 14 | 14
$rg(\sigma,v)$ | 10 | 10
$S$ | 2 | 2
${\cal S}$ | 2 | 2
semantics | 2 | 2
semi-independent | 7 | 7
semilinear | 2 | 2
$(\sigma)$ | 10 | 10
$single(S)$ | 4 | 4
$S^{M}$ | 2 | 2
$snoc$ | 2 | 2
solution of an equation | 14 | 14
sort definitions | 2 | 2
sort equivalence | 2 | 2
sort expressions | 2 | 2
sort names | 2 | 2
sort system | 2 | 2
Sorted Narrowing | 14 | 14
Sorted Rewriting | 6 | 6
$SortName$ | 2 | 2
subsort | 2 | 2
substitution | 2 | 2
$Sum_{x,y,z}$ | 7 | 7
$S^{X}$ | 2 | 2
${\cal T}_{\cal CR}$ | 2 | 2
${\cal T}_{{\cal CR},{\cal F}}/{\mathrel{\begin{picture}(0.37,0.0)\put(0.0,0.05){\makebox(0.0,0.0)[l]{$\leftrightarrow$}} \put(0.2,0.1){\makebox(0.0,0.0)[b]{$\scriptscriptstyle$}} \end{picture}^{*}}}$ | 6 | 6
${\cal T}_{{\cal CR},{\cal F},{\cal V}}$ | 2 | 2
${\cal T}_{{\cal CR},{\cal F},{\cal V},{\cal S}}$ | 2 | 2
${\cal T}_{{\cal CR},{\cal V}}$ | 2 | 2
${\cal T}_{\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}}$ | 3 | 3
${\cal T}_{X}$ | 2 | 2
${\cal T}_{X,Y}$ | 2 | 2
${\cal T}^{*}_{\mathchoice{(V\\!\rightarrow\\!\cal CR)}{(V\\!\rightarrow\\!\cal CR)}{(V\rightarrow\cal CR)}{(V\rightarrow\cal CR)}}$ | 3 | 3
${\cal T}^{*}_{\mathchoice{(V\\!\hookrightarrow\\!\cal CR)}{(V\\!\hookrightarrow\\!\cal CR)}{(V\hookrightarrow\cal CR)}{(V\hookrightarrow\cal CR)}}$ | 5 | 5
terms | 2 | 2
t-sets | 3 | 3
t-substitutions | 3 | 3
tuple | 2 | 2
$u$ | 2 | 2
$use(S)$ | 2 | 2
$v$ | 2 | 2
${\cal V}$ | 2 | 2
$v_{1}\mathrel{\mathop{\sphericalangle}\limits_{\neq}}v_{2}$ | 2 | 2
$v_{1}\mathrel{\sphericalangle}v_{2}$ | 2 | 2
$\langle v_{1},\ldots,v_{n}\rangle$ | 2 | 2
$\langle v_{i}\mid p(v_{i}),\;i=1,\ldots,n\rangle$ | 2 | 2
$val$ | 14 | 14
variables | 2 | 2
$vars(v_{1},\ldots,v_{n})$ | 2 | 2
$w$ | 2 | 2
$((w_{1}:u_{1})\;(w_{2}:u_{2}))^{M}$ | 10 | 10
well-defined | 6 | 6
$w^{M}$ | 10 | 10
$(w:u)^{M}$ | 10 | 10
$x$ | 2 | 2
$[x\\!:=\\!S]$ | 5 | 5
$[x\\!:=\\!u]$ | 5 | 5
$[x_{1}\\!:=\\!v_{1},\ldots,x_{n}\\!:=\\!v_{n}]$ | 2 | 2
$x\raisebox{5.69054pt}{\scriptsize:S}$ | 2 | 2
$y$ | 2 | 2
$z$ | 2 | 2
$Zip_{x,y,z}$ | 7 | 7
|
arxiv-papers
| 2014-04-04T09:54:08 |
2024-09-04T02:50:00.722726
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jochen Burghardt",
"submitter": "Jochen Burghardt",
"url": "https://arxiv.org/abs/1404.1201"
}
|
1404.1316
|
# A note on the spectrum of a two-particle Rashba Hamiltonian
R. Juršėnas Institute of Theoretical Physics and Astronomy of Vilnius
University,
A. Goštauto 12, 01108 Vilnius, Lithuania
###### Abstract
In a series of recent papers it was shown that, when the attractive $s$-wave
interaction is dominant, the spin-orbit coupled fermions form a bound state.
Attributed to a convenient momentum representation, it became a common
condition of agreement to express the bound state as a function of the center-
of-mass momentum $Q$. In this letter we prove that the bound state of Rashba
fermions does not depend on the chosen representation. That is, all the states
characterized by nonzero $Q$ fail to obey the translation symmetry.
###### pacs:
71.70.Ej, 03.65.Ge, 67.85.-d, 05.30.Fk
###### Contents
1. 1 Introduction
2. 2 Translation symmetry
3. 3 Green’s function
4. 4 Nevanlinna function
5. 5 Eigenspace
6. 6 Conclusion and discussion
## 1 Introduction
The role of a bound state is in the spotlight of neutral Fermi atoms. The
motives of research, technical possibilities and important physical issues are
discussed in many papers [1, 2, 3, 4, 5, 6, 7] where—among other celebrated
results—the existence of a bound state is determined by establishing the poles
of the Fourier transform of Green’s function. Also [6], one can explore a free
two-particle Green’s function in deriving the so-called self-consistency
condition which is just the weak (or distributional) solution to the
Schrödinger equation. Although trivial in many applications, the pole itself
is insufficient to determine the bound state. According to the
Aronszajn–Donoghue spectral theory of rank one perturbations [8, 9] [$s$-wave
interaction is exactly rank one], the derivative of the free Green’s function
with respect to (w.r.t.) the eigenvalue must be finite. Otherwise, the
spectral points are in the singular continuous spectrum. This is exactly the
case when one says [6] that the bound state ceases to exist.
The single-particle Rashba Hamiltonian in
$L_{2}(\mathbb{R}^{2};\mathbb{C}^{2})=:\mathscr{X}$, of a particle with mass
$m$ and Rashba spin-orbit coupling $\alpha$ (we use the $\hbar=c=1$ units) is
invariant under Euclidean moves. If parametrized in terms of the center-of-
mass $\vec{R}\in\mathbb{R}^{2}$ and the relative $\vec{r}\in\mathbb{R}^{2}$
coordinates, the same applies to the Hamiltonian in
$(\mathscr{X},d^{2}\vec{R})\text{\large$\otimes$}(\mathscr{X},d^{2}\vec{r})$
which describes the two interacting particles. For the dominant $s$-wave
interaction, with the interaction strength $\gamma$, the Euclidean group is
naturally reduced to the translation group. The latter is represented as a
tensor product of two two-dimensional groups whose irreducible representations
act on $\vec{R}$ and $\vec{r}$, respectively. Due to the interaction, it is
clear that the two-particle Hamiltonian commutes with the generator
$\vec{P}\text{\large$\otimes$}I$, where $i\vec{P}$ is the gradient in
$\vec{R}$ and $I$ is the identity operator in $(\mathscr{X},d^{2}\vec{r})$,
whereas it does not commute with the generator in $\vec{r}$. The commutation
relation indicates that the eigenspace of the Hamiltonian is a dense subset in
the domain of the closure of the generator in the Sobolev norm (apply eg [10,
Corollary A.7.3.6]). Subsequently, the generalized eigenvectors of the
Hamiltonian are labeled by its spectral points $\lambda$ along with the
spectral points $\vec{\Lambda}\in\mathbb{R}^{2}$ of the closure of $\vec{P}$.
In this letter we shall prove that only the translation invariant
eigenvectors, that is those with $\vec{\Lambda}=\vec{0}$, are the eigenvectors
of the two-particle Rashba Hamiltonian with point-interaction. For other two-
particle interactions, this is not necessarily the case though (see discussion
in Sec. 6).
## 2 Translation symmetry
Consider an arbitrary $f=f(\vec{R},\vec{r})$ in the Schwartz space
$\mathscr{D}(\mathbb{R}^{4};\mathbb{C}^{4})$ of smooth functions with compact
support. Then $f$ has a Fourier transform $\chi=\chi(\vec{Q},\vec{k})$, where
$\vec{Q}\in\mathbb{R}^{2}$ denotes the center-of-mass momentum and
$\vec{k}\in\mathbb{R}^{2}$ is the relative momentum.
Since $\mathscr{D}(\mathbb{R}^{4};\mathbb{C}^{4})$ is a dense subspace of the
Hilbert space $L_{2}(\mathbb{R}^{4};\mathbb{C}^{4})$, denoted
$\mathscr{X}_{o}$, and $\mathscr{X}\text{\large$\otimes$}\mathscr{X}$ is
isomorphic to $\mathscr{X}_{o}$, one can restate the irreducible
representation $\exp(i\vec{P}\cdot\vec{a})\text{\large$\otimes$}I$ (all
$\vec{a}\in\mathbb{R}^{2}$) of the translation group by simply writing
$\exp(i\vec{P}\cdot\vec{a})$. Suppose that $\vec{\Lambda}\in\mathbb{R}^{2}$
and that $f$ solves $(\vec{P}-\vec{\Lambda})f=\vec{0}$. Then $f$ is of the
form
$f(\vec{R},\vec{r})=e^{i(\vec{\Lambda}\cdot\vec{R})}\varphi(\vec{r}),\quad\varphi\in\mathscr{D}(\mathbb{R}^{2};\mathbb{C}^{4}).$
(2.1)
Then the Fourier transform reads
$\chi(\vec{Q},\vec{k})=(2\pi)^{2}\delta(\vec{Q}-\vec{\Lambda})\hat{\varphi}(\vec{k}),$
(2.2)
where $\hat{\varphi}\in\mathscr{D}(\mathbb{R}^{2};\mathbb{C}^{4})$ is the
Fourier transform of $\varphi$ and $\delta(\cdot)$ is a two-dimensional Dirac
distribution. It appears from (2.2) that the Fourier transform $\chi$
represents a singular distribution, the existence of which is predetermined by
the relation $\vec{Q}=\vec{\Lambda}$. In particular, (2.1) implies that for
$\vec{\Lambda}=\vec{0}$, $f$ is translation invariant. In general, it also
follows from (2.1) that $f$ is labeled by the spectral point $\vec{\Lambda}$,
while $\varphi$ is independent of $\vec{\Lambda}$.
## 3 Green’s function
The single-particle Rashba Hamiltonian in $\mathscr{X}$ is realized through
the differential expression $-(2m)^{-1}\Delta+\alpha U$, where
$U:=-i(\nabla_{y}\sigma_{x}-\nabla_{x}\sigma_{y})$. Based on the relation
$U^{2}=-\Delta$, the associated Green’s function can be obtained in an
extremely elegant way [11]. In coordinate representation, it is a linear
combination of the Bessel functions of the second kind. It can be shown [12]
that the single-particle Green’s function is sufficient to recover the
spectrum of the two-particle Rashba Hamiltonian with point-interaction. The
bound state $\lambda$ corresponds to the Fourier transform of the Hamiltonian
with $\vec{Q}=\vec{0}$. More precisely, given $m=1/2$, $\alpha>0$, $\gamma<0$,
a single bound state is found to be equal to
$\lambda=-\alpha^{2}/(2\sin^{2}\omega)$, where the parameter $-\pi/2<\omega<0$
solves the transcendental equation
$j_{\gamma}+\ln(\alpha/4)=\ln\lvert\sin\omega\rvert-\omega\lvert\tan\omega\rvert$,
where one defines $j_{\gamma}:=4\pi/\lvert\gamma\rvert-\Psi(1)$, and $\Psi(1)$
is the digamma function. The result is in exact agreement with the associated
self-consistency condition (see the integral in [6, eq. (31)]) obtained by
considering the free two-particle Green’s function in momentum representation.
Moreover, the above equation allows one to evaluate the characteristic radius
of the interaction potential which is found to be $\exp(-\Psi(1))/2\simeq
0.89$ (all $\alpha>0$, all $\gamma<0$).
Let $g_{\mu}^{0}(\vec{Q},\vec{k})$ be the free two-particle Green’s function
in momentum representation projected onto the singlet basis [6];
$\mu\in\mathbb{C}\backslash[-\alpha^{2}m,\infty)$. Then $g_{\mu}^{0}$ is a
bounded everywhere defined function on $\mathbb{R}^{4}$. The function
$g_{\mu}(\vec{Q},\vec{k})$ corresponding to the projected integral kernel of
the two-particle Hamiltonian with point-interaction fulfills the resolvent
identity
$\displaystyle g_{\mu}(\vec{Q},\vec{k})=$ $\displaystyle
g_{\mu}^{0}(\vec{Q},\vec{k})\left(1-\gamma\tilde{g}_{\mu}(Q)\right),\quad\text{where}$
$\displaystyle g_{\mu}^{0}(\vec{Q},\vec{k}):=$
$\displaystyle(\epsilon-\mu)((\epsilon-\mu)^{2}-\alpha^{2}Q^{2})\bigl{(}(\epsilon-\mu)^{4}$
$\displaystyle-4\alpha^{2}m\epsilon(\epsilon-\mu)^{2}+4\alpha^{4}(\vec{Q}\cdot\vec{k})^{2}\bigr{)}^{-1},$
$\displaystyle\epsilon(Q,k):=$
$\displaystyle\frac{k^{2}}{m}+\frac{Q^{2}}{4m},$
$\displaystyle\tilde{g}_{\mu}(Q):=$ $\displaystyle\int_{k\leq
C}\frac{d^{2}\vec{k}}{(2\pi)^{2}}\;g_{\mu}(\vec{Q},\vec{k}).$ (3.1)
The condition $k\leq C$, where $C$ is the UV cutoff, ensures that the integral
exists and the obtained natural logarithm is finite.
It appears from the resolvent identity that the (projected) Green’s function
$g_{\mu}(\vec{Q},\vec{k})=\frac{g_{\mu}^{0}(\vec{Q},\vec{k})}{1+\gamma\tilde{g}_{\mu}^{0}(Q)}$
(3.2)
where $\tilde{g}_{\mu}^{0}$ is defined similar to $\tilde{g}_{\mu}$ but with
$g_{\mu}$ replaced by $g_{\mu}^{0}$. Then the denominator of the $g_{\mu}$
determines the bound state solution unless
$\lvert(\partial/\partial\mu)\tilde{g}_{\mu}^{0}\rvert$ is sufficiently large
at $\mu=\lambda$ (where $\lambda$ solves $1+\gamma\tilde{g}_{\lambda}^{0}=0$).
We use the notion <<sufficiently large>> rather than <<infinite>> because one
accounts for $k\leq C$ but not for $k<\infty$ in the integral in (3.1). By
(3.2), the bound state solution is a function of the center-of-mass momentum,
$\lambda=\lambda(Q)$. The solution agrees with that obtained from the single-
particle Green’s function only if $Q=0$. By using a more general
Aronszajn–Donoghue spectral theory, it will be shown hereafter that the values
of $Q$ other than zero do not represent bound state solutions.
## 4 Nevanlinna function
In virtue of (3.1), equation (3.2) points to
$\tilde{g}_{\mu}=\tilde{g}_{\mu}^{0}/(1+\gamma\tilde{g}_{\mu}^{0})$. In turn,
the expression for $\tilde{g}_{\mu}$ alludes to a well-known Aronszajn–Krein
formula for the symmetric rank one bounded perturbation of a self-adjoint
operator in the Hilbert space. Without referring to the extension theory of
symmetric operators, one can show [8, Theorem 1.1.1] that, if given a self-
adjoint operator $H$, then the subtraction of the resolvents of the perturbed
operator $H_{\gamma}:=H+\gamma\langle\delta,\cdot\rangle\delta$ and that of
$H$ is proportional to $1/(1+\gamma F_{\mu})$, where
$F_{\mu}:=\langle\delta,(H-\mu)^{-1}\delta\rangle$ is known as the Borel
transform of a measure, and it belongs to the Nevanlinna class (that is, the
complex conjugate $\overline{F_{\mu}}=F_{\overline{\mu}}$ and the imaginary
part $\operatorname{Im}F_{\mu}/\operatorname{Im}\mu\geq 0$, provided $\mu$ is
in the resolvent set of $H_{\gamma}$). Here $\delta$ is the Dirac distribution
and $\langle\cdot,\cdot\rangle$ denotes the scalar product in a concrete
Hilbert space.
Let us study the function $F_{\mu}$, applied to our case, in a more detailed
fashion. For this, let $H$ be a free two-particle Rashba Hamiltonian in the
Hilbert space $\mathscr{X}_{o}$. $H$ is parametrized in terms of the center-
of-mass $\vec{R}$ and the relative $\vec{r}$ coordinates. Then the $s$-wave
interaction can be estimated using the Dirac distribution $\delta$ at the
relative coordinate $\vec{r}\in\mathbb{R}^{2}$ (the interaction strength is
$\gamma$). As a result, the form sum $H_{\gamma}$ describes the two-particle
system with point-interaction. To see this explicitly, it suffices to note
that for $f=f(\vec{R},\vec{r})$ in the domain of $H$, it holds
$\langle\delta,f\rangle\delta=f(\vec{R},\vec{0})\delta=f\delta$.
The Aronszajn–Krein formula reads [9, Eq. (11.13)]
$(H_{\gamma}-\mu)^{-1}=(1+\gamma F_{\mu})^{-1}(H-\mu)^{-1}.$ (4.1)
It takes little effort to verify that equation (4.1) eventually leads to the
expression of the Green’s function $g_{\mu}$ in (3.2) but with
$\tilde{g}_{\mu}^{0}$ replaced by $\tilde{F}_{\mu}$, where $\tilde{F}_{\mu}$
denotes the Nevanlinna function $F_{\mu}$ projected onto the singlet basis. By
inspection, the Fourier transform
$\widehat{((H-\mu)^{-1}\delta)}(\vec{Q},\vec{k})=(2\pi)^{2}\delta(\vec{Q})\hat{G}_{\mu}^{0}(\vec{0},\vec{k}),$
(4.2)
where $\hat{G}_{\mu}^{0}$ is the Green’s function of $H$ in momentum
representation. The projection of the Green’s function onto the singlet basis
is given by $g_{\mu}^{0}$. By (4.2) and the fact that the norm of arbitrary
$f\in\mathscr{X}_{o}$ coincides with the norm of its Fourier transform, one
easily deduces that the Nevanlinna function
$\tilde{F}_{\mu}=\tilde{g}_{\mu}^{0}(0).$ (4.3)
Bringing together (3.2) and (4.3), one figures out that the zero-range
interaction $\delta$ admits only the bound state that corresponds to $Q=0$,
while the free Green’s function $g_{\mu}^{0}$ (recall the definition in (3.1))
accepts all possible values $\vec{Q}\in\mathbb{R}^{2}$. The affirmation of the
result for other types of spin-orbit coupling comes from the Nevanlinna
function, for the reason that the Green’s function was not specified when
deriving (4.3).
The obtained result, (4.3), is not accidental, as it is closely related to the
eigenvectors obeying the translation symmetry.
## 5 Eigenspace
The fact that the bound state of two interacting Rashba particles exists only
for the zero-valued center-of-mass momentum can be affirmed by the commutation
property of the Hamiltonian $H_{\gamma}$ with the generator $\vec{P}$ of the
translation group. The commutation relation designates the spectral points
$\vec{\Lambda}\in\mathbb{R}^{2}$ of $\vec{P}$ to the eigenspace of
$H_{\gamma}$. As a result, the eigenvectors of the Hamiltonian are of the form
(2.1). Since the Hamiltonian is defined as the operator in the Hilbert space
$\mathscr{X}_{o}$, it should be emphasized that in this case, equation (2.1)
(as well as (2.2)) is understood in the generalized sense; by the density
result [10], the Schwartz space can be extended to the appropriate domain of
definition of $H_{\gamma}$.
Let $\Omega$ be the projection onto the singlet basis. In virtue of
(2.1)–(2.2), vectors $f$ in the eigenspace
$\operatorname{Ker}(H_{\gamma}-\lambda)$ fulfill the eigenvalue equation
$\hat{\varphi}(\vec{k})+\gamma\hat{G}_{\lambda}^{0}(\vec{\Lambda},\vec{k})\int_{\mathbb{R}^{2}}\frac{d^{2}\vec{k}^{\prime}}{(2\pi)^{2}}\Omega\hat{\varphi}(\vec{k}^{\prime})=0.$
(5.1)
By the fact that $\varphi$ is independent of $\vec{\Lambda}$, equation (5.1)
yields $(\partial/\partial\vec{\Lambda})\hat{G}_{\lambda}^{0}=\vec{0}$. That
is, $\hat{G}_{\lambda}^{0}(\vec{\Lambda},\vec{k})$ is a constant w.r.t.
$\vec{\Lambda}\in\mathbb{R}^{2}$. The only one nontrivial solution to the
latter equation w.r.t. $\vec{\Lambda}\in\mathbb{R}^{2}$ reads $\Lambda=0$. By
(2.1), this implies that $f=\varphi$ or else, the space of translation
invariant eigenvectors is reduced from $L_{2}(\mathbb{R}^{4};\mathbb{C}^{4})$
to $L_{2}(\mathbb{R}^{2};\mathbb{C}^{4})$. The cancellation of one spatial
vector becomes clear recalling that $\mathscr{X}_{o}$ is isomorphic to the
tensor product of spaces
$(\mathscr{X},d^{2}\vec{R})\text{\large$\otimes$}(\mathscr{X},d^{2}\vec{r})$.
The Hamiltonian in the Hilbert space $(\mathscr{X},d^{2}\vec{R})$ is
parametrized in terms of the center-of-mass coordinate $\vec{R}$ and it does
not have bound states, while the Hamiltonian in the space
$(\mathscr{X},d^{2}\vec{r})$ is parametrized in terms of the relative
coordinate $\vec{r}$ and it has a bound state due to the Dirac distribution at
$\vec{r}$. This latter space is exactly the one where the eigenspace of the
two-particle Rashba Hamiltonian is situated.
## 6 Conclusion and discussion
An attempt to clarify the issue regarding the bound state of spin-orbit
coupled fermions was the primary intent of this note. The results of this
letter clearly demonstrate that the eigenvector of the two-particle Rashba
Hamiltonian with point-interaction $\delta$ is translation invariant. The
associated bound state exists only for the zero-valued center-of-mass momentum
$Q$.
The main conclusion of the present report is that, if the two-particle
Hamiltonian $H_{\gamma}$, where $\gamma$ is the strength of interaction,
possesses translation symmetry, then for any two-particle interaction of rank
one, the bound state exists only for the zero-valued center-of-mass momentum
$Q$. The same applies to other types of spin-orbit coupling. On the other
hand, if the Hamiltonian is not translation invariant, the conclusion does not
necessarily hold. To explain this, let us give some examples.
The rank one perturbation to the free Hamiltonian $H$ is characterized by the
term $\gamma\langle\phi,\cdot\rangle\phi$, where $\phi$ need not be the unit
vector in $\mathscr{X}_{o}$; it even need not be in the Hilbert space [8, 9,
13]. In this case the Nevanlinna function $F_{\mu}$ reads
$\langle\phi,(H-\mu)^{-1}\phi\rangle$. For the proper potential (that is, the
vector $\phi$), one can possibly derive bound states with nonzero $Q$. For
example, let $\phi$ be summable on rectangle
$\mathbb{R}^{2}\times\mathbb{R}^{2}$. Then an easy calculation shows that
$\widehat{((H-\mu)^{-1}\phi)}(\vec{Q},\vec{k})=\hat{\phi}(\vec{Q},\vec{k})\hat{G}_{\mu}^{0}(\vec{Q},\vec{k}),$
(6.1)
where $\hat{\phi}$ is the Fourier transform of $\phi=\phi(\vec{R},\vec{r})$.
It is clear from (6.1) that whenever $\phi(\vec{R},\vec{r})=\phi(\vec{r})$,
that is, when the two-particle Hamiltonian is invariant under translations, it
always holds
$\widehat{((H-\mu)^{-1}\phi)}(\vec{Q},\vec{k})=(2\pi)^{2}\delta(\vec{Q})\hat{\phi}(\vec{k})\hat{G}_{\mu}^{0}(\vec{0},\vec{k})$
(6.2)
hence $Q=0$, no matter what type of interaction between two particles is
specified, viz. zero-range or short-range. In particular, choosing
$\phi=\delta$, (6.2) formally coincides with (4.2). On the other hand, if, for
example, given the vector
$\phi(\vec{R},\vec{r})=\exp(i(\vec{a}\cdot\vec{R}))\delta(\vec{r})$ for some
$\vec{a}\in\mathbb{R}^{2}$, one obtains from (6.1) that $\vec{Q}=\vec{a}$, and
thus the bound state depends on the controllable parameter $a$; in this case
the two-particle Hamiltonian is not translation invariant and the (projected)
Nevanlinna function $\tilde{F}_{\mu}=\tilde{g}_{\mu}^{0}(a)$.
### Acknowledgments
The work was supported by the Research Council of Lithuania (No.
VP1-3.1-ŠMM-01-V-02-004). The author thanks B. M. Anderson for enlightening
him with certain aspects related to the self-consistency condition [6]. It is
a pleasure to thank G. Juzeliūnas for discussions. The author is very grateful
to anonymous referee for his careful reading of the manuscript and the remarks
which helped to improve it.
## References
* [1] J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B 83, 094515 (2011)
* [2] Z.-Q. Yu and H. Zhai, Phys. Rev. Lett. 107, 195305 (2011)
* [3] M. Iskin and A. L. Subąsi, Phys. Rev. A 84, 043621 (2011)
* [4] L. Jiang, X.-J. Liu, H. Hu, and H. Pu, Phys. Rev. A 84, 063618 (2011)
* [5] X. Cui, Phys. Rev. A 85, 022705 (2012)
* [6] S. Takei, C.-H. Lin, B. M. Anderson, and V. Galitski, Phys. Rev. A 85, 023626 (2012)
* [7] L. Dong, L. Jiang, H. Hu, and H. Pu, Phys. Rev. A 87, 043616 (2013)
* [8] S. Albeverio and P. Kurasov, _Singular Perturbations of Differential Operators_ (Cambridge University Press (New York), 2000)
* [9] B. Simon, _Trace Ideals and Their Applications_ , 2nd ed. (American Mathematical Society, 2005)
* [10] P. K. Bhattacharyya, _Distributions. Generalized Functions with Applications in Sobolev Spaces_ (Walter de Gruyter GmbH, Berlin, Germany, 2012)
* [11] J. Brüning, V. Geyler, and K. Pankrashkin, J. Phys. A: Math. Theor. 40, F697 (2007)
* [12] R. Juršėnas, in preparation (2014)
* [13] K. Schmüdgen, _Unbounded Self-adjoint Operators on Hilbert Space_ (Springer Dordrecht Heidelberg New York London, 2012)
|
arxiv-papers
| 2014-04-04T17:21:24 |
2024-09-04T02:50:00.751842
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rytis Jursenas",
"submitter": "Rytis Jursenas Dr.",
"url": "https://arxiv.org/abs/1404.1316"
}
|
1404.1567
|
# On the exponent set of nonnegative primitive tensors 111P. Yuan’s research
is supported by the NSF of China (Grant No. 11271142) and the Guangdong
Provincial Natural Science Foundation(Grant No. S2012010009942), L. You’s
research is supported by the Zhujiang Technology New Star Foundation of
Guangzhou (Grant No. 2011J2200090) and Program on International Cooperation
and Innovation, Department of Education, Guangdong Province (Grant
No.2012gjhz0007).
Zilong He222Email address: [email protected]. Pingzhi Yuan333Corresponding author:
[email protected]. Lihua You444Email address: [email protected].
(School of Mathematical Sciences, South China Normal University,
Guangzhou, 510631, P.R. China
)
###### Abstract
In this paper, we present a necessary and sufficient condition for a
nonnegative tensor to be a primitive one, show that the exponent set of
nonnegative primitive tensors with order $m(\geq n)$ and dimension $n$ is
$\\{k|1\leq k\leq(n-1)^{2}+1\\}.$ AMS classification: 05C50; 15A69 Keywords:
tensor; the exponent set; primitive tensor; primitive degree.
## 1 Introduction
An $n\times n$ nonnegative square matrix $A=(a_{ij})$ is nonnegative primitive
(or simply, primitive) if $A^{k}>0$ for some positive integer $k$. The least
such $k$ is called the primitive exponent (or simply, exponent) of $A$ and is
denoted by $\gamma(A)$.
Let $a,b,n$ be positive integers with $b>a$, $[a,b]^{o}=\\{k|k\mbox{ is an
integer and }a\leq k\leq b\\}$, $[n]=[1,n]^{o}=\\{1,2,\ldots,n\\}$,
$E_{n}=\\{k|\mbox{ there exists a primitive matrix }A\mbox{ of order }n\mbox{
such that }$ $\gamma(A)=k\\}.$
In 1950, H. Wielandt [9] first stated the sharp upper bound for $\gamma(A)$,
that is, $\gamma(A)\leq w_{n}=(n-1)^{2}+1$ for all $n\times n$ primitive
matrices and showed that $E_{n}\subseteq[1,w_{n}]^{o}$. In 1964, A. L. Dulmage
and N. S. Mendelsohn [3] revealed the so-called gaps in the exponent set of
primitive matrices, that is, $E_{n}\subset[1,w_{n}]^{o}$, where “gap” is a set
of consecutive integers $[a,b]^{o}(\subset[1,w_{n}]^{o})$, such that no
$n\times n$ matrix $A$ satisfying $\gamma(A)\in[a,b]^{o}$. In 1981, M. Lewin
and Y. Vitek [4] found all gaps in
$[\lfloor\frac{1}{2}w_{n}\rfloor+1,w_{n}]^{o}$, and conjectured that
$[1,\lfloor\frac{1}{2}w_{n}\rfloor]^{o}$ has no gaps, where $\lfloor x\rfloor$
denotes the greatest integer $\leq x$. In 1985, Shao [10] proved that this
Lewin-Vitek Conjecture is true for all sufficiently large $n$, and the
conjecture has one counterexample when $n=11$ since $48\not\in E_{11}$.
Finally, in 1987, Zhang [13] continued and completed the work. He showed that
the Lewin-Vitek Conjecture holds for all $n$ except $n=11$. Thus the exponent
set $E_{n}$ for primitive matrices of order $n$ is completely determined.
Since the work of Qi [8] and Lim [5], the study of tensors and the spectra of
tensors (and hypergraphs) and their various applications have attracted much
attention and interest. In [1] and [2], Chang et al. defined the
irreducibility of tensors, the primitivity of nonnegative tensors, and
extended many important properties of (nonnegative) primitive matrices to
(nonnegative) primitive tensors. As an application of the general tensor
product defined by Shao [11], Shao presented a simple characterization of the
primitive tensors in terms of the zero pattern of the powers of $\mathbb{A}$.
He also proposed a conjecture on the primitive degree $\gamma(\mathbb{A})$.
Recently, the authors [12] confirmed the conjecture of Shao by proving Theorem
1.1.
###### Theorem 1.1.
([12] Theorem 1.2) Let $\mathbb{A}$ be a nonnegative primitive tensor with
order $m$ and dimension $n$. Then its primitive degree
$\gamma(\mathbb{A})\leq(n-1)^{2}+1$, and the upper bound is sharp.
Therefore, it is natural to consider the question to completely determine the
exponent set of primitive tensors with order $m$ and dimension $n$. Let
$m(\geq 2),n$ be positive integers and
$E(m,\,n)=\\{k|\mbox{ there exists a primitive tensor }\mathbb{A}\mbox{ of
order }m\mbox{ and dimension }n\mbox{ such that }k=\gamma(\mathbb{A})\\}.$
The main result of this paper is as follows.
###### Theorem 1.2.
Let $m,n$ be positive integers with $m\geq n\geq 3$. Then
$E(m,\,n)=[1,(n-1)^{2}+1]^{o}$.
The above theorem shows that there are no gaps in tensor case when $m\geq
n\geq 3$. It is well-known that there exist gaps when $m=2$, therefore it is
an interesting open problem to determine whether there exist gaps when $3\leq
m<n$.
The arrangement of the paper is as follows. In Section 2, we will give some
definitions and properties. In Section 3, we present a necessary and
sufficient condition for a nonnegative tensor to be a primitive one (Theorem
3.4). In Section 4, We prove the main result (Theorem 1.2).
## 2 Preliminaries
An order $m$ dimension $n$ tensor $\mathbb{A}=(a_{i_{1}i_{2}\cdots
i_{m}})_{1\leq i_{j}\leq n\hskip 5.69046pt(j=1,\ldots,m)}$ over the complex
field $\mathbb{C}$ is a multidimensional array with all entries
$a_{i_{1}i_{2}\cdots
i_{m}}\in\mathbb{C}\,(i_{1},\ldots,i_{m}\in[n]=\\{1,\ldots,n\\})$. The
majorization matrix $M(\mathbb{A})$ of the tensor $\mathbb{A}$ is defined as
$(M(\mathbb{A}))_{ij}=a_{ij\cdots j}(i,j\in[n])$ by Pearson [6].
Let $\mathbb{A}$ (and $\mathbb{B}$) be an order $m\geq 2$ (and $k\geq 1$),
dimension $n$ tensor, respectively. In [11], Shao defines a general product
$\mathbb{A}\mathbb{B}$ to be the following tensor $\mathbb{D}$ of order
$(m-1)(k-1)+1$ and dimension $n$:
$d_{i\alpha_{1}\cdots\alpha_{m-1}}=\sum\limits_{i_{2},\ldots,i_{m}=1}^{n}a_{ii_{2}\cdots
i_{m}}b_{i_{2}\alpha_{1}}\cdots
b_{i_{m}\alpha_{m-1}}\quad(i\in[n],\,\alpha_{1},\ldots,\alpha_{m-1}\in[n]^{k-1}).$
The tensor product possesses a very useful property: the associative law ([11]
Theorem 1.1).
In [2] and [7], Chang et al. and Pearson define the primitive tensors as
follows.
###### Definition 2.1.
([2, 7]) Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and
dimension $n$, $x=(x_{1},x_{2},\ldots,x_{n})^{T}\in\mathbb{R}^{n}$ a vector
and $x^{[r]}=(x_{1}^{r},x_{2}^{r},\ldots,x_{n}^{r})^{T}$. Define the map
$T_{\mathbb{A}}$ from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ as:
$T_{\mathbb{A}}(x)=(\mathbb{A}x)^{[\frac{1}{m-1}]}$. If there exists some
positive integer $r$ such that $T_{\mathbb{A}}^{r}(x)>0$ for all nonnegative
nonzero vectors $x\in\mathbb{R}^{n}$, then $\mathbb{A}$ is called primitive
and the smallest such integer $r$ is called the primitive degree of
$\mathbb{A}$, denoted by $\gamma(\mathbb{A})$.
In [11], Shao shows the following results and defines the primitive degree by
using the properties of tensor product and the zero patterns.
###### Proposition 2.2.
([11] Theorem 4.1) A nonnegative tensor $\mathbb{A}$ is primitive if and only
if there exists some positive integer $r$ such that $\mathbb{A}^{r}$ is
essentially positive. Furthermore, the smallest such $r$ is the primitive
degree of $\mathbb{A}$, $\gamma(\mathbb{A})$.
In [12], the authors prove some necessary conditions for a nonnegative tensor
to be a primitive one, and they also prove the following result.
###### Proposition 2.3.
([12] Remark 2.6) Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and
dimension $n$. Then $\mathbb{A}$ is primitive if and only if there exists some
positive integer $r$ such that $M(\mathbb{A}^{r})>0.$ Furthermore, the
smallest such $r$ is the primitive degree of $\mathbb{A}$,
$\gamma(\mathbb{A})$.
In [12], the authors introduce some theoretical concepts of digraphs and
matrices.
Let $D=(V,A)$ denote a digraph on $n$ vertices. Loops are permitted, but no
multiple arcs. A $u\rightarrow v$ walk in $D$ is a sequence of vertices
$u,u_{1},\ldots,u_{k}=v$ and a sequence of arcs
$e_{1}=(u,u_{1}),e_{2}=(u_{1},u_{2}),\ldots,e_{k}=(u_{k-1},v)$, where the
vertices and the arcs are not necessarily distinct. We use the notation
$u\rightarrow u_{1}\rightarrow u_{2}\rightarrow\cdots\rightarrow
u_{k-1}\rightarrow v$ to refer to this $u\rightarrow v$ walk. A closed walk is
a $u\rightarrow v$ walk where $u=v$. A path is a walk with distinct vertices.
A cycle is a closed $u\rightarrow v$ walk with distinct vertices except for
$u=v$. The length of a walk $W$ is the number of arcs in $W$, denoted by
$l(W)$. A $k$-cycle is a cycle of length $k$, denoted by $C_{k}$.
###### Definition 2.4.
([12] Definition 2.9) Let $D=(V,A)$ denote a digraph on $n$ vertices. A
digraph $D^{\prime}=(V,A^{\prime})$ is called the reversed digraph of $D$
where $(j,i)\in A^{\prime}$ if and only if $(i,j)\in A$ for any $i,j\in V$,
denoted by $\overleftarrow{D}$.
Let $M=(m_{ij})$ be a square nonnegative matrix of order $n$. The associated
digraph $D(M)=(V,A)$ of $M$ (possibly with loops) is defined to be the digraph
with vertex set $V=\\{1,2,\ldots,n\\}$ and arc set $A=\\{(i,j)|m_{ij}\neq
0\\}$. The associated reversed digraph $\overleftarrow{D(M)}=(V,A^{\prime})$
of $M$ (possibly with loops) is defined to be the digraph with vertex set
$V=\\{1,2,\ldots,n\\}$ and arc set $A=\\{(j,i)|m_{ij}\neq 0\\}$. Clearly, the
associated reversed digraph of $M$ is the reversed digraph of the associated
digraph of $M$.
Let $N^{+}_{D}(i)=\\{j\in V(D)|(i,j)\in E(D)\\}$ denote the out-neighbors of
$i$ and $d^{+}_{i}=|N^{+}_{D}(i)|$ denote the out-degree of the vertex $i$ in
$D$. The following Proposition is the graph theoretical version of Proposition
2.7 in [12].
###### Proposition 2.5.
([12] Proposition 2.7) Let $\mathbb{A}$ be a nonnegative primitive tensor with
order $m$ and dimension $n$, $M(\mathbb{A})$ the majorization matrix of
$\mathbb{A}$. Then in the digraph
$\overleftarrow{D(M(\mathbb{A}))}=(V,A^{\prime})$, we have:
(i) For each $j\in V$, the out-degree of the vertex $j$, $d^{+}_{j}\geq 1$
and $N^{+}_{D}(j)\neq\\{j\\}$.
(ii) There exists at least a vertex $j\in V$, the out-degree of the vertex
$j$, $d^{+}_{j}\geq 2$.
The authors [12] also presented the definition of $j$-primitive degree for a
nonnegative tensor and obtained the following result.
###### Definition 2.6.
([12] Definition 2.13) Let $\mathbb{A}$ be a nonnegative tensor with order $m$
and dimension $n$. For a fixed integer $j\in[n]$, if there exists a positive
integer $k$ such that
$(M(\mathbb{A}^{k}))_{uj}>0,\hskip 5.69046pt{\mbox{f}or\,\,all}\,u\in[n],$
then $\mathbb{A}$ is called $j$-primitive and the smallest such integer $k$ is
called the $j$-primitive degree of $\mathbb{A}$, denoted by
$\gamma_{j}(\mathbb{A})$.
###### Proposition 2.7.
([12] Proposition 2.14) Let $\mathbb{A}$ be a nonnegative primitive tensor
with order $m$ and dimension $n$. Then $\gamma(\mathbb{A})=\max\limits_{1\leq
j\leq n}\\{\gamma_{j}(\mathbb{A})\\}.$
## 3 Necessary and sufficient conditions for nonnegative tensors to be
primitive
In this section, we will present necessary and sufficient conditions for a
nonnegative tensor to be a primitive one. Let $\mathbb{A}$ be a nonnegative
tensor with order $m$ and dimension $n$. For positive integers $k$ and
$j\in[n]$, we put
$S_{k}(\mathbb{A},j)=\\{u\in[n]\hskip 2.27626pt|\hskip
2.27626pt(M(\mathbb{A}^{k}))_{uj}>0\\},k=1,2,\ldots.$
Then $S_{1}(\mathbb{A},j)=\\{u\in[n]\hskip 2.27626pt|\hskip
2.27626pt(M(\mathbb{A}))_{uj}>0\\}=N_{\overleftarrow{D(M(\mathbb{A}))}}^{+}(j)$.
By the definitions of the general tensor product and the majorization matrix
of a tensor, we can show that
$(M(\mathbb{A}^{k+1}))_{uj}=\sum\limits_{j_{2},\ldots,j_{m}=1}^{n}a_{uj_{2}\cdots
j_{m}}(M(\mathbb{A}^{k}))_{j_{2}j}\cdots(M(\mathbb{A}^{k}))_{j_{m}j},$ (3.1)
it follows that $u\in S_{k+1}(\mathbb{A},j)$ if and only if there exist
indices $j_{2},\ldots,j_{m}\in S_{k}(\mathbb{A},j)$ and $a_{uj_{2}\cdots
j_{m}}>0$. Thus
$S_{k+1}(\mathbb{A},j)=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there
exist }j_{2},\ldots,j_{m}\in S_{k}(\mathbb{A},j)\mbox{ and }a_{uj_{2}\cdots
j_{m}}>0\\}.$ (3.2)
By the definition of $S_{k}(\mathbb{A},j)$, we have the following result.
###### Lemma 3.1.
Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$.
(i) Let $k,l,i,j$ be positive integers such that $1\leq i,j\leq n$. Suppose
that $S_{k}(\mathbb{A},i)=S_{l}(\mathbb{A},j)$, then
$S_{k+r}(\mathbb{A},i)=S_{l+r}(\mathbb{A},j)$ holds for every positive integer
$r$.
(ii) For any $j\in[n]$, let $k$ be the least positive integer such that
$S_{k}(\mathbb{A},j)=[n]$. Then for any integer $l\geq k$,
$S_{l}(\mathbb{A},j)=[n]$.
###### Proof.
(i) Since $S_{k}(\mathbb{A},i)=S_{l}(\mathbb{A},j)$, by (3.2), we have
$S_{k+1}(\mathbb{A},i)=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there
exist }j_{2},\ldots,j_{m}\in S_{k}(\mathbb{A},i)\mbox{ and }a_{uj_{2}\cdots
j_{m}}>0\\}$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exist
}j_{2},\ldots,j_{m}\in S_{l}(\mathbb{A},j)\mbox{ and }a_{uj_{2}\cdots
j_{m}}>0\\}$
$=S_{l+1}(\mathbb{A},j).$
Therefore (i) follows by induction on $r$.
(ii) Let $k$ be the least positive integer such that
$S_{k}(\mathbb{A},j)=[n]$. We complete the proof by the following two cases.
Case 1: $k>1$.
Then $S_{k-1}(\mathbb{A},j)\subset[n]$, by (3.2), we have
$[n]=S_{k}(\mathbb{A},j)$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exist
}j_{2},\ldots,j_{m}\in S_{k-1}(\mathbb{A},j)\subset[n]\mbox{ and
}a_{uj_{2}\cdots j_{m}}>0\\}$
$\subseteq\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exist
}j_{2},\ldots,j_{m}\in S_{k}(\mathbb{A},j)=[n]\mbox{ and }a_{uj_{2}\cdots
j_{m}}>0\\}$
$=S_{k+1}(\mathbb{A},j)\subseteq[n]$.
Thus $S_{k+1}(\mathbb{A},j)=[n],$ and we can show $S_{k+r}(\mathbb{A},j)=[n]$
for any nonnegative integer $r$ by induction on $r$.
Case 2: $k=1$.
Then $S_{1}(\mathbb{A},j)=[n]$, i.e., $(M(\mathbb{A}))_{uj}>0$ for all
$u\in[n]$. By taking $j_{2}=j_{3}=\cdots=j_{m}=j$, it is easy to show that
$S_{2}(\mathbb{A},j)=[n]$. Therefore $S_{l}(\mathbb{A},j)=[n]$ for any
positive integer $l$ by induction on $l$. ∎
###### Remark 3.2.
Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$.
Then for any $j\in[n]$, $\gamma_{j}(\mathbb{A})$ is the least positive integer
$k$ satisfying $S_{k}(\mathbb{A},j)=[n]$ by Definition 2.6.
Denote $e_{j}$ the vector (of dimension $n$) whose $j$th component is $1$ and
others are $0$ for any $j=1,\ldots,n$. Let $x_{j}^{(0)}=e_{j}$ and for any
nonnegative integer $k$, we define
$x^{(k+1)}_{j}=\mathbb{A}x^{(k)}_{j}.$
By the definition of the general tensor product by Shao, we know
$x_{j}^{(0)},x_{j}^{(1)},\ldots$ are vectors (of dimension $n$). Let
$T_{k}(\mathbb{A},j)=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{the
$u$th component of }x^{(k)}_{j}\mbox{ is larger than 0}\\}.$
By the definition of $x_{j}^{(1)}$, for $u\in[n]$, we have
$\left(x_{j}^{(1)}\right)_{u}=a_{uj\cdots j}=(M(\mathbb{A}))_{uj},$
it follows that $S_{1}(\mathbb{A},j)=T_{1}(\mathbb{A},j)$ by the definitions
of $S_{1}(\mathbb{A},j)$ and $T_{1}(\mathbb{A},j)$. Since
$\left(x^{(k+1)}_{j}\right)_{u}=\sum_{i_{2},\ldots,i_{m}=1}^{n}a_{ui_{2}\cdots
i_{m}}\left(x^{(k)}_{j}\right)_{i_{2}}\cdots\left(x^{(k)}_{j}\right)_{i_{m}},$
it follows that $\left(x^{(k+1)}_{j}\right)_{u}>0$ if there exist some indices
$i_{2},\ldots,i_{m}\in T_{k}(\mathbb{A},j)$ and $a_{ui_{2}\cdots i_{m}}>0$,
thus $S_{k+1}(\mathbb{A},j)=T_{k+1}(\mathbb{A},j)$. Now by induction on $k$
and the definitions of $S_{k}(\mathbb{A},j)$ and $T_{k}(\mathbb{A},j)$, we see
that $S_{k}(\mathbb{A},j)=T_{k}(\mathbb{A},j)$ for all $k$ and $j\in[n]$.
Therefore, we have
###### Theorem 3.3.
Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$. For
a fixed integer $j\in[n]$, $\mathbb{A}$ is $j$-primitive if and only if there
exists some positive integer $k$ such that $x^{(k)}_{j}>0$ and the smallest
such integer $k$ is the $j$-primitive degree $\gamma_{j}(\mathbb{A})$.
###### Theorem 3.4.
Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$.
Then $\mathbb{A}$ is primitive if and only if there exists some positive
integer $k$ such that $x^{(k)}_{j}>0$ for all $j\in[n]$. Furthermore, the
smallest such integer $k$ is the primitive degree $\gamma(\mathbb{A})$ and
$\gamma(\mathbb{A})\leq(n-1)^{2}+1$.
###### Proof.
By Proposition 2.3, Definition 2.6 and Theorem 3.3, we know that
$\mathbb{A}$ is primitive
$\Longleftrightarrow$ there exists some positive $k$ such that
$M(\mathbb{A}^{k})>0$
$\Longleftrightarrow$ for all $j\in[n]$, there exists some positive $k$ such
that $(M(\mathbb{A}^{k}))_{uj}>0$ for all $u\in[n]$
$\Longleftrightarrow$ for all $j\in[n]$, $\mathbb{A}$ is $j$-primitive
$\Longleftrightarrow$ for all $j\in[n]$, there exists some positive integer
$k$ such that $x^{(k)}_{j}>0$.
Thus by Theorem 1.1, Proposition 2.7, Theorem 3.3 and the above arguments, we
know that the smallest such integer $k$ is the primitive degree
$\gamma(\mathbb{A})$ and $\gamma(\mathbb{A})\leq(n-1)^{2}+1$. ∎
## 4 Proof of the main result
By the relation between matrices and digraphs, we know that
$(A^{k})_{uj}>0\Longleftrightarrow\mbox{ there exists a walk of length $k$
from $u$ to $j$ in the digraph }D(A)$
$\Longleftrightarrow\mbox{ there exists a walk of length $k$ from $j$ to $u$
in the digraph }\overleftarrow{D(A)}.$
Then we obtain the following proposition.
###### Proposition 4.1.
Let $A$ be a nonnegative matrix of order $n$, $j\in[n],k$ be positive
integers, $S_{k}(A,j)=\\{u\in[n]\hskip 2.27626pt|\hskip
2.27626pt(A^{k})_{uj}>0\\}$. Then
$S_{k}(A,j)=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exists a
walk of length $k$ from $j$ to $u$ in the digraph }\overleftarrow{D(A)}\\}.$
###### Lemma 4.2.
Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$ such
that $a_{ii_{2}\cdots i_{m}}=0$ if $i_{2}\cdots i_{m}\not=i_{2}\cdots i_{2}$
for any $i\in[n]$. Then for any positive integers $j\in[n]$ and $k$,
$S_{k}(\mathbb{A},j)=S_{k}(M(\mathbb{A}),j).$ (4.1)
###### Proof.
We prove $S_{k}(\mathbb{A},j)=S_{k}(M(\mathbb{A}),j)$ by induction on $k$.
Clearly, $k=1$ is obvious by the fact $S_{1}(\mathbb{A},j)=\\{u\in[n]\hskip
2.27626pt|\hskip
2.27626pt(M(\mathbb{A}))_{uj}>0\\}=N_{\overleftarrow{D(M(\mathbb{A}))}}^{+}(j)=S_{1}(M(\mathbb{A}),j).$
Assume that (4.1) holds for $k=l\geq 1$. Then by (3.1), (3.2) and Proposition
4.1, we have
$S_{l+1}(\mathbb{A},j)$$=\\{u\in[n]\hskip 2.27626pt|\hskip
2.27626pt(M(\mathbb{A}^{l+1}))_{uj}>0\\}$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exist
}j_{2},\ldots,j_{m}\in S_{l}(\mathbb{A},j)\mbox{ and }a_{uj_{2}\cdots
j_{m}}>0\\}$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exist }v\in
S_{l}(\mathbb{A},j)\mbox{ and }a_{uv\cdots v}>0\\}$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exist }v\in
S_{l}(M(\mathbb{A}),j)\mbox{ and }(M(\mathbb{A}))_{uv}>0\\}$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exists a walk of
length $l$ from $j$ to $v$ and arc $(v,u)$ in
}\overleftarrow{D(M(\mathbb{A}))}\\}.$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exists a walk of
length $l+1$ from $j$ to $u$ in }\overleftarrow{D(M(\mathbb{A}))}\\}.$
$=S_{l+1}(M(\mathbb{A}),j)$.
Thus, for any positive integers $j\in[n]$ and $k$,
$S_{k}(\mathbb{A},j)=S_{k}(M(\mathbb{A}),j)$ holds. ∎
###### Lemma 4.3.
([12] Corollary 3.4) Let $\mathbb{A}$ be a nonnegative primitive tensor with
order $m$ and dimension $n$ such that $a_{ii_{2}\cdots i_{m}}=0$ if
$i_{2}\cdots i_{m}\not=i_{2}\cdots i_{2}$ for any $i\in[n]$. If
$M(\mathbb{A})$ is primitive, then $\gamma(\mathbb{A})=\gamma(M(\mathbb{A})).$
Let $M_{1}=\left(\begin{array}[]{ccccc}0&0&\cdots&1&1\\\ 1&0&\cdots&0&0\\\
0&1&\cdots&0&0\\\ 0&0&\ddots&0&0\\\ 0&0&\cdots&1&0\\\ \end{array}\right)$. It
is well known that $M_{1}$ is primitive, and $\gamma(M_{1})=(n-1)^{2}+1$.
$\textstyle{n}$$\textstyle{\hskip
3.0pt{n-1}}$$\textstyle{\mathrm{1}}$$\textstyle{2}$$\textstyle{\hskip
3.0pt{n-2}}$ $\mbox{ \qquad Figure 1. digraph }\overleftarrow{D(M_{1})}$
Let $\mathbb{A}_{0}$ be the nonnegative primitive tensor with order $m$ and
dimension $n$ such that $a_{ii_{2}\cdots i_{m}}=0$ if $i_{2}\cdots
i_{m}\not=i_{2}\cdots i_{2}$ for any $i\in[n]$, and $M(\mathbb{A}_{0})=M_{1}$.
Then by Lemma 4.2, we have $S_{k}(\mathbb{A}_{0},j)=S_{k}(M_{1},j)$ for any
$j\in[n]$ and any positive integer $k$, and by Lemma 4.3, we have
$\gamma(\mathbb{A}_{0})=(n-1)^{2}+1$.
The following result is well known.
###### Proposition 4.4.
Let $a,b$ be positive integers, if $a,b$ are coprime $(g.c.d.(a,b)=1)$, then
equation $ax+by=ab-a-b$ has no nonnegative integral solutions $(x,y)$.
###### Proposition 4.5.
Let $\mathbb{A}_{0}$ be the nonnegative primitive tensor with order $m$ and
dimension $n$ defined as above. Then
$\gamma_{n-1}(\mathbb{A}_{0})=n^{2}-3n+3$.
###### Proof.
By Remark 3.2, Proposition 4.1 and Lemma 4.2, we know
$\gamma_{n-1}(\mathbb{A}_{0})=\min\\{k\hskip 2.27626pt|\hskip
2.27626ptS_{k}(\mathbb{A}_{0},n-1)=[n]\\}$
$=\min\\{k\hskip 2.27626pt|\hskip 2.27626ptS_{k}(M_{1},n-1)=[n]\\}$
$=\min\\{k\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exists a walk of length
$k$ from $n-1$ to $u$ in }\overleftarrow{D(M_{1})}\mbox{ for all }u\in[n]\\}$.
Let $W$ be any walk of length $n^{2}-3n+2$ from vertex $n-1$ to vertex $n$ in
$\overleftarrow{D(M_{1})}$. Then $W$ is a union of the unique path $P$ from
$n-1$ to $n$ (of length 1) and several cycles of length $n-1$ and several
cycles of length $n$. Let $l(W)$ be the length of $W$. Then there exist two
nonnegative integer $a,b$ such that
$n^{2}-3n+2=l(W)=1+na+(n-1)b,\hskip 2.84544pta\geq 0,\hskip 2.84544ptb\geq 0.$
Note that $n,n-1$ are coprime, by Proposition 4.4, we know that equation
$n^{2}-3n+1=nx+(n-1)y$ has no nonnegative integral solutions $(x,y)$. It is a
contradiction. Therefore there does not exist a walk of length $n^{2}-3n+2$
from $n-1$ to $n$ in $\overleftarrow{D(M_{1})}$, it implies
$S_{n^{2}-3n+2}(M_{1},n-1)\not=[n]$ and thus
$\gamma_{n-1}(\mathbb{A}_{0})>n^{2}-3n+2$.
Let $u\in[n]$ be any vertex in $\overleftarrow{D(M_{1})}$, $P$ be the path of
length $l=l(P)$ from vertex $n-1$ to vertex $u$, then
$l=\left\\{\begin{array}[]{ll}0,&\mbox{ if }u=n-1;\\\ 1,&\mbox{ if }u=n;\\\
u\mbox{ or }u+1,&\mbox{ if }u=1,2,\ldots,n-2.\end{array}\right.$
Let $C_{n-1}$ and $C_{n}$ be the cycles of length $n-1$ and $n$ in
$\overleftarrow{D(M_{1})}$, respectively. Take
$W=\left\\{\begin{array}[]{ll}(n-3)C_{n-1}+C_{n},&\mbox{ if }u=n-1;\\\
1+(n-2)C_{n-1},&\mbox{ if }u=1\mbox{ or }u=n;\\\
l+(l-3)C_{n-1}+(n-l)C_{n},&\mbox{ if }u=2,3,\ldots,n-2,\mbox{ where }3\leq
l\leq n-1.\end{array}\right.$
Clearly, the length of $W$, $l(W)=n^{2}-3n+3$. Therefore there exists a walk
of length $n^{2}-3n+3$ from $n-1$ to any $u\in[n]$ in
$\overleftarrow{D(M_{1})}$, it implies $S_{n^{2}-3n+3}(M_{1},n-1)=[n]$ and
thus $\gamma_{n-1}(\mathbb{A}_{0})\leq n^{2}-3n+3$.
Combining the above arguments, we have
$\gamma_{n-1}(\mathbb{A}_{0})=n^{2}-3n+3$. ∎
###### Remark 4.6.
Similar to the proof of Proposition 4.5, take
$W=\left\\{\begin{array}[]{ll}(n-2)C_{n-1},&\mbox{ if }u=n-1;\\\
l+(l-2)C_{n-1}+(n-l-1)C_{n},&\mbox{ if }u=1,2,\ldots,n-2,\mbox{ where }2\leq
l\leq n-1.\end{array}\right.$
We can show that there exists a walk of length $n^{2}-3n+2$ from $n-1$ to any
$u\in[n-1]$ in $\overleftarrow{D(M_{1})}$, it implies that
$S_{n^{2}-3n+2}(M_{1},n-1)=[n-1]$ and there does not exist a walk of length
$n^{2}-3n+2$ from $n-1$ to $n$ in $\overleftarrow{D(M_{1})}$.
Note that $S_{1}(\mathbb{A}_{0},n-1)=\\{1,n\\}$, then for any positive integer
$k$ with $1\leq k\leq n^{2}-3n+2$, we can show that
$2\leq|S_{k}(\mathbb{A}_{0},n-1)|=|S_{k}(M_{1},n-1)|\leq n-1$ by Proposition
4.1, Proposition 4.5 and Remark 4.6.
###### Proposition 4.7.
([11] Corollary 4.1 ) Let $\mathbb{A}$ be a nonnegative tensor with order $m$
and dimension $n$. If $M(\mathbb{A})$ is primitive, then $\mathbb{A}$ is also
primitive.
###### Proposition 4.8.
Let $m,n,k$ be positive integers with $m\geq n\geq 3$ and $1\leq k\leq
n^{2}-3n+2$, $\mathbb{A}_{k}$ be the nonnegative tensor with order $m$ and
dimension $n$ such that for any $i\in[n]$,
$(\mathbb{A}_{k})_{ii_{2}\cdots i_{m}}=\left\\{\begin{array}[]{ll}1,&\mbox{ if
}\\{i_{2},\ldots,i_{m}\\}=S_{k}(\mathbb{A},n-1);\\\
(\mathbb{A}_{0})_{ii_{2}\cdots i_{m}},&otherwise.\end{array}\right.$
Then $M(\mathbb{A}_{k})=M(\mathbb{A}_{0})=M_{1}$ and $\mathbb{A}_{k}$ is
primitive.
###### Proof.
Clearly, $\mathbb{A}_{k}$ is well-defined since $m\geq n$ and
$|S_{k}(\mathbb{A},n-1)|\leq n-1$.
Since $|S_{k}(\mathbb{A},n-1)|\geq 2$ for any $k$, then $i_{2}\ldots
i_{m}\not=i_{2}\ldots i_{2}$ when
$\\{i_{2},\ldots,i_{m}\\}=S_{k}(\mathbb{A},n-1)$, and thus
$M(\mathbb{A}_{k})=M(\mathbb{A}_{0})=M_{1}$ by the definition of the
majorization matrix of a tensor.
Note that $M_{1}$ is primitive, then $\mathbb{A}_{k}$ is primitive by
Proposition 4.7. ∎
###### Proposition 4.9.
Let $k$ be positive integer with $1\leq k\leq n^{2}-3n+2$. Then
$S_{1}(\mathbb{A}_{0},n-1),S_{2}(\mathbb{A}_{0},n-1),\ldots,S_{k}(\mathbb{A}_{0},n-1)$
are pairwise distinct proper subsets of $[n]$.
###### Proof.
Clearly,
$S_{1}(\mathbb{A}_{0},n-1),S_{2}(\mathbb{A}_{0},n-1),\ldots,S_{k}(\mathbb{A}_{0},n-1)$
are proper subsets of $[n]$ by $2\leq|S_{p}(\mathbb{A}_{0},n-1)|\leq n-1$ for
all $p\in[k]$.
If $S_{p}(\mathbb{A}_{0},n-1)=S_{q}(\mathbb{A}_{0},n-1)$ for $1\leq p<q\leq
k$, then by (i), (ii) of Lemma 3.1 and Proposition 4.5, we have
$S_{n^{2}-3n+2}(\mathbb{A}_{0},n-1)=S_{q+(n^{2}-3n+2-p)}(\mathbb{A}_{0},n-1)=S_{n^{2}-3n+3}(\mathbb{A}_{0},n-1)=[n],$
but $S_{n^{2}-3n+2}(\mathbb{A}_{0},n-1)=[n-1]\not=[n]$ by Remark 4.6, it is a
contradiction. Thus
$S_{1}(\mathbb{A}_{0},n-1),S_{2}(\mathbb{A}_{0},n-1),\ldots,S_{k}(\mathbb{A}_{0},n-1)$
are pairwise distinct. ∎
###### Theorem 4.10.
Let $m,n,k$ be positive integers with $m\geq n\geq 3$ and $1\leq k\leq
n^{2}-3n+2$, $\mathbb{A}_{k}$ be the nonnegative primitive tensor defined as
above. Then $\gamma(\mathbb{A}_{k})=\gamma_{n}(\mathbb{A}_{k})=n+k$.
###### Proof.
Firstly, we show that $S_{t}(\mathbb{A}_{k},n-1)=S_{t}(\mathbb{A}_{0},n-1)$
for any positive integer $t\in[k]$ by induction on $t$.
By Proposition 4.8, we have $S_{1}(\mathbb{A}_{k},n-1)=\\{u\in[n]\hskip
2.27626pt|\hskip 2.27626pt(M(\mathbb{A}_{k}))_{u,n-1}>0\\}=\\{u\in[n]\hskip
2.27626pt|\hskip
2.27626pt(M(\mathbb{A}_{0}))_{u,n-1}>0\\}=S_{1}(\mathbb{A}_{0},n-1)=\\{1,n\\}$.
Then $t=1$ holds.
Assume that $S_{t}(\mathbb{A}_{k},n-1)=S_{t}(\mathbb{A}_{0},n-1)$ for $t=l\geq
1$ holds. Then for $t=l+1\leq k$,
$S_{l}(\mathbb{A}_{0},n-1)\not=S_{k}(\mathbb{A}_{0},n-1)$ by Proposition 4.9.
If $i_{2},\ldots,i_{m}\in S_{l}(\mathbb{A}_{0},n-1)$, we have
$(\mathbb{A}_{k})_{ui_{2}\ldots
i_{m}}>0\Longleftrightarrow(\mathbb{A}_{k})_{ui_{2}\ldots
i_{m}}=(\mathbb{A}_{0})_{ui_{2}\ldots i_{m}}>0$ by the definition of
$\mathbb{A}_{k}$. Thus by (3.2),
$S_{l+1}(\mathbb{A}_{k},n-1)=\\{u\in[n]\hskip 2.27626pt|\hskip
2.27626pt\mbox{there exist }i_{2},\ldots,i_{m}\in
S_{l}(\mathbb{A}_{k},n-1)\mbox{ and }(\mathbb{A}_{k})_{ui_{2}\ldots
i_{m}}>0\\}$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exist
}i_{2},\ldots,i_{m}\in S_{l}(\mathbb{A}_{0},n-1)\mbox{ and
}(\mathbb{A}_{k})_{ui_{2}\ldots i_{m}}>0\\}$
$=\\{u\in[n]\hskip 2.27626pt|\hskip 2.27626pt\mbox{there exist
}i_{2},\ldots,i_{m}\in S_{l}(\mathbb{A}_{0},n-1)\mbox{ and
}(\mathbb{A}_{0})_{ui_{2}\ldots i_{m}}>0\\}$
$=S_{l+1}(\mathbb{A}_{0},n-1)$.
It follows that $S_{t}(\mathbb{A}_{k},n-1)=S_{t}(\mathbb{A}_{0},n-1)$ for any
positive integer $t$ with $1\leq t\leq k$, and thus
$S_{1}(\mathbb{A}_{k},n-1),S_{2}(\mathbb{A}_{k},n-1),\ldots,S_{k}(\mathbb{A}_{k},n-1)$
are pairwise distinct proper subsets of $[n]$ for $1\leq k\leq n^{2}-3n+2$ by
Proposition 4.9.
Since $(\mathbb{A}_{k})_{ii_{2}\cdots i_{m}}=1$ for any $i\in[n]$ when the set
$\\{i_{2},\ldots,i_{m}\\}=S_{k}(\mathbb{A}_{k},n-1)$, we have
$S_{k+1}(\mathbb{A}_{k},n-1)=\\{u\in[n]\hskip 2.27626pt|\hskip
2.27626pt\mbox{there exist }j_{2},\ldots,j_{m}\in S_{k}(\mathbb{A},n-1)\mbox{
and }(\mathbb{A}_{k})_{uj_{2}\cdots j_{m}}>0\\}=[n].$
and thus $\gamma_{n-1}(\mathbb{A}_{k})=k+1$ by Remark 3.2.
By the definition of $\mathbb{A}_{k}$, an easy computation shows that
$S_{1}(\mathbb{A}_{k},n-1)=S_{2}(\mathbb{A}_{k},n-2)=S_{3}(\mathbb{A}_{k},n-3)=\cdots=S_{n-1}(\mathbb{A}_{k},1)=S_{n}(\mathbb{A}_{k},n).$
By (i) of Lemma 3.1 and the definition of $j$-primitive degree, we have
$S_{1}(\mathbb{A}_{k},n-1)=S_{2}(\mathbb{A}_{k},n-2)$
$\Longrightarrow\left\\{\begin{array}[]{ll}([n]\not=)S_{1+r}(\mathbb{A}_{k},n-1)=S_{2+r}(\mathbb{A}_{k},n-2),&\mbox{
if }1\leq r\leq k-1;\\\
([n]=)S_{1+k}(\mathbb{A}_{k},n-1)=S_{2+k}(\mathbb{A}_{k},n-2),&\mbox{ if
}r=k.\end{array}\right.$
$\Longrightarrow\gamma_{n-2}(\mathbb{A}_{k})=k+2$.
Similarly, we can prove
$\gamma_{n-3}(\mathbb{A}_{k})=k+3,\ldots,\gamma_{1}(\mathbb{A}_{k})=n+k-1,\gamma_{n}(\mathbb{A}_{k})=n+k.$
Then following form Proposition 2.7, we have
$\gamma(\mathbb{A}_{k})=\max\limits_{1\leq j\leq
n}\gamma_{j}(\mathbb{A}_{k})=\gamma_{n}(\mathbb{A}_{k})=n+k$.∎
Proof of Theorem 1.2: Let $t$ be any positive integer with $1\leq
t\leq(n-1)^{2}+1$, we will show that there exists a nonnegative primitive
tensor $\mathbb{B}_{t}$ with order $m$ and dimension $n$ such that
$\gamma(\mathbb{B}_{t})=t$. We complete the proof by the following two cases.
Case 1: $1\leq t\leq n$.
It is well known that there exists a primitive matrix $A_{t}$ of order $n$
such that $\gamma(A_{t})=t$. We define the tensor $\mathbb{B}_{t}$ to be the
nonnegative primitive tensor with order $m$ and dimension $n$ such that
$(\mathbb{B}_{t})_{ii_{2}\cdots i_{m}}=0$ if $i_{2}\cdots
i_{m}\not=i_{2}\cdots i_{2}$ for any $i\in[n]$, and $M(\mathbb{B}_{t})=A_{t}$.
Then $\gamma(\mathbb{B}_{t})=\gamma(A_{t})=t$ by Lemma 4.3.
Case 2: $n+1\leq t\leq(n-1)^{2}+1$.
We choose $\mathbb{B}_{t}=\mathbb{A}_{t-n}$. Then
$\gamma(\mathbb{B}_{t})=n+(t-n)=t$ by Theorem 4.10. ∎
## References
* [1] K. C. Chang, K. Pearson, and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci. 6(2008), 507–520.
* [2] K. C. Chang, K. Pearson, and T. Zhang, Primitivity, the convergence of the NQZ method, and the largest eigenvalue tensors, SIAM J. Matrix Anal. Appl. 32(2011), 806–819.
* [3] A. L. Dulmage and N. S. Mendelsohn, Gaps in the exponent set of primitive matrices, Zllitwis J. Math. (1964), 8642–656.
* [4] M. Lewin and Y. Vitek, A system of gaps in the exponent set of primitive matrices, Illinois J. Math. 25(1981), 87–98.
* [5] L.H. Lim, Singular values and eigenvalues of tensors, a rational approach, in proceedings 1st IEEE international workshop on computational advances of adaptive processing (2005), 129–132.
* [6] K. Pearson, Essentially positive tensors, Int. J. Algebra 4(2010),421–427.
* [7] K. Pearson, Primitive tensors and convergence of an iterative process for the eigenvalues of a primitive tensor, arXiv: 1004-2423v1, 2010.
* [8] L. Qi, Eigenvalues of a real supersymmetric tensor, Symbolic Comput. 40(2005), 1302–1324.
* [9] H. Wielandt, Unzerlegbare, nicht negative matrizen, Math. Z. 52(1950), 642–648.
* [10] J.Y. Shao, On a conjecture about the exponent set of primitive matrices, Linear Algebra Appl. 65(1985), 91–123.
* [11] J.Y. Shao, A general product of tensors with applications, Linear Algebra and its Appl. 439(2013), 2350–2366.
* [12] P.Z. Yuan, Z.L. He, and L.H. You, A conjecture on the primitive degree of Tensors, Linear Algebra and its Appl. 450(2014), 175–185.
* [13] K.M. Zhang, On Lewin and Vitek’s conjecture about the exponent set of primitive matrices, Linear Algebra Appl. 96(1987), 101–108.
|
arxiv-papers
| 2014-04-06T11:46:57 |
2024-09-04T02:50:00.775288
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zilong He, Pingzhi Yuan, Lihua You",
"submitter": "Lihua You",
"url": "https://arxiv.org/abs/1404.1567"
}
|
1404.1622
|
# MU-MIMO MAC Protocols for Wireless Local
Area Networks: A Survey
Ruizhi Liao, Boris Bellalta, Miquel Oliver, and Zhisheng Niu R. Liao, B.
Bellalta and M. Oliver are with NeTS Research Group, Department of Information
and Communication Technologies, Universitat Pompeu Fabra, Barcelona, 08018,
Spain (e-mail: {ruizhi.liao, boris.bellalta, miquel.oliver}@upf.edu).Z. Niu is
with NiuLab, Department Of Electronic Engineering, Tsinghua Unisersity,
Beijing, 100084, China (e-mail: [email protected]).
###### Abstract
As wireless devices boom, and bandwidth-hungry applications (e.g., video and
cloud uploading) get popular, today’s Wireless Local Area Networks (WLANs)
become not only crowded but also stressed at throughput. Multi-user Multiple-
Input and Multiple-Output (MU-MIMO), an advanced form of MIMO, has gained
attention due to its huge potential in improving the performance of WLANs.
This paper surveys random access based MAC protocols for MU-MIMO enabled
WLANs. It first provides background information about the evolution and the
fundamental MAC schemes of IEEE 802.11 Standards and Amendments, and then
identifies the key requirements of designing MU-MIMO MAC protocols for WLANs.
After that, the most representative MU-MIMO MAC proposals in the literature
are overviewed by benchmarking their MAC procedures and examining the key
components, such as the channel state information acquisition, de/pre-coding
and scheduling schemes. Classifications and discussions on important findings
of the surveyed MAC protocols are provided, based on which, the research
challenges for designing effective MU-MIMO MAC protocols, as well as the
envisaged MAC’s role in the future heterogeneous networks, are highlighted.
###### Index Terms:
MAC, MU-MIMO, IEEE 802.11, WLANs, MUD, MUIC, CSI, Scheduling.
## I Introduction
IEEE 802.11 is a set of Physical Layer (PHY) and Medium Access Control (MAC)
specifications for the prevalent Wireless Local Area Networks (WLANs). Current
IEEE 802.11 WLANs contribute to approximate $40\%$ of overall Internet traffic
[1]. As wireless devices rapidly increase, and the wireless transmission
evolves towards gigabits per second, IEEE 802.11 WLANs are set to dominate the
way of Internet access at homes and working places in the future.
Multi-user Multiple-Input and Multiple-Output (MU-MIMO), introduced by IEEE
802.11ac [2], is one of the most crucial techniques that lead WLANs towards
the gigabit era. Compared to Single-user MIMO (SU-MIMO), which focuses on
transmitting to a single destination, MU-MIMO holds the following three
advantages: (1) The increased throughput. By employing SU-MIMO, the
theoretical capacity gain can be manifested by a multiplicative factor of
$min\\{N_{\text{t}},N_{\text{r}}\\}$, where $N_{\text{t}}$ and $N_{\text{r}}$
are the number of transmitting and receiving antennas [3][4]; while in the
case of MU-MIMO, the multiplicative factor can be further extended to
$min\\{aN_{\text{t}},bN_{\text{r}}\\}$, where $a$ and $b$ are the number of
simultaneous transmitters and receivers; (2) The increased diversity gain. The
spatially distributed STAs make the MU-MIMO system more immune to the channel
rank loss and the antenna correlation [5], which may affect the SU-MIMO system
performance by limiting the available transmission rates; (3) The reduced
terminal cost. The MU-MIMO system supports multiple spatially separated STAs
(even only equipped with a single antenna) to simultaneously communicate with
the Access Point (AP), which makes the development of compact and low-cost
user terminals possible.
Considerable research efforts have been made to approach the MIMO capacity at
the PHY layer [6][7], and a comprehensive overview can be found in [8].
Following the PHY advance, corresponding MAC enhancements, especially the MU-
MIMO based ones, have sprung up.
MAC, used among multiple stations (STAs) to share a common wireless channel,
can be dated back to the pioneer protocols used in the ALOHA network in the
$1970$s [9] to the current Carrier Sense Multiple Access with Collision
Avoidance (CSMA/CA) based IEEE 802.11 Distributed Coordination Function (DCF).
Since the IEEE 802.11 MAC mechanism only supports one single transmission at a
time, which underutilizes the full potential of the spatial domain of MU-MIMO
transmissions, thus, MU-MIMO MAC proposals have tried to adapt the frame
structure, as well as their operation procedures to control the parallel
transmissions among STAs.
There are two main MAC categories: (1) the fixed-assignment one, where the
channel frequencies, the access time, mutually orthogonal codes or different
polarization is predefined for each STA, namely, Frequency Division, Time
Division, Code Division or Polarization Division Multiple Access (FDMA, TDMA,
CDMA and PDMA); and (2) the random access one, where each STA independently
determines when to compete for the channel, e.g., Aloha and CSMA/CA. It is
worth to note that Space Division Multiple Access (SDMA) is a medium access
scheme by exploring parallel spatial streams. In this sense, the MU-MIMO
transmission, where multiple spatially-separated nodes are involved in
parallel transmissions, is a form of SDMA. Thus, SDMA schemes and MU-MIMO
transmissions are used interchangeably in this paper.
Due to the following three reasons, the random access based CSMA/CA has
dominated the MAC mechanism of WLANs. First, the backward compatibility. The
initial WLAN traffic was sporadic, bursty, and asymmetrical between uplink and
downlink. Although in the past decade, the traffic load has increased
significantly, and the traffic pattern has evolved from mainly web browsing
and file transfers to a wide variety of applications, there are still vast
amount of legacy STAs based on CSMA/CA. Secondly, coexistence with other
networks. Neighbouring WLANs, wireless sensor networks and Bluetooth personal
area networks, that are all operating in the Industrial Scientific Medical
(ISM) band, present significant interference to each other. CSMA/CA offers a
simple but effective solution (i.e., listen before transmitting) to share the
unlicensed spectrum among competing networks. Thirdly, the implementation
simplicity. The implementation of a random access scheme is simpler compared
to others. Since there is no need to use accurate clocks for synchronization,
nor to execute complex functions for scheduling.
The central point of the paper is to study random access based MAC mechanisms
for MU-MIMO enabled WLANs. The main contributions of the paper are threefold.
(1) We report the IEEE bodies’ MAC progress, as well as survey and categorize
the most relevant MU-MIMO MAC proposals in the literature. (2) By doing such
review, we identify key requirements for designing efficient MU-MIMO MAC
protocols, such as the channel state information (CSI) acquisition, de/pre-
coding and scheduling schemes. These requirements are used to benchmark MU-
MIMO MAC proposals in the literature. (3) We provide highlights and
discussions on important findings and research challenges after each
subsection of the literature review, and give our thoughts about possible
future directions on the MU-MIMO MAC design, such as uplink MU-MIMO, new
backoff algorithms and full-duplex transmissions.
The rest of the paper is organized as follows. First, Section II briefly
overviews the evolution of IEEE 802.11 standards/amendments and their
fundamental MAC mechanisms to clarify the MAC development promoted by the IEEE
standard body. Next, Section III identifies the key requirements for designing
MU-MIMO MAC protocols in WLANs. Then, Section IV surveys and classifies the
most prominent MU-MIMO MAC protocols in the literature. Afterwards, Section V
discusses the research challenges and future directions. Finally, Section VI
concludes the paper.
TABLE I: Terms and Abbreviations
Access Category (AC) Additive White Gaussian Noise (AWGN) Network Allocation
Vector (NAV) Access Point (AP) Aggregated MAC Protocol Data Unit (A-MPDU)
Orthogonal Frequency Division Multiplexing (OFDM) Acknowledgement (ACK)
Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) Quadrature
Amplitude Modulation (QAM) Backoff (BO) Channel State Information (CSI)
Quality of Service (QoS) Clear-to-Send (CTS) Code Division Multiple Access
(CDMA) Reduced Inter Frame Space (RIFS) Contention Window (CW) Direct Sequence
Spread Spectrum (DSSS) Short Inter Frame Space (SIFS) DCF Inter Frame Space
(DIFS) Distributed Coordination Function (DCF) Signal-Interference-Noise Ratio
(SINR) Dirty Paper Coding (DPC) Enhanced Distribution Channel Access EDCA
Signal-Noise Ratio (SNR) Explicit Compressed Feedback (ECFB) Frequency
Division Multiple Access (FDMA) Single-user Multiple-Input Multiple-Output
(SU-MIMO) First-in First-out (FIFO) Frequency Hopping Spread Spectrum (FHSS)
Software Defined Radio (SDR) High Throughput (HT) Minimum Mean Square Error
(MMSE) Successive Interference Cancellation (SIC) Maximum Likelihood (ML)
Multi-Packet Reception (MPR) Time Division Multiple Access (TDMA) Medium
Access Control (MAC) Multi-Packet Transmission (MPT) Transmit Opportunity
(TXOP) Physical Layer (PHY) Multi-user Detection (MUD) Very High Throughput
(VHT) Request-to-Send (RTS) Multi-user Interference Cancellation (MUIC)
Wireless Local Area Networks (WLANs) Station (STA) Mingle-user Multiple-Input
Multiple-Output (MU-MIMO) Zero Forcing Beamforming (ZFBF)
## II The Evolution of IEEE 802.11 and MAC Schemes
This section presents an evolutionary overview of IEEE 802.11
standards/amendments, and also introduces how the IEEE 802.11 specified MAC
schemes work. This overview does not go through all aspects of IEEE 802.11
standards/amendments, but focuses on the background information that is
closely related to the topic of the paper, namely, medium access control. Due
to a considerable amount of technical terms and abbreviations have been used,
Table I is given to summarize important terms for the reader’s convenience.
### II-A IEEE 802.11 Standards/Amendments
#### II-A1 Standards
Loosely speaking, both standards and amendments can be interchangeably used to
refer to different variants of IEEE standards or amendments. However, a more
strict nomenclature designates standards as documents with mandatory
requirements (denoted as IEEE 802.11 followed by the published year, e.g.,
IEEE 802.11-2012), and amendments as documents that add to, remove from, or
alter material in a portion of existing standards [10] (denoted as IEEE 802.11
followed by a non-capitalized letter or letters, e.g., IEEE 802.11n or
802.11ac).
Since 1997, IEEE has released four standards: 802.11-1997, 802.11-1999,
802.11-2007 and 802.11-2012. IEEE 802.11-2012 [11] is the latest and the only
version that is currently in publication. Standards are continuously updated
by amendments, e.g., 802.11-2012 is created by integrating ten amendments such
as 802.11n and 802.11p with the base standard 802.11-2007, which was replaced
since the release of 802.11-2012. In other words, each standard will be
superseded by its successor in its entirety.
#### II-A2 Amendments
In 1999, two amendments were first introduced: (1) IEEE 802.11a operates in
the $5$ GHz band using the Orthogonal Frequency Division Multiplexing (OFDM)
modulation with a maximum data rate of $54$ Mbps; (2) IEEE 802.11b operates in
the $2.4$ GHz band using the Direct Sequence Spread Spectrum (DSSS) modulation
with a maximum data rate of $11$ Mbps. Compared to 802.11-1997, 802.11b
substantially increases the data rate (from $2$ Mbps to $11$ Mbps) using the
same modulation technique and the frequency band, which made 802.11b the then-
definitive WLAN technology. In 2003, IEEE 802.11g, a new amendment working in
the $2.4$ GHz band was ratified. It extends 802.11b with a maximum data rate
of $54$ Mbps. IEEE 802.11n [12], ratified in 2009, operates in either $2.4$
GHz or $5$ GHz band, boosting the data rate to $150$ Mbps ($600$ Mbps by $4$
streams) by utilizing MIMO.
802.11ac [2] is the latest IEEE amendment approved in 2013. It is operating
exclusively in the $5$ GHz band. Driven by the need for higher speed, 802.11ac
aims to provide an aggregated multi-station throughput of at least $1$ gigabit
per second, namely, Very High Throughput (VHT) WLANs. Compared to 802.11n,
this significant improvement is achieved by introducing novel PHY and MAC
features, such as wider bandwidths ($80$ and $160$ MHz), a denser modulation
scheme ($256$-QAM: Quadrature Amplitude Modulation), a compulsory frame format
(A-MPDU: Aggregated MAC Protocol Data Unit), and most importantly, downlink
MU-MIMO transmissions (supporting simultaneous transmissions of up to $4$ STAs
with the maximum number of $8$ streams).
Although each amendment is revoked as it is merged into the latest standard,
the sign of IEEE 802.11a/b/g/n/ac is often employed by vendors to denote the
capability and compatibility of their products.
TABLE II: Features of Related IEEE 802.11 Standards/Amendments
Version Description Frequency Max. Data Rate Modulation 802.11-1997 WLAN MAC
and PHY Specifications 20 (MHz) @ 2.4 (GHz) 2 (Mbps) DSSS, FHSS 802.11-1999
Part II WLAN MAC and PHY Specifications 20 @ 2.4 2 DSSS, FHSS a Higher Speed
PHY Extension 20 @ 5 54 OFDM b Higher Speed PHY Extension 20 @ 2.4 11 DSSS g
Further Higher Data Rate Extension 20 @ 2.4 54 OFDM, DSSS 802.11-2007 Standard
Maintenance Revision – – – n High Throughput 20, 40 @ 2.4, 5 150 x 4 OFDM
802.11-2012 Accumulated Maintenance Changes – – – ac Very High Throughput 20,
40, 80, 160 @ 5 866.7 x 8 OFDM ax High Efficiency WLAN (approx. 2019) Below 6
GHz - OFDM
#### II-A3 Next Amendment-802.11ax
In March 2014, IEEE created a new task group-802.11ax [13], aiming at
delivering High Efficiency WLANs (HEW) in both indoor and outdoor high density
scenarios. 802.11ax will operate in frequency bands between $1$ and $6$ GHz.
In terms of performance, the focus of the amendment has shifted from improving
the aggregated system throughput to the throughput observed by each STA,
targeting at at least four-time per-STA throughput increase comparing to the
existing standards and amendments. The ongoing amendment is in its early
stage, and is estimated to be finished in 2019. Currently, the study group is
discussing the 802.11ax usage models focusing on the dense deployment [14],
potential technologies such as full-duplex, uplink MU-MIMO and Massive MIMO
[15], and functional requirements such as the system performance, the spectrum
efficiency, the operation bands and the backward compatibility [16].
The key features of the above mentioned IEEE 802.11 standards/amendments are
given in Table II, where FHSS stands for Frequency Hopping Spread Spectrum.
#### II-A4 New Frequency Bands of IEEE 802.11
Besides the traditional frequency bands ($2.4$ and $5$ GHz), IEEE 802.11 has
extended to support other bands.
IEEE 802.11ad [17], another VHT WLAN amendment, will operate in the $60$ GHz
band and focus on multi-gigabits per second data transmissions in a short
range point-to-point links (around $10$ meters). A typical application
scenario of 802.11ad is the wireless transmission of lightly compressed or
uncompressed high-definition videos for home entertainment systems. Due to
$60$ GHz band’s characteristics of high propagation loss and high attenuation,
directional transmissions and receptions are required.
IEEE 802.11ah [18] will operate in the sub-GHz band. The main purposes of the
amendment are to introduce power saving and STA grouping mechanisms. A typical
use case is a smart metering network with many sensor nodes, where high
collision probability and hidden nodes are expected. 802.11ah will partition
nodes into groups to save power and to reduce the channel contention by
assigning the channel to nodes of a given group at a given time [19][20].
Due to the specific purpose of each amendment and unique features of the
employed frequency, the MU-MIMO MAC schemes designed for WLANs of the
traditional bands can not be directly applied to these new amendments. For
example, 802.11ah relies on the highly centralized medium access scheme, while
802.11ad has to utilize the beam sweeping technique to detect STAs rather than
the omnidirectional carrier sensing adopted by IEEE 802.11 DCF. Therefore,
this paper preserves its focus on MAC proposals for the traditional bands,
i.e., $2.4$ and $5$ GHz. However, the potential collaborations at the MAC
level between protocols of traditional and new bands will be discussed in
Section V-Future Directions.
### II-B IEEE 802.11 Medium Access Control
Although IEEE 802.11 has specified three MAC mechanisms, namely, DCF, Point
Coordination Function (PCF) and Hybrid Coordination Function Controlled Access
(HCCA), this paper only focuses on the distributed and random access based MAC
schemes, because PCF and HCCA (i.e., the centralized schemes) are neither
widely adopted by the industry nor the academia.
#### II-B1 Distributed Coordination Function
DCF is the fundamental medium access scheme of IEEE 802.11 based WLANs. It
relies on CSMA/CA to detect and share the wireless channel among STAs. DCF can
either operate in the basic access scheme (Figure 1(a)) or the optional
Request-to-Send/Clear-to-Send (RTS/CTS, Figure 1(b)) scheme. DCF mandates STAs
to keep sensing the channel. If the channel has been idle for DCF Inter Frame
Space (DIFS), each STA starts decreasing a backoff (BO) timer chosen from its
Contention Window (CW) to compete for the channel. The STA with the lowest BO
wins the channel contention and starts to transmit frames. Collisions occur if
more than one STA happens to choose the same random BO. When a transmitted
frame is successfully received, the receiver waits for a Short Inter Frame
Space (SIFS) and then sends back an Acknowledgement (ACK). Note that as soon
as the winning STA sends out a frame, other STAs will notice the channel has
become busy, therefore immediately freeze their BO timers. These STAs will
wait the channel to be idle for another DIFS, and resume decreasing the
remaining BO timers. The STA who previously succeeded the channel contention
will have a new BO timer at its next transmission attempt.
Examples of a successful transmission for the basic access and the RTS/CTS
schemes are shown in Figure 1, where B denotes the channel is initially busy.
Please refer to the IEEE standard 802.11-2012 [11] for more details about DCF.
(a) A successful transmission of basic access
(b) A successful transmission of RTS/CTS
Figure 1: 802.11 DCF transmission procedures
#### II-B2 Enhanced Distribution Channel Access
IEEE 802.11e [21] proposes an extension to DCF-Enhanced Distribution Channel
Access (EDCA), as a response to the demand of Quality of Service (QoS) for
voice and video applications. The main differences between DCF and EDCA are
twofold. First, the former does not differentiate traffic from different
applications, while the latter classifies traffic into four Access Categories
(ACs) with different priorities: Voice (AC_VO), Video (AC_VI), Best Effort
(AC_BE) and Background (AC_BK). By doing so, EDCA is able to assign ACs with
different parameters. For example, the maximum Transmit Opportunity (TXOP, a
contention-free interval, during which a STA can transmit as many frames as
possible) for AC_VO and AC_VI are $1.504$ ms and $3.008$ ms, respectively.
Secondly, it is also different that the instant of time at which DCF and EDCA
mandate STAs to decrease the BO timer. In DCF, STAs decrease the BO timer at
the end of each slot, while in EDCA, the decrement occurs at the beginning of
each slot. Please refer to [11] and [22] for detailed comparisons of DCF and
EDCA, and [23] for QoS supports in WLANs.
## III Requirements for Designing MU-MIMO MAC Protocols in WLANs
MU-MIMO transmissions in WLANs have two communication paths, the uplink one
(i.e., STAs simultaneously transmit frames to the AP, which is also referred
as the MIMO-MAC channel) and the downlink one (i.e., the AP sends data to a
group of STAs in parallel, which is also referred as the MIMO-broadcast
channel). The MU-MIMO uplink and downlink transmissions face different
challenges, and hence, have different requirements in designing MAC protocols.
### III-A De/Pre-coding Schemes for Simultaneous Receptions/Transmissions
In the uplink, the AP needs to separate the simultaneously transmitted signals
from STAs, which is the Multi-user Detection (MUD) problem. In the downlink,
the AP has to, firstly, select a group of STAs based on a certain criterion
such as the queue occupancy, given that the selected STAs have to be spatially
non-correlated, which is the scheduling problem, and, secondly, precode the
outgoing frames to null the interference among concurrent spatial streams,
which is the Multi-user Interference Cancellation (MUIC) problem. An
illustration of MU-MIMO uplink and downlink transmissions is given in Figure
2.
The design of MUD/MUIC schemes is beyond the topic of the paper. However, some
of the most popular MUD/MUIC schemes adopted in the surveyed papers, as well
as their strong points and drawbacks, are sampled (Table III).
Figure 2: Up/Down-link multi-user transmissions
#### III-A1 MUD Schemes for Simultaneous Uplink Receptions
##### Minimum Mean Square Error (MMSE)
Received signals at each antenna of the AP are multiplied by a complex weight
and then summed up. The weight is adjustable through minimizing the difference
between the summation of the output signal and a reference that is known by
both the AP and STAs. An example of the weight adjustment is to utilize the
steepest descending algorithm. The performance of the MMSE MUD scheme improves
as the number of AP’s antennas increases, and degrades as the network scales
up [24].
##### Maximum Likelihood (ML)
The ML MUD conducts an exhaustive search to extract the transmitted signals.
It provides the best detection performance, but comes with the highest
complexity that increases exponentially with the number of STAs, which makes
it infeasible in practical systems.
##### Sphere Decoding (SD)
Some SD based MUD algorithms have been proposed to reduce the complexity of
the pure ML MUD while to approach the performance of ML MUD. The idea is to
decrease the radius of the search scope by focusing on the vicinity of the ML
solution.
##### Successive Interference Cancellation (SIC)
The SIC MUD is an enhancement to MMSE. A detection algorithm is utilized by
estimating of the received power at the AP. The signal with the highest power,
which is the least interfered by others, is detected. This detected signal is
then subtracted from mixed signals, and the next highest signal is singled out
using the same process until the lowest STA signal is determined. The SIC MUD
tends to be erroneous at the signal classification stage. The false deduction
from composite STA signals may propagate the error to following calculations
[25].
#### III-A2 MUIC Schemes for Simultaneous Downlink Transmissions
Although simultaneous downlink transmissions from the AP to multiple STAs can
be seen as a combination of several single-user transmissions, STAs’ random
and independent locations make it very challenging to jointly null multi-user
interference at the STA side. Therefore, most proposals in the literature
precode outgoing signals at the AP to minimize interference among simultaneous
streams.
##### Zero Forcing (ZF)
In the ZF scheme, the original signal is multiplied by the pseudo-inverse of
the channel matrix to completely null the MUI. The conditions are (1) the AP
has the full CSI, and (2) the channel is invertible. By multiplying the
pseudo-inverse weight, the ZF scheme also increases the error rate (because
the noise vector is amplified). The amplified noise vector indicates that ZF
can only perform well in the high Signal-to-Noise Ratio (SNR) region. In
addition, the ZF scheme requires that the number of total receiving antennas
is not less than that of transmitting antennas [26].
In comparison, the MMSE scheme can minimize the overall error rate without
amplifying the noise. [25] and [26] show that the MMSE scheme performs better
than ZF in the low SNR region, and approaches the performance of ZF in the
high SNR region.
##### Block diagonalization (BD)
BD is a generalized channel inversion technique, especially when receivers
have multiple antennas [6]. Singular value decomposition (SVD) is employed to
remove unitary matrices, which makes the computational complexity of BD higher
than MMSE.
##### Dirty Paper Coding (DPC)
DPC is a non-linear precoding scheme firstly introduced by Costa [27], which
can achieve the optimum performance at the cost of significant computing
complexity. The idea is to add an offset (the negative value of the
interference that is known at the AP) to the transmitted signal, which hints
that (1) the AP has to know the interference in advance, and (2) the AP always
has available codewords, i.e., infinite length of codewords, which make DPC
not suitable for practical use.
TABLE III: Features of MUD/MUIC Schemes
Name Type Main Features Remarks Zero Forcing (ZF) Linear, MUD/MUIC Complete
interference cancellation with full CSI Noise amplified Minimum Mean Square
Error (MMSE) Linear, MUD/MUIC Complete interference cancellation with full CSI
Outperform ZF at low SNR Maximum Likelihood (ML) Nonlinear, MUD Performance
bound of MUD Exhaustive search; Exponential complexity Successive Interference
Cancellation (SIC) Nonlinear, MUD Trade-off between ML & MMSE Error
propagation Block diagonalization (BD) Linear, MUIC More complex than MMSE
SVD; Multi-antenna receivers; Generalized channel inversion Dirty Paper Coding
(DPC) Nonlinear, MUIC Performance bound of MUIC Infinite codewords;
Impractical for use
### III-B Channel State Information Acquisition
MUD and MUIC schemes allow MU-MIMO systems to separate simultaneously
received/transmitted frames, and achieve the spatial multiplexing gain.
However, it is important to point out that, in the above discussion, the
possession of CSI is assumed at the AP.
Most proposals in the literature integrate the CSI acquisition into MAC
operations. There are generally two types of CSI: the statistical CSI and the
instantaneous one. The former employs the statistical characteristics of the
channel (e.g., fading distribution, average channel gain and spatial
correlation) to decide the CSI, which performs well in scenarios where the
channel has a large mean component (i.e., a large Rician factor) or strong
correlation (either in space, time, or frequency) [28].
The instantaneous CSI (or the short-term CSI) means the current channel state
is known, which enables the transmitter to adapt its outgoing signal. Because
wireless channel varies over time, the instantaneous CSI has to be estimated
repeatedly on a short-term basis. The acquisition of CSI can be done by
estimating a training sequence known by both transmitters and receivers. In
the uplink, the AP can easily extract the uplink CSI from the PHY preambles of
received frames. While, for transmissions in the downlink, the acquisition of
the CSI is not that straightforward. Depending on who computes the CSI, there
are two CSI feedback schemes: (1) the implicit feedback (Figure 3(a)), where
the AP computes the CSI by estimating training sequences sent from STAs, and
(2) the explicit one (Figure 3(b)), where STAs calculate the CSI by estimating
the training sequence sent from the AP, and then STAs feedback the calculated
CSI to the AP.
(a) Implicit CSI feedback scheme
(b) Explicit CSI feedback scheme
Figure 3: Implicit & Explicit CSI feedback procedures
By assuming the reciprocity of up/down-link channels, the implicit feedback
scheme produces less overheads compared to the explicit one. However, in a
practical WLAN system, the channel and interference seen by the STAs are
generally not the same as those seen by the AP due to their different
transmitting/receiving filters and PHY paths. Therefore, the antenna
calibration [2] is usually needed to reduce the distortion if implicit
feedback is adopted.
The explicit feedback scheme, i.e., STAs feedback the CSI, provides higher CSI
resolution, but also higher overhead. MAC control frames are usually extended
to support the CSI feedback in the literature, while an explicit compressed
Feedback (ECFB) scheme is introduced by IEEE 802.11ac to schedule and compress
the volume of CSI feedback.
No matter which CSI feedback scheme is applied, the implicit one or the
explicit one, the frequency of CSI feedback would significantly affect the
network performance. It is because the frequent CSI feedback increases
overheads, while the infrequent one results in the outdated CSI that leads to
interference among parallel streams. Please refer to [29] and [30] for more
details about implicit and explicit CSI feedback schemes.
### III-C The Scheduling Scheme
Another key point for designing MU-MIMO MAC protocols is the scheduling
scheme. It is used to select a group of STAs or frames for transmissions,
which can optimize certain aspects of the system performance according to the
specific grouping criteria. The design of the scheduling scheme can be divided
into two parts: the scheduling in the uplink and downlink. The latter can be
easily categorized by different scheduling algorithms (e.g., the destination
based round-robin scheme or the frame based first-in first-out scheme), while
the characterization of the former is not straightforward. The reason and
categorization of the uplink scheduling schemes are as follows.
#### III-C1 Scheduling in the Uplink
In the uplink, it is very challenging to make a joint scheduling decision
among spatially distributed STAs. As shown in Figure 4, only three of surveyed
papers considered scheduling schemes, which will be described in details in
the protocol review Section IV.
Figure 4: Categories of uplink transmissions
Depending on whether the RTS/CTS exchanging process is employed (i.e., whether
the AP has played a coordinating role in exchanging control frames before
transmitting data), uplink transmissions are categorized into the coordinated
and the un-coordinated ones. In the un-coordinated scenario, STAs utilize the
random MAC mechanism to decide who will be allowed for transmissions, which
have two cases: synchronous [31] and asynchronous [32] data transmissions, as
shown in Figure 5. The synchronous scheme lets multiple STAs that
coincidentally choose the same BO to transmit data frames simultaneously,
while the asynchronous one allows STAs to transmit frames along with other
ongoing transmissions.
(a) Synchronous data transmissions
(b) Asynchronous data transmissions
Figure 5: Un-coordinated uplink channel access
In the coordinated scenario, STAs utilize the MAC random mechanism to contend
for the channel, while let the AP to decide who will be involved in the
followed parallel transmissions. The coordinated uplink access scheme implies
the involvement of the AP (as a coordinator) and the employment of RTS/CTS
exchanges [33]. The AP extracts the interested information from RTSs sent by
the contending STAs, and then makes scheduling decisions for simultaneous
frame transmissions (i.e., the scheduled transmissions), or the AP just
responds to the received RTSs to notify who have won the channel contention
(i.e., the un-scheduled transmissions). A general example of the coordinated
uplink access is shown in Figure 6, which can account for both scheduled and
un-scheduled cases depending on whether the CTS is extended to support
scheduling.
Figure 6: Coordinated uplink channel access
Although the un-coordinated uplink channel access requires fewer modifications
compared to the coordinated one, it is likely that the spatial domain will be
underutilized. The reason is that the concurrent uplink access of the un-
coordinated scheme is based on the randomness of the IEEE 802.11 backoff
mechanism. In comparison, the coordinated uplink channel access lets the AP to
mediate the uplink transmissions by either just notifying a group of STAs that
successfully won the channel contention or making a scheduling decision that
aims to optimize the system performance. Obviously, the coordinated scheme
will introduce overheads (e.g., extra fields in RTS/CTS) that are needed for
the AP to best exploit the spatial domain.
#### III-C2 Scheduling in the Downlink
Compared to the uplink, the AP plays a more direct role in the downlink
scheduling, which can be classified into the packet based scheduling and the
STA based one. In principle, the following downlink scheduling schemes can be
applied to uplink transmissions. However, they are adopted in very few
proposals due to the difficulty and overheads to make uplink scheduling
decisions for spatially-distributed STAs.
The packet based scheduling algorithms utilize the packet queueing status at
the AP as the scheduling metric to assemble multiple packets for MU-MIMO
downlink transmissions. The relevant packet based scheduling algorithms
include First-in First-out (FIFO) [34, 35, 36, 37], weighted fair queue (WFQ,
weight based on the priority or the type of packets) [38, 39], earliest
deadline first (EDF, based on the delay or waiting time of packets), greedy
[40, 41], etc.
The STA based scheduling employs some criteria to identify a set of STAs for
simultaneous downlink transmissions. These criteria include the channel state
[42, 43, 44, 45, 46], spatial compatibility, fairness, etc. The pure channel
condition based scheduling is also called opportunistic scheduling that
singles out a set of STAs with best channel conditions by benefiting from the
multi-user diversity. A similar concept is applied by the spatial
compatibility based scheduling, which examines STAs’ spatial correlations to
minimize the interference. Neither of them has taken the fairness into
account, which is considered by the round-robin scheduling.
A combination of the above mentioned scheduling schemes is usually considered.
For example, the proportional fair scheduler usually considers a product of
the channel capacity and fairness. However, as we can see from the above, the
most commonly used scheduling schemes are FIFO and opportunistic, which are
followed by greedy and WFQ. The reasons could be as follows. Comparing to
cellular networks, (1) WLANs consist of fewer STAs, (2) WLAN STAs are more
stable in terms of moving speeds and residing environments, and (3) the AP is
usually less powerful than the base station.
The combination of several criteria hints that parameters from different
layers should be jointly considered [47], which is the concept of cross-layer
scheduling as shown in Figure 7, where $q(t)$ and $H(t)$ represent the
queueing and the channel states at the time $t$.
Figure 7: Queueing and channel based cross-layer scheduling
#### III-C3 Cross-layer Scheduling
The cross-layer scheduling has the promise to achieve the optimal system
performance by sharing and configuring parameters from different layers, such
as the channel information at the PHY layer, the queueing state at the MAC
layer and the routing information at the network layer. Unfortunately, the
cross-layer scheduling remains far more complex than simply combining these
parameters. The reason is that the interaction among the layers breaks the
conventional Open System Interconnection (OSI) layered structures and creates
tensions between the performance and the stability of systems, which could
lead to unexpected consequences as the wireless network scales up [48].
In general, together with the CSI acquisition and other layers’ key
parameters, the following points should be taken into account for the cross-
layer design.
* •
System Complexity: As the cross-layer design breaks the conventional layered
structure, the new wireless system could be incompatible with conventional
ones. Besides, since the maintenance or upgrade of the cross-layer protocol is
no longer isolated within each layer, any parameter changes must be carefully
traced and coordinated [48].
* •
Design Constraints: Various data rates are usually applied to spatially
distributed STAs, which may cause interferences from stronger signals to
weaker ones in the downlink or the near-far effect in the uplink. Therefore, a
power control or a data rate selection scheme needs to be considered with the
MAC design. In addition, some QoS metrics, such as the average delay and the
jitter, are traditionally not in line with the MAC focus (e.g., decreasing
collisions and increasing throughput). Sometimes, maximizing throughput means
sacrificing transmission opportunities of some low-rate STAs. Thus, the
tradeoff of different performance metrics needs to be jointly considered and
given different weights [49] [50].
## IV Survey of MU-MIMO MAC Protocols for WLANs
In this section, we look into prominent MU-MIMO MAC proposals in the
literature by focusing on the required features, performance gains, evaluation
tools, as well as the key assumptions they made.
### IV-A MAC Proposals for The Uplink
#### IV-A1 Un-coordinated Channel Access
##### Synchronous Data Transmissions
Jin et al. in [31] present a simple MU-MIMO MAC scheme that relies on the
simultaneous transmissions from STAs. The MAC procedure is the same as
illustrated in Figure 5(a). The authors assume each STA has an orthogonal
preamble so that the AP can estimate the channel coefficients and
differentiate STAs. Once the AP knows the channel, it employs the ZF scheme to
separate the received signals. The authors extend the Markov chain model
proposed by Bianchi in [51] to analyze the performance of the proposed MU-MIMO
scheme in saturated conditions. Compared with SU-MIMO, the numerical results
show that the proposed MU-MIMO scheme obtains lower collision probability,
shorter delay and higher throughput in the low SNR and small network
conditions.
##### Asynchronous Data Transmissions
Tan et al. in [52] present a practical Spatial Multiple Access (SAM) scheme
for WLANs. SAM relies on a distributed MAC scheme called Carrier Counting
Multiple Access (CCMA) to allow asynchronous concurrent transmissions. A chain
decoding technique to separate simultaneously received frames is adopted. Each
STA maintains a transmission counter by detecting other STAs’ frame preambles,
and decides whether to contend for the channel. SAM is evaluated in SORA, a
Software Defined Radio (SDR) platform developed by Microsoft [53]. Evaluation
results show that the proposal can increase the throughput by $70\%$ over the
default IEEE 802.11 DCF.
Babich et al. in [32] develop a Markov chain analytical model for asynchronous
Multi-Packet Reception (MPR), where a STA is allowed to transmit even if other
STAs are already transmitting. More specifically, a STA is allowed to decrease
its BO counter as long as the channel is empty or the sensed number of ongoing
transmissions is below a threshold. A generic error correction code is assumed
to protect data frames. The MAC procedure is shown in Figure 5(b). Based on
the results obtained from the presented theoretical model and simulations, the
authors claim that the asynchronous MAC scheme can provide considerable
performance gains compared to the synchronous one due to higher utilization of
the channel.
Ettefagh et al. in [54] present a cluster-based MAC protocol for MU-MIMO
transmissions called CB-CSMA/CA. In CB-CSMA/CA, STAs are grouped in clusters,
and the AP is assumed to have MPR capabilities. STAs belonging to the same
cluster share a common backoff counter, which allows simultaneous
transmissions from multiple STAs when the counter reaches zero. As a result of
this, the channel access is competed among clusters. Simulation and analytic
results show that CB-CSMA/CA outperforms a pure STA-based contention approach
since it increases the chance that more than one STA can transmit at the same
slot.
Mukhopadhyay et al. in [55] explore the ACK-delay problem that arises in
asynchronous MPR. The ACK-delay problem refers to that, due to different
transmission durations of asynchronous uplink transmissions, the delayed ACKs
sent by the AP to those earlier-finished STAs can trigger STAs’ ACK time-out
counters, which could interrupt the ACK’s transmission and degrade the network
performance. Since Babich et al. in [32] did not consider the ACK-delay
problem, the authors in [55] propose to change the ACK-waiting STAs’ backoff
timer to be decreased only when the channel has been idle for DIFS. The
simultaneous transmissions are assumed to be decodable by the AP, and a single
data rate is also assumed. By comparing with Babich’s and IEEE 802.11 standard
schemes, the results show that the proposed one not only decreases the frame
collision probability and the average delay, but also increases the
throughput.
Lin et al. in [56] propose a MIMO concurrent uplink transmission scheme called
MIMO/CON, which can support both asynchronous and synchronous data
transmissions. A compressive sensing technique [57] is utilized to estimate
CSI from multiple concurrently received preambles, and ZF is adopted to
separate the data frames. A delayed packet decoding mechanism, namely, using
partially retransmitted information to decode the collided frames, is devised
to avoid the complete retransmission of all corrupted frames. A fixed frame
length is assumed, and the optimal transmission probability is assumed to be
known by STAs. Tan’s CCMA [52] is implemented to compare with the proposed
MIMO/CON. The results show that CCMA outperforms MIMO/CON when the AP has
fewer antennas, while MIMO/CON scales better as the number of antennas at the
AP increases.
Wu et al. in [58] propose a throughput analytical model for asynchronous
uplink transmissions, which is based on the Bianchi Markov chain model [51]. A
beacon sent by the AP will announce the maximum number of STAs that are
allowed to transmit in parallel. The MAC procedure is similar to the one
described above, in [52]. A fixed data rate is assumed and the network is
saturated. Zero-forcing with successive interference cancellation (ZF-SIC) is
employed to decode parallel data streams. By numerically deriving the
contention window size and other network parameters, the uplink throughput is
maximized. With those parameters, the authors derive a threshold of the number
of antennas at the AP, which shows no throughput benefit can be achieved by
adding more antennas. The reasons for that are: (1) the available transmission
time decreases as the number of STAs involved in the parallel transmission
increases; and (2) the collision probability increases as more antennas are
employed at the AP.
Kuo et al. in [59] present a leader-based MAC protocol for uplink MU-MIMO
transmissions. Similar to other asynchronous proposals, each STA counts the
number of concurrent transmissions to decide if it can start a new
transmission. In case there are multiple on-going transmissions, the proposed
protocol requires them to finish at the same time. The main difference
compared with previous proposals is that only the first transmission is
randomly selected (i.e., a leader will follow the CSMA/CA rules). The
remaining STAs will only start a concurrent transmission if they overheard a
transmission from the leader. The STAs’ scheduling is determined by a user
matching solution developed in the paper aiming to both maximize the system
throughput and fairness.
Table IV summarizes the main characteristics of the surveyed un-coordinated
uplink MU-MIMO MAC protocols.
TABLE IV: Un-coordinated uplink MU-MIMO MAC protocols
Remarks Evaluation Tool CSI Scheme MUD Key Assumption Scheduling Jin [31],
compare SU and MU-MIMO, 2008 Analysis Implicit feedback Zero forcing
Orthogonal preambles - Tan [52], carrier counting, 2009 Testbed Implicit
feedback Chain decoding - - Babich [32], asynchronous MPR, 2010 Analysis - -
Code correction scheme - Ettefagh [54], cluster-based MU-MIMO, 2011 Simulation
+ Analysis Implicit feedback - Ideal channel - Mukhopadhyay [55], ACK-aware
MPR, 2012 Simulation + Analysis - - MPR frames decodable - Lin [56], delay
packet decoding, 2013 Simulation + Testbed Compressive sensing Zero forcing
Fixed frame length - Wu [58], throughput model, 2014 Simulation + Analysis
Implicit feedback ZF-SIC Fixed data rate - Kuo [59], leader-based contention,
2014 Simulation + Analysis Implicit feedback ZF-SIC Rayleigh fading User
matching
#### IV-A2 Coordinated Channel Access
##### Scheduled Data Transmissions
Huang et al. in [42] present an MPR MAC protocol that utilizes CDMA to
separate the compound frames. When the STA’s backoff counter reaches zero, a
function that considers the number of STAs in the network, the current channel
state and the MPR capability of the AP is used to schedule STAs. The MAC
procedure is illustrated in Figure 6. The CSI is claimed to be obtained from
the downlink transmission. Data frames are assumed to have the same length.
The analysis is based on the Bianchi’s model [51], and the optimal
transmission probability is obtained by one-dimensional search procedure, such
as Bisection and Newton-Raphson method [60]. Results obtained from the
analytic model and simulations show that the proposed scheme reduces
collisions and avoids considerable transmission errors.
Tandai et al. in [43] propose a synchronized access scheme coordinated by the
AP. On receiving applying-RTSs (A-RTSs) from STAs, the AP responds with a
pilot-requesting CTS (pR-CTS) to expect pilots. Based on the CSI estimated
from the sequential pilots, the AP sends a Notifying-CTS (N-CTS) to inform the
selected STAs for parallel transmissions. A unique subcarrier is assumed to be
allocated to each STA to differentiate A-RTSs, and the MMSE decoder is adopted
to separate the simultaneously received signals. According to the simulation
results, the proposed scheme can reduce the overhead and increase the
throughput.
Li et al. in [61] present a coordinated MAC protocol to improve Channel
Utilization in both Time and Spatial domain (CUTS). In the considered
approach, the AP first sends a grant packet to all STAs. Those backlogged STAs
reply to the AP with a transmission request. Then, the AP analyzes all the
information received, selects and signals those allowed STAs to transmit
simultaneously. The main innovation is that the channel contention is done in
the frequency domain. In CUTS, each STA selects one of the available
subcarriers to transmit a predefined signal, which is used by the AP to
identify the winning STAs. Both experimental and simulation results show the
significant gains achieved by CUTS.
##### Un-scheduled Data Transmissions
Zheng et al. in [33] propose a MU-MIMO MAC protocol called MPR-MAC, which
extends CTS and ACK to accommodate multiple transmitters. The MPR-MAC
procedure is the same as illustrated in Figure 6, where the STAs that won the
channel contention will transmit RTSs at the same time. The AP then replies
with an extended CTS that grants concurrent transmissions to the requesting
STAs. A set of orthogonal training sequences are assigned to STAs by the AP to
facilitate the channel estimation. A Finite Alphabet (FA) based blind
detection scheme is adopted for the frame separation. The network is assumed
to be saturated, and all data frames have the same length. The throughput
analytic model is based on Bianchi’s work [51], and the best transmission
probability is obtained via numerical techniques (i.e., a derivation
function). The authors claim that the throughput increases nearly linearly
with the number of antennas at the AP.
Since the MPR-MAC follows the conventional IEEE DCF access scheme, the
probability of more than one STA choosing the same random BO is low. In order
to increase the number of parallel transmissions, an enhancement called Two-
Round RTS Contention (TRRC, Figure 8) is also proposed in [33]. Compared to
MPR-MAC, TRRC has two RTS contention rounds. Namely, instead of sending a CTS
after the first RTS round, the AP waits for an extra round to recruit more
RTSs. The results show that TRRC obtains a further $7\%$ throughput increase.
Figure 8: TRRC: Two-Round RTS Contention
Barghi et al. in [62] present an MPR-aware MAC protocol for receiving two
concurrent frames by introducing a waiting time window ($t_{\text{w}}$) at the
AP. Specifically, when the AP receives a first RTS, it will wait for a time
period of $t_{\text{w}}$ to recruit a second RTS for double-frame receptions.
CTS and ACK are extended with an extra address field to accommodate two STAs.
A space-time code scheme is adopted to detect multiple frames. The channel is
assumed to be error-free and channel coefficients are assumed to be known.
Based on the obtained results, the authors claim that, by widening
$t_{\text{w}}$, (1) the probability of double-frame transmissions increases,
while the probability of collision (i.e., the probability that the number of
transmitted RTSs is more than two) increases as well; and (2) the performance
of the proposed MPR-aware MAC scheme improves significantly compared to that
of the IEEE 802.11 standard one.
Zhou et al. in [63] propose a two-round channel contention mechanism, which
divides the MAC procedure into two parts. The two parts, namely, the random
access and the data transmission, are illustrated in Figure 9. The random
access finishes as soon as the AP receives $M_{\text{random}}$ (the maximum
number of STAs that can transmit simultaneously) successful RTSs, and then the
data transmission starts. In the random access part, the AP delivers two types
of CTSs: Pending CTS (PCTS) and Final CTS (FCTS). The former responds to RTS
and the latter notifies all STAs about the start of data transmissions. Those
STAs, who have sent RTSs within a predefined time threshold
($T_{\text{timeout}}$), will transmit simultaneously. If the number of
contending users is less than $M_{\text{random}}$, the $T_{\text{timeout}}$
will trigger data transmissions as well. The AP obtains the CSI from RTSs, and
utilizes the MMSE detector to separate STAs’ signals. Data frames are assumed
to be of fixed length. Both simulation and analytic results show that the two-
round contention scheme outperforms the IEEE 802.11 single-round one in terms
of throughput and delay.
(a) Without timeout
(b) With timeout
Figure 9: Two-round MAC procedures, $M_{\text{random}}=3$
A similar work with two contention rounds is presented in [64]. Compared to
[63], [64] devises a shorter second contention round, where a single message
is used to reply all successfully received RTSs. In addition, a special focus
is placed on $2$-nd round Contention Window ($CW_{\text{2nd}}$), a parameter
making the length of the second contention round elastic. By evaluating the
proposal in simulations, a set of optimal $CW_{\text{2nd}}$ values that can
obtain the highest system performance are identified.
Zhang in [65] further extends two contention rounds to multiple rounds, which
give STAs more opportunities to compete for the channel using a threshold
derived from an optimal stopping algorithm [66]. Meanwhile, an auto fall-back
to single-round scheme is also proposed in case the traffic is low and the
single-round scheme can provide higher throughput. Frame arrivals are assumed
to follow the Poisson distribution. Results obtained from simulations and the
analytical model show the multi-round contention can increase the channel
utilization rate in a small to moderate network.
Jung et al. in [67] present a coordinated uplink MPR scheme, which extends the
work in [32] to allow both synchronous and asynchronous transmissions by
employing an additional feedback channel. The proposed MAC procedure is shown
in Figure 10. On receiving an RTS from STA$2$, the AP replies with a CTS that
includes the MPR vacancy (the remaining space for parallel uplink
transmissions). STAs who overhear the MPR vacancy will compete for the channel
to transmit along with STA$2$. Once a STA finishes transmitting ahead of the
other one, the AP immediately sends an ACK with the updated MPR vacancy
information through the additional channel to allow other STAs to compete for
the newly available MPR space. The authors assume an orthogonal training
sequence is included in the preamble of each frame for estimating the channel.
Based on results obtained from the Markov chain analytic model and
simulations, the authors claim that the proposed scheme achieves higher
channel efficiency in scenarios where the frame size and transmission rates
are dynamically varying.
Figure 10: Asynchronous data transmissions with one MPR vacancy
Table V summarizes the main characteristics of the surveyed coordinated uplink
MU-MIMO MAC protocols.
TABLE V: Coordinated Uplink MU-MIMO MAC Protocols
Remarks Evaluation Tool CSI Scheme MUD Key Assumption Scheduling Huang [42],
SNR based MPR, 2008 Simulation + Analysis Downlink estimation CDMA Fixed data
length Optimal SNR Tandai [43], TDMA signalling, 2009 Simulation Implicit
feedback MMSE Unique subcarrier Best CSI Li [61], subcarrier contention, 2014
Simulation + Testbed Implicit feedback - Error-free channel Subcarrier Zheng
[33], DCF based MPR, 2006 Analysis Implicit feedback Blind detection Fixed
data length - Barghi [62], MPR-aware MAC, 2011 Simulation + Analysis - STC
Perfect channel - Zhou [63], two-round contentions, 2010 Simulation + Analysis
Implicit feedback MMSE Fixed data length - Liao [64], elastic $2$-nd round,
2012 Simulation Implicit feedback - Error-free channel - Zhang [65], multi-
round contentions, 2010 Simulation + Analysis - - Poisson arrivals - Jung
[67], asynchronous MPR, 2012 Simulation + Analysis Implicit feedback - Extra
ACK channel -
#### IV-A3 Discussions on Uplink MU-MIMO MAC Proposals
Here, we conclude this subsection by providing highlights, open aspects and
next steps after reviewing the uplink MU-MIMO MAC proposals in the literature.
##### Highlights
* •
In the un-coordinated category, by simply considering the number of summarized
papers, the asynchronous access has got more attention than the synchronous
one. This is because, in the synchronous case, we can only benefit from the
simultaneous multi-frame transmissions when several STAs end their backoff at
the same time. One point that seems not fully solved by asynchronous solutions
is how reliable the distributed counter of concurrent transmissions at each
node is, as which may negatively affect the network performance by causing
collisions. In addition, it is unclear how such carrier sense should be
implemented, as the nodes need to identify the exact number of on-going
transmissions.
* •
In the coordinated category, the AP may schedule (scheduled) or not schedule
(unscheduled) what STAs to transmit. It is surprising to see that there are
more unscheduled proposals. The reason could be (1) the difficulty and
overheads behind the uplink scheduling, and (2) simply keeping the
”decentralized” access philosophy for WLANs.
##### Open aspects
* •
Comparison between different proposals is still difficult because of the
different considered scenarios (e.g., channel models, CSI assumptions and STAs
placement). Therefore, an open challenge is to present a comparative analysis
framework to benchmark the most relevant and promising proposals.
* •
Most of the uplink proposals assume implicit CSI. However, calibration
problems, due to the presence of heterogeneous devices and channel conditions,
make the implicit CSI difficult to be implemented.
* •
It is likely that, in the next generation 802.11 standard, an extended RTS/CTS
exchange will be used to signal and protect MU-MIMO frames, which would mean
the AP-coordinated approach will be adopted. Solutions to further reduce the
handshaking overheads must be considered.
##### Next steps
* •
The IEEE 802.11ax study group is discussing to include uplink MU-MIMO
[68][69]. At the current stage, it is not clear yet how such mechanism will be
implemented. In our perspective, uplink MU-MIMO would follow a coordinated and
scheduled approach controlled by the AP. This judgement is made based on the
following two reasons. First, the AP is playing a central role in collecting
CSI from all STAs for downlink MU-MIMO transmissions in 802.11ac. We believe
this explicit CSI feedback mechanism will be kept in 802.11ax for the backward
compatibility. Additionally, by playing a central role, the AP can ask for
buffer and other information from STAs, enabling it to make the best
scheduling decision in the dense scenario.
* •
Regarding the CSI of the uplink, in our opinion, 802.11ax could extend and
reuse the explicit CSI scheme periodically conducted by 802.11ac for the
downlink MU-MIMO transmissions. For example, the AP can estimate the uplink
channel coefficients from the same packets that carry the downlink channel
coefficients.
* •
802.11ax should place STAs in virtual groups, and poll the STAs in each group
based on the CSI information as well as the estimation of the buffer occupancy
of the STAs to reduce overheads.
### IV-B MAC Proposals for The Downlink
A commonly used MAC procedure for MU-MIMO downlink transmissions is
illustrated in Figure 11. The AP firstly sends out a modified RTS containing a
group of targeted STAs. On receiving the RTS, those listed STAs estimate the
channel, integrate the CSI into the extended CTS and send it back. As soon as
the AP receives all successful CTSs, it precodes the outgoing frames based on
the feedback CSI.
Figure 11: A successful downlink MU-MIMO transmission
#### IV-B1 Downlink MU-MIMO MAC Proposals
Cai et al. in [38] propose a distributed MU-MIMO MAC protocol with extended
RTS/CTS frames. The CSI is obtained from the RTS/CTS exchange. An additive
white Gaussian noise (AWGN) channel is assumed, and a leakage-based precoding
scheme is utilized to cancel interference. The authors adopt a queue based
scheduling scheme, which prioritizes frames with the longer waiting time in
the AP buffer. The results derived from the queue-based unsaturated model [70]
and simulations show the proposed multi-user MAC substantially outperforms the
single-user one.
Kartsakli et al. in [44] propose four multi-user scheduling schemes for
concurrent frame transmissions, namely, MU-Basic, MU-Deterministic, MU-
Threshold Selective, and MU-Probability. The opportunistic beamforming that
selects STAs with the highest signal-to-interference-plus-noise ratio (SINR)
for each randomly generated beam is utilized. The CSI is fedback by STAs
during the channel contention phase. A block fading channel is assumed, which
means the channel remains constant during a frame transmission time. Based on
simulation results, the authors argue that the proposed schemes achieve
notable gains against the single-user case, although there is still
considerable space for improvements compared to the theoretical capacity.
Gong et al. in [39] propose a modified CSMA/CA protocol with three different
ACK-replying mechanisms, namely, the polled ACK response, the scheduled ACK
response and the failed ACK recovery. A weighted queueing mechanism that
associates ACs with the value of contention window is also proposed to address
the fairness concern. The MMSE precoding scheme is adopted and the CSI is
assumed to be known. The simulation results show that the proposed protocol
provides a considerable performance improvement against the IEEE 802.11n
beamforming based approach when the SNR is high.
Zhang et al. in [40] present a One-Sender-Multiple-Receiver (OSMR)
transmission scheme for WLANs. The authors firstly implement an OSMR prototype
using Universal Software Radio Peripheral (USRP) [71] to explore the
feasibility of OSMR at the PHY level. Then, based on the study of the OSMR PHY
characteristics, they modify the RTS/CTS frames to support the channel
estimation. Simulations of the proposed extension are conducted at the MAC
level. A greedy scheduling algorithm is used to transmit as many frames as
possible in a TXOP with the urgent frames being prioritized over the normal
ones. The ZF precoding scheme is adopted, while a flat fading channel is
assumed. The simulation results show that a significant performance gain is
achieved by employing OSMR transmissions.
Liao et al. in [34] present a MAC protocol for downlink MU-MIMO transmissions,
where frames are scheduled to each STA by FIFO. The CSI is obtained through
estimating the training sequence included in the CTS preamble. The channel is
assumed to be error-free and independently fade from frame to frame, which
creates independent channels from the AP to STAs. Simulation results show that
a significant throughput gain is obtained by exploiting the spatial domain of
the channel.
Bellalta et al. in [35] explore the performance of downlink MU-MIMO with the
packet aggregation, and look into the interplay between the buffer size and
the number of antennas at the AP. The impact of these two parameters on the
system throughput and packet delay is also investigated. The proposed
scheduler tries to maximize the number of transmitted packets in each stream,
and also balance the duration of all streams. RTS/CTS control frames are
modified for the CSI estimation, and ZF is used to null the interference. The
analytical model is based on the work presented in [72], assuming packets’
Poisson arrival in a non-saturated network. Both simulation and analytical
results show the performance of the proposed system is close to optimal.
Redieteab et al. in [73] investigate three different transmission schemes in a
PHY and MAC cross-layer platform, which are SU-MIMO, MU-MIMO with multi-user
interference and MU-MIMO without multi-user interference. An IEEE 802.11ac
channel model [74] is utilized to emulate channel variations. The MAC layer is
made to be compliant with the 802.11ac specification draft, and the amendment
defined ECFB protocol is assumed to be employed to obtain the CSI. The ZF
channel-inversion precoding scheme is used to decode frames. Based on the
simulation results, the authors conclude that multi-user interference has
important effects on MU-MIMO transmissions, e.g., it results in less
throughput. Therefore, an automatic switching algorithm between SU-MIMO and
MU-MIMO is suggested by the authors.
Cha et al. in [75] compare the performance of downlink MU-MIMO to Space Time
Block Coding. The ZF precoder is utilized, and the CSI is obtained at the AP
by receivers’ feedback. A Rayeigh fading and error-free channel is assumed.
The results show that the downlink MU-MIMO scheme produces a higher throughput
than the STBC one if transmitted frames are of similar length, while the
results reverse in a fast-varying channel due to the high overheads of the CSI
feedback .
Balan et al. in [76] implement a distributed MU-MIMO system that consists of
several multi-antenna APs, which are connected and assumed to be synchronous
by a coordinating server. The authors employ Zero Forcing Beamforming (ZFBF)
for the frame separation. The CSI is acquired from uplink pilot symbols. Blind
Interference Alignment (BIA) is used when the CSI is unavailable. The system
is evaluated in the Wireless Open-Access Research Platform (WARP) platform
[77]. The experimental results show that the presented MU-MIMO system can
achieve high data rates and approach the theoretical maximum throughput.
Zhu et al. in [78] investigate the required modifications for TXOP to support
multi-user transmissions. The proposed scheme, called multi-user TXOP (MU-
TXOP), enables a STA whose AC won the TXOP to share the transmission period
with MPDUs of other ACs. The authors assume all STAs can be grouped for multi-
user downlink transmissions. Simulation results show that the proposed scheme
not only obtains a higher throughput, but is also more fair compared to the
conventional one.
Ji et al. in [79] present a cooperative transmission scheme that addresses the
redundant Network Allocation Vector (NAV) setting and the outdated SINR
problems. The redundant NAV setting usually occurs at STAs located in the
overlapped areas of neighboring WLANs, while the outdated SINR problem is
caused due to the delay between the channel estimation and the data
transmission. The authors utilize reserved bits in control frames to announce
the last frame in a transmission to synchronize the NAV setting, and employ
STAs’ ACKs to re-estimate and correct the SINR. The analysis model assumes
frames that arrive to STAs to follow the Poisson process. The results obtained
from the model and simulations show the enhanced scheme can achieve noticeable
performance gains compared to the sampled one.
Table VI summarizes the main characteristics of the surveyed downlink MU-MIMO
MAC protocols.
TABLE VI: Downlink MU-MIMO MAC Protocols
Remarks Evaluation Tool CSI Scheme MUIC Key Assumption Scheduling Cai [38],
reduce AP-bottleneck effect, 2008 Simulation + Analysis Explicit feedback
Leakage coding AWGN channel Priority queue Kartsakli [44], $4$ scheduling
schemes, 2009 Simulation Explicit feedback Beamforming Block fading Highest
SINR Gong [39], ACK-replying schemes, 2010 Simulation - MMSE Assume CSI known
Weighted queue Zhang [40], OSMR, 2010 Simulation + Testbed Explicit feedback
Zero Forcing Flat fading Greedy Liao [34], throughput and delay gain, 2011
Simulation Implicit feedback - Error-free channel Per-STA FIFO Bellalta [35],
packet aggregation, 2012 Simulation + Analysis Explicit feedback Zero Forcing
Poisson arrival Per-STA FIFO Redieteab [73], PHY+MAC platform, 2012 Simulation
Explicit feedback Zero Forcing - - Cha [75], STBC & MU-MIMO, 2012 Analysis
Explicit feedback Zero Forcing Error-free channel - Balan [76], multi-AP
system, 2012 Simulation + Testbed Implicit feedback ZFBF, BIA Phase
synchronous - Zhu [78], multi-user TXOP, 2012 Simulation - - All STAs
groupable - Ji [79], outdated NAV and SINR, 2014 Simulation + Analysis
Explicit feedback - Poisson arrival -
#### IV-B2 Discussions on Downlink MU-MIMO MAC Proposals
##### Highlights
* •
Regarding the CSI acquisition, some assume the channel reciprocity [34][76],
which allows the AP to obtain the CSI directly from frames transmitted by the
STAs, while most papers adopt the explicit way, though none of them follow the
approach defined in IEEE 802.11ac. However, it is worth to point out that no
papers discuss or justify whether a SIFS interval is long enough for STAs to
process received preambles and estimate the channel.
* •
In terms of the MUIC scheme, the Zero Forcing approach is the most common one
due to its simplicity and efficiency.
##### Open aspects
* •
Performance comparisons of different solutions, including the approach
considered by IEEE 802.11ac, are still missing. It is of special interest to
compare if it is better to obtain the CSI for each specific transmission,
which guarantees the CSI is fresh, or only periodically, at the risk that the
CSI may be outdated, hence creating higher interference between the different
spatial streams. First works on this direction can be found in [80][81],
though more efforts are still required.
* •
In addition, analytical models on understanding the interactions between the
CSI state, the number of active users, the buffer size, and other specific
mechanisms such as packet aggregation, need more exploration. A first work is
presented in [35], where the authors evaluate the impact of the finite buffer
size with random arrivals at the AP in a donwlink MU-MIMO system.
##### Next steps
Since the IEEE 802.11ac amendment has already been standardized, research on
downlink MU-MIMO should focus on how the 802.11ac approach can be extended or
optimized towards the next amendment-IEEE 802.11ax. Next steps could emphasize
on tuning the downlink MU-MIMO mechanisms of 802.11ac, such as the CSI
acquisition process (e.g., the rate of requests and the set of sampled nodes),
scheduling algorithms considering both the instantaneous traffic and QoS
requirements, and the integration with other IEEE 802.11 mechanisms (e.g.,
multi-cast video communications, as defined by IEEE 802.11aa [82].
### IV-C MAC Proposals for Integrated Up/down-link
Only a few works have considered both MU-MIMO uplink and downlink
transmissions in a single MAC protocol.
#### IV-C1 Integrated Up/down-link MU-MIMO MAC Proposals
Kim et al. in [83] devise a down/up-link back-to-back transmission scheme to
synchronize STAs. The scheme is called Per-flow MAC (PF-MAC, Figure 12), where
RIFS stands for Reduced Inter Frame Space. The AP first sends a Group-RTS
(GRTS) that includes a list of STA addresses to initiate the downlink
transmission. As soon as the AP received expected CTSs, it sends data frames
to the listed STAs. The STAs who received frames then send back ACKs
sequentially. Through the downlink transmission and a Ready to Receive (RTR)
frame from the AP, STAs are synchronized for the parallel uplink transmission.
The CSI is estimated from uplink frames. The limitation of the proposal is
that the uplink transmission can only be started by the downlink one, which
may not be desirable in some scenarios where the uplink access is urgent. Note
that the proposal is just a conceptual model without any simulation or
analytic results.
Figure 12: PF-MAC: Per-flow MAC
Zhao et al. in [45] propose an opportunistic MU-MIMO MAC protocol for Multi-
channel Multi-radio WLANs. In the downlink, a Group RTS is used to signal the
selected STAs, which reply sequentially with a CTS before the AP transmits to
them simultaneously. In the uplink, after the RTS/CTS exchange, the STAs are
signalled to start transmissions using a Group CTS message. Note that both
uplink and downlink transmissions can be done simultaneously as they use
different channels. A common control channel is used to exchange information
about the packet status and the set of available channels for next
transmissions. The CSI is assumed available at the AP. The results show the
proposal is able to outperform other multi-channel MAC schemes.
Li et al. in [46] propose a Multi-user MAC (MU-MAC) protocol, which supports
Multi-Packet Transmission (MPT) in the downlink and multiple control packets
(e.g., CTSs or ACKs) reception in the uplink. OFDM preambles are utilized to
facilitate the CSI extraction from the simultaneously received control frames.
The frame errors are assumed to come from collisions only, and all frames are
transmitted with the same rate. The scheduling scheme jointly considers the
CSI history, the AP’s queueing state and the frames’ application categories.
The analytic model is based on and extended from [38]. By observing the
results, the authors claim that MU-MAC outperforms MPT-only MAC in terms of
the maximum number of supported STAs, and this gain will further increase as
the AP employs more transmitting antennas.
Shen et al. in [41] present a High Throughput MIMO (HT-MIMO) MAC protocol that
utilizes frequency signatures to differentiate simultaneously received control
frames. HT-MIMO works in the PCF mode, hence both uplink and downlink
transmissions can only be initiated by the AP. The CSI is obtained by a
channel measurement method. The uplink and downlink channels are assumed to be
symmetrical. A greedy scheduling algorithm is adopted with the consideration
of fairness and the queue occupancy. By comparing with Cai’s proposal [38],
the analysis results show that the HT-MIMO MAC outperforms the 802.11n SU-MIMO
and Cai’s MU-MIMO schemes.
Liao et al. in [36] propose a unified MU-MIMO MAC protocol (Uni-MUMAC) for
IEEE 802.11ac WLANs by integrating both uplink and downlink enhancements. An
analytic model is developed based on [51]. The implicit CSI acquisition scheme
is adopted by assuming the channel reciprocity. Through adaptively tuning
$CW_{\text{2nd}}$ (a parameter controls the $2$nd uplink contention), and
comparing with Li’s proposal [46], the authors conclude that (1) Uni-MUMAC
performs well in both the traditional downlink-dominant and the emerging
down/up-link balanced traffic scenarios; (2) one-side enhancement (e.g., the
uplink enhancement) does not bring the same benefits to the other side.
Yun et al. in [37] present a multi-point to multi-point MIMO system, where the
uplink multiplexing is implemented in the SORA platform, while the downlink is
implemented in the USRP platform. Multiple APs are coordinated by a
controller, and connected via an Ethernet cable. A leader concept is adopted
for both the uplink and downlink medium access. A trigger frame that includes
co-senders’ addresses is sent out by the leader who first won the channel
contention. Then, the co-senders transmit preambles sequentially as specified
in the trigger frame for the CSI estimation. The downlink scheduling is based
on the packet arrival time and the length of waiting time in the queue, while
the co-senders’ selection in the uplink is randomly made. ZF is employed in
both the uplink and downlink for frames’ de/pre-coding.
Table VII summarizes the main characteristics of the Up/down-link MU-MIMO MAC
protocols.
TABLE VII: Integrated Up/down-link MU-MIMO MAC Protocols
Remarks Evaluation Tool CSI Scheme MUD & MUIC Key Assumption Scheduling Kim
[83], back-to-back transmissions, 2008 Conceptual model Implicit feedback - -
- Zhao [45], multi-radio WLANs, 2009 Simulation - - CSI known Opportunistic Li
[46], MU-MAC, 2010 Simulation + Analysis Explicit feedback - Error-free
channel History CSI Shen [41], control frames’ encoding, 2012 Analysis - -
Symmetrical channel Greedy Liao [36], unified up/downlink MAC, 2013 Simulation
+ Analysis Implicit feedback - Channel reciprocity FIFO Yun [37], multi-APs to
multi-STAs, 2013 Testbed Sequential preambles Zero Forcing - FIFO
#### IV-C2 Discussions on Integrated Up/down-link MAC
* •
The comments detailed in previous subsections are also valid here. Regarding
integrated up/down-link MU-MIMO MAC protocols, we would like to remark that
there are still too few works to allow us to categorize them. A first simple
classification can be made based on if the AP governs both downlink and uplink
communication [41][46][83], or the random access is still considered to
initiate transmissions from STAs [36][37].
* •
Following the IEEE amendment evolution towards 802.11ac, we believe the AP
will also play a coordinating role in scheduling uplink MU-MIMO transmissions.
* •
However, there is another major issue that requires further research, which is
the need for a joint design of up/down-link MU-MIMO MAC protocols, rather than
just simply combining solutions designed independently. The reason is that
one-side enhancement could negatively affect the other side. For instance, as
pointed out in [36], the optimal parameter for uplink MU-MIMO transmissions
does not bring the same benefit to the downlink.
## V Future Directions
The goal of all the surveyed MAC proposals is to improve MAC efficiency in
essence. More specifically, they try to convert the raw data rate brought by
the PHY advance (i.e., MU-MIMO) to the MAC throughput as efficiently as
possible.
With this goal in mind, we discuss possible future research directions for MU-
MIMO MAC protocols through a micro perspective and a macro perspective,
respectively. In the former, we discuss some important performance-affecting
aspects that have not been specified in IEEE 802.11ac; we also identify key
factors that make the conversion from the PHY data rate to the MAC throughput
inefficient, and then give our thoughts on possible ways to improve the MAC
efficiency. In the latter, we envisage the MAC’s role in facilitating the
integration of the future heterogeneous networks.
### V-A The Micro Perspective: Within MU-MIMO Based WLANs
#### V-A1 What Does 802.11ac Not Specify?
The downlink MU-MIMO transmission is one of the most significant features
introduced by IEEE 802.11ac. In order to perform multi-user transmissions, the
amendment proposes the ECFB scheme to feedback the required CSI, and devises a
group identifier (Group-ID) field in the PHY preamble to facilitate grouping
STAs. However, the following two important factors are not specified in the
amendment.
* •
Scope and frequency of CSI feedback: [80] and [81] have shown the significant
impact of the CSI feedback on the system performance. A clear dilemma
regarding the scope and the frequency of the CSI feedback is that a large
scale (e.g., all STAs in the network) or frequent CSI requests will introduce
huge overheads, while the opposite way leads to that the rendered channel
information might be outdated. Therefore, adaptive algorithms are needed to
dynamically adjust the scope and the frequency of CSI feedback.
* •
Conditions to group/re-group STAs: Although it can be argued that the way of
grouping STAs depends on the specific application, a smart grouping algorithm
has to be designed to identify those STAs that can be co-scheduled, or on what
conditions STAs’ re-grouping would be triggered.
#### V-A2 How to Improve MAC Efficiency?
As shown in the paper, significant research efforts have been made to adapt
IEEE 802.11 MAC to advances such as the multi-antenna technique. However, the
MAC throughput is still much lower than the PHY raw rate (lower than $70\%$ in
most cases [84]). The throughput loss mainly comes from the so-called
overheads, which include the management frames (e.g., association
requests/responses), the compulsory idle duration (e.g., the random BO,
DIFS/SIFS), control frames (e.g., RTS/CTS/ACK) and frame headers (e.g., PHY
preambles, PHY/MAC headers). Other non-overhead factors contributing to the
throughput loss include frame collisions and the airtime unfairness caused by
low rate STAs that monopolize the channel. These fundamental IEEE 802.11
mechanisms and features limit the MAC efficiency. In a packet capture test
conducted in a dense area of Tokyo [85], the results show that data frames
only account for $23\%$ of all types of frames ($46\%$ management frames,
$30\%$ control frames, $1\%$ others); moreover, most of management frames are
in the 802.11b format and transmitted at $1$ Mbps to ensure the
interoperability.
The newly created High Efficiency WLAN study group-802.11ax is still at its
early stage. Here, we give our thoughts on possible solutions to improve the
MAC efficiency.
* •
Cooperation among multiple bands: Management frames, control frames and frames
headers are necessary to facilitate correct receptions of data. They can not
be eliminated in the current 802.11 communication architecture. At least the
evolution from IEEE 802.11-1997 to 802.11ac, what we have seen is an ever-
increasing length of PHY preamble. Cooperation among multiple bands could be
an effective way to control overheads. Specifically, a future smart device is
likely to be equipped with multiple interfaces operating in multiple bands
[86]. Thus, 802.11ac at $5$ GHz, could be a candidate for the carrier sense
and data transmissions across rooms; 802.11ah below $1$ GHz, could be utilized
to transmit management and control frames; while 802.11ad at $60$ GHz, could
be used for very high speed data transmissions in the line of sight. [87] has
suggested a possible usage model using 802.11ah for signalling among APs,
while 802.11ac for data frames.
* •
Revision of the backoff scheme: The random BO scheme is a key function
employed by IEEE 802.11 to avoid collisions. Unfortunately, collisions can not
be eliminated and remain as one of the most degrading factors to the system
performance. [88] proposes a collision-free solution, where STAs adopt a
deterministic BO instead of a random one after successful transmissions for
single-antenna based WLANs. On the other side, works in [61] and [89] suggest
to shift the random BO countdown from the time domain to the frequency one by
treating OFDM subcarriers as integers. These innovative ideas can be
considered to support multi-antenna based WLANs in both downlink and uplink.
* •
Uplink MU-MIMO transmissions: The Internet traffic has evolved from web
browsing and file transfers to a wide variety of applications, which include
considerable amount of content-rich files generated by users, such as the
video conferencing, social networks and cloud uploading services. Although the
enhancement for uplink transmissions has recently gained attention, there is
still much space for improvements since the latest amendment-IEEE 802.11ac
does not support uplink MU-MIMO transmissions. In addition, there are only a
few works that unify MU-MIMO downlink and uplink protocols into a single
communication system.
* •
Orthogonal FDMA: OFDMA is currently under consideration for next-generation
WLANs to improve the system efficiency [15]. The interest on OFDMA is
originated from the efficiency loss caused by the 802.11ac channel bonding (up
to $160$ MHz). Transmitting over wider channels reduces the transmission
duration, though it also magnifies the negative impact of the overheads, since
the duration of inter-frame spaces and control packets are constant regardless
of the channel width. OFDMA enables WLANs to multiplex transmissions from/to
different STAs over a single channel by assigning a different OFDM subcarrier
to each transmission. The result is that, transmissions last longer, as they
are transmitted at lower rates, but the efficiency of the system is improved,
as more data are transmitted in parallel. Regarding MU-MIMO transmissions,
using OFDMA would have a positive impact on overheads, since STAs only need to
feedback a lower amount of CSI, which is only proportional to the subcarriers
of the used subchannel.
* •
Full-duplex transmissions: The current form of communications in WLANs is
half-duplex, namely, transmissions and receptions are allocated to different
time slots or frequency bands. The full-duplex transmission has the potential
to double the channel capacity by allowing simultaneous transmissions and
receptions with the same frequency [90][91]. In addition, the coming of full-
duplex transmissions hints us to rethink the IEEE 802.11 fundamental MAC
mechanism-CSMA/CA, which is employed based on the long-held assumptions that
(1) wireless devices can not transmit and receive at the same time, and (2)
wireless devices can not detect collisions while transmitting [92]. The
research on full-duplex transmissions has just started in recent years.
Although the challenging part is to cancel self-interference at the full-
duplex transceiver, the MAC scheme needs to be revised as well [93][94].
* •
Massive MIMO transmissions: Having MU-MIMO already been integrated in the
latest standards (e.g., IEEE 802.11ac and 4G LTE), and in order to further
reap the benefits of MIMO in a much greater scale, Massive MIMO, with orders
of magnitude more antennas equipped at the transmitter, has recently attracted
much research attention. Massive MIMO has the promise to increase the spectral
efficiency to tens of hundreds of bps/Hz, and simultaneously improve the
energy efficiency [95][96], which enables it to be a very good technology
candidate for next-generation wireless standards that target highly-dense user
scenarios. The first challenge of employing Massive MIMO is the limited space
at the AP, since antennas are required to be placed at the least half
wavelength apart. The second challenge is the overhead. IEEE 802.11ac uses
ECFB-the explicit way to feedback CSI. The channel sounding time and the CSI
volume are proportional to the number of AP’s antennas, which would occupy
considerable amount of time and bandwidth, especially in the Massive MIMO
case. The first challenge is out of the scope of the paper, while the second
one needs more research efforts on the precoding, the antenna selection, and
exploring the antenna correlation to reduce the channel sounding time and the
CSI volume.
### V-B The Macro Perspective: Integration with Heterogeneous Networks
It seems to be a trend that in the future there will be a huge and smart
network that integrates heterogeneous networks (e.g., WLANs, broadband mobile
networks and sensor networks), which means individual network will have to
collaborate with others to provide services, rather than just coexist.
Obviously, the integration or the inclusion of WLANs require unique changes at
the MAC layer.
#### V-B1 How, who and when to collaborate?
With the PHY layer’s focus on the air interface and the network layer’s focus
on the routing, the MAC layer plays a role in deciding how to collaborate, who
to collaborate and when to collaborate [97][98]. For example, imagining a
scenario that a sink of a smart metering sensor network requests an AP to
forward the collected data to a user’s mobile phone, the AP first checks
whether the mobile phone is in its vicinity to decide whether to forward the
data by itself or to relay the data to other APs. And then, the AP checks the
channel condition and the queueing status to decide who and when to transmit
the data. Regardless the scale and the type of collaborations, exchanging
management frames and control frames are needed to build connections with
other entities. Note that today’s device-to-device communications already
account for a significant part of total wireless traffic, and they are
regarded as one of the most important challenges for $5$G networks [99][100].
Therefore, the MAC scheme for the integrated network has to take how, who and
when to collaborate into account.
#### V-B2 Multi-hop Cooperative MAC
In addition to the above consideration, it is inevitable for MAC protocols of
integrated networks to support multi-hop indirected links. A typical
cooperative scenario is that a receiver at the edge of the transmitter’s range
can benefit from its neighbours’ relaying. This opens research questions as
follows. First, the selection of optimal relay: (1) who will make the
selection, the source or the destination? (2) what is the criteria to choose
the optimal relay, signal strength, energy level or security issues? (3)
whether the chosen one will agree or be able to relay? Secondly, the timing of
relay: (1) immediately after the source transmission, or (2) compete for the
channel and then relay. Thirdly, the format of relay: (1) amplify-relay,
decode-relay or compress-relay, and (2) single relay or multiple relays [101].
Other issues, such as the hidden-node problem, cooperative diversity, MPT/MPR
functionalities and the joint MAC-routing design, also need to be considered
[102][103].
#### V-B3 Outdoor and mobile WLANs
The inclusion of WLANs to outdoors presents some significant challenges to the
traditional MAC, as which is designed to support limited mobility. First, due
to the user movement and the large scale fading, the outdoor channel is
varying faster than that of indoors, even with the same moving speed [104].
For that reason, the CSI feedback scheme needs to be reconsidered to report
the channel state timely, while maintaining low channel estimation overheads.
Secondly, the cellular operators may offload traffic to public or proprietary
WLANs (e.g., the carrier class Wi-Fi [105]) in a dense public stadium, in
which case, the seamless transfer and the QoS promised by the cellular network
need to be assured at the MAC layer.
## VI Concluding Remarks
The uplink and downlink MU-MIMO MAC protocols for WLANs are investigated and
categorized in the paper. Some typical MUD and MUIC techniques for de/pre-
coding are sampled, and the requirements for designing MU-MIMO MAC protocols
are identified. Based on the study, discussions are carried out to clarify
what challenges and future directions could be for designing effective MU-MIMO
MAC protocols.
Despite considerable research has been conducted, there still exists under-
explored areas toward simple, yet highly efficient MAC protocols for MU-MIMO
based WLANs, especially in the context of the rapid growth of wireless
devices. Therefore, we have given some of our thoughts in that regard. We hope
this survey paper would help the readers to summarize the current research
progress and inspire their future work.
## Acknowledgment
This work has been supported by the Spanish Government and the Catalan
Government under projects TEC2012-32354 (Plan Nacional I+D) and SGR2009#00617.
The authors are also grateful for the useful technical discussions with Xu
Zhang and Shan Zhang at NiuLab of Tsinghua University.
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|
arxiv-papers
| 2014-04-06T21:06:07 |
2024-09-04T02:50:00.784911
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Ruizhi Liao, Boris Bellalta, Miquel Oliver, Zhisheng Niu",
"submitter": "Ruizhi Liao",
"url": "https://arxiv.org/abs/1404.1622"
}
|
1404.1714
|
Characteristics of the Jaco Graph, $J_{\infty}(a),a\in\mathbb{N}$
(Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian
Mukungunugwa)111Affiliation of authors:
Johan Kok (Tshwane Metropolitan Police Department), City of Tshwane, Republic
of South Africa
e-mail: [email protected]
Paul Fisher (Department of Mathematics, University of Botswana), City of
Gaborone, Republic of Botswana
e-mail: [email protected]
Bettina Wilkens (Department of Mathematics, University of Botswana), City of
Gaborone, Republic of Botswana
e-mail: [email protected]
Mokhwetha Mabula (Department of Mathematics, University of Pretoria), City of
Tshwane, Republic of South Africa
e-mail: [email protected]
Vivian Mukungunugwa (Department of Mathematics and Applied Mathematics,
University of Zimbabwe), City of Harare, Republic of Zimbabwe
e-mail: [email protected]
###### Abstract
We introduce the concept of a family of finite directed graphs (_order a_)
which are directed graphs derived from a infinite directed graph (_order a_),
called the $a$-root digraph. The $a$-root digraph has four fundamental
properties which are; $V(J_{\infty}(a))=\\{v_{i}|i\in\mathbb{N}\\}$ and, if
$v_{j}$ is the head of an edge (arc) then the tail is always a vertexc
$v_{i},i<j$ and, if $v_{k}$ for smallest $k\in\mathbb{N}$ is a tail vertex
then all vertices $v_{\ell},k<\ell<j$ are tails of arcs to $v_{j}$ and
finally, the degree of vertex $k$ is $d(v_{k})=ak.$ The family of finite
directed graphs are those limited to $n\in\mathbb{N}$ vertices by lobbing off
all vertices (and edges arcing to vertices) $v_{t},t>n.$ Hence, trivially we
have $d(v_{i})\leq ai$ for $i\in\mathbb{N}.$ We present an interesting
Lucassian-Zeckendorf result and other general results of interest. It is meant
to be an _introductory paper_ to encourage exploratory research.
Keywords: Jaco graph, Hope graph, Directed graph, Jaconian vertex, Jaconian
set, Number of edges, Shortest path, Fibonacci sequence, Zeckendorf
representation
AMS Classification Numbers: 05C07, 05C12, 05C20, 11B39
## 1 Introduction
We introduce the concept of a family of finite Jaco Graphs (_order a_) which
are directed graphs derived from the infinite Jaco Graph (_order a_), called
the $a$-root digraph. The $a$-root digraph has four fundamental properties
which are; $V(J_{\infty}(a))=\\{v_{i}|i\in\mathbb{N}\\}$ and, if $v_{j}$ is
the head of an edge (arc) then the tail is always a vertexc $v_{i},i<j$ and,
if $v_{k}$ for smallest $k\in\mathbb{N}$ is a tail vertex then all vertices
$v_{\ell},k<\ell<j$ are tails of arcs to $v_{j}$ and finally, the degree of
vertex $k$ is $d(v_{k})=ak.$
###### Definition 1.1.
The family of infinite Jaco Graphs denoted by
$\\{J_{\infty}(a)|a\in\mathbb{N}\\}$ is defined by
$V(J_{\infty}(a))=\\{v_{i}|i\in\mathbb{N}\\}$,
$E(J_{\infty}(a))\subseteq\\{(v_{i},v_{j})|i,j\in\mathbb{N},i<j\\}$ and
$(v_{i},v_{j})\in E(J_{\infty}(a))$ if and only if $(a+1)i-d^{-}(v_{i})\geq
j.$
###### Definition 1.2.
For $a\in\mathbb{N},$ we define the series $(c_{a,n})_{n\in\mathbb{N}_{0}}$ by
$c_{a,0}=0,$ $c_{a,1}=1,$ $c_{a,n}=\min\\{k<n|ak+c_{a,k}\geq n\\}$ ($n\geq
2)$.
The connection between the $a$-root digraph $J_{\infty}(a)$ and the series
$(c_{a,n})$ is explained by the following lemma.
###### Lemma 1.1.
Consider the Jaco Graph $J_{\infty}(a)$ and let $n\in\mathbb{N}$ then the
following hold:
(a) $d^{+}(v_{n})+d^{-}(v_{n})=an.$
(b) $d^{-}(v_{n+1})\in\\{d^{-}(v_{n}),\,d^{-}(v_{n})+1\\}.$
(c) If $(v_{i},v_{k})\in E(J_{\infty}(a))$ and $i<j<k,$ then $(v_{j},v_{k})\in
E(J_{\infty}(a)).$
(d) $d^{+}(v_{n})=(a-1)n+c_{a,n}.$
###### Proof.
As $d^{+}(v_{n})=(a+1)n-n-d^{-}(v_{n}),$ result $(a)$ is obvious. We prove
result $(b)$ and $(c)$ simultaneously, using induction on $n.$ First of all,
$d^{-}(v_{1})=0$ implying $d^{-}(v_{2})=1=d^{-}(v_{1})+1.$
Let $n\geq 2$ and assume $(b)$ to hold for $m\leq n$ and $(c)$ to hold for
$m\leq n-1.$ In particular, $d^{-}(v_{n})>0.$ Let $\ell<n$ be minimal with
$(v_{\ell},v_{n})\in E(J_{\infty}(a)),$ i.e. $(a+1)\ell-d^{-}(v_{\ell})\geq
n.$ Let $\ell<j<n.$ By induction,we have $d^{-}(v_{\ell})\leq d^{-}(v_{j})\leq
d^{-}(v_{\ell})+j-\ell$ and $(a+1)j-d^{-}(v_{j})\geq n.$ Hence, and by choice
of $\ell,$ we have $(v_{k},v_{n})\in E(J_{\infty}(a))$ if and only if
$\ell\leq k<n,$ hence result $(c)$ is is valid for $n,$ while
$d^{-}(v_{n})=n-\ell$ and $d^{+}(v_{n})=an-(n-\ell)=(a-1)n+\ell.$
If $(a+1)\ell-d^{-}(v_{\ell})\geq n+1,$ then $\ell$ is minimal with
$(a+1)\ell-d^{-}(v_{\ell})\geq n+1.$ If $(a+1)\ell-d^{-}(v_{\ell})=n,$ we
still, as $\ell+1\leq n,$ have
$d^{-}(v_{\ell+1})\in\\{d^{-}(v_{\ell}),\,d^{-}(v_{\ell)+1}\\}$ and
$(a+1)(\ell+1)-d^{-}(v_{\ell+1})\geq n+1.$ Either way,
$(v_{\ell+1},v_{n+1})\in E(J_{\infty}(a)).$ If $\ell+1<j<n,$ then induction
yields $d^{-}(v_{\ell+1})\leq d^{-}(v_{j})\leq d^{-}(v_{\ell+1})+(j-\ell-1)$
and $(a+1)j-d^{-}(v_{j})\geq n+1.$ As $d^{-}(v_{n})\leq n-1,(v_{n},v_{n+1})\in
E(J_{\infty}(a)),$ so $(v_{k},v_{n+1})\in E(J_{\infty}(a))$ whenever
$\ell+1\leq k\leq n.$ Depending on whether $(v_{\ell},v_{n+1})\in
E(J_{\infty}(a))$ or not, we obtain
$d^{-}(v_{n+1})=n+1-\ell=(n-\ell)+1=d^{-}(v_{\ell})+1$ or
$d^{-}(v_{n+1})=n+1-(\ell+1)=d^{-}(v_{n}).$
Let $n\geq 3,$ and, as before, choose $\ell$ minimal with $(v_{\ell},v_{n})\in
E(J_{\infty}(a)).$ We prove result $(d)$ by induction on $n$ and apply
arguments very similar to the ones already used: First of all,
$d^{-}(v_{1})=0,$ and $d^{+}(v_{1})=a=(a-1)1+c_{a,1}.$ Now let $n>1,$ and, as
before, choose $\ell$ minimal such that $(v_{\ell},v_{n})\in
E(J_{\infty}(a)).$ By $(a)$ and $(c),$ $d^{-}(v_{n})=n-\ell,$ and
$d^{+}(v_{n})=an-n+\ell=(a-1)n+\ell.$ Induction yields that
$d^{+}(v_{k})=a_{k}$ whenever $k<n.$ The definition of $\ell$ says that $\ell$
is minimal with $\ell+d^{+}(v_{\ell})=\ell+(a-1)\ell+c_{a,\ell}\geq n,$ which
means that $\ell=c_{a,n}.$ ∎
###### Corollary 1.2.
Note that $(a)$ and $(b)$ of Lemma 1.1 entail that
$d^{-}(v_{n+1})=(n+1)-c_{a,n+1}\in\\{n-c_{a,n},n-c_{a,n}+1\\}$ and that $(d)$
then implies that the series $(c_{a,n})$ are well-defined and ascending, more
specifically, $c_{a,n+1}\in\\{c_{a,n},c_{a,n}+1\\}$ ($n\in\mathbb{N}_{0}$).
###### Lemma 1.3.
Let $k\in\mathbb{N},$ and $0\leq b<a.$ Then $c_{a,ak+c_{a,k}-b}=k.$
###### Proof.
Let $ak+c_{a,k}-b=\ell.$ Certainly, $ak+c_{a,k}\geq\ell$ i.e. $c_{\ell}\leq
k.$ On the other hand, $a(k-1)+c_{a,\,k}=ak+c_{a,\,k}-a<\ell,$ so Corollary
1.2 says $a(k-1)+c_{a,k-1}<\ell$ and $c_{\ell}=k.$ ∎
Recall that the _generalised Lucas sequence_ $U_{n}(a,\,-1)$ is defined by
$U_{0}=0,$ $U_{1}=1,$
$U_{n+1}=aU_{n}+U_{n-1}.$
It is well known that $U_{n}=\frac{r^{n}-s^{n}}{r-s},$ where
$r=\frac{a}{2}+\sqrt{\frac{a^{2}}{4}+1},$
$s=\frac{a}{2}-\sqrt{\frac{a^{2}}{4}+1}.$
We are going to require a probably well-known (and not hard to prove) theorem,
which in the case $a=1$ is known as _Zeckendorf’s theorem_.
###### Lemma 1.4.
Let $n\in\mathbb{N}$ and let $U_{0},U_{1},\ldots$ be the terms of the Lucas
sequence $U(a,-1).$ Then $n$ may be uniquely expressed by a sum
$n=\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i},$ where
$0\leq\alpha_{1}<a,$ $0\leq\alpha_{i}\leq a$ ($i>2$), and $\alpha_{i}=a$ only
if $\alpha_{i-1}=0$ ($i\in\mathbb{N}).$
###### Proof.
Induction on $n.$ If $b<a,$ then ”$b=bU_{1}$” is a representation of the
required kind, which is clearly unique, as is ”$a=U_{2}.$”
Let $0<b<a,$ and let $n,\,i\in\mathbb{N},i\geq 3.$ If
$bU_{i}<n\leq(b+1)U_{i},$ then $aU_{j}<bU_{i}$ whenever $j<i,$ while
$U_{i+1}>aU_{i}>n.$ If $bU_{2}<n\leq(b+1)U_{2}$ with $b\geq 2,$ then
$aU_{1}<n<U_{3},$ while $(a-1)U_{1}<U_{2}<U_{3}.$ It follows that, if
$a<n\in\mathbb{N},$ then there is a uniquely determined natural number $i$ for
which there exists $\alpha_{i}\in\\{1,\ldots,a\\}$ with
$(\alpha-1)U_{i}<n\leq\alpha_{i}U_{i}.$ Now apply induction to
$m=n-\alpha_{i}U_{i}.$
Let $n\in\mathbb{N},$ and take $n$ as expressed by a sum
$n=\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i}$ where the coefficients satisfy
the conditions of Lemma 1.3. Define the function
$\tau:\mathbb{N}\rightarrow\\{0,1\\}$ as follows:
$\tau(n)=\begin{cases}0&\mbox{if}\,\,\alpha_{1}=0\\\
1&\mbox{if}\,\alpha_{1}>1\\\
\frac{1+(-1)^{i+1}}{2}&\mbox{if}\,\,1=\alpha_{1}=\ldots=\alpha_{i},\,\alpha_{i+1}=0\\\
\frac{1+(-1)^{i}}{2}&\mbox{if}\,\,1=\alpha_{1}=\ldots=\alpha_{i}<\alpha_{i+1}\end{cases}.$
For $m\in\mathbb{N},$ set $c_{a,m}=b_{m}.$ ∎
###### Theorem 1.5.
Let $n\in\mathbb{N},$ $n=\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i}$ where
the requirements of Lemma 1.4 are assumed to be met. Then
$b_{n}=\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i-1}+\tau(n).$
###### Proof.
We proceed by induction on $n.$ Note that $b_{1}=\ldots=b_{a-1}=1=U_{0}+1,$
while $b_{a}=b_{a+1}=U_{1}.$ Now let
$a<n=\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i}.$ First suppose that
$\alpha_{1}=0$ and $\alpha_{2}<a.$ Let
$k=\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i-1}.$ Letting
$\beta_{i}=\alpha_{i+1}(i\in\mathbb{N}),"k=\sum\limits_{j\in\mathbb{N}}\beta_{j}U_{j}$
” is a representation meeting the conditions of Lemma 1.4, hence induction
yields $b_{k}=\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i-2}+\tau(k).$ Now
$n=a(\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i-1})+\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i-2}=k+b_{k}-\tau(k),$
and, as $\tau(k)<a,$ Lemma 1.3 yields $k=b_{n}.$
Now assume that $\alpha_{1}=0$ and $\alpha_{2}=a.$ We consider the case $a=1$
separately and first.If $a=1,$ then $\alpha_{2}=1$ forces $\alpha_{3}=0,$ and
$\sum\limits_{i>3}\alpha_{i}U_{i}$ is the Zeckendorf representation of $n-1.$
Let $k=b_{n-1}.$ Via induction,$k=\sum\limits_{i>3}\alpha_{i}U_{i-1},$ and, as
the sum on the left is the Zeckendorf representation of $k,$ induction yields
$b_{k}=\sum\limits_{i>3}\alpha_{i}U_{i-2}.$ It follows that
$k+b_{k}=\sum\limits_{i>3}\alpha_{i}U_{i}=n-1,$ whence $b_{n}>b_{n-1},$ which,
as we have seen, means that
$b_{n}=b_{n-1}+1=\sum\limits_{i>3}\alpha_{i}U_{i-1}+1=\sum\limits_{i\geq
3}\alpha_{i}U_{i-1}+U_{1}.$ At this stage, the theorem has been proved in the
case $a=1,$ so we will assume $a>1$ from now on.
Suppose that $\alpha_{2}=a$ and let $j$ be maximal with $\alpha_{2j}=a.$ Then
$n=r+s,$ where $r=\sum\limits_{i\geq 2j+1}\alpha_{i}U_{i}$ and
$s=\sum\limits_{i=1}^{j}aU_{2i}=U_{2j+1}-1.$ It follows from our choice of $j$
that both $\alpha_{2j+1}$ and $\alpha_{2(j+1)}$ are less than $a.$
Accordingly, ”$n+1=\sum\limits_{i\geq
2(j+1)}\alpha_{i}U_{i}+(\alpha_{2j+1}+1)U_{2j+1}$” is a sum representation
satisfying the conditions of Lemma 1.4. Let $k=\sum\limits_{i\geq
2(j+1)}\alpha_{i}U_{i-1}+(\alpha_{2j+1}+1)U_{2j}.$ Then $k<n,$ and induction
yields that $b_{k}=\sum\limits_{i\geq
2(j+1)}\alpha_{i}U_{i-2}+(\alpha_{2j+1}+1)U_{2j-1}.$ Now $ak+b_{k}=n+1,$ and
Lemma 1.3 says $b_{n}=b_{n+1}=k.$ On the other hand,
$U_{2j}=\sum\limits_{i=2}^{j}aU_{2i-1},$ so
$k=\sum\limits_{i\in\mathbb{N}}\alpha_{i}U_{i-1}.$
Now assume that $\alpha_{1}>0$ and let $m=\sum\limits_{i\geq
2}\alpha_{i}U_{i-1}.$ As $\alpha_{1}>0,$ we have $\alpha_{2}<a,$ and the
displayed sum representation of $m$ satisfies the conditions of Lemma 1.4.
Via induction, $b_{m}\in\\{\sum\limits_{i\geq
2}\alpha_{i}U_{i-2},\,\sum\limits_{i\geq 2}\alpha_{i}U_{i-2}+1\\},$ hence
$n-\alpha_{1}\leq am+b_{m}\leq n-\alpha_{1}+1$ $(\ast)$
As $\alpha_{1}<a,$ the inequality $(\ast)$ implies that $a(m+1)+b_{m+1}\geq
a(m+1)+b_{m}=a_{m}+b_{m}+a>n.$ Thus $b_{n}\in\\{m,\,m+1\\}.$ If
$\alpha_{1}>1,$ then $\tau(n)=1$ and $n-\alpha_{1}+1<n,$ whence $b_{n}=m+1.$
We are left to deal with the case $\alpha_{1}=1$ which we shall subdivide
further, as follows:
If $\alpha_{1}=1$ and $\alpha_{2}=0,$ then $\tau(n)=1,$ while induction yields
that $b_{m}=\sum\limits_{i\geq 2}\alpha_{i}U_{i-2},$ such that
$am+b_{m}=\sum\limits_{i\geq 2}\alpha_{i}U_{i}=n-1$ and Corollary 1.2 implies
that $b_{n}=b_{n-1}+1=m+\tau(n).$
If $\alpha_{1}=1<\alpha_{2},$ then $\tau(n)=0,$ while induction yields
$b_{m}=\sum\limits_{i\geq 2}\alpha_{i}U_{i-2}+1,$ hence $am+b_{m}=n$ and
$m=b_{n}$ by Lemma 1.3.
If, finally, $\alpha_{1}=1=\alpha_{2},$ then $\tau(m)=0$ if and only if
$\tau(n)=1,$ and induction yields $am+b_{m}=n-1$ if and only if $\tau(n)=1$
and $am+b_{m}=n$ if and only if $\tau(n)=0.$ ∎
## 2 Finite Jaco Graphs $\\{J_{n}(a)|a,n\in\mathbb{N}\\}$
The family of finite Jaco Graphs are those limited to $n\in\mathbb{N}$
vertices by lobbing off all vertices (and edges arcing to vertices)
$v_{t},t>n.$ Hence, trivially we have $d(v_{i})\leq ai$ for $i\in\mathbb{N}.$
###### Definition 2.1.
The family of finite Jaco Graphs denoted by $\\{J_{n}(a)|a,n\in\mathbb{N}\\}$
is the defined by $V(J_{n}(a))=\\{v_{i}|i\in\mathbb{N},i\leq n\\}$,
$E(J_{n}(a))\subseteq\\{(v_{i},v_{j})|i,j\in\mathbb{N},i<j\leq n\\}$ and
$(v_{i},v_{j})\in E(J_{n}(a))$ if and only if $(a+1)i-d^{-}(v_{i})\geq j.$
###### Definition 2.2.
The set of vertices attaining degree $\Delta(J_{n}(a))$ is called the Jaconian
vertices of the Jaco Graph $J_{n}(a),$ and denoted, $\mathbb{J}(J_{n}(a))$ or,
$\mathbb{J}_{n}(a)$ for brevity.
###### Definition 2.3.
The lowest numbered (indiced) Jaconian vertex is called the prime Jaconian
vertex of a Jaco Graph.
###### Definition 2.4.
If $v_{i}$ is the prime Jaconian vertex, the complete subgraph on vertices
$v_{i+1},v_{i+2},\\\ \cdots,v_{n}$ is called the Hope subgraph of a Jaco Graph
and denoted, $\mathbb{H}(J_{n}(a))$ or, $\mathbb{H}_{n}(a)$ for brevity.
Property 1: From the definition of a Jaco Graph $J_{n}(a),$ it follows that,
if for the prime Jaconian vertex $v_{i},$ we have $d(v_{i})=ai$ then in the
underlying Jaco graph we have $d(v_{m})=am$ for all
$m\in\\{1,2,3,\cdots,i\\}.$
Property 2: From the definition of a Jaco Graph $J_{n}(a),$ it follows that
$\Delta(J_{k}(a))\leq\Delta(J_{n}(a))$ for all $k\leq n.$
Property 3: From the definition of a Jaco Graph $J_{n}(a),$ it follows that
the lowest degree attained by all Jaco Graphs is $0\leq\delta(J_{n}(a))\leq
a.$
Property 4: The $d^{-}(v_{k})$ for any vertex $v_{k}$ of a Jaco Graph
$J_{n}(a),~{}n\geq k$ is equal to $d(v_{k})$ in the underlying Jaco Graph
$J_{k}(a).$
###### Lemma 2.1.
For the Jaco Graphs $J_{i}(a),i\in\\{1,2,3,...,a+1\\}$ we have
$\Delta(J_{i}(a))=i-1$ and $\mathbb{J}(J_{i}(a))=\\{v_{k}|1\leq k\leq
i\\}=V(J_{i}(a)).$
###### Proof.
From definition 2.1 it follows that if $m=a+1$ then,
$((a+1)+1).1-d^{-}(v_{1})>(a+1)$ so the edges $(v_{1},v_{i}),i=2,3,...,(a+1)$
exist. It then follows that all edges $(v_{i},v_{j}),i<j$ exist. So the
underlying graph of $J_{a+1}(a)$ is the complete graph $K_{a+1}.$ Since
$\Delta(J_{a+1}(a))=(a+1)-1=a$ and we have $d(v_{i})=a$ for all vertices in
$K_{a+1},$ it follows that $\Delta(J_{a+1}(a))=a$ and
$\mathbb{J}(J_{a+1}(a))=\\{v_{k}|1\leq k\leq a\\}=V(J_{a+1}(a)).$ The result
follows similarly for $m<a+1.$ ∎
###### Lemma 2.2.
If in a Jaco Graph $J_{n}(a),$ and for smallest $i,$ the edge $(v_{i},v_{n})$
is defined, then $v_{i}$ is the prime Jaconian vertex of $J_{n}(a).$
###### Proof.
If in the construction of a Jaco Graph $J_{n}(a),$ and for smallest $i,$ the
edge $(v_{i},v_{n})$ is defined, we have that in the underlying graph of
$J_{n}(a)$, $d(v_{i})\leq ai$ and $d(v_{j})\leq d(v_{i})$ for all $j>i$. So it
follows that $d(v_{i})=\Delta(J_{n}(a))$ so $v_{i}$ is the prime Jaconian
vertex of $J_{n}(a).$ ∎
###### Lemma 2.3.
For all Jaco Graphs $J_{n}(a),~{}n\geq 2$ and, $v_{i},v_{i-1}\in V(J_{n}(a))$
we have that in the underlying graph $|(d(v_{i})-d(v_{i-1})|\leq a.$
###### Proof.
Consider the Jaco Graph $J_{n}(a),~{}n\geq 2.$ The result is trivially true
for all vertices $v_{1},v_{2},v_{3},\cdots,v_{k}$ if $v_{k}$ is the prime
Jaconian vertex of $J_{n}(a).$ Now consider the Hope Graph
$\mathbb{H}(J_{n}(a)).$ All vertices of $\mathbb{H}(J_{n}(a))$ have equal
degree so the result holds for the Hope Graph per se. Furthermore if a vertex
$v_{j},~{}(k+1)\leq j\leq n$ is linked to a vertex $v_{t},~{}1\leq t\leq k$
then all vertices $v_{l},~{}(k+1)\leq l<j$ are linked to $v_{t}$ which implies
$|d(v_{j})-d(v_{l})|=0\leq 1\leq a$ and $|(d(v_{j+1})-d(v_{j})|\leq 1\leq a.$
∎
Note that $\Delta(J_{n}(a))$ might repeat itself as $n$ increases to $n+1$ but
on an increase we always obtain $\Delta(J_{n}(a))+1$ before
$\Delta(J_{n}(a))+2.$
###### Theorem 2.4.
The Jaco Graph $J_{k}(a)$, $k=a(a+1)+1$ is the smallest Jaco Graph in
$\\{J_{n}(a)|a,n\in\mathbb{N}\\}$ which has $\Delta(J_{k}(a))=a(a+1)$ and
$\mathbb{J}(J_{k}(a))=\\{v_{a+1}\\}.$
###### Proof.
For $a=1$, the graph $J_{3}(1)$ is clearly the smallest Jaco Graph for which
$\Delta(J_{3}(1))=1(1+1)=2$ and
$\mathbb{J}(J_{3}(1))=\\{v_{(1+1)}\\}=\\{v_{2}\\}.$ So the result holds for
$a=1.$
Assume the result holds for $a=m.$ So for the Jaco Graph $J_{l}(m),l=m(m+1)+1$
we have that $\Delta(J_{l}(m))=m(m+1)$ and
$\mathbb{J}(J_{l}(m))=\\{v_{(m+1)}\\}.$
Now consider the Jaco Graph $J_{k}(m+1).$ In the Jaco Graph $J_{l}(m)$ the
vertex $v_{(m+1)+1}=v_{(m+2)}$ has $d(v_{(m+2)})=d(v_{(m+1)})-1.$ So in
constructing the Jaco Graph $J_{l}(m+1),$ amongst others the edge
$(v_{1},v_{(m+2)})$ is linked. So at least
$v_{(m+1)},v_{(m+2)}\in\mathbb{J}(J_{l}(m+1)).$ So $d(v_{(m+2)})=m(m+1).$ If
follows that the minimum number of additional vertices (smallest Jaco Graph)
say $t,$ to be added to $J_{l}(m+1)$ to obtain $d(v_{(m+2)})=(m+1)(m+2)$ and
$\mathbb{J}(J_{(l+t)}(m+1))=\\{v_{(m+2)}\\}$ in $J_{(l+t)}(m+1)$ is given by
$t=(m+1)(m+2)-m(m+1)=2(m+1).$ The number of vertices of $J_{l}(m)$ is given by
$m(m+1)+1.$ Now $l+t=(m(m+1)+1)+2(m+1)=(m+1)(m+2)+1.$
Clearly at least $v_{(m+2)}\in\mathbb{J}(J_{k}(m+1)),k=l+t,$ and
$v_{(m+1)}\notin\mathbb{J}(J_{k}(m+1))$ since $m(m+1)<(m+1)(m+2).$ In the
construction of the Jaco Graph $J_{l}(m+1),$ the edge $(v_{1},v_{(m+3)})$ was
not linked so $d(v_{(m+3)})<d(v_{(m+2)})$ in $J_{l}(m+1).$ So it follows that
$d(v_{(m+3)})<d(v_{(m+2)})$ in $J_{k}(m+1),k=l+t.$ The latter implies that
$\mathbb{J}(J_{k}(m+1))=\\{v_{(m+2)}\\}.$
Hence the result holds for $a=m+1$ implying it holds in general. ∎
## 3 Number of Edges of the Finite Jaco Graphs
$\\{J_{n}(a)|n\in\mathbb{N}\\}$
Note that Theorem 3.7 combined with Binet’s formula amounts to a closed
formula for $d^{+}(v_{n}).$ It is hoped that as a special case, a closed
formula can be found for the number of edges of a finite Jaco Graph
$J_{n}(a).$ However, the algorithms discussed in Ahlbach et al.[4] suggest
this might not be possible.
###### Proposition 3.1.
The number of edges of a Jaco Graph $J_{m}(a)=\frac{1}{2}m(m-1)$ if $m\leq
a+1.$
###### Proof.
From definition 2.1 it follows that if $m=a+1$ then,
$((a+1)+1).1-d^{-}(v_{1})>(a+1)$ so the edges $(v_{1},v_{i}),i=2,3,...,(a+1)$
exist. It then follows that all edges $(v_{i},v_{j}),i<j$ exist. So the
underlying graph of $J_{a+1}(a)$ is the complete graph $K_{a+1}$ hence,
$\epsilon(J_{a+1}(a))=\frac{1}{2}a(a+1)=\frac{1}{2}m(m-1).$ The result follows
similarly for $m<a+1.$ ∎
###### Theorem 3.2.
If for the Jaco Graph $J_{n}(a),$ we have $\Delta(J_{n}(a))=k,$ then
$\epsilon(J_{n}(a))=\epsilon(\mathbb{H}(J_{n}(a)))+\sum\limits_{i=1}^{k}d^{+}(v_{i}).$
###### Proof.
For $n=1,$ $d^{+}(v_{1})=0$ and $J_{1}(a)$ is the edgeless graph on vertex
$v_{1}$ whilst $\mathbb{H}(J_{1}(a)),$ is an empty graph so
$E(\mathbb{H}(J_{1}(a)))=\emptyset,$ implying
$\epsilon(\mathbb{H}(J_{1}(a)))=0.$ Thus the result holds.
For $n=2,$ the Jaco Graph $J_{2}(a)$ has the prime Jaconian vertex $v_{1}$ and
$d^{+}(v_{1})=1.$ $\mathbb{H}(J_{2}(a))$ is the null graph on vertex $v_{2}$
so $E(\mathbb{H}(J_{2}(a)))=\emptyset,$ implying
$\epsilon(\mathbb{H}(J_{2}(a)))=0.$ Thus the result holds.
Now assume it holds for all vertices $v_{i},~{}i\leq k-1.$ Thus vertex $v_{k}$
has attained its in-degree $d^{-}(v_{k}).$ To attain
$d(v_{k})=\Delta(J_{n}(a)),$ exactly
$\Delta(J_{n}(a))-d^{-}(v_{k})=d^{+}(v_{k})$ edges can be linked additionally.
So the result holds for vertices $v_{1},v_{2},v_{3},\cdots,v_{k}.$
Clearly we also have $E(\mathbb{H}(J_{n}(a)))\subset E(J_{n}(a)).$ Hence,
$\epsilon(J_{n}(a))=\epsilon(\mathbb{H}(J_{n}(a)))+d^{+}(v_{1})+d^{+}(v_{2})+d^{+}(v_{3})+\cdots+d^{+}(v_{k}).$
Since it is known that
$\epsilon(\mathbb{H}(J_{n}(a)))=1+2+3+\cdots+(n-\Delta(J_{n}(a))-1)$ the
result can be written as
$\epsilon(J_{n}(a))=\frac{1}{2}(n-k)(n-k-1)+\sum\limits_{i=1}^{k}d^{+}(v_{i}).$
∎
###### Corollary 3.3.
The number of edges of a Jaco Graph $J_{n}(a)$ having vertex $v_{i}$ as the
prime Jaconian vertex, can also be expressed recursively as
$\epsilon(J_{n+1}(a))=\begin{cases}\epsilon(J_{n}(a))-i+n&\text{if
$d(v_{i})=ai$}\\\ \epsilon(J_{n}(a))-i+(n+1)&\text{if $d(v_{i})<ai$}\\\
\end{cases}$
###### Proof.
Consider the Jaco Graph $J_{n}(a)$ having vertex $v_{i}$ as the Jaconian
vertex.
Case1: If $d(v_{i})=ai$ and the vertex $v_{(n+1)}$ is added to construct
$J_{(n+1)}(a)$ only the edges $(v_{(i+1)},v_{(n+1)}),....(v_{n},v_{(n+1)})$
can be linked additionally. This amounts to $(n-i)$ edges.
Case 2: If $d(v_{i})<ai$ and the vertex $v_{(n+1)}$ is added to construct
$J_{(n+1)}(a)$ only the edges
$(v_{i},v_{(n+1)}),(v_{(i+1)},v_{(n+1)}),....(v_{n},v_{(n+1)})$ can be linked
additionally. This amounts to $(n-i+1)$ edges. ∎
###### Lemma 3.4.
We find for $a=1$ and the series $(a_{n})_{n\in\mathbb{N}}$ defined by
$a_{0}=0,a_{1}=1,a_{n}=\min\\{k<n|k+a_{k}\geq n\\}$ ($n\geq 2)$.
that Lemma 1.1 changes to:
(a) $d^{+}(v_{n})+d^{-}(v_{n})=n.$
(b) $d^{-}(v_{n+1})\in\\{d^{-}(v_{n}),\,d^{-}(v_{n})+1\\}.$
(c) If $(v_{i},v_{k})\in E(J_{\infty}(1))$ and $i<j<k,$ then $(v_{j},v_{k})\in
E(J_{\infty}(1)).$
(d) $d^{+}(v_{n})=a_{n}.$
###### Corollary 3.5.
Note that $(a))$ and $(c)$ above entail that
$d^{+}(v_{n+1})=n+1-d^{-}(v_{n+1})\in\\{n-d^{-}(v_{n}),n-d^{-}(v_{n})+1\\}$
and that $(d)$ then implies tat the series $(a_{n})$ is well defined and
ascending, more specifically, $a_{n+1}\in\\{a_{n},a_{n}+1\\},$
$(n\in\mathbb{N}_{0}).$
###### Lemma 3.6.
Let $i\in\mathbb{N}.$ Then
$d^{+}(v_{i+d^{+}(v_{i})})=i=d^{+}(v_{i+d^{+}(v_{i+d^{+}(v_{i-1})})}).$
###### Proof.
let $i+d^{+}(v_{i})=k.$ Certainly, $i+d^{+}(v_{i})\geq k,$ so
$d^{+}(v_{k})\leq i.$ From Lemma 1.1, $(a=1)$ it follows that
$i-1+d^{+}(v_{i-1})\leq i-1+d^{+}(v_{i})<i+d^{+}(v_{i})$ so $d^{+}(v_{k})\geq
i.$ Let $\ell=i+d^{+}(v_{i-1}).$ Since $d^{+}(v_{i})\geq d^{+}(v_{i-1})$ we
have, $d^{+}(v_{\ell})\leq i$ and since, $i-1+d^{+}(v_{i-1})<\ell$ we have,
$d^{+}(v_{\ell})=i.$ ∎
###### Theorem 3.7.
(Bettina’s Theorem)222Also see: [5] Kok et al. _Characteristics of Finite Jaco
Graphs, $J_{n}(1),n\in\mathbb{N}.$_: Let
$\mathbb{F}=\\{f_{0},f_{1},f_{2},...\\}$ be the set of Fibonacci numbers and
let $n=f_{i_{1}}+f_{i_{2}}+...+f_{i_{r}},n\in\mathbb{N}$ be the Zeckendorf
representation of $n.$ Then
$d^{+}(v_{n})=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r}-1}.$
###### Proof.
Through induction we have that first of all, $1=f_{2}$ and
$d^{+}(v_{1})=1=f_{1}.$ Let $2\leq n=f_{i_{1}}+f_{i_{2}}+...+f_{i_{r}}$ and
let $k=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r}-1}.$ If $i_{r}\geq 3,$ then
$k=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r}-1}$ is the Zeckendorf representation
of $k$, such that induction yields
$d^{+}(v_{k})=k=f_{i_{1}-2}+f_{i_{2}-2}+...+f_{i_{r}-2}.$ Since
$k+d^{+}(v_{k})=f_{i_{1}-1}+f_{i_{1}-2}+f_{i_{2}-1}+f{i_{2}-2}+...f_{i_{r}-1}+f_{i_{r}-2}=f_{i_{1}}+f_{i_{2}}+...f_{i_{r}}=n,$
read with Lemma 3.6 yields $d^{+}(v_{n})=k.$
Finally consider $n=f_{i_{1}}+f_{i_{2}}+...+f_{i_{r}},i_{r}=2.$ Note that
$n>1$ implies that $i_{r-1}\geq 4$ and that the Zeckendorf representation of
$n-1$ given by $n-1=f_{i_{1}}+f_{i_{2}}+...+f_{i_{r-1}}.$ Let
$k=d^{+}(v_{n-1}).$ Through induction we have that,
$k=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r-1}-1},$ and since $i_{r-1}\leq 4,$ this
is the Zeckendorf representation of $k.$ Accordingly,
$d(v_{k})=f_{i_{1}-2}+f_{i_{2}-2}+...+f_{i_{r-1}-2},$ and
$k+d^{+}(v_{k})=f_{i_{1}-1}+f_{i_{1}-2}+f_{i_{2}-1}+f{i_{2}-2}+...f_{i_{r-1}-1}+f_{i_{r-1}-2}=n-1.$
It follows that $d^{+}(v_{n})>k=d^{+}(v_{n-1}).$ From Corollary 3.4 it follows
that
$d^{+}(v_{n})=k+1=(f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r-1}-1})+f_{1}=f_{i_{1}-1}+f_{i_{2}-1}+...+f_{i_{r}-1}.$
∎
## 4 Number of Shortest Paths in the Finite Jaco Graphs
$\\{J_{n}(a)|n\in\mathbb{N}\\}$
###### Definition 4.1.
Liz numbers are the family of numbers defined by
$\mathbb{L}=\\{\mathbb{L}_{a}|B_{0}=0,B_{1}=1,B_{2}=1,B_{i}=aB_{i-1}+B_{i-2},a,i\in\mathbb{N},i\geq
3\\}.$
###### Definition 4.2.
The set of distance-root vertices of the Jaco Graph $J_{n}(a),$ is the set
$\\{v_{B_{2}},v_{B_{3}},v_{B_{4}},\cdots,v_{B_{j}<n}$,$\cdots,v_{n}\\}$ and
denoted, $\mathbb{D}(J_{n}(a))$ or, $\mathbb{D}_{n}(a)$ for brevity.
Property 5: From definitions 4.1 and 4.2 it follow that besides possibly
$v_{n},$ all other distance-root vertices of the Jaco Graph
$J_{n}(a),~{}n\in\mathbb{N},$ have Liz number indices.
Property 6: The set of Fibonacci numbers $\mathbb{F}\in\mathbb{L},$ since
$a=1$ for $\mathbb{F}.$
Note: Reader should note the subtle difference between the $(a,1)$-Fibonacci
sequence defined in Kalman et al.[3], and the definition of Liz numbers
finding application in this paper.
###### Lemma 4.1.
In a Jaco Graph $J_{n}(a)$ we have for smallest $k,$ such that
$k+d^{+}(v_{k})\geq i,$ that
$d_{J_{n}(a)}(v_{1},v_{i})=d_{J_{n}(a)}(v_{1},v_{k})+1.$
###### Proof.
If in a Jaco Graph $J_{m}(a)$ we have that the edge $(v_{1},v_{i})$ exists,
the result holds because $d_{J_{n}(a)}(v_{1},v_{1})=0.$ Otherwise find
smallest $k$ for which $k+d^{+}(v_{k})\geq i.$ It follows that for all $j<k$
we have $k+d^{+}(v_{j})<i.$ So a path via such $v_{j}$ will have length at
least, $d_{J_{n}(a)}(v_{1},v_{j})+2.$ However, the path via $v_{k}$ has length
$d_{J_{n}(a)}(v_{1},v_{j})+1.$ Hence,
$d_{J_{n}(a)}(v_{1},v_{i})=d_{J_{n}(a)}(v_{1},v_{k})+1.$ ∎
###### Definition 4.3.
Let the number of distinct shortest paths between $v_{1}$ and $v_{n}$ in the
Jaco Graph $J_{n}(1)$ be denoted by $\psi(v_{n}).$
###### Proposition 4.2.
Consider vertex $v_{j},~{}j\geq 1$ in $J_{n}(1)$. The shortest path between
$v_{1}$ and $v_{j}$ is unique if and only if, $d^{+}(v_{j})\in\mathbb{F}.$
###### Proof.
Since any vertex is inherintly linked to itself the result holds for $j=1.$ We
have that the set of Fibonacci numbers are the Liz numbers for $a=1,$ so the
result follows for all $v_{j},j>2$ and $j\in\mathbb{F}$ since definition 4.1
ensures that $d^{+}(v_{j})\in\mathbb{F}.$ and definition 4.2 ensures that the
shortest path is unique.
Assume that there exists a $j\notin\mathbb{F}$ such that the shortest path
between $v_{1}$ and $v_{j}$ is unique. It implies that the smallest $l$ for
which the edge $(v_{l},v_{j})$ exists, is the largest $f_{i}$ for which
$(v_{f_{i}},v_{j})$ exists hence, $l=f_{i}$. Following from Lemma 1.1 ($a=1$),
$d^{+}(v_{l+d^{+}(v_{l})})=l=d^{+}(v_{l+d^{+}(v_{l-1})})$ it follows that
$d^{+}(v_{j})=l\in\mathbb{F}$.
Assume that there exists a $j\notin\mathbb{F}$ such that
$d^{+}(v_{j})\in\mathbb{F}$. It implies that $j$ is the largest natural number
for which $(v_{f_{i}},v_{j})$ exists and $f_{i}$ being the largest Fibonacci
number less than $j$. Clearly, a minimal path between $v_{1}$ and $v_{j}$ via
$v_{k}$, $k<v_{f_{i}}$ or $k>v_{f_{i}}$ will have lenght greater than the
minimal path via $v_{f_{i}}$. Hence, the shortest path between $v_{1}$ and
$v_{j}$ is unique. ∎
###### Proposition 4.3.
Consider vertex $v_{j},~{}j\geq 1$ in $J_{n}(1)$. If and only if
$d^{+}(v_{j})\notin\mathbb{F},$ then
$\psi(v_{j})=\sum\limits_{i=l}^{f_{t}}\psi(v_{i})$, with $f_{t}$ the largest
Fibonacci number less than $j$ and $l$ the smallest integer such that the edge
$(v_{l},v_{j})$ exists.
###### Proof.
Let $d^{+}(v_{j})\notin\mathbb{F},$ then clearly $d(v_{1},v_{l})\leq
d(v_{1},v_{j})$ for $l<j$. So if the edge $(v_{l},v_{j})$ exists Lemma 4.1
states that, $d(v_{1},v_{j})=d(v_{1},v_{l})+1.$ So the number of shortest
paths between $v_{1}$ and $v_{j}$ via vertex $v_{l}$ is given by
$\psi(v_{l})$. Through recursion the result
$\psi(v_{j})=\sum\limits_{i=l}^{f_{t}}\psi(v_{i})$ with $f_{t}$ the largest
Fibonacci number less than $j$ and $l$ the smallest integer such that the edge
$(v_{l},v_{j})$ exists, follows.
The converse follows from Proposition 4.2. ∎
###### Proposition 4.4.
Let $k\geq 7$ and $f_{i}<k\leq f_{i+1}$, and $f_{i},f_{i+1}\in\mathbb{F}.$ If
and only if $d^{+}(v_{k})$ is non-repetitive (meaning $d^{+}(v_{k-1})\neq
d^{+}(v_{k})\neq d^{+}(v_{k+1})$) then $\psi(v_{k})$ is non-repetitive
(meaning $\psi(v_{k-1})\neq\psi(v_{k})\neq\psi(v_{k+1})$).
###### Proof.
(Conjectured) ∎
[Open problem: Does a closed formula exist for the number of edges,
$\epsilon(J_{n}(1)).$]
[Open problem: Prove Proposition 4.4]
[Open problem: Find $Card$ $\mathbb{J}(J_{n}(a))$ in general.]
_Open access:_ This paper is distributed under the terms of the Creative
Commons Attribution License which permits any use, distribution and
reproduction in any medium, provided the original author(s) and the source are
credited.
References
$[1]$ Bondy, J.A., Murty, U.S.R., _Graph Theory with Applications,_ Macmillan
Press, London, (1976).
$[2]$ Zeckendorf, E., _Représentation des nombres naturels par une somme de
nombres de Fibonacci ou de nombres de Lucas,_ Bulletin de la Société Royale
des Sciences de Liège, Vol 41, (1972): pp 179-182.
$[3]$ Kalman, D., Mena, R., _The Fibonacci Number - Exposed,_ Mathematics
Magazine, Vol 76, No. 3, (2003): pp 167-181.
$[4]$ Ahlbach, C., Usatine, J., Pippenger, N., _Efficient Algorithms for
Zerckendorf Arithmetic,_ Fibonacci Quarterly, Vol 51, No. 13, (2013): pp
249-255.
$[5]$ Kok, J., Fisher, P., Wilkens, B., Mabula, M., Mukungunugwa, V.,
_Characteristics of Finite Jaco Graphs, $J_{n}(1),n\in\mathbb{N}$_, arXiv:
1404.0484v1 [math.CO], 2 April 2014.
Acknowledgement will be given to colleagues for preliminary peer review and
other contributions on the content of this paper during the preprint arXiv
publication term:333
[Remark: The concept of Jaco Graphs followed from a dream during the night of
10/11 January 2013 which was the first dream Kokkie could recall about his
daddy after his daddy passed away in the peaceful morning hours of 24 May
2012, shortly before the awakening of Bob Dylan, celebrating Dylan’s 71st
birthday]
|
arxiv-papers
| 2014-04-07T09:55:31 |
2024-09-04T02:50:00.802222
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian\n Mukungunugwa",
"submitter": "Johan Kok",
"url": "https://arxiv.org/abs/1404.1714"
}
|
1404.1717
|
# Riemann hypothesis and the arc length of the Riemann $Z(t)$-curve
Jan Moser Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
[email protected]
###### Abstract.
On Riemann hypothesis it is proved in this paper that the arc length of the
Riemann $Z$-curve is asymptotically equal to the double sum of local maxima of
the function $Z(t)$ on corresponding segment. This paper is English remake of
our paper [9], with short appendix concerning new integral generated by
Jacob’s ladders added.
###### Key words and phrases:
Riemann zeta-function
## 1\. Introduction and result
### 1.1.
Main object of this paper is the study of the integral
(1.1) $\int_{T}^{T+H}\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}{\rm d}t,$
i.e. the study of the arc length of the Riemann curve
$y=Z(t),\ t\in[T,T+H],\quad T\to\infty,$
where (see [13], pp. 79, 329)
(1.2)
$\begin{split}&Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\\\
&\vartheta(t)=-\frac{t}{2}\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right)=\\\
&=\frac{t}{2\pi}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}\left(\frac{1}{t}\right).\end{split}$
###### Remark 1.
Let us remind that the formula
(1.3) $\begin{split}&\\{Z(t)=\\}\
e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)=\\\
&=2\sum_{n\leq\sqrt{\bar{t}}}\frac{1}{\sqrt{n}}\cos\\{\vartheta(t)-t\ln
n\\}+\mathcal{O}(t^{-1/4}),\ \bar{t}=\sqrt{\frac{t}{2\pi}}\end{split}$
was known to Riemann (see [11], p. 60, comp. [12], p. 98).
Next, we will denote the roots of the equations
$Z(t)=0,\ Z^{\prime}(t)=0,\ t_{0}\not=\gamma$
by the symbols
$\\{\gamma\\},\ \\{t_{0}\\},$
correspondingly.
###### Remark 2.
On the Riemann hypothesis, the points of the sequences $\\{\gamma\\}$ and
$\\{t_{0}\\}$ are separated each from other (see [3], Corollary 3), i.e. in
this case we have
$\gamma^{\prime}<t_{0}<\gamma^{\prime\prime},$
where $\gamma^{\prime},\gamma^{\prime\prime}$ are neighboring points of the
sequence $\\{\gamma\\}$. Of course, $Z(t_{0})$ is local extremum of the
function $Z(t)$ located at $t=t_{0}$.
### 1.2.
In this paper we use the Riemann hypothesis together with some synthesis of
properties of the sequences
$\\{t_{0}\\},\ \\{h_{\nu}(\tau)\\},$
where the numbers $h_{\nu}(\tau)$ are defined by the equation (comp. (1.2))
(1.4) $\begin{split}&\vartheta_{1}[h_{\nu}(\tau)]=\pi\nu+\tau+\frac{\pi}{2},\
\nu=1,2,\dots,\ \tau\in[-\pi,\pi],\\\
&\vartheta_{1}(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8},\\\
&\vartheta(t)=\vartheta_{1}(t)+\mathcal{O}\left(\frac{1}{t}\right),\end{split}$
in order to obtain the following theorem.
###### Theorem.
On the Riemann hypothesis we have the asymptotic formula
(1.5) $\begin{split}&\int_{T}^{T+H}\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}{\rm
d}t=2\sum_{T\leq t_{0}\leq T+H}|Z(t_{0})|+\\\ &+\Theta
H+\mathcal{O}\left(T^{\frac{\Delta}{\ln\ln T}}\right),\\\
&\Theta=\Theta(T,H)\in(0,1),\ H=T^{\epsilon},\ T\to\infty\end{split}$
for every fixed $\epsilon>0$.
###### Remark 3.
Geometric meaning of our asymptotic formula (1.5) is as follows: the arc
length of the Riemann curve
$y=Z(t),\ t\in[T,T+H]$
is asymptotically equal to the double of the sum of local maxima of the
function
$|Z(t)|,\ t\in[T,T+H].$
## 2\. Discrete formulae – Lemma 1
### 2.1.
In this part of the paper we use the following formula
(2.1)
$\begin{split}&Z^{\prime}(t)=-2\sum_{n<P}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\sin\\{\vartheta-t\ln
n\\}+\\\ &+\mathcal{O}(T^{-1/4}\ln T),\ P=\sqrt{\frac{T}{2\pi}},\end{split}$
that we have obtained in our work [6], (see (2.1)). Next, we obtain from (2.1)
in the case
$\vartheta\to\vartheta_{1}$
(see (1.4)) that
(2.2)
$\begin{split}&Z^{\prime}(t)=-2\sum_{n<P}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\sin\\{\vartheta_{1}-t\ln
n\\}+\\\ &+\mathcal{O}(T^{-1/4}\ln T),\ H\in(0,\sqrt[4]{T}].\end{split}$
Let $S(a,b)$ denotes elementary trigonometric sum
$S(a,b)=\sum_{a\leq n\leq b}n^{it},\quad 1\leq a<b\leq 2a,\
b\leq\sqrt{\frac{t}{2\pi}}.$
Then we obtain from (2.2) in the case of the sequence $h_{\nu}(\tau)$ (see
(1.4)) the following
###### Lemma 1.
If
(2.3) $|S(a,b)|\leq A(\Delta)\sqrt{a},\ \Delta\in(0,1/6]$
then ($h_{\nu}=h_{\nu}(0)$)
(2.4) $\begin{split}&\sum_{T\leq h_{2\nu}\leq
T+H}Z^{\prime}[h_{2\nu}(\tau)]=-\frac{1}{\pi}H\ln^{2}P\cos\tau+\mathcal{O}(T^{\Delta}\ln^{2}T),\\\
&\sum_{T\leq h_{2\nu+1}\leq
T+H}Z^{\prime}[h_{2\nu+1}(\tau)]=\frac{1}{\pi}H\ln^{2}P\cos\tau+\mathcal{O}(T^{\Delta}\ln^{2}T),\end{split}$
where $\mathcal{O}$-estimates are uniform for $\tau\in[-\pi,\pi]$.
###### Proof.
We obtain from (2.2) by (1.4)
(2.5) $\begin{split}&Z^{\prime}[h_{\nu}(\tau)]=2(-1)^{\nu+1}\ln P\cos\tau-\\\
&-2\sum_{2\leq n\leq P}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\cos\\{\pi\nu-
h_{\nu}(\tau)\ln n+\tau\\}+\\\ &+\mathcal{O}(T^{-1/4}\ln T),\
h_{\nu}(\tau)\in[T,T+H].\end{split}$
∎
### 2.2.
Since (see [5], (23))
(2.6) $\sum_{T\leq h_{\nu}\leq
T+H}1=\frac{1}{2\pi}H\ln\frac{T}{2\pi}+\mathcal{O}(1)=\frac{1}{\pi}H\ln
P+\mathcal{O}(1),$
then we obtain from (2.5) (comp. [4], (59)-(61), [6], (51)-(53)) that
(2.7) $\sum_{T\leq h_{\nu}\leq
T+H}Z^{\prime}[h_{\nu}(\tau)]=-2\bar{w}(T,H;\tau)+\mathcal{O}(\ln^{2}T),$
where
$\begin{split}&\bar{w}=\frac{1}{2}(-1)^{\bar{\nu}}\sum_{n}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\cos\varphi+\\\
&+\frac{1}{2}(-1)^{N+\bar{\nu}}\sum_{n}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\cos(\omega
N+\varphi)+\\\
&+\frac{1}{2}(-1)^{\bar{\nu}}\sum_{n}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\tan\frac{\omega}{2}\sin\varphi+\\\
&+\frac{1}{2}(-1)^{N+\bar{\nu}+1}\sum_{n}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\tan\frac{\omega}{2}\sin(\omega
N+\varphi),\end{split}$
where
$\omega=\pi\frac{\ln n}{\ln P},\ \varphi=h_{\bar{\nu}}(\tau)\ln n-\tau,\
n\in[2,P),$
and
$\bar{\nu}=\min\\{\nu:\ h_{\nu}\in[T,T+H]\\},\ \bar{\nu}+N=\max\\{\nu:\
h_{\nu}\in[T,T+H]\\}.$
Of course, we have
$\sum_{T\leq h_{\nu}(\tau)\leq T+H}1=\sum_{T\leq h_{\nu}\leq
T+H}1+\mathcal{O}(1)$
for any fixed $\tau\in[-\pi,\pi]$. Now, it is clear that the method [6],
(54)-(64) implies by (2.3) that
$\bar{w}=\mathcal{O}(T^{\Delta}\ln^{2}T)$
uniformly for $\tau\in[-\pi,\pi]$, and consequently we obtain (see (2.7)) the
estimate
(2.8) $\sum_{T\leq h_{\nu}\leq
T+H}Z^{\prime}[h_{\nu}(\tau)]=\mathcal{O}(T^{\Delta}\ln^{2}T)$
uniformly for $\tau\in[-\pi,\pi]$.
### 2.3.
Next, we have (see (2.5), (2.6))
$\begin{split}&\sum_{T\leq h_{\nu}\leq
T+H}(-1)^{\nu}Z^{\prime}[h_{\nu}(\tau)]=-\frac{2}{\pi}H\ln^{2}P\cos\tau-2R+\mathcal{O}(\ln^{2}P),\\\
&R=\sum_{2\leq n<P}\frac{1}{\sqrt{n}}\ln\frac{P}{n}\sum_{T\leq h_{\nu}\leq
T+H}\cos\\{h_{\nu}(\tau)\ln n-\tau\\}.\end{split}$
Since by (2.3) and [6], (65)-(79) we have the estimate
$R=\mathcal{O}(T^{\Delta}\ln^{2}T)$
then we obtain the formula
(2.9) $\begin{split}&\sum_{T\leq h_{\nu}\leq
T+H}(-1)^{\nu}Z^{\prime}[h_{\nu}(\tau)]=\\\
&=-\frac{2}{\pi}H\ln^{2}P\cos\tau+\mathcal{O}(T^{\Delta}\ln^{2}T)\end{split}$
uniformly for $\tau\in[-\pi,\pi]$.
Finally, from (2.8), (2.9) formulae (2.4) follow.
## 3\. Integrals over disconnected sets – Lemma 2
Let (comp. [7], (3))
(3.1) $\begin{split}&\mathbb{G}_{2\nu}(x)=\\{t:\ h_{2\nu}(-x)<t<h_{2\nu}(x),\
t\in[T,T+H]\\},\ x\in(0,\pi/2],\\\ &\mathbb{G}_{2\nu+1}(y)=\\{t:\
h_{2\nu+1}(-y)<t<h_{2\nu+1}(y),\ t\in[T,T+H]\\},\ y\in(0,\pi/2],\\\
&\mathbb{G}_{1}(x)=\bigcup_{T\leq h_{2\nu}\leq T+H}\mathbb{G}_{2\nu}(x),\\\
&\mathbb{G}_{2}(y)=\bigcup_{T\leq h_{2\nu+1}\leq
T+H}\mathbb{G}_{2\nu+1}(y).\end{split}$
The following lemma holds true.
###### Lemma 2.
(2.3) implies
(3.2) $\begin{split}&\int_{\mathbb{G}_{1}(x)}Z^{\prime}(t){\rm
d}t=-\frac{2}{\pi}H\ln P\sin x+\mathcal{O}(xT^{\Delta}\ln T),\\\
&\int_{\mathbb{G}_{2}(y)}Z^{\prime}(t){\rm d}t=\frac{2}{\pi}H\ln P\sin
y+\mathcal{O}(yT^{\Delta}\ln T).\end{split}$
###### Proof.
First of all we have (see (1.4), comp. [7], (51))
$\left(\frac{{\rm d}h_{2\nu}(\tau)}{{\rm
d}\tau}\right)^{-1}=\vartheta_{1}^{\prime}[h_{2\nu}(\tau)]=\ln
P+\mathcal{O}\left(\frac{H}{T}\right).$
Next, from (2.2) by (2.3) we obtain the estimate
$Z^{\prime}(t)=\mathcal{O}(T^{\Delta}\ln^{2}T),\ t\in[T,T+H]$
(Abel transformation). Then we have (comp. [7], (52)) that
(3.3) $\begin{split}&\int_{-x}^{x}Z^{\prime}[h_{2\nu}(\tau)]{\rm
d}\tau=\int_{-x}^{x}Z^{\prime}[h_{2\nu}(\tau)]\left(\frac{{\rm
d}h_{2\nu}(\tau)}{{\rm d}\tau}\right)^{-1}\frac{{\rm d}h_{2\nu}(\tau)}{{\rm
d}\tau}{\rm d}\tau=\\\ &=\ln
P\int_{h_{2\nu}(-x)}^{h_{2\nu}(x)}Z^{\prime}(t){\rm
d}t+\mathcal{O}\left(x\frac{H}{T}T^{\Delta}\ln^{2}T\frac{1}{\ln T}\right)=\\\
&=\ln P\int_{\mathbb{G}_{2\nu}(x)}Z^{\prime}(t){\rm
d}t+\mathcal{O}(xHT^{-5/6}\ln T).\end{split}$
Consequently, we obtain from the first formula in (2.4) by (2.6), (3.1), (3.3)
the following asymptotic equality
$\begin{split}&\int_{\mathbb{G}_{1}(x)}Z^{\prime}(t){\rm
d}t=-\frac{2}{\pi}H\ln P\sin x+\\\ &+\mathcal{O}(xT^{\Delta}\ln
T)+\mathcal{O}(xH^{2}T^{-5/6}\ln^{2}T),\end{split}$
i.e. the first integral in (3.2). The second integral can be derived by a
similar way. ∎
## 4\. An estimate from below – Lemma 3
The following lemma holds true.
###### Lemma 3.
From (2.3) the estimate
(4.1) $\int_{T}^{T+H}|Z^{\prime}(t)|{\rm d}t>\frac{4}{\pi}(1-\epsilon)H\ln P,\
P=\sqrt{\frac{T}{2\pi}},\ H\in[T^{\Delta+\epsilon},\sqrt[4]{T}]$
follows, where $\epsilon>0$ is an arbitrarily small number.
###### Proof.
Let (comp. [8], (10))
$\begin{split}&\mathbb{G}_{1}^{+}(x)=\\{t:\ Z^{\prime}(t)>0,\
t\in\mathbb{G}_{1}(x)\\},\\\ &\mathbb{G}_{1}^{-}(x)=\\{t:\ Z^{\prime}(t)<0,\
t\in\mathbb{G}_{1}(x)\\},\\\ &\mathbb{G}_{1}^{0}(x)=\\{t:\ Z^{\prime}(t)=0,\
t\in\mathbb{G}_{1}(x)\\},\end{split}$
and the symbols
$\mathbb{G}_{2}^{+}(y),\mathbb{G}_{2}^{-}(y),\mathbb{G}_{2}^{0}(y)$
have similar meaning. Of course
$m\\{\mathbb{G}_{1}^{0}(x)\\}=m\\{\mathbb{G}_{2}^{0}(y)\\}=0.$
Since the expressions (3.2) in the case
$H\in[T^{\Delta+\epsilon},\sqrt[4]{T}],\quad x,y\in(0,\pi/2]$
are asymptotic formulae then from them we obtain the following inequalities
(4.2) $\begin{split}&\frac{2}{\pi}(1-\epsilon)H\ln
P<-\int_{\mathbb{G}_{1}(\pi/2)}Z^{\prime}(t){\rm d}t\leq\\\
&\leq-\int_{\mathbb{G}_{1}^{-}(\pi/2)}Z^{\prime}(t){\rm
d}t=\int_{\mathbb{G}_{1}^{-}(\pi/2)}|Z^{\prime}(t)|{\rm d}t,\\\
&\frac{2}{\pi}(1-\epsilon)H\ln P<\int_{\mathbb{G}_{2}(\pi/2)}Z^{\prime}(t){\rm
d}t\leq\int_{\mathbb{G}_{2}^{+}(\pi/2)}|Z^{\prime}(t)|{\rm d}t.\end{split}$
Since
$\mathbb{G}_{1}^{-}(\pi/2)\cup\mathbb{G}_{2}^{+}(\pi/2)\subset[T,T+H],\
\mathbb{G}_{1}^{-}(\pi/2)\cap\mathbb{G}_{2}^{+}(\pi/2)=\emptyset$
then by (4.2) needful estimate
$\begin{split}&\int_{T}^{T+H}|Z^{\prime}(t)|{\rm
d}t\geq\int_{\mathbb{G}_{1}^{-}(\pi/2)}|Z^{\prime}(t)|{\rm
d}t+\int_{\mathbb{G}_{2}^{+}(\pi/2)}|Z^{\prime}(t)|{\rm d}t>\\\
&>\frac{4}{\pi}(1-\epsilon)H\ln P.\end{split}$
follows. ∎
## 5\. Quadrature formula – Lemma 4
The following lemma holds true.
###### Lemma 4.
On Riemann hypothesis we have the following asymptotic formula
(5.1) $\begin{split}&\int_{T}^{T+H}|Z^{\prime}(t)|{\rm d}t=2\sum_{T\leq
t_{0}\leq T+H}|Z(t_{0})|+\\\ &+\mathcal{O}\left(T^{\frac{A}{\ln\ln
T}}\right),\quad H\in[T^{\mu},\sqrt[4]{T}],\end{split}$
where $0<\mu$ is an arbitrary small number.
###### Proof.
First of all, we have on Riemann hypothesis the following two Littlewood’s
estimates
(5.2) $\gamma^{\prime\prime}-\gamma^{\prime}<\frac{A}{\ln\ln\gamma^{\prime}},\
\gamma^{\prime}\to\infty$
(see [2], p. 237), and
(5.3) $Z(t)=\mathcal{O}\left(t^{\frac{A}{\ln\ln t}}\right),\ t\to\infty$
(see [13], p. 300). Next, on Riemann hypothesis we have the following basic
configuration (see Remark 2)
(5.4) $\gamma^{\prime}<t_{0}<\gamma^{\prime\prime};\ t_{0}\in[T,T+H].$
Now, there are following possibilities (see (5.4)): either
(5.5) $\begin{split}&Z(t)>0,\ t\in(\gamma^{\prime},\gamma^{\prime\prime})\
\Rightarrow\\\ &Z^{\prime}(t)>0,\ t\in(\gamma^{\prime},t_{0}),\
Z^{\prime}(t)<0,\ t\in(t_{0},\gamma^{\prime\prime}),\end{split}$
or
(5.6) $\begin{split}&Z(t)<0,\ t\in(\gamma^{\prime},\gamma^{\prime\prime})\
\Rightarrow\\\ &Z^{\prime}(t)<0,\ t\in(\gamma^{\prime},t_{0}),\
Z^{\prime}(t)>0,\ t\in(t_{0},\gamma^{\prime\prime}).\end{split}$
Consequently, (5.5) and (5.6) imply that
(5.7) $\int_{\gamma^{\prime}}^{\gamma^{\prime\prime}}|Z^{\prime}(t)|{\rm
d}t=2|Z(t_{0})|,\ \forall t_{0}\in[T,T+H].$
Similarly, we obtain (see (5.2), (5.3)) the estimates
(5.8)
$\int_{\bar{\gamma}^{\prime}}^{\bar{\gamma}^{\prime\prime}}|Z^{\prime}(t)|{\rm
d}t,\
\int_{\bar{\bar{\gamma}}^{\prime}}^{\bar{\bar{\gamma}}^{\prime\prime}}|Z^{\prime}(t)|{\rm
d}t=\mathcal{O}\left(\frac{T^{\frac{A}{\ln\ln T}}}{\ln\ln T}\right)$
in the following cases
$\bar{\gamma}^{\prime}<T\leq t_{0}<\bar{\gamma}^{\prime\prime},\
\bar{\bar{\gamma}}^{\prime}<t_{0}\leq T+H<\bar{\bar{\gamma}}^{\prime\prime}.$
Now, our formula (5.1) follows from (5.7), (5.8). ∎
## 6\. Proof of Theorem
We use the following formula
(6.1) $\begin{split}&\int_{T}^{T+H}\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}{\rm
d}t=\\\ &=\int_{T}^{T+H}|Z^{\prime}(t)|{\rm
d}t+\int_{T}^{T+H}\frac{1}{\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}+|Z^{\prime}(t)|}{\rm
d}t.\end{split}$
Since
$0<\frac{1}{\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}+|Z^{\prime}(t)|}\leq 1$
and
(6.2)
$\left.\frac{1}{\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}+|Z^{\prime}(t)|}\right|_{t=t_{0}}=1,\
t_{0}\in[T,T+H],$
i.e. the inequality (6.2) holds true for the finite set of values, then the
mean-value theorem gives
(6.3)
$\int_{T}^{T+H}\frac{1}{\sqrt{1+\\{Z^{\prime}(t)\\}^{2}}+|Z^{\prime}(t)|}{\rm
d}t=\Theta H,\ \Theta=\Theta(T,H)\in(0,1).$
Next, we obtain by (4.1), (5.1), ($\mu\leq\epsilon$), the inequality
(6.4) $\begin{split}&\frac{4}{\pi}(1-\epsilon)H\ln
P<\int_{T}^{T+H}|Z^{\prime}(t)|{\rm d}t=\\\ &=2\sum_{T\leq t_{0}\leq
T+H}|Z^{\prime}(t_{0})|+\mathcal{O}\left(T^{\frac{A}{\ln\ln
T}}\right).\end{split}$
Hence, by (6.1)-(6.4) the formula (1.5) follows for
(6.5) $H\in[T^{\Delta+\epsilon},\sqrt[4]{T}].$
Since the Riemann hypothesis implies Lindelöf hypothesis a it implies that
$\Delta=\epsilon$ (comp. [1], p. 89), then we obtain from (6.5) that
$H=T^{2\epsilon};\ 2\epsilon\rightarrow\epsilon,$
(see (1.5)).
## Appendix A Influence of Jacob’s ladders
If
$\varphi_{1}\\{[\mathring{T},\widering{T+H}]\\}=[T,T+H],$
then from (1.5) we obtain (see [10], (9.7)) the formula
(A.1)
$\begin{split}&\int_{\mathring{T}}^{\widering{T+H}}\sqrt{1+\\{Z^{\prime}_{\varphi_{1}}[\varphi_{1}(t)]\\}^{2}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim\\\ &\sim\left\\{2\sum_{T\leq t_{0}\leq T+H}|Z(t_{0})|+\Theta
H+\mathcal{O}\left(T^{\frac{A}{\ln\ln T}}\right)\right\\}\ln T,\
T\to\infty.\end{split}$
From (A.1) we obtain by mean-value theorem that
(A.2)
$\begin{split}&\int_{\mathring{T}}^{\widering{T+H}}\sqrt{1+\\{Z^{\prime}_{\varphi_{1}}[\varphi_{1}(t)]\\}^{2}}{\rm
d}t\sim\\\ &\sim\frac{\ln
T}{\left|\zeta\left(\frac{1}{2}+i\alpha\right)\right|^{2}}\left\\{2\sum_{T\leq
t_{0}\leq T+H}\left|\zeta\left(\frac{1}{2}+it_{0}\right)\right|+\Theta
H+\mathcal{O}\left(T^{\frac{A}{\ln\ln T}}\right)\right\\},\\\
&\alpha\in(\mathring{T},\widering{T+H}).\end{split}$
###### Remark 4.
Since we have (see [10], (8.5))
$\rho\\{[T,T+H];[\mathring{T},\widering{T+H}]\\}\sim(1-c)\pi(T)>(1-\epsilon)(1-c)\frac{T}{\ln
T},\ T\to\infty,$
where $\rho$ denotes the distance of corresponding segments and $\pi(T)$ is
the prime-counting function and $c$ is the Euler constant, then the formula
(A.2) gives strongly non-local expression for the integral on the left-hand
side of (A.2).
I would like to thank Michal Demetrian for helping me with the electronic
version of this work.
## References
* [1] A. A. Karatsuba, ‘ _Basic analytic number theory_ ‘, Moscow, (1975), (in Russian).
* [2] J. E. Littlewood, ‘Two notes on the Riemann zeta-function‘, Proc. Cambr. Phil. Soc., 22 (1924), 234-242.
* [3] J. Moser, ‘Some properties of the Riemann zeta-function on the critical line‘, Acta Arith., 26 (1974), 33-39, (in Russian), arXiv: 0710.0943
* [4] J. Moser, ‘On one sum in the theory of the Riemann zeta-function‘, Acta Arith. 31 (1976), 31-43; 40 (1981), 97-107, (in Russian).
* [5] J. Moser, ‘On one theorem of Hardy-Littlewood in the theory of the Riemann zeta-function‘, Acta Arith. 31, (1976), 45-51, (in Russian).
* [6] J. Moser, ‘On the roots of the equation $Z^{\prime}(t)=0$‘, Acta. Arith. 40 (1981), 97-107, (in Russian), arXiv: 1303.0967.
* [7] J. Moser, ‘New consequences of the Riemann-Siegel formula‘, Acta Arith. 42 (1982), 1-10, (in Russian), arXiv: 1312.4767.
* [8] J. Moser, ‘On the behavior of positive and negative values of the function $Z(t)$ in the theory of the Riemann zeta-function‘, Acta Math. Univ. Comen. 46-47, (1985); 41-48 (in Russian), arXiv: 1312.4767.
* [9] J. Moser, ‘Riemann hypothesis and extremal values of $Z(t)$ function‘, Acta Arith. 56, (1990), 225-235, (in Russian).
* [10] J. Moser, ‘Jacob’s ladders, structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations‘, Proc. Stek. Inst., 276, (2011), 208-221, arXiv: 1103..359.
* [11] C. L. Siegel,‘Über Riemann’s Nachlass zur analytischen Zahlentheorie: Quellen und Studien zur Geschichte der Matematik, Astronomie und Physik‘ , Abt. B: Studien, 2, (1932), 45-80.
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|
arxiv-papers
| 2014-04-07T10:02:26 |
2024-09-04T02:50:00.809571
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Moser",
"submitter": "Jan Moser",
"url": "https://arxiv.org/abs/1404.1717"
}
|
1404.1903
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-061 LHCb-PAPER-2014-014 7 April 2014
Observation of the resonant character of the $Z(4430)^{-}$ state
The LHCb collaboration†††Authors are listed on the following pages.
Resonant structures in $B^{0}\rightarrow\psi^{\prime}\pi^{-}K^{+}$ decays are
analyzed by performing a four-dimensional fit of the decay amplitude, using
$pp$ collision data corresponding to $\rm 3~{}fb^{-1}$ collected with the LHCb
detector. The data cannot be described with $K^{+}\pi^{-}$ resonances alone,
which is confirmed with a model-independent approach. A highly significant
$Z(4430)^{-}\rightarrow\psi^{\prime}\pi^{-}$ component is required, thus
confirming the existence of this state. The observed evolution of the
$Z(4430)^{-}$ amplitude with the $\psi^{\prime}\pi^{-}$ mass establishes the
resonant nature of this particle. The mass and width measurements are
substantially improved. The spin-parity is determined unambiguously to be
$1^{+}$.
Submitted to Physical Review Letters
© CERN on behalf of the LHCb collaboration, license
http://creativecommons.org/licenses/by/3.0/ CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L. An3, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1, S.
Belogurov31, K. Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S.
Benson38, J. Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van
Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M.
Bjørnstad54, T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A.
Bondar34, N. Bondar30,38, W. Bonivento15,38, S. Borghi54, A. Borgia59, M.
Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16, T. Brambach9, J. van den
Brand42, J. Bressieux39, D. Brett54, M. Britsch10, T. Britton59, J.
Brodzicka54, N.H. Brook46, H. Brown52, A. Bursche40, G. Busetto22,q, J.
Buytaert38, S. Cadeddu15, R. Calabrese16,f, M. Calvi20,k, M. Calvo Gomez36,o,
A. Camboni36, P. Campana18,38, D. Campora Perez38, A. Carbone14,d, G.
Carboni24,l, R. Cardinale19,38,j, A. Cardini15, H. Carranza-Mejia50, L.
Carson50, K. Carvalho Akiba2, G. Casse52, L. Cassina20, L. Castillo Garcia38,
M. Cattaneo38, Ch. Cauet9, R. Cenci58, M. Charles8, Ph. Charpentier38, S.
Chen54, S.-F. Cheung55, N. Chiapolini40, M. Chrzaszcz40,26, K. Ciba38, X. Cid
Vidal38, G. Ciezarek53, P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J.
Closier38, V. Coco38, J. Cogan6, E. Cogneras5, P. Collins38, A. Comerma-
Montells11, A. Contu15,38, A. Cook46, M. Coombes46, S. Coquereau8, G. Corti38,
M. Corvo16,f, I. Counts56, B. Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz
Torres60, S. Cunliffe53, R. Currie50, C. D’Ambrosio38, J. Dalseno46, P.
David8, P.N.Y. David41, A. Davis57, K. De Bruyn41, S. De Capua54, M. De
Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva57, P. De Simone18, D.
Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach55, O.
Deschamps5, F. Dettori42, A. Di Canto38, H. Dijkstra38, S. Donleavy52, F.
Dordei11, M. Dorigo39, A. Dosil Suárez37, D. Dossett48, A. Dovbnya43, G.
Dujany54, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A. Dziurda26, A.
Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S. Eidelman34, S.
Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I. El Rifai5, Ch.
Elsasser40, S. Ely59, S. Esen11, T. Evans55, A. Falabella16,f, C. Färber11, C.
Farinelli41, N. Farley45, S. Farry52, D. Ferguson50, V. Fernandez Albor37, F.
Ferreira Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M. Fiore16,f, M.
Fiorini16,f, M. Firlej27, C. Fitzpatrick38, T. Fiutowski27, M. Fontana10, F.
Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C. Frei38, M.
Frosini17,38,g, J. Fu21,38, E. Furfaro24,l, A. Gallas Torreira37, D.
Galli14,d, S. Gallorini22, S. Gambetta19,j, M. Gandelman2, P. Gandini59, Y.
Gao3, J. Garofoli59, J. Garra Tico47, L. Garrido36, C. Gaspar38, R. Gauld55,
L. Gavardi9, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38, C.
Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, H. Gordon38, C.
Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E.
Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P.
Griffith45, L. Grillo11, O. Grünberg62, B. Gui59, E. Gushchin33, Yu. Guz35,38,
T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S.
Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N.
Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann62, J. He38, T. Head38, V.
Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E.
van Herwijnen38, M. Heß62, A. Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54,
W. Hulsbergen41, P. Hunt55, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, P.
Jaton39, A. Jawahery58, M. Jezabek26, F. Jing3, M. John55, D. Johnson55, C.R.
Jones47, C. Joram38, B. Jost38, N. Jurik59, M. Kaballo9, S. Kandybei43, W.
Kanso6, M. Karacson38, T.M. Karbach38, M. Kelsey59, I.R. Kenyon45, T. Ketel42,
B. Khanji20, C. Khurewathanakul39, S. Klaver54, O. Kochebina7, M. Kolpin11, I.
Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, A. Kozlinskiy41,
L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P. Krokovny34, F.
Kruse9, M. Kucharczyk20,26,38,k, V. Kudryavtsev34, K. Kurek28, T.
Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38, G. Lafferty54, A. Lai15, D.
Lambert50, R.W. Lambert42, E. Lanciotti38, G. Lanfranchi18, C. Langenbruch38,
B. Langhans38, T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van Leerdam41,
J.-P. Lees4, R. Lefèvre5, A. Leflat32, J. Lefran$c$cois7, S. Leo23, O. Leroy6,
T. Lesiak26, B. Leverington11, Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F.
Lionetto40, B. Liu15, G. Liu38, S. Lohn38, I. Longstaff51, J.H. Lopes2, N.
Lopez-March39, P. Lowdon40, H. Lu3, D. Lucchesi22,q, H. Luo50, A. Lupato22, E.
Luppi16,f, O. Lupton55, F. Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O.
Maev30, S. Malde55, G. Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5,
J.F. Marchand4, U. Marconi14, C. Marin Benito36, P. Marino23,s, R. Märki39, J.
Marks11, G. Martellotti25, A. Martens8, A. Martín Sánchez7, M. Martinelli41,
D. Martinez Santos42, F. Martinez Vidal64, D. Martins Tostes2, A.
Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A. Mazurov16,f, M.
McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B.
Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A. Milanes8, M.-N.
Minard4, N. Moggi14, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,s, J. Moron27, A.-B.
Morris50, R. Mountain59, F. Muheim50, K. Müller40, R. Muresan29, M. Mussini14,
B. Muster39, P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva2, M.
Needham50, N. Neri21, S. Neubert38, N. Neufeld38, M. Neuner11, A.D. Nguyen39,
T.D. Nguyen39, C. Nguyen-Mau39,p, M. Nicol7, V. Niess5, R. Niet9, N.
Nikitin32, T. Nikodem11, A. Novoselov35, A. Oblakowska-Mucha27, V.
Obraztsov35, S. Oggero41, S. Ogilvy51, O. Okhrimenko44, R. Oldeman15,e, G.
Onderwater65, M. Orlandea29, J.M. Otalora Goicochea2, P. Owen53, A.
Oyanguren64, B.K. Pal59, A. Palano13,c, F. Palombo21,t, M. Palutan18, J.
Panman38, A. Papanestis49,38, M. Pappagallo51, C. Parkes54, C.J. Parkinson9,
G. Passaleva17, G.D. Patel52, M. Patel53, C. Patrignani19,j, A. Pazos
Alvarez37, A. Pearce54, A. Pellegrino41, M. Pepe Altarelli38, S.
Perazzini14,d, E. Perez Trigo37, P. Perret5, M. Perrin-Terrin6, L.
Pescatore45, E. Pesen66, K. Petridis53, A. Petrolini19,j, E. Picatoste
Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S. Playfer50,
M. Plo Casasus37, F. Polci8, A. Poluektov48,34, E. Polycarpo2, A. Popov35, D.
Popov10, B. Popovici29, C. Potterat2, A. Powell55, J. Prisciandaro39, A.
Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39, G. Punzi23,r, W.
Qian4, B. Rachwal26, J.H. Rademacker46, B. Rakotomiaramanana39, M. Rama18,
M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G. Raven42, S. Reichert54, M.M.
Reid48, A.C. dos Reis1, S. Ricciardi49, A. Richards53, M. Rihl38, K.
Rinnert52, V. Rives Molina36, D.A. Roa Romero5, P. Robbe7, A.B. Rodrigues1, E.
Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V. Romanovsky35, A. Romero
Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F. Ruffini23, H. Ruiz36, P.
Ruiz Valls64, G. Sabatino25,l, J.J. Saborido Silva37, N. Sagidova30, P.
Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez Mayordomo64, B.
Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37, E.
Santovetti24,l, M. Sapunov6, A. Sarti18,m, C. Satriano25,n, A. Satta24, M.
Savrie16,f, D. Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M.
Schmelling10, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25, M. Seco37, A. Semennikov31, K.
Senderowska27, I. Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11,
M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34,
V. Shevchenko63, A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N.
Skidmore46, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39,
D. Souza46, B. Souza De Paula2, B. Spaan9, A. Sparkes50, F. Spinella23, P.
Spradlin51, F. Stagni38, S. Stahl11, O. Steinkamp40, O. Stenyakin35, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
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Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
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D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H.
Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23,
S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
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Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, F. Zhang3, L. Zhang59, W.C. Zhang12, Y. Zhang3, A.
Zhelezov11, A. Zhokhov31, L. Zhong3, A. Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to 3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated to
31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to 36
65KVI - University of Groningen, Groningen, The Netherlands, associated to 41
66Celal Bayar University, Manisa, Turkey, associated to 38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
The existence of charged charmonium-like states has been a topic of much
debate since the Belle collaboration found evidence for a narrow $Z(4430)^{-}$
peak, with width $\Gamma=45\,{{}^{+18}_{-13}}\,{{}^{+30}_{-13}}$
$\mathrm{\,Me\kern-1.00006ptV}$, in the $\psi^{\prime}\pi^{-}$ mass
distribution ($m_{\psi^{\prime}\pi^{-}}$) in
$B\rightarrow\psi^{\prime}K\pi^{-}$ decays ($K=K^{0}_{s}$ or $K^{+}$) [1].111
The inclusion of charge-conjugate states is implied in this Letter. We use
units in which $c=1$. As the minimal quark content of such a state is
$c\bar{c}d\bar{u}$, this observation could be interpreted as the first
unambiguous evidence for the existence of mesons beyond the traditional
$q\bar{q}$ model [2]. This has contributed to a broad theoretical interest in
this state [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].
Exotic $\chi_{c1,2}\pi^{-}$ structures were also reported by the Belle
collaboration in $B\rightarrow\chi_{c1,2}K\pi^{-}$ decays [20]. Using the
$K^{*}\rightarrow K\pi^{-}$ invariant mass ($m_{K\pi^{-}}$) and helicity angle
($\theta_{K^{*}}$) [21, 22, 23] distributions, the BaBar collaboration was
able to describe the observed $m_{\psi^{\prime}\pi^{-}}$ and
$m_{\chi_{c1,2}\pi^{-}}$ structures in terms of reflections of any $K^{*}$
states with spin $J\leq 3$ ($J\leq 1$ for $m_{K\pi^{-}}<1.2$
$\mathrm{\,Ge\kern-1.00006ptV}$) without invoking exotic resonances [24, 25].
However, the BaBar results did not contradict the Belle evidence for the
$Z(4430)^{-}$ state. The Belle collaboration subsequently updated their
$Z(4430)^{-}$ results with a two-dimensional [26] and later a four-dimensional
(4D) amplitude analysis [27] resulting in a $Z(4430)^{-}$ significance of
$5.2\sigma$, a mass of $M_{Z^{-}}=4485\pm 22\,{{}^{+28}_{-11}}$
$\mathrm{\,Me\kern-1.00006ptV}$, a large width of
$\Gamma_{Z^{-}}=200\,{{}^{+41}_{-46}}\,{{}^{+26}_{-35}}$
$\mathrm{\,Me\kern-1.00006ptV}$, an amplitude fraction (defined further below)
of $f_{Z^{-}}=(10.3\,{{}^{+3.0}_{-3.5}}\,{{}^{+4.3}_{-2.3}})\%$ and spin-
parity $J^{P}=1^{+}$ favored over the other assignments by more than
$3.4\sigma$. Other candidates for charged four-quark states have been reported
in $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}\Upsilon(nS)$ [28, 29],
$e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$[30, 31], $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}h_{c}$[32] and
$e^{+}e^{-}\rightarrow(D^{*}\bar{D}^{*})^{\pm}\pi^{\mp}$[33] processes.
In this Letter, we report a 4D model-dependent amplitude fit to a sample of
$25\,176\pm 174$ $B^{0}\rightarrow\psi^{\prime}K^{+}\pi^{-}$,
$\psi^{\prime}\rightarrow\mu^{+}\mu^{-}$ candidates reconstructed with the
LHCb detector in $pp$ collision data corresponding to $\rm 3~{}fb^{-1}$
collected at $\sqrt{s}=7$ and $8$ TeV. The ten-fold increase in signal yield
over the previous measurement [27] improves sensitivity to exotic states and
allows their resonant nature to be studied in a novel way. We complement the
amplitude fit with a model-independent approach [24].
The LHCb detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<5$, described in detail in Ref. [34]. The $B^{0}$
candidate selection follows that in Ref. [35] accounting for the different
number of final-state pions. It is based on finding
$(\psi^{\prime}\rightarrow\mu^{+}\mu^{-})K^{+}\pi^{-}$ candidates using
particle identification information, transverse momentum thresholds and
requiring separation of the tracks and of the $B^{0}$ vertex from the primary
$pp$ interaction points. To improve modeling of the detection efficiency, we
exclude regions near the $K^{+}\pi^{-}$ vs. $\psi^{\prime}\pi^{-}$ Dalitz plot
boundary, which reduces the sample size by 12%. The background fraction is
determined from the $B^{0}$ candidate invariant mass distribution to be
$(4.1\pm 0.1)\%$. The background is dominated by combinations of
$\psi^{\prime}$ mesons from $B$ decays with random kaons and pions.
Amplitude models are fit to the data using the unbinned maximum likelihood
method. We follow the formalism and notation of Ref. [27] with the 4D
amplitude dependent on
$\Phi=(m_{K^{+}\pi^{-}}^{2},m_{\psi^{\prime}\pi^{-}}^{2},\cos\theta_{\psi^{\prime}},\phi)$,
where $\theta_{\psi^{\prime}}$ is the $\psi^{\prime}$ helicity angle and
$\phi$ is the angle between the $K^{*}$ and $\psi^{\prime}$ decay planes in
the $B^{0}$ rest frame. The signal probability density function (PDF),
$S(\Phi)$, is normalized by summing over simulated events. Since the simulated
events are passed through the detector simulation [36], this approach
implements 4D efficiency corrections without use of a parameterization. We use
$B^{0}$ mass sidebands to obtain a parameterization of the background PDF.
As in Ref. [27], our amplitude model includes all known $K^{*0}\rightarrow
K^{+}\pi^{-}$ resonances with nominal mass within or slightly above the
kinematic limit (1593 $\mathrm{\,Me\kern-1.00006ptV}$) in
$B^{0}\rightarrow\psi^{\prime}K^{+}\pi^{-}$ decays: $K^{*}_{0}(800)$,
$K^{*}_{0}(1430)$ for $J=0$; $K^{*}(892)$, $K^{*}(1410)$ and $K^{*}(1680)$ for
$J=1$; $K^{*}_{2}(1430)$ for $J=2$; and $K^{*}_{3}(1780)$ for $J=3$. We also
include a non-resonant (NR) $J=0$ term in the fits. We fix the masses and
widths of the resonances to the world average values [37], except for the
widths of the two dominant contributions, $K^{*}(892)$ and $K^{*}_{2}(1430)$,
and the poorly known $K^{*}_{0}(800)$ mass and width, which are allowed to
float in the fit with Gaussian constraints. As an alternative $J=0$ model, we
use the LASS parameterization [38, 39], in which the NR and $K^{*}_{0}(800)$
components are replaced with an elastic scattering term (two free parameters)
interfering with the $K^{*}_{0}(1430)$ resonance.
To probe the quality of the likelihood fits, we calculate a binned $\chi^{2}$
variable using adaptive 4D binning, in which we split the data once in
$|\cos\theta_{\psi^{\prime}}|$, twice in $\phi$ and then repeatedly in
$m_{K^{+}\pi^{-}}^{2}$ and $m_{\psi^{\prime}\pi^{-}}^{2}$ preserving any bin
content above 20 events, for a total of $N_{\rm bin}=768$ bins. Simulations of
many pseudoexperiments, each with the same number of signal and background
events as in the data sample, show that the $p$-value of the $\chi^{2}$ test
($p_{\chi^{2}}$) has an approximately uniform distribution assuming that the
number of degrees of freedom (${\rm ndf}$) equals $N_{\rm bin}-N_{\rm par}-1$,
where $N_{\rm par}$ is the number of unconstrained parameters in the fit. Fits
with all $K^{*}$ components and either of the two different $J=0$ models do
not give a satisfactory description of the data; the $p_{\chi^{2}}$ is below
$2\times 10^{-6}$, equivalent to $4.8\sigma$ in the Gaussian distribution. If
the $K^{*}_{3}(1780)$ component is excluded from the amplitude, the
discrepancy increases to $6.3\sigma$.
Figure 1: Background-subtracted and efficiency-corrected
$m_{\psi^{\prime}\pi^{-}}$ distribution (black data points), superimposed with
the reflections of $\cos\theta_{K^{*}}$ moments up to order four allowing for
$J(K^{*})\leq 2$ (blue line) and their correlated statistical uncertainty
(yellow band bounded by blue dashed lines). The distributions have been
normalized to unity.
This is supported by an independent study using the model-independent approach
developed by the BaBar collaboration [24, 25], which does not constrain the
analysis to any combination of known $K^{*}$ resonances, but merely restricts
their maximal spin. We determine the Legendre polynomial moments of
$\cos\theta_{K^{*}}$ as a function of $m_{K^{+}\pi^{-}}$ from the sideband-
subtracted and efficiency-corrected sample of
$B^{0}\rightarrow\psi^{\prime}K^{+}\pi^{-}$ candidates. Together with the
observed $m_{K^{+}\pi^{-}}$ distribution, the moments corresponding to $J\leq
2$ are reflected into the $m_{\psi^{\prime}\pi^{-}}$ distribution using
simulations as described in Ref. [24]. As shown in Fig. 1, the $K^{*}$
reflections do not describe the data in the $Z(4430)^{-}$ region. Since a
$Z(4430)^{-}$ resonance would contribute to the $\cos\theta_{K^{*}}$ moments,
and also interfere with the $K^{*}$ resonances, it is not possible to
determine the $Z(4430)^{-}$ parameters using this approach. The amplitude fit
is used instead.
If a $Z(4430)^{-}$ component with $J^{P}=1^{+}$ (hereafter $Z_{1}^{-}$) is
added to the amplitude, the $p_{\chi^{2}}$ reaches 4% when all the
$K^{*}\rightarrow K^{+}\pi^{-}$ resonances with a pole mass below the
kinematic limit are included. The $p_{\chi^{2}}$ rises to 12% if the
$K^{*}(1680)$ is added (see Fig. 2), but fails to improve when the
$K^{*}_{3}(1780)$ is also included. Therefore, as in Ref. [27] we choose to
estimate the $Z_{1}^{-}$ parameters using the model with the $K^{*}(1680)$ as
the heaviest $K^{*}$ resonance. In Ref. [27] two independent complex
$Z_{1}^{-}$ helicity couplings, $H_{\lambda^{\prime}}^{Z^{-}}$ for
$\lambda^{\prime}=0,+1$ (parity conservation requires
$H_{-1}^{Z^{-}}=H_{+1}^{Z^{-}}$), were allowed to float in the fit. The small
energy release in the $Z_{1}^{-}$ decay suggests neglecting $D$-wave decays. A
likelihood-ratio test is used to discriminate between any pair of amplitude
models based on the log-likelihood difference $\Delta(-2\ln L)$ [40]. The
$D$-wave contribution is found to be insignificant when allowed in the fit,
$1.3\sigma$ assuming Wilks’ theorem222See e.g. Sec. 10.5.2 of Ref. [40] on
asymptotic distribution of $\Delta(-2\ln L)$ for continuous families of
hypotheses.. Thus, we assume a pure $S$-wave decay, implying
$H_{+1}^{Z^{-}}=H_{0}^{Z^{-}}$. The significance of the $Z_{1}^{-}$ is
evaluated from the likelihood ratio of the fits without and with the
$Z_{1}^{-}$ component. Since the condition of the likelihood regularity in
$Z_{1}^{-}$ mass and width is not satisfied when the no-$Z_{1}^{-}$ hypothesis
is imposed, use of Wilks’ theorem is not justified333With the mass and width
floated in the fit a look-elsewhere effect must be taken into account. [41].
Therefore, pseudoexperiments are used to predict the distribution of
$\Delta(-2\ln L)$ under the no-$Z_{1}^{-}$ hypothesis, which is found to be
well described by a $\chi^{2}$ PDF with ${\rm ndf}=7.5$. Conservatively, we
assume ${\rm ndf}=8$, twice the number of free parameters in the $Z_{1}^{-}$
amplitude. This yields a $Z_{1}^{-}$ significance for the default $K^{*}$
model of $18.7\sigma$. The lowest significance among all the systematic
variations to the model discussed below is 13.9$\sigma$.
Figure 2: Distributions of the fit variables (black data points) together
with the projections of the 4D fit. The red solid (brown dashed) histogram
represents the total amplitude with (without) the $Z_{1}^{-}$. The other
points illustrate various subcomponents of the fit that includes the
$Z_{1}^{-}$: the upper (lower) blue points represent the $Z_{1}^{-}$ component
removed (taken alone). The orange, magenta, cyan, yellow, green, and red
points represent the $K^{*}(892)$, total $S$-wave, $K^{*}(1410)$,
$K^{*}(1680)$, $K^{*}_{2}(1430)$ and background terms, respectively.
The default fit gives $M_{Z_{1}^{-}}=4475\pm 7$
$\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma_{Z_{1}^{-}}=172\pm 13$
$\mathrm{\,Me\kern-1.00006ptV}$, $f_{Z_{1}^{-}}=(5.9\pm 0.9)\%$, $f_{\rm
NR}=(0.3\pm 0.8)\%$, $f_{K^{*}_{0}(800)}=(3.2\pm 2.2)\%$,
$f_{K^{*}(892)}=(59.1\pm 0.9)\%$, $f_{K^{*}(1410)}=(1.7\pm 0.8)\%$,
$f_{K^{*}_{0}(1430)}=(3.6\pm 1.1)\%$, $f_{K^{*}_{2}(1430)}=(7.0\pm 0.4)\%$ and
$f_{K^{*}(1680)}=(4.0\pm 1.5)\%$, which are consistent with the Belle
results[27] even without considering systematic uncertainties. Above, the
amplitude fraction of any component $R$ is defined as $f_{R}=\int
S_{R}(\Phi)d\Phi/\int S(\Phi)d\Phi$, where in $S_{R}(\Phi)$ all except the $R$
amplitude terms are set to zero. The sum of all amplitude fractions is not
$100\%$ because of interference effects. To assign systematic errors, we: vary
the $K^{*}$ models by removing the $K^{*}(1680)$ or adding the
$K^{*}_{3}(1780)$ in the amplitude ($f_{K^{*}_{3}(1780)}=(0.5\pm 0.2)\%$); use
the LASS function as an alternative $K^{*}$ $S$-wave representation; float all
$K^{*}$ masses and widths while constraining them to the known values [37];
allow a second $Z^{-}$ component; increase the orbital angular momentum
assumed in the $B^{0}$ decay; allow a $D$-wave component in the $Z_{1}^{-}$
decay; change the effective hadron size in the Blatt-Weisskopf form factors
from the default 1.6 GeV-1 [27] to 3.0 GeV-1; let the background fraction
float in the fit or neglect the background altogether; tighten the selection
criteria probing the efficiency simulation; and use alternative efficiency and
background implementations in the fit. We also evaluate the systematic
uncertainty from the formulation of the resonant amplitude. In the default
fit, we follow the approach of Eq. (2) in Ref. [27] that uses a running mass
$M_{R}$ in the $(p_{R}/M_{R})^{L_{R}}$ term, where $M_{R}$ is the invariant
mass of two daughters of the $R$ resonance; $p_{R}$ is the daughter’s momentum
in the rest frame of $R$ and $L_{R}$ is the orbital angular momentum of the
decay. The more conventional formulation [37, 42] is to use $p_{R}^{L_{R}}$
(equivalent to a fixed $M_{R}$ mass). This changes the $Z_{1}^{-}$ parameters
via the $K^{*}$ terms in the amplitude model: $M_{Z_{1}^{-}}$ varies by $-22$
$\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma_{Z_{1}^{-}}$ by $+29$
$\mathrm{\,Me\kern-1.00006ptV}$ and $f_{Z_{1}^{-}}$ by $+1.7\%$ (the
$p_{\chi^{2}}$ drops to 7%). Adding all systematic errors in quadrature we
obtain $M_{Z_{1}^{-}}=4475\pm 7\,{{}_{-25}^{+15}}$
$\mathrm{\,Me\kern-1.00006ptV}$, $\Gamma_{Z_{1}^{-}}=172\pm
13\,{{}_{-34}^{+37}}$ $\mathrm{\,Me\kern-1.00006ptV}$ and
$f_{Z_{1}^{-}}=(5.9\pm 0.9\,{{}_{-3.3}^{+1.5}})\%$. We also calculate a
fraction of $Z_{1}^{-}$ that includes its interferences with the $K^{*}$
resonances as $f_{Z_{1}^{-}}^{I}=1-\int S_{{\rm no}\mathchar 45\relax
Z_{1}^{-}}(\Phi)d\Phi/\int S(\Phi)d\Phi$, where the $Z_{1}^{-}$ term in
$S_{{\rm no}\mathchar 45\relax Z_{1}^{-}}(\Phi)$ is set to zero. This
fraction, $(16.7\pm 1.6\,{{}_{-5.2}^{+4.5}})\%$, is much larger than
$f_{Z_{1}^{-}}$ implying large constructive interference.
To discriminate between various $J^{P}$ assignments we determine the
$\Delta(-2\ln L)$ between the different spin hypotheses. Following the method
of Ref. [27], we exclude the $0^{-}$ hypothesis in favor of the $1^{+}$
assignment at $25.7\sigma$ in the fits with the default $K^{*}$ model. Such a
large rejection level is expected according to the $\Delta(-2\ln L)$
distribution of the pseudoexperiments generated under the $1^{+}$ hypothesis.
For large data samples, assuming a $\chi^{2}({\rm ndf}=1)$ distribution for
$\Delta(-2\ln L)$ under the disfavored $J^{P}$ hypothesis gives a lower limit
on the significance of its rejection444See Sec. 10.5.7 of Ref. [40] on testing
separate hypotheses.. This method gives more than $17.8\sigma$ rejection.
Since the latter method is conservative and provides sufficient rejection, we
employ it while studying systematic effects. Among all systematic variations
described above, allowing the $K^{*}_{3}(1780)$ in the fit produces the
weakest rejection. Relative to $1^{+}$, we rule out the $0^{-}$, $1^{-}$,
$2^{+}$ and $2^{-}$ hypotheses by at least $9.7\sigma$, $15.8\sigma$,
$16.1\sigma$ and $14.6\sigma$, respectively. This reinforces the $5.1\sigma$
($4.7\sigma$) rejection of the $2^{+}$ ($2^{-}$) hypotheses previously
reported by the Belle collaboration [27], and confirms the $3.4\sigma$
($3.7\sigma$) indications from Belle that $1^{+}$ is favored over $0^{-}$
($1^{-}$).
Figure 3: Fitted values of the $Z_{1}^{-}$ amplitude in six
$m_{\psi^{\prime}\pi^{-}}^{2}$ bins, shown in an Argand diagram (connected
points with the error bars, $m_{\psi^{\prime}\pi^{-}}^{2}$ increases
counterclockwise). The red curve is the prediction from the Breit-Wigner
formula with a resonance mass (width) of 4475 (172)
$\mathrm{\,Me\kern-0.90005ptV}$ and magnitude scaled to intersect the bin with
the largest magnitude centered at (4477 MeV)2. Units are arbitrary. The phase
convention assumes the helicity-zero $K^{*}(892)$ amplitude to be real.
In the amplitude fit, the $Z_{1}^{-}$ is represented by a Breit-Wigner
amplitude, where the magnitude and phase vary with
$m_{\psi^{\prime}\pi^{-}}^{2}$ according to an approximately circular
trajectory in the (Re$\,A^{Z^{-}}$, Im$\,A^{Z^{-}}$) plane (Argand diagram
[37]), where $A^{Z^{-}}$ is the $m_{\psi^{\prime}\pi^{-}}^{2}$ dependent part
of the $Z_{1}^{-}$ amplitude. We perform an additional fit to the data, in
which we represent the $Z_{1}^{-}$ amplitude as the combination of independent
complex amplitudes at six equidistant points in the
$m_{\psi^{\prime}\pi^{-}}^{2}$ range covering the $Z_{1}^{-}$ peak,
$18.0-21.5$ GeV2. Thus, the $K^{*}$ and the $Z_{1}^{-}$ components are no
longer influenced in the fit by the assumption of a Breit-Wigner amplitude for
the $Z_{1}^{-}$. The resulting Argand diagram, shown in Fig. 3, is consistent
with a rapid change of the $Z_{1}^{-}$ phase when its magnitude reaches the
maximum, a behavior characteristic of a resonance.
If a second $Z^{-}$ resonance is allowed in the amplitude with $J^{P}=0^{-}$
($Z_{0}^{-}$) the $p_{\chi^{2}}$ of the fit improves to 26%. the $Z_{0}^{-}$
significance from the $\Delta(-2\ln L)$ is $6\sigma$ including the systematic
variations. It peaks at a lower mass, $4239\pm 18\,{{}^{+45}_{-10}}$
$\mathrm{\,Me\kern-1.00006ptV}$, and has a larger width, $220\pm
47\,{{}^{+108}_{-\phantom{0}74}}$ $\mathrm{\,Me\kern-1.00006ptV}$, with a much
smaller fraction, $f_{Z_{0}^{-}}=(1.6\pm 0.5\,{{}^{+1.9}_{-0.4}})\%$
($f_{Z_{0}^{-}}^{I}=(2.4\pm 1.1\,{{}^{+1.7}_{-0.2}})\%$) than the $Z_{1}^{-}$.
With the default $K^{*}$ model, $0^{-}$ is preferred over $1^{-}$, $2^{-}$ and
$2^{+}$ by $8\sigma$. The preference over $1^{+}$ is only $1\sigma$. However,
the width in the $1^{+}$ fit becomes implausibly large, $660\pm 150$
$\mathrm{\,Me\kern-1.00006ptV}$. The $Z_{0}^{-}$ has the same mass and width
as one of the $\chi_{c1}\pi^{-}$ states reported previously [20] but a $0^{-}$
state cannot decay strongly to $\chi_{c1}\pi^{-}$. Figure 4 compares the
$m_{\psi^{\prime}\pi^{-}}^{2}$ projections of the fits with both $Z_{0}^{-}$
and $Z_{1}^{-}$, or $Z_{1}^{-}$ component only. The model-independent analysis
has a large statistical uncertainty in the $Z_{0}^{-}$ region and shows no
deviations of the data from the reflections of the $K^{*}$ degrees of freedom
(Fig. 1). Argand diagram studies for the $Z_{0}^{-}$ are inconclusive.
Therefore, its characterization as a resonance will need confirmation when
larger samples become available.
Figure 4: Distribution of $m_{\psi^{\prime}\pi^{-}}^{2}$ in the data (black
points) for $1.0<m_{K^{+}\pi^{-}}^{2}<1.8$ GeV2 ($K^{*}(892)$,
$K^{*}_{2}(1430)$ veto region) compared with the fit with two, $0^{-}$ and
$1^{+}$ (solid-line red histogram) and only one $1^{+}$ (dashed-line green
histogram) $Z^{-}$ resonances. Individual $Z^{-}$ terms (blue points) are
shown for the fit with two $Z^{-}$ resonances.
In summary, an amplitude fit to a large sample of
$B^{0}\rightarrow\psi^{\prime}K^{+}\pi^{-}$ decays provides the first
independent confirmation of the existence of the $Z(4430)^{-}$ resonance and
establishes its spin-parity to be $1^{+}$, both with very high significance.
The measured mass, $4475\pm 7\,{{}_{-25}^{+15}}$
$\mathrm{\,Me\kern-1.00006ptV}$, width, $172\pm 13\,{{}_{-34}^{+37}}$
$\mathrm{\,Me\kern-1.00006ptV}$, and amplitude fraction, $(5.9\pm
0.9\,{{}_{-3.3}^{+1.5}})\%$, are consistent with, but more precise than, the
Belle results [27]. An analysis of the data using the model-independent
approach developed by the BaBar collaboration [24] confirms the
inconsistencies in the $Z(4430)^{-}$ region between the data and
$K^{+}\pi^{-}$ states with $J\leq 2$. The $D$-wave contribution is found to be
insignificant in $Z(4430)^{-}$ decays, as expected for a true state at such
mass. The Argand diagram obtained for the $Z(4430)^{-}$ amplitude is
consistent with the resonant behavior. For the first time the resonant
character is demonstrated in this way among all known candidates for charged
four-quark states.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC and the Royal Society (United Kingdom);
NSF (USA). We also acknowledge the support received from EPLANET, Marie Curie
Actions and the ERC under FP7. The Tier1 computing centres are supported by
IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The
Netherlands), PIC (Spain), GridPP (United Kingdom). We are indebted to the
communities behind the multiple open source software packages on which we
depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia).
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* [41] E. Gross and O. Vitells, Trial factors or the look elsewhere effect in high energy physics, http://dx.doi.org/10.1140/epjc/s10052-010-1470-8Eur. Phys. J. C70 (2010) 525, http://arxiv.org/abs/1005.1891arXiv:1005.1891
* [42] D. J. Lange, The EvtGen particle decay simulation package, http://dx.doi.org/10.1016/S0168-9002(01)00089-4Nucl. Instrum. Meth. A462 (2001) 152
|
arxiv-papers
| 2014-04-07T19:52:54 |
2024-09-04T02:50:00.820050
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. An, L.\n Anderlini, J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V.\n Batozskaya, Th. Bauer, A. Bay, L. Beaucourt, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, M. Borsato, T.J.V. Bowcock, E. Bowen, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n J. Brodzicka, N.H. Brook, H. Brown, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, R. Calabrese, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D.\n Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H.\n Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Cassina, L.\n Castillo Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph.\n Charpentier, S. Chen, S.-F. Cheung, N. Chiapolini, M. Chrzaszcz, K. Ciba, X.\n Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic, H.V. Cliff, J. Closier,\n V. Coco, J. Cogan, E. Cogneras, P. Collins, A. Comerma-Montells, A. Contu, A.\n Cook, M. Coombes, S. Coquereau, G. Corti, M. Corvo, I. Counts, B. Couturier,\n G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie, C.\n D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, K. De Bruyn, S. De\n Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone,\n D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D. Derkach, O.\n Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, S. Donleavy, F. Dordei, M.\n Dorigo, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, G. Dujany, F. Dupertuis,\n P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof, L.\n Eklund, I. El Rifai, Ch. Elsasser, S. Ely, S. Esen, T. Evans, A. Falabella,\n C. F\\\"arber, C. Farinelli, N. Farley, S. Farry, D. Ferguson, V. Fernandez\n Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M. Fiore, M.\n Fiorini, M. Firlej, C. Fitzpatrick, T. Fiutowski, M. Fontana, F. Fontanelli,\n R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, J. Fu, E. Furfaro, A.\n Gallas Torreira, D. Galli, S. Gallorini, S. Gambetta, M. Gandelman, P.\n Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld,\n L. Gavardi, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, A. Gianelle, S.\n Giani', V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, C. Gotti, M. Grabalosa G\\'andara, R. Graciani\n Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening,\n S. Gregson, P. Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu.\n Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, B.\n Hamilton, T. Hampson, X. Han, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew,\n J. Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n L. Henry, J.A. Hernando Morata, E. van Herwijnen, M. He{\\ss}, A. Hicheur, D.\n Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, N. Hussain, D.\n Hutchcroft, D. Hynds, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, J.\n Jalocha, E. Jans, P. Jaton, A. Jawahery, M. Jezabek, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, N. Jurik, M. Kaballo, S. Kandybei, W.\n Kanso, M. Karacson, T.M. Karbach, M. Kelsey, I.R. Kenyon, T. Ketel, B.\n Khanji, C. Khurewathanakul, S. Klaver, O. Kochebina, M. Kolpin, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, B. Langhans, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam,\n J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T.\n Lesiak, B. Leverington, Y. Li, M. Liles, R. Lindner, C. Linn, F. Lionetto, B.\n Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, P. Lowdon, H.\n Lu, D. Lucchesi, H. Luo, A. Lupato, E. Luppi, O. Lupton, F. Machefert, I.V.\n Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, M.\n Manzali, J. Maratas, J.F. Marchand, U. Marconi, C. Marin Benito, P. Marino,\n R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M.\n Martinelli, D. Martinez Santos, F. Martinez Vidal, D. Martins Tostes, A.\n Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, A. Mazurov, M. McCann, J.\n McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, N. Moggi, J. Molina Rodriguez,\n S. Monteil, D. Moran, M. Morandin, P. Morawski, A. Mord\\`a, M.J. Morello, J.\n Moron, A.-B. Morris, R. Mountain, F. Muheim, K. M\\\"uller, R. Muresan, M.\n Mussini, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M.\n Needham, N. Neri, S. Neubert, N. Neufeld, M. Neuner, A.D. Nguyen, T.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem,\n A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O.\n Okhrimenko, R. Oldeman, G. Onderwater, M. Orlandea, J.M. Otalora Goicochea,\n P. Owen, A. Oyanguren, B.K. Pal, A. Palano, F. Palombo, M. Palutan, J.\n Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G.\n Passaleva, G.D. Patel, M. Patel, C. Patrignani, A. Pazos Alvarez, A. Pearce,\n A. Pellegrino, M. Pepe Altarelli, S. Perazzini, E. Perez Trigo, P. Perret, M.\n Perrin-Terrin, L. Pescatore, E. Pesen, K. Petridis, A. Petrolini, E.\n Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A. Pistone, S.\n Playfer, M. Plo Casasus, F. Polci, A. Poluektov, E. Polycarpo, A. Popov, D.\n Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H.\n Rademacker, B. Rakotomiaramanana, M. Rama, M.S. Rangel, I. Raniuk, N.\n Rauschmayr, G. Raven, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A.\n Richards, M. Rihl, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe,\n A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky,\n A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P.\n Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B.\n Saitta, V. Salustino Guimaraes, C. Sanchez Mayordomo, B. Sanmartin Sedes, R.\n Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, M. Schiller, H. Schindler, M.\n Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune,\n R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov, K.\n Senderowska, I. Sepp, N. Serra, J. Serrano, L. Sestini, P. Seyfert, M.\n Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, V. Shevchenko,\n A. Shires, R. Silva Coutinho, G. Simi, M. Sirendi, N. Skidmore, T.\n Skwarnicki, N.A. Smith, E. Smith, E. Smith, J. Smith, M. Smith, H. Snoek,\n M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B.\n Spaan, A. Sparkes, F. Spinella, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, O. Stenyakin, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S.\n Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah, L. Sun, W.\n Sutcliffe, K. Swientek, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini,\n F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier,\n S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning,\n M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti,\n A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S.\n Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D.\n Vieira, M. Vieites Diaz, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo{\\ss}, H. Voss, J.A. de Vries, R.\n Waldi, C. Wallace, R. Wallace, J. Walsh, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, G. Wilkinson,\n M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W.\n Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y.\n Xie, Z. Xing, Z. Xu, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M.\n Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A.\n Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Tomasz Skwarnicki",
"url": "https://arxiv.org/abs/1404.1903"
}
|
1404.1932
|
De Rham cohomology with spacelike compact and timelike compact supports
has recently been noticed to be of importance for understanding the
structure of classical and quantum field theories on curved spacetimes.
We compute these cohomology groups for globally hyperbolic spacetimes in
terms of their standard de Rham cohomologies. The calculation exploits
the fact that the de Rham-d'Alambert wave operator can be extended to a
chain map that is homotopic to zero and that its causal Green function
fits into a convenient exact sequence. This method extends also to the
Calabi (or Killing-Riemann-Bianchi) complex and possibly other
differential complexes. We also discuss generalized causal structures
and functoriality.
§ INTRODUCTION
Recently, a number of works on the structure of classical and quantum
field theory on curved
spacetimes [15, 46, 6, 19, 5, 31, 35, 36] have made use
of de Rham cohomology with spacelike compact supports. It appears in the
characterizations of the center of Poisson (or quantum) algebra of
observables of the Maxwell field and also of the degeneracy of the
bilinear pairing between spacelike compactly supported solutions and
compactly supported smearing functions. Similar considerations appear in
more general field theories [36, 35], though involving
cohomologies of complexes that are different from the de Rham one.
It was noticed long ago [1] that non-trivial spacetime topology
can influence in a non-trivial way the construction of the classical and
quantum field theories. However, these effects had not been
systematically investigated until recently. This may explain why neither
the standard literature on differential geometry and topology, nor the
literature on relativity seem to have considered cohomologies with
supports restricted by causal relations (like spacelike or timelike
compactness). So, given their growing importance, they deserve
independent investigation, which is the subject of this work.
In Section <ref>, we briefly outline some well known
geometric properties of the de Rham complex on a Lorentzian spacetime,
as well as some basic facts of homological algebra. These properties are
then used in Section <ref> to express the various cohomologies
with causally restricted supports in terms of the standard de Rham
cohomologies with unrestricted and compact supports. Then,
Section <ref> makes a few remarks and lists some
generalizations of the method of Section <ref>.
Sections <ref> and <ref> deal with the behavior
of the causally restricted cohomology groups under changes of causal
structure and under embeddings. Section <ref> presents the
Calabi differential complex and repeats the calculations of
Section <ref> for it. The Calabi complex plays a role in
linearized gravity on a constant curvature background analogous to that
of the de Rham complex for Maxwell theory. Then, Section <ref>
briefly describes how the methods applied to the de Rham and Calabi
examples could be generalized to other differential complexes that arise
in the study of general field theories with constrains and gauge
invariance [36, 35]. Finally, Section <ref>
concludes with a discussion of our results.
It should be mentioned that results very similar to those in
Section <ref> have been obtained independently in the recent
work [5], though by a different method. On the other hand, the
content of Section <ref> goes beyond [5] in several
§ PRELIMINARIES
Fix an $n$-dimensional smooth manifold $M$ with a Lorentzian metric $g$
such that $(M,g)$ is an oriented, time-oriented space, globally
hyperbolic spacetime [51, 32, 42, 4]. Recall
that, according to the Geroch splitting theorem, there exists a
diffeomorphism $M \cong \R\times \Sigma$ (non-unique, of course) where
the corresponding projection $t\colon M \to \R$ is a Cauchy
temporal function [25, 9, 8].
Let $\Omega^p(M)$ denote the linear space of differential $p$-forms on
$M$ and $\d \colon \Omega^p(M) \to \Omega^{p+1}(M)$ the de Rham
differential, which together form the de Rham complex
\begin{equation}\label{eq:dr-cpx}
\begin{tikzcd}
0 \arrow{r} &
\Omega^0(M) \arrow{r}{\d} &
\Omega^1(M) \arrow{r}{\d} &
\cdots \arrow{r}{\d} &
\Omega^n(M) \arrow{r} &
0 .
\end{tikzcd}
\end{equation}
Its cohomology is denoted by $H^p(M)$. It is well known that this
de Rham cohomology is isomorphic, $H^p(M) \cong H^p(M,\R)$, to the
singular cohomology of $M$ with coefficients in $\R$, to the Čech
cohomology of $M$ with coefficients in $\R$, and to the sheaf cohomology
of $M$ with coefficients in the sheaf of locally constant $\R$-valued
functions, all of which being isomorphic are denoted by $H^p(M,\R)$. If
we replace $\Omega^p(M)$ in (<ref>) with $\Omega_0^p(M)$, the
linear space of differential $p$-forms with compact support, the
corresponding de Rham cohomology of $M$ with compact supports is
isomorphic to the singular homology of $M$ with coefficients in $\R$,
$H_0^p(M) \cong H_p(M,\R)$, under the same hypotheses as
before [11]. There also exists a non-degenerate bilinear
pairing between
$\Omega^p(M)$ and $\Omega_0^{n-p}(M)$,
\begin{equation}
\langle \alpha, \beta \rangle = \int_M \alpha\wedge \beta ,
\end{equation}
which descends to a non-degenerate bilinear pairing between $H^p(M)$ and
$H_0^p(M)$. This result is known as Poincaré duality.
Using the Hodge star operator ${*}\colon \Omega^p(M) \to
\Omega^{n-p}(M)$ associated to the metric $g$, we can define the de Rham
co-differential $\delta = {*}\d{*} \colon \Omega^p(M) \to
\Omega^{p-1}(M)$. Next, we define the so-called
de Rham-d'Alambertian or wave operator $\square\colon
\Omega^p(M) \to \Omega^p(M)$,
\begin{equation}\label{eq:dal-def}
\square = \d\delta + \delta \d .
\end{equation}
This operator differs from the simple tensor d'Alambertian $\nabla_a
\nabla^a$ by terms of lower differential order. From its very
definition, we see that the d'Alambertian is a cochain map from the
de Rham complex to itself, $\d\square = \square\d$, which is moreover
cochain homotopic to zero, with the co-differential $δ$ the
corresponding cochain homotopy. That is, it induces the zero map from
$H^p(M)$ to itself. The following diagram illustrates the discussion:
\begin{equation}\label{eq:dR-homotopy}
\begin{tikzcd}
0 \arrow{r} &
\Omega^0(M) \arrow{r}{\d} \arrow{d}{\square} &
\Omega^1(M) \arrow{r}{\d} \arrow{d}{\square}
\arrow[dashed]{dl}[swap]{\delta} &
\cdots \arrow{r}{\d}
\arrow[dashed]{dl}[swap]{\delta} &
\Omega^n(M) \arrow{r} \arrow{d}{\square}
\arrow[dashed]{dl}[swap]{\delta} &
0 \\
0 \arrow{r} &
\Omega^0(M) \arrow{r}{\d} &
\Omega^1(M) \arrow{r}{\d} &
\cdots \arrow{r}{\d} &
\Omega^n(M) \arrow{r} &
\end{tikzcd} ,
\end{equation}
where the rows constitute (de~Rham) complexes, the solid arrows commute,
and the dashed arrows illustrate the cochain homotopy. This is an
important observation that will be used in an essential way in
Section~\ref{sec:comp}. Note that the formula~\eqref{eq:dal-def} is
analogous to the formula for the Hodge-de~Rham Laplacian in Riemannian
geometry. There, the observation that this Laplacian is homotopic to
zero lies at the foundation of Hodge theory~[27, 33].
The causal structure on $M$ defined by the Lorentzian metric $g$ allows
us to restrict the supports of differential forms in other ways as well.
A closed set $S M$ is said to be \emph{retarded} if $S J^+(K)$
for some compact $K$, \emph{advanced} if $SJ^-(K)$ for some compact
$K$, \emph{spacelike compact} if it $SJ(K)$ for some compact $K$,
\emph{past compact} if $S∩J^-(K)$ is compact for every compact $K$,
\emph{future compact} if $S∩J^+(K)$ is compact for every compact
$K$, and \emph{timelike compact} if $S$ is both past and future
compact~[45, 2]. Timelike compactness is also equivalent to
the property of having compact intersection with every spacelike compact
set. Let $Ω^p_X(M)$, with $X=+,-,sc,pc,fc$ or $tc$, denote the
linear space of differential $p$-forms with, respectively, retarded,
advanced, spacelike compact, past compact, future compact or timelike
compact supports. For brevity, we refer to these spaces as space of
forms with \emph{causally restricted supports}.
Of course, since differential operators preserve supports, it also
restricts to $□Ω_0^p(M) →Ω_0^p(M)$. By the
same reasoning, the spaces of forms with causally restricted supports
are also preserved by both $$̣ and $\square$. We define de Rham
cohomology with causally restricted supports in the obvious way and
denote it by $H^p_X(M)$, with $X=+,-,sc,pc,fc$ or $tc$. Let
$\Omega^p_\square(M)$ and $\Omega^p_{\square,X}(M)$ denote the kernel of
the wave operator $\square$, also known as its solution space, in
the spaces of forms with corresponding supports. Finally, by the
cochain map property, the de Rham differential restricts to the kernel
of the wave operator, hence defining the de Rham cohomology groups
$H^p_\square (M)$ and $H^p_{\square,X}(M)$ of solutions.
The wave operator on a globally hyperbolic Lorentzian manifold is well
known to be Green hyperbolic. That is, it has advanced and
retarded Green functions denoted respectively $\G_+$ and $\G_-$,
$\G_\pm \colon \Omega_0^p(M) \to \Omega_\pm^p(M)$. Since $\square$
commutes with $\d$, then so do $_+$ and $_-$. The form $β=
_±[α]$ is the unique solution of $□β= α$ with,
respectively, retarded or advanced support. The domain of definition of
the Green functions can be extended, in a unique way, to $Ω_X^p(M)$
for $X=+,-,pc$ or $fc$. Then, the maps
\begin{equation}\label{eq:biject}
\square\colon \Omega_Y^p(M) \to \Omega_Y^p(M), \quad
\G_X \colon \Omega_Y^p(M) \to \Omega_Y^p(M)
\end{equation}
are mutually inverse bijections, whenever $X=+$ and $Y=+$ or $pc$, or $X=-$
and $Y=-$ or $fc$. The combination $= _+ - _-$ is known as the
\emph{causal} Green function and fits into the following, in our
terminology Green-hyperbolic, exact
sequences~[3, 26, 36, 35, 2]
% XXX: Why doesn't {gather} work with tikzcd?
\begin{equation}\label{eq:short-sc}
\begin{tikzcd}
0 \arrow{r} &
\Omega_0^p(M) \arrow{r}{\square} &
\Omega_0^p(M) \arrow{r}{\G} &
\Omega_{sc}^p(M) \arrow{r}{\square} &
\Omega_{sc}^p(M) \arrow{r} &
0 ,
\end{tikzcd}
\end{equation}
\begin{equation}\label{eq:short-tc}
\begin{tikzcd}
0 \arrow{r} &
\Omega_{tc}^p(M) \arrow{r}{\square} &
\Omega_{tc}^p(M) \arrow{r}{\G} &
\Omega^p(M) \arrow{r}{\square} &
\Omega^p(M) \arrow{r} &
0 .
\end{tikzcd}
\end{equation}
Note that, according to the above formulas, we can represent the space
of solutions with spacelike compact or unrestricted support either as
\begin{gather}
\Omega^p_{\square,X}(M) = \ker \square \sso \Omega^p_X(M) \\
\text{or}\quad
\Omega^p_{\square,X}(M) = \G[\Omega^p_Y(M)] =
\Omega^p_Y(M)/\square\Omega^p_Y(M),
\end{gather}
with $X=sc$ and $Y=0$, or
$X$ empty and $Y=tc$, respectively. On the other hand, we have trivial
solutions paces $Ω^p_□,X(M) = {0}$ when $X = +,-,pc$ or
The existence of the Green-hyperbolic exact sequences will allow us to
later make use of the following elementary result of homological
algebra. Let $A^∙= (A^p,)̣$ be a cochain complex, and similarly
for $B^∙$ and $C^∙$. It is well known that a short exact
sequence of cochain maps,
\begin{equation}
\begin{tikzcd}
0 \arrow{r} & A^\bullet \arrow{r} & B^\bullet \arrow{r} & C^\bullet
\arrow{r} & 0 ,
\end{tikzcd}
\end{equation}
induces a long exact sequence in cohomology,
\begin{equation}\label{eq:long-ex}
\begin{tikzcd}
0 \arrow{r} &
H^0(A^\bullet,\d) \arrow{r} &
H^0(B^\bullet,\d) \arrow{r} \arrow[draw=none]{d}[name=Z,shape=coordinate]{}&
\arrow[rounded corners,
to path={ -- ([xshift=2ex]\tikztostart.east)
|- (Z) [near end]\tikztonodes
-| ([xshift=-2ex]\tikztotarget.west)
-- (\tikztotarget)}]{dll}[above,pos=1]{\d} \\
H^1(A^\bullet,\d) \arrow{r} &
H^1(B^\bullet,\d) \arrow{r} &
H^1(C^\bullet,\d) \arrow{r}{\d} & \cdots
\end{tikzcd}
\end{equation}
\section{Computation of cohomology groups}\label{sec:comp}
In this section, we state and prove our main results on de~Rham
cohomology with causally restricted supports. We rely essentially on the
properties of the wave operator and its Green functions, as summarized
in Section~\ref{sec:prelim}. The important properties are that the wave
operator $□$ is cochain homotopic to zero, and the way its range
and kernel characterized using the causal Green function $$.
\begin{thm}\label{thm:pf}
De~Rham cohomology $H^p_X(M)$, with $X=+,-,pc$ or $fc$, is trivial.
\end{thm}
\begin{proof}
Let $X=+,-,pc$ or $fc$. Then, as was noted in Section~\ref{sec:prelim},
the wave operator is a cochain map of the corresponding de~Rham complex
into itself, is invertible [Equation~\eqref{eq:biject}] and cochain
homotopic to zero [Equation~\eqref{eq:dal-def}]. Therefore, it is an
elementary consequence of homological algebra~[11] that this
complex is homotopy equivalent to the trivial complex and hence has
vanishing cohomology.
\end{proof}
\begin{thm}\label{thm:sc-tc}
We have the isomorphisms
\begin{align}
H^p_{sc}(M) &\cong H^{p+1}_0(M), &
H^p_{\square,sc} &\cong H^p_0(M) \oplus H^{p+1}_0(M), \\
H^p_{tc}(M) &\cong H^{p-1}(M), & \text{and} ~
H^p_{\square}(M) &\cong H^p(M) \oplus H^{p-1}(M),
\end{align}
with the convention that all cohomologies vanish in degree $p$ for $p<0$
or $p>n$.
\end{thm}
\begin{proof}
Recall again from Section~\ref{sec:prelim} that both the wave operator
$\square$ and its causal Green function $\G$ commute with $\d$ and hence
constitute cochain maps between the de~Rham complexes with appropriate
supports, inducing maps in cohomology. Moreover, since $\square$ is
cochain homotopic to zero [Equation~\eqref{eq:dal-def}], it induces the
zero map in cohomology.
Let us start with spacelike compact supports. We can break the exact
sequence in~\eqref{eq:short-sc} into two short exact sequences of
\begin{equation}
\begin{tikzcd}
0 \arrow{r} &
\Omega_0^p(M) \arrow{r}{\square} &
\Omega_0^p(M) \arrow{r}{\G} &
\Omega_{\square,sc}^p(M) \arrow{r} &
0 ,
\end{tikzcd}
\end{equation}
\begin{equation}
\begin{tikzcd}
0 \arrow{r} &
\Omega_{\square,sc}^p(M) \arrow{r}{\sso} &
\Omega_{sc}^p(M) \arrow{r}{\square} &
\Omega_{sc}^p(M) \arrow{r} &
0 .
\end{tikzcd}
\end{equation}
Because $\square$ always induces the zero map, the corresponding long
exact sequences in cohomology [cf.~Equation~\eqref{eq:long-ex}] break up
into the following short exact sequences:
\begin{equation}
\begin{tikzcd}
0 \arrow{r} &
H^p_0(M) \arrow{r}{\G} &
H^p_{\square,sc}(M) \arrow{r}{\d} &
H^{p+1}_0(M) \arrow{r} &
0 ,
\end{tikzcd}
\end{equation}
\begin{equation}
\begin{tikzcd}
0 \arrow{r} &
H_{sc}^{p-1}(M) \arrow{r}{\d} &
H_{\square,sc}^p(M) \arrow{r}{\sso} &
H_{sc}^p(M) \arrow{r} &
0 ,
\end{tikzcd}
\end{equation}
again with the convention that any $H^p_X(M)$ vanishes for $p<0$ or
$p>n$. By inspection, it is not hard to see that the above short exact
sequences induces the following isomorphisms:
\begin{align}
H^p_{sc}(M) &\cong H^{p+1}_0(M), \\
H^p_{\square,sc}(M) &\cong H^p_{sc}(M) \oplus H^{p-1}_{sc}(M)
\cong H^p_0(M) \oplus H^{p+1}_0(M).
\end{align}
Applying the same argument to the exact sequence~\eqref{eq:short-tc}, we
obtain the isomorphisms
\begin{align}
H^p_{tc}(M) &\cong H^{p-1}(M), \\
H^p_{\square}(M) &\cong H^p_{tc}(M) \oplus H^{p+1}_{tc}(M)
\cong H^p(M) \oplus H^{p-1}(M).
\end{align}
This completes the proof.
\end{proof}
Let $\Sigma\sso M$ be a Cauchy surface. Recall that, by the smooth
Geroch splitting theorem, we can always
smoothly factor $M \cong \R\times \Sigma$. This observation results in
\begin{cor}\label{cor:cauchy}
We have the isomorphisms
\begin{align}
H^p_{sc}(M) &\cong H^p_0(\Sigma), &
H^p_{\square,sc}(M) &\cong H^p_0(\Sigma) \oplus H^{p-1}_0(\Sigma) \\
H^p_{tc}(M) &\cong H^{p-1}(\Sigma), & \text{and} ~
H^p_{\square}(M) &\cong H^p(\Sigma) \oplus H^{p-1}(\Sigma),
\end{align}
with the convention that all cohomologies vanish in degree $p$ for $p<0$
or $p>n$.
\end{cor}
\begin{proof}
The splitting $M \cong \R\times \Sigma$ shows that $M$ is homotopic to
$\Sigma$. Hence, by the homotopy invariance of de~Rham cohomologies with
unrestricted supports, we have the isomorphism $H^p(M) \cong
H^p(\Sigma)$. On the other hand, Poincar\'e duality induces the
isomorphism $H^p_0(M) \cong H^{p-1}_0(\Sigma)$. Therefore, the desired
conclusion follows directly from these identities in combination with
\end{proof}
Finally, knowing the respective de~Rham cohomologies with spacelike and
timelike compact supports, we have the following generalization of the
Poincar\'e lemma.
\begin{cor}\label{cor:pairing}
The non-degenerate bilinear pairing between $\Omega^p_{sc}(M)$ and
$\Omega^{n-p}_{tc}(M)$ descends to a non-degenerate bilinear pairing
between $H^p_{sc}(M)$ and $H^{n-p}_{tc}(M)$. There exists also a
non-degenerate bilinear pairing between $H^p_{\square,sc}(M)$ and
\end{cor}
\begin{proof}
A consequence of Theorem~\ref{thm:sc-tc} is that $H^p_{sc}(M) \cong
H^{p+1}_0(M)$ and $H^{n-p}_{tc}(M) = H^{n-p-1}(M)$. So, the usual
Poincar\'e duality establishes that $H^p_{sc}(M)^* \cong
H^{n-p}_{tc}(M)$. The isomorphism can be exhibited by bilinear pairing,
which descends from the standard bilinear pairing between
$\Omega^p_{sc}(M)$ and $\Omega^{n-p}_{tc}(M)$, tracing its effect
throughout the proof of Theorem~\ref{thm:sc-tc}. Its non-degeneracy is
also a consequence of the Poincar\'e lemma applied to $H^p_0(M)$ and
It also follows from Theorem~\ref{thm:sc-tc} that $H^p_{\square,sc}(M)
\cong H^p_0(M) \oplus H^{p+1}_0(M)$ and $H^{n-p}_{\square}(M) \cong
H^{n-p}(M) \oplus H^{n-p-1}(M)$. Again, the usual Poincar\'e duality
establishes the isomorphism $H^p_{\square,sc}(M) \cong
H^{n-p}_{\square}(M)$. The isomorphism can be exhibited by a bilinear
pairing between $\Omega^p_{\square,sc}(M)$ and
$\Omega^{n-p}_{\square}(M) \cong
\Omega^{n-p}_{tc}/\square\Omega^{n-p}_{tc}(M)$, defined by the latter
identity and the self-adjointness of $\square$ with respect to our
pairing between forms. Again, tracing this pairing through the proof of
Theorem~\ref{thm:sc-tc} and appealing to the standard Poincar\'e duality
establishes its non-degeneracy.
\end{proof}
As already discussed in the introduction, the importance of knowing the
above cohomology groups is important for understanding the
(pre)symplectic and Poisson structure of classical field theories, as
emphasized in~[36, 35, 46, 6, 5]. The same result
as Corollary~\ref{cor:cauchy} was obtained independently
in~[5]. As a matter of fact, the method of~[5] can
be seen as a special case of our homological algebraic calculation, as
discussed more explicitly at the end of Section~\ref{sec:conal-cohom}.
\section{Notes and generalizations}\label{sec:notes}
\subsection{Generalized causal structures}\label{sec:gen-caus}
The notion of a causal structure on a manifold or even a topological
space (in the sense of a partial order on events) can be generalized
quite fare beyond the context of Lorentzian geometry~[37, 22]. We
will stick with the context of differential geometry, where a natural
generalization consists of introducing at every point of a manifold an
arbitrary convex cone in the tangent%
\footnote{One could equally do so in the cotangent bundle, and
produce a tangent cone by convex (or polar) duality.} %
bundle. Such a manifold could be called a \emph{conal
manifold}~[41, 38, 49, 35]. Various notions generated
by the causal structure on Lorentzian manifolds survive almost without
modification on conal manifolds, including spacelike and timelike
compactness. The main question we will try to answer in this section is
the following: Is it possible to use the methods of
Section~\ref{sec:comp} to compute causally restricted cohomologies on a
conal manifold? We shall see that the answer is \emph{yes}, even if the
conal manifold is not Lorentzian.
\subsubsection{Conal manifolds}\label{sec:conal}
Before dealing with spacelike and timelike compactly supported forms,
let us introduce the basics of conal manifolds and causal relations on
them. Let $M$ be a smooth manifold and $C\subset TM$ be an open subset,
such that $C_x = C \cap T_x M$ is an open, convex cone in $T_x M$ that
does not contain any affine line. It can be shown that the interior
$C^\oast_x$ of the polar dual (or convex dual) cone $T^*_x \supset C^*_x
= \{ p \in T^*_xM \mid \forall v\in C_x \colon p\cdot v \ge 0 \}$
satisfies the same conditions, with $C^\oast = \sqcup_{x\in M}
C^\oast_x$. The pair $(M,C)$ or $(M,C^\oast)$ is called a \emph{conal
manifold}, with $C$ (or $C^\oast$) called the tangent (or cotangent)
\emph{cone distribution} or \emph{cone bundle}. For example, the subset
of non-vanishing, future-pointing, timelike vectors on a Lorentzian
manifold with a time orientation satisfies the above conditions. In
general, the cones $C_x$ need not even have elliptic cross sections,
thus not be associated to any Lorentzian metric. The cones of future
pointing timelike vectors of linear symmetric hyperbolic PDE systems
also satisfy the same properties~\cite[Sec.4.1]{kh-big}. Sometimes, it
is also convenient to admit degenerate cases where the cones are not
open or contain some affine lines, but some special care must be taken
in those situations.
Given a conal manifold $(M,C)$ we can define a \emph{chronological
order} relation on the points of $M$. Namely, $x \ll y$ if there exists
a smooth curve $\gamma\colon [0,1] \to M$, such that $\gamma(0) = x$,
$\gamma(1) = y$ and $\dot{\gamma}(t) \in C$ for all $t\in [0,1]$. It can
be shown that the chronological order relation $I^+\sso M\times M$ is
open and transitive. We can also define the \emph{reverse chronological
order}, $I^-$, and \emph{chronological influence}, $I = I^+\cup I^-$,
relations in the obvious way. We avoid defining the analog of the
\emph{causal} order relation usually denoted by $J^+$, simply because we
have not made any hypotheses about the regularity of the set of causal
vectors ($\overline{C}_x \sso T_xM$). Given any set $K\sse M$, we denote
by $I^\pm(K)$ the set of all points of $M$ that respectively
chronologically precede ore are preceded by the points of $K$. In
general, $I^\pm(K)$ is not closed, even if $K$ is. So, for convenience
we define $\overline{I}^\pm(K) = \overline{I^\pm(K)}$. We also use the
notation $I(K) = I^+(K) \cup I^-(K)$ and $\overline{I}(K) =
\overline{I}^+(K) \cup \overline{I}^-(K)$. Note that $\overline{I}^\pm
\sse M\times M$ need not be transitive as relations.
The definition of a Cauchy surface $\Sigma \subset M$ is the usual one,
every inextensible smooth curve with timelike tangents must intersect
$\Sigma$ exactly once. It has recently been shown that the smooth
version of the Geroch splitting theorem~[25, 9, 8]
generalizes to conal manifolds~[17]. So, \emph{globally
hyperbolicity} can be simply characterized by the existence of a Cauchy
surface. Also, the results of~[45] should also directly carry
over to conal manifolds. Finally, we define the notions of
\emph{advanced}, \emph{retarded}, \emph{spacelike compact},
\emph{timelike compact}, \emph{future compact} and \emph{past compact}
exactly in the same way as in Section~\ref{sec:prelim}, with the
exception that we use the relations $\bar{I}^\pm$ and $\bar{I}$ instead
of the relations $J^\pm$ and $J$.%
\footnote{We are not concerned with possible minor inconsistencies
this substitution introduces in the case of Lorentzian manifolds with
ill-behaved causal structures. In any case, we shall only apply these
notions for globally hyperbolic spacetimes, where these differences do
not appear.}
\subsubsection{Cohomology with causally restricted supports}\label{sec:conal-cohom}
Let $M$ be a globally hyperbolic conal manifold and $g$ an auxiliary
globally hyperbolic Lorentzian metric that induces another conal
structure on $M$ that is ``slower'' than the original one. That is,
$\Omega^p_{\pm_g}(M) \sse \Omega^p_{\pm}(M)$, which also implies that
$\Omega^p_{sc_g}(M) \sse \Omega^p_{sc}(M)$, while
$\Omega^p_{fc_g,pc_g}(M) \supseteq \Omega^p_{fc,pc}(M)$, and hence
$\Omega^p_{tc_g}(M) \supseteq \Omega^p_{tc}(M)$. Any conal manifold
admits a nowhere vanishing vector field (contract each cone to a ray and
select a vector from it), which is moreover everywhere future directed.
So, that such an auxiliary Lorentzian metric always exists follows from
general arguments showing the existence of Lorentzian metrics on
manifolds with vanishing Euler characteristic (i.e.,\ admitting a
nowhere vanishing vector field)~[4, 42], which the
``slowness'' requirement implemented by making sure that the Lorentzian
timelike cones closely hug the direction singled out by the above
everywhere timelike vector field.
Let $G_\pm$ denote once again the advanced and retarded Green functions
of the wave operator $\square_g$ defined with respect to $g$. Then it is
easy to see that the Green functions are still well defined and
injective as maps $G_\pm \colon \Omega^p_0(M) \to \Omega^p_\pm(M)$.
Appealing to the same logic as in the standard
proofs~[3, 26, 36, 35, 2], we can extend the Green
functions to bijective maps $G_\pm\colon \Omega^p_\pm(M) \to
\Omega^p_\pm(M)$ and $G_\pm\colon \Omega^p_{fc,pc}(M) \to
\Omega^p_{fc,pc}(M)$, from which it is straightforward to establish
exactness of the following sequences
\begin{equation}\label{eq:short-sc-conal}
\begin{tikzcd}
0 \arrow{r} &
\Omega_{0}^p(M) \arrow{r}{\square} &
\Omega_{0}^p(M) \arrow{r}{\G} &
\Omega_{sc}^p(M) \arrow{r}{\square} &
\Omega_{sc}^p(M) \arrow{r} &
0 ,
\end{tikzcd}
\end{equation}
\begin{equation}\label{eq:short-tc-conal}
\begin{tikzcd}
0 \arrow{r} &
\Omega_{tc}^p(M) \arrow{r}{\square} &
\Omega_{tc}^p(M) \arrow{r}{\G} &
\Omega^p(M) \arrow{r}{\square} &
\Omega^p(M) \arrow{r} &
0 ,
\end{tikzcd}
\end{equation}
where the supports are restricted by the given conal structure on $M$
and not by that induced by the auxiliary Lorentzian metric $g$. Note
that the proofs would make use of the hypothesis that the given conal
structure is globally hyperbolic, specifically in the construction of
explicit splitting maps that demonstrate
exactness~\cite[Lem.2.1]{kh-peierls}. Thus, repeating the arguments
Section~\ref{sec:comp}, we establish the following generalization of
Theorems~\ref{thm:pf} and~\ref{thm:sc-tc}.
\begin{thm}\label{thm:conal}
Consider a globally hyperbolic conal manifold $M$. Its de~Rham
cohomology $H^p_X(M)$ with causally restricted supports $X=+,-,pc$ or
$fc$ is trivial. Moreover, we have the isomorphisms
\begin{align}
H^p_{sc}(M) &\cong H^{p+1}_0(M), &
H^p_{\square,sc} &\cong H^p_0(M) \oplus H^{p+1}_0(M), \\
H^p_{tc}(M) &\cong H^{p-1}(M), & \text{and} ~
H^p_{\square}(M) &\cong H^p(M) \oplus H^{p-1}(M),
\end{align}
with the convention that all cohomologies vanish in degree $p$ for $p<0$
or $p>n$.
\end{thm}
It should be clear from the preceding discussion that there is nothing
inherently special in our use of the d'Alambertian $\square_g$, when it
comes to the calculation of de~Rham cohomologies with causally
restricted supports on a globally hyperbolic conal manifold $M$. It is
merely one of multiple possible auxiliary hyperbolic differential
operators that can serve the same purpose. Here are the key required
properties for such an operator $h$: $h$ must be a cochain map that is
homotopic to zero with respect to the de~Rham complex, it must possess
retarded and advanced Green functions, these Green functions must be
causal with respect to the given conal structure on $M$. In fact, the
conclusion of our Theorem~\ref{thm:sc-tc} was reached independently in
the recent paper~[5] by following an argument structurally
similar to ours, with the d'Alambertian replaced by the Lie derivative
$\Lie_v$ with respect to a complete timelike vector field $v$. It is
clearly (Green) hyperbolic~[3, 2, 35, 36] with Green
functions simply given by integration (into the future or past) along
the flow lines of $v$. Moreover, it is cochain homotopic to zero because
of the well known magic formula of Cartan: $\Lie_v = \iota_v \d + \d
\iota_v$.
\subsection{Functoriality}\label{sec:funct}
Recall that ordinary de~Rham cohomology is defined on any finite
dimensional manifold and the pullback of differential forms along a
smooth map between manifolds induces a map between their cohomologies
(in the direction opposite the original smooth map). This observation
has the following well-known formalisation: de~Rham cohomology in degree
$p$, $H^p(-)$, is a contravariant functor%
\footnote{We shall not delve here into the details of category theory.
It suffices to say that any statement that we shall make involving
functors and categories will be simply a very terse transcription of
some other property that will be spelled out in more elementary terms.
More details about the functorial properties of de~Rham cohomology can
be found in~[11].} %
from the category of smooth manifolds to the category of real vector
spaces. The same cannot be said for de~Rham cohomology with compact
supports, $H_0^p(-)$, because the pullback of a compactly supported form
need not be compactly supported itself. This pullback problem is fixed
by considering only proper%
\footnote{A continuous map is \emph{proper} if the preimage of any compact
set is compact.} %
smooth maps between manifolds. So, given a proper smooth map $f\colon
M\to N$, pullback along it induces a contravariant map between de~Rham
cohomologies in degree $p$ with compact support, $f^*\colon H^p_0(N) \to
H^p_0(M)$. If the map $f$ satisfies an different restrictive condition,
namely that it is an open embedding, it is possible to define a
covariant pushforward map $f_*\colon H^p_0(M) \to H^p_0(N)$: we can
identify $M$ with its image $f(M)$, an open subset of $N$, and extend by
zero any compactly supported form defined $M$ to all of $N$. In short,
de~Rham cohomology with compact supports, $H_0^p(-)$, defines a
contravariant functor on the category of smooth manifolds with proper
maps as morphisms, when paired with the pullback, while it defines a
covariant functor on the category of smooth manifolds with open
embeddings as morphisms, when paired with the pushforward.
A natural question is the following: do similar properties hold, and
under what precise conditions, for de~Rham cohomologies with causally
restricted supports? For instance, this question was briefly raised, but
without any definite answer, in~[5]. In fact, it is straight
forward to present causally restricted cohomologies as functors,
provided we modify the domain category by adding generalized causal
structures to manifolds (as in Section~\ref{sec:gen-caus}) and by
modifying the notion of a proper map with respect to the causal
Consider two conal manifolds $M$ and $N$, with a smooth map $f\colon
M\to N$ between them. We call the map $f$ \emph{reflectively
spacelike-proper} if the preimage of any spacelike compact set is also
spacelike compact, while we call it \emph{reflectively timelike-proper}
if the preimage of any timelike compact set is also timelike compact.
When the map $f$ is an open embedding, we also introduce the terminology
\emph{monotonically spacelike-proper} for the case when the image of any
spacelike compact set is itself spacelike-compact and
\emph{monotonically timelike-proper} for the case when the image of any
timelike compact set is timelike compact. We should note that the above
terminology is partly inspired by some general notions from the theory
of partially ordered sets. A map $f\colon M\to N$ between two partially
ordered sets $(M,\le)$ and $(N,\le)$ is said to be \emph{monotonic} if
$x \le y$ implies $f(x) \le f(y)$ and, on the other hand, it is said to
be \emph{order-reflecting} if $f(x) \le f(y)$ implies $x\le y$. The
following theorem is a straight forward generalization of the previous
arguments for the simpler case of compact supports.
\begin{thm}
Let $\CMan_{sc}$ and $\CMan_{tc}$ be the categories of conal manifolds
with, respectively, reflectively spacelike-proper and reflectively
timelike-proper, smooth maps as morphisms, while the $\CMan_{sc}^{e}$
and $\CMan_{tc}^{e}$ categories have, respectively, monotonically
spacelike-proper and monotonically timelike-proper open embeddings as
morphisms. same where morphisms are also required to be open
embeddings. Then, de~Rham cohomologies with spacelike and timelike
supports, $H^p_{sc}(-)$ and $H^p_{tc}(-)$, are contravariant functors on
$\CMan_{sc}$ and $\CMan_{tc}$, respectively. Similarly, $H^p_{tc}(-)$
and $H^p_{sc}(-)$ are covariant functors on $\CMan_{tc}^{e}$ and
$\CMan_{sc}^{e}$, respectively.
\end{thm}
\begin{proof}
The proof is a direct parallel of the above arguments for the case with
compact supports, since the definitions have been specifically adapted
to that argument.
\end{proof}
To show that the definitions of spacelike- and timelike-proper maps are
in some sense natural, we give a couple of examples.
\begin{lem}
Let $M$ be a manifold and two conal structures on it, $C\sse C' \sse TM$
($C$ is ``slower'' than $C'$) (Section~\ref{sec:gen-caus}). The identity
map is a reflectively spacelike-proper from $(M,C')$ to $(M,C)$ and
reflectively timelike-proper from $(M,C)$ to $(M,C')$.
\end{lem}
\begin{proof}
Let $K\sse M$ be any compact subset. Then, by hypothesis, the
$C$-influence set is smaller than the $C'$-influence set,
$\overline{I}_C(K) \sse \overline{I}_{C'}(K)$. Therefore, any
$C$-spacelike compact set is also $C'$-spacelike and hence the identity
from $(M,C')$ to $(M,C)$ is reflectively spacelike-proper. On the other
hand, if $U\sse M$ is $C'$-timelike compact, then we have the inclusion
$\overline{I}_C(K) \cap U \sse \overline{I}_{C'}(K) \cap U$, the latter
being compact. Therefore, $U$ is also $C$-timelike compact and the
identity from $(M,C)$ to $(M,C')$ is reflectively timelike-proper.
\end{proof}
\begin{lem}
Let $(M,g)$ and $(N,h)$ be two globally hyperbolic Lorentzian manifolds
and $f\colon M\to N$ an open isometric embedding, such that the image of
a Cauchy surface of $M$ is a Cauchy surface of $N$. Then, $f$ is
monotonically timelike-proper.
\end{lem}
\begin{proof}
Let $U\sse M$ be timelike compact. According to~[45], this is
equivalent to $U$ being contained between two Cauchy surfaces in
$(M,g)$, say $\Sigma_1, \Sigma_2 \sso M$. This means that the image,
$f(U)$ is contained between $f(\Sigma_1)$ and $f(\Sigma_2)$, with the
latter, by hypothesis, being Cuachy surfaces in $(N,h)$. Thus, $f(U)$ is
also timelike compact and the map $f$ is monotonically timelike-proper.
\end{proof}
\subsection{Calabi or Killing-Riemann-Bianchi complex}\label{sec:calabi}
In~[36, 35], it was pointed out that the construction of
the symplectic and Poisson structures on the phase space of field
theories with constraints and/or gauge invariance can be done using a
general framework, provided a given field theory satisfies certain
geometric conditions. These conditions include the existence of certain
differential complexes that extend the operators that constitute the
constraints and that generate the gauge transformations. For Maxwell
(and similar) theories, all of these complexes are invariably part of
the de~Rham complex~\cite[Secs.4.2--3]{kh-peierls}. On the other hand,
for linearized gravity, one has to use something different.
Unfortunately, the explicit form of these differential complexes is not
currently known for linearized gravity on an arbitrary
background~\cite[Sec.4.4]{kh-peierls}. However, in the special case of
constant curvature backgrounds, the answer is known and it is the
so-called \emph{Calabi complex}~[13]. It is likely that, once
an explicit understanding of the corresponding differential complexes
for more general backgrounds is achieved, the general framework
of~[36, 35] would supersede recent covariant treatments
of the quantization of linearized gravity
like~[18, 30].
The Calabi complex provides a fine resolution~[12] of the sheaf
of Killing vectors, similarly to how the de~Rham complex provides a fine
resolution of the sheaf of locally constant functions. As such, it has
been studied in some literature on the deformation of constant curvature
structures~[13, 7, 29, 23, 24, 43, 16].
Because its structure is substantially different from the de~Rham
complex, we summarize some of its relevant properties before
concentrating on its causally restricted cohomologies. Many of these
properties are scattered throughout or are simply not available in the
existing literature. We defer a fuller discussion of the Calabi complex,
which collects these properties and their proofs, to~[34].
\subsubsection{Tensor bundles}
We will present later a differential complex whose nodes are
sections of tensor bundles that are not so easy to express in
conventional notation. So, let us introduce the following short-hands.
We denote the cotangent bundle by $VM = T^*M$ and the bundle of metrics
(symmetric, covariant 2-tensors) by $S^2M = S^2T^*M$. Let $RM \sso
(T^*)^4M$ denote the sub-bundle of covariant 4-tensors that satisfy the
algebraic symmetries of the Riemann tensor ($R_{(ab)cd} = R_{ab(cd)} =
R_{abcd} - R_{cdab} = R_{[abc]d} = 0$). Next, we let $BM \sso (T^*)^5M$
denote target bundle of the Bianchi operator $\nabla_{[a} R_{bc]de}$.
At this point it is convenient to notice that the fiber of each of these
bundles carries~[20] an irreducible representation of $\GL(n)$,
with $n = \dim M$. In fact, it is easiest to describe the remaining
tensor bundles in terms of the irreducible $\GL(n)$ representation
carried by their fibers. So let $C_lM \sso (T^*)^{l+2}M$ (with $C$
standing for \emph{Calabi}) denote the sub-bundles of covariant
$(l+2)$-tensors with the corresponding irreducible representations
listed in Table~\ref{tbl:bundles}, which also lists their fiber ranks.
It is consistent for us to assign $C_0M \cong VM$, $C_1M \cong S^2M$ and
$C_2M \cong RM$ and $C_3M \cong BM$. Recall that, on an $n$-dimensional
manifold, the largest rank of a fully antisymmetric tensor is $n$. So
the bundles $C_lM$ become trivial (zero fiber rank) for $l>n$.
\begin{table}
\caption{
It is conventional to label irreducible $\GL(n)$ representations by
\emph{Young diagrams}~[21]. Recall that a Young diagram with
$k$ cells of type $(r_1,r_2,\ldots)$ consists of a number of rows of
non-increasing lengths $r_i$, $r_{i+1} \le r_i$, such that $\sum_i r_i =
k$. Given a Young diagram with $k$ cells, an instance of the
corresponding irreducible $\GL(n)$ representation class can be realized
as the image of the space of covariant $k$-tensors after two
projections: assign an independent tensor index to each cell of the
diagram, symmetrize over each row, anti-symmetrize over each
\hspace{1em}%
The table below lists the tensor bundles of the Calabi complex, the
corresponding irreducible $\GL(n)$ representations (labeled by Young
diagrams), and their fiber ranks, for $\dim M = n$. The rank is given by
the famous \emph{hook formula}, which is the following fraction. The
numerator is the product of the following numbers: place $n$ in the top
left cell, increase by $1$ to the right and decrease by $1$ down, until
all cells are filled. The denominator is the product of the following
numbers: fill a given cell with the number of cells constituting a hook
with vertex at the given location, extending to the right and
\label{tbl:bundles}
\begin{center}
\begin{tabular}{ccc}
bundle & Young diagram & fiber rank \\
\hline \rule[0.5ex]{0pt}{2.5ex} \\[-3ex]
$VM\cong C_0M$
& \yd{1} & $n$
\\ \\[-1ex]
$S^2M \cong C_1M$
& \yd{2} & $\frac{n(n+1)}{2}$ \\ \\[-1ex]
$RM\cong C_2M$
& \yd{2,2} & $\frac{n^2(n^2-1)}{12}$ \\ \\
$BM \cong C_3M$
& \yd{2,2,1} & $\frac{n^2(n^2-1)(n-2)}{24}$ \\ \\[-2ex]
\hline \rule[0.5ex]{0pt}{2.5ex} \\[-3ex]
& \yt{{1}{},{2}{},{\none[\raisebox{-.5ex}{\vdots}]},{l}}
& $\frac{n^2(n^2-1)(n-2)\cdots(n-l+1)}{2(l+1)l(l-2)!}$ \\ \\[-2ex]
\hline \rule[0.5ex]{0pt}{2.5ex}
\end{tabular}
\end{center}
\end{table}
Given two $S^2M$ tensors, we can construct an $RM$ tensor out of them
using the formula
\begin{equation}
(g \odot h)_{abcd}
= g_{ac} h_{bd} - g_{bc} h_{ad} - g_{ad} h_{bc} + g_{bd} h_{ac} .
\end{equation}
In fact, the above formula represents a $\GL(n)$-equivariant map between
$S^2\otimes S^2$ and $R$ (where we use the bundle prefixes to stand in
for the corresponding irreducible representations). The decomposition of
the $S^2\otimes S^2$ tensor product has only one copy of $R$, so by
Schur's lemma such a map is unique, up to an overall rescaling. The same
argument can be repeated for the tensor product $S^2\otimes Y$, where
$Y$ corresponds to any other Young diagram. This tensor product
decomposes into irreducible subrepresentations without multiplicities.
Then the projection onto any of the subrepresentations $Y'$ is
well defined up to a rescaling. If we fix a sections $g$ of $S^2M$ and
$h$ of $YM$, these projections define a bilinear operation between $g$
and $h$ with the result a section of $Y'M$. We use the following
explicit formulas:
\begin{align}
\notag
(g\odot t)_{abc:de}
+ g_{ad} t_{bc:e} + g_{bd} t_{ca:e} + g_{cd} t_{ab:e} \\
&\quad {}
- g_{ae} t_{bc:d} - g_{be} t_{ca:d} - g_{ce} t_{ab:d} , \\
\notag
(g\odot t)_{abcd:ef}
+g_{ae}t_{bcd:f}-g_{be}t_{cda:f}+g_{ce}t_{dab:f}-g_{de}t_{abc:f} \\
&\quad {}
-g_{af}t_{bcd:e}+g_{bf}t_{cda:e}-g_{cf}t_{dab:e}+g_{df}t_{abc:e} .
\end{align}
Note that a tensor with indices written as in $t_{abc:de}$ has the
symmetry type $(2,2,1)$, while $t_{abc:d}$ corresponds to the symmetry
type $(2,1,1)$, and so on.
The metric $g_{ab}$ itself, an $S^2M$ tensor, can now be used to produce
an $RM$ tensor,
\begin{equation}
(g\odot g)_{ab:cd} = 2(g_{ac}g_{bd} - g_{bc}g_{ad}) ,
\end{equation}
which is obviously covariantly constant. In fact, a constant curvature
spacetime must have (covariant) Riemann tensor, Ricci tensor and Ricci
scalar of the following form
\begin{equation}\label{eq:bkg-riem}
\bar{R}_{abcd} = \frac{k}{n(n-1)} (g_{ac}g_{bd} - g_{bc}g_{ad}), \quad
\bar{R}_{ac} = \frac{k}{n} g_{ac} , \quad
\bar{R} = k .
\end{equation}
We have decorated these quantities with a bar to indicate the fact that
we shall fix a constant curvature background metric $g$ and consider
perturbations on it. For our purposes, we also require that the
Lorentzian manifold $(M,g)$ is globally hyperbolic.
We should note that solutions of Einstein equations (including a
possible cosmological constant term) with constant curvature includes
Minkowski space ($k=0$), de~Sitter space ($k>0$) and anti-de~Sitter
space ($k<0$). The latter is not globally hyperbolic, so it is excluded
from part of our discussion. All three examples are simply connected.
Other examples may be obtained by taking quotients thereof with respect
to a discrete subgroup, thus changing the topology. The list of
possibilities is thus exhausted.
\subsubsection{Differential operators}
\label{sec:calabi-ops}
Now, we introduce a number of differential operators between the tensor
bundles that we have defined. These operators fit into the following
(almost) commutative diagram, where the tensor bundles also stand in for
their spaces of sections:
\begin{equation}
\begin{tikzcd}
0 \arrow{r} &
C_0M \arrow{r}{B_1} \arrow{d}{P_{0}} &
C_1M\arrow{r}{B_2} \arrow{d}{P_{1}} \arrow[dashed]{dl}[swap]{E_1} &
C_2M \cdots \arrow{r}{B_{n}} \arrow{d}{P_2} \arrow[dashed]{dl}[swap]{E_2} &
\bar{C}_{n}M \arrow{r} \arrow{d}{P_{n}}
\arrow[dashed]{dl}[swap]{E_{n}} &
0 \\
0 \arrow{r} &
C_0M \arrow{r}{B_1} &
C_1M\arrow{r}{B_2} &
C_2M \cdots \arrow{r}{B_{n}} &
C_{n}M \arrow{r} &
0 \\
\end{tikzcd} ,
\end{equation}
where all the solid arrows commute and the rows constitute (cochain)
complexes. The vertical maps are then necessarily cochain maps. They
happen to satisfy the identities $P_l = E_{l+1}\circ B_{l+1} + B_l\circ
E_l$, which means that they are null-homotopic, with the $E_l$ supplying
the corresponding cochain homotopy.
Below, we give explicit formulas for all these differential operators in
dimension $n=4$. More details can be found in~[34], which
draws from the earlier
works~[13, 7, 29, 23, 24, 43, 16].
As we shall see, for low indices they are well known in the relativity
literature. However, the relations between them in terms of fitting into
the above commutative diagram do not seem to have been fully noted.
The \emph{Calabi differential complex} is given by
\begin{align}
B_1[v]_{ab} &= \nabla_a v_b + \nabla_b v_a , \\
\notag
B_2[h]_{ab:cd} &= \frac{1}{2} \left(
\nabla_{(a}\nabla_{c)} h_{bd} -
\nabla_{(b}\nabla_{c)} h_{ad} -
\nabla_{(a}\nabla_{d)} h_{bc} +
\nabla_{(b}\nabla_{d)} h_{ac} \right) \\
& \qquad {}
+ k\frac{1}{2n(n-1)} (g\odot h)_{ab:cd} , \\
B_3[r]_{abcde} &= 3\nabla_{[a} r_{bc]:de}
= \nabla_a r_{bc:de} + \nabla_b r_{ca:de} + \nabla_c r_{ab:de} , \\
&= 4\nabla_{[a} b_{bcd]:ef} \\
&= \nabla_a b_{bcd:ef} - \nabla_b b_{cda:ef}
- \nabla_c b_{dab:ef} - \nabla_d b_{abc:ef} .
\end{align}
The details showing that these operators have the desired symmetry
properties and indeed define a complex, $B_{l+1}\circ B_l = 0$, which is
moreover elliptic,%
\footnote{A complex of differential operators is \emph{elliptic} if
the corresponding complex of symbol maps is exact for every non-zero
covector.} %
can be found in~[34].
It is interesting to note the following relations with well known
differential operators in relativity. The \emph{Killing} operator is
$K[h] = B_1[h]$. The \emph{linearized Riemann} tensor is $\dot{R}[h] =
-\frac{1}{2} B_2[h] + k \frac{2}{n(n-1)} (g\odot h)$, where the all
covariant non-linear Riemann tensor is expanded as $R[g+\lambda
h]_{ab:cd} = \bar{R}_{ab:cd} + \lambda \dot{R}[h]_{ab:cd}$ (convention
of~[51]). The background \emph{Bianchi} operator is $\bar{B}[r] =
B_3[r]$, with $\bar{B}[\bar{R}] = 0$. Finally, though the name is not
standard, it is meaningful to call $B_4[b]$ a \emph{higher Bianchi}
operator. Thus, it would also make sense to refer to the Calabi complex
as the \emph{Killing-Riemann-Bianchi complex}. This complex also happens
to be locally exact%
\footnote{A differential complex on a manifold $M$ is \emph{locally
exact} if every $x\in M$ has a neighborhood such that the complex
restricted to it becomes exact. For example, this condition is
fulfilled for the de~Rham complex thanks to the Poincar\'e lemma.} %
~[13, 34]. Thus, according to the general machinery of
sheaf theory~[12], the Calabi complex provides a fine
resolution of the \emph{sheaf of Killing vectors} (or \emph{Killing
sheaf}) $\K_g$ on the Lorentzian manifold $(M,g)$. This observation
immediately gains us the following
\begin{prop}[Calabi [13, 34]]
The (unrestricted) cohomology $HC^l(M,g) = \ker B_{l+1}/\im B_l$ of the
Calabi complex is isomorphic to the sheaf cohomology $H^\bullet(M,\K_g)$
of the sheaf of Killing vectors on any spacetime $(M,g)$ of constant
\end{prop}
The homotopy differential operators are given by
\begin{align}
&= \nabla^b h_{ab} - \frac{1}{2} \nabla_a h , \\
&= r_{ac:b}{}^c , \\
\notag
&= \frac{1}{2} \left(\nabla^e b_{e ab:cd} + \nabla^e b_{e cd:ab} \right) \\
& \quad - {} \frac{1}{2} \left(
+ \nabla_a b_{cd e: b}{}^e
- \nabla_b b_{cd e: a}{}^e
+ \nabla_c b_{ab e: d}{}^e
- \nabla_d b_{ab e: c}{}^e \right) , \\
\notag
&= \frac{1}{3} \left(
2 \nabla^f b_{f abc:de}
+ \nabla^f b_{f dea:bc}
+ \nabla^f b_{f deb:ca}
+ \nabla^f b_{f dec:ab} \right) \\
\notag
& \quad {}
+ \frac{1}{6} \left(
2 \nabla_d b_{abc f: e}{}^f - 2 \nabla_e b_{abc f: d}{}^f \right. \\
\notag
& \qquad {}
- \nabla_a b_{deb f: c}{}^f + \nabla_a b_{dec f: b}{}^f \\
\notag
& \qquad {}
- \nabla_b b_{dec f: a}{}^f + \nabla_b b_{dea f: c}{}^f \\
& \qquad \!\! \left. {}
- \nabla_c b_{dea f: b}{}^f + \nabla_c b_{deb f: a}{}^f
\right) .
%, \\
% E_5[b]_{abcd:ef}
% &= \frac{1}{4} \nabla^i \left( 3 b_{i abcd:ef}
% + b_{i efcd:ab} + b_{i ebfd:ac} + b_{i ebcf:ad} \right. \\
% & \qquad \left. {}
% + b_{i aefd:bc} + b_{i aecf:bd} + b_{i abef:cd} \right) \\
% & \quad {}
% - \nabla_{[a} b_{|ef|bc}{}^{i}{}_{:d]i}
% - \frac{1}{2} \nabla_{[e} b_{|abcd|}{}^{i}{}_{:f]i} .
\end{align}
Their desired symmetry properties are demonstrated in~[34].
Again, we find the following relations with classical differential
operators from relativity. The \emph{de~Donder} operator is $D[h] =
E_1[h]$. The trace from the Riemann to the Ricci tensors is given by
$\bar{R}_{ab} = \bar{R}_{ac:b}{}^{c} = E_2[\bar{R}]_{ab}$. The higher
homotopy operators $E_l$ do not seem to be part of the classical
literature. However, they are essentially modified divergence operators
and are thus reminiscent of the de~Rham co-differentials.
Finally, the cochain maps $P_l = E_{l+1}\circ B_{l+1} + B_l\circ E_l$
(with the edge cases $P_0 = E_1\circ B_1$ and $P_n = B_n \circ E_n$) are
given by
\begin{align}
&= \square v_a + k\frac{1}{n} v_a , \\
&= \square h_{ab}
- k \frac{2}{n(n-1)} h_{ab}
+ k \frac{2}{n(n-1)} g_{ab} \tr[h] , \\
&= \square r_{ab:cd}
- k\frac{2}{n} r_{ab:cd}
+ k\frac{2}{n(n-1)} (g\odot \tr[r])_{ab:cd} , \\
&= \square b_{abc:de}
- k\frac{(3n-7)}{n(n-1)} b_{abc:de}
- k\frac{2}{n(n-1)} (g\odot \tr[b])_{abc:de} , \\
&= \square b_{abcd:ef}
- k\frac{2(2n-7)}{n(n-1)} b_{abcd:ef}
+ k\frac{2}{n(n-1)} (g\odot \tr[b])_{abcd:ef} ,
\end{align}
where we have have defined the traces as $\tr[h] = h_e{}^e$,
$\tr[r]_{ab} = r_{a e: b}{}^e$, $\tr[b]_{ab:c} = b_{ab e: c}{}^e$,
$\tr[b]_{abc:d} = b_{abc e: d}{}^e$. The required null-homotopy
identities $P_l = E_{l+1}\circ B_{l+1} + B_l\circ E_l$ (including the
edge cases $P_0 = E_1\circ B_1$ and $P_n = B_n \circ E_n$) are
demonstrated in~[34]. These identities for $P_0[v]$ and
$P_1[v]$ are well known and are tightly linked with the de~Donder gauge
fixing condition in linearized gravity~[51, 18]. The
higher cochain maps and the corresponding identities appear to be new.
Though, the identity for $P_2[r]$ is related to the non-linear wave
equations satisfied by the Riemann and Weyl tensors on any vacuum
background, sometimes known as the \emph{Lichnerowicz
Laplacian}~\cite[Sec.1.3]{lichnerowicz} (see
also~\cite[Sec.7.1]{chr-kl}, \cite[Exr.15.2]{mtw}, \cite[Eq.35]{bcjr}).
\subsubsection{Cohomology with unrestricted and compact supports}
\label{sec:calabi-coh}
Let us denote the cohomology of the Calabi complex by $HC^l_X(M,g)$,
where $X=0,+,-,fc,pc,sc,tc$ or empty, according to the conventions of
Section~\ref{sec:prelim}. As in the case of the de~Rham complex in
Section~\ref{sec:comp}, we will later relate the cohomology with
causally restricted supports to that with unrestricted or compact
supports. It remains still to find a means to calculate these cohomology
groups. We will state some results in that direction below, referring
to~[34] for a fuller discussion.
An important observation is that each of the $P_l$ operators is
wave-like, that is, it has the same principal symbol as the wave
operator $\square_g$ with respect to the background Lorentzian metric
$g$. This observation has a dual role. First, this means that each of
the $P_l$ operators is Green hyperbolic~[3, 2], while being
cochain homotopic to zero, opening the door to using the methods of
Section~\ref{sec:comp} to compute the cohomology with causally
restricted supports.
The second role is more subtle. Note that the principal symbols of the
$B_l$ maps in the Calabi complex are actually $\GL(n)$-equivariant and so
do not actually involve the background metric $g$. On the other hand,
the principal symbols of the cochain maps $P_l$ do depend on $g$. This
dependence comes purely from the cochain homotopy operators $E_l =
E^g_l$ and the identity $P_l = P_l^g = E^g_{l+1}\circ B_{l+1} + B_l\circ
E_l^g$, where we have used the subscript $g$ to indicate that the
background metric was used for covariant differentiation and index
raising. On the other hand, we are completely free to define a different
set of cochain maps $P_l^{g_R} = E^{g_R}_{l+1}\circ B_{l+1} + B_l\circ
E_l^{g_R}$, which now depend on a different metric $g_R$ with Riemannian
signature. It is crucial to note that the principal symbol of
$P^{g_R}_l$ depends only on the principal symbols of the $E_l^{g_R}$ and
$B_l$. So, in fact, it is equal to the principal symbol of $P^g_l$, but
with the Lorentzian metric $g$ replaced by the Riemannian metric $g_R$.
In other words, each of the $P_l^{g_R}$ operators is elliptic, since its
principal symbol coincides with the Laplace operator $\Delta_{g_R}$. Of
course, $P_l^{g_R}$ would differ much more radically from the formulas
we have given for $P_l^g$ in the terms of subleading differential
The existence of such an ``elliptic'' cochain homotopy results in the
\begin{prop}[Calabi [13, 34]]
(a) The cohomology $HC^\bullet(M,g)$ of the Calabi complex
$(\Secs(C_lM),B_l)$ is isomorphic to the cohomology $H^\bullet(M,\K_g)$
of the sheaf $\K_g$ of Killing vectors on $(M,g)$. (b) If $(M,g)$ is a
simply connected, constant curvature Lorentzian manifold, then
$H^\bullet(M,\K_g) \cong H^\bullet(M)\otimes V_g$, where $V_g$ is the
vector space of all Killing vectors and $H^\bullet(M)$ is the de~Rham
cohomology group.
\end{prop}
Killing vectors (or rather covectors in our notation) are solutions
$v\in \Secs(T^*M)$ of the Killing equation $K[v]_{ab} = \nabla_a v_b +
\nabla_b v_a = 0$. On simply connected, constant curvature
$n$-dimensional spacetimes, $\dim V_g = {n+1\choose 2}$. Note also that
the simple connectedness condition implies that $H^1(M) = 0$. The
precise definition of a sheaf and its cohomology is not of particular
importance for the moment. For present purposes, it suffices that the
above result, at the very least, this answers the question of what
$HC^\bullet(M,g)$ is for Minkowski ($\R^n$), de~Sitter ($\R\times
S^{n-1}$) and anti-de~Sitter ($\R^n$) spacetimes. The proof, together
with a partial discussion of the non-simply connected case, can be found
It remains to discuss the cohomologies with compact supports
$HC_0^\bullet(M,g)$. First, we note that the chain complex
$(\Secs(C^*_lM),B^*_l)$ formally adjoint to the Calabi complex has the
interesting property that equation $B^*_n[b] = 0$ is equivalent to the
\emph{(rank-2) Killing-Yano} equation $Y[w]_{abc} = \nabla_a w_{bc} +
\nabla_b w_{ac}$, where a solution with $w_{[ab]} = w_{ab}$ is called a
\emph{(rank-2) Killing-Yano} tensor on $(M,g)$. Since taking formal
adjoints preserves the homotopy identities and ellipticity, appealing to
the same arguments as above we also have
\begin{prop}[[34]]
(a) The homology $HC_l(M,g) = \ker B^*_l/ \im B^*_{l-1}$ of the adjoint
Calabi complex $(\Secs(C^*_lM),B^*_l)$ is isomorphic to the cohomology
$H^\bullet(M,\KY_g)$ of the sheaf $\KY_g$ of Killing-Yano tensors on
$(M,g)$. (b) If $(M,g)$ is a simply connected, constant curvature
Lorentzian manifold, then $H^\bullet(M,\KY_g) \cong H^\bullet(M)\otimes
W_g$, where $W_g$ is the vector space of all Killing-Yano tensors and
$H^\bullet(M)$ is the de~Rham cohomology group.
\end{prop}
On simply connected, constant curvature $n$-dimensional spacetimes,
$\dim W_g = {n+1\choose 3}$~[48]. Furthermore, using some
general results from the theory of elliptic differential complexes (see
Example~5.1.11 of~[50], which relies on the results
of~[47]), we have the following duality isomorphisms.
\begin{prop}
When finite dimensional, the cohomology with compact supports of the
Calabi complex is the linear dual of the homology of the formally
adjoint Calabi complex, $HC_0^l(M,g) = HC_l(M,g)^*$, while the homology
with compact supports of the adjoint Calabi complex is the linear dual
of the cohomology of the Calabi complex, $HC_{l,0}(M,g) = HC^l(M,g)^*$.
In both cases, the duality can be exhibited via the non-degeneracy of
the pairing descended from the natural pairing between the chains and
cochains of corresponding complexes.
\end{prop}
\subsubsection{Cohomology with causally restricted supports}
\label{sec:calabi-caus}
With the above discussion in mind, we can see immediately that we are in
a situation very similar to that of Section~\ref{sec:comp}, with the
de~Rham complex replaced by the Calabi complex and the wave operators
$\square$ replaced by the operators $P_l$, which have wave-like
principal symbols and are Green hyperbolic. So, repeating the arguments
of Section~\ref{sec:comp}, we immediately have the following
\begin{thm}\label{thm:calabi}
Consider a globally hyperbolic, constant curvature Lorentzian manifold
$(M,g)$. Its Calabi cohomology $HC^l_X(M,g)$ with the causally restricted
supports $X=+,-,pc$ or $fc$ is trivial. Moreover, for the cases
$X=sc,tc$, we have the isomorphisms
\begin{align}
HC^l_{sc}(M,g) &{\cong} HC^{l+1}_0(M,g), &
HC^l_{P,sc}(M,g) &{\cong} HC^l_0(M,g) {\oplus} HC^{l+1}_0(M,g), \\
HC^l_{tc}(M,g) &{\cong} HC^{l-1}(M,g), &
HC^l_{P}(M,g) &{\cong} HC^l(M,g) {\oplus} HC^{l-1}(M,g),
\end{align}
with the convention that all cohomologies vanish in degree $l$ for $l<0$
or $l>n$.
\end{thm}
The Calabi cohomology with spacelike compact support in degree $l=1$ is
important in understanding the symplectic and Poisson structure of the
classical field theory (and of course the quantization) of linearized
gravitons on a background of constant curvature. This was pointed out
explicitly in~\cite[Sec.4.4]{kh-peierls} as a special case of a more
general phenomenon (also discussed in~[35]).
\begin{rem}
Using the above theorem and the results of Section~\ref{sec:calabi-coh},
we can assert that for $n$-dimensional Minkowski space $HC^l_sc$
vanishes in all degrees except $l=n-1$, while $HC^l_{P,sc}$ vanishes in
all degrees except $l=n,n-1$. For $n$-dimensional de~Siter space
$HC^l_sc$ vanishes in all degrees except $l=n-1$, while $HC^l_{P,sc}$
vanish in all degrees except $l=0,n-1,n$.
\end{rem}
\subsection{Other differential complexes}\label{sec:other}
Our interest in computing the de~Rham and Calabi cohomologies with
causally restricted supports has was motivated by their importance in
understanding the geometric structure of classical and quantum field
theories~[15, 46, 6, 19, 5, 31, 35, 36]. Namely, for a
general class of linear field theories, one can formulate sufficient
conditions for the non-degeneracy of the theory's Poisson structure and
the completeness of compactly supported smeared fields as physical
observables in terms of the cohomologies of corresponding differential
complexes. Non-linear field theories can be studied in terms of their
linearizations about arbitrary background solutions. To Maxwell
electrodynamics corresponds the de~Rham
complex~\cite[Sec.4.2]{kh-peierls}. To linearized gravity on constant
curvature backgrounds, corresponds the Calabi
complex~\cite[Sec.4.4]{kh-peierls}. Similarly, to Yang-Mills linearized
about a flat connection corresponds a twisted de~Rham complex.
Each of these examples can be treated using the methods presented in
this paper. Few other explicit examples of differential complexes
corresponding to other field theories of physical interest seem to be
known. In particular, they do not seem to be known for linearized
gravity on non-constant curvature backgrounds and, perhaps, not even for
Yang-Mills linearized about non-flat connections. On the other hand,
there are strong abstract reasons to believe that such differential
complexes do indeed exist~[44, 28, 43].
If such a differential complex also shares the apparently crucial
property of admitting cochain homotopies that generate hyperbolic and
elliptic cochain maps (cf.\ the $E_l^g$, $P_l^g$, $E_l^{g_R}$ and
$P_l^{g_R}$ maps of Sections~\ref{sec:calabi-ops}
and~\ref{sec:calabi-coh}), then its causally restricted cohomologies can
be related to those with unrestricted and compactly supported ones, as
in Theorems~\ref{thm:sc-tc} and~\ref{thm:calabi}.
If, in addition, such a differential complex could also be seen as
resolving a locally constant sheaf, its unrestricted cohomologies could
be computed by algebraic means, without actually solving complicated
systems of differential equations, as in Section~\ref{sec:calabi-coh}.
The latter requirement is closely related to the initial differential
operator in the complex having only a finite dimensional space of
solutions (being of \emph{finite type}), as is the case for the
\emph{locally constant} (de~Rham) and \emph{Killing} (Calabi)
The compactly supported cohomologies could also be obtained if the
corresponding formally adjoint complex satisfied similar requirements,
as illustrated in Section~\ref{sec:calabi-coh} by the appearance of the
locally constant sheaf of Killing-Yano tensors.
\section{Discussion}\label{sec:discuss}
We have shown how to compute the de~Rham cohomology with causally
restricted supports (retarded, advanced, past compact, future compact,
spacelike compact and timelike compact) on a globally hyperbolic
Lorentzian spacetime. The result (Theorems~\ref{thm:pf}, \ref{thm:sc-tc}
and Corollary~\ref{cor:pairing}) expresses these causally restricted
cohomologies in terms of the standard de~Rham cohomologies of the
spacetime manifold, with either unrestricted or compact supports. These
results, confirm and generalize the independent similar results of the
recent work~[5], with the generalizations described below.
Further, we showed how the special geometric features of the de~Rham
complex, which we used in the cohomology calculation, can be interpreted
in terms of homological algebra and applied to other complexes of
differential operators that could arise in the investigation of the
geometry of classical and quantum field theories. These applications are
illustrated on the specific example of the Calabi (or
Killing-Riemann-Bianchi) complex, which plays for linearized gravity on
constant curvature backgrounds a role analogous to the de~Rham complex
for Maxwell-like field theories. We have also made comments about the
covariance of causally restricted cohomologies under specific types of
morphisms between spacetimes, adapted to their causal structure, and
under changes of the causal structure itself.
A fuller discussion of the Calabi complex, including its relevant
geometric properties that are difficult to locate in or are absent from
the current literature, is deferred to future work~[34]. In
the future, it will also be interesting to find the analogs of the
Calabi complex on more general Lorentzian backgrounds, which would
consist of differential complexes resolving the locally constant sheaf
of Killing vectors on a given background. It is also a separate
challenge to compute independently and effectively compute the
cohomology of the Killing sheaf based on the topological and geometric
information on a given background.
\section*{Acknowledgments}
The author would like to thank Marco Benini for useful discussions and
also Romeo Brunetti, Claudio Dappiaggi, and Valter Moretti for their
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\end{thebibliography}\endgroup
\end{document}
|
arxiv-papers
| 2014-04-07T20:01:58 |
2024-09-04T02:50:00.831541
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Igor Khavkine",
"submitter": "Igor Khavkine",
"url": "https://arxiv.org/abs/1404.1932"
}
|
1404.2028
|
# Large phase shift of (1+1)-dimensional nonlocal spatial solitons in lead
glass
Qian Shou Miao Wu Qi Guo Corresponding author: [email protected] Laboratory
of Nanophotonic Functional Materials and Devices, South China Normal
University, Guangzhou 510006
###### Abstract
The large phase shift of strongly nonlocal spatial optical soliton(SNSOS) in
the (1+1)-dimensional [(1+1)D] lead glass is investigated using the
perturbation method. The fundamental soliton solution of the nonlocal
nonlinear Schödinger equation(NNLSE) under the second approximation in
strongly nonlocal case is obtained. It is found that the phase shift rate
along the propagation direction of such soliton is proportional to the degree
of nonlocality, which indicates that one can realize $\pi$-phase-shift within
one Rayleigh distance in (1+1)D lead glass. A full comprehension of the
nonlocality-enhancement to the phase shift rate of SNSOS is reached via
quantitative comparisons of phase shift rates in different nonlocal systems.
###### pacs:
(190.6135) Spatial solitons; (190.5940) Self-action effects; (190.4870)
Photothermal effects.
## I Introduction
Nonlocal spatial solitons have been the subject of intensive experimental and
theoretical work Assanto-PRL-2004 ; Rotschild-PRL-2005 ; Bang-PRE-2002 ;
Peccianti-OL-2002 since the pioneering work done by Snyder and Mitchell SM-
Science-97 . The most prominent innovation in their work is that they
transforms the complex nonlocal nonlinear Schödinger equation (NNLSE) into a
simple case of linear propagation of light in a quadratic self-induced index
well SM-Science-97 . Nonlocal nonlinearity is typically the result of certain
transport processes, such as the charge drift in photorefractive crystals
Galvo-EL-2002 and the heat transfer in thermal nonlinear media Rotschild-
PRL-2005 or, long-range interaction, such as the molecular reorientations in
liquid crystals Peccianti-PR-2012 . Due to the nature of the nonlocality,
solitons in nonlocal nonlinear media exhibit several distinct properties that
are not possible in local settings. This includes, on one hand, resulting from
the spatial ‘averaging’ character of the nonlocality, the arrest of
catastrophic collapse Bang-PRE-2002 , the ability to support the formation of
complex optical spatial solitons, such as higher-order solitons Rotschild-
OL-2006 ; Ye-PRA-2008 and vortex solitons Rotschild-PRL-2005 ; Shen-JO-2012 .
On the other hand, out-of-phase solitons attraction Rasmussen-PRE-2005 ; Hu-
APL-2006 , long-range interactions between solitons Rotschild-NP-2006 as well
as the solitons and the boundaries Alfassi-OL-2007 ; Shou-OL-2009 ;
Buccoliero-JOA-2009 in strongly nonlocal media have also been carried out or
predicted due to the fact that the interactions are mediated by the light-
induced refractive index which is ‘enlarged’ by the nonlocal response.
Except for the ‘averaging’ and the ‘enlarging’ features of the nonlocality,
there exists an ‘enhancing’ effect of the nonlocality on the phase shift of
the SNSOS. Although very large in fact, the phase shift of SNSOS is considered
a trivial term, for a long time, and is neglected by the Snyder-Mitchell (SM)
model SM-Science-97 . The first work focused on the phase shift of SNSOS was
done by Guo $et$ $al$. Guo-PRE-2004 ; Xie-oqe-2004 . They predicted a large
phase shift rate of SNSOS, which is $\alpha^{2}$ times ($\alpha$ is the degree
of nonlocality defined as $\alpha=w_{m}/\mu$ where $w_{m}$ is the
characteristic length of the response function and $\mu$ is the beam width),
explicitly 100 times for the lower limit of the strongly nonlocality, larger
than that of the local counterpart. Guo’s conclusion results from a strongly
nonlocal (SN) model in which the large phase shift is included having a
dominating term proportional to the soliton critical power.
SN model can rigorously transform to SM model with a function transformation
involving large phase shift term Guo-2013 . Both of them are derived from a
phenomenological and regular (or at least twice-differentialble at
$\textbf{r}=0$) response function $R(\textbf{r})$. In the nematic liquid
crystal (NLC) and lead glass(LG), the two media found so far in which SNSOSs
can form, the response functions are singular at every source point
(irregular) and therefore one can not obtain accurate solution of NNLSE based
on SN model and SM model even in the strongly nonlocal caseGuo-2013 . Ouyang
$et$ $al$. took the higher order (the forth and the sixth) terms of the light
induced refractive index as the perturbation to the quadratic index well and
obtained considerably accurate analytical soliton solutions in (1+1)D Ouyang-
PRE-2006 and (1+2)D Ouyang-PRA-2007 NLC. The perturbation solution are
different from Gaussian-type solution given by SN modelGuo-2013 , but still
indicated nonlocality-enhanced large phase shifts of SNSOSs. The first
theoretical and experimental study focused on the SNSOS phase shift was
carried out in (1+2)D cylindrical LG by Shou $et$ $al$. Shou-OL-2011 . They
retained the terms of the Taylor expansion of the light-induced refractive
index up to the second order whose coefficient is the on-axis light intensity.
The phase shift rate in (1+2)D LG was predicted to be much smaller than the
result based on SN model, but is still more than one order larger than that in
the local media. More meaningful, Shou $et$ $al$. observed a linear modulation
of the soliton power on the phase shift of the SNSOS Shou-OL-2011 , which
coincides with Guo’s prediction, indicating that the nonlocality enhancement
to the phase shift of SNSOS stems from the fact that the light-induced
refractive index, which directly contributes to the phase shift, is induced
not by the light intensity but by the power of the whole beam.
In this paper, we investigate the phase shift of SNSOS in (1+1)D LG in the
formalism of perturbation theory. The perturbation solution of the fundamental
soliton is obtained under the second approximation. The result indicates that
the phase shift of SNSOS in (1+1)D LG is proportional to the degree of
nonlocality which is at least one order larger than the result for the local
solitons. It will also be shown how the degree of nonlocality affects, or
explicitly speaking, enhances the phase shift rate in different nonlocal
systems.
## II The fundamental strongly nonlocal soliton solution under the second
approximation
We consider a (1+1)D LG with thermal nonlocal nonlinear response occupying the
region $-L\leq x\leq L$. The propagation behavior of a light beam u
propagating along the $z$ axis is governed by the NNLSE, coupled to the
Poisson equation describing the light-induced nonlinear refractive index
variation _N_ ,
$i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial^{2}u}{\partial
x^{2}}+Nu=0,$ (1) $\frac{d^{2}N}{dx^{2}}=-|u|^{2}.$ (2)
The nonlocal response function in (1+1)D LG under the first-kind boundary
condition $N(\pm L)=0$ can be given as Arfken-1985
$G(x,\xi)=\left\\{\begin{array}[]{ll}\frac{(x+L)(\xi-L)}{2L},\qquad(x\leq\xi)\\\
\\\ \frac{(\xi+L)(x-L)}{2L}.\qquad(\xi\leq x)\end{array}\right.$ (3)
According to the Green function method, the nonlinear refractive index in LG
can be written in the form of
$N(x)=-\int_{-L}^{L}G(x,\xi)|u(\xi,z)|^{2}d\xi.$ (4)
It is obvious that the response function in Eq.(3) is not differentiable at
the source point $x=\xi$ (irregular) and therefore cannot be dealt with SN
model Guo-PRE-2004 . We use the perturbation method, previously extended to
solve the NNLSE by Ouyang $et$ $al$. Ouyang-PRE-2006 ; Ouyang-PRA-2007 , to
calculate the fundamental soliton solution of the NNLSE. For the soliton state
$u(x,z)$, we have $|u(-x,z)|^{2}=|u(x,z)|^{2}$ and $u(x,z)=u(x,0)$. On the
analogy of the potential in quantum mechanics which determines the state of
the particle movement, we define the nonlinearity-induced trapping
‘potential’, explicitly the light-induced refractive index, which can
determine the beam propagation behavior,
$V(x)=\int_{-L}^{L}G(x,\xi)|u(\xi,z)|^{2}d\xi.$ (5)
Then Eq.(1) can be reduced to
$i\frac{\partial u}{\partial z}+\frac{1}{2}\frac{\partial^{2}u}{\partial
x^{2}}-V(x)u=0.$ (6)
Taking the Taylor’s expansion of $V(x)$ at $x=0$, we obtain
$V(x)=V_{0}+\frac{1}{2\mu^{4}}x^{2}+\alpha x^{4}+\beta x^{6}+\cdots,$ (7)
where
$V_{0}=V(0),$ (8a) $\frac{1}{\mu^{4}}=V^{(2)}(0),$ (8b)
$\alpha=\frac{1}{4!}V^{(4)}(0),$ (8c) $\beta=\frac{1}{6!}V^{(6)}(0).$ (8d)
In the strongly nonlocal case, $V(x)$ is effective mainly within the beam
region. Consequently the terms $\alpha x^{4}$ and $\beta x^{6}$ are,
respectively, one and two orders of magnitude smaller than the term
$x^{2}/(2\mu^{4})$ Ouyang-PRE-2006 and then can be viewed as the
perturbations. By substituting Eq.(7) into Eq.(6) and neglecting the higher-
order terms, we obtain
$i\frac{\partial u}{\partial z}=\left[-\frac{1}{2}\frac{\partial^{2}}{\partial
x^{2}}+V_{0}+\frac{1}{2\mu^{4}}x^{2}+\alpha x^{4}+\beta x^{6}\right]u.$ (9)
Taking a transformation
$u(x,z)=\phi(x)\exp[-i(\varepsilon+V_{0})z],$ (10)
we arrive at
$\left[-\frac{1}{2}\frac{d^{2}}{dx^{2}}+\frac{1}{2\mu^{4}}x^{2}+\alpha
x^{4}+\beta x^{6}\right]\phi=\varepsilon\phi.$ (11)
If $\alpha=0$ and $\beta=0$, Eq.(11) reduces to the well-known stationary
Schrödinger equation for a harmonic oscillator. Following the perturbation
method we obtain the fundamental soliton solution under the second
approximation
$\displaystyle\phi_{0}(A,\alpha,\beta,x)\approx
A\left(\frac{1}{\pi\mu^{2}}\right)^{1/4}\exp\left(-\frac{x^{2}}{2\mu^{2}}\right)$
$\displaystyle\times\bigg{[}1+\alpha\left(\frac{9\mu^{6}}{16}-\frac{3\mu^{4}}{4}x^{2}-\frac{\mu^{2}}{4}x^{4}\right)$
$\displaystyle+\alpha^{2}\left(-\frac{1247\mu^{12}}{512}+\frac{141\mu^{10}}{64}x^{2}+\frac{53\mu^{8}}{64}x^{4}+\frac{13\mu^{6}}{48}x^{6}+\frac{\mu^{4}}{32}x^{8}\right)$
$\displaystyle+\beta\left(\frac{55\mu^{8}}{32}-\frac{15\mu^{6}}{8}x^{2}-\frac{5\mu^{4}}{8}x^{4}-\frac{\mu^{2}}{6}x^{6}\right)\bigg{]},$
(12)
and
$\varepsilon_{0}\approx\frac{1}{2\mu^{2}}+\frac{3\mu^{4}\alpha}{4}-\frac{21\mu^{10}\alpha^{2}}{8}+\frac{15\mu^{6}\beta}{8}.$
(13)
In the strongly nonlocal case, $\alpha$ and $\beta$ are very small, and
accordingly, so is the difference between the fundamental soliton solution
under the second approximation $\phi_{0}(A,\alpha,\beta,x)$ and that under the
zeroth approximation $\phi_{0}(A,0,0,x)$. $V(x)$ can be approximately given by
$\displaystyle V(x)$ $\displaystyle\approx$
$\displaystyle\int_{-L}^{L}G(x,\xi)\phi_{0}^{2}(A,0,0,\xi)d\xi$ (14)
$\displaystyle=$
$\displaystyle\frac{A^{2}}{2}\Bigg{\\{}\frac{\mu}{\pi}\left[\exp\left(-\frac{x^{2}}{\mu^{2}}\right)-\exp\left(-\frac{L^{2}}{\mu^{2}}\right)\right]$
$\displaystyle-L\textup{erf}\left(\frac{L}{\mu}\right)+x\textup{erf}\left(\frac{x}{\mu}\right)\Bigg{\\}},$
where
$\textup{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-\xi^{2}}d\xi.$ (15)
Combining Eq.(8), we have
$A^{2}\approx\frac{\sqrt{\pi}}{\mu^{3}},$ (16a)
$V_{0}\approx\frac{\sqrt{\pi}}{2\mu^{3}}\Bigg{\\{}\frac{\mu}{\sqrt{\pi}}\bigg{[}1-\exp\left(-\frac{L^{2}}{\mu^{2}}\right)\bigg{]}-L\textup{erf}\left(\frac{L}{\mu}\right)\Bigg{\\}},$
(16b) $\alpha\approx-\frac{1}{12\mu^{6}},$ (16c)
$\beta\approx\frac{1}{60\mu^{8}}.$ (16d)
Inserting Eq.(II) into Eq.(10), we find the fundamental soliton solution in
(1+1)D LG,
$\displaystyle u(x,z)\approx
A\left(\frac{1}{\pi\mu^{2}}\right)^{1/4}\exp\left(-\frac{x^{2}}{2\mu^{2}}\right)\exp(i\gamma
z)$
$\displaystyle\times\bigg{[}0.9649+a\frac{x^{2}}{\mu^{2}}+b\frac{x^{4}}{\mu^{4}}+c\frac{x^{6}}{\mu^{6}}+0.0002\frac{x^{8}}{\mu^{8}}\bigg{]},$
(17)
where
$A^{2}\approx\frac{\sqrt{\pi}}{\mu^{3}},a=0.0386,b=0.0162,c=-0.0009,d=0.0002$,
and the phase shift rate is of the form
$\gamma=-V_{0}-\varepsilon_{0}\approx\frac{1}{2\mu^{2}}\Bigg{[}\frac{\sqrt{\pi}L}{\mu}\textup{erf}\left(\frac{L}{\mu}\right)+\exp\left(-\frac{L^{2}}{\mu^{2}}\right)-1.87\Bigg{]}.$
(18)
It is important to notice that in Eq.(II), $\mu$, defined in Eq. (8b), is
visualized as the beam width. The power of the soliton is approximatively
given by
$P=\int^{+\infty}_{-\infty}|u(x,z)|^{2}dx\approx
A^{2}\approx\frac{\sqrt{\pi}}{\mu^{3}}.$ (19)
In the above equations, $L$ plays the part of the characteristic length
$w_{m}$ of the response function, since $w_{m}$ of the LG modeled by Eq.(2) is
intrinsically infinite but cut off by its boundary Shou-OL-2009 . Therefore
the ratio $L/\mu$ represents the degree of nonlocality $\alpha$. In the
strongly nonlocal limit, $\alpha\gg 1$, we have
$\gamma\approx\frac{\alpha\sqrt{\pi}}{2\mu^{2}}$ (20)
It can be seen that the phase shift rate of SNSOS is proportional to the
degree of nonlocality, which is at least one order larger than that for local
solitons.
Figure 1: (a) The intensity profile of the (1+1)D soliton in LG for
$\alpha=150$. The squares represent the iterative solution and the solid line
represents the perturbative solution expressed in Eq.(II). (b) Simulation of
the soliton propagation where the perturbative solution (solid line in (a))
serves as the incident profile.
Figure 2: Comparison of the phase shifts of SNSOS in (1+1)D LG with different
degree of nonlocality $\alpha$ between the perturbative analytic results
(solid lines) with the numerical results (squares). Inset shows the phase
shift rate as a function of the degree of nonlocality $\alpha$. Circles and
squares are respectively calculated based on the analytical and numerical
results. Solid line is provided as a guide to the eye.
Fig. 1(a) displays the intensity profile of the (1+1)D SNSOS in LG with
$\alpha=150$. In the strongly nonlocal case, the perturbative analytic result
(solid line) is very close to the numerical result (squares) denoted by the
good agreement between them. Fig. 1(b) shows the simulation of the light beam
propagation in the form of soliton with an input amplitude profile described
by Eq.(II). The phase shift of the (1+1)D SNSOS in LG versus the propagation
distance is manifested in Fig. 2. Under different conditions of nonlocalities,
the higher the degree of nonlocality, the faster the phase shift gets. The
phase shift rates of SNSOSs are obtained by calculating the slopes of the data
in Fig.2. It suggests that SNSOS experiences $\pi$ phase shift within one
Rayleigh distance in (1+1)D LG. The inset of Fig. 2 reveals the phase shift
rate changes along with the degree of the nonlocality. It visualizes that the
enhancement-effect of the nonlocality on the phase shift rate acts linearly
and effectively. One can, therefore, obtain faster phase shift by directly
enlarging the size of the LG since the degree of nonlocality of LG is
determined by the glass size.
## III Discussion
In the previous section, we investigate the phase shift of SNSOS in (1+1)D LG
and find that the phase shift rate is proportional to the degree of
nonlocality. A quantitative comparisons of phase shift rates of spatial
solitons in (1+1)D and (1+2)D materials with different response functions are
represented in Table 1 Guo-PRE-2004 ; Ouyang-PRE-2006 ; Ouyang-PRA-2007 ;
Shou-OL-2011 ; Aitchison-OL-1990 . Fig. 3 gives an illustration of the phase
shift rates of solitons versus the degree of nonlocality $\alpha$ in different
local and nonlocal systems. There are several features that should be
emphasized. First of all, solitons propagating in material with nonlocal
nonlinear response have much faster phase shift rate. We call this
‘nonlocality-enhanced phase shift’. In nonlocal media with Gaussian response
functions, nonlocality-enhancing factor is $\alpha^{2}$, which is more than
100, in the strongly nonlocal cases Guo-PRE-2004 . In LG, the nonlocality-
enhancing factors are much smaller but still over 10 for the lower limit of
the strong nonlocality Shou-OL-2011 . Second, the nonlocality-enhancing
factors present different forms in (1+1)D and (1+2)D systems. Generally
speaking, compared with higher dimensional SNSOSs, lower dimensional SNSOSs
have much faster phase shift rates. Specifically, in (1+1)D NLC and LG, phase
shift rates have the same expressions which are proportional to the degree of
nonlocality $\alpha$ Ouyang-PRE-2006 . This indicates the phase shift rates of
SNSOS are more than one order of magnitude faster than those for the local
ones. While in the (1+2)D NLC and LG, the phase shift rate takes the form of
natural logarithm of $\alpha$ Shou-OL-2011 ; Ouyang-PRA-2007 . Smaller
although than that in the lower-dimensional nonlocal media, nonlocality-
enhancing factor is still one order larger than the result for local solitons
in LG, and more that 5 times larger than the result for local solitons in NLC.
The nonlocality-enhancement-effect on the phase shift of SNSOS originates from
the fact that, the refractive index, directly contributing to the phase shift,
is induced not by the light intensity but, thanks to the ‘averaging-effect’ of
the nonlocality, by the power of the whole beam.
3.5in
Table 1: Phase shift rate of spatial solitons in (1+1)D and (1+2)D materials
with different response functions
Dimension | Nonlocality | Material | Phase shift rate
---|---|---|---
(1+1)D | Local | Local media | $1/L_{R}$ Aitchison-OL-1990
Nonlocal | Media with Gaussian response | $\alpha^{2}/L_{R}$ Guo-PRE-2004
NLC | $\sqrt{\pi}\alpha/L_{R}$ Ouyang-PRE-2006
LG | $\sqrt{\pi}\alpha/L_{R}$
(1+2)D | Local | Local media | unstable
Nonlocal | Media with Gaussian response | $\alpha^{2}/L_{R}$ Guo-PRE-2004
NLC | $(8.6{\rm ln}\alpha-4.3)/\pi L_{R}$ Ouyang-PRA-2007
LG | $(2{\rm ln}\alpha+6.24)/L_{R}$ Shou-OL-2011
* 1
$L_{R}$ is the Rayleigh distance.
* 2
$\alpha$ is the degree of nonlocality defined as the ratio of the
characteristic length of the response function to the beam width. In LG, the
characteristic length of the response function is the medium size Shou-OL-2011
.
Figure 3: Comparison of the phase shift rates of solitons versus the degree of
nonlocality $\alpha$ in different local and nonlocal systems. The solid curves
are the phase shift rates, normalized by Rayleigh distance $L_{R}$ in, from
top to bottom, (1+1)D or (1+2)D media with Gaussian response, (1+1)D NLC or
LG, (1+2)D LG, (1+2)D NLC, (1+1)D local media, respectively.
## IV Conclusion
Using the perturbation method, we investigate the large phase shift of SNSOS
in (1+1)D LG. The perturbative solution of the fundamental soliton under the
second approximation suggests that the phase shift rate of (1+1)D SNSOS is
proportional to the degree of the nonlocality, which is at least one order
faster than that of its local counterpart. This facilitates a $\pi$-phase-
shift within one Rayleigh distance in (1+1)D LG. The nonlocality-enhancement-
effect on the phase shift of SNSOS is an important and intrinsic feature of
nonlocality, which, although works differently in different nonlocal systems,
leads to a much faster, generally one order faster, phase shift rate of SNSOS
than that of the local counterpart. Phase shift is very important for
modification, manipulation, and control of optical field based on the
principle of interference. The nonlocality-enhancement to the phase shift
might be of great potential in applications based on the effective generation
of large phase shift.
## Acknowledgment
This research was supported by the National Natural Science Foundation of
China (Grant No. 11274125), and the Natural Science Foundation of Guangdong
Province of China (Grant No. S2012010009178).
## References
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* (3) O. Bang, W. krolikowski, J. Wyller and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619-5 (2002).
* (4) M. Peccianti, K. A. Brzdakiewicz and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460-1462 (2002).
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* (7) M. Peccianti and G. Assanto, “Nematicons,” Phys. Rep. 516, 147-208 (2012).
* (8) C. Rotschild, M. Segev, Z. Xu, Y. V. Kartashov, L. Torner and O. Cohen, “Two-dimensional multipole solitons in nonlocal nonlinear media,” Opt. Lett. 31, 3312-3314 (2006).
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* (11) P. D. Rasmussen, O. Bang and W. Królikowski, “Theory of nonlocal soliton interaction in nematic liquid crystals,” Phys. Rev. E 72, 066611-7 (2005).
* (12) W. Hu, T. Zhang and Q. Guo, “Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals,” Appl. Phys. Lett. 89, 071111-3 (2006).
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* (17) Q. Guo, B. Luo, F. Yi, S. Chi and Y. Xie, “Large phase shift of nonlocal optical spatial solitons,” Phys. Rev. E 69, 016602-8 (2004).
* (18) Y. Xie and Q. Guo, “Phase modulations due to collisions of beam pairs in nonlocal nonlinear media,” Opt. Quant. Electron. 36, 1335-1351 (2004).
* (19) Q. Guo, W. Hu, D. Deng, D. Lu and S. Ouyang, Features of strongly nonlocal spatial solitons, in Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals edited by G. Assanto (John Wiley & Sons, New Jersey, 2012), Chap. 2, 37-69.
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* (22) Q. Shou, X. Zhang, W. Hu and Q. Guo, “Large phase shift of spatial solitons in lead glass,” Opt. Lett. 36, 4194-4196 (2011).
* (23) G. Arfken, Mathematical methods for physics (Academic Press, 1985), Chap. 10.5, 662.
* (24) J. S. Aitchison, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, P. W. E. Smith, “Observation of spatial optical solitons in a nonlinear glass waveguide,” Opt. Lett. 15, 471-473 (1990).
|
arxiv-papers
| 2014-04-08T07:29:12 |
2024-09-04T02:50:00.844345
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Qian Shou, Miao Wu and Qi Guo",
"submitter": "Qi Guo",
"url": "https://arxiv.org/abs/1404.2028"
}
|
1404.2162
|
11institutetext: University of Vienna, Faculty of Computer Science, Austria
CERN
11email: {georg.kaes, juergen.mangler, stefanie.rinderle-ma}@univie.ac.at
11email: [email protected]
## 1 Introduction
This technical report describes a formalization for nursing knowledge found in
the NANDA, NIC and NOC (NNN for short) standards.
Data about nursing diagnoses and treatments consists of the following three
NNN knowledge sources that are widely accepted and used in the field of care
[1], i.e., NANDA111North American Nursing Diagnosis Association, www.nanda.org
containing all basic information about the diagnoses, the Nursing
Interventions Classification (NIC) containing all treatments which can be
executed for improving the patients condition, and the Nursing Outcomes
Classification (NOC) containing all nursing outcomes which can be reached
after a therapy.
Based on the NNN knowledge, our overall goal is to enable the guideline-driven
process development and adaptation in the care domain. Within the
ACaPlan222Adaptive Care Planning:
http://cs.univie.ac.at/research/projects/projekt/infproj/1033/ project, we
work closely together with experts from the care domain. This paper
contributes a first step towards this goal by providing a formalization method
for the NNN knowledge sources in such a way that this information can be
directly utilized for creation and adaptation of individual patient treatment
processes.
The central aspects of designing a corresponding formalization are as follows:
Methodologically, we first analyze which information of the NNN knowledge
sources has to be included in the formalization based on studying NNN
documentation and discussions with experts from the care domain resulting in
the NNN taxonomy (contribution 1). Then we will evaluate existing standards,
primarily from the medical domain, i.e., GLIF [2], Asbru [3], and ARDEN Syntax
[4], with respect to their support for the NNN taxonomy, illustrated by the
use case FATIGUE. The resulting evaluation report and open issues
(contribution 2) will serve as input for the design of the NNN formalization
(contribution 3). The NNN formalization will be evaluated based on use case
FATIGUE, results from discussing with domain experts, and possible application
in other domains.
In detail, NNN knowledge sources are introduced and analyzed in Section 2.
Existing standards from the medical domain are evaluated in Section 3. Section
4 presents the NNN formalization. Section 5 focuses on lessons learned and a
discussion about further advantages which arise from our contribution.
## 2 NANDA, NIC, and NOC Knowledge Sources
The goal of NANDA, NIC and NOC is the development and unification of nursing
diagnoses that are the basis for all processes related to care, as they
contribute a consistent terminology and ease the phrasing and documentation
[5]. NANDA contains 206 nursing diagnoses [6], which are available for care
attendants in various text documents. Based on the nursing diagnoses defined
in NNN, nurses can determine a patients state and in further steps are able to
create therapies and define intended outcomes, which are used for treating the
observed symptoms.
For getting an overview over the different sections and contents of a
diagnosis we studied a variety of these documents ([7, 5, 8, 9, 10]) and
conferred with domain experts who showed us the most relevant parts of the
given taxonomy.
As a fist step, we analyzed and aggregated the necessary building blocks of
NNN, which are depicted in Figure 1. In the following, these building blocks
are shortly described and illustrated by the use case FATIGUE.
Figure 1: Building Blocks for NNN Formalization
* •
, : provide identification of the diagnosis, e.g., FATIGUE, and describes the
most important characteristics of the given diagnosis in one or more natural
language sentences respectively. For FATIGUE, the descripion reads as follows:
“An overwhelming sustained sense of exhaustion and decreased capacity for
physical and mental work at usual level” [9].
* •
: are the symptoms which imply the fact that there is a hardship in the
patients circumstances. Those characteristics, which can be either
subjectively (“I feel a bit dizzy today.”) or objectively (“Your nose is
bleeding.”) determined, are further informational assets for the care
attendant to review or approve the diagnosis [7].
* •
: are possible causes, which may lead to the given nursing diagnosis [7]. It
is a very important fact that care attendants should not concentrate on
treating the symptoms a patient shows, but the causes which are the reasons
for the current circumstances. For FATIGUE, there are several potential risk
factors defined, for example psychological causes such as stress, fear, or
depression [7].
* •
: describe risk factors, defining characteristics for possible problems which
may lead to future hardships for the patient. Diagnoses which contain these
kind of factors only concentrate on risks, and therefore do not implement any
defining characteristics (as there are none yet).
* •
: contains all sources which led to the development of the given taxonomy.
They have to be linked to the appropriate part of the diagnosis.
After defining all information sources related to the current state of the
patient, the second class of information refers to the possible treatments,
their outcome, and their documentation:
* •
: are the concrete tasks which need to be executed for reaching the desired
nursing goals given the current circumstances. They have to be defined in a
standardized terminology like the Nursing Interventions Classification, or
supplemented with resources.
* •
: define the desired outcomes after the execution of the therapy. As with
Nursing Interventions, they either have to be defined within a standardized
terminology like the Nursing Outcomes Classification, or supplemented with
sources.
* •
: are not necessarily part of the NIC taxonomy itself, but nonetheless an
important part of a complete definition of a given diagnosis. They are listed
explicitly in specialized literature such as [7]. The emphases of nursing
documentation define the most important facts for the documentation of the
executed tasks. Based on the necessity of a situational and regular evaluation
of the performed actions, the possibility to totally reconstruct the methods
of the acting care attendants is very important. For the diagnosis of FATIGUE
there exist several categories, which contain the necessary documentational
elements.
* •
and : list the relevant categories from the Nursing Interventions
classification respectively the Nursing Outcomes Classification.
## 3 Evaluation of Existing Standards: GLIF, Asbru, ARDEN Syntax
Similarly to NNN knowledge supporting the work of care personnel, Clinical
Practice Guidelines (CPGs) define how medical staff has to act in certain
situations and are an essential part of modern medicine. During the last
decades several approaches have been developed aiming at formalizing this
medical knowledge in a manner which is also accessible to computers. The goal
of these Computer-Interpretable Guidelines (CIGs) [11] which are a formal
representation of the CPGs, is helping the doctors with their decisions in the
best way possible.
In the following we evaluate three well known approaches - namely the Arden
Syntax, Asbru, and GLIF as representatives for CIG standards. We chose three
well known approaches which we have found in specialized literature [12] [13]
very often. Of course, there do exist more approaches, like the Evicare
project, which focuses on “providing evidence-based medicine at the point of
care” [14], thus increasing quality of patient care. Their guideline
formalizations are based on the DeGeL framework, whose model supports elements
common to clinical guidelines. [15] shows how guidelines defined in Asbru and
GEM can be implemented in DeGeL.
In this section, we describe the design of the Arden Syntax, Asbru and GLIF
and try a formalization of NNN in the particular approach based on the example
of FATIGUE. The result will be a discussion of limitations of existing CIG
standards with respect to formalizing care-related diagnoses.
### 3.1 The Arden Syntax
The Arden Syntax is a guideline formalism implementing a language close to
Pascal with the goal of formalizing medical knowledge. The basic elements are
called Medical Logic Moduls (MLMS), which can be reused in several
applications [16, 17]. Representing knowledge in distinct, separate modules
facilitates the implementation of contents relevant to the respective
institution into their own Electronic Patient Record-System.
Design Principle: Medical Logic Modules are the basic elements of the Arden
Syntax. They contain the structure how knowledge is represented and give the
medical professionals - if they are implemented in a clinical information
system - informations about a patient’s condition by using alerts [4].
MLMs consist of the three section Maintenance, Library, and Knowledge. The
latter contains the actual medical knowledge encoded within different slots,
i.e., key-value pairs where the value can be either text, a coded value, or
structured data. The following MLM excerpt for section Knowledge implementing
NANDA diagnosis FATIGUE demonstrates its basic structure (note that some
simplifications were made).
Listing 1: Component: Knowledge
⬇
type: data-driven;;
data:
listless := READ {select listlessness from results where it occurred within
the past 1 week};
features := READ {select feature from results where TirednessIndicator =
true};
priority: 42;
evoke: ANY OF (listless, features)
logic:
IF features=’very tired’ OR listless > 84 conclude
action:
WRITE ’Assess the patient’s ability to perform activities of daily living’ TO
nurse_infoscreen
The knowledge component contains the medical knowledge. In the data slot we
have defined two variables. The first one saves a fictive indicator for
listlessness, which is going to be read from a database, the second one saves
a corresponding value in the key features. The MLMs priority is set to 42, and
the MLM will be, as defined in the evoke slot, evoked as soon as one of those
two variables is set. The logic slot defines the rule, which has to be
evaluated for executing the action slot, if it returns true. In this example,
the evoke slot returns true, if the feature is “very tired”, or the value on
the listlessness scale is bigger than 84. The action defines to write an
approriate message to an info screen (named nurse_infoscreen).
According to the building block defined in Fig. 1 we summarize as follows:
* •
is best placed into the title slot.
* •
can be defined in the explanation key of the Library section.
* •
can be considered as part of evoke $\rightarrow$ keywords slot
* •
, , and are not supported at all.
* •
has its counterpart in the citations slot.
* •
and are matching with action slots. Although the intention is similar it
should be noted that there is one major conceptual difference: the all-or-
nothing aproach of ARDEN. This makes NNN definitions very complex as each
separate spawns a new MLM.
* •
, although it is similar to the library $\rightarrow$ purpose slot the
difference is still to big to be neglected
### 3.2 Asbru
Asbru is a modeling notation which emphasizes the temporal structures of
medical plans [18]. The underlying skeletal plan leaves room for the temporal
planning of the individual actions, which increases flexibility. This way,
plans can be adapted to changing circumstances, for example if medical
professionals are needed on other stations. Asbru consists of 2 phases. While
during design phase, you can define timely confined actions and alternatives
as reactions for conditions of any patient, during execution phase these
resulting skeleton planes get instantiated for a concrete patient. This
emphasizes the generation of practical plans. The language, which is based on
XML and defined over a DTD shows a syntax close to Lisp.
Design Principles: Plans in Asbru consist of several elements such as
Preferences, Intentions, and Plan Body. Plan hierarchies can be also defined
[17]. With AsbruView a visualization approach for plans has been proposed [3].
For demonstrating the XML structure of Asbru, selected parts of the NNN
diagnosis FATIGUE have been implemented in this notation. The following
listing shows the implementation of an exemplary section of the Intentions
part. The desired nursing outcome of this plan is to increase the energy level
(a fictive scale for measuring the energy a patient currently has) of a
patient to a value of at least 50 over 3 days.
Listing 2: Intentions
⬇
<intentions>
<intention label="PowerEnergy" type="overall-state " verb="achieve"
importance="1">
<temporal-pattern>
<parameter-proposition parameter-name="patientEnergy">
<value-description type="greater-than">
<numerical-constant unit="E" value="50"/>
</value-description>
<context><context-ref name="patientEnergy"/></context>
<time-annotation>
<time-range>
<duration>
<minimum>
<numerical-constant unit="d" value="3"/>
</minimum>
</duration>
</time-range>
<self/>
</time-annotation>
</parameter-proposition>
</temporal-pattern>
</intention>
</intentions>
Although Asbru provides a flexible scheme for formalizing medical guidelines,
it can only be of limited use to care related guidelines. In the following we
list the main differnces according to the building blocks defined in Fig. 1:
* •
can be defined in the title attribute of the plan itself.
* •
can be defined in the preconditions, but it should be noted theat doing so
leads to the loss of support for precautions.
* •
could be put into the comment tag which is applicable to a lot of tags.
* •
and are going to be defined as nested sub plans by using several plan-bodys.
* •
match directly with intentions wich additionally act as a container for .
* •
, , and can not be defined (leaving a misusing of the comment tag aside)
### 3.3 The GuideLine Interchange Format (GLIF)
GLIF (current version GLIF3) [2] has the goal of easing the interchange of
guidelines between different institutions and platforms. It offers structures,
which make it easy for users to understand the purpose of the guideline, as
well as such, which are of use for decision support systems. In GLIF all
classes are represented by UML class diagrams. The actual structure of the
guidelines is defined in RDF. Other constraints are defined by in OCL [19].
Design Principles: GLIF consists of two components: the GLIF model, and the
GLIF syntax. While guidelines are implemented as instances of the classes
defined in the GLIF model, the representation of the knowledge covered by the
guideline is specified in the GLIF syntax. Guideline is the class which
describes a guideline. All general attributes, like the name, the author, the
purpose, selection criteria (which are implemented as criterion objects),
steps, the starting point, and further information (didactics) are part of
this class. The class Guideline_Collection describes a collection of classes
which belong together. The class Supplemental Material List can be used for
describing further information about the class, like the documentation. The
class Guideline_Expressions represents all kinds of expressions, from simple
strings like “weight” to logical expressions, like “weight $<$ 90”. Further
on, GLIF proposes a 3-level-model consisting of the conceptual, computable,
and the implemetable level. All these levels give a different degree of
abstraction, and can therefore be useful to different kinds of users (from
real persons to computer programs).
For showing the structure of GLIF, again we have implemented selected parts of
the NANDA diagnosis FATIGUE. In the following we show how the classes
Guideline, Action Step and Action Specification are being implemented.
The class Guideline contains general information. In this example we have
defined two criteria which have to evaluate to true before starting the
guideline; these are LackOfEnergy and NeedForEnergy. The Steps defines the
steps which are part of the guideline. In this case, only the two simple steps
MedicationWatch and DailyLivingWatch are implemented. First step defines, that
we first have a look on the medication.
Listing 3: Class: Guideline
⬇
name: Fatigue
author: Georg Kaes
intention: Capacity to sustain activity
eligibility\_criteria: LackOfEnergy, NeedForEnergy, […]
didactics: An overwhelming sustained sense of exhaustion and decreased
capacity for physical and mental work at usual level
step: MedicationWatch, DailyLivingWatch, …
first step: MedicationWatch
The Action Steps contain general information about the actions, like which
step follows the current action. Our definition implies that the step
DailyLivingWatch is to be executed after the execution of MedicationWatch.
Listing 4: Class: Action Step
⬇
action: AS\_MedicationWatch
subguideline: null
next\_step: DailyLivingWatch
name: DailyLivingWatch
action: AS\_DailyLiving
subguideline: null
next\_step: […]
Besides that, the Action Specification is defined, where further informations
about the step are given.
Listing 5: Class: Action Specification
⬇
name: AS\_MedicationWatch
patient\_data: Patient\_Medication\_Overview
description: Evaluate the patient’s routine prescription and over-the-counter
medications
didactics: [list of supporting didactic materials]
Listing 6: Class: Action Specification
⬇
name: AS\_DailyLiving
patient\_data: Patient\_DailyLiving\_Overview
description: Assess the patient’s ability to perform activities of daily
living
didactics: [list of supporting didactic materials]
GLIF is a comprehensive approach, which allows medical guidelines to be
modeled from various points of views. But for implementing NANDA in this
formalization, we found that some important features are missing (again we
reuse the building blocks defined in Fig. 1):
* •
match the name property of the Guideline class
* •
is equivalent to the didactics property
* •
can be expressed as a part eligibility_criteria property while is represented
by the intentions property of the same class
* •
and have no match at all
* •
can be implemented either as instance of the Supplemental Material class or in
the corresponding didactics
* •
and are potential instances of the action step class
* •
and can be added to the didactics property of the Action Specification class
### 3.4 Summarizing the Evaluation Results
In this section three representative standards for CIGs have been evaluated
based on their capability to implement the NNN building blocks to as
summarized in Figure 1. Apparently, none of these approaches can implement the
entire structure of the NNN knowledge. Particularly, an implementation for
risks and related factors of a given diagnosis are missing. Hence, in order to
meet our requirement to be able to formalize the NNN knowledge in a complete
manner, we will introduce the NNN formalization for application in the care
domain.
Table 1: Evaluation of CIG Standards along NNN Building Blocks
| | | | | |
---|---|---|---|---|---|---
Arden Syntax | + | + | + | - | - | +
Asbru | + | - | 0 | - | - | +
GLIF | + | + | + | - | - | +
| | | | |
---|---|---|---|---|---
Arden Syntax | + | 0 | + | - | -
Asbru | + | + | + | - | +
GLIF | + | + | + | 0 | 0
+: supported, -: not supported, 0: workaround
## 4 NNN Formalization
The following section describes a way of formalizing NNN. After a short
introduction into the structure, the different parts are described in detail.
### 4.1 Overview
The NNN formalization (based on XML) is divided into the following three
distinct sections.
1. 1.
Meta: Meta-Information about the guideline
2. 2.
Custom: institution-specific preferences for certain tasks
3. 3.
Guideline: information about the guideline itself
Before describing the structure itself, we want to describe some elements that
may reoccur throughout the sections:
* •
$<$hints$>$ can be added to various elements. Each of these elements may
consist of one or more $<$hint$>$ elements, defining its origin (from) and its
purpose (text) in its attributes. Listing 7 defines the RNG schema for hints.
* •
$<$examples$>$ can be added to support nurses when deciding which treaments to
apply. Listing 8 defines the RNG schema for examples.
* •
$<$inputs$>$ are used to enforce comprehensive documentation. A task may may
require more then one parameters of a specific type (e.g. natural numbers,
scales, …) for its comprehensive doumentation, e.g. saving the current blood
pressure (both systolic and diastolic) of a patient. Choosing the right type
further enables to reuse the information defined in the NOC when scoring them.
To enable this, $<$outcome$>$ elements and $<$task$>$ elements support
$<$inputs$>$. To support also complex data structures (e.g. systolic and
diastolic combined as blood pressure), $<$inputs$>$ elements may nest multiple
$<$input$>$ elements, where each of them contains the attributes label
describing the meaning of it. Listing 9 defines the RNG schema for inputs.
* •
Each element containing care relevant information defines a score attribute
containing a natural number between 1 and 10. We intend to use this attribute
to express preferences to decision support systems (DSS).
Listing 7: RNG schema for the hints element
⬇
<define name="refhints">
<optional>
<element name="hints">
<oneOrMore>
<element name="hint">
<optional>
<attribute name="from">
<text/>
</attribute>
</optional>
<attribute name="text">
<text/>
</attribute>
<optional>
<attribute name="score">
<data type="integer"/>
</attribute>
</optional>
</element>
</oneOrMore>
</element>
</optional>
</define>
Listing 8: RNG schema for the examples element
⬇
<define name="refexamples">
<optional>
<element name="examples">
<oneOrMore>
<element name="example">
<attribute name="text">
<text/>
</attribute>
<optional>
<attribute name="score">
<data type="integer"/>
</attribute>
</optional>
<optional>
<ref name="nestedexamples"/>
</optional>
</element>
</oneOrMore>
</element>
</optional>
</define>
<define name="nestedexamples">
<optional>
<element name="examples">
<oneOrMore>
<element name="example">
<attribute name="text">
<text/>
</attribute>
<optional>
<attribute name="score">
<data type="integer"/>
</attribute>
</optional>
</element>
</oneOrMore>
</element>
</optional>
</define>
Listing 9: RNG schema for the inputs element
⬇
<define name="refinputs">
<optional>
<element name="inputs">
<oneOrMore>
<element name="input" ns="http://relaxng.org/ns/structure/1.0">
<attribute name="label">
<text/>
</attribute>
<ref name="any"/>
</element>
</oneOrMore>
</element>
</optional>
</define>
<define name="any">
<element>
<anyName/>
<zeroOrMore>
<choice>
<attribute>
<anyName/>
</attribute>
<text/>
<ref name="any"/>
</choice>
</zeroOrMore>
</element>
</define>
Additionally to the RNG schemes, examples based on the case study of nursing
diagnosis FATIGUE demonstrate several parts of our formalization. The
formalized nursing knowledge can be found in NANDA diagnosis repositories like
[20] and [7]. This example only serves the puropse of demonstrating the
comprehensiveness of our formalization - for real life application, nursing
professionals have to implement a practical formalization including relevant
scales for a full documentation. This care-domain specific knowledge is out of
the scope of this technical report.
### 4.2 Meta
This section contains general information about the guidline. It therefore
includes the bulding blocks , and (see Fig. 1). Information about the state
(i.e research, implementing, testing, running or expired define by [4]), the
author, the validator, the implementer, and various dates of the guidline are
also part of this section.
Listing 10 shows the RNG schema for the meta section, and listing 11
implements the meta section for FATIGUE exemplarily.
Listing 10: RNG schema for the Meta section
⬇
<element name="meta">
<element name="title">
<attribute name="text">
<text/>
</attribute>
</element>
<element name="definition">
<attribute name="text">
<text/>
</attribute>
<attribute name="theme">
<text/>
</attribute>
<ref name="refhints"/>
</element>
<element name="version">
<attribute name="id">
<text/>
</attribute>
</element>
<element name="validation">
<attribute name="status">
<choice>
<value>research</value>
<value>implementing</value>
<value>testing</value>
<value>running</value>
<value>expired</value>
</choice>
</attribute>
</element>
<optional>
<element name="institution">
<attribute name="name">
<text/>
</attribute>
</element>
</optional>
<optional>
<element name="author">
<attribute name="name">
<text/>
</attribute>
</element>
</optional>
<optional>
<element name="validator">
<attribute name="name">
<text/>
</attribute>
</element>
</optional>
<optional>
<element name="implementer">
<attribute name="name">
<text/>
</attribute>
</element>
</optional>
<element name="date">
<attribute name="text">
<data type="date"/>
</attribute>
</element>
</element>
Listing 11: The META section for FATIGUE
⬇
<meta>
<title text="fatigue"/>
<definition text="An overwhelming, sustained sense of exhaustion and decreased
capacity for physical and mental work at usual level"/>
<version id="1.0"/>
<validation status="implementing"/>
<institution name="University of Vienna"/>
<author name="Georg Kaes"/>
<validator name="Stefanie Rinderle-Ma"/>
<implementer name="Juergen Mangler"/>
<date text="2013-04-01"/>
</meta>
### 4.3 Custom
The custom section contains institution specific preferences regarding
$<$tasks$>$ elements (see the next section for more information about tasks).
Listing 12 shows the RNG schema for the custom section, and listing 13
implements the section for FATIGUE exemplarily.
* •
$<$recommended$>$ is used to express not binding priorities for various
treatments of a specific diagnosis. The value of its element (ranging from 1
to 10) influences the order in which arbitrary treatments to a specific
diagnoses are listed.
* •
$<$mandatory$>$ is used to enable care manager to enforce a specific treatment
for a specific diagnoses.
The ID of the respective element references the ID of the recommended or
mandatory task.
Listing 12: RNG scheme of the Custom section
⬇
<element name="custom">
<zeroOrMore>
<element name="recommended">
<attribute name="id">
<text/>
</attribute>
<attribute name="score">
<data type="integer">
<param name="minInclusive">1</param>
<param name="maxInclusive">10</param>
</data>
</attribute>
</element>
</zeroOrMore>
<zeroOrMore>
<element name="mandatory">
<attribute name="id">
<text/>
</attribute>
</element>
</zeroOrMore>
</element>
Listing 13: Exemplary implementation of the Custom section
⬇
<custom>
<recommended id="21" score="7"/>
<recommended id="22" score="4"/>
<mandatory id="30"/>
<mandatory id="31"/>
<mandatory id="32"/>
</custom>
### 4.4 Guideline
The Guideline section, which is defined inside the $<$guideline$>$ Tag,
represents the actual care-specific knowledge and consists of the following
sections:
* •
$<$factors$>$ contain and for each diagnose where they apply. Semantically
speaking, it expresses reasons that may cause symptoms or define the specific
risks. Again, multiple $<$factor$>$ elements (with at least the attributes
text (semantic description), type (either risk () or related ()))are nested
below one $<$factors$>$ element. The optional attributes category is used for
further refinement e.g. psychological, physiological or environmental. Further
do $<$factor$>$ elements support additional elements including $<$hints$>$
and/or $<$examples$>$ elements (desribed above). It should be noted that a
$<$factor$>$ element also may contain $<$factors$>$ elements but that this
nesting is only supported for a depth of 1.
* •
$<$symptoms$>$ represent the building block . They contain multiple
$<$symptom$>$ elements which itself can nest elements of the type
$<$causes$>$, $<$hints$>$, and $<$examples$>$.
* •
$<$outcomes$>$ defining the evaluation criteria for treatments, which is be
done everytime a treatment has ended. This element nests multiple elements of
the type $<$outcome$>$. $<$outcome$>$ elements themselve define three
mandatory attributes, namely text (description), source (), and an id (unique
for the scope of the guidline). Similar to the two above, an $<$outcome$>$
element nests multiple elements of the type $<$hints$>$, $<$examples$>$, and
$<$inputs$>$ (which connects it to for evaluation purposes).
* •
$<$tasks$>$ repesents sequences of activities and are therefore used to
specify , and . During the specification of them, it can be defined to execute
the distinct $<$task$>$ elements, which contain the actual activities, eiter
sequentially (nest them into a $<$sequential-task$>$ element) or in parallel
(nest them into a $<$parallel-task$>$ element). It should be noted that at
this point arbitrary levels of nesting are supported, allowing to define
complex tasks too. We explicitely avoided supporting cycles within guidlines
as this is expressed at the (higher) level of the care plan (which is out of
the scope of this technical report). Further, both $<$sequential-tasks$>$ and
$<$parallel-tasks$>$ elements support the optional attributes name and text.
Each $<$task$>$ element contains the attributes id and text.
Additionally, each tag can contain a source attribute, thus documenting the
scientific source of the formalized knowledge. For example, if this attribute
is applied to a tag like $<$factors$>$, the source has been used for all
factors, if applied to a specific $<$factor$>$ tag, it defines that this
specific factor has been formalized from this source. The same is true for all
other tags in the Guideline section.
The process structure defined by parallel and sequential elements in the
$<$tasks$>$ element specifies the order in which certain $<$task$>$ elements
are executed in a patients therapy plan. Additionally to the tasks, the
elements from the Emphasis of Nursing Documentation section can be added to a
patients therapy plan, thus emphasizing a comprehensive documentation of the
patients state.
The tasks section of the NNN formalization as described in listing 14 and 15
includes all the tasks which are defined in NNN as nursing interventions.
Listing 18 shows an excerpt of the $<$tasks$>$ section of FATIGUE. The
implemented guideline steps have been taken from various sources, including
[8] and german literature ([7] and [5]). Listings 16 and 17 define the RNG
schema for the Emphases of Nursing Documentation, as described in [7].
Listing 14: RNG schema for the definition of tasks
⬇
<element name="tasks">
<element name="labels">
<oneOrMore>
<element name="label">
<attribute name="text">
<text/>
</attribute>
</element>
</oneOrMore>
</element>
<oneOrMore>
<ref name="reftask"/>
</oneOrMore>
</element>
Listing 15: RNG definition for reftask
⬇
<define name="reftask">
<element name="task">
<attribute name="text">
<text/>
</attribute>
<attribute name="id">
<text/>
</attribute>
<optional>
<attribute name="predictedeffort">
<data type="integer"/>
</attribute>
</optional>
<optional>
<attribute name="score">
<data type="integer"/>
</attribute>
</optional>
<ref name="refhints"/>
<ref name="refexamples"/>
<ref name="refinputs"/>
</element>
</define>
Listing 16: RNG schema for the definition of Emphasis of Nursing Documentation
⬇
<element name="documentations">
<oneOrMore>
<ref name="refdocu"/>
</oneOrMore>
</element>
Listing 17: RNG definition for refdocu
⬇
<define name="refdocu">
<element name="documentations">
<attribute name="text">
<text/>
</attribute>
<attribute name="id">
<text/>
</attribute>
<ref name="refhints"/>
<ref name="refexamples"/>
<ref name="refinputs"/>
</element>
</define>
Listing 18: Tasks of the diagnosis FATIGUE in the NNN formalization
⬇
<tasks>
<labels>
<label name="Energy Management"/>
</labels>
<task text="Evaluate medication" id="0">
<!-- examples can support the understanding of the related component \-->
<examples>
<example text="Fatigue can be a byeffect of beta blockers and chemo
therapy."/>
</examples>
<!-- Inputs can be used to define the documentation of a task \-->
<inputs>
<input label="Short summary" xmlns="http://relaxng.org/ns/structure/1.0">
<element name="summary">
<data type="string">
<param name="maxLength">50</param>
</data>
</element>
</input>
<input label="Detailed Medication"
xmlns="http://relaxng.org/ns/structure/1.0">
<element name="eingabe2">
<text/>
</element>
</input>
</inputs>
</task>
<task text="assess physical or psychical medical conditions" id="1">
<examples>
<example text="MS"/>
<example text="Lupus"/>
<example text="chronical pain"/>
<example text="Hepatitis"/>
<example text="AIDS"/>
<example text="fear"/>
</examples>
</task>
<task text="evaluate adequacy of nutrition and sleep" id="2">
<!-- Hints can be used to give other additional information than examples \-->
<hints>
<hint>Sometimes clients with chronic fatigue symptom can sleep excessively and
need support to limit sleeping.</hint>
</hints>
</task>
<task text="assess how the fatigue develops during the day" id="3">
<inputs>
<input label="Detailed Analysis" xmlns="http://relaxng.org/ns/structure/1.0">
<element name="eingabe2">
<text/>
</element>
</input>
</inputs>
</task>
[...]
</tasks>
As stated before, multiple elements necessary for implementing a comprehensive
representation of the NNN guidelines cannot be represented using CPG
approaches. Especially factors related to the diagnosis have no matching
elements in the evaluated CPGs. Listing 21 shows how we model these building
blocks. They also contain a body which will be reused later, so we defined it
separately. This body is shown in listing 19.
Listing 19: RNG schema for the body of factors and symptoms
⬇
<define name="stdbody">
<optional>
<attribute name="category">
<text/>
</attribute>
<optional>
<attribute name="subcategory">
<text/>
</attribute>
</optional>
</optional>
<attribute name="text">
<text/>
</attribute>
<ref name="refhints"/>
<ref name="refexamples"/>
</define>
Listing 20: RNG schema for the factors section
⬇
<element name="factors">
<oneOrMore>
<element name="factor">
<ref name="stdbody"/>
</element>
</oneOrMore>
</element>
Listing 21: Factors of the diagnosis FATIGUE in the NNN formalization
⬇
<factors>
<factor category="psychological" text="Boring lifestyle"/>
<factor category="psychological" text="Stress"/>
<factor category="psychological" text="Anxiety"/>
<factor category="psychological" text="Depression"/>
<factor category="environmental" text="Humidity"/>
<factor category="environmental" text="Humidity"/>
<factor category="environmental" text="Lights"/>
<factor category="environmental" text="Noise"/>
<factor category="environmental" text="Temperature"/>
<factor category="physiologisch" text="changed chemical processes in the
patient’s body">
<factors>
<factor text="medicines"/>
<factor text="drug withdrawal"/>
<factor text="chemotherapy"/>
</factors>
<hints>
<hint text="e.g. because of medicines, drug withdrawal or other reasons"/>
</hints>
</factor>
[...]
</factors>
The defining characteristics are possible symptoms a patient can show when he
suffers the given diagnosis. Listing 23 shows how symptoms can be modeled in
our formalization.
Listing 22: RNG schema for the symptoms section
⬇
<element name="symptoms">
<oneOrMore>
<element name="symptom">
<ref name="stdbody"/>
</element>
</oneOrMore>
</element>
Listing 23: Symptoms of the diagnosis FATIGUE in the NNN formalization
⬇
<symptoms>
<symptom category="subjective" text="inability to restore energy even after
sleep"/>
<symptom category="subjective" text="lack of energy or inability to maintain
usual level of physical activity"/>
<symptom category="subjective" text="increase in the rest requirements"/>
<symptom category="subjective" text="tired"/>
<symptom category="subjective" text="lethargic"/>
<symptom category="subjective" text="listless"/>
<symptom category="subjective" text="perceived need for additional energy to
accomplish routine tasks"/>
<symptom category="subjective" text="introspection"/>
<symptom category="subjective" text="compromised libido"/>
<symptom category="subjective" text="feeling of guilt for not keeping up with
responsibilities"/>
<symptom category="subjective/objective" text="inability to maintain usual
routines"/>
<symptom category="subjective/objective" text="compromised concentration"/>
<symptom category="subjective/objective" text="disinterest in surroundings"/>
<symptom category="subjective/objective" text="drowsy"/>
<symptom category="objective" text="decreased performance"/>
<symptom category="objective" text="increase in physical complaints"/>
<symptom category="objective" text="verbalization of an unremitting and
overwhelming lack of energy"/>
</symptoms>
Outcomes, as defined in the Nursing Outcomes Classification, are seamlessly
integrated into our formalization. Listing 25 shows three exemplary outcomes
and the respective NOC labels. While the first two outcomes implement
documentationary $<$input$>$ items for assessing whether a patient has reached
the goal or not, the last outcome includes the possibility of documenting the
way a patient describes his plan of conserving energy.
Listing 24: RNG schema for the outcome section
⬇
<element name="outcomes">
<element name="labels">
<oneOrMore>
<element name="label">
<attribute name="text">
<text/>
</attribute>
</element>
</oneOrMore>
</element>
<oneOrMore>
<element name="outcome">
<attribute name="goal">
<choice>
<value>achieve</value>
<value>maintain</value>
<value>prevent</value>
</choice>
</attribute>
<attribute name="text">
<text/>
</attribute>
<attribute name="id">
<text/>
</attribute>
<ref name="refhints"/>
<ref name="refexamples"/>
<ref name="refinputs"/>
</element>
</oneOrMore>
</element>
Listing 25: Exemplary outcomes for FATIGUE
⬇
<outcomes>
<labels>
<label text="Endurance"/>
<label text="Concentration"/>
<label text="Energy Conservation"/>
<label text="Nutrition Status: Energy"/>
</labels>
<outcome goal="achieve" text="The patient verbalizes increased energy.">
<inputs>
<input label="Goal reached" xmlns="http://relaxng.org/ns/structure/1.0">
<element name="select">
<element name="yes">Yes</element>
<element name="no">No</element>
</element>
</input>
</inputs>
</outcome>
<outcome goal="achieve" text="The patient verbalizes improved well-being.">
<inputs>
<input label="Goal reached" xmlns="http://relaxng.org/ns/structure/1.0">
<element name="select">
<element name="yes">Yes</element>
<element name="no">No</element>
</element>
</input>
</inputs>
</outcome>
<outcome goal="achieve" text="The patient explains energy conservation plan to
offset fatigue."/>
<inputs>
<input label="Patients explanation"
xmlns="http://relaxng.org/ns/structure/1.0">
<element name="input-pat">
<text/>
</element>
</input>
</inputs>
</outcome>
[...]
</outcomes>
## 5 Discussion
The NNN formalization presented in this paper constitutes an initial step into
the field of ’computer-aided’ nursing in the care domain as envisioned by the
ACaPlan project by providing the basic building blocks required for
comprehensive and formalized care planning and execution.
During development of the NNN formalization, the discrepancies between the
medical and the care domain based on the differences between respective
guidelines as described in chapter 3 became obvious very quickly, although
these fields of research are related on many levels. The fact, that the
revised CIGs do not support any means for implementing risks and related
factors resulted in the need for developing an approach to formalize knowledge
specific for NNN guidelines. The implementation of NNN guidelines in this
technical report covers all relevant parts of a diagnosis, so a patient’s
state can be assessed from various points of view: On the one hand, based on
data about his history, risks and related factors can be used to diagnose
potential threats very early; on the other hand it is possible to use symptoms
a patient shows to determine his diagnoses. These different perspectives
support nursing personnel when creating therapy plans in many situations.
Additionally, by introducing $<$input$>$ tags, different scales and
documentationary forms can be added to support the documentation.
Evaluations with domain experts show that the formalization presented in this
technical report addresses all relevant parts of NANDA, NIC and NOC. Thus,
guidelines formalized using this approach can be applied for supporting nurses
in real life scenarios.
Acknowledgment: We thank Adelheid Beyerl and Martin Zigler from the care
centers St. Pölten Pottenbrunn and Vitacon for the discussions and insights
into the care domain as well as the great cooperation within the ACaPlan
project.
## References
* [1] NANDA International, “NANDA-I, NIC, NOC for safe patient care,” 2012. [Online]. Available: http://www.nanda.org/NNN.aspx
* [2] A. Boxwala, M. Peleg, S. Tu, O. Ogunyemi, Q. Zeng, D. Wang, V. Patel, R. Greenes, E. Shortliffe _et al._ , “Glif3: a representation format for sharable computer-interpretable clinical practice guidelines,” _Journal of biomedical informatics_ , vol. 37, no. 3, pp. 147–161, 2004.
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* [4] T. Pryor and G. Hripcsak, “The arden syntax for medical logic modules,” _International journal of clinical monitoring and computing_ , vol. 10, pp. 215–224, 1993.
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* [6] NANDA International, “Nursing diagnosis frequently asked questions,” 2012. [Online]. Available: http://www.nanda.org/NursingDiagnosisFAQ.aspx
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* [8] NANDA International, “Nursing diagnoses - fatigue — nursing diagnosis for nurses and bsn students,” http://nandanursingdiagnosis.org/nursing-diagnosis-fatigue/, 2013. [Online]. Available: http://nandanursingdiagnosis.org/nursing-diagnosis-fatigue/
* [9] “Fatigue - mynurse,” http://mynurse.weebly.com/fatigue.html, 2013. [Online]. Available: http://mynurse.weebly.com/fatigue.html
* [10] “Nanda glossary of terms,” http://www.nanda.org/nanda-international-glossary-of-terms.html, 2012. [Online]. Available: http://www.nanda.org/nanda-international-glossary-of-terms.html
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* [15] Y. Shahar, O. Young, E. Shalom, A. Mayaffit, R. Moskovitch, A. Hessing, and M. Galperin, “Degel: A hybrid, multiple-ontology framework for specification and retrieval of clinical guidelines,” in _Artificial Intelligence in Medicine_ , ser. Lecture Notes in Computer Science, M. Dojat, E. Keravnou, and P. Barahona, Eds. Springer Berlin Heidelberg, 2003, vol. 2780, pp. 122–131. [Online]. Available: http://dx.doi.org/10.1007/978-3-540-39907-0_18
* [16] G. Hripcsak, P. D. Clayton, T. A. Pryor, P. Haug, O. B. Wigertz, and J. van der lei, “The arden syntax for medical logic modules,” _Symposium on Computer Applications in medical care_ , pp. 200–204, 1990.
* [17] P. A. de Clercq, K. Kaiser, and A. Hasman, “Computer-interpretable guideline formalisms,” _Studies in Health Technology and Informatics_ , vol. 139, pp. 22–43, 2008.
* [18] Y. Shahar, S. Miksch, and P. Johnson, “The asgaard project: a task-specific framework for the application and critiquing of time-oriented clinical guidelines,” _Artificial Intelligence in Medicine_ , vol. 14, no. 1/962, pp. 29 – 51, 1998.
* [19] M. Peleg, A. A. Boxwala, S. Tu, D. Wang, O. Ogunyemi, and Q. Zeng, “Guideline interchange format 3.5 technical specification,” 2004.
* [20] A. K. Aplikasi, “Nanda: Nursing diagnosis: Fatigue application of nanda, noc, nic,” 2012. [Online]. Available: http://asuhankeperawatanonline.blogspot.co.at/2012/03/nursing-diagnosis-fatigue-application.html
|
arxiv-papers
| 2014-04-08T14:50:53 |
2024-09-04T02:50:00.858799
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Georg Kaes, J\\\"urgen Manger, Stefanie Rinderle-Ma, Ralph Vigne",
"submitter": "Georg Kaes",
"url": "https://arxiv.org/abs/1404.2162"
}
|
1404.2281
|
# Probing Asymmetric Structures in the outskirts of Galaxies
Zhang Zheng Wen11affiliation: Purple Mountain Observatory, Chinese Academy of
Sciences, 2 West Beijing Road, Nanjing, 210008, China 22affiliation:
University of Chinese Academy of Sciences, 19A Yuquan Road Beijing, China ,
Xian Zhong Zheng11affiliation: Purple Mountain Observatory, Chinese Academy of
Sciences, 2 West Beijing Road, Nanjing, 210008, China and Fang Xia
An11affiliation: Purple Mountain Observatory, Chinese Academy of Sciences, 2
West Beijing Road, Nanjing, 210008, China
###### Abstract
Upcoming large imaging surveys will allow detailed studies of the structure
and morphology of galaxies aimed at addressing how galaxies form and evolve.
Computational approaches are needed to characterize their morphologies over
large samples. We introduce an automatic method to quantify the outer
structure of galaxies. The key to our approach is the division of a galaxy
image into two sections delineated by the isophote which encloses half the
total brightness of the galaxy. We call the central section the inner half-
flux region (IHR) and the outer section the outer half-flux region (OHR). From
this division, we derive two parameters: $A_{\rm o}$, which measures the
asymmetry of the OHR, and $D_{\rm o}$, which measures the deviation of the
intensity weighted centroid of the OHR from that of the IHR relative to the
effective radius. We derive the two parameters from $HST$/ACS $z_{850}$-band
images for a sample of 764 galaxies with $z_{850}<22$ mag and $0.35<z<0.9$
selected from GEMS and GOODS-South surveys. We show that the sample galaxies
having strong asymmetric structures, in particular tidal tails, are well-
separated from those with regular morphologies in the $A_{\rm o}$-$D_{\rm o}$
space. Meanwhile, the widely used CAS and Gini-$M_{20}$ methods turn out to be
insensitive to such morphological features. We stress that the $A_{\rm
o}$-$D_{\rm o}$ method is an efficient way to select galaxies with significant
asymmetric features like tidal tails and study galaxy mergers in the dynamical
phase traced by these delicate features.
###### Subject headings:
galaxies: interactions — galaxies: fundamental parameters — galaxies: peculiar
— galaxies: structure
## 1\. INTRODUCTION
Major mergers between galaxies of comparable mass are expected to occur
frequently in hierarchical models of galaxy formation and evolution (e.g.,
White & Rees, 1978; Lacey & Cole, 1993). Galaxy merging may be a crucial
process that regulates galaxy mass assembly, galaxy morphology reshaping,
growth of supermassive black holes, and enhancement of star formation (e.g.,
Barnes & Hernquist, 1992; Sanders & Mirabel, 1996; Springel et al., 2005;
Hopkins et al., 2008, 2010; Bundy et al., 2009; Conselice, 2014). Measuring
the galaxy merger rate over cosmic time is thus a central task in determining
the importance of the merging process relative to that of other physical
processes (e.g., feedback and gas accretion) to driving galaxy evolution.
Much effort has been made to measure galaxy merger rate. Interacting or
merging galaxies are rare ($\sim$2%) in the local universe (Athanassoula &
Bosma, 1985; Patton et al., 1997; Xu et al., 2004; Patton & Atfield, 2008;
Domingue et al., 2009), but they become more numerous at higher redshifts.
However, the measurements of the merger rate are inconsistent with each other
and are still under debate. While strong evolution characterized by
$(1+z)^{3-6}$ was often reported (Le Fèvre et al., 2000; Patton et al., 2002;
Conselice et al., 2003; Cassata et al., 2005; Kampczyk et al., 2007;
Kartaltepe et al., 2007; Conselice et al., 2009; Jogee et al., 2009; Xu et
al., 2010; Lotz et al., 2011), mild evolution following $\sim(1+z)^{0.5}$ or
even no evolution was obtained by other studies (Carlberg et al., 2000; Bundy
et al., 2004; Lin et al., 2008; Robaina et al., 2010; Man et al., 2012). The
controversy is likely caused by large uncertainties in current observational
techniques adopted for merger identification (see Lotz et al. 2008 for more
details).
Disturbed morphology is mostly used as the probe of a merger. The violent
tidal forces between merging galaxies can destroy galaxy structures and
produce tidal features. For instance, extended tidal tails can be created when
the mergers involve a disk galaxy (Toomre & Toomre, 1972; Wright, 1972; Barnes
& Hernquist, 1992; Mihos, 2004). The characteristics of the tidal tails in
turn disclose key properties of the parent galaxies such as kinematics, mass
ratio, and orbital parameters (Duc, 2013; Duc & Renaud, 2013). A long tidal
tail is usually seen as evidence for a merger between disk galaxies (e.g.,
Elmegreen et al., 2007; Bridge et al., 2010). Resolving specific features from
mergers helps to provide information on the frequency of mergers, a better
understanding of the merger time scale, and a complete census of various types
of mergers over cosmic time. It further delivers insights on evolutionary
pathways for different galaxy populations. This will be possible with upcoming
deep imaging surveys over large areas (e.g., Euclid, Amendola et al., 2013).
Computational approaches are keenly needed to detect the specific features
tracing given types of mergers.
Non-parametric methods $CAS$ (Conselice et al., 2000; Conselice, 2003) and
Gini-$M_{20}$ (Lotz et al., 2004) are widely used for merger selection. $CAS$
selects mergers mainly according to their morphological asymmetry, but fails
to pick up those with weaker asymmetry in morphology (e.g., double-nucleus).
The Gini coefficient and $M_{20}$ parameters measure the relative distribution
of pixel fluxes and spatial light concentration of a galaxy, respectively.
Mergers and regular galaxies are globally separated in the Gini-$M_{20}$
space. This method favors to select mergers that are in their first pass or
the final stage (Lotz et al., 2008, 2010a, 2010b). Neither $CAS$ nor
Gini-$M_{20}$ provides a complete selection for major mergers (Kampczyk et
al., 2007; Scarlata et al., 2007; Jogee et al., 2009; Kartaltepe et al.,
2010). This is in part because the parameters adopted in the two methods are
flux weighted. These parameters are largely determined by the light
distribution of the bright section of a galaxy, and are insensitive to the
tidal features with lower surface brightness in the outskirts of the galaxy.
In this paper, we present a new approach to quantifying the structure of the
outskirts of a galaxy, aimed at searching for tidal tails. Two parameters are
introduced to accomplish this. In Section 2 we describe how to measure the two
parameters. In Section 3, we verify our method using a visually classified
sample of galaxies. Finally, we compare our method with $CAS$ and
Gini-$M_{20}$ in Section 4. Throughout this paper, we assume $H_{0}=70$ km s-1
Mpc-1, $\Omega_{\rm\Lambda}=0.7$ and $\Omega_{\rm m}=0.3$.
## 2\. METHODOLOGY
### 2.1. Galaxy Division
We divide the image of a galaxy into two sections split by the isophote which
encloses half the total light of the galaxy. The inner section, named as the
inner half-flux region (IHR), usually encloses a smaller area with a higher
surface brightness than the outer section, named as the outer half-flux region
(OHR). In practice, the IHR is identified first, and the rest of the galaxy is
presumed to occupy the OHR. If the galaxy is regular in morphology, the two
sections can be simply separated by the isophote which contains half the total
light. If a galaxy does not have a single, clear core, but has double or even
multiple comparable components, then we fix the center at the brightest
component, and find the half-light isophote around this point.
We develop a technique to deal with image pixels and identify the brightest
group of contiguous pixels as the brightest component in a galaxy image. The
software tool SExtractor (Bertin & Arnouts, 1996) is used to detect pixels
that a galaxy covers in the background-subtracted image (see Section 3.3 for
details). These pixels are sorted in descending order of flux. The pixels with
highest fluxes are selected, yielding $f$, a ratio of the integrated flux to
the total. More pixels within the galaxy will be included if a higher $f$ is
set. Starting from $f$=0.5, the selected pixels may spread into two or more
discrete groups of contiguous pixels for galaxies with complex light
distribution (e.g., late-type spirals with giant clumps). These discrete
groups of pixels are then individually examined to obtain their integrated
fluxes. By increasing $f$, each group is expected to contain more contiguous
pixels and thus a higher fraction of the total flux. The brightest group with
a sum of fluxes higher than 25% of the total is chosen to calculate a flux-
weighted centroid. This position is taken as the center for isophotal fitting
to the galaxy image. For a galaxy with smooth light distribution usually
satisfying a Sérsic function, the selected pixels make up one group, which is
indeed the IHR when $f=0.5$. For a merging system with two nuclei, the
brighter one will be selected. The factor of 25% is decided by this selection.
With the center fixed, elliptical isophotes are fitted to the galaxy image.
Following the algorithms given in SExtractor, the ellipticity and position
angle of an isophote are determined by the maximum and minimum spatial
dispersion of the enclosed pixels by the real isophote. The ellipse containing
50% of the total flux is taken as the boundary for the IHR. And the pixels out
of the isophote make up the OHR. Figure 1 demonstrates the separation between
the IHR and OHR for two representative galaxies.
Figure 1.— $HST$/ACS $z_{850}$-band images (left) and segmentation maps
(right) for a typical spiral galaxy at $z=0.47$ (top) and a merger with tidal
tail at $z=0.58$ (bottom). The ellipses (dashed lines) mark the IHR. The blue
crosses and red diamonds give the centroids of the IHR and OHR of each galaxy,
respectively.
### 2.2. Outer asymmetry
The asymmetry parameter ($A$) is often used to quantify the morphological
disturbance of a galaxy (Abraham et al., 1996; Conselice et al., 2000;
Conselice, 2003). It is calculated as the sum of the absolute residuals of the
galaxy image subtracting itself rotated by 180$\arcdeg$ around its center. The
rotation center is determined iteratively in order to minimize the asymmetry.
We define outer asymmetry, $A_{\rm o}$, as the asymmetry of the OHR of a
galaxy following
$A_{\rm o}=\frac{\sum{|I_{\rm o}-I_{\rm o}^{180^{\circ}}|}}{\sum{I_{\rm
o}}}-\frac{\sum{|B_{\rm o}-B_{\rm o}^{180^{\circ}}|}}{\sum{I_{\rm o}}},$ (1)
where $I_{\rm o}$ is flux distribution of the OHR and $I_{\rm
o}^{180^{\circ}}$ represents the $180^{\circ}$-rotated $I_{\rm o}$. In the
same way, $B_{\rm o}$ is a patch in the background with the same shape as
$I_{\rm o}$. Again, $B_{\rm o}^{180^{\circ}}$ is the $180^{\circ}$-rotated
$B_{\rm o}$.
The rotation center is critical to the estimation of the asymmetry parameter.
For very nearby galaxies, a small shift for the center would cause a
significant change in the estimated asymmetry. For distant galaxies,
determining a rotation center becomes less complicated because of decreasing
resolution (Conselice et al., 2000). We thus adopt the centroid of the OHR as
the rotation center when estimating $A_{\rm o}$.
### 2.3. centroid deviation
The centroid deviation parameter, $D_{\rm o}$, measures the deviation (or
offset) between centroids of the IHR and OHR of a galaxy. The centroid is
determined by
$x_{\rm cen}=\frac{\sum\limits_{i\in S}f_{i}~{}x_{i}}{\sum\limits_{i\in
S}f_{i}}~{}~{}{\rm and}~{}~{}y_{\rm cen}=\frac{\sum\limits_{i\in
S}f_{i}~{}y_{i}}{\sum\limits_{i\in S}f_{i}},$ (2)
where $x_{i}$ and $y_{i}$ define the position of pixel $i$ in the galaxy image
and $f_{\rm i}$ is flux of the pixel. The centroid deviation is calculated
using the formula
$D_{\rm o}=\frac{\sqrt{(x_{\rm o}-x_{\rm c})^{2}+(y_{\rm o}-y_{\rm
c})^{2}}}{R_{\rm e}},$ (3)
where ($x_{\rm c}$,$y_{\rm c}$) and ($x_{\rm o}$,$y_{\rm o}$) refer to the
centroids of the IHR and OHR, respectively. The effective radius of the
galaxy, $R_{\rm e}$, is used for normalization. Here $R_{\rm e}$ is estimated
using $R_{\rm e}=\sqrt{n_{\rm c}/\pi}$, where $n_{\rm c}$ is pixel count of
the IHR.
For regular galaxies with symmetric morphologies, the IHR and OHR share nearly
the same centroid, yielding a centroid deviation near zero. For merging
galaxies with tidal tails, however, the centroid deviation becomes rather
large, as shown in Figure 1. Similarly, the merging galaxies tend to have
higher $A_{\rm o}$ than the regular galaxies. Therefore, galaxies with
asymmetric structures are expected to be away from the regular galaxies in the
diagram of $A_{\rm o}$ versus $D_{\rm o}$. We test this $A_{\rm o}$-$D_{\rm
o}$ method using a sample of galaxies with high-resolution images from the
Hubble Space Telescope ($HST$).
## 3\. VERIFYING THE $A_{\rm o}$-$D_{\rm o}$ METHOD
### 3.1. Galaxy sample
The extended Chandra Deep Field South (ECDFS) is one of the cosmic fields with
the deepest multi-wavelength data over an area of $30\arcmin\times 30\arcmin$,
including $HST$/ACS imaging data from the GEMS (Rix et al., 2004) and the
GOODS-S surveys (Giavalisco et al., 2004), the survey catalog (Caldwell et
al., 2008), photometry and photometric redshifts from the Multiwavelength
Survey by Yale-Chile (MUSYC; Cardamone et al., 2010), and the stellar mass
catalog from the COMBO-17 survey (Borch et al., 2006). The best-fit models of
two-dimensional Sérsic profile derived from $HST$ images are available for
GEMS galaxies (Häussler et al., 2007). We use these data to select a sample of
galaxies and test our method for quantifying morphologies of the OHRs.
We focus on massive galaxies that enable strong tidal features to be produced
in the merging process and detectable at high-$z$. We derive morphological
parameters from $HST$/ACS images in the $z_{850}$ band, which corresponds to
the rest-frame $B$ to $V$ in the redshift range $0.35<z<0.9$. In total, 825
galaxies are selected with stellar mass $\log(M/{\rm M}_{\odot})\geq 10.5$ and
$0.35<z<0.9$ over the 800 square arcminutes area covered by $HST$
observations. Of them, 32 objects are too compact to be resolved in
morphology; and 29 objects are located at image edges. Finally, 764 galaxies
are selected to compose a mass-complete sample for our morphological
investigation. The sample galaxies are mostly with $z_{850}$-band magnitude
$\sim 20-22$, compared to the 5 $\sigma$ depth of $z_{850}=27.1$ mag.
The Sérsic index ($n$) is often used as a morphology indicator (e.g., $n<$2.5
for disk galaxies). However, the two-dimensional structural fitting ignores
irregular structures that are crucial to identifying galaxy mergers, although
the Sérsic models are good proxies for regular galaxies. Visual examination is
still an efficient way to ascertain the tidal features. Our $A_{\rm
o}$-$D_{\rm o}$ method lacks sensitivity to distinguish galaxies with regular
morphologies. We therefore adopt seven morphological types optimized for
verifying the $A_{\rm o}$-$D_{\rm o}$ method, including spheroids (E, S0),
early disks (Sa-b), late disks (Sc-d), edge-on disks, irregulars/minor
mergers, major mergers without tidal tail, and major mergers with tidal tails.
The morphological classification was independently carried out by all three
co-authors. As a result, the 764 galaxies in our sample are classified into
440, 57, 57, 27, 108, 60 and 15 from the first to the last class,
respectively. We are interested in the latter three morphological classes, in
particular major mergers with tidal tails. We note that tidal tails with a
size of tens of kpcs at intermediate redshifts can be resolved with $HST$/ACS
imaging. We explain in detail how to recognize a tidal feature as a tidal
tail.
Figure 2.— A sample of mergers with tidal tails in the ECDFS field. From left
to right, galaxy IDs are 6494, 7357, 11072, 17207 (top), 20158, 24090, 29976,
30004 (the 2nd row), 30076, 44488, 45115, 57822 (the 3rd row), 57896, 60651,
and 61546 (bottom). Figure 3.— The $A_{\rm o}$-$D_{\rm o}$ diagram of our
sample of 764 galaxies with $\log(M/\rm M_{\odot})\geq 10.5$ and $0.35<z<0.9$
in the ECDFS. The two parameters are derived from HST $z_{850}$-band images,
corresponding to the rest-frame optical for the redshift range examined here.
The dashed line is the criterion for selecting tidal tails described as ${\rm
log}\,A_{\rm o}>-1.6\,{\rm log}\,D_{\rm o}-1.1$. The solid line represents the
sequence of regular galaxies best-fitted by ${\rm log}\,A_{\rm o}=0.6\,{\rm
log}\,D_{\rm o}$. Figure 4.— HST $z_{850}$-band images of sample galaxies
satisfying the criterion ${\rm log}\,A_{\rm o}>-1.6\,{\rm log}\,D_{\rm o}-1.1$
and $D_{\rm o}<0.5$. These galaxies are visually classified as spheroids or
disks. From left to right, the galaxy IDs are: 49368, 1032, 29488, 35796
(top); 61789, 2848, 42774, 42874 (the 2nd row); 20664, 30167, 30189, 56918
(the 3rd row); 37335, 59517, and 51329 (bottom).
### 3.2. The properties of tidal tails
The visual investigation by Elmegreen et al. (2007) revealed that the host
merging galaxies of tidal tails in ECDFS can be divided into four
morphological types: diffuse, antennae, M51 and shrimp. Figure 2 shows $HST$
images of merging galaxies with tidal tails selected from the ECDFS field.
The diffuse-type galaxies refer to those with diffuse arc-like or tail-like
intergalactic structures (e.g., Kawata et al., 2006). Such tidal structures
tend to have red color and smooth light profile, suggesting that they are
tidal debris of early-type galaxies with little gas through dry mergers (e.g.,
the top-right panel in Figure 2).
The antennae-type galaxies have disturbed merger cores and extended tidal
tails. They are most likely formed by major mergers between disk galaxies
(e.g., Toomre & Toomre, 1972). The visible timescale for tidal tails can be a
few hundred Myrs (Conselice, 2009) up to a few Gyrs (Hibbard & Mihos, 1995).
About half the merging galaxies with tidal tails shown in Figure 2 are
antennae-type (e.g. ID 7357, 44488, 45115, 57896). Tidal dwarf galaxies can be
formed in these tails, being a result of local gravitational instabilities of
the gas component (Elmegreen et al., 1993; Bournaud & Duc, 2006; Wetzstein et
al., 2007) or at the tip of a tidal tail regulated by the accumulation and
collapse of massive gaseous condensations (Duc et al., 2004). The presence of
long tidal tails or tidal dwarfs dramatically increases $A_{\rm o}$ and
$D_{\rm o}$ (see Section 3.3).
The M51-type galaxies are mergers at their early stages and the stellar bridge
connecting one galaxy to the another is still seen. The galaxies of this type
are rare probably because of the short timescale to be detected (several Myrs;
Bournaud & Duc, 2006). Shown in Figure 2, 11072 is a pair of galaxies of
comparable $z_{850}$-band luminosity. The system has just undergone its first
pericenter passage with a weak tidal bridge. A tail-like structure is seen
with color $V_{606}-z_{850}=1.28$, similar to the color of the central
components ($V_{606}-z_{850}=1.33$). No stellar clump is found on the tidal
tail.
Finally, shrimp-type galaxies are characterized by a highly warped, dominant
arm or tail and have no well-defined central nucleus. This type of merger is
not included in Figure 2.
### 3.3. The $A_{\rm o}$-$D_{\rm o}$ classification
Using $HST$ images, we calculate outer asymmetry $A_{\rm o}$ and centroid
deviation $D_{\rm o}$ for our sample of 764 galaxies. We run SExtractor to
create a segmentation map for every galaxy. The segmentation map is an image
with the same size as the scientific image where objects and their boundaries
are determined. The total flux of a galaxy is then defined as the light
contained in the pixels within its boundary. A detection threshold of 0.8
$\sigma_{\rm bkg}$ above background is chosen. Here $\sigma_{\rm bkg}$ is the
$RMS$ of background noise. Other configuration parameters are set following
Caldwell et al. (2008) for the GEMS catalog. This threshold is lower than
usual settings (e.g., 1.65 $\sigma_{\rm bkg}$, Caldwell et al., 2008) for
source detection, aimed at finding extended tidal features that tend to have
lower surface brightness at larger radii of the OHR of a galaxy. We point out
that the total flux adopted here differs from that given, for example, in the
Third Reference Catalogue of Bright Galaxies (RC3; de Vaucouleurs et al.,
1991), where missing flux is added in by extrapolation of integrated
luminosity profiles to infinite radius. This procedure is impractical for our
analysis since we are not fitting light profiles. The measurements of $A_{\rm
o}$ and $D_{\rm o}$ are nevertheless likely to be only marginally affected by
this difference.
We test the $A_{\rm o}$-$D_{\rm o}$ method for quantifying galaxy morphologies
using our sample of 764 galaxies in comparison with the morphologies visually
classified. Figure 3 illustrates the distribution of the sample in $A_{\rm
o}$-$D_{\rm o}$ diagram. Different morphological types are marked with
different symbols. We can see that all sample galaxies lie on a tight
sequence, where the location is in general correlated with the degree of
morphological disturbance. Galaxies with stronger disturbed morphologies tend
to have higher $A_{\rm o}$ and $D_{\rm o}$. The vast majority of galaxies with
regular morphologies (E, S0, Sa-b, Sc-d, edge-on disks) have $D_{\rm o}<0.5$
and $A_{\rm o}<0.6$, following a tight relation ${\rm log}\,A_{\rm
o}=0.6\,{\rm log}\,D_{\rm o}$. This relation indicates that the two parameters
$A_{\rm o}$ and $D_{\rm o}$ are correlated with each other. This is also
confirmed by the fact that no data points are found in the bottom-right and
top-left regions of the diagram.
However, different morphological types of the sample galaxies are not clearly
separated along the sequence. Spheroids and early disks having symmetric
morphologies reside mostly in the region $D_{\rm o}<0.2$. Edge-on disks and
late disks with non-symmetric substructures like arms, H II regions and
stellar clusters are distributed more widely over the sequence. Such
substructures are unlikely to be responsible for a high $D_{\rm o}$. Irregular
galaxies and major/minor mergers are systematically higher in $A_{\rm o}$ but
spread over the full range of $D_{\rm o}$.
The striking result from Figure 3 is that all galaxies with tidal tails (black
crosses) have apparently higher $D_{\rm o}$ and higher $A_{\rm o}$, compared
with the galaxies of other morphological types, in particular regular ones
(spheroids and disks). The large spread in $D_{\rm o}$ is mostly caused by the
variety of the tidal tails in size and light weight relative to their host
galaxies. The majority (12/15) of these with tidal tails are at $D_{\rm
o}>0.5$. This indicates that $D_{\rm o}$ is sensitive to the tidal tails,
although three objects with weak tidal tails (17207, 20158, 24090) have
$D_{\rm o}<0.5$.
As shown in Figure 3, the criterion ${\rm log}\,A_{\rm o}>-1.6\,{\rm
log}\,D_{\rm o}-1.1$ is sufficient to select all galaxies with tidal tails and
most major mergers (32/60) in our sample. This cut, however, also picks up a
large fraction (43%) of objects with irregular morphologies (irregulars/minor
mergers). We note that some regular galaxies (spheroids and disks) also fall
into the $D_{\rm o}>0.5$ region. We examined their images, and found that
their $A_{\rm o}$ and $D_{\rm o}$ values are significantly affected by the
unmasked residual light from neighboring sources. There are also 45 regular
galaxies satisfying ${\rm log}\,A_{\rm o}>-1.6\,{\rm log}\,D_{\rm o}-1.1$ and
$D_{\rm o}<0.5$. Again, we visually checked their $HST$ images and
segmentation maps. Of the 45 targets, 23 are contaminated by neighboring
sources, and another 22 are isolated with asymmetric outskirts. Again the two
morphological parameters $A_{\rm o}$ and $D_{\rm o}$ are largely overestimated
for the visually-classified regular galaxies (spheroids and disks) due to the
contamination. Figure 4 shows the 15 regular galaxies meeting the selection.
We further investigate their morphological properties in detail for a better
understanding of the selection.
* Galaxy 49368: it has bright stellar clumps of 1 kpc size on one spiral arm, which seriously bias the estimation of $A_{\rm o}$ and $D_{\rm o}$. Such features can be frequently seen in $z>2$ disk galaxies. Minor mergers and instabilities in gas rich, turbulent disks are likely responsible for the formation of such clumps (Genzel et al., 2008; Bournaud & Elmegreen, 2009; Dekel et al., 2009).
* Galaxies 1032, 29488, 35796 and 61789: they are characterized by one prominent spiral arm. Their morphologies are similar to the so-called shrimp-type galaxies defined by Elmegreen et al. (2007). The highly curved spiral arm together with star-forming clumps on it usually imply that they are gas-rich systems disturbed by minor mergers. However, no apparent signature is found for tidal disturbance in the four galaxies.
* Galaxies 2848, 42774 and 42874: they have multiple spiral arms and show strong asymmetry in morphology.
* Galaxies 20664, 30167, 30189 and 56918: they are edge-on disk galaxies with a clear dust lane feature. The dust lane divides the light profile of the parent galaxy into two parts. In such cases, the system is treated as a double core system in our analysis. The centroid of the IHR would significantly deviate from the geometric center if the dust lane is thick and splits the galaxy into two non-equal parts.
* Galaxies 37335 and 59517: they are classified as Sd galaxies. Their surface brightness is relatively low. And their $A_{\rm o}$ and $D_{\rm o}$ are largely biased by clumpy light distribution and background noise.
* Galaxy 51329: it is an edge-on disk galaxy. The disk is apparently warped. Minor mergers/interactions are often proposed as the cause of warped disks (Sanchez-Saavedra et al., 1990; Florido et al., 1991; Reshetnikov & Combes, 1998; Ann & Park, 2006).
It is worth noting that the segmentation strategy used in our analysis relies
on a low detection threshold. This may potentially mistake some noise pixels
for part of the target, resulting in a fuzzy appearance at edges. Since outer
asymmetry and centroid deviation measure the morphological distribution of the
OHR, the background noise may have noticeable effect on the estimate of the
two parameters, in particular for low surface brightness galaxies.
Source deblending is also one issue that may affect the measurements of
$A_{\rm o}$ and $D_{\rm o}$. This is also key to extracting source catalogs
from deep images. We adopted the optimal configuration for Sextractor from the
GEMS catalog to produce segmentation maps for the selected sample galaxies.
While the vast majority of sources can be properly resolved in the $HST$
images with the configuration (see Caldwell et al. 2008 for more details), a
small fraction ($<\sim$5%) of sources may still be contaminated by foreground
or background sources due to projection effects, or over deblended into
multiple sources. Merging galaxies with disturbed morphologies tend to suffer
more from this problem. In this work, we visually examined the deblending for
every sample galaxy and found a few cases with contamination by the
neighboring sources which mislead the targets towards a high outer asymmetry.
Figure 5.— The relation between asymmetry ($A$) and clumpiness ($S$) for our
sample. The solid line $A=0.35\,S+0.02$ gives the relationship of nearby
galaxies that are not involved in mergers while the dashed lines are the
criteria ($A>0.35,A>S$) for selecting merging galaxies (Conselice, 2003). The
data points are given in the same way as in Figure 3.
## 4\. COMPARISON WITH $CAS$ AND Gini-$M_{20}$
The non-parametric methods $CAS$ and Gini-$M_{20}$ are widely used automatic
approaches in the literature to identify merging galaxies. In this section, we
compare our $A_{\rm o}$-$D_{\rm o}$ method with the $CAS$ and Gini-$M_{20}$
methods in finding galaxies with extended tidal features.
### 4.1. $CAS$
The $CAS$ method (Conselice et al., 2000; Conselice, 2003) involves three
morphological parameters: concentration ($C$), asymmetry ($A$) and clumpiness
(or smoothness; $S$). The $C$ parameter describes the intensity of stellar
light contained within the central region in comparison to a larger outer
region of a galaxy. The $A$ quantifies the degree that the surface brightness
profile of the galaxy deviates from a perfectly symmetric distribution. The
$S$ parameter focuses on the flux that is contained in the clumpy part of the
light distribution. We calculate $A$ and $S$ parameters for our sample
galaxies using HST $z_{850}$-band images. The results are shown in Figure 5.
The solid line represents the $S-A$ relationship for regular galaxies in the
local universe (Conselice, 2003). It has been shown that the star formation
activities lead to an increase in both $S$ and $A$ for nearby galaxies. We
find, however, that our sample galaxies at intermediate redshifts deviate from
the relationship of nearby galaxies. This has also been reported and explained
in Conselice et al. (2008). The reason for such a deviation is that both $S$
and $A$ decrease when the resolution and signal-to-noise ratio (S/N) become
lower. And $S$ is affected by a larger degree than $A$.
In $CAS$, a galaxy is identified as a major merger when $A>0.35$ and $A>S$.
This selection works well for those where morphological distortion affects
more than 35% of the total light of the galaxy. The condition $A>S$ ensures
that the asymmetric light is not dominated by clumpy star-forming regions.
Figure 5 shows that only about one quarter (16/60) of the major mergers (green
diamonds) and one third (5/15) of galaxies with tidal tails (black crosses) in
our sample satisfy $A>0.35$. This is not surprising, as $A>0.35$ statistically
accounts for one third of the timescale for a major merger determined by
N-body simulations, while minor mergers with mass ratio below $1:5$ hardly
reach $A>0.35$ (Conselice, 2006). This explains why only about one third of
visually-classified major mergers and those with tidal tails in our sample are
selected in Figure 5. We notice that some regular galaxies also show $A>0.35$.
It is mainly due to the contamination from neighboring sources within the
target’s 1.5 times Petrosian radius. A high $A$ is obtained for galaxy 51329
due to the warped disk as shown in Figure 4.
We emphasize that our $A_{\rm o}$ parameter focuses on the morphological
disturbance in the OHR of a galaxy in comparison with $A$ that is dominated by
the central high S/N light distribution and less sensitive to morphological
disturbance in the OHR. It is clear that galaxies with tidal tails become
largely indistinguishable from disks and irregular galaxies in Figure 5.
Compared with Figure 3, we can conclude that our $A_{\rm o}$-$D_{\rm o}$
method is more efficient than $CAS$ in selecting galaxies with strongly
disturbed morphologies like tidal tails.
Figure 6.— $M_{20}$ versus Gini coefficient for our sample. The dashed line is
the criterion ($G>-0.14\,M_{20}+0.33$) from Lotz et al. (2004) for selecting
merging galaxies. All symbols are given as same as in Figure 3. The majority
of regular galaxies lie below the dashed line and form a well defined
sequence.
### 4.2. Gini-$M_{20}$
The Gini coefficient ($G$) was originally invented in economics and introduced
to describe the light distribution of a galaxy through a Lorentz curve
(Abraham et al., 2003; Lotz et al., 2004). The Gini coefficient ranges from 0
to 1. A lower $G$ means a more uniform flux distribution within a galaxy while
a higher $G$ means that a higher fraction of the total flux is contained in a
smaller fraction of the galaxy pixels. $M_{20}$ is a parameter to measure the
second-order moment of the light distribution of the top 20% brightest pixels
in a galaxy which is then normalized to the light moment for all pixels. Like
parameter $C$, it describes the central concentration of the brightest pixels
of the galaxy depending on their distance from the galaxy center.
Figure 6 shows the distribution of our sample in the Gini-$M_{20}$ diagram. We
can see that our sample galaxies at intermediate redshifts are still
distributed along a tight sequence following the nearby galaxies given in Lotz
et al. (2004). The early-type galaxies have higher $G$ and lower $M_{20}$
(top-right) while late-type galaxies have lower $G$ and higher $M_{20}$ in a
statistical sense. Lotz et al. (2004) suggested to select major mergers by the
criterion $G>-0.14\,M_{20}+0.33$ (the dashed line in Figure 6). However, there
are only 9 of 56 major mergers (green diamonds) and one of 15 galaxies with
tidal tails (black crosses) located above the dashed line. It has been pointed
out that that mergers identified by the Gini-$M_{20}$ method are not complete
(e.g., Kampczyk et al., 2007; Scarlata et al., 2007; Kartaltepe et al., 2010).
Simulations show that the Gini-$M_{20}$ method primarily enables identifying
mergers during the first pass or final stage when the galactic nuclei are
distinguishable (Lotz et al., 2008, 2010a, 2010b). The simulations also
uncover that Gini-$M_{20}$ is insensitive to mergers in the intermediate stage
(i.e., maximal separation and/or second pass) when tidal tails may be formed.
Comparison of Figure 6 with Figure 3 clearly demonstrates that our $A_{\rm
o}$-$D_{\rm o}$ method is better than Gini-$M_{20}$ in identifying major
mergers and galaxies with apparent extended tidal features.
## 5\. DISCUSSION
Two parameters, outer asymmetry and centroid deviation, are developed to
quantify the structure in the OHRs of galaxies, which usually have lower
surface brightness than their IHRs. The tidal features like tidal tails in the
OHRs are signatures of major mergers. Such delicate features are often
extended and faint. Detecting low surface brightness emission in the OHRs is
thus critical to the measurements of the two parameters.
We adopt a low threshold (0.8 $\sigma_{\rm bkg}$) for source detection with
SExtractor in order to probe the extended faint structures around a target.
This may mis-identify some surrounding noise pixels as part of the target,
yielding a fuzzy appearance although the contribution to the total flux is
negligible (see Figure 1). The effect of noise on the structural properties in
the OHR would become significant for low surface brightness galaxies. Such a
detection technique based on surface brightness threshold has been widely used
in studies of galaxy morphologies (e.g., Abraham et al., 2003). However, the
physical size of detection of a given galaxy relies on the depth of the
imaging and redshift due to the cosmological dimming effect. Then the
detection completeness of tidal features needs to be addressed for the
evolution of a population of galaxies with dedicated features. Law et al.
(2007) adopted surface brightness threshold changing as $(1+z)^{3}$ to
compensate for the cosmological dimming effect and ensure a constant surface
brightness cut for all redshifts examined. This is beyond the scope of this
work.
Alternatively, the elliptical or circular Petrosian radius (Petrosian, 1976)
derived from the growth curve of a galaxy can be used to determine the
segmentation map of the galaxy. Taking the pixels within the Petrosian radius
as the segmentation map is able to correct the cosmological dimming effect
(Conselice, 2003; Lotz et al., 2004). Nevertheless, this treatment may
sacrifice the detection sensitivity of extended tidal features in the OHRs.
For instance, the typical tidal tails of merging galaxies can be as extended
as several tens of kpcs (Elmegreen et al., 2007) and beyond roughly 1$\sim$1.5
times Petrosian radius of the host galaxies.
Abraham et al. (2007) developed the so-called quasi-Petrosian segmentation
method and claimed that it enables the probe of extended low surface
brightness structures and the correction for the cosmological dimming effect.
Firstly, all pixels of a galaxy in the SExtractor preliminary segmentation map
are sorted in decreasing order of flux. The method will select pixels from the
highest end of flux to the lowest end until a pixel meets
$f_{i}=\eta(F_{i}/i)$, where $f_{i}$ is flux of pixel $i$, and $\eta$ is scale
factor. $F_{i}/i$ is the cumulative mean surface brightness. However, the
fuzzy sky pixels will still be included when $\eta$ is set lower for the
detection of faint tidal tails.
## 6\. CONCLUSION
We develop a new automatic method to quantify the structure in the outskirts
of galaxies, aiming at probing delicate features like tidal tails. Using the
isophote which encloses half the total light of a galaxy, the division of the
galaxy image into two sections (the IHR and OHR) is the key to our method. Two
parameters are introduced in the method: $A_{\rm o}$, which measures the
asymmetry of the OHR, and $D_{\rm o}$, which measures the deviation of the
intensity weighted centroids of the OHR from that of the IHR relative to the
effective radius. The galaxies with stronger disturbance in morphology are
expected to have higher $A_{\rm o}$ and $D_{\rm o}$. Moreover, the two
parameters are designed to be less affected by the central high surface
brightness section of galaxies, and thus sensitive to low surface brightness
features in the OHR. A sample of 764 galaxies with $\log(M/{\rm
M}_{\odot})>3\times 10^{10}$ and $0.35<z<0.9$ selected from the GEMS and
GOODS-S surveys is used to verify our method. For a comparison, we visually
classify morphologies for the sample using $HST$ $z_{850}$-band images,
following the usual classification scheme given in the literature. The
$z_{850}$ band corresponds to the rest-frame optical over the redshift range
examined here.
Our investigation shows that all sample galaxies fall on a sequence in the
$A_{\rm o}$-$D_{\rm o}$ space. The position along the sequence is in general
correlated with the degree of morphological disturbance. Galaxies with more
disturbed morphologies have higher $A_{\rm o}$ and $D_{\rm o}$ in a
statistical sense. The merging galaxies with tidal tails are well separated
from regular galaxies (spheroids and disks) along the sequence. The regular
galaxies are mostly with $D_{\rm o}<0.5$ and $A_{\rm o}<0.6$, following a
relation described by ${\rm log}\,A_{\rm o}=0.6\,{\rm log}\,D_{\rm o}$. The
galaxies with tidal tails are mostly with $A_{\rm o}>$0.5 and $D_{\rm o}$
ranging from 0.3 to 1.4. The criterion ${\rm log}\,A_{\rm o}>-1.6\,{\rm
log}\,D_{\rm o}-1.1$ is able to select all galaxies with tidal tails and most
major mergers (without apparent tidal tails).
The advantage of this selection is that major mergers with highly asymmetric
morphologies are nearly complete and about 87% of galaxies with regular
morphologies (spheroids and disks) are excluded at the same time. The left 13%
of regular galaxies tend to have higher $A_{\rm o}$ and $D_{\rm o}$
contributed by odd spiral arms, strong dust lanes, warped disks, or
contamination from neighboring sources. Low surface brightness galaxies suffer
more from the contamination by surrounding sources or background noise,
leaving $A_{\rm o}$ and $D_{\rm o}$ highly uncertain. Given that nearly 63% of
the major mergers (including those with tidal tails) can be selected by a
single cut in the $A_{\rm o}$-$D_{\rm o}$ space, out method can be used for a
relatively complete search for major mergers from large scale imaging surveys.
Compared with $CAS$ and Gini-$M_{20}$, our $A_{\rm o}$-$D_{\rm o}$ method is
able to provide a better separation between mergers and regular galaxies. In
particular, our method is unique in probing extended delicate features. In
Gini-$M_{20}$, galaxies with tidal tails mix together with regular galaxies.
And $CAS$ is also insensitive to galaxies of this kind. We point out that our
$A_{\rm o}$ is in a better position than $A$ in $CAS$ for probing asymmetric
structures of galaxies.
Next-generation deep imaging surveys carried out by the Large Synoptic Survey
Telescope (LSST) and Euclid will provide high-quality multi-band imaging data
over large areas of sky, allowing for detailed studies of the structure and
morphology of galaxies to address how galaxies form and evolve. Automatic
approaches like our $A_{\rm o}$-$D_{\rm o}$ method for morphological analysis
are essential in the era of big data.
We are grateful to the referee for the helpful suggestions that improved this
paper. We also thank Yinghe Zhao, Yewei Mao and Cong Ma for stimulating
discussions. This work is supported by National Basic Research Program of
China (973 Program 2013CB834900) and the Strategic Priority Research Program
”The Emergence of Cosmological Structures” of the Chinese Academy of Sciences
(Grant No. XDB09000000).
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|
arxiv-papers
| 2014-04-08T20:00:20 |
2024-09-04T02:50:00.873758
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Z. Z. Wen, X. Z. Zheng, F. X. An (PMO)",
"submitter": "Zhang-Zheng Wen",
"url": "https://arxiv.org/abs/1404.2281"
}
|
1404.2352
|
# Low-complexity Decoding is Asymptotically Optimal in the SIMO MAC
Mainak Chowdhury and Andrea Goldsmith
The authors are with the Department of Electrical Engineering, Stanford
University, Stanford, CA - 94305. Questions or comments can be addressed to
{mainakch,andreag}@stanford.edu. Parts of this work were presented at ISIT,
2013 and Allerton, 2013. This work is supported by the 3Com Corporation
Stanford Graduate Fellowship, the NSF Center for Science of Information
(CSoI): NSF-CCF-0939370 and by a gift from Cablelabs.
###### Abstract
A single input multiple output (SIMO) multiple access channel, with a large
number of transmitters sending symbols from a constellation to the receiver of
a multi-antenna base station, is considered. The fundamental limits of joint
decoding of the signals from all the users using a low complexity convex
relaxation of the maximum likelihood decoder (ML, constellation search) is
investigated. It has been shown that in a rich scattering environment, and in
the asymptotic limit of a large number of transmitters, reliable communication
is possible even without employing coding at the transmitters. This holds even
when the number of receiver antennas per transmitter is arbitrarily small,
with scaling behaviour arbitrarily close to what is achievable with coding.
Thus, the diversity of a large system not only makes the scaling law for coded
systems similar to that of uncoded systems, but, as we show, also allows
efficient decoders to realize close to the optimal performance of maximum-
likelihood decoding. However, while there is no performance loss relative to
the scaling laws of the optimal decoder, our proposed low-complexity decoder
exhibits a loss of the exponential or near-exponential rates of decay of error
probability relative to the optimal ML decoder.
###### Index Terms:
Spatial diversity, Multiuser detection, Convex programming
## I Introduction
Although the capacity-achieving techniques of superposition coding at the
encoder and joint decoding at the decoder [1] promise significantly higher
capacity for multiuser networks, such sophisticated coding schemes often
suffer from practical challenges. Thus the simpler orthogonal schemes to
separate users either in time (TDMA), space (sectorization in cellular
networks) or frequency (FDMA) have remained in widespread use. Moreover, the
capacity benefits of the optimal scheme over orthogonalizing schemes like
time-division have been shown to be negligible in some regimes such as under
asymptotically low-power or for asymptotically many users ([2],[3]).
In this work we consider a multiple access setting where an asymptotically
large number of transmitting users communicate in a rich scattering
environment with a single multi-antenna base station. We look at transmitting
schemes which do not employ coding, but instead transmit symbols from the BPSK
constellation and rely on the diversity inherent in a large system to achieve
reliability. Such a setting may model sensor networks or general distributed
networks with energy/processing power limitations at the transmitters (which
may preclude sophisticated coding schemes) and centralized receivers. A
similar setting was considered in the companion paper [4], where it was shown
that using the optimal maximum likelihood (ML) decoder, the decoding can be
made arbitrarily reliable for an arbitrarily small number of receiver antennas
per transmitter, provided that the number of transmitters is large enough. A
similar setup was also considered in [5] where a relaxation of the maximum
likelihood decoder was shown to be asymptotically reliable (i.e. the
probability of error vanishing to zero) in the number of transmitters,
provided that the number of receiver antennas is more than the number of
transmitters.
In this work, we analyze the same low-complexity decoder proposed in [5] and
modify it to handle underdetermined systems, i.e. systems where the number of
receiver antennas is less than the number of transmitter antennas. In
particular we consider the decoder obtained by expanding the search over
possible symbols to intervals instead of discrete points and then quantizing
the output of the interval search to the nearest constellation point. This
relaxation of the search over integer points to search over intervals allows
more efficient (polynomial time) decoding, but may not be unique in the regime
of underdetermined systems (because of the non-trivial null space for a wide
channel matrix). Thus the above procedure yields a non-singleton solution set
in general. We propose a family of randomization techniques and show that they
can return provably good estimates from this solution set. Henceforth we will
refer to this decoding technique as the randomized interval search and
quantize (r-ISQ) decoder. We obtain analytical bounds on the performance of
the r-ISQ decoder and show that reliable decoding (in a sense made precise in
the later sections) is possible in the asymptotic limit of a large number of
transmitters and receivers, with the per-transmitter number of receiver
antennas being held constant at _any_ arbitrary positive value. Using the same
techniques used in the proof however, the per-transmitter number of receive
antennas can be shown to be arbitrarily close to the theoretically optimal
scaling derived e.g. in [6],[4].
The rest of the paper is organized as follows. We first present the system
model and describe the optimal decoder and the r-ISQ decoder. We then describe
a bound on the error probability of this decoder. Asymptotic analyses of these
bounds are then presented.
## II System Model
Figure 1: System model
Our system model is depicted in Figure 1. We have an uplink system with $n$
single-antenna transmitters and an $m$ antenna receiver. The channel matrix
$\mathbf{H}\in\mathbb{R}^{m\times n}$ is chosen to model a rich scattering
environment, and the entries are assumed to be drawn i.i.d. from
$\mathcal{N}(0,1)$. The $k^{th}$ column of $\mathbf{H}$ is denoted as
$\mathbf{h_{k}}\in\mathbb{R}^{m}$. Thus
$\mathbf{H}=\begin{pmatrix}\mathbf{h_{1}}&\mathbf{h_{2}}&\ldots&\mathbf{h_{n-1}}&\mathbf{h_{n}}\end{pmatrix}.$
We also assume that the users do not cooperate with each other and that they
transmit symbols from the standard unit energy BPSK constellation. The
components of the noise at the receiver ($\nu$ ) are assumed to be i.i.d.
$\mathcal{N}(0,\sigma^{2})$. The received signal at the multi-antenna receiver
is then
$\displaystyle\mathbf{y}=\mathbf{Hx}+\bm{\nu}.$ (1)
The vector $\mathbf{x}\in\\{-1,+1\\}^{n}$, which consists of the transmitted
symbols from the $n$ users, is referred to as the _n-user codeword_ to
indicate that the receiver decodes the block of n-user constellation points
simultaneously. We further assume that the receiver has perfect channel state
information (CSI) and that the transmitters have no CSI.
## III Previous work and results
We now describe a few observations and results about the performance limits of
this system. These results, derived in [6], [4], assume that the receiver
employs ML decoding, i.e. it returns
$\displaystyle\mathbf{\hat{x}}=\operatorname{argmin}_{\mathbf{x}\in\\{-1,+1\\}^{n}}||\mathbf{y}-\mathbf{Hx}||^{2}.$
(2)
With this decoder it has been shown that in the limit of a large number of
transmitters the following holds:
###### Theorem 1.
Under ML decoding, there exists a $d>0$ such that for all sufficiently large
$n$, the probability of error in decoding the _n-user codeword_ satisfies
${P_{\mathrm{error}}}\leq 2^{-dn}.$
In other words the probability of decoding a particular user’s transmitted
symbol in error decreases exponentially with the number of users, even though
the users do not employ any coding across time.
A critical component in [6], [4] to achieve this asymptotic result is the use
of ML decoding. In this work we investigate whether we can achieve reliability
even with lower-complexity decoders. In particular, we ask whether an
efficient polynomial time decoder can realize an asymptotically vanishing
probability of error, as was the case with the ML decoder.
A common approach to relax hard combinatorial optimization problems (such as
ML decoding) is the technique of expanding the search space from discrete
points to intervals or regions [7]. Motivated by this idea, we consider a
convex relaxation of the maximum likelihood decoder search as follows:
$\displaystyle\mathbf{\hat{x}}=\operatorname{sgn}(\operatorname{argmin}_{\mathbf{x}\in[-1,+1]^{n}}||\mathbf{y}-\mathbf{Hx}||^{2}).$
(3)
In the above $\operatorname{sgn}(\mathbf{x})$ for
$\mathbf{x}\in\mathbb{R}^{n}$ refers to the vector obtained by the
coordinatewise application of the signum function defined below for a scalar
$x$.
$\operatorname{sgn}(x)=\begin{cases}1&\text{ if $x>0$}.\\\ -1&\text{
otherwise}.\end{cases}$
The modified decoder in (3) expands the search for a valid _n-user codeword_
to the interval $[-1,1]$ per dimension and then quantizes it to integer values
afterwards, hence we call it an ISQ decoder. This idea of relaxing an integer
program to a box-constrained program is a well known technique and has been
studied in different settings, e.g. in [8], [9], where different asymptotic
properties of this decoder are established. While some of the results
(especially the characterization of the null space of Gaussian random matrices
[8], and the behaviour of approximate message passing (AMP) type algorithms
[9] with the box constraints) from these works do give us insights into the
expected behaviour of the box constrained decoder in some regimes (e.g.
$\frac{m}{n}>0.5$ with BPSK transmissions), the regime of an arbitrarily small
fraction of the per-transmitter number of receiver antennas is still not fully
characterized. In fact, for the AMP decoder, it can be shown that if
$\frac{m}{n}<0.5$, the number of symbol errors in the decoded block would be
$\Theta(n)$, i.e. the number of incorrectly decoded symbols is linear in the
number of transmitting users. We consider a slight modification of the box-
constrained decoder and with this modification, are able to show asymptotic
reliability in a sense made precise below.
Note that in the regime where the channel matrix is underdetermined (i.e.
$m<n$ ) the above procedure may not give a unique solution. If an _n-user_
codeword $\hat{\mathbf{x}}$ is a solution, then any codeword of the form
$\hat{\mathbf{x}}+\bm{Z\beta}$ is also a solution. Here $\mathbf{Z}$ is a
basis for the right null space of $\mathbf{H}$, i.e. $\mathbf{Z}$ is such that
$\mathbf{H}\mathbf{Z}=\mathbf{0}$ and $\bm{\beta}\in\mathbb{R}^{(1-\alpha)n}$.
Thus the ISQ decoder, in this case, cannot uniquely specify a solution by
itself. It would, in general, give an affine subspace as a solution. In order
to specify a unique solution, we propose a randomization step (randomized ISQ
or r-ISQ). Specifically, we propose a family of distributions and show that,
for estimates drawn according to this general family of distributions, we can
achieve reliability in a sense which is made precise in the following
sections. We further show that it is possible to sample efficiently from a
member of this family of distributions. For specificity let us consider the
following decoder.
$\displaystyle\text{r-ISQ:}\,\mathbf{\hat{x}}=\operatorname{sgn}(\operatorname{argmin}_{\mathbf{x}\in
S}||\mathbf{x_{r}}-\mathbf{x}||_{\infty}).$
In the above
$S=\\{\mathbf{x}:\mathbf{x}\in\operatorname{argmin}_{\mathbf{z}}||\mathbf{y}-\mathbf{Hz}||\\}$,
and $\mathbf{x_{r}}\sim\operatorname{Unif}([-1,1]^{n})$. Since both
$\mathbf{x_{r}}$ and $S$ can be computed efficiently (in polynomial time, for
reasons discussed in later sections), we see that there is an efficient
algorithm to achieve asymptotic reliability without employing coding.
However there is a performance hit, relative to ML decoders, when we move to
r-ISQ decoders. This is in terms of the rate of decay of the error probability
with the number of transmitting users. Although the error probability seen by
each user vanishes to zero, we do not have an upper bound for the probability
of having at least one symbol error in the _n-user_ codeword, which is in
contrast to ML, where the block error probability decays exponentially. This
lack of exponential decay with the simpler decoder is primarily due to the
self interference due to the search over intervals. Thus, in particular, we do
seem to lose the exponential fall off in the probability of error that is
achieved with the ML decoder. However, the number of symbol errors in the
decoded block in the asymptotic limit is at most sub linear, i.e. the
probability that a constant fraction of the transmitter symbols are
incorrectly decoded can be made arbitrarily small for a large enough system
size. Defining $P_{e}^{k^{\prime}}$ to be the probability of incorrectly
decoding at least $k^{\prime}n$ out of $n$ transmitted symbols, the following
states a bound on $P_{e}^{k^{\prime}}$.
###### Theorem 2.
Under r-ISQ decoding, for $m=\alpha n$ , $\alpha>0$ and any constant $k>0$,
there exists a $d>0$ such that for all sufficiently large $n$,
$P_{e}^{k}\leq 2^{-dn\log n}.$
We mention that the same proof techniques used to establish the above result
can also be used to get sharper bounds, i.e. for $k=\frac{1}{n^{\gamma}}$ for
some $0<\gamma<1$. Thus the per-user error probability is asymptotically less
than $\frac{1}{n^{\gamma}}$. Also, while in this section we focus on the case
where the ratio ($\alpha$) of the number of receiver antennas to the number of
transmitter antennas is constant, we point out that the same techniques
continue to hold even for
$\alpha_{n}=\frac{1}{(\log n)^{\xi}},\xi<1.$
By making $\xi$ close to $1$ we see that we can come arbitrarily close to the
optimal scaling established for the ML decoder in [6]. Thus we see that by
exploiting the _diversity_ (richness in the scattering environment), one can
not only get arbitrarily reliable communication in different asymptotic
regimes (in this case for a large number of transmitters) without employing
coding or ML decoders, but can also achieve optimal scaling for the per
transmitter number of receiver antennas.
We now use the bound in Theorem 2 to bound the probability of symbol error
seen by each transmitter.
###### Theorem 3.
The probability of error with the r-ISQ decoder seen by any transmitting user
vanishes in the limit of an asymptotically large number of transmitters, with
the per-transmitter number of receive antennas being any constant $\alpha>0$.
The remainder of this paper discusses the proofs of Theorems 2 and 3 and
points out suitable generalizations using the same proof techniques.
## IV An Upper Bound on the Decoding Error
We first present an upper bound on the probability of decoding error. Most of
the steps described in this are similar to what was used in [5], modified to
take into account the fact that there is a non-trivial null space (so the
solution set in general may not be unique). We look at the pairwise error
probability of mistaking the transmitted codeword with one differing in
$k^{\prime}n$ symbols. For (3), the probability of mistaking a codeword
$\mathbf{x_{0}}$ for another differing in $i$ symbol positions is given by
$\displaystyle P_{e,\mathbf{b_{i}}}$ $\displaystyle\leq$ $\displaystyle
Q\left(\min_{{\mathbf{x}}:\operatorname{supp}(\operatorname{sgn}(\mathbf{x})-\mathbf{x_{0}})=\mathbf{b_{i}}}\frac{||\mathbf{H}(\mathbf{x}-\mathbf{x_{0}})||}{2\sigma}\right).$
(4)
Here $Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-x^{2}/2}dx$,
$\mathbf{b_{i}}$ is a vector of size $i$ whose entries are positions where the
codewords differ (arranged in increasing order), and $\mathbf{b_{i}}(j)$ is
the $j^{th}$ symbol position where the codewords differ. We point out here
that $\mathbf{b_{i}}$ has a one-to-one correspondence with a subset of
$\\{1,\ldots,n\\}$ of cardinality $i$. $||\mathbf{x}||_{0}$ refers to the
number of non-zero entries in $\mathbf{x}$. $\operatorname{supp}(\mathbf{x})$
refers to the support (i.e. locations of the non-zero entries of vector
$\mathbf{x}$).
Note that the error probability above is independent of which $\mathbf{x_{0}}$
is chosen, when averaged over the distribution of $\mathbf{H}$. Hence,
choosing $\mathbf{x_{0}}=-\mathbf{1}$, we note that the last expression can be
rewritten as follows.
$\displaystyle P_{e,\mathbf{b_{i}}}\leq Q\left(\min_{\begin{subarray}{c}1\leq
c_{j}\leq 2\,\forall j\in\mathbf{b_{i}}\\\ 0\leq c_{j}\leq 1\,\forall
j\in\mathbf{b_{i}}^{c}\end{subarray}}\frac{||\sum_{j=1}^{n}c_{j}\mathbf{h_{b_{i}(j)}}||}{2\sigma}\right)$
$\displaystyle\overset{(a)}{\leq}\frac{1}{2}\exp{\left(-\min_{\begin{subarray}{c}1\leq
c_{j}\leq 2\,\forall j\in\mathbf{b_{i}}\\\ 0\leq c_{j}\leq 1\,\forall
j\in\mathbf{b_{i}}^{c}\end{subarray}}\frac{||\sum_{j=1}^{n}c_{j}\mathbf{h_{b_{i}(j)}}||^{2}}{8\sigma^{2}}\right)}$
(5)
where (a) follows because $Q(x)\leq\frac{1}{2}\exp(\frac{-x^{2}}{2})$. We
observe now that (5) averaged over the channel realizations is independent of
the particular subset of symbols that are decoded in error and depends only on
the size $i$ of such a subset. Let’s call this averaged probability of error
$P_{i}$. Thus we have
$\displaystyle P_{i}\triangleq
E_{\mathbf{H}}e^{\left(-\min_{\begin{subarray}{c}1\leq c_{j}\leq
2\,\forall\,j\in\\{1,\ldots,i\\}\\\ 0\leq c_{j}\leq 1\,\forall
j\in\\{i+1,\ldots,n\\}\end{subarray}}\frac{||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2}}{8\sigma^{2}}\right)}.$
If $P_{e}^{k^{\prime}}$ is the probability of error of decoding at least
$k^{\prime}n$ transmitter symbols incorrectly, and $S_{i}$ refers to the set
of all vectors representing subsets of size $i$ from $\\{1,\ldots,n\\}$, then
a union bound for the error probability is
$\displaystyle P_{e}^{k^{\prime}}$ $\displaystyle\leq$
$\displaystyle\sum_{k^{\prime}n\leq i\leq n}\sum_{\mathbf{b}\in
S_{i}}\frac{1}{2}P_{i}$ (6) $\displaystyle\leq$
$\displaystyle\sum_{k^{\prime}n\leq i\leq n}\binom{n}{i}\frac{1}{2}P_{i}.$ (7)
Note that, by the symmetry of the system, the probability of error $P_{e}$
seen by each transmitting user is upper bounded by
$P_{e}\leq k^{\prime}+P_{e}^{k^{\prime}}.$
We show that for any small $k^{\prime}$, there exists a large enough system
size for which $P_{e}^{k^{\prime}}$ becomes exponentially small, even with a
convex decoder of much lower complexity. This will establish Theorem 3.
## V Asymptotic analysis of the upper bound
We first prove bounds on the exponent appearing in the bound for
$P_{e,\mathbf{b_{i}}}$ in (5). Specifically, we look at (ignoring a constant
scaling of $8\sigma^{2}$)
$\min_{\begin{subarray}{c}1\leq c_{j}\leq 2\,\forall j\in\\{1,\ldots,i\\}\\\
0\leq c_{j}\leq 1\,\forall
j\in\\{i+1,\ldots,n\\}\end{subarray}}||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2}.$
Before we describe the proof, we define the ($\epsilon,\bm{\delta}$)-grid
inside the hypercube $[-1,+1]^{n}$, for some $0<\epsilon<0.25$. This grid is
simply the set of points
$\mathcal{G}_{n,\epsilon,\bm{\delta}}=\\{\mathbf{x}:x_{i}\bmod\epsilon=\delta_{i},-1\leq
x_{i}\leq 1\,\,\forall i\\}.$
As an illustration, for $\bm{\delta}=\bm{0}$, it may be rewritten as
$\mathcal{G}_{n,\epsilon,\bm{\delta}}=\\{-1,-1+\epsilon,\ldots,1-\epsilon,1\\}^{n}.$
if $\frac{1}{\epsilon}\in\mathbb{N}$.
We now introduce the $(\epsilon,\bm{\delta})$-ISQ decoder, so named because it
replaces the interval search in the ISQ decoder by an
($\epsilon,\bm{\delta}$)-grid search:
$\displaystyle\text{$(\epsilon,\bm{\delta})$-ISQ:}\,\mathbf{\hat{x}}_{\epsilon,\bm{\delta}}=\operatorname{sgn}(\operatorname{argmin}_{\mathbf{x}\in\mathcal{G}_{n,\epsilon,\bm{\delta}}}||\mathbf{y}-\mathbf{Hx}||^{2}).$
The $(\epsilon,\bm{\delta})$-grid error probabilities are defined similar to
the definitions for the ISQ decoder in the previous section, and are indicated
by an $\epsilon,\bm{\delta}$ subscript. We now collect some observations about
the grid error probabilities and use a union bounding argument for
$P_{e,\mathbf{b_{i}},\epsilon,\bm{\delta}}$. Some of these results for the
grid error probabilities have already been derived in [5] and have been
reproduced here for completeness and continuity of presentation. We require
the following lemma about the negative of the exponent in the error
probability $P_{e,\mathbf{b_{i}},\epsilon,\bm{\delta}}$:
$\min_{\begin{subarray}{c}c_{j}\in[1,2],c_{j}\bmod\epsilon=\delta_{j}\,\forall
j\in\\{1,\ldots,i\\}\\\ c_{j}\in[0,1],c_{j}\bmod\epsilon=\delta_{j}\,\forall
j\in\\{i+1,\ldots,n\\}\end{subarray}}||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2}.$
###### Lemma 1.
For any $i>k^{\prime}n$, there exists an $n_{0}$ and an $a>0$, such that for
all $n>n_{0}$,
$P(||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2}<an\log n)\leq\exp(-an\log n).$
###### Proof.
We can show this using Markov’s inequality. Let $a_{1}=\frac{\alpha}{4}.$ Then
$\displaystyle P(||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2}<a_{1}n\log n)$ (12)
$\displaystyle=$ $\displaystyle
P(\exp(-t||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2})>\exp(-ta_{1}n\log n))$
$\displaystyle\overset{(a1)}{\leq}\exp(ta_{1}n\log
n)E\exp(-t||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2})$
$\displaystyle\overset{(a2)}{=}\exp(ta_{1}n\log
n)(1+2t\sum_{j}c_{j}^{2})^{(-\alpha n/2)}$
$\displaystyle\overset{(b)}{\leq}\exp(ta_{1}n\log
n)(1+2tk^{\prime}n)^{(-\alpha n/2)}$
$\displaystyle\overset{(c)}{\leq}\exp(-\tilde{a}n\log n)\,\,\text{ for large
enough n}.$
In the above $(a1)$ follows from Markov’s inequality, $(a2)$ follows from the
moment generating function of a chi-squared random variable, $(b)$ follows
from the fact that for at least $k^{\prime}n$ errors,
$\sum_{j}c_{j}^{2}\geq k^{\prime}n,$
and $(c)$ follows by choosing $t=1$, and defining e.g.
$\tilde{a}=\frac{\alpha}{4}.$ Defining
$a=\min(a_{1},\tilde{a})=\frac{\alpha}{4}$, we get the claim in the lemma.
Thus the claim is established for $\mathbf{H}$ with $\mathcal{N}(0,1)$
entries.
∎
By observing that for a positive r.v., $P(x<d_{0})<\exp(-d_{0})$ implies
$\displaystyle E(\exp(-x))$ (13) $\displaystyle\leq$
$\displaystyle\exp(-d_{0})+(1-\exp(-d_{0}))\exp(-d_{0})$ (14)
$\displaystyle\leq$ $\displaystyle 2\exp(-d_{0}),$ (15)
we get that
$\displaystyle P_{i,\epsilon,\bm{\delta}}$ $\displaystyle\leq$ $\displaystyle
E(\exp(-||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2}))$ (16) $\displaystyle\leq$
$\displaystyle\exp(-an\log n)\,\,\text{ for large enough n}.$ (17)
The probability of the event that there are at least $k^{\prime}n$ symbols
that are decoded incorrectly can then be union bounded as follows.
$\displaystyle P_{e,\epsilon,\bm{\delta}}^{k^{\prime}}$ $\displaystyle\leq$
$\displaystyle\sum_{i=k^{\prime}n}^{n}\binom{n}{i}\left(\frac{1}{\epsilon}\right)^{n}P_{i,\epsilon,\bm{\delta}}$
$\displaystyle\overset{(d1)}{\leq}$ $\displaystyle n2^{n(\max_{k^{\prime}n\leq
i\leq n}H_{2}(\frac{i}{n})-\log(\epsilon)-a\log n)}\,\,\text{ for }$ $a>0$ and
large enough $n$ $\displaystyle\overset{(d2)}{\leq}$ $\displaystyle 2^{-an\log
n}\,\,\text{for a large enough n}.$
In the above,
$H_{2}(x)=-x\log x-(1-x)\log(1-x).$
$(d1)$ follows by noting that
$\binom{n}{i}\leq 2^{H_{2}(i/n)}$
and $(d2)$ follows from the observation that $H_{2}(\cdot)$ is bounded above
by a constant. We now note that by introducing an arbitrary distribution
$f(\bm{\delta})$ on $\bm{\delta}$, i.e. randomizing the grid, there would be a
distribution induced on $\bm{\hat{x}}_{\epsilon,\bm{\delta}}$. Let’s call that
$\hat{f}(\bm{\hat{x}}_{\epsilon,\bm{\delta}})$. Thus statements about the
probability of error associated with $\mathbf{\hat{x}}_{\epsilon,\bm{\delta}}$
would continue to hold even for samples $\mathbf{y}$ drawn from
$\hat{f}(\mathbf{y})$. Note that sampling from this distribution may still be
of exponential complexity. Let’s call this decoder the
r-$(\epsilon,\bm{\delta})$ ISQ decoder, where r stands for randomized.
We now relate the solution from the search over the randomized
$(\epsilon,\bm{\delta})$-grid $\mathcal{G}_{n,\epsilon,\bm{\delta}}$ (i.e. the
output of the r-$(\epsilon,\bm{\delta})$ ISQ decoder) to the solution
($\hat{\mathbf{x}}$) of the r-ISQ decoder. Note that, in general, the ISQ
decoder will not be unique and there is always an uncertainty due to the right
null space of $\mathbf{H}$. Thus if the objective function in the ISQ decoder
attains its infimum at $\hat{\mathbf{x}}$, then it will also attain the same
infimum at all points of the following solution set
$S=\\{\mathbf{x}:\mathbf{x}=\hat{\mathbf{x}}+\mathbf{Z}\bm{\beta},\mathbf{HZ}=\bm{0},\bm{\beta}\in\mathbb{R}^{(1-\alpha)n}\\}.$
We show next that a certain randomized choice of solution from this solution
set will be “good”. Before that however, we introduce some notation. Let the
projection of any vector $\mathbf{y}$ on any set $A$ be defined by
$\displaystyle P_{A}(\mathbf{y})$ $\displaystyle=$
$\displaystyle\operatorname{argmin}_{\mathbf{x}\in
A}||\mathbf{y}-\mathbf{x}||_{\infty}.$ (20)
Note that this can be computed efficiently (using interior point algorithms)
if $A$ is an affine subspace. Thus given
$\mathbf{\hat{x}}_{\epsilon,\bm{\delta}}$, one can compute
$P_{S}(\mathbf{\hat{x}}_{\epsilon,\bm{\delta}})$ efficiently. This is simply
the projection of the solution of the r-$(\epsilon,\bm{\delta})$-ISQ decoder
on the solution space $S$ of the ISQ decoder. Before we proceed we observe a
certain property that this projection enjoys.
###### Lemma 2.
There is at least one point of the solution set $S$ of the ISQ within the
$\epsilon-l_{\infty}$ ball around $\hat{\mathbf{x}}_{\epsilon,\bm{\delta}}$,
i.e.,
$||\mathbf{\hat{x}}_{\epsilon,\bm{\delta}}-P_{S}(\mathbf{\hat{x}}_{\epsilon,\bm{\delta}})||_{\infty}\leq\epsilon$.
###### Proof.
We can show this by contradiction. If the claim is false, we would have that
the function $g(\mathbf{x})=||\mathbf{y}-\mathbf{H}\mathbf{x}||^{2}$ is
strictly convex over the hypercube
$\\{\mathbf{x}:||\mathbf{x}-\mathbf{x}_{\epsilon,\bm{\delta}}||_{\infty}\leq\epsilon\\}$,
with $S$ lying totally outside the hypercube. By observing that, in such a
case, one of the vertices will have a smaller value for $g(\mathbf{x})$ than
$g(\mathbf{\hat{x}}_{\epsilon,\bm{\delta}})$, we arrive at a contradiction. ∎
Figure 2: $(\epsilon,\bm{\delta})$-grid for $n=3,\alpha=2/3$ (dotted grid with
gray grid points) for a $\mathbf{H}\in\mathbb{R}^{2\times 3}$
Thus projecting to the solution space $S$ does not change any entry of the
vector $\mathbf{\hat{x}}_{\epsilon,\bm{\delta}}$ by more than $\epsilon$.
Since $\epsilon<0.25$, if
$|{\mathbf{\hat{x}}}_{\epsilon,\bm{\delta},i}|>\epsilon$, the sign of the
corresponding entry of $P_{S}({\mathbf{\hat{x}}}_{\epsilon,\bm{\delta}})$
would also be the same as that of
$\mathbf{\mathbf{\hat{x}}}_{\epsilon,\bm{\delta}}$. The remaining part of the
proof is to establish that the sampling of the point
$\mathbf{y}=P_{S}({\mathbf{\hat{x}}}_{\epsilon,\bm{\delta}})$ according to
distribution $\hat{f}_{S}(\mathbf{y})$ can be done efficiently, i.e. with
polynomial complexity (this, in general, is not true for arbitrary
multivariate distributions, i.e. [10]). This, together with the fact that the
$|{\mathbf{\hat{x}}}_{\epsilon,\bm{\delta},i}|$ is less than $\epsilon$ at
most at a sublinear number of coordinates $i$ with overwhelming probability,
establishes the fact that $P_{S}({\mathbf{\hat{x}}}_{\epsilon,\bm{\delta}})$
differs from $\mathbf{x_{0}}$ in at most a sublinear number of positions with
high probability.
We now relate this randomized projection to the solution of the r-ISQ decoder.
This simply takes the affine subspace that is a solution to the ISQ decoder
and projects a random point inside the hypercube on it. Thus
$\displaystyle\text{r-ISQ:}\mathbf{\hat{x}}=P_{S}(\mathbf{x_{r}}),\mathbf{x_{r}}\sim\text{Unif}([-1,1]^{n})$
(21)
where $S$ is the solution set of the ISQ decoder. Note that since $S$ is
affine and the sampling is uniform, both can be done efficiently. We now show
that the estimate from this decoder is equal to that of the
r-$(\epsilon,\bm{\delta})$-ISQ decoder for a particular choice of
$f(\bm{\delta})$. This follows from the observation that any distribution on
$\mathbf{x_{r}}$ would induce a distribution on $S$. This distribution belongs
to the family of distributions of the form $\hat{f}_{S}$ induced by a
distribution $f(\bm{\delta})$ on $\bm{\delta}$ because the mapping
$P_{S}(\mathbf{\hat{x}}_{\epsilon,\bm{\delta}})$ from $\bm{\delta}$ to $S$ is
onto (surjective). Also, by following the same union bounding technique used
to bound $P_{e,\epsilon,\bm{\delta}}^{k^{\prime}}$, we get that, for any
$k^{{}^{\prime\prime}}>0$, the probability that
$\hat{\mathbf{x}}_{\epsilon,\bm{\delta}}$ has greater than or equal to
$k^{{}^{\prime\prime}}n$ entries that are close to zero, (i.e. either
$\delta_{i}-\epsilon,\delta_{i},$ or $\delta_{i}+\epsilon$) is upper bounded
by $\exp(-d_{1}n\log n)$, for some $d_{1}>0$. Let
$d_{2}=\min(d_{1},a).$
Thus we conclude that for a large enough $n$, with probability at least
$1-2\exp(-d_{2}n\log n)$, the signs of $P_{S}(\mathbf{x_{r}})$ will match the
signs of $\mathbf{x_{0}}$ (i.e. the correct $n$-user codeword) in at least
$(1-k^{{}^{\prime\prime}}-k^{{}^{\prime}})n$ positions. By choosing
$k^{{}^{\prime}}$ and $k^{{}^{\prime\prime}}$ small enough we see that the
number of mismatches is sublinear in the number of transmitting users with
overwhelming probability. The proof of Theorem 2 is now complete. ∎
To prove Theorem 3, we note that, by the symmetry of the system, the error
probability $P_{e}$ seen by each transmitter is the same. Thus given any
target symbol error rate (SER) $\epsilon_{1}>0$, we can choose
$k<\epsilon_{1}/2$ in Theorem 2. Then there exists an $n_{0}$ depending on $k$
such that
$P_{e}^{k}\leq 2^{-dn\log n}\leq\frac{\epsilon_{1}}{2}\,\,\forall n>n_{0}.$
Then, assuming independent (both temporally and spatially) channel
realizations, we get that the expected number of errors $E(N_{e})$ seen by
_all_ transmitters in $t$ single shot transmissions satisfies
$\displaystyle E(N_{e})$ $\displaystyle\leq$ $\displaystyle
nkt+n(1-k)P_{e}^{k}t$ (22) $\displaystyle\text{or }\frac{E(N_{e})}{t}$
$\displaystyle\leq$ $\displaystyle n\epsilon_{1}\,\,\text{for $n$ large
enough.}$ (23)
Dividing both sides by $n$ we get that the per-transmitter error probability
$P_{e}$ can be made smaller than $\epsilon_{1}$ for a large enough $n$. The
proof is now complete.
We now comment on some of the differences from the analysis in [5]. One
complication is introduced by the fact that the null space is not empty. Thus
the properties of the null space will affect the behaviour of the resulting
estimate. In particular, as seen in [9], there does exist a “bad solution”, in
the sense that it differs from the correct solution in $\Theta(n)$ symbols.
However, we are able to establish that in order to reliably “clean up” the
solution from the ISQ decoder, a random sample would be sufficient. Moreover,
it is possible to sample efficiently from this solution set. Thus although
computing $\hat{\mathbf{x}}_{\epsilon,\bm{\delta}}$ (and thereby
$P_{S}(\hat{\mathbf{x}}_{\epsilon,\bm{\delta}})$) has exponential complexity,
sampling from $\hat{f}_{S}$ does not. We then propose a simple randomized
solution, and show that this belongs to the family of distributions just
mentioned. Note that both the sampling and the projection operation can be
done efficiently in polynomial time. We show that the distribution on the
resulting estimate $\tilde{f_{S}}(\bm{\tilde{y}})$ is within the family of
distributions
$\\{\hat{f}_{S}(\bm{y}):\bm{y}=P_{S}(\hat{\mathbf{x}}_{\epsilon,\bm{\delta}}),\bm{\delta}\sim
f(\bm{\delta})\\}$. This concludes the proof. ∎
## VI Extensions
In this section, we point out several extensions to the same ideas that we
discussed so far, for more general systems. In particular we focus on the more
general kinds of fading distribution, more general constellations, finite
blocklength constellations, and faster decay of the probability of error with
$n$. We also indicate how the Theorem 3 holds not only for a constant
$\alpha>0$, but also for an asymptotically vanishing sequence of $\alpha_{n}$,
i.e.
$\alpha_{n}=\frac{1}{(\log n)^{\xi}}\,\,\textrm{ for }0<\xi<1.$
### VI-A General fading distribution
In the derivation of the proof so far, we assumed i.i.d. $\mathcal{N}(0,1)$
fading for the channel coefficients. We now show how they may be generalized
to a much wider class of fading distributions, namely any distribution
satisfying the Berry-Esseen bounds on the convergence of the cdf of normalized
sums to the gaussian distribution function.
Before we proceed, we state one version of this lemma.
###### Lemma 3.
Berry-Esseen: Given $N$ i.i.d. random variables $U_{1},\ldots,U_{N}$, with
$E[|U_{i}|^{3}]\leq\infty$ and $E[|U_{i}|^{2}]=\sigma^{2}$, the following
holds for all $x$:
$\left|P\left(\frac{\sum_{i=1}^{N}U_{i}}{\sqrt{N}\sigma}\leq
x\right)-\Phi(x)\right|\leq\frac{K\rho}{\sigma^{3}\sqrt{N}}$
In the above $\Phi(x)=1-Q(x)$ is the cumulative distribution function of a
standard normal random variable and $K>0$ is a constant.
With the above we note that the term appearing in (12) can be expressed as
follows:
$\displaystyle E\left(e^{-t||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2}}\right)$
$\displaystyle=$ $\displaystyle
E\left(e^{-tn\left|\left|\frac{\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}}{\sqrt{n}}\right|\right|^{2}}\right)$
(24) $\displaystyle\overset{(a)}{=}$ $\displaystyle
E\left(e^{-tn\left(\sum_{i=1}^{\alpha
n}\left(\sum_{j=1}^{n}c_{j}H_{i,j}/\sqrt{n}\right)^{2}\right)}\right)$ (25)
$\displaystyle\overset{(b)}{=}$ $\displaystyle
E\left(e^{-tn\left(\sum_{j=1}^{n}c_{j}H_{1,j}/\sqrt{n}\right)^{2}}\right)^{\alpha
n}.$ (26)
Here $(a)$ follows by decomposing $||\mathbf{y}||^{2}=\sum_{i=1}^{\alpha
n}y_{i}^{2}$, and $(b)$ follows by noting that $H_{i,j}$ are i.i.d. . We now
observe that an upper bound on the expectation can be written as
$\displaystyle
E\left(-tn\left(\sum_{j=1}^{n}c_{j}H_{1,j}/\sqrt{n}\right)^{2}\right)$
$\displaystyle\overset{(c)}{=}$
$\displaystyle\int_{y=-\infty}^{\infty}e^{-tny^{2}}p(y)dy$ (27)
$\displaystyle\overset{(d)}{=}$
$\displaystyle-\int_{u=-\infty}^{\infty}-2unte^{-tnu^{2}}P(Y\leq u)du$ (28)
$\displaystyle\overset{(e)}{\leq}$
$\displaystyle\int_{u=-\infty}^{\infty}2unte^{-tnu^{2}}\Phi(u)du+\int_{u=-\infty}^{\infty}2unte^{-tnu^{2}}\frac{K\rho}{\sigma^{3}\sqrt{n}}du$
(29) $\displaystyle\overset{(f)}{\leq}$
$\displaystyle(1+2nt)^{-1/2}+C/\sqrt{n}\leq(C+t^{-1/2})n^{-1/2}.$ (30)
In the above $(c)$ follows by defining $p(y)$ to be the density function of
$Y=(\sum_{j=1}^{n}c_{j}H_{1,j}/\sqrt{n}),$ $(d)$ follows by application of
integration by parts, $(e)$ follows from the bound $P(Y\leq
u)\leq\Phi(u)+\frac{K\rho}{\sigma^{3}\sqrt{n}},$ and $(f)$ follows from
actually evaluating the first term and using a trivial bound on the second
term i.e.
$\int_{u=-\infty}^{\infty}2unte^{-tnu^{2}}\frac{K\rho}{\sigma^{3}\sqrt{n}}du\leq
2\int_{u=0}^{\infty}2unte^{-tnu^{2}}\frac{K\rho}{\sigma^{3}\sqrt{n}}du.$
Thus for any fixed $t>0$, we have that
$\displaystyle(E(-tn(\sum_{j=1}^{n}c_{j}H_{1,j}/\sqrt{n})^{2}))^{\alpha
n}\leq((C+t^{-1/2})n^{-1/2})^{\alpha n}\leq e^{\frac{-\alpha n\log
n}{4}}\textrm{ for a large enough $n$.}$ (31)
This, together with the expression in (12), choosing $t=1$ gives us the
precise asymptotic bound in (12). From there on, the remaining claims are the
same.
### VI-B General (i.e. non-BPSK) constellations
For general constellations, the main ideas in the proof remain quite similar,
except that the decoder and the proof analysis needs to be slightly different.
We first present the generalized decoder and then indicate how the ideas used
to establish the result for the BPSK constellation also extend naturally to
more general constellations. Let us refer to such a constellation as
$\mathcal{M}=\\{m_{1},m_{2},\ldots,m_{N}\\}$ where $N$ is the number of
constellation points.
We set up some notation first before describing the decoder.
###### Definition 1.
The quantizer $\operatorname{Q}$ to a constellation point projects any point
$x$ to the nearest constellation point, i.e.
$\operatorname{Q}(x)=\operatorname{argmin}_{m_{i}\in\mathcal{M}}||x-m_{i}||_{2}.$
For simplicity of presentation, $||.||$ refers to the 2-norm unless specified
otherwise. For a vector $\mathbf{x}$ of constellation points,
$\operatorname{Q}(\mathbf{x})$ projects each coordinate of $\mathbf{x}$ to the
nearest constellation point in $\mathcal{M}$, i.e.
$(\operatorname{Q}(\mathbf{x}))_{i}=\operatorname{Q}(x_{i}).$
Given this notation, the ISQ decoder defined earlier, is equivalent to
$\displaystyle\mathbf{\hat{x}}$ $\displaystyle=$
$\displaystyle\operatorname{Q}(\operatorname{argmin}_{||\mathbf{x}||_{\infty}\leq\max_{m_{i}\in\mathcal{M}}||m_{i}||}||\mathbf{y}-\mathbf{Hx}||^{2}).$
(32)
This reduces to (3) when $\mathcal{M}=\\{-1,+1\\}$. The definition of the
randomized ISQ decoder follows along very similar lines. We pick a point
$\mathbf{x_{r}}$ randomly from an uniform distribution over the set
$B_{n}=\\{\mathbf{x}:||\mathbf{x}||_{\infty}\leq\max_{m_{i}\in\mathcal{M}}||m_{i}||\\}.$
Then we project $\mathbf{x}$ on the (possibly non singleton) solution set
$S=\operatorname{argmin}_{\mathbf{x}:||\mathbf{x}||_{\infty}\leq\max_{m_{i}\in\mathcal{M}}||m_{i}||}||\mathbf{y}-\mathbf{Hx}||^{2}$
of the ISQ decoder, i.e. we return
$\hat{\mathbf{x}}=\operatorname{Q}(P_{S}(\mathbf{x_{r}})),\mathbf{x_{r}}\sim\operatorname{Unif}(B_{n}).$
In the above, the projection operation is the same as that introduced in (20).
We can show that with the above decoders, the same conclusions that we derived
in Theorem 3 continue to hold. The proof however needs some generalization of
some of the ingredients involved in the proof. We point out such
generalizations in the following.
A critical step towards obtaining Theorem 3 was the use of the
$r-(\epsilon,\bm{\delta})$-grid detector. For a general constellation
$\mathcal{M}$, define the scalar grid as (using $B_{n}$ as we defined it
earlier)
$\mathcal{G}_{1,\epsilon}=\\{\\{g_{1},g_{2},\ldots,g_{N_{\epsilon}}\\}:\textrm{
For any $x\in B_{1}$},||x-g_{i}||_{\infty}\leq\epsilon\textrm{ for some }i;\\\
||g_{i}-g_{j}||_{\infty}>\epsilon\,\forall i\neq
j;||g_{i}||_{2}\leq\max_{m_{j}\in\mathcal{M}}||m_{j}||_{2}\,\forall i\\}.$
Note that $\mathcal{G}_{1,\epsilon}$ is not unique. We illustrate possible
scalar grids for the BPSK constellation considered earlier and a 2D
constellation (e.g. 4-PSK) in Fig. 3.
-11 (a) $\mathcal{G}_{1,\epsilon}$ (red dots) for $\epsilon=1/3$ for BPSK (blue dots)
-11 (b) $\mathcal{G}_{1,\epsilon}$(red dots) for $\epsilon=1/3$ for $4$-PSK (blue dots)
Figure 3: Possible scalar grids $\mathcal{G}_{1,\epsilon}$ for different
constellations
We define the perturbed grid
$\mathcal{G}_{1,\epsilon,\delta}=\\{{g_{1}+\delta,g_{2}+\delta,\dots,g_{N_{\epsilon}}+\delta}:g_{i}\in\mathcal{G}_{1,\epsilon}\,\,\forall
i\\}$. The perturbed grid in $n$ dimensions is then simply defined as
$\mathcal{G}_{n,\epsilon,\bm{\delta}}=\\{\mathbf{x}:x_{i}\in\mathcal{G}_{1,\epsilon,\delta_{i}}\,\,\forall
i\\}.$
The $(\epsilon,\bm{\delta})$-ISQ decoder would then be
$\displaystyle\text{$(\epsilon,\bm{\delta})$-ISQ:}\,\mathbf{\hat{x}}_{\epsilon,\bm{\delta}}=\operatorname{Q}(\operatorname{argmin}_{\mathbf{x}\in\mathcal{G}_{n,\epsilon,\bm{\delta}}}||\mathbf{y}-\mathbf{Hx}||^{2}).$
A bound on the error event for the general constellation $\mathcal{C}$ can be
had in terms of the minimum distance of the constellation $\mathcal{C}$
defined as $d_{min}=\min_{c_{i},c_{j}\in\mathcal{C},i\neq
j}||c_{i}-c_{j}||_{2}$. A first step towards that is the observation that
Lemma 1 holds by observing that for $k^{{}^{\prime}}n$ errors,
$\sum_{j}||c_{j}||^{2}\geq k^{{}^{\prime}}nd_{min}^{2}.$
Similarly Lemma 2 will follow from the observation that
$g(\mathbf{x})=||\mathbf{y}-\mathbf{Hx}||^{2}$ is convex (independent of the
constellation used).
Combining these two observations, we have the result that the estimate from
the r-$(\epsilon,\bm{\delta})$ ISQ decoder will have greater than or equal to
$k^{{}^{\prime}}n$ errors with probability less than $\exp(-an\log n)$ for
some $a>0$. This, together with the fact that for a finite dimensional
constellation,
$||c_{i}||_{\infty}\leq\epsilon\Rightarrow||c_{i}||_{2}\leq\tilde{K}\epsilon$
for some $\tilde{K}>0$ (and vice versa), gives us the result that
$||Q(P_{S}(\mathbf{x_{r}}))-\mathbf{x_{0}}||_{0}\leq kn,$
with probability at least $1-\exp(-dn\log n)$ with $d>0$, for any $k>0$, and a
large enough $n$.
This establishes Theorem 2 from which Theorem 3 follows by the symmetry in the
system model.
### VI-C Finite blocklength constellation design
A finite blocklength constellation over $T$ time slots (together with a block
fading model i.e. a model with the channel matrix $\mathbf{H}$ remaining
constant over $T$ time slots) can be thought of as a general constellation
within a single shot transmission model. Thus by the generalization of Theorem
3 shown in the previous section for arbitrary constellations, we get that the
same results hold for arbitrary finite blocklength constellations too.
### VI-D Provably faster decay of the error probability
In the proofs so far, we demonstrated that the number of symbol errors in the
_n-user_ codeword is less than $kn$ for any $k>0$, i.e. with high probability,
the number of errors is eventually sublinear. By using the same techniques to
derive the above result, we can also establish the result that the number of
errors is less than $n^{\epsilon}$ for any $0<\epsilon<1.$ This would not
change any of the conclusions of the previously stated theorems.
### VI-E Asymptotically vanishing sequence of $\alpha_{n}$
We restate a version of Theorem 3 for this case.
###### Theorem 4.
The probability of error with the r-ISQ decoder seen by any transmitting user
vanishes in the limit of an asymptotically large number of transmitters, with
the per-transmitter number of receive antennas $\alpha_{n}$ scaling with the
number $n$ of transmitting users like
$\alpha_{n}=\frac{1}{(\log n)^{\xi}},\,\,0<\xi<1.$
The critical step towards proving this theorem is the observation that an
analogue of Lemma 1 continues to hold in this case with the following
modifications
###### Lemma 4.
For any $i>k^{\prime}n$, there exists an $n_{0}$ and an $a>0$, such that for
all $n>n_{0}$,
$P(||\sum_{j=1}^{n}c_{j}\mathbf{h_{j}}||^{2}<n(\log
n)^{1-\xi})\leq\exp(-n(\log n)^{1-\xi}).$
The proof follows along lines very similar to the ones used to derive the
original lemma 1. Based on this observation Theorem 2 holds with the following
modification
###### Theorem 5.
Under r-ISQ decoding, for $m=\alpha_{n}n$ defined earlier, and any constant
$k>0$, there exists a $d>0$ such that for all sufficiently large $n$,
$P_{e}^{k}\leq 2^{-dn(\log n)^{1-\xi}}.$
Theorem 3 would also hold unmodified in this case.
## VII Conclusions and Future Work
We have considered an uplink communication system in a rich scattering
environment with a large number of non-cooperating transmitters and a large
number of antennas at the receiver. The transmitters send bits to the receiver
_without coding_. The receiver does joint decoding of the noisy received
signal from all users using a relaxation of the maximum likelihood (ML)
decoder. We call this technique the interval search and quantize (ISQ)
decoder. Since the solution may not be unique in general, we have proposed an
efficient randomization scheme (i.e. the r-ISQ decoder) which will still allow
us to have reliable estimates from the solution set. Under general assumptions
about the fading distribution of the channel coefficients, we have shown that
with the r-ISQ decoder, for a large enough system size, the error probability
that each user sees is vanishingly small even with the per-transmitter number
of receiver antennas being arbitrarily small.
In spite of these promising asymptotic properties of the efficient box-
constrained decoders, we pay a price in the finite $n$ behaviour with respect
to the rate of decay of the error probability. The decay rates achievable are
at best polynomial. Thus, the question of how large the system size needs to
be for the diversity-induced reliability to kick in remains a pertinent
question and needs further investigation. Also, the issue of how practical
constraints such as limited diversity in large systems or imperfect channel
knowledge would affect these results remains a topic to be investigated.
## Acknowledgements
This work was supported by a 3Com Corporation Stanford Graduate Fellowship, by
the NSF Center for Science of Information (CSoI): NSF-CCF-0939370, and by a
gift from Cablelabs. The authors acknowledge helpful discussions and insights
from Tsachy Weissman, in particular with respect to the proof of Theorem 2.
The first author would also like to acknowledge helpful discussions with Yash
Deshpande, Stefano Rini, Alexandros Manolakos and Nima Soltani.
## References
* [1] T. Cover and J. Thomas, _Elements of Information Theory_. Wiley Online Library, 1991, vol. 6.
* [2] G. Caire, D. Tuninetti, and S. Verdú, “Suboptimality of TDMA in the Low-Power Regime,” _IEEE Transactions on Information Theory_ , vol. 50, no. 4, pp. 608–620, 2004.
* [3] D. Tse and P. Viswanath, _Fundamentals of Wireless Communication_. Cambridge University Press, 2005.
* [4] M. Chowdhury and A. Goldsmith, “Reliable Uncoded Communication in the SIMO MAC,” 2014, submitted to IEEE Transactions on Information Theory.
* [5] M. Chowdhury, A. Goldsmith, and T. Weissman, “Reliable Uncoded Communication in the SIMO MAC Via Low-Complexity Decoding,” in _Proceedings of ISIT_. IEEE, 2013\.
* [6] ——, “The Per-User Number of Receive Antennas in Uncoded Non-Cooperating Transmissions Can Be Arbitrarily Small,” in _Proceedings of the 50th Annual Allerton Conference on Communication,Control, and Computing, Monticello, IL_ , 2012.
* [7] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge University Press, 2004.
* [8] D. L. Donoho and J. Tanner, “Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications,” _Discrete & Computational Geometry_, vol. 43, no. 3, pp. 522–541, 2010.
* [9] M. Bayati and A. Montanari, “The Dynamics of Message Passing on Dense Graphs, with Applications to Compressed Sensing,” _IEEE Transactions on Information Theory_ , vol. 57, no. 2, pp. 764–785, 2011.
* [10] Z. Huang and S. Kannan, “On Sampling from Multivariate Distributions,” in _Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques_. Springer, 2011, pp. 616–627.
|
arxiv-papers
| 2014-04-09T02:06:02 |
2024-09-04T02:50:00.887721
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mainak Chowdhury and Andrea Goldsmith",
"submitter": "Mainak Chowdhury",
"url": "https://arxiv.org/abs/1404.2352"
}
|
1404.2398
|
# Color-Magnitude Distribution of Face-on Nearby Galaxies in SDSS DR7
Shuo-Wen Jin Purple Mountain Observatory, Chinese Academy of Science, 210008,
China Qiusheng Gu School of Astronomy and Space Science, Nanjing University,
Nanjing, 210093, China;
[email protected] Song Huang School of Astronomy and Space Science, Nanjing
University, Nanjing, 210093, China Yong Shi School of Astronomy and Space
Science, Nanjing University, Nanjing, 210093, China Long-Long Feng Purple
Mountain Observatory, Chinese Academy of Science, 210008, China
###### Abstract
We have analyzed the distributions in the color-magnitude diagram (CMD) of a
large sample of face-on galaxies to minimize the effect of dust extinctions on
galaxy color. About 300 thousand galaxies with $log(a/b)<$ 0.2 and redshift
$z<0.2$ are selected from the SDSS DR7 catalog. Two methods are employed to
investigate the distributions of galaxies in the CMD including 1-D Gaussian
fitting to the distributions in individual magnitude bins and 2-D Gaussian
mixture model (GMM) fitting to galaxies as a whole. We find that in the 1-D
fitting only two Gaussians are not enough to fit galaxies with the excess
present between the blue cloud and the red sequence. The fitting to this
excess defines the centre of the green-valley in the local universe to be
$(u-r)_{0.1}=-0.121M_{r,0.1}-0.061$. The fraction of blue cloud and red
sequence galaxies turns over around $M_{r,0.1}\sim-20.1$ mag, corresponding to
stellar mass of $3\times 10^{10}M_{\odot}$. For the 2-D GMM fitting, a total
of four Gaussians are required, one for the blue cloud, one for the red
sequence and the additional two for the green valley. The fact that two
Gaussians are needed to describe the distributions of galaxies in the green
valley is consistent with some models that argue for two different
evolutionary paths from the blue cloud to the red sequence.
###### Subject headings:
fundamental parameters – galaxies: spiral – galaxies: statistics – methods:
data analysis
## 1\. Introduction
The distribution of galaxies in the color-magnitude diagram (CMD) provides a
powerful tool to investigate the evolution of galaxy populations. A remarkable
feature of the CMD is a robust bimodality, which divides the galaxy population
into a “blue cloud” (or blue sequence) and a “red sequence”. The bimodality is
seen in the optical colors (Strateva et al., 2001; Blanton et al., 2003c),
UV$-$optical colors (Wyder et al., 2007), the 4000 Å break ($D_{n}4000$;
Kauffmann et al. 2003a), and spectral type (Madgwick et al., 2002). Galaxies
in “red sequence” are quiescent, bulge-dominated galaxies (Blanton &
Moustakas, 2009), while the “blue cloud” is characterized by star-forming,
disk-dominated galaxies. Between these two sequences, there is a region called
the “green valley”.
Baldry et al. (2004) explored the distribution of galaxies in the $(u-r)$
versus $M_{r}$ diagram for low-redshift SDSS samples. Their galaxies separate
into “blue cloud” and “red sequence”, and the distribution of $(u-r)$ color at
each absolute magnitude bin is well fitted by the sum of two Gaussians.
However, Wyder et al. (2007) showed that the $(NUV-r)$ color distribution at
each $M_{r}$ can not be fitted well by the sum of two Gaussians due to an
excess of galaxies between the blue and red sequences. They utilized Balmer
decrements and the Dust-SFH (star formation history)-Color relation (Johnson
et al., 2006) to correct the extinction of each galaxy, there still remain
galaxies in the green valley region between two sequences. Thus, galaxies in
the green valley region may not be a simple mixture of blue and red galaxies.
The understanding of the CMD color bimodality is also complicated by galaxy
dust extinction. Salim et al. (2009) found that many green valley galaxies are
simply dust-obscured actively star-forming (SF) galaxies. However, there still
exist 24 $\mu$m detected galaxies, some with LIRG-like luminosities, which
have little current SF. They belong to green valley or even the red sequence
because of their SF history, not just dust reddening.
The CMD bimodality is already in place at $z\sim 1$ (Cooper et al., 2006),
with color becoming bluer at higher redshift (Blanton et al., 2006; Willmer et
al., 2006). Based on the DEEP2 and COMBO-17 surveys, Faber et al. (2007)
argued that the number density of blue galaxies is more or less constant from
$z\sim 1$ to 0, while the number density of red galaxies has increased. This
work supports that the red sequence has grown in mass by a factor of 3 since
$z\sim 1$. A plausible scenario is that the growth of red galaxies was
triggered by quenching star formation in blue galaxies, which caused them to
migrate into the red sequence (Bell et al., 2004). In addition, galaxies may
also be moving from the lower end of the red sequence to the blue cloud
through accreting gas-rich dwarf galaxies (Faber et al., 2007).
Studies of galaxy morphologies show that red sequence is dominated by
spheroidal galaxies with Sérsic index n=4, while the blue cloud is occupied by
disk-dominated galaxies with Sérsic index n=1 (Driver et al., 2006). Mendez et
al. (2011) investigated the morphologies of green valley galaxies from the
AEGIS survey and found that most green valley galaxies are not classified as
mergers and that the merger fraction in the green valley is lower than that in
the blue cloud. Lackner & Gunn (2012) presented a set of bulge-disc
decompositions for a sample of 71825 SDSS main-sample galaxies and found that
the majority of green valley galaxies are bulge+disc galaxies, and that the
integrated galaxy color is driven by the color of galaxy disks.
In general, blue galaxies with star formation being quenched will evolve from
the blue cloud to the red sequence, passing through the green valley that thus
represents an intermediate phase of this quench process. Different mechanisms
have been proposed to cease star formation in blue galaxies, such as mergers
(Bell et al., 2004; Hopkins et al., 2010), AGN feedback (Croton et al., 2006;
Martin et al., 2007; Schawinski et al., 2010), morphological quenching (Martig
et al., 2009), cold flows accretion and shock heating (Dekel & Birnboim, 2006;
Cattaneo et al., 2006) (see Peng et al. 2010 for a reccent review).
Peng et al. (2010, 2012) investigated the quenched fraction of galaxies as a
function of local density, stellar mass, and redshift. They parameterized
galaxy quenching as fully separable “environment quenching” and “mass
quenching”, which are directly associated with the quenching processes of
satellite and central galaxies in group. This model is successful in
predicting the mass function of passive and star-forming galaxies. These
effects may also be reflected in the color-magnitude distribution of galaxies,
we will try to find some evidence of these affects in our analysis.
In this paper, we focus on fitting CMD by the use of different methods. Our
study aims at a better understanding of the “green valley” and may answer the
question whether the “green valley” is dominated by one component. Since
different quenching mechanisms will produce different distribution of galaxies
in color-magnitude space, the fine structure of CMDs may also provide valuable
clues about galaxy quenching mechanisms. Previous work always focus on the
whole galaxy population without identification, which may conceal the fine
structure of CMDs due to dust extinction, so we selected a nearly face-on
galaxy sample to minimize the effect of dust.
This paper is organized as follows. Section 2 describes the sample selection
in this work. In Section 3, we investigate the color-magnitude distribution
and show 1D and 2D Gaussian fitting results. Section 4 discuss the implication
of the results. Section 5 is the conclusion. Throughout this paper, we assume
a flat $\Lambda$CDM cosmology with a matter density $\Omega_{m}$=0.3,
cosmological constant $\Lambda$=0.7 and Hubble’s constant
$H_{0}$=100$kms^{-1}Mpc^{-1}$ (i.e., $h=1$).
## 2\. The Sample
The photometric and spectroscopic data used in this paper are taken from the
main galaxy sample in Sloan Digital Sky Survey (SDSS) DR7 (York et al., 2000;
Strauss et al., 2002; Abazajian et al., 2009). The sample selection criteria
are as follows:
(1) $r$ band apparent magnitude $m_{r}<17.77\ mag$ with $ugriz$ flux signal-
to-noise ratio $S/N>5$. The criterion of S/N can reduce outliers. By comparing
the color distribution of bright galaxies and the overall population, this
criterion doesn’t generate significant selection effects.
(2) $log(a/b)<0.2$, $a/b$ is the ratio of major axis to minor axis of de
Vaucouleurs’ profile. Gas and dust tend to reside in the disk of spirals, the
colors of spirals are seriously reddened by dust attenuation. For example, the
total extinction from face-on to edge-on is about 0.7, 0.6, 0.5 and 0.4 $mag$
for the $ugri$ passbands (Masters et al., 2010), so we choose the nearly face-
on galaxies to minimize the dust reddening effect.
In total, our sample consists of 329384 galaxies. To ensure a broader span of
luminosity and the completeness of color in each magnitude bin, we divide
galaxies into 12 bins from $M_{r,0.1}=-18.50\sim-21.5\ mag$ and $0.02<z<0.18$
with bin size of 0.25 $mag$, as shown in Figure 1 ($M_{r,0.1}$ is the absolute
magnitude in $r$ band $k-corrected$ to $z=0.1$). The galaxy number and median
redshift in each bin are shown in Table 1. We will analyse the color
distribution of galaxies in each bin.
Figure 1.— We select galaxies in each red box to ensure the completeness of
color in each magnitude bin. Black dots show a random subsample of SDSS face-
on $[log(a/b)<0.2]$ main galaxy sample. $M_{r,0.1}$ is the absolute magnitude
in $r$ band which have been $k-corrected$ to $z=0.1$. The galaxy number and
median redshift in each bin are shown in Table.1. Table 1The galaxies in each
red bin of Figure.1
$M_{r,0.1}$ | $z$ | $Galaxy\ Number$ | $\overline{z}$
---|---|---|---
$[-18.50,-18.75]$ | $[0.020,0.056]$ | 3541 | 0.042
$[-18.75,-19.00]$ | $[0.020,0.063]$ | 4600 | 0.047
$[-19.00,-19.25]$ | $[0.020,0.071]$ | 6715 | 0.053
$[-19.25,-19.50]$ | $[0.020,0.080]$ | 9800 | 0.061
$[-19.50,-19.75]$ | $[0.022,0.090]$ | 13265 | 0.068
$[-19.75,-20.00]$ | $[0.024,0.100]$ | 16225 | 0.074
$[-20.00,-20.25]$ | $[0.026,0.112]$ | 20116 | 0.083
$[-20.25,-20.50]$ | $[0.029,0.124]$ | 23683 | 0.092
$[-20.50,-20.75]$ | $[0.033,0.137]$ | 27912 | 0.103
$[-20.75,-21.00]$ | $[0.037,0.152]$ | 30836 | 0.114
$[-21.00,-21.25]$ | $[0.042,0.168]$ | 31382 | 0.126
$[-21.25,-21.50]$ | $[0.047,0.186]$ | 29832 | 0.139
* 1
$\overline{z}$ is the galaxies’ median redshift in each bin.
Using the New York University Value-Added Catalog (NYU-VAGC; Blanton et al.
2005) and MPA-JHU catalog, we obtain the physical parameters for our sample,
such as $u,g,r,i,z$ absolute magnitudes k-corrected to $z=0.1$ (Blanton et
al., 2003a), stellar mass, and $D_{n}4000$ (Kauffmann et al., 2003a; Salim et
al., 2007). We use modelMags for galaxy magnitude and color in this paper,
which are derived from the best-fitting exponential or de Vaucouleurs galaxy
profile. modelMags have occasional problems recovering accurate magnitudes for
galaxies with mixed morphologies, and petro magnitude (Petrosian, 1976)
provides a better measure for flux (Taylor et al., 2011; Simard et al., 2011),
but the use of modelMags do not change any of the results that we present in
Section 3.
Although galaxies in each magnitude bin cover certain redshift ranges, the
effect of redshift evolution in color should be negligible given the overall
redshift range of the whole sample from 0.0 to 0.2.
Figure 2.— The $(u-r)$ color distribution in 0.25 $mag$ absolute magnitude
bins. The magnitude range is labeled in the upper left of each figure from
-21.50 to -18.50 $mag$. The black histogram is the $(u-r)$ distribution in
each magnitude bin, the color histogram’s bin is 0.02 $Mag$. The blue and red
dash lines mark the color boundaries when fitting Gaussian distribution to
blue cloud and red sequence. The solid blue and red lines are the Gaussian
fitting results for blue cloud and red sequence, the solid purple line is the
sum of two Gaussians. Finally, the solid green marks the residual distribution
between actual color distribution and the sum of two Gaussians.
## 3\. Results
### 3.1. Color Distribution Fitting
Figure 3.— The top panel plots the color-magnitude diagram (CMD) of face-on
galaxies and the best fit to each sequence in Figure 2. The contour is all
face-on galaxies distribution in CMD, the blue boxes mark the median color of
blue sequence in each magnitude bin, the red diamonds mark the red sequence.
The error bar of each point refers to the 1 $\sigma$ width of the Gaussian.
The lower panel shows the fraction of blue cloud galaxies, red sequence
galaxies and green-valley galaxies, the error bars are calculated by Bootstrap
resampling method.
As shown in Figure 2, the distributions of galaxies in each magnitude bin show
apparent two peaks except for two brightest bins where the distribution
profile is characterized by a red peak plus a blue tail. We fit a single
Gaussian to blue and red peaks. Respectively, we select the left 0.15 $mag$ of
red peaks as the red boundaries and select the right 0.25 $mag$ of blue peaks
as the blue boundaries. For the two brightest bins, since a very small blue
fraction, we artificially adjusted the boundaries until these two peaks are
well fitted, the color boundaries are plotted as the dashed blue and red lines
in each magnitude bin in Figure 2.
As shown in Figure 2, two Gaussians accurately fit the blue cloud and red
sequence, but the color distribution apparently cannot be fitted well by the
sum of 2 Gaussians in each magnitude bin because of the obvious excess between
the two sequences. We define the excess between two sequences as the “green-
valley galaxy” population.
The parameters of the best fit are shown in Figure 3. In the top panel, the
contours illustrate the nearly face-on galaxies distribution in CMD, the blue
boxes mark the median color of blue sequence and the red diamonds mark the red
sequence, the error bar of each point gives the 1 $\sigma$ width of the
Gaussian fitting.
To quantify each sequence, the fraction of blue cloud (red sequence) is
defined as the ratio of blue cloud (red sequence) galaxies number to total
number in each magnitude bin, and the fraction of green-valley galaxies is
defined as the complementary set of them. In the bottom panel of Figure 3, the
fraction of red sequence rises with luminosity while the fraction of blue
cloud galaxies declines, they turn over around $M_{r,0.1}\sim-20.1\ mag$. The
trend of both red and blue fraction in the brightest bin may be artificial
since a very small blue fraction makes two-Gaussian fitting difficult (see
last panel of Figure 2). The fraction of green valley galaxies remains almost
constant around 10% except for the two brightest bins.
Interestingly, we find the mean stellar mass of galaxies in
$M_{r,0.1}\sim-20.1\ mag$ bin is close to the characteristic mass
$10^{10.5}M_{\sun}$ (Kauffmann et al., 2003a) , that is, the fraction of red
sequence and blue cloud cross over at the characteristic mass $M^{*}$.
We also fit the $(u-g)$ color distribution in the same way and find the same
results. For the $(g-r)$ color distribution, the blue cloud and red sequence
are too close to apply this methodology. We have already applied redshift
evolution correction to the data (Blanton et al., 2003b), which makes no
change to the results.
Figure 4.— The contours, boxes and diamonds are the same as the case in the
top panel of Figure 3, the green triangles and their error bars refer to the
green valley location in each magnitude bin which is calculated by fitting the
green residuals in Figure 2 by a Gaussian, the dash black line is the liner
fit of green triangles.
In order to derive the location of green-valley galaxies, we can roughly fit
the excess (green line in Figure 2) by a Gaussian in each magnitude bin, the
median color and 1 $\sigma$ region are plotted as triangles and error bars in
Figure 4. We fit the triangles with a line, which plotted as the black dash
line in the left panel, that is the expression of the centre of the green
valley:
$\centering(u-r)_{0.1}=-0.121M_{r,0.1}-0.061.\@add@centering$ (1)
### 3.2. Color-Magnitude Distribution
Above, we have studied the color distribution in each magnitude bin. However,
the color-magnitude distribution is actually a two dimensional distribution,
and the evolutionary pathway of galaxy may span several magnitude bins. The
two dimensional fitting for CMD may provide us more information, so we apply
Gaussian mixture model (GMM) to fitting the color-magnitude diagram and color-
mass diagram directly. This research made use of astroML, a community-
developed core Python package for Astronomy (VanderPlas et al., 2012).
The discontinuity of the volume limited samples in section 2 makes them
inappropriate for GMM fitting, so we select a subsample from our main sample
with criteria $0.04<z<0.08$, galactic extinction in the $u$ band less than 0.3
$mag$ and in the $r$ band less than 0.5 $mag$, which are used to reduce
outliers. As shown in Figure 5, this subsample contains 67499 galaxies and is
complete to $M_{r,0.1}<-19.2\ mag$.
Figure 5.— A sub sample of the nearly face-on galaxies sample: $0.04<z<0.08$,
galactic extinction at $u$ band less than 0.3 $mag$ and $r$ band less than 0.5
$mag$. Black dots show a random subsample of our main main sample, red dots
show the subsample, this subsample contains 67499 galaxies, which is complete
at $M_{r,0.1}<-19.2$.
The mixture model provides a method of describing more complex probability
distributions, by combining several probability distributions. Gaussian
Mixture Model (GMM) is a parametric probability density function represented
as a weighted sum of Gaussian component densities. GMM is among the most
statistically mature methods for clustering (Xu & Jordan, 1996; Dasgupta,
1999; Verbeek et al., 2003).
When fitting models, it is possible to increase the likelihood by adding more
free parameters, however, which may result in overfitting. To minimize the
number of redundant parameters, we make use of Akaike information criterion
(AIC) and Bayesian information criterion (BIC) for model selection. As shown
in Figure 6, the AIC and BIC have the same trend, but BIC converges faster
than AIC, so we prefer BIC as our criterion, and treat the minimum value of
BIC as the best models for fitting.
The outlier data in each diagram may affect the distribution of AIC and BIC
and produce extra components that occupy very little fraction and spread
distribution, it will generate one or two ellipses that have tiny centre and
large radius, like a “background” in each diagram. This is the intrinsic
problem of BIC, and we can minimize this effect by removing outliers.
The results of GMM fitting are shown in Figure 6. For $(u-r)$ vs. $M_{r}$, the
GMM exactly detects the blue cloud and red sequence. Between these two
sequences, the green valley region contains two Gaussian components: one
located in the faint end of red sequence and extending to the faint end of
blue cloud, and the other in the bright end of blue cloud stretching towards
the bright end of red sequence. For the $(u-g)$ CMD, the red sequence is
decomposed into two components, and the faint component in the green valley is
bluer than the corresponding component in the $(u-r)$ CMD. Whether the number
of components is N$=4$ or N$=5$, there are always two Gaussian components in
the green valley region. In the 4th subgraph of Figure 6, the values of BIC
are almost equal when N$=4$ and N$=5$, it suggests there is no much
improvement for fitting when we chose five components, so we prefer the four
components result as our finding.
Figure 6.— The GMM fitting results of $(u-g)$ and $(u-r)$ CMDs, the first
column plots the input CMDs; the second column shows the best-fitting Gaussian
components, the red ellipse is the 1.5 $\sigma$ region of each Gaussian, the
size of ellipse centre illustrates the fraction of each Gaussian component;
the third column shows the residuals between input distribution and models;
the final column plots the AIC and BIC, we prefer BIC to our criterion. Figure
7.— The results of GMM fitting in color-stellar mass space.
Figure 7 shows the GMM results for color-mass space. The distribution of
Gaussians is the same as that in Figure 6. As shown in Figure 7, the mean
stellar mass of the bright component in the green valley region is close to
the characteristic mass $M^{*}$ $(10^{10.5}M_{\odot})$.
Given that galaxies fainter than $M_{r,0.1}=-19.2$ $mag$ are not complete in
this subsample, we check the results by only fitting the complete
($M_{r,0.1}<-19.2\ mag$) sample only, and find that the AIC and BIC converge
at very large values. But if we set the same number of Gaussian components as
in Figure 6, the ellipses’ centre almost do not change while the size of
ellipses are somewhat different. The completeness thus does not change the GMM
fitting results.
Essentially, the work in section 3.1 is a 1-dimensional (1-D) fitting and GMM
fitting is a 2-dimensional (2-D) fitting. The 1-D color fitting show that the
color distribution of galaxies cannot be fitted by two Gaussians, which may
suggest some potential components in the green valley region, and the GMM
fitting successfully decomposes the color-magnitude distribution.
Figure 8.— The color distribution of GMM result for galaxies in each
magnitude bin. The color solid lines show the models generated by GMM fitting:
the red and blue lines represent the red sequence and blue cloud, the green
and cyan lines mark the faint and bright components of green valley
respectively. The black dashed line show the sum of models, and the plus signs
mark the data.
As small residuals and well-distributed shown in Figure 6, GMM fit the data
very well. We also show the GMM results in terms of 1-D color distribution in
Figure 8, the color solid lines mark the models generated from GMM fitting:
the red and blue lines represent the red sequence and blue cloud, the green
and cyan lines mark the faint and bright components of green valley
respectively. The sum of models (black dashed line) also fit the data (plus
signs) very well too. Distinct from fitting in Section 3.1, the component
number in Figure 8 is selected by BIC, and the model distribution (color solid
line) in each magnitude bin is not a Gaussian, but a skew Gaussian because it
is 1-D projection of an incomplete 2-D Gaussian. These are advantages of GMM
fitting, which make its results reasonable.
## 4\. Discussion
In this study we carried out Gaussian profile fittings to the CMD through two
methods, i.e., the fitting to distributions in individual magnitude bins and
to distributions in 2-D as a whole. The two Gaussian components in the CMD
green valley are homologous to the excess in color distribution fitting, and
explain why we cannot fit the color histogram well only two Gaussians. The
green valley region is not dominated by a single Gaussian, but at least two
Gaussian components. Galaxies in the green valley are composed of these two
components plus the Gaussian tails of blue and red galaxies.
By fitting the $(u-r)$ color distribution in each magnitude bin, our results
are different from those of Baldry et al. (2004), but agree with the $(NUV-r)$
results of Wyder et al. (2007). As we select only nearly face-on galaxies
which are not considered by Baldry et al. (2004), we attribute the difference
to dust extinction, which seriously reddens the color of blue cloud galaxies
and covers the excess in the green valley region.
Kauffmann et al. (2003a) showed that galaxies tend to divide into two distinct
groups below and above a stellar mass of $3\times 10^{10}M_{\odot}$. Galaxies
below this mass limit tend to have younger stellar populations, while more
massive galaxies tend to be older. We find that the fraction of blue cloud and
red sequence are equal to each other around $M_{r,0.1}$$\sim$$-20.1\ mag$,
corresponding to stellar mass about $M^{*}$ ($10^{10.5}M_{\sun}$). As shown in
Figure 3, our result is consistent with their conclusion very well.
If we assume that the density of galaxies represents their evolution time
scale in color-magnitude space, the GMM results would intuitively show us the
evolutionary paths of different galaxies: the components in Figure 6 would
represent galaxies in different evolving phases, and the inclination of
ellipses’ major axes may suggest the galaxy evolving direction in color-
magnitude space. The faint component in the green valley may be the early-
quenching population, which quenched while galaxy are still small and grow
mass along red sequence via “dry” mergers (Faber et al., 2007). In the faint
end of the red sequence, these red low-luminosity galaxies tend to be in
overdense regions (Blanton et al., 2006), environment possibly plays a very
important role in their evolution.
If the two components in the green valley are dominated by two independent
quenching processes, this scene would agree with Peng et al. (2010) very well.
According to the model of Peng et al. (2010), galaxies in the faint component
are dominated by “environment quenching”, and galaxies in the bright one are
dominated by “mass quenching”, which are associated with the quenching
processes of satellite and central galaxies in group (Peng et al., 2012).
Since these two effects are fully independent of each other, they may produce
the two independent Gaussian components we found in the CMD.
As shown in Figure 7, the mean stellar mass of the bright component in the
green valley region is about the characteristic mass $M^{*}$. The
characteristic mass $M^{*}$ also corresponds to a dark halo mass
$M_{shock}\sim 10^{12}M_{\odot}$ based on the model of Dekel & Birnboim
(2006). In their model, galaxies with $M_{halo}\gtrsim 10^{12}M_{\odot}$ will
generate a steady shock in the gas accreting onto dark matter halo, the shock
heats the gas and absolutely quenches star formation when AGNs begin to work.
This process is strongly related to the mass of galaxy, which may dominate the
evolution of bright galaxy component in green valley.
Wong et al. (2012) selected a local post-starburst galaxies (PSGs) sample from
SDSS, those PSGs occupy the low-mass end of the ”green valley” below the
transition mass within the colour-stellar mass diagram (the same position as
the faint component of green valley in Figure 7). They proposed those PSGs
represent a population of galaxies which is rapidly transitioning between the
star-forming and the passively evolving phases. Mendel et al. (2013) select a
sample of young passive galaxies from SDSS, which is identified based on the
contribution of A-type stars to spectra and the relative lack of ongoing star
formation. Most of these recently quenched galaxies have a stellar mass
$>10^{9.5}M_{\odot}$ and are predominantly early-type systems. McIntosh et al.
(2013) studied the recently quenched ellipticals (RQEs) with stellar mass
$>10^{10}M_{\odot}$ and found a number of RQE properties are consistent with
these galaxies being new remnants from a gaseous major merger. Their studies
show that the low- and high- mass galaxies in green valley are very different,
and suggest the green valley are dominated by different quenching processes,
supporting our GMM fitting results.
In Figure 8, it is notable that the blue peak of color bimodality is dominated
by the bright component of green valley (the cyan line) for galaxies
$M_{r}<-20.25$ $mag$, which imply that the blue galaxy population with
$M_{*}>10^{10.5}M_{\odot}$ is distinct from galaxies which traditionally
thought as blue cloud. This case is consistent with Schawinski et al. (2014),
which proposed that the early- and late-type galaxies in green valley have two
different evolutionary pathways. The evolving early-type galaxies generally
have stellar mass $M_{*}>10^{10.5}M_{\odot}$, rapidly quenching star
formation, moving out the blue cloud, into the green valley and to the red
sequence as fast as stellar evolution allows (Schawinski et al., 2014).
It is still unclear about the quenching mechanisms of galaxies,
differentiating such mechanisms requires more evidences which are beyond the
scope of this work.
## 5\. summary
In this paper, we have analyzed the color-magnitude distribution of a sample
of nearly face-on galaxies, selected from SDSS DR7 main galaxy sample. We fit
$(u-g)$ and $(u-r)$ color distribution in each magnitude bin and apply
Gaussian Mixture Models to fit the CMDs and the results are as follows,
(1) The color distributions of galaxies cannot be fitted by two Gaussians,
there is an obvious excess in the green valley region.
(2) With rising luminosity, red galaxies increasing while blue galaxies
decrease, the fraction of blue cloud and red sequence cross at
$M_{r,0.1}$$\sim$$-20.1\ mag$. At this magnitude, the mean stellar mass of
galaxies is about the characteristic mass $M^{*}$.
(3) By fitting the excess between blue cloud and red sequence, we find that
the centre of the green-valley for face-on galaxies in the local universe is:
$(u-r)_{0.1}=-0.121M_{r,0.1}-0.061$.
(4) The GMMs of CMDs accurately illustrate the red sequence and blue cloud,
and yields 2 Gaussian components for the green valley region, which might
suggest two different evolutionary paths from blue cloud to red sequence.
The authors are very grateful to the anonymous referee for his/her thoughtful
and instructive comments that significantly improved the content of this
paper. This work is supported under the National Natural Science Foundation of
China under grants (11273015, 11133001 and 11273060), the National Basic
Research Program (973 program No. 2013CB834905), and Specialized Research Fund
for the Doctoral Program of Higher Education (20100091110009). Funding for the
SDSS and SDSS-II was provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Science Foundation, the U.S.
Department of Energy, the National Aeronautics and Space Administration, the
Japanese Monbukagakusho, the Max Planck Society, and the Higher Education
Funding Council for England. The SDSS was managed by the Astrophysical
Research Consortium for the Participating Institutions. This publication made
extensive use of the Python package AstroML and scikit-learn, which can be
found at http://www.astroml.org/ and http://scikit-learn.org/.
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|
arxiv-papers
| 2014-04-09T09:04:07 |
2024-09-04T02:50:00.897744
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Shuo-Wen Jin, Qiusheng Gu, Song Huang, Yong Shi, Long-Long Feng",
"submitter": "Shuo-Wen Jin",
"url": "https://arxiv.org/abs/1404.2398"
}
|
1404.2533
|
# Additional Evidence Supporting a Model of Shallow, High-Speed
Supergranulation
T.L. Duvall Jr.1S.M. Hanasoge2,3S. Chakraborty4 1 Solar Physics Laboratory,
NASA Goddard Space Flight Center, Greenbelt, MD, 20771, USA
email:[email protected]
2 Tata Institute of Fundamental Research, Mumbai 400005, India
email: [email protected]
3 Max-Planck-Institut fur Sonnensystemforschung, Justus-von-Leibig-Weg 3,
37077 Göttingen, Germany
4W.W. Hansen Experimental Physics Laboratory, Stanford University, Stanford,
CA 94305, USA email: [email protected]
###### Abstract
Recently, Duvall and Hanasoge (Solar Phys. 287, 71-83, 2013) found that large
distance $[\Delta]$ separation travel-time differences from a center to an
annulus $[\delta t_{\rm{oi}}]$ implied a model of the average supergranular
cell that has a peak upflow of $240\rm{\,m\,s^{-1}}$ at a depth of
$2.3\rm{\,Mm}$ and a corresponding peak outward horizontal flow of
$700\rm{\,m\,s^{-1}}$ at a depth of $1.6\rm{\,Mm}$. In the present work, this
effect is further studied by measuring and modeling center-to-quadrant travel-
time differences $[\delta t_{\rm{qu}}]$, which roughly agree with this model.
Simulations are analyzed that show that such a model flow would lead to the
expected travel-time differences. As a check for possible systematic errors,
the center-to-annulus travel-time differences $[\delta t_{\rm{oi}}]$ are found
not to vary with heliocentric angle. A consistency check finds an increase of
$\delta t_{\rm{oi}}$ with the temporal frequency $[\nu]$ by a factor of two,
which is not predicted by the ray theory.
###### keywords:
Helioseismology, Observations; Helioseismology, Direct Modeling; Interior,
Convective Zone; Supergranulation; Velocity Fields, Interior
## 1 Introduction
Supergranulation, first seen as a 30 Mm cellular pattern of horizontal flows
detected by Doppler shifts [Hart (1954), Leighton, Noyes, and Simon (1962)] in
the solar photosphere, continues to puzzle investigators (see review by
Rieutord10). Recent work attempts to understand supergranulation by revealing
its subsurface structure by numerical simulations [Stein _et al._ (2006)] or
by local helioseismology [Gizon, Birch, and Spruit (2010)].
Detailed radiative-hydrodynamic simulations of the outer convection zone and
atmosphere show no excess flow signal at the supergranular scale in the
photosphere, in contrast to the observational results [Nordlund, Stein, and
Asplund (2009)]. These simulations, which match the observations of the solar
granulation so well, would seem to have all of the ingredients required to
reproduce supergranulation. In particular, the early suggestion of Leighton62
that He II ionization could give rise to supergranulation, is tested by the
simulations with a null result. One possibility remaining to be tested is the
simulation of magnetic field, which is known to be present along cell
boundaries.
Local helioseismology has been used extensively to study supergranulation (see
review by Gizon10), although no consensus has emerged about fundamental
questions such as the depth of the peak flow and the existence or not of
counterflows at depth. Some efforts centered on making inversions of
individual realizations of the supergranular flow field [Duvall _et al._
(1997), Zhao and Kosovichev (2003), Woodard (2007), Jackiewicz, Gizon, and
Birch (2008), Švanda _et al._ (2012)]. In some of the work there is great
difficulty in separating a horizontally diverging outflow from an upflow [Zhao
and Kosovichev (2003), Dombroski _et al._ (2013)], although in other work this
may have been solved [Švanda _et al._ (2012)]. To make flow maps of
individual supergranular realizations, it has been necessary to restrict the
measurements to small separations $[\Delta<5^{\circ}]$ for which the signal-
to-noise ratio is large.
To measure the general properties of supergranulation, a large number of cells
needs to be examined (in the present work, $6\times 10^{4}$ supergranules are
analyzed). To increase the signal-to-noise ratio (S/N), spatial averages are
made about cell locations determined from shallow signals such as peaks in the
flow divergence. Such a method was first used by Birch06 and subsequently by
Duvall10 and Svanda12. Weak signals can be separated cleanly from realization
noise, although more attention to systematic errors is required. As noticed by
Svanda12, the present method of defining cells is probably biased towards
larger cells than the average. This might be corrected (in the future) by
directly modeling the spatial autocovariance of the travel-time maps.
The averaging of the signals from many cells makes it possible to use larger
$\Delta$s (up to $24^{\circ}$ in the present study), which would normally not
be feasible for a 12-hour observation because of the increased noise due to
the amplitude reduction from the geometrical spreading of the wavefront [Gizon
and Birch (2004)]. The separation of the horizontal and vertical flow signals
is much better at larger $\Delta$, as the rays are more vertical in the
critical near-surface region. Duvall13 (hereafter Article I) found that the
center-to-annulus travel-time difference $[\delta t_{\rm{oi}}]$ was roughly
constant at $5.1$ seconds in the range $\Delta=10-25^{\circ}$. In a simple
ray-theory interpretation, this requires a vertical upflow considerably larger
than the $10\rm{\,m\,s^{-1}}$ observed at the photosphere [Duvall and Birch
(2010)] and in fact the best-fit model had a peak upflow of
$240\rm{\,m\,s^{-1}}$ at $z=-2.3\rm{\,Mm}$. Plots of this model and the
bracketing models are shown in Figure F-flowmodels. That large vertical
upflows are required was recently confirmed by the analysis of Svanda12 by a
considerably different formalism.
The strategy for obtaining the best model was developed in Article I and is as
follows: We assumed the simplest vertical-flow model that reduces to a
$10\rm{\,m\,s^{-1}}$ vertical flow at the surface and still approaches the
$5.1$ seconds for the asymptotic behavior of the $\delta t_{\rm{oi}}$ signal
at large $\Delta$. This is the gaussian with a single peak. For a particular
choice of depth of the peak vertical flow $[z_{0}]$, the width of the gaussian
and its amplitude are determined uniquely by the $5.1$ seconds $\delta
t_{\rm{oi}}$ signal requirement and the $10\rm{\,m\,s^{-1}}$ upward flow at
the photosphere. With some reasonable choices for the horizontal parameters
$k$ and $R$ (see Article I), the horizontal flow is then determined from the
vertical flow and the continuity equation. Three models were examined that
bracket the observations. These are distinguished by the height of the peak
flow, $z_{0}=-1.15\rm{\,Mm}$, $z_{0}=-2.30\rm{\,Mm}$, and
$z_{0}=-3.45\rm{\,Mm}$. The $\delta t_{\rm{oi}}$ signal is computed from the
ray theory using both the vertical and horizontal flow components. We found
that the $z_{0}=-2.30\rm{\,Mm}$ model was most similar to the observations.
For the $z_{0}=-3.45\rm{\,Mm}$ model (and any with a deeper $z_{0}$), the
horizontal component contributes significantly and leads to a behavior at
large separations that is inconsistent with the observations. We conclude that
if there is a deeper horizontal flow, it must have a small magnitude to not be
observed in the $\delta t_{\rm{oi}}$ signal.
In the present work, the efforts of Article I are extended to include quadrant
analysis in Section sec-quad, an attempt to measure a heliocentric-angle (or
center-to-limb) dependence in Section sec-heliocen, tests with simulations in
Section sec-sim, and an attempt to measure a temporal-frequency $[\nu]$
dependence in Section sec-nu. We give some conclusions in Section sec-dis.
Figure 1.: The flow models from Article I. (a) Velocity vectors for the best
model. This is the model labeled g2 in Table 1 of Article I, with peak upward
flow of $240\rm{\,m\,s^{-1}}$ at $z=-2.3\rm{\,Mm}$ and peak horizontal flow of
$700\rm{\,m\,s^{-1}}$ at $z=-1.6\rm{\,Mm}$ and $x=7\rm{\,Mm}$. The cuts shown
in (b) are taken at the location of the red dashed vertical line in (a). The
cuts in (c) are taken at the location of the turquoise line in (a) at
$x=7\rm{\,Mm}$. (b) Cuts of the vertical flow at cell center for the three
models in Article I, model g1 (green; dashed), model g2 (blue; solid), and g3
(red; dot-dashed). (c) Cuts of the horizontal flow versus height at the
location of the peak flow. Colors and line styles are the same as in (b).
F-flowmodels
## 2 Analysis
### 2.1 Quadrant Analysis
sec-quad In Article I it was shown that the center–annulus travel-time
difference $[\delta t_{\rm{oi}}]$ at large distances $[\Delta]$ is mostly
sensitive to the supergranular vertical-flow signal. We might expect that at
large $\Delta$ that the center-to-quadrant signals $[\delta t_{\rm{qu}}]$,
where $qu$ corresponds to the West–East and North–South quadrant signals
$[\delta t_{\rm{we}}$ and $\delta t_{\rm{ns}}]$, would be mostly sensitive to
the horizontal supergranular flow. To test this idea, travel-time difference
maps were constructed with ray-theory modeling of the average supergranule-
flow model g2 from Article I. The results are shown in Figure F-model_cuts
with the horizontal, vertical, and sum flow contributions to the travel time
differences shown separately. For these relatively large $\Delta$s of
$11.76^{\circ}$ and $20.64^{\circ}$, the center–annulus time differences
$[\delta t_{\rm{oi}}]$ show very little contribution to the peak signal from
the horizontal flow (relative magnitude 0.008 for $\Delta=11.76^{\circ}$ and
0.001 for $\Delta=20.64^{\circ}$). The contribution of the vertical signal to
the peak $\delta t_{\rm{we}}$ is a little larger (0.064 for
$\Delta=11.76^{\circ}$ and 0.050 for $\Delta=20.64^{\circ}$), but still small.
Therefore the center–annulus differences $[\delta t_{\rm{oi}}]$ and quadrant
directional differences $[\delta t_{\rm{qu}}]$ do separate the vertical and
horizontal contributions quite well, if the ray theory can be believed. It
would be very difficult to construct a model with these responses in which the
horizontal flows are leaking into the $\delta t_{\rm{oi}}$ signal to yield the
five-second signal at large $\Delta$.
Figure 2.: Cuts in the east–west direction across model maps for
center–annulus travel time differences $[\delta t_{\rm{oi}}]$ ((a) and (c))
and west–east travel-time differences $[\delta t_{\rm{we}}]$ ((b) and (d)) for
distances $\Delta=11.76^{\circ}$ ((a) and (b)) and $\Delta=20.64^{\circ}$ ((c)
and (d)). Vertical-flow contributions to the travel times are in red (solid in
(a) and (c); dashed in (b) and (d); horizontal contributions are in green
(dashed in (a) and (c); solid in (b) and (d)); and the sum is in blue. The
model is the nominal one from Article I (g2) in which the peak upward vertical
flow is at a depth of $2.3\rm{\,Mm}$ with magnitude of $240\rm{\,m\,s^{-1}}$.
The peak horizontal flow is at a depth of $1.6\rm{\,Mm}$ at a distance
$7\rm{\,Mm}$ from cell center and with velocity $700\rm{\,m\,s^{-1}}$. In
general it is very difficult to separate the sum signal (blue) from the
dominant signal (vertical flow (red) in (a) and (c) and horizontal flow
(green) in (b) and (d)). F-model˙cuts
To compare the quadrant signals with the models, Helioseismic and Magnetic
Imager (HMI) data were analyzed as in Article I with some improvements. The
same 64 12-hour intervals (10 June 2010-10 July 10 2010) were analyzed with
the same constant degree–width filter with width $\Gamma_{\rm{\ell}}=400$.
Cross-correlation maps were constructed for each 12-hour period for the in and
out-annulus signals and for the four quadrant signals (eastward, etc.
direction of waves) for the 14 distance ranges of Article I and two additional
ones at small $\Delta$ (centered at $1.20^{\circ}$ and $1.44^{\circ}$). The
coordinate system used has equal increments in longitude and latitude of
$0.24^{\circ}$. Gizon–Birch travel times [Gizon and Birch (2004)] were
computed for each set of cross correlations and the desired differences:
$\delta t_{\rm{oi}}$, $\delta t_{\rm{we}}$, and $\delta t_{\rm{ns}}$. The
reference correlation was taken as the average over the in and out
correlations averaged over the map, which is of size $96.24^{\circ}$ on a
side. An average of these travel-time differences is made about the
supergranular centers. In this average, the latitude–longitude travel-time
difference maps are transformed locally to a Postel’s coordinate system
centered on the feature. In this way, features at different latitudes are
treated equally and the resultant average maps can be compared more readily
with theory. Note that this was not the case in Article I, where the averages
about the feature locations were done in the latitude–longitude coordinate
system. However, as only the center of the maps where the peak $\delta
t_{\rm{oi}}$ signal was obtained were used, it was acceptable. The
supergranular-averaged 12-hour maps are averaged over the 64 different
intervals. One advantage of doing the analysis in this way is that an estimate
of the error can be made at each map location from the scatter of the 64
different intervals. One might imagine that the scatter would be larger where
the average signal is larger just because of the variability in the
supergranular signal. However, this was not the case, and the maps of errors
showed no distinguishable features.
Some background signals are removed from these superposed images before
further analysis. The $\delta t_{\rm{oi}}$ signal approaches a nonzero
constant far from the central feature. This constant is measured and removed
for each $\Delta$ range from the overall image as in Article I. The $\delta
t_{\rm{we}}$ signal has a relatively large constant (six seconds for the
smallest $\Delta$ to one second for the largest) removed. This signal is due
to the average rotation over the field. This signal could be reduced by
adjusting the tracking rate from the nominal Carrington rate. The $\delta
t_{\rm{ns}}$ signal has both constant offsets and slopes in the North–South
direction at different distances $[\Delta]$. The magnitudes are generally
small ($<$ one second), but they are present in the background and hence were
removed. Of these signals subtracted, only the rotation one is well
understood.
Figure 3.: Comparison of data with models for the distance range
$\Delta=19.1$ – $22.2^{\circ}$. The first column is the center–annulus travel
time difference $[\delta t_{\rm{oi}}]$, The second column is the quadrant
West–East travel time difference $[\delta t_{\rm{we}}]$ and the third column
is North–South $[\delta t_{\rm{ns}}]$. The top row is for the data 32-day
average of all supergranules. The second row is the model and the third row is
the difference of data minus model. The fourth row are cuts through the data
and model images with data shown as blue (with symbols) and model as red.
Where the cuts are taken is shown by the lines across the maps in the top two
rows. The color scales of the images are in a spectral sequence with the first
column covering -5 [seconds] for blue and +5 [seconds] for red. This is
reduced in the second and third columns to -3 [seconds] for blue and +3
[seconds] for red. The dominant blue-green color corresponds to 0 [seconds].
The observed images are made antisymmetric in x about the center point in the
middle column and antisymmetric in y about the center point in the right
column. F-cmp˙model
Comparisons between models and data are shown in Figure F-cmp_model for the
15th (of 16) distance range ($\Delta=19.1$ – $22.2^{\circ}$). $\delta
t_{\rm{oi}}$ are shown in the left column, $\delta t_{\rm{we}}$ in the middle
column, and $\delta t_{\rm{ns}}$ in the right column. The 32-day average data
are shown in the top row; model g2 from Article I images are shown in the
second row; and the residual (data – model) are shown in the third line from
the top. The fourth line shows cuts across important parts of the data and
model images. In general, there is good agreement between the data and the ray
theory modeling. The only apparent systematic difference between the data and
the model is in the $\delta t_{\rm{oi}}$ signal near the center. It would seem
that the width or amplitude of the model could be slightly adjusted.
One might imagine that if the five-second peak signal in $\delta t_{\rm{oi}}$
were due to an incorrect kernel, that the quadrant signals might be very
different from the predicted. However, this is not the case. It does not
preclude the possibility that at these large separations all of the ray
kernels are multiplied by some factor. Birch07 have found a case where the ray
kernel is a factor of two larger than the Born-approximation kernel, although
this was at small $\Delta$ and for the fundamental $f-$mode.
To compare the $\delta t_{\rm{we}}$ and $\delta t_{\rm{ns}}$ signals with
model predictions, the quadrant signals are characterized by the peak signal
in the cuts shown in Figure F-cmp_model. Once the maximum signal is found, the
neighboring two points are used to compute a parabola to refine the value of
the maximum. Model travel-time images are treated the same way as the data.
The three models from Article I are compared with the peak quadrant signals in
Figure F-wens_vs_model. At the upper half of the $\Delta$ range, the observed
signals agree pretty well with the model with peak vertical flow at depth
$2.30\rm{\,Mm}$, the same model that agreed best with the $\delta t_{\rm{oi}}$
signal in Article I. At shorter $\Delta$, the observed signals deviate
significantly from that model and agree better with the more shallow model
with peak vertical flow at depth $1.15\rm{\,Mm}$. Either the precise form of
the model is not right or the kernels are incorrect.
Figure 4.: Measured West–East and North–South quadrant peak travel-time
differences $[\delta t_{\rm{qu}}]$ versus those from the three models from
Article I. The blue symbols and error bars (hardly visible) are for the
$\delta t_{\rm{we}}$ and the red for the $\delta t_{\rm{ns}}$. The central
model (black) is the one that agreed best with the $\delta t_{\rm{oi}}$
signal. $z_{0}$ is the height of the peak vertical flow. On the bottom axis is
shown the separation between center and quadrant $[\Delta]$ and on the top
axis the corresponding turning point depth $[z_{t}]$. F-wens˙vs˙model
### 2.2 Heliocentric Angle Analysis
sec-heliocen Recently it has been found that there are flow artifacts with a
center-to-limb, or heliocentric angle $[\theta]$ dependence [Zhao _et al._
(2012)]. As the center–annulus travel time difference $[\delta t_{\rm{oi}}]$
of $5.1\pm 0.1\rm{\,seconds}$ at distances $\Delta>10^{\circ}$ is a somewhat
surprising signal, it was decided to check whether there is any
$\theta$-dependence, which would indicate an artifact. This is done by
separating the different supergranules into 11 bins based on the heliocentric
angle of the cell center at the central observation time. The bins were chosen
to give roughly equal numbers of features in each bin to attempt to equalize
the errors from the different bins.
The analysis follows the quadrant analysis in Section sec-quad, except that
when the average cross correlations are computed about the supergranular
centers, there are 11 averages computed for the different $\theta$ bins.
Figure 5.: (a) Center–annulus travel-time difference averaged about
supergranule centers for the range of $\Delta=14.4$ – $24^{\circ}$ for
heliocentric angle $\theta<15^{\circ}$. The scale of the colorbar at right is
in seconds. (b) Azimuthal average of (a) (blue; dashed) and for $\theta=48$ –
$54^{\circ}$ (red; solid). The range $r=30$ – $60\rm{\,Mm}$ over which the
offset is averaged is noted by the vertical green lines. Note the offset at
large radii that is smaller (in absolute value) at large $\theta$. This offset
is believed to be an artifact which needs to be removed from the results. (c)
The offset at $r=30$ – $60\rm{\,Mm}$ for the different travel-time definitions
versus $\cos(\theta)$. Blue (dashed) is for the Gabor wavelet phase time
differences. Red (solid) is for the Gizon–Birch phase time differences. (d)
The resultant travel time differences averaged for the 64 12–hour datacubes
corrected for the offset in (c) versus $\cos(\theta)$. The colors are the same
as in (c). In (a) – (d) distances were averaged over the range $\Delta=14.4$ –
$24^{\circ}$. F-tt˙rho
Gizon–Birch travel times are computed for each 12-hour interval and the
results from the 64 intervals are averaged. The reference correlation was
taken as the average over the in and out correlations averaged over the
superposed map, which is of size $9.84^{\circ}$ on a side. There is thus a
separate reference correlation for each $\theta$ bin. This is important as the
reference correlation varies from center to limb, and if it is not taken into
account, spurious results are obtained. With the 11 bins in heliocentric
angle, there was insufficient S/N ratio to compute Gabor-wavelet times for the
individual 12-hour intervals. So, the correlations from the 64 12-hour
intervals were averaged and the Gabor wavelets were fit to the result. With
the travel-time differences computed in this way, the Gabor wavelet and
Gizon–Birch times are almost identical. This should not be too surprising as
the same correlation windows are fit in the two cases. Even the noise in the
resultant travel-time difference maps is highly correlated.
Results for the center–annulus travel-time difference $[\delta t_{\rm{oi}}]$
are presented in Figure F-tt_rho. The average Gabor-wavelet-phase time
difference for the first bin ($\theta<15^{\circ}$) and for the distance range
$\Delta=14.4$ – $24^{\circ}$ is shown in Figure F-tt_rhoa. The normal large
positive signal of $\approx$ five seconds is seen at cell center surrounded by
a negative moat with signal $\approx-$one $\rm{\,seconds}$ corresponding to
the region of downflow of the average cell and also the downflow for
neighboring cells. The overall mottling of the picture has roughly
supergranular scale and is presumably due to incomplete averaging of the
supergranular field. To get a better indication of the average signal, which
we expect to be azimuthally symmetric about cell center, an azimuthal average
of the signal in Figure F-tt_rhoa is shown in Figure F-tt_rhob (blue;dashed).
Also shown in Figure F-tt_rhob in red (solid) is the azimuthal average signal
in the outer $\theta$ bin.
We would expect a zero signal far from cell center. That it is not, at least
for the inner $\theta$ bin, is apparent in Figure F-tt_rhob. This signal,
which we measure as the average on $r=30$ – $60\rm{\,Mm}$, is shown in Figure
F-tt_rhoc for Gabor-wavelet fitting (blue;dashed) and Gizon–Birch times
(red;solid). The variation with $\cos{\theta}$ suggests that this signal is an
artifact that could safely be used to correct the signal at cell center. It
may be related to the center-to-limb flow artifact reported by Zhao12, as a
cell center at disk center will see a $\delta t_{\rm{oi}}$ of roughly the
magnitude shown for the annulus radii used here.
The results in Figure F-tt_rhoc are subtracted from the cell center signal to
yield the corrected cell center signal in Figure F-tt_rhod. Again the Gabor-
wavelet times and the Gizon–Birch times are very close and in addition, no
significant center-to-limb signal is apparent. Fitting a line to the results
yields a slope with an error about equal to the value. The average over
$\theta$, $5.1\rm{\,seconds}$, is consistent with the results of Article I.
### 2.3 Simulations
sec-sim In Article I, a convectively stabilized solar model [Hanasoge and
Duvall (2006)] was used with vertical-flow features with flow peaking at a
depth of $z_{0}=-2.3\rm{\,Mm}$ with Gaussian depth profile with width
$\sigma_{z}=0.82\rm{\,Mm}$ and horizontal Gaussian width
$\sigma_{h}=5.1\rm{\,Mm}$. A global simulation of wave propagation is
performed with wave sources near the surface [Hanasoge, Duvall, and Couvidat
(2007)]. Center to annulus travel-time differences $[\delta t_{\rm{oi}}]$ were
measured from the simulation results as a function of the annulus radius
$[\Delta]$. To obtain travel times similar to the observed $5.1\rm{\,seconds}$
required the peak amplitude of the Gaussian flow to be $338\rm{\,m\,s^{-1}}$.
Ray-theory calculations were made and we found that the $\delta t_{\rm{oi}}$
for the ray theory were 24 % larger, suggesting some problem. However, we had
used the standard Model S [Christensen-Dalsgaard _et al._ (1996)] to do the
ray-theory calculations where we should have been using the convectively
stablilized model, which it turns out makes a significant difference. A
revised version of Figure 2 of Article I is shown in Figure F-sim_vert. On the
distance range $10$ – $24^{\circ}$, the average measured travel-time
difference $[\delta t_{\rm{oi}}]$ is now very close to the ray-theory
prediction. This result shows that vertical flows like the ones suggested in
Article I are correctly modeled by the ray theory, as long as one is using the
correct background model. It does not tell us, however, what might happen to
the corresponding horizontal flows and whether the separation of horizontal
and vertical flows by the $\delta t_{\rm{oi}}$ and $\delta t_{\rm{qu}}$ is
valid.
Figure 6.: Comparison of the center–annulus travel-time differences $[\delta
t_{\rm{oi}}]$ from the linear simulation (blue with symbols) with the travel-
time difference computed from the ray theory with the same flow perturbations
(red line). The model used in the ray-theory computation is the convectively
stablized one used in the simulation. The error bars are computed from the
scatter far from the feature locations. No filtering has been done before the
travel-time measurements. The average travel-time difference in the range
$\Delta=10$ – $24^{\circ}$ has been scaled to match the observationally
determined mean $5.1\rm{\,seconds}$. The same scaling factor is then used to
scale the ray-theory results. F-sim˙vert
To test whether the horizontal- and vertical-flow components are separated by
the measurements of $\delta t_{\rm{oi}}$ and $\delta t_{\rm{qu}}$, a
simulation was done using a flow model identical to model g2 of Article I. The
solar model used is the convectively stabilized one described above and a
Cartesian simulation is done using the SPARC code
(Hanasoge06b,2008;Hanasoge06). 500 features identical to the flow in model g2
are placed randomly in the horizontal plane. Center–annulus and quadrant
travel-time differences are measured as described above. The horizontal
spacing of the simulation is $5.7\rm{\,Mm}$ with $512\times 512$ pixels. The
maximum depth is $104\rm{\,Mm}$. The attempt was to go as deeply as possible
in order to be able to use large distances. In order to be able to put the 500
features over the entire horizontal field and still be able to obtain full
annulus coverage for large separations, the horizontal periodicity of the
simulation was used in the travel-time computation.
The travel-time differences from the simulation and ray-theory computations
are shown in Figure F-sim_oiwe. The travel-time differences are only shown up
to $\Delta=13^{\circ}$, as the simulation was not deep enough to go to larger
$\Delta$. The $\delta t_{\rm{oi}}$ measurements in Figure F-sim_oiwea seem
noisier than expected from the error bars. There is a rough agreement with the
ray theory. The asymptotic limit at $\Delta=24^{\circ}$ is only
$3.1\rm{\,seconds}$, as opposed to the expected $5.1\rm{\,seconds}$. This is
because in the convectively stabilized model the sound speed is modified,
which affects the ray-theory estimate of the travel-time differences in the
integral $\int{{\rm dr}(v/c^{2})}$. There is less general agreement of the
quadrant travel-time differences with the ray theory. Especially for
$\Delta<5^{\circ}$, the ray theory is predicting too large a travel-time
difference, while for $\Delta>10^{\circ}$ the opposite may be the case.
For both the spherical simulation (Figure F-sim_vert) and for the Cartesian
one (Figure F-sim_oiwea), the ray theory has general agreement with the
$\delta t_{\rm{oi}}$ measured from the simulations. This suggests that the ray
theory can adequately predict $\delta t_{\rm{oi}}$ for the range of $\Delta$
examined and for flows at this depth of $z=-2.3\rm{\,Mm}$. For the quadrant
travel times, the situation is more problematic. One question is whether a
more shallow flow peaking at the surface could somehow have its quadrant
signal mimic the $\delta t_{\rm{oi}}$ signal of $5.1\rm{\,seconds}$. This will
need to wait for future work.
Figure 7.: Comparison of travel-time differences measured from a simulation
with ray-theory computations. (a) Center-annulus travel-time differences
$[\delta t_{\rm{oi}}]$ from the simulation (black with crosses, error bars,
and connecting lines) and ray-theory computations of $\delta t_{\rm{oi}}$
(green circles). (b) Quadrant travel-time differences $\delta t_{\rm{we}}$
(blue diamonds) and $\delta t_{\rm{ns}}$ (red crosses) for the simulation and
ray-theory computations for $\delta t_{\rm{we}}$ (green circles). F-sim˙oiwe
### 2.4 Travel Times Versus Temporal Frequency $[\nu]$
sec-nu Another way to test the validity of the ray theory applied to flow
measurements is to measure the travel times versus the temporal frequency
$[\nu]$. To first order, at the same distance $\Delta$, the travel-time
differences due to flows should be constant, according to the ray theory. To
our knowledge, such a test has not been carried out, although the frequency
dependence of travel times has been measured extensively, e.g. Dombroski13.
For the ridge filtering used in that study, a $\nu$-dependence is expected,
however. Such a test was conducted for the 64 12-hour intervals at the largest
distance range used, $\Delta=22.1$ – $24^{\circ}$. The travel-time results
versus $\nu$ are shown in Figure F-vs_freq.
Figure 8.: Peak signals and backgrounds versus the temporal frequency $[\nu]$
for the largest distance band $\Delta=22.1$ – $24^{\circ}$. (a) The
center–annulus travel-time difference $[\delta t_{\rm{oi}}]$ versus the
temporal frequency $[\nu]$ (blue circles). The red horizontal dashed line is
at $5.1\rm{\,seconds}$, the result for the $\nu$-averaged data. The black
crosses and line are ray-path calculations based on model g2 from Article I.
(b) The peak quadrant travel-time difference $[\delta t_{\rm{qu}}]$ (blue
cirlces), where the $\delta t_{\rm{we}}$ and $\delta t_{\rm{ns}}$ signals have
been averaged to reduce the error bar size. The red horizontal dashed line is
at the value for the $\nu$-averaged result, $1.3\rm{\,seconds}$. The black
crosses and curve are ray-path calculations based on model g2 from Article I.
(c) The center–annulus background signal far from the supergranule centers
versus $\nu$ (blue circles). This signal was subtracted from the observed
signal to yield the result in (a). The dashed black line indicates zero. (d)
The background signals for $\delta t_{\rm{we}}$ (blue circles) and $\delta
t_{\rm{ns}}$ (red crosses). These were subtracted from the observed signals
and then the $\delta t_{\rm{we}}$ and $\delta t_{\rm{ns}}$ were averaged to
yield the result in (b). The dashed black line indicates zero. F-vs˙freq
To obtain these results, the datacubes were filtered as before with the phase-
speed filter with constant degree width $\Gamma_{\ell}=400$, and in addition a
frequency filter. Ten separate frequency filters were used. The first seven
had central frequencies from $\nu=2.5\rm{\,mHz}$ to $\nu=5.5\rm{\,mHz}$ in
steps of $0.5\rm{\,mHz}$ with power full width at half maximum of
$0.5\rm{\,mHz}$. The last three filters had central frequencies of
$\nu=6,7,8\rm{\,mHz}$ with power full width at half maximum of
$1.0\rm{\,mHz}$. Center–annulus and quadrant cross-correlation maps were
constructed for each of the 64 12-hour intervals. Average correlations were
used to measure guess times for the different $\nu$ intervals. Gizon–Birch
travel-time maps were computed for the 64 intervals and the results were
averaged over $\Delta$ and the various differences computed. Average maps
about the supergranulation centers were made. Background signals were measured
far from the central feature for each $\nu$ for the different signals. These
are shown in Figure F-vs_freqc and F-vs_freqd. These are subtracted from the
peak signals observed to yield the results for $\delta t_{\rm{oi}}$ in Figure
F-vs_freqa and for the two quadrant signals subsequently averaged to yield
Figure F-vs_freqb.
The center–annulus travel time differences $[\delta t_{\rm{oi}}]$ in Figure
F-vs_freqa are clearly not constant with frequency but increase by a factor of
$\approx$two over the $\nu$ range. For $\nu<\nu_{\rm ac}$, where $\nu_{\rm
ac}\approx 5\rm{\,mHz}$ is the peak acoustic-cutoff frequency of the
atmosphere, the variation is approximately linear with a zero intercept. For
$\nu>\nu_{\rm ac}$, the increase with frequency is somewhat smaller. There
have not been any predictions of flow travel times for $\nu>\nu_{\rm ac}$, but
it was feasible to measure them, and so it was done. Ray-theory calculations
of the travel-time differences are also shown in Figure F-vs_freqa and Figure
F-vs_freqb. This result casts some doubt on the simple interpretation of the
$\delta t_{\rm{oi}}$ measurements in terms of the ray theory. The acoustic-
cutoff frequency used in the ray calculations is Lamb’s definition
$\omega_{c}=c/2H_{p}$ Lamb (1909), where $c$ is the sound speed and $H_{p}$ is
the pressure scale height. It is interesting that the peak quadrant travel
times $[\delta t_{\rm{qu}}]$ in Figure F-vs_freqb are approximately constant
over the $\nu$ range, although with a mean signal of $1.3\rm{\,seconds}$, the
uncertainty in this conclusion is large. The ray-theory estimates are
consistent with the measurements for the quadrant travel times.
The background signal for the $\delta t_{\rm{oi}}$ shown in Figure F-vs_freqc
is very interesting. It is measured far from the central location, but is
basically consistent with taking the average signal over a large area. In the
trapped mode region $[\nu<\nu_{\rm ac}]$, the signal is relatively small and
negative, while for $\nu>\nu_{\rm ac}$, it becomes positive and increases with
$\nu$. We have no particular hypothesis for the source of this signal, but
this additional information on the $\nu$ variation may be important for
understanding it. The background for the quadrant signals $[\delta
t_{\rm{we}}$ and $\delta t_{\rm{ns}}]$ are shown in Figure F-vs_freqd. These
signals are measured far from the central location in the North–South
direction for the $\delta t_{\rm{we}}$ and in the West–East direction for the
$\delta t_{\rm{ns}}$ signal. The $\delta t_{\rm{we}}$ signal has a linear
variation over the frequency range. This background signal is possibly due to
solar rotation, although it would imply that the solar rotation yields a
$\nu$-dependent $\delta t_{\rm{we}}$ while the supergranular signal does not.
Whatever causes background signals, they need to be removed from the
supergranular signal.
## 3 Conclusions
sec-dis
The bulk of the evidence in the present article continues to support a model
of the average supergranulation cell as having an upflow with a velocity much
larger than the surface upflow of $10\rm{\,m\,s^{-1}}$, possibly as large as
$240\rm{\,m\,s^{-1}}$ and a peak flow $2$ – $3\rm{\,Mm}$ below the surface as
seen in the best model g2. In Article I, center–annulus travel time
differences $[\delta t_{\rm{oi}}]$ were shown to agree well with model g2,
while in the present article, the quadrant travel-time differences $[\delta
t_{\rm{we}}$ and $\delta t_{\rm{ns}}]$ also agree mostly with this type of
model. However, there is some disagreement that varies with $\Delta$ for the
$\delta t_{\rm{qu}}$, suggesting that either the functional form of the model
needs to be adjusted or the ray kernels are incorrect, or both.
The apparent disagreement between the present work and the smaller flows seen
before has largely disappeared with the work of Svanda12. That article does an
analysis of average supergranules similar to the averaging done in the present
article. He used $f$-modes and small separation p-modes and finds flows that
largely confirm the present results. There may be a factor of two difference
between the two results, which needs to be resolved.
The lack of a center-to-limb variation of the $\delta t_{\rm{oi}}$ signal is
useful for a general check of systematic errors. However, the $\nu$-dependence
of the $\delta\tau_{\rm oi}$ signal as predicted by ray theory differs from
observations, suggesting that ray theory may be inaccurate in near-surface
layers. The errors may be due to unmodeled finite-frequency effects or
possibly differences in the acoustic-cutoff frequency between the Sun and
Model S (used in ray modelling here). That the acoustic-cutoff frequency may
differ from that derived from Model S was shown by Jefferies94. In a model
with the reflection point that is a significant function of frequency, the
travel-time difference $[\delta t_{\rm{oi}}]$ could then also be a function of
frequency. At a minimum, the variation in $\delta t_{\rm{oi}}$ by a factor of
roughly two over the $\nu$-range observed would seem to make the suggested
flows uncertain by a similar factor.
The simulation results (Section sec-sim) show that the type of model
considered (model g2) does induce the kind of travel-time shifts observed.
These are then seen by both the travel-time shifts measured from the
simulation and by ray theory calculated with the model used. It is unfortunate
that the modification to the solar model to stablize it has such a large
effect on the resulting $\delta t_{\rm{oi}}$.
Some of the earlier work finds the flow velocity peaking very near the surface
with a monotonic decrease with depth Birch _et al._ (2006); Woodard (2007).
These results would appear to be inconsistent with this article and Article I.
It was suggested in Article I that the perturbations to the p-mode spectrum
due to supergranulation are significant in $\ell$, the spherical-harmonic
degree. The idea is that the supergranulation pattern, with a spectrum peaking
near $\ell=120$ would induce a modulation of a $p$-mode ridge that would have
a width in $\ell$ of at least twice this amount. To capture all of the
supergranulation signal requires a filter of at least a full-width-half-max of
$\Gamma_{\ell}=240$ and likely larger. This justifies the value chosen for the
present work of $\Gamma_{\ell}=400$, which clearly captures all of the $\delta
t_{\rm{oi}}$ signal at large $\Delta$. This conclusion is supported by Figure
4 of Article I. Of course, if the modeling is correct, one can use any filter.
However, if much of the supergranulation signal is not being captured, one
becomes much more sensitive to the modeling. In Woodard07, the filters are
narrower than used here, particularly at low frequencies.
Because the acoustic wavelength at the depth of the peak flow is larger than
the depth of the peak flow, ray theory may show inaccuracies. To improve the
quality of the flow model, it would therefore be useful to include finite-
frequency effects Birch and Gizon (2007). Further, the functional dependence
of travel times on the background-flow model may exceed the linear limit if
flow speeds are indeed on the order of $700\rm{\,m\,s^{-1}}$. Therefore a non-
linear inversion for supergranular flow may be necessary to explain the
measured travel times Hanasoge _et al._ (2011).
#### Acknowledgements
The data used here are courtesy of NASA/SDO and the HMI Science Team. We thank
the HMI team members for their hard work. This work is supported by NASA SDO.
## References
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* Jefferies _et al._ (1994) Jefferies, S.M., Osaki, Y., Shibahashi, H., Duvall, T.L. Jr., Harvey, J.W., Pomerantz, M.A.: 1994, Astrophys. J. 434, 795.
* Lamb (1909) Lamb, H.: 1909, Proc. London Math. Soc. (2)7, 122.
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* Stein _et al._ (2006) Stein, R.F., Benson, D., Georgobiani, D., Nordlund, Å.: 2006, in: Fletcher, K., Thompson, M. eds., Proceedings of SOHO 18/GONG 2006/HELAS I, Beyond the spherical Sun ESA SP-624, 79.1
* Švanda _et al._ (2012) Švanda, M.: 2011, Astron. & Astrophys. 530, A148.
* Švanda (2012) Švanda, M.: 2012, Astrophys. J. 759, L29.
* Woodard (2007) Woodard, M.F.: 2007, Astrophys. J. 668, 1189.
* Zhao and Kosovichev (2003) Zhao, J., Kosovichev, A.G.: 2003, In: Sawaya-Lacoste, H. ed., GONG+ 2002. Local and Global Helioseismology: the Present and Future SP-517, ESA, 417.
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|
arxiv-papers
| 2014-04-09T16:25:34 |
2024-09-04T02:50:00.912142
|
{
"license": "Public Domain",
"authors": "T.L. Duvall Jr., S.M. Hanasoge, S. Chakraborty",
"submitter": "Thomas Duvall Jr.",
"url": "https://arxiv.org/abs/1404.2533"
}
|
1404.2574
|
# Cyclic groups with the same Hodge series
Daryl R. DeFord Peter G. Doyle
(Version dated 8 April 2014
No Copyright††thanks: The authors hereby waive all copyright and related or
neighboring rights to this work, and dedicate it to the public domain. This
applies worldwide. )
###### Abstract
The Hodge series of a finite matrix group is the generating function
$\sum_{k,p}x^{k}y^{p}$ for invariant exterior forms of specified order $p$ and
degree $k$. Lauret, Miatello, and Rossetti gave examples of pairs of non-
conjugate cyclic groups having the same Hodge series; the corresponding space
forms are isospectral for the Laplacian on $p$-forms for all $p$, but not for
all natural operators. Here we explain, simplify, and extend their
investigations.
## 0 Terminology and notation
We adopt terminology and notation to avoid some common headaches.
### ‘Just if’.
We follow John Conway in using ‘just if’ in place of the more cumbersome ‘if
and only if’.
### Modular arithmetic.
$a\equiv_{q}b$ means $a$ is equivalent to $b$ mod $q$. We write
$\mathbf{Z}_{q}$ for $\mathbf{Z}/q\mathbf{Z}$, and $\mathbf{Z}^{\star}_{q}$
for its invertible elements, taking $\mathbf{Z}^{\star}_{1}=\\{0\\}$. For
$a\in\mathbf{Z}_{q}$, $b\in\mathbf{Z}^{\star}_{q}$ we write $a/_{q}\,b$ for
the quotient mod $q$.
### Angles; roots of unity.
We use $\tau=2\pi$ in representing angles, because as Vi Hart [5] has so
persuasively argued, _$\pi$ is wrong_. We write
$\omega_{q}=\exp(i\tau/q)$
for the standard $q$th root of unity, so that
$\omega_{q}^{k}=\exp(i\tau k/q)=e^{i\tau\frac{k}{q}}.$
### Unitary and orthogonal groups; conjugacy.
As usual we write $U_{n}\subset GL_{n}(\mathbf{C})$ for the $n$-by-$n$ unitary
matrices, and $O_{n}=U_{n}\cap GL_{n}(\mathbf{R})$ for the orthogonal
matrices.
When we say that two matrices or groups are ‘conjugate’, we mean that they are
conjugate within $GL_{n}(\mathbf{C})$, so that the conjugating matrix can be
any invertible complex matrix. Allowing this generality for the conjugating
matrix is no big deal, because unitary matrices or groups that are conjugate
within $GL_{n}(\mathbf{C})$ are already conjugate within $U_{n}$; real
matrices or groups that are conjugate within $GL_{n}(\mathbf{R})$ are already
conjugate within $GL_{n}(\mathbf{R})$; orthogonal matrices or groups that are
conjugate within $GL_{n}(\mathbf{C})$ are already conjugate within $O_{n}$.
We will be dealing with finite groups of matrices, which we will be interested
in only up to conjugacy. Any finite group of complex matrices is conjugate to
a subgroup of $U_{n}$; any finite group of real matrices is conjugate to a
subgroup of $O_{n}$. So we may take our groups to be unitary—and if real,
orthogonal—without sacrificing generality.
## 1 Hodge series
Let $G\subset U_{n}$ be a finite group of $n$-by-$n$ complex matrices, assumed
to be unitary. Any $g\in G$ is diagonalizable, with the roots
$\lambda_{1},\ldots,\lambda_{n}$ of its characteristic polynomial
$\chi_{g}(x)=\det(I_{n}-xg)=\prod_{i}(x-\lambda_{i})$
being roots of unity.
Define the _Hodge series_
$\displaystyle\Lambda_{G}(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{|G|}\sum_{g}\frac{\det(I_{n}+yg)}{\det(I_{n}-xg)}$
$\displaystyle=$
$\displaystyle\frac{1}{|G|}\sum_{g}\frac{y^{n}\chi_{g}(-1/y)}{x^{n}\chi_{g}(1/x)}.$
This series is a particular kind of _Molien series_ : Crass [1, p. 31] calls
it the ‘exterior Molien series’. By a generalization of Molien’s theorem (cf.
Molien [9], Stanley [11]) this is the generating function for $G$-invariant
exterior forms:
$\Lambda_{G}(x,y)=\sum_{p,k}P_{k}^{p}x^{k}y^{p},$
where $P_{k}^{p}$ is the dimension of the space of $G$-invariant $p$-forms
whose coefficients are homogeneous polynomials of degree $k$ in
$x_{1},\ldots,x_{n}$.
For example we have
$\Lambda_{\\{I_{n}\\}}=\frac{(1+y)^{n}}{(1-x)^{n}}$
and
$\displaystyle\Lambda_{\\{\pm I_{n}\\}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{(1+y)^{n}}{(1-x)^{n}}+\frac{(1-y)^{n}}{(1+x)^{n}}\right)$
$\displaystyle=$
$\displaystyle\frac{\frac{1}{2}((1+x^{n})(1+y^{n})+(1-x^{n})(1-y^{n}))}{(1-x^{2})^{n}}.$
Aside. Here’s a more interesting example. The group $G_{120}$ of proper and
improper symmetries of the icosahedron in Euclidean 3-space has Hodge series
$\Lambda_{G_{120}}=\frac{(1+xy)(1+x^{5}y)(1+x^{9}y)}{(1-x^{2})(1-x^{6})(1-x^{10})}.$
As this might suggest, the algebra of invariant forms is generated by
polynomial invariants of degrees $2,6,10$ and their exterior derivatives of
degrees $1,5,9$. For the index-2 subgroup $G_{60}$ of proper symmetries we
have
$\Lambda_{G_{60}}=\frac{(1+x^{15})(1+y^{3})+(x+x^{5}+x^{6}+x^{9}+x^{10}+x^{14})(y+y^{2})}{(1-x^{2})(1-x^{6})(1-x^{10})}.$
This is a little harder to decipher, though the generating function for
invariant polynomials, obtained by setting $y=0$, is clear enough:
$\Lambda_{G_{60}}(x,0)=\frac{1+x^{15}}{(1-x^{2})(1-x^{6})(1-x^{10})}.$
Here we see the $G_{120}$-invariants of degrees 2,6,10, together with a new
invariant of degree $15$ (the product of the linear forms determining the $15$
planes of symmetry of the icosahedron) whose square is $G_{120}$-invariant,
though it itself is only $G_{60}$-invariant.
###### Exercise 1.
Compute these two Hodge series.
Hint. Resist the temptation to consult Klein [7] or Doyle and McMullen [2]:
You do not need to know the matrix groups explicitly, because the contribution
of a matrix to the Hodge series depends only on its conjugacy class. This fact
is the basis of the notion of ‘almost-conjugacy’ of groups, which we’ll get to
in a jiffy.
## 2 Hodge equivalence
We are interested in pairs of groups $G,H\subset U_{n}$ (and in particular,
pairs of real groups $G,H\subset O_{n}$) having the same Hodge series, meaning
that they have the same dimensions of spaces of invariant forms. We call such
pairs _Hodge-equivalent_ , and write $G\equiv_{\Lambda}H$.
Of course if the groups $G$ and $H$ are conjugate then they are Hodge-
equivalent. More generally, say that $G$ and $H$ are _almost conjugate_ if
there is a bijection $\sigma:G\rightarrow H$ such that $g$ and $\sigma(g)$ are
conjugate. This is the same as requiring that $g$ and $\sigma(g)$ have the
same eigenvalues, so that they make the same contribution to the Hodge series.
Thus almost conjugate groups are Hodge-equivalent.
It is relatively easy to find non-conjugate pairs $(G,H)$ that are almost
conjugate, and hence Hodge-equivalent. (See Gilkey [4], Ikeda [6].) Here we
are interested in pairs (and specifically, real pairs) that are Hodge-
equivalent without being almost conjugate. The first such examples were given
in [8] by Lauret, Miatello, and Rossetti (henceforth ‘LMR’). They exhibited a
multitude of examples arising already among cyclic groups. For cyclic groups,
conjugacy is the same as almost-conjugacy, so their examples can be briefly
described as being Hodge-equivalent without being conjugate. Our goal here is
to explain, simplify, and extend their findings.
## 3 Isospectrality
We discuss here the connection to spectral theory, which is what motivated LMR
to construct their examples. This is meant for background only: In the
approach taken here, spectral theory plays no role. In this section we
restrict to real groups, which we may assume to be orthogonal.
A finite real group $G\subset O_{n}$ is classified up to conjugacy by the
isometry type of the quotient orbifold $Q_{G}=G\backslash S^{n-1}$. According
to Ikeda [6], $G$ and $H$ are Hodge-equivalent just if the quotients $Q_{G}$
and $Q_{H}$ are isospectral for the Hodge Laplacian on $p$-forms for
$p=0,\ldots,n-1$. According to Pesce [10], $Q_{G}$ and $Q_{H}$ are _strongly
isospectral_ (isospectral for all natural operators of a certain kind) just if
$G$ and $H$ are almost conjugate. Using this dictionary, looking for Hodge-
equivalent groups that are not almost conjugate is the same as looking for
Hodge-isospectral orbifolds that are not strongly isospectral. This is why LMR
were interested in this question.
Ikeda, Pesce, and LMR restricted their investigations to the case of groups
whose action on $S_{n-1}$ is fixed-point free (no $g\neq 1$ has $1$ as an
eigenvalue). In this case $Q_{G}$ is a manifold, called a _spherical space
form_. For $n$ odd (i.e. $n-1$ even) we have only the sphere
$\\{I_{n}\\}\backslash S^{n-1}$ and projective space $\\{\pm
I_{n}\\}\backslash S^{n-1}$. So the restriction to fixed-point free actions
effectively limits us to the case of even $n$.
When $G$ is cyclic, as in the LMR examples, $Q_{G}$ is a _lens space_. As
observed above, for cyclic groups almost-conjugacy is the same as conjugacy,
which is the same as isometry of the corresponding lens space. So we can
briefly describe the LMR examples as Hodge-isospectral lens spaces that are
not isometric, hence not almost conjugate, hence not strongly isospectral.
This ends our discussion of isospectrality. The rest is algebra.
## 4 Cyclic groups
For any $q$ and $s=(s_{1},\ldots,s_{n})$, write
$\omega_{q}^{s}=(\omega_{q}^{s_{1}},\ldots,\omega_{q}^{s_{n}}).$
Consider the finite cyclic group
$L(q,s)=\langle\mathrm{diag}(\omega_{q}^{s})\rangle=\\{\mathrm{diag}(\omega_{q}^{ks}):k\in\mathbf{Z}_{q}\\}.$
This group has order $q$ if $\gcd(q,s_{1},\ldots,s_{n})=1$. Up to conjugacy,
any finite cyclic subgroup of $U_{n}$ can be written in this way.
The cyclic group $L(q,s)$ doesn’t change (up to conjugacy) when you rearrange
the entries of $s$, or multiply them all by an element of the multiplicative
group $\mathbf{Z}^{\star}_{q}$. Conversely, the groups $L(q,s)$ and
$L(q,s^{\prime})$ are conjugate just if, when viewed as multisets mod $q$,
$s^{\prime}$ can be obtained from $s$ by multiplying by an invertible element.
Now take $n=2m$, and let $\rho:GL_{m}(\mathbf{C})\mapsto GL_{2m}(\mathbf{R})$
be the standard embedding, so that $\rho(\omega_{q}^{s})$ is the diagonal sum
of the 2-by-2 matrices
$\displaystyle\rho(((\omega_{q}^{s_{i}})))$ $\displaystyle=$
$\displaystyle\exp(\tau\frac{s_{i}}{q}((0,-1),(1,0)))$ $\displaystyle=$
$\displaystyle((\cos(\tau s_{i}/q),-\sin(\tau s_{i}/q)),(\sin(\tau
s_{i}/q),\cos(\tau s_{i}/q))).$
Up to conjugacy in $U_{n}$,
$\rho(L(q,s))\equiv L(q,{s^{\pm}})$
where
${s^{\pm}}=(s_{1},-s_{1},\ldots,s_{m},-s_{m}).$
Let us write
$L^{\pm}(q,s)=L(q,{s^{\pm}})\equiv\rho(L(q,s)).$
For the Hodge series we have
$\displaystyle\Lambda_{L^{\pm}(q,s)}$ $\displaystyle=$
$\displaystyle\frac{1}{q}\sum_{k}\prod_{i}\frac{(1+y\omega_{q}^{ks_{i}})(1+y\omega_{q}^{-ks_{i}})}{(1-x\omega_{q}^{ks_{i}})(1-x\omega_{q}^{-ks_{i}})}$
$\displaystyle=$ $\displaystyle\frac{1}{q}\sum_{k}\prod_{i}\frac{1+2\cos(\tau
ks_{i}/q)y+y^{2}}{1-2\cos(\tau ks_{i}/q)x+x^{2}}.$
## 5 The LMR construction
The LMR examples involve cyclic subgroups of $O_{2m}$ of the form
$\rho(L(r^{2}t,rta+1))\equiv L^{\pm}(r^{2}t,rta+1),$
where $r>2$, $t\geq 1$, $a=(a_{1},\ldots,a_{m})\in\mathbf{Z}^{m}$. Since we
prefer to keep our matrices diagonal we’ll define
$\displaystyle\mathrm{LMR}(r,t,a)$ $\displaystyle=$ $\displaystyle
L^{\pm}(r^{2}t,rta+1)$ $\displaystyle=$ $\displaystyle
L(r^{2}t,(rta_{1}+1,-rta_{1}-1,\ldots,rta_{m}+1,-rta_{m}-1)).$
Note. You may wish to mentally set $t=1$: All evidence indicates that what
works for $t=1$ works in general, and in particular the criterion in Theorem 1
below does not involve $t$.
As we will be seeing, what’s special about the LMR construction is the
following fact:
$(rtc+1)(rtd+1)\equiv_{r^{2}t}rt(c+d)+1.$
Thus the multiplicative subgroup
$\\{rtc+1:c\in\mathbf{Z}_{r}\\}\subset\mathbf{Z}^{\star}_{r^{2}t}$ is cyclic
of order $r$, generated by $rt+1$; the map $rtc+1\mapsto c$ takes the
logarithm of $rtc+1$ base $rt+1$, and gives an isomorphism to the additive
group $\mathbf{Z}_{r}$.
As a first consequence of this, notice that we can add a constant $c$ to the
entries of $a$ without changing the conjugacy class:
$\mathrm{LMR}(r,t,a)\equiv\mathrm{LMR}(r,t,a+c).$
In fact, this characterizes all such coincidences: Let us write
$a\equiv_{S_{m}\times\mathbf{Z}_{r}}a^{\prime}$
if for some $c$, $a+c$ and $a^{\prime}$ are the same as multisets mod $r$.
Then $\mathrm{LMR}(r,t,a)\equiv\mathrm{LMR}(r,t,a^{\prime})$ just if
$a\equiv_{S_{m}\times\mathbf{Z}_{r}}a^{\prime}$.
All the LMR pairs are (conjugate to) pairs of the special form
$(\mathrm{LMR}(r,t,a),\mathrm{LMR}(r,t,-a)).$
Not all such pairs are Hodge-equivalent, however.
## 6 Theorem
In this section we formulate a criterion for Hodge-equivalence of the LMR pair
$(\mathrm{LMR}(r,t,a),\mathrm{LMR}(r,t,-a))$. While this criterion has not
been shown to be necessary, it holds in all the cases (thousands and
thousands!) where the LMR construction has been found to succeed.
###### Definition 1.
Say that $a=(a_{1},\ldots,a_{m})$ is:
* •
_univalent mod $r$_ if its entries are distinct mod $r$;
* •
_reversible mod $r$_ if $a\equiv_{S_{m}\times\mathbf{Z}_{r}}-a$;
* •
_good mod $r$_ if it is univalent or reversible mod $r$;
* •
_hereditarily good mod $r$_ if it is good mod $d$ for all $d$ dividing $r$;
* •
_useful mod $r$_ if it is hereditarily good and irreversible mod $r$.
Any $a$ is reversible (hence good) mod $1$ or $2$. So in checking hereditary
goodness we need only check divisors $d>2$.
In section 8 below we will prove the following:
###### Theorem 1.
If $a$ is hereditarily good mod $r$ then for any $t$,
$\mathrm{LMR}(r,t,a)\equiv_{\Lambda}\mathrm{LMR}(r,t,-a).$
If $a$ is reversible mod $r$ then $a$ is hereditarily good mod $r$, but in
this case $\mathrm{LMR}(r,t,a)$ and $\mathrm{LMR}(r,t,-a)$ are conjugate. So
this result tells us something useful only if $a$ is hereditarily good without
being reversible, which is our definition of ‘useful’.
## 7 Examples
$(0,1,3)$ is:
* •
univalent mod $4,5,6,\ldots$;
* •
reversible mod $1,2,4,5$;
* •
good mod any $r\neq 3$;
* •
hereditarily good mod any $r$ not divisible by $3$;
* •
useful mod any $r\geq 7$ not divisible by $3$.
Putting $r=7,8,10$, $t=1$, we get Hodge-equivalent but non-conjugate pairs of
orders $49,64,100$; Putting $r=7$, $t=2$ we get a pair of order $98$.
$(0,1,4)$ is:
* •
univalent mod $5,6,7,\ldots$;
* •
reversible mod $1,2,5,7$;
* •
good mod any $r\neq 3,4$;
* •
hereditarily good mod any $r$ not divisible by $3$ or $4$;
* •
useful mod any $r\geq 10$ not divisible by $3$ or $4$.
Putting $r=10$, $t=1$ gives a pair of order 100. Together with the four pairs
coming from $(0,1,3)$ above, this gives us all five inequivalent pairs with
$m=3$, $q\leq 100$ (see Table 1 of LMR [8]).
We’ll call the simplest of these pairs the _$49$ -pair_:
$\displaystyle(\mathrm{LMR}(7,1,(0,1,3)),\mathrm{LMR}(7,1,(0,-1,-3)))$
$\displaystyle=$ $\displaystyle(L^{\pm}(49,(1,8,22)),L^{\pm}(49,(1,-6,-20))$
$\displaystyle=$
$\displaystyle(L(49,(1,-1,8,-8,22,-22)),L(49,(1,-1,-6,6,-20,20)))$
$\displaystyle\equiv$
$\displaystyle(L(49,(-6,6,1,-1,15,-15)),L(49,(1,-1,-6,6,-20,20)))$
$\displaystyle\equiv$
$\displaystyle(L^{\pm}(49,(1,6,15)),L^{\pm}(49,(1,6,20))).$
Here at the next-to-last step we’ve multiplied the list $(1,-1,8,-8,22,-22)$
by $-6$ mod $49$ so as to get the lexicographically least representation that
the computer spits out in its search for Hodge-equivalent pairs.
## 8 Proof
It is easy enough to verify that the members of the $49$-pair are Hodge-
equivalent by explicit computation of their Hodge series. The same goes for as
many other pairs as you like, but this only gets you a finite number of
examples.
Using a very explicit representation theory argument, LMR proved Hodge-
equivalence of the 49-pair in a way that extends to cover all pairs of the
form
$(\mathrm{LMR}(r,t,(0,1,3)),\mathrm{LMR}(r,t,(0,-1,-3)))$
with $r$ not divisible by $3$. As we have seen, this infinite family is just
what we get out of Theorem 1 if we take $a=(0,1,3)$. It includes $19$ of the
$62$ examples in the list given by LMR of all pairs with $m=3$ and $q\leq
300$.
To prove Theorem 1 in its full generality, we’re going to show that the two
Hodge series involved are identical as rational functions of $x$ and $y$. This
comes down to a bunch of manipulations with partial fraction expansions. It
all starts with the following familiar identity.
###### Lemma 1.
$\prod_{i=1}^{n}\frac{1}{x-\lambda_{i}}=\sum_{i=1}^{n}\frac{1}{x-\lambda_{i}}\prod_{j\neq
i}\frac{1}{\lambda_{i}-\lambda_{j}}.$
Proof. This follows from the theory of partial fractions.
Alternatively, combine terms on the right over the common denominator
$\prod_{i}(x-\lambda_{i})$. The numerator is
$\sum_{i}\prod_{j\neq i}\frac{x-\lambda_{j}}{\lambda_{i}-\lambda_{j}}.$
This is a polynomial of degree $n-1$ which takes the value $1$ for
$x=\lambda_{1},\ldots,\lambda_{n}$. These $n$ values of $x$ are distinct
(thinking of the $\lambda_{i}$’s as indeterminates), so the numerator is
identically $1$. $\quad\qed$
Proof of Theorem 1. For general $q$, $s\in(\mathbf{Z}_{q})^{m}$ put
$H_{q,s}(x,y)=\sum_{k\in\mathbf{Z}_{q}}\prod_{i}\frac{y-\omega_{q}^{ks_{i}}}{x-\omega_{q}^{ks_{i}}}$
so that
$\Lambda_{L(q,s)}(x,y)=\frac{1}{q}\frac{y^{n}}{x^{n}}H_{q,s}(-1/y,1/x).$
Separate the sum for $H_{q,s}$ into pieces according to $\gcd(k,q)$ by putting
$H^{\star}_{d,s}=\sum_{k\in\mathbf{Z}^{\star}_{d}}\prod_{i}\frac{y-\omega_{d}^{ks_{i}}}{x-\omega_{d}^{ks_{i}}}$
so that
$H_{q,s}=\sum_{d\,|\,q}H^{\star}_{d,s}.$
To prove the theorem, we must show that if $a$ is hereditarily good mod $r$
then
$H_{r^{2}t,{(rta+1)^{\pm}}}=H_{r^{2}t,{(-rta+1)^{\pm}}}.$
Our strategy will be to show that for all $d\,|\,r^{2}t$ we have
$H^{\star}_{d,{(rta+1)^{\pm}}}=H^{\star}_{d,{(-rta+1)^{\pm}}}.$
We dispose first of the case where ${(rta+1)^{\pm}}$ is not univalent mod $d$.
This is taken care of by the assumption that $a$ is hereditarily good, but
things are not quite as straight-forward as you might be expecting, because
that condition deals with divisors of $r$, and here $d$ is any divisor of
$r^{2}t$.
We pass over the trivial cases $d=1,2$. Mod any $d>2$, there is no overlap
between $rta+1$ and $-(rta+1)$, so if ${(rta+1)^{\pm}}$ is not univalent mod
$d$ then neither is $rta$.
We pause for a lemma.
###### Lemma 2.
For $d,\alpha,\beta\in\mathbf{Z}$, suppose $d\,|\,\alpha\beta$. Let
$d^{\prime}=d/\gcd(d,\beta)$. Then $d^{\prime}\,|\,\alpha$ and
$\forall\gamma\in\mathbf{Z}\;(d\,|\,\beta\gamma\iff d^{\prime}\,|\,\gamma).$
Proof. Let $e=\gcd(d,\beta)$, so that $d^{\prime}=d/e$.
$d\,|\,\alpha\beta\Rightarrow d^{\prime}=d/e\,|\,\alpha\beta/e$
and $\gcd(d^{\prime},\beta/e)=1$ so $d^{\prime}\,|\,\alpha$, and for any
$\gamma$
$d\,|\,\beta\gamma\iff d^{\prime}\,|\,\beta/e\gamma\iff
d^{\prime}\,|\,\gamma.$
(Pretty standard stuff, admittedly.) $\quad\qed$
So suppose $d\,|\,r^{2}t$ and $d\,|\,rt(a_{i}-a_{j})$ for $i\neq j$. Putting
$\alpha=r$, $\beta=rt$, $\gamma=a_{i}-a_{j}$ in the lemma we get
$d^{\prime}=d/\gcd(d,rt)\,|\,\alpha=r$
and
$d^{\prime}\,|\,\gamma=a_{i}-a_{j}.$
This tells us that $a$ is not univalent mod $d^{\prime}$, but since by
assumption it is good mod any divisor of $r$, it must be reversible mod
$d^{\prime}$:
$a\equiv_{S_{m}\times\mathbf{Z}_{d^{\prime}}}-a.$
By the lemma, this is equivalent to
$rta\equiv_{S_{m}\times\mathbf{Z}_{d}}-rta$
hence
$rta+1\equiv_{S_{m}\times\mathbf{Z}_{d}}-rta+1.$
From this we get
$H^{\star}_{d,{(ra+1)^{\pm}}}=H^{\star}_{d,{(-ra+1)^{\pm}}}.$
So from here on we may assume that ${(rta+1)^{\pm}}$ (and hence also
${(-rta+1)^{\pm}}$) is univalent mod $d$, with $d\,|\,r^{2}t$.
Returning for a moment to the case of $H^{\star}_{d,s}$ for general $d,s$,
suppose $s\in(\mathbf{Z}^{\star}_{d})^{m}$ with all the $s_{i}$’s distinct mod
$q$, so that the mod-$d$ quotient $s_{j}/_{d}\,s_{i}$ is defined for all
$i,j$, and different from $1$ for $i\neq j$. With this restriction, for
$k\in\mathbf{Z}^{\star}_{d}$ we have
$\prod_{i}\frac{y-\omega_{d}^{ks_{i}}}{x-\omega_{d}^{ks_{i}}}=\sum_{i}\frac{y-\omega_{d}^{ks_{i}}}{x-\omega_{d}^{ks_{i}}}\prod_{j\neq
i}\frac{y-\omega_{d}^{ks_{j}}}{\omega_{d}^{ks_{i}}-\omega_{d}^{ks_{j}}}.$
So
$\displaystyle H^{\star}_{d,s}$ $\displaystyle=$
$\displaystyle\sum_{k\in\mathbf{Z}^{\star}_{d}}\sum_{i}\frac{y-\omega_{d}^{ks_{i}}}{x-\omega_{d}^{ks_{i}}}\prod_{j\neq
i}\frac{y-\omega_{d}^{ks_{j}}}{\omega_{d}^{ks_{i}}-\omega_{d}^{ks_{j}}}$
$\displaystyle=$
$\displaystyle\sum_{l\in\mathbf{Z}^{\star}_{d}}\frac{y-\omega_{d}^{l}}{x-\omega_{d}^{l}}\sum_{i}\prod_{j\neq
i}\frac{y-\omega_{d}^{ls_{j}/s_{i}}}{\omega_{d}^{l}-\omega_{d}^{ls_{j}/_{d}\,s_{i}}}$
$\displaystyle=$
$\displaystyle\sum_{l\in\mathbf{Z}^{\star}_{d}}Y_{d,s}(x,y,\omega_{d}^{l}),$
where
$Y_{d,s}(x,y,w)=\frac{y-w}{x-w}\sum_{i}\prod_{j\neq
i}\frac{y-w^{s_{j}/_{d}\,s_{i}}}{w-w^{s_{j}/_{d}\,s_{i}}}.$
Now recall our notation
${s^{\pm}}=(s_{1},-s_{1},\ldots,s_{m},-s_{m}).$
Assuming the entries of ${s^{\pm}}$ are all invertible and distinct mod $d$,
$Y_{d,{s^{\pm}}}=\frac{(y-w)(y-w^{-1})}{(x-w)(x-w^{-1})}\sum_{i}\prod_{j\neq
i}\frac{(y-w^{s_{j}/_{d}\,s_{i}})(y-w^{-s_{j}/_{d}\,s_{i}})}{(w-w^{s_{j}/_{d}\,s_{i}})(w-w^{-s_{j}/_{d}\,s_{i}})}.$
Specializing finally to the case at hand, take
$s=rta+1=(rta_{1}+1,\ldots,rta_{m}+1)$
so that
${s^{\pm}}=(rta_{1}+1,-rta_{1}-1,\ldots,rta_{m}+1,-rta_{m}-1),$
and assume that these entries are all distinct mod $d$.
Here comes the magic: For any $d\,|\,r^{2}t$ we have
$s_{j}/_{d}\,s_{i}\equiv_{d}rt(a_{j}-a_{i})+1.$
As the entries of ${s^{\pm}}$ are distinct mod $d$, putting
$x_{i}=w^{rta_{i}}$
we have
$w^{s_{j}/_{d}\,s_{i}}=\frac{x_{j}}{x_{i}}w$
and
$w^{-s_{j}/_{d}\,s_{i}}=\frac{x_{i}}{x_{j}}w^{-1}$
so
$Y_{d,{(ra+1)^{\pm}}}=\frac{(y-w)(y-w^{-1})}{(x-w)(x-w^{-1})}\sum_{i}\prod_{j\neq
i}\frac{(y-\frac{x_{j}}{x_{i}}w)(y-\frac{x_{i}}{x_{j}}w^{-1})}{(w-\frac{x_{j}}{x_{i}}w)(w-\frac{x_{i}}{x_{j}}w^{-1})}.$
Setting $u=y/w$, $v=w^{-2}$, we get
$Y_{r^{2}t,{(rta+1)^{\pm}}}=\frac{(y-w)(y-w^{-1})}{(x-w)(x-w^{-1})}F((x_{1},\ldots,x_{m}),u,v)$
where
$F((x_{1},\ldots,x_{m}),u,v)=\sum_{i=1}^{m}\prod_{j\neq
i}\frac{(u-\frac{x_{j}}{x_{i}})(u-\frac{x_{i}}{x_{j}}v)}{(1-\frac{x_{j}}{x_{i}})(1-\frac{x_{i}}{x_{j}}v)}.$
Simultaneously we have
$Y_{r^{2}t,{(-rta+1)^{\pm}}}=\frac{(y-w)(y-w^{-1})}{(x-w)(x-w^{-1})}F((1/x_{1},\ldots,1/x_{m}),u,v).$
In the next section we will prove the identity
$F((x_{1},\ldots,x_{m}),u,v)=F((1/x_{1},\ldots,1/x_{m}),u,v),$
from which we conclude
$H^{\star}_{d,{(rta+1)^{\pm}}}=H^{\star}_{d,{(-rta+1)^{\pm}}}.$
We have now established this last equality for every $d\,|\,r^{2}t$, so
$H_{d,{(ra+1)^{\pm}}}=H_{d,{(-ra+1)^{\pm}}}.\quad\qed$
## 9 The main identity
Define the rational function
$\displaystyle F((x_{1},\ldots,x_{m}),u,v)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\prod_{j\neq
i}\frac{(u-\frac{x_{j}}{x_{i}})(u-\frac{x_{i}}{x_{j}}v)}{(1-\frac{x_{j}}{x_{i}})(1-\frac{x_{i}}{x_{j}}v)}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{m}\prod_{j\neq
i}\frac{(x_{i}u-x_{j})(x_{j}u-x_{i}v)}{(x_{i}-x_{j})(x_{j}-x_{i}v)}.$
Now look at what you get by replacing the variables $x_{1},\ldots,x_{m}$ by
their reciprocals:
$\displaystyle G((x_{1},\ldots,x_{m}),u,v)$ $\displaystyle=$ $\displaystyle
F((1/x_{1},\ldots,1/x_{m}),u,v)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\prod_{j\neq
i}\frac{(u-\frac{x_{i}}{x_{j}})(u-\frac{x_{j}}{x_{i}}v)}{(1-\frac{x_{i}}{x_{j}})(1-\frac{x_{j}}{x_{i}}v)}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{m}\prod_{j\neq
i}\frac{(x_{j}u-x_{i})(x_{i}u-x_{j}v)}{(x_{j}-x_{i})(x_{i}-x_{j}v)}$
###### Proposition 1.
$F=G$.
Proof. The right way to prove this identity is presumably via invariant
theory. (Or maybe it’s just somehow obvious?) But here we are going to prove
it by considering the two sides as rational functions of $v$, expanding their
individual terms in partial fractions, and seeing that the parts on the two
sides agree.
The tricky case turns out to be the polynomial term, corresponding to the pole
at $v=\infty$. We’ll deal with that later, after we address the finite poles.
Let’s look at the case $m=3$, which is sufficient to show what is going on.
Each side of the identity has three terms. On the left the first term is
$\frac{(x_{1}u-x_{2})(x_{2}u-x_{1}v)(x_{1}u-x_{3})(x_{3}u-x_{1}v)}{(x_{1}-x_{2})(x_{2}-x_{1}v)(x_{1}-x_{3})(x_{3}-x_{1}v)}.$
This term is the only one on the left with non-zero residue at
$v=x_{2}/x_{1}$, and its residue there is
$\displaystyle\frac{(x_{1}u-x_{2})(x_{2}u-x_{1}x_{2}/x_{1})(x_{1}u-x_{3})(x_{3}u-x_{1}x_{2}/x_{1})}{(x_{1}-x_{2})(-x_{1})(x_{1}-x_{3})(x_{3}-x_{1}x_{2}/x_{1})}$
$\displaystyle=$
$\displaystyle\frac{(x_{1}u-x_{2})x_{2}(u-1)(x_{1}u-x_{3})(x_{3}u-x_{2})}{(x_{1}-x_{2})(-x_{1})(x_{1}-x_{3})(x_{3}-x_{2})}.$
On the right the only term with a non-zero residue at $v=x_{2}/x_{1}$ is the
second term, namely
$\frac{(x_{1}u-x_{2})(x_{2}u-x_{1}v)(x_{3}u-x_{2})(x_{2}u-x_{3}v)}{(x_{1}-x_{2})(x_{2}-x_{1}v)(x_{3}-x_{2})(x_{2}-x_{3}v)},$
and the residue there is
$\displaystyle\frac{(x_{1}u-x_{2})(x_{2}u-x_{1}x_{2}/x_{1})(x_{3}u-x_{2})(x_{2}u-x_{3}x_{2}/x_{1})}{(x_{1}-x_{2})(-x_{1})(x_{3}-x_{2})(x_{2}-x_{3}x_{2}/x_{1})}$
$\displaystyle=$
$\displaystyle\frac{(x_{1}u-x_{2})x_{2}(u-1)(x_{3}u-x_{2})(x_{1}u-x_{3})}{(x_{1}-x_{2})(-x_{1})(x_{3}-x_{2})(x_{1}-x_{3})},$
which is the same as we found for the left side.
In this way we see that the residues of $v$ at the finite poles all match
between left and right. That leaves the pole at $v=\infty$. Taking the limit
$v\rightarrow\infty$ of
$F((x_{1},\ldots,x_{m}),u,v)=\sum_{i=1}^{m}\prod_{j\neq
i}\frac{(x_{i}u-x_{j})(x_{j}u-x_{i}v)}{(x_{i}-x_{j})(x_{j}-x_{i}v)}$
yields
$\sum_{i=1}^{m}\prod_{j\neq i}\frac{(x_{i}u-x_{j})}{(x_{i}-x_{j})}.$
In the next section, we will prove that this limit is $1+u+\ldots+u^{m-1}$,
which as it is independent of $(x_{1},x_{2},x_{3})$ must agree with the limit
of
$G((x_{1},\ldots,x_{m}),u,v)=F((1/x_{1},\ldots,1/x_{m}),u,v),$
so the residues at $v=\infty$ of the two sides of your identity match, and the
proof is complete. Well, it’s not a proof, exactly, since it doesn’t really
explain what is going on there. Call it a ‘verification’, which persuades us
that the identity is true, at least when coupled with a symbolic computation
checking the identity up through $m=4$. $\quad\qed$
## 10 The subsidiary identity
Define the rational function
$f((x_{1},\ldots,x_{m}),u)=\sum_{i=1}^{m}\prod_{j\neq
i}\frac{u-\frac{x_{j}}{x_{i}}}{1-\frac{x_{j}}{x_{i}}}=\sum_{i=1}^{m}\prod_{j\neq
i}\frac{x_{i}u-x_{j}}{x_{i}-x_{j}}.$
###### Proposition 2.
$f((x_{1},\ldots,x_{m}),u)=1+u+\ldots+u^{m-1}.$
Proof. We use induction on $m$. The cases $m=0,1$ are trivial, and $m=2$ is so
easy as not to illustrate the method. So we will look at the case $m=3$, and
take that as representative. We want to show that
$\frac{(x_{1}u-x_{2})(x_{1}u-x_{3})}{(x_{1}-x_{2})(x_{1}-x_{3})}+\frac{(x_{2}u-x_{1})(x_{2}u-x_{3})}{(x_{2}-x_{1})(x_{2}-x_{3})}+\frac{(x_{3}u-x_{1})(x_{3}u-x_{2})}{(x_{3}-x_{1})(x_{3}-x_{2})}=1+u+u^{2}.$
Expand the terms on the left in partial fractions with respect to the variable
$x_{3}$. The possible poles are at $x_{3}=x_{1}$, $x_{3}=x_{2}$, and
$x_{3}=\infty$. For the coefficient of $\frac{1}{x_{3}-x_{1}}$ we get
$-\frac{(x_{1}u-x_{2})(x_{1}u-x_{1})}{x_{1}-x_{2}}+0+\frac{(x_{1}u-x_{1})(x_{1}u-x_{2})}{x_{1}-x_{2}}.$
So this coefficient vanishes (as it would have to, if our identity is to
hold). Similarly for the coefficient of $\frac{1}{x_{3}-x_{2}}$. This leaves
the pole at $x_{3}=\infty$. Taking the limit of the terms on the left as
$x_{3}\rightarrow\infty$, we get
$\frac{x_{1}u-x_{2}}{x_{1}-x_{2}}+\frac{x_{2}u-x_{1}}{x_{2}-x_{1}}+u^{2}=f((x_{1},x_{2}),u)+u^{2}=1+u+u^{2},$
where in the last step we are using the induction hypothesis. $\quad\qed$
## 11 Open questions
1. 1.
What is the right way to prove these two identities?
2. 2.
LMR showed that in their construction, the full Hodge series agree just if
they agree after setting $w=0$. Surely we can prove this algebraically.
3. 3.
Is the condition in Theorem 1 for Hodge-equivalence of LMR groups necessary as
well as sufficient? If true, this might not be so hard to prove. To start
with, we could prove that what works for $t=1$ works for any $t$.
4. 4.
It seems that the representation-theoretic proof of LMR might give an explicit
matchup between spaces of invariant forms. Can we extract such a matchup from
the algebra in the proof of Theorem 1?
5. 5.
Not all Hodge-isospectral pairs emerge directly from the LMR construction. For
example, you append a $0$ to the list $rta+1$ on both sides, or put in
everything congruent to $2$ mod $rt$. It’s tempting to figure out just what
variations are possible. And then we could ask whether all possible pairs
arise as variations of this kind.
6. 6.
Doyle and Rossetti [3] conjectured that in spherical geometry or hyperbolic
geometry, spaces that are $p$-isospectral for all $p$ are almost conjugate,
and hence isospectral for all natural operators. The LMR examples show that
this is false in spherical geometry, but the hyperbolic case remains open, and
the intuition for this conjecture, born in the hyperbolic case and
incautiously extended to the spherical case, remains more or less intact.
## Acknowledgements
We thank Emilio Lauret for extensive correspondence.
## References
* [1] Scott Crass. Solving the sextic by iteration: A study in complex geometry and dynamics. Experimental Mathematics, 8, 1999, arXiv:math/9903111 [math.DS]. http://arxiv.org/abs/math/9903111.
* [2] Peter Doyle and Curt McMullen. Solving the quintic by iteration. Acta Mathematica, 163:151–180, 1989. http://www.math.harvard.edu/~ctm/papers/home/text/papers/icos/icos.pdf.
* [3] Peter G. Doyle and Juan Pablo Rossetti. Laplace-isospectral hyperbolic 2-orbifolds are representation-equivalent (Version 1), 2011, arXiv:1103.4372v1 [math.DG]. http://arxiv.org/abs/1103.4372v1.
* [4] Peter B. Gilkey. On spherical space forms with metacyclic fundamental groups which are isospectral but not equivariant cobordant. Compositio Math., 56:171–200, 1985.
* [5] Vi Hart. Pi is (still) wrong. https://www.youtube.com/watch?v=jG7vhMMXagQ.
* [6] Akira Ikeda. Riemannian manifolds $p$-isospectral but not $(p+1)$-isospectral. In Geometry of Manifolds, pages 383–417. Academic Press, 1989.
* [7] Felix Klein. Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. B.G. Teubner, 1884. http://www.math.dartmouth.edu/~doyle/docs/ikos/scan/ikos.pdf.
* [8] Emilio A. Lauret, Roberto J. Miatello, and Juan Pablo Rossetti. Lens spaces isospectral on $p$-forms for every $p$, 2013, arXiv:1311.7167v2 [math.DG]. http://arxiv.org/abs/1311.7167v2.
* [9] Theodor Molien. Über die Invarianten der linearen Substitutionsgruppen. Sitzungber. König. Preuss. Akad. Wiss. (J. Berl. Ber.), 52:1152–1156, 1897. http://books.google.com/books?id=EIxK-opAmJYC&pg=PA1152.
* [10] Hubert Pesce. Variétés hyperboliques et elliptiques fortement isospectrales. J. Funct. Anal., 134(2):363–391, 1995.
* [11] Richard P. Stanley. Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.), 1(3):475–511, 1979. http://math.mit.edu/~rstan/pubs/pubfiles/38.pdf.
|
arxiv-papers
| 2014-04-09T18:47:44 |
2024-09-04T02:50:00.921802
|
{
"license": "Public Domain",
"authors": "Daryl R. DeFord, Peter G. Doyle",
"submitter": "Peter G. Doyle",
"url": "https://arxiv.org/abs/1404.2574"
}
|
1404.2622
|
# Serre Multiplicity Question, Mukai Pairing and Hodge theory
Mohammad Reza Rahmati
[email protected]
###### Abstract.
The Serre conjecture on positivity of intersection multiplicity in proper
intersections over general regular rings, is still a challenging open
question. In this article we pretend to apply some general techniques from
geometry to this question. Specificall a discussion of relations with Hodge
theory, Mukai pairing, and Higher residue is proposed to apply to positivity
question.
###### Key words and phrases:
Serre Multiplicity, Chern character, Mukai transform, K-theory, Hochschild-
Konstant-Rosenberg homomorphism, Grothendieck Standard Conjectures, Gamma
class
###### 1991 Mathematics Subject Classification:
14Lxx
## Introduction
In 1950’s J. Serre generalized the definition of Intersection multiplicity
(due to Samuel) of two finitely generated modules over a regular local ring
$A$ as an Euler characteristic, namely Tor-formula. He proved the non-
negativity of this Euler characteristic in several special cases that were
enough for the purpose of geometers, for instance in the case where the ring
$A$ is essentially of finite type over a filed or a discrete valuation ring
$k$. Serre conjectured the vanishing of multiplicity in non-proper
intersections, and its positivity in the proper case, for general regular
local rings, [S]. The vanishing part was proved by H. Gillet and C. Soule,
[GS1], and also independently by P. Roberts, [RO1]. A proof of non-negativity
was given by O. Gabber [RO2]. However, the positivity remained open. I try to
investigate some relations between the positivity and Hodge theory, using
Fourier-Mukai pairing of categories and Hochschild homology. Serre
intersection multiplicity definition, can be lifted to a product on
$K_{0}(A)$. That the $K$-theory of the regular ring $A$ has an intersection
theory. In this way the intersection multiplicity can be written as a cup
product in $K_{0}(A)$. The chern character ch and the Riemann-Roch map
transform this product into the Chow ring of $A$, where one may use a
dimension argument in order to establish the vanishing of multiplicity.
One step further is to try to push the above product into cohomology of
ambient space by integral transform. This allows some more flexible theory to
discuss about positivity. However, it still does not provide a complete
answer, because one needs a type of positivity of Poincare product in some
Weil cohomology theory of a general regular scheme. In this article we
investigate this connection in characterisitic $0$, using de Rham cohomology
and its cup product. In characteristic $p>0$, over a field, we make a
discussion related to Grothendieck Standard Conjectures and also higher
residue pairing.
The Mukai Pairing generally is a non-degenerate pairing on $HH_{*}(X)$.
However it may also be formulated at the level of de Rham cohomologies. Mukai
transform is what that makes the Riemann-Roch map a homomorphism. It is the
toll we have used to express the Serre (Cartan-Eilenberg) Euler characteristic
as a Hodge theory product. To do this one needs to modify Mukai vector by
another class namely Gamma class. The Gamma class appears in the context of
Mirror symmetry as a perturbtive correction term. The original definition of
Mukai vector reflects the Calabi-Yau case, where $\hat{\Gamma}=0$.
In this text we deal with some inter-relations between multiplicity in
algebraic geometry and Hodge theory. We compare the two positivity insights in
the two theories. Although these tools need to be more developed but we think
it opens some windows toward the open questions involved. This paper lie on a
way of inter-relation between Multiplicity in algebraic geometry, Mukai
pairing in Mirror symmetry, and Conjectures in Hodge theory. We propose to
discuss positivity in these 3 areas by one for another.
## 1\. Serre Multiplicity Conjecture
Let $A$ be a regular local ring, and $M,N$ finitely generated $A$-modules such
that $M\otimes_{A}N$ has finite length. J. P. Serre [S], defines the
intersection multiplicity as
(1) $\chi^{A}(M,N):=\displaystyle{\sum(-1)^{i}l(\text{Tor}_{i}^{A}(M,N)})$
He proves the basic fact that in this case $\dim M+\dim N\leq\dim A$, will
hold and makes the following question, known as Serre Multiplicity conjecture.
* (1)
If $\dim M+\dim N<\dim A$, then $\chi^{A}(M,N)=0$
* (2)
In case $\dim M+\dim N=\dim A$, called proper intersection, $\chi^{A}(M,N)>0$.
The vanishing part of the conjecture was proved by P. Roberts and also
independently by H. Gillet-C. Soule. The positivity in general is still open.
The condition, $M,N$ both have finite projective dimensions implies that the
sums in (1) have finitely many terms. Also the condition, $M\otimes_{A}N$ has
finite length, implies, all the $\text{Tor}_{i}^{A}(M,N)$ and hence all
$\text{Ext}_{A}^{i}(M,N)$ have finite length. This makes the former criteria
meaningful. One may explain the Euler characteristic in terms of projective
resolutions. By a perfect $A$-complex we mean a bounded complex of finitely
generated free (Projective) $A$-modules. The support of such complex would be
the closed subspace $\text{Supp}(G_{\bullet})$ where the localization
$(G_{\bullet})_{\mathfrak{p}}$ has non-trivial homology. Then the dimension of
the complex is defined to be $\dim\text{Supp}(G_{\bullet})$.
If $E_{\bullet}$ and $F_{\bullet}$ be free resolutions of the $A$-modules
$M,N$ (which may be taken to be finite, by the regularity of $A$), then
(2) $\chi^{A}(M,N)=\chi(E_{\bullet}\otimes F_{\bullet})=(-1)^{codim\
M}\chi(E_{\bullet}^{*}\otimes F_{\bullet})$
where the right hand side is the usual Euler characteristic of the complex
$E_{\bullet}\otimes F_{\bullet}$. The latter makes sense for the complex is
supported on the maximal ideal of $A$.
## 2\. Mukai Pairing and Hochschild Homology
Let $k$ be a commutative ring. The Hochschild homology and cohomology of an
$k$-algebra $A$ are defined by, $HH_{k}(A):=\text{Tor}_{k}^{A^{e}}(A,A),\
HH^{k}(A):=\text{Ext}_{A^{e}}^{k}(A,A)$, where $A^{e}=A^{op}\otimes_{k}A$.
Note that in case $A$ is commutative $A^{op}=A$. The Hochschild homology and
cohomology of a regular scheme is a sheafification of this definition using
the structure sheaves $\mathcal{O}_{X}$. The Denis trace map
(3) $Den:K_{0}(X)\to HH_{0}(X)$
would play the role of classical chern character, whose composition with
Hochschild-Konstant-Rosenberg homomorphism induces the usual chern character,
(4) $K_{0}(X)\stackrel{{\scriptstyle
ch}}{{\rightarrow}}HH_{0}(X)\stackrel{{\scriptstyle
HKR}}{{\rightarrow}}\displaystyle{\bigoplus_{i}H^{i}(X,\Omega_{X}^{i})}$
The Denis trace can be defined in more general context, even for non-smooth
$X$ by
$\text{ch}([e])=\text{Tr}(\hat{e})$
in terms of the idempotent $e\in M_{n}(A)$, where
$\hat{e}=e+\displaystyle{\sum_{n\geq
1}\dfrac{(2n)!}{(n!)^{2}}(e-\dfrac{1}{2})(de)^{2n}}$
As an element of the completion, $\hat{\Omega}(A)=\prod\Omega^{i}(A)$, [L],
[CST].
We only consider regular rings $A$ and regular schemes. Therefore, we have
$HH_{\bullet}(A)=HH_{\bullet}(\text{perf}(A))$. Thus we identify the theory of
Chern characters for $K$-theory of sheaves or their perfect complexes. The
H-K-R would be the isomorphism induced by the map;
(5) $b_{0}\otimes b_{1}\otimes...\otimes
b_{r}\to\displaystyle{\frac{1}{r!}.b_{0}.db_{1}\wedge...\wedge db_{r}}$
when $X$ is smooth, [RA1]. The philosophy of Mukai-pairing is to modify ch, by
cup product with a class $\sqrt{\text{td}_{X}}$ such that we obtain a
homomorphism in Riemann-Roch theorem. We have
$\text{td}_{X}^{1/2}\in\displaystyle{\bigoplus_{i}H^{i}(X,\Omega_{X}^{i})}$.
Let $X,Y$ be complex manifolds, and let $\mathcal{E}\in D^{b}(X\times Y)$. Let
$\pi_{X},\pi_{Y}$ be the projections. Define the integral transform with
kernel $\mathcal{E}$ by;
(6) $\Phi_{X\to Y}^{\mathcal{E}}:D^{b}(X)\to D^{b}(Y),\qquad\Phi_{X\to
Y}^{\mathcal{E}}(.)=\pi_{Y,*}(\pi_{X}^{*}(.)\otimes\mathcal{E})$
Similarly for $\mu\in H^{*}(X\times Y,\mathbb{Q})$
(7) $\Phi_{X\to Y}^{\mu}:H^{*}(X,\mathbb{Q})\to
H^{*}(Y,\mathbb{Q}),\qquad\Phi_{X\to
Y}^{\mu}(.)=\pi_{Y,*}(\pi_{X}^{*}(.)\otimes\mu)$
called the integral transform in cohomology associated to $\mu$. The
association between objects of $D^{b}(X\times Y)$ or $H^{*}(X\times Y)$ is
functorial. In order to relate the above two functors we use chern character
and Riemann-Roch theorem. The Riemann-Roch theorem states that, if $\pi:X\to
Y$ is a local complete intersection morphism;
(8)
$\pi_{*}(\text{ch}(\bullet)\text{td}(X))=\text{ch}(\pi_{*}(\bullet)).\text{td}(Y)$
This suggest to define the Mukai vector of $\mathcal{E}$ as follows,
(9) $v:D^{b}(X)\to H^{*}(X,\mathbb{Q}),\qquad
v(.)=\text{ch}(.).\sqrt{\text{td}(X)}$
Then the commutativity of the following diagram is straight-forward;
(10) $\begin{CD}D^{b}(X)@>{\Phi_{X\to Y}^{\mathcal{E}}}>{}>D^{b}(Y)\\\
@V{v}V{}V@V{}V{v}V\\\ H^{*}(X,\mathbb{Q})@>{\phi_{X\to
Y}^{v(\mathcal{E})}}>{}>H^{*}(Y,\mathbb{C})\end{CD}$
We will denote $\Phi_{*}=\Phi_{X\to Y}^{v(\mathcal{E})}$, where
$\Phi=\Phi_{X\to Y}^{\mathcal{E}}$, and it satisfies the associativity and
functorial properties naturally. In case $\Phi=\Phi_{X\to Y}^{\mathcal{E}}$ be
an equivalence of categories, $\Phi_{*}=\Phi_{X\to Y}^{v(\mathcal{E})}$ would
be an isomorphism, [CA1], [CA2], [CA3].
When $X$ is a projective smooth manifold, the map $\Phi_{*}$ does respects the
columns of Hodge diamond;
$\Phi_{*}=\phi_{X\to
Y}^{v(\mathcal{E})}:\displaystyle{\bigoplus_{p-q=i}H^{p,q}(X)\to\bigoplus_{p-q=i}H^{p,q}(X)}$
This is for, the class $v(\mathcal{E})$ is a Hodge class. Lets define
$\tau:H^{*}(X,\mathbb{C})\to H^{*}(X,\mathbb{C})$ by
(11) $\tau(v_{0},v_{1},...,v_{2n})=(v_{0},iv_{1},-v_{2},...,i^{2n}v_{2n})$
and set,
(12) $.^{\vee}:H^{*}(X,\mathbb{C})\to H^{*}(X,\mathbb{C}),\qquad
v^{\vee}=\tau(v).\displaystyle{\frac{1}{\sqrt{ch(\omega_{X})}}}$
For
$\text{td}(T_{X}^{\vee})=\text{td}(T_{X}).\exp(-c_{1}(T_{X}))=\text{td}(T_{X}).\text{ch}(\omega_{X})$.
This operator can also be defined more generally on Hochschild homology. If
$X$ is proper and smooth, There is a natural isomorphism $HH_{\bullet}(X)\cong
HH_{\bullet}(\text{perf}(X))$. When $Y$ is of the same type, an object
$\Phi\in\text{perf}(X\otimes Y)$, may be considered as the kernel of an
integral transform $\text{perf}(X)\to\text{perf}(Y)$. Then we would have the
induced map
$\Phi_{*}:HH_{\bullet}(X)\to HH_{\bullet}(Y)$
Using Kunneth quasi-isomorphism ,we get a pairing
$HH_{\bullet}(\text{perf}(X))\otimes HH_{\bullet}(\text{perf}(X))\to
HH_{\bullet}(\text{perf}(X\times
X)){\rightarrow}HH_{\bullet}\text{perf}(\mathbb{C})=\mathbb{C}$
Which is given by shuffle products, [RA1]. A non-trivial fact is that the
above induced map $\Phi_{*}$ will become equivalent to the integral transform
induced by $\Phi$.
The Mukai Pairing can be generalized to Hochschild homology as
$\langle.,.\rangle_{M}:HH_{\bullet}(X)\otimes HH_{\bullet}(X)\to\mathbb{C}$
called generalized Mukai pairing. This generalization can be easily written
using the isomorphism
$D:\text{RHom}(\Delta_{!}\mathcal{O}_{X},\Delta_{*}\mathcal{O}_{X})\cong\text{RHom}(\Delta_{*}\mathcal{O}_{X},\Delta_{*}\omega_{X})$
where $\Delta_{!}\mathcal{O}_{X}\cong\Delta_{*}\omega_{X}^{-1}$ and
$\omega_{X}$ is the dualizing sheaf. Then, the Mukai pairing is
$v\otimes w\to\text{tr}_{X\times X}(D(v)\circ w)$
where tr is Serre duality trace. If
$.^{\vee}:HH_{\bullet}(X)\to HH_{\bullet}(X)$
is the involution induced through H-K-R isomorphism by the similar one to be
$(-1)^{p}$ on $H^{q}(X,\Omega_{p})$, as defined before. Then we would have
###### Theorem 2.1.
[RA1] Suppose $X$ is smooth, then
(13) $\langle b^{\vee},a\rangle_{M}=\langle a,b\rangle,\qquad a,b\in
HH_{\bullet}(X)$
Moreover, the generalized Mukai pairing on the Hochschild homology of $X$
satisfies
$\displaystyle{\langle
a,b\rangle_{M}=\int_{X}I(a)^{\vee}I(b).\text{td}_{X}},\qquad a,b\in
HH_{\bullet}(X)$
where I is the H-K-R isomorphism.
The Euler pairing on $K_{0}(X)$ is defined by
$\chi(\mathcal{E},\mathcal{F}):=\displaystyle{\sum_{i}(-1)^{i}\dim
Ext_{X}^{i}(\mathcal{E},\mathcal{F})}$
Assume $H^{*}(X)$ is equipped with the pairing
$\langle x,y\rangle:=(x\cup y\cup td_{X})\cap[X]$
Then the Riemann-Roch theorem states that, the chern character $ch:K_{0}\to
H^{*}(X)$ is map of inner product spaces. The same fact is true for Hochschild
homology and Denis trace map, where we have the compatibility of the two chern
character by H-K-R homomorphism, [CA1]. The chern character $ch:K_{0}\to
HH_{0}(X)$ is a map of inner product spaces, in other words for
$\mathcal{E},\mathcal{F}\in D(X)$, we have
$\langle ch(E),ch(F)\rangle_{M}=\chi(\mathcal{E},\mathcal{F})$
A modification of Mukai pairing is to use the $\hat{\Gamma}_{X}$-class instead
of the square $\sqrt{td_{X}}$. Then we replace the Mukai vector by the vector
$E\mapsto(2\pi
i)^{\deg(.)/2}\displaystyle{\frac{1}{(2\pi)^{d/2}}}\Gamma(X)\wedge ch(E)$
The cohomology class $\hat{\Gamma}_{X}$ is defined via the identity
$\frac{z}{1-e^{-z}}=e^{i\pi z}\Gamma(1-x)\Gamma(1+x)$ used to share the two
factors of $\sqrt{td_{X}}$ with the other chern classes in the Mukai pairing.
It explicitly is given by the formula,
$\hat{\Gamma}_{X}=\exp(C.ch_{1}(T_{X})+\displaystyle{\sum_{n\geq
2}\dfrac{\zeta(n)}{n}ch_{n}(T_{X})})$
where $C=\lim_{n\to\infty}(1+\frac{1}{2}+...+\frac{1}{n}-ln(n))$ is the Euler
constant, $\zeta$ is the Riemann zeta. Let’s write,
${H^{*}(X,\mathbb{Q})\stackrel{{\scriptstyle\mathfrak{d}}}{{\rightarrow}}H^{*}(X,\mathbb{C})}{\
\stackrel{{\scriptstyle\hat{\Gamma}_{X}\wedge(\bullet)}}{{\rightarrow}}H^{*}(X,\mathbb{C})},\qquad\mathfrak{d}:=(2\pi
i)^{\deg(.)/2}\displaystyle{\frac{1}{(2\pi)^{d/2}}}$
Previously, we defined the Mukai vector as
$\nu(\mathcal{E})=ch(\mathcal{E})\wedge\sqrt{td_{X}}$, and defined the pairing
$\langle v,w\rangle=\displaystyle{\int_{X}v^{\vee}\wedge w,\qquad
v^{\vee}:=\dfrac{\tau(v)}{\sqrt{ch(\omega_{X})}}}$
This vector may also be more modified by setting
$\mu_{\Lambda}(\mathcal{E}):=ch(\mathcal{E})\sqrt{td_{X}}.\exp(i\Lambda)$
where $\tau(\Lambda)=-\Lambda$. Thus the former Mukai vector is the special
case $\Lambda=0$. Then
$\mu_{\Lambda}(\mathcal{E})^{\vee}=ch(\mathcal{E})^{\vee}.\sqrt{td_{X}}.\exp(-i\Lambda)$
Knowing $\tau(td_{X})=ch(\omega_{X})td_{X}$, and we would still have
$\langle\mu_{\Lambda}(\mathcal{E}),\mu_{\Lambda}(\mathcal{F})\rangle=\displaystyle{\int\mu_{\Lambda}(\mathcal{E})^{\vee}\mu_{\Lambda}(\mathcal{F})\\\
=\int ch(\mathcal{E})^{\vee}ch(\mathcal{F}).td_{X}}$
In this way the replacement for the square root of $td_{X}$ is a
multiplicative characteristic class, namely complex Gamma class, [D], [HJLM],
$\hat{\Gamma}_{X}^{\mathbb{C}}=\sqrt{td_{X}}\exp(i\Lambda_{X})$
By the machinery introduced in the former Sections we may easily study the
positivity of Serre multiplicity over $\mathbb{C}$. First we write the Serre-
Cartan-Eilenberg Euler characteristic as,
(14)
$\chi(\mathcal{E},\mathcal{F})=\displaystyle{\int_{X}\mu(\mathcal{E})^{\vee}\wedge\mu(\mathcal{F})},$
In this way we need to study the positivity of the right hand side using
Riemann-Hodge bilinear relations for pure Hodge structures on
$H^{*}(X,\mathbb{C})$.
###### Remark 2.2.
The Mukai pairing $\langle v,w\rangle=\int_{X}v^{\vee}\wedge w$ appears in the
context of Mirror Symmetry as a mirror to polarization form of a PVHS. This
means it is a polarization of the mirror manifold or the PVHS we already have.
The positivity of intersection multiplicity proved in [S], shows when $Y$ and
$Z$ are projective sub-varieties of $X$, then
(15)
$\displaystyle{\int_{X}\mu(\mathcal{O}_{Y})^{\vee}\wedge\mu(\mathcal{O}_{Z})}\geq
0,$
The aforementioned procedure, i.e expressing the Cartan-Eilenberg Euler
characteristic as Hermitian type cup product suggests the idea to prove
positivity in the serre multiplicity conjecture by Hodge theory. Formulas (30)
and (31) directly relate the intersection multiplicity of algebraic cycles in
the chow ring to cup product in homology, and thus to polarization of Hodge
structures. Thus it provides a way to determine the positivity on both sides.
## 3\. Multiplicity question over arbitrary field
We explain two ideas regarding positivity, related to Hodge theory.
(1) Let $X$ be a smooth projective variety $/k$ of dimension $d$, and $L$ an
ample divisor class. $L$ acts on etale cohomology of $X$ and by hard
Lefschetz,
(16) $L^{j}:H^{n-j}(X(\bar{k}),\mathbb{Q}_{l})\cong
H^{n+j}(X(\bar{k}),\mathbb{Q}_{l})$
which implies
(17)
$H^{n-j}(X(\bar{k}),\mathbb{Q}_{l})=\oplus_{k}L^{k}H^{j-2k}(X(\bar{k}),\mathbb{Q}_{l})_{prim}$
that induces a morphism
(18) $\Lambda:H^{j}(X(\bar{k}),\mathbb{Q}_{l})\to
H^{j-2}(X(\bar{k}),\mathbb{Q}_{l}(-1))$
such that for $m\in H^{j}(X(\bar{k}),\mathbb{Q}_{l})^{prim}$,
$\Lambda(L^{k}m)=L^{k-1}.m$, if $k>0$ and $0$ otherwise. The standard
conjecture $B$ asserts that $\Lambda$ is defined algebraically as the action
of a correspondence. If $A^{j}(X)$ is the co-image of the cycle map
(19) $cl:CH^{j}(X)_{\mathbb{Q}}\to H^{2j}(X(\bar{k}),\mathbb{Q}_{l}(j))$
The standard conjecture A asserts that the morphisms
(20) $L^{n-2j}:A^{i}(X)\cong A^{n-i}(X),\qquad i<n/2,$
are isomorphisms. This follows from Conjecture $B$, that says the Lefschetz
decomposition is compatible with $A^{j}$’s,
(21) $A^{j}(X)=\oplus_{k}L^{k}.A^{j-k}(X)^{prim}$
The conjecture $I$ asserts that the pairing,
(22) $(-1)^{j}\langle L^{n-2j}a,b\rangle,\qquad a,b\in A^{j}(X)^{prim}$
is positive definite for $j\leq n/2$. If we assume $I$, then $A$ would be
equivalent to $D$ stating the equivalence of numerical and homological
equivalence for cycles on $X$, [SA]. By the Lefschetz decomposition we have an
isomorphism,
$*:H^{n+j}(X_{\bar{k}},\mathbb{Q}_{l})\to H^{n-j}(X_{\bar{k}},\mathbb{Q}_{l})$
such that for $m\in H^{i}(X_{\bar{k}},\mathbb{Q}_{l})^{prim}$, we have
$*(L^{k}m)=(-1)^{i(i+1)/2}L^{n-i-k}m$
Combined with Poincare duality this defines a pairing on
$H^{*}(X,\mathbb{Q}_{l})$ defined by $(m,*n)$. For a non-zero correspondence
$\lambda\in A^{n}(X\times_{k}X)\subset End(H^{*}(X,\mathbb{Q}_{l}))$ we
consider the transpose $\lambda^{\prime}$ w.r.t this pairing. Then, if the
standard conjecture B and I are satisfied,
$\chi=Tr(\lambda^{\prime}\circ\lambda)>0$
The action of the correspondences always determine the pairing on the
homologies, by composing the action of diagonal $\Delta$ with the product
structure. In this way the positivity of the above trace always implies
positivity of the Mukai pairing and therefore, the intersection multiplicity.
This shows: _If the Grothendieck Standard conjecture I are satisfied then the
intersection multiplicity $\chi(M,N)$ is strictly positive on proper
intersections for projective varieties /$k$_, [SA].
(2) The construction of the higher residue pairing originally belonged to K.
Saito, [SA1], may be re-phrased in terms of an identification of twisted de
Rham complex and the formal complex of poly-vector fields, [LLS]. There are
essentially two type of proof for higher residue pairing over $\mathbb{C}$.
The one cited in [SA1] is mainly a comparison of two construction. One an
application of local (Serre) duality theorem to Brieskorn lattices and their
duals. This amounts to define the Brieskorn modules
$(\mathcal{H}_{f}^{(-k)},\nabla:\mathcal{H}^{(-k-1)}\to\mathcal{H}_{f}^{(-k)})$
together with their duals
$(\check{\mathcal{H}}^{(k)},\check{\nabla}:\check{\mathcal{H}}^{(k)}\to\check{\mathcal{H}}^{(k+1)})$
which satisfy a local duality as
$\mathcal{H}^{(k)}\times\check{\mathcal{H}}^{(k)}\to\mathcal{O}_{S}$
Then this duality is related to the twisted de Rham complex by
$\hat{\alpha}_{k}:\widehat{\mathcal{H}^{(-k)}}\cong
R^{n+1}f_{*}(F^{-k}\Omega,\hat{d}),\qquad k\geq 1$.
where $F$ is the Hodge filtration, [S1]. Specifically
(23) $\hat{\alpha}:\widehat{\mathcal{H}^{(0)}}\cong
R^{n+1}f_{*}(F^{0}\Omega,\hat{d}),\qquad k\geq 1.$
The second method is a duality isomorphism between the twisted de Rham complex
and the twisted differential complex of poly-vector fields. This allows to
formulate higher residue by the trace map, via symplectic pairing;
$K^{f}(\ ,\
):\mathcal{H}_{(0)}^{f}\times\mathcal{H}_{(0)}^{f}\to\mathcal{O}_{S,0}[[t]]$
Both of these constructions are algebraic and can be stated similarly over any
field of characteristic $0$ and can be applied over Witt ring construction.
Thus a proof follows from the formality (algebraicity) of the construction in
[SA1] in characteristic $0$.
A Witt ring over a ring $A$ or the ring of Witt vectors of $A$, is a copy of
the infinite product $A^{\infty}$, with specific sum and products given in
each component by polynomials is $char=p$. Such a ring has characteristic $0$.
The the higher residue pairing is defined over a complete local ring of un-
equal characteristic, where the residue field is perfect of char $>0$, as:
$WK^{f}(\ ,\ ):W\widehat{\mathcal{H}}_{(0)}^{f}\times
W\widehat{\mathcal{H}}_{(0)}^{f}\to W\widehat{\mathcal{O}}_{S,0}[[t]]$
such that the induced pairing on the Jacobi ring of $f$ over the Witt ring is
the Grothendieck pairing. Because the characteristic is $0$, the isomorphism
mentioned proceeds word by word to in this case and we still get a mirror type
identification between these two formal complexes. Then analogous isomorphisms
$(WPV_{S}(X)((t)),Q_{f}=\bar{\partial}_{f}+t\partial)\leftrightarrows(WA_{S}(X)((t)),d+t^{-1}df\wedge\bullet)$
$\imath:(WPV_{S,c}(X)[[t]],Q_{f})\hookrightarrow(WPV_{S}(X)[[t]],Q_{f})$
still hold for $W_{n}$s and also in the limit for $W(k)$, and we can define,
$W\mathcal{H}_{(0)}^{f}:=H^{*}(WPV(X)[[t]],Q_{f})$
By the same method as before we obtain:
$W\widehat{Res}^{f}=\sum_{k}W\widehat{Res}_{k}^{f}(\bullet)t^{k}$
with $\widehat{Res}_{k,N}^{f}$ the higher residues. Similarly, we obtain the
higher residue pairing
$WK^{f}(\ ,\ ):W\mathcal{H}_{(0)}^{f}\times
W\mathcal{H}_{(0)}^{f}\to\mathcal{O}_{S,0}[[t]],\qquad WK^{f}(\
,1):=W\widehat{Res}^{f}$
Now by applying the completion, we obtain
$WK^{f}(\ ,\ ):W\widehat{\mathcal{H}}_{(0)}^{f}\times
W\widehat{\mathcal{H}}_{(0)}^{f}\to W\widehat{\mathcal{O}}_{S,0}[[t]]$
###### Theorem 3.1.
(Higher residue pairing on crystalline site) There exists a
$K=\text{Frac}(W(k))$-sesquilinear form
$WK^{f}(\ ,\ ):W\widehat{\mathcal{H}}_{(0)}^{f}\times
W\widehat{\mathcal{H}}_{(0)}^{f}\to W\widehat{\mathcal{O}}_{S,0}[[t]]$
Let $s_{1},s_{2}$ be local sections of $W\mathcal{H}_{(0)}^{f}$, then;
* •
$WK^{f}(s_{1},s_{2})=\overline{WK^{f}(s_{2},s_{1})}$.
* •
$WK^{f}(v(t)s_{1},s_{2})=WK^{f}(s_{1},v(-t)s_{2})=v(t)WK^{f}(s_{1},s_{2})$,
$v(t)\in\mathcal{O}_{S}[[t]]$.
* •
$\partial_{V}.WK^{f}(s_{1},s_{2})=WK^{f}(\partial_{V}s_{1},s_{2})+WK^{f}(s_{1},\partial_{V}s_{2})$,
for any local section of $T_{S}$.
* •
$(t\partial_{t}+n)WK^{f}(s_{1},s_{2})=WK^{f}(t\partial_{t}.s_{2},s_{1})+WK^{f}(s_{1},t\partial_{t}.s_{2})$
* •
The induced pairing on
$W\mathcal{H}_{(0)}^{f}/t.W\mathcal{H}_{(0)}^{f}\otimes
W\mathcal{H}_{(0)}^{f}/t.W\mathcal{H}_{(0)}^{f}\to\bar{K}$
is the classical Grothendieck residue.
The conjugation is formally done by $\overline{g(t)\otimes\eta}=g(-t).\eta$,
for $g\in W(\mathcal{O}_{S}),\ \eta\in WA_{S}(X)$.
The corollary is just the special case $W(\mathbb{F}_{p})=\mathbb{Z}_{p}$. The
notion of opposite filtration and formal primitive elements, and good sections
may also be generalized to this case easily. By the priod isomorphism one can
state similar formulas as in 3.1 for etale cohomology. This isomorphism
roughly states that crystalline and etal cohomologies each one determines the
other one or they are the same in this sense. The period isomorphism is the
natural filtered quasi-isomorphism,
$R\Gamma_{dR}^{alg}(X)\otimes_{\overline{K}}\mathcal{B}_{dr}\rightarrow
R\Gamma_{et}(X,\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathcal{B}_{dR}$
where $K$ is the field of fractions of $W(k)$ and $\mathcal{B}_{dR}$ is a
discrete valuation field whose valuation ring is called Fontaine ring and its
residue field is $\mathbb{C}_{p}$. It descends to ,
$R\Gamma_{dR}^{alg}(X)\otimes_{\overline{K}}\mathbb{C}_{p}\rightarrow
R\Gamma_{et}(X,\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C}_{p}$
The comparison theorem indicates that for any DVR namely $V$, there exists a
ring $B(V)$, such that for $X$ smooth and proper $V$-scheme, the etale
cohomology of the generic fiber $X/W(k)$ is related to the crystalline
cohomology of $X/W(k)$ by
$H_{et}^{*}(X\otimes_{W(k)}\bar{K})\otimes_{\mathbb{Q}_{p}}B(V)=H_{crys}^{*}(X/W(k))\otimes_{W(k)}B(V)$
with $K=\text{quot}\ W(k)$ a totally ramified extension of degree $e$, [FA].
In fact, for $n,\ i\in\mathbb{N}$, the specialization map induces isomorphisms
compatible with the action of Galois group $G_{K}$:
$H^{i}((X\times_{\mathcal{O}_{K}}\bar{k})_{et},\mathbb{Z}/l^{n}.\mathbb{Z})\cong
H^{i}((X\times_{\mathcal{O}_{K}}\bar{K})_{et},\mathbb{Z}/l^{n}.\mathbb{Z})$
The period isomorphism say that crystalline and etale cohomologies in some way
determine one another. Using the period isomorphism we can state the Higher
residue pairing on the etale site if the ground filed would be
$\mathbb{C}_{p}$. Simply in theorem 3.1 if we tensor every thing with
$\mathbb{C}_{p}$ we obtain the same result on the etale site over
$\mathbb{C}_{p}$.
###### Theorem 3.2.
(Higher residue pairing on etale site) There exists a sesqui-linear form
$K_{p}^{f}(\ ,\
):\widehat{\mathcal{H}}_{(0),p}^{f}\times\widehat{\mathcal{H}}_{(0),p}^{f}\to\widehat{\mathcal{O}}_{S,0}[[t]]$
Let $s_{1},s_{2}$ be local sections of
$\mathcal{H}_{(0,p),p}^{f,\mathbb{C}_{p}}$.
* •
$K_{et}^{f}(s_{1},s_{2})=\overline{K_{et}^{f}(s_{2},s_{1})}$.
* •
$K_{et}^{f}(v(t)s_{1},s_{2})=K_{et}^{f}(s_{1},v(-t)s_{2})=v(t)K_{et}^{f}(s_{1},s_{2})$,
$v(t)\in\mathcal{O}_{S}[[t]]$.
* •
$\partial_{V}.K_{et}^{f}(s_{1},s_{2})=K_{et}^{f}(\partial_{V}s_{1},s_{2})+K_{et}^{f}(s_{1},\partial_{V}s_{2})$,
for any local section of $T_{S}$.
* •
$(t\partial_{t}+n)K_{et}^{f}(s_{1},s_{2})=K_{et}^{f}(t\partial_{t}.s_{2},s_{1})+K_{et}^{f}(s_{1},t\partial_{t}.s_{2})$
* •
The induced pairing on
$\mathcal{H}_{(0),p}^{f}/t.\mathcal{H}_{(0),p}^{f}\otimes\mathcal{H}_{(0),p}^{f}/t.\mathcal{H}_{(0),p}^{f}\to\mathbb{C}_{p}$
is the classical Grothendieck residue.
## 4\. Appendix: Grothendieck Standard Conjectures
We list the Grothendieck Standard conjectures, [CH], [GR]:
* •
A : Hard Lefschetz on cycles
(24) $A(X):.L^{n-2k}:CH^{r}(X)\cong CH^{n-r}(X)$
* •
B : Lefschetz type Standard Conjecture
(25) $B(X):*L:\oplus_{i,r}H^{i}(X)(r)\to\oplus_{i,r}H^{i}(X)(r)\qquad is\
algebraic.$
* •
C : Kunneth type Standard Conjecture
(26) $C(X):\pi_{X}^{i}:H^{\bullet}(X)\twoheadrightarrow
H^{i}(X)\hookrightarrow H^{\bullet}(X)\qquad is\ algebraic$
* •
D : Homological and numerical equivalence coincide
(27) $D(X):\qquad\sim_{hom,\mathbb{Q}}=\sim_{num,\mathbb{Q}}$
* •
I : Hodge type Standard conjecture
I(X): the $\mathbb{Q}$-valued quadratic form
$\alpha\mapsto\langle\alpha,*L(\alpha)\rangle$ on
$Z_{hom}^{\bullet}(X)_{\mathbb{Q}}$ is positive definite.
## References
* [CA1] A. Caldararu, S. Willerton, Mukai pairing, categorical approach, arxiv.math/0707.2052v1, 2007
* [CA2] A. Caldararu, Mukai pairing, Hochschild structure, arxiv.math/0308079v2, 2003
* [CA3] A. Caldararu, Mukai transformation, Hochschild Konstant Rosenberg isomorphism, arxiv:math/0308080v3, 2004
* [CHA] C. Chan, An intersection multiplicity in terms of Ext-modules, Proc. Amer. Math. Soc. 130 (2002) 327-336
* [CH] F. Charles, Remarks on the Lefschetz standard conjecture and hyperkahler varieties, IRMAR – UMR 6625 du CNRS, Université de Rennes 1, Campus de Beaulieu, 35042, RENNES CEDEX, FRANCE
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arxiv-papers
| 2014-04-08T18:38:10 |
2024-09-04T02:50:00.933509
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Mohammad Reza Rahmati",
"submitter": "Mohammad Reza Rahmati",
"url": "https://arxiv.org/abs/1404.2622"
}
|
1404.2873
|
# The set of minimal distances in Krull monoids
Alfred Geroldinger and Qinghai Zhong
###### Abstract.
Let $H$ be a Krull monoid with finite class group $G$. Then every non-unit
$a\in H$ can be written as a finite product of atoms, say
$a=u_{1}\cdot\ldots\cdot u_{k}$. The set $\mathsf{L}(a)$ of all possible
factorization lengths $k$ is called the set of lengths of $a$. If $G$ is
finite, then there is a constant $M\in\mathbb{N}$ such that all sets of
lengths are almost arithmetical multiprogressions with bound $M$ and with
difference $d\in\Delta^{*}(H)$, where $\Delta^{*}(H)$ denotes the set of
minimal distances of $H$. We show that
$\max\Delta^{*}(H)\leq\max\\{\exp(G)-2,\mathsf{r}(G)-1\\}$ and that equality
holds if every class of $G$ contains a prime divisor, which holds true for
holomorphy rings in global fields.
###### Key words and phrases:
non-unique factorizations, sets of distances, Krull monoids, zero-sum
sequences, cross numbers
###### 2010 Mathematics Subject Classification:
11B30, 11R27, 13A05, 20M13
This work was supported by the Austrian Science Fund FWF, Project Number
M1641-N26.
## 1\. Introduction
Let $H$ be a Krull monoid with class group $G$ (we have in mind holomorphy
rings in global fields and give more examples later). Then every non-unit of
$H$ has a factorization as a finite product of atoms (or irreducible
elements), and all these factorizations are unique (i.e., $H$ is factorial) if
and only if $G$ is trivial. Otherwise, there are elements having
factorizations which differ not only up to associates and up to the order of
the factors. These phenomena are described by arithmetical invariants such as
sets of lengths and sets of distances. We first recall some concepts and then
we formulate a main result of the present paper.
For a finite nonempty set $L=\\{m_{1},\ldots,m_{k}\\}$ of positive integers
with $m_{1}<\ldots<m_{k}$, we denote by $\Delta(L)=\\{m_{i}-m_{i-1}\mid
i\in[2,k]\\}$ the set of distances of $L$. Thus $\Delta(L)=\emptyset$ if and
only if $|L|\leq 1$. If a non-unit $a\in H$ has a factorization
$a=u_{1}\cdot\ldots\cdot u_{k}$ into atoms $u_{1},\ldots,u_{k}$, then $k$ is
called the length of the factorization, and the set
$\mathsf{L}_{H}(a)=\mathsf{L}(a)$ of all possible $k$ is called the set of
lengths of $a$. If there is an element $a\in H$ with $|\mathsf{L}(a)|>1$, then
it immediately follows that $|\mathsf{L}(a^{n})|>n$ for every
$n\in\mathbb{N}$. Since $H$ is Krull, every non-unit has a factorization into
atoms and all sets of lengths are finite. The set of distances $\Delta(H)$ is
the union of all sets $\Delta(\mathsf{L}(a))$ over all non-units $a\in H$.
Thus, by definition, $\Delta(H)=\emptyset$ if and only if $|\mathsf{L}(a)|=1$
for all non-units $a\in H$, and $\Delta(H)=\\{d\\}$ if and only if
$\mathsf{L}(a)$ is an arithmetical progression with difference $d$ for all
non-units $a\in H$. The set of minimal distances $\Delta^{*}(H)$ is defined as
$\Delta^{*}(H)=\\{\min\Delta(S)\mid S\subset H\ \text{is a divisor-closed
submonoid with}\ \Delta(S)\neq\emptyset\\}\,.$
By definition, we have $\Delta^{*}(H)\subset\Delta(H)$, and
$\Delta^{*}(H)=\emptyset$ if and only if $\Delta(H)=\emptyset$. If the class
group $G$ is finite, then $\Delta(H)$ is finite and sets of lengths have a
well-defined structure which is given in the next theorem ([13, Chapter 4.7]).
Theorem A. Let $H$ be a Krull monoid with finite class group. Then there is a
constant $M\in\mathbb{N}$ such that the set of lengths $\mathsf{L}(a)$ of any
non-unit $a\in H$ is an AAMP $($almost arithmetical multiprogression$)$ with
difference $d\in\Delta^{*}(H)$ and bound $M$.
The structural description given above is best possible ([32]). The set of
minimal distances $\Delta^{*}(H)$ has been studied by Chapman, Geroldinger,
Halter-Koch, Hamidoune, Plagne, Smith, Schmid, and others and there are a
variety of results. We refer the reader to the monograph [13, Chapter 6.8] for
an overview and mention some results which have appeared since then. Suppose
that $G$ is finite and that every class contains a prime divisor. Then the set
of distances $\Delta(H)$ is an interval ([18]). A simple example shows that
the interval $[1,\mathsf{r}(G)-1]$ is contained in $\Delta^{*}(H)$ (Lemma 3.1)
and thus, by Theorem 1.1 below, $\Delta^{*}(H)$ is an interval too if
$\mathsf{r}(G)\geq\exp(G)-1$. Cyclic groups are in sharp contrast to this.
Indeed, if $G$ is cyclic with $|G|>3$, then
$\max\big{(}\Delta^{*}(H)\setminus\\{|G|-2\\}\big{)}=\lfloor\frac{|G|}{2}\rfloor-1$
([14]). A detailed study of the structure of $\Delta^{*}(H)$ in case of cyclic
groups is given in a recent paper by Plagne and Schmid [23].
The goal of the present paper is to study the maximum of $\Delta^{*}(H)$, and
here is the main direct result.
###### Theorem 1.1.
Let $H$ be a Krull monoid with class group $G$.
1. 1.
If $|G|\leq 2$, then $\Delta^{*}(H)=\emptyset$.
2. 2.
If $2<|G|<\infty$, then
$\max\Delta^{*}(H)\leq\max\\{\exp(G)-2,\mathsf{r}(G)-1\\}$ where
$\mathsf{r}(G)$ denotes the rank of $G$.
3. 3.
Suppose that every class contains a prime divisor. If $G$ is infinite, then
$\Delta^{*}(H)=\mathbb{N}$.
If $2<|G|<\infty$, then
$\max\Delta^{*}(H)=\max\\{\exp(G)-2,\mathsf{r}(G)-1\\}$.
Theorem 1.1 will be complemented by an associated inverse result (Theorem 4.5)
describing how $\max\Delta^{*}(H)$ is realized and disproving a former
conjecture (Remark 4.6). Both the direct as well as the inverse result have
number theoretic relevance beyond the occurrence in Theorem A. Indeed, they
are key tools in the characterization of those Krull monoids whose systems of
sets of lengths are closed under set addition ([17]), in the study of
arithmetical characterizations of class groups via sets of lengths ([13,
Chapter 7.3], [31, 16]), as well as in the asymptotic study of counting
functions associated to periods of sets of lengths ([30] and [13, Theorem
9.4.10]).
In Section 2 we gather the required background from the theory of Krull
monoids and from Additive Combinatorics. In particular, we outline that the
set of minimal distances of $H$ equals the set of minimal distances of an
associated monoid of zero-sum sequences (Lemma 2.1) and that therefore it can
be studied with methods from Additive Combinatorics. The proof of Theorem 1.1
will be given in Section 3 and the associated inverse result will be given in
Section 4.
## 2\. Background on Krull monoids and on Additive Combinatorics
We denote by $\mathbb{N}$ the set of positive integers, and, for
$a,b\in\mathbb{Z}$, we denote by $[a,b]=\\{x\in\mathbb{Z}\mid a\leq x\leq
b\\}$ the discrete, finite interval between $a$ and $b$. We use the convention
that $\max\emptyset=0$. By a monoid, we mean a commutative semigroup with
identity that satisfies the cancellation laws. If $H$ is a monoid, then
$H^{\times}$ denotes the unit group, $\mathsf{q}(H)$ the quotient group, and
$\mathcal{A}(H)$ the set of atoms (or irreducible elements) of $H$. A
submonoid $S\subset H$ is called divisor-closed if $a\in S$, $b\in H$, and $b$
divides $a$ imply that $b\in S$. A monoid $H$ is said to be
* •
atomic if every non-unit can be written as a finite product of atoms.
* •
factorial if it is atomic and every atom is prime.
* •
half-factorial if it is atomic and $|\mathsf{L}(a)|=1$ for each non-unit $a\in
H$ (equivalently, $\Delta(H)=\emptyset$).
* •
decomposable if there exist submonoids $H_{1},H_{2}$ with $H_{i}\not\subset
H^{\times}$ for $i\in[1,2]$ such that $H=H_{1}\times H_{2}$ (and $H$ is called
indecomposable else).
A monoid $F$ is factorial with $F^{\times}=\\{1\\}$ if and only if it is free
abelian. If this holds, then the set of primes $P\subset F$ is a basis of $F$,
we write $F=\mathcal{F}(P)$, and every $a\in F$ has a representation of the
form
$a=\prod_{p\in P}p^{\mathsf{v}_{p}(a)}\quad\text{with}\
\mathsf{v}_{p}(a)\in\mathbb{N}_{0}\quad\text{and}\quad\mathsf{v}_{p}(a)=0\
\text{for almost all}\ p\in P\,.$
A monoid homomorphism $\theta\colon H\to B$ is called a transfer homomorphism
if it has the following properties:
1. 1. (T 1)
$B=\theta(H)B^{\times}$ and $\theta^{-1}(B^{\times})=H^{\times}$.
2. (T 2)
If $u\in H$, $b,\,c\in B$ and $\theta(u)=bc$, then there exist $v,\,w\in H$
such that $u=vw$, $\theta(v)\simeq b$ and $\theta(w)\simeq c$.
If $H$ and $B$ are atomic monoids and $\theta\colon H\to B$ is a transfer
homomorphism, then (see [13, Chapter 3.2])
$\mathsf{L}_{H}(a)=\mathsf{L}_{B}(\theta(a))\ \text{ for all $a\in
H$},\quad\Delta(H)=\Delta(B),\quad\text{and}\quad\Delta^{*}(H)=\Delta^{*}(B)\,.$
Krull monoids. A monoid $H$ is said to be a Krull monoid if it satisfies the
following two conditions:
1. (a)
There exists a monoid homomorphism $\varphi\colon H\to F=\mathcal{F}(P)$ into
a free abelian monoid $F$ such that $a\,|\,b$ in $H$ if and only if
$\varphi(a)\,|\,\varphi(b)$ in $F$.
2. (b)
For every $p\in P$, there exists a finite subset $E\subset H$ such that
$p=\gcd\big{(}\varphi(E)\big{)}$.
Let $H$ be a Krull monoid and $\varphi\colon H\to\mathcal{F}(P)$ a
homomorphism satisfying Properties (a) and (b). Then $\varphi$ is called a
divisor theory of $H$, $G=\mathsf{q}(F)/\mathsf{q}(\varphi(H))$ is the class
group, and $G_{P}=\\{[p]=p\mathsf{q}(\varphi(H)))\mid p\in P\\}\subset G$ the
set of classes containing prime divisors. The class group will be written
additively, and the tuple $(G,G_{P})$ are uniquely determined by $H$. To
provide some examples of Krull monoids, we recall that a domain is a Krull
domain if and only if its multiplicative monoid of nonzero elements is a Krull
monoid, and that a noetherian domain is Krull if and only if it is integrally
closed. Rings of integers, holomorphy rings in algebraic function fields, and
regular congruence monoids in these domains are Krull monoids with finite
class group such that every class contains a prime divisor ([12], [13, Chapter
2.11]). For monoids of modules and monoid domains which are Krull we refer to
[22, 4, 3, 1].
Next we introduce Krull monoids having a combinatorial flavor which are used
to model arbitrary Krull monoids. Let $G$ be an additively written abelian
group and $G_{0}\subset G$ a subset. An element $S=g_{1}\cdot\ldots\cdot
g_{l}\in\mathcal{F}(G_{0})$ is called a sequence over $G_{0}$,
$\sigma(S)=g_{1}+\ldots+g_{l}$ is called its sum, $|S|=l$ its length, and
$\mathsf{h}(S)=\max\\{\mathsf{v}_{g}(S)\mid g\in\operatorname{supp}(S)\\}$ the
maximal multiplicity of $S$. The monoid
$\mathcal{B}(G_{0})=\\{S\in\mathcal{F}(G_{0})\mid\sigma(S)=0\\}$
is a Krull monoid, called the monoid of zero-sum sequences over $G_{0}$. Its
significance for the study of general Krull monoids is summarized in the
following lemma (see [13, Theorem 3.4.10 and Proposition 4.3.13]).
###### Lemma 2.1.
Let $H$ be a Krull monoid, $\varphi\colon H\to D=\mathcal{F}(P)$ a divisor
theory with class group $G$ and $G_{P}\subset G$ the set of classes containing
prime divisors. Let $\widetilde{\boldsymbol{\beta}}\colon
D\to\mathcal{F}(G_{P})$ denote the unique homomorphism defined by
$\widetilde{\boldsymbol{\beta}}(p)=[p]$ for all $p\in P$. Then the
homomorphism
$\boldsymbol{\beta}=\widetilde{\boldsymbol{\beta}}\circ\varphi\colon
H\to\mathcal{B}(G_{P})$ is a transfer homomorphism. In particular, we have
$\Delta^{*}(H)=\Delta^{*}\big{(}\mathcal{B}(G_{P})\big{)}=\big{\\{}\min\Delta\big{(}\mathcal{B}(G_{0})\big{)}\mid
G_{0}\subset G_{P}\ \text{is a subset such that}\ \mathcal{B}(G_{0})\ \text{is
not half-factorial}\big{\\}}\,.$
Thus $\Delta^{*}(H)$ can be studied in an associated monoid of zero-sum
sequences and can thus be tackled by methods from Additive Combinatorics. Such
transfer results to monoids of zero-sum sequences are not restricted to Krull
monoids, but they do exist also from certain seminormal weakly Krull monoids
and from certain maximal orders in central simple algebras over global fields.
We do not outline this here but refer to [33, Theorem 1.1], [15], and [2,
Section 7].
Zero-Sum Theory is a vivid subfield of Additive Combinatorics (see the
monograph [20], the survey [10], and for a sample of recent papers on direct
and inverse zero-sum problems with a strong number theoretical flavor see [19,
8, 21, 34, 9]). We gather together the concepts needed in the sequel.
Let $G$ be a finite abelian group and $G_{0}\subset G$ a subset. Then $\langle
G_{0}\rangle\subset G$ denotes the subgroup generated by $G_{0}$. A family
$(e_{i})_{i\in I}$ of elements of $G$ is said to be independent if $e_{i}\neq
0$ for all $i\in I$ and, for every family $(m_{i})_{i\in
I}\in\mathbb{Z}^{(I)}$,
$\sum_{i\in I}m_{i}e_{i}=0\qquad\text{implies}\qquad
m_{i}e_{i}=0\quad\text{for all}\quad i\in I\,.$
The family $(e_{i})_{i\in I}$ is called a basis for $G$ if $G=\bigoplus_{i\in
I}\langle e_{i}\rangle$. The set $G_{0}$ is said to be independent if the
tuple $(g)_{g\in G_{0}}$ is independent. If for a prime $p\in\mathbb{P}$,
$\mathsf{r}_{p}(G)$ is the $p$-rank of $G$, then
$\mathsf{r}(G)=\max\\{\mathsf{r}_{p}(G)\mid p\in\mathbb{P}\\}\ \ \text{is the
{\it rank} of $G$ and}\ \
\mathsf{r}^{*}(G)=\sum_{p\in\mathbb{P}}\mathsf{r}_{p}(G)\ \text{ is the {\it
total rank} of $G$}\,.$
The monoid $\mathcal{B}(G_{0})$ of zero-sum sequences over $G_{0}$ is a
finitely generated Krull monoid. It is traditional to set
$\mathcal{A}(G_{0}):=\mathcal{A}\big{(}\mathcal{B}(G_{0})\big{)},\
\Delta(G_{0}):=\Delta\big{(}\mathcal{B}(G_{0})\big{)},\ \text{and}\
\Delta^{*}(G_{0}):=\Delta^{*}\big{(}\mathcal{B}(G_{0})\big{)}\,.$
Clearly, the atoms of $\mathcal{B}(G_{0})$ are precisely the minimal zero-sum
sequences over $G_{0}$. The set $\mathcal{A}(G_{0})$ is finite, and
$\mathsf{D}(G_{0})=\max\\{|S|\mid S\in\mathcal{A}(G_{0})\\}$ is the Davenport
constant of $G_{0}$. The set $G_{0}$ is called
* •
half-factorial if the monoid $\mathcal{B}(G_{0})$ is half-factorial
(equivalently, $\Delta(G_{0})=\emptyset$).
* •
non-half-factorial if the monoid $\mathcal{B}(G_{0})$ is not half-factorial
(equivalently, $\Delta(G_{0})\neq\emptyset$).
* •
minimal non-half-factorial if $\Delta(G_{0})\neq\emptyset$ but every proper
subset is half-factorial.
* •
$($in$)$decomposable if the monoid $\mathcal{B}(G_{0})$ is (in)decomposable.
(Maximal) half-factorial and (minimal) non-half-factorial subsets have found a
lot of attention in the literature (see [11, 28, 24, 25, 29, 5, 6]), and cross
numbers are a crucial tool for their study. For a sequence
$S=g_{1}\cdot\ldots\cdot g_{l}\in\mathcal{F}(G_{0})$, we call
$\displaystyle\mathsf{k}(S)$
$\displaystyle=\sum_{i=1}^{l}\frac{1}{\operatorname{ord}(g_{i})}\
\in\mathbb{Q}_{\geq 0}\quad\text{the {\it cross number} of $S$, and }$
$\displaystyle\mathsf{K}(G_{0})$ $\displaystyle=\max\\{\mathsf{k}(S)\mid
S\in\mathcal{A}(G_{0})\\}\quad\text{the {\it cross number} of $G_{0}$}.$
The following simple result ([13, Proposition 6.7.3]) will be used throughout
the paper without further mention.
###### Lemma 2.2.
Let $G$ be a finite abelian group and $G_{0}\subset G$ a subset. Then the
following statements are equivalent :
1. (a)
$G_{0}$ is half-factorial.
2. (b)
$\mathsf{k}(U)=1$ for every $U\in\mathcal{A}(G_{0})$.
3. (c)
$\mathsf{L}(B)=\\{\mathsf{k}(B)\\}$ for every $B\in\mathcal{B}(G_{0})$.
## 3\. Direct results on $\Delta^{*}(H)$
We start with a basic well-known lemma (see [13, Chapter 6.8]).
###### Lemma 3.1.
Let $G$ be a finite abelian group with $|G|>2$.
1. 1.
If $g\in G$ with $\operatorname{ord}(g)>2$, then
$\operatorname{ord}(g)-2\in\Delta^{*}(G)$. In particular,
$\exp(G)-2\in\Delta^{*}(G)$.
2. 2.
If $\mathsf{r}(G)\geq 2$, then $[1,\mathsf{r}(G)-1]\subset\Delta^{*}(G)$.
3. 3.
Let $G_{0}\subset G$ a subset.
1. (a)
If there exists a $U\in\mathcal{A}(G_{0})$ with $\mathsf{k}(U)<1$, then
$\min\Delta(G_{0})\leq\exp(G)-2$.
2. (b)
If $\mathsf{k}(U)\geq 1$ for all $U\in\mathcal{A}(G_{0})$, then
$\min\Delta(G_{0})\leq|G_{0}|-2$.
###### Proof.
1\. Let $g\in G$ with $\operatorname{ord}(g)=n>2$ and set $G_{0}=\\{g,-g\\}$.
Then $\mathcal{A}(G_{0})=\\{g^{n},(-g)^{n},\big{(}(-g)g\big{)}\\}$,
$\Delta(G_{0})=\\{n-2\\}$, and hence $\min\Delta(G_{0})=n-2$.
2\. Let $s\in[2,\mathsf{r}(G)]$. Then there is a prime $p\in\mathbb{P}$ such
that $C_{p}^{s}$ is isomorphic to a subgroup of $G$, and it suffices to show
that $s-1\in\Delta^{*}(C_{p}^{s})$. Let $(e_{1},\ldots,e_{s})$ be a basis of
$C_{p}^{s}$ and set $e_{0}=e_{1}+\ldots+e_{s}$ and
$G_{0}=\\{e_{0},\ldots,e_{s}\\}$. Then a simple calculation (details can be
found in [13, Proposition 6.8.1]) shows that $\Delta(G_{0})=\\{s-1\\}$ and
hence $\min\Delta(G_{0})=s-1$.
3.(a) Let $U=g_{1}\cdot\ldots\cdot g_{l}\in\mathcal{A}(G_{0})$ with
$\mathsf{k}(U)<1$ and $n=\exp(G)$ (note that $\mathsf{k}(U)<1$ implies $U\neq
0$, $l\geq 2$ and $\mathsf{k}(U)>\frac{1}{n}$). Then
$U_{i}=g_{i}^{\operatorname{ord}(g_{i})}\in\mathcal{A}(G_{0})$ for all
$i\in[1,l]$, and
$U^{n}=\prod_{i=1}^{l}U_{i}^{n/\operatorname{ord}(g_{i})}$
implies that
$n\mathsf{k}(U)=\sum_{i=1}^{l}\frac{n}{\operatorname{ord}(g_{i})}\in\mathsf{L}(U^{n})$.
Since $\mathsf{k}(U)<1$, we have $n\mathsf{k}(U)\in[2,n-1]$ and
$\min\Delta(G_{0})\leq n-n\mathsf{k}(U)\in[1,n-2]$.
3.(b) The proof is similar to that of 3.(a), see [13, Lemma 6.8.6] for
details. ∎
Lemma 3.1.3 motivates the following definitions (see [30, 31]). A subset
$G_{0}\subset G$ is called an LCN-set (large cross number set) if
$\mathsf{k}(U)\geq 1$ for each $U\in\mathcal{A}(G_{0})$ and
$\mathsf{m}(G)=\max\big{\\{}\min\Delta(G_{0})\mid G_{0}\subset G\ \text{is a
non-half-factorial LCN-set}\big{\\}}\,.$
Clearly, if $G$ has a non-half-factorial LCN-set, then $|G|\geq 4$. The
following result (due to Schmid [31]) is crucial for our approach.
###### Proposition 3.2.
Let $G$ be a finite abelian group with $|G|>2$. Then
$\max\Delta^{*}(G)=\max\\{\exp(G)-2,\mathsf{m}(G)\\}\ \text{and}\
\mathsf{m}(G)\leq\max\\{\mathsf{r}^{*}(G)-1,\mathsf{K}(G)-1\\}\,.$
If $G$ is a $p$-group, then $\mathsf{m}(G)=\mathsf{r}(G)-1$ and thus
$\max\Delta^{*}(G)=\max\\{\exp(G)-2,\mathsf{r}(G)-1\\}$.
###### Proof.
See [31, Theorem 3.1, Lemma 3.3.(4), and Proposition 3.6]. ∎
###### Lemma 3.3.
Let $G$ be a finite abelian group and $G_{0}\subset G$ a subset.
1. 1.
The following statements are equivalent :
1. (a)
$G_{0}$ is decomposable.
2. (b)
There are nonempty subsets $G_{1},G_{2}\subset G_{0}$ such that
$G_{0}=G_{1}\uplus G_{2}$ and
$\mathcal{B}(G_{0})=\mathcal{B}(G_{1})\negthinspace\times\negthinspace\mathcal{B}(G_{2})$.
3. (c)
There are nonempty subsets $G_{1},G_{2}\subset G_{0}$ such that
$G_{0}=G_{1}\uplus G_{2}$ and
$\mathcal{A}(G_{0})=\mathcal{A}(G_{1})\uplus\mathcal{A}(G_{2})$.
4. (d)
There are nonempty subsets $G_{1},G_{2}\subset G_{0}$ such that $\langle
G_{0}\rangle=\langle G_{1}\rangle\oplus\langle G_{2}\rangle$.
2. 2.
If $G_{0}$ is minimal non-half-factorial, then $G_{0}$ is indecomposable.
###### Proof.
1\. See [26, Lemma 3.7] and [1, Lemma 3.2].
2\. This follows immediately from 1.(b). ∎
###### Lemma 3.4.
Let $G$ be a finite abelian group and $G_{0}\subset G$ a subset.
1. 1.
For each $g\in G_{0}$,
$\displaystyle\gcd\big{(}\\{\mathsf{v}_{g}(B)\mid
B\in\mathcal{B}(G_{0})\\}\big{)}=\gcd\big{(}\\{\mathsf{v}_{g}(A)\mid
A\in\mathcal{A}(G_{0})\\}\big{)}$ $\displaystyle=$
$\displaystyle\min\big{(}\\{\mathsf{v}_{g}(A)\mid\mathsf{v}_{g}(A)>0,A\in\mathcal{A}(G_{0})\\}\big{)}=\min\big{(}\\{\mathsf{v}_{g}(B)\mid\mathsf{v}_{g}(B)>0,B\in\mathcal{B}(G_{0})\\}\big{)}$
$\displaystyle=$ $\displaystyle\min\big{(}\\{k\in\mathbb{N}\mid kg\in\langle
G_{0}\setminus\\{g\\}\rangle\\}\big{)}=\gcd\big{(}\\{k\in\mathbb{N}\mid
kg\in\langle G_{0}\setminus\\{g\\}\rangle\\}\big{)}\,.$
In particular, $\min\big{(}\\{k\in\mathbb{N}\mid kg\in\langle
G_{0}\setminus\\{g\\}\rangle\\}\big{)}$ divides $\operatorname{ord}(g)$.
2. 2.
Suppose that for any $h\in G_{0}$, we have that $h\not\in\langle
G_{0}\setminus\\{h,\ h^{\prime}\\}\rangle$ for any $h^{\prime}\in
G_{0}\setminus\\{h\\}$. Then for any atom $A$ with
$\operatorname{supp}(A)\subsetneq G_{0}$ and any $h\in\operatorname{supp}(A)$,
we have $\gcd(\mathsf{v}_{h}(A),\operatorname{ord}(h))>1$.
3. 3.
If $G_{0}$ is minimal non-half-factorial, then there exists a minimal non-
half-factorial subset $G_{0}^{*}\subset G$ with $|G_{0}|=|G_{0}^{*}|$ and a
transfer homomorphism
$\theta\colon\mathcal{B}(G_{0})\to\mathcal{B}(G_{0}^{*})$ such that the
following properties are satisfied :
1. (a)
For each $g\in G_{0}^{*}$, we have $g\in\langle
G_{0}^{*}\setminus\\{g\\}\rangle$.
2. (b)
For each $B\in\mathcal{B}(G_{0})$, we have
$\mathsf{k}(B)=\mathsf{k}\big{(}\theta(B)\big{)}$.
3. (c)
If $G_{0}^{*}$ has the property that for each $h\in G_{0}^{*}$,
$h\not\in\langle E\rangle$ for any $E\subsetneq G_{0}^{*}\setminus\\{h\\}$,
then $G_{0}$ also has the property.
4. (d)
If $G_{0}^{*}$ has the property that there exists $h\in G_{0}^{*}$, such that
$G_{0}^{*}\setminus\\{h\\}$ is independent, then $G_{0}$ also has the
property.
###### Proof.
1\. Let $g\in G_{0}$ and let $\gamma_{1},\ldots,\gamma_{6}$ denote the six
terms in the given order of the asserted equation. By definition, it follows
that $\gamma_{1}\leq\gamma_{2}\leq\gamma_{3}$. Since $\\{\mathsf{v}_{g}(B)\mid
B\in\mathcal{B}(G_{0})\\}=\\{k\in\mathbb{N}\mid kg\in\langle
G_{0}\setminus\\{g\\}\rangle\\}$, we have that $\gamma_{1}=\gamma_{6}$ and
$\gamma_{4}=\gamma_{5}$. Therefore we only need to show
$\gamma_{3}\leq\gamma_{4}$ and $\gamma_{4}\leq\gamma_{1}$.
To show that $\gamma_{3}\leq\gamma_{4}$, let $B\in\mathcal{B}(G_{0})$ such
that $\mathsf{v}_{g}(B)=\gamma_{4}$. Suppose that $B=A_{1}\cdot\ldots\cdot
A_{s}$ with $s\in\mathbb{N}$ and $A_{1},\ldots,A_{s}\in\mathcal{A}(G_{0})$.
Then $\mathsf{v}_{g}(B)=\mathsf{v}_{g}(A_{1})+\ldots+\mathsf{v}_{g}(A_{s})$.
The minimality of $\mathsf{v}_{g}(B)$ implies that there is precisely one
$i\in[1,s]$ with $\mathsf{v}_{g}(A_{i})=\mathsf{v}_{g}(B)$ and
$\mathsf{v}_{g}(A_{j})=0$ for all $j\in[1,s]\setminus\\{i\\}$. Thus
$\gamma_{3}\leq\mathsf{v}_{g}(A_{i})=\mathsf{v}_{g}(B)=\gamma_{4}$.
Next we show that $\gamma_{4}\leq\gamma_{1}$. There are $s\in\mathbb{N}$,
$r\in[1,s]$, $U_{1},\ldots,U_{s}\in\mathcal{B}(G_{0})$, and
$k_{1},\ldots,k_{s}\in\mathbb{N}$ such that
$\displaystyle\gamma_{1}$
$\displaystyle=k_{1}\mathsf{v}_{g}(U_{s})+\ldots+k_{r}\mathsf{v}_{g}(U_{r})-k_{r+1}\mathsf{v}_{g}(U_{r+1})-\ldots-
k_{s}\mathsf{v}_{g}(U_{s})$
$\displaystyle=\mathsf{v}_{g}(U_{1}^{k_{1}}\cdot\ldots\cdot
U_{r}^{k_{r}})-\mathsf{v}_{g}(U_{r+1}^{k_{r+1}}\cdot\ldots\cdot
U_{s}^{k_{s}})\,.$
Setting $B_{1}=U_{1}^{k_{1}}\cdot\ldots\cdot U_{r}^{k_{r}}$,
$B_{2}=U_{r+1}^{k_{r+1}}\cdot\ldots\cdot U_{s}^{k_{s}}$, and
$B_{3}=\prod_{h\in G_{0}\setminus\\{g\\}}h^{|B_{2}|}$ we obtain that
$B_{1}B_{2}^{-1}B_{3}\in\mathcal{B}(G_{0})$ and
$\gamma_{1}=\mathsf{v}_{g}(B_{1})-\mathsf{v}_{g}(B_{2})=\mathsf{v}_{g}(B_{1}B_{2}^{-1}B_{3})\geq\gamma_{4}\,.$
In particular, $\gamma_{5}=\gamma_{2}$ divides $\operatorname{ord}(g)$ because
$g^{\operatorname{ord}(g)}\in\mathcal{A}(G_{0})$.
2\. Assume to the contrary that there are $A$ and $h$ as above such that
$\gcd(\mathsf{v}_{h}(A),\operatorname{ord}(h))=1$. Choose $h^{\prime}\in
G_{0}\setminus\operatorname{supp}(A)$, then
$h\in\langle\operatorname{supp}(A)\setminus\\{h\\}\rangle\subset\langle
G_{0}\setminus\\{h,h^{\prime}\\}\rangle$, a contradiction.
3\. By [13, Theorem 6.7.11], there are a subset $G_{0}^{*}\subset G$
satisfying Property (a) and a transfer homomorphism
$\theta\colon\mathcal{B}(G_{0})\to\mathcal{B}(G_{0}^{*})$. Moreover, the
transfer homomorphism $\theta$ is a composition of transfer homomorphisms
$\theta^{\prime}$ of the following form:
* •
Let $g\in G_{0}$, $m=\min\bigl{\\{}k\in\mathbb{N}\mid kg\in\langle
G_{0}\setminus\\{g\\}\rangle\bigr{\\}}$,
$G_{0}^{\prime}=G_{0}\setminus\\{g\\}\cup\\{mg\\}$, and
$\theta^{\prime}\colon\mathcal{B}(G_{0})\to\mathcal{B}(G_{0}^{\prime})\,,\quad\text{defined
by}\quad\theta^{\prime}(B)=g^{-\mathsf{v}_{g}(B)}(mg)^{\mathsf{v}_{g}(B)/m}B\,,$
It is outlined that $m\,|\,\mathsf{v}_{g}(B)$ and that
$m\,|\,\operatorname{ord}(g)$.
Therefore it is sufficient to show that $|G_{0}|=|G_{0}^{\prime}|$ and that
$\theta^{\prime}$ satisfies Properties (b) - (d).
(i) By definition, we have $\mathsf{k}(B)=\mathsf{k}(\theta^{\prime}(B))$ for
all $B\in\mathcal{B}(G_{0})$.
(ii) Since $G_{0}$ is a minimal non-half-factorial set, the same is true for
$G_{0}^{\prime}$ by [13, Lemma 6.8.9]. If $mg\in G_{0}\setminus\\{g\\}$, then
$G_{0}^{\prime}\subsetneq G_{0}$ would be non-half-factorial, a contradiction
to the minimality of $G_{0}$. It follows that $mg\not\in
G_{0}\setminus\\{g\\}$, which implies that $|G_{0}^{\prime}|=|G_{0}|$.
(iii) We set $G_{0}=\\{g=g_{1},\ldots,g_{k}\\}$ (note that $k\geq 2$),
$G_{0}^{\prime}=\\{mg,g_{2},\ldots,g_{k}\\}$, and suppose that
$h\not\in\langle E\rangle$ for each $h\in G_{0}^{\prime}$ and for any
$E\subsetneq G_{0}^{\prime}\setminus\\{h\\}$. Assume to the contrary that
there exist $h\in G_{0}$ and $E\subsetneq G_{0}\setminus\\{h\\}$ such that
$h\in\langle E\rangle$. If $h=g$, then $mg\in\langle E\rangle$, a
contradiction.
Suppose that $h\neq g$, say $h=g_{k}\in\langle E\rangle$ with
$E\subsetneq\\{g,g_{2},\ldots,g_{k-1}\\}$. If $g\not\in E$, then $E\subsetneq
G_{0}^{\prime}\setminus\\{mg\\}$, a contradiction. Thus $g\in E$, and we set
$E^{\prime}=E\setminus\\{g\\}\cup\\{mg\\}$. Since $h\in\langle E\rangle$, we
have that $h=\sum_{x\in E\setminus\\{g\\}}t_{x}x+tg$ where
$t_{x},t\in\mathbb{Z}$. Thus $tg=h-\sum_{x\in
E\setminus\\{g\\}}t_{x}x\in\langle
E\cup\\{h\\}\setminus\\{g\\}\rangle\subset\langle
G_{0}\setminus\\{g\\}\rangle.$ By 1., we obtain that $m\,|\,t$ and hence
$h=\sum_{x\in E\setminus\\{g\\}}t_{x}x+\frac{t}{m}mg\in\langle
E^{\prime}\rangle$, a contradiction.
(iv) We set $G_{0}=\\{g=g_{1},\ldots,g_{k}\\}$,
$G_{0}^{\prime}=\\{mg,g_{2},\ldots,g_{k}\\}$, and suppose that there exists
$h\in G_{0}^{\prime}$ such that $G_{0}^{\prime}\setminus\\{h\\}$ is
independent. If $h=mg$, then
$G_{0}\setminus\\{g\\}=G_{0}^{\prime}\setminus\\{h\\}$ is independent. Suppose
that $h\not=mg$, say $h=g_{k}$. Then $\\{mg,g_{2},\ldots,g_{k-1}\\}$ is
independent and assume to the contrary that
$G_{0}\setminus\\{h\\}=\\{g,g_{2},\ldots,g_{k-1}\\}$ is not independent. Then
there exist $t_{1},\ldots,t_{k-1}\in\mathbb{Z}$ such that
$t_{1}g+t_{2}g_{2}+\ldots+t_{k-1}g_{k-1}=0$ but $t_{i}g_{i}\neq 0$ for at
least one $i\in[1,k-1]$. This implies that $t_{1}g\in\langle
g_{2},\ldots,g_{k-1}\rangle\subset\langle G_{0}\setminus\\{g\\}\rangle$. By
1., we obtain that $m\,|\,t_{1}$ and hence
$\frac{t_{1}}{m}mg+t_{2}g_{2}+\ldots+t_{k-1}g_{k-1}=0$, a contradiction to
$\\{mg,g_{2},\ldots,g_{k-1}\\}$ is independent.
∎
###### Lemma 3.5.
Let $G$ be a finite abelian group and $G_{0}\subset G$ a subset with
$|G_{0}|\geq\mathsf{r}(G)+2$ such that the following two properties are
satisfied :
1. (a)
For any $h\in G_{0}$, $G_{0}\setminus\\{h\\}$ is half-factorial and
$h\not\in\langle G_{0}\setminus\\{h,\ h^{\prime}\\}\rangle$ for any
$h^{\prime}\in G_{0}\setminus\\{h\\}$.
2. (b)
There exists an element $g\in G_{0}$ such that $g\in\langle
G_{0}\setminus\\{g\\}\rangle$ and $\operatorname{ord}(g)$ is not a prime
power.
Then $|G_{0}|\leq\exp(G)-2$.
###### Proof.
We set $\exp(G)=n=p_{1}^{k_{1}}\cdot\ldots\cdot p_{t}^{k_{t}}$, where $t\geq
2,k_{1},\ldots,k_{t}\in\mathbb{N}$ and $p_{1},\ldots,p_{t}$ are distinct
primes. By Lemma 3.4.2, we know that for any atom $A$ with
$\operatorname{supp}(A)\subsetneq G_{0}$ and any $h\in\operatorname{supp}(A)$,
we have $\gcd(\mathsf{v}_{h}(A),\operatorname{ord}(h))>1$. In particular,
(3.1) $\mathsf{v}_{h}(A)\geq 2\qquad\text{ for each
$h\in\operatorname{supp}(A)$. }$
We continue with the following assertion.
1. A.
For each $\nu\in[1,t]$ with $p_{\nu}\,|\,\operatorname{ord}(g)$, there is an
atom $U_{\nu}\in\mathcal{A}(G_{0})$ such that
$\mathsf{v}_{g}(U_{\nu})\,|\,\frac{n}{p_{\nu}^{k_{\nu}}}$,
$\mathsf{k}(U_{\nu})=1$, and
$|\operatorname{supp}(U_{\nu})\setminus\\{g\\}|\leq\frac{n-\mathsf{v}_{g}(U_{\nu})}{2}$.
Proof of A. Let $\nu\in[1,t]$ with $p_{\nu}\,|\,\operatorname{ord}(g)$. Since
$g\in\langle G_{0}\setminus\\{g\\}\rangle$ and $t\geq 2$, it follows that
$0\neq\frac{n}{p_{\nu}^{k_{\nu}}}g\in
G_{\nu}=\langle\frac{n}{p_{\nu}^{k_{\nu}}}h\mid h\in
G_{0}\setminus\\{g\\}\rangle$. Obviously, $G_{\nu}$ is a $p_{\nu}$-group. Let
$E_{\nu}\subset G_{0}\setminus\\{g\\}$ be minimal such that
$\frac{n}{p_{\nu}^{k_{\nu}}}g\in\langle\frac{n}{p_{\nu}^{k_{\nu}}}E_{\nu}\rangle$.
The minimality of $E_{\nu}$ implies that
$|E_{\nu}|=|\frac{n}{p_{\nu}^{k_{\nu}}}E_{\nu}|$ and it implies that
$\frac{n}{p_{\nu}^{k_{\nu}}}E_{\nu}$ is a minimal generating set of
$G_{\nu}^{\prime}:=\langle\frac{n}{p_{\nu}^{k_{\nu}}}E_{\nu}\rangle$. Thus
[13, Lemma A.6.2] implies that
$|\frac{n}{p_{\nu}^{k_{\nu}}}E_{\nu}|\leq\mathsf{r}^{*}(G_{\nu}^{\prime})$.
Putting all together we obtain that
$|E_{\nu}|=|\frac{n}{p_{\nu}^{k_{\nu}}}E_{\nu}|\leq\mathsf{r}^{*}(G_{\nu}^{\prime})=\mathsf{r}(G_{\nu}^{\prime})\leq\mathsf{r}(G)\,.$
Let $d_{\nu}\in\mathbb{N}$ be minimal such that $d_{\nu}g\in\langle
E_{\nu}\rangle$. By Lemma 3.4.1, $d_{\nu}\,|\,\frac{n}{p_{\nu}^{k_{\nu}}}$ and
there exists an atom $U_{\nu}$ such that $\mathsf{v}_{h}(U_{\nu})=d_{\nu}$ and
$|\operatorname{supp}(U_{\nu})|\leq|E_{\nu}|+1\leq\mathsf{r}(G)+1\leq|G_{0}|-1$.
Thus Property (a) implies that $\mathsf{k}(U_{\nu})=1$. Let
$U_{\nu}=g^{\mathsf{v}_{g}(U_{\nu})}\prod_{h\in\operatorname{supp}(U_{\nu})\setminus\\{g\\}}h^{\mathsf{v}_{h}(U_{\nu})}\,.$
Since $\mathsf{v}_{h}(U_{\nu})\geq 2$ for each
$h\in\operatorname{supp}(U_{\nu})\setminus\\{g\\}$ by Equation (3.1), it
follows that
$1=\mathsf{k}(U_{\nu})\geq\frac{\mathsf{v}_{g}(U_{\nu})}{n}+|\operatorname{supp}(U_{\nu})\setminus\\{g\\}|\frac{2}{n}\,,$
whence
$|\operatorname{supp}(U_{\nu})\setminus\\{g\\}|\leq\frac{n-\mathsf{v}_{g}(U_{\nu})}{2}$.
∎(Proof of A)
Let $s\in\mathbb{N}$ be minimal such that there exists a nonempty subset
$E\subsetneq G_{0}\setminus\\{g\\}$ with $sg\in\langle E\rangle$ and let
$E\subsetneq G_{0}\setminus\\{g\\}$ be minimal such that $sg\in\langle
E\rangle$. By Lemma 3.4.1, there is an atom $V$ with $\mathsf{v}_{g}(V)=s$ and
$\operatorname{supp}(V)=\\{g\\}\cup E\subsetneq G_{0}$. Then
$1=\mathsf{k}(V)=\frac{s}{\operatorname{ord}(g)}+\sum_{h\in
E}\frac{\mathsf{v}_{h}(V)}{\operatorname{ord}(h)}\,.$
By Equation (3.1), we have that $\mathsf{v}_{h}(V)\geq 2$ for each $h\in E$
and hence the equation above implies that $|E|\leq\frac{n-s}{2}$.
CASE 1: $s$ is a power of a prime, say a power of $p_{1}$.
Let $E_{1}=\operatorname{supp}(U_{1})\setminus\\{g\\}$. Since
$\mathsf{v}_{g}(U_{1})\,|\,\frac{n}{p_{1}^{k_{1}}}$, we have that $g\in\langle
sg,\mathsf{v}_{g}(U_{1})g\rangle\subset\langle E\cup E_{1}\rangle$. Property
(a) implies that $E\cup E_{1}=G_{0}\setminus\\{g\\}$, and thus
$|G_{0}|\leq 1+|E|+|E_{1}|\leq
1+\frac{n-s}{2}+\frac{n-\mathsf{v}_{g}(U_{1})}{2}=1+n-\frac{\mathsf{v}_{g}(U_{1})+s}{2}\,.$
Since $\gcd(\mathsf{v}_{g}(U_{1}),s)=1$, it follows that
$\mathsf{v}_{g}(U_{1})+s\geq 5$, hence $|G_{0}|\leq n-3/2$, and thus
$|G_{0}|\leq n-2$.
CASE 2: $s$ is not a prime power, say $p_{1}p_{2}\,|\,s$.
Then $s\geq 6$. Let $d=\gcd(s,\mathsf{v}_{g}(U_{1}))$ and
$E_{1}=\operatorname{supp}(U_{1})\setminus\\{g\\}$, then $d<s$ and
$dg\in\langle sg,\mathsf{v}_{g}(U_{1})g\rangle\subset\langle E\cup
E_{1}\rangle\subset\langle G_{0}\setminus\\{g\\}\rangle$. The minimality of
$s$ implies that $E\cup E_{1}=G_{0}\setminus\\{g\\}$, and thus
$|G_{0}|\leq 1+|E|+|E_{1}|\leq
1+\frac{n-s}{2}+\frac{n-\mathsf{v}_{g}(U_{1})}{2}=1+n-\frac{\mathsf{v}_{g}(U_{1})+s}{2}\leq
n-3\,.\qed$
###### Lemma 3.6.
Let $G$ be a finite abelian group with $\exp(G)=n$. Let $G_{0}\subset G$ be a
minimal non-half-factorial LCN-set and suppose that there is a subset
$G_{2}\subset G_{0}$ such that $\langle G_{2}\rangle=\langle G_{0}\rangle$ and
$|G_{2}|\leq|G_{0}|-2$. Then $\min\Delta(G_{0})\leq\max\\{1,n-4\\}$.
###### Proof.
Assume to the contrary that $\min\Delta(G_{0})\geq\max\\{2,n-3\\}$. By [27,
Corollary 3.1], the existence of the subset $G_{2}$ implies that
$\mathsf{k}(U)\in\mathbb{N}$ for each $U\in\mathcal{A}(G_{0})$ and
$\min\Delta(G_{0})\,|\,\gcd\big{(}\\{\mathsf{k}(A)-1\mid
A\in\mathcal{A}(G_{0})\\}\big{)}\,.$
We set
$W_{1}=\\{A\in\mathcal{A}(G_{0})\mid\mathsf{k}(A)=1\\}\quad\text{and}\quad
W_{2}=\\{A\in\mathcal{A}(G_{0})\mid\mathsf{k}(A)>1\\}\,.$
Then it follows that, for each $U_{1},U_{2}\in W_{2}$,
(3.2)
$\mathsf{k}(U_{1})\geq\max\\{3,n-2\\}\quad\text{and}\quad\big{(}\text{either}\
\mathsf{k}(U_{1})=\mathsf{k}(U_{2})\ \text{or}\
|\mathsf{k}(U_{1})-\mathsf{k}(U_{2})|\geq\max\\{2,n-3\\}\big{)}\,.$
We choose an element $U\in W_{2}$. Then $\operatorname{supp}(U)=G_{0}$, and we
pick an element $g\in G_{0}\setminus G_{2}$. Then $g\in\langle G_{2}\rangle$
and, by Lemma 3.4.1, there is an atom $A$ with $\mathsf{v}_{g}(A)=1$ and
$\operatorname{supp}(A)\subset G_{2}\cup\\{g\\}\subsetneq G_{0}$. This implies
that $A\in W_{1}$, and
$UA^{\operatorname{ord}(g)-\mathsf{v}_{g}(U)}=g^{\operatorname{ord}(g)}S$
for some zero-sum sequence $S$ over $G$. Since
$\operatorname{supp}(S)=G_{0}\setminus\\{g\\}$ and $G_{0}$ is minimal non-
half-factorial, $S$ has a factorization into a product of atoms from $W_{1}$.
Therefore, for each $U\in W_{2}$, there are $A_{1},\ldots,A_{m}\in W_{1}$,
where $m\leq\operatorname{ord}(g)-\mathsf{v}_{g}(U)\leq n-1$, such that
$UA_{1}\cdot\ldots\cdot A_{m}$ can be factorized into a product of atoms from
$W_{1}$.
We set
$W_{0}=\\{A\in\mathcal{A}(G_{0})\mid\mathsf{k}(A)=\min\\{\mathsf{k}(B)\mid
B\in W_{2}\\}\\}\subset W_{2}\,,$
and we consider all tuples $(U,A_{1},\ldots,A_{m})$, where $U\in W_{0}$ and
$A_{1},\ldots,A_{m}\in W_{1}$, such that $UA_{1}\cdot\ldots\cdot A_{m}$ can be
factorized into a product of atoms from $W_{1}$. We fix one such tuple
$(U,A_{1},\ldots,A_{m})$ with the property that $m$ is minimal possible. Note
that $m\leq n-1$. Let
(3.3) $UA_{1}\cdot\ldots\cdot A_{m}=V_{1}\cdot\ldots\cdot V_{t}\quad\text{
with}\quad t\in\mathbb{N}\quad\text{and}\quad V_{1},\ldots,V_{t}\in W_{1}\,.$
We observe that $\mathsf{k}(U)=t-m$ and continue with the following assertion.
1. A1.
For each $\nu\in[1,t]$, we have $V_{\nu}\nmid UA_{1}\cdot\ldots\cdot A_{m-1}$.
Proof of A1. Assume to the contrary that there is such a $\nu\in[1,t]$, say
$\nu=1$, with $V_{1}\,|\,UA_{1}\cdot\ldots\cdot A_{m-1}$. Then there are
$l\in\mathbb{N}$ and $T_{1},\ldots,T_{l}\in\mathcal{A}(G_{0})$ such that
$UA_{1}\cdot\ldots\cdot A_{m-1}=V_{1}T_{1}\cdot\ldots\cdot T_{\mathit{l}}\,.$
By the minimality of $m$, there exists some $\nu\in[1,l]$ such that
$T_{\nu}\in W_{2}$, say $\nu=1$. Since
$\sum_{\nu=2}^{l}\mathsf{k}(T_{\nu})=\mathsf{k}(U)+(m-1)-1-\mathsf{k}(T_{1})\leq
m-2\leq n-3\,,$
and $\mathsf{k}(T^{\prime})\geq n-2$ for all $T^{\prime}\in W_{2}$, it follows
that $T_{2},\ldots,T_{l}\in W_{1}$, whence
$l=1+\sum_{\nu=2}^{l}\mathsf{k}(T_{\nu})\leq m-1$. We obtain that
$V_{1}T_{1}\cdot\ldots\cdot T_{\mathit{l}}A_{m}=UA_{1}\cdot\ldots\cdot
A_{m}=V_{1}\cdot\ldots\cdot V_{t}\,,$
and thus
$T_{1}\cdot\ldots\cdot T_{\mathit{l}}A_{m}=V_{2}\cdot\ldots\cdot V_{t}\,.$
The minimality of $m$ implies that $\mathsf{k}(T_{1})>\mathsf{k}(U)$. It
follows that
$\mathsf{k}(T_{1})-\mathsf{k}(U)=m-1-{\mathit{l}}\leq m-2\leq
n-3\leq\max\\{n-3,2\\}\leq\mathsf{k}(T_{1})-\mathsf{k}(U).$
Therefore $l=1$, $m=n-1$, $n\geq 5$ and $\mathsf{k}(T_{1})=\mathsf{k}(U)+n-3$.
Thus
$T_{1}A_{n-1}=V_{2}\cdot\ldots\cdot V_{t}\,,\quad\text{and hence}\quad
t-1\leq|A_{n-1}|\,.$
This equation shows that $\mathsf{k}(T_{1})=t-2\leq|A_{n-1}|-1\leq n-1$, and
hence $n-2\leq\mathsf{k}(U)=\mathsf{k}(T_{1})-n+3\leq 2$, a contradiction to
$n\geq 5$. ∎(Proof of A1)
Since $\exp(G)=n$ and $\mathsf{k}(A_{m})=1$, it follows that $|A_{m}|\leq n$.
By A1, for each $\nu\in[1,t]$ there exists an element
$h_{\nu}\in\operatorname{supp}(A_{m})$ such that
$\mathsf{v}_{h_{\nu}}(V_{\nu})>\mathsf{v}_{h_{\nu}}(UA_{1}\cdot\ldots\cdot
A_{m-1})\,.$
For each $h\in\operatorname{supp}(A_{m})$ we define
$F_{h}=\\{\nu\in[1,t]\ \mid\
\mathsf{v}_{h}(V_{\nu})>\mathsf{v}_{h}(UA_{1}\cdot\ldots\cdot
A_{m-1})\\}\subset[1,t]\,.$
Thus
$\bigcup_{h\in\operatorname{supp}(A_{m})}F_{h}=[1,t]\,$
and for each $h\in\operatorname{supp}(A_{m})$, we have
$\mathsf{v}_{h}(A_{m})+\mathsf{v}_{h}(UA_{1}\cdot\ldots\cdot
A_{m-1})=\sum_{i=1}^{t}\mathsf{v}_{h}(V_{i})\geq\sum_{i\in
F_{h}}\mathsf{v}_{h}(V_{i})\geq|F_{h}|\big{(}\mathsf{v}_{h}(UA_{1}\cdot\ldots\cdot
A_{m-1})+1\big{)}\,.$
Since $|A_{m}|>|\operatorname{supp}(A_{m})|$ (otherwise, it would follow that
$A_{m}\,|\,U$, a contradiction), we obtain that
$\displaystyle t$
$\displaystyle=\Big{|}\bigcup_{h\in\operatorname{supp}(A_{m})}F_{h}\Big{|}\leq\sum_{h}|F_{h}|\leq\sum_{h}\frac{\mathsf{v}_{h}(A_{m})+\mathsf{v}_{h}(UA_{1}\cdot\ldots\cdot
A_{m-1})}{\mathsf{v}_{h}(UA_{1}\cdot\ldots\cdot A_{m-1})+1}$
$\displaystyle\leq\sum_{h}\frac{\mathsf{v}_{h}(A_{m})+1}{2}=\frac{|A_{m}|}{2}+\frac{|\operatorname{supp}(A_{m})|}{2}<|A_{m}|\leq
n\,.$
By Equations (3.3) and (3.2), we have
$\max\\{3,n-2\\}\leq\mathsf{k}(U)=t-m\leq n-1-m$ and hence $m=1$, $n\geq 5$,
$t=n-1$, and $\mathsf{k}(U)=n-2$. Therefore
(3.4) $UA_{1}=V_{1}\cdot\ldots\cdot V_{n-1},\ \,|A_{1}|=n,\
\,n-2\leq|\operatorname{supp}(A_{1})|\leq n-1\ \,,$
and
(3.5) $\sum_{h\in\operatorname{supp}(A_{1})}|F_{h}|=n-1,\quad\text{and the
sets $F_{h},h\in\operatorname{supp}(A_{1})$ are pairwise disjoint.}$
Furthermore,
$|F_{h}|\leq\frac{\mathsf{v}_{h}(A_{1})+\mathsf{v}_{h}(U)}{\mathsf{v}_{h}(U)+1}$
for each $h\in\operatorname{supp}(A_{1})$. Then for each
$h\in\operatorname{supp}(A_{1})$, we have that
(3.6) $|F_{h}|\leq 1\quad\text{ when }\mathsf{v}_{h}(A_{1})\leq 2\,\quad\text{
and }\,\quad|F_{h}|\leq 2\quad\text{ when }\mathsf{v}_{h}(A_{1})\leq 4\,.$
Now we consider all atoms $A_{1}\in W_{1}$ such that $UA_{1}$ can be
factorized into a product of $n-1$ atoms from $W_{1}$, and among them the
atoms $A_{1}^{\prime}$ for which $|\operatorname{supp}(A_{1}^{\prime})|$ is
minimal, and among them we choose an atom $A_{1}^{\prime\prime}$ for which
$\mathsf{h}(A_{1}^{\prime\prime})$ is minimal. Changing notation if necessary
we suppose that $A_{1}$ has this property. By Equation (3.4), we distinguish
three cases depending on $|\operatorname{supp}(A_{1})|$ and
$\mathsf{h}(A_{1})$.
CASE 1: $|\operatorname{supp}(A_{1})|=n-1$.
Let $\operatorname{supp}(A_{1})=\\{g_{1},\ldots,g_{n-1}\\}$ and
$A_{1}=g_{1}^{2}g_{2}\cdot\ldots\cdot g_{n-1}$. Since $\mathsf{h}(A_{1})=2$,
Equations (3.6) and (3.5) imply that $|F_{h}|=1$ for each
$h\in\operatorname{supp}(A_{1})$. Note that $Ug_{1}^{2}g_{2}\cdot\ldots\cdot
g_{n-1}=V_{1}\cdot\ldots\cdot V_{n-1}$. After renumbering if necessary we may
suppose that $F_{g_{i}}=\\{i\\}$ for each $i\in[1,n-1]$. Therefore, we have
$\mathsf{v}_{g_{i}}(V_{i})>\mathsf{v}_{g_{i}}(U)\geq 1$ for each
$i\in[1,n-1]$. Hence $\mathsf{v}_{g_{1}}(V_{1})\geq 2$ and we set
$V_{1}=g_{1}^{2}Y_{1}$ for some $Y_{1}$ dividing $U$. Thus
$UY_{1}^{-1}g_{2}\cdot\ldots\cdot g_{n-1}=V_{2}\cdot\ldots\cdot V_{n-1}$ which
implies that $V_{i}=g_{i}Y_{i}$, for $i\in[2,n-1]$, where
$Y_{2}\cdot\ldots\cdot Y_{n-1}=UY_{1}^{-1}$. Summing up we have
(3.7) $U=Y_{1}\cdot\ldots\cdot Y_{n-1}\text{ such that
}V_{i}=g_{i}Y_{i}\,\text{ for }\,i\in[2,\,n-1]\,\text{ and
}\,V_{1}=g_{1}^{2}Y_{1}.$
If $n$ is even and $X\in\mathcal{A}(G)$ such that $X\,|\,A_{1}^{n/2}$, then
$\mathsf{k}(X)\leq(n/2)\mathsf{k}(A_{1})=n/2<n-2$ whence $X\in W_{1}$ and
$\mathsf{k}(X)=1$. This shows that $\mathsf{L}(A_{1}^{n/2})=\\{n/2\\}$.
Similarly, if $n$ is odd, then $\mathsf{L}(A_{1}^{(n+1)/2})=\\{(n+1)/2\\}$.
Therefore,
$A^{\prime}=\left\\{\begin{aligned}
A_{1}^{\frac{n}{2}}&=g_{1}^{n}g_{2}^{\frac{n}{2}}\cdot\ldots\cdot
g_{n-1}^{\frac{n}{2}}\quad\text{can only be written as a product of $n/2$
atoms if $n$ is even, }\\\
A_{1}^{\frac{n+1}{2}}&=g_{1}^{n}g_{1}g_{2}^{\frac{n+1}{2}}\cdot\ldots\cdot
g_{n-1}^{\frac{n+1}{2}}\quad\text{can only be written as product of $(n+1)/2$
atoms if $n$ is odd}\,.\end{aligned}\right.$
Thus we can find an atom $C\,|\,A^{\prime}(g_{1}^{n})^{-1}$ with
$\operatorname{supp}(C)\subset\\{g_{2},\ldots,g_{n-1}\\}$ and
$|\operatorname{supp}(C)|\geq 2$, say $g_{2},g_{3}\in\operatorname{supp}(C)$.
Therefore, we obtain that $V_{2}V_{3}=g_{2}g_{3}Y_{2}Y_{3}\,|\,UC$, say
$UC=V_{2}V_{3}V^{\prime}$ for some $V^{\prime}\in\mathcal{B}(G)$. Since
$\mathsf{k}(UC)=\mathsf{k}(U)+\mathsf{k}(C)=n-1=\mathsf{k}(V_{2})+\mathsf{k}(V_{3})+\mathsf{k}(V^{\prime})\,,$
we obtain that $\mathsf{k}(V^{\prime})=n-3$. Now Equation (3.2) implies that
$V^{\prime}$ is a product of atoms from $W_{1}$, and hence $UC$ can be
factorized into a product of $n-1$ atoms. Since
$|\operatorname{supp}(C)|<n-1=|\operatorname{supp}(A_{1})|$, this is a
contradiction to the choice of $A_{1}$.
CASE 2: $|\operatorname{supp}(A_{1})|=n-2$ and $\mathsf{h}(A_{1})=2$.
Let $\operatorname{supp}(A_{1})=\\{g_{1},\ldots,g_{n-2}\\}$ and
$A_{1}=g_{1}^{2}g_{2}^{2}g_{3}\cdot\ldots\cdot g_{n-2}$. Since
$\mathsf{h}(A_{1})=2$, Equation (3.6) implies that $|F_{h}|\leq 1$ for each
$h\in\operatorname{supp}(A_{1})$. Thus
$\sum_{h\in\operatorname{supp}(A_{1})}|F_{h}|\leq n-2$, a contradiction to
Equation (3.5).
CASE 3: $|\operatorname{supp}(A_{1})|=n-2$ and $\mathsf{h}(A_{1})=3$.
Let $\operatorname{supp}(A_{1})=\\{g_{1},\ldots,g_{n-2}\\}$ and
$A_{1}=g_{1}^{3}g_{2}\cdot\ldots\cdot g_{n-2}$. Since $\mathsf{h}(A_{1})=3$,
the Equations (3.6) and (3.5) imply that $|F_{g_{1}}|=2$ and $|F_{g_{i}}|=1$
for each $i\in[2,\,n-2]$. Note that $Ug_{1}^{3}g_{2}\cdot\ldots\cdot
g_{n-2}=V_{1}\cdot\ldots\cdot V_{n-1}$. After renumbering if necessary we may
suppose that $F_{g_{1}}=\\{1,n-1\\}$ and $F_{g_{i}}=\\{i\\}$ for each
$i\in[2,n-2]$. Therefore we have
$\mathsf{v}_{g_{i}}(V_{i})>\mathsf{v}_{g_{i}}(U)\geq 1$ for each $i\in[1,n-2]$
and $\mathsf{v}_{g_{1}}(V_{n-1})>\mathsf{v}_{g_{1}}(U)\geq 1$. Hence we may
set $V_{n-1}=g_{1}^{2}Y_{n-1}$ for some $Y_{n-1}$ dividing $U$. Thus
$UY_{n-1}^{-1}g_{1}g_{2}\cdot\ldots\cdot g_{n-2}=V_{1}\cdot\ldots\cdot
V_{n-2}$ which implies that $V_{i}=g_{i}Y_{i}$ for each $i\in[1,n-2]$ where
$Y_{1}\cdot\ldots\cdot Y_{n-2}=UY_{n-1}^{-1}$. Summing up we have
(3.8) $U=Y_{1}\cdot\ldots\cdot Y_{n-1}\text{ such that
}V_{i}=g_{i}Y_{i}\,\text{ for }\,i\in[1,\,n-2]\,\text{ and
}\,V_{n-1}=g_{1}^{2}Y_{n-1}.$
As in CASE 1 we obtain that (note $n\geq 5$)
$A^{\prime}=\left\\{\begin{aligned}
A_{1}^{\frac{n}{3}}&=g_{1}^{n}g_{2}^{\frac{n}{3}}\cdot\ldots\cdot
g_{n-2}^{\frac{n}{3}}\quad\ \ \ \,\text{can only be written as a product of
$\,\frac{n}{3}$ atoms if}\ n\equiv 0\mod{3}\\\
A_{1}^{\frac{n+1}{3}}&=g_{1}^{n}g_{1}g_{2}^{\frac{n+1}{3}}\cdot\ldots\cdot
g_{n-2}^{\frac{n+1}{3}}\ \text{can only be written as a product of
$\,\frac{n+1}{3}$ atoms if}\ n\equiv 2\mod{3}\\\
A_{1}^{\frac{n+2}{3}}&=g_{1}^{n}g_{1}^{2}g_{2}^{\frac{n+2}{3}}\cdot\ldots\cdot
g_{n-2}^{\frac{n+2}{3}}\ \text{can only be written as a product of
$\,\frac{n+2}{3}$ atoms if}\ n\equiv 1\mod{3}\,.\end{aligned}\right.$
Let $C\in\mathcal{A}(G)$ be an atom dividing $A^{\prime}(g_{1}^{n})^{-1}$.
Then $\operatorname{supp}(C)\subset\\{g_{1},\ldots,g_{n-2}\\}$ and
$|\operatorname{supp}(C)|\geq 2$, say $g_{i},g_{j}\in\operatorname{supp}(C)$
where $1\leq i<j\leq n-2$. Therefore, we obtain that
$V_{i}V_{j}=g_{i}g_{j}Y_{i}Y_{j}\,|\,UC$ by Equation (3.8). Arguing as in CASE
1 we infer that $UC$ is a product of $n-1$ atoms from $W_{1}$. By the choice
of $A_{1}$, we obtain that $|\operatorname{supp}(C)|=n-2$ and
$\mathsf{h}(C)\geq 3$. Since this holds for all atoms dividing
$A^{\prime}(g_{1}^{n})^{-1}$, we obtain a contradiction to the structure of
$A^{\prime}$. ∎
###### Proof of Theorem 1.1.
Let $H$ be a Krull monoid with class group $G$ and let $G_{P}\subset G$ denote
the set of classes containing prime divisors. If $|G|\leq 2$, then $H$ is
half-factorial by [13, Corollary 3.4.12], and thus
$\Delta^{*}(H)\subset\Delta(H)=\emptyset$. If $G$ is infinite and $G_{P}=G$,
then $\Delta^{*}(H)=\mathbb{N}$ by [7, Theorem 1.1].
Suppose that $2<|G|<\infty$. By Lemma 2.1, it suffices to prove the statements
for the Krull monoid $\mathcal{B}(G_{P})$. If $G$ is finite, then $\Delta(G)$
is finite by [13, Corollary 3.4.13], hence $\Delta^{*}(G)$ is finite, and
Lemma 3.1 shows that $\\{\exp(G)-2,\mathsf{r}(G)-1\\}\subset\Delta^{*}(G)$.
Since $\Delta^{*}(G_{P})\subset\Delta^{*}(G)$, it remains to prove that
$\max\Delta^{*}(G)\leq\max\\{\exp(G)-2,\mathsf{r}(G)-1\\}\,.$
Let $G_{0}\subset G$ be a non-half-factorial subset, $n=\exp(G)$, and
$r=\mathsf{r}(G)$. We need to prove that
$\min\Delta(G_{0})\leq\max\\{n-2,r-1\\}$. If $G_{1}\subset G_{0}$ is non-half-
factorial, then
$\min\Delta(G_{0})=\gcd\Delta(G_{0})\,|\,\gcd\Delta(G_{1})=\min\Delta(G_{1})$.
Thus we may suppose that $G_{0}$ is minimal non-half-factorial. If there is an
$U\in\mathcal{A}(G_{0})$ with $\mathsf{k}(U)<1$, then Lemma 3.1.3 implies that
$\min\Delta(G_{0})\leq n-2$. Suppose that $\mathsf{k}(U)\geq 1$ for all
$U\in\mathcal{A}(G_{0})$, i.e, $G_{0}$ is an LCN-set. Since $G_{0}$ is minimal
non-half-factorial, it follows that $G_{0}$ is indecomposable by Lemma 3.3. By
Lemma 3.4.3, we may suppose that for each $g\in G_{0}$ we have $g\in\langle
G_{0}\setminus\\{g\\}\rangle$. Suppose that the order of each element of
$G_{0}$ is a prime power. Since $G_{0}$ is indecomposable, Lemma 3.3 implies
that each order is a power of a fixed prime $p\in\mathbb{P}$, and thus
$\langle G_{0}\rangle$ is a $p$-group. By Proposition 3.2 we infer that
$\min\Delta(G_{0})\leq\max\Delta^{*}(\langle G_{0}\rangle)=\max\\{\exp(\langle
G_{0}\rangle)-2,\mathsf{r}(\langle G_{0}\rangle)-1\\}\leq\max\\{n-2,r-1\\}\,.$
From now on we suppose that there is an element $g\in G_{0}$ whose order is
not a prime power. Then $n\geq 6$. If $|G_{0}|\leq r+1$, then
$\min\Delta(G_{0})\leq|G_{0}|-2\leq r-1$ by Lemma 3.1.3. Thus we may suppose
that $|G_{0}|\geq r+2$ and we distinguish two cases.
CASE 1: There exists a subset $G_{2}\subset G_{0}$ such that $\langle
G_{2}\rangle=\langle G_{0}\rangle$ and $|G_{2}|\leq|G_{0}|-2$.
Then Lemma 3.6 implies that $\min\Delta(G_{0})\leq n-4\leq n-2$.
CASE 2: Every subset $G_{1}\subset G_{0}$ with $|G_{1}|=|G_{0}|-1$ is a
minimal generating set of $\langle G_{0}\rangle$.
Then for each $h\in G_{0}$, $G_{0}\setminus\\{h\\}$ is half-factorial and
$h\notin\langle G_{0}\setminus\\{h,h^{\prime}\\}\rangle\ \text{for any }\
h^{\prime}\in G_{0}\setminus\\{h\\}$. Thus Lemma 3.5 implies that $|G_{0}|\leq
n-2$ and hence $\min\Delta(G_{0})\leq|G_{0}|-2\leq n-4\leq n-2$ by Lemma
3.1.3. ∎
## 4\. Inverse results on $\Delta^{*}(H)$
Let $G$ be a finite abelian group. In this section we study the structure of
minimal non-half-factorial subsets $G_{0}\subset G$ with
$\min\Delta(G_{0})=\max\Delta^{*}(G)$. These structural investigations were
started by Schmid who obtained a characterization in case
$\exp(G)-2>\mathsf{m}(G)$ (Lemma 4.1.1). Our main result in this section is
Theorem 4.5. All examples of minimal non-half-factorial subsets $G_{0}\subset
G$ with $\min\Delta(G_{0})=\max\Delta^{*}(G)$ known so far are simple, and the
standing conjecture was that all such sets are simple. We provide the first
example of such a set $G_{0}$ which is not simple (Remark 4.6).
###### Lemma 4.1.
Let $G$ be a finite abelian group with $|G|>2$, $\exp(G)=n$,
$\mathsf{r}(G)=r$, and let $G_{0}\subset G$ be a subset with
$\min\Delta(G_{0})=\max\Delta^{*}(G)$.
1. 1.
Suppose that $\mathsf{m}(G)<n-2$. Then $G_{0}$ is indecomposable if and only
$G_{0}=\\{g,-g\\}$ for some $g\in G$ with $\operatorname{ord}(g)=n$.
2. 2.
Suppose that $r\leq n-1$. Then $G_{0}$ is minimal non-half-factorial but not
an LCN-set if and only if $G_{0}=\\{g,-g\\}$ for some $g\in G$ with
$\operatorname{ord}(g)=n$.
###### Proof.
1\. See [30, Theorem 5.1].
2\. Since $n=2$ implies $r=1$ and $|G|=2$, it follows that $n\geq 3$. By
Theorem 1.1, we have that $\min\Delta(G_{0})=n-2$. Obviously, the set
$\\{-g,g\\}$, with $g\in G$ and $\operatorname{ord}(g)=n$, is a minimal non-
half-factorial set with $\min\Delta(\\{-g,g\\})=n-2$ but not an LCN-set.
Conversely, let $G_{0}$ be minimal non-half-factorial but not an LCN-set. Then
there exists an $A\in\mathcal{A}(G_{0})$ with $\mathsf{k}(A)<1$. Since
$\\{n,n\mathsf{k}(A^{n})\\}\subset\mathsf{L}(A^{n})$, it follows that
$n-2\,|\,n(\mathsf{k}(A)-1)$ whence $\mathsf{k}(A)=\frac{2}{n}$. Consequently,
$A=(-g)g$ for some $g$ with $\operatorname{ord}(g)=n$. Thus $\\{-g,g\\}\subset
G_{0}$, and since $G_{0}$ is minimal non-half-factorial, equality follows. ∎
###### Lemma 4.2.
Let $G$ be a finite abelian group with $\exp(G)=n$, $\mathsf{r}(G)=r$, and let
$G_{0}\subset G$ be a minimal non-half-factorial LCN-set with
$\min\Delta(G_{0})=\max\Delta^{*}(G)$.
1. 1.
Then $|G_{0}|=r+1$, $r\geq n-1$ and for each $h\in G_{0}$, $h\not\in\langle
G_{0}\setminus\\{h,\ h^{\prime}\\}\rangle$ for any $h^{\prime}\in
G_{0}\setminus\\{h\\}$.
2. 2.
If $r\leq n-2$, then $\mathsf{m}(G)\leq n-3$.
3. 3.
If $n\geq 5$ and $r\leq n-3$ then $\mathsf{m}(G)\leq n-4$.
###### Proof.
1\. We have that $\min\Delta(G_{0})\leq|G_{0}|-2$ by Lemma 3.1.3 and
$\min\Delta(G_{0})=\max\\{n-2,r-1\\}$ by Theorem 1.1.
By Lemma 3.4.3 (Properties (a) and (c)), we may assume that for each $g\in
G_{0}$ we have $g\in\langle G_{0}\setminus\\{g\\}\rangle$.
CASE 1: There is a subset $G_{2}\subset G_{0}$ such that $\langle
G_{2}\rangle=\langle G_{0}\rangle$ and $|G_{2}|\leq|G_{0}|-2$.
The existence of $G_{2}$ implies that $G$ is neither isomorphic to $C_{3}$ nor
to $C_{2}\oplus C_{2}$ nor to $C_{3}\oplus C_{3}$ (this is immediately clear
for the first two groups; to exclude the case $C_{3}\oplus C_{3}$, use again
[27, Corollary 3.1] which says that $\mathsf{k}(U)\in\mathbb{N}$ for each
$U\in\mathcal{A}(G_{0})$). By Lemma 3.6, we know that
$\min\Delta(G_{0})\leq\max\\{n-4,1\\}<\max\\{n-2,r-1\\}=\min\Delta(G_{0})$, a
contradiction.
CASE 2: Every subset $G_{1}\subset G_{0}$ with $|G_{1}|=|G_{0}|-1$ is a
minimal generating set of $\langle G_{0}\rangle$.
Then for each $h\in G_{0}$, $h\not\in\langle G_{0}\setminus\\{h,\
h^{\prime}\\}\rangle$ for any $h^{\prime}\in G_{0}\setminus\\{h\\}$.
If $|G_{0}|\geq r+2$, then by Lemma 3.5 $|G_{0}|\leq n-2$, it follows that
$\min\Delta(G_{0})\leq|G_{0}|-2\leq n-4$, a contradiction.
If $|G_{0}|\leq r+1$, then
$\max\\{n-2,r-1\\}=\min\Delta(G_{0})\leq|G_{0}|-2\leq r-1$, so we must have
$|G_{0}|=r+1$ and $r\geq n-1$.
2\. Assume to the contrary that $r\leq n-2$ and that $\mathsf{m}(G)\geq n-2$.
Then by Theorem 1.1, $\max\Delta^{*}(G)=\max\\{r-1,n-2\\}=n-2$. Since
$\mathsf{m}(G)\geq n-2$, there is a minimal non-half-factorial LCN-set $G_{0}$
with $\min\Delta(G_{0})=\max\Delta^{*}(G)$, and then 1. implies that $r\geq
n-1$, a contradiction.
3\. Let $G_{0}\subset G$ be a non-half-factorial LCN-subset. We need to prove
that $\min\Delta(G_{0})\leq n-4$. Without restriction we may suppose that
$G_{0}$ is minimal non-half-factorial which implies that $G_{0}$ is
indecomposable by Lemma 3.3. By Lemma 3.4.3, we may suppose that for each
$g\in G_{0}$ we have $g\in\langle G_{0}\setminus\\{g\\}\rangle$. Suppose that
the order of each element of $G_{0}$ is a prime power. Since $G_{0}$ is
indecomposable, Lemma 3.3 implies that each order is a power of a fixed prime
$p\in\mathbb{P}$, and thus $\langle G_{0}\rangle$ is a $p$-group. By
Proposition 3.2, we infer that
$\min\Delta(G_{0})\leq\mathsf{m}(\langle G_{0}\rangle)=\mathsf{r}(\langle
G_{0}\rangle)-1\leq\mathsf{r}(G)-1\leq n-4\,.$
From now on we suppose that there is an element $g\in G_{0}$ whose order is
not a prime power. If $|G_{0}|\leq n-2$, then
$\min\Delta(G_{0})\leq|G_{0}|-2\leq n-4$ by Lemma 3.1.3. Thus we may suppose
that $|G_{0}|\geq n-1\geq r+2$ and we distinguish two cases.
CASE 1: There exists a subset $G_{2}\subset G_{0}$ such that $\langle
G_{2}\rangle=\langle G_{0}\rangle$ and $|G_{2}|\leq|G_{0}|-2$.
Then Lemma 3.6 implies that $\min\Delta(G_{0})\leq n-4$.
CASE 2: Every subset $G_{1}\subset G_{0}$ with $|G_{1}|=|G_{0}|-1$ is a
minimal generating set of $\langle G_{0}\rangle$.
Then for each $h\in G_{0}$, $G_{0}\setminus\\{h\\}$ is half-factorial and
$h\notin\langle G_{0}\setminus\\{h,h^{\prime}\\}\rangle\ \text{for any }\
h^{\prime}\in G_{0}\setminus\\{h\\}$. Thus Lemma 3.5 implies that $|G_{0}|\leq
n-2$, a contradiction. ∎
###### Lemma 4.3.
Let $G$ be a finite abelian group with $\exp(G)=n$, $\mathsf{r}(G)=r$, and let
$G_{0}\subset G$ be a minimal non-half-factorial LCN-set with
$\min\Delta(G_{0})=\max\Delta^{*}(G)$.
1. 1.
If $A\in\mathcal{A}(G_{0})$ with $\mathsf{k}(A)=1$, then
$|\operatorname{supp}(A)|\leq\frac{n}{2}$.
2. 2.
If $A\in\mathcal{A}(G_{0})$ with $\mathsf{k}(A)>1$, then $\mathsf{k}(A)<r$ and
$SA^{-1}$ is also an atom where $S=\prod_{g\in
G_{0}}g^{\operatorname{ord}(g)}$.
###### Proof.
By Lemma 4.2, we have $r\geq n-1$, $|G_{0}|=r+1$, and for each $h\in G_{0}$,
$h\not\in\langle G_{0}\setminus\\{h,\ h^{\prime}\\}\rangle$ for any
$h^{\prime}\in G_{0}\setminus\\{h\\}$. Let $A\in\mathcal{A}(G_{0})$.
1\. Since $\mathsf{k}(A)=1$, it follows that
$|\operatorname{supp}(A)|\leq|A|\leq n$. Assume that
$|\operatorname{supp}(A)|=n$. Then $\mathsf{v}_{g}(A)=1$ for each
$g\in\operatorname{supp}(A)$. Since $G_{0}$ is a minimal non-half-factorial
LCN-set, there is a $V\in\mathcal{A}(G_{0})$ with $\mathsf{k}(V)>1$ and
$\operatorname{supp}(V)=G_{0}$. Therefore $A\,|\,V$, a contradiction. Thus
$|\operatorname{supp}(A)|\leq n-1$ whence $\operatorname{supp}(A)\subsetneq
G_{0}$. Therefore Lemma 3.4.2 implies that
$\gcd(\mathsf{v}_{g}(A),\operatorname{ord}(g))>1$ for each
$g\in\operatorname{supp}(A)$, and hence $|\operatorname{supp}(A)|\leq|A|/2\leq
n/2$.
2\. Let $A\in\mathcal{A}(G_{0})$ with $\mathsf{k}(A)>1$. Then $A\,|\,S$,
$r+1=|G_{0}|=\max\mathsf{L}(S)$, and
$\mathsf{L}(S)\setminus\\{r+1\\}\neq\emptyset$. By Theorem 1.1, we have
$\min\Delta(G_{0})=r-1$, hence $\mathsf{L}(S)=\\{2,r+1\\}$, and thus $SA^{-1}$
is an atom. If $\mathsf{k}(SA^{-1})=1$, then 1. implies that
$|\operatorname{supp}(SA^{-1})|\leq n/2$, but on the other hand we have
$|\operatorname{supp}(SA^{-1})|=|G_{0}|=r+1\geq n$, a contradiction. Therefore
we obtain that $\mathsf{k}(SA^{-1})>1$ and hence
$r+1=\mathsf{k}(S)=\mathsf{k}(A)+\mathsf{k}(SA^{-1})$ implies that
$\mathsf{k}(A)<r$. ∎
###### Lemma 4.4.
Let $G$ be a finite abelian group with $\exp(G)=n$, $\mathsf{r}(G)=r$, and let
$G_{0}\subset G$ be a minimal non-half-factorial LCN-set with
$\min\Delta(G_{0})=\max\Delta^{*}(G)$. Let $g\in G_{0}$ with $g\in\langle
G_{0}\setminus\\{g\\}\rangle$ and $d\in[1,\operatorname{ord}(g)]$ be minimal
such that $dg\in\langle E^{*}\rangle$ for some subset $E^{*}\subsetneq
G_{0}\setminus\\{g\\}$. Then $d\,|\,\operatorname{ord}(g)$, and we have
1. 1.
Let $k\in[1,\,\operatorname{ord}(g)]$. If $kg\not\in\langle E\rangle$ for any
$E\subsetneq G_{0}\setminus\\{g\\}$, then there is an atom $A$ with
$\mathsf{v}_{g}(A)=k$ and $\mathsf{k}(A)>1$.
2. 2.
Let $k\in[1,\,\operatorname{ord}(g)-1]$ with $d\nmid k$. Then there is an atom
$A$ with $\mathsf{v}_{g}(A)=k$ and $\mathsf{k}(A)>1$. In particular, if
$B\in\mathcal{B}(G_{0})$ with $\mathsf{v}_{g}(B)=k$ and $B\,|\,\prod_{g\in
G_{0}}g^{\operatorname{ord}(g)}$, then $B$ is an atom.
3. 3.
If $A_{1},A_{2}$ are atoms with
$\mathsf{v}_{g}(A_{1})\equiv\mathsf{v}_{g}(A_{2})\mod d$, then
$\mathsf{k}(A_{1})=\mathsf{k}(A_{2})$.
###### Proof.
Note that by Lemma 4.2, we have $|G_{0}|=r+1$ and $r\geq n-1$. The minimality
of $d$ and Lemma 3.4.1 imply that $d\,|\,\operatorname{ord}(g)$. We set
$S=\prod_{g\in G_{0}}g^{\operatorname{ord}(g)}$.
1\. Since $kg\in\langle G_{0}\setminus\\{g\\}\rangle$, there is a zero-sum
sequence $A$ such that $\mathsf{v}_{g}(A)=k$, and we choose an $A$ with
minimal length $|A|$. Then $\operatorname{supp}(A)=G_{0}$ by assumption on
$kg$, and we assert that $A$ is an atom. If this holds, then $\mathsf{k}(A)>1$
by Lemma 4.3.1.
Assume to the contrary that $A=A_{1}\cdot\ldots\cdot A_{s}$ with $s\geq 2$ and
atoms $A_{1},\ldots,A_{s}$. The minimality of $|A|$ implies that
$\mathsf{v}_{g}(A_{i})>0$ for each $i\in[1,s]$. If there exists an $i\in[1,s]$
such that $\mathsf{k}(A_{i})>1$, say $A_{1}$, then $S=A_{1}\cdot\ldots\cdot
A_{s}(SA^{-1})$ but $SA_{1}^{-1}=A_{2}\cdot\ldots\cdot A_{s}(SA^{-1})$ is not
an atom, a contradiction to Lemma 4.3.2. Thus, for each $i\in[1,s]$, we have
$\mathsf{k}(A_{i})=1$ and hence $\operatorname{supp}(A_{i})\subsetneq G_{0}$
by Lemma 4.3.1.
For each $i\in[1,\,s]$, we set $t_{i}=\mathsf{v}_{g}(A_{i})$,
$d_{i}=\gcd(\\{t_{1},\ldots,t_{i},\operatorname{ord}(g)\\})$, and let
$E_{i}\subset G_{0}\setminus\\{g\\}$ be minimal such that $d_{i}g\in\langle
E_{i}\rangle$. Note that $k=t_{1}+\ldots+k_{s}$. Since $d_{1}g\in\langle
t_{1}g\rangle\subset\langle\operatorname{supp}(A_{1})\setminus\\{g\\}\rangle\subsetneq\langle
G_{0}\setminus\\{g\\}\rangle$, it follows that $E_{1}\subsetneq
G_{0}\setminus\\{g\\}$. Since $kg\in\langle d_{s}g\rangle\subset\langle
E_{s}\rangle$, it follows that $E_{s}=G_{0}\setminus\\{g\\}$.
Let $l\in[1,s-1]$ be maximal such that $E_{l}\subsetneq
G_{0}\setminus\\{g\\}$. Then $d_{l}g\in\langle E_{l}\rangle$ and
$E_{l+1}=G_{0}\setminus\\{g\\}$. Let $d_{0}\in\mathbb{N}$ be the minimal such
that $d_{0}g\in E_{l}$. Then Lemma 3.4.1 implies that $d_{0}\,|\,d_{l}$ and
there exists an atom $W$ such that $\operatorname{supp}(W)=\\{g\\}\cup E_{l}$,
$\mathsf{v}_{g}(W)=d_{0}$, and $\mathsf{k}(W)=1$. Since $d_{l+1}g\in\langle
d_{l}g,t_{l+1}g\rangle\subset\langle
E_{l}\cup\operatorname{supp}(A_{l+1})\setminus\\{g\\}\rangle$, we have that
$E_{l}\cup\operatorname{supp}(A_{l+1})\setminus\\{g\\}=G_{0}\setminus\\{g\\}$.
Then Lemma 4.3.1 implies that $|G_{0}|\leq
1+|E_{l}|+|\operatorname{supp}(A_{l+1})\setminus\\{g\\}|\leq
1+(n/2-1)+(n/2-1)=n-1$, a contradiction.
2\. If $kg\in\langle E_{1}\rangle$ for some $E_{1}\subsetneq
G_{0}\setminus\\{g\\}$, then $\gcd(d,k)g\in\langle kg\rangle\subset\langle
E_{1}\rangle$, whence the minimality of $d$ implies that $\gcd(d,k)=d$ and
$d\,|\,k$, a contradiction. Therefore, we obtain that $kg\not\in\langle
E\rangle$ for any $E\subsetneq G_{0}\setminus\\{g\\}$. Thus 1. implies that
there is an atom $A$ with $\mathsf{v}_{g}(A)=k$ and $\mathsf{k}(A)>1$.
Let $B\in\mathcal{B}(G_{0})$ with $B\,|\,S$ and $\mathsf{v}_{g}(B)=k$. We set
$B=A_{1}\cdot\ldots\cdot A_{s}$ with $s\in\mathbb{N}$ and atoms
$A_{1},\ldots,A_{s}$. Then
$\mathsf{v}_{g}(A_{1})+\ldots+\mathsf{v}_{g}(A_{s})=\mathsf{v}_{g}(B)=k$.
Since $d\nmid k$, there is an $i\in[1,s]$ with $d\nmid\mathsf{v}_{g}(A_{i})$.
We want to show that $\mathsf{k}(A_{i})>1$, and assume to the contrary that
$\mathsf{k}(A_{i})=1$. Then $|\operatorname{supp}(A_{i})|\leq n/2$ by Lemma
4.3.1. Furthermore, $d^{\prime}=\gcd(d,\mathsf{v}_{g}(A_{i}))<d$, but
$d^{\prime}g\in\langle\mathsf{v}_{g}(A_{i})g\rangle\subset\langle\operatorname{supp}(A_{i})\setminus\\{g\\}\rangle\quad\text{and}\quad\operatorname{supp}(A_{i})\setminus\\{g\\}\subsetneq
G_{0}\setminus\\{g\\}\,,$
a contradiction to the minimality of $d$. Therefore it follows that
$\mathsf{k}(A_{i})>1$. Since $g\,|\,SB^{-1}$, it follows that $S\neq B$. Since
$S=A_{i}\big{(}(BA_{i}^{-1})(SB^{-1})\big{)}$ and $SA_{i}^{-1}$ is an atom by
Lemma 4.3.2, it follows that $B=A_{i}\in\mathcal{A}(G_{0})$.
3\. Let $A_{1}\in\mathcal{A}(G_{0})$. We assert that
$\mathsf{k}(A_{1})=\mathsf{k}(A_{2})$ for all $A_{2}\in\mathcal{A}(G_{0})$
with $\mathsf{v}_{g}(A_{1})\equiv\mathsf{v}_{g}(A_{2})\mod d$. We distinguish
two cases.
CASE 1: $d\,|\,\mathsf{v}_{g}(A_{1})$.
There is an $A\in\mathcal{A}(G_{0})$ with $\mathsf{v}_{g}(A)=d$ and
$\mathsf{k}(A)=1$. It is sufficient to show that $\mathsf{k}(A_{1})=1$. There
are $l\in\mathbb{N}$ and
$V_{1},\ldots,V_{l}\in\mathcal{A}(G_{0}\setminus\\{g\\})$ (hence
$\mathsf{k}(V_{1})=\ldots=\mathsf{k}(V_{l})=1$) such that
$A_{1}A^{\frac{\operatorname{ord}(g)-\mathsf{v}_{g}(A_{1})}{d}}=g^{\operatorname{ord}(g)}V_{1}\cdot\ldots\cdot
V_{l}\quad\text{hence}\quad\mathsf{k}(A_{1})=1+l-\frac{\operatorname{ord}(g)-\mathsf{v}_{g}(A_{1})}{d}\,.$
Furthermore, $\min\Delta(G_{0})=r-1$ divides
$(l+1)-\Big{(}1+\frac{\operatorname{ord}(g)-\mathsf{v}_{g}(A_{1})}{d}\Big{)}=\mathsf{k}(A_{1})-1\,.$
Since $\mathsf{k}(A_{1})<r$ by Lemma 4.3, it follows that
$\mathsf{k}(A_{1})=1$.
CASE 2: $d\nmid\mathsf{v}_{g}(A_{1})$.
Let $d_{0}\in[1,d-1]$ such that $\mathsf{v}_{g}(A_{1})\equiv d_{0}\mod d$. By
2., there are atoms $B_{l}$ such that $\mathsf{v}_{g}(B_{l})=d_{0}+ld$ for all
$l\in\mathbb{N}_{0}$ with $d_{0}+ld<\operatorname{ord}(g)$. Thus by an
inductive argument it is sufficient to prove the assertion for those atoms
$A_{2}$ with $\mathsf{v}_{g}(A_{2})=\mathsf{v}_{g}(A_{1})$ and with
$\mathsf{v}_{g}(A_{2})=\mathsf{v}_{g}(A_{1})+d$.
Suppose that $\mathsf{v}_{g}(A_{1})=\mathsf{v}_{g}(A_{2})$. By 2., there is an
atom $V$ such that
$\mathsf{v}_{g}(V)=\operatorname{ord}(g)-\mathsf{v}_{g}(A_{1})$. Then there
are $l\in\mathbb{N}$ and
$V_{1},\ldots,V_{l}\in\mathcal{A}(G_{0}\setminus\\{g\\})$ such that
$A_{1}V=g^{\operatorname{ord}(g)}V_{1}\cdot\ldots\cdot V_{l}$ and hence
$\mathsf{k}(A_{1})+\mathsf{k}(V)=1+\sum_{i=1}^{l}\mathsf{k}(V_{i})=l+1$. Since
$\min\Delta(G_{0})=r-1$ divides $l-1$, it follows that either $l=r$ or $l\geq
2r-1$. If $l\geq 2r-1$, then $\mathsf{k}(A_{1})\geq r$ or $\mathsf{k}(V)\geq
r$, a contradiction to Lemma 4.3. Therefore
$\mathsf{k}(A_{1})+\mathsf{k}(V)=r+1=\mathsf{k}(A_{2})+\mathsf{k}(V)$ and
hence $\mathsf{k}(A_{1})=\mathsf{k}(A_{2})$.
Suppose that $\mathsf{v}_{g}(A_{1})=\mathsf{v}_{g}(A_{2})+d$. Let $E\subsetneq
G_{0}\setminus\\{g\\}$ such that $dg\in\langle E\rangle$. Then there is an
$A\in\mathcal{A}(E\cup\\{g\\})$ with $\mathsf{v}_{g}(A)=d$, and clearly
$\mathsf{k}(A)=1$. Let $V_{1},\ldots,V_{t}$ be all the atoms with
$V_{\nu}\,|\,A_{2}A$ and $|\operatorname{supp}(V_{\nu})|=1$ for all
$\nu\in[1,\,t]$. Since
$\mathsf{v}_{g}(A_{2}A)=\mathsf{v}_{g}(A_{1})<\operatorname{ord}(g)$, it
follows that $B=A_{2}A(V_{1}\cdot\ldots\cdot V_{t})^{-1}$ divides $S$ and that
$\mathsf{v}_{g}(B)=\mathsf{v}_{g}(A_{1})$. Therefore 2. implies that $B$ is an
atom, and by Step 1 we obtain that $\mathsf{k}(B)=\mathsf{k}(A_{1})$. If
$t\geq 2$, then $A_{2}A=BV_{1}\cdot\ldots\cdot V_{t}$ implies $t\geq
1+\min\Delta(G_{0})=r$, and thus $\mathsf{k}(A_{2})\geq r$, a contradiction to
Lemma 4.3. Therefore we obtain that $t=1$ and thus
$\mathsf{k}(A_{2})+1=\mathsf{k}(B)+1=\mathsf{k}(A_{1})+1$. ∎
###### Theorem 4.5.
Let $G$ be a finite abelian group with $\exp(G)=n$, $\mathsf{r}(G)=r$, and let
$G_{0}\subset G$ be a minimal non-half-factorial set with
$\min\Delta(G_{0})=\max\Delta^{*}(G)$.
1. 1.
If $r<n-1$, then there exists $g\in G$ with $\operatorname{ord}(g)=n$ such
that $G_{0}=\\{g,\>-g\\}$.
2. 2.
Let $r=n-1$. If $G_{0}$ is not an LCN-set, then there exists $g\in G$ with
$\operatorname{ord}(g)=n$ such that $G_{0}=\\{g,\>-g\\}$. If $G_{0}$ is an
LCN-set, then $|G_{0}|=r+1$ and for each $h\in G_{0}$, $h\not\in\langle
G_{0}\setminus\\{h,\ h^{\prime}\\}\rangle$ for any $h^{\prime}\in
G_{0}\setminus\\{h\\}$.
3. 3.
If $r\geq n$, then $G_{0}$ is an LCN-set with $|G_{0}|=r+1$ and for each $h\in
G_{0}$, $h\not\in\langle G_{0}\setminus\\{h,\ h^{\prime}\\}\rangle$ for any
$h^{\prime}\in G_{0}\setminus\\{h\\}$.
4. 4.
If $r\geq n-1$, $G_{0}$ is an LCN-set, and $n$ is odd, then there exists an
element $g\in G_{0}$ such that $G_{0}\setminus\\{g\\}$ is independent.
###### Proof.
1\. Suppose that $r<n-1$. Then Lemma 4.2 implies that $G_{0}$ is not an LCN-
set. Thus Lemma 4.1.2 implies that $G_{0}$ has the asserted form.
2\. If $G_{0}$ is not an LCN-set, then the assertion follows from Lemma 4.1.2.
If $G_{0}$ is an LCN-set, then the assertion follows from Lemma 4.2.1.
3\. Suppose that $r\geq n$. Then Theorem 1.1 implies that
$\min\Delta(G_{0})=\max\Delta^{*}(G)=r-1$. Thus Lemma 3.1.3.(a) imply that
$G_{0}$ is an LCN-set. Hence the assertion follows from Lemma 4.2.1.
4\. Let $r\geq n-1$, $G_{0}$ be an LCN-set, and suppose that $n$ is odd. By
Lemma 3.4.3 (Properties (a) and (d)), we may suppose without restriction that
$g\in\langle G_{0}\setminus\\{g\\}\rangle$ for each $g\in G_{0}$. Lemma 4.2
implies that $|G_{0}|=r+1$ and that for each $g\in G_{0}$ we have
$g\not\in\langle E\rangle$ for any $E\subsetneq G_{0}\setminus\\{g\\}$.
Assume to the contrary that $G_{0}\setminus\\{h\\}$ is dependent for each
$h\in G_{0}$. Then there exist $g\in G_{0}$,
$d\in[2,\,\operatorname{ord}(g)-1]$, and $E\subsetneq G_{0}\setminus\\{g\\}$
such that $dg\in\langle E\rangle$. Now let $d\in\mathbb{N}$ be minimal over
all configurations $(g,E,d)$, and fix $g,E$ belonging to $d$. It follows that
we have an atom $A$ with $\operatorname{supp}(A)\subsetneq G_{0}$ and
$\mathsf{v}_{g}(A)=d$. By Lemma 4.4, we obtain that
$d\,|\,\operatorname{ord}(g)$, and hence $d\geq 3$ because $n$ is odd.
Since $G_{0}\setminus\\{g\\}$ is dependent, there exist atoms
$U^{\prime}\in\mathcal{A}(G_{0}\setminus\\{g\\})$ with
$|\operatorname{supp}(U^{\prime})|>1$. Thus, by Lemma 3.4.1, there exist an
$U\in\mathcal{A}(G_{0}\setminus\\{g\\})$ and an $h\in\operatorname{supp}(U)$
such that $\mathsf{v}_{h}(U)\leq\frac{\operatorname{ord}(h)}{2}$ and
$\mathsf{v}_{h}(U)\,|\,\operatorname{ord}(h)$.
By Lemma 4.4.2, there are atoms $A_{1},\ldots,A_{d-1}$ with
$\mathsf{v}_{g}(A_{i})=i$ and $\mathsf{k}(A_{i})>1$ for each $i\in[1,\,d-1]$,
and we choose each $A_{i}$ in such a way that $\mathsf{v}_{h}(A_{i})$ is
minimal. We continue with the following assertion.
1. A.
For each $i\in[1,d-1]$, we have
$\mathsf{v}_{h}(A_{i})<\mathsf{v}_{h}(U)\leq\frac{\operatorname{ord}(h)}{2}$.
Proof of A. Assume to the contrary that there is an $i\in[1,d-1]$ such that
$\mathsf{v}_{h}(A_{i})\geq\mathsf{v}_{h}(U)$. Then
$h\notin
F=\\{h^{\prime}\in\operatorname{supp}(U)\mid\mathsf{v}_{h^{\prime}}(A_{i})<\mathsf{v}_{h^{\prime}}(U)\\}\quad\text{and}\quad
U\,\big{|}\,A_{i}\prod_{h^{\prime}\in
F}{h^{\prime}}^{\operatorname{ord}(h^{\prime})}\,.$
Hence $A_{i}\prod_{h^{\prime}\in
F}{h^{\prime}}^{\operatorname{ord}(h^{\prime})}=UB_{i}$ for some zero-sum
sequence $B_{i}$. By Lemma 4.4 (items 2. and 3.), $B_{i}$ is an atom with
$i=\mathsf{v}_{g}(A_{i})=\mathsf{v}_{g}(B_{i})$ and with
$\mathsf{k}(B_{i})=\mathsf{k}(A_{i})>1$. Since
$\mathsf{v}_{h}(A_{i})>\mathsf{v}_{h}(B_{i})$, this is a contradiction to the
choice of $A_{i}$. ∎(Proof of A)
Let $j\in[1,d-1]$ be such that
$\mathsf{k}(A_{j})=\min\\{\mathsf{k}(A_{1}),\ldots,\mathsf{k}(A_{d-1})\\}$.
Suppose that $j\geq 2$. Let $V_{1},\ldots,V_{t}$ be all the atoms with
$V_{s}\,|\,A_{1}A_{j-1}$ and $|\operatorname{supp}(V_{s})|=1$ for all
$s\in[1,\,t]$. Then $B=A_{1}A_{j-1}(V_{1}\cdot\ldots\cdot V_{t})^{-1}$ is an
atom by Lemma 4.4.2. Since
$\mathsf{v}_{g}(A_{1}A_{j-1})=j<\operatorname{ord}(g)$,
$\mathsf{v}_{h}(A_{1}A_{j-1})<\operatorname{ord}(h)$, and
$\mathsf{v}_{f}(A_{1}A_{j-1})<2\operatorname{ord}(f)$ for all $f\in
G_{0}\setminus\\{g,h\\}$, it follows that $t\leq|G_{0}|-2\leq r-1$. Since
$\min\Delta(G_{0})=r-1$ and $A_{1}A_{j-1}=V_{1}\cdot\ldots\cdot V_{t}B$, we
must have $t=1$. Therefore
$\mathsf{k}(A_{1})+\mathsf{k}(A_{j-1})=1+\mathsf{k}(B)$ whence
$\mathsf{k}(B)<\mathsf{k}(A_{j-1})$. Since
$\mathsf{v}_{g}(B)=\mathsf{v}_{g}(V_{1}B)=\mathsf{v}_{g}(A_{1}A_{j-1})=j=\mathsf{v}_{g}(A_{j})\,,$
Lemma 4.4.3 implies that
$\mathsf{k}(B)=\mathsf{k}(A_{j})=\min\\{\mathsf{k}(A_{1}),\ldots,\mathsf{k}(A_{d-1})\\}$,
a contradiction.
Suppose that $j=1$. Let $V_{1},\ldots,V_{t}$ be all the atoms with
$V_{s}\,|\,A_{2}A_{d-1}$ and $|\operatorname{supp}(V_{s})|=1$ for all
$s\in[1,\,t]$. Then $B=A_{2}A_{d-1}(V_{1}\cdot\ldots\cdot V_{t})^{-1}$ is an
atom by Lemma 4.4.2. Since
$\mathsf{v}_{g}(A_{2}A_{d-1})=d+1<\operatorname{ord}(g)$,
$\mathsf{v}_{h}(A_{2}A_{d-1})<\operatorname{ord}(h)$, and
$\mathsf{v}_{f}(A_{1}A_{j-1})<2\operatorname{ord}(f)$ for all $f\in
G_{0}\setminus\\{g,h\\}$, it follows that $t\leq|G_{0}|-2\leq r-1$. Since
$\min\Delta(G_{0})=r-1$ and $A_{2}A_{d-1}=V_{1}\cdot\ldots\cdot V_{t}B$, we
must have $t=1$. Therefore
$\mathsf{k}(A_{2})+\mathsf{k}(A_{d-1})=1+\mathsf{k}(B)$ whence
$\mathsf{k}(B)<\mathsf{k}(A_{2})$. Since
$\mathsf{v}_{g}(B)=\mathsf{v}_{g}(V_{1}B)=\mathsf{v}_{g}(A_{2}A_{d-1})=d+1\equiv
1=\mathsf{v}_{g}(A_{1})\mod d\,,$
Lemma 4.4.3 implies that
$\mathsf{k}(B)=\mathsf{k}(A_{1})=\min\\{\mathsf{k}(A_{1}),\ldots,\mathsf{k}(A_{d-1})\\}$,
a contradiction. ∎
In the following remark we provide the first example of a minimal non-half-
factorial subset $G_{0}$ with $\min\Delta(G_{0})=\max\Delta^{*}(G)$ which is
not simple. Furthermore, we provide an example that the structural statement
given in Theorem 4.5.4 does not hold without the assumption that the exponent
is odd.
###### Remarks 4.6.
Following Schmid, we say that a nonempty subset $G_{0}\subset
G\setminus\\{0\\}$ is simple if there exists some $g\in G_{0}$ such that
$G_{0}\setminus\\{g\\}$ is independent, $g\in\langle
G_{0}\setminus\\{g\\}\rangle$ but $g\notin\langle E\rangle$ for any subset
$E\subsetneq G_{0}\setminus\\{g\\}$.
If $G_{0}$ is a simple subset, then $|G_{0}|\leq\mathsf{r}^{*}(G)+1$ and
$G_{0}$ is indecomposable. Moreover, if $G_{1}\subset G$ is a subset such that
any proper subset of $G_{1}$ is independent, then there is a subset $G_{0}$
and a transfer homomorphism
$\theta\colon\mathcal{B}(G_{1})\to\mathcal{B}(G_{0})$ where
$G_{0}\setminus\\{0\\}$ is simple or independent (for all this see [26,
Section 4]). Furthermore, Theorem 4.7 in [26] provides an intrinsic
description of the sets of atoms of a simple set.
In elementary $p$-groups, every minimal non-half-factorial subset is simple
([26, Lemma 4.4]), and so far there are no examples of minimal non-half-
factorial sets $G_{0}$ with $\min\Delta(G_{0})=\max\Delta^{*}(G)$ which are
not simple.
1\. Let $G=C_{9}^{r-1}\oplus C_{27}$ with $r\geq 26$, and let
$(e_{1},\ldots,e_{r})$ be a basis of $G$ with $\operatorname{ord}(e_{i})=9$
for $i\in[1,\,r-1]$ and $\operatorname{ord}(e_{r})=27$. Then
$\max\Delta^{*}(G)=r-1$ by Theorem 1.1. We set
$G_{0}=\\{3e_{1},\ldots,3e_{r-1},e_{r},g\\}$ with $g=e_{1}+\ldots+e_{r}$. Then
$(e_{r},g)$ is not independent, $G_{0}\setminus\\{g\\}$ and
$G_{0}\setminus\\{e_{r}\\}$ are independent, but $g\notin\langle
G_{0}\setminus\\{g\\}\rangle$ and $e_{r}\notin\langle
G_{0}\setminus\\{e_{r}\\}\rangle$. Therefore $G_{0}$ is not simple. It remains
to show that $\min\Delta(G_{0})\geq r-1$. Then $G_{0}$ is minimal non-half-
factorial and $\min\Delta(G_{0})=r-1$ because $\max\Delta^{*}(G)=r-1$.
We have
$\displaystyle W_{1}=\\{A\in\mathcal{A}(G_{0})\mid\mathsf{k}(A)=1\\}=\\{$
$\displaystyle(3e_{1})^{3},\ldots,(3e_{r-1})^{3},e_{r}^{27},g^{27},g^{9}e_{r}^{18},g^{18}e_{r}^{9}\\},$
$\displaystyle W_{2}=\\{A\in\mathcal{A}(G_{0})\mid\mathsf{k}(A)>1\\}=\\{$
$\displaystyle
A_{3}=g^{3}e_{r}^{24}(3e_{1})^{2}\cdot\ldots\cdot(3e_{r-1})^{2},A_{6}=g^{6}e_{r}^{21}(3e_{1})\cdot\ldots\cdot(3e_{r-1}),$
$\displaystyle
A_{12}=g^{12}e_{r}^{15}(3e_{1})^{2}\cdot\ldots\cdot(3e_{r-1})^{2},A_{15}=g^{15}e_{r}^{12}(3e_{1})\cdot\ldots\cdot(3e_{r-1}),$
$\displaystyle
A_{21}=g^{21}e_{r}^{6}(3e_{1})^{2}\cdot\ldots\cdot(3e_{r-1})^{2},A_{24}=g^{24}e_{r}^{3}(3e_{1})\cdot\ldots\cdot(3e_{r-1})\\}$
and $\mathsf{k}(A_{3})=\mathsf{k}(A_{12})=\mathsf{k}(A_{21})=(2r+1)/3$,
$\mathsf{k}(A_{6})=\mathsf{k}(A_{15})=\mathsf{k}(A_{24})=(r+2)/3$. For any
$d\in\Delta(G_{0})$, there exists a $B\in\mathcal{B}(G_{0})$ such that $B$ has
two such factorizations, say
$B=U_{1}\cdot\ldots\cdot U_{s}V_{1}\cdot\ldots\cdot V_{t}W_{1}\cdot\ldots\cdot
W_{u}=X_{1}\cdot\ldots\cdot X_{s^{\prime}}Y_{1}\cdot\ldots\cdot
Y_{t^{\prime}}Z_{1}\cdot\ldots\cdot Z_{u^{\prime}}\,$
where all $U_{i},V_{j},W_{k},X_{i^{\prime}},Y_{j^{\prime}},Z_{k^{\prime}}$ are
atoms, $s,t,u,s^{\prime},t^{\prime},u^{\prime}\in\mathbb{N}_{0}$ with
$d=(s+t+u)-(s^{\prime}+t^{\prime}+u^{\prime})$,
$\mathsf{k}(U_{1})=\ldots=\mathsf{k}(U_{s})=\mathsf{k}(X_{1})=\ldots=\mathsf{k}(X_{s^{\prime}})=\frac{2r+1}{3}$,
$\mathsf{k}(V_{1})=\ldots=\mathsf{k}(V_{t})=\mathsf{k}(Y_{1})=\ldots=\mathsf{k}(Y_{t^{\prime}})=(r+2)/2$,
and
$\mathsf{k}(W_{1})=\ldots=\mathsf{k}(W_{u})=\mathsf{k}(Z_{1})=\ldots=\mathsf{k}(Z_{u^{\prime}})=1$.
This implies that
$\mathsf{k}(B)=s(\frac{2r+1}{3})+t(\frac{r+2}{3})+u=s^{\prime}(\frac{2r+1}{3})+t^{\prime}(\frac{r+2}{3})+u^{\prime}$
and $\mathsf{v}_{3e_{1}}(B)\equiv 2s+t\equiv 2s^{\prime}+t^{\prime}\mod 3$.
Since
$d=(s+t+u)-(s^{\prime}+t^{\prime}+u^{\prime})=\frac{r-1}{3}((t^{\prime}-t)+2(s^{\prime}-s))>0$,
we obtain that $(t^{\prime}-t)+2(s^{\prime}-s)\geq 3$ and hence $d\geq r-1$.
2\. We provide an example of a minimal non-half-factorial LCN-set with
$\min\Delta(G_{0})=\max\Delta^{*}(G)$ in a group $G$ of even exponent which
has no element $g\in G_{0}$ such that $G_{0}\setminus\\{g\\}$ is independent.
In particular, $G_{0}$ is not simple and the assumption in Theorem 4.5.4, that
the exponent of the group is odd, cannot be cancelled.
Let $G=C_{2}^{r-2}\oplus C_{4}\oplus C_{4}$ with $r\geq 3$, and let
$(e_{1},\ldots,e_{r})$ be a basis of $G$ with $\operatorname{ord}(e_{i})=2$
for $i\in[1,\,r-2]$ and
$\operatorname{ord}(e_{r-1})=\operatorname{ord}(e_{r})=4$. We set
$G_{0}=\\{e_{1},\ldots,e_{r-3},e_{r-2}+e_{r-1},e_{r-1},e_{r},g\\}$ with
$g=e_{1}+\ldots+e_{r-2}+e_{r}$. Since $(e_{r-2}+e_{r-1},e_{r-1})$ is dependent
and $(e_{r},g)$ is dependent, we obtain that there is no $h\in G_{0}$ such
that $G_{0}\setminus\\{h\\}$ is independent. We have
$\displaystyle W_{1}=\\{A\in\mathcal{A}(G_{0})\mid\mathsf{k}(A)=1\\}=\\{$
$\displaystyle(e_{1})^{2},\ldots,(e_{r-3})^{2},(e_{r-2}+e_{r-1})^{4},(e_{r-1})^{4},e_{r}^{4},g^{4},$
$\displaystyle(e_{r-2}+e_{r-1})^{2}(e_{r-1})^{2},g^{2}e_{r}^{2}\\},$
$\displaystyle W_{2}=\\{A\in\mathcal{A}(G_{0})\mid\mathsf{k}(A)>1\\}=\\{$
$\displaystyle
A_{1}=ge_{r}^{3}(e_{r-2}+e_{r-1})e_{r-1}^{3}e_{1}\cdot\ldots\cdot e_{r-3},$
$\displaystyle
B_{1}=ge_{r}^{3}(e_{r-2}+e_{r-1})^{3}e_{r-1}e_{1}\cdot\ldots\cdot e_{r-3},$
$\displaystyle
A_{3}=g^{3}e_{r}(e_{r-2}+e_{r-1})e_{r-1}^{3}e_{1}\cdot\ldots\cdot e_{r-3},$
$\displaystyle
B_{3}=g^{3}e_{r}(e_{r-2}+e_{r-1})^{3}e_{r-1}e_{1}\cdot\ldots\cdot e_{r-3}\\}$
and
$\mathsf{k}(A_{1})=\mathsf{k}(A_{3})=\mathsf{k}(B_{1})=\mathsf{k}(B_{3})=(r+1)/2$.
Theorem 1.1 implies that $\max\Delta^{*}(G)=r-1$, and thus it remains to show
that $\min\Delta(G_{0})=r-1$.
For any $d\in\Delta(G_{0})$, there exists a $B\in\mathcal{B}(G_{0})$ such that
$B$ has two such factorizations, say
$B=U_{1}\cdot\ldots\cdot U_{s}V_{1}\cdot\ldots\cdot
V_{t}=X_{1}\cdot\ldots\cdot X_{u}Y_{1}\cdot\ldots\cdot Y_{v}\,$
where all $U_{i},V_{j},X_{k},Y_{l}$ are atoms, $s,t,u,v\in\mathbb{N}_{0}$ with
$d=u+v-(s+t)$,
$\mathsf{k}(U_{1})=\ldots=\mathsf{k}(U_{s})=\mathsf{k}(X_{1})=\ldots=\mathsf{k}(X_{u})=1$,
and
$\mathsf{k}(V_{1})=\ldots=\mathsf{k}(V_{t})=\mathsf{k}(Y_{1})=\ldots=\mathsf{k}(Y_{v})=(r+1)/2$.
This implies that
$\mathsf{k}(B)=s+t\frac{r+1}{2}=u+v\frac{r+1}{2}$
and $\mathsf{v}_{g}(B)\equiv t\equiv v\mod 2$. Since
$d=(v+u)-(s+t)=(t-v)\frac{r-1}{2}>0$, we obtain that $t-v\geq 2$ and hence
$d\geq r-1$.
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* [16] A. Geroldinger and W. A. Schmid, _A characterization of class groups via sets of lengths_ , arXiv:1503.04679.
* [17] by same author, _The system of sets of lengths in Krull monoids under set addition_ , Revista Matematica Iberoamericana, to appear, arXiv:1407.1967v2.
* [18] A. Geroldinger and P. Yuan, _The set of distances in Krull monoids_ , Bull. Lond. Math. Soc. 44 (2012), 1203 – 1208.
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|
arxiv-papers
| 2014-04-10T16:34:06 |
2024-09-04T02:50:00.952182
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alfred Geroldinger and Qinghai Zhong",
"submitter": "Qinghai Zhong",
"url": "https://arxiv.org/abs/1404.2873"
}
|
1404.2920
|
11institutetext: Indian Institute of Astrophysics, II Block, Koramangala,
Bangalore 560 034, India
11email: [email protected] 22institutetext: Physical Research Laboratory,
Navrangpura, Ahmedabad 380 009, India
# $H_{0}$ from ten well-measured time delay lenses
S. Rathna Kumar $H_{0}$ from ten well-measured time delay lenses$H_{0}$ from
ten well-measured time delay lenses$H_{0}$ from ten well-measured time delay
lenses$H_{0}$ from ten well-measured time delay lenses C. S. Stalin $H_{0}$
from ten well-measured time delay lenses$H_{0}$ from ten well-measured time
delay lenses T. P. Prabhu $H_{0}$ from ten well-measured time delay
lenses$H_{0}$ from ten well-measured time delay lenses
(Received 10 April 2014 / Accepted 27 May 2015)
In this work, we present a homogeneous curve-shifting analysis using the
difference-smoothing technique of the publicly available light curves of 24
gravitationally lensed quasars, for which time delays have been reported in
the literature. The uncertainty of each measured time delay was estimated
using realistic simulated light curves. The recipe for generating such
simulated light curves with known time delays in a plausible range around the
measured time delay is introduced here. We identified 14 gravitationally
lensed quasars that have light curves of sufficiently good quality to enable
the measurement of at least one time delay between the images, adjacent to
each other in terms of arrival-time order, to a precision of better than 20%
(including systematic errors). We modeled the mass distribution of ten of
those systems that have known lens redshifts, accurate astrometric data, and
sufficiently simple mass distribution, using the publicly available PixeLens
code to infer a value of $H_{0}$ of 68.1 $\pm$ 5.9 km s-1 Mpc-1 (1$\sigma$
uncertainty, 8.7% precision) for a spatially flat universe having $\Omega_{m}$
= 0.3 and $\Omega_{\Lambda}$ = 0.7. We note here that the lens modeling
approach followed in this work is a relatively simple one and does not account
for subtle systematics such as those resulting from line-of-sight effects and
hence our $H_{0}$ estimate should be considered as indicative.
###### Key Words.:
gravitational lensing: strong – methods: numerical – cosmological parameters –
quasars: general
## 1 Introduction
The Hubble constant at the present epoch ($H_{0}$), the current expansion rate
of the universe, is an important cosmological parameter. All extragalactic
distances, as well as the age and size of the universe depend on $H_{0}$. It
is also an important parameter in constraining the dark energy equation of
state and it is used as input in many cosmological simulations (Freedman &
Madore 2010; Planck Collaboration et al. 2014). Therefore, precise estimation
of $H_{0}$ is of utmost importance in cosmology.
Estimates of $H_{0}$ available in the literature cover a wide range of
uncertainties from $\sim$2% to $\sim$10% and the value ranges between 60 and
75 km s-1 Mpc-1. The most reliable measurements of $H_{0}$ known to date
include
–
the Hubble Space Telescope (HST) Key Project (72 $\pm$ 8 km s-1 Mpc-1;
Freedman et al. 2001),
–
the HST Program for the Luminosity Calibration of Type Ia Supernovae by Means
of Cepheids (62.3 $\pm$ 5.2 km s-1 Mpc-1; Sandage et al. 2006),
–
Wilkinson Microwave Anisotropy Probe (WMAP) (70.0 $\pm$ 2.2 km s-1 Mpc-1;
Hinshaw et al. 2013),
–
Supernovae and $H_{0}$ for the Equation of State (SH0ES) Program (73.8 $\pm$
2.4 km s-1 Mpc-1; Riess et al. 2011),
–
Carnegie Hubble Program (CHP) (74.3 $\pm$ 2.6 km s-1 Mpc-1; Freedman et al.
2012),
–
the Megamaser Cosmology Project (MCP) (68.9 $\pm$ 7.1 km s-1 Mpc-1; Reid et
al. 2013; Braatz et al. 2013),
–
Planck measurements of the cosmic microwave background (CMB) anisotropies
(67.3 $\pm$ 1.2 km s-1 Mpc-1; Planck Collaboration et al. 2014), and
–
Strong lensing time delays (75.2${}^{+4.4}_{-4.2}$ km s-1 Mpc-1; Suyu et al.
2013).
It is worth noting here that the small uncertainties in $H_{0}$ measurements
resulting from WMAP and Planck crucially depend on the assumption of a
spatially flat universe.
Although the values of $H_{0}$ obtained from different methods are consistent
with each other within 2$\sigma$ given the current level of precision, all of
the above methods of determination of $H_{0}$ suffer from systematic
uncertainties. Therefore as the measurements increase in precision, multiple
approaches based on different physical principles need to be pursued so as to
be able to identify unknown systematic errors present in any given approach.
The phenomenon of strong gravitational lensing offers an elegant method to
measure $H_{0}$. For gravitationally lensed sources that show variations in
flux with time, such as quasars, it is possible to measure the time delay
between the various images of the background source. The time delay, which is
a result of the travel times for photons being different along the light paths
corresponding to the lensed images, has two origins: (i) the geometric
difference between the light paths and (ii) gravitational delay due to the
dilation of time as photons pass in the vicinity of the lensing mass. Time
delays, therefore depend on the cosmology, through the distances between the
objects involved, and on the radial mass profile of the lensing galaxies. This
was shown theoretically five decades ago by Refsdal (1964) long before the
discovery of the first gravitational lens Q0957+561 by Walsh et al. (1979).
Estimation of $H_{0}$ through gravitational lens time delays, although it has
its own degeneracies, is based on the well-understood physics of General
Relativity, and compared to distance ladder methods, is free from various
calibration issues. In addition to measuring $H_{0}$, measurement of time
delays between the light curves of a lensed quasar can be used to study the
microlensing variations present in the light curves, and to study the
structure of the quasar (Hainline et al. 2013; Mosquera et al. 2013). However,
these time delay measurements of $H_{0}$ are extremely challenging because of
the need of an intensive monitoring program that offers high cadence and good-
quality photometric data over a long period of time. This type of program
would then be able to cope with the presence of uncorrelated variations
present in the lensed quasar lightcurves, which can interestingly arise due to
microlensing by stars in the lensing galaxy (Chang & Refsdal 1979) or for
mundane reasons, such as the presence of additive flux shifts in the
photometry (Tewes et al. 2013a). Moreover, the estimation of $H_{0}$ from such
high-quality data is hampered by the uncertainty on lens models. Recently,
using time delay measurements from high-quality optical and radio light
curves, deep and high-resolution imaging observations of the lensing galaxies
and lensed AGN host galaxy, and the measurement of stellar velocity dispersion
of the lens galaxy to perform detailed modeling, Suyu et al. (2013) report a
$H_{0}$ of 75.2${}^{+4.4}_{-4.2}$ km s-1 Mpc-1 through the study of two
gravitational lenses namely RX J1131$-$1231 and CLASS B1608+656.
Another approach is to perform simple modeling of a relatively large sample of
gravitational lenses with moderate-precision time delay measurements. In this
way, it should be possible to obtain a precise determination of the global
value of $H_{0}$, even if the $H_{0}$ measurements from individual lenses have
large uncertainties. In addition, when inferring $H_{0}$ from a relatively
large sample of lenses, line-of-sight effects that bias the $H_{0}$
measurements from individual lenses (see Suyu et al. 2013, Sect. 2) should
tend to average out, although a residual systematic error must still remain
(Hilbert et al. 2007; Fassnacht et al. 2011). A pixelized method of lens
modeling is available in the literature and is also implemented in the
publicly available code PixeLens (Saha & Williams 2004). Using this code, Saha
et al. (2006) have found $H_{0}$ = 72${}^{+8}_{-11}$ km s-1 Mpc-1 for a sample
of ten time delay lenses. Performing a similar analysis on an extended sample
of 18 lenses Paraficz & Hjorth (2010) obtained $H_{0}$ = 66${}^{+6}_{-4}$ km
s-1 Mpc-1. Here, we present an estimate of $H_{0}$ using the pixellated
modeling approach on a sample of carefully selected lensed quasars. So far,
time delays have been reported for 24 gravitationally lensed quasars among the
hundreds of such strongly lensed quasars known. However, the quality of the
light curves and the techniques used to infer these time delays vary between
systems. In this work, we apply the difference-smoothing technique, introduced
in Rathna Kumar et al. (2013), to the publicly available light curves of the
24 systems in a homogeneous manner, first to cross-check the previously
measured time delays and then to select a subsample of suitable lens systems
to determine $H_{0}$.
The paper is organized as follows. Section 2 describes the technique used for
time delay determination and introduces a recipe for creating realistic
simulated light curves with known time delays; the simulated light curves are
used in this work to estimate the uncertainty of each measured delay. In Sect.
3, the application of the curve-shifting procedure to the 24 systems is
described. In Sect. 4, we infer $H_{0}$ from the lens-modeling of those
systems that have at least one reliably measured time delay, known lens
redshift, accurate astrometric data, and sufficiently simple mass
distribution. We conclude in Sect. 5.
## 2 Time delay determination
In this section, we briefly describe the previously reported difference-
smoothing technique, which contains one modification to the original version
(see Rathna Kumar et al. 2013 for details). We then introduce a recipe for
simulating realistic light curves having known time delays in a plausible
range around the measured delay in order to estimate its uncertainty. We also
present an approach for tuning the free parameters of the difference-smoothing
technique for a given dataset.
### 2.1 Difference-smoothing technique
Ai and Bi are the observed magnitudes constituting light curves A and B
sampled at epochs $t_{i}$ ($i=1,2,3,...,N$). Light curve A is selected as the
reference. We shift light curve B in time with respect to light curve A by an
amount $\tau$. This shifted version B′ of B is given by
$\displaystyle\mathrm{B}_{i}^{\prime}$ $\displaystyle=$
$\displaystyle\mathrm{B}_{i},$ (1) $\displaystyle t_{i}^{\prime}$
$\displaystyle=$ $\displaystyle t_{i}+\tau.$ (2)
We note here that we do not apply any flux shift to light curve B as in Rathna
Kumar et al. (2013), since we have found that doing so considerably increases
the computational time without significantly changing the results.
For any given estimate of the time delay $\tau$, we form a difference light
curve having points $d_{i}$ at epochs $t_{i}$,
$d_{i}(\tau)=\mathrm{A}_{i}-\frac{\sum_{j=1}^{N}w_{ij}\mathrm{B}_{j}^{\prime}}{\sum_{j=1}^{N}w_{ij}},$
(3)
where the weights $w_{ij}$ are given by
$w_{ij}=\frac{1}{\sigma_{\mathrm{B}_{j}}^{2}}e^{-(t_{j}^{\prime}-t_{i})^{2}/2\delta^{2}}.$
(4)
The parameter $\delta$ is the decorrelation length and
$\sigma_{\mathrm{B}_{j}}$ denotes the photometric error of the magnitude Bj.
We calculate the uncertainty of each $d_{i}$ as
$\sigma_{d_{i}}=\sqrt{\sigma_{\mathrm{A}_{i}}^{2}+\frac{1}{\sum_{j=1}^{N}w_{ij}}},$
(5)
where $w_{ij}$ are given by Eq. 4.
We now smooth the difference curve $d_{i}$ using a Gaussian kernel to obtain a
model $f_{i}$ for the differential extrinsic variability
$f_{i}=\frac{\sum_{j=1}^{N}\nu_{ij}\,d_{j}}{\sum_{j=1}^{N}\nu_{ij}},$ (6)
where the weights $\nu_{ij}$ are given by
$\nu_{ij}=\frac{1}{\sigma_{d_{j}}^{2}}{e^{-(t_{j}-t_{i})^{2}/2s^{2}}}.$ (7)
The smoothing time scale $s$ is another free parameter of this method. The
uncertainty of each $f_{i}$ is computed as
$\sigma_{f_{i}}=\sqrt{\frac{1}{\sum_{j=1}^{N}\nu_{ij}}}.$ (8)
We optimize the time delay estimate $\tau$ to minimize the residuals between
the difference curve $d_{i}$ and the much smoother $f_{i}$. To quantify the
mismatch between $d_{i}$ and $f_{i}$, we define a normalized $\chi^{2}$,
$\overline{\chi}^{2}=\left[\sum_{i=1}^{N}\frac{(d_{i}-f_{i})^{2}}{\sigma_{d_{i}}^{2}+\sigma_{f_{i}}^{2}}\right]/\left[\sum_{i=1}^{N}\frac{1}{\sigma_{d_{i}}^{2}+\sigma_{f_{i}}^{2}}\right],$
(9)
and minimize this $\overline{\chi}^{2}(\tau)$ using a global optimization.
In the above description, since light curves A and B are not interchangeable,
we systematically perform all computations for both permutations of A and B,
and minimize the average of the two resulting values of $\overline{\chi}^{2}$.
### 2.2 Simulation of light curves
In Rathna Kumar et al. (2013), in order to estimate the uncertainty of the
time delay measured using the difference-smoothing technique, we made use of
realistic simulated light curves, which were created following the procedure
introduced in Tewes et al. (2013a). In this work, we introduce an independent
recipe for creating simulated light curves.
We infer the underlying variation $\mathrm{A}(t)$ of the light curve A at the
epoch $t_{i}$ based on the magnitudes Aj for all the epochs as
$\mathrm{A}(t_{i})=\frac{\sum_{j=1}^{N}\frac{1}{\sigma_{\mathrm{A}_{j}}^{2}}e^{-(t_{j}-t_{i})^{2}/2m^{2}}\mathrm{A}_{j}}{\sum_{j=1}^{N}\frac{1}{\sigma_{\mathrm{A}_{j}}^{2}}e^{-(t_{j}-t_{i})^{2}/2m^{2}}},$
(10)
where the value of $m$ is set to equal the mean sampling of the light curves
calculated after excluding the large gaps following a 3$\sigma$ rejection
criterion. For those points having the nearest neighboring points on both
sides separated by a value less than or equal to $m$, we compute the values of
$(\mathrm{A_{i}}-\mathrm{A}(t_{i}))/\sigma_{\mathrm{A}_{i}}$, the standard
deviation of which is multiplied to the error bars $\sigma_{\mathrm{A}_{i}}$
to obtain the rescaled error bars $\hat{\sigma}_{\mathrm{A}_{i}}$. We note
here that the rescaling is applied for all the epochs and not just the epochs
of points used in computing the rescaling factor. Similarly for the B light
curves the rescaled error bars $\hat{\sigma}_{\mathrm{B}_{i}}$ are obtained.
This rescaling inferred from the local scatter properties of the light curves
is done because the magnitudes of the original error bars may suffer from
systematic underestimation or overestimation.
We merge light curves A and B by shifting the B light curve by the time delay
found ($\Delta t$) and subtracting the differential extrinsic variability
$f_{i}$ corresponding to the delay from the A light curve. This merged light
curve $\mathrm{M}_{i}$, whose errors we denote $\sigma_{\mathrm{M}_{i}}$,
consists of the magnitudes $\mathrm{A}_{i}-f_{i}$ at times $t_{i}$ and having
errors $\hat{\sigma}_{\mathrm{A}_{i}}$ and the magnitudes $\mathrm{B}_{i}$ at
times $t_{i}+\Delta t$ and having errors $\hat{\sigma}_{\mathrm{B}_{i}}$. We
now model the quasar brightness variation $\mathrm{M}(t)$ as
$\mathrm{M}(t)=\frac{\sum_{j=1}^{2N}\frac{1}{\sigma_{\mathrm{M}_{j}}^{2}}e^{-(t_{j}-t)^{2}/2m^{2}}\mathrm{M}_{j}}{\sum_{j=1}^{2N}\frac{1}{\sigma_{\mathrm{M}_{j}}^{2}}e^{-(t_{j}-t)^{2}/2m^{2}}}.$
(11)
We then model the quasar brightness variation using only the $\mathrm{A}$
points in $\mathrm{M}_{i}$ as
$\mathrm{M_{A}}(t)=\frac{\sum_{j=1}^{N}\frac{1}{\hat{\sigma}_{\mathrm{A}_{j}}^{2}}e^{-(t_{j}-t)^{2}/2m^{2}}(\mathrm{A}_{j}-f_{j})}{\sum_{j=1}^{N}\frac{1}{\hat{\sigma}_{\mathrm{A}_{j}}^{2}}e^{-(t_{j}-t)^{2}/2m^{2}}}$
(12)
and only the $\mathrm{B}$ points in $\mathrm{M}_{i}$ as
$\mathrm{M_{B}}(t)=\frac{\sum_{j=1}^{N}\frac{1}{\hat{\sigma}_{\mathrm{B}_{j}}^{2}}e^{-(t_{j}+\Delta
t-t)^{2}/2m^{2}}\mathrm{B}_{j}}{\sum_{j=1}^{N}\frac{1}{\hat{\sigma}_{\mathrm{B}_{j}}^{2}}e^{-(t_{j}+\Delta
t-t)^{2}/2m^{2}}}.$ (13)
The residual extrinsic variations present in the A and B light curves can now
be calculated as
$f_{\mathrm{A}_{i}}=\mathrm{M_{A}}(t_{i})-\mathrm{M}(t_{i})$ (14)
and
$f_{\mathrm{B}_{i}}=\mathrm{M_{B}}(t_{i})-\mathrm{M}(t_{i}).$ (15)
We can now simulate light curves $\mathrm{A}_{i}^{simu}$ and
$\mathrm{B}_{i}^{simu}$ having a time delay of $\Delta t+dt$ between them by
sampling $\mathrm{M}(t)$ at appropriate epochs and adding terms for extrinsic
variations and noise,
$\mathrm{A}_{i}^{simu}=\mathrm{M}\left(t_{i}-\frac{dt}{2}\right)+f_{i}+f_{\mathrm{A}_{i}}+N^{*}(0,1)\hat{\sigma}_{\mathrm{A}_{i}}$
(16)
and
$\mathrm{B}_{i}^{simu}=\mathrm{M}\left(t_{i}+\Delta
t+\frac{dt}{2}\right)+f_{\mathrm{B}_{i}}+N^{*}(0,1)\hat{\sigma}_{\mathrm{B}_{i}},$
(17)
where $N^{*}(0,1)$ is a random variate drawn from a normal distribution having
mean 0 and variance 1. These simulated light curves are then assigned the
times $t_{i}$ and the error bars $\sigma_{\mathrm{A}_{i}}$ and
$\sigma_{\mathrm{B}_{i}}$ for the A and B light curves, respectively.
Including the terms $f_{\mathrm{A}_{i}}$ and $f_{\mathrm{B}_{i}}$ in the
calculation of $\mathrm{A}_{i}^{simu}$ and $\mathrm{B}_{i}^{simu}$,
respectively, ensures that our simulated light curves contain extrinsic
variability on all time scales, just as in the real light curves.
Here again in the above description, since light curves A and B are not
interchangeable, we systematically perform all computations for both
permutations of A and B, and average the corresponding values of
$\mathrm{A}_{i}^{simu}$ and $\mathrm{B}_{i}^{simu}$, before adding the noise
terms.
### 2.3 Choice of free parameters
The value chosen for the decorrelation length $\delta$ needs to be equivalent
to the temporal sampling of the light curves. In this work, we set $\delta$
equal to $m$, the mean sampling of the light curves calculated after excluding
the large gaps following a 3$\sigma$ rejection criterion.
The value chosen for the smoothing time scale $s$ needs to be significantly
larger than $\delta$. In this work, its value is optimized such that the
larger of the maximum absolute values of
$\frac{f_{\mathrm{A}_{i}}}{\hat{\sigma}_{\mathrm{A}_{i}}}$ and
$\frac{f_{\mathrm{B}_{i}}}{\hat{\sigma}_{\mathrm{B}_{i}}}$, which quantify the
residual extrinsic variations in units of photometric noise for the A and B
light curves respectively, is equal to 2. This choice ensures that the value
of $s$ is small enough to adequately model the extrinsic variations, so that
the extreme values of residual extrinsic variations are not significantly
larger than the noise in the data.
Again as in the above description, because light curves A and B are not
interchangeable, we systematically perform all the computations for both
permutations of A and B, and average the corresponding maximum absolute
values.
### 2.4 Estimation of uncertainty
We create 200 simulated light curves having a true delay of $\Delta t$ between
them. The difference-smoothing technique is applied on each of them to obtain
200 delay values. The standard deviation of the 200 delay values gives us the
random error, and the systematic error is obtained by the difference between
the mean of the 200 delay values and the true delay. The total error
$\Delta\tau_{0}$ is obtained by adding the random error and the systematic
error in quadrature.
However, as noted by Tewes et al. (2013a), it is important to simulate light
curves that have not only the time delay $\Delta t$ found, but also other time
delays in a plausible range around $\Delta t$, so as to obtain a reliable
estimate of the uncertainty (see also Sect. 3.2 in Rathna Kumar et al. 2013).
To this end, we also simulate 200 light curves for each true delay that
differs from $\Delta t$ by
$\pm\Delta\tau_{0},\pm(\Delta\tau_{0}+\Delta\tau_{1}),...\,,\pm(\Delta\tau_{0}+\Delta\tau_{1}+...+\Delta\tau_{n-1})$,
in each step updating the total error $\Delta\tau_{n}$ by adding the maximum
obtained value of the random error and the maximum obtained absolute value of
the systematic error in quadrature. The value of $n$ is chosen to be the
smallest integer for which
$\Delta\tau_{0}+\Delta\tau_{1}+...+\Delta\tau_{n-1}\geq 2\Delta\tau_{n}.$ (18)
This ensures that we have simulated light curves over a range of delay values
that is at least as wide as or wider than the 95.4% confidence interval
implied by the stated final error $\Delta\tau_{n}$.
### 2.5 Testing the robustness of the procedure
In order to test the robustness of our procedure for estimating the time delay
and its uncertainty, we made use of synthetic light curves from the TDC1 stage
of the Strong Lens Time Delay Challenge111http://timedelaychallenge.org/ (Liao
et al. 2015), which are arranged in five rungs having different sampling
properties (see Liao et al. 2015, Table 1). We applied our procedure on a
sample of 250 light curves, 50 from each rung, selected such that we were able
to reliably measure time delays from them. Comparing our results with the
truth files, we found that all the measured delays agreed with the true delays
to within twice the estimated uncertainties, except in one case. For the
exceptional case, the discrepancy between the measured delay and the true
delay was found to be 2.25$\sigma$. This is still a reasonably good level of
agreement, thus demonstrating the robustness of our procedure. We note here
that this property of robustness also depends on the careful choice of free
parameters as presented here. For instance, setting $\delta$ equal to the mean
sampling of the light curves computed without excluding the large gaps was
found to lead to biased time delay measurements, which was especially
noticeable for light curves having shorter seasons and larger cadence. We show
some plots for the pair of TDC1 light curves corresponding to the exceptional
case mentioned above in Figs. 1–3.
Figure 1: Light curves from the Strong Lens Time Delay Challenge file
“tdc1_rung3_quad_pair9A.txt”. Light curve A is shown in red and light curve B
in blue.
Figure 2: Light curves A and B from Fig. 1 have been merged, with light curve
A as reference, after shifting light curve B by the measured time delay of
$\Delta t$ = $-$20.5 days and subtracting the differential extrinsic
variability from A. $M_{A}(t)$ sampled at the epochs $t_{i}$ and $M_{B}(t)$
sampled at the epochs $t_{i}+\Delta t$ are connected by red and blue lines,
respectively. $M(t)$ sampled at the epochs $t_{i}$ and $t_{i}+\Delta t$ are
connected by black lines. The optimum free parameters for this pair of light
curves were found to be $\delta$ = 3.1 days and $s$ = 139.0 days. The
magnitudes at those epochs corresponding to maximum absolute values of
$\frac{f_{\mathrm{A}_{i}}}{\hat{\sigma}_{\mathrm{A}_{i}}}$ and
$\frac{f_{\mathrm{B}_{i}}}{\hat{\sigma}_{\mathrm{B}_{i}}}$ have been circled.
The negative value of time delay implies that light curve A leads light curve
B. The magnitudes are shown without error bars for convenience of display.
## 3 Time delays of 24 gravitationally lensed quasars
Table 1: Summary of time delay measurements.
Object (Reference for data) | Wavebands | Time delay | Reported valueaa$a$A negative value of time delay implies that the arrival-time order is the reverse of what is implied in the subscript to $\Delta t$. | Our measurementaa$a$A negative value of time delay implies that the arrival-time order is the reverse of what is implied in the subscript to $\Delta t$.
---|---|---|---|---
| | | (days) | (days)
Q0142$-$100 (Koptelova et al. 2012) | $R$ | $\Delta t_{AB}$ | 89 $\pm$ 11 | ?
JVAS B0218+357 (Cohen et al. 2000) | 8 GHz, 15 GHz | $\Delta t_{AB}$ | 10.1${}^{+1.5}_{-1.6}$ (95% CI) | 10.7 $\pm$ 0.8
HE 0435$-$1223 (Courbin et al. 2011, | $R$ | $\Delta t_{AB}$ | 8.4 $\pm$ 2.1 | 9.8 $\pm$ 1.1
Blackburne et al.2014) | | $\Delta t_{AC}$ | 0.6 $\pm$ 2.3 | 3.1 $\pm$ 2.2
| | $\Delta t_{AD}$ | 14.9 $\pm$ 2.1 | 13.7 $\pm$ 1.0
| | $\Delta t_{BC}$ | $-$7.8 $\pm$ 0.8 | $-$8.0 $\pm$ 1.0
| | $\Delta t_{BD}$ | 6.5 $\pm$ 0.7 | 6.2 $\pm$ 1.5
| | $\Delta t_{CD}$ | 14.3 $\pm$ 0.8 | 13.6 $\pm$ 0.8
SBS 0909+532 (Goicoechea et al. 2008, | $r$ | $\Delta t_{AB}$ | $-$50${}^{+2}_{-4}$ | $-$45.9 $\pm$ 3.1
Hainline et al.2013) | | | |
RX J0911.4+0551 (Hjorth et al. 2002) | $I$ | $\Delta t_{(A1+A2+A3)B}$ | $-$146 $\pm$ 8 (2$\sigma$) | $-$141.9 $\pm$ 12.3
FBQ 0951+2635 (Jakobsson et al. 2005) | $R$ | $\Delta t_{AB}$ | 16 $\pm$ 2 | 7.8 $\pm$ 14.0
Q0957+561 (Shalyapin et al. 2012) | $r$, $g$ | $\Delta t_{AB}$ | 417.4 $\pm$ 0.9 | 420.0 $\pm$ 1.4
SDSS J1001+5027 (Rathna Kumar et al. 2013) | $R$ | $\Delta t_{AB}$ | 119.3 $\pm$ 3.3 | 119.7 $\pm$ 1.8
SDSS J1004+4112 (Fohlmeister et al. 2007, | $R$, $r$ | $\Delta t_{AB}$ | $-$40.6 $\pm$ 1.8 | $-$37.2 $\pm$ 3.1
Fohlmeister et al.2008) | | $\Delta t_{AC}$ | $-$821.6 $\pm$ 2.1 | $-$822.5 $\pm$ 7.4
| | $\Delta t_{BC}$ | | $-$777.1 $\pm$ 9.2
SDSS J1029+2623 (Fohlmeister et al. 2013) | $r$ | $\Delta t_{A(B+C)}$ | 744 $\pm$ 10 (90% CI) | 734.3 $\pm$ 3.8
HE 1104$-$1805 (Poindexter et al. 2007) | $R$, $V$ | $\Delta t_{AB}$ | $-$152.2${}^{+2.8}_{-3.0}$ | $-$157.1 $\pm$ 3.6
PG 1115+080 (Tsvetkova et al. 2010) | $R$ | $\Delta t_{(A1+A2)B}$ | 4.4${}^{+3.2}_{-2.5}$ | 8.7 $\pm$ 3.6
| | $\Delta t_{(A1+A2)C}$ | $-$12${}^{+2.5}_{-2.0}$ | $-$12.1 $\pm$ 3.6
| | $\Delta t_{BC}$ | $-$16.4 ${}^{+3.5}_{-2.5}$ | $-$23.9 $\pm$ 5.7
RX J1131$-$1231 (Tewes et al. 2013b) | $R$ | $\Delta t_{AB}$ | 0.7 $\pm$ 1.0 | 0.0 $\pm$ 0.6
| | $\Delta t_{AC}$ | 0.0 $\pm$ 1.3 | $-$1.1 $\pm$ 0.8
| | $\Delta t_{AD}$ | 90.6 $\pm$ 1.4 | 91.7 $\pm$ 0.7
| | $\Delta t_{BC}$ | $-$0.7 $\pm$ 1.5 | $-$1.4 $\pm$ 1.6
| | $\Delta t_{BD}$ | 91.4 $\pm$ 1.2 | 92.4 $\pm$ 1.4
| | $\Delta t_{CD}$ | 91.7 $\pm$ 1.5 | 91.3 $\pm$ 1.3
SDSS J1206+4332 (Eulaers et al. 2013) | $R$ | $\Delta t_{AB}$ | 111.3 $\pm$ 3 | 110.3 $\pm$ 1.9
H1413+117 (Goicoechea & Shalyapin 2010) | $r$ | $\Delta t_{AB}$ | $-$17 $\pm$ 3 | $-$14.3 $\pm$ 5.5
| | $\Delta t_{AC}$ | $-$20 $\pm$ 4 | $-$19.9 $\pm$ 10.9
| | $\Delta t_{AD}$ | 23 $\pm$ 4 | 24.0 $\pm$ 6.8
| | $\Delta t_{BC}$ | | ?
| | $\Delta t_{BD}$ | | ?
| | $\Delta t_{CD}$ | | 28.6 $\pm$ 9.4
JVAS B1422+231 (Patnaik & Narasimha 2001) | 15 GHz | $\Delta t_{AB}$ | $-$1.5 $\pm$ 1.4 | 1.1 $\pm$ 2.1
| | $\Delta t_{AC}$ | 7.6 $\pm$ 2.5 | $-$0.4 $\pm$ 3.0
| | $\Delta t_{BC}$ | 8.2 $\pm$ 2.0 | $-$0.4 $\pm$ 3.2
SBS 1520+530 (Burud et al. 2002b) | $R$ | $\Delta t_{AB}$ | 130 $\pm$ 3 | 124.2 $\pm$ 8.1
CLASS B1600+434 (Burud et al. 2000) | $I$ | $\Delta t_{AB}$ | 51 $\pm$ 4 (95% CI) | ?
CLASS B1600+434 (Koopmans et al. 2000) | 8.5 GHz | $\Delta t_{AB}$ | 47${}^{+5}_{-6}$ | ?
CLASS B1608+656 (Fassnacht et al. 1999, | 8.5 GHz | $\Delta t_{AB}$ | $-$31.5${}^{+2.0}_{-1.0}$ | $-$32.4 $\pm$ 3.0
Fassnacht et al.2002) | | $\Delta t_{AC}$ | | 2.3 $\pm$ 1.2
| | $\Delta t_{AD}$ | | 45.7 $\pm$ 0.9
| | $\Delta t_{BC}$ | 36.0${}^{+1.5}_{-1.5}$ | 37.1 $\pm$ 1.9
| | $\Delta t_{BD}$ | 77.0${}^{+2.0}_{-1.0}$ | 77.6 $\pm$ 3.5
| | $\Delta t_{CD}$ | | 41.3 $\pm$ 1.6
SDSS J1650+4251 (Vuissoz et al. 2007) | $R$ | $\Delta t_{AB}$ | 49.5 $\pm$ 1.9 | 59.2 $\pm$ 15.9
PKS 1830$-$211 (Lovell et al. 1998) | 8.6 GHz | $\Delta t_{AB}$ | 26${}^{+4}_{-5}$ | 28.6 $\pm$ 8.0
WFI J2033$-$4723 (Vuissoz et al. 2008) | $R$ | $\Delta t_{AB}$ | $-$35.5 $\pm$ 1.4 | $-$37.6 $\pm$ 2.1
| | $\Delta t_{AC}$ | | 23.6 $\pm$ 2.5
| | $\Delta t_{BC}$ | 62.6${}^{+4.1}_{-2.3}$ | 65.4 $\pm$ 4.3
HE 2149$-$2745 (Burud et al. 2002a) | $V$ | $\Delta t_{AB}$ | 103 $\pm$ 12 | 72.6 $\pm$ 17.0
HS 2209+1914 (Eulaers et al. 2013) | $R$ | $\Delta t_{AB}$ | $-$20.0 $\pm$ 5 | $-$22.9 $\pm$ 5.3
222
Time delays have been reported for 24 gravitationally lensed quasars. However,
the quality of the data and the curve-shifting procedure followed differs from
system to system. In this section, we present a homogeneous analysis of their
publicly available light curves following the procedure described in the
previous section, with the aim of identifying those systems that have reliable
time delay measurements. In the case of systems with more than two images, we
measured the time delays between all pairs of light curves. The results are
summarized in Table 1. All quoted uncertainties are 1$\sigma$ error bars,
unless stated otherwise. Additional information on some systems listed in
Table 1 and discussion on the possible reasons for our inability to reliably
measure some of the time delays follow. For all the other systems, our time
delay measurements agree with the previously reported values to within
2$\sigma$.
–
Q0142$-$100 (UM673): We were unable to make a reliable time delay measurement
using the light curves presented in Koptelova et al. (2012). This is not
surprising given that the light curves are characterized by large seasonal
gaps and there are no clear variability features that could be matched between
the A and B light curves.
–
JVAS B0218+357: From 8 GHz and 15 GHz VLA observations reported by Cohen et
al. (2000), we measured time delays of 10.4 $\pm$ 1.0 days and 11.4 $\pm$ 1.5
days, respectively. Taking the weighted average of the two results, we find
the time delay to be 10.7 $\pm$ 0.8 days. We note here that Biggs et al.
(1999) monitored this system using VLA during the same period as Cohen et al.
(2000) at the same two frequencies and report a time delay of 10.5 $\pm$ 0.4
days (95% CI).
–
HE 0435$-$1223: We made use of the light curves presented in Courbin et al.
(2011) spanning seven seasons using data from Euler, Mercator, Maidanak, and
SMARTS and the light curves presented in Blackburne et al. (2014) spanning
eight seasons using data from SMARTS. The SMARTS data used by Courbin et al.
(2011) is the same as the first two seasons of data presented in Blackburne et
al. (2014). Hence we excluded the SMARTS data points from the light curves of
Courbin et al. (2011) to make it independent of the light curves of Blackburne
et al. (2014). Owing to the differences in the approaches followed by these
two teams of authors to derive photometry and also the photometric
uncertainties, we avoided merging the two datasets. Our time delay
measurements listed in Table 1 are the weighted averages of the time delays
measured from the two independent sets of light curves. The reported time
delay values in Table 1 are from Courbin et al. (2011). The best-fit time
delay values reported without uncertainties by Blackburne et al. (2014) are
consistent with the values of Courbin et al. (2011) to within 1$\sigma$. In
Table 2, we present our measurements of the time delays of HE 0435$-$1223 from
the two independent sets of light curves and the resulting weighted averages.
For each pair of quasar images, we see that the time delays measured from the
two datasets agree to within 2$\sigma$.
–
SBS 0909+532: For our analysis, we used only the r-band data points obtained
using the Liverpool Robotic Telescope between 2005 January and 2007 January
presented in Goicoechea et al. (2008) and Hainline et al. (2013), based on
homogeneity and sampling considerations.
–
RX J0911.4+0551: We used the light curves presented in Hjorth et al. (2002),
which were made publicly available by Paraficz et al. (2006).
–
FBQ 0951+2635: We used the light curves presented in Jakobsson et al. (2005),
which were made publicly available by Paraficz et al. (2006).
–
Q0957+561: From the $r$-band and $g$-band light curves presented in Shalyapin
et al. (2012), we measured time delays of 420.6 $\pm$ 1.8 days and 419.2 $\pm$
2.2 days, respectively. Taking the weighted average of the two results, we
find the time delay to be 420.0 $\pm$ 1.4 days. The reported delay listed is
the weighted average of the two delays found by Shalyapin et al. (2012).
–
RX J1131$-$1231: Tewes et al. (2013b) measured time delays between all pairs
of light curves using three different numerical techniques. The time delay
value listed in the table for each pair of light curves is for the technique
that resulted in the smallest uncertainty.
–
H1413+117: The light curves presented in Goicoechea & Shalyapin (2010) span
less than one season and display poor variability. Hence our time delay
measurements for the pairs AB, AC, and AD although in good agreement with the
reported values, are of low precision and we could not reliably measure time
delays for the pairs BC and BD.
–
CLASS B1600+434: From both the optical light curves presented in Burud et al.
(2000) (and made publicly available by Paraficz et al. (2006)) and the radio
light curves presented in Koopmans et al. (2000), we were unable to make a
reliable time delay measurement. Although the optical light curves show good
variability, they suffer from poor sampling and thus exclude the possibility
of convincingly matching the variability features between light curves A and
B. The radio light curves spanning one season is well sampled; however, light
curve A displays short time scale fluctuations that are not seen in light
curve B, thus making it difficult to measure the time delay unambiguously.
–
HE 2149$-$2745: We used the light curves presented in Burud et al. (2002a),
which were made publicly available by Paraficz et al. (2006).
Figure 3: Error analysis of the time delay measurement based on delay
estimations on simulated light curves that mimic the light curves displayed in
Fig. 1. The horizontal axis corresponds to the value of the true time delay
used in these simulated light curves. The gray colored rods and 1$\sigma$
error bars show the systematic biases and random errors, respectively. Our
measured time delay of $\Delta t$ = $-$20.5 $\pm$ 1.0 days is discrepant with
the true time delay of 22.75 days listed in the TDC1 truth files at the level
of 2.25$\sigma$. The difference in sign of time delay is simply a matter of
convention. Table 2: Our measurements of the time delays of HE 0435$-$1223
from two independent datasets.
Time delay | Courbin et al. (2011)aa$a$The SMARTS data points were excluded from the light curves of Courbin et al. (2011) so that the measured time delay values were independent of those measured from the SMARTS monitoring light curves of Blackburne et al. (2014) (see discussion in Sect. 3). | Blackburne et al. (2014) | Weighted average
---|---|---|---
| (days) | (days) | (days)
$\Delta t_{AB}$ | 8.4 $\pm$ 1.4 | 12.3 $\pm$ 1.9 | 9.8 $\pm$ 1.1
$\Delta t_{AC}$ | 3.6 $\pm$ 3.4 | 2.7 $\pm$ 2.9 | 3.1 $\pm$ 2.2
$\Delta t_{AD}$ | 13.1 $\pm$ 1.1 | 15.8 $\pm$ 2.1 | 13.7 $\pm$ 1.0
$\Delta t_{BC}$ | $-$8.3 $\pm$ 1.5 | $-$7.7 $\pm$ 1.4 | $-$8.0 $\pm$ 1.0
$\Delta t_{BD}$ | 5.7 $\pm$ 1.7 | 7.9 $\pm$ 3.2 | 6.2 $\pm$ 1.5
$\Delta t_{CD}$ | 13.0 $\pm$ 1.1 | 14.1 $\pm$ 1.1 | 13.6 $\pm$ 0.8
333
## 4 $H_{0}$ from pixellated modeling of ten gravitational lenses
Of the 24 systems analyzed in the last section, 14 of them had light curves of
sufficiently good quality to enable the measurement of at least one time delay
between the images, adjacent to each other in terms of arrival-time order, to
a precision of better than 20% (which corresponds to a 5$\sigma$ detection of
time delay). The ten systems which did not satisfy this criterion are
Q0142$-$100 (UM673), FBQ 0951+2635, PG 1115+080, H1413+117, JVAS B1422+231,
CLASS B1600+434, SDSS J1650+4251, PKS 1830$-$211, HE 2149$-$2745, and HS
2209+1914.
Of the 14 remaining systems, we did not model the mass distribution for four
of them for the following reasons. SDSS J1001+5027 and SDSS J1206+4332 do not
have accurate astrometric data measured from Hubble Space Telescope (HST)
images or ground-based imaging with adaptive optics. Although the astrometry
of JVAS B0218+357, which has a small image separation of 0.33$\arcsec$, has
been measured from HST images by Sluse et al. (2012), the authors warn about
possibly large systematic errors in the published astrometry. SDSS J1029+2623
is a three-image cluster lens with highly complex mass distribution (see Oguri
et al. 2013) and hence not amenable to lens-modeling following the simplistic
approach described below.
To perform mass-modeling of the remaining ten systems – HE 0435$-$1223, SBS
0909+532, RX J0911.4+0551, Q0957+561, SDSS J1004+4112, HE 1104$-$1805, RX
J1131$-$1231, SBS 1520+530, CLASS B1608+656 and WFI J2033$-$4723 – to infer
$H_{0}$, we used the publicly available
PixeLens444http://www.physik.uzh.ch/~psaha/lens/pixelens.php code (Saha &
Williams 2004), which builds an ensemble of pixellated mass maps compatible
with the input data for a given system, which is comprised of the redshifts of
the quasar and the lensing galaxy, the arrival-time order of the images, their
astrometry relative to the center of the main lensing galaxy, and the known
time delays between the images adjacent to each other in terms of arrival-time
order. In case of quadruple lenses in which only some of the time delays are
known, it is still possible to guess the arrival-time order of the images by
following certain simple rules (see Saha & Williams 2003).
We model all lenses, except SDSS J1004+4112, such that their mass profiles
have inversion symmetry about the lens center, including any companion galaxy
to the main lensing galaxy as a point mass. The lensing cluster in SDSS
J1004+4112 consists of several galaxies besides the main lensing galaxy (see
Inada et al. 2005) and hence was modeled without assuming inversion symmetry
about the lens center.
PixeLens builds models such that their projected density profiles are steeper
than $|\boldsymbol{\theta}|^{-\gamma_{min}}$, where $|\boldsymbol{\theta}|$ is
the distance from the center of the lens in angular units, the default value
of $\gamma_{min}$ being 0.5. This is based on the observation that the total
density distribution in the central regions of elliptical galaxies is close to
isothermal (i.e., $r^{-2}$) and also the observation that the total density in
the center of our Galaxy scales is $r^{-1.75}$ (see Saha & Williams 2004,
Sect. 2.2 and references therein). The profiles $r^{-2}$ and $r^{-1.75}$
correspond to projected density profiles of $|\boldsymbol{\theta}|^{-1}$ and
$|\boldsymbol{\theta}|^{-0.75}$, respectively, in the special case of
spherical symmetry. In this work, we relax the restriction of $\gamma_{min}$ =
0.5 and set $\gamma_{min}$ = 0 for those lenses in our sample in which the
largest angular separation between the images is greater than 3$\arcsec$. The
lenses in our sample that satisfy this criterion are RX J0911.4+0551,
Q0957+561, SDSS J1004+4112, HE 1104$-$1805, and RX J1131$-$1231\. A large
image separation implies that there is significant lensing action from the
cluster of which the main lensing galaxy is part, in which case the projected
density profile can be shallower than $|\boldsymbol{\theta}|^{-0.5}$.
For each system, we build an ensemble of 100 models, corresponding to 100
values of $H_{0}$. The mean of the 100 values gives the best estimate of
$H_{0}$, the uncertainty of which is the standard deviation of the 100 values.
This uncertainty includes only the uncertainty in the mass model. PixeLens
assumes that the uncertainty in the input priors to be negligibly small, which
is a reasonable assumption for the redshifts, if they are spectroscopically
measured, and astrometry, if measured from HST or ground-based adaptive optics
imaging. However, the measured time delays have finite uncertainties, which
need to be propagated into the uncertainty of the estimated $H_{0}$. We do
this by remodeling each system after perturbing the time delay by its
1$\sigma$ uncertainty and noticing the deviation of the resulting value of
$H_{0}$ from the original value. For high-precision time delays, the deviation
in $H_{0}$ was found to be the same whether the delays were perturbed upward
or downward. In general, the deviation in $H_{0}$ was found to be slightly
larger when the delays were perturbed downward than when they were perturbed
upward. Hence in this work, to get a conservative estimate of the contribution
of the time delay uncertainty to the uncertainty in $H_{0}$, we decrease the
time delay by its 1$\sigma$ uncertainty and find the resulting increase in
$H_{0}$. This uncertainty in $H_{0}$ resulting from the time delay uncertainty
is added in quadrature to the uncertainty in $H_{0}$ resulting from mass
modeling to find the total uncertainty. In the case of quadruple lenses where
more than one time delay is known, we perturb each delay individually while
leaving the other delays unchanged to infer its uncertainty contribution. The
uncertainty contribution from each independent time delay is then added in
quadrature to the uncertainty in $H_{0}$ resulting from the uncertainty in the
mass model to find the total uncertainty.
In order to include the effects of external shear, an approximate direction of
the shear axis needs to be specified and PixeLens will search for solutions
within 45$\degr$ of the specified direction. Since there is no simple rule to
guess the direction of the external shear for a given system, for each system,
we repeated the modeling specifying the approximate direction of the shear
axis as 90$\degr$, 45$\degr$, 0$\degr$, and $-$45$\degr$ (in this instance,
specifying $\theta$ and $\theta$+180$\degr$ are equivalent). We thus obtain
four estimates of $H_{0}$ and their uncertainties. In each case, we propagate
the uncertainty contributions from the known time delays to the uncertainty in
$H_{0}$, as discussed previously. The final estimate of $H_{0}$ and its
uncertainty are found using maximum likelihood analysis, optimizing their
values so as to maximize the joint posterior probability of these two
parameters for the sample consisting of the four $H_{0}$ values and their
uncertainties (see Barnabè et al. 2011, Eq. 7). In optimizing the value of the
uncertainty, we choose the minimum limit to be the smallest of the four
uncertainties. We note here that for the system HE 1104$-$1805, the choices of
the approximate direction of the shear axis of 90$\degr$ and $-$45$\degr$ were
found to lead to unphysical models involving negative values in the mass
pixels. Hence for this system, the maximum likelihood analysis was carried out
using only the two $H_{0}$ values resulting for the approximate shear
directions of 45$\degr$ and 0$\degr$.
The input priors for each system and the resulting $H_{0}$ estimates are
summarized in Table 3. In Fig. 4 we plot the $H_{0}$ estimates from the ten
lenses, all of which are seen to agree with each other within their error
bars. To combine the ten independent estimates into a best estimate of
$H_{0}$, we again employ maximum likelihood analysis, as described above.
However, in this case, in optimizing the value of the uncertainty of the best
estimate of $H_{0}$, the minimum limit is chosen to be the uncertainty of the
weighted average of the ten values. We infer a value of $H_{0}$ of 68.1 $\pm$
5.9 km s-1 Mpc-1 (1$\sigma$ uncertainty, 8.7% precision) for a spatially flat
universe having $\Omega_{m}$ = 0.3 and $\Omega_{\Lambda}$ = 0.7. The reason
for employing maximum likelihood analysis in this case, rather than taking a
simple weighted average is to detect the presence of any unmodeled
uncertainties. However, as can be seen from Fig. 4, the $H_{0}$ estimates from
the individual systems all agree with each other within their error bars and
hence the $H_{0}$ value inferred above through maximum likelihood analysis is
only marginally different from the weighted average. For the source and lens
redshifts of the current sample, we find the $H_{0}$ estimate to decrease by
7.1% for the Einstein-de Sitter universe ($\Omega_{m}$ = 1.0 and
$\Omega_{\Lambda}$ = 0.0) and increase by 2.3% for an open universe having
$\Omega_{m}$ = 0.3 and $\Omega_{\Lambda}$ = 0.0, thus illustrating the low
level of dependence of the inferred value of $H_{0}$ on the precise values of
$\Omega_{m}$ and $\Omega_{\Lambda}$. In Table 3, we also list the $H_{0}$
estimates obtained without propagating the time delay uncertainties. We see
that the dominant contribution to uncertainty in $H_{0}$ results from the
uncertainty in the mass model.
Figure 4: The $H_{0}$ estimates and their 1$\sigma$ uncertainties for the ten
gravitational lenses – (1) HE 0435$-$1223, (2) SBS 0909+532, (3) RX
J0911.4+0551, (4) Q0957+561, (5) SDSS J1004+4112, (6) HE 1104$-$1805, (7) RX
J1131$-$1231, (8) SBS 1520+530, (9) CLASS B1608+656, and (10) WFI
J2033$-$4723\. The best estimate of $H_{0}$ and its 1$\sigma$ confidence
interval, inferred through maximum-likelihood analysis, are represented by the
horizontal line and the gray shaded region, respectively. Table 3: Summary of
input data to PixeLens and resulting $H_{0}$ estimates.
Object | Redshifts | Imageaa$a$The QSO images are listed in arrival-time order. | $\Delta$RAcc$c$The astrometry of the QSO images and point masses are specified with respect to the center of the main lensing galaxy. | $\Delta$Deccc$c$The astrometry of the QSO images and point masses are specified with respect to the center of the main lensing galaxy. | Delaydd$d$The time delay of a given image is listed (if measured to a precision better than 20%) with respect to the previous image in terms of arrival-time order. | $H_{0}$ee$e$In parentheses we provide the $H_{0}$ estimates and their uncertainties without propagating the uncertainties in time delays. | Referencesff$f$The references are listed for measurements of lens redshift ($z_{l}$), source redshift ($z_{s}$), and astrometry.
---|---|---|---|---|---|---|---
| | / P.M.bb$b$‘P.M.’ is the abbreviation for point mass and refers to secondary lensing galaxies. | ($\arcsec$) | ($\arcsec$) | (days) | (km s-1 Mpc-1) |
HE 0435$-$1223 | $z_{l}$ = 0.4546 | A | 1.1706 | 0.5665 | | 64.1 $\pm$ 21.3 | Morgan et al. (2005)
| $z_{s}$ = 1.689 | C | $-$1.2958 | $-$0.0357 | | (64.1 $\pm$ 19.4) | Wisotzki et al. (2002)
| | B | $-$0.3037 | 1.1183 | 8.0 $\pm$ 1.0 | | Courbin et al. (2011)
| | D | 0.2328 | $-$1.0495 | | |
SBS 0909+532 | $z_{l}$ = 0.830 | B | 0.5228 | $-$0.4423 | | 63.9 $\pm$ 17.3 | Lubin et al. (2000)
| $z_{s}$ = 1.377 | A | $-$0.4640 | 0.0550 | 45.9 $\pm$ 3.1 | (63.9 $\pm$ 16.8) | Kochanek et al. (1997)
| | | | | | | Sluse et al. (2012)
RX J0911.4+0551 | $z_{l}$ = 0.769 | B | $-$2.2662 | 0.2904 | | 80.0 $\pm$ 31.8 | Kneib et al. (2000)
| $z_{s}$ = 2.800 | A2 | 0.9630 | $-$0.0951 | 141.9 $\pm$ 12.3 | (80.0 $\pm$ 31.0) | Bade et al. (1997)
| | A1 | 0.7019 | $-$0.5020 | | | Sluse et al. (2012)
| | A3 | 0.6861 | 0.4555 | | |
| | P.M. | $-$0.7582 | 0.6658 | | |
Q0957+561 | $z_{l}$ = 0.361 | A | 1.408 | 5.034 | | 96.9 $\pm$ 31.3 | Walsh et al. (1979)
| $z_{s}$ = 1.41 | B | 0.182 | $-$1.018 | 420.0 $\pm$ 1.4 | (96.9 $\pm$ 31.3) | Fadely et al. (2010)
SDSS J1004+4112 | $z_{l}$ = 0.68 | C | 3.925 | $-$8.901 | | 89.9 $\pm$ 28.3 | Oguri et al. (2004)
| $z_{s}$ = 1.734 | B | $-$8.431 | $-$0.877 | 777.1 $\pm$ 9.2 | (89.9 $\pm$ 28.1) | Inada et al. (2003)
| | A | $-$7.114 | $-$4.409 | 37.2 $\pm$ 3.1 | | Inada et al. (2005)
| | D | 1.285 | 5.298 | | |
HE 1104$-$1805 | $z_{l}$ = 0.729 | B | 1.9289 | $-$0.8242 | | 104.0 $\pm$ 53.0 | Lidman et al. (2000)
| $z_{s}$ = 2.319 | A | $-$0.9731 | 0.5120 | 157.1 $\pm$ 3.6 | (104.0 $\pm$ 52.9) | Smette et al. (1995)
| | | | | | | Sluse et al. (2012)
RX J1131$-$1231 | $z_{l}$ = 0.295 | C | $-$1.460 | $-$1.632 | | 71.9 $\pm$ 25.6 | Sluse et al. (2003)
| $z_{s}$ = 0.658 | B | $-$2.076 | 0.662 | | (71.9 $\pm$ 25.6) | Suyu et al. (2013)
| | A | $-$2.037 | $-$0.520 | | |
| | D | 1.074 | 0.356 | 91.7 $\pm$ 0.7 | |
| | P.M. | $-$0.097 | 0.614 | | |
SBS 1520+530 | $z_{l}$ = 0.761 | A | $-$1.1395 | 0.3834 | | 59.0 $\pm$ 15.8 | Auger et al. (2008)
| $z_{s}$ = 1.855 | B | 0.2879 | $-$0.2691 | 124.2 $\pm$ 8.1 | (59.0 $\pm$ 15.3) | Chavushyan et al. (1997)
| | | | | | | Sluse et al. (2012)
CLASS B1608+656 | $z_{l}$ = 0.6304 | B | 1.2025 | $-$0.8931 | | 58.7 $\pm$ 11.0 | Myers et al. (1995)
| $z_{s}$ = 1.394 | A | 0.4561 | 1.0647 | 32.4 $\pm$ 3.0 | (58.7 $\pm$ 10.8) | Fassnacht et al. (1996)
| | C | 1.2044 | 0.6182 | | | Sluse et al. (2012)
| | D | $-$0.6620 | $-$0.1880 | 41.3 $\pm$ 1.6 | |
| | P.M. | 0.7382 | 0.1288 | | |
WFI J2033$-$4723 | $z_{l}$ = 0.661 | B | 1.4388 | $-$0.3113 | | 73.7 $\pm$ 12.8 | Eigenbrod et al. (2006)
| $z_{s}$ = 1.66 | A1 | $-$0.7558 | 0.9488 | 37.6 $\pm$ 2.1 | (73.3 $\pm$ 11.6) | Morgan et al. (2004)
| | A2 | $-$0.0421 | 1.0643 | | | Vuissoz et al. (2008)
| | C | $-$0.6740 | $-$0.5891 | 23.6 $\pm$ 2.5 | |
Combined | | | | | | 68.1 $\pm$ 5.9 |
| | | | | | (67.9 $\pm$ 5.6) |
555
## 5 Conclusion
We have presented a homogeneous curve-shifting analysis of the light curves of
24 gravitationally lensed quasars for which time delays have been reported in
the literature so far. Time delays were measured using the difference-
smoothing technique and their uncertainties were estimated using realistic
simulated light curves; a recipe for creating these light curves with known
time delays in a plausible range around the measured delay was introduced in
this work. We identified 14 systems to have light curves of sufficiently good
quality to enable the measurement of at least one time delay between the
images, adjacent to each other in terms of arrival-time order, to a precision
of better than 20% (including systematic errors). Of these 14 systems, we
performed pixellated mass modeling using the publicly available PixeLens
software for ten of them, which have known lens redshifts, accurate
astrometric information, and sufficiently simple mass distributions, to infer
the value of $H_{0}$ to be 68.1 $\pm$ 5.9 km s-1 Mpc-1 (1$\sigma$ uncertainty,
8.7% precision) for a spatially flat universe having $\Omega_{m}$ = 0.3 and
$\Omega_{\Lambda}$ = 0.7. We note here that we have followed a relatively
simple lens modeling approach to constrain $H_{0}$ and our analysis does not
account for biases resulting from line-of-sight effects.
Our measurement closely matches a recent estimate of $H_{0}$ = 69.0 $\pm$ 6
(stat.) $\pm$ 4 (syst.) km s-1 Mpc-1 found by Sereno & Paraficz (2014) using a
method based on free-form modeling of 18 gravitational lens systems. Our value
is also consistent with the recent measurements of $H_{0}$ by Riess et al.
(2011), Freedman et al. (2012) and Suyu et al. (2013); however, it has lower
precision. Increasing the number of lenses with good-quality light curves,
accurate astrometry, and known lens redshift from the current ten used in this
study can bring down the uncertainty in $H_{0}$.
In the future such high-precision time delays will become available from
projects such as COSMOGRAIL (Tewes et al. 2012) involving dedicated medium-
sized telescopes. In addition, the next generation of cosmic surveys such as
the Dark Energy Survey (DES), the Large Synoptic Survey Telescope (LSST;
Ivezic et al. 2008), and the Euclid mission will detect a large sample of
lenses, and time delays might be available for a large fraction of them,
consequently enabling measurement of $H_{0}$ to an accuracy better than 2%.
Furthermore, detection of gravitational wave signals from short gamma-ray
bursts associated with neutron star binary mergers in the coming decade could
constrain $H_{0}$ to better than 1% (Nissanke et al. 2013).
###### Acknowledgements.
We thank Jim Lovell for providing us with the light curves of PKS 1830$-$211\.
We acknowledge useful discussions with Malte Tewes, G. Indu, Leon Koopmans,
Prashanth Mohan and Matthias Bartelmann. We thank Frederic Courbin and Georges
Meylan for carefully reading the manuscript and offering helpful comments. We
thank the organizers of the Strong Lens Time Delay Challenge for enabling a
blind test of our algorithm and subsequently providing the truth files, which
helped us to refine our curve-shifting procedure. We thank the anonymous
referee for constructive reports that helped to improve the presentation of
this work.
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|
arxiv-papers
| 2014-04-10T19:50:57 |
2024-09-04T02:50:00.966528
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "S. Rathna Kumar, C. S. Stalin, and T. P. Prabhu",
"submitter": "S Rathna Kumar",
"url": "https://arxiv.org/abs/1404.2920"
}
|
1404.2981
|
# On free stable distributions
T. Hasebe 111Department of Mathematics, Hokkaido University, Kita 10, Nishi 8,
Kitaku, Sapporo 060-0810, Japan. Email: [email protected] , A.
Kuznetsov 222Department of Mathematics and Statistics, York University, 4700
Keele Street, Toronto, Ontario, M3J 1P3, Canada. Email:
[email protected]
(This version: )
###### Abstract
We investigate the analytical properties of free stable distributions and
discover many connections with the classical stable distributions. Our main
results include an explicit formula for the Mellin transform, series
representations for the characteristic function and for the density of the
free stable distribution; all of these explicit formulas exhibit close
resemblance to the corresponding expressions for classical stable
distributions. As applications of these results we obtain a new derivation of
the duality law obtained in [1] and a factorization of a classical stable
random variable into the independent (in the classical sense) product of a
free stable random variable and a power of Gamma(2) random variable.
Keywords: free stable distribution, stable distribution, Mellin transform
2010 Mathematics Subject Classification : 46L54, 60E07
## 1 Introduction and main results
The class ${\mathfrak{W}}^{*}$ of classical (strictly) stable distributions
(see [7]) is uniquely characterized by the following property:
$\mu\in{\mathfrak{W}}^{*}$ if and only if for any $c_{1},c_{2}>0$ there exists
$c_{3}>0$ such that
$\mu_{c_{1}}*\mu_{c_{2}}=\mu_{c_{3}},$ (1)
where $\mu_{c}({\textnormal{d}}y)$ is obtained from measure
$\mu({\textnormal{d}}x)$ by scaling $y=cx$ (equivalently,
$\mu_{c}((-\infty,y]):=\mu((-\infty,y/c])$) and the binary operator “$*$”
denotes the classical convolution. Similarly, the class
${\mathfrak{W}}^{\boxplus}$ of free (strictly) stable distributions (see [1,
5]) is characterized by the scaling property (1) with the classical
convolution “$*$” replaced by the free convolution “$\boxplus$”. The
similarities end at this stage, and in all other respects these two families
of distributions seem to be quite different. For example, the main tool for
working with a classical stable distribution is its characteristic function,
while free stable distributions are described by their Cauchy transform.
Another difference is that classical stable distributions have explicit
formulas for the Mellin transform and their densities have explicit series
expansions (see Sections 2.4, 2.5 and Theorem 2.6.3 in [7]), whereas the
density of the free stable distributions enjoys a representation as an inverse
function of a rather simple explicit function (Propositions A.1.2-A.1.4 in
[1]) (a result which does not have an analogue in the case of classical stable
distributions).
At the same time, several results in the recent papers by Demni [3] and
Haagerup and Möller [4] have offered glimpses of possible deeper similarities
and connections between the ${\mathfrak{W}}^{*}$ and
${\mathfrak{W}}^{\boxplus}$ distributions. Demni [3] has investigated the
appearance of the Kanter random variable both in classical and free stable
distributions, while the Mellin transform of the positive free stable
distributions which was computed by Haagerup and Möller in [4] bears close
resemblance to the Mellin transform of the classical stable distributions, see
[7, Theorem 2.6.3].
Our main goal in this paper is to investigate analytical properties of free
stable distributions, and to demonstrate their close connections with the
classical stable distributions. First, let us introduce notations and
definitions and present a new parameterization of free stable distributions.
It is known that (up to a scaling parameter) any strictly stable distribution
is uniquely characterized by two parameters $\alpha$ and $\rho$ belonging to
the set of admissible parameters
${\mathcal{A}}:=\\{\alpha\in(0,1),\;\rho\in[0,1]\\}\cup\\{\alpha\in(1,2],\;\rho\in[1-\alpha^{-1},\alpha^{-1}]\\}.$
(2)
The characteristic function of a distribution
$\mu_{\alpha,\rho}\in{\mathfrak{W}}^{*}$ is given by
${\mathfrak{g}}_{\alpha,\rho}(z):=\int_{{\mathbb{R}}}e^{{\textnormal{i}}zx}\mu_{\alpha,\rho}({\textnormal{d}}x)=\exp\left(-z^{\alpha}e^{\pi{\textnormal{i}}\alpha(\frac{1}{2}-\rho)}\right),\;\;\;z>0.$
(3)
Similarly, free stable distributions can be characterized by two parameters
$\alpha$ and $\tilde{\rho}$, belonging to the set
${\mathcal{B}}:=\\{\alpha\in(0,1)\cup(1,2],\;\tilde{\rho}\in[0,1]\\}.$ (4)
A free stable distribution
$\nu_{\alpha,\tilde{\rho}}\in{\mathfrak{W}}^{\boxplus}$ is characterized by
the Voiculescu transform
$\displaystyle\phi_{\alpha,\tilde{\rho}}(z)=\begin{cases}-e^{\pi{\textnormal{i}}\alpha\tilde{\rho}}z^{-\alpha+1},\;&\textnormal{
if }\;\alpha\in(0,1),\\\
e^{\pi{\textnormal{i}}(\alpha-2)\tilde{\rho}}z^{-\alpha+1},\;&\textnormal{ if
}\;\alpha\in(1,2],\end{cases}$ (5)
where $\operatorname{Im}(z)>0$. In the above formula and everywhere else in
this paper we use the principal branch of the logarithm $\ln(z)$ and the power
function $z^{\alpha}=\exp(\alpha\ln(z))$. The Voiculescu transform is
connected to the measure $\nu_{\alpha,\tilde{\rho}}$ via the Cauchy transform
$G_{\alpha,\tilde{\rho}}(z):=\int_{{\mathbb{R}}}\frac{\nu_{\alpha,\tilde{\rho}}({\textnormal{d}}x)}{z-x},\;\;\;\operatorname{Im}(z)>0,$
(6)
by requiring $G_{\alpha,\tilde{\rho}}(z)$ to be the inverse function of
$z\mapsto 1/z+\phi_{\alpha,\tilde{\rho}}(1/z)$. The parameterization
$(\alpha,\tilde{\rho})$ which is used to define the Voiculescu transform via
(5) goes back to the seminal work of Bercovici, Pata and Biane [1]. In order
to present our formulas in a more compact way and to highlight close
connections with the classical stable distributions, we have to introduce a
new parameterization for the class of free stable distributions. Instead of
using the pair $(\alpha,\tilde{\rho})\in{\mathcal{B}}$ we will use parameters
$(\alpha,\rho)=\left(\alpha,\left(1-(2-\alpha)\tilde{\rho}\right)\tfrac{1}{\alpha}\right).$
(7)
Note that the map $(\alpha,\tilde{\rho})\mapsto(\alpha,\rho)$ is a bijection
between the set ${\mathcal{B}}$ and the set of admissible parameters
${\mathcal{A}}$. Using this new parameterization
$(\alpha,\rho)\in{\mathcal{A}}$ we see that the Voiculescu transform of a
distribution $\nu_{\alpha,\rho}\in{\mathfrak{W}}^{\boxplus}$ is given by a
single expression
$\phi_{\alpha,\rho}(z)=-e^{\pi{\textnormal{i}}\alpha\rho}z^{-\alpha+1},$ (8)
whereas the original parameterization (5) requires different formulas for
$\alpha\in(0,1)$ and $\alpha\in(1,2]$.
For $(\alpha,\rho)\in{\mathcal{A}}$ we will denote by $X_{\alpha,\rho}$
($Y_{\alpha,\rho}$) an ${\mathbb{R}}$-valued random variable having
distribution $\nu_{\alpha,\rho}\in{\mathfrak{W}}^{\boxplus}$ (respectively,
$\mu_{\alpha,\rho}\in{\mathfrak{W}}^{*}$).
Now we are ready to present our results, and the proofs are provided in the
next section.
###### Theorem 1.
Assume that $(\alpha,\rho)\in{\mathcal{A}}$. Then for $s\in(-1,\alpha)$
${\mathbb{E}}\left[(X_{\alpha,\rho})^{s}{\bf
1}_{\\{X_{\alpha,\rho}>0\\}}\right]=\frac{1}{\pi}\sin(\pi\rho
s)\frac{\Gamma(s)\Gamma\left(1-\frac{s}{\alpha}\right)}{\Gamma\left(2+s-\frac{s}{\alpha}\right)}.$
(9)
The expression for the Mellin transform of $X_{\alpha,\rho}$ in (9) is
remarkably similar to the corresponding result for the classical stable
distributions (see [7, Theorem 2.6.3]):
${\mathbb{E}}\left[(Y_{\alpha,\rho})^{s}{\bf
1}_{\\{Y_{\alpha,\rho}>0\\}}\right]=\frac{1}{\pi}\sin(\pi\rho
s)\Gamma(s)\Gamma\left(1-\frac{s}{\alpha}\right),\;\;\;\textnormal{ for
}\;-1<s<\alpha.$ (10)
Taking the limit $s\to 0$ in (9) we obtain the following
###### Corollary 1.
Assume that $(\alpha,\rho)\in{\mathcal{A}}$. Then
${\mathbb{P}}(X_{\alpha,\rho}>0)=\rho$.
The above result clearly demonstrates the benefit of our new parameterization
for the free stable distributions: The parameter $\rho$ (unlike
$\tilde{\rho}$) has a very natural interpretation as the positivity parameter
$\rho={\mathbb{P}}(X_{\alpha,\rho}>0)$, which is consistent with its
definition for the classical stable random variables $Y_{\alpha,\rho}$.
As we will see later, Theorem 1 implies that the free stable distribution
$\nu_{\alpha,\rho}$ is absolutely continuous, with a smooth density
$\psi_{\alpha,\rho}(x)$ (a fact that was first established in [1]). The
following duality result was first established in [1], and in this paper we
give a new derivation of this result as a simple corollary of Theorem 1. We
would like to emphasize that the same duality relation also holds for the
densities of classical stable distributions.
###### Corollary 2.
Assume that $\alpha\geq\frac{1}{2}$ and $(\alpha,\rho)\in{\mathcal{A}}$. Then
for $x>0$
$\psi_{\alpha,\rho}(x)=x^{-\alpha-1}\psi_{\frac{1}{\alpha},\alpha\rho}(x^{-\alpha}).$
(11)
The above duality law can also be written in terms of random variables. If
$\xi$ is a random variable such that ${\mathbb{P}}(\xi>0)>0$, we will denote
by $\hat{\xi}$ the cutoff of $\xi$, which is a positive random variable with
distribution given by
${\mathbb{P}}(\hat{\xi}\in A)={\mathbb{P}}(\xi\in A\;|\;\xi>0),$
for all Borel sets $A\subseteq(0,\infty)$. It is easy to see from
${\mathbb{P}}(X_{\alpha,\rho}>0)=\rho$ that the identity (11) is equivalent to
$\hat{X}_{\alpha,\rho}\stackrel{{\scriptstyle
d}}{{=}}\left(\hat{X}_{\frac{1}{\alpha},\alpha\rho}\right)^{-\frac{1}{\alpha}}.$
(12)
Theorem 1 also gives the following distributional identity between stable and
free stable distributions.
###### Corollary 3.
Let $Z$ be a Gamma(2) random variable (that is, a positive random variable
with the density $p_{Z}(x)=xe^{-x}$, $x>0$). For
$(\alpha,\rho)\in{\mathcal{A}}$ we have
$Y_{\alpha,\rho}\stackrel{{\scriptstyle d}}{{=}}X_{\alpha,\rho}\times
Z^{1-\frac{1}{\alpha}},$ (13)
where the random variables $X_{\alpha,\rho}$ and $Z$ are assumed to be
independent.
###### Corollary 4.
Assume that $(\alpha,\rho_{i})\in{\mathcal{A}}$, $i=1,2$. Then
$\frac{X_{\alpha,\rho_{1}}}{X_{\alpha,\rho_{2}}}\stackrel{{\scriptstyle
d}}{{=}}\frac{X_{\alpha,\rho_{2}}}{X_{\alpha,\rho_{1}}}\;\;\;\textnormal{ and
}\;\;\;\frac{\hat{X}_{\alpha,\rho_{1}}}{\hat{X}_{\alpha,\rho_{2}}}\stackrel{{\scriptstyle
d}}{{=}}\frac{\hat{X}_{\alpha,\rho_{2}}}{\hat{X}_{\alpha,\rho_{1}}},$ (14)
where all random variables are assumed to be independent.
The next theorem is our second main result, where we establish a convergent
series representation for $\psi_{\alpha,\rho}(x)$.
###### Theorem 2.
Assume that $\alpha\in(0,1)$ and $\rho\in[0,1]$ and denote
$x^{*}:=\alpha(1-\alpha)^{\frac{1}{\alpha}-1}$. Then
$\displaystyle\psi_{\alpha,\rho}(x)$
$\displaystyle=\frac{1}{\pi}\sum\limits_{n\geq
1}(-1)^{n-1}\frac{\Gamma(1+\alpha
n)}{n!\Gamma(2+(\alpha-1)n)}\sin(n\alpha\rho\pi)x^{-\alpha n-1},\;\;\;x\geq
x^{*},$ (15) $\displaystyle\psi_{\alpha,\rho}(x)$
$\displaystyle=\frac{1}{\pi}\sum\limits_{n\geq
1}(-1)^{n-1}\frac{\Gamma\left(1+\frac{n}{\alpha}\right)}{n!\Gamma\left(2+\left(\frac{1}{\alpha}-1\right)n\right)}\sin(n\rho\pi)x^{n-1},\;\;\;0\leq
x\leq x^{*}.$ (16)
The corresponding series expansions for $\psi_{\alpha,\rho}(x)$ when
$\alpha\in(1,2]$ and $\rho\in[1-\alpha^{-1},\alpha^{-1}]$ can be obtained
using (11), (15) and (16), and the expressions for $x<0$ follow from
$\psi_{\alpha,\rho}(x)=\psi_{\alpha,1-\rho}(-x)$.
The series expansions (15) and (16) have direct analogues in the case of
classical stable distribution. The following result is given in [7, Theorem
2.4.2, Theorem 2.5.1]: Let $p_{\alpha,\rho}(x)$ denote the density of the
random variable $Y_{\alpha,\rho}$, then for $\alpha\in(0,1)$ and
$\rho\in[0,1]$
$p_{\alpha,\rho}(x)=\frac{1}{\pi}\sum\limits_{n\geq
1}(-1)^{n-1}\frac{\Gamma(1+\alpha n)}{n!}\sin(n\alpha\rho\pi)x^{-\alpha
n-1},\;\;\;x>0,$ (17)
and at the same time $p_{\alpha,\rho}$ has the asymptotic expansion
$p_{\alpha,\rho}(x)\sim\frac{1}{\pi}\sum\limits_{n\geq
1}(-1)^{n-1}\frac{\Gamma\left(1+\frac{n}{\alpha}\right)}{n!}\sin(n\rho\pi)x^{n-1},\;\;\;x\to
0.$ (18)
When $x<0$ we have $p_{\alpha,\rho}(x)=p_{\alpha,1-\rho}(x)$. In the case
$\alpha>1$ the situation is reversed: the series (18) gives a convergent
series representation while (17) provides a complete asymptotic series for
$p_{\alpha,\rho}(x)$ as $x\to+\infty$.
Our third main result is the series expansion for the characteristic function
of the free stable distribution, defined as
${\mathfrak{f}}_{\alpha,\rho}(z):=\int_{{\mathbb{R}}}e^{{\textnormal{i}}zx}\psi_{\alpha,\rho}(x){\textnormal{d}}x.$
(19)
###### Theorem 3.
Assume that $(\alpha,\rho)\in{\mathcal{A}}$. Then for $z>0$
${\mathfrak{f}}_{\alpha,\rho}(z)=\sum\limits_{n\geq
0}(-1)^{n}\frac{e^{\pi{\textnormal{i}}\alpha(\frac{1}{2}-\rho)n}}{n!\Gamma(2+(\alpha-1)n)}z^{\alpha
n}.$ (20)
Note that for $z\in{\mathbb{R}}$ the value of
${\mathfrak{f}}_{\alpha,\rho}(z)$ is the conjugate of
${\mathfrak{f}}_{\alpha,\rho}(-z)$, thus the series representation for
${\mathfrak{f}}_{\alpha,\rho}(z)$ when $z<0$ follows at once from (20).
Formula (20) is a direct analogue of the corresponding series for the
characteristic function of the classical stable distribution
${\mathfrak{g}}_{\alpha,\rho}(z)=\sum\limits_{n\geq
0}(-1)^{n}\frac{e^{\pi{\textnormal{i}}\alpha(\frac{1}{2}-\rho)n}}{n!}z^{\alpha
n},\;\;\;\;\;z>0,$
which can be easily obtained from (3).
## 2 Proofs
We denote the upper-half plane
${\mathbb{C}}^{+}:=\\{z\in{\mathbb{C}}\;:\;\operatorname{Im}(z)>0\\}$ and
similarly for the lower half-plane ${\mathbb{C}}^{-}$. For $s\in{\mathbb{C}}$
lying on the vertical line $\operatorname{Re}(s)=0$ we define
$M_{\alpha,\rho}(s):={\mathbb{E}}\left[(X_{\alpha,\rho})^{s}{\bf
1}_{\\{X_{\alpha,\rho}>0\\}}\right]=\int\limits_{0}^{\infty}x^{s}\psi_{\alpha,\rho}(x){\textnormal{d}}x.$
(21)
Proof of Theorem 1: Our first goal is to establish identity (9) for
$s\in(-1,0)$. Assume that $(\alpha,\rho)\in{\mathcal{A}}$. Let
$\gamma_{\alpha,\rho}\subset\bar{\mathbb{C}}^{-}$ be the the curve given in
polar coordinates
$\gamma_{\alpha,\rho}:=\\{r(\theta)e^{-{\textnormal{i}}\theta}\;:\;0<\theta<\pi\\}\cup\\{0\\},$
where
$r(\theta):=\left(\frac{\sin(\theta)}{\sin[(1-\alpha\rho)\pi+(\alpha-1)\theta)]}\right)^{\frac{1}{\alpha}}.$
As explained in Lemma A1.1 in [1], $\gamma_{\alpha,\rho}$ is a closed simple
curve lying in the lower half-plane. We consider this curve as the boundary of
a Jordan domain $\Omega_{\alpha,\rho}$. As was shown in Lemma A1.1 in [1], the
function $G_{\alpha,\rho}(z)$ is a one-to-one conformal transformation from
${\mathbb{C}}^{+}$ onto $\Omega_{\alpha,\rho}$, which extends continuously to
the boundary. Thus $G_{\alpha,\rho}$ gives a homeomorphism of
${\mathbb{C}}^{+}\cup{\mathbb{R}}\cup\\{\infty\\}$ with
$\overline{\Omega}_{\alpha,\rho}$. Let us define
$z^{*}:=\exp(-\pi{\textnormal{i}}\rho).$
One can check that $1/z^{*}+\phi_{\alpha,\rho}(1/z^{*})=0$. This shows that
$G_{\alpha,\rho}(z)$ (being the inverse of $1/z+\phi_{\alpha,\rho}(1/z)$) maps
the positive real line $(0,\infty)$ onto the curve
$\gamma^{+}:=\\{r(\theta)e^{-{\textnormal{i}}\theta}\;:\;0\leq\theta\leq\pi\rho\\}.$
Since $G_{\alpha,\rho}(0)=z^{*}$ and $G_{\alpha,\rho}(+\infty)=0$, the curve
$\gamma^{+}$ is traversed in the direction from $z^{*}$ to $0$.
The fact that $G_{\alpha,\rho}(z)$ is the inverse of
$1/z+\phi_{\alpha,\rho}(1/z)$ implies
$w=G_{\alpha,\rho}(x)\Longleftrightarrow
x=w^{-1}-e^{\pi{\textnormal{i}}\alpha\rho}w^{\alpha-1}.$ (22)
We know that $G_{\alpha,\rho}(x)\to 0$ as $x\to\infty$, and formula (22) shows
that
$G_{\alpha,\rho}(x)=w=\frac{1}{x}\left(1-e^{\pi{\textnormal{i}}\alpha\rho}w^{\alpha}\right)=O(1/x),\;\;\;x\to\infty.$
(23)
At the same time, since the function $1/z+\phi_{\alpha,\rho}(1/z)$ is analytic
and has a non-vanishing derivative at $z=z^{*}$, its inverse function
$G_{\alpha,\rho}(z)$ is analytic in the neighborhood of zero. By the inversion
formula for the Cauchy transform we have
$\psi_{\alpha,\rho}(x)=-\frac{1}{\pi}\operatorname{Im}(G_{\alpha,\rho}(x)),\;\;\;x>0,$
therefore for $s\in(-1,0)$
$M_{\alpha,\rho}(s)=-\frac{1}{\pi}\operatorname{Im}\left[\int_{(0,\infty)}x^{s}G_{\alpha,\rho}(x){\textnormal{d}}x\right].$
(24)
The above integral exists for all $s\in(-1,0)$ due to (23) and the fact that
$G_{\alpha,\rho}(x)$ is analytic in the neighborhood of $x=0$. The main step
of the proof is to make a change of variables $w=G_{\alpha,\rho}(x)$ in the
above integral. From (22) we find
$\frac{{\textnormal{d}}x}{{\textnormal{d}}w}=-w^{-2}-(\alpha-1)e^{\pi{\textnormal{i}}\alpha\rho}w^{\alpha-2}.$
Using the above result and the fact that
$G_{\alpha,\rho}((0,\infty))=\gamma^{+}$, we rewrite the integral in (24) as
follows:
$\displaystyle\int_{(0,\infty)}x^{s}G_{\alpha,\rho}(x){\textnormal{d}}x$
$\displaystyle=\int_{\gamma^{+}}\left(w^{-1}-e^{\pi{\textnormal{i}}\alpha\rho}w^{\alpha-1}\right)^{s}w\left(-w^{-2}-(\alpha-1)e^{\pi{\textnormal{i}}\alpha\rho}w^{\alpha-2}\right){\textnormal{d}}w$
(25)
$\displaystyle=-\int_{\gamma^{+}}\left(1-\left(\frac{w}{z^{*}}\right)^{\alpha}\right)^{s}w^{-s-1}{\textnormal{d}}w-(\alpha-1)e^{\pi{\textnormal{i}}\alpha\rho}\int_{\gamma^{+}}\left(1-\left(\frac{w}{z^{*}}\right)^{\alpha}\right)^{s}w^{-s+\alpha-1}{\textnormal{d}}w.$
Note that both integrals in the right-hand side of (25) are finite for
$s\in(-1,0)$. Let us compute the first integral in the right-hand side of
(25): we make a substitution $w=uz^{*}$ and obtain
$\displaystyle
I_{1}:=\int_{\gamma^{+}}\left(1-\left(\frac{w}{z^{*}}\right)^{\alpha}\right)^{s}w^{-s-1}{\textnormal{d}}w=e^{\pi{\textnormal{i}}\rho
s}\int_{L}(1-u^{\alpha})^{s}u^{-s-1}{\textnormal{d}}u,$
where $L:=\gamma^{+}/z^{*}$ ($L$ is the curve $\gamma^{+}$ rotated by the
angle $\pi\rho$). It is clear that $L$ connects points $0$ and $1$ (in the
direction $1\mapsto 0$) and $L\subset{\mathbb{C}}^{+}$ (except for the
endpoints). The function $w\mapsto f(w)=(1-w^{\alpha})^{s}w^{-s-1}$ extends to
${\mathbb{C}}^{+}$ analytically, and therefore we can deform the contour of
integration $L$ into an interval $[0,1]$, and we finally obtain
$\displaystyle I_{1}$ $\displaystyle=e^{\pi{\textnormal{i}}\rho
s}\int_{1}^{0}(1-u^{\alpha})^{s}u^{-s-1}{\textnormal{d}}u=-\frac{1}{\alpha}e^{\pi{\textnormal{i}}\rho
s}\int_{0}^{1}(1-x)^{s}x^{-\frac{s}{\alpha}-1}{\textnormal{d}}x$
$\displaystyle=-\frac{1}{\alpha}e^{\pi{\textnormal{i}}\rho
s}\frac{\Gamma(s+1)\Gamma(-\frac{s}{\alpha})}{\Gamma(1+s-\frac{s}{\alpha})},$
(26)
where in the second step we made a substitution $u=x^{\frac{1}{\alpha}}$ and
in the third step we have used the well-known integral representation for the
beta function. In exactly the same way we deal with the second integral in the
right-hand side of (25) and find that for $s\in(-1,0)$
$I_{2}:=\int_{\gamma^{+}}\left(1-\left(\frac{w}{z^{*}}\right)^{\alpha}\right)^{s}w^{-s+\alpha-1}{\textnormal{d}}w=-\frac{1}{\alpha}e^{\pi{\textnormal{i}}\rho(s-\alpha)}\frac{\Gamma(s+1)\Gamma(1-\frac{s}{\alpha})}{\Gamma(2+s-\frac{s}{\alpha})}.$
(27)
Combining formulas (24), (25), (2) and (27) we obtain
$\displaystyle M_{\alpha,\rho}(s)$
$\displaystyle=-\frac{1}{\pi}\operatorname{Im}\Bigg{[}\frac{1}{\alpha}e^{\pi{\textnormal{i}}\rho
s}\frac{\Gamma(s+1)\Gamma(-\frac{s}{\alpha})}{\Gamma(1+s-\frac{s}{\alpha})}$
$\displaystyle\qquad\qquad-(\alpha-1)e^{-\pi{\textnormal{i}}(1-\alpha\rho)}\frac{1}{\alpha}e^{\pi{\textnormal{i}}\rho(s-\alpha)}\frac{\Gamma(s+1)\Gamma(1-\frac{s}{\alpha})}{\Gamma(2+s-\frac{s}{\alpha})}\Bigg{]}$
$\displaystyle=-\frac{1}{\pi\alpha}\sin\left(\pi\rho
s\right)\left[\frac{\Gamma(s+1)\Gamma(-\frac{s}{\alpha})}{\Gamma(1+s-\frac{s}{\alpha})}+(\alpha-1)\frac{\Gamma(s+1)\Gamma(1-\frac{s}{\alpha})}{\Gamma(2+s-\frac{s}{\alpha})}\right]$
$\displaystyle=-\frac{1}{\pi\alpha}\sin\left(\pi\rho
s\right)\frac{\Gamma(s)\Gamma(1-\frac{s}{\alpha})}{\Gamma(2+s-\frac{s}{\alpha})}\left[-\alpha(1+s-\tfrac{s}{\alpha})+(\alpha-1)s\right]$
$\displaystyle=\frac{1}{\pi}\sin\left(\pi\rho
s\right)\frac{\Gamma(s)\Gamma(1-\frac{s}{\alpha})}{\Gamma(2+s-\frac{s}{\alpha})}.$
This ends the proof of (9) for $s\in(-1,0)$. The extension of the result for
$s\in(-1,\alpha)$ follows by analytic continuation.
$\sqcap\kern-8.0pt\hbox{$\sqcup$}$
###### Remark 1.
The proof of Theorem 1 is similar in spirit to the proof of Lemma 10 in [4].
One can also give an alternative proof of Theorem 1, based on the following
steps:
1. (i)
The Mellin transform $M_{\alpha,1}(z)$ for $\alpha\in(0,1)$ is known due to
Theorem 3 in [4].
2. (ii)
Assume that $\alpha\in(0,1)$ and $\rho\in[0,1]$. One can prove that
$X_{\alpha,\rho}\stackrel{{\scriptstyle d}}{{=}}X_{\alpha,1}\times K_{\rho},$
(28)
where the random variable $K_{\rho}$ is independent of $X_{\alpha,\rho}$ and
follows the Cauchy distribution
${\mathbb{P}}(K_{\rho}\in{\textnormal{d}}x)=\frac{1}{\pi}\times\frac{\sin(\pi\rho)}{(x+\cos(\pi\rho))^{2}+\sin^{2}(\pi\rho)}{\textnormal{d}}x.$
(29)
The factorization (28) can be established as follows. Let $G_{X}(z)$ denote
the Cauchy transform ${\mathbb{E}}[1/(z-X)]$ of a random variable $X$. First
note the fact $G_{K_{\rho}}(z)=1/(z+e^{\pi{\textnormal{i}}\rho})$ which can be
proved by the residue theorem (this formula and the Cauchy transform inversion
formula give the density (29)). Then we have
$\begin{split}G_{X_{\alpha,1}\times
K_{\rho}}(z)&={\mathbb{E}}\\!\left[\frac{1}{z-X_{\alpha,1}\times
K_{\rho}}\right]={\mathbb{E}}\\!\left[\frac{1}{X_{\alpha,1}}\times\frac{1}{z/X_{\alpha,1}-K_{\rho}}\right]\\\
&={\mathbb{E}}\\!\left[\frac{1}{X_{\alpha,1}}\times\frac{1}{z/X_{\alpha,1}+e^{\pi{\textnormal{i}}\rho}}\right]={\mathbb{E}}\\!\left[\frac{-e^{-\pi{\textnormal{i}}\rho}}{-ze^{-\pi{\textnormal{i}}\rho}-X_{\alpha,1}}\right]\\\
&=-e^{-\pi{\textnormal{i}}\rho}G_{X_{\alpha,1}}(-e^{-\pi{\textnormal{i}}\rho}z),\qquad
z\in{\mathbb{C}}^{+}.\end{split}$ (30)
Recall the fact that for any random variable $X$ and any $a>0$, there exists
$b>0$ such that $G_{X}$ has the right compositional inverse $G_{X}^{-1}$
defined in
$\Delta_{a,b}=\\{z\in{\mathbb{C}}^{-}\mid\operatorname{Im}z\in(-b,0),a|\operatorname{Re}z|\leq-\operatorname{Im}z\\}$
and that
$G_{X}^{-1}(z)=1/z+\phi_{X}(1/z),\qquad z\in\Delta_{a,b};$ (31)
see [2]. From (30) and (31) one has
$\phi_{X_{\alpha,1}\times
K_{\rho}}(z)=-e^{{\textnormal{i}}\pi\rho}\phi_{X_{\alpha,1}}(-e^{-{\textnormal{i}}\pi\rho}z)=\phi_{X_{\alpha,\rho}}(z)$
in a common domain. This shows the factorization (28).
3. (iii)
The Mellin transform of the Cauchy distribution is given by (see the case
$\alpha=1$ of [7, 2.11.11])
${\mathbb{E}}[(K_{\rho})^{s}{\bf 1}_{\\{K_{\rho}>0\\}}]=\frac{\sin\pi\rho
s}{\sin\pi s},\qquad-1<s<1.$
4. (iv)
The explicit expression for $M_{\alpha,\rho}(z)$ for $\alpha\in(0,1)$ and
$\rho\in[0,1]$ follows from the factorization
$X_{\alpha,\rho}\stackrel{{\scriptstyle d}}{{=}}X_{\alpha,1}\times K_{\rho}$.
We then use the duality identity (11) to obtain $M_{\alpha,\rho}(z)$ for
$\alpha\in(1,2]$ and $\rho\in[1-\frac{1}{\alpha},\frac{1}{\alpha}]$.
Proof of Corollary 2: Formula (9) implies that for all
$(\alpha,\rho)\in{\mathcal{A}}$ and $-1<s<\alpha^{-1}$ we have
$M_{\alpha,\rho}(-\alpha s)=\alpha^{-1}M_{\alpha^{-1},\alpha\rho}(s)$, which
is equivalent to (11) and (12). $\sqcap\kern-8.0pt\hbox{$\sqcup$}$
Proof of Corollary 3: Assume that $\alpha\in(0,1)$ and $\rho\in[0,1]$. It is
easy to see that
${\mathbb{E}}\left[Z^{(1-\frac{1}{\alpha})s}\right]=\Gamma\left(2+\left(1-\tfrac{1}{\alpha}\right)s\right).$
Using the above identity and formulas (9) and (10) we see that
${\mathbb{E}}\left[\left(Y_{\alpha,\rho}\right)^{s}{\bf
1}_{\\{Y_{\alpha,\rho}>0\\}}\right]={\mathbb{E}}\left[Z^{(1-\frac{1}{\alpha})s}\right]\times{\mathbb{E}}\left[\left(X_{\alpha,\rho}\right)^{s}{\bf
1}_{\\{X_{\alpha,\rho}>0\\}}\right],$
and the uniqueness of the Mellin transform implies that
${\mathbb{P}}(Y_{\alpha,\rho}\in A)={\mathbb{P}}(X_{\alpha,\rho}\times
Z^{1-\frac{1}{\alpha}}\in A),$ (32)
for all Borel sets $A\subset(0,\infty)$. The fact that (32) holds also for all
Borel sets $A\subset(-\infty,0)$ follows by using the symmetry condition
$-X_{\alpha,\rho}\stackrel{{\scriptstyle d}}{{=}}X_{\alpha,1-\rho}$ and
$-Y_{\alpha,\rho}\stackrel{{\scriptstyle d}}{{=}}Y_{\alpha,1-\rho}$. This ends
the proof of (13). $\sqcap\kern-8.0pt\hbox{$\sqcup$}$
###### Lemma 1.
Assume that $X,Y,Z$ and $W$ are independent random variables, such that (i)
$X>0$ and $Y>0$ a.s., (ii) $X\stackrel{{\scriptstyle d}}{{=}}Y$, (iii)
$X\times Z\stackrel{{\scriptstyle d}}{{=}}Y\times W$ and (iv) there exists
$\epsilon>0$ such that ${\mathbb{E}}[|Z|^{s}]<\infty$ and
${\mathbb{E}}[|W|^{s}]<\infty$ for all $s\in(-\epsilon,\epsilon)$. Then
$Z\stackrel{{\scriptstyle d}}{{=}}W$.
###### Proof.
Recall that we denote by $\hat{\xi}$ the cutoff of a random variable $\xi$.
Assume that ${\mathbb{P}}(Z>0)>0$. Condition (iii) implies that
${\mathbb{P}}(Z>0)={\mathbb{P}}(W>0)$ and
$X\times\hat{Z}\stackrel{{\scriptstyle d}}{{=}}Y\times\hat{W}$. For
$t\in{\mathbb{R}}$ we denote
$f(t)={\mathbb{E}}[X^{{\textnormal{i}}t}]={\mathbb{E}}[Y^{{\textnormal{i}}t}]$
and obtain
$f(t)\times{\mathbb{E}}[\hat{Z}^{{\textnormal{i}}t}]={\mathbb{E}}[(X\hat{Z})^{{\textnormal{i}}t}]={\mathbb{E}}[(Y\hat{W})^{{\textnormal{i}}t}]=f(t)\times{\mathbb{E}}[\hat{W}^{{\textnormal{i}}t}].$
Since $f(t)$ is continuous (as a characteristic function of $\ln(X)$) and
$f(0)=1$ we conclude that for some $\delta$ small enough, the function $f(t)$
is non-zero for $t\in(-\delta,\delta)$. Therefore, we can divide both sides of
the above identity by $f(t)$ and conclude that
${\mathbb{E}}[\hat{Z}^{{\textnormal{i}}t}]={\mathbb{E}}[\hat{W}^{{\textnormal{i}}t}]$
for $t\in(-\delta,\delta)$. Condition (iv) implies that both functions
${\mathbb{E}}[\hat{Z}^{{\textnormal{i}}t}]$ and
${\mathbb{E}}[\hat{W}^{{\textnormal{i}}t}]$ are analytic in the strip
$\operatorname{Im}(t)\in(-\epsilon,\epsilon)$, and since they are equal on the
set $t\in(-\delta,\delta)$, they must be equal for all $t$ in the strip
$\operatorname{Im}(t)\in(-\epsilon,\epsilon)$, which implies
$\hat{Z}\stackrel{{\scriptstyle d}}{{=}}\hat{W}$.
In the case if ${\mathbb{P}}(Z<0)>0$ we can apply the same argument as above
and show that the cutoff of $-Z$ has the same distribution as the cutoff of
$-W$. $\sqcap\kern-8.0pt\hbox{$\sqcup$}$
Proof of Corollary 4 follows at once from Corollary 3, Lemma 1, and the
following result: the identity (14) is true for classical stable random
variables $Y_{\alpha,\rho}$ and their cutoffs $\hat{Y}_{\alpha,\rho}$ (see
formulas (3.2.3) and (3.2.5) in [7]). $\sqcap\kern-8.0pt\hbox{$\sqcup$}$
Proof of Theorem 2: First of all, let us check that the series in (15) and
(16) converge for $x=x^{*}$. We recall Stirling’s asymptotic formula for the
gamma function: For every $\epsilon>0$
$\ln(\Gamma(z))=\left(z-\tfrac{1}{2}\right)\ln(z)-z+\tfrac{1}{2}\ln(2\pi)+O_{\epsilon}(z^{-1}),\;\;\;|\arg(z)|<\pi-\epsilon,\;|z|\to\infty.$
(33)
Using (33) and the reflection formula for the gamma function we check that
$\frac{\Gamma(1+\alpha
n)}{n!\Gamma(2+(\alpha-1)n)}=\frac{1}{\pi}\frac{\Gamma(1+\alpha
n)}{n!}\sin(\pi(\alpha-1)n)\Gamma(-1+(1-\alpha)n)=O\left(n^{-\frac{3}{2}}(x^{*})^{\alpha
n}\right),\;\;\;n\to+\infty,$
which shows that the series in (15) converges for $x=x^{*}$ (therefore, it
converges uniformly on $x\in[x^{*},\infty)$). In the same way we check that
$\frac{\Gamma\left(1+\frac{n}{\alpha}\right)}{n!\Gamma\left(2+\left(\frac{1}{\alpha}-1\right)n\right)}=O\left(n^{-\frac{3}{2}}(x^{*})^{-n}\right),\;\;\;n\to+\infty,$
therefore the series in (16) converges for $x=x^{*}$ (and it converges
uniformly for $x\in[-x^{*},x^{*}]$).
Let us prove identity (15). The function $M_{\alpha,\rho}(s)$ given by (9) has
simple poles at points
$s_{n}=\alpha n,\;n\geq 1\;\;\;\textnormal{ and }\;\;\;\hat{s}_{m}=-m,\;m\geq
1.$
The poles at $s_{n}$ {resp. $\hat{s}_{m}$} come from the factor
$\Gamma(1-s/\alpha)$ {resp. $\Gamma(s)$} in (9). The residues are given by
$\displaystyle{\textnormal{Res}}(M_{\alpha,\rho}(s):s=s_{n})=\frac{1}{\pi}(-1)^{n}\frac{\Gamma(1+\alpha
n)}{n!\Gamma(2+(\alpha-1)n)}\sin(n\alpha\rho\pi),$ (34)
$\displaystyle{\textnormal{Res}}(M_{\alpha,\rho}(s):s=\hat{s}_{m})=\frac{1}{\pi}(-1)^{m-1}\frac{\Gamma\left(1+\frac{m}{\alpha}\right)}{m!\Gamma\left(2+\left(\frac{1}{\alpha}-1\right)m\right)}\sin(m\rho\pi).$
(35)
Using the reflection formula for the Gamma function we check that
$M_{\alpha,\rho}(s)\equiv f_{1}(s)\times f_{2}(s)$, where
$f_{1}(s):=-\frac{1}{\pi}\frac{\sin(\pi\rho
s)\sin\left(\pi\left(\frac{1}{\alpha}-1\right)s\right)}{\sin\left(\frac{\pi
s}{\alpha}\right)},\;\;\;f_{2}(s):=\frac{\Gamma(s)\Gamma\left(-1-s+\frac{s}{\alpha}\right)}{\Gamma\left(\frac{s}{\alpha}\right)}.$
Let us denote $B_{r}(w):=\\{z\in{\mathbb{C}}\mid|z-w|<r\\}$ and
$\delta:=\alpha/4$. The function $f_{1}(s)$ has poles at points $s=n\alpha$,
$n\in{\mathbb{Z}}$, and it satisfies
$f_{1}(s)=O(\exp(-\pi(1-\rho)|\operatorname{Im}(s)|))$ as
$\operatorname{Im}(s)\to\infty$. This implies that for some $C_{1}>0$
$|f_{1}(s)|\leq C_{1},\qquad
s\notin\bigcup_{n=0}^{\infty}B_{\delta}(n\alpha),\quad\operatorname{Re}(s)\geq
0.$ (36)
From Stirling’s formula (33) we find that there exist $C_{2}>0$ and $C_{3}>0$
such that
$|f_{2}(s)|<C_{2}|s|^{-\frac{3}{2}}(x^{*})^{\operatorname{Re}(s)},\;\;\;|s|>C_{3},~{}\operatorname{Re}(s)>0.$
(37)
Formulas (36) and (37) show that we may use the Mellin transform inversion
formula
$\psi_{\alpha,\rho}(x)=\frac{x^{-1}}{2\pi{\textnormal{i}}}\int_{{\textnormal{i}}{\mathbb{R}}}M_{\alpha,\rho}(s)x^{-s}{\textnormal{d}}s.$
Let us define $b_{k}:=\alpha(2k+1)/2$. Shifting the contour of integration
${\textnormal{i}}{\mathbb{R}}\mapsto b_{k}+{\textnormal{i}}{\mathbb{R}}$ we
obtain
$\psi_{\alpha,\rho}(x)=-\sum\limits_{n=1}^{k}{\textnormal{Res}}(M_{\alpha,\rho}(s):s=s_{n})\times
x^{-s_{n}-1}+\frac{x^{-1}}{2\pi{\textnormal{i}}}\int_{b_{k}+{\textnormal{i}}{\mathbb{R}}}M_{\alpha,\rho}(s)x^{-s}{\textnormal{d}}s.$
(38)
Let us denote the integral in the right-hand side of (38) by $I_{k}(x)$.
Changing the variable of integration $s=b_{k}+{\textnormal{i}}u$ we find
$|I_{k}(x)|=\left|{\textnormal{i}}\int_{{\mathbb{R}}}M_{\alpha,\rho}(b_{k}+{\textnormal{i}}u)x^{-b_{k}-{\textnormal{i}}u}{\textnormal{d}}u\right|\leq
x^{-b_{k}}\int_{{\mathbb{R}}}|f_{1}(b_{k}+{\textnormal{i}}u)|\times|f_{2}(b_{k}+{\textnormal{i}}u)|{\textnormal{d}}u.$
(39)
Note that the vertical line $b_{k}+{\textnormal{i}}{\mathbb{R}}$ does not
intersect the collection of circles
$\bigcup_{n=0}^{\infty}B_{\delta}(n\alpha)$, thus the estimate (36) holds true
for $s\in b_{k}+{\textnormal{i}}{\mathbb{R}}$. Estimates (36),(37) and (39)
show that for all $b_{k}>C_{3}$ we have
$|I_{k}(x)|\leq
C_{1}C_{2}\left(\frac{x^{*}}{x}\right)^{b_{k}}\int_{{\mathbb{R}}}(b_{k}^{2}+u^{2})^{-\frac{3}{4}}{\textnormal{d}}u,$
which implies that $I_{k}(x)\to 0$ as $k\to+\infty$ as long as $x\geq x^{*}$.
Combining this statement with (34) and (38) gives us (15).
The proof of (16) can be obtained in exactly the same way, except that now one
would have to shift the contour of integration in the opposite direction,
taking into account the contribution from the simple poles at points
$\hat{s}_{m}$. The details are left to the reader.
$\sqcap\kern-8.0pt\hbox{$\sqcup$}$
###### Lemma 2.
Assume that $(\alpha,\rho)\in{\mathcal{A}}$. For $s>0$ we have
$\int_{0}^{\infty}{\mathfrak{f}}_{\alpha,\rho}(z)z^{s-1}{\textnormal{d}}z=\frac{1}{\alpha}e^{\pi{\textnormal{i}}s\left(\rho-\frac{1}{2}\right)}\frac{\Gamma\left(\frac{s}{\alpha}\right)}{\Gamma\left(2-s+\frac{s}{\alpha}\right)}.$
(40)
###### Proof.
We will use the following result (see Lemma 1 in [6]): Let $s\in(0,1)$ and $X$
be a random variable such that ${\mathbb{E}}[|X|^{-s}]<\infty$. Then
$\displaystyle\int_{0}^{\infty}z^{s-1}{\mathbb{E}}[\cos(zX)]{\textnormal{d}}z$
$\displaystyle=\Gamma(s)\cos\left(\tfrac{\pi
s}{2}\right){\mathbb{E}}[|X|^{-s}],$
$\displaystyle\int_{0}^{\infty}z^{s-1}{\mathbb{E}}[\sin(zX)]{\textnormal{d}}z$
$\displaystyle=\Gamma(s)\sin\left(\tfrac{\pi
s}{2}\right){\mathbb{E}}[|X|^{-s}{\textnormal{sign}}(X)].$
Combining the above two identities we obtain for $s\in(0,1)$
$\displaystyle\int_{0}^{\infty}{\mathfrak{f}}_{\alpha,\rho}(z)z^{s-1}{\textnormal{d}}y=\Gamma(s)\cos\left(\tfrac{\pi
s}{2}\right)\left[M_{\alpha,\rho}(-s)+M_{\alpha,1-\rho}(-s)\right]+{\textnormal{i}}\Gamma(s)\sin\left(\tfrac{\pi
s}{2}\right)\left[M_{\alpha,\rho}(-s)-M_{\alpha,1-\rho}(-s)\right],$
where $M_{\alpha,\rho}(s)$ is defined by (21). The required result (40)
follows from the above identity, formula (9) and the following trigonometric
identity
$\cos\left(\tfrac{\pi s}{2}\right)\left[\sin(\pi\rho
s)+\sin(\pi(1-\rho)s)\right]+{\textnormal{i}}\sin\left(\tfrac{\pi
s}{2}\right)\left[\sin(\pi\rho s)-\sin(\pi(1-\rho)s)\right]=\sin(\pi
s)e^{\pi{\textnormal{i}}s\left(\rho-\frac{1}{2}\right)}.$
Extension of (40) for all $s>0$ is achieved by analytic continuation, since
the right-hand side of (40) is analytic in the half-plane
$\operatorname{Re}(s)>0$. $\sqcap\kern-8.0pt\hbox{$\sqcup$}$
Proof of Theorem 3: The proof follows the Mellin transform inversion
technique, and is very similar to the proof of Theorem 2. We will only sketch
the main steps and will leave the details to the reader. Let us denote the
function in the right-hand side of (40) by $m_{\alpha,\rho}(s)$. Assume that
$\alpha\in(0,1)$ and $\rho\in(0,1)$. Using the reflection formula for the
Gamma function we find that $m_{\alpha,\rho}(s)=f_{1}(s)\times f_{2}(s)$,
where
$f_{1}(s):=e^{\pi{\textnormal{i}}s\left(\rho-\frac{1}{2}\right)}\frac{\sin(\frac{1}{\alpha}-1)\pi
s}{\sin\frac{\pi
s}{\alpha}},\;\;\;f_{2}(s):=\frac{\Gamma(-1-(\frac{1}{\alpha}-1)s)}{\Gamma(1-\frac{s}{\alpha})}.$
Using Stirling’s asymptotic formula (33) we check that the function
$u\in{\mathbb{R}}\mapsto m_{\alpha,\rho}({\textnormal{i}}u)$ decays
exponentially as $u\to\infty$, thus we can express
${\mathfrak{f}}_{\alpha,\rho}(x)$ as the inverse Mellin transform
${\mathfrak{f}}_{\alpha,\rho}(z)=\frac{1}{2\pi{\textnormal{i}}}\int_{1+{\textnormal{i}}{\mathbb{R}}}m_{\alpha,\rho}(s)z^{-s}{\textnormal{d}}s.$
(41)
We define $b_{k}:=\alpha(2k+1)/2$ and shift the contour of integration in (41)
$1+{\textnormal{i}}{\mathbb{R}}\mapsto-b_{k}+{\textnormal{i}}{\mathbb{R}}$.
Taking into account the residues of the integrand at points $s=-\alpha n$
(coming from the factor $\Gamma(s/\alpha)$ in (40)) we obtain
${\mathfrak{f}}_{\alpha,\rho}(z)=\sum\limits_{0\leq n\leq
k}(-1)^{n}\frac{e^{\pi{\textnormal{i}}\alpha(\frac{1}{2}-\rho)n}}{n!\Gamma(2+(\alpha-1)n)}z^{\alpha
n}+\frac{1}{2\pi{\textnormal{i}}}\int_{-b_{k}+{\textnormal{i}}{\mathbb{R}}}m_{\alpha,\rho}(s)x^{-s}{\textnormal{d}}s.$
(42)
It is easy to see that for some $C_{1}>0$ we have for all $k\geq 0$
$|f_{1}(-b_{k}+{\textnormal{i}}u)|<C_{1}\exp\left(-\pi(1-|\rho-1/2|)|u|\right),\;\;\;u\in{\mathbb{R}}.$
Stirling’s asymptotic formula (33) shows that there exist constants
$C_{2}\in{\mathbb{R}}$ and $C_{3}>0$ such that
$\displaystyle\ln(f_{2}(s))=s\ln(-s)+Cs+O(1)$
as $s\to\infty$, uniformly in the half-plane $\operatorname{Re}(s)<-C_{3}$.
The above two estimates can be used to show that the integral in the right-
hand side of (42) converges to zero as $k\to+\infty$, which ends the proof of
(20) for $\alpha\in(0,1)$ and $\rho\in(0,1)$. The extension of the result in
the case $\rho\in\\{0,1\\}$ follows by considering the limit of (20) as
$\rho\to 0^{+}$ or $\rho\to 1^{-}$. The proof in the case $\alpha\in(1,2]$ can
be obtained along the same lines. $\sqcap\kern-8.0pt\hbox{$\sqcup$}$
## Acknowledgement
The first-named author was supported by Marie Curie Actions – International
Incoming Fellowships (Project 328112 ICNCP). The research of the second-named
author was supported by the Natural Sciences and Engineering Research Council
of Canada.
## References
* [1] H. Bercovici, V. Pata, and P. Biane. Stable laws and domains of attraction in free probability theory. Annals of Mathematics, 149(3):1023 – 1060, 1999.
* [2] H. Bercovici and D.-V. Voiculescu. Free convolution of measures with unbounded support. Indiana University Mathematics Journal, 42(3):733 – 773, 1993.
* [3] N. Demni. Kanter random variable and positive free stable distributions. Electronic Communications in Probability, 16:137 – 149, 2011.
* [4] U. Haagerup and S. Möller. The law of large numbers for the free multiplicative convolution. In Operator Algebra and Dynamics, volume 58 of Springer Proceedings in Mathematics & Statistics, pages 157 – 186. Springer, Berlin, Heidelberg, 2013.
* [5] V. Pata. Lévy type characterization of stable laws for free random variables. Transactions of the American Mathematical Society, 347(7):2457 – 2472, 1995.
* [6] C. Profeta and T. Simon. Persistence of integrated stable processes. arXiv:1403.1064, 2014.
* [7] V. M. Zolotarev. One-dimensional stable distributions, volume 65 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1986.
|
arxiv-papers
| 2014-04-11T02:12:51 |
2024-09-04T02:50:00.980697
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Takahiro Hasebe and Alexey Kuznetsov",
"submitter": "Alexey Kuznetsov",
"url": "https://arxiv.org/abs/1404.2981"
}
|
1404.3009
|
# Unconventional pairings of spin-orbit coupled attractive degenerate Fermi
gas in a one-dimensional optical lattice
Junjun Liang1 Xiaofan Zhou1 Pak Hong Chui2 Kuang Zhang1 Shi-jian Gu2 Ming
Gong2,∗ Gang Chen1,∗ Suotang Jia1
###### Abstract
Understanding novel pairings in attractive degenerate Fermi gases is crucial
for exploring rich superfluid physics. In this report, we reveal
unconventional pairings induced by spin-orbit coupling (SOC) in a one-
dimensional optical lattice, using a state-of-the-art density-matrix
renormalization group method. When both bands are partially occupied, we find
a strong competition between the interband Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO) and intraband Bardeen-Cooper-Schrieffer (BCS) pairings. In particular,
for the weak and moderate SOC strengths, these two pairings can coexist,
giving rise to a new phase called the FFLO-BCS phase, which exhibits a unique
three-peak structure in pairing momentum distribution. For the strong SOC
strength, the intraband BCS pairing always dominates in the whole parameter
regime, including the half filling. We figure out the whole phase diagrams as
functions of filling factor, SOC strength, and Zeeman field. Our results are
qualitatively different from recent mean-field predictions. Finally, we
address that our predictions could be observed in a weaker trapped potential.
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute
of Laser spectroscopy, Shanxi University, Taiyuan 030006, P. R. China
Department of Physics and Center for Quantum Coherence, The Chinese University
of Hong Kong, Shatin, N.T., Hong Kong, China
∗Corresponding authors, e-mail: [email protected] or [email protected]
Ultracold atoms have become standard toolboxes for simulating fundamental
physics with strong interactions[1, 2]. Recently, these systems are used to
mimic the spin-orbit coupling (SOC)[3], which is one of the most intriguing
interaction in nature. In particular, the one-dimensional (1D) SOC—the
simplest non-Abelian gauge potential[4]—has been realized experimentally in
fermionic 40K[5, 6, 7] and 6Li[8] atoms, by using a similar scheme achieved in
bosonic 87Rb atom[9]. This remarkable progress opens an immediate possibility
for exploring nontrivial quantum phases of degenerate Fermi gases[10]. Some
intriguing phases, including the topological Bardeen-Cooper-Schrieffer
(BCS)[11, 12, 13, 14, 15, 16, 17, 18] and topological Fulde-Ferrell-Larkin-
Ovchinnikov (FFLO)[19, 20, 21] phases, have been revealed. The defects in
these topological phases are expected to host self-Hermitian Majorana
fermions, which are the major building blocks for achieving topological
quantum computation[22]. The basic picture for realizing these nontrivial
topological superfluids is that SOC, Zeeman field, and $s$-wave interaction
can induce triplet $p$-wave pairing, when the chemical potential just occupies
one single band[23, 24, 25]. In this regard, understanding the true pairing(s)
in the spin-orbit-coupled systems is essential for achieving these novel
phases.
All the previous predictions, both in free space and optical lattice, are
demonstrated in the framework of mean-field theory[11, 13, 17, 12, 14, 15, 16,
18, 19, 20, 21]. In the detailed calculations, the pairing is simply assumed
to take place between two fermions with a total center-of-mass momentum $Q$,
which serves as a parameter to minimize the total free energy. $Q=0$ and
$Q\neq 0$ correspond to the BCS and FFLO pairings, respectively. This
fundamental picture is also widely used even in 1D systems[21, 26, 27]. In
fact, in 1D the effect of quantum fluctuation becomes significant and the
mean-field results are, in principle, unreliable[28]. This means that the true
pairings in this new platform need to be examined more seriously, which is,
however, still lacking. This work is devoted to addressing this fundamental
issue in a 1D spin-orbit coupled optical lattice, using a state-of-the-art
density matrix renormalization group (DMRG) method[29, 30].
Our numerical results demonstrate that the relevant physics in this model is
completely modified by the SOC-induced triplet pairing[23, 24, 25]. (I) When
both bands are partially occupied, the SOC can lead to a strong competition
between the interband FFLO and intraband BCS pairings, due to the induced
momentum-dependent spin polarizations. (II) For the weak and moderate SOC
strengths, these two pairings can coexist, leading to a new phase called the
FFLO-BCS phase. This new phase is characterized by a unique three-peak
structure in pairing momentum distribution. (III) For the strong SOC strength,
the system is dominated by the intraband BCS pairing in the whole parameter
regime, including the half filling. (IV) We figure out the whole phase
diagrams as functions of filling factor, SOC strength, and Zeeman field, in
terms of the properties of pairing correlations in both real and momentum
spaces. All the results predicted are qualitatively different from the recent
mean-field predictions[27]. (V) Finally, we address the effect of the trapped
potential on pairing correlations and local density. We show that our
predictions could be observed in a weaker trapped potential, which is easily
prepared in experiments.
## Results
### 0.1 Model and Hamiltonian.
We consider the following 1D Fermi-Hubbard model with a synthetic SOC[26, 27]:
$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle-t\sum_{l,s=\uparrow,\downarrow}(c_{ls}^{\dagger}c_{l+1s}+\text{H.c.})+h\sum_{l}(n_{l\uparrow}-n_{l\downarrow})+U\sum_{l}(n_{l\uparrow}-{\frac{1}{2}})(n_{l\downarrow}-{\frac{1}{2}})$
(1)
$\displaystyle+\lambda\sum_{l}(c_{l\uparrow}^{\dagger}c_{l+1\downarrow}-c_{l\downarrow}^{\dagger}c_{l+1\uparrow}+\text{H.c.}),$
where $c_{ls}^{\dagger}$ and $c_{ls}$ are the creation and annihilation
operators, with spin $s=\uparrow,\downarrow$ (encoded by the hyperfine
states), at lattice site $l$, $n_{ls}=c_{ls}^{\dagger}c_{ls}$ is the number
operator, $t$ is the spin-independent hopping, $h$ is the Zeeman field along
$z$ direction, $U$ is the on-site attractive interaction, $\lambda$ is the SOC
strength, and H.c. denotes the Hermitian conjugate.
Recently, the spin-orbit coupled Bose-Einstein condensate in a 1D optical
lattice has been realized experimentally[31]. Using a similar technique, the
Hamiltonian (1) could also be achieved in 1D degenerate Fermi gases[32, 33].
Moreover, the corresponding parameters can be tuned widely. For example, the
3D optical lattice can be prepared by the interference of three pairs of
counter-propagating laser beams[34]. The corresponding periodic potential is
$V_{\text{lattice}}=V_{0}\cos^{2}(k_{w}x)+V_{0}\cos^{2}(k_{w}y)+V_{0}\cos^{2}(k_{w}z)$,
where $V_{0}$ is the lattice depth, $k_{w}=\lambda_{w}/2\pi$ is the wave
vector, and $\lambda_{w}$ is wavelength. By further using a large harmonic
transverse confinement $V_{\text{2D}}=m\omega_{\bot}^{2}r^{2}/2$ in 3D optical
lattice, i.e., the 2D harmonic potential frequency $\omega_{\bot}$ is far
larger than the trapped frequency $\omega_{z}$ along the weakly-confining
axis, the required 1D optical lattice can be generated[32, 33]. In such case,
the 1D effective interaction is described by[35, 36]
$U(z)=-\frac{2\hbar^{2}}{m_{0}a_{1\text{D}}}\delta(z),$ (2)
with the 1D $s$-wave scattering length
$a_{1\text{D}}=-\frac{a_{\bot}^{2}}{2a_{3\text{D}}}(1-C\frac{a_{3\text{D}}}{a_{\bot}}),$
(3)
where $C\simeq 1.46$, $a_{\bot}=(2\hbar/m_{0}\omega_{\bot})^{1/2}$,
$a_{3\text{D}}$ is the 3D $s$-wave scattering length, and $m_{0}$ is the
atomic mass. Equations (2) and (3) show that the 1D on-site attractive
interaction can be tuned by Feshbach resonance[37]. In addition, for 40K[5] or
6Li[8] systems, two spin states are chosen respectively as
$\left|\uparrow\right\rangle=\left|9/2,-9/2\right\rangle$ and
$\left|\downarrow\right\rangle=\left|9/2,-7/2\right\rangle$, or
$\left|\uparrow\right\rangle=\left|3/2,-3/2\right\rangle$ and
$\left|\downarrow\right\rangle=\left|3/2,-1/2\right\rangle$. By using a pair
of counter-propagating Raman lasers, the 1D SOC in the Hamiltonian (1) can
also be realized[5, 6, 7, 8, 9]. Moreover, the SOC strength can be tuned
through a fast and coherent modulation of the Raman beams[38]; see also recent
experiment[39]. For a typical optical lattice, the strong SOC strength,
$\lambda\sim t$, can be achievable[40].
Since the effect of quantum fluctuation in 1D becomes significant, here we
perform a state-of-the-art DMRG method to discuss the Hamiltonian (1). Notice
that the similar Hamiltonian has been discussed by means of the same
method[41]. In their work, they focused on the effect of the many-body
interaction on topological phase and related Majorana fermions. In their
calculation, the pairing term is preassigned to be the BCS pairing, i.e., no
FFLO pairing can be driven from the many-body interaction. Here we mainly
explore fundamental pairings induced by SOC, including the FFLO and BCS
pairings.
In the following calculations, the basic energy scale is chosen as $t=1$, the
on-site interaction is set to $U/t=-4$, and the lattice lengths are chosen as
$L=60$ and $100$. In addition, the open boundary condition is taken into
account and $20$ sweeps are always used. In Fig. 1(a), we plot the scaled
ground-state energy $E_{g}/(Lt)$ as a function of the number of states kept.
It can be seen that the scaled ground-state energy $E_{g}/(Lt)$ tends to a
stable value, when increasing the number of states kept. This indicates that
the number of states kept can be chosen as $150$ per DMRG block. In Fig. 1(b),
we show the truncation error as a function of the number of states kept. When
the number of states kept is chosen as $150$, the truncation error is smaller
than $10^{-5}$, which is sufficient for the numerically-reliable results.
### 0.2 Basic physical picture for unconventional pairings.
Before proceeding, we first illustrate the basic physical picture of the
Hamiltonian (1) with or without SOC. In the absence of SOC ($\lambda/t=0$),
the Hamiltonian $\mathcal{H}$ reduces to the well-studied Fermi-Hubbard
model[42, 43], in which the Zeeman field breaks the degeneracy of each band
[the breaking of symmetry from $SO(4)$[44] at the half filling and otherwise
$SU(2)$ to $U(1)\otimes U(1)$]; however, in both bands spin are still fully
polarized along $z$ direction. Consequently, the pairing can only be formed
between two fermions at different bands, due to the interspecies interaction.
This pairing gives rise to the well-known FFLO phase[45, 46], which can be
observed at any nonzero population imbalance and has two nonzero center-of-
mass momenta; see Fig. 2(a). However, this picture is completely modified by
SOC, since it leads to momentum-dependent spin polarizations
$\mathbf{S}_{\pm}=\pm\frac{1}{\sqrt{4\lambda^{2}\sin(k)^{2}+h^{2}}}(2\lambda\sin(k),0,h),$
(4)
where $\pm$ denote the upper ($+$) and lower ($-$) bands. Since no spin
polarizations can be found in the $y$ component, the corresponding spin-
polarized angles can be defined, using only one variable, as
$\tan(\theta_{\pm})=|\frac{2\lambda k_{\text{F}\pm}}{h}|.$ (5)
When both bands are partially occupied, the physics is extremely interesting.
In this case, the SOC-induced triplet pairing can lead to the intraband BCS
pairing, which can compete with the original interband FFLO pairing. This
competition depends crucially on the spin-polarized angles $\theta_{\pm}$ at
the Fermi points [in the mean-field level, the effective pairing is $p$-wave
type with pairing strength
$\Delta_{\text{eff}}(k)\simeq\Delta\sin(\theta_{\pm})k$[47]], as shown below.
For a weaker SOC strength, i.e., $|2\lambda k_{\text{F}\pm}|\ll|h|$, the spin
is still fully polarized along $z$ direction, and hence the spin-polarized
angles $\theta_{\pm}\sim 0$. It indicates that the interband FFLO pairing
dominates. When increasing the SOC strength, the spin will gradually polarize
towards the $x$ direction. For the weak and moderate SOC strengths, the spin-
polarized angles $\theta_{\pm}$ are typically of the order of $\pi/10$ (see
Supplementary Information). In such case, both the intraband BCS and interband
FFLO pairings are allowed and govern simultaneously the true pairings of the
Hamiltonian (1); see Fig. 2(b). For the strong SOC strength, the spin is
almost polarized along $x$ direction ($\theta_{\pm}\sim\pi/2$), and the
intraband BCS pairing thus dominates; see Fig. 2(c). Therefore, we can expect
a crossover from the interband FFLO pairing to the intraband BCS pairing, when
the spin-polarized angles $\theta_{\pm}$ exceeds the critical values. We find
that this transition is nontrivial because these two pairings can coexist in
some parameter regimes; see below. Hereafter, all these pairings, which arise
from strong competition between the two pairing channels induced by SOC and
the Zeeman field, are called unconventional pairings. These unconventional
pairings can lead to rich superfluid phases, which can be captured by
considering pairing correlations in both real and momentum spaces.
### 0.3 Pairing correlation in real space.
The pairing correlation function in real space is defined as[48, 49, 50, 51]
$P(l,j)=\langle
c_{l\downarrow}^{\dagger}c_{l\uparrow}^{\dagger}c_{j\uparrow}c_{j\downarrow}\rangle.$
(6)
Without SOC, this pairing function can be used to identify the BCS and FFLO
pairings. Physically, for the FFLO pairing,
$P(l,j)\sim\exp\left[iQ(l-j)\right]$, which oscillates in real space and
exhibits two peaks at $k=\pm Q$ in momentum space[48, 49, 50, 51]; see also
discussions below. For the BCS pairing, $Q=0$, and no oscillation can thus be
found in real space. In the presence of SOC, we also use this function as an
important tool (but not a unique tool) to identify the unconventional pairings
of the Hamiltonian (1).
In Fig. 3, we plot the pairing correlation functions $P(l,j)$ and the local
densities $n(l)$ for the different SOC strengths. Without SOC ($\lambda/t=0$),
the pairing correlation in real space exhibits strong oscillations in both
magnitude and sign; see Fig. 3(a). Moreover, the local spin polarization also
exhibits a similar oscillating behavior. This indicates the emergence of a
FFLO phase[48, 49, 50, 51]. For the moderate SOC strength [see, for example,
$\lambda/t=0.16$ and $\lambda/t=0.2$ in Figs. 3(b) and 3(c)], the intraband
BCS pairing increases, and has a strong competition with the interband FFLO
pairing. In this case, the pairing correlations also have similar oscillating
behaviors. For the strong SOC strength [see, for example, $\lambda/t=0.4$ in
Fig. 3(d)], the pairing correlation exhibits a power decay with respect to
$|l-j|$ without node, and no obvious oscillation of spin polarization can be
identified (the oscillation of spin polarization near the two ends is
attributed to the finite-size effect). This means that a BCS phase emerges, as
expected. In Fig. 4, we plot the off-diagonal pairing correlation functions
$P(l,L-l)$ for the different SOC strengths. This figure also shows clearly the
oscillations of the pairing correlation in real space, when the SOC strength
is not very strong. This oscillation is gradually suppressed by increasing the
SOC strength.
### 0.4 Pairing momentum distribution.
Although the pairing correlation functions at the weak and moderate SOC
strengths exhibit the similar behaviors as those in the FFLO phase, their
corresponding pairing momentum distributions $P(k)$ have quite different
behaviors. The pairing momentum distribution—the Fourier transformation of
$P(l,j)$—is given by
$P(k)=\frac{1}{2L}\sum_{l,j}P(l,j)e^{ik(l-j)}.$ (7)
Without SOC ($\lambda/t=0$), the polarization angles $\theta_{\pm}=0$, and two
nonzero center-of-mass momenta $\pm Q$($\neq 0$) can be found explicitly; see
Figs. 5(a)-5(b). This is a direct consequence of inversion symmetry in our
model, $P(k)=P(-k)$ is thus expected. The corresponding phase is referred as
the FFLO phase[48, 49, 50, 51]. When the SOC strength $\lambda/t=0.16$, the
polarization angles $\theta_{\pm}\simeq\pi/16$ (see Supplementary
Information). In such case, the dip at zero momentum of the pairing momentum
distributions $P(k)$ turns to a peak, while the other two peaks at $\pm Q$
change slightly (the detailed discussions are shown below). This indicates
that the pairing momentum distribution $P(k)$ has a unique three-peak
structure, which demonstrates clearly that the intraband BCS and interband
FFLO pairings can coexist. This result goes beyond the recent mean-field
prediction[27]. We call the corresponding phase the _FFLO-BCS phase_. When the
SOC strength $\lambda/t=0.20$, the polarization angles
$\theta_{\pm}\simeq\pi/13$, and the pairing mechanism is still similar to that
of $\lambda/t=0.16$. However, the peak of zero momentum is higher than that of
nonzero momenta, which implies that the intraband BCS pairing is stronger than
the interband FFLO pairing. For the strong SOC strength (see, for example,
$\lambda/t=0.4$), the polarization angles $\theta_{\pm}>\pi/10$, and the
intraband BCS pairing dominates. The corresponding phase is referred as the
BCS phase, in which the pairing momentum distribution $P(k)$ only has a peak
at zero momentum. We need to emphasize that SOC affects significantly the
pairing momentum distribution $P(k)$ at the small momentum regime. For the
large momentum regime, the system’s properties are determined mainly by the
short-range interaction, and the pairing momentum distribution is thus
unaffected by SOC; see also Figs. 5(a) and 5(c).
Since the pairing momentum distribution $P(k)$ can be measured by the time-of-
flight imaging[52, 53], the predicted three phases can be observed directly in
experiments. The corresponding boundary between the FFLO and FFLO-BCS phases
can be determined by
${\frac{d^{2}P(k)}{dk^{2}}}|_{k=0}=0,$ (8)
whereas the boundary between the FFLO-BCS and BCS phases can be determined by
${\frac{dP(k)}{dk}}|_{k=Q}=0.$ (9)
We now explain why the intraband BCS and interband FFLO pairings can coexist
in the FFLO-BCS phase. In the presence of SOC, there are two bands (see Fig.
2), which contain spin-up and spin-down fermions. In the lower band, there are
lots of fermions, which, however, become less in the upper band. Due to the
existence of different spin-component fermions in the same band, the pairings
can, in principle, be formed in the same or different bands, i.e., both the
intraband BCS and interband FFLO pairings are allowed. For the small spin-
polarized angles, the interband FFLO pairing is favored, while for the
relative large spin-polarized angles, the intraband BCS pairing is favored.
More importantly, the corresponding ground-state energies for both the
intraband BCS and interband FFLO pairings are degenerate in the FFLO-BCS phase
(see Fig. 6), which confirms the coexistence of these pairings.
In Figs. 7(a) and 7(c), we plot the center-of-mass momentum $Q$ as a function
of the SOC strength $\lambda/t$. Numerically, $Q$ is determined by
$dP(k)/dk=0$ and $d^{2}P(k)/dk^{2}<0$. We find that $Q$ is a non-monotonic
function of the SOC strength $\lambda/t$. Here we develop a simple model to
understand the relevant behavior. We assume that the Fermi points for two
bands have momenta $\pm k_{1}$ and $\pm k_{2}$, respectively. These values are
governed by the following equations:
$n=\frac{(k_{1}+k_{2})}{\pi},\quad
nm={\frac{1}{\pi}}\int_{k_{1}}^{k_{2}}\frac{h}{\sqrt{4k^{2}\lambda^{2}+h^{2}}}dk,$
(10)
where $m=(N_{\uparrow}-N_{\downarrow})/N$ is the experimentally-measurable
population imbalance[54, 55]. The center-of-mass momentum is determined by
$Q=|k_{1}-k_{2}|$.
For simplicity, we adopt the simplified model in free space, with which the
analytical expression can be obtained perturbatively. We do not observe
quantitatively modification of our conclusion by replacing $k$ with $\sin(k)$
for a lattice model. For the weak SOC strength, we employ the Taylor expansion
of Eq. (10) (up to the leading term) to obtain
$nm\pi=Q\left[1+{\frac{2(k_{1}^{3}-k_{2}^{3})}{3Qh^{2}}}\lambda^{2}+{\frac{6(k_{2}^{5}-k_{1}^{5})}{5h^{4}Q}}\lambda^{4}\right],$
(11)
where $k_{1}=(n\pi-Q)/2$ and $k_{2}=(n\pi+Q)/2$. We assume the solution of $Q$
has the following term
$Q=nm\pi\left[1+\mathcal{A}_{2}\lambda^{2}-\mathcal{A}_{4}\lambda^{4}+\mathcal{O}(\lambda^{6})\right].$
(12)
If letting the coefficient of $\lambda^{2}$ and $\lambda^{4}$ to be zero by
the Taylor expansion of Eq. (11), we can immediately find
$\mathcal{A}_{2}=\frac{(3+m^{2})n^{2}\pi^{2}}{6h^{2}}>0,$ (13)
$\mathcal{A}_{4}=\frac{(15+50m^{2}-m^{4})n^{4}\pi^{4}}{120h^{4}}>0.$ (14)
We find that Eqs. (12)–(14) can well describe the evolution of the center-of-
mass momentum $Q$ in the presence of a weak SOC; see Figs. 7(a) and 7(c).
Moreover, without SOC ($\lambda/t=0$), Eq. (12) reduces to the well-known
result[48, 49, 50, 51]: $Q=nm\pi$.
From Eq. (12), we also see that without SOC, any nonzero population imbalance
can give rise to the FFLO phase[48, 49, 50, 51]. However, this basic
conclusion is completely modified by SOC. In Figs. 7(b) and 7(d), we plot the
center-of-mass momentum $Q$ as a function of the population imbalance $m$,
when the SOC strength $\lambda/t=0.06$. We find that a finite population
imbalance is required to realize the interband FFLO pairing in our model.
Moreover, in both the FFLO-BCS and FFLO phases, $Q=nm(\lambda)\pi$, where
$m(\lambda)$ is obtained from the state-of-the-art DMRG calculations. In Fig.
8, we plot the relationship between the critical population imbalance $m_{c}$
for the different phases and the SOC strength. Obviously,
$m_{c}(\lambda=0)=0$, as expected.
In Figs. 9(a) and 9(c), we plot the population imbalances $m$ as functions of
the Zeeman field for the different SOC strengths, when $L=60$ and $L=100$. In
the absence of SOC ($\lambda/t=0$), the population imbalances $m$ exhibit step
behaviors for the finite-size lattice strengths. The corresponding step gap is
given by $\Delta=2/(Ln)$. When the lattice strength increases, this step gap
becomes small and especially $\Delta\rightarrow 0$ for $L\rightarrow\infty$.
In the presence of SOC ($\lambda/t\neq 0$), the finite-size step behaviors
still exist but become smoother, since SOC can make fermions hop between the
nearest-neighbor sites with spin flipping and thus has a strong effect on the
population imbalances $m$. Similarly, the finite-size step behaviors also
exist when the population imbalances $m$ vary as the SOC strength; see the
insets of Figs. 9(b) and 9(d). In terms of Eq. (12), we find straightforwardly
that the finite-size step behaviors with respect to the SOC strength can lead
to the similar behaviors of the center-of-mass momentum $Q$; see Figs. 9(b)
and 9(d). Apart from the finite-size effect, the step behaviors of the center-
of-mass momentum $Q$ depend strongly on the Zeeman field and the SOC strength.
For some parameter regimes, we find numerically that the corresponding steps
become unobvious; see, for example, the red dash-dot line for $h/t=1.5$ and
$L=60$ in Figs. 9(b).
It should be pointed out that the boundary condition may influence the spin
polarizations at the two ends; however, it does not affect our main
predictions about spin polarizations in both real and momentum spaces, as
demonstrated in Figs. 4 and 5 with $L=60$ and $L=100$. We also do not observe
phase separation in the open boundary condition. So we can exclude the
possibility of three peaks in the FFLO-BCS phase from the phase-separation
effect. In Fig. 10, we plot the critical SOC strengths $\lambda_{c}$, which
govern the phase boundaries, as functions of the lattice length, when the
Zeeman field $h/t=1.5$ (the dash line of Fig. 12). In terms of this finite-
size-scaling analysis, we find that when increasing the lattice length, our
predicted FFLO-BCS phase, with a unique three-peak structure, still exists,
although the center-of-mass momentum $Q$ and the phase boundaries change
slightly.
### 0.5 Phase diagram.
Having identified three superfluid phases, including the FFLO-BCS, FFLO, and
BCS phases, we now figure out the corresponding phase diagram as a function of
the filling factor, the Zeeman field, and the SOC strength. Numerically, the
FFLO-BCS, FFLO, and BCS phases are characterized by three, two, and one
peak(s) in the pairing momentum distribution $P(k)$, respectively. In
addition, when the fermions are fully polarized, i.e., $m=1$, no pairing can
occur. The corresponding phase is referred as the fully-polarized (FP)
phase[32]. The boundary between the FFLO and FFLO-BCS phases can be determined
by Eq. (8), whereas the boundary between the FFLO-BCS and BCS phases can be
determined by Eq. (9).
In Fig. 11, we plot the phase diagram in the $n-h$ plane. In the absence of
SOC ($\lambda/t=0$), the FP, BCS, and FFLO phases can be found[42]; see Fig.
11(a). For the weak SOC strength (see, for example, $\lambda/t=0.05$), the
FFLO-BCS phase can be found, and the FFLO phase is suppressed; see Fig. 11(b).
When the SOC strength $\lambda/t=0.1$, the FFLO phase vanishes, and the FFLO-
BCS phase is enhanced; see Fig. 11(c). For the strong SOC strength (see, for
example, $\lambda/t=0.4$), the FFLO-BCS phase almost disappears, and the BCS
phase dominates in the whole parameter regime; see Fig. 11(d). These results
demonstrate that for the weak SOC strength, a large regime for the FFLO phase
can always be observed. However, for the strong SOC strength, the interband
FFLO pairing are completely suppressed and the intraband BCS pairing always
dominates in the whole parameter regime. This result is in contrast to that
from mean-field prediction[27], in which the FFLO phase always exists even for
a stronger SOC strength ($\lambda/t>1$).
In addition, all the phase diagrams in Fig. 11 are symmetric about the half
filling ($n=1$), which can be understood from the following particle-hole
transformation:
$c_{i\uparrow}\rightarrow-(-1)^{i}d_{i\downarrow}^{\dagger},\quad
c_{i\downarrow}\rightarrow(-1)^{i}d_{i\uparrow}^{\dagger}.$ (15)
Under the transformation (15), we find (see Methods section)
$\mathcal{H}(t,\mu,h,U,\lambda)\rightarrow\mathcal{H}(t,-\mu,h,U,\lambda),$
(16)
Here we have introduced a chemical potential $\mu$ to the original Hamiltonian
(1), which equals exactly to zero at the half filling. Equation (16)
demonstrates that the Hamiltonian (1) has the particle-hole symmetry. This
symmetry ensures that the relevant physics in the low filling factor regime
($n<1$) is identical to that in the high filling factor regime ($n>1$), i.e.,
we have the observation in Fig. 11.
Figure 12 shows the phase diagram in the $h-\lambda$ plane at the half filling
($n=1$), which further confirms that the interband FFLO pairing can be
suppressed by the intraband BCS pairing. However, the situation for the FFLO-
BCS phase is quite different. Since this phase requires not only an
appropriate spin polarization but also a finite energy difference between
$\varepsilon_{\text{F}\pm}$ (see Supplementary Information), we see that it is
more likely to be observed at a finite SOC strength and a stronger Zeeman
field. Obviously, without the Zeeman field ($h/t=0$), the spin is fully
polarized along $x$ direction ($\theta_{\pm}=\pi/2$), and only BCS phase can
be observed. In the presence of SOC, a stronger Zeeman field is thus required
to bring the polarization along $z$ direction (smaller than the critical
polarization angle), so as to favor the interband FFLO pairing. We choose the
results of $h/t=1.5$ as an example to illustrate this point. In the absence of
SOC ($\lambda/t=0$), the polarization angles $\theta_{\pm}=0$, and we can only
observe the FFLO phase. When $\lambda/t<0.07$ ($\theta_{\pm}\simeq\pi/40$),
this interband FFLO pairing always dominates. However, when
$0.07<\lambda/t<0.21$ ($\theta_{\pm}\simeq\pi/13$), we find the FFLO-BCS
phase. Finally, when $\lambda/t>$ $0.21$, the intraband BCS pairing dominates.
Strikingly, we find that these critical angles are generally of the order of
$\pi/10$, thus it is very easily to drive the FFLO phase to the BCS phase by a
weak SOC.
## Discussion
In real experiments, a harmonic trapped potential usually exists, and the
Hamiltonian (1) should be added an extra term
$\mathcal{H}_{\text{trap}}=V\left(\frac{2}{L-1}\right)^{2}\sum\limits_{l}\left(l-\frac{L+1}{2}\right)^{2}n_{l},$
(17)
where $V$ is the trapped frequency. In Fig. 13, we plot the pairing
correlation functions $P(l,j)$, the local densities $n(l)$, and the pairing
momentum distributions $P(k)$ for the different trapped frequencies, when
$h/t=1.5$, $\lambda/t=0.16$, and $L=100$. The results for the other lattice
length (such as $L=60$) are similar and thus not plotted here.
Without the trapped potential, the system is located at the FFLO-BCS phase, in
which the pairing correlation function $P(l,j)$ has an oscillating behavior
and the pairing momentum distribution $P(k)$ has a unique three-peak
structure; see Figs. 5 and 7. For a weaker trapped frequency (see, for
example, $V/t=0.2$), the oscillation of the pairing correlation function
$P(l,j)$ and especially three peaks of the pairing momentum distribution
$P(k)$ still exist; see Fig. 13(a). In addition, the corresponding density
profile is almost the same as that without trapped potential; see Fig. 3(f).
It means that no obvious phase separation in real space occurs. Thus, the
predicted phase diagrams in Figs. 11 and 12, including the FFLO-BCS phase,
also remain, although the corresponding phase boundaries change slightly.
However, due to the existence of the trapped potential, the particle-hole
symmetry of the inhomogeneous Hamiltonian
$\mathcal{H}+\mathcal{H}_{\text{trap}}$ is broken, and the phase diagrams in
Fig. 11 are not symmetric about the half filling ($n=1$). When increasing the
trapped frequency (see, for example, $V/t=2.0$ and $6.0$), the phase
separation in real space occurs, since in this case the number of the fermions
in the different sites is not same[48, 50, 51, 54, 55, 56, 57]. When the
trapped frequency $V/t=2.0$, the pairing correlation function $P(l,j)$
exhibits an oscillation in $5<l<55$, and the pairing momentum distribution
$P(k)$ has three peaks. However, the local density $n(l)$ shows that the sites
are fully polarized in two sides. It means that the FFLO-BCS and FP phases are
mixed, and the system is thus located at the FFLO-BCS phase core with the FP
phase wings; see Fig. 13(b). When the trapped frequency $V/t=6.0$, the
oscillation regime of the pairing correlation function $P(l,j)$ turns into
$10<l<20$ and $40<l<50$, and moreover, the pairing momentum distribution
$P(k)$ becomes smoother, i.e., no obvious peaks can be found. In addition, the
local density $n(l)$ shows the emergence of five phases, including the vacuum,
FP, partly-polarized, metal, and band insulator phases (from left to center of
the lattice); see Fig. 13(c). In the metal phase, all spin-down fermions can
move freely in a uniform background of the spin-up fermions, and the band
insulator is fully occupied by the spin-up and spin-down fermions[58]. Due to
the phase separation in real space, the phase boundaries and the phase
diagrams are hardly to be determined[48]. For a large trapped frequency (see,
for example, $V/t=40.0$), the physics is quite different, since in such case
the term $\mathcal{H}_{\text{trap}}$ dominates in the inhomogeneous
Hamiltonian. As a consequence, all fermions are forced to the center of the
trap and there is only the band insulator without any moving fermions; see
Fig. 13(d). From above discussions, it can be seen that our predictions could
be observed for a weaker trapped potential, which is easily prepared in
experiments.
In summary, we have shown, using the state-of-the-art DMRG calculations, that
the true pairings in a 1D optical lattice can be completely modified by SOC,
due to the induced triplet pairing. Especially, this system admits an exotic
coexistence of the interband FFLO and intraband BCS pairings for the weak and
moderate SOC strengths. However, for the strong SOC strength, the intraband
BCS pairing always dominates, and the relevant physics is thus the BCS
superfluid in the whole parameter regime. This yields a new picture to
understand the true pairings in 1D spin-orbit coupled degenerate Fermi gases.
The last conclusion (III) should be useful for searching the topological
superfluids in this model. Finally, we have addressed the effect of the
trapped potential on the pairing correlations and the local density. We have
shown that our predictions could be observed in a weaker trapped potential,
which is easily prepared in experiments.
## Methods
By means of the transformation (15), we find that the kinetic energy
$\sum_{is}c_{is}^{\dagger}c_{is}\rightarrow-\sum_{is}d_{is}d_{i+1s}^{\dagger}=\sum_{i}d_{is}^{\dagger}d_{is}$,
the chemical potential and Zeeman field
$\mu(n_{i\uparrow}+n_{i\downarrow})+h(n_{i\uparrow}-n_{i\downarrow})\rightarrow
2\mu-\mu(d_{i\downarrow}^{\dagger}d_{i\downarrow}+d_{i\uparrow}^{\dagger}d_{i\uparrow})-h(d_{i\uparrow}^{\dagger}d_{i\uparrow}-d_{i\downarrow}^{\dagger}d_{i\downarrow})$,
the on-site attractive interaction
$(n_{i\uparrow}-{\frac{1}{2}})(n_{i\downarrow}-{\frac{1}{2}})\rightarrow(1-d_{i\uparrow}^{\dagger}d_{i\uparrow}-{\frac{1}{2}})(1-d_{i\downarrow}^{\dagger}d_{i\downarrow}-{\frac{1}{2}})=(d_{i\uparrow}^{\dagger}d_{i\uparrow}-{\frac{1}{2}})(d_{i\downarrow}^{\dagger}d_{i\downarrow}-{\frac{1}{2}})$,
and the SOC term
$c_{l\uparrow}^{\dagger}c_{l+1\downarrow}-c_{l\downarrow}^{\dagger}c_{l+1\uparrow}+c_{l+1\downarrow}^{\dagger}c_{l\uparrow}-c_{l+1\uparrow}^{\dagger}c_{l\downarrow}\rightarrow-(-1)^{2l+1}(d_{l\downarrow}d_{l+1\uparrow}^{\dagger}-d_{l\uparrow}d_{l+1\downarrow}^{\dagger}+d_{l+1\uparrow}d_{l\downarrow}^{\dagger}-d_{l+1\downarrow}d_{l\uparrow}^{\dagger})=d_{l\uparrow}^{\dagger}d_{l+1\downarrow}-d_{l\downarrow}^{\dagger}d_{l+1\uparrow}+d_{l+1\downarrow}^{\dagger}d_{l\uparrow}-d_{l+1\uparrow}^{\dagger}d_{l\downarrow}$.
As a result, we derive
$\mathcal{H}(t,\mu,h,U,\lambda)\rightarrow\mathcal{H}(t,-\mu,h,U,\lambda)$,
i.e., Eq. (16).
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## Acknowledgments
We acknowledge Profs. Wei Yi, An-chun Ji, and Qing Sun, and Dr. Chunlei Qu for
their valuable discussions. This work is supported partly by the 973 program
under Grant No. 2012CB921603; the NNSFC under Grants No. 11422433 and No.
61275211; the NCET under Grant No. 13-0882; the FANEDD under Grant No. 201316;
the OIT under Grant No. 2013804; OYTPSP; and SSCC. M.G. is supported by Hong
Kong RGC/GRF Projects (No. 401011 and No. 2130352), University Research Grant
(No. 4053072) and The Chinese University of Hong Kong (CUHK) Focused
Investments Scheme.
## Author Contributions
S.G., M.G., G.C., and S.J. conceived the idea, J.L. and X.Z. performed the
numerical calculations, C.P.H. and K.Z. performed the theoretical
calculations, M.G. and G.C. wrote the manuscript, M.G., G.C. and S.J.
supervised the whole research project. J.L. and X.Z. contributed equally to
this work.
## Additional information
Supplementary information accompanies this paper at
http://www.nature.com/scientificreports
Competing financial interests: The authors declare no competing financial
interests.
Figure 1: (a) The scaled ground-state energy $E_{g}/(Lt)$ and (b) the
truncation error as functions of the number of states kept (SK). In all
subfigures, $n=1$, $\lambda/t=0.16$, and $h/t=1.5$.
Figure 2: A schematic picture for illustrating unconventional pairings in a 1D
optical lattice. (a) Without SOC, (b) the weak and moderate SOC strengths, and
(c) the strong SOC strength. When both bands are partially occupied, there are
two Fermi surfaces, denoted by $\varepsilon_{\text{F}\pm}$, which give rise to
four Fermi points $\pm k_{1}$ and $\pm k_{2}$ (see Supplementary Information).
The corresponding spin-polarized angles at the two Fermi surfaces are defined
as $\theta_{\pm}$, which are determined by the SOC strength and the Zeeman
field; see Eq. (5). These polarizations are essential for describing the true
pairings of the Hamiltonian (1). In (a), the spin is fully polarized along $z$
direction ($\theta_{\pm}=0$), and only the interband FFLO pairing is thus
formed. For the weak and moderate SOC strengths, $\theta_{\pm}$ are typically
of the order of $\pi/10$ (see Supplementary Information). In such case, both
the interband FFLO and intraband BCS pairings are allowed; see (b). More
importantly, these two pairings can coexist, leading to a new phase called the
FFLO-BCS phase. This new phase is characterized by a unique three-peak
structure in pairing momentum distribution. For the strong SOC strength, the
spin is almost polarized along $x$ direction ($\theta_{\pm}\sim\pi/2$), and
the intraband BCS pairing thus dominates; see (c).
Figure 3: The pairing correlation functions $P(l,j)$ and local densities
$n(l)$ for the different SOC strengths. Left two columns for $L=60$ and right
two columns for $L=100$. In the local density, the solid line marks the local
spin difference (diff.), which is defined as $s_{z}=\langle
n_{l\uparrow}-n_{l\downarrow}\rangle$. In all subfigures, $n=1$ and $h/t=1.5$.
Figure 4: The off-diagonal pairing correlation functions $P(l,L-l)$. (a)
$P(l,L-l)$ for the different SOC strengths, when $L=60$, $n=1$, and $h/t=1.5$.
(b) shows the zoomed images of the center $20$ sites of (a). (c) and (d) are
the same as those of (a) and (b), but with $L=100$.
Figure 5: The pairing momentum distributions $P(k)$. (a) $P(k)$ for the
different SOC strengths, when $L=60$, $n=1$, and $h/t=1.5$. (b) shows the
zoomed image of (a). (c) and (d) are the same as those of (a) and (b), but
with $L=100$.
Figure 6: The ground-state energy $E_{g}/t$ as a function of the SOC strength.
(a) $L=60$ and (b) $L=100$. In all subfigures, $n=1$ and $h/t=1.5$.
Figure 7: The center-of-mass momentum $Q$. (a) $Q$, which is derived
respectively from the state-of-the-art DMRG calculations (Symbols) and
analytical Eq. (12) (Solid line), as a function of the SOC strength, when
$h/t=1.5$, $n=1$, and $L=60$. In the analytical result, the population
imbalance $m$ is also obtained from the state-of-the-art DMRG calculations.
(b) $Q$ as a function of the population imbalance, when $\lambda/t=0.06$,
$n=1$, and $L=60$. (c) and (d) are the same as those of (a) and (b), but with
$L=100$.
Figure 8: The critical population imbalance $m_{c}$ as a function of the SOC
strengths. In this figure, $n=1$.
Figure 9: The population imbalance $m$ and the center-of-mass momentums $Q$
for the different Zeeman fields. (a) $m$ as a function of the Zeeman field for
the different SOC strengths, when $n=1$ and $L=60$. (b) $Q$ as a function of
the SOC strength for the different Zeeman fields, when $n=1$ and $L=60$. The
inset of (b) shows $m$ as a function of the SOC strength. (c) and (d) are the
same as those of (a) and (b), but with $L=100$.
Figure 10: The critical SOC strengths $\lambda_{c}$ as functions of the
lattice length. In this figure, $n=1$ and $h/t=1.5$.
Figure 11: Phase diagrams in the $h-n$ plane for the different SOC strengths.
(a)-(d) $L=60$ and (e)-(h) $L=100$.
Figure 12: Phase diagram in the $h-\lambda$ plane for the different lattice
lengths $L=60$ and $L=100$. In this figure, $n=1$.
Figure 13: The pairing correlation functions $P(l,j)$ (left column), the local
densities $n(l)$ (center column), and the pairing momentum distributions
$P(k)$ (right column) for the different trapped frequencies. In this figure,
$h/t=1.5$, $\lambda/t=0.16$, and $L=100$.
Figure 1: (a) The scaled ground-state energy $E_{g}/(Lt)$ and (b) the
truncation error as functions of the number of states kept (SK). Figure 2: A
schematic picture for illustrating unconventional pairings in a 1D optical
lattice. Figure 3: The pairing correlation functions $P(l,j)$ and local
densities $n(l)$ for the different SOC strengths. Figure 4: The off-diagonal
pairing correlation functions $P(l,L-l)$. Figure 5: The pairing momentum
distributions $P(k)$. Figure 6: The ground-state energy $E_{g}/t$ as a
function of the SOC strength. Figure 7: The center-of-mass momentum $Q$.
Figure 8: The critical population imbalance $m_{c}$ as a function of the SOC
strengths. Figure 9: The population imbalance $m$ and the center-of-mass
momentums $Q$ for the different Zeeman fields. Figure 10: The critical SOC
strengths $\lambda_{c}$ as functions of the lattice length. Figure 11: Phase
diagrams in the $h-n$ plane for the different SOC strengths. Figure 12: Phase
diagram in the $h-\lambda$ plane for the different lattice lengths $L=60$ and
$L=100$. Figure 13: The pairing correlation functions $P(l,j)$ (left column),
the local densities $n(l)$ (center column), and the pairing momentum
distributions $P(k)$ (right column) for the different trapped frequencies.
|
arxiv-papers
| 2014-04-11T06:06:36 |
2024-09-04T02:50:00.990154
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Junjun Liang, Xiaofan Zhou, Pak Hong Chui, Kuang Zhang, Shi-jian Gu,\n Ming Gong, Gang Chen, and Suotang Jia",
"submitter": "Gang Chen",
"url": "https://arxiv.org/abs/1404.3009"
}
|
1404.3044
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# The wronskian solution of the constrained discrete KP hierarchy
Maohua Li1,2,3, Jingsong He1∗ 1\. Department of Mathematics, Ningbo
University, Ningbo, 315211 Zhejiang, China
2\. School of Mathematical Sciences, USTC, Hefei, 230026 Anhui, China
3\. Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001
Leuven, Belgium
[email protected]
[email protected]
###### Abstract.
From the constrained discrete KP (cdKP) hierarchy, the Ablowitz-Ladik lattice
has been derived. By means of the gauge transformation, the Wronskian solution
of the Ablowitz-Ladik lattice have been given. The $u_{1}$ of the cdKP
hierarchy is a Y-type soliton solution for odd times of the gauge
transformation, but it becomes a dark-bright soliton solution for even times
of the gauge transformation. The role of the discrete variable $n$ in the
profile of the $u_{1}$ is discussed.
∗ Corresponding author
Keywords: Constrained discrete KP hierarchy, Gauge transformation, Wronskian
solution
Mathematics Subject Classification (2000): 37K10, 37K40, 35Q51, 35Q55.
## 1\. Introduction
In the past few years, lots of attention have been given to the study of
Kadomtsev-Petviashvili (KP) hierarchy [1, 2] in the field of integrable
systems. The Lax pairs, Hamiltonian structures, symmetries and conservation
laws, the $N$-soliton, tau function, the gauge transformation, reductions etc.
of the KP hierarchy and its sub-hierarchies have been discussed. There are
several sub-hierarchies of the KP by considering different reduction
conditions on the Lax operator $L$. One of them is called constrained KP (cKP)
hierarchy [3, 4, 5] by setting the Lax operator as
$L=\partial+\sum_{i=1}^{m}\Phi_{i}\partial^{-i}\Psi_{i}$. The cKP hierarchy
contains a large number of interesting soliton equations. The basic idea of
this procedure is so-called symmetry constraint [3, 4, 5]. The negative part
of the Lax operator of the constrained KP, i.e.
$\sum_{i=1}^{m}\Phi_{i}\partial^{-i}\Psi_{i}$, is a generator [2] of the
additional symmetry [6] of the KP hierarchy. And the additional symmetry of
BKP hierarchy and CKP hierarchy have been given [7, 8]. Very recently, by a
further modification of the additional flows, the additional symmetries of the
constrained BKP and constrained CKP hierarchies are given in references [9,
10].
It is well known that a continuous integrable system has a discrete analogue
in general. The famous 3-dimensional difference equation is known to provide a
canonical integrable discretization for most important types of soliton
equations. There are several different kinds of the discrete hierarchies
including differential-difference KP (dKP) hierarchy [11, 12], semi-discrete
integrable systems, full discrete equations and so on. The differential-
difference KP hierarchy, defined by the difference operator $\Delta$, is one
interesting object of the discrete integrable systems. Note that, the
additional symmetry of dKP hierarchy and it’s Sato Bäcklund transformations
have been given in reference [13]. Moreover, gauge transformation is one kind
of powerful method to construct the solutions of the integrable systems for
both the continuous KP hierarchy [14, 15, 16, 17, 18, 19, 20, 21, 22] and the
dKP hierarchy [23, 24]. It is discussed to reduce the gauge transformation of
the dKP hierarchy to the constrained discrete KP(cdKP) hierarchy [25]. And the
algebraic structure of the additional symmetry of the cdKP hierarchy also has
been found [26], which is same for the cKP hierarchy [8].
A crucial observation [12] about the KP hierarchy and the dKP hierarchy is
that the $\tau$ function of the discrete KP hierarchy can be constructed by
shift of the $t$ of the $\tau$ function of the continuous KP hierarchy. It is
an interesting question to find any other difference among the two
hierarchies. In this direction, the correspondence between the solutions of
discrete and continuous hierarchy can be used to explore the difference
between them. In particular, a key step is to demonstrate how the discrete
variable $n$ affects the profile of the solutions of the dKP hierarchy.
The purpose of this paper is to find the the correspondence between the
solutions of the KP hierarchy and the dKP hierarchy by means of the multi-
channel gauge transformation. The paper is organized as follows. Some basic
results of the dKP hierarchy and the cdKP hierarchy are summarized in Section
2. The main theorem about the solution of cdKP hierarchy are give in Section
3. An example is give in section 4. We find that the odd kinds of gauge
transformation of cdKP hierarchy can change to a new profile of solution of
the cdKP hierarchy. Section 5 is devoted to conclusions and discussions.
## 2\. the cdKP hierarchy
Let $L$ be a general first-order pseudo difference operator(PDO)
$L(n)=\Delta+\sum_{i=1}^{\infty}u_{i}(n)\Delta^{-i},$ (2.1)
the cdKP hierarchy [26] is defined by the following Lax equation
$\frac{\partial L}{\partial t_{l}}=[B_{l},L],B_{l}:=(L^{l})_{+},l=1,2,\cdots,$
(2.2)
associated with a constrained Lax operator
$L^{l}_{-}=\sum_{i=1}^{m}q_{i}(t)\Delta^{-1}r_{i}(t),$ (2.3)
which is $m$-components Lax operator of the cdKP hierarchy. It has relation
between the dynamical variables $q_{i},r_{i}$ and $u_{i}$. Specially,
$u_{1}=q_{1}\Lambda^{-1}(r_{1})$, where $\Delta=\Lambda-I$. The eigenfunction
and adjoint eigenfunction $q_{i}(t),r_{i}(t)$ are important dynamical
variables in the cdKP hierarchy. It can be checked that the Lax equation (2.2)
is consistent with the evolution equations of the eigenfunction (or adjoint
eigenfunction)
$\displaystyle\begin{cases}q_{i,t_{m}}=B_{m}q_{i},\\\
r_{i,t_{m}}=-B_{m}^{*}r_{i},\quad B_{m}=(L^{m})_{+},\forall m\in
N.\end{cases}$ (2.4)
Therefore the cdKP hierarchy in eq.(2.2) is well defined.
From the Lax equation (2.2), we get the first nontrival $t_{2}$ flow equations
of the cdKP hierarchy for $m=1,l=2$ as
$\displaystyle\begin{cases}q_{1,t_{2}}=\Delta^{2}q_{1}+2q_{1}^{2}r_{1}=q_{1}(n+2)-2q_{1}(n+1)+q_{1}(n)+2q_{1}^{2}r_{1},\\\
r_{1,t_{2}}=-{\Delta^{*}}^{2}r_{1}+2q_{1}r_{1}^{2}=r_{1}(n)-2r_{1}(n-1)+r_{1}(n-2)+2q_{1}(n)r_{1}(n)^{2}.\end{cases}$
(2.5)
It is nothing but the Ablowitz-Ladik lattice [27]. It can be reduced to the
discrete non-linear Schrödinger (DNLS) equation [28] by letting
$r_{1}=q_{1}^{*}$ and a scaling transformation $t_{2}=it_{2}$.
The Lax operator in eq.(2.3) can be generated by the dressing action
$L=W\circ\Delta\circ W^{-1},$ (2.6)
with a dressing operator
$W(n;t)=1+\sum^{\infty}_{j=1}w_{j}(n;t)\Delta^{-j}.$ (2.7)
Further the flow equation (2.2) is equivalent to the so-called Sato equation,
$\partial_{t_{l}}W=-(L^{l})_{-}\circ W.$ (2.8)
Denote the exponential function as following
$Exp(n;t,z)=(1+z)^{n}exp(\sum_{i=1}^{\infty}t_{i}z^{i})=exp(\sum_{i=1}^{\infty}(t_{i}+n\frac{(-1)^{i-1}}{i})z^{i}),$
(2.9)
then
$\Delta Exp(n;t,z)=zExp(n;t,z),\Delta^{*}Exp^{-1}(n;t,z)=zExp^{-1}(n;t,z).$
(2.10)
There are the wave function $w(n;t,z)$ and the adjoint wave function
$w^{*}(n;t,z)$ for the dKP hierarchy as the following forms:
$w(n;t,z)=W(n;t)Exp(n;t,z)=(1+\frac{w_{1}(n;t)}{z}+\frac{w_{2}(n;t)}{z^{2}}+\cdots)exp(\sum_{i=1}^{\infty}(t_{i}+n\frac{(-1)^{i-1}}{i})z^{i})$
(2.11)
and
$\displaystyle w^{*}(n;t,z)$ $\displaystyle=$
$\displaystyle(W^{-1}(n-1;t))^{*}Exp^{-1}(n;t,z)$ (2.12) $\displaystyle=$
$\displaystyle(1+\frac{w_{1}^{*}(n;t)}{z}+\frac{w_{2}^{*}(n;t)}{z^{2}}+\cdots)exp(\sum_{i=1}^{\infty}-(t_{i}+n\frac{(-1)^{i-1}}{i})z^{i}).$
There also exists a $\tau$ function $\tau(n;t)$ for the dKP hierarchy [12]
such that the wave function is expressed by
$\displaystyle w(n;t,z)=\frac{\tau(n,t-[z^{-1}])}{\tau(n,t)}Exp(n;t,z),$
(2.13)
and the adjoint wave function is expressed by
$\displaystyle
w^{*}(n;t,z)=\frac{\tau(n,t+[z^{-1}])}{\tau(n,t)}Exp^{-1}(n;t,z),$ (2.14)
where $[z]=(z,z^{2}/2,x^{3}/3,\cdots).$
The difference $\Delta-$Wronskian [24]
$\tau_{\Delta}(n)=W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{m})=\left|\begin{array}[]{cccc}q_{1}&q_{2}&\cdots&q_{m}\\\
\Delta q_{1}&\Delta q_{2}&\cdots&\Delta q_{m}\\\
\vdots&\vdots&\ddots&\vdots\\\
\Delta^{m-1}q_{1}&\Delta^{m-1}q_{2}&\cdots&\Delta^{m-1}q_{m}\end{array}\right|$
(2.15)
is a $\tau$ function of dKP hierarchy. In this section, we will reduce
$\tau_{\Delta}(n)$ in (2.15) to a $\tau$ function of the constrained discrete
KP hierarchy.
Now we consider a chain of gauge transformation operator of multi-channel
difference type $T_{d}$ [19, 21, 25] starting from the initial $m$-component
Lax operator $L^{(0)}=L=L_{+}+\sum_{i=1}^{m}q_{i}(t)\Delta^{-1}r_{i}(t)$,
$L^{[0]}\xrightarrow{T_{d}^{[1]}(q_{1}^{[0]})}L^{[1]}\xrightarrow{T_{d}^{[2]}(q_{2}^{[1]})}L^{[2]}\rightarrow\dots\rightarrow
L^{[n-1]}\xrightarrow{T_{d}^{[n]}(q_{n}^{[n-1]})}L^{[n]}.$ (2.16)
Here the index $i$ in the gauge transformation operator
$T_{d}^{[i]}(q_{j}^{[j-1]})$ $(j>i)$means the $i$-th gauge transformation, and
$q_{j}^{[j-1]}$ (or $r_{j}^{[j-1]}$) is transformed by $(j-1)$-steps gauge
transformations from $q_{j}$ (or $r_{j}$), $L^{[k]}$ is transformed by
$k$-steps gauge transformations from the initial Lax operator $L$.
Now we firstly consider successive gauge transformations in (2.16). We define
the operator as
$T_{m}=T_{d}^{[m]}(q_{m}^{[m-1]})\circ\dots\circ T_{d}^{[2]}(q_{2}^{[1]})\circ
T_{d}^{[1]}(q_{1}^{[0]}),$ (2.17)
in which
$\displaystyle q_{i}^{[j]}=T_{d}^{[j]}(q_{j}^{[j-1]})\circ\dots\circ
T_{d}^{[2]}(q_{2}^{[1]})\circ T_{d}^{[1]}(q_{1}^{[0]})q_{i},i,j=1,\cdots,m;$
(2.18) $\displaystyle
r_{k}^{[j]}=((T_{d}^{[j]})^{-1})^{*}(q_{j}^{[j-1])})\circ\dots\circ((T_{d}^{[2]})^{-1})^{*}(q_{2}^{[1]})\circ((T_{d}^{[1]})^{-1})^{*}(q_{1}^{[0]})r_{k},j,k=1,\cdots,m.$
(2.19)
It means that $q_{i}^{[0]}=q_{i}$, $r_{i}^{[0]}=r_{i}$. We shall find another
criterion for the Wronskian entries $f_{1},f_{2},\cdots,f_{n}$ leading to cdKP
flows. The following theorem can be easily got from the Ref. [25].
###### Theorem 2.1.
The gauge transformation operator $T_{m}$ and $T_{m}^{-1}$ have the following
determinant representation:
$\displaystyle T_{m}$ $\displaystyle=$ $\displaystyle
T_{d}^{[m]}(q_{m}^{[m-1]})\circ\dots\circ T_{d}^{[2]}(q_{2}^{[1]})\circ
T_{d}^{[1]}(q_{1}^{[0]})$ (2.25) $\displaystyle=$
$\displaystyle\frac{1}{W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{m})}\left|\begin{array}[]{ccccc}q_{1}&q_{2}&\cdots&q_{m}&1\\\
\Delta q_{1}&\Delta q_{2}&\cdots&\Delta q_{m}&\Delta\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\Delta^{m-1}q_{1}&\Delta^{m-1}q_{2}&\cdots&\Delta^{m-1}q_{m}&\Delta^{m-1}\\\
\Delta^{m}q_{1}&\Delta^{m}q_{2}&\cdots&\Delta^{m}q_{m}&\Delta^{m}\end{array}\right|,$
and
$\displaystyle T_{m}^{-1}$ $\displaystyle=$
$\displaystyle\left|\begin{array}[]{ccccc}q_{1}\circ\Delta^{-1}&\Lambda(q_{1})&\Lambda(\Delta
q_{1})&\cdots&\Lambda(\Delta^{m-2}q_{1})\\\
q_{2}\circ\Delta^{-1}&\Lambda(q_{2})&\Lambda(\Delta
q_{2})&\cdots&\Lambda(\Delta^{m-2}q_{2})\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
q_{m}\circ\Delta^{-1}&\Lambda(q_{m})&\Lambda(\Delta
q_{m})&\cdots&\Lambda(\Delta^{m-2}q_{m})\end{array}\right|\frac{(-1)^{m-1}}{\Lambda(W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{m}))}$
(2.30) $\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\phi_{i}\circ\Delta^{-1}b_{i}$ (2.31)
with
$b_{i}=(-1)^{m+i}\Lambda(\frac{W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{i-1},\hat{i},q_{i+1},\dots,q_{m})}{W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{i-1},q_{i},q_{i+1},\dots,q_{m})}).$
(2.32)
Here $\hat{i}$ means that the column containing $q_{i}$ is delete from
$W_{m}^{\Delta}(q_{1},q_{2},\dots,q_{i-1},q_{i},q_{i+1},\dots,q_{m})$ and the
last row is also deleted. Here the determinant of $T_{m}$ is expanded by the
last column and collecting all sub-determinants on the left side of the
$\Delta^{i}$ with the action $"\circ"$. And $T_{m}^{-1}$ is expanded by the
first column and all the sub-determinants are on the right side with the
action $"\circ"$.
## 3\. Wronskian solution of constrained discrete KP hierarchy
Similar to reference [29], it has [26]
$(K\circ q\circ\Delta^{-1}\circ r)_{-}=K(q)\circ\Delta^{-1}\circ
r,(q\circ\Delta^{-1}\circ r\circ K)_{-}=-q\circ\Delta^{-1}\circ K^{*}(r),$
(3.1)
for a pure-difference operator $K$ and two arbitrary smooth functions ($q,r$).
An important fact is that there exist two $m$-th order $\Delta$-differential
operators
$A=\Delta^{m}+a_{m-1}\Delta^{m-1}+\dots+a_{0},B=\Delta^{m}+b_{m-1}\Delta^{m-1}+\dots+b_{0},$
(3.2)
such that $AL^{l}$ and $L^{l}B$ are differential operators. From
$(AL^{l})_{-}=0$ and $(L^{l}B)_{-}=0$, we get that $A$ and $B$ annihilate the
functions $q^{i}$ and $r_{j}$ , i.e., $A(q_{1})=\dots=A(q_{m})=0,$
$B^{*}(r_{1})=\dots=B^{*}(r_{m})=0$, that implies $q_{i}\in Ker(A)$, $r_{i}\in
Ker(B^{*})$, $i=1,\dots,m$. The dimension of $Ker(A)$ is $m$.
The following theorem provides a criterion for reducing the $\Delta$-Wronskian
$\tau$ function in (2.15) of dKP hierarchy to the $\Delta$-Wronskian $\tau$
function of the cdKP hierarchy defined by (2.2).
###### Theorem 3.1.
The constrained discrete KP hierarchy has a solution
$L=(L)_{+}+\sum_{j=1}^{m}f_{j}\circ\Delta\circ g_{j}$ generated by the $\tau$
function $\tau_{\Delta}(n)=W_{m}^{\Delta}(f_{1},\cdots,f_{m})\neq 0$ satisfies
the $k$-constrained with some suitable functions $q_{1},q_{2},\cdots,q_{M}$
and $r_{1},r_{2},\cdots,r_{M}$ if and only if
$W_{m+M+1}^{\Delta}(f_{1},\cdots,f_{m},\Delta^{k}f_{i_{1}},\cdots,\Delta^{k}f_{i_{M+1}})=0$
(3.3)
for all $(M+1)$ indices $1\leq i_{1}<i_{2}<\cdots<i_{M+1}\leq m,$ which is
equivalent to
$\mbox{}W_{M+1}^{\Delta}(\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{1}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})},\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{2}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})},\cdots,\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{M+1}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})})=0.$
(3.4)
Here $f_{i}$ satisfied linear $\Delta-$difference equations
$\frac{\partial f_{i}}{\partial t_{k}}=\Delta^{k}f_{i},\quad
i=1,2,\cdots,m;k=1,2,\cdots.$ (3.5)
###### Proof.
Similar to case of KP hierarchy [30] and $q$-KP hierarchy [31], there has the
following Wronskian identity
$\displaystyle
W_{m+M+1}^{\Delta}(\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{1}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})},$
$\displaystyle\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{2}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})},\cdots,\frac{W_{m+1}^{\Delta}(f_{1},\dots,f_{m},\Delta^{k}f_{i_{M+1}})}{W_{m}^{\Delta}(f_{1},\dots,f_{m})})$
(3.6)
$\displaystyle=W_{m}(f_{1},\cdots,f_{m},\Delta^{k}f_{i_{1}},\cdots,\Delta^{k}f_{i_{M+1}})$
which provides the equivalence between (3.3) and (3.4).
Using Theorem 2.1 and the relation of operator identities in (3.1), one finds
$(L^{k})_{-}=(W\circ\Delta^{k}\circ
W^{-1})_{-}=\sum_{j=1}^{m}W(\Delta^{k}f_{j})\Delta^{-1}g_{j}$ (3.7)
As it was pointed out in the beginning of this section, there exists an $m$-th
order differential operator $A$ such that $AL^{k}$ is a difference operator.
Application to $W(f_{j})=0$ yields
$0=AL^{k}(W(f_{j}))=AW\circ\Delta^{k}(f_{j})=AW(\partial_{t_{k}}(f_{j})).$
(3.8)
So,
$W(\partial_{t_{k}}(f_{j}))=\frac{W_{m}^{\Delta}(f_{1},\cdots,f_{m},\Delta^{k}f_{j})}{W_{m}^{\Delta}(f_{1},\cdots,f_{m})}\in
Ker(A).$ (3.9)
Since the kernel of $A$ has dimension $m$, at most $m$ of these functions
$\Delta^{k}f_{j}$ can be linearly independent. So, (3.4) is deduced.
Conversely, if (3.4) holds, then there exists one $M$-component of cdKP
$(M<m)$ constrained from (3.7). The equation (3.4) implies that at most $M$ of
functions $W(\Delta^{k}(f_{j}))$ are linearly independent, here $f_{j}$
satisfy (3.5). Then we can find suitable $M$ functions
${q_{1},q_{2},\dots,q_{M}}$, which are linearly independent, to express
functions $W(\Delta^{k}(f_{j}))$ as
$W(\partial_{t_{k}}(f_{j}))=\frac{W_{m}^{\Delta}(f_{1},\cdots,f_{m},\Delta^{k}f_{j})}{W_{m}^{\Delta}(f_{1},\cdots,f_{m})}=\sum_{i=1}^{M}c_{ij}q_{i},j=1,\dots,m.$
(3.10)
with some constant $c_{ij}$. Taking it back into the (3.7), it becomes
$(L^{k})_{-}=\sum_{j=1}^{m}(\sum_{i=1}^{M}c_{ij}q_{i})\circ\Delta^{-1}\circ
g_{j}=\sum_{i=1}^{M}q_{i}\circ\Delta^{-1}\circ(\sum_{j=1}^{m}c_{ij}g_{j})=\sum_{i=1}^{M}q_{i}\circ\Delta^{-1}\circ
r_{i},$ (3.11)
then a $m$-component cdKP hierarchy is reduced to a $M$-component cdKP
hierarchy. ∎
Remark: This theorem is a difference version of the corresponding theorem of
the Ref. [30].
The Wronskian solution of the cdKP hierarchy can be got by the Theorem 3.1
under the gauge transformation. If the initial Lax operator of the constrained
discrete KP hierarchy is a ”free” operator $\Delta$, then $L=\Delta$ means
that the initial $\tau$ function $\tau_{\Delta}$ is $1$.
## 4\. Example of reducing dKP hierarchy to cdKP hierarchy
In this section, we use the method in Theorem 3.1 to find the solution of the
multi-component cdKP hierarchy. We discuss the cdKP hierarchy generated by
$T_{i}\mid_{i=2}$, possesses a $\tau$ function
$\tau_{\Delta}^{(2)}=W_{2}^{\Delta}(f_{1},f_{2})=\left|\begin{array}[]{cc}f_{1}&f_{2}\\\
\Delta f_{1}&\Delta f_{2}\end{array}\right|=f_{1}\circ\Delta
f_{2}-f_{2}\circ\Delta f_{1},$ (4.1)
with
$f_{1}=f_{11}(z_{1},n,t)+f_{12}(z,n,t),f_{2}=f_{21}(z_{2},n,t)+f_{22}(z_{3},n,t).$
(4.2)
Here
$\displaystyle
f_{11}(z_{1},n,t)=(1+z_{1})^{n}e^{\xi_{1}},f_{12}(z,n,t)=(1+z)^{n}e^{\xi},$
$\displaystyle
f_{21}(z_{2},n,t)=(1+z_{2})^{n}e^{\xi_{2}},f_{22}(z_{3},n,t)=(1+z_{3})^{n}e^{\xi_{3}},$
where $\xi_{i}=c_{i}+z_{i}t_{1}+z_{i}^{2}t_{2}+z_{i}^{3}t_{3},i=1,2,3$ and
$\xi=d+zt_{1}+z^{2}t_{2}+z^{3}t_{3},$ $c_{i}$ and $d$ are arbitrary constants.
These functions $f_{1}$ and $f_{2}$ satisfy linear equations (3.5) for
$k=1,2,3$. By (3.5), the cdKP hierarchy generated by $T_{2}$ is in the form of
$\displaystyle L^{l}$ $\displaystyle=$
$\displaystyle(L^{l})_{+}+(T_{2}(\Delta^{k}f_{1}))\circ\Delta^{-1}\circ
g_{1}+(T_{2}(\Delta^{k}f_{2}))\circ\Delta^{-1}\circ g_{2}$ (4.3)
$\displaystyle\stackrel{{\scriptstyle constraint}}{{====}}$
$\displaystyle(L^{l})_{+}+q_{1}\circ\Delta^{-1}\circ r_{1},$ (4.4)
where $q_{1},r_{1}$ are undetermined, which can be expressed by $f_{1}$ and
$f_{2}$ as follows. $\tau_{\Delta}^{(2)}$ possesses a form as
$\displaystyle\tau_{\Delta}^{(2)}$ $\displaystyle=$ $\displaystyle
W_{2}^{\Delta}(f_{1},f_{2})$ (4.5) $\displaystyle=$
$\displaystyle(z_{2}-z_{1})(1+z_{1})^{n}(1+z_{2})^{n}e^{\xi_{1}+\xi_{2}}+(z_{2}-z)(1+z)^{n}(1+z_{2})^{n}e^{\xi+\xi_{2}}$
$\displaystyle+(z_{3}-z_{1})(1+z_{1})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{3}}+(z_{3}-z)(1+z)^{n}(1+z_{3})^{n}e^{\xi+\xi_{3}}.$
According to (3.3) in Theorem 3.1, the restriction for $f_{1}$ and $f_{2}$ to
reduce (4.3) to (4.4) is given by
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
W_{4}^{\Delta}(f_{1},f_{2},f_{1}^{(k)},f_{2}^{(k)})$ (4.6) $\displaystyle=$
$\displaystyle(z^{k}-z_{1}^{k})(z_{2}^{k}-z_{3}^{k})V(z_{1},z_{2},z_{3},z)F(n;z_{i},t),$
with Vandermonde determinant
$V(z_{1},z_{2},z_{3},z)=\left|\begin{array}[]{cccc}1&1&1&1\\\
z_{1}&z_{2}&z_{3}&z\\\ z_{1}^{2}&z_{2}^{2}&z_{3}^{2}&z^{2}\\\
z_{1}^{3}&z_{2}^{3}&z_{3}^{3}&z^{3}\end{array}\right|,$ (4.7)
and
$F(n;z_{i},t)=(1+z_{1})^{n}(1+z_{2})^{n}(1+z_{3})^{n}(1+z)^{n}e^{\xi_{1}+\xi_{2}+\xi_{3}+\xi}.$
(4.8)
Obviously, $f_{1}$ and $f_{2}$ satisfy (4.6) by setting $z=z_{2}$ and
$d=c_{2}$. Then the $\tau$ function of a single component $k$-constrained cdKP
hierarchy defined by
$\displaystyle\tau_{cdKP}^{\Delta}$ $\displaystyle=$
$\displaystyle(z_{2}-z_{1})(1+z_{1})^{n}(1+z_{2})^{n}e^{\xi_{1}+\xi_{2}}+(z_{3}-z_{1})(1+z_{1})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{3}}$
(4.9)
$\displaystyle+(z_{3}-z_{2})(1+z_{2})^{n}(1+z_{3})^{n}e^{\xi_{2}+\xi_{3}},$
which is deduced by (4.5) with $\xi_{2}=\xi$. It means that we indeed reduced
the $\tau$ function $\tau_{\Delta}^{(2)}$ in (4.5) of the dKP hierarchy to the
$\tau$ function $\tau_{cdKP}^{\Delta}$ of the $1$-component cdKP hierarchy.
We would like to get the explicit forms of $q_{1}$ and $r_{1}$ of cdKP
hierarchy in (4.4). With the determinant representation of $T_{2}$ and
$T_{2}^{-1}$, one can have
$\displaystyle f_{1}^{\Delta}$ $\displaystyle\triangleq
T_{2}(\Delta^{k}f_{1})=\frac{(z_{1}^{k}-z_{2}^{k})V(z_{1},z_{2},z_{3})(1+z_{1})^{n}(1+z_{2})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{2}+\xi_{3}}}{\tau_{cdKP}^{\Delta}}$
(4.10a) $\displaystyle f_{2}^{\Delta}$ $\displaystyle\triangleq
T_{2}(\Delta^{k}f_{2})=\frac{(z_{3}^{k}-z_{2}^{k})V(z_{1},z_{2},z_{3})(1+z_{1})^{n}(1+z_{2})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{2}+\xi_{3}}}{\tau_{cdKP}^{\Delta}}$
(4.10b) $\displaystyle g_{1}^{\Delta}$
$\displaystyle\triangleq(T_{2}^{-1})^{*}(\Delta^{k}g_{1})=-\Lambda(\frac{f_{2}}{W_{2}^{\Delta}(f_{1},f_{2})})=-\Lambda(\frac{(1+z_{2})^{n}e^{\xi_{2}}+(1+z_{3})^{n}e^{\xi_{3}}}{\tau_{cdKP}^{\Delta}})$
(4.10c) $\displaystyle g_{2}^{\Delta}$
$\displaystyle\triangleq(T_{2}^{-1})^{*}(\Delta^{k}g_{1})=\Lambda(\frac{f_{1}}{W_{2}^{\Delta}(f_{1},f_{2})})=\Lambda(\frac{(1+z_{1})^{n}e^{\xi_{1}}+(1+z_{2})^{n}e^{\xi_{2}}}{\tau_{cdKP}^{\Delta}})$
(4.10d)
with the Vandermonde determinant
$V(z_{1},z_{2},z_{3})=\left|\begin{array}[]{ccc}1&1&1\\\ z_{1}&z_{2}&z_{3}\\\
z_{1}^{2}&z_{2}^{2}&z_{3}^{2}\end{array}\right|.$ (4.11)
It is clearly
$(z_{3}^{k}-z_{2}^{k})f_{1}^{\Delta}=(z_{1}^{k}-z_{2}^{k})f_{2}^{\Delta}.$
So the $q_{1}$ in (4.4) is
$\displaystyle q_{1}\mbox{}$
$\displaystyle\triangleq(z_{3}^{k}-z_{2}^{k})f_{1}^{\Delta}=(z_{1}^{k}-z_{2}^{k})f_{2}^{\Delta}$
(4.12) $\displaystyle=$
$\displaystyle\mbox{}\frac{(z_{3}^{k}-z_{2}^{k})(z_{1}^{k}-z_{2}^{k})(z_{2}-z_{1})(z_{3}-z_{1})(z_{3}-z_{2})(1+z_{1})^{n}(1+z_{2})^{n}(1+z_{3})^{n}e^{\xi_{1}+\xi_{2}+\xi_{3}}}{\tau_{cdKP}^{\Delta}}.$
And the (4.3) is reduced to
$\displaystyle(L^{l})_{-}$ $\displaystyle=$ $\displaystyle
f_{1}^{\Delta}\circ\Delta^{-1}\circ
g_{1}^{\Delta}+f_{2}^{\Delta}\circ\Delta^{-1}\circ g_{2}^{\Delta}$ (4.13)
$\displaystyle=$
$\displaystyle(z_{3}^{k}-z_{2}^{k})f_{1}^{\Delta}\circ\Delta^{-1}\circ\frac{g_{1}^{\Delta}}{(z_{3}^{k}-z_{2}^{k})}+(z_{1}^{k}-z_{2}^{k})f_{2}^{\Delta}\circ\Delta^{-1}\circ\frac{g_{2}^{\Delta}}{z_{1}^{k}-z_{2}^{k}}$
$\displaystyle=$ $\displaystyle q_{1}\circ\Delta^{-1}\circ r_{1},$
where
$\displaystyle r_{1}$ $\displaystyle\triangleq$
$\displaystyle\frac{g_{1}^{\Delta}}{(z_{3}^{k}-z_{2}^{k})}+\frac{g_{2}^{\Delta}}{z_{1}^{k}-z_{2}^{k}}$
(4.14) $\displaystyle=$
$\displaystyle\Lambda(\frac{(z_{3}^{k}-z_{2}^{k})(1+z_{1})^{n}e^{\xi_{1}}+(z_{3}^{k}-z_{1}^{k})(1+z_{2})^{n}e^{\xi_{2}}+(z_{2}^{k}-z_{1}^{k})(1+z_{3})^{n}e^{\xi_{3}}}{(z_{3}^{k}-z_{2}^{k})(z_{1}^{k}-z_{2}^{k})\tau_{cdKP}^{\Delta}}).$
For simplicity, denote $t_{1}=x;t_{2}=y$. In particular, choosing $z_{1}=z$,
$z_{2}=0$, $z_{3}=-z$, $c_{1}=c$, $c_{2}=0$, $c_{3}=-c$ then $\xi_{2}=0$,
$\xi_{3}=-\xi_{1}$ and
$q_{1}=\frac{(-1)^{k+2}z^{2k+2}(1+z)^{n}(1-z)^{n}e^{2z^{2}y}}{(1+z)^{n}(1-z)^{n}+\frac{(1+z)^{n}e^{\eta}+(1-z)^{n}e^{-\eta}}{2}e^{z^{2}y}}.$
(4.15)
Base on above choice,
$r_{1}=-\frac{1}{z^{k+1}}\Lambda(\frac{\frac{[(1+z)^{n}e^{\eta}+(1-z)^{n}e^{-\eta}]}{2}+e^{-z^{2}y}}{\frac{[(1+z)^{n}e^{\eta}+(1-z)^{n}e^{-\eta}]}{2}+(1-z^{2})^{n}e^{z^{2}y}})$
(4.16)
and if $k$ is odd, and
$r_{1}=-\frac{1}{z^{k+1}}\Lambda(\frac{\frac{(1+z)^{n}e^{\eta}-(1-z)^{n}e^{-\eta}}{2}}{\frac{(1+z)^{n}e^{\eta}+(1-z)^{n}e^{-\eta}}{2}+(1-z^{2})^{n}e^{z^{2}y}})$
(4.17)
if $k$ is even. Here $\eta\triangleq c+zx+z^{3}t_{3}.$
So the dynamical variable $u_{1}=q_{1}\Lambda^{-1}(r_{1})$ of the Lax operator
$L$ of the cdKP hierarchy
$u_{1}=2z^{k+1}\frac{(1-(-1)^{k})+e^{-(c+zx+z^{3}t_{3})+z^{2}y}(1-z)^{n}-e^{(c+zx+z^{3}t_{3})+z^{2}y}(-1)^{k}(1+z)^{n}}{(1-z^{2})^{n}(\frac{e^{c+zx+z^{3}t_{3}}}{(1-z)^{n}}+2e^{z^{2}y}+\frac{e^{-(c+zx+z^{3}t_{3})}}{(1+z)^{n}})^{2}}.$
(4.18)
An example is
$u_{1}=2z^{k+1}\frac{(1-(-1)^{k})+e^{-zx+z^{2}y}(1-z)^{n}-e^{zx+z^{2}y}(-1)^{k}(1+z)^{n}}{(1-z^{2})^{n}(\frac{e^{zx}}{(1-z)^{n}}+2e^{z^{2}y}+\frac{e^{-zx}}{(1+z)^{n}})^{2}},$
(4.19)
by setting $t_{3}=0,c=0.$ For this case
$\displaystyle
u_{1}=2z^{k+1}\frac{2+e^{-zx+z^{2}y}(1-z)^{n}+e^{zx+z^{2}y}(1+z)^{n}}{(1-z^{2})^{n}(\frac{e^{zx}}{(1-z)^{n}}+2e^{z^{2}y}+\frac{e^{-zx}}{(1+z)^{n}})^{2}},\text{$k$
is odd},$ (4.20) $\displaystyle
u_{1}=2z^{k+1}\frac{e^{-zx+z^{2}y}(1-z)^{n}-e^{zx+z^{2}y}(1+z)^{n}}{(1-z^{2})^{n}(\frac{e^{zx}}{(1-z)^{n}}+2e^{z^{2}y}+\frac{e^{-zx}}{(1+z)^{n}})^{2}},\text{$k$
is even.}$ (4.21)
Remark: Actually, (4.6) also can be satisfied by other two choices $z=z_{1}$
or $z=z_{3}$. But $u_{1}$ (4.18) will be only one-soliton solution because the
$f_{1}^{\Delta}=0$ in (4.10a) or $f_{2}^{\Delta}=0$ in (4.10b) separately.
The graph of $q_{1}=q_{1}(x,y,n),r_{1}=r_{1}(x,y,n),u_{1}=u_{1}(x,y,n)$ were
plotted in below for fixed $k=1,2$. We shall discuss the function of the gauge
transformation for the the cdKP hierarchy to emphasize two sides about the
discrete variable $n$ of it and the times variable $k$ of the gauge
transformation of it. The profile of $q_{1},r_{1},u_{1}$ are plotted according
to the value of discrete variable $n$ from $0$ to $2$ and the value of time
$k$ of the gauge transformations from $1$ to $2$. The five conditions of the
profile of $q_{1},r_{1},u_{1}$ are $\\{n=0,k=1\\}$,$\\{n=1,k=1\\}$,
$\\{n=2,k=1\\}$, $\\{n=0,k=2\\}$, $\\{n=1,k=2\\}$ and $\\{n=2,k=2\\}$ as
following.
(1).The profile of $q_{1}$ are plotted with $k=1,n=0,1,2$ in Figure 1, Figure
2 and Figure 3 respectively.
(2).The profile of $r_{1}$ are plotted with $k=1,n=0,1,2$ in Figure 4, Figure
5 and Figure 6 respectively.
(3).The Y-type soliton profile of $u_{1}$ are plotted with $k=1,n=0,1,2$ in
Figure 7 , Figure 8 and Figure 9 respectively.
(4).The bright-dark soliton profile of $u_{1}$ are plotted with $k=2$ and
$n=0,1,2$ in Figure 10, Figure 11 and Figure 12 respectively.
From the graphs of the solution of cdKP hierarchy, it can be found that:
(1) The profile of the solution $q_{1}$ of the cdKP hierarchy is decreasing to
the one of the classical KP hierarchy in Ref. [30] when $n\to 0$ see Figure 1
($k=1$). For $r_{1},u_{1}$ of the cdKP hierarchy, the profile of its are also
decreasing the analogues of the classical KP hierarchy (see Figure 4 and
Figure 7).
(2)When the times $k$ of gauge transformation is an odd number, the profiles
of $u_{1}$ become the Y-type soliton, see Figure 7, Figure 8 and Figure 9.
(3)When the times $k$ of gauge transformation is an even number, the profiles
of $u_{1}$ become bright-dark soliton, see Figure 10, Figure 11 and Figure 12.
For the end of showing more detail about dependence of $u_{1},q_{1}$ on $n$,
it is necessary to define $n$-effect quantity $\Delta
u_{1}(z,x,y,n)=u_{1}(z,x,y,n)-u_{1}(z,x,y,n=0)=u_{1}(n)-u_{1}(0)$ for fixed
$z=0.5$. Figure 13 are plotted for the $\Delta u_{1}(z,x,y,n)$ where $n=1,2,3$
respectively, which shows the dependence of $u_{1}$ on $n$. It was obviously
they are decreasing to almost zero when $n$ goes from $3$ to $1$ with fixed
$z=0.5$. They also demonstrate that discretization of the cdKP hierarchy keeps
the profile of the soliton though it has discrete variable $n$. These figures
give us again an opportunity to observe the role of discrete variable $n$ in
the Wronskian solution of the cdKP hierarchy.
## 5\. Conclusions
In this paper, the Wronskian solutions of the equation in the cdKP hierarchy
have been given by means of the multi-channel gauge transformation. Based on
the results of our previous papers [25, 26], Theorem 3.1 provides a necessary
and sufficient condition of the $k$-constrained discrete KP hierarchy with $m$
components. As an example, the reduction from $2$-cdKP hierarchy to $1$-cdKP
hierarchy is presented. It can be found that the profiles of solution $u_{1}$
of cdKP hierarchy can be the Y-type solition by the odd number times gauge
transformation, but the solution of cdKP hierarchy becomes bright-dark
solition by even times gauge transformation. From these profiles, it can be
find that the solution $u_{1}$ of the cdKP hierarchy is decreasing to the
analogues of the classical KP hierarchy when $n\to 0$.
Acknowledgments This work is supported by the National Natural Science
Foundation of China under Grant Nos.11271210 and 11201251, K.C.Wong Magna Fund
in Ningbo University, the Natural Science Foundation of Zhejiang Province
under Grant No. LY12A01007 and Science Fund of Ningbo University
(No.XYL14028). One of the authors (MH) is supported by Erasmus Mundus Action 2
EXPERTS III and would like to thank Prof. Antoine Van Proeyen for many helps.
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* [31] J. S. He, Y. H. Li and Y. Cheng, $Q$-Deformed KP Hierarchy and $q$-Deformed Constrained KP Hierarchy, SIGMA 2(2006), 1-32.
Figure 1. The profile of the solution $q_{1}$(4.15) of equation (2.5) (left)
and its density plot (right)with $c=0,t_{3}=0,z=0.5,k=1$ and $n=0$ .
Figure 2. The profile of the solution of $q_{1}$(4.15) of equation (2.5)
(left) and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$
and $n=1$.
Figure 3. The profile of the solution of $q_{1}$(4.15) of equation (2.5)
(left) and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$
and $n=2$.
Figure 4. The profile of the solution $r_{1}$ (4.16) (left) of equation (2.5)
and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$ and
$n=0$. The vertical axis $r$ denotes the $r_{1}$.
Figure 5. The profile of $r_{1}$ (4.16) (left) and its density plot (right)
with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=1$. The vertical axis $r$
denotes the $r_{1}$.
Figure 6. The profile of $r_{1}$ (4.16) (left) and its density plot (right)
with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=2$. The vertical axis $r$
denotes the $r_{1}$.
Figure 7. The profile of type Y soliton of $u_{1}$ (4.20) (left) and its
density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=0$. The
vertical axis $u$ denotes the $u_{1}$.
Figure 8. The profile of type Y soliton of $u_{1}$ (4.20) (left) and its
density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=1$. The
vertical axis $u$ denotes the $u_{1}$.
Figure 9. The profile of type Y soliton of $u_{1}$ (4.20) (left) and its
density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=1$ and $n=2$. The
vertical axis $u$ denotes the $u_{1}$.
Figure 10. The profile of bright-dark type soliton of $u_{1}$ (4.21) (left)
and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=2$ and $n=0$
. The vertical axis $u$ denotes the $u_{1}$.
Figure 11. The profile of bright-dark type soliton of $u_{1}$ (4.21) (left)
and its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=2$ and $n=1$
. The vertical axis $u$ denotes the $u_{1}$.
Figure 12. The profile of bright-dark soliton of $u_{1}$ (4.21) (left) and
its density plot (right) with parameters $c=0,t_{3}=0,z=0.5,k=2$ and $n=2$.
The vertical axis $u$ denotes the $u_{1}$.
(a) (b) (c)
Figure 13. $\Delta u_{1}(z,x,y,n)$ with $c=0,t_{3}=0,z=0.5,k=1$ and $n=1$ in
(a), $2$ in (b) and $3$ in (c).
|
arxiv-papers
| 2014-04-11T09:21:00 |
2024-09-04T02:50:00.999612
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Maohua Li, Jingsong He",
"submitter": "Maohua Li",
"url": "https://arxiv.org/abs/1404.3044"
}
|
1404.3084
|
# Bibliometric Indicators of Young Authors in Astrophysics:
Can Later Stars be Predicted?
Frank Havemann111Institut für Bibliotheks- und Informationswissenschaft,
Humboldt-Universität zu Berlin, D 10099 Berlin, Dorotheenstr. 26, Germany
Birger Larsen222Department of Communication, Aalborg University, Copenhagen,
Denmark
###### Abstract
We test 16 bibliometric indicators with respect to their validity at the level
of the individual researcher by estimating their power to predict later
successful researchers. We compare the indicators of a sample of astrophysics
researchers who later co-authored highly cited papers before their first
landmark paper with the distributions of these indicators over a random
control group of young authors in astronomy and astrophysics. We find that
field and citation-window normalisation substantially improves the predicting
power of citation indicators. The two indicators of total influence based on
citation numbers normalised with expected citation numbers are the only
indicators which show differences between later stars and random authors
significant on a 1 % level. Indicators of paper output are not very useful to
predict later stars. The famous $h$-index makes no difference at all between
later stars and the random control group.
## 1 Introduction
Any indicator should actually indicate what it is made for. If an indicator is
used for evaluation it should not provide an incentive for an unwanted
behaviour. In scholarly publishing we know salami and multiple publications,
unjustified assignment of co-authorship, and different practices of tactical
citation behaviour. Bibliometricians should strive to develop valid research
indicators which have no unwanted adverse effects [Kreiman and MaunsellKreiman
and Maunsell2011].
Most bibliometric indicators are not developed for the evaluation of
individual researchers [Costas, van Leeuwen, and BordonsCostas et al.2010, p.
1565], however individuals are increasingly being evaluated using such
indicators. We test selected indicators with respect to their validity at the
level of the individual researcher by estimating their power to predict later
successful researchers. For this reason, we compare bibliometric indicators of
a sample of astrophysics researchers who later co-authored highly cited papers
(later stars, for short) before their first landmark paper with the
distributions of these indicators over a random control group of young authors
in astronomy and astrophysics.
Results obtained with some standard basic indicators have been presented on a
poster at ISSI 2013.333 14th International Society of Scientometrics and
Informetrics Conference in Vienna, Austria, 15th to 20th July 2013 [Havemann
and LarsenHavemann and Larsen2013] Here we extend the study to more
sophisticated measures with the aim to find the best indicators for predicting
later stars. We imagine that later stars apply for a job in an astrophysical
research institute five years after their first paper in a journal indexed in
Web of Science (WoS). Do they perform better bibliometrically than the average
of applicants with the same period of publishing?
## 2 Data and method
### 2.1 Sampling of authors
We inspected 64 astronomy and astrophysics journals to find researchers who
started publishing after 1990 and had published for a period of at least five
years in WoS journals. We excluded those who had more than 50 co-authors on
average because evaluating those big-science authors cannot be supported by
bibliometrics. We draw a random sample of 331 authors mainly publishing in
this field and affiliated longer in Europe then elsewhere. The latter
criterion contradicts with the international character of astrophysics
research but makes the sample more homogenous with respect to the educational
and cultural background of the researchers.
To find authors with highly cited papers, for each journal considered we
ranked papers with more than four citations per year and less than ten authors
according to their citations per year. We excluded papers with ten or more
authors because we want to have later stars whose contributions to the
successful papers are not too small. From the top 20 percent of these paper
rank-lists we extracted all European authors of highly cited papers. We
obtained 362 candidates who published their first highly cited paper at least
five years after their first paper in one the 64 journals.
We ranked these later-star candidates according to their number of highly
cited papers. We went through this list and checked whether the authors had
really five years or more to wait for the breakthrough paper if all their
papers in WoS-journals are taken into account. We chose the first 40 authors
to keep the effort manageable. For all WoS-papers of the 40 later stars and of
the 331 random authors (downloaded at Humboldt-University, Berlin) all citing
papers were determined by CWTS, Leiden. All bibliometric indicators presented
below are based on papers and their citations within the first five years of
the author. To compare only authors with similar collaboration behaviour we
restricted both samples to authors with less than four and more than one co-
author on average ending up with 30 later stars and 179 random authors.
We further restricted both samples to authors starting before 1999 because
there is only one star starting later (in 2002) but many random authors (more
than 100). By this restriction to 29 stars and 74 authors in the control group
we take into account that the citation behaviour of astrophysicists has
changed remarkably during the last 25 years. The numbers of references have
increased. The median of reference numbers of the 448 papers published in the
1986 volume of the Monthly Notices of the Royal Astronomical Society was 24.
Till the year 2010 the median of reference numbers has doubled (calculated
with 2,006 papers, data source: WoS).444 cf. [p. 5]henneken2011ads Longer
lists of references induce higher citation numbers of papers. Thus, both
samples still have a time variance of expected citation numbers. This time
variance increases the overlap between the citation-indicator distributions of
the samples when citation numbers are not normalised. In other aspects the
union of our samples is surely more homogenous than many real groups of
applicants (career duration, collaboration behaviour, geographical
background).
An alternative data source for astrophysics publications and their citations
is the Astrophysics Data System (ADS)555 http://adsabs.harvard.edu delivered
jointly by the US National Aeronautics and Space Administration (NASA) and the
Smithsonian Astrophysical Observatory [Henneken, Kurtz, and AccomazziHenneken
et al.2011]. ADS includes also non-refereed publications. Any user can obtain
a whole slew of bibliometric indicators for any set of selected publications.
### 2.2 Statistics
Figure 1: The authors in the two samples have similar distributions of
collaboration behaviour.
For each bibliometric indicator considered, we test whether both samples
behave like random samples drawn from the same population by applying a one-
sided Wilcoxon rank sum test with continuity correction. We test the null
hypothesis that for both samples we have the same probability of drawing an
author with a larger value in the other sample. The alternative hypothesis is
that indicator values of later stars exceed the values of random authors.666
cf. the Wikipedia article http://en.wikipedia.org/wiki/Mann-Whitney-
Wilcoxon_test
We have also tested the hypothesis that for both samples we have the same
probability of drawing an author with a larger value of the collaborative
coefficient [Ajiferuke, Burrell, and TagueAjiferuke et al.1988, cf. also our
Table 1, p. 1] in the other sample. In both samples we have a similar
collaboration behaviour (cf. Figure 1). If we would refuse the null hypothesis
we would fail in about one half of possible cases (test probability $p=.516$).
This result ensures that differences between both groups are not due to
different typical team sizes.
All work was done using the free open-source statistics software R (which
includes a graphics package).777 http://www.r-project.org (R-scripts for
indicator calculation and sample data can be obtained from the first author of
this paper.)
### 2.3 Selection of indicators
Table 1: List of author indicators: $a_{i}$ is the number of authors of paper $i$; $c_{i}$ is the number of citations of paper $i$; $E(c_{i})$ is the expected number of citations of paper $i$ (cf. Appendix 2); we assume that papers of an author are ordered according to $c_{i}$ and denote the paper’s rank with $r$; the effective rank is defined as $r_{\mathrm{eff}}(r)=\sum_{i}^{r}1/a_{i}$. name | definition
---|---
productivity: |
nr. of papers | $\sum_{i}1=n$
fractional score | $\sum_{i}1/a_{i}=f$
total influence: |
nr. of citations | $\sum_{i}c_{i}$
norm. nr. cit. | $\sum_{i}c_{i}/\mathrm{E}(c_{i})$
$j$-index | $\sum_{i}\sqrt{c_{i}}$
fract. citations | $\sum_{i}c_{i}/a_{i}$
fract. norm. cit. | $\sum_{i}c_{i}/(\mathrm{E}(c_{i})a_{i})$
typical infl.: |
mean cit. nr. | $\sum_{i}c_{i}/n$
mean fract. cit. | $\sum_{i}(c_{i}/a_{i})/n$
med. fract. cit. | $\mathrm{median}(c_{i}/a_{i}$)
max. fract. cit. | $\max(c_{i}/a_{i})$
h-type indices: |
Hirsch index | $\max(r|c_{r}\geq r)$
$g$-index | $\max(r|\sum_{i}^{r}c_{i}\geq r^{2})$
fract. h-type: |
$h_{\mathrm{m}}$-index | $\max(r_{\mathrm{eff}}|c_{r(r_{\mathrm{eff}})}\geq r_{\mathrm{eff}})$
$g_{\mathrm{f}}$-index | $\max(r|\sum_{i}^{r}c_{i}/a_{i}\geq r^{2})$
$g_{\mathrm{m}}$-index | $\max(r_{\mathrm{eff}}|\sum_{i}^{r(r_{\mathrm{eff}})}c_{i}/a_{i}\geq r_{\mathrm{eff}}^{2})$
collaboration: |
collab. coeff. | $1-f/n$
The indicators analysed here are listed together with their mathematical
definitions in Table 1. In Appendix A.1 we discuss the definition of each of
these indicators.
We have calculated and tested two simple output indicators and nine indicators
of influence. Beside pure numbers of papers and their citations within the
first five publishing years of the authors we use fractionally counted papers
and citations as the input for indicators of productivity and of influence.
The use of fractional counting in evaluation penalises unjustified assignment
of co-authorship to friends.
If we compare papers published in fields with different citation behaviour any
citation indicator should be field normalised with expected citation numbers.
Here we consider only one field but—as mentioned above—the citation behaviour
of astrophysicists has changed dramatically within the last decades. That
means, distributions of unnormalised citation indicators of the two samples
overlap partly due to the changing citation behaviour.
Another wanted effect of normalising with expected citation numbers is that we
account for different citation windows of papers. Thus, citations to papers
published in the beginning of a period obtain a lower weight than those to
papers published in the last year. The estimation of expected citation numbers
of papers is described in Appendix 2.
Another method to deal with varying citation behaviour is to determine each
paper’s percentile in the citation distribution of a control sample of papers.
bornmann_which_2013 compare five approaches to this promising method.
Percentile ranking avoids the use of arithmetic means of heavily skewed
citation distributions. We minimise the influence of skewness by calculating
expected citation numbers by a linear regression over all years considered (s.
Appendix 2). We have to leave a test of the percentile method with our samples
to further work due to a lack of citation data of control samples.
Recently, several authors tested a third approach to field normalisation of
citation numbers. Here data on the citing side are normalised. [s. also
references of this paper]waltman_systematic_2013 discuss three variants of
this method. Also this approach cannot be tested with the data we have at
hand. We could test the simplest variant where each citation of a paper is
divided by the number of all references of the citing paper [Zhou and
LeydesdorffZhou and Leydesdorff2011, Pepe and KurtzPepe and Kurtz2012].
waltman_systematic_2013 and also radicchi_testing_2012 found that this
fractional counting of references does not properly normalise for field and
subfield differences. A further drawback of this variant is that citation
numbers are not corrected for the age of the cited paper. We therefore did not
test it.
In addition to the eleven indicators of productivity and of influence we
calculated the widely used Hirsch or $h$-index [HirschHirsch2005], a number
combining influence and output performance in an uncontrolled and arbitrary
manner, and four variants of it which have been introduced to avoid
disadvantages of the Hirsch index.
We did not consider any indicator based on the number of highly cited papers
because this contradicts our sampling procedure: we selected later stars who
have no highly cited paper in their first five years of publishing.
## 3 Results
Medians of all 16 indicators of both samples are given in Table 2. In the next
to last column of Table 2 we list the failure probability $p$ of rejecting the
null hypothesis that both samples behave like random samples drawn from the
same population. In the last column we give the rank $R$ according to $p$. For
all but the two indicators on least ranks (Hirsch index and median of
fractional citation numbers) the stars’ sample has a higher median than the
random sample.
The boxplots in Appendix A.3 allow a comparison of indicator distributions for
both samples. The figures are ordered according to the ranking $R$. That means
that $p$ -values increase from the first to the last boxplot. The boxplots
have a logarithmic scale because all indicator distributions are highly
skewed. All citation indicators have zero values for some uncited authors in
the control sample. Therefore we display the logarithm of indicator values +
1.
The two indicators based on normalised citation numbers are the most useful
among the 16 indicators considered (s. Figure 3). With respect to normalised
numbers of citations and to fractional normalised citations both samples
behave not like random samples from the same population. In both cases,
rejecting the null hypothesis has a failure probability below 1 %.
Table 2: Median indicators of samples, test probability $p$, and rank $R$ (according to $p$) indicator | stars | random | $p$ | $R$
---|---|---|---|---
productivity: | | | |
nr. of papers | 8 | 6 | .076 | 12
fractional score | 2.67 | 1.86 | .095 | 13
total influence: | | | |
nr. of citations | 36 | 22.5 | .028 | 6
norm. nr. cit. | 6.03 | 3.83 | .003 | 1
$j$-index | 11.86 | 8.76 | .031 | 9
fract. citations | 10.00 | 6.57 | .030 | 7
fract. norm. cit. | 1.82 | 1.10 | .008 | 2
typical infl.: | | | |
mean cit. nr. | 5.25 | 4.00 | .117 | 14
mean fract. cit. | 1.23 | 0.99 | .062 | 11
med. fract. cit. | 0.50 | 0.67 | .260 | 16
max. fract. cit. | 4.67 | 3.00 | .030 | 8
h-type indices: | | | |
Hirsch index | 3 | 3 | .210 | 15
$g$-index | 5 | 4 | .037 | 10
fract. h-type: | | | |
$h_{\mathrm{m}}$-index | 1.32 | 1.00 | .020 | 3
$g_{\mathrm{f}}$-index | 3 | 2 | .024 | 4
$g_{\mathrm{m}}$-index | 2.38 | 1.68 | .025 | 5
collaboration: | | | |
collab. coeff. | .683 | .683 | .516 | 17
The distributions of eight further indicators differ at least on a 5 %
significance level (s. Figures 4–7). For the remaining six indicators there is
no significant difference between distributions of later stars and of authors
in the control group (s. Figures 8–10). The Hirsch-index has very similar
distributions for both samples ($p=21\,\%$, rank 15, s. Figure 10).
## 4 Discussion
Our results underline the necessity to correct citation indicators for the age
of the cited papers and also for varying citation behaviour.888 It would be
interesting—from a theoretical point of view—to determine the influence of
each of both corrections separately. The two indicators of total influence
based on citation numbers normalised with expected citation numbers are the
only indicators among a total of 16 which show significant differences between
later stars and random authors on a 1 % level. Thus, normalised citation
indicators of total influence can indeed help to predict later successful
authors. Despite this relative good performance of normalised citation
indicators of total influence we cannot recommend to use them as the only
basis for an evaluation of young authors in astrophysics and in similar fields
of natural sciences. Normalisation at the field level cannot correct for a
variability in citation numbers between different topics.
opthof_differences_2011 analysed the citation density in different topics of
cardiovascular research papers and concluded that even normalised citation
indicators “should not be used for quality assessment of individual
scientists” (cf. his abstract).999 Topics in physics as in astrophysics also
differ substantially in citation density [Radicchi and CastellanoRadicchi and
Castellano2011, Pepe and KurtzPepe and Kurtz2012]. In each case, bibliometrics
can only support evaluation and cannot replace individual peer review.
None of the two output indicators have a significant difference below the 5 %
level.101010 This is in accordance with the result obtained by [cf. p.
9]neufeld_peer_2013 when comparing successful with non-successful applicants
of a funding programme for young researchers. Thus, it is very unlikely to
discover a later star in astrophysics by comparing her productivity with the
productivity of a random author (Figures 8 and 9). The Hirsch index makes no
difference at all ($p=21\,\%$, Figure 10). This is in agreement with
conclusions drawn by lehmann_measures_2006 and also by
kosmulski_calibration_2012 who analysed small samples of mature scientists and
found that the number of publications “is rather useless” as a tool of
assessment and that also the $h$-index is not really helpful. In contrast to
these findings, pudovkin_research_2012 found that $h$-index and number of
papers are indicators which differ most significantly between group leaders
and other scientists at a medical research institution. This can surely be
explained by real output differences of elder and younger researchers but
maybe partly also by the assumption that group leaders have more often been
working at the institute over the whole analysed 5-years period than other
researchers.
We could have analysed the generalised $h$-index proposed by
radicchi_universality_2008 who use normalised citation and paper numbers. We
did not because $h$ performs much worse than indicators of total influence.
The $g$-index proposed by egghe_improvement_2006 to improve the $h$-index
performs indeed better than the original ($p=3.7\,\%$, Figure 7). The same
holds for the analysed three $h$-type indices which are based on fractional
counting. They have been introduced by egghe_mathematical_2008 and by
Schreiber schreiber2008share,schreiber_fractionalized_2009 to account for
varying collaboration behaviour.
There is no significant difference between the two samples when we compare
citation indicators which are designed to reflect the mean influence of an
author’s papers. We calculated three of them: the arithmetic mean of citation
numbers ($p=11.7\,\%$, Figure 9), fractionally counted citations per paper
($p=6.2\,\%$, Figure 8), and the median of the fractionally counted citations
($p=26\,\%$, Figure 10). We wondered whether for a later star a large maximum
of (fractional) citations is more typical than a large value of any measure of
central tendency of citation numbers. The answer is yes. The maximum of
fractional citations is a better indicator of typical influence ($p=3\,\%$,
Figure 6). We could have analysed normalised indicators of typical influence,
too. We did not because indicators of typical influence do not perform better
than those of total influence.
We do not exclude self-citations when calculating citation indicators. There
are arguments for their exclusion in evaluative bibliometrics but we assume
that it would be difficult for young authors to massively cite their own
papers within their first five years of publishing.
We expect that weighting (fractional) paper numbers with a measure of journal
reputation would improve the predictive power of output indicators. We did not
test this because the only journal-reputation indicator available for us was
the journal impact factor which is not useful here—albeit often used for
weighting paper numbers [SeglenSeglen1997, Lozano, Larivière, and
GingrasLozano et al.2012, s. also the references of these papers].
Analysing 85 researchers in oncology honekopp2012future found that “a linear
combination of past productivity and the average paper’s citation” is a better
predictor of future publication success than any of the single indicators they
had studied. We did not consider combinations of indicators of productivity
and of mean influence because the simpler indicators of total influence also
reflect productivity—as far as the produced papers have been cited. Neglecting
uncited papers is a wanted effect that is also quoted in favour of the
$h$-index.
hornbostel_funding_2009 found only small differences in numbers of
publications and citations between approved and rejected applicants to a
German funding programm for young researchers. In an earlier study,
nederhof_peer_1987 compared 19 PhD graduates in physics with best degrees to
119 other graduates with lower grade. They considered the total number of
papers before and after graduation and their total and average (short time)
impact. The 19 best graduates performed significantly better but,
interestingly, the impact of their papers declined and reached the level of
the control-group papers a few years after graduation. The authors speculate
about the reason of this phenomenon and suggest that better students could
have been engaged for hot and therefore highly cited research projects. They
conclude, that maybe “the quality of the research project, and not the quality
of the particular graduate is the most important determinant of both
productivity and impact figures” [Nederhof and van RaanNederhof and van
Raan1987, p. 348]. This hypothesis could also hold for the young
astrophysicists analysed by us. Its confirmation would further diminish the
weight of bibliometric indicators in the evaluation of young researchers.
## Acknowledgements
We thank Jesper Schneider for helpful discussions of an early draft and Paul
Wouters at CWTS in Leiden for providing citation data. The analysis was done
for the purposes of the ACUMEN project, financed by the European Commission,
cf. http://research-acumen.eu/.
## Appendix A Appendix
### A.1 Descriptions of indicators
#### A.1.1 Productivity indicators
##### Number of papers:
This elementary indicator of productivity belongs to a bygone era when co-
authorship was the exception and not the rule. It has the unwanted adverse
effects of multiple publishing of the same results and of honorary
authorships.
##### Fractional score:
Each paper $i$ is divided into $a_{i}$ fractions where $a_{i}$ is the number
of authors. These fractions are summed up for the papers of the evaluated
author. We use the simplest variant where all fractions of a paper are equal:
$f=\sum_{i}1/a_{i}$. This indicator penalises honorary authorships and takes
into account that larger teams can be more productive.
#### A.1.2 Total influence
All indicators of total influence tend to increase with the author’s number of
papers. That means, they are also indicating productivity.
##### Number of citations:
Each citation of a paper indicates that it has influenced the citing
author(s). The sum $\sum_{i}c_{i}$ of raw numbers $c_{i}$ of citations of an
author’s papers is highly field dependent. The paper’s number of citations
$c_{i}$ depends on the age of a paper at the time of evaluation. Highly cited
papers have surely some quality but less cited ones can also be of high
quality.
##### Normalised numbers of citations:
We normalise each paper’s number of citations $c_{i}$ by an expected number of
citations $\mathrm{E}(c_{i})$ which takes into account the paper’s age and the
citation behaviour in astrophysics during the first five (calendar) years in
the paper’s lifetime (cf. Appendix 2). After normalising each paper’s citation
number we sum the ratios of observed and expected citation numbers:
$\sum_{i}c_{i}/\mathrm{E}(c_{i})$. Some bibliometricians do not calculate the
sum of ratios but the ratio of sums $\sum_{i}c_{i}/\sum_{i}\mathrm{E}(c_{i})$
[Schubert and BraunSchubert and Braun1986]. This procedure is thought to
evaluate the whole oeuvre of an author but has been criticised recently for
being not “consistent” [Opthof and LeydesdorffOpthof and Leydesdorff2010,
Waltman, van Eck, van Leeuwen, Visser, and van RaanWaltman et al.2011].111111
The $h$-index is also not consistent [MarchantMarchant2009, Waltman and van
EckWaltman and van Eck2012].
##### The j-index:
The $j$-index is the sum of the square roots of citation numbers of the
author’s papers $\sum_{i}\sqrt{c_{i}}$. It was proposed by
levene_bibliometric_2012 to downgrade the influence of highly cited papers in
the sum of citation numbers.
##### Fractional citations:
Analogously to the fractional score described above we distribute citations of
each paper equally to its authors: $\sum_{i}c_{i}/a_{i}.$
##### Fractional normalised citations:
The normalised numbers of citations can also be distributed among the authors
involved [Radicchi and CastellanoRadicchi and Castellano2011]:
$\sum_{i=1}^{n}\dfrac{c_{i}}{\mathrm{E}(c_{i})a_{i}}.$
#### A.1.3 Typical influence
##### Mean citation number:
The arithmetic mean of citations of an author’s papers $\sum_{i}c_{i}/n$ is
the simplest indicator of influence which does not tend to increase with the
author’s productivity.
##### Mean fractional citations:
The arithmetic mean of fractionally counted citations of an author’s papers:
$\sum_{i}(c_{i}/a_{i})/n$.
##### Median of fractional citations:
The median of fractionally counted citations of an author’s papers
$\mathrm{median}(c_{i}/a_{i})$ is considered because citation distributions
are skewed.
##### Maximum of fractional citations:
We wondered whether for a later star a large maximum of (fractional) citations
$\max(c_{i}/a_{i})$ is more typical than a large value of any measure of
central tendency of citation numbers [Lehmann, Jackson, and LautrupLehmann et
al.2008, cf. p. 375].
#### A.1.4 Indices of h-type
##### Hirsch index:
The $h$-index was introduced by hirsch2005iqi “to quantify an individual’s
scientific research output.” It is defined as the maximum rank $r$ in a rank
list of an author’s papers according to their citation numbers $c_{i}$ which
is less than or equal to the citation number $c_{r}$ of the paper with rank
$r$: $h=\max(r|c_{r}\geq r).$ The $h$-index has been criticised for its
arbitrariness [van Eck and Waltmanvan Eck and Waltman2008]. It is arbitrary
because in the definition Hirsch “assumes an equality between incommensurable
quantities” [Lehmann, Jackson, and LautrupLehmann et al.2008, p. 377], namely
a rank and a citation number. Hirsch himself stated that his index depends on
field-specific citation and collaboration behaviour [HirschHirsch2005, p.
16571].
##### Egghe’s g-index:
egghe_improvement_2006 criticised the $h$-index for being insensitive to the
citation frequency of an author’s highly cited papers. His $g$-index can be
defined as the maximum rank $r$ which is less than or equal to the mean
citation number $(\sum_{i}^{r}c_{i})/r$ of papers till rank $r$
[SchreiberSchreiber2008b]. This condition is equivalent to
$\sum_{i}^{r}c_{i}\geq r^{2}$. That means, $g$ can also be defined as
$g=\max(r|\sum_{i=1}^{r}c_{i}\geq r^{2}).$
#### A.1.5 Fractional indices of h-type
##### Schreiber’s $h_{\mathrm{m}}$-index:
Fractional counting of papers or of citations could be applied to define an
$h$-index which takes multi-authorship into account [EggheEgghe2008,
SchreiberSchreiber2008c]. schreiber_modification_2008 argued that fractionally
counted citations could remove highly cited papers from the $h$-core if they
have a lot of authors. This led him to define the $h_{\mathrm{m}}$-index as
the maximal effective rank $r_{\mathrm{eff}}(r)=\sum_{i}^{r}1/a_{i}$ which is
less than or equal to the number of citations $c_{r}$:
$h_{\mathrm{m}}=\max(r_{\mathrm{eff}}|c_{r(r_{\mathrm{eff}})}\geq
r_{\mathrm{eff}}).$
##### Egghe’s $g_{\mathrm{f}}$-index:
egghe_mathematical_2008 proposed to define a fractional $g$-index
$g_{\mathrm{f}}$ as
$g_{\mathrm{f}}=\max(r|\sum_{i=1}^{r}\frac{c_{i}}{a_{i}}\geq r^{2}).$
Here the citations are counted fractionally.
##### Schreiber’s $g_{\mathrm{m}}$-index:
schreiber_fractionalized_2009 proposed a fractional $g$-index $g_{\mathrm{m}}$
where both, papers and citations, are counted fractionally:
$g_{\mathrm{m}}=\max(r_{\mathrm{eff}}|\sum_{i=1}^{r(r_{\mathrm{eff}})}\frac{c_{i}}{a_{i}}\geq
r_{\mathrm{eff}}^{2}).$
### A.2 Expected citation numbers
Figure 2: Linear regressions and averages of citation numbers of papers of
random authors in astrophysics after the first (the publication) year (red),
the second year (orange), the third year (yellow), the fourth year (green),
and the fifth year (blue).
Usually, for field normalisation expected citation numbers of papers are
calculated as arithmetic means of citation numbers of all papers (of the same
document type) published in all journals of the field in the same year. There
are two main technical problems with this method, the rough delineation of
fields and the skewness of citation distributions.
We do not evaluate single authors but only want to show the influence of field
normalisation on distributions of citation indicators of authors. Therefore we
can use a random sample of papers (for which we have already the citation
data) instead of all papers in the field. This sample contains papers
published in the years 1991–2009 by all 331 random authors of our initial
control sample. We only consider those 2342 papers with at most 20 authors.
Figure 2 shows the average cumulated citation numbers in the publication year,
one year later, two years later etc. Due to the skewness of citation
distributions these arithmetic means fluctuate. Therefore we made a linear
regression for each of the five time series of citation numbers of papers (not
of the averages) but restricted the analysis to the years 1995–2007 (coloured
part of the regression lines) where we have more than 100 papers in each year.
The interpolated citation numbers obtained by linear regression are used as
expected citation numbers $\mathrm{E}(c_{i})$ of papers published in the
corresponding years.
From these data we estimate a doubling of citation numbers in astrophysics in
the two decades around the millennium.
Calculating expected citation numbers as field averages is problematic because
the arithmetic mean is not a good measure for the central tendency of skewed
citation distributions. lundberg_lifting_2007 therefore proposed to determine
expected citation numbers as geometric means of citation numbers of papers in
the field. Because papers can have zero citations he adds 1 to be able to
calculate the geometric mean. This can be justified by saying that publishing
a paper is the first citation of the published results.
### A.3 Boxplots of indicators
On this page and the next pages you find boxplots of distributions of all 16
indicators both of the sample of 29 later stars and of the control sample of
74 random young astrophysicists.
Figure 3: The two indicators with best $p$-values: $p<1$ %
Figure 4: The indicators on rank 3 and 4 according to $p$-values: $p<5$ %
Figure 5: The indicators on rank 5 and 6 according to $p$-values: $p<5$ %
Figure 6: The indicators on rank 7 and 8 according to $p$-values: $p<5$ %
Figure 7: The indicators on rank 9 and 10 according to $p$-values: $p<5$ %
Figure 8: The indicators on rank 11 and 12 according to $p$-values: $p<10$ %
Figure 9: The indicators on rank 13 and 14 according to $p$-values
Figure 10: The indicators on rank 15 and 16 according to $p$-values
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* [Waltman, van Eck, van Leeuwen, Visser, and van RaanWaltman et al.2011] Waltman, L., N. J. van Eck, T. N. van Leeuwen, M. S. Visser, and A. F. J. van Raan (2011, February). Towards a new crown indicator: an empirical analysis. Scientometrics 87(3), 467–481.
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arxiv-papers
| 2014-04-11T12:01:09 |
2024-09-04T02:50:01.008773
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Frank Havemann and Birger Larsen",
"submitter": "Frank Havemann",
"url": "https://arxiv.org/abs/1404.3084"
}
|
1404.3104
|
See pages - of INFOCOM.pdf
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arxiv-papers
| 2014-04-11T13:28:38 |
2024-09-04T02:50:01.016959
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Siyi Wang, Weisi Guo, Mark D. McDonnell",
"submitter": "Siyi Wang",
"url": "https://arxiv.org/abs/1404.3104"
}
|
1404.3118
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# Yamabe type equations with a sign-changing nonlinearity, and the prescribed
curvature problem
Bruno Bianchini and Luciano Mari and Marco Rigoli Bruno Bianchini
Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova
Via Trieste 63, I-35121 Padova (Italy) [email protected] Luciano Mari
Departamento de Matemática, Universidade Federal do Ceará
Av. Humberto Monte s/n, Bloco 914, 60455-760 Fortaleza (Brazil)
[email protected] Marco Rigoli
Dipartimento di Matematica, Università degli studi di Milano
Via Saldini 50, I-20133 Milano (Italy) [email protected]
(Date: August 29, 2024)
###### Abstract.
In this paper, we investigate the prescribed scalar curvature problem on a
non-compact Riemannian manifold $(M,\langle\,,\,\rangle)$, namely the
existence of a conformal deformation of the metric $\langle\,,\,\rangle$
realizing a given function $\widetilde{s}(x)$ as its scalar curvature. In
particular, the work focuses on the case when $\widetilde{s}(x)$ changes sign.
Our main achievement are two new existence results requiring minimal
assumptions on the underlying manifold, and ensuring a control on the
stretching factor of the conformal deformation in such a way that the
conformally deformed metric be bi-Lipschitz equivalent to the original one.
The topological-geometrical requirements we need are all encoded in the
spectral properties of the standard and conformal Laplacians of $M$. Our
techniques can be extended to investigate the existence of entire positive
solutions of quasilinear equations of the type
$\Delta_{p}u+a(x)u^{p-1}-b(x)u^{\sigma}=0$
where $\Delta_{p}$ is the $p$-Laplacian, $\sigma>p-1>0$, $a,b\in
L^{\infty}_{\mathrm{loc}}(M)$ and $b$ changes sign, and in the process of
collecting the material for the proof of our theorems, we have the opportunity
to give some new insight on the subcriticality theory for the Schrödinger type
operator
$Q_{V}^{\prime}\ :\
\varphi\longmapsto-\Delta_{p}\varphi-a(x)|\varphi|^{p-2}\varphi.$
In particular, we prove sharp Hardy-type inequalities in some geometrically
relevant cases, notably for minimal submanifolds of the hyperbolic space.
###### Key words and phrases:
Yamabe equation, Schrödinger operator, subcriticality, p-Laplacian, spectrum,
prescribed curvature
###### 2010 Mathematics Subject Classification:
primary 58J05, 35B40; secondary 53C21, 34C11, 35B09.
###### Contents
1. 1 Introduction, I: existence for the generalized Yamabe problem
2. 2 Introduction, II: our main results in their general setting
3. 3 Preliminaries
4. 4 Criticality theory for $Q_{V}$, capacity and Hardy weights
5. 5 Hardy weights and comparison geometry
1. 5.1 Hardy weights on manifolds with a pole
2. 5.2 Hardy weights on minimally immersed submanifolds
3. 5.3 Specializing our main theorems: an example
6. 6 Proofs of Theorems 2.1 and 2.2
7. 7 Proofs of our geometric corollaries, and concluding comments
## 1\. Introduction, I: existence for the generalized Yamabe problem
Generalizations of the classical Yamabe problem on a Riemannian manifold have
been the focus of an active area of research over the past 30 years. Among
these, the prescribed scalar curvature problem over non-compact manifolds
appears to be challenging: briefly, given a non-compact Riemannian manifold
$(M^{m},\langle\,,\,\rangle)$ with scalar curvature $s(x)$ and a smooth
function $\widetilde{s}\in C^{\infty}(M)$, the problem asks under which
conditions there exists a conformal deformation of $\langle\,,\,\rangle$,
(1.1) $\widetilde{\langle\,,\,\rangle}=\varphi^{2}\langle\,,\,\rangle,\qquad
0<\varphi\in C^{\infty}(M),$
realizing $\widetilde{s}(x)$ as its scalar curvature. When the dimension $m$
of $M$ is at least $3$, writing $\varphi=u^{\frac{2}{m-2}}$, the problem
becomes equivalent to determining a positive solution $u\in C^{\infty}(M)$ of
the Yamabe equation
(1.2) $\displaystyle\Delta
u-\frac{s(x)}{c_{m}}u+\frac{\widetilde{s}(x)}{c_{m}}u^{\frac{m+2}{m-2}}=0,\qquad
c_{m}=\frac{4(m-1)}{m-2}.$
Here, $\Delta$ is the Laplace-Beltrami operator of the background metric
$\langle\,,\,\rangle$. For $m=2$, setting $\varphi=e^{u}$ one substitutes
(1.2) with
$2\Delta u-s(x)+\widetilde{s}(x)e^{2u}=0$
where now $u\in C^{\infty}(M)$ may change sign (see [43]). Hereafter, we will
confine ourselves to dimension $m\geq 3$, and $M$ will always be assumed to be
connected. Agreeing with the literature, we will call the linear operator in
(1.2):
(1.3)
$L_{\langle\,,\,\rangle}\doteq-\Delta-\frac{s(x)}{c_{m}}=-\Delta+\frac{m-2}{4(m-1)}s(x)$
the conformal Laplacian of $(M,\langle\,,\,\rangle)$.
The original Yamabe problem is a special case of the prescribed scalar
curvature problem, namely that when $\widetilde{s}(x)$ is a constant, and for
this reason, in the literature, the prescribed scalar curvature problem is
often called the generalized Yamabe problem. Besides establishing existence of
a positive solution $u$ of (1.2), it is also useful to investigate its
qualitative behaviour since this reflects into properties of
$\widetilde{\langle}\,,\,\rangle$. For instance, $u\in L^{\frac{2m}{m-2}}(M)$
is equivalent to the fact that $\widetilde{\langle}\,,\,\rangle$ has finite
volume. Also, if $u$ is bounded between two positive constants, the identity
map
(1.4) $i\ \ :\ \
(M,\langle\,,\,\rangle)\longrightarrow(M,\widetilde{\langle}\,,\,\rangle)$
is globally bi-Lipschitz, and thus $\widetilde{\langle}\,,\,\rangle$ inherits
some fundamental properties of $\langle\,,\,\rangle$. For instance, geodesic
completeness, parabolicity, Gromov-hyperbolicity, etc. (see [36, 35]).
Agreeing with the literature, when
$C^{-1}\langle\,,\,\rangle\leq\widetilde{\langle\,,\,\rangle}\leq
C\langle\,,\,\rangle$ for some constant $C>0$ we will say that
$\widetilde{\langle}\,,\,\rangle$ and $\langle\,,\,\rangle$ are uniformly
equivalent.
Given the generality of the geometrical setting, it is reasonable to expect
that existence or non-existence of the desired conformal deformation heavily
depends on the topological and metric properties of $M$ and their relations
with $\widetilde{s}(x)$. As we shall explain in awhile, a particularly
intriguing (and difficult) case is when $\widetilde{s}(x)$ is allowed to
change sign. In this situation, with the exception of a few special cases, a
satisfactory answer to the prescribed curvature problem is still missing. To
properly put our results into perspective, first we describe some of the main
technical problems that arise when looking for solutions of (1.2) for sign-
changing $\widetilde{s}(x)$. Then, we briefly comment on some classical and
more recent approaches. In particular, we pause to describe in detail four
results that allow us to grasp the situation in the relevant examples of
Euclidean and hyperbolic spaces and to underline the key features of our new
achievements. We stress that, when $\widetilde{s}(x)\leq 0$, there is a vast
literature and the interaction between topology and geometry is better
understood. Among the various references on the existence problem, we refer
the reader to [8, 72, 71, 16, 48].
If $\widetilde{s}(x)$ is positive somewhere, basic tools to produce solutions
are in general missing. More precisely, uniform $L^{\infty}$-estimates fail to
hold on regions where $\widetilde{s}(x)$ is non-negative, and comparison
theorems are not valid where $\widetilde{s}(x)$ is positive. This suggests
why, in the literature, equation (1.2) in a non-compact ambient space has
mainly been studied via variational and concentration-compactness techniques
([37, 80]) or radialization techniques ([57, 56, 42, 8]). We also quote the
interesting method developed in [72, 71, 9].
To the best of our knowledge, up to now there have been few attempts to adapt
the variational approach to (non-compact, of course) spaces other than
$\mathbb{R}^{m}$, [37, 80]. In this respect, a particularly interesting result
is the next one due to Q.S. Zhang [80].
###### Theorem 1.1 ([80], Thm. 1.1).
Let $(M^{m},\langle\,,\,\rangle)$ be a complete manifold with dimension $m\geq
3$ and scalar curvature $s(x)\geq 0$. Suppose that $\mathrm{vol}(B_{r}(x))\leq
Cr^{m}$ for some uniform $C$ independent of $x$, and that $M$ has positive
Yamabe invariant $Y(M)$:
(1.5)
$Y(M)=\inf\left\\{\int_{M}\Big{[}|\nabla\phi|^{2}+\frac{s(x)}{c_{m}}\phi^{2}\Big{]}\
\ :\ \ \phi\in\mathrm{Lip}_{c}(M),\ \int_{M}\phi^{\frac{2m}{m-2}}=1\right\\},$
$c_{m}$ as in (1.2). Assume further that
* -
$\widetilde{s}(x)\geq 0,\not\equiv 0$ and $\widetilde{s}(x)\rightarrow 0$ as
$r(x)\rightarrow+\infty$,
* -
$\widetilde{s}(x)$ is sufficiently flat around at least one of its maximum
points.
Then, there exists a solution $u\in L^{\frac{2m}{m-2}}(M)$ of (1.2) such that
(1.6) $u\leq C\big{(}1+r(x)\big{)}^{-\frac{m-2}{2}},$
for some $C>0$. In particular,
$\widetilde{\langle\,,\,\rangle}=u^{\frac{4}{m-2}}\langle\,,\,\rangle$ has
finite volume and is geodesically incomplete.
###### Remark 1.1.
_The flatness condition above is the one usually required for the compact
Yamabe problem, see[28, 29]. _
The above theorem is not, indeed, the most general statement of Zhang’s
result, but however the version here is a good compromise between generality
and simplicity, and it is enough for the sake of comparison with our main
theorems. On the positive side, topological conditions on $M$ are not so
demanding. However, we underline that the polynomial volume growth assumption
is essential for Zhang’s method to work, hence this excludes the case of
negatively curved manifolds like the hyperbolic space
$\mathbb{H}^{m}_{\kappa}$ of sectional curvature $-\kappa^{2}$. On the
contrary, as the recent [13] highlights, the radialization methods developed
by W.M. Ni, M. Naito and N. Kawano in [57, 56, 42] on $\mathbb{R}^{m}$, and by
P. Aviles and R. McOwen in [8] for $\mathbb{H}^{m}$ are very flexible with
respect to curvature control on $M$, but on the other hand they require $M$ to
possess a pole (that is, a point $o$ for which the exponential map $\exp_{o}$
is a diffeomorphism), a quite restrictive topological assumption. We quote the
two results, starting from Ni-Naito-Kawano’s theorem.
###### Theorem 1.2 ([57, 56, 42]).
Let $\widetilde{s}(x)\in C^{\infty}(\mathbb{R}^{m})$, $m\geq 3$, and suppose
that there exists $B\in C^{0}(\mathbb{R})$ such that
(1.7) $|\widetilde{s}(x)|\leq B\big{(}r(x)\big{)}\qquad\text{and}\qquad
rB(r)\in L^{1}(+\infty).$
Then, there exists a small $\gamma_{0}>0$ such that, for each
$\gamma\in(0,\gamma_{0})$, there exists a conformal deformation
$\widetilde{\langle}\,,\,\rangle$ of the flat metric $\langle\,,\,\rangle$
such that
(1.8)
$\widetilde{\langle\,,\,\rangle}_{x}\rightarrow\gamma\langle\,,\,\rangle_{x}\qquad\text{as
}\,r(x)\rightarrow+\infty.$
###### Theorem 1.3 ([8], Thm 4).
Let $(M^{m},\langle\,,\,\rangle)$ be a complete manifold with a pole and
dimension $m\geq 3$, and suppose that there exist constants
$\bar{\kappa}\geq\kappa>0$ such that sectional curvature $K$ of $M$ be pinched
as follows:
(1.9) $-\bar{\kappa}^{2}\leq K\leq-\kappa^{2}<0,\qquad\text{with
}\quad\bar{\kappa}^{2}<\frac{(m-1)^{2}}{m(m-2)}\kappa^{2}.$
Suppose also that $\widetilde{s}(x)\in C^{\infty}(M)$ satisfies
(1.10) $-C_{1}\leq\widetilde{s}(x)\leq-C_{2}<0\qquad\text{outside of a compact
set,}$
for some constants $C_{1},C_{2}>0$. Then, there exists $\delta>0$ sufficiently
small such that, if
(1.11) $\widetilde{s}(x)\leq\delta\qquad\text{on }\,M,$
there exists a conformal deformation $\widetilde{\langle\,,\,\rangle}$
realizing $\widetilde{s}(x)$ and satisfying
(1.12) $C^{-1}\langle\,,\,\rangle\leq\widetilde{\langle\,,\,\rangle}\leq
C\langle\,,\,\rangle\qquad\text{on }\,M,$
for some positive constant $C$.
###### Remark 1.2.
_Theorem 1.3 has later been improved in [72] with a different technique:
however, the main Theorem 0.1 in [72] still requires (1.11) and a couple of
conditions on the curvatures of $M$ that, though more general than (1.9),
nevertheless are more demanding than (1.23), (1.24) appearing in our Corollary
1.2 below. _
###### Remark 1.3.
_For the special case of the Hyperbolic space, in[71] (see Theorem 1.1
therein) the authors were able to guarantee the existence of a solution for
the Yamabe equation giving rise to a complete metric even when (1.10) is
replaced by the weaker_
$-Cr(x)^{2}\leq\widetilde{s}(x)<0\qquad\text{outside of a compact set.}$
_The counterpart of this improvement is that a control of the type ( 1.12) is
no longer available. We remark that, in Theorem 1.1 of [71], condition (1.11)
still appears. _
Inspired by Ni-Naito-Kawano’s approach, in [13] we have obtained sharp
existence theorems for (1.2) (and, more generally, for (2.1) below) on
manifolds possessing a pole $o$ via mild assumptions on the radial sectional
curvature $K_{\mathrm{rad}}$ (the sectional curvature restricted to $2$-planes
containing $\nabla r$, with $r(\cdot)=\mathrm{dist}(\cdot,o)$). In the
particular case of manifolds close to the hyperbolic space, our outcome has
been the following result. Observe that condition 1.15 below guarantees the
existence of solutions even when $\widetilde{s}(x)$ is strongly oscillating.
On the other hand, (1.16) implies that the conformally deformed metric is
incomplete and has finite volume.
###### Theorem 1.4 ([13], Thm 2).
Let $(M,\langle\,,\,\rangle)$ be a complete manifold of dimension $m\geq 3$,
with a pole $o$ and sectional curvature $K$ satisfying
(1.13) $-\kappa^{2}-\mathcal{K}\big{(}r(x)\big{)}\leq K(x)\leq-\kappa^{2},$
for some constant $\kappa>0$ and some non-negative $\mathcal{K}\in
C^{0}(\mathbb{R}^{+}_{0})\cap L^{1}(\mathbb{R}^{+})$. Suppose that the scalar
curvature $s(x)$ of $M$ is such that
(1.14) $s(x)\geq-\frac{(m-1)^{3}\kappa^{2}}{m-2}\qquad\text{on }M.$
Then, for each $\widetilde{s}(x)\in C^{\infty}(M)$ satisfying, for some $B\in
C^{0}(\mathbb{R}^{+}_{0})$,
(1.15) $|\widetilde{s}(x)|\leq B(r(x)),\qquad e^{-2\kappa r}B(r)\in
L^{1}(+\infty),$
the metric $\langle\,,\,\rangle$ can be conformally deformed to a smooth
metric $\widetilde{\langle\,,\,\rangle}$ of scalar curvature
$\widetilde{s}(x)$, satisfying
(1.16) $\Gamma_{1}e^{-2\kappa
r(x)}\langle\,,\,\rangle_{x}\leq\widetilde{\langle\,,\,\rangle}_{x}\leq\Gamma_{2}e^{-2\kappa
r(x)}\langle\,,\,\rangle_{x}\qquad\forall\,x\in M,$
for some $0<\Gamma_{1}\leq\Gamma_{2}$. In particular,
$\widetilde{\langle}\,,\,\rangle$ is incomplete and has finite volume.
Furthermore, $\Gamma_{2}$ and consequently $\Gamma_{1}$ can be chosen to be as
small as we wish.
###### Remark 1.4.
_The growth conditions ( 1.7) and (1.15) are sharp: it is proved in [21] (for
$\mathbb{R}^{m}$) and [15] (for $\mathbb{H}^{m}_{\kappa}$) that no conformal
deformation exists whenever $\widetilde{s}(x)\leq 0$ on $\mathbb{R}^{m}$
(respectively, on $\mathbb{H}^{m}_{\kappa}$) and_
$\widetilde{s}(x)\leq-\frac{C}{r(x)^{2}\log
r(x)}\qquad\left(\text{respectively, }\ \
\widetilde{s}(x)\leq-\frac{Ce^{2\kappa r(x)}}{r(x)\log r(x)}\right)$
_for some $C>0$ and large $r(x)$. _
The above four results are, to the best of our knowledge, an up-to-date
account of what is known on the prescribed scalar curvature problem, in
dimension $m\geq 3$ and with sign-changing $\widetilde{s}(x)$, on non-compact
manifolds close to $\mathbb{R}^{m}$ and $\mathbb{H}^{m}_{\kappa}$. Figures 1
and 2 below summarize Theorems 1.1 to (1.4) when assumptions overlap.
Figure 1. Euclidean space, (1.7) and Zhang’s assumptions on $\widetilde{s}(x)$
in force. Figure 2. Manifolds close to $\mathbb{H}^{m}_{\kappa}$,
$-C_{1}\leq\widetilde{s}(x)\leq-C_{2}<0$ for large $x$.
A first step in the direction of removing the pole requirement has been taken
in [14] by adapting some ideas of [13] via the use of Green functions.
Unfortunately, even though the requirements on $s(x)$ and $\widetilde{s}(x)$
in Theorem 5 of [14] are sharp, they express in a form that is generally
difficult to check. In summary, the task of obtaining results of the type
above but with a substantial weakening of the geometric assumptions calls for
new ideas, and this is the objective of the present work. More precisely, we
have a twofold concern in this paper. First, we aim to produce an existence
theorem for sign-changing $\widetilde{s}(x)$ where topological and geometrical
conditions are confined to a minimum. Second, we also want to keep control on
the conformally deformed metric, in particular in such a way that
$\langle\,,\,\rangle$ and $\widetilde{\langle\,,\,\rangle}$ are uniformly
equivalent. Our contributions are Theorems 1.5, 1.6 and 2.3 below, a special
case of Theorems 2.1, 2.2 which we are going to describe in awhile.
###### Notation.
_Hereafter, given $b\in L^{\infty}_{\mathrm{loc}}(M)$, we respectively denote
with $b_{+}$ and $b_{-}$ its positive and negative parts, so that
$b=b_{+}-b_{-}$. For $a\in L^{\infty}_{\mathrm{loc}}(M)$, we will write_
$a(x)=O\big{(}b(x)\big{)}\qquad\big{(}\text{respectively,}\quad a(x)\asymp
b(x)\ \big{)}\quad\text{as $x$ diverges}$
_to indicate that there exists a constant $C>0$ and a compact set $\Omega$
such that_
$a(x)\leq Cb(x)\qquad\big{(}\text{respectively,}\quad C^{-1}b(x)\leq a(x)\leq
Cb(x)\ \big{)}$
_on $M\backslash\Omega$. _
Our first result deals with the case of non-parabolic manifolds with non-
negative scalar curvature. We recall that a manifold $M$ is said to be non-
parabolic if it admits a positive, non-constant solution of $\Delta u\leq 0$.
The notion of non-parabolicity will be recalled later in a more general
setting (see Proposition 2.1 and the subsequent discussion), here we limit to
refer the interested reader to [34] for deepening. The following theorem shall
be compared to Theorems 1.1 and 1.2. In particular, to compare Theorem 1.5
below with Zhang’s Theorem 1.1 we need some tools that will be defined in the
next introduction, and therefore we postpone the analysis to Remark 2.3.
###### Theorem 1.5.
Let $(M,\langle\,,\,\rangle)$ be a non-parabolic manifold of dimension $m\geq
3$ and scalar curvature $s(x)$ satisfying
(1.17) $s(x)\geq 0\text{ on M,}\qquad s\in L^{1}(M),$
and let $\widetilde{s}\in C^{\infty}(M)$ with the following properties:
(1.18) $\text{$\widetilde{s}_{+}$ has compact support,}\qquad\widetilde{s}\in
L^{1}(M).$
Then, for each constant $C>0$, $\langle\,,\,\rangle$ can be pointwise
conformally deformed to a new metric $\widetilde{\langle\,,\,\rangle}$ of
scalar curvature $\widetilde{s}(x)$ such that
$\widetilde{\langle\,,\,\rangle}\leq C\langle\,,\,\rangle$. Moreover, if $s$
and $\widetilde{s}$ have compact support, each such
$\widetilde{\langle\,,\,\rangle}$ can be chosen to be uniformly equivalent to
$\langle\,,\,\rangle$.
###### Remark 1.5.
_Non-parabolicity is a very mild requirement, and it is necessary to guarantee
existence in all the cases investigated in Theorem 1.5. In fact, if $M$ is
scalar flat and taking $\widetilde{s}(x)$ to be compactly supported, non-
negative and not identically zero, any eventual solution $u$ of the Yamabe
equation (1.2) would be a (non-constant) positive solution of $\Delta
u=-\widetilde{s}(x)/c_{m}u^{\frac{m+2}{m-2}}\leq 0$, showing that $M$ must be
non-parabolic. _
Theorem 1.5 applies, for instance, to the physically relevant setting of
asymptotically flat spaces. According to [47], $(M^{m},\langle\,,\,\rangle)$
is called asymptotically flat if
* -
its scalar curvature $s(x)$ satisfies (1.17),
* -
there exists a compact set $K\subset M$ such that each connected component
$U_{j}$ of $M\backslash K$ has a global chart
$\Psi_{j}:(\mathbb{R}^{m}\backslash
B_{R}(0),\langle\,,\,\rangle_{\mathrm{can}})\rightarrow U_{j}$ for which the
local expression $g_{ij}$ of $\langle\,,\,\rangle$ satisfies
(1.19)
$|g_{ij}-\delta_{ij}|=O\big{(}r^{-p}\big{)},\quad|\partial_{k}\,g_{ij}|=O\big{(}r^{-p-1}\big{)},\quad|\partial^{2}_{kl}\,g_{ij}|=O\big{(}r^{-p-2}\big{)}$
as $r(x)=|x|\rightarrow+\infty$, for some $p>(m-2)/2$ and for each $1\leq
i,j,k,l\leq m$.
###### Corollary 1.1.
Let $(M,\langle\,,\,\rangle)$ be an asymptotically flat manifold of dimension
$m\geq 3$. Then, for each smooth function $\widetilde{s}(x)$ satisfying
(1.20) $\text{$\widetilde{s}_{+}$ has compact
support}\qquad\text{and}\qquad\widetilde{s}\in L^{1}(M),$
and for each constant $C>0$, $\widetilde{s}(x)$ is realizable via a conformal
deformation $\widetilde{\langle}\,,\,\rangle$ of $\langle\,,\,\rangle$
satisfying $\widetilde{\langle}\,,\,\rangle\leq C\langle\,,\,\rangle$.
Furthermore, if $\widetilde{s}\equiv 0$ outside some compact set, then
$\widetilde{\langle}\,,\,\rangle$ can be chosen to be uniformly equivalent to
$\langle\,,\,\rangle$.
Now, we deal with manifolds whose original scalar curvature can be somewhere
negative.
###### Theorem 1.6.
Let $(M^{m},\langle\,,\,\rangle)$ be a non-parabolic Riemannian manifold of
dimension $m\geq 3$ with scalar curvature $s(x)$. Suppose that the conformal
Laplacian (1.3) admits a positive Green function on $M$.
Let $\widetilde{s}(x)\in C^{\infty}(M)$ be such that
(1.21) $\text{$\widetilde{s}_{+}$ has compact
support,}\qquad\widetilde{s}(x)\asymp s(x)\,\text{ as $x$ diverges.}$
Then, there exists $\delta>0$ such that if
(1.22) $\widetilde{s}(x)\leq\delta\qquad\text{on }M,$
then $\widetilde{s}$ is realizable via a uniformly equivalent, conformal
deformation $\widetilde{\langle}\,,\,\rangle$ of $\langle\,,\,\rangle$.
###### Remark 1.6.
_Observe that ( 1.21) implies that the original scalar curvature $s(x)$ is
non-positive outside a compact set. _
Note that Theorems 1.5 and 1.6 seem to be new even in the simpler case
$\widetilde{s}(x)\leq 0$ on $M$. In this respect, these are skew with the main
theorem in [48] and with Theorem 2.30 of [12].
The requirement (1.3) shows the central role played by the conformal Laplacian
$L_{\langle}\,,\,\rangle$ for the Yamabe equation. The relevance of
$L_{\langle\,,\,\rangle}$ for the original Yamabe problem is well-known and
highlighted, for instance, in the comprehensive [47]. We will spend a
considerable part of the paper to discuss on assumptions like (1.3). Clearly,
Theorem 1.6 is tightly related to Aviles-McOwen’s Theorem 1.3, a parallel
which is even more evident in view of the next
###### Corollary 1.2.
Let $(M,\langle\,,\,\rangle)$ be a complete manifold of dimension $m\geq 3$
with a pole $o$ and sectional curvature $K$ satisfying
(1.23) $K\leq-\kappa^{2},$
for some constant $H>0$. Suppose further that
(1.24) $s(x)\geq-\frac{(m-1)^{3}}{(m-2)}\kappa^{2}$
on $M$. Let $\widetilde{s}(x)\in C^{\infty}(M)$ satisfying
(1.25) $\displaystyle-C_{1}\leq\widetilde{s}(x)\leq-C_{2}<0\qquad\text{outside
some compact set,}$
for some constants $0<C_{2}<C_{1}$. Then, there exists $\delta>0$ such that if
(1.26) $\widetilde{s}(x)\leq\delta\qquad\text{on }M,$
then $\widetilde{s}(x)$ is realizable by a conformal deformation
$\widetilde{\langle\,,\,\rangle}$ of $\langle\,,\,\rangle$ which is uniformly
equivalent to $\langle\,,\,\rangle$.
###### Remark 1.7.
_Corollary 1.2 improves on Theorem 1.3, since requirement (1.9) in Theorem 1.3
implies (1.23), (1.24). In particular, the hyperbolic space
$\mathbb{H}^{m}_{\kappa}$ of sectional curvature $-\kappa^{2}$ satisfies all
the assumptions of Corollary 1.2, being $s(x)=-m(m-1)\kappa^{2}$. Moreover, as
the proof in [8] shows, (1.9) is essential to ensure the existence of
$\delta>0$ in (1.11); the case of equality in (1.24) seems, therefore, hardly
obtainable with the approach described in [8]. It is worth to observe that the
existence of a pole and the pinching assumption (1.9) on the sectional
curvature are needed in [8] to apply both the Laplacian comparison theorems
(from above and below) for the distance function $r(x)=\mathrm{dist}(x,o)$ in
order to find suitable radial sub- and supersolutions. On the contrary, here
the weaker (1.23) and (1.24) are just used to ensure that the conformal
Laplacian has a positive Green function. _
We pause for a moment to comment on assumption (1.22). Both Theorems 1.5 and
1.6 will be consequences of Theorem 2.3 below, and thus they will be proved
via a common technique. The reason why assumption (1.22) is required in
Theorem 1.6 but not in Theorem 1.5 can be summarized in the existence, in the
second case, of a global, positive supersolution for the conformal Laplacian
(that is, a solution $w$ of $L_{\langle\,,\,\rangle}w\geq 0$ on $M$) which is
bounded both from below and from above by positive constants; one can take,
for instance, $u\equiv 1$. Such a function is not possible to construct in the
general setting of Theorem 1.6 (see Remark 6.2 below for deepening). We stress
that, unfortunately, the value of $\delta$ in Corollary 1.2 is not explicit:
indeed, it depends on a uniform $L^{\infty}$ bound for solutions of some
suitable PDEs, which is shown to exist via an indirect method.
The need of (1.22) to obtain existence for $\widetilde{\langle\,,\,\rangle}$
is investigated in Remark 6.4. It is very interesting that the same condition
(1.22) appears both in our theorem and in Aviles-McOwen’s one, as well as in
Theorems 0.1 in [72] and 1.1 in [71], although the techniques to prove them
are different. This may suggest that, in general, (1.22) could not be
removable. However, at present we still have no counterexample showing that
(1.22) is necessary. For future work, we thus feel interesting to investigate
the next
###### Question.
Can assumption (1.22) in Theorem 1.6 be removed, even without expecting the
new metric to be uniformly equivalent to $\langle\,,\,\rangle$?
## 2\. Introduction, II: our main results in their general setting
Although the prescribed scalar curvature problem is the main focus of our
investigation, the techniques developed here allow us to study more general
classes of PDEs, namely nonlinear extensions (described in (2.4)) of the
equation
(2.1) $\Delta u+a(x)u-b(x)u^{\sigma}=0\qquad\text{on }M,\qquad u>0\ \text{ on
}M$
with $\sigma>1$, $a,b\in C^{\infty}(M)$ and sign-changing $b$. Note that the
signs of $a,b$ are reversed with respect to those of $s,\widetilde{s}$ in
(1.2), and that $\sigma$ can be greater than $\frac{m+2}{m-2}$, preventing a
direct use of variational techniques. However, when
$\sigma\leq\frac{m+2}{m-2}$ and $b(x)<0$ on $M$, the investigation of (2.1) on
Euclidean space is still the core of a very active area of research. In this
respect, we quote the seminal [17] and, for sign-changing $b(x)$ (and singular
$a(x)$), the recent [30].
As a matter of fact, even for (2.1) the spectral properties of the linear part
$L=-\Delta-a(x)$ play a prominent role, in particular the analysis of the
fundamental tone $\lambda_{1}^{L}(M)$ of the Friedrichs extension of
$\left(L,C^{\infty}_{c}(M)\right)$. We recall that $\lambda^{L}_{1}(M)$ is
characterized via the Rayleigh quotient as follows:
(2.2)
$\lambda_{1}^{L}(M)=\inf_{0\not\equiv\varphi\in\mathrm{Lip}_{c}(M)}\frac{\int_{M}\big{[}|\nabla\varphi|^{2}-a(x)\varphi^{2}\big{]}\mathrm{d}x}{\|\varphi\|^{2}_{L^{2}(M)}}.$
For instance, if $\lambda_{1}^{L}(M)<0$, the situation is somewhat rigid:
* (a)
if $b(x)\leq 0$, then (2.1) has no positive solutions. This follows from a
direct spectral argument111Indeed, assume the contrary and let $u>0$ solves
(2.1); then, $u$ is a positive solution of $Lu\geq 0$, and a result of [32,
54, 3] implies that $\lambda_{1}^{L}(M)\geq 0$, contradicting our assumption.;
* (b)
if $b(x)\geq 0$ and the zero set of $b$ is small in a suitable spectral sense,
then there always exist a minimal and a maximal (possibly coinciding) positive
solutions of (2.1); see [48, 63] and Section 2.4 in [12].
It is important to underline that, in both cases, the geometry of $M$ only
reveals via the spectral properties of $L$. In other words, no _a-priori_
assumptions of completeness of $M$, nor curvature nor topological requests are
made. As suggested by $(a)$ and $(b)$ above, it seems that the subtler case is
that of investigating existence under the assumption $\lambda_{1}^{L}(M)\geq
0$. This condition is often implicitly met in the literature and it is
automatically satisfied in many geometric situations. This happens, for
instance, for Theorems 1.1 to 1.4.
There is another aspect of the above picture which is worth mentioning.
Partial differential equations similar to (2.1) are of interest even for
quasilinear operators more general than the Laplacian. Just to give an
example, of a certain importance in Physics, we can consider the general
equation for radiative cooling
(2.3) $\kappa^{-1}\mathrm{div}\big{(}\kappa|\nabla u|^{p-2}\nabla
u\big{)}-\big{(}\tau\kappa^{-1}\big{)}u^{4}=0\qquad\text{on }\mathbb{R}^{m},$
where $\kappa>0$ is the coefficient of the heat conduction and $\tau$ is a
function describing the radiation (see [70], p.9). The existence problem for
this type of quasilinear PDEs when the coefficient of the nonlinearity changes
sign seems to be quite open. This suggests to extend our investigation to the
existence of positive solutions to the quasilinear, Yamabe-type equation
(2.4) $\Delta_{p,f}u+a(x)u^{p-1}-b(x)F(u)=0\qquad\text{on }M,$
where $f\in C^{\infty}(M)$,
(2.5) $\Delta_{p,f}u=e^{f}\mathrm{div}\big{(}e^{-f}|\nabla u|^{p-2}\nabla
u\big{)}$
and $F(u)$ is a nonlinearity satisfying the following assumptions:
(2.6) $\left\\{\begin{array}[]{l}F\in C^{0}(\mathbb{R}),\quad F(0)=0,\quad
F>0\ \text{ on }\mathbb{R}^{+},\\\\[5.69046pt]
\displaystyle\frac{F(t)}{t^{p-1}}\ \text{ is strictly increasing on
$\mathbb{R}^{+}$},\\\\[8.5359pt] \displaystyle\lim_{t\rightarrow
0^{+}}\frac{F(t)}{t^{p-1}}=0,\qquad\lim_{t\rightarrow+\infty}\frac{F(t)}{t^{p-1}}=+\infty\end{array}\right.$
The prototype example of $F(t)$ is $F(t)=t^{\sigma}$ for $\sigma>p-1$ and
$t\geq 0$. Of course (2.1) is recovered by choosing $p=2$ and $f$ constant. We
underline that, even for (2.4) in the Euclidean space and with
$F(t)=t^{\sigma}$, there seems to be no result covering the cases described in
Theorems 2.1 and 2.2. The family of operators in (2.5) above encompasses two
relevant geometrical cases: $\Delta_{p,0}$ the standard $p$-Laplacian
$\Delta_{p}$, and $\Delta_{2,f}$ the drifted Laplacian, $\Delta_{f}$,
appearing, for instance, in the analysis of Ricci solitons and quasi-Einstein
manifolds. Note that the radiative cooling equation is of type (2.4) provided
$1<p\leq 5$. Note also that, since the definition of $\Delta_{p,f}$ is
intended in the weak sense, solutions will be, in general, only of class
$C^{1,\mu}_{\mathrm{loc}}(M)$ by [78].
###### Notation.
_Hereafter, with a slight abuse of notation, with
$C^{1,\mu}_{\mathrm{loc}}(M)$ we mean that for each relatively compact open
set $\Omega\subset M$, there exists $\mu=\mu(\Omega)\in(0,1)$ such that $u\in
C^{1,\mu}(\overline{\Omega})$. _
###### Remark 2.1.
_We stress that, with possibly the exception of Theorem 1.1 when
$F(t)=t^{\sigma}$ and $\sigma\leq p^{*}-1$, the techniques used to prove
Theorems 1.1 to 1.4 seem hard to extend to deal with (2.5) for nonradial $f$,
even for $p=2$. For constant $f$, it seems also very difficult to adapt them
to investigate (2.5) when $p\neq 2$. In particular, the transformation
performed in [13] for $p=2$ to absorb the linear term $a(x)u$ can only be
applied when the driving operator is linear. _
To state our main result, Theorem 2.1 below, we need to introduce some
terminology. Let $\mathrm{d}\mu_{f}$ be the weighted measure
$e^{-f}\mathrm{d}x$, with $\mathrm{d}x$ the Riemannian volume element on $M$.
For $V\in L^{\infty}_{\mathrm{loc}}(M)$, we consider the functional $Q_{V}$
defined on $\mathrm{Lip}_{c}(M)$ by
(2.7)
$Q_{V}(\varphi)=\frac{1}{p}\left[\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}-\int_{M}V|\varphi|^{p}\mathrm{d}\mu_{f}\right].$
Its Gateaux derivative $Q^{\prime}_{V}$ is given by
(2.8) $Q_{V}^{\prime}(w)=-\Delta_{p,f}w-V|w|^{p-2}w\qquad\text{for }w\in
W^{1,p}_{\mathrm{loc}}(M).$
When $M=\mathbb{R}^{m}$, the spectral properties of $Q_{V}$ have been
investigated in [5, 4, 33] and in a series of papers by Y. Pinchover and K.
Tintarev (see in particular [67, 68]). From now on, we follow the notation and
terminology in [68]. In the linear case $p=2$, $f\equiv 0$, that is, for the
Schrödinger operator $Q^{\prime}_{V}=-\Delta-V$, we refer the reader to [54,
3, 60, 59, 55].
To begin with, and according to [55, 67], we recall the following:
###### Definition 2.1.
For $V\in L^{\infty}_{\mathrm{loc}}(M)$, define $Q_{V}$ as in (2.7) and let
$\Omega\subseteq M$ be an open set.
* i)
$Q_{V}$ is said to be _non-negative_ on $\Omega$ (shortly, $Q_{V}\geq 0$) if
and only if $Q_{V}(\varphi)\geq 0$ for each
$\varphi\in\mathrm{Lip}_{c}(\Omega)$, that is, if and only if the Hardy type
inequality
(2.9)
$\int_{M}V(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\
\varphi\in\mathrm{Lip}_{c}(\Omega).$
holds.
* ii)
$Q_{V}$ is said to be _subcritical_ (or _non-parabolic_) on $\Omega$ if and
only if $Q_{V}\geq 0$ there, and there exists $w\in
L^{1}_{\mathrm{loc}}(\Omega)$, $w\geq 0$, $w\not\equiv 0$ on $\Omega$, such
that
(2.10) $\int_{M}w(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq
Q_{V}(\varphi)\qquad\forall\ \varphi\in\mathrm{Lip}_{c}(\Omega).$
Sometimes, especially in dealing with the prescribed scalar curvature problem
and when no possible confusion arises, we also say that $Q_{V}^{\prime}$, and
not $Q_{V}$, is non-negative (or subcritical). The term “non-parabolic" is
justified by the following statement for $Q_{0}$ (that is, $Q_{V}$ with
$V(x)\equiv 0$), which is part of Proposition 4.4 below:
###### Proposition 2.1.
Let $(M,\langle\,,\,\rangle)$ be Riemannian, $f\in C^{0}(M)$ and $p>1$. Then,
$Q_{0}$ is subcritical on $M$ if and only if there exists a non-constant,
positive weak solution $g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of
$\Delta_{p,f}g\leq 0$.
According to the literature, the existence of such $g$ is one of the
equivalent conditions that characterize $M$ as being not $p$-parabolic; there
are various other characterizations of $p$-parabolicity, given in terms of
Green kernels, $p$-capacity of compact sets, Ahlfors’ type maximum principles,
and so on. We refer to the survey [34] for deepening in the linear case $p=2$,
and to (see [79, 64, 41, 40, 38]) for $p\neq 2$. The equivalence in
Proposition 2.1 has been observed, in the linear setting, by [7, 19, 49], and
for $p\neq 2$ it has also recently been proved in [22] with a technique
different from our.
In fact, all of these characterizations of the non-parabolicity of
$-\Delta_{p,f}$ can be seen as a special case of a theory developed in [55,
66] (when $p=2$) and in [67, 68] for operators $Q_{V}$ with potential. In
Section 4, we recall the main result in [67, 68], the ground state
alternative, and we give a proof of it by including a further equivalent
condition, see Theorem 4.1 below; as a corollary, we prove Propositions 4.4
and 2.1.
###### Remark 2.2.
_In the prescribed scalar curvature problem, the role of $\langle\,,\,\rangle$
and $\widetilde{\langle}\,,\,\rangle$ can be exchanged. Such a symmetry
suggests that those geometric conditions which are invariant with respect to a
conformal change of the metric turn out to be more appropriate to deal with
the Yamabe equation. This is the case for the non-negativity and the
subcriticality of the conformal Laplacian $L_{\langle\,,\,\rangle}$ of
$(M,\langle\,,\,\rangle)$ in (1.3). In fact, the covariance of $L$ with
respect to the conformal deformation
$\displaystyle\widetilde{\langle\,,\,\rangle}=u^{\frac{4}{m-2}}\langle\,,\,\rangle$
of the metric:_
$L_{\widetilde{\langle}\,,\,\rangle}(\cdot)=u^{-\frac{m+2}{m-2}}L_{\langle\,,\,\rangle}(u\cdot)$
_implies that, for each $0\leq w\in L^{1}_{\mathrm{loc}}(M)$ and
$\varphi\in\mathrm{Lip}_{c}(M)$,_
$\displaystyle\int_{M}\Big{[}\|\widetilde{\nabla}\varphi\|^{2}+\frac{\widetilde{s}}{c_{m}}\varphi^{2}\Big{]}\mathrm{d}\widetilde{x}-\int_{M}\widetilde{w}\varphi^{2}\mathrm{d}\widetilde{x}=\int_{M}\Big{[}|\nabla(u\varphi)|^{2}+\frac{s}{c_{m}}(u\varphi)^{2}\Big{]}\mathrm{d}x-\int_{M}w(u\varphi)^{2}\mathrm{d}x,$
_where $\sim$ superscript indicates quantities referred to
$\widetilde{\langle}\,,\,\rangle$, $\|\cdot\|$ is its induced norm, and_
$\widetilde{w}=wu^{-\frac{4}{m-2}}\in
L^{1}_{\mathrm{loc}}(M),\quad\widetilde{w}\geq 0.$
_Consequently, $L_{\langle\,,\,\rangle}$ is non-negative (resp. subcritical)
if and only if so is $L_{\widetilde{\langle}\,,\,\rangle}$. _
###### Remark 2.3.
_As a direct consequence of the ground state alternative, the positivity of
the Yamabe invariant $Y(M)$ in Zhang’s Theorem 1.1 implies that
$L_{\langle\,,\,\rangle}$ is subcritical (see Remark 4.3). On the other hand,
in our Theorem 1.5 the subcriticality of $L_{\langle\,,\,\rangle}$ follows
combining the non-parabolicity of $M$ and $s(x)\geq 0$. Although, in general,
the positivity of $Y(M)$ might not imply the non-parabolicity of $M$, this is
so if $M$ is scalar flat outside a compact set and $\mathrm{vol}(M)=+\infty$.
Indeed, if $s(x)\equiv 0$ outside a compact set $K$, then $Y(M)>0$ gives the
validity of an $L^{2}$-Sobolev inequality on $M\backslash K$, and coupling
with $\mathrm{vol}(M)=+\infty$ the non-parabolicity of $M$ follows by a result
in [20, 64]. We underline that, in the same assumptions, again by [20, 64]
property $\mathrm{vol}(M)=+\infty$ is automatic when $M$ is geodesically
complete. Summarizing, if the manifold in Zhang’s Theorem 1.1 is scalar flat
near infinity, the geometric requirements there properly contain those of our
Theorem 1.5. _
We are now ready to state
###### Theorem 2.1.
Let $M^{m}$ be a Riemannian manifold, $f\in C^{\infty}(M)$ and
$p\in(1,+\infty)$. Suppose that $Q_{0}$ is subcritical on $M$, and let $a\in
L^{\infty}_{\mathrm{loc}}(M)$ be such that $Q_{a}$ is subcritical on $M$.
Consider $b\in L^{\infty}_{\mathrm{loc}}(M)$, and assume
* $i)$
$b_{-}(x)$ has compact support;
* $ii)$
$a(x)=O\big{(}b(x)\big{)}$ as $x$ diverges;
* $iii)$
for some $\theta>0$, $\big{(}a(x)-\theta b_{+}(x)\big{)}_{-}\in
L^{1}(M,\mathrm{d}\mu_{f})$.
Fix a nonlinearity $F(t)$ satisfying (2.6). Then, there exists $\delta>0$ such
that if
(2.11) $b(x)\geq-\delta\qquad\text{on }M,$
there exists a weak solution $u\in C^{1,\mu}_{\mathrm{loc}}(M)$ of
(2.12)
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}u+a(x)u^{p-1}-b(x)F(u)=0\qquad\text{
on M}\\\\[5.69046pt] 0<u\leq\|u\|_{L^{\infty}(M)}<+\infty.\end{array}\right.$
If we replace $ii)$ and $iii)$ by the stronger condition
$iv)\qquad b_{+}(x)\asymp a(x)\qquad\text{as }x\text{ diverges,}$
and we keep the validity of (2.11), then $u$ can also be chosen to satisfy
(2.13) $\inf_{M}u>0.$
In the next theorem, we remove requirement (2.11); see also Remark 6.4 for a
related discussion.
###### Theorem 2.2.
Let $M^{m}$ be a Riemannian manifold, $f\in C^{\infty}(M)$ and
$p\in(1,+\infty)$. Suppose that $Q_{0}$ is subcritical on $M$ and let $a\in
L^{\infty}_{\mathrm{loc}}(M)$ be such that $Q_{a}$ is subcritical on $M$.
Consider $b\in L^{\infty}_{\mathrm{loc}}(M)$, and assume
* $i)$
$b_{-}(x)$ has compact support;
* $ii^{\prime})$
$a(x)\leq 0$ outside a compact set;
* $iii^{\prime})$
$a(x),b(x)\in L^{1}(M,\mathrm{d}\mu_{f})$.
Fix a nonlinearity $F(t)$ satisfying (2.6). Then, there exists a sequence
$\\{u_{k}\\}\subset C^{1,\mu}_{\mathrm{loc}}(M)$ of distinct weak solutions of
(2.14)
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}u_{k}+a(x)u_{k}^{p-1}-b(x)F(u_{k})=0\qquad\text{
on M}\\\\[5.69046pt]
0<u_{k}\leq\|u_{k}\|_{L^{\infty}(M)}<+\infty,\end{array}\right.$
such that $\|u_{k}\|_{L^{\infty}(M)}\rightarrow 0$ as $k\rightarrow+\infty$.
If we replace $ii^{\prime})$ and $iii^{\prime})$ by the stronger condition
$iv^{\prime})\qquad a(x),b(x)\quad\text{have compact support,}$
then each $u_{k}$ also satisfies $\inf_{M}u_{k}>0$.
One of the main features in the proof of Theorems 2.1 and 2.2 above is a new
flexible technique, which is based on a direct use of the non-negativity and
subcriticality assumptions on $Q_{a}$ and $Q_{0}$. Consequently, all the
geometric information needed on $M$ is encoded in the spectral behaviour of
$Q_{0}$ and $Q_{a}$. For this reason, in Sections 4 and 5 we concentrate on
operators $Q_{V}$ to show that the assumption on $Q_{0},Q_{a}$ in Theorems 2.1
and 2.2 can be made explicit and easily verifiable in various relevant cases.
We now come to the strategy to prove Theorems 2.1 and 2.2. The lack of tools
to produce solutions in the present generality forces us to proceed along very
simple, general schemes. In particular, the argument can be roughly divided
into three parts:
* (1)
For a big relatively compact domain $\Omega$, we solve locally (2.4) with
boundary condition 1 and $b(x)$ replaced by $b_{+}(x)$. Call $z_{\Omega}$ the
solution. This is the easiest part, and is addressed in Lemma 6.1.
* (2)
We find uniform $L^{\infty}$ estimates from below and above for $z_{\Omega}$,
independent of $\Omega$. According to our geometric assumptions, these
estimates can be on the whole $M$ or on a relatively compact set $\Lambda$.
The proof of this step combines Lemmas 6.2 and 6.3, and Proposition 6.1.
* (3)
Making use of the results in Step (2), we “place" $b_{-}$ in the Dirichlet
problem for (2.4) on a domain $\Omega$ via an iterative procedure, to produce
a local solution of (2.4) that possesses uniform upper and lower bounds. The
desired global solution is then obtained by passing to the limit along an
exhaustion $\\{\Omega_{j}\\}$. Note that this is the point where a distinction
between Theorems 2.1 and 2.2 appears.
Among the lemmas, which are of independent interest, we underline and briefly
comment on the next uniform $L^{\infty}$-estimate, Lemma 6.3. This result is a
cornerstone both for steps (2) and (3).
###### Lemma 2.1 (Uniform $L^{\infty}$-estimate).
Let $M$ be a Riemannian manifold, $f\in C^{\infty}(M)$, $p\in(1,+\infty)$. Let
$A,B\in L^{\infty}_{\mathrm{loc}}(M)$ with $B\geq 0$ a.e. on $M$. Assume that
either
* $(i)$
$B\equiv 0$ and $Q_{A}$ is subcritical, or
* $(ii)$
$B\not\equiv 0$ and $Q_{A}$ is non-negative.
Suppose that there exist a smooth, relatively compact open set $\Lambda\Subset
M$ and a constant $c>0$ such that
(2.15) $A\leq cB\qquad\text{a.e. on }M\backslash\Lambda,$
and fix a smooth, relatively compact open set $\Lambda^{\prime}$ such that
$\Lambda\Subset\Lambda^{\prime}$, and a nonlinearity $F(t)$ satisfying (2.6).
Then, there exists a constant $C_{\Lambda}>0$ such that, for each smooth,
relatively compact open set $\Omega$ with $\Lambda^{\prime}\Subset\Omega$, the
solution $0<z\in C^{1,\mu}(\overline{\Omega})$ of
(2.16)
$\left\\{\begin{array}[]{ll}\Delta_{p,f}z+A(x)z^{p-1}-B(x)F(z)=0&\quad\text{on
}\Omega,\\\\[5.69046pt] z=1&\quad\text{on }\partial\Omega.\end{array}\right.$
satisfies
(2.17) $z\leq C_{\Lambda}\qquad\text{on }\Omega.$
The proof of the above estimate is accomplished by using non-negativity (resp.
subcriticality) of $Q_{A}$ alone. As far as we know, the argument in the proof
seems to be new and applicable beyond the present setting. Clearly, when $z$
is $C^{2}$ and $B\in C^{0}(M)$, $B>0$ on $M$, possibly evaluating (2.16) at a
interior maximum point $x_{0}$ we get
(2.18) $\frac{F(z)}{z^{p-1}}(x_{0})\leq\sup_{M}\left(\frac{A_{+}}{B}\right),$
whence by (2.6) $z(x_{0})$ is uniformly bounded from above. Taking into
account the boundary condition for $z$, in this case the uniform $L^{\infty}$
estimate is trivial with no assumption on $Q_{A}$. On the other hand, even a
single point at which $B(x)=0$ makes this simple argument to fail, and
actually Lemma 2.1 will be applied in cases when we have no control at all on
the zero-set of $B$. Observe that (2.15) is just assumed to hold outside of a
compact set, hence $A$ is not required to be non-positive on the set where
$B=0$. This suggests that the validity of (2.18) cannot be recovered “in the
limit" by using approximating positive functions $B_{\varepsilon}$ for $B$ and
related solutions $z_{\varepsilon}$ for $z$. Note also that, when $B\equiv 0$,
Proposition 3.4 below shows that $Q_{A}$ is necessarily non-negative, for
otherwise $z$ might not exist for sufficiently large $\Omega$’s. Therefore, in
the present generality at least the non-negativity of $Q_{A}$ on the whole $M$
needs to be assumed in any case.
We pause for a moment to comment on the subcriticality of $Q_{V}$. By its very
definition, a sufficient condition for $Q_{V}$ to be subcritical is the
coupling of the following two:
* -
$Q_{0}$ is subcritical, thus there exists $w\in L^{1}_{\mathrm{loc}}(M)$,
$w\geq 0$, $w\not\equiv 0$ such that
(2.19)
$\int_{M}w(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\
\varphi\in\mathrm{Lip}_{c}(M),\ \text{ and}$
* -
$V\leq w$, $V\not\equiv w$.
Therefore, when $Q_{0}$ is subcritical, we can state simple, explicit
conditions guaranteeing the subcriticality of $Q_{V}$ provided that we know
explicit $w\in L^{1}_{\mathrm{loc}}(M)$, $w\geq 0$, $w\not\equiv 0$ satisfying
(2.19). We define each of these $w$ a Hardy weight for $\Delta_{p,f}$.
In the literature, there are conditions to imply the subcriticality of $Q_{0}$
that involve curvature bounds, volume growths, doubling properties and Sobolev
type inequalities. For example, when $f\equiv 0$, in [41] it is proved that a
complete, non-compact manifold $M$ with non-negative Ricci curvature outside a
compact set is not $p$-parabolic (i.e. $Q_{0}$ is subcritical) if and only if
$p<m$. The interested reader can also consult [40, 73]. However, it seems
challenging to obtain explicit Hardy weights in the setting of [41, 40, 73].
Nevertheless, Hardy weights have been found in some interesting cases,
starting with the famous Hardy type inequality for Euclidean space
(2.20)
$\left(\frac{m-p}{p}\right)^{p}\int_{\mathbb{R}^{m}}\frac{|\varphi|^{p}}{r^{p}}\mathrm{d}x\leq\int_{\mathbb{R}^{m}}|\nabla\varphi|^{p}\mathrm{d}x\qquad\forall\
\varphi\in\mathrm{Lip}_{c}(\mathbb{R}^{m}),$
where $r(x)=|x|$ and $m>p$. In recent years ([19, 49, 12, 10, 1, 22, 23, 24])
it has been observed how Hardy weights are related to positive Green kernels
for $\Delta_{p,f}$. By exploiting the link established in Proposition 4.4
below, we will devote Section 5 to produce explicit Hardy weights in various
geometrically relevant cases, see Theorems 5.1, 5.2, 5.3 below: in fact, a
typical construction of Hardy weights via the Green kernel is compatible with
comparison results for the Laplacian of the distance function, and thus Hardy
weights can be transplanted from model manifolds to general manifolds, as
observed in [12], Theorem 4.15 and subsequent discussion. Moreover, the set of
Hardy weights is convex in $L^{1}_{\mathrm{loc}}(M)$, thus via simple
procedures one can produce new weights, such as multipole Hardy weights or
weights blowing up along a fixed submanifold of $M$. Hardy weights can also be
transplanted to submanifolds, but this procedure is more delicate and requires
extra care. Let $N^{n}$ be a Cartan-Hadamard manifold (i.e. a simply
connected, complete manifold of non-positive sectional curvature), and let
$M^{m}$ be a minimal submanifold of $N^{n}$. Suppose that the sectional
curvature $\bar{K}$ of $N$ satisfies $\bar{K}\leq-\kappa^{2}$, for some
constant $\kappa\geq 0$. In [19, 49], the authors proved the following Hardy
type inequality:
(2.21)
$\left(\frac{m-2}{2}\right)^{2}\int_{M}\frac{\varphi^{2}}{\rho^{2}}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{2}\mathrm{d}x\qquad\forall\
\varphi\in\mathrm{Lip}_{c}(M),$
$\rho(x)$ being the extrinsic distance in $N$ from a fixed origin $o\in N$.
The Hardy weight in (2.21) is sharp if $\kappa=0$ (in particular, if $N$ is
the Euclidean space), but not if $\kappa>0$. Here, we will prove (2.21) as a
particular case of Theorem 5.3 below, which also strengthen (2.21) to a sharp
inequality when $\kappa>0$, in particular for minimal submanifolds of
hyperbolic spaces. We stress that our Hardy weight for $\kappa>0$ is skew with
the one found in [49].
Using the Hardy inequalities mentioned before, we can rewrite the
subcriticality assumption for $Q_{0}$ and $Q_{V}$ in Theorems 2.1, 2.2 in
simple form for a wide class of manifolds; by a way of example, see Corollary
5.1 in Section 7. We conclude by rephrasing Theorems 2.1, 2.2 in the setting
of the generalized Yamabe problem.
###### Theorem 2.3.
Let $(M,\langle\,,\,\rangle)$ be a non-parabolic Riemannian manifold of
dimension $m\geq 3$ and scalar curvature $s(x)$. Suppose that the conformal
Laplacian $L_{\langle\,,\,\rangle}$ in (1.3) is subcritical, and let
$\widetilde{s}\in C^{\infty}(M)$.
* $(I)$
Assume that
* $i)$
$\widetilde{s}_{+}$ has compact support;
* $ii)$
$s_{-}(x)=O\big{(}\widetilde{s}_{-}(x)\big{)}$ as $x$ diverges;
* $iii)$
for some $\theta>0$, $\big{(}\theta\widetilde{s}_{-}(x)-s_{-}(x)\big{)}_{+}\in
L^{1}(M)$.
Then, there exists $\delta>0$ such that if
(2.22) $\widetilde{s}(x)\leq\delta,$
the metric $\langle\,,\,\rangle$ can be pointwise conformally deformed to a
new metric $\widetilde{\langle\,,\,\rangle}$ with scalar curvature
$\widetilde{s}(x)$ and satisfying
(2.23) $\widetilde{\langle\,,\,\rangle}\leq C\langle\,,\,\rangle\qquad\text{on
}M,$
for some constant $C>0$. Moreover, if $ii)$ and $iii)$ are replaced by the
stronger
$iv)\qquad s(x)\asymp\widetilde{s}(x)\qquad\text{as $x$ diverges,}$
then (under the validity of (2.22)) there exists a pointwise conformal
deformation $\widetilde{\langle\,,\,\rangle}$ of $\langle\,,\,\rangle$ as
above and satisfying
(2.24) $C_{1}\langle\,,\,\rangle\leq\widetilde{\langle\,,\,\rangle}\leq
C_{2}\langle\,,\,\rangle\qquad\text{on }M,$
for some constants $0<C_{1}\leq C_{2}$. In particular,
$\widetilde{\langle}\,,\,\rangle$ is non-parabolic, and it is complete
whenever $\langle\,,\,\rangle$ is complete.
* $(II)$
If $ii)$ and $iii)$ are replaced with
* $ii^{\prime})$
$s(x)\geq 0$ outside a compact set;
* $iii^{\prime})$
$s(x),\widetilde{s}(x)\in L^{1}(M)$,
then the existence of the desired conformal deformation is guaranteed without
the requirement (2.22), and moreover the constant $C$ in (2.23) can be chosen
as small as we wish (so that, indeed, there exist infinitely many conformal
deformations realizing $\widetilde{s}$). If $ii^{\prime})$ and $iii^{\prime})$
are replaced with
$iv^{\prime})\qquad s(x),\widetilde{s}(x)\qquad\text{have compact support,}$
each of these conformally deformed metrics $\widetilde{\langle}\,,\,\rangle$
satisfies (2.24).
The paper is organized as follows. In Section 3 we collect some basic material
on $Q_{V}$ and $Q_{V}^{\prime}$. Section 4 will then be devoted to the
criticality theory for $Q_{V}$, its link with Hardy weights and with a
$Q_{V}$-capacity theory. In Section 5, we use comparison geometry to produce
sharp Hardy inequalities. Section 6 contains the proof of Lemma 2.1 and of our
main Theorems 2.1, 2.2. Then, in Section 7 we derive our geometric
corollaries, and we place them among the existing literature. Finally, in the
Appendix we give a full proof of the pasting lemma, an important technical
result for the $Q_{V}$-capacity theory. Besides the presence of new results, a
major concern of Sections 3 to 5 is to help the reader to get familiar with
various aspects of the theory of Schrödinger type operators $Q^{\prime}_{V}$.
For this reason, the experienced reader may possibly skip them and go directly
to Section 6.
## 3\. Preliminaries
In this section we recall some general facts for the operators
$\Delta_{p,f},Q_{V},Q^{\prime}_{V}$ that are extensions, to a quasilinear
setting, of some classical results of spectral theory (see [55, 66, 54, 3]).
The interested reader may consult [5, 4, 33, 67, 68] for further information.
###### Notation.
_Hereafter, given two open subsets $\Omega,U$, with $\Omega\Subset U$ we
indicate that $\Omega$ has compact closure contained in $U$. We say that
$\\{\Omega_{j}\\}$ is an exhaustion of $M$ if it is a sequence of relatively
compact, connected open sets $\Omega_{j}$ with smooth boundary and such that
$\Omega_{j}\Subset\Omega_{j+1}\Subset M$, $M=\bigcup_{j}\Omega_{j}$. The
symbol $1_{U}$ denotes the characteristic function of a set $U$, and the
symbol $\doteq$ is used to define an object. _
###### Definition 3.1.
Let $\Omega\subset M$ be an open set and, for $V\in
L^{\infty}_{\mathrm{loc}}(\Omega)$, let $Q_{V},Q_{V}^{\prime}$ be as in (2.7),
(2.8). We say that $w\in W^{1,p}_{\mathrm{loc}}(\Omega)$ is a supersolution on
$\Omega$ (respectively, subsolution, solution) if $Q_{V}^{\prime}(w)\geq 0$
weakly on $\Omega$ (resp. $\leq 0$, $=0$) that is, if
$\int_{\Omega}|\nabla w|^{p-2}\langle\nabla
w,\nabla\varphi\rangle\mathrm{d}\mu_{f}-\int_{\Omega}V|w|^{p-2}w\varphi\geq
0\qquad\text{(resp. }\leq 0,\,=0)$
for each non-negative $\varphi\in\mathrm{Lip}_{c}(\Omega)$.
The basic technical material that is necessary for our purposes is summarized
in the following
###### Theorem 3.1.
Let $\Omega\Subset M$ be a relatively compact, open domain with $C^{1,\alpha}$
boundary for some $0<\alpha<1$. Let $f\in C^{\infty}(M)$, $p\in(1,+\infty)$,
$V\in L^{\infty}_{\mathrm{loc}}(M)$ and define $Q_{V},Q^{\prime}_{V}$ as in
(2.7), (2.8). Let $g\in L^{\infty}(\Omega)$, $\xi\in
C^{1,\alpha}(\partial\Omega)$ and suppose that $u\in W^{1,p}(\Omega)$ is a
solution of
(3.1) $\left\\{\begin{array}[]{ll}Q^{\prime}_{V}(u)=g&\quad\text{on
}\Omega,\\\\[5.69046pt] u=\xi&\quad\text{on
}\partial\Omega.\end{array}\right.$
Then,
* $(1)$
_[Boundedness]_ $u\in L^{\infty}_{\mathrm{loc}}(\Omega)$, and for any
relatively compact, open domains $B\Subset B^{\prime}\Subset\Omega$ there
exists a positive constant
$C=C(p,f,m,g,\xi,\Omega,\|u\|_{L^{p}(B^{\prime},\mathrm{d}\mu_{f})})$ such
that
$\|u\|_{L^{\infty}(B)}\leq C.$
If $\xi\in C^{2,\alpha}(\partial\Omega)$, $C$ can be chosen globally on
$\Omega$, and thus $u\in L^{\infty}(\Omega)$.
* $(2)$
_[ $C^{1,\mu}$-regularity]_ When $u\in L^{\infty}(\Omega)$, there exists
$\mu\in(0,1)$ depending on $p,f,m,g,\alpha$ and on upper bounds for
$\|u\|_{L^{\infty}},\|g\|_{L^{\infty}},\|\xi\|_{C^{1,\alpha}},\|V\|_{L^{\infty}}$
on $\Omega$ such that
$\|u\|_{C^{1,\mu}(\overline{\Omega})}\leq C$
for some constant $C$ depending on $\alpha,p$, the geometry of $\Omega$ and
upper bounds for $\|u\|_{L^{\infty}}$,
$\|g\|_{L^{\infty}},\|\xi\|_{C^{1,\alpha}},\|V\|_{L^{\infty}}$ on $\Omega$.
* $(3)$
_[Harnack inequality]_. For any relatively compact open sets $B\Subset
B^{\prime}\Subset\Omega$ there exists $C=C(f,p,m,B,B^{\prime})>0$ such that,
for each $u\geq 0$, $u\in W^{1,p}(\Omega)$ solution of $Q^{\prime}_{V}(u)=0$
on $\Omega$,
(3.2) $\sup_{B}u\leq C\inf_{B^{\prime}}u.$
In particular, either $u>0$ on $\Omega$ or $u\equiv 0$ on $\Omega$.
* $(3a)$
_[Half-Harnack inequalities]_ For any relatively compact, open sets $B\Subset
B^{\prime}\Subset\Omega$ the following holds:
* _(Subsolutions)_ for each $s>p-1$, there exists $C=C(f,p,m,B,B^{\prime},V,s)>0$ such that for each $u\geq 0$ , $u\in W^{1,p}(\Omega)$ solution of $Q^{\prime}_{V}(u)\leq 0$ on $\Omega$
(3.3) $\sup_{B}u\leq C\|u\|_{L^{s}(B^{\prime})};$
* _(Supersolutions)_ for each
$s\in\left(0,\frac{(p-1)m}{m-p}\right)\quad\text{if }p<m,\qquad
s\in(0,+\infty)\quad\text{if }p\geq m,$
there exists $C=C(f,p,m,B,B^{\prime},V,s)>0$ such that for each $u\geq 0$ ,
$u\in W^{1,p}(\Omega)$ solution of $Q^{\prime}_{V}(u)\geq 0$ on $\Omega$
(3.4) $\|u\|_{L^{s}(B^{\prime})}\leq C\inf_{B}u.$
* $(4)$
_[Hopf lemma]_ Suppose that $\xi\geq 0,g\geq 0$ and let $u\in
C^{1}(\overline{\Omega})$ be a solution of (3.1) with $u\geq 0$, $u\not\equiv
0$. If $x\in\partial\Omega$ is such that $u(x)=\xi(x)=0$, then, indicating
with $\nu$ the inward unit normal vector to $\partial\Omega$ at $x$ we have
$\langle\nabla u,\nu\rangle(x)>0$.
###### Remark 3.1.
__
* (1)
The local boundedness of $u$ is a particular case of Serrin’s theorem, see
[70], Theorem 7.1.1, and does not need the boundary condition. When $\xi\in
C^{2,\alpha}(\partial\Omega)$, global boundedness can be reached via a
reflection technique described at page 54 of [33], see also [77, 11].
* (2)
is a global version, Theorem 1 of [50], of a local regularity result in [78]
and [27].
* (3)
is due to J. Serrin, see Theorem 7.2.1 in [70] for $p<m$, the discussion at
the beginning of Section 7.4 therein for $p=m$, and Theorem 7.4.1 for $p>m$.
* (3a)
The half-Harnack for subsolutions can be found in Theorem 7.1.1 of [70], the
one for supersolutions in the subsequent Theorems 7.1.2 (case $p<m$) and 7.4.1
(case $p>m$) of [70]. Again, see the discussion at the beginning of Section
7.4 of [70].
* (4)
The Hopf lemma can be found in Corollary 5.5 of [69].
An important tool for our investigation is the following Lagrangian
representation in [67].
###### Proposition 3.1.
For each $\varphi,g\in W^{1,p}_{\mathrm{loc}}(M)$, with $\varphi/g$ a.e.
finite on $M$, the Lagrangian
(3.5)
$\mathcal{L}(\varphi,g)=|\nabla\varphi|^{p}+(p-1)\left(\frac{\varphi}{g}\right)^{p}|\nabla
g|^{p}-p\left(\frac{\varphi}{g}\right)^{p-1}|\nabla g|^{p-2}\langle\nabla
g,\nabla\varphi\rangle$
satisfies $\mathcal{L}(\varphi,g)\geq 0$ on $M$, and
$\mathcal{L}(\varphi,g)\equiv 0$ on some connected open set $U$ if and only if
$\varphi$ is a constant multiple of $g$ on $U$.
Moreover, suppose that $g\in W^{1,p}_{\mathrm{loc}}(M)$ is a positive solution
of $Q_{V}^{\prime}(g)=0$ (resp. $Q_{V}^{\prime}(g)\geq 0$) on $M$. Then, for
$\varphi\in L^{\infty}_{c}(M)\cap W^{1,p}(M)$, $\varphi\geq 0$ it holds
(3.6)
$Q_{V}(\varphi)=\int_{M}\mathcal{L}(\varphi,g)\mathrm{d}\mu_{f}\qquad(\text{resp,
}\geq).$
###### Proof.
The non-negativity of $\mathcal{L}(\varphi,g)$ follows by applying Cauchy-
Schwarz and Young inequalities on the third addendum in (3.5), and analyzing
the equality case, $\mathcal{L}(\varphi,g)\equiv 0$ if and only if
$\varphi=cg$ on $M$, for some constant $c\in\mathbb{R}$.
We now prove the integral (in)equality in (3.6). By Harnack inequality, $g$ is
locally essentially bounded from below on $M$. This, combined with our
regularity requirement on $\varphi$, guarantees that $\varphi^{p}/g^{p-1}\in
W^{1,p}(M)$ and is compactly supported. Thus, we integrate on $M$ the
pointwise identity
$|\nabla\varphi|^{p}-|\nabla g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\varphi^{p}}{g^{p-1}}\right)\rangle=\mathcal{L}(\varphi,g),$
and couple with the weak definition of $Q_{V}^{\prime}(g)=0$ (resp. $\geq 0$)
applied to the test function $\varphi^{p}/g^{p-1}$:
$0=\int_{M}|\nabla g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\varphi^{p}}{g^{p-1}}\right)\rangle\mathrm{d}\mu_{f}-\int_{M}Vg^{p-1}\left(\frac{\varphi^{p}}{g^{p-1}}\right)\mathrm{d}\mu_{f}\qquad\text{(resp.
}\leq\text{)}$
to deduce (3.6). ∎
Rewriting the expression of $\mathcal{L}(\varphi,g)$ we deduce the next useful
Picone type inequality due to [4, 26, 6].
###### Proposition 3.2.
Let $M$ be a Riemannian manifold, and let $\Omega\Subset M$ be a relatively
compact, connected open set. Then, the functional
(3.7) $I(w,z)=\int_{\Omega}|\nabla w|^{p-2}\left\langle\nabla
w,\nabla\frac{w^{p}-z^{p}}{w^{p-1}}\right\rangle\mathrm{d}\mu_{f}-\int_{\Omega}|\nabla
z|^{p-2}\left\langle\nabla
z,\nabla\frac{w^{p}-z^{p}}{z^{p-1}}\right\rangle\mathrm{d}\mu_{f}$
is non-negative on the set
${\cal D}_{\Omega}=\left\\{(w,z)\in W^{1,p}(\Omega)\times W^{1,p}(\Omega)\ :\
w,z\geq 0\ {\rm on}\ \Omega\ ,\ \frac{w}{z},\frac{z}{w}\in
L^{\infty}(\Omega)\right\\}.$
Furthermore, $I(w,z)=0$ if and only if $w=Cz$ on $\Omega$, for some constant
$C>0$.
###### Proof.
Since $w,z\in\mathcal{D}_{\Omega}$, it is easy to see that
$\frac{w^{p}}{z^{p-1}},\frac{z^{p}}{w^{p-1}}\in W^{1,p}(\Omega)$. We can
therefore expand the integrand in (3.7) and rearrange to deduce that
$I(w,z)=\int_{\Omega}\big{[}\mathcal{L}(w,z)+\mathcal{L}(z,w)\big{]}\mathrm{d}\mu_{f},$
with $\mathcal{L}$ as in (3.5). The first part of previous proposition then
gives the desired inequality. ∎
Now, we investigate property $Q_{V}\geq 0$ and its consequences. By its very
definition, $Q_{V}\geq 0$ on an open set $\Omega\subset M$ is equivalent to
the non-negativity of the fundamental tone
(3.8)
$\lambda_{V}(\Omega)=\inf_{0\not\equiv\varphi\in\mathrm{Lip}_{c}(\Omega)}\frac{pQ_{V}(\varphi)}{\|\varphi\|^{p}_{L^{p}(\Omega,\mathrm{d}\mu_{f})}}.$
If $\Omega$ is a relatively compact domain with smooth boundary, then it is
well-known that the infimum (3.8) is attained by a first eigenfunction
$\phi\not\equiv 0$ solving Euler-Lagrange equation
$\left\\{\begin{array}[]{l}\displaystyle
Q_{V}^{\prime}(\phi)=\lambda_{V}(\Omega)|\phi|^{p-2}\phi\qquad\text{on
}\,\Omega,\\\\[5.69046pt] \displaystyle\phi=0\qquad\text{on
}\,\partial\Omega,\end{array}\right.$
and $\phi>0$ on $\Omega$ up to changing its sign 222Briefly, $|\phi|$ still
minimizes the Rayleigh quotient in (3.8), thus it satisfies the Euler-Lagrange
equation $Q_{V}^{\prime}(|\phi|)=\lambda_{V}(\Omega)|\phi|^{p-1}$, hence
$|\phi|>0$ on $\Omega$ by Harnack inequality in Theorem 3.1, $(3)$..
Furthermore, by Harnack inequality, if $\Omega\subset\Omega^{\prime}$ are two
relatively compact open sets and $\Omega^{\prime}\backslash\Omega$ has non-
empty interior, then $\lambda_{V}(\Omega)>\lambda_{V}(\Omega^{\prime})$.
The next comparison result will be used throughout the paper, and improves on
Theorem 5 of [33].
###### Proposition 3.3.
Let $M^{m},p,f$ be as above and, for $A\in L^{\infty}_{\mathrm{loc}}(M)$,
define $Q_{A},Q_{A}^{\prime}$ as in (2.7), (2.8) with $V(x)=A(x)$. Consider a
relatively compact, open set $\Omega\Subset M$ with smooth boundary, and let
$u_{1},u_{2}\in C^{1,\mu}(\overline{\Omega})$, for some $\mu\in(0,1)$.
Furthermore, suppose that, for some non-negative $B\in L^{\infty}(\Omega)$ and
a nonlinearity $F(t)$ satisfying (2.6),
(3.9)
$\left\\{\begin{array}[]{l}\Delta_{p,f}u_{1}+A|u_{1}|^{p-2}u_{1}-BF(u_{1})\geq
0,\\\\[5.69046pt] \Delta_{p,f}u_{2}+A|u_{2}|^{p-2}u_{2}-BF(u_{2})\leq
0,\\\\[5.69046pt] u_{1}\leq u_{2}\ \text{ on }\partial\Omega,\qquad u_{1}\geq
0,\,u_{2}>0\ \text{ on }\Omega.\end{array}\right.$
Then, either
* $i)$
$u_{1}\leq u_{2}$ on $\Omega$, or
* $ii)$
$B(x)\equiv 0$ on $\Omega$, $u_{2}$ satisfies $Q_{A}^{\prime}(u_{2})=0$,
$u_{2}\equiv 0$ on $\partial\Omega$, and $\lambda_{A}(\Omega)=0$.
###### Proof.
We let $\displaystyle\xi={u_{2}}_{|\partial\Omega}\in
C^{1,\mu}(\partial\Omega)$ and let
$V=A(x)-B(x)\frac{F(u_{2})}{u_{2}^{p-1}}.$
Note that $V\in L^{\infty}(\Omega)$ by (2.6). For $x\in\partial\Omega$ such
that $u_{2}(x)=0$, we let $\nu$ be the inward unit normal to $\partial\Omega$
at $x$. Then, applying Theorem 3.1 $(4)$ we deduce that
$\langle\nabla u_{2}(x),\nu\rangle(x)>0,$
by continuity, there thus exists a constant $C>0$ such that
(3.10) $u_{1}\leq Cu_{2}\qquad\text{on }T(\partial\Omega),$
for some tubular neighbourhood $T(\partial\Omega)$ of $\partial\Omega$. Using
assumption $u_{2}>0$ on $\Omega$ we can suppose that (3.10) is true on all of
$\overline{\Omega}$ with $C>1$. Because of (3.9) and since $B(x)\geq 0$, $C>1$
and, by (2.6), $F(t)/t^{p-1}$ is increasing on $\mathbb{R}^{+}$, $Cu_{2}$ is
still a supersolution:
$\begin{array}[]{l}\displaystyle\Delta_{p,f}(Cu_{2})+A(Cu_{2})^{p-1}-BF(Cu_{2})=C^{p-1}\big{[}\Delta_{p,f}u_{2}+Au_{2}^{p-1}\big{]}-BF(Cu_{2})\\\\[5.69046pt]
\displaystyle\leq C^{p-1}BF(u_{2})-BF(Cu_{2})\leq
B(Cu_{2})^{p-1}\left[\frac{F(u_{2})}{u_{2}^{p-1}}-\frac{F(Cu_{2})}{(Cu_{2})^{p-1}}\right]\leq
0.\end{array}$
Using $u_{1}\geq 0$ as a subsolution, by (3.10) and applying the method of
sub- and supersolutions, see Theorem 4.14, page 272, in [25], we find a
solution $v$ of
(3.11)
$\left\\{\begin{array}[]{l}\Delta_{p,f}v+A|v|^{p-2}v-BF(v)=0,\\\\[5.69046pt]
v=u_{2}\qquad\text{on }\partial\Omega,\end{array}\right.$
satisfying
(3.12) $u_{1}\leq v\leq Cu_{2}\qquad{\rm on}\ \Omega$
By the $C^{1,\mu}$-regularity of Theorem 3.1, $v\in
C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$. If we show that
$v\leq u_{2}$ then (3.12) implies $u_{1}\leq u_{2}$ on $\Omega$ which is the
conclusion $i)$ of the Proposition.
Suppose that this is not the case, that is, assume that the open set
$U=\\{v>u_{2}\\}$ is non-empty. We are going to prove that $ii)$ holds. Since
$v\geq u_{1}\geq 0$ and $v$ is positive on $U$, then $v>0$ on $\Omega$ as a
consequence of the Harnack inequality in Theorem 3.1 (use
$V=A(x)-B(x)F(v)/v^{p-1}$, which by (2.6) is bounded on $\Omega$).
Alternatively, one can use the version of the strong maximum principle in
Theorem 5.4.1 in [70]. Now, again by the Hopf Lemma of Theorem 3.1,
$\langle\nabla v(x),\nu(x)\rangle>0$, $\nu$ the inward unit normal to
$\partial\Omega$ at $x$, at each point $x\in\partial\Omega$ where
$v(x)=u_{2}(x)=0$. Hence, the ratio $u_{2}/v$ is well defined at $x$ along the
half line determined by $\nu$. This shows that $u_{2}/v$ and similarly
$v/u_{2}$ are in $L^{\infty}(\Omega)$. Applying Proposition 3.2 on $U$ we
deduce $I(u_{2},v)\geq 0$, and $I(u_{2},v)=0$ if and only if $u_{2}$ and $v$
are proportional on $U$. However, the positivity of the test function
$(v^{p}-u_{2}^{p})/u_{2}^{p-1}>0$ on $U$ implies, by (3.9) and (3.11), that
$0\leq\displaystyle
I(u_{2},v)\leq\displaystyle-\int_{U}B\big{(}u_{2}^{p}-v^{p}\big{)}\left(\frac{F(u_{2})}{u_{2}^{p-1}}-\frac{F(v)}{v^{p-1}}\right)\mathrm{d}\mu_{f}.$
Being $F(t)/t^{p-1}$ strictly increasing on $\mathbb{R}^{+}$ and $B\geq 0$, we
deduce
(3.13)
$B\big{(}u_{2}^{p}-v^{p}\big{)}\left(\frac{F(u_{2})}{u_{2}^{p-1}}-\frac{F(v)}{v^{p-1}}\right)\geq
0$
on $U$, whence $I(u_{2},v)=0$. We therefore conclude that $u_{2}=cv$ on $U$,
for some constant $c$ which, because of the definition of $U$, satisfies
$c>1$. Using that $v=u_{2}$ on $\partial U$, we necessarily have $v=u_{2}=0$
on $\partial U$, hence $U\equiv\Omega$. Substituting $u_{2}=cv$ on $\Omega$
into (3.13) we deduce
$B\left(c^{p}-1\right)v^{p-1}\left(\frac{F(cv)}{(cv)^{p-1}}-\frac{F(v)}{v^{p-1}}\right)\equiv
0.$
Since $v>0$ on $\Omega$ and $F(t)/t^{p-1}$ is strictly increasing, $B\equiv 0$
and, from (3.11), $v>0$ solves
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}v+A(x)|v|^{p-2}v=0\qquad\text{on
}\,\Omega\\\\[5.69046pt] \displaystyle v=0\qquad\text{on
}\,\partial\Omega.\end{array}\right.$
Consequently, $0$ admits a positive eigenfunction of $Q_{A}^{\prime}$. By a
result in [6], $\lambda_{A}(\Omega)=0$, showing the validity of $ii)$. ∎
###### Remark 3.2.
_We underline that, in the above proposition, the non-negativity of $Q_{A}$ is
not required. However, if $B\equiv 0$, $u_{2}$ turns out to be a positive
solution of $Q_{A}(u)\geq 0$, and using Proposition 3.4 below we automatically
have $\lambda_{A}(\Omega)\geq 0$. _
In what follows we shall frequently use the next formula: for
$I\subset\mathbb{R}$ and $\alpha\in C^{2}(I)$ with $\alpha^{\prime}>0$ on $I$,
and for $u\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ with $u(M)\subset I$ we
have, weakly on $M$,
(3.14)
$\Delta_{p,f}\alpha(u)=\alpha^{\prime}(u)\left|\alpha^{\prime}(u)\right|^{p-2}\Delta_{p,f}u+(p-1)\alpha^{\prime\prime}(u)\left|\alpha^{\prime}(u)\right|^{p-2}|\nabla
u|^{p}.$
A second ingredient is the following existence result that goes under the name
of the Allegretto-Piepenbrink theorem, see [5, 4, 33]. We include a proof of
the next slightly more general version, for the sake of completeness.
###### Proposition 3.4.
Let $(M,\langle\,,\,\rangle)$ be a non-compact Riemannian manifold, $f\in
C^{\infty}(M)$, $p\in(1,+\infty)$ and, for $V\in
L^{\infty}_{\mathrm{loc}}(M)$, set $Q^{\prime}_{V},Q_{V}$ as in (2.8) and
(2.7). Then, the following statements are equivalent:
* $i)$
There exists $w\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$, $w>0$ weak
solution of
(3.15) $Q^{\prime}_{V}(w)\geq 0\qquad\text{on }\,M;$
* $ii)$
There exists $u\in C^{1,\mu}_{\mathrm{loc}}(M)$, $u>0$ weak solution of
(3.16) $Q^{\prime}_{V}(u)=0\qquad\text{on }\,M;$
* $iii)$
$Q_{V}\geq 0$ on $M$.
* $iv)$
For each relatively compact domain $\Omega\Subset M$ with $C^{1,\alpha}$
boundary for some $\alpha\in(0,1)$, and for each $\xi\in
C^{1,\alpha}(\partial\Omega)$, $\xi\geq 0$, there exists a unique solution
$\varphi\in C^{1,\mu}(\overline{\Omega})$ of
(3.17) $\left\\{\begin{array}[]{ll}Q_{V}^{\prime}(\varphi)=0&\quad\text{on
}\,\Omega,\\\\[5.69046pt] \varphi=\xi&\quad\text{on
}\,\partial\Omega\end{array}\right.$
satisfying $\varphi\geq 0$ on $\Omega$. Moreover, if $\xi\not\equiv 0$, then
$\varphi>0$ on $\Omega$.
###### Proof.
The scheme of proof is $ii)\Rightarrow i)\Rightarrow iii)\Rightarrow
iv)\Rightarrow ii);$ Note that the first implication is trivial.
$i)\Rightarrow iii)$. It follows immediately from Proposition 3.1 and the non-
negativity of $\mathcal{L}$.
$iii)\Rightarrow iv)$. By assumption $\lambda_{V}(M)\geq 0$; it follows that,
for $\Omega$ as in $iv)$, by the monotonicity property for eigenvalues
$\lambda_{V}(\Omega)>0$. Hence, the variational problem associated to (3.17)
is coercive and sequentially weakly lower-semicontinuous (see also Theorem 7.1
in Appendix). Therefore (3.17) admits a weak solution $\varphi\in
W^{1,p}(\Omega)$. By the $C^{1,\mu}$-regularity of Theorem 3.1 we have that
$\varphi\in C^{1,\mu}(\overline{\Omega})$ for some $\mu\in(0,1)$. Moreover, by
the local Harnack inequality of item $(3)$, $\varphi>0$ on $\overline{\Omega}$
whenever $\phi\geq 0$ on $\Omega$, unless $\varphi\equiv 0$.
By contradiction suppose that $\varphi$ is somewhere negative in $\Omega$.
Since $\xi\geq 0$, on $\partial\Omega$,
$\varphi_{-}=-\min\\{\varphi,0\\}\in\mathrm{Lip}_{0}(\Omega)$ and it is thus
an admissible test function for (3.17) on $\Omega$. We have
$0=\displaystyle
Q_{V}^{\prime}(\varphi)[-\varphi_{-}]\doteq-\int_{\Omega}\left\\{|\nabla\varphi|^{p-2}\langle\nabla\varphi,\nabla\varphi_{-}\rangle-V|\varphi|^{p-2}\varphi\varphi_{-}\right\\}\mathrm{d}\mu_{f}\equiv
pQ_{V}(\varphi_{-})$
and therefore $\lambda_{V}(\Omega)\leq 0$, a contradiction.
$iv)\Rightarrow ii)$. Choose an exhaustion $\\{\Omega_{j}\\}$ of $M$. Let
$u_{j}>0$, $u_{j}\in C^{1,\mu_{j}}(\overline{\Omega}_{j})\subset
W^{1,p}(\Omega_{j})$ be a solution of
(3.18) $\left\\{\begin{array}[]{l}Q_{V}^{\prime}(u_{j})=0\qquad\text{on
}\Omega_{j}\\\\[5.69046pt] u_{j}=1\qquad\text{on
}\partial\Omega_{j}.\end{array}\right.$
Fix $x_{0}\in\Omega_{1}$ and rescale $u_{j}$ in such a way that
$u_{j}(x_{0})=1$ for every $j$. By Theorem 3.1 $3)$, $\\{u_{j}\\}$ is
uniformly locally bounded in $\Omega$, thus by Theorem 3.1 $2)$ $\\{u_{j}\\}$
is uniformly locally bounded in $C^{1,\mu}(\Omega)$. It follows that
$\\{u_{j}\\}$ has a subsequence converging weakly and pointwise to a weak
solution $u\in C^{1,\mu}_{\mathrm{loc}}(M)$ of
$\left\\{\begin{array}[]{l}Q_{V}^{\prime}(u)=0\qquad\text{on }\
M\\\\[5.69046pt] u(x_{0})=1.\end{array}\right.$
Since $u\geq 0$ and $u\not\equiv 0$, again by $3)$ of Theorem 3.1 we deduce
$u>0$ on $M$. This shows the validity of $ii)$. ∎
Next, we need a gluing result which we will call the pasting lemma. Although
for $V\equiv 0$ this is somehow standard (a simple proof can be given by
adapting Lemma 2.4 in [65]), the presence of a nonzero $V$ makes things more
delicate. First, we introduce some definitions. We recall that, given an open
subset $\Omega\subset M$ possibly with non-compact closure, the space
$W^{1,p}_{\mathrm{loc}}(\overline{\Omega})$ is the set of all functions $u$ on
$\Omega$ such that, for every relatively compact open set $U\Subset M$ with
$U\cap\Omega\not=\emptyset$, $u\in W^{1,p}(\Omega\cap U)$. A function $u$ in
this space is thus well behaved on relatively compact portions of
$\partial\Omega$, while no global control is assumed on the $W^{1,p}$ norm of
$u$.
###### Lemma 3.1 (The pasting lemma).
Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold, $f\in C^{\infty}(M)$,
$p\in(1,\infty)$, $V\in L^{\infty}_{\mathrm{loc}}(M)$. Let $\Omega_{1}$,
$\Omega_{2}$ be open sets such that $\Omega_{1}\subset\Omega_{2}$. For
$j=1,2$, let $u_{j}\in C^{0}(\overline{\Omega}_{j})\cap
W^{1,p}_{\mathrm{loc}}(\overline{\Omega}_{j})$ be a positive supersolution of
$Q_{V}^{\prime}$ on $\Omega_{j}$, that is, $Q_{V}^{\prime}(u_{j})\geq 0$ on
$\Omega_{j}$. If
(3.19) $u_{2}\leq u_{1}\qquad\text{on }\partial\Omega_{1}\cap\Omega_{2},$
then the positive function
(3.20)
$u\doteq\left\\{\begin{array}[]{ll}\displaystyle\min\\{u_{1},u_{2}\\}&{\rm
on}\ \overline{\Omega}_{1}\\\\[11.38092pt] \displaystyle u_{2}&{\rm on}\
\Omega_{2}\backslash\Omega_{1}\end{array}\right.$
is in $W^{1,p}_{\mathrm{loc}}(\overline{\Omega}_{2})$ and it satisfies
$Q^{\prime}_{V}(u)\geq 0$ on $\Omega_{2}$.
When $\Omega_{1}\equiv\Omega_{2}$, a general result of V.K. Le, [46],
guarantees that $\min\\{u_{1},u_{2}\\}$ is a supersolution. The pasting lemma
can then be deduced by an approximation argument, along the lines described in
[2], and we leave the details to the interested reader. In the Appendix below,
we give a quite different proof by using the obstacle problem for $Q_{V}$ and
the minimizing properties of its solutions, that might have an independent
interest.
## 4\. Criticality theory for $Q_{V}$, capacity and Hardy weights
The criticality theory for $Q_{V}$ reveals an interesting scenario, and
extends in a nontrivial way the parabolicity theory for the standard Laplacian
and for the $p$-Laplacian (developed, among others, in [34, 64, 79]). Although
a thorough description goes beyond the scope of this paper, nevertheless the
validity of the pasting lemma gives us the opportunity to complement known
results (especially those in [67, 68]) by relating them to a capacity theory
for $Q_{V}$, see Theorem 4.1 below. We underline that, although the
$Q_{V}$-capacity theory is investigated by following the same lines as those
for the standard $p$-Laplacian, as in the previous results the presence of a
nontrivial $V$ makes things subtler.
Let $Q_{V}\geq 0$ on $M$, and fix a positive _supersolution_ $g\in
C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of $Q^{\prime}_{V}$, that is, a
solution of
(4.1) $Q_{V}^{\prime}(g)\geq 0.$
For each $K\Subset\Omega\Subset M$, $K$ compact, $\Omega$ open, let
$\mathcal{D}(K,\Omega,g)=\Big{\\{}\varphi\in C^{0}(\overline{\Omega})\cap
W^{1,p}_{0}(\Omega)\ :\ \varphi\geq g\ \text{ in a neighbourhood of
}K\Big{\\}}$
and define the $Q_{V}$-capacity
$\displaystyle\mathrm{cap}_{Q_{V}}(K,\Omega,g)\doteq\displaystyle\inf_{\varphi\in\mathcal{D}(K,\Omega,g)}Q_{V}(\varphi)$
Clearly, $\mathrm{cap}_{Q_{V}}(K,\Omega,g)$ grows if we decrease $\Omega$, as
well as if we increase $K$. If $V\equiv 0$, it is customary to choose $g\equiv
1$ as solution of $\Delta_{p,f}g=0$, and we recover the classical definition
of capacity. We however underline that, for the next arguments to work, it is
essential that the fixed $g$ solves $Q_{V}^{\prime}(g)\geq 0$ on $M$, for
otherwise the basic properties needed in the next results could not hold.
###### Proposition 4.1.
Let $K$ be the closure of an open domain and suppose that
$\partial\Omega,\,\partial K$ are of class $C^{1,\alpha}$ for some
$\alpha\in(0,1)$. Then
(4.2) $\mathrm{cap}_{Q_{V}}(K,\Omega,g)=Q_{V}(u)$
where $u$ is the unique positive solution $u\in C^{0}(\overline{\Omega})\cap
W^{1,p}_{0}(\Omega)\cap C^{1,\mu}(\overline{\Omega}\backslash K)$ of
(4.3) $\left\\{\begin{array}[]{l}Q_{V}^{\prime}(u)=0\qquad\text{on
}\Omega\backslash K,\\\\[5.69046pt] u=g\quad\text{on }K,\qquad
u=0\quad\text{on }\partial\Omega.\end{array}\right.$
We call such a solution $u$ the $Q_{V}$-capacitor of $(K,\Omega,g)$.
###### Remark 4.1.
_The existence, uniqueness and positivity of $u$ is granted by $iv)$ in
Proposition 3.4. As for regularity, the interior estimate $u\in
C^{1,\mu}_{\mathrm{loc}}(\overline{\Omega}\backslash K)$ follows by [78, 27],
and the boundary continuity at $\partial K\cup\partial\Omega$ by Theorem 5.4,
page 235 in [52]. The fact that $u\in W^{1,p}(\Omega)$ follows by standard
theory of Sobolev functions333In fact, $u-g\in W^{1,p}(\Omega\backslash K)$
has zero trace on $\partial K$, thus it is the $W^{1,p}$-limit (and also, up
to extracting a subsequence, the pointwise limit) of some sequence
$\\{\varphi_{j}\\}\subset C^{\infty}(\overline{\Omega\backslash K})$ where
$\varphi_{j}\equiv 0$ in a neighbourhood of $\partial K$. Extending
$\varphi_{j}$ to be zero on $K$ we have that $g+\varphi_{j}$ is Cauchy in
$W^{1,p}$ and pointwise convergent to $u$, thus $g+\varphi_{j}\rightarrow u\in
W^{1,p}(\Omega)$.. _
###### Proof.
Let $\varphi\in\mathcal{D}(K,\Omega,g)$. First, we claim that
$\hat{\varphi}=\min\\{\varphi,g\\}\in\mathcal{D}(K,\Omega,g)$ solves
$Q_{V}(\hat{\varphi})\leq Q_{V}(\varphi)$, whence we can assume, without loss
of generality, that $\varphi\leq g$ on $\Omega$ (and hence $\varphi=g$ on a
neighbourhood of $K$).
Consider the open set $U=\\{\varphi>g\\}\Subset\Omega$. We test
$Q_{V}^{\prime}(g)\geq 0$ with the non-negative function
$(\varphi^{p}-g^{p})_{+}/g^{p-1}\in W^{1,p}_{0}(\Omega)$ and use the non-
negativity of the Lagrangian in (3.5) to deduce
(4.4) $\begin{array}[]{lcl}0&\leq&\displaystyle\displaystyle\int_{U}|\nabla
g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\varphi^{p}}{g^{p-1}}\right)\rangle\mathrm{d}\mu_{f}-\int_{U}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}g_{|U}\big{)}\\\\[11.38092pt]
&=&\displaystyle p\int_{U}\left(\frac{\varphi}{g}\right)^{p-1}|\nabla
g|^{p-2}\langle\nabla
g,\nabla\varphi\rangle\mathrm{d}\mu_{f}-(p-1)\int_{U}\left(\frac{\varphi}{g}\right)^{p}|\nabla
g|^{p}\mathrm{d}\mu_{f}\\\\[11.38092pt]
&&\displaystyle-\int_{U}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}g_{|U}\big{)}\\\\[11.38092pt]
&=&\displaystyle\int_{U}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}-\int_{U}\mathcal{L}(\varphi,g)\mathrm{d}\mu_{f}-\int_{U}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}g_{|U}\big{)}\\\\[11.38092pt]
&\leq&\displaystyle
pQ_{V}\big{(}\varphi_{|U}\big{)}-pQ_{V}\big{(}g_{|U}\big{)},\end{array}$
hence $Q_{V}(\hat{\varphi}_{|U})=Q_{V}(g_{|U})\leq Q_{V}(\varphi_{|U})$. Since
$\varphi\equiv\hat{\varphi}$ on $\Omega\backslash U$, the claim follows. Let
now $\varphi\in\mathcal{D}(K,\Omega,g)$ be such that $\varphi=g$ on $K$. By
density, we can assume that $\varphi\in\mathrm{Lip}_{0}(\Omega)$. We therefore
have $u-\varphi\in\mathrm{Lip}_{0}(\Omega\backslash K)$. Again by density, we
can further assume that $\varphi=0$ in a neighbourhood of $\partial\Omega$.
Thus, testing $Q_{V}^{\prime}(u)=0$ with
$(\varphi^{p}-u^{p})/u^{p-1}\in\mathrm{Lip}_{0}(\Omega\backslash K)$ and
proceeding as above we obtain
(4.5)
$\begin{array}[]{lcl}0&=&\displaystyle\displaystyle\int_{\Omega\backslash
K}|\nabla u|^{p-2}\langle\nabla
u,\nabla\left(\frac{\varphi^{p}}{u^{p-1}}\right)\rangle\mathrm{d}\mu_{f}-\int_{\Omega\backslash
K}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}u_{|\Omega\backslash
K}\big{)}\\\\[11.38092pt] &=&\displaystyle p\int_{\Omega\backslash
K}\left(\frac{\varphi}{u}\right)^{p-1}|\nabla u|^{p-2}\langle\nabla
u,\nabla\varphi\rangle\mathrm{d}\mu_{f}-(p-1)\int_{\Omega\backslash
K}\left(\frac{\varphi}{u}\right)^{p}|\nabla
u|^{p}\mathrm{d}\mu_{f}\\\\[11.38092pt] &&\displaystyle-\int_{\Omega\backslash
K}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}u_{|\Omega\backslash
K}\big{)}\\\\[11.38092pt] &=&\displaystyle\int_{\Omega\backslash
K}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}-\int_{\Omega\backslash
K}\mathcal{L}(\varphi,u)\mathrm{d}\mu_{f}-\int_{\Omega\backslash
K}V\varphi^{p}\mathrm{d}\mu_{f}-pQ_{V}\big{(}u_{|\Omega\backslash
K}\big{)}\\\\[11.38092pt] &\leq&\displaystyle
pQ_{V}\big{(}\varphi_{|\Omega\backslash
K}\big{)}-pQ_{V}\big{(}u_{|\Omega\backslash K}\big{)}.\end{array}$
As $u=\varphi=g$ on $K$, we conclude $Q_{V}(u)\leq Q_{V}(\varphi)$ and whence
$Q_{V}(u)\leq\mathrm{cap}(K,\Omega,g)$. Since $u$ lies in the $W^{1,p}$
closure of $\mathcal{D}(K,\Omega,g)$, equality (4.2) follows. ∎
###### Remark 4.2.
_By the pasting Lemma 3.1, note that $u$ solving (4.3) is a supersolution on
the whole $\Omega$, that is, $Q_{V}^{\prime}(u)\geq 0$ on $\Omega$. _
###### Proposition 4.2.
In the assumptions of the previous theorem, suppose that $Q_{V}^{\prime}(g)=0$
on a neighbourhood of $K$, and that $\partial K$ is smooth. Then,
(4.6) $Q_{V}(u)=\frac{1}{p}\int_{\partial K}g\left[|\nabla
g|^{p-2}\frac{\partial g}{\partial\nu}-|\nabla u|^{p-2}\frac{\partial
u}{\partial\nu}\right]\mathrm{d}\sigma_{f},$
where $\nu$ is the unit normal to $\partial K$ pointing outward of $K$.
###### Proof.
Let $T\approx\partial K\times(-\varepsilon_{0},\varepsilon_{0})\Subset\Omega$
be a tubular neighbourhood of $\partial K$ where Fermi coordinates are
defined, and let $\rho(x)$ be the smooth signed distance from $\partial K$,
that is, $\rho(x)=\mathrm{dist}(x,\partial K)$ if $x\not\in K$, and
$\rho(x)=-\mathrm{dist}(x,\partial K)$ if $x\in K$. Let
$h\in\mathrm{Lip}(\mathbb{R}^{+}_{0})$ be such that $h(t)=0$ if $t\leq 0$,
$h(t)=t$ for $t\in[0,1]$ and $h(t)=1$ for $t\geq 1$ and, for small
$\varepsilon\in(0,\varepsilon_{0})$, set $h_{\varepsilon}(t)\doteq
h(t/\varepsilon)$. Applying (4.3) on $\Omega\backslash K$ to the test function
$h_{\varepsilon}(\rho)u\in\mathrm{Lip}_{0}(\Omega\backslash K)$, using the
coarea’s formula and letting $\varepsilon\rightarrow 0$, since $g$ is $C^{1}$
we deduce
(4.7) $\displaystyle 0=\lim_{\varepsilon\rightarrow
0}Q_{V}^{\prime}(u)\big{[}h_{\varepsilon}(\rho)u\big{]}=p\displaystyle
Q_{V}\big{(}u_{\Omega\backslash K}\big{)}+\int_{\partial K}u|\nabla
u|^{p-2}\frac{\partial u}{\partial\nu}\mathrm{d}\sigma_{f}.$
In a similar way, applying $Q_{V}^{\prime}(g)=0$ on $K$ to the non-negative
test function $h_{\varepsilon}(-\rho)g\in\mathrm{Lip}_{0}(K)$ and letting
$\varepsilon\rightarrow 0$ we deduce that
(4.8) $\displaystyle\displaystyle 0=\lim_{\varepsilon\rightarrow
0}Q_{V}^{\prime}(u)\big{[}h_{\varepsilon}(-\rho)g\big{]}=\displaystyle
pQ_{V}\big{(}u_{K}\big{)}-\int_{\partial K}g|\nabla g|^{p-2}\frac{\partial
g}{\partial\nu}\mathrm{d}\sigma_{f}.$
Subtracting the two identities and using $u=g$ on $\partial K$ yields (4.6). ∎
Next, we consider the $Q_{V}$-capacity of $K$ in the whole $M$:
(4.9) $\begin{array}[]{c}\displaystyle\mathcal{D}(K,g)=\Big{\\{}\varphi\in
C^{0}_{c}(M)\cap W^{1,p}_{\mathrm{loc}}(M)\ :\ \varphi\geq g\ \text{ in a
neighbourhood of }K\Big{\\}}\\\\[11.38092pt]
\displaystyle\mathrm{cap}_{Q_{V}}(K,g)\doteq\displaystyle\inf_{\varphi\in\mathcal{D}(K,g)}Q_{V}(\varphi).\end{array}$
Let $\\{\Omega_{j}\\}$ be an exhaustion of $M$ with $K\Subset\Omega_{1}$.
Then, from the definitions it readily follows that
$\mathrm{cap}_{Q_{V}}(K,g)=\inf_{j}\mathrm{cap}_{Q_{V}}(K,\Omega_{j},g)=\lim_{j\rightarrow+\infty}\mathrm{cap}_{Q_{V}}(K,\Omega_{j},g).$
If $K$ is the closure of a open set and $\partial K$ is of class
$C^{1,\alpha}$ for some $\alpha\in(0,1)$, let $u_{j}$ be the $Q_{V}$-capacitor
of $(K,\Omega_{j},g)$. By Proposition 4.1,
(4.10) $\mathrm{cap}_{Q_{V}}(K,g)=\lim_{j\rightarrow+\infty}Q_{V}(u_{j}).$
Proposition 3.3 implies that $0\leq u_{j}\leq u_{j+1}\leq g$ for each $j$,
whence, by Dini theorem and elliptic estimates, $u_{j}$ converges locally
uniformly on $M$, in $W^{1,p}_{\mathrm{loc}}(M)$ and in the $C^{1}$ topology
on $M\backslash K$ to a weak solution $u\in C^{0}(M)\cap
W^{1,p}_{\mathrm{loc}}(M)\cap C^{1,\mu}_{\mathrm{loc}}(M\backslash K)$ of
(4.11) $\left\\{\begin{array}[]{l}Q_{V}^{\prime}(u)=0\qquad\text{on
}M\backslash K,\\\\[5.69046pt] u=g\quad\text{on }K,\qquad 0<u\leq
g\quad\text{on }M\backslash K.\end{array}\right.$
The pasting Lemma 3.1 guarantees that $Q_{V}^{\prime}(u)\geq 0$ on the whole
$M$. We call such a $u$ the $Q_{V}$-capacitor of $(K,g)$.
###### Proposition 4.3.
In the assumptions of Proposition 4.2, if $Q_{V}^{\prime}(g)=0$ on a
neighbourhood of $K$,
(4.12) $\mathrm{cap}_{Q_{V}}(K,g)=\frac{1}{p}\int_{\partial K}g\left[|\nabla
g|^{p-2}\frac{\partial g}{\partial\nu}-|\nabla u|^{p-2}\frac{\partial
u}{\partial\nu}\right]\mathrm{d}\sigma_{f},$
where $\nu$ is the unit normal to $\partial K$ pointing outward of $K$.
###### Proof.
By Proposition 4.1, $\mathrm{cap}_{Q_{V}}(K,g)=\lim_{j}Q_{V}(u_{j})$, where
$u_{j}$ is the $Q_{V}$-capacitor of $(K,\Omega_{j},g)$. Now, since
$u_{j}\rightarrow u$ in $C^{1}(\partial K)$, it is enough to pass to the limit
in 4.6. ∎
Next Theorem, the core of this section, relates the subcriticality of $Q_{V}$
and the $Q_{V}$-capacity with other basic properties, which we will define
below. It is due to Y. Pinchover and K. Tintarev (see [67]), and it is known
in the literature as the ground state alternative. The authors state it for
$f$ constant and $M=\mathbb{R}^{m}$. Our contribution here is to include the
$Q_{V}$-capacity properties to the above picture. However, since at some point
of [67] the authors use inequalities for which we found no counterpart in a
manifold setting, we prefer to provide a full proof which sometimes uses
arguments that differ from those in [67, 68], though keeping the same
guidelines.
###### Definition 4.1.
For $V\in L^{\infty}_{\mathrm{loc}}(M)$, define $Q_{V}$ as in (2.7) and let
$\Omega\subseteq M$ be an open set.
* iii)
$Q_{V}$ has a _weighted spectral gap_ on $\Omega$ if there exists $W\in
C^{0}(\Omega)$, $W>0$ on $\Omega$ such that
(4.13) $\displaystyle\int_{\Omega}W(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq
Q_{V}(\varphi)\qquad\forall\ \varphi\in\mathrm{Lip}_{c}(\Omega).$
* iv)
A sequence $\\{\eta_{j}\\}\in L^{\infty}_{c}(\Omega)\cap W^{1,p}(\Omega)$ is
said to be a _null sequence_ if $\eta_{j}\geq 0$ a.e. for each $j$,
$Q_{V}(\eta_{j})\rightarrow 0$ as $j\rightarrow+\infty$ and there exists a
relatively compact open set $B\Subset M$ and $C>1$ such that
$C^{-1}\leq\|\eta_{j}\|_{L^{p}(B)}\leq C$ for each $j$.
* v)
A function $0\leq\eta\in W^{1,p}_{\mathrm{loc}}(\Omega)$, $\eta\geq 0$,
$\eta\not\equiv 0$ is a _ground state_ for $Q_{V}$ on $\Omega$ if it is the
$L^{p}_{\mathrm{loc}}(\Omega)$ limit of a null sequence.
###### Theorem 4.1.
Let $(M,\langle\,,\,\rangle)$ be connected and non-compact, and for $V\in
L^{\infty}_{\mathrm{loc}}(M)$ consider an operator $Q_{V}\geq 0$ on $M$. Then,
either $Q_{V}$ has a weighted spectral gap or a ground state on $M$, and the
two possibilities mutually exclude. Moreover, the following properties are
equivalent:
* $(i)_{\mathrm{S}}$
$Q_{V}$ has a weighted spectral gap.
* $(ii)_{\mathrm{S}}$
$Q_{V}$ is subcritical on $M$.
* $(iii)_{\mathrm{S}}$
There exist two positive solutions $u_{1},u_{2}\in C^{0}(M)\cap
W^{1,p}_{\mathrm{loc}}(M)$ of $Q^{\prime}_{V}(u)\geq 0$ which are not
proportional.
* $(iv)_{\mathrm{S}}$
For some (any) $K\Subset M$ compact with non-empty interior, and for some
(any) $0<g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ solving
$Q_{V}^{\prime}(g)\geq 0$, $\mathrm{cap}(K,g)>0$.
When $Q_{V}$ has a ground state $\eta$, $\eta$ solves
$Q_{V}^{\prime}(\eta)=0$, and in particular $\eta\in
C^{1,\mu}_{\mathrm{loc}}(M)$, $\eta>0$ on $M$. Furthermore, the next
properties are equivalent:
* $(i)_{\mathrm{GS}}$
$Q_{V}$ has a ground state.
* $(ii)_{\mathrm{GS}}$
All positive solutions $g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of
$Q_{V}^{\prime}(g)\geq 0$ are proportional; in particular, each positive
supersolution is indeed a solution (hence, a ground state).
* $(iii)_{\mathrm{GS}}$
For some (any) $K\Subset M$ compact with non-empty interior, and for some
(any) $0<g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ solving
$Q_{V}^{\prime}(g)\geq 0$, $\mathrm{cap}(K,g)=0$.
###### Proof.
Hereafter, $L^{p}$ and $W^{1,p}$ spaces will be considered with respect to the
measure $\mathrm{d}\mu_{f}$. We begin with the following fact.
Claim 1: Fix an open set $U\Subset M$. If $\\{\phi_{j}\\}\subset
L^{\infty}_{c}(M)\cap W^{1,p}(M)$ is such that
$\|\phi_{j}\|_{L^{p}(U)}+Q_{V}(\phi_{j})\leq C$ for some $C>0$ independent of
$j$, then $\\{\phi_{j}\\}$ is locally bounded in $W^{1,p}(M)$.
Proof of Claim 1. Up to replacing $\phi_{j}$ with $|\phi_{j}|$, we can assume
that $\phi_{j}\geq 0$ a.e. on $M$. Using that $Q_{V}$ is non-negative on $M$,
choose a positive solution $g\in C^{1,\mu}_{\mathrm{loc}}(M)$ of
$Q_{V}^{\prime}(g)=0$ and consider the Lagrangian representation of
$Q_{V}(\phi_{j})$:
$Q_{V}(\phi_{j})=\int_{M}\mathcal{L}(\phi_{j},g)\mathrm{d}\mu_{f}.$
Fix $\Omega\Subset M$ containing $\overline{U}$. Since $Q_{V}(\phi_{j})\leq C$
and $\mathcal{L}(\phi_{j},g)\geq 0$ on $M$, it holds
(4.14)
$0\leq\limsup_{j\rightarrow+\infty}\int_{\Omega}\mathcal{L}(\phi_{j},g)\mathrm{d}\mu_{f}\leq
C.$
By using Cauchy-Schwarz and Young inequalities on the third addendum of the
expression of $\mathcal{L}(\phi_{j},g)$ we deduce that, for each
$\varepsilon>0$,
(4.15)
$\mathcal{L}(\phi_{j},g)\geq(1-\varepsilon^{p})|\nabla\phi_{j}|^{p}+(p-1)\left(1-\varepsilon^{-\frac{p}{p-1}}\right)\left(\frac{\phi_{j}}{g}\right)^{p}|\nabla
g|^{p}.$
In our assumptions, $|\nabla\log g|\in L^{\infty}(\Omega)$. Setting
$\varepsilon=1/2$ in (4.15), integrating and using (4.14) we get
(4.16)
$\limsup_{j\rightarrow+\infty}\left[\left(1-2^{-p}\right)\int_{\Omega}|\nabla\phi_{j}|^{p}\mathrm{d}\mu_{f}+(p-1)\left(1-2^{\frac{p}{p-1}}\right)\|\nabla\log
g\|_{L^{\infty}(\Omega)}\int_{\Omega}|\phi_{j}|^{p}\mathrm{d}\mu_{f}\right]\leq
C.$
From this inequality we argue the existence of constants $C_{1},C_{2}>0$
independent of $j$ such that
(4.17) $\|\nabla\phi_{j}\|_{L^{p}(\Omega)}\leq
C_{1}\|\phi_{j}\|_{L^{p}(\Omega)}+C_{2}.$
Suppose now, by contradiction, that $\\{\phi_{j}\\}$ is not bounded in
$W^{1,p}(\Omega)$. By (4.17), $\|\phi_{j}\|_{L^{p}(\Omega)}$ diverge. Set
$\bar{\phi}_{j}=\phi_{j}/\|\phi_{j}\|_{L^{p}(\Omega)}$. Then, by (4.17)
$\\{\bar{\phi}_{j}\\}$ is bounded in $W^{1,p}(\Omega)$, thus it has a
subsequence (still called $\\{\bar{\phi}_{j}\\}$) converging weakly in
$W^{1,p}$, strongly in $L^{p}$ and pointwise almost everywhere to some non-
negative function $\Psi$ satisfying
$\|\Psi\|_{L^{p}(\Omega)}=\lim_{j}\|\bar{\phi}_{j}\|_{L^{p}(\Omega)}=1$. Since
$|\nabla\log g|\in L^{\infty}(\Omega)$ we straightforwardly have that
(4.18) $\begin{array}[]{rll}(i)&\bar{\phi}_{j}^{p}|\nabla\log
g|^{p}\rightarrow\Psi^{p}|\nabla\log g|^{p}&\quad\text{in
}L^{1}(\Omega),\\\\[5.69046pt] (ii)&\bar{\phi}_{j}^{p-1}|\nabla\log
g|^{p-1}\rightarrow\Psi^{p-1}|\nabla\log g|^{p-1}&\quad\text{in
}L^{\frac{p}{p-1}}(\Omega).\end{array}$
We just prove $(ii)$, the other being a consequence of the proof. By the
elementary inequalities
(4.19) $(x-y)^{\beta}\leq x^{\beta}-y^{\beta}\leq
Cx^{\beta-1}(x-y)\qquad\text{for }y\in[0,x]\text{ and }\beta>1,$
for some $C$ depending on $\beta$, with a final application of Hölder
inequality we deduce that
(4.20)
$\begin{array}[]{l}\displaystyle\|(\bar{\phi}_{j}^{p-1}-\Psi^{p-1})|\nabla\log
g|^{p-1}\|_{L^{p/(p-1)}(\Omega)}^{p/(p-1)}\leq\|\nabla\log
g\|_{L^{\infty}(\Omega)}^{p}\int_{\Omega}\big{|}\bar{\phi}_{j}^{p-1}-\Psi^{p-1}\big{|}^{\frac{p}{p-1}}\mathrm{d}\mu_{f}\\\\[8.5359pt]
\displaystyle\leq\|\nabla\log
g\|_{L^{\infty}(\Omega)}^{p}\int_{\Omega}\big{|}\bar{\phi}_{j}^{p}-\Psi^{p}\big{|}\mathrm{d}\mu_{f}\leq
C\|\nabla\log
g\|^{p}_{L^{\infty}(\Omega)}\int_{\Omega}\big{(}\max\\{\bar{\phi}_{j},\Psi\\}\big{)}^{p-1}\big{|}\bar{\phi}_{j}-\Psi|\mathrm{d}\mu_{f},\\\\[8.5359pt]
\displaystyle\leq C\|\nabla\log
g\|^{p}_{L^{\infty}(\Omega)}\left\|\bar{\phi}_{j}+\Psi\right\|_{L^{p}(\Omega)}^{(p-1)/p}\|\bar{\phi}_{j}-\Psi\|_{L^{p}(\Omega)},\end{array}$
and this latter goes to zero $\bar{\phi}_{j}\rightarrow\Psi$ in
$L^{p}(\Omega)$. Now, coupling $(ii)$ with the weak convergence of
$\bar{\phi}_{j}$ to $\Psi$ in $W^{1,p}(\Omega)$, we get
$\int_{\Omega}\left(\frac{\bar{\phi}_{j}}{g}\right)^{p-1}|\nabla
g|^{p-2}\langle\nabla
g,\nabla\bar{\phi}_{j}\rangle\mathrm{d}\mu_{f}\longrightarrow\int_{\Omega}\left(\frac{\Psi}{g}\right)^{p-1}|\nabla
g|^{p-2}\langle\nabla g,\nabla\Psi\rangle\mathrm{d}\mu_{f},$
so that, combining $(i),(ii)$ and the weak lower semicontinuity of
$\|\cdot\|_{W^{1,p}(\Omega)}$,
$0\leq\int_{\Omega}\mathcal{L}(\Psi,g)\mathrm{d}\mu_{f}\leq\liminf_{j\rightarrow+\infty}\int_{\Omega}\mathcal{L}(\bar{\phi}_{j},g)\mathrm{d}\mu_{f}=\|\phi_{j}\|^{-p}_{L^{p}(\Omega)}\int_{\Omega}\mathcal{L}(\phi_{j},g)\mathrm{d}\mu_{f}\rightarrow
0$
as $j\rightarrow+\infty$. Hence $\mathcal{L}(\Psi,g)=0$ on $\Omega$, so by
Proposition 3.1 $\Psi=cg$ for some constant $c\geq 0$. However, since
$\|\phi_{j}\|_{L^{p}(U)}\leq C$ for each $j$,
$0\leq\|\Psi\|_{L^{p}(U)}=\lim_{j\rightarrow+\infty}\|\bar{\phi}_{j}\|_{L^{p}(U)}=\lim_{j\rightarrow+\infty}\frac{\|\phi_{j}\|_{L^{p}(U)}}{\|\phi_{j}\|_{L^{p}(\Omega)}}\leq\limsup_{j\rightarrow+\infty}\frac{C}{\|\phi_{j}\|_{L^{p}(\Omega)}}=0,$
hence $c=0$ and $\Psi\equiv 0$ on $\Omega$, contradicting the fact that
$\|\Psi\|_{L^{p}(\Omega)}=1$. This concludes the proof of the claim.
Claim 2: Either $Q_{V}$ has a weighted spectral gap or a ground state, but
not both.
Proof of Claim 2. For a relatively compact open set $U$, define the
$L^{p}$-capacity $c$ as follows:
$\mathcal{D}(U)=\Big{\\{}\varphi\in L^{\infty}_{c}(M)\cap
W^{1,p}(M):\|\varphi\|_{L^{p}(U)}=1\Big{\\}},\qquad
c(U)=\inf_{\varphi\in\mathcal{D}(U)}Q_{V}(\varphi),$
where, as usual, the $L^{p}$-norm is computed with respect to
$\mathrm{d}\mu_{f}$. Then, two mutually exclusive cases may occur: ether
$c(U)>0$ for each $U$, or $c(U)=0$ for some $U$. In the first case, it is easy
to see that $Q_{V}$ has a weighted spectral gap. Indeed, let $\\{U_{j}\\}$ be
a locally finite covering of $M$ via relatively compact, open sets, set
$c_{j}=c(U_{j})>0$ and let $\\{t_{j}\\}$ be a sequence of positive numbers
such that $\sum_{j}t_{j}=1$. For each $\varphi\in\mathrm{Lip}_{c}(M)$, by the
definition of $c_{j}$ we get $Q_{V}(\varphi)\geq
c_{j}\|\varphi\|_{L^{p}(U_{j})}^{p}$, and summing up we get
$Q_{V}(\varphi)=\left[\sum_{j=1}^{+\infty}t_{j}\right]Q_{V}(\varphi)\geq\sum_{j=1}^{+\infty}t_{j}c_{j}\int_{U_{j}}|\varphi|^{p}\mathrm{d}\mu_{f}=\int_{M}\hat{W}|\varphi|^{p}\mathrm{d}\mu_{f},$
where
$\hat{W}(x)=\sum_{j=1}^{+\infty}t_{j}c_{j}1_{U_{j}}(x)>0\qquad\text{on }M.$
We can thus choose a weighted spectral gap $W$ by taking a positive,
continuous function $W$ not exceeding $\hat{W}$.
Now, suppose that $c(U)=0$ for some $U$. We show that there exists a ground
state. Indeed, by the definition of $c(U)$ there exists $\\{\eta_{j}\\}\subset
L^{\infty}_{c}(M)\cap W^{1,p}(M)$ such that $\|\eta_{j}\|_{L^{p}(U)}=1$ and
$Q_{V}(\eta_{j})\rightarrow 0$. Up to replacing $\eta_{j}$ with $|\eta_{j}|$,
we can suppose that $\eta_{j}\geq 0$ a.e. on $M$. By Claim 1, $\eta_{j}$ is
locally bounded in $W^{1,p}$, and a Cantor type argument on an increasing
exhaustion of $M$ ensures the existence of a subsequence, still called
$\\{\eta_{j}\\}$, converging weakly in $W^{1,p}_{\mathrm{loc}}(M)$ and
strongly in $L^{p}_{\mathrm{loc}}(M)$ to some function $\eta\in
W^{1,p}_{\mathrm{loc}}(M)$. By definition, $\eta$ is a ground state.
We now show our desired equivalences. To establish those involving
$(iv)_{\mathrm{S}}$ and $(iii)_{\mathrm{GS}}$, where the “some/all"
alternative appears, then we will always assume the weakest alternative and
prove the strongest one.
$\mathbf{(i)_{\mathrm{GS}}\Rightarrow(ii)_{\mathrm{GS}}}$. Let $\eta\geq 0$ be
a ground state, and let $\\{\eta_{j}\\}\subset L^{\infty}_{c}(M)\cap
W^{1,p}(M)$ be a null sequence converging in $L^{p}_{\mathrm{loc}}$ to $\eta$.
Then, by Claim 1 $\\{\eta_{j}\\}$ is locally bounded in $W^{1,p}$, thus up to
passing to a subsequence we can assume that also $\eta_{j}\rightarrow\eta$
weakly in $W^{1,p}_{\mathrm{loc}}$. Consider a positive solution $g\in
C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of $Q_{V}^{\prime}(g)\geq 0$. Fix
$\Omega\Subset M$. Up to multiply $g$ by a large positive constant, we can
suppose that $\\{x\in\Omega:\eta(x)<g(x)\\}$ has positive measure. Let
$\bar{\eta}_{j}=\min\\{\eta_{j},g\\}$, and note that $\\{\bar{\eta}_{j}\\}$ is
still a null sequence, converging weakly in $W^{1,p}_{\mathrm{loc}}$ to
$\bar{\eta}=\min\\{\eta,g\\}$. Consider the Lagrangian representation
(4.21)
$Q_{V}(\bar{\eta}_{j})\geq\int_{M}\mathcal{L}(\bar{\eta}_{j},g)\mathrm{d}\mu_{f}$
guaranteed by Proposition 3.1. We claim that
(4.22)
$\frac{\bar{\eta}_{j}^{p}}{g^{p-1}}\rightarrow\frac{\bar{\eta}^{p}}{g^{p-1}}\qquad\text{weakly
in }W^{1,p}(\Omega).$
To see this, we follows arguments analogous to those yielding (4.18). Choose a
constant $c_{\Omega}>0$ large enough to satisfy $g\geq c_{\Omega}^{-1}$ on
$\Omega$ and $\bar{\eta}_{j}\leq c_{\Omega}$. By a direct computation,
$\|\bar{\eta}_{j}^{p}/g^{p-1}\|_{W^{1,p}(\Omega)}$ is uniformly bounded so
that, by density, it is enough to check the weak convergence with test
function $\varphi\in\mathrm{Lip}_{0}(\Omega)$. From
$\left|\int_{\Omega}\frac{\bar{\eta}_{j}^{p}-\bar{\eta}^{p}}{g^{p-1}}\varphi\mathrm{d}\mu_{f}\right|\leq
c_{\Omega}^{p-1}\|\varphi\|_{L^{\infty}(\Omega)}\int_{\Omega}\big{|}\bar{\eta}_{j}^{p}-\bar{\eta}^{p}\big{|}\mathrm{d}\mu_{f},$
applying the inequalities in (4.20) from the second line to the end we deduce
that $\bar{\eta}_{j}^{p}/g^{p-1}\rightarrow\bar{\eta}^{p}/g^{p-1}$ weakly in
$L^{p}(\Omega)$. Regarding the gradient part,
$\left|\int_{\Omega}\langle\nabla\left(\frac{\bar{\eta}_{j}^{p}-\bar{\eta}^{p}}{g^{p-1}}\right),\nabla\varphi\rangle\mathrm{d}\mu_{f}\right|\leq\mathrm{(I)}+\mathrm{(II)},$
where
$\begin{array}[]{rcl}\mathrm{(I)}&=&\displaystyle(p-1)\int_{\Omega}\big{|}\bar{\eta}_{j}^{p}-\bar{\eta}^{p}\big{|}|\nabla
g|\frac{|\nabla\varphi|}{g^{p}}\\\\[8.5359pt] \mathrm{(II)}&=&\displaystyle
p\left|\int_{\Omega}\langle\left(\frac{\bar{\eta}_{j}}{g}\right)^{p-1}\nabla\varphi,\nabla\bar{\eta}_{j}\rangle\mathrm{d}\mu_{f}-\int_{\Omega}\langle\left(\frac{\bar{\eta}}{g}\right)^{p-1}\nabla\varphi,\nabla\bar{\eta}\rangle\mathrm{d}\mu_{f}\right|.\end{array}$
As for (I), by Hölder and both the inequalities in (4.19) we deduce
$\begin{array}[]{rcl}\frac{1}{p-1}\mathrm{(I)}&\leq&\displaystyle
c_{\Omega}^{p}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\|\nabla
g\|_{L^{p}(\Omega)}\left(\int_{\Omega}\big{|}\bar{\eta}_{j}^{p}-\bar{\eta}^{p}\big{|}^{\frac{p}{p-1}}\mathrm{d}\mu_{f}\right)^{\frac{p-1}{p}}\\\\[8.5359pt]
&\leq&\displaystyle
c_{\Omega}^{p}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\|\nabla
g\|_{L^{p}(\Omega)}\left(\int_{\Omega}\left|\bar{\eta}_{j}^{\frac{p^{2}}{p-1}}-\bar{\eta}^{\frac{p^{2}}{p-1}}\right|\mathrm{d}\mu_{f}\right)^{\frac{p-1}{p}}\\\\[8.5359pt]
&\leq&\displaystyle
c_{\Omega}^{p}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\|\nabla
g\|_{L^{p}(\Omega)}C^{\frac{p-1}{p}}\left(\int_{\Omega}\max\\{\bar{\eta}_{j},\bar{\eta}\\}^{\frac{p^{2}}{p-1}-1}\left|\bar{\eta}_{j}-\bar{\eta}\right|\mathrm{d}\mu_{f}\right)^{\frac{p-1}{p}}\\\\[8.5359pt]
&\leq&\displaystyle
c_{\Omega}^{p+\frac{p}{p-1}-\frac{p-1}{p}}\|\nabla\varphi\|_{L^{\infty}(\Omega)}\|\nabla
g\|_{L^{p}(\Omega)}C^{\frac{p-1}{p}}\left(\int_{\Omega}\left|\bar{\eta}_{j}-\bar{\eta}\right|\mathrm{d}\mu_{f}\right)^{\frac{p-1}{p}}.\end{array}$
Again by Hölder inequality, the last integral goes to zero since
$\bar{\eta}_{j}\rightarrow\bar{\eta}$ in $L^{p}(\Omega)$, which shows that
(I)$\rightarrow 0$ as $j\rightarrow+\infty$. Finally, we consider (II). To
show that (II)$\rightarrow 0$, using the weak convergence if $\bar{\eta}_{j}$
to $\bar{\eta}$ in $W^{1,p}(\Omega)$ and standard estimates it is enough to
prove that
$\left(\frac{\bar{\eta}_{j}}{g}\right)^{p-1}\nabla\varphi\rightarrow\left(\frac{\bar{\eta}}{g}\right)^{p-1}\nabla\varphi\qquad\text{strongly
in }L^{\frac{p}{p-1}}(\Omega).$
This follows from
$\int_{\Omega}|\nabla\varphi|^{\frac{p}{p-1}}\left|\left(\frac{\bar{\eta}_{j}}{g}\right)^{p-1}-\left(\frac{\bar{\eta}}{g}\right)^{p-1}\right|^{\frac{p}{p-1}}\mathrm{d}\mu_{f}\leq
c_{\Omega}^{p}\|\nabla\varphi\|_{L^{\infty}(\Omega)}^{\frac{p}{p-1}}\int_{\Omega}\left|\bar{\eta}_{j}^{p-1}-\bar{\eta}^{p-1}\right|^{\frac{p}{p-1}}\mathrm{d}\mu_{f}$
and inequalities analogous to those in the second and third lines of (4.20).
This concludes the proof of (4.22).
Now, (4.22) implies that
$\int_{\Omega}|\nabla g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\bar{\eta}_{j}^{p}}{g^{p-1}}\right)\rangle\mathrm{d}\mu_{f}\longrightarrow\int_{\Omega}|\nabla
g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\bar{\eta}^{p}}{g^{p-1}}\right)\rangle\mathrm{d}\mu_{f},$
so that integrating on $\Omega$ the Lagrangian identity
$\mathcal{L}(\bar{\eta}_{j},g)=|\nabla\bar{\eta}_{j}|^{p}-|\nabla
g|^{p-2}\langle\nabla
g,\nabla\left(\frac{\bar{\eta}_{j}^{p}}{g^{p-1}}\right)\rangle$
and using the weak lower semicontinuity of the $W^{1,p}$ norm we deduce
(4.23)
$0\leq\int_{\Omega}\mathcal{L}(\bar{\eta},g)\mathrm{d}\mu_{f}\leq\liminf_{j\rightarrow+\infty}\int_{\Omega}\mathcal{L}(\bar{\eta}_{j},g)\mathrm{d}\mu_{f}.$
Inequalities (4.21) and $\mathcal{L}(\bar{\eta}_{j},g)\geq 0$ on $M$ then
imply
$0\leq\int_{\Omega}\mathcal{L}(\bar{\eta}_{j},g)\mathrm{d}\mu_{f}\leq\int_{M}\mathcal{L}(\bar{\eta}_{j},g)\mathrm{d}\mu_{f}\leq
Q_{V}(\bar{\eta}_{j})\longrightarrow 0$
as $j\rightarrow+\infty$, so we conclude by (4.23) that
$\mathcal{L}(\bar{\eta},g)\equiv 0$ on $\Omega$. By Proposition 3.1,
$\bar{\eta}=cg$ on $\Omega$ for some $c\geq 0$. In fact, $c>0$ since
$\|\eta\|_{L^{p}(U)}=\lim_{j}\|\eta_{j}\|_{L^{p}(U)}\neq 0$, and $c<1$ since
$\\{x\in\Omega:\eta(x)<g(x)\\}$ has positive measure. Therefore,
$\bar{\eta}\equiv\eta$ on $\Omega$, showing that $g$ is a positive multiple of
$\eta$ on $\Omega$, that is, $(ii)_{\mathrm{GS}}$ holds.
Claim 3. When $Q_{V}$ has a ground state $\eta$, $\eta\in
C^{1,\mu}_{\mathrm{loc}}(M)$, is positive and solves $Q_{V}^{\prime}(\eta)=0$.
Proof of Claim 3. By $(i)_{\mathrm{GS}}\Rightarrow(ii)_{\mathrm{GS}}$, all
solutions $g\in C^{0}(M)\cap W^{1,p}_{\mathrm{loc}}(M)$ of
$Q_{V}^{\prime}(g)\geq 0$ are proportional. Choosing $g$ to be a positive
solution of $Q_{V}^{\prime}(g)=0$ (which exists by Proposition 3.4), we get
that $\eta>0$ solves $Q_{V}^{\prime}(\eta)=0$ and $\eta\in
C^{1,\mu}_{\mathrm{loc}}(M)$, proving the claim.
$\mathbf{(iii)_{\mathrm{GS}}\Rightarrow(i)_{\mathrm{GS}}}$. Since
$\mathrm{Int}(K)\neq\emptyset$ we select a closed smooth geodesic ball $B$
contained in $\mathrm{Int}(K)$. By the monotonicity of capacity,
$\mathrm{cap}_{Q_{V}}(B,g)=0$. Fix an exhaustion $\\{\Omega\\}_{j}$ of $M$
with $B\Subset\Omega_{1}$, let $\eta_{j}$ be the $Q_{V}$-capacitor of
$(B,\Omega_{j},g)$ extended with zero outside $\Omega_{j}$, and let $\eta$ be
the capacitor of $(B,g)$. Then, (4.10) ensures that
$Q_{V}(\eta_{j})\rightarrow\mathrm{cap}_{Q_{V}}(B,g)=0$, and since
$\eta_{j}\rightarrow\eta$ in $L^{p}_{\mathrm{loc}}(M)$ we deduce that $\eta$
is the desired ground state.
$\mathbf{(iv)_{\mathrm{S}}\Rightarrow(iii)_{\mathrm{S}}}$. Up to enlarging $K$
(capacity increases), we can assume that $K$ is the closure of a relatively
compact open set with smooth boundary. Now, consider a solution $\bar{g}\in
C^{1,\mu}_{\mathrm{loc}}(M)$ of $Q_{V}^{\prime}(\bar{g})=0$ on $M$. Up to
multiplying $\bar{g}$ by a small positive constant, we can suppose that
$\bar{g}\leq g$ on $K$. By the very definition of $Q_{V}$-capacity,
$\mathrm{cap}_{Q_{V}}(K,g)\geq\mathrm{cap}_{Q_{V}}(K,\bar{g}).$
We claim that $\mathrm{cap}_{Q_{V}}(K,\bar{g})>0$. Indeed, otherwise, by
$(iii)_{\mathrm{GS}}\Rightarrow(i)_{\mathrm{GS}}$ just proved, $Q_{V}$ has a
ground state and so all solutions $u\in C^{0}(M)\cap
W^{1,p}_{\mathrm{loc}}(M)$ of $Q_{V}^{\prime}(u)\geq 0$ are proportional. In
particular, $\bar{g}=cg$ for some constant $c>0$, which implies
$\mathrm{cap}_{Q_{V}}(K,g)=c^{-p}\mathrm{cap}_{Q_{V}}(K,\bar{g})=0$
contradicting our assumptions. Now, since $\bar{g}$ solves
$Q_{V}^{\prime}(\bar{g})=0$ and $\mathrm{cap}_{Q_{V}}(K,\bar{g})>0$, applying
formula (4.12) in Proposition 4.3 with $\bar{g}$ replacing $g$ we deduce that
necessarily the $Q_{V}$-capacitor $u$ of $(K,\bar{g})$ is different from
$\bar{g}$. As $u$ solves $Q_{V}^{\prime}(u)\geq 0$ on $M$ and $u=g$ on $K$,
$u$ and $\bar{g}$ are two non-proportional solutions of $Q_{V}^{\prime}(v)\geq
0$, proving $(iii)_{\mathrm{S}}$.
$\mathbf{(ii)_{\mathrm{GS}}\Rightarrow(iii)_{\mathrm{GS}}}$. If, by
contradiction, $\mathrm{cap}_{Q_{V}}(K,g)>0$ for some $(K,g)$, then by
$(iv)_{\mathrm{S}}\Rightarrow(iii)_{\mathrm{S}}$ above there would exists two
non-proportional solutions of $Q_{V}^{\prime}(g)\geq 0$, contradicting
$(ii)_{\mathrm{GS}}$.
Now, we have concluded the equivalence
$(i)_{\mathrm{GS}}\Leftrightarrow(ii)_{\mathrm{GS}}\Leftrightarrow(iii)_{\mathrm{GS}}$
in the “ground state" part of the theorem. Combining with Claim $2$, we
automatically have the validity of
$\mathbf{(i)_{\mathrm{S}}\Leftrightarrow(iii)_{\mathrm{S}}\Leftrightarrow(iv)_{\mathrm{S}}}$.
We are thus left to show $(i)_{\mathrm{S}}\Leftrightarrow(ii)_{\mathrm{S}}$.
Having observed that $(i)_{\mathrm{S}}\Rightarrow(ii)_{\mathrm{S}}$ is obvious
(set $w=W$), to conclude we prove that
$\mathbf{(ii)_{\mathrm{S}}\Rightarrow(i)_{\mathrm{S}}}$. Suppose by
contradiction that $(i)_{\mathrm{S}}$ is not true. Then, by Claim $2$, $Q_{V}$
has a ground state $\eta$, which is positive on $M$ and solves
$Q_{V}^{\prime}(\eta)=0$. Fix a smooth open set $U$ such that $w\not\equiv 0$
on $U$, let $\\{\Omega_{j}\\}\uparrow M$ be an exhaustion of $M$ with
$U\Subset\Omega_{1}$, and let $\eta_{j}$ be the $Q_{V}$-capacitor of
$(U,\Omega_{j},\eta)$. Then, by the equivalence
$(i)_{\mathrm{GS}}\Leftrightarrow(iii)_{\mathrm{GS}}$ (in particular, by the
proof of $(iii)_{\mathrm{GS}}\Rightarrow(i)_{\mathrm{GS}}$), $\\{\eta_{j}\\}$
is a null sequence. By the subcriticality assumption and since $\eta_{j}=\eta$
on $U$,
$\int_{U}w|\eta|^{p}\mathrm{d}\mu_{f}\leq\int_{M}w|\eta_{j}|^{p}\mathrm{d}\mu_{f}\leq
Q_{V}(\eta_{j})\rightarrow 0$
as $j\rightarrow+\infty$, contradicting the fact that $\eta>0$ on $M$ and
$w\not\equiv 0$ on $U$. ∎
###### Remark 4.3.
_We can now give a simple proof of the fact that the positivity of the Yamabe
invariant $Y(M)$ in Theorem 1.1 implies the subcriticality of the conformal
Laplacian $L_{\langle\,,\,\rangle}$. Indeed, suppose the contrary. Then, by
Theorem 4.1, there exist a ground state $\eta>0$ and a null sequence
$\\{\eta_{j}\\}$ locally $L^{2}$-converging to $\eta$. From the chain of
inequalities_
$Y(M)\left(\int_{M}|\eta_{j}|^{\frac{2m}{m-2}}\right)^{\frac{m-2}{m}}\leq\int_{M}\left[|\nabla\eta_{j}|^{2}+\frac{s(x)}{c_{m}}\eta_{j}^{2}\right]\rightarrow
0\quad\text{as }\,j\rightarrow+\infty,$
_we deduce that $\eta_{j}\rightarrow 0$ in $L^{\frac{2m}{m-2}}(M)$, hence
locally in $L^{2}(M)$, a contradiction. _
It is now immediate to prove the equivalence partly mentioned in the
Introduction (Proposition 2.1). We suggest the interested reader to consult
the very recent [23, 24] for an investigation on the optimality of the Hardy
weights given in items $5)$ and $6)$ below.
###### Proposition 4.4.
Let $M^{m}$ be a Riemannian manifold of dimension $m\geq 2$, $p\in(1,+\infty)$
and $f\in C^{\infty}(M)$. The following properties are equivalent:
* $1)$
There exists a positive, non-constant solution $g\in C^{0}(M)\cap
W^{1,p}_{\mathrm{loc}}(M)$ of $\Delta_{p,f}g\leq 0$.
* $2)$
For some (any) compact $K\subset M$ with non-empty interior, and for some
(any) solution $g$ of $\Delta_{p,f}g\leq 0$, $\mathrm{cap}(K,g)>0$.
* $3)$
$Q_{0}$ is subcritical on $M$: there exists $w\in L^{1}_{\mathrm{loc}}(M)$,
$w\geq 0$, $w\not\equiv 0$ on $M$ such that
(4.24)
$\displaystyle\int_{M}w(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$
* $4)$
$Q_{0}$ has a weighted spectral gap on $M$: there exists $W\in C^{0}(M)$,
$W>0$ on $M$ such that
(4.25)
$\displaystyle\int_{M}W(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$
* $5)$
For each non-constant, positive weak solution $u\in W^{1,p}_{\mathrm{loc}}(M)$
of $\Delta_{p,f}u\leq 0$ the following Hardy type inequality holds:
(4.26) $\left(\frac{p-1}{p}\right)^{p}\int_{M}\frac{|\nabla
u|^{p}}{u^{p}}|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$
* $6)$
For each $y\in M$, $-\Delta_{p,f}$ has a positive Green kernel
$\mathcal{G}(x,y)$ and the following Hardy inequality holds:
(4.27)
$\left(\frac{p-1}{p}\right)^{p}\int_{M}\frac{|\nabla_{x}\mathcal{G}|^{p}}{\mathcal{G}^{p}}|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$
###### Remark 4.4.
_Here, a Green kernel $\mathcal{G}(x,y)$ means a distributional solution of_
$\Delta_{p,f}\mathcal{G}(\cdot,y)=-\delta_{y},$
_where $\delta_{y}$ is the Dirac delta function at $y$. _
###### Proof.
The equivalence $1)\Leftrightarrow 2)\Leftrightarrow 3)\Leftrightarrow 4)$ is
the particular case $V\equiv 0$ of Theorem 4.1. Indeed, as any positive
constant is a solution of $\Delta_{p,f}u\leq 0$, $1)$ can be rephrased as the
existence of two non-proportional solutions of $\Delta_{p,f}u\leq 0$. Clearly
$5)\Rightarrow 3)$, thus $5)$ yields that $Q_{0}$ is subcritical on $M$;
therefore, we just need to show that $1)\Rightarrow 5)$. Towards this aim
observe that if $g\in C^{0}(M)\cap\in W^{1,p}_{\mathrm{loc}}(M)$, $g>0$ is a
positive, non-constant solution of $\Delta_{p,f}g\leq 0$, then by (3.14)
$\displaystyle z=g^{\frac{p-1}{p}}$ is a positive weak solution of
$\displaystyle\Delta_{p,f}z+\left(\frac{p-1}{p}\right)^{p}\frac{|\nabla
u|^{p}}{u^{p}}z^{p-1}\leq 0\qquad\text{on }M.$
Proposition 3.1 and the non-negativity of $\mathcal{L}$ then imply that
$Q_{V}$ is non-negative for
$V=\left(\frac{p-1}{p}\right)^{p}\frac{|\nabla u|^{p}}{u^{p}},$
which gives (4.26).
To prove $6)\Rightarrow 1)$, take a positive Green kernel $\mathcal{G}(x,y)$
for $-\Delta_{p,f}$ at $y$. For large $c>0$, the function
$G_{c}(x)=\min\\{c,\mathcal{G}(x,y)\\}$ is a non-constant, positive weak
solution of $\Delta_{p,f}G_{c}\leq 0$ by the pasting Lemma 3.1, showing $1)$.
Vice versa, if $1)$ holds, then by the equivalence $1)\Leftrightarrow 2)$ we
deduce that each compact set has positive capacity with respect to the
“standard" supersolution $g\equiv 1$. This implies, by results in [38] and
[39], that for each $y\in M$ there exists a positive Green kernel
$\mathcal{G}(x,y)$ for $-\Delta_{p,f}$. Defining again $G_{c}$ as above,
$\Delta_{p,f}G_{c}\leq 0$ and the equivalence $1)\Leftrightarrow 5)$ ensures
that the Hardy inequality
(4.28) $\left(\frac{p-1}{p}\right)^{p}\int_{M}\frac{|\nabla
G_{c}|^{p}}{G_{c}^{p}}|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\text{holds
for each }\varphi\in\mathrm{Lip}_{c}(M),$
and (4.27) follows by letting $c\rightarrow+\infty$ and using the monotone
convergence theorem. ∎
###### Remark 4.5.
_As a consequence of[75, 76] and Theorem 1.1 in [44] (see also [45]), if
$M=\mathbb{R}^{m}$ each positive Green kernel $\mathcal{G}$ satisfies_
(4.29)
$\frac{|\nabla_{x}\mathcal{G}|^{p}}{\mathcal{G}^{p}}(x,y)\sim\left\\{\begin{array}[]{ll}C\,\mathrm{dist}(x,y)^{-m}\log^{-m}\mathrm{dist}(x,y)&\quad\text{if
}p=m,\\\\[8.5359pt] C\,\mathrm{dist}(x,y)^{-p}&\quad\text{if
}p<m\end{array}\right.$
_as $\mathrm{dist}(x,y)\rightarrow 0$, for some explicit $C>0$. In the linear
case $p=2$, the first order expansions in [75, 76] guarantee the validity of
(4.29) on each Riemannian manifold. On the contrary, when $p\neq 2$, the
scaling arguments used in [44] are typical of the Euclidean setting and,
although we believe (4.29) to be true in general, to the best of our knowledge
there is still no proof of (4.29) in a manifold setting. _
The above proposition gives a useful, simple criterion to check the
subcriticality of some $Q_{V}$.
###### Proposition 4.5.
Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold, $f\in C^{0}(M)$ and
$p>1$. Let $V\in L^{\infty}_{\mathrm{loc}}(M)$. Suppose that $Q_{0}$ is
subcritical on $M$. If, for some Hardy weight $\hat{w}$, it holds
$V\leq\hat{w}$ and $V\not\equiv\hat{w}$, then $Q_{V}$ is subcritical.
###### Proof.
Indeed, using (4.24),
$Q_{V}(\varphi)\doteq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}-\int_{M}V|\varphi|^{p}\mathrm{d}\mu_{f}\geq\int_{M}(\hat{w}-V)|\varphi|^{p}\mathrm{d}\mu_{f};$
consequently, $w$ in (2.10) can be chosen to be $\hat{w}-V$, proving the
subcriticality of $Q_{V}$. ∎
###### Remark 4.6.
_The alternative in Theorem 4.1 is also related to the existence of a global
positive Green kernel for $Q_{V}^{\prime}$. It has been shown in [67]
(Theorems 5.4 and 5.5) that, on $\mathbb{R}^{m}$, $Q_{V}$ has a weighted
spectral gap if and only if $Q_{V}^{\prime}$ admits a global positive Green
kernel. The result depends on Lemma 5.1 therein, which has been proved via a
rescaling argument typical of the Euclidean space, and calls for a different
strategy in a manifold setting. However, in the linear case we can easily
obtain the above equivalence on general manifolds by relating $Q_{V}^{\prime}$
to a weighted Laplacian via a standard transformation. In fact, suppose that
$p=2$ and that $Q_{V}$ is non-negative. Let $g$ be a positive solution of
$Q_{V}^{\prime}(g)=0$. Then, setting $h=f-2\log g$, by a simple computation
the following formula holds weakly for $\varphi\in C^{2}(M)$:_
(4.30)
$-\Delta_{h}\left(\frac{\varphi}{g}\right)=g^{-1}Q_{V}^{\prime}(\varphi).$
_Note that, according to our notation, for smooth $\phi$_
$\Delta_{h}\phi=g^{-2}e^{f}\mathrm{div}(e^{-f}g^{2}\nabla\phi)$
_Integrating by parts, we infer_
$\frac{1}{2}\int_{M}\left|\nabla\left(\frac{\varphi}{g}\right)\right|^{2}g^{2}\mathrm{d}\mu_{f}=Q_{V}(\varphi)\qquad\forall\,\varphi\in
C^{2}_{c}(M).$
_Therefore, it is readily seen that $Q_{V}^{\prime}$ is subcritical if and
only if so is $-\Delta_{h}$ with respect to the measure
$g^{2}\mathrm{d}\mu_{f}$. Now, Proposition 4.4 guarantees that this happens if
and only if $-\Delta_{h}$ has a positive Green kernel $\mathcal{G}(x,y)$.
Coming back with the aid of (4.30), it is easy to see that
$\mathcal{G}(x,y)g(x)g(y)$ is a global positive Green kernel for
$Q_{V}^{\prime}$. _
## 5\. Hardy weights and comparison geometry
On a manifold $M$ for which $Q_{0}$ is subcritical, the criterion in
Proposition 4.5 shifts the problem of the subcriticality of $Q_{V}$ to the one
of finding explicit Hardy weights. As we will see in a moment, the
construction of weights given in $4)$ and $5)$ of Proposition 4.4 is
compatible with the usual geometric comparison theorems. Therefore, it gives a
useful way to produce simple Hardy weights on manifolds satisfying suitable
curvature assumptions. In this section, we describe some examples to
illustrate the method.
First, we underline the following simple fact. By its very definition, the set
of Hardy weights $w\in L^{1}_{\mathrm{loc}}(M)$, $w\geq 0$, $w\not\equiv 0$ is
convex in $L^{1}_{\mathrm{loc}}(M)$. More generally, given a family
$\\{w_{\alpha}\\}_{\alpha\in A}$ of Hardy weights on $M$ whose index $\alpha$
lies in a measurable space $(A,\mathscr{F})$ ($\mathscr{F}$ a
$\sigma$-algebra), and such that the map
$w\ \ :\ \ (x,\alpha)\in M\times A\longrightarrow w_{\alpha}(x)\in[0,+\infty]$
is measurable, for each measure $\lambda$ on $A$ such that $0<\lambda(A)\leq
1$ the function
(5.1) $\chi(x)=\int_{A}w(x,\alpha)\mathrm{d}\lambda(\alpha)$
is still a Hardy weight. Indeed, it is enough to apply Tonelli’s theorem: for
each $\varphi\in\mathrm{Lip}_{c}(M)$,
(5.2)
$\begin{array}[]{lcl}\displaystyle\int_{M}\chi|\varphi|^{p}\mathrm{d}\mu_{f}&=&\displaystyle\int_{A}\left[\int_{M}w_{\alpha}(x)|\varphi(x)|^{p}\mathrm{d}\mu_{f}\right]\mathrm{d}\lambda(\alpha)\\\\[11.38092pt]
&\leq&\displaystyle\int_{A}\left[\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\right]\mathrm{d}\lambda(\alpha)\leq\displaystyle\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}.\end{array}$
Clearly, this construction makes sense also if for $\lambda$-almost all
$\alpha\in A$, $w_{\alpha}$ is a Hardy weight.
###### Remark 5.1.
_By item $5)$ of Proposition 4.4, each Green kernel $\mathcal{G}$ generates a
family $w$ indexed by $A=M$:_
(5.3)
$w(x,y)\doteq\left(\frac{p-1}{p}\right)^{p}\frac{|\nabla_{x}\mathcal{G}(x,y)|^{p}}{\mathcal{G}(x,y)^{p}}\
\ \ :\ \ \ M\times M\longrightarrow[0,+\infty],$
_provided that $w$ is measurable. If $p=2$, the standard construction of a
Green kernel in [34] produces a symmetric kernel, and measurability is
obvious. However, measurability seems to be a subtle issue if $p\neq 2$, since
$\mathcal{G}(x,y)$ is constructed in [38] and [39] by fixing $y$ and finding a
solution of $\Delta_{p,f}\mathcal{G}(x,y)=-\delta_{y}$. The dependence of
$\mathcal{G}(x,y)$ from $y$ could be, a priori, wild. _
We now describe two simple measures $\lambda$ that have been considered in the
literature when $M=\mathbb{R}^{m}$, and the corresponding Hardy inequalities.
###### Example 5.1.
[Multipole Hardy weights]
_Fix a possibly infinite sequence $\\{y_{j}\\}\subset M$, $j\in
I\subset\mathbb{N}$, let $\\{t_{j}\\}_{j\in I}\subset(0,1]$ be such that
$\sum_{j}t_{j}\leq 1$ and define_
$\lambda=\sum_{j}t_{j}\delta_{y_{j}},$
_where $\delta_{y_{j}}$ is the Dirac delta function at $y_{j}$. Being
$\lambda$ discrete, measurability of $w$ follows automatically and thus, by
(5.2),_
(5.4) $\left(\frac{p-1}{p}\right)^{p}\int_{M}\left[\sum_{j\in
I}\frac{t_{j}|\nabla_{x}\mathcal{G}(x,y_{j})|^{p}}{\mathcal{G}(x,y_{j})^{p}}\right]\big{|}\varphi(x)\big{|}^{p}\mathrm{d}\mu_{f}\leq\int_{M}\big{|}\nabla\varphi(x)\big{|}^{p}\mathrm{d}\mu_{f}$
_holds for $\varphi\in\mathrm{Lip}_{c}(M)$. _
###### Example 5.2.
[Hardy weights that blow-up along a submanifold]
_If $w(x,y)$ in (5.3) is measurable, for each rectifiable subset
$\Sigma\hookrightarrow M^{m}$ of finite non-zero $k$-dimensional Hausdorff
measure $\mathcal{H}^{k}$ we can set_
$\lambda=\frac{\mathcal{H}^{k}\llcorner\Sigma}{\mathcal{H}^{k}(\Sigma)},$
_Then, we have the Hardy inequality_
(5.5)
$\left(\frac{p-1}{p}\right)^{p}\int_{M}\left[\frac{1}{\mathcal{H}^{k}(\Sigma)}\int_{\Sigma}\frac{|\nabla_{x}\mathcal{G}(x,y)|^{p}}{\mathcal{G}(x,y)^{p}}\mathrm{d}\mathcal{H}^{k}(y)\right]\big{|}\varphi(x)\big{|}^{p}\mathrm{d}x\leq\int_{M}\big{|}\nabla\varphi(x)\big{|}^{p}\mathrm{d}x$
_for each $\varphi\in\mathrm{Lip}_{c}(M)$.
Hardy weights of this type have been considered, for instance, in [31]: in
Theorem 1.1 therein, $M=\mathbb{R}^{m}=\mathbb{R}^{2}\times\mathbb{R}^{m-2}$,
$p=2$, $f\equiv 0$ and $\Sigma\subset\mathbb{R}^{m}$ is a round circle
$S_{\rho}\doteq\mathbb{S}^{1}(\rho)\times\\{0\\}\subset\mathbb{R}^{2}\times\mathbb{R}^{m-2}$
centered at the origin and of radius $\rho$. We remark that Hardy weights of
different type but still depending on the distance from a submanifold have
been investigated, among others, in [10]. The reader is also suggested to see
the references therein for deepening. _
To introduce the results below, we first recall comparison geometry, starting
with the definition of a model manifold. Briefly, fix a point
$o\in\mathbb{R}^{m}$. Given $g\in C^{2}(\mathbb{R}^{+}_{0})$ such that $g>0$
on some open interval $(0,R)\subset\mathbb{R}^{+}$, $g(0)=0$,
$g^{\prime}(0)=1$, a model $(M^{m}_{g},\mathrm{d}s_{g}^{2})$ is the manifold
$B_{R}(o)\subset\mathbb{R}^{m}$ equipped with a radially symmetric $C^{2}$
metric $\mathrm{d}s_{g}^{2}$ whose expression, in polar geodesic coordinates
$(\rho,\theta)$ centered at $o$ (where $\theta\in\mathbb{S}^{m-1}$), is given
by
$\mathrm{d}s_{g}^{2}=\mathrm{d}\rho^{2}+g(\rho)^{2}\mathrm{d}\theta^{2},$
$\mathrm{d}\theta^{2}$ being the standard metric on the unit sphere
$\mathbb{S}^{m-1}$. Clearly, $\rho$ is the distance function from $o$, and
$M_{g}$ is complete if and only if $R=+\infty$. At a point $x=(\rho,\theta)$,
the radial sectional curvature $K_{\mathrm{rad}}$ of $M_{g}$ (that is, the
sectional curvature restricted to planes containing $\nabla\rho(x)$), the
Hessian and the Laplacian of $\rho$ are given by
(5.6) $\displaystyle
K_{\mathrm{rad}}(x)=-\frac{g^{\prime\prime}(\rho)}{g(\rho)},\quad\displaystyle\nabla\mathrm{d}\rho(x)=\frac{g^{\prime}(\rho)}{g(\rho)}\Big{(}\mathrm{d}s_{g}^{2}-\mathrm{d}\rho\otimes\mathrm{d}\rho\Big{)},\quad\Delta\rho(x)=(m-1)\frac{g^{\prime}(\rho)}{g(\rho)}.$
By the first formula, in (5.6), a model can also be given by prescribing the
radial sectional curvature $-G\in C^{0}(\mathbb{R}^{+}_{0})$ and recovering
$g\in C^{2}(\mathbb{R}^{+}_{0})$ as the solution of
(5.7) $\left\\{\begin{array}[]{l}g^{\prime\prime}-Gg=0\qquad\text{on
}\mathbb{R}^{+},\\\\[5.69046pt] g(0)=0,\quad
g^{\prime}(0)=1,\end{array}\right.$
on the maximal interval where $g>0$. A sharp condition on $G$ that ensures the
positivity of $g$ on the whole $\mathbb{R}^{+}$ is given by
$G_{-}\in L^{1}(\mathbb{R}^{+}),\qquad
t\int_{t}^{+\infty}G_{-}(s)\mathrm{d}s\leq\frac{1}{4}\qquad\text{on
}\mathbb{R}^{+},$
see Proposition 1.21 in [12]. In particular, if $G\equiv\kappa^{2}$ for some
constant $\kappa\geq 0$, we will denote with $g_{\kappa}$ the solution $g$ of
(5.7):
(5.8) $g_{\kappa}(\rho)=\left\\{\begin{array}[]{ll}\rho&\ \text{if
}\kappa=0,\\\\[5.69046pt] \kappa^{-1}\sinh(\kappa\rho)&\ \text{if
}\kappa>0.\end{array}\right.$
When $\kappa=0$, $M_{g}$ is the Euclidean space $\mathbb{R}^{m}$, while if
$\kappa>0$ our model is the hyperbolic space $\mathbb{H}^{m}_{\kappa}$ of
sectional curvature $-\kappa^{2}$. Clearly, the two examples are radially
symmetric with respect to any chosen origin $o$. Hereafter, we will always
consider geodesically complete models.
Given $p\in(1,+\infty)$, $-\Delta_{p}$ is subcritical on $M_{g}$ if and only
if
(5.9) $g(\rho)^{-\frac{m-1}{p-1}}\in L^{1}(+\infty)$
(the case $p=2$ can be found, for instance, in [34], and for $p\neq 2$ the
argument of the proof goes along the same lines). In fact, under the validity
of (5.9), up to an unessential constant, the function
$G(x,o)=G\big{(}(\rho,\theta),o\big{)}=\int_{\rho}^{+\infty}g(s)^{-\frac{m-1}{p-1}}\mathrm{d}s$
is the minimal positive Green kernel for $-\Delta_{p}$ with singularity at
$o$.
A simplified version of the Hessian and Laplacian comparison theorems from
below (see [58, 62, 12]) can be stated as follows: suppose that
$(M,\langle\,,\,\rangle)$ has a pole $o$ and, denoting with
$r(x)=\mathrm{dist}(x,o)$, that the radial sectional curvature
$K_{\mathrm{rad}}$ satisfies
(5.10) $K_{\mathrm{rad}}(x)\leq-G\big{(}r(x)\big{)}$
(i.e., for each plane $\pi\leq T_{x}M$ containing $\nabla r(x)$,
$K(\pi)\leq-G\big{(}r(x)\big{)}$). Denote with $g$ the solution of (5.7), and
let $(0,R)$ be the maximal interval where $g>0$. Then, in the sense of
quadratic forms,
(5.11)
$\nabla\mathrm{d}r(x)\geq\frac{g^{\prime}(r(x))}{g(r(x))}\Big{(}\langle\,,\,\rangle-\mathrm{d}r\otimes\mathrm{d}r\Big{)}$
pointwise on $B_{R}(o)\backslash\\{o\\}$, and tracing
(5.12) $\Delta r(x)\geq(m-1)\frac{g^{\prime}(r(x))}{g(r(x))},$
whose validity holds weakly on the geodesic ball $B_{R}(o)\subset M$. As it is
apparent from (5.6), $M$ is compared with a model $M_{g}$ of radial sectional
curvature $-G$. Indeed, via Sturm comparison, for (5.12) to hold it is enough
that $g\in C^{2}(\mathbb{R}^{+}_{0})$ solves the inequality
(5.13) $\left\\{\begin{array}[]{l}g^{\prime\prime}-Gg\leq 0\qquad\text{on
}\mathbb{R}^{+},\\\\[5.69046pt] g(0)=0,\quad
g^{\prime}(0)=1,\end{array}\right.$
so that the model to which $M$ is compared has radial sectional curvature
greater than or equal to $-G$. In [13], we have collected a number of examples
of $G$ for which explicit positive solutions $g$ of (5.13) can be found. These
also include manifolds whose sectional curvature can be positive but in a
controlled way, loosely speaking manifolds whose compared model is some sort
of paraboloid.
With this preparation, we are now ready to discuss the next cases.
### 5.1. Hardy weights on manifolds with a pole
###### Theorem 5.1.
Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold with a pole $o$.
Denoting with $r(x)=\mathrm{dist}(x,o)$, assume that the radial sectional
curvature satisfies
$K_{\mathrm{rad}}\leq-G\big{(}r(x)\big{)},$
for some $G\in C^{2}(\mathbb{R}^{+}_{0})$. Suppose that $g$ solving (5.13) is
positive on $\mathbb{R}^{+}$, and, for $p\in(1,+\infty)$, assume that
(5.14) $g(r)^{-\frac{m-1}{p-1}}\in L^{1}(+\infty).$
Then, the Hardy inequality
(5.15) $\int_{M}(\chi\circ
r)|\varphi|^{p}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}x$
holds for each $\varphi\in\mathrm{Lip}_{c}(M)$, where
(5.16)
$\chi(t)=\left(\frac{p-1}{p}\right)^{p}\left[g(t)^{\frac{m-1}{p-1}}\int_{t}^{+\infty}g(s)^{-\frac{m-1}{p-1}}\mathrm{d}s\right]^{-p}$
###### Proof.
Consider the transplanted Green kernel of the model $M_{g}$ with pole at $o$:
(5.17) $G(x)=\int_{r(x)}^{+\infty}g(s)^{-\frac{m-1}{p-1}}.$
Then, a computation using (5.11) shows that $\Delta_{p}G\leq 0$ on
$M\backslash\\{o\\}$.
Furthermore, $G$ satisfies the asymptotic behaviour
(5.18) $G(x)\sim\left\\{\begin{array}[]{ll}C>0&\quad\text{if
}p>m,\\\\[2.84544pt] |\log r|&\quad\text{if }p=m,\\\\[2.84544pt]
\frac{p-1}{m-p}r^{-\frac{m-p}{p-1}}&\quad\text{if
}p<m,\end{array}\right.\qquad\text{as }r\rightarrow 0^{+},$
for some constant $C>0$. It thus follows that, for each $c\in(0,C)$ if $p>m$
and $c\in\mathbb{R}^{+}$ if $p\leq m$, by the pasting Lemma 3.1 the function
$G_{c}=\min\\{G,c\\}$ is a solution of $\Delta_{p}G_{c}\leq 0$, hence by (4.4)
(5.19) $\left(\frac{p-1}{p}\right)^{p}\int_{M}\frac{|\nabla
G_{c}|^{p}}{G_{c}^{p}}|\varphi|^{p}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}x\qquad\text{holds
for each }\varphi\in\mathrm{Lip}_{c}(M).$
Letting $c\rightarrow C^{-}$ when $p>m$, $C\rightarrow+\infty$ when $p\leq m$,
and since
$\left(\frac{p-1}{p}\right)^{p}\frac{|\nabla
G|^{p}}{G^{p}}(x)=\chi\big{(}r(x)\big{)},$
we conclude the validity of (5.15). ∎
The function $\chi\circ r$ has the following asymptotics as $r=r(x)\rightarrow
0^{+}$:
(5.20)
$\chi(r)\sim\left\\{\begin{array}[]{ll}\left(\frac{p-1}{Cp}\right)^{p}r^{-\frac{p(m-1)}{p-1}}&\quad\text{if
}p>m,\\\\[8.5359pt]
\left(\frac{m-1}{m}\right)^{m}\frac{1}{r^{m}\log^{m}r}&\quad\text{if
}p=m,\\\\[8.5359pt]
\left(\frac{m-p}{p}\right)^{p}\frac{1}{r^{p}}&\quad\text{if
}p<m,\end{array}\right.$
where, for $p>m$, the constant $C$ is the same as in (5.18). Note that, on
each case, $\chi\circ r\in L^{1}_{\mathrm{loc}}(M)$, in particular the
singularity at $o$ is integrable.
###### Remark 5.2.
_Under the same assumptions on $M$, this example can be extended to deal with
the weighted operator $\Delta_{p,f}$ provided that $f=f(r(x))$ is a radial
function. In this case, it can be seen that (5.14) must be replaced by_
$\frac{e^{f(r)}}{g(r)^{\frac{m-1}{p-1}}}\in L^{1}(+\infty),$
_and that_
$G(x)=\int_{r(x)}^{+\infty}\frac{e^{f(s)}\mathrm{d}s}{g(s)^{\frac{m-1}{p-1}}}$
_is a solution of $\Delta_{p,f}G\leq 0$ on $M\backslash\\{o\\}$ that gives
rise to a Hardy weight analogous to (5.15). _
The Hardy weight in (5.15) is explicit once we have an explicit $g$ solving
(5.13), for example those related to the families of $G$ described in the
Appendix of [13]. We refer the reader to the above mentioned paper also for a
detailed study of the corresponding Hardy weight (called the “critical curve"
therein) for $p=2$. The modifications needed to deal with general $p$ are
straightforward. Here, we just focus on the Euclidean and hyperbolic settings.
###### Example 5.3.
[Hardy weights for Euclidean and hyperbolic spaces]
_Consider the Euclidean space, where $g(r)=g_{0}(r)=r$. Condition (5.14) is
met if and only if $p<m$, and the Hardy weight (5.15) has the simple
expression_
(5.21)
$\chi\big{(}r(x)\big{)}=\left(\frac{m-p}{p}\right)^{p}\frac{1}{r(x)^{p}}.$
_Consequently, the Hardy inequality_
(5.22)
$\left(\frac{m-p}{p}\right)^{p}\int_{M}\frac{|\varphi|^{p}}{r^{p}}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}x\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M)$
_holds on each manifold with a pole and radial sectional curvature
$K_{\mathrm{rad}}\leq 0$. Inequality (5.22) is classical and well-known in
$\mathbb{R}^{m}$, see [10].
On the hyperbolic space $\mathbb{H}^{m}_{\kappa}$ of sectional curvature
$-\kappa^{2}$, where $g_{\kappa}(r)=\kappa^{-1}\sinh(\kappa r)$, condition
(5.14) is met for each $m\geq 2$ independently of $p$, but the expression of
$\chi(r(x))$ is not so neat. However, an iterative argument allows us to
explicitly compute the integral in the expression of $\chi$ in some relevant
cases. For $\alpha>0$, set_
$I_{\alpha}(r)\doteq\int_{r}^{+\infty}\frac{\mathrm{d}s}{g_{\kappa}(s)^{\alpha}}\qquad\text{and}\qquad\chi_{\alpha}(r)\doteq\left(\frac{p-1}{p}\right)^{p}\left[g_{\kappa}(r)^{\alpha}I_{\alpha}(r)\right]^{-p}.$
_In view of ( 5.16), our case of interest is $\alpha=\frac{m-1}{p-1}$.
Writing_
$\frac{I_{\alpha+2}}{\kappa^{\alpha+2}}=\int_{r}^{+\infty}\frac{\cosh(\kappa
s)\cosh(\kappa s)}{\sinh^{\alpha+2}(\kappa
s)}\mathrm{d}s-\frac{I_{\alpha}}{\kappa^{\alpha}},$
_and integrating by parts, we obtain the recursion formula_
$\alpha I_{\alpha}(r)=\frac{\cosh(\kappa
r)}{\kappa^{2}g_{\kappa}(r)^{\alpha+1}}-\frac{\alpha+1}{\kappa^{2}}I_{\alpha+2}(r)$
_that yields_
$\alpha\chi_{\alpha}^{-1/p}=\frac{p\coth(\kappa
r)}{(p-1)\kappa}-\frac{\alpha+1}{\kappa^{2}g_{\kappa}(r)^{2}}\chi_{\alpha+2}^{-1/p}.$
_Therefore, one may inductively recover $\chi_{\alpha}$. In particular, if
$\alpha=1$ (i.e., in our case of interest, $p=m$), by explicit integration of
$I_{1}(r)$ we get_
(5.23) $\chi_{1}(r)=\left(\frac{(m-1)\kappa}{m}\right)^{m}\left[\sinh(\kappa
r)\log\left(\frac{e^{\kappa r}+1}{e^{\kappa r}-1}\right)\right]^{-m},$
_while if $\alpha=2$ (i.e. $m=2p-1$), again by explicit integration of
$I_{2}(r)$ we deduce_
(5.24)
$\chi_{2}(r)=\left(\frac{2(m-1)\kappa}{m+1}\right)^{\frac{m+1}{2}}\big{(}1-e^{-2\kappa
r}\big{)}^{-\frac{m+1}{2}},$
_see Example 3.15 in[12]. An important feature of $\chi_{\alpha}(r)$ is the
following:_
(5.25)
$\chi_{\alpha}(r)\geq\left(\frac{p-1}{p}\right)^{p}\alpha^{p}\kappa^{p},\qquad\chi_{\alpha}(r)\rightarrow\left(\frac{p-1}{p}\right)^{p}\alpha^{p}\kappa^{p}\quad\text{as
}r\rightarrow+\infty.$
_Indeed, the limit is straightforwardly computable. As for the first relation,
it follows from the following property. To state it, for fixed $\alpha>0$ and
for $g$ satisfying $1/g^{\alpha}\in L^{1}(+\infty)$, write_
$\chi_{g}(r)\doteq\left(\frac{p-1}{p}\right)^{p}\left(g(r)^{\alpha}\int_{r}^{+\infty}\frac{\mathrm{d}s}{g(s)^{\alpha}}\right)^{-p}.$
_Then, the next comparison result holds:_
> if $g_{1}/g_{2}$ is non-decreasing on $\mathbb{R}^{+}$ (respectively, non-
> increasing),
> then $\chi_{g_{1}}\geq\chi_{g_{2}}$ on $\mathbb{R}^{+}$ (resp, $\leq$).
_The proof of this fact goes along the same lines as in Proposition 3.12
in[12], and is left to the interested reader. Using this with $g_{1}(r)\doteq
g_{\kappa}(r)$ and $g_{2}(r)\doteq\exp\\{\kappa r\\}$ we get_
$\chi_{g_{1}}(r)\geq\chi_{g_{2}}(r)\equiv\left(\frac{p-1}{p}\right)^{p}\alpha^{p}\kappa^{p},$
_as claimed._
When each point of the manifold $M$ is a pole, we can construct multipole
Hardy weights by the standard procedure described at the beginning of Section
5. This is the case if, for example, $M$ is a Cartan-Hadamard manifold, that
is, a simply-connected, complete manifold with non-positive sectional
curvature.
###### Theorem 5.2.
For $m\geq 2$, let $M^{m}$ be a Cartan-Hadamard manifold satisfying
$K\leq-\kappa^{2}$, for some constant $H\geq 0$. Given the solution
$g_{\kappa}$ of (5.8), let
(5.26) $\left\\{\begin{array}[]{ll}p\in(1,m)&\quad\text{if
}\kappa=0,\\\\[2.84544pt] p\in(1,+\infty)&\quad\text{if
}\kappa>0.\end{array}\right.$
Then, for each unit measure $\lambda$ on $M$, the Hardy inequality
(5.27)
$\int_{M}\left[\int_{M}\big{(}\chi\big{(}\mathrm{dist}(x,y)\big{)}\big{)}\mathrm{d}\lambda(y)\right]\big{|}\varphi(x)\big{|}^{p}\mathrm{d}x\leq\int_{M}\big{|}\nabla\varphi(x)\big{|}^{p}\mathrm{d}x$
holds for each $\varphi\in\mathrm{Lip}_{c}(M)$, where
(5.28)
$\chi(t)=\left(\frac{p-1}{p}\right)^{p}\left[g_{\kappa}(t)^{\frac{m-1}{p-1}}\int_{t}^{+\infty}g_{\kappa}(s)^{-\frac{m-1}{p-1}}\mathrm{d}s\right]^{-p}.$
In particular, when $\kappa=0$, for each $\varphi\in\mathrm{Lip}_{c}(M)$
(5.29)
$\left(\frac{m-p}{p}\right)^{p}\int_{M}\left[\int_{M}\frac{\mathrm{d}\lambda(y)}{\mathrm{dist}(x,y)^{p}}\right]\big{|}\varphi(x)\big{|}^{p}\mathrm{d}x\leq\int_{M}\big{|}\nabla\varphi(x)\big{|}^{p}\mathrm{d}x.$
###### Proof.
By (5.26), (5.14) is met for $g=g_{\kappa}$. Since each $y\in M$ is a pole of
$M$, setting $r_{y}(\cdot)=\mathrm{dist}(\cdot,y)$ one can apply Theorem 5.1
to deduce that (5.15) holds for each fixed $y$, namely,
(5.30) $\int_{M}(\chi\circ
r_{y})|\varphi|^{p}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}x\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M),$
where $\chi$ is as in (5.28). The generalization in (5.27) follows from the
argument at the beginning of Section 5. ∎
### 5.2. Hardy weights on minimally immersed submanifolds
###### Theorem 5.3.
Let $F:(M^{m},\langle\,,\,\rangle)\rightarrow(N^{n},(\,,\,))$ be an immersed,
minimal submanifold of a Cartan-Hadamard ambient space $N^{n}$, and suppose
that the sectional curvature $\bar{K}$ of $N$ satisfies
$\bar{K}\leq-\kappa^{2}$, for some constant $H\geq 0$. If $\kappa=0$, we
assume that $m\geq 3$. Then, given the solution $g_{\kappa}$ of (5.8), and
denoting with $\bar{r}_{q}$ the extrinsic distance from a point $q\in N$
evaluated along the immersion $F$, the following Hardy inequality holds:
(5.31)
$\int_{M}(\chi\circ\bar{r}_{q})\varphi^{2}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{2}\mathrm{d}x\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M),$
where
(5.32)
$\chi(t)=\frac{1}{4}\left[g_{\kappa}(t)^{m-1}\int_{t}^{+\infty}g_{\kappa}(s)^{1-m}\mathrm{d}s\right]^{-2}.$
In particular, if $\kappa=0$,
(5.33)
$\left(\frac{m-2}{2}\right)^{2}\int_{M}\frac{\varphi^{2}}{\bar{r}_{q}^{2}}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{2}\mathrm{d}x,\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M),$
while, if $\kappa>0$,
* -
when $m=2$, that is, $M$ is a surface, for each
$\varphi\in\mathrm{Lip}_{c}(M)$
(5.34)
$\left(\frac{\kappa}{2}\right)^{2}\int_{M}\left[\sinh(\kappa\bar{r}_{q})\log\left(\frac{e^{\kappa\bar{r}_{q}}+1}{e^{\kappa\bar{r}_{q}}-1}\right)\right]^{-2}\varphi^{2}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{2}\mathrm{d}x;$
* -
when $m=3$, for each $\varphi\in\mathrm{Lip}_{c}(M)$
(5.35)
$\kappa^{2}\int_{M}\frac{\varphi^{2}}{(1-e^{-2\kappa\bar{r}_{q}})^{2}}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{2}\mathrm{d}x.$
###### Remark 5.3.
_The inequality in ( 5.33) has been proved in [19, 49] by combining the
comparison for the Hessian of the extrinsic distance function with an
integration by parts argument. The case $\kappa>0$ has been considered in [49,
Example 1.8]. However, the Hardy weight found there is skew with (5.34) and
(5.35), in particular it is quite smaller if $\bar{r}_{q}$ is close to zero.
As before, other Hardy weights can be constructed in the way described in
Examples (5.1) and (5.2). _
###### Proof.
We mark with a bar each quantity when referred to $N$, so that, for example,
$\bar{\nabla},\overline{\mathrm{dist}}$ are the Riemannian connection and the
distance function of $N$. For simplicity, we denote with $\bar{r}_{q}$ the
distance function from $q$ in the manifold $N$, i.e.
$\bar{r}_{q}(\cdot)=\overline{\mathrm{dist}}(\cdot,q)$, so that the function
$\bar{r}_{q}$ in the statement of the theorem is, indeed, $\bar{r}_{q}\circ
F$. By the Hessian comparison theorem (5.11), for each $q\in N$ it holds
$\bar{\nabla}\mathrm{d}\bar{r}_{q}\geq\frac{g_{\kappa}^{\prime}(\bar{r}_{q})}{g_{\kappa}(\bar{r}_{q})}\Big{(}(\,,\,)-\mathrm{d}\bar{r}_{q}\otimes\mathrm{d}\bar{r}_{q}\Big{)}\qquad\text{on
}N\backslash\\{q\\}.$
For $x\in M$, $q\in N$ define
(5.36)
$G_{q}(x)=h\big{(}\bar{r}_{q}\big{(}F(x)\big{)}\big{)},\qquad\text{where
}h(t)=\int_{t}^{+\infty}\frac{\mathrm{d}s}{g_{\kappa}(s)^{m-1}}.$
Observe that the integral in $h(t)$ converges for each $m\geq 2$ when
$\kappa>0$, and for each $m\geq 3$ if $\kappa=0$, which accounts for our
dimensional restrictions. Denote with $\mathrm{II}$ the second fundamental
form of $F$. By the chain rule and using $h^{\prime}<0$, for each vector field
$X$ on $M$ (identified with $F_{*}(X))$ we get
$\begin{array}[]{lcl}\displaystyle\nabla\mathrm{d}G_{q}(X,X)&=&\displaystyle\displaystyle
h^{\prime\prime}\big{(}\bar{\nabla}\bar{r}_{q},X\big{)}^{2}+h^{\prime}\bar{\nabla}\mathrm{d}\bar{r}_{q}(X,X)+h^{\prime}\big{(}\bar{\nabla}\bar{r}_{q},\mathrm{II}(X,X)\big{)}\\\\[5.69046pt]
&\leq&\displaystyle\displaystyle\left(h^{\prime\prime}-\frac{g_{\kappa}^{\prime}}{g_{\kappa}}h^{\prime}\right)\big{(}\bar{\nabla}\bar{r}_{q},X\big{)}^{2}+h^{\prime}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}|X|^{2}+h^{\prime}\big{(}\bar{\nabla}\bar{r}_{q},\mathrm{II}(X,X)\big{)}\\\\[11.38092pt]
&=&\displaystyle\displaystyle-
mh^{\prime}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}\big{(}\bar{\nabla}\bar{r}_{q},X\big{)}^{2}+h^{\prime}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}|X|^{2}+h^{\prime}\big{(}\bar{\nabla}\bar{r}_{q},\mathrm{II}(X,X)\big{)}\\\\[11.38092pt]
&=&\displaystyle\displaystyle
h^{\prime}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}\Big{[}|X|^{2}-m\big{(}\bar{\nabla}\bar{r}_{q},X\big{)}^{2}\Big{]}+h^{\prime}\big{(}\bar{\nabla}\bar{r}_{q},\mathrm{II}(X,X)\big{)}\end{array}$
Tracing with respect to an orthonormal frame $\\{e_{i}\\}$ of $M$, using
minimality and again $h^{\prime}<0$, we obtain
(5.37) $\displaystyle\Delta G_{q}\leq
mh^{\prime}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}\left(1-|\bar{\nabla}^{T}\bar{r}_{q}|^{2}\right)\leq
0$
where $\bar{\nabla}^{T}$ is the component of the gradient in $N$ which is
tangent to $M$. By Proposition 4.4,
(5.38) $\frac{|\nabla
G_{q}(x)|^{2}}{4G_{q}(x)^{2}}=\left(\frac{h^{\prime}(\bar{r}_{q})}{2h(\bar{r}_{q})}\right)^{2}|\bar{\nabla}^{T}\bar{r}_{q}|^{2}=\chi\big{(}\bar{r}_{q}\big{(}F(x)\big{)}\big{)}|\bar{\nabla}^{T}\bar{r}_{q}|^{2}.$
is a Hardy weight. Unfortunately, such a weight is not effective, since we
cannot control the size of $\bar{\nabla}^{T}\bar{r}_{q}$. However, we can
improve (5.38) to the effective Hardy weight $\chi(\bar{r}_{q})$ by using the
full information coming from (5.37). In fact, since $h^{\prime}<0$, by (5.37)
the function $u_{1}\doteq\sqrt{G_{q}}$ solves, on $M\backslash f^{-1}\\{q\\}$,
(5.39) $\begin{array}[]{lcl}\Delta u_{1}&=&\displaystyle\left[\frac{\Delta
G_{q}}{2G_{q}}-\chi(\bar{r}_{q})|\bar{\nabla}^{T}\bar{r}_{q}|^{2}\right]u_{1}\leq\left[m\frac{h^{\prime}}{2h}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}\big{(}1-|\bar{\nabla}^{T}\bar{r}_{q}|^{2}\big{)}-\chi(\bar{r}_{q})|\bar{\nabla}^{T}\bar{r}_{q}|^{2}\right]u_{1}\\\\[14.22636pt]
&\leq&\displaystyle\left[-m\sqrt{\chi(\bar{r}_{q})}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}+\sqrt{\chi(\bar{r}_{q})}|\bar{\nabla}^{T}\bar{r}_{q}|^{2}\left(m\frac{g_{\kappa}^{\prime}}{g_{\kappa}}-\sqrt{\chi(\bar{r}_{q})}\right)\right]u_{1}.\end{array}$
Now, we claim that
(5.40) $\zeta(t)\doteq
m\frac{g_{\kappa}^{\prime}(t)}{g_{\kappa}(t)}-\sqrt{\chi(t)}\geq
0\qquad\text{for }t\in\mathbb{R}^{+}.$
Indeed, in the Euclidean case $\kappa=0$, $g_{\kappa}(t)=t$ and explicit
computation gives
$\zeta(t)=\frac{m+2}{2t}>0\qquad\text{on }\mathbb{R}^{+}.$
When $\kappa>0$, $g_{\kappa}(t)=\kappa^{-1}\sinh(\kappa t)$. A computation
gives
(5.41) $\zeta(t)\sim\frac{m+2}{2t}\quad\text{as }\,t\rightarrow
0,\qquad\zeta(t)\sim\frac{(m+1)\kappa}{2}\quad\text{as
}\,t\rightarrow+\infty.$
Now, $y(t)=\sqrt{\chi(t)}$ solves
(5.42)
$y^{\prime}=2y^{2}-y(m-1)\frac{g_{\kappa}^{\prime}}{g_{\kappa}}\qquad\text{on
}\mathbb{R}^{+},$
Suppose that $\zeta(\bar{t})\leq 0$ for some $\bar{t}>0$. Then, an inspection
of (5.42) and the fact that $g_{\kappa}^{\prime}/g_{\kappa}$ is decreasing
show that $y^{\prime}>0$ on $[\bar{t},+\infty)$, whence there exists $c>0$
such that
$y>\frac{m-1}{2}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}+c\qquad\text{on
}[\bar{t},+\infty).$
But then
$\lim_{t\rightarrow+\infty}\zeta(t)=m\kappa-\lim_{t\rightarrow+\infty}y(t)\leq
m\kappa-c-\frac{m-1}{2}\kappa=\frac{m+1}{2}\kappa-c,$
contradicting (5.41) and proving the claim. Next, by (5.40) we can use the
estimate $|\bar{\nabla}^{T}\bar{r}_{q}|\leq 1$ to conclude
(5.43) $\Delta
u_{1}\leq\left[-m\sqrt{\chi(\bar{r}_{q})}\frac{g_{\kappa}^{\prime}}{g_{\kappa}}+\sqrt{\chi(\bar{r}_{q})}\left(m\frac{g_{\kappa}^{\prime}}{g_{\kappa}}-\sqrt{\chi(\bar{r}_{q})}\right)\right]u_{1}=-\chi(\bar{r}_{q})u_{1}.$
For $\varepsilon>0$, consider the truncated function $\chi_{\varepsilon}(t)$
given by $\chi_{\varepsilon}(t)=\chi(t)$ if $t\geq 2\varepsilon$, and $0$
otherwise. Then, $\chi_{\varepsilon}(\bar{r}_{q}\circ F)\in
L^{\infty}_{\mathrm{loc}}(M)$ and by (5.43)
(5.44) $\Delta u_{1}+\chi_{\varepsilon}(\bar{r}_{q})u_{1}\leq 0\qquad\text{on
}M\backslash F^{-1}\\{q\\}.$
If $F^{-1}\\{q\\}\neq\emptyset$ observe that the constant function
$u_{2}\doteq\sqrt{G_{q}(\varepsilon)}$ solves (5.44) on
$F^{-1}\\{B_{2\varepsilon}(q)\\}$. By the pasting Lemma 3.1 with the choices
$\Omega_{1}\doteq F^{-1}\\{B_{\varepsilon}(q)\\}\backslash F^{-1}\\{q\\}$,
$\Omega_{2}\doteq F^{-1}\\{B_{2\varepsilon}(q)\\}\backslash F^{-1}\\{q\\}$,
and since $h^{\prime}<0$, we deduce that
$u\doteq\left\\{\begin{array}[]{ll}\sqrt{G_{q}(\varepsilon)}&\quad\text{on
}F^{-1}\\{B_{\varepsilon}(q)\\},\\\\[5.69046pt] \sqrt{G_{q}}&\quad\text{on
}M\backslash F^{-1}\\{B_{\varepsilon}(q)\\}\end{array}\right.$
solves $\Delta u+\chi_{\varepsilon}(\bar{r}_{q})u\leq 0$ on
$\Omega_{2}=F^{-1}\\{B_{2\varepsilon}(q)\\}\backslash F^{-1}\\{q\\}$. Since
the pasting region $F^{-1}\\{\partial B_{\varepsilon}(q)\\}$ is internal to
$\Omega_{2}$ and $u$ is smooth in a neighbourhood of $F^{-1}\\{q\\}$, then
clearly $u$ solves
(5.45) $\Delta u+\chi_{\varepsilon}(\bar{r}_{q})u\leq 0\qquad\text{on the
whole }M.$
By Proposition 3.4 with $p=2$, $f=1$ and $V=\chi_{\varepsilon}(\bar{r}_{q})$
we deduce that $\lambda_{V}(M)\geq 0$, that is
$\int_{M}\chi_{\varepsilon}(\bar{r}_{q}\circ
F)\varphi^{2}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{2}\mathrm{d}x\qquad\text{for
each }\varphi\in\mathrm{Lip}_{c}(M),$
whence, letting $\varepsilon\rightarrow 0$ and using monotone convergence we
deduce (5.31). The cases (5.33), (5.34), (5.35) follow by computing $\chi(r)$
according to Example 5.3, in particular see (5.21), (5.23), (5.24). ∎
### 5.3. Specializing our main theorems: an example
To illustrate the results of the last two sections, by way of an example we
specialize our Theorem 2.1 to the case of quasilinear Yamabe type equations on
Cartan-Hadamard manifolds. We underline that all the assumptions in the next
Corollary are explicit and easy to check. Clearly, analogous results can be
stated using any of Theorems 5.1, 5.2, 5.3, as well as Theorem 2.2 instead of
Theorem 2.1.
###### Corollary 5.1.
Let $M$ be a Cartan-Hadamard manifold of dimension $m\geq 3$, and let
$p\in(1,m)$. Let $a,b\in L^{\infty}_{\mathrm{loc}}(M)$, and suppose that, for
some countable set of points $\\{y_{j}\\}_{j\in I}\subset M$ and
$\\{t_{j}\\}_{j\in I}\subset[0,1]$ with $\displaystyle\sum_{j}t_{j}\leq 1$,
(5.46) $a(x)\leq\left(\frac{m-p}{p}\right)^{p}\,\sum_{j\in
I}\frac{t_{j}}{\mathrm{dist}(x,y_{j})^{p}}.$
Furthermore, suppose that
* $i)$
$b_{-}(x)$ has compact support;
* $ii)$
$a(x)=O\big{(}b(x)\big{)}$ as $x$ diverges
* $iii)$
for some $\theta>0$, $\big{(}a(x)-\theta b_{+}(x)\big{)}_{-}\in L^{1}(M)$.
Fix a nonlinearity $F(t)$ satisfying (2.6). Then, there exists $\delta>0$ such
that if
$b(x)\geq-\delta\qquad\text{on }M,$
there exists a weak solution $u\in C^{1,\mu}_{\mathrm{loc}}(M)$ of
(5.47)
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p}u+a(x)u^{p-1}-b(x)F(u)=0\qquad\text{on
}M,\\\\[5.69046pt] 0<u\leq\|u\|_{L^{\infty}(M)}<+\infty.\end{array}\right.$
###### Proof.
In view of Theorem 2.1, it is enough to prove that $-\Delta_{p}$ and $Q_{a}$
are subcritical on $M$. By Theorem 5.2, the Hardy inequality
$\left(\frac{m-p}{p}\right)^{p}\,\int_{M}\sum_{j\in
I}\frac{t_{j}}{\mathrm{dist}(x,y_{j})^{p}}|\varphi(x)|^{p}\mathrm{d}x\leq\int_{M}|\nabla\varphi(x)|^{p}\mathrm{d}x$
holds for $\varphi\in\mathrm{Lip}_{c}(M)$. Therefore, by Proposition 4.4,
$-\Delta_{p}$ is subcritical. Next, the fact that the inequality in (5.46) is
strict on a set of positive measure (the right-hand side is essentially
unbounded at each $y_{j}$) assures that $Q_{a}$ be subcritical by Proposition
4.5. ∎
## 6\. Proofs of Theorems 2.1 and 2.2
We first address the local solvability of the Dirichlet problem for
$\Delta_{p,f}u+A(x)u^{p-1}-B(x)F(u)=0$
when $B\geq 0$.
###### Lemma 6.1.
Let $M$ be a Riemannian manifold, $p\in(1,+\infty)$ and $f\in C^{\infty}(M)$.
Let $\Omega\Subset M$ be a smooth relatively compact open set, and let
$A(x),B(x)\in L^{\infty}(\Omega)$ satisfy $\lambda_{A}(\Omega)>0$ and $B\geq
0$ a.e. on $\Omega$. Then, for each nonlinearity $F(t)$ that satisfies (2.6),
and for each $\varphi\in C^{1,\alpha}(\partial\Omega)$, $\alpha\in(0,1)$ such
that $\varphi\geq 0$, $\varphi\not\equiv 0$, there exist $\mu\in(0,1)$ and a
unique $0<z\in C^{1,\mu}(\overline{\Omega})$ solving
(6.1)
$\left\\{\begin{array}[]{ll}\Delta_{p,f}z+A(x)z^{p-1}-B(x)F(z)=0&\quad\text{on
}\Omega,\\\\[5.69046pt] z=\varphi&\quad\text{on
}\partial\Omega.\end{array}\right.$
###### Proof.
Take the positive solution $z_{0}\in C^{1,\mu}(\overline{\Omega})$ of
($P_{0}$)
$\left\\{\begin{array}[]{ll}\displaystyle\Delta_{p,f}z_{0}+A|z_{0}|^{p-2}z_{0}=0&\quad\text{on
}\Omega,\\\\[5.69046pt] z_{0}=\varphi&\quad\text{on
}\partial\Omega,\end{array}\right.$
which exists by Proposition 3.4. Since $B\geq 0$, $z_{0}$ gives rise to a
supersolution for (6.1). To construct a subsolution with boundary value
$\varphi$, we solve
($P_{n}$)
$\left\\{\begin{array}[]{ll}\displaystyle\Delta_{p,f}z_{1}+(A-B_{1})|z_{1}|^{p-2}z_{1}=0&\quad\text{on
}\Omega,\displaystyle\qquad B_{1}\doteq
B\frac{F(z_{0})}{z_{0}^{p-1}}\\\\[5.69046pt] z_{1}=\varphi&\quad\text{on
}\partial\Omega.\end{array}\right.$
Since $A-B_{1}\leq A$, $\lambda_{A-B_{1}}(\Omega)>0$, thus $z_{1}$ exists and,
by comparison, $z_{1}\leq z_{0}$. Since $F(t)/t^{p-1}$ is increasing,
$\displaystyle\Delta_{p,f}z_{1}+\left(A-B\frac{F(z_{1})}{z_{1}^{p-1}}\right)z_{1}^{p-1}\geq\displaystyle\Delta_{p,f}z_{1}+\left(A-B\frac{F(z_{0})}{z_{0}^{p-1}}\right)z_{1}^{p-1}=0.$
Hence $z_{1}$ is a subsolution for (6.1). Applying the subsolution-
supersolution method (see [25], Theorem 4.14 page 272) we obtain the existence
of $z$ satisfying (6.1). The local Harnack inequality in Theorem 3.1 gives
$z>0$ on $\overline{\Omega}$, and uniqueness follows from the comparison
Proposition 3.3. ∎
Next, we investigate the existence of local uniform lower bounds for solutions
of the Dirichlet problem when $\Omega$ varies.
###### Lemma 6.2.
Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold, $f\in C^{\infty}(M)$,
$p\in(1,+\infty)$, and assume that $Q_{0}$ is subcritical on $M$. Let $A,B\in
L^{\infty}_{\mathrm{loc}}(M)$ such that
(6.2) $\left\\{\begin{array}[]{l}Q_{A}\,\text{ is non-
negative},\\\\[5.69046pt] (A(x)-\theta B(x)\big{)}_{-}\in
L^{1}(M,\mathrm{d}\mu_{f}),\qquad\text{for some constant
}\,\theta>0.\end{array}\right.$
Fix $\varepsilon>0$. Consider a triple of relatively compact, open sets
$\Lambda\Subset\Lambda^{\prime}\Subset\Omega\Subset M$, with $\partial\Omega$
smooth, and let $z\in C^{1,\mu}(\overline{\Omega})$ be a positive solution of
(6.3) $\left\\{\begin{array}[]{ll}\Delta_{p,f}z+A(x)z^{p-1}-B(x)F(z)\leq
0&\quad\text{on }\Omega;\\\\[5.69046pt] z=\varepsilon&\quad\text{on
}\partial\Omega.\end{array}\right.$
Then, there exists $C>0$ depending on $\varepsilon$, but independent of
$\Omega$, such that
(6.4) $\inf_{\Lambda}z\geq C.$
###### Proof.
Observe that, by the half-Harnack inequality in Theorem 3.1, $z>0$ on
$\Omega$. By comparison, we can also suppose that $z$ solves (6.3) with the
equality sign, whence $z\in C^{1,\mu}(\overline{\Omega})$. Fix
$\delta\in(0,\varepsilon)$ small enough that
(6.5) $\frac{F(t)}{t^{p-1}}<\theta\qquad\text{if }\,t\in(0,\delta).$
This is possible by (2.6). Let
$\eta\in\mathrm{Lip}_{\mathrm{loc}}(\mathbb{R})$, $\eta(\log\varepsilon)=0$ to
be specified later, and define $u=\log z$ on $\Omega$, so that, weakly,
(6.6)
$\left\\{\begin{array}[]{lr}\displaystyle\Delta_{p,f}u=-A+B\frac{F(z)}{z^{p-1}}-(p-1)|\nabla
u|^{p}&{\rm on}\ \Omega\\\\[5.69046pt] \displaystyle u=\log\varepsilon&{\rm
on}\ \partial\Omega.\end{array}\right.$
The function $\eta(u)\in\mathrm{Lip}_{0}(\Omega)$ can be used as a test
function for (6.6) to obtain
(6.7)
$\displaystyle\int_{\Omega}\big{[}(p-1)\eta(u)-\eta^{\prime}(u)\big{]}|\nabla
u|^{p}\mathrm{d}\mu_{f}=\int_{\Omega}\left[-A+B\frac{F(z)}{z^{p-1}}\right]\eta(u)\mathrm{d}\mu_{f}.$
Let $L(\delta)\doteq\\{x:u(x)<\log\delta\\}$. Choose
$\eta(t)=\left[1-\frac{e^{(p-1)t}}{\delta^{p-1}}\right]1_{(-\infty,\log\delta)}(t)\in\mathrm{Lip}(\mathbb{R}).$
Then, from (6.7) and (6.2), and since $\delta\in(0,\varepsilon)$,
$\eta(\log\varepsilon)=0$. Plugging in (6.7) and using (6.5) we deduce
(6.8) $\int_{L(\delta)}|\nabla
u|^{p}\mathrm{d}\mu_{f}\leq\frac{1}{p-1}\int_{L(\delta)}\left[-A+B\frac{F(z)}{z^{p-1}}\right]\eta(u)\mathrm{d}\mu_{f}\leq\frac{1}{p-1}\|(A-\theta
B)_{-}\|_{L^{1}(M,\mathrm{d}\mu_{f})}$
Since $Q_{0}$ is subcritical on $M$, by Proposition 4.4 there exists $W\in
C^{0}(M)$, $W>0$ on $M$ such that
(6.9)
$\int_{M}W|\varphi|^{p}\mathrm{d}\mu_{f}\leq\int_{M}|\nabla\varphi|^{p}\mathrm{d}\mu_{f}\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$
Now the function $\varphi(x)=\big{(}u(x)-\log\delta\big{)}1_{L(\delta)}(x)$,
extended with $0$ outside $\Omega$, is an admissible test function for (6.9)
and with this choice of $\varphi$ we obtain
(6.10) $\displaystyle\int_{L(\delta)}|\nabla
u|^{p}\mathrm{d}\mu_{f}\geq\int_{L(\delta)}W|u-\log\delta|^{p}\mathrm{d}\mu_{f}.$
By contradiction assume that there exists a sequence of relatively compact
open sets $\\{\Omega_{j}\\}$ with smooth boundary such that
$\Lambda^{\prime}\Subset\Omega_{j}$ and a sequence of associated solutions
$\varphi_{j}$ of (6.3), such that $\inf_{\Lambda}\varphi_{j}\rightarrow 0^{+}$
as $j\rightarrow+\infty$. Using Harnack inequality of Theorem 3.1 $3)$,
$\phi_{j}\rightarrow 0$ uniformly on $\Lambda$ as $j\to+\infty$ (note that, to
infer $\phi_{j}\rightarrow 0$, we need that each $\Omega_{j}$ contains a fixed
domain larger than $\Lambda$, which accounts for the presence of
$\Lambda^{\prime}$). Having fixed $N>0$, we choose $j_{0}$ large enough that
$u_{j}=\log\varphi_{j}<-N+\log\delta$ on $\Lambda$ when $j\geq j_{0}$.
Consequently, $\Lambda\subset\\{u_{j}<\log\delta\\}\doteq L_{j}(\delta)$, and
from (6.8) and (6.10) we deduce
$\begin{array}[]{lcl}\displaystyle
N^{p}\int_{\Lambda}W\mathrm{d}\mu_{f}&\leq&\displaystyle\int_{L_{j}(\delta)}W|u_{j}-\log\delta|^{p}\mathrm{d}\mu_{f}\leq\int_{L_{j}(\delta)}|\nabla
u_{j}|^{p}\mathrm{d}\mu_{f}\\\\[11.38092pt]
&\leq&\displaystyle\frac{1}{p-1}\|(A-\theta
B)_{-}\|_{L^{1}(M,\mathrm{d}\mu_{f})}.\end{array}$
This gives a contradiction provided $N$ is large enough, proving the validity
of (6.4). ∎
###### Remark 6.1.
_To guarantee ( 6.4), the second condition in (6.2) cannot be relaxed too
much. To see this, let us suppose that $B\equiv 0$, for which the second in
(6.2) reads $A_{-}\in L^{1}(M,\mathrm{d}\mu_{f})$. We first observe that the
validity of (6.4) is granted provided that there exists a positive, bounded
solution of $Q^{\prime}_{A}(u)=0$ on $M$. Indeed, if such a $u$ exists,
comparing a solution $z$ of (6.3) with $\varepsilon u/\|u\|_{L^{\infty}(M)}$
with the aid of Proposition 3.3 yields (6.4). If, on the other hand, there
exists a positive solution of $Q_{A}^{\prime}(u)=0$ with
$u(x)\rightarrow+\infty$ as $x$ diverges, (6.4) fails: indeed, if we consider
a smooth exhaustion $\\{\Omega_{j}\\}$ with
$\partial\Omega_{j}\subset\\{u\in[j,j+1)\\}$, we compare the solution of (6.3)
on $\Omega_{j}$ (with the equality sign) and the function $\varepsilon u/j$,
and we let $j\rightarrow 0$, it is easy to see that the left-hand side of
(6.4) is zero.
Now, consider the case $p=2$, $f\equiv 0$. In Theorem 3 of [13] we have shown
that, if $M$ is a manifold with a pole $o$, dimension $m\geq 3$ and radial
sectional curvature $K_{\rm rad}\leq 0$, and for $A\in
L^{\infty}_{\mathrm{loc}}(M)$, conditions_
(6.11)
$\left\\{\begin{array}[]{l}\displaystyle|A(x)|\leq\bar{A}\big{(}r(x)\big{)}\leq\frac{(m-2)^{2}}{4}\frac{1}{r(x)^{2}}\qquad\text{on
}M,\\\\[8.5359pt] \displaystyle r\bar{A}(r)\in
L^{1}(+\infty),\end{array}\right.$
_imply the existence of a positive bounded solution $u\in
C^{1,\mu}_{\mathrm{loc}}(M)$ of $Q_{A}^{\prime}(u)=0$. Thus, in this case
(6.4) holds true. On the other hand, by Theorem 11 of [13] if, for $A\in
L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{m})$_
(6.12)
$\left\\{\begin{array}[]{l}\displaystyle\bar{A}_{1}\big{(}r(x)\big{)}\leq
A(x)\leq\bar{A}_{2}\big{(}r(x)\big{)}\leq\frac{(m-2)^{2}}{4}\frac{1}{r(x)^{2}}\qquad\text{on
}\mathbb{R}^{m},\\\\[11.38092pt] \displaystyle
r\left[\bar{A}_{j}(r)-k\frac{(m-2)^{2}}{4r^{2}}\right]\in
L^{1}(+\infty),\end{array}\right.$
_for some $k<0$ and $j\in\\{1,2\\}$, then there exists a positive solution $u$
of $Q_{A}^{\prime}(u)=0$ on $\mathbb{R}^{m}$ such that
$u(x)\rightarrow+\infty$ as $x\rightarrow\infty$ and so (6.4) cannot hold.
Note that (6.11) is weaker than $A_{-}\in L^{1}(M)$. However, we stress that
it seems hard to find conditions analogous of (6.11) and (6.12) on more
general manifolds and for nonlinear operators. _
We now investigate the existence of uniform upper bounds, i.e. independent of
$\Omega$, for the solutions of (6.1) with boundary data $\varphi=1$. The next
lemma, Lemma 2.1 of the Introduction, ensures _global_ $L^{\infty}$-estimates.
In view of a subtle asymmetry between bounds from below and above, to reach
our goal we had to find a new strategy.
###### Lemma 6.3.
Let $M$ be a Riemannian manifold, $f\in C^{\infty}(M)$, $p\in(1,+\infty)$ and
fix $F(t)$ satisfying (2.6). Let $A,B\in L^{\infty}_{\mathrm{loc}}(M)$ with
$B\geq 0$ a.e. on $M$. Assume that either
* $(i)$
$B\equiv 0$ and $Q_{A}$ is subcritical, or
* $(ii)$
$B\not\equiv 0$ and $Q_{A}$ is non-negative.
Suppose that there exist a smooth, relatively compact open set $\Lambda\Subset
M$ and a constant $c>0$ such that
(6.13) $A\leq cB\qquad\text{a.e. on }M\backslash\Lambda,$
and fix a smooth, relatively compact open set $\Lambda^{\prime}$ such that
$\Lambda\Subset\Lambda^{\prime}$, and a constant $\varepsilon>0$.
Then, there exists a constant $C_{\Lambda}>0$ depending on
$\varepsilon,p,f,F,c,A,B,\Lambda,\Lambda^{\prime}$ but not on $\Omega$ such
that for each smooth, relatively compact open set $\Omega$ with
$\Lambda^{\prime}\Subset\Omega$, the solution $0<z\in
C^{1,\mu}(\overline{\Omega})$ of
(6.14)
$\left\\{\begin{array}[]{ll}\Delta_{p,f}z+A(x)z^{p-1}-B(x)F(z)=0&\quad\text{on
}\Omega,\\\\[5.69046pt] z=\varepsilon&\quad\text{on
}\partial\Omega.\end{array}\right.$
satisfies
(6.15) $z\leq C_{\Lambda}\qquad\text{on }\Omega.$
###### Proof.
Without loss of generality, we can suppose that $c>1$. Using that, by (2.6),
$F(t)/t^{p-1}\rightarrow+\infty$ as $t\rightarrow+\infty$, we can fix
$\alpha>\varepsilon$ such that
(6.16) $\frac{F(t)}{t^{p-1}}\geq c\qquad\text{for }\,t\geq\alpha.$
Consider the open set
$U=\left\\{x\in\Omega\ :\ z(x)>\alpha\right\\}.$
We note that $U\Subset\Omega$ and that, by (6.16), $z$ solves
(6.17)
$\left\\{\begin{array}[]{ll}\displaystyle\Delta_{p,f}z+(A-cB)z^{p-1}\geq
0&\quad\text{on }\,U\\\\[5.69046pt] \displaystyle z=\alpha&\quad\text{on
}\,\partial U,\end{array}\right.$
If $U=\emptyset$, for each $\Omega,z$, then (6.15) trivially holds with
$C_{\Lambda}=\alpha$. Therefore, suppose that $U\neq 0$ for some $\Omega,z$.
By (6.17) and (6.13), $\Delta_{p,f}z\geq 0$ on $U\backslash\Lambda$. Thus, in
the case $U\cap\Lambda=\emptyset$, $\Delta_{p,f}z\geq 0$ on $U$. For
$\varepsilon>0$ small, choose a smooth increasing sequence
$\\{U_{\varepsilon}\\}$ exhausting $U$ as $\varepsilon\rightarrow 0$ and such
that $z\leq\alpha+\varepsilon$ on $\partial U_{\varepsilon}$. Applying
Proposition 3.3 we infer that $\displaystyle z\leq\alpha+\varepsilon$ on
$U_{\varepsilon}$, thus letting $\varepsilon\rightarrow 0$ we get
$\displaystyle z\leq\alpha$ on $U$, contradicting its very definition. Hence,
we conclude that $U\cap\Lambda\neq\emptyset$. Note that
$\sup_{U\cap\Lambda}z\geq\alpha=z_{|\partial U},$
and since $\Delta_{p,f}z\geq 0$ on $U\backslash\Lambda$, again via Proposition
3.3 we obtain
$\sup_{U\backslash\Lambda}z=\sup_{\partial(U\backslash\Lambda)}z\leq\max\left\\{\sup_{\partial
U}z,\sup_{\partial\Lambda}z\right\\}=\max\left\\{\alpha,\sup_{\partial\Lambda\cap
U}z\right\\}\leq\sup_{U\cap\Lambda}z.$
It follows that
(6.18) $\text{if
}\,U=\\{z>\alpha\\}\neq\emptyset,\quad\text{then}\qquad\sup_{U}z\equiv\sup_{\Lambda}z.$
To prove (6.15) we proceed by contradiction: if it does not hold, and in view
of (6.18) there exists a sequence of triples $(\Omega_{j},z_{j},U_{j})$, where
$\Omega_{j}$ is a smooth, relatively compact open set such that
$\Lambda^{\prime}\Subset\Omega_{j}$, $z_{j}$ is a solution of
(6.19)
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}z_{j}+\left(A-B\frac{F(z_{j})}{z_{j}^{p-1}}\right)z_{j}^{p-1}=0\qquad\text{on
}\Omega_{j},\\\\[5.69046pt] z_{j}=\varepsilon\quad\text{on
}\partial\Omega_{j},\end{array}\right.$
$U_{j}=\left\\{x\in\Omega_{j}\ :\ z_{j}(x)>\alpha\right\\}$, and
(6.20)
$\|z_{j}\|\doteq\|z_{j}\|_{L^{\infty}(\Lambda)}\rightarrow+\infty\qquad{\rm
as}\ j\rightarrow+\infty.$
Up to taking $j$ large enough, we can suppose that $\|z_{j}\|>2\alpha$.
Consider the rescaled functions
(6.21) $\displaystyle h_{j}\doteq\frac{z_{j}}{\|z_{j}\|}\qquad{\rm on}\
\Omega_{j}$
and note that they all solve
(6.22)
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}h_{j}+Ah_{j}^{p-1}\geq
0\qquad\text{on }\Lambda^{\prime},\\\\[5.69046pt]
\displaystyle\sup_{\Lambda}h_{j}=1,\qquad
h_{j}\leq\frac{\alpha}{\|z_{j}\|}<\frac{1}{2}\qquad\text{on
}\Omega_{j}\backslash U_{j}.\end{array}\right.$
By (6.22) and the half-Harnack inequality of Theorem 3.1 $(3a)$ there exists
$C_{H}>0$ such that
(6.23) $1=\|h_{j}\|_{L^{\infty}(\Lambda)}\leq
C_{H}\|h_{j}\|_{L^{p}(\Lambda^{\prime},\mathrm{d}\mu_{f})}\qquad\forall\,j.$
Next we define
(6.24)
$\eta_{j}(x)=\left\\{\begin{array}[]{ll}\displaystyle\frac{z_{j}(x)-\alpha}{\|z_{j}\|}&\quad\text{on
}U_{j},\\\\[8.5359pt] 0&\quad\text{on }M\backslash U_{j}.\end{array}\right.$
Then, $0\leq\eta_{j}\in\mathrm{Lip}_{c}(M)$. Let $V=A-cB$. Thus, $V\leq 0$ on
$M\backslash\Lambda$ and, by (6.17), $Q_{V}^{\prime}(z_{j})\leq 0$ on $U_{j}$,
therefore
(6.25) $\begin{array}[]{lcl}\displaystyle 0\leq
Q_{V}(\eta_{j})&=&\displaystyle\frac{1}{p}Q^{\prime}_{V}(\eta_{j})[\eta_{j}]\\\\[8.5359pt]
\displaystyle&=&\displaystyle\frac{1}{p\|z_{j}\|^{p-1}}\int_{U_{j}}\left[|\nabla
z_{j}|^{p-2}\langle\nabla
z_{j},\nabla\eta_{j}\rangle-V(z_{j}-\alpha)^{p-1}\eta_{j}\right]\mathrm{d}\mu_{f}\\\\[11.38092pt]
&\leq&\displaystyle\frac{1}{p\|z_{j}\|^{p}}\int_{U_{j}}V\left\\{z_{j}^{p-1}-(z_{j}-\alpha)^{p-1}\right\\}(z_{j}-\alpha)\mathrm{d}\mu_{f}\\\\[11.38092pt]
&\leq&\displaystyle\frac{1}{p\|z_{j}\|^{p}}\int_{U_{j}\cap\Lambda}V\left[z_{j}^{p-1}-(z_{j}-\alpha)^{p-1}\right](z_{j}-\alpha)\mathrm{d}\mu_{f}\\\\[11.38092pt]
&\leq&\displaystyle\frac{1}{p\|z_{j}\|^{p}}\int_{U_{j}\cap\Lambda}V_{+}(x)\left[z_{j}^{p}-(z_{j}-\alpha)^{p}\right]\mathrm{d}\mu_{f}.\end{array}$
We observe that on $[\alpha,+\infty)$, the function
$\rho(t)=t^{p}-(t-\alpha)^{p}$
is increasing and
$\rho(t)\sim p\alpha t^{p-1}\qquad{\rm as}\ t\rightarrow+\infty.$
We thus infer the existence of constants $c_{1},c_{2}>0$ just depending on
$\alpha$ and such that
$\rho(t)\leq c_{1}t^{p-1}+c_{2}\qquad\text{on }\,[\alpha,+\infty).$
Hence
$\begin{array}[]{l}\displaystyle\frac{1}{p\|z_{j}\|^{p}}\int_{U_{j}\cap\Lambda}V_{+}(x)\left[z_{j}^{p}-(z_{j}-\alpha)^{p}\right]\mathrm{d}\mu_{f}\leq\displaystyle\frac{\|V_{+}\|_{L^{\infty}(\Lambda)}}{p\|z_{j}\|^{p}}\int_{U_{j}\cap\Lambda}\big{(}c_{1}z_{j}^{p-1}+c_{2}\big{)}\mathrm{d}\mu_{f}\\\\[14.22636pt]
\displaystyle\leq\frac{\|V_{+}\|_{L^{\infty}(\Lambda)}}{p\|z_{j}\|}\mathrm{vol}_{f}(\Lambda)c_{1}+\frac{\|V_{+}\|_{L^{\infty}(\Lambda)}}{p\|z_{j}\|^{p}}\mathrm{vol}_{f}(\Lambda)c_{2}\
\ \longrightarrow 0\qquad\text{as }\,j\rightarrow+\infty.\end{array}$
This fact, together with inequality (6.25), implies that
$Q_{V}(\eta_{j})\rightarrow 0$ as $j\rightarrow+\infty$. Now, in both cases
$(i)$ and $(ii)$ in the statement of the lemma, $Q_{V}$ is subcritical on $M$.
In fact, if $(i)$ holds, $V\equiv A$ and $Q_{A}$ is assumed to be subcritical,
while under the validity of $(ii)$ the subcriticality property (2.10) follows
from
$Q_{V}(\varphi)=Q_{A}(\varphi)+\int_{M}(cB)|\varphi|^{p}\mathrm{d}\mu_{f}\geq\int_{M}(cB)|\varphi|^{p}\mathrm{d}\mu_{f}.$
By Theorem 4.1, $Q_{V}$ has thus a weighted spectral gap, and in particular
$\eta_{j}\rightarrow 0$ in $L^{p}_{\mathrm{loc}}(M)$. Since, by definition
(6.21),
$h_{j}=\eta_{j}+\frac{\alpha}{\|z_{j}\|}\qquad\text{on }U_{j},$
we deduce that
$\displaystyle\|h_{j}\|_{L^{p}(\Lambda^{\prime}\cap
U_{j},\mathrm{d}\mu_{f})}\rightarrow 0\qquad\text{as }\,j\rightarrow+\infty$
and since $\displaystyle h_{j}\leq\frac{\alpha}{\|z_{j}\|}$ on
$\Lambda^{\prime}\backslash U_{j}$ we conclude that
$h_{j}\rightarrow 0\quad\text{in
}\,L^{p}(\Lambda^{\prime},\mathrm{d}\mu_{f})\quad\text{as
}\,j\rightarrow+\infty.$
This contradicts (6.23) and proves the claimed (6.15). ∎
Lemmas 6.2 and 6.3 enable us to prove
###### Proposition 6.1.
Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold, $f\in C^{\infty}(M)$,
$p\in(1,+\infty)$ and suppose that $Q_{0}$ is subcritical on $M$. Let $V\in
L^{\infty}(M)$ have compact support and assume that $Q_{V}$ is subcritical on
$M$. Then, there exists a positive solution $g\in C^{1,\mu}_{\mathrm{loc}}(M)$
of
(6.26) $-Q^{\prime}_{V}(g)=\Delta_{p,f}g+V|g|^{p-2}g=0\qquad{\rm on}\ M$
satisfying
(6.27) $C^{-1}\leq g(x)\leq C\qquad{\rm on}\ M$
for some constant $C>1$.
###### Proof.
Let $\Lambda$ be a smooth, relatively compact open set such that $V\equiv 0$
on $M\backslash\Lambda$, and consider an exhaustion $\\{\Omega_{j}\\}$ of $M$
by smooth, relatively compact open sets with
$\overline{\Lambda}\subset\Omega_{1}$. We let $\varphi_{j}\in
C^{1,\mu_{j}}(\overline{\Omega}_{j})$ be the positive solution of
(6.28) $\left\\{\begin{array}[]{l}\displaystyle
Q_{V}^{\prime}(\varphi_{j})=0\qquad{\rm on}\ \Omega_{j},\\\\[5.69046pt]
\varphi_{j}=1\qquad\qquad{\rm on}\ \partial\Omega_{j},\end{array}\right.$
whose existence is granted by $iv)$ of Proposition 3.4. Our assumptions enable
us to apply Lemma 6.2 and 6.3 and to conclude the existence of a constant
$C>1$ independent of $j$, for which
(6.29) $C^{-1}\leq\varphi_{j}(x)\leq C\qquad{\rm on}\ \Lambda.$
On the other hand, because of (6.28) and $V\equiv 0$ on $M\backslash\Lambda$,
we have
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}\varphi_{j}=0\qquad{\rm
on}\ \Omega_{j}\backslash\overline{\Lambda}\\\\[5.69046pt]
\varphi_{j}=1\quad{\rm on}\ \partial\Omega_{j},\quad C^{-1}\leq\varphi_{j}\leq
C\quad{\rm on}\ \partial\Lambda,\end{array}\right.$
and thus, by the comparison Proposition 3.3,
$\displaystyle C^{-1}\leq\varphi_{j}(x)\leq C\qquad{\rm on}\ \Omega_{j}.$
Hence, the $\varphi_{j}$’s are uniformly bounded from above and below. By
elliptic estimates $\varphi_{j}\rightarrow g$ for some $g\in
C^{1,\mu}_{\mathrm{loc}}(M)$ solving $Q^{\prime}_{V}(g)=0$ and satisfying
(6.27). ∎
We are ready to prove Theorem 2.1. We rewrite its statement for the
convenience of the reader.
###### Theorem 6.1.
Let $M^{m}$ be a Riemannian manifold, $f\in C^{\infty}(M)$, $p\in(1,+\infty)$,
and suppose that $Q_{0}$ is subcritical on $M$. Let $a,b\in
L^{\infty}_{\mathrm{loc}}(M)$. Assume that $Q_{a}$ is subcritical and that
* $i)$
$b_{-}$ has compact support;
* $ii)$
for some $\theta>0$, $(a-\theta b_{+})_{-}\in L^{1}(M,\mathrm{d}\mu_{f})$;
* $iii)$
$a(x)=O\big{(}b_{+}(x)\big{)}$ as $x$ diverges.
Then, there exists $\delta>0$ such that, if
(6.30) $b(x)\geq-\delta\qquad\text{on }M,$
we can find a solution $u\in C^{1,\mu}_{\mathrm{loc}}(M)$ of
(6.31)
$\left\\{\begin{array}[]{l}\Delta_{p,f}u+a(x)u^{p-1}-b(x)F(u)=0\qquad\text{on
}M,\\\\[5.69046pt] 0<u\leq\|u\|_{L^{\infty}(M)}<+\infty\qquad\text{on
}M.\end{array}\right.$
Moreover, if $ii)$ and $iii)$ are replaced by the stronger condition
(6.32) $a(x)\asymp b_{+}(x)\qquad\text{as $x$ diverges,}$
then $u$ can be constructed with the further property that $\inf_{M}u>0$.
###### Proof.
By assumption $iii)$, there exists a relatively compact, open subset $\Lambda$
containing $\operatorname{supp}(b_{-})$, and a constant $C\geq\theta$ such
that
(6.33) $a(x)\leq Cb_{+}(x)\qquad\text{on }M\backslash\Lambda.$
The subcriticality of $Q_{a}$ on $M$ implies, via Theorem 4.1, that $Q_{a}$
has a weighted spectral gap, that is, there exists $W\in C^{0}(M)$, $W>0$ on
$M$ such that
(6.34) $\displaystyle\int_{M}W(x)|\varphi|^{p}\mathrm{d}\mu_{f}\leq
Q_{a}(\varphi)\qquad\forall\,\varphi\in\mathrm{Lip}_{c}(M).$
In particular, if $A(x)=a(x)+W(x)1_{\Lambda}(x)$ then $Q_{A}$ is still
subcritical on $M$. Fix $\varepsilon>0$ and a relatively compact subset
$\Lambda^{\prime}$ with $\Lambda\Subset\Lambda^{\prime}$. Next, consider a
smooth, relatively compact open set $\Omega$ with
$\Lambda^{\prime}\Subset\Omega$. Applying Lemma 6.1 with the choices $A=a$
(respectively, $A=a+W1_{\Lambda}$) and $B=b_{+}$, we produce solutions
$\varphi_{\infty}$, $\varphi_{0}$ of
(6.35)
$\left\\{\begin{array}[]{ll}\Delta_{p,f}\varphi_{\infty}+a\varphi_{\infty}^{p-1}-b_{+}F(\varphi_{\infty})=0&\quad\text{on
}\Omega,\\\\[5.69046pt] \varphi_{\infty}=\varepsilon&\quad\text{on
}\partial\Omega,\end{array}\right.$ (6.36)
$\qquad\left\\{\begin{array}[]{ll}\Delta_{p,f}\varphi_{0}+(a+W1_{\Lambda})\varphi_{0}^{p-1}-b_{+}F(\varphi_{0})=0&\quad\text{on
}\Omega,\\\\[5.69046pt] \varphi_{0}=\varepsilon&\quad\text{on
}\partial\Omega.\end{array}\right.$
By comparison, $\varphi_{\infty}\leq\varphi_{0}$. By Lemma 6.2, because of
assumption $ii)$ we can guarantee the existence of a constant
$\bar{C}_{\Lambda}(\varepsilon)>0$ (we emphasize its dependence on
$\varepsilon$), independent of $\Omega$, such that
(6.37) $\varphi_{\infty}\geq\bar{C}_{\Lambda}(\varepsilon)\qquad\text{on
}\,\Lambda.$
On the other hand, (6.33) implies that $a+W1_{\Lambda}\leq Cb_{+}$ outside of
$\Lambda$. Hence, Lemma 6.3 ensures the existence of
$C_{\Lambda}(\varepsilon)$ independent of $\Omega$ and such that
(6.38) $\big{(}\varphi_{\infty}\leq\big{)}\,\varphi_{0}\leq
C_{\Lambda}(\varepsilon)\qquad\text{on }\Omega,$
and we can also consider the sharpest one, that is,
(6.39)
$C_{\Lambda}(\varepsilon)\doteq\sup_{\footnotesize\begin{array}[]{c}\Omega:\Omega\text{
open smooth,}\\\ \Lambda^{\prime}\Subset\Omega\Subset
M\end{array}}\|\varphi_{0}\|_{L^{\infty}(\Omega)},$
where $\varphi_{0}$ solves (6.36). As a consequence of the comparison
Proposition 3.3, $C_{\Lambda}(\varepsilon)$ is non-increasing as a function of
$\varepsilon$.
Define
(6.40)
$\delta=\big{(}\min_{\overline{\Lambda}}W\big{)}\left[\frac{C_{\Lambda}(\varepsilon)^{p-1}}{F(C_{\Lambda}(\varepsilon))}\right],$
and observe that, by assumption (6.30), our definition (6.40) of $\delta$ and
the fact that $F(t)/t^{p-1}$ is increasing,
(6.41) $b_{-}\left[\frac{F(\varphi_{0})}{\varphi_{0}^{p-1}}\right]\leq\delta
1_{\Lambda}\left[\frac{F(C_{\Lambda}(\varepsilon))}{C_{\Lambda}(\varepsilon)^{p-1}}\right]=\big{(}\min_{\overline{\Lambda}}W\big{)}1_{\Lambda}\leq
W1_{\Lambda},$
on $\Lambda$, and the same relation clearly holds on $\Omega\backslash\Lambda$
since there $b_{-}\equiv 0$. Inserting into (6.36) we obtain
$\Delta_{p,f}\varphi_{0}+a\varphi_{0}^{p-1}-bF(\varphi_{0})\leq
0\qquad\text{on }\,\Omega,$
that is, $\varphi_{0}$ is a supersolution for the problem
(6.42)
$\left\\{\begin{array}[]{l}\Delta_{p,f}u_{\Omega}+au_{\Omega}^{p-1}-bF(u_{\Omega})=0\qquad\text{on
}\Omega,\\\\[5.69046pt] \displaystyle u_{\Omega}=\varepsilon\quad\text{on
}\partial\Omega.\end{array}\right.$
On the other hand, $\varphi_{\infty}$ is a subsolution for (6.42), and the
subsolution-supersolution method (see [25], Theorem 4.14 page 272) gives the
existence of $u_{\Omega}$ solving (6.42), satisfying
(6.43) $\varphi_{\infty}\leq u_{\Omega}\leq\varphi_{0}\leq
C_{\Lambda}(\varepsilon),$
and, by (6.37),
(6.44) $u_{\Omega}\geq\bar{C}_{\Lambda}(\varepsilon)\qquad\text{on
}\,\Lambda.$
Indeed, in our setting we can describe a simple iteration scheme to produce
$u_{\Omega}$: set $V_{0}=a+W1_{\Lambda}$. For $n\geq 1$, we inductively define
$\varphi_{n}\in C^{1,\mu}(\overline{\Omega})$ as the positive solution of
($P_{n}$)
$\left\\{\begin{array}[]{ll}\Delta_{p,f}\varphi_{n}+V_{n}\varphi_{n}^{p-1}-b_{+}F(\varphi_{n})=0&\quad\text{on
}\Omega,\\\\[5.69046pt] \varphi_{n}=\varepsilon&\quad\text{on
}\partial\Omega\end{array}\right.$
where
(6.45) $V_{n}\doteq
a+b_{-}\left(\frac{F(\varphi_{n-1})}{\varphi_{n-1}^{p-1}}\right).$
We claim that each $\varphi_{n}$ exists and that $\\{\varphi_{n}\\}$ is a non-
increasing sequence bounded below by $\varphi_{\infty}$. Indeed, by (6.41),
$V_{1}\leq V_{0}$ and so $\lambda_{V_{1}}(\Omega)>0$, which ensures the
existence of $\varphi_{1}$ by Proposition 6.1. Moreover, $\varphi_{1}$ solves
(6.46)
$\left\\{\begin{array}[]{ll}\Delta_{p,f}\varphi_{1}+V_{0}\varphi_{1}^{p-1}-b_{+}F(\varphi_{1})\geq
0&\quad\text{on }\Omega;\\\\[5.69046pt] \varphi_{1}=\varepsilon&\quad\text{on
}\partial\Omega,\end{array}\right.$
whence by comparison $\varphi_{1}\leq\varphi_{0}$ on $\Omega$. Now, this last
inequality (and the monotonicity of $F(t)/t^{p-1}$) gives $V_{2}\leq V_{1}$,
so that $\lambda_{V_{2}}(\Omega)>0$ and $(P_{2})$ admits a unique positive
solution $\varphi_{2}$. Again from $V_{2}\leq V_{1}$, $\varphi_{2}$ turns out
to be a subsolution of $(P_{1})$, hence $\varphi_{2}\leq\varphi_{1}$ by
comparison. Repeating the argument above shows the monotonicity of
$\\{\varphi_{n}\\}$. The positivity of $\varphi_{n-1}$ ensures that $V_{n}\geq
V_{\infty}$, so by comparison $\varphi_{n}\geq\varphi_{\infty}$ for each $n$.
The inequalities $a\leq V_{n}\leq V$ and
$\varphi_{\infty}\leq\varphi_{n}\leq\varphi_{0}$ for each $n$ then guarantee,
via Theorem 3.1 $(2)$, that there exists $\mu\in(0,1)$ such that the
$C^{1,\mu}$-norm of $\varphi_{n}$ on $\Omega$ is uniformly bounded. Therefore,
a subsequence of $\\{\varphi_{n_{k}}\\}_{k}\subset\\{\varphi_{n}\\}_{n}$
converges in the $C^{1}$-norm, as $k\rightarrow+\infty$, to some non-negative
$v_{\Omega}\in C^{1}(\overline{\Omega})$, and since $\\{\varphi_{n}\\}$ is a
non-increasing sequence the whole $\\{\varphi_{n}\\}$ converges to
$v_{\Omega}$ uniformly. By letting $k\rightarrow+\infty$ in the weak
definition of ($P_{n}$) along the subsequence $\\{n_{k}\\}$, we deduce that
$v_{\Omega}$ is a weak solution of (6.42). Now, we choose an exhaustion
$\Omega_{j}$, and let $u_{j}=u_{\Omega_{j}}$ be the solution of (6.42) on
$\Omega_{j}$ constructed above. Note that
$\varphi_{\infty,j}\leq u_{j}\leq C_{\Lambda}(\varepsilon)\ \text{ on
}\Omega_{j},\qquad u_{j}\geq\bar{C}_{\Lambda}(\varepsilon)\ \ \text{ on
}\,\Lambda.$
where $\varphi_{\infty,j}$ solves (6.35) on $\Omega_{j}$. Hence, by local
elliptic estimates the sequence $\\{u_{j}\\}$ subconverges to some global
solution $u\geq 0$ of (6.31) on the whole $M$, satisfying
$u\leq C_{\Lambda}(\varepsilon)\quad\text{on }\,M,\qquad
u\geq\bar{C}_{\Lambda}(\varepsilon)\quad\text{on }\,\Lambda.$
By Harnack inequality, $u>0$ on $M$, and we have proved the first part of our
theorem. The final step is to guarantee that, when $a\asymp b_{+}$,
$\inf_{M}u>0$.
This will be accomplished by proving corresponding lower bounds for
$\varphi_{\infty,j}$. Up to reducing $\theta$, we can suppose that
$a-\theta b_{+}\geq 0$
outside some compact set. Using the assumption that $F(t)/t^{p-1}\rightarrow
0$ as $t\rightarrow 0$, fix $\alpha\in(0,\varepsilon)$ small enough that
$\frac{F(t)}{t^{p-1}}\leq\theta\qquad\text{for }\,t\in[0,\alpha].$
Inspecting problem (6.35) and noting that $\alpha<\varepsilon$, we deduce that
on the subset $U_{j}\doteq\\{\varphi_{\infty,j}<\alpha\\}\Subset\Omega_{j}$
the function $\varphi_{\infty,j}$ solves
$\Delta_{p,f}\varphi_{\infty,j}+(a-\theta b_{+})\varphi_{\infty,j}^{p-1}\leq
0$, hence in particular
(6.47) $\left\\{\begin{array}[]{ll}\Delta_{p,f}\varphi_{\infty,j}-(a-\theta
b_{+})_{-}\varphi_{\infty,j}^{p-1}\leq 0&\quad\text{on }U_{j},\\\\[5.69046pt]
\varphi_{\infty,j}=\alpha&\quad\text{on }\partial U_{j},\end{array}\right.$
In view of the boundary regularity requirements to apply Proposition 3.3, we
fix a smooth open set $S_{j}$ satisfying
$\big{\\{}x:\varphi_{\infty,j}\leq\alpha/2\big{\\}}\Subset S_{j}\Subset
U_{j},$
so that
(6.48) $\left\\{\begin{array}[]{ll}\Delta_{p,f}\varphi_{\infty,j}-(a-\theta
b_{+})_{-}\varphi_{\infty,j}^{p-1}\leq 0&\quad\text{on }S_{j},\\\\[5.69046pt]
\varphi_{\infty,j}\geq\frac{\alpha}{2}&\quad\text{on }\partial
S_{j},\end{array}\right.$
Next, we use the fact that $V\doteq-(a(x)-\theta b_{+}(x))_{-}$ is compactly
supported and $Q_{V}$ is subcritical on $M$ (being $Q_{0}$ subcritical by
assumption, and $V\leq 0$): by Proposition 6.1, there exists $g\in
C^{1,\mu}_{\mathrm{loc}}(M)$ solution of
(6.49) $\left\\{\begin{array}[]{l}\Delta_{p,f}g-(a-\theta
b_{+})_{-}g^{p-1}=0\qquad\text{on }M,\\\\[5.69046pt]
0<\inf_{M}g\leq\|g\|_{L^{\infty}(M)}<+\infty.\end{array}\right.$
Rescaling $g$ by multiplying by a positive constant we can assume that
(6.50) $\displaystyle 0<\inf_{M}g\leq\sup_{M}g\leq\frac{\alpha}{2}.$
Comparing (6.48) and (6.49) on $S_{j}$, by Proposition 3.3 we infer that
$\varphi_{\infty,j}\geq g$ on $S_{j}$, and thus by (6.50)
$u_{j}\geq\displaystyle\varphi_{\infty,j}\geq\min\left\\{\frac{\alpha}{2},g\right\\}=g\qquad\text{on
}\,\Omega_{j}$
Passing to the limit, we finally get
$\displaystyle u(x)\geq g(x)\quad\text{on }\,M,$
and $\inf_{M}u>0$ follows since $\inf_{M}g>0$. ∎
In the proof of the above result, the parameter $\varepsilon$ plays no role.
However, a judicious choice of $\varepsilon$ is crucial in the following proof
of Theorem 2.2.
###### Theorem 6.2.
Let $M^{m}$ be a Riemannian manifold, $f\in C^{\infty}(M)$ and
$p\in(1,+\infty)$. Suppose that $Q_{0}$ is subcritical on $M$ and let $a\in
L^{\infty}_{\mathrm{loc}}(M)$ be such that $Q_{a}$ is subcritical on $M$.
Consider $b\in L^{\infty}_{\mathrm{loc}}(M)$, and assume
* $i)$
$b_{-}(x)$ has compact support;
* $ii^{\prime})$
$a(x)\leq 0$ outside a compact set;
* $iii^{\prime})$
$a(x),b(x)\in L^{1}(M,\mathrm{d}\mu_{f})$.
Fix a nonlinearity $F(t)$ satisfying (2.6). Then, there exists a sequence
$\\{u_{k}\\}\subset C^{1,\mu}_{\mathrm{loc}}(M)$ of distinct weak solutions of
(6.51)
$\left\\{\begin{array}[]{l}\displaystyle\Delta_{p,f}u_{k}+a(x)u_{k}^{p-1}-b(x)F(u_{k})=0\qquad\text{
on M}\\\\[5.69046pt]
0<u_{k}\leq\|u_{k}\|_{L^{\infty}(M)}<+\infty,\end{array}\right.$
such that $\|u_{k}\|_{L^{\infty}(M)}\rightarrow 0$ as $k\rightarrow+\infty$.
If we replace $ii^{\prime})$ and $iii^{\prime})$ by the stronger condition
$iv^{\prime})\qquad a(x),b(x)\quad\text{have compact support,}$
then each $u_{k}$ also satisfies $\inf_{M}u_{k}>0$.
###### Proof.
We first observe that our set of assumptions on $a$ and $b$ is a special case
of the one in Theorem 6.1, namely we just include the requirement that $a\leq
0$ outside some compact set. Hence, the constructions in the previous theorem
hold as well in our setting, and we will refer to the proof of Theorem 6.1
also for relevant definitions.
Up to enlarging $\Lambda$ we can assume that $a\leq 0$ on
$M\backslash\Lambda$. Let $\Omega$ satisfy $\Lambda^{\prime}\Subset\Omega$. By
Lemma 6.3 applied with $A=a+W1_{\Lambda}$, $B\equiv 0$ there exists a uniform
constant $\hat{C}$, independent of $\Omega$, such that the solution $\psi$ of
(6.52)
$\left\\{\begin{array}[]{ll}\Delta_{p,f}\psi+(a+W1_{\Lambda})\psi^{p-1}=0&\quad\text{on
}\Omega,\\\\[5.69046pt] \psi=1&\quad\text{on
}\partial\Omega\end{array}\right.$
satisfies $\psi\leq\hat{C}$ on $\Omega$. By comparison and recalling our
definition of $\varphi_{0}$ in (6.36), we get
$\varphi_{0}\leq\varepsilon\psi$. Therefore, by our definition of
$C_{\Lambda}(\varepsilon)$,
(6.53) $C_{\Lambda}(\varepsilon)\leq\varepsilon\hat{C}\rightarrow
0\qquad\text{as }\varepsilon\rightarrow 0.$
Combining (6.53), the monotonicity of $C_{\Lambda}(\varepsilon)$ and property
$F(t)/t^{p-1}\rightarrow 0$ as $t\rightarrow 0$ in (2.6), we deduce that,
given any $b$ with $b_{-}$ compactly supported, there exists $\varepsilon_{0}$
sufficiently small such that
(6.54)
$b_{-}\leq\big{(}\min_{\overline{\Lambda}}W\big{)}\left[\frac{C_{\Lambda}(\varepsilon)^{p-1}}{F(C_{\Lambda}(\varepsilon))}\right]\qquad\text{for
each }\varepsilon\leq\varepsilon_{0}.$
Fix such $\varepsilon_{0}$ and follow the arguments in Theorem 6.1 with
$\varepsilon=\varepsilon_{0}$. Note that, as in (6.41),
$b_{-}(x)\left(\frac{F(\varphi_{0}(x))}{\varphi_{0}(x)^{p-1}}\right)\leq
b_{-}(x)\left[\frac{F(C_{\Lambda}(\varepsilon))}{C_{\Lambda}(\varepsilon)^{p-1}}\right]\leq
1_{\Lambda}(x)\big{(}\min_{\overline{\Lambda}}W\big{)}\leq
W(x)1_{\Lambda}(x),$
which is the key step to produce the local solutions of (6.42), and now it
does not require (6.30). Proceeding with the construction, we get a solution
$u_{0}$ of
(6.55)
$\left\\{\begin{array}[]{l}\Delta_{p,f}u_{0}+a(x)u_{0}^{p-1}-b(x)F(u_{0})=0\qquad\text{on
}M,\\\\[5.69046pt] 0<u_{0}\leq C_{\Lambda}(\varepsilon_{0})\qquad\text{on
}M,\end{array}\right.$
and $\inf_{M}u_{0}>0$ when $(iv^{\prime})$ holds. Next, choose
$\varepsilon_{1}<\varepsilon_{0}$ small enough that
$C_{\Lambda}(\varepsilon_{1})<\min_{\overline{\Lambda}}u_{0}.$
Proceeding as above with $\varepsilon_{1}$ replacing $\varepsilon_{0}$ we get
a solution $u_{1}$ of
(6.56)
$\left\\{\begin{array}[]{l}\Delta_{p,f}u_{1}+a(x)u_{1}^{p-1}-b(x)F(u_{1})=0\qquad\text{on
}M,\\\\[5.69046pt] 0<u_{1}\leq C_{\Lambda}(\varepsilon_{1})\qquad\text{on
}M,\end{array}\right.$
and $\inf_{M}u_{1}>0$ when $(iv^{\prime})$ is in force. By our choice of
$\varepsilon_{1}$, $u_{1}<u$ on $\Lambda$, thus in particular the two
solutions are different. We can now repeat the procedure inductively by
choosing, at each step, $\varepsilon_{k}<\varepsilon_{k-1}$ such that
$C_{\Lambda}(\varepsilon_{k})<\min_{\overline{\Lambda}}u_{k-1},$
obtaining a solution $u_{k}$ of
$\Delta_{p,f}u_{k}+a(x)u_{k}^{p-1}-b(x)F(u_{k})=0\qquad\text{on }M$
satisfying
(6.57) $0<u_{k}\leq C_{\Lambda}(\varepsilon_{k})\ \text{ on }M,\quad
u_{k}<u_{k-1}\ \text{ on }\Lambda,\quad\inf_{M}u_{k}>0\ \text{ when
}(iv^{\prime})\text{ holds.}$
By construction, $C_{\Lambda}(\varepsilon_{k})\rightarrow 0$ as
$k\rightarrow+\infty$, hence $\\{u_{k}\\}$ is the desired sequence. ∎
###### Remark 6.2.
_The key point that allows, in Theorem 6.2, to get rid of (6.30) is the
validity of the asymptotic relation_
(6.58) $C_{\Lambda}(\varepsilon)\rightarrow 0^{+}\qquad\text{as
}\varepsilon\rightarrow 0^{+},$
_which is granted via the presence of an uniform $L^{\infty}$-bound for
solutions $\psi$ of (6.52). For general $a,b$, just satisfying the assumptions
of Proposition 6.1, (6.58) may not hold. As an example, consider the
hyperbolic space $\mathbb{H}^{m}$ of sectional curvature $-1$. For each
$\tau\geq 1$, the radial functions_
$u_{\tau}(x)=\left(2\cosh^{2}\left(\frac{r(x)}{2}\right)\right)^{-\frac{m-2}{2}}\beta_{\tau}\left(\tanh\left(\frac{r(x)}{2}\right)\right),$
_where_
$\beta_{\tau}(t)=\frac{(\tau^{2}-t^{2})^{-\frac{m-2}{2}}}{m(m-2)\tau^{2}}$
_are all solutions of_
$\Delta u+\frac{m(m-2)}{4}u-u^{\frac{m+2}{m-2}}=0\qquad\text{on
}\mathbb{H}^{m}.$
_Moreover, they are decreasing functions of $r(x)$, and the sequence
$\\{u_{\tau}\\}$ is monotone decreasing. For each $\varepsilon>0$, consider
$\Omega=u_{1}^{-1}\\{(\varepsilon,+\infty)\\}$. Then, for $\varepsilon$ small
enough in such a way that $\Omega\neq\emptyset$, by the definition of
$C_{\Lambda}(\varepsilon)$ in (6.39) we deduce_
$C_{\Lambda}(\varepsilon)\geq\|u_{1}\|_{L^{\infty}(\Omega)}=u_{1}(0)=\frac{2^{-\frac{m-2}{2}}}{m(m-2)},$
_preventing from the validity of ( 6.58). _
We briefly comment on the sharpness of the subcriticality assumption for
$Q_{a}$ in Theorem 2.1, and for this reason we now state a result that
improves on a theorem in [48]. First of all, we extend definition (3.8) to
arbitrary subsets $\Lambda\subset M$, that is, we define the fundamental tone
$\lambda_{V}(\Lambda)$ by setting
(6.59) $\lambda_{V}(\Lambda)=\sup\lambda_{V}(\Omega)$
where the supremum is taken over all open subsets $\Omega\subset M$ with
smooth boundary such that $\overline{\Lambda}\subset\Omega$.
###### Proposition 6.2.
Let $(M,\langle\,,\,\rangle)$ be a Riemannian manifold $f\in C^{\infty}(M)$,
$p\in(1,\infty)$ and let $a(x)\in L^{\infty}_{\mathrm{loc}}(M)$, $b(x)\in
C^{0}(M)$. Define
$\qquad B_{0}\doteq\\{x\in M\ :\ b(x)\leq 0\\},$
Let $\Omega$ be an open domain containing $\overline{B}_{0}$ and such that
there exists a positive, bounded solution $\displaystyle u\in
C^{0}(\overline{\Omega})\cap W^{1,p}_{\mathrm{loc}}(\Omega)$ of
(6.60) $\Delta_{p,f}u+a(x)u^{p-1}-b(x)F(u)\leq 0\qquad{\rm on}\ \Omega,$
for some nonlinearity $F(t)$ that satisfies (2.6). Then
(6.61) $\lambda_{a}(B_{0})\geq 0.$
###### Remark 6.3.
_Let us consider the case $p=2$, $f\equiv 0$. The first Dirichlet eigenvalue
of the Laplacian on a geodesic ball $B_{r}$ grows like $r^{-2}$ as
$r\rightarrow 0^{+}$, thus $\lambda_{a}(B_{r})>0$ provided $r$ is sufficiently
small and one may think that condition (6.61) expresses the fact that $b_{-}$
is, loosely speaking, small in a spectral sense. _
###### Proof.
Let $u$ be as above and by contradiction assume that
$\lambda_{a}(B_{0})\doteq\lambda<0$. Then, by definition (6.59) we can find a
sequence $\\{U_{i}\\}$ of open sets with smooth boundaries such that
$\overline{B}_{0}\subset
U_{i}\subset\overline{U}_{i}\subset\Omega,\qquad\lambda_{a}(U_{i})\rightarrow\lambda\quad\text{as
}i\rightarrow+\infty.$
Take a nested sequence $\\{V_{i}\\}$ of smooth open sets shrinking to $B_{0}$
such that:
$V_{i}\subset\left\\{x:b(x)<\frac{1}{i}\right\\},\qquad\overline{B}_{0}\subset
V_{i+1}\subset\overline{V}_{i+1}\subset
V_{i}\subset\Omega,\qquad\bigcap_{i=1}^{+\infty}V_{i}=\overline{B}_{0}.$
Then, up to replacing $U_{i}$ with $V_{i}\cap U_{i}$ and smoothing corners, by
the monotonicity of eigenvalues we can further suppose that $\\{U_{i}\\}$
satisfies
(6.62) $\bigcap_{i=1}^{+\infty}U_{i}=\overline{B}_{0},\qquad b<\frac{1}{i}\
\text{ on }U_{i}.$
By the definition of $\lambda_{a}(U_{i})$, there exists a smooth, relatively
compact open set $\Omega_{i}\Subset U_{i}$ for which
$\lambda_{i}\doteq\lambda_{a}(\Omega_{i})\leq\lambda_{a}(U_{i})+1/i$, so that
clearly $\lambda_{i}\rightarrow\lambda$ as $i$ diverges. Corresponding to
$\lambda_{i}$ there exists a positive eigenfunction $\varphi_{i}\in
C^{1,\mu}(\overline{\Omega}_{i})$ satisfying
$\left\\{\begin{array}[]{ll}\displaystyle
Q_{a}^{\prime}(\varphi_{i})=\lambda_{i}|\varphi_{i}|^{p-2}\varphi_{i}&\quad\text{on
}\Omega_{i}\\\\[5.69046pt] \displaystyle\varphi_{i}=0&\quad\text{on
}\partial\Omega_{i}.\end{array}\right.$
Setting $\displaystyle h=\log u$ using (3.14) and (6.60) we see that $h$
solves
$\displaystyle\Delta_{p,f}h\leq-a(x)+b(x)\frac{F(u)}{u^{p-1}}-(p-1)\left|\nabla
h\right|^{p}.$
Integrating on $\Omega_{i}$ against $\varphi_{i}^{p}$ and proceeding as in the
proof of $i)\Rightarrow iii)$ in Proposition 3.4 we obtain
$\displaystyle\int_{\Omega_{i}}a(x)\varphi_{i}^{p}\mathrm{d}\mu_{f}-\int_{\Omega_{i}}b(x)\frac{F(u)}{u^{p-1}}\varphi_{i}^{p}\mathrm{d}\mu_{f}\leq\int_{\Omega_{i}}|\nabla\varphi_{i}|^{p}\mathrm{d}\mu_{f}.$
Now, since the $\varphi_{i}$’s are eigenfunctions, $\displaystyle
pQ_{a}(\varphi)=\lambda_{i}\|\varphi_{i}\|^{p}_{L^{p}(\Omega_{i},\mathrm{d}\mu_{f})}$.
Therefore, inserting into the above, observing that $b<1/i$ on $\Omega_{i}$
and using the monotonicity of $f(t)/t^{p-1}$ we deduce
(6.63)
$0\geq\int_{\Omega_{i}}\left[-\lambda_{i}-b(x)\frac{F(u)}{u^{p-1}}\right]\varphi_{i}^{p}\mathrm{d}\mu_{f}\geq\int_{\Omega_{i}}\left[-\lambda_{i}-\frac{F(\|u\|_{L^{\infty}(\Omega)})}{i\|u\|_{L^{\infty}(\Omega)}^{p-1}}\right]\varphi_{i}^{p}\mathrm{d}\mu_{f}.$
Concluding, as $\lambda_{i}\rightarrow\lambda<0$, taking $i$ sufficiently
large the previous inequality yields the desired contradiction. ∎
In view of Proposition 6.2, and since we made no assumptions on the size of
the set $\\{x:b(x)=0\\}$ in Theorem 2.1, the existence of a bounded solution
of (2.12) on $M$ when $b$ changes sign requires at least that $Q_{a}\geq 0$ on
the whole $M$. We feel interesting to investigate the validity of Theorem 2.1
when assumption $i)$ is replaced with the requirement that $Q_{a}$ be non-
negative and with a ground state.
###### Remark 6.4.
_We conclude this section with a remark on the role of ( 2.11) in Theorem 2.1.
that is, $\|b_{-}\|_{L^{\infty}(M)}\leq\delta$. Denote with $B_{0}$ the
relatively compact set $\\{x:b(x)<0\\}$, and for each $\varepsilon\in(0,1)$
choose a smooth relatively compact set
$\Omega_{\varepsilon}\subset\\{x:b_{-}(x)\geq(1-\varepsilon)\|b_{-}\|_{L^{\infty}(M)}\\}$.
Then we let $\varphi_{\varepsilon}$ be the positive eigenfunction of_
$\left\\{\begin{array}[]{l}Q^{\prime}_{a}(\varphi_{\varepsilon})=\lambda_{a}(\Omega_{\varepsilon})\varphi_{\epsilon}^{p-1}\qquad\text{on
}\,\Omega_{\varepsilon}\\\\[5.69046pt]
\displaystyle\varphi_{\varepsilon}=0\quad\text{on
}\,\partial\Omega_{\varepsilon},\qquad\varphi_{\varepsilon}>0\quad\text{on
}\,\Omega_{\varepsilon}.\end{array}\right.$
_Note that $\lambda_{a}(\Omega_{\varepsilon})>0$ by assumption. Reasoning as
in Proposition 6.2 to get (6.63) we obtain_
$\lambda_{a}(\Omega_{\varepsilon})\|\varphi_{\varepsilon}\|^{p}_{L^{p}(\Omega_{\varepsilon})}\geq\int_{\Omega_{\varepsilon}}b_{-}\frac{F(u)}{u^{p-1}}\varphi_{\varepsilon}^{p}\mathrm{d}\mu_{f},$
_hence using the definition of $\Omega_{\varepsilon}$_
$\lambda_{a}(\Omega_{\varepsilon})\|\varphi_{\varepsilon}\|^{p}_{L^{p}(\Omega_{\varepsilon})}\geq(1-\varepsilon)\|b_{-}\|_{L^{\infty}(M)}\left[\inf_{B_{0}}\frac{F(u)}{u^{p-1}}\right]\|\varphi_{\varepsilon}\|^{p}_{L^{p}(\Omega_{\varepsilon})}$
_in other words_
(6.64)
$\|b_{-}\|_{L^{\infty}(M)}\left[\inf_{B_{0}}\frac{F(u)}{u^{p-1}}\right]\leq\inf_{\varepsilon\in(0,1)}\frac{\lambda_{a}(\Omega_{\varepsilon})}{1-\varepsilon}.$
_The above inequality helps to understand the relationship between the
$L^{\infty}$-norm of the negative part of $b$ and a lower bound for $u$ on
$B_{0}$. In particular, in view of the relation $F(t)/t^{p-1}\rightarrow 0$ as
$t\rightarrow 0$ in (2.6), a larger size of $b_{-}$ forces $u$ to squeeze
towards zero on $B_{0}$. _
## 7\. Proofs of our geometric corollaries, and concluding comments
In this section, we prove Theorems 2.3, 1.5, 1.6 and their Corollaries 1.1 and
1.2.
###### Proof of Theorem 2.3.
Taking into account that the conformal factor $u$ in the deformation
$\widetilde{\langle\,,\,\rangle}=u^{\frac{4}{m-2}}\langle\,,\,\rangle$
shall satisfy (1.2), the theorem follows immediately from Theorems 2.1 and
2.2. When the two-sided bound
(7.1) $\displaystyle
C^{-1}\langle\,,\,\rangle\leq\widetilde{\langle\,,\,\rangle}\leq
C\langle\,,\,\rangle$
holds, $(M,\widetilde{\langle\,,\,\rangle})$ is complete if and only if
$\displaystyle(M,\langle\,,\,\rangle)$ is so. Furthermore, because of (7.1)
since $\langle\,,\,\rangle$ is non-parabolic the same holds for
$\widetilde{\langle\,,\,\rangle}$. Indeed, it is easy to see that (7.1)
induces a similar two-sided bound between the capacities $\mathrm{cap}$ and
$\widetilde{\mathrm{cap}}$ of the Laplace-Beltrami operators of the two
metrics (with, say, supersolution $g=1$), whence the preservation of
parabolicity follows from Theorem 4.1. This concludes the proof. ∎
###### Proof of Theorem 1.5.
It follows directly from case $(II)$ of Theorem 2.3: it is enough to observe
that, if $s(x)\geq 0$ on the whole $M$ and $M$ is non-parabolic (i.e.,
$-\Delta$ is subcritical), then the conformal Laplacian
$L_{\langle\,,\,\rangle}$ is subcritical. ∎
###### Proof of Corollary 1.1.
We begin with performing a “reduction" argument that goes back to the original
works of Schoen-Yau on the positive mass theorem, see [47], pp. 82-83. Via a
cut-off procedure and a careful analysis of Schrödinger operators on weighted
spaces, they showed that there exists a conformal deformation
(7.2) $\langle\,,\,\rangle_{1}=u_{1}^{\frac{4}{m-2}}\langle\,,\,\rangle$
of the original asymptotically flat metric in such a way that
$(M,\langle\,,\,\rangle_{1})$ is still asymptotically flat and has zero scalar
curvature outside a compact set. Moreover, $\langle\,,\,\rangle_{1}$ is
uniformly equivalent to $\langle\,,\,\rangle$ (actually much more is true, but
this is enough for our purposes). Next, by the very definition of asymptotic
flatness, the metric $\langle\,,\,\rangle_{1}$ on each end $U_{j}$ with
respect to the compact set $K$ is bi-Lipschitz equivalent to the Euclidean one
on $\mathbb{R}^{m}\backslash B_{r}(0)$, hence proceeding as in the proof of
Theorem 2.3 we deduce that $(M,\langle\,,\,\rangle_{1})$ is non-parabolic.
Consequently, since $\langle\,,\,\rangle_{1}$ has non-negative scalar
curvature, the conformal Laplacian $L_{\langle\,,\,\rangle_{1}}$ is
subcritical. Applying previous Theorem 1.5 to the background manifold
$(M,\langle\,,\,\rangle_{1})$, we get the existence of a family of conformal
deformations to scalar curvature $\widetilde{s}(x)$ which, after composing
with the deformation (7.2), concludes the proof of the corollary. ∎
###### Proof of Theorem 1.6.
It follows directly from $(I)$ of Theorem 2.3. Note that, according to Remark
4.6, $L_{\langle\,,\,\rangle}$ is subcritical if and only if it admits a
positive Green kernel. ∎
###### Proof of Corollary 1.2.
Condition $K\leq-\kappa^{2}$ implies, via Theorem 5.1, that the Hardy
inequality
$\int_{M}(\chi\circ
r)\varphi^{2}\mathrm{d}x\leq\int_{M}|\nabla\varphi|^{2}\mathrm{d}x,$
holds for each $\varphi\in\mathrm{Lip}_{c}(M)$, where
$\chi(r)=\frac{1}{4}\left(g_{\kappa}(r)^{m-1}\int_{r}^{+\infty}\frac{\mathrm{d}s}{g_{\kappa}(s)^{m-1}}\right)^{-2}.$
Now, since by (5.25) the Hardy weight satisfies
$\chi(r)\geq\frac{(m-1)^{2}\kappa^{2}}{4}\qquad\text{on }\mathbb{R}^{+},$
using assumption (1.24) we deduce that
$-\frac{s(x)}{c_{m}}=-\frac{m-2}{4(m-1)}s(x)\leq\frac{(m-1)^{2}\kappa^{2}}{4}\leq\chi\big{(}r(x)\big{)},$
thus the conformal Laplacian $L_{\langle\,,\,\rangle}=-\Delta+s/c_{m}$ is
subcritical by Proposition 4.5 (clearly, $-s/c_{m}\not\equiv(\chi\circ r)$
since this latter tends to infinity at $o$). Now, if (1.25) holds outside of a
compact set, assumption $i)$ of Theorem 2.3 is met. Tracing $K\leq-\kappa^{2}$
we deduce
$s(x)\leq-m(m-1)\kappa^{2},$
which coupled with (1.24) and (1.25) implies $s(x)\asymp\widetilde{s}(x)$ as
$x$ diverges. Applying Theorem 1.6 we eventually have the desired conformal
deformation to a uniformly equivalent metric
$\widetilde{\langle\,,\,\rangle}$. ∎
We conclude with a couple of remarks. In the Introduction, the prototype cases
of Euclidean and hyperbolic space helped us to have a picture of the variety
of phenomena concerning the prescribed curvature problem. We have seen that
the uniqueness of the conformal deformation in Theorem 2.1 fails to hold for
sign-changing $\widetilde{s}(x)$, and that fastly decaying solutions coexist
with solutions bounded from below and above by positive constants. In
particular, assumptions like (1.25) do not imply a control of the decay of the
conformal factor from both sides by two comparable quantities. However, when
$\widetilde{s}<0$ on the whole $M$, something more precise can be said about
uniqueness and asymptotic behaviour of solutions $u$ of the Yamabe equation
(1.2). As above, consider the prototype case of $\mathbb{H}^{m}_{\kappa}$, and
suppose that
(7.3) $-C_{1}\leq\widetilde{s}(x)\leq-C_{2}<0\qquad\text{on
}\,\mathbb{H}^{m}_{\kappa}.$
By Theorem 3.4 in [74] with the choice $\beta=0$ (or even by Theorem 4 in
[8]), assumption (7.3) guarantees that the conformal deformation given in
Aviles-McOwen’s Theorem 1.3 and in Corollary 1.2 is the _unique_ conformal
deformation realizing $\widetilde{s}(x)$ and such that the conformal factor
satisfies $\inf_{\mathbb{H}^{m}_{\kappa}}u>0$. Moreover, by [71, 61] (a
simpler form can also be found in Theorem 2.3 of [74]), in the same
assumptions each solution of (1.2) satisfies
$\sup_{\mathbb{H}^{m}_{\kappa}}u<+\infty$. On the other hand, since
$\widetilde{s}<0$ on $\mathbb{H}^{m}_{\kappa}$, estimates from below for
positive solutions of the Yamabe type equation (2.1) have been provided in
[74]. Applying Theorem 2.4 of [74] with the choices
$m>4,\ \ \delta=0,\ \ \beta=-1-\epsilon,\ \ \alpha<0\ \text{ arbitrary,}\ \
\sigma=\frac{m+2}{m-2},\ \ \gamma=\frac{m-2}{2}H>H$
with $\epsilon>0$, we deduce that any solution $u$ of (1.2) satisfies
(7.4) $u(x)\geq Ce^{-\frac{m-2}{2}\kappa r(x)}\qquad\text{on
}\mathbb{H}^{m}_{\kappa},$
for some $C>0$. Indeed, the right-hand side in (7.4) is exactly the asymptotic
decay of the solutions that create the conformally deformed metrics in Theorem
1.4, and in fact it is also the decay of a radial solution of
$L_{\langle\,,\,\rangle}u=0$ on $\mathbb{H}^{m}_{\kappa}$. In summary, when
(7.3) holds, the decay of the solutions $\\{u_{j}\\}$ in Theorem 1.4 is the
minimal one that a solution of the Yamabe equation with (7.3) in force can
have (if $m>4$), while the function $u$ produced in Corollary 1.2 (we call it
$\hat{u}$) is the unique solution which is bounded below by a positive
constant, and indeed it also has the maximal possible order at infinity, as
each solution shall be bounded above by a constant. This intriguing scenario
is enriched by the fact that, by Theorem 1.1 in [71], the Yamabe equation on
$\mathbb{H}^{m}_{\kappa}$ also admits a solution $u_{c}$ giving rise to a
complete metric $\widetilde{\langle}\,,\,\rangle$ whenever
(7.5) $\widetilde{s}<0\ \text{ on }M,\ \text{and }\quad\widetilde{s}(x)\geq-
Cr(x)^{2}\quad\text{as }r\rightarrow+\infty,$
for some $C>0$. It is still not clear whether $u_{c}$ coincides with $\hat{u}$
or not, or even if (7.5) ensures the existence of a whole infinite family of
solutions, distinct from $\\{u_{j}\\}$ and $\hat{u}$, giving rise to complete
metrics.
###### Remark 7.1.
_Based on the hints in[72], we conjecture that there exists a conformal
deformation of the hyperbolic metric that gives rise to a complete metric of
scalar curvature $\widetilde{s}(x)$ whenever $|\widetilde{s}(x)|\leq
Cr(x)^{2}$, where $r(x)$ is the distance from a fixed origin of
$\mathbb{H}^{m}_{\kappa}$ and $C>0$. See also [13] for some comments. _
## Appendix: the obstacle problem and the pasting lemma
The aim of this section is to present a proof of the pasting Lemma 3.1. The
argument is divided in three steps. First, observe that our assumptions in
Lemma 3.1 imply $\lambda_{V}(\Omega_{2})\geq 0$. Therefore, the obstacle
problem that we shall consider below is solvable on relatively compact open
subsets of $\Omega_{2}$. Secondly, the minimizing properties of its solutions
yield a quick proof of the fact that the minimum of two positive
supersolutions is still a supersolution. Finally we obtain Lemma 3.1 by
refining the argument used in the second step. The idea of the proof is close
to that in Section 3 of [53]. Hereafter, each $W^{1,p}$-norm is intended to be
with respect to the measure $\mathrm{d}\mu_{f}$
Let $\Omega\Subset M$ be a relatively compact, open subset and $V\in
L^{\infty}(\Omega)$. Given $\psi$ measurable and $\theta\in W^{1,p}(\Omega)$
such that $\psi\leq\theta$ a.e. on $\Omega$, we define the non-empty, closed,
convex set
(7.6) $\mathcal{K}_{\psi,\theta}\doteq\Big{\\{}\varphi\in W^{1,p}(\Omega)\ |\
\ \varphi\geq\psi\ \text{ a.e. and }\ \varphi-\theta\in
W^{1,p}_{0}(\Omega)\Big{\\}}.$
We say that $u\in\mathcal{K}_{\psi,\theta}$ solves the obstacle problem if
(7.7) $\displaystyle Q_{V}^{\prime}(u)[\varphi-u]\geq 0\qquad\text{for each
}\varphi\in\mathcal{K}_{\psi,\theta},$
that is, weakly,
(7.8) $\int_{\Omega}|\nabla u|^{p-2}\langle\nabla
u,\nabla(\varphi-u)\rangle\mathrm{d}\mu_{f}-\int_{\Omega}V|u|^{p-2}u(\varphi-u)\mathrm{d}\mu_{f}\geq
0$
Note that, for each non-negative $\widehat{\varphi}\in C^{1}_{c}(\Omega)$, the
function $\varphi=u+\widehat{\varphi}\in\mathcal{K}_{\psi,\theta}$, and
putting into (7.7) we get that $u$ solving (7.7) satisfies
$Q_{V}^{\prime}(u)\geq 0$, that is, $u$ is a supersolution. We address the
solvability of the obstacle problem in the next
###### Theorem 7.1.
Let $M$ be a Riemannian manifold, $f\in C^{\infty}(M)$, $p\in(1,+\infty)$ and
$V\in L^{\infty}_{\mathrm{loc}}(M)$. Let $\Omega\Subset M$ be a relatively
compact open set with Lipschitz boundary for which $\lambda_{V}(\Omega)>0$.
Suppose that the obstacle $\psi$ satisfies $0\leq\psi\leq\theta$ a.e. on
$\Omega$, for some $\theta\in W^{1,p}(\Omega)$. Then, there exists a solution
$u\in\mathcal{K}_{\psi,\theta}$ of (7.7).
###### Proof.
We consider the translated set
$\bar{\mathcal{K}}_{\psi,\theta}\doteq\mathcal{K}_{\psi,\theta}-\theta=\big{\\{}\bar{g}\
:\ \bar{g}+\theta\in\mathcal{K}_{\psi,\theta}\big{\\}}\subset
W^{1,p}_{0}(\Omega).$
Note that, since $\psi\geq 0$, $\forall\
\bar{g}\in\bar{\mathcal{K}}_{\psi,\theta}$ we have $\bar{g}+\theta\geq 0$.
We now define the functional $\mathcal{F}:\bar{\mathcal{K}}_{\psi,\theta}\to
W^{1,p}_{0}(\Omega)^{*}$ by setting:
$\forall\,\bar{u}\in\bar{\mathcal{K}}_{\psi,\theta}$ and $\varphi\in
W^{1,p}_{0}(\Omega)$,
(7.9) $\mathcal{F}(\bar{u})[\varphi]\doteq
Q_{V}^{\prime}(\bar{u}+\theta)[\varphi]=\mathcal{A}(\bar{u})[\varphi]-\mathcal{B}(\bar{u})[\varphi]$
with
$\mathcal{A}(\bar{u})[\varphi]=\int_{\Omega}\left|\nabla(\bar{u}+\theta)\right|^{p-2}\langle\nabla(\bar{u}+\theta),\nabla\varphi\rangle\mathrm{d}\mu_{f}$
and
$\mathcal{B}(\bar{u})[\varphi]=\int_{\Omega}V\left|\bar{u}+\theta\right|^{p-2}\left(\bar{u}+\theta\right)\varphi\mathrm{d}\mu_{f}.$
Clearly, $\bar{u}\in\bar{\mathcal{K}}_{\psi,\theta}$ solves the obstacle
problem
(7.10) $\mathcal{F}(\bar{u})[\bar{\varphi}-\bar{u}]\geq
0\qquad\forall\,\bar{\varphi}\in\bar{\mathcal{K}}_{\psi,\theta}$
if and only if $u=\bar{u}+\theta\in\mathcal{K}_{\psi,\theta}$ is a solution of
the obstacle problem (7.7).
According to Theorem 8.2, p. 247 in [51], to solve the obstacle problem it is
enough to verify that:
* 1.
$\mathcal{F}$ is pseudo-monotone on $\bar{\mathcal{K}}_{\psi,\theta}$, that
is,
* i)
$\mathcal{F}\ :\
\left(W^{1,p}_{0}(\Omega),\|\cdot\|_{W^{1,p}_{0}(\Omega)}\right)\rightarrow\left(W^{1,p}_{0}(\Omega)^{*},\|\cdot\|_{W^{1,p}_{0}(\Omega)^{*}}\right)$
is bounded.
* ii)
if $u_{i},u\in\bar{\mathcal{K}}_{\psi,\theta}$ and $u_{i}\rightharpoonup u$ in
$\left(W^{1,p}_{0}(\Omega),{\rm weak}\right)$ as $i\rightarrow+\infty$ and
(7.11) $\limsup_{i\to+\infty}\mathcal{F}(u_{i})[u_{i}-u]\leq 0$
then
(7.12)
$\liminf_{i\to+\infty}\mathcal{F}(u_{i})[u_{i}-\varphi]\geq\mathcal{F}(u)[u-\varphi]\qquad\forall\,\varphi\in
W^{1,p}_{0}(\Omega).$
* 2.
$\mathcal{F}$ is coercive on $\bar{\mathcal{K}}_{\psi,\theta}$, that is,
* iii)
there exists $\bar{\varphi}\in\bar{\mathcal{K}}_{\psi,\theta}$ for which
$\displaystyle\frac{\mathcal{F}(\bar{u})[\bar{u}-\bar{\varphi}]}{\|\bar{u}\|_{W^{1,p}_{0}(\Omega)}}\rightarrow+\infty$
if $\|\bar{u}\|_{W^{1,p}_{0}(\Omega)}\rightarrow+\infty$ with
$\bar{u}\in\bar{\mathcal{K}}_{\psi,\theta}$.
We first address the pseudo-monotonicity of $\mathcal{F}$.
* i)
Boundedness follows since, for each $\varphi\in W^{1,p}_{0}(\Omega)$, by
Hölder inequality we have
$\left|\mathcal{F}(\bar{u})[\varphi]\right|\leq
C\left(1+\|\bar{u}\|_{W^{1,p}(\Omega)}^{p-1}\right)\|\varphi\|_{W^{1,p}(\Omega)}$
for some constant $C>0$.
* ii)
Let $u_{i}\rightharpoonup u$ weakly in $W^{1,p}_{0}(\Omega)$ with
$u_{i},u\in\bar{\mathcal{K}}_{\psi,\theta}$ and suppose (7.11). Since
$u_{i}\rightharpoonup u$, $\\{u_{i}\\}$ is bounded in $W^{1,p}_{0}(\Omega)$;
since $\partial\Omega$ is Lipschitz, by Rellich-Kondrachov compactness theorem
$\|u_{i}-u\|_{L^{p}(\Omega)}\rightarrow 0$ and almost everywhere. Then, it is
easy to see that $\mathcal{B}(u_{i})[u_{i}-u]\rightarrow 0$. Therefore, by
(7.11)
(7.13) $\limsup_{i}\mathcal{A}(u_{i})[u_{i}-u]\leq 0.$
Since $\mathcal{A}$ is a monotone operator and $u_{i}\rightharpoonup u$, from
(7.13) we have
$\displaystyle\displaystyle 0$
$\displaystyle\leq\big{(}\mathcal{A}(u_{i})-\mathcal{A}(u)\big{)}[u_{i}-u]=\mathcal{A}(u_{i})[u_{i}-u]-\mathcal{A}(u)[u_{i}-u]$
$\displaystyle=\mathcal{A}(u_{i})[u_{i}-u]+o(1)$
as $i\rightarrow+\infty$, and we therefore deduce that
$\lim_{i}\big{(}\mathcal{A}(u_{i})-\mathcal{A}(u)\big{)}[u_{i}-u]=0$. By
Browder lemma, see Lemma 3 in [18], since $\mathcal{A}$ is strictly monotone
we obtain that $u_{i}\rightarrow u$ strongly in $W^{1,p}_{0}(\Omega)$. Now we
show that $\mathcal{F}$ is sequentially continuous from
$(W^{1,p}_{0}(\Omega),\|\cdot\|_{W^{1,p}_{0}(\Omega)})$ to
$(W^{1,p}_{0}(\Omega)^{*},{\rm weak})$. Let $u_{i}$ be a sequence in
$W^{1,p}_{0}(\Omega)$ which converges strongly to $u$. Up a subsequence we
have $(u_{k}(x),\nabla u_{k}(x))\rightarrow(u(x),\nabla u(x))$ for a.e.
$x\in\Omega$. Set for convenience
$X_{k}(x)=\nabla\left(u_{k}(x)+\theta(x)\right),\qquad
X(x)=\nabla\left(u(x)+\theta(x)\right).$
If $k\to\infty$,
$\left(X_{k}(x),\left|X_{k}(x)\right|\right)\to\left(X(x),\left|X(x)\right|\right)$
for a.e. $x\in\Omega$. If we set $\nabla\varphi=Y$, we have
$\begin{array}[]{l}\displaystyle\big{(}{\cal A}(u_{k})-{\cal
A}(u)\big{)}[\varphi]=\displaystyle\int_{\Omega}\langle
Y,X_{k}\left|X_{k}\right|^{p-2}-X\left|X\right|^{p-2}\rangle\mathrm{d}\mu_{f}=(I)+(II),\end{array}$
where
$\begin{array}[]{rcl}\displaystyle(I)&=&\displaystyle\int_{\Omega}\frac{\left|X_{k}\right|^{p-2}\langle
Y,X_{k}\rangle}{1+\left|X_{k}\right|^{p-1}}\left[\left|X_{k}\right|^{p-1}-\left|X\right|^{p-1}\right]\mathrm{d}\mu_{f};\\\\[11.38092pt]
\displaystyle(II)&=&\displaystyle\int_{\Omega}\langle
Y,\frac{\left|X_{k}\right|^{p-2}X_{k}}{1+\left|X_{k}\right|^{p-1}}-\frac{\left|X\right|^{p-2}X}{1+\left|X\right|^{p-1}}\rangle\left(1+\left|X\right|^{p-1}\right)\mathrm{d}\mu_{f}.\end{array}$
Since the integrand in $(II)$ is bounded by
$2|Y|\left(1+\left|X\right|^{p-1}\right)\in L^{1}(\Omega)$, by Lebesgue
theorem
(7.14) $(II)\rightarrow 0\qquad\text{as }k\rightarrow+\infty.$
We now consider $(I)$. Fix $\epsilon>0$. Then, by Egoroff theorem there exists
$\Omega_{\epsilon}\subset\Omega$ such that
$\mu_{f}(\Omega\backslash\Omega_{\epsilon})<\epsilon$ and
$|X_{k}|\rightarrow|X|$ uniformly on $\Omega_{\epsilon}$ as
$k\rightarrow+\infty$. Since
$\left|\frac{\left|X_{k}\right|^{p-2}\langle
Y,X_{k}\rangle}{1+\left|X_{k}\right|^{p-1}}\right|\leq|Y|\qquad{\rm on}\
\Omega$
we therefore obtain
$\int_{\Omega_{\epsilon}}\left|\frac{\left|X_{k}\right|^{p-2}\langle
Y,X_{k}\rangle}{1+\left|X_{k}\right|^{p-1}}\left(\left|X_{k}\right|^{p-1}-\left|X\right|^{p-1}\right)\right|\mathrm{d}\mu_{f}\to
0\qquad\text{as }k\rightarrow+\infty.$
On the other hand, using Hölder inequality we deduce
$\begin{array}[]{l}\displaystyle\left|\int_{\Omega\backslash\Omega_{\epsilon}}\frac{\left|X_{k}\right|^{p-2}\langle
Y,X_{k}\rangle}{1+\left|X_{k}\right|^{p-1}}\left(\left|X_{k}\right|^{p-1}-\left|X\right|^{p-1}\right)\right|\leq\|Y\|_{L^{p}(\Omega)}\left\|\left|X_{k}\right|^{p-1}-\left|X\right|^{p-1}\right\|_{L^{\frac{p}{p-1}}(\Omega\backslash\Omega_{\epsilon})}\\\\[11.38092pt]
\displaystyle\leq
C\|Y\|_{L^{p}(\Omega)}\Big{\|}\left|X_{k}\right|^{p}+\left|X\right|^{p}\Big{\|}^{\frac{p-1}{p}}_{L^{1}(\Omega\backslash\Omega_{\epsilon})}\leq
C\|Y\|_{L^{p}(\Omega)}\Big{\|}\big{|}|X_{k}-X|+|X|\big{|}^{p}+\left|X\right|^{p}\Big{\|}^{\frac{p-1}{p}}_{L^{1}(\Omega\backslash\Omega_{\epsilon})}\\\\[11.38092pt]
\leq\displaystyle C\|Y\|_{L^{p}(\Omega)}\left(\displaystyle
2^{p}\Big{\|}\left|X_{k}-X\right|^{p}\Big{\|}_{L^{1}(\Omega)}+\left(2^{p}+1\right)\Big{\|}\left|X\right|^{p}\Big{\|}_{L^{1}(\Omega\backslash\Omega_{\varepsilon})}\right)^{\frac{p-1}{p}}.\end{array}$
For some constant $C>0$ only depending on $p$. The first integral converges to
$0$ because $\|X_{k}-X\|_{p}\to 0$ as $k\to\infty$, while the second is
infinitesimal, if $\epsilon\to 0$, by the absolute continuity of the integral.
Thus
$0\leq\limsup_{k\rightarrow+\infty}|(I)|\leq C(\epsilon),$
where $C(\epsilon)\rightarrow 0^{+}$ as $\epsilon\rightarrow 0$. By the
arbitrariness of $\varepsilon$, $(I)\rightarrow 0$ as $k\rightarrow+\infty$
and, combining with (7.14),
$\left(\mathcal{A}(u_{k})-\mathcal{A}(u)\right)[\varphi]\rightarrow 0$ as
$k\rightarrow+\infty$. Next, in an analogous way, it can be shown that
$\big{(}\mathcal{B}(u_{k})-\mathcal{B}(u)\big{)}[\varphi]\rightarrow
0\qquad\text{as }k\rightarrow+\infty.$
We have thus proved that
${\cal F}(u_{k})\rightharpoonup{\cal F}(u)\qquad{\rm in}\ \
W_{0}^{1,p}(\Omega)^{*}\quad{\rm as}\ \ k\to\infty,$
for some subsequence $\\{u_{k}\\}$ of the original $\\{u_{i}\\}$. A simple
reasoning by contradiction thus shows that the whole
$\mathcal{F}(u_{i})\rightharpoonup\mathcal{F}(u)$ weakly on
$W^{1,p}_{0}(\Omega)^{*}$, proving the sequential continuity of $\mathcal{F}$.
Therefore, since $\|u_{i}-u\|_{W^{1,p}_{0}(\Omega)}\rightarrow 0$ as
$i\to+\infty$, for each $\varphi\in W^{1,p}_{0}(\Omega)$, we have:
$\mathcal{F}(u_{i})[u_{i}-\varphi]=\mathcal{F}(u_{i})[u_{i}-u]+\mathcal{F}(u_{i})[u-\varphi]\rightarrow
0+\mathcal{F}(u)[u-\varphi]$
as $i\rightarrow+\infty$, which because of (7.11) proves the pseudo-
monotonicity of $\mathcal{F}$.
* iii)
We are left to prove the coercivity of $\mathcal{F}$. This is a more or less
predictable consequence of the assumption $\lambda_{V}(\Omega)>0$, but some
technical details suggest to provide a full proof. We shall prove the validity
of $iii)$ with the choice $\bar{\varphi}\equiv
0\in\bar{\mathcal{K}}_{\psi,\theta}$.
First we observe that, by Cauchy-Schwarz inequality,
(7.15)
$\displaystyle\mathcal{A}(\bar{u})[\bar{u}]\geq\displaystyle\int_{\Omega}|\nabla(\bar{u}+\theta)|^{p}\mathrm{d}\mu_{f}-\int_{\Omega}|\nabla(\bar{u}+\theta)|^{p-1}|\nabla\theta|\mathrm{d}\mu_{f}.$
Next, using
$|X+Y|^{p}\geq|X|^{p}-p|X|^{p-1}|Y|,\qquad|X+Y|^{p-1}\leq
2^{p-1}\big{(}|X|^{p-1}+|Y|^{p-1}\big{)}$
and Hölder inequality into (7.15) we obtain
(7.16)
$\displaystyle\mathcal{A}(\bar{u})[\bar{u}]\geq\|\nabla\bar{u}\|_{L^{p}(\Omega)}^{p}-C_{3}\|\nabla\bar{u}\|_{L^{p}(\Omega)}^{p-1}-C_{2}$
for some constants $C_{2},C_{3}>0$ independent of $\bar{u}$.
To deal with $\mathcal{B}(\bar{u})[\bar{u}]$ we first observe that, since
$\theta\geq\psi\geq 0$ and $\bar{u}+\theta\geq 0$, we have
(7.17)
$\begin{array}[]{lcl}\displaystyle-\mathcal{B}(\bar{u})[\bar{u}]&=&\displaystyle-\int_{\Omega}V|\bar{u}|^{p}\mathrm{d}\mu_{f}-\int_{\Omega}V\left[|\bar{u}+\theta|^{p-2}(\bar{u}+\theta)-|\bar{u}|^{p-2}\bar{u}\right]\bar{u}\mathrm{d}\mu_{f}\\\\[11.38092pt]
\displaystyle&\geq&\displaystyle-\int_{\Omega}V|\bar{u}|^{p}\mathrm{d}\mu_{f}-\displaystyle\|V\|_{L^{\infty}(\Omega)}\int_{\\{\bar{u}\geq\theta\\}}\left[|\bar{u}+\theta|^{p-1}-|\bar{u}|^{p-1}\right]\bar{u}\mathrm{d}\mu_{f}\\\\[11.38092pt]
&&-\displaystyle\|V\|_{L^{\infty}(\Omega)}\int_{\\{\bar{u}<\theta\\}}\left[|\bar{u}+\theta|^{p-1}-|\bar{u}|^{p-1}\right]\bar{u}\mathrm{d}\mu_{f}.\end{array}$
Since $\bar{u}\geq-\theta$, on the set $\\{x\in\Omega\ :\
\bar{u}(x)\leq\theta(x)\\}$ we have $\bar{u}(x)\in[-\theta,\theta]$; hence the
third integral in the right-hand side of the above is bounded above by
$\displaystyle
C_{5}=3^{p-1}2\|V\|_{L^{\infty}(\Omega)}\|\theta\|_{L^{p}(\Omega)}^{p}$. As
for the second integral, on $\\{\bar{u}\geq\theta\\}$, the following
elementary inequality holds:
$(\bar{u}+\theta)^{p-1}-\bar{u}^{p-1}\leq\left\\{\begin{array}[]{ll}\displaystyle\bar{u}^{p-1}+\theta^{p-1}-\bar{u}^{p-1}=\theta^{p-1}&\quad\text{if
}p-1\leq 1,\\\\[8.5359pt]
\displaystyle\bar{u}^{p-1}\left[\left(1+\frac{\theta}{\bar{u}}\right)^{p-1}-1\right]\leq\bar{u}^{p-1}\left[p2^{p-1}\frac{\theta}{\bar{u}}\right]&\quad\text{if
}p-1>1.\end{array}\right.$
Therefore,
$\begin{array}[]{l}\displaystyle\int_{\\{\bar{u}\geq\theta\\}}\big{[}(\bar{u}+\theta)^{p-1}-\bar{u}^{p-1}\big{]}\bar{u}\mathrm{d}\mu_{f}\\\\[11.38092pt]
\leq\left\\{\begin{array}[]{ll}\displaystyle\int_{\\{\bar{u}\geq\theta\\}}\theta^{p-1}\bar{u}\mathrm{d}\mu_{f}\leq\|\theta\|_{L^{p}(\Omega)}^{p-1}\|\bar{u}\|_{L^{p}(\Omega)}&\quad\text{if
}p-1\leq 1,\\\\[8.5359pt] \displaystyle
p2^{p-1}\int_{\\{\bar{u}\geq\theta\\}}\theta\bar{u}^{p-1}\mathrm{d}\mu_{f}\leq
p2^{p-1}\|\theta\|_{L^{p}(\Omega)}\|\bar{u}\|_{L^{p}(\Omega)}^{p-1}&\quad\text{if
}p-1>1.\end{array}\right.\end{array}$
Inserting the obtained estimates into (7.17) we finally get
(7.18)
$\displaystyle-\mathcal{B}(\bar{u})[\bar{u}]\geq-\int_{\Omega}V|\bar{u}|^{p}\mathrm{d}\mu_{f}-C_{4}\|\bar{u}\|^{\max\\{p-1,1\\}}_{L^{p}(\Omega)}-C_{5}$
for some constants $C_{4},C_{5}>0$ independent of $\bar{u}$.
Combining (7.16) and (7.18) we obtain
(7.19)
$\mathcal{F}(\bar{u})[\bar{u}]\geq\|\nabla\bar{u}\|^{p}_{L^{p}(\Omega)}-C_{3}\|\nabla\bar{u}\|^{p-1}_{L^{p}(\Omega)}-C_{6}-C_{4}\|\bar{u}\|_{L^{p}(\Omega)}^{\max\left\\{p-1,1\right\\}}-\int_{\Omega}V|\bar{u}|^{p}\mathrm{d}\mu_{f}.$
On the other hand, using Rayleigh characterization of $\lambda_{V}(\Omega)$,
since $\bar{u}\in W^{1,p}_{0}(\Omega)$ we get
$\displaystyle\|\nabla\bar{u}\|^{p}_{L^{p}(\Omega)}-\int_{\Omega}V|\bar{u}|^{p}\mathrm{d}\mu_{f}\geq\lambda_{V}(\Omega)\|\bar{u}\|^{p}_{L^{p}(\Omega)},$
thus,
(7.20)
$\displaystyle\mathcal{F}(\bar{u})[\bar{u}]\geq\lambda_{V}(\Omega)\|\bar{u}\|_{L^{p}(\Omega)}^{p}-C_{4}\|\bar{u}\|^{\max\\{p-1,1\\}}_{L^{p}(\Omega)}-C_{3}\|\nabla\bar{u}\|_{L^{p}(\Omega)}^{p-1}-C_{6}$
for some constants $C_{4},C_{3},C_{6}>0$ and independent of $\bar{u}$.
Since $\bar{u}\in W^{1,p}_{0}(\Omega)$, by Poincaré inequality on $\Omega$,
there exists a constant $C_{P}>0$ independent of $\bar{u}$ such that
$\|\bar{u}\|_{L^{p}(\Omega)}\leq C_{P}\|\nabla\bar{u}\|_{L^{p}(\Omega)}.$
Now, let $\bar{u}_{k}\in\bar{\mathcal{K}}_{\psi,\theta}$ be any sequence such
that $\|\bar{u}_{k}\|_{W^{1,p}(\Omega)}\rightarrow+\infty$ as
$k\rightarrow+\infty$. Again by Poincaré inequality,
$\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}\rightarrow+\infty$ as
$k\rightarrow+\infty$ , and two cases may occur: either
$(a)\quad\frac{\|\bar{u}_{k}\|_{L^{p}(\Omega)}}{\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}}\rightarrow
0,\qquad\text{or}\quad(b)\quad\limsup_{k\rightarrow+\infty}\frac{\|\bar{u}_{k}\|_{L^{p}(\Omega)}}{\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}}=c>0.$
In the case $(a)$, using (7.19), we deduce
$\begin{array}[]{l}\displaystyle\frac{\mathcal{F}(\bar{u}_{k})[\bar{u}_{k}]}{\|\bar{u}_{k}\|_{W^{1,p}_{0}(\Omega)}}\geq\\\\[11.38092pt]
\displaystyle\geq\frac{\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}^{p}-\|V\|_{L^{\infty}(\Omega)}\|\bar{u}_{k}\|_{L^{p}(\Omega)}^{p}-C_{4}\|\bar{u}_{k}\|^{\max\\{p-1,1\\}}_{L^{p}(\Omega)}-C_{3}\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}^{p-1}-C_{6}}{(1+C_{P})\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}}\rightarrow+\infty\end{array}$
as $\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}\rightarrow+\infty$.
In case $(b)$, for each subsequence (still denoted with $\\{\bar{u}_{k}\\}$)
satisfying
$\frac{\|\bar{u}_{k}\|_{L^{p}(\Omega)}}{\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}}\rightarrow\bar{c}\in(0,c]\qquad\text{for}\
k\to+\infty$
using (7.20) and $\lambda_{V}(\Omega)>0$ we get
$\displaystyle\frac{\mathcal{F}(\bar{u}_{k})[\bar{u}_{k}]}{\|\bar{u}_{k}\|_{W^{1,p}_{0}(\Omega)}}\geq\displaystyle\frac{\lambda_{V}(\Omega)\|\bar{u}_{k}\|_{L^{p}(\Omega)}^{p}-C_{4}\|\bar{u}_{k}\|^{\max\\{p-1,1\\}}_{L^{p}(\Omega)}-C_{3}\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}^{p-1}-C_{6}}{(1+C_{P})\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}}\rightarrow+\infty$
as $\|\nabla\bar{u}_{k}\|_{L^{p}(\Omega)}\rightarrow+\infty$. This concludes
the proof of the coercivity of $\mathcal{F}$.
∎
For $\theta=\psi$ we set ${\cal K}_{\psi}$ for ${\cal K}_{\psi,\psi}$. Next,
we prove a minimizing properties for solutions of the obstacle problem.
###### Proposition 7.1.
Let $u\in\mathcal{K}_{\psi,\theta}$, $u\not\equiv 0$ be a solution of the
obstacle problem with $\psi\geq 0$. Suppose that $w\in W^{1,p}(\Omega)$,
$w\not\equiv 0$ solves $Q_{V}^{\prime}(w)\geq 0$ on $\Omega$, such that
$\min\\{u,w\\}\in\mathcal{K}_{\psi,\theta}$. If
(7.21) $\frac{u}{w},\,\,\frac{w}{u}\in L^{\infty}(\Omega),$
then $u\leq w$ on $\Omega$.
###### Proof.
Since $w,u$ are non-negative, nonzero supersolutions ($u$ being a solution of
the obstacle problem), by the half-Harnack inequality we have $u>0$ and $w>0$
on $\Omega$. Combining with (7.21) and since $\Omega$ is relatively compact,
we deduce that the following set is non-empty.
$A\doteq\\{c>1:cw(x)\geq u(x)\quad\text{for a.e. }x\in\Omega\\}.$
Let $c_{0}=\inf A$, so that $c_{0}w\geq u$ a.e. on $\Omega$. We shall show
that $c_{0}=1$. By contradiction suppose $c_{0}>1$, and choose $1<c<c_{0}$
close enough to $c_{0}$ to have
(7.22) $\frac{(cw)^{p}}{u^{p-1}}\geq\psi\quad\text{on }\Omega.$
This is possible since, for $c\geq c_{0}^{\frac{p-1}{p}}$ we have
$\frac{(cw)^{p}}{u^{p-1}}\geq\frac{(cw)^{p}}{(c_{0}w)^{p-1}}=\frac{c^{p}}{c_{0}^{p-1}}w\geq
w\geq\psi,\qquad\text{on}\qquad\overline{\Omega}.$
Consider the non-empty set $U=\\{x\in\Omega\ :\ u(x)>cw(x)\\}$, and define
$\varphi=u+\min\left\\{\frac{(cw)^{p}-u^{p}}{u^{p-1}},0\right\\}=\min\left\\{u,\frac{(cw)^{p}}{u^{p-1}}\right\\}.$
Because of (7.22), $\varphi\geq\psi$ and $\varphi-\theta=u-\theta=0$ on
$\partial\Omega$; in other words, $\varphi\in\mathcal{K}_{\psi,\theta}$.
Since $u$ solves the obstacle problem (7.7), using the above $\varphi$ we
deduce
(7.23) $\begin{array}[]{lcl}\displaystyle 0\leq
Q_{V}^{\prime}(u)[\varphi-u]&=&\displaystyle\int_{U}|\nabla
u|^{p-2}\langle\nabla
u,\nabla\left(\frac{(cw)^{p}-u^{p}}{u^{p-1}}\right)\rangle\mathrm{d}\mu_{f}\\\\[11.38092pt]
&&-\displaystyle\int_{U}V\big{(}(cw)^{p}-u^{p}\big{)}\mathrm{d}\mu_{f}.\end{array}$
On the other hand, applying the definition of supersolution to $cw$ with the
non-negative test function
$\displaystyle\widetilde{\varphi}=\frac{\big{(}u^{p}-(cw)^{p}\big{)}_{+}}{(cw)^{p-1}}\in
W^{1,p}_{0}(\Omega),$
we get
(7.24) $\begin{array}[]{lcl}\displaystyle 0\leq
Q_{V}^{\prime}(cw)[\widetilde{\varphi}]&=&\displaystyle\int_{U}|\nabla(cw)|^{p-2}\langle\nabla(cw),\nabla\left(\frac{u^{p}-(cw)^{p}}{(cw)^{p-1}}\right)\rangle\mathrm{d}\mu_{f}\\\\[11.38092pt]
&&-\displaystyle\int_{U}V\left(u^{p}-(cw)^{p}\right)\mathrm{d}\mu_{f}.\end{array}$
Summing up (7.23) and (7.24) we get
(7.25)
$\int_{U}|\nabla(cw)|^{p-2}\langle\nabla(cw),\nabla\left(\frac{u^{p}-(cw)^{p}}{(cw)^{p-1}}\right)\rangle\mathrm{d}\mu_{f}-\int_{U}|\nabla
u|^{p-2}\langle\nabla
u,\nabla\left(\frac{u^{p}-(cw)^{p}}{u^{p-1}}\right)\rangle\mathrm{d}\mu_{f}\geq
0.$
Set $z=\max\\{u,cw\\}$, and note that (7.25) is equivalent to $I(cw,z)\leq 0$.
Indeed, in the definition (3.7) of $I$, the part of the integral outside $U$
is zero since $z\equiv cw$. To conclude, applying Proposition 3.2 we deduce
$I(cw,z)=0$, so that $cw$ and $z$ (hence $u$) are proportional. Since $cw=u$
on $\partial U$ we conclude that $u\equiv cw$ on $U$, contradicting the
definition of this latter. ∎
###### Remark 7.2.
_A typical case when ( 7.21) is automatically met is when the data
$\psi,\theta$ satisfy $\psi,\theta\in W^{1,p}(\Omega)\cap
C^{0}(\overline{\Omega})$ and $\theta>0$ on $\partial\Omega$, which we will
frequently use. In fact, when $\psi,\theta\in C^{0}(\overline{\Omega})$, the
solution $u\in\mathcal{K}_{\psi,\theta}$ of the obstacle problem is continuous
on $\overline{\Omega}$. In this respect, see Theorem 5.4, page 235 of [52]. _
###### Remark 7.3.
_Although it is not explicitly stated, condition $\lambda_{V}(\Omega)\geq 0$
is automatic by assuming the existence of a $u$ solving the obstacle problem
and $\psi\geq 0$, $\psi\not\equiv 0$. Indeed, $u$ is a positive solution of
$Q_{V}^{\prime}(u)\geq 0$ on $M$, and $\lambda_{V}(\Omega)\geq 0$ follows from
Proposition 3.4. _
In particular from the above proposition we deduce the following
characterization.
###### Corollary 7.1.
Let $\Omega\Subset M$ be a relatively compact open set, and let
$0\leq\theta\in W^{1,p}(\Omega)\cap C^{0}(\overline{\Omega})$ be such that
$\theta>0$ on $\partial\Omega$. Then, a solution $u$ of the obstacle problem
on $\mathcal{K}_{\psi,\theta}$ with obstacle $0\leq\psi\leq\theta$,
$\psi\not\equiv 0$ is the minimal $w\in\mathcal{K}_{\psi,\theta}$ satisfying
$Q_{V}^{\prime}(w)\geq 0$ on $\Omega$. Consequently, such a solution is
unique.
A second important consequence of Proposition 7.1 is
###### Proposition 7.2.
Let $w_{1},w_{2}\in W^{1,p}_{\mathrm{loc}}(M)\cap C^{0}(M)$ be positive
solutions of $Q_{V}^{\prime}(w_{j})\geq 0$ on $M$, $j=1,2$. Then,
$w\doteq\min\\{w_{1},w_{2}\\}$
solves
(7.26) $Q_{V}^{\prime}(w)\geq 0\qquad{\rm on}\ \ M.$
###### Proof.
Fix any relatively compact open set $\Omega$ with smooth boundary and consider
a solution $s$ of the obstacle problem ${\cal K}_{w}$ on $\Omega$. This is
possible since $w\in W^{1,p}(\Omega)$. From $s\in{\cal K}_{w}$ we have $s\geq
w$, and, being a solution of the obstacle problem, $Q_{V}^{\prime}(s)\geq 0$.
Next, since $w_{j}>0$ on $\overline{\Omega}$ and $Q^{\prime}_{V}(w_{j})\geq
0$, we can apply Proposition 7.1 to obtain $s\leq w_{j}$ for each $j$, whence
$s\leq w$. This shows that $s\equiv w$, so that $w$ solves (7.26) as claimed.
∎
We are now ready for the
###### Proof of Lemma 3.1.
First we show that $u\in W^{1,p}_{\mathrm{loc}}(\overline{\Omega}_{2})$. Let
$U\subset\overline{\Omega}_{2}$ be a relatively compact open set. Without loss
of generality we can assume that $\Omega_{1}\cap U\neq\emptyset$, for
otherwise $u\equiv u_{2}$ on $U$ and the sought is immediate. Since
$z=\min\\{u_{1}-u_{2},0\\}\in W^{1,p}(\Omega_{1})$ is zero on
$\partial\Omega_{1}\cap\Omega_{2}$, there exists a sequence
$\\{\varphi_{j}\\}\subset C^{0}(\overline{U\cap\Omega}_{1})\cap
W^{1,p}(U\cap\Omega_{1})$, $\varphi_{j}\equiv 0$ on some neighbourhood of
$\partial\Omega_{1}\cap\Omega_{2}$, converging in the
$W^{1,p}(U\cap\Omega_{1})$ norm to $z$. Using that $z\leq 0$ on $\Omega_{1}$,
one can take, for instance,
(7.27) $\varphi_{j}=\left(z+\frac{1}{j}\right)_{-}.$
We extend each $\varphi_{j}$ to a continuous function on
$U\backslash\Omega_{1}$ by setting $\varphi_{j}\equiv 0$ on
$U\backslash\Omega_{1}$, so that $\varphi_{j}\in W^{1,p}(U)$ and clearly
$\varphi_{j}\rightarrow z1_{\Omega_{1}}$ in $W^{1,p}(U)$, which shows that
$z1_{\Omega_{1}}\in W^{1,p}(U)$. It follows that $u=u_{2}+\varphi_{j}\in
W^{1,p}(U)$ converges to $u=u_{2}+z1_{\Omega_{1}}$, whence $u\in W^{1,p}(U)$.
To prove that $Q_{V}^{\prime}(u)\geq 0$ on $\Omega_{2}$, we shall reduce
ourselves to Proposition 7.1. We take a smooth open set $U\Subset\Omega_{2}$
which, without loss of generality, intersects $\Omega_{1}$. Since
$\lambda_{V}(U)>0$ by monotonicity of eigenvalues, Theorem 7.1 guarantees the
existence of a solution $s$ of the obstacle problem in ${\cal K}_{u}$ on $U$
which, applying Theorem 5.4, page 235 in [52] is continuous on $\overline{U}$.
We want to show that $s\equiv u$ on $U$. First, since $u>0$ on $\overline{U}$,
using Proposition 7.2 we get
(7.28) $s\leq u_{2}\qquad\text{on }U.$
Hence, $s=u_{2}=u$ on $U\backslash\Omega_{1}$. Consequently, since also
$s\in\cal K_{u}$, $s-u=0$ on $\partial U\cup(\partial\Omega_{1}\cap
U)=\partial(\Omega_{1}\cap U)$, and so by standard theory
(7.29) $s-u\in W_{0}^{1,p}(\Omega_{1}\cap U).$
(one can construct an approximating sequence as in (7.27) above). Therefore,
$s$ is also a solution of the obstacle problem on the closed convex set
$\hat{\cal K}_{u}=\Big{\\{}\varphi\in W^{1,p}(\Omega_{1}\cap U)\ |\ \
\varphi\geq u\ \text{ a.e. and }\ \varphi-u\in W^{1,p}_{0}(\Omega_{1}\cap
U)\Big{\\}}.$
As $u_{1}$ is a positive supersolution on $\Omega_{1}\cap U$ and
$\min\\{s,u_{1}\\}\in\hat{\cal K}_{u}$, by Proposition 7.1
(7.30) $s\leq u_{1}\qquad\text{on }\Omega_{1}\cap U.$
Coupling with (7.28) we get $s\leq u$ on $\Omega_{1}\cap U$, and combining
with $s\geq u$ on $U$ ($s\in\cal K_{u}$), $s=u=u_{1}$ on
$U\backslash\Omega_{1}$, we conclude $s\equiv u$ on $U$. This shows that $u$
solves $Q_{V}^{\prime}(u)\geq 0$ on $U$. As $U\Subset\Omega_{2}$ is arbitrary,
$Q_{V}^{\prime}(u)\geq 0$ on $\Omega_{2}$, concluding the proof. ∎
Acknowledgements: The second author is supported by the grant PRONEX - Núcleo
de Análise Geométrica e Aplicacões Processo nº PR2-0054-00009.01.00/11.
It is a pleasure to thank Y. Pinchover for having suggested us, after we
posted a first version of this work on arXiv, the very recent [23, 24] which
contain results tightly related to those in Section 4.
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|
arxiv-papers
| 2014-04-11T14:31:37 |
2024-09-04T02:50:01.025666
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Bruno Bianchini, Luciano Mari, Marco Rigoli",
"submitter": "Luciano Mari",
"url": "https://arxiv.org/abs/1404.3118"
}
|
1404.3201
|
# Detail study of the medium created in Au+Au collisions with high $p_{T}$
probes by the PHENIX experiment at RHIC
Takao Sakaguchi, for the PHENIX Collaboration Brookhaven National Laboratory,
Upton, NY 11973, USA.
###### Abstract
Recent results on high $p_{T}$ identified hadrons in Au+Au collisions from the
PHENIX experiment are presented. The $R_{\rm AA}$ for $\pi^{0}$ and $\eta$ are
found to be very consistent. The second and fourth order collective flow of
$\pi^{0}$’s have been measured and found that $v_{4}/v_{2}^{2}$ is consistent
with the one observed in lower $p_{T}$ region. Assuming the suppression of the
$\pi^{0}$ yield at highest $p_{T}$ arises from energy loss of partons, we
found that the energy loss is $L^{3}$ dependent, where $L$ is the path length
of the partons in the medium. The $\delta\mbox{$p_{T}$}/\mbox{$p_{T}$}$’s of
high $p_{T}$ hadrons are computed from 39 GeV Au+Au over to 2.76 TeV Pb+Pb,
and found that they vary by a factor of six. We have seen a smooth trend in
$\delta\mbox{$p_{T}$}/\mbox{$p_{T}$}$ from RHIC energy to LHC energy when
plotting against charged multiplicity of the systems.
###### keywords:
QGP , high $p_{T}$ hadrons , energy loss
###### PACS:
25.75.-q , 25.75.Bh , 25.75.Ld
## 1 Introduction
The interaction of hard scattered partons with the medium created by heavy ion
collisions (i.e., quark-gluon plasma, QGP) has been of interest since the
beginning of the RHIC running [1]. A large suppression of the yields of high
transverse momentum ($p_{T}$) hadrons which are the fragments of such partons
was observed, suggesting that the matter is sufficiently dense to cause
parton-energy loss prior to hadronization [2]. The PHENIX experiment [3] has
been exploring the highest $p_{T}$ region with single $\pi^{0}$ and $\eta$
mesons, which are leading hadrons of jets, and thus provide a good measure of
momentum of hard scattered partons. Here, we present the recent results
obtained from Au+Au collisions in the Year-2007 run (0.81 nb-1). Fig. 1(a)
shows the nuclear modification factors $R_{\rm AA}$ ($\equiv(dN_{\rm
AA}/dydp_{T})/(\langle T_{\rm AA}\rangle d\sigma_{pp}/dydp_{T})$) for
$\pi^{0}$ and $\eta$’s in 200 GeV Au+Au collisions [4].
Figure 1: (a, left) $R_{\rm AA}$ for $\pi^{0}$ and $\eta$ in minimum bias
Au+Au collisions. (b, right) $R_{\rm AA}$ for $\pi^{0}$ from the RHIC
Year-2004 run and Year-2007 run.
They are very consistent each other in spite of hidden strangeness contents in
$\eta$ mesons. This also implies that the fragmentation function is not
modified by the medium for the $p_{T}$ range we measured. Because $\eta$ has
four times larger mass compared to that of $\pi^{0}$, one can resolve two
photons decaying from $\eta$ up to four times larger $p_{T}$ of $\pi^{0}$,
resulting in a higher $p_{T}$ reach with smaller systematic errors with
$\eta$. The $R_{\rm AA}$’s in Fig. 1(b) demonstrates that the $\pi^{0}$ yield
from the Year-2007 run has smaller errors and is consistent with that from the
Year-2004 run [5]. We used the same $p+p$ reference for $R_{\rm AA}$’s from
both Au+Au running.
## 2 Anisotropy of high $p_{T}$ $\pi^{0}$ yield
### 2.1 Collective flow
The transition from anisotropy driven by hydrodynamic flow to anisotropy
driven by jet quenching can be probed by the ratio of $v_{4}/v_{2}^{2}$, where
$v_{4}$ is the fourth order and $v_{2}$ is the second order flow. Perfect
fluid hydrodynamics predicts a value of 0.5 for this ratio [6]. The
geometrical fluctuations and other dynamical fluctuations, as well as viscous
damping, can increase the magnitude of the ratio, especially in central
collisions [7]. At high $p_{T}$, the directions that maximize collective flow
and jet quenching may not be the same [8, 9]. Therefore, this ratio could
change in the $p_{T}$ region where jet quenching begins to dominate. With the
large statistics, we were able to measure the $v_{2}$ and $v_{4}$ of
$\pi^{0}$’s with the same second-order event plane ($\Psi_{2}$) over a wide
$p_{T}$ range. Fig. 2(a) shows the $v_{4}(p_{T})$ of $\pi^{0}$ in 200GeV Au+Au
collisions [10].
Figure 2: (a, left) $v_{4}$ of $\pi^{0}$, and (b,right) $v_{4}/v_{2}^{2}$ for
various centralities in 200GeV Au+Au collisions.
We can see significant $v_{4}$ values even for $p_{T}>5$ GeV/$c$. Fig. 2(b)
shows the $v_{4}/v_{2}^{2}$ ratios for $\pi^{0}$ obtained in several
centrality ranges [10]. The ratios are approximately independent of $p_{T}$,
with values of $\sim$0.8-1.0 depending on centrality selections. The constant
ratios over the $p_{T}$ are not trivial at all given several physics processes
are involved, and may put additional constraint on dynamical description of
the medium. The values for $p_{T}<\sim 5$ GeV/$c$ are consistent with our
prior observations of this ratio for inclusive charged hadron measurements
[11].
### 2.2 Path-length dependence of yield suppression
Using the event plane information, we were also able to measure the $R_{\rm
AA}$ of $\pi^{0}$ as a function of $\Delta\phi$ with respect to the event
plane. Since the path length in the medium that partons traverse changes by
the emission angle with respect to the event plane (especially in peripheral
collision case), the angle dependence of the yields can be associated with the
path length dependent energy loss of partons. We show the $R_{\rm AA}$ for in-
and out-of event planes for $\pi^{0}$s in 20-30 % central 200 GeV Au+Au
collisions in Fig. 3 [5].
Figure 3: $R_{\rm AA}(p_{T},\Delta\phi)$ of $\pi^{0}$ in 20-30 % centrality
for in-plane and out-of-plane. Data are compared with a pQCD-inspired model
(left), and an AdS/CFT-inspired model (right).
Depending on the energy loss models, the powers of the path-length dependence
change. The data favors an AdS/CFT-inspired (strongly coupled) model rather
than a pQCD-inspired (weakly coupled) model, implying that the energy loss is
$L^{3}$ dependent rather than $L^{2}$ dependence, where $L$ denotes the path-
length of partons in the medium.
## 3 Fractional momentum loss of hadrons in A+A collisions
Experiments have been looking at the suppression of the yield at a given
$p_{T}$ to quantify the energy loss effect. However, the suppression is
primarily the consequence of the reduction of momentum of hadrons which have
exponential $p_{T}$ distributions. We have statistically extracted the
fractional momentum loss ($S_{\rm loss}\equiv\delta p_{T}/p_{T}$, $\delta
p_{T}\equiv p_{T}-p_{T}^{\prime}$, where $p_{T}$ is the transverse momentum of
$p+p$ data, and $p_{T}^{\prime}$ is that of Au+Au data) of the partons using
the hadron $p_{T}$ spectra measured in $p+p$ and Au+Au collisions [5]. Fig.
4(a) depicts the method to compute the $S_{\rm loss}$. Using this method, we
computed the $S_{\rm loss}$ in Au+Au collisions at $\sqrt{s_{NN}}=$39, 62, and
200 GeV as shown in Fig. 4(b) [12]. We also computed the $S_{\rm loss}$ in
2.76 TeV Pb+Pb collisions using charged hadron spectra measured by the ALICE
experiment [13] as shown in Fig. 4(c). $S_{\rm loss}$’s vary by a factor of
six from 39 GeV Au+Au to 2.76 TeV Pb+Pb collisions.
Figure 4: (a, left) Method of calculating average $S_{\rm loss}$. We scaled
the $p+p$ yield by $T_{\rm AA}$ corresponding to centrality selection of Au+Au
data, shifted the $p+p$ points closest to Au+Au in yield, and calculated
momentum difference of $p+p$ and Au+Au points. (b, middle) $S_{\rm loss}$ for
$\pi^{0}$ for 0-10 % centrality 39, 62, and 200 GeV Au+Au collisions. (c,
right) $S_{\rm loss}$ for $\pi^{0}$ in 200 GeV Au+Au collisions and charged
hadrons in 2.76 TeV Pb+Pb collisions.
Naively, one expects that the energy loss is energy density dependent. We
plotted the $S_{\rm loss}$ against charged multiplicity, $dN_{\rm ch}/d\eta$,
at $p_{T}(p+p)=7$ GeV/c, which is reasonably in hard scattering regime as
shown in Fig. 5. We assume $dN_{\rm ch}/d\eta$ well represents the energy
density of the system.
Figure 5: $\delta p_{T}/p_{T}$ as a function of $dN_{\rm ch}/d\eta$ for
$\pi^{0}$ in 200 GeV and 62.4 GeV Au+Au collisions measured by PHENIX and
charged hadrons in 2.76 TeV Pb+Pb collisions measured by ALICE.
It is interesting to note that the trend of $S_{\rm loss}$ in Au+Au collisions
points to the most central points in Pb+Pb collisions at LHC. The 62 GeV Au+Au
point and the most peripheral ALICE point are off the trend. These features
have not been found by looking at $R_{\rm AA}$’s. In order to cross-check the
new result, we have performed a power-law fit to the points on $\delta
p_{T}/p_{T}$ vs $dN_{\rm ch}/d\eta$, and compared the power with the result
obtained from a different method [14]. We fitted the points of this work with
$\delta p_{T}/p_{T}=\beta(dN_{\rm ch}/d\eta)^{\alpha}$ assuming $dN_{\rm
ch}/d\eta$ $\propto$$N_{\rm part}$, and obtained $\alpha$ as 0.55$\pm$0.06.
Assuming the spectra shape is power-law with the power $n$, one can write the
relation between $S_{\rm loss}$ and $R_{\rm AA}$ as:
$S_{\rm loss}\equiv\delta p_{T}/p_{T}=\beta N_{\rm part}^{\alpha},\\\ R_{\rm
AA}=(1-S_{\rm loss})^{n-2}=(1-\beta N_{\rm part}^{\alpha})^{n-2}$
Following this relation, we obtained the power $\alpha$ as 0.57$\pm$0.13 from
the fit to the integrated $R_{\rm AA}$ as a function of $N_{\rm part}$ in the
literature [14]. Thus, we confirmed that the powers obtained by two methods
are very consistent.
## 4 Summary
We presented the recent results on high $p_{T}$ identified hadrons in Au+Au
collisions from the PHENIX experiment. The $R_{\rm AA}$ for $\pi^{0}$ and
$\eta$ are found to be very consistent. The second and fourth order collective
flow of $\pi^{0}$’s have been measured and found that $v_{4}/v_{2}^{2}$ is
consistent with the one observed in lower $p_{T}$ region, which is not trivial
given several physics processes are involved. We found that the energy loss is
$L^{3}$ dependent, where $L$ is the path length of the partons in the medium.
The $\delta p_{T}/p_{T}$’s of high $p_{T}$ hadrons are computed from 39 GeV
Au+Au over to 2.76 TeV Pb+Pb, and found that they vary by a factor of six. We
have seen a smooth trend in $\delta p_{T}/p_{T}$ from RHIC energy to LHC
energy when plotting against charged multiplicity of the systems. We performed
power-law fit to the $\delta p_{T}/p_{T}$ vs $dN_{\rm ch}/d\eta$, and obtained
a power that is very consistent with the one obtained from the fitting to the
integrated $R_{\rm AA}$. We are going to add points from other systems to
systematically investigate the $\delta p_{T}/p_{T}$.
## References
* [1] X. -N. Wang, Phys. Rev. C 58, 2321 (1998).
* [2] S. S. Adler et al. [PHENIX Collaboration], Phys. Rev. Lett. 91, 072301 (2003).
* [3] K. Adcox et al. [PHENIX Collaboration], Nucl. Instrum. Meth. A 499, 469 (2003).
* [4] A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 82, 011902 (2010).
* [5] A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 87, 034911 (2013).
* [6] N. Borghini and J. -Y. Ollitrault, Phys. Lett. B 642, 227 (2006).
* [7] C. Gombeaud and J. -Y. Ollitrault, Phys. Rev. C 81, 014901 (2010).
* [8] J. Jia, Phys. Rev. C 87, 061901 (2013).
* [9] X. Zhang and J. Liao, Phys. Rev. C 87, 044910 (2013).
* [10] A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 88, 064910 (2013).
* [11] A. Adare et al. [PHENIX Collaboration], Phys. Rev. Lett. 105, 062301 (2010).
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|
arxiv-papers
| 2014-04-11T19:43:57 |
2024-09-04T02:50:01.053217
|
{
"license": "Public Domain",
"authors": "Takao Sakaguchi (for the PHENIX collaboration)",
"submitter": "Takao Sakaguchi",
"url": "https://arxiv.org/abs/1404.3201"
}
|
1404.3203
|
# Compressive classification and the rare eclipse problem
Afonso S. Bandeira Program in Applied and Computational Mathematics (PACM),
Princeton University, Princeton, NJ 08544, USA ([email protected]). ,
Dustin G. Mixon Department of Mathematics and Statistics, Air Force Institute
of Technology, Wright-Patterson AFB OH, USA ([email protected]). and
Benjamin Recht Department of Electrical Engineering and Computer Science,
Department of Statistics, University of California, Berkeley CA, USA
([email protected]).
###### Abstract.
This paper addresses the fundamental question of when convex sets remain
disjoint after random projection. We provide an analysis using ideas from
high-dimensional convex geometry. For ellipsoids, we provide a bound in terms
of the distance between these ellipsoids and simple functions of their
polynomial coefficients. As an application, this theorem provides bounds for
compressive classification of convex sets. Rather than assuming that the data
to be classified is sparse, our results show that the data can be acquired via
very few measurements yet will remain linearly separable. We demonstrate the
feasibility of this approach in the context of hyperspectral imaging.
## 1\. Introduction
A decade of powerful results in compressed sensing and related fields have
demonstrated that many signals that have low-dimensional latent structure can
be recovered from very few compressive measurements. Building on this work,
many researchers have shown that classification tasks can also be run on
compressive measurements, provided that either the data or classifier is
sparse in an appropriate basis [14, 9, 11, 4, 13, 24]. However, classification
is a considerably simpler task than reconstruction, as there may be a large
number of hyperplanes which successfully cleave the same data set. The
question remains:
> Can we successfully classify data from even fewer compressive measurements
> than required for signal reconstruction?
Prior work on compressive classification has focused on preserving distances
or inner products between data points. Indeed, since popular classifiers
including the support vector machine and logistic regression only depend on
dot products between data points, it makes sense that if dot products are
preserved under a compressive measurement, then the resulting decision
hyperplane should be close to the one computed on the uncompressed data.
In this paper, we take a different view of the compressive classification
problem, and for some special cases, we are able to show that data can be
classified with extremely few compressive measurements. Specifically, we
assume that our data classes are circumscribed by disjoint convex bodies, and
we seek to avoid intersection between distinct classes after projection. By
studying the set of separating hyperplanes, we provide a general way to
estimate the minimal dimension under which two bodies remain disjoint after
random projection. In Section 3, we specialize these results to study
ellipsoidal classes and give our main theoretical result—that $k$ ellipsoids
of sufficient pairwise separation remain separated after randomly projecting
onto $O(\log k)$ dimensions. Here, the geometry of the ellipsoids plays an
interesting and intuitive role in the notion of sufficient separation. Our
results differ from prior work insofar as they can be applied to full
dimensional data sets and are independent of the number of points in each
class. We provide a comparison with principal component analysis in Section 4
by considering different toy examples of classes to illustrate strengths and
weaknesses, and then by applying both approaches to hyperspectral imaging
data. We conclude in Section 5 with a discussion of future work.
## 2\. Our Model and Related Work
In this section, we discuss our model for the classes as well as the
underlying assumptions we apply throughout this paper. Consider an ensemble of
classes $C_{i}\subseteq\mathbb{R}^{N}$ that we would like to classify. We
assume that these classes are pairwise linearly separable, that is, for every
pair $i,j$ with $i\neq j$, there exists a hyperplane in $\mathbb{R}^{N}$ which
separates $C_{i}$ and $C_{j}$. Equivalently, we assume that the convex hulls
$S_{i}:=\operatorname{hull}(C_{i})$ are disjoint, and for simplicity, we
assume these convex hulls are closed sets.
Linear separability is a particularly useful property in the context of
classification, since to demonstrate non-membership, it suffices to threshold
an inner product with the vector normal to a separating hyperplane. Of course,
in many applications, classes do not enjoy this (strong) property, but the
property can be weakened to near linear separability, in which there exists a
hyperplane that mostly distinguishes of a pair of classes. One may also lift
to a tensored version of the vector space and find linear separability there.
Since linear separability is so useful, we use this property as the basis for
our notion of distortion: We seek to project the classes
$\\{C_{i}\\}_{i=1}^{k}$ in such a way that their images are linearly
separable.
Our assumptions on the $C_{i}$’s and our notion of distortion both lead to a
rather natural problem in convex geometry (see Figure 1 for an illustration):
###### Rare Eclipse Problem.
Given a pair of disjoint closed convex sets $A,B\subseteq\mathbb{R}^{N}$ and
$\eta>0$, find the smallest $M$ such that a random $M\times N$ projection $P$
satisfies $PA\cap PB=\emptyset$ with probability $\geq 1-\eta$.
Figure 1. Two sufficiently separated convex sets remain separated when
projected onto a subspace. The Rare Eclipse Problem asks for the smallest $M$
such that this happens when projecting onto a random subspace of dimension
$M$. Solving this problem for a given ensemble of classes enables
dimensionality reduction in a way that ensures linear separability for
classification.
At this point, we discuss some related work in the community. It appears that
compressive classification was studied as early as 2006, when [14] considered
a model in which each class is a point in Euclidean space. Interestingly, this
bears some resemblance to the celebrated work in [16, 19], which used random
projections to quickly approximate nearest-neighbor search. The work in [9,
11] considered a more exotic family of classes, namely low-dimensional
manifolds—this is particularly applicable to the classification of images
according to the primary object featured in each image. Along these lines of
low-dimensional classes, there has since been some work in the case where
classes are low-dimensional subspaces [21, 26], or unions thereof [3].
Specifically, [26] considers a Gaussian mixture model in which each Gaussian
is supported on a different subspace. From a slightly dual view, researchers
have also shown that if the classifier is known to be sparse, then we can
subsample the data itself, and the separating hyperplane can be determined
from a number of examples roughly proportional to the sparsity of the
hyperplane [4, 13, 24].
It is striking that, to date, all of the work in compressive classification
has focused on classes of low dimension. This is perhaps an artifact of the
mindset of compressed sensing, in which the projection preserves all
information on coordinate planes of sufficiently small dimension. However,
classification should not require nearly as much information as signal
reconstruction does, and so we expect to be able to compressively classify
into classes of full dimension; indeed, we allow two points in a common class
to be mapped to the same compressive measurement, as this will not affect the
classification. A Boolean version of this idea is studied in [1], which
considers both random and optimality constructed projections. In the
continuous setting, the closest existing work is that of Dasgupta [6, 7],
which uses random projections to learn a mixture of Gaussians. In particular,
Dasgupta shows that sufficiently separated Gaussians stay separated after
random projection. In the next section, we prove a similar result about
ellipsoids, but with a sharper notion of separation.
## 3\. Theoretical Results
Given two disjoint closed convex bodies $A,B\subseteq\mathbb{R}^{N}$ and a
projection dimension $M$, the Rare Eclipse Problem asks whether a random
$M\times N$ projection $P$ of these bodies avoids collision, i.e., whether
$PA\cap PB$ is typically empty. This can be recast as a condition on the
$(N-M)$-dimensional null space of $P$:
$PA\cap
PB=\emptyset\qquad\Longleftrightarrow\qquad\operatorname{Null}(P)\cap(A-B)=\emptyset,$
where $A-B$ denotes the Minkowski difference of $A$ and $B$. Of course, the
null space of $P$ is closed under scalar multiplication, and so avoiding $A-B$
is equivalent to avoiding the normalized versions of the members of $A-B$.
Indeed, if we take $S$ to denote the intersection between the unit sphere in
$\mathbb{R}^{N}$ and the cone generated by $A-B$, then
$PA\cap PB=\emptyset\qquad\Longleftrightarrow\qquad\operatorname{Null}(P)\cap
S=\emptyset.$
Now suppose $P$ is drawn so that its entries are iid $\mathcal{N}(0,1)$. Then
by rotational invariance, the distribution of its null space is uniform over
the Grassmannian. As such, the Rare Eclipse Problem reduces to a classical
problem in convex geometry: Given a “mesh” (a closed subset of the unit
sphere), how small must $K$ be for a random $K$-dimensional subspace to
“escape through the mesh,” i.e., to avoid collision? It turns out that for
this problem, the natural way to quantify the size of a mesh is according to
its Gaussian width:
$w(S):=\mathbb{E}_{g}\bigg{[}\sup_{z\in S}\langle z,g\rangle\bigg{]},$
where $g$ is a random vector with iid $\mathcal{N}(0,1)$ entries. Indeed,
Gaussian width plays a crucial role in the following result, which is an
improvement to the original (Corollary 3.4 in [12]); the proof is given in the
appendix, and follows the proof of Corollary 3.3 in [5] almost identically.
###### Gordon’s Escape Through a Mesh Theorem.
Take a closed subset $S$ of the unit sphere in $\mathbb{R}^{N}$, and denote
$\lambda_{M}:=\mathbb{E}\|g\|_{2}$, where $g$ is a random $M$-dimensional
vector with iid $\mathcal{N}(0,1)$ entries. If $w(S)<\lambda_{M}$, then an
$(N-M)$-dimensional subspace $Y$ drawn uniformly from the Grassmannian
satisfies
$\operatorname{Pr}\Big{(}Y\cap S=\emptyset\Big{)}\geq
1-\exp\bigg{(}-\frac{1}{2}\Big{(}\lambda_{M}-w(S)\Big{)}^{2}\bigg{)}.$
It is straightforward to verify that $\lambda_{M}\geq\sqrt{M-1}$, and so
rearranging leads to the following corollary:
###### Corollary 3.1.
Take disjoint closed convex sets $A,B\subseteq\mathbb{R}^{N}$, and let
$w_{\cap}$ denote the Gaussian width of the intersection between the unit
sphere in $\mathbb{R}^{N}$ and the cone generated by the Minkowski difference
$A-B$. Draw an $M\times N$ matrix $P$ with iid $\mathcal{N}(0,1)$ entries.
Then
$\begin{array}[]{lcl}M>\Big{(}w_{\cap}+\sqrt{2\log(1/\eta)}\Big{)}^{2}+1&\Longrightarrow&\operatorname{Pr}\big{(}PA\cap
PB=\emptyset\big{)}\geq 1-\eta.\end{array}$
Now that we have a sufficient condition on $M$, it is natural to wonder how
tight this condition is. Recent work by Amelunxen, Lotz, McCoy and Tropp [2]
shows that the Gordon’s results are incredibly tight. Indeed, by an immediate
application Theorem I and Proposition 10.1 in [2], we achieve the following
characterization of a phase transition for the Rare Eclipse Problem:
###### Corollary 3.2.
Take disjoint closed convex sets $A,B\subseteq\mathbb{R}^{N}$, and let
$w_{\cap}$ denote the Gaussian width of the intersection between the unit
sphere in $\mathbb{R}^{N}$ and the cone generated by the Minkowski difference
$A-B$. Draw an $M\times N$ matrix $P$ with iid $\mathcal{N}(0,1)$ entries.
Then
$\begin{array}[]{lcl}M\geq
w_{\cap}^{2}+\sqrt{16N\log(4/\eta)}+1&\Longrightarrow&\operatorname{Pr}\big{(}PA\cap
PB=\emptyset\big{)}\geq 1-\eta,\\\ M\leq
w_{\cap}^{2}-\sqrt{16N\log(4/\eta)}&\Longrightarrow&\operatorname{Pr}\big{(}PA\cap
PB=\emptyset\big{)}\leq\eta,\end{array}$
Considering the second part of Corollary 3.2, the bound in Corollary 3.1 is
essentially tight. Also, since Corollary 3.2 features an additional $\sqrt{N}$
factor in the error term of the phase transition, the bound in Corollary 3.1
is stronger than the first part of Corollary 3.2 when
$w_{\cap}\ll\sqrt{N}-\sqrt{\log(1/\eta)}$, which corresponds to the regime
where we can compress the most: $M\ll N$.
### 3.1. The case of two balls
Corollaries 3.1 and 3.2 demonstrate the significance of Gaussian width to the
Rare Eclipse Problem. In this subsection, we observe these quantities to solve
the Rare Eclipse Problem in the special case where $A$ and $B$ are balls.
Since each ball has its own parameters (namely, its center and radius), in
this subsection, it is more convenient to write $A=S_{1}$ and $B=S_{2}$. The
following lemma completely characterizes the difference cone $S_{1}-S_{2}$:
###### Lemma 3.3.
For $i=1,2$, take balls $S_{i}:=\\{c_{i}+r_{i}x:x\in\mathcal{B}\\}$, where
$c_{i}\in\mathbb{R}^{N}$, $r_{i}>0$ such that $r_{1}+r_{2}<\|c_{1}-c_{2}\|$,
and $\mathcal{B}$ denotes the ball centered at $0$ of radius $1$. Then the
cone generated by the Minkowski difference $S_{1}-S_{2}$ is the circular cone
$\operatorname{Circ}(\alpha):=\\{z:\langle
z,c_{1}-c_{2}\rangle\geq\|z\|\|c_{1}-c_{2}\|\cos\alpha\\},$
where $\alpha\in(0,\pi/2)$ is the angle such that
$\sin\alpha=(r_{1}+r_{2})/\|c_{1}-c_{2}\|$.
In three dimensions, the fact that the difference cone is circular makes
intuitive sense. The proof of Lemma 3.3 is routine and can be found in the
appendix.
Considering the beginning on this section, it now suffices to bound the
Gaussian width of the circular cone’s intersection with the unit sphere
$\mathbb{S}^{N-1}$. Luckily, this computation is already available as
Proposition 4.3 in [2]:
$\Big{(}w(\operatorname{Circ}(\alpha))\cap\mathbb{S}^{N-1}\Big{)}^{2}=N\sin^{2}\alpha+O(1).$
See Figure 2 for an illustration of the corresponding phase transition. By
Lemma 3.3 (and Corollaries 3.1 and 3.2), this means a random $M\times N$
projection will keep two balls from colliding provided
$M\geq N\bigg{(}\frac{r_{1}+r_{2}}{\|c_{1}-c_{2}\|}\bigg{)}^{2}+O(\sqrt{N}).$
Note that there is a big payoff in the separation $\|c_{1}-c_{2}\|$ between
the balls. Indeed, doubling the separation decreases the required projection
dimension by a factor of $4$.
$0$$\pi/8$$\pi/4$$3\pi/8$$\pi/2$$\alpha$: Angle of
cone$0$$25$$50$$75$$100$$M$: Rank of random projection Figure 2. Phase
transition for a random null space to avoid a circular cone. Fixing the
ambient dimension to be $N=100$, then for each $\alpha=1:\pi/200:\pi/2$ and
$M=1:100$, we randomly drew $100$ $M\times N$ matrices with iid
$\mathcal{N}(0,1)$ entries and plotted the proportion whose null spaces
avoided the circular cone with angle $\alpha$. As expected, if $\alpha$ is
large, then so must $M$ so that the null space is small enough to avoid the
cone. In red, we plot the curve $M=N\sin^{2}\alpha+\cos 2\alpha$, which
captures the phase transition by Theorem I and Proposition 4.3 in [2]. By
Lemma 3.3, the circular cone is precisely the difference cone of two balls,
and so this phase transition solves the Rare Eclipse Problem in this special
case.
### 3.2. The case of two ellipsoids
Now that we have solved the Rare Eclipse Problem for balls, we consider the
slightly more general case of ellipsoids. Actually, this case is somewhat
representative of the general problem with arbitrary convex sets. This can be
seen by appealing to the following result of Paouris [22]:
###### Theorem 3.4 (Concentration of Volume).
There is an absolute constant $c>0$ such that the following holds: Given a
convex set $K\subseteq\mathbb{R}^{N}$, draw a random vector $X$ uniformly from
$K$. Suppose $K$ has the property that $\mathbb{E}[X]=0$ and
$\mathbb{E}[XX^{\top}]=I$. Then
$\operatorname{Pr}\Big{(}\|X\|_{2}>r\Big{)}\leq e^{-cr}\qquad\forall
r\geq\sqrt{N}.$
In words, the above theorem says that the volume of an isotropic convex set is
concentrated in a round ball. The radius of the ball of concentration is
$O(\sqrt{N})$, which corresponds to the fact that
$\mathbb{E}\|X\|_{2}^{2}=\mathbb{E}\operatorname{Tr}[XX^{\top}]=N$. This
result can be modified to describe volume concentration of any convex set
(isotropic or not). To see this, consider any convex set
$K\subseteq\mathbb{R}^{N}$ of full dimension (otherwise the volume is zero).
Then taking $Y$ to be a random vector drawn uniformly from $K$, we define the
centroid $c:=\mathbb{E}[Y]$. Also, since $K$ has full dimension, the inertia
matrix $\mathbb{E}[(Y-c)(Y-c)^{\top}]$ is symmetric and positive definite, and
we can take $A_{0}:=(\mathbb{E}[(Y-c)(Y-c)^{\top}])^{1/2}$. It is
straightforward to verify that $X:=A_{0}^{-1}(Y-c)$ is distributed uniformly
over $K^{\prime}:=A_{0}^{-1}(K-c)$, and that $K^{\prime}$ satisfies the
hypotheses of Theorem 3.4. We claim that $Y$ is concentrated in an ellipsoid
defined by
$S_{r}:=\\{c+rA_{0}x:x\in\mathcal{B}\\}$
for some $r\geq\sqrt{N}$, where $\mathcal{B}$ denotes the ball centered at $0$
of radius $1$. Indeed, Theorem 3.4 gives
$\operatorname{Pr}\Big{(}Y\not\in
S_{r}\Big{)}=\operatorname{Pr}\Big{(}\|A_{0}^{-1}(Y-c)\|_{2}>r\Big{)}\leq
e^{-cr}.$
Overall, the vast majority of any convex set is contained in an ellipsoid
defined by its centroid and inertia matrix, and so two convex sets are nearly
linearly separable if the corresponding ellipsoids are linearly separable. (A
similar argument relates the case of two ellipsoids to a mixture of two
Gaussians.)
Note that any ellipsoid has the following convenient form:
$\\{c+Ax:x\in\mathcal{B}\\},$
where $c\in\mathbb{R}^{N}$ is the center of the ellipsoid, $A$ is some
$N\times N$ symmetric and positive semidefinite matrix, and $\mathcal{B}$
denotes the ball centered at the origin of radius $1$. Intuitively, the
difference cone of any two ellipsoids will not be circular in general, as it
was in the case of two balls. Indeed, the oblong shape of each ellipsoid
(determined by its shape matrix $A$) precludes most of the useful symmetries
in the difference cone, and as such, the analysis of the size of the cone is
more difficult. Still, we established the following upper bound on the
Gaussian width in the general case, which by Corollaries 3.1 and 3.2,
translates to a sufficient number of rows for a random projection to typically
maintain separation:
###### Theorem 3.5.
For $i=1,2$, take ellipsoids $S_{i}:=\\{c_{i}+A_{i}x:x\in\mathcal{B}\\}$,
where $c_{i}\in\mathbb{R}^{N}$, $A_{i}$ is symmetric and positive
semidefinite, and $\mathcal{B}$ denotes the ball centered at $0$ of radius
$1$. Let $w_{\cap}$ denote the Gaussian width of the intersection between the
unit sphere in $\mathbb{R}^{N}$ and the cone generated by the Minkowski
difference $S_{1}-S_{2}$. Then
$w_{\cap}\leq\frac{\|A_{1}\|_{F}+\|A_{2}\|_{F}}{\zeta-\big{(}\|A_{1}e\|_{2}+\|A_{2}e\|_{2}\big{)}}+\frac{1}{\sqrt{2\pi}}$
provided $\zeta>\|A_{1}e\|_{2}+\|A_{2}e\|_{2}$; here, $\zeta:=\|c_{2}-c_{1}\|$
and $e:=(c_{1}-c_{2})/\|c_{1}-c_{2}\|$.
The proof is technical and can be found in the appendix, but the ideas behind
the proof are interesting. There are two main ingredients, the first of which
is the following result:
###### Proposition 3.6 (Proposition 3.6 in [5]).
Let $\mathcal{C}$ be any non-empty convex cone in $\mathbb{R}^{N}$, and let
$g$ be an $N$-dimensional vector with iid $\mathcal{N}(0,1)$ entries. Then
$w(\mathcal{C}\cap\mathbb{S}^{N-1})\leq\mathbb{E}_{g}\Big{[}\|g-\Pi_{\mathcal{C}^{*}}(g)\|_{2}\Big{]},$
where $\Pi_{\mathcal{C}^{*}}$ denotes the Euclidean projection onto the dual
cone $\mathcal{C}^{*}$ of $\mathcal{C}$.
Proposition 3.6 is essentially a statement about convex duality, and while it
provides an upper bound on $w_{\cap}$, in our case, it is difficult to find a
closed form expression for the right-hand side. However, the bound is in terms
of distance to the dual cone, and so any point in this cone provides an upper
bound on this distance. This leads to the second main ingredient in our
analysis: We choose a convenient mapping $\widetilde{\Pi}$ that sends any
vector $g$ to a point in $\mathcal{C}^{*}$ (but not necessarily the closest
point), while at the same time allowing the expectation of
$\|g-\widetilde{\Pi}(g)\|_{2}$ to have a closed form. Since
$\|g-\Pi_{\mathcal{C}^{*}}(g)\|_{2}\leq\|g-\widetilde{\Pi}(g)\|_{2}$ for every
possible instance of $g$, this produces a closed-form upper bound on the bound
in Proposition 3.6.
$c_{1}-c_{2}$$S_{1}$$S_{2}$ | $g$$\widetilde{\Pi}(g)$$e$$\mathcal{C}_{-}$$\mathcal{C}_{-}^{*}$
---|---
$c_{1}-c_{2}$$S_{1}$$S_{2}$ | $g$$e$$\mathcal{C}_{-}$$\mathcal{C}_{-}^{*}$
Figure 3. Two examples of two ellipsoids along with their difference cone and
dual cone. For each pair, on the left, the vector $c_{1}-c_{2}$ is depicted,
as are the extreme difference directions between the ellipsoids—these form the
boundary of the difference cone $\mathcal{C}_{-}$, which is illustrated on the
right along with its dual cone $\mathcal{C}_{-}^{*}$, i.e., the cone of
separating hyperplanes. The vector $e\in\mathcal{C}_{-}$ is a normalized
version of $c_{1}-c_{2}$. In the first example, the pseudoprojection
$\widetilde{\Pi}$ sends any point $g$ to the closest point in the dual cone
$\mathcal{C}_{-}^{*}$ along the line spanned by $e$. Interestingly, in cases
where the ellipsoids are far apart, the cone $\mathcal{C}_{-}$ will be narrow,
and so the boundary of the dual cone will essentially be the orthogonal
complement of $e$. As such, the pseudoprojection is close the true projection
onto the polar cone in this limiting case. For this pseudoprojection to be
well-defined, we require that for every $g$, the line which passes through $g$
in the direction of $e$ hits the dual cone at some point. This is not always
the case, as the second example illustrates. It is straightforward to show
that this pseudoprojection is well-defined if and only if the ellipsoids
remain separated when projecting onto the line spanned by $e$.
Figure 3 illustrates how we chose the pseudoprojection $\widetilde{\Pi}$.
Interestingly, this pseudoprojection behaves more like the true projection
when the ellipsoids are more distant from each other. At the other extreme,
note that Theorem 3.5 does not hold if the ellipsoids are too close, i.e., if
$\|c_{1}-c_{2}\|\leq\|A_{1}e\|_{2}+\|A_{2}e\|_{2}$. This occurs, for example,
if the two ellipsoids collide after projecting onto the span of $e$; indeed,
taking $x$ and $y$ to be unit-norm vectors such that
$e^{\top}(c_{1}+A_{1}x)=e^{\top}(c_{2}+A_{2}y)$, then rearranging gives
$\|c_{1}-c_{2}\|=e^{\top}c_{1}-e^{\top}c_{2}=-e^{\top}A_{1}x+e^{\top}A_{2}y\leq|e^{\top}A_{1}x|+|e^{\top}A_{2}y|\leq\|A_{1}e\|_{2}+\|A_{2}e\|_{2}.$
As Figure 3 illustrates, our pseudoprojection even fails to be well-defined
when the ellipsoids collide after projecting onto the span of $e$. So why
bother using a random projection to maintain linear separability when there is
a rank-$1$ projection available? There are two reasons: First, calculating
this rank-$1$ projection requires access to the centers of the ellipsoids,
which are not available in certain applications (e.g., unsupervised or semi-
supervised learning, or if the projection occurs blindly during the data
collection step). Second, the use of a random projection is useful when
projecting multiple ellipsoids simultaneously to preserve pairwise linear
separability—as we will detail in the next subsection, randomness allows one
to appeal to the union bound in a way that permits several ellipsoids to be
projected simultaneously using particularly few projected dimensions.
At this point, we compare Theorem 3.5 to the better understood case of two
balls. In this case, $A_{1}=r_{1}I$ and $A_{2}=r_{2}I$, and so Theorem 3.5
gives that
$w_{\cap}\leq\sqrt{N}\cdot\frac{r_{1}+r_{2}}{\|c_{1}-c_{2}\|_{2}-(r_{1}+r_{2})}+\frac{1}{\sqrt{2\pi}}.$
If we consider the regime in which
$r_{1}+r_{2}\leq\frac{1}{2}\|c_{1}-c_{2}\|_{2}$, then we recover the case of
two balls to within a factor of $2$, suggesting that the analysis is tight (at
least in this case). For a slightly more general lower bound, note that a
projection maintains separation between two ellipsoids only if it maintains
separation between balls contained in each ellipsoid. The radius of the
largest ball in the $i$th ellipsoid is equal to the smallest eigenvalue
$\lambda_{\textrm{min}}(A_{i})$ of the shape matrix $A_{i}$, and the center of
this ball coincides with the center $c_{i}$ of its parent ellipsoid. As such,
we can again appeal to the case of two balls to see that Theorem 3.5 is
reasonably tight for ellipsoids of reasonably small eccentricity
$\lambda_{\textrm{max}}(A_{i})/\lambda_{\textrm{min}}(A_{i})$. Closed form
bounds for general ellipses with high eccentricity are unwieldy, but Figure 4
illustrates that our bound is far from tight when the ellipsoids are close to
each other. Still, the bound improves considerably as the distance increases.
As such, we leave improvements to Theorem 3.5 as an open problem (in
particular, finding a closed-form characterization of the phase transition in
terms of the $c_{i}$’s and $A_{i}$’s).
$0$$100$$200$$300$$400$$\zeta$: Distance between ellipsoid centers$0$$10$$20$$30$$40$$M$: Rank of random projection | $0$$100$$200$$300$$400$$\zeta$: Distance between ellipsoid centers$0$$10$$20$$30$$40$
---|---
Figure 4. Phase transition for a random projection to keep ellipsoids
separated. (a) Fixing the ambient dimension to be $N=40$, then for each
$\zeta=1:400$ and $M=1:40$, we conducted $10$ trials. For each trial, we
randomly drew $A_{1}$ and $A_{2}$ as iid standard Wishart-distributed $N\times
N$ matrices with $N$ degrees of freedom (i.e., $A_{i}=XX^{\top}$, where $X$ is
$N\times N$ with iid $\mathcal{N}(0,1)$ entries), along with an $M\times N$
matrix $P$ with iid $\mathcal{N}(0,1)$ entries. Plotted is the proportion of
trials for which the ellipsoids are disjoint after applying $P$ (we did not
record whether the ellipsoids were separated before projection). For each of
the 160,000 trials, the shape matrices satisfied
$\zeta\leq\|A_{1}e\|_{2}+\|A_{2}e\|_{2}$, thereby rendering Theorem 3.5
irrelevant. (b) Next, we performed the same experiment, except we changed the
distribution of $A_{1}$ and $A_{2}$ so that $e$ is in the null space of both,
and in the orthogonal complement of $e$, they are iid standard Wishart-
distributed $(N-1)\times(N-1)$ matrices with $N-1$ degrees of freedom. As
such, the corresponding ellipsoids resided in parallel hyperplanes, and
$\|A_{1}e\|_{2}+\|A_{2}e\|_{2}=0$ so that Theorem 3.5 applies. For each trial,
we stored the bound on $w_{\cap}$ from Theorem 3.5 and calculated the sample
average of the squares of these bounds corresponding to each $\zeta=1:400$.
The red curve plots these sample averages (or 40, whichever is smaller)—think
of this as an upper bound on the phase transition. As one might expect, this
bound appears to sharpen as the distance increases.
### 3.3. The case of multiple convex sets
Various classification tasks require one to distinguish between several
different classes, and so one might ask for a random projection to maintain
pairwise linear separability. For a fixed projection dimension $M$, let
$\eta_{ij}$ denote the probability that convex classes $S_{i}$ and $S_{j}$
collide after projection. Then the union bound gives that the probability of
maintaining separation is $\geq 1-\sum_{i,j:i<j}\eta_{ij}$.
This use of the union bound helps to illustrate the freedom which comes with a
random projection. Recall that Theorem 3.5 requires that projecting the
ellipsoids onto the line spanned by the difference $c_{1}-c_{2}$ of their
centers maintains separation. In the case of multiple ellipsoids, one might
then be inclined to project onto the span of $\\{c_{i}-c_{j}\\}_{i,j:i<j}$.
Generically, such a choice of projection puts $M=\binom{K}{2}=\Omega(K^{2})$,
where $K$ is the total number of classes. On the other hand, suppose each
pairwise distance $\|c_{i}-c_{j}\|$ is so large that the $(i,j)$th Gaussian
width satisfies
$w_{\cap}<\sqrt{2\log\bigg{(}\frac{1}{p}\binom{K}{2}\bigg{)}}.$
Then by Corollary 3.1, taking $M=8\log(\binom{K}{2}/p)+1=O_{p}(\log K)$
ensures that classes $S_{i}$ and $S_{j}$ collide after projection with
probability $\eta_{ij}\leq p/\binom{K}{2}$, and so the probability of
maintaining overall separation is $\geq 1-p$. Of course, we will not save so
much in the projection dimension when the convex bodies are closer to each
other, but we certainly expect $M<K^{2}$ in reasonable cases.
At this point, we note the similarity between the performance $M=O(\log K)$
and what the Johnson–Lindenstrauss lemma guarantees when the classes are each
a single point. Indeed, a random projection of $M=\Omega_{\epsilon}(\log K)$
dimensions suffices to ensure that pairwise distances are preserved to within
a factor of $1\pm\epsilon$ with constant probability; this in turn ensures
that pairwise separated points remain pairwise separated after projection. In
fact, the proof technique for the Johnson–Lindenstrauss lemma is similar:
First prove that a random projection typically preserves the norm of any
vector, and then perform a union bound over all $\binom{K}{2}$ difference
vectors. One might be inspired to use Johnson–Lindenstrauss ideas to prove a
result analogous to Theorem 3.5 (this was actually an initial attempt by the
authors). Unfortunately, since Johnson–Lindenstrauss does not account for the
shape matrices $A_{i}$ of the ellipsoids, one is inclined to consider worst-
case orientations, and so terms like $\|A_{i}e\|_{2}$ are replaced by spectral
norms $\|A_{i}\|_{2}$ in the analysis, thereby producing a strictly weaker
result. Dasgupta [6] uses this Johnson–Lindenstrauss approach to project a
mixture of Gaussians while maintaining some notion of separation.
## 4\. Random projection versus principal component analysis
In this section, we compare the performance of random projection and principal
component analysis (PCA) for dimensionality reduction. First, we should
briefly review how to perform PCA. Consider a collection of data points
$\\{x_{i}\\}_{i=1}^{p}\subseteq\mathbb{R}^{N}$, and define the empirical mean
by $\bar{x}:=\frac{1}{p}\sum_{i=1}^{p}x_{i}$. Next, consider the empirical
inertia matrix
$\widehat{\Sigma}:=\frac{1}{p}\sum_{i=1}^{p}(x_{i}-\overline{x})(x_{i}-\bar{x})^{\top}=\frac{1}{p}\sum_{i=1}^{p}x_{i}x_{i}^{\top}-\bar{x}\bar{x}^{\top}.$
The eigenvectors of $\widehat{\Sigma}$ with the largest eigenvalues are
identified as the principal components, and the idea of PCA is to project
$\\{x_{i}\\}_{i=1}^{p}$ onto the span of these components for dimensionality
reduction.
In this section, we will compare random projection with PCA in a couple of
ways. First, we observe some toy examples of data sets that illustrate when
PCA is better, and when random projection is better. Later, we make a
comparison using a real-world hyperspectral data set.
### 4.1. Comparison using toy examples
Here, we consider a couple of extreme data sets which illustrate when PCA
outperforms random projection and vice versa. Our overarching model for the
data sets will be the following: Given a collection of disjoint balls
$\\{S_{i}\\}_{i=1}^{K}$ in $\mathbb{R}^{N}$, we independently draw $p$ data
points uniformly from $S:=\bigcup_{i=1}^{K}S_{i}$. When $p$ is large, we can
expect $\widehat{\Sigma}$ to be very close to
$\Sigma:=\frac{1}{\operatorname{vol}(S)}\sum_{i=1}^{K}\int_{S_{i}}xx^{\top}dx-\mu\mu^{\top}$
by the law of large numbers; here, $\mu\in\mathbb{R}^{N}$ denotes the mean of
the distribution. Recall that the projection dimension for PCA is the number
of large eigenvalues of $\widehat{\Sigma}$. Since the operator spectrum is a
continuous function of the operator, we can count large eigenvalues of
$\Sigma$ to estimate this projection dimension. The following lemma will be
useful to this end:
###### Lemma 4.1.
Consider a ball of the form $S:=c+r\mathcal{B}$, where
$\mathcal{B}\subseteq\mathbb{R}^{N}$ denotes the ball centered at $0$ of
radius $1$. Define the operator
$W:=\int_{S}xx^{\top}dx.$
Then the span of $c$ and its orthogonal complement form the eigenspaces of $W$
with eigenvalues
$\lambda_{c}=r^{N}\|c\|^{2}\operatorname{vol}(\mathcal{B})+Cr^{N+2},\qquad\lambda_{c^{\perp}}=Cr^{N+2},$
respectively, where $C$ is some constant depending on $N$.
###### Proof.
Pick any vector $v\in\mathbb{R}^{N}$ of unit norm. Then
$v^{\top}Wv=\int_{\mathcal{B}}v^{\top}(c+ry)(c+ry)^{\top}vr^{N}dy=(v^{\top}c)^{2}\cdot
r^{N}\operatorname{vol}(\mathcal{B})+r^{N+2}v^{\top}\bigg{(}\int_{\mathcal{B}}yy^{\top}dy\bigg{)}v.$
Notice that the operator $\int_{\mathcal{B}}yy^{\top}dy$ is invariant under
conjugation by any rotation matrix. As such, this operator is a constant $C$
multiple of the identity operator. Thus, $v^{\top}Wv$ is maximized at
$\lambda_{c}$ when $v$ is a normalized version of $c$, and minimized at
$\lambda_{c^{\perp}}$ whenever $v$ is orthogonal to $c$. ∎
We start by considering the case where $S$ is composed of two balls, namely
$S_{1}:=c+r\mathcal{B}$ and $S_{2}:=-c+r\mathcal{B}$. As far as random
projection is concerned, in this case, we are very familiar with the required
projection dimension: $\Omega_{\eta}(Nr^{2}/\|c\|^{2})$. In particular, as
$\|c\|$ approaches $r$, a random projection cannot provide much dimensionality
reduction. To compare with PCA, note that in this case, $\Sigma$ is a scalar
multiple of $W_{1}+W_{2}$, where
$W_{i}:=\int_{S_{i}}xx^{\top}dx.$
Moreover, it is easy to show that $W_{1}=W_{2}$. By Lemma 4.1, the dominant
eigenvector of $W_{i}$ is $c$, and so PCA would suggest to project onto the
one-dimensional subspace spanned by $c$. Indeed, this projection always
preserves separation, and so in this case, PCA provides a remarkable savings
in projection dimension.
Now consider the case where $S$ is composed of $2N$ balls
$\\{S_{n,1}\\}_{n=1}^{N}\cup\\{S_{n,2}\\}_{n=1}^{N}$ defined by
$S_{n,1}:=e_{n}+r\mathcal{B}$ and $S_{n,2}:=-e_{n}+r\mathcal{B}$, where
$e_{n}$ denotes the $n$th identity basis element. Then $\Sigma$ is a scalar
multiple of $\sum_{n=1}^{N}(W_{n,1}+W_{n,2})$, where
$W_{n,i}:=\int_{S_{n,i}}xx^{\top}dx.$
Recall that $W_{n,1}=W_{n,2}$. Then $\Sigma$ is simply a scalar multiple of
$\sum_{n=1}^{N}W_{n,1}$. By Lemma 4.1, the $W_{n,1}$’s are all diagonal, and
their diagonals are translates of each other. As such, their sum (and
therefore $\Sigma$) is a scalar multiple of the identity matrix—in this case,
PCA would choose to not project down to fewer dimensions. On the other hand,
if we take
$M>\left(\sqrt{N\bigg{(}\frac{2r}{\sqrt{2}}\bigg{)}^{2}+1}+\sqrt{2\log\bigg{(}\frac{1}{p}\binom{2N}{2}\bigg{)}}\right)^{2}+1,$
then by Corollary 3.1, a random projection maintains separation between any
fixed pair of balls from $\\{S_{n,1}\\}_{n=1}^{N}\cup\\{S_{n,2}\\}_{n=1}^{N}$
with probability $\geq 1-p/\binom{2N}{2}$, and so by the union bound, the
balls are pairwise separated with probability $\geq 1-p$. In particular, if
$r=O(N^{-1/2})$, then we can take $M=O_{p}(\log N)$.
Overall, random projection performs poorly when the classes are close, but
when there are multiple sufficiently separated classes, you can expect a
dramatic dimensionality reduction. As for PCA, we have constructed a toy
example for which PCA performs well (the case of two balls), but in general,
the performance of PCA seems difficult to describe theoretically. Whereas the
performance of random projection can be expressed in terms of “local”
conditions (e.g., pairwise separation), as the last example illustrates, the
performance of PCA can be dictated by more “global” conditions (e.g., the
geometric configuration of classes). In the absence of theoretical guarantees
for PCA, the following subsection provides simulations with real-world
hyperspectral data to illustrate its performance compared to random
projection.
### 4.2. Simulations with hyperspectral data
One specific application of dimensionality reduction is the classification of
hyperspectral data. For this application, the idea is to distinguish materials
by observing them across hundreds of spectral bands (like the red, green and
blue bands that the human eye detects). Each pixel of a hyperspectral image
can be viewed as a vector of spectral information, capturing how much light of
various frequencies is being reradiated from that portion of the scene. A
hyperspectral image is naturally represented as a data cube with two spatial
indices and one spectral index, and a common task is to identify the material
observed at each pair of spatial indices. To do this, one might apply per-
pixel classification, in which a classifier simply identifies the material in
a given pixel from its spectral content, ignoring any spatial context. Since
the spectral information is high-dimensional, it is natural to attempt
dimensionality reduction before classification. A popular choice for this task
is PCA [17, 27], and in this subsection, we provide some preliminary
simulations to compare its performance with random projection.
All experiments described in this subsection were conducted using the Indian
Pines hyperspectral data set [15]. This data set consists of a hyperspectral
image with $145\times 145$ pixels, each containing spectral reflectance data
represented by a vector of length $N=200$. Each pixel corresponds to a
particular type of vegetation or crop, such as corn or wheat, with a total of
$17$ different classes (see Figure 5 for an illustration).
Figure 5. The Indian Pines hyperspectral data set [15]. Each pixel
corresponds to a different type of vegetation or crop. The ground truth image
of labels is depicted on the left, and a sample band of the data set is
displayed on the right.
For our simulations, the task will consist of using the known labels of a
training set (a small subset of the $21,025=145\times 145$ pixels) to make
accurate predictions for the remainder of the pixels. To keep the simulations
fast, each simulation considers a small patch of pixels. More precisely, given
a patch of $p$ pixels and a prescribed training ratio $r$, we pick a random
subset of the pixels of size $rp$ to be the training set. We use the labels
from this training set to train a classifier that will then attempt to guess
the label of each of the other $(1-r)p$ pixels from the location of its
spectral reflectance in $200$-dimensional space. The classifier we use is
MATLAB’s built-in implementation of multinomial logistic regression.
Performance is measured by classification error and runtime.
Given this setting, for different values of projection dimension $M$, we draw
an $M\times N$ matrix $P$ with iid $\mathcal{N}(0,1)$ entries and replace
every spectral reflectance data point $x$ by $Px$. In the degenerate case
$M=N$, we simply take $P$ to be the identity matrix. For comparison, we also
use principal component analysis (PCA) for dimensionality reduction, which
will interrogate the training set to identify $M$ principal components before
projecting the data set onto the span of these components. An immediate
advantage of random projection is that it allows the sensing mechanism to
blindly compress the data, as it does not need a training set to determine the
compression function.
Figure 6 uses different patches of the Indian Pines dataset and different
training ratios to compare both the classification accuracy and runtime of
multinomial logistic regression when applied to various projections of the
data set. The first experiment focuses on a small patch of $225$ pixels, and
the second considers a patch of $3481$ pixels. These experiments reveal a few
interesting phenomena. First of all, dimensionality reduction leads to
impressive speedups in runtime. Perhaps more surprising is the fact that there
seems to be an improvement in classification performance after projecting the
data. We are far from completely understanding this behavior, but we suspect
it has to do with regularization and overfitting.
Figure 6. The performance of classification by multinomial logistic
regression after projecting onto subspaces of various dimensions $M$. Depicted
are two particular patches of the entire Indian Pines data set—the top uses a
patch of $225$ pixels, while the bottom uses a patch of $3481$ pixels. In each
case, the first two plots in the first row depict the ground truth labels in
the patch, as well as the random training set we selected. The third plot
compares, for different values of projection dimension $M$, the classification
error incurred with random projection and with principal component analysis.
The fourth plot shows the runtime (in seconds) for different values of $M$.
The second and third rows depict the classification outcomes when using random
projection and PCA, respectively. One can see that dimensionality reduction
not only speeds up the algorithm, but also improves the classification
performance by discouraging overfitting.
It is also interesting how similar random projection and PCA perform. Note
that the PCA method has an unfair advantage since it is data-adaptive, meaning
that it uses the training data to select the projection, and in practical
applications for which the sensing process is expensive, one might be
interested in projecting in a non-adaptive way, thereby allowing for less
sensing. Our simulations suggest that the expense is unnecessary, as a random
projection will provide almost identical performance. As indicated in the
previous subsection, random projection is also better understood as a means to
maintain linear separability, and so there seems to be little benefit in
choosing PCA over random projection (at least for this sort of classification
task).
## 5\. Future work
One of the main points of this paper is that random projections can maintain
separation between sufficiently separated sets, and this is useful for
classification in the projected domain. Given the mindset of compressed
sensing, it is impressive that the sets need not be low-dimensional to enjoy
separation in the projected domain. What this suggests is a more general
notion of simplicity that is at play, of which low-dimensionality and
sufficient separation are mere instances. Obviously, understanding this
general notion is a worthy subject of future work.
From a more applied perspective, it would be worth investigating alternative
notions of distortion. Indeed, linear separability is the best-case scenario
for classification, but it is not at all necessary. After identifying any
worthy notion of distortion, one might study how much distortion is incurred
by random projection, and hopefully some of the ideas contained in this paper
will help.
One of our main results (Theorem 3.5) provides a sufficient number of rows for
a random projection to maintain separation between ellipsoids. However, as
illustrated in Figure 4, this bound is far from optimal. Considering this case
of two ellipsoids is somewhat representative of the more general case of two
convex sets (as we identified using Theorem 3.4), improvements to Theorem 3.5
would be rather interesting. In particular, it would be nice to characterize
the phase transition in terms of the ellipsoids’ parameters, as we already
have in the case of two balls.
Finally, the random projections we consider here all have iid
$\mathcal{N}(0,1)$ entries, but real-world sensing systems may not enjoy this
sort of flexibility. As such, it would be interesting to extend the results of
this paper to more general classes of random projections, in particular,
random projections which can be implemented with a hyperspectral imager (say).
## 6\. Appendix: Proofs
### 6.1. Proof of Gordon’s Escape Through a Mesh Theorem
This proof is chiefly based on the following result, which appears as
Corollary 1.2 in [12]:
###### Gordon’s Comparison Theorem.
Let $S$ be a closed subset of $\mathbb{S}^{n-1}$. Draw an $M\times N$ matrix
$P$ with iid $\mathcal{N}(0,1)$ entries. Then
$\mathbb{E}\bigg{[}\min_{x\in S}\|Px\|_{2}\bigg{]}\geq\lambda_{M}-w(S),$
where $\lambda_{M}:=\mathbb{E}\|g\|_{2}$, and $g$ is a random $M$-dimensional
vector with iid $\mathcal{N}(0,1)$ entries.
To prove the escape theorem, consider the function
$f_{S}\colon P\mapsto\min_{x\in S}\|Px\|_{2}.$
Gordon’s Comparison Theorem gives that
$\mathbb{E}[f_{S}]\geq\lambda_{M}-w(S)$, and so
$\displaystyle\operatorname{Pr}\Big{(}Y\cap S=\emptyset\Big{)}$
$\displaystyle=\operatorname{Pr}\Big{(}\min_{x\in S}\|Px\|_{2}>0\Big{)}$
$\displaystyle=\operatorname{Pr}\Big{(}\min_{x\in
S}\|Px\|_{2}>\big{(}\lambda_{M}-w(S)\big{)}-\big{(}\lambda_{M}-w(S)\big{)}\Big{)}$
(1) $\displaystyle\geq\operatorname{Pr}\Big{(}\min_{x\in
S}\|Px\|_{2}>\mathbb{E}[f_{S}]-\big{(}\lambda_{M}-w(S)\big{)}\Big{)}.$
Next, we note that $f_{S}$ is Lipschitz with respect to the Frobenius norm
with constant $1$, and so we can appeal to (1.6) of [20] to get
(2) $\operatorname{Pr}\Big{(}f_{S}(P)>\mathbb{E}[f_{S}]-t\Big{)}\geq
1-e^{-t^{2}/2}\qquad\forall t>0.$
Taking $t=\lambda_{M}-w(S)$ and applying (2) to (1) then gives the result.
### 6.2. Proof of Lemma 3.3
Let $\mathcal{C}_{-}$ denote the cone generated by the Minkowski difference
$S_{1}-S_{2}$. We will show $\mathcal{C}_{-}=\operatorname{Circ}(\alpha)$ by
verifying both containments.
We begin by finding the smallest $\alpha\in[0,\pi/2]$ for which
$\mathcal{C}_{-}\subseteq\operatorname{Circ}(\alpha)$. By the definition of
$\operatorname{Circ}(\alpha)$, this containment is equivalent to
(3) $\cos\alpha\leq\inf_{z\in\mathcal{C}_{-}}\frac{\langle
z,c_{1}-c_{2}\rangle}{\|z\|\|c_{1}-c_{2}\|}=\min_{z\in
S_{1}-S_{2}}\frac{\langle z,c_{1}-c_{2}\rangle}{\|z\|\|c_{1}-c_{2}\|}.$
To find the smallest such $\alpha$, we solve this optimization problem. Taking
$d:=c_{1}-c_{2}$, then $S_{1}-S_{2}=(r_{1}+r_{2})\mathcal{B}+d$, and so we
seek to
$\mbox{minimize}\quad f(y)=\frac{\langle
y+d,d\rangle}{\|y+d\|\|d\|}\quad\mbox{subject to}\quad\|y\|\leq r_{1}+r_{2}.$
Quickly note that the objective function is well defined over the feasibility
region due to the assumption $r_{1}+r_{2}<\|d\|$. We first claim that $f(y)$
is minimized on the boundary, i.e., where $\|y\|=r_{1}+r_{2}$. To see this,
suppose $\|y\|<r_{1}+r_{2}$, and letting $P_{d^{\perp}}$ denote the orthogonal
projection onto the orthogonal complement of the span of $d$, take $t>0$ such
that $\|y+tP_{d^{\perp}}y\|=r_{1}+r_{2}$. Then $y+tP_{d^{\perp}}y$ lies on the
boundary and
$f(y+tP_{d^{\perp}}y)=\frac{\langle
y+tP_{d^{\perp}}y+d,d\rangle}{\|y+tP_{d^{\perp}}y+d\|\|d\|}=\frac{\langle
y+d,d\rangle}{\|y+tP_{d^{\perp}}y+d\|\|d\|}<\frac{\langle
y+d,d\rangle}{\|y+d\|\|d\|}=f(y).$
As such, it suffices to minimize subject to $\|y\|=r_{1}+r_{2}$. At this
point, the theory of Lagrange multipliers can be applied since the equality
constraint $g(y):=\|y\|^{2}=(r_{1}+r_{2})^{2}$ is a level set of a function
whose gradient $\nabla g(y)=2y$ does not vanish on the level set. Thus, the
minimizers of $f$ with $g(y)=(r_{1}+r_{2})^{2}$ satisfy $\nabla
f(y)=-\lambda\nabla g(y)=-2\lambda y$ for some Lagrange multiplier
$\lambda\in\mathbb{R}$.
To continue, we calculate $\nabla f(y)$. It is actually easier to calculate
the gradient of $h(u):=\langle u,d\rangle/\|u\|\|d\|$:
$\nabla
h(u)=\frac{1}{\|u\|^{2}}\bigg{(}d-\bigg{\langle}\frac{u}{\|u\|},d\bigg{\rangle}\frac{u}{\|u\|}\bigg{)}.$
Note that $\nabla h(u)=0$ only if $u$ is a nontrivial multiple of $d$, i.e.,
only if $u$ maximizes $h$ (by Cauchy–Schwarz). Also, it is easy to verify that
$\langle u,\nabla h(u)\rangle=0$. Overall, changing variables $u\leftarrow
y+d$ gives that any minimizer $y^{\natural}$ of $f$ subject to
$\|y\|=r_{1}+r_{2}$ satisfies
(4) $\displaystyle\nabla f(y^{\natural})$ $\displaystyle=-2\lambda
y^{\natural}\qquad\mbox{for some }\lambda\in\mathbb{R},$ (5)
$\displaystyle\nabla f(y^{\natural})$ $\displaystyle\neq 0,$ (6)
$\displaystyle\langle y^{\natural}+d,\nabla f(y^{\natural})\rangle$
$\displaystyle=0.$
At this point, (4) and (5) together imply that $\nabla f(y^{\natural})$ is a
nontrivial multiple of $y^{\natural}$, and so combining with (6) gives
$\langle y^{\natural}+d,y^{\natural}\rangle=0.$
As such, $0$, $d$ and $y^{\natural}+d$ form vertices of a right triangle with
hypotenuse $\|d\|$, and the smallest $\alpha$ satisfying (3) is the angle
between $d$ and $y^{\natural}+d$. Thus,
$\sin\alpha=\|y^{\natural}\|/\|d\|=(r_{1}+r_{2})/\|c_{1}-c_{2}\|$.
It remains to prove the reverse containment,
$\operatorname{Circ}(\alpha)\subseteq\mathcal{C}_{-}$, for this particular
choice of $\alpha$. Define
$G:=\\{z:\langle z,d\rangle=\|z\|\|d\|\cos\alpha,\leavevmode\nobreak\
\|z\|=\|y^{\natural}+d\|\\}.$
Then $\operatorname{Circ}(\alpha)$ is the cone generated by $G$, and so it
suffices to show that $G\subseteq S_{1}-S_{2}=(r_{1}+r_{2})\mathcal{B}+d$. To
this end, pick any $z\in G$, and consider the triangle with vertices $0$, $d$
and $z$. By definition, the angle between $d$ and $z$ is $\alpha$, and the
side $z$ has length $\|y^{\natural}+d\|$. As such, by the side-angle-side
postulate, this triangle is congruent to the triangle with vertices at $0$,
$d$ and $y^{\natural}+d$. This implies that the side between $z$ and $d$ has
length $\|z-d\|=\|y^{\natural}\|=r_{1}+r_{2}$, and so
$z=(z-d)+d\in(r_{1}+r_{2})\mathcal{B}+d$, as desired.
### 6.3. Proof of Theorem 3.5
This proof makes use of the following lemma:
###### Lemma 6.1.
Take an $n\times n$ matrix $A$ and let $g$ have iid $\mathcal{N}(0,1)$
entries. Then
$\sqrt{\frac{2}{\pi}}\|A\|_{F}\leq\mathbb{E}\|Ag\|_{2}\leq\|A\|_{F}.$
###### Proof.
Let $A=UDV$ be the singular value decomposition of $A$. Since the Gaussian is
isotropic, $\mathbb{E}\|Ag\|_{2}=\mathbb{E}\|Dg\|_{2}$, and since the function
$x\mapsto x^{2}$ is convex, Jensen’s inequality gives
$\mathbb{E}\|Dg\|_{2}\leq\sqrt{\mathbb{E}\|Dg\|_{2}^{2}}=\sqrt{\sum_{i=1}^{n}D_{ii}^{2}\mathbb{E}g_{i}^{2}}=\|D\|_{F}=\|A\|_{F}.$
Similarly, since $x\mapsto\|x\|_{2}$ is convex, we can also use Jensen’s
inequality to get
$\mathbb{E}\|Dg\|_{2}=\mathbb{E}\sqrt{\sum_{i=1}^{n}D_{ii}^{2}g_{i}^{2}}\geq\sqrt{\sum_{i=1}^{n}\left(\mathbb{E}|D_{ii}g_{i}|\right)^{2}}=\mathbb{E}|g_{1}|\sqrt{\sum_{i=1}^{n}D_{ii}^{2}}=\sqrt{\frac{2}{\pi}}\|A\|_{F},$
which completes the proof. ∎
To prove Theorem 3.5, let $\mathcal{C}_{-}$ denote the cone generated by the
Minkowski difference $S_{1}-S_{2}$. We will exploit Proposition 3.6, which
gives the following estimate in terms of the polar cone
$\mathcal{C}_{-}^{*}:=\\{w:\langle w,z\rangle\leq 0\leavevmode\nobreak\
\forall z\in\mathcal{C}_{-}\\}$:
$w_{\cap}\leq\mathbb{E}_{g}\Big{[}\|g-\Pi_{\mathcal{C}_{-}^{*}}(g)\|_{2}\Big{]},$
where $g$ has iid $\mathcal{N}(0,1)$ entries and $\Pi_{\mathcal{C}_{-}^{*}}$
denotes the Euclidean projection onto $\mathcal{C}_{-}^{*}$. Instead of
directly computing the distance between $g$ and its projection onto
$\mathcal{C}_{-}^{*}$, we will construct a mapping $\widetilde{\Pi}$ which
sends $g$ to some member of $\mathcal{C}_{-}^{*}$, but for which distances are
easier compute; indeed $\|g-\widetilde{\Pi}(g)\|_{2}$ will be an upper bound
on $\|g-\Pi_{\mathcal{C}_{-}^{*}}(g)\|_{2}$. Consider the polar decomposition
$c_{2}-c_{1}=\zeta e$, where $\zeta>0$. Then we can decompose
$g=g_{1}e+g_{2}$, and we define $\widetilde{\Pi}(g)$ to be the point in
$\mathcal{C}_{-}^{*}$ of the form $\alpha e+g_{2}$ which is closest to $g$.
With this definition, we have
$\|g-\Pi_{\mathcal{C}_{-}^{*}}(g)\|_{2}\leq\|g-\widetilde{\Pi}(g)\|_{2}=\min|g_{1}-\alpha|\quad\mbox{s.t.}\quad\alpha
e+g_{2}\in\mathcal{C}_{-}^{*}.$
To simplify this constraint, we find a convenient representation of the polar
cone:
$\displaystyle\mathcal{C}_{-}^{*}$ $\displaystyle=\\{w:\langle w,z\rangle\leq
0\leavevmode\nobreak\ \forall z\in\mathcal{C}_{-}\\}$
$\displaystyle=\\{w:\langle w,u-v\rangle\leq 0\leavevmode\nobreak\ \forall
u\in S_{1},v\in S_{2}\\}$ $\displaystyle=\\{w:\langle
w,c_{2}-c_{1}\rangle\geq\langle w,A_{1}x\rangle-\langle
w,A_{2}y\rangle\leavevmode\nobreak\ \forall x,y\in\mathcal{B}\\}$
$\displaystyle=\Big{\\{}w:\langle
w,c_{2}-c_{1}\rangle\geq\max_{x\in\mathcal{B}}\langle
w,A_{1}x\rangle+\max_{y\in\mathcal{B}}\langle w,-A_{2}y\rangle\Big{\\}}$
$\displaystyle=\Big{\\{}w:\langle
w,c_{2}-c_{1}\rangle\geq\max_{x\in\mathcal{B}}\langle
A_{1}^{\top}w,x\rangle+\max_{y\in\mathcal{B}}\langle-
A_{2}^{\top}w,y\rangle\Big{\\}}$ $\displaystyle=\\{w:\langle
w,c_{2}-c_{1}\rangle\geq\|A_{1}w\|_{2}+\|A_{2}w\|_{2}\\},$
where the last step uses the fact that each $A_{i}$ is symmetric. The
constraint $\alpha e+g_{2}\in\mathcal{C}_{-}^{*}$ is then equivalent to
$\alpha\zeta\geq\|A_{1}(\alpha e+g_{2})\|_{2}+\|A_{2}(\alpha e+g_{2})\|_{2}.$
At this point, we focus on the case in which the projection $e^{\top}S_{1}$ is
disjoint from $e^{\top}S_{2}$. In this case, we have the following strict
inequality:
$\max_{x\in\mathcal{B}}\langle c_{1}+A_{1}x,e\rangle=\max_{u\in S_{1}}\langle
u,e\rangle<\min_{v\in S_{2}}\langle v,e\rangle=\min_{y\in\mathcal{B}}\langle
c_{2}+A_{2}y,e\rangle,$
and rearranging then gives
$\displaystyle\zeta=\langle c_{2}-c_{1},e\rangle$
$\displaystyle>\max_{x\in\mathcal{B}}\langle
A_{1}x,e\rangle+\max_{y\in\mathcal{B}}\langle-A_{2}x,e\rangle$
$\displaystyle=\max_{x\in\mathcal{B}}\langle
x,A_{1}^{\top}e\rangle+\max_{y\in\mathcal{B}}\langle
x,-A_{2}^{\top}e\rangle=\|A_{1}e\|_{2}+\|A_{2}e\|_{2}.$
As such, taking
(7)
$\alpha\geq\alpha^{*}:=\frac{\|A_{1}g_{2}\|_{2}+\|A_{2}g_{2}\|_{2}}{\zeta-\big{(}\|A_{1}e\|_{2}+\|A_{2}e\|_{2}\big{)}}$
produces a point $\alpha e+g_{2}\in\mathcal{C}_{-}^{*}$, considering
$\alpha\zeta\geq\alpha\big{(}\|A_{1}e\|_{2}+\|A_{2}e\|_{2}\big{)}+\|A_{1}g_{2}\|_{2}+\|A_{2}g_{2}\|_{2}\geq\|A_{1}(\alpha
e+g_{2})\|_{2}+\|A_{2}(\alpha e+g_{2})\|_{2},$
where the last step follows from the triangle inequality. Note that if
$g_{1}\geq\alpha^{*}$, then we can take $\alpha=g_{1}$ to get
$\|g-\widetilde{\Pi}(g)\|_{2}=0$. Otherwise,
$\|g-\widetilde{\Pi}(g)\|_{2}\leq|g_{1}-\alpha^{*}|=\alpha^{*}-g_{1}$.
Overall, we have
$\|g-\Pi_{\mathcal{C}_{-}^{*}}(g)\|_{2}\leq\|g-\widetilde{\Pi}(g)\|_{2}\leq(\alpha^{*}-g_{1})_{+}.$
By the monotonicity of expectation, we then have
(8)
$w_{\cap}\leq\mathbb{E}_{g}\Big{[}\|g-\Pi_{\mathcal{C}_{-}^{*}}(g)\|_{2}\Big{]}\leq\mathbb{E}_{g}(\alpha^{*}-g_{1})_{+}=\mathbb{E}_{g_{2}}\Big{[}\mathbb{E}_{g_{1}}\Big{[}(\alpha^{*}-g_{1})_{+}\Big{|}g_{2}\Big{]}\Big{]}.$
To estimate the right-hand side, we first have
(9)
$\mathbb{E}_{g_{1}}\Big{[}(\alpha^{*}-g_{1})_{+}\Big{|}g_{2}\Big{]}=\int_{-\infty}^{\infty}(\alpha^{*}-z)_{+}d\Phi(z)=\alpha^{*}\Phi(\alpha^{*})+\frac{1}{\sqrt{2\pi}}e^{-(\alpha^{*})^{2}/2},$
which lies between $\alpha^{*}/2$ and $\alpha^{*}+1/\sqrt{2\pi}$ since
$\alpha\geq 0$.
Let $P_{e^{\perp}}$ denote the $n\times n$ orthogonal projection onto the
orthogonal complement of the span of $e$. Appealing to Lemma 6.1 with
$A:=A_{i}P_{e^{\perp}}$ then gives
$\mathbb{E}\|A_{i}g_{2}\|_{2}=\mathbb{E}\|A_{i}P_{e^{\perp}}g\|_{2}\leq\|A_{i}P_{e^{\perp}}\|_{F}\leq\|A_{i}\|_{F},$
where the last inequality follows from the fact that each row of
$A_{i}P_{e^{\perp}}$ is a projection of the corresponding row in $A_{i}$, and
therefore has a smaller $2$-norm. Considering (7), this implies
$\mathbb{E}_{g_{2}}\alpha^{*}\leq\frac{\|A_{1}\|_{F}+\|A_{2}\|_{F}}{\zeta-\big{(}\|A_{1}e\|_{2}+\|A_{2}e\|_{2}\big{)}},$
which combined with (8) and (9) then gives
$w_{\cap}\leq\mathbb{E}_{g_{2}}\Big{[}\mathbb{E}_{g_{1}}\Big{[}(\alpha^{*}-g_{1})_{+}\Big{|}g_{2}\Big{]}\Big{]}\leq\frac{\|A_{1}\|_{F}+\|A_{2}\|_{F}}{\zeta-\big{(}\|A_{1}e\|_{2}+\|A_{2}e\|_{2}\big{)}}+\frac{1}{\sqrt{2\pi}}.$
### Acknowledgments
The authors thank Matthew Fickus and Katya Scheinberg for insightful
discussions. A. S. Bandeira was supported by AFOSR award FA9550-12-1-0317, D.
G. Mixon was supported by NSF award DMS-1321779, and B. Recht was supported by
ONR award N00014-11-1-0723 and NSF awards CCF-1139953 and CCF-11482. The views
expressed in this article are those of the authors and do not reflect the
official policy or position of the United States Air Force, Department of
Defense, or the U.S. Government.
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|
arxiv-papers
| 2014-04-11T19:49:05 |
2024-09-04T02:50:01.059634
|
{
"license": "Public Domain",
"authors": "Afonso S. Bandeira and Dustin G. Mixon and Benjamin Recht",
"submitter": "Afonso S. Bandeira",
"url": "https://arxiv.org/abs/1404.3203"
}
|
1404.3234
|
# Equivalence of optimal $L^{1}$-inequalities on Riemannian Manifolds 1112010
Mathematics Subject Classification: 35A09, 35B44 222Key words: sharp Sobolev
inequalities, best constant, extremal maps
Jurandir Ceccon 333E-mail addresses: [email protected] (J. Ceccon)
Departamento de Matemática, Universidade Federal do Paraná,
Caixa Postal 019081, 81531-990, Curitiba, PR, Brazil
Leandro Cioletti 444E-mail addresses: [email protected] (L. Cioletti)
Departamento de Matemática, Universidade de Brasília,
UnB, 70910-900, Brasília, Brazil
Abstract
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$.
This paper concerns to the validity of the optimal Riemannian $L^{1}$-Entropy
inequality
${\bf Ent}_{dv_{g}}(u)\leq n\log\left(A_{opt}\|Du\|_{BV(M)}+B_{opt}\right)$
for all $u\in BV(M)$ with $\|u\|_{L^{1}(M)}=1$ and existence of extremal
functions. In particular, we prove that this optimal inequality is equivalent
a optimal $L^{1}$-Sobolev inequality obtained by Druet [7].
## 1 Introduction
The isoperimetric problem on the Euclidean space $\mathbb{R}^{n}$ consist in
finding among all the domains with a given fixed volume one that has the
lowest surface area. The solution in this case is a sphere. This property is
precisely expressed in terms of the Isoperimetric inequality for domains in
$\mathbb{R}^{n}$, that is, if $\Omega$ is a domain with volume $|\Omega|$ and
surface area $|\partial\Omega|$ then
$\frac{|\partial\Omega|}{|\Omega|^{\frac{n-1}{n}}}\geq\frac{1}{K(n,1)}\;,$ (1)
where $K(n,1)^{-1}=n^{(1-1/n)}(\omega_{n-1})^{\frac{1}{n}}$ and $\omega_{n-1}$
denotes the volume of the unit ball in the Euclidean space $\mathbb{R}^{n-1}$.
The equality is attained iff $\Omega$ is a sphere. Observe that the last
statement implies that $K(n,1)$ is the best constant for the inequality (1).
The Isoperimetric inequality shows up on other branchs of mathematics. For
example, it has been used to prove the non-uniqueness of the Gibbs measures of
the Ising model on the lattice
$\mathbb{Z}^{n}\equiv\mathbb{Z}\times\ldots\times\mathbb{Z}$. In this context,
the inequality is generalized as follows. We consider $\mathbb{Z}^{n}$ as a
metric space, where the distance between $x,y\in\mathbb{Z}^{n}$ is defined in
terms of the $\ell_{1}$ norm. We also look at $\mathbb{Z}^{n}$ as a graph
$(\mathbb{Z}^{n},\mathbb{E}^{n})$, where $\mathbb{Z}^{n}$ is the vertex set
and
$\mathbb{E}^{n}\equiv\\{\\{x,y\\}\in\mathbb{Z}^{n}\times\mathbb{Z}^{n};|x-y|_{1}=1\\}$
is the edge set. The discrete Isoperimetric inequality says that for any fixed
integer $n\geq 2$ and any finite subset $\Omega\subset\mathbb{Z}^{n}$, we have
that
$\frac{|\partial\Omega|}{|\Omega|^{\frac{n-1}{n}}}\geq\frac{1}{2n},$
where $\partial\Omega=\\{\\{i,j\\}\in\
\mathbb{E}^{d}:i\in\Omega,j\in\Omega^{c}\\}$ and $|\Omega|$ and
$|\partial\Omega|$ denotes the cardinality of $\Omega$ and $\partial\Omega$,
respectively. Note that $2n$ is exactly the volume of the unit sphere on the
$\ell_{1}$ norm. It is worth pointing out that the proof of this discrete
inequality is similar on spirit of our proof on the continuous setting and is
based on the entropy inequalities. It is possible that the equivalences
obtained here can be extended to the discrete case but we will not develop
this point here. More information about this connection can be found in [16].
For more details about the Isoperimetric inequality see the excellent
Osserman’s work [15] and references therein.
Now we consider a more analytic context. We say that a function $u\in
L^{1}(\mathbb{R}^{n})$ has bounded variation if
$\|Du\|_{BV(\mathbb{R}^{n})}=\sup\left\\{\int_{\mathbb{R}^{n}}u\,\mbox{div}(\varphi)\,dx;\,\varphi\in
C_{0}^{1}(\mathbb{R}^{n},\mathbb{R}^{n}),|\varphi|\leq 1\right\\}<\infty\;.$
The space of all bounded variation functions is denoted by
$BV(\mathbb{R}^{n})$. The optimal Euclidean $L^{1}$-Sobolev inequality in
$BV(\mathbb{R}^{n})$ states that for all $u\in BV(\mathbb{R}^{n})$ we have
$\|u\|_{L^{1^{*}}(\mathbb{R}^{n})}\leq K_{0}\|Du\|_{BV(\mathbb{R}^{n})}\;,$
(2)
where $1^{*}=\frac{n}{n-1}$ is the critical Sobolev exponent and
$K_{0}^{-1}=\inf\\{\|Du\|_{BV(\mathbb{R}^{n})};u\in
BV(\mathbb{R}^{n}),\|u\|_{L^{1^{*}}(\mathbb{R}^{n})}=1\\}\;$
is the best constant for this inequality. This inequality it was studied by
Federer and Fleming [9], Fleming and Rishel [10] and Maz′ja [14]. In this
case, the characteristic functions of the balls are extremal functions for the
optimal $L^{1}$-Sobolev inequality and explicit value for best constant is
given by
$K_{0}=K(n,1)\;.$
This inequalities gains in interest if we realize that the geometric
inequality (1) and the analytic inequality (2) are equivalents. This relation
was pointed out independently by Federer and Fleming [9] and Maz′ja [14].
Let’s move on to the manifold setting.
Let $(M,g)$ be a smooth compact Riemannian manifold without boundary of
dimension $n\geq 2$. We say that $u\in L^{1}(M)$ is a bounded variation
function if
$\|D_{g}u\|_{BV}=\sup\left\\{\int_{M}u\,\mbox{div}_{g}(X)\,dv_{g};X\in\Gamma(TM),|X|_{g}(x)\leq
1\hskip 5.69046pt\mbox{for all}\hskip 5.69046ptx\in M\right\\}<\infty\;,$
where $\Gamma(TM)$ is the set of all vector fields over $M$ with divergent in
$n$-fold Cartesian product $C^{1}(M)\times\cdots\times C^{1}(M)$. We denote by
$BV(M)$ the space of bounded variation functions. The optimal Riemannian
$L^{1}$-Sobolev inequality was obtained by Druet [7]. He proved that for all
$u\in BV(M)$ the following inequality holds
$\|u\|_{L^{1^{*}}(M)}\leq K(n,1)\|D_{g}u\|_{BV(M)}+B(1)\|u\|_{L^{1}(M)}\;,$
(3)
where $K(n,1)$ is the first best constant and $B(1)$ is the second best
constant for the optimal Riemannian $L^{1}$-Sobolev inequality.
Moreover, Druet also proved that the only extremal functions for (3) are the
characteristic function of some $\Omega_{0}\in\Sigma$, where
$\Sigma=\\{\Omega\subset M;\chi_{\Omega}\in BV(M)\\}\;.$ Finally, Druet
observed that for $\Omega\in\Sigma$, the optimal inequality (3) is equivalent
to the following isoperimetric inequality
$|\Omega|^{\frac{n-1}{n}}\leq K(n,1)|\partial\Omega|+B(1)|\Omega|,$ (4)
where $|\Omega|$ and $|\partial\Omega|$ are the Riemannian volume and area of
$\Omega$ and $\partial\Omega$, respectively and $B(1)$ is the best constant in
this inequality. For more details about this equivalence and additional
references see Druet [7].
The aim of this work is to show that these inequalities are also equivalent to
the Entropy and Gagliardo-Nirenberg inequalities on both context Euclidean and
Riemannian, and determine the equivalence among its extremal functions.
## 2 Euclidean inequality
For any function $u\in BV(\mathbb{R}^{n})$ we have that its entropy, with
respect to the Lebesgue measure, is well defined and given by the following
expression
${\bf Ent}_{dx}(u)=\int_{\mathbb{R}^{n}}|u|\log|u|dx\;.$
The optimal Euclidean entropy inequality states that for all $u\in
BV(\mathbb{R}^{n})$ with $\|u\|_{L^{1}(\mathbb{R}^{n})}=1$,
${\bf Ent}_{dx}(u)\leq n\log\left(L(n,1)\|\nabla
u\|_{BV(\mathbb{R}^{n})}\right)\;,$ (5)
where $L(n,1)$ is the best constant for this inequality. Note that the
existence of this best constant is guaranteed by the same argument we use in
(2). This inequality it was studied by Beckner [5] and Ledoux [13]. For the
optimal entropy inequality, Beckner showed that the extremal functions for (5)
are the characteristic functions of the Euclidean balls
$B(x_{0},r)=\\{x\in\mathbb{R}^{n};\|x-x_{0}\|<r\\}$, with $|B(x_{0},r)|=1$. So
in this case, it is easy to check that
$K(n,1)=L(n,1)\;.$ (6)
In addition, if $\Omega$ is a domain in $\mathbb{R}^{n}$ and $\lambda>0$ is
chosen so that
$\lambda\int_{M}\chi_{\Omega}dx=1$
then by using $\lambda\chi_{\Omega}$ in (5), we obtain the Isoperimetric
inequality (1).
Now we show that the entropy inequality can be obtained by the Gagliardo-
Nirenberg inequality. Let $1\leq q<r<1^{*}$. By using the interpolation
inequality and the optimal Euclidean $L^{1}$-Sobolev inequality, we obtain the
Euclidean Gagliardo-Nirenberg inequality, which says that for any function
$u\in BV(\mathbb{R}^{n})$ we have
$\|u\|_{L^{r}(\mathbb{R}^{n})}^{\frac{1}{\theta}}\leq
K(n,1)\|Du\|_{BV(\mathbb{R}^{n})}\|u\|_{L^{q}(\mathbb{R}^{n})}^{\frac{1-\theta}{\theta}}\,,$
(7)
where $\theta=\frac{n(r-q)}{r(q(1-n)+n)}\in(0,1)$. Let
$A(n,q,r)^{-1}=\inf\left\\{\|Du\|_{BV(\mathbb{R}^{n})}\|u\|_{L^{q}(\mathbb{R}^{n})}^{\frac{1-\theta}{\theta}};u\in
BV(\mathbb{R}^{n}),\|u\|_{L^{r}(\mathbb{R}^{n})}=1\right\\}$
be the best possible constant for this inequality. If $\chi_{B(0,r)}$ denotes
the characteristic function of the Euclidean ball of radius $r>0$, then we
have that $\chi_{B(0,r)}\in BV(\mathbb{R}^{n})$ and an easy computation shows
that
$\|\chi_{B(0,r)}\|_{L^{r}(\mathbb{R}^{n})}^{\frac{1}{\theta}}=K(n,1)\|D\chi_{B(0,r)}\|_{BV(\mathbb{R}^{n})}\cdot\|\chi_{B(0,r)}\|_{L^{q}(\mathbb{R}^{n})}^{\frac{1-\theta}{\theta}}\;.$
Note that the above equality actually proved that
$A(n,q,r)=K(n,1)\;,$ (8)
for all $1\leq q<r<1^{*}$. Therefore the inequality (7) is in fact the optimal
Euclidean $L^{1}$-Gagliardo-Nirenberg inequality and characteristic function
of the balls are extremal functions for (7).
Proceeding, with minor modifications, as in [4] (see also Section 3) we can
verify that the optimal Euclidean $L^{1}$-Gagliardo-Nirenberg inequality
implies the optimal Euclidean $L^{1}$-entropy inequality (5). Piecing together
these information we conclude that (1) $\Rightarrow$ (2) $\Rightarrow$ (7)
$\Rightarrow$ (5) $\Rightarrow$ (1).
Before proceed, we remark that the equality (8) is the key point in Section 3
to prove the main result of this paper.
## 3 The Riemannian inequality and the main result
Using the interpolation inequality and the optimal Riemannian $L^{1}$-Sobolev
inequality we get for any function $u\in BV(M)$, $1\leq q<r<1^{*}$ that
$\|u\|_{L^{r}(M)}^{\frac{1}{\theta}}\leq\left(K(n,1)\|D_{g}u\|_{BV(M)}+B(1)\|u\|_{L^{1}(M)}\right)\|u\|_{L^{q}(M)}^{\frac{1-\theta}{\theta}}\,,$
($I_{q,r}(K(n,1),B(1))$)
where $\theta=\frac{n(r-q)}{r(q(1-n)+n)}$ is the interpolation parameter. The
first Riemannian $L^{1}$-Gagliardo-Nirenberg best constant is defined by
$A_{opt}=\inf\\{A\in\mathbb{R}:\;\mbox{there exists}\hskip
5.12128ptB\in\mathbb{R}\hskip 5.12128pt\mbox{such that}\hskip
5.12128ptI_{q,r}(A,B)\hskip 5.12128pt\mbox{is valid}\\}\,.$
Using a partition unity and a similar argument as in [8] together with (8), we
can verify that
${\cal A}_{opt}=A(n,q,r)=K(n,1)\;,$
for all $1\leq q<r<1^{*}$. So this equality shows that the first optimal
Riemannian $L^{1}$-Gagliardo-Nirenberg inequality ($I_{q,r}(K(n,1),B(1))$) is
valid for all $u\in BV(M)$ and $K(n,1)$ is the first best constant.
It follows from [7] that every extremal function for the Sobolev inequality is
of the form $u_{0}=\lambda\chi_{\Omega_{0}}\;,$ for some
$\lambda\in\mathbb{R}$ and $\Omega_{0}\in\Sigma$. We see at once that for such
functions the equality in $I_{q,r}(K(n,1),B(1))$ is verified. Consequently the
second Riemannian $L^{1}$-Gagliardo-Nirenberg best constant is given by
$B(1)=\inf\\{B\in\mathbb{R};I_{q,r}(K(n,1),B)\hskip 5.69046pt\mbox{is
valid}\\}\;.$
So we have that optimal Riemannian $L^{1}$-Gagliardo-Nirenberg inequality
$\|u\|_{L^{r}(M)}^{\frac{1}{\theta}}\leq\left(K(n,1)\|D_{g}u\|_{BV(M)}+B(1)\|u\|_{L^{1}(M)}\right)\|u\|_{L^{q}(M)}^{\frac{1-\theta}{\theta}}\,,$
(9)
is valid for all $u\in BV(M)$ and $1\leq q<r<1^{*}$. Notice that the Sobolev
extremal functions are also Gagliardo-Nirenberg extremal functions.
In the sequel we show how to use the inequality (9) (with $q=1$) to obtain the
optimal Riemannian $L^{1}$-entropy inequality. The proof is based on the
Bakry, Coulhon, Ledoux and Sallof-Coste argument given in [4]. Consider the
optimal Riemannian $L^{1}$-Gagliardo-Nirenberg inequality
$\|u\|_{L^{r}(M)}^{\frac{1}{\theta}}\left(\leq
K(n,1)\|D_{g}u\|_{BV(M)}+B(1)\|u\|_{L^{1}(M)}\right)\|u\|_{L^{1}(M)}^{\frac{1-\theta}{\theta}}\;,$
for all $u\in BV(M)$. By taking the logarithm on both sides above and use the
definition of $\theta$, we get
$\frac{r(1-n+n)}{n}\frac{1}{r-1}\log\left(\frac{\|u\|_{L^{r}(M)}}{\|u\|_{L^{1}(M)}}\right)\leq\log\left(K(n,1)\frac{\|D_{g}u\|_{BV(M)}}{\|u\|_{L^{1}(M)}}+B(1)\right).$
Taking the limit when $r\to 1^{+}$, on the above expression, we obtain
$\frac{1}{n}\lim_{r\rightarrow
1^{+}}\frac{1}{r-1}\log\left(\frac{\|u\|_{L^{r}(\mathbb{R}^{n})}}{\|u\|_{L^{1}(\mathbb{R}^{n})}}\right)\leq\log\left(K(n,1)\frac{\|D_{g}u\|_{L(M)}}{\|u\|_{L^{1}(M)}}+B(1)\right)\;.$
To evaluate the remainder limit, we first observe that
$\displaystyle\log\left(\frac{\|u\|_{L^{r}(M)}}{\|u\|_{L^{1}(M)}}\right)$
$\displaystyle=\frac{1}{r}\log(\|u\|_{L^{r}(M)}^{r})-\log(\|u\|_{L^{1}(M)})$
$\displaystyle=\frac{1-r}{r}\log(\|u\|_{L^{1}(M)})+\frac{1}{r}\left(\log(\|u\|_{L^{r}(M)}^{r})-\log(\|u\|_{L^{1}(M)})\right).$
Next, we apply two times the mean value theorem, obtaining
$\lim_{r\rightarrow
1^{+}}\frac{1}{r-1}\log\left(\frac{\|u\|_{L^{r}(M)}}{\|u\|_{L^{1}(M)}}\right)=\int_{M}\frac{|u|}{\|u\|_{L^{1}(M)}}\log\left(\frac{|u|}{\|u\|_{L^{1}(M)}}\right)dx.$
From the above equation it follows that
$\int_{M}|u|\log(|u|)dv_{g}\leq
n\log\left(K(n,1)\|D_{g}u\|_{BV(M)}+B(1)\right)\ ,$
for all $u\in BV(M)$ with $\|u\|_{L^{1}(M)}=1$. As in the previous section we
define
${\bf Ent}_{dv_{g}}(u)=\int_{M}|u|\log|u|dv_{g}\;.$
As a consequence of the previous inequality we have that
${\bf Ent}_{dv_{g}}(u)\leq n\log\left(L(n,1)\|D_{g}u\|_{BV(M)}+B(1)\right)\;,$
($Ent(L(n,1),B(1)$)
for all $u\in BV(M)$ with $\|u\|_{L^{1}(M)}=1$. We shall remember that
$K(n,1)=L(n,1)$. Now we consider the optimal Riemannian $L^{1}$-entropy
inequality
${\bf Ent}_{dv_{g}}(u)\leq n\log\left(L_{opt}\|D_{g}u\|_{BV(M)}+B\right)$
where $u\in BV(M)$ with $\|u\|_{L^{1}(M)}=1$, $B\in\mathbb{R}$ and the first
Riemannian $L^{1}$-entropy best constant is defined by
$L_{opt}=\inf\\{A\in\mathbb{R}:\;\mbox{there exists}\hskip
5.12128ptB\in\mathbb{R}\hskip 5.12128pt\mbox{such that}\hskip
5.12128ptEnt(A,B)\hskip 5.12128pt\mbox{is valid}\\}\,.$
From the definition of best constant and the validity of $Ent(L(n,1),B(1))$
follows that $L_{opt}\leq L(n,1)$. On the other hand, the equality between
these two constants requires a proof.
Assuming that $Ent(L_{opt},B)$ is valid for some $B\in\mathbb{R}$, one can
define the second Riemannian $L^{1}$-Entropy best constant by
$B_{opt}=\inf\\{B\in\mathbb{R};Ent(L_{opt},B)\hskip 5.69046pt\mbox{is
valid}\\}\;.$
Our main result states that $Ent(L_{opt},B_{opt})$ is valid, moreover the
Riemannian first best constant is equals to the Euclidean first best constant.
We also prove that the second best constant for Sobolev and entropy
inequalities are the same.
###### Theorem 1.
Let $(M,g)$ be a smooth compact Riemannian manifold without boundary of
dimension $n\geq 2$. Then $Ent(L_{opt},B(1))$ is valid. In addition,
$L_{opt}=L(n,1)$ and $B_{opt}=B(1)$.
Remark. Together with the results in [7] the Theorem 1 can be used to complete
the equivalence of the four $L^{1}$ optimal inequalities considered on this
work.
Indeed, in [7] we have that isoperimetric $\Rightarrow$ Sobolev. In the
Section 3 we shown that Sobolev $\Rightarrow$ Gagliardo-Nirenberg and
Gagliardo-Nirenberg + Theorem 1 $\Rightarrow$ entropy. Finally, by taking
$\Omega\in\Sigma$ and $\lambda>0$ such that
$\int_{M}\lambda\;\chi_{\Omega}dv_{g}=1\;,$
and replacing $\lambda\;\chi_{\Omega}$ in $Ent(L(n,1),B(1))$ we get the
optimal isoperimetric inequality.
So the four $L^{1}$-optimal inequalities considered here are all equivalent
and we have that (4) $\Rightarrow$ (3) $\Rightarrow$ ($I_{q,r}(K(n,1),B(1))$)
$\Rightarrow$ ($Ent(L(n,1),B(1)$) $\Rightarrow$ (4) .
Proof of Theorem 1: We proceed to show that
$L_{opt}=L(n,1)\;.$
From this equality it may be concluded that $Ent(L(n,1),B(1))$ is the best
$L^{1}$-entropy. To prove the above equality, we use the definition of
$L_{opt}$. That is, by definition, for all $s>0$ there exists $B(s)$ such that
$\int_{M}|u|\log(|u|)dv_{g}\leq
n\log\left((L_{opt}+s)\|D_{g}u\|_{BV(M)}+B(s)\right)\ ,$
for all $u\in BV(M)$ with $\|u\|_{L^{1}(M)}=1$. This is equivalent to the
following inequality
$\frac{1}{\|u\|_{L^{1}(M)}}\int_{M}|u|\log(|u|)dv_{g}+(n-1)\log(\|u\|_{L^{1}(M)})\leq
n\log\left((L_{opt}+s)\|D_{g}u\|_{BV(M)}+B(s)\int_{M}|u|dv_{g}\right)$ (10)
for all $u\in BV(M)$. Let $r>0$ be such that
$\int_{B(0,r)}dx=1\;,$
where $B(0,r)=\\{x\in\mathbb{R}^{n};\|x\|<r\\}$ is the Euclidean ball. For
$\varepsilon>0$, we define
$\chi_{\varepsilon}=(r\varepsilon)^{-n}\chi_{B_{g}(0,\varepsilon)}$, where
$\chi_{B_{g}(0,\varepsilon)}$ is the characteristic function of the Riemannian
ball $B_{g}(0,\varepsilon)=\\{x\in M;d_{g}(x_{0},x)\leq\varepsilon\\}$ and
$x_{0}\in M$. In the normal coordinates in around $x_{0}$, we have
$\int_{M}\chi_{\varepsilon}dv_{g}=(\varepsilon
r)^{-n}\int_{B(x_{0},\varepsilon)}dv_{g}=\int_{B(0,r)}dv_{g_{\varepsilon}}$
and $dv_{g_{\varepsilon}}=\sqrt{(g_{ij}(\exp_{x_{0}}(\varepsilon
x)))}dx\rightarrow dx$ when $\varepsilon\rightarrow 0$, where $g_{ij}$ are the
coefficients of the metric $g$ in the normal coordinates. We also have
$\|D_{g}\chi_{\varepsilon}\|_{BV(M)}=(\varepsilon r)^{-n}\int_{\partial
B(x_{0},\varepsilon)}dv_{g}=(\varepsilon r)^{-1}\int_{\partial
B(0,r)}dv_{g_{\varepsilon}}\;.$
By replacing this identity in (10) we get
$-n\log(\varepsilon
r)+(n-1)\log|B(0,r)|_{g_{\varepsilon}}\leq-n\log(\varepsilon
r)+n\log\left((L_{opt}+s)|\partial B(0,r)|_{g_{\varepsilon}}+\varepsilon
rB(s)|B(0,r)|_{g_{\varepsilon}}\right)\;.$
Because $|B(0,r)|_{\xi}=1$ ($\xi$ denotes the Euclidean metric) and by taking
the limits when $\varepsilon\rightarrow 0$ and $s\rightarrow 0$ in this order,
we obtain
$0\leq n\log(L_{opt}|\partial B(0,r)|_{\xi})\;.$
By the choice of $r$ and remembering again that the characteristic function of
the Euclidean ball $B(0,r)$ is the extremal function in (5), we have now
reached the following identity
$0=n\log(L(n,1)|\partial B(0,r)|_{\xi})\;.$
From where we conclude that $L(n,1)\leq L_{opt}$, which proves the desired
equality. Thus ($Ent(L(n,1),B(1)$) is the optimal Riemannian $L^{1}$-entropy.
Now we compute the second best constant. Consider $\Omega_{0}\in\Sigma$ and
$k>0$, such that
$\int_{M}k\chi_{\Omega_{0}}dv_{g}=1\;,$
such that, $\chi_{\Omega_{0}}$ is the extremal function in the optimal Sobolev
inequality (2). We can immediately check that the optimal inequality
$Ent(L(n,1),B(1))$ becomes an equality when we evaluate in the function
$k\chi_{\Omega_{0}}$. Therefore $B(1)=B_{opt}$.
## 4 Equivalence between extremal functions
As we already observed, the extremal functions for the optimal $L^{1}$-Sobolev
inequality (3) are characteristic functions. We also remarked that these
functions are also extremal for the optimal $L^{1}$-Gagliardo-Nirenberg
inequality ($I_{q,r}(K(n,1),B(1))$). We can also prove, using the limit
process employed in the Section 3, that the extremal functions for the optimal
$L^{1}$-Gagliardo-Nirenberg ($I_{q,r}(K(n,1),B(1))$) are the extremal
functions for the optimal $L^{1}$-Entropy inequality ($Ent(L(n,1),B(1)$).
If we prove that the extremal functions for the optimal $L^{1}$-Entropy
inequality ($Ent(L(n,1),B(1)$) are the extremal functions for the optimal
$L^{1}$-Sobolev inequality (3), we have that the set of the extremal functions
for the four optimal $L^{1}$ inequalities considered here are the same.
Claim. If $u_{0}$ is an extremal function for ($Ent(L(n,1),B(1)$)
$\Longrightarrow$ $u_{0}$ is an extremal function for (2). In fact, from the
Jensen Inequality for any $u_{0}$ such that $\|u_{0}\|_{L^{1}(M)}=1$ we get
$\log\int_{M}|u_{0}|^{1^{*}}dv_{g}=\log\int_{M}|u_{0}|^{1^{*}-1}|u_{0}|dv_{g}\geq\int_{M}\log(|u_{0}|^{\frac{1}{n-1}})|u_{0}|dv_{g}=\frac{1}{n-1}\int_{M}\log(|u_{0}|)|u_{0}|dv_{g}.$
By using ($Ent(L(n,1),B(1)$) it follows that
$\log\int_{M}|u|^{1^{*}}dv_{g}\geq
1^{*}\log\left(K(n,1)\|Du_{0}\|_{BV(M)}+B(1)\right),$
that is,
$\|u_{0}\|_{L^{1^{*}}(M)}\geq K(n,1)\|Du_{0}\|_{BV(M)}+B(1).$
This inequality shows that $u_{0}$ is an extremal function for (3).
Acknowledgments. The first author was partially supported by CAPES through
INCTmat and second author is supported by FEMAT.
## References
* [1]
* [2] R. Adams - General logarithmic Sobolev inequalities and Orlicz embedding, J. Funct. Anal. 34, 292-303 (1979).
* [3] T. Aubin - Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11, 573-598 (1976).
* [4] D. Bakry, T. Coulhon, M. Ledoux, L. Sallof-Coste, Sobolev inequalities in disguise, Indiana J. Math., 44 (4), 1033-1074 (1995).
* [5] W. Beckner - Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math. 11, 105-137 (1999).
* [6] E. Carlen - Superadditivity of Fisher’s information and logarithmic Sobolev inequalities, J. Funct. Anal. 101, 194-211 (1991).
* [7] O. Druet - Isoperimetric Inequalities on Compact Manifolds, Geom. Dedicata 90, 217-236 (2002).
* [8] O. Druet, E. Hebey, M. Vaugon - Optimal Nash’s inequalities on Riemannian manifolds: the influence of geometry, Int. Math. Res. Not. 14, 735-779 (1999).
* [9] H. Federer, W. H. Fleming - Normal and integral currents, Ann. of Math. 72, 458-520 (1960).
* [10] W. H. Fleming, R. Rishel - An integral formula for total gradient variation, Arch. Math. 11, 218-222 (1960).
* [11] L. Gross - Logarithmic Sobolev inequalities, Amer. J. Math. 97, 1061-1083 (1975).
* [12] E. Hebey, M. Vaugon - Meilleures constantes dans le théorème d’inclusion de Sobolev, Ann. Inst. H. Poincaré. 13, 57-93 (1996).
* [13] M. Ledoux - Isoperimetry and Gaussian analysis, Lectures on Probability Theory and Statistics (Saint-Flour, 1994), Lecture Notes in Mathematics, v. 1648, Springer, Berlin, 165-294, (1996).
* [14] V. G. Maz′ja - Classes of domains and imbedding theorems for function spaces, Soviet Math. Dokl. 1, 882-885 (1960).
* [15] R. Osserman - The isoperimetric inequality, Bull. Amer. Math. Soc. 84, 1182-1238 (1978).
* [16] L. Saloff-Coste Sobolev inequalities in familiar and unfamiliar settings, Sobolev spaces in mathematics. I, Springer 299–343 (2009).
* [17] F. Weissler - Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. Amer. Math. Soc. 237, 255-269 (1978).
|
arxiv-papers
| 2014-04-11T21:57:23 |
2024-09-04T02:50:01.073196
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jurandir Ceccon and Leandro Cioletti",
"submitter": "Leandro Cioletti",
"url": "https://arxiv.org/abs/1404.3234"
}
|
1404.3296
|
Large Even Number Represent The Sum Of Odd Primes
Jin Li
College of Mathematics,Sichuan University,Chengdu, PRC
Abstract
In this paper, I proved that
$N=p_{1}+p_{2}+2p_{3},p_{1}\sim N/2,p_{2}\sim N/2,p_{3}=o(N),$
where $N$ is a large even number, and $p_{i}\ (i=1,2,3)$ are odd primes.
## 1 Introduction
In 1930, Shnirel’ man proved that every integer greater than one is the sum of
a bounded number of primes. This is the first significant result on the
Goldbach conjecture. In 1937, I. M. Vinogradov proved that every large odd
integer is the sum of three primes. Now I prove an analogue result on even
number by Vinogradov’s method.
## 2 Notation
$p_{i},p$– prime number.
$\varepsilon$– any sufficient small positive constant.
$N$– a sufficiently large even number.
$a,q,r,n$– positive integers.
$\alpha,\beta,t$–real variables.
$c,c_{i}$– some positive constant.
$\lambda$– a suitably choosed positive number.
$A=N\cdot e^{-\varepsilon\sqrt{\log N}},Q=\log^{\lambda}N,\ \ \ \
\tau=A^{2}N^{-1}Q^{-1}.$
$\varphi(q)$– the Euler function
$\mu(q)$– the M$\ddot{\rm o}$bius function
$e(x)={\rm exp}\\{2\pi ix\\}.$
$C_{q}(m)$–the Ramanujan sum
$\sum\limits^{q}_{a=1,(a,q)=1}e\left(\frac{ma}{q}\right).$
$f=O(g)$ or $f\ll g$ or $g\gg f$, if there exists a constant $c>0$ s.t.
$|f(x)|\leq cg(x)$ for all $x$ in the domain of $f$. We write $f=o(g)$, if
$\lim\limits_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0$.
## 3 Main result and proof
### 3.1 Main theorem
Theorem 1 There exists an arithmetic function $D(N)$ and positive constant
$c_{1},c_{2}$ and $c$ such that $c_{1}<D(N)<c_{2}$ for all sufficiently large
even integers $N$, and
$R(N,A)=2D(N)A^{2}+O\left(\frac{A^{2}}{\log^{c}N}\right),$
where $R(N,A)=\sum\limits_{\tiny{\begin{array}[]{l}p_{1}+p_{2}+2p_{3}=N,\\\
\frac{N}{2}-A<p_{1},p_{2}\leq\frac{N}{2}+A\\\ 2<p_{3}\leq A\end{array}}}\log
p_{1}\cdot\log p_{2}\cdot\log p_{3}$
### 3.2 The singular series
We begin by studying the arithmetic functions
$\displaystyle
D(N)=:\sum\limits^{\infty}_{q=1,q{\rm\,is\,odd}}\frac{\mu(q)C_{q}(-N)}{\varphi^{3}(q)}-\sum\limits^{\infty}_{q=1,q{\rm\,is\,even}}\frac{\mu(q)C_{q}(-N)}{\varphi^{3}(q)},$
$\displaystyle
G(N)=:\sum\limits^{\infty}_{q=1}\frac{\mu(q)C_{q}(-N)}{\varphi^{3}(q)}.$
The functions $D(N),G(N)$ are called the singular series.
Theorem 2 The singular series $D(N),G(N)$ converge absolutely and uniformly in
$N$ and have the Euler product
$\displaystyle
D(N)=2\prod\limits_{p|N,p{\rm\,is\,odd}}\left(1-\frac{1}{(p-1)^{2}}\right)\cdot\prod\limits_{p\nmid
N}\left(1+\frac{1}{(p-1)^{3}}\right).$ $\displaystyle
G(N)=\prod\limits_{p|N}\left(1-\frac{1}{(p-1)^{2}}\right)\prod\limits_{p\nmid
N}\left(1+\frac{1}{(p-1)^{3}}\right)$
Moreover, for any $q>0$.
$\displaystyle D(N,Q)$ $\displaystyle=$
$\displaystyle:\sum\limits^{Q}_{q=1,q{\rm\,is\,odd}}\frac{\mu(q)C_{q}(-N)}{\varphi^{3}(q)}-\sum\limits^{Q}_{q=1,q{\rm\,is\,oven}}\frac{\mu(q)C_{q}(-N)}{\varphi^{3}(q)}$
$\displaystyle=$ $\displaystyle
D(N)+O\left(\frac{1}{Q^{1-\varepsilon}}\right).$
Proof Clearly,
$|C_{q}(-N)|\leq\varphi(q),$
and so
$\left|\frac{\mu(q)C_{q}(-N)}{\varphi^{3}(q)}\right|\leq\frac{1}{\varphi^{2}(q)}\ll\frac{\log^{2}\log
q}{q^{2}}.$
Thus the singular series converges absolutely and uniformly in $N$. Moreover,
$D(N)-D(N,Q)\ll\sum\limits_{q>Q}\frac{1}{\varphi^{2}(q)}\ll\sum\limits_{q>Q}\frac{1}{q^{2-\varepsilon}}\ll\frac{1}{Q^{1-\varepsilon}}.$
For $C_{q}(-N)$ is a multiplicative function of $q$ and
$C_{p}(-N)=\left\\{\begin{array}[]{ll}p-1,&p|N\\\ -1,&p\nmid
N\end{array}\right.$, By Euler product
$\displaystyle G(N)$ $\displaystyle=$
$\displaystyle\prod\limits_{p}\left(1+\sum\limits^{\infty}_{j=1}\frac{\mu(p^{j})C_{p^{j}}(-N)}{\varphi^{3}(p^{j})}\right)$
$\displaystyle=$
$\displaystyle\prod\limits_{p}\left(1-\frac{C_{p}(-N)}{\varphi^{3}(p)}\right)=\prod\limits_{p|N}\left(1-\frac{1}{(p-1)^{2}}\right)\cdot\prod\limits_{p\nmid
N}\left(1+\frac{1}{(p-1)^{3}}\right),$
since $N$ is even integer $p=2|N$, then $G(N)=0$, and
$D(N)=2\sum\limits^{\infty}_{q=1,q{\rm\,is\,odd}}\frac{\mu(q)C_{q}(-N)}{\varphi^{3}(q)}=2\prod\limits_{p|N,p{\rm\,is\,odd}}\left(1-\frac{1}{(p-1)^{2}}\right)\cdot\prod\limits_{p\nmid
N}\left(1+\frac{1}{(p-1)^{3}}\right).$
So there exist positive constant $c_{1},c_{2}$ such that $c_{1}<D(N)<c_{2}$.
### 3.3 Decomposition into major and minor arcs.
For $1\leq q\leq Q,0\leq a\leq q-1$ and $(a,q)=1$ the Major arc
$\mathcal{M}(q,a)$ is the interval consisting of all real numbers
$\alpha\in\left[-\frac{1}{\tau},1-\frac{1}{\tau}\right]$ such that
$\left|\alpha-\frac{a}{q}\right|\leq\frac{1}{\tau}.$
The set of major arcs is
$\mathcal{M}=\cup^{Q}_{q=1}\cup^{q-1}_{a=0,(a,q)=1}\mathcal{M}(q,a)\subseteq\left[-\frac{1}{\tau},1-\frac{1}{\tau}\right],$
and the set of minor arcs is
$m=\left[-\frac{1}{\tau},1-\frac{1}{\tau}\right]\backslash\mathcal{M}.$
Let
$F(\alpha,A)=\sum\limits_{\frac{N}{2}-A<p\leq\frac{N}{2}+A}(\log
p)e(\alpha\cdot p),$ $\tilde{F}(\alpha,A)=\sum\limits_{2<p\leq A}(\log
p)e(\alpha\cdot 2p).$
By the circle method, we can express $R(N,A)$ as the integal of a
trigonometric polynomial over the major and minor arcs.
$\displaystyle R(N,A)$ $\displaystyle=$
$\displaystyle\int^{1-\frac{1}{\tau}}_{-\frac{1}{\tau}}F^{2}(\alpha,A)\tilde{F}(\alpha,A)e(-N\alpha)d\alpha$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}F^{2}(\alpha,A)\tilde{F}(\alpha,A)e(-N\alpha)d\alpha+$
$\displaystyle\int_{m}F^{2}(\alpha,A)\tilde{F}(\alpha,A)e(-N\alpha)d\alpha.$
### 3.4 The integal over the major arcs
Theorem 3 Let
$u(\beta,A)=\sum\limits_{\frac{N}{2}-A<n\leq\frac{N}{2}+A}e(\beta
n),\tilde{u}(\beta,A)=\sum\limits_{2<n\leq A}e(\beta\cdot 2n).$
Then
$\displaystyle J(N,A)$ $\displaystyle=$
$\displaystyle\int^{1-\frac{1}{\tau}}_{-\frac{1}{\tau}}u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\beta)d\beta$
$\displaystyle=$ $\displaystyle 2A^{2}+O(A).$
Proof By circle method,
$\displaystyle J(N,A)$ $\displaystyle=$
$\displaystyle\sum\limits_{\tiny{\begin{array}[]{l}N=n_{1}+n_{2}+2n_{3}\\\
\frac{N}{2}-A<n_{1},n_{2}\leq\frac{N}{2}+A\\\ 2<n_{3}\leq A\end{array}}}1$
$\displaystyle=$
$\displaystyle\sum\limits_{\frac{N}{2}-A<n_{1}\leq\frac{N}{2}}\cdot\sum\limits_{\tiny\begin{array}[]{l}n_{2}+2n_{3}=N-n_{1}\\\
2<n_{3}\leq
A\end{array}}1+\sum\limits_{\frac{N}{2}<n_{1}\leq\frac{N}{2}+A}\cdot\sum\limits_{\tiny\begin{array}[]{l}n_{2}+2n_{3}=N-n_{1}\\\
2<n_{3}\leq A\end{array}}1$ $\displaystyle=$ $\displaystyle\sum\limits_{0\leq
d_{1}<A}\sum\limits_{\tiny\begin{array}[]{l}d_{2}+2n_{3}=d_{1}\\\ -A<d_{2}\leq
A\\\ 2<n_{3}\leq A\end{array}}1+\sum\limits_{0<d_{1}\leq
A}\sum\limits_{\tiny\begin{array}[]{l}d_{2}+2n_{3}=-d_{1}\\\ -A<d_{2}\leq A\\\
2<n_{3}\leq A\end{array}}1$ $\displaystyle=$ $\displaystyle\sum\limits_{0\leq
d_{1}<A}(A+d_{1})+\sum\limits_{0<d_{1}\leq A}(A-d_{3})+O(A)$ $\displaystyle=$
$\displaystyle 2A^{2}+O(A).$
Theorem 4(Siegel-Walfisz)
If $1\leq q\leq Q=\log^{\lambda}N$ and $(q,r)=1$, then there exists a constant
$c_{3}$ depends only on $\lambda$ such that.
$\theta(t;q,r)=:\sum\limits_{\tiny\begin{array}[]{l}p\leq t\\\ p\equiv r({\rm
mod}\ q)\end{array}}\log p=\frac{t}{\varphi(q)}+r(t),\ \ \ \ t\geq 2.$
where $r(t)\ll t\cdot e^{-c_{3}\sqrt{\log t}}$.
Lemma 5 If $\alpha\in\mathcal{M}(q,a)$ and $\beta=\alpha-\frac{a}{q}$, then
$F(\alpha,A)=\frac{\mu(q)}{\varphi(q)}\sum\limits_{\frac{N}{2}-A<n\leq\frac{N}{2}+A}e(\beta
n)+O(A\cdot e^{-c_{4}\sqrt{\log N}})$
$\tilde{F}(\alpha,A)=\frac{C_{q}(2a)}{\varphi(q)}\sum\limits_{2<n\leq
A}e(\beta\cdot 2n)+O(A\cdot e^{-c_{4}\sqrt{\log N}}).$
Proof Let $p\equiv r({\rm mod}\ q)$. Then $p|q$ is and only if $(r,q)>1$, and
so
$\sum\limits^{q}_{\tiny\begin{array}[]{l}r=1\\\
(r,q)=1\end{array}}\sum\limits_{\tiny\begin{array}[]{l}\frac{N}{2}-A<p\leq\frac{N}{2}+A\\\
p\equiv r({\rm mod}\ q)\end{array}}e(\alpha p)\cdot\log
p=\sum\limits_{\tiny\begin{array}[]{l}\frac{N}{2}-A<p\leq\frac{N}{2}+A\\\
p|q\end{array}}e(\alpha p)\cdot\log p\ll\sum\limits_{p|q}\log p\ll\log q.$
So
$\displaystyle F(\alpha,A)$ $\displaystyle=$
$\displaystyle\sum\limits^{q}_{\tiny\begin{array}[]{l}r=1\\\
(r,q)=1\end{array}}\cdot\sum\limits_{\tiny\begin{array}[]{l}\frac{N}{2}-A<p\leq\frac{N}{2}+A\\\
p\equiv r({\rm mod}\ q)\end{array}}e\left(\left(\frac{a}{q}+\beta\right)\cdot
p\right)\cdot\log p+O(\log q)$ $\displaystyle=$
$\displaystyle\sum\limits^{q}_{\tiny\begin{array}[]{l}r=1\\\
(r,q)=1\end{array}}e(\frac{a}{q}r)\sum\limits_{\tiny\begin{array}[]{l}\frac{N}{2}-A<p\leq\frac{N}{2}+A\\\
p\equiv r({\rm mod}\ q)\end{array}}e(\beta p)\log p+O(\log q)$
$\displaystyle=$ $\displaystyle\sum\limits^{q}_{\tiny\begin{array}[]{l}r=1\\\
(r,q)=1\end{array}}e(\frac{a}{q}r)\int^{\frac{N}{2}+A}_{\frac{N}{2}-A}e(\beta
t)d\theta(t;q,r)+O(\log q)$
By theorem 4,
$F(\alpha,A)=\frac{\mu(q)}{\varphi(q)}\int^{\frac{N}{2}+A}_{\frac{N}{2}-A}e(\beta
t)dt+O(Ae^{-c_{3}\sqrt{\log N}}).$
$\displaystyle\int^{\frac{N}{2}+A}_{\frac{N}{2}-A}e(\beta
t)dt-\sum\limits_{\frac{N}{2}-A<n\leq\frac{N}{2}+A}e(\beta n)$
$\displaystyle=$
$\displaystyle\sum\limits_{\frac{N}{2}-A<n\leq\frac{N}{2}+A}\int^{n+1}_{n}(e(\beta
t)-e(\beta n))dt+O(1)$ $\displaystyle=$
$\displaystyle\sum\limits_{\frac{N}{2}-A<n\leq\frac{N}{2}+A}\int^{n+1}_{n}\left(\int^{t}_{n}de(\beta
u)\right)dt+O(1)$ $\displaystyle=$
$\displaystyle\sum\limits_{\frac{N}{2}-A<n\leq\frac{N}{2}+A}\int^{n+1}_{n}\left(\int^{t}_{n}2\pi
i\beta e(\beta u)du\right)dt+O(1)$ $\displaystyle\ll$
$\displaystyle\sum\limits_{\frac{N}{2}-A<n\leq\frac{N}{2}+A}|\beta|+O(1),$
since
$|\beta|\leq\frac{1}{\tau}\ll A\cdot\frac{1}{\tau}=\frac{NQ}{A}\ll
e^{c_{4}\sqrt{\log N}};$
Similarly, we have
$\tilde{F}(\alpha,A)=\frac{c_{q}(2a)}{\varphi(q)}\sum\limits_{2<n\leq
A}e(\beta\cdot 2n)+O(A\cdot e^{-c_{4}\sqrt{\log N}}).$
Theorem 6 There is a positive constant $0<\varepsilon<1$, the integal over the
major arcs is
$\int_{\mathcal{M}}F^{2}(\alpha,A)\tilde{F}(\alpha,A)e(-N\alpha)d\alpha=2D(N)A^{2}+O\left(\frac{A^{2}}{\log^{(1-\varepsilon)\lambda}N}\right).$
Proof
$\displaystyle\int_{\mathcal{M}}[F^{2}(\alpha,A)\tilde{F}(\alpha,A)-\frac{\mu^{2}(q)}{\varphi^{3}(q)}C_{q}(2a)u^{2}(\beta,A)\tilde{u}(\beta,A)]e(-N\alpha)d\alpha$
$\displaystyle=$
$\displaystyle\sum\limits^{Q}_{q=1}\sum\limits^{q-1}_{\tiny\begin{array}[]{l}a=0\\\
(a,q)=1\end{array}}\int_{\mathcal{M}(q,a)}[F^{2}(\alpha,A)\tilde{F}(\alpha,A)-\frac{\mu^{2}(q)}{\varphi^{3}(q)}C_{q}(2a)u^{2}(\beta,A)\tilde{u}(\beta,A)]e(-N\alpha)d\alpha$
$\displaystyle\ll$
$\displaystyle\sum\limits^{Q}_{q=1}\sum\limits^{q-1}_{\tiny\begin{array}[]{l}a=0\\\
(a,q)=1\end{array}}\int_{\mathcal{M}(q,a)}A^{3}\cdot e^{-c_{5}\sqrt{\log
N}}d\alpha$ $\displaystyle\leq$ $\displaystyle Q^{2}A^{3}e^{-c_{5}\sqrt{\log
N}}\cdot\frac{1}{\tau}$ $\displaystyle=$ $\displaystyle
NAQ^{3}e^{-c_{5}\sqrt{\log N}}$ $\displaystyle\leq$ $\displaystyle
A^{2}e^{-c_{6}\sqrt{\log N}}.$
$\displaystyle\int_{\mathcal{M}}\frac{\mu^{2}(q)}{\varphi^{3}(q)}C_{q}(2a)u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\alpha)d\alpha$
$\displaystyle=$
$\displaystyle\sum\limits^{Q}_{q=1}\sum\limits^{q-1}_{\tiny\begin{array}[]{l}a=0\\\
(a,q)=1\end{array}}\frac{\mu^{2}(q)C_{q}(2a)}{\varphi^{2}(q)}\int^{\frac{a}{q}+\frac{1}{\tau}}_{\frac{a}{q}-\frac{1}{\tau}}u^{2}(\alpha-\frac{a}{q},A)\tilde{u}(\alpha-\frac{a}{q},A)e(-N\alpha)d\alpha$
$\displaystyle=$ $\displaystyle\sum\limits^{Q}_{\tiny\begin{array}[]{l}q=1\\\
q\
{\rm\,is\,odd}\end{array}}\frac{\mu(q)}{\varphi^{3}(q)}\sum\limits^{q-1}_{\tiny\begin{array}[]{l}a=0\\\
(a,q)=1\end{array}}e(-\frac{a}{q}N)\int^{\frac{1}{\tau}}_{-\frac{1}{\tau}}u^{2}(\beta,A)$
$\displaystyle\tilde{u}(\beta,A)e(-N\beta)d\beta-\sum\limits^{Q}_{\tiny\begin{array}[]{l}q=1\\\
q\
{\rm\,is\,even}\end{array}}\frac{\mu(q)}{\varphi^{3}(q)}\sum\limits^{q-1}_{\tiny\begin{array}[]{l}a=0\\\
(a,q)=1\end{array}}e(-\frac{a}{q}N)$
$\displaystyle\int^{\frac{1}{\tau}}_{-\frac{1}{\tau}}u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\beta)d\beta$
$\displaystyle=$ $\displaystyle
D(N,Q)\int^{\frac{1}{\tau}}_{-\frac{1}{\tau}}u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\beta)d\beta,$
because of
$C_{q}(a)=\mu\left(\frac{q}{(a,q)}\right)\varphi(q)\varphi^{-1}\left(\frac{q}{(a,q)}\right).$
If $|\beta|\leq\frac{1}{2}$,then
$u(\beta,A)\ll\frac{1}{|\beta|},\tilde{u}(\beta,A)\ll\frac{1}{|\beta|}$
and
$\int^{\frac{1}{2}}_{\frac{1}{\tau}}u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\beta)d\beta\ll\int^{\frac{1}{\tau}}_{\frac{1}{\tau}}\frac{d\beta}{\beta^{3}}\ll\tau^{2}=A^{2}Q^{-2}e^{-2\varepsilon\sqrt{\log
N}}.$
Similarly,
$\int^{-\frac{1}{\tau}}_{-\frac{1}{2}}u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\beta)d\alpha\ll
A^{2}Q^{-2}e^{-2\varepsilon\sqrt{\log N}}.$
By theorem 3,
$\displaystyle\int^{\frac{1}{\tau}}_{-\frac{1}{\tau}}u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\beta)d\beta$
$\displaystyle=$
$\displaystyle\int^{\frac{1}{2}}_{-\frac{1}{2}}u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\beta)d\beta+O(A^{2}e^{-2\varepsilon\sqrt{\log
N}})$ $\displaystyle=$ $\displaystyle 2A^{2}+O(A^{2}e^{-2\varepsilon\sqrt{\log
N}}).$
By theorem 2,
$D(N,Q)=D(N)+O\left(\frac{1}{Q^{1-\varepsilon}}\right).$
Therefore,
$\displaystyle\int_{\mathcal{M}}F^{2}(\alpha,A)\tilde{F}(\alpha,A)e(-N\alpha)d\alpha$
$\displaystyle=$ $\displaystyle
D(N,Q)\int^{\frac{1}{\tau}}_{-\frac{1}{\tau}}u^{2}(\beta,A)\tilde{u}(\beta,A)e(-N\beta)d\beta+O(A^{2}e^{-c_{6}\sqrt{\log
N}})$ $\displaystyle=$ $\displaystyle
2D(N)A^{2}+O(\frac{A^{2}}{\log^{(1-\varepsilon)\lambda}N}).$
This completes the proof.
### 3.5 Proof of theorem
Theorem 7(I. M. Vinogradov) If
$\left|\alpha-\frac{a}{q}\right|\leq\frac{1}{q^{2}}$, where $a$ and $q$ are
integers such that $1\leq q\leq A$ and $(a,q)=1$, then
$\tilde{F}(\alpha,A)=\sum\limits_{2<p\leq A}e(\alpha\cdot 2p)\log p\ll
A\log^{4}N\left(\sqrt{\frac{q}{A}}+\sqrt{\frac{1}{q}}+\frac{1}{H}\right),$
where
$H=e^{\frac{1}{2}\sqrt{\log N}}.$
Lemma 8(Dirichlet) If $\alpha\in m$, then there must exist integer $q,a$ such
that
$\frac{a}{q}\in\left[-\frac{1}{\tau},1-\frac{1}{\tau}\right],(a,q)=1,Q<q\leq\tau$
and
$\left|\alpha-\frac{a}{q}\right|<\frac{1}{q\tau}.$
Theorem 9 For any $\lambda>0$, we have
$\int_{m}F^{2}(\alpha,A)\tilde{F}(\alpha,A)e(-\alpha
N)d\alpha\ll\frac{A^{2}}{\log^{\frac{\lambda}{2}-5}}.$
Proof If $\alpha\in m$, then $Q<q\leq\tau$. By theorem 7,
$\displaystyle\tilde{F}(\alpha,A)$ $\displaystyle\ll$ $\displaystyle
A\log^{4}N\left(\sqrt{\frac{q}{A}}+\sqrt{\frac{1}{q}}+\frac{1}{H}\right)\ll
A\log^{4}N$
$\displaystyle\left(\sqrt{\frac{\tau}{A}}+\sqrt{\frac{1}{Q}}+\frac{1}{H}\right)$
$\displaystyle=$ $\displaystyle
A\log^{4}N\left(e^{-\frac{\varepsilon}{2}\sqrt{\log
N}}\log^{-\frac{\lambda}{2}}N+\log^{-\frac{\lambda}{2}}N+e^{-\frac{1}{2}\sqrt{\log
N}}\right)$ $\displaystyle\ll$
$\displaystyle\frac{A}{\log^{\frac{\lambda}{2}-4}N}.$
Since
$\theta(N,A)=:\sum\limits_{\frac{N}{2}-A<p\leq\frac{N}{2}+A}\log p\ll A,$
we have
$\int^{1-\frac{1}{\tau}}_{-\frac{1}{\tau}}|F(\alpha,A)|^{2}d\alpha=\sum\limits_{\frac{N}{2}-A<p\leq\frac{N}{2}+A}\log^{2}p\ll\log
N\cdot\sum\limits_{\frac{N}{2}-A<p\leq\frac{N}{2}+A}\log p\ll A\log N,$
and so
$\displaystyle\int_{m}|F(\alpha,A)|^{2}|\tilde{F}(\alpha,A)|d\alpha$
$\displaystyle\ll$ $\displaystyle\sup\\{|\tilde{F}(\alpha,A)|:\alpha\in
m\\}\int_{m}|F(\alpha,A)|^{2}d\alpha$ $\displaystyle\ll$
$\displaystyle\frac{A}{\log^{\frac{\lambda}{2}-4}N}\int^{1}_{0}|F(\alpha,A)|^{2}d\alpha$
$\displaystyle\ll$ $\displaystyle\frac{A^{2}}{\log^{\frac{\lambda}{2}-5}N}$
This completes the proof.
Proof of theorem 1
$\displaystyle R(N,A)$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}F^{2}(\alpha,A)\tilde{F}(\alpha,A)e(-N\alpha)d\alpha+\int_{m}F^{2}(\alpha,A)\tilde{F}(\alpha,A)e(-N\alpha)d\alpha$
$\displaystyle=$ $\displaystyle
2D(N)A^{2}+O\left(\frac{A^{2}}{\log^{(1-\varepsilon)\lambda}N}\right)+O\left(\frac{A^{2}}{\log^{\frac{\lambda}{2}-5}}\right),$
let
$c=min\\{(1-\varepsilon)\lambda,\frac{\lambda}{2}-5\\},\lambda>10.$
Acknowledgment. I am grateful to my parents. I also wish to thank professor
shiqing zhang, from whom I learned a lot.
## References
* [1] G. H. Hardy and S. Ramanujan. Asymptotic formular in combinatory analysis. Proc. London Math. Soc. 17: 75-115, 1918.
* [2] M. B. Nathanson. Additive Number Theory: The Classica lBases, Volume 164 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1996.
* [3] Pang Chengdong. Pan chengbiao. Analytic number theory. Sci. Pub.1991
* [4] Pang chengdong. Pangchengbiao. Goldbach conjecture. Sci. Pub. 2011.
* [5] L. G. Shnirel’man. On the additive properties of integers. Izv. Donskovo Politakh. Inst. Novocherkasske, 14: 3-27, 1930.
* [6] I. M. Vinogradov. Representation of an odd number as the sum of three primes. Doklady Akad. Nauk SSSR, 15(6-7): 291-294, 1937.
* [7] I. M. Vinogradov. The Method of Trigonometric sums in Number Theory. Nauka, Moscow. 1980. E-mail:[email protected]
|
arxiv-papers
| 2014-04-12T15:29:23 |
2024-09-04T02:50:01.083547
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jin Li",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1404.3296"
}
|
1404.3308
|
# Measurement of strong coupling constant by using event shape moments in
perturbative theory
L. Khajooee T. Kalalian R. Saleh-Moghaddam A. Sepehri M. E. Zomorrodian
Department of Physics, Faculty of sciences, Ferdowsi university of Mashhad,
91775-1436, Mashhad, Iran
###### Abstract
We measure the strong coupling constant at NNLO corrections. We do this
analysis with moments of event shape variables: thrust, C parameter, heavy
hemisphere mass, wide and total jet broadening, by fitting the L3 and DELPHI
data with NNLO model. Our real data are consistent with NNLO calculations,
because it involves higher order terms in QCD calculations.
13.66.Bc; 11.15.Me; 12.38.Bx Keywords: Hadron production in e+ e-
interactions; Strong coupling constant; Perturbative theory
## 1 Introduction
Analyses of events originating from ${e}^{+}e{}^{-}$ annihilation into hadrons
allow for studies [1, 2, 3, 4, 5, 6] of Quantum Chromo Dynamics (QCD), the
theory of the strong interaction [7, 8, 9, 10, 11, 12]. Comparison of
observables such as jet production rates or event shapes with theoretical
predictions provides access to the determination of the strong coupling
($\alpha_{s}$). Recently significant progress in the theoretical calculations
of event shape moments and three jet rates has been made [13].
Event shape variables are interesting for studying the interplay between
perturbative and non-perturbative dynamics. Apart from distributions of these
observables one can study mean values and higher moments. The $n\ th$ moment
of an event shape observable $Y$ is defined by [14]:
$\left\langle
y^{n}\right\rangle=\,\frac{1}{\sigma_{had}}\int_{\,\,\,0}^{\,\,\,y_{\max}}y^{n}\,\frac{d\sigma}{dy}dy$
(1)
where ${y}_{\max}$ is the kinematically allowed upper limit of the observable.
Measurements of the strong coupling ($\alpha_{s}$) of QCD, the theory of
strong interaction, using different observables and different analysis methods
serve as an important consistency test of QCD.
In section 2, we perform a new extraction of $\alpha_{s}$ from L3 and DELPHI
data with Next to Next Leading Order (NNLO) model and measure the strong
coupling constant at NNLO then we compare the moment observables with other
experiments. The last section includes our conclusions.
## 2 NNLO corrections to event shape moments in electron positron
annihilation
### 2.1 Definition of the observables
The properties of hadronic events may be characterized by a set of event shape
observables. In this subsection 2.1 briefly recall the definitions of the
relevant event shape observables. The event shape variables are defined by the
following sentences.
Thrust (T) defined by the expression [15, 16, 17, 18]:
$T=\max\left(\frac{\sum_{i}\left|\vec{P}_{i.}\vec{n}\right|}{\sum_{i}\left|p_{i}\right|}\right)$
(2)
The thrust axis $\vec{n}_{T}$ is the direction $\vec{n}$ which maximizes the
expression in parentheses the value of the thrust can vary between $0.5$ and
$1$.
A plane through the origin and perpendicular to $\vec{n}_{T}$ divides the
event into two hemispheres ${H}_{1}$ and ${H}_{2}$.
C-Parameter: The linearized momentum tensor $\theta^{ij}$ is defined by [19,
20]:
$\theta^{ij}=\frac{1}{\sum_{1}\left|\vec{P}_{L}\right|}\sum_{k}\frac{P_{k}^{i}P_{k}^{i}}{\left|\vec{P}_{k}\right|},i,j=1,2,3$
(3)
Where the sum runs over all final state particles and ${P}_{k}^{i}$ is the $i\
th$ component of the three – momentum $\vec{P}_{k}$of particle $k$ in the
center of mass system. The tensor $\theta$ is normalized to have unit trace.
In terms of the eigenvalues of the $\theta$ tensor,
$\lambda_{1},\lambda_{2},\lambda_{3},with\lambda_{1}+\lambda_{2}+\lambda_{3}=1,$
one defines
$C=3(\lambda_{1}\lambda_{2}+\lambda_{2}\lambda_{3}+\lambda_{3}\lambda_{1})$
(4)
The c-parameter exhibits in perturbation theory a singularity at the three-
parton boundary $c=\frac{3}{4}$
Heavy hemisphere mass ( $\rho$): The hemisphere masses are defined by [21]
$M_{i}^{2}=\,\left(\begin{array}[]{l}{\sum Pj}\\\ {j\in
H_{i}}\end{array}\right)^{2},i=1,2$ (5)
where ${P}_{j}$ denotes the four-momentum of particle j. The heavy hemisphere
mass ${M}_{H}$ is then defined by:
$M_{H}^{2}=\,\max\,\left(M_{1}^{2}\,,\,M_{2}^{2}\right)$ (6)
It is convenient to introduce the dimensionless quantity
$\rho=\frac{M_{H}^{2}}{Q^{2}}$ (7)
where $Q$ is the centre of mass energy. In leading order the distribution of
the heavy hemisphere mass ($\rho$) is identical to the distribution of
$(1-T)$.
Jet broadening observables ${B}_{T}$ and ${B}_{W}$:
The hemisphere broadening [22, 23, 24, 25, 26] are defined by
$B_{K}=\,\left(\frac{\sum_{i\in\,H_{K}}\left|\vec{p}_{i}\times\vec{n}_{T}\right|}{2\sum_{i}\left|\vec{p}_{i}\right|}\right)$
(8)
For each of the two event hemispheres, ${H}_{K}$ defined above. The two
observables are defined by:
$B_{T}=B_{1}+B_{2}\,\,\,\,\,\,\,\,\,\,\,\,,\,and\,\,\,\,\,\,B_{W}=\max\,\left(B_{1},B_{2}\right)$
(9)
where ${B}_{T}$ is the total and ${B}_{W}$ is the wide jet broadening.
### 2.2 Theoretical framework
The perturbative QCD expansion for the moment of the event shape observable
$y$ up to NNLO at centre-of- mass energy $\sqrt{S}$ for renormalization scale
$\mu^{2}=s$ and $\alpha_{s}\equiv\,\alpha_{s}\left(S\right)$ is given by [14]:
$\left\langle
y^{n}\right\rangle\,\,\left(s,\,\mu^{2}=s\right)=\,\left(\frac{\alpha_{s}}{2\pi}\right)^{1}\,\overline{A}_{y,n}+\,\left(\frac{\alpha_{s}}{2\pi}\right)^{2}\,\overline{B}_{y,n}+\,\left(\frac{\alpha_{s}}{2\pi}\right)^{3}\,\overline{C}_{y,n}+O\left(\alpha_{s}\right)^{4}$
(10)
The detailed calculations of the coefficients ${A}_{y,n}$ , ${B}_{y,n}$ and
${C}_{y,n}$ was achieved by Gehrmann et.al, [14, 27]. In addition
$\overline{A}_{y,n},\,\,\overline{B}_{y,n}\,\,and\,\,\,\overline{C}_{y,n}$ are
related to ${A}_{y,n}$ , ${B}_{y,n}$ and ${C}_{y,n}$ according to;
$\displaystyle\overline{A}_{y,n}$ $\displaystyle=\,A_{y,n}$ (11)
$\displaystyle\overline{B}_{y,n}$
$\displaystyle=\,B_{y,n}-\,\frac{3}{2}\,C_{F}\,\,A_{y,n},$ (12)
$\displaystyle\overline{C}_{y,n}$
$\displaystyle=\,C_{y,n}-\frac{3}{2}\,C_{F}B_{y,n}+\left(\frac{9}{4}C_{F}^{2}-K_{2}\right)A_{y,n}$
(13)
The constant ${K}_{2}$ is given by [28, 29]
$K_{2}=\frac{1}{4}\left[-\frac{3}{2}C_{F}^{2}+C_{F}\,C_{A}\left(\frac{123}{2}-44\zeta_{3}\right)+C_{F}T_{R}N_{F}\left(-22+16\zeta_{3}\right)\right]$
(14)
Where the QCD color factors are:
$C_{A}=N\,,\,\,C_{F}=\frac{N^{2}-1}{2N},\,\,\,\,T_{R}=\frac{1}{2}$ (15)
For $N=3$ colours and ${N}_{F}$ light quark flavors. The coefficients $A$, $B$
and $C$ have been computed for several event-shape variables In terms of the
running coupling $\alpha_{s}\left(\mu^{2}\right)$ the NNLO expression for an
event shape moment measured at centre-of-mass energy squared s becomes [14]:
$\left\langle
y^{n}\right\rangle\,\left(s,\mu^{2}\right)=\,\left(\frac{\alpha_{s}\left(\mu\right)}{2\pi}\right)\,\overline{A}_{y,n}+\,\left(\frac{\alpha_{s}\left(\mu\right)}{2\pi}\right)^{2}\left(\overline{B}_{y,n}+\overline{A}_{y,n}\,\beta_{\circ}\,\log\,\frac{\mu^{2}}{s}\right)+\left(\frac{\alpha_{s}\left(\mu\right)}{2\pi}\right)^{3}$
$\left(\overline{C}_{y,n}+2\overline{B}_{y,n}\,\beta_{\circ}\,\log\,\frac{\mu^{2}}{s}+\overline{A}_{y,n}\left(\beta_{\circ}^{2}\,\log^{2}\,\frac{\mu^{2}}{s}+\beta_{1}\,\log\,\frac{\mu^{2}}{s}\right)\right)+o\left(\alpha_{s}^{4}\right).$
(16)
In which
$\begin{array}[]{l}{\beta_{\circ}=\,\frac{11C_{A}-4T_{R}N_{F}}{6}}\\\
{\beta_{1}=\,\frac{17C_{A}^{2}\,\,-10C_{A}\,T_{R}\,N_{F}\,-6C_{F}\,T_{R}\,N_{F}}{6}}\end{array}$
(17)
## 3 physics results
Moments of event shapes have been measured by various ${e}^{+}e^{-}$ collider
experiments at centre-of- mass energies ranging from $40$ GeV to $207$ GeV
[30, 31].
To determine $\alpha_{s}$ at each energy point, the measured distributions are
fitted in the ranges of energies. In this paper we use the experimental data
for the five variables at $\left\langle\sqrt{s}\right\rangle=200.2\ GeV$. The
scale uncertainly is obtained by repeating the fit for different values of the
renormalization scale in the interval $0.5\sqrt{s}\leq\mu\leq 2\sqrt{s}$ [32].
The moments of the five standard event shapes are displayed in Figures 1 and
2. The predictions are compared to ${L}_{3}$ and DELPHI data. Our real data
are consistent with NNLO, compared with NLO or LO calculations, because it
involves higher order terms in QCD calculations [14].
Figure 1: First moments of Five event shape variables fitted with eq (16), the
data are from the $L3$ and DELPHI experiments, taken from refs [30, 31].
Figure 2: Second moments of Five event shape variables fitted with eq(16), the
data are from the L3 and DELPHI experiments, taken from refs [30, 31].
The value of $\alpha_{s}$ was estimated by fitting the data [30, 31] with NNLO
expression for an event shape moment (16). Separate fits were performed to
each of the five observables at centre-of- mass energies ranging from $40$ GeV
to $207$ GeV [30, 31]. The fitted values for $\alpha_{s}$ change for different
choices of the renormalization scale. This is demonstrated for the moments of
event shapes in tables 1-5. The given errors are the statistical errors. We
also have indicated on each table the value of ${\alpha}_{{\rm
S}}{\rm(}M_{Z}{\rm)}$ extracted at $M_{{\rm Z}}$ energy. We don’t observe any
significant change between the values of strong coupling constant for
different event shape moments and renormalization scales.
Table 1: Measurement of $\alpha_{s}$ from $\left\langle{(B}_{T})\right\rangle$ and $\left\langle{{(B}_{T})}^{2}\right\rangle$ moments for different choices of the renormalization scale. Event shape variable | $\mu$ | $\alpha_{s}$
---|---|---
$\left\langle{(B}_{T})\right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1327\pm 0.0008$
$M_{{\rm Z}}$ | $0.1268\pm 0.0008$
${\sqrt{s}}/{3}$ | $0.1254\pm 0.0009$
${\sqrt{s}}/{2}$ | 0.1188$\pm 0.0009$
$\sqrt{s}$ | $0.1131\pm 0.0009$
| $2\sqrt{s}$ | $0.1104\pm 0.0009$
$\left\langle{{(B}_{T})}^{2}\right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1345\pm 0.0022$
$M_{{\rm Z}}$ | $0.1294\pm 0.0020$
${\sqrt{s}}/{3}$ | $0.1285\pm 0.0021$
${\sqrt{s}}/{2}$ | 0.1230$\pm$0.0021
$\sqrt{s}$ | $0.1182\pm 0.0021$
| $2\sqrt{s}$ | $0.1159\pm 0.0020$
Table 2: Measurement of $\alpha_{s}$ from $\left\langle 1-T\right\rangle$ and $\left\langle{\left(1-T\right)}^{2}\right\rangle$ moments for different choices of the renormalization scale. Event shape variable | $\mu$ | $\alpha_{s}$
---|---|---
$\left\langle 1-T\right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1339\pm 0.0024$
$M_{{\rm Z}}$ | $0.1298\pm 0.0021$
${\sqrt{s}}/{3}$ | $0.1273\pm 0.0023$
${\sqrt{s}}/{2}$ | $0.1211\pm 0.0022$
$\sqrt{s}$ | $0.1158\pm 0.0021$
| $2\sqrt{s}$ | $0.1133\pm 0.0021$
$\left\langle{\left(1-T\right)}^{2}\right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1377\pm 0.0049$
$M_{{\rm Z}}$ | $0.1364\pm 0.0039$
${\sqrt{s}}/{3}$ | $0.1316\pm 0.0047$
${\sqrt{s}}/{2}$ | $0.1260\pm 0.0045$
$\sqrt{s}$ | $0.1210\pm 0.0043$
| $2\sqrt{s}$ | $0.1186\pm 0.0042$
Table 3: Measurement of $\alpha{}_{s}$ from $\left\langle{(B}_{W})\right\rangle$ and $\left\langle{{(B}_{W})}^{2}\right\rangle$ moments for different choices of the renormalization scale. Event shape variable | $\mu$ | $\alpha_{s}$
---|---|---
$\left\langle{(B}_{W})\right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1297\pm 0.0009$
$M_{{\rm Z}}$ | $0.1249\pm 0.0006$
${\sqrt{s}}/{3}$ | $0.1228\pm 0.0008$
${\sqrt{s}}/{2}$ | $0.1159\pm 0.0007$
$\sqrt{s}$ | $0.1097\pm 0.0006$
| $2\sqrt{s}$ | $0.1068\pm 0.0006$
$\left\langle{{(B}_{W})}^{2}\right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1231\pm 0.0016$
$M_{{\rm Z}}$ | $0.1195\pm 0.0013$
${\sqrt{s}}/{3}$ | $0.1166\pm 0.0015$
${\sqrt{s}}/{2}$ | $0.1107\pm 0.0014$
$\sqrt{s}$ | $0.1055\pm 0.0014$
| $2\sqrt{s}$ | $0.1031\pm 0.0013$
Table 4: Measurement of $\alpha_{s}$ from $\left\langle\ C\ \right\rangle$ and $\left\langle\ C^{2}\ \right\rangle$ moments for different choices of the renormalization scale. Event shape variable | $\mu$ | $\alpha_{s}$
---|---|---
$\left\langle\ C\ \right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1324\pm 0.0019$
$M_{{\rm Z}}$ | $0.1272\pm 0.0017$
${\sqrt{s}}/{3}$ | $0.1258\pm 0.0019$
${\sqrt{s}}/{2}$ | $0.1197\pm 0.0018$
$\sqrt{s}$ | $0.1144\pm 0.0018$
| $2\sqrt{s}$ | $0.1120\pm 0.0017$
$\left\langle\ C^{2}\ \right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1364\pm 0.0039$
$M_{{\rm Z}}$ | $0.1335\pm 0.0029$
${\sqrt{s}}/{3}$ | $0.1305\pm 0.0038$
${\sqrt{s}}/{2}$ | $0.1249\pm 0.0036$
$\sqrt{s}$ | $0.1200\pm 0.0035$
| $2\sqrt{s}$ | $0.1177\pm 0.0035$
Table 5: Measurement of $\alpha_{s}$ from $\left\langle\ \rho\ \right\rangle$ and $\left\langle\ {\rho}^{2}\ \right\rangle$ moments of the renormalization scale. Event shape variable | $\mu$ | $\alpha_{s}$
---|---|---
$\left\langle\ \rho\ \right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1367\pm 0.0021$
$M_{{\rm Z}}$ | $0.1304\pm 0.0020$
${\sqrt{s}}/{3}$ | $0.1287\pm 0.0020$
${\sqrt{s}}/{2}$ | $0.1215\pm 0.0019$
$\sqrt{s}$ | $0.1153\pm 0.0018$
| $2\sqrt{s}$ | $0.1124\pm 0.0018$
$\left\langle\ {\rho}^{2}\ \right\rangle$ | ${\sqrt{s}}/{4}$ | $0.1317\pm 0.0035$
$M_{{\rm Z}}$ | $0.1260\pm 0.0033$
${\sqrt{s}}/{3}$ | $0.1246\pm 0.0033$
${\sqrt{s}}/{2}$ | $0.1180\pm 0.0030$
$\sqrt{s}$ | $0.1124\pm 0.0029$
| $2\sqrt{s}$ | $0.1098\pm 0.0028$
We observe that our obtained values for coupling constant considering the NNLO
corrections for different event shape variables are in good agreement with the
other experiments [10].
## 4 Conclusions
In this paper we have presented measurements of the strong coupling constant
for hadronic events produced at $L3$ and DELPHI in the centre-of-mass energies
$40$ GeV to $207$GeV. We measured the coupling constant considering the
moments of the event shape observables using next-to-next leading order (NNLO)
calculations for different choices of the renormalization scale. Our results
are consistent with those obtained from other experiments, as well as with the
QCD theory.
## References
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|
arxiv-papers
| 2014-04-12T17:49:55 |
2024-09-04T02:50:01.090540
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Khajooee, T. Kalalian, R. Saleh-Moghaddam, A. Sepehri, M. E.\n Zomorrodian",
"submitter": "R Salehmoghaddam",
"url": "https://arxiv.org/abs/1404.3308"
}
|
1404.3340
|
CHEBYSHEV POLYNOMIALS on a SYSTEM of CONTINUA
Vladimir V. Andrievskii
###### Abstract
The estimates of the uniform norm of the Chebyshev polynomial associated with
a compact set $K$ consisting of a finite number of continua in the complex
plane are established. These estimates are exact (up to a constant factor) in
the case where the components of $K$ are either quasismooth (in the sense of
Lavrentiev) arcs or closed Jordan domains bounded by a quasismooth curve.
††footnotetext: Date received: . Communicated by AMS classification: 30A10,
30C10, 30C62, 30E10 Key words and phrases: Chebyshev polynomial, Equilibrium
measure, Quasismooth curve.
1\. Introduction and Main Results
Let $K\subset{\bf C}$ be a compact set in the complex plane consisting of
disjoint closed connected sets (continua) $K^{j},j=1,2,\ldots,m$, i.e.,
$K=\cup_{j=1}^{m}K^{j};\quad K^{j}\cap K^{k}=\emptyset\mbox{ for }j\neq
k;\quad\mbox{ diam}(K^{j})>0.$
Here
$\mbox{diam}(S):=\sup_{z,\zeta\in S}|z-\zeta|,\quad S\subset{\bf C}.$
Denote by $T_{n}(z)=T_{n}(z,K),n\in{\bf N}:=\\{1,2,\ldots\\}$ the $n$-th
Chebyshev polynomial associated with $K$, i.e.,
$T_{n}(z)=z^{n}+c_{n-1}z^{n-1}+\ldots+c_{0},c_{k}\in{\bf C}$ is the (unique)
monic polynomial which minimizes the supremum norm $||T_{n}||_{K}:=\sup_{z\in
K}|T_{n}(z)|$ among all monic polynomials of the same degree.
It is well-known (see, for example, [12, Theorem 5.5.4 and Corollary 5.5.5])
that
$||T_{n}||_{K}\geq\mbox{
cap}(K)^{n}\quad\mbox{and}\quad\lim_{n\to\infty}||T_{n}||_{K}^{1/n}=\mbox{
cap}(K),$
where$\mbox{ cap}(S)$ denotes the logarithmic capacity of a compact set
$S\subset{\bf C}$ (see [12, 13]).
We are interested in finding the estimates of $||T_{n}||_{K}$ from above. This
problem attracted attention of many mathematicians; for the complete survey of
results and further citations, see [14]-[21]. Without loss of generality we
always assume that $\Omega:={\bf C}\setminus K$ is connected. Denote by
$c,c_{1},\ldots$ positive constants (different in different sections) that are
either absolute or they depend only on $K$; otherwise, the dependence on other
parameters is explicitly stated.
###### Theorem 1
Under the above assumptions,
(1.1) $||T_{n}||_{K}\leq c_{1}\log(n+1){\em{\mbox{ cap}}}(K)^{n},\quad
n\in{\bf N}.$
The estimate (1.1) is surprising. It is unexpected even in the case where $K$
is a continuum, i.e., $m=1$. In this case, one of the major sources for
estimates of $||T_{n}||_{K}$ are Faber polynomials $F_{n}(z)=F_{n}(z,K)$
associated with $K$ (see [14, 16]). According to Koevary and Pommerenke [7,
Theorem 1] (see also [16, Chapter IX, §4]), there exist a continuum $K_{0}$
with$\mbox{ cap}(K_{0})=1$ and an infinite set $\Lambda\subset{\bf N}$ such
that for the (monic) polynomial $F_{n}(z)=F_{n}(z,K_{0})$ and $n\in\Lambda$ we
have
$||F_{n}||_{K_{0}}>n^{\alpha},\quad\alpha=0.138.$
It is interesting to compare this classic estimate with (1.1).
If more information is known about the geometry of $K$, (1.1) can be improved
in the following way. A Jordan curve $L\subset{\bf C}$ is called quasismooth
(in the sense of Lavrentiev) (see [11, p. 163]) if for every $z_{1},z_{2}\in
L$,
(1.2) $|L(z_{1},z_{2})|\leq c_{2}|z_{2}-z_{1}|,$
where $L(z_{1},z_{2})$ is the shorter arc of $L$ between $z_{1}$ and $z_{2}$,
a constant $c_{2}$ depends on $L$, and $|S|$ is the linear measure (length) of
a (Borel) set $S\subset{\bf C}$ (see [11, p. 129]). Any subarc of a
quasismooth curve is called a quasismooth arc.
###### Theorem 2
Let each $K^{j}$ in the definition of $K$ be either a quasismooth arc or a
closed Jordan domain bounded by a quasismooth curve. Then
(1.3) $||T_{n}||_{K}\leq c_{3}{\em{\mbox{ cap}}}(K)^{n},\quad n\in{\bf N}.$
In the case of sufficiently smooth components $K^{j}$, the estimate (1.3) was
proved in [21, Theorem 11.4] and recently in [19, Theorem 1.3]. Comparing
(1.3) with known estimates for the Faber polynomials for a domain with the
quasismooth boundary (see [10]) we also see the improvement by a logarithmic
factor.
Our constructions below are based on the method of discretization of the
equilibrium measure due to Totik [17]-[19], representation of the Green
function via special conformal mappings due to Widom [21], and distortion
properties of conformal mappings which can be found, for example, in [11] or
[5].
We use the following notation. For the functions $a>0$ and $b>0$, we write
$a\preceq b$ (order inequality) if $a\leq cb$. The expression $a\asymp b$
means that $a\preceq b$ and $b\preceq a$ simultaneously. Moreover, for
$z\in{\bf C}$ and $\delta>0$ we let
$C(\delta):=\\{z:|z|=\delta\\},\quad{\bf D}:=\\{z:|z|<1\\},\quad{\bf
D}^{*}:={\bf C}\setminus\overline{{\bf D}},$ $d(S_{1},S_{2}):=\inf_{z_{1}\in
S_{1},z_{2}\in S_{2}}|z_{2}-z_{1}|,\quad S_{1},S_{2}\subset{\bf C}.$
Let $m_{2}(S)$ be the two-dimensional Lebesgue measure (area) of a (Borel) set
$S\subset{\bf C}$. For a bounded Jordan curve $J\subset{\bf C}$, denote by
int$(J)$ the bounded component of ${\bf C}\setminus J$.
2\. Totik-Type Polynomials
In this section we review (in more general setting) the construction of the
monic polynomials suggested in [17]-[19]. Let $K=\cup_{j=1}^{m}K^{j}$ be as
above. We recall some general facts from potential theory which can be found,
for example, in [20, 21, 12, 13]. Denote by
$g(z)=g_{\Omega}(z,\infty),z\in\Omega$ Green’s function for $\Omega$ with pole
at infinity. It has a multiple-valued harmonic conjugate $\tilde{g}(z)$. Let
$\Phi(z):=\exp(g(z)+i\tilde{g}(z)),$ $K_{s}:=\\{z\in\Omega:g(z)=s\\},\quad
s>0.$
Note that
(2.1) $\mbox{ cap}(K_{s})=e^{s}\mbox{ cap}(K).$
Let $s_{0}>0$ be such that for $0<s<2s_{0}$, the set
$K_{s}=\cup_{j=1}^{m}K_{s}^{j}$ consists of $m$ mutually disjoined curves,
where $K_{s}^{j}$ is the curve surrounding $K^{j}$. Moreover, we can fix
$s_{0}$ so small that
(2.2) $d(\zeta,K^{j})=d(\zeta,K),\quad\zeta\in\mbox{int}(K^{j}_{2s_{0}}).$
Let $\mu=\mu_{K}$ be the equilibrium measure of $K$. According to Gauss’
Theorem (see [13, p. 83]), for the net-change of the function $\tilde{g}$, we
obtain
$\Delta_{K_{s}^{j}}\tilde{g}:=\int_{K_{s}^{j}}\frac{\partial\tilde{g}(\zeta)}{\partial
t_{\zeta}}|d\zeta|=2\pi\omega_{j}$
(see [21, p. 140]), where $0<s<2s_{0}$, $t_{\zeta}$ is the tangent vector to
the curve $K_{s}^{j}$ (traversed in the positive, i.e., counterclockwise
direction) at $\zeta$, and $\omega_{j}:=\mu(K^{j}).$
Therefore, the function $\phi_{j}:=\Phi^{1/\omega_{j}}(\zeta)$ is a conformal
and univalent mapping of $\Omega^{j}:=\mbox{int}(K_{2s_{0}}^{j})\setminus
K^{j}$ onto the annulus $A^{j}:=\\{w:1<|w|<e^{2s_{0}/\omega_{j}}\\}$ as well
as
$K_{s}^{j}=\\{\zeta\in\Omega^{j}:|\phi_{j}(\zeta)|=e^{s/\omega_{j}}\\},\quad
0<s<2s_{0}.$
Let $\mu_{s}:=\mu_{K_{s}}$ be the equilibrium measure of $K_{s}.$ By Gauss’
Theorem (see [13, p. 83]),
$\mu_{s}(K_{s}^{j})=\mu(K^{j})=\omega_{j},\quad 0<s<2s_{0}.$
Moreover, by virtue of [13, p. 90, Theorem 1.4], for any arc
$\gamma=\\{\zeta\in
K_{s}^{j}:\theta_{1}\leq\arg\phi_{j}(\zeta)\leq\theta_{2}\\},\quad
0<\theta_{2}-\theta_{1}\leq 2\pi$
we have
(2.3) $\mu_{s}(\gamma)=\frac{(\theta_{2}-\theta_{1})\omega_{j}}{2\pi}\,.$
Assuming that $n\in{\bf N}$ is sufficiently large, i.e.,
$n>n_{1}:=10/(\min_{j}\omega_{j})$ we let
$n_{j}:=[n\omega_{j}],\quad j=1,2,\ldots,m-1,$
$n_{m}:=n-(n_{1}+\ldots+n_{m-1}),$
where $[a]$ means the integer part of a real number $a$.
Therefore,
(2.4) $0\leq n_{m}-n\omega_{m}=\sum_{j=1}^{m-1}(n\omega_{j}-n_{j})\leq m-1.$
Next, for $0<s<s_{0}/2$, we represent each $K^{j}_{s}$ as the union of closed
subarcs $I^{j}_{k},k=1,\ldots,n_{j}$ such that their interiors do not
intersect,
$I_{k}^{j}\cap I_{k+1}^{j}=:\xi_{k}^{j},\quad k=1,\ldots,n_{j}-1,$
and $I_{n_{j}}^{j}\cap I_{1}^{j}=:\xi_{n_{j}}^{j}$ are points of $K_{s}^{j}$
ordered in a positive direction, and
$\mu_{s}(I_{k}^{j})=\frac{\omega_{j}}{n_{j}},\quad k=1,\ldots,n_{j}.$
Let $\xi_{0}^{j}:=\xi^{j}_{n_{j}}$ and let
$\eta_{0}^{j}<\eta_{1}^{j}<\ldots<\eta_{n_{j}}^{j}=\eta_{0}^{j}+2\pi$ be
determined by $\phi_{j}(\xi_{k}^{j})=\exp((s/\omega_{j})+i\eta_{k}^{j})$,
i.e., $\eta_{k}^{j}-\eta_{k-1}^{j}=2\pi/n_{j}$. Consider
$\tilde{B}_{k}^{j}:=\left\\{w=re^{i\eta}:\eta_{k-1}^{j}\leq\eta\leq\eta_{k}^{j},0\leq
r-e^{s/\omega_{j}}\leq\frac{2\pi}{n_{j}}\right\\}.$
Further, we assume that
(2.5)
$n>n_{2}:=\frac{2\pi}{\min_{j}\left(e^{s_{0}/\omega_{j}}-e^{s_{0}/(2\omega_{j})}\right)},$
which implies $\tilde{B}_{k}^{j}\subset
E^{j}:=\\{w:1<|w|<e^{s_{0}/\omega_{j}}\\}.$ Let
$B_{k}^{j}:=\psi_{j}(\tilde{B}_{k}^{j})$, where $\psi_{j}:=\phi_{j}^{-1}$.
###### Lemma 1
There exist constants $c$ and $n_{0}=n_{0}(K,c)\in{\bf N}$ such that for
$s=s(n)=c/n<s_{0}/2$ and $n>n_{0}$ we have
(2.6) $d(I_{k}^{j},K)\preceq|\xi_{k}^{j}-\xi_{k-1}^{j}|\leq\mbox{\em
diam}(I_{k}^{j})\leq|I_{k}^{j}|\leq 0.1d(I_{k}^{j},K)$ (2.7)
$|\xi_{k}^{j}-\xi_{k-1}^{j}|^{2}\asymp m_{2}(B_{k}^{j})\asymp\mbox{\em
diam}(B_{k}^{j})^{2}\leq\frac{1}{4}d(B_{k}^{j},K)^{2}.$
For the proof of the lemma, see Section 3.
For sufficiently large $n>n_{0}$ and $s:=c/n$ as in Lemma 1, consider the
points
$\zeta_{k}^{j}:=\frac{1}{\mu_{s}(I_{k}^{j})}\int_{I_{k}^{j}}\xi d\mu_{s}(\xi)$
and the polynomial
$P_{n}(z):=\prod_{j=1}^{m}\prod_{k=1}^{n_{j}}(z-\zeta_{k}^{j}).$
According to Lemma 1,
$|\zeta_{k}^{j}-\xi_{k}^{j}|=\left|\frac{1}{\mu_{s}(I_{k}^{j})}\int_{I_{k}^{j}}(\xi-\xi_{k}^{j})d\mu_{s}(\xi)\right|\leq
0.1d(I_{k}^{j},K)$
and for $\xi\in I_{k}^{j}$,
(2.8)
$|\xi-\zeta_{k}^{j}|\leq|\xi-\xi_{k}^{j}|+|\xi_{k}^{j}-\zeta_{k}^{j}|\leq
0.2d(I_{k}^{j},K).$
For $z\in K$ we have
(2.9) $\displaystyle n\log\mbox{ cap}(K_{s})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m}\sum_{k=1}^{n_{j}}\left(n-\frac{n_{j}}{\omega_{j}}\right)\int_{I_{k}^{j}}\log|z-\xi|d\mu_{s}(\xi)$
$\displaystyle+\sum_{j=1}^{m}\sum_{k=1}^{n_{j}}\frac{1}{\mu_{s}(I_{k}^{j})}\int_{I_{k}^{j}}\log|z-\xi|d\mu_{s}(\xi)$
$\displaystyle=:$ $\displaystyle\Sigma_{1}(z)+\Sigma_{2}(z).$
Since for $\xi\in K_{s}$ and $z\in K$,
$\displaystyle\int_{K_{s}}|\log|z-\xi||d\mu_{s}(\xi)$ $\displaystyle\leq$
$\displaystyle|\log\mbox{diam}(K_{s})|+\int_{K_{s}}\log\frac{\mbox{diam}(K_{s})}{|z-\xi|}d\mu_{s}(\xi)$
$\displaystyle\leq$ $\displaystyle 2\log^{+}\mbox{diam}(K_{s})-\log\mbox{
cap}(K_{s})\preceq 1,$
by virtue of (2.4) we obtain
(2.10) $|\Sigma_{1}(z)|\preceq\int_{K_{s}}|\log|z-\xi||d\mu_{s}(\xi)\preceq
1.$
Furthermore,
(2.11)
$\log|P_{n}(z)|-\Sigma_{2}(z)=\sum_{j=1}^{m}\sum_{k=1}^{n_{j}}\frac{1}{\mu_{s}(I_{k}^{j})}\int_{I_{k}^{j}}\log\left|\frac{z-\zeta_{k}^{j}}{z-\xi}\right|d\mu_{s}(\xi)$
which, together with (2.6) and (2.8), imply that for $z\in K$ and $\xi\in
I_{k}^{j}$,
(2.12) $\displaystyle\log\left|\frac{z-\zeta_{k}^{j}}{z-\xi}\right|$
$\displaystyle=$
$\displaystyle-\Re\log\left(1+\frac{\zeta_{k}^{j}-\xi}{z-\zeta_{k}^{j}}\right)$
$\displaystyle=$
$\displaystyle\Re\left(\frac{\xi-\zeta_{k}^{j}}{z-\zeta_{k}^{j}}\right)+A(\xi),$
where
$|A(\xi)|\preceq\left(\frac{\mbox{diam}(I_{k}^{j})}{d(z,I_{k}^{j})}\right)^{2}.$
Since by the definition of $\zeta_{k}^{j}$,
$\int_{I_{k}^{j}}(\xi-\zeta_{k}^{j})d\mu_{s}(\xi)=0,$
according to (2.9)-(2.12) we have
(2.13) $\displaystyle|\log|P_{n}(z)|-n\log\mbox{ cap}(K_{s})|$
$\displaystyle\preceq$ $\displaystyle
1+\sum_{j=1}^{m}\sum_{k=1}^{n_{j}}\frac{1}{\mu_{s}(I_{k}^{j})}\int_{I_{k}^{j}}\left(\frac{\mbox{diam}(I_{k}^{j})}{d(z,I_{k}^{j})}\right)^{2}d\mu_{s}(\xi)$
$\displaystyle=$ $\displaystyle
1+\sum_{j=1}^{m}\sum_{k=1}^{n_{j}}\left(\frac{\mbox{diam}(I_{k}^{j})}{d(z,I_{k}^{j})}\right)^{2}=:1+\Sigma_{3}(z).$
Next, we formulate a statement which is proved in Section 3. Let
$\Omega_{s}:=\\{\zeta\in\Omega:s\leq g(\zeta)\leq 2s\\},\quad s>0.$
###### Lemma 2
For $z\in K$ and $n>n_{0}$ we have
(2.14)
$\Sigma_{3}(z)\preceq\int_{K_{c/n}}\frac{d(\zeta,K)}{|\zeta-z|^{2}}|d\zeta|,$
(2.15)
$\Sigma_{3}(z)\preceq\int_{\Omega_{c/n}}\frac{dm_{2}(\zeta)}{|\zeta-z|^{2}}.$
Thus, (2.1), (2.13), and Lemma 2 imply the following result which is of
independent interest.
###### Theorem 3
There exist constants $c,c_{1},c_{2}$, and $n_{0}=n_{0}(K,c)$ such that for
$n>n_{0}$ we have
(2.16) $||T_{n}||_{K}\leq c_{1}{\em{\mbox{ cap}}}(K)^{n}\sup_{z\in\partial
K}\int_{K_{c/n}}\frac{d(\zeta,K)}{|\zeta-z|^{2}}|d\zeta|,$ (2.17)
$||T_{n}||_{K}\leq c_{2}{\em{\mbox{ cap}}}(K)^{n}\sup_{z\in\partial
K}\int_{\Omega_{c/n}}\frac{dm_{2}(\zeta)}{|\zeta-z|^{2}}.$
3\. Distortion Properties of $\phi_{j}$
By [5, p. 23, Lemma 2.3], which is an immediate consequence of Koebe’s one-
quarter theorem, we have the following statement. Recall that
$\psi_{j}:=\phi_{j}^{-1}$ is defined in
$A^{j}:=\\{w:1<|w|<e^{2s_{0}/\omega_{j}}\\}$ and
$E^{j}:=\\{w:1<|w|<e^{s_{0}/\omega_{j}}\\}$.
###### Lemma 3
For $w\in E^{j}$ and $z=\psi_{j}(w)$,
$c_{1}^{-1}\frac{d(z,K^{j})}{|w|-1}\leq|\psi_{j}^{\prime}(w)|\leq
c_{1}\frac{d(z,K^{j})}{|w|-1}.$
Moreover, if $|\tau-w|\leq(|w|-1)/2$ and $\zeta=\psi_{j}(\tau)$, then
(3.1) $c_{2}^{-1}\frac{|\tau-w|)}{|w|-1}\leq\frac{|\zeta-z|}{d(z,K^{j})}\leq
c_{2}\frac{|\tau-w|}{|w|-1}.$
In addition to the mapping $\phi_{j}$, we consider a conformal (and univalent)
mapping $\Phi_{j}:{\bf C}\setminus K^{j}\to{\bf D}^{*}$ normalized by the
condition $\Phi_{j}(\infty)=\infty$ Then $h_{j}:=\Phi_{j}\circ\psi_{j}$ is a
conformal mapping of $A^{j}$ onto a doubly connected domain bounded by a unit
circle and the curve $h_{j}(C(e^{2s_{0}/\omega_{j}}))\subset{\bf D}^{*}$.
According to the Carathéodory prime end theorem (see [11, p. 30, Theorem
2.15]), $h_{j}$ can be extended continuously to the unit circle $C(1)$.
Moreover, by the Schwarz reflection principle (see [11, p. 4]) $h_{j}$ can be
extended analytically into
$\\{w:e^{-2s_{0}/\omega_{j}}<|w|<e^{2s_{0}/\omega_{j}}\\}$. This implies that
(3.2) $\left|\frac{h_{j}(w_{2})-h_{j}(w_{1})}{w_{2}-w_{1}}\right|\asymp
1,\quad w_{1},w_{2}\in E^{j}.$
Since $\psi_{j}=\Phi_{j}^{-1}\circ h_{j}$ in $E^{j}$, according to (3.2), many
known distortion properties of $\Phi_{j}$ and $\Phi_{j}^{-1}$ imply the
analogous properties of $\phi_{j}$ and $\psi_{j}$. We describe some of them.
Let for $0<\delta\leq\delta_{j}:=e^{s_{0}/(2\omega_{j})}-1$,
(3.3)
$L_{\delta}^{j}:=K_{\omega_{j}\log(1+\delta)}=\\{z:|\phi_{j}(z)|=1+\delta\\}.$
For the points $z_{k}\in L_{\delta}^{j}$ and
$w_{k}:=\phi_{j}(z_{k})=(1+\delta)e^{i\theta_{k}},k=1,2$, such that
$0<\theta_{2}-\theta_{1}<2\pi$, consider
$\hat{L}_{\delta}^{j}(z_{1},z_{2}):=\\{z=\psi_{j}((1+\delta)e^{i\theta}):\theta_{1}\leq\theta\leq\theta_{2}\\},$
$B_{\delta}^{j}(z_{1},z_{2}):=\\{z=\psi_{j}(re^{i\theta}):\theta_{1}\leq\theta\leq\theta_{2},0\leq
r-1-\delta\leq\min(\theta_{2}-\theta_{1},e^{s_{0}/\omega_{j}}-1-\delta)\\}.$
By virtue of Lemma 3, for $z_{1},z_{2}\in L^{j}_{\delta}$ and $\zeta\in
B_{\delta}^{j}(z_{1},z_{2})$ such that
$\theta_{2}-\theta_{1}\leq\delta/c_{3}$, where
$c_{3}=10c_{2}(1+\max_{j}\delta_{j})$, we have
$\theta_{2}-\theta_{1}<\delta\leq
e^{s_{0}/(2\omega_{j})}-1<e^{s_{0}/\omega_{j}}-e^{s_{0}/(2\omega_{j})}\leq
e^{s_{0}/\omega_{j}}-1-\delta,$ (3.4) $\frac{|\zeta-
z_{1}|}{d(z_{1},K^{j})}\leq
c_{2}\frac{2(\theta_{2}-\theta_{1})(1+\delta_{j})}{\delta}\leq\frac{1}{5},$
i.e., $|\psi_{j}^{\prime}(\zeta)|\asymp d(z_{1},K^{j})/\delta$ and, therefore,
(3.5) $\displaystyle\frac{\theta_{2}-\theta_{1}}{c_{4}\delta}d(z_{1},K^{j})$
$\displaystyle\leq$
$\displaystyle|z_{2}-z_{1}|\leq|\hat{L}_{\delta}^{j}(z_{1},z_{2})|$
$\displaystyle=$
$\displaystyle(1+\delta)\int_{\theta_{1}}^{\theta_{2}}|\psi_{j}^{\prime}((1+\delta)e^{i\theta})|d\theta\leq
c_{4}\frac{\theta_{2}-\theta_{1}}{\delta}d(z_{1},K^{j}),$
as well as
(3.6) $\displaystyle
c_{5}^{-1}\left(\frac{\theta_{2}-\theta_{1}}{\delta}\right)^{2}d(z_{1},K^{j})^{2}$
$\displaystyle\leq$ $\displaystyle
m_{2}(B_{\delta}^{j}(z_{1},z_{2}))=\int_{\theta_{1}}^{\theta_{2}}\int_{1+\delta}^{1+\delta+\theta_{2}-\theta_{1}}|\psi_{j}^{\prime}(re^{i\theta})|^{2}rdrd\theta$
$\displaystyle\leq$ $\displaystyle
c_{5}\left(\frac{\theta_{2}-\theta_{1}}{\delta}\right)^{2}d(z_{1},K^{j})^{2}.$
Proof of Lemma 1. First, we note that by (3.3) $K_{s}^{j}=L_{\delta}^{j}$,
where
(3.7) $\frac{s}{\omega_{j}}\leq\delta=e^{s/\omega_{j}}-1\leq
e^{s_{0}/(2\omega_{j})}\frac{s}{\omega_{j}},\quad 0<\delta<\delta_{j}.$
By (2.2), (2.3), and (3.5) if $2\pi/n_{j}\leq\delta/c_{3}$ and
$n>n_{1}=10/(\min_{j}\omega_{j})$, then
$|I_{k}^{j}|\leq
c_{4}\frac{2\pi}{n_{j}\delta}d(\xi_{k-1}^{j},K^{j})\leq\frac{4\pi
c_{4}}{ns}d(\xi_{k-1}^{j},K).$
Therefore, if we let $c=100\pi c_{4}$, then, for
$n>n_{0}:=n_{1}+n_{2}+2c/s_{0}$, where $n_{2}$ is defined in (2.5), and
$s=c/n$ we obtain the last inequality in (2.6). Moreover, (2.2) and the left-
hand side of (3.1) with $z=\xi_{k-1}^{j}$ and $\zeta=\xi_{k}^{j}$ yield the
first inequality in (2.6).
Furthermore, (2.6), (3.4), (3.6), and (3.7) imply first two order equivalences
in (2.7). The last inequality in (2.7) follows from (2.2) and (3.4) with
$z_{1}=\xi_{k-1}^{j}$.
$\Box$
Proof of Lemma 2. By (2.6) for $z\in K$ we have
$\sum_{k=1}^{n_{j}}\left(\frac{\mbox{diam}(I_{k}^{j})}{d(z,I_{k}^{j})}\right)^{2}\preceq\sum_{k=1}^{n_{j}}\frac{|I_{k}^{j}|d(I_{k}^{j},K^{j})}{d(z,I_{k}^{j})^{2}}\preceq\int_{K_{s}^{j}}\frac{d(\zeta,K)}{|\zeta-z|^{2}}|d\zeta|,$
which implies (2.14).
Furthermore, by Lemma 1 for $z\in K$ we obtain
$\sum_{k=1}^{n_{j}}\left(\frac{\mbox{diam}(I_{k}^{j})}{d(z,I_{k}^{j})}\right)^{2}\preceq\sum_{k=1}^{n_{j}}\frac{m_{2}(B_{k}^{j})}{d(z,B_{k}^{j})^{2}}\preceq\int_{\Omega_{s}^{j}}\frac{dm_{2}(\zeta)}{|\zeta-z|^{2}},$
where $\Omega_{s}^{j}:=\\{\zeta\in\mbox{int}(K_{2s_{0}}^{j}):s\leq
g(\zeta)\leq 2s\\}$, which yields (2.15).
$\Box$
Proof of Theorem 1. Without loss of generality we can assume that $n$ is
sufficiently large. By virtue of (3.2) and the Loewner inequality (see [9] or
[5, p. 27, Lemma 2.5]) for $0<\delta<\delta_{j}$ we obtain
$d(K^{j},L_{\delta}^{j})\succeq\delta^{2},$ which, together with (3.7), imply
(3.8) $d_{s}:=d(K,K_{s})\succeq s^{2},\quad 0<s<\frac{s_{0}}{2}.$
Therefore, for $z\in\partial K$,
$\int_{\Omega_{s}}\frac{dm_{2}(\zeta)}{|\zeta-z|^{2}}\leq\int_{0}^{2\pi}\int_{d_{s}}^{c_{6}}\frac{rdrd\theta}{r^{2}}=2\pi\log\frac{c_{6}}{d_{s}}$
and (1.1) follows from (2.17) and (3.8).
$\Box$
4\. Quasismooth Components
We need the concept of quasiconformality which can be found in [2] or [8].
Moreover, almost all facts, necessary for our consideration, can be derived
from a simple statement on the change of the relative positions of three
points under a quasiconformal mapping. We formulate it in a slightly more
general form than, for example, in [4, 5]. However, it is clear how the proof
from there has to be modified to prove the following result (cf. [1]).
###### Lemma 4
([5, p. 29, Theorem 2.7]). Let $w=F(\zeta)$ be a $Q$-quasiconformal mapping of
a domain $G\subset{\bf C}$ and let $\zeta_{k}\in E\subset
G,w_{k}:=F(\zeta_{k}),k=1,2,3$, where $E$ is a compact set. Then, the
following is true.
(i) The conditions $|\zeta_{1}-\zeta_{2}|\leq c_{1}|\zeta_{1}-\zeta_{3}|$ and
$|w_{1}-w_{2}|\leq c_{2}|w_{1}-w_{3}|$ are equivalent. Besides, the constants
$c_{1}$ and $c_{2}$ are mutually dependent and depend on $Q,G,$ and $E$.
(ii) If $|\zeta_{1}-\zeta_{2}|\leq c_{1}|\zeta_{1}-\zeta_{3}|$, then
$c_{3}^{-1}\left|\frac{w_{1}-w_{3}}{w_{1}-w_{2}}\right|^{1/Q}\leq\left|\frac{\zeta_{1}-\zeta_{3}}{\zeta_{1}-\zeta_{2}}\right|\leq
c_{3}\left|\frac{w_{1}-w_{3}}{w_{1}-w_{2}}\right|^{Q},$
where $c_{3}\geq 1$ depends on $c_{1},Q,G$, and $E$.
First, we consider the case where a component $K^{j}$ of $K$ is a domain
bounded by a quasismooth curve $L^{j}:=\partial K^{j}$. By (1.2) and Ahlfors’
theorem (see [8, p. 100, Theorem 8.6]) $L^{j}$ is quasiconformal and
$\phi_{j}$ can be extended to a quasiconformal homeomorphism of a neighborhood
$N^{j}=:G$ of $\psi_{j}(\overline{E^{j}})=:E$. Here
$E^{j}=\\{w:1<|w|<e^{s_{0}/\omega_{j}}\\}$. For $\zeta\in E$ denote by
$\zeta_{0}\in L^{j}$ any point with $|\zeta-\zeta_{0}|=d(\zeta,L^{j})$ and let
$\zeta^{*}:=\psi_{j}(\phi_{j}(\zeta)/|\phi_{j}(\zeta)|)$. Furthermore, for
$0<\delta\leq\delta_{j}=e^{s_{0}/(2\omega_{j})}-1$ and $z\in L^{j}$ denote by
$z_{\delta}\in L_{\delta}^{j}$ any point with
$|z-z_{\delta}|=d(z,L^{j}_{\delta})$ and let
$z_{\delta}^{*}:=\psi_{j}((1+\delta)\phi_{j}(z))$. Applying Lemma 4(i) to the
triplets $z,z_{\delta}^{*},z_{\delta}$ and $\zeta,\zeta^{*},\zeta_{0}$ we
obtain
$d(z,L^{j}_{\delta})=|z-z_{\delta}|\asymp|z-z^{*}_{\delta}|,$
$d(\zeta,L^{j})=|\zeta-\zeta_{0}|\asymp|\zeta-\zeta^{*}|.$
Moreover, repeated application of Lemma 4 implies that for $z\in L^{j}$ and
$\zeta\in L^{j}_{\delta}$,
$|\zeta-z|\leq|\zeta-\zeta^{*}|+|\zeta^{*}-z|\preceq|z_{\delta}^{*}-\zeta^{*}|$
and
(4.1) $\displaystyle\frac{d(\zeta,L^{j})}{|\zeta-z|}$ $\displaystyle\leq$
$\displaystyle\left|\frac{\zeta-\zeta^{*}}{\zeta-z}\right|\preceq\left|\frac{\phi_{j}(\zeta)-\phi_{j}(\zeta^{*})}{\phi_{j}(\zeta)-\phi_{j}(z)}\right|^{1/Q}=\left|\frac{\phi_{j}(z^{*}_{\delta})-\phi_{j}(z)}{\phi_{j}(z^{*}_{\delta})-\phi_{j}(\zeta^{*})}\right|^{1/Q}$
$\displaystyle\preceq$
$\displaystyle\left|\frac{z_{\delta}^{*}-z}{z_{\delta}^{*}-\zeta^{*}}\right|^{1/Q^{2}}\preceq\left(\frac{d(z,L^{j}_{\delta})}{|\zeta-z|}\right)^{1/Q^{2}}.$
We claim that $L^{j}_{\delta}$ are “uniformly” quasismooth. That is, for
$\zeta_{1},\zeta_{2}\in L^{j}_{\delta}$, denote by
$L_{\delta}^{j}(\zeta_{1},\zeta_{2})$ the shorter arc of $L_{\delta}^{j}$
between $\zeta_{1}$ and $\zeta_{2}$.
###### Lemma 5
For $\zeta_{1},\zeta_{2}\in L^{j}_{\delta}$ and $0<\delta<\delta_{j}$,
(4.2) $|L_{\delta}^{j}(\zeta_{1},\zeta_{2})|\preceq|\zeta_{2}-\zeta_{1}|.$
Proof. Let
$\gamma:=L_{\delta}^{j}(\zeta_{1},\zeta_{2})=\\{\psi_{j}((1+\delta)e^{i\theta}):\theta_{1}\leq\theta\leq\theta_{2}\\},\quad
0<\theta_{2}-\theta_{1}<2\pi,$
$\phi_{j}(\zeta_{k})=:\tau_{k}=(1+\delta)e^{i\theta_{k}},\quad k=1,2.$
According to (3.1) and (3.5) there exists a sufficiently large constant
$c_{4}$ such that if
$\theta_{2}-\theta_{1}\leq\delta/c_{4}=:\varepsilon\delta$ then
(4.3) $|\gamma|\preceq
d(\zeta_{1},L^{j})\frac{|\tau_{2}-\tau_{1}|}{\delta}\asymp|\zeta_{2}-\zeta_{1}|.$
Therefore, we can assume that $\theta_{2}-\theta_{1}>\varepsilon\delta$. Let
$\gamma^{*}:=\\{\psi_{j}(e^{i\theta}):\theta_{1}\leq\theta\leq\theta_{2}\\}$.
Consider the points
$\theta_{1}=:\eta_{1}<\eta_{2}<\ldots<\eta_{p}:=\theta_{2}$ such that
$\frac{\varepsilon\delta}{2}\leq\eta_{k+1}-\eta_{k}\leq\varepsilon\delta,\quad
k=1,\ldots,p-1.$
Let
$\xi_{k}:=\psi_{j}((1+\delta)e^{i\eta_{k}}),\quad\xi_{k}^{*}:=\psi_{j}(e^{i\eta_{k}}),$
$\gamma_{k}:=\\{\zeta=\psi_{j}((1+\delta)e^{i\theta}):\eta_{k}\leq\theta\leq\eta_{k+1}\\},$
$\gamma_{k}^{*}:=\\{z=\psi_{j}(e^{i\theta}):\eta_{k}\leq\theta\leq\eta_{k+1}\\},$
By virtue of Lemma 4(i), (4.3), and our assumption that $L^{j}$ is quasismooth
we have
$|\zeta_{2}-\zeta_{1}|\asymp|\zeta_{2}-\xi_{1}^{*}|\asymp|\xi_{p}^{*}-\xi_{1}^{*}|\asymp|\gamma^{*}|$
as well as
$|\gamma_{k}|\asymp|\xi_{k+1}-\xi_{k}|\asymp|\xi_{k}^{*}-\xi_{k}|\asymp|\xi_{k}^{*}-\xi_{k+1}^{*}|\asymp|\gamma_{k}^{*}|.$
Hence,
$|\gamma|=\sum_{k=1}^{p-1}|\gamma_{k}|\asymp\sum_{k=1}^{p-1}|\gamma_{k}^{*}|=|\gamma^{*}|\asymp|\zeta_{2}-\zeta_{1}|.$
$\Box$
Further, we claim that for $z\in\partial K$ and $0<\delta\leq\delta_{j}$,
(4.4)
$\int_{L_{\delta}^{j}}\frac{d(\zeta,L^{j})}{|\zeta-z|^{2}}|d\zeta|\preceq 1.$
Indeed, the only nontrivial case is where $z\in L^{j}$. In this case,
according to Lemma 5 and [3, (3.20)], for $\alpha>0$ we have
(4.5) $\int_{L_{\delta}^{j}}\frac{|d\zeta|}{|\zeta-z|^{1+\alpha}}\leq
c_{5}d(z,L_{\delta}^{j})^{-\alpha},$
where $c_{5}$ depends on $K$ and $\alpha$. Furthermore, by virtue of (4.1) and
(4.5),
$\int_{L_{\delta}^{j}}\frac{d(\zeta,L^{j})}{|\zeta-z|^{2}}|d\zeta|\preceq\int_{L_{\delta}^{j}}\frac{d(z,L^{j}_{\delta})^{1/Q^{2}}}{|\zeta-z|^{1+1/Q^{2}}}|d\zeta|\preceq
1$
which yields (4.4).
Next, let $K^{j}=:L^{j}$ be a quasismooth arc. Below, we show how to modify
the above reasoning to obtain (4.4) as well. We only give the main ideas of
the proof. Denote by $z_{1}$ and $z_{2}$ the endpoints of $L^{j}$. For
$k=1,2$, let $w_{k}:=\phi_{j}(z_{k})$,
$E^{j}_{1}:=\\{w:1<|w|<e^{s_{0}/\omega_{j}},\arg w_{1}<\arg w<\arg w_{2}\\},$
$E^{j}_{2}:=E^{j}\setminus\overline{E_{1}^{j}},\quad
R_{k}^{j}:=\psi_{j}(E^{j}_{k}),\quad
L^{j}_{\delta,k}:=L_{\delta}^{j}\cap\overline{R_{k}^{j}}.$
According to the Ahlfors criterion (see [8, p. 100, Theorem 8.6]) $\partial
E^{j}_{k}$ is a quasiconformal curve. Moreover, repeating the proof of [5, p.
30, Lemma 2.8] practically word for word, we can show that $\partial
R^{j}_{k}$ is also a quasiconformal curve. Therefore, $\psi_{j}$ can be
extended quasiconformally into a neighborhood $N_{k}^{j}$ of
$\overline{E_{k}^{j}}$. Denote this extension by $\psi_{j,k}$. Next, we can
use Lemma 4 with $F=\psi_{j,k},E=\overline{E^{j}_{k}}$, and $G=N_{k}^{j}$ to
prove the analogues of (4.1), (4.2), and (4.5) with $L_{\delta,k}^{j}$ instead
of $L_{\delta}^{j}$ (for more detail, see the proof of [6, Lemma 2.3]). This
implies (4.4) in the case of a quasismooth arc $L^{j}$ as well. Summarizing,
we have the following result.
###### Lemma 6
If the components of $K$ are either closed Jordan domains bounded by a
quasismooth curve or quasismooth arcs, then for $z\in\partial K$ and $0<s\leq
s_{0}/2$,
$\int_{K_{s}}\frac{d(\zeta,K)}{|\zeta-z|^{2}}|d\zeta|\preceq 1.$
Proof of Theorem 2. The inequality (1.3) follows immediately from (2.16) and
Lemma 6.
$\Box$
5\. Acknowledgements
The author would like to warmly thank M. Nesterenko for many useful remarks.
## References
* [1] F. G. Abdullaev (1986): On Orthogonal Polynomials in Domains with Quasiconformal Boundary. Dissertation, Donetsk (Russian).
* [2] L. V. Ahlfors (1966): Lectures on Quasiconformal Mappings. Princeton, N.J.: Van Nostrand.
* [3] V. V. Andrievskii (2012): Weighted $L_{p}$ Bernstein-type inequalities on a quasismooth curve in the complex plane. Acta Math. Hungar., 135(1-2):8–23.
* [4] V. V. Andrievskii, V. I. Belyi, V. K. Dzjadyk (1995): Conformal Invariants in Constructive Theory of Functions of Complex Variable. Atlanta, Georgia: World Federation Publisher.
* [5] V. V. Andrievskii, H.- P. Blatt (2002): Discrepancy of Signed Measures and Polynomial Approximation. Berlin/New York: Springer-Verlag.
* [6] V. V. Andrievskii, V. V. Maimeskul (1995): Constructive description of certain classes of functions on quasismooth arcs. Russian Acad. Sci. Izv. Math., 44:193–206.
* [7] T. Kövary, Ch. Pommerenke (1967): On Faber polynomials and Faber expansions. Mathem. Zeitschr., 99:193–206.
* [8] O. Lehto, K. I. Virtanen (1973): Quasiconformal Mappings in the Plane, 2nd ed., New York: Springer-Verlag.
* [9] K. Löwner (1919): Über Extremumsätze der konformen Abbildung des Äusseren des Einheitskreises. Math. Z., 3:65–77.
* [10] F. D. Lesley, V. S. Vinge, S. E. Warschawski (1974): Approximation by polynomials for a class of Jordan domains. Mathem. Zeitschr., 138:225–237.
* [11] Ch. Pommerenke (1992): Boundary Behaviour of Conformal Maps. Berlin/New York: Springer-Verlag.
* [12] T. Ransford (1995): Potential Theory in the Complex Plane, Cambridge: Cambridge University Press.
* [13] E. B. Saff , V. Totik (1997): Logarithmic Potentials with External Fields, New York/Berlin: Springer-Verlag.
* [14] V. I. Smirnov, N. A. Lebedev (1968): Functions of a Complex Variable. Constructive Theory, Cambridge: Mass. Institute of Technology.
* [15] M. L. Sodin, P. M. Yuditskii (1993): Functions least deviating from zero on closed subsets of the real line. St. Petersburg Math. J., 4:201–249.
* [16] P. K. Suetin (1998): Series of Faber Polynomials, Amsterdam: Gordon and Breach Science Publishers.
* [17] V. Totik (2012): Chebyshev polynomials on a system of curves. Journal D’Analyse Mathématique, 118:317–338.
* [18] V. Totik (2013): Chebyshev polynomials on compact sets. Potential Anal., http://dx.doi.org/10.1007/s11118-013-9357-6.
* [19] V. Totik (2014): Asymptotics of Christoffel functions on arcs and curves. Advances in Mathematics, 252:114–149.
* [20] J. L. Walsh (1969): Interpolation and Approximation by Rational Functions in the Complex Plane, 5th ed. Providence, American Mathematical Society.
* [21] H. Widom (1969): Extremal polynomials assosiated with a system of curves in the complex plane. Adv. Math., 3:127–232.
V. V. Andrievskii
Department of Mathematical Sciences
Kent State University
Kent, OH 44242
USA
e-mail: [email protected]
tel: 330-672-9029
|
arxiv-papers
| 2014-04-13T02:27:02 |
2024-09-04T02:50:01.097037
|
{
"license": "Public Domain",
"authors": "V. V. Andrievskii",
"submitter": "Vladimir Andrievskii V",
"url": "https://arxiv.org/abs/1404.3340"
}
|
1404.3362
|
# Beyond the $\Lambda$CDM cosmology: complex composition of dark matter.
M. Demiański1,2, A.G. Doroshkevich3,
$1$Institute of Theoretical Physics, University of Warsaw, 00-681 Warsaw,
Poland
$2$Department of Astronomy, Williams College, Williamstown, MA 01267, USA
$3$Astro Space Center of Lebedev Physical Institute of Russian Academy of
Sciences, 117997 Moscow, Russia
(Accepted …, Received …, in original form … .)
###### Abstract
The mass and composition of dark matter (DM) particles and the shape of the
power spectrum of density perturbations are estimated using recent
observations of the DM dominated relaxed objects – dSph, THINGs and LSB
galaxies and clusters of galaxies. We consider the most extensive available
sample of observed objects with masses $10^{6}\leq M_{vir}/M_{\odot}\leq
10^{15}$ which includes $\sim 60$ DM dominated galaxies and $\sim 40$ clusters
of galaxies. We show that the observed characteristics of these objects are
inconsistent with expectations of the standard $\Lambda$CDM cosmological
model. However, they are well reproduced by a mixed CDM+WDM model with a
significant contribution of the HDM–like power spectrum with a relatively
large damping scale. We show that the central pressure of DM dominated objects
is surprisingly weakly dependent upon their virial mass but it is very
sensitive to the efficiency of cooling of the baryonic component. In contrast,
the central entropy of both DM and baryonic components strongly depends upon
the virial mass of halo and the period of halo formation. Unfortunately the
available data prevent our qualitative approach to reach more reliable and
definite conclusions which requires confrontation of more representative
observational data with high resolution numerical simulations.
###### keywords:
cosmology: composition of dark matter–formation of DM halos, galaxies and
clusters of galaxies.
## 1 Introduction
The nature of dark matter (DM) particles is one of the intriguing questions of
modern physics. These particles are an important element of the Standard
Cosmology, they represent $\sim 20-25\%$ of the mean matter-energy density and
explain some observed properties of the Universe (see, e.g., Komatsu 2011;
Larson 2011; Burenin & Vikhlinin 2012; Saro 2013; Ade et al. 2013; Samushia et
al. 2014). At the same time various candidates of DM particles are widely
discussed as a very important element of high energy physics. This dual role
of DM particles (see, e.g., Rubakov 2011) explains the great attention which
is recently devoted to these problems. Many possible candidates of the DM
particles are now considered. Thus, these particles may have masses ranging
from massive gravitons with $m\sim 10^{-19}eV$ and up to the supersymmetric
WIMPs with $m\sim 10^{13}GeV$. So wide range of possible masses is a result of
very weak observational restrictions and implies similar wide range of other
particle properties. In particular it is possible to note specially such
traditional candidates as axino (Choi, Kim, Roszkowski 2013), or black holes
(Carr 2014), and such exotic ones as Atomic DM (Cyr-Racine & Sigurdson, 2013)
and the flavor-mixed two component DM models (Medvedev 2014) or reincarnation
of massive neutrino models (Costanzi et al. 2013; Villaescusa-Navarro et al.
2013). Very detailed discussion of various aspects of contribution of
neutrinos to dark matter in the context of latest observations of Planck
mission, baryonic oscillations and cluster properties can be found in Verde et
al. (2013) and Wyman et al. (2014).
In turn the more and more refined observations determine evolution in time of
the DM models. Historically in cosmology DM particles were introduced in
Doroshkevich et al. (1980) and Bisnovaty-Kogan & Novikov (1980), as the Hot
Dark Matter (HDM) model with the massive neutrino as the DM particles. Soon
after the DM models were extended by introduction of the cold DM (CDM) and
warm (WDM) models (Bond, Efstathiou & Silk, 1980; Bond & Szalay, 1983;
Primack, 1984; Blumenthale et al., 1984; Bardeen et al. 1986) and even more
complex models of multicomponent (MDM) and unstable DM particles (UDM)
(Doroshkevich, Khlopov, 1984; Turner, Steigmann, Krauss, 1984; Doroshkevich,
Khlopov, Kotok, 1986; Doroshkevich, Klypin, Khlopov, 1988).
All these models were solving some actual cosmological problems but their
potential was always limited and none of them survived confrontation with
observations. The scientific progress generates more problems and poses new
questions what requires continuous modifications and development of new models
of DM particles. Thus, early in this century observations of CMB fluctuations
by the WMAP mission, SPT and other ground telescopes established the
$\Lambda$CDM model as the best cosmological model (Bennet et al. 2003; Komatsu
2011; Larson 2011; Saro 2013). This inference was supported by Planck
measurements (Ade et al. 2013), observations of clusters of galaxies (see,
e.g., Burenin & Vikhlinin 2012) and baryonic oscillations (Eisenstein & Hu
1998; Meiksin et al. 1999; Samushia et al., 2014). At that time the main hope
of identifying the “missing” cosmological components had been focused on links
of possible distortions of the kinetic of recombination and the corresponding
CMB fluctuations (see, e.g., Peebles et al. 2000; Doroshkevich et al. 2003).
On the other hand, for many years the emerging conflict between the Standard
$\Lambda$CDM theory and observations of clustering on subgalactic scales is
widely discussed. First, it is believed that the $\Lambda$CDM model predicts
an excess of low–mass satellites of Milky Way, next is the core–cusp problem
seen as a discrepancy between the observed and simulated shape of the density
profiles in central regions of relaxed objects (see, e.g., Bovill & Ricotti,
M., 2009; Koposov et al., 2009; Walker, & Penarrubia, 2011; Boylan-Kolchin et
al., 2012; Penarrubia et al. 2012; Governato et al. 2012; Sawala, 2013;
Teyssier et al. 2013; Laporte et al. 2013; Collins et al. 2014). Significance
of these conflicts is quite moderate as objects with very different masses,
densities and evolutionary histories are compared (see, e.g., Penarrubia et
al. 2008). Moreover limited reliability of these contradictions is enhanced by
limited resolution of both the observations and simulations (see, e.g.,
Mikheeva, Doroshkevich, Lukash 2007; Doroshkevich, Lukash & Mikheeva 2012;
Pilipenko et al. 2012).
During last years the analysis of the $Ly-\alpha$ forest becomes very popular
again (see, e.g. Boyarsky at al. 2009a,b, d; Dipak et al. 2012; Viel et al.
2013; Marcovi$\breve{c}$ & Viel 2013; Borde et al. 2014). However information
obtained in this way is also indirect and unreliable because there are many
problems with measurements and especially with selection and interpretation of
weak lines. Moreover properties of the $Ly-\alpha$ forest are very sensitive
to the spatial variations of the poorly known UV background (Demiański et al.
2006; Kollmeier et al. 2014)).
Attempts of direct detection of DM particles by DAMA (Bernabei 2008, 2010),
CRESST-II (Angloher et al. 2012), and SuperCDMS (Agnese 2013) experiments and
others (see review in Gaitskill 2014) have not yet produced reliable positive
results. Hence up to now we have no reliable estimates of the mass, the nature
and properties of DM particles. However, the large amount of data already
accumulated by the LHC could soon lead to detection of DM particles.
Now one of the most popular candidate for the DM particle is the sterile
neutrino with a mass in the keV range (see, e.g., reviews of Feng 2010;
Boyarsky et al., 2009c, 2013; Kusenko 2009; Kusenko & Rosenberg 2013; Dreves
2013; Horiuchi et al. 2013; Marcovi$\breve{c}$ & Viel 2013; Pontzen &
Governato 2014). Sterile neutrinos can be produced during the inflation period
or later via various processes. In particular a possible decay of some sort of
sterile neutrinos is discussed (e.g., Ferrer & Hunter 2013; Bulbul et al.
2014; Boyarsky et al. 2014). So great diversity of possible properties and
processes of generation of sterile neutrinos – from inflation and up to decays
at some redshifts – eliminates correlations between the masses and velocities
of these particles and increases uncertainties in expectations of their impact
on the power spectrum and in particular on estimates of the damping scales.
Let us note that the model with unstable DM particles implies also the
multicomponent composition of DM.
It is believed that by modeling the three mentioned above effects namely, the
$Ly-\alpha$ forest, density profile of DM dominated galaxies, the number of
observed satellites of Milky Way, and observations of the high-z gamma-ray
bursts (see e.g., reviews Boyarsky at al. 2009; Viel et al. 2013;
Marcovi$\breve{c}$ & Viel 2013; de Souza et al. 2013), it is possible to solve
the problem of sterile neutrino and in particular to restrict its mass by
$m_{\nu}\geq 10-20keV$. However these estimates restrict the damping scales
and the shape of power spectrum rather than masses of DM particles (see, e.g.,
Tremain & Gunn 1979; Ruffini et al. 2014). Recently X-ray emission with the
energy $E\sim 3.5keV$ was detected from 73 galaxy clusters what can be
interpreted either as a radiative decay of DM particles or as a recombination
line of Ar (Bulbul et al. 2014; Boyarsky et al. 2014). A spatially extended
excess of 1 – 3 GeV gamma rays from the Galactic Center could be related to
annihilation of DM particles (Daylan et al. 2014; see also Modak et al. 2013).
This discussion shows that now we do not have any reliable estimates of
properties of DM particles.
The observations of DM dominated halos are very well complemented by numerical
simulations which allow to trace and investigate the early stages of halos
formation, as well as the process of halos virialization and formation of
their internal structure. The formation of virialized DM halos begins as the
anisotropic collapse in accordance with the Zel’dovich theory of gravitational
instability (Zel’dovich 1970). During later stages the evolution of such
objects becomes again more complex because it is influenced by their
anisotropic environment (see, e.g., real cluster representations in Pratt et
al. 2009). Moreover all the time the evolution goes through the process of
violent relaxation and merging what is well reproduced by simulations.
Analysis of simulated halos can be performed in a wide range of halo masses
and redshifts what allows to improve the description of properties of relaxed
halos of galactic and cluster scales and to link them with the power spectrum
of initial perturbations. Thus it is established that after a period of rapid
evolution the main characteristics of majority of the high density virialized
DM halos become frozen and their properties are only weakly changing owing to
the accretion of diffuse matter and/or the evolution of their baryonic
component. The basic properties of the relaxed DM halos are described in many
papers (see, e.g., Tasitsiomi et al. 2004; Nagai et al. 2007; Croston et al.
2008; Pratt et al., 2009, 2010; Vikhlinin et al. 2006, 2009; Arnaud et al.
2010; Klypin et al. 2011; Kravtsov & Borgani 2012).
For the WDM model the available simulations (see, e.g., Maccio 2012, 2013;
Angulo, Hahn, Abel, 2013; Schneider, Smith & Reed, 2013; Wang et al. 2013;
Libeskind et al. 2013; Marcovi$\breve{c}$ & Viel 2013; Schultz et al. 2014;
Schneider et al. 2014; Dutton et al. 2014) show that in accordance with
expectations the number of low mass halos decreases and the central cusp in
the density profile is transformed into the core. For larger halos the
standard density profile is formed again but formation of high density objects
is accompanied by appearance of some unexpected phenomena. Thus Maccio et al.
(2012, 2013), Schneider et al. (2014) confirm the decrease of matter
concentration in the WDM model in comparison with the CDM model but they
inferred that ’standard’ WDM model is not able to reproduce the density
profile of low mass galaxies. This inference is enhanced in Libeskind et al.
(2013) where in contrast with the CDM model their simulation with 1 keV WDM
particles cannot reproduce the formation of the Local Group. In turn, Schultz
et al. (2014) note that in their simulations with 3keV WDM particles formation
of objects at large redshifts and reionization are oversuppressed. This means
that the simulations of WDM and especially the multicomponent DM models
require further detailed analysis and it is necessary to put special attention
to reproduce links between the mass function of halos and the power spectra
with the free streaming cut-off.
Without doubt, one can expect a rapid progress in simulations of more complex
cosmological models. However for preliminary discussions of such models we can
use the semi analytical description of DM dominated objects proposed in our
previous paper (Demiański & Doroshkevich 2014). It is based on the approximate
analytical description of the structure of collapsed halos formed by
collisionless DM particles. During the last fifty years similar models have
been considered and applied to study various aspects of nonlinear matter
evolution (see, e.g., Peebles 1967; Zel’dovich & Novikov 1983; Fillmore and
Goldreich 1984; Gurevich & Zybin 1995; Bryan & Norman 1998; Lithwick & Dalal
2011). Of course it ignores many important features of the process of halos
formation and is based on the assumption that the virialized DM halo is formed
during a short period of the spherical collapse at $z\approx z_{f}$ and later
on its parameters vary slowly owing to the successive matter accretion (see,
e.g., discussion in Bullock et al. 2001; Diemer et al. 2013).
Of course a spherical model can not adequately describe the real process of
halos formation. However properties of the steady state virialized DM objects
are mainly determined by the integral characteristics of protoobjects and are
only weakly sensitive to details of their evolution. This is clearly seen in
numerous simulations which show that the Navarro – Frenk – White (NFW) density
profile (Navarro et al. 1995, 1996, 1997) very stably appears in majority of
simulated DM halos.
The same simulations show also that properties of the central cores of
virialized DM halos are established during the early period of halos formation
and later on the slow pseudo– evolution of cores dominates. This means that
properties of halo cores only weakly depend on the halo periphery and are
determined mainly by their mass and the redshift of formation (Klypin et al.
2011). Using these results we formulate a rough two parametric description of
all the basic properties of virialized DM halos. These two basic parameters
are the virial mass of halos and the redshift of their formation. Of course,
they actually characterize the initial entropy of compressed DM particles and
its growth in the process of violent relaxation of the compressed DM
component. But this approach allows us also to reveal a close correlation
between the central pressure and density of DM halos and the initial power
spectrum of density perturbations.
Of course this approach is applied for the DM dominated halos only as the
dissipative evolution of the baryonic component distorts properties of the
cores of DM halos. Thus we can use this model in two ways:
1. 1.
We can use the central density and pressure of the DM component and redshift
of the DM halos formation, $z_{f}$, as a parameter that characterizes the
’frozen’ properties of the central region of DM halos. This redshift of
formation correlates also with the virial mass of halos and with the initial
power spectrum.
2. 2.
Application of the Press – Schechter (1974) approach allows us to link
together data obtained for a wide range of mass of virialized objects and thus
to restrict the shape of the initial power spectrum and some characteristics
of the DM particles.
However potential of this approach should not be overestimated. As usual we
can only determine the probability of object formation and therefore they have
statistical character rather than strict constrains or predictions. Moreover
in our discussion we use observational data of only limited quality and
representativity. Thus we can use only small number ($\leq 100$) of the
observed DM dominated objects, their observed characteristics – even so
important as the virial mass – are known only with very limited precision and
significant scatter. None the less the potential of the proposed approach is
significant as it considers objects in a wide range of masses. We hope that
further accumulation of observational data and their comparison with the high
resolution simulations will allow to essentially improve presented results.
This paper is organized as follows: In Sec. 2 the basic model, relations and
assumptions of our approach are formulated. In Sec. 3 a short description of
an approximate model of DM dominated halo is presented. Characteristics of the
observed DM dominated objects – galaxies and clusters of galaxies – are
described in Secs. 4 and 5, and in Sec. 6 properties of the central entropy
and pressure of DM dominated virialized objects are discussed. Some possible
MDM models are presented in Sec. 7. Discussion and conclusions can be found in
Sec. 8.
## 2 Cosmological models with the CDM, WDM and multi component DM
### 2.1 Cosmological parameters
In this paper we consider the spatially flat $\Lambda$ dominated model of the
Universe with the Hubble parameter, $H(z)$, the mean critical density
$\langle\rho_{cr}\rangle$, the mean density of non relativistic matter (dark
matter and baryons), $\langle\rho_{m}(z)\rangle$, and the mean density and
mean number density of baryons, $\langle\rho_{b}(z)\rangle\,\&\,\langle
n_{b}(z)\rangle$, given by Komatsu et al. (2011), Hinshaw et al. (2013):
$H^{2}(z)=H_{0}^{2}[\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}],\quad
H_{0}=100h\,{\rm km/s/Mpc}\,,$ $\langle\rho_{m}(z)\rangle=2.5\cdot
10^{-27}z_{10}^{3}\Theta_{m}\frac{g}{cm^{3}}=3.4\cdot
10^{4}z_{10}^{3}\Theta_{m}\frac{M_{\odot}}{kpc^{3}}\,,$
$\displaystyle\langle\rho_{b}(z)\rangle={3H_{0}^{2}\over 8\pi
G}\Omega_{b}(1+z)^{3}\approx 4\cdot
10^{-28}z_{10}^{3}\Theta_{b}\frac{g}{cm^{3}}\,,$ (1)
$\langle\rho_{cr}\rangle=\frac{3H^{2}}{8\pi G},\quad
z_{10}=\frac{1+z}{10},\quad\Theta_{m}=\frac{\Omega_{m}h^{2}}{0.12},\quad\Theta_{b}=\frac{\Omega_{b}h^{2}}{0.02}\,.$
Here $\Omega_{m}=0.24\,\&\,\Omega_{\Lambda}=0.76$ are the mean dimensionless
density of non relativistic matter and dark energy, $\Omega_{b}\approx 0.04$
and $h=0.7$ are the dimensionless mean density of baryons, and the Hubble
constant measured at the present epoch. Cosmological parameters presented in
the recent publication of the Planck collaboration (Ade et al. 2013) slightly
differ from those used above (1).
For this model the evolution of perturbations can be described with sufficient
precision by the expression
$\displaystyle\delta\rho/\rho\propto B(z),\quad
B^{-3}(z)\approx\frac{1-\Omega_{m}+2.2\Omega_{m}(1+z)^{3}}{1+1.2\Omega_{m}}\,,$
(2)
(Demiański, Doroshkevich, 1999, 2004, 2014; Demiański et al. 2011) and for
$\Omega_{m}\approx 0.25$ we get
$\displaystyle B^{-1}(z)\approx\frac{1+z}{1.35}[1+1.44/(1+z)^{3}]^{1/3}\,.$
(3)
For $z=0$ we have $B=1$ and for $z\geq 1,\,B(z)$ is reproducing the exact
value with accuracy better than 90%.
For $z\gg 1$ these relations simplify. Thus, for the Hubble constant and the
function $B(z)$ we get
$\displaystyle H^{-1}(z)\approx\frac{2.7\cdot
10^{16}}{\sqrt{\Theta_{m}}}s\left[\frac{10}{1+z}\right]^{3/2},\quad
B(z)\approx\frac{1.35}{1+z}\,.$ (4)
### 2.2 Power spectrum in the WDM models
The transfer function for the WDM model with thermalized DM particles was
obtained by Bode, Ostriker and Turok (2001) and more recently in Viel et al.
(2005) (see also Polisensky & Ricotti 2011; Marcovi$\breve{c}$ & Viel 2013).
In these papers the transfer function was written as
$\displaystyle T_{WDM}\approx[1+(\alpha_{w}q)^{2.25}]^{-4.46}\,,$ (5)
$q=\frac{k}{\Omega_{m}h^{2}}Mpc,\quad\alpha_{w}=6\cdot
10^{-3}\left(\frac{\Omega_{m}h^{2}}{0.12}\right)^{1.4}\left(\frac{1keV}{m_{w}}\right)^{1.1}\,,$
where $k$ is the comoving wave number.
Figure 1: Correlation functions of the matter density $\sigma_{m}$ are plotted
vs. the virial mass of objects, $M_{vir}/M_{\odot}$, for the CDM power
spectrum (Bardeen et al. 1986) (points) and for the WDM power spectrum (5)
with $m_{w}=1,\,0.1,\,0.05,\,\&\,0.03keV$ (rhombus, squares, triangles and
stars). Fits (9), and (11) are plotted by solid lines, fits (8) and (10) are
plotted by dashed and long dashed lines.
The correlation function of the density perturbations
$\displaystyle\sigma_{m}^{2}(R)=4\pi\int_{0}^{\infty}p(k)W^{2}(kR)k^{2}dk\,,$
(6)
with the standard top–hat window function $W(kR)$ has been discussed already
many years ago (see, e.g., Loeb & Barkana, 2001). Following the Press –
Schechter approach (Press & Schechter 1974; Peebles 1974; Peacock & Heavens
1990; Bond et al. 1991; Mantz et al. 2010) we can link the redshift of
formation $z_{f}$ of virialized objects with mass $M_{vir}$ with the
correlation function $\sigma_{m}(M_{vir})$ by the condition
$\displaystyle B(z_{f})\,\sigma_{m}(M_{vir})\approx const\,.$ (7)
However this approach does not allow us to obtain an independent estimate of
the small scale amplitude of perturbations. More detailed comparison of the
mass dependence of the redshift of formation of galaxies and clusters of
galaxies requires much more precise estimates of the observational parameters
of both galaxies and clusters of galaxies. In particular thus defined
$\sigma_{m}$ depends upon the shape of the transfer function (5) and for the
MDM model more accurate determination of this function is required.
For the transfer function (5) the correlation function of the density
fluctuations is plotted in Fig. 1 for four values of
$m_{w}=1,\,0.1,\,0.05,\,\&\,0.03keV$ and can be fitted with reasonable
precision by the expressions
$\displaystyle\sigma_{m}(M)=2.5/(1+0.12M_{12}^{0.45}),\quad m_{w}=0.1keV\,,$
(8) $\displaystyle\sigma_{m}(M)=1.8/(1+0.05M_{12}^{0.45}),\quad
m_{w}=0.05keV\,,$ (9)
$\displaystyle\sigma_{m}(M)=1.4/(1+0.03M_{12}^{0.45}),\quad m_{w}=0.03keV\,.$
(10)
For comparison in the same Figure the correlation function for the standard
CDM model is also plotted. It is well fitted by the expression (Klypin et al.,
2011):
$\displaystyle\sigma_{m}(M)=3.9M_{12}^{-0.077}/(1+0.18M_{12}^{0.133}+0.14M_{12}^{0.333})\,.$
(11)
As is seen from Fig. 1 for the mass of the WDM particles $m_{w}\geq 1keV$
functions $\sigma_{m}$ differ from the CDM one for objects with moderate mass
$M_{vir}\leq 10^{9}M_{\odot}$ only. For the WDM model with less massive
particles $1keV\geq m_{w}\geq 0.03keV$ the damping mass falls in the range
$10^{10}\leq M_{vir}/M_{\odot}\leq 10^{14}$. Such damping strongly decelerates
formation of objects with mass $M_{obj}\leq M_{dmp}$ (see, e.g., Schultz et
al. 2014).
### 2.3 Power spectrum in MDM models
The evolution of perturbations in MDM models was discussed many times in
different approximations (see e.g. Grishchuk & Zel’dovich 1981; Turner et al.
1984; Doroshkevich et al. 1984; Boyarsky et al. 2009b; Anderhalden et al.
2012).
It is important that in contrast with the CDM or WDM models with one mass of
particles in MDM models the shape of transfer function is time dependent.
Indeed even relatively small fraction of WDM particles with $m_{w}\leq
m_{cdm}$ significantly decelerates the growth of CDM perturbations at small
scales as compared with the standard CDM model. This deceleration decreases
with time the height of plateau in the transfer function of MDM model for
larger $k$ what leads to the progressive damping of the power spectrum at such
scales. This problem was briefly discussed by Boyarsky et al. (2009b) where
the contribution of the CDM power spectrum, $g_{cdm}$, to the full power
spectrum at redshifts $z\leq 1$ is roughly linked with the fraction of CDM
matter $f_{cdm}$ as
$\displaystyle g_{cdm}\approx f_{cdm}10^{2.58(1-f_{cdm})}\,.$ (12)
This fit neglects the time dependence of the height of plateau in the MDM
transfer function what misrepresents the shape of the MDM power spectrum and
the function $\sigma_{m}(M_{vir})$ and thus increases uncertainties of our
consideration. Non the less allowing for qualitative character of our
approach, further on, we will use this relation to roughly link the spectral
and matter fractions, $g_{cdm}$ and $f_{cdm}$. Unfortunately we do not see a
simple way to improve on these disadvantages and in what follows we will
determine the MDM correlation function by the relation
$\displaystyle\sigma_{m}(M)=\sqrt{g_{cdm}\sigma^{2}_{cdm}+g_{wdm}\sigma^{2}_{wdm}}\,.$
(13)
This relation implies statistical independence of perturbations in the CDM and
WDM mediums what is only approximately correct. More precise conclusions can
be obtained with high resolution numerical simulations.
## 3 Physical model of halos formation
Properties of both simulated and observed virialized objects – galaxies and
clusters of galaxies – are usually described in the framework of spherical
models such as Navarro – Frenk – White (NFW) (Navarro et al. 1995, 1996, 1997;
Ludlow et al. 2013), Burkert (1995) or isothermal models. In this paper we
link the virial mass of DM halos $M_{vir}$ with the redshift of their
formation, $z_{f}$. For this purpose we use the spherical model of DM halos
formation which was discussed in many papers (see, e.g., Peebles 1967, Umemura
et al., 1993, Bryan & Norman 1998). Here we will briefly describe the main
properties of this model.
It is commonly accepted that in the course of complex nonlinear condensation
the DM forms stable virialized halos with a more or less standard density
profile. Numerical simulations show that the virialized DM halos are formed
from initial perturbations after a short period of rapid complex evolution.
For example such virialized objects are observed as isolated galaxies and/or
as high density galaxies embedded within clusters of galaxies, filaments,
superclusters or other elements of the Large Scale Structure of the Universe.
This approach also allows us to estimate the redshift when the observed DM
dominated objects such us the dSph galaxies and clusters of galaxies were
formed. Of course, this model ignores all details of the complex process of
halos formation. But it allows to obtain a very simple, though rough,
description of this process and introduces some order of objects formation.
Our simple physical model of halos formation is based on the following
assumptions:
1. 1.
We assume that at redshift $z=z_{f}$ the evolution of DM perturbations results
in the formation of spherical virialized DM halos with masses
$M_{vir}=M_{13}\cdot 10^{13}M_{\odot}$ and the central densities $\rho_{c}$.
2. 2.
We do not discuss the dynamics of DM halos evolution which is accompanied by
the progressive matter accretion, the growth of the halos masses and
corresponding variations of other halos parameters. The real process of halos
formation is extended in time what causes some ambiguity in their parameters
such as the halos masses and the redshift of their formation (see, e.g.
discussion in Diemand, Kuhlen & Madau 2007; Kravtsov & Borgani 2012). In the
proposed model the redshift of halo formation, $z_{f}$, is identified with the
redshift of collapse of the homogeneous spherical cloud with the virial mass
$M_{vir}$,
$\displaystyle 1+z_{f}\approx 0.63(1+z_{tr})\,,$ (14)
where $z_{tr}$ is the redshift corresponding to the turn around moment of the
dust cloud evolution (see discussion of spherical model in Umemura, Loeb &
Turner 1993).
3. 3.
We assume that in the course of DM halo formation the main fraction of the
baryonic component is heated by the accompanied shock waves up to the
temperature and pressure comparable with the virialized values of the DM
component. These processes are responsible for the formation of equilibrium
distribution of the baryonic component.
4. 4.
We assume that some (random) fraction of the compressed baryons is collected
into a system of subclouds which are rapidly cooled and transformed into high
density subclouds. Thus, the virialized halo configuration is composed of the
DM particles, the hot low density baryonic gas, and cold high density baryonic
subclouds.
5. 5.
Transformation of less massive DM halos into the first observed galaxies with
some fraction of stars was discussed in (Demiański & Doroshkevich 2014).
The evolution of the cooled subclouds can be very complex. It can be
approximated by the isobaric mode of the thermal instability and therefore it
does not preserve the compact shape of the cooled subclouds. As was discussed
in Doroshkevich and Zel’dovich (1981) the motion of such subclouds within the
hot gas leads to their deformation and less massive subclouds could be
disrupted and even dissipated. The complex aspherical shape of such subclouds
makes their survival problematic and requires very detailed investigation to
estimate their evolution even in simulations. These problems are however
beyond the scope of this paper.
The basic parameters of the discussed model – the virial mass, $M_{vir}$,
central density, $\rho_{c}$, and concentration, $C$, connect the central and
mean densities of objects by expressions
$\displaystyle
M_{vir}=4\pi/3R_{vir}^{3}\Delta_{v}\langle\rho_{cr}(z_{f})\rangle=4\pi\rho_{c}R_{vir}^{3}C^{-3}f_{m}(C)\,,$
(15)
$\displaystyle\frac{C^{3}(z_{f},M_{vir})}{f_{m}(C)}=\frac{3\rho_{c}}{\Delta_{v}\langle\rho_{cr}(z_{f})\rangle},\quad
f_{m}=\int_{0}^{C}dx\,x^{2}\frac{\rho(x)}{\rho_{c}}\,,$ (16)
$C=R_{vir}/r_{c},\quad x=r/r_{c},\quad\Delta_{v}=18\pi^{2}\approx 200\,.$
Here $R_{vir}$ and $r_{c}$ are the virial radius of halo and the radius of
central core (for the NFW or Burkert (1995) models) or size of the isothermal
core, $\rho_{c}$ and $\langle\rho_{cr}(z_{f})\rangle$ are the central density
of halo and the critical density of the Universe at redshift $z_{f}$ (1). The
value of mean overdensity, $\Delta_{v}$, was derived from the simple model of
spherical collapse that ignores the influence of complex anisotropic halos
environment (see, e.g., Bryan & Norman 1998; Vikhlinin et al. 2009;
Lloyd–Davies et al. 2011). The well known factor $f_{m}(C)\sim 1$ links the
virial mass of an object with its concentration. Impact of the factor $f(C)$
can be determined by the method of successive approximations using, for
example, the NFW density profile with
$f_{m}(C)=\ln(1+C)-C/(1+C)\,,$
or quite similar expression for the Burkert (1995) model.
Application of this approach is possible for a given concentration
$C(z_{f},M_{vir})$. Below in section 4 we will use suitable fits for
$C(z_{f},M_{vir})$ given by Klypin et al. (2011).
Of course, this approach provides the qualitative description only and has
limited predictive power. Thus, it ignores the complex anisotropic successive
matter compression within filaments and walls before formation of compact
halos, it ignores the effects produced by mergers, by anisotropic halo
environment and so on. More detailed description can be achieved in the
framework of more complex aspherical model which would take into account
possible impact of such ignored effects as a random scatter of redshift of
halos formation and other halo characteristics for a given virial mass.
However use of such more complex models is not verified and direct comparison
of observed and simulated objects seems to be more successful.
In central regions of halos the gas pressure is supported by adiabatic inflow
of high entropy gas from outer regions of halos what leads to progressive
concentration of baryonic component within central regions of halos and to
formation of massive baryonic cores (see, e.g. Wise & Abel 2008; Pratt et al.
2009; McDonald et al. 2013).
## 4 Observed characteristics of clusters of galaxies
For our analysis we use more or less reliable observational data for $\sim
150$ DM dominated clusters of galaxies, for $\sim$40 dSph, and $\sim$20 other
DM dominated galaxies. The analysis of observations of the dSph galaxies in
the framework of accepted model was performed in Demiański & Doroshkevich
(2014). Here we continue this analysis and compare observational results with
theoretical expectations.
### 4.1 Redshift of formation of clusters of galaxies
Now there are more or less reliable observational data at least for $\sim 300$
clusters of galaxies (Pointecouteau et al. 2005; Arnaud et al., 2005; Pratt et
al., 2006; Zhang et al., 2006; Branchesi et al., 2007; Vikhlinin et al., 2009;
Pratt et al. 2010; Suhada et al. 2012; Moughan et al. 2012; Foe̋x et al. 2013;
Bhattacharya et al. 2013). However, the central cluster characteristics are
not directly observed and are obtained by a rather complex procedure (see,
e.g., Bryan & Norman 1998; Vikhlinin et al. 2009; Lloyd–Davies et al. 2011;
McDonald et al., 2013).
In spite of the speedy progress in investigations of the clusters of galaxies
recent publications discuss mainly the observations of general cluster
characteristics such as their redshift $z_{obs}$, virial mass, $M_{vir}$,
radius, $R_{vir}$, and average temperature, $T_{x}$. It is important that only
the observed redshift of clusters, and their averaged temperature are really
measured while other characteristics are usually derived using empirical
correlations (see, e.g., Pointecouteau et al. 2005; Vikhlinin et al., 2006;
Nulsen, Powell, & Vikhlinin, 2010). It is interesting that some of these
correlations can be expressed in the standard form used for description of
virialized objects,
$\beta=\frac{U}{W}\approx\frac{GM_{vir}}{R_{vir}T_{x}}=const\,,$
where $U\,\&\,W$ are the gravitational and internal energy of the object. Thus
for 179 observed clusters with masses $2\leq
M_{13}=M_{vir}/10^{13}M_{\odot}\leq 250$ we have
$\displaystyle\langle\beta\rangle=\left\langle\frac{M_{13}}{R_{Mpc}T_{kev}}\right\rangle\approx
7.16(1\pm 0.08)\,.$ (17)
In spite of limited precision of these determinations small scatter of $\beta$
demonstrates both stability of observational methods and similarity of
internal structure of clusters.
However in this paper we are mainly interested in discussion of more stable
central regions of clusters and, in particular, in concentrations and in the
central pressure of clusters. Unfortunately the body of such data is very
limited.
In this section we consider properties of the central regions of virialized DM
halos using the approximation presented in Klypin et al. (2011), Prada et al.
(2012), Angulo et al. (2012). We characterize halos by their virial mass
$M_{vir}$ and redshift of formation, $z=z_{f}$. We assume that at $z\leq
z_{f}$ the halos mass and mean temperature and density remain almost the same
and as usual we take
$\displaystyle\langle\rho_{cl}(z_{f})\rangle\simeq 500\rho_{m}(z_{f})\approx
1.25\cdot 10^{-27}(1+z_{f})^{3}g/cm^{3}\,.$ (18)
The mean baryonic number density of relaxed halos is
$\langle n_{b}(z_{f})\rangle=1.5\cdot
10^{-4}(1+z_{f})^{3}\Theta_{b}cm^{-3}\,.$
### 4.2 Observed characteristics of clusters formed at small redshifts
For clusters of galaxies with mass $M_{vir}=M_{13}\cdot 10^{13}M_{\odot}$
formed at redshifts $z\leq 1$ the concentration $C(M_{vir},z_{f})$ is given by
the expression (Klypin et al. 2011)
$\displaystyle C(M_{vir},z_{f})\approx 7.5B^{4/3}(z_{f})M_{13}^{-0.09}\,,$
(19)
and for such clusters determination of the redshift $z_{f}$ is difficult.
Indeed, comparison of the central density $\rho_{c}(M_{vir},z_{f})$ with the
mean density shows that,
$\rho_{c}(M,z_{f})\approx\frac{\langle\rho_{cl}\rangle
C^{3}}{3f_{m}(C)}\approx 1.8\cdot
10^{-25}\frac{g}{cm^{3}}\frac{D^{3}(z_{f})}{M_{13}^{0.27}f_{m}(C)}\,\,,$
$\displaystyle n_{c}(M,z_{f})\approx
0.1M_{13}^{-0.27}cm^{-3}D^{3}(z_{f})/f_{m}(C),$ (20)
$D(z_{f})=(1+z_{f})B^{4/3}(z_{f})\,,$
and the function $f_{m}(C)\sim 1$ was introduced by (16). It is important that
$D(z)$ only very weakly depends on the redshift, $D(z)\sim 1.1$ for $0\leq
z\leq 1$ and the precision of available data set makes it difficult to reveal
evolution of clusters and to use them for discussion of cosmological problems.
For 18 nearby clusters observed at $\langle z_{obs}\rangle=0.095$ and with
masses $8\leq M_{13}\leq 120$ (Pointecouteau 2005; Vikhlinin 2006;
Bhattacharya et al. 2013) the measured values of concentration are
$\langle C\rangle=3.46(1\pm 0.15),\quad f_{m}(C)\approx 0.73\,.$
These data allow us to roughly estimate the central pressure, $P_{c}$, baryon
number density, $n_{c}$, and entropy, $S_{c}$ in clusters and the redshift of
their formation as
$\displaystyle\langle P_{c}\rangle\approx 23.1(1\pm 0.5)eV/cm^{3}\,,$ (21)
$\langle n_{c}\rangle\approx 0.5\cdot 10^{-2}(1\pm
0.5)cm^{-3},\quad\left\langle\frac{1+z_{f}}{1+z_{obs}}\right\rangle\approx
1.66\,,$ $\displaystyle\langle S_{b}\rangle=\langle
P_{c}/n_{c}^{5/3}\rangle\approx 180(1\pm 0.7)cm^{2}keV\,,$ (22)
$\displaystyle\langle 1+z_{f}\rangle\approx 1.8(1\pm 0.2),\quad\langle
B^{-1}(z_{f})\rangle\approx 1.5(1\pm 0.14)\,.$ (23)
These results confirm that clusters are formed earlier than they are observed
and it is necessary to bear in mind this difference in the course of
interpretation of cluster parameters.
### 4.3 Properties of the DM halos formed at larger redshifts
Figure 2: For 83 clusters from the SPT–sample the distribution functions of
the central pressure, $F_{p}(p_{c}/\langle p_{c}\rangle)$ and central entropy
$F_{s}(0.5s_{c}/\langle s_{c}\rangle)$ are plotted.
#### 4.3.1 Theoretical expectations
For description of galactic scale halos or earlier formed DM halos of cluster
scale it is convenient to use other approximation of halo parameters (Klypin
et al. 2011; Demiański & Doroshkevich, 2014)
$\displaystyle C\approx
0.18M_{13}^{1/6}(1+z_{f})^{7/3}=0.18M_{13}^{-1/15}\eta^{7/3}\,,$ (24)
$\eta=(1+z_{f})M_{13}^{0.1}\,.$
Comparison with observations shows that for the DM dominated relaxed objects
$\displaystyle\eta\sim 3.2-3.3\,,$ (25)
remains almost the same in a wide range of virial masses and redshifts
$z_{f}$. For such halos using (16) and (24) we expect to have for the central
density of the DM matter
$\displaystyle\rho_{c}(M,z)\approx\rho_{0}(1+z_{f})^{10}M_{13}^{1/2}=\rho_{0}\eta^{10}M_{13}^{-1/2}\,,$
(26) $\rho_{0}=1.1\cdot
10^{-8}\Theta_{\rho}\frac{M_{\odot}}{pc^{3}},\quad\Theta_{\rho}=\frac{\delta_{r}}{f_{m}(C)}\frac{\Delta_{v}}{200}\Theta_{m}\,,$
and for the central baryonic number density
$\displaystyle
n_{b}=0.14cm^{-3}(\eta/3.3)^{10}M_{13}^{-1/2}\Theta_{b}\Theta_{\rho}\,.$ (27)
Here $\Theta_{\rho}$ describes the random variations of the central density
($\delta_{r}$) and uncertainties in determination of the observed parameters.
As was shown in Demiański & Doroshkevich (2014) for the concentration (24) the
central pressure depends only upon $\eta(M_{vir},z_{f})$ (25) and we get
$\displaystyle P_{c}(M_{vir},z_{f})\approx P_{0}(\eta/3.3)^{40/3},\quad
P_{0}\approx 28eV/cm^{3}\,.$ (28)
It can be expected that it is only weakly sensitive to other parameters of
clusters. For the central entropy we get
$\displaystyle S_{c}=P_{c}/n_{b}^{5/3}\approx
0.76(3.3/\eta)^{10/3}M_{13}^{5/6}cm^{2}keV\,.$ (29)
#### 4.3.2 Properties of SPT clusters
These expectations can be compared with parameters of 83 clusters selected by
the South Pole Telescope (Reinhardt et al. 2013; McDonald et al. 2013; Ruel et
al. 2013; Saliwanchik et al. 2013). For this SPT-sample of clusters the
central baryonic density, temperature and entropy are given at radius $r\leq
0.012R_{500}$ where $R_{500}$ is the radius of a sphere within which the
average density is 500$\varrho_{crit}(z_{abs})$ . For 31 clusters also
standard X-ray masses derived from Chandra observations (Ruel et al. 2013) are
known.
For this sample the distribution functions of the central pressure and entropy
are plotted in Fig. 2 where
$\displaystyle\langle P_{c}\rangle\approx 146eV/cm^{3},\quad\langle
S_{c}\rangle\approx 145keVcm^{2}\,.$ (30)
As is seen from this Figure this sample is clearly divided into two groups.
One of them contains 39 clusters with higher central pressure and low entropy
$\displaystyle\langle P_{col}\rangle=270eV/cm^{3},\quad\langle
S_{col}\rangle=100keVcm^{2}\,.$ (31)
For these clusters both the baryonic density and pressure are extremely high,
while the entropy is small. This is explained by strong cooling and clumping
of the observed gaseous component. It can be expected that in these clusters
owing to the thermal instability the baryonic matter forms two fractions, one
of which is represented by a system of high density low temperature clouds and
the other is composed of high temperature low density gas. In this case the
measured density relates to the denser fraction while the temperature relates
to the hot gas and random velocities of clouds (see, e.g., Khedekar et al.
2013). If this interpretation is correct then the measured $P_{c}$ and $S_{c}$
(31) are artificial and the real central pressure and entropy of the hot
component are close to that measured for the other 44 clusters presented below
while the entropy of the cold component is less than (31).
For the subsample of 44 clusters with $1\leq M_{13}\leq 100$ and moderate
central pressure $P_{c}\leq 70eV/cm^{3}$, we have
$\displaystyle\langle P_{c}\rangle\approx 36.1(1\pm
0.37)eV/cm^{3},\quad\eta\approx 3.36(1\pm 0.04)\,,$ (32)
$\kappa(n_{c},T_{c})\approx-0.67\,,$
where $\kappa(f_{1},f_{2})$ is the standard correlation coefficient
$\displaystyle\kappa(f_{1},f_{2})=(\langle f_{1}f_{2}\rangle-\langle
f_{1}\rangle\langle f_{2}\rangle)/\sigma_{1}/\sigma_{2}\,.$ (33)
However this subsample is naturally divided into two groups. Hotter group
accumulates 24 clusters with
$\langle n_{c}\rangle=0.5\cdot 10^{-2}(1\pm
0.4)cm^{-3},\quad\kappa(n_{c},T_{c})\approx-0.65,$ $\displaystyle\langle
P_{c}\rangle\approx 36.2(1\pm 0.35)eV/cm^{3},\quad\langle
T_{c}\rangle=7.8(1\pm 0.3)keV\,,$ (34) $\langle S_{b}\rangle=305(1\pm
0.5)cm^{2}keV,\quad\kappa(P_{c},S_{b})=-0.2\,.$
For 20 colder clusters we get
$\langle n_{c}\rangle=1.8\cdot 10^{-2}(1\pm
0.6)cm^{-3},\quad\kappa(n_{c},T_{c})\approx-0.54\,,$ $\displaystyle\langle
P_{c}\rangle\approx 35.9(1\pm 0.39)eV/cm^{3},\quad\langle
T_{c}\rangle=2.3(1\pm 0.4)keV\,,$ (35) $\langle S_{b}\rangle=41(1\pm
0.6)cm^{2}keV,\quad\kappa(P_{c},S_{b})=-0.1\,.$
The noticeable correlation of the cluster temperature and density together
with negligible correlation between cluster pressure and entropy allows to
obtain more detailed description of the process of DM compression and
successive violent relaxation of the compressed matter. As is seen from (34,
35) both the central pressure and entropy contain some regular term depending
upon characteristics of clusters (for (34)) and upon cooling of baryonic
component (for (35)). The random fluctuations of the pressure and entropy,
$\delta P_{c}/P_{c}\,\&\,\delta S_{c}/S_{c}$, with
$\sigma_{p}^{2}=\langle(\delta P_{c}/P_{c})^{2}\rangle\approx
0.16,\quad\sigma_{s}^{2}=\langle(\delta S_{c}/S_{c})^{2}\rangle\approx 0.3\,,$
are (almost) independent. Neglecting their correlation,
$\kappa(P_{c},S_{b})=0$, it is easy to see that the dispersions of the central
density, $\sigma_{n}$, and temperature, $\sigma_{T}$, and their correlation
coefficient, $\kappa(n_{c},T_{c})$, are simple functions of
$\sigma_{p}\,\&\,\sigma_{s}$. Indeed, for $P_{c}=S_{b}n_{c}^{5/3}=T_{c}n_{c}$
$\sigma_{n}=0.6\sqrt{\sigma_{p}^{2}+\sigma_{s}^{2}}\sim
0.45,\quad\sigma_{T}=0.6\sqrt{0.44\sigma_{p}^{2}+\sigma_{s}^{2}}\sim 0.36,$
$\kappa(n_{c},T_{c})\approx-0.6,\,{\rm for}\,\,\sigma_{p}/\sigma_{s}\approx
0.7\,,$
what is comparable with (34, 35). These properties of perturbations seem to
suggest that uncorrelated pressure and entropy perturbations are fundamental
while perturbations of density and temperature can be considered as their
consequence.
The differences in properties of 24 hotter and 20 colder clusters could be
mainly caused by cooling of the baryonic component. Indeed, the isobaric
thermal instability results in formation of a two phase medium – cold denser
clouds moving within hot gas – but it does not perturb the gas pressure. Weak
scatter of the central pressure of these clusters (32) shows that this
pressure as well as the entropy of 24 hot clusters characterize the regular
process of violent relaxation of the dominant DM component while the decrease
of entropy for 20 colder clusters is naturally explained by the cooling of
baryonic component. The random uncorrelated scatter of the central pressure
and entropy are naturally related to random uncorrelated variations in the
initial state of the compressed matter.
Figure 3: For 9 clusters from the SPT–sample the virial factor
$\beta=M_{13}/R_{vir}/T$, redshift creation, $1+z_{f}$ and central pressure,
$P_{c}eV/cm^{3}$ are plotted vs. the masses $M_{14}$. Fits (36, 37,& 38) are
plotted by dashed lines
For 9 clusters of this subsample also their X-ray mass was determined. For
these clusters with $10\leq M_{13}\leq 80$ and
$\displaystyle\langle\beta\rangle=\left\langle\frac{M_{13}}{R_{x}T_{x}}\right\rangle\approx
8(1\pm 0.18),\quad\langle 1+z_{obs}\rangle=1.7(1\pm 0.12)\,,$ (36)
we get for the pressure, $P_{c}$ & $\eta$, baryonic number density, $n_{b}$,
and the entropy, $S_{b}$:
$\langle n_{b}\rangle\approx 0.7\cdot 10^{-2}(1\pm 0.4)cm^{-3}\,,$
$\displaystyle\langle P_{c}\rangle\approx 34.2(1\pm 0.24)eV/cm^{3}\,,$ (37)
$\langle\eta\rangle\approx 3.35(1\pm 0.02)\,,$ $\langle S_{b}\rangle=200(1\pm
0.7)cm^{2}keV\,.$
It is important that if the central pressures of both subsamples (34) and (35)
are close to the expected one (28) then the observed central entropies (34)
and (35) noticeably exceed the expected value (29). Reasons of these
divergences are unknown but it can be expected that the progressive growth of
small scale perturbations and their following randomization can sufficiently
increase the large scale entropy of matter compressed in clusters of galaxies.
This problem requires further analysis in simulations.
The redshift of cluster formation, $z_{f}$ can be determined by two methods.
Firstly, we could use the measurements of the baryonic number density and find
the concentration, $C$, and $z_{f}$ from (16) and (26). However, the
$n_{c}\,\&\,T_{c}$ are observed at the radius $r\sim 0.012R_{vir}$, where
radial variations of density and temperature can be significant, what
generates additional uncertainties in the estimated value of $z_{f}$. One can
also use measurements of pressure and parameter $\eta$ from (28). In central
regions of DM halos radial variations of pressure are small, scatter of $\eta$
is negligible and therefore precision of so determined $z_{f}$ depends mainly
on the precision of measurements of the cluster mass, $M_{13}$, in Eq. (25).
Using the second method we get for these 9 clusters
$\displaystyle\langle 1+z_{f}\rangle\approx 2.35(1\pm 0.1)\approx 3.3(1\pm
0.02)M_{13}^{-0.1}\,,$ (38) $\langle B^{-1}(z_{f})\rangle\approx 1.74(1\pm
0.1)\,.$
To improve representativity of the subsample of 9 SPT clusters (37, 38) we can
extend it up to 31 objects setting $\eta=3.3$ for all objects with measured
X-ray masses. This assumption agrees with inference (32) that for all clusters
the pressure in central regions is almost the same and it weakly depends upon
the virial masses and redshifts of formation of clusters. Thus for subsample
of 31 clusters with $1\leq M_{13}\leq 150$ we get
$\langle 1+z_{obs}\rangle=1.66(1\pm 0.15)\,,$ $\displaystyle\langle
1+z_{f}\rangle=\langle 3.3M_{13}^{-0.1}\rangle\approx 2.1(1\pm 0.05)\,,$ (39)
$\langle B^{-1}(z_{f})\rangle=1.66(1\pm 0.1)\,.$
Similarity of these results and those obtained above for the subsample of 9
clusters (38) confirms validity of this approach.
#### 4.3.3 Properties of REXCESS and Bolocam clusters
It is interesting to compare the pressure, density and redshift of formation,
(37 & 38) with the same values obtained for 9 clusters of REXCESS survey
(Arnaud et al. 2010; Pratt et al. 2010) with $10\leq M_{13}\leq 75$
$\displaystyle\langle P_{c}\rangle\approx 21.2(1\pm 0.51)eV/cm^{3}\,,$ (40)
$\langle n_{b}\rangle\approx 0.33\cdot 10^{-2}(1\pm
0.54)cm^{-3},\quad\langle\eta\rangle\approx 3.2(1\pm 0.05)\,,$
$\displaystyle\langle 1+z_{f}\rangle\approx 2.3(1\pm 0.1),\quad\langle
B^{-1}(z_{f})\rangle\approx 1.7(1\pm 0.1)\,.$ (41)
Moderate differences of the pressure $\langle P_{c}\rangle$ observed in very
different clusters (21, 32, 37, 40) demonstrate the high stability of these
parameters. Unfortunately, for these clusters the central temperature was not
measured so we have to use our estimates (26, 28) instead.
This comparison can be continued for the sample of 45 massive galaxy clusters
imaged using the Bolocam for which pressure profiles were measured (Sayers et
al. 2013). These clusters with masses $23\leq M_{13}\leq 420$, temperature
$4.4keV\leq\langle kT\rangle\leq 14keV$ and outer pressure $2.8eV/cm^{3}\leq
P_{500}\leq 14.9eV/cm^{3}$ are situated at redshifts $0.151\leq z\leq 0.888$.
For this sample the central pressure at $r\sim 0.07R_{500}$ is
$\displaystyle\langle P_{c}\rangle\sim 50(1\pm
0.2)eV/cm^{3}M_{15}^{2/3}E^{4/3}(z)\,,$ (42)
with $E(z)=(1+(1+z)^{3}/3)$. For clusters with the mass $M_{15}\leq 1$ this
result is also close to (32), (37), and (40). For these clusters the central
temperature is also unknown and the central pressure is estimated with (26,
28).
### 4.4 Observed density profile of SPT clusters of galaxies
For all 83 objects of the SPT–catalogs the baryonic density slope $\alpha$ at
a distance $r\simeq 0.04R_{500}$ is also measured and it is justly linked with
the process of cooling of the compressed gas. Here we confirm that this slope
strongly correlates with the density of central regions of clusters,
$\displaystyle\kappa(\alpha,n_{b})\approx 0.76\,,$ (43)
where the correlation coefficient $\kappa$ is obtained according to (33). So
strong correlation indicates that the steep profile is caused by the limited
resolution of observations. Indeed the cold high density gaseous clouds
naturally arise in the central regions of many clusters owing to significant
density fluctuations that are enhanced by isobaric modes of the thermal
instability. As was demonstrated in Doroshkevich, Zel’dovich (1981) peculiar
motions of such clouds sometimes lead to their deformation and even complete
disruption.
These results are fully consistent with conclusions of Arnaud et al. (2010)
where similar entropy and density gradients were found for the set of REXCESS
clusters. The close link between the power index and the baryonic density is
indicated by their correlation coefficient.
## 5 Observed properties of the DM dominated galaxies
Several DM dominated objects of galactic scale are known. These are the 41
dSph galaxies, 14 THING and 10 LSB galaxies. Analysis of these objects can be
performed in the same manner as done above.
### 5.1 Observed properties of the dSph galaxies
During last years properties of dSph galaxies were discussed in detail in many
papers. Thus, the main observed parameters of 28 dSph galaxies are listed and
discussed in Walker et al. (2009, 2011), Penarrubia et al., (2010), and 13 And
galaxies with similar properties are listed in Tollerud et al. (2012). These
samples include objects in a wide range of masses, $0.1\leq
M_{6}=M_{gal}/10^{6}M_{\odot}\leq 100$, what allows us to reveal more reliably
the mass dependence of their redshift of formation (Demiański & Doroshkevich
2014). In this case we have to deal with parameters of the central regions at
the projected half–light radius but their reliability is limited and scatter
is large. In spite of this it is interesting to compare characteristics of
these galaxies with characteristics of clusters of galaxies presented in this
Section and with theoretical expectations.
Figure 4: For 23 dSph galaxies functions $P,\,S/M_{6}^{0.87},$ and
$\eta_{6}=z_{f}M_{6}^{0.1}$ are plotted vs. the masses $M_{6}$. Fits (44) &
(45) are plotted by dashed lines
Thus for these galaxies we have for the central pressure, $P_{c}$, baryonic
density, $n_{b}$, and entropy, $S_{b}$
$\langle n_{b}\rangle\approx 28(1\pm
0.78)M_{6}^{-0.5}cm^{-3},\quad\langle\eta\rangle\approx 3.2(1\pm 0.1)\,,$
$\displaystyle\langle P_{c}\rangle\approx 28(1\pm 0.9)eV/cm^{3}\,,$ (44)
$\langle S_{b}\rangle=36(1\pm 0.35)M_{6}^{0.87}eV\cdot cm^{2}\,.$
It is important that in spite of large scatter of measured characteristics for
these galaxies the pressure $P_{c}$ (44) is quite similar to the pressure (21,
32, 37 & 40) found above for clusters of galaxies. As was discussed in
Demiański & Doroshkevich (2014) these results agree well with expectations of
simulations (Klypin et al., 2011) and reflect some important intrinsic
properties of violent relaxation and formation of DM dominated virialized
objects.
For the redshift of formation of the dSph galaxies we get (Demiański &
Doroshkevich 2014)
$\displaystyle\langle 1+z_{f}\rangle\approx 3.3(1\pm 0.12)M_{13}^{-0.1}\,,$
(45) $\langle B^{-1}(z_{f})\rangle\approx 2.4(1\pm 0.12)M_{13}^{-0.1}\,.$
### 5.2 Direct estimates of mass of the DM particles
For clusters of galaxies the thermal velocities in central regions are
$v_{c}\sim 100-1000km/s$ and they are clearly generated in the course of
violent relaxation of the compressed matter. In contrast, the observed
velocity dispersion of the dSph galaxies is not so large, $\sigma_{obs}\leq
10km/s$, what allows to obtain direct rough estimates of the velocity,
free–streaming scale and mass of DM particles accumulated by these galaxies.
One example of such estimates can be found in Boyarsky et al. (2009d) where
for the mass of WDM particle four values were obtained in the range
$\displaystyle m_{wdm}\geq 0.4-2.8keV\,.$ (46)
Here we can get similar estimates using properties of low mass dSph galaxies
with minimal velocity dispersions $\sigma_{obs}\sim 3-4km/s$.
Assuming that formation of these galaxies is accompanied by the adiabatic
compression of weakly perturbed DM we can estimate the random velocities of
the same population of DM particles before compression, $\sigma_{homo}$. For
the four low mass dSph galaxies this velocity dispersion is
$\displaystyle\sigma_{homo}(z=0)=\sigma_{obs}\left(\frac{\langle\rho_{m}(z=0)\rangle}{\rho_{c}}\right)^{1/3}\sim
0.01km/s\,,$ (47)
and their comoving radius is
$\displaystyle R_{homo}=(3M/4\pi\langle\rho_{m}(z=0)\rangle)^{1/3}\approx
11kpc\,.$ (48)
The mass of these DM particles can be estimated from the temperature at the
redshift when these particles become non relativistic (Doroshkevich et al.
1980):
$\displaystyle m_{v}c^{2}\sim 3.5kT_{\gamma}\frac{c}{\sigma_{homo}}\sim
22keV\,.$ (49)
According to the estimates of Bardeen et al. (1986) for such particles the
free–streaming scale is
$\displaystyle R_{f}\sim(50-100)kpc(1keV/m_{v})\sim(2-5)kpc\,,$ (50)
what is even less than (48). The low precision of measurements of both
$\sigma_{obs}$ and $\rho_{c}$ as well as a possible growth of $\sigma_{obs}$
owing to the violent relaxation makes the estimates (47) and (49) very rough.
Non the less, they demonstrate that probably among the population of
cosmological DM particles there is a subpopulation with the mass (49) and the
damping scale (50).
### 5.3 Observed properties of the THING and LSB galaxies
The number of observed DM dominated galaxies is very small and therefore we
will use observations of all objects for which the influence of DM component
is significant.
Here we consider 14 THING galaxies (de Blok et al. 2008) for which the
contribution of DM component seems to be important. The virial mass of these
galaxies can be roughly found from published rotation curves while estimates
of their central density are given in de Blok et al. (2008). Similar estimates
of mass and density of these galaxies are presented also in Chemin et al.
(2011). However the central temperature of these galaxies is not known and
further analysis is based on estimates (26, 28). Using relation (26) we can
estimate the parameter $\eta$ and redshift $z_{f}$ for these 14 galaxies with
the virial masses $5\cdot 10^{9}\leq M_{vir}/M_{\odot}\leq 7\cdot 10^{11}\,:$
$\langle\rho_{c}\rangle\approx 3.2\cdot 10^{-2}(1\pm
0.8)M_{\odot}/pc^{3},\quad\langle\eta\rangle\approx 3.0(1\pm 0.1)\,,$
$\displaystyle\langle 1+z_{f}\rangle\approx 5.0(1\pm 0.2),\quad\langle
B^{-1}(M_{vir})\rangle\approx 3.7(1\pm 0.2)\,.$ (51)
The central pressure and entropy for this sample can be found with (28) with
large scatter
$\displaystyle\langle P_{c}\rangle\approx 37(1\pm 0.9)eV/cm^{3}\,,$ (52)
$\langle S_{c}\rangle\approx 27cm^{2}eV(1\pm 0.9)\,,$
which can be partly related to the stronger influence of the cooling process
of baryonic component.
For the sample of LSB galaxies the masses and central densities are discussed
in Kuzio de Naray et al. (2008). As before for these galaxies the central
temperature was not measured and our analysis is based on estimates (26, 28).
For 10 LSB galaxies with $10^{9}\leq M_{vir}/M_{\odot}\leq 2\cdot 10^{11}$ we
get
$\langle\rho_{c}\rangle\approx 2.7\cdot 10^{-2}M_{\odot}/pc^{3}(1\pm
0.75),\quad\langle\eta\rangle\approx 3.1(1\pm 0.1)\,,$ $\displaystyle\langle
1+z_{f}\rangle\approx 5.9(1\pm 0.1),\quad\langle B^{-1}(M_{vir})\rangle\approx
4.4(1\pm 0.1)\,.$ (53)
Here the parameter $\eta$ is determined by (25) and it coincides with the
expected one. For this sample the central pressure and entropy can also be
found with (28) with large scatter
$\displaystyle\langle P_{c}\rangle\approx 14(1\pm 0.8)eV/cm^{3}\,,$ (54)
$\langle S_{c}\rangle\approx 19cm^{2}eV(1\pm 0.7)\,,$
which also can be partly related to the stronger random influence of the
cooling process of baryonic component.
It is necessary to bear in mind that for these objects the virial mass is
under estimated while the central density and $z_{f}$ are over estimated owing
to the possible excess of baryons. Non the less the obtained value of
$\langle\eta\rangle$ (51) and (53) shows that the final results are
sufficiently reasonable.
These results for both THING and LSB galaxies are plotted in Fig. 5 & 6.
## 6 Central pressure and entropy in DM dominated virialized objects
Our analysis revealed some unexpected peculiarities in the internal structure
of DM dominated virialized objects. First of them is a very weak dependence of
the central pressure of such halos (21, 32, 37, 40, 42, 44, 52, & 54) on the
virial mass, redshift of formation and other characteristics of these objects.
In contrast, the central entropy of clusters of galaxies (22, 34, 35)
significantly exceeds the entropy of DM dominated objects of galactic scale
(44, 52, 54). For the dSph, THING and LSB galaxies and for 9 SPT (37) and 18
nearby (21, 22) clusters of galaxies the central observed pressure, $P_{c}$,
density, $\rho_{c}$, and entropy, $S_{c}$ are plotted in Fig. 5 and fitted by
(55) :
$\displaystyle P_{c}\approx 36eV/cm^{3},\quad\rho_{c}\approx 2.\cdot
10^{-3}M_{12}^{-0.4}M_{\odot}/pc^{3}\,,$ (55) $S_{c}\approx
3M_{12}^{0.7}keVcm^{2},\quad M_{12}=M_{vir}/10^{12}M_{\odot}\,.$
The strong correlation of $n_{c}\&T_{c}$ together with weak correlation of
$P_{c}\,\&\,S_{c}$ discussed in Sec. 4.3.2 confirm the objective character of
these inferences
Figure 5: The central pressure, density and entropy of virialized DM dominated
objects are plotted vs. mass for dSph (points), THING (triangle) and LSB
(rhombus) galaxies and for 9 SPT and 18 nearby (21, 22) clusters of galaxies
(stars) with x-ray masses and hot baryonic component. Fits (55) are plotted by
dashed lines.
Of course these effects have only statistical significance what implies
natural scatter of measured central pressures and entropy and the parameter
$\eta$ (25). However both the limited precision of observations and the impact
of cooling process of baryonic component significantly increase the random
scatter of real measurements.
In order to explain so different behavior of central pressure and entropy it
is necessary to remind that if the central pressure is determined mainly by
the dynamical equilibrium of compressed DM dominated component then the
central entropy includes two components, namely, the initial entropy of
compressed matter and entropy generated in the course of compression and
relaxation. The relative contribution of these components depends upon the
halo mass and redshift of formation and our results indicate that the
contribution of the first component progressively increases together with the
virial mass of formed halos. This inference agrees with the well known regular
growth of entropy with radius within virialized objects, $S\propto M_{vir}r$,
what indicates the progressive generation of entropy in the course of matter
relaxation. It is quite interesting to trace this behavior in more details in
both observed and simulated clusters.
Thus first galaxies such as dSph ones are formed from cold DM particles and
baryonic component with very low entropy (see, e.g. Demiański & Doroshkevich
2014). This implies that for these objects the central entropy of both DM and
baryonic components (44) is mainly generated in the course of objects
formation. In contrast, for later formed massive clusters of galaxies the
contribution of initial entropy progressively increases and becomes more and
more essential.
The initial entropy of baryonic component can be partly related to the
progressive heating of intergalactic gas by ionizing UV background. For
redshifts $z\leq 3$ when strong photo ionization of HeI and HeII is caused by
the hard UV radiation of quasars the entropy for slightly perturbed baryons
and 3D Hubble expansion can be estimated as
$T_{b}\sim 0.7eV(1+z)^{6/7},\quad S_{b}\sim 18(1+z)^{-8/7}cm^{2}keV\,.$
This entropy strongly exceeds the entropy of the THING and LSB galaxies and is
more similar to the observed entropy of low mass clusters of galaxies. The
Jeans mass of such baryonic component increases up to
$M_{J}\approx
10^{6}M_{\odot}\left(\frac{T}{10^{4}K}\right)^{3/2}\left(\frac{1cm^{-3}}{\langle
n_{bar}\rangle}\right)^{1/2}\sim 10^{10}M_{\odot}(1+z)^{0.2}\,,$
and the formation of less massive objects is sharply decelerated.
At redshifts $z_{f}\geq 3$ the quasars contribution to ionizing UV radiation
is small and the generated entropy of baryons depends upon the shape of more
soft spectrum of the UV background. Thus the observed entropy of THING (52)
and LSB (54) galaxies is similar to the entropy of dSph galaxies (44) what
shows that for them the contribution of initial entropy is small. This means
that probably at redshifts $z_{f}\geq 3$ the ionization of intergalactic gas
is not accompanied by its essential heating.
On the other hand extraordinary efforts are required in order to increase the
much more conservative entropy of DM component. As was noted above, perhaps,
the progressive growth of small scale perturbations and their following
randomization is the most promising way to increase the large scale entropy of
both DM and baryonic components and to explain the observed high entropy of
clusters of galaxies. The problem requires further analysis in simulations.
The high stability of the central pressure for DM dominated objects was
already noted in our previous paper (Demiański & Doroshkevich 2014) and here
it is confirmed with wider observational base. It can be related to the
combined influence of the violent relaxation and of the regular shape of the
initial power spectrum of density perturbations and so, also velocity
perturbations.
It is important that for the CDM model simulations show similarity of the
dimensionless characteristics of DM halos such as the density and pressure
profiles. In particular the density profiles are found to be close to NFW or
Burkert (1995) ones with moderate variations of concentration $C\sim 3-5$. On
the other hand the regular character of the CDM (Bardeen et al. 1986) and WDM
(5) initial power spectra links together the virial mass of DM objects with
their redshifts of formation and mean densities as this is demonstrated by
Eqs. (25, 26, 28) and is discussed in the next Section. So the impact of the
growth with time of the virial mass of formed objects is compensated
approximately by corresponding drop of their mean density (see, e.g., (18,
26)).
## 7 MDM cosmological models
Simulations show that characteristics of the virialized DM halos are much more
stable than the characteristics of baryonic component and after formation at
$z=z_{f}$ of virialized DM halos with $\langle\rho_{vir}\rangle\approx
18\pi^{2}\langle\rho(z_{f})\rangle$ slow matter accretion only moderately
changes their characteristics (see, e.g., Diemer et al. 2013). Because of
this, we can observe earlier formed high density galaxies with moderate masses
even within later formed more massive but less dense clusters of galaxies,
filaments and other elements of the Large Scale Structure. This means that
using the model presented in Sec. 3 for description of the observed dSph,
THING and LSB galaxies and clusters of galaxies dominated by DM component we
can find one–to–one correspondence between their observed parameters and the
so called redshift of object formation, $z_{f}$. Of course according to the
Press – Schechter approach (Press, Schechter, 1974; Peebles 1974; Bond et al.
1991) these redshifts characterize the power spectrum of the density
perturbation rather than the real period of the object formation.
Following the Press – Schechter ideas (7) we compare the function
$\sigma_{m}(M)$ for various DM models with the observed function
$B^{-1}(M_{vir})$ obtained in the previous Sections in a wide range of masses
and redshifts. Such comparison allows us to quantify the influence of the DM
particles on the rate of DM halos formation. Thus in Fig. 6 the function
$B^{-1}(M_{vir})$ is plotted for the dSph, THING and LSB galaxies and for the
sample of 31 clusters of galaxies (39) together with the function
$\sigma_{m}(M_{vir})$ calculated for the standard CDM power spectrum, and for
the MDM power spectrum (56) with $\sigma_{m}$ obtained according to (13) for
low massive WDM particles
$\displaystyle p(k)=0.27p_{cdm}+0.73p_{wdm}(m_{w}),\quad m_{w}\approx 30eV\,,$
(56)
and corresponding damping scale $M_{dmp}\sim 10^{14}M_{\odot}$. The functions
$\sigma_{m}(M_{vir})$ are normalized using the cluster points (39).
Figure 6: Correlation function of the matter density $\sigma_{m}$ for the CDM
power spectrum (11) and the combined spectrum (56) are plotted by solid and
dashed lines. Function $B^{-1}(M_{vir})$ is plotted for the 41 dSph galaxies
(left group of points) and for 31 clusters of galaxies from the SPT observed
sample (right group of points). For 14 THINGS galaxies function
$B^{-1}(M_{vir})$ is plotted by squares and for 10 LSB galaxies - by
triangles.
Of course the observational base used in this discussion is very limited and
it should be extended by adding observations of new objects with masses
$M\simeq 10^{10}-10^{12}M_{\odot}$, what may be crucial for determination of
the real composition of dark matter. Unfortunately more or less appropriate
estimates of the redshift $z_{f}$ can be obtained mainly for DM dominated
objects or for objects with clearly discriminated impact of DM and baryonic
components. Here we use results obtained for 14 THINGS galaxies (de Blok et
al. 2008) and 10 LSB galaxies (Kuzio de Naray et al. 2008). We hope that the
list of possible appropriate candidates can be extended.
Next important problem is the reliability of the approach used in our analysis
and obtained results. It depends upon the representativity of the
observational data and is low because of the very limited available data and
their significant scatter. Progress achieved during last years allows to begin
discussion of MDM models but the low reliability of available observations is
causing only qualitative character of our discussion. Indeed, the problem of
estimates of the mass, density and other parameters of the observed objects is
quite complex, methods used for such estimates are very rough and model
dependent while their reliability is limited. Moreover the influence of
baryonic component increases scatter of the measured parameters and makes it
difficult to estimate the real precision of our results. Let us hope that
because of the importance of this problem such observations will be extended
and their precision improved.
On the other hand the simulated data are focused on clusters of galaxies, what
is caused by the finite resolution of simulations. This limitation also does
not increase reliability of our inferences. Non the less let us note that the
analysis discussed in Sec. 4 is based on results of very large and high
quality simulations of the standard $\Lambda$CDM model (Klypin et al. 2011)
which embraces relatively large accessible range of object masses and
redshifts of formation.
As is seen from Figure 6 the standard CDM model cannot describe the system of
observed points and, so, should be rejected. Quality of the WDM model with
$m_{w}\approx 3keV$ is also limited and simulations (Shultz et al. 2014) show
that in this model the objects formation at high redshifts is oversuppressed.
Better description of observations is presented in Fig. 6 by the two component
DM model with parameters (56).
The important result of our analysis is the demonstration of limited
applicability of cosmological models with only one component power spectrum
and great promises of models with more complex structure of the power
spectrum. It is also important that the basic element of such complex power
spectrum is the large contribution ($\sim 70\%$) of the low mass WDM spectrum
with $m_{w}\sim 30eV$. It is noteworthy that after 30 years of absolute
domination of the CDM model we return to more complex versions of the HDM
models.
In turn such models imply existence of at least two damping scales one of
which corresponds to the mass of clusters of galaxies, $M_{dmp}\sim
10^{13}M_{\odot}$. However such partial damping of the power spectrum leads
only to a decrease of the rate of formation of less massive objects relatively
to the standard CDM–like power spectrum.
Of course all these inferences are very preliminary. Thus, here we use the WDM
power spectra with the transfer function (5) (Viel et al. 2005; Polisensky &
Ricotti 2011; Marcovi$\breve{c}$ & Viel 2013). More refined description of the
MDM power spectrum and specially further progress in observations of DM
dominated objects will change the best model parameters and estimates of
masses and composition of the MDM model. Non the less even today replacement
of some fraction of CDM particles by heavy WDM particles with $m_{w}\sim
10keV$ can be considered. Spectrum of such particles identified now (according
to majority preference) as the sterile neutrinos can be included in (56) as a
third component without noticeable changes of Fig. 6. Indeed, the available
sample of observed DM dominated objects does not yet allow us to make any
far–reaching conclusions about the actual properties of massive DM particles.
However, the noticeable contribution of low mass WDM spectrum in (56) is
crucial for the considered models
At the same time it is very important that the observed characteristics of
objects are determined mainly by the linear combination of power spectra and
so by the damping scales which in turn depend both upon particle masses and
velocities. This means that the construction of the adequate complex
cosmological model should include discussion of full DM evolution beginning
from the period of inflation with estimates of the actual damping scales and
transfer functions for all components allowing also for linear evolution of
perturbations at $z\leq z_{eq}$. Simple example of such evolution is discussed
by Boyarsky et al. (2009b) and in Sec. 2.3 .
Thus, allowing for the decay of the separate components of power spectra
according to (12) we can estimate the matter fractions of CDM and WDM
components, $f_{cdm}\,\&\,f_{wdm}$, for the model (56) with $g_{cdm}=0.27$
$\displaystyle f_{cdm}\approx 0.82,\quad f_{wdm}\approx 0.18\,.$ (57)
Unexpectedly in spite of the relatively small value of $g_{cdm}$ the fraction
of the CDM particles, $f_{cdm}$, remains significant as well as their
influence upon the evolution of small scale objects. However the widely
discussed controversial characteristics of DM objects such as the core–cusp
problem, or number of low mass satellites depend upon the dissipative scale
and the power spectrum rather than directly upon the mass or fraction of the
CDM component. This supports the hope that these simpler problems also will be
successfully resolved in the framework of the discussed MDM cosmological
models.
These discussed complex intercorrelations produce additional problems for
numerical simulations which now practically cannot simultaneously provide
suitable size of computational box and required high mass resolution. This
means that direct simulations with realistic complex power spectrum encounter
many problems and require to use model simulations with subsequent rescaling
procedures what makes difficult further comparison with observations. As usual
the additional problem is the accurate description of possible impact of
baryonic component.
So, more accurate simulations with various compositions of dark matter are
required before we will have reliable inferences about properties of the DM
component. These doubts are supported by moderate results of the first
published simulations of the WDM cosmological models (Maccio 2012, 2013;
Angulo, Hahn, Abel, 2013; Schneider, Smith & Reed, 2013; Wang et al. 2013;
Libeskind et al. 2013; Marcovi$\breve{c}$ & Viel 2013; Schultz et al. 2014;
Schneider et al. 2014; Dutton et al. 2014).
## 8 Conclusions
In this paper we discuss two important problems of modern cosmology:
1. 1.
the composition of the dark matter,
2. 2.
the internal structure of virialized DM dominated objects – the high stability
of their central pressure and dependence of the central entropy upon mass and
period of halos formation.
Summing up it is necessary to note that the proposed approach allows us to
consider and to compare properties of DM dominated objects in an
unprecedentedly wide range of masses $10^{5}\leq M_{vir}/M_{\odot}\leq
10^{15}$. This comparison unexpectedly favors MDM models for which the
domination of massive DM component is accompanied by a significant
contribution of low mass DM particles. In turn in these models the power
spectra of density perturbations at galactic scale significantly differ from
the standard CDM–like ones.
Here we consider as a quite promising the MDM model for which the power
spectrum is composed of fraction $g_{cdm}\sim 0.3$ of the CDM spectrum and
fraction $g_{wdm}\sim 0.7$ of the WDM spectrum with low mass thermalized WDM
particles and transfer function (5). The rough estimates of the mass fraction
of these components are given by (57). Further progress can be achieved with
more complex models with more realistic transfer functions instead of (5, 13)
and/or with replacement of some fraction of CDM particles by heavy WDM
particles with mass $m_{w}\geq 3keV$ (such as the sterile neutrino).
We want to emphasize that unexpectedly in the spectrum (56) of this more
promising MDM model the spectrum of low mass WDM particles with relatively
large damping scale dominates. This can be considered as reincarnation in a
new version of the earlier rejected HDM model. It is important that in the MDM
model the impact of such low mass WDM particles decelerates the growth of
perturbations and the rate of the objects formation for all objects with
masses less than the low mass clusters of galaxies or massive galaxies,
$M_{vir}\leq 10^{12}M_{\odot}$. However the contribution of CDM–like spectrum
provides successful formation of low mass halos and other structure elements.
Further development of this approach could result in further complication of
power spectra and in particular in introduction of an excess of power
localized at small scale. Such modifications allow to essentially extend
possibilities of the MDM models and even can allow to link the possible
unexpected observed properties of some set of objects of galactic scales with
peculiarities of the power spectrum. In some respect such an excess of power
reminds the isocurvature models (see, e.g. Savelainen et al. 2013) with
similar predictions and problems. However all such problematic multi
parametric proposals should be considered in the context of general
cosmological and inflationary models.
During the last decade the most reliable and interesting information about the
power spectrum comes from observations of the perturbations at the period of
recombination which are seen as the CMB fluctuations. Such observations are
performed both with satellites (Komatsu et al. 2011; Larson et al. 2011; Ade
et al. 2013) and the SPT and other ground telescopes (see, e.g., Saro et al.
2013). But they relate to large scales $L\geq 10Mpc$ only. Limited information
about the power spectrum at smaller scales can be obtained from observations
of absorption spectra of distant quasars. But properties of such absorbers are
very sensitive to spatial variations of poorly known ionizing UV background,
what makes difficult interpretation of these observations and strongly limits
their reliability.
The approach used in this paper for discussion of composition and properties
of the DM particles is very indirect. We consider effects of strongly
nonlinear multistep evolution of perturbations resulting in observed
properties of the relaxed objects. The nonlinear evolution already leads to a
strong loss of information about the primeval perturbations and composition of
the DM component. These losses are further enhanced by the masking effects of
dissipative evolution of the baryonic matter. To seek out the missing impact
of the DM composition and primeval perturbations we have to compare
observational data with numerical simulations majority of which are now
focused on studying evolution of the standard $\Lambda$CDM models.
We used the general theory of gravitational instability for the DM objects to
justify the expression (7) and Figure 6 and to quantify correlations between
the virial mass of objects, $M_{vir}$, their redshift of formation, $z_{f}$,
and the shape of the primordial power spectrum of perturbations (13) in a wide
range of masses. In this respect our approach seems to be more helpful and
informative than the earlier mentioned contradictions between the observed and
simulated characteristics of the DM halos. Non the less it provides us with
only preliminary qualitative inferences about the nature of DM particles and
cosmological models.
It can be expected that another manifestation of the same interactions is the
well known similarity of the internal structure of DM dominated virialized
objects and, in particular, the high stability of the central pressure of
these objects discussed in Secs. 4 & 6.
As is seen from Figs. 1 and 6 deviations between the CDM and MDM power spectra
appear at scales $M_{vir}/M_{\odot}\sim 10^{12}-10^{13}$. This means that
properties of massive galaxies and clusters of galaxies such as their central
pressure, entropy and specially the rate of formation are more sensitive to
the compositio n of the dark matter and for impact of low mass particles.
These possibilities were discussed in Sec. 6. Unfortunately published – and
discussed in Secs. 4 and 5 – observations of such objects are limited and
their interpretation is unreliable what do not allow us to reveal the expected
effects. Additional problem is the impact of cooling of the baryonic component
which masks the possible impact of the power spectrum and DM composition.
We believe that more detailed conclusions will be made on the basis of special
simulations and after accumulation of more representative observational sample
of high precision measurements of properties of DM dominated objects.
### 8.1 Acknowledgments
This paper was supported in part by the grant of president RF for support of
scientific schools NS4235.2014.2. We thank S.Pilipenko, B.Komberg, A.Saburova
and R.Rufini for useful comments. We wish to thank the anonymous referee for
many valuable comments and suggestions.
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|
arxiv-papers
| 2014-04-13T09:47:39 |
2024-09-04T02:50:01.105387
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Demianski, A.Doroshkevich",
"submitter": "Doroshkevich Andrei",
"url": "https://arxiv.org/abs/1404.3362"
}
|
1404.3458
|
# Novel Polynomial Basis and Its Application to Reed-Solomon Erasure Codes
Sian-Jheng Lin, Wei-Ho Chung Research Center for Information Technology
Innovation
Academia Sinica
Taipei City, Taiwan
Email: [email protected]; [email protected] Yunghsiang S. Han
Department of Electrical Engineering
National Taiwan University of Science and Technology
Taipei City, Taiwan
Email: [email protected]
###### Abstract
In this paper, we present a new basis of polynomial over finite fields of
characteristic two and then apply it to the encoding/decoding of Reed-Solomon
erasure codes. The proposed polynomial basis allows that $h$-point polynomial
evaluation can be computed in $O(h\log_{2}(h))$ finite field operations with
small leading constant. As compared with the canonical polynomial basis, the
proposed basis improves the arithmetic complexity of addition, multiplication,
and the determination of polynomial degree from
$O(h\log_{2}(h)\log_{2}\log_{2}(h))$ to $O(h\log_{2}(h))$. Based on this
basis, we then develop the encoding and erasure decoding algorithms for the
$(n=2^{r},k)$ Reed-Solomon codes. Thanks to the efficiency of transform based
on the polynomial basis, the encoding can be completed in $O(n\log_{2}(k))$
finite field operations, and the erasure decoding in $O(n\log_{2}(n))$ finite
field operations. To the best of our knowledge, this is the first approach
supporting Reed-Solomon erasure codes over characteristic-2 finite fields
while achieving a complexity of $O(n\log_{2}(n))$, in both additive and
multiplicative complexities. As the complexity leading factor is small, the
algorithms are advantageous in practical applications.
## I Introduction
For a positive integer $r\geq 1$, let $\mathbb{F}_{2^{r}}$ denote a
characteristic-2 finite field containing $2^{r}$ elements. A polynomial over
$\mathbb{F}_{2^{r}}$ is defined as
$a(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dots+a_{h-1}x^{h-1},$
where each $a_{i}\in\mathbb{F}_{2^{r}}$. A fundamental issue is to reduce the
computational complexities of arithmetic operations over polynomials. Many
fast polynomial-related algorithms, such as Reed-Solomon codes, are based on
fast Fourier transforms (FFT). However, it is algorithmically harder as the
traditional fast Fourier transform (FFT) cannot be applied directly over a
characteristic-2 finite fields. To the best of our knowledge, no existing
algorithm for characteristic-2 finite field FFT/polynomial multiplication has
provably achieved $O(h\lg(h))$ operations111Throughout this paper, the
notation $\lg(x)$ represents the logarithm to the base $2$. (see Section VII
for more details).
In algorithmic viewpoint, FFT is a polynomial evaluations at a period of
consecutive points, where the polynomial is in monomial basis. This viewpoint
gives us the ability to design fast polynomial-related algorithms. In this
paper, we present a new polynomial basis in the polynomial ring
$\mathbb{F}_{2^{r}}[x]/(x^{2^{r}}-x)$. Then a transform in the new basis is
defined to compute the polynomial evaluations. The new basis possesses a
recursive structure which can be exploited to compute the polynomial
evaluations at a period of $h$ consecutive points in time $O(h\lg(h))$ with
small leading constant. Furthermore, the recursive structure also works in
formal derivative with time complexity $O(h\lg(h))$.
An application of the proposed polynomial basis is in erasure codes, that is
an error-correcting code by converting a message of $k$ symbols into a
codeword with $n$ symbols such that the original message can be recovered from
a subset of the $n$ symbols. An $(n,k)$ erasure code is called Maximum
Distance Separable (MDS) if any $k$ out of the $n$ codeword symbols are
sufficient to reconstruct the original message. A typical class of MDS codes
is Reed-Solomon (RS) codes [1]. Nowadays, RS codes have been applied to many
applications, such as RAID systems [2, 3], distributed storage codes [4, 5],
and data carousel [6]. Hence, the computational complexity of RS erasure code
is considered crucial and has attracted substantial research attention. Based
on the new polynomial basis, this paper presents the encoding/decoding
algorithms for RS erasure codes. The proposed algorithms use the structure [7]
that requires evaluating a polynomial and it’s derivatives, while the
polynomial used in the structure is in the new polynomial basis, rather than
the monomial basis.
The rest of this paper is organized as follows. The proposed polynomial basis
is defined in Section II. Section III gives the definition and algorithm of
the transform to compute the polynomial evaluations based on the proposed
polynomial basis. Section IV shows the formal derivative of polynomial.
Section V presents the encoding and erasure decoding algorithm for Reed-
Solomon codes. The discussions and comparisons are placed in Section VI.
SectionVII reviews some related literature. Concluding remarks are provided in
Section VIII.
## II A new polynomial basis over $\mathbb{F}_{2^{r}}$
### II-A Finite field arithmetic
Let $\mathbb{F}_{2^{r}}$ be an extension finite field with dimension $r$ over
$\mathbb{F}_{2}$. The elements of $\mathbb{F}_{2^{r}}$ are represented as a
set $\\{\omega_{i}\\}_{i=0}^{2^{r}-1}$. We order those elements as follows.
Assume that $V$ be the $r$-dimensional vector space spanned by
$v_{0},v_{1},\dots,v_{r-1}\in\mathbb{F}_{2^{r}}$ over $\mathbb{F}_{2}$. For
any $0\leq i<2^{r}$, its binary representation is given as
$i=i_{0}+i_{1}\cdot 2+i_{2}\cdot 2^{2}+\dots+i_{r-1}\cdot 2^{r-1},\forall
i_{j}\in\\{0,1\\}.$ (1)
Then $\omega_{i}$ is defined as
$\omega_{i}=i_{0}\cdot v_{0}+i_{1}\cdot v_{1}+i_{2}\cdot
v_{2}+\dots+i_{r-1}\cdot v_{r-1}.$
A polynomial $f(x)$ defined over $\mathbb{F}_{2^{r}}$ is a polynomial whose
coefficients are from $\mathbb{F}_{2^{r}}$.
### II-B Subspace vanishing polynomial
The subspace vanishing polynomial defined in [8, 9, 10] is expressed as
$W_{j}(x)=\prod_{i=0}^{2^{j}-1}(x+\omega_{i}),$ (2)
where $0\leq j\leq r-1$. It can be seen that $deg(W_{j}(x))=2^{j}$.
Next we present properties of $W_{j}(x)$ without proof.
###### Lemma 1 ([9]).
$W_{j}(x)$ is an $\mathbb{F}_{2}$-linearlized polynomial for which
$W_{j}(x)=\sum_{i=0}^{j}a_{j,i}x^{2^{i}},$ (3)
where each $a_{j,i}\in\mathbb{F}_{2^{r}}$ is a constant. Furthermore,
$W_{j}(x+y)=W_{j}(x)+W_{j}(y),\forall x,y\in\mathbb{F}_{2^{r}}.$ (4)
### II-C Polynomial basis
In this work, we consider the polynomial ring
$\mathbb{F}_{2^{r}}[x]/(x^{2^{r}}-x)$. A form of polynomial basis we work with
is denoted as
$\mathbb{X}(x)=\left(X_{0}(x),X_{1}(x),\dots,X_{2^{r}-1}(x)\right)$ over
$\mathbb{F}_{2^{r}}$. Each polynomial $X_{i}(x)$ is defined as the product of
subspace vanishing polynomials. For each polynomial $X_{i}(x)$, $i$ is written
in binary representation as
$i=i_{0}+i_{1}\cdot 2+\dots+i_{r-1}\cdot 2^{r-1},\forall i_{j}\in\\{0,1\\}.$
(5)
The polynomial $X_{i}(x)$ is then defined as
$X_{i}(x)=\prod_{j=0}^{r-1}\left(\frac{W_{j}(x)}{W_{j}(\omega_{2^{j}})}\right)^{i_{j}},$
(6)
for $0\leq i<2^{r}$. Notice that
$\left(\frac{W_{j}(x)}{W_{j}(\omega_{2^{j}})}\right)^{i_{j}}=1$, if $i_{j}=0$.
It can be seen that $deg(X_{i}(x))=i$.
Then a form of polynomial expression $[\bullet](x)$ is given as follows.
###### Definition 1.
A form of polynomial expression over $\mathbb{F}_{2^{r}}$ is defined as
$[D_{h}](x)=\sum_{i=0}^{h-1}d_{i}X_{i}(x),$ (7)
where
$D_{h}=(d_{0},d_{1},\dots,d_{h-1})$ (8)
is an $h$-element vector denoting the polynomial coefficients and $h\leq
2^{r}$. Consequently, $deg([D_{h}](x))\leq h-1$.
## III Fast transform $\Psi_{h}^{l}[\bullet]$
In this section, we define a $h$-point transformation $\Psi_{h}^{l}[\bullet]$
that computes the evaluations of $[\bullet](x)$ at $h$ successive points, for
$h$ a power of two. Given a $h$-element input vector $D_{h}$, the polynomial
$[D_{h}](x)$ can be constructed accordingly. The transform outputs a
$h$-element vector
$\hat{D}_{h}^{l}=\Psi_{h}^{l}[D_{h}],$
where
$\hat{D}_{h}^{l}=([D_{h}](\omega_{0}+\omega_{l}),[D_{h}](\omega_{1}+\omega_{l}),\dots,[D_{h}](\omega_{h-1}+\omega_{l})),$
and $l$ denotes the amount of shift in the transform.
Oppositely, the inversion, denoted as $(\Psi_{h}^{l})^{-1}[\bullet]$, can
convert $\hat{D}_{h}^{l}$ into $D_{h}$, and we have
$(\Psi_{h}^{l})^{-1}[\hat{D}_{h}^{l}]=D_{h}$. Here, we omit to provide the
close form for inversion. Instead, an algorithm for transform
$\Psi_{h}^{l}[\bullet]$ and the inverse algorithm will be presented later.
### III-A Recursive structure in polynomial basis
This subsection shows that the polynomial $[D_{h}](x)$ can be formulated as a
recursive function $[D_{h}](x)=\Delta_{0}^{0}(x)$, where the function
$\Delta_{i}^{m}(x)$ is defined as
$\displaystyle\Delta_{i}^{m}(x)=\Delta_{i+1}^{m}(x)+\frac{W_{i}(x)}{W_{i}(\omega_{2^{i}})}\Delta_{i+1}^{m+2^{i}}(x)$
(9) $\displaystyle,\textup{ for }0\leq i\leq\lg(h)-1;$
$\Delta_{\lg(h)}^{m}(x)=d_{m},\textup{ for }0\leq m\leq h-1.$ (10)
Note that $m$ in $\Delta_{i}^{m}(x)$ represents a $\lg(h)$-bits binary integer
$m=m_{0}+m_{1}\cdot 2+\dots+m_{i-1}\cdot 2^{i},\forall m_{j}\in\\{0,1\\}.$
(11)
By induction, it can be seen that $deg(\Delta_{i}^{m}(x))\leq h/2^{i}-1$. For
example, if $h=8$, we have
$\displaystyle[D_{8}](x)=\sum_{i=0}^{7}d_{i}X_{i}(x)$ (12) $\displaystyle=$
$\displaystyle
d_{0}+d_{1}\frac{W_{0}(x)}{W_{0}(\omega_{1})}+d_{2}\frac{W_{1}(x)}{W_{1}(\omega_{2})}+d_{3}\frac{W_{0}(x)}{W_{0}(\omega_{1})}\frac{W_{1}(x)}{W_{1}(\omega_{2})}$
$\displaystyle+d_{4}\frac{W_{2}(x)}{W_{2}(\omega_{4})}+d_{5}\frac{W_{0}(x)}{W_{0}(\omega_{1})}\frac{W_{2}(x)}{W_{2}(\omega_{4})}+d_{6}\frac{W_{1}(x)}{W_{1}(\omega_{2})}\frac{W_{2}(x)}{W_{2}(\omega_{4})}$
$\displaystyle+d_{7}\frac{W_{0}(x)}{W_{0}(\omega_{1})}\frac{W_{1}(x)}{W_{1}(\omega_{2})}\frac{W_{2}(x)}{W_{2}(\omega_{4})}$
$\displaystyle=$
$\displaystyle\left(d_{0}+d_{4}\frac{W_{2}(x)}{W_{2}(\omega_{4})}+\frac{W_{1}(x)}{W_{1}(\omega_{2})}\left(d_{2}+d_{6}\frac{W_{2}(x)}{W_{2}(\omega_{4})}\right)\right)$
$\displaystyle+\frac{W_{0}(x)}{W_{0}(\omega_{1})}\left(d_{1}+d_{5}\frac{W_{2}(x)}{W_{2}(\omega_{4})}+\frac{W_{1}(x)}{W_{1}(\omega_{2})}\left(d_{3}+d_{7}\frac{W_{2}(x)}{W_{2}(\omega_{4})}\right)\right)$
$\displaystyle=$
$\displaystyle\left(\Delta_{2}^{0}(x)+\frac{W_{1}(x)}{W_{1}(\omega_{2})}\Delta_{2}^{2}(x)\right)$
$\displaystyle+\frac{W_{0}(x)}{W_{0}(\omega_{1})}\left(\Delta_{2}^{1}(x)+\frac{W_{1}(x)}{W_{1}(\omega_{2})}\Delta_{2}^{3}(x)\right)$
$\displaystyle=$
$\displaystyle\Delta_{1}^{0}(x)+\frac{W_{0}(x)}{W_{0}(\omega_{1})}\Delta_{1}^{1}(x)=\Delta_{0}^{0}(x).$
The $\Delta_{i}^{m}(x)$ possesses the following equality that will be utilized
in the algorithm:
###### Lemma 2.
$\Delta_{i}^{m}(x+y)=\Delta_{i}^{m}(x),\forall
y\in\\{\omega_{b}\\}_{b=0}^{2^{i}-1}.$ (13)
###### Proof.
By Lemma 1, we have
$W_{i}(x+y)=W_{i}(x)+W_{i}(y)=W_{i}(x),\forall
y\in\\{\omega_{b}\\}_{b=0}^{2^{i}-1}.$ (14)
The proof follows mathematical induction on $i$. In the base case, we consider
(9) at $i=\lg(h)-1$:
$\displaystyle\Delta_{\lg(h)-1}^{m}(x)$ $\displaystyle=$
$\displaystyle\Delta_{\lg(h)}^{m}(x)+\frac{W_{\lg(h)-1}(x)}{W_{\lg(h)-1}(\omega_{2^{\lg(h)-1}})}\Delta_{\lg(h)}^{m+2^{\lg(h)-1}}(x)$
$\displaystyle=$ $\displaystyle
d_{m}+\frac{W_{\lg(h)-1}(x)}{W_{\lg(h)-1}(\omega_{2^{\lg(h)-1}})}d_{m+2^{\lg(h)-1}}.$
From (14), we have
$\displaystyle\Delta_{\lg(h)-1}^{m}(x+y)$ $\displaystyle=$ $\displaystyle
d_{m}+\frac{W_{\lg(h)-1}(x+y)}{W_{\lg(h)-1}(\omega_{2^{\lg(h)-1}})}d_{m+2^{\lg(h)-1}}$
$\displaystyle=$ $\displaystyle
d_{m}+\frac{W_{\lg(h)-1}(x)}{W_{\lg(h)-1}(\omega_{2^{\lg(h)-1}})}d_{m+2^{\lg(h)-1}}$
$\displaystyle=$ $\displaystyle\Delta_{\lg(h)-1}^{m}(x),\forall
y\in\\{\omega_{b}\\}_{b=0}^{h/2-1}.$
Thus (13) holds for $i=\lg(h)-1$.
Assume (13) holds for $i=c+1$. When $i=c$, we have
$\displaystyle\Delta_{c}^{m}(x+y)$ $\displaystyle=$
$\displaystyle\Delta_{c+1}^{m}(x+y)+\frac{W_{c}(x+y)}{W_{c}(\omega_{2^{c}})}\Delta_{c+1}^{m+2^{c}}(x+y)$
$\displaystyle=$
$\displaystyle\Delta_{c+1}^{m}(x+y)+\frac{W_{c}(x)}{W_{c}(\omega_{2^{c}})}\Delta_{c+1}^{m+2^{c}}(x+y)$
$\displaystyle=$
$\displaystyle\Delta_{c+1}^{m}(x)+\frac{W_{c}(x)}{W_{c}(\omega_{2^{c}})}\Delta_{c+1}^{m+2^{c}}(x)$
$\displaystyle=$ $\displaystyle\Delta_{c}^{m}(x),\forall
y\in\\{\omega_{b}\\}_{b=0}^{2^{c}-1}.$
This completes the proof. ∎
### III-B Proposed algorithm
Let
$\displaystyle\Psi(i,m,l)=\\{\Delta_{i}^{m}(\omega_{c}+\omega_{l})|c\in\\{b\cdot
2^{i}\\}_{b=0}^{h/2^{i}-1}\\}$ (15) $\displaystyle,\textup{ for }0\leq
i\leq\lg(h)-1;$ $\Psi(\lg(h),m,l)=\\{d_{m}\\}.$ (16)
The objective of algorithm is to compute the values in set $\Psi(0,0,l)$. In
the following, we rearrange the set $\Psi(i,m,l)$ into two parts:
$\Psi(i+1,m,l)$ and $\Psi(i+1,m+2^{i},l)$, by taking around $h/2^{i}$
additions and $h/2^{i+1}$ multiplications.
In (15), $\Psi(i,m,l)$ can be divided into two individual subsets:
$\\{\Delta_{i}^{m}(\omega_{c}+\omega_{l})|c\in\\{b\cdot
2^{i+1}\\}_{b=0}^{h/2^{i+1}-1}\\}$ (17)
and
$\\{\Delta_{i}^{m}(\omega_{c}+\omega_{l}+\omega_{2^{i}})|c\in\\{b\cdot
2^{i+1}\\}_{b=0}^{h/2^{i+1}-1}\\}.$ (18)
In (17), we have
$\displaystyle\Delta_{i}^{m}(\omega_{c}+\omega_{l})$ (19) $\displaystyle=$
$\displaystyle\Delta_{i+1}^{m}(\omega_{c}+\omega_{l})+\frac{W_{i}(\omega_{c}+\omega_{l})}{W_{i}(\omega_{2^{i}})}\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l}).$
It can be seen that $\Delta_{i+1}^{m}(\omega_{c}+\omega_{l})\in\Psi(i+1,m,l)$,
and $\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l})\in\Psi(i+1,m+2^{i},l)$. The
factor $\frac{W_{i}(\omega_{c}+\omega_{l})}{W_{i}(\omega_{2^{i}})}$ can be
precomputed and stored. Hence, for each element of the set given in (17), the
calculation requires a multiplication and an addition. Note that when
$\omega_{c}+\omega_{l}=0$, we have
$\Delta_{i}^{m}(0)=\Delta_{i+1}^{m}(0),$ (20)
which does not involve any arithmetic operations.
Next we consider the computation in (18), and we have
$\displaystyle\Delta_{i}^{m}(\omega_{c}+\omega_{l}+\omega_{2^{i}})=\Delta_{i+1}^{m}(\omega_{c}+\omega_{l}+\omega_{2^{i}})$
(21)
$\displaystyle+\frac{W_{i}(\omega_{c}+\omega_{l}+\omega_{2^{i}})}{W_{i}(\omega_{2^{i}})}\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l}+\omega_{2^{i}}).$
By Lemma 2, we have
$\Delta_{i+1}^{m}(\omega_{c}+\omega_{l}+\omega_{2^{i}})=\Delta_{i+1}^{m}(\omega_{c}+\omega_{l});$
$\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l}+\omega_{2^{i}})=\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l}).$
Furthermore, the factor can be rewritten as
$\displaystyle\frac{W_{i}(\omega_{c}+\omega_{l}+\omega_{2^{i}})}{W_{i}(\omega_{2^{i}})}$
$\displaystyle=$
$\displaystyle\frac{W_{i}(\omega_{c}+\omega_{l})+W_{i}(\omega_{2^{i}})}{W_{i}(\omega_{2^{i}})}$
$\displaystyle=$
$\displaystyle\frac{W_{i}(\omega_{c}+\omega_{l})}{W_{i}(\omega_{2^{i}})}+1.$
With above results, (21) can be rewritten as
$\displaystyle\Delta_{i}^{m}(\omega_{c}+\omega_{l}+\omega_{2^{i}})$ (22)
$\displaystyle=$
$\displaystyle\Delta_{i+1}^{m}(\omega_{c}+\omega_{l})+\left(\frac{W_{i}(\omega_{c}+\omega_{l})}{W_{i}(\omega_{2^{i}})}+1\right)\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l})$
$\displaystyle=$
$\displaystyle\Delta_{i+1}^{m}(\omega_{c}+\omega_{l})+\frac{W_{i}(\omega_{c}+\omega_{l})}{W_{i}(\omega_{2^{i}})}\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l})$
$\displaystyle+\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l})$ $\displaystyle=$
$\displaystyle\Delta_{i}^{m}(\omega_{c}+\omega_{l})+\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l}).$
Hence, the element requires an addition.
### III-C Inverse transform
The inversion is a transform converts $\Psi(i,m,l)$ into polynomial
coefficients $\\{d_{m}\\}_{m=0}^{h-1}$. The inversion can be done through
backtracking the transform algorithm. As mentioned previously, $\Psi(i,m,l)$
can be rearranged into two parts: $\Psi(i+1,m,l)$ and $\Psi(i+1,m+2^{i},l)$.
Assume the set $\Psi(i,m,l)$ is given, we present the method to compute
$\Psi(i+1,m,l)$ and $\Psi(i+1,m+2^{i},l)$, respectively.
To construct $\Psi(i+1,m+2^{i},l)$, (22) is reformulated as
$\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l})=\Delta_{i}^{m}(\omega_{c}+\omega_{l})+\Delta_{i}^{m}(\omega_{c}+\omega_{l}+\omega_{2^{i}}).$
(23)
Since
$\Delta_{i}^{m}(\omega_{c}+\omega_{l}),\Delta_{i}^{m}(\omega_{c}+\omega_{l}+\omega_{2^{i}})\in\Psi(i,m,l)$,
each $\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l})\in\Psi(i+1,m+2^{i},l)$ can
be calculated with taking an addition.
To construct $\Psi(i+1,m,l)$, (19) is reformulated as
$\displaystyle\Delta_{i+1}^{m}(\omega_{c}+\omega_{l})$ (24) $\displaystyle=$
$\displaystyle\Delta_{i}^{m}(\omega_{c}+\omega_{l})+\frac{W_{i}(\omega_{c}+\omega_{l})}{W_{i}(\omega_{2^{i}})}\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l}).$
Since $\Delta_{i}^{m}(\omega_{c}+\omega_{l})\in\Psi(i,m,l)$ and
$\Delta_{i+1}^{m+2^{i}}(\omega_{c}+\omega_{l})\in\Psi(i+1,m+2^{i},l)$ are
known, each $\Delta_{i+1}^{m}(\omega_{c}+\omega_{l})\in\Psi(i+1,m,l)$ can be
calculated with taking an addition and a multiplication.
Figure 1 depicts an example of the proposed transform $\Psi_{h}^{l}[\bullet]$
of length $h=8$. Figure 1(a) shows the flow graph of the transform. The dotted
line arrow denotes that the element should be multiplied with a scalar factor
$\hat{W}_{i}^{j}$ upon adding together with other element, where the scalar
factor is denoted as
$\hat{W}_{i}^{j}=\frac{W_{i}(\omega_{j})}{W_{i}(\omega_{2^{i}})}.$
Figure 1(b) shows the flow graph of inversion. Also, it would be of interest
to compare Figure 1 with the butterfly diagram of radix-2 FFT.
(a) The transform.
(b) The inverse transform.
Figure 1: Data flow diagram of proposed transform of length $h=8$.
### III-D Computational complexity
Clearly, the proposed transform and its inversion have the same computational
complexity. Thus, we only consider the computational complexity on transform.
By the recursive structure, the number of arithmetic operations can be
formulated as recursive functions. Let $A(h)$ and $M(h)$ respectively denote
the number of additions and multiplications used in the algorithm. By (19) and
(22), the recursive formula is given by
$\displaystyle A(h)=2A(h/2)+h;A(1)=0;$ $\displaystyle
M(h)=2M(h/2)+h/2;M(1)=0.$
The solution is
$A(h)=h\lg{(h)};\ M(h)=\frac{h}{2}\lg{(h)}.$
Notice that when the amount of shift $\omega_{l}=0$, the number of operations
can be reduced slightly (see (20)). In this case, we have
$A_{0}(h)=h\lg{(h)}-h+1;\ M_{0}(h)=\frac{h}{2}\lg{(h)}-h+1.$
### III-E Space complexity
In a $h$-point transform, we need $h$ units of space for the input data and an
array to store the factors used in the computation of (17). From (19), the
factors are
$\frac{W_{i}(\omega_{c}+\omega_{l})}{W_{i}(\omega_{2^{i}})}=\frac{W_{i}(\omega_{c})}{W_{i}(\omega_{2^{i}})}+\frac{W_{i}(\omega_{l})}{W_{i}(\omega_{2^{i}})},\forall
c\in\\{b\cdot 2^{i+1}\\}_{b=0}^{h/2^{i+1}-1}.$
As $0\leq i\leq\lg(h)$, a $h$-point transform requires a total of
$\frac{h}{2}+\frac{h}{4}+\dots+\frac{h}{h}=h-1$
units of space to store the factors. Hence, the space complexity is $O(h)$.
## IV Formal derivative
In this section, we consider the formal derivative over the proposed basis.
Section IV-A gives the closed form of the formal derivative. SectionIV-B
presents a computation method that has lower multiplicative complexity than
the original approach.
### IV-A Closed-form expression of formal derivative of $[D_{h}](x)$
###### Lemma 3.
The formal derivative of $W_{i}(x)$ is a constant given by
$W_{i}^{\prime}(x)=\prod_{j=1}^{2^{i}-1}\omega_{j}.$ (25)
###### Proof.
Let
$C(x)=c\cdot x^{j},$
where $c\in\mathbb{F}_{2^{r}}$. Its formal derivative is defined as
$C^{\prime}(x)=\left\\{\begin{matrix}0&\textrm{if $j$ is even;}\\\
cx^{j-1}&\textrm{otherwise.}\end{matrix}\right.$
From Lemma 1, $W_{i}(x)$ has terms in the degrees of $1,2,4,\dots,2^{i}$, so
the formal derivative of $W_{i}(x)$ is a constant that is the coefficient of
$W_{i}(x)$ at degree $1$. The value is
$\sum_{l=0}^{2^{i}-1}\prod_{j\neq
l}\omega_{j}=\prod_{j=1}^{2^{i}-1}\omega_{j}.$
This completes the proof. ∎
By Lemma 3, the formal derivative of $X_{i}(x)$ is shown to be
$\displaystyle X_{i}(x)=$
$\displaystyle\sum_{l=0}^{r-1}i_{l}\frac{W_{l}^{\prime}(x)}{W_{l}(\omega_{2^{l}})}\prod_{j\neq
l}\left(\frac{W_{j}(x)}{W_{j}(\omega_{2^{j}})}\right)^{i_{j}}$ (26)
$\displaystyle=$ $\displaystyle\sum_{l\in I_{i}}W^{\prime}_{l}\cdot
X_{i-2^{l}}(x),$
where
$W^{\prime}_{l}=\frac{W_{l}^{\prime}(x)}{W_{l}(\omega_{2^{l}})}=\frac{\prod_{j=1}^{2^{l}-1}\omega_{j}}{W_{l}(\omega_{2^{l}})},$
(27)
and $I_{i}$ is a set including all the non-zero indices in the binary
representation of $i$, given by
$I_{i}=\\{j|i_{j}=1,j=0,1,\dots,r-1\\}.$
For example, if $i=13=2^{0}+2^{2}+2^{3}$, we have
$\displaystyle X^{\prime}_{13}(x)$ (28) $\displaystyle=$ $\displaystyle
W^{\prime}_{0}W_{2}(x)W_{3}(x)+W^{\prime}_{2}W_{0}(x)W_{3}(x)+W^{\prime}_{3}W_{0}(x)W_{2}(x)$
$\displaystyle=$ $\displaystyle
W^{\prime}_{0}X_{12}(x)+W^{\prime}_{2}X_{9}(x)+W^{\prime}_{3}X_{5}(x).$
From (26), the formal derivative of $[D_{h}](x)$ is given by
$[D_{h}]^{\prime}(x)=\sum_{i=0}^{h-1}d_{i}\sum_{l\in I_{i}}W^{\prime}_{l}\cdot
X_{i-2^{l}}(x),$ (29)
We move the term $X_{j}(x)$ out of the summation operator to get
$[D_{h}]^{\prime}(x)=\sum_{j=0}^{h-1}X_{j}(x)\sum_{l\in
I_{j}^{c}}W^{\prime}_{l}\cdot d_{j+2^{l}},$ (30)
where $I_{j}^{c}$ is the complement of $I_{j}$ defined as
$I_{j}^{c}=\\{i\\}_{i=0}^{\lg(h)-1}\setminus I_{j}.$
From (30), when $W^{\prime}_{l}$ given in (27) are pre-computed and stored,
each coefficient of $X_{j}(x)$ requires at most $\lg(h)-1$ additions and
$\lg(h)$ multiplications. Thus a native way to compute the formal derivation
of $[D_{h}](x)$ requires $O(h\lg(h))$ operations, in both additive complexity
and multiplicative complexity.
### IV-B Computation method with lower multiplicative complexity
We present an alternative approach whose multiplicative complexity is lower
than the above approach. Define
$d_{i}^{\mathrm{d}}=d_{i}\prod_{j\in I_{i}}W_{j}^{\prime},$ (31)
for $0\leq i\leq h-1$. By substituting (31) into (30), we have
$[D_{h}]^{\prime}(x)=\sum_{j=0}^{h-1}X_{j}(x)\sum_{l\in
I_{j}^{c}}\frac{W_{l}^{\prime}\cdot d_{j+2^{l}}^{\mathrm{d}}}{\prod_{m\in
I_{j+2^{l}}}W_{m}^{\prime}}.$ (32)
As
$\prod_{m\in I_{j+2^{l}}}W_{m}^{\prime}=W_{l}^{\prime}\prod_{m\in
I_{j}}W_{m}^{\prime},$
(32) can be rewritten as
$\displaystyle\left[D_{h}\right]^{\prime}(x)$
$\displaystyle=\sum_{j=0}^{h-1}X_{j}(x)\sum_{l\in
I_{j}^{c}}\frac{d_{j+2^{l}}^{\mathrm{d}}}{\prod_{m\in I_{j}}W_{m}^{\prime}}$
(33) $\displaystyle=\sum_{j=0}^{h-1}X_{j}(x)\frac{\sum_{l\in
I_{j}^{c}}d_{j+2^{l}}^{\mathrm{d}}}{\prod_{m\in I_{j}}W_{m}^{\prime}}.$
By the above formulas, the method of computing $[D_{h}]^{\prime}(x)$ consists
of two steps. In the first step, we compute (31). Here, the set of factors
$B=\\{\prod_{j\in I_{i}}W_{j}^{\prime}|i=0,1,\dots,h-1\\}$ (34)
can be pre-computed and stored, and this step only requires $h$
multiplications. In the second step, we compute the coefficients through (33).
Notice that the denominator is an element of $B$. Thus, this step needs around
$\frac{1}{2}h\lg(h)$ additions and $h$ multiplications.
Next we use an example to demonstrate how to obtain $[D_{h}]^{\prime}(x)$. If
$h=8$ and the set $B$ includes $8$ elements defined as
$\displaystyle
B_{0}=1;B_{1}=W_{0}^{\prime};B_{2}=W_{1}^{\prime};B_{3}=W_{0}^{\prime}W_{1}^{\prime};$
$\displaystyle
B_{4}=W_{2}^{\prime};B_{5}=W_{0}^{\prime}W_{2}^{\prime};B_{6}=W_{1}^{\prime}W_{2}^{\prime};B_{7}=W_{0}^{\prime}W_{1}^{\prime}W_{2}^{\prime}.$
From (31), each $d_{i},0\leq i\leq 7$ is computed via
$d_{i}^{\mathrm{d}}=d_{i}B_{i}.$
From (33), the formal derivative of $[D_{8}](x)$ is shown to be
$\displaystyle[D_{8}]^{\prime}(x)$ $\displaystyle=$ $\displaystyle
X_{0}(x)\frac{d_{1}^{\mathrm{d}}+d_{2}^{\mathrm{d}}+d_{4}^{\mathrm{d}}}{B_{0}}+X_{1}(x)\frac{d_{3}^{\mathrm{d}}+d_{5}^{\mathrm{d}}}{B_{1}}+X_{2}(x)\frac{d_{3}^{\mathrm{d}}+d_{6}^{\mathrm{d}}}{B_{2}}$
$\displaystyle+X_{3}(x)\frac{d_{7}^{\mathrm{d}}}{B_{3}}+X_{4}(x)\frac{d_{5}^{\mathrm{d}}+d_{6}^{\mathrm{d}}}{B_{4}}+X_{5}(x)\frac{d_{7}^{\mathrm{d}}}{B_{5}}+X_{6}(x)\frac{d_{7}^{\mathrm{d}}}{B_{6}}.$
## V Algorithms of Reed-Solomon erasure codes
Based on the new polynomial basis, this section presents the encoding and
decoding algorithms for $(n,k)$ Reed-Solomon (RS) erasure codes over
characteristic-2 fields. There are two major approaches on the encoding of
Reed-Solomon codes, termed as polynomial evaluation approach and generator
polynomial approach. In this paper, we follow the polynomial evaluation
approach, which treats the codeword symbols as the evaluation values of a
polynomial $F(x)$ of degree less than $k$. Let
$M_{k}=(m_{0},m_{1},\dots,m_{k-1})$
denote the vector of message, for each $m_{i}\in\mathbb{F}_{2^{r}}$. In the
systematic construction, $F(x)$ is a polynomial of degree less than $k$ such
that
$F(\omega_{i})=m_{i},\textup{ for }0\leq i\leq k-1.$ (35)
By the set of equations (35), $F(x)$ can be uniquely constructed via
polynomial interpolation. Then we use this $F(x)$ to calculate the codeword
$F_{n}=(F(\omega_{0}),F(\omega_{1}),\dots,F(\omega_{n-1})).$
In decoding, assume the received codeword has $n-k$ erasures $\\{F(y):y\in
E\\}$, where $E$ denotes the set of evaluation points of erasures. With the
$k$ un-erased symbols, $F(x)$ can be uniquely reconstructed via polynomial
interpolation, and thus the erasures can be computed accordingly.
In the following, we illustrate the algorithms of encoding and erasure
decoding for Reed-Solomon codes. The proposed algorithm is for $k$ a power of
two, and $n=2^{r}$. The codes for other $k$ can be derived through code
shortening strategy; i.e., appending zeros to message vector so that the
length of the vector is power of two.
### V-A Encoding algorithm
Algorithm 1 Reed-Solomon encoding algorithm.
1:A $k$-element message vector $M_{k}$ over $\mathbb{F}_{2^{r}}$.
2:An $n$-element systematic codeword $F_{n}$.
3: $\bar{M}_{k}=(\Psi_{k}^{0})^{-1}[M_{k}]$
4:for $i=1$ to $(n/k-1)$ do
5: $\bar{F}_{i}=\Psi_{k}^{i\cdot k}[\bar{M}_{k}]$
6:end for
7:return $F_{n}=(M_{k},\bar{F}_{1},\bar{F}_{2},\dots,\bar{F}_{\left\lceil
n/k\right\rceil-1})$.
Algorithm 1 illustrates the pseudocode of the $(n,k)$ RS encoding algorithm.
In Line 1, we compute the vector
$\bar{M}_{k}=(\bar{m}_{0},\bar{m}_{1},\dots,\bar{m}_{k-1}),$
which can be formed as a polynomial
$[\bar{M}_{k}](x)=\sum_{i=0}^{k-1}\bar{m}_{i}X_{i}(x).$
Since $deg([\bar{M}_{k}](x))\leq k-1$ and
$[\bar{M}_{k}](\omega_{i})=m_{i},\textup{ for }0\leq i\leq k-1$ (36)
we conclude that $[\bar{M}_{k}](x)=F(x)$. Thus, the parity-check symbols can
be computed by applying the proposed transform on $\bar{M}_{k}$ (see Lines
2-4). The parity-check symbols are obtained in blocks with size $k$ and there
are $n/k-1$ blocks.222Since $k$ and $n$ are both powers of $2$, $n$ is
divisible by $k$. For each block, the vector $\bar{F}_{i}$ includes $k$
elements and each element is
$\bar{F}_{i}[j]=[\bar{M}_{k}](\omega_{j+(i\cdot
k)})=[\bar{M}_{k}](\omega_{j}+\omega_{i\cdot k}),\textup{ for }0\leq j\leq
k-1.$
In Line 5, we assemble those vectors to get the codeword vector $F_{n}$.
In summary, the encoding algorithm requires a $k$-element inversion
$(\Psi_{k}^{0})^{-1}[\bullet]$ and $(n/k-1)$ times of $k$-element transform
$\Psi_{k}^{i}[\bullet]$. Thus, the encoding algorithm has the complexity
$O((n/k)k\lg{(k)})=O(n\lg{(k)}).$
### V-B Erasure decoding algorithm
Algorithm 2 Framework of Reed-Solomon erasure decoding algorithm.
1:Received codeword $\bar{F}_{n}$, and the positions of erasures
$E=\\{e_{i}\\}_{i=0}^{n-k-1}$.
2:The erasures $\\{F(j)|j\in E\\}$.
3:Compute two sets of values $\bar{\Pi}$ and $\Pi^{\prime}$, defined in (40)
and (42).
4:From (39), compute
$\Phi=(\hat{F}(\omega_{0}),\hat{F}(\omega_{1}),\dots,\hat{F}(\omega_{n-1})).$
5:Apply $n$-point fast inverse transform on $\Phi$ to get
$\bar{\Phi}_{n}=(\Psi_{n}^{0})^{-1}[\Phi].$
6:Compute the formal derivative of $\bar{\Phi}_{n}$. The result is denoted as
$\bar{\Phi}^{\mathrm{d}}_{n}$.
7:Apply $n$-point fast transform on $\bar{\Phi}^{\mathrm{d}}_{n}$ to get
$\Phi^{\mathrm{d}}_{n}=\Psi_{n}^{0}[\bar{\Phi}^{\mathrm{d}}_{n}].$
8:Compute the erasures via
$F(j)=\frac{\Phi^{\mathrm{d}}_{n}[j]}{\Pi^{\prime}(j)},\forall j\in E.$
The decoding algorithm follows our previous work [7] that requires evaluating
a polynomial and it’s derivatives. The code proposed in [7] is based on Fermat
number transforms (FNT). In this paper, we replace the role of FNT over
$\mathbb{F}_{2^{r}+1}$ with the proposed transform over $\mathbb{F}_{2^{r}}$.
However, since the proposed transform is not Fourier transform, some
arithmetic operations involved in the transform should be modified
accordingly.
Assume the received codeword $\bar{F}_{n}$ has $n-k$ erasures. The set of
evaluation points of erasures are denoted as
$E=\\{\omega_{e_{i}}\\}_{i=0}^{n-k-1}.$
Let
$\Pi(x)=\prod_{y\in E}(x+y)$
denote the error locator polynomial having zeros at all erased symbols. It can
be seen that $\Pi(j)=0,\forall j\in E$. Define
$\hat{F}(x)=F(x)\Pi(x),$
and the polynomial degree is $deg(\hat{F}(x))=deg(F(x))+deg(\Pi(x))\leq n-1$.
The formal derivative of $\hat{F}(x)$ is
$\hat{F}^{\prime}(x)=F^{\prime}(x)\Pi(x)+F(x)\Pi^{\prime}(x).$ (37)
By substituting $x=j\in E$ into (37), we have
$\hat{F}^{\prime}(j)=F(j)\Pi^{\prime}(j),\forall j\in E.$
Hence the erasures can be computed by
$F(j)=\frac{\hat{F}^{\prime}(j)}{\Pi^{\prime}(j)},\forall j\in E.$ (38)
Based on above formulas, the decoding procedure consists of three major
stages: First, compute the coefficients of $\hat{F}(x)$; second, compute the
formal derivative of $\hat{F}(x)$; and third, compute the erasures by (38).
The details are elaborated as follows.
In the first stage, we need to compute the coefficients of $\hat{F}(x)$. It
can be shown that
$\hat{F}(j)=F(j)\Pi(j)=\left\\{\begin{matrix}0&\forall j\in E;\\\
F(j)\Pi(j)&\mathrm{otherwise.}\end{matrix}\right.$ (39)
Here, we define
$\bar{\Pi}=\\{\Pi(j)|j\in\mathbb{F}_{2^{r}}\backslash E\\}.$ (40)
Appendix shows the algorithm of computing $\bar{\Pi}$ proposed by [11]. Since
$F(j),\ j\in\mathbb{F}_{2^{r}}\backslash E$ are elements of the received
vector, the result of (39) can computed with $n$ multiplications after
$\bar{\Pi}$ is obtained and is denoted as a vector
$\Phi=(\hat{F}(\omega_{0}),\hat{F}(\omega_{1}),\dots,\hat{F}(\omega_{n-1})).$
Then we compute
$\bar{\Phi}_{n}=(\Psi_{n}^{0})^{-1}[\Phi].$ (41)
Here, the resulting vector
$\bar{\Phi}_{n}=(\bar{\phi}_{0},\bar{\phi}_{1},\dots,\bar{\phi}_{n-1})$ can be
formed as a polynomial
$[\bar{\Phi}_{n}](x)=\sum_{i=0}^{n-1}\bar{\phi}_{i}X_{i}(x),$
where $[\bar{\Phi}_{n}](\omega_{j})=\hat{F}(\omega_{j})$, for $0\leq j\leq
n-1$. That is, $[\bar{\Phi}_{n}](\omega_{j})-\hat{F}(\omega_{j})=0$, for
$0\leq j\leq n-1$. Since the degree of $[\bar{\Phi}_{n}](x)-\hat{F}(x)$ is at
most $n-1$, it must be the zero polynomial when it has $n$ roots. Hence, we
conclude $[\bar{\Phi}_{n}](x)=\hat{F}(x)$.
The second stage is to compute the formal derivative of $\hat{F}(x)$. Since
$[\bar{\Phi}_{n}](x)$ is under the polynomial basis given by Definition 1, we
compute the formal derivative of $[\bar{\Phi}_{n}](x)$ by the method presented
in Section IV. Then we can obtain the result vector
$\bar{\Phi}^{\mathrm{d}}_{n}=(\bar{\phi}^{\mathrm{d}}_{0},\bar{\phi}^{\mathrm{d}}_{1},\dots,\bar{\phi}^{\mathrm{d}}_{n-1})$,
and the polynomial
$[\bar{\Phi}^{\mathrm{d}}_{n}](x)=\sum_{i=0}^{n-1}\bar{\phi}_{i}^{\mathrm{d}}X_{i}(x)$
is the formal derivative of $[\bar{\Phi}_{n}](x)$.
In the final stage, we need to compute the erasures via (38). Here, the
denominators in (38) are defined as a set
$\Pi^{\prime}=\\{\Pi^{\prime}(j)|j\in E\\},$ (42)
which can be constructed by the algorithm introduced in Appendix. We then
compute
$\Phi^{\mathrm{d}}_{n}=\Psi_{n}^{0}[\bar{\Phi}^{\mathrm{d}}_{n}],$ (43)
where the resulting vector includes the evaluations of $\hat{F}^{\prime}(j)$
for $j\in\mathbb{F}_{2^{r}}$; i.e., the $\Phi^{\mathrm{d}}_{n}$ is denoted as
$\Phi^{\mathrm{d}}_{n}=(\hat{F}^{\prime}(\omega_{0}),\hat{F}^{\prime}(\omega_{1}),\dots,\hat{F}^{\prime}(\omega_{n-1})).$
Then the erasures can be computed through (38).
The decoding procedure is summarized in Algorithm 2. The complexity of this
algorithm is dominated by Steps 1, 3, 4 and 5, whereas Steps 2 and 6 only
require $O(n)$ multiplications. By the proposed fast transform algorithm,
Steps 3 and 5 can be done with $O(n\lg{(n)})$ additions and $O(n\lg{(n)})$
multiplications. By the method in Section IV, Step 4 requires $O(n\lg(n))$
additions and $O(n)$ multiplications. In Step 1, we use the algorithm shown in
Appendix, and it can be done with $O(n\lg(n))$ modulus operations. In summary,
the proposed decoding algorithm has the complexity of order $O(n\lg{(n)})$.
## VI Discussions and comparisons
### VI-A Complexities of operations in polynomial basis
We consider some polynomial operations in this section. Table I tabulates the
complexities of some polynomial operations in the monomial basis and the
proposed basis over characteristic-2 finite fields. In particular, the
polynomial addition is simple by adding the coefficients of two polynomials.
Hence, the complexity is $O(h)$ in both basis. For the polynomial
multiplication, an algorithm with order $O(h\lg(h)\lg\lg(h))$ is proposed by
[12], in 1977. To compute $[A_{h}](x)\times[B_{h}](x)$ in the proposed basis,
the result polynomial is computed via
$(\Psi_{2h}^{l})^{-1}[\Psi_{2h}^{l}[A_{2h}]\star\Psi_{2h}^{l}[B_{2h}]],$
where $A_{2h}$( and $B_{2h}$) is $2h$-point vector by appending zeros to
$A_{h}$( and $B_{h}$), and $\star$ denotes the operation of pairwise
multiplication. Hence, the complexity is $O(h\lg(h))$.
To determine the degree polynomial in proposed basis, we scan the coefficients
of $[D_{h}](x)$ to determine the highest degree term $d_{j}X_{j}(x),d_{j}\neq
0$, and thus the complexity is $O(h\lg(h))$; and so does the polynomial in
monomial basis.
The formal derivative in proposed basis requires $O(h\lg(h))$ field operations
shown in Section IV. In contrast, the formal derivative in monomial basis only
requires $O(h)$ operations.
TABLE I: Complexities of operations in polynomial basis over characteristic-2 finite fields Operations | Monomial basis | Proposed basis
---|---|---
Addition | $O(h)$ | $O(h)$
Multiplication | $O(h\lg(h)\lg\lg(h))$ | $O(h\lg(h))$
Polynomial degree | $O(h)$ | $O(h)$
Formal derivative | $O(h)$ | $O(h\lg(h))$
### VI-B Comparisons with Didier’s approach
In 2009, Didier [11] present an erasure decoding algorithm for Reed-Solomon
codes based on fast Walsh-Hadamard transforms. The algorithm [11] consists of
two major parts: the first part is to compute the polynomial evaluations of
error locator polynomial, and the second part is to decompose the Lagrange
polynomial into several logical convolutions, which are then respectively
computed with fast Walsh-Hadamard transforms. The first part requires
$O(n\lg(n))$ time, and the second part requires $O(n\lg^{2}(n))$ time, so the
complexity [11] is $O(n\lg^{2}(n))$. In contrast, the proposed approach
employs the first part in [11]; in the second part, we design another decoding
structure based on the proposed basis. The proposed transform only requires
$O(n\lg(n))$ time, so that the proposed approach can reduce the complexity
from $O(n\lg^{2}(n))$ to $O(n\lg(n))$.
We also implement the proposed algorithm in C and run the program on Intel
core i7-950 CPU. While $n=2^{16}$, $k/n=1/2$, the program took about $1.12$
seconds to generate a codeword, and $3.06$ seconds to decode an erased
codeword on average. On the other hand, we also ran the program [11] written
by the author on the same platform. In our simulation, the program [11] took
about $52.91$ seconds in both encoding and erasure decoding under the same
parameter configuration. Thus, the proposed erasure decoding is around $17$
times faster than [11], while $n=2^{16}$.
## VII Literature review
In the original view of [1], the codeword of the RS code is a sequence of
evaluation values of a polynomial interpreted by message. By this viewpoint,
the encoding process can be treated as an oversampling process through
discrete Fourier transform (DFT) over finite fields. Some studies [13, 14, 15]
indicate that, if a $O(n\lg(n))$ finite field FFT is available, the error-
correction decoding can be reduced to $O(n\lg^{2}(n))$. An $n$-point radix-2
FFT butterfly diagram requires $n\lg(n)$ additions and $\frac{n}{2}\lg(n)$
multiplications. This FFT butterfly diagram can be directly applied on Fermat
prime fields $\mathbb{F}_{2^{r}+1},r\in\\{1,2,4,8,16\\}$. In this case, the
transform, referred to as Fermat number transform (FNT), requires $n\lg(n)$
finite field additions and $\frac{n}{2}\lg(n)$ finite field multiplications.
By employing FNT, a number of fast approaches [16, 17, 13] had been presented
to reduce the complexity of encoding and decoding of RS codes. Some FNT-based
RS erasure decoding algorithms had been proposed [18, 7, 19] in $O(n\lg(n))$
finite field operations. Thus far, no existing algorithm for $(n,k)$ RS codes
has decoding complexity achieving lower than $\Omega(n\lg(n))$ operations, in
a context of a fixed coding rate $k/n$. However, the major drawback of FNT is
that it needs more space to store one extra symbol in practical implementation
such that the FNT-based codes are impractical in general applications.
On the other hand, FFTs over characteristic-2 finite fields require higher
complexities than $O(n\lg(n))$. Table II tabulates the arithmetic complexities
of FFT algorithms over characteristic-2 finite fields. As shown in Table II,
Gao and Mateer [10] gave two versions of additive FFTs over
$\mathbb{F}_{2^{r}}$ that are most likely the most efficient FFTs by far. The
first is for arbitrary $r$, and the second is for $r$ a power of two. Notably,
Wu’s approach [20] has very low multiplicative complexity
$O(n\lg^{\lg(3/2)}(n))$, but the additive complexity is higher with complexity
$O(n^{2}/\lg^{\lg(8/3)}(n))$. This implies that when the polynomial
representation in RS codes are in monomial basis, the complexity will fail to
reach $O(n\lg(n))$.
There exist faster encoding and erasure decoding approaches in some non-MDS
codes. Such codes, termed as fountain codes [6], require a little more than
$k$ codeword symbols to recover the original message. Two famous classes of
fountain codes are LT code [21] and Raptor code [22]. Due to the low
complexity, fountain codes have significant merits in many applications.
However, because of the randomly generated generator matrices, the hardware
parallelization of fountain code is not trivial.
TABLE II: Complexities of $n$-point FFT algorithms over $\mathbb{F}_{2^{r}}$, where $n=2^{r}-1$ Algorithm | Restriction | Additive complexity | Multiplicative complexity
---|---|---|---
Gao [10] | $r$ is a power of two | $O(n\lg(n)\lg\lg(n))$ | $O(n\lg(n))$
Cantor [8] | $r$ is a power of two | $O(n\lg^{\lg(3)}(n))$ | $O(n\lg(n))$
Gao [10] | | $O(n\lg^{2}(n))$ | $O(n\lg(n))$
Wang [23], Gathen [9] | | $O(n\lg^{2}(n))$ | $O(n\lg^{2}(n))$
Pollard [24] | $r$ is even | $O(n^{3/2})$ | $O(n^{3/2})$
Wu [20] | | $O(n^{2}/\lg^{\lg(8/3)}(n))$ | $O(n\lg^{\lg(3/2)}(n))$
Sarwate [25] | | $O(n^{2})$ | $O(n\lg(n))$
Naive approach | | $O(n^{2})$ | $O(n^{2})$
## VIII Concluding remarks
Based on the proposed polynomial basis, we can compute the polynomial
evaluations in the complexity of order $O(h\lg(h))$ with a small leading
constant. This enables our capability to encode/erasure decode $(n,k)$ Reed-
Solomon codes over characteristic-2 finite field in $O(n\lg(n))$ time. As the
complexity leading factor is small, the algorithms are advantageous in
practical applications. To the best of our knowledge, this is the first
approach supporting Reed-Solomon erasure codes on characteristic-2 finite
fields to achieve complexity of $O(n\lg(n))$. In addition, all the transforms
employed in the Reed-Solomon algorithms can be easily implemented in parallel
processing. Hence, the proposed algorithms substantially facilitate practical
applications. While this paper has demonstrated the polynomial basis and
operations over characteristic-2 finite fields, it is of interest to consider
the case over fields with arbitrary characteristics.
## Acknowledgment
This work was supported in part by Ministry of Science and Technology of
Taiwan, under grants NSC 101-2221-E-011-069-MY3, NSC 102-2221-E-001-006-MY2,
and MOST 103-3113-E-110-002.
In [11], Didier present an efficient algorithm to compute the elements in two
sets (40) and (42). The method is presented here for the purpose of
completeness. Consider the construction of $\Pi^{\prime}$. The formal
derivative of $\Pi(x)$ is given by
$\Pi^{\prime}(x)=\sum_{j\in E}\prod_{y\in E,y\neq j}(x+y).$
By substituting $x=j\in E$ into $\Pi^{\prime}(x)$, we have
$\Pi^{\prime}(j)=\prod_{y\in E,y\neq
j}(j+y)=\prod_{y\in\mathbb{F}_{2^{r}},y\neq j}(j+y)^{R(y)},$ (44)
where $R(x)$ is a function defined as
$R(y)=\left\\{\begin{matrix}1&\textrm{if $y\in E$;}\\\
0&\textrm{otherwise.}\end{matrix}\right.$ (45)
Define $Log(x)$ as the discrete logarithm function. For each
$i\in\mathbb{F}_{2^{r}}^{*}$, we denote $Log(i)=j$ iff $i=\alpha^{j}$, where
$\alpha$ is the primitive element of $\mathbb{F}_{2^{r}}^{*}$. Then (44) can
be reformulated as
$Log(\Pi^{\prime}(j))=\biguplus_{y\in\mathbb{F}_{2^{r}},y\neq
j}R(y)Log(j+y),\forall j\in E.$
Note that the symbol $\biguplus$ means the summation with normal additions. By
setting $Log(0)=0$, the above equation can be rewritten as
$Log(\Pi^{\prime}(j))=\biguplus_{y\in\mathbb{F}_{2^{r}}}R(y)Log(j+y),\forall
j\in E.$ (46)
Upon describing the algorithm to compute (46), we consider the construction of
another set $\Pi$. In the same way, the elements of $\Pi$ can be formulated as
$Log(\Pi(j))=\biguplus_{y\in\mathbb{F}_{2^{r}}}R(y)Log(j+y),\forall
j\in\mathbb{F}_{2^{r}}\setminus E.$ (47)
With combining (46) and (47), the objective of algorithm is to compute
$Log(\Pi(j))=\biguplus_{y\in\mathbb{F}_{2^{r}}}R(y)Log(j+y),\forall
j\in\mathbb{F}_{2^{r}}.$ (48)
In (48), the operation $+$ is the $\mathbb{F}_{2^{r}}$ addition, that can be
treated as exclusive-or operation. Hence, (48) is namely the logical
convolution [26][27], that can be efficiently computed with fast Walsh-
Hadamard transform [28]. The algorithm is elaborated as follows.
Let $FWT_{h}[\bullet]$ denote the $h$-point fast Walsh-Hadamard transform
(FWHT). A $h$-point FWHT requires $h\lg(h)$ additions. Define
$R_{2^{r}}=(R(0),R(1),\dots,R(2^{r}-1)),$
$L_{2^{r}}=(0,Log(\omega_{1}),Log(\omega_{2}),\dots,Log(\omega_{2^{r}-1})).$
The result of (48) is computed by
$R_{2^{r}}^{\mathrm{w}}=\mathrm{FWHT}_{2^{r}}[\mathrm{FWHT}_{2^{r}}[R_{2^{r}}]\star\mathrm{FWHT}_{2^{r}}[L_{2^{r}}]],$
(49)
where the operation $\star$ denotes pairwise multiplication. To further reduce
the complexity, the $\mathrm{FWHT}_{2^{r}}[L_{2^{r}}]$ can be pre-computed and
stored, and thus (49) can be done with performing two fast Walsh transforms of
length $2^{r}$. We remark that all the above computation can be performed over
modulo $2^{r}-1$. After obtaining $R_{2^{r}}^{\mathrm{w}}$, we compute the
exponent for each element of $R_{2^{r}}^{\mathrm{w}}$, and this step can be
done via table lookup. In summary, the algorithm requires $O(2^{r}\lg(2^{r}))$
modulus additions, $O(2^{r})$ modulus multiplications, and $O(2^{r})$
exponentiations.
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|
arxiv-papers
| 2014-04-14T04:23:37 |
2024-09-04T02:50:01.121292
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sian-Jheng Lin, Wei-Ho Chung, Yunghsiang S. Han",
"submitter": "Sian-Jheng Lin",
"url": "https://arxiv.org/abs/1404.3458"
}
|
1404.3640
|
André Chailloux, Laura Mančinska, Giannicola Scarpa and Simone Severini09th
Conference on the Theory of Quantum Computation, Communication, and
Cryptography111 10.4230/LIPIcs.xxx.yyy.p
# Graph-theoretical Bounds on the Entangled Value of Non-local Games
André Chailloux SECRET Project Team, INRIA Paris-Rocquencourt,
Paris, France
[email protected] Laura Mančinska Center for Quantum Technologies,
Singapore
[email protected] Giannicola Scarpa Universitat Autònoma de Barcelona,
Barcelona, Spain
[email protected] Simone Severini University College London,
London, United Kingdom
[email protected]
###### Abstract.
We introduce a novel technique to give bounds to the entangled value of non-
local games. The technique is based on a class of graphs used by Cabello,
Severini and Winter in 2010. The upper bound uses the famous Lovász theta
number and is efficiently computable; the lower one is based on the quantum
independence number, which is a quantity used in the study of entanglement-
assisted channel capacities and graph homomorphism games.
###### Key words and phrases:
Graph theory, non-locality, entangled games
###### 1991 Mathematics Subject Classification:
G.2.3 Applications
## 1\. Introduction
In non-local games, two non-communicating players cooperate in order to
achieve a task. Each player receives an input and produces an output, and they
must satisfy the task’s requirements.
In physics, this class of games is also known as “entangled games”. They are
mostly used to investigate the power of entanglement, by designing intuitive
Bell inequalities. One designs a non-local game and proves an upper bound on
the winning probability of the classical players (the Bell inequality). Later,
one shows that there exists a quantum strategy that by using entanglement can
beat that winning probability. Two famous examples of such approach are the
CHSH game (based on [3]) and the magic square game (based on [14]).
Non-local games are also important in computer science, where they are usually
called “two-prover one-round games”. Their intuitive nature has been used in
complexity theory to approach the difficult problem of P vs. NP, by defining
probabilistically checkable proofs and ultimately leading to the famous unique
games conjecture [9, 10].
Estimating or bounding the value of a game given its description is an
important task, and much effort has been devoted to the question. For example,
the entangled value for the class of XOR games has been shown to be easy to
compute with a semidefinite program by Cirel’son [4]. Also the entangled value
of unique games turns out to be easy to compute, therefore falsifying the
unique games conjecture in the quantum world [11].
Here, we propose a general approach to bound the value of a non-local game
based on graph theory. Given the description of a game, we construct a graph
that contains all the information about the game, and we call it the “game
graph”. The construction is based on the techniques in [7]. Such techniques
have also been extended and used in [1].
We first show that the classical value of any game is equal to the
independence number of its game graph (renormalized). This reflects the fact
that computing exactly the classical value of a game is NP-hard. We then show
an efficiently computable upper bound on the quantum value of a game (and
therefore on the classical value), given by the celebrated Lovász theta
number. We then give lower bound for the games on the uniform distribution
given by the quantum independence number, a graph parameter introduced in [5]
and futher discussed in [13, 15]. To conclude, we give a class of games for
which this upper bound is tight.
We believe this graph-theoretical approach is an important and a fertile field
for improvements. We discuss these in the conclusions section.
## 2\. Preliminaries
### 2.1. Non-local games
We now briefly describe the setting of a non-local game $\mathcal{G}$.
Alice and Bob are separated and forbidden to communicate. They receive inputs
$x$ and $y$ from some input sets $X$ and $Y$, according to some fixed and
known probability distribution $\pi$, and are required to produce outputs $a$
and $b$ from output sets $A$ and $B$, respectively. The game rules are encoded
in a predicate $\lambda:X\times Y\times A\times B\rightarrow\\{0,1\\}$, which
specifies which outputs $a,b$ are correct on inputs $x,y$. In other words,
players win the game on inputs $x,y$ if they output some $a,b$ such that
$\lambda(x,y,a,b)=1$. The goal of the players is to maximize the winning
probability.
A _classical strategy_ for the game is without loss of generality a pair of
functions, $f_{A}:X\to A$ for Alice and $f_{B}:Y\to B$ for Bob. (Shared
randomness between the two players is easily seen not to be beneficial.) The
winning probability of a strategy is calculated as follows:
$\sum_{x,y}\pi(x,y)\lambda(x,y,f_{A}(x),f_{B}(y)).$
The _classical value_ $\omega(\mathcal{G})$ of the game is the maximum winning
probability among all classical strategies.
In _entangled strategies_ (a.k.a. quantum strategies), players share a fixed
(_i.e._ , independent of the inputs) entangled state $|\psi\rangle$. For each
input $x$, Alice has a projective measurement $\\{P^{x}_{a}\\}_{a\in A}$, and
for each input $y$, Bob has a projective measurement $\\{Q^{y}_{b}\\}_{b\in
B}$. Upon receiving the inputs, they apply the corresponding measurements to
their parts of the entangled state and produce classical outputs $a$ and $b$,
respectively. The winning probability of a strategy is calculated as follows:
$\sum_{xy}\pi(x,y)\lambda(x,y,a,b)\langle\psi|P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle.$
The _entangled value_ $\omega^{*}(\mathcal{G})$ of the game is the supremum of
the winning probability, taken over all entangled strategies.
A Bell inequality for a game is an upper bound on its classical value. We have
a Bell inequality violation for a game $\mathcal{G}$ if the entangled value is
strictly larger than the classical one. The violation is quantified by the
ratio $\omega^{*}(\mathcal{G})/\omega(\mathcal{G})$.
The CHSH game is one particularly famous example [3]. Here, the inputs
$x\in\\{0,1\\}$ and $y\in\\{0,1\\}$ are uniformly distributed, and Alice and
Bob win the game if their respective outputs $a\in\\{0,1\\}$ and
$b\in\\{0,1\\}$ satisfy $a\oplus b=x\wedge y$; in other words, $a$ should
equal $b$ unless $x=y=1$. The classical value of this game is easily seen to
be $\omega(\mathcal{G})=3/4$, while the entangled value is known to be
$\omega^{*}(\mathcal{G})=1/2+1/(2\sqrt{2})\approx 0.85$.
A non-local game is said to be a _pseudo-telepathy_ game if the quantum value
is 1 while the classical value is strictly less than 1.
### 2.2. Notions of graph theory
A _simple graph_ $G=(V,E)$ consists of a finite vertex set $V$ and its edge
set $E\subsetneq V\times V$ (the inclusion here is strict because there are no
edges of the form $(v,v)$). Two vertices $(v,w)\in E$ are “adjacent” or
equivalently “form an edge”. All graphs considered here are simple graphs. For
a graph $G=(V,E)$, we also denote its vertex set with $V(G)$ and its edge set
with $E(G)$ whenever confusion has to be avoided.
An _independent set_ of a graph is a subset $I$ of $V(G)$ such that no two
elements of $I$ are adjacent. The _independence number_ of a graph $G$,
denoted by $\alpha(G)$, is the maximum size of an independent set of $G$.
A $d$-dimensional orthogonal representation of $G=(V,E)$ is a map
$\phi:V\rightarrow\mathbb{C}^{d}$ such that for all $(v,w)\in E$,
$\langle{\phi(v)}|{\phi(w)}\rangle=0$. (If all the vectors have unit norm,
this is called orthonormal representation.)
We finally introduce an important graph parameter: the _theta number_ (a.k.a.
Lovász number or theta function). It was originally defined by Lovász [12] to
solve a long-standing problem posed by Shannon [16]: computing the Shannon
capacity of the five-cycle. There are many equivalent formulations of the
theta number (see [8] for a detailed survey). The one that we use in this
paper is the following:
$\vartheta(G)=\max\sum_{v\in V(G)}|\langle\psi|\psi_{v}\rangle|^{2},$ (1)
where the maximum is over unit vectors $\psi$ and orthonormal representations
$\\{\psi_{v}\\}_{v\in V(G)}$. Lovász [12] proved that
$\alpha(G)\leq\vartheta(G)$ holds (this inequality is part of the so-called
“sandwich theorem” [8]). The theta number can be approximated to within
arbitrary precision in polynomial time, hence it gives a tractable and in many
cases useful bound for $\alpha$.
### 2.3. Quantum Independence Number
In this section we define the quantum independence number and state some of
its properties.
First, let us briefly give some historical background. In [13] the concept of
quantum independence number is presented in the context of zero-error
information theory. This quantity is usually called in literature “one-shot
zero-error entanglement-assisted channel capacity” and denoted as
$\alpha^{*}$. A new definition of quantum independence number, denoted as
$\alpha_{q}$, came in [15], in the context of graph homomorphisms. As of
today, it is not known if the two quantities are equal for all graphs. In this
paper we use the second quantity, but for simplicity we omit the details about
homomorphisms and provide a direct definition.
As with the quantum chromatic number (see [6]), the quantum independence
number can be defined in terms of a non-local game. Informally, the
_independent set game_ with parameter $t$ for a graph $G=(V,E)$ is as follows.
Two players, Alice and Bob, claim that they know an independent set $I$ of $G$
consisting of $t$ vertices. A referee wants to test this claim with a non-
local game. He forbids communication between the players, generates two
uniformly random numbers $x,y\in[t]$ and separately asks Alice to provide the
$x$-th vertex of $I$ and Bob to provide the $y$-th vertex of $I$. The players
are required to output the same vertex if $x=y$, and to output non-adjacent
vertices if $x\neq y$. A formal definition follows.
###### Definition 2.1.
The independent set game with parameter $t$ on the graph $G=(V,E)$ is a non-
local game with input sets $X=Y=[t]$, output sets $A=B=V$. The probability
distribution $\pi$ is the uniform distribution on the input pairs. Alice gets
input $x$ and outputs $v$, Bob gets input $y$ and outputs $w$. The players
_lose the game_ in the following two cases:
1. (1)
$x=y$ and $v\neq w$
2. (2)
$x\neq y$ and $(v,w)\in E$ or $v=w$
A classical strategy consists w.l.o.g. of two deterministic functions
$f_{A}:[t]\rightarrow V$ for Alice and $f_{B}:[t]\rightarrow V$ for Bob.
Shared randomness, as seen for the coloring game, is not beneficial. A little
thought will show that to win with probability 1, we must have $f_{A}=f_{B}$
(to avoid the first losing condition) and that $\\{f_{A}(1),\dots,f_{A}(t)\\}$
must be a valid independent set of the graph of size $t$ (to avoid the second
losing condition). It follows that the classical players cannot win the game
with probability $1$ when $t>\alpha(G)$.
It is proven in [15] that w.l.o.g. quantum strategies for the independent set
game consist of projective measurements on a maximally entangled state, that
the projective measurements of Alice and Bob are the same and that all the
projectors can be real-valued. Therefore we can define a _quantum independent
set_ of size $t$ as a collection of $t$ projective measurements
$\\{P_{v}^{x}\\}_{v\in V}$ for all $x\in[t]$ that have the whole vertex set as
outputs, with the following consistency condition:
$\mbox{for all }(u,v)\in E\mbox{ or }u=v\mbox{ and for all }x\neq
x^{\prime},\quad P_{u}^{x}P_{v}^{x^{\prime}}=0.$ (2)
###### Definition 2.2.
For all graphs $G$, the _quantum independence number_ $\alpha_{q}(G)$ is the
maximum number $t$ such that there exists a quantum independent set of $G$ of
size $t$.
## 3\. Game graphs
### 3.1. Definition and relation to $\omega(\mathcal{G})$
Consider a non-local game $\mathcal{G}$ with input sets $X,Y$, output sets
$A,B$, predicate $\lambda:X\times Y\times A\times B\rightarrow\\{0,1\\}$ and
uniform distribution on the inputs.
###### Definition 3.1.
A graph $G=\left(V,E\right)$ _associated_ to the game $\mathcal{G}$ has:
1. (1)
$V=\\{xyab\mid x\in X,y\in Y,a\in A,b\in B\mbox{ and }\lambda(x,y,a,b)=1\\}$,
2. (2)
$E=\\{\\{xyab,x^{\prime}y^{\prime}a^{\prime}b^{\prime}\\}\mid(x=x^{\prime}\wedge
a\neq a^{\prime})\vee(y=y^{\prime}\wedge b\neq b^{\prime})\\}$.
This definition is inspired by a construction in [7] in the framework of
contextuality of physical theories. The authors used something similar to
Definition 3.1 for the special case of the CHSH game. Here we generalize to
all games.
For simplicity, we prove the results in this section for the case where the
game has the uniform distribution on the inputs and $\lambda$ is a boolean
function. It is straightforward to generalize to games with real-valued
predicate and any probability distribution $\pi$ of the inputs, as follows.
Consider the (vertex) weighted graph with all the quadruples $xyab$ in the
vertex set, labelled with $\mbox{weight}(xyab)=\lambda(x,y,a,b)\cdot\pi(x,y)$,
and the same edge set as before. The classical bound and the Lovász theta
bound that we will prove later can be adapted by considering the weighted
versions of these parameters. However, we do not know how to generalize our
last result because we do not define the quantum independence number for a
weighted graph.
Now we prove that that the classical value of a game can be expressed in terms
of the independence number of its game graph.
###### Theorem 3.2.
Let $\mathcal{G}$ be a non-local game with input sets $X$ and $Y$, uniform
input distribution and associated graph $G$. Then
$\omega(\mathcal{G})=\frac{\alpha(G)}{|X\times Y|}.$
###### Proof 3.3.
Let $k=|X\times Y|.$ We begin by proving that
$\omega(\mathcal{G})\geq\alpha(G)/k$. Namely, we show that given a maximal
independent set $I\subseteq V$ of size $\ell$, we can exhibit a strategy for
$\mathcal{G}$ that answers correctly to at least $\ell$ of the $k$ questions.
By the structure of $G$, the independent set $I$ cannot contain vertices
$xyab$ and $xy^{\prime}a^{\prime}b^{\prime}$ such that $a\neq a^{\prime}$.
Similarly, $I$ cannot contain vertices $xyab$ and
$x^{\prime}ya^{\prime}b^{\prime}$ such that $b\neq b^{\prime}$. Hence, we have
the following strategy: on input $x$, Alice outputs the unique $a$ determined
by the vertices in the independent set $I$. Bob behaves similarly. Since $V$
contains only winning quadruples $xyab$, the size $\ell$ of the independent
set means Alice and Bob answer correctly to at least $\ell$ input pairs.
Hence, $\omega(\mathcal{G})\geq\ell/k$.
Now we show that $\omega(\mathcal{G})\leq\alpha(G)/k$, _i.e._ , if there
exists a strategy that wins on $\ell$ of the $k$ input pairs, then there
exists an independent set with weight $\ell$. We have that w.l.o.g. classical
strategies consist of a pair of functions. Fix Alice and Bob’s functions
$f_{A}$ and $f_{B}$ that win on $\ell$ input pairs. Now take the set of
quadruples $S=\\{(x,y,f_{A}(x),f_{B}(y))\\}_{x\in X,y\in Y}$. We have that
$I=S\cap V$ is a set of $\ell$ vertices of $G$. Since $f_{A}$ and $f_{B}$ are
deterministic, $I$ cannot contain vertices $xyab$ and
$xy^{\prime}a^{\prime}b^{\prime}$ such that $a\neq a^{\prime}$ nor vertices
$xyab$ and $x^{\prime}ya^{\prime}b^{\prime}$ such that $b\neq b^{\prime}$.
Therefore, there cannot be an edge between any pair of the elements of $I$ and
we have that $I$ is an independent set of $G$ of size $\ell$. Hence,
$\alpha(G)\geq\ell$. Combining the two directions proves the theorem.
### 3.2. Bounds on the entangled value of a game
Cabello, Severini and Winter [7] observe that the quantum value of the CHSH
game is equal to the theta number of its associated graph divided by the
number of questions. We have found by direct calculation that this is not
always true for general games, for example in the case of the 2-fold parallel
repetition of CHSH. The same conclusion follows from the results of Acín _et
al._ in [1]. Here we prove the upper bound directly for our specific
constructions.
###### Theorem 3.4.
Let $\mathcal{G}$ be a non-local game with input sets $X$ and $Y$, uniform
input distribution and associated graph $G=(V,E)$. Then
$\omega^{*}(\mathcal{G})\leq\frac{\vartheta(G)}{|X\times Y|}.$
###### Proof 3.5.
Let $k=|X\times Y|$. Consider a quantum strategy for $\mathcal{G}$ that
achieves the value $\omega^{*}(\mathcal{G})$. It consists of a shared
entangled state $|\psi\rangle$ and a collection of projective measurements
$\\{P^{x}_{a}\\},\\{Q^{y}_{b}\\}$, such that
$\sum_{xyab}\frac{1}{k}\lambda(x,y,a,b)\langle\psi|P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle=\frac{1}{k}\sum_{xyab\in V}\langle\psi|P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle=\omega^{*}(\mathcal{G}).$
For each quadruple $xyab$ let $|\psi_{xyab}\rangle=P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle.$ This is an orthogonal representation of $G$, since for
every edge $(xyab,x^{\prime}y^{\prime}a^{\prime}b^{\prime})$ either
$P^{x}_{a}P^{x^{\prime}}_{a^{\prime}}=0$ or
$Q^{y}_{b}Q^{y^{\prime}}_{b^{\prime}}=0$. Now for each $xyab$ consider the
normalized vector
$|\psi^{\prime}_{xyab}\rangle=\frac{|\psi_{xyab}\rangle}{||\psi_{xyab}||}=\
\frac{|\psi_{xyab}\rangle}{\sqrt{\langle\psi|P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle}}.$
We have that $\\{\psi^{\prime}_{xyab}\\}_{xyab\in V}$ and $\psi$ are a
feasible solution for the formulation (1) of $\vartheta(G)$.
We conclude
$\displaystyle\vartheta(G)$ $\displaystyle\geq\sum_{xyab\in
V}|\langle\psi|\psi_{xyab}\rangle|^{2}$ $\displaystyle=\sum_{xyab\in
V}\left|\frac{\langle\psi|\psi_{xyab}\rangle}{||\psi_{xyab}||}\right|^{2}$
$\displaystyle=\sum_{xyab\in V}\frac{\langle\psi|P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle^{2}}{\langle\psi|P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle}$ $\displaystyle=\sum_{xyab\in
V}\langle\psi|P^{x}_{a}\otimes Q^{y}_{b}|\psi\rangle$
$\displaystyle=k\cdot\omega^{*}(\mathcal{G}).$
We now have the following lower bound in terms of the quantum independence
number.
###### Theorem 3.6.
Let $\mathcal{G}$ be a non-local game with input sets $X$ and $Y$, uniform
input distribution and associated graph $G=(V,E)$. Then
$\omega^{*}(\mathcal{G})\geq\frac{\alpha_{q}(G)}{|X\times Y|}$
To prove the theorem, we will use the following lemma.
###### Lemma 3.7.
Let $M,N$ be positive semidefinite matrices. Then for any vector $|v\rangle$,
we have that
$\langle v|\operatorname{supp}(M+N)|v\rangle\geq\langle
v|\operatorname{supp}(M)|v\rangle,$
where $\operatorname{supp}(M)$ denotes the projector onto the support (_i.e.,_
the column space) of $M$.
###### Proof 3.8.
If $P$ is a projector onto a subspace $\Pi$ then $\langle v|P|v\rangle$ is the
squared length of the projection of $|v\rangle$ into $\Pi$. Hence, to prove
the lemma it suffices to show that
$\operatorname{supp}(M)\subseteq\operatorname{supp}(M+N)$, where by abusing
the notation we use $\operatorname{supp}$ to denote the support itself (rather
than the projection onto it).
For contradiction, suppose that
$\operatorname{supp}(M)\not\subseteq\operatorname{supp}(M+N)$. Then the
orthogonal complement of $\operatorname{supp}(M)$ (i.e. the nullspace
$\operatorname{Null}(M))$ does not contain $\operatorname{Null}(M+N)$. Hence
we can pick a vector $|w\rangle$ such that $(M+N)|w\rangle=0$ but
$M|w\rangle\neq 0$. This further implies that
$\langle w|N|w\rangle=\langle w|(M+N)|w\rangle-\langle w|M|w\rangle=-\langle
w|M|w\rangle<0,$
since $M$ is positive semidefinite and $M|w\rangle\neq 0$. This completes the
proof as we have reached a contradiction with the initial assumption that $N$
is positive semidefinite.
###### Proof 3.9 (Proof of Theorem 3.6.).
Given a quantum strategy $\\{P_{xyab}^{i}\\}$ for the independent set game on
$G$ with parameter $t$, we construct a strategy to win the game $\mathcal{G}$
with probability at least $t/|X\times Y|$, as follows.
Players share a maximally entangled state with local dimension $d$ (which is
the dimension of the projectors above). On input $x,$ Alice measures her half
of the state using the projective measurement$\\{P^{x}_{a}\\}_{a\in
A}\bigcup\\{I-\sum_{a}P^{x}_{a}\\}$, where the individual elements are defined
as follows:
$P^{x}_{a}=\operatorname{supp}\left(\sum_{\stackrel{{\scriptstyle
yb}}{{xayb\in V}}}\sum_{i}P_{xayb}^{i}\right),$
where we use $\operatorname{supp}(M)$ to denote the projector onto the image
of $M$. We show that this is a valid projective measurement. For all
$y,b,y^{\prime},b^{\prime}$ there is an edge
$(xyab,xy^{\prime}a^{\prime}b^{\prime})\in E$. Therefore in the strategy for
the independent set game we have that for all $i,j$ each projector
$P^{i}_{xyab}$ is orthogonal to $P^{j}_{xy^{\prime}a^{\prime}b^{\prime}}$.
Hence, for all $a\neq a^{\prime}$ we have $P^{x}_{a}\cdot
P^{x}_{a^{\prime}}=0$. Bob constructs projectors $P^{y}_{b}$ similarly.
Now we lower bound the quantum value of $\mathcal{G}$ as follows:
$\displaystyle|X\times Y|\cdot\omega^{*}(\mathcal{G})$
$\displaystyle\geq\sum_{xyab\in V}\langle\psi|P^{x}_{a}\otimes
P^{y}_{b}|\psi\rangle$ $\displaystyle=\sum_{xyab\in
V}\langle\psi|\operatorname{supp}\Big{(}\sum_{i,j}\sum_{\stackrel{{\scriptstyle
y^{\prime}b^{\prime}}}{{xay^{\prime}b^{\prime}\in
V}}}\sum_{\stackrel{{\scriptstyle
x^{\prime}a^{\prime}}}{{x^{\prime}a^{\prime}yb\in
V}}}P_{xay^{\prime}b^{\prime}}^{i}\otimes
P_{x^{\prime}a^{\prime}yb}^{j}\Big{)}|\psi\rangle,$
where we have used the fact that $\operatorname{supp}(M\otimes
N)=\operatorname{supp}(M)\otimes\operatorname{supp}(N)$ for all matrices $M,N$
to obtain the last equality. Now by applying Lemma 3.7, we drop all the terms
except the ones with $i=j,a=a^{\prime},b=b^{\prime},x=x^{\prime}$ and
$y=y^{\prime}$, and we have that
$\displaystyle|X\times Y|\cdot\omega^{*}(\mathcal{G})$
$\displaystyle\geq\sum_{xyab\in
V}\langle\psi|\operatorname{supp}\Big{(}\sum_{i}P_{xayb}^{i}\otimes
P_{xayb}^{i}\Big{)}|\psi\rangle$ (3) $\displaystyle=\sum_{xyab\in
V}\langle\psi|\Big{(}\sum_{i}P_{xayb}^{i}\otimes
P_{xayb}^{i}\Big{)}|\psi\rangle$ (4) $\displaystyle=\sum_{xyab\in
V}\sum_{i}\frac{1}{d}\mbox{\rm Tr}(P_{xayb}^{i})$ (5)
$\displaystyle=\sum_{i}\frac{1}{d}\mbox{\rm Tr}(I_{d})$ (6)
$\displaystyle=\alpha_{q}(G).$ (7)
In the above we have observed that $\operatorname{supp}(P+Q)=P+Q$ for mutually
orthogonal projectors $P$ and $Q$ to get Expression (4). We have used
properties of $|\psi\rangle=\frac{1}{\sqrt{d}}\sum_{i}|i,i\rangle$ to obtain
Expression (5). We have used the fact that, for all $i$,
$\\{P^{i}_{xayb}:\lambda(x,a,y,b)=1\\}$ forms a measurement to obtain
Expression (6).
#### 3.2.1. Tightness of the lower bound
Here we obtain an equality relation between the value of the game and the
quantum independence number of the game graph, for a class of pseudo-telepathy
games.
###### Theorem 3.10.
Let $\mathcal{G}$ be a pseudo-telepathy game with a 0-1 valued predicate
$\lambda$, admitting a quantum strategy consisting of a maximally entangled
state $|\psi\rangle$ and pairwise commuting projectors. Let $G$ be the
corresponding game graph. Then,
$\omega^{*}(\mathcal{G})=\frac{\alpha_{q}(G)}{|X\times Y|}=1.$
###### Proof 3.11.
From Theorem 3.6 we have $\alpha_{q}(G)\leq|X\times
Y|\cdot\omega^{*}(\mathcal{G})$. We need to prove the other direction.
Let $\\{P^{x}_{a}\\},\\{Q^{y}_{b}\\}$ be the strategies that win the game
$\mathcal{G}$ on $|\psi\rangle$. We have:
$\sum_{xy}\pi(x,y)\sum_{ab:\lambda(xyab)=1}\langle\psi|P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle=1,$
so for all $(x,y)$ we must have
$\sum_{ab:\lambda(xyab)=1}\langle\psi|P^{x}_{a}\otimes
Q^{y}_{b}|\psi\rangle=1$
and for all quadruples $(x,y,a,b)$ such that $\lambda(xyab)=0$ we have
$P^{x}_{a}Q^{y}_{b}=0$.
Let $\Pi_{xyab}=P^{x}_{a}Q^{y}_{b}.$ These are projectors thanks to the
commutativity assumption. We observe:
1. (1)
For all $(x,y)$ we have
$\sum_{ab:\lambda(xyab)=1}P^{x}_{a}Q^{y}_{b}=\sum_{\stackrel{{\scriptstyle
ab}}{{}}}P^{x}_{a}Q^{y}_{b}=\sum_{a}P^{x}_{a}\sum_{b}Q^{y}_{b}=I,$
where the second equality follows from
$Q^{y}_{b}Q^{y}_{b^{\prime}}=\delta_{bb^{\prime}}$.
2. (2)
For each edge $(x,y,a,b),(x^{\prime},y^{\prime},a^{\prime},b^{\prime})$ we
have a collection of $t$ real-valued projective measurements
$\\{P_{v}^{x}\\}_{v\in V}$ for all $x\in[t]$ that have the whole vertex set as
outputs,
$\Pi_{xyab}\Pi_{x^{\prime}y^{\prime}a^{\prime}b^{\prime}}=0,$
because if $x=x^{\prime}$ and $a\neq a^{\prime}$ then
$P^{x}_{a}P^{x}_{a^{\prime}}=0$, and if $y=y^{\prime}$ and $b\neq b^{\prime}$
then $Q^{y}_{b}Q^{y}_{b^{\prime}}=0$.
Therefore, we can construct $|X\times Y|$ projective measurements that are a
winning strategy for the independent set game with $t=|X\times Y|$ as follows.
For each pair $(x,y)$ consider the projective measurement
$\\{\Pi_{xyab}\\}_{a,b:\lambda(xyab)=1}$ (and zero matrices for the other
vertices of the graph). The first observation above proves that those are
valid projective measurements; the second observation shows that they respect
the consistency condition (2).
## 4\. Concluding remarks and open problems
We have formalized and discussed a novel approach for the study of non-local
game in a combinatorial fashion. Work in progress on this approach relate to
the easy generalization to more than 2 players, and the less-easy computation
of graphs for the parallel repetition of games.
Our approach has ample room for improvement. Open questions include:
1. (1)
Can we find a tighter lower bound for the entangled value of all games by
using some variant of the quantum independence number, such as the one in [2]?
Alternatively, can we prove tightness of the current lower bound?
2. (2)
Can we find better lower bounds, for example using one of the variants of
Lovász theta number?
3. (3)
Can we characterize the class of games for which the Lovász bound is tight? We
know that the value of CHSH is exactly the theta number of its game graph (see
[7]). Is this true for all the XOR games? This would reflect the fact that
their value is easy to compute.
4. (4)
Are there other graph parameters related to the classical and entangled values
of specific classes of games, for example unique games?
5. (5)
We have shown that for a class of pseudo-telepathy games that quantum players
can win using commutative projective measurements on maximally entangled
state, this bound is tight. A similar class of games is shown in [13] to be in
one-to-one correspondence with a generalization of Kochen-Specker sets. It is
not clear to us if those two results together could be used to prove something
stronger. Perhaps the whole class could be interpreted as pseudo-telepathy
games based on some graph parameter (maybe the homomorphism games in [15]) and
the relationship to the quantum independence number would be a consequence of
this.
##### Acknowledgements
The authors thank the referees of TQC 2014 for useful comments. Laura
Mančinska is supported by the MOE Tier 3 Grant “Random numbers from quantum
processes” (MOE2012-T3-1-009). G. Scarpa is supported by the European
Commission project RAQUEL(323 970). Part of this work was done while G. Scarpa
was a PhD student at CWI, supported by Ronald de Wolf’s VIDI grant from NWO.
S. Severini is supported by the Royal Society and EPSRC.
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arxiv-papers
| 2014-04-14T16:09:30 |
2024-09-04T02:50:01.133134
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Andr\\'e Chailloux, Laura Man\\v{c}inska, Giannicola Scarpa and Simone\n Severini",
"submitter": "Giannicola Scarpa",
"url": "https://arxiv.org/abs/1404.3640"
}
|
1404.3757
|
††thanks: Electronic address: [email protected]
# Inheritance patterns in citation networks reveal scientific memes
Tobias Kuhn Chair of Sociology, in particular of Modeling and Simulation, ETH
Zurich, 8092 Zurich, Switzerland Matjaž Perc Faculty of Natural Sciences and
Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor,
Slovenia CAMTP – Center for Applied Mathematics and Theoretical Physics,
University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia Dirk Helbing
Chair of Sociology, in particular of Modeling and Simulation, ETH Zurich, 8092
Zurich, Switzerland Risk Center, ETH Zurich, 8092 Zurich, Switzerland
###### Abstract
Memes are the cultural equivalent of genes that spread across human culture by
means of imitation. What makes a meme and what distinguishes it from other
forms of information, however, is still poorly understood. Our analysis of
memes in the scientific literature reveals that they are governed by a
surprisingly simple relationship between frequency of occurrence and the
degree to which they propagate along the citation graph. We propose a simple
formalization of this pattern and we validate it with data from close to 50
million publication records from the Web of Science, PubMed Central, and the
American Physical Society. Evaluations relying on human annotators, citation
network randomizations, and comparisons with several alternative approaches
confirm that our formula is accurate and effective, without a dependence on
linguistic or ontological knowledge and without the application of arbitrary
thresholds or filters.
The evaluation of scientific output and the study of patterns of scientific
collaboration have received increasing attention by researchers. From citation
distributions 1, 2, coauthorship networks 3 and the formation of research
teams 4, 5, to the ranking of researchers 6, 7, 8 and the quantification and
prediction of scientific success 9, 10 — how we do science has become a
science in its own right. While the famous works of Derek de Solla Price 11
and Robert Merton 12 from the mid 1960s marked the beginning of a popular and
long-lasting research field, the rapid progress made in recent years is
largely due to the increasing availability of vast amounts of digitized data.
Massive publication and citation databases, also referred to as
“metaknowledge” 13, along with leaps of progress in the theory and modeling of
complex systems, fuel large-scale explorations of human culture that were
unimaginable even a decade ago 14. The “science of science” is scaling up
massively as well, with studies on global citation and collaboration networks
15, the “scientific food web” 16, and phylomemetic patterns in the evolution
of science 17, culminating in the visually compelling atlases of science 18
and knowledge 19.
Science is a key pillar of modern human culture, and the general concept of
memes has proved to be very insightful for the study of culture. The term
“meme” was coined by Richard Dawkins in his book _The Selfish Gene_ 20, where
he argues that cultural entities such as words, melodies, recipes, and ideas
evolve similarly as genes, involving replication and mutation but using human
culture instead of the gene pool as their medium of propagation. Recent
research on memes has enhanced our understanding of the dynamics of the news
cycle 21, the tracking of information epidemics in the blogspace 22, and the
political polarization on Twitter 23. It has been shown that the evolution of
memes can be exploited effectively for inferring networks of diffusion and
influence 24, and that information contained in memes is evolving as it is
being processed collectively in online social media 25. The question of how
memes compete with each other for the limited and fluctuating resource of user
attention has also amassed the attention of scientists, demonstrating that
social network structure is crucial to understand the diversity of memes 26,
which suggests that social contagion mechanisms 27 play an important role. It
has also been shown that the competition among memes can bring the network at
the brink of criticality 28, where even minute disturbances can lead to
avalanches of events that make a certain meme go viral 29.
While the study of memes in mass media and popular culture has been based
primarily on their aggregated bursty occurrence patterns, the citation network
of scientific literature allows for more sophisticated and fine-grained
analyses. _Quantum_ , _fission_ , _graphene_ , _self-organized criticality_ ,
and _traffic flow_ are examples of well-known memes from the field of physics,
but what exactly makes such memes different from other words and phrases found
in the scientific literature? As an answer to this question, we propose the
following definition that is a modified version of Dawkins’ definition of the
word “gene” 20: _A scientific meme is a short unit of text in a publication
that is replicated in citing publications and thereby distributed around in
many copies; the more likely a certain sequence of words is to be broken
apart, altered, or simply not present in citing publications, the less it
qualifies to be called a meme._ Publications that reproduce words or phrases
from cited publications are thus the analogue to offspring organisms that
inherit genes from their parents. In contrast to existing work on scientific
memes, our approach is therefore grounded in the “inheritance mechanisms” of
memes and not just their accumulated frequencies. The above definition covers
memes made up of exact words and phrases, but the same methods apply just as
well to more abstract forms of memes.
Figure 1: Citation networks of the Web of Science and the American Physical
Society (APS) datasets reveal community structures that nicely align with
scientific disciplines, journals covering particular subfields, and
occurrences of memes. The generation of the visualizations was based on Gephi
30 and the OpenOrd plugin 31, which implements a force-directed layout
algorithm that is able to handle very large graphs.
For our analyses, we rely on 47.1 million publication records from three
sources. Due to its representative long-term coverage of a specific field of
research, we focus mainly on the titles and abstracts from the dataset of the
American Physical Society, consisting of almost half a million publications
from the Physical Review journals published between July 1893 and December
2009. We also present results for the over 46 million publications indexed by
the comprehensive Web of Science database, and for the over 0.6 million
publications from the open access subset of PubMed Central that covers
research mostly from the biomedical domain and mostly from recent years.
Fig. 1 shows visualizations of these citation graphs. The leftmost network
depicts the entire giant component of the citation graph of the Web of
Science, consisting of more than 33 million publications. Different scientific
disciplines form relatively compact communities: The physical sciences (cyan)
are close to engineering and technology (magenta) in the top right corner of
the network, but rather far from the social sciences and humanities (green) as
well as the medical and health sciences (red), which take up the majority of
the left hand side of the network, with the natural and agricultural sciences
in between (blue). Zooming in on the physical sciences and switching to the
dataset from the American Physical Society, we get the picture shown in the
middle. The colors now encode the five most important special-focus journals
of Physical Review, each covering a particular subfield of physics (general-
coverage and smaller journals are shown in gray). We see a complex structure
with many small and large clusters. Importantly, even though the employed
layout algorithm 31 did not take the scientific disciplines and the journal
information explicitly into account, the different communities can be clearly
inferred in the citation graphs. Following our general meme-centric
perspective, the rightmost network highlights the above-mentioned memes from
physics, which mostly appear in publications that form compact communities in
the citation graph. The meme _quantum_ is widely but by no means uniformly
distributed, pervading several large clusters. Publications containing the
meme _fission_ form a few connected clusters limited to an area that makes up
the journal Physical Review C covering nuclear physics. Similarly, the memes
_graphene_ , _self-organized criticality_ , and _traffic flow_ (see enlarged
area) are each concentrated in their own medium-sized or small communities.
## Results
All words and phrases that occur frequently in the literature can be
considered important memes, but many frequent words like “method” are not
particularly interesting for any given scientific field. To quantify the
degree to which a meme is interesting, we define the propagation score
$P_{m}$, which determines the alignment of the occurrences of a given meme
with the citation graph. $P_{m}$ is high for memes that frequently appear in
publications that cite meme-carrying publications (“sticking”) but rarely
appear in publications that do not cite a publication that already contains
the meme (“sparking”). Formally, we define the propagation score for a given
meme $m$ as its _sticking factor_ divided by its _sparking factor_. The
sticking factor quantifies the degree to which a meme replicates in a
publication that cites a meme-carrying publication. Concretely, it is defined
as $d_{m{\rightarrow}m}/d_{{\rightarrow}m}$, where $d_{m{\rightarrow}m}$ is
the number of publications that carry the meme and cite at least one
publication carrying the meme, while $d_{{\rightarrow}m}$ is the number of all
publications (meme-carrying or not) that cite at least one publication that
carries the meme. Similarly, the sparking factor quantifies how often a meme
appears in a publication without being present in any of the cited
publications. It is thus defined as
$d_{m{\rightarrow}\cancel{m}}/d_{{\rightarrow}\cancel{m}}$, where
$d_{m{\rightarrow}\cancel{m}}$ is the number of meme-carrying publications
that do _not_ cite publications that carry the meme, and
$d_{{\rightarrow}\cancel{m}}$ is the number of _all_ publications (meme-
carrying or not) that do _not_ cite meme-carrying publications. For the
propagation score $P_{m}$, we thus obtain
$P_{m}=\left.\frac{d_{m{\rightarrow}m}}{d_{{\rightarrow}m}}\middle/\frac{d_{m{\rightarrow}\cancel{m}}}{d_{{\rightarrow}\cancel{m}}}\right..$
(1)
Based on the propagation score $P_{m}$ and the frequency of occurrence $f_{m}$
(which is simply the ratio of publications carrying the meme) of a particular
meme $m$, we define the _meme score_ $M_{m}$ as
$M_{m}=f_{m}P_{m}.$ (2)
The propagation score, as defined in Eq. 1, can be improved by adding a small
amount of controlled noise $\delta$, thus obtaining
$P_{m}=\left.\frac{d_{m{\rightarrow}m}}{d_{{\rightarrow}m}+\delta}\middle/\frac{d_{m{\rightarrow}\cancel{m}}+\delta}{d_{{\rightarrow}\cancel{m}}+\delta}\right..$
(3)
This corrects for the fact that any of the four basic terms can be zero, and
it also prevents that phrases with a very low frequency get a high score by
chance. The controlled noise corresponds to $\delta$ fictitious publications
that carry all memes and cite none, plus another $\delta$ publications that
carry no memes and cite all. This decreases the sticking factors and increases
the sparking factors of all memes, thereby reducing all meme scores — very
slightly so for frequent memes but heavily for rare ones. Our tests show that
a small value of $\delta$ (e.g. $\delta=3$ as used throughout this work unless
stated otherwise) is sufficient. Another matter that deserves attention is the
potential “free-riding” of shorter memes on longer ones. For example, the
multi-token meme “the littlest Higgs model,” contains the specific token
“littlest” that rarely occurs otherwise. The meme “littlest” therefore gets
about the same propagation score as the long meme, yet the larger meme is
clearly more interesting. This can be addressed by discounting for free-riding
by redefining the term $d_{m{\rightarrow}m}$ in Eq. 1 to exclude publications
where the given meme appears in the publication and its cited publications
only within the same larger meme. If “littlest,” for example, is always
followed by “Higgs” in a given publication and all its cited publications,
then this publication shall not contribute to the $d_{m{\rightarrow}m}$ term
for $m=\mbox{``littlest''}$.
The meme score considers whether a meme is important ($f_{m}$) and whether it
is interesting ($P_{m}$), and it has additionally a number of desirable
properties: (i) it can be calculated exactly without the introduction of
arbitrary thresholds, such as a minimum number of occurrences, without
limiting the length of $n$-grams to consider, and without filtering out words
containing special characters; (ii) it does not depend on external resources,
such as dictionaries or other linguistic data; (iii) it does not depend on
filters, like stop-word lists, to remove the most common words and phrases;
and (iv) it is very simple with only one parameter ($\delta$).
Figure 2: Universality in the distribution patterns of scientific memes
across datasets. Heat maps encode the density of all $n$-grams with $M\neq 0$
($N$ being the number of such $n$-grams) with respect to their propagation
score and frequency. Maps A, C and D each show a broad band with a downward
slope for the datasets from the American Physical Society (APS), the open
access subset of PubMed Central, and the Web of Science, respectively. The
99.9%-quantile with respect to the meme score distribution ($M_{0.999}$) is
depicted as a white line. Memes are located mostly around the very edge of the
top-right side of the band (in the vicinity of the 99.9%-quantile line). Heat
map B shows the results obtained with a time-preserving randomization of the
APS citation graph (see Methods for details).
Calculating the meme score for all $n$-grams in the three datasets considered
gives us the results presented in Fig. 2. Their relatively frequency and their
propagation score are plotted against each other in the form of heat maps with
logarithmic scales. There is no upper limit to the length of $n$-grams, and
the presented maps cover without exception all $n$-grams with a non-zero meme
score ($N$ being the number of such $n$-grams). Meme scores are increasing
towards the top-right and decreasing towards the bottom-left corner. Maps A, C
and D feature a broad band with a downward slope, indicating that, in general,
more frequent memes tend to propagate less via the citation graph. In the
lower half of each map, we see a wedge of very high densities that follows the
larger band on the bottom-left edge, but getting narrower towards the middle
where it ends. Though this wedge has a somewhat rounder and broader shape for
the Web of Science database, overall these patterns look remarkably similar
across all datasets despite their differences with respect to topic, coverage,
and size. This is an indication of universality in the distribution patterns
of scientific memes. The 99.9%-quantile line ($M_{0.999}$) is also
surprisingly stable, considering that the underlying values range over five
orders of magnitude or more. Localizing the previously mentioned physics memes
in the APS dataset (map A), we see that they are located on the very edge of
the top-right side of the band, where the density of $n$-grams is very low.
(Very frequent words like “of” or “the” are found in the faint spike at the
top of the plot where $P\approx 1$ and the frequency is close to 100%.)
The heat map B in Fig. 2 illustrates a typical case of what happens when the
APS citation graph is randomized but the time ordering of publications is
preserved. The number of terms with a non-zero meme score decreases
dramatically (from $\approx\,$1.4 million in map A to just 89,356 in map B),
the universal distribution pattern of scientific memes vanishes, and the top-
right part, where the top-ranked memes should be located, disappears
completely. Naturally, if the APS citation graph is randomized without
preserving the time ordering, the overlap with the original results presented
in map A is even smaller (see Supplementary Material). Statistical analysis
reveals that median values of the meme score obtained with the randomized
networks differ by more than one order of magnitude from those obtained with
the original citation graph, with very little variation between different
randomization runs. These results show that topology and time structure alone
fail to account for the reported universality in the distribution patterns,
and that thus the top memes get their high meme scores based on intricate
processes and conventions that underlie the dynamics of scientific progress
and the way credit is given to previous work.
Table 1 shows the $50$ top-ranked memes from the APS dataset, also indicating
their agreement with human annotation and whether they can be found under a
subcategory of physics in Wikipedia. Most of these memes are noun phrases
denoting real and reasonable physics concepts, which is remarkable given that
the computation of the simple meme score formula uses no linguistic or
ontological knowledge whatsoever. The dominance of noun phrases is consistent
with the finding that (scientific) concepts are typically captured by noun
phrases when represented as keywords in terminologies 32, 33. The extracted
memes consist of one, two or three words, which indicates that the meme score
does not favor short or long phrases, again without applying explicit measures
to balance $n$-gram lengths. A further observation is that chemical formulas
such as MgB2 and CuGeO3 are relatively frequent, which we investigate in more
detail below.
1. | loop quantum cosmology+* | 14. | strange nonchaotic | 27. | NaxCoO2+ | 38. | inspiral*
---|---|---|---|---|---|---|---
2. | unparticle+* | 15. | in NbSe3 | 28. | the unparticle+ | 39. | spin Hall effect+*
3. | sonoluminescence+* | 16. | spin Hall+ | 29. | black | 40. | PAMELA
4. | MgB2+ | 17. | elliptic flow+* | 30. | electromagnetically induced | 41. | BaFe2As2+
5. | stochastic resonance+* | 18. | quantum Hall+* | | transparency+* | 42. | quantum dots+*
6. | carbon nanotubes+* | 19. | CeCoIn5+ | 31. | light-induced drift+ | 43. | Bose-Einstein condensates+
7. | NbSe3+ | 20. | inflation+ | 32. | proton-proton bremsstrahlung+ | 44. | X(3872)*
8. | black hole+* | 21. | exchange bias+* | 33. | antisymmetrized molecular | 45. | relaxor+
9. | nanotubes+ | 22. | Sr2RuO4+ | | dynamics+ | 46. | blue phases+
10. | lattice Boltzmann+* | 23. | traffic flow+* | 34. | radiative muon capture+ | 47. | black holes+*
11. | dark energy+* | 24. | TiOCl | 35. | Bose-Einstein+ | 48. | PrOs4Sb12+
12. | Rashba | 25. | key distribution+ | 36. | C60+ | 49. | the Schwinger multichannel method+
13. | CuGeO3+ | 26. | graphene+* | 37. | entanglement+ | 50. | Higgsless+
Table 1: Top 50 memes with respect to their meme score from the APS dataset. The symbol + indicates memes where the human annotators agreed that this is an interesting and important physics concept, while the symbol * indicates memes that are also found on the list of memes extracted from Wikipedia (see Methods for details). method | main class | annotator | (annotator | classified as main class: | $p$-value for difference to:
---|---|---|---|---|---
| | | agreement) | _individually_ | _in agreement_ | _random_ | _weighted random_
| physics concept | A1 | (90.0%) | 85.3% | 81.3% | ${<}10^{-15}$* | ${<}10^{-15}$*
meme score | A2 | 87.3%
noun phrase | A1 | (93.3%) | 86.0% | 82.7% | ${<}10^{-15}$* | ${<}10^{-15}$*
| A2 | 86.0%
| physics concept | A1 | (85.3%) | 32.7% | 25.3% | – | 0.123
random | A2 | 32.7%
noun phrase | A1 | (86.0%) | 39.3% | 33.3% | – | 0.163
| A2 | 36.0%
| physics concept | A1 | (90.7%) | 20.7% | 19.3% | 0.123 | –
weighted random | A2 | 27.3%
noun phrase | A1 | (86.0%) | 28.7% | 25.3% | 0.163 | –
| A2 | 28.7%
Table 2: The two annotators (A1 and A2) classified more than 80% of the memes
with the highest meme scores as relevant physics concepts and noun phrases.
The differences involving the meme score are highly significant (*). See
Methods and Supplementary Material for details.
In Table 2, we present results of a manual annotation of terms identified by
meme score as compared to randomly selected terms (see also Supplementary
Material). Each of the annotators considered around 86% of the meme score
terms to be important physics concepts, agreeing on this in 81% of the cases.
With respect to their linguistic categories, each annotator considered 86% of
the meme score terms to be noun phrases, and the two annotators agreed on that
for 83% of the terms. The respective values are much lower for the randomly
extracted terms. Only 25% (non-weighted) and 19% (weighted) of terms were, in
agreement, found to be important physics concepts, and only 33% (non-weighted)
and 25% (weighted) to be noun phrases. The reported differences between the
meme score and the two random selection methods are highly significant
($p<10^{-15}$ using Fisher’s exact test on the number of agreed
classifications). These results confirm that the meme score strongly favors
noun phrases and important concepts.
Figure 3: The meme score outperforms alternative metrics. The box plot shows
the agreement with the ground-truth list of physics terms extracted from
Wikipedia as achieved by the different metrics. Agreement is measured as the
area under the curve $A$, as shown for the meme score in the embedded graph on
the top right. The curves are defined as the percentage from the $x$ top-
ranked terms that also appear on the ground-truth list for the different
parameter settings ($1\leq\delta\leq 10$ for the meme score; the thick line
stands for $\delta=4$, which has the largest area $A$; see Methods for
details).
Next we compare the meme score to a number of possible alternative metrics, as
defined in the Methods section, and align the identified words and phrases
with a ground-truth list of terms extracted from physics-related Wikipedia
titles. Fig. 3 summarizes the results, showing on the top right that
$\approx\,$70% of the top $10$ memes identified by meme score correspond to
terms extracted from Wikipedia, $\approx\,$55% of the top $20$, $\approx\,$40%
of the top $50$, and $\approx\,$26% of the top $100$. The largest area under
the curve $A$ is obtained for a controlled noise level $\delta=4$, which is
highlighted by the thick blue line. The box plot compares the outcomes of
different metrics with respect to $A$, as described in the Methods section.
The isolated outlier at 32.3% on the lower end of the distribution for the
meme score originates from the parameter setting $\delta=1$. All other
parameter settings $2\leq\delta\leq 10$ lead to results in a narrow band
between 40.9% and 44.8% (meaning that the performance of the metric is not
sensitive to movements within this range of the parameter space). In contrast,
all alternative metrics score considerably worse, consistently below 22%
(including outliers).
Figure 4: Plot A shows that phrases with a high meme score (i.e. around the
99.9%-quantile $M_{0.999}$) tend to be found as titles of Wikipedia articles
on physics, while other phrases tend not to. Plot B shows that phrases of
relatively low frequency but high meme score have the highest density of terms
containing chemical formulas (e.g. “MgB2”). The data points of the two plots
are the same as in plot A of Fig. 2, but colors showing the relative density
of the specific type of terms. Note that the colors are log scaled: white
stands for 100% terms of the given type, yellow for $\approx\,$30%, orange for
$\approx\,$10%, light red for $\approx\,$2%, dark red for $<\,$1%, and black
for $<\,$0.1%.
The two plots of Fig. 4 show the same data points as plot A of Fig. 2 but with
colors standing for the relative density of Wikipedia terms (A) and of terms
containing chemical formulas (B). Plot A confirms that phrases in the area of
a high meme score (towards the top right) tend to show up as titles of
Wikipedia articles on physics. Additionally, the plot shows that this is the
_only_ such area. There are a few scattered outliers, but the only significant
area with a high density of Wikipedia terms is found around the
99.9%-quantile. Plot B shows that phrases containing chemical formulas (such
as “BaFe2As2”) tend to have a relatively low frequency (individually) but high
propagation score. The area with the highest density can again be found along
the 99.9%-quantile, which is consistent with the expectation of chemical
compounds to be important and interesting entities for physics research. The
fact that they are standardized and compressed representations reduces
moreover their “vulnerability” to synonyms or spelling variants, making them
stronger memes on the level of pure character sequences. (We come back to the
issue of memes on different levels of abstraction below.)
Figure 5: Time history of top physics memes based on their meme scores
obtained from the American Physical Society dataset. The time axis is scaled
by publication count. Bars and labels are shown for all memes that top the
rankings for at least ten out of the displayed $911$ points in time. The gray
area represents the second-ranked meme at a given time.
Fig. 5 shows the top memes over time, revealing bursty dynamics, akin to the
one reported previously in humans dynamics 34 and the temporal distribution of
words 35. These bursts might be a reflection of the fast rise and fall of many
scientific memes in terms of their popularity. As new scientific paradigms
emerge, the old ones seem to quickly lose their appeal, and only a few memes
manage to top the rankings over extended periods of time. The bursty dynamics
also support the idea that both the rise and fall of scientific paradigms is
driven by robust principles of self-organization 36.
## Discussion
By going back to the original analogy with genes put forward by Richard
Dawkins 20, we investigated the relation between the occurrence frequency of
scientific memes and the degree to which they propagate along the citation
graph. We found that scientific memes are indeed governed by a surprisingly
simple relationship of these two factors. This is formalized by the meme score
— a metric to characterize and identify scientific memes — defined as the
product of the frequency of occurrence and the propagation score.
We have shown that the meme score can be calculated exactly and exhaustively
without the introduction of arbitrary thresholds or filters and without
relying on any kind of linguistic or ontological knowledge. The method is fast
and reliable, and it can be applied to massive databases. We have demonstrated
the effectiveness of the meme score on more than 47 million publication
records from the Web of Science, PubMed Central, and the American Physical
Society. Moreover, we have evaluated the accuracy of the proposed meme score
by means of full and time-preserving randomizations of the citation graphs, by
means of manual annotation of publications, as well as by means of several
alternative metrics. We have provided statistical evidence for the agreement
between human annotators and the meme-score results, and we have shown that it
is superior to alternative metrics. We have also confirmed that the observed
patterns cannot be explained by topological or temporal features alone, but
are grounded in more intricate processes that determine the dynamics of the
scientific progress and the way credit is given to preceding publications.
Furthermore, the top-ranking scientific memes reveal a bursty time dynamics,
which might be a reflection of the fierce competition among memes for the
limited and fluctuating resource of scientists’ attention.
We have only considered fixed character sequences as potential memes, but it
is clear that memes do not only exist on this low level, and it is reasonable
to expect that the inclusion of additional layers of processing using
linguistic and ontological resources would lead to even better results and
would let us capture memes on a more abstract level. Such memes might consist
of sets of morphological variants, co-occurrences of words, compositions of
multiple memes, grammatical constructions, or even argumentation schemes and
rhetorical styles. We deliberately kept the meme score as simple as possible
to emphasize that it is surprisingly precise on its own. At the same time,
there are many ways to improve the metric in the future with more
sophisticated processing to capture memes on a higher level. In general, we
believe that the presented approach by allowing to study memes in a
comprehensive manner opens up the field for a wide range of future research on
topics such as information diffusion, complex systems, innovation, scientific
progress, social dynamics, ecosystems, cultural evolution, and of course the
study of memes themselves.
## Methods
### Graph randomization
The analyzed randomized networks have exactly the same topology as the
original ones but the article texts (i.e. titles and abstracts with their
memes) are randomly assigned to the nodes. Each node therefore owes its
position in the network to one particular publication but has text attached
that comes from a different one. For the time-preserving randomizations, we
shuffle only publications that were published within narrow consecutive time
windows. Concretely we use time windows of 1000 publications, meaning that —
after shuffling — no publication has moved more than 1000 positions forward or
backward from the original chronological order.
### Human annotation
For the first part of the manual annotation, we use the following two
categories: (i) the phrase is not a meaningful term or not an important
concept of physics, and (ii) the phrase is an important concept or entity of
physics — it could appear as the title of an entry of a comprehensive
encyclopedia of physics. For the second part, we defined the following
linguistic phrase types: (i) noun phrase, (ii) verb, (iii) adjective or
adverb, and (iv) other.
The set of phrases used for this evaluation consisted of the top 150 memes
with respect to their meme score, extracted from the American Physical Society
dataset, plus another two sets for comparison of 150 randomly drawn phrases
each. For the two comparison sets, we considered all phrases that appear in at
least 100 publications. From these, 150 terms were drawn randomly without
taking into account their frequency, i.e., frequent terms had the same chance
of being selected as infrequent ones, whereas the 150 terms of the second set
were drawn with a weight that corresponded to their frequency. Moreover, to
rule out effects of different $n$-gram lengths, we made sure that the two
batches of random terms followed exactly the same length distribution as the
main sample extracted based on the meme score. The resulting 450 terms were
shuffled and given to two human annotators, both PhD students with a degree in
physics, who independently annotated the terms.
### Metrics for comparison
We have used the following metrics with different parameter settings as
alternatives to the meme score (see Supplementary Material for details): (i)
frequency — the most frequent terms, optionally skipping the first $x$ terms;
(ii) maximum absolute change over time — the highest-scoring terms with
respect to maximum _absolute_ change in frequency; (iii) maximum relative
change over time — the same as (ii) but based on _relative_ changes; (iv)
maximum absolute difference across journals — the highest-scoring terms with
respect to maximum _absolute_ difference in frequency between journals; (v)
maximum relative difference across journals — the same as (iv) but based on
_relative_ changes. Metric (i) is based on the assumption that important memes
are frequent but not as frequent as the small class of general words that can
be found in all types of texts. Metrics (ii) and (iii) are based on an idea
proposed in 36, being that interesting memes exhibit trends over time. Metrics
(iv) and (v) are based on the intuition that phrases occurring mostly in
specific journals but not in others must be specific concepts of the
particular field of research.
As a ground-truth list of memes, we automatically extracted $5178$ terms from
Wikipedia. We collected the titles of all articles — and terms redirecting to
them — from the categories “physics”, “applied and interdisciplinary physics”,
“theoretical physics”, “emerging technologies”, and their direct sub-
categories, but filtering out terms that appear in less than $10$ publications
of the American Physical Society dataset. To quantify the agreement between
the top memes identified by a particular metric and the Wikipedia list, we use
the normalized area $A$ under the curve as shown on Fig. 3. The step-shaped
curved has a log-scaled $x$-axis running up to the number of terms $s$ on the
ground-truth meme list ($s=5178$ in our case) and an $y$-axis running from $0$
(no overlap) to $1$ (perfect overlap). Limiting cases are $A=1$, representing
perfect agreement, and $A=0$, representing no agreement at all between the two
compared lists.
###### Acknowledgements.
This research was supported by the European Commission through the ERC
Advanced Investigator Grant “Momentum” (Grant No. 324247) and by the Slovenian
Research Agency through the Program P5-0027. In addition, we would like to
thank Karsten Donnay, Matthias Leiss, Christian Schulz, and Olivia Woolley-
Meza for their useful feedback and help with the realization of the
evaluations.
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|
arxiv-papers
| 2014-04-14T21:19:23 |
2024-09-04T02:50:01.146458
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tobias Kuhn, Matjaz Perc, Dirk Helbing",
"submitter": "Tobias Kuhn",
"url": "https://arxiv.org/abs/1404.3757"
}
|
1404.3792
|
# Photon-induced sideband transitions in a many-body Landau-Zener process
Honghua Zhong1,2 Qiongtao Xie1,3 Jiahao Huang1 Xizhou Qin1 Haiming Deng1 Jun
Xu1 Chaohong Lee1, Corresponding author. Email: [email protected] 1State Key
Laboratory of Optoelectronic Materials and Technologies, School of Physics and
Engineering, Sun Yat-Sen University, Guangzhou 510275, China 2Department of
Physics, Jishou University, Jishou 416000, China 3School of Physics and
Electronic Engineering, Hainan Normal University, Haikou 571158, China
###### Abstract
We investigate the many-body Landau-Zener (LZ) process in a two-site Bose-
Hubbard model driven by a time-periodic field. We find that the driving field
may induce sideband transitions in addition to the main LZ transitions. These
photon-induced sideband transitions are a signature of the photon-assisted
tunneling in our many-body LZ process. In the strong interaction regime, we
develop an analytical theory for understanding the sideband transitions, which
is confirmed by our numerical simulation. Furthermore, we discuss the
quantization of the driving field. In the effective model of the quantized
driving field, the sideband transitions can be understood as the LZ
transitions between states of different “photon” numbers.
###### pacs:
67.85.-d, 03.75.Lm, 03.75.Kk, 05.30.Jp
## I Introduction
The Landau-Zener (LZ) model has long been a physical paradigm for the
tunneling process in a two-level quantum system subject to a linearly time-
dependent external field Landau ; Zener ; Stueckelberg ; Majorana . Despite
its simplicity, this model has been applied to a great variety of physical
systems, including atomic and molecular systems mgong ; longhi , semiconductor
superlattices betthau , and superconducting devices william . In these
systems, the two-level LZ model acts as a good starting point to investigate
more realistic situations.
In recent years, ultracold atomic gases have provided new opportunities for
investigating a many-body extension of the simple two-level LZ tunneling
process Wu ; Zobay ; Liu ; Witthaut ; Tomadin ; Lee ; Smith ; korsch ; lignier
; Kasztelan . The remarkable controllability in these systems allows a clear
study of the effect of the interatomic interaction on the LZ tunneling
process. For weakly interacting ultracold atomic gases in optical lattices,
the LZ tunneling between the two lowest Bloch bands has been observed
experimentally Arimondo ; Salger ; Kling . It is shown that due to the
presence of the interatomic interaction, the LZ tunneling can be enhanced if
the system is initially in the ground Bloch band, and be suppressed if the
system is initially in the higher Bloch band. In addition, the many-body LZ
tunneling in the Mott-insulating regime has been addressed in experiments
Chena .
Periodic driving fields have been extensively used to control quantum
tunneling and transport grifoni98 . Along this line, periodic driving fields
have also been used to control LZ processes. For the simple two-level LZ
process, the effect of an additional periodic driving field in the bias or the
coupling has been discussed Kayanuma ; Ful ; malossi ; wubs . The driving
field can induce interesting quantum-interference effects between two well-
separated LZ sub-processes, and the probability of LZ transitions depends
sensitively on the parameter values of the driving field. Recently, the
problem of how periodic driving field affect the nonlinear two-level LZ
processes has been investigated qzhang ; qzhang08 . Dependent on the
nonlinearity strength, the final transition probability shows a shifted phase-
dependence on the driving field. However, for the LZ process in an interacting
many-body quantum system, the effects of the periodic driving field is still
unclear.
In the present paper, we use a two-site Bose-Hubbard model to study the effect
of the periodic driving field on the many-body LZ process. In this model, the
energy bias is subject to the usual linear change with time superimposed by a
time-periodic driving field. We find that the periodic driving field can
modify the condition for the occurrence of the many-body LZ tunneling. In
addition to the original LZ transitions without periodic driving field,
sideband LZ transitions are induced. In the high-frequency limit and strong
interaction regime, we obtain an effective system without periodic driving
field to understand these photon-assisted LZ transitions. The parametric
dependence of this photon-assisted LZ tunneling is analyzed. In addition, by
quantizing the periodic driving field, we introduce the fully quantum
mechanical model to understand the sideband LZ transitions. Our results show
that the photon-assisted tunneling due to the periodic driving field can
provide an efficient way to control the many-body LZ tunneling process.
The structure of this article is as following. In section II, we give a
physical description of the two-mode Bose-Hubbard model where the energy bias
is subject to a linear change with time and a time-periodic driving field. In
section III, we discuss the effect of the periodic driving on the many-body LZ
process in the high-frequency limit and the strong interaction regime. In
section IV, we discuss the quantization of the periodic driving field. In the
last section, we briefly summarize our results.
## II Model of many-body LZ processes in a diagonal periodic driving field
The system under consideration is a gaseous BEC of bosonic atoms in a double-
well potential smerzi97 ; corney ; smerzi ; martin ; gati2007 ; Lee2012 , see
its schematic diagrams in Fig. 1. The double-well potential is driven by
external field which is composed of both a linearly time-dependent change with
the sweep rate $S_{0}$ and a time-periodic driving field with the amplitude
$S_{1}$ and the frequency $\omega$, $S(t)=S_{0}t+S_{1}\cos(\omega t)$ Kayanuma
; Ful ; malossi . Then the total system is described by a second quantized
Hamiltonian
$\displaystyle H(t)$ $\displaystyle=$ $\displaystyle H_{0}+H_{int}.$
The Hamiltonian $H(t)$ includes two parts. $H_{0}$ is non-interacting part
$\displaystyle
H_{0}=\int\hat{\Psi}^{+}(x)[h_{0}+S(t)V_{1}(x)]\hat{\Psi}(x)dx,$ (1)
with
$\displaystyle h_{0}=-\frac{\hbar^{2}\nabla^{2}}{2m_{s}}+V_{0}(x).$
Here $\hat{\Psi}^{(+)}(x)$ are the bosonic field operators which annihilate
(create) a particle at position $x$, and $m_{s}$ is the single-atom mass.
$V_{0}(x)$ is of a symmetric double-well structure, and $V_{1}(-x)=-V_{1}(x)$
is a anti-symmetric, which can be realized in optical double-well experiment
della ; kartashov . The Hamiltonian $H_{int}$ describes the two-body
collisions between atoms, and is given by
$\displaystyle
H_{int}=\frac{1}{2}g\int\hat{\Psi}^{+}(x)\hat{\Psi}^{+}(x)\hat{\Psi}(x)\hat{\Psi}(x)dx.$
(2)
Here $g=4\pi\hbar^{2}a_{s}/m_{s}$ measures the interaction strength between
atoms, where $a_{s}$ is the corresponding s-wave scattering length. If the
depth of the symmetric double-well $V_{0}(x)$ is enough large, so that the
dynamics is only involved in the two lowest states localized to each well, we
can apply the standard two-mode approximation smerzi ; martin
$\hat{\Psi}(x)=\hat{a}_{1}u_{1}(x)+\hat{a}_{2}u_{2}(x),$ (3)
where $\hat{a}_{j}^{(\dagger)}\;(j=1,2)$ are the atomic annihilation
(creation) operators for the $i$-th well,
$u_{1}(x)=[\phi_{g}(x)+\phi_{e}(x)]/\sqrt{2}$ and
$u_{2}(x)=[\phi_{g}(x)-\phi_{e}(x)]/\sqrt{2}$ are localized waves in the wells
1 and 2, in which $\phi_{e}(x)$ and $\phi_{g}(x)$ are the two lowest energy
eigenstates of $h_{0}$, $h_{0}\phi_{e,g}(x)=E_{e,g}\phi_{e,g}(x)$. The total
number of atoms $N$ corresponding to the atom number operator
$\hat{N}=\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2}$ is
a conserved quantity. This two-mode approximation eventually simplifies Eq.
(1) to
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle-J(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})$
(4) $\displaystyle+\frac{1}{2}[\alpha t+\delta_{1}\cos(\omega
t)](\hat{a}_{2}^{\dagger}\hat{a}_{2}-\hat{a}_{1}^{\dagger}\hat{a}_{1}),$
Figure 1: (Color online) Schematic diagram for BEC in a double-well potential
where the time-dependent bias $\delta(t)$ is induced by an external field.
with the parameters
$\displaystyle J$ $\displaystyle=$ $\displaystyle-\int
dx[u_{1}^{\ast}(x)h_{0}u_{2}(x)],$ $\displaystyle\alpha$ $\displaystyle=$
$\displaystyle 2S_{0}\int dx[u_{1}^{\ast}(x)V_{1}(x)u_{1}(x)],$
$\displaystyle\delta_{1}$ $\displaystyle=$ $\displaystyle 2S_{1}\int
dx[u_{1}^{\ast}(x)V_{1}(x)u_{1}(x)].$
The term proportional to $J$ describes tunneling of particles from one to the
other well, and we have assumed it to be real. In the same way, we stipulate
that the overlap of the functions $u_{1}(x)$ and $u_{2}(x)$ be only minute,
which implies that the condensates in the two wells are merely weakly coupled,
and ignore the high-order overlaps between two functions martin ; gati2007 ;
Lee2012 . Then the interaction between atoms $H_{int}$ is given as
$\displaystyle
H_{int}=\frac{U_{11}}{4}\hat{a}_{1}^{\dagger}\hat{a}_{1}^{\dagger}\hat{a}_{1}\hat{a}_{1}+\frac{U_{22}}{4}\hat{a}_{2}^{\dagger}\hat{a}_{2}^{\dagger}\hat{a}_{2}\hat{a}_{2},$
(5)
with
$\displaystyle U_{jj}=2g\int dx|u_{j}(x)|^{4},\ \ j=1,2.$
After omitting the constant terms $O(N)$ and $O(N^{2})$, the total Hamiltonian
now can be rewritten as a two-mode Bose-Hubbard Hamiltonian
$\displaystyle H(t)$ $\displaystyle=$
$\displaystyle-J(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})+\frac{E_{c}}{8}(\hat{a}_{2}^{\dagger}\hat{a}_{2}-\hat{a}_{1}^{\dagger}\hat{a}_{1})^{2}$
(6) $\displaystyle+$
$\displaystyle\frac{\delta(t)}{2}(\hat{a}_{2}^{\dagger}\hat{a}_{2}-\hat{a}_{1}^{\dagger}\hat{a}_{1}),$
with $E_{c}=U_{11}+U_{22}$ and time-dependent energy bias
$\displaystyle\delta(t)=\alpha t+\delta_{1}\cos(\omega t).$ (7)
In addition, by introducing the angular momentum operators
$S_{x}=(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{1}\hat{a}_{2}^{\dagger})/2$,
$S_{y}=(\hat{a}_{2}^{\dagger}\hat{a}_{1}-\hat{a}_{1}^{\dagger}\hat{a}_{2})/2i$,
and
$S_{z}=(\hat{a}_{2}^{\dagger}\hat{a}_{2}-\hat{a}_{1}^{\dagger}\hat{a}_{1})/2$
with the Casimir invariant
Figure 2: (Color online) Occupation probability of the system in the
instantaneous eigenstates of the Hamiltonian $H(t)_{\delta_{1}=0}$ for three
different values of the sweep rate $\alpha$ with $N=2$, $J=1$, $E_{c}=100$,
and $\delta_{1}=0$. The system initially starts from its instantaneous ground
state with a large negative bias. Here the solid lines are for $\alpha=0.1$,
the dashed lines are for $\alpha=5$, and the solid lines with circles are for
$\alpha=10$. Here the labels $|1\rangle,|2\rangle$, and $|3\rangle$ represent
the three lowest instantaneous eigenstates of the Hamiltonian
$H(t)_{\delta_{1}=0}$.
$S^{2}=(N/2)(N/2+1)$, the Hamiltonian (6) also can be rewritten as
$\displaystyle H(t)=-2JS_{x}+\frac{E_{c}}{2}S_{z}^{2}+\delta(t)S_{z}.$ (8)
Clearly, the problem with $\delta_{1}=0$ is reduced to the usual many-body LZ
problem korsch . We note that in the mean-field approximation where the system
can be described by a nonlinear two-level model, the nonlinear LZ process in a
periodic driving field has been studied qzhang .
## III Photon-induced sideband transitions
In the following, we discuss the effect of the additional periodic driving on
the many-body LZ tunneling process. By expanding the state vector
$|\psi(t)\rangle$ as a liner combination of the Fock states
$|N/2-n,N/2+n\rangle$, denoting the state with $N/2-n$ bosons on the first
well and $N/2+n$ on the second well,
$\displaystyle|\psi(t)\rangle=\sum_{n=-N/2}^{N/2}\exp[-i\theta_{n}(t)]c_{n}(t)|N/2-n,N/2+n\rangle,$
with
$\displaystyle\theta_{n}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}[E_{c}n^{2}/2+(\alpha\tau+\delta_{1}\cos(\omega\tau))n]d\tau$
$\displaystyle=$ $\displaystyle E_{c}n^{2}t/2+\alpha
nt^{2}/2+\delta_{1}n\sin(\omega t)/\omega,$
from the time-dependent Schrödinger equation
$i\partial|\psi(t)\rangle/\partial t=H(t)|\psi(t)\rangle$, we can get $N+1$
coupled first-order differential equations for the coefficients $c_{n}(t)$
$\displaystyle i\frac{dc_{n}}{dt}=$
$\displaystyle-J\sqrt{(\frac{N}{2}-n)(\frac{N}{2}+n+1)}e^{-i\Delta\theta_{n+1}^{n}}c_{n+1}$
(9)
$\displaystyle-J\sqrt{(\frac{N}{2}+n)(\frac{N}{2}-n+1)}e^{i\Delta\theta_{n}^{n-1}}c_{n-1},$
with $\Delta\theta_{n+1}^{n}=\theta_{n+1}-\theta_{n}=E_{c}(2n+1)t/2+\alpha
t^{2}/2+\delta_{1}\sin(\omega t)/\omega$. By applying the generating function
of the Bessel functions, $\exp[\pm i\delta_{1}\sin(\omega
t)/\omega]=\sum_{m=-\infty}^{\infty}J_{m}(\delta_{1}/\omega)\exp[\pm im\omega
t]$, we have
$\displaystyle i\frac{dc_{n}}{dt}=$
$\displaystyle-\sum_{m=-\infty}^{\infty}\widetilde{J}_{n,eff}^{m}e^{-i\varphi_{n}^{m}}c_{n+1}$
(10)
$\displaystyle-\sum_{m=-\infty}^{\infty}\widetilde{J}_{n-1,eff}^{m}e^{i\varphi_{n-1}^{m}}c_{n-1},$
with
$\displaystyle\widetilde{J}_{n,eff}^{m}$ $\displaystyle=$ $\displaystyle
J\sqrt{(\frac{N}{2}-n)(\frac{N}{2}+n+1)}J_{m}(\delta_{1}/\omega),$ (11)
$\displaystyle\varphi_{n}^{m}$ $\displaystyle=$ $\displaystyle
E_{c}(2n+1)t/2+\alpha t^{2}/2+m\omega t.$ (12)
In our study, we mainly focus on the strong interaction regime where the
interaction energy $E_{c}$ dominates the tunneling coupling $E_{c}/J>>1$ Lee ;
Chena ; Kasztelan and the high-frequency limit $E_{c}/J>>\omega/J>>1$.
Therefore, these terms on the right hand side of Eq. (10) are rapidly
oscillating, and make important contributions only if $\varphi_{n}^{m}$ has a
stationary phase at certain instant determined by the following condition
Kayanuma ; Ful
$\displaystyle\frac{d\varphi_{n}^{m}}{dt}=E_{c}(2n+1)/2+\alpha t+m\omega,$
(13)
thereby resulting in
$\alpha t_{n}^{m}=-(n+1/2)E_{c}-m\omega.$ (14)
In the absence of the periodic driving field, $\delta_{1}=0$, the last term
$m\omega$ vanishes. We note that in the case of $\delta_{1}=0$, the ground
state undergoes $N$ LZ transitions at $\alpha t_{n}=-(n+1/2)E_{c}$. So the LZ
transitions at $t_{n}^{0}=t_{n}$ just correspond to the usual many-body LZ
process without periodic driving. Naturally, a problem arises, what is the
dynamics of the system near $t_{n}^{m}$ with $m\neq 0$?
Figure 3: (Color online) Occupation probability of the system in the two
lowest instantaneous eigenstates of the Hamiltonian $H(t)_{\delta_{1}=0}$ for
different driving amplitudes $\delta_{1}$ with $N=2$, $J=1$, $E_{c}=100$,
$\alpha=0.01$, and $\omega=10$. Here the labels $|1\rangle$ and $|2\rangle$
represent the two lowest instantaneous eigenstates of the Hamiltonian
$H(t)_{\delta_{1}=0}$.
To answer this problem, we first make a change of variable $\tau=t-t_{n}^{m}$
near $t_{n}^{m}$, and then assume a high-frequency driving $\omega/J>>1$ for
simplifying our discussions. Because of $\omega/J>>1$, we may retain the
$m$-th term, and approximately replace the Eq. (10) by two first-order
differential equations Kayanuma ; Ful
$\displaystyle i\frac{dc_{n}}{d\tau}$ $\displaystyle=$
$\displaystyle-\widetilde{J}_{n,eff}^{m}e^{-i\phi_{n}^{m}}e^{-i\frac{\alpha\tau^{2}}{2}}c_{n+1},$
(15) $\displaystyle i\frac{dc_{n+1}}{d\tau}$ $\displaystyle=$
$\displaystyle-\widetilde{J}_{n,eff}^{m}e^{i\phi_{n}^{m}}e^{i\frac{\alpha\tau^{2}}{2}}c_{n},$
(16)
where $\phi_{n}^{m}=\alpha(t_{n}^{m})^{2}/2+E_{c}(2n+1)t_{n}^{m}/2+m\omega
t_{n}^{m}$. Here $n$ takes from $-N/2$ to $N/2-1$. We note that similar
coupled equation between $c_{n}$ and $c_{n-1}$ can also be obtained where $n$
starts from $N/2$ to $-N/2+1$.
To clearly see the physics described by the above equations, we make the
transformation $c_{n}=\widetilde{c}_{n}\exp[-i\alpha\tau^{2}/4]$ and
$c_{n+1}=\widetilde{c}_{n+1}\exp[i\alpha\tau^{2}/4]$, and get
$\displaystyle i\frac{d\widetilde{c}_{n}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\alpha\tau}{2}\widetilde{c}_{n}-\widetilde{J}_{n,eff}^{m}e^{-i\phi_{n}^{m}}\widetilde{c}_{n+1},$
(17) $\displaystyle i\frac{d\widetilde{c}_{n+1}}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\alpha\tau}{2}\widetilde{c}_{n+1}-\widetilde{J}_{n,eff}^{m}e^{i\phi_{n}^{m}}\widetilde{c}_{n}.$
(18)
These results tell us that the change of the coefficients $c_{n}$ due to a
linear sweep across $t=t_{n}^{m}$ is nothing but that of the LZ-type level
crossing in which the coupling is renormalized effectively by a factor of the
Bessel function.
Therefore, the LZ transitions at $\alpha t_{n}^{0}$ just correspond to the
usual LZ transitions without a periodic driving field, while the sideband LZ
transitions at $\alpha t_{n}^{m}$ with $m\neq 0$ arise from the periodic
driving field of the time-dependent bias. In principle, the index $m$ can take
arbitrary integer values. However, from the well-know fact that
$J_{m}(\delta_{1}/\omega)\rightarrow 0$ with $m\neq 0$ if
$\delta_{1}/\omega\rightarrow 0$ or $m\rightarrow\pm\infty$, it follows that
for the very small driving amplitude, the sideband LZ transitions are so small
that they are actually not visible. If the driving amplitude is chosen
suitably, they become important and visible.
To confirm our analytical results with numerical simulations, we use the usual
many-body LZ Hamiltonian $H(t)_{\delta_{1}=0}$ as a reference system. We solve
numerically the time-dependent Schrödinger equation
$i\partial|\psi(t)\rangle/\partial t=H(t)|\psi(t)\rangle$ starting from the
ground state of Hamiltonian (6) with a large negative bias
$\delta(t=-T)\rightarrow-\infty$, and at the end of the linear sweep
$\delta(t=T)\rightarrow\infty$ of the bias, and compute the occupation
probability of the system in the lowest instantaneous eigenstates of the
Hamiltonian $H(t)_{\delta_{1}=0}$. In Fig. 2, we first display the numerical
results without the periodic driving field for the small atom number $N=2$.
Here the other parameters are given by $J=1$ and $E_{c}=100$. This situation
corresponds to the usual many-body LZ problem. As is expected, if the sweep
rate $\alpha$ is low enough, the system is still in the instantaneous ground
state during the linear sweep. However, the presence of the periodic driving
field modifies this physical picture. To only show the effect of the periodic
driving, we take a small sweep rate $\alpha=0.01$, for which the evolution of
the system without periodic driving field is adiabatic. In Fig. 3, we display
the occupation probability of the system in the two lowest instantaneous
eigenstates of the Hamiltonian $H(t)_{\delta_{1}=0}$ for different driving
amplitudes with $N=2$, $J=1$, $\alpha=0.01$, $\omega=10$ and $E_{c}=100$. We
find from these numerical results that the occupation probability displays a
series of steplike changes at particular values of $\alpha t_{n}^{m}$ with
$m\neq 0$. In this situation, for the small driving amplitudes
$\delta_{1}=0.1$ and $\delta_{1}=0.5$, the LZ transitions with $m=\pm 1$ are
clearly visible, while for a larger driving amplitudes $\delta_{1}=2.0$, the
LZ transitions with $m=\pm 2$ become also visible.
## IV Quantization of the periodic driving field
In this section, we show how to understand the sideband transitions in our
many-body LZ process by quantizing the periodic driving field. Usually, the
quantization of a classical field is achieved by introducing a harmonic
oscillator and then quantizing the harmonic oscillator. We introduce a hybrid
quantum-classical system composed of a quantum subsystem and a classical
harmonic oscillator, and then show that the coupling between the quantum
subsystem and the classical harmonic oscillator can act as the periodic
driving for the quantum subsystem. Therefore, the quantization of the periodic
driving field corresponds to the quantization of the classical harmonic
oscillator in the hybrid quantum-classical system.
We consider a hybrid quantum-classical system
$H_{hy}=H_{q}+H_{ho}+H_{q}^{ho},$ (19)
with
$H_{q}=-2JS_{x}+\frac{E_{c}}{2}S_{z}^{2}+\alpha tS_{z},$ (20)
for the quantum subsystem,
$H_{ho}=\frac{P^{2}}{2M}+\frac{1}{2}M\omega^{2}Q^{2}$ (21)
for the classical harmonic oscillator, and
$H_{q}^{ho}=kQS_{z},$ (22)
for the coupling between the quantum subsystem and the classical harmonic
oscillator. Here $k$ is the coupling strength, $Q$ is the oscillator position,
and $P$ is the oscillator momentum. If the mass $M$ is large enough, the
harmonic oscillator will not be affected by the quantum subsystem and the
quantum subsystem feels a periodic driving produced by the harmonic oscillator
makarov ; makri .
Through introducing the destruction and creation operators $b$ and
$b^{\dagger}$ for the momentum $P$ and the position $Q$ of a harmonic
oscillator, the fully quantum model for the hybrid quantum-classical system
may be written as
$\displaystyle H_{fq}$ $\displaystyle=$
$\displaystyle-2JS_{x}+\frac{E_{c}}{2}S_{z}^{2}+\alpha
tS_{z}+\lambda(b^{\dagger}+b)S_{z}+\omega b^{\dagger}b.$
Here $\lambda=k/\sqrt{2}$ is the rescaled coupling strength. For the case of
$N=1$ and $E_{c}=0$, the resulting model can be used to describe the LZ
process in a quantum two-level system coupled to a photon mode nori . By
employing a unitary transformation $U=e^{i\omega tb^{\dagger}b}$, the quantum
Hamiltonian (IV) is equivalent to an effective Hamiltonian
$\displaystyle H_{fq}^{eff}$ $\displaystyle=$
$\displaystyle-2JS_{x}+\frac{E_{c}}{2}S_{z}^{2}+\alpha tS_{z}$ (24)
$\displaystyle+\lambda(e^{i\omega t}b^{\dagger}+e^{-i\omega t}b)S_{z}.$
Clearly, if the harmonic oscillator stays in a coherent state
$\left|\beta\right\rangle$, the quantum subsystem just feels a periodic
driving field induced by the coupling term $H_{q}^{ho}$ and so that it obeys,
$\displaystyle H_{q}^{eff}$ $\displaystyle=$
$\displaystyle-2JS_{x}+\frac{E_{c}}{2}S_{z}^{2}+\alpha tS_{z}$ (25)
$\displaystyle+\lambda\langle\beta|(e^{i\omega t}b^{\dagger}+e^{-i\omega
t}b)|\beta\rangle S_{z},$ $\displaystyle=$
$\displaystyle-2JS_{x}+\frac{E_{c}}{2}S_{z}^{2}+\alpha tS_{z}$
$\displaystyle+2\lambda\beta\cos(\omega t)S_{z}.$
Usually, we can assume that $\beta$ is a real number. Then the amplitude of
the periodic driving $\delta_{1}$ is related to $2\lambda\beta$.
Figure 4: (Color online) (a) Instantaneous energy spectra of the many-body LZ
Hamiltonian $H_{fq}$ for $J=0$ (dashed lines) and $J=1$ (solid liens) with
$\lambda=\omega=0$. Here the total atom number is $N=2$, and the interaction
strength is $E_{c}=100$. (b) Instantaneous energy spectra of the many-body LZ
Hamiltonian $H_{fq}$ with $\lambda=1$ and $\omega=10$. Here for simplicity, we
only take a relatively small basis set for the photon mode. The labels of
-2,-1,+1 and +2 denote the avoided level-crossings at $\alpha t=\pm
E_{c}/2-m\omega$ with $m=-2,-1,+1,+2$. In (b), the inset shows the enlarged
region near the avoided level-crossings labeled by $m=+2,+1$.
To compute the energy spectrum of the Hamiltonian $H_{fq}$ with a fixed
$\alpha t$, we use the basis $|n_{b}\rangle\otimes|n\rangle$ satisfying
$b^{\dagger}b|n_{b}\rangle=n_{b}|n_{b}\rangle$ and
$S_{z}|n\rangle=n|n\rangle$. For a small value of $\beta$, we may take a
finite basis set, $n_{b}=0,1,2$, and diagonalize the Hamiltonian numerically.
In Fig. 4, we display instantaneous energy spectra of the Hamiltonian $H_{fq}$
as a function of $\alpha t$ for two different cases (a) $\lambda=\omega=0$ and
(b) $\lambda=1$ and $\omega=10$. In the two situations, the atom number is
$N=2$, and the other parameters are given as $J=1$ and $E_{c}=100$. For the
energy spectrum without the photon mode, the avoided level-crossings of the
ground state only appear around $\alpha t=\pm E_{c}/2$ and these avoided
level-crossings dominate the population transfer in the LZ process, as shown
in Fig. 4 (a). For the energy spectrum in the presence of the photon mode,
because of the many photon effects, the avoided level-crossings can appear
around $\alpha t=\pm Ec/2-m\omega$ with $m=0,\pm 1,\pm 2$. It is observed that
the gap around $\alpha t=\pm Ec/2-m\omega$ with $m=\pm 1$ is lager than those
for $m=\pm 2$, as illustrated in Fig. 4 (b). This results explain why the
sideband transitions with $m=\pm 1$ are observed in Fig. 3 for a relative
small driving with $\delta_{1}=0.1,0.5$. When the driving amplitude is
increased to a larger value related to a larger $\beta$, we need to include
more photon numbers to compute the energy spectrum, and thus more sideband
transitions may be observed. For example, in Fig. 3, the sideband transitions
with $m=\pm 2$ are observed in the case of $\delta_{1}=2$.
## V Conclusions
In summary, we have investigated the many-body LZ process in the two-site
Bose-Hubbard model where the energy bias between sites is subject to a linear
change with time and a periodic driving field. It is revealed that the
periodic driving can modify the conditions for the many-body LZ transitions,
and induce sideband LZ transitions under certain parameter conditions. In the
high-frequency limit and strong interaction regime, we have applied an
analytical method to understanding these sideband LZ transitions. In addition,
by quantizing the periodic driving field, we introduce the fully quantum
mechanical model to understand the sideband LZ transitions, in which the
sideband transitions can be understood as the LZ transitions between states of
different “photon” numbers. Our results show that the periodic driving field
can provide an efficient way for controlling the many-body LZ tunneling
process.
Our results of sideband transitions in many-body LZ problem offer an
alternative route to manipulating many-body quantum systems. With currently
avaliable experimental techniques for observing many-body LZ tunneling Chena ,
it is possible to test our theoretical predictions. It is also possible to
apply our analysis for treating the case of more complex driving fields, such
as multi- frequency driving field, and the driving field with time-dependent
amplitude.
###### Acknowledgements.
H.-H. Zhong and Q.-T. Xie made equal contributions. This work is supported by
the NBRPC under Grants No. 2012CB821305, the NNSFC under Grants No. 11374375,
11147021, 11375059, the Hunan Provincial Natural Science Foundation under
Grant No. 12JJ4010, the Scientific Research Fund of Hunan Provincial Education
Department under Grant No. 13A058, and the Ph.D. Programs Foundation of
Ministry of Education of China under Grant No. 20120171110022.
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|
arxiv-papers
| 2014-04-15T01:08:51 |
2024-09-04T02:50:01.156202
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Honghua Zhong, Qiongtao Xie, Jiahao Huang, Xizhou Qin, Haiming Deng,\n Jun Xu and Chaohong Lee",
"submitter": "Honghua Zhong",
"url": "https://arxiv.org/abs/1404.3792"
}
|
1404.3847
|
[[email protected]]7 Vavilova Str., Moscow, Russia, 117312
# Teichmüller spaces, ergodic theory
and global Torelli theorem
Misha Verbitsky
###### Abstract
A Teichmüller space $\operatorname{Teich}$ is a quotient of the space of all
complex structures on a given manifold $M$ by the connected components of the
group of diffeomorphisms. The mapping class group $\Gamma$ of $M$ is the group
of connected components of the diffeomorphism group. The moduli problems can
be understood as statements about the $\Gamma$-action on
$\operatorname{Teich}$. I will describe the mapping class group and the
Teichmüller space for a hyperkähler manifold. It turns out that this action is
ergodic. We use the ergodicity to show that a hyperkähler manifold is never
Kobayashi hyperbolic.
###### :
P
###### keywords:
Torelli theorem, hyperkähler manifold, moduli space, mapping class group,
Teichmüller space, ergodicity
rimary 32G13, Secondary 53C26.
###### Contents
1. 1 Teichmüller spaces
1. 1.1 Teichmüller spaces and period maps
2. 1.2 Marked moduli spaces
2. 2 Torelli theorem
1. 2.1 Torelli theorem: an introduction
2. 2.2 Birational Teichmüller space
3. 3 Hyperkähler manifolds and Bogomolov-Beauville-Fujiki form
1. 3.1 Hyperkähler manifolds: definition and examples
2. 3.2 Bogomolov-Beauville-Fujiki form and the mapping class group
4. 4 Global Torelli theorem
1. 4.1 Period map
2. 4.2 Moduli of hyperkähler structures and twistor curves
5. 5 Teichmüller spaces and ergodic theory
1. 5.1 Ergodic complex structures
2. 5.2 Ergodicity of the monodromy group action
6. 6 Applications of ergodicity
1. 6.1 Ergodic complex structures, Gromov-Hausdorff closures, and semicontinuity
2. 6.2 Kobayashi non-hyperbolicity of hyperkähler manifolds
3. 6.3 Symplectic packing and ergodicity
This talk is based on two papers, [V:2013] and [V]. In these papers one can
find details, examples, and rigorous proofs omitted here.
## 1 Teichmüller spaces
### 1.1 Teichmüller spaces and period maps
The notion of Teichmüller spaces has a long history since its discovery by
Teichmüller in 1944 ([Te:1944]) and further development by Ahlfors, Bers and
others. However, it is rarely applied to complex manifolds of dimension $>1$.
It turns out that this notion is interesting and useful for many purposes of
complex geometry in any dimension.
###### Definition 1.1.
Let $M$ be a smooth manifold. An almost complex structure is an operator
$I:\;TM{\>\longrightarrow\>}TM$ which satisfies
$I^{2}=-\operatorname{Id}_{TM}$. An almost complex structure is integrable if
$\forall X,Y\in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$. In this case $I$ is
called a complex structure operator. A manifold with an integrable almost
complex structure is called a complex manifold.
###### Definition 1.2.
The space of almost complex structures is an infinite-dimensional Fréchet
manifold $X_{M}$ of all tensors $I\in\operatorname{End}(TM)$ satisfying
$I^{2}=-\operatorname{Id}_{TM}$. Similarly, one considers the group of
diffeomorphisms as a Fréchet Lie group.
###### Remark 1.3.
Definition of Fréchet manifolds and Fréchet spaces and many results on the
geometry of infinite-dimensional manifolds can be found in [Ha].
###### Definition 1.4.
Let $M$ be a compact complex manifold, and $\operatorname{Diff}_{0}(M)$ a
connected component of its diffeomorphism group (the group of isotopies).
Denote by $\operatorname{Comp}$ the space of complex structures on $M$,
considered with the topology induced from a Fréchet manifold of almost complex
structures, and let
$\operatorname{Teich}:=\operatorname{Comp}/\operatorname{Diff}_{0}(M)$ be the
quotient space with the quotient topology. We call it the Teichmüller space.
###### Remark 1.5.
When the complex manifold $M$ admits a certain geometric structure, such as
Kähler, or hyperkähler structure, it is natural to consider the Teichmüller
space of complex structure compatible with (say) Kähler structure. Consider
the open subset $\operatorname{Comp}_{K}\subset\operatorname{Comp}$ of all
complex structures $I$ such that $(M,I)$ admits a Kähler structure. The
corresponding Teichmüller space is
$\operatorname{Teich}_{K}:=\operatorname{Comp}_{K}/\operatorname{Diff}_{0}$.
When working with the Teichmüller space of hyperkähler manifolds, or a torus,
we shall always restrict ourselves to $\operatorname{Comp}_{K}$ and
$\operatorname{Teich}_{K}$.
Results of Kuranishi about local structure of deformation spaces can be
summarized as a statement about local structure of $\operatorname{Comp}$ as
follows ([Ku1, Ku2, Dou]).
###### Theorem 1.6
Let $M$ be a compact complex manifold and $I\in\operatorname{Comp}$. Then
there exists an open neighbourhood $U\ni I$ in $\operatorname{Comp}$ and a
neighbourhood $R$ of unit in $\operatorname{Diff}_{0}$ satisfying the
following. Consider the quotient $U/R$ of $U$ by an equivalence relation
generated by $x\sim wy$, for all $x,y\in U$ and $w\in R$. Then $U/R$ is a
complex variety, equipped with a natural holomorphic embedding to $H^{1}(TM)$.
###### Remark 1.7.
The quotient space $U/R$ obtained by Kuranishi is called the Kuranishi space.
Let $\operatorname{Teich}(U)$ be an image of $U$ in the Teichmüller space.
Clearly, the Kuranishi space admits a surjective, continuous map to
$\operatorname{Teich}(U)$. It is not entirely clear whether this map is always
a homeomorphism. However, if it is always a homeomorphism, for a given $M$,
the space $\operatorname{Teich}$ acquires a structure of a complex variety.
As shown by F. Catanese ([C, Proposition 15]), for Kähler manifolds with
trivial canonical bundle, e.g. for the hyperkähler manifolds, the Teichmüller
space is locally isomorphic to the Kuranishi moduli space, hence it is a
complex variety. In this case it is actually a complex manifold, by Bogomolov-
Tian-Todorov theorem ([B:1981, Ti, Tod2]).
It is not clear if this is true for a general complex manifold; in the present
work we deal with hyperkähler manifolds, which are Calabi-Yau.
###### Question 1.8.
Consider a compact complex manifold $M$, and let $\operatorname{Teich}$ be its
Teichmüller space. Can we equip $\operatorname{Teich}$ with a structure of a
complex variety (possibly non-Hausdorff), in a way which is compatible with
the local charts obtained from the Kuranishi theorem?
When $M$ is a torus, or a hyperkähler manifold, $\operatorname{Teich}$ is a
complex manifold which can be described explicitly (Theorem 2.1, Theorem 4.5).
However, even for a hyperkähler manifold, $\operatorname{Teich}$ is not
Hausdorff.
###### Claim 1.9
Assume that $M$ is Kähler, and $\operatorname{Teich}$ the Teichmüller space of
all complex structures of Kähler type on $M$. For a given
$I\in\operatorname{Teich}$, choose a representative
$\tilde{I}\in\operatorname{Comp}$. Then the Hodge decomposition
$H^{*}(M)=\bigoplus H^{p,q}(M,\tilde{I})$ is independent from the choice of
$\tilde{I}$.
Proof: The ambiguity of a choice of $\tilde{I}$ lies in
$\operatorname{Diff}_{0}$. However, $\operatorname{Diff}_{0}$ acts trivially
on $H^{*}(M)$.
This elementary claim allows one to define the period map.
###### Definition 1.10.
Lety $M$ be a Kähler manifold, $\operatorname{Teich}$ its Teichmüller space,
and $\operatorname{\sf Per}$ the map associating to $I$ the Hodge
decomposition $H^{*}(M)=\bigoplus H^{p,q}(M,I)$. Then $\operatorname{\sf Per}$
is called the period map of $M$.
###### Remark 1.11.
Consider the product $\operatorname{Comp}\times M$ trivially fibered over
$\operatorname{Comp}$. The fibers of
$\pi:\;M\times\operatorname{Comp}{\>\longrightarrow\>}\operatorname{Comp}$ can
be considered as complex manifolds, with complex structure at
$I\in\operatorname{Comp}$ given by $I$. This complex structure is clearly
$\operatorname{Diff}_{0}$-invariant, giving a complex structure on the fibers
of the quotient fibration
$(M\times\operatorname{Comp})/\operatorname{Diff}_{0}{\>\longrightarrow\>}\operatorname{Teich}$.
At each $I\in\operatorname{Teich}$, the fiber of this fibration (called the
universal fibration) is isomorphic to $(M,I)$.
### 1.2 Marked moduli spaces
A more conventional approach to the moduli problem goes as follows. Given a
complex manifold $M$, one defines the deformation functor from marked complex
spaces to sets as a functor mapping a complex space $(B,x)$ to the set of
equivalence classes of deformations $\pi:\;{\cal X}{\>\longrightarrow\>}B$ of
$M$ over $B$ with $M$ identified with the fiber of $\pi$ at $x$.
If the deformation functor is representable by a complex space, this space is
called the fine moduli space of deformations of $M$.
Usually, the fine moduli space does not exist. In this case, one considers the
category of natural transformations from the deformation functor to
representable functors. The initial object in this category is called the
coarse moduli space. The points of coarse moduli are identified with
equivalence classes of deformations of $M$.
In this setup, an analogue of Teichmüller space can be defined as follows. Fix
an abelian group which is isomorphic to $H^{*}(M,{\mathbb{Z}})$, and define a
marked manifold as a pair
$(M,\varphi:\;V\tilde{\>\longrightarrow\>}H^{*}(M,{\mathbb{Z}}))$, where $M$
is a complex manifold, and $\varphi$ a group isomorphism. In the same way as
above, one defines a coarse moduli space of deformations of marked manifolds.
To compare this space with Teichmüller space, consider a subgroup group
$\Gamma_{0}$ of mapping class group which acts trivially on cohomology.
Clearly, the points of $\operatorname{Teich}/\Gamma_{0}$ are in bijective
correspondence with the equivalence classes of marked complex structures on
$M$.
Given a coarse marked moduli space $W$, one obtains the tautological map
$W{\>\longrightarrow\>}\operatorname{Teich}/\Gamma_{0}$, by construction
continuous. For hyperkähler manifolds (or compact tori), this map is a
diffeomorphism on each connected component ([V:2013, Corollary 4.31]).
## 2 Torelli theorem
### 2.1 Torelli theorem: an introduction
Torelli theorems are a broad class of results which describe the Teichmüller
spaces in terms of the period maps (Definition 1.10).
The name originates with Ruggiero Torelli, who has shown that it is possible
to reconstruct a Riemann surface from its Jacobian ([To:1913]). The term
“Torelli theorems” is due to André Weil ([W:1957]), who gave a modern proof of
this classical result, and explained its possible generalizations.
One may distinguish between the “local Torelli theorem”, where a local
structure of deformation space is described in terms of periods, and “global
Torelli”, where the Teichmüller space is described globally.
Weil, who was the first to define and study K3 surfaces, spent much time
trying to prove the Torelli theorem for K3 surfaces, but it was notoriously
difficult. Its local version is due to Tjurina, Piatetski-Shapiro and
Shafarevich ([Tj, PS]). The local Torelli was generalized by Bogomolov to
hyperkähler manifolds ([B:1978]) and by Bogomolov-Tian-Todorov to Calabi-Yau
manifolds ([B:1981, Ti, Tod2]), building foundation for the theory of Mirror
Symmetry.
In dimension $>1$, the global Torelli theorem was known only for compact tori
(where it is essentially trivial) and the K3 surfaces, where it was proven by
Kulikov in 1977 ([Kul]), and then improved many times during the 1980-ies
([Tod1, Bea2, L, Si, Fr]).
### 2.2 Birational Teichmüller space
In what follows, a hyperkähler manifold is a compact complex manifold
admitting a Kähler structure and a holomorphic symplectic form.
In generalizing global Torelli to more general hyperkähler manifolds, two
problems were apparent. First of all, bimeromorphic hyperkähler manifolds have
the same periods, hence the period map cannot distinguish between them.
However, for $\dim_{\mathbb{C}}M>0$, birational holomorphically symplectic
manifolds can be non-isomorphic ([De]).
Another (mostly psychological) difficulty is based on attachment to moduli
spaces, as opposed to marked moduli or Teichmüller spaces. For a K3, one can
reconstruct a K3 from its Hodge structure, and this gives an identification
between the moduli and the space of Hodge structures. In bigger dimension, one
has to use the Teichmüller space. Indeed, for some classes of hyperkähler
manifolds, the group $O(H^{2}(M,{\mathbb{Z}})$ of Hodge isometries of
cohomology is strictly bigger than the image of the mapping class group. This
gives elements $\gamma\in O(H^{2}(M,{\mathbb{Z}})$ acting non-trivially on the
Teichmüller space in such a way that the complex manifolds $(M,I)$ and
$(M,\gamma(I))$ are not birationally equivalent ([Na, Ma2]). However, their
Hodge structures are equivalent, by construction. This example explains the
necessity of using the Teichmüller spaces (or marked moduli) to state the
Torelli theorem: its Hodge-theoretic version is often false.
For Teichmüller spaces, the Torelli theorem is a statement about the period
map (Definition 1.10). Ideally, we want the period map to give a
diffeomporphism between $\operatorname{Teich}$ and the corresponding space of
Hodve structures. This is what happens for a compact torus.
###### Theorem 2.1
Let $M$ be a compact torus, $\dim_{\mathbb{R}}M=2n$, and
$\operatorname{Teich}$ the Teichmüller space of all complex structures of
Kähler type on $M$. Denote by $\operatorname{{\mathbb{P}}\sf er}$ the space
$SL(2n,{\mathbb{R}})/SL(n,{\mathbb{C}})$ of all Hodge structures of weight one
on $H^{1}(M,{\mathbb{C}})$, that is, the space of all complex operators on
$H^{1}(M,{\mathbb{R}})$ compatible with the orientation. Then the period map
$\operatorname{\sf
Per}:\;\operatorname{Teich}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ is a diffeomorphism on each connected component of
$\operatorname{{\mathbb{P}}\sf er}$.
Unfortunately, this ideal situation is almost never realized. Even in the
simplest cases (such as for hyperkähler manifolds), the Teichmüller space is
no longer Hausdorff. However, in some situations it is still possible to deal
with non-Hausdorff points.
###### Remark 2.2.
A non-Hausdorff manifold is a topological space locally diffeomorphic to
${\mathbb{R}}^{n}$ (but not necessarily Hausdorff).
###### Definition 2.3.
Let $X$ be a topological space, and
$X\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}X_{0}$ a continuous
surjection. The space $X_{0}$ is called a Hausdorff reduction of $X$ if any
continuous map $X{\>\longrightarrow\>}X^{\prime}$ to a Hausdorff space is
factorized through $\varphi$.
###### Definition 2.4.
Let $M$ be a topological space. We say that $x,y\in M$ are non-separable
(denoted by $x\sim y$) if for any open sets $V\ni x,U\ni y$, one has $U\cap
V\neq\emptyset$.
###### Remark 2.5.
Suppose that $\sim$ is an equivalence relation, and the quotient $M/\sim$ is
Hausdorff. Then $M/\sim$ is a Hausdorff reduction of $M$.
Unfortunately, this notion cannot be applied universally. Firstly, $\sim$ is
not always an equivalence relation; and secondly, even if $\sim$ is
equivalence, the $M/\sim$ is not always Hausdorff. Fortunately, for
Teichmüller space of a hyperkähler manifold, Hausdorff reduction can be
defined, using the following theorem due to D. Huybrechts ([Hu1]).
###### Theorem 2.6
If $I_{1}$, $I_{2}\in\operatorname{Teich}$ are non-separate points, then
$(M,I_{1})$ is birationally equivalent to $(M,I_{2})$.
Using this result and geometry of the period map (Bogomolov’s local Torelli
theorem), it is elementary to show that the quotient
$\operatorname{Teich}_{b}:=\operatorname{Teich}/\sim$ is a Hausdorff manifold.
This quotient is called the birational Teichmüller space of a hyperkähler
manifold.
Global Torelli theorem implies that for hyperkähler manifolds the period map
induces a diffeomorphism between the Hausdorff reduction of the Teichmüller
space and the appropriate period domain.
## 3 Hyperkähler manifolds and Bogomolov-Beauville-Fujiki form
### 3.1 Hyperkähler manifolds: definition and examples
The standard definition of hyperkähler manifolds is rather differential
geometric. It is, indeed, synonymous with “holomorphic symplectic”, but this
synonymity follows from Calabi-Yau theorem. For more details about hyperkähler
manifolds, please see [Bea1] or [Bes].
###### Definition 3.1.
A hyperkähler structure on a manifold $M$ is a Riemannian structure $g$ and a
triple of complex structures $I,J,K$, satisfying quaternionic relations
$I\circ J=-J\circ I=K$, such that $g$ is Kähler for $I,J,K$.
###### Remark 3.2.
This is equivalent to $\nabla I=\nabla J=\nabla K=0$: the parallel translation
along the connection preserves $I,J,K$.
###### Remark 3.3.
A hyperkähler manifold has three symplectic forms:
$\omega_{I}:=g(I\cdot,\cdot)$, $\omega_{J}:=g(J\cdot,\cdot)$,
$\omega_{K}:=g(K\cdot,\cdot)$.
###### Definition 3.4.
Let $M$ be a Riemannian manifold, $x\in M$ a point. The subgroup of
$GL(T_{x}M)$ generated by parallel translations (along all paths) is called
the holonomy group of $M$.
###### Remark 3.5.
A hyperkähler manifold can be defined as a manifold which has holonomy in
$Sp(n)$ (the group of all endomorphisms preserving $I,J,K$).
###### Definition 3.6.
A holomorphically symplectic manifold is a complex manifold equipped with non-
degenerate, holomorphic $(2,0)$-form.
###### Remark 3.7.
Hyperkähler manifolds are holomorphically symplectic. Indeed,
$\Omega:=\omega_{J}+{\sqrt{1}}\omega_{K}$ is a holomorphic symplectic form on
$(M,I)$.
###### Theorem 3.8 ((Calabi-Yau))
A compact, Kähler, holomorphically symplectic manifold admits a unique
hyperkähler metric in any Kähler class.
###### Remark 3.9.
For the rest of this talk, a hyperkähler manifold means a compact complex
manifold admitting a Kähler structure and a holomorphically symplectic
structure.
###### Definition 3.10.
A hyperkähler manifold $M$ is called simple, or IHS, if $\pi_{1}(M)=0$,
$H^{2,0}(M)={\mathbb{C}}$.
The rationale for this terminology comes from Bogomolov’s decomposition
theorem.
###### Theorem 3.11 ((Bogomolov, [B:1974]))
Any hyperkähler manifold admits a finite covering which is a product of a
torus and several simple hyperkähler manifolds.
Further on, all hyperkähler manifolds are assumed to be simple.
###### Remark 3.12.
A hyperkähler manifold is simple if and only if its holonomy group is $Sp(n)$,
and not a proper subgroup of $Sp(n)$ ([Bes]).
###### Example 3.13.
Take a 2-dimensional complex torus $T$, then the singular locus of $T/{\pm 1}$
is 16 points locally of form ${\mathbb{C}}^{2}/{\pm 1}$. Its resolution by
blow-up is called a Kummer surface. It is not hard to see that it is
holomorphically symplectic.
###### Definition 3.14.
A K3 surface is a hyperkähler manifold which is diffeomorphic to a Kummer
surface.
In real dimension 4, the only compact hyperkähler manifolds are tori and K3
surfaces, as follows from the Kodaira-Enriques classification.
###### Definition 3.15.
A Hilbert scheme $M^{[n]}$ of a complex surface $M$ is a classifying space of
all ideal sheaves $I\subset{\cal O}_{M}$ for which the quotient ${\cal
O}_{M}/I$ has dimension $n$ over ${\mathbb{C}}$.
###### Remark 3.16.
A Hilbert scheme is obtained as a resolution of singularities of the symmetric
power $\operatorname{Sym}^{n}M$.
###### Theorem 3.17 ((Fujiki, Beauville))
A Hilbert scheme of a hyperkähler manifold of real dimension 2 is hyperkähler.
###### Example 3.18.
Let $T$ be a torus. Then $T$ acts on its Hilbert scheme freely and properly by
translations. For $n=2$, the quotient $T^{[n]}/T$ is a Kummer K3-surface. For
$n>2$, a universal covering of $T^{[n]}/T$ is called a generalized Kummer
variety.
###### Remark 3.19.
There are 2 more “sporadic” examples of compact hyperkähler manifolds,
constructed by K. O’Grady ([O]). All known simple hyperkaehler manifolds are
these 2 and the two series: Hilbert schemes of K3 and generalized Kummer.
### 3.2 Bogomolov-Beauville-Fujiki form and the mapping class group
###### Theorem 3.20
(Fujiki, [Fu])
Let $\eta\in H^{2}(M)$, and $\dim M=2n$, where $M$ is hyperkähler. Then
$\int_{M}\eta^{2n}=cq(\eta,\eta)^{n}$, for some primitive integer quadratic
form $q$ on $H^{2}(M,{\mathbb{Z}})$, and $c>0$ a positive rational number,
called Fujiki constant.
###### Definition 3.21.
This form is called Bogomolov-Beauville-Fujiki form. It is defined by the
Fujiki’s relation uniquely, up to a sign. The sign is determined from the
following formula (Bogomolov, Beauville)
$\displaystyle\lambda q(\eta,\eta)$
$\displaystyle=\int_{X}\eta\wedge\eta\wedge\Omega^{n-1}\wedge\bar{\Omega}^{n-1}-$
$\displaystyle-\frac{n-1}{n}\left(\int_{X}\eta\wedge\Omega^{n-1}\wedge\bar{\Omega}^{n}\right)\left(\int_{X}\eta\wedge\Omega^{n}\wedge\bar{\Omega}^{n-1}\right)$
where $\Omega$ is the holomorphic symplectic form, and $\lambda>0$.
###### Remark 3.22.
The BBF form $q$ has signature $(b_{2}-3,3)$. It is negative definite on
primitive forms, and positive definite on
$\langle\Omega,\bar{\Omega},\omega\rangle$, where $\omega$ is a Kähler form.
Using the BBF form, it is possible to describe the automorphism group of
cohomology in a very convenient way.
###### Theorem 3.23
Let $M$ be a simple hyperkähler manifold, and $G\subset GL(H^{*}(M))$ a group
of automorphisms of its cohomology algebra preserving the Pontryagin classes.
Then $G$ acts on $H^{2}(M)$ preserving the BBF form. Moreover, the map
$G{\>\longrightarrow\>}O(H^{2}(M,{\mathbb{R}}),q)$ is surjective on a
connected component, and has compact kernel.
Proof. Step 1: Fujiki formula $v^{2n}=q(v,v)^{n}$ implies that $\Gamma_{0}$
preserves the Bogomolov-Beauville-Fujiki up to a sign. The sign is fixed, if
$n$ is odd.
Step 2: For even $n$, the sign is also fixed. Indeed, $G$ preserves
$p_{1}(M)$, and (as Fujiki has shown in [Fu]), $v^{2n-2}\wedge
p_{1}(M)=q(v,v)^{n-1}c$, for some $c\in{\mathbb{R}}$. The constant $c$ is
positive, because the degree of $c_{2}(B)$ is positive for any non-trivial
Yang-Mills bundle with $c_{1}(B)=0$.
Step 3: ${\mathfrak{o}}(H^{2}(M,{\mathbb{R}}),q)$ acts on
$H^{*}(M,{\mathbb{R}})$ by derivations preserving Pontryagin classes
([V:1996]). Therefore $\operatorname{Lie}(G)$ surjects to
${\mathfrak{o}}(H^{2}(M,{\mathbb{R}}),q)$.
Step 4: The kernel $K$ of the map
$G{\>\longrightarrow\>}G{\left|{}_{{H^{2}(M,{\mathbb{R}})}}\right.}$ is
compact, because it commutes with the Hodge decomposition and Lefschetz
${\mathfrak{s}l}(2)$-action, hence preserves the Riemann-Hodge form, which is
positive definite.
Using this result, the mapping class group can also be computed. We use a
theorem of D. Sullivan, who expressed the mapping group in terms of the
rational homotopy theory, and expressed the rational homotopy in terms of the
algebraic structure of the de Rham algebra.
###### Theorem 3.24
(Sullivan, [Su, Theorem 10.3, Theorem 12.1, Theorem 13.3])
Let $M$ be a compact, simply connected Kähler manifold,
$\dim_{\mathbb{C}}M\geqslant 3$. Denote by $\Gamma_{0}$ the group of
automorphisms of an algebra $H^{*}(M,{\mathbb{Z}})$ preserving the Pontryagin
classes $p_{i}(M)$. Then the natural map
$\operatorname{Diff}(M)/\operatorname{Diff}_{0}{\>\longrightarrow\>}\Gamma_{0}$
has finite kernel, and its image has finite index in $\Gamma_{0}$.
As a corollary of this theorem, we obtain a similar result about hyperkähler
manifolds.
###### Theorem 3.25
Let $M$ be a simple hyperkähler manifold, and $\Gamma_{0}$ the group of
automorphisms of an algebra $H^{*}(M,{\mathbb{Z}})$ preserving the Pontryagin
classes $p_{i}(M)$. Then
(i)
$\Gamma_{0}{\left|{}_{{H^{2}(M,{\mathbb{Z}})}}\right.}$ is a finite index
subgroup of $O(H^{2}(M,{\mathbb{Z}}),q)$.
(ii)
The map $\Gamma_{0}{\>\longrightarrow\>}O(H^{2}(M,{\mathbb{Z}}),q)$ has finite
kernel.
We obtained that the mapping group is arithmetic (commensurable to a subgroup
of integer points in a rational Lie group).
As follows from [Hu2, Theorem 2.1], there are only finitely many connected
components of $\operatorname{Teich}$. Let $\Gamma^{I}$ be the group of
elements of mapping class group preserving a connected component of
Teichmüller space containing $I\in\operatorname{Teich}$. Then $\Gamma^{I}$ is
also arithmetic. Indeed, it has finite index in $\Gamma$.
###### Definition 3.26.
The image of $\Gamma^{I}$ in $GL(H^{2}(M,{\mathbb{Z}}))$ is called monodromy
group of a manifold.
###### Remark 3.27.
The monodromy group can also be obtained as a group generated by monodromy of
all Gauss-Manin local system for all deformations of $M$ ([V:2013, Theorem
7.2]). This explains the term. This notion was defined and computed in many
special cases by E. Markman ([Ma1], [Ma2]).
## 4 Global Torelli theorem
### 4.1 Period map
To study the moduli problem, one should understand the mapping class group
(described above) and the Teichmüller space. It turns out that the birational
Teichmüller space has a very simple description in terms of the period map,
inducing a diffeomorphism
$\operatorname{Teich}_{b}{\>\longrightarrow\>}\frac{SO(b_{2}-3,3)}{SO(b_{2}-3,1)\times
SO(2)}$ on each connected component of $\operatorname{Teich}_{b}$.
###### Definition 4.1.
Let $\operatorname{\sf
Per}:\;\operatorname{Teich}{\>\longrightarrow\>}{\mathbb{P}}H^{2}(M,{\mathbb{C}})$
map $J$ to a line $H^{2,0}(M,J)\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$. The map
$\operatorname{\sf
Per}:\;\operatorname{Teich}{\>\longrightarrow\>}{\mathbb{P}}H^{2}(M,{\mathbb{C}})$
is called the period map.
###### Remark 4.2.
$\operatorname{\sf Per}$ maps $\operatorname{Teich}$ into an open subset of a
quadric, defined by
$\operatorname{{\mathbb{P}}\sf er}:=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})\
\ |\ \ q(l,l)=0,q(l,\bar{l})>0\\}.$
The manifold $\operatorname{{\mathbb{P}}\sf er}$ is called the period space of
$M$.
As follows from Proposition 4.8 below, $\operatorname{{\mathbb{P}}\sf
er}=\frac{SO(b_{2}-3,3)}{SO(b_{2}-3,1)\times SO(2)}$.
###### Theorem 4.3
(Bogomolov, [B:1978])
Let $M$ be a simple hyperkähler manifold, and $\operatorname{Teich}$ its
Teichmüller space. Then the period map $\operatorname{\sf
Per}:\;\operatorname{Teich}{\>\longrightarrow\>}\operatorname{\sf Per}$ is
etale (has invertible differential everywhere).
###### Remark 4.4.
Bogomolov’s theorem implies that $\operatorname{Teich}$ is smooth. It is non-
Hausdorff even in the simplest examples.
Now the global Torelli theorem can be stated as follows. Recall that the
birational Teichmüller space $\operatorname{Teich}_{b}$ is a Hausdorff
reduction of the Teichmüller space of the holomorphic symplectic manifolds of
Kähler type.
###### Theorem 4.5
Let $M$ be a simple hyperkähler manifold, and $\operatorname{\sf
Per}:\;\operatorname{Teich}_{b}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}$ the period map. Then $\operatorname{\sf Per}$ is a diffeomorphism on each
connected component.
The following proposition is proven in a straghtforward manner using 1950-ies
style arguments of geometric topology.
###### Proposition 4.6 ((The Covering Criterion))
Let $X\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}Y$ be an etale
map of smooth manifolds. Suppose that each $y\in Y$ has a neighbourhood $B\ni
y$ diffeomorphic to a closed ball, such that for each connected component
$B^{\prime}\subset\varphi^{-1}(B)$, $B^{\prime}$ projects to $B$ surjectively.
Then $\varphi$ is a covering.
Now, the Global Torelli implied by the following result, which is proven in
Subsection 4.2 using hyperkähler structures.
###### Proposition 4.7
In assumptions of Theorem 4.5, the period map satisfies the conditions of the
Covering Criterion.
### 4.2 Moduli of hyperkähler structures and twistor curves
###### Proposition 4.8
The period space
$\operatorname{{\mathbb{P}}\sf er}:=\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})\
\ |\ \ q(l,l)=0,q(l,\bar{l})>0\\}$
is identified with $\frac{SO(b_{2}-3,3)}{SO(2)\times SO(b_{2}-3,1)}$, which is
a Grassmannian of positive oriented 2-planes in $H^{2}(M,{\mathbb{R}})$.
Proof. Step 1: Given $l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$, the space
generated by $\operatorname{Im}l,\operatorname{Re}l$ is 2-dimensional, because
$q(l,l)=0,q(l,\bar{l})$ implies that $l\cap H^{2}(M,{\mathbb{R}})=0$.
Step 2: This 2-dimensional plane is positive, because
$q(\operatorname{Re}l,\operatorname{Re}l)=q(l+\bar{l},l+\bar{l})=2q(l,\bar{l})>0$.
Step 3: Conversely, for any 2-dimensional positive plane $V\in
H^{2}(M,{\mathbb{R}})$, the quadric $\\{l\in
V\otimes_{\mathbb{R}}{\mathbb{C}}\ \ |\ \ q(l,l)=0\\}$ consists of two lines;
a choice of a line is determined by orientation.
###### Remark 4.9.
Two hyperkähler structures $(M,I,J,K,g)$ and
$(M,I^{\prime},J^{\prime},K^{\prime},g)$ are called equivalent if there exists
a unitary quaternion $h$ such that $I^{\prime}=hIh^{-}1$,
$J^{\prime}=hJh^{-}1$, $K^{\prime}=hKh^{-}1$. From the holonomy
characterization of simple hyperkähler manifolds (Remark 3.12) it follows that
two hyperkähler structures are isometric if and only if they are equivalent.
###### Definition 4.10.
Let $(M,I,J,K,g)$ be a hyperkähler manifold. A hyperkähler 3-plane in
$H^{2}(M,{\mathbb{R}})$ is a positive oriented 3-dimensional subspace $W$,
generated by three Kähler forms $\omega_{I},\omega_{J},\omega_{K}$.
###### Definition 4.11.
Similarly to the Teichmüller space and period map of complex structures, one
can define the period space of hyperkähler metrics. Denote it by
$\operatorname{Teich}_{H}$. The corresponding period map is
$\operatorname{\sf
Per}:\;\operatorname{Teich}_{H}{\>\longrightarrow\>}\operatorname{{\mathbb{P}}\sf
er}_{H},$
where $\operatorname{{\mathbb{P}}\sf er}_{H}=\frac{SO(b_{2}-3,3)}{SO(3)\times
SO(b_{2}-3)}$ is the space of positive, oriented 3-planes, and
$\operatorname{\sf Per}$ maps a hyperkähler structure to the corresponding
hyperkähler 3-plane.
###### Remark 4.12.
There is one significant difference between $\operatorname{Teich}$ and the
hyperkähler Teichmüller space $\operatorname{Teich}_{H}$: the latter is
Hausdorff, and, in fact, metrizable. Indeed, we could equip the space
$\operatorname{Teich}_{H}$ of hyperkähler metrics with the Gromov-Hausdorff
metric.
Let $I\in\operatorname{Teich}$ be a complex structure, and ${\cal K}(I)$ its
Kähler cone. The set of hyperkähler metrics compatible with $I$ is
parametrized by ${\cal K}(I)$, by Calabi-Yau theorem. The corresponding
3-dimensional subspaces are generated by $\operatorname{\sf Per}(I)+\omega$,
where $\omega\in{\cal K}(I)$. The local Torelli theorem implies that locally
$I\in\operatorname{Teich}$ is uniquely determined by the 2-plane generated by
$\omega_{J}$ and $\omega_{K}$; Calabi-Yau theorem implies that the hyperkähler
metric is uniquely determined by the complex structure and the Kähler
structure. This gives the following hyperkähler version of the local Torelli
theorem.
###### Theorem 4.13
Let $M$ be a simple hyperkähler manifold, and $\operatorname{Teich}_{H}$ its
hyperkähler Teichmüller space. Then the period map $\operatorname{\sf
Per}:\;\operatorname{Teich}{\>\longrightarrow\>}\operatorname{\sf Per}_{H}$
mapping an equivalence class of hyperkähler structures to is its 3-plane is
etale (has invertible differential everywhere).
###### Remark 4.14.
Let $W\subset H^{2}(M,{\mathbb{R}})$ be a positive 3-dimensional plane. The
set $S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$ of oriented 2-dimensional
planes in $W$ is identified with $S^{2}={\mathbb{C}}P^{1}$. When $W$ is a
hyperkähler 3-plane, $S_{W}$ is called the twistor family of a hyperkähler
structure. A point in the twistor family corresponds to a complex structure
$aI+bJ+cK\in{\mathbb{H}}$, with $a^{2}+b^{2}+c^{2}=1$. We call the
corresponding rational curves ${\mathbb{C}}P^{1}\subset\operatorname{Teich}$
the twistor lines. It is not hard to see that the twistor lines are
holomorphic.
###### Definition 4.15.
Let $W\in\operatorname{{\mathbb{P}}\sf er}_{H}$ be a positive 3-plane,
$S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$ the corresponding rational
curve, and $x\in S_{W}$ be a point. It is called liftable if for any point
$y\in\operatorname{\sf Per}^{-1}(x)\subset\operatorname{Teich}$ there exists
${\cal H}\in\operatorname{Teich}_{H}$ such that the corresponding twistor line
contains $y$.
When $W$ is generic, the corresponding line $S_{W}$ is liftable, as indicated
below.
###### Definition 4.16.
The Neron-Severi lattice $NS(I)$ of a hyperkähler manifold $(M,I)$ is
$H^{1,1}(M,I)\cap H^{2}(M,{\mathbb{Z}})$.
The following theorem, based on results of [DP], was proven by D. Huybrechts.
###### Theorem 4.17 (([Hu1]))
Let $M$ be a hyperkaehler manifold with $\operatorname{\sf NS}(M)=0$. Then its
Kaehler cone is one of two components of the set
$\\{\nu\in H^{1,1}(M,{\mathbb{R}})\ |\ q(\nu,\nu)\geqslant 0\\}.$
###### Definition 4.18.
Let $S\subset\operatorname{Teich}$ be a ${\mathbb{C}}P^{1}$ associated with a
twistor family. It is called generic if it passes through a point
$I\in\operatorname{Teich}$ with $\operatorname{\sf NS}(M,I)=0$. Clearly, a
hyperkähler 3-plane $W\subset H^{2}(M,{\mathbb{R}})$ corresponds to a generic
twistor family if and only if its orthogonal complement $W^{\bot}\subset
H^{2}(M,{\mathbb{R}})$ does not contain rational vectors. A 3-plane
$W\in\operatorname{{\mathbb{P}}\sf er}_{H}$ is called generic if
$W^{\bot}\subset H^{2}(M,{\mathbb{R}})$ does not contain rational vectors. The
corresponding rational curve $S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$
is called a GHK line. GHK lines are liftable, which is very useful for many
purposes, including the proof of Torelli theorem (see also [AV], where GHK
lines were used to study Kähler cones of hyperkähler manifolds).
The following theorem immediately follows from the Calabi-Yau theorem and the
description of the Kähler cone given in Theorem 4.17.
###### Theorem 4.19
Let $W\in\operatorname{{\mathbb{P}}\sf er}_{H}$ be a generic plane,
$S_{W}\subset\operatorname{{\mathbb{P}}\sf er}$ the corresponding rational
curve, and $x\in S_{W}$ a generic point. Then $(S_{W},x)$ is liftable.
Assumptions of the covering criterion (Proposition 4.7) immediately follow
from Theorem 4.19. Indeed, it is not hard to see that any two points on a
closed ball $B\subset\operatorname{{\mathbb{P}}\sf er}$ can be connected
inside $B$ by a sequence of GHK curves intersecting in generic points of $B$.
Since these curves are liftable, any connected component of $\operatorname{\sf
Per}^{-1}(B)$ is mapped to $B$ surjectively.
## 5 Teichmüller spaces and ergodic theory
### 5.1 Ergodic complex structures
After the Teichmüller space and the mapping class group are understood, it is
natural to consider the quotient space
$\operatorname{Comp}/\operatorname{Diff}=\operatorname{Teich}/\Gamma$ of the
Teichmüller space by the mapping class group
$\Gamma:=\operatorname{Diff}/\operatorname{Diff}_{0}$.
###### Claim 5.1
Let $M$ be a simple hyperkähler manifold, $\Gamma$ its mapping class group,
and $\operatorname{Teich}_{b}$ the birational Teichmüller space. Then the
quotient $\operatorname{Teich}_{b}/\Gamma$ parametrizes the birational classes
of deformations of $M$.
One could call the quotient $\operatorname{Teich}_{b}/\Gamma$ “the moduli
space”, but, unfortunately, this is not a space in any reasonable sense.
Indeed, as we shall see, non-trivial closed subsets of
$\operatorname{Teich}_{b}/\Gamma$ are at most countable, making
$\operatorname{Teich}_{b}/\Gamma$ terribly non-Hausdorff. This means that the
concept of “moduli space” has no meaning, and all interesting information
about moduli problems is hidden in dynamics of $\Gamma$-action on
$\operatorname{Teich}$.
Let $I\in\operatorname{Teich}$ be a point, and
$\operatorname{Teich}^{I}\subset\operatorname{Teich}$ its connected component.
Since $\operatorname{Teich}$ has finitely many components, a subgroup mapping
class group fixing $\operatorname{Teich}$ has finite index. Its image in
$\operatorname{Aut}(\operatorname{Teich}^{I})$ is called monodromy group and
denoted $\Gamma^{I}$ (Definition 3.26). It is a finite index subgroup in
$SO(H^{2}(M,{\mathbb{Z}}))$.
All that said, we find that the moduli problem for hyperkähler manifold is
essentially reduced to the dynamics of the $\Gamma^{I}$-action on the space
$\operatorname{{\mathbb{P}}\sf er}$, which is understood as a Grassmannian of
positive, oriented 2-planes in $H^{2}(M,{\mathbb{R}})$.
It is natural to study the dynamics of a group action from the point of view
of ergodic theory, ignoring measure zero subsets. However, the quotient map
$\operatorname{Teich}{\>\longrightarrow\>}\operatorname{Teich}_{b}$ is
bijective outside of a union of countably many divisors, corresponding to
complex structures $I$ with $NS(M,I)$ non-zero. This set has measure 0.
Therefore, the quotient map
$\operatorname{Teich}{\>\longrightarrow\>}\operatorname{Teich}_{b}$ induces an
equivalence of measured spaces. For the purposes of ergodic theory, we shall
identify $\operatorname{Teich}^{I}$ with the corresponding homogeneous space
$\operatorname{{\mathbb{P}}\sf er}$.
Ths first observation, based on a theorem of C. Moore, implies that the
monodromy action on $\operatorname{{\mathbb{P}}\sf er}$ is ergodic.
###### Definition 5.2.
Let $(M,\mu)$ be a space with measure, and $G$ a group acting on $M$. This
action is ergodic if all $G$-invariant measurable subsets $M^{\prime}\subset
M$ satisfy $\mu(M^{\prime})=0$ or $\mu(M\backslash M^{\prime})=0$.
###### Claim 5.3
Let $M$ be a manifold, $\mu$ a Lebesgue measure, and $G$ a group acting on
$(M,\mu)$ ergodically. Then the set of non-dense orbits has measure 0.
Proof: Consider a non-empty open subset $U\subset M$. Then $\mu(U)>0$, hence
$M^{\prime}:=G\cdot U$ satisfies $\mu(M\backslash M^{\prime})=0$. For any
orbit $G\cdot x$ not intersecting $U$, one has $x\in M\backslash M^{\prime}$.
Therefore, the set of such orbits has measure 0.
###### Definition 5.4.
Let $I\in\operatorname{Comp}$ be a complex structure on a manifold. It is
called ergodic if its $\operatorname{Diff}$-orbit is dense in its connected
component of $\operatorname{Comp}$.
###### Remark 5.5.
This is equivalent to density of $\Gamma$-orbit of $I$ in its Teichmüller
component.
### 5.2 Ergodicity of the monodromy group action
###### Definition 5.6.
Let $G$ be a Lie group, and $\Gamma\subset G$ a discrete subgroup. Consider
the pushforward of the Haar measure to $G/\Gamma$. We say that $\Gamma$ has
finite covolume if the Haar measure of $G/\Gamma$ is finite. In this case
$\Gamma$ is called a lattice subgroup.
###### Remark 5.7.
Borel and Harish-Chandra proved that an arithmetic subgroup of a reductive
group $G$ is a lattice whenever $G$ has no non-trivial characters over
${\mathbb{Q}}$ (see e.g. [VGS]). In particular, all arithmetic subgroups of a
semi-simple group are lattices.
###### Theorem 5.8
(Calvin C. Moore, [Mo:1966, Theorem 7])
Let $\Gamma$ be an arithmetic subgroup in a non-compact simple Lie group $G$
with finite center, and $H\subset G$ a non-compact subgroup. Then the left
action of $\Gamma$ on $G/H$ is ergodic.
###### Theorem 5.9
Let ${\operatorname{Teich}}$ be a connected component of a Teichmüller space,
and $\Gamma^{I}$ its monodromy group. Then the set of all non-ergodic points
of $\operatorname{Teich}$ has measure 0.
Proof: Global Torelli theorem identifies $\operatorname{Teich}$ (as a measured
space) and $G/H$, where $G=SO(b_{2}-3,3)$, $H=SO(2)\times SO(b_{2}-3,1)$.
Since $\Gamma^{I}$ is an arithmetic lattice, $\Gamma^{I}$-action on $G/H$ is
ergodic, by Moore’s theorem.
Moore’s theorem implies that outside of a measure zero set, all complex
structures on $\operatorname{Teich}$ are ergodic. If we want to determine
which exactly complex structures are ergodic, we have to use Ratner’s theorem,
giving precise description of a closure of a $\Gamma^{I}$-orbit in a
homogeneous space.
Now I will state some basic results of Ratner theory. For more details, please
see [KSS] and [Mor].
###### Definition 5.10.
Let $G$ be a Lie group, and $g\in G$ any element. We say that $g$ is unipotent
if $g=e^{h}$ for a nilpotent element $h$ in its Lie algebra. A group $G$ is
generated by unipotents if $G$ is multiplicatively generated by unipotent
elements.
###### Theorem 5.11
([Mor, 1.1.15 (2)])
Let $H\subset G$ be a Lie subroup generated by unipotents, and $\Gamma\subset
G$ a lattice. Then a closure of any $H$-orbit in $G/\Gamma$ is an orbit of a
closed, connected subgroup $S\subset G$, such that $S\cap\Gamma\subset S$ is a
lattice.
When this lattice is arithmetic, one could describe the group $S$ very
explicitly.
###### Claim 5.12
( [KSS, Proposition 3.3.7] or [Sh, Proposition 3.2])
Let $x\in G/H$ be a point in a homogeneous space, and $\Gamma\cdot x$ its
$\Gamma$-orbit, where $\Gamma$ is an arithmetic lattice. Then its closure is
an orbit of a group $S$ containing stabilizer of $x$. Moreover, $S$ is a
smallest group defined over rationals and stabilizing $x$.
For the present purposes, we are interested in a pair $SO(3,k)\supset
SO(1,k)\times SO(2)\subset G$ (or, rather, their connected components
$G=SO^{+}(3,k)$ and $H=SO(1,k)\times SO(2)\subset G$). In this case, there are
no intermediate subgroups.
###### Claim 5.13
Let $G=SO^{+}(3,k)$, and $H\cong SO^{+}(1,k)\times SO(2)\subset G$. Then any
closed connected Lie subgroup $S\subset G$ containing $H$ coincides with $G$
or with $H$.
###### Corollary 5.14
Let $J\in\operatorname{{\mathbb{P}}\sf er}=G/H$. Then either $J$ is ergodic,
or its $\Gamma$-orbit is closed in $\operatorname{{\mathbb{P}}\sf er}$.
By Ratner’s theorem, in the latter case the $H$-orbit of $J$ has finite volume
in $G/\Gamma$. Therefore, its intersection with $\Gamma$ is a lattice in $H$.
This brings
###### Corollary 5.15
Let $J\in\operatorname{{\mathbb{P}}\sf er}$ be a point such that its
$\Gamma$-orbit is closed in $\operatorname{{\mathbb{P}}\sf er}$. Consider its
stabilizer $\operatorname{St}(J)\cong H\subset G$. Then
$\operatorname{St}(J)\cap\Gamma$ is a lattice in $\operatorname{St}(J)$.
###### Corollary 5.16
Let $J$ be a non-ergodic complex structure on a hyperkähler manifold, and
$W\subset H^{2}(M,{\mathbb{R}})$ be a plane generated by
$\operatorname{Re}\Omega,\operatorname{Im}\Omega$. Then $W$ is rational.
Equivalently, this means that $Pic(M)$ has maximal possible dimension.
Similar results are true for a torus of dimension $>1$; we refer the reader to
[V] for precise statements and details of the proof.
## 6 Applications of ergodicity
### 6.1 Ergodic complex structures, Gromov-Hausdorff closures, and
semicontinuity
The ergodicity theorem (Subsection 5.2) has some striking and even paradoxical
implications. For instance, consider a Kähler cone $\operatorname{Kah}$ of a
hyperkähler manifold (or a torus of dimension $>1$) equipped with an ergodic
complex structure. By Calabi-Yau theorem, each point of $\operatorname{Kah}$
corresponds to a Ricci-flat metric on $M$. If we restrict ourselves to those
metrics which satisfy $\operatorname{\sf diam}(M,g)\leqslant d$ (with bounded
diameter), then, by Gromov’s compactness theorem ([Gr]), the set $X_{d}$ of
such metrics is precompact in the Gromov’s space of all metric spaces,
equipped with the Gromov-Hausdorff metric. It is instructive to see what kind
of metric spaces occur on its boundary (that is, on $\bar{X}_{d}\backslash
X_{d}$). To see this, let $\nu_{i}$ be a sequence of diffeomorphisms
satisfying $\lim_{i}\nu_{i}(I)=I^{\prime}$. By Kodaira stability theorem, the
Kähler cone of $(M,I)$ is lower continuous on $I$. Therefore, there exists a
family of Kähler classes $\omega_{i}$ on $(M,\nu_{i}(I))$ which converge to a
given Kähler class $\omega^{\prime}$ on $(M,I^{\prime})$. This implies
convergence of the corresponding Ricci-flat metrics. We obtain that any Ricci-
flat metric on $(M,I^{\prime})$ (for any $I^{\prime}$ in the same deformation
class as $I$) is obtained as a limit of Ricci-flat metrics on $(M,I)$.
This gives the following truly bizzarre theorem.
###### Theorem 6.1
Let $(M,I)$ be an ergodic complex structure on a hyperkähler manifold,
$X\cong\operatorname{Kah}$ the set of all Ricci-flat Kähler metrics on
$(M,I)$, and $g^{\prime}$ another Ricci-flat metric on $M$ in the same
deformation class. Then $g^{\prime}$ lies in the closure of $X$ with respect
to the Gromov topology on the space of all metrics.
This result is very strange, because $\operatorname{Kah}$ is a smooth manifold
of dimension $b_{2}(M)-2$. By Theorem 4.13, the space
$\operatorname{Teich}_{H}$ of all hyperkähler metrics is a smooth manifold of
dimension $\frac{b_{2}(b_{2}-1)(b_{2}-2)}{6}$, clearly much bigger than
$\dim\operatorname{Kah}$. Obviously, the boundary of $X$ is highly irregular
and chaotic.
For another application, consider some numerical quantity $\mu(I)$ associated
with an equivalence class of complex structures. Suppose that $\mu$ is
continuous or semi-continuous on $\operatorname{Teich}$. Then $\mu$ is
constant on ergodic complex structures. To see this, suppose that $\mu$ is
upper semicontinuous, giving
$\mu(\lim_{k}I_{k})\geqslant\lim_{k}(\mu(I_{k})).$ (6.1)
Given an ergodic complex structire $I$, find a sequence $I_{k}=\nu_{k}(I)$
converging to a complex structure $I^{\prime}$. Then (6.1) gives
$\mu(I)\leqslant\mu(I^{\prime})$. This implies that any ergodic complex
structure satisfies
$\mu(I)=\inf_{I^{\prime}\in\operatorname{Teich}}\mu(I^{\prime})$.
This observation can be applied to Kobayashi pseudometric and Kobayashi
hyperbolicity.
### 6.2 Kobayashi non-hyperbolicity of hyperkähler manifolds
###### Definition 6.2.
Pseudometric on $M$ is a function $d:\;M\times
M{\>\longrightarrow\>}{\mathbb{R}}^{\geqslant 0}$ which is symmetric:
$d(x,y)=d(y,x)$ and satisfies the triangle inequality $d(x,y)+d(y,z)\geqslant
d(x,z)$.
###### Remark 6.3.
Let ${\mathfrak{D}}$ be a set of pseudometrics. Then $d_{{\operatorname{\sf
max}}}(x,y):=\sup_{d\in{\mathfrak{D}}}d(x,y)$ is also a pseudometric.
###### Definition 6.4.
The Kobayashi pseudometric on a complex manifold $M$ is $d_{\operatorname{\sf
max}}$ for the set ${\mathfrak{D}}$ of all pseudometrics such that any
holomorphic map from the Poincaré disk to $M$ is distance-non-increasing.
In other words, a Kobayashi pseudo-distance between two points $x,y$ is an
infimum of distance from $x$ to $y$ in Poincare metric for any sequence of
holomorphic disks connecting $x$ to $y$.
The following observation is not difficult to see.
###### Claim 6.5
Let $\pi:\;{\cal M}{\>\longrightarrow\>}X$ be a smooth holomorphic family,
which is trivialized as a smooth manifold: ${\cal M}=M\times X$, and $d_{x}$
the Kobayashi metric on $\pi^{-1}(x)$. Then $d_{x}(m,m^{\prime})$ is upper
continuous on $x$.
###### Corollary 6.6
Denote the diameter of the Kobayashi pseudometric by $\operatorname{\sf
diam}(d_{x}):=\sup_{m,m^{\prime}}d_{x}(m,m^{\prime})$. Then the Kobayashi
diameter of a fiber of $\pi$ is an upper continuous function:
$\operatorname{\sf diam}:\;X{\>\longrightarrow\>}{\mathbb{R}}^{\geqslant 0}$.
For a projective K3 surface, the Kobayashi pseudometric vanishes ([Vo, Lemma
1.51]). However, all non-projective K3 surfaces are ergodic (Corollary 5.16).
This proves the vanishing of Kobayashi pseudodistance for all K3 surfaces. A
more general version of this result is due to due to Kamenova-Lu-Verbitsky.
###### Theorem 6.7 (([KLV]))
Let $M$ be a Hilbert scheme of K3. Then the Kobayashi pseudometric on $M$
vanishes
###### Definition 6.8.
A complex manifold is called Kobayashi hyperbolic if the Kobayashi
pseudometric is a metric.
###### Definition 6.9.
An entire curve is a non-constant map ${\mathbb{C}}{\>\longrightarrow\>}M$.
Brody has shown that a compact manifold is Kobayashi hyperbolic if and only if
it admits no entire curves. The same argument also proves semicontinuity.
###### Theorem 6.10
([Br:1978]) Let $I_{i}$ be a sequence of complex structures on $M$ which are
not hyperbolic, and $I$ its limit. Then $(M,I)$ is also not hyperbolic.
With ergodicity, this can be used to prove that all hyperkähler manifolds are
non-hyperbolic.
Recall that a twistor family of complex structures on a hyperkähler manifold
$(M,I,J,K)$ is a family of complex structures of form
$S^{2}\cong\\{L:=aI+bJ+cK,\ \ \ a^{2}+b^{2}+c^{2}=1\\}$. F. Campana has
obtained a remarkable partial result towards non-hyperbolicity.
###### Theorem 6.11
([Cam]) Let $M$ be a hyperkähler manifold, and $S\subset\operatorname{Teich}$
a twistor family. Then there exists an entire curve in some $I\in S$.
###### Claim 6.12
There exists a twistor family which has only ergodic fibers.
Proof: There are only countably many complex structures which are not ergodic;
however, twistor curves move freely through the Teichmüller space of a
hyperkähler manifold, as seen from Theorem 4.19.
Applying Campana’s theorem to the family constructed in Claim 6.12, we obtain
an ergodic complex structure which is non-hyperbolic. Then the Brody’s theorem
implies that all complex structures in the same deformation class are non-
hyperbolic.
###### Theorem 6.13
All hyperkähler manifolds are non-hyperbolic.
### 6.3 Symplectic packing and ergodicity
I will finish this talk with a list of open problems of hyperkähler and
holomorphically symplectic geometry which might be solvable with ergodic
methods.
###### Question 6.14.
Let $M$ be a hyperkähler manifold, and $\operatorname{Teich}$ its Teichmüller
space. Consider the universal fibration ${\cal
X}{\>\longrightarrow\>}\operatorname{Teich}$ (Remark 1.11). The mapping class
group $\Gamma$ acts on ${\cal X}$ in a natural way. Is this action ergodic?
This question (suggested by Claire Voisin) seems to be related to the
following conjecture.
###### Conjecture 6.15
Let $M$ be a K3 surface. Then for each $x\in M$ and $v\in T_{x}M$ there exists
an entire curve $C\ni x$ with $T_{x}C\ni v$.
The symplectic packing problem is a classical subject of symplectic geometry
([MP]). However, its holomorphically symplectic version seems to be completely
unexplored.
###### Definition 6.16.
A holomorphic symplectic ball $B_{r}$ of radius $r$ is a complex
holomorphically symplectic manifold admitting a holomorphic symplectomorphism
to an open ball in ${\mathbb{C}}^{2n}$ of radius $r$ with the standard
holomorphic symplectic form $\sum_{i=1}^{n}dz_{2i-1}\wedge dz_{2i}$.
Notice that by a holomorphic symplectic version of Darboux theorem, any
holomorphically symplectic manifold is locally symplectomorphic to a
holomorphic symplectic ball.
###### Definition 6.17.
Let $M$ be a holomorphically symplectic manifold. Symplectic packing of radii
$r_{1},...,r_{k}$ of $M$ is a set of holomorphic symplectomorphisms
$\varphi_{i}:\;B_{r_{i}}\hookrightarrow M$ with images of $\varphi_{i}$ not
intersecting.
Obviously, in these assumptions,
$\sum\operatorname{Vol}(B_{r_{i}})\leqslant\operatorname{Vol}_{M}$, where
$\operatorname{Vol}$ denotes the symplectic volume of a holomorphic symplectic
manifold $(M,\Omega_{M})$:
$\operatorname{Vol}(M)=\int_{M}(\Omega_{M}\wedge\bar{\Omega}_{M}),\ \
2n=\dim_{\mathbb{C}}M.$
The volume inequality puts certain restrictions on the possible symplectic
packing. Are there any other restrictions?
For the usual (smooth) symplectic packing, some additional restrictions are
obtained from the Gromov’s symplectic capacity theorem and from the study of
pseudoholomorphic curves. However, it seems that in holomorphic symplectic
situation these restrictions are also trivial. For a general compact torus of
real dimension 4, volume is known to be the only restriction to existence of
symplectic packing ([LMS]). It seems that a similar result about the smooth
symplectic packings is true for K3 surfaces as well, and, possibly, for any
hyperkähler manifold.
The arguments used to treat the usual (smooth) symplectic packings don’t work
for the holomorphic symplectic case. However, the set of possible radii for
symplectic packing is obviously semicontinous, hence it can be studied by
ergodic methods, in the same way as one studies the Kobayashi pseudometric.
The following classical question was treated Buzzard and Lu in [BL].
###### Definition 6.18.
A complex manifold $M$ of dimension $n$ is called dominated by
${\mathbb{C}}^{n}$ if there exists a holomorphic map
$\varphi:\;{\mathbb{C}}^{n}{\>\longrightarrow\>}M$ which has non-degenerate
differential in generic point.
Buzzard and Lu proved that Kummer K3 surfaces are dominated by
${\mathbb{C}}^{2}$. So far, there is not a single example of a hyperkähler
manifold $M$ for which it is proven that $M$ is not dominated. This leads to
the following conjecture
###### Conjecture 6.19
Any compact hyperkähler manifold is dominated by ${\mathbb{C}}^{n}$.
There is no semicontinuity in dominance, because Brody lemma fails to produce
dominating maps ${\mathbb{C}}^{n}{\>\longrightarrow\>}M$ for $n>1$ as limits
of sequences of dominating maps. In the proof of Brody’s lemma (showing that a
limit of a sequence of entire curves contains an entire curve) one takes a
reparametrizations of each of the curves in the sequence. Starting from a
sequence of dominating maps, one could apply the same argument, but each
subsequent reparametrization can lead to smaller Jacobian of the differential,
and the differential of the limit could be zero.
It seems that more of the Brody’s argument can be retained if we restrict
ourselves to symplectomorphisms.
###### Question 6.20.
Consider a flat holomorphically symplectic structure on ${\mathbb{C}}^{2}$. Is
there a holomorphic map ${\mathbb{C}}^{2}{\>\longrightarrow\>}M$ to a K3
surface which is compatible with the holomorhic symplectic form?
Probably not. However, a quantitative version of this question makes sense.
Let $M$ be a hyperkähler manifold, and $K(M)$ the supremum of all $r$ such
that there exists a symplectic immersion from a symplectic ball of radius $r$
to $M$. It is not hard to see that $K(M)$ is semicontinuous in families, hence
constant on ergodic complex structures.
###### Question 6.21.
For a given hyperkähler manifold, find $K(M)$.
It is not clear if $K(M)$ is finite or infinite, even for a K3 surface (it is
clearly infinite for a torus).
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|
arxiv-papers
| 2014-04-15T08:47:16 |
2024-09-04T02:50:01.171833
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Misha Verbitsky",
"submitter": "Misha Verbitsky",
"url": "https://arxiv.org/abs/1404.3847"
}
|
1404.3852
|
# Moments of Riesz measures on Poincaré disk and homogeneous tree – a
comparative study
Tetiana BOIKO and Wolfgang WOESS Institut für Mathematische Strukturtheorie
(Math C), Technische Universität Graz, Steyrergasse 30, A-8010 Graz, Austria
[email protected], [email protected]
(Date: 31 March 2014)
###### Abstract.
One of the purposes of this paper is to clarify the strong analogy between
potential theory on the open unit disk and the homogeneous tree, to which we
dedicate an introductory section. We then exemplify this analogy by a study of
Riesz measures. Starting from interesting work by Favorov and Golinskii [10],
we consider subharmonic functions on the open unit disk, resp. on the
homogenous tree. Supposing that we can control the way how those functions may
tend to infinity at the boundary, we derive moment type conditions for the
Riesz measures. One one hand, we generalise the previous results of [10] for
the disk, and on the other hand, we show how to obtain analogous results in
the discrete setting of the tree.
###### Key words and phrases:
Euclidean disk, hyperbolic plane, homogeneous tree, Laplace operators,
boundaries, harmonic and subharmonic functions, Riesz measure
###### 2000 Mathematics Subject Classification:
05C05, 30F45, 31C05, 60J50
Supported by Austrian Science Fund projects FWF W1230-N13 and FWF P24028-N18
and by NAWI Graz
## 1\. Introduction
The homogeneous tree $\mathbb{T}=\mathbb{T}_{q}$ with degree $q+1$ is in many
respects a discrete analogue of the hyperbolic plane. These are the two basic
examples of Gromov-hyperbolic metric spaces. In the Poincaré metric, the
hyperbolic plane is the open unit disk $\mathbb{D}$ as a topological space.
Its natural geometric compactification is obtained by passing from the
hyperbolic to the Euclidean metric and taking the closure, i.e., the closed
unit disk. Analogously, the end compactification of $\mathbb{T}$ is obtained
by passing from the original graph metric to a new (bounded) metric and taking
the completion.
Various objects, formulas, properties, theorems, etc., of geometric,
algebraic, analytic, potential theoretic, or stochastic nature on $\mathbb{D}$
have counterparts on $\mathbb{T}$ and vice versa. It is not always immediately
apparent that looking at $\mathbb{D}$ both with Euclidean and with hyperbolic
eyeglasses may provide additional insight. But this _is_ true when one wants
to understand the analogies between $\mathbb{T}$ and $\mathbb{D}$. The purpose
of this note is to exhibit some potential theoretic aspects of that
correspondence. The starting point is a classical theorem of Blaschke [5]:
_A set $\\{z_{k}:k\in\mathbb{N}\\}\subset\mathbb{D}$ is the set of zeroes of a
bounded analytic function $f$ on $\mathbb{D}$ if and only if
$\quad\displaystyle\sum_{k}(1-|z_{k}|)<\infty\,$._
This also allows for the case where each $z_{k}$ is counted according to its
multiplicity $\mathsf{mult}(z_{k})$ as a zero of $f$. We interpet this theorem
in terms of the subharmonic function $v:\mathbb{D}\to[-\infty\,,\,\infty)$
given by $v(z)=\log|f(z)|$. We let $\mu_{v}$ be the Riesz measure of $v$ in
its Riesz decomposition (see below for details). Being bounded above, $v$ has
a harmonic majorant, which leads to finiteness of the “moment”
(1.1) $\int_{\mathbb{D}}(1-|z|)\,d\mu_{v}(z)<\infty\,.$
Since $\mu_{v}=\sum_{k}\mathsf{mult}(z_{k})\cdot\delta_{z_{k}}\,$, this just
means finiteness of $\sum_{k}(1-|z_{k}|)\,\mathsf{mult}(z_{k})$, so that the
Blaschke condition takes the form (1.1).
A change of the viewpoint is now suggestive. We start directly with a
subharmonic function $v$, and instead of assuming that it is bounded above, we
admit that it tends to $\infty$ in some controlled way when approaching a
subset $E\subset\partial\mathbb{D}=\mathbb{S}$, the boundary of the disk (the
unit circle). From properties of $E$ and the way how $v$ tends to infinity at
$E$, we then want to deduce properties of the Riesz measure $\mu_{v}\,$. This
approach was undertaken in two substantial papers by Favorov and Golinskii
[10], [11], which were the main inspiration for the present note. We shall
provide a more general version of one of their results on the Riesz measure of
subharmonic functions on the disk, and an example that shows sharpness.
On the homogeneous tree $\mathbb{T}$, the geometrical habit is converse as
compared to the disk: on one hand, one is used to look at the _Euclidean_ unit
disk $\mathbb{D}$ and its closure, which in the spirit of the present note
arises by a change from the “original” hyperbolic metric to the “new”
Euclidean metric which is the one of its compactification. On the other hand,
one is used to look at the tree with its habitual integer-valued graph metric
– this is our _hyperbolic_ object, and when we introduce the end
compactification, we pass to a suitable new, maybe less habitual metric which
is the one that corresponds to the Euclidean metric of $\mathbb{D}$.
(Sub)harmonic functions on $\mathbb{T}$ are defined via the discrete Laplacian
$P-I$ (or $I-P$, if one desires a positive semidefinite operator), where $P$
is the transition matrix of the simple random walk. Of course, here we also
have the Riesz decomposition theorem. We shall see that once we understand the
correspondence between tree and disk completely, we can obtain the same type
of moment condition for the Riesz measure of a subharmonic function as in
(1.1): we need to realise that the term $1-|z|$ in (1.1) is the distance from
$z$ to the boundary in the metric of the respective compactification.
In the next Section 2, we provide an expository description of the basic
potential theoretic features of $\mathbb{D}$ and $\mathbb{T}$. On purpose
slightly beyond the scope of the subsequent sections, it aims at providing a
good understanding of part of the many common features of those two
structures. Subsequently, in section 3, we shall prove our basic moment
conditions for subharmonic functions on those two spaces.
## 2\. Basic potential theory on disk and tree
A. Euclidean and hyperbolic disk
The Euclidean unit disk
$\mathbb{D}=\\{z=x+\mathfrak{i}\,y\in\mathbb{C}:|z|=\sqrt{x^{2}+y^{2}}<1.$
carries the Euclidean metric $\mathsf{d}_{\mathbb{D}}\,$, induced by the
absolute value, resp. length element
(2.1) $\mathsf{d}_{\mathbb{D}}(z,w)=|z-w|\quad\text{and}\quad
d_{\mathbb{D}}s=\sqrt{dx^{2}+dy^{2}}\,.$
The standard measure is Lebesgue measure – for which here we sometimes write
$\mathsf{m}_{\mathbb{D}}$ – with area element $d_{\mathbb{D}}z=dz=dx\,dy$. The
Euclidean Laplace operator is
$\Delta_{\mathbb{D}}=\partial_{x}^{2}+\partial_{y}^{2}\,.$
A _harmonic function_ is a real-valued function $h\in C^{2}(\mathbb{D})$ such
that $\Delta_{\mathbb{D}}=0$. For the definition of a _subharmonic function_ ,
see e.g. Helms [16, p.58], who rather considers superharmonic functions: the
correspondence is just by a change of the sign. A function
$u:\mathbb{D}\to[-\infty,+\infty)$ is subharmonic on $\mathbb{D}$ if it is
upper semicontinuous, and for every $z\in\mathbb{D}$ and $r<1-|z|$, one has
$\mathsf{A}_{r}u(z)\geq u(z)$, where
$\mathsf{A}_{r}u(z)=\mathsf{A}_{r}^{\mathbb{D}}u(z)=\frac{1}{2\pi}\int_{0}^{2\pi}u(z+re^{\mathfrak{i}\,t})\,dt$
is the (Euclidean) average of $u$ over the circle with radius $r$ centred at
$z$. In addition, we require that the set $\\{z:u(z)=-\infty\\}$ has Lebesgue
measure $0$. It is well known that $u\in C^{2}(\mathbb{D})$ is subharmonic if
and only if $\Delta_{\mathbb{D}}u\geq 0$, see [16, Thm. 4.8]. If $u$ is not
smooth, then $\Delta_{\mathbb{D}}u$ is defined in the sense of distributions.
Subharmonicity means that this is a non-negative Radon measure. The _Riesz
measure_ associated with $u$ is then
(2.2) $\mu^{u}=\frac{1}{2\pi}\Delta_{\mathbb{D}}u\,,\quad\text{that
is,}\quad\int_{\mathbb{D}}f\,d\mu^{u}=\frac{1}{2\pi}\int_{\mathbb{D}}u(z)\,\Delta_{\mathbb{D}}f(z)\,d\mathsf{m}_{\mathbb{D}}z$
for every $C^{\infty}$-function $f$ on $\mathbb{D}$ with compact support in
$\mathbb{D}$. If $u\in C^{2}(\mathbb{D})$ then the ordinary function
$\frac{1}{2\pi}\Delta_{\mathbb{D}}u$ is the density of $\mu^{u}$ with respect
to Lebesgue measure $\mathsf{m}_{\mathbb{D}}\,$. Furthermore, $h\in
C^{2}(\mathbb{D})$ is harmonic if and only if $\mathsf{A}_{r}h(z)=h(z)$ for
every $z\in\mathbb{D}$ and $r<1-|z|$.
The _Green function_ of $\Delta_{\mathbb{D}}$ is
(2.3) $G_{\mathbb{D}}(z,w)=\log\frac{|1-z\bar{w}|}{|z-w|}\,,\quad
z,w\in\mathbb{D}\,.$
For any signed measure $\mu$ on $\mathbb{D}$, the _potential_
$G_{\mathbb{D}}\mu$ is the function on $\mathbb{D}$ defined by
$G_{\mathbb{D}}\mu(z)=\int_{\mathbb{D}}G_{\mathbb{D}}(z,w)\,d\mu(w)\,,$
if the integral is finite for some ($\\!\\!\iff\\!$ every) $z\in\mathbb{D}$.
When $\mu$ is non-negative, $-G_{\mathbb{D}}\mu$ is is a subharmonic function.
If $u$ is subharmonic and, in addition, posseses some harmonic majorant on
$\mathbb{D}$, then it possesses its smallest harmonic majorant $h$. In this
case, the _Riesz decomposition_ of $u$ has the form
(2.4) $u=h-G_{\mathbb{D}}\mu^{u}\,.$
See e.g. Ransford [17, Thm. 4.5.4]. In absence of a harmonic majorant, for the
general Riesz decomposition theorem see [17, Thm. 3.7.9] or [16, Thm. 6.18].
We now consider the hyperbolic plane ${\mathbb{H}}$. Basic hyperbolic
potential theory appears rather to be “common knowledge” than being accessible
in a comprehensive treatise, with the exception of Stoll [19]. See also the
introductory chapter of Helgason [15]. We use the Poincaré disk model; see
e.g. Beardon [4, Chapter 7]. ${\mathbb{H}}$ coincides with $\mathbb{D}$ as a
set and topologically, but the hyperbolic length element and metric are
(2.5)
$d_{{\mathbb{H}}}s=\frac{2\sqrt{dx^{2}+dy^{2}}}{1-|z|^{2}}\quad\text{and}\quad\rho_{{\mathbb{H}}}(z,w)=\log\frac{|1-z\bar{w}|+|z-w|}{|1-z\bar{w}|-|z-w|}.$
The hyperbolic measure $\mathsf{m}_{{\mathbb{H}}}$ has area element
(2.6)
$d\mathsf{m}_{{\mathbb{H}}}(z)=d_{{\mathbb{H}}}z=\frac{4dz}{(1-|z|^{2})^{2}}=4\cosh^{4}\frac{\rho_{{\mathbb{H}}}(z,0)}{2}\,dz\,.$
This means conversely that we can express Lebesgue measure
$\mathsf{m}_{\mathbb{D}}$ on ${\mathbb{H}}$ as
(2.7)
$d\mathsf{m}_{\mathbb{D}}(z)=\frac{1}{4\cosh^{4}\bigl{(}\rho_{{\mathbb{H}}}(z,0)/2\bigr{)}}\,d\mathsf{m}_{{\mathbb{H}}}(z)\approx
e^{-2\rho_{{\mathbb{H}}}(z,0)}\,d\mathsf{m}_{{\mathbb{H}}}(z)\,,\quad\text{as}\;\rho_{{\mathbb{H}}}(z,0)\to\infty\,.$
The hyperbolic Laplace operator in the variable $z=x+\mathfrak{i}\,y$ is
(2.8) $\Delta_{{\mathbb{H}}}=\frac{(1-|z|)^{2}}{4}\Delta_{\mathbb{D}}\,.$
In particular, its harmonic functions are the same as the
$\Delta_{\mathbb{D}}$-harmonic functions. Above, we defined the Euclidean
average over a circle in $\mathbb{D}$. Now, we let $r>0$ and
$z\in{\mathbb{H}}$ and consider the hyperbolic circle
$C^{{\mathbb{H}}}(z,r)=\\{w\in{\mathbb{H}}:\rho_{{\mathbb{H}}}(z,w)=r\\}$.
This is also a Euclidean circle:
$\;C^{{\mathbb{H}}}(z,r)=C^{\mathbb{D}}(z^{\prime},r^{\prime})\,,$ where
$z^{\prime}=\frac{1-\tanh^{2}(r/2)}{1-|z|^{2}\tanh^{2}(r/2)}\,z\quad\text{and}\quad
r^{\prime}=\frac{1-|z|^{2}}{1-|z|^{2}\tanh^{2}(r/2)}\,\tanh(r/2)\,.$
Its hyperbolic length is $2\pi\sinh r$, see [4, page 132].
Now, a function $u:{\mathbb{H}}\to[-\infty,+\infty)$ is _subharmonic on
${\mathbb{H}}$_ if it is lower semicontinuous,
$\mathsf{m}_{{\mathbb{H}}}(\\{z:u(z)=-\infty\\})=0$, and for every
$z\in{\mathbb{H}}$ and $r>0$, one has $\mathsf{A}^{{\mathbb{H}}}_{r}u(z)\geq
u(z)$, where
$\mathsf{A}_{r}^{{\mathbb{H}}}u(z)=\frac{1}{2\pi\sinh
r}\int_{C^{{\mathbb{H}}}(z,r)}u\,\,d_{{\mathbb{H}}}s\,.$
###### (2.9) Lemma.
A function $u$ is hyperbolically superharmonic if and only if it is
superharmonic on $\mathbb{D}$ in the Euclidean sense.
The Green function of $\Delta_{{\mathbb{H}}}$ is the same as the one for
$\Delta_{\mathbb{D}}$ given in (2.3), and will henceforth also be denoted by
$G_{{\mathbb{H}}}(\cdot,\cdot)$. Using the hyperbolic metric,
(2.10)
$G_{{\mathbb{H}}}(z,w)=-\log\,\tanh\bigl{(}\rho_{{\mathbb{H}}}(z,w)/2\bigr{)}.$
Consequently, the hyberbolic Riesz decomposition and the Riesz measure of a
superharmonic function $u$ are the same as the Euclidean one.
The natural hyperbolic _compactification_ $\widehat{\mathbb{H}}$ of
${\mathbb{H}}$ arises from the identification of ${\mathbb{H}}$ with
$\mathbb{D}$ and taking the Euclidean closure. The _boundary at infinity_
$\partial{\mathbb{H}}$ of ${\mathbb{H}}$ is then the unit circle $\mathbb{S}$.
It is instructive to interpret this as follows: we first transform the metric
$\rho_{{\mathbb{H}}}$ of the hyperbolic plane into a new metric, namely the
Euclidean metric. For use in the subsection on trees, note that on the large
scale, the change of the metric is quantified by
(2.11) $\displaystyle\mathsf{d}_{\mathbb{D}}(z,\mathbb{S})$
$\displaystyle=1-|z|=\frac{2}{1+e^{\rho_{{\mathbb{H}}}(z,0)}}$
$\displaystyle\approx 2e^{-\rho_{{\mathbb{H}}}(z,0)}\quad\text{as}\;|z|\to
1\,,\;\text{or equivalently, as}\;\rho_{{\mathbb{H}}}(z,0)\to\infty\,.$
In order to get used to the two geometric views on the same object, we shall
freely switch back and forth: $\mathbb{D}\leftrightarrow{\mathbb{H}}$ and
$\mathbb{S}\leftrightarrow\partial{\mathbb{H}}$.
The _Poisson kernel_ on
${\mathbb{H}}\times\partial{\mathbb{H}}=\mathbb{D}\times\mathbb{S}$ is defined
for $z\in{\mathbb{H}}$, $\xi\in\mathbb{S}$ as
(2.12)
$P(z,\xi)=\frac{1-|z|^{2}}{|\xi-z|^{2}}=\lim_{w\to\xi}\frac{G_{{\mathbb{H}}}(z,w)}{G_{{\mathbb{H}}}(0,w)}=e^{-\mathfrak{h}_{{\mathbb{H}}}(z,\xi)}.$
with the _Busemann function_
(2.13)
$\mathfrak{h}_{{\mathbb{H}}}(z,\xi)=\lim_{w\to\xi}\Bigl{(}\rho_{{\mathbb{H}}}(w,z)-\rho_{{\mathbb{H}}}(w,0)\Bigr{)}.$
It also has a probabilistic interpretation: we start Euclidean Brownian motion
(BM) at $z\in\mathbb{D}$ and consider its hitting distribution $\nu_{z}$ on
the boundary $\mathbb{S}$. That is, if $B\subset\mathbb{S}$ is a Borel set,
then $\nu_{z}(B)$ is the probability that the first visit of BM to
$\mathbb{S}$ occurs in a point of $B$. Denoting by $\lambda_{\mathbb{S}}$ the
normalized Lebesgue arc measure on the unit circle, we have
(2.14)
$\frac{d\nu_{z}}{d\lambda_{\mathbb{S}}}(\xi)=P(z,\xi)\,,\quad\xi\in\mathbb{S}\,.$
Note that $\nu_{0}=\lambda_{\mathbb{S}}\,$.
###### (2.15) Theorem.
_(a)_ For every $\xi\in\mathbb{S}$, the function $z\mapsto P(z,\xi)$ is
harmonic on $\mathbb{D}\equiv{\mathbb{H}}$.
_(b)_ _[Poisson representation]_ For every positive harmonic function $h$ on
$\mathbb{D}\equiv{\mathbb{H}}$, there is a unique Borel measure $\nu^{h}$ on
$\mathbb{S}\equiv\partial{\mathbb{H}}$ such that
$h(z)=\int_{\mathbb{S}}P(z,\cdot)\,d\nu^{h}\,.$
_(c)_ For every continuous function $\varphi$ on
$\mathbb{S}\equiv\partial{\mathbb{H}}$,
$h(z)=\int_{\mathbb{S}}P(z,\cdot)\,\varphi\,d\lambda_{\mathbb{S}}$
is the unique harmonic function $h$ on $\mathbb{D}\equiv{\mathbb{H}}$ such
that
$\lim_{z\to\xi}h(z)=\varphi(\xi)\quad\text{for every}\;\xi\in\mathbb{S}\,.$
B. Homogeneous tree
We think of a graph as a set of vertices, equipped with a symmetric
neighbourhood relation $\sim$. An edge is a pair (usually considered un-
oriented) $e=[x,y]$ with $x\sim y$. Now, we consider the homogeneous tree
$\mathbb{T}=\mathbb{T}_{\mathsf{q}}\,$, where every vertex has
$\mathsf{q}+1\geq 3$ neighbours. The discrete Laplacian $\Delta_{\mathbb{T}}$
acts on functions $f:\mathbb{T}\to\mathbb{R}$ by
(2.16) $\Delta_{\mathbb{T}}f(x)=\frac{1}{\mathsf{q}+1}\sum_{y\sim
x}\bigl{(}f(y)-f(x)\bigr{)}\,.$
It is related with _simple random walk_ (SRW) on $\mathbb{T}$ in the same way
as the above Laplacians are the infinitesimal generators of Euclidean and
hyperbolic _Brownian motion_. (Hyperbolic BM is Euclidean BM slowed down close
to the boundary of the hyperbolic disk.) SRW is the Markov chain
$(Z_{n})_{n\geq 0}$ on $\mathbb{T}$ where $Z_{n}$ is the random position at
discrete time $n$ of the particle, which moves from the current vertex $x$ to
any of its neighbours $y$ with equal probability $p(x,y)=1/(\mathsf{q}+1)$,
while $p(x,y)=0$ if $x\not\sim y$. This gives rise to the transition operator
$Pf(x)=\sum_{y}p(x,y)f(y)$, and $\Delta_{\mathbb{T}}=P-I$, where $I$ is the
identity operator. For potential theory on trees, see e.g. Woess [20].
For any pair of vertices $x,y$, there is a _geodesic path_ $\pi(x,y)$ from $x$
to $y$ without repetitions. The number of edges of that path is the graph
distance
(2.17) $\mathsf{d}_{\mathbb{T}}(x,y)\,.$
The standard measure on (the vertex set of) $\mathbb{T}$ is the counting
measure $\mathsf{m}_{\mathbb{T}}(A)=|A|$ ($A\subset\mathbb{T}$). In comparing
$\mathbb{T}$ with ${\mathbb{H}}$, $\mathsf{d}_{\mathbb{T}}$ and
$\mathsf{m}_{\mathbb{T}}$ correspond to the hyperbolic distance and area
element $\rho_{{\mathbb{H}}}$ and $\mathsf{m}_{{\mathbb{H}}}$ on
${\mathbb{H}}$, respectively.
Functions $h,u:\mathbb{T}\to\mathbb{R}$ are _harmonic_ , resp. _subharmonic_ ,
if $\Delta_{\mathbb{T}}h=0$, resp. $\Delta_{\mathbb{T}}u\geq 0$. Equivalently,
$Ph=h$, resp. $Pu\geq u$, a definition in terms of the arithmetic averages
over spheres with radius $1$. Here, it makes no sense to allow for value
$-\infty$, since sets of $\mathsf{m}_{\mathbb{T}}$-measure $0$ are empty.
The Green function of $\Delta_{\mathbb{T}}$ is
(2.18)
$G_{\mathbb{T}}(x,y)=\frac{\mathsf{q}}{\mathsf{q}-1}\,\mathsf{q}^{-\mathsf{d}_{\mathbb{T}}(x,y)}\,,\quad
x,y\in\mathbb{T}\,.$
The potential of a function $f:\mathbb{T}\to\mathbb{R}$ is
$G_{\mathbb{T}}f(x)=\sum_{y\in\mathbb{T}}G_{\mathbb{T}}(x,y)\,f(y)\,.$
Since $\mathbb{T}$ is countable, measures on $\mathbb{T}$ are defined by their
atoms, that is, they can be identified with non-negative functions. Thus, the
Riesz measure of a subharmonic function $u$ can be identified with the
function
(2.19) $\mu^{u}=\Delta_{\mathbb{T}}u=Pu-u\,.$
More precisely, the function $Pu-u$ should be understood as the density of the
Riesz measure $\mu^{u}$ with respect to the counting measure
$\mathsf{m}_{\mathbb{T}}$. If $u$ has a harmonic majorant, then its Riesz
decomposition reads
$u=h-G_{\mathbb{T}}\mu^{u}\,,\quad\text{where}\quad
h(x)=\lim_{n\to\infty}P^{n}u(x)$
is the smallest harmonic majorant of $u$. Again, if there is no harmonic
majorant, then the statement of the Riesz decomposition theorem is a little
bit more involved. In the Markov chain (= Discrete Potential Theory)
literature, the only source for the latter seems to be [21], but it will not
be needed here.
We next describe the _end compactification_ of $\mathbb{T}$. A _ray_ or
_geodesic ray_ is a one-sided infinite sequence
$\pi=[x_{0},x_{1},x_{2},\dots]$ of vertices such that $x_{n}\sim x_{n-1}$ and
$x_{n}\neq x_{m}$ for all $n,m$, $n\neq m$. Two rays are called _equivalent_ ,
if they differ only by finite initial pieces. An end of $\mathbb{T}$ is an
equivalence class of rays. The set of all ends is the boundary
$\partial\mathbb{T}$. We set
$\widehat{\mathbb{T}}=\mathbb{T}\cup\partial\mathbb{T}$. For every vertex
$x\in\mathbb{T}$ and every $\xi\in\partial\mathbb{T}$, there is precisely one
geodesic ray $\pi(x,\xi)$ starting at $x$ that represents $\xi$. Analogously,
for any two disctinct ends $\xi,\eta$, there is a unique two-sided _geodesic_
$\pi(\xi,\eta)=[\dots,-x_{2},-x_{1},x_{0},x_{1},x_{2},\dots]$ such that
$[x_{k},x_{k-1},x_{k-2},\dots]$ and $[x_{k},x_{k+1},x_{k+2},\dots]$ are rays
representing $\xi$ and $\eta$, respectively.
We now pursue the line followed above by an exponential change the metric of
${\mathbb{H}}$, see (2.11). A natural choice is as follows. We fix a root
vertex $o\in\mathbb{T}$. For $z\in\widehat{\mathbb{T}}$, we denote
$|z|=\mathsf{d}_{\mathbb{T}}(o,z)$, with value $\infty$ if
$z\in\partial\mathbb{T}$. For $w,z\in\widehat{\mathbb{T}}$ , we define their
_confluent_ $w\wedge z$ with respect to $o$ as the last common element on the
geodesics $\pi(o,w)$ and $\pi(o,z)$. This is a vertex, unless
$z=w\in\partial\mathbb{T}$. We let
(2.20) $\rho_{\mathbb{T}}(w,z)=\begin{cases}q^{-|w\wedge
z|}\,,&\text{if}\;z\neq w\,,\\\ 0\,,&\text{if}\;z=w\,.\end{cases}$
This is an ultra-metric. In the induced topology, $\widehat{\mathbb{T}}$ is
compact, and $\mathbb{T}$ is discrete and dense. Convergence in this topology
is as follows: if $\xi\in\partial\mathbb{T}$ then a sequence $(z_{n})$ in
$\widehat{\mathbb{T}}$ converges to $\xi$ if and only if $|\xi\wedge
z_{n}|\to\infty\,$.
At this point, we underline that in the “translation” from disk to tree, the
graph metric $\mathsf{d}_{\mathbb{T}}$ corresponds to the hyperbolic metric
$\rho_{{\mathbb{H}}}\,$, while the metric $\rho_{\mathbb{T}}$ is the one that
may be interpreted to correspond to the Euclidean metric
$\mathsf{d}_{\mathbb{D}}\,$. The next identity should be compared with (2.11).
(2.21)
$\rho_{\mathbb{T}}(x,\partial\mathbb{T})=\mathsf{q}^{-|x|}\quad\text{for}\;x\in\mathbb{T}\,.$
The _Martin kernel_ on $\mathbb{T}\times\partial\mathbb{T}$ is defined for
$x\in\mathbb{T}$, $\xi\in\partial\mathbb{T}$ as
(2.22)
$K(x,\xi)=\lim_{y\to\xi}\frac{G_{\mathbb{T}}(x,y)}{G_{\mathbb{T}}(o,y)}=\mathsf{q}^{-\mathfrak{h}_{\mathbb{T}}(x,\xi)}$
with the _Busemann function_
(2.23)
$\mathfrak{h}_{\mathbb{T}}(x,\xi)=\lim_{y\to\xi}\Bigl{(}\mathsf{d}_{\mathbb{T}}(y,x)-\mathsf{d}_{\mathbb{T}}(y,o)\Bigr{)}=\mathsf{d}_{\mathbb{T}}(x\wedge\xi,x)-\mathsf{d}_{\mathbb{T}}(x\wedge\xi,o).$
Again, we have a probabilistic interpretation. It is a well-known exercise to
show that SRW on $\mathbb{T}$ converges alsmost surely in the topology of
$\widehat{\mathbb{T}}$ to a limit random variable $Z_{\infty}$ that takes its
values in $\partial\mathbb{T}$. Let $\nu_{x}$ be the distribution of
$Z_{\infty}\,$, when SRW starts at vertex $x$. Then
$\nu_{o}=\lambda_{\partial\mathbb{T}}$ is the tree-analogue of the normalized
Lebesgue measure $\lambda_{\mathbb{S}}$ on the unit circle:
$\lambda_{\partial\mathbb{T}}$ is the unique probability measure on
$\partial\mathbb{T}$ which is invariant under “rotations” of $\mathbb{T}$,
that is, self-isometries of the graph $\mathbb{T}$ which fix the root vertex
$o$. Connectedness of $\mathbb{T}$ implies that $\nu_{x}$ is absolutely
continuous with respect to $\lambda_{\partial\mathbb{T}}\,$, and the Radon-
Nikodym-derivative is (realised by) the Martin kernel:
(2.24) $\frac{d\nu_{x}}{d\lambda_{\partial\mathbb{T}}}(\xi)=K(x,\xi)\,.$
We have a perfect analogy with Theorem 2.15.
###### (2.25) Theorem.
_(a)_ For every $\xi\in\partial\mathbb{T}$, the function $x\mapsto K(x,\xi)$
is harmonic on $\mathbb{T}$.
_(b)_ For every positive harmonic function $h$ on $\mathbb{T}$, there is a
unique Borel measure $\nu^{h}$ on $\partial\mathbb{T}$ such that
$h(x)=\int_{\partial\mathbb{T}}K(x,\cdot)\,d\nu^{h}\,.$
_(c) [Solution of the Dirichlet problem]_ For every continuous function
$\varphi$ on $\partial\mathbb{T}$,
$h(x)=\int_{\partial\mathbb{T}}\varphi\,d\nu_{x}=\int_{\partial\mathbb{T}}K(x,\cdot)\,\varphi\,d\lambda_{\partial\mathbb{T}}\,.$
is the unique harmonic function $h$ on $\mathbb{T}$ such that
$\lim_{x\to\xi}h(x)=\varphi(\xi)\quad\text{for
every}\;\xi\in\partial\mathbb{T}\,.$
C. A table of correspondences
As a general “rule” of translation from ${\mathbb{H}}$ to $\mathbb{T}$, we
note that base $e$ (Eulerian number) has to be replaced by base $\mathsf{q}$
(branching number of the tree).
hyperbolic plane ${\mathbb{H}}$ | ref. number | homogeneous tree $\mathbb{T}$ | ref. number
---|---|---|---
| | |
hyperbolic metric $\rho_{{\mathbb{H}}}$ | (2.5) | graph metric $\mathsf{d}_{\mathbb{T}}$ | (2.17)
Euclidean metric $\mathsf{d}_{\mathbb{D}}$ | (2.1) | length metric $\rho_{\mathbb{T}}$ | (2.20)
boundary $\mathbb{S}$ (unit circle) | | boundary $\partial\mathbb{T}$ |
${\displaystyle\text{compactification}\atop\displaystyle\widehat{\mathbb{H}}=\mathbb{D}^{-}=\mathbb{D}\cup\mathbb{S}}$ | | ${\displaystyle\text{compactification}\atop\displaystyle\widehat{\mathbb{T}}=\mathbb{T}\cup\partial\mathbb{T}}$ |
hyperbolic measure $\mathsf{m}_{{\mathbb{H}}}$ | (2.6) | counting measure $\mathsf{m}_{\mathbb{T}}$ |
Lebesgue measure $\mathsf{m}_{\mathbb{D}}$ | (2.7) | $\mathsf{q}^{-2|x|}\,d\mathsf{m}_{\mathbb{T}}(x)$ | (2.7)
${\displaystyle\text{normalised arc measure}}\atop{\displaystyle\lambda_{\mathbb{S}}\text{ on }\mathbb{S}}$ | | ${\displaystyle\text{rotation invariant measure}}\atop{\displaystyle\lambda_{\partial\mathbb{T}}\text{ on }\partial\mathbb{T}}$ |
hyperbolic Laplacian $\Delta_{{\mathbb{H}}}$ | (2.8) | discrete Laplacian $\Delta_{\mathbb{T}}$ | (2.16)
Green function $G_{{\mathbb{H}}}=G_{\mathbb{D}}$ | (2.3) | Green function $G_{\mathbb{T}}$ | (2.18)
Poisson kernel | (2.12) | Martin kernel | (2.22)
There are many further analogies between analysis, probability, group actions,
etc. on $\mathbb{D}$ and $\mathbb{T}$. The present introduction is not
intended to cover all those aspects. For further tips of the iceberg, see e.g.
Casadio Tarabusi, Cohen, Korányi and Picardello [7], Rigoli, Salvatori and
Vignati [18], Cohen, Colonna and Singman [9], Atanasi and Picardello [3] or
Casadio Tarabusi and Figà-Talamanca [8], and the references given there.
## 3\. Moment conditions and harmonic majorants
Let $\mathbb{X}=\mathbb{D}$ or $\mathbb{X}=\mathbb{T}$, with respective
boundary $\partial\mathbb{X}$ and compactification $\widehat{\mathbb{X}}$ (see
the above table). The boundary carries the metric $\mathsf{dist}$ and measure
$\lambda$, where $\mathsf{dist}=\mathsf{d}_{\mathbb{D}}$ and
$\lambda=\lambda_{\mathbb{S}}$ in case of the disk, while
$\mathsf{dist}=\rho_{\mathbb{T}}$ and $\lambda=\lambda_{\mathbb{T}}$ in case
of the tree. Given a subharmonic function $u$ on $\mathbb{X}$ and its Riesz
measure $\mu^{u}$, we are interested in finiteness of its _first (boundary)
moment_
(3.1) $\int_{\mathbb{X}}\mathsf{dist}(x,\partial\mathbb{X})\,d\mu^{u}(x)$
and variants thereof. One principal tool is the following lemma.
###### (3.2) Lemma.
The subharmonic function $u$ has a harmonic majorant on $\mathbb{X}$ if and
only if $\mu^{u}$ has finite first moment (3.1).
###### Proof.
Our function $u$ has a harmonic majorant if and only if
$G_{\mathbb{X}}\mu^{u}(x)<\infty$ for all $x\in\mathbb{X}$.
If $\mathbb{X}=\mathbb{D}$ and there is a harmonic majorant then we choose
$x=0$ and get
$\infty>G_{\mathbb{D}}\mu^{u}(0)=-\int_{\mathbb{D}}\log|z|\,d\mu^{u}(z)\geq\int_{\mathbb{D}}(1-|z|)\,d\mu^{u}(z).$
Conversely, if the first moment is finite, then $G_{\mathbb{D}}\mu^{u}$ is
finite on $\mathbb{D}$ by Armitage and Gardiner [2, Thm. 4.2.5].
If $\mathbb{X}=\mathbb{T}$ then by (2.18), $G_{\mathbb{T}}(x,y)\leq
q^{|x|}G_{\mathbb{T}}(o,y)$ for all $x,y$, so that $G_{\mathbb{T}}\mu^{u}$ is
finite on $\mathbb{T}$ if and only if $G_{\mathbb{T}}\mu^{u}(o)<\infty$. Now
∎
$G_{\mathbb{T}}\mu^{u}(o)=\sum_{x\in\mathbb{T}}\frac{q}{q-1}q^{-|x|}\mu^{u}(x)=\frac{q}{q-1}\int_{\mathbb{T}}\rho_{\mathbb{T}}(x,\partial\mathbb{T})\,d\mu^{u}(x)\,.$
So in fact what we are going to do is to exhibit a sufficient condition for a
subharmonic function on $\mathbb{X}=\mathbb{D}$, resp.
$\mathbb{X}=\mathbb{T}$, to possess a (global or restricted) harmonic
majorant, even if it is not bounded above.
###### (3.3) Theorem.
Let $u$ be a subharmonic function on $\mathbb{X}$ and consider the closed set
$E=\Bigl{\\{}\xi\in\partial\mathbb{X}:\limsup_{\mathbb{X}\ni
x\to\xi}u(x)=\infty\Bigr{\\}}.$
Suppose that $\Psi:[0\,,\,\mathsf{diam}(\mathbb{X})]\to[0\,,\,\infty]$ is a
continuous, decreasing function with
$\Psi(t)=\infty\iff t=0\quad\text{and}\quad\lim_{t\to 0}\Psi(t)=\infty\,,$
and that
$u(x)\leq\Psi\bigl{(}\mathsf{dist}(x,E)\bigr{)}\quad\text{for
all}\;x\in\mathbb{X}\,.$
If
(3.4)
$\int_{\partial\mathbb{X}}\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)<\infty\,,$
then $u$ has a finite harmonic majorant, and the Riesz measure $\mu^{u}$ has
finite first boundary moment.
We note that for condition (3.4) it is necessary that $\lambda(E)=0$. For the
proof of the theorem, we shall work with the function
(3.5)
$h=\int_{\partial\mathbb{X}}K_{\mathbb{X}}(\cdot,\xi)\,\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)\,,$
where $K_{\mathbb{X}}$ is the Poisson kernel (2.12) when
$\mathbb{X}=\mathbb{D}$, and the Martin kernel (2.22) when
$\mathbb{X}=\mathbb{T}$. Since for fixed $x\in\mathbb{X}$, the function
$\xi\mapsto K_{\mathbb{X}}(x,\xi)$ is continuous on $\partial\mathbb{X}$
(whence bounded), the function $h$ is finite and harmonic on $\mathbb{X}$
under condition (3.4).
We need some preparations. We let
$0<t\leq\max\\{\mathsf{dist}(x,E):x\in\mathbb{X}$ and consider the sets
$E^{(t)}=\\{\xi\in\partial\mathbb{X}:\mathsf{dist}(\xi,E)\leq
t\\}\quad\text{and}\quad
E^{(t)}_{*}=\\{\xi\in\partial\mathbb{X}:\mathsf{dist}(\xi,E)>t\\}\,,$
and, for $0<t<1$, the set $\mathbb{X}^{(t)}$ which is the component of the
origin of the set $\\{x\in\mathbb{X}:\mathsf{dist}(x,E)>t\\}\,.$
Disk case: $\mathbb{D}^{(t)}$ (denoted $\Omega_{t}$ in [10]) is an open
domain, and its boundary is
$\partial\mathbb{D}^{(t)}=\partial_{\infty}\mathbb{D}^{(t)}\,\cup\,\Gamma^{(t)},\quad\text{where
}\;\partial_{\infty}\mathbb{D}^{(t)}\subset\overline{E^{(t)}_{*}}\;\text{ and
}\;\Gamma^{(t)}\\!=\Gamma^{(t)}_{\mathbb{D}}=\\{z\in\mathbb{D}:\mathsf{d}_{\mathbb{D}}(z,E)=t\\}.$
The sets $E^{(t)}$ and $\partial_{\infty}\mathbb{D}^{(t)}$ are both unions of
finitely many closed arcs on $\mathbb{S}$ and meet at finitely many endpoints
of those arcs. $\partial_{\infty}\mathbb{D}^{(t)}$ may be a strict subset of
the closure of $E^{(t)}_{*}\,$, because some arcs of the latter set can be the
boundary of a different component of
$\\{z\in\mathbb{D}:\mathsf{d}_{\mathbb{D}}(z,E)>t\\}$. (The latter can arise
as “triangular” regions bounded by an arc of $\mathbb{S}$ and of arcs of two
intersecting circles $\\{z:|z-\zeta_{j}|=t\\}$, where $\zeta_{j}\in E$,
$j=1,2$.)
Tree case: The origin is of course the root vertex of $\mathbb{T}$. The metric
$\mathsf{dist}=\rho_{\mathbb{T}}$ takes only the countably many values
$\mathsf{q}^{-k}$, $k\geq 0$ (integer). For $0<t<1$ let $k\geq 1$ be the
integer such that
(3.6) $\mathsf{q}^{-k}\leq t<\mathsf{q}^{-(k-1)}\,,\quad k=k(t).$
For any vertex $y\in\mathbb{T}$, we consider the _branch_ of $\mathbb{T}$ at
$y$. This is the subtree (induced by)
$\mathrm{T}_{y}=\\{u\in\mathbb{T}:y\in\pi(o,u)\\}.$
Its boundary $\partial\mathrm{T}_{y}\subset\partial\mathbb{T}$ consists of
those ends which are represented by geodesics that lie entirely within
$\mathrm{T}_{y}\,$. Note that the open-compact sets
$\partial\mathrm{T}_{y}\,$, $y\in\mathbb{T}$, are a basis of the topology of
$\partial\mathbb{T}$. Given $t$, let $k=k(t)$ and consider the set
$\Gamma^{(t)}=\Gamma^{(t)}_{T}=\\{y\in\mathbb{T}:|y|=k\,,\;\partial\mathrm{T}_{y}\cap
E\neq\emptyset\\}\,.$
We have
$E^{(t)}=E^{(t)}_{\mathbb{T}}=\bigcup_{y\in\Gamma^{(t)}}\partial\mathrm{T}_{y}\,.$
For small $t$ $\equiv$ large $k=k(t)\,$, only few vertices $y$ with $|y|=k$
belong to $\Gamma^{(t)}$: as $t\to 0$ $\equiv$ $k\to\infty$, we have
$\frac{|\Gamma^{(t)}|}{|\\{y\in\mathbb{T}:|y|=k(t)\\}|}=\lambda_{\partial\mathbb{T}}\bigl{(}E^{(t)}\bigr{)}\to\lambda_{\partial\mathbb{T}}(E)=0.$
When $\mathbb{X}=\mathbb{T}$, the set $\mathbb{T}^{(t)}$ is the subtree of
$\mathbb{T}$ obtained by chopping off each branch $\mathrm{T}_{y}\,$,
$y\in\Gamma^{(t)}$, that is,
$\mathbb{T}^{(t)}=\mathbb{T}\setminus\bigcup\nolimits_{y\in\Gamma^{(t)}}\mathrm{T}_{y}\,.$
The _boundary_ of this truncated tree is
$\partial\mathbb{T}^{(t)}=\partial_{\infty}\mathbb{T}^{(t)}\cup\Gamma^{(t)}_{\mathbb{T}}\,,\quad\text{where}\quad\partial_{\infty}\mathbb{T}^{(t)}=E^{(t)}_{*}\,,$
while $\Gamma^{(t)}$ is the outer vertex boundary of $\mathbb{T}^{(t)}$: it
consists of those vertices in the complement that have a neighbour (here:
precisely one neighbour) in $\mathbb{T}^{(t)}$. In the topology of
$\widehat{\mathbb{T}}$, we have the compact subspaces
$\widehat{\mathbb{T}}^{(t)}=\mathbb{T}^{(t)}\cup\partial\mathbb{T}^{(t)}$ and
the boundary $\partial\mathbb{T}^{(t)}$.
We shall need the following simple estimate.
###### (3.7) Lemma.
For $x\in\mathbb{X}$, consider the harmonic measure $\nu_{x}$ on
$\partial\mathbb{X}\,$; see (2.14), resp. (2.24). Then
$\text{for }\;y\in\Gamma^{(t)}\,,\quad\nu_{y}(E^{(t)})\geq
1/c_{\mathbb{X}}\;=\begin{cases}1/3\,,&\text{if}\;\
\mathbb{X}=\mathbb{D}\,,\\\ \mathsf{q}/(\mathsf{q}+1)\,,&\text{if}\;\
\mathbb{X}=\mathbb{T}\,.\end{cases}$
###### Proof.
A. Disk case. For $y\in\Gamma^{(t)}_{\mathbb{D}}$ there is $\zeta=\zeta_{y}\in
E$ such that $|y-\zeta_{y}|=\mathsf{d}(y,E)=t$. Consider the arc
$\gamma_{\zeta}=\\{\xi\in\mathbb{S}:|\xi-\zeta|\leq t\\}\subset E^{(t)}$, as
well as the circle $\\{z\in\mathbb{C}:|z-\zeta|=t\\}$. At any of the two
intersection points of that circle with $\mathbb{S}$, the angle $\alpha$
between the tangents to the two circles is such that $\pi/2>\alpha>\pi/3$, as
$0<t<1$. By [13, p. 13, Fig.1.1], $\nu_{y}(\gamma_{\zeta})=\alpha/\pi>1/3$.
(In [10], the lower estimate $1/6$ is used, but apparently also $1/3$ works.)
B. Tree case. For $y\in\Gamma^{(t)}_{\mathbb{T}}$, we have that
$\partial\mathrm{T}_{y}\subset E^{(t)}$. We note that $\nu_{y}$ gives equal
mass to the boundaries of each of the $\mathsf{q}+1$ branches of $\mathbb{T}$
that are emanating from $y$. Among those, $\mathsf{q}$ branches are part of
$\mathrm{T}_{y}\,$, that is,
$\nu_{y}(\partial\mathrm{T}_{y})=\mathsf{q}/(\mathsf{q}+1)$, providing the
lower bound. ∎
###### Proof of Theorem 3.3.
Consider the continuous function
$\psi^{(t)}(\xi)=\min\bigl{\\{}\Psi(t),\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\bigr{\\}}$
on $\partial\mathbb{X}$ and the harmonic function
$h^{(t)}(x)=\int_{\partial\mathbb{X}}K(x,\cdot)\,\psi^{(t)}\,d\lambda=\int_{\partial\mathbb{X}}\psi^{(t)}\,d\nu_{x}\,.$
We know from theorems 2.15, resp. 2.25 that it is the solution of the
Dirichlet problem on $\mathbb{X}$ with boundary function $\psi^{(t)}$. We have
$\psi^{(t)}(\xi)=\Psi(t)$ on $E^{(t)}$, while $\psi^{(t)}(\xi)\leq\Psi(t)$ on
$E^{(t)}_{*}\supset\partial_{\infty}\mathbb{X}^{(t)}$. Thus,
(3.8)
$h^{(t)}(x)=\int_{E^{(t)}_{*}}\Psi\bigl{(}\mathsf{dist}(\cdot,E)\bigr{)}\,d\nu_{x}+\Psi(t)\,\nu_{x}(E^{(t)})\,.$
Taking boundary limits for points $x$ within $\widehat{\mathbb{X}}^{(t)}$, and
using Lemma 3.7,
(3.9) $\displaystyle\lim_{x\to\xi}h^{(t)}(x)$
$\displaystyle=\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,,\quad\text{for
}\;\xi\in\partial_{\infty}\mathbb{X}^{(t)}\,,\quad\text{and}\quad$
$\displaystyle\lim_{x\to y}h^{(t)}(x)$
$\displaystyle=h^{(t)}(y)\geq\Psi(t)\,\nu_{y}(E^{(t)})\geq\Psi(t)/c_{\mathbb{X}}\quad\text{for
}\;y\in\Gamma^{(t)}.$
(In the tree case, since $y$ is an isolated point, the last limit just means
stabilisation at $y$.) On the other hand, by assumption our subharmonic
function $u$ satisfies
(3.10) $\displaystyle\limsup_{x\to\xi}u(x)$
$\displaystyle\leq\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\quad\text{for}\;\xi\in\partial_{\infty}\mathbb{X}^{(t)}\,,\quad\text{and}\quad$
$\displaystyle\limsup_{x\to y}u(y)$
$\displaystyle\leq\Psi(t)\quad\text{for}\;y\in\Gamma^{(t)}\,.$
Therefore, again taking boundary limits within $\widehat{\mathbb{X}}^{(t)}$,
$\limsup_{x\to\eta}\Bigl{(}u(x)-c_{\mathbb{X}}\,h^{(t)}(x)\Bigr{)}\leq
0\quad\text{for every}\;\eta\in\partial\mathbb{X}^{(k)}.$
Thus, by the maximum principle (which also holds on the tree because
$\mathbb{T}^{(t)}$ is a connected graph, a simple excercise),
(3.11) $u(x)\leq c_{\mathbb{X}}\,h^{(t)}(x)\quad\text{for
every}\;x\in\mathbb{X}^{(t)}.$
Having this, we obtain the proposed first moment: let
$h(x)=\int_{\partial\mathbb{X}}K(x,\cdot)\,\Psi\bigl{(}\mathsf{dist}(\cdot,E)\bigr{)}\,d\lambda$
be the harmonic function proposed in (3.5). Then $h^{(t)}\leq h$ on
$\mathbb{X}^{(t)}$ for any $t$. Given any $x\in\mathbb{X}$, we can choose
$t<\mathsf{dist}(x,E)$ to see that $c_{\mathbb{X}}\cdot h$ is a (finite)
harmonic majorant for our subharmonic function $u$. ∎
There is a simple converse to Theorem 3.3.
###### (3.12) Proposition.
Let $u$ be a subharmonic function on $\mathbb{X}$, and let $E$ and $\Psi$ be
as in Theorem 3.3. If
$u(x)\geq\Psi\bigl{(}\mathsf{dist}(x,E)\bigr{)}\quad\text{for
all}\;x\in\mathbb{X}$
and
(3.13)
$\int_{\partial\mathbb{X}}\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)=\infty$
then $u$ has no harmonic majorant on $\mathbb{X}$, and the first moment of
$\mu^{u}$ is infinite.
###### Proof.
We give a combined proof for $\mathbb{X}=\mathbb{D}$ and
$\mathbb{X}=\mathbb{T}$. Suppose that the first moment of $\mu^{u}$ is finite.
Then by Lemma 3.2, $u$ has a (finite) harmonic majorant $h$. Consider the
continuous function $\Psi_{M}=\min\\{\Psi,M\\}$. Then for all
$x\in\mathbb{X}$,
$h(x)\geq u(x)\geq\Psi_{M}\bigl{(}\mathsf{dist}(x,E)\bigr{)}$
The function
$g_{M}(x)=\int_{\partial\mathbb{X}}K_{\mathbb{X}}(\cdot,\xi)\,\psi_{M}\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)\,,$
defined analogously to (3.5), provides the solution of the Dirichlet problem
on $\mathbb{X}$ with boundary data
$\psi_{M}\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}$. We have
$\liminf_{x\to\xi}\bigl{(}h(x)-g_{M}(x)\bigr{)}\geq 0\quad\text{for every
}\;\xi\in\partial\mathbb{X}\,.$
By the minimum principle, $h\geq g_{M}$ on $\mathbb{X}$, and in particular,
$h(o)\geq g_{M}(o)$. Letting $M\to\infty$, monotone convergence yields
$h(o)=\infty$, contradicting finiteness of $h$. ∎
Next, in a similar spirit to [10], we want to extend Theorem 3.3 to a
situation where the integral in (3.4) is infinite. For that purpose, we shall
need an estimate of the Green function
$G_{\mathbb{X}^{(t)}}(x,y)=G_{\mathbb{X}^{(t)}}(y,x)$ of $\mathbb{X}^{(t)}$.
On the disk, this function is of course well described in the classical
potential theory literature.
On the tree, for $x,y\in\mathbb{T}^{(k)}$, it is the expected number of visits
to $y$ of the random walk starting at $x$ before it hits $\Gamma^{(t)}$. It is
natural to define $G_{\mathbb{T}^{(t)}}(x,y)=0$ when one of $x,y$ lies in
$\Gamma^{(t)}$ and the other in $\mathbb{T}^{(t)}$. In potential theoretic
terms, $f=G_{\mathbb{T}^{(t)}}(\cdot,y)$ is the smallest non-negative function
on $\mathbb{T}^{(t)}\cup\Gamma^{(t)}$ satisfying
$\Delta_{\mathbb{T}}f(x)=-\delta_{y}(x)$ for $x\in\mathbb{T}^{(k)}$. This
corresponds directly to the disk situation.
###### (3.14) Theorem.
Define $r=r_{\mathbb{X}}\,$, $a=a_{\mathbb{X}}$ and $b=b_{\mathbb{X}}$ and for
$\mathbb{X}=\mathbb{D}$ or $=\mathbb{T}$ by
$r_{\mathbb{D}}=7\quad\text{and}\quad
a_{\mathbb{D}}=b_{\mathbb{D}}=18\,,\quad\text{resp.}\quad
r_{\mathbb{T}}=1\,,\;a_{\mathbb{T}}=\mathsf{q}/(\mathsf{q}-1)\quad\text{and}\quad
b_{\mathbb{T}}=1\,.$
Let $0<t<1/r$. Then for any $x\in\mathbb{X}^{(rt)}$, we have
$G_{\mathbb{X}}(x,o)\geq
G_{\mathbb{X}^{(t)}}(x,o)\geq\frac{1}{a}\,G_{\mathbb{X}}(x,o)\geq\frac{1}{b}\,\mathsf{dist}(x,\partial
X)\,,$
where $o$ is the origin (root) of $\mathbb{X}$.
###### Proof.
The first inequality is clear in both cases. The third inequality is also
clear, and it is an equality in the tree case. We need to prove the second
inequality separately for tree and disk, and begin this time with the tree.
A. Tree case. Let $\nu_{x}^{(t)}$ be the harmonic measure of
$\mathbb{T}^{(t)}$ on its boundary. In particular, for $y\in\Gamma^{(t)}$, the
probability that the random walk starting at $x$ first hits $\Gamma^{(t)}$ in
$y$ is $\nu_{x}^{(t)}(y)$. The function
$g^{(t)}(x)=G_{\mathbb{T}}(x,o)-G_{\mathbb{T}^{(k)}}(x,o)$ is positive
harmonic on $\mathbb{T}^{(t)}$. We have $\lim_{x\to\xi}g^{(t)}(x)=0$ for
$\xi\in\partial_{\infty}\mathbb{T}^{(t)}$ (because this holds for
$G_{\mathbb{T}}(x,o)$), while $g^{(t)}(y)=G_{\mathbb{T}}(y,o)$ for
$y\in\Gamma^{(t)}$. Since the Dirichlet problem on
$\widehat{\mathbb{T}}^{(t)}$ admits solution (a straightforward adaptation of
[6, Thm.4], including in that argument vertices which are boundary points), we
get that
$g^{(t)}(x)=\sum_{y\in\Gamma^{(t)}}G_{\mathbb{T}}(y,o)\nu_{x}^{(t)}(y)=\frac{\mathsf{q}}{\mathsf{q}-1}\mathsf{q}^{-k}\,\nu_{x}^{(t)}(\Gamma^{(t)})\,,$
where $k=k(t)$, as defined in (3.6). In the last identity (which can of course
also be derived probabilistically), (2.18) was used. Now let
$x\in\mathbb{T}^{(t)}$ and let $x_{0}$ be the last point on the geodesic
$\pi(o,x)$ that lies on some $\pi(o,y)$ with $y\in\Gamma^{(t)}$. Note that
$|x_{0}|\leq k-1$. In order to reach $\Gamma^{(t)}$, the random walk starting
at $x$ needs to pass through $x_{0}$. Unless $x=x_{0}$, this is unrestricted
random walk on $\mathbb{T}$ before the first visit in $x_{0}$, because up to
that time it evolves on a branch of $\mathbb{T}$ that contains no element of
$\Gamma^{(t)}$. It is well known and easy to see that
$\mathsf{Pr}[\exists n:Z_{n}=x_{0}\mid Z_{0}=x]=G(x,x_{0})/G(x_{0},x_{0}),$
see e.g. [20, Thm.1.38]. Thus (compare with [20, Prop.9.23]),
$\nu_{x}^{(t)}(\Gamma^{(t)})=\mathsf{Pr}[\exists n:Z_{n}=x_{0}\mid
Z_{0}=x]\underbrace{\nu_{x_{0}}^{(t)}(\Gamma^{(t)})}_{\displaystyle\leq
1}\leq\mathsf{q}^{-\mathsf{d}_{\mathbb{T}}(x,x_{0})}=\mathsf{q}^{|x_{0}|-|x|}\leq\mathsf{q}^{k-1-|x|}\,.$
We infer that
$g_{k}(x)\leq\frac{\mathsf{q}}{\mathsf{q}-1}\mathsf{q}^{-k}\mathsf{q}^{k-1-|x|}=\frac{1}{\mathsf{q}-1}\mathsf{q}^{-|x|}$
Consequently,
$G_{\mathbb{T}^{(k)}}(x,o)=G_{\mathbb{T}}(x,o)-g_{k}(x)=\frac{\mathsf{q}}{\mathsf{q}-1}\mathsf{q}^{-|x|}-g_{k}(x)\geq\mathsf{q}^{-|x|}\,,$
and in view of (2.18), the proposed estimate is proved for the tree.
B. Disk case. The proof follows [10], but we re-elaborate it to get the
constant $a_{\mathbb{D}}=7$ and to have $G_{\mathbb{D}}(z,0)$ in the lower
bound. As before, we prefer to write $z$ instead of $x$ for the elements of
$\mathbb{D}$. We start in the same way as for the tree. We know that
$G_{\mathbb{D}}(z,0)=\log\frac{1}{|z|}\,$, and we can decompose
$G_{\mathbb{D}^{(t)}}(z,0)=G_{\mathbb{D}}(z,0)-g^{(t)}(z)\,,\quad
z\in\mathbb{D}^{(t)}\,,$
where $g^{(t)}$ is harmonic on $\mathbb{D}^{(t)}$ with boundary values $0$ at
$\partial_{\infty}\mathbb{D}^{(t)}$. For $z\in\Gamma^{(t)}$, there is
$\zeta\in E$ with $|z-\zeta|=t$, whence $|z|\geq 1-t$. Thus, using (3.7),
(3.15) $g^{(t)}(z)=G_{\mathbb{D}^{(t)}}(z,0)\leq\log\frac{1}{1-t}\leq
3\,\log\frac{1}{1-t}\,\nu_{z}(E^{(t)}).$
The right hand side is a harmonic function of $z$ on the whole of
$\mathbb{D}$. By the maximum principle, (3.15) holds on all of
$\mathbb{D}^{(t)}$.
We now choose real parameters $r>s>1$ with $r-s>1$. We assume that $t<1/r$.
Let $z\in\mathbb{D}$.
_Case 1._ Let $|z|<(1-t)^{s}$. Then
$g^{(t)}(z)\leq\log\frac{1}{1-t}\leq\frac{1}{s}\log\frac{1}{|z|}$, and
$G_{\mathbb{D}^{(t)}}(z,0)\geq\frac{s-1}{s}G_{\mathbb{D}}(z,0)\,.$
_Case 2._ Let $z\in\mathbb{D}^{(rt)}$ with $|z|\geq(1-t)^{s}$. By the
Bernoulli inequality, $|z|\geq 1-st$. Following [10], we write
$z=|z|e^{i\theta}$ and
$\nu_{z}(E^{(t)})=\int_{E^{(t)}}P(z,\xi)\,d\lambda_{\mathbb{D}}(\xi)=\bigl{(}1-|z|^{2}\bigr{)}\frac{1}{2\pi}\int_{\\{\varphi:e^{i\varphi}\in
E^{(t)}\\}}\frac{d\varphi}{(1-|z|)^{2}+4|z|\sin^{2}\frac{\varphi-\theta}{2}}\,.$
Then for $\varphi\in(-\pi\,,\pi]$ with $e^{i\varphi}\in E^{(t)}$, using
$rt\leq\mathsf{dist}(z,E)\leq 1-|z|+\mathsf{dist}(e^{i\theta},E)$,
$\pi\geq|\phi-\theta|\geq
2\big{|}\sin\tfrac{\varphi-\theta}{2}\big{|}=|e^{i\theta}-e^{i\varphi}|\geq\mathsf{dist}(e^{i\theta},E)-t\geq
rt-(1-|z|)-t\geq\tau\,t\,,$
where $\tau=r-s-1$. Combining these estimates with (3.15),
$\displaystyle g^{(t)}(z)$ $\displaystyle\leq
3\Bigl{(}\log\frac{1}{1-t}\Bigr{)}\bigl{(}1-|z|^{2}\bigr{)}\,\frac{1}{2\pi}\int_{\\{\varphi\,:\,\tau\,t\leq|\varphi-\theta|\leq\pi\\}}\frac{d\varphi}{(1-|z|)^{2}+4|z|\sin^{2}\frac{\varphi-\theta}{2}}$
$\displaystyle=\frac{6}{\pi}\Bigl{(}\log\frac{1}{1-t}\Bigr{)}\bigl{(}1-|z|^{2}\bigr{)}\int_{\tau\,t/2}^{\pi/2}\frac{d\varphi}{(1-|z|)^{2}+4|z|\sin^{2}\varphi}$
$\displaystyle=\frac{6}{\pi}\Bigl{(}\log\frac{1}{1-t}\Bigr{)}\arctan\\!\left(\frac{1-|z|}{1+|z|}\cot\Bigl{(}\frac{\tau\,t}{2}\Bigr{)}\right)$
$\displaystyle\leq\frac{6}{\pi}\Bigl{(}\log\frac{1}{1-t}\Bigr{)}\Bigl{(}\cot\frac{\tau\,t}{2}\Bigr{)}\bigl{(}1-|z|\bigr{)}\,.$
Since $rt<1<\pi/3$, we have $\tau\,t/2<\pi/6$, whence $\cot(\tau\,t/2)\leq
2\pi/(3\tau\,t)$. Also, for $0<t<1/r$, we have $\log 1/(1-t)\leq r\,t/(r-1)$.
Therefore
$g^{(t)}(z)\leq\frac{4}{\tau\,t}\Bigl{(}\log\frac{1}{1-t}\Bigr{)}\bigl{(}1-|z|\bigr{)}\leq\frac{4r}{(r-1)(r-s-1)}\log\frac{1}{|z|}$
Thus, in Case 2,
$G_{\mathbb{D}^{(t)}}(z,0)\geq\Bigl{(}1-\frac{4r}{(r-1)(r-s-1)}\Bigr{)}G_{\mathbb{D}}(z,0)\,.$
Choosing $r=7$ and $s=18/17$, we get the proposed estimate. ∎
At the cost of increasing $r$, one can get a better (bigger) lower bound on
the disk. For our purpose, smaller $r_{\mathbb{D}}$ will be better. The proof
allows to take any number $r>(7+\sqrt{41})/2$.
With $u$ and $\Psi$ as in Theorem 3.3, we would like to have a more general
type of boundary moment to be finite, even when the integral in (3.4) is
infinite. To this end, we consider a continuous, increasing function
$\Phi:\bigl{[}0\,,\,\mathsf{diam}(\mathbb{X})\bigr{]}\to[0\,,\,\infty)$ with
$\Phi(0)=0$. With $\Phi$ as well as with $\Psi$, we associate the continuous,
non-negative measures $d\Phi$ and $d\Psi$ on
$\bigl{(}0\,,\,\mathsf{diam}(\mathbb{X})\bigr{]}$ which give mass
$\Phi(b)-\Phi(a)$, resp. $\Psi(a)-\Psi(b)$ to any interval
$(a\,,\,b]\subset\bigl{(}0\,,\,\mathsf{diam}(\mathbb{X})\bigr{]}$.
Furthermore, we consider the decreasing, continuous function
(3.16)
$\Upsilon:\bigl{[}0\,,\,\mathsf{diam}(\mathbb{X})\bigr{]}\to[0\,,\,\infty]\,,\quad\Upsilon(t)=\int_{t}^{\mathsf{diam}(\mathbb{X})}\Phi(s)\,d\Psi(s)\,.$
It will (typically) occur that $\Upsilon(0)=\infty$. We should consider
$\Upsilon$ as a downscaling of $\Psi$; indeed,
$\Upsilon(t)\leq\|\Phi\|_{\infty}\,\Psi(t)$. If $\Psi$ is differentiable on
$\bigl{(}0\,,\,\mathsf{diam}(\mathbb{X})\bigr{)}$, then
$d\Psi(t)=-\Psi^{\prime}(t)\,dt$, and
$\Upsilon^{\prime}(t)=\Phi(t)\,\Psi^{\prime}(t)$. The case considered in [10]
is the one where $\Psi(t)=t^{-q}$ and $\Phi(t)=t^{\alpha}$, where
$0<\alpha<q$, so that $\Upsilon(t)\asymp t^{\alpha-q}$.
###### (3.17) Theorem.
Let the subharmonic function $u$ on $\mathbb{X}$, the “singular” set
$E\subset\partial\mathbb{X}$ and the function $\Psi$ be as in Theorem 3.3, but
with infinite integral in (3.4). For continuous, increasing
$\Phi:\bigl{[}0\,,\,\mathsf{diam}(\mathbb{X})\bigr{]}\to[0\,,\,\infty)$ with
$\Phi(0)=0$ and the associated function $\Upsilon(t)$ according to (3.16),
suppose that
$\int_{\partial\mathbb{X}}\Upsilon\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)<\infty\,.$
Then the Riesz measure $\mu^{u}$ satisfies the extended boundary moment
condition
(3.18)
$\int_{\mathbb{X}}\mathsf{dist}(x,\partial\mathbb{X})\,\Phi\bigl{(}\mathsf{dist}(x,E)/R\bigr{)}\,d\mu^{u}(x)<\infty\,,$
where $R=R_{\mathbb{X}}$ is given by $R_{\mathbb{D}}=14$, resp.
$R_{\mathbb{T}}=1$.
For the disk case, when $\Psi(t)=t^{-q}$ and $\Phi(t)=t^{\alpha}$
($0<\alpha<q$), this boils down to Theorem 1-(ii)-(7) of [10].
In typical instances, $\Phi$ will have the _doubling property_ $\Phi(t/2)\geq
C\cdot\Phi(t)$ for a fixed $C>0$. In this case, division by $R$ can be omitted
in (3.18) even on the disk.
###### (3.19) Corollary.
Consider the disk. Under the assumptions of Theorem 3.17, if $1/\Psi$ is
doubling and
$\int_{\mathbb{S}}\Psi\bigl{(}\mathsf{d}_{\mathbb{D}}(\xi,E)\bigr{)}^{1-\varepsilon}\,d\lambda_{\mathbb{S}}(\xi)<\infty\,,$
then
$\int_{\mathbb{D}}\mathsf{d}_{\mathbb{D}}(x,\mathbb{S})\,\Psi\bigl{(}\mathsf{d}_{\mathbb{D}}(x,E)\bigr{)}^{-\varepsilon}\,d\mu^{u}(x)<\infty\,.$
###### Proof of Theorem 3.17.
Once again, the proof works in similar ways on disk and tree. We should keep
in mind that on the tree, integrals with respect to the Riesz measure are
infinite sums.
For most of the proof, we assume that $u(o)$ is finite. On the tree, this is
always required, but on the disk, one may have $u(z)=-\infty$ on a set of
measure $0$. We shall briefly explain at the end how to handle the case
$u(0)=-\infty$.
We take up the thread from the end of the proof of Theorem 3.3, in particular
(3.11). That inequality tells us that $u$ has $c_{\mathbb{X}}\,h^{(t)}$ as a
harmonic majorant on $\mathbb{X}^{(t)}$. Thus, it has its least harmonic
majorant $v^{(t)}$ on that set, and we have the Riesz decomposition
$u(x)=v^{(t)}(x)-G_{\mathbb{X}^{(t)}}\mu^{u}(x)\,,\quad x\in\mathbb{X}^{(t)}.$
We have $G_{\mathbb{D}}(z,0)\geq 1-|z|=\mathsf{d}_{\mathbb{D}}(z,\mathbb{S})$
on the disk, and
$G_{\mathbb{T}}(x,o)=b_{\mathbb{T}}\,\rho_{\mathbb{T}}(x,\partial\mathbb{T})$.
Using Theorem 3.14, we get for $0<t<1/r$ ($r=r_{\mathbb{X}}$)
$\displaystyle\int_{\mathbb{X}^{(rt)}}\mathsf{dist}(x,$
$\displaystyle\partial\mathbb{X})\,d\mu^{u}(x)\leq
b_{\mathbb{X}}\,G_{\mathbb{X}^{(t)}}\mu^{u}(o)$
$\displaystyle=b_{\mathbb{X}}\,\bigl{(}v^{(t)}(o)-u(o)\bigr{)}\leq
b_{\mathbb{X}}\,c_{\mathbb{X}}\,h^{(t)}(o)-b_{\mathbb{X}}\,u(o)$
$\displaystyle=b_{\mathbb{X}}\,c_{\mathbb{X}}\,\int_{E^{(t)}_{*}}\Psi\bigl{(}\mathsf{dist}(\cdot,E)\bigr{)}\,d\lambda+b_{\mathbb{X}}\,c_{\mathbb{X}}\,\Psi(t)\,\lambda(E^{(t)})-b_{\mathbb{X}}\,u(o).$
(In the disk case, $o$ stands once more for the origin.) For the next
computation, we note that $\max\\{\mathsf{dist}(x,E):x\in\mathbb{X}\\}$ has
value $1$ for the tree, but may be between $1$ and $2$ for the disk. Tacitly
using continuity of the involved measures, and using monotonicity of $\Psi$,
for $0<t<1$
$\displaystyle\int_{E^{(t)}_{*}}$
$\displaystyle\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)=\int_{E^{(1)}\cap
E^{(t)}_{*}}\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)+\int_{E^{(1)}_{*}}\Psi\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)$
$\displaystyle\leq\int_{\mathsf{dist}(\xi,E)}^{1}\,d\Psi(s)\,d\lambda(\xi)\;\;+\;\;\Psi(1)\,\lambda(E^{(1)}\cap
E^{(t)}_{*})+\Psi(1)\,\lambda(E^{(1)}_{*})$
$\displaystyle=\int_{t}^{1}\lambda\bigl{(}\\{\xi\in\partial\mathbb{D}:t<\mathsf{dist}(\xi,E)\leq
s\\}\bigr{)}\,d\Psi(s)\;\;+\;\ \Psi(1)\,\lambda(E^{(t)}_{*}))$
$\displaystyle=\int_{t}^{1}\lambda(E^{(s)})\,d\Psi(s)\;\;-\;\;\lambda(E^{(t)})\,\Psi(t)\;+\;\Psi(1)\,.$
Combining this with the previous inequality, we get for $0<t<1$
(3.20)
$\int_{x\in\mathbb{X}^{(t)}}\mathsf{dist}(x,\partial\mathbb{X})\,d\mu^{u}(x)\leq
b_{\mathbb{X}}\,c_{\mathbb{X}}\,\int_{t/r}^{1}\lambda(E^{(s)})\,d\Psi(s)+C_{1}\,,$
where $\;C_{1}=b_{\mathbb{X}}\,c_{\mathbb{X}}\,\Psi(1)-b_{\mathbb{X}}\,u(o)$.
Because of several smaller subtleties, we now conclude the proofs separately.
A. Tree case. Recalling that $b_{\mathbb{T}}=r_{\mathbb{T}}=R_{\mathbb{T}}=1$,
$\displaystyle\sum_{x\in\mathbb{T}}$
$\displaystyle\rho_{\mathbb{T}}(x,\partial\mathbb{T})\,\Phi\bigl{(}\rho_{\mathbb{T}}(x,E)\bigr{)}\,\mu^{u}(x)=\sum_{x\in\mathbb{T}}\rho_{\mathbb{T}}(x,\partial\mathbb{T})\int_{0}^{\rho_{\mathbb{T}}(x,E)}d\Phi(t)\,\mu^{u}(x)$
$\displaystyle=\int_{0}^{1}\Biggl{(}\sum_{\,x\in\mathbb{T}^{(t)}}\rho_{\mathbb{T}}(x,\partial\mathbb{T})\,\mu^{u}(x)\Biggr{)}d\Phi(t)$
$\displaystyle[\text{by \eqref{eq:ineq2}}]$ $\displaystyle\leq
c_{\mathbb{T}}\int_{0}^{1}\int_{t}^{1}\lambda_{\mathbb{T}}(E^{(s)})\,d\Psi(s)\,d\Phi(t)+C_{1}\,\Phi(1)$
$\displaystyle[\text{Fubini}]$
$\displaystyle=c_{\mathbb{T}}\int_{0}^{1}\lambda_{\mathbb{T}}(E^{(s)})\,\Phi(s)\,d\Psi(s)+C_{2}=c_{\mathbb{T}}\int_{\partial\mathbb{T}}\Upsilon\bigl{(}\rho_{\mathbb{T}}(\xi,E)\bigr{)}\,d\lambda_{\mathbb{T}}(\xi)+C_{2}\,,$
which is finite by assumption.
B. Disk case. Note that the maximum possible value of
$\mathsf{d}_{\mathbb{D}}(z,E)$ is $2$. We refer to a simple observation of
[10]: if $0<t<2$ then for every $z\in\mathbb{D}$ and $\alpha\in[0\,,\,1]$, we
have $\mathsf{d}_{\mathbb{D}}(z,E)\leq 2\mathsf{d}_{\mathbb{D}}(\alpha\,z,E)$.
In particular, if $\mathsf{d}_{\mathbb{D}}(z,E)>t$ then
$\mathsf{d}_{\mathbb{D}}(\alpha\,z,E)>t/2$, so that $z$ lies in the component
of $0$ of the set $\\{w\in\mathbb{D}:\mathsf{d}_{\mathbb{D}}(w,E)>t/2\\}$.
This means that
(3.21)
$\\{z\in\mathbb{D}:\mathsf{d}_{\mathbb{D}}(z,E)>t\\}\subset\mathbb{D}^{(t/2)}\,.$
Using this, we now compute
$\displaystyle\int_{\mathbb{D}}\mathsf{d}_{\mathbb{D}}(z,\mathbb{S})\,\Phi\bigl{(}\mathsf{d}_{\mathbb{D}}(z,E)/14\bigr{)}\,d\mu^{u}(z)$
$\displaystyle=\int_{\mathbb{D}}\,\,\int_{0}^{\mathsf{d}_{\mathbb{D}}(z,E)/14}\mathsf{d}_{\mathbb{D}}(z,\mathbb{S})\,d\Phi(t)\,d\mu^{u}(z)$
$\displaystyle[\text{since}\;\mathsf{d}(z,E)<2]$
$\displaystyle=\int_{0}^{1/7}\int_{\\{z\in\mathbb{D}\,:\,\mathsf{d}_{\mathbb{D}}(z,E)>14t\\}}\mathsf{d}_{\mathbb{D}}(z,\mathbb{S})\,d\mu^{u}(z)\,d\Phi(t)$
$\displaystyle[\text{by \eqref{eq:subset}}]$
$\displaystyle\leq\int_{0}^{1/7}\int_{\mathbb{D}^{(7t)}}\mathsf{d}_{\mathbb{D}}(z,\mathbb{S})\,d\mu^{u}(z)\,d\Phi(t)$
$\displaystyle[\text{by \eqref{eq:ineq2}}]$ $\displaystyle\leq
b_{\mathbb{X}}\,c_{\mathbb{X}}\,\int_{0}^{1}\int_{t}^{1}\lambda(E^{(s)})\,d\Psi(s)+C_{1}\,\Phi(1)\,,$
which is seen to be finite by the same calculation as in the tree case.
The case when $u(0)=-\infty$ can be treated exactly as in [10, p.43] (where
the subharmonic function is denoted $v$) and is omitted here. ∎
Finally, we want to prove a converse to Theorem 3.17 analogous to Proposition
3.12.
###### (3.22) Theorem.
Let the set $E\subset\partial\mathbb{X}$ and the function $\Psi$ be as in
Theorem 3.3, but with infinite integral in (3.4). Let
$\Phi:[0,1]\to[0\,,\,\infty)$ be continuous and increasing with $\Phi(0)=0$
and $\Phi(t)>0$ for $t>0$. For the associated function $\Upsilon(t)$ according
to (3.16), suppose that
$\int_{\partial\mathbb{X}}\Upsilon\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}\,d\lambda(\xi)=\infty\,.$
If $u$ is a subharmonic function on $\mathbb{X}$ such that
$u(x)\geq\Psi\bigl{(}\mathsf{dist}(x,E)\bigr{)}$
then the Riesz measure $\mu^{u}$ is such that
(3.23)
$\int_{\mathbb{X}}\mathsf{dist}(x,\partial\mathbb{X})\,\Phi\bigl{(}\mathsf{dist}(x,E)\bigr{)}\,d\mu^{u}(x)=\infty\,.$
###### Proof.
First of all, we note that (3.23) hold if and only if
(3.24)
$\int_{\mathbb{X}}G(x,o)\,\Phi\bigl{(}\mathsf{dist}(x,E)\bigr{)}\,d\mu^{u}(x)=\infty\,.$
On the tree, this is obvious, because
$G_{\mathbb{T}}(x,o)=\frac{\mathsf{q}}{\mathsf{q}-1}\,\rho_{\mathbb{T}}(x,\partial\mathbb{T})$.
On the disk, it is clear that (3.23) implies (3.24). Conversely,
$\int_{|z|<1/2}G(z,0)\,\Phi\bigl{(}\sf_{\mathbb{D}}(z,E)\bigr{)}\,d\mu^{u}(z)\leq\|\Phi\|_{\infty}\int_{|z|<1/2}G(z,0)\,d\mu^{u}(z)<\infty\,,$
while for $|z|\geq 1/2$, we have $G(z,0)=\log\frac{1}{|z|}\leq(2\log
2)(1-|z|)$, so that (3.24) implies
$2\log 2\int_{|z|\geq
1/2}(1-|z|)\,\Phi\bigl{(}\mathsf{d}_{\mathbb{D}}(z,E)\bigr{)}\,d\mu^{u}(z)\geq\int_{|z|\geq
1/2}G(z,0)\,\Phi\bigl{(}\sf_{\mathbb{D}}(z,E)\bigr{)}\,d\mu^{u}(z)=\infty\,.$
_Case 1._ Suppose that there is $t\in(0\,,\,1)$ such that $u$ has no harmonic
majorant on the set $\mathbb{X}^{(t)}$. Then $G_{\mathbb{X}^{(t)}}\mu^{u}$ is
infinite on that set. Thus,
$\displaystyle\int_{\mathbb{X}}G(x,o)\,\Phi\bigl{(}\mathsf{dist}(x,E)\bigr{)}\,d\mu^{u}(x)$
$\displaystyle\geq\int_{\mathbb{X}^{(t)}}G_{\mathbb{X}^{(t)}}(x,o)\,\Phi\bigl{(}\mathsf{dist}(x,E)\bigr{)}\,d\mu^{u}(x)$
$\displaystyle\geq\Phi(t)\,G_{\mathbb{X}^{(t)}}\mu^{u}(o)=\infty\,,$
and the equivalence of (3.23) with (3.24) implies the result.
_Case 2._ We are left with the case when for each $t\in(0\,,\,1)$ there is the
(finite) least harmonic majorant $v^{(t)}$ of $u$ on $\mathbb{X}^{(t)}$.
Recall the function $h^{(t)}$ of (3.8). Then for every
$\eta\in\partial\mathbb{X}^{(t)}$,
$\limsup_{x\to\eta}v^{(t)}(x)\geq\limsup_{x\to\eta}v^{(t)}(x)\geq\Psi\bigl{(}\mathsf{dist}(\eta,E)\bigr{)}=\lim_{x\to\eta}h^{(t)}(x)\,.$
By the minimum principle, applied to the harmonic function
$v^{(t)}-h^{(t)}\,$, we have $v^{(t)}\geq h^{(t)}$ on $\mathbb{X}^{(t)}$. Now
we can replace the computations of the proof of Theorem 3.17 with similar
inequalities in the reverse direction.
$\displaystyle\int_{\mathbb{X}^{(t)}}$ $\displaystyle
G(x,o)\,d\mu^{u}(x)\geq\int_{\mathbb{X}^{(t)}}G_{\mathbb{X}^{(t)}}\mu^{u}(o)=v^{(t)}(o)-u(o)\geq
h^{(t)}(o)-u(o)$ $\displaystyle=\int_{E^{(1)}\cap
E^{(t)}_{*}}\Psi\bigl{(}\mathsf{dist}(\cdot,E)\bigr{)}\,d\lambda+\int_{E^{(1)}_{*}}\Psi\bigl{(}\mathsf{dist}(\cdot,E)\bigr{)}\,d\lambda+\Psi(t)\,\lambda(E^{(t)})-u(o)$
$\displaystyle\geq\int_{t}^{1}\lambda(E^{(s)}\setminus
E^{(t)})\,d\Psi(s)+\Psi(1)\,\lambda(E^{(1)}\setminus
E^{(t)})+\Psi(1)\,\lambda(E^{(1)}_{*})+\Psi(t)\,\lambda(E^{(t)})-u(o)$
$\displaystyle=\int_{t}^{1}\lambda(E^{(s)})\,d\Psi(s)+C_{3}\,,\hskip
56.9055pt\text{where}\quad C_{3}=\Psi(1)-u(o)\,.$
Now let $0<\varepsilon<1$. Let
$\Phi_{\varepsilon}(s)=\max\\{\Phi(s)-\Phi(\varepsilon)\,,\,0\\}$. Since $u$
has a harmonic majorant on $\mathbb{X}^{(\varepsilon)}$, the first integral in
the following computation is finite. The above estimate is used in the third
line.
$\displaystyle\int_{\mathbb{X}^{(\varepsilon)}}G(x,o)$
$\displaystyle\Phi\bigl{(}\mathsf{dist}(x,E)\bigr{)}\,d\mu^{u}(x)\geq\int_{\mathbb{X}^{(\varepsilon)}}G(x,o)\int_{\varepsilon}^{\mathsf{dist}(x,E)}\,d\Phi(t)\,d\mu^{u}(x)$
$\displaystyle\geq\int_{\varepsilon}^{1}\int_{\mathbb{X}^{(t)}}G(x,o)\,d\mu^{u}(x)\,d\Phi(t)$
$\displaystyle\geq\int_{\varepsilon}^{1}\int_{t}^{1}\lambda(E^{(s)})\,d\Psi(s)\,d\Phi(t)+(1-\varepsilon)C_{3}$
$\displaystyle=\int_{\varepsilon}^{1}\lambda(E^{(s)})\int_{\varepsilon}^{s}d\Phi(t)\,d\Psi(s)+(1-\varepsilon)C_{3}$
$\displaystyle=\int_{0}^{1}\biggl{(}\int_{\\{\xi\in\partial\mathbb{X}:\mathsf{dist}(\xi,E)\leq
s}d\lambda(\xi)\biggr{)}\Phi_{\varepsilon}(s)\,d\Psi(s)+(1-\varepsilon)C_{3}$
$\displaystyle=\int_{E^{(1)}}\int_{\mathsf{dist}(\xi,E)}^{1}\Phi_{\varepsilon}(s)\,d\Psi(s)\,d\lambda(\xi)+(1-\varepsilon)C_{3}$
As $\varepsilon\to 0$, by monotone convergence, the double integral in the
last line tends to
$\int_{E^{(1)}}\Bigl{(}\Upsilon\bigl{(}\mathsf{dist}(\xi,E)\bigr{)}-\Upsilon(1)\Bigr{)}\,d\lambda(\xi)\,,$
which is infinite by assumption. ∎
###### (3.25) Remarks.
(a) _[Hyperbolic versus Euclidean.]_ In the introduction and in Section 2 we
insisted on a hyperbolic “spirit” inherent in the material presented here.
After all, this was not dominant in most of our computations. Not only on the
disk, we always used the Euclidean metric $\mathsf{d}_{\mathbb{D}}$, but also
on the tree, the dominant role was played by the metric $\rho_{\mathbb{T}}$
which is the tree-analogue of the Euclidean metric. One point is that to see
the latter analogy, one should first understand that the graph metric on the
tree corresponds to the hyperbolic one on the disk.
One result where hyperbolicity is strongly present is Theorem 3.14. The proof
in the tree case relies directly on the fact that the tree with its graph
metric is $\delta$-hyperbolic in the sense of Gromov [14], with $\delta=0$:
every vertex is a cut-point (it disconnects the tree). Analogously, one might
try to prove that theorem in the disk case using $\delta$-hyperbolicity with
$\delta=\log(1+\sqrt{2}\,)$. Indeed, this is related with the inequalities of
Ancona [1] which say that the Green kernel of the open disk is almost
submultiplicative along hyperbolic geodesics. (For the disk, this can be seen
by direct inspection via the explicit formulas for the Green kernel.) Now, for
points $z\in\mathbb{D}^{(rt)}$ and $\xi\in E^{(t)}$, the hyperbolic geodesic
from $z$ to $\xi$ must be at bounded hyperbolic distance from the origin
(depending on $r$ and $t$), similarly to the (simpler) tree case. However,
this idea is more vague than the down-to-earth proof following [10].
(b) In view of the equivalence (3.23) $\iff$ (3.24), in all the results
presented here, one can replace the distance to the boundary
$\mathsf{dist}(x,\partial\mathbb{X})$ with the Green kernel $G(x,o)$.
(c) Among the common features of disk and tree which allowed us to formulate
and prove the results in very similar ways, the key facts are
* •
comparability of $G(x,o)$ with $\mathsf{dist}(x,\partial\mathbb{X})$ (the
metric is “intrinsic” in this sense),
* •
solvability of the Dirichlet problem for continuous functions on
$\partial\mathbb{X}$, and in particular, vanishing of the Green kernel at the
boundary, and
* •
the Green kernel estimate of Theorem 3.14.
###### (3.26)
An extension for trees. Instead of the homogeneous tree, we can take an
arbitrary locally finite tree $\mathbb{T}$ and equip its edges with
_conductances_ $a(x,y)=a(y,x)>0\iff x\sim y$. Letting $m(x)=\sum_{y}a(x,y)$,
the transition probabilities $p(x,y)=a(x,y)/m(y)$ give rise to a nearest
neighbour random walk $(Z_{n})_{n\geq 0}$ and to the associated Laplacian
$\Delta_{\mathbb{T}}f(x)=\sum_{y\sim x}p(x,y)\bigl{(}f(y)-f(x)\bigr{)}.$
We asssume the following.
* (i)
Strong irreducibility: $\;0<m_{0}\leq m(x)\leq M_{0}<\infty\;$ and
$\;a(x,y)\geq a_{0}>0\;$ for all $x$ and all $y\sim x$.
* (ii)
Strong transience: $\;F(x,y)\leq\delta<1\;$ for all $x$ and all $y\sim x$,
where for arbitrary $x,y\in\mathbb{T}$,
$F(x,y)=\mathsf{Pr}[\exists n\geq 0:Z_{n}=y\mid Z_{0}=x]$
The associated Green kernel
$G(x,y)=\sum_{n=0}^{\infty}p^{(n)}(x,y)\,,\quad\text{where}\quad
p^{(n)}(x,y)=\mathsf{Pr}[Z_{n}=y\mid Z_{0}=x]\,,\quad x,y\in X$
is finite and tends to $0$ at infinity by assumption (ii). Note that in our
notation, $G(x,y)=F(x,y)G(y,y)$.
We can adapt all the above results regarding the homogenous tree to this more
general situation. The main issue is to define a suitable metric on the
compactification $\widehat{\mathbb{T}}$ in the right way: for
$z,w\in\widehat{\mathbb{T}}$,
$\rho_{\mathbb{T}}(w,z)=\begin{cases}F(w\wedge z,o)\,,&\text{if}\;z\neq
w\,,\\\ 0\,,&\text{if}\;z=w\,.\end{cases}$
[For simple random walk on the homogeneous tree, as considered above, this is
just the metric of (2.20).]
In this setting, the tree-versions of theorems 3.3, 3.17 and 3.22 remain true.
This applies, in particular, to arbitrary symmetric nearest neighbour random
walks on the free group ($\equiv$ homogeneous tree with even degree).
In conclusion, we remark that the very recent note by Favorov and Radchenko
[12] was written in parallel to the present article without mutual knowledge.
The results of [12] concern the disk case and are a bit less general than
ours. We want to point out that here, our main focus has been on elaborating
some aspects of the very strong analogies of the potential theory on disk and
tree, respectively, via focussing on properties of Riesz measures.
Acknowledement. The authors acknowledge email exchanges with M. Stoll
(Columbia, SC) and with S. Favorov (Kharkov).
## References
* [1] Ancona, A.: _Negatively curved manifolds, elliptic operators, and the Martin boundary._ Ann. of Math. 125 (1987) 495–536.
* [2] Armitage, D. H., and Gardiner, S. J.: _Classical Potential Theory._ Springer-Verlag, London, 2001.
* [3] Atanasi, L., and Picardello, M. A.: _The Lusin area function and local admissible convergence of harmonic functions on homogeneous trees._ Trans. Amer. Math. Soc. 360 (2008) 3327–3343.
* [4] Beardon, A. F.: _The Geometry of Discrete Groups._ Graduate Texts in Mathematics, 91. Springer-Verlag, New York, 1983.
* [5] Blaschke, W.: _Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen._ Berichte Math.-Phys. Kl., Sächs. Gesell. der Wiss. Leipzig 67 (1915) 194–200.
* [6] Cartwright, D. I., Soardi, P. M., and Woess, W.: _Martin and end compactifications for non-locally finite graphs._ Trans. Amer. Math. Soc. 338 (1993) 679–693.
* [7] Casadio Tarabusi, E., Cohen, J. M., Korányi, A., and Picardello, M. A.: _Converse mean value theorems on trees and symmetric spaces._ J. Lie Theory 8 (1998) 229–254.
* [8] Casadio Tarabusi, E., and Figà-Talamanca, A.: _Poisson kernels of drifted Laplace operators on trees and on the half-plane._ Colloq. Math. 118 (2010) 147–159.
* [9] Cohen, J. M., Colonna, F., and Singman, D.: _A global Riesz decomposition theorem on trees without positive potentials._ J. Lond. Math. Soc. 75 (2007) 1–17; corrigendum in J. Lond. Math. Soc. 83 (2011).
* [10] Favorov, S., and Golinskii, L.: _A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators._ Linear and complex analysis, pp. 37–47, Amer. Math. Soc. Transl. (2) 226, Amer. Math. Soc., Providence, RI, 2009.
* [11] Favorov, S., and Golinskii, L.: _Blaschke-type conditions for analytic and subharmonic functions in the unit disk: local analogs and inverse problems._ Comput. Methods Funct. Theory 12 (2012) 151–166.
* [12] Favorov, S. Ju., and Radchenko, L. D.: _On analytic and subharmonic functions in unit disc growing near a part of the boundary._ Zh. Mat. Fiz. Anal. Geom. 9 (2013) 304–315, 423, 427.
* [13] Garnett, J. B.: _Bounded Analytic Functions._ Revised first edition. Graduate Texts in Math. 236, Springer, New York, 2007.
* [14] Gromov, M.: _Hyperbolic groups._ In _Essays in Group Theory,_ 75–263, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987.
* [15] Helgason, S.: _Groups and Geometric Analysis._ Corrected reprint of the 1984 original. Math. Surveys and Monographs 83. American Math. Soc., Providence, RI, 2000.
* [16] Helms, L. L.: _Introduction to Potential Theory._ Wiley, New York, 1969.
* [17] Ransford, Th.: _Potential Theory in the Complex Plane._ London Math. Soc. Student Texts 28, Cambridge University Press, Cambridge, 1995.
* [18] Rigoli, M., Salvatori, M., and Vignati, M.: _Strongly subharmonic functions, graphs, and their asymptotic growth._ Math. Annalen 331 (2005) 21–39.
* [19] Stoll, M.: _Harmonic Function Theory on Real Hyperbolic Space._ Notes (Univ. South Carolina) available at http://www.math.sc.edu/people/faculty/stoll/hyperbolic.pdf
* [20] Woess, W.: _Denumerable Markov Chains. Generating Functions, Boundary Theory, Random Walks on Trees._ European Math. Soc. Publishing House, 2009.
* [21] Woess, W.: _On the Riesz decomposition for Markov chains._ Unpublished Note (TU Graz, 2011), available at http://www.math.tu-graz.ac.at/$\sim$woess/papers/rieszdecomp.pdf
|
arxiv-papers
| 2014-04-15T09:37:18 |
2024-09-04T02:50:01.181061
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Tetiana Boiko and Wolfgang Woess",
"submitter": "Wolfgang Woess",
"url": "https://arxiv.org/abs/1404.3852"
}
|
1404.3955
|
# The centre of the extended Haagerup subfactor has 22 simple objects
Scott Morrison and Kevin Walker
We explain a technique for discovering the number of simple objects in
$Z({\mathcal{C}})$, the center of a fusion category ${\mathcal{C}}$, as well
as the combinatorial data of the induction and restriction functors at the
level of Grothendieck rings. The only input is the fusion ring
$K({\mathcal{C}})$ and the dimension function $K({\mathcal{C}})\to\mathbb{C}$.
The method is not guaranteed to succeed (it may give spurious answers besides
the correct one, or it may simply take too much computer time), but it seems
it often does. We illustrate by showing that there are 22 simple objects in
the center of the extended Haagerup subfactor [BMPS12], and that the induction
functors from the $6$-object and $8$-object fusion categories arising as the
even parts of the subfactor are given by
$\displaystyle I_{EH1}$
$\displaystyle=\left(\begin{array}[]{cccccccccccccccccccccc}1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
2&1&1&1&0&0&1&1&1&1&1&1&1&1&0&0&0&0&1&1&1&1\\\
1&2&1&1&0&3&2&2&2&2&2&2&2&2&1&1&1&1&1&1&1&1\\\
2&4&4&1&0&2&4&4&4&4&3&3&3&3&3&3&3&3&1&1&1&1\\\
4&5&2&1&0&3&5&5&5&5&3&3&3&3&4&4&4&4&1&1&1&1\\\
1&3&2&1&0&2&3&3&3&3&1&1&1&1&2&2&2&2&1&1&1&1\end{array}\right)$ $\displaystyle
I_{EH2}$
$\displaystyle=\left(\begin{array}[]{cccccccccccccccccccccc}2&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&1&1&1&0&1&1&1&1&1&1&1&1&1&0&0&0&0&1&1&1&1\\\
2&2&1&1&0&2&2&2&2&2&2&2&2&2&1&1&1&1&1&1&1&1\\\
1&4&4&1&0&3&4&4&4&4&3&3&3&3&3&3&3&3&1&1&1&1\\\
3&4&2&1&0&2&4&4&4&4&2&2&2&2&3&3&3&3&1&1&1&1\\\
3&4&2&1&0&2&4&4&4&4&2&2&2&2&3&3&3&3&1&1&1&1\\\
1&1&0&0&0&1&1&1&1&1&1&1&1&1&1&1&1&1&0&0&0&0\\\
1&1&0&0&0&1&1&1&1&1&1&1&1&1&1&1&1&1&0&0&0&0\end{array}\right)$
Observe that the fifth column corresponds to $\mathbf{1}\in Z({\mathcal{C}})$,
and that there are four sets of four objects in $Z({\mathcal{C}})$ which each
restrict the same way to both $EH1$ and $EH2$.
Of course, it would also be interesting to compute all of
$K(Z({\mathcal{C}}))$, and even more interesting to compute the $S$ and $T$
matrices. We don’t address those questions here.
### Acknowledgements
We’d like to thank Brendan McKay for suggesting the reduction used in Theorem
2.2, and Pinhas Grossman and Noah Snyder for interesting discussions while we
were doing this work. Scott Morrison was supported by an Australian Research
Council ‘Discovery Early Career Researcher Award’, DE120100232 and ‘Discovery
Project’ DP140100732. Scott Morrison and Kevin Walker were supported by DOD-
DARPA grant HR0011-12-1-0009. We would like to thank the Erwin Schrödinger
Institute and its 2014 programme on “Modern Trends in Topological Quantum
Field Theory” for their hospitality.
## 1 Fusion categories and centers
Given a fusion 2-category ${\mathcal{C}}$, we use the following facts about
its center, the modular tensor category $Z({\mathcal{C}})$.
###### Fact 1.
For each simple object $X\in Z({\mathcal{C}})$, $\dim(X)$ divides
$\dim({\mathcal{C}})$ as an algebraic integer.
(This follows from Lemma 1.2 of [EG98] and
$\dim(Z({\mathcal{C}}))=\dim({\mathcal{C}})^{2}$.)
For a fusion 2-category $\dim({\mathcal{C}})$ can be computed as the sum of
$\dim(X)^{2}$ over $X$ in the collection of simple endomorphisms of any chosen
object.
###### Fact 2.
For each simple object $X\in Z({\mathcal{C}})$, $\dim(X)$ is a $d$-number in
the sense of Ostrik. [Ost09]
Recall that an algebraic integer with minimal polynomial
$p(x)=\sum_{i=0}^{n}a_{i}x^{i}$ is a $d$-number if $a_{0}^{i}$ divides
$a_{n-i}^{n}$ for each $i$.
###### Fact 3.
For each object $a\in{\mathcal{C}}$, there is an induction functor
$I:\operatorname{End}_{\mathcal{C}}(a)\to Z({\mathcal{C}})$ which is a pivotal
functor. In particular, it induces a ring homomorphism
$K(\operatorname{End}_{\mathcal{C}}(a))\to K(Z({\mathcal{C}}))$, and it
preserves dimensions.
The center may be realized as $\operatorname{Rep}{\mathcal{C}}(S^{1})$, the
representation category of the annular category of ${\mathcal{C}}$, and the
induction functor is given by the inclusion of the rectangle in the annulus.
###### Fact 4.
Given $X\in\operatorname{End}_{\mathcal{C}}(a)$ and
$Y\in\operatorname{End}_{\mathcal{C}}(b)$, the space
$\operatorname{Hom}_{Z({\mathcal{C}})}(I(X),I(Y))$ has a basis
$Z$$v$$X$$u$$Y$$W$
where $W,Z\in\operatorname{Hom}_{\mathcal{C}}(b,a)$ and $u$ runs over a basis
of $\operatorname{Hom}^{2}_{\mathcal{C}}(W,Y\otimes Z)$ while $v$ runs over a
basis of $\operatorname{Hom}^{2}_{\mathcal{C}}(Z\otimes X,W)$.
In particular,
$\displaystyle\dim\operatorname{Hom}_{(Z({\mathcal{C}})}(I(X),I(Y))$
$\displaystyle=$
$\displaystyle\sum_{W,Z\in\operatorname{Hom}_{\mathcal{C}}(b,a)}\dim\operatorname{Hom}^{2}_{\mathcal{C}}$
$\displaystyle(W,Y\otimes Z)\dim\operatorname{Hom}^{2}_{\mathcal{C}}(Z\otimes
X,W).$ (1.1)
(This follows from the apparently folkloric spine lemma, c.f. [Mor13].)
Equivalent to this fact is that the composition of induction from
$\operatorname{End}_{\mathcal{C}}(a)$ to $Z({\mathcal{C}})$ followed by
restriction from $Z({\mathcal{C}})$ to $\operatorname{End}_{\mathcal{C}}(b)$
is given on objects by $X\mapsto\oplus_{V}V^{*}XV$, where the sum is over
simple objects $V\in\operatorname{Hom}_{\mathcal{C}}(a,b)$. (See, e.g. [ENO05,
Proposition 5.4].)
With respect to the basis of simples,
$I_{a}:K(\operatorname{End}_{\mathcal{C}}(a))\to K(Z({\mathcal{C}}))$ has a
matrix $A_{a}$, with rows indexed by simples in
$\operatorname{End}_{\mathcal{C}}(a)$ and columns indexed by simples in
$Z({\mathcal{C}})$. We order the simples in
$\operatorname{End}_{\mathcal{C}}(a)$ by dimension. We order the simples in
$Z({\mathcal{C}})$ so that the columns of $A_{a}$ appear in reverse
lexicographic order.
We denote by $M_{ab}$ the matrix whose $ij$ entry is
$\dim\operatorname{Hom}_{(Z({\mathcal{C}})}(I(X),I(Y))$ computed as in
Equation (4), for $X$ the $i$-th simple in
$\operatorname{End}_{\mathcal{C}}(a)$ and $Y$ the $j$-th simple in
$\operatorname{End}_{\mathcal{C}}(b)$. Equation (4) tells us that
$M_{ab}=A_{a}A_{b}^{t}$. We denote by ${\mathcal{M}}$ the block matrix whose
$ab$ block is $M_{ab}$, and ${\mathcal{A}}$ the matrix made by stacking the
matrices $A_{a}$ above each other. Then
${\mathcal{M}}={\mathcal{A}}{\mathcal{A}}^{t}$.
Our task now is to compute all possible forms for the matrix ${\mathcal{A}}$.
## 2 Combinatorics
We begin with a symmetric $n$-by-$n$ matrix ${\mathcal{M}}$ with non-negative
integer entries. A decomposition of ${\mathcal{M}}$ is a $n$-by-$m$ matrix
${\mathcal{A}}$ (for some $m$) with non-negative integer entries, so
${\mathcal{M}}={\mathcal{A}}{\mathcal{A}}^{t}$.
Let $d$ be some algebraic number. Fix some collection of vectors
$v_{i}\in\mathbb{Q}(d)^{n}$. Further fix an algebraic number
${\mathcal{D}}\in\mathbb{Q}(d)$. We wish to find all $n$-by-$m$ matrices
${\mathcal{A}}$ so that ${\mathcal{M}}={\mathcal{A}}{\mathcal{A}}^{t}$, and
for each column $w$ of ${\mathcal{A}}$ and each $i$, $v_{i}.w$ is an Ostrik
$d$-number and divides ${\mathcal{D}}$ as an algebraic integer. We call such a
decomposition for $({\mathcal{M}},\\{v_{i}\\},{\mathcal{D}})$ an algebraic
decomposition.
Because we work in the fixed number field $\mathbb{Q}(d)$, we can easily
compute the minimal polynomial of $w.v_{i}$, and hence determine if it is an
Ostrik $d$-number.
For each object $a$ of ${\mathcal{C}}$, take $v_{a}$ to be the vector of
dimensions of simple endomorphisms of $a$. If there are $n_{a}$ simple
endomorphisms of $a$, $v_{a}\in\mathbb{Q}(d)^{n_{a}}$. We abuse notation and
also think of $v_{a}$ as the corresponding vector in
$\mathbb{Q}(d)^{n}\cong\oplus_{a}\mathbb{Q}(d)^{n_{a}}$ by padding with
zeroes.
We can summarize the facts from the previous section as
###### Theorem 2.1.
Let ${\mathcal{C}}$ be a fusion 2-category, ${\mathcal{M}}$ be the matrix of
inner products defined by Equation (4), and ${\mathcal{A}}$ be the induction
matrix defined above.
Then ${\mathcal{A}}$ is an algebraic decomposition of
$({\mathcal{M}},\\{v_{a}\\},\dim({\mathcal{C}}))$.
Observe that we can take $d$ to be the Frobenius-Perron eigenvalue of
${\mathcal{M}}$, and in fact $d=k{\mathcal{D}}$, where $k$ is the number of
simple objects of ${\mathcal{C}}$.
Since a decomposition ${\mathcal{A}}$ being algebraic implies strong
conditions on the columns, we intend to enumerate algebraic decompositions by
building up the matrix column at a time. First however, we perform a reduction
of the problem (replacing ${\mathcal{M}}$ above with another, smaller
${\mathcal{M}}^{\prime}$) which will be essential for reasonable runtimes on
intended examples.
###### Theorem 2.2.
Suppose ${\mathcal{M}}$ has rank $r$, and the top left $r$-by-$r$ minor
${\mathcal{M}}^{\prime}$ is nonsingular. There is a unique $n$-by-$r$ matrix
${\mathcal{R}}$ (with rational entries), so
${\mathcal{M}}={\mathcal{R}}{\mathcal{M}}^{\prime}{\mathcal{R}}^{t}$, and the
(not necessarily algebraic) decompositions of ${\mathcal{M}}$ are exactly
those ${\mathcal{R}}{\mathcal{A}}^{\prime}$, for ${\mathcal{A}}^{\prime}$ is a
decomposition of ${\mathcal{M}}^{\prime}$, where
${\mathcal{R}}{\mathcal{A}}^{\prime}$ has non-negative integer entries.
Moreover, for any collection of vectors $\\{v_{i}\\}\subset\mathbb{Q}(d)^{n}$,
the algebraic decompositions of $({\mathcal{M}},\\{v_{i}\\},{\mathcal{D}})$
correspond in the same way to those algebraic decompositions of
$({\mathcal{M}}^{\prime},\\{v_{i}{\mathcal{R}}\\},{\mathcal{D}})$ with non-
negative integer entries.
###### Proof.
Certainly for any decomposition ${\mathcal{M}}={\mathcal{A}}{\mathcal{A}}^{t}$
taking ${\mathcal{A}}^{\prime}$ to be the first $r$ rows of ${\mathcal{A}}$
gives a decomposition
${\mathcal{M}}^{\prime}={\mathcal{A}}^{\prime}{{\mathcal{A}}^{\prime}}^{t}$.
In the opposite direction, first note that there is some $n$-by-$r$ matrix
${\mathcal{R}}$ (with rational entries) so that
${\mathcal{M}}={\mathcal{R}}{\mathcal{M}}^{\prime}{\mathcal{R}}^{t}$. Now
given a decomposition
${\mathcal{M}}^{\prime}={\mathcal{A}}^{\prime}{{\mathcal{A}}^{\prime}}^{t}$,
we obtain a not necessarily integral decomposition
${\mathcal{M}}=({\mathcal{R}}{\mathcal{A}}^{\prime})({\mathcal{R}}{\mathcal{A}}^{\prime})^{t}$.
Finally, if we obtained ${\mathcal{A}}^{\prime}$ as the first $r$ rows of an
${\mathcal{A}}$ satisfying ${\mathcal{M}}={\mathcal{A}}{\mathcal{A}}^{t}$,
this reconstructs the original ${\mathcal{A}}$.
Thus we see that it suffices to search for decompositions
${\mathcal{M}}^{\prime}={\mathcal{A}}^{\prime}{{\mathcal{A}}^{\prime}}^{t}$,
and take exactly those ${\mathcal{R}}{\mathcal{A}}^{\prime}$ which are
integral.
For the last part, we see that a column $w$ of ${\mathcal{A}}^{\prime}$
satisfies the algebraic conditions with respect to $\\{v_{i}{\mathcal{R}}\\}$
exactly if ${\mathcal{R}}w$ (the corresponding column of
${\mathcal{R}}{\mathcal{A}}^{\prime}$) satisfies the algebraic conditions with
respect to $\\{v_{i}\\}$, since
$v_{i}.({\mathcal{R}}w)=(v_{i}{\mathcal{R}}).w$. ∎
A partial algebraic decomposition of
$({\mathcal{M}},\\{v_{i}\\},{\mathcal{D}})$ is a matrix ${\mathcal{B}}$ so
that ${\mathcal{M}}-{\mathcal{B}}{\mathcal{B}}^{t}$ is a non-negative matrix,
and the columns of ${\mathcal{B}}$ satisfy the same conditions, determined by
$\\{v_{i}\\}$ and ${\mathcal{D}}$, as the columns of an algebraic
decomposition.
In particular an algebraic decomposition is a partial algebraic decomposition.
###### Lemma 2.3.
Deleting a column from a partial algebraic decomposition gives another partial
algebraic decomposition.
We say a new column for a partial algebraic decomposition ${\mathcal{B}}$ is a
vector $w$, such that
1. 1.
each $w_{i}\geq 0$,
2. 2.
if $p$ is the greatest number such that the top left $p$-by-$p$ minor of
${\mathcal{M}}-{\mathcal{B}}{\mathcal{B}}^{t}$ is exactly zero, $w_{p+1}>0$,
3. 3.
writing $u$ for the last column of ${\mathcal{B}}$, if the first $k$ entries
of the $u$ agree with the first $k$ entries of $w$, then $w_{k+1}\leq
u_{k+1}$,
4. 4.
$w_{i}\leq({\mathcal{M}}-{\mathcal{B}}{\mathcal{B}}^{t})_{ij}/w_{j}$, for each
$j\leq i$,
5. 5.
$w$ satisfies the algebraic conditions determined by $\\{v_{i}\\}$ and
${\mathcal{D}}$, and
6. 6.
${\mathcal{M}}-{\mathcal{B}}{\mathcal{B}}^{t}-ww^{t}$ is a non-negative
matrix.
We can clearly enumerate all possible new columns for ${\mathcal{B}}$; in
practice for the last condition, we numerically estimate all the eigenvalues,
accepting $w$ if they are all at least $-0.001$. Condition (4) is redundant
with (6); we include it as a token optimization.
We now have
###### Theorem 2.4.
Every partial algebraic decomposition with $k$ columns in reverse
lexicographic order may be obtained by appending a new column to some partial
algebraic decomposition with $k-1$ columns in reverse lexicographic order.
This theorem gives a relatively efficient mechanism for enumerating all
algebraic decompositions of a given
$({\mathcal{M}},\\{v_{i}\\},{\mathcal{D}})$. It is implemented in a
Mathematica notebook available with the arXiv sources of this article. That
notebook relies on the FusionAtlas package introduced in [MS12, MPPS12,
IJMS12, PT12], although only to prepare the fusion rings of (and calculate the
dimesions for) some familiar examples. It should be easy to see how to run it
without this dependency.
## 3 Calculations
For the extended Haagerup principal graphs
$\left({\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[height=17.07164pt]{diagrams/graphs/CA1952BED6C8EB18}}\end{array}},{\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[height=17.07164pt]{diagrams/graphs/C53E32050C862F37}}\end{array}}\right)$
there are a unique fusion rings for the two even parts, given by
$\displaystyle\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\ 1&1&1&0&0&0\\\
0&1&1&1&0&0\\\ 0&0&1&1&1&1\\\ 0&0&0&1&2&1\\\ 0&0&0&1&1&0\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&0&1&0&0&0\\\ 0&1&1&1&0&0\\\
1&1&1&1&1&1\\\ 0&1&1&2&3&1\\\ 0&0&1&3&3&2\\\ 0&0&1&1&2&1\\\
\end{array}\right),$
$\displaystyle\qquad\qquad\left(\begin{array}[]{cccccc}0&0&0&1&0&0\\\
0&0&1&1&1&1\\\ 0&1&1&2&3&1\\\ 1&1&2&4&5&3\\\ 0&1&3&5&6&3\\\ 0&1&1&3&3&2\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&0&0&0&1&0\\\ 0&0&0&1&2&1\\\
0&0&1&3&3&2\\\ 0&1&3&5&6&3\\\ 1&2&3&6&7&4\\\ 0&1&2&3&4&2\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&0&0&0&0&1\\\ 0&0&0&1&1&0\\\
0&0&1&1&2&1\\\ 0&1&1&3&3&2\\\ 0&1&2&3&4&2\\\ 1&0&1&2&2&1\\\
\end{array}\right)$ and
$\displaystyle\left(\begin{array}[]{cccccccc}1&0&0&0&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&1\\\
\end{array}\right),\left(\begin{array}[]{cccccccc}0&1&0&0&0&0&0&0\\\
1&1&1&0&0&0&0&0\\\ 0&1&1&1&0&0&0&0\\\ 0&0&1&1&1&1&0&0\\\ 0&0&0&1&1&1&0&1\\\
0&0&0&1&1&1&1&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&0&0&1&0&0&0\\\
\end{array}\right),\left(\begin{array}[]{cccccccc}0&0&1&0&0&0&0&0\\\
0&1&1&1&0&0&0&0\\\ 1&1&1&1&1&1&0&0\\\ 0&1&1&2&2&2&1&1\\\ 0&0&1&2&2&2&1&0\\\
0&0&1&2&2&2&0&1\\\ 0&0&0&1&1&0&0&0\\\ 0&0&0&1&0&1&0&0\\\
\end{array}\right),\left(\begin{array}[]{cccccccc}0&0&0&1&0&0&0&0\\\
0&0&1&1&1&1&0&0\\\ 0&1&1&2&2&2&1&1\\\ 1&1&2&4&4&4&1&1\\\ 0&1&2&4&3&4&1&1\\\
0&1&2&4&4&3&1&1\\\ 0&0&1&1&1&1&0&1\\\ 0&0&1&1&1&1&1&0\\\ \end{array}\right),$
$\displaystyle\qquad\qquad\left(\begin{array}[]{cccccccc}0&0&0&0&1&0&0&0\\\
0&0&0&1&1&1&1&0\\\ 0&0&1&2&2&2&0&1\\\ 0&1&2&4&3&4&1&1\\\ 1&1&2&3&4&3&1&1\\\
0&1&2&4&3&3&1&1\\\ 0&1&0&1&1&1&1&0\\\ 0&0&1&1&1&1&0&0\\\
\end{array}\right),\left(\begin{array}[]{cccccccc}0&0&0&0&0&1&0&0\\\
0&0&0&1&1&1&0&1\\\ 0&0&1&2&2&2&1&0\\\ 0&1&2&4&4&3&1&1\\\ 0&1&2&4&3&3&1&1\\\
1&1&2&3&3&4&1&1\\\ 0&0&1&1&1&1&0&0\\\ 0&1&0&1&1&1&0&1\\\
\end{array}\right),\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&1&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&1&0&1&0&0\\\ 0&0&1&1&1&1&0&1\\\ 0&0&1&1&1&1&0&0\\\
0&1&0&1&1&1&1&0\\\ 0&0&0&1&0&0&0&0\\\ 1&0&0&0&1&0&0&0\\\
\end{array}\right),\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&1\\\
0&0&0&0&0&1&0&0\\\ 0&0&0&1&1&0&0&0\\\ 0&0&1&1&1&1&1&0\\\ 0&1&0&1&1&1&0&1\\\
0&0&1&1&1&1&0&0\\\ 1&0&0&0&0&1&0&0\\\ 0&0&0&1&0&0&0&0\\\ \end{array}\right).$
(Here, the $j$, $k$ entry of the $i$-th matrix gives the multiplicity of
$X_{k}$ in $X_{i}X_{j}$.) The bimodule category between the even parts has
left module structure
$\displaystyle\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
\end{array}\right),\left(\begin{array}[]{cccccc}1&1&0&0&0&0\\\ 1&1&1&0&0&0\\\
0&1&1&1&0&0\\\ 0&0&1&2&1&1\\\ 0&0&0&1&1&0\\\ 0&0&0&1&0&1\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&1&1&0&0&0\\\ 1&1&1&1&0&0\\\
1&1&1&2&1&1\\\ 0&1&2&4&2&2\\\ 0&0&1&2&0&1\\\ 0&0&1&2&1&0\\\
\end{array}\right),$
$\displaystyle\qquad\qquad\left(\begin{array}[]{cccccc}0&0&1&1&0&0\\\
0&1&1&2&1&1\\\ 1&1&2&4&2&2\\\ 1&2&4&8&3&3\\\ 0&1&2&3&1&2\\\ 0&1&2&3&2&1\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&0&0&1&1&1\\\ 0&0&1&3&1&1\\\
0&1&3&5&2&2\\\ 1&3&5&9&4&4\\\ 1&1&2&4&2&1\\\ 1&1&2&4&1&2\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&0&0&1&0&0\\\ 0&0&1&1&1&1\\\
0&1&1&3&1&1\\\ 1&1&3&5&2&2\\\ 0&1&1&2&1&1\\\ 0&1&1&2&1&1\\\
\end{array}\right)$
and right module structure
$\displaystyle\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\ 1&1&0&0&0&0\\\
0&1&1&0&0&0\\\ 0&0&1&1&0&0\\\ 0&0&0&1&1&0\\\ 0&0&0&1&0&1\\\ 0&0&0&0&0&1\\\
0&0&0&0&1&0\\\ \end{array}\right),\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\\
1&1&1&0&0&0\\\ 1&1&1&1&0&0\\\ 0&1&1&2&1&1\\\ 0&0&1&2&1&1\\\ 0&0&1&2&1&1\\\
0&0&0&1&0&0\\\ 0&0&0&1&0&0\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&0&1&0&0&0\\\ 0&1&1&1&0&0\\\
1&1&1&2&1&1\\\ 1&1&2&4&2&2\\\ 0&1&2&4&1&2\\\ 0&1&2&4&2&1\\\ 0&0&1&1&1&0\\\
0&0&1&1&0&1\\\ \end{array}\right),$
$\displaystyle\qquad\qquad\left(\begin{array}[]{cccccc}0&0&0&1&0&0\\\
0&0&1&2&1&1\\\ 0&1&2&4&2&2\\\ 1&2&4&8&3&3\\\ 1&2&4&7&3&3\\\ 1&2&4&7&3&3\\\
0&1&1&2&1&1\\\ 0&1&1&2&1&1\\\
\end{array}\right),\left(\begin{array}[]{cccccc}0&0&0&0&1&0\\\ 0&0&0&1&0&1\\\
0&0&1&2&1&0\\\ 0&1&2&3&1&2\\\ 1&1&1&3&2&1\\\ 0&1&2&3&1&1\\\ 1&0&0&1&0&1\\\
0&0&1&1&0&0\\\ \end{array}\right),\left(\begin{array}[]{cccccc}0&0&0&0&0&1\\\
0&0&0&1&1&0\\\ 0&0&1&2&0&1\\\ 0&1&2&3&2&1\\\ 0&1&2&3&1&1\\\ 1&1&1&3&1&2\\\
0&0&1&1&0&0\\\ 1&0&0&1&1&0\\\ \end{array}\right).$
From this, we calculate
${\mathcal{M}}=\left(\begin{array}[]{cccccccccccccc}6&5&8&13&15&9&6&5&8&13&12&12&3&3\\\
5&19&26&45&52&28&7&17&28&43&41&41&11&11\\\
8&26&56&93&110&60&6&28&54&95&84&84&26&26\\\
13&45&93&181&211&115&13&45&93&181&163&163&48&48\\\
15&52&110&211&259&138&16&51&111&210&199&199&60&60\\\
9&28&60&115&138&79&8&29&59&116&108&108&30&30\\\
6&7&6&13&16&8&8&5&8&11&13&13&3&3\\\ 5&17&28&45&51&29&5&17&28&45&40&40&11&11\\\
8&28&54&93&111&59&8&28&54&93&85&85&26&26\\\
13&43&95&181&210&116&11&45&93&183&162&162&48&48\\\
12&41&84&163&199&108&13&40&85&162&154&154&45&45\\\
12&41&84&163&199&108&13&40&85&162&154&154&45&45\\\
3&11&26&48&60&30&3&11&26&48&45&45&15&15\\\
3&11&26&48&60&30&3&11&26&48&45&45&15&15\\\ \end{array}\right)$
and find
${\mathcal{D}}=50\zeta_{13}^{11}+50\zeta_{13}^{10}-125\zeta_{13}^{9}-125\zeta_{13}^{7}-125\zeta_{13}^{6}-125\zeta_{13}^{4}+50\zeta_{13}^{3}+50\zeta_{13}^{2}+170,$
and
$\displaystyle v_{EH1}=\Big{\\{}$ $\displaystyle 1,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}+\zeta_{13}^{3}+\zeta_{13}^{2}+2,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}-\zeta_{13}^{9}-\zeta_{13}^{7}-\zeta_{13}^{6}-\zeta_{13}^{4}+\zeta_{13}^{3}+\zeta_{13}^{2}+3,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}-3\zeta_{13}^{9}-3\zeta_{13}^{7}-3\zeta_{13}^{6}-3\zeta_{13}^{4}+\zeta_{13}^{3}+\zeta_{13}^{2}+4,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}-4\zeta_{13}^{9}-4\zeta_{13}^{7}-4\zeta_{13}^{6}-4\zeta_{13}^{4}+\zeta_{13}^{3}+\zeta_{13}^{2}+4,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}-2\zeta_{13}^{9}-2\zeta_{13}^{7}-2\zeta_{13}^{6}-2\zeta_{13}^{4}+\zeta_{13}^{3}+\zeta_{13}^{2}+2,$
$\displaystyle 0,0,0,0,0,0,0,0\Big{\\}}$ $\displaystyle v_{EH2}=\Big{\\{}$
$\displaystyle 0,0,0,0,0,0,1,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}+\zeta_{13}^{3}+\zeta_{13}^{2}+2,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}-\zeta_{13}^{9}-\zeta_{13}^{7}-\zeta_{13}^{6}-\zeta_{13}^{4}+\zeta_{13}^{3}+\zeta_{13}^{2}+3,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}-3\zeta_{13}^{9}-3\zeta_{13}^{7}-3\zeta_{13}^{6}-3\zeta_{13}^{4}+\zeta_{13}^{3}+\zeta_{13}^{2}+4,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}-3\zeta_{13}^{9}-3\zeta_{13}^{7}-3\zeta_{13}^{6}-3\zeta_{13}^{4}+\zeta_{13}^{3}+\zeta_{13}^{2}+3,$
$\displaystyle\zeta_{13}^{11}+\zeta_{13}^{10}-3\zeta_{13}^{9}-3\zeta_{13}^{7}-3\zeta_{13}^{6}-3\zeta_{13}^{4}+\zeta_{13}^{3}+\zeta_{13}^{2}+3,$
$\displaystyle-\zeta_{13}^{9}-\zeta_{13}^{7}-\zeta_{13}^{6}-\zeta_{13}^{4}+1,$
$\displaystyle-\zeta_{13}^{9}-\zeta_{13}^{7}-\zeta_{13}^{6}-\zeta_{13}^{4}+1\Big{\\}}$
where $\zeta_{13}=\exp(2\pi i/13)$ is a primitive $13$-th root of unity.
This ${\mathcal{M}}$ has rank $6$. However its leading $6$-by-$6$ minor is
singular; we need to need permute the rows and columns before we can apply
Theorem 2.2. In fact, the computational difficulty of the subsequent
calculations depends on the choice made here. The rule of thumb we use is to
first permute rows and columns so that the diagonal entries are increasing,
and then take the lexicographically least 6 element subset of the rows and
columns so that the corresponding minor is non-singular.
We obtain
${\mathcal{M}}^{\prime}=\left(\begin{array}[]{cccccc}6&6&3&5&9&13\\\
6&8&3&5&8&13\\\ 3&3&15&11&30&48\\\ 5&5&11&17&29&45\\\ 9&8&30&29&79&115\\\
13&13&48&45&115&181\\\ \end{array}\right)$
with
${\mathcal{R}}=\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\ -1&1&0&1&0&0\\\
1&-1&1&1&0&0\\\ 0&0&0&0&0&1\\\ -1&1&2&0&1&0\\\ 0&0&0&0&1&0\\\ 0&1&0&0&0&0\\\
0&0&0&1&0&0\\\ 0&0&1&1&0&0\\\ 1&-1&0&0&0&1\\\ -1&1&1&0&1&0\\\ -1&1&1&0&1&0\\\
0&0&1&0&0&0\\\ 0&0&1&0&0&0\\\ \end{array}\right)$
In about 2 minutes of computer time, we find that ${\mathcal{M}}^{\prime}$ has
a unique algebraic decomposition. Preparing the corresponding unique algebraic
decomposition of ${\mathcal{M}}$ according to Theorem 2.2, we find the
combinatorial induction functors for the extended Haagerup subfactor given on
the first page of this article.
This method also uniquely finds the combinatorial data of the induction
functor for the Haagerup subfactor (which has appeared already in [Izu01]) and
for the Asaeda-Haagerup subfactor, where it is given by
$\displaystyle I_{AH1}$
$\displaystyle=\left(\begin{array}[]{cccccccccccccccccccccc}1&1&0&0&0&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&0&1&1&1&2&1&1&0&1&1&1&1&1&1&1&1&1&0&0&0&0\\\
1&2&1&3&1&1&2&2&0&1&2&2&2&2&2&2&2&2&2&2&0&2\\\
0&1&2&2&1&1&1&2&0&1&1&1&1&1&1&1&1&1&2&2&1&1\\\
1&3&3&2&1&1&3&3&0&1&3&3&3&3&3&3&3&3&3&3&2&1\\\
2&2&2&1&1&0&1&0&0&1&1&1&1&1&1&1&1&1&1&1&1&0\\\ \end{array}\right)$
$\displaystyle I_{AH2}$
$\displaystyle=\left(\begin{array}[]{cccccccccccccccccccccc}1&0&0&0&0&2&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&1&1&1&1&1&1&1&0&0&1&1&1&1&1&1&1&1&0&0&0&0\\\
1&1&1&3&1&2&2&2&0&2&2&2&2&2&2&2&2&2&2&2&0&2\\\
0&2&2&2&1&1&2&3&0&0&2&2&2&2&2&2&2&2&2&2&1&1\\\
1&3&3&2&1&0&2&2&0&1&2&2&2&2&2&2&2&2&3&3&2&1\\\
1&1&1&0&0&0&1&0&0&1&1&1&1&1&1&1&1&1&1&1&1&0\\\
1&1&1&0&0&0&1&0&0&1&1&1&1&1&1&1&1&1&1&1&1&0\\\
1&1&1&1&1&1&1&1&0&0&1&1&1&1&1&1&1&1&0&0&0&0\\\
1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \end{array}\right)$
For some fusion categories, however, for example the even part of the 4442
subfactor described in [MP12], this method seems to be insufficient, or to at
least require a faster implementation. (We stopped the search after a day of
computer time.)
This approach to computing the combinatorial induction functor does not appear
to be useful for ruling out candidate fusion rings; so far we haven’t found an
interesting example. In fact, often the method does not produce a unique
answer. A simple example is for the principal graphs
$\left({\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[height=17.07164pt]{diagrams/graphs/E69D51A0185FD82C}}\end{array}},{\begin{array}[]{c}\raisebox{-2.5pt}{\includegraphics[height=17.07164pt]{diagrams/graphs/E69D51A0185FD82C}}\end{array}}\right)$
which have a unique compatible fusion ring, but four different compatible
combinatorial induction functors, with either 4, 6, 7, or 12 simple objects in
the centre.
## References
* [BMPS12] Stephen Bigelow, Scott Morrison, Emily Peters, and Noah Snyder. Constructing the extended Haagerup planar algebra. Acta Math., 209(1):29–82, 2012. arXiv:0909.4099 MR2979509 DOI:10.1007/s11511-012-0081-7.
* [EG98] Pavel Etingof and Shlomo Gelaki. Some properties of finite-dimensional semisimple Hopf algebras. Math. Res. Lett., 5(1-2):191--197, 1998. MR1617921 DOI:10.4310/MRL.1998.v5.n2.a5.
* [ENO05] Pavel Etingof, Dmitri Nikshych, and Viktor Ostrik. On fusion categories. Ann. of Math. (2), 162(2):581--642, 2005. arXiv:math.QA/0203060 MR2183279 DOI:10.4007/annals.2005.162.581.
* [IJMS12] Masaki Izumi, Vaughan F. R. Jones, Scott Morrison, and Noah Snyder. Subfactors of index less than 5, Part 3: Quadruple points. Comm. Math. Phys., 316(2):531--554, 2012. arXiv:1109.3190 MR2993924 DOI:10.1007/s00220-012-1472-5.
* [Izu01] Masaki Izumi. The structure of sectors associated with Longo-Rehren inclusions. II. Examples. Rev. Math. Phys., 13(5):603--674, 2001. MR1832764 DOI:10.1142/S0129055X01000818.
* [Mor13] Scott Morrison. Provide a citation for the ‘‘spine lemma’’? MathOverflow, 2013. http://mathoverflow.net/q/142456 (version: 2013-09-19).
* [MP12] Scott Morrison and David Penneys. Constructing spoke subfactors using the jellyfish algorithm, 2012. arXiv:1208.3637, to appear in Transactions of the American Mathematical Society.
* [MPPS12] Scott Morrison, David Penneys, Emily Peters, and Noah Snyder. Classification of subfactors of index less than 5, part 2: triple points. International Journal of Mathematics, 23(3):1250016, 2012. arXiv:1007.2240 MR2902285 DOI:10.1142/S0129167X11007586.
* [MS12] Scott Morrison and Noah Snyder. Subfactors of index less than 5, part 1: the principal graph odometer. Communications in Mathematical Physics, 312(1):1--35, 2012. arXiv:1007.1730 MR2914056 DOI:10.1007/s00220-012-1426-y.
* [Ost09] Victor Ostrik. On formal codegrees of fusion categories. Math. Research Letters, 16(5):895--901, 2009. arXiv:0810.3242 MR2576705.
* [PT12] David Penneys and James Tener. Classification of subfactors of index less than 5, part 4: vines. International Journal of Mathematics, 23(3):1250017 (18 pages), 2012\. arXiv:1010.3797 MR2902286 DOI:10.1142/S0129167X11007641.
|
arxiv-papers
| 2014-04-15T15:22:27 |
2024-09-04T02:50:01.197576
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Scott Morrison and Kevin Walker",
"submitter": "Scott Morrison",
"url": "https://arxiv.org/abs/1404.3955"
}
|
1404.4018
|
# On the blow-up results for a class of strongly perturbed semilinear heat
equations
V. T. Nguyen [email protected] Université Paris 13, Sorbonne
Paris Cité,
LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France.
###### Abstract
We consider in this work some class of strongly perturbed for the semilinear
heat equation with Sobolev sub-critical power nonlinearity. We first derive a
Lyapunov functional in similarity variables and then use it to derive the
blow-up rate. We also classify all possible asymptotic behaviors of the
solution when it approaches to singularity. Finally, we describe precisely the
blow-up profiles corresponding to these behaviors.
###### keywords:
Finite-time blow-up , asymptotic behavior of solutions , nonlinear parabolic
equations.
## 1 Introduction
We are interested in the following nonlinear parabolic equation:
$\left\\{\begin{array}[]{rcl}u_{t}&=&\Delta u+|u|^{p-1}u+h(u),\\\
u(0)&=&u_{0}\in L^{\infty}(\mathbb{R}^{n}),\end{array}\right.$ (1)
where $u$ is defined for $(x,t)\in\mathbb{R}^{n}\times[0,T)$, $p$ is a sub-
critical nonlinearity,
$1<p,\quad(n-2)p<n+2.$ (2)
The function $h$ is in $\mathcal{C}^{1}(\mathbb{R},\mathbb{R})$ satisfying
$j=0,1,\;|h^{(j)}(z)|\leq
M\left(\dfrac{|z|^{p-j}}{\log^{a}(2+z^{2})}+1\right),\quad|h^{\prime\prime}(z)|\leq
M\dfrac{|z|^{p-2}}{\log^{a}(2+z^{2})},$ (3)
where $a>1$, $M>0$. Typically, $h(z)=\frac{\mu|z|^{p-1}z}{\log^{a}(2+z^{2})}$
with $\mu\in\mathbb{R}$.
By standard results, the problem (1) has a unique classical solution $u(x,t)$
in $L^{\infty}(\mathbb{R}^{n})$, which exists at least for small times. The
solution $u(x,t)$ may develop singularities in some finite time. We say that a
function $u:\mathbb{R}^{n}\times[0,T)\mapsto\mathbb{R}$ is a solution of (1)
if $u$ solves (1) and satisfies
$u,u_{t},\nabla u,\nabla^{2}u\;\text{are bounded and continuous
on}\;\mathbb{R}^{n}\times[0,\tau],\;\forall\tau<T.$ (4)
It is said that $u(x,t)$ blows up in a finite time $T<+\infty$ if $u(x,t)$
satisfies (1), (4) and
$\lim_{t\to T}\|u(t)\|_{L^{\infty}(\mathbb{R}^{n})}=+\infty.$
Here we call $T$ the blow-up time of $u(x,t)$. In such a blow-up case, a point
$x_{0}\in\mathbb{R}^{n}$ is called a blow-up point of $u(x,t)$ if and only if
there exist $(x_{n},t_{n})\to(x_{0},T)$ such that $|u(x_{n},t_{n})|\to+\infty$
as $n\to+\infty$.
Consider $v$ a positive blow-up solution of the associated ODE of (1). It is
clear that $v$ is given by
$v^{\prime}=v^{p}+h(v),\quad v(T)=+\infty,\quad\text{for some $T>0$.}$ (5)
Since the blow-up solution of (5) satisfies (see Lemma A.1)
$v(t)\sim\kappa(T-t)^{-\frac{1}{p-1}}\quad\text{as $t\to T$, where
$\kappa=(p-1)^{-\frac{1}{p-1}}$},$ (6)
it is natural to ask whether the blow-up solution $u(t)$ of (1) has the same
blow-up rate as $v(t)$ does. More precisely, are there constants $c,C>0$ such
that
$c(T-t)^{-\frac{1}{p-1}}\leq\|u(t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C(T-t)^{-\frac{1}{p-1}},\quad\forall t\in(0,T)?$ (7)
By a simple argument based on Duhamel’s formula, we can show that the lower
bound in (7) is always satisfied (see [19]). For the upper blow-up rate
estimate, it is much less simple and requires more work. Practically, we
define for all $x_{0}\in\mathbb{R}^{n}$ ($x_{0}$ may be a blow-up point of $u$
or not) the following _similarity variables_ introduced in Giga and Kohn [4,
5, 6]:
$y=\frac{x-x_{0}}{\sqrt{T-t}},\quad s=-\log(T-t),\quad
w_{x_{0},T}=(T-t)^{\frac{1}{p-1}}u(x,t).$ (8)
Hence $w_{x_{0},T}$ satisfies for all $s\geq-\log{T}$ and for all
$y\in\mathbb{R}^{n}$:
$\partial_{s}w_{x_{0},T}=\frac{1}{\rho}\text{div}(\rho\nabla
w_{x_{0},T})-\frac{w_{x_{0},T}}{p-1}+|w_{x_{0},T}|^{p-1}w_{x_{0},T}+e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}w_{x_{0},T}\right),$
(9)
where
$\rho(y)=\left(\frac{1}{4\pi}\right)^{n/2}e^{-\frac{|y|^{2}}{4}}.$ (10)
Here, we say that $w:\mathbb{R}^{n}\times[-\log T,+\infty)\mapsto\mathbb{R}$
is a solution of (9) if $w$ solves (9) and satisfies
$w,w_{s},\nabla w,\nabla^{2}w\;\text{are bounded and continuous
on}\;\mathbb{R}^{n}\times[-\log T,S],\;\forall S<+\infty.$ (11)
We can see that the study of $u$ in the neighborhood of $(x_{0},T)$ is
equivalent to the study of the long-time behavior of $w_{x_{0},T}$ and each
result for $u$ has an equivalent formulation in term of $w_{x_{0},T}$. In
particular, the proof of the upper bound in (7) is now equivalent to showing
that there exists a time $\hat{s}\geq-\log T$ large enough such that
$\|w_{x_{0},T}(s)\|_{L^{\infty}(\mathbb{R}^{n})}\leq C,\quad\forall
s\geq\hat{s}.$ (12)
We remark that the perturbation term added to equation (9) satisfies the
following inequality,
$j=0,1,\quad
e^{-\frac{(p-j)s}{p-1}}\left|h^{(j)}\left(e^{\frac{s}{p-1}}z\right)\right|\leq\frac{C_{0}}{s^{a}}\left(|z|^{p-j}+1\right),\quad\forall
s\geq s_{0},$ (13)
for some $C_{0}>0$ and $s_{0}>0$ (see Lemma A.2 for a proof of this fact).
When $h\equiv 0$, Giga and Kohn proved (12) in [5] for $1<p<\frac{3n+8}{3n-4}$
or for non-negative initial data (so that the solution is positive everywhere)
with sub-critical $p$. Estimate (12) is extended for all $p$ satisfying (2)
without assuming non-negativity for initial data $u_{0}$ by Giga, Matsui and
Sasayama in [7]. The proof written in [7] is strongly based on the existence
of the following Lyapunov functional:
$\mathcal{E}_{0}[w](s)=\int_{\mathbb{R}^{n}}\left(\frac{1}{2}|\nabla
w|^{2}+\frac{1}{2(p-1)}|w|^{2}-\frac{1}{p+1}|w|^{p+1}\right)\rho dy.$ (14)
Based on this functional, some energy estimates related to this structure and
a bootstrap argument given in [14], the authors in [7] have established the
following key integral estimate
$\sup_{s\geq
s^{\prime}}\int_{s}^{s+1}\|w_{x_{0},T}(s)\|_{L^{p+1}(\mathbf{B}_{R})}^{(p+1)q}ds\leq
C_{q,s^{\prime}},\quad\forall q\geq 2,\quad s^{\prime}>-\log T.$ (15)
Since this estimate holds for all $q\geq 2$, we obtain an upper bound for
$w_{x_{0},T}$ which yields (12).
When $h\not\equiv 0$, we wonder whether a perturbation of the method of [7]
would work for our problem. A key step is to find a Lyapunov functional for
equation (9). Following the method introduced by Hamza and Zaag in [9, 8] for
perturbations of the semilinear wave equation, we introduce
$\mathcal{J}[w](s)=\mathcal{E}[w](s)e^{\frac{\gamma}{a-1}s^{1-a}}+\theta
s^{1-a},$ (16)
where $\gamma=8C_{0}\left(\frac{p+1}{p-1}\right)^{2}$ and $\theta>0$ is
sufficiently large constant which will be determined later,
$\mathcal{E}[w]=\mathcal{E}_{0}[w]+\mathcal{I}[w],\quad\mathcal{I}[w](s)=-e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy,$ (17)
with $H(z)=\int_{0}^{z}h(\xi)d\xi$.
With this introduction, we derive that the functional $\mathcal{J}[w]$ is a
decreasing function of time for equation (9), provided that $s$ is large
enough. More precisely, we have the following:
###### Theorem 1 (Existence of a Lyapunov functional for equation (9)).
Let $a,p,n,M$ be fixed, consider $w$ a solution of equation (9) satisfying
(11). Then there exist $\hat{s}_{0}=\hat{s}_{0}(a,p,n,M)\geq s_{0}$ and
$\hat{\theta}_{0}=\hat{\theta}_{0}(a,p,n,M)$ such that if
$\theta\geq\hat{\theta}_{0}$, then $\mathcal{J}$ satisfies the following
inequality, for all $s_{2}>s_{1}\geq\max\\{\hat{s}_{0},-\log T\\}$,
$\mathcal{J}[w](s_{2})-\mathcal{J}[w](s_{1})\leq-\frac{1}{2}\int_{s_{1}}^{s_{2}}\int_{\mathbb{R}^{n}}(\partial_{s}w)^{2}\rho
dyds.$ (18)
As mentioned above, the existence of this Lyapunov functional $\mathcal{J}$ is
a crucial step in the derivation of the blow-up rate for equation (1). Indeed,
with the functional $\mathcal{J}$ and some more work, we are able to adapt the
analysis in [7] for equation (1) in the case $h\equiv 0$ and get the following
result:
###### Theorem 2 (Blow-up rate for equation (1)).
Let $a,p,n,M$ be fixed, $p$ satisfy (2). There exists
$\hat{s}_{1}=\hat{s}_{1}(a,p,n,M)\geq\hat{s}_{0}$ such that if $u$ is a blow-
up solution of equation (1) with a blow-up time $T$, then
$(i)$ for all $s\geq s^{\prime}=\max\\{\hat{s}_{1},-\log T\\}$,
$\|w_{x_{0},T}(y,s)\|_{L^{\infty}(\mathbb{R}^{n})}\leq C,$ (19)
where $w_{x_{0},T}$ is defined in (8) and $C$ is a positive constant depending
only on $n,p,M$ and a bound of $\|w_{x_{0},T}(\hat{s}_{0})\|_{L^{\infty}}$.
$(ii)$ For all $t\in[t_{1},T)$ where $t_{1}=T-e^{-s^{\prime}}$,
$\|u(x,t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq C(T-t)^{-\frac{1}{p-1}}.$ (20)
###### Remark 1.
The proof of Theorem 2 is far from being a straightforward adaptation of [7].
Indeed, three major difficulties arise in our case and make the heart of our
contribution:
\- the existence of a Lyapunov functional in similarity variables (see Theorem
1 above),
\- the control of the $L^{2}$-norm in terms of the energy (see $(ii)$ of
Proposition 8, where we rely on a new blow-up criterion greatly simplifying
the approach in [5]),
\- the proof of a nonlinear parabolic result (see Proposition 12 below).
The estimate obtained in Theorem 2 is a fundamental step in studying the
asymptotic behavior of blow-up solutions. When $h\equiv 0$, Giga and Kohn in
[5, 6] (see also [4]) obtained the following result: For a given blow-up point
$x_{0}$, it holds that
$\lim_{s\to+\infty}w_{x_{0},T}(y,s)=\lim_{t\to
T}(T-t)^{\frac{1}{p-1}}u(x_{0}+y\sqrt{T-t},t)=\pm\kappa,$
where $\kappa=(p-1)^{\frac{1}{p-1}}$, uniformly on compact subsets of
$\mathbb{R}^{n}$. The result is pointwise in $x_{0}$. Besides, for a.e. $y$,
$\lim_{s\to+\infty}\nabla w_{x_{0},T}(y,s)=0$.
For our problem, when $h\not\equiv 0$ and $h$ is given in (3), we also derive
an analogous result on the behavior of $w_{x_{0},T}$ as $s\to+\infty$. We
claim the following:
###### Theorem 3 (Behavior of $w_{x_{0},T}$ as $s\to+\infty$).
Let $a,p,n,M$ be fixed, $p$ satisfy (2). Consider $u(t)$ a solution of
equation (1) which blows up at time $T$ and $x_{0}$ a blow-up point. Then
$\lim_{t\to
T}(T-t)^{\frac{1}{p-1}}u(x_{0}+y\sqrt{T-t},t)=\lim_{s\to+\infty}\,w_{x_{0},T}(y,s)=\pm\kappa,$
holds in $L^{2}_{\rho}$ ($L^{2}_{\rho}$ is the weighted $L^{2}$ space
associated with the weight $\rho$ (10)), and also uniformly on each compact
subset of $\mathbb{R}^{n}$.
Up to changing $u_{0}$ in $-u_{0}$ and $h$ in $-h$, we may assume that
$w\to\kappa$ in $L^{2}_{\rho}$ as $s\to+\infty$. Let us consider $\phi$ a
positive solution of the associated ordinary differential equation of equation
(9)
$\phi_{s}=-\frac{\phi}{p-1}+\phi^{p}+e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}\phi\right)$
(21)
such that
$\phi(s)=\kappa+\mathcal{O}\left(\frac{1}{s^{a}}\right)\quad\text{as}\quad
s\to+\infty,$ (22)
(see Lemma A.3 for a proof of the existence of $\phi$).
Let us introduce $v_{x_{0},T}=w_{x_{0},T}-\phi(s)$, then
$\|v_{x_{0},T}(y,s)\|_{L^{2}_{\rho}}\to 0$ as $s\to+\infty$ and $v_{x_{0},T}$
(or $v$ for simplicity) satisfies the following equation:
$\partial_{s}v=(\mathcal{L}+\omega(s))v+F(v)+H(v,s),\quad\forall
y\in\mathbb{R}^{n},\;\forall s\in[-\log T,+\infty),$
where $\mathcal{L}=\Delta-\frac{y}{2}\cdot\nabla+1$ and $\omega$, $F$, $H$
satisfy
$|\omega(s)|=\mathcal{O}(s^{-a})\quad\text{and}\quad|F(v)|+|H(v,s)|=\mathcal{O}(|v|^{2})\quad\text{as
$s\to+\infty$},$
(see the beginning of Section 3 for the proper definitions of $\omega$, $F$
and $G$).
Since the linear part will play an important role in our analysis, let us
point out its properties. The operator $\mathcal{L}$ is self-adjoint on
$L^{2}_{\rho}(\mathbb{R}^{n})$. Its spectrum is given by
$spec(\mathcal{L})=\\{1-\frac{m}{2},\;m\in\mathbb{N}\\},$
and it consists of eigenvalues. The eigenfunctions of $\mathcal{L}$ are
derived from Hermite polynomials:
\- For $n=1$, the eigenfunction corresponding to $1-\frac{m}{2}$ is
$h_{m}(y)=\sum_{k=0}^{\left[\frac{m}{2}\right]}\frac{m!}{k!(m-2k)!}(-1)^{k}y^{m-2k},$
(23)
\- For $n\geq 2$: we write the spectrum of $\mathcal{L}$ as
$spec(\mathcal{L})=\\{1-\frac{|m|}{2},\;|m|=m_{1}+\dots+m_{n},\;(m_{1},\dots,m_{n})\in\mathbb{N}^{n}\\}.$
For $m=(m_{1},\dots,m_{n})\in\mathbb{N}^{n}$, the eigenfunction corresponding
to $1-\frac{|m|}{2}$ is
$H_{m}(y)=h_{m_{1}}(y_{1})\dots h_{m_{n}}(y_{n}),$ (24)
where $h_{m}$ is defined in (23).
By studying the behavior of $v$ as $s\to+\infty$, we obtain the following
result:
###### Theorem 4 (Classification of the behavior of $w$ as $s\to+\infty$).
Consider $u(t)$ a solution of equation (1) which blows-up at time $T$ and
$x_{0}$ a blow-up point. Let $w(y,s)$ be a solution of equation (9). Then one
of the following possibilities occurs:
$i)\;$ $w(y,s)\equiv\phi(s)$,
$ii)$ There exists $l\in\\{1,\dots,n\\}$ such that up to an orthogonal
transformation of coordinates, we have
$w(y,s)=\kappa-\frac{\kappa}{4ps}\left(\sum_{j=1}^{l}y_{j}^{2}-2l\right)+\mathcal{O}\left(\frac{1}{s^{a}}\right)+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right)\quad\text{as}\quad s\to+\infty.$
$iii)$ There exist an integer number $m\geq 3$ and constants $c_{\alpha}$ not
all zero such that
$w(y,s)=\phi(s)-e^{-\left(\frac{m}{2}-1\right)s}\sum_{|\alpha|=m}c_{\alpha}H_{\alpha}(y)+o\left(e^{-\left(\frac{m}{2}-1\right)s}\right)\quad\text{as}\quad
s\to+\infty.$
The convergence takes place in $L^{2}_{\rho}$ as well as in
$\mathcal{C}^{k,\gamma}_{loc}$ for any $k\geq 1$ and some $\gamma\in(0,1)$.
###### Remark 2.
Applying our result to a space-independent solution of (9), we get the
uniqueness of the solution of the ODE (21) that converges to $\kappa$ as
$s\to+\infty$.
###### Remark 3.
Since both the perturbed ($h\not\equiv 0$) and the unperturbed ($h\equiv 0$)
cases in equation (1) share the same convergence stated in Theorem 4, we
wonder whether the perturbation $h$ may have an influence on further terms of
the expansion of $w$. From our result, if case $(ii)$ occurs, we see no
difference in the following term of the expansion. On the contrary, if case
$(i)$ or $(iii)$ occurs, with $h(x)=\mu\frac{|x|^{p-1}x}{\log(2+x^{2})}$, we
see from Lemma A.3 that
$w(y,s)-\kappa\sim\frac{C_{0}(a,p,\mu)}{s^{a}}\quad\text{as $s\to+\infty$},$
which is clearly different from the unperturbed case when in case $(i)$, we
have $w\equiv\kappa$ and case $(iii)$, we have $w-\kappa=\mathcal{O}(e^{-s})$,
(see [10], [18]).
###### Remark 4.
If we linearize $w$ around $\kappa$, which is an explicit profile, we then
fall in logarithmic scales $\mu=\frac{1}{|\log\epsilon|}$ with $\epsilon=T-t$.
Further refinements in this direction should give an expansion of $w-\kappa$
in terms of powers of $\mu$, i.e in logarithmic scales of $\epsilon$.
Therefore, we can not reach significantly small error terms in the expansion
of the solution $w$ as $(iii)$ of Theorem 4 describes. In order to escape this
situation, a relevant approximation is required in order to go beyond all
logarithmic scales, i.e approximations up to lower order terms such as
$\epsilon^{\alpha}$ for some $\alpha>0$. Our idea to capture such relevant
terms is to abandon the explicit profile obtained as a first order
approximation, namely $\kappa$, and take an implicit profile function as a
first order description of the singular behavior, namely $\phi(s)$ introduced
in (21) and (22). A similar idea was used by Zaag [20] where the solution was
linearized around a less explicit profile function in order to go beyond all
logarithmic scales. For our problem, we particularly take $\phi(s)$ as the
implicit profile function, which is a solution of the associated ODE of
equation (9) in $w$ such that $\phi(s)\to\kappa$ as $s\to+\infty$. By
linearizing the solution $w$ around $\phi$, we can get to error terms of
polynomial order $\epsilon^{\left(\frac{m}{2}-1\right)}$, as stated in $(iii)$
of Theorem 4.
###### Remark 5.
When $h(x)=|x|^{q}$ with $q\in(1,p)$, we see that
$\phi(s)-\kappa\sim C_{0}^{\prime}(p,q)e^{-\lambda s}\quad\text{as}\quad
s\to+\infty.$
If case $(ii)$ in Theorem 4 holds, we then recover the same expansion as in
the unperturbed case $(h\equiv 0)$. On the contrary, if case $(i)$ or $(iii)$
occurs, then
$w(y,s)-\kappa\sim C_{0}^{\prime}(p,q)e^{-\lambda s}\quad\text{as}\quad
s\to+\infty.$
Moreover, if case $(iii)$ in Theorem 4 holds, we have new terms in the
expansion of $w$ which was not available in the unperturbed case, namely
$w(y,s)=\kappa-\sum_{k=1}^{K}C_{k}e^{-k\lambda
s}-e^{-\left(\frac{m}{2}-1\right)s}\sum_{|\alpha|=m}c_{\alpha}H_{\alpha}(y)+o\left(e^{-\left(\frac{m}{2}-1\right)s}\right)\;\text{as
$s\to+\infty$},$
where $C_{k},k=1,2,\dots,K$ are some constants depending on $p$ and $q$, and
$K\in\mathbb{N}$ is the integer part of
$\frac{1}{\lambda}\left(\frac{m}{2}-1\right)$.
In the last section, we will extend the asymptotic behavior of $w$ obtained in
Theorem 4 to larger regions. Particularly, we claim the following:
###### Theorem 5 (Convergence extension of $w_{a}$ to larger regions).
For all $K_{0}>0$,
$i)$ if $ii)$ of Theorem 4 occurs, then
$\sup_{|\xi|\leq
K_{0}}\left|w(\xi\sqrt{s},s)-f_{l}(\xi)\right|=\mathcal{O}\left(\frac{1}{s^{a-1}}\right)+\mathcal{O}\left(\frac{\log
s}{s}\right)\quad\text{as}\;s\to+\infty,$ (25)
where
$f_{l}(\xi)=\kappa\left(1+\frac{p-1}{4p}\sum_{j=1}^{l}\xi_{j}^{2}\right)^{-\frac{1}{p-1}},\quad\forall\xi\in\mathbb{R}^{n},$
with $l$ the same as in $ii)$ of Theorem 4.
$ii)$ if $iii)$ of Theorem 4 occurs, then $m\geq 4$ is even, and
$\sup_{|\xi|\leq K_{0}}\left|w\left(\xi
e^{\left(\frac{1}{2}-\frac{1}{m}\right)s}\right)-\psi_{m}(\xi)\right|\to
0\quad\text{as}\;s\to+\infty,$ (26)
where
$\psi_{m}(\xi)=\kappa\left(1+\kappa^{-p}\sum_{|\alpha|=m}c_{\alpha}\xi^{\alpha}\right)^{-\frac{1}{p-1}},\quad\forall\xi\in\mathbb{R}^{n},$
with $c_{\alpha}$ the same as in Theorem 4.
Let us mention briefly the structure of the paper. In Section 2, we prove the
existence of Lyapunov functional for equation (9) (Theorem 1), we then get
Theorem 2 and Theorem 3. In Section 3, we follow the method of [3] and [18] to
prove Theorem 4. Finally, the section 4 is devoted to the proof of Theorem 5.
Acknowledgement: The author is grateful to H. Zaag for his dedicated advice,
suggestions and remarks during the preparation of this paper.
## 2 A Lyapunov functional
This section is divided in four subsections: we first prove the existence of a
Lyapunov functional for equation (9) (Theorem 1); after that, we derive a
blow-up criterion for equation (9) and some energy estimates based on this
Lyapunov functional. Following the method of [7], we prove the boundedness of
solution in similarity variables which determines the blow-up rate for
solution of (1) (Theorem 2). Finally, we derive the limit of $w$ as
$s\to+\infty$, which concludes Theorem 3.
In what follows, we denote by $C_{i},i=0,1,\dots$ positive constants depending
only on $a,n,p,M$, and by $L^{q}_{\rho}(\Omega)$ the weighted $L^{q}(\Omega)$
space endowed with the norm
$\|\varphi\|_{L^{q}_{\rho}(\Omega)}=\left(\int_{\Omega}|\varphi(y)|^{q}\rho(y)dy\right)^{\frac{1}{q}},$
and by $H^{1}_{\rho}(\Omega)$ the space of function $\varphi\in
L^{2}_{\rho}(\Omega)$ satisfying $\nabla\varphi\in L^{2}_{\rho}(\Omega)$,
endowed with the norm
$\|\varphi\|_{H^{1}_{\rho}(\Omega)}=\left(\|\varphi\|_{L^{2}_{\rho}(\Omega)}^{2}+\frac{1}{p-1}\left\|\nabla\varphi\right\|_{L^{2}_{\rho}(\Omega)}^{2}\right)^{\frac{1}{2}}.$
We denote by $\mathbf{B}_{R}(x)$ the open ball in $\mathbb{R}^{n}$ with center
$x$ and radius $R$, and set $\mathbf{B}_{R}:=\mathbf{B}_{R}(0)$.
### 2.1 Existence of a Lyapunov function
In this part, we aim at proving that the functional $\mathcal{J}$ defined in
(16) is a Lyapunov functional for equation (9). Note that that functional is
far from being trivial and it is our main contribution. We first claim the
following lemma:
###### Lemma 6.
Let $a,p,n,M$ be fixed and $w$ be solution of equation (9) satisfying (11).
There exists $\tilde{s}_{0}=\tilde{s}_{0}(a,p,n,M)\geq s_{0}$ such that the
functional of $\mathcal{E}$ defined in (17) satisfies the following
inequality, for all $s\geq\max\\{\tilde{s}_{0},-\log T\\}$,
$\frac{d}{ds}\mathcal{E}[w](s)\leq-\frac{1}{2}\int_{\mathbb{R}^{n}}w_{s}^{2}\rho
dy+\gamma s^{-a}\mathcal{E}[w](s)+Cs^{-a},$ (27)
where $\gamma=8C_{0}\left(\frac{p+1}{p-1}\right)^{2}$, $C_{0}$ is introduced
in (13) and $C$ is a positive constant depending only on $a,p,n,M$.
Let us first derive Theorem 1 from Lemma 6 which will be proved later.
###### Proof of Theorem 1 admitting Lemma 6.
Differentiating the functional $\mathcal{J}$ defined in (16), we obtain
$\displaystyle\frac{d}{ds}\mathcal{J}[w](s)$
$\displaystyle=\frac{d}{ds}\left\\{\mathcal{E}[w](s)e^{\frac{\gamma}{a-1}s^{1-a}}+\theta
s^{1-a}\right\\}$
$\displaystyle=\frac{d}{ds}\mathcal{E}[w](s)e^{\frac{\gamma}{a-1}s^{1-a}}-\gamma
s^{-a}\mathcal{E}[w](s)e^{\frac{\gamma}{a-1}s^{1-a}}-(a-1)\theta s^{-a}$
$\displaystyle\leq-\frac{1}{2}e^{\frac{\gamma}{a-1}s^{1-a}}\int_{\mathbb{R}^{n}}w_{s}^{2}\rho
dy+\left[Ce^{\frac{\gamma}{a-1}s^{1-a}}-(a-1)\theta\right]s^{-a}\quad\text{(use
\eqref{equ:estimateDE}).}$
Choosing $\theta$ large enough such that
$Ce^{\frac{\gamma}{a-1}{\tilde{s}_{0}}^{1-a}}-(a-1)\theta\leq 0$ and noticing
that $e^{\frac{\gamma}{a-1}s^{1-a}}\geq 1$ for all $s>0$, we derive
$\frac{d}{ds}\mathcal{J}[w](s)\leq-\frac{1}{2}\int_{\mathbb{R}^{n}}w_{s}^{2}\rho
dy,\quad\forall s\geq\tilde{s}_{0}.$
This implies inequality (18) and concludes the proof of Theorem 1, assuming
that Lemma 6 holds. ∎
It remains to prove Lemma 6 in order to conclude the proof of Theorem 1.
###### Proof of Lemma 6 .
Multiplying equation (9) with $w_{s}\rho$ and integrating by parts:
$\displaystyle\int_{\mathbb{R}^{n}}|w_{s}|^{2}\rho=-\frac{d}{ds}\left\\{\int_{\mathbb{R}^{n}}\left(\frac{1}{2}|\nabla
w|^{2}+\frac{1}{2(p-1)}|w|^{2}-\frac{1}{p+1}|w|^{p+1}\right)\rho dy\right\\}$
$\displaystyle+e^{-\frac{ps}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)w_{s}\rho
dy$ .
For the last term of the above expression, denoting
$H(z)=\int_{0}^{z}h(\xi)d\xi$, we write in the following:
$\displaystyle
e^{-\frac{ps}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)w_{s}\rho
dy=e^{-\frac{(p+1)s}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)\left(e^{\frac{s}{p-1}}w_{s}+\frac{e^{\frac{s}{p-1}}}{p-1}w\right)\rho
dy$
$\displaystyle-\frac{1}{p-1}e^{-\frac{ps}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)w\rho
dy$
$\displaystyle=e^{-\frac{p+1}{p-1}s}\frac{d}{ds}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy-\frac{1}{p-1}e^{-\frac{ps}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)w\rho
dy$ .
This yields
$\displaystyle\int_{\mathbb{R}^{n}}|w_{s}|^{2}\rho
dy=-\frac{d}{ds}\left\\{\int_{\mathbb{R}^{n}}\left(\frac{1}{2}|\nabla
w|^{2}+\frac{1}{2(p-1)}|w|^{2}-\frac{1}{p+1}|w|^{p+1}\right)\rho dy\right\\}$
$\displaystyle+\frac{d}{ds}\left\\{e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy\right\\}$
$\displaystyle+\frac{p+1}{p-1}e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy$
$\displaystyle-\frac{1}{p-1}e^{-\frac{ps}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)w\rho
dy$ .
From the definition of the functional $\mathcal{E}$ given in (17), we derive a
first identity in the following:
$\displaystyle\frac{d}{ds}\mathcal{E}[w](s)=-\int_{\mathbb{R}^{n}}|w_{s}|^{2}\rho
dy+\frac{p+1}{p-1}e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy$
$\displaystyle-\frac{1}{p-1}e^{-\frac{ps}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)w\rho
dy$ . (28)
A second identity is obtained by multiplying equation (9) with $w\rho$ and
integrating by parts:
$\displaystyle\frac{d}{ds}\int_{\mathbb{R}^{n}}|w|^{2}\rho
dy=-4\left\\{\int_{\mathbb{R}^{n}}\left(\frac{1}{2}|\nabla
w|^{2}+\frac{1}{2(p-1)}|w|^{2}-\frac{1}{p+1}|w|^{p+1}\right)\rho dy\right.$
$\displaystyle\left.-e^{-\frac{(p+1)s}{p-1}}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy\right\\}$
$\displaystyle+\left(2-\frac{4}{p+1}\right)\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy-4e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy$
$\displaystyle+2e^{-\frac{ps}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)w\rho
dy$ .
Using again the definition of $\mathcal{E}$ given in (17), we derive the
second identity in the following:
$\displaystyle\frac{d}{ds}\int_{\mathbb{R}^{n}}|w|^{2}\rho dy$
$\displaystyle=-4\mathcal{E}[w](s)+2\frac{p-1}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy$
$\displaystyle-4e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy+2e^{-\frac{ps}{p-1}}\int_{\mathbb{R}^{n}}h\left(e^{\frac{s}{p-1}}w\right)w\rho
dy.$ (29)
From (28), we estimate
$\displaystyle\frac{d}{ds}\mathcal{E}[w](s)$
$\displaystyle\leq-\int_{\mathbb{R}^{n}}|w_{s}|^{2}\rho dy$
$\displaystyle+\frac{p+1}{p-1}\int_{\mathbb{R}^{n}}\left\\{\left|e^{-\frac{p+1}{p-1}s}H\left(e^{\frac{s}{p-1}}w\right)\right|+\left|e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}w\right)w\right|\right\\}\rho
dy.$
From (13) and using the fact that $|w|\leq|w|^{p+1}+1$, we obtain for all
$s\geq s_{0}$,
$\left|e^{-\frac{p+1}{p-1}s}H\left(e^{\frac{s}{p-1}}w\right)\right|+\left|e^{-\frac{p}{p-1}s}h\left(e^{\frac{s}{p-1}}w\right)w\right|\leq
2C_{0}s^{-a}\left(|w|^{p+1}+1\right).$ (30)
Using (30) yields
$\frac{d}{ds}\mathcal{E}[w](s)\leq-\int_{\mathbb{R}^{n}}|w_{s}|^{2}\rho
dy+C_{1}s^{-a}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho dy+C_{1}s^{-a},$ (31)
where $C_{1}=2C_{0}\frac{p+1}{p-1}$.
From (29), we have
$\displaystyle\int_{\mathbb{R}^{n}}|w|^{p+1}\rho dy$
$\displaystyle\leq\frac{2(p+1)}{p-1}\mathcal{E}[w](s)+\frac{p+1}{p-1}\int_{\mathbb{R}^{n}}|w_{s}w|\rho
dy$
$\displaystyle\quad+\frac{2(p+1)}{p-1}\int_{\mathbb{R}^{n}}\left(\left|e^{-\frac{p+1}{p-1}s}H\left(e^{\frac{s}{p-1}}w\right)\right|+\left|e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}w\right)w\right|\rho
dy\right).$
Using the fact that
$|w_{s}w|\leq\epsilon(|w_{s}|^{2}+|w|^{p+1})+C_{2}(\epsilon)$ for all
$\epsilon>0$ and (30), we obtain
$\displaystyle\int_{\mathbb{R}^{n}}|w|^{p+1}\rho dy$
$\displaystyle\leq\frac{2(p+1)}{p-1}\mathcal{E}[w](s)+\epsilon^{\prime}\int_{\mathbb{R}^{n}}|w_{s}|^{2}\rho
dy$
$\displaystyle\quad+\left(\epsilon^{\prime}+2C_{1}s^{-a}\right)\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy+2C_{1}s^{-a}+C_{3},$
where $\epsilon^{\prime}=\epsilon\frac{p+1}{p-1}$,
$C_{3}=2C_{1}+C_{2}\frac{p+1}{p-1}$.
Taking $\epsilon=\frac{p-1}{4(p+1)}$ and $s_{1}$ large enough such that
$2C_{1}s^{-a}\leq\frac{1}{4}$ for all $s\geq s_{1}$, we see that
$\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy\leq\frac{4(p+1)}{p-1}\mathcal{E}[w](s)+\frac{1}{2}\int_{\mathbb{R}^{n}}|w_{s}|^{2}\rho
dy+C_{4},\quad\forall s>s_{1},$ (32)
with $C_{4}=\frac{C_{3}}{2}+\frac{1}{8}$.
Substituting (32) into (31) yields (27) with
$\tilde{s}_{0}=\max\\{s_{0},s_{1}\\}$. This concludes the proof of Lemma 6.
Since we have already showed that Theorem 1 is a direct consequence of Lemma
6, this is also the conclusion of Theorem 1. ∎
### 2.2 A blow-up criterion for the equation in similarity variables
In this part, we give a new blow-up criterion for equation (9). Then, we will
use it to control the $L^{2}$-norm in terms of the energy (see $(ii)$ of
Proposition 8). We claim the following:
###### Lemma 7.
Let $a,p,n,M$ be fixed and $w$ be solution of equation (9) satisfying (11). If
there exists $\tilde{s}_{1}=\tilde{s}_{1}(a,p,n,M)\geq\max\\{\hat{s}_{0},-\log
T\\}$ such that
$-4\mathcal{J}[w](\bar{s})+\frac{p-1}{p+1}\left(\int_{\mathbb{R}^{n}}|w(y,\bar{s})|^{2}\rho
dy\right)^{\frac{p+1}{2}}>0\quad\text{for some}\;\bar{s}\geq\tilde{s}_{1},$
(33)
then $w$ is not defined for all
$(y,s)\in\mathbb{R}^{n}\times[\bar{s},+\infty)$.
###### Proof.
We proceed by contradiction and suppose that $w$ is defined for all
$s\in[\bar{s},+\infty)$. From definition of $\mathcal{J}$ in (16) and from
(29), (30), we have for all $s\geq s_{0}$,
$\displaystyle\frac{d}{ds}\int_{\mathbb{R}^{n}}|w|^{2}\rho dy$
$\displaystyle\geq-4e^{-\frac{\gamma}{a-1}s^{1-a}}\left(\mathcal{J}[w](s)-\theta
s^{1-a}\right)$
$\displaystyle+2\left(\frac{p-1}{p+1}-4C_{0}s^{-a}\right)\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy-8C_{0}s^{-a}.$ (34)
We take $s_{1}$ large enough such that
$4C_{0}s^{-a}\leq\frac{p-1}{2(p+1)}\quad\text{and}\quad
e^{-\frac{\gamma}{a-1}s^{1-a}}-\frac{2C_{0}}{s}>0\quad\text{for all}\quad
s\geq s_{1}.$
Then, using Jensen’s inequality and noting that
$e^{-\frac{\gamma}{a-1}s^{1-a}}\leq 1$ for all $s>0$, we get from (34) the
following: for all $s\geq\max\\{0,s_{0},s_{1}\\}$,
$\frac{d}{ds}\int_{\mathbb{R}^{n}}|w|^{2}\rho
dy\geq-4\mathcal{J}[w](s)+\frac{p-1}{p+1}\left(\int_{\mathbb{R}^{n}}|w|^{2}\rho
dy\right)^{\frac{p+1}{2}}.$ (35)
Setting $f(s)=\int_{\mathbb{R}^{n}}|w(y,s)|^{2}\rho dy$,
$A=-4\mathcal{J}[w](\bar{s})$ and $B=\frac{p-1}{p+1}$, then using the fact
that $\mathcal{J}$ is decreasing in time to get that
$f^{\prime}(s)\geq A+Bf(s)^{\frac{p+1}{2}},\quad\forall s\geq\bar{s}.$
The hypothesis reads $A+Bf(\bar{s})^{\frac{p+1}{2}}>0$ which implies that
$f^{\prime}(s)>0\quad\text{and}\quad A+Bf(s)^{\frac{p+1}{2}}>0,\quad\forall
s\geq\bar{s}.$
By a direct integration, we obtain
$\forall s\geq\bar{s},\quad
s-\bar{s}\leq\int_{f(\bar{s})}^{f(s)}\frac{dz}{A+Bz^{\frac{p+1}{2}}}\leq\int_{f(\bar{s})}^{+\infty}\frac{dz}{A+Bz^{\frac{p+1}{2}}}<+\infty,$
which is a contradiction and Lemma 7 is proved.
∎
As a consequence of Theorem 1 and Lemma 7, we obtain the following estimates
which will be useful for getting Theorem 2:
###### Proposition 8.
Let $w$ be solution of equation (9) satisfying (11), it holds that
$-Q_{0}\leq\mathcal{E}[w](s)\leq 2J_{0},\quad\forall
s\geq\tilde{s}_{2}=\max\\{\hat{s}_{0},-\log T\\},$
where $J_{0}=\mathcal{J}[w](\tilde{s}_{2})$ and
$Q_{0}=\theta\tilde{s}_{2}^{1-a}$. Moreover, there exists a time
$\tilde{s}_{3}\geq\max\\{\hat{s}_{0},-\log T\\}$ such that for all
$s\geq\tilde{s}_{3}$
$\displaystyle(i)\quad\int_{s}^{s+1}\left\|w_{\tau}(\tau)\right\|_{L^{2}_{\rho}(\mathbb{R}^{n})}^{2}d\tau\leq
2J_{0},$
$\displaystyle(ii)\quad\|w(s)\|_{L_{\rho}^{2}(\mathbb{R}^{n})}^{2}\leq J_{1},$
$\displaystyle(iii)\quad\|w(s)\|^{p+1}_{L^{p+1}_{\rho}(\mathbb{R}^{n})}\leq
J_{2}\left(1+\|w(s)\|^{2}_{H^{1}_{\rho}(\mathbb{R}^{n})}\right),$
$\displaystyle(iv)\quad\|w(s)\|^{2}_{H^{1}_{\rho}(\mathbb{R}^{n})}\leq
J_{3}\left(1+\|w_{s}(s)\|_{L_{\rho}^{2}(\mathbb{R}^{n})}\right),$
$\displaystyle(v)\quad\int_{s}^{s+1}\left\|w(\tau)\right\|_{L^{p+1}_{\rho}(\mathbb{R}^{n})}^{2(p+1)}d\tau\leq
J_{4},$
$\displaystyle(vi)\quad\int_{s}^{s+1}\left\|w(\tau)\right\|_{H^{1}_{\rho}(\mathbb{R}^{n})}^{2}d\tau\leq
J_{5},$
where $J_{i},\,i=1,\dots,5$ depend only on $J_{0},Q_{0},a,p,n,M$.
###### Proof.
The upper and lower bounds of $\mathcal{E}$, $(i)$ and $(ii)$ obviously follow
from Theorem 1 and Lemma 7 (in fact, since $w$ is defined for all
$s\geq\tilde{s}_{1}$, condition (33) is never satisfied).
$(iii)$ By definition of $\mathcal{E}$ given in (17) and (30), we get for all
$s\geq\max\\{s_{0},-\log T\\}$,
$\displaystyle 2\mathcal{E}[w](s)$
$\displaystyle\leq\int_{\mathbb{R}^{n}}\left(|\nabla
w|^{2}+\frac{1}{p-1}|w|^{2}\right)\rho dy$
$\displaystyle-2\left(\frac{1}{p-1}-C_{0}s^{-a}\right)\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy+2C_{0}s^{-a}.$
Let $s_{1}$ large enough such that for all $s\geq s_{1}$,
$C_{0}s^{-a}\leq\frac{1}{2(p-1)}$, then for all $s\geq\max\\{s_{0},s_{1},-\log
T\\}$,
$2\mathcal{E}[w](s)\leq\int_{\mathbb{R}^{n}}\left(|\nabla
w|^{2}+\frac{1}{p-1}|w|^{2}\right)\rho
dy-\frac{1}{p-1}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho dy+\frac{2}{p-1}.$
This follows that for all $s\geq\max\\{s_{0},s_{1},-\log T\\}$,
$\left\|w(s)\right\|_{L_{\rho}^{p+1}(\mathbb{R}^{n})}^{p+1}\leq-2(p-1)\mathcal{E}[w](s)+(p-1)\left\|w(s)\right\|_{H_{\rho}^{1}(\mathbb{R}^{n})}^{2}+1.$
Since $\mathcal{E}$ is bounded from below, then $(iii)$ follows.
$(iv)$ From the definition $\mathcal{E}$ in (17), (29) and (30), we have
$\forall s\geq\max\\{s_{0},-\log T\\}$,
$\displaystyle\left\|w(s)\right\|_{H_{\rho}^{1}(\mathbb{R}^{n})}^{2}$
$\displaystyle\leq\frac{1}{p-1}\frac{d}{ds}\int_{\mathbb{R}^{n}}|w|^{2}\rho
dy+\frac{2(p+1)}{p-1}\mathcal{E}[w](s)$
$\displaystyle\quad+\frac{4C_{0}(p+1)}{p-1}s^{-a}\left\|w(s)\right\|_{L_{\rho}^{p+1}(\mathbb{R}^{n})}^{p+1}+\frac{4C_{0}(p+1)}{p-1}s^{-a}.$
Using $(iii)$, we have for all $s\geq\tilde{s}_{3}$,
$\displaystyle\left\|w(s)\right\|_{H_{\rho}^{1}(\mathbb{R}^{n})}^{2}$
$\displaystyle\leq\frac{1}{p-1}\frac{d}{ds}\int_{\mathbb{R}^{n}}|w|^{2}\rho
dy+\frac{2(p+1)}{p-1}\mathcal{E}[w](s)$
$\displaystyle\quad+\frac{4C_{0}J_{2}(p+1)}{p-1}s^{-a}\left(1+\left\|w(s)\right\|_{H_{\rho}^{1}(\mathbb{R}^{n})}^{2}\right)+\frac{4C_{0}(p+1)}{p-1}s^{-a}.$
Let $s_{2}$ large enough such that
$\frac{4C_{0}J_{2}(p+1)}{p-1}s^{-a}\leq\frac{1}{2}$ for all $s\geq s_{2}$ and
noting that $\mathcal{E}(s)$ is bounded from above, we obtain for all
$s\geq\max\\{s_{2},\tilde{s}_{2}\\}$,
$\left\|w(s)\right\|_{H_{\rho}^{1}(\mathbb{R}^{n})}^{2}\leq\frac{4}{p-1}\int_{\mathbb{R}^{n}}|ww_{s}|\rho
dy+C_{1},$
where $C_{1}=\frac{4J_{0}(p+1)}{p-1}+\frac{1}{J_{2}}$.
Using Schwarz’s inequality and $(ii)$ yields
$\left\|w(s)\right\|_{H_{\rho}^{1}(\mathbb{R}^{n})}^{2}\leq\frac{4}{p-1}\|w(s)\|_{L^{2}_{\rho}(\mathbb{R}^{n})}\|w_{s}(s)\|_{L^{2}_{\rho}(\mathbb{R}^{n})}+C_{1}\leq\frac{4\sqrt{J_{1}}}{p-1}\|w_{s}(s)\|_{L^{2}_{\rho}(\mathbb{R}^{n})}+C_{1},$
which follows $(iv)$.
Since $(v)$ and $(vi)$ follows directly from $(i)$ and $(iii),(iv)$, we end
the proof of Proposition 8. ∎
### 2.3 Boundedness of the solution in similarity variables
This section is devoted to the proof of Theorem 2, which is a direct
consequence of the following theorem:
###### Theorem 9.
Let $a,p,n,M$ be fixed, $p$ satisfy (2). There exists
$\hat{s}_{1}=\hat{s}_{1}(a,p,n,M)\geq\hat{s}_{0}$ such that if $u$ is a blow-
up solution of equation (1) with a blow-up time $T$, then for all $s\geq
s^{\prime}=\max\\{\hat{s}_{1},-\log T\\}$,
$\|w_{x_{0},T}(y,s)\|_{L^{\infty}(\mathbf{B}_{R})}\leq C,$ (36)
where $C$ is a positive constant depending only on $n,p,M,R$ and a bound of
$\|w_{x_{0},T}(\hat{s}_{0})\|_{L^{\infty}}$.
Let us show that Theorem 2 follows from Theorem 9.
###### Proof of Theorem 2 admitting Theorem 9.
We have from (36) that
$|w_{x_{0},T}(0,s)|\leq C,\quad\forall s\geq s^{\prime},$
with $C$ independent on $x_{0}\in\mathbb{R}^{n}$. Therefore, we get from (8)
that
$|u(x_{0},t)|\leq C(T-t)^{-\frac{1}{p-1}},\quad\forall
x_{0}\in\mathbb{R}^{n},\forall t\in[T-e^{-s^{\prime}},T),$
which is the conclusion of Theorem 2, assuming that Theorem 9 holds. ∎
Following the method in [7], the proof of Theorem 9 requests the following key
integral estimate:
###### Lemma 10 (Key integral estimate).
Let $a,p,n,M$ be fixed and $w$ be solution of equation (9) satisfying (11).
For all $q\geq 2$ and $R>0$, there exists $\hat{s}_{2}\geq\tilde{s}_{3}$ and a
positive constant $K_{q}$ such that,
$\int_{s}^{s+1}\|w(\tau)\|_{L^{p+1}(\mathbf{B}_{R})}^{q(p+1)}d\tau\leq
K_{q},\quad\forall s\geq\hat{s}_{2},$ (37)
where $K_{q}$ depends only on $J_{0},Q_{0},a,n,p,q,R,\hat{s}_{2}$.
Let us first show that how Theorem 9 follows from Lemma 10, then we will prove
it later. In order to derive uniform bound in Theorem 9 for all $p$ satisfying
(2), we need two following techniques. The first one is an interpolation
result from Cazenave and Lions [1]:
###### Lemma 11 (Interpolation technique, Cazenave and Lions [1]).
Assume that
$v\in L^{\alpha}\left((0,\infty);L^{\beta}(\mathbf{B}_{R})\right),\;v_{t}\in
L^{\gamma}\left((0,\infty);L^{\delta}(\mathbf{B}_{R})\right)$
for some $1<\alpha,\beta,\gamma,\delta<\infty$. Then
$v\in\mathcal{C}\left([0,\infty);L^{\lambda}(\mathbf{B}_{R})\right)$
for all
$\lambda<\lambda_{0}=\frac{(\alpha+\gamma^{\prime})\beta\delta}{\gamma^{\prime}\beta+\alpha\delta}$
with $\gamma^{\prime}=\frac{\gamma}{\gamma-1}$, and satisfies
$\sup_{t\geq 0}\|v(t)\|_{L^{\lambda}(\mathbf{B}_{R})}\leq
C\int_{0}^{\infty}\left(\|v(\tau)\|_{L^{\beta}(\mathbf{B}_{R})}^{\alpha}+\|v_{\tau}(\tau)\|_{L^{\delta}(\mathbf{B}_{R})}^{\gamma}\right)d\tau$
for $\lambda<\lambda_{0}$. The positive constant $C$ depends only on
$\alpha,\beta,\gamma,\delta,n$ and $R$.
The second one is an interior regularity result for a nonlinear parabolic
equation:
###### Proposition 12 (Interior regularity).
Let $v(x,t)\in L^{\infty}\big{(}(0,+\infty),L^{2}(\mathbf{B}_{R})\big{)}\cap
L^{2}\big{(}(0,+\infty),H^{1}(\mathbf{B}_{R})\big{)}$ which satisfies
$v_{t}-\Delta v+b.\nabla v=F,\quad(x,t)\in
Q_{R}=\mathbf{B}_{R}\times(0,+\infty),$ (38)
where $R>0$, $|b(x,t)|\leq\mu_{1}$ in $Q_{R}$ and $|F(x,t,v)|\leq
g(x,t)(|v|+1)$ with
$\int_{t}^{t+1}\left\|g(\tau)\right\|^{\beta^{\prime}}_{L^{\alpha^{\prime}}(\mathbf{B}_{R})}d\tau\leq\mu_{2},\quad\forall
t\in(0,+\infty),$ (39)
and $\frac{1}{\beta^{\prime}}+\frac{n}{2\alpha^{\prime}}<1$ and
$\alpha^{\prime}\geq 1$. If
$\int_{t}^{t+1}\|v(\tau)\|^{2}_{L^{2}(\mathbf{B}_{R})}d\tau\leq\mu_{3},\quad\forall
t\in(0,+\infty),$ (40)
and $\mu_{1}$, $\mu_{2}$ and $\mu_{3}$ are uniformly bounded in $t$, then
there exists a positive constant $C$ depending only on $\mu_{1}$, $\mu_{2}$,
$\mu_{3}$, $\alpha^{\prime}$, $\beta^{\prime}$, $n$, $R$ and $\tau\in(0,1)$
such that
$|v(x,t)|\leq C,\quad\forall(x,t)\in\mathbf{B}_{R/4}\times(\tau,+\infty).$
###### Proof.
Since the argument of the proof is analogous as in the corresponding part in
[12], we then leave the proof to Appendix B.1. ∎
Let us now use Lemma 10 to derive the conclusion of Theorem 9, then we will
prove it later.
###### Proof of Theorem 9 admitting Lemma 10.
Let us recall the equation in $w$:
$w_{s}-\Delta w+\frac{1}{2}y.\nabla
w=-\frac{w}{p-1}+|w|^{p-1}w+e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}w\right),$
where $h$ is given in (3).
We now apply Proposition 12 to $w$ with $b=\frac{y}{2}$ and
$F=-\frac{w}{p-1}+|w|^{p-1}w+e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}w\right).$
From (13), we see that
$|F|\leq C^{\prime}(C_{0},p)(|w|^{p-1}+1)(|w|+1),\quad\forall s\geq s_{0}.$
Thus, the first identity in (39) holds with $g=C^{\prime}(|w|^{p-1}+1)$ and
the second condition in (39) turns into
$\int_{s}^{s+1}\left(\int_{\mathbf{B}_{R}}|w(y,\tau)|^{\alpha^{\prime}(p-1)}dy\right)^{\frac{\beta^{\prime}}{\alpha^{\prime}}}d\tau\leq
C_{1}\quad\text{for some $C_{1}>0$,}$
for some $\alpha^{\prime}$ and $\beta^{\prime}$ satisfying
$\frac{1}{\beta^{\prime}}+\frac{n}{2\alpha^{\prime}}<1$.
For this bound, we first use $(i)$ of Proposition 8, (37) and apply Lemma 11
with $\alpha=q(p+1)$, $\beta=p+1$, $\gamma=\delta=\gamma^{\prime}=2$ to get
that
$\sup_{s\geq\hat{s}_{2}}\|w(s)\|_{L^{\lambda}(\mathbf{B}_{R})}\leq
C_{2}(R,K_{q}),\quad\forall\lambda<\lambda_{1}=p+1-\frac{p-1}{q+1}.$ (41)
Next, applying Proposition 12 with $\alpha^{\prime}(p-1)=\lambda$,
$\beta^{\prime}$ and $q$ large (note that the condition
$\frac{1}{\beta^{\prime}}+\frac{n}{2\alpha^{\prime}}<1$ turns into
$p<\frac{n+2}{n-2}$), we obtain
$\int_{s}^{s+1}\left(\int_{\mathbf{B}_{R}}|w(y,\tau)|^{\alpha^{\prime}(p-1)}dy\right)^{\frac{\beta^{\prime}}{\alpha^{\prime}}}d\tau\leq
C_{2}^{\beta^{\prime}(p-1)}.$
Hence, condition (39) holds. Therefore, $|w(y,s)|$ is bounded for all
$(y,s)\in\mathbf{B}_{R/4}\times(\tau+\hat{s}_{2},+\infty)$ for some
$\tau\in(0,1)$, which concludes the proof of Theorem 9, assuming that Lemma 10
holds. ∎
###### Remark 6.
If we use $(v)$ of Proposition 8, we already have for all
$s\geq\tilde{s}_{3}$,
$\int_{s}^{s+1}\left(\int_{\mathbf{B}_{R}}|w(y,\tau)|^{p+1}dy\right)^{2}d\tau\leq
C(R)K_{1}.$
Applying Proposition 12 with $\alpha^{\prime}=\frac{p+1}{p-1}$ and
$\frac{\beta^{\prime}}{\alpha^{\prime}}=2$ (noting that the condition
$\frac{1}{\beta^{\prime}}+\frac{n}{2\alpha^{\prime}}<1$ turns into
$p<\frac{n+3}{n-1}$), we obtain $w$ is uniformly bounded with
$p\in\left(1,\frac{n+3}{n-1}\right)$.
If we use $(i)$ and $(v)$ in Proposition 8, Lemma 11 with
$\alpha=2(p+1),\quad\beta=p+1,\quad\gamma=\delta=\gamma^{\prime}=2$, then we
obtain
$\sup_{s\geq\tilde{s}_{3}}\|w(s)\|_{L^{\lambda}(\mathbf{B}_{R})}\leq
C(R),\quad\forall\lambda<\lambda_{1}=\frac{2(p+2)}{3}$
Next, Proposition 12 applies with $\alpha^{\prime}(p-1)=\lambda$ with
$\lambda$ approaches to $\frac{2(p+1)}{3}$ and $\beta^{\prime}$ very large,
then the condition $\frac{1}{\beta^{\prime}}+\frac{n}{2\alpha^{\prime}}<1$ now
becomes
$\exists\lambda<\frac{2(p+1)}{3},\quad\text{such
that}\quad\frac{n}{2\alpha^{\prime}}<1.$
This turns into $p<\frac{3n+8}{3n-4}$. This result was proved by Giga and Kohn
in [5].
Relying on a bootstrap argument, [7] improved the input estimate of
Proposition 12 covering this way the whole subcritical range
$p<\frac{n+2}{n-2}$. Here, we extend their approach to a larger class of
equation.
Let us now give the proof of Lemma 10 in order to complete the proof of
Theorem 9 and Theorem 2 also. To this end, let
$\psi\in\mathcal{C}^{2}(\mathbb{R}^{n})$ be a bounded function, we introduce
the following local functional, which is a perturbed version of the function
of [7],
$\displaystyle\mathcal{E}_{\psi}[w](s)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{n}}\psi^{2}\left(|\nabla
w|^{2}+\frac{1}{p-1}|w|^{2}\right)\rho dy$
$\displaystyle\quad-\frac{1}{p+1}\int_{\mathbb{R}^{n}}\psi^{2}|w|^{p+1}\rho
dy-e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}\psi^{2}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy.$ (42)
We get the following bound on the local functional $\mathcal{E}_{\psi}$:
###### Proposition 13.
Let $a,p,n,M$ be fixed and $w$ be solution of equation (9) satisfying (11).
For $\psi\in\mathcal{C}^{2}(\mathbb{R}^{n})$ bounded, there exist positive
constants $Q^{\prime},K^{\prime}$ such that
$-Q^{\prime}\leq\mathcal{E}_{\psi}[w](s)\leq K^{\prime},\quad\forall
s\geq\tilde{s}_{3},$ (43)
where $\tilde{s}_{3}$ is given in Proposition 8 and $Q^{\prime},K^{\prime}$
depend on $a$, $p$, $n$, $M$, $\|\psi\|^{2}_{L^{\infty}}$,
$\|\nabla\psi\|^{2}_{L^{\infty}}$ and $J_{0}$.
###### Proof.
The proof is essentially the same as the corresponding part in [7], except for
the control of the last term in (42). Since that control is a bit long and
technical, we leave the proof to B.2. ∎
Let $R>0$, we fix $\psi(y)$ so that it satisfies
$\psi(y)\in\mathcal{C}_{0}^{\infty}(\mathbb{R}^{n}),\quad 0\leq\psi(y)\leq
1,\quad\psi(y)=\left\\{\begin{array}[]{lcl}1&\quad\text{on}&\quad\mathbf{B}_{R}\\\
0&\quad\text{on}&\quad\mathbb{R}^{n}\setminus\mathbf{B}_{2R}\end{array}\right..$
(44)
We claim the following:
###### Lemma 14.
Let $a,p,n,M$ be fixed and $w$ be solution of equation (9) satisfying (11).
Then there exists $\tilde{s}_{5}\geq\tilde{s}_{3}$ such that
$\|w\|_{L^{p+1}_{\rho}(\mathbf{B}_{R})}^{p+1}\leq
K_{1}\left(1+\|w\|^{2}_{H^{1}_{\rho}(\mathbf{B}_{2R})}\right),\quad\forall
s\geq\tilde{s}_{5},$ (45)
where $K_{1}=K_{1}(a,p,n,M,Q^{\prime})$.
###### Proof.
From (30) and the definition of $\mathcal{E}_{\psi}$ in (42), we have $\forall
s\geq\max\\{s_{0},s_{1}\\}$,
$\displaystyle\int_{\mathbb{R}^{n}}\psi^{2}|w|^{p+1}\rho dy$
$\displaystyle\leq-2(p+1)\mathcal{E}_{\psi}[w](s)$
$\displaystyle+(p+1)\int_{\mathbb{R}^{n}}\psi^{2}\left(|\nabla
w|^{2}+\frac{1}{p-1}|w|^{2}\right)\rho dy+1,$ (46)
where $s_{1}$ is large enough such that $2C_{0}s^{-a}\leq\frac{1}{2(p+1)}$ for
all $s\geq s_{1}$.
Thus, (45) follows from the lower bound of $\mathcal{E}_{\psi}$ and the
property of $\psi$. This ends the proof of Lemma 14. ∎
###### Remark 7.
By (45), the proof of estimate (37) is equivalent to showing that
$\int_{s}^{s+1}\|w(\tau)\|_{H^{1}_{\rho}(\mathbf{B}_{R})}^{2q}d\tau\leq
K_{q},\quad\forall s\geq\hat{s}_{2}.$ (47)
Note from $(i)$ and $(iv)$ in Proposition 8 that (47) already holds in the
case $q=2$.
In order to derive (47) for all $q\geq 2$, we need the following result:
###### Lemma 15.
Let $a,p,n,M$ be fixed and $w$ be solution of equation (9) satisfying (11).
Then there exists $\tilde{s}_{6}\geq\tilde{s}_{3}$ such that
$\|w\|_{H^{1}_{\rho}(\mathbf{B}_{R})}^{2}\leq
K_{2}\left(1+\|\psi^{2}ww_{s}\|^{2}_{L^{1}_{\rho}(\mathbf{B}_{2R})}\right),\quad\forall
s\geq\tilde{s}_{6},$ (48)
where $K_{2}=K_{2}(a,p,n,M,Q^{\prime},K^{\prime})$.
###### Proof.
Multiplying equation (9) with $\psi^{2}w\rho$, integrating over
$\mathbb{R}^{n}$, using the definition of $\mathcal{E}_{\psi}$ and estimate
(30), we have
$\displaystyle\int_{\mathbb{R}^{n}}\psi^{2}\left(|\nabla
w|^{2}+\frac{1}{p-1}|w|^{2}\right)\rho dy$
$\displaystyle\leq\frac{2}{p-1}\int_{\mathbb{R}^{n}}\psi^{2}ww_{s}\rho
dy+\frac{2(p+1)}{p-1}\mathcal{E}_{\psi}[w](s)$
$\displaystyle+\frac{4}{p-1}\int_{\mathbb{R}^{n}}\psi w\nabla\psi.\nabla w\rho
dy$
$\displaystyle+\frac{4(p+1)C_{0}}{(p-1)s^{a}}\int_{\mathbb{R}^{n}}\psi^{2}(|w|^{p+1}+1)\rho
dy,\;\;\forall s\geq s_{0}.$
Using (46), then taking $s_{2}$ large such that
$\frac{4(p+1)^{2}C_{0}}{(p-1)s^{a}}\leq\frac{1}{2}$ and noting that
$\mathcal{E}$ is bounded, we have for all $s\geq\max\\{s_{0},s_{1},s_{2}\\}$,
$\int_{\mathbb{R}^{n}}\psi^{2}\left(|\nabla
w|^{2}+\frac{1}{p-1}|w|^{2}\right)\rho dy\leq
C\left(\int_{\mathbb{R}^{n}}\psi^{2}ww_{s}\rho dy+\int_{\mathbb{R}^{n}}\psi
w\nabla\psi.\nabla w\rho dy+1\right).$
Let $J_{\psi}[w](s)=\int_{\mathbb{R}^{n}}\psi w\nabla\psi.\nabla w\rho dy$,
then one can show that $J_{\psi}[w](s)\leq C_{1}$ (see (107) for a proof of
this fact). Hence, we have for all $s\geq\max\\{s_{0},s_{1},s_{2}\\}$,
$\int_{\mathbb{R}^{n}}\psi^{2}\left(|\nabla
w|^{2}+\frac{1}{p-1}|w|^{2}\right)\rho dy\leq
C_{2}\left(\int_{\mathbb{R}^{n}}\psi^{2}ww_{s}\rho dy+1\right).$
Thus, (48) follows from the property of $\psi$, and Lemma 15 is proved. ∎
Since the estimate (47) already holds in the case $q=2$, we now use a
bootstrap argument in order to get (47) for all $q\geq 2$.
###### Proof of (47) for all $q\geq 2$ by a bootstrap argument.
This part is the same as in [7]. We give it here for the sake of completeness.
Suppose that (47) holds for some $q\geq 2$, let us show that (47) holds for
all $\tilde{q}\in[q,q+\epsilon]$ for some $\epsilon>0$ independent from $q$.
We start with Holder’s inequality,
$\|\psi^{2}ww_{s}\|_{L^{1}_{\rho}(\mathbf{B}_{2R})}\leq\|\psi
w\|_{L^{\lambda}_{\rho}(\mathbf{B}_{2R})}\times\|\psi
w_{s}\|_{L^{\lambda^{\prime}}_{\rho}(\mathbf{B}_{2R})},\quad\frac{1}{\lambda}+\frac{1}{\lambda^{\prime}}=1.$
Using (37) and applying Lemma 11, we obtain
$\|w\|_{L^{\lambda}(\mathbf{B}_{2R})}\leq
C_{q}^{\prime},\quad\forall\lambda<\lambda_{1}(q)=p+1-\frac{p-1}{q+1}.$
Let us now bound $\|\psi
w_{s}\|_{L^{\lambda^{\prime}}_{\rho}(\mathbf{B}_{2R})}$. We remark that for
$q$ large then $\lambda$ approaches to $p+1$ and $\lambda^{\prime}$ approaches
to $p_{1}=\frac{p+1}{p}$. Let $f=\psi w_{s}$ and make use Holder’s inequality,
$\|f\|_{L^{\lambda^{\prime}}}\leq\|f\|_{L^{2}}^{1-\theta}\times\|f\|_{L^{p_{1}}}^{\theta},\quad\frac{1}{\lambda^{\prime}}=\frac{1-\theta}{2}+\frac{\theta}{p_{1}},\quad\theta\in[0,1].$
From now on, we take $\lambda\geq 2$ and fix
$\theta=\frac{(\lambda-2)(p+1)}{\lambda(p-1)}$ (note that with this choice,
$\theta\in[0,1]$). From Lemma 15, we have
$\|w(s)\|_{H^{1}_{\rho}(\mathbf{B}_{R})}^{2}\leq K_{2}^{\prime}\left(1+\|\psi
w_{s}\|_{L^{2}_{\rho}(\mathbf{B}_{2R})}^{1-\theta}\times\|\psi
w_{s}\|_{L^{p_{1}}_{\rho}(\mathbf{B}_{2R})}^{\theta}\right).$
This follows that
$\int_{s}^{s+1}\|w(s)\|_{H^{1}_{\rho}(\mathbf{B}_{R})}^{2\tilde{q}}d\tau\leq
C_{\tilde{q}}\left[1+\underbrace{\int_{s}^{s+1}\|\psi
w_{s}\|_{L^{2}_{\rho}(\mathbf{B}_{2R})}^{\tilde{q}(1-\theta)}\times\|\psi
w_{s}\|_{L^{p_{1}}_{\rho}(\mathbf{B}_{2R})}^{\tilde{q}\theta}d\tau}_{\mathbf{G}}\right],$
(49)
for some $\tilde{q}>q$.
Let $\alpha=\frac{2}{(1-\theta)\tilde{q}}$ and use Holder’s inequality in time
to $\mathbf{G}$, we obtain
$\displaystyle\mathbf{G}$ $\displaystyle\leq\left(\int_{s}^{s+1}\|\psi
w_{s}\|_{L^{2}_{\rho}(\mathbf{B}_{2R})}^{2}d\tau\right)^{\frac{1}{\alpha}}\left(\int_{s}^{s+1}\|\psi
w_{s}\|_{L^{p_{1}}_{\rho}(\mathbf{B}_{2R})}^{\tilde{q}\theta\alpha^{\prime}}d\tau\right)^{\frac{1}{\alpha^{\prime}}}$
$\displaystyle\leq(2J_{0})^{\frac{1}{\alpha}}\left(\int_{s}^{s+1}\|\psi
w_{s}\|_{L^{p_{1}}_{\rho}(\mathbf{B}_{2R})}^{\tilde{q}\theta\alpha^{\prime}}d\tau\right)^{\frac{1}{\alpha^{\prime}}}\equiv\mathbf{G}_{1},$
where we used $(i)$ in Proposition 8.
Let us bound $\mathbf{G}_{1}$. To this end, we use the $L^{p}-L^{q}$ estimate
for the heat equation (see Lemmas 6.3 and 6.4 in [7]) to get
$\displaystyle\int_{s}^{s+1}\|\psi
w_{s}\|_{L^{p_{1}}_{\rho}(\mathbf{B}_{2R})}^{\tilde{q}\theta\alpha^{\prime}}d\tau$
$\displaystyle\leq
C_{\tilde{q}}^{\prime}\left(1+\int_{s}^{s+1}\left\||w|^{p}\right\|_{L^{p_{1}}_{\rho}(\mathbf{B}_{2R})}^{\tilde{q}\theta\alpha^{\prime}}d\tau\right)$
$\displaystyle=C_{\tilde{q}}^{\prime}\left(1+\int_{s}^{s+1}\left\|w\right\|_{L^{p+1}_{\rho}(\mathbf{B}_{2R})}^{p\tilde{q}\theta\alpha^{\prime}}d\tau\right)$
$\displaystyle\leq
C_{\tilde{q}}^{\prime\prime}\left(1+\int_{s}^{s+1}\left\|w\right\|_{H^{1}_{\rho}(\mathbf{B}_{4R})}^{\frac{2p\tilde{q}\theta\alpha^{\prime}}{p+1}}d\tau\right)\quad\text{(using
Lemma \ref{eq:tmpEu12}).}$
By Proposition 6.2 in [7], we have
$\frac{2p\tilde{q}\theta\alpha^{\prime}}{p+1}<2q$ for all
$\tilde{q}\in[q,q+\frac{2}{p+1}]$. Then, applying Holder’s inequality again
yields
$\displaystyle\int_{s}^{s+1}\|w(s)\|_{H^{1}_{\rho}(\mathbf{B}_{R})}^{2\tilde{q}}d\tau\leq
C_{\tilde{q}}^{\prime\prime\prime}\left[1+\left(\int_{s}^{s+1}\|w(s)\|_{H^{1}_{\rho}(\mathbf{B}_{4R})}^{2q}d\tau\right)^{\frac{1}{2q\alpha^{\prime}}}\right]\leq\bar{C}_{\tilde{q}}.$
Thus, inequality (47) is valid for all $\tilde{q}\in[q,q+\frac{2}{p+1}]$.
Repeating this argument, we would obtain that (47) holds for all $q\geq 2$.
This concludes the proof of Lemma 10, Theorem 9 and Theorem 2 too. ∎
### 2.4 Limit of $w$ as $s\to+\infty$
This section is devoted to the proof of Theorem 3. Note in the unperturbed
case ($h\equiv 0$) that Theorem 3 was proved in [6] (see also [4], [5]). The
proof is divided into two steps. The first step is to show that the limit of
solution in similarity variables exists and belongs to the set of solutions of
the following equation,
$0=\Delta w-\frac{1}{2}y.\nabla w-\frac{1}{p-1}w+|w|^{p-1}w,$ (50)
Then, by using a nondegeneracy result (Lemma 19), the blow-up criterion (Lemma
7) and suitable energy arguments, we shall show that the possibility of
$w_{a}\to 0$ as $s\to+\infty$ is excluded if $a$ is a blow-up point. Let us
restate Theorem 3 in below:
###### Proposition 16 (Limit of $w$ as $s\to+\infty$).
Let $a,p,n,M$ be fixed, $p$ be a sub-critical non-linearity given in (2).
Consider $u(t)$ a solution of equation (1) which blows up at time $T$ and $a$
a blow-up point. Then
$\lim_{s\to+\infty}\,w_{a}(y,s)=\pm\kappa,\quad\text{uniformly on each compact
subset of $\mathbb{R}^{n}$.}$
Before going into the proof of Proposition 16, let us first derive some
elementary results. The first one concerns the stationary solutions in
$\mathbb{R}^{n}$ of equation (50). Particularly, we have the following:
###### Lemma 17 (Stationary solutions, Giga and Kohn [4]).
Let $p$ satisfy (2), then all bounded solutions of (50) are constants:
$w\equiv 0$ or $w\equiv\pm\kappa$.
###### Proof.
The proof is given in Proposition 2 of [4]. For the reader’s interest, we
mention that the proof relies on a clever use of multiplying factors, together
with a Pohozaev technique, resulting in the following identity:
$\left(\frac{n}{p+1}-\frac{2-n}{2}\right)\int_{\mathbb{R}^{n}}|\nabla
w|^{2}\rho
dy+\frac{1}{2}\left(\frac{1}{2}-\frac{1}{p+1}\right)\int_{\mathbb{R}^{n}}|y|^{2}|\nabla
w|^{2}\rho dy=0.$ (51)
From (51) and the fact that $p$ is Sopolev subcritical, it follows that
$\frac{n}{p+1}-\frac{2-n}{2}>0$ and $\frac{1}{2}-\frac{1}{p+1}>0$, hence
$\nabla w\equiv 0$. This implies that $w$ is actually a constant. This
concludes the proof of Lemma 17. ∎
The second one is due to parabolic estimates:
###### Lemma 18 (Parabolic estimates).
Let $u$ be a solution to equation (1). Assume that $T=T_{\max}(u_{0})<+\infty$
and that $u$ satisfies (20). Then, there is a positive constant $C$ such that
for all $t\in[T/2,T)$,
$\|\nabla u(t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C(T-t)^{-\frac{1}{p-1}-\frac{1}{2}}\quad\text{and}\quad\|\nabla^{2}u(t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C(T-t)^{-\frac{1}{p-1}-1}.$ (52)
In similarity variables, we have for all $s\in[-\log(T/2),+\infty)$ and
$x_{0}\in\mathbb{R}^{n}$,
$\|\nabla w_{x_{0},T}(s)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C\quad\text{and}\quad\|\nabla^{2}w_{x_{0},T}(s)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
C.$ (53)
###### Proof.
Since $|h(z)|\leq C(|z|^{p}+1)$ and $|h^{\prime}(z)|\leq C(|z|^{p-1}+1)$ from
(3), the proof given in Proposition 23.15, page 189 of Souplet and Quittner
[15] in the case $h\equiv 0$ extends with no difficulty in this case. ∎
The last one is the nondegeneracy result from Giga and Kohn [6]:
###### Lemma 19 (Nondegeneracy, Giga and Kohn [6]).
Let $p>1$, $T>0$, $r>0$, $\sigma\in(0,1)$, $a\in\mathbb{R}^{n}$ and denote
$Q_{r,\sigma}(a)=\mathbf{B}_{r}(a)\times(T-\sigma,T)$. There exists
$\epsilon=\epsilon(n,p)>0$ such that if $u$ is a classical solution of
$u_{t}-\Delta u=F(u),\quad(x,t)\in Q_{r,\sigma}(a),$ (54)
where $|F(u)|\leq M(|u|^{p}+1)$ for some $M>0$. Assume that $u$ satisfies
$|u(x,t)|\leq\epsilon(T-t)^{-\frac{1}{p-1}},\quad(x,t)\in Q_{r,\sigma}(a),$
(55)
then $u$ is uniformly bounded in a neighborhood of $(a,T)$.
###### Proof.
See Theorem 2.1, page 850 in Giga and Kohn [6]. ∎
Let us now give the proof of Proposition 16.
###### Proof of Proposition 16.
Consider $a$ a blow-up point and write $w$ instead of $w_{a}$ for simplicity.
By Lemma 18 and equation (9), we see that $|w_{s}(y,s)|\leq C(|y|+1)$ for some
$C>0$. Therefore, $w$, $\nabla w$, $\nabla^{2}w$ and $w_{s}$ are bounded for
all $|y|\leq R$ and $s\geq s^{\prime}$ for some $R>0$ and
$s^{\prime}\in\mathbb{R}$. Let $\\{s_{j}\\}$ be a sequence tending to
$+\infty$ and $w_{j}(y,s)=w(y,s+s_{j})$. By the Arzela-Ascoli theorem, there
is a subsequence of $s_{j}$ (still denoted $s_{j}$) such that $w_{j}$
converges uniformly on compact sets to some $w^{\infty}$, $\nabla
w_{j}\to\nabla w^{\infty}$, $\Delta w_{j}\to\Delta w^{\infty}$ and $w_{js}\to
w^{\infty}_{s}$. On the other hand, by $(i)$ and $(vi)$ of Proposition 8, we
see that as $j\to+\infty$,
$\displaystyle\int_{\tilde{s}_{3}}^{+\infty}\int_{\mathbf{B}_{R}}|w_{js}|^{2}dyds=\int_{\tilde{s}_{3}+s_{j}}^{+\infty}\int_{\mathbf{B}_{R}}|w_{s}|^{2}dyds\to
0.$
This implies that $w^{\infty}_{s}=0$ and $w^{\infty}$ satisfies (50). Hence,
by Lemma 17, $w^{\infty}\equiv 0$ or $w^{\infty}\equiv\pm\kappa$.
It remains to show that $w(\cdot,s_{j})\nrightarrow 0$ as $j\to+\infty$. We
proceed by contradiction. Let us assume that $w(\cdot,s_{j})\to 0$ as
$j\to+\infty$. We observer that if $w(\cdot,s_{j})\to 0$, then by the
definition of $\mathcal{J}$ given in (16), the bound of $w$ and $\nabla w$ and
dominated convergence, then $\mathcal{J}[w](s_{j})\to 0$. Since $\mathcal{J}$
is a Lyapunov functional, it follows that the whole sequence
$\mathcal{J}[w](s)\to 0\quad\text{as}\quad s\to+\infty.$ (56)
Let $b\in\mathbb{R}^{n}$, then by (53), we have $w_{b}(y,s)$ and $\nabla
w_{b}(y,s)$ are bounded for all $y\in\mathbb{R}^{n}$ and $s\geq s^{\prime}$.
We now use the interpolation inequality which reads
$|w_{b}(0,s)|\leq C\left(\|w_{b}\|_{L^{2}(\mathbf{B}_{R})}^{\theta}\|\nabla
w_{b}\|^{1-\theta}_{L^{\infty}(\mathbf{B}_{R})}+\|w_{b}\|_{L^{2}{(\mathbf{B}_{R})}}\right),$
where $\theta\in(0,\frac{2}{n+2})$ if $n\geq 2$ and $\theta=1/2$ if $n=1$.
By Lemma 7, we see that $\|w_{b}(s)\|_{L^{2}(\mathbf{B}_{R})}\leq
C(p)\big{(}\mathcal{J}[w_{b}](s)\big{)}^{\frac{1}{p+1}}$ for all
$s\geq\tilde{s}_{1}$. Hence,
$|w_{b}(0,s)|\leq
C^{\prime}\left(\big{(}\mathcal{J}[w_{b}](\tilde{s}_{1})\big{)}^{\frac{\theta}{p+1}}+\big{(}\mathcal{J}[w_{b}](\tilde{s}_{1})\big{)}^{\frac{1}{p+1}}\right),\quad\forall
s\geq\tilde{s}_{1}.$
Consider some $\epsilon>0$ small. From (56), there is $s^{\prime}(\epsilon)$
such that $\mathcal{J}[w](s)\leq\epsilon$ for all $s\geq
s^{\prime}(\epsilon)$. Therefore, by continuity depending of
$\mathcal{J}[w_{b}](s)$ on $b$ and the monotonicity of $\mathcal{J}[w_{b}](s)$
in time $s$, we infer that $\mathcal{J}[w_{b}](s)\leq 2\epsilon$ for all
$s\geq s^{\prime}$ and $|b-a|$ small. This implies that
$|w_{b}(0,s)|\leq\epsilon^{\prime\prime}$ for all $s\geq s^{\prime}$, or
$|u(b,t)|\leq\epsilon^{\prime\prime}(T-t)^{-\frac{1}{p-1}}$ for $(b,t)$ close
to $(a,T)$, where
$\epsilon^{\prime\prime}=\epsilon^{\prime\prime}(\epsilon)\to 0$ as
$\epsilon\to 0$. Thus, $a$ is not a blow-up point by Lemma 19, and this is a
contradiction. Therefore, this concludes the proof of Proposition 16 and the
proof of Theorem 3 also. ∎
## 3 Classification of the behavior of $w$ as $s\to+\infty$ in $L_{\rho}^{2}$
This section is devoted to the proof of Theorem 4. Consider $a$ a blow-up
point and write $w$ instead of $w_{a}$ for simplicity. From Theorem 3 and up
to changing the signs of $w$ and $h$, we may assume that
$\|w(y,s)-\kappa\|_{L^{2}_{\rho}}\to 0$ as $s\to+\infty$, uniformly on compact
subsets of $\mathbb{R}^{n}$. As mentioned in the introduction, by setting
$v(y,s)=w(y,s)-\phi(s)$ ($\phi$ is a positive solution of (21) such that
$\phi(s)\to\kappa$ as $s\to+\infty$), we see that
$\|v(y,s)\|_{L^{2}_{\rho}}\to 0$ as $s\to+\infty$ and $v$ solves the following
equation:
$\partial_{s}v=(\mathcal{L}+\omega(s))v+F(v)+H(v,s),\quad\forall
y\in\mathbb{R}^{n},\;\forall s\in[-\log T,+\infty),$ (57)
where $\mathcal{L}=\Delta-\frac{y}{2}\cdot\nabla+1$ and $\omega$, $F$, $H$ are
given by
$\displaystyle\omega(s)=p\left(\phi^{p-1}-\kappa^{p-1}\right)+e^{-s}h^{\prime}\left(e^{\frac{s}{p-1}}\phi\right),$
$\displaystyle F(v)=|v+\phi|^{p-1}(v+\phi)-\phi^{p}-p\phi^{p-1}v,$
$\displaystyle
H(v,s)=e^{-\frac{ps}{p-1}}\left[h\left(e^{\frac{s}{p-1}}(v+\phi)\right)-h\left(e^{\frac{s}{p-1}}\phi\right)-e^{\frac{s}{p-1}}h^{\prime}\left(e^{\frac{s}{p-1}}\phi\right)v\right].$
We remark from (22) and (13) that
$|\omega(s)|=\mathcal{O}\left(\frac{1}{s^{a}}\right)\quad\text{as}\quad
s\to+\infty.$ (58)
Let us introduce for all $y\in\mathbb{R}^{n}$, for all $s\in[-\log
T,+\infty)$,
$\beta(s)=e^{-\int_{s}^{+\infty}\omega(\tau)d\tau}\quad\text{and}\quad
V(y,s)=\beta(s)v(y,s),$ (59)
(note that $\beta(s)\to 1$ as $s\to+\infty$).
By multiplying equation (57) to $\beta(s)$, we find the following equation
satisfied by $V$:
$\partial_{s}V=\mathcal{L}V+\bar{F}(V,s),\quad\forall
y\in\mathbb{R}^{n},\;\forall s\in[-\log T,+\infty),$ (60)
where $\bar{F}(V,s)=\beta(s)(F(v)+H(v,s))$ satisfying
$|\bar{F}(V,s)|\leq CV^{2}.$ (61)
Since $\|w(s)\|_{L^{\infty}}\leq C$ from Theorem 2, we may use a Taylor
expansion, (13), (22) and the fact that
$\beta(s)=1+\mathcal{O}\left(\frac{1}{s^{a-1}}\right)$ as $s\to+\infty$ to
write
$\left|\bar{F}(V,s)-\frac{p}{2\kappa}V^{2}\right|=\mathcal{O}(|V|^{3})+\mathcal{O}\left(\frac{V^{2}}{s^{a-1}}\right)\quad\text{as
$s\to+\infty$},$ (62)
(see Lemma C.1 for the proof of (62), and note that (61) follows from (62)).
Since the eigenfunctions of $\mathcal{L}$ constitute a total orthonormal
family of $L^{2}_{\rho}$, we can expand $V$ as follows:
$V(y,s)=\sum_{k=1}^{\infty}\pi_{k}(V)(y,s)=V_{+}(y,s)+V_{null}(y,s)+V_{-}(y,s),$
(63)
where $\pi_{k}(V)$ is the orthogonal projector of $v$ on the eigenspace
associated to $\lambda_{k}=1-\frac{k}{2}$,
$\displaystyle V_{+}(y,s)$
$\displaystyle=\pi_{+}(V)(y,s)=\sum_{k=0}^{1}\pi_{k}(V)(y,s),$ $\displaystyle
V_{-}(y,s)$
$\displaystyle=\pi_{-}(V)(y,s)=\sum_{k=3}^{\infty}\pi_{k}(V)(y,s),$
$\displaystyle V_{null}(y,s)$
$\displaystyle=\pi_{2}(V)(y,s)=V_{2}(s)\centerdot H_{2}(y),$ (64)
where $H_{2}(y)=\left(H_{2,ij},i\leq j\right)$, with $H_{2,ii}=h_{2}(y_{i})$
and $H_{2,ij}=h_{1}(y_{i})h_{1}(y_{j})$ if $i\neq j$, $h_{m}$ is introduced in
(24); $V_{2}(s)=\left(V_{2,ij},i\leq j\right)$, with $V_{2,ij}$ being the
projection of $V$ on $H_{2,ij}$.
We claim that Theorem 4 is a direct consequence of the following:
###### Proposition 20 (Classification of the behavior of $V$ as
$s\to+\infty$).
One of the following possibilities occurs:
$i)\;$ $V(y,s)\equiv 0$,
$ii)$ There exists $l\in\\{1,\dots,n\\}$ such that up to an orthogonal
transformation of coordinates, we have
$V(y,s)=-\frac{\kappa}{4ps}\left(\sum_{j=1}^{l}y_{j}^{2}-2l\right)+\mathcal{O}\left(\frac{1}{s^{a}}\right)+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right)\quad\text{as}\quad s\to+\infty.$
$iii)$ There exist an integer number $m\geq 3$ and constants $c_{\alpha}$ not
all zero such that
$V(y,s)=-e^{\left(1-\frac{m}{2}\right)s}\sum_{|\alpha|=m}c_{\alpha}H_{\alpha}(y)+o\left(e^{\left(1-\frac{m}{2}\right)s}\right)\quad\text{as}\quad
s\to+\infty.$
The convergence takes place in $L^{2}_{\rho}$ as well as in
$\mathcal{C}^{k,\gamma}_{loc}$ for any $k\geq 1$ and $\gamma\in(0,1)$.
###### Remark 8.
Let us insist on the fact that the linearizing of $w$ around $\kappa$ would
generate some terms of the size $\frac{1}{s^{a}}$, and prevent us from
reaching exponentially small terms.
Let us first derive Theorem 4 assuming Proposition 20 and then we will prove
it later.
###### Proof of Theorem 4 assuming that Proposition 20 holds.
By the definition (59) of $V$, we see that $i)$ of Proposition 20 directly
follows that $v(y,s)\equiv\phi(s)$ which is $i)$ of Theorem 4. Using $ii)$ of
Proposition 20 and the fact that $\beta(s)=1+\mathcal{O}(\frac{1}{s^{a-1}})$
as $s\to+\infty$, we see that as $s\to+\infty$,
$\displaystyle w(y,s)$
$\displaystyle=\phi(s)+V(y,s)\left(1+\mathcal{O}(\frac{1}{s^{a-1}})\right)$
$\displaystyle=\phi(s)-\frac{\kappa}{4ps}\left(\sum_{j=1}^{l}y_{j}^{2}-2l\right)+\mathcal{O}\left(\frac{1}{s^{a}}\right)+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right)$
$\displaystyle=\kappa-\frac{\kappa}{4ps}\left(\sum_{j=1}^{l}y_{j}^{2}-2l\right)+\mathcal{O}\left(\frac{1}{s^{a}}\right)+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right),$
which yields $ii)$ of Theorem 4.
Using $iii)$ of Proposition 20 and again the fact that
$\beta(s)=1+\mathcal{O}(\frac{1}{s^{a-1}})$ as $s\to+\infty$, we have
$w(y,s)=\phi(s)-e^{\left(1-\frac{m}{2}\right)s}\sum_{|\alpha|=m}c_{\alpha}H_{\alpha}(y)+o\left(e^{\left(1-\frac{m}{2}\right)s}\right)\quad\text{as}\quad
s\to+\infty.$
This concludes the proof of Theorem 4 assuming that Proposition 20 holds. ∎
The proof of Proposition 20 will be very close to that in [3] and [18], thanks
to (61) and (62). It happens that the proofs written in Filippas, Kohn, Liu,
Herrero and Velázquez [2],[3], [10], [18] in the unperturbed case ($h\equiv
0$) hold for equation (60) under the general assumptions (61) and (62). For
that reason, we only give the sketch of the proof below and refer to these
papers for details of the proofs.
Following [3] and [18], we divide the proof into 3 steps which are given in
separated subsections:
\- Step 1: deriving the fact that either
$\|V_{+}(s)\|_{L^{2}_{\rho}}+\|V_{-}(s)\|_{L^{2}_{\rho}}=o\left(\|V_{null}(s)\|_{L^{2}_{\rho}}\right)$,
or $\|V(s)\|_{L^{2}_{\rho}}=\mathcal{O}(e^{-\mu s})$ for some $\mu>0$.
\- Step 2: assuming that
$\|V(y,s)\|_{L^{2}_{\rho}}\sim\|V_{null}(y,s)\|_{L^{2}_{\rho}}$, we find an
equation satisfied by $V_{null}(s)$ as $s\to+\infty$. Solving this equation,
we find that $\|V(s)\|_{L^{2}_{\rho}}$ behaves like $\frac{1}{s}$ as
$s\to+\infty$. Using this information, we can get a more accurate equation for
$V_{null}(s)$ as $s\to+\infty$ and then $ii)$ of Proposition 20 follows.
\- Step 3: assuming $\|V(s)\|_{L^{2}_{\rho}}=\mathcal{O}(e^{-\mu s})$ for some
$\mu>0$ as $s\to+\infty$, we derive $i)$ or $iii)$ of Proposition 20.
### 3.1 Finite dimension reduction of the problem.
We claim the following proposition:
###### Proposition 21 (Competition between $V_{+},V_{-}$ and $V_{null}$).
As $s\to+\infty$,
either $\displaystyle i)\;\|V(s)\|_{L^{2}_{\rho}}=\mathcal{O}\left(e^{-\mu
s}\right),\quad\text{for some $\mu>0$,}\qquad\qquad\qquad$ (65) or
$\displaystyle
ii)\;\|V_{+}(s)\|_{L^{2}_{\rho}}+\|V_{-}(s)\|_{L^{2}_{\rho}}=o\left(\|V_{null}(s)\|_{L^{2}_{\rho}}\right).\qquad\qquad$
(66)
###### Proof.
Let us denote
$Z(s)=\|V_{+}(s)\|_{L^{2}_{\rho}},\quad
X(s)=\|V_{null}(s)\|_{L^{2}_{\rho}},\quad Y(s)=\|V_{-}(s)\|_{L^{2}_{\rho}},$
(67)
then the following lemma is claimed:
###### Lemma 22.
Let $\epsilon>0$, there exists $s^{*}=s^{*}(\epsilon)\in\mathbb{R}$ such that
for all $s\geq s^{*}$,
$\displaystyle Z^{\prime}$
$\displaystyle\geq\left(\frac{1}{2}-\epsilon\right)Z-\epsilon(X+\bar{Y})$
$\displaystyle\left|X^{\prime}\right|$
$\displaystyle\leq\epsilon(X+\bar{Y}+Z)$ $\displaystyle\bar{Y}^{\prime}$
$\displaystyle\leq-\left(\frac{1}{2}-\epsilon\right)\bar{Y}+\epsilon\left(X+Z\right)$
where $\bar{Y}(s)=Y(s)+r(s)$ with
$r(s)=\left\||y|^{\frac{k}{2}}V^{2}(s)\right\|_{L^{2}_{\rho}}$ for a fixed
integer $k$.
###### Proof.
From the fact that $|\bar{F}(V,s)|\leq CV^{2}$ for $s$ large, the proof is the
same as the proof of Theorem A, pages 842-847 in Filippas and Kohn [2]. ∎
The following lemma allows us to conclude Proposition 21:
###### Lemma 23.
Let $\xi(t),\nu(t),\zeta(t)$ be absolutely continuous, real-valued functions
that are nonnegative and satisfy:
$i)$ $(\xi(t),\nu(t),\zeta(t))\to 0$ as $t\to+\infty$,
$ii)$ For all $\epsilon>0$, there exists $t_{0}\in\mathbb{R}$ such that for
all $t\geq t_{0}$,
$\displaystyle\zeta^{\prime}$ $\displaystyle\geq c_{0}\zeta-\epsilon(\xi+\nu)$
$\displaystyle|\xi^{\prime}|$ $\displaystyle\leq\epsilon(\xi+\nu+\zeta)$
$\displaystyle\nu^{\prime}$ $\displaystyle\leq-c_{0}\nu+\epsilon(\xi+\zeta),$
for some $c_{0}>0$.
Then either $\xi+\zeta=o(\nu)$ or $\nu+\zeta=o(\xi)$ as $t\to+\infty$.
###### Remark 9.
In the first case, we clearly see that $\nu^{\prime}\leq-\frac{c_{0}}{2}\nu$
for $t$ large, hence $\xi,\upsilon,\zeta$ tend to zero exponentially fast.
###### Proof.
The original proof is due to Filippas and Kohn [2]. For this particular
statement, see Lemma A.1, page 3425 [13] for the proof. ∎
Since $\|V(s)\|_{L^{\infty}_{loc}}\to 0$ as $s\to+\infty$, we have
$X(s),\bar{Y}(s),Z(s)\to 0$ as $s\to+\infty$. Thus, Lemma 23 applies to
$X(s),\bar{Y}(s),$ and $Z(s)$ and yields the desired result (use the remark
after the statement). This ends the proof of Proposition 21. ∎
### 3.2 Deriving conclusion $ii)$ of Proposition 20
In this part, we recall from Filippas and Liu the proof of $ii)$ of
Proposition 20. We focus on the case $ii)$ of Proposition 21, namely that
$\|V_{+}(s)\|_{L^{2}_{\rho}}+\|V_{-}(s)\|_{L^{2}_{\rho}}=o\left(\|V_{null}(s)\|_{L^{2}_{\rho}}\right)\quad\text{as}\quad
s\to+\infty,$ (68)
and show that it leads to case $ii)$ of Proposition 20.
We first claim the following proposition:
###### Proposition 24 (An ODE satisfied by $V_{null}(s)$ as $s\to+\infty$).
If
$\|V_{+}(s)\|_{L^{2}_{\rho}}+\|V_{-}(s)\|_{L^{2}_{\rho}}=o\left(\|V_{null}(s)\|_{L^{2}_{\rho}}\right)$,
then
$i)$ for all $i,j\in\\{1,...,n\\}$ and as $s\to+\infty$,
$V_{2,ij}^{\prime}(s)=\frac{p}{2\kappa}\int_{\mathbb{R}}V_{null}^{2}(y,s)\frac{H_{2,ij}(y)}{\|H_{2,ij}(y)\|_{L^{2}_{\rho}}^{2}}\rho(y)dy+o\left(\|V_{null}(s)\|^{2}_{L^{2}_{\rho}}\right).$
(69)
$ii)$ There exist a symmetric $n\times n$ matrix $A(s)$ such that for all
$s\in\mathbb{R}$,
$\displaystyle V_{null}(y,s)=y^{T}A(s)y-2tr(A(s))$ and $\displaystyle
c_{1}\|A(s)\|\leq\|V_{null}(s)\|_{L^{2}_{\rho}}\leq c_{2}\|A(s)\|$ (70)
where $c_{1},c_{2}$ are some positive constant and $\|A\|$ stands for any norm
on the space of $n\times n$ symmetric matrices. Moreover,
$A^{\prime}(s)=\frac{4p}{\kappa}A^{2}(s)+o\left(\|A(s)\|^{2}\right)\quad\text{as}\quad
s\to+\infty.$ (71)
###### Proof.
Let us remark that $ii)$ follows directly from $i)$. Here, one has to use (62)
which is more accurate than (61), in order to isolate the $\mathcal{O}(V^{2})$
term in the nonlinear term. Using properties of Hermites polynomials, we may
project that term and obtain (69). ∎
In the next step, we show that although we can not derive directly from (69)
the asymptotic behavior of $V_{null}(s)$, we can use it to show that
$\|V(s)\|_{L^{2}_{\rho}}$ decays like $\frac{1}{s}$ as $s\to+\infty$. More
precisely, we have the following proposition:
###### Proposition 25.
If
$\|V_{+}(s)\|_{L^{2}_{\rho}}+\|V_{-}(s)\|_{L^{2}_{\rho}}=o\left(\|V_{null}(s)\|_{L^{2}_{\rho}}\right)$,
then for $s$ large, we have
$\frac{c_{1}}{s}\leq\|V(s)\|_{L^{2}_{\rho}}\leq\frac{c_{2}}{s},$ (72)
for some positive constants $c_{1}$ and $c_{2}$.
###### Proof.
Since $\|V(s)\|_{L^{2}_{\rho}}\sim\|V_{null}(s)\|_{L^{2}_{\rho}}$ and because
of (70), it is enough to show that
$\frac{c_{1}}{s}\leq\|A(s)\|\leq\frac{c_{2}}{s},\quad\text{for $s$ large}.$
(73)
Since the proof of (73) is totally given in Section 3 of Filippas and Liu [3],
we just give its steps of the proof below. The following Lemma asserts that
$A(s)$ has continuously differential eigenvalues:
###### Lemma 26 ([16, 11]).
Suppose that $A(s)$ is a $n\times n$ symmetric and continuously differentiable
matrix-function in some interval $I$, then there exists continuously
differentiable functions $\lambda_{1}(s),\dots,\lambda_{n}(s)$ in $I$ such
that for all $i\in\\{1,\dots,n\\}$,
$A(s)\Phi^{(i)}(s)=\lambda_{i}(s)\Phi^{(i)}(s),$
for some orthonormal system of vector-functions
$\Phi^{(1)}(s),\dots,\Phi^{(n)}(s)$.
Let $\lambda_{1}(s),\dots,\lambda_{n}(s)$ be the eigenvalues of $A(s)$. We can
derive from (71) an equation satisfied by
$\lambda_{i}(s),\,i\in\\{1,\dots,n\\}$:
###### Lemma 27 (Filippas and Liu [3]).
The eigenvalues of $A(s)$ satisfy for all $i\in\\{1,\dots,n\\}$,
$\lambda^{\prime}_{i}(s)=\frac{4p}{\kappa}\lambda^{2}_{i}(s)+o\left(\sum_{i=1}^{n}\lambda^{2}_{i}(s)\right).$
(74)
Using (74), one can show that (see the end of Section 3 in [3])
$\frac{c_{1}}{s}\leq\sum_{i=1}^{n}|\lambda_{i}(s)|\leq\frac{c_{2}}{s},\quad\text{for
$s$ large}.$ (75)
Since $\|A(s)\|=\sum_{i=1}^{n}|\lambda_{i}(s)|$, this concludes the proof of
(73) and Proposition 25 also. ∎
Using the fact that $\|V(s)\|_{L^{2}_{\rho}}$ decays like $\frac{1}{s}$, we
will show that $\|V_{-}(s))\|_{L^{2}_{\rho}}+\|V_{+}(s))\|_{L^{2}_{\rho}}$ is
in fact $\mathcal{O}(\|V_{null}(s))\|^{2}_{L^{2}_{\rho}}$ and not only
$o(\|V_{null}(s))\|_{L^{2}_{\rho}})$. This new estimate will be used then to
derive a more accurate equation satisfied by $V_{null}$.
###### Proposition 28.
If
$\|V_{+}(s)\|_{L^{2}_{\rho}}+\|V_{-}(s)\|_{L^{2}_{\rho}}=o\left(\|V_{null}(s)\|_{L^{2}_{\rho}}\right)$,
then we have
$\displaystyle V_{2,ij}^{\prime}(s)$
$\displaystyle=\frac{p}{2\kappa}\int_{\mathbb{R}^{n}}V^{2}_{null}(y,s)\frac{H_{2,ij}(y)}{\|H_{2,ij}\|^{2}_{L^{2}_{\rho}}}\rho(y)dy$
$\displaystyle\qquad\qquad+\mathcal{O}\left(\|V_{null}(s)\|^{3}_{L^{2}_{\rho}}\right)+\mathcal{O}\left(\frac{\|V_{null}(s)\|^{2}_{L^{2}_{\rho}}}{s^{a-1}}\right),$
(76)
and
$A^{\prime}(s)=\frac{4p}{\kappa}A^{2}(s)+\mathcal{O}\left(\frac{1}{s^{3}}\right)+\mathcal{O}\left(\frac{1}{s^{a+1}}\right),$
(77)
where $A(s)$ is given in (70).
###### Proof.
The proof corresponds to Section 4 in [3]. Let us mention that the proof
relies on the following priori estimate of solutions of (60) shown by Herrero
and Velázquez in [10]. Although they proved their result in the case $N=1$,
their proof holds in higher dimensions under the general assumption (61).
###### Lemma 29 (Herrero and Valázquez [10]).
Assume that $V$ solves (60) and $|V|\leq M<+\infty$. Then for any $r>1$, $q>1$
and $L>0$, there exist $s_{0}^{*}=s_{0}^{*}(q,r)$ and $C=C(q,r,L)>0$ such that
$\left(\int_{\mathbb{R}^{n}}|V(y,s+\tau)|^{r}\rho(y)dy\right)^{\frac{1}{r}}\leq
C\left(\int_{\mathbb{R}^{n}}|V(y,s)|^{q}\rho(y)dy\right)^{\frac{1}{q}},$
for any $s\geq 0$ and any $\tau\in[s_{0}^{*},s_{0}^{*}+L]$.
From Proposition 25, we have $\|V(s)\|_{L^{2}_{\rho}}$ decays like
$\frac{1}{s}$. Then Lemma 29 implies that
$\left(\int_{\mathbb{R}^{n}}|V(y,s)|^{r}\rho(y)dy\right)^{\frac{1}{r}}\leq
C\left(\int_{\mathbb{R}^{n}}|V(y,s)|^{q}\rho(y)dy\right)^{\frac{1}{q}},$ (78)
for any $r>1$, $q>1$ and for $s$ large.
Using estimate (78), we derive the fact that
$\|V_{+}(s)\|_{L^{2}_{\rho}}+\|V_{-}(s)\|_{L^{2}_{\rho}}=\mathcal{O}\left(\|V_{null}(s)\|^{2}_{L^{2}_{\rho}}\right).$
(79)
Then, projecting (60) onto the null space of $\mathcal{L}$ and using (79),
(78), we would obtain (76). Since
$\|V(s)\|_{L^{2}_{\rho}}\sim\|V_{null}(s)\|_{L^{2}_{\rho}}\sim\frac{1}{s}$, we
then obtain (77) from (76). This ends the proof of Proposition 28. ∎
Let us now use 28 to derive conclusion $ii)$ of Proposition 20. Using Lemma
26, we get from (77) that the eigenvalues of $A(s)$ satisfy
$\forall
i\in\\{1,\dots,n\\},\;\;\lambda^{\prime}_{i}(s)=\frac{4p}{\kappa}\lambda_{i}^{2}(s)+\mathcal{O}\left(\frac{1}{s^{a+1}}\right)+\mathcal{O}\left(\frac{1}{s^{3}}\right),\quad\text{as}\;\;s\to+\infty,$
then Lemma C.2 yields
either
$\displaystyle\;\;\lambda_{i}(s)=-\frac{\kappa}{4ps}+\mathcal{O}\left(\frac{1}{s^{a}}\right)\;\;\text{or}\;\;\lambda_{i}(s)=\mathcal{O}\left(\frac{1}{s^{a}}\right),\;\;\text{if}\;a\in(1,2),$
(80) either
$\displaystyle\;\;\lambda_{i}(s)=-\frac{\kappa}{4ps}+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right)\;\;\text{or}\;\;\lambda_{i}(s)=\mathcal{O}\left(\frac{1}{s^{2}}\right),\;\;\text{if}\;a\geq
2.$ (81)
Therefore, Proposition 5.1 in [3] yields the existence of
$l\in\\{1,\dots,n\\}$ and a $n\times n$ orthonormal matrix $Q$ such that
$\displaystyle A(s)$
$\displaystyle=-\frac{\kappa}{4ps}A_{l}+\mathcal{O}\left(\frac{1}{s^{a}}\right),\;\;\text{if}\;a\in(1,2),$
$\displaystyle A(s)$
$\displaystyle=-\frac{\kappa}{4ps}A_{l}+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right),\;\;\text{if}\;a\geq 2,$
where
$A_{l}=Q\left(\begin{array}[]{cc}\mathbf{I}_{l}&O\\\
O&O\end{array}\right)Q^{-1}.$
Combining this with (70), it yields the behavior of $V_{null}(y,s)$ and
$V(y,s)$ announced in $ii)$ of Proposition 20. The convergence in
$\mathcal{C}^{k,\gamma}_{loc}$ follows from standard parabolic regularity (see
section 5 in [3] for a brief demonstration). This completes the proof of $ii)$
of Proposition 20.
### 3.3 Deriving conclusions $i)$ and $iii)$ of Proposition 20
In this part, we recall the proof given by Velázquez [18]. We focus on the
case $i)$ of Proposition 21, namely
$\|V(s)\|_{L^{2}_{\rho}}=\mathcal{O}(e^{-\mu s})$ for some $\mu>0$, and we
will show that it leads to either $i)$ or $iii)$ of Proposition 20. Let us
start the first step. From equation (60), we write $V(y,s)$ in the integration
form
$V(y,s)=S_{\mathcal{L}}(s)V(s_{0})+\int_{s_{0}}^{s}S_{\mathcal{L}}(s-\tau)\bar{F}(V(\tau),\tau)d\tau,\quad\text{with}\quad
s_{0}=-\log T,$
where $S_{\mathcal{L}}(s)$ is the linear semigroup corresponding to the heat-
type equation $\partial V=\mathcal{L}V$ given by
$S_{\mathcal{L}}(s)V(y,\tau)=\sum_{|\alpha|=0}^{\infty}a_{\alpha}(\tau)e^{\left(1-\frac{|\alpha|}{2}\right)(s-\tau)}H_{\alpha}(y),$
with
$a_{\alpha}(\tau)=\langle
V(\tau),H_{\alpha}\rangle:=\int_{\mathbb{R}^{n}}V(y,\tau)H_{\alpha}(y)\rho(y)dy.$
Let us fix a integer $k_{0}>2$ such that
$\frac{k_{0}}{2}-1<2\mu<\frac{k_{0}+1}{2}-1$ and write $V(y,s)$ as follow:
$\displaystyle V(y,s)$ $\displaystyle=\sum_{|\alpha|\leq
k_{0}}a_{\alpha}(s_{0})e^{\left(1-\frac{|\alpha|}{2}\right)(s-s_{0})}H_{\alpha}(y)+\sum_{|\alpha|\geq
k_{0}+1}a_{\alpha}(s_{0})e^{\left(1-\frac{|\alpha|}{2}\right)(s-s_{0})}H_{\alpha}(y)$
$\displaystyle+\sum_{|\alpha|\leq
k_{0}}H_{\alpha}(y)\int_{s_{0}}^{s}e^{\left(1-\frac{|\alpha|}{2}\right)(s-\tau)}\langle\bar{F}(V(y,\tau),\tau),H_{\alpha}(y)\rangle
d\tau$ $\displaystyle+\sum_{|\alpha|\geq
k_{0}+1}H_{\alpha}(y)\int_{s_{0}}^{s}e^{\left(1-\frac{|\alpha|}{2}\right)(s-\tau)}\langle\bar{F}(V(y,\tau),\tau),H_{\alpha}(y)\rangle
d\tau$ $\displaystyle:=I+II+III+IV.$
Since $|\bar{F}(V,s)|\leq C|V|^{2}$ and $\|V(s)\|_{L^{2}_{\rho}}\leq Ce^{-\mu
s}$, we derive from Lemma 29 that
$\|\bar{F}(V(\cdot,\tau))\|_{L^{2}_{\rho}}\leq Ce^{-2\mu\tau}.$ (82)
By a direct computation, we find that
$\|II\|_{L^{2}_{\rho}}+\|IV\|_{L^{2}_{\rho}}\leq Ce^{-2\mu s},\quad\text{for
some $C>0$}.$
For $III$, we write
$\displaystyle\int_{s_{0}}^{s}e^{-\left(1-\frac{|\alpha|}{2}\right)\tau}\langle\bar{F}(V(y,\tau),\tau),H_{\alpha}(y)\rangle
d\tau$
$\displaystyle\qquad\qquad=\beta_{\alpha}-\int_{s}^{+\infty}e^{-\left(1-\frac{|\alpha|}{2}\right)\tau}\langle\bar{F}(V(y,\tau),\tau),H_{\alpha}(y)\rangle
d\tau.$
Using (82), we can bound the last term of the above expression by $Ce^{-2\mu
s}$. Hence,
$V(y,s)=\sum_{|\alpha|\leq
k_{0}}(a_{\alpha}+\beta_{\alpha})e^{\left(1-\frac{|\alpha|}{2}\right)s}H_{\alpha}(y)+Q(y,s),$
where $\|Q(y,s)\|_{L^{2}_{\rho}}=\mathcal{O}(e^{-2\mu s})$.
Since $\|V(y,s)\|_{L^{2}_{\rho}}=\mathcal{O}(e^{-\mu s})$, it requires
$a_{\alpha}+\beta_{\alpha}=0$ for $|\alpha|\leq 2$. Thus, we have two
possibilities: if there exists an integer $m\in[3,k_{0}]$ such that
$a_{\alpha}+\beta_{\alpha}\neq 0$ for $|\alpha|=m$ and
$a_{\alpha}+\beta_{\alpha}=0$ for all $|\alpha|<m$, then we obtain $iii)$ of
Proposition 20 for some $m\in[3,k_{0}]$. If this is not the case, we get
$\|V(y,s)\|_{L^{2}_{\rho}}=\mathcal{O}(e^{-2\mu s})$. Using this new estimate
and repeating the process in a finite number of steps, we may obtain either
$iii)$ of Proposition 20 for some $m\geq 3$ or
$\|V(y,s)\|_{L^{2}_{\rho}}=\mathcal{O}(e^{-Rs})$ for any $R>0$. For the second
case, we use the following nondegeneracy result from Herrero and Velázquez
[10] in order to conclude that $V(y,s)\equiv 0$, which is $i)$ of Proposition
20,
###### Lemma 30 (Herrero and Velázquez [10]).
Let $V$ be a solution to equation (60). Assume that $|V(y,s)|$ is bounded, and
that for any $R>0$ there exists $C=C(R)$ such that
$\|V(s)\|_{L^{2}_{\rho}}\leq Ce^{-Rs}\quad\text{if}\quad s\geq 0,$
then $V(y,s)\equiv 0$.
###### Proof.
Since the proof written in [10] holds under general assumption (61), we then
refer the reader to Lemma 3.5, page 144 of [10] for detail of the proof. ∎
Since the convergence in $\mathcal{C}^{k,\gamma}_{loc}$ for any $k\geq 1$ and
$\gamma\in(0,1)$ follows from a standard parabolic regularity, we end the
proof of Proposition 20 here. This also concludes the proof of Theorem 4.
## 4 Bow-up profile for equation (1) in extended spaces regions
We give the proof of Theorem 5 in this section. Note that the derivation of
Theorem 5 from Theorem 4 in the unperturbed case ($h\equiv 0$) was done by
Velázquez in [17]. The idea to extend the convergence up to sets of the type
$\\{|y|\leq K_{0}\sqrt{s}\\}$ or $\\{|y|\leq
K_{0}e^{\left(\frac{1}{2}-\frac{1}{m}\right)s}\\}$ is to estimate the effect
of the convective term $-\frac{y}{2}\cdot\nabla w$ in the equation (9) in
$L^{q}_{\rho}$ spaces with $q>1$. Since the proof of Theorem 5 is actually in
spirit by the method given in [17], all that we need to do is to control the
strong perturbation term in equation (9). We therefore give the main steps of
the proof and focus only on the new arguments. The proof will be separated
into two parts: the first part concerns case $ii)$ in Theorem 4 and gives the
asymptotic behavior of $w$ in the $\frac{y}{\sqrt{s}}$ variable, and the
second part concerns case $iii)$ in Theorem 4 and gives the asymptotic
behavior of $w$ in the $ye^{-\left(\frac{1}{2}-\frac{1}{m}\right)s}$ variable.
In Part 1, we stick to the method of Velázquez [17], whereas, in Part 2, where
we work in the scale $e^{-\mu s}$ for $\mu>0$, we need new ideas to get rid of
the term in the scale $\frac{1}{s}$ coming from the strong perturbation.
Part 1: Case $ii)$ in Theorem 4 and asymptotic behavior in the
$\frac{y}{\sqrt{s}}$ variable.
Let us restate $i)$ of Theorem 5 in the following proposition:
###### Proposition 31 (Asymptotic behavior in the $\frac{y}{\sqrt{s}}$
variable).
Assume that $w$ is a solution of equation (9) which satisfies $ii)$ of Theorem
4. Then, for all $K>0$,
$\sup_{|\xi|\leq
K}\left|w(\xi\sqrt{s},s)-f_{l}(\xi)\right|=\mathcal{O}\left(\frac{1}{s^{a-1}}\right)+\mathcal{O}\left(\frac{\log
s}{s}\right),\quad\text{as}\quad s\to+\infty,$
where
$f_{l}(\xi)=\kappa\left(1+\frac{p-1}{4p}\sum_{j=1}^{l}\xi_{j}^{2}\right)^{-\frac{1}{p-1}}$.
###### Proof.
Following the method in [17], we define $q=w-\varphi$, where
$\varphi(y,s)=\kappa\left(1+\frac{p-1}{4ps}\sum_{j=1}^{l}y_{j}^{2}\right)^{-\frac{1}{p-1}}+\frac{\kappa
l}{2ps}.$ (83)
Using Taylor’s formula in (83) and $ii)$ of Theorem 4, we find that
$\|q(y,s)\|_{L^{2}_{\rho}}=\mathcal{O}\left(\frac{1}{s^{a}}\right)+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right),\quad\text{as}\quad s\to+\infty.$ (84)
Straightforward calculations based on equation (9) yield
$\partial_{s}q=(\mathcal{L}+\omega)q+F(q)+G(q,s)+R(y,s),\quad\forall(y,s)\in\mathbb{R}^{n}\times[-\log
T,+\infty),$ (85)
where
$\displaystyle\omega(y,s)$
$\displaystyle=p(\varphi^{p-1}-\kappa^{p-1})+e^{-s}h^{\prime}\left(e^{\frac{s}{p-1}}\varphi\right),$
$\displaystyle F(q)$
$\displaystyle=|q+\varphi|^{p-1}(q+\varphi)-\varphi^{p}-p\varphi^{p-1}q,$
$\displaystyle G(q,s)$
$\displaystyle=e^{-\frac{ps}{p-1}}\left[h\left(e^{\frac{s}{p-1}}(q+\varphi)\right)-h\left(e^{\frac{s}{p-1}}\varphi\right)-e^{\frac{s}{p-1}}h^{\prime}\left(e^{\frac{s}{p-1}}\varphi\right)q\right],$
$\displaystyle R(y,s)$
$\displaystyle=-\partial_{s}\varphi+\Delta\varphi-\frac{y}{2}\cdot\nabla\varphi-\frac{\varphi}{p-1}+\varphi^{p}+e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}\varphi\right).$
Let $K_{0}>0$ be fixed, we consider first the case $|y|\geq 2K_{0}\sqrt{s}$
and then $|y|\leq 2K_{0}\sqrt{s}$ and make a Taylor expansion for
$\xi=\frac{y}{\sqrt{s}}$ bounded. Simultaneously, noticing from (13), we then
obtain for all $s\geq s_{0}$,
$\omega(y,s)\leq\frac{C_{1}}{s},$ $|F(q)|+|G(q,s)|\leq
C_{1}(q^{2}+\mathbf{1}_{\\{|y|\geq 2K_{0}\sqrt{s}\\}}),$ $|R(y,s)|\leq
C_{1}\left(\frac{|y|^{2}}{s^{2}}+\frac{1}{s^{2}}+\frac{1}{s^{a}}+\mathbf{1}_{\\{|y|\geq
2K_{0}\sqrt{s}\\}}\right),$
where $C_{1}=C_{1}(M_{0},K_{0})>0$, $M_{0}$ is the bound of $w$ in
$L^{\infty}$-norm.
Let $Q=|q|$, we then use the above estimates and Kato’s inequality, i.e
$\Delta f\cdot\text{sign}(f)\leq\Delta(|f|)$, to derive from equation (85) the
following: for all $K_{0}>0$ fixed, there are $C_{*}=C_{*}(K_{0},M_{0})>0$ and
a time $s^{\prime}>0$ large enough such that for all $s\geq
s_{*}=\max\\{s^{\prime},-\log T\\}$,
$\partial_{s}Q\leq\left(\mathcal{L}+\frac{C_{*}}{s}\right)Q+C_{*}\left(Q^{2}+\frac{|y|^{2}}{s^{2}}+\frac{1}{s^{2}}+\frac{1}{s^{a}}+\mathbf{1}_{\\{|y|\geq
2K_{0}\sqrt{s}\\}}\right),\quad\forall y\in\mathbb{R}^{n}.$ (86)
Since
$\left|w(y,s)-f_{l}\left(\frac{y}{\sqrt{s}}\right)\right|\leq Q+\frac{\kappa
l}{2ps},$
the conclusion of Proposition 31 follows if we show that
$\forall K_{0}>0,\quad\sup_{|y|\leq K_{0}\sqrt{s}}Q(y,s)\to
0\quad\text{as}\quad s\to+\infty.$ (87)
Let us now focus on the proof of (87) in order to conclude Proposition 31. For
this purpose, we introduce the following norm: for $r\geq 0$, $q>1$ and $f\in
L^{q}_{loc}(\mathbb{R}^{n})$,
$L_{\rho}^{q,r}(f)\equiv\sup_{|\xi|\leq
r}\left(\int_{\mathbb{R}^{n}}|f(y)|^{q}\rho(y-\xi)dy\right)^{\frac{1}{q}}.$
Following the idea in [17], we shall make estimates on solution of (86) in the
$L^{2,r(\tau)}_{\rho}$ norm where $r(\tau)=K_{0}e^{\frac{\tau-\bar{s}}{2}}\leq
K_{0}\sqrt{\tau}$. Particularly, we have the following:
###### Lemma 32.
Let $s$ be large enough and $\bar{s}$ is defined by $e^{s-\bar{s}}=s$. Then
for all $\tau\in[\bar{s},s]$ and for all $K_{0}>0$, it holds that
$g(\tau)\leq
C_{0}\left(e^{\tau-\bar{s}}\epsilon(\bar{s})+\int_{\bar{s}}^{(\tau-2K_{0})_{+}}\frac{e^{(\tau-t-2K_{0})}g^{2}(t)}{\left(1-e^{-(\tau-t-2K_{0})}\right)^{1/20}}dt\right)$
where $g(\tau)=L^{2,r(K_{0},\tau,\bar{s})}_{\rho}(Q(\tau))$,
$r(K_{0},\tau,\bar{s})=K_{0}e^{\frac{\tau-\bar{s}}{2}}$,
$\epsilon(s)=\mathcal{O}\left(\frac{1}{s^{a}}\right)+\mathcal{O}\left(\frac{\log
s}{s^{2}}\right)$, $C_{0}=C_{0}(C_{*},M_{0},K_{0})$ and $z_{+}=\max\\{z,0\\}$.
###### Proof.
Multiplying (86) by $\alpha(\tau)=e^{\int_{\bar{s}}^{\tau}\frac{C_{*}}{t}}dt$,
then we write $Q(y,\tau)$ for all $(y,\tau)\in\mathbb{R}^{n}\times[\bar{s},s]$
in the integration form:
$\displaystyle Q(y,\tau)$
$\displaystyle=\alpha(\tau)S_{\mathcal{L}}(\tau-\bar{s})Q(y,\bar{s})$
$\displaystyle+C_{*}\int_{\bar{s}}^{\tau}\alpha(\tau)S_{\mathcal{L}}(\tau-t)\left(Q^{2}+\frac{|y|^{2}}{t^{2}}+\frac{1}{t^{2}}+\frac{1}{t^{a}}+\mathbf{1}_{\\{|y|\geq
2K_{0}\sqrt{t}\\}}\right)dt,$
where $S_{\mathcal{L}}$ is the linear semigroup corresponding to the operator
$\mathcal{L}$.
Next, we take the $L^{2,r(K_{0},\tau,\bar{s})}_{\rho}$-norms both sides in
order to get the following:
$\displaystyle g(\tau)$ $\displaystyle\leq
C_{0}L^{2,r}_{\rho}\big{[}S_{\mathcal{L}}(\tau-\bar{s})Q(\bar{s})\big{]}+C_{0}\int_{\bar{s}}^{\tau}L^{2,r}_{\rho}\big{[}S_{\mathcal{L}}(\tau-t)Q^{2}(t)\big{]}dt$
$\displaystyle+C_{0}\int_{\bar{s}}^{\tau}L^{2,r}_{\rho}\left[S_{\mathcal{L}}(\tau-t)\left(\frac{|y|^{2}}{t^{2}}+\frac{1}{t^{2}}+\frac{1}{t^{a}}\right)\right]dt$
$\displaystyle+C_{0}\int_{\bar{s}}^{\tau}L^{2,r}_{\rho}\big{[}S_{\mathcal{L}}(\tau-t)\mathbf{1}_{\\{|y|\geq
2K_{0}\sqrt{t}\\}}\big{]}dt$ $\displaystyle\equiv J_{1}+J_{2}+J_{3}+J_{4}.$
Proposition 2.3 in [17] (with a slight modification for the estimate of
$J_{3}$) yields
$|J_{1}|\leq
C_{0}e^{\tau-\bar{s}}\|Q(\bar{s})\|_{L^{2}_{\rho}}=e^{\tau-\bar{s}}\mathcal{O}(\epsilon(\bar{s}))\quad\text{as}\quad\bar{s}\to+\infty,$
$|J_{2}|\leq\frac{C_{0}}{\bar{s}^{2}}e^{\tau-\bar{s}}+C_{0}\int_{\bar{s}}^{(\tau-2K_{0})_{+}}\frac{e^{(\tau-t-2K_{0})}}{\left(1-e^{-(\tau-t-2K_{0})}\right)^{1/20}}\left[L_{\rho}^{2,r(K_{0},t,\bar{s})}Q(t)\right]^{2}dt,$
$|J_{3}|\leq
C_{0}e^{\tau-\bar{s}}\left(\frac{1}{\bar{s}^{2}}+\frac{1}{\bar{s}^{a}}\right)(1+(\tau-\bar{s})),$
$|J_{4}|\leq
C_{0}e^{-\delta\bar{s}},\quad\text{where}\quad\delta=\delta(K_{0})>0.$
Putting together the estimates on $J_{i},i=1,2,3,4$, we conclude the proof of
Lemma 32. ∎
We now use the following Gronwall lemma from Velázquez [17]:
###### Lemma 33 (Velázquez [17]).
Let $\epsilon,C,R$ and $\delta$ be positive constants, $\delta\in(0,1)$.
Assume that $H(\tau)$ is a family of continuous functions satisfying
$\mathcal{H}(\tau)\leq\epsilon
e^{\tau}+C\int_{0}^{(\tau-R)_{+}}\frac{e^{\tau-s}\mathcal{H}^{2}(s)}{\left(1-e^{-(\tau-
s-R)}\right)^{\delta}}ds,\quad\text{for $\tau>0$}.$
Then there exist $\theta=\theta(\delta,C,R)$ and
$\epsilon_{0}=\epsilon_{0}(\delta,C,R)$ such that for all
$\epsilon\in(0,\epsilon_{0})$ and any $\tau$ for which $\epsilon
e^{\tau}\leq\theta$, we have
$\mathcal{H}(\tau)\leq 2\epsilon e^{\tau}.$
Applying Lemma 33 with $\mathcal{H}\equiv g$, we see from Lemma 32 that for
$s$ large enough,
$g(\tau)\leq
2C_{0}e^{\tau-\bar{s}}\epsilon(\bar{s}),\quad\forall\tau\in[\bar{s},s].$
If $\tau=s$, then $e^{s-\bar{s}}=s$, $r=K_{0}\sqrt{s}$ and
$g(s)\equiv
L^{2,K_{0}\sqrt{s}}_{\rho}\big{(}Q(s)\big{)}=\mathcal{O}\left(\frac{1}{s^{a-1}}\right)+\mathcal{O}\left(\frac{\log
s}{s}\right),\;\text{as}\quad s\to+\infty.$
By using the regularizing effects of the semigroup $S_{\mathcal{L}}$ (see
Proposition 2.3 in [17]), we then obtain
$\sup_{|y|\leq\frac{K_{0}\sqrt{s}}{2}}Q(y,s)\leq
C^{\prime}(C_{*},K_{0},M_{0})L^{2,K_{0}\sqrt{s}}_{\rho}(Q(s))=\mathcal{O}\left(\frac{1}{s^{a-1}}\right)+\mathcal{O}\left(\frac{\log
s}{s}\right),$
as $s\to+\infty$, which concludes the proof of Proposition 31. ∎
Part 2: Case $iii)$ in Theorem 4 and the asymptotic behavior in the
$ye^{-\left(\frac{1}{2}-\frac{1}{m}\right)s}$ variable.
We give the proof of $ii)$ of Theorem 5 in this part. Since we work in the
scale $e^{-\mu s}$ for $\mu>0$ in the case where $iii)$ in Theorem 4 occurs,
we need new ideas to get rid of the term in the scale $\frac{1}{s}$ coming
from the strong perturbation.
Let us restate $ii)$ of Theorem 5 in the following proposition:
###### Proposition 34 (Asymptotic behavior in the
$ye^{-\left(\frac{1}{2}-\frac{1}{m}\right)s}$ variable).
Assume that $w$ is a solution of equation (9) and satisfies $iii)$ of Theorem
4. Then, for all $K>0$,
$\sup_{|\xi|\leq K}\left|w(\xi
e^{\left(\frac{1}{2}-\frac{1}{m}\right)s},s)-\psi_{m}(\xi)\right|\to
0,\quad\text{as}\quad s\to+\infty,$ (88)
where
$\psi_{m}(\xi)=\kappa\left(1+\kappa^{-p}\sum\limits_{|\alpha|=m}c_{\alpha}\xi^{\alpha}\right)^{-\frac{1}{p-1}}$,
and $m\geq 4$ is an even integer.
###### Proof.
Note that the proof will proceed in the same way as for the proof of
Proposition 31. Let us introduce $q=w-\varphi$, where
$\varphi(y,s)=\frac{\phi(s)}{\kappa}J(y,s),$ (89)
with
$J(y,s)=\frac{\phi(s)}{\kappa}\left[G\left(ye^{-\left(\frac{1}{2}-\frac{1}{m}\right)s}\right)+e^{-\left(\frac{m}{2}-1\right)s}\left(\sum_{|\alpha|=m}c_{\alpha}y^{\alpha}-\sum_{|\alpha|=m}c_{\alpha}H_{\alpha}(y)\right)\right],$
and
$G(\xi)=\kappa\left(1+\kappa^{-p}\sum\limits_{|\alpha|=m}c_{\alpha}\xi^{\alpha}\right)^{-\frac{1}{p-1}}$
satisfying
$-\frac{\xi}{m}\cdot\nabla G(\xi)+G^{p}(\xi)=\frac{G(\xi)}{p-1}.$ (90)
Note that Velázquez [17] takes $\varphi=J$, and if we do the same, we will
obtain some terms in the scale of $\frac{1}{s}$, much stronger than the
$e^{-\mu s}$ scale that we intended to work in.
Using Taylor’s formula in (89) and recalling from Lemma A.2 that fact that
$\frac{\phi(s)}{\kappa}=1+\mathcal{O}(s^{-a})$ as $s\to+\infty$, we have by
$iii)$ of Theorem 4,
$\|q(y,s)\|_{L^{2}_{\rho}}=o\left(e^{-\left(\frac{m}{2}-1\right)s}\right),\quad\text{as
$s\to+\infty$}.$ (91)
Straightforward calculations based on equation (9) yield
$\partial_{s}q=(\mathcal{L}+\omega)q+F(q)+G(q,s)+R(y,s),\quad\forall(y,s)\in\mathbb{R}^{n}\times[-\log
T,+\infty),$ (92)
where
$\displaystyle\omega(y,s)$
$\displaystyle=p(\varphi^{p-1}-\kappa^{p-1})+e^{-s}h^{\prime}\left(e^{\frac{s}{p-1}}\varphi\right),$
$\displaystyle F(q)$
$\displaystyle=|q+\varphi|^{p-1}(q+\varphi)-\varphi^{p}-p\varphi^{p-1}q,$
$\displaystyle G(q,s)$
$\displaystyle=e^{-\frac{ps}{p-1}}\left[h\left(e^{\frac{s}{p-1}}(q+\varphi)\right)-h\left(e^{\frac{s}{p-1}}\varphi\right)-e^{\frac{s}{p-1}}h^{\prime}\left(e^{\frac{s}{p-1}}\varphi\right)q\right],$
$\displaystyle R(y,s)$
$\displaystyle=-\partial_{s}\varphi+\Delta\varphi-\frac{y}{2}\cdot\nabla\varphi-\frac{\varphi}{p-1}+\varphi^{p}+e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}\varphi\right).$
Fix now $K_{0}>0$ and define $\chi(y,s)=1$ if $|y|\geq
2K_{0}e^{\left(\frac{1}{2}-\frac{1}{m}\right)s}$ and $\chi(y,s)=0$ otherwise.
Then, using Taylor’s formula for
$\xi=ye^{-\left(\frac{1}{2}-\frac{1}{m}\right)s}$ bounded, and noticing from
(13), we then obtain for all $s\geq s_{0}$,
$\omega(y,s)\leq\frac{C_{1}}{s},$ $|F(q)|+|G(q,s)|\leq
C_{1}\left(q^{2}+\chi(y,s)\right),$
where $C_{1}=C_{1}(M_{0},K_{0})>0$.
To estimate $R(y,s)$, we write $R(y,s)$ as follow:
$\displaystyle R(y,s)$
$\displaystyle=\frac{\phi(s)}{\kappa}\left(-\partial_{s}J+\Delta
J-\frac{y}{2}\cdot\nabla J-\frac{J}{p-1}+J^{p}\right)$
$\displaystyle+\left(-\frac{\phi^{\prime}(s)}{\kappa}J-\frac{\phi(s)}{\kappa}J^{p}+\varphi^{p}+e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}\varphi\right)\right)\equiv\frac{\phi(s)}{\kappa}I+II.$
By using Taylor’s formula, (90) and Hermite’s equation, i.e.
$\mathcal{L}H_{\alpha}(y)=\left(1-\frac{|\alpha|}{2}\right)H_{\alpha}(y),$
it was proved in [17] (see Proposition 2.4) that
$I\leq Ce^{-(m-2)s}(|y|^{2m-2}+1)(1-\chi(y,s))+C\chi(y,s),\quad\text{for some
$C>0$}.$
It remains to estimate $II$. To do so, we write $J(y,s)$ for
$|y|e^{\left(\frac{1}{2}-\frac{1}{m}\right)s}$ bounded in the form:
$J(y,s)=\kappa-e^{-\left(\frac{m}{2}-1\right)s}\sum_{|\alpha|=m}c_{\alpha}H_{\alpha}(y)+\mathcal{O}\left(e^{-(m-2)s}|y|^{2m}\right).$
We then use Taylor’s formula in $II$, (13), and the fact that $\phi(s)$
satisfies (21) to find that
$II\leq\frac{C}{s^{a}}e^{-\left(\frac{m}{2}-1\right)s}(|y|^{m}+1)(1-\chi(y,s))+C\chi(y,s),\quad\text{for
some $C>0$}.$
Note that
$e^{-(m-2)s}|y|^{2m-2}(1-\chi(y,s))\leq\frac{1}{s^{a}}e^{-\left(\frac{m}{2}-1\right)s}|y|^{m}(1-\chi(y,s))$
for $s$ large, we then obtain
$|R(y,s)|\leq
C\left(\frac{1}{s^{a}}e^{-\left(\frac{m}{2}-1\right)s}(|y|^{m}+1)(1-\chi(y,s))+\chi(y,s)\right),\;\;\text{for
some $C>0$}.$
Let $Q=|q|$ and use Kato’s inequality, we obtain from (92) and from the above
estimates that: for all $K_{0}>0$ fixed, there are
$C_{*}=C_{*}(K_{0},M_{0})>0$ and a time $s^{\prime}>0$ large enough such that
for all $s\geq s_{*}=\max\\{s^{\prime},-\log T\\}$,
$\partial_{s}Q\leq\left(\mathcal{L}+\frac{C_{*}}{s}\right)Q+C_{*}\left(Q^{2}+\frac{1}{s^{a}}e^{-\left(\frac{m}{2}-1\right)s}(|y|^{m}+1)+\chi(y,s)\right),\;\;\forall
y\in\mathbb{R}^{n}.$ (93)
We claim the following:
###### Lemma 35.
Let $s$ be large enough and $\bar{s}=\frac{2s}{m}$. Then for all
$\tau\in[\bar{s},s]$, $\tau-\bar{s}\geq 2$ and for all $K_{0}>0$, it holds
that
$g(\tau)\leq
e^{\tau-\bar{s}}\left(o\left(e^{-\left(\frac{m}{2}-1\right)\bar{s}}\right)+C^{\prime}\int_{\bar{s}}^{(\tau-2K_{0})_{+}}\frac{e^{(\tau-t-2K_{0})}g^{2}(t)}{\left(1-e^{-(\tau-t-2K_{0})}\right)^{1/20}}dt\right)$
where $g(\tau)=L^{2,r(K_{0},\tau,\bar{s})}_{\rho}(Q(\tau))$,
$r(K_{0},\tau,\bar{s})=K_{0}e^{\frac{\tau-\bar{s}}{2}}$,
$C^{\prime}=C^{\prime}(C_{*},M_{0},K_{0})$ and $z_{+}=\max\\{z,0\\}$.
###### Proof.
Proceeding as in the proof of Lemma (32), we write
$\displaystyle L^{2,r}_{\rho}(Q)$ $\displaystyle\leq
C_{0}L^{2,r}_{\rho}\big{[}S_{\mathcal{L}}(\tau-\bar{s})Q(\bar{s})\big{]}+C_{0}\int_{\bar{s}}^{\tau}L^{2,r}_{\rho}\big{[}S_{\mathcal{L}}(\tau-t)Q^{2}(t)\big{]}dt$
$\displaystyle+C_{0}\int_{\bar{s}}^{\tau}L^{2,r}_{\rho}\left[S_{\mathcal{L}}(\tau-t)\left(\frac{1}{t^{a}}e^{-\left(\frac{m}{2}-1\right)t}(|y|^{m}+1)\right)\right]dt$
$\displaystyle+C_{0}\int_{\bar{s}}^{\tau}L^{2,r}_{\rho}\big{[}S_{\mathcal{L}}(\tau-t)\chi(y,t)\big{]}dt$
$\displaystyle\equiv J_{1}+J_{2}+J_{3}+J_{4}.$
One can show that for $\bar{s}$ large enough (see Proposition 2.4 in [17]),
$|J_{1}|=e^{\tau-\bar{s}}o\left(e^{(1-m/2)\bar{s}}\right),$ $|J_{2}|\leq
Ce^{2\tau-2(m-1)\bar{s}}+C\int_{\bar{s}}^{(\tau-2K_{0})_{+}}\frac{e^{(\tau-t-2K_{0})}g^{2}(t)}{\left(1-e^{-(\tau-t-2K_{0})}\right)^{1/20}}dt,$
$|J_{3}|\leq
Ce^{\tau-\bar{s}}\frac{e^{(1-m/2)\bar{s}}}{\bar{s}^{a}}\left(1+\tau-\bar{s}\right)=e^{\tau-\bar{s}}o\left(e^{(1-m/2)\bar{s}}\right),$
$|J_{4}|\leq Ce^{-\theta e^{(1-2/m)\bar{s}}}\quad\text{for some $\theta>0$}.$
Putting together the above estimates yields the desired result. This ends the
proof of Lemma 35. ∎
Applying now Lemma 33 and Lemma 35, we obtain for $s$ large enough,
$g(\tau)\leq
e^{\tau-\bar{s}}o\left(e^{-(m/2-1)\bar{s}}\right),\quad\forall\tau\in[\bar{s},s].$
Since $\bar{s}=\frac{2s}{m}$, if we set $\tau=s$, then
$r=K_{0}e^{\left(\frac{1}{2}-\frac{1}{m}\right)s}$ and
$g(s)\equiv L^{2,r(s)}_{\rho}(Q(s))=o(1)\quad\text{as}\quad s\to+\infty.$
By the regularizing effects of the semigroup $S_{\mathcal{L}}$, we then obtain
$\sup_{|y|\leq\frac{K_{0}}{2}e^{\left(\frac{1}{2}-\frac{1}{m}\right)s}}Q(y,s)\leq
C^{\prime}(C_{*},K_{0},M_{0})L^{2,r(s)}_{\rho}(Q(s))\to
0,\;\text{as}\;s\to+\infty,$
From (89), we see that for all
$|y|\leq\frac{K_{0}}{2}e^{\left(\frac{1}{2}-\frac{1}{m}\right)s}$,
$\left|w(y,s)-\frac{\phi(s)}{\kappa}G\left(ye^{-\left(\frac{1}{2}-\frac{1}{m}\right)s}\right)\right|\leq
Q(y,s)+Ce^{-\left(1-\frac{2}{m}\right)s},$
Noticing from Lemma A.3 that $\frac{\phi(s)}{\kappa}=1+\mathcal{O}(s^{-a})$ as
$s\to+\infty$, we obtain
$\sup_{|y|\leq\frac{K_{0}}{2}e^{\left(\frac{1}{2}-\frac{1}{m}\right)s}}\left|w(y,s)-G\left(ye^{-\left(\frac{1}{2}-\frac{1}{m}\right)s}\right)\right|=o(1),\quad\text{as}\quad
s\to+\infty.$
It remains to show that $m$ is even. Indeed, from (88), we can see that if $m$
is not even, there would exist $\xi_{0}\in\mathbb{R}^{n}$ such that
$w\left(\xi_{0}e^{\left(\frac{1}{2}-\frac{1}{m}\right)s},s\right)\to\psi_{m}(\xi_{0})\to+\infty$
as $s\to+\infty$, which contradicts the fact that $w$ is bounded as stated in
(19). Therefore, $m$ must be even. This concludes the proof of Proposition 34
and Theorem 5 too. ∎
## Appendix A Appendix A
The following lemma shows the asymptotic behavior of the solution of the
associated ODE of equation 1:
###### Lemma A.1.
Let $v$ be a blow-up solution of the following ordinary differential equation:
$v^{\prime}(t)=v^{p}(t)+h(v),\quad v(T)=+\infty\quad\text{for some $T>0$},$
(94)
where $h$ is defined in (3). Then $v$ satisfies
$v(t)\sim\kappa(T-t)^{-\frac{1}{p-1}}\quad\text{as $t\to T$, where
$\kappa=(p-1)^{-\frac{1}{p-1}}$}.$
###### Proof.
Divide (94) by $v^{p}$ and note that $\frac{h(v)}{v^{p}}\to 0$ as
$v\to+\infty$, we see that for all $\varepsilon>0$, there exists a number
$\delta=\delta(\epsilon)>0$ such that
$\left|\frac{v^{\prime}}{v^{p}}-1\right|\leq\varepsilon,\quad\forall
t\in[T-\delta,T).$ (95)
Solving (95) with noting that $v(T)=+\infty$ yields
$\displaystyle(1+\varepsilon)^{-\frac{1}{p-1}}\kappa(T-t)^{-\frac{1}{p-1}}$
$\displaystyle\leq
v(t)\leq(1-\varepsilon)^{-\frac{1}{p-1}}\kappa(T-t)^{-\frac{1}{p-1}},\quad\forall
t\in[T-\delta,T).$
This concludes the proof of Lemma A.1. ∎
The following lemma gives us an estimation of the perturbation term in
equation (9):
###### Lemma A.2.
Let $h$ be the function defined in (3), then it holds that
$j=0,1,\quad
e^{-\frac{(p-j)s}{p-1}}\left|h^{(j)}\left(e^{\frac{s}{p-1}}w\right)\right|\leq
Cs^{-a}\left(|w|^{p-j}+1\right),\quad\forall s\geq\hat{s},$
where $C=C(a,p,\mu,M)>0$ and $\hat{s}=\hat{s}(a,p)>0$ such that $\frac{\log
s}{s}\leq\frac{p}{a(p-1)}$ for all $s\geq\hat{s}$.
###### Proof.
We have from (3) that for $j=0,1,$
$e^{-\frac{(p-j)s}{p-1}}\left|h^{(j)}\left(e^{\frac{s}{p-1}}w\right)\right|\leq
C^{\prime}(M,\mu)\left(\frac{|w|^{p-j}}{\log^{a}\left(2+e^{\frac{2s}{p-1}}w^{2}\right)}+e^{-\frac{(p-j)s}{p-1}}\right).$
Considering the first case $w^{2}e^{\frac{s}{p-1}}\geq 4$, we have
$\frac{|w|^{p-j}}{\log^{a}\left(2+e^{\frac{2s}{p-1}}w^{2}\right)}\leq\frac{|w|^{p-j}}{\log^{a}\left(4e^{\frac{s}{p-1}}\right)}\leq\frac{(p-1)^{a}}{s^{a}}|w|^{p-j}.$
Now, considering the second case $w^{2}e^{\frac{s}{p-1}}\leq 4$, we have
$|w|^{p-j}\leq 2^{p-j}e^{-\frac{(p-j)s}{2(p-1)}}$ which yields
$\frac{|w|^{p-j}}{\log^{a}\left(2+e^{\frac{2s}{p-1}}w^{2}\right)}\leq\frac{|w|^{p-j}}{\log^{a}(2)}\leq\frac{2^{p-j}}{\log^{a}2}e^{-\frac{(p-j)s}{2(p-1)}}.$
Taking $C=\max\left\\{C^{\prime},\frac{2^{p}}{\log^{a}2},(p-1)^{a}\right\\}$
and $\hat{s}>0$ such that $e^{-\frac{(p-j)s}{p-1}}\leq s^{-a}$ for all
$s\geq\hat{s}$, we have the conclusion. This ends the proof of Lemma A.2. ∎
The following lemma shows us the existence of solutions of the associated ODE
of equation (9):
###### Lemma A.3.
Let $\phi$ be a positive solution of the following ordinary differential
equation:
$\phi_{s}=-\frac{\phi}{p-1}+\phi^{p}+e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}\phi\right).$
(96)
Then $\phi(s)\to\kappa$ as $s\to+\infty$ and $\phi(s)$ is given by
$\phi(s)=\kappa(1+\eta_{a}(s))^{-\frac{1}{p-1}},\quad\text{where}\quad\eta_{a}(s)=\mathcal{O}\left(\frac{1}{s^{a}}\right).$
(97)
If $h(x)=\mu\frac{|x|^{p-1}x}{\log^{a}(2+x^{2})}$, we have
$\eta_{a}(s)\sim
C_{0}\int_{s}^{+\infty}\frac{e^{s-\tau}}{\tau^{a}}d\tau=\frac{C_{0}}{s^{a}}\left(1+\sum_{j\geq
1}\frac{b_{j}}{s^{j}}\right),$
where $C_{0}=\mu\left(\frac{p-1}{2}\right)^{a}$ and
$b_{j}=(-1)^{j}\prod_{i=0}^{j-1}(a+i)$.
###### Proof.
By the following transformation
$v(t)=(T-t)^{-\frac{1}{p-1}}\phi(s),\quad s=-\log(T-t),$
equation (94) is transformed into (96). From Lemma A.1, we see that
$\phi(s)\to\kappa$ as $s\to+\infty$.
By dividing equation (96) by $\phi^{p}$, we find that
$\left(\frac{1}{\phi^{p-1}}\right)^{\prime}=\frac{1}{\phi^{p-1}}-(p-1)(1+g(s)),\quad
g(s)=\frac{1}{\phi^{p}}e^{-\frac{ps}{p-1}}h\left(e^{\frac{s}{p-1}}\phi\right).$
(98)
Since $\phi(s)\to\kappa$ as $s\to+\infty$, we have from Lemma A.2 that
$g(s)=\mathcal{O}\left(\frac{1}{s^{a}}\right)$ as $s\to+\infty$. Solving
equation (98) yields
$\phi(s)=\kappa\left(1+\eta_{a}(s)\right)^{-\frac{1}{p-1}},\quad\text{where}\quad\eta_{a}(s)=\int_{s}^{+\infty}e^{s-\tau}g(\tau)d\tau.$
By integration by part, we find that
$\int_{s}^{+\infty}e^{s-\tau}\tau^{-a}d\tau=\frac{1}{s^{a}}\left(1+\sum_{j\geq
1}\frac{b_{j}}{s^{j}}\right),\quad b_{j}=(-1)^{j}\prod_{i=0}^{j-1}(a+i).$ (99)
This follows that $\eta_{a}(s)=\mathcal{O}(s^{-a})$ as $s\to+\infty$. In
particular case $h(x)=\mu\frac{|x|^{p-1}x}{\log^{a}(2+x^{2})}$, we have
$g(s)=\mu\log^{-a}\left(2+e^{\frac{2s}{p-1}}\phi^{2}(s)\right)\sim\mu\left(\frac{p-1}{2s}\right)^{a}$
as $s\to+\infty$. By (99), we get the desired result and finish the proof of
Lemma A.3. ∎
## Appendix B Appendix B
### B.1 Proof of Proposition 12
We give the proof of Proposition 12 here.
###### Proof.
The idea of the proof is given in Ladyženskaja and al. [12]. Note that we
still get interior regularity even if we know nothing about the initial or
boundary data. Indeed, let $\tau\in(0,1)$ and fix $t_{0}$ such that
$t_{0}-\tau>0$, we denote
$Q_{\tau}(t_{0})=\mathbf{B}_{R/2}\times(t_{0}-\tau,t_{0})\subset Q_{R}$, and
let $\varphi(x,t)$ be a smooth function defined in $Q_{R}$ such that
$0\leq\varphi(x,t)\leq 1$ and $\varphi(x,t)=0$ for all $(x,t)\in
Q_{R}\setminus Q_{\tau}(t_{0})$. Let $k\geq 1$, define
$v_{k}(x,t)=\max\\{v(x,t)-k,0\\}\quad\text{and}\quad
A_{k}(t)=\\{x\in\mathbf{B}_{R}:v(x,t)>k\\}.$
Then, multiplying equation (38) by $v_{k}\varphi^{2}$ and integrating over
$Q_{\tau}(t_{0})$, we find that
$\displaystyle\frac{1}{2}\int_{\mathbf{B}_{R}}v_{k}^{2}\varphi^{2}dx|_{t_{0}-\tau}^{t_{0}}+\int_{t_{0}-\tau}^{t_{0}}\int_{\mathbf{B}_{R}}\left|\nabla
v_{k}\right|^{2}\varphi^{2}dxdt$
$\displaystyle=-\int_{t_{0}-\tau}^{t_{0}}\int_{\mathbf{B}_{R}}v_{k}^{2}\varphi\varphi_{t}dxdt+2\int_{t_{0}-\tau}^{t_{0}}\int_{\mathbf{B}_{R}}\left(\nabla
v_{k}\cdot\nabla\varphi\right)v_{k}\varphi dxdt$
$\displaystyle\quad-\int_{t_{0}-\tau}^{t_{0}}\int_{\mathbf{B}_{R}}\left(b\cdot\nabla
v_{k}\right)v_{k}\varphi^{2}dxdt+\int_{t_{0}-\tau}^{t_{0}}\int_{A_{k}(t)}Fv_{k}\varphi^{2}dxdt.$
Using the assumption $|F|\leq g(|v|+1)$ and some elementary inequalities with
noticing that $\varphi(\cdot,t_{0}-\tau)=0$, we then obtain
$\displaystyle\max_{t_{0}-\tau\leq t\leq
t_{0}}\left\|v_{k}(t)\varphi(t)\right\|^{2}_{L^{2}(\mathbf{B}_{R})}$
$\displaystyle+\int_{t_{0}-\tau}^{t_{0}}\int_{\mathbf{B}_{R}}\left|\nabla
v_{k}\right|^{2}\varphi^{2}dxdt$ $\displaystyle\leq
2\int_{t_{0}-\tau}^{t_{0}}\int_{\mathbf{B}_{R}}\left(4|\nabla\varphi|+\varphi|\varphi_{t}|\right)v_{k}^{2}dxdt$
$\displaystyle+2\int_{t_{0}-\tau}^{t_{0}}\int_{A_{k}(t)}(\mu_{1}^{2}+2g)\left(v_{k}^{2}+k^{2}\right)\varphi^{2}dxdt.$
(100)
For the last term in the right-hand side (denote by $I$), we use Holder’s
inequality and (39), which reads
$\displaystyle|I|$
$\displaystyle\leq\left(\int_{t_{0}-\tau}^{t_{0}}\|2\mu_{1}^{2}+4g\|_{L^{\alpha^{\prime}}(A_{k}(t))}^{\beta^{\prime}}dt\right)^{\frac{1}{\beta^{\prime}}}\left(\int_{t_{0}-\tau}^{t_{0}}\left\|\left(v_{k}^{2}+k^{2}\right)\varphi^{2}\right\|_{L^{\alpha_{1}}(A_{k}(t))}^{\beta_{1}}dt\right)^{\frac{1}{\beta_{1}}},$
$\displaystyle\leq\gamma\left(\int_{t_{0}-\tau}^{t_{0}}\left\|\left(v_{k}^{2}+k^{2}\right)\varphi^{2}\right\|_{L^{\alpha_{1}}(A_{k}(t))}^{\beta_{1}}dt\right)^{\frac{1}{\beta_{1}}}=\gamma
II,$
where $\gamma=\gamma(\mu_{1},\mu_{2},R,\alpha^{\prime},\beta^{\prime})>0$,
$\alpha_{1}=\frac{\alpha^{\prime}}{\alpha^{\prime}-1}$ and
$\beta_{1}=\frac{\beta^{\prime}}{\beta^{\prime}-1}$.
From pages 184 and 185 in [12], we have the following interpolation identity:
$II\leq\beta\theta_{k}^{\frac{2\epsilon}{r}}\left(\max_{t_{0}-\tau\leq t\leq
t_{0}}\left\|v_{k}(t)\varphi(t)\right\|^{2}_{L^{2}(A_{k}(t))}+\int_{t_{0}-\tau}^{t_{0}}\int_{A_{k}(t)}\left|\nabla
v_{k}\right|^{2}\varphi^{2}dxdt\right)+k^{2}\sigma_{k}^{\frac{2(1+\epsilon)}{r}},$
where $\epsilon\in(0,1)$, $r\geq 2$, $\beta>0$ are constants,
$\theta_{k}=\int_{t_{0}-\tau}^{t_{0}}\left|A_{k}(t)\right|^{\beta_{1}/\alpha_{1}}dt,\quad\sigma_{k}=\int_{t_{0}-\tau}^{t_{0}}\|\varphi(t)\|^{\beta_{1}}_{L^{\alpha_{1}}(A_{k}(t))}dt.$
Since $\theta_{k}\leq\tau R^{\beta_{1}/\alpha_{1}}$, we can take $\tau$ small
enough such that
$\gamma\beta(\tau
R^{\beta_{1}/\alpha_{1}})^{\frac{2\epsilon}{r}}\leq\frac{1}{2}.$
Then from (100), we have
$\displaystyle\max_{t_{0}-\tau\leq t\leq
t_{0}}\left\|v_{k}(t)\varphi(t)\right\|^{2}_{L^{2}(\mathbf{B}_{R})}$
$\displaystyle+\int_{t_{0}-\tau}^{t_{0}}\int_{\mathbf{B}_{R}}\left|\nabla
v_{k}\right|^{2}\varphi^{2}dxdt$
$\displaystyle\leq\gamma^{\prime}\left[\int_{t_{0}-\tau}^{t_{0}}\int_{\mathbf{B}_{R}}\left(|\nabla\varphi|+\varphi|\varphi_{t}|\right)v_{k}^{2}dxdt+k^{2}\sigma_{k}^{\frac{2(1+\epsilon)}{r}}\right].$
(101)
By Remark 6.4, page 109 and Theorem 6.2, page 103 in [12], we know that if $v$
satisfies (101) for any $k\geq 1$, then for all
$(x,t)\in\mathbf{B}_{R/4}\times(t_{0}-\tau/2,t_{0})$,
$\displaystyle|v(x,t)|\leq\gamma^{\prime\prime}\left[\left(\frac{R}{2}\right)^{-\frac{n+2}{2}}\left(1+\frac{R}{2\sqrt{\tau}}\right)\left(\int_{t_{0}-\tau}^{t_{0}}\|v(t)\|^{2}_{L^{2}(\mathbf{B}_{R})}dt\right)^{1/2}\right.$
$\displaystyle+\left.\left(1+\frac{4\tau}{R}\right)^{\frac{1+\epsilon}{r}}\right]<+\infty$
. (102)
Analogous arguments with the function $-v$ would yield the same estimate.
Since $\mu_{1}$, $\mu_{2}$ and $\mu_{3}$ are uniformly bounded in $t_{0}$,
this implies that estimate (102) holds for all
$(x,t)\in\mathbf{B}_{R/4}\times(\tau/2,+\infty)$. This concludes the proof of
Proposition 12. ∎
### B.2 Proof of Proposition 13
We prove Proposition 13 here. Let us first derive the upper bound for
$\mathcal{E}_{\psi}$.
###### Proof of the upper bound for $\mathcal{E}_{\psi}$.
Multiplying equation (9) with $\psi^{2}w_{s}$ and integrating on
$\mathbb{R}^{n}$ yield
$\displaystyle\int_{\mathbb{R}^{n}}\psi^{2}w_{s}^{2}\rho dy$
$\displaystyle=-\frac{1}{2}\frac{d}{ds}\int_{\mathbb{R}^{n}}\psi^{2}|\nabla
w|^{2}\rho dy-2\int_{\mathbb{R}^{n}}\psi w_{s}\nabla\psi.\nabla w\rho dy$
$\displaystyle\quad-\frac{1}{2(p-1)}\frac{d}{ds}\int_{\mathbb{R}^{n}}\psi^{2}|w|^{2}\rho
dy+\frac{1}{p+1}\frac{d}{ds}\int_{\mathbb{R}^{n}}\psi^{2}|w|^{p+1}\rho dy$
$\displaystyle+e^{-\frac{p+1}{p-1}}\frac{d}{ds}\int_{\mathbb{R}^{n}}\psi^{2}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy$
$\displaystyle-\frac{1}{p-1}e^{-\frac{p}{p-1}s}\int_{\mathbb{R}^{n}}\psi^{2}h\left(e^{\frac{s}{p-1}}w\right)w\rho
dy.$
We derive the following identity from the definition (42) of the local
functional $\mathcal{E}_{\psi}$,
$\displaystyle\frac{d}{ds}\mathcal{E}_{\psi}[w](s)$
$\displaystyle=-\int_{\mathbb{R}^{n}}\psi^{2}|w_{s}|^{2}\rho
dy-2\int_{\mathbb{R}^{n}}\psi w_{s}\nabla\psi.\nabla w\rho dy$
$\displaystyle\quad+\frac{p+1}{p-1}e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}\psi^{2}H\left(e^{\frac{1}{p-1}s}w\right)\rho
dy$
$\displaystyle-\frac{1}{p-1}e^{-\frac{p}{p-1}s}\int_{\mathbb{R}^{n}}\psi^{2}h\left(e^{\frac{1}{p-1}s}w\right)w\rho
dy.$ (103)
Using the fact that $2ab\leq\frac{a^{2}}{2}+2b^{2}$, we obtain
$2\psi w_{s}\nabla\psi.\nabla
w\leq\frac{1}{2}\psi^{2}w_{s}^{2}+2|\nabla\psi|^{2}|\nabla w|^{2}.$
Combining with (30), we get an estimate for (103) as follows:
$\displaystyle\frac{d}{ds}\mathcal{E}_{\psi}[w](s)$
$\displaystyle\leq-\frac{1}{2}\int_{\mathbb{R}^{n}}\psi^{2}|w_{s}|^{2}\rho
dy+2\|\nabla\psi\|^{2}_{L^{\infty}}\int_{\mathbb{R}^{n}}|\nabla w|^{2}\rho dy$
$\displaystyle\quad+Cs^{-a}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho dy+Cs^{-a},$
where $C=C(a,p,n,M,\|\psi\|^{2}_{L^{\infty}})$.
Using $(iii)$ and $(iv)$ of Proposition 8, we see that
$\frac{d}{ds}\mathcal{E}_{\psi}[w](s)\leq
C_{1}\left(1+\|w_{s}\|_{L^{2}_{\rho}(\mathbb{R}^{n})}\right),\quad\forall
s\geq\tilde{s}_{3},$ (104)
where
$C_{1}=C_{1}\left(a,p,n,N,J_{3},J_{4},\|\psi\|^{2}_{L^{\infty}},\|\nabla\psi\|^{2}_{L^{\infty}}\right)$
and $J_{i}$ is introduced in Proposition 8.
From the definition of $\mathcal{E}_{\psi}$ given in (42), we have
$\displaystyle\mathcal{E}_{\psi}[w](s)$
$\displaystyle\leq\|\psi\|^{2}_{L^{\infty}}\left\\{\frac{1}{2}\int_{\mathbb{R}^{n}}\left(|\nabla
w|^{2}+\frac{1}{p-1}|w|^{2}\right)\rho
dy-e^{-\frac{p+1}{p-1}}\int_{\mathbb{R}^{n}}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy.\right\\}$
$\displaystyle=\|\psi\|^{2}_{L^{\infty}}\left\\{\mathcal{E}[w](s)+\frac{1}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy\right\\}$
$\displaystyle\leq\|\psi\|^{2}_{L^{\infty}}\left\\{J_{0}+\frac{1}{p+1}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy\right\\}.\quad\forall s\geq\tilde{s}_{3}.$
Integrating on $[s,s+1]$, we obtain
$\displaystyle\int_{s}^{s+1}\mathcal{E}_{\psi}[w](\tau)d\tau$
$\displaystyle\leq\|\psi\|^{2}_{L^{\infty}}\left\\{J_{0}+\frac{1}{p+1}\int_{s}^{s+1}\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dyd\tau\right\\}$
$\displaystyle\leq\|\psi\|^{2}_{L^{\infty}}\left\\{J_{0}+\frac{1}{p+1}\left[\int_{s}^{s+1}\left(\int_{\mathbb{R}^{n}}|w|^{p+1}\rho
dy\right)^{2}d\tau\right]^{\frac{1}{2}}\right\\}$ $\displaystyle\leq
C_{2}\left(\|\psi\|^{2}_{L^{\infty}},J_{0},J_{5}\right)\quad\text{(use $(v)$
of Proposition \ref{prop:boundEpsi})}.$
Hence,
$\int_{s}^{s+1}\mathcal{E}_{\psi}[w](\tau)d\tau\leq C_{2},\quad\forall
s\geq\tilde{s}_{3}.$ (105)
Thus, there exists $\tau(s)\in[s,s+1]$ such that
$\mathcal{E}_{\psi}[w](\tau(s))=\int_{s}^{s+1}\mathcal{E}_{\psi}[w](\tau^{\prime})d\tau^{\prime}\leq
C_{2}.$
We then have
$\displaystyle\mathcal{E}_{\psi}[w](s)$
$\displaystyle=\mathcal{E}_{\psi}[w](\tau(s))+\int_{\tau(s)}^{s}\frac{d}{ds}\mathcal{E}_{\psi}[w](\tau^{\prime})d\tau^{\prime}$
$\displaystyle\leq
C_{2}+\int_{s}^{s+1}C_{1}\left(1+\|w_{s}\|_{L^{2}_{\rho}(\mathbb{R}^{n})}\right)d\tau^{\prime}\leq
C_{2}^{\prime}.\;\;\text{(use $(i)$ of Proposition \ref{prop:boundEpsi})}$
This concludes the proof of the upper bound for $\mathcal{E}_{\psi}$.
It remains to prove the lower bound in order to conclude the proof of
Proposition (8). ∎
###### Proof of the lower bound for $\mathcal{E}_{\psi}$.
Multiplying equation (9) with $\psi^{2}w$ and integrating on $\mathbb{R}^{n}$
yield
$\displaystyle\frac{1}{2}\frac{d}{ds}\int_{\mathbb{R}^{n}}(\psi w)^{2}\rho dy$
$\displaystyle=-2\mathcal{E}_{\psi}[w](s)+\frac{p+1}{p-1}\int_{\mathbb{R}^{n}}\psi^{2}|w|^{p+1}\rho
dy$ $\displaystyle\quad-2\int_{\mathbb{R}^{n}}\psi w\nabla\psi.\nabla w\rho
dy$
$\displaystyle\quad-2e^{-\frac{p+1}{p-1}s}\int_{\mathbb{R}^{n}}\psi^{2}H\left(e^{\frac{s}{p-1}}w\right)\rho
dy$
$\displaystyle\quad+e^{-\frac{p}{p-1}s}\int_{\mathbb{R}^{n}}\psi^{2}h\left(e^{\frac{s}{p-1}}w\right)w\rho
dy.$ (106)
We now control the new term $J_{\psi}[w](s)=2\int_{\mathbb{R}^{n}}\psi
w\nabla\psi.\nabla w\rho dy$ as follows:
$\displaystyle J_{\psi}[w](s)$
$\displaystyle=-2\int_{\mathbb{R}^{n}}w\nabla.\left(\psi
w\nabla\psi\rho\right)dy$
$\displaystyle=-2\int_{\mathbb{R}^{n}}|w|^{2}|\nabla\psi|^{2}\rho
dy-2\int_{\mathbb{R}^{n}}\psi w\nabla\psi.\nabla w\rho dy$
$\displaystyle\quad-2\int_{\mathbb{R}^{n}}\psi|w|^{2}\Delta\psi\rho
dy+\int_{\mathbb{R}^{n}}\psi|w|^{2}y.\nabla\psi\rho dy.$
$\displaystyle=-\int_{\mathbb{R}^{n}}|w|^{2}|\nabla\psi|^{2}\rho
dy-\int_{\mathbb{R}^{n}}\psi|w|^{2}\Delta\psi\rho
dy+\frac{1}{2}\int_{\mathbb{R}^{n}}\psi|w|^{2}y.\nabla\psi\rho dy.$
$\displaystyle\leq\left[\|\psi\|_{L^{\infty}}\left(\|\Delta\psi\|_{L^{\infty}}+\frac{1}{2}\|y.\nabla\psi\|_{L^{\infty}}\right)\right]\int_{\mathbb{R}^{n}}|w|^{2}\rho
dy$ $\displaystyle\leq J_{2}C_{1}(\psi),\quad\forall
s\geq\tilde{s}_{3}\quad\text{(use $(ii)$ of Proposition
\ref{prop:boundEpsi})}.$ (107)
Using (30) and (106), we obtain
$\displaystyle\frac{1}{2}\frac{d}{ds}\int_{\mathbb{R}^{n}}(\psi w)^{2}\rho dy$
$\displaystyle\geq-2\mathcal{E}_{\psi}-J_{2}C_{1}(\psi)-C_{2}s^{-a}$
$\displaystyle\qquad\qquad+\left(\frac{p+1}{p-1}-C_{2}s^{-a}\right)\int_{\mathbb{R}^{n}}\psi^{2}|w|^{p+1}\rho
dy.$
Taking $S$ large enough such that $C_{2}s^{-a}\leq\frac{p+1}{2(p-1)}$ for all
$s\geq S$, we obtain for all $s\geq\max\\{S,\tilde{s}_{3}\\}$,
$\frac{1}{2}\frac{d}{ds}\int_{\mathbb{R}^{n}}(\psi w)^{2}\rho
dy\geq-(2\mathcal{E}_{\psi}+C_{3})+\frac{p+1}{2(p-1)}\int_{\mathbb{R}^{n}}\psi^{2}|w|^{p+1}\rho
dy,$
where $C_{3}=J_{2}C_{1}+\frac{p+1}{2(p-1)}$.
Let $g(s)=2\mathcal{E}_{\psi}+C_{3}$ and
$f(s)=\frac{1}{2}\int_{\mathbb{R}^{n}}\psi^{2}|w|^{2}\rho dy$. Using Jensen’s
inequality, we have
$\displaystyle f(s)^{\frac{p+1}{2}}$
$\displaystyle=2^{-\frac{p+1}{2}}\left(\int_{\mathbb{R}^{n}}\psi^{2}|w|^{2}\rho
dy\right)^{\frac{p+1}{2}}$ $\displaystyle\leq
2^{-\frac{p+1}{2}}\int_{\mathbb{R}^{n}}(\psi|w|)^{p+1}\rho dy\leq
2^{-\frac{p+1}{2}}\|\psi\|_{L^{\infty}}^{p-1}\int_{\mathbb{R}^{n}}\psi^{2}|w|^{p+1}\rho
dy.$
We therefore obtain for all $s\geq S_{1}=\max\\{S,\tilde{s}_{3}\\}$,
$f^{\prime}(s)\geq-g(s)+C_{4}f(s)^{\frac{p+1}{2}}.$ (108)
From (104), we also have
$g^{\prime}(s)\leq C_{5}+h(s),\quad\forall s\geq\tilde{s}_{3},$ (109)
where $h(s)=C_{5}\|w_{s}\|_{L^{2}_{\rho}(\mathbb{R}^{n})}$ and
$m=\int_{\tilde{s}_{3}}^{+\infty}h(s)ds\leq C_{6}$ by using $(i)$ of
Proposition 8, where $C_{5},C_{6}$ are some positive constants.
We claim that the function of $g$ is bounded from below by some constant $M$.
Arguing by contradiction, we suppose that there exists a time $s^{*}\geq
S_{1}$ such that $g(s^{*})\leq-M$. Then for all $s\geq s^{*}$, we write
$\displaystyle g(s)$
$\displaystyle=g(s^{*})+\int_{s^{*}}^{s}g^{\prime}(\tau)d\tau\leq-M+\int_{s^{*}}^{s}(C_{5}+h(\tau))d\tau$
$\displaystyle\leq-M+m+C_{5}(s-s^{*}).$
Thus, we have by (108),
$f^{\prime}\geq M-m-C_{5}(s-s^{*})+C_{4}f^{\frac{p+1}{2}},\quad f(s^{*})\geq
0.$
On the other hand, we know that the solution of the following equation
$f^{\prime}\geq 1+C_{5}f^{\frac{p+1}{2}},\quad f(s^{*})\geq 0$
blows up in finite time before
$s=s^{*}+\int_{0}^{+\infty}\frac{d\xi}{1+C_{4}f^{\frac{p+1}{2}}}=s^{*}+T^{*}.$
On the interval $[s^{*},s^{*}+T^{*}]$, we have
$M-m-C_{5}(s-s^{*})\geq M-m-C_{5}T^{*}.$
Thus, we fix $M=m+C_{5}T^{*}+1$ to get $M-m-C_{5}(s-s^{*})\geq 1$ for all
$s\in[s^{*},s^{*}+T^{*}]$.
Therefore, $f$ blows up in some finite time before $s^{*}+T^{*}$. But this
contradicts with the existence global of $w$. This follows (43) and we
complete the proof of Proposition 13. ∎
## Appendix C Appendix C
We claim the following:
###### Lemma C.1 (Estimate on $\bar{F}$).
For $s$ large enough, we have
$\left|\bar{F}(V,s)-\frac{p}{2\kappa}V^{2}\right|\leq
C|V|^{3}+\frac{C|V|^{2}}{s^{a-1}},$
where $C=C(a,p,M,\mu)>0$.
###### Proof.
Consider the Taylor expansion of the nonlinear terms $F$ and $H$, we have
$\displaystyle F(v)$
$\displaystyle=\frac{1}{2}p(p-1)\phi^{p-2}v^{2}+\gamma_{1}v^{3},\quad
H(v,s)=\gamma_{2}v^{2},$
where
$\gamma_{1}=\frac{1}{6}p(p-1)(p-2)|\phi+\theta_{1}v|^{p-3},\quad\gamma_{2}=\frac{1}{2}e^{-\frac{(p-2)s}{p-1}}h^{\prime\prime}\left(e^{\frac{s}{p-1}}(\phi+\theta_{2}v)\right),$
with $\theta_{i}\in[0,1]$, $i=1,2$.
We claim the following: for $s$ large,
$|\gamma_{1}|\leq C\quad\text{and}\quad|\gamma_{2}|\leq\frac{C}{s^{a}}.$ (110)
Let us leave the proof of (110) later and continue the proof of Lemma C.1.
Recalling from Lemma A.3 that $\phi(s)=\kappa+\mathcal{O}(s^{-a})$ as
$s\to+\infty$, we derive
$\left|F(v)+H(v,s)-\frac{p}{2\kappa}v^{2}\right|=\mathcal{O}\left(\frac{|v|^{2}}{s^{a}}\right)+\mathcal{O}(|v|^{3}),\quad\text{as}\;s\to+\infty.$
From the definition of $\bar{F}$ and the fact that
$\beta(s)=1+\mathcal{O}(\frac{1}{s^{a-1}})$ as $s\to+\infty$, we have for $s$
large enough,
$\displaystyle\left|\bar{F}(V,s)-\frac{p}{2\kappa}V^{2}\right|$
$\displaystyle=\left|\beta(s)\left(F(v)+H(v,s)\right)-\frac{p}{2\kappa}v^{2}\beta^{2}\right|$
$\displaystyle\leq\left|F(v)+H(v,s)-\frac{p}{2\kappa}v^{2}\right|+\frac{C|v|^{2}}{s^{a-1}}$
$\displaystyle\leq\frac{C|v|^{2}}{s^{a}}+C|v|^{3}+\frac{C|v|^{2}}{s^{a-1}}\leq
C|V|^{3}+\frac{C|V|^{2}}{s^{a-1}},$
which concludes the proof of Lemma C.1, assuming that (110) holds.
Let us now give the proof of (110). Since $\phi(s)\to\kappa$ as $s\to+\infty$,
we can take $s_{*}>0$ such that
$\frac{3\kappa}{4}\leq\phi(s)\leq\frac{5\kappa}{4},\quad\forall s\geq s_{*}.$
Let us bound $|\gamma_{1}|$. If $p\geq 3$, by the boundedness of $|\phi|$ and
$|v|$, then $|\gamma_{1}|$ is already bounded. If $p\in(1,3)$, we consider the
case $|\theta_{1}v|\leq\frac{\kappa}{2}$, then the case
$|\theta_{1}v|>\frac{\kappa}{2}$. In the first case, we have
$|\phi+\theta_{1}v|\geq\frac{\kappa}{4}$ for all $s\geq s_{*}$, then
$|\gamma_{1}|\leq C|\phi+\theta_{1}v|^{p-3}\leq
C\left(\frac{\kappa}{4}\right)^{p-3}$ for all $s\geq s_{*}$ . Now, considering
the second case where $|\theta_{1}v|>\frac{\kappa}{2}$, note that in this
case, we have $\theta_{1}\neq 0$ and $\phi<\frac{5}{2}|\theta_{1}v|$ for all
$s\geq s_{*}$. From the definition of $F(v)$, we have
$\displaystyle|\gamma_{1}v^{3}|$
$\displaystyle=\left||\phi+v|^{p-1}(\phi+v)-\phi^{p}-p\phi^{p-1}v-\frac{1}{2}p(p-1)\phi^{p-2}v^{2}\right|$
$\displaystyle\leq C(|v|^{p}+v^{2}),\quad\forall s\geq s_{*}.$
This yields $|\gamma_{1}|\leq C(|v|^{p-3}+|v|^{-1})\leq
C\left((\kappa/2\theta_{1})^{p-3}+(\kappa/2\theta_{1})^{-1}\right)$ for all
$s\geq s_{*}$. This concludes the proof of the first estimate of (110).
Let us now prove that $|\gamma_{2}|\leq Cs^{-a}$ for $s$ large enough. From
(3), we have
$|\gamma_{2}|\leq
M\frac{|\phi+\theta_{2}v|^{p-2}}{\log^{a}(2+e^{\frac{s}{p-1}}(\phi+\theta_{2}v)^{2})}.$
(111)
If $p>2$, by the same technique given in the proof of Lemma A.2, we can show
that (111) implies
$|\gamma_{2}|\leq\frac{C}{s^{a}}(|\phi+\theta_{2}v|^{p-2}+1)\leq\frac{2C}{s^{a}},\quad\forall
s\geq s^{\prime}(a,p).$
If $p\in(1,2]$, we consider the first case
$|\theta_{2}v|\leq\frac{\kappa}{2}$, which implies
$|\phi(s)+\theta_{2}v|\geq\frac{\kappa}{4}$ for all $s\geq s_{*}$. From (111),
we derive
$|\gamma_{2}|\leq\frac{C(\kappa/4)^{p-2}}{\log^{a}(2+e^{\frac{s}{p-1}}(\kappa/4)^{2})}\leq\frac{2C}{s^{a}},\quad\text{for
$s$ large}.$
In the case where $|\theta_{2}v|>\frac{\kappa}{2}$, we note that
$\theta_{2}\neq 0$ and $\phi(s)\leq\frac{5}{2}|\theta_{2}v|$ for all $s\geq
s_{*}$. Using the definition of $H(v,s)$ and (3), we find that
$\displaystyle|\gamma_{2}v^{2}|$ $\displaystyle\leq
C\left(\frac{|\phi+v|^{p}}{\log^{a}\left(2+e^{\frac{2s}{p-1}}(\phi+v)^{2}\right)}+\frac{\phi^{p}}{\log^{a}\left(2+e^{\frac{2s}{p-1}}\phi^{2}\right)}\right.$
$\displaystyle\qquad\qquad\left.+\frac{\phi^{p-1}v}{\log^{a}\left(2+e^{\frac{2s}{p-1}}\phi^{2}\right)}+e^{-\frac{ps}{p-1}}+e^{-s}\right)$
$\displaystyle\leq\frac{C}{s^{a}}\left(|\phi+v|^{p}+\phi^{p}+\phi^{p-1}|v|+1\right)\leq\frac{2C}{s^{a}}(|v|^{p}+1),$
for $s$ large. This yields
$|\gamma_{2}|\leq\frac{2C}{s^{a}}(|v|^{p-2}+|v|^{-2})\leq\frac{2C}{s^{a}}\left((\frac{\kappa}{2\theta_{2}})^{p-2}+(\frac{\kappa}{2\theta_{2}})^{-2}\right)\leq\frac{3C}{s^{a}}$.
This concludes the proof of (110) and the proof of Lemma C.1 also. ∎
###### Lemma C.2.
Let $\alpha(s)$ be a solution of
$\alpha^{\prime}(s)=\alpha^{2}(s)+\mathcal{O}\left(\frac{1}{s^{q}}\right),\quad
q\in(2,3],$ (112)
which exists for all time. Then
either
$\displaystyle\quad\alpha(s)=-\dfrac{1}{s}+\mathcal{O}\left(\dfrac{1}{s^{q}}\right)\quad\text{or
}\quad\alpha(s)=\mathcal{O}\left(\dfrac{1}{s^{q}}\right),\quad\text{if}\quad
q\in(2,3),$ (113) either
$\displaystyle\quad\alpha(s)=-\dfrac{1}{s}+\mathcal{O}\left(\dfrac{\log{s}}{s^{2}}\right)\quad\text{or
}\quad\alpha(s)=\mathcal{O}\left(\dfrac{1}{s^{2}}\right),\quad\text{if}\quad
q=3.$ (114)
###### Proof.
Let us first show that
$\text{either}\;\;\alpha(s)=\mathcal{O}\left(\frac{1}{s^{1+\sigma}}\right)\;\;\text{or}\;\;\alpha(s)=-\frac{1}{s}+\mathcal{O}\left(\frac{1}{s^{1+\sigma^{\prime}}}\right)\;\;\text{as}\;s\to+\infty,$
(115)
for some $\sigma\in\left(0,\frac{q-2}{2}\right)$ and
$\sigma^{\prime}=q-2-2\sigma$.
Fix $s_{0}$ large enough et let $\sigma\in\left(0,\frac{q-2}{2}\right)$. If
$|\alpha(s)|\leq\frac{1}{s^{1+\sigma}}$, for all $s\geq s_{0}$, then we are
done. If not, namely there exists a time $s_{1}>s_{0}$ such that
$|\alpha(s_{1})|>\frac{1}{s^{1+\sigma}}$, we have two possibilities:
$|\alpha(s)|>\frac{1}{s^{1+\sigma}},\quad\forall s\geq s_{1},$ (116)
or there exists a time $s_{2}>s_{1}$ such that
$|\alpha(s_{2})|=\frac{1}{s_{2}^{1+\sigma}}\quad\text{and}\quad|\alpha(s_{2})|\leq\frac{1}{s^{1+\sigma}},\quad\forall
s\in(s_{2},s_{2}+\delta),\;\delta>0.$ (117)
If (116) is the case, then we have by equation (112),
$\left(\frac{1}{\alpha}\right)^{\prime}=1+\mathcal{O}\left(\frac{1}{s^{q-2-2\sigma}}\right),\quad\forall
s\geq s_{1},$
which yields (115) by integration.
If (117) is the case, we assume that $\alpha(s_{2})>0$, then
$\alpha^{\prime}(s_{2})\leq-\frac{1+\sigma}{s_{2}^{2+\sigma}}<0$. By equation
(112) and note that $2+2\sigma<q$, we have $\alpha^{\prime}(s_{2})>0$ and a
contradiction follows. If $\alpha(s_{2})<0$, then
$\alpha^{\prime}(s_{2})\geq\frac{1+\delta}{s_{2}^{2+\delta}}$, by equation
(112), we get
$\frac{1+\delta}{s_{2}^{2+\delta}}\leq\alpha^{\prime}(s_{2})\leq\frac{1}{s^{2+2\sigma}_{2}}+\frac{1}{s_{2}^{q}}.$
Since $2+\delta<2+2\delta<q$, we have a contradiction and (115) follows.
We now use (115) in order to conclude Lemma C.2. Let us give the proof in the
case $q=3$. Assume $\alpha(s)=\mathcal{O}\left(\frac{1}{s^{1+\sigma}}\right)$
for some $\sigma\in(0,\frac{1}{2})$, then (112) yields
$\alpha^{\prime}(s)=\mathcal{O}\left(\frac{1}{s^{2+2\sigma}}\right)+\mathcal{O}\left(\frac{1}{s^{3}}\right)=\mathcal{O}\left(\frac{1}{s^{2+2\sigma}}\right).$
By integration, we get
$\alpha(s)=\mathcal{O}\left(\frac{1}{s^{1+2\sigma}}\right)$. Using this
estimate, we obtain
$\alpha^{\prime}(s)=\mathcal{O}\left(\frac{1}{s^{3}}\right)$ and the
conclusion follows.
Let us consider
$\alpha(s)=-\frac{1}{s}+\beta(s),\quad\text{with}\quad\beta(s)=\mathcal{O}\left(\frac{1}{s^{1+\sigma^{\prime}}}\right),\quad\sigma^{\prime}=1-2\sigma.$
Substituting this into (112) yields
$\beta^{\prime}(s)=\frac{2\beta(s)}{s}+\beta^{2}(s)+\mathcal{O}\left(\frac{1}{s^{3}}\right).$
Multiplying this equation by $s^{2}$, we find
$\left[s^{2}\beta(s)\right]^{\prime}=s^{2}\beta^{2}+\mathcal{O}\left(\frac{1}{s}\right)=\mathcal{O}\left(\frac{1}{s^{2\sigma^{\prime}}}\right)+\mathcal{O}\left(\frac{1}{s}\right).$
If $\sigma^{\prime}\geq\frac{1}{2}$, then
$\left[s^{2}\beta(s)\right]^{\prime}=\mathcal{O}\left(\frac{1}{s}\right)$
which follows $\beta(s)=\mathcal{O}\left(\frac{\log s}{s^{2}}\right)$. If
$\sigma^{\prime}<\frac{1}{2}$, then
$\beta(s)=\mathcal{O}\left(\frac{1}{s^{1+2\sigma^{\prime}}}\right)$. Using
this estimate and repeating the process again, we would obtain
$\beta(s)=\mathcal{O}\left(\frac{\log s}{s^{2}}\right)$ and (114) then
follows. Since the argument is similar in the case $q\in(2,3)$, we escape here
and concludes the proof of Lemma C.2. ∎
## References
* [1] T. Cazenave and P. L. Lions. Solutions globales d’équations de la chaleur semi linéaires. Comm. Partial Differential Equations, 9(10):955–978, 1984.
* [2] S. Filippas and R. V. Kohn. Refined asymptotics for the blowup of $u_{t}-\Delta u=u^{p}$. Comm. Pure Appl. Math., 45(7):821–869, 1992.
* [3] S. Filippas and W. X. Liu. On the blowup of multidimensional semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10(3):313–344, 1993.
* [4] Y. Giga and R. V. Kohn. Asymptotically self-similar blow-up of semilinear heat equations. Comm. Pure Appl. Math., 38(3):297–319, 1985.
* [5] Y. Giga and R. V. Kohn. Characterizing blowup using similarity variables. Indiana Univ. Math. J., 36(1):1–40, 1987.
* [6] Y. Giga and R. V. Kohn. Nondegeneracy of blowup for semilinear heat equations. Comm. Pure Appl. Math., 42(6):845–884, 1989.
* [7] Y. Giga, S. Matsui, and S. Sasayama. Blow up rate for semilinear heat equations with subcritical nonlinearity. Indiana Univ. Math. J., 53(2):483–514, 2004.
* [8] M. A. Hamza and H. Zaag. Lyapunov functional and blow-up results for a class of perturbations of semilinear wave equations in the critical case. J. Hyperbolic Differ. Equ., 9(2):195–221, 2012.
* [9] M. A. Hamza and H. Zaag. A Lyapunov functional and blow-up results for a class of perturbed semilinear wave equations. Nonlinearity, 25(9):2759–2773, 2012.
* [10] M. A. Herrero and J. J. L. Velázquez. Blow-up behaviour of one-dimensional semilinear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire, 10(2):131–189, 1993.
* [11] T. Kato. Perturbation theory for linear operators. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.
* [12] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I., 1968.
* [13] N. Nouaili and H. Zaag. A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up. Trans. Amer. Math. Soc., 362(7):3391–3434, 2010.
* [14] P. Quittner. A priori bounds for global solutions of a semilinear parabolic problem. Acta Math. Univ. Comenian. (N.S.), 68(2):195–203, 1999.
* [15] P. Quittner and P. Souplet. Superlinear parabolic problems. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2007. Blow-up, global existence and steady states.
* [16] F. Rellich. Perturbation theory of eigenvalue problems. Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz. Gordon and Breach Science Publishers, New York, 1969.
* [17] J. J. L. Velázquez. Higher-dimensional blow up for semilinear parabolic equations. Comm. Partial Differential Equations, 17(9-10):1567–1596, 1992\.
* [18] J. J. L. Velázquez. Classification of singularities for blowing up solutions in higher dimensions. Trans. Amer. Math. Soc., 338(1):441–464, 1993.
* [19] F. B. Weissler. Existence and nonexistence of global solutions for a semilinear heat equation. Israel J. Math., 38(1-2):29–40, 1981.
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Address: Université Paris 13, Sorbonne Paris Cité, Institut Galilée, LAGA, 99
avenue J.B. Clément, 93430 Villetaneuse, France.
E-mail: [email protected]
|
arxiv-papers
| 2014-04-15T18:44:59 |
2024-09-04T02:50:01.205762
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Van Tien Nguyen",
"submitter": "Nguyen Van Tien",
"url": "https://arxiv.org/abs/1404.4018"
}
|
1404.4021
|
# A $d$-dimensional extension of Christoffel words††thanks: With the support
of NSERC (Canada)
S. Labbé and C. Reutenauer
LIAFA, Université Paris Diderot - Paris 7,
Case 7014, 75205 Paris Cedex 13, France
[email protected]
Laboratoire de Combinatoire et d’Informatique Mathématique,
Université du Québec à Montréal,
C. P. 8888 Succursale ‘‘Centre-Ville’’, Montréal (QC), CANADA H3C 3P8
[email protected]
(Mathematics Subject Classifications: 05C75, 52C35, 68R15.)
###### Abstract
In this article, we extend the definition of Christoffel words to directed
subgraphs of the hypercubic lattice in arbitrary dimension that we call
Christoffel graphs. Christoffel graphs when $d=2$ correspond to well-known
Christoffel words. Due to periodicity, the $d$-dimensional Christoffel graph
can be embedded in a $(d-1)$-torus (a parallelogram when $d=3$). We show that
Christoffel graphs have similar properties to those of Christoffel words:
symmetry of their central part and conjugation with their reversal. Our main
result extends Pirillo’s theorem (characterization of Christoffel words which
asserts that a word $amb$ is a Christoffel word if and only if it is conjugate
to $bma$) in arbitrary dimension. In the generalization, the map $amb\mapsto
bma$ is seen as a flip operation on graphs embedded in $\mathbb{Z}^{d}$ and
the conjugation is a translation. We show that a fully periodic subgraph of
the hypercubic lattice is a translate of its flip if and only if it is a
Christoffel graph.
## 1 Introduction
This article is a contribution to the study of discrete planes and hyperplanes
in any dimension $d$. We study only rational hyperplanes, that is, those which
are defined by an equation with rational coefficients. We extract from such an
hyperplane a finite pattern that we call, for $d=3$, a _Christoffel
parallelogram_. We show that they are a generalization of Christoffel words.
Discrete planes were introduced by [Rev91] and further studied [Deb95, Fra96,
ARC97, Vui99]. Recognition algorithms were proposed in [Rev95, PBDR06, FST96].
See [BCK07] for a complete review about many aspects of digital planarity,
such as characterizations in arithmetic geometry, periodicity, connectivity
and algorithms. Discrete planes can be seen as an union of square faces. Such
stepped surface, introduced in [IO94, IO93] as a way to construct
quasiperiodic tilings of the plane, can be generated from multidimensional
continued fraction algorithms by introducing substitutions on square faces
[ABEI01, ABI02].
While discrete planes are a satisfactory generalization of Sturmian words, it
is still unclear what is the equivalent notion of Christoffel words in higher
dimension. In [Fer07, Fig. 6.6 and 6.7], fundamental domain of rational
discrete planes are constructed from the iteration of generalized
substitutions on the unit cube. Recently [DV12] generalized central words to
arbitrary dimension using palindromic closure. In both cases the
representation is nonconvex and has a boundary like a fractal.
In this article, we propose to extend the definition of Christoffel words to
directed subgraphs of the hypercubic lattice in arbitrary dimension that we
call Christoffel graphs. A similar construction, called _roundwalk_ , but
serving a different purpose was given in [BT04] producing multi-dimensional
words that are closely related to $k$-dimensional Sturmian words. Christoffel
graphs when $d=2$ correspond to Christoffel words. Due to its periods, the
$d$-dimensional Christoffel graph can be embedded in a $(d-1)$-torus and when
$d=3$, the torus is a parallelogram. This extension is motivated by Pirillo’s
theorem which asserts that a word $amb$ is a Christoffel word if and only if
it is conjugate to $bma$. In the generalization, the map $amb\mapsto bma$ is
seen as a flip operation on graphs embedded in $\mathbb{Z}^{d}$ and the
conjugation is replaced by some translation. When $d=3$, our flip corresponds
to a flip in a rhombus tiling [BFRR11, BFR08, ABFJ07]. We show that these
Christoffel graphs have similar properties to those of Christoffel words:
symmetry of their central part (Lemma 22) and conjugation with their reversal
(Corollary 25 and 26). Our main result is Theorem 36 which extends Pirillo’s
theorem in arbitrary dimension.
We recall in Section 2 the basic notion on Christoffel words and discrete
planes. The discrete hyperplane graphs are defined in Section 3. The operation
on them (flip, reversal and translation) are introduced in Section 4. We show
that the flip of a Christoffel graph is a translate of itself in Section 5.
This is the sufficiency of the Pirillo’s theorem. In Section 6, we consider
the necessity and obtain a $d$-dimensional Pirillo’s theorem, our main result.
Finally, we construct in the Section 7 in appendix, the mathematical framework
for the definition of discrete hyperplanes, since we could not find explicit
and complete references.
## 2 Christoffel words and discrete planes
### 2.1 Christoffel words
Recall that Christoffel words are obtained by discretizing a line segment in
the plane as follows: let $(p,q)\in\mathbb{N}^{2}$ with $\gcd(p,q)=1$, and let
$S$ be the line segment with endpoints $(0,0)$ and $(p,q)$.
Figure 1: The lower Christoffel word $w=aabaababaabab$.
The word $w$ is a _lower Christoffel word_ if the path induced by $w$ is under
$S$ and if they both delimit a polygon with no integral interior point. An
_upper Christoffel word_ is defined similarly, by taking the path which is
above the segment. A _Christoffel word_ is a lower Christoffel word. See
Figure 1 and reference [BLRS08]. An astonishing result about Christoffel words
is the following characteristic property given by Pirillo [Pir01].
###### Theorem 1 (Pirillo).
A word $w=amb\in\\{a,b\\}^{*}$ is a Christoffel word if and only if $amb$ and
$bma$ are conjugate.
It is even known that the two words $amb$ and $bma$ are conjugate by
palindromes [Chu97] Theorem 3.1 (see also [BR06] Proposition 6.1): for
example, the Christoffel word in Figure 1 can be factorized as a product of
two palindromes, but also as a letter, a central word $m$ and a last letter:
$w=aabaa\cdot babaabab=a\cdot abaababaaba\cdot b=amb,$
and the conjugate word $w^{\prime}$ of $w$ obtained by exchanging of the two
palindromes can also be factorized as the product of a letter, the same
central word $m$ and a last letter:
$w^{\prime}=babaabab\cdot aabaa=b\cdot abaababaaba\cdot a=bma.$
Centrals words are the words $m$ such that $amb$ is a Christoffel word. They
can be defined independently of Christoffel words: a word $m$ is a _central
word_ if and only if for some coprime integers $p$ and $q$, the length of $m$
is $p+q-2$ and $p$ and $q$ are periods of $m$. In this case, the Christoffel
word $amb$ is associated as above to the vector $(p,q)$. See [CL05] for more
informations and [Ber07] for fourteen different characterizations of central
words. There are also some properties which are satisfied by Christoffel words
but do not characterize them.
###### Lemma 2.
Let $w=amb$ be a Christoffel word of vector $(p,q)$. Then,
1. (i)
the central word $m$ is a palindrome: $\widetilde{m}=m$;
2. (ii)
$p$ is a period of $am$ and $q$ is a period of $mb$;
3. (iii)
the reversal $\widetilde{w}$ of a Christoffel word $w$ is conjugate to $w$.
The proof of (iii) follows from (i) and from Theorem 1. Words conjugate to
their reversal were studied in [BHNR04], are product of two palindromes and
are not necessarily Christoffel words. Moreover, not every palindrome is a
central word. In this article, we generalize Theorem 1 to dimension 3. We also
show that properties like the one enumerated in Lemma 2 hold.
### 2.2 Discrete planes
Given ${\vec{a}}=(a_{1},a_{2},a_{3})\in\mathbb{R}^{3}$ and
$\mu,\omega\in\mathbb{R}$, the _lower arithmetical discrete plane_ [Rev91]
${\mathcal{P}}$ is the set of point $x=(x_{1},x_{2},x_{3})\in\mathbb{Z}^{3}$
satisfying
$\mu\leq a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}<\mu+\omega.$
The parameter $\omega$ is called the (arithmetic) _width_. If
$\omega=\|{\vec{a}}\|_{1}=|a_{1}|+|a_{2}|+|a_{3}|$, then the discrete plane is
said to be _standard_. Standard arithmetical discrete plane can be furnished
with a canonical structure of a two-dimensional, connected, orientable
combinatoric manifold without boundary, whose faces are quadrangles and whose
vertices are points on the plane [Fra96]. See the appendix in Section 7 where
we provide the mathematical framework for the definition of discrete
hyperplanes ${\mathcal{P}}$ and _stepped surfaces_ ${\mathcal{S}}$ [ABFJ07].
Let $k$ be an integer such that $0\leq k<d$. We say that
$u,v\in\mathbb{Z}^{d}$ are _$k$ -neighbor_ if and only if
$\|v-u\|_{\infty}=1\text{ and }\|v-u\|_{1}\leq d-k.$
In this article, we are interested in the graph representing the
$2$-neighboring relation for the discrete plane ${\mathcal{P}}$ and in general
the $(d-1)$-neighboring relation for the discrete hyperplane in
$\mathbb{Z}^{d}$. Note that $u,v\in\mathbb{Z}^{d}$ are $(d-1)$-neighbors if
and only if their difference is $\pm e_{i}$ for some $i$ such that $1\leq
i\leq d$.
## 3 Discrete hyperplane graphs
Let $a_{1},\ldots,a_{d}$ be relatively prime positive integers and
$s=\|{\vec{a}}\|_{1}=\sum a_{i}$ be their sum. We denote
${\vec{a}}=(a_{1},a_{2},\ldots,a_{d})\in\mathbb{N}^{d}$. We define the mapping
${\mathcal{F}}_{\vec{a}}:\mathbb{Z}^{d}\to\mathbb{Z}/s\mathbb{Z}$ sending each
integral vector $(x_{1},\ldots,x_{d})$ onto $\sum_{i}a_{i}x_{i}\bmod s$. We
identify $\mathbb{Z}/s\mathbb{Z}$ and $\\{0,1,\ldots,s-1\\}$. A total order on
$\mathbb{Z}/s\mathbb{Z}$ is defined correspondingly; it is this order that is
used in the definition of ${\mathcal{H}}_{\vec{a}}$ below. The map
${\mathcal{F}}_{\vec{a}}$ induces a $\mathbb{Z}^{d}$-action $x\cdot
g=g+{\mathcal{F}}_{\vec{a}}(x)$ on the cyclic group $\mathbb{Z}/s\mathbb{Z}$,
so that it is a rational case of the $\mathbb{Z}^{2}$-action on the torus as
studied in [BV00, ABI02]. We consider
${\mathbb{E}}_{d}=\\{(u,u+e_{i}):u\in\mathbb{Z}^{d}\,\text{and}\,1\leq i\leq
d\\}$, the set of oriented edges of the hypercubic lattice. Note that the set
${\mathbb{E}}_{d}$ also corresponds to the Cayley graph of $\mathbb{Z}^{d}$
with generators $e_{i}$ for all $i$ with $1\leq i\leq d$.
### 3.1 The Christoffel graph ${\mathcal{H}}_{\vec{a}}$
The _Christoffel graph_ ${\mathcal{H}}_{\vec{a}}$ of normal vector ${\vec{a}}$
is the subset of edges of ${\mathbb{E}}_{d}$ increasing for the function
${\mathcal{F}}_{\vec{a}}$:
${\mathcal{H}}_{\vec{a}}=\\{(u,u+e_{i})\in{\mathbb{E}}_{d}:{\mathcal{F}}_{\vec{a}}(u)<{\mathcal{F}}_{\vec{a}}(u+e_{i})\\}.$
An example of the graph ${\mathcal{H}}_{\vec{a}}$ when $d=2$ and
${\vec{a}}=(a_{1},a_{2})=(2,5)$ is shown in Figure 2 (left) where the edges
are represented in blue and a small red circle surrounds the origin.
Figure 2: Left: the graph ${\mathcal{H}}_{\vec{a}}$ with ${\vec{a}}=(2,5)$.
Right: Standard discrete line ${\mathcal{P}}$ of normal vector
${\vec{a}}=(2,5)$.
A first observation is stated in the next lemma.
###### Lemma 3.
The graph ${\mathcal{H}}_{\vec{a}}$ is invariant under the translation by the
vector $\sum_{i=1}^{d}e_{i}=(1,1,\ldots,1)$.
The proof is postponed at Lemma 10 where we show that the graph
${\mathcal{H}}_{\vec{a}}$ is invariant under all translations $t\in{\rm
Ker}\,{\mathcal{F}}_{\vec{a}}$. Because of this invariance, a question is to
find a good representent for the equivalence class
$x+(1,1,\ldots,1)\mathbb{Z}$ for each $x\in\mathbb{Z}^{d}$. It is natural to
choose $\bar{x}\in x+(1,1,\ldots,1)\mathbb{Z}$ such that
$0\leq\sum a_{i}\bar{x}_{i}<s.$ (1)
If $(u,v)$ is an edge of ${\mathcal{H}}_{\vec{a}}$ such that $v-u=e_{i}$ then
$(\bar{u},\bar{v})$ is a pair of points which are (d-1)-neighbors satisfying
Equation (1) and $\bar{v}-\bar{u}=e_{i}$. Thus the vertices satisfying
Equation (1) are a set of representents for the vertices of
${\mathcal{H}}_{\vec{a}}$, see Figure 2 (right). Thus, each connected
component of the graph ${\mathcal{H}}_{\vec{a}}$ corresponds exactly to a
_standard discrete plane_ ${\mathcal{P}}$ with the $(d-1)$-neighbor relation.
The advantage of ${\mathcal{H}}_{\vec{a}}$ over the discrete hyperplane
${\mathcal{P}}$ is its algebraic structure. The next lemma gives an equivalent
definition of the edges of the graph ${\mathcal{H}}_{\vec{a}}$. It will be
useful in the sequel.
Let $a,b\in[0,s[$ be two integers. If $a<b$ then $]a,b]$ is a subinterval of
$[0,s[$. If $a>b$ then $]a,b]=]a,s[\,\cup\,[0,b]$ is defined as the union of
two subintervals of $[0,s[$.
###### Lemma 4.
Let $(u,v)\in{\mathbb{E}}_{d}$ such that $v=u+e_{i}$ for some $1\leq i\leq d$.
Then,
$\displaystyle(u,v)\in{\mathcal{H}}_{\vec{a}}\iff{\mathcal{F}}_{\vec{a}}(u)\in[0,s-a_{i}-1]\iff{\mathcal{F}}_{\vec{a}}(v)\in[a_{i},s-1]\iff
0\notin\,]{\mathcal{F}}_{\vec{a}}(u),{\mathcal{F}}_{\vec{a}}(v)],$ (2)
$\displaystyle(u,v)\notin{\mathcal{H}}_{\vec{a}}\iff{\mathcal{F}}_{\vec{a}}(u)\in[s-a_{i},s-1]\iff{\mathcal{F}}_{\vec{a}}(v)\in[0,a_{i}-1]\iff
0\in\,]{\mathcal{F}}_{\vec{a}}(u),{\mathcal{F}}_{\vec{a}}(v)].$ (3)
For each permutation $\sigma$ of the set $\\{1,2,\cdots,d\\}$, there exists a
_$\sigma$ -path_
$(u,u+e_{\sigma(1)}),(u+e_{\sigma(1)},u+e_{\sigma(1)}+e_{\sigma(2)}),\cdots,(u+\sum_{i=1}^{d-1}e_{\sigma(i)},u+\sum_{i=1}^{d}e_{\sigma(i)})$
made of $d$ edges of ${\mathbb{E}}_{d}$ going from the vertex
$u\in\mathbb{Z}^{d}$ to the vertex $u+\sum_{i=1}^{d}e_{i}$.
###### Lemma 5.
All of the $d$ edges of a $\sigma$-path but one belong to
${\mathcal{H}}_{\vec{a}}$.
###### Proof.
To each edge $(u+\sum_{i=1}^{k-1}e_{\sigma(i)},u+\sum_{i=1}^{k}e_{\sigma(i)})$
corresponds an interval (or an union of two intervals according to the above
remark)
$[{\mathcal{F}}_{\vec{a}}(u)+\sum_{i=1}^{k-1}a_{\sigma(i)},{\mathcal{F}}_{\vec{a}}(u)+\sum_{i=1}^{k}a_{\sigma(i)}[$.
Since $\sum_{i=1}^{d}a_{\sigma(i)}=\sum_{i=1}^{d}a_{i}=s$, those $d$ sets,
with $1\leq k\leq d$, are a partition of $[0,s[$. Therefore, only one of them
contains $0$. From Lemma 4, only one edge of the $\sigma$-path do not belong
to ${\mathcal{H}}_{\vec{a}}$. ∎
Let $R\subseteq\\{1,2,\cdots,d\\}$ and $u\in\mathbb{Z}^{d}$. An _hypercube
graph from vertex $u$ to vertex $u+\sum_{i\in R}e_{i}$_ with $2^{{\rm
Card\,}R}$ vertices is the subgraph of ${\mathbb{E}}_{d}$ defined by
$\left\\{\left(u+\sum_{i\in P}e_{i},u+\sum_{i\in
Q}e_{i}\right)\in{\mathbb{E}}_{d}\,\middle|\,P\subset Q\subseteq R\text{ and
}{\rm Card\,}Q\setminus P=1\right\\}.$
Each nonedge of ${\mathcal{H}}_{\vec{a}}$ implies the presence of a hypercube
graph with $2^{d-1}$ vertices orthogonal and incident to it. For example,
$(-e_{1},0)\notin{\mathcal{H}}_{\vec{a}}$ and
$(0,e_{2})\in{\mathcal{H}}_{\vec{a}}$ when ${\vec{a}}=(2,5)$. This is proved
in the next lemma.
###### Lemma 6.
If $(u,v)\in{\mathbb{E}}_{d}\setminus{\mathcal{H}}_{\vec{a}}$, then the
hypercube graph from vertex $v$ to vertex $u+\sum_{i=1}^{d}e_{i}$ with
$2^{d-1}$ vertices is a subgraph of ${\mathcal{H}}_{\vec{a}}$.
###### Proof.
From Lemma 5, the last $d-1$ edges of every $\sigma$-path starting with the
edge $(u,v)$ and ending in $u+\sum_{i=1}^{d}e_{i}$ are in
${\mathcal{H}}_{\vec{a}}$. The set of last $d-1$ edges of these paths
generates an hypercube graph from vertex $v$ to vertex
$u+\sum_{i=1}^{d}e_{i}$. ∎
A line containing some point $x\in\mathbb{Z}^{d}$ parallel to $e_{i}$ in the
hypercubic lattice ${\mathbb{E}}_{d}$ is a set
$L_{x,i}=\\{(x+ke_{i},x+(k+1)e_{i}):k\in\mathbb{Z}\\}\subset{\mathbb{E}}_{d}.$
The intersection $L_{x,i}\cap{\mathcal{H}}_{\vec{a}}$ of such a line with a
discrete hyperplane graph ${\mathcal{H}}_{\vec{a}}$ is made of consecutive
edges and nonedges. The next Lemma states that Christoffel words appear in
this sequence.
###### Lemma 7.
The sequence of consecutive edges and nonedges in
$L_{x,i}\cap{\mathcal{H}}_{\vec{a}}$ is periodic and the period is a
Christoffel word.
###### Proof.
Each subset $L_{x,i}\cap{\mathcal{H}}_{\vec{a}}$ can be described by the
subgroup of $\mathbb{Z}/s\mathbb{Z}$ generated by
${\mathcal{F}}_{\vec{a}}(e_{i})$, i.e.,
$(x+ke_{i},x+(k+1)e_{i})\in L_{x,i}\cap{\mathcal{H}}_{\vec{a}}\iff
0\leq{\mathcal{F}}_{\vec{a}}(x+ke_{i})<s-a_{i}$
This corresponds to the well-known construction of Christoffel words from the
labelling of Cayley graphs of $\mathbb{Z}/s\mathbb{Z}$ with the generator
$a_{i}$ [BLRS08, Section 1.2 Cayley graph definition]. ∎
For example, in the discrete hyperplane graph $H_{(2,5)}$ shown in Figure 2,
coding an edge by the letter $a$ and a nonedge by letter $b$, we get the
periods $aaabaab$ and $abbabbb$ for the lines
$L_{x,i}\cap{\mathcal{H}}_{\vec{a}}$ for $i=1,2$ respectively. Both are
Christoffel words.
###### Definition 8 (Image).
Let $f:\mathbb{Z}^{d}\to S$ be an homomorphism of $\mathbb{Z}$-module. For
some subset of edges $X\subseteq{\mathbb{E}}_{d}$, we define the image by $f$
of the edges $X$ by
$f(X)=\\{(f(u),f(v))\mid(u,v)\in X\\}.$
This definition allows to define the graphs $I_{\vec{a}}$ and
${\mathcal{G}}_{\vec{a}}$ as projections of ${\mathcal{H}}_{\vec{a}}$ in the
sections below.
### 3.2 The graph $I_{\vec{a}}$
Let $\pi$ be the orthogonal projection from $\mathbb{R}^{d}$ onto the
hyperplane $\cal D$ of equation $\sum x_{i}=0$. Its restriction to the stepped
surface ${\mathcal{S}}$ of the discrete plane ${\mathcal{P}}$ of normal vector
${\vec{a}}\in\mathbb{Z}^{d}$ is a bijection onto ${\mathcal{D}}$. It maps
${\mathcal{P}}$, the integral points in ${\mathcal{S}}$, onto a lattice $L$
[ABI02, section 2.2] in ${\mathcal{D}}$ spanned by the vectors
$h_{i}=\pi(e_{i})$; they satisfy $\sum_{i}h_{i}=0$. Note that
$\pi(\mathbb{Z}^{d})$ is also equal to $L$, since each point in
$\mathbb{Z}^{d}$ is congruent to some point in ${\mathcal{P}}$ modulo the
kernel of the projection. We may identify the set $L$ and
$\mathbb{Z}^{d}/(1,1,\cdots,1)\mathbb{Z}$, since two integral points are
projected by $\pi$ onto the same point if and only if their difference is a
multiple of the vector $(1,1,\ldots,1)$ and since this multiple is necessarily
an integral multiple. Since ${\mathcal{F}}_{\vec{a}}(1,1,\ldots,1)=0$, the
mapping ${\mathcal{F}}_{\vec{a}}$ induces a mapping
${\mathcal{F}}_{\vec{a}}^{\prime}:L\to\mathbb{Z}/s\mathbb{Z}$. We have the
following commuting diagram:
$\mathbb{Z}^{d}$$\mathbb{Z}/s\mathbb{Z}$$L=\mathbb{Z}^{d}/(1,1,\ldots,1)\mathbb{Z}$${\mathcal{F}}_{\vec{a}}$${\mathcal{F}}_{\vec{a}}^{\prime}$$\pi$
We consider the directed graph whose set of edges is
$I_{\vec{a}}=\pi({\mathcal{H}}_{\vec{a}})$. The graphs $I_{\vec{a}}$ for
${\vec{a}}=(a_{1},a_{2})=(2,5)$ and ${\vec{a}}=(a_{1},a_{2},a_{3})=(2,3,5)$
are shown in Figure 3. Note that the orientation of an edge is redundant when
$d=3$, since each edge is oriented as one of the vector $h_{i}$.
Figure 3: Left: the graph $I_{\vec{a}}$ when ${\vec{a}}=(2,5)$. Right: the
graph $I_{\vec{a}}$ when ${\vec{a}}=(2,3,5)$. The label at each vertex is its
image under ${\mathcal{F}}_{\vec{a}}^{\prime}$.
Lemma 6 can be seen on $I_{\vec{a}}$ when $d=3$ by the fact that each nonedge
is the short diagonal of a rhombus. For example, if ${\vec{a}}=(2,3,5)$ then
$(-h_{1},0)\notin I_{\vec{a}}$. From the lemma, the paths
$(0,h_{2}),(h_{2},h_{2}+h_{3})$ and $(0,h_{3}),(h_{3},h_{2}+h_{3})$ are in
$I_{\vec{a}}$.
The next lemma is a generalization of the fact that $I_{\vec{a}}$ is a tiling
of rhombus when $d=3$ proved in [Fra96] and [ABI02]. Indeed, each rhombus is
the projection under $\pi$ of one of three types of square in
$\mathbb{R}^{3}$. Below, the projection under $\pi$ of the convex hull of the
$2^{k}$ vertices of a $k$-dimensional hypercube graph in ${\mathbb{E}}_{d}$ is
called a _$k$ -dimensional parallelotope_. The edges of such a parallelotope
have equal length. When $d=3$, a $(d-1)$-dimensional parallelotope is a
rhombus.
###### Proposition 9.
The graph $I_{\vec{a}}$ produces a tiling of ${\mathcal{D}}$ by $d$ types of
$(d-1)$-dimensional parallelotopes.
In the following proof the fractional part of a real number $x\in\mathbb{R}$
is denoted by $\\{x\\}=x-\lfloor x\rfloor$.
###### Proof.
Each real point $x=(x_{1},\cdots,x_{d})\in\mathbb{R}^{d}$ of the hyperplane
${\mathcal{D}}$ is contained in a $(d-1)$-simplex with vertices
$\\{\pi(u)+\sum_{i=1}^{k}h_{\sigma(i)}:0\leq k\leq d-1\\}$ for $u=(\lfloor
x_{1}\rfloor,\cdots,\lfloor x_{d}\rfloor)\in\mathbb{Z}^{d}$ and permutation
$\sigma$ of $\\{1,2,\cdots,d\\}$ such that
$\\{x_{\sigma(1)}\\}\geq\\{x_{\sigma(2)}\\}\geq\cdots\geq\\{x_{\sigma(d)}\\}$.
We illustrate this on an example. Suppose $d=4$ and $x$ is such that $\sigma$
is the identity permutation on $\\{1,2,3,4\\}$. We have
$\displaystyle x$ $\displaystyle=\left\\{\begin{array}[]{l}u\\\
+\\{x_{1}\\}e_{1}\\\ +\\{x_{2}\\}e_{2}\\\ +\\{x_{3}\\}e_{3}\\\
+\\{x_{4}\\}e_{4}\\\ \end{array}\right.=\left\\{\begin{array}[]{l}u\\\
+\left(\\{x_{1}\\}-\\{x_{2}\\}\right)e_{1}\\\
+\left(\\{x_{2}\\}-\\{x_{3}\\}\right)(e_{1}+e_{2})\\\
+\left(\\{x_{3}\\}-\\{x_{4}\\}\right)(e_{1}+e_{2}+e_{3})\\\
+\\{x_{4}\\}(e_{1}+e_{2}+e_{3}+e_{4})\end{array}\right.=\left\\{\begin{array}[]{l}\left(1-\\{x_{1}\\}\right)u\\\
+\left(\\{x_{1}\\}-\\{x_{2}\\}\right)(u+e_{1})\\\
+\left(\\{x_{2}\\}-\\{x_{3}\\}\right)(u+e_{1}+e_{2})\\\
+\left(\\{x_{3}\\}-\\{x_{4}\\}\right)(u+e_{1}+e_{2}+e_{3})\\\
+\\{x_{4}\\}(u+e_{1}+e_{2}+e_{3}+e_{4}).\end{array}\right.$
Therefore $x$ is in the convex hull of the $3$-simplex with vertices
$\\{\pi(u),\pi(u)+h_{1},\pi(u)+h_{1}+h_{2},\pi(u)+h_{1}+h_{2}+h_{3}\\}$ since
$x=\pi(x)=\left\\{\begin{array}[]{l}(1+\\{x_{4}\\}-\\{x_{1}\\})\pi(u)\\\
+(\\{x_{1}\\}-\\{x_{2}\\})(\pi(u)+h_{1})\\\
+(\\{x_{2}\\}-\\{x_{3}\\})(\pi(u)+h_{1}+h_{2})\\\
+(\\{x_{3}\\}-\\{x_{4}\\})(\pi(u)+h_{1}+h_{2}+h_{3}).\end{array}\right.$
Consider the $\sigma$-path in ${\mathbb{E}}_{d}$ starting at vertex $u$ and
ending at $u+\sum_{i=1}^{d}e_{i}$. From Lemma 5, there is an edge
$(u^{\prime},v^{\prime})$ of the $\sigma$-path that is not in
${\mathcal{H}}_{\vec{a}}$. From Lemma 6, the hypercube graph from vertex
$v^{\prime}$ to vertex $u^{\prime}+\sum_{i=1}^{d}e_{i}$ with $2^{d-1}$
vertices is a subgraph of ${\mathcal{H}}_{\vec{a}}$. Therefore, the hypercube
graph (projected in $\pi({\mathbb{E}}_{d})$) going from vertex
$\pi(v^{\prime})$ to vertex $\pi(u^{\prime})$ with $2^{d-1}$ vertices is a
subgraph of $I_{\vec{a}}$. The convex hull of this graph contains the
$(d-1)$-simplex with vertices $\\{\pi(u)+\sum_{i=1}^{k}h_{\sigma(i)}:0\leq
k\leq d-1\\}$ which in turns contains $x$. Therefore the point $x$ of the
hyperplane ${\mathcal{D}}$ is contained in the image under $\pi$ of the convex
hull of a $(d-1)$-dimensional hypercube graph in ${\mathbb{E}}_{d}$. For
almost every $x$, the inequalities
$\\{x_{\sigma(1)}\\}>\\{x_{\sigma(2)}\\}>\cdots>\\{x_{\sigma(d)}\\}$ are
strict and the parallelotope is unique. We conclude that ${\mathcal{D}}$ is
tiled by $(d-1)$-dimensional parallelotopes. ∎
### 3.3 Kernel of ${\mathcal{F}}_{\vec{a}}$ and
${\mathcal{F}}_{\vec{a}}^{\prime}$
###### Lemma 10.
The discrete hyperplane graph ${\mathcal{H}}_{\vec{a}}$ is invariant under any
translation $t\in{\rm Ker}\,{\mathcal{F}}_{\vec{a}}$.
###### Proof.
Let $u\in\mathbb{Z}^{d}$ and $t\in{\rm Ker}\,{\mathcal{F}}_{\vec{a}}$. We have
${\mathcal{F}}_{\vec{a}}(u+t)={\mathcal{F}}_{\vec{a}}(u)+{\mathcal{F}}_{\vec{a}}(t)={\mathcal{F}}_{\vec{a}}(u)$.
From Lemma 4, $(u,u+e_{i})\in{\mathcal{H}}_{\vec{a}}$ if and only if
${\mathcal{F}}_{\vec{a}}(u)\in[0,s-a_{i}-1]$ if and only if
${\mathcal{F}}_{\vec{a}}(u+t)\in[0,s-a_{i}-1]$ if and only if
$(u+t,u+e_{i}+t)\in{\mathcal{H}}_{\vec{a}}$. ∎
We can find generators of the kernel of ${\mathcal{F}}_{\vec{a}}$ when $d=3$.
###### Proposition 11.
If $d=3$, the kernel of ${\mathcal{F}}_{\vec{a}}$ is
${\rm
Ker}\,{\mathcal{F}}_{\vec{a}}=\langle(a_{3},0,-a_{1}),(0,a_{3},-a_{2}),(a_{2},-a_{1},0),(1,1,1)\rangle.$
The result is based on the following well-known lemma.
###### Lemma 12.
Let $K$ be a subgroup of $\mathbb{Z}^{n}$ generated by the rows of a $s\times
n$ matrix $M\in\mathbb{Z}^{s\times n}$ of rank $n$. The index
$[\mathbb{Z}^{n}:K]$ is equal to the $\gcd$ of the $n$-minors of the matrix
$M$.
###### Proof.
By the rank condition, we have $s\geq n$. Suppose first that $M$ is in
diagonal form; that is, the diagonal elements of $M$ are $d_{1},\ldots,d_{n}$
and that the other elements are 0; by the rank condition, the $d_{i}$ are all
nonzero. Then the subgroup is
$K={d_{1}}\mathbb{Z}\times\cdots\times{d_{n}}\mathbb{Z}$, the quotient group
is $\mathbb{Z}/{d_{1}}\mathbb{Z}\times\cdots\times\mathbb{Z}/d_{n}\mathbb{Z}$,
and therefore the index is $d_{1}\cdots d_{n}$. Moreover the only nonzero
$n$-minor is $d_{1}\cdots d_{n}$.
In the general case, it is well-known that the matrix $M$ may be brought into
diagonal form by row and column operations within $\mathbb{Z}^{s\times n}$;
moreover, these operations do not change the subgroup, up to change of basis
in $\mathbb{Z}^{n}$; and finally, the $\gcd$ of the $n$-minors is invariant
under these operations. Thus the general case follows from the diagonal case.
∎
###### Proof.
(of the proposition) The $\supseteq$ part. The kernel of
$F={\mathcal{F}}_{\vec{a}}$ contains the four vectors, because
$\begin{array}[]{l}F(a_{3},0,-a_{1})=ac+b0+a_{3}(-a_{1})=ac-ca=0,\\\
F(0,a_{3},-a_{2})=a0+bc+a_{3}(-a_{2})=bc-cb=0,\\\
F(a_{2},-a_{1},0)=ab+a_{2}(-a_{1})+c0=ab-ba=0.\\\ \end{array}$
and
$F(1,1,1)=m(a_{1}+a_{2}+a_{3})=0.$
The $\subseteq$ part. Let
$K=\langle(a_{3},0,-a_{1}),(0,a_{3},-a_{2}),(a_{2},-a_{1},0),(1,1,1)\rangle$.
$K$ is a subgroup of $\mathbb{Z}^{3}$. By showing that the index
$[\mathbb{Z}^{3}:K]$ is exactly the size $a_{1}+a_{2}+a_{3}$ of the image of
$F$, we conclude that $K={\rm Ker}\,{\mathcal{F}}_{\vec{a}}$. The subgroup $K$
is generated by the lines of the matrix
$M=\left(\begin{array}[]{rrr}1&1&1\\\ a_{3}&0&-a_{1}\\\ 0&a_{3}&-a_{2}\\\
a_{2}&-a_{1}&0\end{array}\right).$
From Lemma 12, the index is equal to the $\gcd$ of the four $3$-minors of the
matrix $M$:
$\begin{array}[]{ll}\det\left(\begin{array}[]{rrr}1&1&1\\\ a_{3}&0&-a_{1}\\\
0&a_{3}&-a_{2}\\\
\end{array}\right)=a_{3}(a_{1}+a_{2}+a_{3}),&\det\left(\begin{array}[]{rrr}1&1&1\\\
a_{3}&0&-a_{1}\\\
a_{2}&-a_{1}&0\end{array}\right)=-a_{1}(a_{1}+a_{2}+a_{3})\\\
\det\left(\begin{array}[]{rrr}1&1&1\\\ 0&a_{3}&-a_{2}\\\
a_{2}&-a_{1}&0\end{array}\right)=-a_{2}(a_{1}+a_{2}+a_{3}),&\det\left(\begin{array}[]{rrr}a_{3}&0&-a_{1}\\\
0&a_{3}&-a_{2}\\\ a_{2}&-a_{1}&0\end{array}\right)=0.\end{array}$
That is the index is
$[\mathbb{Z}^{3}:K]=\gcd(a_{3}(a_{1}+a_{2}+a_{3}),-a_{1}(a_{1}+a_{2}+a_{3}),-a_{2}(a_{1}+a_{2}+a_{3}),0)=a_{1}+a_{2}+a_{3}.\qed$
###### Corollary 13.
The kernel of ${\mathcal{F}}_{\vec{a}}^{\prime}$ is spanned by the vectors
$a_{3}h_{1}-a_{1}h_{3},a_{3}h_{2}-a_{2}h_{3},a_{2}h_{1}-a_{1}h_{2}$.
###### Proof.
This is because $\pi({\rm Ker}\,{\mathcal{F}}_{\vec{a}})={\rm
Ker}\,{\mathcal{F}}_{\vec{a}}^{\prime}$. Indeed,
${\mathcal{F}}_{\vec{a}}={\mathcal{F}}_{\vec{a}}^{\prime}\circ\pi$ and
$\pi(1,1,1)=0$. ∎
It is likely that the proposition and its corollary have an evident extension
to any dimension. The proof requires a higher minor calculation. We leave this
to the interested reader.
### 3.4 The graph ${\mathcal{G}}_{\vec{a}}$
Let $d\geq 2$ be an integer and ${\vec{a}}$ as before. The graph
${\mathcal{G}}_{\vec{a}}$ of normal vector ${\vec{a}}\in\mathbb{Z}^{d}$ is the
directed graph
${\mathcal{G}}_{\vec{a}}={\mathcal{F}}_{\vec{a}}({\mathcal{H}}_{\vec{a}})$. It
is also equal to
${\mathcal{G}}_{\vec{a}}=\\{(k,k+a_{i})\mid k\in\mathbb{Z}/s\mathbb{Z},1\leq
i\leq d\text{ and }k<k+a_{i}\\}.$
Two examples are shown at Figure 4.
Figure 4: The Christoffel graphs ${\mathcal{G}}_{\vec{a}}$ for
${\vec{a}}=(2,5)$ and ${\vec{a}}=(2,3,5)$.
Since the graph ${\mathcal{G}}_{\vec{a}}$ is isomorphic to the quotients
${\mathcal{H}}_{\vec{a}}/{\rm Ker}\,{\mathcal{F}}_{\vec{a}}$ (and also
$I_{\vec{a}}/{\rm Ker}\,{\mathcal{F}}_{\vec{a}}^{\prime}$), we call it
_Christoffel graph_ as well, because ${\mathcal{G}}_{\vec{a}}$ is the part of
${\mathcal{H}}_{\vec{a}}$ in its fundamental domain. The graph
${\mathcal{G}}_{\vec{a}}$ can be embedded in a torus. Indeed, the graph
$I_{\vec{a}}$ lives in the diagonal plane
${\mathcal{D}}\simeq\mathbb{R}^{d-1}$ and is invariant under the group ${\rm
Ker}\,{\mathcal{F}}_{\vec{a}}^{\prime}$. The quotient ${\mathcal{D}}/{\rm
Ker}\,{\mathcal{F}}_{\vec{a}}^{\prime}$ is a torus and contains the graph
$I_{\vec{a}}/{\rm
Ker}\,{\mathcal{F}}_{\vec{a}}^{\prime}\simeq{\mathcal{G}}_{\vec{a}}$ (see
Figure 5).
${\mathcal{G}}_{(2,5)}$ | | ${\mathcal{G}}_{(2,3,5)}$ |
---|---|---|---
Figure 5: The Christoffel graphs ${\mathcal{G}}_{\vec{a}}$ for
${\vec{a}}=(2,5)$ and ${\vec{a}}=(2,3,5)$ can be embedded in the torus
${\mathcal{D}}/{\rm Ker}\,{\mathcal{F}}_{\vec{a}}^{\prime}$.
| |
---|---|---
${\mathcal{G}}_{(2,3,3)}$ | ${\mathcal{G}}_{(2,3,4)}$ | ${\mathcal{G}}_{(4,6,7)}$
|
---|---
${\mathcal{G}}_{(6,10,15)}$ | ${\mathcal{G}}_{(12,15,20)}$
Figure 6: Some Christoffel graphs in dimension $d=3$. The body is blue, legs
are in red.
The vertices of the Christoffel graph ${\mathcal{G}}_{\vec{a}}$ and their
image under the function ${\mathcal{F}}_{\vec{a}}$ corresponds to what is
called _roundwalk_ in [BT04]. Their contribution allows to construct larger
and larger domain of roundwalks by iteration of extension rules.
The Christoffel graph has a natural representation inside $I_{\vec{a}}$. We
define this for $d=3$, leaving the generalizations for elsewhere. Recall that
the lattice $L$, defined in Section 3, is a free abelian group of rank 2,
spanned by the 3 vectors $h_{1},h_{2},h_{3}$ with $h_{1}+h_{2}+h_{3}=0$.
Moreover, the homomorphism
${\mathcal{F}}_{\vec{a}}^{\prime}:L\to\mathbb{Z}/s\mathbb{Z}$ maps $h_{i}$
onto $a_{i}$, with $s=a_{1}+a_{2}+a_{3}$, and the $a_{i}$ are relatively
prime; therefore, the mapping is surjective.
Choose some parallelogram in the plane ${\mathcal{D}}$ which is a fundamental
domain for its discrete subgroup ${\rm
Ker}\,{\mathcal{F}}_{\vec{a}}^{\prime}$. We may assume that $O$ is a vertex of
this parallelogram. Then ${\mathcal{F}}_{\vec{a}}^{\prime}$ induces a
bijection between $\mathbb{Z}/s\mathbb{Z}$ and the integral points inside the
parallelogram, excluding those lying on the two edges not containing $O$. It
is such a parallelogram, with the part of the edges of $I_{\vec{a}}$ which lie
inside him, that we may call a Christoffel parallelogram. This we may consider
as the generalization in dimension 3 of Christoffel words. Such a
parallelogram tiles the plane ${\mathcal{D}}$ and completely codes the graph
$I_{\vec{a}}$. Furthermore it is in bijection with the Christoffel graph, as
is easily verified. Examples are seen in Figure 6.
###### Remark 14.
The graphs ${\mathcal{H}}_{\vec{a}},I_{\vec{a}},{\mathcal{G}}_{\vec{a}}$ are
compatible, in the sense that $I_{\vec{a}}$ is the image under $\pi$ of
${\mathcal{H}}_{\vec{a}}$, ${\mathcal{G}}_{\vec{a}}$ is the image under
$F_{\vec{a}}$ of ${\mathcal{H}}_{\vec{a}}$ and also the image of $I_{\vec{a}}$
under $F^{\prime}_{\vec{a}}$.
${\mathcal{H}}_{\vec{a}}$${\mathcal{G}}_{\vec{a}}$$I_{\vec{a}}$${\mathcal{F}}_{\vec{a}}$${\mathcal{F}}_{\vec{a}}^{\prime}$$\pi$
### 3.5 The graph ${\mathcal{H}}_{{\vec{a}},\omega}$
In this section, we extend the definition of Christoffel graphs to discrete
plane such that the width $\omega$ is smaller than $s=\|{\vec{a}}\|_{1}=\sum
a_{i}$ where ${\vec{a}}\in\mathbb{N}^{d}$ is a vector of relatively prime
positive integers as before. We consider only width $\omega$ such that
$s/\omega$ is a positive integer strictly smaller than the dimension $d$:
$0<s/\omega<d$. We define the mapping
${\mathcal{F}}_{{\vec{a}},\omega}:\mathbb{Z}^{d}\to\mathbb{Z}/\omega\mathbb{Z}$
sending each integral vector $(x_{1},\ldots,x_{d})$ onto
$\sum_{i}a_{i}x_{i}\bmod\omega$. We identify $\mathbb{Z}/\omega\mathbb{Z}$ and
$\\{0,1,\cdots,\omega-1\\}$. A total order on $\mathbb{Z}/\omega\mathbb{Z}$ is
defined correspondingly. The _Christoffel graph of normal vector
${\vec{a}}\in\mathbb{N}^{d}$ of width $\omega$_ is the subset of edges
${\mathcal{H}}_{{\vec{a}},\omega}\subseteq{\mathbb{E}}_{d}$ defined by
${\mathcal{H}}_{{\vec{a}},\omega}=\\{(u,v)\in{\mathbb{E}}_{d}\mid{\mathcal{F}}_{{\vec{a}},\omega}(u)<{\mathcal{F}}_{{\vec{a}},\omega}(v)\\}.$
This graph is related but does not correspond exactly to discrete plane of
width $\omega$. In fact, ${\mathcal{H}}_{{\vec{a}},\omega}$ can be obtained by
the superposition of $s/\omega$ discrete plane of width $\omega$. The
definition of ${\mathcal{H}}_{{\vec{a}},\omega}$ is motivated by Pirillo’s
theorem, because this is what allows to generalize Pirillo’s theorem in
arbitrary dimension (see Theorem 36). Of course if $\omega=s$, then
${\mathcal{H}}_{{\vec{a}},\omega}={\mathcal{H}}_{\vec{a}}$ is the Christoffel
graph of normal vector ${\vec{a}}$. Also, if $d=2$ then $s=\omega$. If $d=3$,
then either $\omega=s$ or $\omega=s/2$. If $d=4$, then either $\omega=s$,
$\omega=s/2$ or $\omega=s/3$ and so on for $d\geq 5$. If $s$ is a prime
number, then $\omega=s$.
As earlier, we define the projected graphs
${\mathcal{G}}_{{\vec{a}},\omega}:={\mathcal{F}}_{{\vec{a}},\omega}({\mathcal{H}}_{{\vec{a}},\omega})$
and $I_{{\vec{a}},\omega}:=\pi({\mathcal{H}}_{{\vec{a}},\omega})$. The
Christoffel graph ${\mathcal{G}}_{{\vec{a}},\omega}$ for the vector
${\vec{a}}=(15,11,10)$ of width $\omega=s=36$ is shown at Figure 7 (left). The
Christoffel graph ${\mathcal{G}}_{{\vec{a}},\omega}$ for the vector
${\vec{a}}=(15,11,10)$ of width $\omega=18=s/2$ is shown at Figure 7 (right)
and a larger part is shown at Figure 8.
|
---|---
${\mathcal{G}}_{(15,11,10)}={\mathcal{G}}_{(15,11,10),36}$ | ${\mathcal{G}}_{(15,11,10),18}$
Figure 7: Left: the Christoffel graph ${\mathcal{G}}_{\vec{a}}$ for the vector
${\vec{a}}=(15,11,10)$. Right: the Christoffel graph
${\mathcal{G}}_{{\vec{a}},\omega}$ of width $\omega=18$ for the vector
${\vec{a}}=(15,11,10)$.
---
$I_{(15,11,10),18}$
Figure 8: The Christoffel graph $I_{{\vec{a}},\omega}$ of width $\omega=18$
for the vector ${\vec{a}}=(15,11,10)$. It corresponds to the union of two
discrete planes of width $\omega$.
The next lemma gives an equivalent definition of the edges of the graph
${\mathcal{H}}_{{\vec{a}},\omega}$.
###### Lemma 15.
Let $(u,v)\in{\mathbb{E}}_{d}$ such that $v-u=e_{i}$ for some $1\leq i\leq d$.
Then,
$\displaystyle(u,v)\in{\mathcal{H}}_{{\vec{a}},\omega}\iff{\mathcal{F}}_{{\vec{a}},\omega}(u)\in[0,\omega-
a_{i}-1]\iff{\mathcal{F}}_{{\vec{a}},\omega}(v)\in[a_{i},\omega-1],$ (4)
$\displaystyle(u,v)\notin{\mathcal{H}}_{{\vec{a}},\omega}\iff{\mathcal{F}}_{{\vec{a}},\omega}(u)\in[\omega-
a_{i},\omega-1]\iff{\mathcal{F}}_{{\vec{a}},\omega}(v)\in[0,a_{i}-1].$ (5)
## 4 Flip, reversal and translation
In this short section, we define the flip, reversal and translate of set of
edges. We define the operations for set of edges $X\subseteq{\mathbb{E}}_{d}$
but they extend naturally to set of edges of the form $\pi(X)$ and
${\mathcal{F}}_{\vec{a}}(X)$ (see Definition 20 below). In order to define the
flip operation, we need to define the edges incident to zero.
###### Definition 16 (edges of ${\mathbb{E}}_{d}$ incident to zero).
Let $d\geq 2$ be an integer and ${\vec{a}}\in\mathbb{Z}^{d}$ be a vector of
relatively prime positive integers. The set of _edges of ${\mathbb{E}}_{d}$
incident to zero_ is
${\mathcal{Q}}=\\{(u,v)\in{\mathbb{E}}_{d}:{\mathcal{F}}_{\vec{a}}(u)=0\text{
or }{\mathcal{F}}_{\vec{a}}(v)=0\\}.$
###### Definition 17 (body, legs).
Let $X\subseteq{\mathbb{E}}_{d}$. The set $X\setminus{\mathcal{Q}}$ is the
_body_ and the edges of $X\cap{\mathcal{Q}}$ are the _legs_ of $X$.
See Figure 6 where the legs of graphs ${\mathcal{G}}_{\vec{a}}$ are
represented in red, and the body in black. The flip is an operation which
generalizes the function $amb\mapsto bma$ defined for Christoffel words. While
we define the flip on graphs, it can also be seen as a flip in a rhombus
tiling when $d=3$ [BFRR11, BFR08, ABFJ07].
###### Definition 18 (flip).
For a subset of edges $X\subseteq{\mathbb{E}}_{d}$, we define the _flip_
operation which exchanges edges incident to zero:
$\textsc{flip}:X\mapsto(X\setminus{\mathcal{Q}})\cup({\mathcal{Q}}\setminus
X).$
We see that $\textsc{flip}(X)$ exchanges the legs of $X$ and keeps the body of
$X$ invariant. If $(u,v)\in{\mathbb{E}}_{d}$, then the reversal edge
$(-v,-u)\in{\mathbb{E}}_{d}$ is also an edge of the hypercubic lattice and
similarly for the translated edge $(u+t,v+t)\in{\mathbb{E}}_{d}$ for all
$t\in\mathbb{Z}^{d}$. The reversal and translate operations extend on subsets
of edges as follows:
###### Definition 19 (Reversal, Translate).
Let $X\subseteq{\mathbb{E}}_{d}$ be a subset of edges. We define the
_reversal_ $-X$ of $X$ and the _translate_ $X+t$, for some
$t\in\mathbb{Z}^{d}$, of $X$ as
$-X=\\{(-v,-u)\mid(u,v)\in X\\}\quad\quad\text{and}\quad\quad
X+t=\\{(u+t,v+t)\mid(u,v)\in X\\}.$
###### Definition 20 (flip, Reversal, Translate).
Let $X\subseteq{\mathbb{E}}_{d}$. The flip, reversal and translate of set of
edges of the form $\pi(X)$ and ${\mathcal{F}}_{\vec{a}}(X)$ are defined
naturally by commutativity:
$\begin{array}[]{lll}\textsc{flip}(\pi(X)):=\pi(\textsc{flip}(X)),&-(\pi(X)):=\pi(-X),&\pi(X)+\pi(t):=\pi(X+t),\\\
\textsc{flip}({\mathcal{F}}_{\vec{a}}(X)):={\mathcal{F}}_{\vec{a}}(\textsc{flip}(X)),&-({\mathcal{F}}_{\vec{a}}(X)):={\mathcal{F}}_{\vec{a}}(-X),&{\mathcal{F}}_{\vec{a}}(X)+{\mathcal{F}}_{\vec{a}}(t):={\mathcal{F}}_{\vec{a}}(X+t).\end{array}$
Therefore, statements proven for ${\mathcal{H}}_{\vec{a}}$ using flip,
reversal and translate operations are also true for $I_{\vec{a}}$ and
${\mathcal{G}}_{\vec{a}}$. For example, the goal of next section is to show
that ${\mathcal{H}}_{\vec{a}}+t=\textsc{flip}({\mathcal{H}}_{\vec{a}})$ for
some $t\in\mathbb{Z}^{d}$. If such an equation is satisfied for
${\mathcal{H}}_{\vec{a}}$, it is clear from Definition 20 that
$I_{\vec{a}}+\pi(t)=\textsc{flip}(I_{\vec{a}})$ and
${\mathcal{G}}_{\vec{a}}+{\mathcal{F}}_{\vec{a}}(t)=\textsc{flip}({\mathcal{G}}_{\vec{a}})$.
## 5 Flipping is translating
In this section, we show that the flip of the Christoffel graph
${\mathcal{H}}_{\vec{a}}$ is a translate of ${\mathcal{H}}_{\vec{a}}$; this is
a generalization of one implication of Theorem 1. We also show that the body
of ${\mathcal{H}}_{\vec{a}}$ is symmetric and as a consequence we obtain that
a Christoffel graph is a translate of its reversal. The results stated in this
section are stated and proved for ${\mathcal{H}}_{\vec{a}}$ but they are valid
for $I_{\vec{a}}$ and ${\mathcal{G}}_{\vec{a}}$ by Definition 20.
The following lemma describes the legs of ${\mathcal{H}}_{\vec{a}}$.
###### Lemma 21 (Legs of ${\mathcal{H}}_{\vec{a}}$).
An edge $(u,v)$ is a leg of ${\mathcal{H}}_{\vec{a}}$ if and only if
${\mathcal{F}}_{\vec{a}}(u)=0$.
###### Proof.
We have $(-e_{i},0)\notin{\mathcal{H}}_{\vec{a}}$ because there is no
$u\in\mathbb{Z}^{d}$ such that
${\mathcal{F}}_{\vec{a}}(u)<{\mathcal{F}}_{\vec{a}}(0)=0$. Moreover
$(0,e_{i})\in E$ for each $i$, $1\leq i\leq d$, because
${\mathcal{F}}_{\vec{a}}(0)=0<a_{i}={\mathcal{F}}_{\vec{a}}(e_{i})$. ∎
We now show that the body of a Christoffel graph is symmetric, i.e., it is
equal to its reversal. This generalizes the fact that central words are
palindromes.
###### Lemma 22.
The body of ${\mathcal{H}}_{\vec{a}}$ is symmetric, i.e.,
$-({\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}})={\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}}$.
###### Proof.
It is sufficient to prove
$-({\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}})\supseteq{\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}}$,
the other inclusion being equivalent, since symmetry is involutive. Let
$(u,v)\in{\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}}$. Then $v-u=e_{i}$ for
some $1\leq i\leq d$. Then ${\mathcal{F}}_{\vec{a}}(u)\in[0,s-a_{i}-1]$ by
Lemma 4 and ${\mathcal{F}}_{\vec{a}}(u)\notin\\{0,s-a_{i}\\}$ so that
${\mathcal{F}}_{\vec{a}}(u)\in[1,s-a_{i}-1]$. Thus
${\mathcal{F}}_{\vec{a}}(-u)=s-{\mathcal{F}}_{\vec{a}}(u)\in[a_{i}+1,s-1]$. We
obtain that $(-v,-u)\in{\mathcal{H}}_{\vec{a}}$ by Lemma 4 because
$-u-(-v)=v-u=e_{i}$. Since ${\mathcal{Q}}=-{\mathcal{Q}}$,
$(u,v)\notin{\mathcal{Q}}$ implies that $(-v,-u)\notin{\mathcal{Q}}$. We
conclude $(-v,-u)\in{\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}}$ and
$(u,v)\in-({\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}})$. ∎
Now we show that the reversal is equal to the flip of a Christoffel graph.
This generalizes the fact that the reversal $\widetilde{amb}$ of a Christoffel
word is equal to $bma$.
###### Lemma 23.
The reversal of ${\mathcal{H}}_{\vec{a}}$ is equal to its flip, i.e.,
$-{\mathcal{H}}_{\vec{a}}=\textsc{flip}({\mathcal{H}}_{\vec{a}})$.
###### Proof.
For ${\mathcal{H}}_{\vec{a}}$, we have to show that
$-{\mathcal{H}}_{\vec{a}}=({\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}})\cup({\mathcal{Q}}\setminus{\mathcal{H}}_{\vec{a}})$.
We prove the result in two parts since
$-{\mathcal{H}}_{\vec{a}}=\left((-{\mathcal{H}}_{\vec{a}})\setminus{\mathcal{Q}}\right)\cup\left((-{\mathcal{H}}_{\vec{a}})\cap{\mathcal{Q}}\right)$
is the disjoint union of a part outside of ${\mathcal{Q}}$ and a part inside
of ${\mathcal{Q}}$. Outside of ${\mathcal{Q}}$: since ${\mathcal{Q}}$ is
symmetric and because ${\mathcal{H}}_{\vec{a}}$ is symmetric from Lemma 22, we
have
$-({\mathcal{H}}_{\vec{a}})\setminus{\mathcal{Q}}=-({\mathcal{H}}_{\vec{a}})\setminus-{\mathcal{Q}}=-({\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}})={\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}}$.
Inside of ${\mathcal{Q}}$: we have
$(-{\mathcal{H}}_{\vec{a}})\cap{\mathcal{Q}}=-({\mathcal{H}}_{\vec{a}}\cap{\mathcal{Q}})=\\{(u,v)\in{\mathbb{E}}_{d}:{\mathcal{F}}_{\vec{a}}(v)=0\\}={\mathcal{Q}}\setminus{\mathcal{H}}_{\vec{a}}$
by Lemma 21. ∎
Now we show that the flip of a Christoffel graph ${\mathcal{H}}_{\vec{a}}$ is
equal to a translate of ${\mathcal{H}}_{\vec{a}}$. It generalizes one
implication of Theorem 1. It corresponds to the fact that a Christoffel word
$amb$ is conjugate to to $bma$. Proposition 24 is illustrated in Figure 9 and
Figure 10.
###### Proposition 24.
Let $t\in\mathbb{Z}^{d}$ be such that ${\mathcal{F}}_{\vec{a}}(t)=1$. The
translate by $t$ of ${\mathcal{H}}_{\vec{a}}$ is equal to its flip, i.e.,
${\mathcal{H}}_{\vec{a}}+t=\textsc{flip}({\mathcal{H}}_{\vec{a}})$.
Note that, since ${\mathcal{F}}_{\vec{a}}$ is surjective, there exists indeed
$t\in\mathbb{Z}^{d}$ such that ${\mathcal{F}}_{\vec{a}}(t)=1$. Also,
${\mathcal{F}}_{\vec{a}}(-t)=-{\mathcal{F}}_{\vec{a}}(t)=-1$.
###### Proof.
We prove the result in two parts since ${\mathcal{H}}_{\vec{a}}+t$ is the
disjoint union of a part outside of ${\mathcal{Q}}$ and a part inside of
${\mathcal{Q}}$:
${\mathcal{H}}_{\vec{a}}+t=\left(({\mathcal{H}}_{\vec{a}}+t)\setminus{\mathcal{Q}}\right)\cup\left(({\mathcal{H}}_{\vec{a}}+t)\cap{\mathcal{Q}}\right).$
1\.
$({\mathcal{H}}_{\vec{a}}+t)\setminus{\mathcal{Q}}\supseteq{\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}}$.
Suppose that $(u,v)\in{\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}}$. Thus
$v-u=e_{i}$ for some $1\leq i\leq d$. Then,
${\mathcal{F}}_{\vec{a}}(u)\in[0,s-a_{i}-1]$ by Lemma 4 and, since the edge is
not a leg, ${\mathcal{F}}_{\vec{a}}(u)\notin\\{0,s-a_{i}\\}$. Hence,
${\mathcal{F}}_{\vec{a}}(u)\in[1,s-a_{i}-1]$. Then
${\mathcal{F}}_{\vec{a}}(u-t)={\mathcal{F}}_{\vec{a}}(u)-1\in[0,s-a_{i}-2]$
which implies that $(u-t,v-t)\in{\mathcal{H}}_{\vec{a}}$. Then
$(u,v)\in{\mathcal{H}}_{\vec{a}}+t$.
2\.
$({\mathcal{H}}_{\vec{a}}+t)\setminus{\mathcal{Q}}\subseteq{\mathcal{H}}_{\vec{a}}\setminus{\mathcal{Q}}$.
Let $(u+t,v+t)\in({\mathcal{H}}_{\vec{a}}+t)\setminus{\mathcal{Q}}$ for some
edge $(u,v)\in{\mathcal{H}}_{\vec{a}}$. Then,
${\mathcal{F}}_{\vec{a}}(u+t)\notin\\{0,s-a_{i}\\}$. From Lemma 4,
${\mathcal{F}}_{\vec{a}}(u+t)={\mathcal{F}}_{\vec{a}}(u)+1\in[1,s-a_{i}]$.
Therefore, ${\mathcal{F}}_{\vec{a}}(u+t)\in[1,s-a_{i}-1]$ and we conclude that
$(u+t,v+t)\in{\mathcal{H}}_{\vec{a}}$.
3\.
$({\mathcal{H}}_{\vec{a}}+t)\cap{\mathcal{Q}}\supseteq{\mathcal{Q}}\setminus{\mathcal{H}}_{\vec{a}}$.
Let $(u,v)\in{\mathcal{Q}}\setminus{\mathcal{H}}_{\vec{a}}$. Then,
${\mathcal{F}}_{\vec{a}}(v)=0$ which implies that
${\mathcal{F}}_{\vec{a}}(v-t)=s-1$. By Lemma 4, we have
$(u-t,v-t)\in{\mathcal{H}}_{\vec{a}}$ so that
$(u,v)\in{\mathcal{H}}_{\vec{a}}+t$.
4\.
$({\mathcal{H}}_{\vec{a}}+t)\cap{\mathcal{Q}}\subseteq{\mathcal{Q}}\setminus{\mathcal{H}}_{\vec{a}}$.
Let $(u+t,v+t)\in({\mathcal{H}}_{\vec{a}}+t)\cap{\mathcal{Q}}$ for some edge
$(u,v)\in{\mathcal{H}}_{\vec{a}}$. Either ${\mathcal{F}}_{\vec{a}}(u+t)=0$ or
${\mathcal{F}}_{\vec{a}}(v+t)=0$. If ${\mathcal{F}}_{\vec{a}}(u+t)=0$, then
${\mathcal{F}}_{\vec{a}}(u)=s-1$ which implies that
$(u,v)\notin{\mathcal{H}}_{\vec{a}}$ by Lemma 4, a contradiction. One must
have ${\mathcal{F}}_{\vec{a}}(v+t)=0$, which implies that
$(u+t,v+t)\notin{\mathcal{H}}_{\vec{a}}$. ∎
Figure 9: Left: the graph ${\mathcal{H}}_{\vec{a}}$ with ${\vec{a}}=(2,5)$.
Right: $\textsc{flip}({\mathcal{H}}_{\vec{a}})$.
Figure 10: Left: the graph $I_{\vec{a}}$ with ${\vec{a}}=(4,6,7)$. Right:
$\textsc{flip}(I_{\vec{a}})$. Consider the Christoffel parallelogram $P$ with
vertices labeled by $0$ embedded in $I_{\vec{a}}$. The parallelogram $P$ also
appears in the graph $\textsc{flip}(I_{\vec{a}})$ with vertices labeled by
$1$.
The previous proposition proves that the body of a Christoffel graph
${\mathcal{H}}_{\vec{a}}$ has a period. This generalizes the fact that central
words of length $p+q-2$ have periods $p$ and $q$ (remark that $p$ and $-q$ is
the same period mod the length of the Christoffel words $|w|=p+q$). Indeed,
let $P$ be some parallelogram and $M$ be an inner point. Consider the $4$
vectors with origin equal to one of the vertices of $P$ and with end $M$.
Then, for each point $X$ in $P$, there is one of these vector, $\vec{v}$ say,
such that the segment $[X,X+\vec{v}]$ is contained in $P$. We leave the
verification of this to the reader. It follows that a Christoffel
parallelogram may be reconstructed from the edges incident to zero by applying
translations which stay completely in the parallelogram. This is completely
analoguous to the fact that a central word is completely determined by its two
periods.
The next result generalizes the fact that the reversal $\widetilde{w}$ of a
Christoffel word $w$ is conjugate to $w$. This is not a characteristic
property of Christoffel words, because it is satisfied for all words that are
the product of two palindromes.
###### Corollary 25.
Let $t\in\mathbb{Z}^{d}$ be such that ${\mathcal{F}}_{\vec{a}}(t)=1$. Then
$-{\mathcal{H}}_{\vec{a}}={\mathcal{H}}_{\vec{a}}+t$.
###### Proof.
Follows from Lemma 23 and Proposition 24. ∎
###### Corollary 26.
(i) The body of a Christoffel parallelogram is symmetric with respect to its
center.
(ii) Consider a Christoffel parallelogram $P$ embedded in $I_{\vec{a}}$. The
parallelogram obtained by symmetry of $P$ with respect to its center appears
as a translate of $P$ within $I_{\vec{a}}$, and is also equal to the flip of
$P$.
We thus have obtained a generalization of: (i) a central word is a palindrome;
(ii) the reversal of a Christoffel word $amb$ is conjugate to it, and equal to
$bma$. The corollary can be checked on Figures 6, 9 and 10
## 6 Higher-dimensional Pirillo’s theorem
In this section, we study the converse of Proposition 24. In other words, does
the fact of being a translate of its flip is a characteristic property of
Christoffel graphs as it is the case for Christoffel words? We show that it
must a Christoffel graph ${\mathcal{H}}_{{\vec{a}},\omega}$ for some vector
${\vec{a}}\in\mathbb{Z}^{d}$ and width $\omega$. If $d=3$, we show that
parallelograms that are translate to their flip are Christoffel parallelograms
or their edge-complement.
Let $K$ be a subgroup of $\mathbb{Z}^{d}$ for some integer $d\geq 2$ such that
the index $[\mathbb{Z}^{d}:K]$ is finite and
$\sum_{i=1}^{d}e_{i}=(1,1,\ldots,1)\in K$. Let ${\mathcal{Q}}$ be the set of
_edges of ${\mathbb{E}}_{d}$ incident to zero mod $K$_:
${\mathcal{Q}}=\\{(u,v)\in{\mathbb{E}}_{d}\mid u\in K\text{ or }v\in K\\}.$
For a subset of edges $X\subseteq{\mathbb{E}}_{d}$, we redefine the flip
operation according to the above set ${\mathcal{Q}}$:
$\textsc{flip}:X\mapsto(X\setminus{\mathcal{Q}})\cup({\mathcal{Q}}\setminus
X).$
In what follows, we assume that $M\subseteq{\mathbb{E}}_{d}$ is a set of edges
such that
* •
$M$ is invariant for the group of translations $K$;
* •
$\textsc{flip}(M)=M+t$ for some $t\in\mathbb{Z}^{d}$.
If $\textsc{flip}(M)=M+t$, then for each $i$, $(0,e_{i})\in M$ or
$(-e_{i},0)\in M$ but not both. Otherwise the number of edges parallel to the
vector $e_{i}$ is not preserved by the flip and the equation can not be
satisfied. Therefore, we suppose that for each $i$, $1\leq i\leq d$,
$(0,e_{i})\in M$ and $(-e_{i},0)\notin M$. In other words, the legs of $M$
are:
* •
${\mathcal{Q}}\cap M=\\{(u,u+e_{i})\in{\mathbb{E}}_{d}\mid u\in K\\}$.
The question we consider in this section is: for which set of edges
$M\subseteq{\mathbb{E}}_{d}$ satisfying the above three conditions does there
exist a translation $t\in\mathbb{Z}^{d}$ such that $\textsc{flip}(M)=M+t$ (see
Figure 11).
Figure 11: Left: $M$. Right: $\textsc{flip}(M)$ for the subgroup
$K=\langle(0,4,1),(-2,0,3),(1,1,1)\rangle$.
###### Lemma 27.
Let $t\in\mathbb{Z}^{d}$ be a translation. Let $X\subseteq{\mathbb{E}}_{d}$
and $h\in{\mathbb{E}}_{d}$. We have
1. (i)
If $h\in X$, then $h+t\notin{\mathcal{Q}}$ if and only if
$h+t\in\textsc{flip}(X+t)$.
2. (ii)
If $h\notin X$, then $h+t\in{\mathcal{Q}}$ if and only if
$h+t\in\textsc{flip}(X+t)$.
###### Proof.
We have
$\textsc{flip}(X+t)=((X+t)\setminus{\mathcal{Q}})\cup({\mathcal{Q}}\setminus(X+t))$.
(i) If $h\in X$ then $h+t\in X+t$. If $h+t\notin{\mathcal{Q}}$, then
$h+t\in(X+t)\setminus{\mathcal{Q}}\subseteq\textsc{flip}(X+t)$. If
$h+t\in{\mathcal{Q}}$, then $h+t\in(X+t)\cap{\mathcal{Q}}$. Therefore
$h+t\notin\textsc{flip}(X+t)$.
(ii) If $h\notin X$ then $h+t\notin X+t$. If $h+t\in{\mathcal{Q}}$, then
$h+t\in{\mathcal{Q}}\setminus(X+t)\subseteq\textsc{flip}(X+t)$. If
$h+t\notin{\mathcal{Q}}$, then
$h+t\notin(X+t)\cup{\mathcal{Q}}\supseteq\textsc{flip}(X+t)$. Therefore
$h+t\notin\textsc{flip}(X+t)$. ∎
###### Proposition 28.
For all $i$, with $1\leq i\leq d$, there exists a unique integer $b_{i}$,
$0<b_{i}<\omega$, such that $e_{i}+b_{i}t\in K$ where $\omega$ is the order of
$t$ in the group $\mathbb{Z}^{d}/K$. Moreover,
$(0,e_{i})+kt\begin{cases}\in M&\text{if}\quad 0\leq k<b_{i},\\\ \notin
M&\text{if}\quad b_{i}\leq k<\omega,\end{cases}$
In the following proof, for two elements $u,u^{\prime}\in\mathbb{Z}^{d}$ the
notation $u\equiv u^{\prime}$ is used when $u^{\prime}-u\in K$. The notation
is also used for two edges $(u,v),(u^{\prime},v^{\prime})\in{\mathbb{E}}_{d}$:
$(u,v)\equiv(u^{\prime},v^{\prime})$ if and only if
$u^{\prime}-u=v^{\prime}-v\in K$.
###### Proof.
Let $\omega$ be the order of $t$ in $\mathbb{Z}^{d}/K$. In this proof, we
denote by $\vec{0}$ the zero of $\mathbb{Z}^{d}/K$. Thus let
$\omega=\min\\{k>0|kt\in K\\}={\rm order}_{\mathbb{Z}^{d}/K}(t)$
Consider the orbit under the translation $t$ of the edge
$h=(\vec{0},e_{i})\in{\mathcal{Q}}\cap M$. We have that $h+\omega t\equiv
h\in{\mathcal{Q}}\cap M$. We want to show that there exists $b_{i}$, such that
$0<b_{i}<\omega$ and $h+b_{i}t\equiv(-e_{i},\vec{0})\in{\mathcal{Q}}$.
Suppose (by contradiction) that $h+kt\notin{\mathcal{Q}}$ for all
$0<k<\omega$. From Lemma 27 (i), $h\in M$ and $h+t\notin{\mathcal{Q}}$, then
$h+t\in\textsc{flip}(M+t)=M$. Recursively, we have
$h+kt\in\textsc{flip}(M+t)=M$ for all $0<k<\omega$. This is summarized in the
following graph:
$\begin{array}[]{cccccccccccc}h&\xrightarrow{+t}&h+t&\xrightarrow{+t}&h+2t&\xrightarrow{+t}&\cdots&\xrightarrow{+t}&h+(\omega-1)t&\xrightarrow{+t}&h+\omega
t\equiv h\\\ \\\ \in{\mathcal{Q}}\cap M&&\in M\setminus{\mathcal{Q}}&&\in
M\setminus{\mathcal{Q}}&&&&\in
M\setminus{\mathcal{Q}}&&\in{\mathcal{Q}}\end{array}$
But then, $h+(\omega-1)t\in M$ and $h+\omega t\in{\mathcal{Q}}$, so that
$h+\omega t\notin\textsc{flip}(M+t)=M$ from Lemma 27 (i). This is a
contradiction because $h+\omega t\equiv h\in M$. Hence, there must exist some
$b_{i}$, $0<b_{i}<\omega$ such that $h+b_{i}t\in{\mathcal{Q}}$. Since $h$ is
an edge parallel to the vector $e_{i}$, then either
$h+b_{i}t\equiv(\vec{0},e_{i})$ or $h+b_{i}t\equiv(-e_{i},\vec{0})$. The first
option contradicts the minimality of $\omega$. We conclude that
$e_{i}+b_{i}t\equiv\vec{0}$.
The number $b_{i}$ is also unique. Indeed, suppose there exist $b_{i}$ and
$b_{i}^{\prime}$ with $0<b_{i}<b_{i}^{\prime}<\omega$ such that
$h+b_{i}t\equiv h+b_{i}^{\prime}t\equiv(-e_{i},\vec{0})$. Then
$(b_{i}^{\prime}-b_{i})t=(\vec{0}+b_{i}^{\prime}t)-(\vec{0}+b_{i}t)\equiv(-e_{i})-(-e_{i})=\vec{0}$.
This contradicts the minimality of $\omega$ since
$0<b_{i}^{\prime}-b_{i}<\omega$.
From the above paragraph, we have that $h+kt\notin{\mathcal{Q}}$ for all $k$
such that $0<k<b_{i}$ or $b_{i}<k<\omega$. Using Lemma 27 (i), if $h+(k-1)t\in
M$ and $h+kt\notin{\mathcal{Q}}$, then $h+kt\in\textsc{flip}(M+t)=M$. Thus by
recursion $h+kt\in M$ for all $k$ with $0<k<b_{i}$. Also
$h+b_{i}t\equiv(-e_{i},\vec{0})\in{\mathcal{Q}}\setminus M$. Using Lemma 27
(ii), if $h+b_{i}t\notin M$ and $h+(b_{i}+1)t\notin{\mathcal{Q}}$, then
$h+(b_{i}+1)t\notin\textsc{flip}(M+t)=M$. Thus by recursion $h+kt\notin M$ for
all $k$ with $b_{i}<k<\omega$. ∎
###### Lemma 29.
$\mathbb{Z}^{d}/K$ is cyclic and generated by $t$.
###### Proof.
Let $u=(x_{1},x_{2},\cdots,x_{d})\in\mathbb{Z}^{d}$. Using Proposition 28, we
have
$u=\sum x_{i}e_{i}\equiv\sum x_{i}(-b_{i}t)=-\sum(b_{i}x_{i})t$
Let $k=-\sum(b_{i}x_{i})\mod\omega$. Then, $0\leq k<\omega$ and $u=kt$. ∎
###### Lemma 30.
$M=\\{(0,e_{i})+kt:1\leq i\leq d\text{ and }0\leq k<b_{i}\\}+K.$
###### Proof.
($\supseteq$) If $0\leq k<b_{i}$, then $(0,e_{i})+kt\in M$ by Proposition 28.
($\subseteq$) Let $(u,u+e_{i})\in M$ with $u\in\mathbb{Z}^{d}$. From Lemma 29,
$(u,u+e_{i})=(0,e_{i})+u\equiv(0,e_{i})+kt$ for some integer $k$ such that
$0\leq k<\omega$. From Proposition 28, $0\leq k<b_{i}$. ∎
For all $i$ with $1\leq i\leq d$, let $a_{i}$ be such that
$a_{i}+b_{i}=\omega$. Also let
${\vec{b}}=(b_{1},b_{2},\cdots,b_{d})\quad\text{and}\quad{\vec{a}}=(a_{1},a_{2},\cdots,a_{d})$
We have $a_{i}t=(\omega-b_{i})t=\omega t-b_{i}t\equiv e_{i}$. The next result
shows that $\omega$ is a divisor of $\sum a_{i}$ and $\sum b_{i}$.
###### Lemma 31.
There exist integers $q$ and $\ell$, with $0<q<d$ and $0<\ell<d$, such that
$\omega\cdot q=a_{1}+a_{2}+\cdots+a_{d}$ and
$\omega\cdot\ell=b_{1}+b_{2}+\cdots+b_{d}$. Moreover $d=q+\ell$.
###### Proof.
For all $1\leq i\leq d$, we have $e_{i}=a_{i}t=-b_{i}t$. Thus,
$-(b_{1}+b_{2}+\cdots+b_{d})t$ is an overall translation of
$e_{1}+e_{2}+\cdots+e_{d}\in K$, i.e., the identity. Similarly,
$(a_{1}+a_{2}+\cdots+a_{d})t=e_{1}+e_{2}+\cdots+e_{d}\in K$. Therefore, the
order of $t$ ($=\omega$) must divide both $a_{1}+a_{2}+\cdots+a_{d}$ and
$b_{1}+b_{2}+\cdots+b_{d}$. Then, there exist integers $q$ and $\ell$ such
that $\omega\cdot q=a_{1}+a_{2}+\cdots+a_{d}$ and
$\omega\cdot\ell=b_{1}+b_{2}+\cdots+b_{d}$. But $a_{i}<\omega$ for each $i$ so
that $a_{1}+a_{2}+\cdots+a_{d}<d\omega$ and $q<d$. Similarly, $\ell<d$.
Finally, $\omega q+\omega\ell=\sum(a_{i}+b_{i})=\omega d$ and therefore
$d=q+\ell$. ∎
If the sum of the $a_{i}$ or the sum of the $b_{i}$ is $\omega$, then the next
two theorems claim that $M$ is closely related to the Christoffel graph.
###### Theorem 32.
(i) If $\sum a_{i}=\omega$, then $M={\mathcal{H}}_{\vec{a}}$;
(ii) if $\sum b_{i}=\omega$, then the complement
$M^{c}={\mathbb{E}}_{d}\setminus M$ of $M$ is equal to $-H_{\vec{b}}$.
###### Proof.
(i) For all $u=(x_{1},x_{2},\cdots,x_{d})\in\mathbb{Z}^{d}$, we have $u=kt$
with
$k=\sum(-b_{i}x_{i})\bmod\omega=\sum(a_{i}-\omega)x_{i}\bmod\omega=\sum
a_{i}x_{i}\bmod\sum a_{i}={\mathcal{F}}_{\vec{a}}(u)$
We want to show that $M={\mathcal{H}}_{\vec{a}}$. We have that
$(u,u+e_{i})=(0,e_{i})+kt\in M$ if and only if $0\leq k<b_{i}$ if and only if
${\mathcal{F}}_{\vec{a}}(u)\in[0,\omega-a_{i}-1]$ if and only if
$(u,u+e_{i})\in{\mathcal{H}}_{\vec{a}}$.
(ii) For all $u=(x_{1},x_{2},\cdots,x_{d})\in\mathbb{Z}^{d}$, we have $u\equiv
kt$ with
$k=\sum(-b_{i}x_{i})\bmod\omega=-\sum b_{i}x_{i}\bmod\sum
b_{i}=-{\mathcal{F}}_{\vec{b}}(u)$
We want to show that $M^{c}=-{\mathcal{H}}_{\vec{b}}$. We have that
$(u,v)=(u,u+e_{i})=(0,e_{i})+kt\in{\mathbb{E}}_{d}\setminus M$ if and only if
$b_{i}\leq k<\omega$ if and only if
${\mathcal{F}}_{\vec{b}}(-u)\in[b_{i},\omega-1]$ if and only if
$(-u-e_{i},-u)\in{\mathcal{H}}_{\vec{b}}$ if and only if
$(u,v)\in-{\mathcal{H}}_{\vec{b}}$. ∎
###### Corollary 33.
Let $d=3$. $M$ is the Christoffel graph ${\mathcal{H}}_{\vec{a}}$ or $M$ is
the complement of the reversal of the Christoffel graph
${\mathcal{H}}_{\vec{b}}$.
Note that the complement of the reversal is equal to the reversal of the
complement.
###### Proof.
From Lemma 31 there exist integers $0<q<3$ and $0<\ell<3$ such that
$\omega\cdot q=a_{1}+a_{2}+a_{3}$ and $\omega\cdot\ell=b_{1}+b_{2}+b_{3}$.
Therefore, there are two cases, either $q=1$ and $\ell=2$ or $q=2$ and
$\ell=1$. If $q=1$, then Theorem 32 (i) applies. Therefore, $M$ is a
Christoffel graph $M={\mathcal{H}}_{\vec{a}}$ for the vector
${\vec{a}}=(a_{1},a_{2},a_{3})$. If $\ell=1$, then Theorem 32 (ii) applies.
Therefore, the complement of $M$ is the reversal of a Christoffel graph. More
precisely, $M^{c}=-{\mathcal{H}}_{\vec{b}}$ for the vector
${\vec{b}}=(b_{1},b_{2},b_{3})$. ∎
The previous result has also a counterpart in the triangular lattice $L$.
###### Corollary 34.
Let $M^{\prime}\subset\pi({\mathbb{E}}_{d})$ such that
$\textsc{flip}(M^{\prime}+t^{\prime})=M^{\prime}$ for some $t^{\prime}\in L$,
that is invariant under some subgroup of finite index of $L$ and that
satisfies ${\mathcal{Q}}\cap M=\\{(0,h_{1}),(0,h_{2}),\ldots,(0,h_{d})\\}+K$.
If $d=3$, then $M^{\prime}$ is equal to a graph $I_{\vec{a}}$ or to the
reversal of its edge-complement.
###### Proof.
All we have to do is to lift $M^{\prime}$ to a set $M\subset{\mathbb{E}}_{d}$
using the projection $\pi:\mathbb{R}^{d}\to{\mathcal{D}}$ in such a way that
$M=\textsc{flip}(M+t)$, with $\pi(t)=t^{\prime}$, and to show that $M$ is
invariant under the subgroup $K=\pi^{-1}(K^{\prime})$ and satisfies
${\mathcal{Q}}\cap M=\\{(0,e_{1}),(0,e_{2}),\ldots,(0,e_{d})\\}+K$. Then the
corollary follows from the previous one. The details are left to the reader. ∎
Finally, we give the similar result for Christoffel parallelograms. We
consider some parallelogram $P$ whose vertices are in $L$, and edges (which
are in ${\mathbb{E}}^{\prime}_{d}$) within it; these edges must be torally
compatible, in the sense that such an edge hits some edge of the
parallelogram, then it reappears on the opposite edge of the parallelogram.
Such a parallelogram defines a subgroup of finite index $K^{\prime}$ of $L$
(spanned by the edges of the parallelogram) and tiles the whole hyperplane
$D$. We say that $\textsc{flip}(P)=P+t^{\prime}$, for some $t^{\prime}\in L$,
if $P+t^{\prime}$ is the parallelogram obtained by flipping the edges of $P$
incident to zero mod $K$.
###### Corollary 35.
Under the previous hypothesis, $P$ is a Christoffel parallelogram or the
reversal of its edge-complement.
###### Proof.
It suffices to verify that $P$ defines a subset $M^{\prime}$ of
$\pi({\mathbb{E}}_{d})$ satisfying the hypothesis of the previous corollary. ∎
The result is illustrated in Figure 12.
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${\mathcal{H}}_{(3,7,8)}$ | ${\mathcal{H}}_{(15,11,10),18}$
Figure 12: Left: the Christoffel graph ${\mathcal{H}}_{\vec{a}}$ for the
vector ${\vec{a}}=(3,7,8)$. It satisfies the equation $M=\textsc{flip}(M+t)$
for the translation vector $t=e_{3}-e_{2}$. Right: the complement of the
reversal of the Christoffel graph for the vector ${\vec{b}}=(3,7,8)$, i.e.
$-{\mathcal{H}}_{\vec{b}}^{c}$. It corresponds to the Christoffel graph
${\mathcal{H}}_{{\vec{a}},\omega}$ for the vector ${\vec{a}}=(15,11,10)$ and
width $\omega=18$. It satisfies the equation $M=\textsc{flip}(M+t)$ for the
translation vector $t=e_{2}-e_{3}$. They represent the only two possibilities
for a pattern $M$ satisfying $M=\textsc{flip}(M+t)$ when $d=3$ and $K$ is the
subgroup of $\mathbb{Z}^{3}$ given by
$\langle(0,4,1),(-2,0,3),(1,1,1)\rangle$.
We are now ready for the main result of this article which generalizes
Pirillo’s theorem (Theorem 1) to arbitrary dimension: a graph
$M\subseteq{\mathbb{E}}_{d}$ is a translate of its flip if and only if it is a
Christoffel graph of width $\omega$.
###### Theorem 36 ($d$-dimensional Pirillo’s theorem).
Let $K$ be a subgroup of finite index of $\mathbb{Z}^{d}$. Let
$M\subseteq{\mathbb{E}}_{d}$ be a subset of edges invariant for the group of
translations $K$ such that the edges of $M$ incident to zero mod $K$ are
${\mathcal{Q}}\cap M=\\{(0,e_{i})\mid 1\leq i\leq d\\}+K$. There exists
$t\in\mathbb{Z}^{d}$ such that $M=\textsc{flip}(M+t)$ if and only if
$M={\mathcal{H}}_{{\vec{a}},\omega}$ is the Christoffel graph of normal vector
${\vec{a}}$ and width $\omega$.
###### Proof.
Suppose $M=\textsc{flip}(M+t)$ for some $t\in\mathbb{Z}^{d}$. From Lemma 29,
for all $u=(x_{1},x_{2},\cdots,x_{d})\in\mathbb{Z}^{d}$ there exists an
integer $k$ such that $u=kt$ with
$k=\sum(-b_{i}x_{i})\bmod\omega=\sum(a_{i}-\omega)x_{i}\bmod\omega=\sum
a_{i}x_{i}\bmod\omega={\mathcal{F}}_{{\vec{a}},\omega}(u).$
We want to show that $M={\mathcal{H}}_{{\vec{a}},\omega}$. We have that
$(u,u+e_{i})=(0,e_{i})+kt\in M$ if and only if $0\leq k<b_{i}$ if and only if
${\mathcal{F}}_{{\vec{a}},\omega}(u)\in[0,\omega-a_{i}-1]$ if and only if
$(u,u+e_{i})\in{\mathcal{H}}_{{\vec{a}},\omega}$ from Lemma 15.
Reciprocally, suppose ${\mathcal{H}}_{{\vec{a}},\omega}$ is the Christoffel
graph of normal vector ${\vec{a}}$ of width $\omega$. We can show that
${\mathcal{H}}_{{\vec{a}},\omega}+t=\textsc{flip}({\mathcal{H}}_{{\vec{a}},\omega})$
where $t\in\mathbb{Z}^{d}$ is such that
${\mathcal{F}}_{{\vec{a}},\omega}(t)=1$. The proof goes along the same lines
as Proposition 24 using Lemma 15 instead of Lemma 4. ∎
## 7 Appendix: Discrete planes
In this section, we show some results on standard discrete planes. Discrete
planes were introduced in [Rev91] and standard discrete planes were further
studied in [Fra96]. The projection of a standard discrete plane gives a tiling
of ${\mathcal{D}}$ by three kinds of rhombus [BV00] thus yielding a coding of
it by $\mathbb{Z}^{2}$-actions by rotations on the unit circle [ABI02,
ABFJ07]. Our construction of the discretized hyperplane is equivalent, for the
dimension 3, to that in [ABI02]. Our point of view is slightly different from
the classical one; inspired by the 2-dimensional case (discrete lines), we
define a discrete hyperplane by “what the observer sees”: the observer is at
$-\infty$ in the direction $(1,1,\ldots,1)$ and he looks towards the
“transparent” hyperplane all the unit hypercubes which are located on the
other side. This may be modelled mathematically; all the results are
intuitively clear, but require a proof. We prove them, since we could not find
precise references. We recover some known results.
Imagine the $d$-dimensional space filled with unit hypercubes with opaque
faces. Consider a transverse hyperplane generated by its integer points
(formally, of equation $\sum a_{i}x_{i}=0$, $a_{i}>0$ coprime integers). As an
observer, we install in the open half-space $H_{-}$ bounded by the plane.
Then, we remove all the cubes in this half-space, including the cubes
intersecting this half-space; in other words, we keep only the cubes contained
in $H_{+}$. Figure 13 illustrates this construction for $d=2$.
Figure 13: Observation in dimension 2. What the observer sees can be projected
parallel to the vector $(1,1)$ on the line $x+y=0$.
For $d=3$, when we look towards $H_{+}$ parallely to the vector $(1,1,1)$,
then we see something like in Figure 14.
Figure 14: What the observer sees in dimension 3. The surface of cubes was
projected parallel to the vector $(1,1,1)$ on the plane $x+y+z=0$.
Let $s$ be the sum $s=\sum_{i}a_{i}$. We denote
${\vec{a}}=(a_{1},a_{2},\ldots,a_{d})\in\mathbb{Z}^{d}$. The complement of $H$
has two connected components $H_{-}$ and $H_{+}$, where the first is
determined by the inequation $\sum_{i}a_{i}x_{i}<0$. We consider the unit
cubes in $\mathbb{R}^{d}$ and their facets. Such a facet is a subset of
$\mathbb{R}^{d}$ of the form $M+\sum_{j\neq i}[0,1]e_{j}$, for some coordinate
$i\in\\{1,\ldots,d\\}$ and some integral point $M\in\mathbb{Z}^{d}$. Denote by
$\cal C$ the standard unit cube.
Consider the unit hypercubes that are contained in the closed half-space
$H\cup H_{+}$ and their facets; denote by $\cal U_{+}$ the union of all these
facets. Note that a unit cube $M+\cal C$ ($M\in\mathbb{Z}^{d}$) is contained
in $H\cup H_{+}$ if and only if $M\in H\cup H_{+}$ if and only if
$\sum_{j}a_{j}m_{j}\geq 0$.
We say that a point $M$ in $\mathbb{R}^{d}$ is visible if the open half-line
$M+]-\infty,0[(1,1,\ldots,1)$ does not contain any point in $\cal U_{+}$.
Intuitively, this means that, all facets being opaque, that an observer
located at infinity in the direction of the vector $-(1,1,\ldots,1)$ can see
this point $M$, because no point in $\cal U_{+}$ hides this point.
Now, we consider the set of visible points which belong to $\cal U_{+}$. This
we may call the discretized hyperplane associated to $H$. Intuitively, it is
the set of facets that the observer can sees, as is explained in the
introduction.
We characterize now the discretized hyperplane. For this, we denote by $R$ the
following subset of $\mathbb{R}^{d}$: $R=\\{(x_{i})\mid
0\leq\sum_{i}a_{i}x_{i}<s\\}$. Note that $R\subset H\cup H_{+}$.
Denote by ${\mathcal{S}}$ the union of the facets that are contained in $R$.
In other words,
${{\mathcal{S}}}=\bigcup_{M\in\mathbb{Z}^{d},1\leq i\leq d,M+\sum_{j\neq
i}[0,1]e_{j}\subset R}(M+\sum_{j\neq i}[0,1]e_{j}).$
Observe that the condition $M+\sum_{j\neq i}[0,1]e_{j}\subset R$ is equivalent
to: $\sum_{j}a_{j}m_{j}\geq 0$ and $a_{i}m_{i}+\sum_{j\neq
i}a_{j}(m_{j}+1)<s$. Note also that ${\mathcal{S}}\subset\cal U_{+}$.
###### Theorem 37.
The discretized hyperplane is equal to ${\mathcal{S}}$.
Observe that if we project ${\mathcal{S}}$ onto the hyperplane perpendicular
to the vector $(1,1,\ldots,1)$, we obtain exactly what the observer sees.
An example of this, for $d=3$, is given in Figure 14. This observation
motivates the introduction of the graph $I_{\vec{a}}$ in Section 3.2.
We first give an simple characterization of ${\mathcal{S}}$.
###### Proposition 38.
Let $X=(x_{i})\in\mathbb{R}^{d}$. Then $X$ is in ${\mathcal{S}}$ if and only
if the three conditions below hold:
(i) some coordinate of $X$ is an integer;
(ii)$\sum_{i}a_{i}\lfloor x_{i}\rfloor\geq 0$;
(iii) $\sum_{i}a_{i}\lceil x_{i}\rceil<s$.
We recover Proposition 1 of [ABI02].
###### Corollary 39.
Let $X=(x_{i})\in\mathbb{Z}^{d}$. Then $X$ is in ${\mathcal{S}}$ if and only
if $0\leq\sum_{i}a_{i}x_{i}<s$.
###### Proof of the proposition..
Suppose that $X\in{\mathcal{S}}$. Then $X\in M+\sum_{j\neq
i}[0,1]e_{j}\subset{\mathcal{S}}$ and the coordinates $m_{j}$ of $M$ are
integers. Thus, by an observation made previously,
$0\leq\sum_{j}a_{j}m_{j}\leq\sum_{j}a_{j}\lfloor x_{j}\rfloor$, since
$x_{j}=m_{j}+\theta_{j}$, with $0\leq\theta_{j}\leq 1$ and $\theta_{i}=0$.
Moreover, $\lceil x_{i}\rceil=m_{i}$, and $\lceil x_{j}\rceil\leq m_{j}+1$ if
$j\neq i$. Thus, $\sum_{j}a_{j}\lceil x_{j}\rceil\leq a_{i}m_{i}+\sum_{j\neq
i}(m_{j}+1)<s$, by the same observation.
Conversely, suppose that the three conditions of the proposition hold. Without
restricting the generality (by permutation of the coordinates), we may assume
that for some $i\in\\{1,\ldots,d\\}$, one has
$x_{i},\ldots,x_{i}\in\mathbb{Z}$ and $x_{i+1},\ldots,x_{d}\notin\mathbb{Z}$.
Let $0\leq p\leq i$ be maximum subject to the condition $\sum_{j\leq
p}a_{j}(x_{j}-1)+\sum_{j>p}a_{j}\lfloor x_{j}\rfloor\geq 0$ (note that $p$
exists since the inequality is satisfied for $p=0$). Suppose that $p=i$; then
$\sum_{j}a_{j}\lceil x_{j}\rceil=\sum_{j\leq
i}a_{j}x_{j}+\sum_{j>i}a_{j}(\lfloor
x_{j}\rfloor+1)=a_{1}+\cdots+a_{d}+\sum_{j\leq
i}a_{j}(x_{j}-1)+\sum_{j>i}a_{j}\lfloor x_{j}\rfloor\geq a_{1}+\cdots+a_{d}$
(since $p=i$) $=s$; thus we obtain a contradiction with condition (iii).
Thus $p<i$ and $p+1\leq i$. We have by maximality the inequality $\sum_{j\leq
p+1}a_{j}(x_{j}-1)+\sum_{j>p+1}a_{j}\lfloor x_{j}\rfloor<0$. Let
$M=(m_{j})=(x_{1}-1,\ldots,x_{p}-1,\lfloor x_{p+1}\rfloor,\ldots,\lfloor
x_{d}\rfloor\in\mathbb{Z}^{d}$. We have $\sum_{j}a_{j}m_{j}\geq 0$ (by
definition of $p$) and $a_{p+1}m_{p+1}+\sum_{j\neq
p+1}a_{j}(m_{j}+1)=\sum_{j\leq
p}(a_{j}(x_{j}-1)+a_{j})+a_{p+1}(x_{p+1}-1)+a_{p+1}+\sum_{j>p+1}(a_{j}\lfloor
x_{j}\rfloor+a_{j})=s+\sum_{j\leq p+1}a_{j}(x_{j}-1)+\sum_{j>p+1}a_{j}\lfloor
x_{j}\rfloor<s$, by the previous inequality. Thus $M+\sum_{j\neq
i}[0,1]e_{j}\subset{\mathcal{S}}$, by the observation made above. Moreover
$X\in M+\sum_{j\neq i}[0,1]e_{j}$ since $p+1\leq i$. ∎
###### Corollary 40.
For each point $X$ in $\mathbb{R}^{d}$, there is a unique point $Y$ in
${\mathcal{S}}$ such that $XY$ is parallel to the vector $(1,1,\ldots,1)$.
Denote by $f$ the function such that $Y=f(X)$ with the notations of the
corollary. This function is a kind of projection onto ${\mathcal{S}}$,
parallely to the vector $(1,1,\ldots,)$. Denote also by $t(X)$ the real-valued
function defined by $X=f(X)+t(X)(1,1,\ldots,1)$, and by $t(X)=0$ if and only
if $X\in{\mathcal{S}}$.
###### Proof.
We prove first unicity. By contradiction: we have $Y,Z\in{\mathcal{S}}$ and
$Z=Y+t(1,1,\ldots,1)$ with $t>0$. Then $z_{i}=y_{i}+t$. Thus $\lceil
z_{i}\rceil\geq\lfloor y_{i}\rfloor+1$. Hence $\sum_{i}a_{i}\lceil
z_{i}\rceil\geq s+\sum_{i}a_{i}\lfloor y_{i}\rfloor$. Since by the
proposition, applied to $Y$, the last sum is $\geq 0$, we obtain
$\sum_{i}a_{i}\lceil z_{i}\rceil\geq s$, which contradicts the proposition,
applied to $Z$.
We prove now the existence of $Y$. We may assume that
$L(X)=\sum_{i}a_{i}\lfloor x_{i}\rfloor\geq 0$, by adding to $X$ some positive
multiple of $(1,1,\ldots,1)$ if necessary. We prove existence of $Y$ by
induction on the sum $U(X)=\sum_{i}a_{i}\lceil x_{i}\rceil$.
Let $\epsilon=\min_{i}(x_{i}-\lfloor x_{i}\rfloor)$. Observe that if we
replace $X$ by $X-\epsilon(1,1,\ldots,1)$, then $L(X)$ does not change, $U(X)$
does not increase and moreover some $x_{i}$ is now an integer.
If $U(X)$ is $<s$, this observation implies the existence of $Y$.
Suppose now that $U(X)\geq s$. By the observation, we may assume that at least
one of the $x_{i}$ is an integer. Without restricting the generality, we may
also assume that $x_{1},\ldots,x_{i}\in\mathbb{Z}$ and that
$x_{i+1},\ldots,x_{d}\notin\mathbb{Z}$, with $i\geq 1$.
If $i=d$, then the $x_{j}$ are all integers, $L(X)=U(X)$, we replace $X$ by
$X-(1,1,\ldots,1)$ and we conclude by induction, since $L(X)$ is replaced by
$L(X)-s$.
Suppose now that $i<d$. Let $\epsilon=\min_{j>i}(x_{j}-\lfloor x_{j}\rfloor)$;
then $\epsilon>0$. We have $s\leq\sum_{j}a_{j}\lceil x_{j}\rceil=\sum_{j\leq
i}a_{j}x_{j}+\sum_{j>i}a_{j}(\lfloor
x_{j}\rfloor+1)=L(X)+a_{i+1}+\ldots+a_{d}$, hence $L(X)\geq
a_{1}+\cdots+a_{i}$. Note that $\sum_{j}a_{j}(\lfloor
x_{j}-\epsilon\rfloor)=\sum_{j\leq i}a_{j}(x_{j}-1)+\sum_{j>i}a_{j}\lfloor
x_{j}\rfloor=L(X)-a_{1}-\cdots-a_{i}\geq 0$. We replace $X$ by
$X-\epsilon(1,1,\ldots,1)$, and we may conclude by induction, since $U(X)$
strictly decreases and since $L(X)$ remains $\geq 0$. ∎
###### Proof.
(of the theorem) Let $X$ be a point on the discretized hyperplane associated
to $H$. Suppose that $t(X)>0$. Then $X=f(X)+t(X)(1,1,\ldots,1)$ so that $X$ is
hidden by $f(X)$: formally, $f(X)$ is on the open half-line
$X+]-\infty,0[(1,1,\ldots,1)$ and since $f(X)$ is in ${\mathcal{S}}$, it is a
point in $\cal U_{+}$. We conclude that we must have $t(X)\leq 0$. Suppose
that $t(X)<0$. We know that $X$ is in $\cal U_{+}$, so that $X$ belongs to a
hypercube $M+\cal C$ with $\sum_{j}a_{j}m_{j}\geq 0$, and therefore $x_{j}\geq
m_{j}$. Let $Y=f(X)$. Then $X=Y+t(X)(1,1,\ldots,1)$ so that $y_{j}>x_{j}\geq
m_{j}$ which implies $\sum_{j}a_{j}\lceil
y_{j}\rceil\geq\sum_{j}a_{j}(m_{j}+1)\geq s$, a contradiction with Proposition
38. Thus $t(X)=0$ and $X\in{\mathcal{S}}$.
Conversely suppose that $X\in{\mathcal{S}}$. Suppose that $X$ is not on the
discretized hyperplane associated to $H$. This implies that there is some
point $Y\in\cal U_{+}$ on the open half-line $X+]-\infty,0[(1,1,\ldots,1)$. We
have $Y\in M+\cal C$ with $\sum_{j}a_{j}m_{j}\geq 0$. Thus $x_{j}>y_{j}\geq
m_{j}$ which implies that $\sum_{j}a_{j}\lceil
x_{j}\rceil\geq\sum_{j}a_{j}(m_{j}+1)\geq s$, a contradiction with Proposition
38. ∎
###### Corollary 41.
Let $d\geq 2$. Let $M\in{{\mathcal{S}}}\cap\mathbb{Z}^{d}$. Let
$i=1,2,\ldots,d$ and $N=M+e_{i}$.
(i) $N\in{\mathcal{S}}$ if and only if $\sum_{j}a_{j}n_{j}<s$; in this case,
the segment $M+[0,1]e_{i}$ is contained in ${\mathcal{S}}$.
(ii) If $N\notin{\mathcal{S}}$, then the only point in
$(M+[0,1]e_{i})\cap{{\mathcal{S}}}$ is $M$.
###### Proof.
The fact that $N\in{\mathcal{S}}$ if and only if $\sum_{j}a_{j}n_{j}<s$ is a
consequence of the proposition.
Suppose that $N\in{\mathcal{S}}$ and let $X$ be on the segment $M+[0,1]e_{i}$.
Then $0\leq\sum_{j}a_{j}m_{j}\leq\sum_{j}a_{j}\lfloor x_{j}\rfloor$ and
$\sum_{j}a_{j}\lceil x_{j}\rceil\leq\sum_{j}a_{j}n_{j}<s$. Thus the corollary
follows from the proposition.
Suppose no that $N\notin{\mathcal{S}}$ and let $X$ be on this segment. Since
$0\leq\sum_{j}a_{j}m_{j}$, we have also $0\leq\sum_{j}a_{j}n_{j}$. Since
$N\notin{\mathcal{S}}$, we must have $\sum_{j}a_{j}n_{j}\geq s$. Moreover, if
$X\neq M$, we have $\lceil x_{j}\rceil=n_{j}$, so that $\sum_{j}a_{j}\lceil
x_{j}\rceil\geq s$ and $X\notin{\mathcal{S}}$. ∎
The next result, which is not needed in this article, is of independent
interest, and intuitively clear (but it requires a proof).
###### Proposition 42.
The function $f:X\mapsto Y$, with the notations of Corollary 40, is
continuous. The open set $\mathbb{R}^{d}\setminus{\mathcal{S}}$ has two
connected components.
###### Lemma 43.
Let ${\mathcal{S}}$ be a closed subset of $\mathbb{R}^{d}$such that for each
$X$ in $\mathbb{R}^{d}$, there is a unique $Y$ in ${\mathcal{S}}$ such that
$XY$ is parallel to $(1,1,\ldots,1)$. If the mapping $X\mapsto Y$ is bounded,
then it is continuous.
###### Proof.
Recall that a bounded sequence in $\mathbb{R}^{d}$, converges if for any two
convergent subsequences, they have the same limit. Let $(X_{n})$ be a sequence
in $\mathbb{R}^{d}$, with limit $l$. It is enough to show that $(f(X_{n}))$
converges; note that this sequence is bounded. Consider two subsequences of
$(X_{n})$ such that their images under $f$ have limits, $l_{1}$ and $l_{2}$
say. Since ${\mathcal{S}}$ is closed, $l_{1},l_{2}\in{\mathcal{S}}$. Let
$\epsilon>0$. For $n$ large enough, $\mid X_{n}-l\mid<\epsilon$; hence
$f(X_{n})$ is in the open cylinder of diameter $\epsilon$ and with axis the
line $l+(1,1,\ldots,1)$. This implies that $l_{1},l_{2}$ are in this cylinder
and consequently, $\epsilon$ being arbitrary, $l_{1},l_{2}$ are on the
previous line. By unicity, $l_{1}=l_{2}$ ($=f(l)$). We conclude using the
remark at the beginning of the proof. ∎
###### Proof.
(of the proposition) The mapping $f$ is continuous: by the lemma, it is enough
to show that ${\mathcal{S}}$ is closed and that the mapping is bounded. Since
each convergent sequence is contained in some compact set, it is enough to
show that for each compact set $K$, $K\cap{\mathcal{S}}$ is closed; but this
is clear, since the latter set is the union of finitely many $K\cap F$, $F$
facet of unit hypercube. The mapping is bounded since its image is between the
two hyperplanes of equations $\sum_{i}a_{i}x_{i}=0$ and
$\sum_{i}a_{i}x_{i}=s$, so that the image of each bounded set is contained in
a cylinder of axis parallel to $(1,1,\ldots)$ and limited by these two
hyperplanes.
Now, we show that the set $\mathbb{R}^{d}\setminus{\mathcal{S}}$ has two
connected components. Note that for each point $X$, one has
$X=f(X)+t(X)(1,1,\ldots,1)$ for some continuous real-valued function $t$.
Since $f(1,1,\ldots,1)=(0,0,\ldots,0)=f(-1,-1,\ldots,-1)$, one has
$t(1,1,\ldots,1)=1$ and $t(-1,-1,\ldots,-1)=-1$. Moreover $t(X)=0$ if and only
if $X\in{\mathcal{S}}$. Thus $t(\mathbb{R}^{d}\setminus{\mathcal{S}})$ is not
connected and neither is $\mathbb{R}^{d}\setminus{\mathcal{S}}$.
Now, if $t(X)>0$, one may connect $X$ by a piece of the line
$X+\mathbb{R}(1,1,\ldots,1)$ to a point of the half-space
$\sum_{i}a_{i}x_{i}>0$ and this implies that the set of points $X$ with
$t(X)>0$ is connected. Similarly, the set of points with $t(X)<0$ is
connected, and $\mathbb{R}^{d})\setminus{\mathcal{S}}$ has therefore two
connected components. ∎
We recover Proposition 2 of [ABI02] and Proposition 4 of [ABFJ07].
###### Corollary 44.
The restriction of $f$ to ${\mathcal{D}}$ is a homeomorphism of
${\mathcal{D}}$ onto ${\mathcal{S}}$.
###### Proof.
Indeed, the inverse mapping is the projection onto the hyperplane
${\mathcal{D}}$ parallely to the vector $(1,1,\ldots,1)$. ∎
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|
arxiv-papers
| 2014-04-15T18:55:27 |
2024-09-04T02:50:01.219396
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S\\'ebastien Labb\\'e and Christophe Reutenauer",
"submitter": "S\\'ebastien Labb\\'e",
"url": "https://arxiv.org/abs/1404.4021"
}
|
1404.4101
|
# Frequency Shifts in NIST Cs Primary Frequency Standards
Due To Transverse RF Field Gradients
Neil Ashby [email protected] National Institute of Standards and
Technology Stephan Barlow [email protected] National Institute of
Standards and Technology Thomas Heavner [email protected] National
Institute of Standards and Technology Steven Jefferts
[email protected] National Institute of Standards and Technology
###### Abstract
A single-particle Green’s function (propagator) is introduced to study the
deflection of laser-cooled Cesium atoms in an atomic fountain due to RF field
gradients in the Ramsey TE011 cavity. The deflection results in a state-
dependent loss of atoms at apertures in the physics package, resulting in a
frequency bias. A model accounting only for motion in one dimension transverse
to the symmetry axis of the fountain is discussed in detail and then
generalized to two transverse dimensions. Results for fractional frequency
shifts due to transverse field gradients are computed for NIST F-1 and F-2
Cesium fountains. The shifts are found to be negligible except in cases of
higher RF power applied to the cavities.
###### pacs:
06.20fb,31.30Gs,37.30+1
## I 1\. Introduction
Frequency shifts in atomic clocks are of fundamental importance in the
accuracy determination of the SI second, which is presently realized using
laser-cooled Cs fountains operated by many standards laboratories around the
worldWynands and Weyers (2005). The potential systematic bias due to momentum-
changing interactions between the microwave interrogation field and the atoms
undergoing Ramsey excitation have long been a source of concern,
investigation, and conjecture. Bordè and Wolf estimated the fractional
frequency shift due to microwave recoil in atomic standards to be on the order
of $\frac{\delta f}{f}\approx 10^{-16}$ Wolf and Bordé (2004).
Recently, GibbleLi _et al._ (2011)Gibble (2006) published a theory
reinvestigating the microwave recoil shift along lines originally used by
CookCook (1978)Cook (1987). Reference Gibble (2006) uses Cook’s methods in the
optical domain to quantify the state-dependent deflection in the atomic
trajectories due to gradients in the microwave field and the resulting
frequency bias. This work contains several unphysical results and predicts a
frequency shift of order $10^{-16}$Gibble (2006); several Primary Frequency
Standard Groups (NPL, PTB, SYRTE) are correcting for this bias. In light of
problematic results in Gibble (2006) (e.g., the nonvanishing nature of the
shift in the absence of microwave excitation), here we present a new
theoretical treatment that extends the work by Cook or Gibble.
The operation of the NIST fountains has been described in Heavner _et al._
(2005). Figure 1 shows a simplified configuration of the NIST-F1 Cesium
fountain. A cloud of atoms is collected and laser-cooled to $\approx 0.5\mu$K
in optical molasses, then launched upwards through a state-selection region
that results in a sample of atoms in the $\left<3,0|\right.$ state. The atoms
pass into a Ramsey cavity (a cylindrical TE011 cavity), where they are subject
to a $\pi/2$ pulse of resonant RF radiation of frequency $f_{0}=9.192631770$
GHz that puts the atoms into a superposition of the two clock states
$\left<4,0|\right.$ and $\left<3,0|\right.$. The atoms then coast upwards into
a drift region and fall back down through the cavity where they are subjected
to a second $\pi/2$ pulse. This causes some of the atoms to make transitions
to the upper hyperfine state. The atoms then fall through a detection region
which measures the numbers of atoms in each state; from the detected atom
numbers in the two hyperfine states a relative transition probability is
measured.
Figure 1: Schematic diagram of the main components of NIST F-2. See text for
discussion.
Here we use a full wave-packet description of the atoms undergoing Ramsey
excitation along with a full time-dependent solution to the Schrödinger
equation that propagates the wave-packets through the two microwave
interactions up through the detection step. We find frequency shifts of order
$10^{-17}$ in our atomic fountainsHeavner _et al._ (2005) with our theory.
(Because the microwave cavity is essentially identical in the two fountains at
NIST and the geometry of the standards is similar, the results given here are
representative of those for both fountains.)
The organization of this paper is as follows. In Sect. 2 we introduce the
Schrödinger equation for a two-state model of the hyperfine states of 133Cs,
coupled to the axial component of the RF magnetic field in a cylindrically
symmetric cavity. The Hamiltonian of this problem includes kinetic energy,
internal energy of the hyperfine states, and magnetic interaction energy of
the spins with the magnetic field. The RF field is assumed to be almost
exactly resonant, a slight detuning is modeled by allowing the phase of the RF
field to be different when the atoms pass through the cavity a second time. In
the presence of transverse microwave magnetic field gradients, which are
dictated by vanishing of the transverse magnetic field at the cylindrical
cavity boundaries, the transverse motion of the atoms is slightly affected.
This is described by the introduction of a propagator that exactly solves the
Schrödinger equations after they are decoupled by a series of exact
transformations. We consider mainly the case of a $\pi/2$ pulse applied in
each cavity but discuss up to $7\pi/2$ pulses in a later section. The
propagator accounts for the transverse recoil of the atoms due to the
interaction of the spins with the field gradients. Time development of the
phases of the spinor components are discussed in Sect. 3. Sections 4, 5, and 6
discuss construction of atom wavepackets and their quantum mechanical
spreading during propagation through the apparatus. In Section 7 a prototype
one-dimension model of passage of a ball of laser-cooled atoms at temperature
$T\approx 1\mu$K, with spatial extent $\sigma_{n}$ in the transverse
dimension, is developed in detail and solutions of the Schrödinger equation
for balls entering the detector are presented. Sect. 8 discusses the method of
detection and Sect. 9 describes results for the one-dimensional case. Sect. 10
generalizes the model to the full two-dimensional case of transverse motion.
The results for this case are described in Sect. 11.
## II 2\. Equations of Motion
In this section we derive decoupled equations of motion for the atoms in the
fountain and introduce a propagator (Green’s function) that solves the spatial
equations of motion for the atoms in a transverse magnetic field gradient. The
$\left<F,m_{F}|\right.=\left<4,0|\right.$ and
$\left<F,m_{F}|\right.=\left<3,0|\right.$ hyperfine states of 133Cs, of
intrinsic energies $\hbar\omega_{a}$ and $\hbar\omega_{b}$, respectively, are
coupled only by the $z-$component of an applied microwave magnetic
field.Scully _et al._ (1989) We consider a two-state model in which the
energy separation of the hyperfine states is
$\hbar\Delta=\hbar(\omega_{a}-\omega_{b})$ and the zero of energy is halfway
in between $\omega_{a}$ and $\omega_{b}$. $\psi_{a}$ and $\psi_{b}$ denote the
wavefunctions of atoms in the upper and lower hyperfine states, respectively.
In the presence of an applied RF field the equations of motion are:
$\displaystyle i\hbar\frac{\partial\psi_{a}}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi_{a}+\frac{\hbar\Delta}{2}\psi_{a}+\mu_{{}_{B}}gB_{z}\psi_{b};$
(1) $\displaystyle i\hbar\frac{\partial\psi_{b}}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi_{b}-\frac{\hbar\Delta}{2}\psi_{b}+\mu_{{}_{B}}gB_{z}\psi_{a}.$
(2)
For a TE011 mode in a cylindrical cavity the applied high-frequency RF field
may be represented by
$\displaystyle\mu_{B}gB_{z}=\hbar\pi b\cos(\omega t+\theta_{1})\hbox
to72.26999pt{}$ $\displaystyle=\hbar\pi b_{0}\cos(\omega
t+\theta_{1})J_{0}(x_{1}r/d)\sin(Kz),$ (3)
where $K=(\omega^{2}/c^{2}-x_{1}^{2}/d^{2})^{(1/2)}$, $b_{0}$ is a
conveniently chosen measure of the amplitude, $x_{1}=3.83171...$ is the first
zero of the ordinary Bessel function of order 1, $d$ is the cavity radius, and
$c$ is the speed of light. The applied field is assumed to be almost exactly
on resonance: $\omega\approx\Delta$.
In NIST F-1 and F-2, the symmetry axis of the cylindrical cavity is along the
$z$-direction. After entering the cavity at the reference level $z=0$, atoms
are subject to a half-sine wave pulse of RF energy (the $\sin(Kz)$ in Eq.
(II)), coast upwards to height $h$, then fall back down through the cavity
where another pulse is applied, then fall into a detector. We allow the phases
of the RF fields during cavity passage to be different: $\theta_{1}$ and
$\theta_{2}$, respectively. Time of passage through the cavities is denoted by
$\tau$, time in the drift region by $T$, and time $T_{d}$ to fall from the
bottom of the cavity into the detector.
The cavity has entry and exit apertures of radius $r_{a}<d/2$ and an atom can
enter the cavity at some off-axis position ${x_{c},y_{c}}$ where we think of
this position as the center of a gaussian wave packet whose spread is small
compared to the scale of distance over which the microwave magnetic field
changes in the transverse direction. A one-dimensional gaussian wave packet
initially centered at $x_{c}$ can be constructed by superposing plane waves:
$\displaystyle\phi(x,t)=\sqrt{\frac{\sigma}{2\pi^{3/2}}}\int_{-\infty}^{\infty}dke^{i(k(x-x_{c})-\frac{\hbar
k^{2}t}{2m}-\sigma^{2}(k-k_{0})^{2}/2}$
$\displaystyle=\frac{1}{\pi^{1/4}}\frac{e^{i(k_{0}(x-x_{c})-\hbar
k_{0}^{2}t/(2m))}}{\sqrt{\sigma+i\hbar t/(\sigma m)}}e^{-\frac{(x-x_{c}-\hbar
k_{0}t/m)^{2}}{2(\sigma^{2}+i\hbar t/m)}}.\hbox to14.45377pt{}$ (4)
The packet (II) is normalized to unity and satisfies the Schrödinger equation
for a free particle,
$i\hbar\frac{\partial\phi(x,t)}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\phi(x,t).$ (5)
Quantum mechanical packet spreading occurs due to the terms proportional to
$\hbar t/m$ in the denominators of (II).
It will be seen that all of the integrations performed as we follow the
trajectory of an atom involve gaussian exponentials; such integrals can be
performed in any order and the packets are normalized by means of the
weighting functions in (II) so it will be convenient to simply assume that the
packet entering the first cavity can be represented by a plane wave at $t=0$:
$e^{i(k_{x}(x-x_{c})+k_{y}(y-y_{c}))}.$ (6)
At an off-axis position ${x_{c},y_{c}}$ there is a transverse field gradient
arising from the Bessel function in (II). Expanding in a Taylor series about
such a position,
$\displaystyle\hbar\pi b_{0}J_{0}(x_{1}r/d)=\hbar\pi
b_{0}J_{0}(x_{1}\sqrt{x_{c}^{2}+y_{c}^{2}}/d)$
$\displaystyle+2\gamma_{x}(x-x_{c})+2\gamma_{y}(y-y_{c}),$ (7)
where for example,
$\gamma_{x}=\frac{\hbar\pi b_{0}}{2}\frac{\partial J_{0}(x_{1}r/d)}{\partial
x}\Bigr{|}_{x_{c},y_{c}}$ (8)
These gradients exert transverse forces on the spins and result in transverse
displacements which are, however, quite tiny as will be seen. That the
gradients occur as a sum of terms in $x-x_{c}$ and $y-y_{c}$ in the equations
of motion, makes it possible to solve (1) and (2) by separating variables.
The time dependence in the equations of motion is simplified by passing to the
“rotating phase approximation.” We introduce a transformation of functions by
means of
$\left[\begin{array}[]{c}\psi_{a}\\\
\psi_{b}\end{array}\right]=\left[\begin{array}[]{c}e^{-i(\omega
t/2+\theta_{1}/2)}D_{a}\\\ e^{i(\omega
t/2+\theta_{1}/2)}D_{b}\end{array}\right].$ (9)
Then introducing the exponential form for the $\cos(\omega t+\theta_{1})$
appearing in (II) and neglecting terms that oscillate with twice the hyperfine
resonance frequency the equations of motion become
$\displaystyle i\hbar\frac{\partial D_{a}}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}D_{a}+\frac{\hbar\pi
b}{2}D_{b}+\frac{\hbar(\Delta-\omega)}{2}D_{a};$ $\displaystyle
i\hbar\frac{\partial D_{b}}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}D_{b}-\frac{\hbar\pi
b}{2}D_{a}+\frac{\hbar(\Delta-\omega)}{2}D_{b}.$ (10)
We shall neglect the detuning terms in Eqs. (II) above. In the case where no
spatial dependence is considered, it can be shownShirley _et al._ (2001) that
these terms cause a very small change in the width of the central Ramsey
fringe; detuning does not itself cause a frequency shift. We shall however
keep the exponential phase terms in (9) as they play an important role in the
discussion of detection. The equations can then be decoupled by introducing
the following linear combinations of wavefunctions:
$f_{+}=\frac{1}{\sqrt{2}}(D_{a}+D_{b});\hbox
to14.45377pt{}f_{-}=\frac{1}{\sqrt{2}}(D_{a}-D_{b}).$ (11)
The equations of motion become:
$i\hbar\frac{\partial f_{\pm}}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}f_{\pm}\pm\frac{\hbar\pi b}{2}f_{\pm}.$ (12)
Such linear combinations were introduced by CookCook (1978) to treat motion of
electric dipoles under the influence of a laser beam; the $f_{+}$ and $f_{-}$
packets are displaced in opposite directions, but consist at all times of
equal contributions from the $a$ and $b$ spinor states, but with different
phases. The solution for $f_{-}$ can be obtained from the $f_{+}$ solution by
changing the sign of $b$, so we shall consider only the $+$ sign and for
convenience will drop the subscript.
Launched atoms entering the cavity aperture arrive with a thermal distribution
of velocities and hence with a distribution of arrival times; also the atom
cloud may be clipped by the aperture edges. Passage of the cloud through the
cavity may thus entail a distribution of positions within the aperture as well
as a distribution of times $\tau$ required to pass through the cavity. We
shall consider a particular packet centered at $(x_{c},y_{c})$ with a velocity
$\hbar{\bf k}/m$ on entry to the cavity. Transition probabilities for such an
atom will be averaged over position and velocity at the end of the
calculation. As the packet traverses the cavity, it samples the half-sine wave
dependence of the resonant RF field; this field in effect becomes a slowly
varying time-dependent field because the packet will be well-localized
compared to the cavity length. The packet falls back through the cavity some
time later; we assume that the second time of passage is the same as that
during the first passage. We therefore seek a reasonable approximation that
allows description of the motion in the $z-$direction to be separated off;
effects of field gradients in the $z-$direction cancel out and are not of
interest in the present paper. Therefore we assume
$f(x,y,z,t)=g(x,y,t)\phi(z,t);$ (13)
where $\phi(z,t)$ is of the general form (II). Then (12) becomes
$\phi(z,t)i\hbar\frac{\partial g}{\partial
t}=-\frac{\hbar^{2}}{2m}\phi(z,t)\bigg{(}\frac{\partial^{2}g}{\partial
x^{2}}+\frac{\partial^{2}g}{\partial y^{2}}\bigg{)}+\frac{\hbar\pi
b}{2}\phi(z,t)g.$ (14)
If we multiply by $\phi(z,t)^{*}$ the $z-$dependence can be simplified since
integrating over all $z$,
$\displaystyle\int|\phi(z,t)|^{2}\sin(Kz)\,dz=\sin\bigg{(}\frac{K\hbar
k_{z0}t}{m}\bigg{)}$ $\displaystyle\times
e^{-\frac{K^{2}\hbar^{2}k_{z0}^{2}t^{2}}{m^{2}}\bigg{(}\sigma^{2}+\hbar^{2}t^{2}/(\sigma^{2}m^{2})\bigg{)}}.$
(15)
The quantity appearing in the exponent in (II) is extremely small during
cavity passage and cannot affect the resonant frequency, so we shall neglect
it and retain the time-dependent sine function. Eq. (II) then becomes
$\displaystyle i\hbar\frac{\partial g}{\partial
t}=-\frac{\hbar^{2}}{2m}\bigg{(}\frac{\partial^{2}g}{\partial
x^{2}}+\frac{\partial^{2}g}{\partial y^{2}}\bigg{)}\hbox to36.135pt{}$
$\displaystyle+\sin(\kappa t)\bigg{(}\frac{\hbar\pi
b_{1}}{2}+\gamma_{x}(x-x_{c})+\gamma_{y}(y-y_{c})\bigg{)}g,$ (16)
where $b_{1}=(\hbar\pi b_{0}/2)J_{0}(x_{1}\sqrt{x_{c}^{2}+y_{c}^{2}}/d)$,
$\gamma_{x}$ is given in Eq. (8) and where $\kappa=K\hbar k_{0}/m$. The first
time-dependent term in (II) term can be eliminated by lettingShirley _et al._
(2001):
$g(x,y,t)=e^{-ia(t)}h(x,y,t).$ (17)
Then if
$a(t)=\frac{\pi b_{1}}{2}\int_{0}^{t}\sin{\kappa t^{\prime}}dt^{\prime},$ (18)
Eq. (II) reduces to
$\displaystyle i\hbar\frac{\partial h}{\partial
t}=-\frac{\hbar^{2}}{2m}\bigg{(}\frac{\partial^{2}h}{\partial
x^{2}}+\frac{\partial^{2}h}{\partial y^{2}}\bigg{)}\hbox to36.135pt{}$
$\displaystyle+\sin(\kappa
t)\bigg{(}\gamma_{x}(x-x_{c})+\gamma_{y}(y-y_{c})\bigg{)}h.$ (19)
The effective Hamiltonian on the right of (II) is a sum of terms that permit a
solution by separation of variables into a product of factors of similar form.
If we let $h(x,y,t)=\alpha(x,t)\beta(y,t)$, then a solution of (II) is found
if $\alpha$ satisfies
$i\hbar\frac{\partial\alpha}{\partial
t}=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}\alpha}{\partial
x^{2}}+\gamma_{x}(x-x_{c})\sin(\kappa t)\alpha,$ (20)
with an equation of motion of similar form for $\beta(y,t)$. (There will be
another set of solutions with opposite signs for $a(t)$ and $\gamma_{x}$,
corresponding to selecting the “-” option in (13).) We shall drop the
subscript on $\gamma_{x}$ as long as we are discussing a one-dimensional
model.
Separation of variables usually involves a “separation constant;” in the
present case this quantity becomes a function of time. However its effect can
be shown to cancel out of the product $\alpha(x,t)\beta(y,t)$.
At the boundary $z=0$, we take $t=0$ and require that the wavefunction be a
plane wave. Then (20) can be solved with the propagator (Green’s function)
$G_{\gamma}(x,t;x^{\prime},0)=\sqrt{\frac{m}{2\pi i\hbar t}}e^{iS/\hbar},$
(21)
where
$\displaystyle S=\frac{m}{2t}(x-x^{\prime})^{2}-\hbox to158.99377pt{}$
$\displaystyle\frac{\gamma(x-x_{c})}{\kappa}\bigg{(}-\cos(\kappa
t)+\frac{\sin(\kappa t)}{\kappa t}\bigg{)}\hbox to97.56493pt{}$ (22)
$\displaystyle-\frac{\gamma(x^{\prime}-x_{c})}{\kappa}\bigg{(}1-\frac{\sin(\kappa
t)}{\kappa t}\bigg{)}\hbox to79.49744pt{}$
$\displaystyle-\gamma^{2}\bigg{(}\frac{-2+2\kappa^{2}t^{2}+2\cos(2\kappa
t)+\kappa t\sin(2\kappa t)}{8\kappa^{4}mt}\bigg{)}.$
It is easily verified by straightforward calculation that,
$\displaystyle i\hbar\frac{\partial G_{\gamma}(x,t;x^{\prime},0)}{\partial
t}+\frac{\hbar^{2}}{2m}\frac{\partial^{2}G_{\gamma}(x,t;x^{\prime},0)}{\partial
x^{2}}\hbox to65.04256pt{}$ (23)
$\displaystyle=i\hbar\delta(x-x^{\prime})\delta(t)+\gamma(x-x_{c})\sin(\kappa
t)G_{\gamma}(x,t;x^{\prime},0).$
For a plane wave of the form $\psi(x,0)=e^{ik(x-x_{c})}$ at $t=0$ entering the
cavity, the solution of the Schrödinger equation in the cavity at time $t$ can
be obtained from the propagator by integrating over the variable $x^{\prime}$
at the initial time:
$\displaystyle\psi_{\gamma}(x,t)=\int_{-\infty}^{\infty}dx^{\prime}G(x,t;x^{\prime},0)e^{ik(x^{\prime}-x_{c})}\hbox
to54.2025pt{}$ $\displaystyle={\rm Exp}\bigg{(}-\frac{i\hbar
k^{2}t}{2m}+i\big{(}k-\frac{\gamma}{\hbar\kappa}(1-\cos(\kappa
t)\big{)})(x-x_{c})$ $\displaystyle+\frac{ik\gamma t}{\kappa
m}\big{(}1-\frac{\sin(\kappa t)}{\kappa t}\big{)}\hbox to93.95122pt{}$ (24)
$\displaystyle-\frac{i\gamma^{2}}{8\hbar\kappa^{3}m}\big{(}6\kappa
t-8\sin(\kappa t)+\sin(2\kappa t)\big{)}\bigg{)};$
a normalization constant would not be changed. Similar propagators have been
used by Scully, Schwinger, and Englert to describe the Stern-Gerlach effect in
a static magnetic field,Scully _et al._ (1989),Schwinger _et al._
(1988),Englert _et al._ (1988). The propagator introduced in Eq. (21) is one
of a class of time dependent propagators that can be constructed by path-
integral methodsShulman (1981). A propagator for an electron in a static
electric field gradient was first constructed by KennardKennard (1927).
In the present application, the transverse velocities of atoms are small
compared with the launch velocity. The launch velocity determines the total
amount of time spent in the cavity by an atom. The atoms are narrowly
distributed about the launch velocity in the $z$ direction, so the values of
the most likely $t$ that occurs in Eq. (II) will be narrowly distributed about
a value $\tau$ determined by the on-axis Ramsey pulse, $v_{0}\tau=L$ where $L$
is the cavity length and $v_{0}$ is the central velocity at the entry
aperture. For a complete half-sine wave pulse on axis, $\kappa\tau=\pi$ and
the exponentials in Eq. (II), which are independent of $x$ and $x_{c}$, may be
simplified to give:
$\displaystyle\psi_{\gamma}(x,\tau)={\rm Exp}\bigg{(}-\frac{i\hbar
k^{2}\tau}{2m}+i\big{(}k-\frac{2\gamma\tau}{\hbar\pi}\big{)}(x-x_{c})$
$\displaystyle+\frac{ik\gamma\tau^{2}}{m\pi}-\frac{i3\gamma^{2}\tau^{3}}{4\hbar\pi^{2}m}\big{)}\bigg{)}.\hbox
to36.135pt{}$ (25)
Thus the propagator takes a plane wave into another plane wave at the entry
into the drift region, with another phase determined by the kinetic energy of
the particle, the wavenumber, and the field gradient, without changing the
plane wave normalization. The integrations over spatial variables can be
performed by completing the squares in the exponents. For example, to
illustrate that order of integration is immaterial, a packet such as (II) is
propagated through the cavity by calculating
$\displaystyle\int
dx^{\prime}G_{\gamma}(x,\tau;x^{\prime},0)\phi(x^{\prime}-x_{c},0)\hbox
to54.2025pt{}$
$\displaystyle=\frac{1}{\pi^{1/4}}\frac{e^{i(k_{0}-2\gamma\tau/(\hbar\pi))(x-x_{c})-i\hbar
k_{0}^{2}\tau/(2m)}}{\sqrt{\sigma+i\hbar t/(\sigma m)}}\hbox to18.06749pt{}$
$\displaystyle\times e^{-{(x-x_{c}-\hbar
k_{0}\tau/m+\gamma\tau^{2}/(m\pi))^{2}}/{(2(\sigma^{2}+i\hbar\tau/m))}}$ (26)
On the other hand if the packet is constructed at the end of the cavity using
Eq. (II), precisely the same result is obtained. This justifies extending the
range of integration over $x^{\prime}$ to infinity, since a well-localized
packet is small in size relative to the cavity aperture.
After exiting from the cavity, the wavefunction can be propagated to the end
of the drift region with a free-particle propagator obtained from Eqs. (21-II)
by setting $\gamma=0$. Thus, without constructing packets,
$\displaystyle\psi(x,\tau+T)|_{\gamma}=\int
dx^{\prime}G_{\gamma=0}(x,T;x^{\prime},0)\psi_{\gamma}(x^{\prime},\tau)$
$\displaystyle=e^{-\frac{i\hbar
k^{2}(\tau+T)}{2m}+i(k-\frac{2\gamma\tau}{\hbar\pi})(x-x_{c})}\hbox
to50.58878pt{}$ $\displaystyle\times
e^{+\frac{ik\gamma\tau(\tau+2T)}{m\pi}-\frac{i\gamma^{2}\tau^{2}(3\tau+8T)}{4\hbar
m\pi^{2}})}.\hbox to43.36243pt{}$ (27)
Such integrals may be performed with the help of a convergence factorKennard
(1927); this is discussed in the Appendix.
Concatenation of these propagators can help in understanding the phase factors
that enter the calculation. For example, a propagator can be constructed that
takes a particle from the entry aperture to the end of the drift region; thus
$\displaystyle G_{free,\gamma,x_{c}}(x,\tau+T;x^{\prime},0)\hbox
to108.405pt{}$ $\displaystyle=\int
dx^{\prime\prime}G_{\gamma=0}(x,T;x^{\prime\prime},0)G_{\gamma}(x^{\prime\prime},\tau;x^{\prime},0)$
$\displaystyle=\sqrt{\frac{m}{2\pi
i\hbar(\tau+T)}}e^{\frac{im(x-x^{\prime})^{2}}{2\hbar(\tau+T)}-\frac{i\gamma^{2}\tau^{3}(\tau+3T)}{4\hbar
m\pi^{2}(\tau+T)}}$ $\displaystyle\times
e^{-\frac{i\gamma\tau(\tau(x-2x_{c}+x^{\prime})+2T(x^{\prime}-x_{c}))}{\hbar\pi(\tau+T)}}.\hbox
to21.68121pt{}$ (28)
Propagation from the end of the drift region to the detector, through a cavity
with a field gradient $\gamma_{p}$ with a packet centered at $x_{p}$ upon
entry, can be calculated with a propagator similar to (II):
$\centering{G_{free,\gamma_{p},x_{p}}(x,\tau+T_{d};x^{\prime},0).}\@add@centering$
(29)
Then concatenating these two propagators will take an incoming plane wave all
the way to the detector:
$\displaystyle G_{final}(x,2\tau+T+T_{d};x^{\prime},0)=\hbox to72.26999pt{}$
$\displaystyle\int
dx^{\prime\prime}G_{free,\gamma_{p},x_{p}}(x,\tau+T_{d};x^{\prime\prime},0)\hbox
to72.26999pt{}$ $\displaystyle\times
G_{free,\gamma,x_{c}}(x^{\prime\prime},\tau+T,x^{\prime},0)\hbox
to54.2025pt{}$ $\displaystyle=\sqrt{\frac{m}{2\pi i\hbar
T_{t}}}e^{\frac{im(x-x^{\prime})^{2}}{2\hbar(T_{t})}-\frac{i\gamma\gamma_{p}\tau^{3}(\tau+2T_{d})}{\hbar
m\pi^{2}T_{t}}}\hbox to65.04256pt{}$ $\displaystyle\times
e^{-\frac{i\gamma^{2}\tau^{3}(4\tau+3T+3T_{d})}{4\hbar
m\pi^{2}T_{t}}-\frac{i\gamma_{p}^{2}\tau^{2}(4\tau^{2}+3\tau T+11\tau
T_{d}+8TT_{d})}{4\hbar m\pi^{2}T_{t}}}\hbox to21.68121pt{}$
$\displaystyle\times
e^{-\frac{i\gamma\tau(-2(T+T_{d})(x_{c}-x^{\prime})+\tau(x-4x_{c}+3x^{\prime}))}{\hbar\pi
T_{t}}}\hbox to57.81621pt{}$ $\displaystyle\times
e^{-\frac{i\gamma_{p}\tau(2T(x-x_{p})+\tau(3x-4x_{p}+x^{\prime})+2T_{d}(-x_{p}+x^{\prime}))}{\hbar\pi
T_{t}}}.\hbox to36.135pt{}$ (30)
where
$T_{t}=2\tau+T+T_{d}.$ (31)
Then if we insert the initial plane wave (centered at $x_{c}$), at the
detector we find the plane wave
$\displaystyle\int
dx^{\prime\prime}G_{final}(x,2\tau+T+T_{d};x^{\prime\prime},0)e^{ik(x"-x_{c})}$
$\displaystyle=e^{-\frac{i\hbar
k^{2}(T_{t})}{2m}+i(k-\frac{2\gamma\tau}{\hbar\pi})(x-x_{c})-\frac{2i\gamma_{p}\tau(x-x_{p})}{\hbar\pi}}\hbox
to14.45377pt{}$ $\displaystyle\times
e^{\frac{ik\gamma_{p}\tau(\tau+2T_{d})}{m\pi}+\frac{ik\gamma\tau(3\tau+2T+2T_{d})}{m\pi}-\frac{2i\gamma\gamma_{p}\tau^{2}(\tau+2T_{d})}{\hbar
m\pi^{2}}}\hbox to7.22743pt{}$ $\displaystyle\times
e^{-\frac{i\gamma^{2}\tau^{2}(11\tau+8T+8T_{d})}{4\hbar
m\pi^{2}}-\frac{i\gamma_{p}^{2}\tau^{2}(3\tau+8T_{d})}{4\hbar m\pi^{2}}}.\hbox
to43.36243pt{}$ (32)
Of course this is only part of the solution since it only accounts for the
dynamical particle motion. The internal states of the spinors will be treated
in the next section. To compress the equations we write the exponential in the
result of Eq. (II) as
$e^{i\Phi_{f}(\gamma,\gamma_{p})}.$ (33)
The function (II) is an eigenstate of the momentum operator, since operating
on (II) with the momentum operator $i\hbar\partial/\partial x$, the momentum
at the detector is
$\hbar k-\frac{2\gamma\tau}{m}-\frac{2\gamma_{p}\tau}{m}.$ (34)
Other components of the wavefunction will differ in the signs of the
contributions from $\gamma$ and $\gamma_{p}$. Plane waves remain plane; the
wavefronts acquire no curvature. Thus there is no focusing of these solutions.
This is a consequence of the linear approximations for the field gradients in
the dynamical equations of motion. We next consider the development of the
internal phases of the spinors.
## III 3\. Boundary Conditions; Spinor Phases
In this section we discuss boundary conditions appropriate for the spinor part
of the $f_{\pm}$ functions, given initial prepraration of the wavefunctions in
an arbitrary superposition of $a$ and $b$ states. In this section we consider
here only the phase development of the spinors due to their internal energy;
the dynamical phases have been treated above. This discussion also applies to
boundary conditions on the wavefunctions at the beginning of the second cavity
passage.
Suppose that upon first entry into the cavity the atomic spinor is in an
arbitrary superposition of hyperfine states:
$\left[\begin{array}[]{c}e^{-i\theta_{1}/2}D_{a}\\\
e^{i\theta_{1}/2}D_{b}\end{array}\right]\Bigg{|}_{t=0}=\left[\begin{array}[]{c}u_{a0}\\\
u_{b0}\end{array}\right].$ (35)
Solving for the amplitudes $D_{a}$ and $D_{b}$ (Eq. (9)) and substituting into
Eq. (12) gives initial conditions for $f_{\pm}$:
$f_{\pm}(0)=\frac{1}{\sqrt{2}}\big{(}u_{a0}e^{i\theta_{1}/2}\pm
u_{b0}e^{-i\theta_{1}/2}\big{)}e^{ik(x^{\prime}-x_{c})},$ (36)
where $x^{\prime}$ has been inserted in place of $x$ in anticipation of an
integration. If we were to assume the atoms are prepared in the lower
hyperfine state before entering the first cavity, then $u_{a0}=0$ and
$u_{b0}=1$ so the boundary condition (36) would become
$f_{\pm}(0)=\pm\frac{1}{\sqrt{2}}\big{(}e^{-i\theta_{1}/2}\big{)}e^{ik(x^{\prime}-x_{c})}.$
(37)
Combining (9) and (37) in the general case, at the end of the first cavity
passage the spinor functions are:
$\displaystyle
u_{a}(\tau)=\frac{e^{-i(\omega\tau/2+\theta_{1}/2+a(\tau))}}{\sqrt{2}}f_{+}(0)\hbox
to21.68121pt{}$
$\displaystyle+\frac{e^{-i(\omega\tau/2+\theta_{1}/2-a(\tau))}}{\sqrt{2}}f_{-}(0);\hbox
to7.22743pt{}$ (38) $\displaystyle
u_{b}(\tau)=\frac{e^{i(\omega\tau/2+\theta_{1}/2-a(\tau))}}{\sqrt{2}}f_{+}(0)\hbox
to21.68121pt{}$
$\displaystyle-\frac{e^{i(\omega\tau/2+\theta_{1}/2+a(\tau))}}{\sqrt{2}}f_{-}(0).\hbox
to7.22743pt{}$ (39)
In the drift region the spinor states acquire additional phase factors due to
their internal energy. These phase factors are respectively
$e^{\mp i\Delta T/2}.$ (40)
Thus the spinor wavefunctions at the end of the drift region can be expressed
as
$\displaystyle u_{a}(\tau+T)=\frac{e^{-i(\omega\tau+\Delta
T+\theta_{1})/2-ia}}{2}\big{(}u_{a0}e^{i\theta_{1}/2}+u_{b0}e^{-i\theta_{1}/2}\big{)}$
$\displaystyle+\frac{e^{-i(\omega\tau+\Delta
T+\theta_{1})/2+ia}}{2}\big{(}u_{a0}e^{i\theta_{1}/2}-u_{b0}e^{-i\theta_{1}/2}\big{)};\hbox
to28.90755pt{}$ (41) $\displaystyle u_{b}(\tau+T)=\frac{e^{i(\omega\tau+\Delta
T+\theta_{1})/2-ia}}{2}\big{(}u_{a0}e^{i\theta_{1}/2}+u_{b0}e^{-i\theta_{1}/2}\big{)}\hbox
to7.22743pt{}$ $\displaystyle-\frac{e^{i(\omega\tau+\Delta
T+\theta_{1})/2+ia}}{2}\big{(}u_{a0}e^{i\theta_{1}/2}-u_{b0}e^{-i\theta_{1}/2}\big{)}.\hbox
to36.135pt{}$ (42)
The signs of $\gamma$ and $a(\tau)$ occur in a given term with opposite signs,
thus at the end of the calculation the dynamical phase factors can be matched
with the spinor phase factors.
In Eq. (36), linear combinations of the spinor wavefunctions are combined to
give boundary conditions for the decoupled functions $f_{\pm}$. Similarly,
linear combinations of Eqs. (III) and (III) give boundary conditions for
solution of the decoupled wave equations for second cavity passage. In the
second cavity, the phase factor $\exp(\pm i\theta_{1}/2)$ is replaced by
$\exp(\pm i\theta_{2}/2)$; the value of $\theta_{2}$ will be discussed in
Sect. 8.
In order to simplify further development, to form the needed combinations we
depart from a general treatment and assume the spinors are prepared in the
lower hyperfine state, so $u_{a0}=0$, $u_{b0}=1.$ We therefore need the
following linear combinations:
$\displaystyle
u_{a}(\tau+T)e^{i\theta_{2}/2}+\phi_{b}(\tau+T)e^{-i\theta_{2}/2}\hbox
to65.04256pt{}$ (43)
$\displaystyle=(\cos\Theta)e^{-ia-i\theta_{1}/2}+(+i\sin\Theta)e^{-ia-i\theta_{1}/2},$
$\displaystyle
u_{a}(\tau+T)e^{i\theta_{2}/2}-\phi_{b}(\tau+T)e^{-i\theta_{2}/2}\hbox
to65.04256pt{}$ (44)
$\displaystyle=-(-i\sin\Theta)e^{ia-i\theta_{1}/2}-(-\cos\Theta)e^{ia-i\theta_{1}/2},$
where
$\Theta=\frac{1}{2}(\theta_{2}-\theta_{1}-\omega\tau-\Delta T).$ (45)
The Ramsey fringe is determined by $\Theta$, which depends on the time $\tau$
through the first cavity and $T$ through the drift region.
Eqs. (III) and (III) give the spinors at the entry to the second cavity in
terms of their initial values. Equations of the same form must hold for the
spinors at the detector in terms of their values at the entry to the second
cavity. Thus the spinors at the detector can be obtained by iterating (III)
and (III). Such initial values have been given in Eqs. (43-44). Therefore we
can immediately write down the spinors at the detector. The principal changes
are: $\gamma$ and $a$ are replaced by $\gamma_{p}$ and $a_{p}$, respectively,
$\theta_{1}$ is replaced by $\theta_{2}$, and $T$ is replaced by $T_{d}$. Thus
at the detector we have
$\displaystyle u_{a}(2\tau+T+T_{d})=\frac{e^{-i(\omega\tau+\Delta
T_{d})/2-i\theta_{1}/2-i\theta_{2}/2}}{2}\hbox to43.36243pt{}$
$\displaystyle\times\bigg{(}e^{-ia_{p}-ia}(\cos\Theta)+e^{ia_{p}-ia}(i\sin\Theta)\hbox
to54.2025pt{}$
$\displaystyle+e^{-ia_{p}+ia}(-i\sin\Theta)+e^{ia_{p}+ia}(-\cos\Theta)\bigg{)};\hbox
to36.135pt{}$ (46) $\displaystyle
u_{b}(2\tau+T+T_{d})=\frac{e^{i(\omega\tau+\Delta
T_{d})/2-i\theta_{1}/2+i\theta_{2}/2}}{2}\hbox to72.26999pt{}$
$\displaystyle\times\bigg{(}e^{-ia_{p}-ia}(\cos\Theta)+e^{ia_{p}-ia}(-i\sin\Theta)\hbox
to54.2025pt{}$
$\displaystyle+e^{-ia_{p}+ia}(-i\sin\Theta)-e^{ia_{p}+ia}(-\cos\Theta)\bigg{)}.\hbox
to36.135pt{}$ (47)
The spinor components in (III) and (III) are listed in the order
$(\gamma,\gamma_{p}),(\gamma,-\gamma_{p}),(-\gamma,\gamma_{p}),(-\gamma,-\gamma_{p})$
corresponding to $(a,a_{p}),(a,-a_{p}),(-a,a_{p}),(-a,-a_{p})$.
With the solution for the dynamical phase (33), the wavefunctions at the
detector can now be assembled:
$\displaystyle u_{a}(2\tau+T+T_{d})=\frac{e^{-i(\omega\tau+\Delta
T_{d})/2-i\theta_{1}/2-i\theta_{2}/2}}{2}\hbox to18.06749pt{}$ (48)
$\displaystyle\times\bigg{(}e^{-ia_{p}-ia+i\Phi_{f}(\gamma,\gamma_{p})}(\cos\Theta)\hbox
to72.26999pt{}$
$\displaystyle+e^{ia_{p}-ia+i\Phi_{f}(\gamma,-\gamma_{p})}(i\sin\Theta)\hbox
to72.26999pt{}$
$\displaystyle+e^{-ia_{p}+ia+i\Phi_{f}(-\gamma,\gamma_{p})}(-i\sin\Theta)\hbox
to54.2025pt{}$
$\displaystyle+e^{ia_{p}+ia+i\Phi_{f}(-\gamma,-\gamma_{p})}(-\cos\Theta)\bigg{)};\hbox
to36.135pt{}$ $\displaystyle u_{b}(2\tau+T+T_{d})=\frac{e^{i(\omega\tau+\Delta
T_{d})/2-i\theta_{1}/2+i\theta_{2}/2}}{2}\hbox to18.06749pt{}$ (49)
$\displaystyle\times\bigg{(}e^{-ia_{p}-ia+i\Phi_{f}(\gamma,\gamma_{p})}(\cos\Theta)\hbox
to72.26999pt{}$
$\displaystyle+e^{ia_{p}-ia+i\Phi_{f}(\gamma,-\gamma_{p})}(-i\sin\Theta)\hbox
to72.26999pt{}$
$\displaystyle+e^{-ia_{p}+ia+i\Phi_{f}(-\gamma,\gamma_{p})}(-i\sin\Theta)\hbox
to54.2025pt{}$
$\displaystyle+e^{ia_{p}+ia+i\Phi_{f}(-\gamma,-\gamma_{p})}(+\cos\Theta)\bigg{)};\hbox
to36.135pt{}$
Eqs. (48) and (49) provide the complete solution for plane wave spinors
passing through the apparatus. In the next section we discuss the dynamical
phases arising from transverse particle motion.
Figure 2: Schematic illustration of trajectories of packets through the
system; each cavity causes separation into two packets, each with equal
numbers of $a$ and $b$ spinor states. The ordinate represents the direction
transverse to the cavity axis.
## IV 4\. Construction of Wave Packets
We construct wavepackets at the detector by multiplying by the weighting
function $\exp(-\sigma^{2}(k-k_{0})^{2}/2)$, as in (II), and integrating over
$k$. Every term in (48) and (49) acquires a normalization factor
$N_{p}=\frac{1}{\sqrt{\sqrt{\pi}\big{(}\sigma+\frac{i\hbar T_{t}}{\sigma
m}}\big{)}}.$ (50)
The expressions become quite cumbersome. We shall illustrate the result in
only one case, the first term in (48). The terms in the exponent involving the
wave vector $k$ are
$\displaystyle-\sigma^{2}(k-k_{0})^{2}/2-\frac{i\hbar
k^{2}T_{t}}{2m}+ik(x-x_{c})$
$\displaystyle+\frac{ik\gamma_{p}\tau(\tau+2T_{d})}{m\pi}+\frac{ik\gamma\tau(3\tau+2T+2T_{d})}{m\pi}.\hbox
to14.45377pt{}$ (51)
This term then becomes
$\frac{N_{p}}{2}e^{-ia_{p}-ia+i\Phi_{packet}(k_{0},\gamma,\gamma_{p})}(\cos\Theta)$
(52)
where
$\displaystyle\Phi_{packet}(k_{0},\gamma,\gamma_{p})=-\frac{\hbar
k_{0}^{2}T_{t}}{2m}-\frac{2\gamma\tau}{\hbar\pi}(x-x_{c})\hbox to28.90755pt{}$
$\displaystyle-\frac{2\gamma_{p}\tau}{\hbar\pi}(x-x_{p})-\frac{2\gamma\gamma_{p}\tau^{2}(\tau+2T_{d})}{4\hbar
m\pi^{2}}\hbox to28.90755pt{}$ (53)
$\displaystyle+k_{0}\big{(}x-x_{c}+\frac{\gamma\tau(3\tau+2T+T_{d})}{m\pi}+\frac{\gamma_{p}\tau(\tau+2T_{d})}{m\pi}\big{)}$
$\displaystyle+i\frac{(x-x_{c}-\frac{\hbar
k_{0}T_{t}}{m}+\frac{\gamma\tau(3\tau+2T+3T_{d})}{m\pi}+\frac{\gamma_{p}\tau(\tau+2T_{d}))}{m\pi})^{2}}{2(\sigma^{2}+i\hbar
T_{t}/m)}.$
The quadratic term in $(x-x_{c}...)^{2}$ in (IV) clearly shows where the
packet is centered. One can also backtrack and obtain the packet center at the
exit aperture by setting $T_{d}=0$.
The packets for the hyperfine states at the detector are therefore:
$\displaystyle\Psi_{a}(2\tau+T+T_{d})\hbox to144.54pt{}$
$\displaystyle=\frac{N_{p}}{2}\times\bigg{(}e^{-ia_{p}-ia+i\Phi_{packet}(k_{0},\gamma,\gamma_{p})}(\cos\Theta)\hbox
to72.26999pt{}$
$\displaystyle+e^{ia_{p}-ia+i\Phi_{packet}(k_{0},\gamma,-\gamma_{p})}(i\sin\Theta)\hbox
to72.26999pt{}$
$\displaystyle+e^{-ia_{p}+ia+i\Phi_{packet}(k_{0},-\gamma,\gamma_{p})}(-i\sin\Theta)\hbox
to54.2025pt{}$
$\displaystyle+e^{ia_{p}+ia+i\Phi_{packet}(k_{0},-\gamma,-\gamma_{p})}(-\cos\Theta)\bigg{)};\hbox
to36.135pt{}$ (54) $\displaystyle\Psi_{b}(2\tau+T+T_{d})\hbox to144.54pt{}$
$\displaystyle=\frac{N_{p}}{2}\times\bigg{(}e^{-ia_{p}-ia+i\Phi_{packet}(k_{0},\gamma,\gamma_{p})}(\cos\Theta)\hbox
to72.26999pt{}$
$\displaystyle+e^{ia_{p}-ia+i\Phi_{packet}(k_{0},\gamma,-\gamma_{p})}(-i\sin\Theta)\hbox
to72.26999pt{}$
$\displaystyle+e^{-ia_{p}+ia+i\Phi_{packet}(k_{0},-\gamma,\gamma_{p})}(-i\sin\Theta)\hbox
to54.2025pt{}$
$\displaystyle+e^{ia_{p}+ia+i\Phi_{packet}(k_{0},-\gamma,-\gamma_{p})}(+\cos\Theta)\bigg{)}.\hbox
to36.135pt{}$ (55)
To save writing, we shall refer to these terms in order as
$\psi_{1a},\psi_{2a}...\psi_{3b},\psi_{4b}$ respectively. The probability of
finding a particle in the upper hyperfine state at the detector will be
$\big{<}\big{|}\Psi_{a}\big{|}^{2}\big{>}$ (56)
where $<>$ means appropriately averaged over thermal velocities and integrated
over the apertures.
In computing probabilities such as in (56), each of the squared terms gives 16
contributions. The forms for the solutions given above allow us to cancel many
terms even without knowing much about the functions $\Phi_{packet}$. We are
going to compress the notation by labeling the terms in the solutions with
subscripts 1 through 4. Thus for example, for the contribution from the square
of the lower hyperfine state, the product of the first term, times the complex
conjugate of the third term, will be denoted by
$\displaystyle ie^{-2ia}\cos\Theta\sin\Theta P_{13b}=N_{p}^{2}\hbox
to93.95122pt{}$ $\displaystyle\times\int\big{<}(\cos\Theta e^{-ia-
ia_{p}}e^{i\Phi_{packet}(\gamma,\gamma_{p})})\hbox to54.2025pt{}$
$\displaystyle\times(i\sin\Theta
e^{-ia+ia_{p}}e^{i\Phi_{packet}^{*}(-\gamma,\gamma_{p})})\big{>}.\hbox
to36.135pt{}$ (57)
Then normalization of the wavefunction at the detector is computed by means of
$\displaystyle\int\big{<}\big{|}\Psi_{a}\big{|}^{2}\big{>}=\frac{1}{4}\bigg{(}(\cos\Theta)^{2}(P_{11a}+P_{44a})\hbox
to79.49744pt{}$
$\displaystyle-(\cos\Theta)^{2}(e^{-2ia_{p}-2ia}P_{14a}+e^{2ia_{p}+2ia}P_{41a})\hbox
to65.04256pt{}$ $\displaystyle+(\sin\Theta)^{2}(P_{22a}+P_{33a})\hbox
to137.31255pt{}$
$\displaystyle-(\sin\Theta)^{2}(e^{2ia_{p}-2ia}P_{23a}+e^{-2ia_{p}+2ia}P_{32a})\hbox
to50.58878pt{}$
$\displaystyle+i\cos\Theta\sin\Theta\big{(}-e^{-2ia_{p}}P_{12a}+e^{2ia_{p}}P_{21a}\hbox
to50.58878pt{}$ $\displaystyle+e^{-2ia}P_{13a}-e^{2ia}P_{31a}\hbox
to65.04256pt{}$ $\displaystyle-e^{-2ia}P_{24a}+e^{2ia}P_{42a}\hbox
to65.04256pt{}$
$\displaystyle+e^{-2ia_{p}}P_{34a}-e^{2ia_{p}}P_{43a}\big{)}\bigg{)}\hbox
to50.58878pt{}$ (58)
and
$\displaystyle\int\big{<}\big{|}\Psi_{b}\big{|}^{2}\big{>}=\frac{1}{4}\bigg{(}(\cos\Theta)^{2}(P_{11b}+P_{44b})\hbox
to79.49744pt{}$
$\displaystyle+(\cos\Theta)^{2}(e^{-2ia_{p}-2ia}P_{14b}+e^{2ia_{p}+2ia}P_{41b})\hbox
to65.04256pt{}$ $\displaystyle+(\sin\Theta)^{2}(P_{22b}+P_{33b})\hbox
to137.31255pt{}$
$\displaystyle+(\sin\Theta)^{2}(e^{2ia_{p}-2ia}P_{23b}+e^{-2ia_{p}+2ia}P_{32b})\hbox
to50.58878pt{}$
$\displaystyle+i\cos\Theta\sin\Theta\big{(}e^{-2ia_{p}}P_{12b}-e^{2ia_{p}}P_{21b}\hbox
to50.58878pt{}$ $\displaystyle+e^{-2ia}P_{13b}-e^{2ia}P_{31b}\hbox
to65.04256pt{}$ $\displaystyle-e^{-2ia}P_{24b}+e^{2ia}P_{42b}\hbox
to65.04256pt{}$
$\displaystyle-e^{-2ia_{p}}P_{34b}+e^{2ia_{p}}P_{43b}\big{)}\bigg{)}\hbox
to50.58878pt{}$ (59)
These integrals have been defined so that when $``\int"$ in (IV) or (IV) is
interpreted as an integration over all space,
$P_{iia}=P_{iib}=1,\quad i=1,2,3,4.$ (60)
It can then be shown that for all $i$ and $j$,
$P_{ija}=P_{ijb}.$ (61)
The normalization condition reduces to
$\displaystyle 1=1+\frac{2}{4}i\cos\Theta\sin\Theta\hbox to93.95122pt{}$
$\displaystyle\times\int\big{(}e^{-2ia}P_{13a}-e^{2ia}P_{31a}\hbox
to36.135pt{}$ $\displaystyle-e^{-2ia}P_{24a}+e^{2ia}P_{42a}\big{)}.$ (62)
In can be shown that in addition to the above symmetry properties of the
integrals, we have when integrating over all space
$P_{13a}=P_{24a};\quad P_{31a}=P_{24a}.$ (63)
Therefore all terms proportional to $\cos\Theta\sin\Theta$ on the right side
of the normalization condition cancel and the solution is correctly
normalized. In order for the integrals to converge, it is necessary to
construct wavepackets as in (II) before squaring.
## V 5\. Packet centers
“It is the mark of an educated mind to rest satisfied with the degree of
precision which the nature of the subject admits and not to seek exactness
where only an approximation is possible.–Aristotle The factors multiplying the
terms in large parentheses in (48) and (49) cannot affect any probability
computed from the squares of the wavefunctions; these factors will therefore
be dropped.
Wave packets can be constructed at the beginning or end of the drift region by
taking weighted superpositions of $e^{ik(x-x_{c})}$ as in (II). Figure 1
illustrates, greatly exaggerated, what happens to the wave packet
trajectories. The decoupled components travel in opposite directions in the
first cavity, with an effective acceleration $\pm 2\gamma/(m\pi)$. For a
typical off-axis field gradient, the separation in position is only about a
nanometer. The velocity difference is about 15 nm/second, so at the end of the
drift region if $T=1$ sec, the packet separation is only about 30 nm. On the
other hand a typical transverse velocity due to a finite temperature of
$\approx.5\mu K$ is 0.008 m/sec. The displacements due to the thermal velocity
distribution are $\approx 3\times 10^{5}$ larger than the displacements due to
transverse field gradients. The latter are also small on the scale of changes
in the transverse field gradients themselves. We therefore use a value of
$\gamma$, denoted by $\gamma_{p}$, in the second cavity that accounts for the
change in the central position of the packet during time $\tau+T$. A packet
entering the first cavity with position $x_{c}$ and average velocity $\hbar
k_{0x}/m$ will end up, on average, at the entry to the second cavity with
position
$x_{p}=x_{c}+\hbar k_{0x}(\tau+T)/m.$ (64)
The field gradient at this position will be denoted by
$\displaystyle\gamma_{p}=-\frac{\hbar\pi bx_{1}^{2}x_{p}}{4d^{2}}\hbox
to86.72377pt{}$ $\displaystyle=-\bigg{(}\frac{\hbar\pi
bx_{1}^{2}x_{c}}{4d^{2}}+\frac{\hbar^{2}\pi
bx_{1}^{2}k_{0x}(\tau+T)}{4d^{2}m}\bigg{)},$ (65)
and the second term will be accounted for when performing thermal averages. On
exiting the first cavity, the packet will be centered at
$x=x_{c}+\frac{\hbar k_{0x}\tau}{m}\mp\frac{\gamma\tau^{2}}{m\pi};$ (66)
at the end of the drift region, the center will be at
$x=x_{c}+\frac{\hbar k_{0x}(\tau+T)}{m}\mp\frac{\gamma\tau(\tau+2T)}{m\pi};$
(67)
after exiting from the second cavity, the center will be at
$x=x_{c}+\frac{\hbar
k_{0x}(2\tau+T)}{m}\mp\frac{\gamma\tau(3\tau+2T)}{m\pi}\mp\frac{\gamma_{p}\tau^{2}}{m\pi};$
(68)
and finally at the detector the center will be
$\displaystyle x=x_{c}+\frac{\hbar
k_{0x}(2\tau+T+T_{d})}{m}\mp\frac{\gamma\tau(3\tau+2T+2T_{d})}{m\pi}$
$\displaystyle\mp\frac{\gamma_{p}\tau(\tau+2T_{d})}{m\pi}\hbox
to86.72377pt{}.$ (69)
The parameter $a_{p}(\tau)$ in the second cavity will also be affected by
transversFe forces. If the on-axis value of $a(\tau)$ is given by a normal
$\pi/2$ pulse, then for an atom on the axis
$a_{axis}=b\tau=\pi/4.$ (70)
At the entry to the first cavity we have
$a=a_{axis}\bigg{(}1-\frac{x_{1}^{2}x_{c}^{2}}{4d^{2}}\bigg{)},$ (71)
and therefore at the entry to the second cavity
$a_{p}=a_{axis}\bigg{(}1-\frac{x_{1}^{2}(x_{c}+\hbar
k_{0x}(\tau+T)/m)^{2}}{4d^{2}}\bigg{)}.$ (72)
## VI 6\. Launch conditions; Thermal Averaging
In NIST F-2, atoms are collected, cooled, and launched from a trap some dozens
of centimeters below the cavity entry aperture. We let $z_{L}<0$, $v_{L}>0$
and $t_{L}<0$ be the launch position, velocity and time of launch, such that
the atom clouds are centered at the entry aperture at $z=0,\ t=0$ with entry
velocity $v_{0}=v_{L}+gt_{L}$. Here $z$ is positive upwards and the
acceleration of gravity is $-g$. We treat the cloud of atoms as collisionless,
characterized by an initial spread $\sigma_{n}$ and temperature $T_{n}$ in the
transverse directions, and spread $\sigma_{p}$ and temperature $T_{p}$ in the
z-dimension. The atom distribution is described by a product of exponential
distribution functions of the following form:
$f_{n}(x,v_{x},t)=\frac{1}{2\pi\sigma_{n}}\sqrt{\frac{m}{k_{B}T_{n}}}e^{-\frac{(x-v_{x}(t+t_{L}))^{2}}{2\sigma_{n}^{2}}}e^{-\frac{mv_{x}^{2}}{2k_{B}T_{n}}},$
(73) $\displaystyle
f_{p}(z,v_{z},t)=\frac{1}{2\pi\sigma_{p}}\sqrt{\frac{m}{k_{B}T_{p}}}e^{-\frac{m(v_{z}-v_{L}+g(t+t_{L}))^{2}}{2k_{B}T_{p}}}$
$\displaystyle\times
e^{-\frac{(z+d_{L}-v_{z}(t+t_{L})-g(t+t_{L})^{2}/2)^{2}}{2\sigma_{p}^{2}}}.$
(74)
Then the complete distribution function is
$f({\bf r},{\bf v},t)=f_{n}(x,v_{x},t)f_{n}(y,v_{y},t)f_{p}(z,v_{z},t).$ (75)
This distribution function satisfies the collisionless Boltzmann equation,
$\frac{\partial f}{\partial t}+{\bf v}\cdot\nabla f-g\frac{\partial
f}{\partial v_{z}}=0.$ (76)
At $t=t_{L}$, the cloud is centered at $z=z_{L}$. In the transverse direction,
the half-width of the cloud is $\sigma_{n}$ at $t=-t_{L}$ (in a single
tranverse dimension) and spreads to
$(\sigma_{n}^{2}+k_{B}(t+t_{L})^{2}T_{n}/m)^{1/2}$ after time $t$.
There are many contributions to the width of the wavepackets and to the atom
balls; generally these combine in quadrature. First, there is the initial
half-width $\sigma$ at launch, which is not known precisely. Due to quantum
mechanical spreading of a packet, there is a second contribution to the half-
width that increases with time but is inversely proportional to $\sigma$. If
$\sigma$ is small, quantum mechanical spreading will become large and
contribute significantly to clipping. The value of $\sigma$ has been set equal
to the approximate thermal wavelength of a Cs133 atom, about 200 nanometers.
In Figure 9 we illustrate the effect of changing this assumption, and verify
that the results do not depend significantly on the choice of this parameter.
Third, there is a contribution arising from thermal averaging over the
transverse velocities. At temperatures $0.5\mu K$ and higher, and after a
Ramsey time in the neighborhood of 1 second, this contribution dominates the
spreading of the ball. There are numerous additional contributions to
spreading, of the order of a nanometer or less, arising from transverse field
gradients and off-axis positions of the atoms. These contributions are
automatically included in the calculations.
## VII 7\. One-Dimensional Model
The detector measures the numbers of particles in the cloud in each of the
hyperfine states and the following ratio is computed:
$\frac{\int\big{<}|\Psi_{a}|^{2}\big{>}}{\int\big{<}|\Psi_{a}|^{2}\big{>}+\int\big{<}|\Psi_{b}|^{2}\big{>}},$
(77)
where the integrals go over the initial aperture with a weight function
determined by the Boltzmann distribution, after forming wave packets. Thus we
average with
$\int_{-D/2}^{D/2}f_{n}(x_{c},v_{x},0)dx_{c}dv_{x}|\Psi(x)|^{2}$ (78)
where $D=.005$ meters is the aperture radius. We also integrate over the exit
aperture:
$\int\big{<}|\Psi_{a}|^{2}\big{>}=\int_{-D/2}^{D/2}dx\int_{-D/2}^{D/2}f_{n}(x_{c},v_{x},0)dx_{c}dv_{x}|\Psi_{a}(x)|^{2}$
(79)
with the understanding that the integrals now go over a finite region, we
still denote the contributions with notations such as $P_{11a}$, etc. The
dependence of the spinor part of the wavefunctions is not affected by such
restrictions.
The numerator of (77) is of the form
$\int\big{<}|\Psi_{a}|^{2}\big{>}=A_{a}+B_{a}\cos 2\Theta+C_{a}\sin 2\Theta,$
(80)
where
$\displaystyle A_{a}=\frac{1}{8}\bigg{(}P_{11a}+P_{22a}+P_{33a}+P_{44a}$
$\displaystyle-e^{-2ia_{p}-2ia}P_{14a}-e^{2ia_{p}+2ia}P_{41a}$
$\displaystyle-e^{2ia_{p}-2ia}P_{23a}-e^{-2ia_{p}+2ia}P_{32a}\bigg{)}$ (81)
$\displaystyle B_{a}=\frac{1}{8}\bigg{(}P_{11a}-P_{22a}-P_{33a}+P_{44a}$
$\displaystyle-e^{-2ia_{p}-2ia}P_{14a}-e^{2ia_{p}+2ia}P_{41a}$
$\displaystyle+e^{2ia_{p}-2ia}P_{23a}+e^{-2ia_{p}+2ia}P_{32a}\bigg{)}$ (82)
$\displaystyle
C_{a}=\frac{i}{8}\bigg{(}-e^{-2ia_{p}}P_{12a}+e^{2ia_{p}}P_{21a}\hbox
to14.45377pt{}$ $\displaystyle+e^{-2ia}P_{13a}-e^{2ia}P_{31a}\hbox
to28.90755pt{}$ $\displaystyle-e^{-2ia}P_{24a}+e^{2ia}P_{42a}\hbox
to28.90755pt{}$
$\displaystyle+e^{-2ia_{p}}P_{34a}-e^{2ia_{p}}P_{43a}\bigg{)}.\hbox
to14.45377pt{}$ (83)
Similarly for the other hyperfine state,
$\displaystyle A_{b}=\frac{1}{8}\bigg{(}P_{11b}+P_{22b}+P_{33b}+P_{44b}$
$\displaystyle+e^{-2ia_{p}-2ia}P_{14b}+e^{2ia_{p}+2ia}P_{41b}$
$\displaystyle+e^{2ia_{p}-2ia}P_{23b}+e^{-2ia_{p}+2ia}P_{32b}\bigg{)}$ (84)
$\displaystyle B_{b}=\frac{1}{8}\bigg{(}P_{11b}-P_{22b}-P_{33b}+P_{44b}$
$\displaystyle+e^{-2ia_{p}-2ia}P_{14b}+e^{2ia_{p}+2ia}P_{41b}$
$\displaystyle-e^{2ia_{p}-2ia}P_{23b}-e^{-2ia_{p}+2ia}P_{32b}\bigg{)}$ (85)
$\displaystyle
C_{b}=\frac{i}{8}\bigg{(}e^{-2ia_{p}}P_{12b}-e^{2ia_{p}}P_{21b}\hbox
to14.45377pt{}$ $\displaystyle+e^{-2ia}P_{13b}-e^{2ia}P_{31b}\hbox
to28.90755pt{}$ $\displaystyle-e^{-2ia}P_{24b}+e^{2ia}P_{42b}\hbox
to28.90755pt{}$
$\displaystyle-e^{-2ia_{p}}P_{34b}+e^{2ia_{p}}P_{43b}\bigg{)}.\hbox
to14.45377pt{}$ (86)
The denominator of (77) is the sum of contributions from both hyperfine states
and is therefore
$A+B\cos 2\Theta+C\sin 2\Theta$ (87)
where
$\displaystyle
A=A_{a}+A_{b}=\frac{1}{4}\big{(}P_{11a}+P_{22a}+P_{33a}+P_{44a}\big{)};$
$\displaystyle
B=B_{a}+B_{b}=\frac{1}{4}\big{(}P_{11a}-P_{22a}-P_{33a}+P_{44a}\big{)};$ (88)
$\displaystyle C=C_{a}+C_{b}=\frac{i}{4}\big{(}e^{-2ia}P_{13a}-e^{2ia}P_{31a}$
$\displaystyle-e^{-2ia}P_{24a}+e^{2ia}P_{42b}\big{)}.\hbox to72.26999pt{}$
(89)
and where we have made use of the relations (60, 61).
## VIII 8\. Detection
Let the numbers of particles detected in the $a$ and $b$ hyperfine states be
$\displaystyle n_{a}=\big{<}\int|\Psi_{a}|^{2}\big{>};$ (90) $\displaystyle
n_{b}=\big{<}\int|\Psi_{b}|^{2}\big{>}.$ (91)
where it is understood that the spatial integrals only go over the apertures,
and that averages over transverse velocity are performed with weight functions
derived from (75).
The transition probabilities found in the preceding calculations depend on the
phase difference $\theta_{2}-\theta_{1}$ only through the quantity
$2\Theta=\theta_{2}-\theta_{1}-(\omega\tau+\Delta T).$ (92)
At the instant the atom enters the cavity for the second time, the angle
$\theta_{2}$ includes the advance of phase of the microwave field, as well as
any additional phase angle $\alpha$ that is imposed on the RF field during the
drift time. Thus at resonance the phase of the microwave field satisfies
$\displaystyle\theta_{2}=\theta_{1}+\omega(\tau+T)+\alpha;$ $\displaystyle
2\Theta=\alpha+(\omega-\Delta)T.\hbox to14.45377pt{}$ (93)
Due to the contibutions of the coefficients $C_{a},C_{b}$ to the transition
probabilities, the center of the line will be slightly shifted from its
nominal value at $\omega=\Delta$.
In the F-2 fountain, the central Ramsey fringe is located, corresponding
ideally to $\alpha=0,\ \omega=\Delta$. The numbers of particles in the upper
hyperfine state are measured part way down the line profile, on opposite sides
of the line in successive balls; this corresponds to setting
$(\omega-\Delta)T=\pm\pi/2$ or if $\omega=\Delta$, $\alpha=\pm\pi/2$. A servo
locates the line center in such a way that the numbers observed on the two
sides of the line are equal. A frequency shift error $\delta\omega$ would
appear through an additional term $\Delta\rightarrow\Delta+\delta\omega$ in
(VIII):
$2\Theta_{\pm}=\pm\pi/2-\delta\omega T.$ (94)
After thermal averages and averages over the aperture are taken, theory gives
the following predictions for the observables:
$\frac{n_{+}}{n_{+}+n_{-}}\bigg{|}_{+\alpha}=\frac{A_{a}+B_{a}\cos(2\Theta_{+})+C_{a}\sin(2\Theta_{+})}{A+B\cos(2\Theta_{+})+C\sin(2\Theta_{+})};$
(95)
$\frac{n_{+}}{n_{+}+n_{-}}\bigg{|}_{-\alpha}=\frac{A_{a}+B_{a}\cos(2\Theta_{-})+C_{a}\sin(2\Theta_{-})}{A+B\cos(2\Theta_{-})+C\sin(2\Theta_{-})}.$
(96)
If the coefficients $C_{a},C_{b}$ vanished, there could be no frequency shift
since the transition probabilities would be symmetric with respect to
reflection about the line center.
We consider two cases. First, setting the ratios (95) and (96) equal,
expanding for small $\delta\omega$ and solving we find that $\delta\omega$ is
proportional to the factor
$\cos(\alpha)(CB_{a}-BC_{a})-(AC_{a}-CA_{a}).$ (97)
If $\alpha$ were chosen to satisfy the above relation, the shift of the center
of the line would be unobservable. For NIST F-2, $\alpha\approx 1.7$ radians,
whereas the value of $\alpha$ actually used in making measurements is
$\pi/2=1.57$ radians. Figure 3 shows the calculated shift as a function of the
angle $\alpha$ for a reasonable set of parameters. The measured shift
decreases from the actual shift at $\alpha=0$ to a smaller value at
$\alpha=\pm\pi/2$.
Figure 3: Calculated fractional frequency shift as a function of the angle
$\alpha$ in the two-dimensional case. The values of toss height and
temperature are 1.1 meters and 0.5 $\mu$K, respectively.
Second, for the actual operating conditions of NIST F-1 and F-2,
$\alpha=\pm\pi/2$. Solving for the shift as in the previous case,
$\delta\omega=\frac{AC_{a}-CA_{a}}{AB_{a}T}$ (98)
the fractional frequency error is thus
$\frac{\delta\omega}{\omega}=\frac{AC_{a}-CA_{a}}{2\pi\times(9.192\times
10^{9})TB_{a}A}.$ (99)
## IX 9\. Results: One-Dimensional Case
Figure 4 plots the fractional frequency shift as a function of toss height
$h$, at a temperature of $0.5\mu{\rm K}$. The fractional frequency shift as a
function of initial atom cloud temperature is plotted in Figure 5 for a toss
height 0.75 meters.
Figure 4: Fractional shift as a function of toss height, at temperature
$0.5\mu{\rm K}$; one-dimensional calculation Figure 5: Fractional frequency
shift for various atom cloud temperatures at a toss height 0.75 meters; one-
dimensional calculation
## X 10\. Two-dimensional model
The theory may be extended to two dimensions in a straightforward way. Eq.
(II) admits a solution that is a product $\alpha(x,t)\beta(y,t)$. The function
$\beta$ is formally identical to $\alpha$ with replacements $x\rightarrow y$,
$x_{c}\rightarrow y_{c}$, $\gamma_{x}\rightarrow\gamma_{y}$, $x_{p}\rightarrow
y_{p}$, $\gamma_{xp}\rightarrow\gamma_{yp}$, $k_{0}=k_{0x}\rightarrow k_{0y}$,
$|N_{p}|^{2}\rightarrow|N_{p}|^{4}$. Factors such as $e^{\pm a\pm
a_{p}}\cos(\Theta)$ are formally unchanged, but $a$ and $a_{p}$ are evaluated
at an off-axis point corresponding to $(x_{c},y_{c})$. Thus, Eqs. (71) and
(72) become
$a=a_{axis}\bigg{(}1-\frac{x_{1}^{2}(x_{c}^{2}+y_{c}^{2})}{4d^{2}}\bigg{)};$
(100) $\displaystyle a_{p}=a_{axis}\bigg{(}1-\frac{x_{1}^{2}(x_{c}+\hbar
k_{0x}(\tau+T)/m)^{2}}{4d^{2}}\hbox to18.06749pt{}$
$\displaystyle-\frac{x_{1}^{2}(y_{c}+\hbar
k_{0y}(\tau+T)/m)^{2}}{4d^{2}}\bigg{)}.\hbox to7.22743pt{}$ (101)
Cavity phase factors involving $a$ and $a_{p}$ are sums in the exponent, which
becomes a product of exponential phase factors. The dynamical phase factors in
Eqs. (IV) and (IV) are augmented by factors of the form
$e^{i\Phi_{packet}(k_{0y},\pm\gamma_{y},\pm\gamma_{yp})}.$ (102)
Let
$\displaystyle\Phi_{pkt}(k_{0x},\gamma_{x},\gamma_{xp},k_{0y},\gamma_{y},\gamma_{yp})\hbox
to108.405pt{}$ (103)
$\displaystyle=\Phi_{packet}(k_{0x},\gamma_{x},\gamma_{xp})+\Phi_{packet}(k_{0y},\gamma_{y},\gamma_{yp}).$
(We suppress the dependence on $x,x_{c},x_{p},y,y_{c},y_{p}$ to save writing.)
Then the wavefunctions at the detector are
$\displaystyle\Psi_{a}(2\tau+T+T_{d})=\frac{N_{p}^{2}}{2}\hbox to108.405pt{}$
(104)
$\displaystyle\times\bigg{(}e^{-ia_{p}-ia+i\Phi_{pkt}(k_{0x},\gamma_{x},\gamma_{xp},k_{0y},\gamma_{y},\gamma_{yp})}(\cos\Theta)\hbox
to28.90755pt{}$
$\displaystyle+e^{ia_{p}-ia+i\Phi_{pkt}(k_{0x},\gamma_{x},-\gamma_{xp},k_{0y},\gamma_{y},-\gamma_{yp})}(i\sin\Theta)\hbox
to14.45377pt{}$
$\displaystyle+e^{-ia_{p}+ia+i\Phi_{pkt}(k_{0x},-\gamma,_{x}\gamma_{xp},k_{0y},-\gamma_{y},\gamma_{yp})}(-i\sin\Theta)\hbox
to7.22743pt{}$
$\displaystyle+e^{ia_{p}+ia+i\Phi_{pkt}(k_{0x},-\gamma_{x},-\gamma_{xp},k_{0y},-\gamma_{y},-\gamma_{yp})}(-\cos\Theta)\bigg{)};\hbox
to7.22743pt{}$ $\displaystyle\Psi_{b}(2\tau+T+T_{d})=\frac{N_{p}^{2}}{2}\hbox
to108.405pt{}$ (105)
$\displaystyle\times\bigg{(}e^{-ia_{p}-ia+i\Phi_{pkt}(k_{0x},\gamma_{x},\gamma_{xp},k_{0y},\gamma_{y},\gamma_{yp})}(\cos\Theta)\hbox
to28.90755pt{}$
$\displaystyle+e^{ia_{p}-ia+i\Phi_{pkt}(k_{0x},\gamma,_{x},-\gamma_{xp},k_{0y},\gamma_{y},-\gamma_{yp})}(-i\sin\Theta)\hbox
to14.45377pt{}$
$\displaystyle+e^{-ia_{p}+ia+i\Phi_{pkt}(k_{0x},-\gamma,_{x},\gamma_{xp},k_{0y},-\gamma_{y},\gamma_{yp})}(-i\sin\Theta)\hbox
to7.22743pt{}$
$\displaystyle+e^{ia_{p}+ia+i\Phi_{pkt}(k_{0x},-\gamma,_{x},-\gamma_{xp},k_{0y},-\gamma_{y},-\gamma_{yp})}(+\cos\Theta)\bigg{)}.\hbox
to7.22743pt{}$
The Boltzmann distribution function for motion in the $y$ direction is
formally the same as that for motion in the $x$ direction and the net particle
distribution function that depends on $x$ and $y$ is just a product of two
similar exponential functions. Similarly, the quantum mechanical probability
that depends on the two transverse coordinates $(x,y)$ is just a product of
two functions of the same form. If we let $P_{ija}$ or $P_{ijb}$ denote
integrals over $x$ as in Eq. (IV), and $Q_{ija}$ or $Q_{ijb}$ denote
corresponding integrals over $y$, then Eqs. (IV-IV) are valid when we make the
replacements
$P_{ija}\rightarrow P_{ija}Q_{ija};\quad P_{ijb}\rightarrow P_{ijb}Q_{ijb}.$
(106)
The discussion of detection is unchanged.
When numerically averaging over a circular entry aperture, $x$ and $y$ are
restricted to
$(x^{2}+y^{2})^{1/2}\leq r_{a};\quad(x_{c}^{2}+y_{c}^{2})^{1/2}\leq r_{a}.$
(107)
For thermal averaging over the initial velocity distributions, the transverse
part of the Boltzmann distribution takes the following two-dimensional form:
$\displaystyle f(x,v_{x},y,v_{y},0)=f_{n}(x,v_{x},0)f_{n}(y,v_{y},0)$
$\displaystyle=\frac{m}{(2\pi\sigma_{n})^{2}k_{B}T}e^{-\frac{m}{2k_{B}T_{n}}(v_{x}^{2}+v_{y}^{2})}$
$\displaystyle\times
e^{-\frac{(x-v_{x}t_{L})^{2}}{2\sigma_{n}^{2}}}e^{-\frac{(y-v_{y}t_{L})^{2}}{2\sigma_{n}^{2}}}.$
(108)
## XI 11\. Results: Two-Dimensional Model
In Figure 6 we plot the results for transmission through circular apertures of
radius 5 mm for temperature $T=0.5\mu$K, as a function of toss height. This
should be compared with the one-dimensional results plotted in Figure 3.
Figure 6: Fractional frequency shift vs. toss height for NIST F-2 at
temperature $0.5\mu$K; full two-dimensional calculation.
Figure 7 plots the fractional frequency shift at a fixed height, as a function
of the temperature.
Figure 7: Fractional frequency shift for NIST F-2 at a toss height 0.75
meters, as a function of temperature.
In Figure 8 we plot the probability of arrival at the detector, of a state-
selected and launched atom, as a function of toss height. The fraction
decreases as toss height increases because the atom ball has more time to
spread out due to the distribution of thermal velocities. Figure 9 shows the
frequency shift dependence on initial packet width for a toss height of 0.47 m
at temperature $0.5\mu$K.
Figure 8: Fraction of atoms arriving at the detector as a function of toss
height; two dimensional case.
The wavepacket widths at launch are unknown; if chosen to be too small,
quantum mechanical spreading will become very important and loss of atoms due
to clipping by the apertures will become significant. In Figure 9 we plot the
fractional frequency shift as a function of the width parameter $\sigma$. (The
de Broglie wavelength of a Cesium atom at $0.5\mu$K is about 400 nm.) If the
assumed width is larger than about 200 nm, which is the value we have used in
all our other calculations, the shift is essentially independent of $\sigma$.
Figure 10 plots the fractional frequency shifts as a function of toss height
for NIST F-2 for $\pi/2$, $3\pi/2$, and $5\pi/2$ pulses.
Figure 9: Fractional frequency shift for a toss height of 0.47 m and
temperature .5$\mu$K as a function of initial packet width. Figure 10:
Fractional frequency shift as a function of toss height for $\pi/2,\ 3\pi/2,\
5\pi/2$ pulses.
Table I provides values of the fractional frequency shift as a function of RF
amplitude corresponding to $b=\pi/4\tau,\ 3\pi/4\tau,\ 5\pi/4\tau$, and
$7\pi/4\tau$ pulses for different toss heights. For $b=3\pi/4\tau$ and
$7\pi/4\tau$, the $n\pi/2$ the shifts are negative. The dependence on applied
RF amplitude is similar to that of the microwave leakage frequency shift and
is accounted for during normal evaluation of the NIST fountains. The
fractional frequency shifts at zero applied rf amplitude, calculated using the
method described in this paper, are zero at all launch heights.
Table 1: Fractional shifts $\times 10^{17}$ for various launch heights as a function of applied microwave field amplitude. The last line contains the values extrapolated to zero applied rf amplitude. | height(m)
---|---
b | 0.47 | 0.75 | 1.00
$\pi/4\tau$ | 2.78 | 2.07 | 1.69
$3\pi/4\tau$ | -8.56 | -6.39 | -5.22
$5\pi/4\tau$ | 15.1 | 11.3 | 9.20
$7\pi/4\tau$ | -23.3 | -17.4 | -14.2
0 (fit) | -1.2 | -0.9 | -0.7
## XII 12\. Conclusions
For the parameters for NIST F-1 and F-2 Cesium fountains–e.g., cavity radius,
configuration of detectors, aperture size, etc., we have not found any
conditions such that frequency shifts due to transverse field gradients are
large enough to be significant in the systematic error budget. We have
accounted for many factors that make the present calculation realistic. These
include temperature and spatial distributions in the launched atom balls,
quantum mechanical spreading of the atomic wave packets, clipping of
probability distributions at aperture boundaries, distances between trapping
regions, cavities, and detectors that affect times of passage of atoms through
the cavities and the time spent in the drift region, transverse motion of
atoms in the drift region that results in their sampling different values of
the transverse field gradients, off-axis values of the Rabi pedestal, as well
as a full two-dimensional theory of atomic trajectories through the cavities
based on an exact Green’s function solution of the Schrödinger equation in the
presence of a transverse field gradient.
If there is a weakness in this approach, it is that use of the Green’s
function solutions require integrations from $-\infty$ to $+\infty$ in the
transverse direction whereas such directions are limited by the cavity
apertures. However wavepackets are concentrated in a very small region and
their distributions rapidly approach zero away from their centroids;
consequently the contributions to such integrals outside of the apertures are
extremely small, but have not been neglected
## XIII APPENDIX
The integral (29) may be evaluated with the aid of a convergence factor that
is allowed to approach zero at the end of the calculation. Displaying only the
factors that participate in the integration, we have
$\displaystyle\lim_{\alpha\to 0}\sqrt{\frac{m}{2\pi i\hbar T}}\int
dx^{\prime}e^{\frac{im(x-x^{\prime})^{2}}{2\hbar
T}-\alpha^{2}(x-x^{\prime})^{2}+i(k\mp\frac{2\gamma\tau}{\hbar\pi})(x-x^{\prime})}$
$\displaystyle=\lim_{\alpha\to 0}\sqrt{\frac{m}{2\pi i\hbar
T}}\sqrt{2\pi}\sqrt{\frac{\hbar T}{-im+2\alpha^{2}\hbar T}}\hbox
to72.26999pt{}$ $\displaystyle\times
e^{(k\mp\frac{2\gamma\tau}{\hbar\pi})^{2}/(2im/(\hbar T)-2\alpha^{2})}\hbox
to93.95122pt{}$ $\displaystyle=\exp\big{(}-i\hbar
T(k\mp\frac{2\gamma\tau}{\hbar\pi})^{2}/(2m)\big{)}\hbox to36.135pt{}.$ (109)
When combined with the other phase factors in (27) this yields (30).
### XIII.1
#### XIII.1.1
## References
* Wynands and Weyers (2005) R. Wynands and S. Weyers, Metrologia 42, S64 (2005).
* Wolf and Bordé (2004) P. Wolf and C. J. Bordé, arXiv:quant-ph/0403194 (2004).
* Li _et al._ (2011) R. Li, K. Gibble, and K. Szymaniec, Metrologia 48, 283 (2011).
* Gibble (2006) K. Gibble, Phys. Rev. Letts. 97, 073002 (2006).
* Cook (1978) R. J. Cook, Phys. Rev. Letts. 41, 1788 (1978).
* Cook (1987) R. H. Cook, Phys. Rev. A 35, 3844 (1987).
* Heavner _et al._ (2005) T. P. Heavner, S. R. Jefferts, E. A. Donley, J. H. Shirley, and T. E. Parker, Metrologia 42, 411 (2005).
* Scully _et al._ (1989) M. O. Scully, B.-O. Englert, and J. Schwinger, Phys. Rev. A 40, 1775 (1989).
* Shirley _et al._ (2001) J. H. Shirley, W. D. Lee, and R. E. Drullinger, Metrologia 38, 427 (2001).
* Schwinger _et al._ (1988) J. Schwinger, M. O. Scully, and B.-G. Englert, Zeits. Phys. D 10, 135 (1988).
* Englert _et al._ (1988) B.-G. Englert, J. Schwinger, and M. O. Scully, Found. Phys. 18, 1045 (1988).
* Shulman (1981) L. S. Shulman, _Techniques and Applications of Path-Integral Methods_ (Wiley, 1981) p. 38.
* Kennard (1927) E. H. Kennard, Zeits. Phys. 44, 326 (1927).
|
arxiv-papers
| 2014-04-15T22:43:34 |
2024-09-04T02:50:01.230929
|
{
"license": "Public Domain",
"authors": "Neil Ashby, Stephan Barlow, Thomas Heavner, Steven Jefferts",
"submitter": "Neil Ashby",
"url": "https://arxiv.org/abs/1404.4101"
}
|
1404.4148
|
∎
11institutetext: A. Bensoussan 22institutetext: International Center for
Decision and Risk Analysis,Jindal School of Management, The University of
Texas at Dallas
Department of Systems Engineering and Engineering Management, College of
Science and Engineering, City University of Hong Kong
22email: [email protected] 33institutetext: M. H. M. Chau, S. C. P.
Yam44institutetext: Department of Statistics, The Chinese University of Hong
Kong
44email: [email protected], [email protected]
# Mean Field Games with a Dominating Player
A. Bensoussan M. H. M. Chau S. C. P. Yam
(Received: date / Accepted: date)
###### Abstract
In this article, we consider mean field games between a dominating player and
a group of representative agents, each of which acts similarly and also
interacts with each other through a mean field term being substantially
influenced by the dominating player. We first provide the general theory and
discuss the necessary condition for the optimal controls and equilibrium
condition by adopting adjoint equation approach. We then present a special
case in the context of linear-quadratic framework, in which a necessary and
sufficient condition can be asserted by stochastic maximum principle; we
finally establish the sufficient condition that guarantees the unique
existence of the equilibrium control. The proof of the convergence result of
finite player game to mean field counterpart is provided in Appendix.
Keywords: Mean field games; Dominating player; Wasserstein Metric; Adjoint
equation approach/Stochastic maximum principle; Stochastic Hamilton-Jacobi-
Bellman equations; Linear quadratic; Separation principle; Banach fixed point
theorem.
## 1 Introduction
For long, modeling the joint interactive behaviour of individual
objects(agents) in a large population in various dynamic systems has been one
of the major problems. For instance, physicists often apply the traditional
variational methods in Lagrangian and/or Hamiltonian mechanics to study
interacting particle systems, which left a shortcoming of extremely high
computational cost that made this microscopic approach almost mathematically
intractable. To resolve this matter, a completely different macroscopic
approach from statistical physics had been gradually developed, which
eventually leads to the primitive notion of mean field theory. The novelty of
this approach is that particles interact through a medium, namely the mean
field term, which aggregates by action and reaction on other particles.
Moreover, by passing the number of particles to the infinity in these
macroscopic models, the mean field term become a functional of the density
function which represents the whole population of particles. This leads to
mathematical problems of much less computational complexity.
From the economic perspective, due to the dramatic population growth and rapid
urbanization, urgent needs of in-depth understanding of collective strategic
interactive behavior of a huge group of decision makers is crucial in order to
maintain a sustainable economic growth. Since the vector of fair prices is
determined by both demand and supply, it is natural to utilize the aggregation
effect from the players’ states as a canonical candidate of mean-field term,
and then we employ the mean-field models in place of the corresponding
classical equilibrium models; moreover, as the decision makers control the
evolution of a dynamic system, it is necessary to also incorporate the theory
of stochastic differential games (SDGs) in these mean-field models. Over the
past few decades, the theory of SDGs has been a major research topic in
control theory and financial economics, especially in studying the continuous-
time decision making problem between non-cooperative investors; in regard to
the one-dimensional setting the theory of two person zero-sum games is quite
well-developed via the notion of viscosity solutions, see for example Fleming
and Souganidis (1989). Unfortunately, most interesting SDGs are $N$-player
non-zero sum SDGs; see Bensoussan and Frehse BF1 ; BF2 and Bensoussan et al.
BFV , yet there are still relatively few results in the literature.
As a macroscopic equilibrium model, et al. HCM1 ; HCM2 investigated
stochastic differential game problems involving infinitely many players under
the name “Large Population Stochastic Dynamic Games”; and independently, Lasry
and Lions LL1 ; LL2 ; LL3 studied similar problems from the viewpoint of the
mean-field theory in physics and termed “Mean-Field Games (MFGs)”. As an
organic combination of mean field theory and theory of stochastic differential
games, MFGs provide a more realistic interpretation of individual dynamics at
the microscopic level, so that each player will be able to optimize his
prescribed objectives, yet with the mathematical tractability in a macroscopic
framework. To be more precise, the general theory of MFGs has been built by
combining various consistent assumptions on the following modeling aspects:
(1) a continuum of players; (2) homogeneity in strategic performance of
players; and (3) social interactions through the impact of mean field term.
The first aspect is describing the approximation of a game model with a huge
number of players by a continuum one yet with a sufficient mathematical
tractability. The second aspect is assuming that all players obey the same set
of rules of the interactive game, which provide guidance on their own behavior
that potentially leads them to optimal decisions. Finally, due to the
intrinsic complexity of the society in which the players participate in, the
third aspect is explaining the fact that each player is so negligible and can
only affect others marginally through his own infinitesimal contribution to
the society. In a MFG, each player will base his decision making purely on his
own criteria and certain summary statistics (that is, the mean field term)
about the community; in other words, in explanation of their interactions, the
pair of personal and mean-field characteristics of the whole population is
already sufficient and exhaustive. Mathematically, each MFG will possess the
following forward-backward structure: (1) a forward dynamic describes the
individual strategic behavior; (2) a backward equation describes the evolution
of individual optimal strategy, such as those in terms of the individual value
function via the usual backward recursive techniques. For the detail of the
derivation of this system of equations with forward-backward feature, one can
consult from the works of Huang et al. HCM2 , Lasry and Lions LL1 ; LL2 ; LL3
and Bensoussan et al. BFY .
In this article, we consider a class of MFG problems, in which there is a
‘significantly big’ player playing together with a huge group of ‘small’
players. The first work along this direction under a Linear Quadratic setting
has been investigated by Huang H0 . In their following work HN1 , the authors
regard the mean field term, represented by the conditional expectation of the
small agent, as exogenous to the whole control problem for both the big (the
authors called it,‘major’) and small (minor) players. Nourian and Caines NC
consider a similar problem under a generalized framework. However, the authors
also consider the mean field term, which is represented by a conditional
probability measure, as exogenous to the control problem for the major player.
In contrast, we here consider the mean field term as endogenous for the big
(we rephrase as ‘dominating’ in order to emphasize our distinction from the
previous works) player. That is to say, changes in the control of the big
(dominating) player would directly affect and even essentially determine the
mean field term. Our present setting appears to be natural in the economic
literature related to ‘actual’ governance, as the governor can often take up
the initiative or key role on setting up rubrics and regulations to be
followed by citizens. To avoid ambiguity, we here regard the ‘dominating’
major player as a “Dominating Player”, and all other minor players as
“Representative Agents” throughout the whole paper. In our work, we assume
that this dominating player can influence both the mean field term and
representative agents directly. We first discuss the necessary condition for
the optimality under the most general setting in which both the state
coefficients and the objective functions are sufficiently regular (e.g.
differentiable); we then consider the Linear-Quadratic case by applying the
results obtained in the general theory, which results in three adjoint
equations. It is noted that Huang et al. HN2 also considered the non-
stationary case and obtained the intermediary result with only two adjoint
equations, which represents a particular case of our present theory. Besides,
concerning the related fixed point issue in any standard MFG problem in order
to achieve the equilibrium strategy, we here only need to involve one single
affine map, that simplifies much than that in HN2 , in which the authors need
a couple of two similar mappings; apart from the simplicity of the sufficient
condition provided here, it is also directly expressed in terms of the data
(coefficients) of the underlying model.
The paper is organized as follows: In Section 2, we present the general theory
of the Mean Field Games in the presence of a dominating Player, in which both
the state coefficients and the objective functions are sufficiently regular.
The necessary condition for optimality and equilibrium is also provided there.
Firstly, solving for the control problem of the representative agent, and then
the equilibrium condition leads to a coupled Hamilton-Jacobi-Bellman and
Fokker Planck equations. As the mean field term is endogenous to the
dominating Player, in order to achieve an optimal control, he/she should take
into account of the coupled equations when deciding his own controlling
strategy. The related fixed point problem is described by six equations. In
Section 3, we study a special case with linear states together with linear
quadratic objective functions. Due to natural coerciveness of the problem
formulation, a necessary and sufficient condition for the optimality can be
guaranteed. We write down both the stochastic maximum principle and the
corresponding adjoint equations. In Section 4, the corresponding fixed point
problem is then tackled by considering the related Riccati equation, with
which the equilibrium could be achieved. We then provide a ‘practical’
sufficient condition, which only involves the data (coefficients) of the model
without referring to any specific solution of any Riccati equations, for the
existence of the equilibrium strategy. In Appendix, proof of the approximate
Nash equilibrium for the general setting is also provided.
## 2 General Theory
Consider a probability space $(\Omega,\mathcal{F},P)$, a fixed terminal time
$T$ and two independent standard Brownian motion $W_{0}(t)$ and $W_{1}(t)$
taking values in $\mathbb{R}^{d_{0}}$ and $\mathbb{R}^{d_{1}}$ respectively.
Also consider two independent initial square integrable random variables
$\xi_{0}\in\mathbb{R}^{n_{0}}$ and $\xi_{1}\in\mathbb{R}^{n_{1}}$, which are
also assumed to be independent of both $W_{0}(t)$ and $W_{1}(t)$. Define the
filtrations as follows, in which $\mathcal{F}_{t}^{0}$ and
$\mathcal{F}_{t}^{1}$ are clearly independent to each other,
$\begin{split}\mathcal{F}_{t}^{0}&:=\sigma(\xi_{0},W_{0}(s),s\leq t),\\\
\mathcal{F}_{t}^{1}&:=\sigma(\xi_{1},W_{1}(s),s\leq t),\\\
\mathcal{G}_{t}&:=\mathcal{F}_{t}^{0}\vee\mathcal{F}_{t}^{1}.\end{split}$
Let $\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$ be the space of probability measures
equipped with the $2^{nd}$ Wasserstein metric (for example, see villani ),
$W_{2}(\cdot,\cdot)$ such that for any $\mu$ and $\nu$ in
$\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$,
$W_{2}(\nu_{1},\nu_{2}):=\inf_{\gamma\in\Gamma(\nu_{1},\nu_{2})}\bigg{(}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}|x-y|^{2}d\gamma(x,y)\bigg{)}^{\frac{1}{2}},$
where the infimum is taken over the family $\Gamma(\nu_{1},\nu_{2})$, the
collection of all joint measures with respective marginals $\nu_{1}$ and
$\nu_{2}$. Denote $d\lambda$ to be the Lebesgue measure on
$\mathbb{R}^{n_{1}}$.
Denote $x_{0}(t)\in\mathbb{R}^{n_{0}}$ and $x_{1}(t)\in\mathbb{R}^{n_{1}}$ the
state evolutions for the dominating player and a representative agent
respectively whose dynamics are given by the following stochastic differential
equations (SDEs),
$\begin{split}\left\\{\begin{array}[]{rcl}dx_{0}&=&g_{0}\Big{(}x_{0}(t),\mu(t),u_{0}(x_{0}(t),t)\Big{)}dt+\sigma_{0}\Big{(}x_{0}(t)\Big{)}dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\
dx_{1}&=&g_{1}\Big{(}x_{1}(t),x_{0}(t),\mu(t),u_{1}(x_{1}t)\Big{)}dt+\sigma_{1}\Big{(}x_{1}(t)\Big{)}dW_{1}(t),\\\
x_{1}(0)&=&\xi_{1}.\end{array}\right.\end{split}$ (1)
The functional coefficients are defined as follows:
$\left\\{\begin{array}[]{rl}&g_{0}:\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathbb{R}^{m_{0}}\rightarrow\mathbb{R}^{n_{0}},\\\
&g_{1}:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathbb{R}^{m_{1}}\rightarrow\mathbb{R}^{n_{1}},\\\
&\sigma_{0}:\mathbb{R}^{n_{0}}\rightarrow\mathbb{R}^{n_{0}\times d_{0}},\\\
&\sigma_{1}:\mathbb{R}^{n_{1}}\rightarrow\mathbb{R}^{n_{1}\times d_{1}}.\\\
\end{array}\right.$ (2)
The dominating player and the representative agents also possess the following
objective functionals respectively:
$\begin{array}[]{rcl}J_{0}(u_{0})&=&\mathbb{E}\left[\int_{0}^{T}f_{0}\Big{(}x_{0}(t),\mu(t),u_{0}(t)\Big{)}dt+h_{0}\Big{(}x_{0}(T),\mu(T)\Big{)}\right],\\\
J_{1}(u_{1},x_{0},\nu)&=&\mathbb{E}\left[\int_{0}^{T}f_{1}\Big{(}x_{1}(t),x_{0}(t),\mu(t),u_{1}(t)\Big{)}dt+h_{1}\Big{(}x_{1}(T),x_{0}(T),\mu(T)\Big{)}\right].\end{array}$
The functions are defined as follows:
$\left\\{\begin{array}[]{rl}&f_{0}:\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathbb{R}^{m_{0}}\rightarrow\mathbb{R},\\\
&f_{1}:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\times\mathbb{R}^{m_{1}}\rightarrow\mathbb{R},\\\
&h_{0}:\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\rightarrow\mathbb{R},\\\
&h_{1}:\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{0}}\times\mathcal{P}_{2}(\mathbb{R}^{n_{1}})\rightarrow\mathbb{R}.\\\
\end{array}\right.$ (3)
Here $u_{0}\in\mathbb{R}^{m_{0}}$ and $u_{1}\in\mathbb{R}^{m_{1}}$ represent
the respective controls of the dominating player and the representative agent.
The controls $u_{0}$ and $u_{1}$ are respectively adapted to the filtrations
$\mathcal{F}_{t}^{0}$ and $\mathcal{G}_{t}$. We further assume that the
functional form (being a function of $(x_{1}(t),t)$) of $u_{1}$ is adapted to
$\mathcal{F}_{t}^{0}$ and uniformly Lipschitz in $x_{1}(t)$, even though its
value evaluated at $x_{1}(t)$ would be adapted to $\mathcal{G}_{t}$ instead.
Loosely speaking the dominating player takes his own privilege of setting up
the framework to be followed by the representative agent. We shall then define
the classes of admissible controls for the dominating player and the
representative agent by $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ respectively,
where $\mathcal{A}_{0}$ (resp. $\mathcal{A}_{1}$) is a subset of
$\mathcal{F}^{0}-$ (resp. $\mathcal{G}-$)progressively measurable process
which are in $\mathcal{L}^{2}(\Omega\times[0,T];\mathbb{R}^{m_{0}})$ (resp.
$\mathcal{L}^{2}(\Omega\times[0,T];\mathbb{R}^{m_{1}})$).
The mean field term, $\mu(t)\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$, is the
probability measure of the state of the representative agent at time $t$.
Indeed, the dominating player sets rules for representative agent to take into
account. One natural consideration is that the dominating player is incapable
of tracing the state of each individual’s evolution, but only takes account of
the overall performance of the community subject to the rules he set, that is
his own flow of information, $\mathcal{F}^{0}_{t}$. By the same token, each
agent cannot fully keep track of any other agents’ states and they can only
rely on the summarized information of the community provided by the dominating
player. Thus it is justifiable to assume that the mean field term, $\mu(t)$,
is adapted to $\mathcal{F}^{0}_{t}$. The dominating player can directly
influence both the representative agent and the mean field term, thus we
consider $\mu(t)$ as endogenous in the consideration of optimal behavior of
$x_{0}(t)$ rather than as an exogenous variable commonly found in the
literature such as that of H0 ; NC .
For any probability measure $\mu\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$, we
write $M_{2}(\mu)=(\int_{\mathbb{R}^{n_{1}}}|x|^{2}d\mu(x))^{\frac{1}{2}}$. We
first give the following assumptions on the functional coefficients:
(A.1) Lipschitz Continuity
$g_{0}$, $\sigma_{0}$, $g_{1}$ and $\sigma_{1}$ are globally Lipschitz
continuous in all arguments. In particular, there exists $K>0$, such that
$\begin{array}[]{rcl}|g_{0}(x_{0},\mu,u_{0})-g_{0}(x_{0}^{\prime},\mu^{\prime},u_{0}^{\prime})|&\leq&K\bigg{(}|x_{0}-x_{0}^{\prime}|+W_{2}(\mu,\mu^{\prime})+|u_{0}-u_{0}^{\prime}|\bigg{)};\\\
|\sigma_{0}(x_{0})-\sigma_{0}(x_{0}^{\prime})|&\leq&K|x_{0}-x_{0}^{\prime}|.\\\
|g_{1}(x_{1},x_{0},\mu,u_{1})-g_{1}(x_{1}^{\prime},x_{0}^{\prime},\mu^{\prime},u_{1}^{\prime})|&\leq&K\bigg{(}|x_{1}-x_{1}^{\prime}|+|x_{0}-x_{0}^{\prime}|+W_{2}(\mu,\mu^{\prime})+|u_{1}-u_{1}^{\prime}|\bigg{)};\\\
|\sigma_{1}(x_{1})-\sigma_{1}(x_{1}^{\prime})|&\leq&K|x_{1}-x_{1}^{\prime}|.\\\
\end{array}$
(A.2) Linear Growth
$g_{0}$, $\sigma_{0}$, $g_{1}$ and $\sigma_{1}$ are of linear growth in all
arguments. In particular, there exists $K>0$, such that
$\begin{split}|g_{0}(x_{0},\mu,u_{0})|&\leq
K(1+|x_{0}|+M_{2}(\mu)+|u_{0}|);\\\ |\sigma_{0}(x_{0})|&\leq K(1+|x_{0}|).\\\
|g_{1}(x_{1},x_{0},\mu,u_{1})|&\leq
K(1+|x_{0}|+|x_{1}|+M_{2}(\mu)+|u_{1}|);\\\ |\sigma_{1}(x_{1})|&\leq
K(1+|x_{1}|).\\\ \end{split}$
(A.3) Quadratic Condition on the Cost Functional (See (A.5) in Carmona and
Delarue CAR_PROB .)
There exists $K>0$, such that
$\begin{split}|f_{1}(x_{1},x_{0},\mu,u_{1})-f_{1}(x_{1}^{\prime},x_{0}^{\prime},\mu^{\prime},u_{1}^{\prime})|\leq&K\Big{[}1+|x_{1}|+|x_{1}^{\prime}|+|x_{0}|+|x_{0}^{\prime}|+|M_{2}(\mu)|+|M_{2}(\mu^{\prime})|+|u_{1}|+|u_{1}^{\prime}|\Big{]}\\\
&\qquad\cdot\Big{[}|x_{1}-x_{1}^{\prime}|+|x_{0}-x_{0}^{\prime}|+W_{2}(\mu,\mu^{\prime})+|u_{1}-u_{1}^{\prime}|\Big{]};\\\
|h_{1}(x_{1},x_{0},\mu)-h_{1}(x_{1}^{\prime},x_{0}^{\prime},\mu^{\prime})|\leq&K\Big{[}1+|x_{1}|+|x_{1}^{\prime}|+|x_{0}|+|x_{0}^{\prime}|+|M_{2}(\mu)|+|M_{2}(\mu^{\prime})|\Big{]}\\\
&\qquad\cdot\Big{[}|x_{1}-x_{1}^{\prime}|+|x_{0}-x_{0}^{\prime}|+W_{2}(\mu,\mu^{\prime})\Big{]}.\end{split}$
(4)
Under the assumptions A.1-A.3, we show in the Appendix that if we have the
mean field term coincides with the probability measure of $x_{1}(t)$
conditioning on $\mathcal{F}^{0}_{t}$, then the optimization problem for the
representative agent constitutes to a Mean Field Game. In general, it is more
convenient to compare two probability measures if they possess density
functions on $\mathbb{R}^{n_{1}}$. We define the second order operator $A_{1}$
and its adjoint $A_{1}^{*}$ by
$\begin{split}A_{1}\varphi(x,t)&=-\text{tr}\Big{(}a_{1}(x)D^{2}\varphi(x,t)\Big{)},\\\
A_{1}^{*}\varphi(x,t)&=-\sum_{i,j=1}^{n_{1}}\frac{\partial^{2}}{\partial{x_{i}}\partial{x_{j}}}\Big{(}a_{1}^{ij}(x)\varphi(x,t)\Big{)},\end{split}$
where $a_{1}(x)=\frac{1}{2}\sigma_{1}(x)\sigma_{1}(x)^{*}$ is a positive
definite matrix. Let $x_{1}=x_{1}^{u_{1}}$ be the solution of the SDE for the
representative agent with respect to control $u_{1}$. For any test function
$f$, by Itô’s lemma,
$\mathbb{E}^{\mathcal{F}^{0}_{t}}[f(x_{1}^{u_{1}}(t))]=\mathbb{E}^{\mathcal{F}^{0}_{t}}\bigg{[}f(\xi_{1})+\displaystyle\int_{0}^{t}\big{(}\partial_{t}+g\cdot
D+A_{1}\big{)}fds\bigg{]},$ (5)
The conditional density function $p^{u_{1}}(\cdot,t)$ of $x_{1}^{u_{1}}(t)$
(if exists), i.e.
$\mathbb{E}^{\mathcal{F}^{0}_{t}}[f(x_{1}^{u_{1}}(t))]=\int_{\mathbb{R}^{n_{1}}}f(x)p^{u_{1}}(x,t)dx$,
would be given by the Fokker-Planck (FP) equation
$\begin{split}\left\\{\begin{array}[]{rl}\dfrac{\partial p^{u_{1}}}{\partial
t}&=-A_{1}^{*}p^{u_{1}}(x,t)-\text{div}\Big{(}g_{1}\Big{(}x,x_{0}(t),\mu(t),u_{1}(x,t)\Big{)}p^{u_{1}}(x,t)\Big{)},\\\
p^{u_{1}}(x,0)&=\omega(x),\end{array}\right.\end{split}$ (6)
where $\omega(x)$ is the initial density function of $\xi_{1}$. We will
justify the existence and regularities of the conditional density function
$p^{u_{1}}$ later. We first assume
$p^{u_{1}}(\cdot,t)\in\mathcal{L}^{2}(\mathbb{R}^{n_{1}})$ and
$p^{u_{1}}(\cdot,t)d\lambda\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$. For any
density $m$, we may write $md\lambda=m$ if no ambiguity arises. We impose
further assumptions on the functional coefficients:
(A.4)
$g_{0},f_{0},h_{0},g_{1},f_{1}$ and $h_{1}$ are continuously differentiable in
(if the argument exist) $x_{0}\in\mathbb{R}^{n_{0}}$,
$x_{1}\in\mathbb{R}^{n_{1}}$ $u_{0}\in\mathbb{R}^{m_{0}}$,
$u_{1}\in\mathbb{R}^{m_{1}}$ with bounded derivatives. We will denote, for
example, the derivative of $g_{0}$ with respect to $x_{0}$ by $g_{0,x_{0}}$.
They are also Gâteaux differentiable in
$\mu=md\lambda\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$, for example, for
$m\in\mathcal{L}^{2}(\mathbb{R}^{n_{1}})$,
$\and{\dfrac{d}{d\theta}}{\theta=0}g_{0}(x_{i},(m+\theta\tilde{m})d\lambda,u_{i})=\int_{\mathbb{R}^{n}}\frac{\partial
g_{0}}{\partial m}(x_{i},md\lambda,u_{i})(\xi)\tilde{m}(\xi)d\xi,$
for some $\frac{\partial g_{0}}{\partial
m}(x_{i},md\lambda,u_{i})\in\mathcal{L}^{2}(\mathbb{R}^{n_{1}})$.
(A.5)
$\sigma_{0}$ (resp. $\sigma_{1}$) is twice continuously differentiable in
$x_{0}\in\mathbb{R}^{n_{0}}$ (resp. $x_{1}\in\mathbb{R}^{n_{1}}$) with bounded
first order and second order derivative.
###### Remark
With the regularities on the coefficients, if we have the initial density
$\omega(x)\in\mathcal{L}^{2}\cap\mathcal{L}^{\infty}(\mathbb{R}^{n_{1}})$,
then the FP equation (6) admits a unique solution
$p^{u_{1}}\in\mathcal{L}^{\infty}([0,T],\mathcal{L}^{2}\cap\mathcal{L}^{\infty}(\mathbb{R}^{n_{1}}))$.
See Proposition $4$ and $5$ in Le Bris and Lions BL for details.
Define then a pair of mutually dependent control problems for the dominating
player and the representative agent as below:
###### Problem 2.1
Control of Representative Agent
Given the process $x_{0}$ and an exogenous probability measure-valued process
$\nu$ (adapted to $\mathcal{F}^{0}_{t}$), find a control
$u_{1}\in\mathcal{A}_{1}$ which minimizes the cost functional
$J_{1}(u_{1},x_{0},\nu):=\mathbb{E}\left[\int_{0}^{T}f_{1}\Big{(}x_{1}(t),x_{0}(t),\nu(t),u_{1}(t)\Big{)}dt+h_{1}\Big{(}x_{1}(T),x_{0}(T),\nu(T)\Big{)}\right].$
(7)
###### Problem 2.2
Equilibrium Condition
Given an exogenous probability measure-valued process $\nu$, let
$\mathcal{M}(\nu)(t)$ be the measure induced by the corresponding optimal
$x_{1}(t)$ found in Problem 2.1 conditioning on $\mathcal{F}_{t}^{0}$. Find
the probability measure-valued process $\mu$ such that the fixed point
property is satisfied: $\mathcal{M}(\mu)(\cdot)=\mu(\cdot)$.
###### Problem 2.3
Control of the Dominating Player
Find a control $u_{0}\in\mathcal{A}_{0}$ which minimizes the cost functional
$J_{0}(u_{0}):=\mathbb{E}\left[\int_{0}^{T}f_{0}\Big{(}x_{0}(t),\mu(t),u_{0}(t)\Big{)}dt+h_{0}\Big{(}x_{0}(T),\mu(T)\Big{)}\right],$
(8)
where $\mu$ is the solution given in Problem 2.2.
###### Remark
The setting of our problem is different from those mean field related problems
with a major player (not a dominating player) commonly found in the
literature, such as that in H0 ; NC . For example, in H0 , the corresponding
objective functions for the major player and i-th minor player are
respectively
$\begin{array}[]{rcl}J_{0}(u_{0},z)&=&\mathbb{E}\int_{0}^{T}\left\\{|x_{0}-H_{0}z-\eta|^{2}_{Q_{0}}+u_{0}^{*}R_{0}u_{0}\right\\}dt,\\\
J_{i}(u_{i},z)&=&\mathbb{E}\int_{0}^{T}\left\\{|x_{i}-Hx_{0}-\hat{H}_{0}z-\eta|^{2}_{Q}+u^{*}Ru\right\\}dt,\\\
\end{array}$
where the mean field term $z$ is exogenous to both control optimization
problems for $J_{0}$ and $J_{i}$. Instead, we here consider the mean field
term $\nu$, as established in Problem 2.2, as endogenous for the dominating
player in Problem 2.3. In particular, changes in control $u_{0}$ would affect
and even completely determine the mean field term $\nu$ accordingly. Our
setting appears to be natural in the economic context related to governance,
as the governor can sometimes take the initiative to set-up rubrics to be
obeyed and followed by citizens; this latter notion is covered in ORIGINCRISIS
.
We first establish the necessary condition of optimality for the
representative agent (Problem 2.1) by the adjoint equation approach. After
resolving Problem 2.1, we solve for the fixed point in Problem 2.2. Recall
that $x_{0}(t)$, the functional form of $u_{1}$ (conditioning on
$\mathcal{F}_{0}^{t}$, $u_{1}$ is a function of $(x_{1}(t),t)$) and now
together with the input measure-valued process $\nu$ are all adapted to
$\mathcal{F}_{t}^{0}$. We can then rewrite the cost functional (7) for the
representative agent as
$\begin{split}J_{1}(u_{1},x_{0},\nu)=&\mathbb{E}\bigg{[}\int_{0}^{T}\mathbb{E}^{\mathcal{F}_{t}^{0}}f_{1}\Big{(}x_{1}(t),x_{0}(t),\nu(t),u_{1}(x_{1}(t),t)\Big{)}dt+\mathbb{E}^{\mathcal{F}_{T}^{0}}h_{1}\Big{(}x_{1}(T),x_{0}(T),\nu(T)\Big{)}\bigg{]}\\\
=&\mathbb{E}\bigg{[}\int_{0}^{T}\int_{\mathbb{R}^{n_{1}}}p_{u_{1}}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}dxdt+\int_{\mathbb{R}^{n}}p_{u_{1}}(x,T)h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}dx\bigg{]}.\end{split}$
###### Lemma 2.4
(Necessary condition for Problem 2.1)
Given $x_{0}$ and $\nu$ as in Problem 2.1, the control
$\hat{u}_{1}\in\mathcal{A}_{1}$ is optimal only if it satisfies the following
SHJB:
$\left\\{\begin{array}[]{rl}-\partial_{t}\Psi=&\Big{(}H_{1}(x,x_{0}(t),\nu(t),D\Psi(x,t))-A_{1}\Psi(x,t)\Big{)}dt-
K_{\Psi}(x,t)dW_{0}(t),\\\ \Psi(x,T)=&h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)},\\\
\end{array}\right.$ (9)
where
$\begin{split}H_{1}(x,x_{0},\nu,q)&=\inf_{u_{1}}L(x,x_{0},\nu,u_{1},q),\\\
L(x,x_{0},\nu,u_{1},q)&=f_{1}(x,x_{0},\nu,u_{1})+qg_{1}(x,x_{0},\nu,u_{1}).\\\
\end{split}$
and the infimum is uniquely attained at $\hat{u}_{1}$, i.e.
$H_{1}(x,x_{0},\nu,q)=L(x,x_{0},\nu,\hat{u}_{1},q)$.
###### Remark
As in the work in Carmona et al. CAR_PROB , one convenient set of assumptions
on $g_{1}$,$f_{1}$ and $h_{1}$ which ensures the unique existence of the
minimizer, $\hat{u}_{1}(x,x_{0},\nu,q)={\arg\min}_{u}L(x,x_{0},\nu,u,q)$, is
the affine and convexity assumption. See Lemma $2.1$ in CAR_PROB for more
details. In particular, for
$x_{1}\in\mathbb{R}^{n_{1}};x_{0}\in\mathbb{R}^{n_{0}};\mu\in\mathcal{P}_{2}(\mathbb{R}^{n_{1}})$
and $u_{1},u_{1}^{\prime}\in\mathbb{R}^{m_{1}}$, there exists $K>0$ such that
1. 1.
$g_{1}(x_{1},x_{0},\mu,u_{1})=g_{1,1}(x_{1},x_{0},\mu)+g_{1,2}\cdot u_{1}$,
2. 2.
$f_{1}(x_{1}^{\prime},x_{0},\mu,u_{1}^{\prime})\geq
f_{1}(x_{1},x_{0},\mu,u_{1})+f^{*}_{1,x_{1}}(x_{1},x_{0},\mu,u_{1})(x_{1}^{\prime}-x_{1})+f^{*}_{1,u_{1}}(x_{1},x_{0},\mu,u_{1})(u_{1}^{\prime}-u_{1})+K|u_{1}^{\prime}-u_{1}|^{2}$.
Moreover, the minimizer $(x,q)\mapsto\hat{u}_{1}(x,x_{0},\mu,q)$ is Lipschitz
continuous, uniformly in $(x_{0},\mu)$. Similar conditions for $g_{0}$,
$f_{0}$ and $h_{0}$ can be assumed to guarantee a unique minimizer for the
Lagrangian of the control problem for the dominating player in Lemma 2.6.
###### Proof
To express a necessary condition for optimality, we adopt the stochastic
maximum principle. In particular, for any $\tilde{u}_{1}\in\mathcal{A}_{1}$,
$\begin{array}[]{rcl}0&=&\and{\dfrac{d}{d\theta}}{\theta=0}J_{1}(\hat{u}_{1}+\theta\tilde{u}_{1},x_{0},\nu)\\\
&=&\and{\dfrac{d}{d\theta}}{\theta=0}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n_{1}}}p_{\hat{u}_{1}+\theta\tilde{u}_{1}}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)+\theta\tilde{u}_{1}(x,t)\Big{)}dxdt\\\
&&\qquad\qquad\qquad+\displaystyle\int_{\mathbb{R}^{n}}p_{\hat{u}_{1}+\theta\tilde{u}_{1}}(x,T)h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}dx\bigg{]}\\\
&=&\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)\Big{)}+p_{\hat{u}_{1}}(x,t)f_{1,u_{1}}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)\Big{)}\tilde{u}_{1}(x,t)dxdt\\\
&&\qquad\qquad\qquad+\displaystyle\int_{\mathbb{R}^{n}}\tilde{p}(x,T)h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}dx\bigg{]},\end{array}$
(10)
where
$\tilde{p}=\and{\frac{d}{d\theta}}{\theta=0}p_{\hat{u}_{1}+\theta\tilde{u}_{1}}$.
By taking derivative with respect to $\theta$ in the FP equation (6), we have
$\begin{split}\left\\{\begin{array}[]{rl}\dfrac{\partial\tilde{p}}{\partial
t}&=-A_{1}^{*}\tilde{p}(x,t)-\text{div}\Big{(}\tilde{u}_{1}(x,t)g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}p_{\hat{u}_{1}}(x,t)\Big{)}\\\
&\qquad\qquad-\text{div}\Big{(}g_{1}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}\tilde{p}(x,t)\Big{)},\\\
\tilde{p}(x,0)&=0.\end{array}\right.\end{split}$
As an adjoint process, we consider the backward stochastic differential
equation
$\left\\{\begin{array}[]{rcl}-\partial_{t}\Psi&=&\Big{\\{}f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}+D\Psi(x,t)g_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}-A_{1}\Psi(x,t)\Big{\\}}dt\\\
&&\qquad-K_{\Psi}(x,t)dW_{0}(t),\\\
\Psi(x,T)&=&h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}.\end{array}\right.$
We consider the inner product
$\begin{array}[]{rcl}&&d\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)\Psi(x,t)dx\\\
&=&\displaystyle\int_{\mathbb{R}^{n_{1}}}\Big{\\{}-A_{1}^{*}\tilde{p}(x,t)-\text{div}\Big{(}\tilde{u}_{1}(x,t)g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}p_{\hat{u}_{1}}(x,t)\Big{)}\\\
&&\qquad\qquad-\text{div}\Big{(}g_{1}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}\tilde{p}(x,t)\Big{)}\Big{\\}}\Psi(x,t)dxdt\\\
&&-\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)\Big{\\{}f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}+D\Psi(x,t)g_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}-A_{1}\Psi(x,t)\Big{\\}}dxdt\\\
&&+\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)K_{\Psi}(x,t)dxdW_{0}(t)\\\
&=&\displaystyle\int_{\mathbb{R}^{n_{1}}}\Big{(}\tilde{u}_{1}(x,t)g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}p_{\hat{u}_{1}}(x,t)\Big{)}D\Psi(x,t)dxdt\\\
&&-\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}dxdt+\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)K_{\Psi}(x,t)dxdW_{0}(t).\\\
\end{array}$
Taking integration on $[0,T]$ and expectation on both sides yields
$\begin{array}[]{rcl}&&\mathbb{E}\bigg{[}\displaystyle\int_{\mathbb{R}^{n}}\tilde{p}(x,T)h_{1}\Big{(}x,x_{0}(T),\nu(T)\Big{)}dx\bigg{]}\\\
&=&\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\displaystyle\int_{\mathbb{R}^{n_{1}}}\Big{(}\tilde{u}_{1}(x,t)g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}p_{\hat{u}_{1}}(x,t)\Big{)}D\Psi(x,t)dxdt\\\
&&\qquad-\displaystyle\int_{0}^{T}\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{p}(x,t)f_{1}\Big{(}x,x_{0}(t),\nu(t),u_{1}(x,t)\Big{)}dxdt\bigg{]}.\end{array}$
Together with the first order condition (10), we have
$0=\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\displaystyle\int_{\mathbb{R}^{n_{1}}}\tilde{u}_{1}(x,t)\Big{[}g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}D\Psi(x,t)+f_{1,u_{1}}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)\Big{)}\Big{]}p_{\hat{u}_{1}}(x,t)dxdt\bigg{]}.$
Recall that the $p_{\hat{u}_{1}}(\cdot,t)$ is a conditional probability
density function and hence non-negative, and $\tilde{u}_{1}$ is an arbitrary
Markovian control, we have $\hat{u}_{1}$ is optimal only if
$g_{1,u_{1}}\Big{(}x,x_{0}(t),\mu(t),\hat{u}_{1}(x,t)\Big{)}D\Psi(x,t)+f_{1,u_{1}}\Big{(}x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t)\Big{)}=0,a.e.(x,t).$
With the definition of $L$ in the theorem, the condition becomes
$L_{u_{1}}(x,x_{0}(t),\nu(t),\hat{u}_{1}(x,t),D\Psi(x,t))=0,a.e.(x,t),$
which provides a necessary condition for the minimization problem. As the
minimizer is assumed to be attained at $\hat{u}_{1}$, which depends on $x$,
$x_{0}$, $\nu$, and $D\Psi$, we arrive for the SHJB Equation. ∎
Replace the arbitrary measure $\nu$ by the mean field measure $\mu$. Equating
$\mu:=m_{x_{0}}d\lambda$ with $p_{\hat{u}_{1}}d\lambda$, the measure of the
optimal state of the representative agent conditioning on
$\mathcal{F}^{0}_{t}$; the couple (6) and (9) give the following corollary.
###### Corollary 2.5
(Necessary condition for Problems 2.1 and 2.2)
The control for the representative agent is optimal and the equilibrium
condition holds only if the SHJB-FP coupled equations are satisfied
$\left\\{\begin{array}[]{rcl}-\partial_{t}\Psi&=&\Big{(}H_{1}\Big{(}x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t)\Big{)}-A_{1}\Psi(x,t)\Big{)}dt-
K_{\Psi}(x,t)dW_{0}(t),\\\ \Psi(x,T)&=&h_{1}(x,x_{0}(T),m_{x_{0}}(x,T)).\\\
\dfrac{\partial m_{x_{0}}}{\partial
t}&=&-A_{1}^{*}m_{x_{0}}(x,t)-{\text{div}}\Big{(}G_{1}\Big{(}x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t)\Big{)}m_{x_{0}}(x,t)\Big{)},\\\
m_{x_{0}}(x,0)&=&\omega(x),\\\ \end{array}\right.$ (11)
where $G_{1}(x,x_{0},m,q)=g_{1}(x,x_{0},m,\hat{u}_{1}(x,x_{0},m,q))$.
The SHJB-FP coupled equations (11) allow us to obtain the control of the
representative agent in terms of a given trajectory of the dominating player
$x_{0}$ while the equilibrium condition also holds.
We then turn to the optimal problem for the dominating player. As $m_{x_{0}}$
is not external to the dominating player, the dominating player has to
consider both its own dynamic evolution and (11).
###### Lemma 2.6
Necessary condition for Problem 2.3
The control for the dominating player $\hat{u}_{0}$ is optimal only if
$\begin{split}f_{0}(x_{0},m_{x_{0}},\hat{u}_{0})+p\cdot
g_{0}(x_{0},m_{x_{0}},\hat{u}_{0})&=\inf_{u_{0}}\Big{\\{}f_{0}(x_{0},m_{x_{0}},u_{0})+p\cdot
g_{0}(x_{0},m_{x_{0}},u_{0})\Big{\\}}\\\ :&=H_{0}(x_{0},m_{x_{0}},p)\\\
\end{split}$
where $p(t)$ satisfies the following adjoint processes
$\begin{array}[]{rcl}-dp&=&\bigg{[}g_{0,x_{0}}^{*}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))p(t)+f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\\\
&&+\displaystyle\int
G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)dx\\\
&&+\displaystyle\int
r(x,t)H_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})dx\bigg{]}dt\\\
&&-\sum_{l=1}^{d_{0}}K_{p}^{l}(t)dW_{0}^{l}(t)+\sum_{l=1}^{d_{0}}\sigma_{0,x_{0}}^{l*}(x_{0}(t))K_{p}^{l}(t)dt,\\\
p(T)&=&h_{0,x_{0}}(x_{0}(T),m_{x_{0}}(x,T))+\displaystyle\int{r}(x,T)h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))dx;\\\
\end{array}$
$\begin{array}[]{rcl}-\partial_{t}{q}&=&\bigg{[}-A_{1}{q}(x,t)+p(t)\dfrac{\partial
g_{0}}{\partial m}(x_{0}(t),m_{x_{0}}(\xi,t),\hat{u}_{0}(t))(x)\\\
&&+D{q}(x,t)G_{1}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})\\\
&&+\displaystyle\int D{q}(\xi,t)\dfrac{\partial G_{1}}{\partial
m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)m_{x_{0}}(\xi,t)d\xi\\\
&&+\displaystyle\int{r}(\xi,t)\dfrac{\partial H_{1}}{\partial
m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)d\xi\\\ &&+\dfrac{\partial
f_{0}}{\partial m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(x)\bigg{]}dt-
K_{q}(x,t)dW_{0}(t),\\\ {q}(x,T)&=&\dfrac{\partial h_{0}}{\partial
m}(x_{0}(T),m_{x_{0}}(\xi,T))(x)+\displaystyle\int{r}(\xi,T)\dfrac{\partial
h_{1}}{\partial m}(\xi,x_{0}(T),m_{x_{0}}(\xi,T))(x)d\xi;\end{array}$
$\begin{array}[]{rcl}\dfrac{\partial{r}}{\partial
t}&=&-A_{1}^{*}{r}(x,t)-\text{\rm
div}\bigg{[}{r}(x,t)H_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})\\\
&&+G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)\bigg{]},\\\
{r}(x,0)&=&0.\end{array}$
###### Proof
Again we consider the Gâteaux derivative
$\begin{split}0=&\and{\dfrac{d}{d\theta}}{\theta=0}J_{0}(\hat{u}_{0}+\theta\tilde{u}_{0})\\\
=&\mathbb{E}\left\\{\int_{0}^{T}[f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{x}_{0}(t)+\int\dfrac{\partial
f_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\right.\\\
&+f_{0,u_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)]dt\\\
&+\left.h_{0,x_{0}}(x_{0}(T),m_{x_{0}}(x,T))\tilde{x}_{0}(T)+\int\dfrac{\partial
h_{0}}{\partial
m}(x_{0}(T),m_{x_{0}}(x,T))(\xi)\tilde{m}_{x_{0}}(\xi,T)d\xi\right\\},\end{split}$
(12)
where
$\tilde{x}_{0}=\and{\frac{d}{d\theta}}{\theta=0}x_{0}(\hat{u}_{0}+\theta\tilde{u}_{0})$;
$\tilde{m}_{x_{0}}=\and{\frac{d}{d\theta}}{\theta=0}m_{x_{0}}(\hat{u}_{0}+\theta\tilde{u}_{0})$;
$\tilde{\Psi}=\and{\frac{d}{d\theta}}{\theta=0}\Psi(\hat{u}_{0}+\theta\tilde{u}_{0})$,
satisfy
$\begin{array}[]{rcl}d\tilde{x}_{0}&=&\bigg{[}g_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{x}_{0}(t)+\displaystyle\int\dfrac{\partial
g_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\\\
&&+g_{0,u_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)\bigg{]}dt+\sum_{l=1}^{d_{0}}\sigma_{0,x_{0}}^{l}(x_{0}(t))\tilde{x}_{0}(t)dW_{0}^{l}(t),\\\
\tilde{x}_{0}(0)&=&0;\\\ \end{array}$
$\begin{array}[]{rcl}\dfrac{\partial\tilde{m}_{x_{0}}}{\partial
t}&=&-A_{1}^{*}\tilde{m}_{x_{0}}(x,t)-\text{\rm
div}\bigg{\\{}\Big{[}G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{x}_{0}(t)\\\
&&+\displaystyle\int\dfrac{\partial G_{1}}{\partial
m}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\\\
&&+G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))D\tilde{\Psi}(x,t)\Big{]}m_{x_{0}}(x,t)\\\
&&+G_{1}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{m}_{x_{0}}(x,t)\bigg{\\}},\\\
\tilde{m}_{x_{0}}(x,0)&=&0;\\\ \end{array}$
$\begin{array}[]{rcl}-\partial_{t}\tilde{\Psi}&=&\bigg{[}H_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{x}_{0}(t)\\\
&&+\displaystyle\int\dfrac{\partial H_{1}}{\partial
m}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\\\
&&+H_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))D\tilde{\Psi}(x,t)-A_{1}\tilde{\Psi}(x,t)\bigg{]}dt-
K_{\tilde{\Psi}}(x,t)dW_{0}(t),\\\
\tilde{\Psi}(x,T)&=&h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))\tilde{x}_{0}(T)+\displaystyle\int\dfrac{\partial
h_{1}}{\partial
m}(x,x_{0}(T),m_{x_{0}}(x,T))(\xi)\tilde{m}_{x_{0}}(\xi,T)d\xi.\\\
\end{array}$
Introduce the adjoint processes $p(t)$, ${q}(x,t)$ and ${r}(x,t)$ as stated in
the lemma statement and consider the following differentials
$\begin{array}[]{rcl}&&d(p^{*}\tilde{x}_{0})\\\
&=&-\tilde{x}_{0}^{*}(t)\bigg{[}f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))+\displaystyle\int{r}(x,t)H_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))dx\\\
&&+\displaystyle\int
G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)dx\bigg{]}dt\\\
&&+p^{*}(t)\bigg{[}\displaystyle\int\dfrac{\partial g_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\\\
&&+g_{0,u_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)\bigg{]}dt+\left\\{\dots\right\\}dW_{0}(t).\\\
\end{array}$
$\begin{array}[]{rcl}&&d\displaystyle\int{q}(x,t)\tilde{m}_{x_{0}}(x,t)dx\\\
&=&\displaystyle\int{q}(x,t)\bigg{[}-\text{\rm
div}\Big{[}[G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{x}_{0}(t)\\\
&&+G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))D\tilde{\Psi}]m_{x_{0}}(x,t)\Big{]}\bigg{]}dxdt\\\
&&-\displaystyle\int\Big{[}p(t)\dfrac{\partial g_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(\xi,t),\hat{u}_{0}(t))(x)+\displaystyle\int{r}(\xi,t)\dfrac{\partial
H_{1}}{\partial m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)d\xi\\\
&&+\dfrac{\partial f_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(x)\Big{]}\tilde{m}_{x_{0}}(x,t)dxdt+\left\\{\dots\right\\}dW_{0}(t).\\\
\end{array}$
$\begin{array}[]{rcl}&&d\displaystyle\int{r}(x,t)\tilde{\Psi}(x,t)dx\\\
&=&\displaystyle\int\bigg{[}-\text{\rm
div}\Big{[}G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)\Big{]}\bigg{]}\tilde{\Psi}(x,t)dxdt\\\
&&-\displaystyle\int
r(x,t)\bigg{[}H_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))\tilde{x}_{0}(t)\\\
&&+\displaystyle\int\dfrac{\partial H_{1}}{\partial
m}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))(\xi)\tilde{m}_{x_{0}}(\xi,t)d\xi\bigg{]}dxdt+\left\\{\dots\right\\}dW_{0}(t).\\\
\end{array}$
Using the results above, we have
$\begin{array}[]{rcl}&&d\displaystyle\Big{\\{}p^{*}\tilde{x}_{0}+\int{q}(x,t)\tilde{m}_{x_{0}}(x,t)dx-\int{r}(x,t)\tilde{\Psi}(x,t)dx\Big{\\}}\\\
&=&\bigg{[}\displaystyle-\tilde{x}_{0}^{*}(t)f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))+p^{*}(t)g_{0,u_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)\\\
&&-\displaystyle\int\dfrac{\partial f_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(x)\tilde{m}_{x_{0}}(x,t)dx\bigg{]}dt+\left\\{\dots\right\\}dW_{0}(t).\\\
\end{array}$
Integrating and taking expectation on both sides gives
$\begin{array}[]{rl}&\mathbb{E}\bigg{[}(h_{0,x_{0}}(x_{0}(T),m_{x_{0}}(x,T))+\displaystyle\int{r}(x,T)h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))dx)\tilde{x}_{0}(T)dx\\\
&\displaystyle+\int\Big{[}\dfrac{\partial h_{0}}{\partial
m}(x_{0}(T),m_{x_{0}}(x,T))(x)+\displaystyle\int{r}(\xi,T)\dfrac{\partial
h_{1}}{\partial
m}(\xi,x_{0}(T),m_{x_{0}}(x,T))(x)d\xi\Big{]}\tilde{m}_{x_{0}}(x,T)dx\\\
&\displaystyle-\int
r(x,T)\Big{[}h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))\tilde{x}_{0}(T)+\displaystyle\int\dfrac{\partial
h_{1}}{\partial
m}(x,x_{0}(T),m_{x_{0}}(x,T))(\xi)\tilde{m}_{x_{0}}(\xi,T)d\xi\Big{]}dx\bigg{]}\\\
=&\displaystyle\mathbb{E}\int_{0}^{T}\bigg{\\{}\displaystyle-\tilde{x}_{0}^{*}(t)f_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))+p^{*}(t)g_{0,\hat{u}_{0}}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))\tilde{u}_{0}(t)\\\
&-\displaystyle\int\dfrac{\partial f_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(x,t),\hat{u}_{0}(t))(x)\tilde{m}_{x_{0}}(x,t)dx\bigg{\\}}dt.\\\
\end{array}$
Finally we consider (12) and obtain
$\begin{array}[]{rcl}0&=&\mathbb{E}\displaystyle\int_{0}^{T}\Big{\\{}f_{0,\hat{u}_{0}}(x_{0},m_{x_{0}},\hat{u}_{0})+p^{*}g_{0,u_{0}}(x_{0},m_{x_{0}},\hat{u}_{0})\Big{\\}}\tilde{u}_{0}dt.\end{array}$
Since $\tilde{u}_{0}$ is arbitrary, the control is optimal for the dominating
player only if
$\begin{split}f_{0,u_{0}}(x_{0},m_{x_{0}},\hat{u}_{0})+p^{*}g_{0,u_{0}}(x_{0},m_{x_{0}},\hat{u}_{0})=0,\text{a.e.
t.}\end{split}$
Again, we are considering a minimization problem with the first order
condition given above. Since the unique existence of a minimizer,
$\hat{u}_{0}$ is assumed, we conclude that $\hat{u}_{0}$ satisfies the infimum
$f_{0}(x_{0},m_{x_{0}},\hat{u}_{0})+p\cdot
g_{0}(x_{0},m_{x_{0}},\hat{u}_{0})=\inf_{u_{0}}\Big{\\{}f_{0}(x_{0},m_{x_{0}},u_{0})+p\cdot
g_{0}(x_{0},m_{x_{0}},u_{0})\Big{\\}}.\\\ $
∎
Let
$G_{0}(x,m_{x_{0}},p)=g_{0}(x_{0},m_{x_{0}},\hat{u}_{0}(x_{0},m_{x_{0}},p))$.
We then conclude the main result in this section.
###### Theorem 2.7
The necessary condition for Problems 2.1, 2.2 and 2.3 is provided by the
following six equations
$\begin{array}[]{l}\left\\{\begin{array}[]{rll}dx_{0}&=&G_{0}(x_{0}(t),m_{x_{0}(t)}(t),p(t))dt+\sigma_{0}(x_{0}(t))dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\ \\\ \dfrac{\partial m_{x_{0}}}{\partial
t}&=&-A_{1}^{*}m_{x_{0}}(x,t)-{\text{div}}\Big{(}G_{1}\Big{(}x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t)\Big{)}m_{x_{0}}(x,t)\Big{)},\\\
m_{x_{0}}(x,0)&=&\omega(x),\\\ \\\
-\partial_{t}\Psi&=&\Big{(}H_{1}\Big{(}x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t)\Big{)}-A_{1}\Psi(x,t)\Big{)}dt-
K_{\Psi}(x,t)dW_{0}(t),\\\ \Psi(x,T)&=&h_{1}(x,x_{0}(T),m_{x_{0}}(x,T)).\\\
\end{array}\right.\\\ \\\
\left\\{\begin{array}[]{rll}-dp&=&\bigg{[}H_{0,x_{0}}(x_{0}(t),m_{x_{0}}(x,t),p(t))+\displaystyle\int{r}(x,t)H_{x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))dx\\\
&&+\displaystyle\int
G_{1,x_{0}}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi(x,t))D{q}^{*}(x,t)m_{x_{0}(t)}(x,t)dx\bigg{]}dt\\\
&&-\sum_{l=1}^{d_{0}}K_{p}^{l}(t)dW_{0}^{l}(t)+\sum_{l=1}^{d_{0}}\sigma_{0,x_{0}}^{l*}(x_{0}(t))K_{p}^{l}(t)dt,\\\
p(T)&=&h_{0,x_{0}}(x_{0}(T),m_{x_{0}}(x,T))+\displaystyle\int{r}(x,T)h_{1,x_{0}}(x,x_{0}(T),m_{x_{0}}(x,T))dx;\\\
\\\ -\partial_{t}{q}&=&\bigg{[}-A_{1}{q}(x,t)+\dfrac{\partial H_{0}}{\partial
m}(x_{0}(t),m_{x_{0}}(\xi,t),p(t))(x)\\\
&&+D{q}(x,t)G_{1}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})\\\
&&+\displaystyle\int D{q}(\xi,t)\dfrac{\partial G_{1}}{\partial
m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)m_{x_{0}}(\xi,t)d\xi\\\
&&+\displaystyle\int{r}(\xi,t)\dfrac{\partial H_{1}}{\partial
m}(\xi,x_{0}(t),m_{x_{0}}(\xi,t),D\Psi{(\xi,t)})(x)d\xi\bigg{]}dt-
K_{q}(x,t)dW_{0}(t),\\\ {q}(x,T)&=&\dfrac{\partial h_{0}}{\partial
m}(x_{0}(T),m_{x_{0}}(\xi,T))(x)+\displaystyle\int{r}(\xi,T)\dfrac{\partial
h_{1}}{\partial m}(\xi,x_{0}(T),m_{x_{0}}(\xi,T))(x)d\xi;\\\ \\\
\dfrac{\partial{r}}{\partial t}&=&-A_{1}^{*}{r}(x,t)-\text{\rm
div}\bigg{[}{r}(x,t)H_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})\\\
&&+G_{1,q}(x,x_{0}(t),m_{x_{0}}(x,t),D\Psi{(x,t)})D{q}(x,t)m_{x_{0}}(x,t)\bigg{]},\\\
{r}(x,0)&=&0.\end{array}\right.\end{array}$
## 3 Linear Quadratic Case
In this section we present a special case of the problem in the Linear
Quadratic setting in which both necessary and sufficient condition could be
established. Suppose that the state evolutions of the processes
$x_{0}(t),x_{1}(t)$ are described by
$\left\\{\begin{array}[]{rcl}dx_{0}&=&\Big{(}A_{0}x_{0}(t)+B_{0}z(t)+C_{0}u_{0}(x_{0}(t),t)\Big{)}dt+\sigma_{0}dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\
dx_{1}&=&\Big{(}A_{1}x_{1}(t)+B_{1}z(t)+C_{1}u_{1}(x_{1}(t),t)+Dx_{0}(t)\Big{)}dt+\sigma_{1}dW_{1}(t),\\\
x_{1}(0)&=&\xi_{1}.\\\ \end{array}\right.$
To simplify, the matrices $A_{0},C_{0},B_{0},\sigma_{0}$;
$A_{1},C_{1},B_{1},D,\sigma_{1}$ are assumed to be constant though the case
with time dependent and deterministic function are similar. For if the
dominating player did not exist, it is customary to consider $z(t)$ as
deterministic, and the equilibrium condition is
$z(t)=\mathbb{E}x_{1}(t).$ (13)
We can first find an optimal stochastic control for the representative agent
given $z$, then solve for the fixed point equation (13). However, one cannot
assume $z(t)$ to be deterministic when the dominating player exists, which
induces a two-layer problem. Since the dominating player can directly
influence the mean field term in the present setting, $z(t)$ should be adapted
to the filtration $\mathcal{F}_{t}^{0}$. The equilibrium condition hence is
$z(t)=\mathbb{E}^{\mathcal{F}_{t}^{0}}x_{1}(t)=\int\xi
m_{x_{0}(t)}(\xi,t)d\xi.$ (14)
We define control problems for both the dominating player and the
representative agent and fist solve for the control problem of the
representative agent as if both $x_{0}(t)$ and $z(t)$ as exogenous. The next
problem is to solve the equilibrium condition (14) as a fixed point property.
Finally we solve for the control problem of the dominating player, but now
$z(t)$ is regarded as endogenous.
For any vector $v$ and matrix $M$ with appropriate dimensions, we write the
inner product $v^{*}Mv$ as $|v|^{2}_{M}$ for simplicity. Problems 2.1, 2.2 and
2.3 can now be rewritten in the present Linear Quadratic framework as follows:
###### Problem 3.1
Control of the Representative Agent
Given the process $x_{0}$ and $\kappa$, find a control
$u_{1}\in\mathcal{A}_{1}$ which minimizes the cost functional
$\begin{split}J_{1}(u_{1},x_{0},\kappa)&=\mathbb{E}\bigg{[}\int_{0}^{T}(|x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1}|_{Q_{1}}^{2}+u^{*}_{1}(t)R_{1}u_{1}(t))dt\\\
&\qquad\qquad+|x_{1}(T)-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1}|_{\bar{Q}_{1}}^{2}\bigg{]}.\end{split}$
###### Problem 3.2
Equilibrium Condition
Let $x_{1}:=x_{1,\kappa}$ be the trajectory of the representative agent with
the optimal control found in Problem 3.1. Find the process $z(t)$ such that
the fixed point property is satisfied
$z(t)=\mathbb{E}^{\mathcal{F}_{t}^{0}}x_{1,z}(t),$
###### Problem 3.3
Control of the Dominating Player
Find a control $u_{0}\in\mathcal{A}_{0}$ which minimizes the cost functional
$\begin{split}J_{0}(u_{0})&=\mathbb{E}\bigg{[}\int_{0}^{T}(|x_{0}(t)-E_{0}z(t)-\zeta_{0}|_{Q_{0}}^{2}+u_{0}^{*}(t)R_{0}u_{0}(t))dt\\\
&\qquad\qquad+|x_{0}(T)-\bar{E}_{0}z(T)-\bar{\zeta}_{0}|_{\bar{Q}_{0}}^{2}\bigg{]},\end{split}$
where $z$ is the solution given in Problem 3.2.
For simplicity, $E_{0},F,R_{0},R_{1},E_{1},\zeta_{0},\zeta_{1},Q_{0},Q_{1}$
are constant matrices and vectors; $Q_{0},Q_{1},R_{0},R_{1}$ are positive
symmetric and invertible. Note that Problems 3.1 and 3.3 are strictly convex
quadratic and coercive, we can write the stochastic principle.
###### Lemma 3.4
(Control of Representative Agent)
Problem 3.1 is uniquely solvable and the optimal control $\hat{u}_{1}(t)$ is
$-R_{1}^{-1}C_{1}^{*}n(t)$, where $n$ satisfies the adjoint process
$\begin{split}\left\\{\begin{array}[]{rlc}-dn&=\Big{(}A_{1}^{*}n(t)+Q_{1}(x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1})\Big{)}dt-
Z_{n,0}(t)dW_{0}(t)-Z_{n,1}(t)dW_{1}(t),\\\
n(T)&=\bar{Q}_{1}\Big{(}x_{1}(T)-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1}\Big{)}.\end{array}\right.\end{split}$
###### Proof
Consider a perturbation of the optimal control
$\hat{u}_{1}+\theta\tilde{u}_{1}$, where $\tilde{u}_{1}$ is adapted to the
filtration $\mathcal{G}_{t}$. The original state $x_{1}$ becomes
$x_{1}+\theta\tilde{x}_{1}$ with
$\begin{split}\left\\{\begin{array}[]{rlc}d\tilde{x}_{1}&=\Big{(}A_{1}\tilde{x}_{1}(t)+C_{1}\tilde{u}_{1}(t)\Big{)}dt,\\\
\tilde{x}_{1}(0)&=0.\end{array}\right.\end{split}$
the optimality of $\hat{u}_{1}$ is expressed by the following Euler condition
$\begin{split}0&=\and{\dfrac{d}{d\theta}}{\theta=0}J_{1}(\hat{u}_{1}+\theta\tilde{u}_{1},x_{0},\kappa)\\\
&=\mathbb{E}\bigg{[}\int_{0}^{T}[\tilde{x}_{1}^{*}(t)Q_{1}(x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1})+\tilde{u}_{1}^{*}(t)R_{1}\hat{u}_{1}(t)]dt\\\
&\qquad\qquad+\tilde{x}_{1}^{*}(T)\bar{Q}_{1}(x_{1}(T)-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1})\bigg{]}\\\
\end{split}$
On the other hand, we have
$\begin{split}d(n^{*}\tilde{x}_{1})&=\Big{(}n^{*}(t)C_{1}\tilde{u}_{1}(t)-\tilde{x}_{1}^{*}(t)Q_{1}(x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1})\Big{)}dt\\\
&\qquad+\tilde{x}_{1}(t)Z_{n,0}(t)dW_{0}(t)+\tilde{x}_{1}(t)Z_{n,1}(t)dW_{1}(t).\end{split}$
Integrate both side and take expectation, combining with the Euler condition,
it becomes
$\mathbb{E}\int_{0}^{T}(n^{*}(t)C_{1}+\hat{u}_{1}^{*}(t)R_{1})\tilde{u}_{1}(t)dt=0,\quad\forall\tilde{u}_{1}(\cdot)\text{
adpated to }\mathcal{G},$
which implies, together with an application of tower property,
$\hat{u}_{1}(t)=-R_{1}^{-1}C_{1}^{*}n(t).$ (15)
Therefore the stochastic maximum principle for the representative agent is
expressed by the system
$\left\\{\begin{array}[]{rlc}dx_{1}&=\Big{(}A_{1}x_{1}(t)+B_{1}\kappa(t)-C_{1}R_{1}^{-1}C_{1}^{*}n(t)+Dx_{0}(t)\Big{)}dt+\sigma_{1}dW_{1}(t),\\\
x_{1}(0)&=\xi_{1}.\\\
-dn&=\Big{(}A_{1}^{*}n(t)+Q_{1}(x_{1}(t)-E_{1}\kappa(t)-Fx_{0}(t)-\zeta_{1})\Big{)}dt-
Z_{n,0}(t)dW_{0}(t)-Z_{n,1}(t)dW_{1}(t),\\\
n(T)&=\bar{Q}_{1}\Big{(}x_{1}(T)-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1}\Big{)}.\end{array}\right.$
(16)
and hence for every exogenous pair $(x_{0},\kappa)$, this system defines a
unique pair $(x_{1},n)$. By the convexity and coerciveness of the cost
functional, $\hat{u}_{1}$ is uniquely defined and the sufficient condition is
automatically satisfied. ∎
###### Remark
The optimal control has the representation
$-R_{1}^{-1}C_{1}^{*}n(t)=-R_{1}^{-1}C_{1}^{*}(P_{t}x_{1}(t)+g(t))$, where $P$
satisfies the symmetric Riccati equation
$\left\\{\begin{array}[]{rl}&\dot{P_{t}}+P_{t}A_{1}+A_{1}^{*}P_{t}-P_{t}C_{1}R_{1}^{-1}C_{1}^{*}P_{t}+Q_{1}=0,\\\
&P_{T}=\bar{Q}_{1};\\\ \end{array}\right.\\\ $
and $g$ satisfies the BSDE
$\left\\{\begin{array}[]{rcl}-dg&=&\Big{(}(A_{1}-C_{1}R_{1}^{-1}C_{1}^{*}P_{t})^{*}g(t)+(P_{t}B_{1}-Q_{1}E_{1})+(P_{t}D-Q_{1}F)x_{0}(t)\kappa(t)-Q_{1}\zeta_{1}\Big{)}dt\\\
&&\qquad-Z_{g,0}(t)dW_{0}(t)-Z_{g,1}(t)dW_{1}(t),\\\
g(T)&=&\bar{Q}_{1}\Big{(}-\bar{E}_{1}\kappa(T)-\bar{F}x_{0}(T)-\bar{\zeta}_{1}\Big{)}.\\\
\end{array}\right.$
To obtained the equilibrium condition stated in Problem (3.2), we simply take
expectation conditional on $\mathcal{F}_{t}^{0}$ on both sides of Equation
(16). By requiring $\mathbb{E}^{\mathcal{F}^{0}_{t}}x_{1}(t)=z(t)$, and
replace $\kappa(t)$ by $z(t)$, we have the following system, which is
analogical to the SHJB-FP (11) in Section 2.
$\left\\{\begin{array}[]{rlc}dz&=\Big{(}(A_{1}+B_{1})z(t)-C_{1}R_{1}^{-1}C_{1}^{*}m(t)+Dx_{0}(t)\Big{)}dt+\sigma_{1}dW_{1}(t),\\\
z_{1}(0)&=\mathbb{E}[\xi_{1}].\\\
-dm&=\Big{(}A_{1}^{*}m(t)+Q_{1}(I-E_{1})z(t)-Q_{1}Fx_{0}(t)-Q_{1}\zeta_{1}\Big{)}dt-
Z_{m}(t)dW_{0}(t),\\\
m(T)&=\bar{Q}_{1}(I-\bar{E}_{1})z(T)-\bar{Q}_{1}\bar{F}x_{0}(T)-\bar{Q}_{1}\bar{\zeta}_{1}.\end{array}\right.$
(17)
We next proceed on the problem of the dominating player.
###### Lemma 3.5
(Control of the Dominating Player)
Problem 3.3 is uniquely solvable and the optimal control $\hat{u}_{0}(t)$ is
$-R_{0}^{-1}C_{0}^{*}p(t)$, where $p$ satisfies
$\left\\{\begin{array}[]{rcl}-dp&=&\Big{(}A_{0}^{*}{p}(t)+D^{*}q(t)-(Q_{1}F)^{*}r(t)+Q_{0}(x_{0}(t)-E_{0}z(t)-\zeta_{0})\Big{)}dt-
Z_{p}(t)dW_{0}(t),\\\
{p}(T)&=&\bar{F}^{*}\bar{Q}_{1}r(T)+\bar{Q}_{0}(x_{0}(T)-\bar{E}_{0}z(T)-\bar{\zeta}_{0}).\\\
-dq&=&\Big{(}(A_{1}+B_{1})^{*}q(t)+B_{0}^{*}{p}(t)+(I-E_{1})^{*}Q_{1}^{*}r(t)-E_{0}^{*}Q_{0}(x_{0}(t)-E_{0}z(t)-\zeta_{0})\Big{)}dt-
Z_{q}(t)dW_{0}(t),\\\
q(T)&=&(I-\bar{E}_{1})^{*}\bar{Q}_{1}r(T)-\bar{E}_{0}^{*}\bar{Q}_{0}(x_{0}(t)-\bar{E}_{0}z(t)-\bar{\zeta}_{0}).\\\
dr&=&\Big{(}A_{1}r(t)-C_{1}R_{1}^{-1}C_{1}^{*}q(t)\Big{)}dt,\\\
{r}(0)&=&0.\end{array}\right.$ (18)
###### Proof
Consider $\hat{u}_{0}+\theta\tilde{u}_{0}$ the perturbation of the optimal
control, where $\tilde{u}_{0}$ is adapted to the filtration
$\mathcal{F}_{t}^{0}$. The original states $x_{0}$, $z$, $m$ become
$x_{0}+\theta\tilde{x}_{0}$, $z+\theta\tilde{z}$, $m+\theta\tilde{m}$ with
$\left\\{\begin{array}[]{rcl}d\tilde{x}_{0}&=&\Big{(}A_{0}\tilde{x}_{0}(t)+B_{0}\tilde{z}(t)+C_{0}\tilde{u}_{0}(t)\Big{)}dt,\\\
\tilde{x}(0)&=&0.\\\
d\tilde{z}&=&\Big{(}(A_{1}+B_{1})\tilde{z}(t)-C_{1}R_{1}^{-1}C_{1}^{*}\tilde{m}(t)+D\tilde{x}_{0}(t)\Big{)}dt,\\\
\tilde{z}(0)&=&0.\\\
-d\tilde{m}&=&\Big{(}A_{1}^{*}\tilde{m}(t)+Q_{1}(I-E_{1})\tilde{z}(t)-Q_{1}F\tilde{x}_{0}(t)\Big{)}dt-
Z_{\tilde{m}}dW_{0}(t)\\\
\tilde{m}(T)&=&\bar{Q}_{1}(I-\bar{E}_{1})\tilde{z}(T)-\bar{Q}_{1}\bar{F}\tilde{x}_{0}(T).\\\
\end{array}\right.$
The corresponding maximum principle for $\hat{u}_{0}$ is
$\begin{split}0&=\and{\dfrac{d}{d\theta}}{\theta=0}J_{0}(\hat{u}_{0}+\theta\tilde{u}_{0})\\\
&=\mathbb{E}\bigg{[}\int_{0}^{T}[(\tilde{x}_{0}(t)-E_{0}\tilde{z}(t))^{*}Q_{0}(x_{0}(t)-E_{0}z(t)-\zeta_{0})+\hat{u}_{0}^{*}R_{0}\tilde{u}_{0}]dt\\\
&\qquad\qquad+(\tilde{x}_{0}(T)-\bar{E}_{0}\tilde{z}(T))^{*}Q_{0}(x_{0}(T)-\bar{E}_{0}z(T)-\bar{\zeta_{0}})\bigg{]}\end{split}$
(19)
On the other hand, we can easily check that
$\begin{split}d({p}^{*}\tilde{x}_{0}+q^{*}\tilde{z}-r^{*}\tilde{m})=\Big{(}{p}^{*}(t)C_{0}\tilde{u}_{0}(t)-(\tilde{x}_{0}(t)-E_{0}\tilde{z}(t))^{*}Q_{0}(x_{0}(t)-E_{0}z(t)-\zeta_{0})\Big{)}dt+\\{\dots\\}dW_{0}(t).\end{split}$
Integrating and also taking expectation on both sides of the last equation,
together with an application of (19), we deduce that
$\begin{split}\mathbb{E}\int_{0}^{T}({p}^{*}C_{0}+\hat{u}_{0}^{*}R_{0})\tilde{u}_{0}dt=0,\quad\forall\tilde{u}_{0}\text{
adapted to }\mathcal{F}_{t}^{0},\end{split}$
which implies the desired result by again the application of tower property,
$\hat{u}_{0}(t)=-R_{0}^{-1}C_{0}^{*}p(t)(t).$ (20)
∎
Summarizing the results we obtained so far, we present the main theorem in
this section.
###### Theorem 3.6
The necessary and sufficient conditions for the unique existence of the
solution to Problems 3.1, 3.2 and 3.3 are described by the following six
equations in matrix form
$\begin{array}[]{rl}&\left\\{\begin{array}[]{rcl}d\begin{pmatrix}x_{0}\\\ r\\\
z\end{pmatrix}&=&\left\\{\begin{pmatrix}A_{0}&0&B_{0}\\\ 0&A_{1}&0\\\
D&0&A_{1}+B_{1}\end{pmatrix}\begin{pmatrix}x_{0}(t)\\\ r(t)\\\
z(t)\end{pmatrix}\right.\\\
&&\left.-\begin{pmatrix}C_{0}R_{0}^{-1}C_{0}^{*}&0&0\\\
0&C_{1}R_{1}^{-1}C_{1}^{*}&0\\\
0&0&C_{1}R_{1}^{-1}C_{1}^{*}\end{pmatrix}\begin{pmatrix}p(t)\\\ q(t)\\\
m(t)\end{pmatrix}\right\\}dt+\begin{pmatrix}\sigma_{0}\\\ 0\\\
0\end{pmatrix}dW_{0},\\\ \begin{pmatrix}x_{0}(0)\\\ r(0)\\\
z(0)\end{pmatrix}&=&\begin{pmatrix}\xi_{0}\\\ 0\\\
\mathbb{E}[\xi_{1}]\end{pmatrix}.\\\ \end{array}\right.\\\ \\\
&\left\\{\begin{array}[]{rcl}-d\begin{pmatrix}p\\\ q\\\
m\end{pmatrix}&=&\left\\{\begin{pmatrix}A_{0}^{*}&D^{*}&0\\\
B_{0}^{*}&A_{1}^{*}+B_{1}^{*}&0\\\
0&0&A_{1}^{*}\end{pmatrix}\begin{pmatrix}p(t)\\\ q(t)\\\
m(t)\end{pmatrix}\right.\\\
&&\left.+\begin{pmatrix}Q_{0}&-(Q_{1}F)^{*}&-Q_{0}E_{0}\\\
-(Q_{0}E_{0})^{*}&(Q_{1}(I-E_{1}))^{*}&E_{0}^{*}Q_{0}E_{0}\\\
-Q_{1}F&0&Q_{1}(I-E_{1})\end{pmatrix}\begin{pmatrix}x_{0}(t)\\\ r(t)\\\
z(t)\end{pmatrix}+\begin{pmatrix}-Q_{0}\zeta_{0}\\\ E_{0}^{*}Q_{0}\zeta_{0}\\\
-Q_{1}\zeta_{1}\end{pmatrix}\right\\}dt-\begin{pmatrix}Z_{p}(t)\\\ Z_{q}(t)\\\
Z_{m}(t)\end{pmatrix}dW_{0}(t),\\\ \begin{pmatrix}p(T)\\\ q(T)\\\
m(T)\end{pmatrix}&=&\begin{pmatrix}\bar{Q}_{0}&-(\bar{Q}_{1}\bar{F})^{*}&-\bar{Q}_{0}\bar{E}_{0}\\\
-\bar{Q}_{0}\bar{E}_{0})^{*}&(\bar{Q}_{1}(I-\bar{E}_{1}))^{*}&\bar{E}_{0}^{*}\bar{Q}_{0}\bar{E}_{0}\\\
\bar{Q}_{1}F&0&\bar{Q}_{1}(I-\bar{E}_{1})\end{pmatrix}\begin{pmatrix}x_{0}(t)\\\
r(t)\\\ z(t)\end{pmatrix}+\begin{pmatrix}-\bar{Q}_{0}\bar{\zeta}_{0}\\\
\bar{E}_{0}^{*}\bar{Q}_{0}\bar{\zeta}_{0}\\\
-\bar{Q}_{1}\bar{\zeta}_{1}\end{pmatrix}.\\\ \end{array}\right.\end{array}$
(21)
###### Remark
One can easily compare these six equations with those stated in Theorem 2.7.
We obtain the same results by applying the general theory, however, it is more
convenient to acquire these six equations directly under the Linear Quadratic
setting, which also illuminates the power of using the principle of
separation. On comparison with the intermediary result obtained in HN2 . The
latter work did not take account of the third adjoint equation $r$ since it
fails to consider the impact on $m$ with respect to the change of the control
of the dominating player.
## 4 Fixed Point Problem
In this section, we provide a sufficient condition, which solely depends on
the coefficients of the mean field game system, for the unique existence of
the solution to Problems 3.1, 3.2 and 3.3 by means of tackling a non-symmetric
Riccati equation. To facilitate our argument, we define
${\bf x}:=\begin{pmatrix}x_{0}\\\ r\\\ z\end{pmatrix};\quad{\bf
p}:=\begin{pmatrix}p\\\ q\\\ m\end{pmatrix};\quad{\bf
Z}:=\begin{pmatrix}Z_{p}\\\ Z_{q}\\\ Z_{m}\end{pmatrix}.$
Hence we can write (21) as
$\begin{array}[]{rl}&\left\\{\begin{array}[]{rcl}d{\bf
x}&=&\Big{(}(\mathcal{A}+\mathcal{B}){\bf x}(t)-\mathcal{C}{\bf
p}(t)\Big{)}dt+\sigma dW_{0}(t),\\\ {\bf x}(0)&=&\xi.\\\ \end{array}\right.\\\
&\left\\{\begin{array}[]{rcl}-d{\bf p}&=&\Big{(}\mathcal{A}^{*}{\bf
p}(t)+(\mathcal{Q}+\mathcal{S}){\bf x}(t)+{\bf k}\Big{)}dt-{\bf
Z}(t)dW_{0}(t),\\\ {\bf p}(T)&=&(\bar{\mathcal{Q}}+\bar{\mathcal{S}}){\bf
x}(T)+\bar{\bf k},\\\ \end{array}\right.\end{array}$ (22)
where
$\begin{array}[]{l}\mathcal{A}:=\begin{pmatrix}A_{0}&B_{0}&0\\\
D&A_{1}+B_{1}&0\\\
0&0&A_{1}\end{pmatrix};\quad\mathcal{B}:=\begin{pmatrix}0&-B_{0}&B_{0}\\\
-D&-B_{1}&0\\\
D&0&B_{1}\end{pmatrix};\quad\mathcal{C}:=\begin{pmatrix}C_{0}R_{0}^{-1}C_{0}^{*}&0&0\\\
0&C_{1}R_{1}^{-1}C_{1}^{*}&0\\\ 0&0&C_{1}R_{1}^{-1}C_{1}^{*}\end{pmatrix};\\\
{\sigma}:=\begin{pmatrix}\sigma_{0}\\\ 0\\\
0\end{pmatrix};\quad\xi:=\begin{pmatrix}\xi_{0}\\\ 0\\\
0\end{pmatrix};\quad\mathcal{Q}+\mathcal{S}:=\begin{pmatrix}Q_{0}&-F^{*}Q_{1}&-Q_{0}E_{0}\\\
-E_{0}^{*}Q_{0}&\quad(I-E_{1})^{*}Q_{1}^{*}&E_{0}^{*}Q_{0}E_{0}\\\
-Q_{1}F&0&\quad Q_{1}(I-E_{1})\end{pmatrix};\quad{\bf
k}:=\begin{pmatrix}-Q_{0}\zeta_{0}\\\ E_{0}^{*}Q_{0}\zeta_{0}\\\
-Q_{1}\zeta_{1}\end{pmatrix};\\\
\bar{\mathcal{Q}}+\bar{\mathcal{S}}:=\begin{pmatrix}\bar{Q}_{0}&-(\bar{Q}_{1}\bar{F})^{*}&-\bar{Q}_{0}\bar{E}_{0}\\\
-\bar{Q}_{0}\bar{E}_{0})^{*}&(\bar{Q}_{1}(I-\bar{E}_{1}))^{*}&\bar{E}_{0}^{*}\bar{Q}_{0}\bar{E}_{0}\\\
\bar{Q}_{1}F&0&\bar{Q}_{1}(I-\bar{E}_{1})\end{pmatrix};\quad\bar{\bf
k}:=\begin{pmatrix}-\bar{Q}_{0}\bar{\zeta}_{0}\\\
\bar{E}_{0}^{*}\bar{Q}_{0}\bar{\zeta}_{0}\\\
-\bar{Q}_{1}\bar{\zeta}_{1}\end{pmatrix},\end{array}$
where $\mathcal{Q}$ and $\bar{\mathcal{Q}}$ are positive matrices. Consider
the following non-symmetric Riccati equation
$\left\\{\begin{array}[]{l}\dot{\Gamma}+\mathcal{A}^{*}\Gamma_{t}+\Gamma_{t}(\mathcal{A}+\mathcal{B})-\Gamma_{t}\mathcal{C}\Gamma_{t}+(\mathcal{Q}+\mathcal{S})=0,\\\
\Gamma_{T}=\bar{\mathcal{Q}};\\\ \end{array}\right.$ (23)
and the backward ODE
$\left\\{\begin{array}[]{rcl}-d{\bf
g}&=&\Big{(}(\mathcal{A}^{*}-\Gamma_{t}\mathcal{B}){\bf g}(t)+{\bf
k}\Big{)}dt,\\\ {\bf g}(T)&=&\bar{\bf k}.\end{array}\right.$ (24)
It is easy to check that ${\bf p}(t)=\Gamma_{t}{\bf x}(t)+{\bf g}(t)$. With
respect to this affine form, the forward backward equation (22) admits a
unique solution if and only if (23) admits a unique solution. In accordance
with Theorem 2.4.3 in Ma and Young FBSDE or AB , we have the following
proposition.
###### Proposition 4.1
Suppose the following forward-backward ordinary differential equations
$\begin{array}[]{rl}&\left\\{\begin{array}[]{rcl}\dfrac{dX}{dt}&=&(\mathcal{A}+\mathcal{B})X(t)-\mathcal{C}Y(t)\\\
X(0)&=&0.\\\ \end{array}\right.\\\
&\left\\{\begin{array}[]{rcl}-\dfrac{dY}{dt}&=&\mathcal{A}^{*}Y(t)+(\mathcal{Q}+\mathcal{S})X(t),\\\
Y(T)&=&(\bar{\mathcal{Q}}+\bar{\mathcal{S}})X(T).\\\
\end{array}\right.\end{array}$
admits a unique solution for any $t_{0}\in[0,T]$. Then there is a unique
solution of (23).
Our next theorem concludes the results in this section.
###### Theorem 4.2
Let $\phi(s,t)$ be the fundamental solution to $\mathcal{A}$. Suppose that
$\Big{(}1+\sqrt{T}\|\phi\|_{T}\cdot\|\mathcal{B}\mathcal{Q}^{-\frac{1}{2}}\|\Big{)}\Big{(}1+N(S)\Big{)}<2,$
where $\|\cdot\|$ stands for usual Euclidean norm. Then there exists a unique
solution of equation (21), and hence a unique (mean field) equilibrium exists.
Here,
$\|\phi\|_{T}:=\sup_{0\leq t\leq
T}\sqrt{\|\phi^{*}(T,t)\bar{\mathcal{Q}}^{\frac{1}{2}}\|^{2}+\int_{t}^{T}\|\phi^{*}(s,t)\mathcal{Q}^{\frac{1}{2}}\|^{2}ds}$
and
$N(S)=\max\\{\|\bar{\mathcal{Q}}^{-\frac{1}{2}}\bar{\mathcal{S}}\bar{\mathcal{Q}}^{-\frac{1}{2}}\|,\|\mathcal{Q}^{-\frac{1}{2}}\mathcal{S}\mathcal{Q}^{-\frac{1}{2}}\|\\}$
###### Proof
Let $x,y$ be elements in the Hilbert Space
$\mathcal{H}^{2}([0,T];\mathbb{R}^{n_{0}+n_{1}+n_{1}})$ endowed with the inner
product
$\langle
x,y\rangle_{\mathcal{H}}=x^{*}(T)\bar{\mathcal{Q}}y(T)+\int_{0}^{T}x^{*}(s)\mathcal{Q}y(s)ds$
Furthermore,
$\|\cdot\|_{\mathcal{H}}:=|\langle\cdot,\cdot\rangle|_{\mathcal{H}}^{\frac{1}{2}}$
stands for the induced norm under this inner product. We consider the forward
backward ordinary differential equation
$\begin{array}[]{rl}&\left\\{\begin{array}[]{rcl}\dfrac{dX}{dt}&=&\mathcal{A}X(t)-\mathcal{C}Y(t)+\mathcal{B}x(t)\\\
X(0)&=&0.\\\ \end{array}\right.\\\
&\left\\{\begin{array}[]{rcl}-\dfrac{dY}{dt}&=&\mathcal{A}^{*}Y(t)+\mathcal{Q}X(t)+\mathcal{S}x(t),\\\
Y(T)&=&\bar{\mathcal{Q}}X(T)+\bar{\mathcal{S}}x(T).\\\
\end{array}\right.\end{array}$ (25)
Observe that both $\mathcal{C}$ and $\mathcal{Q}$ are positive definite,
Equation (25) corresponds a well-defined (deterministic) control problem.
Hence, $x\mapsto X$ is well defined in $\mathcal{H}^{2}$. It suffices to show
that this mapping is indeed a contraction. Consider the inner product
$\dfrac{d}{dt}(X^{*}Y)=-Y^{*}(t)\mathcal{C}Y(t)+Y^{*}(t)\mathcal{B}x(t)-X^{*}(t)\mathcal{Q}X(t)-X^{*}(t)\mathcal{S}x(t).$
Taking integration on $[0,T]$ yields
$X^{*}(T)\bar{\mathcal{Q}}X(T)+\int_{0}^{T}X^{*}(t)\mathcal{Q}X(t)dt=X(T)^{*}\bar{\mathcal{S}}x(T)+\int_{0}^{T}-Y^{*}(t)\mathcal{C}Y(t)+Y^{*}(t)\mathcal{B}x(t)-X^{*}(t)\mathcal{S}x(t)dt.$
By the positivity of $\mathcal{C}$, and Cauchy-Schwarz inequality, we have
$\begin{array}[]{rcl}\|X\|_{\mathcal{H}}^{2}&\leq&\|\bar{\mathcal{Q}}^{-\frac{1}{2}}\bar{\mathcal{S}}\bar{\mathcal{Q}}^{-\frac{1}{2}}\|\cdot\Big{(}X^{*}(T)\bar{\mathcal{Q}}^{\frac{1}{2}}\cdot\bar{\mathcal{Q}}^{\frac{1}{2}}x(T)\Big{)}\\\
&&\quad+\|\mathcal{Q}^{-\frac{1}{2}}\mathcal{S}\mathcal{Q}^{-\frac{1}{2}}\|\cdot\displaystyle\int_{0}^{T}X^{*}(t)\mathcal{Q}^{\frac{1}{2}}\cdot\mathcal{Q}^{\frac{1}{2}}x(t)dt+\displaystyle\int_{0}^{T}Y^{*}(t)\mathcal{B}x(t)dt\\\
&\leq&N(S)\Big{(}\|X\|_{\mathcal{H}}\cdot\|x\|_{\mathcal{H}}\Big{)}+\displaystyle\int_{0}^{T}Y^{*}(t)\mathcal{B}x(t)dt.\\\
\end{array}$ (26)
On the other hand, for $t\in[0,T]$, we have
$\begin{split}Y(t)&=\phi^{*}(T,t)\Big{(}\bar{\mathcal{Q}}X(T)+\bar{\mathcal{S}}x(T)\Big{)}+\int_{t}^{T}\phi^{*}(s,t)\Big{(}\mathcal{Q}X(s)+\mathcal{S}x(s)\Big{)}ds\\\
&=\phi^{*}(T,t)\bar{\mathcal{Q}}^{\frac{1}{2}}\Big{(}\bar{\mathcal{Q}}^{\frac{1}{2}}X(T)+\bar{\mathcal{Q}}^{-\frac{1}{2}}\bar{\mathcal{S}}x(T)\Big{)}+\int_{t}^{T}\phi^{*}(s,t)\mathcal{Q}^{\frac{1}{2}}\Big{(}\mathcal{Q}^{\frac{1}{2}}X(s)+\mathcal{Q}^{-\frac{1}{2}}\mathcal{S}x(s)\Big{)}ds\\\
\end{split}$
which implies
$\sup_{t\in[0,T]}\|Y_{t}\|\leq\|\phi\|_{T}\Big{(}\|X\|_{\mathcal{H}}+N(S)\|x\|_{\mathcal{H}}\Big{)}.$
(27)
Combining Equations (26) and (27) yields
$\begin{array}[]{rcl}\|X\|_{\mathcal{H}}^{2}&\leq&N(S)\Big{(}\|X\|_{\mathcal{H}}\cdot\|x\|_{\mathcal{H}}\Big{)}+\sqrt{T}\|\phi\|_{T}\Big{(}\|X\|_{\mathcal{H}}+N(S)\|x\|_{\mathcal{H}}\Big{)}\|\mathcal{B}Q^{\frac{1}{2}}\|\|x\|_{\mathcal{H}}\end{array}$
which shows that $x\mapsto X$ is a contraction if
$\Big{(}1+\sqrt{T}\|\phi\|_{T}\cdot\|\mathcal{B}\mathcal{Q}^{-\frac{1}{2}}\|\Big{)}\Big{(}1+N(S)\Big{)}<2.$
∎
## 5 Conclusion
In this paper, by adopting adjoint equation approach, we provide the general
theory and discuss the necessary condition for optimal controls for both the
dominating player and the representative agent, and study the corresponding
fixed point problem in relation to the equilibrium condition. A convenient
necessary and sufficient condition has been provided under the Linear
Quadratic setting; in particular, a illuminative sufficient condition, which
only involves the coefficient of the mean field game system, for the unique
existence of the equilibrium control has been given. Finally, proof of the
convergence result of finite player game to mean field counterpart is provided
in Appendix. Applications of the present model in connection with central bank
lending and systematic risk in financial context will be provided in the
future work.
## 6 Acknowledgements
The first author-Alain Bensoussan acknowledges the financial support of the
Hong Kong RGC GRF 500113 and the National Science Foundation under grant DMS
1303775. The second author-Michael Chau acknowledges the financial support
from the Chinese University of Hong Kong, and the present work constitutes a
part of his work for his postgraduate dissertation. The third author-Phillip
Yam acknowledges the financial support from The Hong Kong RGC GRF 404012 with
the project title: Advanced Topics In Multivariate Risk Management In Finance
And Insurance, The Chinese University of Hong Kong Direct Grants 2010/2011
Project ID: 2060422 and 2011/2012 Project ID: 2060444. Phillip Yam also
expresses his sincere gratitude to the hospitality of both Hausdorff Center
for Mathematics and Hausdorff Research Institute for Mathematics of the
University of Bonn during the preparation of the present work.
## Appendix A Appendix
### A.1 $\epsilon$-Nash Equilibrium
We now establish that the solutions of Problems 2.1 and 2.2 is an
$\epsilon$-Nash Equilibrium. Suppose that there are $N$ representative agents
behaving in similar manner, so that the state of the dominating player and the
$i$-th agent satisfies the following SDE respectively:
$\begin{array}[]{l}\left\\{\begin{array}[]{rcl}dy_{0}&=&g_{0}\Big{(}y_{0}(t),\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{y_{1}^{j}(t)},u_{0}(t)\Big{)}dt+\sigma_{0}\Big{(}y_{0}(t)\Big{)}dW_{0}(t),\\\
y_{0}(0)&=&\xi_{0}.\\\ \end{array}\right.\\\ \\\
\left\\{\begin{array}[]{rcl}dy_{1}^{i}&=&g_{1}\Big{(}y_{1}^{i}(t),y_{0}(t),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{y_{1}^{j}(t)},u_{1}^{i}(t)\Big{)}dt+\sigma_{1}\Big{(}y_{1}^{i}(t)\Big{)}dW_{1}^{i}(t),\\\
y_{1}^{i}(0)&=&\xi_{1}^{i}.\\\ \end{array}\right.\end{array}$ (28)
where $\delta_{y}$ is Dirac measure with a unit mass at $y$. We call Equation
(28) the empirical system. The corresponding objective functional for the
$i$-th agent is:
$\begin{array}[]{rcl}\mathcal{J}^{N,i}({\bf
u})=\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}f_{1}\Big{(}y_{1}^{i}(t),y_{0}(t),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{y_{1}^{j}(t)},u^{i}(t)\Big{)}dt+h_{1}\Big{(}y_{1}^{i}(T),y_{0}(T),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{y_{1}^{j}(T)}\Big{)}\bigg{]},\end{array}$
where ${\bf u}=(u_{1}^{1},u_{1}^{2},\dots,u_{1}^{N})$. We expect that when
$N\rightarrow\infty$, the hypothetical approximation is described by (1), that
is:
$\begin{array}[]{l}\left\\{\begin{array}[]{rcl}dx_{0}&=&g_{0}\Big{(}x_{0}(t),m_{x_{0}(t)},u_{0}(t)\Big{)}dt+\sigma_{0}\Big{(}x_{0}(t)\Big{)}dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\ \end{array}\right.\\\ \\\
\left\\{\begin{array}[]{rcl}dx_{1}^{i}&=&g_{1}\Big{(}x_{1}^{i}(t),x_{0}(t),m_{x_{0}(t)},u_{1}^{i}(t)\Big{)}dt+\sigma_{1}\Big{(}x_{1}^{i}(t)\Big{)}dW_{1}^{i}(t),\\\
x_{1}^{i}(0)&=&\xi_{1}^{i}.\\\ \end{array}\right.\end{array}$ (29)
We call Equation (29) the mean field system. The corresponding limiting
objective functional for the $i$-th player is
$\begin{array}[]{rcl}\mathcal{J}^{i}(u_{1}^{i})=\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}f_{1}\Big{(}x_{1}^{i}(t),x_{0}(t),m_{x_{0}(t)},u_{1}^{i}(t)\Big{)}dt+h_{1}\Big{(}x_{1}^{i}(T),x_{0}(T),m_{x_{0}(T)}\Big{)}\bigg{]}.\end{array}$
(30)
Using Corollary 2.5, the necessary condition for optimality is described by
the SHJB-FP coupled equation (11). To proceed, we assume that the optimal
control $\hat{\bf u}=(\hat{u}_{1}^{1},\hat{u}_{1}^{2},\dots,\hat{u}_{1}^{N})$
exists. To avoid ambiguity, denote $\hat{x}_{1}^{i}$ and $\hat{y}_{1}^{i}$ the
states dynamics of $x_{1}^{i}$ and $y_{1}^{i}$ corresponding to the optimal
control $\hat{u}_{1}^{i}$. The mean field term $m_{x_{0}(t)}$, is the
probability measure of the optimal trajectory $\hat{x}_{1}^{i}$ at time $t$,
conditioning on $\mathcal{F}^{0}_{t}$. Under this construction, being
conditional on $\mathcal{F}^{0}_{t}$, $\\{\hat{x}_{1}^{i}\\}_{i}$ are
identical and independent processes; while $\\{\hat{y}_{1}^{i}\\}_{i}$ are
dependent on each other through the empirical distribution. For simplicity,
for two density functions $m$ and $m^{\prime}$, we write
$W_{2}(md\lambda,m^{\prime}d\lambda)=W_{2}(m,m^{\prime})$.
###### Lemma A.1
Suppose the assumptions (A.1-A.3) hold. If $m_{x_{0}(t)}$ is chosen to be the
density function of $\hat{x}_{1}^{i}$ conditional on $\mathcal{F}_{0}^{t}$,
then
$\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|y_{0}(u)-x_{0}(u)|^{2}\Big{]}+\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\Big{]}=\mathcal{O}(\dfrac{1}{N}).\\\
$
###### Proof
Observe that for any $t\in[0,T]$
$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}&\leq&C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\bigg{|}g_{0}\Big{(}y_{0}(s),\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)},u_{0}(s)\Big{)}-g_{0}\Big{(}x_{0}(s),m_{x_{0}(s)},u_{0}(s)\Big{)}\bigg{|}^{2}ds\\\
&&\qquad\qquad+\mathbb{E}\displaystyle\int_{0}^{t}\bigg{|}\sigma_{0}\Big{(}y_{0}(s)\Big{)}-\sigma_{0}\Big{(}x_{0}(s)\Big{)}\bigg{|}^{2}ds\bigg{\\}},\end{array}$
and
$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}&\leq&C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\bigg{|}g_{1}\Big{(}\hat{y}_{1}^{i}(s),y_{0}(s),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\hat{u}_{1}^{i}(s)\Big{)}-g_{1}\Big{(}\hat{x}_{1}^{i}(s),x_{0}(s),m_{x_{0}(s)},\hat{u}_{1}^{i}(s)\Big{)}\bigg{|}^{2}ds\\\
&&\qquad\qquad+\mathbb{E}\displaystyle\int_{0}^{t}\bigg{|}\sigma_{1}\Big{(}\hat{y}_{1}^{i}(s)\Big{)}-\sigma_{1}\Big{(}\hat{x}_{1}^{i}(s)\Big{)}\bigg{|}^{2}ds\bigg{\\}},\end{array}$
By the Lipschitz assumptions, we have
$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}ds+\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}ds\bigg{\\}}\\\
&&\qquad+C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2}+\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},m_{x_{0}(t)}\Big{)}ds+\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{1}^{i}(s)-x_{1}^{i}(s)\Big{|}^{2}ds\bigg{\\}}\\\
&\leq&C\bigg{\\{}\mathbb{E}\displaystyle\int_{0}^{t}\displaystyle\sup_{u\leq
s}\Big{|}y_{0}(u)-x_{0}(u)\Big{|}^{2}+\displaystyle\sup_{u\leq
s}\Big{|}\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)\Big{|}^{2}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}ds\bigg{\\}},\\\
\end{array}$ (31)
where $C>0$ is a constant, changing line by line, depends only on $T$ and $K$.
By definition, for any Dirac measures $\delta_{y}$ on $\mathbb{R}^{n_{1}}$ and
density function $m$, we have
$W_{2}^{2}(\delta_{y},m)=\int_{\mathbb{R}^{n_{1}}}|y-x|^{2}dm(x).$
Also observe that the joint measure
$\frac{1}{N}\sum_{j=1}^{N}\delta_{(\hat{y}_{1}^{j}(s),\hat{x}_{1}^{j}(s))}$ on
${\mathbb{R}^{n_{1}}\times{\mathbb{R}^{n_{1}}}}$ has respective marginals
$\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)}$ and
$\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)}$ on
${\mathbb{R}^{n_{1}}}$. Using the definition of Wasserstein metric, we
evaluate
$\begin{array}[]{rcl}\mathbb{E}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}\bigg{]}&\leq&\mathbb{E}\bigg{[}\displaystyle\int_{\mathbb{R}^{n_{1}}\times{\mathbb{R}^{n_{1}}}}|y-x|^{2}d\bigg{(}\dfrac{1}{N}\sum_{j=1}^{N}\delta_{(\hat{y}_{1}^{j}(s),\hat{x}_{1}^{j}(s))}(y,x)\bigg{)}\bigg{]}\\\
&\leq&\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\mathbb{E}\Big{|}\hat{y}_{1}^{j}(s)-\hat{x}_{1}^{j}(s)\Big{|}^{2}\\\
&=&\mathbb{E}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2},\end{array}$
(32)
where the last equality results from the fact that
$\\{\hat{y}^{j}-\hat{x}^{j}\\}_{j=1}^{N}$ are symmetric. Similarly, we also
have
$\begin{array}[]{rcl}\mathbb{E}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}\bigg{]}&\leq&\mathbb{E}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2}.\end{array}$
Combining with (31) and applying Gronwall’s inequality, we have
$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&Ce^{Ct}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{t}W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}ds\bigg{]}.\\\
\end{array}$ (33)
By definition of the Wasserstein metric, we have
$\begin{array}[]{rcl}&&W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\\\
&=&\displaystyle\inf_{\Gamma}\displaystyle\int_{\mathbb{R}^{n_{1}}}\displaystyle\int_{\mathbb{R}^{n_{1}}}|x-y|^{2}d\Gamma_{\Big{(}\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}}(x,y)\\\
&\leq&\displaystyle\int_{\mathbb{R}^{n_{1}}}\displaystyle\int_{\mathbb{R}^{n_{1}}}|x|^{2}-2x\cdot
y+|y|^{2}d\Big{(}\frac{1}{N}\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}(x)dm_{x_{0}(s)}(y)\\\
&=&\Big{|}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\hat{x}_{1}^{j}(s)\Big{|}^{2}-2\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\hat{x}_{1}^{j}(s)\Big{)}\cdot\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)+\mathbb{E}^{\mathcal{F}^{0}_{s}}|\hat{x}_{1}^{j}(s)|^{2}\\\
&=&\Big{|}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}\\\
\end{array}$
Hence,
$\begin{array}[]{rcl}&&\mathbb{E}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\bigg{]}\\\
&\leq&\dfrac{1}{N^{2}}\mathbb{E}\bigg{[}\displaystyle\sum_{j=1}^{N}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}+2\displaystyle\sum_{j<k}^{N}[\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)]\cdot[\hat{x}_{1}^{k}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{k}(s)]\bigg{]}\\\
\end{array}$
Recall that given $\mathcal{F}^{0}_{t}$, $\\{\hat{x}_{1}^{j}\\}_{j}$ are
identically and independently distributed, we thus get
$\begin{array}[]{rcl}&&\mathbb{E}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\displaystyle\sum_{j=1}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\bigg{]}\\\
&\leq&\dfrac{1}{N^{2}}\mathbb{E}\bigg{[}\displaystyle\sum_{j=1}^{N}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}+2\displaystyle\sum_{j<k}^{N}[\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)]\cdot[\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{k}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{k}(s)]\bigg{]}\\\
&=&\dfrac{1}{N^{2}}\mathbb{E}\bigg{[}\displaystyle\sum_{j=1}^{N}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}+0\bigg{]}\\\
&=&\dfrac{1}{N}\mathbb{E}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}.\\\
\end{array}$
Similar estimate applies on the second term in Equation (33). Put $t=T$, we
finally have
$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
T}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
T}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&\dfrac{Ce^{CT}}{N}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\Big{|}\hat{x}_{1}^{j}(s)-\mathbb{E}^{\mathcal{F}^{0}_{s}}\hat{x}_{1}^{j}(s)\Big{|}^{2}ds\bigg{]}\\\
&\leq&\dfrac{4Ce^{CT}}{N}\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\Big{|}\hat{x}_{1}^{j}(s)\Big{|}^{2}ds\bigg{]}\end{array}$
(34)
With the linear growth assumptions, for any $t\in[0,T]$, we easily get the
estimates
$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq t}|x_{0}(t)|^{2}\leq
C\mathbb{E}\bigg{\\{}|\xi_{0}|^{2}+\displaystyle\int_{0}^{t}\displaystyle\sup_{u\leq
s}|x_{0}(u)|^{2}+\mathbb{E}^{\mathcal{F}_{s}^{0}}|\hat{x}_{1}^{j}(s)|^{2}+|u_{0}(s)|^{2}ds\bigg{\\}}\end{array}$
and
$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{x}_{1}^{j}(t)|^{2}\leq
C\mathbb{E}\bigg{\\{}|\xi_{1}^{j}|^{2}+\displaystyle\int_{0}^{t}\displaystyle\sup_{u\leq
s}|\hat{x}_{1}^{j}(u)|^{2}+\displaystyle\sup_{u\leq
s}|x_{0}(u)|^{2}+\mathbb{E}^{\mathcal{F}_{s}^{0}}|\hat{x}_{1}^{j}(s)|^{2}+|\hat{u}_{1}^{j}(s)|^{2}ds\bigg{\\}}\end{array}$
Applying the Tower property of expectation and the Gronwall’s inequality on
the sum of the two inequalities above yields
$\begin{array}[]{rcl}\mathbb{E}\displaystyle\sup_{u\leq
t}|x_{0}(t)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{x}_{1}^{j}(t)|^{2}\leq
Ce^{Ct}\mathbb{E}\bigg{\\{}|\xi_{0}|^{2}+|\xi_{1}^{j}|^{2}+\displaystyle\int_{0}^{t}|\hat{u}_{1}^{j}(s)|^{2}+|u_{0}(s)|^{2}ds\bigg{\\}}<\infty\end{array}$
We have the order for the estimate (34)
$\mathbb{E}\displaystyle\sup_{u\leq
T}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
T}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}=\mathcal{O}(\dfrac{1}{N}).$ (35)
∎
We also have approximation for the cost functionals.
###### Lemma A.2
$\begin{array}[]{rcl}&&\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}_{1}^{i})=O(\dfrac{1}{\sqrt{N}}).\\\ \end{array}$
###### Proof
With the quadratic assumptions (4), we have
$\begin{array}[]{rcl}|\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}^{i})|&\leq&\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}f_{1}\Big{(}\hat{y}_{1}^{i}(t),y_{0}(t),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(t)},\hat{u}^{i}(t)\Big{)}-f_{1}\Big{(}\hat{x}_{1}^{i}(t),x_{0}(t),m_{x_{0}(t)},\hat{u}_{1}^{i}(t)\Big{)}dt\\\
&&\qquad\qquad+h_{1}\Big{(}\hat{y}_{1}^{i}(T),y_{0}(T),\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(T)}\Big{)}-h_{1}\Big{(}\hat{x}_{1}^{i}(T),\hat{x}_{0}(T),m_{x_{0}(T)}\Big{)}\bigg{]}\\\
&\leq&C\mathbb{E}\bigg{[}\displaystyle\int_{0}^{T}\Big{[}1+|\hat{y}_{1}^{i}(t)|+|\hat{x}_{1}^{i}(t)|+|y_{0}(t)|+|x_{0}(t)|+\Big{(}\dfrac{\sum_{j=1,j\neq
i}^{N}|\hat{y}_{1}^{j}(t)|^{2}}{N-1}\Big{)}^{\frac{1}{2}}+\Big{(}\mathbb{E}^{\mathcal{F}_{t}^{0}}|\hat{x}_{1}^{i}(t)|^{2}\Big{)}^{\frac{1}{2}}+2|\hat{u}^{i}(t)|\Big{]}\\\
&&\qquad\qquad\qquad\cdot\Big{[}|\hat{y}_{1}^{i}(t)-\hat{x}_{1}^{i}(t)|+|y_{0}(t)-x_{0}(t)|+W_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(t)},m_{x_{0}(t)}\Big{)}\Big{]}dt\\\
&&\qquad+\Big{[}1+|\hat{y}_{1}^{i}(T)|+|\hat{x}_{1}^{i}(T)|+|y_{0}(T)|+|x_{0}(T)|+\Big{(}\dfrac{\sum_{j=1,j\neq
i}^{N}|\hat{y}_{1}^{j}(T)|^{2}}{N-1}\Big{)}^{\frac{1}{2}}+\Big{(}\mathbb{E}^{\mathcal{F}_{t}^{0}}|\hat{x}_{1}^{i}(T)|^{2}\Big{)}^{\frac{1}{2}}\Big{]}\\\
&&\qquad\qquad\qquad\cdot\Big{[}|\hat{y}_{1}^{i}(T)-\hat{x}_{1}^{i}(T)|+|y_{0}(T)-x_{0}(T)|+W_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(T)},m_{x_{0}(T)}\Big{)}\Big{]}\bigg{]}.\\\
\end{array}$
An application of Hölder’s inequality, and the symmetry on
$\\{\hat{x}_{1}^{j}\\}_{j}$ gives
$\begin{array}[]{rcl}|\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}^{i})|&\leq&C\bigg{\\{}\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}1+|\hat{y}_{1}^{i}(t)|^{2}+|\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)|^{2}+|x_{0}(t)|^{2}+\dfrac{\sum_{j=1,j\neq
i}^{N}|\hat{y}_{1}^{j}(t)|^{2}}{N-1}+\mathbb{E}^{\mathcal{F}_{t}^{0}}|\hat{x}_{1}^{i}(t)|^{2}+|\hat{u}^{i}(t)|^{2}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad\qquad\qquad\cdot\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}|\hat{y}_{1}^{i}(t)-\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)-x_{0}(t)|^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(t)},m_{x_{0}(t)}\Big{)}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad+\bigg{[}\mathbb{E}\Big{[}1+|\hat{y}_{1}^{i}(T)|^{2}+|\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)|^{2}+|x_{0}(T)|^{2}+\dfrac{\sum_{j=1,j\neq
i}^{N}|\hat{y}_{1}^{j}(T)|^{2}}{N-1}+\mathbb{E}^{\mathcal{F}_{t}^{0}}|\hat{x}_{1}^{i}(T)|^{2}\Big{]}\bigg{]}^{\frac{1}{2}}\\\
&&\qquad\qquad\qquad\cdot\bigg{[}\mathbb{E}\Big{[}|\hat{y}_{1}^{i}(T)-\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)-x_{0}(T)|^{2}+W^{2}_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(T)},m_{x_{0}(T)}\Big{)}\Big{]}\bigg{]}^{\frac{1}{2}}\bigg{\\}}\\\
&=&C\bigg{\\{}\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}1+|\hat{y}_{1}^{i}(t)|^{2}+|\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)|^{2}+|x_{0}(t)|^{2}+|\hat{u}^{i}(t)|^{2}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad\qquad\qquad\cdot\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}|\hat{y}_{1}^{i}(t)-\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)-x_{0}(t)|^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(t)},m_{x_{0}(t)}\Big{)}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad+\bigg{[}\mathbb{E}\Big{[}1+|\hat{y}_{1}^{i}(T)|^{2}+|\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)|^{2}+|x_{0}(T)|^{2}\Big{]}\bigg{]}^{\frac{1}{2}}\\\
&&\qquad\qquad\qquad\cdot\bigg{[}\mathbb{E}\Big{[}|\hat{y}_{1}^{i}(T)-\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)-x_{0}(T)|^{2}+W^{2}_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(T)},m_{x_{0}(T)}\Big{)}\Big{]}\bigg{]}^{\frac{1}{2}}\bigg{\\}}\\\
\end{array}$
By the linear growth assumptions on $g_{0}$, $\sigma_{0}$, $g_{1}$ and
$\sigma_{1}$, it is easy to show that
$\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}1+|\hat{y}_{1}^{i}(t)|^{2}+|\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)|^{2}+|x_{0}(t)|^{2}+|\hat{u}^{i}(t)|^{2}\Big{]}dt$
and
$\mathbb{E}\Big{[}1+|\hat{y}_{1}^{i}(T)|^{2}+|\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)|^{2}+|x_{0}(T)|^{2}\Big{]}$
are bounded (independent of $N$). We finally arrive at the estimates
$\begin{array}[]{rcl}|\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}^{i})|&\leq&C\bigg{\\{}\bigg{[}\mathbb{E}\displaystyle\int_{0}^{T}\Big{[}|\hat{y}_{1}^{i}(t)-\hat{x}_{1}^{i}(t)|^{2}+|y_{0}(t)-x_{0}(t)|^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(t)},m_{x_{0}(t)}\Big{)}\Big{]}dt\bigg{]}^{\frac{1}{2}}\\\
&&\qquad+\bigg{[}\mathbb{E}\Big{[}|\hat{y}_{1}^{i}(T)-\hat{x}_{1}^{i}(T)|^{2}+|y_{0}(T)-x_{0}(T)|^{2}+W^{2}_{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=1,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(T)},m_{x_{0}(T)}\Big{)}\Big{]}\bigg{]}^{\frac{1}{2}}\bigg{\\}},\\\
\end{array}$
which goes to $0$ as $N\rightarrow\infty$, as shown in Lemma A.1. Hence
$|\mathcal{J}^{N,i}(\hat{\bf
u})-\mathcal{J}^{i}(\hat{u}_{1}^{i})|=O(\dfrac{1}{\sqrt{N}}).$
∎
In the previous lemmas, we assumed that all players adopt their corresponding
mean field optimal controls. By symmetry, the convergences of state dynamics
and the cost functionals are then established. To show that the mean field
optimal controls ${\bf u}$ indeed constitute a $\epsilon$-Nash equilibrium on
the empirical system, without loss of generality, we assume that the first
player did not obey the mean field optimal control. In particular, let
$u_{1}^{1}$ be an arbitrary control in $\mathcal{A}_{1}$, define ${\bf
u}:=(u_{1}^{1},\hat{u}_{1}^{2},\dots,\hat{u}_{1}^{N})$. We then have the
following empirical and mean field SDEs for the dominating player, the $1$-st
player and the $i$-th player ($i>1$) respectively:
$\left\\{\begin{array}[]{rcl}dy_{0}&=&g_{0}\Big{(}y_{0}(t),\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(t)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(t)}\Big{)},u_{0}(t)\Big{)}dt+\sigma_{0}\Big{(}y_{0}(t)\Big{)}dW_{0}(t),\\\
y_{0}(0)&=&\xi_{0}.\\\
dx_{0}&=&g_{0}\Big{(}x_{0}(t),m_{x_{0}(t)},u_{0}(t)\Big{)}dt+\sigma_{0}\Big{(}x_{0}(t)\Big{)}dW_{0}(t),\\\
x_{0}(0)&=&\xi_{0}.\\\ \end{array}\right.$ (36)
$\left\\{\begin{array}[]{rcl}dy_{1}^{1}&=&g_{1}\Big{(}y_{1}^{1}(t),y_{0}(t),\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(t)},u_{1}^{1}(t)\Big{)}dt+\sigma_{1}\Big{(}y_{1}^{1}(t)\Big{)}dW_{1}^{1}(t),\\\
y_{1}^{1}(0)&=&\xi_{1}^{1}.\\\
dx_{1}^{1}&=&g_{1}\Big{(}x_{1}^{1}(t),x_{0}(t),m_{x_{0}(t)},u_{1}^{1}(t)\Big{)}dt+\sigma_{1}\Big{(}x_{1}^{1}(t)\Big{)}dW_{1}^{1}(t),\\\
x_{1}^{1}(0)&=&\xi_{1}^{1}.\\\ \end{array}\right.$ (37)
$\left\\{\begin{array}[]{rcl}d\hat{y}_{1}^{i}&=&g_{1}\Big{(}\hat{y}_{1}^{i}(t),y_{0}(t),\dfrac{1}{N-1}\Big{(}\delta_{y_{1}^{1}(t)}+\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(t)}\Big{)},\hat{u}_{1}^{i}(t)\Big{)}dt+\sigma_{1}\Big{(}\hat{y}_{1}^{i}(t)\Big{)}dW_{1}^{i}(t),\\\
\hat{y}_{1}^{i}(0)&=&\xi_{1}^{i}.\\\
d\hat{x}_{1}^{i}&=&g_{1}\Big{(}\hat{x}_{1}^{i}(t),x_{0}(t),m_{x_{0}(t)},\hat{u}_{1}^{i}(t)\Big{)}dt+\sigma_{1}\Big{(}\hat{x}_{1}^{i}(t)\Big{)}dW_{1}^{i}(t),\\\
\hat{x}_{1}^{i}(0)&=&\xi_{1}^{i}.\\\ \end{array}\right.$ (38)
We claim that if $m_{x_{0}}$ is the density function of $\hat{x}_{1}^{i}$
conditioning on $\mathcal{F}^{0}$, then we have the convergence
$y_{0}\rightarrow x_{0}$, $y_{1}^{1}\rightarrow x_{1}^{1}$ and
$\hat{y}_{1}^{i}\rightarrow\hat{x}_{1}^{i}$ in the sense of the following
lemma
###### Lemma A.3
$\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|y_{0}(u)-x_{0}(u)|^{2}\Big{]}+\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|y_{1}^{1}(u)-x_{1}^{1}(u)|^{2}\Big{]}+\mathbb{E}\Big{[}\displaystyle\sup_{u\leq
T}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\Big{]}\rightarrow 0,\quad\text{
as }N\rightarrow\infty.\\\ $
###### Proof
We first show the convergence of the dominating player and the $i$-th player.
Similar to the proof of Lemma A.1, we first have
$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},m_{x_{0}(s)}\Big{)}ds+\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}\bigg{\\}}ds\\\
&&\qquad+C\bigg{\\{}t\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2}+\Big{|}y_{0}(s)-x_{0}(s)\Big{|}^{2}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},m_{x_{0}(s)}\Big{)}ds+\mathbb{E}\displaystyle\int_{0}^{t}\Big{|}\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}(s)\Big{|}^{2}\bigg{\\}}ds\\\
&\leq&C\mathbb{E}\displaystyle\int_{0}^{t}\bigg{[}\displaystyle\sup_{u\leq
s}\Big{|}y_{0}(u)-x_{0}(u)\Big{|}^{2}+\displaystyle\sup_{u\leq
s}\Big{|}\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)\Big{|}^{2}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}+W_{2}^{2}(\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)})\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\bigg{]}ds\\\
\end{array}$ (39)
By the same argument used in Equation (32) in Lemma A.1, we have
$\mathbb{E}\Big{[}W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{x}_{1}^{j}(s)}\Big{)}\Big{]}+\mathbb{E}\Big{[}W_{2}^{2}(\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)},\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)})\Big{]}\leq
2\mathbb{E}|\hat{y}_{1}^{i}(s)-\hat{x}_{1}^{i}|^{2}.$
Hence, by applying Gronwall’s inequality on Equation (39), we have
$\begin{array}[]{rcl}&&\mathbb{E}\displaystyle\sup_{u\leq
t}|y_{0}(u)-x_{0}(u)|^{2}+\mathbb{E}\displaystyle\sup_{u\leq
t}|\hat{y}_{1}^{i}(u)-\hat{x}_{1}^{i}(u)|^{2}\\\
&\leq&Ce^{Ct}\mathbb{E}\displaystyle\int_{0}^{t}\bigg{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\\\
&&\qquad+W_{2}^{2}\Big{(}\dfrac{1}{N-1}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}+W_{2}^{2}\Big{(}\dfrac{1}{N-2}\displaystyle\sum_{j=2,j\neq
i}^{N}\delta_{\hat{x}_{1}^{j}(s)},m_{x_{0}(s)}\Big{)}\bigg{]}ds\\\
\end{array}$ (40)
For the first term in (40), consider the following joint measure on
$\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{1}}$
$\begin{split}\mu(x,y)&=\dfrac{1}{N}\displaystyle\sum_{j=2}^{N}\delta_{(\hat{y}_{1}^{j}(s),\hat{y}_{1}^{j}(s))}(x,y)+\dfrac{1}{N(N-1)}\displaystyle\sum_{j=2}^{N}\delta_{(y_{1}^{1}(s),\hat{y}_{1}^{j}(s))}(x,y),\\\
\end{split}$
which has respective marginals
$\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}\quad\text{and}\quad\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}.$
By the definition of Wasserstein metric,
$\begin{array}[]{rcl}&&\mathbb{E}\Big{[}W_{2}^{2}\Big{(}\dfrac{1}{N}\Big{(}\delta_{y_{1}^{1}(s)}+\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)},\dfrac{1}{N-1}\displaystyle\sum_{j=2}^{N}\delta_{\hat{y}_{1}^{j}(s)}\Big{)}\Big{]}\\\
&\leq&\mathbb{E}\Big{[}\displaystyle\int_{\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{1}}}|x-y|^{2}d\mu(x,y)\Big{]}\\\
&=&\mathbb{E}\Big{[}\dfrac{1}{N(N-1)}\displaystyle\sum_{j=2}^{N}|y_{1}^{1}(s)-\hat{y}_{1}^{j}(s)|^{2}\Big{]}\\\
&=&\dfrac{1}{N}\mathbb{E}|y_{1}^{1}(s)-\hat{y}_{1}^{2}(s)|^{2},\end{array}$
where the last equality results from symmetry on $\\{y_{1}^{j}\\}_{j}$,
clearly goes to $0$ as $N\rightarrow\infty$. Similar argument applies for the
third term in (40). For the convergence of the second and the forth term, we
refer to the argument in the last part of Lemma A.1 and the results follow.
For the convergence of the $1$-st player, the procedure are similar and we do
not provide here. ∎
We conclude from the similar procedures to show the convergence of the cost
functional. In particular, we have
$|\mathcal{J}^{N,1}({\bf
u})-\mathcal{J}^{1}(u_{1}^{1})|=O(\dfrac{1}{\sqrt{N}}).$
###### Theorem A.4
$\hat{\bf u}$ is an $\epsilon$-Nash equilibrium.
###### Proof
Summarizing all the obtained results in this section, we can conclude
$\begin{array}[]{rcl}&&|\mathcal{J}^{N,1}(\hat{\bf
u})-\mathcal{J}^{1}(\hat{u}_{1}^{1})|=O(\dfrac{1}{\sqrt{N}});\\\
&&|\mathcal{J}^{N,1}({\bf
u})-\mathcal{J}^{1}(u_{1}^{1})|=O(\dfrac{1}{\sqrt{N}}).\\\ \end{array}$
Since $\hat{u}_{1}^{1}$ is optimal control, we have
$\mathcal{J}^{1}(\hat{u}_{1}^{1})\leq\mathcal{J}^{1}(u_{1}^{1})$. We deduce
$\mathcal{J}^{N,i}(\hat{\bf u})\leq\mathcal{J}^{N,1}({\bf
u})+O(\dfrac{1}{\sqrt{N}}).$
Hence, $\hat{\bf u}$ is an $\epsilon$-Nash equilibrium. ∎
## References
* (1) A. Bensoussan, K.C.J. Sung, S.C.P. Yam and S.P. Yung. Linear-Quadratic Mean Field Games. Submitted, 2011.
* (2) A. Bensoussan and J. Frehse. Stochastic Games for $N$ players. Journal of Optimization Theory and Applications, 105(3), 543-565, 2000.
* (3) A. Bensoussan and J. Frehse. On Diagonal Elliptic and Parabolic Systems with Super-quadratic Hamiltonians. Communications on Pure and Applied Analysis, 8, 83-94, 2009.
* (4) A. Bensoussan, J. Frehse and J. Vogelgesang. Systems of Bellman Equations to Stochastic Differential Games with Non-compact Coupling. Discrete and Continuous Dynamical Systems, 27(4), 1375-1389, 2010.
* (5) A. Bensoussan, J. Frehse and S.C.P. Yam. Mean Field Games and Mean Field Type Control Theory. Springer Briefs in Mathematics, 2013.
* (6) C. Le Bris and P.L. Lions. Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Communications in Partial Differential Equations, 33 (7), 1272-1317, 2008.
* (7) R. Carmona and F. Delarue. Probabilistic Analysis of Mean-Field Games. SIAM Journal on Control and Optimization 51 (4), 2705-2734, 2013.
* (8) G. Cooper. The Origin of Financial Crises: Central Banks, Credit Bubbles, and the Efficient Market Fallacy. Harriman House Limited, 2008.
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* (11) M. Huang, R.P. Malhamé and P.E. Caines. Large Population Stochastic Dynamic Games: Closed-loop McKean-Vlasov Systems and the Nash Certainty Equivalence Principle. Communications in Information and Systems, 6 (3), 221-252, 2006.
* (12) J.M. Lasry and P.L. Lions. Jeux á champ moyen I - Le cas stationnaire. Comptes Rendus de l’Académie des Sciences, Series I, 343, 619-625, 2006a.
* (13) J.M. Lasry and P.L. Lions. Jeux á champ moyen II. Horizon fini et contrôle optimal. Comptes Rendus de l’Académie des Sciences, Series I, 343, 679-684, 2006b.
* (14) J.M. Lasry and P.L. Lions. Mean Field Games. Japanese Journal of Mathematics 2(1), 229-260, 2007.
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* (18) M. Nourian and P.E. Caines. $\epsilon$-Nash Mean Field Game Theory for Nonlinear Stochastic Dynamical Systems with Major and Minor Agents. SIAM Journal on Control and Optimization 51 (4), 3302-3331, 2013.
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|
arxiv-papers
| 2014-04-16T06:10:13 |
2024-09-04T02:50:01.241926
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Alain Bensoussan, Michael Chau, Phillip Yam",
"submitter": "Man Ho Chau",
"url": "https://arxiv.org/abs/1404.4148"
}
|
1404.4160
|
# Spatial Coherence Properties of Organic Molecules Coupled to Plasmonic
Surface Lattice Resonances in the Weak and Strong Coupling Regimes
L. Shi COMP Centre of Excellence, Department of Applied Physics, Aalto
University, FI-00076 Aalto, Finland T. K. Hakala COMP Centre of Excellence,
Department of Applied Physics, Aalto University, FI-00076 Aalto, Finland H.
T. Rekola COMP Centre of Excellence, Department of Applied Physics, Aalto
University, FI-00076 Aalto, Finland J.-P. Martikainen COMP Centre of
Excellence, Department of Applied Physics, Aalto University, FI-00076 Aalto,
Finland R. J. Moerland COMP Centre of Excellence, Department of Applied
Physics, Aalto University, FI-00076 Aalto, Finland Department of Imaging
Physics, Faculty of Applied Sciences, Delft University of Technology,
Lorentzweg 1, NL-2628 CJ, Delft, The Netherlands P. Törmä Electronic
address: [email protected] COMP Centre of Excellence, Department of
Applied Physics, Aalto University, FI-00076 Aalto, Finland
###### Abstract
We study spatial coherence properties of a system composed of periodic silver
nanoparticle arrays covered with a fluorescent organic molecule (DiD) film.
The evolution of spatial coherence of this composite structure from the weak
to the strong coupling regime is investigated by systematically varying the
coupling strength between the localized DiD excitons and the collective,
delocalized modes of the nanoparticle array known as surface lattice
resonances. A gradual evolution of coherence from the weak to the strong
coupling regime is observed, with the strong coupling features clearly visible
in interference fringes. A high degree of spatial coherence is demonstrated in
the strong coupling regime, even when the mode is very excitonlike (80 $\%$),
in contrast to the purely localized nature of molecular excitons. We show that
coherence appears in proportion to the weight of the plasmonic component of
the mode throughout the weak-to-strong coupling crossover, providing evidence
for the hybrid nature of the normal modes.
###### pacs:
33.80.-b, 73.20.Mf, 42.50.Nn
Spatial coherence properties of waves can be probed by passing a wave front
through distant slits and observing interference. Inspired by this phenomenon
well known for classical radiation, interference experiments were crucial in
establishing the wave-particle nature of single photons, as well as massive
particles Davisson and Germer (1927); Arndt et al. (1999); Juffmann et al.
(2012), within quantum mechanics. In the experiments Davisson and Germer
(1927); Arndt et al. (1999); Juffmann et al. (2012) the quantum mechanical
wave properties of matter became visible at low temperatures. Here, we
consider a different question: the spatial coherence properties of objects, or
modes, that are hybrids of wavelike and particlelike components. Mixing a
localized matter component with light may possibly give the hybrid object a
nontrivial spatial coherence length.
Examples of light-matter hybrids include coherent superpositions of atoms and
cavity photons Rempe et al. (1987); Thompson et al. (1992), semiconductor
cavity polaritons, which have been brought to quantum degeneracy and
condensation Kasprzak et al. (2006), and cavity photon mediated strong
coupling between spatially separated localized molecular excitons Lidzey et
al. (2000). Recently, delocalized electromagnetic modes supported by metal
surfaces (surface plasmon polaritons) or periodic arrays of metallic
nanoparticles [surface lattice resonances (SLRs) Zou et al. (2004); García de
Abajo (2007); Auguié and Barnes (2008); Zhou et al. (2013)] have been shown to
strongly couple with localized emitters Bellessa et al. (2004); Dintinger et
al. (2005); Hakala et al. (2009); Gomez et al. (2010); Schwartz et al. (2011);
Vasa et al. (2013); Rodriguez and Gomez Rivas (2013); Väkeväinen et al.
(2013). The strong coupling in these plasmonic systems involves a large number
$N$ of emitters. The normal mode splittings observed are consistent both with
classical linear dispersion theory and with the vacuum Rabi splitting obtained
as the low excitation limit of the Dicke model, similarly to the early
experiments on many atoms in cavities Zhu et al. (1990). The collective
behavior of many emitters has been clearly demonstrated in these systems,
manifested as the $\sqrt{N}$ dependence of the splitting. The observed
splittings in dispersions strongly support the interpretation that the new
normal modes are hybrid modes formed by strong coupling of lightlike (the
surface plasmon polariton/SLR) and matterlike (the molecular excitation)
components. Observations of the dispersions alone, however, cannot directly
test whether the new modes carry all the essential properties of the original
modes, as should be the case if the hybrids are linear, coherent combinations
of the original modes. In particular, spatial coherence is the specific
characteristic of an extended light mode: in order to prove that the new modes
carry this property, interference experiments are needed. To be conclusive, it
is necessary to show that the coherence appears in proportion to the weight of
the light mode in the hybrid. This in turn requires a systematic study of
coherence throughout the weak-to-strong coupling crossover. This is the goal
of the present work.
The spatial interference effects of light-matter hybrids have been studied in
a few experiments in the context of exciton-polariton condensates Richard et
al. (2005); Kasprzak et al. (2006); Deng et al. (2007). In plasmonic systems,
only one experiment has been reported Aberra Guebrou et al. (2012): signatures
of coherence were observed in the strong coupling regime in a planar metal
surface—molecular film system. However, that work does not prove the
connection of the spatial coherence with the weight of the light component
since there was no study of the weak-to-strong coupling crossover (a different
system, namely quantum dots, was given as the weak coupling reference). Here,
we study the spatial coherence properties of a system composed of periodic
silver nanoparticle arrays covered with fluorescent organic molecules (DiD) by
employing a double slit experiment. We gradually increase the molecule
concentration to investigate both the strong and the weak coupling coherence
properties within the same system.
Figure 1(a) shows a scanning electron micrograph (SEM) of a typical array (for
fabrication details, see Supplemental Material Supplementarynote ). The
$d_{y}$ = 50 nm, $p_{y}$ = 200 nm, while $d_{x}$ and $p_{x}$ were varied
between 133–400 nm and 380–500 nm, respectively. The DiD concentration in
poly(methyl methacrylate) film was varied between 20–800 mM.
The measurement setup is depicted in Fig. 1(b). $y$-polarized white light was
incident on the sample; see Fig. 1(a). Angle and wavelength-resolved
transmission spectra $T=I_{\textrm{Structure}}/I_{\textrm{Reference}}$ [Fig.
1(b)] were measured and subsequently used for calculating the dispersion for
each array. The entrance slit of the spectrometer and the in-plane wave vector
$k$ is parallel to the $x$ axis of the sample with magnitude
$k=2\pi/\lambda\sin(\theta)$, where $\lambda$ is the wavelength in the medium
and $\theta$ is the angle between the optical axis and the light propagation
direction.
Figure 1: (a) A SEM of a typical sample. The scale bar is 1 $\mu$m (200 nm for
the inset). (b) The measurement setup. Angle resolved transmission spectra for
each array were measured by placing the back focal plane of the sample at the
entrance slit of the spectrometer. For spatial coherence measurements, a
double slit was placed at the first intermediate image plane of the system.
Figure 2: The dispersions of three different nanoparticle arrays with
inreasing DiD concentration. (a)–(e) Array 1, $(d_{x})\times(d_{y})$ = 50 nm
$\times$ 220 nm, $p_{x}$ = 500 nm. (f)–(j) Array 2, $(d_{x})\times(d_{y})$ =
50 nm $\times$ 355 nm, $p_{x}$ = 500 nm. (k)–(o) Array 3,
$(d_{x})\times(d_{y})$ = 50 nm $\times$ 167 nm, $p_{x}$ = 380 nm. The first
column corresponds to a case without DiD molecules, while the second, third,
fourth, and fifth columns have 20, 200, 400, and 800 mM concentrations of DiD,
respectively. White areas correspond to maximum extinction. The blue
horizontal lines depict the absorption maximum of the DiD film. The yellow
lines correspond to peak positions obtained from fitting a Gaussian curve to
the line cuts of dispersions while keeping $k$ constant, and the red lines are
obtained from the coupled oscillator model. (p) The SLR-exciton coupling
strength as a function of square root of concentration. The blue plus signs,
red crosses, and green circles correspond to arrays 1, 2 and 3, respectively.
(q)-(s) The relative SLR-exciton weights of the arrays 1-3, respectively. The
solid (dashed) line corresponds to exciton (SLR) percentage and black, orange
and purple to concentrations of 200, 400, and 800 mM, respectively.
In Figs. 2(a)–2(o) are shown the measured angle resolved extinction ($1-T$)
spectra for different nanoparticle arrays. Several observations can be made
from these figures. First, the energy of the $\Gamma$ point ($k=0$) can be
changed by changing the periodicity [see for example Figs. 2(a) and 2(k)].
Second, upon coupling of the <+1, 0> and <-1, 0> diffractive orders Barnes et
al. (1996), a band gap is formed in Fig. 2(f) and the associated new modes can
be made either dark or bright by changing the filling fraction [$d_{x}/p_{x}$,
see Fig. 1(a)]. For details, see Supplemental Material Supplementarynote .
The dispersions in Fig. 2(b)–2(e) illustrate how the system gradually evolves
from the weak to the strong coupling regime with increasing molecular
concentration. A clear modification of the system energies is observed in
Figs. 2(b)–2(d), which then in Fig. 2(e) develops into a distinctive band
bending and anticrossing at the energy corresponding to the absorption maximum
of the molecule, a behavior that is characteristic for the strong coupling
regime. Similar evolution from weak to strong coupling regime can readily be
identified for arrays 2 [Figs. 2(f)–2(j)] and 3 [Figs. 2(k)–2(o)], but now the
system energies are drastically different due to different filling fraction
(array 2) and periodicity (array 3). These results demonstrate how the choice
of geometry and molecular concentration provides excellent control over the
system properties.
In the strong coupling theory, the new modes are linear combinations of the
uncoupled SLRs and the molecular excitations. To describe such hybrid modes,
we employ a coupled oscillator model satisfying the equation
$\left(\begin{array}[]{cc}E_{SLR}(k)+i\gamma_{SLR}&\Omega\\\
\Omega&E_{DiD}+i\gamma_{DiD}\end{array}\right)\left(\begin{array}[]{c}\alpha\\\
\beta\end{array}\right)=0,$ (1)
where $E$ and $\gamma$ are the energies and the widths of the uncoupled modes,
$\Omega$ is the coupling strength between the SLR and DiD, and $\alpha$ and
$\beta$ are the coefficients of the linear combination of SLR and the DiD
exciton (for details see Supplemental Material Supplementarynote ). The SLR-
exciton coupling strength $\Omega$ and the linewidth $\gamma_{\mathrm{DiD}}$
of the exciton are used as fitting parameters. The resulting mode energies are
plotted in Figs. 2(c)–2(e), 2(h)–2(j), and 2(m)–2(o) for different arrays and
are in good agreement with the experimentally observed mode energies. The SLR-
exciton coupling is expected to scale as $\sqrt{N/V}$, where $N$ is the number
of molecules and $V$ is the mode volume Agranovich et al. (2003); González-
Tudela et al. (2013): this is confirmed in Fig. 2(p). Notably, the size of the
observed splitting is in reasonable agreement with microscopic theory
Agranovich et al. (2003) (see Supplemental Material Supplementarynote ). Note
that spectrally broad emitters coupled to spectrally selective (plasmon) modes
can produce luminescence spectra reminiscent of those observed in strongly
coupled systems (see, e.g., Ref. Gruber et al. (2013)). That we observe strong
coupling instead of this phenomenon is proven by the series of different
concentrations that we studied, showing the $\sqrt{N/V}$ dependence expected
for strong coupling.
Figure 3: (a)–(d) The spatial coherence images for the array 2 with
concentrations 0, 20, 400, and 800 mM, respectively. Here white areas
correspond to transmission maximum. The yellow lines have the same meaning as
in Fig. 2. (e) A sample having a random distribution of nanoparticles with 800
mM DiD concentration. Two transmission minima are seen at 1.85 eV (yellow
line) and 2.25 eV, corresponding to DiD absorption and the single particle
plasmon resonance, respectively.
In Figs. 2(q)–2(s), we plot the relative weights of the hybrid modes as
functions of the in-plane wave vector $k$ for arrays 1–3, respectively, with
molecular concentrations of 200, 400, and 800 mM. For arrays 1 and 2, the SLR-
exciton hybrid is mostly SLR-like for $k\sim 0$, and becomes increasingly
excitonlike for higher $k$ values. The relative exciton contribution at $k\sim
0$ increases with concentration due to stronger hybridization of the SLR with
the exciton. Note, however, that for array 3 [Fig. 2(s)] the mode is
excitonlike at $k\sim 0$, and then gradually evolves to SLR-like mode at
higher $k$. This is due to the SLR $\Gamma$-point energy being above the
molecular excitation energy [compare, for example, Figs. 2(g) and 2(l)). These
results demonstrate how the relative weights of the hybrid mode at a given
energy and wave vector can be tailored by choice of geometry and molecular
concentration.
To investigate coherence, angle resolved transmission spectra are recorded
with a double slit placed on the image plane of the sample; see Fig. 1(b).
This forms the crucial test for the presence of spatial coherence in the new
modes: if the spatial coherence length of the mode is greater than the
interslit distance, a distinctive fringe pattern would be expected in the
Fourier plane of the imaging system. In Fig. 3(a)–3(d) are shown the
wavelength-resolved spatial coherence images obtained from array 2 and with
molecular concentrations ranging from 0 to 800 mM. Intriguingly, bending of
the interference pattern is observed towards the strong coupling regime. In
other words, one of the destructive interference fringes in spatial coherence
images always overlaps with the extinction maxima of the dispersion (yellow
symbols); see Figs. 2(g)-2(j). This allows to make an important connection
with the original modes: If a spatially coherent light source (i.e., the
sample) is radiating through a double slit, the interference fringes can be
interpreted as replicas of the original dispersion (Fig. 2) created by the
diffracted orders from the double slit. At high frequencies the interference
pattern becomes complex due to the close spacing of the crossing points of the
replicas (see also Supplemental Material Supplementarynote ). Thus, the fact
that band bending with increasing concentration is seen both in the
dispersions and the spatial coherence images provides a clear signature that
the interference fringes are directly related to the modes of interest and are
not due to any secondary reason. We have thus conclusively shown that the
system modes have prominent spatial coherence throughout the crossover, also
deep in the strong coupling regime.
We want to point out the important role of the array periodicity, i.e the
existence of the dispersive SLR modes, for the emergence of long-range
coherence. Figure 3(e) shows a spatial coherence image of a sample having a
random interparticle spacing (for a SEM image, see Supplemental Material
Supplementarynote ) while the molecular concentration, nanoparticle size,
orientation and number are the same as in the sample in Fig. 3(d). Evidently,
no interference fringes are present in this case. Also, they are absent in DiD
films without nanoparticles.
Notably, the fringes become less visible with increasing concentration at
energies above 1.8 eV; see Figs. 3(b)–3(d). Higher molecular concentration
induces stronger hybridization between the delocalized SLR and localized
molecular excitons. At higher energies, these hybrid modes become increasingly
excitonlike and localized as the energy gets closer to DiD dye absorption,
reducing the spatial coherence length below the interslit distance. Note,
however, that the fringe pattern persists below 1.8 eV energies, even with 800
mM concentration. We have thus demonstrated that the SLR-exciton hybrid modes
display long-range coherence even when the mode is very excitonlike: from Fig.
2(r) the exciton weight can be deduced to be $80\%$ at high $k$-vector values.
In the rest of this Letter, we consider the crucial question of whether there
is a systematic, quantitative connection between the spatial coherence and the
expected weight of the light component in a hybrid mode. First, we want to
show that detailed structure of the interference fringes can be produced by
assuming hybrid modes, with weights of the light and matter parts as obtained
by fitting the experimental dispersion with the coupled oscillator model Eq.
(1) (the obtained dispersion was then used to provide the energy and wave
vector specific information of the mode radiating through the double slit, see
Supplemental Material Supplementarynote ). In Fig. 4(a) we show a close-up of
the spatial coherence image of Fig. 3(d) (800 mM concentration) and in Fig.
4(b) we show the interference image obtained from calculations based on the
coupled oscillators model. While the intensities in both Figs. 4(a) and 4(b)
are of comparable magnitude, at high energies, the experimental data have less
transmission intensity. This can be due to additional absorption of the
molecules that are not contributing to strong coupling Agranovich et al.
(2003); Bellessa et al. (2004). In general, however, the correspondence of the
model with the most prominent features of the experimental data is excellent.
This is the first step of systematically proving the connection between the
hybrid structure and the coherence: the model with weights of matter and light
parts in the hybrid as given by strong coupling theory indeed reproduces the
interference pattern observed experimentally.
Figure 4: (a) A close-up of the spatial coherence image (800 mM
concentration). (b) The interference image obtained from the coupled
oscillator model. (c) The $\Delta k$ obtained from the experiments (red empty
circles) and from the coupled oscillator model (blue crosses). Dashed and
solid lines correspond to the SLR and exciton weights of the mode,
respectively. (d) The spatial coherence length obtained from the experiments
(red circles) and from the coupled oscillator model (blue empty circles). The
dashed line is the effective interslit distance at the sample plane.
Second, we consider the important connection between the interference fringes,
mode delocalization, the width of the mode $\Delta k$, and the relative
weights of the strongly coupled modes. In Fig. 4(c) we show the $\Delta k$ of
the mode as a function of the energy obtained from the experiments (Fig. 2)
and from the model. The $\Delta k$ was obtained as FWHM of constant-energy
line cuts from the dispersions. Also shown are the relative SLR and exciton
weights of the hybrid mode. In Fig. 4(d) we show the spatial coherence lengths
of the mode obtained as $L_{x}=2\pi/\Delta k$ Mandel and Wolf (1965). Because
the momentum and position are Fourier related, a small $\Delta k$ at energies
around 1.6 eV [see Fig. 4(c)] suggests a delocalized mode and large spatial
coherence length. The delocalization is also evident from the high SLR
fraction (80 $\%$) of the mode. In the spatial coherence image, the
delocalization manifests itself as a distinct interference pattern [Figs. 4(a)
and 4(b)]. As $\Delta k$ increases at energies $E>1.65$ eV, however, the
hybrid mode becomes more localized and more excitonlike, which gradually
yields a less prominent interference pattern in accordance with the increasing
weight of the matter component. At energies above 1.7 eV, the spatial
coherence length decreases below the interslit distance [Fig. 4(d)], and,
consequently the interference pattern disappears; see Fig. 3(d).
In both classical optics and quantum mechanics, modes are characterized not
only by their energies, observable in dispersions, but also by the coherent
modes or wave functions forming linear superpositions. Both aspects should be
considered in identifying physical phenomena, cf. the observation of Bose-
Einstein condensation by evidence in momentum distribution Anderson et al.
(1995) and in interference patterns Andrews et al. (1997). The strong coupling
regime of various types of surface plasmon modes and emitters has been widely
studied by observing dispersion relations. Splittings in the dispersions have
been attributed to hybridization of plasmonic and excitonlike modes. Here we
provide the first systematic study of the evolution of the spatial coherence
in a plasmonic-molecule system when transiting from the weak to the strong
coupling regime. The evolution of spatial coherence is shown to be directly
connected to the hybrid mode structure. Significant spatial coherence lengths
in the strongly coupled system are observed even when the mode is very
excitonlike. Complementing the energy dispersions and dynamics observed
earlier, our interference results provide conclusive evidence for the hybrid
nature of the normal modes in strongly coupled surface plasmon—emitter
systems. In general, our results demonstrate the potential of hybridization in
creating nanosystems with designed properties, in this case long range
coherence for modes that are largely matterlike.
###### Acknowledgements.
We thank Dr. Shaoyu Yin for useful discussions. This work was supported by the
Academy of Finland through its Centres of Excellence Programme (Projects No.
251748, No. 263347, No. 135000, and No. 141039) and by the European Research
Council (ERC-2013-AdG-340748-CODE). Part of the research was performed at the
Micronova Nanofabrication Centre, supported by Aalto University.
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|
arxiv-papers
| 2014-04-16T07:47:05 |
2024-09-04T02:50:01.252282
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "L. Shi, T. K. Hakala, H. T. Rekola, J.-P. Martikainen, R. J. Moerland,\n and P. T\\\"orm\\\"a",
"submitter": "Heikki Rekola",
"url": "https://arxiv.org/abs/1404.4160"
}
|
1404.4189
|
# Factor Complexity of $S$-adic sequences generated by the Arnoux-Rauzy-
Poincaré Algorithm
V. Berthé and S. Labbé
LIAFA, Université Paris Diderot - Paris 7,
Case 7014, 75205 Paris Cedex 13, France
[email protected]
[email protected]
(Mathematics Subject Classifications: 68R15, 37B10.)
###### Abstract
The Arnoux-Rauzy-Poincaré multidimensional continued fraction algorithm is
obtained by combining the Arnoux-Rauzy and Poincaré algorithms. It is a
generalized Euclidean algorithm. Its three-dimensional linear version consists
in subtracting the sum of the two smallest entries to the largest if possible
(Arnoux-Rauzy step), and otherwise, in subtracting the smallest entry to the
median and the median to the largest (the Poincaré step), and by performing
when possible Arnoux-Rauzy steps in priority. After renormalization it
provides a piecewise fractional map of the standard $2$-simplex. We study here
the factor complexity of its associated symbolic dynamical system, defined as
an $S$-adic system. It is made of infinite words generated by the composition
of sequences of finitely many substitutions, together with some restrictions
concerning the allowed sequences of substitutions expressed in terms of a
regular language. Here, the substitutions are provided by the matrices of the
linear version of the algorithm. We give an upper bound for the linear growth
of the factor complexity. We then deduce the convergence of the associated
algorithm by unique ergodicity.
## 1 Introduction
Multidimensional continued fraction algorithms aim at providing good rational
approximations of a given vector. There exist many different types of
continued fraction algorithms. Among them, piecewise fractional ones in the
sense of [Bre81, Sch00] have been widely studied whereas for their arithmetic
or for their ergodic properties. The viewpoint we take here on these
algorithms is issued from word combinatorics and symbolic dynamics. It is
indeed possible to generate with such algorithms infinite words with
prescribed letter frequencies: the letter frequency vector is indeed the
vector on which the algorithm is applied. We recall that a substitution is a
morphism of the free monoid that replaces letters by finite words. A piecewise
fractional continued fraction algorithm produces (unimodular) matrices with
non-negative entries that we consider as incidence matrices of substitutions.
We then iterate these substitutions in an $S$-adic way, that is, as the
(inverse) limit of an infinite product of substitutions (see e.g. [BD14, CN10,
DLR13, Ler12]). We thus obtain an infinite word $\mathbf{u}$ of the form
$\mathbf{u}=\lim_{n\to\infty}\sigma_{0}\circ\sigma_{1}\circ\cdots\circ\sigma_{n}(a^{\infty}).$
As an illustration consider the generation of Sturmian words with the
classical continued fraction algorithm (see [Lot02, Fog02] for more details).
The Diophantine approximation properties of the underlying continued fraction
algorithm are reflected in the generic behaviour of the balance function of
the generated word $\mathbf{u}$, where the balance function counts, for each
given letter, the difference between the numbers of occurrences of this letter
in any two words of the same length that occur in $\mathbf{u}$. It is also
closely related to the notion of symbolic discrepancy such as considered in
[Ada03]. We also would like the combinatorics of the generated infinite word
$\mathbf{u}$ to be “simple” in the sense that the factor complexity of
$\mathbf{u}$ is expected to be of linear growth, were the factor complexity
counts the number of factors of a given length.
Observe that there exist several methods for producing infinite words with
prescribed letter frequencies having a linear factor complexity $p(n)$ and/or
a bounded balance. The Sturmian words form a well-known family of infinite
balanced words over a two-letter alphabet having a linear factor complexity
($p(n)=n+1$ for all $n$). Nevertheless the situation is more contrasted for
words defined on alphabets having at least three letters concerning the
possibility of having simultaneously prescribed letter frequencies, a linear
factor complexity and a bounded balance. Typical generalizations of Sturmian
words are natural codings of interval exchanges and the billiard words in the
$d$-dimensional cube. However, billiard words have quadratic factor complexity
[Bar95, Bed03] and codings of interval exchanges are not balanced [Zor97].
Other approaches were considered in digital geometry where arithmetic
definitions of $3$D discrete lines were proposed. The standard model of
[And03] is one of them and can be encoded as a word on a three-letter
alphabet. It turns out that this model corresponds to the one of billiard
words [Lab12], thus also having a quadratic factor complexity in general.
The experimentations described in [BL11, Lab12] indicate that some
multidimensional continued fraction algorithms generate $S$-adic words having
a linear factor complexity and a bounded balance for almost every letter
frequencies vector. In particular, Brun multidimensional continued fraction
algorithm as well as the Arnoux-Rauzy-Poincaré algorithm seem to be the two
best choices in terms of balance properties. In this article, we focus on the
Arnoux-Rauzy-Poincaré algorithm which performs experimentally a bit better
than does Brun algorithm. This algorithm (under its linear form) consists in
subtracting the sum of the two smallest entries to the largest if possible and
otherwise, in subtracting the smallest entry to the median and the median to
the largest. In order to generate infinite words, we introduce an
$\mathcal{S}$-adic system associated with the nine possible matrices of the
algorithm that thus provide a set $\mathcal{S}$ of nine substitutions. Three
of them are substitutions known under the name of Arnoux-Rauzy substitutions
[AR91], and the other six are named Poincaré substitutions after Poincaré
algorithm [Nog95]. Moreover, the execution of the Arnoux-Rauzy-Poincaré
algorithm yields restrictions to the allowed infinite sequences of
substitutions, expressed in terms of a regular language. We then have a
bijection (up to a set of zero measure) between the infinite words in the
corresponding $\mathcal{S}$-adic system and the standard $2$-simplex
$\Delta=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}_{+}\mid
x_{1}+x_{2}+x_{3}=1\\}$ (the vectors of letter frequencies). The main result
of the present paper is that these words have a linear factor complexity
$p(n)$.
###### Theorem 1 (Factor Complexity).
Let $\mathbf{u}$ be an $\mathcal{S}$-adic word generated by the Arnoux-Rauzy-
Poincaré algorithm applied to a totally irrational vector
$\mathbf{x}\in\Delta$. Then the factor complexity of $\mathbf{u}$ is such that
$p(n+1)-p(n)\in\\{2,3\\}$ and $2n+1\leq p(n)\leq\frac{5}{2}n+1$ for all $n\geq
0$.
The proof relies on a careful study of bispecial factors of $\mathbf{u}$, that
is, of factors having several left and right extensions in $\mathbf{u}$. We
prove that weak and strong bispecial factors are alternating in the sequence
(ordered by increasing length) of non-neutral bispecial factors. The
restriction for the directive sequences of the $\mathcal{S}$-adic words to the
regular language provided by the Arnoux-Rauzy-Poincaré algorithm is clearly
important; indeed quadratic factor complexity can be reached otherwise (see
Section 4.5).
Then, by using a result of Boshernitzan [Bos85], we deduce unique ergodicity
and thus, the existence of (uniform) frequency of any factor, and in
particular of the letters. This also provides a combinatorial proof of
convergence for this multidimensional continued fraction algorithm.
###### Theorem 2 (Frequencies and Convergence).
Let $\mathbf{u}$ be an $\mathcal{S}$-adic word generated by the Arnoux-Rauzy-
Poincaré algorithm applied to a totally irrational vector
$\mathbf{x}\in\Delta$. Then the symbolic dynamical system generated by
$\mathbf{u}$ is uniquely ergodic. As a consequence, the frequencies of factors
and letters exist in $\mathbf{u}$, the latter being equal to the coordinates
of $\mathbf{x}$.
Furthermore, the Arnoux-Rauzy-Poincaré algorithm is a weakly convergent
algorithm, that is, for Lebesgue almost every $\mathbf{x}\in\Delta$, if
$(M_{n})_{n}$ stands for the sequence of matrices produced by the Arnoux-
Rauzy-Poincaré algorithm, then one has $\cap_{n}M_{0}\cdots
M_{n}({\mathbb{R}}_{+}^{3})={\mathbb{R}}_{+}\mathbf{x}.$
Let us sketch the content of the present paper. The Arnoux-Rauzy-Poincaré
multidimensional continued fraction algorithm is introduced in Section 2. We
also define the associated Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic system
based on its nine substitutions (provided by its linear version) together with
a rational restriction on the directive sequences of substitutions that are
iterated. In Section 3, we introduce the basic notions used to compute the
factor complexity, namely languages, bispecial factors and extension types. In
Section 4, we study the life of bispecial factors under Arnoux-Rauzy and
Poincaré substitutions (with no restriction on the order of application of
substitutions). In Section 5, we prove the upper bound on the factor
complexity stated in Theorem 1. The convergence of the algorithm together with
unique ergodicity is lastly considered in Section 6.
This article is an extended version of [BL13]. The present paper provides the
upper bound $p(n)\leq\frac{5}{2}n+1$, whereas the upper bound in [BL13] was
$p(n)\leq 3n+1$.
Acknowledgements We are thankful to Pierre Arnoux, Srecko Brlek, Julien
Cassaigne, Julien Leroy and Thierry Monteil for many fruitful discussions on
the subject. This work was supported by Agence Nationale de la Recherche and
the Austrian Science Fund through project Fractals and Numeration
ANR-12-IS01-0002 and project Dyna3S ANR-13-BS02-0003. The second author is
supported by NSERC (Canada).
## 2 The Arnoux-Rauzy-Poincaré Algorithm
### 2.1 The algorithm
The Arnoux-Rauzy-Poincaré (ARP) is a multidimensional continued fraction
algorithm in the sense of [Bre81, Sch00], defined by piecewise fractional maps
acting on the standard $2$-simplex
$\Delta=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}_{+}:x_{1}+x_{2}+x_{3}=1\\}$.
It is a fusion algorithm such as introduced in [BL11, Lab12] which combines
the two classical algorithms that are Poincaré (P) algorithm and Arnoux-Rauzy
(AR) algorithm, which are respectively defined (under their linear form) in
dimension 3 as follows: Poincaré algorithm acts on a triple of non-negative
entries by subtracting the smallest entry to the median and the median to the
largest, whereas Arnoux-Rauzy algorithm acts by subtracting the sum of the two
smallest entries to the largest, when possible. Our algorithm privilegiates an
Arnoux-Rauzy step if possible, otherwise it perfoms a Poincaré step.
The simplex $\Delta$ admits as vertices the vectors
$\mathbf{e}_{1}=(1,0,0)^{\top}$, $\mathbf{e}_{2}=(0,1,0)^{\top}$ and
$\mathbf{e}_{3}=(0,0,1)^{\top}$. In order to partition $\Delta$, we consider
the following fifteen matrices, namely
$\footnotesize\begin{array}[]{lllll}A_{1}=\left(\begin{array}[]{rrr}1&1&1\\\
0&1&0\\\ 0&0&1\end{array}\right),&P_{21}=\left(\begin{array}[]{rrr}1&1&1\\\
0&1&1\\\ 0&0&1\end{array}\right),&P_{31}=\left(\begin{array}[]{rrr}1&1&1\\\
0&1&0\\\ 0&1&1\end{array}\right),&H_{21}=\left(\begin{array}[]{rrr}1&0&0\\\
0&1&0\\\ 1&0&1\end{array}\right),&H_{31}=\left(\begin{array}[]{rrr}1&0&0\\\
1&1&0\\\ 0&0&1\end{array}\right),\\\ A_{2}=\left(\begin{array}[]{rrr}1&0&0\\\
1&1&1\\\ 0&0&1\end{array}\right),&P_{12}=\left(\begin{array}[]{rrr}1&0&1\\\
1&1&1\\\ 0&0&1\end{array}\right),&P_{32}=\left(\begin{array}[]{rrr}1&0&0\\\
1&1&1\\\ 1&0&1\end{array}\right),&H_{12}=\left(\begin{array}[]{rrr}1&0&0\\\
0&1&0\\\ 0&1&1\end{array}\right),&H_{32}=\left(\begin{array}[]{rrr}1&1&0\\\
0&1&0\\\ 0&0&1\end{array}\right),\\\ A_{3}=\left(\begin{array}[]{rrr}1&0&0\\\
0&1&0\\\ 1&1&1\end{array}\right),&P_{13}=\left(\begin{array}[]{rrr}1&1&0\\\
0&1&0\\\ 1&1&1\end{array}\right),&P_{23}=\left(\begin{array}[]{rrr}1&0&0\\\
1&1&0\\\ 1&1&1\end{array}\right),&H_{13}=\left(\begin{array}[]{rrr}1&0&0\\\
0&1&1\\\ 0&0&1\end{array}\right),&H_{23}=\left(\begin{array}[]{rrr}1&0&1\\\
0&1&0\\\ 0&0&1\end{array}\right),\end{array}$
whose column vectors define partitions by triangles of the simplex such as
illustrated at Figure 1 (left).
Figure 1: Left: the partition provided by three Arnoux-Rauzy matrices, the six
Poincaré matrices and the six half triangles. Right: the partition of Arnoux-
Rauzy-Poincaré algorithm.
Then, the column vectors of $A_{1}$, $A_{2}$, $A_{3}$, $P_{31}H_{31}$,
$P_{13}H_{13}$, $P_{23}H_{23}$, $P_{32}H_{32}$, $P_{12}H_{12}$ and
$P_{21}H_{21}$ describe a partition of $\Delta$ depicted in Figure 1 (right).
Partitions are considered here up to a set of zero measure. This partition
allows one to associate with almost every point of $\Delta$ a matrix as
follows:
$\begin{array}[]{rcl}M:\Delta&\to&GL(3,\mathbb{Z})\\\
\mathbf{x}&\mapsto&\begin{cases}A_{k}&\text{ if }\mathbf{x}\in A_{k}\Delta,\\\
P_{jk}&\text{ else if }\mathbf{x}\in
P_{jk}H_{jk}\Delta.\end{cases}\end{array}$
We say that $\mathbf{x}=(x_{1},x_{2},x_{3})\in\Delta$ is _totally irrational_
if $x_{1}$, $x_{2}$, $x_{3}$ are linearly independent over $\mathbb{Q}$. When
$\mathbf{x}$ is not a totally irrational vector, there might be more than one
choice for the matrix $M(\mathbf{x})$ in the previous definition.
Nevertheless, the matrix $M(\mathbf{x})$ is uniquely defined for a totally
irrational vector.
Then, the Arnoux-Rauzy-Poincaré algorithm is defined (by renormalizing with
respect to the simplex $\Delta$) the linear map $M$:
$\begin{array}[]{rcl}T:\Delta&\to&\Delta\\\
\mathbf{x}&\mapsto&\displaystyle\frac{M(\mathbf{x})^{-1}\cdot\mathbf{x}}{\left\|M(\mathbf{x})^{-1}\cdot\mathbf{x}\right\|_{1}}\,\cdot\end{array}$
Each totally irrational vector $\mathbf{x}\in\Delta$ defines an orbit under
the map $T$ and a sequence of matrices
$(M_{n}(\mathbf{x}))_{n\in{\mathbb{N}}}$:
$M_{0}(\mathbf{x})=\mathrm{Id},\quad M_{n}(\mathbf{x})=M(T^{n-1}(\mathbf{x}))\
\mbox{ for all }n.$
###### Example 3.
Consider $\mathbf{x}=(1,\pi,\sqrt{2})$. The first $5$ points of the orbit of
$\mathbf{x}$ under the map $T$ are
$\mathbf{x}\in A_{2}\Delta,\quad T(\mathbf{x})\in P_{13}H_{13}\Delta,\quad
T^{2}(\mathbf{x})\in A_{2}\Delta,\quad T^{3}(\mathbf{x})\in A_{3}\Delta,\quad
T^{4}(\mathbf{x})\in A_{1}\Delta,\cdots$
One has $M_{0}(\mathbf{x})=\mathrm{Id}$, $M_{1}(\mathbf{x})=A_{2}$,
$M_{2}(\mathbf{x})=P_{13}$, $M_{3}(\mathbf{x})=A_{2}$,
$M_{4}(\mathbf{x})=A_{3}$ and $M_{5}(\mathbf{x})=A_{1}$.
### 2.2 Arnoux-Rauzy-Poincaré $S$-adic words
We now associate with the Arnoux-Rauzy-Poincaré algorithm a finite set
$\mathcal{S}$ of substitutions as well as $\mathcal{S}$-adic words.
We first start with some terminology. We consider a finite set of letters
${\cal A}$, called alphabet. Here ${\mathcal{A}}=\\{1,2,3\\}$. A (finite) word
is an element of the free monoid ${\cal A}^{*}$ generated by ${\cal A}$. The
unique word of length $0$ is the _empty word_ and we let it be denoted as
$\varepsilon$. We let the set of all (finite) words over $\mathcal{A}$ be
denoted by $\mathcal{A}^{*}$. With the concatenation of words as product
operation, ${\mathcal{A}}^{*}$ is the free monoid with $\varepsilon$ as
identity element. A substitution on the alphabet ${\mathcal{A}}$ is a non-
erasing morphism of the free monoid wich replaces letters by words. Let
$\sigma$ be a substitution. Its incidence matrix (also called abelianized
matrix) $M_{\sigma}=\left(m_{i,j}\right)_{1\leq i,j\leq d}$ is defined as the
square matrix whose entry of index $(i,j)$ is equal to the number of
occurrences of the letter $i$ in $\sigma(j)$. If a word $u$ can be factorized
as $pvs$, with $p,v,s\in\mathcal{A}^{*}$, then we say that $p$ is a _prefix_ ,
$v$ is a _factor_ and $s$ is a _suffix_ of $u$. The factor $v$ is said
_proper_ if $p$ and $s$ are non-empty. This notion extends to any infinite
word $\mathbf{u}$. The set ${\mathcal{A}}^{\mathbb{N}}$ is equipped with the
product topology of the discrete topology on each copy of ${\mathcal{A}}$;
this topology is induced by the following distance: for two distinct infinite
words ${\bf u}$ and ${\bf v}$ in ${\mathcal{A}}^{\mathbb{N}}$,
$\operatorname{d}({\bf u},{\bf v})=2^{-\min\\{n\in{\mathbb{N}}\ \mid\
u_{n}\neq v_{n}\\}}$.
The infinite word $\mathbf{u}\in{\mathcal{A}}^{\mathbb{N}}$ is said to admit
an $S$-adic representation if there exist a finite set $S$ of substitutions
defined on the alphabet $\mathcal{A}$, a sequence
$s=(\sigma_{n})_{n\in\mathbb{N}}\in S^{\mathbb{N}}$ of substitutions that all
belong to $S$, and $(a_{n})_{n\in\mathbb{N}}$ a sequence of letters in
${\mathcal{A}}$ such that
$\mathbf{u}=\lim_{n\to\infty}\sigma_{0}\sigma_{1}\cdots\sigma_{n}(a_{n}^{\infty}),$
with the notation $a_{n}^{\infty}$ standing for the infinite constant word
taking the value $a_{n}$. The word $\mathbf{u}$ is said to be $S$-adic, and
the sequence $s$ is called the _directive sequence_. We will use the following
notation: for all $m\in{\mathbb{N}}$
$\mathbf{u}^{(m)}=\lim_{n\to\infty}\sigma_{m}\sigma_{m+1}\cdots\sigma_{n}(a_{n}^{\infty}).$
An $S$-adic expansion with directive sequence
$(\sigma_{n})_{n\in{\mathbb{N}}}$ is said _weakly primitive_ if, for each $n$,
there exists $r$ such that the substitution $\sigma_{n}\cdots\sigma_{{n+r}}$
is positive, that is, its incidence matrix has only positive entries. If an
infinite word $\mathbf{u}$ admits a weakly primitive $S$-adic representation,
then it is _uniformly recurrent_ , that is, all its factors occur infinitely
often and with bounded gaps [Dur03]. An infinite word $\mathbf{u}$ is said
_recurrent_ if all its factors occur infinitely often in $\mathbf{u}$. For
more on $S$-adic words, see [BD14, CN10, DLR13, Ler12].
We now associate substitutions with the matrices defining the Arnoux-Rauzy-
Poincaré algorithm. Let $i,j,k$ be such that $\\{i,j,k\\}=\\{1,2,3\\}$. A
Poincaré substitution is a substitution of the form $\pi_{jk}:i\mapsto
ijk,j\mapsto jk,k\mapsto k$. An Arnoux-Rauzy substitution is given by
$\alpha_{k}:i\mapsto ik,j\mapsto jk,k\mapsto k$. For each
$\\{i,j,k\\}=\\{1,2,3\\}$, $P_{jk}$ is the incidence matrix of the
substitution $\pi_{jk}$ and $A_{k}$ is the incidence matrix of $\alpha_{k}$.
There are thus 6 Poincaré and 3 distinct Arnoux-Rauzy substitutions:
$\begin{array}[]{lll}\pi_{23}=\left\\{\begin{array}[]{l}1\mapsto 123\\\
2\mapsto 23\\\ 3\mapsto 3\\\
\end{array}\right.,&\pi_{13}=\left\\{\begin{array}[]{l}1\mapsto 13\\\ 2\mapsto
213\\\ 3\mapsto 3\\\
\end{array}\right.,&\alpha_{3}=\left\\{\begin{array}[]{l}1\mapsto 13\\\
2\mapsto 23\\\ 3\mapsto 3\\\ \end{array}\right.,\\\
\pi_{12}=\left\\{\begin{array}[]{l}1\mapsto 12\\\ 2\mapsto 2\\\ 3\mapsto
312\\\ \end{array}\right.,&\pi_{32}=\left\\{\begin{array}[]{l}1\mapsto 132\\\
2\mapsto 2\\\ 3\mapsto 32\\\
\end{array}\right.,&\alpha_{2}=\left\\{\begin{array}[]{l}1\mapsto 12\\\
2\mapsto 2\\\ 3\mapsto 32\\\ \end{array}\right.,\\\
\pi_{31}=\left\\{\begin{array}[]{l}1\mapsto 1\\\ 2\mapsto 231\\\ 3\mapsto
31\\\ \end{array}\right.,&\pi_{21}=\left\\{\begin{array}[]{l}1\mapsto 1\\\
2\mapsto 21\\\ 3\mapsto 321\\\
\end{array}\right.,&\alpha_{1}=\left\\{\begin{array}[]{l}1\mapsto 1\\\
2\mapsto 21\\\ 3\mapsto 31\\\ \end{array}\right..\end{array}$
Let
$\mathcal{S}:=\\{\alpha_{1},\alpha_{2},\alpha_{3},\pi_{12},\pi_{13},\pi_{21},\pi_{23},\pi_{31},\pi_{32}\\}.$
We also denote by $\mathcal{S}_{\alpha}$, $\mathcal{S}_{\pi}$, respectively,
the following sets of substitutions:
$\mathcal{S}_{\alpha}=\\{\alpha_{1},\alpha_{2},\alpha_{3}\\},\
\mathcal{S}_{\pi}=\\{\pi_{12},\pi_{13},\pi_{23},\pi_{21},\pi_{31},\pi_{32}\\},\mbox{
with }\mathcal{S}=\mathcal{S}_{\alpha}\cup\mathcal{S}_{\pi}.$
The substitutions in $\mathcal{S}$ are such that for any letter
$i\in\\{1,2,3\\}$, $\sigma(i)$ admits $i$ as a prefix. This yields the
convergence of any $\mathcal{S}$-adic representation in
${\mathcal{A}}^{\mathbb{N}}$ if the sequence of letters $(a_{n})_{n}$ is
constant. More precisely, for any sequence of substitutions $(\sigma_{n})_{n}$
with values in $\mathcal{S}$ and for every letter $a\in\\{1,2,3\\}$ then the
following limit exists
$\lim_{n\to\infty}\sigma_{0}\sigma_{1}\cdots\sigma_{n}(a^{\infty}).$
###### Definition 4 (Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word).
An _Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word_ is an infinite word of the
form
$\mathbf{u}=\lim_{n\to\infty}\sigma_{0}\sigma_{1}\cdots\sigma_{n}(a^{\infty}),$
where $a\in\mathcal{A}$ and $\sigma_{n}\in\mathcal{S}$ for all $n\geq 0$. Its
directive sequence is the sequence $s=(\sigma_{n})_{n}$.
### 2.3 The Arnoux-Rauzy-Poincaré $S$-adic system
The aim of this section is to associate with the Arnoux-Rauzy-Poincaré
algorithm an $\mathcal{S}$-adic symbolic dynamical system by taking into
account the restrictions provided by the algorithm which is not complete but
Markovian. We first recall the definition of an $S$-adic system. An $S$-adic
system is obtained by adding restrictions on the set of allowed directive
sequences: it is given by a finite directed strongly connected graph
$\mathcal{G}$ labeled by the substitutions, with each infinite path giving
rise to a directive sequence [BD14].
The partition of $\Delta$ allows to associate with almost any point of
$\Delta$ a substitution of $\mathcal{S}$:
$\begin{array}[]{rcl}\sigma:\Delta&\to&\mathcal{S}\\\
\mathbf{x}&\mapsto&\begin{cases}\alpha_{k}&\text{ if }\mathbf{x}\in
A_{k}\Delta,\\\ \pi_{jk}&\text{ else if }\mathbf{x}\in
P_{jk}H_{jk}\Delta,\end{cases}\end{array}$
and a directive sequence $s=(\sigma_{n})_{n}$ with
$\sigma_{n}=\sigma(T^{n}(\mathbf{x}))$ for all $n$. Observe that the
substitution $\sigma(\mathbf{x})$ has for incidence matrix $M(\mathbf{x})$
such as defined in Section 2.1.
###### Definition 5.
An _$\mathcal{S}$ -adic word $\mathbf{u}$ generated by the Arnoux-Rauzy-
Poincaré algorithm applied to the totally irrational vector
$\mathbf{x}\in\Delta$_ is an infinite word of the form
$\mathbf{u}=\lim_{n\to\infty}\left(\sigma(\mathbf{x})\cdot\sigma(T(\mathbf{x}))\cdot\sigma(T^{2}(\mathbf{x}))\cdot\ldots\cdot\sigma(T^{n-1}(\mathbf{x}))\right)(a^{\infty})$
where $a\in\\{1,2,3\\}$. Its directive sequence is the sequence
$s=(\sigma_{n})_{n}$ with $\sigma_{n}=\sigma(T^{n}(\mathbf{x}))$ for all $n$.
Let us show that the factors of the directive sequences produced by the
Arnoux-Rauzy-Poincaré algorithm belong to a rational language strictly
included in $\mathcal{S}^{*}$. We consider the automaton
$\mathcal{G}=(Q,\mathcal{S},\delta,I,F)$ defined by the states
$Q=\\{\Delta,H_{12},H_{13},\\\ H_{21},H_{23},H_{31},H_{32}\\},$
the alphabet $\mathcal{S}$, with the transitions $\delta\subset
Q\times\mathcal{S}\times Q$ being defined by
$\displaystyle\delta=\bigcup_{\\{i,j,k\\}=\\{1,2,3\\}}\\{(\Delta,\alpha_{k},\Delta),$
$\displaystyle(\Delta,\pi_{jk},H_{jk}),(H_{jk},\alpha_{j},H_{jk}),$
$\displaystyle(H_{jk},\alpha_{i},\Delta),(H_{jk},\pi_{ij},H_{ij}),(H_{jk},\pi_{ki},H_{ki}),(H_{jk},\pi_{ji},H_{ji})\\},$
and with initial state $I=\\{\Delta\\}$ and final state $F=Q$ (see Figure 2).
We consider the $\mathcal{S}$-adic system associated with the regular language
$\mathcal{L}(\mathcal{G})$.
Figure 2: The deterministic automaton $\mathcal{G}$. To avoid crossing arrows,
the initial state $\Delta$ is drawn at three places. The indices of $\pi$
transitions are not written since they are determined by the indices of the
arrival state: $\xrightarrow{\pi}H_{jk}$ means $\xrightarrow{\pi_{jk}}H_{jk}$.
This language corresponds to directive sequences for which the sequence of
incidence matrices is generated by the execution of the Arnoux-Rauzy-Poincaré
algorithm.
###### Proposition 6 (ARP regular language).
The set of directive sequences produced by the Arnoux-Rauzy-Poincaré algorithm
is included in the set of labeled infinite paths in the automaton
$\mathcal{G}$.
The proof of the proposition is provided in the appendix.
###### Remark 7.
We can even prove that the closure of the set of directive sequences produced
by the Arnoux-Rauzy-Poincaré algorithm is equal to the set $X_{\mathcal{G}}$
of labeled infinite paths starting in the automaton $\mathcal{G}$, as a
consequence of the convergence of the algorithm proved in Section 6. Let
$\Sigma\colon\Delta\rightarrow X_{\mathcal{G}}$ be the map that associates
with a (totally irrational vector) $\mathbf{x}$ the directive sequence
$(\sigma_{n})_{n}$ where $\sigma_{n}=\sigma(T^{n}(\mathbf{x}))$ for all $n$.
One has the following diagram and measure-theoretical isomorphism, where
$\Sigma$ is a.e. one-to-one and where the shift associates with the label of
an infinite path the label of the path deprived of its first edge:
$\begin{array}[]{ccc}\Delta&\stackrel{{\scriptstyle
T}}{{\longrightarrow}}&\Delta\\\
\Big{\downarrow}\scriptstyle{\Sigma}&&\Big{\downarrow}\scriptstyle{\Sigma}\\\
\
X_{\mathcal{G}}&\underset{\mathrm{shift}}{\longrightarrow}&X_{\mathcal{G}}\end{array}$
We now can define the Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic system from the
multidimensional continued fraction algorithm.
###### Definition 8 (Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic system).
The _Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic system_ is the set of
$\mathcal{S}$-adic words
$\mathbf{u}=\lim_{n\to\infty}\sigma_{0}\sigma_{1}\cdots\sigma_{n}(a^{\infty}),$
whose directive sequence $(\sigma_{n})_{n}$ is an infinite path in
$\mathcal{G}$. We distinguish three types of directive sequences together with
some restrictions on the chosen letter $a$:
1. 1.
if $(\sigma)_{n}\in\mathcal{S}^{*}\\{\alpha_{k}\\}^{\mathbb{N}}$, for
$k\in\\{1,2,3\\}$, then $a=k$ (Type 1);
2. 2.
else if
$(\sigma)_{n}\in\mathcal{S}^{*}\\{\alpha_{k},\alpha_{j}\\}^{\mathbb{N}}$, then
$a\in\\{j,k\\}$, for some $\\{i,j,k\\}=\\{1,2,3\\}$ (Type 2);
3. 3.
otherwise, take any $a\in\\{1,2,3\\}$ (Type 3).
The requirements in this definition concerning the choice of the letter $a$
will be clearer with Proposition 13 below: they aim at working with recurrent
words which will be used in the computation of the factor complexity function.
According to Proposition 6, any $\mathcal{S}$-adic word $\mathbf{u}$ generated
by the Arnoux-Rauzy-Poincaré algorithm applied to a totally irrational vector
$\mathbf{x}\in\Delta$, according to Definition 5, belongs to the Arnoux-Rauzy-
Poincaré $\mathcal{S}$-adic system. Furthermore, they correspond to Type 3 in
Definition 8.
###### Remark 9.
We stress the following terminology: by Arnoux-Rauzy-Poincaré
$\mathcal{S}$-adic word, we mean an $\mathcal{S}$-adic word with no other
restriction on the directive sequence that the fact that it belongs to
$\mathcal{S}^{\mathbb{N}}$ (see Definition 4), whereas for a word in the
Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic system, the restrictions of
Proposition 6 are taken into account.
###### Example 10.
We continue Example 3. The word generated by the Arnoux-Rauzy-Poincaré
algorithm applied to $\mathbf{x}=(1,\pi,\sqrt{2})$ is:
$\displaystyle\mathbf{u}$
$\displaystyle=\alpha_{2}\pi_{13}\alpha_{2}\alpha_{3}\alpha_{1}\pi_{31}\pi_{23}\pi_{31}\pi_{12}(\alpha_{3})^{8}\alpha_{1}(\alpha_{2})^{6}\pi_{21}\alpha_{3}\alpha_{3}\alpha_{1}\pi_{32}\cdots(1)$
$\displaystyle=1232212323221232212323221232123221232322123232212321232212323\cdots$
Note that the substitutions shown on the above line determine the prefix of
$\mathbf{u}$ of length $1453060$. The first prefixes are
$\alpha_{2}(1)=12,\quad\alpha_{2}\pi_{13}(1)=1232,\quad\alpha_{2}\pi_{13}\alpha_{2}(1)=123221232,\quad\alpha_{2}\pi_{13}\alpha_{2}\alpha_{3}(1)=1232212323221232.$
Observe that due to its $S$-adic construction, the infinite word $\mathbf{u}$
can be decomposed on three-block codes (that is, on codes consisting of three
finite words) in many ways:
$\displaystyle 12322|1232322|12322|1232322|1232|12322|1232322|1232322|\cdots$
$\displaystyle 12|32|2|12|32|32|2|12|32|2|12|32|32|2|12|32|12|\cdots$
$\displaystyle 123|22123|23|22123|22123|23|22123|2123|22123|23|22123|\cdots$
The blocks are in each case respectively $\\{12322,1232322,1232\\}$,
$\\{12,2,32\\}$ and $\\{23,22123,2123\\}$ (they are obtained as
$\sigma_{1}\cdots\sigma_{n}(i)$, for $i=1,2,3$, or else, as return words on
the letter $1$ in $\mathbf{u}$, where a return word on $1$ is a finite word
$v$ that does not contain the letter $1$, but that is such that $v1$ is a
factor of $\mathbf{u}$). For comparison, the billiard word of direction
$(1,\pi,\sqrt{2})$ starting at $(0,0,0)$ is:
$2321232212322312232123221322231223212322321223212322132232123\cdots$
It has quadratic factor complexity. It cannot be decomposed on a three-factor
code (its has too much return words on each letter).
### 2.4 Totally irrational vectors and weak primitivity
The next lemma provides a characterization of weakly primitive
$\mathcal{S}$-adic expansions. Indeed weak primitivity fails if and only if
the directive sequence $(\sigma_{n})_{n\in{\mathbb{N}}}$ contains finitely
many Poincaré substitutions and takes ultimately at most two values (that thus
are Arnoux-Rauzy substitutions).
###### Lemma 11.
Let
$\mathbf{u}=\lim_{n\to\infty}\sigma_{0}\sigma_{1}\cdots\sigma_{n}(a_{n}^{\infty})$
be an $\mathcal{S}$-adic word generated by the Arnoux-Rauzy-Poincaré algorithm
applied to the vector $\mathbf{x}=(x_{1},x_{2},x_{3})\in\Delta$. Its
associated $\mathcal{S}$-adic expansion is weakly primitive if and only if
$(\sigma_{n})_{n\in{\mathbb{N}}}\not\in\mathcal{S}^{*}\cdot\left(\\{\alpha_{1},\alpha_{2}\\}^{\mathbb{N}}\cup\\{\alpha_{1},\alpha_{3}\\}^{\mathbb{N}}\cup\\{\alpha_{2},\alpha_{3}\\}^{\mathbb{N}}\cup\\{\alpha_{1}\\}^{\mathbb{N}}\cup\\{\alpha_{2}\\}^{\mathbb{N}}\cup\\{\alpha_{3}\\}^{\mathbb{N}}\right).$
###### Proof.
If
$(\sigma_{n})_{n\in{\mathbb{N}}}\in\mathcal{S}^{*}\cdot\left(\\{\alpha_{1},\alpha_{2}\\}^{\mathbb{N}}\cup\\{\alpha_{1},\alpha_{3}\\}^{\mathbb{N}}\cup\\{\alpha_{2},\alpha_{3}\\}^{\mathbb{N}}\cup\\{\alpha_{1}\\}^{\mathbb{N}}\cup\\{\alpha_{2}\\}^{\mathbb{N}}\cup\\{\alpha_{3}\\}^{\mathbb{N}}\right)$,
then it is easily seen that $(\sigma_{n})_{n\in{\mathbb{N}}}$ is not weakly
primitive.
Now, let $(\sigma_{n})_{n\in{\mathbb{N}}}$ be the directive sequence of an
$\mathcal{S}$-adic expansion which is not weakly primitive in the Arnoux-
Rauzy-Poincaré $\mathcal{S}$-adic system. Being not weakly primitive means
that there exists $m$ such that for all $p$ with $m\leq p$ the substitution
$\sigma_{m}\cdots\sigma_{{p}}$ is not positive, that is, one of the entries of
its incidence matrix is zero. Moreover, for all $p$ and $r$ such that $m\leq
p\leq r$ the incidence matrix of the substitution
$\sigma_{p}\cdots\sigma_{{r}}$ is not positive.
Note that since the incidence matrix of every substitution in $\mathcal{S}$
has entries $1$ on the diagonal, the positivity of entries is preserved by
left and right multiplication. Therefore, if
$\sigma_{1}\sigma_{2}\cdots\sigma_{n}\in\mathcal{S}^{*}$ is positive, then
$\varphi$ is positive for every
$\varphi\in\mathcal{S}^{*}\sigma_{1}\mathcal{S}^{*}\sigma_{2}\mathcal{S}^{*}\cdots\mathcal{S}^{*}\sigma_{n}\mathcal{S}^{*}$.
Assume first that $(\sigma_{n})_{n\geq m}$ contains no Poincaré substitution.
If $(\sigma_{n})_{n\geq m}$ contains three distinct Arnoux-Rauzy
substitutions, there are some values of $p$ and $r$ with $m\leq p\leq r$ such
that $\sigma_{p}\cdots\sigma_{{r}}$ contains three distinct Arnoux-Rauzy
substitutions. One verifies that $\alpha_{i}\alpha_{j}\alpha_{k}$ is positive
for all possible values of $i$, $j$, $k$ with $\\{i,j,k\\}=\\{1,2,3\\}$. Then
$\sigma_{p}\cdots\sigma_{{r}}$ is positive which is a contradiction.
Therefore, we conclude that
$(\sigma_{n})_{n\in{\mathbb{N}}}\in\mathcal{S}^{*}\cdot\left(\\{\alpha_{1},\alpha_{2}\\}^{\mathbb{N}}\cup\\{\alpha_{1},\alpha_{3}\\}^{\mathbb{N}}\cup\\{\alpha_{2},\alpha_{3}\\}^{\mathbb{N}}\right)$.
Assume $(\sigma_{n})_{n\geq m}$ contains at least one Poincaré substitution.
We may suppose that $(\sigma_{n})_{n\geq p}$ starts with a Poincaré
substitution $\sigma_{p}=\pi_{jk}$ for $p\geq m$. Since $(\sigma_{n})_{n\geq
p}\in\mathcal{L}(\mathcal{G})$, then
$(\sigma_{n})_{n\geq
p}\in\left(\pi_{jk}\alpha_{j}^{\infty}\right)\cup\left(\pi_{jk}\alpha_{j}^{t}\\{\alpha_{i},\pi_{ki},\pi_{ji}\\}\mathcal{S}^{\mathbb{N}}\right)\cup\left(\pi_{jk}\alpha_{j}^{t}\pi_{ij}\alpha_{i}^{\infty}\right)\cup\left(\pi_{jk}\alpha_{j}^{t}\pi_{ij}\alpha_{i}^{s}\\{\alpha_{k},\pi_{jk},\pi_{ik},\pi_{ki}\\}\mathcal{S}^{\mathbb{N}}\right),$
for some non-negative integers $s$ and $t$ and $\\{i,j,k\\}=\\{1,2,3\\}$. But
$\pi_{jk}\alpha_{i}$, $\pi_{jk}\pi_{ki}$ and $\pi_{jk}\pi_{ji}$ are positive.
Also $\pi_{jk}\pi_{ij}\alpha_{k}$, $\pi_{jk}\pi_{ij}\pi_{jk}$,
$\pi_{jk}\pi_{ij}\pi_{ik}$ and $\pi_{jk}\pi_{ij}\pi_{ki}$ are positive.
Therefore,
$(\sigma_{n})_{n\geq
p}\in\left(\pi_{jk}\alpha_{j}^{\infty}\right)\cup\left(\pi_{jk}\alpha_{j}^{t}\pi_{ij}\alpha_{i}^{\infty}\right)$
and we have shown that
$(\sigma_{n})_{n\in{\mathbb{N}}}\in\mathcal{S}^{*}\cdot\left(\\{\alpha_{1}\\}^{\mathbb{N}}\cup\\{\alpha_{2}\\}^{\mathbb{N}}\cup\\{\alpha_{3}\\}^{\mathbb{N}}\right)$.
∎
###### Proposition 12.
Let $\mathbf{u}$ be an $\mathcal{S}$-adic word generated by the Arnoux-Rauzy-
Poincaré algorithm applied to the totally irrational vector
$\mathbf{x}\in\Delta$. Then the associated $S$-adic expansion is weakly
primitive. In particular, $\mathbf{u}^{(m)}$ is of Type $3$, uniformly
recurrent and proper, for all $m$.
###### Proof.
The conclusion follows from Lemma 11 by noticing that if
$(\sigma_{n})_{n\in{\mathbb{N}}}\in\mathcal{S}^{*}\cdot\left(\\{\alpha_{1},\alpha_{2}\\}^{\mathbb{N}}\cup\\{\alpha_{1},\alpha_{3}\\}^{\mathbb{N}}\cup\\{\alpha_{2},\alpha_{3}\\}^{\mathbb{N}}\cup\\{\alpha_{1}\\}^{\mathbb{N}}\cup\\{\alpha_{2}\\}^{\mathbb{N}}\cup\\{\alpha_{3}\\}^{\mathbb{N}}\right)$
then $\mathbf{x}$ cannot be totally irrational. ∎
Observe that not every word of the Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic
system is uniformly recurrent. Nevertheless, one easily checks that words of
this system are all recurrent.
###### Proposition 13.
Any infinite word ${\bf u}$ in the Arnoux-Rauzy-Poincaré system is recurrent
as well as $\mathbf{u}^{(m)}$ for any $m$.
###### Example 14.
The infinite word $\alpha_{1}^{\infty}(2^{\infty})=21^{\infty}$ is not
recurrent whereas $\alpha_{1}^{\infty}(1^{\infty})=1^{\infty}$ is recurrent.
The restriction of the infinite words under study to the case where each
letter always appears as proper factor will also be useful to prove the main
result of this article.
###### Definition 15 (Proper word ).
A word $\mathbf{u}\in\\{1,2,3\\}^{\mathbb{N}}$ is said _proper_ if each letter
$i\in\\{1,2,3\\}$ is a proper factor of $\mathbf{u}$, or equivalently, for
each letter $i\in\\{1,2,3\\}$, there exists a letter $e$ such that $ei$ is a
factor of $\mathbf{u}$.
## 3 Factor complexity
In this section, we define the terminology relative to languages, bispecial
factors, extension types and factor complexity. We adopt the notation of
[CN10].
### 3.1 Language and complexity function $p(n)$
Let $\mathcal{A}=\\{1,2,\ldots,d\\}$ be an alphabet. The length of a word
$u\in\mathcal{A}^{n}$ is denoted by $|u|$ and is equal to $n$, whereas the
notation $|u|_{i}$ stands for the number of occurrences of the letter $i$ in
$u$. A _language_ is a subset of the free monoid $\mathcal{A}^{*}$. A language
$L$ is _factorial_ if for any $w\in L$, then any factor $u$ of $w$ belongs to
$L$. The abelianized of a finite word $w\in\mathcal{A}^{*}$ is the vector
$\overrightarrow{w}=(|w|_{1},|w|_{2},\ldots,|w|_{d})\in\mathbb{N}^{d}.$
We consider an infinite word
$\mathbf{u}=u_{0}u_{1}u_{2}u_{3}\cdots\in\mathcal{A}^{\mathbb{N}}$. For each
$n\in\mathbb{N}$, $\mathcal{L}_{n}(\mathbf{u})$ is the set of factors of
length $n$ in $\mathbf{u}$, while $\mathcal{L}(\mathbf{u})$ is the set of all
factors in $\mathbf{u}$, and is called the _language of $\mathbf{u}$_. The
language of $\mathbf{u}$ is factorial. For each $n\in\mathbb{N}$, let
$p_{\mathbf{u}}(n)$ be the cardinality of $\mathcal{L}_{n}(\mathbf{u})$. Then
$p_{\mathbf{u}}:\mathbb{N}\to\mathbb{N}$ is a function called the _factor
complexity function_ of $\mathbf{u}$. When no confusion is possible, we omit
$\mathbf{u}$ and just write $p$.
### 3.2 Bispecial Factors and Extension Types
Let $w$ be a factor of either a recurrent infinite word or of a finite word
$\mathbf{u}$. We let $E^{+}(w)=\\{x\in\mathcal{A}\mid
wx\in\mathcal{L}(\mathbf{u})\\}$ denote the set of right extensions of $w$ in
$\mathbf{u}$. The _right valence_ $d^{+}(w)=\mathrm{Card}\,E^{+}(w)$ of $w$
(in $\mathbf{u}$) is defined as the number of distinct right extensions of
$w$. _Left extensions_ $E^{-}(w)$ and _left valence_ $d^{-}(w)$ are defined in
a a similar way. A factor whose right valence is at least $2$ is called _right
special_. A factor whose left valence is at least $2$ is called _left
special_. A factor which is both left and right special is called _bispecial_.
The _extension type_ $E_{\mathbf{u}}(w)$ of a factor $w$ of $\mathbf{u}$ is
the set of pairs $(a,b)$ of $\mathcal{A}\times\mathcal{A}$ such that $w$ can
be extended in both directions as $awb$:
$E_{\mathbf{u}}(w)=\\{(a,b)\in\mathcal{A}\times\mathcal{A}\mid
awb\in\mathcal{L}(\mathbf{u})\\}.$
We also use the notation $E_{\mathbf{u}}(w)$ by $E(w)$ when the context is
clear. The _bilateral multiplicity_ of a factor $w$ is the number
$m(w)=\mathrm{Card}\,E(w)-d^{-}(w)-d^{+}(w)+1.$
We have the following fact (see e.g. [CN10, Proposition 4.5.1]) which links
bilateral multiplicity to the notion of bispecial factor: let $w$ be a factor
of a recurrent infinite word such that $m(w)\neq 0$; then, $w$ is bispecial. A
bispecial factor is said _strong_ if $m(w)>0$, _weak_ if $m(w)<0$ and
_neutral_ if $m(w)=0$. A bispecial factor is _ordinary_ if there exist letters
$a,b\in\mathcal{A}$ such that
$\\{(a,b)\\}\subseteq
E(w)\subseteq\left(\\{a\\}\times\mathcal{A}\right)\cup\left(\mathcal{A}\times\\{b\\}\right).$
(1)
An ordinary bispecial factor is neutral, but the converse is not true for
$|\mathcal{A}|>2$. We will use this notion in particular in Section 4.4.
###### Lemma 16.
If a bispecial factor is ordinary, then it is neutral.
###### Proof.
If $w$ is ordinary, then
$\mathrm{Card}\,E(w)=\mathrm{Card}\,E^{-}(w)+\mathrm{Card}\,E^{+}(w)-1$
because $(a,b)\in E(w)$. Thus, we have
$m(w)=\mathrm{Card}\,E(w)-d^{-}(w)-d^{+}(w)+1=0$. ∎
It is convenient to represent the extension type $E(w)$ of a bispecial factor
$w$ in a graphical way. It is often represented as a bipartite graph, but we
choose here a table representation: a cross ($\times$) is drawn at the
intersection of row $a$ and column $b$ if and only if $(a,b)\in E(w)$ (see
Figure 3).
| 1 | 2 | 3
---|---|---|---
1 | | $\times$ |
2 | | $\times$ |
3 | $\times$ | $\times$ | $\times$
$m(w)=0$
neutral and ordinary
| 1 | 2 | 3
---|---|---|---
1 | | $\times$ |
2 | | | $\times$
3 | $\times$ | $\times$ | $\times$
$m(w)=0$
neutral but not ordinary
| 1 | 2 | 3
---|---|---|---
1 | | $\times$ |
2 | | |
3 | | | $\times$
$m(w)=-1$
weak
| 1 | 2 | 3
---|---|---|---
1 | | |
2 | | $\times$ | $\times$
3 | $\times$ | $\times$ | $\times$
$m(w)=1$
strong
Figure 3: Examples of tables representing the extension type $E(w)$ of a
bispecial factor $w$.
###### Definition 17 (Left equivalence).
Let $w$ and $w^{\prime}$ be two bispecial factors defined on the alphabet
$\mathcal{A}$. We say that their extension types are _left equivalent_ if
there exists a permutation $\tau$ acting on $\mathcal{A}$ such that
$E(w^{\prime})=\\{(\tau(a),b)\mid(a,b)\in E(w)\\}$.
Right equivalence is defined similarly. Left equivalence can be interpreted on
the table representation of the extension type as follows. Indeed one
representation can be obtained from the other by a permutation of the rows:
$\scriptsize E(w)=\begin{array}[]{c|ccc}&1&2&3\\\ \hline\cr 1&&&\times\\\
2&&&\\\ 3&\times&\times&\times\end{array}\quad\quad\quad\text{\normalsize
and}\quad\quad\quad E(w^{\prime})=\begin{array}[]{c|ccc}&1&2&3\\\ \hline\cr
1&\times&\times&\times\\\ 2&&&\times\\\ 3&&&\end{array}$
Substitutions considered in this article preserve the first letter and thus
preserve the right extensions. Then, the notion of left equivalence is
sufficient for our need. But in general, we have the following definition. Of
course if the extension type of $w$ and $w^{\prime}$ are left or right
equivalent, then they are also equivalent. When the extension type of two
words are equivalent, they share common properties. In particular, being
ordinary, strong or weak is preserved under equivalence.
###### Lemma 18.
Let $w$ and $w^{\prime}$ be two bispecial factors such that the extension type
of $w$ and $w^{\prime}$ are equivalent, then
1. •
$w$ is ordinary (neutral, strong, weak resp.) if and only if $w^{\prime}$ is
ordinary (neutral, strong, weak resp.),
2. •
$\mathrm{Card}E(w)=\mathrm{Card}E(w^{\prime})$, $d^{-}(w)=d^{-}(w^{\prime})$,
$d^{+}(w)=d^{+}(w^{\prime})$, $m(w)=m(w^{\prime})$,
3. •
if the extension type of $w$ and $w^{\prime}$ are left equivalent, then
$E^{+}(w)=E^{+}(w^{\prime})$,
4. •
if the extension type of $w$ and $w^{\prime}$ are right equivalent, then
$E^{-}(w)=E^{-}(w^{\prime})$.
### 3.3 Factor Complexity
Let $p(n)$ be the factor complexity function of the infinite word
$\mathbf{u}$. Two other functions derived from the factor complexity are
useful, namely the sequences of _finite differences of order $1$ and $2$_
respectively of $p(n)$:
$\displaystyle s(n)$ $\displaystyle=$ $\displaystyle p(n+1)-p(n),$ (2)
$\displaystyle b(n)$ $\displaystyle=$ $\displaystyle s(n+1)-s(n).$ (3)
Of course, we have
$\displaystyle p(n)$ $\displaystyle=$ $\displaystyle
p(0)+\sum_{\ell=0}^{n-1}s(\ell),$ (4) $\displaystyle s(n)$ $\displaystyle=$
$\displaystyle s(0)+\sum_{\ell=0}^{n-1}b(\ell).$ (5)
These equations are very useful to compute the complexity function $p(n)$ when
its growth is slow (for example in the case of a linear growth), since in this
case functions $s$ and $b$ take small values. For example, we have $p(n)=n+1$
for all $n$ ($\mathbf{u}$ is thus a Sturmian word) if and only if $s(n)$ is
always equal to $1$, which is also equivalent to the fact that exactly two
letters occur ($p(1)=2$, $s(0)=1$) and that $b(n)$ always takes the value $0$.
In this article, one of our main results is to show that some infinite words
on a three-letter alphabet have complexity $p(n)<3n$. In order to achieve
this, we use the next lemma.
###### Lemma 19.
Suppose $|\mathcal{A}|=3$. Then, $p(n+1)-p(n)\in\\{2,3\\}$ if and only if
$\sum_{\ell=0}^{n-1}b(\ell)\in\\{0,1\\}$. Furthermore, if the sequence of
finite differences of order $2$ is such that
$(b(\ell))_{\ell}=0,\ldots,0,1,0,\ldots,0,-1,0,\ldots,0,1,0,\ldots,0,-1,\ldots$
then $\sum_{\ell=0}^{n-1}b(\ell)\in\\{0,1\\}$.
###### Proof.
Since $|\mathcal{A}|=3$, then $p(1)=3$ and $s(0)=p(1)-p(0)=3-1=2$. We have
$p(n+1)-p(n)=s(n)=s(0)+\sum_{\ell=0}^{n-1}b(\ell)=2+\sum_{\ell=0}^{n-1}b(\ell),$
which yields the proof of the first statement.
The proof of the second one comes from the fact that the first non-zero term
of the sequence $(b(\ell))_{\ell}$ is $+1$. ∎
The finite differences of order $1$ and $2$ of $p(n)$ are related to special
and bispecial factors as explained in [Cas97a]. We state a weaker form (for
recurrent words) of a result of [CN10]. Indeed, as we are interested in the
factor complexity of some recurrent words, we do not need to consider
unioccurrent or exceptional prefixes.
###### Theorem 20.
[CN10, Theorem 4.5.4] Let $\mathbf{u}\in\mathcal{A}^{\mathbb{N}}$ be an
infinite recurrent word. Then, for all $n\in\mathbb{N}$:
$\displaystyle s(n)$ $\displaystyle=$
$\displaystyle\sum_{w\in{\mathcal{L}}_{n}(u)}(d^{+}(w)-1)=\sum_{w\in{\mathcal{L}}_{n}(u)}(d^{-}(w)-1)$
(6) $\displaystyle b(n)$ $\displaystyle=$
$\displaystyle\sum_{w\in{\mathcal{L}}_{n}(u)}m(w).$ (7)
## 4 Bispecial Factors under Arnoux-Rauzy and Poincaré Substitutions
The goal of the next sections is to describe factors of Arnoux-Rauzy-Poincaré
$\mathcal{S}$-adic words. The key ingredient is a synchronization lemma that
allows the desubstitution with respect to the substitutions in $\mathcal{S}$
(Section 4.1). As a consequence for bispecial factors, antecedents (they are
uniquely defined and always bispecial) and bispecial images together with
their possible extensions are described in details in Section 4.2 and 4.3,
respectively. We then can consider the notion of life of a bispecial factor
produced by an $\mathcal{S}$-adic expansion (Section 4.4). So far we still do
not use the restrictions of Proposition 6 on the possible directive sequences
in $\mathcal{S}^{\mathbb{N}}$ (they will be considered only in Section 5).
Section 4.5 illustrates the fact that a quadratic factor complexity can be
reached without these restrictions. We end this section with the introduction
of notions of order on vectors allowing the comparison of abelianized vectors
under the application of substitutions in $\mathcal{S}$ (Section 4.6).
We recall that a Poincaré substitution is of the form $\pi_{jk}:i\mapsto
ijk,j\mapsto jk,k\mapsto k$. An Arnoux-Rauzy substitution is given by
$\alpha_{k}:i\mapsto ik,j\mapsto jk,k\mapsto k$.
### 4.1 Synchronization lemma
From now on, the alphabet is set to $\mathcal{A}=\\{1,2,3\\}$. The next lemma
describes the preimage of a factor under Arnoux-Rauzy (AR) and Poincaré (P)
substitutions. Such statements are classical tools when computing the factor
complexity of fixed points of substitutions.
###### Lemma 21 (Synchronization).
Let $u\in\mathcal{A}^{*}$ and $w$ be a factor of $\alpha_{k}(u)$ for some
$\\{i,j,k\\}=\\{1,2,3\\}$.
1. (i)
If $w$ is empty or if the first letter of $w$ is $i$ or $j$, then there exist
a unique $v\in\mathcal{A}^{*}$ and a unique $s\in\\{\varepsilon,i,j\\}$ such
that $w=\alpha_{k}(v)\cdot s$.
2. (ii)
If the first letter of $w$ is $k$, then there exist a unique
$v\in\mathcal{A}^{*}$ and a unique $s\in\\{\varepsilon,i,j\\}$ such that
$w=k\cdot\alpha_{k}(v)\cdot s.$
Let $u\in\mathcal{A}^{*}$ and $w$ be a factor of $\pi_{jk}(u)$ for some
$\\{i,j,k\\}=\\{1,2,3\\}$.
1. (iii)
If $w$ is empty or if the first letter of $w$ is $i$, then there exist a
unique $v\in\mathcal{A}^{*}$ and a unique $s\in\\{\varepsilon,i,j,ij\\}$ such
that $w=\pi_{jk}(v)\cdot s$.
2. (iv)
If $w=j$, then there exist a unique $v(=\varepsilon)$ such that
$w=j\cdot\pi_{jk}(v)$.
3. (v)
If the first letter of $w$ is $j$ and $|w|>1$, then there exist a unique
$v\in\mathcal{A}^{*}$ and a unique $s\in\\{\varepsilon,i,j,ij\\}$ such that
$w=jk\cdot\pi_{jk}(v)\cdot s$.
4. (vi)
If the first letter of $w$ is $k$, then there exist a unique
$v\in\mathcal{A}^{*}$ and a unique $s\in\\{\varepsilon,i,j,ij\\}$ such that
$w=k\cdot\pi_{jk}(v)\cdot s$.
###### Proof.
The sets $\\{ik,jk,k\\}$ and $\\{ijk,jk,k\\}$ form a prefix code. ∎
###### Definition 22 (Antecedent, extended image).
Let $\sigma=\alpha_{k}$ or $\sigma=\pi_{jk}$, $u\in\mathcal{A}^{*}$ and $w$ be
a factor of $\sigma(u)$. We say that the _antecedent of $w$ under $\sigma$_ is
the unique word $v$ as defined by Lemma 21. If $v$ is the antecedent of a word
$w$, then we say that the word $w$ is an _extended image_ of $v$.
Note that the antecedent is unique, but that a word $v$ may have more than one
extended image. Consider for instance $w_{1}=23\pi_{23}(11)1=231231231$ and
$w_{2}=3\pi_{23}(11)2=31231232$ which are two distinct extended images of
$v=11$. This is why the situation becomes here quite intricate especiallly for
bispecial factors. In fact, it happens that strong and weak bispecial words
appear in pairs: the image of a neutral bispecial factor $v$ can have two
extended images that are bipsecial, with one of them being strong, and the
other one being weak. For more details, see Lemma 36 and Remark 37 below.
We now consider images and antecedents of bispecial factors.
###### Definition 23 (Bispecial extended image).
Let $u\in\mathcal{A}^{*}\cup\mathcal{A}^{\mathbb{N}}$ and $v$ be a factor of
$u$. We shall say that a _bispecial extended image_ $w$ of $v$ under $\sigma$
is a bispecial word of $\sigma(u)$ which is an extended image of $v$ under
$\sigma$.
For example, let $v$ be a bispecial factor and suppose
$E(v)=\\{(1,2),(2,3),(3,1),(3,2),(3,3)\\}$. Then $w=3\pi_{23}(v)$ and
$w^{\prime}=23\pi_{23}(v)$ are both bispecial extended images of $v$ under
$\pi_{23}$. Indeed, we have
$\pi_{23}\left(\\{1v2,2v3,3v1,3v2,3v3\\}\right)=\\{123\pi_{23}(v)23,23\pi_{23}(v)3,3\pi_{23}(v)123,3\pi_{23}(v)23,3\pi_{23}(v)3\\}$
and the extension types are $E(w)=\\{(2,2),(2,3),(3,1),(3,2),(3,3)\\}$ and
$E(w^{\prime})=\\{(1,2),(3,3)\\}$.
The next lemma allows one to relate every bispecial factor to a shorter one
and eventually to the empty word.
###### Lemma 24 (Bispecial extended image growth).
Let $\sigma=\alpha_{k}$ or $\sigma=\pi_{jk}$ and $w\neq\varepsilon$ be a non-
empty bispecial extended image of $v$ under $\sigma$. Then, $|v|<|w|$.
###### Proof.
Suppose that $\sigma=\alpha_{k}$ for some $k\in\\{1,2,3\\}$. Since $w$ is non-
empty, $w$ starts and ends with letter $k$ and from Lemma 21 (ii), the unique
antecedent $v$ of $w$ is such that $w=k\alpha_{k}(v)$. We conclude that
$|v|<|w|$.
Suppose that $\sigma=\pi_{jk}$ for some $\\{i,j,k\\}=\\{1,2,3\\}$. Since $w$
is non-empty, $w$ starts with letter $j$ or $k$ and ends with letter $k$. From
Lemma 21 (iv) and (v), the unique antecedent $v$ of $w$ is such that
$w=k\pi_{jk}(v)$ or $w=jk\pi_{jk}(v)$. In both cases, $|v|<|w|$. ∎
### 4.2 Arnoux-Rauzy substitutions
The case of Arnoux-Rauzy substitutions is particularly convenient to handle,
both for bispecial extended images or for antecedents of bispecial factors.
###### Lemma 25 (AR - Bispecial extended image).
Let $u\in\mathcal{A}^{*}$ and let $v$ be a bispecial factor of $u$. There is a
unique bispecial extended image $w=k\alpha_{k}(v)$ of $v$ in $\alpha_{k}(u)$.
###### Proof.
Let $w$ and $w^{\prime}$ be two extended images of $v$ under $\alpha_{k}$.
Since they are bispecial factors, one deduces from Lemma 21 that both $w$ and
$w^{\prime}$ start and end with letter $k$. Hence
$w=k\alpha_{k}(v)=w^{\prime}$. ∎
###### Lemma 26 (AR - Antecedent of a bispecial).
Let $u\in\\{1,2,3\\}^{*}$ and $w\neq\varepsilon$ be a bispecial factor of
$\alpha_{k}(u)$. Let $v$ be the unique antecedent of $w$ under $\alpha_{k}$.
One has $w=k\alpha_{k}(v)$. Furthermore, $v$ is bispecial and it has the same
extension type $E_{\alpha_{k}(u)}(w)=E_{u}(v)$ and same multiplicity
$m(w)=m(v)$ as $w$.
Figure 4: The preimage of the bispecial word $w$ under $\alpha_{k}$.
###### Proof.
One checks that $(a,b)\in E(v)$ if and only if $(a,b)\in E(k\alpha_{k}(v))$
(see Figure 4). Then $E(k\alpha_{k}(v))=E(v)$. We deduce that
$E^{+}(k\alpha_{k}(v))=E^{+}(v)$ and $E^{-}(k\alpha_{k}(v))=E^{-}(v)$. From
this we conclude that $m(k\alpha_{k}(v))=m(v)$. ∎
### 4.3 Poincaré substitutions
The case of Poincaré substitutions is more delicate to handle as already
illustrated by the following result. We loose here unicity for the bispecial
extended images.
###### Lemma 27 (P - Bispecial extended images).
Let $i,j,k$ such that $\\{i,j,k\\}=\\{1,2,3\\}$. Let $u\in\\{1,2,3\\}^{*}$ and
let $v$ be a bispecial factor of $u$. There are at most two distinct bispecial
extended images of $v$ under $\pi_{jk}$. They are either $k\pi_{jk}(v)$ or
$jk\pi_{jk}(v)$.
###### Proof.
Let $w$ be a bispecial extended image of $v$ under $\pi_{jk}$. Since $w$ is a
bispecial factor, it must start with letter $j$ or $k$ and end with letter
$k$. From Lemma 21, one gets $w\in\\{jk\pi_{jk}(v),k\pi_{jk}(v)\\}$. ∎
The “at most two" of Lemma 27 will be made more precise later in Lemma 30
where conditions will be given for when a bispecial factor has one or two
bispecial extended images under a Poincaré substitution.
In order to get a similar result concerning the antecedent of a bispecial
factor under Poincaré substitutions (see Lemma 29 below), we first need the
following result stated for factors in general which is also used for proving
Lemma 30 and 36.
###### Lemma 28 (P - Extensions).
Let $i,j,k$ such that $\\{i,j,k\\}=\\{1,2,3\\}$. Let $u\in\\{1,2,3\\}^{*}$ and
$v$ be a factor of $u$. We assume that for all $(a,b)\in E(v)$, there exists a
letter $e$ such that $eavb$ is also a factor of $u$. The extensions of $v$ in
$u$ are related to the extensions of $k\pi_{jk}(v)$ and $jk\pi_{jk}(v)$
considered as factors of $\pi_{jk}(u)$:
$\begin{array}[]{l}(i,b)\in E(v)\iff(j,b)\in
E(k\pi_{jk}(v))\quad\text{and}\quad(i,b)\in E(jk\pi_{jk}(v)),\\\ (j,b)\in
E(v)\iff(j,b)\in E(k\pi_{jk}(v))\quad\text{and}\quad(k,b)\in
E(jk\pi_{jk}(v)),\\\ (k,b)\in E(v)\iff(k,b)\in E(k\pi_{jk}(v)).\end{array}$
###### Proof.
First note that $i\notin E^{-}(k\pi_{jk}(v))$ and $j\notin
E^{-}(jk\pi_{jk}(v))$. Note also that the right extensions are preserved by
$\pi_{jk}$ because $\pi_{jk}$ preserves the first letter of words. Let
$(a_{0},b)\in E(v)$, $(a_{1},b)\in E(k\pi_{jk}(v))$ and $(a_{2},b)\in
E(jk\pi_{jk}(v))$ and let us consider each case $a_{0}=i$, $a_{0}=j$ and
$a_{0}=k$ separately (see Figure 5).
Figure 5: The preimage of $k\pi_{jk}(v)$ and $jk\pi_{jk}(v)$ under $\pi_{jk}$.
According to the assumption made on $v$, one checks that if $a_{0}=i$, then
$a_{1}=j$ and $a_{2}=i$; if $a_{0}=j$, then $a_{1}=j$ and $a_{2}=k$; if
$a_{0}=k$, then $a_{1}=k$. The reciprocals are also verified. ∎
In the next lemma, we show that bispecial factors are preserved under
desubstitution by the Poincaré substitution.
###### Lemma 29 (P - Antecedent of a bispecial).
Let $u\in\\{1,2,3\\}^{*}$ and $w\neq\varepsilon$ be a bispecial factor of
$\pi_{jk}(u)$. Let $v$ be the unique antecedent of $w$ under $\pi_{jk}$. One
has either $w=k\pi_{jk}(v)$, or $w=jk\pi_{jk}(v)$. Furthermore, $v$ is a
bispecial factor of $u$.
###### Proof.
The result is a direct consequence of Lemma 28. Since right extensions are
preserved by $\pi_{jk}$, we only need to check that if $w$ has at least two
left extensions then so does $v$.
Suppose that $w=k\pi_{jk}(v)$. Remark that $i\notin E^{-}(w)$. Thus $j,k\in
E^{-}(w)$ since $w$ is bispecial. From Lemma 28, $k\in E^{-}(w)$ implies $k\in
E^{-}(v)$. Also, $j\in E^{-}(w)$ implies that $i\in E^{-}(v)$ or $j\in
E^{-}(v)$. Thus $v$ is bispecial.
Suppose that $w=jk\pi_{jk}(v)$. Since $j\notin E^{-}(w)$, then $i,k\in
E^{-}(w)$. Or course, the existence of $w$ implicitly suppose $j\in
E^{-}(k\pi_{jk}(v))$. Then, $i,j\in E^{-}(v)$. We conclude that $v$ is
bispecial. ∎
Now we want to describe more precisely under which conditions a bispecial word
$v$ has a unique bispecial extended image and provide its extension type as we
were able to do in Lemma 26 for Arnoux-Rauzy substitutions. In general (see
Table 1 and 2), this depends on its left extensions $E^{-}(v)$. However, if
the left valence satisfies $d^{-}(v)=2$, we deduce the unicity of the
bispecial extended image as well as important information on the extension
type of the extended image. Recall that the notion of left equivalence for
extension types was defined in Section 3.2 in Definition 17.
###### Lemma 30 (P - Bispecial extended images in details).
Let $i,j,k$ such that $\\{i,j,k\\}=\\{1,2,3\\}$. Let $u\in\\{1,2,3\\}^{*}$ and
let $v$ be a bispecial factor of $u$. We assume that for all $(a,b)\in E(v)$,
there exists a letter $e$ such that $eavb$ is also a factor of $u$.
1. (i)
If $d^{-}(v)=2$, $v$ admits a unique bispecial extended image
$w\in\\{k\pi_{jk}(v),jk\pi_{jk}(v)\\}$ under $\pi_{jk}$ and $d^{-}(w)=2$.
Moreover, the extension types $E(v)$ and $E(w)$ (in $\pi_{jk}(u)$) are left
equivalent and are related according to Table 1.
2. (ii)
If $d^{-}(v)=3$, then $v$ admits either one, or two bispecial extended images
$w\in\\{k\pi_{jk}(v),jk\pi_{jk}(v)\\}$ under $\pi_{jk}$. In any case,
$d^{-}(w)=2$ and the two non-empty rows of $E(w)$ are obtained by projection
of rows of $E(v)$. Furthermore, they are related according to Table 2.
###### Proof.
For each $a\in\\{1,2,3\\}$, let $R_{a}\subseteq\\{1,2,3\\}$ be such that
$E(v)=\bigcup_{a\in\\{1,2,3\\}}\\{a\\}\times R_{a}.$
The set $R_{a}$ denotes the right extensions associated with the left
extension $a\in E^{-}(v)$.
(i) If $d^{-}(v)=2$, then $E^{-}(v)$ is equal to either $\\{i,j\\}$,
$\\{i,k\\}$ or $\\{j,k\\}$. We proceed case by case. If $E^{-}(v)=\\{i,j\\}$,
then $k\pi_{jk}(v)$ is not left special and $jk\pi_{jk}(v)$ is the unique
bispecial extended image of $v$. If $E^{-}(v)=\\{i,k\\}$ or $\\{j,k\\}$, then
$jk\pi_{jk}(v)$ is not left special and $k\pi_{jk}(v)$ is the unique bispecial
extended image of $v$. This is summarized in Table 1 where the information
follows from Lemma 28.
$\begin{array}[]{c|c|c}E(v)&E(k\pi_{jk}(v))&E(jk\pi_{jk}(v))\\\
\hline\cr(\\{i\\}\times R_{i})\cup(\\{j\\}\times
R_{j})&\\{j\\}\times(R_{i}\cup R_{j})&(\\{i\\}\times R_{i})\cup(\\{k\\}\times
R_{j})\\\ (\\{i\\}\times R_{i})\cup(\\{k\\}\times R_{k})&(\\{j\\}\times
R_{i})\cup(\\{k\\}\times R_{k})&\\{i\\}\times R_{i}\\\ (\\{j\\}\times
R_{j})\cup(\\{k\\}\times R_{k})&(\\{j\\}\times R_{j})\cup(\\{k\\}\times
R_{k})&\\{k\\}\times R_{j}\end{array}$ Table 1: If $d^{-}(v)=2$, then exactly
one extended image of $v$ amongst $k\pi_{jk}(v)$ and $jk\pi_{jk}(v)$ is
bispecial. This only depends on the left extensions as the right extensions
are preserved.
In each case, the extension type $E(v)$ is left equivalent to the extension
type of the unique bispecial extended image $w$ of $v$. Moreover $d^{-}(w)=2$.
(ii) If $d^{-}(v)=3$, i.e., $E^{-}(v)=\\{i,j,k\\}$, then
$E^{-}(k\pi_{jk}(v))=\\{j,k\\}$ and $E^{-}(jk\pi_{jk}(v))=\\{i,k\\}$. Thus,
both extended images can be bispecial but their left valence is at most $2$.
This is summarized in Table 2.
$\begin{array}[]{c|c|c}E(v)&E(k\pi_{jk}(v))&E(jk\pi_{jk}(v))\\\
\hline\cr(\\{i\\}\times R_{i})\cup(\\{j\\}\times R_{j})\cup(\\{k\\}\times
R_{k})&(\\{j\\}\times R_{i}\cup R_{j})\cup(\\{k\\}\times R_{k})&(\\{i\\}\times
R_{i})\cup(\\{k\\}\times R_{j})\end{array}$
Table 2: If $d^{-}(v)=3$, then one or both extended images of $v$ amongst
$k\pi_{jk}(v)$ and $jk\pi_{jk}(v)$ are bispecial. In each case, their left
valence is $2$.
∎
Note that Table 1 and 2 provide much more information than does the statement
of Lemma 30 and they will be used to prove a more general result in Lemma 36.
For example, in Table 2, if $v$ is a bispecial factor such that $d^{-}(v)=3$,
$R_{i}=R_{j}$ and $|R_{i}|=|R_{j}|=1$, then $jk\pi_{jk}(v)$ is a left special
factor but not a right special factor, it is thus not bispecial.
### 4.4 Life of a bispecial factor under ARP substitutions
In this section, the life of a bispecial factor is analyzed more precisely
under the application of Arnoux-Rauzy and Poincaré substitutions in the spirit
of [Cas97a, Section 4.2.2] where bispecial factors are described under the
image of circular morphisms. To achieve this, we need to understand exactly
the left extensions which will give information about the multiplicity of the
bispecial factors.
Let $\mathcal{S}=\mathcal{S}_{\alpha}\cup\mathcal{S}_{\pi}.$ Let $w$ be a
factor of an infinite word Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word. Let
$w_{0}=w$ and $w_{i+1}$ be the unique antecedent of $w_{i}$ under $\sigma_{i}$
for $i\geq 0$. In particular, $w_{1}$ is the antecedent of $w_{0}$ under
$\sigma_{0}$ and $w_{2}$ is the antecedent of $w_{1}$ under $\sigma_{1}$. If
$|w_{i}|>0$, then $|w_{i+1}|<|w_{i}|$ by Lemma 24. There thus exists $n$ such
that $w_{n}=\varepsilon$.
###### Definition 31 (Age, History, Life).
Let $w$ be a factor of an Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word. Let
$w_{0}=w$ and $w_{i+1}$ be the unique antecedent of $w_{i}$ under $\sigma_{i}$
for $i\geq 0$. The smallest of the integers $n$ for which $w_{n}=\varepsilon$
is called the _age_ of $w$ and is denoted as $\mathrm{age}(w)$. Furthermore,
we say that the finite sequence $\sigma_{0}\sigma_{1}\cdots\sigma_{n}$ is the
_history_ and the sequence $(w_{i})_{0\leq i\leq n}$ is the _life_ of the word
$w$.
Figure 6: Life and history of a factor $w$.
The above definition is illustrated in Figure 6. According to Lemma 26 and 29,
all the words $w_{i}$ of the history of $w$ are bispecial factors when $w$ is
bispecial. We will consider from now on recurrent Arnoux-Rauzy-Poincaré
$\mathcal{S}$-adic words $\mathbf{u}$, with $\mathbf{u}^{(m)}$ being also
recurrent, in order to apply the assumptions of Lemma 28 and 30. According to
Proposition 13, note that this assumption applies in particular to all the
words of the Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic system.
###### Lemma 32.
Let ${\bf u}$ be a recurrent Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word
such that $\mathbf{u}^{(m)}$ is also recurrent for all $m$. Let $n\geq 0$ be
an integer. Let $B_{n}$ be the set of all bispecial factors of age $n$ in
$\mathbf{u}$. Then $\mathrm{Card}\,B_{n}\leq 2$.
###### Proof.
Let $w\in B_{n}$, $\sigma_{0}\sigma_{1}\cdots\sigma_{n}$ be its history, and
let $(w_{i})_{0\leq i\leq n}$ be its life.
Suppose first that
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}_{\alpha}^{*}\mathcal{S}$,
that is, the substitutions of the history of $w$ are all Arnoux-Rauzy
substitutions except possibly $\sigma_{n}$ which may be a Poincaré
substitution. From Lemma 25, $w_{i}$ is the unique extended image of $w_{i+1}$
for all $0\leq i\leq n-1$. Then $\mathrm{Card}B_{n}=1$.
Suppose now that
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\mathcal{S}$.
Let $\ell$ be the largest index smaller than $n$ of occurrence of $\pi_{jk}$,
that is,
$\sigma_{0}\sigma_{1}\cdots\sigma_{\ell}\in\mathcal{S}^{*}\,\pi_{jk}\quad\text{
and
}\quad\sigma_{\ell+1}\sigma_{\ell+2}\cdots\sigma_{n}\in\mathcal{S}_{\alpha}^{*}\mathcal{S}.$
Then, from Lemma 25, $w_{i}$ is the unique extended image of $w_{i+1}$ for all
$\ell+1\leq i\leq n-1$. Also, from Lemma 30, $w_{\ell+1}$ has at most two
extended images $w_{\ell}$ and $w^{\prime}_{\ell}$ in
$\\{k\pi_{jk}(w_{\ell+1}),jk\pi_{jk}(w_{\ell+1})\\}$. But then,
$d^{-}(w^{\prime}_{\ell})=d^{-}(w_{\ell})=2$ (still by Lemma 30). Therefore
both $w^{\prime}_{\ell}$ has a unique extended image $w^{\prime}_{\ell-1}$ and
$w_{\ell}$ has a unique extended image $w_{\ell-1}$ (by Lemma 30 (i)).
Recursively, we get $d^{-}(w^{\prime}_{i})=d^{-}(w_{i})=2$ for all $0\leq
i\leq\ell$, $w^{\prime}_{i}$ has a unique extended image $w_{i-1}$, and
$w_{i}$ has a unique extended image $w_{i-1}$ for all $1\leq i\leq\ell$. We
thus get $\mathrm{Card}B_{n}\leq 2$. ∎
The life $(w_{i})_{0\leq i\leq n}$ of bispecial factors “starts” (when read
backwards with decreasing indices) as the empty word $\varepsilon$ at $i=n$.
The word $w_{i}$ for $i<n$ is then obtained as the concatenation of one or two
letters concatenated with $\sigma_{i}(w_{i+1})$. These letters depend on the
extension type $E(w_{i+1})$ and recursively on the extension type $E(w_{n})$
of $w_{n}=\varepsilon$. Furthermore, $w_{n}$ is the antecedent of $w_{n-1}$
under $\sigma_{n-1}$ and the extension type $E(w_{n})$ of $w_{n}=\varepsilon$
depends on $\sigma_{n}$. Thus, it is important to understand properly what are
the possible extension types of the empty word under the application of
Arnoux-Rauzy and Poincaré substitutions. Below, the extension type
$E(\varepsilon)$ of the empty word considered as a bispecial factor in the
language of $\sigma(u)$ is denoted by $E_{\sigma(u)}(\varepsilon)$.
###### Lemma 33.
Let $\mathbf{u}\in\mathcal{A}^{*}\cup\mathcal{A}^{\mathbb{N}}$ be a proper
word. Considered as a bispecial factor of the language of the word
$\alpha_{k}(\mathbf{u})$, the empty word $\varepsilon$ is ordinary. Considered
as a bispecial factor of the language of the word $\pi_{jk}(\mathbf{u})$, the
empty word $\varepsilon$ is neutral but not ordinary:
$\scriptsize
E_{\alpha_{k}(\mathbf{u})}(\varepsilon)=\begin{array}[]{c|ccc}&i&j&k\\\
\hline\cr i&&&\times\\\ j&&&\times\\\
k&\times&\times&\times\end{array}\quad\quad\text{\normalsize and}\quad\quad
E_{\pi_{jk}(\mathbf{u})}(\varepsilon)=\begin{array}[]{c|ccc}&i&j&k\\\
\hline\cr i&&\times&\\\ j&&&\times\\\ k&\times&\times&\times\end{array}.$
###### Proof.
We need to consider the set of pairs of consecutive letters appearing in the
language $\alpha_{k}(u)$. These can be consecutive letters inside
$\alpha_{k}(1)$, $\alpha_{k}(2)$ or $\alpha_{k}(3)$, i.e., $\\{ik,jk\\}$.
Alternatively, it may be the last letter of a word $\alpha_{k}(a)$ with the
first letter of a word $\alpha_{k}(b)$: $\\{ki,kj,kk\\}$.
Similarly for the language $\pi_{jk}(u)$, consecutive letters inside
$\pi_{jk}(i)$, $\pi_{jk}(j)$ and $\pi_{jk}(k)$ are $\\{ij,jk\\}$ and pairs
made of the last letter of a word $\pi_{jk}(a)$ with the first letter of a
word $\pi_{jk}(b)$ are $\\{ki,kj,kk\\}$. ∎
From now on, we assume that the Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic words
$\mathbf{u}^{(m)}$ are all proper for all $m$ in order to apply Lemma 33 for
the bispecial factors of all ages. Note that being recurrent does not imply
the fact of being proper: indeed an infinite word can be recurrent on the
alphabet $\\{1,2\\}$ while each letter of the alphabet $\\{1,2,3\\}$ must
appear for this word to be proper.
###### Lemma 34.
Let ${\bf u}$ be an Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word such that
$\mathbf{u}^{(m)}$ is proper and recurrent for all $m$. Let $w$ be a bispecial
factor of ${\bf u}$. Then $|E(w)|\leq 5$.
###### Proof.
Let $n=\mathrm{age}(w)$ and $(w_{i})_{i}$ be the life of $w$. From Lemmas 26,
30 and 33, we have
$|E(w_{0})|\leq|E(w_{1})|\leq|E(w_{2})|\leq\cdots\leq|E(w_{n})|\leq 5.\qed$
The following lemma shows that the histories of bispecial factors in a same
infinite word ${\bf u}$ are related.
###### Lemma 35.
Let ${\bf u}$ be an Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word such that
$\mathbf{u}^{(m)}$ is proper and recurrent for all $m$. Let $w$ and $z$ be two
bispecial factors of ${\bf u}$.
1. (i)
If $\mathrm{age}(w)<\mathrm{age}(z)$, then the history of $w$ is a prefix of
the one of $z$.
2. (ii)
If $\mathrm{age}(w)=\mathrm{age}(z)$, then $w$ and $z$ have the same history.
###### Proof.
Statement (i) follows from the definition and statement (ii) follows from (i).
∎
In the next lemma, we describe exactly what are the bispecial factors
associated with each possible history. We recall that there are at most two
bispecial factors of the same age for a given history according to Lemma 32.
It has the same history as $w$ according to Lemma 35.
###### Lemma 36.
Let $\mathbf{u}$ be an Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word such that
$\mathbf{u}^{(m)}$ is proper and recurrent for all $m$. Let $w$ be a bispecial
factor of $\mathbf{u}$ and let $n=\mathrm{age}(w)$. Let $w^{\prime}$ be the
other bispecial factor of the same age as $w$ if it exists. Then the common
history $\sigma_{0}\sigma_{1}\cdots\sigma_{n}$ of $w$ and $w^{\prime}$
determines the left valence, the multiplicity and the extension type of both
$w$ and $w^{\prime}$. More precisely, the multiplicity and the extension type
are described in Table 3, whereas extension types are provided in Figures 7,
8, 9 and 10.
$\begin{array}[]{l|ccc|ccc}\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in&d^{-}(w)&m(w)&\text{ordinary}&d^{-}(w^{\prime})&m(w^{\prime})&\text{ordinary}\\\
\hline\cr\mathcal{S}_{\alpha}^{*}\mathcal{S}_{\alpha}&3&0&\mathrm{yes}\\\
\mathcal{S}_{\alpha}^{*}\mathcal{S}_{\pi}&3&0&\mathrm{no}\\\
\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\alpha_{k}\\}&2&0&\mathrm{yes}\\\
\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\alpha_{i},\alpha_{j}\\}&2&0&\mathrm{yes}&2&0&\mathrm{yes}\\\
\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\pi_{ji},\pi_{ki},\pi_{ij},\pi_{kj}\\}&2&0&\mathrm{yes}&2&0&\mathrm{yes}\\\
\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\pi_{jk},\pi_{ik}\\}&2&+1&\mathrm{no}&2&-1&\mathrm{no}\\\
\end{array}$ Table 3: Left valence and multiplicity for the (at most two)
bispecial factors of the same age.
###### Remark 37.
Recall that the occurrence of strong and weak bispecial factors has an impact
on the factor complexity. According to Lemma 36, strong and weak bispecial
words appear in pairs under the application of Poincaré substitutions each
time $\pi_{jk}$ is followed by either $\pi_{jk}$ or $\pi_{ik}$ for
$\\{i,j,k\\}=\\{1,2,3\\}$ with possibly some Arnoux-Rauzy substitutions
$\alpha_{k}$, $k\in\\{1,2,3\\}$, in between.
###### Proof.
In the following proof, elements $(j,k)$ of $E(w)$ are noted $jk$ for short.
We refer below to the lines of Table 3.
Line 1. Assume
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}_{\alpha}^{*}\mathcal{S}_{\alpha}$.
According to Lemma 26, the extension type is preserved by Arnoux-Rauzy
substitutions, which yields $E(w)=E_{\sigma_{n}}(\varepsilon)$, so that
$d^{-}(w)=3$. Moreover, since $\sigma_{n}\in\mathcal{S}_{\alpha}$, then
$E_{\sigma_{n}}(\varepsilon)$ is ordinary and the multiplicity is $m(w)=0$ (by
Lemma 33). Also, the bispecial extended images are unique under the
application of each substitution $\sigma_{i}\in\mathcal{S}_{\alpha}$, by Lemma
25.
Line 2. Assume
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}_{\alpha}^{*}\mathcal{S}_{\pi}$.
The proof is the same as for Line 1 except that the extension type of the
empty word $E_{\sigma_{n}}(\varepsilon)$ is not ordinary because
$\sigma_{n}\in\mathcal{S}_{\pi}$ (by Lemma 33).
Line 3-6. We assume
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\mathcal{S}$.
Let $\ell$ be the largest index of occurrence smaller than $n$ of $\pi_{jk}$,
that is,
$\sigma_{0}\sigma_{1}\cdots\sigma_{\ell}\in\mathcal{S}^{*}\,\pi_{jk}\quad\text{
and
}\quad\sigma_{\ell+1}\sigma_{\ell+2}\cdots\sigma_{n}\in\mathcal{S}_{\alpha}^{*}\mathcal{S}.$
The bispecial antecedent of $w$ under the substitution
$\sigma_{0}\sigma_{1}\cdots\sigma_{\ell-1}\in\mathcal{S}^{*}$ is $w_{\ell}$,
and $w_{\ell+1}$ is the bispecial antecedent of $w_{\ell}$ under the
substitution $\sigma_{\ell}=\pi_{jk}$. Since
$\sigma_{\ell+1}\sigma_{\ell+2}\cdots\sigma_{n}\in\mathcal{S}_{\alpha}^{*}\mathcal{S}$,
then $d^{-}(w_{\ell+1})=3$ and $w_{\ell+1}$ has two extended images under
$\sigma_{\ell}=\pi_{jk}$. Moreover, let $w^{\prime}_{\ell}$, with
$w^{\prime}_{\ell}\neq w_{\ell}$, be the other extended image of $w_{\ell+1}$.
One has $w_{\ell},w^{\prime}_{\ell}\in\\{k\pi_{jk}(w_{\ell+1}),\
jk\pi_{jk}(w_{\ell+1})\\}$. Note that the factor $w^{\prime}_{\ell}$ may be
bispecial or not (see e.g. the proof of the case of Line 3 below). The end of
the proof for lines 3-6 follows the same pattern. In fact, the first part
$\sigma_{0}\sigma_{1}\cdots\sigma_{\ell-1}\in\mathcal{S}^{*}$ is always
applied on a bispecial factor $w_{\ell}$ or $w^{\prime}_{\ell}$ with left
valence satisfying $d^{-}(w_{\ell})=d^{-}(w^{\prime}_{\ell})=2$. Therefore,
from Lemma 30 (i) the extension types of $w=w_{0}$ and $w_{\ell}$ are left-
equivalent. Similarly, the extension types of $w^{\prime}=w^{\prime}_{0}$ and
$w^{\prime}_{\ell}$ are left-equivalent (where the $w^{\prime}_{i}$ are
inductively defined as extended images). From Lemma 18, the multiplicity, the
left valence and the fact of being strong, weak or ordinary is preserved by
left-equivalence. Below, we suppose $w_{\ell}=k\pi_{jk}(w_{\ell+1})$ and
$w^{\prime}_{\ell}=jk\pi_{jk}(w_{\ell+1})$.
Figure 7: Life of a bispecial word if
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\alpha_{k}\\}$.
Line 3. We assume
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\alpha_{k}\\}$
(see Figure 7). If $\sigma_{n}=\alpha_{k}$, then
$E(w_{\ell+1})=E_{\sigma_{n}(u)}(\varepsilon)=E_{\alpha_{k}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ik,jk,ki,kj,kk\\}$, $E(w_{\ell})=\\{jk,ki,kj,kk\\}$ and
$E(w^{\prime}_{\ell})=\\{ik,kk\\}$. Then $w_{\ell}$ is bispecial ordinary and
$w^{\prime}_{\ell}$ is not bispecial.
Figure 8: Life of a bispecial word if
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\alpha_{i},\alpha_{j}\\}$.
The extension types depicted represent the case $\sigma_{n}=\alpha_{j}$.
Line 4. Assume
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\alpha_{i},\alpha_{j}\\}$
(see Figure 8). If $\sigma_{n}=\alpha_{i}$, then
$E(w_{\ell+1})=E_{\sigma_{n}(u)}(\varepsilon)=E_{\alpha_{i}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ii,ij,ik,ji,ki\\}$, $E(w_{\ell})=\\{ji,jj,jk,ki\\}$, and
$E(w^{\prime}_{\ell})=\\{ii,ij,ik,ki\\}$. Then $w_{\ell}$ and
$w^{\prime}_{\ell}$ are both bispecial ordinary.
If $\sigma_{n}=\alpha_{j}$, then
$E(w_{\ell+1})=E_{\sigma_{n}(u)}(\varepsilon)=E_{\alpha_{j}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ij,ji,jj,jk,kj\\}$, $E(w_{\ell})=\\{ji,jj,jk,kj\\}$ and
$E(w^{\prime}_{\ell})=\\{ij,ki,kj,kk\\}$. Then $w_{\ell}$ and
$w^{\prime}_{\ell}$ are both bispecial ordinary.
Figure 9: Life of a bispecial word if
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\pi_{ji},\pi_{ki},\pi_{ij},\pi_{kj}\\}$.
The extension types depicted represent the case $\sigma_{n}=\pi_{kj}$.
Line 5. Assume
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\pi_{ji},\pi_{ki},\pi_{ij},\pi_{kj}\\}$
(see Figure 9). If $\sigma_{n}=\pi_{ji}$, then
$E(w_{\ell+1})=E_{\sigma_{n}(u)}(\varepsilon)=E_{\pi_{ji}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ii,ij,ik,ji,kj\\}$, $E(w_{\ell})=\\{ji,jj,jk,kj\\}$ and
$E(w^{\prime}_{\ell})=\\{ii,ij,ik,ki\\}$. Then $w_{\ell}$ and
$w^{\prime}_{\ell}$ are both bispecial ordinary.
If $\sigma_{n}=\pi_{ki}$, then
$E(w_{\ell+1})=E_{\sigma_{n}(u)}(\varepsilon)=E_{\pi_{ki}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ii,ij,ik,jk,ki\\}$, $E(w_{\ell})=\\{ji,jj,jk,ki\\}$ and
$E(w^{\prime}_{\ell})=\\{ii,ij,ik,kk\\}$. Then $w_{\ell}$ and
$w^{\prime}_{\ell}$ are both bispecial ordinary.
If $\sigma_{n}=\pi_{ij}$, then
$E(w_{\ell+1})=E_{\sigma_{n}(u)}(\varepsilon)=E_{\pi_{ij}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ij,ji,jj,jk,ki\\}$, $E(w_{\ell})=\\{jk,jj,jk,ki\\}$ and
$E(w^{\prime}_{\ell})=\\{ij,ki,kj,kk\\}$. Then $w_{\ell}$ and
$w^{\prime}_{\ell}$ are both bispecial ordinary.
If $\sigma_{n}=\pi_{kj}$, then
$E(w_{\ell+1})=E_{\sigma_{n}(u)}(\varepsilon)=E_{\pi_{kj}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ik,ji,jj,jk,kj\\}$, $E(w_{\ell})=\\{ji,jj,jk,kj\\}$ and
$E(w^{\prime}_{\ell})=\\{ik,ki,kj,kk\\}$. Then $w_{\ell}$ and
$w^{\prime}_{\ell}$ are both bispecial ordinary.
Figure 10: Life of a bispecial word if
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\pi_{jk},\pi_{ik}\\}$.
The extension types shown represent the case $\sigma_{n}=\pi_{jk}$.
Line 6. Assume
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\\{\pi_{jk},\pi_{ik}\\}$
(see Figure 10). If $\sigma_{n}=\pi_{jk}$, then
$E(w_{\ell+1})=E_{\sigma_{n}(u)}(\varepsilon)=E_{\pi_{jk}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ij,jk,ki,kj,kk\\}$, $E(w_{\ell})=\\{jj,jk,ki,kj,kk\\}$ and
$E(w^{\prime}_{\ell})=\\{ij,kk\\}$. Then $w_{\ell}$ is bispecial strong and
$w^{\prime}_{\ell}$ is bispecial weak.
If $\sigma_{n}=\pi_{ik}$, then
$E(w_{\ell+1})=E_{\sigma_{n}}(u)(\varepsilon)=E_{\pi_{ik}(u)}(\varepsilon)$.
Then from Lemma 30 (ii) and Table 2, we have
$E(w_{\ell+1})=\\{ik,ji,ki,kj,kk\\}$, $E(w_{\ell})=\\{ji,jk,ki,kj,kk\\}$ and
$E(w^{\prime}_{\ell})=\\{ik,ki\\}$. Then $w_{\ell}$ is bispecial strong and
$w^{\prime}_{\ell}$ is bispecial weak. ∎
### 4.5 Quadratic complexity is achievable
According to Remark 37, each time $\pi_{jk}$ and $\pi_{ik}$ are found one next
to the other in a certain $\mathcal{S}$-adic sequence, a new pair of strong
and weak bispecial factors is created (see Lemma 36) and the length of a newly
created weak bispecial factor can be larger than the length of an older strong
bispecial factor. Therefore, the complexity can increase by more than $3$,
i.e., $p(n+1)-p(n)>3$ for some values of $n$. Let us illustrate it on the
following example. Let
$\displaystyle u$
$\displaystyle=\pi_{23}\pi_{23}\pi_{13}\pi_{23}\pi_{23}\alpha_{1}\alpha_{3}\alpha_{2}(1)$
$\displaystyle=1232333233123233332331232333333123233323\cdots$
The bispecial factors of $u$ of age $\leq 5$ and their life are shown in
Figure 11. We see that some weak bispecial factors are longer than older
strong bispecial factors. Because of this fact and of Equation (7), the non-
zero values of the sequence $(b(n))_{n}$ do not alternate in the set
$\\{+1,-1\\}$. Therefore, there are values of $n$ for which $s(n)>3$. The
complete computation of $b(n)$, $s(n)$ and $p(n)$ for $n\leq 10$ is given in
Table 4. The complexity $p(n)$ of the finite word $u$ satisfies
$p(n+1)-p(n)=4$ for some values of $n$ and $p(n)>3n+1$ for $n$ such that
$7\leq n\leq 17$ (recall Equations (2), (3), (4), (5), (7) and in particular
that $s(n)=2+\sum_{\ell=0}^{n-1}b(\ell)$ when the size of alphabet is $3$).
Figure 11: The young bispecial factors of
$u=\pi_{23}\pi_{23}\pi_{13}\pi_{23}\pi_{23}\alpha_{1}\alpha_{3}\alpha_{2}(1)$
of age $\leq 5$ and their life. The factors $3$, $33$, $333$, $3333$ are
strong while $33333$ is neutral. For the age $k$ equal to $2$ and $3$, the
weak factor of age $k$ is longer than the strong factor of age $k+1$. The
quantity under parenthesis indicates the value of $m$.
$\begin{array}[]{c|c|c|c|c|c|c}n&\displaystyle\sum_{\begin{subarray}{c}w\in\mathcal{L}_{n}(u)\\\
w\,\text{is
strong}\end{subarray}}m(w)&\displaystyle\sum_{\begin{subarray}{c}w\in\mathcal{L}_{n}(u)\\\
w\,\text{is weak}\end{subarray}}m(w)&b(n)&s(n)&p(n)&3n+1\\\ \hline\cr
0&0&0&0&2&1&1\\\ 1&+1&0&1&2&3&4\\\ 2&+1&-1&0&3&5&7\\\ 3&+1&0&1&3&8&10\\\
4&+1&-1&0&4&11&13\\\ 5&0&0&0&4&15&16\\\ 6&0&0&0&4&19&19\\\ 7&0&0&0&4&23&22\\\
8&0&0&0&4&27&25\\\ 9&0&-1&-1&4&31&28\\\ 10&0&0&0&3&35&31\\\ \end{array}$
Table 4: The lengths of strong bispecial factors of $u$ are $1$, $2$, $3$, $4$
while the lengths of weak bispecial factors of $u$ are $2$, $4$, $9$, $14$.
Since there are two more strong bispecial factors of length $\leq n$ than the
number of weak bispecial factors of length $\leq n$ for all $n$ such that
$3\leq n\leq 8$, then $s(n)=4$ for each $n$ with $4\leq n\leq 9$. For example,
$s(4)=p(5)-p(4)=s(0)+\sum_{\ell=0}^{3}b(\ell)=4$. Moreover, $p(7)=23>22=3\cdot
7+1$.
In fact the complexity can get higher. It follows from Theorem 4.7.66 of
[CN10, p. 214] that the fixed point of $\pi_{23}\pi_{13}$ starting with letter
$1$ has a quadratic factor complexity because it has infinitely many distinct
factors, namely the factors $3^{n}$, that are bounded (in fact fixed) under
$\pi_{23}\pi_{13}$.
### 4.6 Partial and strict partial order on $\mathbb{R}^{3}$
In this section, we consider two distinct partial orders on $\mathbb{R}^{3}$
and consider how these partial orders are preserved by the application of
Arnoux-Rauzy and Poincaré substitutions. The results allow the understanding
of the growth of bispecial factors and are used in the proof of Theorem 1 in
the next section.
Let
$\overrightarrow{u}=(u_{1},u_{2},u_{3}),\overrightarrow{v}=(v_{1},v_{2},v_{3})\in\mathbb{N}^{3}$
be two abelianized vectors (for two words $u,v$). We define $<$ as the strict
partial order (irreflexive, transitive and thus asymmetric) defined coordinate
per coordinate on $\mathbb{N}^{3}$ by:
$\overrightarrow{u}<\overrightarrow{v}\iff u_{1}<v_{1}\quad\text{and}\quad
u_{2}<v_{2}\quad\text{and}\quad u_{3}<v_{3}.$
Also, we define $\leq$ as the partial order (reflexive, transitive and
antisymmetric) defined coordinate per coordinate on $\mathbb{N}^{3}$:
$\overrightarrow{u}\leq\overrightarrow{v}\iff u_{1}\leq
v_{1}\quad\text{and}\quad u_{2}\leq v_{2}\quad\text{and}\quad u_{3}\leq
v_{3}.$
Moreover, we say that the inequality
$\overrightarrow{u}\leq\overrightarrow{v}$ is _strict on the index $i$_ if
$u_{i}<v_{i}$. Note that $\leq$ is not the reflexive closure of $<$ since it
includes more relations.
The next lemma shows that the relation $<$ is preserved by Arnoux-Rauzy and
Poincaré substitutions and that some stronger conditions are satisfied. These
stronger conditions are used to show at Lemma 39 that the relation $<$ is also
preserved for extended images of factors. In the next lemma and the next
sections, we fix $\mathbf{e}_{1}=(1,0,0)$, $\mathbf{e}_{2}=(0,1,0)$ and
$\mathbf{e}_{3}=(0,0,1)$.
###### Lemma 38.
Let $v,v^{\prime}\in\mathcal{A}^{*}$ be such that
$\overrightarrow{v}<\overrightarrow{v^{\prime}}$. For all
$\\{i,j,k\\}=\\{1,2,3\\}$,
1. (i)
$\overrightarrow{\alpha_{k}(v)}+2\mathbf{e}_{k}<\overrightarrow{\alpha_{k}(v^{\prime})}$,
2. (ii)
$\overrightarrow{\pi_{jk}(v)}+\mathbf{e}_{j}+2\mathbf{e}_{k}<\overrightarrow{\pi_{jk}(v^{\prime})}$.
In particular, if $\overrightarrow{v}<\overrightarrow{v^{\prime}}$ then
$\overrightarrow{\alpha_{k}(v)}<\overrightarrow{\alpha_{k}(v^{\prime})}$ and
$\overrightarrow{\pi_{jk}(v)}<\overrightarrow{\pi_{jk}(v^{\prime})}$.
The proof is in the appendix.
The next lemma shows that the relation $<$ is preserved by Arnoux-Rauzy and
Poincaré substitutions from a pair of factors to their extended images.
###### Lemma 39.
Let $\sigma\in\mathcal{S}$. Let $v,v^{\prime},w,w^{\prime}\in\mathcal{A}^{*}$
and suppose $w$ (resp. $w^{\prime}$) is an extended image of $v$ (resp.
$v^{\prime}$) under $\sigma$. If
$\overrightarrow{v}<\overrightarrow{v^{\prime}}$, then
$\overrightarrow{w}<\overrightarrow{w^{\prime}}$.
The proof is in the appendix.
###### Remark 40.
The previous lemma is false for the order $\leq$. Indeed $\pi_{jk}$ does not
preserve the relation $\leq$ for extended images. For example, if
$v=\varepsilon$ and $v^{\prime}=3$, then
$\overrightarrow{v}=(0,0,0)\leq(0,0,1)=\overrightarrow{v^{\prime}}$ but
$\overrightarrow{13\pi_{13}(v)}=\overrightarrow{13}=(1,0,1)\not\leq(0,0,2)=\overrightarrow{33}=\overrightarrow{3\pi_{13}(v^{\prime})},$
and this may even lead after some more substitutions to an inversion of the
order:
$\overrightarrow{3\pi_{23}(13)}=\overrightarrow{31233}=(1,1,3)\geq(0,0,3)=\overrightarrow{333}=\overrightarrow{3\pi_{23}(33)}.$
This example can be seen between age $3$ and $4$ in Figure 11.
## 5 Proof of Theorem 1
We now consider $\mathcal{S}$-adic words $\mathbf{u}$ generated by the Arnoux-
Rauzy-Poincaré algorithm applied to a totally irrational vector
$\mathbf{x}\in\Delta$. By Proposition 12, $\mathbf{u}^{(m)}$ is proper and
uniformly recurrent for all $m$ so the hypothesis introduced in the previous
section is satisfied. Such sequences are in the Arnoux-Rauzy-Poincaré
$\mathcal{S}$-adic system (Type $3$), that is, we take into account the
restrictions on the directive sequences provided by Proposition 6. The
examples in Section 4.5 show that Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic
sequences can lead in general to quadratic factor complexity. Nevertheless we
show that the factor complexity $p(n)$ of $\mathcal{S}$-adic words
$\mathbf{u}$ generated by the Arnoux-Rauzy-Poincaré algorithm applied to a
totally irrational vector satisfy $p(n+1)-p(n)\in\\{2,3\\}$. Thus, their
factor complexity is bounded below and above, that is, $2n+1\leq p(n)\leq
3n+1$ for all $n$. In fact, we even prove that $p(n+1)-p(n)$ is equal to $2$
more often than it is equal to $3$ which implies that
$p(n)\leq\frac{5}{2}n+1$. More precisely, we will show that strong and weak
bispecial words alternate when the length increases in Section 5.1. We then
consider more closely the lengths of consecutive values of $2$ and $3$ in the
sequence $(p(n+1)-p(n))_{n}$ in Section 5.2. By making use of Lemma 19
together with Lemma 44 (see Figure 14), we will be able to prove Theorem 1 in
Section 5.3.
### 5.1 Alternance of strong and weak bispecial factors
We first gather the lemmas required in the proof (see Section 5.3) of the fact
that the $\mathcal{S}$-adic words $\mathbf{u}$ (with $\mathbf{u}^{(m)}$
recurrent for all $m$) such that
$\sigma_{k}\sigma_{k+1}\cdots\sigma_{\ell}\in\mathcal{L}(\mathcal{G})$ (for
all $k,\ell$) provide words that satisfy $p(n+1)-p(n)\in\\{2,3\\}$.
Restricted to the language of the automaton $\mathcal{G}$, illustrated in
Figure 2, the history of a strong or weak bispecial factor necessarily
contains Arnoux-Rauzy substitutions.
###### Lemma 41.
Let $\mathbf{u}=\lim_{n\to\infty}\sigma_{0}\sigma_{1}\cdots\sigma_{n}(a_{n})$
be an $\mathcal{S}$-adic word generated by the Arnoux-Rauzy-Poincaré algorithm
applied to a totally irrational vector $\mathbf{x}\in\Delta$. Let $w$ be a
bispecial factor of $\mathbf{u}$ and let $n=\mathrm{age}(w)$.
If $w$ is weak or strong and the history of $w$ is in the regular language
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{L}(\mathcal{G})$, then
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\\{\alpha_{j}\\}^{*}\,\alpha_{i}\,\mathcal{S}_{\alpha}^{*}\,\,\\{\pi_{ik},\pi_{jk}\\}$
for some $\\{i,j,k\\}=\\{1,2,3\\}$.
###### Proof.
From Lemma 36, we have
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\,\mathcal{S}_{\alpha}^{*}\,\\{\pi_{ik},\pi_{jk}\\}$
for some $\\{i,j,k\\}=\\{1,2,3\\}$. Let $p\in\mathcal{S}^{*}\,\pi_{jk}$ and
$q\in\mathcal{S}_{\alpha}^{*}\,\\{\pi_{ik},\pi_{jk}\\}$ such that
$pq=\sigma_{0}\sigma_{1}\sigma_{2}\cdots\sigma_{n}$. The word $p$ starts at
the initial state $\Delta$ and ends in the state $H_{jk}$, the word $q$ starts
from the state $H_{jk}$ and ends in state $H_{jk}$ or $H_{ik}$ (see Figure
12).
Figure 12: The subautomaton of $\mathcal{G}$ describing a path
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}=pq$ such that
$p\in\mathcal{S}^{*}\,\pi_{jk}$ and
$q\in\mathcal{S}_{\alpha}^{*}\,\\{\pi_{ik},\pi_{jk}\\}$.
In the automaton $\mathcal{G}$, the possible transitions issued from state
$H_{jk}$ are $\pi_{ij}$, $\pi_{ji}$, $\pi_{ki}$, $\alpha_{j}$ and $\alpha_{i}$
where only $\alpha_{j}$ (looping on state $H_{jk}$) and $\alpha_{i}$ (going to
state $\Delta$) are allowed by
$q\in\mathcal{S}_{\alpha}^{*}\,\\{\pi_{ik},\pi_{jk}\\}$. Once in state
$\Delta$, $q$ allows loops for each symbol in $\mathcal{S}_{\alpha}$, and
finally the transitions $\pi_{jk}$ or $\pi_{ik}$ (see Figure 12). It follows
from this that
$q\in\\{\alpha_{j}\\}^{*}\,\alpha_{i}\,\mathcal{S}_{\alpha}^{*}\,\,\\{\pi_{ik},\pi_{jk}\\}$
which was to be proved. ∎
###### Lemma 42.
Let $w$ be a bispecial factor of an Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic
word. If for some $\\{i,j,k\\}=\\{1,2,3\\}$,
$\mathrm{history}(w)\in\pi_{jk}\,\mathcal{S}^{*}\alpha_{i}\,\mathcal{S}^{*}\mathcal{S},$
then $\overrightarrow{w}\geq(1,1,1)$.
Figure 13: We suppose here that
$\mathrm{history}(w)\in\pi_{jk}\,\mathcal{S}^{*}\alpha_{i}\,\mathcal{S}^{*}\mathcal{S}$.
###### Proof.
Let $w_{1}$ be the ancestor of $w$ under $\pi_{jk}$. Let $r$ and $n$ be
integers such that $1\leq r<n=\mathrm{age}(w)$ and $w_{r+1}$ is the ancestor
of $w_{r}$ under substitution $\alpha_{i}$ as depicted in Figure 13. We have
that $\overrightarrow{w_{n}}=(0,0,0)$. Also,
$\overrightarrow{w_{r+1}}\geq(0,0,0)$ but $w_{r}=i\alpha_{i}(w_{r+1})$
contains at least one occurence of the letter $i$. Then, $w_{1}$ also contains
at least one occurrence of the letter $i$. Therefore
$\overrightarrow{w}\geq(1,1,1)$, because $\pi_{jk}$ maps $i$ to $ijk$. ∎
In order to prove that $p(n+1)-p(n)\in\\{2,3\\}$ for Arnoux-Rauzy-Poincaré
$\mathcal{S}$-adic words $\mathbf{u}$ such that $\mathbf{u}^{(m)}$ is proper
and recurrent for all $m$, it is sufficient that strong and weak bispecial
words alternate when the length increases because of Lemma 19. More precisely,
if $z_{1}$ and $z_{3}$ are two strong (with multiplicity $+1$) bispecial
factors of a word $\mathbf{u}$ such that $|z_{1}|<|z_{3}|$, then there exists
a weak (with multiplicity $-1$) bispecial factor $z_{2}$ such that
$|z_{1}|<|z_{2}|\leq|z_{3}|$. Note that the notion of alternance was also used
to prove Theorem 4.11.2 in [CN10, p. 238].
###### Lemma 43.
Let $\mathbf{u}$ be an Arnoux-Rauzy-Poincaré $\mathcal{S}$-adic word such that
$\mathbf{u}^{(m)}$ is proper and recurrent for all $m$. Let $z^{+}$ and
$z^{-}$ be two bispecial factors of $\mathbf{u}$ of the same age. Suppose that
$z^{-}$ is weak and $z^{+}$ is strong. Then $|z^{+}|<|z^{-}|$.
###### Proof.
In this proof, we denote by
$\overrightarrow{z^{+}}\leq_{j}\overrightarrow{z^{-}}$ when
$\overrightarrow{z^{+}}\leq\overrightarrow{z^{-}}$ is strict on the coordinate
$j\in\\{1,2,3\\}$.
We prove by induction on the age of bispecial factors that
$\overrightarrow{z^{+}}\leq_{j}\overrightarrow{z^{-}}$ is strict on at least
one coordinate $j\in\\{1,2,3\\}$ with $j\in E^{-}(z^{+})$.
Let us prove the base step of the induction. Suppose that $z^{+}$ and $z^{-}$
have a common neutral bispecial antecedent $v$ thus under the substitution
$\pi_{jk}$ for some $\\{i,j,k\\}=\\{1,2,3\\}$. Then, $z^{+}=k\pi_{jk}(v)$ and
$z^{-}=jk\pi_{jk}(v)$ so that
$\overrightarrow{z^{+}}\leq_{j}\overrightarrow{z^{-}}$ is strict on the
coordinate $j$. Moreover $E^{-}(z^{+})=\\{j,k\\}$ and $E^{-}(z^{-})=\\{i,k\\}$
and hence $j\in E^{-}(z^{+})$.
Suppose now that $z^{+}_{h}$ and $z^{-}_{h}$ are two respectively strong and
weak bispecial factors of a word $u$ of the same age such that
$\overrightarrow{z^{+}_{h}}\leq_{k}\overrightarrow{z^{-}_{h}}$ is strict on at
least one coordinate $k\in E^{-}(z_{h}^{+})$. Let $z^{+}_{h-1}$ and
$z^{-}_{h-1}$ be respectively the unique bispecial extended images of
$z^{+}_{h}$ and $z^{-}_{h}$ under the application of $\sigma_{h-1}$. We want
to show the following implication for proving the induction:
$\overrightarrow{z^{+}_{h}}\leq_{k}\overrightarrow{z^{-}_{h}}\text{ and }k\in
E^{-}(z_{h}^{+})\implies\text{there exists $j$ such that
}\overrightarrow{z^{+}_{h-1}}\leq_{j}\overrightarrow{z^{-}_{h-1}}\text{ and
}j\in E^{-}(z_{h-1}^{+}).$
Since the letters prepended to the left of bispecial extended images depend on
the left extensions by Table 1, if $E^{-}(z^{-}_{h})=E^{-}(z^{+}_{h})$, it is
clear that $E^{-}(z^{-}_{h-1})=E^{-}(z^{+}_{h-1})$ and
$\overrightarrow{z^{+}_{h-1}}\leq_{j}\overrightarrow{z^{-}_{h-1}}$ is strict
for some letter $j\in E^{-}(z^{+}_{h-1})$. Suppose now that
$E^{-}(z^{-}_{h})\neq E^{-}(z^{+}_{h})$ and suppose without lost of generality
that $E^{-}(z^{+}_{h})=\\{2,3\\}$ and $E^{-}(z^{-}_{h})=\\{1,3\\}$. The
possible cases depending on $\sigma_{h-1}$ are described in the following
table:
$\begin{array}[]{c|cc|cc|cc}&&&&&\text{if }k=2&\text{if }k=3\\\
\sigma_{h-1}&z^{+}_{h-1}&E^{-}(z^{+}_{h-1})&z^{-}_{h-1}&E^{-}(z^{-}_{h-1})&\\{j\mid\overrightarrow{z^{+}_{h-1}}\leq_{j}\overrightarrow{z^{-}_{h-1}}\\}&\\{j\mid\overrightarrow{z^{+}_{h-1}}\leq_{j}\overrightarrow{z^{-}_{h-1}}\\}\\\
\hline\cr\alpha_{1}&1\alpha_{1}(z^{+}_{h})&\\{2,3\\}&1\alpha_{1}(z^{-}_{h})&\\{1,3\\}&\\{1,2\\}&\\{1,3\\}\\\
\alpha_{2}&2\alpha_{2}(z^{+}_{h})&\\{2,3\\}&2\alpha_{2}(z^{-}_{h})&\\{1,3\\}&\\{2\\}&\\{2,3\\}\\\
\alpha_{3}&3\alpha_{3}(z^{+}_{h})&\\{2,3\\}&3\alpha_{3}(z^{-}_{h})&\\{1,3\\}&\\{2,3\\}&\\{3\\}\\\
\pi_{12}&2\pi_{12}(z^{+}_{h})&\\{1,2\\}&12\pi_{12}(z^{-}_{h})&\\{2,3\\}&\\{1,2\\}&\\{1,2,3\\}\\\
\pi_{32}&2\pi_{32}(z^{+}_{h})&\\{2,3\\}&32\pi_{32}(z^{-}_{h})&\\{1,2\\}&\\{2,3\\}&\\{2,3\\}\\\
\pi_{13}&3\pi_{13}(z^{+}_{h})&\\{1,3\\}&3\pi_{13}(z^{-}_{h})&\\{1,3\\}&\\{1,2,3\\}&\\{3\\}\\\
\pi_{23}&3\pi_{23}(z^{+}_{h})&\\{2,3\\}&3\pi_{23}(z^{-}_{h})&\\{2,3\\}&\\{2,3\\}&\\{3\\}\\\
\pi_{21}&21\pi_{21}(z^{+}_{h})&\\{1,3\\}&1\pi_{21}(z^{-}_{h})&\\{1,2\\}&\\{1\\}&\\{1,3\\}\\\
\pi_{31}&31\pi_{31}(z^{+}_{h})&\\{1,2\\}&1\pi_{31}(z^{-}_{h})&\\{1,3\\}&\\{1,2\\}&\\{1\\}\end{array}$
We check that for all nine possible values of $\sigma_{h-1}\in\mathcal{S}$, we
always have that
$\overrightarrow{z^{+}_{h-1}}\leq_{j}\overrightarrow{z^{-}_{h-1}}$ is strict
for some $j\in E^{-}(z^{+}_{h-1})$. Since we proved
$\overrightarrow{z^{+}}\leq_{j}\overrightarrow{z^{-}}$ is strict on at least
one coordinate $j$, then we conclude that $|z^{+}|<|z^{-}|$. ∎
###### Lemma 44.
Let $\mathbf{u}=\lim_{n\to\infty}\sigma_{0}\sigma_{1}\cdots\sigma_{n}(a_{n})$
be an $\mathcal{S}$-adic word generated by the Arnoux-Rauzy-Poincaré algorithm
applied to a totally irrational vector $\mathbf{x}\in\Delta$. Let $z^{-}$ and
$w^{+}$ be two bispecial factors of ${\bf u}$ such that $z^{-}$ is weak,
$w^{+}$ is strong, and the history of both $z^{-}$ and $w^{+}$ are in the
regular language
$\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{L}(\mathcal{G})$. If
$\mathrm{age}(z^{-})<\mathrm{age}(w^{+})$, then $|z^{-}|<|w^{+}|$.
Figure 14: Lifes of two pairs of strong and weak bispecial factors: $z^{+}$,
$z^{-}$ and $w^{+}$, $w^{-}$.
###### Proof.
In this proof, we denote a bispecial factor $w$ as $w^{+}$ if it is strong,
$w^{-}$ if it is weak, and with no sign if it is neutral. Let
$m=\mathrm{age}(z^{-})$, $z^{-}_{0}=z^{-}$ and $z_{i+1}$ be the unique
antecedent of $z_{i}$ under $\sigma_{i}$ for $0\leq i\leq m-1$. Let
$n=\mathrm{age}(w^{+})$, $w^{+}_{0}=w^{+}$ and $w_{i+1}$ be the unique
antecedent of $w_{i}$ under $\sigma_{i}$ for $0\leq i\leq n-1$. From Lemma 41,
we have
$\begin{array}[]{l}\mathrm{history}(w^{+})=\sigma_{0}\sigma_{1}\cdots\sigma_{n}\in\mathcal{S}^{*}\,\pi_{jk}\\{\alpha_{j}\\}^{*}\,\alpha_{i}\,\mathcal{S}_{\alpha}^{*}\,\,\\{\pi_{ik},\pi_{jk}\\},\\\
\mathrm{history}(z^{-})=\sigma_{0}\sigma_{1}\cdots\sigma_{m}\in\mathcal{S}^{*}\,\pi_{j^{\prime}k^{\prime}}\\{\alpha_{j^{\prime}}\\}^{*}\,\alpha_{i^{\prime}}\,\mathcal{S}_{\alpha}^{*}\,\,\\{\pi_{i^{\prime}k^{\prime}},\pi_{j^{\prime}k^{\prime}}\\},\end{array}$
for some $\\{i,j,k\\}=\\{1,2,3\\}$ and some other values of
$\\{i^{\prime},j^{\prime},k^{\prime}\\}=\\{1,2,3\\}$. We want to show that
$\overrightarrow{w^{+}}>\overrightarrow{z^{-}}$ in order to conclude that
$|w^{+}|>|z^{-}|$. Let $\ell$ ($h$ resp.) be the largest integer such that
$w^{+}_{\ell}$ ($z^{-}_{h}$ resp.) is strong (weak resp.). The situation is
illustrated in Figure 14.
We have $m\leq\ell$. Indeed, suppose on the contrary that $m>\ell$. We know
that $\sigma_{m}\in\mathcal{S}_{\pi}$. Also,
$\sigma_{i}\in\mathcal{S}_{\alpha}$ for all $\ell+1\leq i\leq n-1$. This
implies that $m\geq n$ which is a contradiction. Hence, $h<m\leq\ell<n$. Since
$\mathrm{history}(w^{+}_{\ell})\in\pi_{jk}\\{\alpha_{j}\\}^{*}\,\alpha_{i}\,\mathcal{S}_{\alpha}^{*}\,\,\\{\pi_{ik},\pi_{jk}\\}$,
from Lemma 42 we have that $\overrightarrow{w^{+}_{\ell}}\geq(1,1,1)$. Then,
$\overrightarrow{z_{m}}=(0,0,0)<(1,1,1)\leq\overrightarrow{w^{+}_{\ell}}\leq\overrightarrow{w^{+}_{m}}.$
Using induction and Lemma 39, we obtain that
$\overrightarrow{z^{-}}<\overrightarrow{w^{+}}$. Then, $|w^{+}|>|z^{-}|$. ∎
### 5.2 Ranges of $2$ and $3$ in the sequence $(p(n+1)-p(n))_{n}$
The next lemma, whose proof requires a deeper understanding of the abelianized
vectors of bispecial factors, will allow us in Section 5.3 to get a more
precise information concerning the alternance of weak and strong bispecial
factors.
Figure 15: Three phases of the lifes of two pairs of strong and weak bispecial
factors: $z^{+}$, $z^{-}$ and $w^{+}$, $w^{-}$.
###### Lemma 45.
Let $\mathbf{u}$ be an $\mathcal{S}$-adic word generated by the Arnoux-Rauzy-
Poincaré algorithm applied to a totally irrational vector
$\mathbf{x}\in\Delta$. Let $w^{+}$ and $w^{-}$ be two bispecial factors of the
same age of $\mathbf{u}$ such that $w^{+}$ is strong and $w^{-}$ is weak. If
there exists a younger weak bispecial factor $z^{-}$ of ${\bf u}$, i.e.,
$\mathrm{age}(z^{-})<\mathrm{age}(w^{+})=\mathrm{age}(w^{-})$, then
$|w^{+}|-|z^{-}|>|w^{-}|-|w^{+}|$. If there is no younger weak bispecial
factor, then $|w^{+}|\geq|w^{-}|-|w^{+}|$.
###### Proof.
The proof is divided into three phases according to the lifes of the bispecial
factors (see Figure 15).
1. (i)
At the end of Phase A, we have
$\overrightarrow{w_{m}^{+}}-\overrightarrow{z_{m}}\geq\overrightarrow{w_{m}^{-}}-\overrightarrow{w_{m}^{+}}$
is strict on two letters in $\\{1,2,3\\}$.
2. (ii)
At the end of Phase B, we have
$\overrightarrow{w_{h}^{+}}-\overrightarrow{z_{h}^{-}}>\overrightarrow{w_{h}^{-}}-\overrightarrow{w_{h}^{+}}$
and the words $w_{h}^{-}$ and $w_{h}^{+}$ have the same left extensions.
3. (iii)
At the end of Phase C, we have
$\overrightarrow{w^{+}}-\overrightarrow{z^{-}}>\overrightarrow{w^{-}}-\overrightarrow{w^{+}}$.
Phase A. Let $\ell$ be the largest index such that $w^{+}_{\ell}\neq
w^{-}_{\ell}$. One has that $w^{+}_{\ell}$ is strong and $w^{-}_{\ell}$ is
weak and their antecent $w^{+}_{\ell+1}=w^{-}_{\ell+1}=w_{\ell+1}$ are equal
and neutral. The bispecial factor $w_{\ell}^{+}$ contains each of the letters
in $\\{1,2,3\\}$ because of Lemma 41 and Lemma 42. Also
$\overrightarrow{w_{\ell}^{-}}-\overrightarrow{w_{\ell}^{+}}=\mathbf{e}_{a}$
for some $a\in\\{1,2,3\\}$. Thus
$\overrightarrow{w_{\ell}^{+}}\geq\overrightarrow{w_{\ell}^{-}}-\overrightarrow{w_{\ell}^{+}}$
is a strict inequality on at least two coordinates. One checks that this
property is preserved by each of the nine possible substitutions. This implies
that
$\overrightarrow{w_{m}^{+}}\geq\overrightarrow{w_{m}^{-}}-\overrightarrow{w_{m}^{+}}$
is strict on at least two letters in $\\{1,2,3\\}$ as well. This also proves
the last part of the lemma, concerning the case where there is no younger weak
bispecial factor.
Phase B. Each of the inequality below is implied by the precedent one. The
substitution $\alpha_{i}$ brings the inequality (by at least two units) on the
coordinate $i$. Then, the substitution $\pi_{jk}$ spreads the strict
inequality on every coordinate:
$\overrightarrow{w_{m}^{+}}-\overrightarrow{z_{m}}\geq\overrightarrow{w_{m}^{-}}-\overrightarrow{w_{m}^{+}}$
is strict on at least two letters in $\\{1,2,3\\}$,
---
$\overrightarrow{w_{t+1}^{+}}-\overrightarrow{z_{t+1}}\geq\overrightarrow{w_{t+1}^{-}}-\overrightarrow{w_{t+1}^{+}}$
is strict on at least two letters in $\\{1,2,3\\}$,
$\overrightarrow{w_{t}^{+}}-\overrightarrow{z_{t}}-2\mathbf{e}_{i}\geq\overrightarrow{w_{t}^{-}}-\overrightarrow{w_{t}^{+}}$,
$\overrightarrow{w_{h+1}^{+}}-\overrightarrow{z_{h+1}}-2\mathbf{e}_{i}\geq\overrightarrow{w_{h+1}^{-}}-\overrightarrow{w_{h+1}^{+}}$,
$\overrightarrow{\pi_{jk}(w_{h+1}^{+})}-\overrightarrow{\pi_{jk}(z_{h+1})}-(2,2,2)\geq\overrightarrow{\pi_{jk}(w_{h+1}^{-})}-\overrightarrow{\pi_{jk}(w_{h+1}^{+})}$,
$\overrightarrow{w_{h}^{+}}-\overrightarrow{z_{h}^{-}}+\mathbf{e}_{j}-(2,2,2)\geq\overrightarrow{w_{h}^{-}}-\overrightarrow{w_{h}^{+}}$,
$\overrightarrow{w_{h}^{+}}-\overrightarrow{z_{h}^{-}}>\overrightarrow{w_{h}^{-}}-\overrightarrow{w_{h}^{+}}$.
The left extensions of $w_{m}^{+}$ and $w_{m}^{-}$ are $\\{j,k\\}$ or
$\\{i,k\\}$. Arnoux-Rauzy substitutions preserve the extensions so that
$E^{-}(w_{h+1}^{+})=E^{-}(w_{m}^{+})$ and
$E^{-}(w_{h+1}^{-})=E^{-}(w_{m}^{-})$. Finally, according to Table 1,
$\pi_{jk}$ projects those left extensions onto the same set
$E^{-}(w_{h+1}^{+})=E^{-}(w_{h+1}^{-})=\\{j,k\\}$.
Phase C. We have
$\overrightarrow{w_{h}^{+}}-\overrightarrow{z_{h}^{-}}>\overrightarrow{w_{h}^{-}}-\overrightarrow{w_{h}^{+}}$.
Since the words $w_{h}^{-}$ and $w_{h}^{+}$ have the same left extensions,
then so do $w_{h-1}^{-}$ and $w_{h-1}^{+}$ for all
$\sigma_{h-1}\in\mathcal{S}_{\pi}$. But the left extensions of $z_{h}^{-}$ can
be different from the one of $w_{h}^{-}$ and $w_{h}^{+}$. This can lead to
$z_{h-1}^{-}=jk\pi_{jk}(z_{h}^{-})$ while $w_{h-1}^{-}=k\pi_{jk}(w_{h}^{-})$
and $w_{h-1}^{+}=k\pi_{jk}(w_{h}^{+})$. Thus, the proof of Phase C relies on
the following recurrences on the age of bispecial factors (all other cases for
left extensions are easier and follow from the same recurrences):
1. (i)
(Recurrence AR) If
$\overrightarrow{w^{+}}-\overrightarrow{z^{-}}>\overrightarrow{w^{-}}-\overrightarrow{w^{+}}$,
then
$\overrightarrow{\alpha_{k}(w^{+})}-\overrightarrow{\alpha_{k}(z^{-})}>\overrightarrow{\alpha_{k}(w^{-})}-\overrightarrow{\alpha_{k}(w^{+})}$.
2. (ii)
(Recurrence P) If
$\overrightarrow{w^{+}}-\overrightarrow{z^{-}}>\overrightarrow{w^{-}}-\overrightarrow{w^{+}}$,
then
$\overrightarrow{\pi_{jk}(w^{+})}-\overrightarrow{\pi_{jk}(z^{-})}-\mathbf{e}_{j}>\overrightarrow{\pi_{jk}(w^{-})}-\overrightarrow{\pi_{jk}(w^{+})}$.
Let $\overrightarrow{z^{-}}=(x,y,z)$, $\overrightarrow{w^{+}}=(a,b,c)$,
$\overrightarrow{w^{-}}=(d,e,f)$ where the convention
$\mathbf{e}_{i}=(1,0,0)$, $\mathbf{e}_{j}=(0,1,0)$, $\mathbf{e}_{k}=(0,0,1)$
is used. For the Arnoux-Rauzy recurrence, we have
$\overrightarrow{\alpha_{k}(z^{-})}=(x,y,x+y+z),\quad\overrightarrow{\alpha_{k}(w^{+})}=(a,b,a+b+c),\quad\overrightarrow{\alpha_{k}(w^{-})}=(d,e,d+e+f).$
Then
$\displaystyle\overrightarrow{\alpha_{k}(w^{+})}-\overrightarrow{\alpha_{k}(z^{-})}$
$\displaystyle=$
$\displaystyle\overrightarrow{w^{+}}-\overrightarrow{z^{-}}+(0,0,a-x)+(0,0,b-y)$
$\displaystyle>$
$\displaystyle\overrightarrow{w^{-}}-\overrightarrow{w^{+}}+(0,0,a-x)+(0,0,b-y)$
$\displaystyle\geq$
$\displaystyle\overrightarrow{w^{-}}-\overrightarrow{w^{+}}+(0,0,d-a+1)+(0,0,e-b+1)$
$\displaystyle=$
$\displaystyle\overrightarrow{w^{-}}-\overrightarrow{w^{+}}+(0,0,d-a)+(0,0,e-b)+(0,0,2)$
$\displaystyle=$
$\displaystyle\overrightarrow{w^{-}}-\overrightarrow{w^{+}}+(0,0,d-a)+(0,0,e-b)+(0,0,2)$
$\displaystyle=$ $\displaystyle(d,e,d+e+f)-(a,b,a+b+c)+(0,0,2)$
$\displaystyle=$
$\displaystyle\overrightarrow{\alpha_{k}(w^{-})}-\overrightarrow{\alpha_{k}(w^{+})}+2\mathbf{e}_{k}$
For the Poincaré recurrence, we have
$\overrightarrow{\pi_{jk}(z^{-})}=(x,x+y,x+y+z),\quad\overrightarrow{\pi_{jk}(w^{+})}=(a,a+b,a+b+c),\quad\overrightarrow{\pi_{jk}(w^{-})}=(d,d+e,d+e+f).$
Then
$\displaystyle\overrightarrow{\pi_{jk}(w^{+})}-\overrightarrow{\pi_{jk}(z^{-})}-\mathbf{e}_{j}$
$\displaystyle=$
$\displaystyle\overrightarrow{w^{+}}-\overrightarrow{z^{-}}+(0,a-x,a-x)+(0,0,b-y)-(0,1,0)$
$\displaystyle>$
$\displaystyle\overrightarrow{w^{-}}-\overrightarrow{w^{+}}+(0,a-x,a-x)+(0,0,b-y)-(0,1,0)$
$\displaystyle\geq$
$\displaystyle\overrightarrow{w^{-}}-\overrightarrow{w^{+}}+(0,d-a+1,d-a+1)+(0,0,e-b+1)-(0,1,0)$
$\displaystyle=$
$\displaystyle\overrightarrow{w^{-}}-\overrightarrow{w^{+}}+(0,d-a,d-a)+(0,0,e-b)+(0,0,2)$
$\displaystyle=$
$\displaystyle\overrightarrow{w^{-}}-\overrightarrow{w^{+}}+(0,d-a,d-a)+(0,0,e-b)+(0,0,2)$
$\displaystyle=$ $\displaystyle(d,d+e,d+e+f)-(a,a+b,a+b+c)+(0,0,2)$
$\displaystyle=$
$\displaystyle\overrightarrow{\pi_{jk}(w^{-})}-\overrightarrow{\pi_{jk}(w^{+})}+2\mathbf{e}_{k}.\qed$
### 5.3 Linear growth for the factor complexity
We now have gathered all the elements for proving Theorem 1.
###### Proof of Theorem 1.
Since $\mathbf{x}$ is totally irrational, Proposition 12 certifies that lemmas
of the previous two sections can be applied since the $\mathcal{S}$-adic words
$\mathbf{u}^{(m)}$ are proper and uniformly recurrent for all $m$. The set of
bispecial factors of length $n$ contains at most one weak or strong bispecial
factor. Indeed, suppose on the contrary that it contains two of them: $w$ and
$z$. They cannot have the same age according to Lemma 43 since this would
otherwise imply $|w|\neq|z|$. Also, if one is older, e.g.
$\mathrm{age}(w)>\mathrm{age}(z)$, then $|w|>|z|$ from Lemma 44. Then
$b(n)\in\\{-1,0,+1\\}$ according to Equation (7) of Theorem 20. Finally, it
remains to prove that the assumptions of Lemma 19 are satisfied. The first
non-zero value of $b(n)$ is $+1$ because strong and weak bispecial factors
come in pairs and the strong one is smaller than the weak one from Lemma 43.
Moreover, non-zero values are alternating. Indeed, let $z^{+}$ and $w^{+}$ be
two strong bispecial factors such that
$\mathrm{age}(w^{+})>\mathrm{age}(z^{+})$. Let $z^{-}$ be the weak bispecial
factor such that $\mathrm{age}(z^{-})=\mathrm{age}(z^{+})$. From Lemma 43 and
Lemma 44, $|z^{+}|<|z^{-}|<|w^{+}|$. Hence, there is always a $-1$ between two
$+1$ in the sequence $(b(n))_{n\geq 0}$. This alternance of non-zero values in
the sequence $(b(n))_{n}$ shows that $p(n+1)-p(n)\in\\{2,3\\}$ (Lemma 19), so
that $2n+1\leq p(n)\leq 3n+1$ for $n\geq 0$.
Now we show that $p(n)\leq\frac{5}{2}n+1$. We prove by recurrence that
$p(q+1)\leq\frac{5}{2}(q+1)+1$ for each $q$ such that $b(q)=-1$. By assuming
that $b(-1)=-1$, we remark that the statement is valid for $q=-1$ because
$p(0)=1\leq 1$. Suppose $q$ and $t$ are two consecutive occurrences of $-1$ in
the sequence $(b(\ell))_{\ell}$, that is, $b(q)=b(t)=-1$ and $b(\ell)\neq-1$
for all $\ell$ such that $q<\ell<t$. We show that if
$p(q+1)\leq\frac{5}{2}(q+1)+1$ then $p(n+1)\leq\frac{5}{2}(n+1)+1$ for each
$n$ such that $q<n\leq t$. From the alternance of non-zero values $+1$ and
$-1$ in the sequence $(b(\ell))_{\ell}$, there exists an integer $r$ with
$q<r<t$ such that $b(r)=+1$ and such that for all integers $r^{\prime}\neq r$
with $q<r^{\prime}<t$ then $b(r^{\prime})=0$. Since the first non-zero value
of $(b(\ell))_{\ell\geq 0}$ is $+1$, then $\sum_{\ell=0}^{q}b(\ell)=0$. The
consequence of Lemma 45 is that $r-q>t-r$ which is true if and only if
$\frac{t-q}{2}>t-r$. Note that if $n$ is such that $q<n\leq r$, then
$s(n)=s(0)+\sum_{\ell=0}^{n-1}b(\ell)=2+\sum_{\ell=0}^{q}b(\ell)=2$. Also, if
$n$ is such that $r<n\leq t$, then
$s(n)=s(0)+\sum_{\ell=0}^{n-1}b(\ell)=2+\sum_{\ell=0}^{q}b(\ell)+b(r)=2+1=3$.
Therefore, for each $n$ such that $r<n\leq t$ we have
$\displaystyle p(n+1)-p(q+1)$ $\displaystyle=$
$\displaystyle\sum_{\ell=q+1}^{r}s(\ell)+\sum_{\ell=r+1}^{n}s(\ell)=2(r-q)+3(n-r)$
$\displaystyle=$ $\displaystyle
2(n-q)+(n-r)<2(n-q)+\frac{n-q}{2}=\frac{5}{2}(n-q).$
But since $p(q+1)\leq\frac{5}{2}(q+1)+1$ we conclude that
$p(n+1)\leq\frac{5}{2}(n+1)+1$. We get the same conclusion for each $n$ such
that $q<n\leq r$. From this we conclude that $p(n)\leq\frac{5}{2}n+1$ and
$\limsup_{n\to\infty}\frac{p(n)}{n}\leq\frac{5}{2}$. ∎
We in fact prove the more general result.
###### Theorem 46.
Let $\mathbf{u}$ be a word of the Arnoux-Rauzy-Poincaré system.
* •
If $\mathbf{u}$ is of Type $1$, then it has a bounded factor complexity.
* •
If $\mathbf{u}$ is of Type $2$, then its factor complexity satisfies
ultimately $p(n)=n+k$ for some constant $k$.
* •
If $\mathbf{u}$ is of Type $3$, then $p(n+1)-p(n)\in\\{2,3\\}$ and $2n+1\leq
p(n)\leq\frac{5}{2}n+1$ for all $n\geq 0$.
###### Proof.
Words $\mathbf{u}$ of Type $1$ are periodic and thus have a bounded factor
complexity. A word $\mathbf{u}$ of Type $2$ is an image by a substitution of a
Sturmian sequences. Then, according to [Cas97b], its factor complexity
satisfies ultimately $p(n)=n+k$ for some constant $k$. A word $\mathbf{u}$ of
Type $3$ is weakly primitive, $\mathbf{u}^{(m)}$ is recurrent and proper for
all $m$, and its factor complexity was proven to satisfy the desired bounds in
Theorem 1. ∎
## 6 Convergence and unique ergodicity
We start with some terminology. Let $\mathbf{u}$ be an infinite word in
${\mathcal{A}}^{\mathbb{N}}$. Let $X_{\mathbf{u}}$ be the orbit closure of the
infinite word $\mathbf{u}$ under the action of the shift $S$, that is,
$X_{\mathbf{u}}$ is the closure in ${\mathcal{A}}^{\mathbb{N}}$ of the set
$\\{S^{n}(\mathbf{u})\mid n\in\mathbb{N}\\}=\\{(u_{k})_{k\geq n}\mid
k\in\mathbb{N}\\}$, where the shift $S$ satisfies $S((u_{n})_{n})=(u_{n+1})$.
The set $X_{\mathbf{u}}$ coincides with the set of infinite words whose
language is contained in ${\mathcal{L}}({\mathbf{u}})$, and is called the
_symbolic dynamical system_ generated by $\mathbf{u}$. The topological
dynamical system $(X_{\bf u},S)$ can be endowed with a structure of a measure-
theoretic dynamical system $(X_{u},T,\mu,{\cal B})$, where ${\cal B}$ is a
$\sigma$-algebra, by taking any probability measure $\mu$ preserved by $T$,
that is, for all $B\in\mathcal{B}$, $\mu(S^{-1}(B))=\mu(B)$. The system
$X_{\mathbf{u}}$ is said to be _uniquely ergodic_ if there exists a unique
shift-invariant probability measure on $X$.
One natural way for getting an $S$-invariant measure is to consider factor
frequencies (for more details, see [FM10]). The _frequency_ of a letter $i$ in
$\mathbf{u}$ is defined as the limit when $n$ tends towards infinity, if it
exists, of the number of occurrences of $i$ in $u_{0}u_{1}\cdots u_{n-1}$
divided by $n$. The infinite word $\mathbf{u}$ has _uniform letter
frequencies_ if, for every letter $i$ of $u$, the number of occurrences of $i$
in $u_{k}\cdots u_{k+n-1}$ divided by $n$ has a limit when $n$ tends to
infinity, uniformly in $k$. Similarly, we can define the frequency and the
uniform frequency of a factor, and we say that $u$ has _uniform frequencies_
if all its factors have uniform frequency. The property of having uniform
factor frequencies for a shift $X$ is actually equivalent to unique ergodicity
(see e.g. [FM10]).
Factor complexity is a priori a topological notion. However it may yield (in
particular when it has a linear growth order) measure-theoretical information
on the the symbolic dynamical system $X_{\bf u}$. Indeed, according to
[Bos85], if $\mathbf{u}$ is assumed to be uniformly recurrent, and if $\limsup
p(n)/n<3$, then $(X_{\mathbf{u}},S)$ is uniquely ergodic.
###### Proof of Theorem 2.
We now have gathered all the elements for observing that Theorem 2 is a direct
consequence of Theorem 1 together with the above mentioned result of [Bos85]
and Proposition 12. ∎
## 7 Conclusion
Given a totally irrational vector of frequencies
$\mathbf{x}=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}_{+}$ (with $\sum x_{i}=1$),
we thus have shown how to construct an infinite word $\mathbf{u}$ over the
alphabet $\mathcal{A}=\\{1,2,3\\}$ such that the frequency of each letter
$i\in\mathcal{A}$ exists and is equal to $x_{i}$, with this word $\mathbf{u}$
having a linear factor complexity. This word is contructed by translating
symbolically within the $S$-adic formalism a multidimensional continued
fraction algorithm, namely the Arnoux-Rauzy-Poincaré algorithm.
Observe that usual proofs of convergence for multidimensional continued
fraction algorithms rely on linear algebra and on the use of the Hilbert
projective metric (see e.g. [Sch00]). Let us stress the fact that we provide
here a purely combinatorial proof of convergence for a two-dimensional
continued fraction algorithm based on the unique ergodicity.
The restriction to the regular language $\mathcal{L}(\mathcal{G})$ is clearly
important; there exist examples of $\mathcal{S}$-adic words constructed with
the alphabet of substitutions $\mathcal{S}$ for which the upper bound
$\frac{5}{2}n+1$ does not hold. Moreover, a quadratic complexity is even also
achievable (see Section 4.5). Hence, the present study gives some more insight
on a statement of the $S$-adic conjecture (it rather should be qualified of
problem) which is to find conditions for which $S$-adic sequences have a
linear complexity (see e.g. [DLR13, Ler12]). Note that any uniformly recurrent
word $\mathbf{u}$ whose complexity function $p(n)$ satisfies
$p(n+1)-p_{u}(n)\leq k$, for all $n$, is $S_{k}$-adic, with a set $S_{k}$ of
substitutions that depends on $k$ ([Fer96]).
The upper bound $\limsup_{n\to\infty}\frac{p(n)}{n}\leq\frac{5}{2}$ is not
sharp. Numerical experimentations tend to indicate that the worst case in the
language $\mathcal{L}(\mathcal{G})$ of the Arnoux-Rauzy-Poincaré algorithm is
obtained with the fixed point of $\pi_{23}\alpha_{1}$ for which the value is
approximately $\limsup_{n\to\infty}\frac{p(n)}{n}\approx 2.26079201$.
Factor complexity of Poincaré and Arnoux-Rauzy substitutions can be described
exactly by considering left and right extensions of length one. It is not
always the case, and the study of Brun substitutions (provided by the Brun
multidimensional continued fraction algorithm) seems to be an example for
which extensions of length longer than one are necessary to describe bispecial
factors. Recently, Klouda [Klo12] described bispecial factors in fixed points
of morphisms where extensions of length longer than one were considered.
Extending this work to $S$-adic words deserves further research.
Balance properties of the Poincaré and Arnoux-Rauzy $S$-adic system have also
nice properties and their study should be done more deeply. An infinite word
$\mathbf{u}\in{\mathcal{A}}^{\mathbb{N}}$ is said to be $C$-balanced if for
any pair $v,w$ of factors of the same length of $\mathbf{u}$, and for any
letter $i\in\mathcal{A}$, one has $||v|_{i}-|w|_{i}|\leq C$. It is said
balanced if there exists $C>0$ such that it is $C$-balanced. For example, it
was proven in [DHS13] that words generated by Brun algorithm gives almost
everywhere balanced sequences. Balance properties are intimately connected
with Diophantine properties of the algorithm. Indeed, an infinite word
$\mathbf{u}\in{\mathcal{A}}^{\mathbb{N}}$ is balanced if and only if it has
uniform letter frequencies and there exists a constant $B$ such that for any
factor $w$ of $u$, we have $||w|_{i}-f_{i}|w||\leq B$ for all letter $i$ in
${\mathcal{A}}$, where $f_{i}$ is the frequency of $i$.
## 8 Appendix
###### Proof of Proposition 6.
We define
$\widetilde{\mathcal{P}}=\\{A_{j}H_{jk}:\\{i,j,k\\}=\\{1,2,3\\}\\}\cup\\{P_{jk}H_{jk}:\\{i,j,k\\}=\\{1,2,3\\}\\}$
which describes another partition of $\Delta$ into 12 triangles shown in
Figure 16.
First, we show that the transformation $T$ is a Markov transformation for the
partition $\widetilde{\mathcal{P}}$.
Figure 16: The Markov partition $\widetilde{\mathcal{P}}$ of Arnoux-Rauzy-
Poincaré algorithm.
Let $\\{i,j,k\\}=\\{1,2,3\\}$. The image of $A_{j}H_{jk}$ and of
$P_{jk}H_{jk}$ under $T$ are the same and are equal to the half triangle
$H_{jk}$:
$T(A_{j}H_{jk})=T(P_{jk}H_{jk})=H_{jk}.$
But the half triangle $H_{jk}$ is a union of elements of
$\widetilde{\mathcal{P}}$:
$H_{jk}=A_{i}H_{ik}\cup A_{i}H_{ij}\cup A_{j}H_{jk}\cup P_{ij}H_{ij}\cup
P_{ji}H_{ji}\cup P_{ki}H_{ki}.$
Thus, the transformation $T$ is a Markov transformation for the partition
$\widetilde{\mathcal{P}}$. This defines an automaton
$\widetilde{G}=(\widetilde{\mathcal{P}},\widetilde{\Sigma},\widetilde{\delta},\widetilde{I},\widetilde{F})$
where the alphabet is
$\widetilde{\Sigma}=\\{A_{1}^{-1},A_{2}^{-1},A_{3}^{-1},P_{31}^{-1},P_{13}^{-1},P_{23}^{-1},P_{32}^{-1},P_{12}^{-1},P_{21}^{-1}\\},$
the transitions are
$\widetilde{\delta}=\\{(p,M,q)\in\widetilde{\mathcal{P}}\times\widetilde{\Sigma}\times\widetilde{\mathcal{P}}:q\subseteq
M\cdot p=T(p)\\},$
or, more precisely,
$\widetilde{\delta}=\left\\{\begin{array}[]{ll}A_{j}H_{jk},A_{j}^{-1}\to
A_{i}H_{ik},&P_{jk}H_{jk},P_{jk}^{-1}\to A_{i}H_{ik},\\\
A_{j}H_{jk},A_{j}^{-1}\to A_{i}H_{ij},&P_{jk}H_{jk},P_{jk}^{-1}\to
A_{i}H_{ij},\\\ A_{j}H_{jk},A_{j}^{-1}\to
A_{j}H_{jk},&P_{jk}H_{jk},P_{jk}^{-1}\to A_{j}H_{jk},\\\
A_{j}H_{jk},A_{j}^{-1}\to P_{ij}H_{ij},&P_{jk}H_{jk},P_{jk}^{-1}\to
P_{ij}H_{ij},\\\ A_{j}H_{jk},A_{j}^{-1}\to
P_{ji}H_{ji},&P_{jk}H_{jk},P_{jk}^{-1}\to P_{ji}H_{ji},\\\
A_{j}H_{jk},A_{j}^{-1}\to P_{ki}H_{ki},&P_{jk}H_{jk},P_{jk}^{-1}\to
P_{ki}H_{ki}\end{array}\text{for each }\,\\{i,j,k\\}=\\{1,2,3\\}\right\\},$
the initial states and final states are all of the twelve states, i.e.,
$\widetilde{I}=\widetilde{F}=\widetilde{\mathcal{P}}$. The automaton
$\widetilde{G}$ recognize all the expansions of the Arnoux-Rauzy-Poincaré
continued fraction algorithm. It is clearly not deterministic. A minimized and
deterministic version of it is the automaton $\mathcal{G}$ shown in Figure 2
where the alphabet considered is
$\mathcal{S}=\\{\alpha_{1},\alpha_{2},\alpha_{3},\pi_{31},\pi_{13},\pi_{23},\pi_{32},\pi_{12},\pi_{21}\\}$
instead of $\widetilde{\Sigma}$. In fact, amongst all the elements of
$2^{\widetilde{\mathcal{P}}}$ considered in the determinization process, only
the states in the set
$Q=\\{\Delta,H_{12},H_{13},H_{21},H_{23},H_{31},H_{32}\\}$ survive the
minimization. ∎
###### Proof of Lemma 38.
(i) Let
$\overrightarrow{z}=(z_{1},z_{2},z_{3})=\overrightarrow{\alpha_{k}(v)}$. Let
$\overrightarrow{z^{\prime}}=(z^{\prime}_{1},z^{\prime}_{2},z^{\prime}_{3})=\overrightarrow{\alpha_{k}(v^{\prime})}$.
We have
$\left\\{\begin{array}[]{l}z_{i}=v_{i}\\\ z_{j}=v_{j}\\\
z_{k}=v_{i}+v_{j}+v_{k}\end{array}\right.\quad\text{and}\quad\left\\{\begin{array}[]{l}z^{\prime}_{i}=v^{\prime}_{i}\\\
z^{\prime}_{j}=v^{\prime}_{j}\\\
z^{\prime}_{k}=v^{\prime}_{i}+v^{\prime}_{j}+v^{\prime}_{k}.\end{array}\right.$
Then
$z_{k}+2=v_{i}+v_{j}+v_{k}+2=(v_{i}+1)+(v_{j}+1)+(v_{k}+1)-1\leq
v^{\prime}_{i}+v^{\prime}_{j}+v^{\prime}_{k}-1<z^{\prime}_{k}$
and $\overrightarrow{z}+2e_{k}<\overrightarrow{z^{\prime}}$.
(ii) Let
$\overrightarrow{z}=(z_{1},z_{2},z_{3})=\overrightarrow{\pi_{jk}(v)}$. Let
$\overrightarrow{z^{\prime}}=(z^{\prime}_{1},z^{\prime}_{2},z^{\prime}_{3})=\overrightarrow{\pi_{jk}(v^{\prime})}$.
We have
$\left\\{\begin{array}[]{l}z_{i}=v_{i}\\\ z_{j}=v_{i}+v_{j}\\\
z_{k}=v_{i}+v_{j}+v_{k}\end{array}\right.\quad\text{and}\quad\left\\{\begin{array}[]{l}z^{\prime}_{i}=v^{\prime}_{i}\\\
z^{\prime}_{j}=v^{\prime}_{i}+v^{\prime}_{j}\\\
z^{\prime}_{k}=v^{\prime}_{i}+v^{\prime}_{j}+v^{\prime}_{k}\end{array}\right.$
As above we have $z_{k}+2<z^{\prime}_{k}$. Moreover,
$z_{j}+1=v_{i}+v_{j}+1=(v_{i}+1)+(v_{j}+1)-1\leq
v^{\prime}_{i}+v^{\prime}_{j}-1<z^{\prime}_{j}.$
Then $\overrightarrow{z}+e_{j}+2e_{k}<\overrightarrow{z^{\prime}}$. ∎
###### Proof of Lemma 39.
(i) Under Arnoux-Rauzy substitution, the extended image of $v$ and
$v^{\prime}$ are uniquely determined: $w=k\alpha_{k}(v)$ and
$w^{\prime}=k\alpha_{k}(v^{\prime})$. From Lemma 38,
$\overrightarrow{\alpha_{k}(v)}<\overrightarrow{\alpha_{k}(v^{\prime})}$. Then
$\overrightarrow{w}=\overrightarrow{\alpha_{k}(v)}+e_{k}<\overrightarrow{\alpha_{k}(v^{\prime})}+e_{k}=\overrightarrow{w^{\prime}}.$
(ii) The proof is divided into four disjoint cases depending on the values of
$w\in\\{jk\pi_{jk}(v),k\pi_{jk}(v)\\}$ and
$w^{\prime}\in\\{jk\pi_{jk}(v^{\prime}),k\pi_{jk}(v^{\prime})\\}$. The proof
relies on the fact that
$\overrightarrow{\pi_{jk}(v)}<\overrightarrow{\pi_{jk}(v^{\prime})}$ but only
the fourth case makes a stronger use of Lemma 38, i.e.,
$\overrightarrow{\pi_{jk}(v)}+e_{j}<\overrightarrow{\pi_{jk}(v^{\prime})}$.
(ii.i) If $w=jk\pi_{jk}(v)$ and $w^{\prime}=jk\pi_{jk}(v^{\prime})$, then
$\overrightarrow{w}=\overrightarrow{\pi_{jk}(v)}+e_{j}+e_{k}<\overrightarrow{\pi_{jk}(v^{\prime})}+e_{j}+e_{k}=\overrightarrow{w^{\prime}}.$
(ii.ii) If $w=k\pi_{jk}(v)$ and $w^{\prime}=k\pi_{jk}(v^{\prime})$, then
$\overrightarrow{w}=\overrightarrow{\pi_{jk}(v)}+e_{k}<\overrightarrow{\pi_{jk}(v^{\prime})}+e_{k}=\overrightarrow{w^{\prime}}.$
(ii.iii) If $w=k\pi_{jk}(v)$ and $w^{\prime}=jk\pi_{jk}(v^{\prime})$, then
$\overrightarrow{w}=\overrightarrow{\pi_{jk}(v)}+e_{k}<\overrightarrow{\pi_{jk}(v^{\prime})}+e_{j}+e_{k}=\overrightarrow{w^{\prime}}.$
(ii.iv) If $w=jk\pi_{jk}(v)$ and $w^{\prime}=k\pi_{jk}(v^{\prime})$, then
$\overrightarrow{w}=\overrightarrow{\pi_{jk}(v)}+e_{j}+e_{k}<\overrightarrow{\pi_{jk}(v^{\prime})}+e_{k}=\overrightarrow{w^{\prime}}.\qed$
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|
arxiv-papers
| 2014-04-16T10:17:55 |
2024-09-04T02:50:01.260755
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Val\\'erie Berth\\'e and S\\'ebastien Labb\\'e",
"submitter": "S\\'ebastien Labb\\'e",
"url": "https://arxiv.org/abs/1404.4189"
}
|
1404.4258
|
An Analysis of State-Relevance Weights and Sampling Distributions
on $L_1$-Regularized Approximate Linear Programming Approximation Accuracy
Gavin [email protected]
Connor Geer
David Piekut
United States Naval Academy,
572M Holloway Rd., Stop 9F, Annapolis, MD 21402-5002
Reinforcement Learning, Feature Selection, L1 Regularization,
Value Function Approximation
Recent interest in the use of $L_1$ regularization in the use of value
function approximation includes Petrik et al.'s introduction of
$L_1$-Regularized Approximate Linear Programming (RALP). RALP is unique among
$L_1$-regularized approaches in that it approximates the optimal value
function using off-policy samples. Additionally, it produces policies which
outperform those of previous methods, such as LSPI. RALP's value function
approximation quality is affected heavily by the choice of state-relevance
weights in the objective function of the linear program, and by the
distribution from which samples are drawn; however, there has been no
discussion of these considerations in the previous literature. In this
paper, we discuss and explain the effects of choices in the state-relevance
weights and sampling distribution on approximation quality, using both
theoretical and experimental illustrations. The results provide insight not
only onto these effects, but also provide intuition into the types of MDPs
which are especially well suited for approximation with RALP.
§ INTRODUCTION
In recent years, the Reinforcement Learning community has paid considerable
attention to creating value function approximation approaches which perform
automated feature selection while approximating a value
function [Kolter & Ng, 2009, Johns et al., 2010, Mahadevan & Liu, 2012, Liu et al., 2012]. This approach frees
researchers from hand-selecting and -tuning feature sets, while greatly
increasing approximation accuracy. One of these approaches, $L_1$-Regularized
Approximate Linear Programming (RALP) [Petrik et al., 2010, Taylor & Parr, 2012] is unique in that
it results in an approximation of the optimal value function and makes use of
off-policy samples.
However, important aspects of RALP have not been fully explored or explained.
In particular, the objective function of the linear program offers an
opportunity to create a better approximation of the value function in some
regions of the state space at the expense of others. Optimal policies of many
realistic reinforcement learning problems heavily traffic some parts of the
state space while avoiding others, making an understanding of this flexibility
Therefore, in Section <ref>, we derive a new error bound for the RALP
approximation, which is tighter than those presented by Petrik et
al. icml10 and provides new insight into the result of changing the
state-relevance weights in the objective function. In addition, this bound
provides an insight into the types of MDPs particularly well suited to the
RALP approach. Finally, this section provides evidence that rather than
weighting all states equally, as was done by Petrik et al. icml10 and
Taylor and Parr taylor12, states should instead be weighted in
proportion to the stationary distribution under the optimal policy.
Additionally, if the sampling distribution is not uniform across the state
space, the approximation can be heavily affected. In realistic domains,
sampling is rarely uniform. Therefore, for RALP to be appropriate for these
domains, it is important to address this lack of analysis and understand how
the approximation is likely to be altered due to the problem's available
sampling scheme.
Section <ref> then discusses the impact on the approximation if
learning is performed using samples drawn other than uniformly from the state
space. We demonstrate that sampling from a distribution acts as a de
facto alteration of the objective function. We also discuss the effects of
sampling distributions in light of the bounds from Section <ref>.
The intuition provided by Sections <ref> and <ref> are then
demonstrated experimentally in Section <ref>. Using a simple,
easily visualized domain, we demonstrate the effect of the various parameters
on approximation quality.
§ NOTATION AND PROBLEM STATEMENT
In this section, we formally define Markov decision processes and linear value
function approximation. A Markov decision process (MDP) is a tuple
$(\sS,\sA,P,R,\gamma)$, where $\sS$ is the measurable, possibly infinite set
of states, and $\sA$ is the finite set of actions. $P: \sS \times \sS \times
\sA \mapsto [0,1]$ is the transition function, where $P(s'|s,a)$ represents
the probability of transitioning from state $s$ to state $s'$, given action
$a$. The function $R: \sS \mapsto \Re$ is the reward function, and $\gamma,$ a
number between 0 and 1, is the discount factor, representing the comparative
desire for reward at the current time step to the desire for reward at the
next time step.
We are concerned with finding a value function $V$ that maps each state
$s\in\sS$ to the expected total $\gamma$-discounted reward for the
process. Value functions can be useful in creating or analyzing a policy
$\pi: \sS \times \sA \rightarrow [0,1]$ such that for all
$s\in\sS$, $\sum_{a\in\sA} \pi(s,a) = 1$. The transition and reward
functions for a given policy are denoted by $P_\pi$ and $R_\pi$. We denote
the Bellman operator for a given policy as $T_\pi,$ and the max
Bellman operator simply as $T$. That is, for some state $s\in\sS$:
\[
T_\pi V(s)=R(s)+\gamma\int_{\sS}P(ds'|s,\pi(s))V(s')
\]
\[
TV(s)=\max_{\pi\in\Pi}T_\pi V(s).
\]
We additionally denote the Bellman operator for selecting a particular action
$a$ as
\[
T_a V(s)=R(s)+\int_{\sS}P(ds'|s,a)V(s').
\]
The optimal value function $V^*$ satisfies $T V^*(s) = V^*(s)$ for all
For simplicity, in this paper we will assume no noise exists in the MDP;
results can be easily extended to noisy domains using Taylor and
Parr's taylor12 approach of local smoothing.
Sets of samples, therefore, are defined as $\Sigma \subseteq \{(s,a,r,s' |
s,s'\in\sS, a\in\sA\}$, where $s'$ is the state the agent arrived at given
that it started in state $s$ and took action $a$, and $r=R(s)$. An individual
sample in the set $\Sigma$ will be denoted $\sigma$, and an element of a
sample $\sigma$ will be denoted with superscripts; that is, the $s$ component
of a $\sigma=(s,a,r,s')$ sample will be denoted $\sigma^s$.
We focus on linear value function approximation for discounted
infinite-horizon problems, in which the value function is represented as a
linear combination of possibly nonlinear basis functions (vectors). For each
state $s$, we define a vector $\Phi(s)$ of features. The rows of the basis
matrix $\Phi$ correspond to $\Phi(s)$, and the approximation space is
generated by the columns of the matrix. That is, the basis matrix $\Phi$, and
the approximate value function $\hat V$ are represented as:
\[
\Phi = \begin{pmatrix} - & \Phi(s_1) & - \\ &
\vdots & \end{pmatrix} \qquad \hat V = \Phi w .
\]
This form of linear representation allows for the calculation of an
approximate value function in a lower-dimensional space, which provides
significant computational benefits over using a complete basis; if the number
of features is small and the environment is noisy, this framework can also
guard against overfitting any noise in the samples.
If we define $\Phi$ to be overcomplete, with potentially far more features
than sampled states, then to receive the above benefits we must perform
feature selection. In this process, a few features are chosen from the set, the
span of which will represent the available linear approximation space. We can
use $L_1$ regularization to calculate a sparse $w$, in which nearly all
features receive a weight of 0, thereby performing automated feature selection.
§ PREVIOUS WORK
RALP was introduced by Petrik et al. icml10 to extend the
capabilities of the linear programming approach to value function
approximation [d'Epenoux, 1963, Schweitzer & Seidmann, 1985, de Farias & Van Roy, 2003]. Given a set of
samples $\Sigma$, the linear program is defined as follows:
\begin{equation}
\label{eqn:ralp}
\begin{array}{rl}
\displaystyle\min_{w} &\rho^T \Phi w\\
\mbox{s.t.} &T_{\sigma^a}\Phi(\sigma^s) w\leq
\Phi(\sigma^s) w ~~\forall \sigma\in\Sigma \\ &\norm{w}_{-1}
\leq \psi,
\end{array}
\end{equation}
where $\rho$ is a distribution, which we call the state-relevance weights, in
keeping with the (unregularized) Approximate Linear Programming (ALP)
terminology of de Farias and Van Roy deFarias03.
$\norm{w}_{-1}$ is the $L_1$ norm of the vector consisting of all weights
excepting the one corresponding to the constant feature.
This final constraint, which contributes $L_1$ regularization, provides
several benefits. First, regularization in general ensures the linear program
is bounded, and produces a smoother value function. Second, $L_1$
regularization in particular produces a sparse solution, producing automated
feature selection from an overcomplete feature set. Finally, the sparsity
results in few of the constraints being active, speeding the search for a
solution by a linear program solver, particularly if constraint generation is
Other techniques have used $L_1$ regularization in similar ways.
LARS-TD [Kolter & Ng, 2009] and LC-MPI [Johns et al., 2010] both approximate the fixed
point of the $L_1$-regularized LSTD problem. Mahadevan and
Liu mahadevan12 introduced the use of mirror descent, which has a
computation complexity which allows it to be better suited than many other
approaches for online reinforcement learning problems. These above approaches
are most reliable when samples are collected on-policy. Liu et
al. liu12 introduced RO-TD, which converges to an approximation of
the value function of a given policy, even when trained on off-policy samples.
In contrast to the above approaches, RALP provides an approximation to the
value function of the optimal policy, even when samples are drawn from
non-optimal or random policies. Approximations produced by RALP have bounded
error, and have performed well experimentally in comparison to other
approaches. Finally, in noisy domains, the well-known weakness of linear
programming approaches to value function approximation can be mitigated or
eliminated using local smoothing [Taylor & Parr, 2012].
This previous work on RALP has largely ignored the state-relevance weights
$\rho$ in the objective function, setting $\rho=\bone$ without discussion.
However, a change in the objective function would obviously affect the
solution of the linear program. In certain practical situations this would be
useful to understand.
For example, consider the task of calculating a value function for an aircraft
in flight. The space of possible flight attitudes, velocities, etc. is very
large. However, the percentage of this space trafficked in non-catastrophic
flight is small; it is likely worthwhile to improve the approximation quality
in this relevant portion, at the expense of accuracy in the remainder of the
Some previous results exist regarding the effects of changing the
state-relevance weights in the closely-related ALP [de Farias & Van Roy, 2003].
However, the assumptions and types of appropriate problems are very different
between ALP and RALP, making these previous results insufficient. First, ALP
assumes a sample is drawn from every state-action pair, an assumption which is
not required for RALP. This means it was not necessary with ALP to consider
the behavior of the approximation between samples or in a continuous space.
Furthermore, it was not necessary to consider the effects of sampling
distributions at all. This assumption was later weakened by a followup
paper [de Farias & Van Roy, 2004], but not to a degree necessary for large or continuous
problems, particularly when large numbers of samples are not available. The
second difference is ALP is unregularized, simplifying the definition of the
feasible space of the linear program. Despite these differences, these
previous results will serve as a useful guide.
Additionally, previous work does not cover the effects of sampling schemes on
RALP. If states are sampled heavily in one portion of the state space, the
linear program will choose a solution which tightens constraints in that
portion of the space over others. In realistic settings, it can be difficult
to sample uniformly, making it especially important to understand the effect
of other sampling distributions on the resulting approximate value function.
The remainder of this document fills in these gaps in the previous work.
§ STATE-RELEVANCE WEIGHTS
The theoretical results presented on RALP in the literature thus far offer no
insights into the behavior of the approximation as the state-relevance weights
are altered. Therefore, it is necessary to derive new bounds for RALP which
contain $\rho$ to understand its effects.
The approach we take follows the example of a proof introduced by de Farias
and Van Roy deFarias03 to bound the ALP approximation, but we extend
it match the weaker assumptions of RALP, along with the requirement that the
weights be $\lone$ regularized.
We begin by defining the relevant notation. In the following definitions, we
will use $\Re_+$ to refer to the set of non-negative real numbers.
We introduce an operator $H$, defined by
\[
\]
for all $L: \sS\rightarrow\Re_+$.
Therefore, $(HL)(s)$ represents the expected value of $L$ of the next
state if actions are chosen to maximize $L$.
A non-negative function $L:\sS\rightarrow \Re_+$ is a Lyapunov
function if there exists a subset of states $\sB$ and a $\beta_L<1$
such that for all $s\in \sS\setminus\sB$, $\gamma(HL)(s)\leq \beta_L
For an example of a Lyapunov function defined over a MDP, consider the simple
case of a random walk along the non-negative number line, so that
$s\in\ints_+$. Assume a single action, in which the probability
\[
\]
\[
$L(s)=s$ and $\sB=\{0\}$, then $L$ is a valid Lyapunov function, because
$L(s)$ is expected to decrease for all $s\neq 0$.
Lyapunov functions are often used to prove stability of Markov processes.
Definition <ref> differs from the definition commonly used for
stability analysis in a few ways. First, in stability analysis, it is
required that $\sS$ be countable, and that $\sB$ be finite. We have made
neither of these assumptions, though our bounds will be tightest when $\sB$ is
small. The second difference is we have added a multiplicative term of
$\gamma$. Because of these differences, a Lyapunov function as defined in
Definition <ref> may not strictly evidence stability.
Besides stability analysis, Lyapunov functions have also previously appeared
in Reinforcement Learning literature, though in different contexts from our
application [Perkins & Barto, 2003, Rohanimanesh et al., 2004].
In the remainder of this section, we will occasionally refer to the weighted
max-norm, where for a vector $U$ and a function $F$,
$\norm{U}_{\infty,F}=\max_i \abs{U(s_i)\cdot F(s_i)}$, and the weighted $L_1$
norm, where $\norm{U}_{1,F}=\sum_i \abs{U(s_i)\cdot F(s_i)}$.
We will start with the following Lemma. To conserve space, and because the
proof is similar to one presented by de Farias and Van
Roy deFarias03, we reserve the proof for Appendix <ref>.
Assume samples have been drawn from every possible state-action pair. Let
$\sW=\{w:\norm{w}_{-1}\leq\psi\},$ and let $w^*=\min_{w\in\sW}\norm{V^*-\Phi
w}_\infty$. Additionally, for a given Lyapunov function $\Phi w_L$, let
\[
\bar w = w^*+\lnorm{V^*-\Phi w^*}\betafrac w_L.
\]
If a Lyapunov function $\Phi w_L$ is constructed such that $\bar w\in\sW$,
\[
\norm{V^*-\Phi \tilde w}_{1,\rho}\leq \frac{2\rho^T\Phi
w_L}{1-\beta_{\Phi w_L}} \min_{w\in\sW} \lnorm{V^*-\Phi
\]
We note that proving the existence of a Lyapunov function as required in the
above lemma is trivial. First we construct a weight vector $w_L$ with all
zeros but for a positive weight corresponding to the bias feature; this
results in $\Phi w_L$ being a valid Lyapunov function. Second, we note that
in this case $\norm{\bar w}_{-1}=\norm{w^*}_{-1}$, meeting the requirement
that $\bar w\in\sW$.
We must now remove the assumption that a sample exists for every state-action
pair. To enable us to bound the behavior of the value function between
samples, we make the following assumption, similar to the sufficient sampling
assumption made by Petrik et al. icml10:
Assume sufficient sampling, that is, for all $s\in\sS$ and $a\in\sA$, there
exists a $\sigma\in\Sigma$ such that $\sigma^a=a$ and:
\begin{align*}
\norm{\phi(\sigma^s)-\phi(s)}_\infty\leq&\delta_\phi\\
\norm{R(\sigma^s)-R(s)}_\infty\leq&\delta_R\\
\norm{p(s'|\sigma^s,a)-p(s'|s,a)}_\infty\leq&\delta_P~\forall s'\in\sS
\end{align*}
This assumption is not unrealistic. For example, if the reward function,
basis functions, and transition functions are Lipschitz continuous, then
appropriate values of $\delta_\phi, \delta_R,$ and $\delta_P$ are easily
calculated given the greatest distance between any point in the state space
and a sampled point.
We describe the maximum difference between the RALP solution and the true
solution by using the limits from Assumption <ref> to
demonstrate the following Lemma:
Let $M_1$ be an MDP with optimal value function $V^*_1$, and let $\Sigma$ be
an incomplete set of samples drawn from $M_1$ such that not all state-action
pairs are sampled, but Assumption <ref> is fulfilled.
Therefore, the RALP for $M_1$ has the constraint
$T_{\sigma^a}\Phi(\sigma^s)w\leq\Phi(\sigma^s)w$ for all $\sigma\in\Sigma$,
and the bounded $L_1$ constraint, but is missing all other possible RALP
There exists an MDP $M_2$ with an optimal value function $V^*_2$, identical
in every way to $M_1$ but for the reward function, such that the RALP
solution with no missing constraints is equal to the RALP solution
constructed on $\Sigma$, and $\norm{V^*_1-V^*_2}_{\infty}\leq
\frac{2(\delta_\phi\psi+\delta_R+\delta_P\psi)}{1-\gamma}$.
Proof Sketch: We first show that if $R_1$ and $R_2$ are the
respective reward functions of $M_1$ and $M_2$,
\[
\norm{R_1-R_2}_\infty\leq 2(\delta_\phi\psi + \delta_R+\delta_P\psi).
\]
We then show that if
$\norm{R_1-R_2}_\infty\leq \delta$,
\[
\norm{V^*_1-V^*_2}_{\infty}\leq\frac{\delta}{1-\gamma}.
\]
We leave the details of the proof for Appendix <ref>.
We are now prepared to present the first result of this paper.
Let $\Phi w_L$ be a Lyapunov function as required by Lemma
<ref>. Define $\Sigma$, MDPs $M_1$ and $M_2$ and their
respective optimal value functions $V^*_1$ and $V^*_2$ as in Lemma
<ref>. Define $\epsilon_p=\delta_\phi\psi+\delta_R+\delta_P\psi$.
Let $\tilde w$ be the RALP solution to $M_1$.
\[
\norm{V_1^*-\Phi \tilde w}_{1,\rho}\leq \frac{2\rho^T\Phi
w_L}{1-\beta_{\Phi w_L}} \min_{w\in\sW} \lnorm{V^*_2-\Phi
\]
Because $\tilde w$ is an optimal solution given all samples from $M_2$, Lemmas
<ref> and <ref> allow us
\[
\norm{V_2^*-\Phi \tilde w}_{1,\rho}\leq \frac{2\rho^T\Phi
w_L}{1-\beta_{\Phi w_L}} \min_{w\in\sW} \lnorm{V_2^*-\Phi
\]
Additionally, Lemma <ref> gave us
Because $\rho$ is a probability distribution,
\[
\norm{V_1^*-V_2^*}_{1,\rho}\leq\norm{V_1^*-V_2^*}_\infty
\leq\frac{2\epsilon_p}{1-\gamma}.
\]
Due to the triangle inequality, Theorem <ref> follows.
§.§ Discussion
This bound is not only tighter than those presented in previous literature,
but also allows us to analyze the RALP approximation quality in new ways.
First, we can observe which Lyapunov functions would result in a better
approximation, and discuss the characteristics of MDPs which allow for those
Lyapunov functions, and therefore lend themselves particularly well to value
function approximation by RALP. Second, as $\rho$ now appears in our bound,
the bound provides a way of relating our choice of $\rho$ to approximation
quality, allowing for more intuitive and successful parameter assignments. We
address these in turn.
The Lyapunov function $\Phi w_L$ appears in the first term of our bound in
three places, namely the dot product with $\rho$, the definition of
$\beta_{\Phi w_L}$, and in the norm defining the “optimal" $w$ to which we
compare our approximation. We first note that the bound becomes smaller as
$\beta_{\Phi w_L}$ decreases. This suggests that the more stable the MDP, the
better RALP can approximate the value function.
This interpretation of Theorem <ref> leads to other intuitive
explanations. Consider an MDP with only $L(s)=1~\forall s\in\sS$ as a
Lyapunov function. Now assume two nearby samples, one where a “good" action
is taken, in the direction of positive reward, and another where a “bad"
action is taken, in the direction of negative reward. When the linear program
is solved, due to the proximity of the two samples, the constraint
corresponding to the “bad" sample is nearly certain to be loose, and may as
well be removed. However, if the MDP is highly stable, then these two
extremely different samples would be unlikely, and both constraints are
candidates for being tight. Therefore, more samples from an MDP with a small
$\beta_L$ are likely to be involved in defining the feasible space,
potentially resulting in an improved approximation.
The appearance of the Lyapunov function in the norm of the bound indicates the
bound is tighter when the feature space $\Phi$ allows for a close
approximation in areas where the Lyapunov function is small. Bertsimas et
al. bertsimas98 demonstrated that a small Lyapunov function value
correlates in expectation with a higher probability in the stationary
distribution of a Markov chain. This is particularly interesting when
considering the appearance of the dot product between the Lyapunov function
and $\rho$. This dot product makes it apparent that the approximation
improves when $\rho$ is large only where $\Phi w_L$ is small. This provides
evidence that the stationary distribution of the MDP under the optimal policy
may be an advantageous setting for $\rho.$ This evidence meshes well with the
intuition that greater accuracy is most useful in frequently-visited
§ SAMPLING DISTRIBUTION
Imagine an MDP with a small finite state space and a single action. Ideal
sampling would provide a single sample from each state, giving us an objective
function of $\sum_{s\in\sS}\rho(s)\Phi(s)w$. However, if sampling from a
distribution across the state space, this ideal situation would be unlikely;
some states would go unsampled, while others would be sampled multiple times.
Because the objective function is defined on samples, this means states that
were sampled multiple times would appear in the objective function multiple
times, causing the linear program to tighten constraints at those states at
the expense of accuracy in other states.
Of course, a similar scenario occurs in infinite state spaces as well.
Multiple states near to each other may be sampled, while other regions have
very few samples; this encourages the linear program to choose features and
an approximate value function which tightens constraints in heavily-sampled
regions at the expense of sparsely-sampled regions. In this section we
discuss the effects of sampling from an arbitrary distribution over the state
space $\mu$ and ways this can help the researcher understand how to design
sampling methods.
Let $\Sigma_1$ and $\Sigma_\mu$ be sample sets of equal cardinality $N$
drawn from the state space from the uniform distribution and an arbitrary
distribution $\mu$, respectively. Let $\Phi_1$ and $\Phi_\mu$ be the
feature matrices defined over the states of sets $\Sigma_1$ and
$\Sigma_\mu$. For any weight vector $w$,
\[
\expect{\mu\Phi_1 w}=\expect{\bone\Phi_\mu w}.
\]
This observation is easy to support; both expectations equal $\int_\sS
\mu(s)\Phi(s) w~ds$.
This means that for any $w$, sampling from a non-uniform distribution $\mu$
provides an equivalent objective function in expectation to sampling from a
uniform distribution but setting the state-relevance weights equal to $\mu$.
We note this is not equivalent to expecting the same approximate value
function as the constraints remain different; this makes the effect of
altering the sampling distribution greater than that of altering the objective
function alone.
Additionally, the bound presented in Theorem <ref> offers an
interpretation of results from sampling from a distribution. As sampling
becomes less uniform, $\epsilon_p$ likely increases, due to the existence of
larger unsampled regions. For example, this would happen in the
Lipschitz-continuous case for fulfilling Assumption <ref>.
However, for a sampling distribution which is dense where the Lyapunov value
is small, then the total Lyapunov value in the numerator of the first addend
is small, as well. Therefore, it may be that the ideal sampling distribution
is one which is dense where the Lyapunov value is low, but still provides
sufficient coverage for $\epsilon_p$ to be small.
§ EXPERIMENTAL RESULTS
In this section, we demonstrate experimentally the conclusions drawn in the
previous sections. Previous literature has already demonstrated RALP's
effectiveness in common benchmark domains; the purpose of this section
therefore, is to clearly illustrate the conclusions of the previous sections.
In a simple, easily visualized domain, we make a series of comparisons.
First, we compare the approximation accuracy of sampling from a domain with a
stable Lyapunov function to the accuracy resulting from sampling from a domain
without such a function. Next, we compare the accuracy of the approximation
resulting from sampling uniformly to the accuracy of the approximation
resulting from sampling from two different nonuniform distributions. Finally,
we compare the approximation accuracy of calculating an approximation with
$\rho=\bone$ to the approximation accuracy of calculating an approximation
when $\rho$ is nonuniform. This is demonstrated using two different,
nonuniform distributions.
The results were obtained by drawing samples and calculating an approximation
500 times for each compared approach; the error $\abs{V^*-\Phi w}$
was then calculated, and averaged across all 500 trials. Finally, we
calculate and display the difference between the average errors from the two
approaches. So, if $\hat V_i^A$ is the approximation from the $i$-th run on
approach A, then when comparing two approaches, A and B, a point on the graphs
of Figure <ref> equals
\[
\sum_{i=1}^{500}\frac{\abs{V^*(s)-\hat
V_i^A(s)}}{500}-\sum_{i=1}^{500}\frac{\abs{V^*(s)-\hat V_i^B(s)}}{500}.
\]
§.§ Domain
Room Domain
All of the experiments were run on a domain defined by a 25 by 25 grid world.
This world included four reward regions in the corners of the grid, each of
which consisted of 9 states, as can be seen in Figure <ref>.
Two reward regions (colored gold) had a reward of 1, while the others had a
reward of -1 (colored red). The remaining states had a reward of 0. Actions
were to move one square in any of the four directions, unless constrained by a
wall, in which case that action would result in no movement. The discount
factor $\gamma$ was set to 0.95. For all trials, the feature set consisted of
symmetric Gaussian features centered around each $\sigma^s$ with variances of
2, 5, 10, 15, 25, 50, and 75, plus the bias feature, resulting in $9n+1$
features for $n$ samples. The optimal value function can be seen in Figure
This value function is an easy one for RALP to approximate with proper
settings of the regularization parameter and sufficient sampling. The number
of samples and choices of the regularization parameter $\psi$ were therefore
chosen to illustrate the differences between the results of the compared
methods, not to optimize performance.
[Average error from sampling from stable domain - Average error
from sampling from unstable domain]
[Average error from sampling from $\bone$ - Average error from
sampling from $\zeta$]
[Average error from $\rho=\bone$ - Average error from
[Average error from sampling from $\bone$ - Average error from
sampling from $(\bone-\zeta)$]
[Average error from $\rho=\bone$ - Average error from
Difference in Average Error
§.§ Lyapunov Stable Domain
First, we demonstrate the improvement in RALP's approximation when the domain
has a stable Lyapunov function. A stable Lyapunov function can be created by
forcing the actor into a defined area in the state space. In order to keep
the representational difficulty of the optimal value functions the same, we
created a Lyapunov function by eliminating actions which move the actor
further from the nearest positive reward. This preserves the optimal policy
of the unaltered domain, keeping the optimal value functions identical, making
approximation accuracy a fair comparison.
In the domain without a stable Lyapunov function, the actor was free to move
in the state space based on a random choice among the four actions. However,
in the domain with a stable Lyapunov function, the actor was only allowed to
move in the two directions which would not move it further from the nearest
goal. We note that not all remaining actions are optimal, so sampling still
includes off-policy samples.
This creates a Lyapunov function where $L(s)$ equals the Manhattan distance
from $s$ to the nearest of state (1,1) or (25,25), and
$\sB=\{(1,1),(25,25)\}$. In each trial, we uniformly sampled 20 samples. The
regularization parameter $\psi$ was set to 0.2 and $\rho$ was $\bone$.
The result of subtracting the average errors of the approximation from the
domain with a Lyapunov function from the domain without a Lyapunov function
can be seen in Figure <ref>. Therefore, positive values
indicate higher error from the domain without a Lyapunov function.
The results show that an approximation learned from samples drawn from a
stable domain is more accurate than an approximation learned from a less
stable domain everywhere except for where $s$ is near equal distance from the
two goal states. This reinforces the intuition from Subsection
<ref> that samples drawn from a
stable domain are more effective than those that are not, particularly near
the most heavily-visited regions.
§.§ Sampling from a Nonuniform Distribution
Next we illustrate the change in approximation accuracy when sampling from a
nonuniform distribution $\mu$. Section <ref> presents evidence that
a distribution which is most dense where $L(s)$ is smallest may be
advantageous. To create such a distribution, an agent was started at a random
state, and was allowed to take the optimal policy for 25 steps. This was done
10,000 times, and the number of visits to each state was tabulated and
normalized. This defined our distribution, which was heaviest on the edges
and reward corners, and otherwise slightly increasing with increasing
proximity to the positive reward regions. We will refer to this distribution
as $\zeta$.
For these trials, 20 samples were drawn per run from the domain with the
stable Lyapunov function as discussed in Subsection <ref>.
The regularization parameter $\psi$ was set to 1.5, and $\rho=\bone$. The
average error from the 500 uniformly sampled runs was subtracted from the
average error from the 500 runs with $\mu=\zeta$; the result can be seen
in Figure <ref>. Because the error from the nonuniform sampling
was subtracted from the error from uniform sampling, the positive difference
indicates the results from sampling from $\zeta$ were superior. The results show the distribution met the goal from Section <ref>;
sampling was varied enough to keep $\epsilon_p$ low, while dense enough in the
areas where $L(s)$ was small.
We then subtracted $\zeta$ from $\bone$ and normalized, making a distribution
we will refer to as $\bone-\zeta$, which was largest where an agent was least
likely to traverse in the stable domain. We subtracted the average error from
sampling from $\mu=\bone-\zeta$ from the average error from sampling
uniformly, to produce Figure <ref>. A positive value would
indicate larger error from the approximations on uniformly sampled states.
However, through the entirety of the state space, and particularly in the most
trafficked areas, sampling from $\bone-\zeta$ gave us an inferior result.
This provides evidence for the conclusions of Section <ref> that a
sampling distribution which is densest in the areas where the Lyapunov
function is smallest would produce the best approximations.
§.§ Changing the State-Relevance Weights
Lastly, we illustrate the effect on the approximation of changing the
state-relevance weights. 200 samples were drawn uniformly from the state
space; whereas the effects from the previous two experiments are most
pronounced when samples are sparse, the effects from altering $\rho$ are most
pronounced when a number of constraints can be tightened in a given region.
We would prefer to set $\rho$ to the stationary distribution; however, because
the domain is not recurrent, this is not an option. However, the distribution
$\zeta$ created for sampling in Subsection <ref> is large
where the Lyapunov function is small, making it a reasonable replacement. In
one set of trials, $\rho(s)$ was set to the value of this distribution at that
state; in the other, $\rho=\bone$. $\psi$ was set to 4. Average error from
the approximation resulting from a nonuniform $\rho$ was subtracted from
average error from the approximation resulting from a uniform $\rho$.
Therefore, a positive value indicates a better approximation from the
nonuniform $\rho.$ Sampling was done uniformly from the stable domain.
Figure <ref> shows that nearly the entire state space was more
accurate with a nonuniform $\rho$, except for where $\zeta(s)\approx 0$. The
difference is small because with a large number of samples, both
approximations were quite accurate.
In addition, we compared the uniform $\rho$ approximation to an approximation
using $\rho=\bone-\zeta$, which is large where the Lyapunov value is large,
resulting in an increased dot product $\rho\Phi w$ in Theorem <ref>.
Again, we subtracted the error of the approximation resulting from
$\rho=\zeta-1$ from the error of the approximation resulting from
$\rho=\bone$, producing Figure <ref>. A positive value
indicates the nonuniform $\rho$ approximated that state better than did using
a uniform $\rho$. However, there are few positive values as the use of
$\rho=\bone-\zeta$ resulted in a dramatically inferior approximation, particularly
in areas where $\rho$ was small.
From both figures, it is clear that a higher $\rho$ value in a given portion
of the state space resulted in an improved approximation, particularly if
designed with Theorem <ref> in mind.
§ CONCLUSION
The experimental success of RALP in previous literature, along with its
easily-fulfilled assumptions, suggests promise for its application to
real-life, complicated, and complex domains. Despite this promise, and
despite the evidence that the effects are dramatic, no theory had been
produced to analyze changes in the approximation quality given changes to the
objective function parameter $\rho,$ or due to differences in sampling
strategies. These considerations are essential to the use of RALP in the real
world; it is rarely possible to sample uniformly, and the importance of
accuracy across the state space is rarely consistent.
In this paper, we demonstrate the importance of understanding these ideas, and
produce a bound on the approximation error of RALP which is tighter and more
informative than previous bounds. This bound provides intuition into the
quality of the RALP approximation as a function of state-relevance weights and
sampling distributions. In addition, we demonstrated that the quality of a
RALP approximation is particularly good when the domain is stable and has a
Lyapunov function with a small $\beta_L$.
Future work remains, particularly in the area of solving the linear program
quickly in the presence of large amount of data. Though convex optimization
solvers are considered “fast," with large amounts of data, the memory and
time requirements may be too large for realistic use. Fortunately, it may be
that the structure of the problem, the small percentage of tight constraints,
and small percentage of active features will avail itself to faster, but
equivalent, approaches.
§ ACKNOWLEDGEMENTS
Thank you to the anonymous reviewers for their help in improving this paper.
Additionally, we are grateful for support from the Naval Research Laboratory
Information Management & Decision Architecture Branch (Code 5580), as well as
financial support by the Office of Naval Research, grant numbers
N001613WX20992 and N0001414WX20507.
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§ PROOF OF LEMMA 1
As stated in Section 4, this proof is very similar to Theorem 3 by de Farias
and van Roy (2003), but we include it for clarity nonetheless. The Lemma
constructs a point in the feasible space of the linear program which provides
an approximation with bounded error, and then shows the point chosen by RALP
must be no further than that constructed point. The proof first requires a
series of additional Lemmas.
For any functions $V$ and $\bar V$,
\[
\abs{T\bar V-TV}\leq\gamma \max_\pi P_\pi \abs{\bar V-V}.
\]
For any $V$ and $\bar V$,
\begin{align*}
T\bar V - TV =& \max_\pi(R+\gamma P_\pi \bar V)-\max_\pi(R+\gamma P_\pi V)\\
=&R+\gamma P_{\pi_{\bar V}} \bar V-R-\gamma P_{\pi_V} V \\
\leq&\gamma \max_\pi P_\pi (\bar V - V)\\
\leq&\gamma \max_\pi P_\pi \abs{\bar V - V},
\end{align*}
where $\pi_V$ and $\pi_{\bar V}$ represent the greedy policies with respect to
value functions $V$ and $\bar V$. By reversing the terms, we can show
$TV-T\bar V\leq\gamma \max_\pi P_\pi \abs{\bar V - V}$, leading to our result.
For any vector $L$ with positive components and any vector $V$,
\[
TV\leq V+(\gamma HL+L)\|V-V^*\|_{\infty,\frac{1}{L}}.
\]
Note that
\[
\abs{V^*(s)-V(s)}\leq\|V-V^*\|_{\infty,\frac{1}{L}}V(s).
\]
Because of Lemma <ref>,
\begin{align*}
\leq&\gamma\max_\pi\sum_{s'\in\sS}P_\pi(s,s')\abs{V(s')-V^*(s')}\\
\leq&\gamma\|V-V^*\|_{\infty,\frac{1}{L}}\max_{a\in\sA}\sum_{s'\in\sS}P_a(s,s')L(s')\\
\end{align*}
Define $\epsilon=\|V-V^*\|_{\infty,\frac{1}{L}}$.
\begin{align*}
TV(s)\leq& V^*(s)+\gamma\epsilon(HL)(s)\\
\leq& V(s)+\epsilon L(s)+\gamma\epsilon(HL)(s).
\end{align*}
Let $w_L$ be a weight vector such that $\Phi w_L$ is a Lyapunov function, $w$
be an arbitrary weight vector, and
\[
\bar w = w+\lnorm{V^*-\Phi w}\betafrac w_L.
\]
Then, $T\Phi\bar w\leq \Phi \bar w$.
Let $\epsilon=\lnorm{V^*-\Phi w}$. For any state
\begin{align*}
|(T\Phi&\bar w)(s)-(T\Phi w)(s)|\\
=&\abs{\left(T\left[(\Phi w + \epsilon\betafrac \Phi
w_L\right]\right)(s)-(T\Phi w)(s)}\\
\leq&\gamma\max_\pi \sum_{s'\in\sS}P_\pi(s,s')\\
&\cdot\abs{(\Phi w(s') +
\epsilon\betafrac \Phi w_L(s'))-\Phi w(s')}\\
\leq&\gamma\max_\pi \sum_{s'\in\sS}P_\pi(s,s')\epsilon\betafrac(\Phi
=&\gamma\epsilon\betafrac(H\Phi w_L)(s).
\end{align*}
The first line is a replacement of $\bar w$ with its definition, the second is
due to Lemma <ref>, the third is due to the cancellation of the
two $\Phi w$ terms and the fact that because $\Phi w_L$ is a Lyapunov
function, $\betafrac>0$ (note this is true for states in sets $\sB$ and
$\sS\setminus\sB$ as defined in Definition 1), and the final line is due to
Definition 2.
From this, we can conclude
\[
T\Phi\bar w\leq T\Phi w + \gamma \epsilon\betafrac H\Phi w_L.
\]
We can apply Lemma <ref> to get
\[
T\Phi \bar w \leq \Phi w + \epsilon(\gamma H\Phi w_L+\Phi w_L),
\]
and therefore,
\begin{align*}
T\Phi\bar w \leq& \Phi w + \epsilon(\gamma H\Phi w_L+\Phi w_L) +
\gamma\epsilon\betafrac H\Phi w_L\\
=&\Phi w+\epsilon\betafrac \Phi w_L-\epsilon\betafrac \Phi w_L\\
&+ \epsilon(\gamma H\Phi w_L+\Phi w_L)+ \gamma\epsilon\betafrac H\Phi w_L\\
=&\Phi \bar w-\epsilon\betafrac \Phi w_L+\epsilon(\gamma H\Phi w_L+\Phi
&+ \gamma\epsilon\betafrac H\Phi w_L\\
=&\Phi \bar w+ \epsilon(\gamma H\Phi w_L+\Phi w_L)- \epsilon\betafrac(\Phi w_L-\gamma H\Phi w_L)\\
\leq&\Phi \bar w+ \epsilon(\gamma H\Phi w_L+\Phi w_L)-\epsilon(\Phi w_L+\gamma H\Phi w_L)\\
=&\Phi \bar w.
\end{align*}
The penultimate line can be shown given that $\Phi w_L-\gamma H\Phi w_L >0$
\begin{align*}
\frac{2}{1-\beta_{\Phi
w_L}}-1=&\frac{2}{1-\max_{s\in\sS\setminus\sB}((\gamma(H\Phi w_L)(s))/((\Phi
=&\max_{s\in\sS\setminus\sB}\frac{(\Phi w_L)(s)+\gamma(H\Phi w_L)(s)}{(\Phi w_L)(s)-\gamma(H\Phi w_L)(s)}
\end{align*}
Lemma <ref> demonstrates that all constraints in RALP will be
satisfied by $\bar w$, with the exception of the constraint enforcing the
$L_1$ regularization. However, we have required even this constraint to be
satisfied by requiring $\bar w\in\sW$. Therefore, $\bar w$ lies in the
feasible region for RALP.
If every state-action pair is represented with a constraint in the RALP, a
vector $\tilde w$ solves the RALP if and only if it solves
\begin{align*}
\argmin_w \hspace{.3cm}&\norm{V^*-\Phi w}_{1,\rho}\\
s.t. \hspace{.5cm}&T_a\Phi(s) w\leq\Phi(s) w
~~\forall s\in\sS, a\in\sA\\
\hspace{.5cm}&\norm{w_{-1}}_1\leq \psi
\end{align*}
For any policy $\pi$, the Bellman operator $T\pi$ is a contraction in max
norm. If the Bellman error is one-sided, $T$ is also monotonic. Therefore,
for any $V$ such that $V\geq TV$,
\[
V\geq TV \geq T^2V \geq V^*.
\]
Therefore, any $w$ that is a feasible solution to a RALP satisfies
$\Phi w\geq V^*$. From this, we can conclude
\begin{align*}
\norm{V^*-\Phi w}_{1,\rho}=&\sum_{x\in S}\rho(x)\abs{V^*(x)-\Phi(x) w}\\
=&\rho^T\Phi w-\rho^T V^*.
\end{align*}
Because $V^*$ is constant, minimizing $\rho^T\Phi w$ with RALP
constraints is equivalent to minimizing $\norm{V^*-\Phi w}_{1,\rho}$
with RALP constraints.
Given Lemmas <ref> and <ref>, we can finally prove
Lemma 1.
\begin{align*}
\norm{V^*-\Phi \tilde w}_{1,\rho}\leq& \norm{V^*-\Phi\bar w}_{1,\rho}\\
=&\sum_{s\in\sS}\rho(s)\abs{V^*-(\Phi\bar w)(s)}\\
=&\sum_{s\in\sS}\rho(s)(\Phi w_L)(s)\frac{\abs{V^*-(\Phi\bar
w)(s)}}{(\Phi w_L)(s)}\\
\leq&\left(\sum_{s\in\sS}\rho(s)(\Phi w_L)(s)\right)
\max_{s'\in\sS}\frac{\abs{V^*-(\Phi\bar w)(s')}}{(\Phi w_L)(s')}\\
=&\rho^T\Phi w_L\norm{V^*-\Phi\bar w}_{\infty,1/\Phi w_L}\\
\leq&\rho^T\Phi w_L\left(\norm{V^*-\Phi w^*}_{\infty,1/\Phi
w_L}+\norm{\Phi\bar w-\Phi w^*}_{\infty,1/\Phi w_L}\right)\\
\leq&\rho^T\Phi w_L (\norm{V^*-\Phi w^*}_{\infty,1/\Phi
&+\norm{V^*-\Phi w^*}_{\infty,1/\Phi w_L} \left(\frac{2}{1-\beta_{\Phi
w_L}}-1\right)\norm{\Phi w_L}_{\infty,1/\Phi w_L})\\
=&\frac{2\rho^T\Phi w_L}{1-\beta_{\Phi w_L}}\norm{V^*-\Phi
w^*}_{\infty,1/\Phi w_L}.
\end{align*}
The penultimate line is due to the definition of $\bar w$, and the final line
occurs because $\norm{\Phi w_L}_{\infty,1/\Phi w_L}=1$.$\blacksquare$
§ PROOF OF LEMMA 2
When a constraint does not exist in RALP for some state, this does not mean
the value at that state is completely unconstrained; because we bounded the
rate of change of all components of the approximate and true value functions
in Assumption 1, the existence of a constraint constructed on a nearby state
means the existence of what we will call an implied constraint.
This lemma explicitly constructs these implied constraints, and quantifies the
maximum distance from the true constraint which would have existed had that
state been sampled. It does this by building two MDPs, identical in every
way, but for the reward function. $M_1$ has been incompletely sampled, with
sample set $\Sigma$. Every state-action pair therefore has either an explicit
or implied constraint in the corresponding RALP. $M_2$, however, has been
completely sampled. Every state-action pair in the set $\Sigma$ is identical
in $M_2$, but all state-action pairs not in $\Sigma$ have a sample producing a
constraint identical to the implied constraints of $M_1$. Because the
constraints are the same, the RALP solution is the same. We demonstrate the
difference in the reward functions $R_1$ and $R_2$ is bounded, and thus, the
difference in the optimal value functions $V^*_1$ and $V^*_2$ is bounded.
Given an MDP $M_1$ such that Assumption 1 is true and incomplete sample set
$\Sigma$, an MDP $M_2$ exists such that constructing the RALP with constraints
for all state-action pairs results in an identical RALP solution to that of
the RALP constructed from $\Sigma$, and
Consider an arbitrary state-action pair $s,a$, which is not represented by a
sample in $\Sigma$. This means we are missing the constraint
\begin{equation}
\label{eqn:wish}
R_1(s)+\gamma \sum_{x\in\sS}\left[p(x|s,a)\Phi(x)\right] w \leq
\Phi(s) w.
\end{equation}
Let us refer to the sample in $\Sigma$ which fulfills the sampling assumption
with $s$ and $a$ as $\sigma$. We can now construct a bound for how incorrect
each component of this constraint can be if we use the constraint at $\sigma$
and our sampling assumption to replace the missing constraint. For instance,
the reward function $R(s)$ is easily bounded.
\begin{equation*}
\label{eqn:rewbound}
R_1(\sigma^s)-\delta_R\leq R_1(s) \leq R_1(\sigma^s)+\delta_R
\end{equation*}
We now bound $\Phi(s) w$. Because the sampling assumption allows each basis
function to change only a finite amount, and because $\norm{ w_{-1}}_1\leq
\psi$, and $\bone(s)=\bone(\sigma^s)$,
\begin{equation*}
\label{eqn:phibound}
\Phi(\sigma^s) w-\delta_\Phi\psi
\leq\Phi(s) w\leq\Phi(\sigma^s) w+\delta_\Phi \psi
\end{equation*}
The final component is $\gamma \sum_{x\in \sS}p(x|s,a)\Phi(x) w$,
which expresses our expected value at the next state. It will be convenient
to separate the bias feature $\bone$ from the rest of $\Phi$. We will denote
the remainder of the design matrix as $\Phi_{-1}$, and the weights that
correspond to $\Phi_{-1}$ as $w_{-1}$. Similarly, we will denote the weight
corresponding to $\bone$ as $w_1$.
\begin{align*}
\sum_{x\in \sS}p(x|s,a)\Phi(x) w=&
\sum_{x\in\sS}p(x|s,a) w_1
+ \sum_{x\in\sS}p(x|s,a)\Phi_{-1}(x) w_{-1}\\
=& w_1+\sum_{x\in\sS}p(x|s,a)
\Phi_{-1}(x) w_{-1}
\end{align*}
Again, we have bounded the allowable change in our expression of probability.
\begin{align*}
w_1+&\sum_{x\in\sS}p(x|s,a) \Phi_{-1}(x) w_{-1}\\
\leq& w_1+\sum_{x\in\sS}\left[p(x|\sigma^s,\sigma^a)+
\delta_P\right]\Phi_{-1}(x) w_{-1}\\ =&
w_1+\sum_{x\in\sS}p(x|\sigma^s,\sigma^a)\Phi_{-1}(x) w_{-1}+
\delta_P\sum_{x\in\sS}\Phi_{-1}(x) w_{-1}
\end{align*}
Because each basis function $\Phi$ is can be standardized such that
$\norm{\Phi}_1=1$, and because $\norm{w_{-1}}_1\leq \psi$, the second
summation can be at most $\psi$. So,
\begin{align*}
\label{eqn:pbound}
\sum_{x\in\sS}p(x|\sigma^s,\sigma^a)\Phi(x) w-\delta_P \psi&\leq
\sum_{x\in\sS}\left[p(x|s,a)\Phi(x)\right] w\\
&\leq \sum_{x\in\sS}p(x|\sigma^s,\sigma^a)\Phi(x) w+\delta_P\psi.
\end{align*}
We now combine these results, and construct our implied
constraint to take the place of the missing constraint expressed by Equation
<ref>. We see that the maximum possible change by the
approximate value function is $\delta_\Phi \psi + \delta_R + \delta_P \psi$.
So, the total cumulative error in the constraint is at most $2(\delta_\Phi
\psi + \delta_R + \delta_P \psi)$. So, we effectively have the following
\begin{equation*}
\label{eqn:qer}
R_1(s)+q- \gamma \sum_{x\in\sS}\left[p(x|s,a) \Phi(x)\right] w \geq
\Phi(s) w,
\end{equation*}
where $\abs{q}\leq 2(\delta_\Phi \psi + \delta_R + \delta_P \psi)$.
Let $M_2$ be an MDP which is identical in every way to $M_1$, except
$R_2(s)=R_1(s)+q$. The RALP solution for $M_1$ will be equivalent to the RALP
solution for $M_2$, and $\norm{R_1-R_2}_\infty\leq
An illustration of Lemma 2. The blue bars are constraints at sampled
points $s_1,s_2\in\sS$. The red and purple lines indicate the maximum rate
of change of the value function, given our settings of $\psi, \delta_\phi,
\delta_R,$ and $\delta_P$. The center diamond is therefore the feasible
area for the approximate value function, and the red bar is the
implied constraint at some novel point $s'\in\sS$. Because
$\epsilon_p=\delta_\Phi\psi+\delta_R+\delta_P\psi$ is the maximum change, we
see that the difference between the best possible setting of $\Phi(s') w$
and the worst possible setting of
$\Phi(s') w$ is at most $2\epsilon_p$.
Let $M_1$ and $M_2$ be MDPs that differ only in their reward vectors $R_1$ and
$R_2$. Let $V_1^*$ and $V_2^*$ be their optimal value functions. Then, for
Let $s$ be an arbitrary point in the sets $\sS_1$ and $\sS_2$,
and define $r_{1i}(s)$ and $r_{2i}(s)$ to be the $i$-th reward received in
exploring $M_1$ and $M_2$ from state $s$, respectively. Note that
\begin{equation*}
\end{equation*}
\begin{align*}
\end{align*}
\begin{align*}
\left(\sum_{i=0}^\infty\gamma^i\expect{r_{1i}(s)}+
\sum_{i=0}^\infty\gamma^i\delta\right)\\
\end{align*}
Because this is true for an arbitrary $s$,
Lemma 2 is trivially proven by combining Lemmas <ref> and
<ref>, and is illustrated by Figure <ref>.
|
arxiv-papers
| 2014-04-16T14:15:43 |
2024-09-04T02:50:01.276895
|
{
"license": "Public Domain",
"authors": "Gavin Taylor and Connor Geer and David Piekut",
"submitter": "Gavin Taylor",
"url": "https://arxiv.org/abs/1404.4258"
}
|
1404.4325
|
# On the spectrum of the discrete $1d$ Schrödinger operator with an arbitrary
even potential
S. B. Rutkevich Fakultät für Physik, Universität Duisburg-Essen, D-47058
Duisburg, Germany [email protected]
###### Abstract
The discrete one-dimensional Schrödinger operator is studied in the finite
interval of length $N=2M$ with the Dirichlet boundary conditions and an
arbitrary potential even with respect to the spacial reflections. It is shown,
that the eigenvalues of such a discrete Schrödinger operator (Hamiltonian),
which is represented by the $2M\times 2M$ tridiagonal matrix, satisfy a set of
polynomial constrains. The most interesting constrain, which is explicitly
obtained, leads to the effective Coulomb interaction between the Hamiltonian
eigenvalues. In the limit $M\to\infty$, this constrain induces the
requirement, which should satisfy the scattering date in the scattering
problem for the discrete Schrödinger operator in the half-line. We obtain such
a requirement in the simplest case of the Schrödinger operator, which does not
have bound and semi-bound states, and which potential has a compact support.
###### pacs:
03.65.Aa,03.65.Nk,05.30.-d
## 1 Introduction
Consider the discrete Schrödinger eigenvalue problem in the one-dimensional
chain having even number of sites $N=2M$, with an arbitrary real even
potential
$V=\\{v_{j}\\}_{j=1}^{N}$:
$\displaystyle
v_{j}\,\psi_{l}(j)+\left[2\psi_{l}(j)-\psi_{l}(j-1)-\psi_{l}(j+1)\right]=\lambda_{l}\psi_{l}(j),$
(1.1) $\displaystyle v_{N+1-j}=v_{j},$ (1.2) $\displaystyle
j=1,\ldots,N,\quad\quad\lambda_{1}<\lambda_{2}<\ldots<\lambda_{N}.$
Eigenstates of (1.1) are subjected to the Dirichlet boundary conditions
$\psi_{l}(0)=\psi_{l}(N+1)=0.$ (1.3)
The discrete Sturm-Liouvelle problem (1.1)-(1.3) without the parity constrain
(1.2) plays an important role in the theory of Anderson localization [1, 2].
The problem (1.1)-(1.3) with an even potential (1.2) naturally arrises in the
context of the theory of the thermodynamic Casimir effect [3].
It is proved in this paper, that the eigenvalues of the discrete Sturm-
Liouville problem (1.1)-(1.3) satisfy the following equality:
$\prod_{m=1}^{M}\prod_{n=1}^{M}(\lambda_{2m-1}-\lambda_{2n})=2^{M}\,(-1)^{M(M+1)/2}.$
(1.4)
One can easily check, that (1.4) is satisfied for small $M=1,2,\ldots$ For
arbitrary natural $M$, relation (1.4) is proved in Section 2. Section 3
contains some well-known basic facts about the scattering problem in the half-
line for the discrete Schrödinger operator. In the limit $M\to\infty$,
equality (1.4) leads to certain constrains on the scattering data in such a
problem, which are derived in Section 4. Concluding remarks are given in
Section 5. Proof of (1.4) for the free case $v_{j}=0$, $j=1,\ldots,M$ is
presented in A.
## 2 Discrete Sturm-Liouville problem in the finite interval
In the case of zero potential $v_{j}=0$, the solution of (1.1)-(1.3) reads as
$\displaystyle\psi_{l}(j)=\sin(k_{l}j),$ (2.1)
$\displaystyle\lambda_{l}=\omega(k_{l}),$ (2.2) $\displaystyle k_{l}=\frac{\pi
l}{N+1},$ (2.3)
with
$\omega(p)=4\sin^{2}(p/2),$ (2.4)
and $l=1,\ldots,N$.
For a general real even potential $v_{j}$, the eigenstates $\psi_{2m-1}(j)$,
$m=1,\ldots,M$ are even with respect to the reflection
$\psi_{2m-1}(N+1-j)=\psi_{2m-1}(j),$ (2.5)
whereas eigenstates $\psi_{2m}(j)$, $m=1,\ldots,M$ are odd:
$\psi_{2m}(N+1-j)=-\psi_{2m}(j).$ (2.6)
It is useful to consider two associated eigenvalue problems for the even and
odd states, which are restricted to the half-chain $j=1,\ldots,M$. The
eigenvectors $\psi_{2m-1}(j)$, $j=1,\ldots,M$, are the eigenstates of the
tridiagonal $M\times M$ matrix $H^{(ev)}$:
$\displaystyle
H^{(ev)}=\left(\begin{array}[]{ccccccc}b_{1}&-1&0&0&0&\dots&0\\\
-1&b_{2}&-1&0&0&\dots&0\\\ 0&-1&b_{3}&-1&0&\dots&0\\\
\dots&\dots&\dots&\dots&\dots&\dots&\dots\\\ 0&0&\dots&0&-1&b_{M-1}&-1\\\
0&0&\dots&0&0&-1&b_{M}-1\end{array}\right),$ (2.13) $\displaystyle
b_{j}=v_{j}+2,$
with eigenvalues $\mu_{m}=\lambda_{2m-1}$, $m=1,\ldots,M$. Similarly, the
eigenvectors $\psi_{2m}(j)$, $j=1,\ldots,M$, are the eigenstates of the
tridiagonal $M\times M$ matrix $H^{(od)}$:
$H^{(od)}=\left(\begin{array}[]{ccccccc}b_{1}&-1&0&0&0&\dots&0\\\
-1&b_{2}&-1&0&0&\dots&0\\\ 0&-1&b_{3}&-1&0&\dots&0\\\
\dots&\dots&\dots&\dots&\dots&\dots&\dots\\\ 0&0&\dots&0&-1&b_{M-1}&-1\\\
0&0&\dots&0&0&-1&b_{M}+1\end{array}\right),$ (2.14)
with eigenvalues $\nu_{m}=\lambda_{2m}$, $m=1,\ldots,M$. Note, that the
matrices $H^{(ev)}$ and $H^{(od)}$ are simply related
$H^{(od)}-H^{(ev)}=2P,$ (2.15)
with the projecting matrix
$P_{m,m^{\prime}}=\delta_{m,M}\delta_{m^{\prime},M}$,
$\;\;m,m^{\prime}=1,\ldots,M$, and ${\rm rank}\,P=1$.
It is convenient to allow the potential $\\{b_{j}\\}_{j=1}^{M}$ in the
diagonal of the matrices (2.13), (2.14) to take complex values.
###### Lemma 2.1
The matrices $H^{(od)}$ and $H^{(ev)}$ defined by (2.13), (2.14) have no
common eigenvalues for arbitrary complex potential $\\{b_{j}\\}_{j=1}^{M}$.
* Proof
We will assume that the matrices $H^{(ev)}$ and $H^{(od)}$ have a common
eigenvalue $\Lambda$ and come to contradiction.
So, let us suppose that
$\sum_{j^{\prime}=1}^{M}H^{(ev)}_{j,j^{\prime}}x_{j^{\prime}}=\Lambda\,x_{j},\quad\sum_{j^{\prime}=1}^{M}H^{(od)}_{j,j^{\prime}}y_{j^{\prime}}=\Lambda\,y_{j},$
with nonzero vectors $\\{x_{j}\\}_{j=1}^{M}$, $\\{y_{j}\\}_{j=1}^{M}$. We get
$\displaystyle\Lambda\sum_{j=1}^{M}y_{j}\,x_{j}=\sum_{j=1}^{M}\sum_{j^{\prime}=1}^{M}y_{j}H^{(ev)}_{j,j^{\prime}}x_{j^{\prime}}=\sum_{j=1}^{M}\sum_{j^{\prime}=1}^{M}y_{j}(H^{(od)}_{j,j^{\prime}}-2P_{j,j^{\prime}})x_{j^{\prime}}=$
$\displaystyle\sum_{j=1}^{M}\sum_{j^{\prime}=1}^{M}y_{j}H^{(od)}_{j,j^{\prime}}x_{j^{\prime}}-2\sum_{j=1}^{M}\sum_{j^{\prime}=1}^{M}y_{j}P_{j,j^{\prime}}x_{j^{\prime}}=-2y_{M}\,x_{M}+\Lambda\sum_{j=1}^{M}y_{j}\,x_{j}.$
Here we have taken into account, that the matrix $H^{(od)}$ is symmetric.
Thus,
$y_{M}\,x_{M}=0,$
which means that at least one of the numbers $y_{M}$ and $x_{M}$ is zero.
However, if $y_{M}=0$, we conclude immediately111Really, if
$0=y_{M}\equiv\psi(j=M)$, then $\psi(j=M+1)=-\psi(j=M)=0$, since
$\psi(j)=-\psi(2M+1-j)$ for all $j=1,\dots,2M$. And since the wave-function
$\psi(j)$ takes zero values at two neighbor sites $\psi(M)=\psi(M+1)=0$, one
can check recursively from (1.1), that
$\psi(M-1)=0,\;\psi(M-2)=0,\;\dots,\;\psi(1)=0$, and, therefore, $\psi(j)=0$
for all $j=1,\dots,2M$. , that $y_{m}=0$ for all $m=1,\ldots,M$, providing
that $\Lambda$ is not an eigenvalue of $H^{(od)}$. Similarly, if $x_{M}=0$, we
conclude, that $x_{m}=0$ for all $m=1,\ldots,M$, providing that $\Lambda$ is
not an eigenvalue of $H^{(ev)}$. This contradiction with the initial
assumption proofs the Lemma.
The sets of eigenvalues $\\{\mu_{m}\\}_{m=1}^{M}$ and
$\\{\nu_{m}\\}_{m=1}^{M}$ of the matrices $H^{(ev)}$ and $H^{(od)}$ are not
independent, but are subjected to certain polynomial constrains following from
(2.15):
$\displaystyle\sum_{m=1}^{M}\mu_{m}={\rm Tr}H^{(ev)}=-2+{\rm
Tr}H^{(od)}=-2+\sum_{m=1}^{M}\nu_{m},$ (2.16)
$\displaystyle\sum_{m=1}^{M}\mu_{m}^{2}={\rm Tr}(H^{(ev)})^{2}={\rm
Tr}(-2P+H^{(od)})^{2}=-4b_{M}+\sum_{m=1}^{M}\nu_{m}^{2},$
$\displaystyle\sum_{m=1}^{M}\mu_{m}^{3}={\rm Tr}(H^{(ev)})^{3}={\rm
Tr}(-2P+H^{(od)})^{3}=-8-6b_{M}^{2}+\sum_{m=1}^{M}\nu_{m}^{3},$
$\displaystyle\sum_{m=1}^{M}\mu_{m}^{4}={\rm Tr}(H^{(ev)})^{4}={\rm
Tr}(-2P+H^{(od)})^{4}=-8b_{M-1}-24b_{M}-8b_{M}^{2}+\sum_{m=1}^{M}\nu_{m}^{4},$
$\displaystyle\sum_{m=1}^{M}\mu_{m}^{5}={\rm Tr}(H^{(ev)})^{5}={\rm
Tr}(-2P+H^{(od)})^{5}=$
$\displaystyle-32-10b_{M-1}^{2}-20b_{M-1}b_{M}-50b_{M}^{2}-10b_{M}^{4}+\sum_{m=1}^{M}\nu_{m}^{5},$
$\displaystyle\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots.$
* Remark
* (1) Equations (2.16) provide a simple way to solve the inverse spectral problem, i.e. to determine one by one the potential $b_{M},\,b_{M-1},\ldots,b_{1}$, if the both sets of eigenvalues $\\{\mu_{m}\\}_{m=1}^{M}$ and $\\{\nu_{m}\\}_{m=1}^{M}$ are known.
* (2) Excluding one by one the potential $b_{M},\,b_{M-1},\ldots,b_{1}$ from equations (2.16), one can obtain the infinite set of polynomial constrains of increasing degrees on the eigenvalues $\\{\mu_{m}\\}_{m=1}^{M}$, $\\{\nu_{m}\\}_{m=1}^{M}$. No more than $M$ of constrains in this set can be independent, since the above mentioned eigenvalues are determined by $M$ parameters $\\{v_{j}\\}_{j=1}^{M}$ as zeroes of the characteristic polynomials of the matrices (2.13) and (2.14).
* (3) One can easily see from (2.16), that the symmetric polynomials of eigenvalues $\\{\mu_{m}\\}_{m=1}^{M}$, as well as the symmetric polynomials of eigenvalues $\\{\nu_{m}\\}_{m=1}^{M}$, can be written as polynomial functions of the potential $\\{b_{j}\\}_{j=1}^{M}$.
Now we are ready to prove the main
###### Theorem 2.2
For arbitrary complex numbers $\\{b_{j}\\}_{j=1}^{M}$, the eigenvalues
$\\{\mu_{m}\\}_{m=1}^{M}$ and $\\{\nu_{m}\\}_{m=1}^{M}$ of the matrices
$H^{(od)}$ and $H^{(ev)}$ defined by (2.13), (2.14) satisfy the equality:
$\prod_{m=1}^{M}\prod_{n=1}^{M}(\mu_{m}-\nu_{n})=2^{M}\,(-1)^{M(M+1)/2}.$
(2.17)
By restriction of this result to real $\\{b_{j}\\}_{j=1}^{M}$, we arrive to
(1.4).
* Proof
The left-hand side of (2.17) is a symmetric polynomial function of
$\\{\mu_{m}\\}_{m=1}^{M}$. It is also a a symmetric polynomial function of
$\\{\nu_{m}\\}_{m=1}^{M}$. Due to Remark (3), we can conclude, that the left-
hand side of (2.17) can be written as a polynomial function $Q_{M}(b)$ of the
potential $\\{b_{j}\\}_{j=1}^{M}$:
$\prod_{m=1}^{M}\prod_{n=1}^{M}(\mu_{m}-\nu_{n})=Q_{M}(b).$ (2.18)
It follows from this relation and Lemma 2.1, that the polynomial function
$Q_{M}(b)$ of $M$ complex variables $\\{b_{j}\\}_{j=1}^{M}$ has no zeros. This
means, that this function is a constant, $Q_{M}(b)\equiv C_{M}$, which must
not depend on the potential $\\{b_{j}\\}_{j=1}^{M}$.
One can now determine this constant $C_{M}$ using an appropriate convenient
choice of the potential. To this end, we put
$b_{j}=2,\quad j=1,\ldots,M,$ (2.19)
which corresponds to $v_{j}=0$, $j=1,\ldots,M$. Reminding (2.2), we get
$\mu_{m}=4\sin^{2}\frac{(2m-1)\pi}{2(2M+1)},\quad\nu_{m}=4\sin^{2}\frac{2m\pi}{2(2M+1)},$
(2.20)
and equality we need to prove for all natural $M$ takes the form
$\prod_{m=1}^{M}\prod_{n=1}^{M}\left[4\sin^{2}\frac{(2m-1)\pi}{2(2M+1)}-4\sin^{2}\frac{2n\pi}{2(2M+1)}\right]=2^{M}\,(-1)^{M(M+1)/2}.$
(2.21)
Proof of this formula is given in A.
It is interesting to note, that equality (1.4) allows the electrostatic
interpretation. Really, let us take the logarithm of the absolute values of
the both sides of (1.4), and rewrite the result in the form
$-\sum_{m=1}^{M}\sum_{n=1}^{M}\ln|x_{m}^{(A)}-x_{n}^{(B)}|=-M\,\ln 2,$ (2.22)
where $x_{m}^{(A)}=\lambda_{2m-1}$, and $x_{n}^{(B)}=\lambda_{2n}$, with
$m,n=1,\ldots,M$ will be treated as space coordinates of two different sets of
$M$ particles of types $A$ and $B$, which are distributed along the $x$-axis
in the two-dimensional plane. Particles of the $A$ type interlace with
particles of the $B$ type, $x_{m}^{(A)}<x_{m}^{(B)}<x_{m+1}^{(A)}.$ If
particles of the same type do not interact with each other, and particles of
different types interact via the pair $2d$ Coulomb potential
$u(x^{(A)},x^{(B)})=-\ln|x^{(A)}-x^{(B)}|$, then equation (2.22) states
simply, that the total Coulomb energy of this system of $2M$ particles should
be equal to $-M\ln 2$.
## 3 Scattering problem for the discrete Schrödinger operator in the half-
line
In this Section we briefly summarize some well-known basic results from the
scattering theory in the half-line (see, for example [4, 5]) adapted for the
the discrete Schrödinger operator [1, 2].
Consider the discrete Schrödinger equation (1.1) in the half-line
$j\in\mathbb{N}$
$\displaystyle\left(H\psi\right)_{j}=\lambda\,\psi(j),$ (3.1)
$\displaystyle\left(H\psi\right)_{j}=v_{j}\,\psi(j)+\left[2\psi(j)-\psi(j-1)-\psi(j+1)\right],$
(3.2) $\displaystyle j=1,2,\ldots,\infty,$
supplemented with the Dirichlet boundary condition
$\psi(0)=0.$ (3.3)
The potential $V=\\{v_{j}\\}_{j=1}^{\infty}$ in (3.1) is the infinite sequence
of real numbers. In the scattering theory, the potential should vanish fast
enough at infinity. It is usually required [4], that
$\sum_{j=1}^{\infty}j|v_{j}|<\infty.$ (3.4)
For such a potential, the spectrum $\sigma[H]$ of the operator $H$ defined by
(3.1)-(3.3) consists of the continuous part $\sigma_{cont}[H]=(0,4)$ and a
finite number of discrete eigenvalues.
At a given $\lambda$, equations (3.1), (3.2) with omitted boundary condition
(3.3) have two linearly independent solutions, and the general solution of
(3.1), (3.2) can be written as their linear combination. For two sequences
$\\{\psi_{1}(j)\\}_{j=0}^{\infty}$, and $\\{\psi_{2}(j)\\}_{j=0}^{\infty}$,
one can define the Wronskian
$W[\psi_{1},\psi_{2}]_{j}=\psi_{1}(j)\psi_{2}(j+1)-\psi_{1}(j+1)\psi_{2}(j),\quad
j=0,1,2,\ldots$ (3.5)
It is straightforward to check, that the Wronskian of two solutions of
equations (3.1), (3.2) does not depend on $j$.
Let us turn now to the scattering problem associated with
equations(3.1)-(3.3), which corresponds to the case $0<\lambda<4$. Instead of
parameter $\lambda\in\sigma_{cont}[H]$, it is also convenient to use the
momentum $p$ and the related complex parameter $z=\rm\rme^{\rmi p}$:
$\lambda=2-2\cos p=2-z-z^{-1}.$
Three solutions of (3.1), (3.3) are important for the scattering problem.
* •
The regular solution $\varphi(j,p)$, which is fixed by the boundary condition
$\varphi(0,p)=0,\quad\varphi(1,p)=1.$ (3.6)
* •
Two Jost solutions $f(j,p)$, and $f(j,-p)$, which are determined by their
behavior at large $j\to\infty$, and describe the out- and in-waves,
respectively,
$f(j,\pm p)\to\exp(\pm\rmi pj)=z^{\pm j},\quad{\rm{at}}\quad j\to\infty.$
(3.7)
The regular solution $\varphi(j,p)$ can be represented as a linear combination
of two Jost solutions,
$\varphi(j,p)=\frac{\rmi}{2\sin p}\left[F(p)f(j,-p)-F(-p)f(j,p)\right],\quad
0<p<\pi.$ (3.8)
The complex coefficient $F(p)$ in the above equation is known as the Jost
function. It is determined by (3.8) for real momenta $p$ in the interval
$p\in(-\pi,\pi)$, where it satisfies the relation
$F(-p)=[F(p)]^{*},$ (3.9)
and can be written as
$F(p)=\exp[{\sigma(p)-\rmi\eta(p)}].$ (3.10)
At large $j\to\infty$, the regular solution behaves as
$\varphi(j,p)\to\frac{A(p)}{\sin p}\sin[p\,j+\eta(p)],\quad j\to+\infty,$
where $A(p)=\exp[\sigma(p)]$ is the scattering amplitude, and $\eta(p)$ is the
scattering phase. The latter can be defined in such a way, that
$\eta(-p)=-\eta(p)$ for $-\pi<p<\pi$.
The following exact representation holds for the Jost function $F(p)$ in terms
of the regular solution $\varphi(j,p)$:
$F(p)=1+\sum_{j=1}^{\infty}\rme^{\rmi pj}v_{j}\varphi(j,p),$ (3.11)
cf. equation (1.4.4) in [4] in the continuous case.
For $|z|=1$, denote by $\hat{F}(z)$ the Jost function $F(p)$ expressed in the
complex parameter $z$: $F(p)=\hat{F}(z=\rme^{\rmi p})$. The function
$\hat{F}(z)$ can be analytically continued into the circle $|z|<1$, where it
has finite number of zeros $\\{a_{n}\\}_{n=1}^{\mathfrak{N}}$. These zeroes
determine the discrete spectrum $\\{\lambda_{n}\\}_{n=1}^{\mathfrak{N}}$ of
the problem (3.1)-(3.3):
$\lambda_{n}=2-a_{n}-a_{n}^{-1},\quad{\rm for}\quad n=1,\ldots,\mathfrak{N}.$
(3.12)
Of course, these eigenvalues are real in the boundary problem with a real
potential.
To simplify further analysis, we shall consider in the sequel the potentials
which satisfy the following requirements:
1. 1.
The potential $V$ should have a compact support, i.e.
$v_{j}=0,\quad{\rm for\;all}\quad j>J,$ (3.13)
with some natural $J$.
2. 2.
The corresponding Jost function $\hat{F}(z)$ should not have zeroes inside the
circle $|z|<1$, i.e. $\mathfrak{N}=0$. In other words, the spectrum
$\sigma[H]$ should be purely continuous.
3. 3.
The Jost function $\hat{F}(z)$ should take non-zero values at $z=\pm 1$:
$\hat{F}(1)\neq 0$, and $\hat{F}(-1)\neq 0$.
Conditions (2) and (3) imply, that the operator $H$ does not have bound and
semi-bound states [4], respectively.
For the potential satisfying (3.13), only $J$ initial terms survive in the sum
in the right-hand side of (3.11). Since $\varphi(j,p)$ is a polynomial of the
spectral parameter $\lambda=2-z-z^{-1}$ of the order $j-1$, the Jost function
(3.11) expressed in the parameter $z$ is a polynomial of the degree $2J-1$:
$\hat{F}(z)=1+\sum_{j=1}^{2J-1}c_{j}(V)\,z^{j}=\prod_{n=1}^{2J-1}\left[1-\frac{z}{a_{n}(V)}\right],$
(3.14)
where the coefficients $c_{j}(V)$ polynomially depend on the potential
$v_{j}$, $j=1,\ldots,J$. Due to the constrains (2), (3), we get
$|a_{n}(V)|>1,\quad{\rm for\;all}\quad n=1,\ldots,J.$ (3.15)
Let us periodically continue the scattering phase $\eta(p)$ from the interval
$(-\pi,\pi)$ to the whole real axis $p\in\mathbb{R}$. It follows from (3.15),
that for a potential satisfying (1)-(3), the scattering phase is an analytical
odd $2\pi$-periodical function in the whole real axis: $\eta(p)\in
C^{\infty}(\mathbb{R}/2\pi\mathbb{Z})$.
The notation $\delta(\lambda)$ will be used for the scattering phase $\eta(p)$
expressed in terms of the spectral parameter $\lambda$:
$\eta(p)=\delta(\lambda=2-2\cos p)$, for $0\leq\lambda\leq 4$, and $0\leq
p\leq\pi$. Conditions (2), (3) guarantee, that
$\delta(0)=\delta(4)=0.$ (3.16)
## 4 Constrains on the scattering data in the discrete Schrödinger scattering
problem in the half-line
It is shown in this Section, that the scattering phase $\delta(\lambda)$ in
the boundary problem (3.1)-(3.3) for the discrete Schrödinger operator in the
half-line with an arbitrary potential $V$ obeying (1)-(3) should satisfy the
constrain
$\int_{0}^{4}\rmd\lambda\,\delta(\lambda)\,\frac{\lambda-2}{\lambda(\lambda-4)}+\frac{1}{\pi}\int_{0}^{4}\rmd\lambda_{1}\,\delta(\lambda_{1})\,\mathcal{P}\\!\\!\,\int_{0}^{4}\rmd\lambda_{2}\,\frac{\delta^{\prime}(\lambda_{2})}{\lambda_{2}-\lambda_{1}}=0,$
(4.1)
where $\mathcal{P}\\!\\!\int$ indicates the principal value integral. It is
straightforward to rewrite the above constrain in the equivalent form in terms
of the Jost function $\hat{F}(z)$:
${\ln\hat{F}(z=1)+\ln\hat{F}(z=-1)}+\oint_{|z|=1}\frac{\rmd
z}{2\pi\rmi}\,\ln[\hat{F}(1/z)]\frac{\rmd\ln[\hat{F}(z)]}{\rmd z}=0,$ (4.2)
where the integration path in the right-hand side is gone in the counter-
clockwise direction.
To prove (4.1), let us consider the discrete Schrödinger eigenvalue problem
(1.1)-(1.3) in the finite interval $1\leq j\leq N=2M$, with $M>J$, and with
the even potential $V^{(M)}=\\{v_{j}^{(M)}\\}_{j=1}^{2M}$, which restriction
to the interval $[1,M]$ coincides with that of the potential
$V=\\{v_{j}\\}_{j=1}^{\infty}$:
$v_{j}^{(M)}=\cases{v_{j},\quad{\rm if}\quad j\leq J,\\\ 0,\quad{\rm if}\quad
J<j\leq 2M-J,\\\ v_{2M+1-j},\quad{\rm if}\quad 2M-J<j\leq 2M.}$ (4.3)
It is easy to see, that the spectrum $\\{\lambda_{l}\\}_{l=1}^{2M}$ of the
problem (1.1)-(1.3) with such a potential can be expressed in terms of the
scattering phase $\eta(p)$ of the corresponding semi-infinite problem
(3.1)-(3.3) by the relations
$\displaystyle\lambda_{l}=\omega(p_{l}),\quad{l=1,\ldots,2M},$ (4.4)
$\displaystyle(2M+1)\,p_{l}+2\,\eta(p_{l})=(2M+1)\,k_{l},$ (4.5)
where $\omega(p)=2-2\cos p$, and $k_{l}=\pi l/(2M+1)$. Solving equation (4.5)
with respect to $p_{l}$ one obtains at large $M$:
$p_{l}=k_{l}-\frac{2\,\eta(k_{l})}{2M+1}+\frac{4\,\eta(k_{l})\,\eta^{\prime}(k_{l})}{(2M+1)^{2}}+O(M^{-3}).$
(4.6)
For an arbitrary $M>J$, we get from (1.4):
$\sum_{m=1}^{M}S_{m}=0,$ (4.7)
where
$S_{m}=\sum_{n=1}^{M}\left[\ln|\omega(p_{2n-1})-\omega(p_{2m})|-\ln|\omega(k_{2n-1})-\omega(k_{2m})|\right].$
(4.8)
Proceeding to the large-$M$ limit, one finds after substitution of (4.6) into
(4.8) and expansion the result in $1/(2M+1)$:
$S_{m}=S_{m}^{(0)}+S_{m}^{(1)}+O(M^{-2}),$ (4.9)
where
$\displaystyle
S_{m}^{(0)}=\frac{2}{2M+1}\sum_{n=1}^{M}\frac{\omega^{\prime}(k_{2m})\,\eta(k_{2m})-\omega^{\prime}(k_{2n-1})\,\eta(k_{2n-1})}{\omega(k_{2n-1})-\omega(k_{2m})},$
(4.10) $\displaystyle
S_{m}^{(1)}=\frac{2}{(2M+1)^{2}}\sum_{n=1}^{M}\Bigg{\\{}\frac{\left[\omega^{\prime}(k_{2n-1})\,\eta^{2}(k_{2n-1})\right]^{\prime}-\left[\omega^{\prime}(k_{2m})\,\eta^{2}(k_{2m})\right]^{\prime}}{\omega(k_{2n-1})-\omega(k_{2m})}$
$\displaystyle-\left[\frac{\omega^{\prime}(k_{2m})\,\eta(k_{2m})-\omega^{\prime}(k_{2n-1})\,\eta(k_{2n-1})}{\omega(k_{2n-1})-\omega(k_{2m})}\right]^{2}\Bigg{\\}}.$
(4.11)
In the right-hand side of the second equation we can safely [up to the terms
of order $O(M^{-2})$] replace the sum in $n$ by the integral over the momentum
$q$:
$\displaystyle S_{m}^{(1)}=\frac{1}{\pi(2M+1)}\int_{0}^{\pi}\rmd
q\,\Bigg{\\{}\frac{\left[\omega^{\prime}(q)\,\eta^{2}(q)\right]^{\prime}-\left[\omega^{\prime}(k_{2m})\,\eta^{2}(k_{2m})\right]^{\prime}}{\omega(q)-\omega(k_{2m})}$
$\displaystyle-\left[\frac{\omega^{\prime}(k_{2m})\,\eta(k_{2m})-\omega^{\prime}(q)\,\eta(q)}{\omega(q)-\omega(k_{2m})}\right]^{2}\Bigg{\\}}+O(M^{-2}).$
(4.12)
Calculation of the large-$M$ asymptotics of $S_{m}^{(0)}$ is more delicate.
First, we extend summation in (4.10) in the index $n$ from 1 till $2M+1$
$\displaystyle S_{m}^{(0)}=\frac{2}{2M+1}\sum_{n=1}^{M}R_{m}(k_{2n-1})=$
$\displaystyle\frac{2}{2M+1}\left[-\frac{R_{m}(k_{2M+1})}{2}+\frac{1}{2}\sum_{n=1}^{2M+1}R_{m}(k_{2n-1})\right],$
(4.13)
where
$R_{m}(q)=\frac{\omega^{\prime}(k_{2m})\,\eta(k_{2m})-\omega^{\prime}(q)\,\eta(q)}{\omega(q)-\omega(k_{2m})}.$
(4.14)
In (4.13) we have taken into account the reflection symmetry
$R_{m}(q)=R_{m}(2\pi-q)$ of the function (4.14), providing
$R_{m}(k_{2n-1})=R_{m}(k_{2(2M+1-n)+1})$.
Since $k_{2M+1}=\pi$, and $\eta(\pi)=0$, $\omega(\pi)=4$, we get from (4.14)
$R_{m}(k_{2M+1})=\frac{\omega^{\prime}(k_{2m})\,\eta(k_{2m})}{4-\omega(k_{2m})}.$
(4.15)
The sum in the second line of (4.13) reads as
$\sum_{n=1}^{2M+1}R_{m}(k_{2n-1})=\sum_{n=1}^{2M+1}R_{m}\left(2\pi\,\frac{n-1/2}{2M+1}\right).$
(4.16)
Since $R_{m}(q)\in C^{\infty}(\mathbb{R}/2\pi\mathbb{Z})$, this sum can be
replaced with exponential accuracy by the integral at large $M\to\infty$:
$\sum_{n=1}^{2M+1}R_{m}\left(2\pi\,\frac{n-1/2}{2M+1}\right)=\frac{2M+1}{2\pi}\int_{0}^{2\pi}\rmd
q\,R_{m}(q)+o(M^{-\mu}),$ (4.17)
where $\mu$ is an arbitrary positive number, see formula 25.4.3 in [6]. Taking
into account (4.14), the integral in the right-hand side can be written as
$\displaystyle\int_{0}^{2\pi}\rmd
q\,R_{m}(q)=\omega^{\prime}(k_{2m})\,\eta(k_{2m})\,\mathcal{P}\\!\int_{0}^{2\pi}\frac{\rmd
q}{\omega(q)-\omega(k_{2m})}-\mathcal{P}\\!\int_{0}^{2\pi}\rmd
q\,\frac{\omega^{\prime}(q)\,\eta(q)}{\omega(q)-\omega(k_{2m})}=$
$\displaystyle
2\omega^{\prime}(k_{2m})\,\eta(k_{2m})\,\mathcal{P}\\!\int_{0}^{\pi}\frac{\rmd
q}{\omega(q)-\omega(k_{2m})}-2\,I[\omega(k_{2m})]=-2\,I[\omega(k_{2m})],$
(4.18)
where
$I(\Lambda)=\mathcal{P}\\!\int_{0}^{4}\rmd\lambda\,\frac{\delta(\lambda)}{\lambda-\Lambda},\quad{\rm
with}\quad 0<\Lambda<4.$ (4.19)
In the second line of (4.18) we have taken into account the equality
$\displaystyle\mathcal{P}\\!\int_{0}^{\pi}\,\frac{\rmd
q}{[\omega(q)-\omega(k)]^{\nu}}\equiv$
$\displaystyle\frac{1}{2}\lim_{\epsilon\to+0}\left\\{\int_{0}^{\pi}\,\frac{\rmd
q}{[\omega(q+\rmi\epsilon)-\omega(k))]^{\nu}}+\int_{0}^{\pi}\,\frac{\rmd
q}{[\omega(q-\rmi\epsilon)-\omega(k)]^{\nu}}\right\\}=0,$ (4.20)
with $0<k<\pi$, and $\nu=1$.
Collecting (4.13)-(4.19), we get
$S_{m}^{(0)}=-\frac{I[\omega(k_{2m})]}{\pi}-\frac{1}{2M+1}\frac{\omega^{\prime}(k_{2m})\,\eta(k_{2m})}{4-\omega(k_{2m})}+O(M^{-2}).$
(4.21)
Thus, we obtain from (4.21) the equality
$\lim_{M\to\infty}\sum_{m=1}^{M}S_{m}^{(0)}+\lim_{M\to\infty}\sum_{m=1}^{M}S_{m}^{(1)}=0,$
(4.22)
where $S_{m}^{(0)}$ and $S_{m}^{(1)}$ are given by equations (4.21), and
(4.11), respectively.
In the second term, we can replace with sufficient accuracy the summation in
$m$ by integration in the momentum $k$:
$\displaystyle\sum_{m=1}^{M}S_{m}^{(1)}=\frac{1}{2\pi^{2}}\int_{0}^{\pi}\rmd
k\int_{0}^{\pi}\rmd
q\,\frac{\left[\omega^{\prime}(q)\,\eta^{2}(q)\right]^{\prime}-\left[\omega^{\prime}(k)\,\eta^{2}(k)\right]^{\prime}}{\omega(q)-\omega(k)}$
$\displaystyle-\frac{1}{2\pi^{2}}\int_{0}^{\pi}\rmd k\int_{0}^{\pi}\rmd
q\,\left[\frac{\omega^{\prime}(k)\,\eta(k)-\omega^{\prime}(q)\,\eta(q)}{\omega(q)-\omega(k)}\right]^{2}+O(M^{-1}).$
(4.23)
The first integral in the right-hand side vanishes due to equality (4.20) with
$0<k<\pi$, and $\nu=1$. The second line in (4.23) can be transformed as
follows
$\displaystyle-\frac{1}{2\pi^{2}}\int_{0}^{\pi}\rmd k\int_{0}^{\pi}\rmd
q\,\left[\frac{\omega^{\prime}(k)\,\eta(k)-\omega^{\prime}(q)\,\eta(q)}{\omega(q)-\omega(k)}\right]^{2}=$
(4.24) $\displaystyle-\frac{1}{\pi^{2}}\int_{0}^{\pi}\\!\rmd
k\,[\omega^{\prime}(k)\,\eta(k)]^{2}\,\mathcal{P}\\!\\!\int_{0}^{\pi}\,\frac{\rmd
q}{[\omega(q)-\omega(k)]^{2}}+\frac{1}{\pi^{2}}\int_{0}^{\pi}\\!\rmd
k\,\mathcal{P}\\!\\!\int_{0}^{\pi}\\!\rmd
q\,\frac{\omega^{\prime}(k)\,\eta(k)\omega^{\prime}(q)\,\eta(q)}{[\omega(q)-\omega(k)]^{2}}.$
The first term in the right-hand side vanishes due to equality (4.20) with
$\nu=2$. Then, after a simple algebra we obtain from the second term in the
right-hand side of (4.24)
$\lim_{M\to\infty}\sum_{m=1}^{M}S_{m}^{(1)}=\frac{1}{\pi^{2}}\int_{0}^{4}\rmd\lambda_{1}\,\delta(\lambda_{1})\,\mathcal{P}\\!\\!\,\int_{0}^{4}\rmd\lambda_{2}\,\frac{\delta^{\prime}(\lambda_{2})}{\lambda_{2}-\lambda_{1}}.$
(4.25)
Let us turn now to calculation of the first term in the left-hand side of
equality (4.22). At large $M$, one obtains
$\displaystyle\sum_{m=1}^{M}S_{m}^{(0)}=-\frac{1}{2\pi}\int_{0}^{\pi}\rmd
k\,\frac{\omega^{\prime}(k)\,\eta(k)}{4-\omega(k_{2m})}-\frac{1}{\pi}\sum_{m=1}^{M}I[\omega(k_{2m})]+O(M^{-1}).$
(4.26)
The first term in the right-hand side equals to $I(4)/(2\pi)$. The large-$M$
asymptotics of the sum in the right-hand side can be found as follows
$\displaystyle\sum_{m=1}^{M}I[\omega(k_{2m})]=-\frac{I(0)}{2}+\frac{1}{2}\sum_{m=1}^{2M+1}I[\omega(k_{2m})]=$
$\displaystyle-\frac{I(0)}{2}+\frac{2M+1}{4\pi}\int_{0}^{2\pi}\rmd
k\,I[\omega(k)]+O(M^{-\mu}),$ (4.27)
where $\mu$ is an arbitrary positive number. In deriving (4.27) we have taken
into account, that $I[(\omega(k)]$ is the $2\pi$-periodical function of $p$ in
$\mathbb{R}$, which is continuous with all its derivatives [i.e.,
$I[(\omega(k)]\in C^{\infty}(\mathbb{R}/2\pi\mathbb{Z})$], and formula 25.4.3
in [6]. The integral in the right-hand side vanishes due to equality (4.20)
with $\nu=1$:
$\int_{0}^{2\pi}\rmd k\,I[\omega(k)]=\int_{0}^{2\pi}\rmd
k\,\mathcal{P}\\!\\!\int_{0}^{4}\rmd\lambda\,\frac{\delta(\lambda)}{\lambda-\omega(k)}=\int_{0}^{4}\rmd\lambda\,\delta(\lambda)\,\mathcal{P}\\!\\!\int_{0}^{2\pi}\frac{\rmd
k}{\lambda-\omega(k)}=0.$ (4.28)
Collecting (4.26)-(4.28), we come to the simple formula
$\lim_{M\to\infty}\sum_{m=1}^{M}S_{m}^{(0)}=\frac{I(0)+I(4)}{2\pi}.$ (4.29)
From (4.19), (4.22), (4.25), (4.29), we arrive to the final result (4.1).
Similarly to (1.4), equation (4.1) also admits the electrostatic
interpretation. Let us treat the function
$\rho(\lambda)=2\delta^{\prime}(\lambda)/\pi$ as the electric charge density,
which is distributed in the linear interval $0<\lambda<4$. Requirement (3.16)
implies, that the total electric charge of this distribution is zero,
$\int_{0}^{4}\rmd\lambda\,\rho(\lambda)=0.$
It is straightforward to rewrite (4.1) in terms of the function
$\rho(\lambda)$:
$\displaystyle-\frac{1}{2}\int_{0}^{4}\rmd\lambda_{1}\,\rho(\lambda_{1})\int_{0}^{4}\rmd\lambda_{2}\,\rho(\lambda_{2})\ln|\lambda_{1}-\lambda_{2}|-$
$\displaystyle\int_{0}^{4}\rmd\lambda\,\rho(\lambda)\left[q_{1}\,\ln\lambda+q_{2}\,\ln(4-\lambda)\right]-q_{1}\,q_{2}\ln
4=-\frac{\ln 2}{2},$ (4.30)
where $q_{1}=q_{2}=-1/2$. The left-hand side of the above equality represents
the Coulomb energy of the continuous charge distribution $\rho(\lambda)$
located in the interval $(0,\lambda)$ in the two-dimensional plane, and two
point charges $q_{1}=q_{2}=-1/2$ located at the points $\lambda_{1}=0$ and
$\lambda_{2}=4$.
## 5 Conclusions
We have studied some general spectral properties of the one-dimensional Sturm-
Liouville problem for the discrete Schrödinger equation with the Dirichlet
boundary conditions. Both cases of the finite-interval and semi-infinite
problems were considered.
For the finite-interval problem with even number of sites $2M$ and an
arbitrary even potential, it was shown, that its eigenvalues should satisfy
the infinite set of polynomial constrains of increasing degrees. Though the
number of these constrains is infinite, no more than $M$ of them are
independent. It is simple to find few initial small-degree polynomials in this
set, but explicit calculation of subsequent polynomials of higher degrees
becomes more and more difficult. Nevertheless, we have obtained in the
explicit form one polynomial constrain from this set, which has the degree
$M$, see equation (1.4). It leads to the effective Coulomb interaction between
the eigenvalues, which correspond to even and odd eigenstates.
The scattering problem for the discrete one-dimensional Schrödinger equation
in the half-line has been analysed as the $M\to\infty$ limit of the described
above $2M$-site discrete Sturm-Liouville problem in the finite-interval. It
was proved, that the scattering phase of the discrete scattering problem
(3.1)-(3.3) should satisfy condition (4.1), if: (i) the potential has a
compact support, (ii) the spectrum of the Hamiltonian is purely continuous,
$\sigma[H]=(0,4)$, and (iii) the Jost function takes nonzero values on its end
points $\lambda=0$ and $\lambda=4$. Constrain (4.1) admits the electrostatic
interpretation (4.30), as its finite-interval counterpart (1.4).
We did not try to prove (4.1) for the most general case of the discrete semi-
infinite scattering problem. It is natural to expect, however, that it should
hold for some more general class of potentials, which vanish fast enough at
infinity, though do not have a compact support. On the other hand, in the case
of the potentials with discrete spectrum and/or semi-bound states (the latter
appear if the Jost function has zeroes at the end points of the continuous
spectrum, see [4]), some modified forms of (4.1) should also exist.
We believe, that obtained results could be useful for the theory of Anderson
localisation and for the theory of random matrices.
I am thankful to H. W. Diehl for fruitful discussions. Support of this work by
Deutsche Forschungsgemeinschaft (DFG) via grant Ru 1506/1 is also gratefully
acknowledged.
## Appendix A Proof of equality (2.21)
Let us start from the following auxiliary
###### Lemma A.1
The following equality holds for all natural $M$ and integer $n$:
$\prod_{m=1}^{2M+1}\left\\{4\sin^{2}\left[\frac{(2m-1)\pi}{2(2M+1)}-\alpha\right]-4\sin^{2}\left[\frac{2n\pi}{2(2M+1)}\right]\right\\}=4\cos^{2}[\alpha(2M+1)].$
(1.1)
* Proof
Denote
$g(\alpha)=\prod_{m=1}^{2M+1}\left\\{4\sin^{2}\left[\frac{(2m-1)\pi}{2(2M+1)}-\alpha\right]-4\sin^{2}\left[\frac{2n\pi}{2(2M+1)}\right]\right\\}.$
(1.2)
The symmetry properties of the function $g(\alpha)$
$\displaystyle g(-\alpha)=g(\alpha),$ $\displaystyle
g\left(\alpha+\frac{\pi}{2M+1}\right)=g(\alpha)$
follow from (1.2).
Function $g(\alpha)$ is analytical in the complex $\alpha$-plane and has the
second order zeroes at the points
$\alpha_{l}=\frac{\pi}{2(2M+1)}+\frac{\pi l}{2M+1},\quad\quad l=0,\pm 1,\pm
2,\ldots$ (1.3)
It follows from (1.3) that the function
$R(\alpha)=\frac{g(\alpha)}{4\cos^{2}[\alpha(2M+1)]}$ (1.4)
is analytical and has no zeroes in the complex $\alpha$-plane. Furthermore,
this function is rational in the variable $z=\rme^{\rmi\alpha}$ and approaches
to 1 at $z\to\infty$ and at $z\to 0$. Therefore, $R(\alpha)=1$.
Putting $\alpha=0$ in (1.1) we find
$\prod_{m=1}^{2M+1}\left\\{4\sin^{2}\left[\frac{(2m-1)\pi}{2(2M+1)}\right]-4\sin^{2}\left[\frac{2n\pi}{2(2M+1)}\right]\right\\}=4.$
(1.5)
Since
$\displaystyle\prod_{j=1}^{2M+1}\left\\{4\sin^{2}\left[\frac{(2m-1)\pi}{2(2M+1)}\right]-4\sin^{2}\left[\frac{2n\pi}{2(2M+1)}\right]\right\\}=$
$\displaystyle\left[\prod_{m=1}^{M}\left\\{4\sin^{2}\left[\frac{(2m-1)\pi}{2(2M+1)}\right]-4\sin^{2}\left[\frac{2n\pi}{2(2M+1)}\right]\right\\}\right]^{2}\,4\left(1-\sin^{2}\left[\frac{2n\pi}{2(2M+1))}\right]\right),$
we get
$\left[\prod_{m=1}^{M}\left\\{4\sin^{2}\left[\frac{(2m-1)\pi}{2(2M+1)}\right]-4\sin^{2}\left[\frac{2n\pi}{2(2M+1)}\right]\right\\}\right]^{2}=\left[\cos\frac{\pi
n}{2M+1}\right]^{-2},$
or
$\prod_{m=1}^{M}\left\\{4\sin^{2}\left[\frac{(2m-1)\pi}{2(2M+1)}\right]-4\sin^{2}\left[\frac{2n\pi}{2(2M+1)}\right]\right\\}=(-1)^{n}\left[\cos\frac{\pi
n}{2M+1}\right]^{-1}.$ (1.6)
To fix the sign of the right-hand side of (1.6), we have taken into account
that just the first $n$ factors in the product in the left-hand side are
negative at $n=1,\ldots,M$.
Thus,
$\displaystyle\prod_{n=1}^{M}\prod_{m=1}^{M}\left[4\sin^{2}\frac{(2m-1)\pi}{2(2M+1)}-4\sin^{2}\frac{2n\pi}{2(2M+1)}\right]=\prod_{n=1}^{M}\frac{(-1)^{n}}{\cos\frac{\pi
n}{2M+1}}.$ (1.7)
To determine the product in the right-hand side we use formula 1.392.1 in Ref.
[7]:
$\sin nx=2^{n-1}\prod_{k=0}^{n-1}\sin\left(x+\frac{\pi k}{n}\right).$ (1.8)
For $n=2M+1$, $x=\pi/2$, we get from (1.8)
$\prod_{k=0}^{2M}\cos\left(\frac{\pi k}{2M+1}\right)=2^{-2M}(-1)^{M},$
providing
$\prod_{k=1}^{M}\cos\left(\frac{\pi k}{2M+1}\right)=2^{-M}.$ (1.9)
Substitution of (1.9) into (1.7) leads finally to (2.21).
## References
## References
* [1] Carmona R and Lacroix J 1990 Spectral Theory of Random Schrödinger Operators (Probability and its Applications) (Birkhäuser Boston)
* [2] Pastur L and Figotin A 1992 Spectra of Random and Almost-Periodic Operators. (Grundlehren Der Mathematischen Wissenschaften) (New York: Springer Verlag)
* [3] Diehl H W, Grüneberg D, Hasenbusch M, Hucht A, Rutkevich S B and Schmidt F M 2014 Large-$n$ approach to thermodynamic Casimir effects in slabs with free surfaces Phys. Rev. E at press (arXiv:1402.3510)
* [4] Chadan K and Sabatier P C 1989 Inverse problems in quantum scattering theory (Texts and Monographs in Physics) 2nd edn (New York: Springer-Verlag)
* [5] Case K M 1973 On discrete inverse scattering problems. II. Journal of Mathematical Physics 14 916
* [6] Abramowitz M and Stegun I 1972 Handbook of Mathematical Functions 10th edn (Dover Publications)
* [7] Gradshteyn I S and Ryshik I M 1965 Table of Integrals, Series, and Products (London: Academic Press)
|
arxiv-papers
| 2014-04-16T17:53:47 |
2024-09-04T02:50:01.286035
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sergei B. Rutkevich",
"submitter": "S. B. Rutkevich",
"url": "https://arxiv.org/abs/1404.4325"
}
|
1404.4670
|
RBC and UKQCD Collaborations
# $B$-meson decay constants from 2+1-flavor lattice QCD with domain-wall
light quarks and relativistic heavy quarks
N. H. Christ Physics Department, Columbia University, New York, NY 10027, USA
J. M. Flynn School of Physics and Astronomy, University of Southampton,
Southampton SO17 1BJ, UK T. Izubuchi RIKEN-BNL Research Center, Brookhaven
National Laboratory, Upton, NY 11973, USA Physics Department, Brookhaven
National Laboratory, Upton, NY 11973, USA T. Kawanai Theoretical Research
Division, Nishina Center, RIKEN, Wako 351-0198, Japan Physics Department,
Brookhaven National Laboratory, Upton, NY 11973, USA C. Lehner Physics
Department, Brookhaven National Laboratory, Upton, NY 11973, USA A. Soni
Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA R.
S. Van de Water Theoretical Physics Department, Fermi National Accelerator
Laboratory, Batavia, IL 60510, USA O. Witzel Center for Computational
Science, Boston University, Boston, MA 02215, USA
###### Abstract
We calculate the $B$-meson decay constants $f_{B}$, $f_{B_{s}}$, and their
ratio in unquenched lattice QCD using domain-wall light quarks and
relativistic $b$-quarks. We use gauge-field ensembles generated by the RBC and
UKQCD collaborations using the domain-wall fermion action and Iwasaki gauge
action with three flavors of light dynamical quarks. We analyze data at two
lattice spacings of $a\approx 0.11,0.086$ fm with unitary pion masses as light
as $M_{\pi}\approx 290$ MeV; this enables us to control the extrapolation to
the physical light-quark masses and continuum. For the $b$-quarks we use the
anisotropic clover action with the relativistic heavy-quark interpretation,
such that discretization errors from the heavy-quark action are of the same
size as from the light-quark sector. We renormalize the lattice heavy-light
axial-vector current using a mostly nonperturbative method in which we compute
the bulk of the matching factor nonperturbatively, with a small correction,
that is close to unity, in lattice perturbation theory. We also improve the
lattice heavy-light current through ${\mathcal{O}}(\alpha_{s}a)$. We
extrapolate our results to the physical light-quark masses and continuum using
SU(2) heavy-meson chiral perturbation theory, and provide a complete
systematic error budget. We obtain $f_{B^{0}}=199.5(12.6)$ MeV,
$f_{B^{+}}=195.6(14.9)$ MeV, $f_{B_{s}}=235.4(12.2)$ MeV,
$f_{B_{s}}/f_{B^{0}}=1.197(50)$, and $f_{B_{s}}/f_{B^{+}}=1.223(71)$, where
the errors are statistical and total systematic added in quadrature. These
results are in good agreement with other published results and provide an
important independent cross check of other three-flavor determinations of
$B$-meson decay constants using staggered light quarks.
###### pacs:
11.15.Ha 12.38.Gc 13.20.He 14.40.Nd
††preprint: FERMILAB-PUB-14-100-T
## I Introduction
Leptonic decays of bottom mesons probe the quark-flavor-changing transitions
$b\to u$ and $b\to s$, and therefore play an important role in constraining
and searching for new physics in the flavor sector.
In the Standard Model, the decay rate for $B^{+}\to\ell^{+}\nu_{\ell}$ is
given by
$\Gamma(B\to\ell\nu_{\ell})=\frac{m_{B}}{8\pi}G_{F}^{2}f_{B}^{2}|V_{ub}|^{2}m_{\ell}^{2}\left(1-\frac{m_{\ell}^{2}}{m_{B}^{2}}\right)^{2}\;,$
(1)
where $f_{B}$ is the leptonic decay constant that parameterizes
nonperturbative QCD contributions to the electroweak decay process, and we use
the convention $f_{\pi}\sim 130$ MeV. The decay rate in Eq. (1) is suppressed
by the small value of the CKM matrix element $|V_{ub}|$, which is of
${\mathcal{O}}(10^{-3})$, and is further helicity suppressed for light final-
state charged leptons. When combined with an experimental measurement of the
decay rate, a lattice-QCD calculation of $f_{B}$ enables the determination of
the CKM matrix element $|V_{ub}|$ within the Standard Model. This is
particularly important given the long-established $\sim 3\sigma$ disagreement
between $|V_{ub}|$ obtained from exclusive $B\to\pi\ell\nu$ semileptonic decay
and inclusive $B\to X_{u}\ell\nu$ decay CKMfitter ; UTfit ; Antonelli:2009ws ;
Laiho:2009eu ; Beringer:2012zz ; Aoki:2013ldr . Thus far only the charged-
current decay $B^{+}\to\tau^{+}\nu_{\tau}$ has been observed experimentally.
The experimental measurements from Belle and BaBar have $\sim 30\%$ errors
Aubert:2009wt ; Hara:2010dk ; Lees:2012ju ; Adachi:2012mm , but no individual
measurement has $5\sigma$ significance. The precision will improve, however,
with additional data collected by Belle II, which is expected to begin running
in around 2016. At this point the independent determination of $|V_{ub}|$ from
$B^{+}\to\tau^{+}\nu_{\tau}$ may be sufficiently precise to provide some
insight into the current $|V_{ub}|$ puzzle.
The leptonic decays of neutral $B_{d}^{0}$ and $B_{s}^{0}$ mesons proceed via
flavor-changing neutral currents. Thus they are loop suppressed in the
Standard Model, and potentially more sensitive to new physics than $B^{+}$
leptonic decays. The decay rate for these neutral-current processes is given
by:
$\Gamma(B_{q}\to\ell^{+}\ell^{-})=\frac{G_{F}^{2}}{\pi}\,Y\,\left(\frac{\alpha}{4\pi\sin^{2}\Theta_{W}}\right)^{2}m_{B_{q}}f_{B_{q}}^{2}|V_{tb}^{*}V_{tq}|^{2}m_{\ell}^{2}\sqrt{1-4\frac{m_{\ell}^{2}}{m_{B}^{2}}}\;,$
(2)
where $q=d,s$ and the loop function $Y$ includes next-to-leading-order short-
distance QCD and electroweak corrections Buchalla:1993bv . Here lattice-QCD
calculations of the decay constants $f_{B}$ and $f_{B_{s}}$ are needed to
calculate predictions for $B_{d,s}\to\ell^{+}\ell^{-}$ both within the
Standard Model and in beyond-the-Standard Model theories (see, e.g. Refs.
Buras:2012ru ; Bobeth:2013uxa ). Evidence for $B_{s}\to\mu^{+}\mu^{-}$ decay
has been seen at the $\sim 4\sigma$ level by both LHCb Aaij:2012nna ;
Aaij:2013aka and CMS Chatrchyan:2013bka , while LHCb has also seen $\sim
2\sigma$ evidence for $B_{d}^{0}\to\mu^{+}\mu^{-}$ decay Aaij:2013aka . The
statistical significance of both of these results will increase in the next
few years. Many uncertainties cancel, or are at least suppressed, in the
Standard-Model prediction for
${\mathcal{B}}(B_{s}\to\mu^{+}\mu^{-})/{\mathcal{B}}(B_{d}^{0}\to\mu^{+}\mu^{-})$,
which is proportional to the (squared) ratio of decay constants. Therefore,
once the experimental measurements are more precise, this $SU(3)$-breaking
ratio will provide an especially clean test of the Standard Model given a
similarly accurate lattice-QCD calculation of $f_{B_{s}}/f_{B_{d}^{0}}$.
In this work we present a new calculation of the leptonic decay constants
$f_{B}$, $f_{B_{s}}$, and the ratio $f_{B_{s}}/f_{B}$ in (2+1)-flavor lattice
QCD. We use the gauge-field ensembles generated by the RBC and UKQCD
collaborations with the domain-wall fermion action and Iwasaki gluon action
which include the effects of dynamical $u,d$, and $s$ quarks Allton:2008pn ;
Aoki:2010dy . For the bottom quarks, we use the relativistic heavy-quark (RHQ)
action introduced by Christ, Li, and Lin in Ref. Christ:2006us , with the
parameters of the action that were obtained nonperturbatively in Ref.
Aoki:2012xaa . We improve the lattice heavy-light axial vector current through
${\mathcal{O}}(\alpha_{s}a)$, and renormalize the current using the mostly
nonperturbative method introduced in Ref. ElKhadra:2001rv . We analyze data
with several values of the light-quark mass (down to $\approx$ 290 MeV) and
two lattice spacings of $a\approx$ 0.11 and 0.086 fm. We then extrapolate our
numerical simulation to the physical light-quark mass and continuum limit
using next-to-leading order SU(2) heavy-light meson chiral perturbation theory
(HM$\chi$PT) Goity:1992tp ; Arndt:2004bg ; Aubin:2005aq ; Albertus:2010nm .
This work is the first application of the RHQ action to weak-matrix element
calculations relevant for phenomenology. The general relativistic heavy-quark
framework was introduced by El Khadra, Kronfeld, and Mackenzie in Ref.
ElKhadra:1996mp , and can be used to simulate systems with both light quarks
$am_{0}\ll 1$ (where $a$ is the lattice spacing and $m_{0}$ is the bare quark
mass) and heavy quarks with $am_{0}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
318\relax$}}1$ with controlled discretization errors. This method takes
advantage of the fact that, in the rest frame of the heavy-light bound states,
the spatial momentum carried by the heavy quark is smaller than the mass of
the heavy quark and of order of $\Lambda_{\text{QCD}}$
$\left|\vec{p}_{hl}\right|\sim\Lambda_{\text{QCD}}.$ (3)
Performing a Symanzik-like expansion in powers of the spatial derivative
$D_{i}$ and keeping all orders of the mass $m_{0}a$ and the temporal
derivative $D_{0}$, one arrives at an anisotropic action which breaks the
axis-interchange symmetry between spatial and temporal directions. There are
several implementations of the relativistic heavy-quark framework. Here we use
the RHQ action introduced by Christ, Li, and Lin in Refs. Christ:2006us .
These authors showed that if the three coefficients in the anisotropic
Sheikholeslami-Wohlert (clover) action — the bare quark mass $m_{0}a$,
anisotropy $\zeta$, and clover coefficient $c_{P}$ — are suitably tuned, one
can eliminate errors of $O(|\vec{p}|a)$, $O([m_{0}a]^{n})$, and
$O(|\vec{p}|a[m_{0}a]^{n})$ from on-shell Green’s functions. Thus the RHQ
action allows us to simulate heavy quarks such as bottom with discretization
errors of similar size to those of light-quark systems. In this work we use
the nonperturbatively-determined values of $\\{m_{0}a,c_{P},\zeta\\}$ on the
RBC/UKQCD domain-wall + Iwasaki ensembles corresponding to the physical
$b$-quark. The values of these parameters were fixed using masses in the
$B_{s}$ system, and validated by comparison with the experimentally-measured
low-lying masses and mass-splittings in the bottomonium system.
There are several (2+1)-flavor and (2+1+1) calculations of the $B_{(s)}$-meson
decay constants and their ratio in the literature using a variety of actions
for the bottom and light quarks Albertus:2010nm ; McNeile:2011ng ;
Bazavov:2011aa ; Na:2012kp ; Dowdall:2013tga ; Carrasco:2013naa ;
Bazavov:2013wia . Of these, our calculation is most similar to that of the
Fermilab Lattice and MILC collaborations, who also use the relativistic heavy
quark framework. Their calculation uses the Fermilab interpretation of the
isotropic clover action ElKhadra:1996mp with the tadpole-improved tree-level
value of the clover coefficient $c_{SW}$. They also ${\mathcal{O}}(a)$-improve
the heavy-light axial-vector current at tree level. Thus, for similar values
of the lattice spacing, their calculation suffers from larger heavy-quark
discretization errors than ours. All of the published $N_{f}\geq 3$ results
for $f_{B}$, $f_{B_{s}}$, and $f_{B_{s}}/f_{B}$ use staggered light quarks;
the three $N_{f}=2+1$ calculations use the same asqtad-improved ensembles
generated by the MILC Collaboration. Our calculation using domain-wall light
quarks therefore provides a valuable independent check for these
phenomenologically-important quantities.
This paper is organized as follows. In Sec. II we describe the lattice actions
and simulation parameters used in this work. Next we present the determination
of the $B_{(s)}$ meson decay amplitudes in Sec. III. First we discuss the
operator renormalization and improvement, followed by the two-point correlator
fits, the interpolation to the tuned $b$-quark mass, and finally (for $B_{s}$
meson quantities) the interpolation to the physical $s$ quark. In Sec. IV we
extrapolate the numerical simulation data to the physical light-quark masses
and the continuum limit using SU(2) HM$\chi$PT. Section V presents our
complete uncertainty budget; for clarity, we discuss each source of systematic
uncertainty in a separate subsection. Finally, we conclude in Sec. VI with a
comparison of our results with other lattice determinations, and with an
outlook for the future. This paper also has two appendices describing our
determination of the heavy-heavy current renormalization factor $Z_{V}^{bb}$
(App. A) and our estimate of heavy-quark discretization errors (App. B).
## II Lattice actions and parameters
In this section we describe the setup of our numerical lattice simulations,
which is the same in our earlier work on tuning the parameters of the RHQ
action Aoki:2012xaa . Sec. II.1 summarizes the parameters of the light-quark
and gluon actions, while II.2 summarizes those of the heavy $b$-quark action.
### II.1 Light-quark and gluon actions
We use the dynamical “2+1”-flavor domain-wall Iwasaki ensembles generated by
the RBC and UKQCD Collaborations with two lattice spacings of $a\approx 0.11$
fm ($a^{-1}=1.729$ GeV) and $a\approx 0.08$ ($a^{-1}=2.281$ GeV) Allton:2008pn
; Aoki:2010dy . These ensembles were generated with three dynamical quarks:
the two lighter sea quarks have equal masses which are denoted by $m_{l}$,
while the heavier sea quark mass is tuned to within 10% of the physical
strange-quark mass and is denoted by $m_{h}$. The lattices employ the five-
dimensional Shamir domain-wall action Shamir:1993zy ; Furman:1994ky for the
fermions in combination with the Iwasaki gauge action Iwasaki:1983ck . This
combination allows for sufficient tunneling between topological sectors
Antonio:2008zz . For the calculation of the $B_{(s)}$-meson decay constants,
we analyze five ensembles with unitary pion masses as light as $\approx$ 290
MeV. All spatial volumes are about $2.5$ fm, such that $M_{\pi}L\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise
2.0pt\hbox{$\mathchar 318\relax$}}4$. Table 1 summarizes the parameters of the
gauge-field ensembles used in this analysis. Throughout this work, we refer to
the coarser ensembles with $a\approx 0.11$ fm as the “$24^{3}$” ensembles and
the finer ($a\approx 0.08$ fm) ensembles as the “$32^{3}$” ensembles.
Table 1: Lattice ensemble parameters. The columns list the lattice volume, approximate lattice spacing, light ($m_{l}$) and strange ($m_{h}$) sea-quark masses, residual chiral symmetry breaking parameter $m_{\rm res}$, physical $u/d$\- and $s$-quark mass, unitary pion mass, and number of configurations analyzed. The tildes over $a\widetilde{m}_{u/d}$ and $a\widetilde{m}_{s}$ denote that these values include the residual quark mass. $\left(\frac{L}{a}\right)^{3}\times\left(\frac{T}{a}\right)$ | $\approx a$(fm) | $a^{-1}$ [GeV] | $am_{l}$ | $am_{h}$ | $am_{\rm res}$ | $a\widetilde{m}_{u/d}$ | $a\widetilde{m}_{s}$ | $M_{\pi}$[MeV] | # configs.
---|---|---|---|---|---|---|---|---|---
$24^{3}\times 64$ | 0.11 | 1.729(25) | 0.005 | 0.040 | 0.003152 | 0.00136(4) | 0.0379(11) | 329 | 1636
$24^{3}\times 64$ | 0.11 | 1.729(25) | 0.010 | 0.040 | 0.003152 | 0.00136(4) | 0.0379(11) | 422 | 1419
$32^{3}\times 64$ | 0.086 | 2.281(28) | 0.004 | 0.030 | 0.0006664 | 0.00102(5) | 0.0280(7) | 289 | 628
$32^{3}\times 64$ | 0.086 | 2.281(28) | 0.006 | 0.030 | 0.0006664 | 0.00102(5) | 0.0280(7) | 345 | 889
$32^{3}\times 64$ | 0.086 | 2.281(28) | 0.008 | 0.030 | 0.0006664 | 0.00102(5) | 0.0280(7) | 394 | 544
For the light valence quarks we use the same fermion action and parameters as
in the sea sector. Hence we can use RBC-UKQCD’s earlier determinations of the
unitary pion masses, residual quark mass $m_{\rm res}$, and values of the
physical $u/d$\- and $s$-quark masses from Ref. Aoki:2010dy . In particular,
we use $L_{s}=16$ for the extent of the fifth dimension, a domain-wall height
of $M_{5}=1.8$, and periodic boundary conditions in all directions. With these
choices the size of residual chiral symmetry breaking is small: $am_{\rm res}$
is approximately $3\times 10^{-3}$ or less on all ensembles. For the
calculation of the $B_{(s)}$-meson decay constants, we generated point-source
valence quark propagators with six different masses including approximately
the physical strange quark and the unitary point; their values are listed in
Tab. 2. These point-source domain-wall propagators were saved and are
available for non-competing projects upon request.
Table 2: Partially quenched light-quark masses analyzed. On the $32^{3}$ ensembles, two propagators were generated on each configuration with sources separated by $T/2a$. | $a^{-1}$ [GeV] | $am_{q}$
---|---|---
$24^{3}$ | 1.729(25) | 0.005, 0.01, 0.02, 0.03, 0.0343, 0.04
$32^{3}$ | 2.281(28) | 0.004, 0.006, 0.008, 0.025, 0.0272, 0.03
### II.2 Heavy-quark action
We simulate the heavy $b$-quarks (denoted by $Q(x)$) with the anisotropic
Sheikholeslami-Wohlert (clover) action Sheikholeslami:1985ij :
$\displaystyle S_{\rm RHQ}$
$\displaystyle=a^{4}\sum_{x,x^{\prime}}\overline{Q}(x^{\prime})\left(m_{0}+\gamma_{0}D_{0}+\zeta\vec{\gamma}\cdot\vec{D}-\frac{a}{2}(D^{0})^{2}-\frac{a}{2}\zeta(\vec{D})^{2}+\sum_{\mu,\nu}\frac{ia}{4}c_{P}\sigma_{\mu\nu}F_{\mu\nu}\right)_{x^{\prime}x}Q(x)\,,$
(4) where $\displaystyle D_{\mu}Q(x)$
$\displaystyle=\frac{1}{2a}\left[U_{\mu}(x)Q(x+\hat{\mu})-U_{\mu}^{\dagger}(x-\hat{\mu})Q(x-\hat{\mu})\right]$
(5) $\displaystyle D^{2}_{\mu}Q(x)$
$\displaystyle=\frac{1}{a^{2}}\left[U_{\mu}(x)Q(x+\hat{\mu})+U_{\mu}^{\dagger}(x-\hat{\mu})Q(x-\hat{\mu})-2Q(x)\right]$
(6) $\displaystyle F_{\mu\nu}Q(x)$
$\displaystyle=\frac{1}{8a^{2}}\sum_{s,s^{\prime}=\pm
1}ss^{\prime}\left[U_{s\mu}(x)U_{s^{\prime}\nu}(x+s\hat{\mu})U_{s\mu}^{\dagger}(x+s^{\prime}\nu)U_{s^{\prime}\nu}^{\dagger}(x)-\textrm{h.c.}\right]Q(x)$
(7)
and $\gamma_{\mu}=\gamma_{\mu}^{\dagger}$ ,
$\\{\gamma_{\mu},\gamma_{\nu}\\}=2\delta_{\mu\nu}$ and
$\sigma_{\mu\nu}=\frac{i}{2}[\gamma_{\mu},\gamma_{\nu}]$. In Reference
Aoki:2012xaa we nonperturbatively determined the values of the three
parameters $m_{0}a$, $c_{P}$, and $\zeta$ that correspond to the physical
$b$-quark mass using the same set of gauge field configurations as in this
work. We follow the same approach for our computation of the decay constants
so that we can propagate statistical uncertainties from the tuning procedure
directly to the decay constants. Here we briefly summarize the aspects of the
tuning procedure needed to understand the error propagation; further details
can be found in Ref. Aoki:2012xaa .
The RHQ parameters were tuned using two experimental inputs from the
$B_{s}$-meson system – the spin-averaged mass
$\overline{M}_{B_{s}}=(M_{B_{s}}+3M_{B_{s}^{*}})/4$ and the hyperfine-
splitting $\Delta_{M_{B_{s}}}=M_{B_{s}^{*}}-M_{B_{s}}$ – along with the
constraint that the lattice rest mass (measured from the exponential decay of
meson correlators) equals the kinetic mass (measured from the meson dispersion
relation). They were obtained nonperturbatively via an iterative procedure as
follows. We began with an initial guess for the tuned values of $\\{m_{0}a$,
$c_{P}$, $\zeta\\}$, and computed $B_{s}$-meson two-point correlation
functions for seven sets of parameters centered on these values, as depicted
in Fig. 1. For each of the seven parameter sets we computed
$\overline{M}_{B_{s}}$, $\Delta_{M_{B_{s}}}$, and $M_{1}/M_{2}$, and then
linearly interpolated/extrapolated to the values of $\\{m_{0}a,c_{P},\zeta\\}$
that reproduced the experimental meson masses from the 2010 PDG
Nakamura:2010zzi and $M_{1}/M_{2}=1$. We repeated this procedure, re-
centering the seven parameter sets each time, until all of the tuned parameter
values remained inside the “box” depicted in Fig. 1, and thus were the result
of an interpolation rather than an extrapolation. We confirmed the assumption
that the meson masses depend linearly on $\\{m_{0}a$, $c_{P}$, $\zeta\\}$ with
additional simulations using larger box sizes.
Figure 1: Parameter sets used to obtain the tuned coefficients of the RHQ
action. The seven sets of $\\{m_{0}a,c_{P},\zeta\\}$ are located on a cube at
the centers of the six faces and at the midpoint.
The nonperturbatively tuned RHQ parameters determined in Aoki:2012xaa and
used in this work are presented in Tab. 3 and 4. These tables list the final
choice for the seven parameter sets used in the interpolation and the tuned
results on the individual ensembles. Because we do not observe any
statistically significant dependence on the sea-quark mass, we can average the
values on the different ensembles. The tuned RHQ parameters on the $24^{3}$
and $32^{3}$ lattice spacings obtained from the weighted averages of different
sea-quark ensembles are given in Table 5, along with our estimate of the
systematic uncertainties in these values as estimated in our earlier work,
Ref. Aoki:2012xaa . These values of $\\{m_{0}a,c_{P},\zeta\\}$ are used in our
calculation of the renormalization factor $Z_{V}^{bb}$ in Appendix A, our
estimation of heavy-quark discretization errors in Appendix B, and our
companion calculation of the $B\to\pi\ell\nu$ form factor Kawanai:2013qxa .
Table 3: Tuned RHQ parameters $m_{0}a$, $c_{P}$, and $\zeta$ corresponding to the physical $b$-quark obtained on the same $24^{3}$ gauge field configurations used in this work Aoki:2012xaa . We used the same seven sets of parameters for the final interpolation to the tuned values on both $24^{3}$ ensembles. Only statistical uncertainties are quoted. | $am_{l}$ | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---|---
tuning box | | $8.40\pm 0.15$ | $5.80\pm 0.45$ | $3.20\pm 0.30$
tuned values | 0.005 | 8.43(7) | 5.7(2) | 3.11(9)
tuned values | 0.010 | 8.47(9) | 5.8(2) | 3.1(1)
Table 4: Tuned RHQ parameters $m_{0}a$, $c_{P}$, and $\zeta$ corresponding to the physical $b$ quark obtained on the same $32^{3}$ gauge field configurations used in this work Aoki:2012xaa . We used the same seven sets of parameters for the final interpolation to the tuned values on all $32^{3}$ ensembles. Only statistical uncertainties are quoted. | $am_{l}$ | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---|---
tuning box | | $3.98\pm 0.10$ | $3.60\pm 0.30$ | $1.97\pm 0.15$
tuned values | 0.004 | 4.07(6) | 3.7(1) | 1.86(8)
tuned values | 0.006 | 3.97(5) | 3.5(1) | 1.94(6)
tuned values | 0.008 | 3.95(6) | 3.6(1) | 1.99(8)
Table 5: Tuned values of the RHQ parameters on the $24^{3}$ and $32^{3}$ ensembles Aoki:2012xaa . The central values and statistical errors are from a weighted average of the results on the individual sea-quark ensembles given in Tables 3 and 4. The errors listed in $m_{0}a$, $c_{P}$, and $\zeta$ are from left to right: statistics, heavy-quark discretization errors, the lattice scale uncertainty, and the uncertainty in the experimental measurement of the $B_{s}$-meson hyperfine splitting, respectively. Details on the error estimation can be found in Ref. Aoki:2012xaa . | $m_{0}a$ | $c_{P}$ | $\zeta$
---|---|---|---
$a\approx 0.11$ fm | 8. | 45(6)(13)(50)(7) | 5. | 8(1)(4)(4)(2) | 3. | 10(7)(11)(9)(0)
$a\approx 0.086$ fm | 3. | 99(3)(6)(18)(3) | 3. | 57(7)(22)(19)(14) | 1. | 93(4)(7)(3)(0)
## III Lattice calculation of $B$-meson decay amplitudes
In QCD the $B_{q}$-meson decay constant is defined by the vacuum-to-meson
matrix element of the heavy-light axial-vector current
${\mathcal{A}}_{\mu}=\overline{b}\gamma_{\mu}\gamma_{5}q$:
$\displaystyle\langle
0|{\mathcal{A}}_{\mu}|B_{q}(p)\rangle=if_{B_{q}}p_{\mu},$ (8)
where $q$ denotes the light quark and $p_{\mu}$ is the $B_{q}$-meson four-
momentum. Because $f_{B_{q}}$ behaves as $1/\sqrt{M_{B_{q}}}$ when $M_{B_{q}}$
is large, it is advantageous to compute the decay amplitude,
$\displaystyle\Phi_{B_{q}}=f_{B_{q}}\sqrt{M_{B_{q}}},$ (9)
which is proportional to $f_{B_{q}}$.
In this section we describe the numerical computation of the $B$-meson decay
amplitudes on the five sea-quark ensembles listed in Table 1. We first
describe the lattice axial-current operator renormalization and improvement,
then the two-point correlator calculations and fits, and finally the a
posteriori interpolation to the physical strange-quark mass.
### III.1 Operator renormalization and improvement
The lattice version of the axial-current operator, $A_{\mu}$, is related to
the continuum current as follows:
$\displaystyle Z_{A_{\mu}}A_{\mu}$ $\displaystyle\doteq$
$\displaystyle{\mathcal{A}}_{\mu}+{\mathcal{O}}\left(\alpha_{s}^{2}a\Lambda_{\rm
QCD}f_{i}(m_{0}a,c_{P},\zeta)\right)$ (10) $\displaystyle+$
$\displaystyle{\mathcal{O}}\left(a^{2}\Lambda_{\rm
QCD}^{2}f_{j}(m_{0}a,c_{P},\zeta)\right)\,,$
where $\doteq$ denotes the equality of on-shell matrix elements, and where the
${\mathcal{O}}(\alpha_{s}^{2}a,a^{2})$ discretization errors on the right-
hand-side are specific to our choice of operator improvement, discussed below.
We calculate the matching factor for the temporal component of the axial
current, hereafter called $Z_{\Phi}$, using the mostly nonperturbative method
introduced by El-Khadra et al. in Reference ElKhadra:2001rv . This approach
takes advantage of rewriting $Z_{\Phi}$ as the following product:
$\displaystyle Z_{\Phi}=\rho_{A}^{bl}\sqrt{Z_{V}^{ll}Z_{V}^{bb}}.$ (11)
Because the flavor-conserving renormalization factors $Z_{V}^{bb}$ and
$Z_{V}^{ll}$ can be obtained nonperturbatively from standard heavy-light and
light-light meson charge normalization conditions, only the residual
correction $\rho_{A}^{bl}$ needs to be computed perturbatively. The flavor-
conserving factors $Z_{V}^{bb}$ and $Z_{V}^{ll}$ account for most of the
operator renormalization, while $\rho_{A}^{bl}$ is expected to be close to
unity because most of the radiative corrections, including contributions from
tadpole graphs, cancel in the ratio $Z_{\Phi}/\sqrt{Z_{V}^{bb}Z_{V}^{ll}}$
Harada:2001fi . Therefore $\rho_{A}^{bl}$ has a more convergent series
expansion in $\alpha_{s}$ than $Z_{\Phi}$ and can be computed in lattice
perturbation theory to few-percent precision.
In practice, we calculate the flavor off-diagonal correction $\rho_{A}^{bl}$
at 1-loop in tadpole-improved lattice perturbation theory. The results
corresponding to $\alpha_{s}^{\overline{\rm MS}}(1/a)$ are given in Table 6.
Details on the calculation will be provided in a forthcoming publication
CLehnerPT . The light-light renormalization factor $Z_{V}^{ll}$ has already
been obtained by the RBC/UKQCD Collaborations (see Ref. Aoki:2010dy ), where
we use the fact that $Z_{A}=Z_{V}$ for domain-wall fermions up to corrections
of ${\cal O}(am_{\rm res})$. We use the determinations in the chiral limit
given in Tab. 6. We calculate the heavy-heavy renormalization factor
$Z_{V}^{bb}$ as part of this project. Details of the calculation are provided
in Appendix A; the results are given in Tab. 6. As a cross-check of our use of
lattice perturbation theory for $\rho_{A}^{bl}$, we can compare our
nonperturbatively determined values of $Z_{V}^{bb}$ with those computed at one
loop in perturbation theory,
$\displaystyle\left(Z_{V}^{bb}\right)^{\textrm{PT}}_{24c}=10.72\,,\quad\left(Z_{V}^{bb}\right)^{\textrm{PT}}_{32c}=5.725\,.$
(12)
We find agreement to better than 10% percent, which is consistent with
expectations of perturbative errors of ${\mathcal{O}}(\alpha_{s}^{2})$.
Table 6: Matching factors and improvement coefficients. The light-light flavor conserving renormalization factor $Z_{V}^{ll}=Z_{A}$ for domain-wall fermions up to corrections of $O(m_{\text{res}})$ Aoki:2010dy ; results quoted here are in the chiral limit. Errors shown on $Z_{V}^{ll}$ and $Z_{V}^{bb}$ are statistical only. The flavor-diagonal matching factor $\rho_{A}^{bl}$ and improvement coefficient $c_{A}$ are both computed at 1-loop in mean-field improved lattice perturbation theory Lehner:2012bt . $a^{-1}$ [GeV] | $Z_{V}^{ll}$ | $Z_{V}^{bb}$ | $\alpha_{s}^{\overline{\rm MS}}(a^{-1})$ | $\rho_{A}^{bl}$ | $c_{A}$
---|---|---|---|---|---
1.729(25) | 0.71689(51) | 10.039(25) | 0.23 | 1.02658 | 0.066
2.281(28) | 0.74469(13) | 5.256(8) | 0.22 | 1.01661 | 0.064
To reduce lattice discretization errors we improve the axial-vector current
${\mathcal{O}}(a)$ at one-loop in mean field improved lattice perturbation
theory. At this order, only one additional matrix element needs to be
computed:
$\displaystyle\Phi_{B_{q}}^{(1)}$ $\displaystyle=\langle
0|\overline{b}\gamma_{0}\gamma_{5}\sum_{i}\gamma_{i}\left(2\overleftarrow{D}_{i}\right)q|B_{q}(p)\rangle/\sqrt{M_{B_{q}}},$
(13)
where the symmetric covariant derivative $\overleftarrow{D}_{\mu}$ acts on
fields to the left:
$\displaystyle\overline{b}(x)\overleftarrow{D}_{\mu}$
$\displaystyle=\frac{1}{2}\left(\overline{b}(x+\hat{\mu})U_{\mu}^{\dagger}(x)-\overline{b}(x-\hat{\mu})U_{\mu}(x-\hat{\mu})\right)\,.$
(14)
The $O(\alpha_{s}a)$-improved decay amplitude is then given by
$\displaystyle\Phi_{B_{q}}^{\text{imp}}=\Phi_{B_{q}}+c_{A}\Phi_{B_{q}}^{(1)},$
(15)
with values of the coefficient $c_{A}$ given in Table 6. Finally, we obtain
the improved, renormalized decay amplitude as follows:
$\displaystyle\Phi_{B_{q}}^{\text{ren}}=Z_{\Phi}\left(\Phi_{B_{q}}+c_{A}\Phi_{B_{q}}^{(1)}\right).$
(16)
Discretization errors in our simulations from the heavy-light axial-vector
current are therefore of ${\mathcal{O}}(\alpha_{s}^{2}a,a^{2})$.
### III.2 Two-point correlator fits
To obtain the decay amplitudes we first cacluate the following two-point
correlation functions:
$\displaystyle C_{AP}(t,t_{0})$ $\displaystyle=\sum_{\vec{y}}\langle{\cal
O}_{A}^{\dagger}(\vec{y},t)\widetilde{\cal O}_{P}(\vec{0},t_{0})\rangle\,,$
(17) $\displaystyle C_{A^{(1)}P}(t,t_{0})$
$\displaystyle=\sum_{\vec{y}}\langle{\cal
O}_{A^{(1)}}^{\dagger}(\vec{y},t)\widetilde{\cal
O}_{P}(\vec{0},t_{0})\rangle\,,$ (18) $\displaystyle C_{PP}(t,t_{0})$
$\displaystyle=\sum_{\vec{y}}\langle{\cal
O}_{P}^{\dagger}(\vec{y},t)\widetilde{\cal O}_{P}(\vec{0},t_{0})\rangle\,,$
(19) $\displaystyle\widetilde{C}_{PP}(t,t_{0})$
$\displaystyle=\sum_{\vec{y}}\langle\widetilde{\cal
O}_{P}^{\dagger}(\vec{y},t)\widetilde{\cal O}_{P}(\vec{0},t_{0})\rangle\,.$
(20)
where ${\cal O}_{P}=\overline{b}\gamma_{5}q$ is a pseudoscalar interpolating
operator, ${\cal O}_{A}=\overline{b}\gamma_{0}\gamma_{5}q$ is the leading
axial-current operator, and ${\cal
O}_{A^{(1)}}=\overline{b}\gamma_{0}\gamma_{5}\sum_{i}\gamma_{i}\left(D_{i}^{+}+D_{i}^{-}\right)q$
is the ${\mathcal{O}}(a)$ axial-current operator. We use point sources for the
light quarks in the correlation functions and gauge-invariant Gaussian-smeared
sources Alford:1995dm ; Lichtl:2006dt for the $b$-quark propagators. The
parameters for the Gaussian smearing were optimized in our earlier work to
suppress excited-state contamination Aoki:2012xaa . We use point sinks for the
$b$-quarks in order to minimize the statistical errors, except in
$\widetilde{C}_{PP}(t,t_{0})$ which is used to obtain the wavefunction
renormalization. The tildes above the operators in Eqs. (17)–(20) indicate
that a smeared source or sink was used for the $b$-quark.
To reduce autocorrelations between results on consecutive configurations, we
place the sources of our propagators at the origin of the lattice after
translating the gauge field by a random four-vector $(\vec{x},\,t)$. This is
equivalent to selecting a random source position for each configuration, but
simplifies the subsequent analysis. We double our statistics on all ensembles
by folding the correlators at the temporal midpoint of the lattice, which
allows us to use both forward and backward propagating states. In the case of
the $32^{3}$ ensembles we also double the statistics by placing a second
source on each shifted configuration located at the temporal midpoint of the
lattice, $(\vec{x},\,t)=(\vec{0},\,T/2)$. After folding and averaging the
correlators with the two source positions, the entire subsequent analysis
chain, including the chiral-continuum extrapolations, is then carried out
using a single-elimination jackknife error analysis.
At sufficiently large times, the two-point correlators are dominated by the
contribution from the ground-state meson. We can then extract the masses and
renormalized, ${\mathcal{O}}(\alpha_{s}a)$-improved decay amplitudes from
simple ratios of correlators:
$\displaystyle M_{B_{q}}$ $\displaystyle=\lim_{t\gg
t_{0}}\cosh^{-1}\left(\frac{C_{PP}(t,t_{0})+C_{PP}(t+2,t_{0})}{{2}\,C_{PP}(t+1,t_{0})}\right)\,,$
(21) $\displaystyle\Phi_{B_{q}}$ $\displaystyle=\sqrt{2}Z_{\Phi}\lim_{t\gg
t_{0}}\frac{\left|C_{AP}(t,t_{0})+c_{A}C_{A^{(1)}P}(t,t_{0})\right|}{\sqrt{\widetilde{C}_{PP}(t,t_{0})e^{-M_{B_{q}}(t-t_{0})}}},$
(22)
where we use the values of the renormalization factor
$Z_{\Phi}=\rho_{A}^{bl}\left(Z_{V}^{ll}Z_{V}^{bb}\right)^{1/2}$ and
improvement coefficient $c_{A}$ given in Table 6.
On each ensemble, for each of the seven sets of RHQ parameters
$\\{m_{0}a,c_{P},\zeta\\}$ listed in in the first rows of Tables 3 and 4 and
six valence-quark masses listed in Table 2, we obtain the meson masses and
decay amplitudes from correlated plateau fits to the above ratios. We use the
same range of time slices as in our tuning procedure Aoki:2012xaa : $[t_{\rm
min},t_{\rm max}]=[10,25]$ on the $24^{3}$ ensembles and $[t_{\rm min},t_{\rm
max}]=[11,21]$ on the $32^{3}$ ensembles. Figures 2 and 3 show effective mass
and decay amplitude plots for the six different light valence-quark masses and
the central RHQ parameter set on the $32^{3}$ ensemble with $m_{l}=0.006$ in
units of $M_{B_{s}}$. The plateaus for other ensembles and sets of RHQ
parameters look similar.
To check for residual autocorrelations, on each ensemble we compare the fit
results for the masses and decay amplitudes of $B_{q}$ mesons with unitary and
close-to-strange valence quarks after blocking the configurations. We consider
binning the data in groups of 2 to 8 configurations, and find no significant
change in the statistical errors with bin size. We therefore use the unbinned
results in our subsequent analysis.
Figure 2: Effective masses for all valence-quark masses we use on the
$32^{3}$, $am_{l}=0.006$ ensemble. The triangles show the data points with
jackknife statistical errors, while the horizontal bands show the result of a
correlated constant fit to the data on those time slices.
Figure 3: Effective decay amplitudes for all valence-quark masses we use on
the $32^{3}$, $am_{l}=0.006$ ensemble. The triangles show the data points with
jackknife statistical errors, while the horizontal bands show the result of a
correlated constant fit to the data on those time slices.
Our computation is carried out using the Chroma software library
Edwards:2004sx supplemented by our own code for measuring matrix elements for
the $O(a)$-improvement and the three-point correlation functions needed for
the determination of $Z_{V}^{bb}$.
### III.3 Interpolation to the tuned RHQ parameters
As mentioned above, the extraction of $B$-meson decay amplitudes is performed
for each of the seven sets of RHQ parameters. We must then interpolate these
results to the tuned values of $\\{m_{0}a,c_{P},\zeta\\}$ that correspond to
the physical $b$-quark.
We first interpolate the seven different masses $M_{B_{q}}^{r}$, where the
index $r$ runs over the seven parameter sets, to the mass of the $B_{q}$-meson
via a jackknife procedure, in which we utilize the jackknife blocks for the
RHQ parameters created as part of our tuning procedure Aoki:2012xaa . We
assume that the masses depend linearly on $\\{m_{0}a,c_{P},\zeta\\}$:
$\displaystyle M_{B_{q}}^{\text{RHQ}}$
$\displaystyle=J_{M}\times\left[\begin{matrix}m_{0}a\\\ c_{P}\\\
\zeta\end{matrix}\right]^{\text{RHQ}}+A_{M},$ (23)
where $J_{M}$ is a three component vector and $A_{M}$ a constant for each
jackknife block,
$\displaystyle J_{M}$
$\displaystyle=\left[\frac{M_{B_{q}}^{3}-M_{B_{q}}^{2}}{2\sigma_{m_{0}a}},\frac{M_{B_{q}}^{5}-M_{B_{q}}^{4}}{2\sigma_{c_{P}}},\frac{M_{B_{q}}^{7}-M_{B_{q}}^{6}}{2\sigma_{\zeta}}\right],$
(24) $\displaystyle A_{M}$
$\displaystyle=M_{B_{q}}^{1}-J_{M}\times\left[m_{0}a,c_{P},\zeta\right]^{T}\,,$
(25)
and the $\sigma$’s are the variations of the parameters listed in Tables 3 and
4. This procedure allows us to directly propagate statistical uncertainties
from the tuning procedure to the meson masses and later also into the decay
amplitudes. We list the values for all meson masses interpolated to the
physical $b$-quark in Table 7. We follow the same procedure for the decay
amplitudes, but with $M_{B_{q}}\to\Phi_{B_{q}}^{\text{ren}}$ in Eqs.
(23)–(25).
The renormalized decay amplitudes for all valence-quark masses and ensembles
are also listed in Table 7. We present the results as dimensionless ratios in
units of the $B_{s}$-meson mass, and perform the subsequent chiral-continuum
extrapolation using these ratios. Because the RHQ parameters are tuned such
that $M_{B_{s}}$ reproduces the experimental value, this enables us to avoid
two potential sources of uncertainty associated with the lattice-scale
determination: (1) in the joint chiral-continuum fits to the data on both
lattice spacings we do not need to know the ratio $\left(a_{24}/a_{32}\right)$
to relate the overall normalizations of the decay amplitudes $\Phi_{B_{q}}$ on
the different ensembles, and (2) when we convert the final results for the
decay constants to physical units we can simply multiply by the experimental
value of $M_{B_{s}}$.
Table 7: Masses and renormalized decay amplitudes on the $24^{3}$ ensembles (upper two panels) and $32^{3}$ ensembles (lower three panels) with statistical errors. $a^{-1}$ [GeV] | $am_{l}$ | $am_{q}$ | $aM_{B_{q}}$ | $\Phi_{B_{q}}^{\text{ren}}/M_{B_{s}}^{3/2}$
---|---|---|---|---
1.729(25) | 0.005 | 0.0050 | 3.0644(16) | 0.03999(64)
1.729(25) | 0.005 | 0.0100 | 3.0715(11) | 0.04107(60)
1.729(25) | 0.005 | 0.0200 | 3.0849(5) | 0.04323(58)
1.729(25) | 0.005 | 0.0300 | 3.0978(2) | 0.04532(58)
1.729(25) | 0.005 | 0.0343 | 3.1034(2) | 0.04619(58)
1.729(25) | 0.005 | 0.0400 | 3.1106(3) | 0.04733(58)
1.729(25) | 0.010 | 0.0050 | 3.0656(19) | 0.04001(75)
1.729(25) | 0.010 | 0.0100 | 3.0723(12) | 0.04105(70)
1.729(25) | 0.010 | 0.0200 | 3.0854(6) | 0.04315(67)
1.729(25) | 0.010 | 0.0300 | 3.0983(3) | 0.04520(67)
1.729(25) | 0.010 | 0.0343 | 3.1038(3) | 0.04607(67)
1.729(25) | 0.010 | 0.0400 | 3.1111(3) | 0.04718(68)
2.281(28) | 0.004 | 0.0040 | 2.3231(13) | 0.03961(61)
2.281(28) | 0.004 | 0.0060 | 2.3252(10) | 0.04005(59)
2.281(28) | 0.004 | 0.0080 | 2.3275(8) | 0.04054(57)
2.281(28) | 0.004 | 0.0250 | 2.3497(2) | 0.04504(60)
2.281(28) | 0.004 | 0.0272 | 2.3526(2) | 0.04560(60)
2.281(28) | 0.004 | 0.0300 | 2.3564(2) | 0.04632(61)
2.281(28) | 0.006 | 0.0040 | 2.3233(10) | 0.03930(51)
2.281(28) | 0.006 | 0.0060 | 2.3254(8) | 0.03971(49)
2.281(28) | 0.006 | 0.0080 | 2.3277(6) | 0.04016(48)
2.281(28) | 0.006 | 0.0250 | 2.3496(1) | 0.04447(49)
2.281(28) | 0.006 | 0.0272 | 2.3526(1) | 0.04502(50)
2.281(28) | 0.006 | 0.0300 | 2.3563(1) | 0.04572(50)
2.281(28) | 0.008 | 0.0040 | 2.3236(14) | 0.03961(67)
2.281(28) | 0.008 | 0.0060 | 2.3257(11) | 0.03997(65)
2.281(28) | 0.008 | 0.0080 | 2.3281(9) | 0.04041(63)
2.281(28) | 0.008 | 0.0250 | 2.3495(1) | 0.04448(64)
2.281(28) | 0.008 | 0.0272 | 2.3523(1) | 0.04500(64)
2.281(28) | 0.008 | 0.0300 | 2.3560(1) | 0.04565(65)
### III.4 Interpolation to the physical strange-quark mass
For the determination of $f_{B_{s}}$ and the ratio $f_{B_{s}}/f_{B}$ we must
slightly interpolate our data with close-to-strange valence-quark masses to
the physical strange quark mass as determined in Aoki:2010dy . We perform a
linear, uncorrelated fit to interpolate the three heaviest masses on each
ensemble: $am_{q}=0.03,\,0.0343,\,\text{and}\,0.04$ on the $24^{3}$ ensembles
and $am_{q}=0.025,\,0.0272,\,\text{and}\,0.03$ on the $32^{3}$ ensembles.
Figure 4 shows an example determination of $\Phi_{B_{s}}^{\text{ren}}$ on the
$32^{3}$ ensemble with $am_{l}=0.006$. Table 8 lists the
$\Phi_{B_{s}}^{\text{ren}}$ values for all five ensembles. We then use the
interpolated values for $\Phi_{B_{s}}^{\text{ren}}$ to obtain the ratio of the
SU(3)-breaking ratio $\Phi_{B_{s}}^{\text{ren}}/\Phi_{B_{q}}^{\text{ren}}$ for
all six light valence-quark masses on each ensemble. We include statistical
correlations between the numerator and denominator via a jackknife, and list
the results in Tab. 9.
Table 8: Interpolated decay amplitudes $\Phi_{B_{s}}^{\text{ren}}$ with statistical errors. $a^{-1}$ [GeV] | $am_{l}$ | $\Phi_{B_{s}}^{\text{ren}}/M_{B_{s}}^{3/2}$
---|---|---
1.729(25) | 0.005 | 0.04627(58)
1.729(25) | 0.010 | 0.04615(67)
2.281(28) | 0.004 | 0.04563(61)
2.281(28) | 0.006 | 0.04505(50)
2.281(28) | 0.008 | 0.04503(64)
Table 9: Decay-amplitude ratios $\Phi_{B_{s}}^{\text{ren}}/\Phi_{B_{q}}^{\text{ren}}$ at the physical strange-quark mass with statistical errors. $a^{-1}$ [GeV] | $am_{l}$ | $am_{q}$ | $\Phi_{B_{s}}^{\text{ren}}/\Phi_{B_{q}}^{\text{ren}}$
---|---|---|---
1.729(25) | 0.005 | 0.0050 | 1.1573(94)
1.729(25) | 0.005 | 0.0100 | 1.1266(60)
1.729(25) | 0.005 | 0.0200 | 1.0705(25)
1.729(25) | 0.005 | 0.0300 | 1.02113(61)
1.729(25) | 0.005 | 0.0343 | 1.001807(41)
1.729(25) | 0.005 | 0.0400 | 0.97777(59)
1.729(25) | 0.010 | 0.0050 | 1.153(11)
1.729(25) | 0.010 | 0.0100 | 1.1242(67)
1.729(25) | 0.010 | 0.0200 | 1.0694(27)
1.729(25) | 0.010 | 0.0300 | 1.02086(65)
1.729(25) | 0.010 | 0.0343 | 1.001789(44)
1.729(25) | 0.010 | 0.0400 | 0.97803(63)
2.281(28) | 0.004 | 0.0040 | 1.1522(81)
2.281(28) | 0.004 | 0.0060 | 1.1393(62)
2.281(28) | 0.004 | 0.0080 | 1.1255(49)
2.281(28) | 0.004 | 0.0250 | 1.01329(28)
2.281(28) | 0.004 | 0.0272 | 1.000694(18)
2.281(28) | 0.004 | 0.0300 | 0.98529(30)
2.281(28) | 0.006 | 0.0040 | 1.1465(64)
2.281(28) | 0.006 | 0.0060 | 1.1347(49)
2.281(28) | 0.006 | 0.0080 | 1.1218(38)
2.281(28) | 0.006 | 0.0250 | 1.01308(22)
2.281(28) | 0.006 | 0.0272 | 1.000691(11)
2.281(28) | 0.006 | 0.0300 | 0.98551(23)
2.281(28) | 0.008 | 0.0040 | 1.1367(85)
2.281(28) | 0.008 | 0.0060 | 1.1264(66)
2.281(28) | 0.008 | 0.0080 | 1.1143(52)
2.281(28) | 0.008 | 0.0250 | 1.01235(30)
2.281(28) | 0.008 | 0.0272 | 1.000666(19)
2.281(28) | 0.008 | 0.0300 | 0.98629(32)
Figure 4: Linear interpolation to determine $\Phi_{B_{s}}$ at the physical
strange-quark mass on the $32^{3}$, $am_{l}=0.006$ ensemble. The black
vertical line with error band shows the physical strange-quark mass with
errors from Ref. Aoki:2010dy . The red sloped line with error band shows the
interpolation of the three strange-ish data points with jackknife statistical
errors from the fit.
## IV Chiral and continuum extrapolations
In this section we present the extrapolation to the physical light-quark
masses and to the continuum limit of the numerical lattice data presented in
the previous section and summarized in Tabs. 7–9. All extrapolations are
performed for dimensionless ratios of decay amplitudes in units of the
$B_{s}$-meson mass; we obtain the physical decay constants in GeV after the
chiral-continuum extrapolation by multiplying by the appropriate power of
$M_{B_{s}}$.
### IV.1 Chiral-continuum extrapolations of $f_{B}$ and $f_{B_{s}}/f_{B}$
We obtain the decay constant $f_{B}$ and the ratio $f_{B_{s}}/f_{B}$ at the
physical pion mass and in the continuum limit using theoretical knowledge of
the light-quark-mass and lattice-spacing dependence from chiral perturbation
theory for heavy-light mesons (HM$\chi$PT) to guide the extrapolation. Chiral
perturbation theory provides a model-independent, low-energy effective
description of QCD in terms of the light-pseudoscalar-meson degrees-of-
freedom, provided that the mesons are sufficiently light. The RBC and UKQCD
collaborations find that next-to-leading order (NLO) SU(3) $\chi$PT does not
describe their data for light pseudoscalar-meson masses and decay constants
near the physical strange-quark mass, but that NLO SU(2) $\chi$PT can be
applied to their heavier data and leads to reasonable estimates for the NNLO
corrections Allton:2008pn . Other collaborations also find that, within its
range of validity, SU(2) $\chi$PT converges more quickly than SU(3) $\chi$PT
Aoki:2008sm ; Kadoh:2008sq ; Bazavov:2009ir . Thus, in this work, we perform
the combined chiral- and continuum extrapolation using NLO SU(2) HM$\chi$PT.
In the SU(2) theory, the strange-quark mass is integrated out, and only the
light-quarks’ degrees-of-freedom are included. The SU(2) low-energy constants
therefore depend upon the value of $m_{s}$, as well as on the value of $m_{b}$
for heavy-light quantities. In Ref. Albertus:2010nm we derived the HM$\chi$PT
expressions for $B_{(s)}$-meson decay constants in the context of our
calculation using domain-wall light quarks and static heavy quarks, which we
quote here:
$\displaystyle\Phi_{B_{x}}$
$\displaystyle=\Phi_{0}\Bigg{\\{}1-\frac{1+3g_{b}^{2}}{(4\pi
f_{\pi})^{2}}\cdot M_{xl}^{2}\ln(M_{xl}^{2}/\Lambda_{\chi}^{2})$
$\displaystyle-\frac{1+3g_{b}^{2}}{(4\pi
f_{\pi})^{2}}\frac{1}{4}\cdot\Big{[}(M_{ll}^{2}-M_{xx}^{2})\cdot(\ln(M_{xx}^{2}/\Lambda_{\chi}^{2})+1)$
$\displaystyle\qquad-M_{xx}^{2}\ln(M_{xx}^{2}/\Lambda_{\chi}^{2})\Big{]}$
$\displaystyle+c_{\text{sea}}\cdot\frac{2Bm_{l}}{(4\pi
f_{\pi})^{2}}+c_{\text{val}}\cdot\frac{2Bm_{x}}{(4\pi f_{\pi})^{2}}$
$\displaystyle+c_{\text{a}}\cdot\frac{a^{2}}{(4\pi
f_{\pi})^{2}a_{32}^{4}}\Bigg{\\}},$ (26)
$\displaystyle\frac{\Phi_{B_{s}}}{\Phi_{B_{x}}}$
$\displaystyle=R_{\Phi^{(2)}}\Bigg{\\{}1+\frac{1+3g_{b}^{2}}{(4\pi
f_{\pi})^{2}}\cdot M_{xl}^{2}\ln(M_{xl}^{2}/\Lambda_{\chi}^{2})$
$\displaystyle{+}\frac{1+3g_{b}^{2}}{(4\pi
f_{\pi})^{2}}\frac{1}{4}\cdot\Big{[}(M_{ll}^{2}-M_{xx}^{2})\cdot(\ln(M_{xx}^{2}/\Lambda_{\chi}^{2})+1)$
$\displaystyle\qquad-
M_{xx}^{2}\ln(M_{xx}^{2}/\Lambda_{\chi}^{2})\Big{]}\Bigg{\\}}$
$\displaystyle+d_{\text{sea}}^{(2)}\cdot\frac{2B}{(4\pi
f)^{2}}m_{l}+d_{\text{val}}^{(2)}\cdot\frac{2Bm_{x}}{(4\pi f)^{2}}$
$\displaystyle+d_{\text{a}}^{(2)}\cdot\frac{a^{2}}{(4\pi f)^{2}a_{32}^{4}}.$
(27)
where $x$ denotes the light valence quark in the $B_{x}$ meson, $l$ the light
sea quark, and $M_{xy}$ denotes a “pion” composed of two domain-wall valence
quarks with flavors $x$ and $y$. At tree level, the light pseudoscalar pion
masses are given in terms of the constituent quark masses $m_{x}$ and $m_{y}$
by
$\displaystyle M_{xy}^{2}=B(m_{x}+m_{y}+2m_{\rm res}).$ (28)
The fit functions Eq. (IV.1) and (IV.1) incorporate discretization errors due
to the light-quark and gluon actions via the residual-quark mass in Eq. (28)
and the analytic term in $a^{2}$.
From simple power-counting, we estimate that discretization errors in the
decay amplitudes on the $32^{3}$ ensembles from the light-quark and gluon
actions are of ${\mathcal{O}}\left(a\Lambda_{\rm QCD}\right)^{2}\sim 5\%$,
using $\Lambda_{\rm QCD}=500$ MeV. There are also light-quark and gluon
discretization errors in the heavy-light current, and heavy-quark
discretization errors from both the action and current. In Secs. V.5–V.6 and
App. B we estimate the size of these other discretization errors to be below
2%. Thus we expect light-quark and gluon discretization errors from the action
to dominate the scaling behavior of the decay amplitudes, and that including
an $a^{2}$ term in the fit will largely remove these contributions. Heavy-
quark discretization errors as well as light-quark and gluon discretization
errors in the current will be estimated using power-counting and added a
posteriori to the systematic error budget.
Several parameters enter the expressions in Eq. (IV.1) and (IV.1). We take the
values of the lattice spacings and low-energy constant $B$ from the RBC/UKQCD
analysis of light pseudoscalar meson masses and decay constants in Ref.
Aoki:2010dy . We use the experimental value of $f_{\pi}=130.4$ MeV from the
PDG Beringer:2012zz , and use $\Lambda_{\chi}=1$ GeV for the scale in the
chiral logarithms. We take the $B^{\ast}B\pi$-coupling constant,
$g_{b}=0.57(8)$ from our companion analysis Flynn:2013kwa using the same
actions and ensembles. The constant parameters used in our chiral fits are
compiled in Table 10.
Table 10: Constants used in the chiral and continuum extrapolations of $\Phi_{B}^{\text{ren}}$, $\Phi_{B_{s}}^{\text{ren}}$ and $\Phi_{B_{s}}^{\text{ren}}/\Phi_{B}^{\text{ren}}$ Aoki:2010dy ; Beringer:2012zz ; Flynn:2013kwa . | $24^{3}$ | $32^{3}$
---|---|---
$a^{-1}$ | 1.729 GeV | 2.28 GeV
$aB$ | 2.348 | 1.826
$f_{\pi}$ | 130.4 MeV
$g_{b}$ | 0.57
$\Lambda_{\chi}$ | 1 GeV
We cannot obtain a good fit (as measured by the $\chi^{2}/{\rm dof}$ or
$p$-value) to our entire data set using the NLO SU(2) HM$\chi$PT expressions
above. This is not surprising given that our heaviest pseudoscalar mesons, in
which the valence-quark masses are close to that of the physical strange
quark, have masses around 600 MeV. We therefore tried removing the heaviest
points from our fits. We find that we can obtain good fits of $\Phi_{B}$ with
NLO HM$\chi$PT while including as much of our data as possible when we impose
the following cut: $M_{\pi}^{\text{sea}}{\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
316\relax$}}}425$ MeV and $M_{\pi}^{\text{val}}{\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
316\relax$}}}350$ MeV. To obtain an acceptable fit for the decay-constant
ratio $\Phi_{B_{s}}/\Phi_{B}$, however, we must make an equally-stringent cut
on the sea-pion masses:
$M_{\pi}^{\text{val}},M_{\pi}^{\text{sea}}\mathrel{\hbox to0.0pt{\lower
3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar
316\relax$}}350$ MeV. We also tried adding higher-order terms analytic in the
pion mass in order to extend the reach of the HM$\chi$PT expressions. We find,
however, that multiple NNLO analytic terms are needed to improve the
$p$-value, at which point the errors on the extrapolated values of $f_{B}$ and
$f_{B_{s}}/f_{B}$ become uncontrolled because we cannot sufficiently constrain
the coefficients with data at only two lattice spacings and a narrow range of
light-quark masses. Finally, we tried NLO SU(2) fits to only the five unitary
data points with $M_{\pi}^{\text{val}}=M_{\pi}^{\text{sea}}$, all of which
satisfy $M_{\pi}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}425$ MeV. Without the
partially-quenched data, which are strongly correlated on a given ensemble, we
obtain good $p$-values for fits of both $\Phi_{B}$ and
$\Phi_{B_{s}}/\Phi_{B}$. Thus we take the unitary NLO SU(2) HM$\chi$PT fit
results as our central values for $f_{B}$ and $f_{B_{s}}/f_{B}$.
Our preferred fits are shown in Fig. 5. The decay amplitudes of the neutral
and charged $B$ mesons at the physical light-quark masses and in the continuum
are obtained by setting the lattice spacing to zero and the light-quark mass
to $m_{d}$ and $m_{u}$, respectively, in the chiral-continum fit
function.111Technically, the light sea-quark mass should be fixed to
$(m_{u}+m_{d})/2$, but we cannot change the sea- and valence-quark masses
independently in our unitary chiral-continuum extrapolation. We expect the
sea-quark mass dependence of the decay constants, however, to be much smaller
than the valence-quark mass dependence from the partially-quenched HM$\chi$PT
expressions in Eqs. (IV.1)–(IV.1). Further, we do not observe any
statistically-significant sea-quark mass dependence in our data. Thus fixing
both the valence- and sea-quark masses to either $m_{u}$ or $m_{d}$ provides a
good approximation. Our results for the decay constants are
$f_{B^{0}}=199.5(6.2)$ MeV and $f_{B^{+}}=195.6(6.4)$ MeV, and for the ratios
are $f_{B_{s}}/f_{B^{0}}=1.197(13)$ and $f_{B_{s}}/f_{B^{+}}=1.223(14)$, where
all errors are statistical only. We find about a 1.5% difference between
$f_{B^{0}}$ and $f_{B^{+}}$, which is consistent with the recent four-flavor
lattice calculation of this splitting by HPQCD using NRQCD $b$ quarks
Dowdall:2013tga .
Figure 5: Chiral and continuum extrapolation of $\Phi_{B_{q}}$ (left) and
$\Phi_{B_{s}}/\Phi_{B_{q}}$ (right) from a correlated fit using NLO SU(2)
HM$\chi$PT. The different colors/symbols distinguish our data points on the
five different ensembles. For better visibility data points on the
$am_{l}=0.004,\,0.008,\,0.01$ ensembles are plotted with a small horizontal
offset. We plot all partially-quenched data, but the fit only includes the
five unitary points (filled). The colored fit curves show the extrapolation in
light-quark mass: the fit function is evaluated in full QCD with $m_{x}=m_{l}$
at the nonzero lattice spacings on the different ensembles, such that the
curves should approximately go through the filled data points of similar
color. The chiral extrapolation in full QCD and the continuum is shown by the
black line with grey error band. The physical values of $\Phi_{B^{+}}$
($\Phi_{B^{0}}$) and $\Phi_{B_{s}}/\Phi_{B^{+}}$ ($\Phi_{B_{s}}/\Phi_{B^{0}}$)
correspond to the intersection of this curve with the dashed (dot-dashed)
vertical line on the left-hand side indicating the physical $u$-quark
($d$-quark) mass. The right-hand solid, vertical line indicates the $s$-quark
mass. Only statistical errors are shown.
In Fig. 5, the colored fit curves show the extrapolation in light-quark mass
at fixed lattice spacing. They go approximately through the unitary data
points included in the fit, and curve downward for $\Phi_{B}$ (upward for
$\Phi_{B_{s}}/\Phi_{B}$) as they approach chiral limit due to the chiral
logarithms in the SU(2) HM$\chi$PT fit functions, Eqs. (IV.1)–(IV.1). Because
the coefficients of the NLO chiral logarithms are fixed in terms of $g_{b}$
and $f_{\pi}$, the fit yields large chiral logarithms at pion masses below
$\sim 200$ MeV despite the fact that our data is too heavy for us to observe
their onset. Towards the right-hand sides of the plots in Fig. 5, the valence
quarks become appreciably heavier than the light sea quarks ($m_{x}\gg m_{l}$)
and the full-QCD fit curves deviate substantially from the partially-quenched
data points. Our decay-constant data displays no significant dependence on
either the light sea-quark mass or the lattice spacing. We find that the
coefficients of the $a^{2}$ terms are $\sim 0.01$ or smaller for both fits in
Fig. 5, and that $c_{a}$ is in fact statistically consistent with zero in the
left-hand fit. In alternate fits that include some partially-quenched data,
the coefficients of the sea-quark mass terms are closer to ${\mathcal{O}}(1)$,
but with $\sim 50$% or larger errors.
We also tried fits using SU(3) HM$\chi$PT, in which the strange-quark mass is
explicit in the fit functions. This in principle has the advantage of building
in the constraint that $\Phi_{B_{s}}/\Phi_{B_{q}}=1$ in the SU(3) limit
$m_{q}=m_{s}$. Although we can obtain reasonable NLO SU(3) fits of $\Phi_{B}$
when $M_{\pi}^{\text{sea}}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}425$ MeV and
$M_{\pi}^{\text{val}}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}400$ MeV, we were
unable to obtain any acceptable NLO SU(3) fit of the ratio
$\Phi_{B_{s}}^{\text{ren}}/\Phi_{B_{q}}^{\text{ren}}$ because the data are so
precise. This is consistent with observations by the RBC and UKQCD
collaborations in earlier analyses of light pseudoscalar meson masses and
decay constants Allton:2008pn , in which they concluded that NLO SU(3)
$\chi$PT was incompatible with their heaviest data. We can only successfully
describe the data for $\Phi_{B_{s}}^{\text{ren}}/\Phi_{B_{q}}^{\text{ren}}$
with SU(3) HM$\chi$PT after adding several NNLO analytic terms and restricting
the masses to $M_{\pi}^{\text{sea}}<425$ MeV and $M_{\pi}^{\text{val}}<400$
MeV. Although we obtain results that are consistent with our preferred NLO
SU(2) fits, the statistical errors from the NNLO SU(3) fits are larger because
of the increased number of fit parameters, several of which are not well-
constrained by the data.
We use the alternative SU(3) fits with good $p$-values, as well as fits with
only analytic dependence on the quark masses and lattice spacings, to estimate
the systematic uncertainty due to the chiral-continuum extrapolation in Sec.
V.1.
### IV.2 Decay constant $f_{B_{s}}$
After interpolating the decay amplitude $\Phi_{B_{s}}^{\text{ren}}$ to the
physical strange-quark mass, we only need to extrapolate to the continuum. We
do not observe any sea-quark mass dependence in our data, so we use a simple
linear function in $a^{2}$,
$\displaystyle\Phi_{B_{s}}=\varrho\,a^{2}+\varphi,$ (29)
which captures the leading scaling behavior from the light-quark and gluon
actions. Again, discretization errors from the heavy-quark action will be
estimated via heavy-quark power-counting in Sec. V.5 and added to the
systematic error budget. We show the continuum extrapolation of $\Phi_{B_{s}}$
in Fig. 6; our result is $\Phi_{B_{s}}=0.158(3)$, which corresponds to
$f_{B_{s}}=235.4(5.2)$ MeV (statistical errors only).
Figure 6: Continuum extrapolation of $\Phi_{B_{s}}/M_{B_{s}}^{3/2}$ from a
linear fit in $a^{2}$. We plot the five data points for
$\Phi_{B_{s}}/M_{B_{s}}^{3/2}$ interpolated to the physical strange-quark mass
on each ensemble using the same colors/symbols as in Fig. 5. The extrapolation
is shown by the black line with gray error band. For better visibility data
points on the $am_{l}=0.004,\,0.008,\,0.01$ ensembles are plotted with a small
horizontal offset. Only statistical errors are shown.
## V Estimation of systematic errors
We now discuss the sources of systematic uncertainties in our determinations
of the $B_{(s)}$-meson decay constants and their ratio. Each uncertainty is
discussed in a separate subsection and the total error budgets are provided in
Table 11.
Table 11: Error budgets for the $B_{(s)}$-meson decay constants and their ratios. Errors that were considered but found to be negligible are listed as “0.0.” Errors are given in %. The total error is obtained by adding the individual errors in quadrature. | $f_{B^{0}}$(%) | $f_{B^{+}}$(%) | $f_{B_{s}}$(%) | $f_{B_{s}}/f_{B^{0}}$(%) | $f_{B_{s}}/f_{B^{+}}$(%)
---|---|---|---|---|---
statistics | 3.1 | 3.3 | 2.2 | 1.1 | 1.1
chiral-continuum extrapolation | 4.4 | 5.9 | 3.1 | 3.9 | 5.5
lattice-scale uncertainty | 1.5 | 1.5 | 1.5 | 0.1 | 0.1
light- and strange-quark mass uncertainty | 0.1 | 0.1 | 0.9 | 0.8 | 0.9
RHQ parameter tuning | 1.2 | 1.2 | 1.2 | 0.1 | 0.1
HQ discretization errors | 1.7 | 1.7 | 1.7 | 0.3 | 0.3
LQ and gluon discretization errors | 1.1 | 1.1 | 1.2 | 0.6 | 0.6
renormalization factor | 1.7 | 1.7 | 1.7 | 0.0 | 0.0
finite volume | 0.4 | 0.5 | 0.0 | 0.5 | 0.5
isospin-breaking and EM | 0.7 | 0.7 | 0.7 | 0.1 | 0.7
total | 6.3 | 7.6 | 5.2 | 4.2 | 5.8
### V.1 Chiral- and continuum extrapolation
We estimate the systematic uncertainty due to the chiral- and continuum
extrapolation of the decay constants by varying the chiral-continuum
extrapolation fit inputs and Ansätze. From the alternative fits tried, we take
the largest difference of the central value to be the chiral-continuum
extrapolation error. We do not include fits with poor $p$-values in looking
for the largest difference because such fits are not compatible with the data.
For $f_{B}$ and the ratio $f_{B_{s}}/f_{B}$, we vary the inputs to our
preferred NLO SU(2) HM$\chi$PT fits in the following ways:
* •
excluding the heaviest unitary data point with $M_{\pi}\sim$ 425 MeV,
* •
including some or all partially-quenched data with
$M_{\pi}^{\text{val,sea}}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}$ 425 MeV,
* •
varying the value of $f_{\pi}$ in the coefficients of the chiral logarithms
from $f_{0}$ in the chiral limit (112 MeV Aoki:2010dy ) to $f_{K}=155.5$ MeV
Beringer:2012zz ,
* •
varying the $B^{*}$-$B$-$\pi$ coupling in the coefficient of the chiral
logarithms $g_{b}=0.57(8)$ by plus or minus one standard deviation
Flynn:2013kwa .
For the partially-quenched fits, we obtain good $p$-values for $f_{B}$ when
$M_{\pi}^{\rm val}$ is below $\sim$ 350 MeV, and for $f_{B_{s}}/f_{B}$ when
both $M_{\pi}^{\rm sea}$ and $M_{\pi}^{\rm val}$ are below $\sim$ 350 MeV. We
also consider the following alternate chiral-continuum extrapolation fit
functions:
* •
analytic fits in which we omit the chiral logarithms in Eqs. (IV.1) and
(IV.1),
* •
analytic fits in which we remove the chiral logarithms and also remove either
the terms linear in the sea-quark mass $m_{l}$, squared lattice-spacing
$a^{2}$, or both,
* •
NLO SU(3) HM$\chi$PT,
* •
SU(2) or SU(3) HM$\chi$PT with NNLO analytic terms.
For these fits we include partially-quenched data with
$M_{\pi}^{\text{val,sea}}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}$ 425 MeV. The
additional constraints are needed for the NNLO fits because they have more
free parameters.
For both, $f_{B}$ and $f_{B_{s}}/f_{B}$, we find that analytic fits produce
the largest changes with respect to the preferred NLO SU(2) HM$\chi$PT fit
results. Specifically, for $f_{B}$ the largest difference is obtained from a
fit to all partially-quenched data with
$M_{\pi}^{\text{val,sea}}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}$ 425 MeV and a
single analytic term proportional to the light valence-quark mass $m_{x}$. For
$f_{B_{s}}/f_{B}$, a fit to only the unitary data with $M_{\pi}\mathrel{\hbox
to0.0pt{\lower 3.0pt\hbox{$\mathchar 536\relax$}\hss}\raise
2.0pt\hbox{$\mathchar 316\relax$}}$ 425 MeV and terms proportional to both the
light-quark mass $m_{x}$ and squared lattice spacing $a^{2}$ leads to the
largest shift in the central value. We show these fits in Fig. 7. Our data
shows no evidence of curvature and the analytic fits have excellent
$p$-values. Once our pion masses are sufficiently light, however, HM$\chi$PT
predicts the onset of chiral logarithms that will reduce the value of $f_{B}$
(and increase $f_{B_{s}}/f_{B}$) relative to the results of the analytic fits.
Thus we expect the true value of $f_{B}$ to be lower than that from the
analytic fit (and of $f_{B_{s}}/f_{B}$ to be higher). Nevertheless, for both
quantities we conservatively take the full difference between the SU(2)
HM$\chi$PT and analytic fits as the systematic error due to the chiral
extrapolation, so our error estimates cover the results for $f_{B}$ and
$f_{B_{s}}/f_{B}$ from the analytic fits.
Figure 7: Alternate chiral-continuum extrapolations for $\Phi_{B}$ (upper) and
$\Phi_{B_{s}}/\Phi_{B}$ (lower) used to obtain our chiral-continuum
extrapolation error estimate. Both fits use analytic fit Ansätze. The
$\Phi_{B}$ fit includes all partially-quenched data with
$M_{\pi}^{\text{val,sea}}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}$ 425 MeV and a
single analytic term linear in the valence-quark mass $m_{x}$. The
$\Phi_{B_{s}}/\Phi_{B}$ fit uses only the unitary data with
$M_{\pi}\mathrel{\hbox to0.0pt{\lower 3.0pt\hbox{$\mathchar
536\relax$}\hss}\raise 2.0pt\hbox{$\mathchar 316\relax$}}$ 425 MeV and
includes terms proportional to both $m_{x}$ and $a^{2}$. Colors/symbols are
the same as in Fig. 5, and for better visibility data points on the
$am_{l}=0.004,\,0.008,\,0.01$ ensembles are plotted with a small horizontal
offset. Only statistical errors are shown. For comparison, the results for
$f_{B^{+}}$ ($f_{B^{0}}$) from our preferred unitary NLO SU(2) HM$\chi$PT fits
are shown as open (filled) black circles.
For $f_{B_{s}}$, the preferred continuum extrapolation is from a fit linear in
$a^{2}$. Our ability to perform alternate fits, however, is limited by the
fact that we only have two values of the lattice spacing and therefore do not
have sufficient data to add quadratic or even higher-order terms. Because the
lattice-spacing dependence of our data is quite mild (the results on of all
our five ensembles are statistically consistent), we take as an alternative
the weighted average of the finer $32^{3}$ data points.
### V.2 Lattice-scale uncertainty
We exploit the procedure that we use to tune the parameters of the RHQ action
to minimize the uncertainty due to the input lattice scale in our final
results for the decay constants. By construction, at the tuned point the
$B_{s}$-meson mass is fixed to the experimentally-measured value. We therefore
choose to perform our decay-constant analysis in terms of dimensionless ratios
over $M_{B_{s}}$. We can then obtain the decay constants in GeV by multiplying
the ratio by $M_{B_{s}}=5.366$ GeV from the PDG Nakamura:2010zzi . Hence our
decay-constant results have no explicit dependence on the lattice-scale; we
do, however, still need to consider the implicit dependence on the lattice
spacing through the RHQ parameters.
We estimate this source of scale uncertainty by measuring the slope of the
decay constants and ratios with respect to the RHQ parameters {$m_{0}a$,
$c_{P}$, $\zeta$}. These are shown for the $32^{3}$, $am_{l}=0.006$ ensemble
in Fig. 8. We then multiply each of these slopes by the uncertainty in the
corresponding RHQ parameter due to the lattice scale as provided in Tab. 5,
e.g. $\Delta(\Phi)/\Delta(c_{P})\times\sigma(c_{P})_{a}$. Finally, for each of
our data points on the $24^{3}$ and $32^{3}$ ensembles, we add the three
contributions for each data point in quadrature. For each physical quantity,
we take the largest estimated total as the uncertainty due to the lattice
scale, which gives 1.5% for the decay constants and 0.1% for the ratios.
Figure 8: Dependence of the decay amplitudes $\Phi_{B_{q}}/M_{Bs}^{3/2}$ and
the ratio $\Phi_{B_{s}}/\Phi_{B_{q}}$ on our RHQ parameters. The plots on the
left show the decay amplitude for $am_{q}=0.006$ (red triangles) and
$am_{q}=0.0272$ (green circles) vs. the RHQ parameters $m_{0}a$, $c_{P}$, and
$\zeta$ (from top to bottom). The plots on the right show dependence of the
ratio $\Phi_{Bs}/\Phi_{B}$ obtained for $am_{s}=0.0272$ and $am_{q}=0.006$ on
$m_{0}a$, $c_{P}$ and $\zeta$. Only statistical errors are shown.
### V.3 Light- and strange-quark mass uncertainties
Here we estimate the uncertainties due to the input quark masses in the
chiral-continuum extrapolations, as well as the mis-tuning of the strange sea
quark. We discuss each source of uncertainty in a separate subsection for
clarity. Because most of the individual uncertainty estimates turn out to be
small relative to other errors, we add the numbers from the three subsections
in quadrature and quote a single error due to to the light- and strange-quark
mass uncertainties in Table 11.
#### V.3.1 Light $u$\- and $d$-quark mass uncertainties
In the chiral-continuum extrapolations of $\Phi_{B_{x}}$ and
$\Phi_{B_{s}}/\Phi_{B_{x}}$, we set the light-quark mass to the physical
$d$-quark mass to obtain the neutral-meson decay constant $f_{B^{0}}$ and the
corresponding ratio $f_{B_{s}}/f_{B^{0}}$, and to the physical $u$-quark mass
to obtain $f_{B^{+}}$ and $f_{B_{s}}/f_{B^{+}}$. We use the preliminary values
of the quark masses $a_{32}m_{d}=1.327(13)\times 10^{-3}$ and
$a_{32}m_{u}=6.06(24)\times 10^{-4}$ from simulations by the RBC/UKQCD
collaborations including both QCD and QED. We estimate the uncertainty due to
the determination of the light-quark masses by repeating the chiral-continuum
extrapolation with $m_{d}$($m_{u}$) shifted by plus and minus one sigma. We
observe small changes in the central values between 0.0–0.1%.
For $f_{B_{s}}$, we study the dependence of our three (two) $32^{3}$
($24^{3}$) data points on the light sea-quark mass. Because we cannot resolve
any sea-quark mass dependence within our statistical uncertainties, we take
this error to be negligible.
#### V.3.2 Valence $s$-quark mass uncertainty
We estimate the errors in $f_{B_{s}}$ and $f_{B_{s}}/f_{B}$ due to the
uncertainty in the valence strange quark mass by repeating the interpolation
to $m_{s}$ described in Sec. III.4 and then using these new values as inputs
to the chiral and continuum extrapolations. We vary independently the values
for $a_{32}m_{s}=0.0280(7)$ and $a_{24}m_{s}=0.0369(11)$ on the $32^{3}$ and
$24^{3}$ ensembles again by one sigma Aoki:2010dy . We find shifts in the
central values due to varying $m_{s}$ of 0.8–0.9% for $f_{B_{s}}$,
$f_{B_{s}}/f_{B^{0}}$, and $f_{B_{s}}/f_{B^{+}}$.
#### V.3.3 Strange sea-quark mistuning
Our ensembles were generated with the heavy sea-quark mass $m_{h}$
approximately 10% heavier than that of the physical strange quark, and with
only a single value of $m_{h}$ at each lattice spacing. Thus we cannot
directly study the strange sea-quark mass dependence with our data, and must
use the light sea-quark mass dependence as a proxy. Because, however, we use
SU(2) HM$\chi$PT for the chiral-continuum extrapolations of $f_{B}$ and
$f_{B_{s}}/f_{B}$, and a linear-in-$a^{2}$ continuum extrapolation for
$f_{B_{s}},$ the fit functions do not have any explicit strange-quark
dependence, so we cannot interpolate to the correct strange sea-quark mass a
posteriori.
We therefore study the data for the decay constants at fixed valence-quark
mass on the three $32^{3}$ ensembles, and on the two $24^{3}$ ensembles. In
the case of the decay amplitudes $\Phi_{B}$ and $\Phi_{B_{s}}$, the
statistical uncertainties are too large to resolve any sea-quark mass
dependence; hence we quote for these quantities an error of 0.0% in Tab. 11.
The statistical errors in the ratio $\Phi_{B}/\Phi_{B_{s}}$ are sufficiently
small that we can resolve the light sea-quark mass dependence. We therefore
perform a linear fit in $m_{l}$ to the three (two) data points to obtain the
slope with respect to $m_{l}$. Because the leading sea-quark mass dependence
enters as $(2m_{l}+m_{h})$ in SU(3) $\chi$PT, we expect the slope with respect
to $m_{h}$ to be roughly half this value. Using this slope to correct the
central value of $f_{B_{s}}/f_{B}$ leads to a change of 0.5%. We take this
entire shift to be the error in $f_{B_{s}}/f_{B}$ due to strange sea-quark
mass mistuning, but it is clearly a conservative upper bound.
### V.4 RHQ parameter uncertainty
Although we tune the values of the RHQ parameters to correspond to the
physical $b$-quark, the tuned parameter values have both statistical and
systematic uncertainties.
As described in Sec. II.2, we use seven sets of RHQ parameters and then
interpolate to the physical $b$-quark mass using the jackknife blocks from our
tuning analysis. The advantage of this method is that the statistical
uncertainties in the RHQ parameters are automatically included in the
statistical errors of the decay amplitudes.
In Ref. Aoki:2012xaa we presented a systematic error budget for each of the
parameters $m_{0}a$, $c_{P}$, and $\zeta$; the results are shown in Table 5.
We already estimated the uncertainty in the decay constants due to the scale
uncertainty in the RHQ parameters in Sec. V.2. Hence we need only consider the
errors in the RHQ parameters due to the other two sources: heavy-quark
discretization errors and experimental inputs used in the tuning procedure. As
in the case of the lattice-scale uncertainty, we use the slopes of the decay
constants with respect to $\\{m_{0}a,c_{P},\zeta\\}$. For each of the three
RHQ parameters $\\{m_{0}a,c_{P},\zeta\\}$ and two sources of uncertainty {HQ,
exp.}, we estimate the error in the decay amplitudes as, e.g.,
$\Delta(\Phi_{B})/\Delta(m_{0}a)\times\sigma_{\rm HQ}$. We then add these six
individual contributions in quadrature. We do this for all five sea-quark
ensembles, and take the largest total to be the error in the decay constants
due to the systematic errors associated with the RHQ tuning procedure. For the
decay constants $f_{B}$ and $f_{B_{s}}$, we obtain about 1.2%, and for the
ratio $f_{B_{s}}/f_{B}$ about 0.1%.
### V.5 Heavy-quark discretization errors
The RHQ action gives rise to nontrivial lattice-spacing dependence in the
decay constants in the region $m_{0}a\sim 1$. Thus, instead of including
additional functions of $m_{0}a$ in the combined chiral-continuum
extrapolation, we estimate the size of heavy-quark discretization errors using
power-counting. We follow the approach developed by El Khadra, Kronfeld, and
Mackenzie ElKhadra:1996mp for lattice calculations using the anisotropic
Clover action for heavy quarks, and later extended to include dimension 6 and
7 operators in Oktay and Kronfeld Oktay:2008ex .
Heavy-quark discretization errors in our decay-constant calculation arise from
two sources: operators in the heavy-quark action and in the heavy-light axial-
vector current. We tune the parameters of the dimension-5 RHQ action
nonperturbatively; therefore the leading discretization errors from the action
are of ${\mathcal{O}}(a^{2})$. We use an ${\cal O}(a)$-improved current
operator with the improvement coefficient computed at 1-loop in $\alpha_{s}$;
therefore the leading discretization errors from the current are of
${\mathcal{O}}(\alpha_{s}^{2}a)$ and ${\mathcal{O}}(a^{2})$.222This general
approach for error estimation is also used in similar calculations of heavy-
light meson decay constants and form factors using the Fermilab action for
$b$-quarks by the Fermilab Lattice and MILC collaborations Bailey:2008wp ;
Bazavov:2011aa . The primary differences stem the fact that (1) we remove
${\mathcal{O}}(a)$ errors from the action to all orders in $\alpha_{s}$ by
tuning the parameters of the anisotropic Clover action nonperturbatively, and
(2) we calculate the ${\mathcal{O}}(a)$-improvement coefficient of the heavy-
light axial-vector current to one higher order in $\alpha_{s}$. Thus the
heavy-quark discretization errors in our calculation are smaller than those in
the Fermilab/MILC work.
To obtain the numerical error estimates we first consider a nonrelativistic
description of the heavy-quark action. Both the lattice and continuum theories
can be described by effective Lagrangians built from the same operators, and
discretization errors arise due to mismatches between the short-distance
coefficients of higher-dimension operators in the two theories. For each
operator ${\cal O}_{i}$ in the heavy-quark effective Lagrangian or current,
the associated discretization error is given by
$\displaystyle\textrm{error}_{i}=\left({\cal C}_{i}^{\textrm{lat}}-{\cal
C}_{i}^{\textrm{cont}}\right)\langle{\cal O}_{i}\rangle\,.$ (30)
For heavy-light meson systems, the sizes of matrix elements can be estimated
using Heavy-Quark Effective Theory (HQET) power-counting. Continuum HQET is an
expansion in the spatial momentum of the heavy quark, $\vec{p}/m_{b}$. The
$b$-quarks in $B$ hadrons typically carry a spatial momentum
$\vec{p}\approx\Lambda_{\rm QCD}$, the scale of the strong interactions. The
lattice introduces an additional scale, $a$. The relative error contribution
to a physical quantity such as the $B$-meson decay constant from an operator
with dimension $d$ can then be estimated as $\langle{\cal
O}_{i}\rangle\sim(a\Lambda_{\rm QCD})^{d-4+n_{\Gamma}}$, where
$n_{\Gamma}=0\,(1)$ if the operator commutes (anticommutes) with
$\gamma_{4}$.333Operators in the Symanzik effective Lagrangian that
anticommute with $\gamma_{4}$ are suppressed because they connect large upper
spinor components with small lower spinor components. Hence they are promoted
to operators of one dimension higher in the heavy-quark effective Lagrangian.
The details of our numerical error estimation are provided in Appendix B. We
compute the sizes of the mismatch coefficients using the tuned parameters of
the RHQ action; their values on the $24^{3}$ and $32^{3}$ ensembles are given
in Table 12. We take $\Lambda_{\rm QCD}=500$ MeV as suggested by fits to
moments of inclusive $B$-decays Buchmuller:2005zv . We add the contributions
from the individual operators in quadrature to obtain the total uncertainty.
Finally, we take the size of heavy-quark discretization errors in our
calculation of the $B_{(s)}$-meson leptonic decay constants to be the estimate
on our finer $a^{-1}=2.281$ GeV lattices (see Table 13), which is 1.7%. For
the ratios we estimate that discretization errors will be suppressed by the
SU(3)-breaking factor $\left(m_{s}-m_{d}\right)/\Lambda_{\text{QCD}}$. Using
the quark-mass determinations from FLAG Aoki:2013ldr we estimate the
uncertainty in $f_{B_{s}}/f_{B}$ from heavy-quark discretization errors to be
about 0.3%.
### V.6 Light-quark and gluon discretization errors
The dominant discretization errors from the light-quark and gluon sectors are
of ${\mathcal{O}}\left(a\Lambda_{\rm QCD}\right)^{2}$ from the action, which
we estimate to be $\sim 5\%$ on the finer $32^{3}$ ensembles. We remove these
errors in our chiral-continuum extrapolation by including an analytic term
proportional to $a^{2}$ in the fit function. We estimate the light-quark and
gluon discretization errors in the heavy-light axial-vector current, which are
subleading, with power-counting and add them in quadrature with the other
uncertainties in the error budget.
The leading light-quark and gluon discretization errors in the current are of
${\mathcal{O}}\left(\alpha_{s}a\widetilde{m}_{q},\,(a\widetilde{m}_{q})^{2},\,\alpha_{s}^{2}a\Lambda_{\rm
QCD}\right)$, where $a\widetilde{m}_{q}$ denotes the bare lattice mass. On the
$32^{3}$ ensembles, the first term leads to an $\sim 0.6\%$ uncertainty in
$f_{B_{s}}$ (using $a\widetilde{m}_{s}$) and uncertainties below $0.1\%$ in
$f_{B^{+}}$ and $f_{B^{0}}$ (using $a\widetilde{m}_{ud}$). The second term is
significantly smaller, about $\sim 0.1\%$ in $f_{B_{s}}$ and negligible in
$f_{B^{+}}$ and $f_{B^{0}}$. The third term leads to uncertainties of $\sim
1.1\%$ in all three decay constants. Adding these three contributions in
quadrature, we estimate the total uncertainty from light-quark and gluon
discretization errors in the heavy-light current to be about $1.2\%$ in
$f_{B_{s}}$ and about $1.1\%$ in $f_{B^{+}}$ and $f_{B^{0}}$. For the decay-
constant ratios, we estimate the error from the larger quark-mass dependent
term to be of
${\mathcal{O}}\left(\alpha_{s}(a\widetilde{m}_{s}-a\widetilde{m}_{ud})\right)\sim
0.6\%$. This contribution is not suppressed because the strange-quark mass is
so much larger than the light up- and down-quark masses. We estimate the error
from the lattice-spacing dependent term to be of
${\mathcal{O}}\left(\alpha_{s}^{2}a(m_{s}-m_{ud})\right)\sim 0.2\%$, which is
about five times smaller than in the individual decay constants. Again, adding
the contributions in quadrature, we estimate the total uncertainty from light-
quark and gluon discretization errors in the heavy-light current to be about
$0.6\%$ in all three decay-constant ratios.
### V.7 Renormalization factor
In our computation we divide the heavy-light current renormalization factor
into three contributions $Z_{\Phi}=\rho_{A}^{bl}\sqrt{Z_{V}^{ll}Z_{V}^{bb}}$;
we consider the errors from each of these factors in turn. For
$\rho_{A}^{bl}$, we must estimate the uncertainty due to truncating the
perturbative series in $\alpha_{s}$. We conservatively take the full size of
the 1-loop correction on the fine lattice, 1.7%, as the estimate. For
$Z_{V}^{ll}$ we use the nonperturbative determination of the axial-current
renormalization factor $Z_{A}$ in the chiral limit from Ref. Aoki:2010dy . The
statistical uncertainty in $Z_{A}$ on the finer ensemble is 0.02%, which is
negligible compared to our other sources of error. The renormalization factors
$Z_{V}$ and $Z_{A}$ for domain-wall fermions differ, however, by
${\mathcal{O}}(am_{\rm res})$, which is about $3\times 10^{-3}$ on our finer
ensembles. Thus we take 0.3% to be the systematic uncertainty in $Z_{V}^{ll}$
due to chiral symmetry breaking. Further, because we use the values
$Z_{V}^{ll}$ and $\rho_{A}^{bl}$ in the chiral limit, we must consider the
errors due to the nonzero physical up, down, and strange-quark masses. The
leading quark-mass dependent errors in $Z_{V}^{ll}$ are of
${\mathcal{O}}\left((a\widetilde{m}_{q})^{2}\right)$, and in $\rho_{A}^{bl}$
are of ${\mathcal{O}}\left(\alpha_{s}a\widetilde{m}_{q}\right)$. These
contributions are already included in the estimate of light-quark and gluon
discretization errors in Sec. V.6 above, so we do not count them again here.
For $Z_{V}^{bb}$, we use the weighted average of the two (three)
determinations on the $24^{3}$ ($32^{3}$) ensembles. The statistical
uncertainty in $Z_{V}^{bb}$ on the finer ensemble is again small, 0.15%.
Adding the contributions from $\rho_{A}^{bl}$, $Z_{V}^{ll}$, and $Z_{V}^{bb}$
in quadrature, we estimate the total error in the decay constants from the
renormalization factor to be 1.7%. Because we use the values $Z_{V}^{ll}$ and
$\rho_{A}^{bl}$ in the chiral limit, the renormalization factor $Z_{\Phi}$
cancels exactly in our computation of the ratio $f_{B_{s}}/f_{B}$, so we quote
an error of “0.0%” in the error budget.
### V.8 Finite volume errors
We estimate the error due to the finite spatial lattice volume using one-loop
finite-volume SU(2) HM$\chi$PT. In the finite-volume theory, the loop
integrals become sums over lattice sites, such that the chiral logarithms in
the fit functions in Eqs. (IV.1) and (IV.1) become bessel functions. For
$f_{B}$ and $f_{B_{s}}/f_{B}$ we repeat the combined chiral- and continuum
extrapolation using the finite volume SU(2) HM$\chi$PT expressions. These lead
to shifts in the central values of 0.4–0.5% for $f_{B}$ and 0.5% for
$f_{B_{s}}/f_{B}$. For $f_{B_{s}}$ we do not perform a chiral extrapolation,
but we can still calculate the size of the corrections to the data points
using finite-volume HM$\chi$PT. The use of SU(2) $\chi$PT at the strange-quark
mass may in general be questionable, but we expect it to be good enough to
obtain a rough estimate of the systematic error. We find that the finite-
volume corrections to $f_{B_{s}}$ are below $0.05\%$ on all of our ensembles,
and hence quote an uncertainty of 0.0% in the error budget.
### V.9 Isospin breaking and electromagnetism
The decay constants of the charged and neutral $B$ mesons $f_{B^{+}}$ and
$f_{B^{0}}$ differ due to both the masses and the charges of the constituent
light $u$ and $d$ quarks. The quark-mass contribution to this difference comes
from the valence-quark masses, and the leading term is of
${\mathcal{O}}\left(\Delta m_{ud}/\Lambda_{\rm QCD}\right)$, where $\Delta
m_{ud}\equiv(m_{d}-m_{u})$. We account for this effect by extrapolating the
light valence quark to either the physical $u$\- or $d$-quark mass in the
chiral-continuum extrapolation, and find that this leads to a difference of
1.5% between $f_{B^{+}}$ and $f_{B^{0}}$. This observed size is somewhat
larger than the power-counting estimate of 0.5% obtained using the
determination of the quark masses from FLAG Aoki:2013ldr and $\Lambda_{\rm
QCD}=500$ MeV, but is close to the 2% difference observed by HPQCD in Ref.
Dowdall:2013tga . The electromagnetic contribution to the difference between
$f_{B^{+}}$ and $f_{B^{0}}$ is expected to be the typical size of 1-loop QED
corrections, or ${\mathcal{O}}(\alpha_{QED})\sim 0.7\%$. Thus we estimate that
the uncertainties in $f_{B^{+}}$ and $f_{B^{0}}$ due to isospin breaking and
electromagnetism are $\sim 0.7\%$. Because only the omission of
electromagnetic effects contributes significantly to the error, this estimate
also applies to $f_{B_{s}}$. For the ratio of neutral-meson decay constants
$f_{B_{s}}/f_{B^{0}}$, the electromagnetic contribution is suppressed due to
the equal charges of the down and strange valence quarks. We estimate its size
to be of ${\mathcal{O}}\left(\alpha_{\rm EM}(m_{s}-m_{d})/\Lambda_{\rm
QCD}\right)\sim 0.1\%$. This cancellation does not occur when the valence-
quark charges are different, so electromagnetic effects in
$f_{B_{s}}/f_{B^{+}}$ are still of ${\mathcal{O}}(\alpha_{QED})\sim 0.7\%$.
We note that the difference between the $u$\- and $d$-quark masses in the sea
sector cannot lead to a difference between $f_{B^{+}}$ and $f_{B^{0}}$ because
the sea quarks couple to the valence quark in the $B$ meson through $I=0$
gluon exchange. The use of degenerate $u$ and $d$ sea quarks does lead to
identical shifts in $f_{B^{+}}$, $f_{B^{0}}$, and $f_{B_{s}}$. Such
contributions, however, are negligible because they must be symmetric under
the interchange $m_{u}\leftrightarrow m_{d}$ and are of of
${\mathcal{O}}\left((\Delta m_{ud}/\Lambda_{\rm QCD})^{2}\right)\sim 0.003\%$.
## VI Results and conclusions
After adding the systematic error estimates from Table 11 in quadrature, our
final results for the $B_{(s)}$-meson decay constants and their ratios are:
$\displaystyle f_{B^{0}}$ $\displaystyle={199.5(6.2)(12.6)}\;\text{MeV}$ (31)
$\displaystyle f_{B^{+}}$ $\displaystyle={195.6(6.4)(14.9)}\;\text{MeV}$ (32)
$\displaystyle f_{B_{s}}$ $\displaystyle=235.4(5.2)(11.1)\;\text{MeV}$ (33)
$\displaystyle f_{B_{s}}/f_{B^{0}}$ $\displaystyle={1.197(13)(49)}$ (34)
$\displaystyle f_{B_{s}}/f_{B^{+}}$ $\displaystyle={1.223(14)(70)}\,,$ (35)
where the errors are statistical and total systematic, respectively.
Figure 9 compares our results with other unquenched determinations. For all
quantities, they agree well with the other $N_{f}>2$ determinations in the
literature. Our result for $f_{B_{s}}/f_{B}$ is more precise than the
published $N_{f}=2+1$ RBC/UKQCD result Albertus:2010nm using static
$b$-quarks because we include domain-wall ensembles with much lighter pions
and a finer lattice spacing. For both the decay constants and their ratio, our
uncertainties are comparable to the results of ETM Carrasco:2012de ;
Carrasco:2013naa ALPHA Bernardoni:2014fva , as well as the similar
calculation by the Fermilab Lattice and MILC collaborations Bazavov:2011aa
using the Fermilab relativistic heavy-quark interpretation. Our results are
not as precise as those by HPQCD using HISQ $b$-quarks on the MILC asqtad
staggered ensembles McNeile:2011ng , which include ensembles as fine as
$a\approx 0.045$ fm, or using NRQCD $b$-quarks and HISQ sea quarks
Dowdall:2013tga , which include ensembles with the physical pion mass.
Figure 9: Lattice determinations of $f_{B}$, $f_{B_{s}}$, and
$f_{B_{s}}/f_{B}$ using 2, 2+1, and 2+1+1 dynamical sea-quarks Albertus:2010nm
; McNeile:2011ng ; Bazavov:2011aa ; Na:2012kp ; Carrasco:2012de ;
Dowdall:2013tga ; Carrasco:2013naa ; Bernardoni:2014fva . The gray error bands
show the FLAG averages Aoki:2013ldr , which were obtained from the FNAL/MILC
and HPQCD determinations for $N_{f}=2+1$ and equal the ETMC result for
$N_{f}=2$. No FLAG average was presented for the 2+1+1-flavor data. Most of
the results shown are for the decay constants in the isospin limit $f_{B}$ and
$f_{B_{s}}/f_{B}$ except for the determination by FNAL/MILC who reported
results for $f_{B^{+}}$ and $f_{B_{s}}/f_{B^{+}}$; our results refer to the
determination of $f_{B^{+}}$ and $f_{B_{s}}/f_{B^{+}}$. The description in
parentheses show the light- and heavy-quark actions used, with the exception
of the $N_{f}=2+1$ HPQCD calculations. The HPQCD works use Asqtad sea quarks,
and the determination of $f_{B}$ labeled “HISQb/NRQCD” is obtained by
combining $f_{B_{s}}$ using HISQ $b$ quarks from Ref. McNeile:2011ng with the
ratio $f_{B_{s}}/f_{B}$ using NRQCD $b$-quarks from Na:2012kp .
The largest source of uncertainty in our calculations of the $B_{(s)}$-meson
decay constants is from the chiral extrapolation to the physical light-quark
masses and the extrapolation to the continuum limit. We are currently
generating light and strange-quark propagators on the RBC/UKQCD Möbius domain-
wall + Iwasaki ensembles Brower:2004xi ; Brower:2012vk ; MawhinneyLat13Talk
with the same lattice spacing as the $24^{3}$ ensembles used in our current
analysis, but with $m_{\pi}\approx 140$ MeV. The inclusion of data at the
physical pion mass will significantly reduce our chiral-continuum
extrapolation errors. Statistical errors are the next-largest source of
uncertainty in our current analysis. Shortening the distance of the chiral
extrapolation will reduce the statistical errors at the physical point. We are
also investigating the use of all-mode averaging Blum:2012uh ;
Shintani:2014vja to reduce the statistical errors on the individual numerical
data points. All of the other systematic uncertainties are estimated in Table
11 to be much smaller; thus we anticipate obtaining significantly smaller
errors in the future. There has been no difference observed between
calculations of the $B$-meson decay constants from two-, three-, and four-
flavor lattice simulations (see Fig. 9). Nevertheless, the inclusion of the
dynamical charm quark will be important once calculations reach even higher
precision. The RBC and UKQCD collaborations are currently generating
$N_{f}=2+1+1$ domain-wall ensembles which will allow a direct study of the
effects of charm-quark loops on the $B$\- (and $D$-) meson decay constants.
Our results for the $B_{(s)}$-meson decay constants in Eqs. (31)–(35) are the
first from simulations with domain-wall light quarks and relativistic heavy
quarks, and also the first application of the RHQ action to weak-matrix
elements relevant for phenomenology. They provide valuable independent cross-
checks of the published unquenched calculations using staggered sea quarks.
The good agreement with other works bolsters confidence in lattice-QCD
calculations of the $B_{(s)}$-meson decay constants, and provides further
support that the RHQ action can be used to obtain accurate results for bottom
systems with competitive and reliable uncertainties. We are also undertaking
companion calculations of $B$-meson semileptonic form factors Kawanai:2013qxa
, $B^{0}$-$\overline{B^{0}}$ mixing matrix elements Witzel:2013sla , and
$B^{*}$-$B$-$\pi$ coupling Flynn:2013kwa using the same lattice actions and
ensembles. These will enable determinations of $|V_{ub}|$ from both leptonic
and semileptonic decays and place an important constraint on the apex of the
CKM unitarity triangle. Finally, we note that rare decays such as $B\to
K\ell^{+}\ell^{-}$ and $B\to\pi\ell^{+}\ell^{-}$ provide potentially sensitive
probes of new physics, and are therefore future possible applications of the
RHQ action.
## Acknowledgments
We thank the referee for invaluable comments on the manuscript that led to
improvements in both the presentation and analysis.
Computations for this work were carried out in part on facilities of the USQCD
Collaboration, which are funded by the Office of Science of the U.S.
Department of Energy. We thank BNL, Columbia University, Fermilab, RIKEN, and
the U.S. DOE for providing the facilities essential for the completion of this
work.
This work was supported in part by the U.S. Department of Energy under grant
No. DE-FG02-92ER40699 (N.H.C.), by the UK Science and Technology Facilities
Council (STFC) grant ST/J000396/1 (J.M.F.), and by the Grant-in-Aid of the
Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT
Grant) Nos. 26400261, 22540301, and 23105715 (T.I). T.K. is supported by the
JSPS Strategic Young Researcher Overseas Visits Program for Accelerating Brain
Circulation (No. R2411). O.W. ackknowledges support at Boston University by
the U.S. DOE grant DE-SC0008814. This manuscript has been authored by
employees of Brookhaven Science Associates, LLC under Contract No. DE-
AC02-98CH10886 with the U.S. Department of Energy.
## Appendix A Determination of $Z_{V}^{bb}$
The flavor conserving heavy-heavy renormalization factor $Z_{V}^{bb}$ is
obtained from the matrix element of the $b\to b$ vector current between two
$B_{q}$-mesons:
$\displaystyle Z_{V}^{bb}\times\langle
B_{q}|V^{bb,0}|B_{q}\rangle=2M_{B_{q}}.$ (36)
We compute the three-point correlation function shown in Fig. 10 by fixing the
locations of the two $B_{q}$ mesons at $t_{0}$ and $t_{\rm sink}$ and varying
the location $t$ of the operator over all time slices in between:
$\displaystyle
C_{PVP}(t_{0},t,t_{\text{sink}})=\sum_{\vec{x},\vec{y}}\langle\widetilde{{\cal
O}}_{P}(\vec{x},t_{\text{sink}}){\cal O}_{V_{0}}(\vec{y},t)\widetilde{{\cal
O}}_{P}(\vec{0},t_{0})\rangle\,,$ (37)
where ${\widetilde{\cal O}}_{P}$ are pseudoscalar interpolating operators for
the $B_{q}$ mesons and ${\cal O}_{V_{0}}=\overline{b}\gamma_{0}b$ is the
leading temporal vector-current operator. The ${\mathcal{O}}(a)$-improvement
of $Z_{V}^{bb}$ does not require the computation of any additional matrix
elements because we are only interested in the temporal component of the
vector-current operator without momentum injected. In this situation, the
equations of motion can be used to parameterize the ${\mathcal{O}}(a)$
improvement as an overall multiplicative factor that depends upon the
$b$-quark mass. We can then absorb this correction into the values of the
perturbative contribution to the renormalization factor $\rho_{A}^{bl}$ given
in Table 6.
We use a point-source propagator for the light spectator quark, and Gaussian
smeared sources for the $b$-quarks in the two $B_{q}$ mesons.
Figure 10: Three-point correlation function used to compute the flavor-
conserving renormalization factor $Z_{V}^{bb}$. The locations of the $B_{s}$
mesons are fixed and the location of the vector current is varied over all
time slices in between.
The desired renormalization factor is then given by the ratio
$\displaystyle Z_{V}^{bb}(t_{0},t,t_{\text{sink}})=\lim_{t_{0}\ll t\ll
t_{\text{sink}}}\frac{\widetilde{C}_{PP}(t,t_{0})}{C_{PVP}(t_{0},t,t_{\text{sink}})}\,,$
(38)
where $\widetilde{C}_{PP}$ is the pseudoscalar-pseudoscalar correlator with a
Gaussian smeared heavy quark source and sink [see Eq. (20)]. For the
computation of $Z_{V}^{bb}$ we double the statistics by computing the
sequential $b$-quark propagator for both the forward- and the backward-
propagating light spectator quark. Similarly, we fold the 2-point correlation
function about the temporal midpoint of the lattice. After testing several
source-sink separations
$\Delta_{t}\equiv\left(t_{\text{sink}}-t_{0}\right)=\\{18,20,22\\}$ on the
$24^{3}$ ensemble with $am_{l}=0.005$, we found that $\Delta_{t}=20$ led to
the best signal-to-noise. We scaled this value by $a^{-1}_{32c}/a^{-1}_{24c}$
to obtain $\Delta_{t}=26$ on the $32^{3}$ ensembles.
Figure 11 shows an example determination of $Z_{V}^{bb}$ via Eq. (38) for two
different values of the spectator-quark mass on the $am_{l}=0.005$, $24^{3}$
ensemble; results on other ensembles look similar.
Figure 11: Determination of $Z_{V}^{bb}$ from three-point correlators with
unitary (open red circles) and strange (filled blue squares) spectator quarks
on the $24^{3}$ ensemble with $am_{l}=0.005$. The $B\to B$ data points are
shown with a slight horizontal offset for clarity.
The data display long plateaus with small error bars over almost the entire
time range. We do not observe any spectator-quark mass dependence within
statistical uncertainties, but the statistical errors increase with lighter
spectator-quark mass. We therefore determine $Z_{V}^{bb}$ using a strange
spectator quark ($m_{q}\sim m_{s}$) in Eq. (38). The values for $Z_{V}^{bb}$
are extracted by performing a correlated constant fit over time slices [7:13]
on the $24^{3}$ ensembles and [9:17] on the $32^{3}$. The fits are shown in
Fig. LABEL:Fig:ZvbbFits and the results with jackknife statistical errors are
summarized in Tab. 6.
## Appendix B Numerical estimate of heavy-quark discretization errors
Here we provide the explicit forms of the relevant operators and mismatch
functions for the heavy-quark action in Sec. B.1, and for the heavy-light
current in Sec. B.2. Then, in Sec. B.3, we present numerical estimates of
heavy-quark discretization errors in our calculation of the $B_{(s)}$-meson
leptonic decay constants on the $24^{3}$ and $32^{3}$ ensembles. For the
discretization errors from the current, we compare our estimates from heavy-
quark power counting with ones based on the observed sizes of the
${\mathcal{O}}(a)$ and ${\mathcal{O}}(\alpha_{s}a)$ contributions to the decay
amplitudes, and find good agreement.
### B.1 ${\mathcal{O}}(a^{2})$ errors from the action
Oktay and Kronfeld present the complete set of bilinears and four-quark
operators that can appear in the Symanzik effective Lagrangian through
dimension 7 in Ref. Oktay:2008ex . At dimension 6, there are two bilinears
$\overline{h}\\{\bm{\gamma}\cdot\bm{D},\bm{\alpha}\cdot\bm{E}\\}h$ and
$\overline{h}\gamma_{4}(\bm{D}\cdot\bm{E}-\bm{E}\cdot\bm{D})h$ and many four-
quark operators. At tree-level, the mismatch coefficients of all of the four-
quark operators vanish. The tree-level mismatch coefficients of the two
bilinears are the same, and are given by:
$f_{E}(m_{0}a,c_{P},\zeta)=\frac{1}{8m_{E}^{2}a^{2}}-\frac{1}{8m_{2}^{2}a^{2}},$
(39)
where
$\displaystyle\frac{1}{m_{2}a}$ $\displaystyle=$
$\displaystyle\frac{2\zeta^{2}}{m_{0}a(2+m_{0}a)}+\frac{\zeta}{1+m_{0}a},$
(40) $\displaystyle\frac{1}{4m_{E}^{2}a^{2}}$ $\displaystyle=$
$\displaystyle\frac{\zeta^{2}}{[m_{0}a(2+m_{0}a)]^{2}}+\frac{\zeta
c_{P}}{m_{0}a(2+m_{0}a)}\,.\quad$ (41)
The size of the relative error from each of the dimension 6 bilinears is then
estimated to be
${\rm error}_{E}\sim f_{E}(m_{0}a,c_{P},\zeta)\left(a\Lambda_{\rm
QCD}\right)^{2}\,.$ (42)
### B.2 ${\mathcal{O}}(\alpha_{s}^{2}a,a^{2})$ errors from the current
Harada et al. present the complete set of operators needed to improve the
vector and axial-vector heavy-light currents to all orders in
${\mathcal{O}}(a)$ in Ref. Harada:2001fi . There are six such operators; two
for the temporal current and four for the spatial current. Although their
coefficients have been computed numerically at one-loop for the perturbative
matching used in this work, the expressions are not known analytically. We
therefore use the tree-level mismatch functions as a guide. At tree-level, all
of the operators have the same mismatch coefficient:
$\displaystyle f_{3}^{[0]}(m_{0}a,c_{P},\zeta)$
$\displaystyle=\frac{\zeta(1+m_{0}a)}{m_{0}a(2+m_{0}a)}-\frac{1}{2m_{2}a}-d_{1}$
(43) $\displaystyle=\frac{\zeta}{m_{0}a(2+m_{0}a)}+\frac{\zeta}{(2+m_{0}a)}$
$\displaystyle-\frac{\zeta}{2(1+m_{0}a)}-\frac{\zeta^{2}}{m_{0}a(2+m_{0}a)}-d_{1}\,,$
(44)
where at tree-level $d_{1}^{[0]}$ is defined such that $f_{3}^{[0]}=0$. For
the 2-loop mismatch function(s), we multiply the above expression by
$\alpha_{s}^{2}$ and set $d_{1}^{[2]}=0$. The result, however, approaches
infinity in the $m_{0}a\to 0$ limit. We therefore instead consider several
functions similar to $f_{3}^{[0]}$ that have the expected parametric
dependence on the strong coupling and $\zeta$, as well as the correct
asymptotic behavior in both the chiral and static limits. For our final
estimate, we use the simple ansatz
$f_{3}^{[2]}(m_{0}a,c_{P},\zeta)=\alpha_{s}^{2}\zeta\frac{2}{(2+m_{0}a)}\,,$
(45)
where the factor of two in the numerator allows for more than one term of this
size in the true mismatch function. In our numerical simulations, the
parameter $\zeta$ is of ${\mathcal{O}}(1)$, so the small size of $f_{3}^{[2]}$
is primarily due to the perturbative factor $\alpha_{s}^{2}$. The exact
dependence on $m_{0}a$ in the denominator does not impact the size of
$f_{3}^{[2]}$ significantly, but we conservatively take the function that
leads to the largest value of the mismatch function. The size of the relative
error from the ${\mathcal{O}}(a)$ heavy-light current operators is then
estimated to be
${\rm error}_{3}\sim f_{3}^{[2]}(m_{0}a,c_{P},\zeta)\left(a\Lambda_{\rm
QCD}\right)\,.$ (46)
El Khadra, Kronfeld, and Mackenzie present the expression for the tree-level
${\mathcal{O}}(a^{2})$-improved heavy-light electroweak current in Eq. (A.17)
of Ref. ElKhadra:1996mp . At ${\mathcal{O}}(a^{2})$, there are three relevant
operators — $\overline{q}\Gamma\mathbf{D}^{2}h$, $\overline{q}\Gamma
i\mathbf{\Sigma}\cdot\mathbf{B}h$, and
$\overline{q}\Gamma\mathbf{\alpha}\cdot\mathbf{E}h$ — where $q$ and $h$ denote
the light- and heavy-quark fields, respectively. Their tree-level coefficients
are given in Eq. (A.19) of the same paper, from which the mismatch functions
can be inferred:
$\displaystyle f_{X_{1}}(m_{0}a,c_{P},\zeta)$
$\displaystyle=-\frac{1}{2}\left[d_{1}^{2}-\frac{\zeta}{2(1+m_{0}a)}\right]$
(47) $\displaystyle f_{X_{2}}(m_{0}a,c_{P},\zeta)$
$\displaystyle=-\frac{1}{2}\left[d_{1}^{2}-\frac{c_{P}}{2(1+m_{0}a)}\right]$
(48) $\displaystyle f_{Y}(m_{0}a,c_{P},\zeta)$
$\displaystyle=-\frac{1}{2}\left[\frac{(\zeta-
c_{P})(1+m_{0}a)}{m_{0}a(2+m_{0}a)}-\frac{d_{1}}{m_{2}a}\right]\,,$ (49)
with
$d_{1}=(m_{0}a,c_{P},\zeta)=\frac{\zeta(1+m_{0}a)}{m_{0}a(2+m_{0}a)}-\frac{1}{2m_{2}a}\,.$
(50)
The sizes of the relative errors from the three operators are then estimated
to be
$\displaystyle{\rm error}_{X_{1}}$ $\displaystyle\sim$ $\displaystyle
f_{X_{1}}(m_{0}a,c_{P},\zeta)\left(a\Lambda_{\rm QCD}\right)^{2}$ (51)
$\displaystyle{\rm error}_{X_{2}}$ $\displaystyle\sim$ $\displaystyle
f_{X_{2}}(m_{0}a,c_{P},\zeta)\left(a\Lambda_{\rm QCD}\right)^{2}$ (52)
$\displaystyle{\rm error}_{Y}$ $\displaystyle\sim$ $\displaystyle
f_{Y}(m_{0}a,c_{P},\zeta)\left(a\Lambda_{\rm QCD}\right)^{2}\,.$ (53)
### B.3 Numerical estimates
Table 12 presents the numerical values of the mismatch functions at the tuned
values of the RHQ parameters given in Table 5. Table 13 presents the estimated
size of heavy-quark discretization errors in $f_{B_{(s)}}$ from operators of
${\mathcal{O}}(a^{2})$ in the action and of
${\mathcal{O}}(\alpha_{s}^{2}a,a^{2})$ in the axial-vector current taking
$\Lambda_{\rm QCD}=500$ MeV. The last column gives the sum of the errors from
the individual operators added in quadrature.
Table 12: Mismatch functions for the nonperturbatively-tuned parameters of the RHQ action on the $24^{3}$ and $32^{3}$ ensembles given in Table 5. The tree-level coefficients $f_{E}$, $f_{X_{i}}$, and $f_{Y}$ are known exactly. The two-loop coefficient $f_{3}^{[2]}$ is not known, so we use an ansatz based on the tree-level expression. | $f_{E}$ | $f_{X_{1}}$ | $f_{X_{2}}$ | $f_{Y}$ | $f_{3}^{[2]}$
---|---|---|---|---|---
$a\approx$ 0.11 fm | 0.0652 | 0.0803 | 0.1517 | 0.1605 | 0.0659
$a\approx$ 0.086 fm | 0.0864 | 0.0953 | 0.1774 | 0.1900 | 0.0312
Table 13: Percentage errors from mismatches in the action and current for the bottom quark on the $24^{3}$ and $32^{3}$ ensembles. For this estimate, we calculate the mismatch functions for the nonperturbatively-tuned parameters of the RHQ action from Table 5. We estimate the size of operators using HQET power counting with $\Lambda_{\rm QCD}=500$ MeV. To obtain the total, we add the individual errors in quadrature, counting contributions $E$ and 3 twice because they each arise from two operators in the Symanzik effective Lagrangian. | | ${\mathcal{O}}(a^{2})$ error | ${\mathcal{O}}(a^{2})$ errors | ${\mathcal{O}}(\alpha_{s}^{2}a)$ error |
---|---|---|---|---|---
| | from action | from current | from current |
| $\alpha_{s}^{\overline{\rm MS}}(1/a)$ | $E$ | $X_{1}$ | $X_{2}$ | $Y$ | 3 | Total (%)
$a\approx$ 0.11 fm | 1/3 | 0.55 | 0.67 | 1.27 | 1.34 | 1.91 | 3.42
$a\approx$ 0.086 fm | 0.22 | 0.42 | 0.46 | 0.85 | 0.91 | 0.68 | 1.75
We can also estimate the size of heavy-quark discretization errors from the
current, i.e. those corresponding to operators $X_{1},X_{2},Y$, and $3$ in
Table 13, by looking at the known ${\mathcal{O}}(a)$ and
${\mathcal{O}}(\alpha_{s}a)$ contributions to the decay amplitudes in our
data. We estimate the size of the omitted ${\mathcal{O}}(a^{2})$ contributions
by assuming that they are approximately $a\Lambda_{\rm QCD}$ times the size of
the tree-level ${\mathcal{O}}(a)$ contribution, which is 0.75% on the finer
$32^{3}$ ensembles. This is of the same size as the estimates of the
contributions from ${\mathcal{O}}_{X_{1},X_{2},Y}$ in Table 13 based on heavy-
quark power counting. We estimate the size of the omitted
${\mathcal{O}}(\alpha_{s}^{2}a)$ contributions in two ways: by assuming that
they are approximately $\alpha_{s}^{2}$ times the size of the tree-level
contribution, or that they are $\alpha_{s}$ times the size of the 1-loop
contribution. On the $32^{3}$ ensembles, the first approach leads to an
estimate of 0.17%, while the second leads to one of 0.59%. These are both
similar in magnitude to the estimated contribution from ${\mathcal{O}}_{3}$ in
Table 13. Given the overall consistency between the two error estimation
approaches, we use the values obtained from heavy-quark power counting in our
systematic error budget. This, in fact, leads to a slightly larger quoted
heavy-quark discretization error than what we would obtain if we used the
data-driven numbers.
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|
arxiv-papers
| 2014-04-17T23:47:53 |
2024-09-04T02:50:01.303624
|
{
"license": "Public Domain",
"authors": "Norman H.Christ, Jonathan M. Flynn, Taku Izubuchi, Taichi Kawanai,\n Christoph Lehner, Amarjit Soni, Ruth S. Van de Water, Oliver Witzel",
"submitter": "Oliver Witzel",
"url": "https://arxiv.org/abs/1404.4670"
}
|
1404.4726
|
# Cohomology in nonunitary representations of semisimple Lie groups (the group
$U(2,2)$)
A. M. Vershik St. Petersburg Department of Steklov Institute of Mathematics
and St. Petersburg State University, St. Petersburg, Russia. Email:
[email protected]. Supported by the RFBR grants 11-01-00677-a and
13-01-12422-ofi-m. M. I. Graev Institute for System Studies, Moscow, Russia.
Email: [email protected]. Supported by the RFBR grant 13-01-00190 .
###### Abstract
We suggest a method of constructing special nonunitary representations of
semisimple Lie groups using representations of Iwasawa subgroups. As a typical
example, we study the group $U(2,2)$.
To the 100th birthday of our teacher Israel Moiseevich Gelfand
## 1 Introduction: a survey of the theory of special representations
### 1.1 Groups of currents and special representations
The group of currents $G^{X}$, where $X$ is a topological space equipped with
a Borel probability measure $m$ and $G$ is an arbitrary locally compact group,
is the group of continuous maps $X\rightarrow G$ with pointwise multiplication
and with some integrability condition with respect to $m$. The study of
representations of such groups is inspired both by representation theory
itself and applications to mathematical physics. A well-known model of
irreducible representations of current groups $G^{X}$ is the Fock, or
Gaussian, model, in which a crucial role is played by nonzero first cohomology
of the group of coefficients $(G$) with values in irreducible unitary
representations of this group. Such an irreducible representation $T$ of the
group $G$ in a Hilbert space $H$ has a nontrivial 1-cocycle, i.e., a
continuous map $b:G\rightarrow H$ satisfying the condition
$b(g_{1}g_{2})=T(g_{1})b(g_{2})+b(g_{1})\quad\mbox{for any}\quad
g_{1},g_{2}\in G$
and the following nontriviality condition: there is no vector $\xi\in H$ such
that $b(g)=T(g)\xi-\xi$ for every $g\in G$. Such representations are called
special. The trivial representation is special for all groups having
nontrivial complex characters, since an additive complex character is exactly
a nontrivial 1-cocycle with values in the one-dimensional trivial complex
representation.
In other terms, a special representation is a representation in which there
exist almost invariant vectors; the latter term means, for a representation of
a group $G$ in a space $H$, that for every $\epsilon>0$ and every compact
subset $K$ in $G$, there exists a vector $h$ in $H$ such that
$\|U_{k}h-h\|<\epsilon$ for every $k\in K$, where $U_{k}$ is the operator in
the representation corresponding to the element $k$.
By a theorem from [25], every special representation of a compactly generated
group is not Hausdorff separated from the trivial representation in the Fell
topology. The Fell topology on the space of unitary representations of a
locally compact group $G$ is defined as follows: an open neighborhood of a
representation $\pi$ in a Hilbert space $H$ is determined by a number
$\epsilon$, a finite collection $h_{1},\dots h_{k}$ of elements of $H$, and a
compact subset $K$ in $G$; it consists of all unitary representations $\rho$
of $G$ such that the space of $\rho$ contains elements $f_{1},\dots,f_{k}$
such that
$\sup_{i=1,\dots,k;\;g\in
K}\left\\{|\langle\pi(g)h_{i},h_{i}\rangle-\langle\rho(g)f_{i},f_{i}\rangle|<\epsilon\right\\}.$
The converse is not true: a representation that is not separated from the
trivial representation is not necessarily special, i.e., does not necessarily
have non vanishing first cohomology.
In [13], Y. Shalom proved the conjecture stated in [25]: if for a locally
compact group the trivial representation is not isolated in the space of all
irreducible unitary representations (i.e., the group does not have Kazhdan’s
property (T)), then it has at least one special representation. The proof of
the main theorem in [13] (see also [2, Theorem 3.2.1]) is not constructive,
and hence does not provide a direct method of finding a special
representation: we know very little about how to select special
representations from the set of all representations that are not separated
from (or “glued” to) the trivial representation.
We use the following terminology. The core of a given representation $\pi$ is
the set of all representations $\rho$ such that arbitrary open neighborhoods
of $\pi$ and $\rho$ have a nonempty intersection. The core of the regular
representation of an amenable group contains all irreducible representations,
and this is a characteristic property of amenability. This notion is of
special importance for irreducible representations and, in particular, for the
trivial representation. Unless otherwise stated, by the core of a group we
mean the core of its one-dimensional trivial representation.111A more detailed
terminology is as follows. Representations lying in the core of the trivial
representation are called infinitesimal (in a Leibniz-like sense, see [25]);
the subcore of a given representation $\pi$ is the set of all representations
$\rho$ whose closure contains $\pi$ (correspondingly, the subcore of a group
is the subcore of its trivial representation). In other terms, this means that
$\rho$ weakly contains $\pi$. The subcore is, obviously, a subset of the core.
The interpretation of these notions in terms of Hausdorff’s separation axioms
is as follows: if the trivial representation lies in the core of $\rho$, then
$T_{2}$ does not hold for these two elements; if $\rho$ lies in the subcore,
even $T_{1}$ does not hold; $T_{0}$ always holds, since the trivial
representation is closed.
One says that nonvanishing cohomology with values in a representation is
reduced (see [13]) if the corresponding cocycle is not a limit of trivial
cocycles. This can be the case only for representations lying in the core, but
not in the subcore. For the semisimple groups $O(n,1),U(n,1)$, the non
vanishing cohomology is reduced. The authors do not know whether there exist
solvable groups satisfying this condition. Finally, for completeness we
mention another interesting notion — that of groups with the Haagerup property
(see [2, 8]): these are groups for which the trivial representation is not
isolated (i.e., which do not have Kazhdan’s property) but cocycles with values
in a special representation satisfy a certain nondegeneracy condition:
regarded as a map
$\beta:G\rightarrow H$
from the group to a Hilbert space, the cocycle is proper, i.e., the preimages
of bounded sets in $H$ are precompact in the group. This property holds for
all amenable groups, free groups ([8]), the groups $O(n,1),U(n,1)$, and others
(see [4]). Examples of groups that satisfy neither Kazhdan’s property nor the
Haagerup property are not yet sufficiently studied.
### 1.2 Special representations of rank $1$ groups and their solvable
subgroups
In the most important class of Lie groups — that of semisimple groups — only
the groups $O(n,1)$ and $U(n,1)$, which are of rank $1$, have special
representations. The other semisimple groups (including, as proved by Kostant
[10], even rank $1$ groups $Sp(n,1)$) have Kazhdan’s property (T): their
trivial representations are isolated, and the Fock model of constructing
representations of the corresponding groups of currents does not apply. Note
that Kostant’s theorem on $Sp(n,1)$ still has no geometric proof.
One may say that the analysis of representations of the groups of currents for
$O(n,1)$ and $U(n,1)$ is developed quite well. Studies in this direction began
from the pioneer work by I. M. Gelfand and the authors of this paper, see [17,
18, 6]. The general scheme of the Fock model, regardless the concrete group,
was earlier considered by Araki [1] (see [12]), however, before the paper [17]
there were no examples of semisimple groups for which the core is nontrivial.
In these papers, as well as in [9, 5, 7, 3], irreducibility conditions for
representations of current groups were found, and other properties of these
groups were established. The key role was played by the ideas of the paper
[6], in which a method was suggested, for semisimple groups of rank $1$, of
reducing the problem to a solvable subgroup, on the example of $SL(2,R)$. The
elaboration of this idea by the authors of this paper during the last 10 years
has led to new models of representations of current groups which are
equivalent to the Fock one but are constructed from other (non-Gaussian) Lévy
measures (see [19, 20]). This has led to constructing the integral, and then
Poisson and quasi-Poisson, models of representations of current groups ([21,
22]). There has also appeared the so-called infinite-dimensional Lebesgue
measure [14, 15, 16], which is the most precise continual analog of the
Lebesgue measure on a finite-dimensional positive octant. This measure is
closely related to the gamma process and is of great interest in itself.
### 1.3 Solvable subgroups of semisimple groups, and a refinement of the
Iwasawa decomposition
It is well known that for commutative and nilpotent groups the special
representations are exhausted by the trivial representation (see [7, 5]). But
even for solvable groups, this problem is not sufficiently studied. We are
interested in a concrete class of solvable groups, which are subgroups of
$O(p,q),U(p,q)$, and our first example is the group $U(2,2)$. We are working
with a solvable subgroup of $U(2,2)$ which should be called the Iwasawa group.
Such a subgroup can be defined in an arbitrary connected semisimple real Lie
group (see [11], where it was called the “maximal connected triangular
subgroup”). The Iwasawa decomposition means that this subgroup is
“complementary” to the maximal compact subgroup.
A solvable subgroup (like any amenable group) has a special representation, so
that we obtain the following strategy of constructing a representation of the
group of currents of a semisimple group $G$: first find a special
representation, unitary or not, of this solvable subgroup and construct a
representation of the group of its currents, and then try to extend the
special representation to the whole semisimple group $G$ and construct an
extension of the representation to the group of currents of $G$. As mentioned
above, for groups of rank $1$, this trick was first used in [6] in the case of
$SL(2,R)$, and then studied in detail in a recent series of papers of the
authors ([21, 22]). When passing from groups of rank $1$ to higher ranks, this
idea is still working.
Here we consider this problem on the concrete example of the group $U(2,2)$,
keeping in mind the more general situation, which will be considered
elsewhere. We describe in detail the Iwasawa subgroup (denoted by $P$ in what
follows) for this case. The first question is, what are its special
representations? But here we encounter a new problem: for our plan to be
viable, this special representation of the Iwasawa subgroup must be
faithful.222A faithful (or nondegenerate) representation is a representation
whose kernel is trivial. The authors do not know for what groups this is the
case. For example, nilpotent groups have no faithful special representations.
This fact is of interest already for the Heisenberg group, and it is
equivalent to a version of the uncertainty principle.333From discussions with
V. P. Khavin, the first author inferred that this fact about the Heisenberg
group apparently still has no purely analytical (rather than representation
theoretic) proof. For the affine group $\mbox{Aff}(\mathbb{R})$, the infinite-
dimensional representation (which is quasi-equivalent to the regular one) is a
faithful special representation. But even for simplest three-dimensional
solvable groups (in particular, for the group $S$, see below), the question is
not trivial.
Problem. What groups (in particular, what solvable groups) have a faithful
irreducible unitary special representation? More precisely: when does such a
representation exist for the Iwasawa subgroup of a semisimple real Lie group?
Thus the main difficulty is to construct a faithful special representation,
unitary or not, of the Iwasawa subgroup. A construction of a nonunitary
faithful special representation of the group $U(2,2)$ is the main result of
this paper, and the authors have no doubts that such a construction can be
carried out for every real semisimple group. This makes it possible to extend
the cocycle to the whole semisimple group and construct a representation of
the group of currents, since, as follows from the results of the above-
mentioned papers, there are models (for instance, the Poisson model, unlike
the Fock one) of representations of groups of currents that do not in any way
rely on the unitarity of the original representation. The question of whether
a cocycle can be constructed with values in a unitary representation of the
Iwasawa subgroup is yet open.
In conclusion of this survey, we mention a somewhat different approach, which
is closer to the original work on groups of currents and classical work on the
representation theory of semisimple groups. Namely, it is well known that
special representations of semisimple groups of rank $1$ lie in the “tail” of
the complementary series, and complementary series exist for every semisimple
group. However, for groups of rank greater than 1, unitary representations do
not “reach” the trivial representation; more exactly, unitarity gets lost
under deformations of the trivial representation. On the other hand, it is
known that in some cases one can find an indefinite bilinear form invariant
under the action of the group in such a nonunitary representation. The
properties of the corresponding space with an indefinite metric are poorly
studied, and the problem of the existence of a special representation, perhaps
nonunitary, apparently has not been stated. Our alternative strategy relies on
the analytic continuation of unitary representations of the Iwasawa subgroups
and, in particular, construction of a nonunitary special representation within
this framework. It is conceivable that these two approaches to the
representation theory of the groups of currents for semisimple groups of rank
greater than 1 may lead to different classes of representations.
## 2 The group $U(2,2)$ and its Iwasawa subgroup
### 2.1 Description of the group $U(2,2)$
As a first example, we consider the group $U(2,2)$ of linear transformations
of ${\mathbb{C}}^{4}$ preserving a fixed Hermitian form with signature
$(2,2)$; here we choose the Hermitian form
$x_{1}\bar{x}_{3}+\bar{x}_{1}x_{3}+x_{2}\bar{x}_{4}+\bar{x}_{2}x_{4}.$
The group $U(2,2)$ is one of the simplest examples of a semisimple Lie group
whose real rank is greater than one (it is equal to 2). The groups of the form
$U(p,q)$ are called pseudo-unitary. In this section, we refine the Iwasawa
decomposition for this group and study the structure of the key object, the
solvable Iwasawa subgroup $P$.
We will write elements of the group $U(2,2)$ as $2\times 2$ block matrices
with $2\times 2$ blocks:
$\left(\begin{array}[]{cc}g_{11}&g_{12}\\\ g_{21}&g_{22}\\\
\end{array}\right),$
where $g_{ij}$ are complex $2\times 2$ matrices satisfying the relation
$g\sigma
g^{*}=\sigma,\qquad\mbox{with}\quad\sigma=\left(\begin{array}[]{cc}0&e_{2}\\\
e_{2}&0\\\ \end{array}\right).$
Here $e_{2}$ is the $2\times 2$ identity matrix and $*$ stands for the
conjugate transpose in the complex case, and for the transpose in the real
case.
These relations are equivalent to the following relations between the blocks
of the matrix $g$:
$\displaystyle g_{12}g^{*}_{21}+g_{11}g^{*}_{22}$ $\displaystyle=$
$\displaystyle e_{2},$ $\displaystyle g_{11}g^{*}_{12}+g_{12}g^{*}_{11}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
g_{22}g^{*}_{21}+g_{21}g^{*}_{22}$ $\displaystyle=$ $\displaystyle 0.$
The real dimension of the group $U(2,2)$ is equal to 16. In what follows, the
key role is played by the solvable subgroup $P$ of $U(2,2)$ generated by the
following two subgroups $N$ and $S$:
* •
the additive (commutative) group $N$ of skew-Hermitian block matrices of the
form
$\left(\begin{array}[]{cc}e_{2}&0\\\ n&e_{2}\\\ \end{array}\right),$
where $n$ is a skew-Hermitian $2\times 2$ matrix: $n+n^{*}=0$;
* •
the solvable subgroup $S$ (of derived length 2) of block matrices of the form
$\left(\begin{array}[]{cc}s^{*{-1}}&0\\\ 0&s\\\ \end{array}\right),$
where $s$ is a lower triangular complex matrix with positive diagonal entries:
$\left(\begin{array}[]{cc}r_{1}&0\\\ r&r_{2}\\\ \end{array}\right),\qquad
r_{1},r_{2}>0,\;r\in\mathbb{C}.$
The real dimension of the groups $S$ and $N$ is equal to 4, and that of the
group $P$ is equal to 8.
A general element of the group $P$ (a pair $(s,X)$) is as follows:
$\left(\begin{array}[]{cc}s^{*{-1}}&0\\\ X&s\\\ \end{array}\right),$
where $X$ is a $2\times 2$ matrix satisfying a condition that could be called
the relative skew-Hermiticity (with respect to the matrix $s$):
$sX^{*}+Xs^{*}=0.$
The following assertion can be checked directly.
###### Proposition 1.
The group $U(2,2)$ is algebraically generated by the elements of the group $P$
and the involution $\sigma$; the intersection of the groups $N$ and $S$
consists of the identity element.
The homogeneous space $U(2,2)/K$ where $K$ is the maximal compact subgroup of
$U(2,2)$ is exactly the space of the subgroup $P$, which justifies calling it
the Iwasawa subgroup. (This makes it possible to extend a cocycle from $P$ to
the whole group $U(2,2)$, see below.)
Since the group $S$ acts in an obvious way on the additive group $N$ of skew-
Hermitian matrices according to the rule
$n\mapsto sns^{*},\qquad s\in S,\;n\in N,$
we can define the semidirect product $Q=S\rightthreetimes N$ of $S$ and $N$.
One can directly check the following important assertion.
###### Theorem 1.
The groups $P$ and $Q$ are canonically isomorphic. The isomorphism
$I:P\rightarrow Q$ is given by the formula
$I:(s,X)\rightarrow(s,Xs^{*})$
(the left-hand side is an element of the subgroup $P\subset U(2,2)$, and the
right-hand side is an element of the semidirect product $Q$, the matrix
$Xs^{*}$ being obviously skew-Hermitian). The inverse isomorphism is as
follows: $(s,n)\rightarrow(s,{ns^{*}}^{-1}),$ where $n$ is skew-Hermitian.
###### Proof.
We check that the multiplication is homomorphic:
$I((s_{1},X_{1})\circ(s_{2},X_{2}))=I(s_{1}s_{2},X_{1}{s_{2}^{*}}^{-1}+s_{1}X_{2})=$
$=(s_{1}s_{2},X_{1}s_{1}^{*}+s_{1}X_{2}s_{2}^{*}s_{1}^{*})=I(s_{1},X_{1})I(s_{2},X_{2}).$
∎
The map $I$ sends $P$ to the group whose matrix realization consists of the
collection of pairs $(s,X)$ satisfying the above condition. Hence the
isomorphism of the semidirect product $Q$ of $S$ and $N$ with its matrix
realization, i.e., the group $P$, does not coincide with the direct product of
the identity isomorphism of the subgroup $S$ and the identity isomorphism of
the normal subgroup $N$. Nevertheless, the group $P$ is the semidirect product
$S\rightthreetimes N$.444In the case of a group of rank $1$, the matrix
realization of the Iwasawa subgroup is somewhat simpler; for example, for the
group $SL(2)$ the Iwasawa subgroup $P$ is the subgroup of triangular matrices
$\left(\begin{array}[]{cc}s^{-1}&o\\\ n&s\\\ \end{array}\right),\qquad
s\in{\mathbb{R}}_{+},\;n\in\mathbb{R},$ and the structure of the semidirect
product agrees with the ordinary matrix representation, since the group
regarded as a set is the direct product of the normal subgroup
$\left(\begin{array}[]{cc}1&0\\\ n&1\\\ \end{array}\right)$ and the subgroup
$\left(\begin{array}[]{cc}s^{-1}&0\\\ 0&s\\\ \end{array}\right).$
A general element of the commutant of the group $S$ in the above notation is
as follows:
$\left(\begin{array}[]{cc}1&0\\\ r&1\\\ \end{array}\right),$
where $r$ is a complex number. The derived length of the solvable group $S$ is
equal to 2. Thus the commutant of the group $P$ is the semidirect product of
$\mathbb{C}$ and the commutative group of skew-Hermitian matrices (with a
nontrivial action of $\mathbb{C}$).
###### Corollary 1.
The derived length of the solvable group $P$ is equal to $3$.
### 2.2 Representations of the Iwasawa subgroup and almost invariant measures
Our aim is to study the special representation of the subgroup $P$ and then
extend it to the whole group $U(2,2)$. We will study representations of the
group $P$ regarded as a semidirect product.
The group $N$ is isomorphic to the group $\hat{N}$ of all its continuous
characters, and we have the conjugate action of the group $S$ on $\hat{N}$;
moreover, the direct and conjugate actions on the group $N$ (identified with
its dual group) coincide:
$n\rightarrow
sns^{*};\qquad\chi\rightarrow\chi_{s}\quad\mbox{where}\quad\chi_{s}(n)=\chi(sns^{*}).$
###### Proposition 2.
The action of the group $S$ by automorphisms on the group $\hat{N}$ (and on
the group $N$) has four orbits of positive measure; they are parametrized by
the signs of the imaginary parts of the diagonal entries, i.e., are the orbits
of the elements
$\left(\begin{array}[]{cc}\pm i&0\\\ 0&\pm i\\\ \end{array}\right),$
respectively. On every orbit, the action of $S$ is free and faithful, so that
each orbit can be identified with the group $S$; then the action coincides
with the action of $S$ on itself by right translations.
Note that there are also orbits of smaller dimension, which have zero measure,
but we will not need them.
It is clear that all four actions of the group $S$ on the nondegenerate orbits
are topologically isomorphic and the corresponding representations of the
semidirect product differ only by a ${\mathbb{Z}}_{2}$-valued cocycle which
acts as a multiplication, so that it suffices to consider only one (any)
orbit.
A unitary representation of the semidirect product of a group and a
commutative group (in our case, $P=S\rightthreetimes N$) has the following
canonical realization. Consider a probability measure $\mu$ on the group of
characters (i.e., on $\hat{N}$) that is quasi-invariant with respect to the
action (of the group $S$). All such measures are equivalent, since $S$ is
locally compact and its action is transitive. Hence in the Hilbert space
$L^{2}_{\mu}(\hat{N})$ we can define the unitary representation of the group
$P$ induced by the above action of $S$ and the representation of $N$ in which
an element of $N$ acts as the multiplication by the corresponding character or
$\hat{N}$. General representations are realized in the vector-valued space
$L^{2}_{\mu}(\hat{N})$, but we do not consider them.
The irreducibility of the above representation of the semidirect product is
equivalent to the ergodicity of the measure $\mu$, and, by the above, we could
assume that the orbit is the group $S$ itself, i.e., consider the
representation of $S$ in the space $L^{2}_{\mu}(S)$ over a measure $\mu$
quasi-invariant with respect to the right action of the group.
These representations belong to the core of the group $P$; this follows from
the fact that each of them is quasi-equivalent to the regular representation
of the group, which (by the amenability of $S$) weakly contains the trivial
representation. Besides, the core contains the trivial representation, as well
as the special representations of the group $S$, which can be regarded as
representations of $P$, since $S=P/N$. It is not known whether the core is
exhausted by these representations, nor whether the four constructed
representations are special for the group $P$, or, in other words, whether
they have an almost invariant vector. But first we find out the structure of
the special representations of the group $S$.
###### Lemma 1.
The group $S$ has a continuum of unitary representations parametrized by the
points of $\mathbb{C}$ lying on the unit circle (characters). It has no
faithful special representations.
###### Proof.
The affine group $\mbox{Aff}(\mathbb{R})$, i.e., the group of matrices
$\left(\begin{array}[]{cc}e^{a}&0\\\ b&e^{-a}\\\ \end{array}\right),\qquad
a,b\in{\mathbb{R}},$
is the Iwasawa subgroup for $SL(2,R)$ and plays the same role as the group $P$
does for $U(2,2)$. Note that, regarded as a subgroup of $SL(2,R)$, it is the
semidirect product of ${\mathbb{R}}_{+}$ and $\mathbb{R}$ that agrees with the
matrix representation; as mentioned above, in our situation this is not the
case. It has (two) faithful special unitary representations, and they can be
extended to a special representation of $SL(2,R)$.
Consider the group $S$:
$\left(\begin{array}[]{cc}r_{1}&0\\\ r&r_{1}\\\ \end{array}\right),\qquad
r\in{\mathbb{C}},\quad r_{1},r_{2}>0;$
note that if we fix a value of the determinant ($=r_{1}\cdot r_{2}$) and a
unitary character on the group ${\mathbb{C}}={\mathbb{R}}^{2}$, the group $S$
can be mapped isomorphically to the group $\mbox{Aff}(\mathbb{R})$, and hence
all special representations of $\mbox{Aff}(\mathbb{R})$ can be lifted to
representations of $S$; all of them are not faithful; the group $S$ has no
other special representations. ∎
However, we are interested not as much in the group $S$, but in the group $P$.
Conjecturally, special unitary representations of $P$ can be constructed using
special (nonfaithful) representations of $S$ together with the representations
of $P$ constructed above. At the moment, the question of whether such
representations exist is open. For completeness, we describe a model of the
Hilbert space of a special unitary representation for the group $SL(2,R)$ and
its triangular subgroup ($P$). Consider the space
$L^{2}_{m}({\mathbb{R}}_{+})$ (where $m$ is the Lebesgue measure on the half-
line). It is more convenient to pass to the Fourier transform, and then the
required representation of the triangular subgroup (written as the group of
transformations $x\mapsto e^{\beta}x+a$ with $\beta,a\in\mathbb{R}$) can be
realized in $L^{2}_{\hat{m}}(\mathbb{R})$ (where ${\hat{m}}$ is the Lebesgue
measure on the line) as follows:
$(U_{a,\beta}F)(z)=\exp\\{iae^{z}\\}F(z+\beta),\quad z\in\mathbb{R}.$
An almost invariant vector in this model is an arbitrary function $f$
satisfying, for any $t,a,b\in\mathbb{R}$, the conditions
$f(x)=0\mbox{ if }x>t\in{\mathbb{R}};$ $f\notin
L^{2};\quad(1-\exp\\{ie^{z}b\\})f\in L^{2};\quad[f(\cdot)-f(\cdot+a)]\in
L^{2}.$
Another, more popular, description of the special representation (see [17]) in
a space of analytic functions is related to the limit of complementary series
representations as they tend to the trivial representation.
### 2.3 Almost invariant measures and nonunitary representations
We say that a measure $\nu$ on a group $S$ is (right) almost invariant if it
is infinite, absolutely continuous with respect to the right Haar measure on
$S$ (and hence quasi-invariant with respect to the right translations
$s\mapsto ss_{0}$), and its derivatives $\frac{d\nu(ss_{0})}{d\nu(s)}$ are
defined and bounded for every $s_{0}\in S$. (By the above isomorphism
$S\rightarrow H$, where $H$ is an arbitrary nondegenerate $S$-orbit on the
group of characters $\hat{N}$, this definition can be translated to measures
concentrated on any of the nondegenerate orbits of the group $\hat{N}$).
The almost invariance condition is obviously satisfied for the Lebesgue
measure on $S$, i.e.,
$ds=ds_{11}ds_{22}ds_{21}d\bar{s}_{21},$
since $d(ss_{0})=\pi(s)ds$, where $\pi(s)=s_{11}^{3}s_{22}$. It follows that
this condition holds for any measure of the form $d\nu(s)=u(s)ds$, where
$u(s)$ is an arbitrary function such that the ratio $\frac{u(ss_{0})}{u(s)}$
is a bounded function for every $s_{0}\in S$. In particular, it holds for the
measure $\mu$ that is invariant under the right translations (the Haar
measure):
$d\mu(s)=\pi^{-1}(s)ds,\quad\pi(s)=s^{3}_{11}s_{22}.$
However, for our purposes it is convenient to consider another measure.
Assume that a group $G$ acts on a space $X$, and we are given two equivalent
$G$-quasi-invariant measures $\mu$ and $\nu$ on $X$. Assume that the density
of one measure with respect to the other one is bounded away from zero and
infinity. In the spaces $L^{2}_{\mu}(X)$ and $L^{2}_{\nu}(X)$ we consider the
representation of the group $G$ by the substitutions
$(U_{g}f)(\cdot)=f(g\cdot)$ and the natural representation of the group of
multiplicators with absolute value equal to one. The well-known isometry
between these spaces, which multiplies a function by the square root of the
density of one measure with respect to the other one, commutes with the
multiplicators, but, in general, does not commute with the action of the
group. This isometry is widely used to correct the action; for instance, if
one of the measures is invariant, and thus determines a unitary representation
of the cross product, then the corrected action also becomes unitary.
Let $\nu$ be an almost invariant measure on $H$; consider a nonunitary
representation of the group $P$ in the Hilbert space $L^{2}(S,\nu)$, i.e., the
space of functions $F$ on $S$ with the norm
$\|F\|^{2}=\int_{S}|F(s)|^{2}d\nu(s)<\infty$. By definition, the operators
corresponding to the elements of the subgroups $N$ and $S$ are given by the
following formulas:
$\displaystyle(T(n)F)(s)$ $\displaystyle=$
$\displaystyle\chi_{k}(n,s)F(s)\quad\mbox{ for }n\in N;$ (1)
$\displaystyle(T(s_{0})F)(s)$ $\displaystyle=$ $\displaystyle
F(ss_{0})\quad\mbox{ for }s_{0}\in S.$ (2)
Here $\chi_{k}(n,s)$ is the image of a character $\chi(\cdot)\in\hat{N}$
regarded as a function on $S$ under the (unique) isomorphism between the orbit
of $S$ in $\hat{N}$ indexed by $k=1,2,3,4$ and $S$ preserving the action of
$S$; the difference between the four orbits reduces to multiplying the image
by $\pm i$ in each of the variables. It follows from the definition that the
operators $T(n)$ are unitary and the operators $T(s_{0})$ are bounded, by the
almost invariance of the measure $\nu$. One can rewrite the formulas in a more
compact form:
$\displaystyle(T(n)f)(s)$ $\displaystyle=$
$\displaystyle\chi(sns^{*})f(s)\quad\mbox{ for }n\in N;$ (3)
$\displaystyle(T(s_{0})f)(s)$ $\displaystyle=$ $\displaystyle
f(ss_{0})\quad\mbox{ for }s_{0}\in S,$ (4)
where $\chi$ is a fixed character of the $S$-orbit $H$ on the group of
characters $\hat{N}$.
It is not difficult to verify that the operators of the subgroups $N$ and $S$
together generate a representation of the whole group $P$ in the space
$L^{2}(S,\nu)$. In particular, if $\nu=\mu$ is the Haar measure on $S$, this
representation is unitary. Denote these representations of the group $P$ by
$\pi_{k}$, $k=1,2,3,4$; since all four representations essentially differ from
one another only by a choice of a character on the orbits, we omit the index
$k$.
###### Theorem 2.
The nonunitary representations $\pi$ of the group $P$ defined above are
operator irreducible and space irreducible. The representations corresponding
to different measures and different indices $k$ are space (unitarily)
equivalent if and only if the measures $\nu$ differ by a factor
$(\nu^{\prime}=c\nu)$ and the indices $k$ coincide.
An important question for us is how the cohomology depends on the measure. We
emphasize that changing the measure and, in particular, the unitarization of
representations (see above) does not induce an isomorphism of the cohomology
groups $H^{1}(G,\pi_{\mu})$ and $H^{1}(G,\pi_{\nu})$ where $\pi_{\mu}$,
$\pi_{\nu}$ are the representations corresponding to the measures $\mu$ and
$\nu$, since a space isometry does not in general send a cocycle of a group
with values in one space to a cocycle with values in another space. In other
words, for a given action of the group, the cohomologies with values in the
Hilbert space $L^{2}_{\mu}(X)$ for various almost invariant measures $\mu$ are
in general different. In the next section we choose an almost invariant
measure for which the cohomology is nontrivial.
### 2.4 A faithful nonunitary special irreducible representation of the
Iwasawa group
An important question for us is how the cohomology with values in
$L_{\nu}^{2}(X)$ depends on the almost invariant measure $\nu$. We emphasize
that changing the replacement of the measure by an equivalent one and, in
particular, the unitarization of representations (see above) does not induce
an isomorphism of the cohomology groups $H^{1}(G,\pi_{\mu})$ and
$H^{1}(G,\pi_{\nu})$, where $\pi_{\mu}$, and $\pi_{\nu}$ are the
representations corresponding to the measures $\mu$ and $\nu$, since a space
isometry does not in generally send a cocycle of a group with values in one
space to a cocycle with values in another the other space. In other words, for
a given action of the group, the cohomologies with values in the Hilbert space
$L^{2}_{\mu}(X)$ for various almost invariant measures $\mu$ are in generally
different. This explains our choice of an almost invariant measure in what
follows. In the next section we choose an almost invariant measure for which
the cohomology is nontrivial.
Let us fix an almost invariant measure $\nu$ on the space $S$ and introduce
the space $Z_{\nu}$ of measurable functions on $S$ satisfying the two
conditions
$\displaystyle\int_{S}|f(ss_{0})-f(s)|^{2}\,d\nu(s)$
$\displaystyle<\infty\quad\text{for any}\,s_{0}\in S,$
$\displaystyle\int_{S}|(\chi(sns^{*})-1)f(s)|^{2}\,d\nu(s)$
$\displaystyle<\infty\quad\text{for any}\,n\in N.$
Obviously, $L^{2}_{\nu}(S)\subset Z_{\nu}$.
We treat the elements of $f\in Z_{\nu}$ as coboundaries and the space of
functions of the form $b(g)=T(g)f-f$ as the space of cocycles. If $f\in
L^{2}_{\nu}(S)$, then $b(g)=T(g)f-f$ is a cocycle cohomologous to zero. This
implies the following assertion.
###### Lemma 2.
A representation of the group $P$ in the space $L^{2}_{\nu}(S)$ is special if
and only if
$L^{2}_{\nu}(S)\varsubsetneq Z_{\nu},$
i.e., if there exists a coboundary not lying in $L^{2}_{\nu}(S)$. Thus, the
first cohomology group has the form
$H^{1}(P;L^{2}_{\nu}(S)=Z_{\nu}/Z_{\nu}L^{2}_{\nu}(S).$
A measure $\nu$ is said to be special if $H^{1}(P;L^{2}_{\nu}(S))\neq 0$,
i.e., if there exist coboundaries not lying in $L^{2}_{\nu}(S)$. The authors
do not know whether the Haar measure $m$ on $S$ is special and, thereby,
whether the natural unitary representation of the group $P$ on $L^{2}_{m}(S)$
is special. For this reason, we use a different almost invariant measure to
construct a special, but not unitary, representation of $P$.
In what follows, we fix an almost invariant measure $\nu$ of the following
form:
$d\nu(s)=|s|^{-4}\,ds,\quad\text{where
}\,|s|^{2}=tr(s^{*}s)=s_{11}^{2}+s_{22}^{2}+|s_{21}|^{2}.$
It is convenient to write this measure in polar coordinates on $S$. To do this
end, note that the variety of elements $\omega\in S$ with norm $|\omega|=1$ is
equivalent to a domain on the unit sphere in $\mathbb{R}^{4}$. We define
spherical coordinates of a matrix $s\in S$ we mean in equation 3 as the number
$r=|s|$ and the matrix $\omega=|s|^{-1}s$. Then $s=r\omega$, and the
expression for $\nu$ in polar coordinates has the form
$d\nu(s)=r^{-1}\,dr\,d\omega,$
where $d\omega$ is the invariant measure on the sphere.
The following assertion can be checked verified directly.
###### Theorem 3.
The representation $\pi$ of the group $P$ in the Hilbert space $L^{2}(S,\nu)$,
where $d\nu(s)=|s|^{-4}ds$, is special and has a nontrivial cocycle of the
form
$b(g)=T(g)f-f,\quad\text{where }\,f(s)=e^{-|s|/2}.$ (4)
### 2.5 Extending the special nonunitary representation of the subgroup $P$
to the whole group $U(2,2)$
It remains to check that the special representation can be extended to a
nonunitary representation of the whole group $U(2,2)$. The construction of the
required extension is based on the following property of the group $U(2,2)$.
Every element $g\in U(2,2)$ can be uniquely written as a product $g=pk$, $p\in
P$, $k\in K$, where $K$ is the maximal compact subgroup, which consists of the
elements $k\in U(2,2)$ satisfying the relation $kk^{*}=e$, i.e., the subgroup
of block matrices of the form
$k=\left(\begin{array}[]{cc}\alpha&\beta\\\ \beta&\alpha\\\
\end{array}\right)$
where $\alpha\alpha^{*}+\beta\beta^{*}=e_{p}$ and
$\alpha\beta^{*}+\beta\alpha^{*}=0$ (the Iwasawa decomposition).
Let $T$ be the special representation of the subgroup $P$ in the Hilbert space
$H=L^{2}(S,\nu)$ defined in Theorem 2, and $b$ be the nontrivial cocycle
defined by (8). Denote by $H_{0}$ the linear invariant subspace in $H$ spanned
by the vectors $b(p)$, $p\in P$.
###### Lemma 3.
1. The subspace $H_{0}$ is dense in $H$.
2. The vectors $b(p)$, $p\neq e$, are linearly independent (the nondegeneracy property).
Let $T(k)$, $k\in K$, be the operators defined on the set of vectors $b(p)$,
$p\in P$, by the formula
$T(k)b(p)=b(p^{\prime}),$
where $p^{\prime}\in P$ is defined by the relation $kp=p^{\prime}k^{\prime}$,
$k^{\prime}\in K$.
3. The operators $T(k)$ satisfy the group relation $T(k_{1}k_{2})b(p)=T(k_{1})T(k_{2})b(p)$ for any $k_{1},k_{2}\in K$ and $p\in P$, and hence generate a representation of the subgroup $K$ in the subspace $H_{0}$.
###### Theorem 4.
The operators $T(k)$, $k\in K$, together with the operators $T(p)$, $p\in P$,
generate a representation of the whole group $U(2,2)$ in the subspace $H_{0}$,
in which the operator corresponding to the involution
$\sigma=\left(\begin{array}[]{cc}0&e_{p}\\\ e_{p}&0\\\ \end{array}\right)$ is
defined on the set of vectors $b(p)$ by the following formula:
$T(\sigma)b(p)=b(\hat{p}),\qquad\hat{p}\in P,$ (5)
where $\hat{p}$ is uniquely determined by the relation
$\hat{p}\hat{p}^{*}=\sigma pp^{*}\sigma.$ In this extension, the operators
corresponding to the elements of the subgroup $P$ are unitary, the operators
corresponding to the elements of the subgroup $K$ are bounded, and the
operator corresponding to the involution is unbounded and cannot be extended
to the whole space $H$. The extension of the $1$-cocycle $b$ from the group
$P$ to the group $U(2,2)$ is given by the formula
$b(pq)=b(p)\mbox{ for any }p\in P\mbox{ and }q\in Q.$ (6)
Applications of the described constructions to representations of the group of
currents will be described elsewhere. In this paper we restrict ourselves to
the group $U(2,2)$ just for methodological reasons: our aim is to give a
simple example of the general theory, which, in the authors’ opinion, covers a
wide class of semisimple Lie groups and the corresponding groups of currents.
## References
* [1] H. Araki, Factorisable representations of current algebra. Publ. RIMS, Ser. A 5, No. 3, 361 422 (1970).
* [2] B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s Property $(T)$. Cambridge Univ. Press, 2007.
* [3] F. A. Berezin, Representations of the continuous direct product of universal coverings of the group of motions of a complex ball. Trans. Moscow Math. Soc. 36, 281–298 (1979).
* [4] P.-A. Cherix et al., Groups With the Haagerup Property: Gromov’s A-T-Menability. Springer, 2001.
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* [6] I. M. Gelfand, M. I. Graev, and A. M. Vershik, Models of representations of current groups. In: Representations of Lie Groups and Lie Algebras, Akad. Kiado, Budapest, 1985, pp. 121–179.
* [7] A. Guichardet, Sur la cohomologie des groupe topologiques. II, Bull. Soc. Math. France 96, 305–332 (1972).
* [8] U. Haagerup, An example of nonnuclear $C^{*}$-algebra, which has the metric approximation property, Inv. Math. 50, 279–293 (1978).
* [9] R. S. Ismagilov, Representations of Infinite-Dimensional Groups, Transl. Math. Monogr., 152, Amer. Math. Soc., Providence, RI, 1996.
* [10] B. Kostant, On existence and irreducibility of certain series of representations, Bull Amer. Math Soc. 75, 627–642 (1969).
* [11] A. L. Onishchik and E. B. Vinberg, Lie Groups and Algebraic Groups. Springer-Verlag, 1990.
* [12] K. Parthasarathy and K. Schmidt, Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory. Springer, 1972.
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* [14] A. M. Vershik, Does there exist the Lebesgue measure in the infinite-dimensional space? Proc. Steklov Institute of Mathematics 259, 256–281 (2007).
* [15] A. M. Vershik, Invariant measures for the continual Cartan subgroup. J. Funct. Anal. 255, No. 9, 2661–2682 (2008).
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* [17] A. M. Vershik, I. M. Gelfand, and M. I. Graev, Representations of the group $SL(2,R)$, where $R$ is a ring of functions. Russ. Math. Surv. 28, No. 5, 82–128 (1973).
* [18] A. M. Vershik, I. M. Gelfand, and M. I. Graev, Irreducible representations of the group $G^{X}$ and cohomology. Funct. Anal. Appl. 8, 151–153 (1974).
* [19] A. M. Vershik and M. I. Graev, The basic representation of the current group $O(n,1)^{X}$ in the $L^{2}$ space over the generalized Lebesgue measure. Indag. Math. 16, No. 3/4, 499–529 (2005).
* [20] A. M. Vershik and M. I. Graev, Special representations of the groups $SO(n,1)$ and $SU(n,1)$. Russ. Math. Surv. 61, No. 5, 799–884 (2006).
* [21] A. M. Vershik and M. I. Graev, Integral representations of current groups. Funct. Anal. Appl. 42, No. 1, 22–32 (2008).
* [22] A. M. Vershik and M. I. Graev, Integral models of representations of the current groups of simple Lie groups. Russ. Math. Surv. 64, No. 2, 205–271 (2009).
* [23] A. M. Vershik and M. I. Graev, The Poisson model of the Fock space and representations of current groups, St. Petersburg Math. J. 23, No. 3, 459–510 (2012).
* [24] A. M. Vershik and M. I. Graev, Special representations of the groups $U(\infty,1)$ and $O(\infty,1)$ and the associated representations of the current groups $U(\infty,1)^{X}$ and $O(\infty,1)^{X}$ in quasi-Poisson spaces. Funct. Anal. Appl. 46, No. 1, 1–10 (2012).
* [25] A. M. Vershik and S. I. Karpushev, Cohomology of groups in unitary representations, the neighborhood of the identity, and conditionally positive definite functions. Math. in USSR 47, 513–526 (1984).
Translated by N.Tsilevich.
|
arxiv-papers
| 2014-04-18T09:18:00 |
2024-09-04T02:50:01.314784
|
{
"license": "Public Domain",
"authors": "A.M.Vershik, M.I.Graev",
"submitter": "Anatoly Vershik M",
"url": "https://arxiv.org/abs/1404.4726"
}
|
1404.4727
|
# Signatures of new $d$-wave vortex physics in overdoped Tl2Ba2CuO6+x revealed
by TF-$\mu^{+}$SR
Jess H. Brewer [email protected] Scott L. Stubbs Department of Physics and
Astronomy, The University of British Columbia, Vancouver, BC, Canada V6T 1Z1
Ruixing Liang D. A. Bonn W. N. Hardy Department of Physics and Astronomy,
The University of British Columbia, Vancouver, BC, Canada V6T 1Z1 Canadian
Institute for Advanced Research, Toronto, Ontario, Canada M5G 1Z8 J. E.
Sonier Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G
1Z8 Department of Physics, Simon Fraser University, Burnaby, BC, Canada V5A
1S6 W. Andrew MacFarlane Department of Chemistry, The University of British
Columbia, Vancouver, BC, Canada V6T 1Z1 Darren C. Peets [email protected]
Department of Physics and Astronomy, The University of British Columbia,
Vancouver, BC, Canada V6T 1Z1 Center for Correlated Electron Systems,
Institute for Basic Science, Seoul National University, Seoul 151-747, Korea
###### Abstract
The spontaneous expulsion of applied magnetic field, the Meissner effect, is a
defining feature of superconductors; in Type-II superconductors above the
lower critical field, this screening takes the form of a lattice of magnetic
flux vortices. Using implanted spin-1/2 positive muons, one can measure the
vortex lattice field distribution through the spin precession and deduce key
parameters of the superconducting ground state, and thereby fundamental
properties of the superconducting pairing. Muon spin rotation/relaxation
($\mu$SR) experiments have indeed revealed much interesting physics in the
underdoped cuprates, where superconductivity is closely related to, or
coexistent with, disordered or fluctuating magnetic and charge excitations.
Such complications should be absent in overdoped cuprates, which are believed
to exhibit conventional Fermi liquid behaviour. These first transverse field
(TF)-$\mu^{+}$SR experiments on heavily-overdoped single crystals reveal a
superfluid density exhibiting a clear inflection point near 0.5$T_{\text{c}}$,
with a striking doping-independent scaling. This reflects hitherto
unrecognized physics intrinsic to $d$-wave vortices, evidently generic to the
cuprates, and may offer fundamentally new insights into their still-mysterious
superconductivity.
###### pacs:
74.25.Ha, 74.72.Gh, 76.75.+i, 74.25.Uv
## Introduction
Charge doping of the CuO2 planes tunes the occurrence of superconductivity in
the high-temperature hole-doped cuprate superconductors between the limits of
an undoped insulating antiferromagnet and a possible conventional Fermi liquid
at high dopings. It is appealing to try to understand how the unconventional
superconductivity evolves out of these more conventional electronic ground
states. However, hole doping is typically effected chemically, in the best
case via the composition of a distinct, well-separated subunit of the layered
crystal structure, to leave the planes themselves little altered structurally
and the dopant site well-shielded when a hole is promoted to the planes. One
such example is oxygen doping in the CuO chain layer of YBa2Cu3O7-δ (YBCO).
Unfortunately, compositional tuning is limited by (thermodynamic) phase
stability and can seldom be used to traverse the entire superconducting phase
diagram in a single system. In this context overdoped cuprates, those nearer
the apparent Fermi liquid regime, are rare and, moreover, relatively few have
highly-ordered CuO2 planes. For example, doping by cation substitution in
LaxSr2-xCuO4 (LSCO) introduces substantial disorder directly adjacent to the
CuO2 planes Eisaki _et al._ (2004). In contrast, Tl2Ba2CuO6+x (Tl-2201)
offers tunability throughout the overdoped regime with highly-ordered,
isolated, and flat CuO2 planes, doped via dilute interstitial oxygen in the
distant TlO layers Wagner _et al._ (1997), although Cu/Tl substitution in
this layer Shimakawa (1993); Peets _et al._ (2007) may contribute an offset
in doping. The overdoping also appears to eliminate a predicted electron Fermi
surface (FS) pocket at the $\Gamma$ point, leaving only a single, large, FS
sheet Platé _et al._ (2005).
Pure $d_{x^{2}-y^{2}}$-wave symmetry of the superconducting order in Tl-2201
has been conclusively established by observation of half-integer flux quanta
at crystal boundaries in films Tsuei _et al._ (1997); line nodes are evident
in microwave Broun _et al._ (1997); Özcan _et al._ (2006); Deepwell _et
al._ (2013) and thermal transport Hawthorn _et al._ (2007) measurements; and
the admixture of another pairing symmetry is unlikely because it would require
spontaneous breaking of the crystal symmetry. However, some $\mu$SR
measurements have suggested an additionial transition at low temperatures
within the vortex state Khasanov _et al._ (2007a, 2008); Blasius _et al._
(1999), which has been interpreted in terms of a multiple-component order
parameter. Here, we extend $\mu$SR studies of the vortex state of the cuprates
deep into the overdoped regime with the first transverse-field muon spin
rotation (TF-$\mu^{+}$SR) results on single-crystalline Tl-2201, in the form
of high-quality single crystal mosaics at a range of dopings. Our measurements
were performed at low magnetic fields, in a doping regime free from competing
charge density wave order Ghiringhelli _et al._ (2012); Chang _et al._
(2012a), and are thus sensitive to the intrinsic structure of $d$-wave
vortices. We show that the unusual temperature dependence is real, and generic
to the cuprates, but also demonstrate that it is a signature of $d$-wave
vortex physics rather than a multicomponent order parameter.
Figure 1: Example of $\mu^{+}$SR data. (a) Complex TF-$\mu^{+}$SR time
spectrum (red circles: real part; blue triangles: imaginary part) in a
rotating reference frame (RRF) at 0.1 T and 10 K on the $T_{\text{c}}\approx
56$ K Tl-2201 mosaic, including time-domain best fit, the residual errors of
which are shown in (b) for the first 4 $\mu$s where the statistics are
highest. (c) Fourier transforms at several temperatures. The relatively sharp
peak at 13.55 MHz arises from muons stopping outside the sample.
## Results
Fig. 1 shows an example of the data and time domain fit at 10 K on a
$T_{\text{c}}=56$ K mosaic. Fits on all mosaics at all fields and temperatures
converged very well, and fully reproduce the data. Fourier transforms
corresponding to the field distribution are also shown for a selection of
temperatures; the additional peak just above the cusp is attributed to muons
stopping outside the sample and precessing about the applied field, and is
accounted for in the fits. The high-field (high-frequency) cutoff in the
lineshape is indistinct, precluding a quantitative analysis of the in-plane
coherence length $\xi_{ab}$, but the in-plane magnetic penetration depth
$\lambda_{ab}$, which controls the linewidth, may be reliably extracted.
Varying the fit parameters indicated that the absolute $1/\lambda_{ab}^{2}$ is
accurate to within $\sim 10$%, while its temperature dependence is robust;
after a few global fits with different choices of
$\kappa_{ab}\equiv\lambda_{ab}/\xi_{ab}$ we chose a fixed value,
$\kappa_{ab}=100$, for all remaining fits.
One strength of TF-$\mu^{+}SR$ in a Type-II superconductor is its ability to
determine the absolute $\lambda$ and its inverse square, which is proportional
to the density of superconducting carriers Sonier _et al._ (2000).
Circumstances are not yet as good for Tl2Ba2CuO6+x as for high-quality YBCO;
in particular, only small improvements of global $\chi^{2}$ minimization
distinguish the broadening due to vortex lattice disorder, $\sigma_{d}$, which
should scale with $\lambda^{-2}(T)$, from $T$-independent broadening due to
nuclear magnetic dipoles and crystal defects, $\sigma_{0}$. The amplitude
$A_{B}$ of the background signal due to muons stopping outside the sample is
also known only from the best fit; like $\sigma_{0}$, it can be subtly coupled
to $\lambda^{-2}$. These uncertainties do not alter the temperature
dependence, and have been incorporated into the quoted $\sim 10$% uncertainty
in $1/\lambda_{ab}^{2}$.
Figure 2: (a) Temperature dependence of fitted $\lambda_{ab}^{-2}$ at $H=0.1$ T for all Tl-2201 mosaics; A and B denote two different mosaics with the same $T_{\text{c}}$. Absolute microwave data (curve) at zero field on a $T_{\text{c}}=25$ K crystal at 2.497 GHz Deepwell _et al._ (2013), included for comparison, follow a qualitatively different form. Curves are provided for two mosaics as a guide to the eye. (b) Normalized values $\lambda_{ab}^{-2}(T)/\lambda_{ab}^{-2}(0)$ vs. reduced temperature $T/T_{\text{c}}$ for all Tl-2201 mosaics. All dopings exhibit essentially the same temperature dependence, and differ from the microwave results (solid curve). Table 1: Zero-temperature in-plane magnetic penetration depths in 0.1 T for overdoped Tl-2201 mosaics having various $T_{\text{c}}$s, from a linear extrapolation of $\lambda_{ab}^{-2}(T)$ at low temperatures, with estimated uncertainties in parentheses. The variations in $\lambda_{ab}(0)$ are most likely dominated by the degree of order in the samples, rather than any systematic doping dependence, as discussed in the text. Uncertainties in $T_{\text{c}}$ represent primarily the variation in $T_{\text{c}}$ among the crystals comprising the mosaic. $T_{\text{c}}$ (K) | 46(1), A | 46(1), B | 56(1) | 60(1) | 72(1) | 75(1)
---|---|---|---|---|---|---
$\lambda_{ab}(0)$ (nm) | 187(2) | 165(2) | 166(1) | 175(1) | 182(2) | 153(2)
Figure 2 shows the vortex-state $1/\lambda_{ab}^{2}(T)$, which is proportional
to the superconducting carrier density, for the six mosaics measured, and
Table 1 reports the zero-temperature penetration depth $\lambda_{ab}(0)$ from
linear extrapolations of $\lambda_{ab}^{-2}(T)$. A highly unusual
$T$-dependence, common to all dopings, is immediately apparent. The extent of
this similarity is more striking when $\lambda_{ab}^{-2}(T)$ is normalized to
its extrapolated $T=0$ value and plotted against reduced temperature
$T/T_{\text{c}}$ — the relative temperature dependence is identical. The most
intriguing feature, exhibited in all six mosaics, is upward curvature between
$\frac{1}{3}T_{\text{c}}$ and an inflection point around 0.5$T_{\text{c}}$.
This unusual temperature dependence is robust and evident in any measure of
the linewidth, but is absent in zero-field (Meissner state) microwave surface
resistance at higher and lower dopings Özcan _et al._ (2006); Deepwell _et
al._ (2013), the former included for comparison. The intrinsic $T$-dependence
of the superconducting carrier density (or $1/\lambda_{ab}^{2}$) in a single-
gap $s$\- or $d$-wave superconductor exhibits downward curvature over the
entire temperature range 0–$T_{\text{c}}$.
## Discussion
$\mu$SR reports of the cuprates’ temperature-dependent in-plane penetration
depth typically exhibit the shape associated with a pure $d$-wave order
parameter Sonier _et al._ (1994, 1999a). However, this has not been the case
in all data. Upward curvature in the $\mu$SR penetration depth can be
recognized in relatively disordered overdoped LSCO Khasanov _et al._ (2007a);
at high dopings in cleaner YBa2Cu3O7-δ Sonier _et al._ (2007), in lightly
underdoped YBa2Cu4O8 Khasanov _et al._ (2008); and in optimally and overdoped
Bi2Sr2CaCu2O8+δ Blasius _et al._ (1999). This unusual temperature dependence
has appeared most clearly near and above optimal doping Sonier _et al._
(1999a, 2007), with some limited evidence that it strengthens on overdoping
Blasius _et al._ (1999). It is most evident at relatively low applied fields
Harshman _et al._ (2004); Amin _et al._ (2000).
With no phase transition expected at the fields and temperatures in question,
no such feature unambiguously visible in most published data or any microwave
penetration depth studies, and based on a small number of data points in many
of these cases, it has not been widely accepted as a real effect. Its now-
confirmed appearance in a variety of systems, and its particularly conspicuous
appearance in Tl-2201, implies that it is real and generic, at least to high
doping ranges. The contrast with zero-field microwave data argues against the
few interpretations floated thus far, instead pointing toward an origin in an
unexpected property of the cuprates’ vortex state. We first briefly dispense
with some alternative explanations before returning to vortex physics.
First, multiband superconductors, those with more than one band crossing the
Fermi level, can exhibit unconventional temperature dependence in the vortex
state Zehetmayer (2013); a two-component order parameter, e.g. $d+s$, with
separate order parameters on distinct Fermi surface sheets, has been advanced
to explain the LSCO Khasanov _et al._ (2007a) and YBa2Cu4O8 Khasanov _et
al._ (2008) results. However, Tl-2201 has only one FS sheet, and the pure
$d$-wave symmetry and dissimilar microwave penetration depth at both lower and
higher dopings Broun _et al._ (1997); Deepwell _et al._ (2013) exclude such
an origin. Second, the remarkable scaling seen in Fig. 2 argues against an
electronic phase transition within the superconducting dome, such as an
extension of the pseudogap crossover temperature $T^{*}$. Third, dilute
paramagnetic impurities would yield an additional broadening scaling with
$H/T$, contrary to the observed $T$ dependence. Some type of magnetically
frozen state might account for the observed temperature dependence (since
$\mu^{+}$SR is a local probe, macroscopic phase separation would not affect
the superconducting component’s lineshape). However, one would not expect the
onset of a competing magnetic phase to track $T_{\text{c}}$ with doping
Niedermayer _et al._ (1998). Furthermore, higher fields should enhance any
competing magnetic order Wu _et al._ (2011), particularly in the vortex cores
Sonier (2007), but in YBCO they instead suppress the exotic upward curvature
in the temperature dependence of the linewidth Harshman _et al._ (2004);
Khasanov _et al._ (2007a). Proximity-induced chain superconductivity has been
advanced as an explanation for the inflection point in YBCO Sonier _et al._
(2007), but this cannot explain its appearance in chain-free Tl-2201.
Having excluded several alternative explanations, we return to physics of the
vortex phase, which would be absent in the Meissner-phase microwave
experiments and previous work on Tl-2201 powder Uemura _et al._ (1993), and
may thus offer a natural explanation. First, the resistive upper critical
field of Tl-2201 (actually the irreversibility field Lundqvist _et al._
(1999); Zheng _et al._ (2000)) exhibits unusual upward curvature Mackenzie
_et al._ (1993) and stays very close to $H_{c2}(T)$, far from the required
temperature regime at the low fields relevant here. A dimensionality crossover
within the frozen vortex state, as in the much more anisotropic Bi2Sr2CaCu2O8
Bernhard _et al._ (1995); Blasius _et al._ (1999), would produce a symmetric
lineshape, and the reduced field inhomogeneity would narrow the linewidth at
high temperatures, in contrast to our results. Trapping of vortices by
preferred pinning sites at low temperature has been advanced to explain the
inflection in YBCO Harshman _et al._ (2004), but the linewidth changes in
Tl-2201 would require very significant disorder and change the lineshape
significantly. This is not seen in the FFT spectra; moreover, allowing the
temperature-independent $\sigma_{0}$ broadening to vary produced no systematic
trend with temperature. To further exclude vortex disorder, it will be
essential to quantify it independently using, for instance, small-angle
neutron scattering or scanning probe techniques.
Another relevant feature of the vortex state is the symmetry of the vortex
lattice itself. An unusual square vortex lattice, as found in optimally and
overdoped LSCO Gilardi _et al._ (2002); Chang _et al._ (2012b) and in YBCO
White _et al._ (2011), produces a broader but qualitatively similar field
distribution with a more pronounced low-field tail. However, the vortex
lattice in fully-oxygenated YBCO gradually transforms from triangular to
square as the field increases from 4 to 11 T, while strong upward curvature in
the second moment of the TF-$\mu^{+}$SR lineshape is strongest at much lower
fields $\sim 0.1$ T Khasanov _et al._ (2007b) and is completely suppressed by
4 T Sonier _et al._ (1999a), excluding this interpretation, at least for
YBCO. The $\mu$SR data for the $T_{\text{c}}=56$ K Tl-2201 mosaic were fit to
a simple square vortex lattice model as a trial, but no crossover, in the form
of a systematic improvement in the quality of fit, was evident.
We are thus drawn to the conclusion that the upward curvature must arise from
some fundamental property intrinsic to the $d$-wave vortices themselves. In
this scenario, the difference between the zero-field microwave measurements
and these vortex-state $\mu$SR results arises because the measurements probe
different phases. In microwave measurements, only the surface contributes,
while the bulk is shielded. In $\mu$SR, however, vortices also contribute, and
these can overlap and interact in certain field and temperature regimes.
The scaling with $T_{\text{c}}$ and absence in underdoped samples point to an
explanation in terms of the electronic structure expected for a vortex in a
$d$-wave superconductor. Theoretical calculations Ichioka _et al._ (1999a, b)
show that such vortex cores shrink with increasing magnetic field due to an
enhanced transfer of quasiparticles between neighbouring vortices. This effect
has been observed in YBCO by $\mu$SR Sonier _et al._ (1999b, a, 2007), with
the core size rapidly shrinking and saturating at fields above $H\sim 4$ T. At
low fields where the vortices are well separated, the quasiparticle transfer
is minimal and the vortices are expected to behave essentially as if they were
isolated. A vortex in a $d$-wave superconductor is fourfold symmetric at low
temperatures, with extended low-energy quasiparticle states along the 45∘
(nodal) directions in the CuO2 plane. With increasing thermal population of
the higher energy states, the vortex core size grows, and calculations show
that above $\sim 0.5T_{\text{c}}$ the fourfold-symmetric magnetic field
profile about the vortex core becomes nearly cylindrical Ichioka _et al._
(1996).
As previously stressed Sonier (2007), $\lambda_{ab}$ as measured by $\mu$SR
may be regarded as the in-plane magnetic penetration depth only in the
$T\rightarrow 0$ and $H\rightarrow 0$ limit, but is otherwise an effective
length scale partially influenced by changes to the field profile outside the
vortex core by extended quasiparticle states. Consequently, below $\sim
0.5T_{\text{c}}$ and at low fields the temperature dependence of
$\lambda_{ab}$ is expected to be influenced by the evolving quasiparticle
states that extend far beyond the vortex core. At higher temperatures, where
the fourfold symmetry of the vortex is essentially gone, the behavior of
$\lambda_{ab}$ should closely resemble that of the magnetic penetration depth,
unless the vortex lattice melts. Since the vortex core radius is directly
proportional to the gap magnitude Sonier (2004), which in turn is proportional
to $T_{\text{c}}$ in overdoped cuprates, the scaling with $T_{\text{c}}$ is
naturally explained.
The inflection point near $0.5T_{\text{c}}$ being less prominent or absent in
underdoped cuprates is likely due to the stabilization of competing charge-
density-wave (CDW) order localized in the vicinity of the vortex cores. STM
measurements on optimally- and slightly overdoped Bi2Sr2CaCu2O8+δ Hoffman _et
al._ (2002); Levy _et al._ (2005) and nuclear magnetic resonance (NMR)
measurements on underdoped YBCO Wu _et al._ (2011, 2013) show static CDW
order in the vortex core region, where superconductivity is suppressed. While
this induced static CDW order is observed only at low temperatures in
optimally and slightly overdoped samples, in underdoped YBCO with $p\sim
0.12$, NMR shows that static CDW order occurs over much of the temperature
range below $T_{\text{c}}$. The occurrence of static CDW order in the vortex
core region constitutes a significant modification of the electronic structure
of the $d$-wave vortex Agterberg and Garaud (2015), and consequently the loss
of the inflection point in the temperature dependence of $\lambda_{ab}$ is not
surprising. The CDW competes with superconductivity Chang _et al._ (2012a);
Ghiringhelli _et al._ (2012), so it will preferentially inhabit — and likely
gap out — the regions of momentum space in the nodal direction at the Fermi
surface to maximally avoid competition between the two orders. As a result,
the extended quasiparticle core states along the nodal directions will become
bound. This should lead to isotropic $s$-wave-like vortex behaviour throughout
much of the underdoped side of the phase diagram, while the behaviour seen in
the overdoped regime reflects the intrinsic physics of $d$-wave vortices
without such complications. Supplementary Fig. S2 shows that the fit
parameters, based on an $s$-wave model, fail at the lowest temperatures
(beginning well below the inflection point). The $s$-wave model’s inability to
reproduce the observed field distribution provides evidence of its failure to
adequately describe the vortex phase, particularly at low temperatures.
Figure 3: Fitted values of $\lambda_{ab}^{-2}$ vs. $T$ for the earlier (“A”)
and later (“B”) $T_{\text{c}}=46$ K mosaics. Inset: same data in normalized
form.
Finally, our use of mosaics with similar $T_{\text{c}}$s has important
implications for techniques relying on $\mu$SR for values of the zero-
temperature penetration depth. Mosaics grown and annealed under very similar
conditions, having the same or similar $T_{\text{c}}$, exhibit quite different
absolute penetration depths, as shown in Fig. 3 for $T_{\text{c}}=46$ K. The
normalized linewidths $\lambda_{ab}^{-2}(T)/\lambda_{ab}^{-2}(0)$, however,
are almost identical. Mosaics with $T_{\text{c}}$s of 46 K (“A”) and 72 K were
prepared several years before the 46 K (“B”) and 75 K mosaics. The crystal
growth was still being optimized when the early mosaics were assembled,
demonstrating that suppression of $T_{\text{c}}$ by disorder is not equivalent
to its suppression by carrier overdoping. That the zero-temperature
$\lambda_{ab}^{-2}$ can apparently increase by $\sim 30$% due to increasing
crystalline perfection means that the $\lambda_{ab}(0)$ values in Table 1
should not be regarded as intrinsic. A variety of other techniques rely upon
$\mu$SR to obtain absolute penetration depth values, but our work indicates
that in the overdoped regime, the values are only valid for $T,H\rightarrow
0$, and even then are highly susceptible to disorder and must be treated with
caution.
The nature of superconductivity in the cuprates remains one of the most
important open questions in condensed matter physics, and the overdoped regime
proffers the promising prospect of understanding the normal state from which
high-temperature superconductivity emerges. However, our $\mu^{+}$SR data
indicate exotic physics survives to high dopings. A striking universal
temperature dependence unambiguously confirms an unusual upturn in the in-
plane penetration depth, now seen in at least four distinct material families,
establishing that it is new physics generic to the cuprates. Fundamental
differences from zero-field microwave measurements of nominally the same
quantity Deepwell _et al._ (2013) imply it is intrinsic to the $d$-wave
vortices themselves. Aside from the substantial impacts of this result on
vortex physics and what light this may shed on cuprate superconductivity,
there are important ramifications for other techniques. $\mu$SR is uniquely
suited to extracting the absolute penetration depth from the vortex phase, and
the values thus obtained underpin results of other techniques which can’t
measure $\lambda$ in absolute terms or at all. While the $\mu$SR temperature
evolution should be robust Sonier _et al._ (2000), the absolute values are
model-dependent. Our data indicate that a more complex model is required when
$d$-wave vortices are present, which will necessitate revisiting some previous
results. First, however, the details and origin of the exotic temperature
dependence must be conclusively confirmed. Sensitive scanning probe
microscopies may provide insights into the current and quasiparticle
distribution in the vortex cores, while confirmation that this is a vortex
state phenomenon awaits Meissner-state field distribution measurements, such
as low-energy $\mu$SR Ofer _et al._ (2012) or $\beta$-NMR Hossain _et al._
(2009). The regimes in which these $d$-wave vortices manifest their unique
$d$-wave physics may provide crucial new insight on the cuprates’ order
parameter and still-mysterious pairing.
## Methods
Single crystals of Tl-2201 were grown in gold-sealed alumina crucibles by an
encapsulated copper-rich self-flux method as described elsewhere Peets _et
al._ (2010). The oxygen content (which determines hole content and Tc) was set
by annealing under controlled oxygen partial pressures and temperatures
Opagiste _et al._ (1993); two different annealing schemes were employed
depending on the desired oxygen content Peets _et al._ (2010). Crystals were
assembled in mosaics on substrates of aluminized mylar or GaAs to minimize the
background signal, with the crystallographic $c$-axis perpendicular to the
substrate (parallel to the applied field). In this geometry, the applied field
is shielded by supercurrents running within the $ab$-plane, thus the in-plane
penetration depth $\lambda_{ab}$ governs the field distribution. The large
number of small crystals to be mutually aligned precluded measuring the
superconducting transition of every individual crystal, but care was taken to
construct each mosaic from a small number of annealing batches, each drawing
crystals from only one growth run, and $T_{\text{c}}$ was measured on a
selection of crystals sampled from each annealing run. Examples of
magnetization data on three mosaics are included in Supplementary Fig. S1.
Quoted uncertainties in $T_{\text{c}}$ reflect transition widths of individual
crystals, as determined by DC magnetization in applied fields of 0.1–0.2 mT,
and the expected variation within the mosaic based on the crystals sampled.
These fields were used because the lower critical field $H_{c1}$ in this
material is quite low.
Spin-polarized positive muons from the M15 muon channel at TRIUMF were
injected into the mosaics at an energy of 3-4 MeV in one of several different
$\mu^{+}$SR spectrometers. While a range of magnetic fields were used, the
lowest field for which the rotating reference frame transformation worked
reliably was 0.1 T, and this field was used for all data presented here. In
muon spin rotation/resonance/relaxation ($\mu$SR) Brewer (1994); Sonier _et
al._ (2000), implanted muons settle into specific preferred crystallographic
sites, where their spins precess around the local magnetic field
$\vec{B}_{loc}$ with frequency $\omega=\gamma B_{loc}$, where the muon
gyromagnetic ratio $\gamma\approx 2\pi\times 135.54$ MHz/T. The precession is
detected via a decay positron (in the case of $\mu^{+}$) emitted
preferentially along the muon’s spin direction. The experimental $\beta^{+}$
decay asymmetry reflects the precession of the ensemble of $\sim 10^{7}$
randomly implanted muons and thereby the distribution of the local magnetic
field. A Fourier transform of the time spectrum, which can be useful for
visualizing the field distribution characteristic of the vortex state, shows a
low-field cutoff corresponding to the midpoint of a triad of near neighbor
vortices, a Van Hove cusp from saddle points in the vortex lattice, and a
high-field tail and cutoff corresponding to the maximum $B_{loc}$ in the
vortex cores. However, fitting is best performed on the original time spectra:
statistics decay with the muon lifetime (2.197 $\mu$s), and mixing high- with
low-statistics data (as in an FFT) is undesirable, but later times determine
the frequency resolution and thereby the reliability of fit parameters. All
$\lambda_{ab}$ values were therefore extracted from fits in the time domain to
the lineshape described in Ref. Sonier _et al._ , 2004 as calculated
numerically for a triangular vortex lattice with $\lambda_{ab}$ and $\xi_{ab}$
as fitted parameters. A test with a square vortex lattice used the same
approach.
## Acknowledgements
This work was supported by the Canadian Institute for Advanced Research and
the Natural Sciences and Engineering Research Council of Canada. The authors
are indebted to Ch. Bernhard, D.M. Broun, M. Franz, and K. Machida for
stimulating discussions, D. Deepwell and D.M. Broun for microwave data, M.D.
Le and A.C. Shockley for critical reading of the manuscript, and the TRIUMF
CMMS staff for their assistance. A portion of this work was supported by
IBS-R009-G1.
## Author Contributions
This study was conceived, planned and supervised by JHB, DAB, WNH, and RL.
Crystals were grown, annealed and characterized by DCP and RL, then aligned
into mosaics by SLS, JHB and RL. $\mu^{+}$SR experiments were performed by SLS
and JHB, and JHB was pestered by DCP and WAM into analysing the data. The data
were interpreted and the manuscript written by WAM, DCP, JHB, and JES.
## Additional Information
The authors declare no competing financial interests.
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## Appendix A Supplementary Information
The field-cooled magnetization data collected for three mosaics — 46 K (B), 56
K, and 75 K — are shown in Fig. S1. Data were taken on cooling in applied
fields of 0.1–0.2 mT to minimize broadening due to the lower critical field.
To aid comparison, the data have been normalized. All data were collected with
the field applied along the $c$-axis.
Figure S1: Example field-cooled magnetization data for three of the mosaics
measured in this study. Figure S2: Anomalous temperature-dependence of fit
parameters at $H=0.1$ T for the $T_{\text{c}}=75$ K mosaic. The frequency of
the background peak, superconducting cusp, and the mean frequency (first
moment) of the fitted vortex lattice lineshape are shown, with the frequency
corresponding to the applied field subtracted. Rescaled $\lambda_{ab}^{-2}(T)$
values from Fig. 2 are included for reference, again with a curve to serve as
a guide to the eye.
Figure S2 shows anomalous behaviour in the extracted vortex lattice lineshape,
using data collected on the $T_{\text{c}}=75$ K mosaic at $H=0.1$ T as a
representative example. Because the field is applied by a driven
superconducting magnet and can drift by up to 0.5% over the course of a full
temperature sweep, frequency values here are corrected by subtracting the
frequencies corresponding to the actual applied fields as measured by a Hall
sensor. The background peak only drifts significantly near $T_{\text{c}}$,
where it becomes difficult to distinguish from the superconducting signal. The
mean frequency (first moment) of the fitted superconducting lineshape drifts
higher at low temperature, while the cusp corresponding to saddle points
between vortices departs from the background on cooling as expected before
coming back toward it. In both cases, the departure from expected behaviour
occurs at temperatures well below the inflection point in the extracted
$\lambda_{ab}^{-2}(T)$. Other fit parameters exhibited no systematic
temperature dependence. The anomaly reflects a subtle change in shape of the
superconducting field distribution, indicating the breakdown at low
temperatures of the $s$-wave vortex model used, and may provide guidance as to
the temperature regime in which $d$-wave vortex physics must be taken into
account.
Raw data for all $\mu$SR measurements performed at TRIUMF are freely available
at http://musr.physics.ubc.ca/mud. Data used in this study may be found by
searching for Experiment 958. The current version of LSHfit, used to analyse
the data, is available from Jeff Sonier on request.
|
arxiv-papers
| 2014-04-18T09:28:46 |
2024-09-04T02:50:01.322910
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jess H. Brewer, Scott L. Stubbs, Ruixing Liang, D. A. Bonn, W. N.\n Hardy, J. E. Sonier, W. Andrew MacFarlane, and Darren C. Peets",
"submitter": "Darren Peets",
"url": "https://arxiv.org/abs/1404.4727"
}
|
1404.4823
|
# Newly Discovered RR Lyrae Stars in the SDSS$\times$Pan-
STARRS1$\times$Catalina Footprint
M. A. Abbas1 , E. K. Grebel1, N. F. Martin2,3, W. S. Burgett4, H.
Flewelling4,R. J. Wainscoat4
1Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität
Heidelberg, Mönchhofstr. 12–14, D-69120 Heidelberg, Germany
2Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg,
Germany
3Observatoire astronomique de Strasbourg, Université de Strasbourg, CNRS, UMR
7550, 11 rue de l’Université, F-67000 Strasbourg, France
4Institute for Astronomy, University of Hawaii at Manoa, Honolulu, HI 96822,
USA Member of the IMPRS for Astronomy & Cosmic Physics at the University of
Heidelberg and of the Heidelberg Graduate School for Fundamental PhysicsE-
mail: [email protected]
###### Abstract
We present the detection of 6,371 RR Lyrae (RRL) stars distributed across
$\sim$14,000 deg2 of the sky from the combined data of the Sloan Digital Sky
Survey (SDSS), the Panoramic Survey Telescope and Rapid Response System 1
(PS1), and the second photometric catalogue from the Catalina Survey (CSDR2),
out of these, $\sim$2,021 RRL stars ($\sim$572 RRab and 1,449 RRc) are new
discoveries. The RRL stars have heliocentric distances in the 4–28 kpc
distance range. RRL-like color cuts from the SDSS and variability cuts from
the PS1 are used to cull our candidate list. We then use the CSDR2 multi-epoch
data to refine our sample. Periods were measured using the Analysis of
Variance technique while the classification process is performed with the
Template Fitting Method in addition to the visual inspection of the light
curves. A cross-match of our RRL star discoveries with previous published
catalogs of RRL stars yield completeness levels of $\sim$50$\%$ for both RRab
and RRc stars, and an efficiency of $\sim$99$\%$ and $\sim$87$\%$ for RRab and
RRc stars, respectively. We show that our method for selecting RRL stars
allows us to recover halo structures. The full lists of all the RRL stars are
made publicly available.
###### keywords:
stars: variables: RR Lyrae - Galaxy: halo - Galaxy: structure - Galaxy:
formation - Galaxy: evolution.
††pagerange: Newly Discovered RR Lyrae Stars in the SDSS$\times$Pan-
STARRS1$\times$Catalina Footprint–References††pubyear: 2013
## 1 Introduction
Studying stars with ages approaching the age of the Universe is of a great
importance since they can serve as tracers of the formation and early
evolution of galaxies. In particular, they allow us to study the stellar halo
which is mainly composed of old stars (e.g. Johnston et al. 2008; Schlaufman
et al. 2009). It is believed from observations and simulations that mergers
and accretions of smaller systems contributed to the formation of the outer
halo (e.g. Bullock, Kravtsov, & Weinberg 2001; Bullock & Johnston 2005;
Carollo et al. 2007; McCarthy et al. 2012; Beers et al. 2012) while the inner
halo is a result of accretion of a few massive systems in addition to in situ
star formation processes (e.g. Yanny et al. 2003; Jurić et al. 2008; De Lucia
& Helmi 2008; Zolotov et al. 2010; Font et al. 2011; Schlaufman et al. 2012).
These accretion events and mergers leave signatures in the structure and
kinematics of the stellar halo, usually in the form of stellar streams,
substructures, and overdensities (Ibata, Gilmore, & Irwin, 1995; Newberg et
al., 2003; Duffau et al., 2006; Schlaufman et al., 2009; Sesar et al., 2010).
It is easier to detect substructures and overdensities of stars at larger
Galactocentric distances where the dynamical time scales are longer (Bullock,
Kravtsov, & Weinberg, 2001; Bell et al., 2008). If the theoretical picture is
correct, we expect an inhomogeneous outer halo that is full of streams and
substructures from accreted systems (e.g. Johnston 1998; Johnston et al. 2008;
Cooper et al. 2010).
The absence of massive and luminous stars, the old main-sequence turn-offs,
the prevalence of horizontal branch stars, and the low metallicities of the
halo stars indicate that halo stars are predominantly old. However, it is
still unclear whether these stars were mainly formed in situ during the early
phase of the collapse of the Milky Way, or whether they were formed outside
the Milky Way in satellite galaxies only to be accreted by the Milky Way at a
later date (e.g. Vivas et al. 2004; Carollo et al. 2007; Bell et al. 2008).
Answers to such questions may be found by identifying and characterizing the
streams that the satellite galaxies have left in the halo (e.g. Zolotov et al.
2010) of the Milky Way where the contamination of foreground stars makes the
mapping of stellar structures difficult. Additionally, these streams can serve
as very sensitive probes to deduce the shape of the Milky Way’s potential
(Newberg et al., 2002; Law, Majewski, & Johnston, 2009).
### 1.1 RR Lyrae stars as halo tracers
One way of finding streams is by identifying and mapping RR Lyrae (RRL) stars
in the halo. RRL stars are low-mass, core helium burning pulsating stars that
fall on the horizontal branch of a stellar population’s color-magnitude
diagram. RRL stars have a mean absolute $V$-band magnitude of $\langle
M_{V}\rangle$ $=$ 0.6 $\pm$ 0.1 (Layden et al., 1996), which makes them very
good distance indicators. These variable stars are still bright enough to be
detected at large distances such as in the halo (Ivezić et al., 2000). They
have been used as tracers of the chemical and dynamical properties of old
stellar populations (e.g. Kinman et al. 2007; Bernard et al. 2008; Keller et
al. 2008; Morrison et al. 2009; Kinman, Morrison, & Brown 2009; Haschke et al.
2012) and have served as test objects for theories of the evolution of low-
mass stars and for theories of stellar pulsation (Smith, 1995). Many of the
substructures that were discovered in the Milky Way were re-confirmed using
RRL stars (e.g. Duffau et al. 2006; Kepley et al. 2007; Watkins et al. 2009;
Sesar et al. 2010).
The best and most reliable way to detect RRL stars is by using multi-band time
series observations of a sufficiently high cadence and over a sufficiently
long period. RRL stars can be divided into fundamental-mode (RRab stars) and
first-overtone (RRc stars) pulsators. Since RRL stars are short-period
pulsating stars with typical mean periods of $\sim$ 0.57 and $\sim$ 0.34 days
for RRab and RRc stars (Smith, 1995), respectively, the time between
observations is preferred to be short in order to sample the magnitudes of the
stars at each phase. In addition to that, monitoring an RRL star over a long
period of time will result in a more accurate and reliable classification and
period determination. Over the past two decades, colors, variability, and
light curve properties of RRL stars have been well studied and characterized
(e.g. Smith 1995; Pojmanski 2002; Moody et al. 2003; Vivas & Zinn 2006; Wils,
Lloyd, & Bernhard 2006; Sesar et al. 2010).
### 1.2 Our approach
In this paper, we use and combine data from different sky surveys out of which
each survey has a distinctive advantage that helps in identifying RRL stars
with high efficiency (fraction of true RRL stars in the candidate sample),
completeness (the fraction of selected RRL stars), and reliability levels.
First, we apply color cuts to the Sloan Digital Sky Survey (SDSS; Fukugita et
al. 1996; York et al. 2000; Abazajian et al. 2009), 8th data release (DR8;
Aihara et al. 2011). Second, we apply variability cuts using data from the
Panoramic Survey Telescope and Rapid Response System 1 3$\pi$ survey
(hereafter PS1; Kaiser et al. 2002) . Finally, we plot light curves and find
the periods of the RRL stars using the second photometric catalogue from the
Catalina Survey (CSDR2; Drake et al. 2009, 2013a) which is based on seven
years of multi-epoch observations. Using one of the surveys without the others
to find RRL stars results in low efficiency and completeness levels (see
Section 2).
In order to define the SDSS color selection threshold limits and the PS1 and
CSDR2 variability threshold limits, we use the color and variability
properties of the Quasar Equatorial Survey Team (QUEST) RRL star catalogue
(QRRL; Vivas et al. 2004; Vivas & Zinn 2006). To compute our efficiency and
completeness levels, we compare our results with the RRL stars found in Stripe
82. Stripe 82 ($-50\,^{\circ}\textless$ R.A. $\textless 59\,^{\circ}$,
$-1.25\,^{\circ}\textless$ Dec. $\textless 1.25\,^{\circ}$, where both right
ascension (R.A.) and declination (Dec.) are given in decimal degrees) covers
$\sim$ 270 deg2 of the celestial equator and was observed $\sim$ 80 times by
the SDSS.
Watkins et al. (2009) and Sesar et al. (2007, 2010) independently searched for
RRL stars in Stripe 82 using the SDSS data. Because Sesar et al.’s (2010)
catalog is 100$\%$ efficient and complete, we use it as a test catalog to
compute the efficiency and completeness levels of our method.
#### 1.2.1 Previous Studies
Drake et al. (2013a) had full access to the first photometric catalogue of the
Catalina Survey (CSDR1; Drake et al. 2009), which allowed them to look for
RRab stars in the whole $\sim$ 20,000 deg2 area of the sky that was covered by
the CSDR1. On the other hand, we had to manually do a multiple object cone
search for at most 100 objects at a time as more is not permitted by the
public data interface. The CSDR1 RRL star catalogue (Drake et al., 2013a)
contains 12,227 RRab stars found in the CSDR1 (Drake et al., 2009) database
and covers stars with heliocentric distances ($d_{h}$) up to 60 kpc. Because
these authors were only interested in RRab stars and to avoid spurious
detections, they removed all stars with periods outside the 0.43–0.95 days
range. While our paper was nearing completion, Drake et al. (2013b) announced
the discovery of $\sim$ 2,700 additional RRab stars in a re-analyses of the
CSDR1 photometry and using additional data from the CSDR2.
In this paper, we use different variability statistics techniques than the
ones used by Drake et al. (2013a) and Drake et al. (2013b) to find RRab stars.
Unlike the latter two studies, we use template fitting techniques to help us
in the classification process in addition to the visual inspection of all of
the RRL candidate light curves. This allowed us to discover 646 additional
RRab stars as compared to Drake et al. (2013a) and Drake et al. (2013b). We
also found 1,571 RRc stars, of which $\sim$ 1,449 stars are new discoveries.
The properties of the different surveys used in this study are described in
Section 2. In Section 3, we describe our method for selecting RRL candidates
within the overlapping area between the PS1 and the SDSS. Using the QUEST RRL
stars, we define and apply our SDSS color cuts in Section 3.1 and our PS1
variability cuts in Section 3.2. In Section 4, we use the multi-epoch data
from the CSDR2 database to look for stellar variability. The method used to
find the periods of the RRL stars is described in Section 4.1. In Section 4.2,
light curves are plotted and the methods used to distinguish RRL from
contaminant (non-RRL) stars are described. Section 5 summarizes our results
and provides our catalogue of RRL stars. In Section 6, we compare our RRL star
discoveries with the catalogue of RRL stars in Stripe 82 (Sesar et al., 2010)
to compute the efficiency and completeness levels of our method and periods.
The properties of the RRL stars that we missed are also described in the same
section. In Section 7, we compare our RRL star discoveries with the catalog of
RRab stars from Drake et al. (2013a) and Drake et al. (2013b) and with the La
Silla QUEST (LSQ) catalog of RRL stars (Zinn et al., 2014). We discuss our
newly discovered RRL stars in Section 8 and we compare them with stars found
in the General Catalogue of Variable Stars (GCVS111Published in 2012 and
available from VizieR via http://cdsarc.u-strasbg.fr/viz-bin/Cat?B/gcvs; Samus
et al. 2009). In Section 9, we find the distances for our RRL stars and we use
these distances to recover previously known halo substructures. The results of
the paper are summarized in Section 10.
Table 1: The SDSS color cuts. $(g-r)_{0}$ $\textless$ 0.4*$(u-g)_{0}$ $-$ 0.16
---
$(g-r)_{0}$ $\textgreater$ 0.4*$(u-g)_{0}$ $-$ 0.67
$(g-r)_{0}$ $\textless$ $-$2.5*$(u-g)_{0}$ $+$ 3.42
$(g-r)_{0}$ $\textgreater$ $-$2.5*$(u-g)_{0}$ $+$ 2.70
$-0.25$ $\textless$ $(g-r)_{0}$ $\textless$ $0.40$
$-0.20$ $\textless$ $(r-i)_{0}$ $\textless$ $0.20$
$-0.30$ $\textless$ $(i-z)_{0}$ $\textless$ $0.30$
## 2 Used Surveys
Our method for searching for RRL stars exploits the strong points of different
surveys (e.g. SDSS colors of RRL stars) to mitigate other weak points of the
same surveys (e.g. lack of SDSS multi-epoch data). As we will show in this
study, the synergy between data from different surveys results in an efficient
and systematic method to find RRL stars. The different surveys used are
described below.
### 2.1 The SDSS
The SDSS (Fukugita et al., 1996; York et al., 2000; Abazajian et al., 2009) is
a photometric and spectroscopic survey containing more than 900,000 galaxies
and 110,000 quasars. The SDSS (York et al., 2000) uses a 2.5-m telescope
located at Apache Point Observatory in New Mexico to image $\sim$ 14,000 deg2
of the sky over a period of 8 years. One part of the SDSS is a multi-
wavelength imaging survey (in $u$, $g$, $r$, $i$, and $z$) that goes as deep
as 22.3, 23.3, 23.1, 22.3, and 20.8 in $u$, $g$, $r$, $i$, and $z$,
respectively (Morganson et al., 2012). Most of the SDSS data (Stoughton et
al., 2002; Abazajian et al., 2009) are based on single epoch observations with
the exception of the overlapping regions of adjacent scans and of Stripe 82,
which was observed $\sim$ 80 times.
Watkins et al. (2009) searched for RRL stars in Stripe 82 using the public
archive of light-motion curves in Stripe 82, published by Bramich et al.
(2008). Watkins et al.’s (2009) catalog contains 407 RRL stars that lie 5–115
kpc from the Galactic Centre. Additionally, Sesar et al. (2010) used the SDSS-
Stripe 82 multi-epoch data to discover 483 RRL stars. The light curves,
periods, amplitudes, and properties of these stars are all discussed in their
paper. Sesar et al. (2010) used their discoveries to map the spatial
distribution of the halo RRL stars in the 5–120 kpc Galactocentric distance
range and were able to detect halo stellar streams and overdensities.
Because Sesar et al. (2010) used wider color ranges than the ones used by
Watkins et al. (2009), Sesar et al.’s (2010) catalog of RRL stars is more
complete. According to Sesar et al.’s (2010) study, their RRL star discoveries
are not contaminated by any other type of stars and are 100$\%$ complete.
Thus, we use the latter catalogue in this study to check and test the
reliability of our method that is aiming to find RRL stars in the halo.
#### 2.1.1 The SDSS Observing Technique
The SDSS telescope uses the drift scanning technique and images the same patch
of the sky using its 5 different filters almost simultaneously. This technique
yields the true instantaneous colors of the observed sky objects unless they
are variable on time scales of less than a few minutes. Since RRL stars have
periods between $\sim$ 0.2 and 1 days, the SDSS colors reflect the true colors
of the RRL stars. Consequently, since the true colors of RRL stars are
available in the SDSS photometric system, we decided to adopt the color cuts
of the RRL stars based on the SDSS colors to eliminate most of the non-RRL
stars from our sample (see Section 3.1).
Contaminant stars that have colors similar to the colors of the RRL stars will
still be present in our sample. Potential contaminant stars include non-
variable stars (e.g. main-sequence stars with colors at the edge of the color
range of the RRL stars), non-RRL variable stars like Ursae Majoris (W UMa)
contact binary stars, Algol eclipsing binary stars, $\delta$ Scuti stars, SX
Phe stars (Palaversa et al., 2013), and stars with large photometric errors.
In order to eliminate the non-variable contaminant stars, multi-epoch data are
needed. Because most of the SDSS data (Stoughton et al., 2002; Abazajian et
al., 2009) are based on single epoch observations, we use multi-epoch data
from the PS1.
### 2.2 The PS1 3$\pi$ Survey
The PS1 (Kaiser et al., 2002) 3$\pi$ Survey began operating from Hawaii in
2010. It uses a 1.8-m telescope that patrols $\sim$ 30,000 deg2 of the sky
(north of declination $-30^{\circ}$) between $\sim$ 10 and 50 times (Aller et
al., 2013) during its three and a half years period of operations. The PS1
uses 5 bandpasses ($g_{P1}$, $r_{P1}$, $i_{P1}$, $z_{P1}$, and $y_{P1}$) that
cover the optical and near-infrared spectral range (4,000 Å$<\lambda<$ 10,500
Å; Tonry et al. 2012). At the end of the survey, the PS1 is predicted to go as
deep as 23.1, 23.0, 22.7, 21.9, and 20.9 in $g_{P1}$, $r_{P1}$, $i_{P1}$,
$z_{P1}$, and $y_{P1}$, respectively, in co-added images (Morganson et al.,
2012). Individual $g_{P1}$, $r_{P1}$, $i_{P1}$, $z_{P1}$, and $y_{P1}$
exposures that we use in this study have limiting magnitudes of 21.9, 21.8,
21.5, 20.7, and 19.7, respectively (Morganson et al., 2012). The PS1 exposure
times are filter-dependent and vary between 30s in $z_{P1}$ and $y_{P1}$ and
$\sim$ 42s in $g_{P1}$, $r_{P1}$, and $i_{P1}$ (Aller et al., 2013).
#### 2.2.1 The PS1 3$\pi$ Observing Technique
Unlike the SDSS, the PS1 images a selected patch of sky using one filter only
before moving to the next patch. The PS1 then re-visits the same patch of the
sky at a different time to image it using a different filter. This observing
technique does not reflect the true colors of the short period variable
objects (e.g. RRL stars) as their magnitudes in different filters correspond
to different phases. Hence, we favor using the SDSS color cuts to find RRL
stars.
At the same time, the current average number of PS1 clean detections in two
(the $g_{P1}$ and $r_{P1}$ bands) out of its five bands are $\sim$ 5 (in each
band). We use variability cuts from the PS1 multi-epoch data to distinguish
possible variable from non-variable stars of all the stars that pass the SDSS
color cuts.
Applying the SDSS color cuts and the PS1 variability cuts does not result in a
clean sample of RRL stars. First, variable contaminant stars with colors close
to the colors of the RRL stars will still be present. Second, although the PS1
variability cuts will reduce the number of non-variable contaminant stars,
they will fail to eliminate all of them because they are based only on $\sim$
5 epochs in two different filters. With such a small number of epochs, a
single outlier can bias the variability statistics. Third, it is not possible
to obtain well-sampled light curves and correct periods for RRL stars using
only $\sim$ 5 PS1 epochs. Hence, we use the multi-epoch data from the CSDR2
database to study the light curves of our RRL candidates. The CSDR2 light
curves allow us to find the periods and subtypes (ab or c) of the RRL
candidates very efficiently.
### 2.3 The Catalina Survey
Aiming to discover rare and interesting transient phenomena (e.g. optical
transients, Near Earth Objects, etc.), the Catalina Survey (Drake et al.,
2009, 2013a) uses three different surveys and telescopes: the Catalina Sky
Survey (CSS), the Mt. Lemmon Survey (MLS), and the Siding Spring Survey (SSS).
While the CSS and MLS are carried out with two different telescopes located in
Tuscon, Arizona, the SSS uses a third telescope in Siding Spring, Australia.
Each telescope is equipped with an unfiltered 4k $\times$ 4k CCD. 2,500 deg2
of the sky are covered by these telescopes every night (Drake et al., 2013a).
In total, the Catalina Survey telescopes observe around 33,000 deg2 of the sky
($-75\,^{\circ}\textless$ Dec. $\textless 70\,^{\circ}$ and $\mid$b$\mid$
$\textgreater$ 10∘).
Using the SExtractor photometry software, the CSDR2 was released and is now
available online222http://nesssi.cacr.caltech.edu/DataRelease/.. The CSDR2
contains seven years of observations taken between 2005 and 2011 using the
CSS, MLS, and SSS telescopes. The CSDR2 data are available for $\sim$ 500
million objects with $V$-band magnitudes between 11.5 and 21.5 mag.
The CSS uses a 0.7-m Schmidt telescope, which started operating in April 1998
and is located $\sim$ 2,500 meters above the sea level. It uses an unfiltered
CCD with a 2$\arcsec$.5 pixel scale providing an 8 deg2 field of view (Drake
et al., 2013a). The CSS bright and faint magnitude cut-offs are $\sim$ 11.5
and 19.5 mag, respectively. Its typical exposure time is 30 seconds.
The SSS telescope is a 0.5-m Schmidt telescope with a computing system
identical to that of the CSS telescope. The SSS telescope began operating in
April 2004 and can detect objects with $V$-band magnitudes between $\sim$ 11.5
and 19.0 mag using a CCD camera with a 1$\arcsec$.8 pixel scale and 4.2 deg2
field of view.
The largest telescope used in the Catalina Survey is the MLS telescope. It is
a 1.5-m Cassegrain reflector telescope equipped with an unfiltered CCD (1 deg2
field of view and 1$\arcsec$ pixel scale). The MLS can detect objects as
bright as $V$ $\sim$ 11.5 mag and as faint as $V$ $\sim$ 21.5 mag, which makes
it more sensitive than the CSS and SSS. However, the CSS and SSS cover a much
larger area of the sky than the MLS.
The CSDR1 consists of photometry taken by the CSS telescope only and covers an
area of $\sim$ 24,000 deg2 of the sky while the CSDR2 consists of photometry
taken from all three surveys (CSS, SSS, and MLS) that cover $\sim$ 33,000 deg2
of the sky. The covered areas of the SSS and CSS surveys overlap in the
$-20\,^{\circ}\textless$ Dec. $\textless 0\,^{\circ}$ region, while the MLS
survey overlaps the CSS and SSS surveys along the ecliptic. It should be kept
in mind that the faint limits of the three contributing surveys differ.
Thus, the CSDR2 covers more area than the CSDR1 and at the same time offers a
larger number of repeated observations per object when two or more of its
three surveys overlap. We use the CSDR2 data in this study to plot the light
curves of the RRL candidates in order to correctly classify them and find
their periods.
It would have been inefficient to only use the CSDR2 data to find RRL stars in
this study because the CSDR2 database allows one to search for only up to 100
point sources in a given query. If we did not apply our SDSS color and PS1
variability cuts we would have had to manually download a large number of data
points (mostly for non-RRL stars) from the CSDR2 database before plotting
their light curves. Hence, combining data from the SDSS, PS1, and CSDR2
allowed us to find RRL stars in the halo with a higher efficiency level.
### 2.4 The Quasar Equatorial Survey Team RRL Stars Survey
QUEST used a 1-m Schmidt telescope located at 3,610 meters elevation at the
Llano del Hato Observatory in Venezuela and covered 380 deg2 of the sky (Vivas
et al., 2004). The QRRL used the QUEST camera, which is a 16 CCDs mosaic with
2,048 $\times$ 2,048 pixels per CCD (1$\arcsec$.02 pixel scale). These CCDs
were arranged in a 4 $\times$ 4 array and the entire camera had a field of
view of $2^{\circ}.3$ $\times$ $2^{\circ}.3$. The QUEST camera allows
simultaneous multi-filter photometry as it is designed to operate in a drift-
scanning mode. Stars would then pass through the four different filters in the
different CCD chips almost simultaneously, similar to the drift-scan imaging
technique used in the SDSS. The QRRL is based on $V$-band observations to a
limiting magnitude of $V$ $\sim$ 19.5 mag, taken over $\sim$ 2.3 years (Vivas
et al., 2004).
The QUEST catalogue of RRL stars contains 498 RRL stars (Vivas et al., 2004).
According to Vivas & Zinn (2006), 41 out of the 498 stars are not true
variables. These 41 stars were found in crowded regions and the photometric
pipeline that was used by QUEST did not include a de-blending algorithm at
that time. Consequently, only the remaining 457 stars are used in the analyses
in our study.
### 2.5 The La Silla QUEST Southern Hemisphere Variability Survey
The La Silla QUEST (LSQ) Southern Hemisphere Variability Survey used a 1-m
Schmidt Telescope at the La Silla Observatory in Chile to observe $\sim$ 1,000
deg2 per night (Hadjiyska et al., 2012). It was mainly designed to study
supernovae, RRL stars, quasars, and trans-Neptunian objects. The survey is
equipped with a broad-band filter (4,000–7,000 Å) and a camera that consists
of 112 CCD detectors. The camera covers an area of $4.\,^{\circ}6\times
3.\,^{\circ}6$ on the sky. Every one or two days, the LSQ Southern Hemisphere
Variability Survey observes the same patch of the sky for 60 seconds twice.
These exposures are separated by $\sim$ 2 hours (Hadjiyska et al., 2012; Zinn
et al., 2014).
Using its multi-epoch data, Zinn et al. (2014) discovered 1,372 RRL stars
(1,013 RRab and 359 RRc) with $d_{h}$ in the 5–80 kpc distance range. These
stars are distributed across $\sim$ 840 deg2 of the sky in the 150∘–210∘ R.A.
and $-10^{\circ}$ to $10^{\circ}$ Dec. range and have been observed between 11
and 300 times.
Figure 1: The $(u-g)$ $vs.$ $(g-r)$ color-color diagram displaying QRRL stars
with DR8 magnitudes in blue dots. The black rhomboidal box indicates our
color-color selection cut. Stars that are plotted with red dots and are
located inside the black rhomboidal box are considered as RRL candidates and
are retained for further analyses. These colors are corrected for the line-of-
sight interstellar extinction using the Schlegel, Finkbeiner, & Davis (1998)
dust map.
## 3 Identifying RRL stars
We start by using data from the overlapping area between the PS1 and the SDSS
that cover $\sim$ 14,000 deg2 of the sky. We first select RRL candidates based
on the SDSS $(u-g)$, $(g-r)$, $(r-i)$, and $(i-z)$ colors. As a variability
cut, we use the multi-epoch data from the PS1 to distinguish variable from
non-variable stars. Applying the SDSS color and PS1 variability cuts in this
Section is necessary to reduce the number of light curves to be requested from
the CSDR2 as the CSDR2 allows the retrieval of only 100 sky objects at a time.
### 3.1 The SDSS Color Cuts
Having different filters is a great advantage as it allows us to construct
diagnostic color-color diagrams and consequently helps in having several color
constraints for selecting and distinguishing RRL from contaminant stars.
Although applying the SDSS color cuts will eliminate a large fraction of
contaminant stars, contaminant stars with colors similar to the colors of the
RRL stars will still be present (e.g. main-sequence stars with colors at the
edge of the color range of the RRL stars, W UMa contact binary stars, Algol
eclipsing binary stars, $\delta$ Scuti and SX Phe stars)
We study and identify the $(u-g)$-$(g-r)$ SDSS colors of QUEST RRL stars as
this color-color diagram is the most sensitive and efficient in selecting RRL
stars. Other single-epoch SDSS colors of RRL stars in the remaining SDSS
filters were adopted from Sesar et al. (2010). In order to increase the
efficiency of our color cuts, we define and use a $(u-g)$-$(g-r)$ rhomboidal
cut instead of a rectangular cut like the one used by Sesar et al. (2010).
Additionally, we chose not to use the $(u-g)$-$(g-r)$ colors of the RRL stars
found in the catalogue of RRL stars in Stripe 82 (Sesar et al., 2010) because
we wanted to use the latter catalogue as a test catalogue to determine our
efficiency and completeness levels, especially because that catalogue is
100$\%$ efficient and complete (Sesar et al., 2010). Using the $(u-g)$-$(g-r)$
colors of Sesar et al.’s (2010) RRL stars in Stripe 82 would have biased the
computation of our efficiency and completeness levels in Section 6. At the
same time, we cannot accurately compute our efficiency and completeness levels
by comparing our results with the QUEST catalogue of RRL stars as the latter
catalogue is not complete (Vivas et al., 2004).
By positionally cross-matching the 457 QUEST RRL stars with the DR8 database,
we obtained the $u$, $g$, $r$, $i$, and $z$ magnitudes of these stars. We used
a circle of 3$\arcsec$ centered at the QUEST’s positions. Out of the 457 RRL
stars, 216 stars are recovered in the DR8 database having $12.0\textless
u\textless 19.0$. Most of the remaining missed RRL stars have magnitudes
beyond our magnitude cut or do not have clean photometry in all of the 5 SDSS
filters.
We adopted the above magnitude cut because the PS1 variability of faint
($u\textgreater 19.0$) and bright ($12.0\textless u$) stars can be easily
biased by the small number of the PS1 repeated observations currently
available. Some of the PS1 data are affected by de-blending, cosmic rays,
saturation, or non-photometric conditions as the PS1 final calibrated
catalogues have not been produced yet. We will investigate much deeper areas
of the sky when more PS1 epochs are available and the final PS1 calibrated
catalogues are available.
Based on the SDSS colors of the 216 QUEST RRL stars, we define a color-color
rhomboidal cut in the $(u-g)$ $vs.$ $(g-r)$ diagram, which is presented in
Fig. 1. These colors are corrected for the line-of-sight interstellar
extinction using the Schlegel, Finkbeiner, & Davis (1998) dust map. We believe
that such color corrections can be applied to our stars as these stars are
found at high Galactic latitudes in the halo where the overall extinction is
small. QRRL stars with DR8 magnitudes are indicated by blue dots and the black
rhomboidal box indicates our RRL candidates color-corrected selection box in
the $(u-g)$ $vs.$ $(g-r)$ diagram. Only stars that are plotted with red dots
and that are found inside the black rhomboidal box are considered as RRL
candidates and are retained for further analyses.
Fig. 1 illustrates how RRL stars are distinguished in such a color-color
diagram, which demonstrates the usefulness of applying such color cuts. RRL
stars follow a trend in this color-color diagram. Their colors always spread
out around the blue end of the main stellar locus. The farther away we go from
the main stellar locus, the fewer contaminant stars we have. In addition to
the $(u-g)$-$(g-r)$ rhomboidal cut that we computed using the 216 QUEST RRL
stars, we adopt other SDSS rectangular color cuts from Sesar et al. (2010).
All our color cuts are listed in Table 1.
Since we use the catalogue of RRL stars in Stripe 82 (Sesar et al., 2010) as a
reference catalogue to compute our completeness and efficiency levels in
Section 6, computing our own $(u-g)$-$(g-r)$ SDSS color cut using the QUEST
RRL stars and then applying them to the stars found in Stripe 82 ensures that
the completeness and efficiency levels we achieve are unbiased. Additionally,
Sesar et al. (2010) use rectangular cuts while we use a $(u-g)$-$(g-r)$
rhomboidal cut. The SDSS $(u-g)$ color serves as a surface gravity indicator
for these stars. The range ($\sim$ 0.3 mag) and the $rms$ scatter ($\sim$ 0.06
mag) are the smallest in this color (Ivezić et al., 2005).
Within the overlapping area between the PS1 and DR8, 308,342 stars passed all
the color cuts listed in Table 1.
### 3.2 The PS1 Variability Cuts
Because many stars passed the SDSS color cuts and because the CSDR2 allows us
to do a manual search of only 100 sky objects at a time, we use the PS1
preliminary variability cuts to reduce the number of RRL candidates light
curves to be retrieved from the CSDR2.
On average and at the end of the survey, the number of the PS1 pointings per
object will be around 12 per filter. Taking chip gaps and dead cells into
account, the PS1 observations per object will likely amount to about 9 times
in each filter. Since this survey is still continuing, the average number of
clean detections in the $g_{P1}$ and $r_{P1}$ filters are currently only
$\sim$ 5\. These are the “good” detections, which means that these detections
were not saturated or blended, and were not flagged as cosmic rays (Morganson
et al., 2012). We will include more epochs from the PS1 in future studies when
they are available.
The PS1 photometric catalogue contains the average $g_{P1}$, $r_{P1}$,
$i_{P1}$, $z_{P1}$, and $y_{P1}$ magnitudes for each point source that were
computed using the multi-epoch data available. It also contains the standard
deviations ($\sigma$) that show the scatter of the single-epoch data about the
average magnitudes in each filter. As a preliminary variability cut, we select
stars with a standard deviation greater than 0.05 in $g_{P1}$
($\sigma_{g_{P1}}$) or in $r_{P1}$ ($\sigma_{r_{P1}}$). The mean $g_{P1}$ and
$r_{P1}$ errors are $\sim$ 0.02 mag in both filters. Although this preliminary
variability cut does not result in a clean sample of variable stars, it
eliminates a large fraction of non-variable contaminant stars (e.g. main-
sequence stars) and reduces the number of CSDR2 light curves to be requested.
The PS1 variability statistics are based on $\sim$ 5 epochs in two different
filters where a single outlier can bias the statistics. Potential contaminant
stars that can still be present after the preliminary variability cut include
non-variable stars (e.g. main-sequence stars) with relatively large $g_{P1}$
or $r_{P1}$ photometric errors and non-RRL variable stars with colors similar
to the colors of RRL stars.
Around 34,200 stars (11$\%$ of the 308,342 stars selected in Section 3.1)
passed these two variability selection cuts. Most of the stars that passed our
SDSS color cuts but not the PS1 variability cuts are main-sequence stars with
colors close to the edge of the color range of the RRL stars. We use the CSDR2
multi-epoch data in the next section to obtain a clean list of variable stars
and to carry out a light curve analysis.
## 4 The CSDR2 Light-Curves
We extract the CSDR2 light curves for the stars that passed the SDSS color and
the PS1 variability cuts. We searched for all of our $\sim$ 34,200 RRL
candidates and found $\sim$ 21,050 stars in the CSDR2 database. The remaining
stars were either not observed with the Catalina Survey, or were found in
crowded regions. We used a circle of 3$\arcsec$ centered at the DR8 positions.
The mean number of epochs for these stars in the CSDR2 database is 270. More
than 90$\%$ of these stars were observed more than 100 times. The number of
CSDR2 observations per star as a function of equatorial J2000.0 R.A. and Dec.
is illustrated in Fig. 2 where the values are color-coded according to the
legend. Because 90$\%$ of the stars have more than 100 epochs, the CSDR2
variability statistics and light curve analyses were sufficient and very
accurate to reliably distinguish RRL from contaminant stars.
Figure 2: The number of CSDR2 observations per star as a function of
equatorial J2000.0 right ascension and declination. These are only the stars
that are in the footprint of the SDSS covering $\sim$ 14,000 deg2 of the sky.
The values are color-coded according to the legend. The mean number of epochs
of these stars in the CSDR2 database is 270. 90$\%$ of these stars were
visited more than 100 times.
In order to get rid of possible outliers, we omit data points that are more
than 3$\sigma$ from the CSDR2 mean magnitudes ($Mag$) for each star. This step
ensures a reliable variability statistics and better phased light curves
(light curves folded using a specific period). This is because some of the
CSDR2 observations were taken under non-photometric conditions and sometimes
possibly due to the recalibration process itself. Potential outliers can also
be contaminated by cosmic rays.
Using data from the CSDR2, we calculated variability statistics like the
weighted standard deviation ($W_{\sigma}$), $\chi^{2}$, variability index
($\alpha$), variance ($Var$), amplitude ($Amp$), and the skewness of the light
curves ($\gamma$) for a better separation of variable and non-variable stars.
We list the definitions of these quantities below.
$W_{<m>}=\frac{\sum\limits_{i=1}^{N}\frac{x_{i}}{x_{ierr}^{2}}}{\sum\limits_{i=1}^{N}\frac{1}{x_{ierr}^{2}}}$
(1)
$W_{\sigma}=\sqrt{\frac{\sum\limits_{i=1}^{N}\frac{(x_{i}-W_{<m>})^{2}}{x_{ierr}^{2}}}{\frac{N-1}{N}\sum\limits_{i=1}^{N}\frac{1}{x_{ierr}^{2}}}}$
(2)
$\chi^{2}=\frac{1}{N-1}\sum\limits_{i=1}^{N}\frac{(x_{i}-\bar{x})^{2}}{x_{ierr}^{2}}$
(3)
$\alpha=\frac{\sum\limits_{i=1}^{N}(x_{i}-\bar{x})^{2}-x_{i_{err}}}{N-1}$ (4)
$\gamma=\frac{N}{(N-1)(N-2)}\frac{1}{\zeta^{3}}\sum\limits_{i=1}^{N}(x_{i}-\bar{x})^{3}$
(5)
$\zeta=\sqrt{\frac{1}{N-1}\sum\limits_{i=1}^{N}(x_{i}-\bar{x})^{2}}$ (6)
where $x_{i}$ and $x_{i_{err}}$ represent the CSDR2 single-epoch magnitude and
the error corresponding to this magnitude, respectively. $\bar{x}$ is the mean
magnitude ($Mag$), and $N$ is the number of the CSDR2 epochs for each star.
Our RRL candidates are the stars that passed our SDSS color cuts in Section
3.1, the PS1 variability cuts in Section 3.2, and that have $W_{\sigma}$,
$\chi^{2}$, $\alpha$, $Var$, and $Amp$ greater than 0.1, 1.0, 0.002, 0.006,
and 0.4, respectively, and $-1.0\textless\gamma\textless 1.0$. These
variability threshold limits were defined using the CSDR2 variability
statistics of the QUEST RRL stars (Vivas & Zinn, 2006) discussed in Section
2.4. We did not define the CSDR2 variability statistics threshold limits using
the catalogue of RRL stars in Stripe 82 (Sesar et al., 2010) because we later
use the latter catalogue to test our efficiency and completeness levels.
Of the $\sim$ 21,050 stars with the CSDR2 information, 8,351 stars ($\sim$
40$\%$) passed the CSDR2 additional variability cuts applied in this section.
These stars are retained for further analyses.
### 4.1 The Analysis of Variance
We used the Analysis of Variance (AoV; Schwarzenberg-Czerny 1989) technique to
find the periods of the 8,351 RRL candidates. Using Fourier methods, the AoV
technique calculates variances of the light curves using different periods and
tries to detect sharp signals (Schwarzenberg-Czerny, 1989). Each calculated
signal corresponding to a different trial period (in the 0.1–1.1 days range)
is then represented in a periodogram (period vs. power). For each light curve,
we chose the period with the highest signal. Having low signal reflects a non-
periodic behavior for the phased light curve while a large signal reflects a
good periodic behavior.
Figure 3: Panels (a), (b), (c), and (d) illustrate the four best-fitted
templates by the TFM for CSDR2 star-id 1157029004107 (period of 0.52853 days).
The phased light curves and the best-fitted templates are plotted in red and
green, respectively. All of the four best-fitted templates belong to RRab
stars with an $rms$ ranging between 0.0479 and 0.0607.
### 4.2 Template Fitting Method
After estimating the best-fitted periods using the AoV technique, we used the
Template Fitting Method (TFM; Layden 1998; Layden et al. 1999) in order to
determine the types of our RRL candidates.
In total, the TFM uses a set of 10 different template light curves
representing different variable stars. Six of these template light curves are
for RRab stars, two for RRc stars, one for W UMa contact binary stars, and one
for Algol eclipsing binary stars. The CSDR2 light curves are folded using
periods from the AoV technique and are fitted by each of the 10 templates. The
$\chi^{2}_{TFM}$ and $rms$ scatter of each fit are then calculated. Small
$\chi^{2}_{TFM}$ and $rms$ values reflect a good template fit, which in turn
reflects the correct type of the star.
In order to insure that this classification process was correct, we visually
inspected the four best-fitted templates for each star. The four best-fitted
templates for CSDR2 star-id 1157029004107 (period of 0.52853 days) are shown
in panels (a), (b), (c), and (d) of Fig. 3, respectively. The phased light
curves and the best-fitted templates are shown in red and green, respectively.
All of the four best-fitted templates belong to RRab stars with $rms$ ranging
between 0.0479 and 0.0607 for the first and fourth best-fitted template,
respectively. The asymmetric, steep rise, and slow decrease in brightness of
the phased light curves and templates suggest that this is an RRab star with
an amplitude of $\sim$ 1.0 mag. This visual inspection was done for all of the
RRL candidates.
## 5 Results
After applying the AoV and the TFM methods and after the visual inspection of
the four best-fitted templates by the TFM, we were able to detect 4,800 RRab
stars and 1,571 RRc stars (6,371 RRL stars in total). These are the stars that
passed the variability and color cuts and that were well fitted with the
templates provided by the TFM. The positions (R.A. and Dec.), CSDR2 mean
magnitudes ($Mag$), CSDR2 amplitudes, subtypes, periods, ephemeris (MJDmax;
time at maximum light), and the heliocentric distances ($d_{h}$, see Section
9) of our RRab and RRc stars are found in Table 2.
Table 2: The CSDR2 catalogue of RRL stars. Both equatorial J2000.0 R.A. and Dec. are given in decimal degrees. A portion of the table is shown here for guidance regarding its form and content. The table is available in its entirety in the electronic version of the paper, and from the Centre de Données Astronomiques de Strasbourg (CDS). SDSS NAME333The official SDSS designation for an object where the coordinates are truncated, not rounded, given by the format: JHHMMSS.ss$+$DDMMSS.s | R.A. | Dec. | $Mag$444The CSDR2 mean magnitude | $Amp$555The CSDR2 amplitude range | Type | Period666Period in days | MJDmax777Ephemeris of the stars (time at maximum light) | $d_{h}$888Heliocentric distances in kpc
---|---|---|---|---|---|---|---|---
SDSS J140016.30+155821.4 | 210.0679 | 15.9726 | 17.54 | 1.03 | ab | 0.4860 | 55676.21067 | 27.6
SDSS J170343.25+115155.5 | 255.9302 | 11.8654 | 15.61 | 1.43 | ab | 0.4695 | 54591.39047 | 11.7
SDSS J143301.30+181254.3 | 218.2554 | 18.2150 | 16.32 | 0.93 | ab | 0.4865 | 54228.29835 | 15.9
SDSS J103555.83+382214.0 | 158.9826 | 38.3705 | 16.25 | 1.15 | ab | 0.4875 | 54566.25784 | 14.8
SDSS J153604.59+210746.7 | 234.0191 | 21.1296 | 16.88 | 0.8 | ab | 0.6701 | 53866.34446 | 16.1
SDSS J134644.45+454526.2 | 206.6852 | 45.7572 | 15.76 | 1.2 | ab | 0.4738 | 54138.43831 | 11.8
SDSS J201518.96-124928.8 | 303.8290 | -12.8246 | 16.61 | 0.75 | ab | 0.5559 | 54286.39649 | 15.5
SDSS J172732.63-133844.0 | 261.8859 | -13.6455 | 15.20 | 0.91 | ab | 0.5558 | 53986.47201 | 5.3
SDSS J144412.93+203641.2 | 221.0538 | 20.6114 | 14.37 | 0.47 | c | 0.3479 | 54884.44511 | 7.4
SDSS J145428.85+501007.7 | 223.6202 | 50.1687 | 17.03 | 0.53 | c | 0.3662 | 56126.21641 | 21.2
The upper and lower panels of Fig. 4 illustrate the phased light curves of one
of our RRab and RRc stars, respectively. The asymmetrical shape, steep rise,
and slow decrease of the RRab stars’ phased light curves have been observed in
all of our RRab stars. On the other hand, the more symmetrical phased light
curves with relatively smaller amplitudes were observed in all of our RRc
stars. The unique phased light curve shape of RRab stars makes it relatively
easy to identify them with high efficiency, compared to RRc stars that have
phased light curves that can be confused with other types of variable stars
(e.g. W UMa, $\delta$ Scuti, and SX Phe stars, see Section 8.3).
The period-amplitude diagram for the RRab (red dots) and RRc (blue dots) stars
is shown in Fig. 5. RRab stars tend to have higher amplitudes and periods than
RRc stars, as expected. Fig. 5 also shows that most of the RRab stars are
concentrated in a narrow period-amplitude range (periods less than $\sim$ 0.65
days), these stars belong to the Oosterhoff I group (Oo I). Some RRab stars
lie toward the longer period region and belong to the Oosterhoff II group (Oo
II). The different Oosterhoff groups (Oosterhoff, 1939) depend on the stellar
metallicities, horizontal branch morphology, and ages (Catelan, 2009). RRL
stars in the Oo I group are more metal rich compared to RRL stars in the Oo II
group. More than 75$\%$ of our RRab stars have periods $\textless 0.65$ days
and thus are more likely to belong to the Oo I group. This result agrees well
with several studies that have demonstrated that $\geq 70\%$ of the RRab stars
in the halo belong to the Oo I group (e.g. Miceli et al. 2008; Zinn et al.
2014). Using the CSDR2 light curves, we are planning to conduct a detailed
study in the near future about the properties and characteristics of this
phenomenon.
Figure 4: Illustration of one of our RRab and RRc stars phased light curves
shown in panels (a) and (b), respectively. Figure 5: The period-amplitude
distribution for the RRab (red dots) and RRc (blue dots) stars.
## 6 Comparison with Stripe 82
It is important to check how complete, efficient, and reliable our catalogue
is. Accordingly, we compare our RRL star discoveries with the catalogue of RRL
stars in Stripe 82 (Sesar et al., 2010). We also discuss the properties of the
RRL stars that we missed.
### 6.1 Completeness
Our catalogue contains 184 RRL stars (137 RRab and 47 RRc stars) in the Stripe
82 area, out of which 177 (136 RRab and 41 RRc stars) are also found in the
catalogue of RRL stars in Stripe 82 (Sesar et al., 2010). The 7 extra RRL
stars are found in our catalogue only. The presence of the extra RRL stars can
either prove that our catalog is slightly contaminated by non-RRL stars or
that Sesar et al.’s (2010) catalog is not 100$\%$ complete.
Our achieved completeness level is distance-dependent because of the magnitude
cut ($12.0\textless u\textless 19.0$) applied and discussed in Section 3.1.
All of the common 177 RRL stars in Stripe 82 that are found in our and in
Sesar et al.’s (2010) catalogue have $d_{h}$ between $\sim$ 4 and $\sim$ 28
kpc. These distances are provided in the latter catalogue. Accordingly, the
completeness level will be studied and analyzed in the mentioned distance
range.
Sesar et al.’s (2010) catalogue contains 337 RRL stars (255 and 82 of type ab
and c, respectively) with $d_{h}$ in the 4–28 kpc distance range. Since we
recovered 136 and 41 RRab and RRc stars, respectively, our completeness level
is then $\sim$ 50$\%$ for RRab and RRc stars.
### 6.2 Missed RRL Stars
With our 50$\%$ completeness level computed in the previous section, we know
that we missed $\sim$ 50$\%$ of the RRL stars in Stripe 82 with $d_{h}$ in the
4–28 kpc distance range. These missed stars either did not have CSDR2
information or did not pass the variability cuts in Section 3.2
($\sigma_{g_{P1}}\textgreater 0.05$ or $\sigma_{r_{P1}}\textgreater 0.05$).
For example, only $\sim$ 60$\%$ of the RRL candidates that passed the SDSS
color and the PS1 variability cuts had CSDR2 data. The remaining missed stars
were either not observed with the Catalina Survey, or they are located in
crowded regions (e.g. de-blending problems) where they had a small number of
clean CSDR2 detections.
Another reason for why we missed 50$\%$ of the RRL stars are the PS1
variability cuts applied in Section 3.2. Although most of the missed RRL stars
were observed $\sim$ 4 times in $g_{P1}$ or $r_{P1}$, the variability of the
missed RRL stars did not appear in the PS1 database. RRL stars are short
period variable stars and are repeating their cycles between $\sim$ 1 and
$\sim$ 5 times a day. Hence, it is likely that some RRL stars were repeatedly
observed at the same or a close phase. Nevertheless, we were still able to
recover more than 50$\%$ of the RRL stars with the small number of epochs
available from the PS1 at the moment. The completeness level will be
significantly higher when more PS1 epochs are available. Nevertheless, the
variability cuts were necessary to distinguish a possible variable from a non-
variable star.
### 6.3 Efficiency
Among the 184 RRL stars we found in Stripe 82, 137 and 47 are of type ab and
c, respectively. Out of the 137 RRab stars, 136 are classified as RRab stars
in Sesar et al.’s (2010) catalogue.
Of our 47 RRc stars, 41 are found and classified as RRc stars in the latter
catalogue. Assuming that Sesar et al.’s (2010) catalogue is complete, our
efficiency levels are then $\sim$ 99$\%$ and $\sim$ 87$\%$ for RRab and RRc
stars, respectively.
The phased light curves of the extra RRab star and 3 out of the 6 RRc stars
are plotted in red in Fig. 6. These are the RRL stars in Stripe 82 region that
are found in our catalogue but not in Sesar et al.’s (2010) catalogue. The
best-fitted templates from the TFM are shown in green for stars with the CSDR2
star-ids 1001118058621, 2101029003452, 1101120009185, and 1101010021310 in
panels (a), (b), (c), and (d) of Fig. 6, respectively. These stars passed the
color and variability cuts that are well defined for RRL stars, have been well
fitted with RRL star templates by the TFM, and were observed by the CSDR2
between $\sim$ 250 and $\sim$ 380 times. We believe that the phased light
curve shown in Fig. 6a belong to an RRab star that was missed by Sesar et al.
(2010). However, we are less confident about the 6 extra RRc stars that we
found in Stripe 82 as their light curves can be confused with other types of
variable stars (see Section 8.3). In the worst-case scenario, if we assume
that all of the 6 RRc stars are non-RRL stars, our efficiency level would be
$\sim$ 87$\%$.
Figure 6: The phased light curves of the RRab star and 3 out of the 6 RRc
stars that are found in our catalogue but not in Sesar et al.’s (2010) Stripe
82 catalogue. The phased light curves and the best-fitted templates from the
TFM are plotted in red and green, respectively.
### 6.4 Period Testing
Finally, $\sim$ 95$\%$ of our periods differ on average by only 0.009$\%$ from
the periods found by Sesar et al. (2010). The maximum percentage difference
was $\sim$ 0.17$\%$. Periods in Sesar et al.’s (2010) catalogue were obtained
using the Supersmoother routine (Reimann, 1994) which is a smoothing routine
that fits data points as a function of phase to a range of frequencies. It
uses a running mean or running linear regression on the data points. The small
percentage difference between our and Sesar et al.’s (2010) periods
demonstrates the reliability of our method.
## 7 Comparison With the CSDR and the La Silla QUEST catalog of RRL stars
### 7.1 The CSDR catalog of RRL stars
In total, there are $\sim$ 14,500 RRab stars found in both the first catalog
of RRL stars in the CSDR1 database (Drake et al., 2013a), and in its re-
analysis study (Drake et al., 2013b). These stars were chosen using the Welch-
Stetson variability index ($I_{WS}$; Welch & Stetson 1993), the Lomb-Scargle
periodogram analysis (LS; Lomb 1976; Scargle 1982), and the M-Test (Kinemuchi
et al., 2006). While the $I_{WS}$ measures the variability and tries to
separate variable from non-variable stars, the LS looks for periodicity in a
specific period range, and the M-Test measures the percentage time spent by
the object below the mean magnitude. Drake et al. (2013a) used the AoV
technique and the Adaptive Fourier Decomposition (AFD) method (G. Torrealba et
al., in preparation) to find the periods of their RRab stars.
Of the 14,500 RRab stars found in the CSDR catalogue of RRab stars, $\sim$
7,500 RRab stars are located in our $d_{h}$ distance range (4–28 kpc) and area
(SDSS$\times$PS1$\times$CSDR2 footprint).
We were able to recover $\sim$ 55$\%$ ($\sim$ 4,150 stars) of the 7,500 RRab
stars that are found in the CSDR catalogue of RRL stars (Drake et al., 2013a,
b). Missing the remaining $\sim$ 45$\%$ RRab stars was expected as these stars
did not show any sign of variability in the PS1 data due to the small number
of detections in $g_{P1}$ and $r_{P1}$ that we discussed in Section 6.2.
Comparing our periods with periods from the CSDR catalogue of RRL stars for
the 4,150 common RRL stars made us trust our results as 99$\%$ of the matched
periods had percentage difference less than 0.009$\%$.
Panels (a) and (b) of Fig. 7 show the phased light curves of CSDR2 star-id
1109090090390 and 1129076074116, respectively. The light curves phased to our
periods (P1) and to Drake et al.’s (2013a) periods (P2) are shown in red and
gray, respectively. Light curves phased in red are less scattered and look
more like RRab stars than those plotted with gray implying that our periods
are more accurate. We applied the TFM on the phased light curves twice (using
P1 and P2). The $rms$ values for the P1-phased light curves were smaller than
those corresponding to P2-phased light curves, in both cases. Hence, we trust
our claimed periods in the mis-matched cases.
Figure 7: Panels (a) and (b) show the phased light curves of CSDR2 star-id
1109090090390 and 1129076074116, respectively. The light curves phased to our
periods (P1) and to Drake et al.’s (2013a) periods (P2) are shown in red and
gray, respectively.
### 7.2 La Silla QUEST catalog of RRL stars
The LSQ catalog of RRL stars (Zinn et al., 2014) contains 1,372 RRL stars
(1,013 RRab and 359 RRc) with $d_{h}\textless 80$ kpc. Around 70$\%$ of the
RRab stars found in the LSQ catalog of RRL stars are also found in the CSDR
catalogue of RRL stars (Drake et al., 2013a, b) that we already compared to
our catalog in the previous section. We recovered 355 out of the 1,013 LSQ
RRab stars of which 336 are also found in the CSDR catalogue of RRL stars.
Only 84 RRc stars are found in both our and LSQ catalog of RRL stars. Most of
the remaining LSQ RRL stars that we missed are located beyond our covered
distance range ($d_{h}\textgreater$ 28 kpc).
Our catalog contains $\sim$ 230 RRL stars that are not found in the LSQ
catalog of RRL stars. Zinn et al. (2014) computed their completeness ($\sim$
70$\%$) level by comparing their catalog with the CSDR catalogue of RRL stars
(Drake et al., 2013a, b). We believe that their completeness level is slightly
lower than 70$\%$ as we have shown in this study that the CSDR catalogue of
RRL stars is not complete. The additional visual inspection that we performed
for the light curves of the $\sim$ 230 RRL stars suggests that these are
indeed RRL stars.
## 8 Newly Discovered RRL Stars
Although Drake et al. (2013a, b) searched for RRab stars in the whole CSDR
database, we were still able to find 646 new RRab stars that were missed by
them. Of these 646 RRab stars, 19 stars were found also in the LSQ catalog of
RRL stars (but not in the CSDR catalogue of RRL stars). Around 572 of our RRab
stars are new discoveries. Additionally, we present the discovery of 1,571 RRc
stars of which $\sim$ 1,449 are new discoveries. Based on our analyses in
Section 6, we estimate that only $\sim$ 2$\%$ and $\sim$ 13$\%$ of our RRab
and RRc star discoveries are non-RRL stars (contaminant stars). We discuss the
nature of our contaminant stars in Section 8.3.
### 8.1 RRab Stars
We identified 627 additional RRab stars that are neither found in the CSDR
(Drake et al., 2013a, b) nor in the LSQ catalogs of RRL stars (Zinn et al.,
2014). The phased light curves, TFM and AoV analyses, and variability
statistics of these stars are similar to the other common RRL stars that we
and Drake et al. (2013a, b) and Zinn et al. (2014) have found.
Out of the 627 newly discovered RRab stars, 55 are found in the GCVS (Samus et
al., 2009). 42 out of the 55 RRL stars are classified as RRab stars in the
GCVS, 12 as RRL stars without giving the sub-type and one as an RRc star.
Our period for the RRc star mentioned earlier (CSDR2 star-id 2122228003249)
does not match the period found in the GCVS. The two phased light curves (in
red) and the best-fitted templates by the TFM (in green) corresponding to our
period (P1 = 0.54288 days) and to the GCVS’s period (PGCVS = 0.34784 d) are
shown in panels (a) and (b) of Fig. 8, respectively. The two cycles observed
in one phase and the large scatter ($rms$ = 0.5451) around the TFM template in
Fig. 8b suggest that this is not an RRc star, even if we assume that PGCVS’s
period is correct. However, the RRab-like phased light curve and the small
scatter ($rms$ = 0.1178) around the best-fitted RRab template from the TFM in
Fig. 8a indicate that our classification and period are more accurate.
Figure 8: A comparison between the two phased light curves (in red) and the
best-fitted templates by the TFM (in green) representing our (P1 = 0.54288
days) and the GCVS (PGCVS = 0.34784 d) periods are represented in panels (a)
and (b) for CSDR2 star-id 2122228003249.
### 8.2 RRc Stars
Our catalogue contains 1,571 RRc stars, of which 84 are found in the LSQ
catalog of RRL stars (Zinn et al., 2014). Additionally, only 38 of our RRc
stars are found in the GCVS out of which 25 are classified as RRc stars, four
as RRL stars without a sub-type, six as RRab stars, and three as stars in
eclipsing binary systems (EW stars).
Four out of the six RRc stars that were classified as RRab stars have periods
in the GCVS. The phased light curves corresponding to our (left part of each
panel) and to the GCVS (right part of each panel) periods (PGCVS) are shown in
red and gray in Fig. 9, respectively. The green fits indicate the best-fitted
templates from the TFM, all corresponding to RRc stars. It is clear that
folding the light curves to the PGCVS periods (in gray) did not produce
periodic signals in the panels (a), (b), and (c) of the latter figure. On the
other hand, periodic signals (in red) and well fitted RRc templates (in green)
were observed when folding the light curves to our periods, indicating that
these are indeed RRc stars.
Although periodic signals are observed when folding the data to our and to the
PGCVS period in Fig. 9d, we believe that our period and classification are
more accurate because the AoV and TFM analyses are based on 274 CSDR2 epochs
for this star.
Figure 9: The phased light curves corresponding to our (left part of each
panel) and to the GCVS (right part of each panel) periods are shown in red and
gray, respectively. The green fits indicate the best-fitted templates from the
TFM, all corresponding to RRc stars.
### 8.3 Contaminant Stars
It is not surprising that some of our RRc stars are contaminated by other type
of stars (e.g. W UMa stars, $\delta$ Scuti, etc.) as these stars have
moderately symmetric light curves (e.g. sinusoidal) and share the same period
range. However, we used color cuts that are well characterized for RRL stars.
Additionally, each light curve was fitted to both RRc and W UMa templates by
the TFM. The fits worked better for the RRc templates for all of our RRc
stars.
After comparing our RRc star discoveries with the GCVS in the previous
section, and with Stripe 82 in Section 6.3, we believe that the RRc
contamination level by eclipsing binaries is less than $\sim$15$\%$, and is
caused mainly due to W UMa stars.
Finally, $\delta$ Scuti and SX Phe stars have colors similar to the colors of
RRL stars (Palaversa et al., 2013) and have periods less than $\sim$ 0.3
days999http://www.aavso.org/types-variables. Because almost half of our RRc
stars have periods less than 0.3 days, $\sim$ half of our RRc stars are prone
to contamination from $\delta$ Scuti and SX Phe stars.
## 9 Halo Sub-Structure
In this section, we check whether the completeness and efficiency levels of
our catalogue are good enough to detect previously known and possibly new halo
overdensities. First, we derive the $d_{h}$ of the RRL stars in our catalogue
using Equation 7:
$d_{h}=10^{(\langle V_{0}\rangle-M_{v}+5)/5}$ (7)
where $\langle V_{0}\rangle$ magnitudes are calculated using Equation 8 which
was adopted from Ivezić et al. (2005).
$\langle V_{0}\rangle=r-2.06(g-r)+0.355$ (8)
where the $g$ and $r$ SDSS magnitudes were corrected for the line-of-sight
interstellar extinction using the recalibration of Schlegel, Finkbeiner, &
Davis’s (1998) dust map by Schlafly & Finkbeiner (2011). This equation
corrects a bias in single-epoch SDSS measurements due to the unknown phase and
introduces a minimal rms scatter of 0.12 mag.
Like Sesar et al. (2010), we adopt $\langle M_{V}\rangle$ = 0.60 mag for the
absolute magnitude of RRL stars; a value that was calculated by (Cacciari &
Clementini, 2003) using Equation 9.
$\displaystyle M_{V}=(0.23\pm 0.04)\mathrm{[Fe/H]}+(0.93\pm 0.12)$ (9)
where the mean halo metallicity of [Fe/H] = $-1.5$ $\pm$ 0.32 dex is used
(Ivezić et al., 2008). Adopting [Fe/H] = $-1.5$ dex introduces $rms_{M_{v}}$
of $\sim$ 0.1 mag because of the actual dispersion of [Fe/H] and their
corresponding uncertainties. Taking the uncertainties of [Fe/H], $\langle
V_{0}\rangle$, and ${M_{v}}$ into account, $d_{h}$ is calculated with at least
$\sim$ 7% fractional error.
To test if our completeness and efficiency levels are good enough to detect
substructures (and to possibly find new ones), we plotted the number density
distribution of the 184 RRL stars we found in Stripe 82 in Fig. 10. Assuming
that Sesar et al.’s (2010) catalogue is 100$\%$ efficient and complete implies
that the contamination level of our 184 RRL stars is $\sim$ 4$\%$. The density
of the points that is accentuated by the white contours is shown in scaled
density levels. The smoothed surface regions with high and low numbers of
stars are represented in red and dark blue, respectively.
The Hercules-Aquila cloud (Belokurov et al., 2007) halo substructure appears
at R.A.101010Add 360 ∘ to obtain the correct values of R.A. when R.A.
$\textless$ 0∘. Negative values of R.A. were used for better visualization
only. $\sim$ $-40\,^{\circ}$ and spans $d_{h}$ in the 8 to 24 kpc distance
range while the arm of Sagittarius dwarf spheroidal (dSph) tidal stream
(Majewski et al., 2003; Law & Majewski, 2010) appears at R.A. $\sim$
$30\,^{\circ}$ and $d_{h}$ $\sim$ 23 kpc.
Our RRL stars that are found in the Northern Galactic hemisphere section of
the celestial equator ($-1.25\,^{\circ}\textless$ Dec. $\textless
1.25\,^{\circ}$) are plotted in Fig. 11. The Virgo overdensity (Vivas et al.,
2001) was detected at R.A. $\sim$ $190\,^{\circ}$ and $d_{h}$ $\sim$ 19 kpc
while the Hercules-Aquila cloud at R.A. $\sim$ $240\,^{\circ}$ and $d_{h}$
$\sim$ 10 kpc.
Some parts of the well-defined narrow tidal tails of the extended, low-
concentration globular cluster Palomar 5 (Pal 5; Odenkirchen et al. 2001,
2003; Grillmair & Dionatos 2006) overlap in projection with the clump
appearing at R.A. $\sim$ $235\,^{\circ}$ and $d_{h}$ $\sim$ 20 kpc (see Fig.
11). Pal 5 is a faint halo cluster that is currently undergoing tidal
disruption due to disc shocks (Dehnen et al., 2004). This globular cluster has
a very sparsely populated red giant branch and horizontal branch (Odenkirchen
et al., 2003), with only very few RRL stars (five, see Vivas & Zinn 2006). Two
additional RRL stars have been suggested to be associated with Pal 5’s tidal
tails (Vivas & Zinn, 2006). Although the clump appearing at R.A. $\sim$
$235\,^{\circ}$ and $d_{h}$ $\sim$ 20 kpc can indeed be associated with Pal 5,
we do not confirm this association because of the small number of RRL stars
(5–7) associated with Pal 5 in addition to possible stars that are
contaminating our catalog. We suggest radial velocity studies of our RRL stars
that belong to this clump in order confirm the association of this clump to
Pal 5.
Figure 10: The number density distribution of the 184 RRL stars found in our
catalogue in the Stripe 82 area. The density of the points that is accentuated
by the white contours is shown in scaled density levels. The smoothed surface
regions with high and low numbers of stars are represented in red and dark
blue, respectively. The Hercules-Aquila cloud halo structure appears at R.A.
$\sim$ $-40\,^{\circ}$ and $d_{h}$ in the 8 kpc to 24 kpc distance range while
the arm of the Sagittarius dwarf spheroidal (dSph) tidal stream appears at
R.A. $\sim$ $30\,^{\circ}$ and $d_{h}$ $\sim$ 23 kpc. Negative values of R.A.
were used for better visualization only (R.A. = R.A. + 360∘ when R.A.
$\textless$ 0∘). Figure 11: The number density distribution of the RRL stars
found in the Northern Galactic hemisphere section of the celestial equator
($-1.25\,^{\circ}\textless$ Dec. $\textless 1.25\,^{\circ}$). Two main
structures are detected: the Virgo overdensity at R.A. $\sim$ $190\,^{\circ}$
and $d_{h}$ $\sim$ 19 kpc, the Hercules-Aquila cloud at R.A. $\sim$
$240\,^{\circ}$ and $d_{h}$ $\sim$ 10 kpc.
## 10 Summary
We have combined data from different sky surveys (the SDSS, the PS1, and the
Catalina Survey) to look for RRL stars in the Milky Way halo. The search
resulted in the discovery of 6,371 RRL stars (4,800 RRab and 1,571 RRc)
distributed around 14,000 deg2 of the sky and with $d_{h}$ in the 4–28 kpc
distance range. Around 2,021 ($\sim$ 572 RRab and 1,449 RRc) of these stars
are new discoveries.
In this paper, RRL stars were discovered using the SDSS color and the PS1
variability cuts in Section 3. We define the threshold limits of these cuts
using the QUEST catalogue of RRL stars (Vivas & Zinn, 2006) rather than using
the catalogue of RRL stars in Stripe 82 (Sesar et al., 2010) as we use the
latter catalogue to test the efficiency and completeness levels of our method.
Additional variability cuts were applied and light curves were plotted using
the CSDR2 multi-epoch data. Periods were obtained using the AoV technique
while the classification process was done by the TFM and by visual inspection.
The comparison of our RRL star discoveries with the RRL stars in Stripe 82
from the SDSS shows that our completeness levels are $\sim$ 50$\%$ for RRab
and RRc stars and that our efficiency levels are $\sim$ 99$\%$ and $\sim$
87$\%$ for RRab and RRc stars, respectively.
Additional comparison of our RRL star discoveries with the GCVS, the LSQ
catalog of RRL stars, and the 14,500 RRab stars found previously in the
Catalina Survey (Drake et al., 2009, 2013b) suggests the reliability of our
method. Additionally, the Virgo overdensity, Hercules-Aquila cloud, and
Sagittarius stream were recovered after plotting the number density
distribution of our RRL stars in the Stripe 82 and Northern Galactic
hemisphere areas. This indicates that our method is capable of identifying
halo overdensities. In a forthcoming paper, we will present a more detailed
analysis of halo substructure as traced by RRL stars.
## Acknowledgments
We thank the referee for comments and constructive suggestions that helped to
improve the manuscript. We thank E. Bernard, J. Vanderplas, S. Duffau, and A.
Huxor for helpful discussion that improved the quality of this paper. M.A.,
E.K.G., and N.F.M acknowledge support by the Collaborative Research Center
“The Milky Way System" (SFB 881, subproject A3) of the German Research
Foundation (DFG). The Pan-STARRS1 Surveys (PS1) have been made possible
through contributions of the Institute for Astronomy, the University of
Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its
participating institutes, the Max Planck Institute for Astronomy, Heidelberg
and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns
Hopkins University, Durham University, the University of Edinburgh, Queen’s
University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las
Cumbres Observatory Global Telescope Network Incorporated, the National
Central University of Taiwan, the Space Telescope Science Institute, the
National Aeronautics and Space Administration under Grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation under Grant No. AST-1238877, the
University of Maryland, and Eotvos Lorand University (ELTE). Funding for SDSS-
III has been provided by the Alfred P. Sloan Foundation, the Participating
Institutions, the National Science Foundation, and the US Department of
Energy. The SDSS-III Web site is http://www.sdss3.org/. SDSS-III is managed by
the Astrophysical Research Consortium for the Participating Institutions of
the SDSS-III Collaboration including the University of Arizona, the Brazilian
Participation Group, Brookhaven National Laboratory, University of Cambridge,
University of Florida, the French Participation Group, the German
Participation Group, the Instituto de Astrofisica de Canarias, the Michigan
State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence
Berkeley National Laboratory, Max Planck Institute for Astrophysics, New
Mexico State University, New York University, Ohio State University,
Pennsylvania State University, University of Portsmouth, Princeton University,
the Spanish Participation Group, University of Tokyo, University of Utah,
Vanderbilt University, University of Virginia, University of Washington, and
Yale University. CRTS is supported by the U.S. National Science Foundation
under grants AST-0909182 and CNS-0540369. The work at Caltech was supported in
part by the NASA Fermi grant 08-FERMI08-0025 and by the Ajax Foundation. The
CSS survey is funded by the National Aeronautics and Space Administration
under grant No. NNG05GF22G issued through the Science Mission Directorate
Near-Earth Objects Observations Program.
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|
arxiv-papers
| 2014-04-17T17:48:57 |
2024-09-04T02:50:01.386804
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. A. Abbas, E. K. Grebel, N. F. Martin, W. S. Burgett, H. Flewelling,\n and R. J. Wainscoat",
"submitter": "Mohamad Abbas",
"url": "https://arxiv.org/abs/1404.4823"
}
|
1404.4945
|
# A Criterion for Irreducibility of Parabolic Baby Verma Modules of Reductive
Lie Algebras
Yi-Yang Li, Bin Shu and Yu-Feng Yao School of Fundamental Studies, Shanghai
University of Engineering Science, Shanghai 201620, China.
yiyang$\\[email protected] Department of Mathematics, East China Normal
University, Shanghai 200241, China. [email protected] Department of
Mathematics, Shanghai Maritime University, Shanghai 201306, China.
[email protected]
###### Abstract.
Let $G$ be a connected, reductive algebraic group over an algebraically closed
field $k$ of prime characteristic $p$ and ${\mathfrak{g}}=\mbox{\rm Lie}(G)$.
In this paper, we study representations of ${\mathfrak{g}}$ with a
$p$-character $\chi$ of standard Levi form. When ${\mathfrak{g}}$ is of type
$A_{n},B_{n},C_{n}$ or $D_{n}$, a sufficient condition for the irreducibility
of standard parabolic baby Verma ${\mathfrak{g}}$-modules is obtained. This
partially answers a question raised by Friedlander and Parshall in
[Friedlander E. M. and Parshall B. J., Deformations of Lie algebra
representations, Amer. J. Math. 112 (1990), 375-395]. Moreover, as an
application, in the special case that ${\mathfrak{g}}$ is of type $A_{n}$ or
$B_{n}$, and $\chi$ lies in the sub-regular nilpotent orbit, we recover a
result of Jantzen in [Jantzen J. C., Subregular nilpotent representations of
$sl_{n}$ and $so_{2n+1}$, Math. Proc. Cambridge Philos. Soc. 126 (1999),
223-257].
###### Key words and phrases:
parabolic baby Verma module, nilpotent orbit, sub-regular nilpotent orbit,
support variety, standard Levi form
###### 2000 Mathematics Subject Classification:
17B10; 17B20; 17B35; 17B50
This work is supported by the National Natural Science Foundation of China
(Grant Nos. 11201293 and 11271130) and the Innovation Program of Shanghai
Municipal Education Commission (Grant Nos. 12ZZ038 and 13YZ077).
## 1\. Introduction and main results
The modular representations of reductive Lie algebras in prime characteristic
have been developed over the past decades with intimate connections to
algebraic groups (cf. [7], [1, 2], [3, 4], [10], [8], [11] etc.).
Let $k$ be an algebraically closed field of prime characteristic $p$ and $G$
be a connected, reductive algebraic group over $k$ with
${\mathfrak{g}}=\hbox{Lie}(G)$. Fix a maximal torus $T$ of $G$ and let $X(T)$
be the character group of $T$. Assume that the derived group $G^{(1)}$ of $G$
is simply connected, $p$ is a good prime for the root system of
${\mathfrak{g}}$, and ${\mathfrak{g}}$ has a non-degenerated $G$-invariant
bilinear form. Associated with any given linear form
$\chi\in{\mathfrak{g}}^{*}$, the $\chi$-reduced enveloping algebra
$U_{\chi}({\mathfrak{g}})$ is defined to be the quotient of the universal
enveloping algebra $U({\mathfrak{g}})$ by the ideal generated by all
$x^{p}-x^{[p]}-\chi(x)^{p}$ with $x\in\mathfrak{g}$. Each isomorphism class of
irreducible representations of ${\mathfrak{g}}$ corresponds to a unique
$p$-character $\chi$. Furthermore, a well-known result of Kac-Weisfeiler shows
that there is a Morita equivalence between $U_{\chi}({\mathfrak{g}})$-module
category and $U_{\chi}(\mathfrak{l})$-module category, where $\mathfrak{l}$ is
a certain reductive subalgebra of ${\mathfrak{g}}$ such that
$\chi|_{[\mathfrak{l},\,\mathfrak{l}]}$ is nilpotent (cf. [7] and [1]). This
enables us to study representations of $U_{\chi}({\mathfrak{g}})$ just with
nilpotent $\chi$.
We say a $p$-character $\chi$ has standard Levi form if $\chi$ is nilpotent
and if there exists a subset $I$ of $\Pi$ such that
$\chi({\mathfrak{g}}_{-\alpha})\neq 0$ for $\alpha\in{I}$ and
$\chi({\mathfrak{g}}_{-\alpha})=0$ for $\alpha\in{R^{+}\backslash I}$, where
$\Pi$ is the set of all simple roots, $R^{+}$ is the set of all positive roots
and $|R^{+}|=N$ (cf. [3, §10]). In this paper, we study representations of
${\mathfrak{g}}$ with a $p$-character $\chi$ of standard Levi form. Fix a
triangular decomposition
${\mathfrak{g}}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}$ and
let $\mathfrak{b}^{+}=\mathfrak{h}\oplus\mathfrak{n}^{+}$. For a subset $I$ in
$\Pi$, set $J=\Pi\setminus I$. Let
${\mathfrak{g}}_{J}=\mathfrak{n}_{J}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{J}^{+}$
and $\mathfrak{p}_{J}={\mathfrak{g}}_{J}\oplus\mathfrak{u}_{J}^{+}$ be the
Levi subalgebra and the parabolic subalgebra of ${\mathfrak{g}}$ corresponding
to $J$, respectively. Denote by
$\widehat{L}_{\mathfrak{{\mathfrak{g}}}_{J}}(\lambda)$ the irreducible
$X(T)/\mathbb{Z}I$-graded $U_{\chi}({\mathfrak{g}}_{J})$-module with “highest”
weight $\lambda$ (note that
$U_{\chi}({\mathfrak{g}}_{J})=U_{0}({\mathfrak{g}}_{J})$). Then
$\widehat{L}_{\mathfrak{{\mathfrak{g}}}_{J}}(\lambda)$ can be extended to a
$U_{\chi}({\mathfrak{p}}_{J})$-module with trivial
$\mathfrak{u}_{J}^{+}$-action. The induced module
$U_{\chi}(\mathfrak{g})\otimes_{U_{\chi}(\mathfrak{p}_{J})}\widehat{L}_{\mathfrak{p}_{J}}(\lambda)$
is called a parabolic baby Verma module, denoted by
$\widehat{{\mathcal{Z}}}_{P}(\lambda)$.
In [2, § 5.1], Friedlander and Parshall put forward the following open
question:
###### Question 1.1.
Can one give necessary and sufficient conditions on an irreducible module for
a parabolic subalgebra ${\mathfrak{p}}_{J}$ to remain irreducible upon
induction to ${\mathfrak{g}}$?
When $\chi$ is regular nilpotent, Friedlander-Parshall answered this question
in [1]. They showed that all such inductions remain irreducible. When
${\mathfrak{g}}$ is of type $A_{2}$, and $\chi(\neq 0)$ is of standard Levi
form, then each irreducible ${\mathfrak{g}}$-module with a $p$-character
$\chi$ is a parabolic baby Verma module. Quite recently, the authors of the
present paper obtained a necessary and sufficient condition for irreducibility
of parabolic baby Verma modules of $\mathfrak{sl}(4,k)$ in [9].
Let $C_{0}=\\{\lambda\in X(T)_{\mathbb{R}}\mid
0\leq\langle\lambda+\rho,\alpha^{\vee}\rangle<p\mbox{ for all }\alpha\in
R^{+}\\}$ be the first dominant alcove of $X(T)_{\mathbb{R}}$. Let
$X_{1}(T)=\\{\lambda\in X(T)\mid
0\leq\langle\lambda+\rho,\alpha^{\vee}\rangle<p\mbox{ for all
}\alpha\in\Pi\\}$ and $X^{\prime}_{1}(T)\subset X_{1}(T)$ be a system of
representatives for $X(T)/pX(T)$. Each $\lambda\in X(T)$ has a unique
decomposition $\lambda=\lambda_{0}+p\lambda_{1}$ with $\lambda_{0}\in
X^{\prime}_{1}(T)$ and $\lambda_{1}\in X(T)$ (cf. [5, II § 9.14]). For each
$\lambda\in X(T)$, the map $\lambda\mapsto d\lambda$ induces a bijection
$X(T)/pX(T)\cong\Lambda=\\{\mu\in{\mathfrak{h}}^{*}\mid\mu(h)^{p}-\mu(h^{[p]})=0,\,\forall\,h\in{\mathfrak{h}}\\}$
(cf. [3, § 11.1]). So we can regard $X^{\prime}_{1}(T)$ as a system of
representatives for $\Lambda$. We call $\lambda\in X(T)$ $p$-regular if the
stabilizer of $\lambda$ in $W_{p}$ is trivial, where $W_{p}$ is the affine
Weyl group of ${\mathfrak{g}}$.
In the present paper, we give a sufficient condition on the irreducibility of
some parabolic baby Verma modules, which partially answers Question 1.1.
###### Theorem 1.2.
Let $\mathfrak{g}$ be of type $A_{n},B_{n},C_{n}$ or $D_{n}$ satisfying the
hypotheses (H1)-(H3) in Section 2.1. Let $\lambda=\lambda_{0}+p\lambda_{1}\in
X(T)$ such that $\lambda_{0}\in C_{0}$ and $\lambda_{1}\in X(T)$. Let
$\chi\in{\mathfrak{g}}^{*}$ be of standard Levi form and
$I=\\{\alpha\in\Pi\mid\chi({\mathfrak{g}}_{-\alpha})\neq 0\\}$ in the Dynkin
diagram of ${\mathfrak{g}}$ is one of the following forms
$\begin{cases}\text{\rm(i)\,\,or\,\, (ii)},&\text{\rm
for}\,\,{\mathfrak{g}}=A_{n},\\\ \text{\rm(iii)},&\text{\rm
for}\,\,{\mathfrak{g}}=B_{n},\\\ \text{\rm(iv)},&\text{\rm
for}\,\,{\mathfrak{g}}=C_{n},\\\ \text{\rm(v)\,\,or\,\, (vi)},&\text{\rm
for}\,\,{\mathfrak{g}}=D_{n},\\\ \end{cases}$
where
${\rm(i)}\quad\underbrace{\bullet-\cdot\cdot\cdot-\bullet-\bullet}\limits_{I}-\circ-\cdot\cdot\cdot-\circ,$
${\rm(ii)}\quad\circ-\circ-\cdot\cdot\cdot-\underbrace{\bullet-\cdot\cdot\cdot-\bullet-\bullet}\limits_{I}.$
${\rm(iii)}\quad\circ-...-\circ-\underbrace{\bullet-...-\bullet\Rightarrow\bullet}_{I}.$
${\rm(iv)}\quad\underbrace{\bullet-\bullet-...-\bullet}_{I}-\circ-...-\circ\Leftarrow\circ.$
${\rm(v)}\quad\begin{matrix}\circ&\cdot\cdot\cdot&-\circ-&\bullet&-&\cdot\cdot\cdot&-&\bullet&-&\bullet\\\
&&&&&&&|&&\\\ &&&&&&&\bullet&&\end{matrix}$
${\rm(vi)}\quad\begin{matrix}\bullet-&\cdot\cdot\cdot&-&\bullet&-&\cdot\cdot\cdot&-&\bullet&-&\bullet\\\
&&&&&&&|&&\\\ &&&&&&&\circ&&\end{matrix}$
Set $J=\Pi\setminus I$. Then the parabolic baby Verma module
$\widehat{{\mathcal{Z}}}_{P}(\lambda)$ is irreducible, provided that $\lambda$
is $p$-regular.
As a consequence of Theorem 1.2, we have
###### Corollary 1.3.
Maintain the notations as in Theorem 1.2. The following statements hold.
* (1)
Assume that $\mathfrak{g}$ is of type $A_{n}$, and that
$\chi\in{\mathfrak{g}}^{*}$ is sub-regular nilpotent and has standard Levi
form. Then each irreducible ${\mathfrak{g}}$-module is a parabolic baby Verma
module.
* (2)
Assume that $\mathfrak{g}$ is of type $B_{n}$, and that
$\chi\in{\mathfrak{g}}^{*}$ is sub-regular nilpotent and has standard Levi
form. Let $\\{\widehat{L}_{\chi}(\lambda_{i})\mid 1\leq i\leq 2n\\}$ be the
set of isomorphism classes of simple ${\mathfrak{g}}$-modules in the block
containing $\widehat{L}_{\chi}(\lambda_{1})$ described as in [6, Proposition
3.13]. Then $\widehat{{\mathcal{Z}}}_{P}(\lambda_{i})$, $i\neq n,2n$, is
irreducible with dimension $r_{i}p^{N-1}$ for $1\leq i\leq n-1$ and
$r_{2n-i}p^{N-1}$ for $n+1\leq i\leq 2n-1$.
###### Remark 1.4.
Corollary 1.3 coincides with the results by Jantzen in [6, Theorem 2.6,
Proposition 3.13].
## 2\. Preliminaries
### 2.1. Notations and assumptions
Throughout this paper, we always assume that $k$ is an algebraically closed
field of prime characteristic $p$. We use notations in [3].
Let $G$ be a connected, reductive algebraic group over $k$ and
${\mathfrak{g}}=\hbox{Lie}(G)$. Then ${\mathfrak{g}}$ carries a natural
restricted mapping $[p]$: $x\mapsto x^{[p]}$. We assume that the following
three hypotheses are satisfied ([3, § 6.3]):
1. (H1)
The derived group $\mathcal{D}G$ of $G$ is simply connected;
2. (H2)
The prime $p$ is good for ${\mathfrak{g}}$;
3. (H3)
There exists a $G$-invariant non-degenerate bilinear form on ${\mathfrak{g}}$.
Let $T$ be a maximal torus of $G$ and $X(T)$ be the character group of $T$.
Denote respectively by $R^{\pm}$ the sets of all positive roots and all
negative roots. For each $\alpha\in R$, let ${\mathfrak{g}}_{\alpha}$ denote
the root subspace of ${\mathfrak{g}}$ corresponding to $\alpha$ and
$\mathfrak{n}^{+}=\sum_{\alpha\in
R^{+}}{\mathfrak{g}}_{\alpha},\mathfrak{n}^{-}=\sum_{\alpha\in
R^{-}}{\mathfrak{g}}_{\alpha}$. We have the triangular decomposition:
${\mathfrak{g}}=\mathfrak{n}^{+}\oplus\mathfrak{h}\oplus\mathfrak{n}^{-}$ with
$\mathfrak{h}$ being the Cartan subalgebra of ${\mathfrak{g}}$ with rank $l$.
Take a Chevalley basis $\\{x_{\alpha},y_{\alpha},h_{i}\mid\alpha\in
R^{+},1\leq i\leq l\\}$ of ${\mathfrak{g}}$. Let
$\mathfrak{b}^{+}=\mathfrak{h}\oplus\mathfrak{n}^{+}$ be the Borel subalgebra
of ${\mathfrak{g}}$. For each $\alpha\in R$, let $\alpha^{\vee}$ denote the
coroot of $\alpha$, $W$ the Weyl group generated by all $s_{\alpha}$ with
$\alpha\in R$, and $W_{p}$ the affine Weyl group generated by
$s_{\alpha,r}\,(r\in\mathbb{Z})$, where $s_{\alpha,r}$ is the affine
reflection defined by $s_{\alpha,r}(\mu)=\mu-(\langle\mu,\alpha^{\vee}\rangle-
rp)\alpha$ for any $\mu\in\mathfrak{h}^{*}$. Define the dot action of $w$ on
$\lambda$ by $w.\lambda=w(\lambda+\rho)-\rho$ for $w\in W$ and
$\lambda\in{\mathfrak{h}}^{*}$, where $\rho$ is half the sum of all positive
roots.
### 2.2. Baby Verma modules
Modulo Morita equivalence of representations, we can assume that
$\chi({\mathfrak{b}}^{+})=0$ without loss of generality (cf. [7, 1]). Set
$\Lambda:=\\{\lambda\in{\mathfrak{h}}^{*}\mid\lambda(h)^{p}=\lambda(h^{[p]})\\}$.
Any simple $U_{0}({\mathfrak{h}})$-module corresponds to a unique
$\lambda\in\Lambda$ (cf. [3]) and is one-dimensional, denoted by
$k_{\lambda}=kv_{\lambda}$, with $h\cdot v_{\lambda}=\lambda(h)v_{\lambda}$
for any $h\in{\mathfrak{h}}$. Since $k_{\lambda}$ can be extended to a
$U_{0}({\mathfrak{b}}^{+})$-module with trivial ${\mathfrak{n}}^{+}$-action,
we have an induced module
${Z}_{\chi}(\lambda)=U_{\chi}({\mathfrak{g}})\otimes_{U_{0}(\mathfrak{b}^{+})}k_{\lambda}$
which is called a baby Verma module. Each simple
$U_{\chi}({\mathfrak{g}})$-module is the homomorphic image of some baby Verma
module ${Z}_{\chi}(\lambda),\lambda\in\Lambda$ (cf. [3] or [4]).
### 2.3. Standard Levi forms
We say a $p$-character $\chi$ has standard Levi form if $\chi$ is nilpotent
and if there exists a subset $I$ of all simple roots such that
(2.1) $\chi({\mathfrak{g}}_{-\alpha})=\begin{cases}\neq 0,&\text{if
$\alpha\in{I}$,}\\\ 0,&\text{if $\alpha\in{R^{+}\backslash I}$}.\end{cases}$
As in [3, § 10.4; § 10.5], when $I$ is the full set of all simple roots, we
call $\chi$ a regular nilpotent element in ${\mathfrak{g}}^{*}$. When
$I=\varnothing$, then $\chi=0$. We denote by $R_{I}$ the root system
corresponding to the subset $I$, and $W_{I}$ the Weyl group generated by all
the $s_{\alpha}$ with $\alpha\in I$.
### 2.4. Graded module category
Assume that $\chi$ is of standard Levi form with
$I=\\{\alpha\in\Pi\mid\chi({\mathfrak{g}}_{-\alpha})\neq 0\\}$. Following
Jantzen [3, § 11], we can define a refined subcategory of the
$U_{\chi}({\mathfrak{g}})$-module category, which is the
$X(T)/\mathbb{Z}I$-graded $U_{\chi}({\mathfrak{g}})$-module category, denoted
by $\mathcal{C}$. For $\lambda\in X(T)$, the graded baby Verma module
$\widehat{{Z}}_{\chi}(\lambda)$ has a unique irreducible quotient, denoted by
$\widehat{L}_{\chi}(\lambda)$. The latter is also a graded simple module.
### 2.5. Parabolic baby Verma module
Assume that $\chi$ has standard Levi form associated with a subset $I$ of the
full set $\Pi$ of simple roots. Set $J=\Pi\setminus I$. Let
$\mathfrak{g}_{J}=\mathfrak{h}\oplus\bigoplus_{\alpha\in
R\cap\mathbb{Z}J}\mathfrak{g}_{\alpha}$,
$\mathfrak{u}_{J}^{+}=\bigoplus_{\alpha>0,\alpha\notin\mathbb{Z}J}\mathfrak{g}_{\alpha}$,
and $\mathfrak{p}_{J}=\mathfrak{g}_{J}\oplus\mathfrak{u}_{J}^{+}$. Then
$U_{\chi}(\mathfrak{p}_{J})=U_{0}(\mathfrak{p}_{J})$. For $\mu\in X(T)$, let
$\widehat{L}_{\mathfrak{p}_{J}}(\mu)$ be the graded irreducible
$U_{0}(\mathfrak{p}_{J})$-module, which is indeed a graded irreducible
$U_{0}(\mathfrak{{\mathfrak{g}}}_{J})$-module with trivial $u_{J}^{+}$-action.
The parabolic baby Verma module is defined as the following induced module
$\widehat{\mathcal{Z}}_{P}(\lambda):=U_{\chi}(\mathfrak{g})\otimes_{U_{0}(\mathfrak{p}_{J})}\widehat{L}_{\mathfrak{p}_{J}}(\lambda),\lambda\in
X(T).$
Then $\widehat{\mathcal{Z}}_{P}(\lambda)$ is a quotient module of
$\widehat{Z}_{\chi}(\lambda)$. Let $\varphi$:
$\widehat{Z}_{\chi}(\lambda)\twoheadrightarrow\widehat{\mathcal{Z}}_{P}(\lambda)$
be the canonical surjective morphism.
### 2.6.
We apply the translation principle to $\widehat{\mathcal{Z}}_{P}(\lambda)$.
###### Lemma 2.1.
Let $\lambda=(0,0,...,0)$ and $\mu\in C_{0}\cap X(T)$ be a regular weight.
Then we have
$T_{\lambda}^{\mu}\widehat{\mathcal{Z}}_{P}(\lambda)=\widehat{\mathcal{Z}}_{P}(\mu)$
where $T_{\lambda}^{\mu}$ is the so-called translation functor defined in [3,
§ 11].
###### Proof.
By the definition of the translation factor, we have
$T_{\lambda}^{\mu}\widehat{\mathcal{Z}}_{P}(\lambda)=\mbox{pr}_{\mu}(L(\nu)\otimes\widehat{\mathcal{Z}}_{P}(\lambda))$
where $L(\nu)$ is the simple $G$-module with highest weight $\nu$ in
$W(\mu-\lambda)$ (cf. [3, § 11.20]).
Since $\widehat{L}_{\chi}(\lambda)$ is the head of
$\widehat{\mathcal{Z}}_{P}(\lambda)$ and
$T_{\lambda}^{\mu}\widehat{L}_{\chi}(\lambda)\cong\widehat{L}_{\chi}(\mu)$
(cf. [3, Proposition 11.21]), we have
$T_{\lambda}^{\mu}\widehat{\mathcal{Z}}_{P}(\lambda)\neq 0.$
Let $\Xi$ be the set of weights in $L(\nu)$. By [5, Lemma 7.7, Proposition
7.11], there exists a unique weight $\xi\in\Xi$ with $\xi+\lambda\in
W_{p}.\mu$ and $\xi+\lambda=\mu$. Note that $\lambda$ is trivial, then
$\xi=\mu$. Hence, $\mu$ is the unique weight of $L(\nu)\otimes k_{\lambda}$
lying in $W_{p}.\mu$, where $k_{\lambda}$ is the one-dimensional trivial
$U_{0}({\mathfrak{g}}_{J})$-module.
The generalized tensor identity (cf. [4, §1.12]) yields the following
isomorphism
(2.2)
$T_{\lambda}^{\mu}\widehat{\mathcal{Z}}_{P}(\lambda):=\mbox{pr}_{\mu}(L(\nu)\otimes\widehat{\mathcal{Z}}_{P}(\lambda))\cong\mbox{pr}_{\mu}\big{(}U_{\chi}(\mathfrak{g})\otimes_{U_{0}(\mathfrak{p}_{J})}(L(\nu)\otimes\widehat{L}_{\mathfrak{p}_{J}}(\lambda))\big{)}.$
Since $\lambda=(0,0,...,0)$, the simple module
$\widehat{L}_{\mathfrak{p}_{J}}(\lambda)$ is trivial, i.e.,
$\widehat{L}_{\mathfrak{p}_{J}}(\lambda)=k_{\lambda}$. Then
$L(\nu)\otimes\widehat{L}_{\mathfrak{p}_{J}}(\lambda)=L(\nu)\otimes
k_{\lambda}.$ The set of weights in $L(\nu)\otimes k_{\lambda}$ is just $\Xi$.
We have the following composition series of $L(\nu)\otimes k_{\lambda}$
$0=M_{0}\subset M_{1}\subset M_{2}\subset\cdot\cdot\cdot\subset
M_{r}=L(\nu)\otimes k_{\lambda}$
where the factors
$M_{j}/M_{j-1}\cong\widehat{L}_{\mathfrak{p}_{J}}(\lambda_{j})$ with
$\lambda_{j}\in\Xi$. Since
$T_{\lambda}^{\mu}\widehat{\mathcal{Z}}_{P}(\lambda)\neq 0$ and $\mu$ is the
unique weight of $L(\nu)\otimes\lambda$ which lies in $W_{p}.\mu$, there
exists a unique $l\leq r$ with $\lambda_{l}=\mu$. Then
$\widehat{L}_{\mathfrak{p}_{J}}(\mu)\cong M_{l}/M_{l-1}$ is the unique
composition factor of $L(\nu)\otimes\lambda$ whose highest weight lies in
$W_{p}.\mu$.
Since the short sequence $0\rightarrow M_{l}\rightarrow M_{r}\rightarrow
M_{r}/M_{l}\rightarrow 0$ is exact and the functor
$\mbox{pr}_{\mu}\big{(}U_{\chi}(\mathfrak{g})\otimes_{U_{0}(\mathfrak{p}_{J})}(-)\big{)}$
is exact, we have the following isomorphisms
(2.3)
$\displaystyle\mbox{pr}_{\mu}\big{(}U_{\chi}(\mathfrak{g})\otimes_{U_{0}(\mathfrak{p}_{J})}(L(\nu)\otimes\widehat{L}_{\mathfrak{p}_{J}}(\lambda))\big{)}$
(2.4) $\displaystyle=$
$\displaystyle\mbox{pr}_{\mu}\big{(}U_{\chi}(\mathfrak{g})\otimes_{U_{0}(\mathfrak{p}_{J})}M_{r}\big{)}$
(2.5) $\displaystyle\cong$
$\displaystyle\mbox{pr}_{\mu}\big{(}U_{\chi}(\mathfrak{g})\otimes_{U_{0}(\mathfrak{p}_{J})}M_{l}\big{)}$
(2.6) $\displaystyle\cong$
$\displaystyle\mbox{pr}_{\mu}\big{(}U_{\chi}(\mathfrak{g})\otimes_{U_{0}(\mathfrak{p}_{J})}\widehat{L}_{\mathfrak{p}_{J}}(\mu)\big{)}$
(2.7) $\displaystyle\cong$
$\displaystyle\mbox{pr}_{\mu}\big{(}\widehat{\mathcal{Z}}_{P}(\mu)\big{)}.$
As $\widehat{\mathcal{Z}}_{P}(\mu)$ is the quotient of the baby Verma module
$\widehat{Z}_{\chi}(\mu)$, then $\widehat{\mathcal{Z}}_{P}(\mu)$ has a simple
head and is indecomposable. Furthermore, we have
(2.8)
$\mbox{pr}_{\mu}\big{(}\widehat{\mathcal{Z}}_{P}(\mu)\big{)}\cong\widehat{\mathcal{Z}}_{P}(\mu).$
It follows from (2.2), (2.3) and (2.8) that
$T_{\lambda}^{\mu}\widehat{\mathcal{Z}}_{P}(\lambda)=\widehat{\mathcal{Z}}_{P}(\mu).$
The proof is completed. ∎
Let $\widehat{M}\in\mathcal{C}$. A weight vector $m\in\widehat{M}$ is called a
maximal weight vector if $x_{\alpha}.m=0$ for any $\alpha\in R^{+}$.
### 2.7.
We need the following lemma for later use.
###### Lemma 2.2.
Assume that $\chi(\mathfrak{b}^{+})=0$. Then every submodule of
$\widehat{\mathcal{Z}}_{P}(\lambda)$ contains a maximal weight vector.
###### Proof.
Let $\widehat{M}$ be a submodule of $\widehat{\mathcal{Z}}_{P}(\lambda)$.
Since $U_{0}(\mathfrak{b}^{+})$ is a subalgebra of $U_{\chi}(\mathfrak{g})$,
$\widehat{M}$ is also a $U_{0}(\mathfrak{b}^{+})$-module. Then there exists an
irreducible $U_{0}(\mathfrak{b}^{+})$-submodule in $\widehat{M}$ which is one-
dimensional annihilated by ${\mathfrak{n}}^{+}$. This completes the proof. ∎
## 3\. Proof of Theorem 1.2 for type $A_{n}$
### 3.1.
In this section, we always assume that $\mathfrak{g}$ is a simple Lie algebra
of type $A_{n}$ and $\chi\in{\mathfrak{g}}^{*}$ has standard Levi form, and
$I=\\{\alpha\in\Pi\mid\chi({\mathfrak{g}}_{-\alpha})\neq 0\\}$ in the Dynkin
diagram is described as case (i) in Theorem 1.2.
We assume that $I=\\{\alpha_{1},\alpha_{2},...,\alpha_{s}\\}$ with $s\leq n$.
Then $J=\\{\alpha_{s+1},...,\alpha_{n}\\}$. If $s=n$, i.e., $\chi$ is regular
nilpotent, it’s well known that
$\widehat{\mathcal{Z}}_{P}(\lambda)=\widehat{Z}_{\chi}(\lambda)$ is
irreducible (cf. [3, § 10]). When $s<n$, we have
###### Claim 3.1.
The parabolic baby Verma module $\widehat{\mathcal{Z}}_{P}(\lambda)$ has only
one maximal weight vector (up to scalars) that generate
$\widehat{\mathcal{Z}}_{P}(\lambda)$.
###### Remark 3.2.
It follows from Claim 3.1 and Lemma 2.2 that
$\widehat{\mathcal{Z}}_{P}(\lambda)$ is irreducible.
In the following subsections, we first prove Claim 3.1.
### 3.2.
Thanks to Lemma 2.1, it suffices to prove Claim 3.1 for the special case that
$\lambda=(0,0,...,0)$. In this case, the irreducible
$\widehat{L}_{{\mathfrak{p}}_{J}}(\lambda)$-module is one-dimensional. This
means that this module is ${\mathfrak{p}}_{J}$-trivial. Then
$\widehat{\mathcal{Z}}_{P}(\lambda)=U_{\chi}({\mathfrak{u}}_{J}^{-})$ as
vector spaces, where ${\mathfrak{u}}_{J}^{-}$ is the negative counterpart of
${\mathfrak{u}}_{J}^{+}$ such that
${\mathfrak{g}}={\mathfrak{u}}_{J}^{-}\oplus{\mathfrak{p}}_{J}$.
### 3.3.
In the remainder of this section, we always take $\lambda=(0,0,\cdots,0)$.
Assume that $w=u\otimes v_{\lambda}$ is a maximal weight vector of
$\widehat{\mathcal{Z}}_{P}(\lambda)$ of weight $\mu$. We aim at proving that
$w$ generates the whole $\widehat{\mathcal{Z}}_{P}(\lambda)$.
Suppose the submodule generated by $w$ is proper, then $\mu\in W_{p}.\lambda$,
and
$\mu=\lambda-\sum\limits_{i=1}^{n}k_{i}\alpha_{i},~{}k_{i}\in\mathbb{Z}_{+}$
(cf. [4, Proposition 4.5]). Fix an order of the Chevalley basis in
${\mathfrak{u}}_{J}^{-}$ as follows
$\displaystyle y_{\alpha_{1}};$ $\displaystyle
y_{\alpha_{1}+\alpha_{2}},y_{\alpha_{2}};$ $\displaystyle
y_{\alpha_{1}+\alpha_{2}+\alpha_{3}},y_{\alpha_{2}+\alpha_{3}},y_{\alpha_{3}};$
$\displaystyle\cdots$ $\displaystyle
y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{s}},\cdots,y_{\alpha_{s-1}+\alpha_{s}},y_{\alpha_{s}};$
$\displaystyle
y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{s+1}},\cdots,y_{\alpha_{s}+\alpha_{s+1}};$
$\displaystyle\cdots$ $\displaystyle
y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}},\cdots,y_{\alpha_{s}+\cdots+\alpha_{n}}.$
Then $U_{\chi}({\mathfrak{u}}_{J}^{-})$ has the following basis
(3.1) $\displaystyle{\underline{y}}_{1}^{\bf a_{1}}\underline{y}_{2}^{\bf
a_{2}}\cdots\underline{y}_{s}^{\bf a_{s}}\underline{y}_{s+1}^{\bf
a_{s+1}}\cdots\underline{y}_{n}^{\bf a_{n}},$
where $\bf a=(a_{1},a_{2},\cdots,a_{n})$,
$\underline{y}_{j}^{\bf
a_{j}}=y_{\alpha_{1}+\alpha_{2}+...+\alpha_{j}}^{a_{j}(1)}y_{\alpha_{2}+...+\alpha_{j}}^{a_{j}(2)}\cdots
y_{\alpha_{j-1}+\alpha_{j}}^{a_{j}(j-1)}y_{\alpha_{j}}^{a_{j}(j)}\cdot\cdot\cdot\mbox{
for }j=1,2,\cdots,s$
and
$\underline{y}_{j}^{\bf
a_{j}}=y_{\alpha_{1}+\cdots+\alpha_{j}}^{a_{j}(1)}y_{\alpha_{2}+\cdots+\alpha_{j}}^{a_{j}(2)}\cdot\cdot\cdot
y_{\alpha_{s}+\alpha_{s+1}+\cdots+\alpha_{j}}^{a_{j}(s)}\mbox{ for
}j=s+1,\cdots,n$
with $0\leq a_{j}(k)\leq p-1,\,\forall\,j,k$. Hence $u\otimes v_{\lambda}$ can
be uniquely written as follows
(3.2) $\displaystyle u\otimes v_{\lambda}=\sum\limits_{\bf a}l_{\bf
a}{\underline{y}}_{1}^{\bf a_{1}}\underline{y}_{2}^{\bf
a_{2}}\cdots\underline{y}_{s}^{\bf a_{s}}\underline{y}_{s+1}^{\bf
a_{s+1}}\cdots\underline{y}_{n}^{\bf a_{n}}\otimes v_{\lambda}.$
For $\widehat{M}\in\mathcal{C}$, we have a decomposition
$\widehat{M}=\bigoplus\limits_{\nu\in X(T)/\mathbb{Z}I}\widehat{M}^{\nu}$ with
$x_{\alpha}.\widehat{M}^{\nu}\subset\widehat{M}^{\nu+\alpha}$ for all
$\alpha\in R$. By [3, § 11.7], we have
$\widehat{Z}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}$.
Hence,
$\widehat{\mathcal{Z}}_{P}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}.$
Since $u\otimes v_{\lambda}$ generates a proper submodule of
$\widehat{\mathcal{Z}}_{P}(\lambda)$, it follows that
(3.3) $\displaystyle u\otimes
v_{\lambda}\notin\widehat{\mathcal{Z}}_{P}(\lambda)^{\lambda+\mathbb{Z}I}\mbox{
and }\mu\in W_{p}.\lambda\setminus W_{I,p}.\lambda.$
### 3.4.
It follows from (3.1) and (3.2) that $k_{s}\geq k_{s+1}$ for a weight vector
$u\otimes v_{\lambda}\in\widehat{\mathcal{Z}}_{P}(\lambda)$ of weight
$\mu=\lambda-\sum\limits_{i=1}^{n}k_{i}\alpha_{i},~{}k_{i}\in\mathbb{Z}_{+}$.
Furthermore, we have
###### Lemma 3.3.
Assume that $u\otimes v_{\lambda}$ is a maximal weight vector of
$\widehat{\mathcal{Z}}_{P}(\lambda)$ with weight
$\mu=\lambda-\sum\limits_{i=1}^{n}k_{i}\alpha_{i},~{}k_{i}\in\mathbb{Z}_{+}$,
and that $u\otimes v_{\lambda}$ generates a proper submodule of
$\widehat{\mathcal{Z}}_{P}(\lambda)$. Then $k_{s}>k_{s+1}$.
###### Proof.
Suppose $k_{s}=k_{s+1}\neq 0$. Then the factor $\underline{y}_{s}^{\bf a_{s}}$
does not appear in any monomial summand of (3.2). The expression (3.2) can be
written as
(3.4) $\displaystyle u\otimes v_{\lambda}=$ $\displaystyle\sum\limits_{\bf
a}l_{\bf
a}{\underline{y}}_{1}^{a_{1}(1)}\underline{y}_{2}^{(a_{2}(1),a_{2}(2))}\cdot\cdot\cdot\underline{y}_{s-1}^{(a_{s-1}(1),a_{s-1}(2),\cdots,a_{s-1}(s-1))}$
$\displaystyle\cdot\underline{y}_{s+1}^{(a_{s+1}(1),a_{s+1}(2),\cdots,a_{s+1}(s))}\cdot\cdot\cdot\underline{y}_{n}^{(a_{n}(1),a_{n}(2),\cdots,a_{n}(s))}\otimes
v_{\lambda}.$
Since $k_{s+1}\neq 0$, there exists some factor $\underline{y}_{i}^{\bf
a_{i}}\neq 0$ with $s+1\leq i\leq n$. Without loss of generality, we may
assume that $\underline{y}_{s+1}^{\bf a_{s+1}}\neq 0$, then
$\displaystyle
x_{\alpha_{s+1}}\cdot\underline{y}_{s+1}^{(a_{s+1}(1),a_{s+1}(2),\cdots,a_{s+1}(s))}$
$\displaystyle=$
$\displaystyle\sum\limits_{t=1}^{s}a_{s+1}(t)N_{\alpha_{s+1},-(\alpha_{t}+\cdots+\alpha_{s+1})}y_{\alpha_{t}+\cdots+\alpha_{s}}\underline{y}_{s+1}^{(a_{s+1}(1),a_{s+1}(2),\cdots,a_{s+1}(t)-1,\cdots,a_{s+1}(s))}$
$\displaystyle+\underline{y}_{s+1}^{(a_{s+1}(1),a_{s+1}(2),\cdots,a_{s+1}(s))}x_{\alpha_{s+1}}$
where $N_{\alpha_{s+1},-(\alpha_{t}+\cdots+\alpha_{s+1})}$ is a structure
constant of ${\mathfrak{g}}$ relative to the Chevalley basis. Since
$x_{\alpha_{s+1}}$ commutes with $\underline{y}_{t}$ for any $t$ with $t\neq
s$, and annihilates $v_{\lambda}$, it follows that
(3.5) $\displaystyle x_{\alpha_{s+1}}\cdot u\otimes v_{\lambda}$
$\displaystyle=$ $\displaystyle(\sum\limits_{\bf a}\sum\limits_{t=1}^{s}l_{\bf
a}a_{s+1}(t)N_{\alpha_{s+1},-(\alpha_{t}+...+\alpha_{s+1})}{\underline{y}}_{1}^{a_{1}(1)}\cdot\cdot\cdot\underline{y}_{s-1}^{(a_{s-1}(1),a_{s-1}(2),...,a_{s-1}(s))}y_{\alpha_{t}+...+\alpha_{s}}$
$\displaystyle\cdot\underline{y}_{s+1}^{(a_{s+1}(1),a_{s+1}(2),...,a_{s+1}(t)-1,...,a_{s+1}(s))}\cdot\cdot\cdot\underline{y}_{n}^{(a_{n}(1),a_{n}(2),...,a_{n}(t),\cdots,a_{n}(s))})\otimes
v_{\lambda}.$
Note that ${\mathcal{Z}}_{P}(\lambda)$ is free over
$U_{\chi}({\mathfrak{u}}_{J}^{-})$. Since $u\otimes v_{\lambda}$ is a maximal
weight vector, it follows from (3.5) and (3.1) that $u\otimes v_{\lambda}=0$,
a contradiction. Hence, $k_{s}>k_{s+1}$. ∎
###### Lemma 3.4.
Maintain the notations as in Lemma 3.3. Assume that
$\underline{y}_{s}^{\bf
m_{s}}=y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{s}}^{m_{s}(1)}y_{\alpha_{2}+\cdots+\alpha_{s}}^{m_{s}(2)}\cdot\cdot\cdot
y_{\alpha_{s-1}+\alpha_{s}}^{m_{s}(s-1)}y_{\alpha_{s}}^{m_{s}(s)}$
is a factor of one monomial summand of $u$ such that $m_{s}(s)$ is maximal
among all the powers of $y_{\alpha_{s}}$ in monomial summands of $u$. Then
neither $y_{\alpha+\alpha_{s}}$ nor $y_{\alpha_{s}+\beta}$ with $\alpha\in
R_{I}^{+}$, $\beta\in R_{J}^{+}$, appear in a monomial summand of $u$
containing the factor $y_{\alpha_{s}}^{m_{s}(s)}$.
###### Proof.
For the weight $\mu=\lambda-\sum\limits_{i=1}^{n}k_{i}\alpha_{i}$ of $u\otimes
v_{\lambda}$, we have $k_{s}>k_{s+1}$ by Lemma 3.3. Hence, there exists some
nontrivial factor $\underline{y}_{s}^{\bf a_{s}}$ in each monomial summand of
$u$.
(1) We first claim that there exists at least one monomial summand of $u$
which contains a factor $y_{\alpha_{s}}^{m}$ with $m>0$.
Otherwise, the factor $~{}\underline{y}_{s}^{\bf a_{s}}$ of each monomial
summand of $u$ can be written as $\underline{y}_{s}^{\bf
a_{s}}=y_{\alpha_{1}+\cdots+\alpha_{s}}^{a_{s}(1)}\cdot\cdot\cdot
y_{\alpha_{s-1}+\alpha_{s}}^{a_{s}(s-1)}$. Since $\bf a_{s}\neq 0$, there
exist at least one $a_{s}(t)\neq 0$ for some $t\leq s-1$. Consider the action
of $x_{\alpha_{t}+\cdots+\alpha_{s-1}}$ on $u\otimes v_{\lambda}$. By
assumption, there does not exist monomial summand of $u$ which contains a
factor $y_{\alpha_{s}}^{m}$ with $m>0$. Since
$x_{\alpha_{t}+\cdots+\alpha_{s-1}}y_{\alpha_{t}+...+\alpha_{s}}=y_{\alpha_{t}+...+\alpha_{s}}x_{\alpha_{t}+\cdots+\alpha_{s-1}}+N_{\alpha_{t}+\cdots+\alpha_{s-1},-(\alpha_{t}+...+\alpha_{s})}y_{\alpha_{s}}$,
it follows that $x_{{\alpha_{t}+\cdots+\alpha_{s-1}}}\cdot u\otimes
v_{\lambda}$ contains the following monomial summand
(3.6)
${\underline{y}}_{1}^{a_{1}(1)}\underline{y}_{2}^{(a_{2}(1),a_{2}(2))}\cdot\cdot\cdot\underline{y}_{s}^{(a_{s}(1),\cdots,a_{s}(t)-1,\cdots,a_{s}(s))}\cdots\underline{y}_{n}^{(a_{n}(1),\cdots,a_{n}(s))}$
with $a_{s}(s)=1$. Note that there does not exist a similar item as (3.6)
among all the monomial summands of $x_{{\alpha_{t}+\cdots+\alpha_{s-1}}}\cdot
u\otimes v_{\lambda}$, then $x_{\alpha_{t}+\cdots+\alpha_{s-1}}\cdot u\otimes
v_{\lambda}\neq 0$. This is a contradiction. So, there exists at least one
monomial summand of $u\otimes v_{\lambda}$ containing $y_{\alpha_{s}}^{m}$
with $m>0$.
(2) Suppose that $y_{\alpha_{t}+...+\alpha_{s}}^{m_{s}(t)}\cdot\cdot\cdot
y_{\alpha_{s-1}+\alpha_{s}}^{m_{s}(s-1)}y_{\alpha_{s}}^{m_{s}(s)}$ with
$m_{s}(t)\geq 1$ is a factor of a monomial summand of $u\otimes v_{\lambda}$.
Consider the action of $x_{\alpha_{t}+...+\alpha_{s-1}}$ on $u\otimes
v_{\lambda}$. A direct computation implies that
$y_{\alpha_{t}+...+\alpha_{s}}^{m_{s}(t)-1}\cdot\cdot\cdot
y_{\alpha_{s-1}+\alpha_{s}}^{m_{s}(s-1)}y_{\alpha_{s}}^{m_{s}(s)+1}$ is a
factor of a monomial summand of $x_{\alpha_{t}+...+\alpha_{s-1}}\cdot u\otimes
v_{\lambda}$. Since $m_{s}(s)$ is maximal among all the powers of
$y_{\alpha_{s}}$ in monomial summands of $u$, there does not exist a similar
item as $y_{\alpha_{t}+...+\alpha_{s}}^{m_{s}(t)-1}\cdot\cdot\cdot
y_{\alpha_{s-1}+\alpha_{s}}^{m_{s}(s-1)-1}y_{\alpha_{s}}^{m_{s}(s)+1}$ among
all the monomial summands of $x_{{\alpha_{t}+\cdots+\alpha_{s-1}}}\cdot
u\otimes v_{\lambda}$. Then $x_{\alpha_{t}+...+\alpha_{s-1}}\cdot u\otimes
v_{\lambda}$ is not zero and this is contrary to the fact that $u\otimes
v_{\lambda}$ is maximal. So $y_{\alpha_{t}+...+\alpha_{s}}^{m_{s}({t})}$ with
$m_{s}(t)\geq 1$ does not appear in the same monomial summand of $u$
containing $y_{\alpha_{s}}^{m_{s}(s)}$, i.e., $y_{\alpha+\alpha_{s}}$ with
$\alpha\in R_{I}^{+}$ do not appear in the monomial summand of $u$ containing
$y_{\alpha_{s}}^{m_{s}(s)}$. Similarly, $y_{\alpha_{s}+\beta}$ with $\beta\in
R_{J}^{+}$ do not appear in the monomial summand of $u$ containing
$y_{\alpha_{s}}^{m_{s}(s)}$. ∎
### 3.5. Proof of Claim 3.1
By Lemma 3.4, neither $y_{\alpha+\alpha_{s}},\alpha\in R_{I}^{+}$ nor
$y_{\alpha_{s}+\beta},\beta\in R_{J}^{+}$ appear in the monomial summand of
$u$ provided it contains the factor $y_{\alpha_{s}}^{m_{s}(s)}$, where
$m_{s}(s)$ is maximal.
Denote by $u_{s}$ the sum of all those monomial summands of $u$ containing
$y_{\alpha_{s}}^{m_{s}(s)}$. By Lemma 3.4, $u_{s}$ can be written as follows
(3.7) $\displaystyle u_{s}=$ $\displaystyle\sum\limits_{\bf a}l_{\bf
a}{\underline{y}}_{1}^{a_{1}(1)}\underline{y}_{2}^{(a_{2}(1),a_{2}(2))}\cdot\cdot\cdot\underline{y}_{s-1}^{(a_{s-1}(1),a_{s-1}(2),\cdots,a_{s-1}({s-1}))}y_{\alpha_{s}}^{m_{s}(s)}$
(3.8) $\displaystyle\cdot
y_{s+1}^{(a_{s+1}(1),a_{s+1}(2),\cdots,a_{s+1}({s-1}))}\cdot\cdot\cdot\underline{y}_{n}^{(a_{n}(1),a_{n}(2),\cdots,a_{n}({s-1}))}.$
If $k_{s-1}\neq 0$, by a similar argument as in the proof of Lemma 3.4, there
exists at least one monomial summand of $u$ which contains
$y_{\alpha_{s-1}}^{m}$ with $m>0$. Let $m_{s-1}(s-1)$ be maximal among all
those $a_{s-1}(s-1)$ appearing in (3.7). Then there exists a monomial summand
of $u_{s}$ with $a_{s-1}(s-1)=m_{s-1}(s-1)$ which does not contain
$y_{\alpha_{s}+\beta}$ and $y_{\alpha+\alpha_{s}}$. By the same method as in
the proof of Lemma 3.4, those $y_{\alpha+\alpha_{s-1}},~{}\alpha\in R^{+}$ and
$y_{\alpha_{s-1}+\beta},~{}\beta\in R^{+}$ do not appear in the monomial
summand of $u_{s}$ in which $a_{s-1}(s-1)=m_{s-1}(s-1)$.
Denote by
$\displaystyle u_{s-1}=$ $\displaystyle\sum\limits_{\bf a}l_{\bf
a}{\underline{y}}_{1}^{a_{1}(1)}\underline{y}_{2}^{(a_{2}(1),a_{2}(2))}\cdot\cdot\cdot\underline{y}_{s-2}^{(a_{s-2}(1),a_{s-2}(2),\cdots,a_{s-2}({s-2}))}y_{\alpha_{s-1}}^{m_{s-1}(s-1)}y_{\alpha_{s}}^{m_{s}(s)}$
$\displaystyle\cdot\underline{y}_{s+1}^{(a_{s+1}(1),a_{s+1}(2),\cdots,a_{s+1}({s-2}),0,0)}\cdot\cdot\cdot\underline{y}_{n}^{(a_{n}(1),a_{n}(2),\cdots,a_{n}({s-2}),0,\cdots,0)}.$
Similar to the discussion above, if $k_{s-2}\neq 0$, then
$y_{\alpha+\alpha_{s-2}},~{}\alpha\in R^{+}$ and
$~{}y_{\alpha_{s-2}+\beta},~{}\beta\in R^{+}$, do not appear in the monomial
summand of $u_{s-1}$ which contains $y_{\alpha_{s-2}}^{m_{s-2}(s-2)}$, where
$m_{s-2}(s-2)$ is maximal among all those $a_{s-2}(s-2)$ appearing in (3.7).
Since $I$ is connected, we can repeat the process above. Finally, we obtain
that
$u_{1}=y_{\alpha_{1}}^{m_{1}({1})}y_{\alpha_{2}}^{m_{2}({2})}\cdot\cdot\cdot
y_{\alpha_{s-1}}^{m_{s-1}({s-1})}y_{\alpha_{s}}^{m_{s}({s})}$ with
$m_{s}({s})>0$ and $m_{s-i}({s-i})\geq 0,1\leq i\leq s-1$. As $u_{1}$ is a
summand of $u\otimes v_{\lambda}$, then $u\otimes
v_{\lambda}\in\widehat{\mathcal{Z}}_{P}(\lambda)^{\lambda+\mathbb{Z}I}\cong_{{\mathfrak{g}}_{I}}\widehat{L}_{\chi}(\lambda)^{\lambda+\mathbb{Z}I}$.
This is contrary to the fact that $u\otimes v_{\lambda}$ generates the proper
submodule of $\widehat{\mathcal{Z}}_{P}(\lambda)$. Therefore the Claim 3.1
holds in the case that $I=\\{\alpha_{1},\cdots,\alpha_{s}\\}$.
When $I=\\{\alpha_{t},\alpha_{t+1},\cdots,\alpha_{n}\\},~{}1<t\leq n$, the
proof is similar. We complete the proof of Claim 3.1.
### 3.6.
For the decomposition $\lambda=\lambda_{0}+p\lambda_{1}$ with $\lambda_{0}\in
X^{\prime}_{1}(T)$ and $\lambda_{1}\in X(T)$, we have
$\mathcal{F}(\widehat{\mathcal{Z}}_{P}(\lambda))\cong{\mathcal{Z}}_{P}(\lambda_{0})$
where $\mathcal{F}$ is the forgetful functor (cf. [3, §11]). So the non-graded
parabolic baby Verma module ${\mathcal{Z}}_{P}(\lambda_{0})$ is also
irreducible.
### 3.7.
Assume that $I=\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{s}\\}$ with $s\leq n$.
From the discussions above, if one replaces the condition
$\lambda=(0,0,\cdots,0)$ by $(m_{1}-1,\cdots,m_{s}-1,0,\cdots,0)$ with
$0<m_{i}<p,\forall~{}i$, Lemma 2.1 and Claim 3.1 still hold. We have the
following consequence.
###### Corollary 3.5.
Keep the same notations as above. Assume that $I$ is connected in the Dykin
diagram with $\alpha_{1}\in I$ (resp. $\alpha_{n}\in I$) and
$\lambda_{0}+\rho=(m_{1},\cdots,m_{s},1,\cdots,1)$ (resp.
$\lambda_{0}+\rho=(1,\cdots,1,m_{t},\cdots,m_{n})$), $0<m_{i}<p,\forall~{}i$.
Then $\widehat{\mathcal{Z}}_{P}(\lambda)$ is irreducible.
###### Remark 3.6.
For $\lambda=\lambda_{0}+p\lambda_{1}$ with $\lambda_{0}\in X^{\prime}_{1}(T)$
and $\lambda_{1}\in X(T)$, by Lemma 2.1, when $\lambda_{0}$ lies in the
alcoves which contain the weight $(m_{1},\cdots,m_{s},1,\cdots,1)-\rho$ (resp.
$(1,\cdots,1,m_{t},\cdots,m_{n})-\rho$), $0<m_{i}<p,\forall~{}i$, the
parabolic baby Verma module $\widehat{\mathcal{Z}}_{P}(\lambda)$ is
irreducible.
## 4\. Proof of Theorem 1.2 for types $B_{n}$, $C_{n}$ and $D_{n}$
In this section, we give the proof of Theorem 1.2 for types $B_{n}$, $C_{n}$
and $D_{n}$ cases by cases.
### 4.1. Proof of Theorem 1.2 for type $B_{n}$
Let $\mathfrak{g}$ be of type $B_{n}$ and $\chi\in{\mathfrak{g}}^{*}$ be of
standard Levi form with
$I=\\{\alpha\in\Pi\mid\chi({\mathfrak{g}}_{-\alpha})\neq 0\\}$. Assume that
$I=\\{\alpha_{s},\cdots,\alpha_{n}\\}$ for some $1\leq s\leq n$ (note that
$\alpha_{n}$ is a short root), i.e., we have the following Dynkin diagram
$\circ-\cdots-\circ-\underbrace{\bullet-\cdots-\bullet\Rightarrow\bullet}_{I}$
Fix an order of the Chevalley basis in ${\mathfrak{u}}_{J}^{-}$ as follows
$\displaystyle y_{\alpha_{n}};$ $\displaystyle
y_{\alpha_{n-1}+\alpha_{n}},y_{\alpha_{n-1}+2\alpha_{n}},y_{\alpha_{n-1}};$
$\displaystyle\cdots$ $\displaystyle
y_{\alpha_{s}+\alpha_{s+1}},y_{\alpha_{s}+\alpha_{s+1}+\alpha_{s+2}},\cdots,y_{\alpha_{s}+2\alpha_{s+1}+\cdots+2\alpha_{n-1}+2\alpha_{n}},y_{\alpha_{s}};$
$\displaystyle
y_{\alpha_{s-1}+\alpha_{s}},y_{\alpha_{s-1}+\alpha_{s}+\alpha_{s+1}},\cdots,y_{\alpha_{s-1}+2\alpha_{s}+\cdots+2\alpha_{n-1}+2\alpha_{n}};$
$\displaystyle\cdots$ $\displaystyle
y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{s}},\cdots,y_{\alpha_{1}+\alpha_{2}+\alpha_{3}+\cdots+\alpha_{n}}\cdots,y_{\alpha_{1}+\alpha_{2}+2\alpha_{3}+\cdots+2\alpha_{n}},y_{\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\cdots+2\alpha_{n}}.$
Suppose that $u\otimes v_{\lambda}$ is a maximal weight vector of
$\widehat{\mathcal{Z}}_{P}(\lambda)$ such that the submodule of
$\widehat{\mathcal{Z}}_{P}(\lambda)$ generated by $u\otimes v_{\lambda}$ is
proper. Then the weight $\mu$ of $u\otimes v_{\lambda}$ belongs to
$W_{p}.\lambda$ and can be written as
$\mu=\lambda-\sum\limits_{i=1}^{n}k_{i}\alpha_{i},~{}k_{i}\in\mathbb{Z}_{+}$
(cf. [4, Proposition 4.5]). We can use similar discussion as the case of type
$A_{n}$. We enumerate the strategy as follows with the details omitted.
(i) Assume that $\lambda=(0,0,\cdots,0)$. Then $k_{s}>0$.
(ii) There exists $y_{\alpha_{s}}^{m}$ with $m>0$ as a factor of some monomial
summand of $u$. Assume that the power $m_{s}(s)$ of $y_{\alpha_{s}}$ is
maximal among all the powers of $y_{\alpha_{s}}$ in monomial summands of $u$.
First, we prove that $y_{\alpha_{s+1}+\alpha_{s}}$ do not appear in the same
monomial summand of $u$ which contains $y_{\alpha_{s}}^{m_{s}(s)}$. Next, we
can prove that $y_{\gamma}$ does not appear in the same monomial summand of
$u$ which contains $y_{\alpha_{s}}^{m_{s}(s)}$ as a factor, where $\gamma\in
R^{+}$ and $\alpha_{s}$ is the first or last summand of $\gamma$.
(iii) $u$ has a monomial summand
$y_{\alpha_{n}}^{m_{n}({n})}y_{\alpha_{n-1}}^{m_{n-1}({n-1})}\cdot\cdot\cdot
y_{\alpha_{s+1}}^{m_{s+1}({s+1})}y_{\alpha_{s}}^{m_{s}({s})}.$
Furthermore, we have
###### Remark 4.1.
Let $\lambda=\lambda_{0}+p\lambda_{1}$ with $\lambda_{0}\in X^{\prime}_{1}(T)$
and $\lambda_{1}\in X(T)$. Assume that $\lambda_{0}$ lies in the alcoves which
contain the weight $(1,\cdots,1,m_{s},\cdots,m_{n})-\rho$,
$0<m_{i}<p,\forall~{}i$, then the parabolic baby Verma module
$\widehat{\mathcal{Z}}_{P}(\lambda)$ is irreducible by Lemma 2.1.
### 4.2. Proof of Theorem 1.2 for type $C_{n}$
Let $\mathfrak{g}$ be of type $C_{n}$ and $\chi\in{\mathfrak{g}}^{*}$ be of
standard Levi form with
$I=\\{\alpha\in\Pi\mid\chi({\mathfrak{g}}_{-\alpha})\neq 0\\}$. Assume that
$I=\\{\alpha_{1},\cdots,\alpha_{s}\\},1\leq s\leq n$ (note that $\alpha_{n}$
is a long root), i.e., we have the following Dynkin diagram
$\underbrace{\bullet-\bullet-\cdots-\bullet}_{I}-\circ-\cdots-\circ\Leftarrow\circ$
Fix an order of the Chevalley basis in ${\mathfrak{u}}_{J}^{-}$ as follows
$\displaystyle y_{\alpha_{1}};$ $\displaystyle
y_{\alpha_{1}+\alpha_{2}},y_{\alpha_{2}};$ $\displaystyle\cdots$
$\displaystyle
y_{\alpha_{s-1}+\alpha_{s}},y_{\alpha_{s-2}+\alpha_{s-1}+\alpha_{s}},\cdots,y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{s}},y_{\alpha_{s}};$
$\displaystyle\cdots$ $\displaystyle
y_{\alpha_{s}+\alpha_{s+1}+\cdots+\alpha_{n-1}},y_{\alpha_{s-1}+\alpha_{s}+\cdots+\alpha_{n-1}},\cdots,y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n-1}};$
$\displaystyle
y_{\alpha_{s}+\alpha_{s+1}+\cdots+\alpha_{n}},\cdots,y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}}\cdots,y_{\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{n-1}+\alpha_{n}},y_{2\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{n-1}+\alpha_{n}}.$
By a similar argument as the case of type $B_{n}$, we can prove Theorem 1.2 in
the case of type $C_{n}$.
Furthermore, let $\lambda=\lambda_{0}+p\lambda_{1},\lambda_{0}\in
X^{\prime}_{1}(T),\lambda_{1}\in X(T)$. Assume that $\lambda_{0}$ lies in the
alcoves which contain the weight $(m_{1},\cdots,m_{s},1,\cdots,1)-\rho$,
$0<m_{i}<p,\forall~{}i$. Then the parabolic baby Verma module
$\widehat{\mathcal{Z}}_{P}(\lambda)$ is irreducible by Lemma 2.1.
### 4.3. Proof of Theorem 1.2 for type $D_{n}$
Let $\mathfrak{g}$ be of type $D_{n}$ and $\chi\in{\mathfrak{g}}^{*}$ be of
standard Levi form with
$I=\\{\alpha\in\Pi\mid\chi({\mathfrak{g}}_{-\alpha})\neq 0\\}$.
(i) Assume that
$I=\\{\alpha_{s},\cdots,\alpha_{n-2},\alpha_{n-1},\alpha_{n}\\},1\leq s\leq
n-2$, i.e., we have the following Dynkin diagram
$\begin{matrix}\circ-&\cdot\cdot\cdot&-\circ-&\bullet&-&\cdot\cdot\cdot&-&\bullet&-&\bullet\\\
&&&&&&&|&&\\\ &&&&&&&\bullet&&\end{matrix}$
Fix an order of the Chevalley basis in ${\mathfrak{u}}_{J}^{-}$ as follows
$\displaystyle y_{\alpha_{n}},y_{\alpha_{n-1}};$ $\displaystyle
y_{\alpha_{n}+\alpha_{n-2}},y_{\alpha_{n-1}+\alpha_{n-2}},y_{\alpha_{n}+\alpha_{n-1}+\alpha_{n-2}},y_{\alpha_{n-2}};$
$\displaystyle\cdots$ $\displaystyle
y_{\alpha_{s}+\alpha_{s+1}},\cdot\cdot\cdot,y_{\alpha_{s}+\cdots+\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}},\cdot\cdot\cdot,y_{\alpha_{s}+2\alpha_{s+1}+2\alpha_{s+2}+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}},y_{\alpha_{s}};$
$\displaystyle
y_{\alpha_{s-1}+\alpha_{s}},\cdot\cdot\cdot,y_{\alpha_{s-1}+\cdots+\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}},\cdot\cdot\cdot,y_{\alpha_{s-1}+2\alpha_{s}+2\alpha_{s+1}+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}};$
$\displaystyle\cdots$ $\displaystyle
y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{s}},\cdots,y_{\alpha_{1}+\cdots+\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}}\cdots,y_{\alpha_{1}+2\alpha_{2}+2\alpha_{3}+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}}.$
By a similar argument as the case of type $B_{n}$, we can prove Theorem 1.2 in
this case for type $D_{n}$.
Furthermore, let $\lambda=\lambda_{0}+p\lambda_{1}$ with $\lambda_{0}\in
X^{\prime}_{1}(T)$ and $\lambda_{1}\in X(T)$. Assume that $\lambda_{0}$ lies
in the alcoves which contain the weight
$(1,\cdots,1,m_{s},\cdots,m_{n})-\rho$, $0<m_{i}<p,\forall~{}i$. Then the
parabolic baby Verma module $\widehat{\mathcal{Z}}_{P}(\lambda)$ is
irreducible by Lemma 2.1.
(ii) Assume that $I=\\{\alpha_{1},\cdots,\alpha_{n-2},\alpha_{n-1}\\}$, i.e.,
we have the following Dynkin diagram
$\begin{matrix}\bullet-&\cdot\cdot\cdot&-&\bullet&-&\cdot\cdot\cdot&-&\bullet&-&\bullet\\\
&&&&&&&|&&\\\ &&&&&&&\circ&&\end{matrix}$
Fix an order of the Chevalley basis in ${\mathfrak{u}}_{J}^{-}$ as follows
$\displaystyle y_{\alpha_{1}};$ $\displaystyle
y_{\alpha_{1}+\alpha_{2}},y_{\alpha_{2}};$ $\displaystyle\cdots$
$\displaystyle
y_{\alpha_{n-2}+\alpha_{n-1}},y_{\alpha_{n-3}+\alpha_{n-2}+\alpha_{n-1}},\cdot\cdot\cdot,y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n-1}},y_{\alpha_{n-1}};$
$\displaystyle
y_{\alpha_{n-2}+\alpha_{n}},y_{\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}},\cdot\cdot\cdot,y_{\alpha_{1}+\cdots+\alpha_{n-2}+\alpha_{n}},y_{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n-1}+\alpha_{n}},$
$\displaystyle
y_{\alpha_{1}+\cdots+\alpha_{n-3}+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}},\cdot\cdot\cdot,y_{\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}}.$
By a similar argument as the case of type $B_{n}$, we can prove Theorem 1.2 in
this case.
Furthermore, let $\lambda=\lambda_{0}+p\lambda_{1}$ with $\lambda_{0}\in
X^{\prime}_{1}(T)$ and $\lambda_{1}\in X(T)$, and $\lambda_{0}$ lies in the
alcoves which contain the weight
$(m_{1},\cdots,m_{n-2},m_{n-1},1)-\rho,0<m_{i}<p,\forall~{}i$, then
$\widehat{\mathcal{Z}}_{P}(\lambda)$ is irreducible by Lemma 2.1.
## 5\. Proof of Corollary 1.3
### 5.1.
Assume that ${\mathfrak{g}}$ is of type $A_{n}$ and
$\chi\in{\mathfrak{g}}^{*}$ has standard Levi form associated with
$I=\\{\alpha\in\Pi\mid\chi({\mathfrak{g}}_{-\alpha})\neq
0\\}=\\{\alpha_{1},\alpha_{2},\cdots,\alpha_{n-1}\\}$. Set
$\sigma=s_{1}s_{2}\cdots s_{n}$ where $s_{i}$ is the simple reflection
corresponding to $\alpha_{i}$. Assume $\lambda_{0}\in C_{0}$ with
$\lambda_{0}+\rho=(r_{1},r_{2},\cdots,r_{n})$. Then
$0\leq\sum\limits_{i=1}^{n}r_{i}\leq p$.
Set $\lambda_{i}=\sigma^{i}.\lambda_{0}$ for $1\leq i\leq n$. Then
$\lambda_{i}+\rho=(r_{n-i+2},r_{n-i+3},\cdots,r_{n},-(r_{1}+\cdots+r_{n}),r_{1},r_{2},\cdots,r_{n-i})$.
Each $\widehat{L}_{\chi}(w.\lambda_{0})$ with $w\in W$ is isomorphic to some
$\widehat{L}_{\chi}(\lambda_{i})$ with $0\leq i\leq n$ (cf. [6, § 2.3]). Then
$\\{\widehat{L}_{\chi}(\lambda_{i})\mid 0\leq i\leq n\\}$ is the set of
isomorphism classes of simple modules in the block containing
$\widehat{L}_{\chi}(\lambda_{0})$.
For the decomposition $\lambda_{i}=\lambda_{i,0}+p\lambda_{i,1}$ with
$\lambda_{i,0}\in X^{\prime}_{1}(T)$ and $\lambda_{i,1}\in X(T)$, since
$0\leq\sum\limits_{j=1}^{n}r_{j}\leq p$, we have $\lambda_{i,0}\in
C_{0},~{}\forall~{}i$. Therefore $\widehat{\mathcal{Z}}_{P}(\lambda_{i})$ is
irreducible by Theorem 1.2. Then $\widehat{\mathcal{Z}}_{P}(\lambda_{i})$ has
dimension $r_{n-i}p^{N-1}$, i.e., we get part (1) of Corollary 1.3 which
coincides with [6, Theorem 2.6].
### 5.2.
Assume that ${\mathfrak{g}}$ is of type $B_{n}$ and
$\chi\in{\mathfrak{g}}^{*}$ has standard Levi form associated with
$I=\\{\alpha\in\Pi\mid\chi({\mathfrak{g}}_{-\alpha})\neq
0\\}=\\{\alpha_{2},\alpha_{3},\cdots,\alpha_{n}\\}$ (where $\alpha_{n}$ is a
short root). Assume $\lambda_{1}\in C_{0}$ with
$\lambda_{1}+\rho=(r_{1},r_{2},\cdots,r_{n})$. Then we have
$0\leq\sum\limits_{i=1}^{n-1}2r_{i}+r_{n}\leq p$. Let
$w_{i}=\begin{cases}s_{1}s_{2}\cdots s_{i-1},&\text{for $1\leq i\leq n$,}\\\
s_{1}s_{2}\cdots s_{n}s_{n-1}\cdots s_{2n+1-i},&\text{ for $n+1\leq i\leq
2n$}.\end{cases}$
Set $\lambda_{i}=w_{i}.\lambda_{1}$ for $1\leq i\leq 2n$. Then
$\\{\widehat{L}_{\chi}(\lambda_{i})|1\leq i\leq 2n\\}$ is the set of
isomorphism classes of simple modules in the block containing
$\widehat{L}_{\chi}(\lambda_{1})$ (cf. [6, § 3.8]).
Moreover, we have
$\lambda_{2}+\rho=(-r_{1},r_{1}+r_{2},r_{3},\cdots,r_{2n}),$
$\lambda_{3}+\rho=(-(r_{1}+r_{2}),r_{1},r_{2}+r_{3},\cdots,r_{2n}),$
$\cdots\cdots\cdots\cdots\cdots\cdots,$
$\lambda_{2n}+\rho=(-(r_{1}+2r_{2}+2r_{3}+\cdots+2r_{n-1}+r_{n}),r_{2},r_{3},\cdots,r_{2n}).$
Let
$\lambda_{i}^{\prime}=\begin{cases}s_{2}s_{3}\cdots
s_{i}.\lambda_{i},&\text{for $2\leq i\leq n-1$,}\\\ s_{2}s_{3}\cdots
s_{n}s_{n-1}\cdots s_{2n+1-i}.\lambda_{i},&\text{ for $n+1\leq i\leq
2n-1$}.\end{cases}$
Then
$\lambda_{2}^{\prime}+\rho=(r_{2},-(r_{1}+r_{2}),r_{1}+r_{2}+r_{3},\cdots,r_{n}),$
$\lambda_{3}^{\prime}+\rho=(r_{3},-(r_{1}+r_{2}+r_{3}),r_{1},r_{2}+r_{3}+r_{4},r_{5},\cdots,r_{n}),$
$\cdots\cdots\cdots\cdots\cdots\cdots,$
$\lambda_{2n-1}^{\prime}+\rho=(r_{1},-(r_{1}+r_{2}+2r_{3}+\cdots+2r_{n-1}+r_{n}),r_{3},\cdots,r_{n}).$
It is obvious that the first component of $\lambda_{i}^{\prime}+\rho$ is
$r_{i}$ for $1\leq i\leq n-1$ and $r_{2n-i}$ for $n+1\leq i\leq 2n-1$. By [3,
Proposition 11.9], we have
$\widehat{L}_{\chi}(\lambda_{i})\cong\widehat{L}_{\chi}(\lambda_{i}^{\prime})$
for $i\neq n,2n$.
Since $0\leq r_{i}\leq p$ and $0\leq\sum\limits_{i=1}^{n-1}2r_{i}+r_{n}\leq
p$, it follows from Remark 4.1 that
$\widehat{\mathcal{Z}}_{P}(\lambda^{\prime}_{i})$ ($i\neq n,\,2n$) is
irreducible with dimension $r_{i}p^{N-1}$ for $1\leq i\leq n-1$ and
$r_{2n-i}p^{N-1}$ for $n+1\leq i\leq 2n-1$, i.e., we get part (2) of Corollary
1.3 which coincides with [6, Proposition 3.13].
## References
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* [2] Friedlander E. M. and Parshall B. J., Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), 375-395.
* [3] Jantzen J. C., Representations of Lie algebras in prime characteristic, in: A. Borel (Ed.), Representation Theories and Algebraic Geometry, Proceedings, Montreal, 1997, in: NATO ASI Series, Vol. C514, Kluwer, Dordrecht, 1998, pp. 185-235.
* [4] Jantzen J. C., Modular representations of reductive Lie algebras, J. Pure Appl. Algebra 152 (2000), 133-185.
* [5] Jantzen J. C., Reresentations of Algebraic Groups, 2nd edn, American Mathematical Society, Providence, RI, 2003.
* [6] Jantzen J. C., Subregular nilpotent representations of $sl_{n}$ and $so_{2n+1}$, Math. Proc. Cambridge Philos. Soc. 126 (1999), 223-257.
* [7] Kac V. and Weisfeiler B., The irreducible representations of Lie p-algebras, Funct. Anal. Appl. 5 (1971), 111-117.
* [8] Li Y. Y. and Shu B., Filtrations in modular representations of reductive Lie algebras, Algebra Colloq. 17 (2010), 265-282.
* [9] Li Y. Y., Shu B. and Yao Y. F., A necessary and sufficient condition for irreducibility of parabolic baby Verma modules of $\mathfrak{sl}(4,k)$, J. Pure Appl. Algebra, to appear, 2014.
* [10] Premet A., Irreducible representations of Lie algebras of reductive groups and the KacWeisfeiler conjecture, Invent. Math. 121 (1995), 79 C117.
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|
arxiv-papers
| 2014-04-19T11:10:09 |
2024-09-04T02:50:01.465955
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yi-Yang Li, Bin Shu and Yu-Feng Yao",
"submitter": "Yu-Feng Yao",
"url": "https://arxiv.org/abs/1404.4945"
}
|
1404.4963
|
# Functional dependencies with null markers
Antonio Badia [email protected] CECS Department, University of
Louisville, Louisville KY 40292, USA Daniel Lemire LICEF, Université du
Québec, 5800 Saint-Denis, Montreal, QC, H2S 3L5 Canada
###### Abstract
Functional dependencies are an integral part of database design. However, they
are only defined when we exclude null markers. Yet we commonly use null
markers in practice. To bridge this gap between theory and practice,
researchers have proposed definitions of functional dependencies over
relations with null markers. Though sound, these definitions lack some
qualities that we find desirable. For example, some fail to satisfy
Armstrong’s axioms—while these axioms are part of the foundation of common
database methodologies. We propose a set of properties that any extension of
functional dependencies over relations with null markers should possess. We
then propose two new extensions having these properties. These extensions
attempt to allow null markers where they make sense to practitioners. They
both support Armstrong’s axioms and provide _realizable_ null markers: at any
time, some or all of the null markers can be replaced by actual values without
causing an anomaly. Our proposals may improve database designs.
###### keywords:
Functional Dependencies; Database Design; Missing Information
## 1 Introduction
Functional dependencies (hereafter, FDs) are the basis of good relational
database design. Database designers use FDs to define and enforce consistency;
for example, if each user account has only one corresponding primary email
address, we say that the user account determines the email address
($\textrm{user-account}\to\textrm{email}$). Though database designers may not
always work directly with FDs, almost all of them work with normal forms which
exist to support FDs.
Meanwhile, relational databases use null markers to cope with incomplete
information. Indeed, Codd introduced null markers in the relational model
along with a 3-value logic: a comparison between a value and a null marker has
an unknown truth value [1]. A null marker can indicate anything from an
applicable but unknown value to a non-applicable attribute; it may even
indicate that we have _no information_ [2, 3, 4]. We refer the reader to
Libkin [5] for an early survey on incomplete information within databases.
Unfortunately, the concept of functional dependence, even though central to
relational database design, is not part of the SQL standard. Hence, FDs are
not _directly_ supported by relational databases. Admittedly, given the
procedural extensions of the SQL standard, SQL is computationally complete and
therefore can enforce FDs using checks, assertions or triggers. However, the
problem is more fundamental: FDs are not _defined_ in the presence of null
markers. Thus, there is no clear semantics to enforce when we have FDs and
null markers.
We believe that the fact that FDs are not defined in the presence of null
markers undermines the role of FDs in database design [6]. While there are
proposed definitions of FDs in relations with null markers, such proposals may
not be practical.
Our work is organized as follows. In the next section, we argue for a minimal
set of desirable properties that FDs in relations with null markers should
have to be of practical use. After briefly introducing our formal notation in
§ 3, we review two existing extensions of FDs to null markers in § 4: weak and
strong FDs. We show that neither extension is satisfactory in our context, as
they do not have the desired properties. We introduce two novel extensions in
§ 5: literal and super-reflexive functional dependencies, and show that they
that possess our properties. In § 6 we compare our extensions to weak and
strong FDs, in order to give the reader a context for interpretation. Next in
§ 7 we address the issue of whether the newly proposed concepts can be
efficiently supported. In § 8, we show how to extend logical database design
to include null markers using one of the proposed extensions. Finally, we
close in § 9 with some comments and a discussion of further research.
## 2 Desirable properties
The SQL standard allows null markers, but does not allude to FDs [7]. One
could point out that the concept of _key_ is explicitly present in the SQL
standard; given that the concepts of key and FD are related, it would seem
that we support FDs by enforcing the _key constraint_ [8]. However, when we
enforce FDs by key constraints, we assume that all the tables in the database
have attained Boyce-Codd normal form (BCNF). Yet such a normal form excludes
the commonly used null markers.
Codd was well aware of the perceived problems with null markers. Yet he was
unconcerned by our inability to enforce FDs in the presence of null markers.
He believed that only when the null markers where replaced by actual values
would concepts such as keys, normalization and FDs be applicable:
> It should be clear that, because nulls (or, as they are now called, marks)
> are NOT database values, the rules of functional dependence—and of multi-
> valued dependence—do not apply to them. (E. F. Codd [1])
However, consider the relation in Table 1 subject to the FDs
$\text{professor}\to\text{chair}$ and $\text{department} \to\text{chair}$.
Given that Jill and Arthur are distinct people, it is not possible to replace
the null marker by an actual value. Going back to Codd’s vision, we have that
keys, normalization and FDs are never going to be applicable to all tuples of
such relation. This might reasonably be considered anomalous.
To avoid such problems, we establish as a first goal that if a FD is enforced
in a relation with null markers, it should always be possible to replace some
or all of the null markers by actual values without violating the FD (G1). In
such a case, we say that such FDs are _realizable_ under null markers.
Moreover, and since we consider _sets_ of FDs for design (as opposed to single
FDs in isolation), we posit that FDs should be defined so that if a _set_ of
FDs is enforced in a relation with null markers, it should be possible to
replace some or all of the null markers without violating any of the FDs in
the set (strong G1).
Table 1: Example relation with attributes professor, chair and department professor | chair | department
---|---|---
Joe | null | Mathematics
Joe | Jill | Computer Science
Bill | Arthur | Mathematics
When defining FDs in the presence of null markers, we are also interested in
Armstrong’s three axioms, especially transitivity:
Reflexivity: If $Y\subseteq X$, then $X\to Y$.
Augmentation: we have that $X\to Y$ implies $XZ\to YZ$.
Transitivity: If $X\to Y$ and $Y\to Z$, then $X\to Z$. E.g., we might say that
your social security number (SSN) determines your income
($\text{SSN}\to\text{Income}$), and that your income determines your tax
bracket ($\text{Income}\to\text{Taxation}$). If transitivity holds, then your
SSN determines your tax bracket ($\text{SSN}\to\text{Taxation}$).
Table 2: Example relation with attributes SSN, income and taxation SSN | income | taxation
---|---|---
1112233 | null | 15%
1112233 | null | 25%
Codd’s interpretation, that FD do not apply when there are null markers fails
to enforce transitivity in the following sense: given the FD
$\textrm{SSN}\to\textrm{Income}$ and $\textrm{Income}\to\textrm{Taxation}$,
both tuples in Table 2 are allowable, even though one would expect not to see
such data in the database. It implies that $\textrm{SSN}\to\textrm{Taxation}$
does not hold (whereas it should under transitivity) even though there is no
null marker over attributes SSN and Taxation.
Codd would no doubt reply that there is no violation of transitivity since FDs
do not apply in the presence null markers. But we wish to consider null
markers as an integral part of the database.
Failing to enforce Armstrong’s axioms has significant consequences. For one
thing, without these axioms, normalization is no longer sufficient to enforce
FDs. In practical terms, any redefinition of the FDs that fails to satisfy
Armstrong’s axioms cannot be enforced through normalization. Indeed, given a
database $D$ and a set of FDs ${\cal F}$ on $D$, before we can use ${\cal F}$
to determine normal forms for the relations in $D$, we need to make sure that
${\cal F}$ is in minimal or canonical form [9]. But minimizing ${\cal F}$
depends _crucially_ on FDs respecting transitivity, as covered in standard
database textbooks.
We might be willing to forgo normalization and enforce FDs through other
means. In such a case, it might seem like Armstrong’s axioms are no longer
required. For example, we might think that it is possible to build a database
design without assuming that FDs are transitive. However, such watered-down
FDs might be impractical for other reasons. The first problem that we
encounter is that standard database design methodologies, like the entity-
relationship model, implicitly assume Armstrong’s axioms and transitivity in
particular [10].
We believe that database designers would have a hard time coping with the lack
of transitivity (see, for instance, the example of Table 2, where intuitively
one would expect to see, for the same SSN, the same Taxation), and hence we
require that Armstrong’s axioms hold, even though such a requirement could be
seen as too strong. Of course, we could help designers with additional tools
and methodologies [11] to compensate for the added constraint. Nevertheless,
everything else being equal, we view as desirable that a new definition of FDs
in the presence of null markers should respect Armstrong’s axioms (G2), as
well as enforce realizable null markers.
Of course, our goals so far can be accomplished by being very restrictive on
the use of null markers. One might even take the stance that null markers
should always be forbidden [12]. But we also want to allow common uses seen in
production-quality applications. For example, we have observed that many
database schemas allow null markers on attributes that do not determine other
attributes. Thus, it is another objective (G3) of this research to allow null
markers when this can be done without violating other goals. At a minimum, we
should allow null markers without having to come up with contrived examples.
Finally, any enforcement of FDs is going to be considered, from the point of
view of transaction or query processing, as overhead—just like enforcing
primary and foreign key constraints. Thus, any definition should be
computationally inexpensive (G4). In practice, this means that we exclude
elegant but challenging models such as v-tables [13]. For example, we should
be able to determine whether the FD $X\to Y$ holds by considering _only_ the
attributes in $X$ and $Y$.
To summarize, we seek to extend FDs to include null markers in such a way
that:
G1: FDs enforce realizable null markers. Further, this should hold for sets of
FDs (strong G1).
G2: Armstrong’s axioms are satisfied.
G3: FDs should not restrict the use of null markers unnecessarily.
G4: Enforcing FDs should be computationally practical.
## 3 Basic concepts
Let a _relation_ $r$ be as in SQL: a finite multiset of tuples over a given
schema $\mathop{\mathrm{sch}}(r)$, with the caveat that a tuple may contain
null markers. Two tuples are considered _duplicates_ if all non-null
attributes are equal and any null marker in one tuple is matched by a null
marker in the other tuple; otherwise the tuples are _distinct_.
We assume that there is an infinite set $V$ of values, from where all the
constants in any relation are drawn. These values have a relation ’=’ defined
on them, which is reflexive, symmetric and transitive: given any $x,y,z\in V$,
we have than $x=x$, $(x=y)\Rightarrow(y=x)$ and $(x=y,y=z)\Rightarrow x=z$.
Hence, the relation “=” is an equivalence relation.
Given an attribute $A$ in the schema of relation $r$ and a tuple $t$, we use
$t[A]$ as is customary, to denote the value of $t$ for $A$. This is extended
to sets of attributes $X\subseteq\mathop{\mathrm{sch}}(r)$ as usual. We then
say, for two tuples $t$, $t^{\prime}$, that $t[A]=t^{\prime}[A]$ is _true_ if
both $t[A]$ and $t^{\prime}[A]$ are equal non-null values; _false_ if both
$t[A]$ and $t^{\prime}[A]$ are values, but they are different; and _unknown_
otherwise (i.e., if either one of $t[A]$ or $t^{\prime}[A]$, possibly both,
are null markers). Again, this is extended to sets of attributes
$X\subseteq\mathop{\mathrm{sch}}(r)$ in the usual way: for two tuples $t$,
$t^{\prime}$, $t[X]=t^{\prime}[X]$ is _true_ if $t[A]=t^{\prime}[A]$ is true
for every $A\in X$; _false_ if $t[A]=t^{\prime}[A]$ is false for some $A\in
X$, and _unknown_ otherwise. As a shorthand, we write $t=t^{\prime}$ for
$t[\mathop{\mathrm{sch}}(r)]=t^{\prime}[\mathop{\mathrm{sch}}(r)]$. We write
$\pi_{X}(r)$ for the projection of all tuples in $r$ on $X$: starting from
$\\{t[X]|t\in r\\}$, all duplicates are removed.
Let $r$ be a fixed relation. If we disallow null markers in $r$, then a FD
$X\to Y$ is satisfied if $t[Y]=t^{\prime}[Y]$ when two tuples $t,t^{\prime}$
are such that $t[X]=t^{\prime}[X]$. In this context (where null markers are
forbidden), FDs satisfy Armstrong’s axioms. We can also formalize the concept
of key if there is no null marker in $r$. A set of attributes $K$ is a
_superkey_ iff $K\to A$ holds for any attribute
$A\in\mathop{\mathrm{sch}}(r)$; and a _key_ if it is a minimal superkey.
_Primary keys_ in SQL are keys with attributes where null markers are
forbidden; SQL allows null markers in other keys.
We say that an attribute $A$ is _non- null_ in a relation $r$ if for all $t\in
r$ we have that $t[A]$ is non-null. Given a set of attributes $X$, we say that
the null marker appearing in a tuple $t$ at attribute $A$
($t[A]=\texttt{null}{}$) is _in $X$_ if $A\in X$.
## 4 Strong and weak functional dependencies
Levene and Loizou [14] propose one of the few extensions of FDs over null
markers. We formalize their definitions as follows. A _valuation_ $\varphi$
for a relation $r$ assigns to each null marker in a tuple of $r$ a value from
$V$—each null marker may receive a different value—while leaving non-null
values from $V$ unchanged. Given a relation $r$, each $\varphi(r)$ is called a
_possible world_ for $r$. The semantics of FDs with null markers, as defined
by Levene and Loizou, follows the idea of modal logic [15]: we have two
distinct readings, depending on whether the FD holds in some or all possible
worlds.
###### Definition .
A FD $F$ holds weakly in relation $r$ iff $F$ holds in a possible world for
$r$—i.e., there exists a valuation $\varphi$ such that $F$ holds in
$\varphi(r)$. $F$ is called a _weak FD_ (WFD).
###### Definition .
A FD $F$ holds strongly in relation $r$ iff $F$ holds in all possible worlds
for $r$—i.e., for every valuation $\varphi$ for $r$, $F$ holds in
$\varphi(r)$. $F$ is called a _strong FD_ (SFD).
Looking back at Table 1, consider the following FDs:
$\textrm{chair}\to\textrm{professor}$ and
$\textrm{professor}\to\textrm{chair}$. Both hold weakly, while neither holds
strongly (see Table 3).
Table 3: Various FDs applied to the relation of Table 1 and whether they hold strongly, weakly, super-reflexively or literally | SFD | WFD | SRFD | LFD
---|---|---|---|---
$\textrm{chair}\to\textrm{professor}$ | no | yes | no | yes
$\textrm{professor}\to\textrm{chair}$ | no | yes | yes | no
A strong FD is always also a weak FD. Strong FDs satisfy Armstrong’s axioms
while weak FDs do not. When a FD holds strongly, we can replace any null
marker by a value, and the FD still holds. In fact, strong FDs enforce
realizable null markers.
The Levene and Loizou model has a substantial formal appeal, but it also has
some drawbacks from a pragmatic point of view:
* •
Weak FDs might be too weak. Even if each FD in a set $\mathcal{F}$ of FDs
holds weakly, there might be no single possible world $\varphi(r)$ where all
of the FDs in the set $\mathcal{F}$ hold. See Table 1 for a counterexample:
both FDs ($\textrm{professor}\to\textrm{chair}$ and
$\textrm{chair}\to\textrm{professor}$) hold weakly, but there is no possible
world where they both hold. That is, we can substitute some value for the null
marker to satisfy the first FD, and substitute _another_ value to satisfy the
second FD, but no single value makes both FDs true at the same time (thus,
$\mathcal{F}$ does not have property strong G1). This is true even when each
attribute appears only once on the right-hand-side of a FD: given the schema
$A,B,C$ and the FDs $A\to B$ and $B\to C$, the set of tuples
$(a,\texttt{null}{},b)$ and $(a,\texttt{null}{},c)$ satisfy both FD weakly,
but there is no world where they both hold. This last example also illustrates
that weak FDs are not transitive: $A\to B$ and $B\to C$ can hold weakly while
$A\to C$ may not. That is, weak FDs do not satisfy Armstrong’s axioms (thus
failing G2).
We could _fix_ weak FDs to ensure that they enforce, for example, realizable
null markers by requiring that there exists a valuation corresponding to the
set of all FDs. However, this may prove computationally challenging (hence
failing G4).
* •
Strong FDs might be too strong (thus failing G3). Indeed, it seems unnecessary
to always require that FDs should hold in all possible worlds. For example,
consider the schema $A,B,C$ and the FDs $A\to B$ and $B\to C$, and the set of
tuples $(a,b,\texttt{null}{})$ and $(c,b,\texttt{null}{})$. Though it appears
like a reasonable relation, the FD $B\to C$ fails to hold strongly.
In some sense, the weak and strong FDs are two extremes, whereas the right
solution might be somewhere in between. Indeed, any form of FD that supports
realizable null markers (G1) on a per FD basis, is going to be equivalent to
or stronger than weak FDs. Meanwhile, strong FDs have the properties we seek,
except that they are too restrictive (G3).
## 5 Literal and Super-reflexive functional dependencies
We propose two alternative definitions of the concept of FD in the presence of
null markers. The first one is a natural extension of the 3-value logic
proposed by Codd and used by SQL.
###### Definition .
A FD $X\to Y$ holds super-reflexively if, for any two tuples $t,t^{\prime}$,
when $t[X]=t^{\prime}[X]$ is not false (i.e., it is either true or unknown),
then $t[Y]=t^{\prime}[Y]$ is also not false. We say $X\to Y$ is a _super-
reflexive_ FD (SRFD).
In this first definition, the null marker is effectively equal to any other
value (i.e., $\texttt{null}{}=a$ is treated as true for any value $a$), hence
the term _super-reflexive_ (SR).
As an illustration, consider Table 1. We have that
$\textrm{professor}\to\textrm{chair}$ hold super-reflexively whereas
$\textrm{chair}\to\textrm{professor}$ does not (see Table 3).
Our second definition is reminiscent of how languages such as JavaScript
handle null markers. As a first approximation, they consider null to be
effectively a regular value, with $\texttt{null}{}=\texttt{null}{}$ is always
true (i.e., “=” remains a reflexive relation111In fact, ’=’ remains an
equivalence relation, something that does not hold for Codd’s 3-value logic if
you consider null markers to be part of the value domain.), but
$\texttt{null}{}=a$ always false for $a$ non-null.
We say that $t[A]$ and $t^{\prime}[A]$ are _identical_ if both contain null or
both contain the same value; this is also extended to set of attributes $X$
and to whole tuples as usual: $t[X]$ is identical to $t^{\prime}[X]$ if and
only if $t[A]$ and $t^{\prime}[A]$ are identical for all $A\in X$.
###### Definition .
A FD $X\to Y$ holds literally if, for any two tuples $t,t^{\prime}$, when
$t[X]$ and $t^{\prime}[X]$ are identical then $t[Y]$ and $t^{\prime}[Y]$ are
also identical. We say $X\to Y$ is a _literal_ FD (LFD).
Consider again Table 1. In contrast with the super-reflexive case, we have
that the FDs $\textrm{chair}\to\textrm{professor}$ holds literally whereas
$\textrm{professor}\to\textrm{chair}$ does not. (See again Table 3.)
There are alternative definitions that we could have used. For example, we
could have defined FDs $X\to Y$ to hold if when $t[X]=t^{\prime}[X]$ is true
(as per Codd’s 3-value logic) then $t[Y]=t^{\prime}[Y]$ must be true. Or, we
could have defined an FD $X\to Y$ to hold if when $t[X]=t^{\prime}[X]$ is not
false then $t[Y]=t^{\prime}[Y]$ must be true; or if when $t[X]=t^{\prime}[X]$
is true then $t[Y]=t^{\prime}[Y]$ must be not false. However, these
alternative definitions are unsatisfying: they fail to satisfy Armstrong’s
axioms (G2), or do not allow null markers where they are commonly used (G4).
By contrast, Definitions 5.1 and 5.2 have the properties that we required in §
2 starting with Armstrong’s axioms (which follows by inspection). For example,
regarding transitivity, consider the two FDs $X\to Y$ and $Y\to Z$ that hold
super-reflexively (resp. literally). Given two tuples $t,t^{\prime}$ such that
$t[X]=t^{\prime}[X]$ is not false (resp. $t[X]$ and $t^{\prime}[X]$ are
identical), then $t[Y]=t^{\prime}[Y]$ is not false (resp. $t[Y]$ and
$t^{\prime}[Y]$ are identical) which implies that $t[Z]=t^{\prime}[Z]$ is not
false (resp. $t[Z]$ and $t^{\prime}[Z]$ are identical). Thus we have that
$X\to Z$, proving transitivity.
###### Lemma .
Super-reflexive and literal FDs respect Armstrong’s axioms (G2).
Before we proceed to establish other properties, we need a technical result
that makes other proofs easier. First, we can verify that all FDs can be
decomposed into FDs where the right-hand-side contains a single attribute. For
example, the FD
$\textrm{professor}\to\\{\textrm{chair},\textrm{department}\\}$ is equivalent
to the following two FDs:
* •
$\textrm{professor}\to\textrm{chair}$, and
* •
$\textrm{professor}\to\textrm{department}$.
###### Lemma .
We have that $X\to Y$ for $Y=\\{A_{1},\ldots A_{n}\\}$ is equivalent to
$(X-\\{A_{1}\\}\to\\{A_{1}\\})\land\cdots\land(X-\\{A_{n}\\}\to\\{A_{n}\\})$
whether we consider weak, strong, literal or super-reflexive FDs.
The proof of this lemma follows by inspection. Hence, it is enough to consider
FDs $X\to Y$ where $Y$ is a singleton disjoint from $X$.
First, we must show that SRFDs and LFDs satisfy condition G1: null markers are
realizable. We begin with SRFDs. We show that given the SRFD $X\to Y$, all
null markers in $X\cup Y$ are realizable with respect to $X\to Y$. To
illustrate this result, consider Table 1 where $\textrm{professor}
\to\textrm{chair}$ holds super-reflexively: we can substitute Jill for the
null marker without violating the FD.
###### Lemma .
Consider a SRFD $X\to Y$ in a relation such that $X$ and $Y$ are disjoint and
$Y$ is a singleton.
1. 1.
We can replace any null marker in $X$ by any actual value without violating
the SRFD $X\to Y$.
2. 2.
We can replace any null marker in $r$ in $Y$ by at least one actual value
without violating the SRFD $X\to Y$, assuming that attributes in $X$ are non-
null.
###### Proof .
Assume that the SRFD is initially satisfied over $r$.
(1) Suppose we replace a null marker in attribute $B\in X$. Regarding the SRFD
$X\to Y$, the following might happen for two tuples $t,t^{\prime}$:
* •
if $t[X]=t^{\prime}[X]$ was false, then it is still false: this cannot affect
the SRFD;
* •
if $t[X]=t^{\prime}[X]$ was true, then it is still true: this cannot affect
the SRFD;
* •
if $t[X]=t^{\prime}[X]$ was unknown, then it might become true or false.
Because of the SRFD $X\to Y$, and because $t[X]=t^{\prime}[X]$ was not false,
we have that $t[Y]=t^{\prime}[Y]$ is not false. Therefore, if the tuples
$t,t^{\prime}$ satisfied the SRFD before the update, they must satisfy it
after the update as well.
Because all pairs of tuples satisfy the conditions of SRFD after the update,
the SRFD still hold, proving the first part of the result.
(2) Assume that attributes in $X$ are non-null. We want to show that we can
replace any null marker in $Y$ by an actual value. Pick a tuple $t$ in $r$
where $t[Y]$ contains a null marker. Consider the set $\tau$ of $t^{\prime}$
such that $t^{\prime}[X]=t[X]$ is not false. (Because attributes in $X$ are
non-null, we have that “$t^{\prime}[X]=t[X]$ is not false” is equivalent to
“$t^{\prime}[X]=t[X]$ is true”.) We have that the projection of $\tau$ over
$Y$ contains at most one actual value. (Suppose it does not, then you can find
tuples $t^{\prime\prime}$ and $t^{\prime\prime\prime}$ in $\tau$ such that
$t^{\prime\prime}[X]=t^{\prime\prime\prime}[X]$ is not false but
$t^{\prime\prime}[Y]=t^{\prime\prime\prime}[Y]$ is false.) If there is one
actual value, set $t[Y]$ to this value; if not, pick a value at random. This
modification clearly does not violate the SRFD $X\to Y$ but it eliminates one
null marker.
To see why Lemma 5.5 implies that SRFDs satisfy G1, consider any SRFD $X\to Y$
over a given relation. We can substitute actual values for any null marker in
an attribute of $X$ by the first part of the lemma. As a second step, since
attributes in $X$ have become non-null, we can substitute actual values for
any null marker in $Y$.
As for LFDs, we are going to prove something stronger: that they support
strong G1.
###### Lemma .
LFDs strongly enforces realizable null markers (strong G1).
###### Proof .
To prove G1, it suffices to replace all the null markers with a single $v\in
V$ not already in the relation. By inspection, this extends to sets of FDs, so
we get also strong G1 with this method.
We have that both LFDs and SRFDs enforce realizable null markers. That is,
given a relation with a set of FDs, we can always replace null markers with
some actual values without violating the FDs. In fact, if we add an extra
constraint on SRFDs, they both _strongly_ enforce realizable null markers (in
the sense of strong G1). In this context, we adopt the practical convention
that some attributes are allowed to contain null markers while others may not:
this is motivated by the SQL standard. Given a set of FDs $\mathcal{F}$, we
say that an attribute $B$ _determines_ another attribute $A$ under
$\mathcal{F}$ if there is a FD $B\in X\to Y\ni A$ in the transitive closure of
$\mathcal{F}$. Naturally, this property is transitive: if $A$ determines $B$
and $B$ determines $C$ then $A$ determines $C$. (By convention, we omit loops
in $\mathcal{F}$: $X\to X$.)
###### Condition 1RHS.
Consider a set of FDs $\mathcal{F}$ over a relation. Consider any attribute
$A$ allowed to contain null markers. Then $A$ must appear on the right-hand-
side of at most one FD in the set of FDs $\mathcal{F}$ of the relation.
Moreover, given two distinct attributes allowed to contain null marker, $A$
and $B$, if $A$ determines $B$, $B$ cannot determine $A$.
We stress that Condition 1RHS only applies to attributes allowed to contain
null markers: no constraint is required on other attributes.
Fig. 1 gives an example of a set of FDs satisfying the condition 1RHS even if
all attributes are allowed to contain null markers:
$\\{E,D\\}\to\\{A\\},\\{A\\}\to\\{F\\},\\{A,B\\}\to\\{F\\}$. However, if we
replaced the single FD $\\{E,D\\}\to\\{A\\}$ by two FDs such as
$\\{E\\}\to\\{A\\}$ and $\\{D\\}\to\\{A\\}$, we would need to add the
requirement that $A$ is non-null to satisfy 1RHS. Similarly, if we added the
FD $\\{F\\}\to\\{A\\}$ in addition to the existing FD $\\{A\\}\to\\{F\\}$, we
would need to require that both $A$ and $F$ are non-null since $F$ would
determine $A$ while $A$ determines $F$.
###### Lemma .
SRFDs strongly enforces realizable null markers whenever the 1RHS condition is
satisfied (strong G1).
###### Proof .
Assume without loss of generality that all SRFDs in the set are of the form
$X\to Y$ where $Y$ is a singleton and $X,Y$ are disjoint.
$A$$B$$C$$D$$E$$F$$\\{A\\}\to\\{F\\}$$\\{A,B\\}\to\\{F\\}$$\\{E,D\\}\to\\{A\\}$$\\{E,D\\}\to\\{A\\}$$\\{A\\}\to\\{F\\}$
Figure 1: Illustration used by the proof of Lemma 5.9 for the set of FDs
$\\{\\{E,D\\}\to\\{A\\},\\{A\\}\to\\{F\\},\\{A,B\\}\to\\{F\\}\\}$.
Construct a graph where each attribute allowed to contain null markers is a
node, and there is an edge between two attributes $A,B$ if and only if $A$
determines $B$. We illustrate such a graph in Fig. 1 where, for simplicity, we
omitted some of the edges that are implied by transitivity.
Temporarily assume that the graph is not empty. Because of condition 1RHS, the
graph must be cycle-free and, therefore, some of the nodes must have a zero
in-degree (e.g., $E$ and $D$ in Fig. 1). Call this set of nodes/attributes
$\mathcal{A}_{0}$.
As per Lemma 5.5, we can substitute actual values for any null marker they
contain. Indeed, consider such an attribute $A\in\mathcal{A}_{0}$. This
attribute appears on the right-hand-side of at most one FD in the set (call it
$F_{A}$), however $A$ may appear on the left-hand-side of several FDs. These
FDs are not a concern: replacing a null marker in attribute $A$ may never
violate a super-reflexive FD as per the first part of Lemma 5.5. Meanwhile,
because $F_{A}$ is such that no attribute on its left-hand-side contains a
null marker, then the second part of Lemma 5.5 tells us that the null markers
of $A$ are realizable.
After substituting actual values for any null marker in the attributes of
$\mathcal{A}_{0}$, remove these nodes from the graph. There must again be
nodes with zero in-degree (e.g., node $A$ in Fig. 1) or the graph is empty.
Repeat the process until the graph is empty.
Attributes that either do not appear as part of any FD, or that are not
allowed to contain null markers, are not a concern.
## 6 Comparing functional dependencies
We are now in a position to characterize LFDs and SRFDs properly. We can
relate LFDs and SRFDs to each other, and with Levene and Loizou’s definitions.
All are _conservative_ extensions of the classical concept: in a table without
any null, they coincide with classical FDs. However, in general, LFDs and
SRFDs are _incomparable_ as illustrated by Table 3.
We show that $\mbox{strong}\Rightarrow\mbox{super-
reflexive}\Rightarrow\mbox{weak}$ and
$\mbox{strong}\Rightarrow\mbox{literal}\Rightarrow\mbox{weak}$ (see Fig. 2).
That is, both LFDs and SRFDs are stronger than weak FDs, whereas strong FDs
are stronger than both LFDs and SRFDs.
Figure 2: Venn diagram illustrating Lemma 6.1.
###### Lemma .
The following holds:
1. 1.
If the FD $X\to Y$ holds literally, then it holds weakly.
2. 2.
If the FD $X\to Y$ holds super-reflexively, then it holds weakly.
3. 3.
If the FD $X\to Y$ holds strongly then it must hold literally and super-
reflexively.
###### Proof .
Assume without loss of generality that $X$ and $Y$ are disjoint and that $Y$
is a singleton.
(1) If the FD $X\to Y$ holds literally then we can replace any null marker by
any one $v\in V$ not already present in the relation, and the FD still holds
by inspection. This shows that literal FDs are stronger than weak FDs.
(2) By Lemma 5.5, we can construct a valuation such that the super-reflexive
FD is valid in the conventional sense. Because of the existence of the
valuation, we have that $X\to Y$ holds weakly.
(3) We first prove that strong implies super-reflexive. Suppose that $X\to Y$
holds strongly. Consider two tuples $t,t^{\prime}$. If $t[X]=t^{\prime}[X]$ is
not false (in Codd’s 3-value logic), then in some possible world,
$t[X]=t^{\prime}[X]$ must hold. In such a possible world $t[Y]=t^{\prime}[Y]$
must hold which implies that $t[Y]=t^{\prime}[Y]$ must be not false (in Codd’s
3-value logic). This prove that a strong FD implies a super-reflexive FD.
We prove that strong implies literal. Suppose that the FD $X\to Y$ holds
strongly and that $t[X]$ and $t^{\prime}[X]$ are identical. (We assume that
$X$ and $Y$ are disjoint and $Y$ is a singleton as previously stated.) There
are some worlds where $t[X]=t^{\prime}[X]$ is true. If $t[Y]$ and
$t^{\prime}[Y]$ are not identical, then in at least one of these worlds, they
would differ. That is, if one has a null marker in $t[Y]$ and a value $a$ in
$t^{\prime}[Y]$ (or vice versa) we can replace the null marker by a value that
differs from $a$. Hence, we must have that $t[Y]$ and $t^{\prime}[Y]$ are
identical given that $t[X]$ and $t^{\prime}[X]$ are identical. Therefore
strong FDs imply literal FDs. This completes the proof.
We can verify that LFDs and SRFDs are strictly weaker than SFD. Recall that
given the schema $A,B,C$ and the FDs $A\to B$ and $B\to C$, the set of tuples
$(a,b,\texttt{null}{})$ and $(c,b,\texttt{null}{})$ violates the strong FD
$B\to C$. However, it satisfies the FD $A\to B$ and $B\to C$ literally and
super-reflexively. The fact that LFDs and SRFDs allow this use of null markers
support our claim that they do not unnecessarily forbid null markers where
they might make sense (G3).
## 7 Computational efficiency
The remaining question is the efficient implementation of our proposed
concepts (G4). Here we show that both LFDs and SRFDs can be enforced with a
cost similar to enforcing _regular_ FDs.
Since SQL does not enforce FDs directly, we compare with a closely associated
property, the cost of enforcing the key constraint in a relation. This assumes
that the database is in an appropriate normal form such that enforcing keys is
equivalent to enforcing FDs. Given relation $r$, if $X\subseteq r$ is declared
as the primary key for $r$, any insertion $t\in r$ is checked to make sure
that $X$ remains a key. In practice, relational database systems forbid the
occurrence of two distinct tuples $t,t^{\prime}$ agreeing on $X$
($t[X]=t^{\prime}[X]$ must be false). This can be achieved by creating an
index (traditionally, a tree-based index) on $X$: on inserting $t$, we search
for any $t^{\prime}\in r$ with $t[X]=t^{\prime}[X]$ (note that null markers
are forbidden in a primary key). Call the set of tuples resulting from the
search $S$. When inserting, if $S=\emptyset$, the insertion can proceed; else,
for each $t^{\prime}\in S$ we check whether $t^{\prime}$ is identical to $t$.
If this is so, the insertion can proceed; else it is forbidden. Updates are
handled in a similar manner. The complexity of this procedure is considered
$\mathcal{O}(\log\mid r\mid)$ when using a tree-based index, since it is
expected that $\mid S\mid\leq 1$. (Of course, we can get an expected constant
time complexity by using a hash-based index.)
We can also enforce FDs directly, whether they are literal or super-reflexive.
Given relation $r$ and arbitrary FD $X\to Y$ on it, we create an index on
$X$—for concreteness, we assume a $B^{*}$-tree index since these are available
on almost any database system.
To enforce FDs with an index, it suffices to check whether the insertion of a
new tuple $t$ is allowed. Indeed, an update can be viewed as a deletion
followed by an insertion, and deletions cannot violate a FD.
Given an index, enforcing LFDs is not difficult. We build a single index on
$X$ by considering a concatenation of all attributes—lexicographic order is
followed, with null markers in individual attributes kept together at the
beginning or end of their positions. That is, in an index for attributes
$ABC$, tuple $(a_{1},b_{1},\texttt{null}{})$ would go before or after all
tuples $(a_{1},b_{1},c)$, for $c$ any value of $C$; tuple
$(a_{1},\texttt{null}{},b_{1})$ would go before or after any tuples
$(a_{1},c,b_{1})$ for $c$ any value of $C$; and so on. This can be achieved by
considering null markers as either strictly larger than any other value, or
strictly smaller. Suppose that we insert a new tuple $t$. We can find in time
$\mathcal{O}(\mid S\mid+\log\mid r\mid)$ the set $S$ of all tuples
$t^{\prime}$ such that $t[X]$ and $t^{\prime}[X]$ are identical:
$\displaystyle S=\bigcap_{A\in X}\\{t^{\prime}\in r|t^{\prime}[A]=t[A]
\enskip\lor\enskip\text{$t^{\prime}[A]$ and $t[A]$ are {null}{}}\\}$
We can then check whether $t[Y]$ and $t^{\prime}[Y]$ are identical for all
$t^{\prime}$ in linear time $\mathcal{O}(\mid S\mid)$. Note that we consider
the cardinality of the set $X$ ($\mid X\mid$) to be a small constant.
We can also enforce $X\to Y$ as an SRFD efficiently, though the total cost is
probably larger. Index each one of the attribute $A\in X$ using, as before,
the convention that y considering null markers as either strictly larger than
any other value, or strictly smaller. Given $t$, first we check whether $t[Y]$
contains a null marker (assume $Y$ is a singleton disjoint from $X$), in which
case no work is needed. Otherwise, we find all $t^{\prime}$ such that “$t[X]
=t^{\prime}[X]$ is not false” by computing
$\displaystyle S=\bigcap_{A\in X|t[A]\text{~{}not {null}{}}}\\{t^{\prime}\in
r|t^{\prime}[A]=t[A] \lor t^{\prime}[A]\text{~{}is {null}{}}\\}$
with the convention that the result is $r$ if $t[X]$ only contains null
markers. Computing the intersection $S$ between several sets
$S_{1},S_{2},\ldots,S_{\mid X\mid}$ requires only complexity
$\mathcal{O}(\sum_{i=1}^{\mid X\mid}|S_{i}|+\mid S\mid)$ or better [16]. Each
set $S_{i}$ can be generated in time $\mathcal{O}(\mid S_{i}\mid+\log\mid
r\mid)$ for an overall complexity of $\mathcal{O}(\sum_{i=1}^{\mid
X\mid}|S_{i}|+\mid S\mid+\log\mid r\mid)$. We then consider $t^{\prime}[Y]$
for all $t^{\prime}\in S$: if there is an actual value, check that $t[Y]$ is
equal to $t^{\prime}[Y]$. That is, we return
$\displaystyle\bigwedge_{t^{\prime}\in S}t^{\prime}[Y]\
=t[Y]\enskip\lor\enskip t^{\prime}[Y]\text{~{}is {null}{}}.$
This can be computed in time $\mathcal{O}(\mid S\mid)$.
To sum up, enforcing a LFD or SRFD under an update or insertion can be done in
time
* •
$\mathcal{O}(\mid S\mid+\log\mid r\mid)$ or
* •
$\mathcal{O}(\sum_{i=1}^{\mid X\mid}|S_{i}|+\mid S\mid+\log\mid r\mid)$.
Our sketched implementations only require tree look-ups and set intersections,
both of which are well supported by all database systems. This is sufficient
to conclude that they are computationally practical (G4).
## 8 Extending logical database design to include null markers
As we discussed in § 5, it is possible to enforce FDs with null markers using
either SRFDs or LFDs—without any particular effort on the part of the database
designer. However, we commonly enforce FDs using logical database design. That
is, we decompose relations into normal forms and identify keys.
We would like to remain as close as possible to the spirit of SQL. Thus, we
ask whether we can use logical design with SRFDs and LFDs. This would allow an
extension of logical database design to include null markers. Unfortunately,
while both LFDs and SRFDs fulfill all our desiderata (G1 to G4), SRFDs are not
compatible with conventional logical design. But we have better luck with
LFDs.
Recall that in a relation $r_{1}$, a set of attributes
$X\subseteq\mathop{\mathrm{sch}}(r_{1})$ is a _key_ if the (standard) FD
$X\to\mathop{\mathrm{sch}}(r_{1})$ holds and if $X$ is minimal. Moreover, in
relation $r_{2}$, a set of attributes $Y\subseteq\mathop{\mathrm{sch}}(r_{2})$
is a _foreign key_ for $r_{1}$ if $\pi_{Y}(r_{2})\subseteq\pi_{X}(r_{1})$ for
all extensions of $r_{1}$ and $r_{2}$.
A join $r=r_{1}\Join_{X=Y}r_{2}$ is a new relation made by combining all
tuples $t_{1}\in r_{1}$ and all tuples $t_{2}\in r_{2}$ such that
$t_{1}[X]=t_{2}[Y]$ into a new tuple $t$ equal to $t_{1}$ on
$\mathop{\mathrm{sch}}(r_{1})$ and equal to $t_{2}$ on
$\mathop{\mathrm{sch}}(r_{2})-Y$. A join $r=r_{1}\Join_{X=Y}r_{2}$ is lossless
when $\pi_{\mathop{\mathrm{sch}}(r_{1})}(r)=r_{1}$ and
$\pi_{\mathop{\mathrm{sch}}(r_{2})}(r)=r_{2}$.
We extend the concepts of keys and joins in the context of LFDs as follows.
* •
We say that $X\subseteq\mathop{\mathrm{sch}}(r)$ is a _literal superkey_ for
$r$ if $X\to Y$ holds literally for any $Y\subseteq\mathop{\mathrm{sch}}(r)$
and a _literal key_ iff it is a minimal literal superkey. A _literal foreign
key_ is an integrity constraint between two relations: a set of attributes $X$
in relation $r_{1}$ must match a set of attributes $X$ in a relation $r_{2}$
such that for every tuple $t$ in $r_{1}$, there must be a tuple $t^{\prime}$
in $r_{2}$ such that $t[X]$ and $t^{\prime}[X]$ are identical and such that
$X$ contains a literal key in $r_{2}$.
* •
As in SQL, each literal foreign key constraint supports a corresponding join.
The literal join of $r_{1}$ and $r_{2}$ on $X$, a foreign key in $r_{1}$,
noted $\widehat{\Join}{}$ is defined as follows: given any two tuples
$t_{1},t_{2}$ from $r_{1},r_{2}$ such that $t_{1}[X]$ and $t_{2}[X]$ are
identical, we generate the tuple $t$ such that $t[A]=t_{1}[A]$ for all
$A\in\mathop{\mathrm{sch}}(r_{1})$ and $t[A]=t_{2}[A]$ for all
$A\in\mathop{\mathrm{sch}}(r_{2})-\mathop{\mathrm{sch}}(r1)$.
With these definitions, we have that lossless joins are supported, even with
null markers. Formally, given relation $r$ with
$\mathop{\mathrm{sch}}(r)=Z\cup W$ and such that $Z\cap W\to W$ holds
literally, $\widehat{\Join}{}$ is a _lossless join_ :
$r=\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$.
###### Proposition .
(Lossless join) If $\mathop{\mathrm{sch}}(r)=Z\cup W$ and $Z\cap W\to W$ holds
literally then $r=\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$.
###### Proof .
For the purpose of the literal join, the null marker can be treated like any
other value. That is, the set of values $V$ extended with the null marker is
effectively reflexive, symmetric and transitive. To conclude the proof, we
have to show that $r=\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$.
1. 1.
Suppose that $t\in r$. There will be a tuple $t^{(Z)}$ in $\pi_{Z}(r)$ such
that $t[Z]$ and $t^{(Z)}$ are identical. Similarly, there will be a tuple
$t^{(W)}$ in $\pi_{W}(r)$ such that $t[W]$ and $t^{(W)}$ are identical. We
have that $t^{(W)}[Z\cap W]$ is identical to $t^{(Z)}[Z\cap W]$. Thus we have
that $t\in\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$.
2. 2.
Suppose that $t\in\pi_{Z}(r)\widehat{\Join}{}\pi_{W}(r)$. There must be
$t^{\prime}\in r$ such that $t^{\prime}[Z]=t[Z]$. We have that $t[Z\cap W]$
and $t^{\prime}[Z\cap W]$ must be identical because $Z\cap W\subset Z$.
Because $Z\cap W\to W$, we have that $t[W]$ and $t^{\prime}[W]$ are identical.
Thus we have that $t[Z\cup W]$ and $t^{\prime}[Z\cup W]$ which shows that
$t\in r$.
This concludes the proof.
In fact, we can show that logical design is sound over LFDs: as long as we can
normalize a relation in the conventional sense, then we can normalize it in
the sense of LFDs.
## 9 Conclusion and Further Research
We have reviewed the concept of FD in the presence of null markers, and have
specified a set of properties that we believe any definition of the concept
should satisfy. We have proposed two new definitions of what it means for an
FD to hold in this situation for which the properties do hold: our definitions
satisfy Armstrong’s axioms for relations with null markers, allow null markers
to be used in practice (not only with contrived examples), and at the same
time allows those null markers to be updated to real values consistently.
These FDs can also be enforced efficiently in computational terms despite null
markers. Both definitions have slightly different properties (LFDs enforce
lossless join, in addition to database consistency), but they both satisfy our
axioms.
Clearly, our requirements are tied to our goal of obtaining a definition that
can be used in practical situations. An open question is whether the
properties put forth here as desirable are the only ones—or at least the
_important_ ones, in some sense of _important_. We believe that our properties
satisfy intuitions that make the concept of FD usable in practical situations.
However, additional properties should be explored to get a better
understanding of the _desired_ or _expected_ behavior of FDs in the presence
of null markers; perhaps a set of alternative properties, each giving raise to
a concept of FD, can be developed for different scenarios. Agreeing on some
set or sets of basic requirements would provide researchers with an explicit
milestone by which to judge different formalizations.
A narrower question is whether the definitions proposed here are the only ones
that can satisfy all the requirements put forth, or whether alternatives
exist. While the authors have considered many alternative definitions, and
found them missing some requirement, it is not known whether other definitions
could exist that would still satisfy all properties, and if so, what would the
relationships between different definitions be.
## 10 Funding
This work was supported by the Natural Sciences and Engineering Research
Council of Canada [26143].
## 11 Acknowledgements
We thank Carles Farré and Andre Vellino for their helpful comments.
## References
* [1] Codd, E. F. (1986) Missing information (applicable and inapplicable) in relational databases. SIGMOD Rec., 15, 53–53.
* [2] Zaniolo, C. (1984) Database relations with null values. J. Comput. Syst. Sci., 28, 142–166.
* [3] Atzeni, P. and Morfuni, N. M. (1984) Functional dependencies in relations with null values. Inf. Process. Lett., 18, 233–238.
* [4] Hartmann, S. and Link, S. (2012) The implication problem of data dependencies over SQL table definitions: Axiomatic, algorithmic and logical characterizations. ACM Trans. Database Syst., 37, 13:1–13:40.
* [5] Libkin, L. (1994) Aspects of partial information in databases. PhD thesis University of Pennsylvania.
* [6] Badia, A. and Lemire, D. (2011) A call to arms: revisiting database design. SIGMOD Rec., 40, 61–69.
* [7] ISO 9075-1:2008 (2008) Information technology: database languages – SQL– Part 1 Framework, 3rd edition. ISO, Geneva.
* [8] Hartmann, S., Leck, U., and Link, S. (2011) On Codd families of keys over incomplete relations. Comput. J., 54, 1166–1180.
* [9] Bernstein, P. (1976) Synthesizing third normal form relations from functional dependencies. ACM Trans. Database Syst., 1, 277–298.
* [10] Chen, P. P.-S. (1976) The entity-relationship model—toward a unified view of data. ACM Trans. Database Syst., 1, 9–36.
* [11] Le, V., Link, S., and Ferrarotti, F. (2013) Effective recognition and visualization of semantic requirements by perfect sql samples. Conceptual Modeling, Lecture Notes in Computer Science, 8217, pp. 227–240. Springer, Berlin Heidelberg.
* [12] Date, C. J. (2009) SQL and Relational Theory: How to Write Accurate SQL Code. O’Reilly Media, Inc., Sebastopol, California.
* [13] Imieliński, T. and Lipski, W., Jr. (1984) Incomplete information in relational databases. J. ACM, 31, 761–791.
* [14] Levene, M. and Loizou, G. (1998) Axiomatisation of functional dependencies in incomplete relations. Theor. Comput. Sci., 206, 283–300.
* [15] Chellas, B. F. (1980) Modal logic: an introduction. Cambridge Univ Press, Cambridge.
* [16] Ding, B. and König, A. C. (2011) Fast set intersection in memory. Proc. VLDB Endow., 4, 255–266.
|
arxiv-papers
| 2014-04-19T15:46:01 |
2024-09-04T02:50:01.498490
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Antonio Badia and Daniel Lemire",
"submitter": "Daniel Lemire",
"url": "https://arxiv.org/abs/1404.4963"
}
|
1404.5002
|
# A Geometric Distance Oracle for Large Real-World Graphs
Deepak Ajwani, W. Sean Kennedy, Alessandra Sala∗, Iraj Saniee† Bell Labs,
Alcatel-Lucent, Dublin, IrelandBell Labs, Alcatel-Lucent, Murray Hill, NJ, USA
###### Abstract
Many graph processing algorithms require determination of shortest-path
distances between arbitrary numbers of node pairs. Since computation of exact
distances between all node-pairs of a large graph, e.g., 10M nodes and up, is
prohibitively expensive both in computational time and storage space, distance
approximation is often used in place of exact computation. A distance oracle
is a data structure that answers inter-point distance queries more efficiently
than in standard $O(n^{2})$ in time or storage space for an $n$ node graph,
e.g., in $O(n\log{n})$. In this paper, we present a novel and scalable
distance oracle that leverages the hyperbolic core of real-world large graphs
for fast and scalable distance approximation via spanning trees. We show
empirically that the proposed oracle significantly outperforms prior oracles
on a random set of test cases drawn from public domain graph libraries. There
are two sets of prior work against which we benchmark our approach. The first
set, which often outperforms all other oracles, employs embedding of the graph
into low dimensional Euclidean spaces with carefully constructed hyperbolic
distances, but provides no guarantees on the distance estimation error. The
second set leverages Gromov-type tree contraction of the graph with the
additive error guaranteed not to exceed $2\delta\log{n}$, where $\delta$ is
the hyperbolic constant of the graph. We show that our proposed oracle 1) is
significantly faster than those oracles that use hyperbolic embedding (first
set) with similar approximation error and, perhaps surprisingly, 2) exhibits
substantially lower average estimation error compared to Gromov-like tree
contractions (second set). We substantiate our claims through numerical
computations on a collection of a dozen real world networks and synthetic test
cases from multiple domains, ranging in size from 10s of thousand to 10s of
millions of nodes.
## 1 Introduction
The explosion of available information in the past decade, in part due to the
rapid shift towards online media and interactions has led many research,
business and marketing communities to store and analyze very large data sets.
Mining these data sets promises to reveal a wealth of information about the
interests of and the kind of interactions between subscribers, groups, people,
objects and even ideas. These interactions are often naturally represented by
graphs, where for example, nodes correspond to the entities of interest and
the (weighted) links represent the strength of the interaction between them.
Graphs extracted for data mining are often massive, comprising of millions to
billions of connections. At this scale, graph algorithms requiring
$\Omega(n^{2})$ computational steps or storage reach their useful limit in
terms of run time and memory requirements. There is clearly a need for
implementations of graph computational primitives at this scale.
Computing shortest path distance between arbitrary nodes of a graph is among
such fundamental computational primitives. Many data mining schemes invoke
this computational primitive in the scale of the number of nodes, and
therefore it is imperative that this computation can be carried out very
rapidly and with limited memory consumption. Distance oracles are among the
many approaches that have been proposed and used for simplification of
shortest distance computation for large-scale graphs. A distance oracle
involves an auxiliary data structure which is cheaper to compute and fast to
query. It should ideally satisfy the following four properties:
1. 1.
(Initial Processing Speed) the computation involved in the creation of the
auxiliary data structure should be scalable, e.g. be $O(n)$ or $O(n\log{n})$
(and not $O(n^{2})$ or more complex),
2. 2.
(Storage) the auxiliary data structure should be represented in much smaller
space (storage or memory) compared to storing shortest path lengths between
all node pairs,
3. 3.
(Fidelity) path length (estimation) queries using the auxiliary data-structure
should return distances which are as close as possible (if not equal) to the
actual distances,
4. 4.
(Query Speed) the time required to query the distance between any two nodes
should be very small (e.g., small fraction of a second).
In this paper, we focus on a distance oracle for the large- scale graphs that
models interactions arising naturally in the real-world, as in online social
networks, call graph, co-authorship, citations, hyperlinks in world wide web
and similar graphs. We refer to these graphs as real-world graphs. In recent
years, considerable effort has been expended in developing approximate
distance oracles on these graphs, e.g., see [40, 37, 32, 39, 10, 5, 31],
however these heuristics lack a theoretical foundation that would explain
their observed accuracy in practice. On the other hand, there are theoretical
results [19, 12, 14, 11, 1, 20, 13] which provide guaranteed approximation
bounds for specific graph classes. However, the accuracy of many of these
methods has not been empirically evaluated on real-world graphs.
Our Contribution. We present a novel distance approximation oracle that
leverages the notion of graph hyperbolicity [20] observed in real-world
networks [27, 26, 22, 15] and specifically uses the ‘hyperbolic core’ of the
graph [27, 21] for a spanning tree approximation. Hyperbolicity captures the
geometric notion of negative curvature in smooth geometry, which we formally
define in Section 3 in the context of a graph; intuitively, and crucially, as
observed recently [27, 21], it expresses the case that a fixed fraction
$\Theta(n^{2})$ of all shortest paths traverse a small set of nodes in the
graph, thus relative to this small set of nodes, the graph is tree-like in
some fundamental ways, a property which we exploit. Our approach also bridges
the gap between a) the theoretical understanding of tree approximations for
hyperbolic graphs [20, 19, 12, 14, 11, 1, 13] and b) the recent practical
distance approximation solutions [40, 37, 32, 39, 10, 5, 31] which exhibit
high accuracy.
Our approach is distinct from both sets of prior work since we use neither 1)
any kind of hyperbolic embedding [40, 39], nor 2) any form of Gromov-type tree
contraction and labeling schemes [12, 19, 14]. We construct a breadth-first
search spanning tree prioritized on the nodal betweenness centrality and its
approximation by nodal degree, as detailed in Section 4.1. The height of these
trees is almost always $O(\log{n})$111This is due to the well-cited small
world property of real-world graphs where the diameter is observed to scale
like $O(\log{n})$ or even faster (unless the graph has a dominant line in it).
and we use this fact to encode distances in trees with $O(n\log{n})$ bits (or
$O(n)$ words) to support $O(\log{n})$ distance queries. Thus, our proposed
oracle 1) creates tree approximations rapidly, 2) uses $O(n)$ words of space
for storage for an $n$-node graph, 3) has high fidelity and low distortion
(due to the tree root being in the hyperbolic core) and 4) returns queries
very rapidly, meeting the key criteria we set out for an effective oracle. We
also demonstrate empirically that the accuracy of the proposed oracle on large
real-world graphs is significantly better than the theoretical bounds and
competitive with the best practical solutions.
Empirical and Synthetic Datasets. Our empirical comparison is carried out on
a wide range of real-world and synthetic graphs, representing a variety of
different topological structures from the irregular connectivity of small
world graphs to the geometric symmetry and regularity of grid and hypergrid
graphs. Specifically, we experiment on two online social graphs from Facebook,
i.e., Santa Barbara and New York (from [38]), two call graphs from anonymous
telecom operators, two collaboration networks [24], a Google news graph, a
Peer-to-Peer network, a web graph, and two synthetic networks, that we call
FlatGrid and HyperGrid. FlatGrid is a square lattice. HyperGrid is a bona fide
hyperbolic locally planar graph with degree 7 for interior nodes, lower
degrees for boundary nodes, and triangular faces. In Table 1 we summarize the
topological and geometric characteristics of these datasets.
Outline of Paper Section 2 summarizes the relevant prior work on distance
oracles. In Section 3, we describe the notion of graph hyperbolicity that is
at the core of our proposed geometric oracle. Next, in Section 4, we describe
the proposed distance approximation oracle and also present various
alternative oracles for benchmarking. Subsequently in Section 5, we describe
the experimental methodology and summarize our results.
Table 1: $9$ Real and $2$ Synthetic Benchmark Graphs Dataset | #Nodes | #Edges | Avg. | Topological
---|---|---|---|---
| | Deg. | Deg. | Structure
GoogleNews | 15.8K | 164.1K | 21 | News
AstroPhysics | 18.7K | 216.8K | 23 | Collab.
Facebook SB | 26.6K | 968K | 73 | Social
Gnutella | 62.6K | 210.5K | 7 | Peer-to-Peer
Call Graph I | 631.6K | 822.9K | 3 | Call Graph
BerkStanford | 685.2K | 7.3M | 21 | Web
Facebook NY | 905.7K | 10.6M | 23 | Social
DBLP | 1.1M | 4.7M | 9 | Collab.
Call Graph II | 47.2M | 329.3M | 14 | Call Graph
FlatGrid | 10.0K | 19.8K | 4 | Grid
HyperGrid | 29.3K | 51.6K | 4 | Hyperbolic
## 2 Related Work
Theoretical Bounds on General Graphs. There is a rich body of literature on
distance oracles for general graphs, including the special case of distance
labeling scheme where the distance for query node pairs is estimated by merely
using labels associated with the query nodes, and the related problems of
graph spanners. The seminal work of Thorup and Zwick [36] described a distance
oracle that gives $2k-1$ approximation with $O(k)$ query time, $O(kn^{1+1/k})$
space and $O(kmn^{1/k})$ preprocessing time on an arbitrary weighted
undirected graph with $n$ nodes and $m$ edges, for any integer $k\geq 2$. The
preprocessing time and the query time of this distance oracle were
subsequently improved (c.f. [9, 8, 7, 25]), but the space versus approximation
factor trade-off has remained almost the same. In fact, various lower bounds
have been proved under plausible conjectures (c.f. [28] and the references
therein) for the space versus worst-case approximation factor trade-off. These
lower bound results suggest that it is unlikely that a distance oracle can
result in a significantly better trade-off for general weighted graphs.
Empirical Work on Road Networks. In contrast to the theoretical work on
general graphs, there has been considerable algorithm engineering and
experimentation work on road networks for navigation applications using global
positioning systems (GPS). These solutions (e.g., [6, 17, 18, 33]) crucially
rely on many specific characteristic properties of road networks such as the
existence of small natural cuts, a grid-like structure, highway hierarchies,
guiding the search towards the target using the latitude and longitude of the
target location, etc. On other graph classes such as those from online social
networks, equivalent solutions are not generally known to produce equally good
approximations.
Approximate Distances on Real-World Networks.
Distance oracles have been investigated both from theoretical and practical
perspectives. As described earlier, our focus is on distance oracles that
provide accurate distance estimates on large real-world graphs rapidly (a few
microseconds) while having scalable preprocessing and near-linear storage
requirement. This category includes a number of recent practical heuristics
[39, 40, 10, 37, 32, 31]) that aim to estimate distance by embedding the graph
in geometric spaces, like Euclidean or Hyperbolic, or by extracting different
kinds of approximating trees from the real graph. Other heuristics, such as,
[37, 10] or the landmark based approaches with diverse seeding strategies [30]
use variants of breadth first search (BFS) trees. We also remark that the
sketch-based distance oracle by Sarna et al. [34], that engineers a distance
oracle with provable multiplicative guarantees, is similar in nature to other
approaches cited above. On the other side, there are some theoretical
approaches [11, 1, 20, 13] that prove worst-case bounds on accuracy for
specific graph classes such as those with power-law degree distribution or
those with small graph hyperbolicity [12, 19, 14]. These techniques have not
been evaluated on large real-world graphs. We evaluate some of these
techniques for benchmarking of our oracle’s performance.
Exact Distance Oracles. Since exact distance oracles require the computation
or storage of all pair shortest path, we do not consider these (e.g., [5, 2]
and the references therein) in our comparison. We observe that even with the
best combination of engineering insights, all-pairs shortest path computation
remains far too slow for large graphs with $\sim 10^{7}$ nodes or edges and
beyond, and this is especially unmanageable when these graphs do not fit in
the main memory of the computing device. Also, we do not consider the oracles
that have $\Omega(n^{\epsilon}),\epsilon>1$ query complexity (e.g., [4, 3,
29]), as these are unlikely to yield efficient solutions.
## 3 A Geometric Distance Oracle
In this section, we present an overview of graph hyperbolicity and discuss how
graph hyperbolicity may be exploited to design an effective distance oracle.
Recent studies [27, 15, 22] show that large-scale networks, from IP-layer
connectivity, citation, collaboration, co-authorship and friendship graphs,
exhibit strong intrinsic hyperbolicity.
(a) 3-point condition
(b) 4-point condition
(c) 2$\delta$ distortion relative to trees
Figure 1: (a) A schematic representation of the simple-to-visualize 3-point
condition. (b) Simpler to compute is the the equivalent 4-point condition,
where for any four points $w,x,y,z$, the largest sum $L$ of opposite pairs
($xz$+$yw$ in this picture) minus the middle sum $M$ (=$xy+wz$ here) is no
more than twice $\delta$. (c) The 4-point condition implies a $2\delta$
deviation from a tree graph, in which the largest among the three sums of
opposite pairs of distances of four points is always equal to the medium sum.
$\delta$-Hyperbolicity Intrinsic hyperbolicity, by which is meant
hyperbolicity without any embedding of the graph into some Euclidean and other
space, captures the geometric notion of negative curvature in smooth geometry.
In the past two decades, the notion of negative curvature has been
successfully exported to metric spaces, which are more general and less
restrictive than Euclidean spaces. This generalized notion of curvature lends
itself naturally to the investigation of (large-scale) curvature in graphs
[27, 22]. A simple description of $\delta$-hyperbolicity in a (path) metric
space is that the three sides of _any_ shortest-path triangle $X,Y,Z$ always
come within a certain fixed distance $\delta$ of each other, where $\delta$ is
a fixed minimal constant associated with the graph. In other words, the union
of two $\delta$-neighborhoods of any two sides of a shortest-path triangle
includes the third, as depicted in Figure 1(a).
An easier to compute equivalent condition is as follows. Given a graph
$G(V,E)$ and any four nodes $w,x,y,z\in V$, consider the three sums of
distances between opposite pairs of nodes. Specifically, let $S\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=S(w,x,y,z)=d(w,x)+d(y,z)\leq
M\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=M(w,x,y,z)=d(x,y)+d(w,z)\leq
L\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=L(w,x,y,z)=d(x,z)+d(w,y)$
as shown in Figure 1(b). Here, $d(x,y)$ is the shortest path distance between
nodes $x$ and $y$ in $G(V,E)$; often we will assume the standard ‘hop’ metric
on $G$. The hyperbolicity $\delta$ of a graph may be defined [20] as follows:
###### Definition 1.
A graph $G(V,E)$ is said to be $\delta$-hyperbolic for some $\delta\geq 0$ if
for every four nodes $w,x,y,z\in V$, $(L-M)/2\leq\delta$ is always satisfied
(i.e., the largest two of the three sums of opposite side distances differ by
no more than $2\delta$).
We refer to $\delta$ as the (4-point) hyperbolic constant of the graph222Even
though $\delta$ is used to refer to both the 3-point and 4-point constants,
for the same graph these two need not have the same value. From here on we use
$\delta$ to denote the 4-point constant of the graph.. Notice that any finite
graph with diameter $\Delta$ is trivially $\Delta$-hyperbolic. Thus this
definition is insightful only when $\delta$ is considerably smaller than
$\Delta$, as argued, for example, in [27, 22]. In Table 2 we list the
estimated $\delta$ values on our benchmark graphs to show that indeed on real
world networks the $\delta$ values are actually very small. For a detailed
description of how $\delta$ may be estimated for large graphs, see [27, 22,
15].
Table 2: Key Geometric Characteristics of the $9$ Real and $2$ Synthetic Benchmark Graphs (based on large samples). Dataset | Clust Coeff. | $\max\delta$ | avg $\delta$ | Diam
---|---|---|---|---
GoogleNews | 0.45 | 1.0 | 0.03 | 4
AstroPhysics | 0.50 | 2.0 | 0.24 | ~5
Facebook SB | 0.22 | 2.0 | 0.23 | 14
Gnutella | 0.01 | 1.5 | 0.08 | ~5
Call Graph I | 0.14 | 4.5 | 0.39 | ~47
BerkStanford | 0.50 | 2.0 | 0.16 | ~5
Facebook NY | 0.16 | 2.0 | 0.22 | ~19
DBLP | 0.65 | 2.5 | 0.24 | ~23
Call Graph II | 0.09 | $<4$ | 0.27 | ~27
FlatGrid | 0.00 | 87.0 | 7.40 | 198
HyperGrid | 0.33 | 1.0 | 0.27 | 18
$\delta$-Approximation of Hyperbolic Graphs with Trees. It has been observed
that the distance metric of a $\delta$-hyperbolic graph can be viewed as a
$\delta$-approximation of a tree metric, since for any four points $w,x,y,z\in
V$ in a tree $T(V,E^{\prime})$, (for an appropriate set $E^{\prime}$ of edges)
one necessarily has that $S\leq M=L$, thus resulting in $\delta=0$.333Note
that another notion of tree-likeness, tree-width, is an independent property
and unlike $\delta$-hyperbolicity its existence in real-world graphs is not
common. For instance, snapshots of the internet at autonomous system level
have large tree-width, but low hyperbolicity [26].
While trees and block graphs are $0$-hyperbolic and chordal graphs have
$\delta\leq 1$, cycles $C_{n}$ are $O(n)$-hyperbolic and square grids with $n$
nodes are $O(\sqrt{n})$-hyperbolic.
It turns out that similar to trees, $\delta$-hyperbolic graphs have a non-
empty core of nodes whose betweenness centrality is maximal possible, that is,
they have nodes whose betweenness centrality scales as $\Theta(n^{2})$ (where
$n:=|V|$ is the size of the graph), as argued in [27, 21]. In other words,
$\delta$-hyperbolicity also captures the notion that there is always a non-
empty set of vertices, its ‘core’, where a fraction of _all_ shortest paths
pass through.444Note that our usage of the term ‘core’ here is different from
the core of online social networks (that contains most of the graph nodes) and
the notion of coresets used in streaming algorithms literature. This is a
crucial property that we shall leverage in our distance approximation oracle.
Theoretical Bounds via Tree Approximations. We focus on three classes of
distance oracles that approximate distances on graphs via trees. First,
Gromov-like techniques, principally [20, 19, 14], yield an additive guarantee
on the distortion on distances via tree approximations of the graph, where the
resulting trees are not sub-graphs of the original graph but are typically
(based on) contractions. Specifically, Gromov’s notion of hyperbolicity leads
to approximations of a graph $G$ with trees $T$ that satisfy the following
bound: For any two points $x,y\in V(G)$, the shortest path distance
$d_{GT}(x,y)$ between them in $T$ has distortion $|d_{G}(x,y)-d_{GT}(x,y)|\leq
2\delta\log{n}$. Second, the tree embedding proposed by Abraham et al [2].
yields a multiplicative guarantee on the distortion for distances in the tree.
A metric space $X$ is defined to be $\varepsilon$-hyperbolic,
$\varepsilon\in[0,1]$, if every set of four points $w,x,y,z$ in $X$ ordered so
that $d(w,x)+d(y,z)\leq d(w,y)+d(x,z)\leq d(w,z)+d(x,y)$ satisfies
$d(w,z)+d(x,y)-d(w,y)-d(x,z)\leq 2\varepsilon\min\\{d(w,x),d(y,z)\\}.$
For such a metric space the authors give an algorithm to construct an
embedding $X$ into a tree $T$ where for any two points $x,y\in X$ the shortest
path distance $d_{T}(x,y)$ between them in $T$ satisfies
$\frac{\max_{x,y}(d_{T}(x,y)/d_{G}(x,y))}{\min_{x,y}(d_{T}(x,y)/d_{G}(x,y))}\leq(1+\varepsilon)^{c_{1}\log|X|}$.
Further, up to the constant $c_{1}$, they show that there exists an example
where this inequality is tight. Finally, Chepoi and Dragan [12] construct
contraction trees (i.e., many nodes of $G$ may map to the same node in $T$)
that approximate the graph distances with an additive distortion not exceeding
$|c(G)/2|+2$ where $c(G)$ is the length of the longest chordless cycle in $G$.
This kind of approximation is also appropriate to consider when real-world
graphs only have relatively short chordless cycles.
## 4 Algorithmic Design
In this section, we present an algorithmic framework for designing distance
oracles that leverage the small hyperbolicity of real-world graphs. We exploit
the hyperbolic core [27, 21] to construct a spanning tree BFS sub-graph
approximation for the graph, and not tree contractions with non-graph edges as
done in the above-referenced three approaches. Since, as cited, for
$\delta$-hyperbolic graphs, asymptotically a fixed fraction of all shortest
paths in the graph pass through its non-empty hyperbolic core, picking a node
from the core as the tree root will give us zero distortion for all node-pair
distances whose shortest paths traverse that node. The node with the highest
betweenness centrality is thus ideal as the root of the spanning tree: The
smaller the hyperbolic constant $\delta$ of the graph, the tighter the core,
and the larger the fraction of all node-pair paths that traverse a typical
node in the core, until for $\delta=0$ we have a single node to pick as the
root to ensure a fixed fraction of all node-pair paths traverse it. The
proposed tree oracle therefore starts from nodes with the highest
(betweenness) centrality. The tree then expands as a BFS tree by adding
unvisited nodes prioritized by their (betweenness) centrality values until all
nodes are included in the tree.
Now, since computation of centrality is too expensive on large-scale graphs,
we need a good proxy for centrality that is easy to compute. Empirically, it
has been observed on real-world graphs that nodal degrees correlate
(significantly) well with (betweenness) centrality [27], and Figure 2 shows a
typical correlation chart for the FacebookSB network. A degree-based
prioritization is then employed both during the selection of initial root of
the tree as well as during its expansion.
Figure 2: Degree and betweenness centrality of nodes in the Facebook
SantaBarbara graph.
Four Benchmarked Oracles. In the remainder of this section, we describe three
distance oracles that either have proven bounds on distance distortion or are
best-of-class empirically. We compare the results with our proposed geometric
oracle on the eleven samples of real networks and the two synethetic graphs,
as shown in Table 2. More specifically, we examine 1) a representative
approach from Gromov-type contraction-based tree approximation with proven
bounds: Gromov Tree, see Figure 3(c), 2) an oracle based on Steiner trees with
proven multiplicative bound: $\varepsilon$-approximate Steiner Tree, 3) an
empirical best-of-class landmark-based approach that exploits hyperbolic
embedding: Rigel, and 4) our proposed centrality-based spanning tree oracle:
HyperBFS, see Figure 3(e), as outlined above.
### 4.1 Hyperbolicity-based Tree Oracle
A $\delta$-hyperbolic graph may be viewed as an approximation to a tree. We
thus aim to extract a ‘backbone spanning tree’ (or a small collection of such
trees for better distance approximation) from the input graph to construct our
geometric oracle. As stated earlier, we expect a BFS spanning tree with a
highly central vertex as root would closely approximate distances in a
$\delta$-hyperbolic real- world graph (cf. Section 5 for more details). Also
as argued earlier, we may use degree as a proxy for centrality for
computational efficiency and select the highest degree node as the root.
Figure 3: (a) An example graph, (b) its layering partitioning [12], (c) the
resulting “Gromov Tree” approximation, (d) the associated tree spanner [14],
and (e) our Hyper BFS spanning tree. As it may be seen from this simple
example, Hyper BFS is distinct from (c) and (d), while (c) and (d) are very
closely associated.
Therefore, in constructing the Hyper BFS spanning tree $T$, we choose the
order in which new nodes are added to the BFS tree strictly based on their
degree. Algorithm 1 gives pseudocode for constructing the Hyper BFS tree based
on a general vertex ordering $\pi$, which in our implementation, and unless
stated otherwise, is based on degree.
Input: Graph $G$, vertex ordering $\pi$
Output: Tree $T$
let $r$ be the first vertex in $\pi$
set $Q=\\{r\\}$ and $T=r$.
while $Q$ is not empty:
… set $v$ top of $Q$ (remove $v$ from $Q$)
… let $N=N(v)$ ordered by $\pi$
… for each $n\in N$ not already in $T$:
…… push $n$ on $Q$
…… add vertex $n$ and edge $nv$ to $T$
return T
Algorithm 1 – Hyper BFS with Vertex Ordering
To improve the accuracy of distance estimates, and as practiced previously, we
use a small number of different trees, each with a distinct node as root. We
construct a collection of such trees rooted at distinct nodes in the
hyperbolic core of the graph, which as stated, we approximate by nodal degree.
Finally, the distance between two nodes $x$ and $y$ is answered by returning
the minimum of the distances between $x$ and $y$ in the (small set of)
different trees we construct.
### 4.2 Contraction Trees with Bounded Approximation
We start with the ‘Gromov tree’ technique which yields an approximation with
guaranteed additive distance distortion based on a contraction tree. Next we
outline the approach due to Abraham et al. which yields a multiplicative
guarantee on the distance distortion in an approximating tree.
Gromov tree approximation. Gromov’s notion of hyperbolicity [20] naturally
leads to a design for a linear time algorithm for approximating a graph $G$
with a contraction tree $T$. For any two points $x,y\in V(G)$, the unique
shortest path distance $d_{GT}(x,y)$ between them in the Gromov tree $T$
satisfies $|d_{G}(x,y)-d_{GT}(x,y)|\leq 2\delta\log{n}$. Following [12], for a
connected graph $G$, we fix a root $r\in V(G)$ and for $i\geq 0$, let
$N_{i}(r)$ be the vertices at distance exactly $i$ from $r$ in $G$, and
$B_{i}(r)$ be the vertices at distance at most $i$ from $r$ in $G$. A Gromov
tree $T$ is then obtained by contracting all node pairs $u,v\in N_{k}(r)$ (for
all $k\geq 1$), such that there exists a path between $u$ and $v$ entirely
contained outside $B_{k-1}(r)$. For convenience, we define a $i$-level
connected component as a maximal subset $C$ of vertices in $N_{i}(r)$ such
that each pair of vertices in $C$ is connected in $G\setminus B_{(i-1)}(r)$.
This construction (‘layer partition’ of a graph), proposed in [12, 14] and
also used by [19], determines a tree $T$ satisfying the above properties in
linear time by dealing first with those nodes furthest from $r$; if we think
of trees with the root at the top, we can say ‘in a bottom-up manner’. In each
level, $N_{i}(r)$, we contract the $i$-level connected components into a
single node. The bottom-up approach yields that in order to find these
components we need only consider the edges completely contained in $N_{i}(r)$
and those between $N_{i}(r)$ and $N_{i+1}(r)$. It follows that the total time
checking $i$-level connectivity throughout the execution is linear, and
therefore, the algorithm also runs in linear time.
Although this Gromov-type tree construction (per [12, 14, 19]) allows
arbitrary nodes to be picked as root, in our implementation we pick a high
centrality (with degree as proxy) node. We found that this modification
considerably improves the accuracy of distance estimates via ‘Gromov type’
contraction trees. Hereafter, we refer to our modified and improved
implementation of prior work (such as [12, 14, 19]) as the Gromov Tree.
Pseudocode of our implementation is given in Algorithm 2.
Input: Graph $G$
Output: Tree $T$
let $r$ be high degree node
determine sets $N_{0}(r),\ldots,N_{\ell}(r)$
set $T=G$
for $i=\ell$ downto $0$:
… for each $i$-level connected component $C$ of $N_{i}(r)$:
…… contract $C$ in $T$.
return T
Algorithm 2 – Gromov Tree
In the construction of ‘Gromov tree’, the distance $d_{GT}(x,y)\leq
d_{G}(x,y)$. This allows us to use the above technique for a lower bound on
the distance estimate and, thereby, compute an approximation range around the
real distance (more on this in Section 5.5).
###### Lemma 1.
For any two nodes $x,y$ in $G$, $d_{GT}(x,y)\leq d_{G}(x,y)$
###### Proof.
A key observation here is that in the construction, we only contract nodes and
do not delete any edge. This contraction of nodes may coalesce multiple edges
into a single edge. Thus, for each edge $\\{u,v\\}\in E(G)$, we either have
that $u$ and $v$ are contracted to the same node in $T$ or $\\{C(u),C(v)\\}\in
E(T)$, where $C(u)$ is the contracted nodes of $u$ in $T$. In particular, this
means that for $\\{u,v\\}\in E(G)$, we have $d_{GT}(u,v)\mathrel{\hbox
to0.0pt{\raisebox{1.29167pt}{$\cdot$}\hss}\raisebox{-1.29167pt}{$\cdot$}}=d_{GT}(C(u),C(v))\leq
1$.
Consider the shortest path $P(x,y)$ in $G$ (with number of edges
$|P(x,y)|=d_{G}(x,y)$). For each edge $\\{u,v\\}$ in $P(x,y)$, it holds that
$d_{GT}(u,v)\leq 1$. Thus, $d_{GT}(x,y)\leq\sum_{\\{u,v\\}\in
P(x,y)}d_{GT}(u,v)\leq\sum_{\\{u,v\\}\in P(x,y)}1=|P(x,y)|=d_{G}(x,y)$. ∎
We note in passing that [14] further refines this contraction tree by
expanding each partition back to the original number of nodes by connecting
these nodes back to a single node in the lower layer partition, as shown in
Figure 3. We shall not consider this “layer partition spanner trees” further
as these only add/subtract a fixed amount to the “Gromov Tree” approximation
(zero distance for nodes in the same partition change to 2).
$\varepsilon$-approximate Steiner tree. A slightly different version of
hyperbolicity was studied by Abraham et al. in [1] and it too leads to a tree
approximation but with a guaranteed multiplicative distortion. As in the
Gromov construction, their algorithm fixes a root $r\in V(G)$ and consider
$N_{i}(r)$. The tree $T$ is then constructed on vertex set $V(G)\cup S$, where
$S$ is a set of Steiner points, one for each connected component in the graph
induced on each $N_{i}(r)$, $i>0$. The edge set is defined as follows. For a
connected component $C$ in the graph induced on $N_{i}(r)$, where $s_{C}$ is
its corresponding Steiner point, we add edges $cs_{C}$ of weight $\frac{1}{2}$
for each $c$ in $C$ and edge $s_{C}x$ of weight $\frac{1}{2}$, where $x$ is
some vertex of $N_{i-1}(r)$ connected to some vertex of $C$. Pseudocode is
found in Algorithm 3.
Input: Graph $G$
Output: Tree $T$
let $r$ be high degree node
determine sets $N_{0}(r),\ldots,N_{\ell}(r)$
set $T=\emptyset$
for $i=\ell$ downto $0$:
… for each connected component $C$ of $N_{i}(r)$:
…… add vertex $s_{C}$ to $T$
…… add edge $s_{C}x$ of weight $1/2$ to $T$ (for one $x\in N_{i-1}(r)$ which
is adjacent to $c\in C$)
…… for each $c\in C$:
……… add vertex $c$ and edge $cs_{C}$ of weight $1/2$ to $T$
return T
Algorithm 3 – $\varepsilon$-approximate Steiner Tree
### 4.3 Embedding in Geometric Space
In this section, we present oracles that embed a graph into some multi-
dimensional geometric space. We pick these oracles for special attention since
they typically outperform other distance approximation oracles and thus help
benchmark the fidelity of our oracle. Hyperbolic embedding involves explicitly
mapping the nodes of the graph into points in the hyperbolic space. (We
observe here once more that intrinsic hyperbolicity is not the same as
hyperbolic embedding used in these distance oracles based on [23, 40].)
Some oracles in this category, such as Orion [39] and that of Qi et al. [31]
use the $L_{2}$ norm distance function in $\mathbb{R}^{10}$. Others such as
Rigel [40] use a Hyperboloid model [35] with curvature $c$, where the distance
between two $d$-dimension points $x=(x_{1},x_{2},\ldots,x_{d})$ and
$y=(y_{1},y_{2},\ldots,y_{d})$ is defined as follows:
$arccosh\left(\sqrt{(1+\sum_{i=1}^{d}{x_{i}^{2}})(1+\sum_{i=1}^{d}{y_{i}^{2}})}-\sum_{i=1}^{d}{x_{i}y_{i}}\right)\cdot|c|$
Input: Graph $G$
Output: Coordinates for all nodes in $\mathbb{R}^{10}$
Select a set of high degree nodes $L$
for each $v\in L$:
… BFS$(v)$
Compute coordinates for $v\in L$ minimizing distortion from each other
for each $v\in V\setminus L$:
… Select a subset $L_{v}$ of $L$
… $Ord(v)=$ Coordinates that minimize distortion from $L_{v}$
return Ord
Algorithm 4 – Rigel & Hyperbolic Embedding of Graph
These oracles embed a graph $G$ by first identifying a small set of nodes with
high degrees that are called landmarks. Using BFS from all landmark nodes,
they compute the distance of all nodes in the graph from these landmark nodes.
Then, a linear program is defined to compute an embedding of these landmark
nodes in $\mathbb{R}^{10}$ such that the difference between the actual
pairwise distance of these landmarks and their distance estimated using the
defined distance function in the embedded space is minimized. Solving this
linear program (e.g., via the simplex method) provides the coordinates of
landmark nodes.
The remaining nodes in $G$ are then given a coordinate using another linear
program. The objective of this linear program is to minimize the difference
between the actual distance of the node to a subset $L_{v}$ of landmark nodes
and the distance estimated using the distance function in the embedded space.
Once again, the simplex method is used to solve the linear program. Algorithm
4 provides a pseudocode for the Rigel approach.
The approximate distance among each pair of nodes is then their distance in
the embedding. Thus, the query time and space per node only depends on the
dimensionality of the embedding and for a fixed dimension, it reduces to
$O(1)$ (possibly with a high value for the prefactor).
We remark that a plausible reason for the success of these hyperbolic
embedding techniques is that in real-world graphs, the set of ‘core nodes’ of
high centrality (as implied by small graph hyperbolicity) and the set of
landmark nodes (computed based on degree) includes many of these same nodes as
shown in Figure 2.
Note that these coordinate-based systems can both underestimate and
overestimate the real distances. Rigel has been shown to be significantly more
accurate than high-dimensional Euclidean embeddings [40] and our preliminary
experiments also confirmed the same. Hence, we only consider Rigel from this
category in our empirical comparison.
(a) Santa Barbara, Facebook
(b) Call Graph
(c) Hypergrid Graph
(d) P2P Gnutella
(e) Web Berk-Stan
(f) Google News
(g) CaAstro-Ph
(h) DBLP
(i) Facebook NY
Figure 4: Average absolute error of various approximation techniques on
various synthetic and real-world graphs.
## 5 Experimental Evaluation
In this section we describe the methodology used in our experiments, followed
by a detailed analysis. The goal of our benchmarking is to determine if the
observed $\delta$-hyperbolicity of real-world or synthetic networks yields a
low cost and effective distance approximation competitive with or better than
the 1) best-of-class or 2) theoretically guaranteed approaches, as discussed.
### 5.1 Experimental Setup
The networks we study range from 10s of thousands to 100s of millions of nodes
and edges. Storing $(|V|(|V|-1))/2$ distances between all node pairs results
in Gigabytes of storage for small size graphs and quickly hits hundreds of
Terabytes for our larger graphs, which is impractical. To ensure that accuracy
is measured over a large enough sample of all node pairs in a graph, we use
two different approaches. When the studied graph is small enough, we compute
the exact distortion of distance for each pair of nodes. For large graphs we
compute the distance distortion by sampling a large enough set of node pairs
to ensure that the mean error due to sampling is sufficiently small.
A few observations are in order. First, for each of the Hyper BFS and Gromov
Tree techniques, we construct not just one tree but a small collection of
trees rooted at nodes in the hyperbolic core of the graph, i.e., those with
the highest centrality, approximated with degree centrality for computational
efficiency. Once we obtain this collection of trees, the distance between two
nodes $x$ and $y$ is answered by returning the minimum of the distances
between $x$ and $y$ in the different trees of the Hyper BFS. For the Gromov
Tree, the maximum distance gives a better lower bound to the graph distance,
since Gromov Trees are contractions. Second, we have verified experimentally
that $10$ such trees are enough to provide very good distance approximations
for the Hyper BFS and our implementation of Gromov Trees. This is an
improvement relative to prior work [10], where it was observed experimentally
that $20$ of their spanning trees were needed for a similar level of accuracy.
This improvement is likely due to our selection of roots from the centrality
core of the graph. Finally, although we have computed and carefully examined
all four oracles on all 11 benchmarked graphs, in what follows we present only
the most representative plots.
Measures of Distortion. We use three measures for evaluating the performance
of the various distances approximations on each graph. Each of these measures
compares the ground truth graph distance $d_{G}$ with the approximate distance
given by the proxy $d_{A}$ and captures a different performance feature.
###### Definition 2.
Let $x,y$ be vertices of a graph $G$ and let $d_{A}$ be the distance
approximated by a distance oracle.
* •
The additive distortion between $x$ and $y$ with respect to $d_{A}$ is
$d_{G}-d_{A}$.
* •
The absolute distortion between $x$ and $y$ with respect to $d_{A}$ is
$|d_{G}-d_{A}|$.
* •
The multiplicative distortion between $x$ and $y$ with respect to $d_{A}$ is
$\frac{|d_{G}-d_{A}|}{d_{G}}$.
The selection of proper distortion metrics is critical in assessing the
quality of each distance approximation model since each metric captures
distinct aspects of the distortion error and may have different consequences
for different applications. The choice of a measure is also important in the
context of different kind of bounds known for different approaches. For
instance, the Gromov tree approach has additive guarantees while
$\varepsilon$-approximate Steiner tree has multiplicative guarantees on
distance errors. We visualize each of these distortion metrics by plotting its
expected value conditioned on the value of $d_{G}$ being fixed. This allows us
to study the accuracy of various techniques for node pairs at different
distances. This is important as applications, like graph segmentation, SVD
decomposition, recommendation systems or influential node detection, may
require different accuracy guarantees across the range of node distances, i.e.
neighbor nodes, diametrically opposite nodes or node pairs with distances in
between the two extremes. For instance, recommendation systems may require
highly accurate short range distances around a particular user to detect
similar users in a particular radius; on the contrary, influential node
detection would require accurate long distances to determine the centrality of
a particular node.
Figure 5: Average absolute error of various approximation techniques on a
SquareGrid network
### 5.2 Methods’ Fidelity
Comparing Oracles Based on Absolute Error. Our first key observation is that
the error due to our oracle in estimating shortest path distances is very
small. The 10-Hyper BFS approximation results in an absolute error of less
than 2 in all instances. Figure 4 shows results on eight real graphs and one
synthetic graph, HyperGrid. We next compare the accuracy of the Hyper BFS
approach with the best practical oracle system, i.e. Rigel. This system is
particularly optimized for networks with Power-Law degree distribution like
Santa Barbara(Figure 4(a)) and a Call Graph (Figure 4(b)). However, it
significantly loses accuracy for pure hyperbolic graphs like Hypergrid(Figure
4(c)) for short distances. We note that our proposed Hyper BFS technique shows
intriguingly low mean error on real-world graphs: on the same range as Rigel
on real-world graphs and better for others. Figure 4 also shows that the other
techniques, 10 Gromov and Steiner trees, achieve poor results, particularly
for node pairs at large distances. Compared to 10-Hyper BFS and Rigel, the
Gromov tree contraction technique results in large errors, not only on the
call graph and SantaBarbara Facebook graph shown in these figures, but also on
the other real- world graphs that we considered. This technique contracts many
nodes into the same high degree nodes, thereby underestimating their
distances. For instance, on node pairs at a distance of 13 from each other on
the call graph, it has an average absolute error of around 6.5, i.e. around
50%.
Short-Long Distance Approximation Accuracy. Here we study how different
topological structures impact the fidelity of various techniques on different
distance lengths. An important observation that follows from Figure 4, is that
Rigel and 10-Hyper BFS result in better accuracy for node pairs with long
distances while Gromov performs consistently poorer for such node pairs. We
expect that the consistent accuracy of 10-Hyper BFS and Rigel, for farther
node pairs, is due to their having nodes with high betweenness centrality (as
approximated by degree centrality) at the root or center of their embedding.
This ensures that the distances for a large number of node pairs whose
shortest path passes through the core nodes are correctly estimated.
Interestingly enough, doing the same for ‘Gromov Tree’ did not seem to improve
its accuracy by much, possibly due to its large number of nodal contractions.
For short distances instead, Rigel accuracy, in particular on the HyperGrid
(Figure 4(c)), leads to a mean absolute error of 8 for the real distance of 3,
but gets more accurate for node-pairs at larger distances. This is clearly due
to the fact the a planar graph hardly fills a hyperbolic space and hyperbolic
embedding is a poor fit. Finally, on the SquareGrid network (Figure 5), which
is typical of planar graphs and close to road networks, some techniques like
Rigel and Steiner have very different behaviors compared to the previous
networks. Rigel, for instance, estimates long distances with a very large
absolute errors. On the contrary, Steiner shows a very high error for
distances less than 100 and then for some reason it dramatically improves for
very long distances. The fact that these techniques perform well on real-world
graphs and synthetic HyperGrid graphs but not on SquareGrid graphs, suggests
that the reported success of these techniques most likely relies on the
intrinsic hyperbolicity of real world graphs and use of root nodes in the
hyperbolic core, two properties not reported heretofore. However, 10-Hyper BFS
seems to cope with flatness well, most likely due to its BFS spanning tree
structure than its other features such as use of highly central nodes as root,
which a square grid does not posses.
(a) Average Additive Error
(b) Average Multiplicative Error
Figure 6: Comparing accuracy of the various approximation techniques on the
SantaBarbara Facebook graph.
Fidelity Versus Theoretical Bounds. Figures 6(a) and 6(b) show the additive
and multiplicative errors of the various techniques on SantaBarbara Facebook
graph, to provide a quantitative comparison against the respective theoretical
bounds provided by the Gromov and the Steiner constructions. For
$\delta$-hyperbolic graphs, the theoretical worst-case bound for the Gromov
tree based approach is an additive factor of $\delta\log{n}$. For the social
networks we considered, $\delta$ was a small constant, but owing to large
sizes of the graph, this theoretical bound of $\delta\log{n}$ is more than the
diameter of the graph. However, in Figures 6(a) we observe that the average
additive error of Hyper BFS is much smaller than the theoretical bound, and
the Gromov tree contraction is evidently the most erroneous compared to other
techniques considered. Similarly, the theoretical guarantee for
$\varepsilon$-approximate Steiner tree approach is quite large for the studied
graphs, in Figure 6(b) we observe a comparatively small multiplicative error
of less than 0.6 for most node pairs on this graph. This provides further
evidence that the worst-case bounds for the accuracy of these oracles are
pessimistic and the average error on real-world graphs is actually much less.
Rigel and 10-Hyper BFS approaches are significantly better the theoretical
bounds, by exhibiting both a smaller additive and multiplicative errors
compared to the Gromov and Steiner techniques. These results confirm that the
shortest paths in these graphs are well-captured by ‘backbone trees’, which is
in line with our understanding of the intrinsic hyperbolicity of the real-
world graphs.
Exploiting General Hyperbolic Topologies. To reduce the dependence of Hyper
BFS on the correlation between degree and betweenness (centrality) of nodes
and, thereby, to make this technique more accurate on a wider range of graph
classes, we consider alternative strategies for selecting the root nodes for
our tree oracle. We identify a node with high closeness555A high closeness
central node is a node with low shortest path distances to all other nodes in
the graph. centrality by selecting a random node $u$, finding the node $m_{u}$
that is at maximum distance from $u$, selecting a node $m_{m_{u}}$ at maximum
distance from $m_{u}$ and returning the node in the middle of a path between
$m_{u}$ and $m_{m_{u}}$ in the graph. We note that similar techniques have
been used to approximate diameter of large graphs (see e.g., [16]).
We aim for a technique that would work well on a general graph topologies,
particularly real-world social interaction graphs. To this end, we consider a
diverse seeding strategy, in which the first seed node is random, the second
seed is selected at maximum distance from first seed, the third seed is
selected based on closeness centrality (as described before) and the remaining
seeds are selected from among high degree nodes. We found that this diverse
seeding strategy performs well on various graph classes, and in particular on
HyperGrid, its accuracy is close to the best results from degree based root
selection and closeness based root selection methods (cf. Figure 7).
Figure 7: Average absolute error of various Hyper BFS approaches with diverse
seeding on the HyperGrid network.
### 5.3 Role of Hyperbolic Core
The above benchmarking tests show that Hyper BFS is at least as fast as other
tree-based oracles and gives distance estimates comparable to the best. We
recall that Hyper BFS constructs a spanning tree of the graph based on the
ordering of nodes by their centrality, with a root in the hyperbolic core of
the graph. What aspect of Hyper BFS is key to it performance? Here we argue
that the selection of the root node of the BFS spanning tree in the hyperbolic
core is likely the most critical element in making Hyper BFS a strong distance
oracle for real-world networks. We show this by two sets of experiments; The
first set keeps the root node in the hyperbolic core but changes the rest of
the spanning BFS tree, and the second set changes the root but keeps
everything else the same.
(a) DBLP
(b) Santa Barbara, Facebook
(c) DBLP
(d) Santa Barbara, Facebook
Figure 8: Top (a)-(b): Comparison of Hyper BFS and Hyper BFS(inc) with
increasing nodal degree, on two representative networks both with highest
centrality node as BFS root. Bottom (c)-(d): Two instances of Hyper BFS on the
same two networks, one with highest centrality node as root (green) and the
other with a random node as root (red).
Figures 8(a)-8(b) show comparisons on two sets of benchmark real-world
networks between Hyper BFS and another implementation of Algorithm 1, which we
label Hyper BFS(inc). In this implementation of Hyper BFS, the root node is
still selected from the hyperbolic core (i.e., has the highest centrality) but
the ordering of the subsequent nodes in the BFS spanning tree is in reverse of
Hyper BFS, that is, in order of increasing degrees. As it can be seen, the
results are almost identical in terms of average distortion compared to Hyper
BFS. However, when we start the Hyper BFS tree with a random root node as
shown in Figures 8(c)-8(d), then we see a factor 2-4 increase in the absolute
error compared to Hyper BFS where the root has highest centrality. This
result, perhaps initially surprising, is consistent with our understanding of
the structure of real-world graphs, since the key property of
$\delta$-hyperbolic networks is the existence of a core whose centrality is
$O(n^{2})$, thus ensuring that a large fraction of all shortest paths
automatically traverse this small set and for which distance error is zero for
that many node pairs.
### 5.4 Scalability of Hyper BFS
The Call Graph II has $\sim$50 million nodes and $\sim$300 million edges. But
as it can be seen from Figure 9 and Table 3, the total run time for the Hyper
BFS on this network was under one minutes (under 10 minute for 10 Hyper BFS),
including one million node-pair distance queries that were completed in 25
seconds, all on a 2.4 GHz Intel(R) Xeon(R) processor with 190 GB of RAM. We do
not know of faster distance approximation techniques with error as small as
shown in Figure 9. In our implementation of Hyper BFS, we used a standard tree
labeling scheme whereby we store the list of parent nodes of each node to the
root, $nk$ indices for $k$ Hyper BFS trees on $n$ nodes. Since the height of
the Hyper BFS tree is $O(\log{n})$, a query has $O(k\log{n})$ complexity. We
did not run the other three oracles on this data set as competitive
implementations for a graph of this size required substantial optimization of
the code and investment of time. We expect Rigel to require several hours for
computation of its embedding phase but from there it is likely competitive
with Hyper BFS in completing one million queries in 20-30 seconds and based on
our observations from other sample real networks, we expect similar or
marginally better accuracy.
Figure 9: Hyper BFS accuracy on a 50M node call-graph with different number of trees, i.e. from 1 to 20. Table 3: Computational Time of Hyper BFS on Call Graph II. Loading Graph | Hyper BFS Tree | 1M Queries
---|---|---
250 sec | 50 sec | 25 sec
### 5.5 Bounding the Hyper BFS Error
Since the Hyper BFS is a spanning tree of the graph, the distance $d_{H}(x,y)$
in the Hyper BFS tree is an upper bound on the distance $d_{G}(x,y)$ in the
graph for any node pair $(x,y)$. On the other hand, a Gromov tree is a
contraction of the input graph and the distance in the Gromov tree
$d_{GT}(x,y)$ is a lower bound on the graph distance $d_{G}(x,y)$ (cf. Lemma
1). Thus, we can use these approaches together to get an approximation range
$[l(x,y),u(x,y)]$ with guaranteed upper $(u(x,y))$ and lower $(l(x,y))$ bound
for the actual distance $d_{G}(x,y)$ in the graph, i.e., $l(x,y)\leq
d_{G}(x,y)\leq u(x,y)$. Since $d_{G}(x,y)\leq d_{GT}(x,y)+2\delta\log{n}$, we
can use $u(x,y)=\min\\{d_{H}(x,y),d_{GT}(x,y)+2\delta\log{n}\\}$ and
$l(x,y)=d_{GT}(x,y)$. By the above definition of the upper and lower bound, it
follows that the width of the approximation range $u(x,y)-l(x,y)$ is less than
equal to $2\delta\log{n}$.
We can reduce the width of this approximation range at the cost of increasing
the precomputation time and the storage space by running multiple runs of
Hyper BFS tree and the Gromov Tree from different root nodes. Thus, the new
lower bound is the maximum over all Gromov Trees. The new upper bound is the
minimum over all the hyper BFS trees and the Gromov trees with additive error
bound, i.e.,
$u(x,y)=\min\\{\min_{H}\\{d_{H}(x,y)\\},\min_{GT}\\{d_{GT}(x,y)+2\delta\log{n}\\}\\}$.
Note that both in the Hyper BFS tree and in the Gromov tree, the distance from
the root $r$ to any node $x\in V$ is exact, i.e.,
$d_{H}(r,x)=d_{GT}(x,y)=d_{G}(x,y)$. Thus, if we use $n$ different hyper BFS
trees with different root nodes, the minimum over them will yield the exact
graph distance. Similarly, if we use $n$ different Gromov trees with different
root nodes, the maximum over them will give the exact graph distance,
resulting in a approximation range width of zero. However, this extreme point
of the solution space requires $O(n^{2})$ preprocessing time and $O(n^{2})$
storage space.
Figure 10: Upper and lower bounds and the width of the approximation range by
using 10-Hyper BFS tree and 20 Gromov trees on the SantaBarbara Facebook
graph.
The accuracy of the Hyper BFS approach is seen to be much smaller than the
$2\delta\log{n}$ theoretical bound for Gromov trees, on the real-world graphs
that we tested. This implies that we can obtain a fairly small approximation
range by combining the Hyper BFS and the Gromov tree approach. In fact, as
Figure 10 shows, the upper and lower bounds on SantaBarbara Facebook graph are
quite close and the resultant range with just 10-Hyper BFS trees and 20 Gromov
trees is quite small.
## 6 Conclusion
In this work, we construct a novel distance oracle that is based on the
intrinsic hyperbolicity of the graph and specifically leverages its hyperbolic
core. Rooting a breadth-first-search spanning tree within the hyperbolic core
of the graph ensures that a fixed fraction of all shortest paths have no
distance estimation error, and the remaining ones have small error.
Furthermore, the smaller the hyperbolic constant $\delta$ of the graph, the
better such a tree approximation is expected to be.
We tested an implementation of this theory-based framework on a dozen real and
synthetic large graphs, from 10s of thousand to 10s of millions of nodes, and
found that in practice it leads to surprisingly fast and accurate results,
significantly better than anticipated by existing theoretical error bounds
that have been proven for alternative tree approximations of the graph. This
theoretical framework, especially the existence of the hyperbolic core of the
graph, together with evidence from our experiments also suggest that the
success of prior heuristic distance oracles may also be due to the same
reasons: 1) the underlying hyperbolicity of the real-world graphs and 2) good
correlation between nodal degrees and their betweenness centrality in many
real-world graphs. In particular, $\delta$-hyperbolicity implies that in these
graphs, shortest paths can be well- approximated by appropriately rooted
‘backbone spanning trees’.
Another interesting observations is that a simple approach of computing a few
BFS trees from high centrality/degree seed nodes provides quite accurate
results. To improve this technique further, we considered two strategies: 1)
selecting a diverse set of seed nodes for growing BFS trees and 2) to grow the
tree by expanding along high degree nodes first. We found that our first
strategy helped make the technique more robust – it helped the 20 BFS provide
good accuracy even on hypergrid graphs where there was no degree distribution
as proxies for betweenness centrality of nodes in the BFS expansion. The
second strategy helped on some synthetic graphs, such as hypergrids, but it
did not improve distance approximation on real-world networks: it makes a big
difference to root the spanning tree approximation at a vertex in the
hyperbolic core. We note additionally that compared to the best performing
distance approximation oracles, such as Rigel, our approach has a very low
computing cost for creating the oracle and thus lends itself well to settings
where the graph connectivity changes frequently, such as in large scale
dynamic graphs.
We contend that further theoretical understanding of the geometry of
$\delta$-hyperbolic graphs can lead to even better distance oracles for real-
world graphs. For example, it would be helpful to know how large the
hyperbolic core of a $\delta$-hyperbolic graph can be. Another interesting
open problem is to find alternative proxies for betweenness centrality in
real-world graphs which, as with nodal degrees, is easy to compute.
Acknowledgment. The work of Iraj Saniee and Sean Kennedy was supported by the
AFOSR grant no. FA9550-11-1-0278 and the NIST grant no. 60NANB10D128.
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|
arxiv-papers
| 2014-04-20T02:05:34 |
2024-09-04T02:50:01.534481
|
{
"license": "Public Domain",
"authors": "Deepak Ajwani, W. Sean Kennedy, Alessandra Sala, Iraj Saniee",
"submitter": "Iraj Saniee",
"url": "https://arxiv.org/abs/1404.5002"
}
|
1404.5035
|
# $n$-widths and Approximation theory on Compact Riemannian Manifolds
###### Abstract.
We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of
unit balls in Sobolev and Besov norms in $L_{p}$-spaces on compact Riemannian
manifolds. The proofs rely on estimates for the near-diagonal localization of
the kernels of elliptic operators. We also summarize some of our previous
results about approximations by eigenfunctions of elliptic operators on
manifolds.
Daryl Geller 111Department of Mathematics, Stony Brook University, Stony
Brook, NY 11794-3651; (12/26/1950-01/27/2011)
Isaac Z. Pesenson 222 Department of Mathematics, Temple University,
Philadelphia, PA 19122; [email protected]. The author was supported in part
by the National Geospatial-Intelligence Agency University Research Initiative
(NURI), grant HM1582-08-1-0019.
Keywords and phrases: Compact manifold, Laplace-Beltrami operator, kernels,
Sobolev space, Besov space, eigenfunctions, polynomials, best approximation,
$n$-widths.
Subject classifications[2000] 43A85; 42C40; 41A17; 41A10
## 1\. Introduction and the main results
Daryl Geller and I started to work on this paper during the Summer of 2010.
Sadly, Daryl Geller passed away suddenly in January of 2011. I will always
remember him as a good friend and a wonderful mathematician.
Approximation theory on compact manifolds is an old subject [36], [37], [35],
[38], [24]-[27], [16], [17]. However it attracted considerable interest during
last years [4]-[7] due to numerous applications of function theory on
$S^{2},\>S^{3},$ and $SO(3)$ to seismology, weather prediction, astrophysics,
texture analysis, signal analysis, computer vision, computerized tomography,
neuroscience, and statistics [2], [7], [23], [33].
In the classical approximation theory of functions on Euclidean spaces the so
called Kolmogorov width $d_{n}$ and linear width $\delta_{n}$ are of primary
importance. The width $d_{n}$ characterizes the best approximative
possibilities by approximations by $n$-dimensional subspaces, the width
$\delta_{n}$ characterizes the best approximative possibilities of any
$n$-dimensional linear method. The width $d_{n}$ was introduced by A.N.
Kolmogorov in [19] and $\delta_{n}$ was introduced by V.M. Tikhomirov in [39].
The goal of the paper is of two fold. We determine asymptotic estimates of
Kolmogorov and linear $n$-widths of unit balls in Sobolev and Besov norms in
$L_{p}({\bf M})$-spaces on a compact Riemannian manifold ${\bf M}$ and we give
a brief account of our previous results about approximations by eigenfunctions
of elliptic operators on manifolds.
Let us recall [21] that for a given subset $H$ of a normed linear space $Y$,
the Kolmogorov $n$-width $d_{n}(H,Y)$ is defined as
$d_{n}(H,Y)=\inf_{Z_{n}}\sup_{x\in H}\inf_{z\in Z_{n}}\|x-z\|_{Y}$
where $Z_{n}$ runs over all $n$-dimensional subspaces of $Y$. The linear
$n$-width $\delta_{n}(H,Y)$ is defined as
$\delta_{n}(H,Y)=\inf_{A_{n}}\sup_{x\in H}\|x-A_{n}x\|_{Y}$
where $A_{n}$ runs over all bounded operators $A_{n}:Y\rightarrow Y$ whose
range has dimension $n$. In our paper the notation $S_{n}$ will be used for
either $d_{n},$ or $\delta_{n}$.
One has the following relation (see [21], pp. 400-403,):
(1.1) $S_{n}(H,Y)\leq S_{n}(H,Y_{1}),\>\>H\subset Y_{1}\subset Y,$
where $Y_{1}$ is a subspace of $Y$.
If $\gamma\in\bf R$, we write $S_{n}(H,Y)\ll n^{\gamma}$ to mean that one has
the upper estimate $S_{n}(H,Y)\leq Cn^{\gamma}$ for $n>0$ where $C$ is
independent of $n$. Let $L_{q}=L_{q}({\bf M}),\>1\leq q\leq\infty,$ be the
regular Lebesgue space constructed with the Riemannian density. Let $L$ be an
elliptic smooth second-order differential operator $L$ which is self-adjoint
and positive definite in $L_{2}({\bf M})$, such as the Laplace-Beltrami
operator $\Delta$. For such an operator all the powers $L^{r},\>\>r>0,$ are
well defined on $C^{\infty}({\bf M})\subset L_{2}({\bf M})$ and continuously
map $C^{\infty}({\bf M})$ into itself. Using duality every operator
$L^{r},\>\>r>0,$ can be extended to distributions on ${\bf M}$. The Sobolev
space $W_{p}^{r}=W_{p}^{r}({\bf M}),\>1\leq p\leq\infty,\>\>r>0,$ is defined
as the space of all $f\in L_{p}({\bf M}),1\leq p\leq\infty$ for which the
following graph norm is finite
(1.2) $\|f\|_{W^{r}_{p}({\bf M})}=\|f\|_{p}+\|L^{r/2}f\|_{p}.$
If $p\neq 1,\infty$, this graph norm is independent of $L$, up to equivalence,
by elliptic regularity theory on compact manifolds. If $p=1$ or $\infty$ we
will need to specify which operator $L$ we are using; some of our results will
apply for $L$ general. In fact, for our results which apply to general ${\bf
M}$, we can use any $L$.
Our objective is to obtain asymptotic estimates of $S_{n}(H,L_{q}({\bf M}))$,
where $H$ is the unit ball $B^{r}_{p}({\bf M})$ in the Sobolev space
$W_{p}^{r}=W_{p}^{r}({\bf M}),1\leq p\leq\infty,\>\>r>0,$ Thus,
$B^{r}_{p}=B^{r}_{p}({\bf M})=\left\\{f\in W_{p}^{r}({\bf
M})\>:\>\|f\|_{W_{p}^{r}({\bf M})}\leq 1\right\\}.$
It is important to remember that in all our considerations the inequality
$r>s\left(\frac{1}{p}-\frac{1}{q}\right)_{+}$
with $s=dim\>{\bf M}$ will be satisfied. Thus, by the Sobolev embedding
theorem the set $B^{r}_{p}({\bf M})$ is a subset of $L_{q}({\bf M})$.
Moreover, since ${\bf M}$ is compact by the Rellich-Kondrashov theorem the
embedding of $B^{r}_{p}({\bf M})$ into $L_{q}({\bf M})$ will be compact.
We set $s=\dim{\bf M}$. Let as usual $p^{\prime}=\frac{p}{p-1}$. Our main
result is the following theorem..
###### Theorem 1.1.
(Upper estimate) For any compact Riemannian manifold, any $L$, and for any
$1\leq p,q\leq\infty,\>r>0$, if $S_{n}$ is either of $d_{n}$ or $\delta_{n}$
then the following holds
(1.3) $S_{n}(B^{r}_{p}({\bf M}),L_{q}({\bf M}))\ll
n^{-\frac{r}{s}+(\frac{1}{p}-\frac{1}{q})_{+}},$
provided that $-\frac{r}{s}+(\frac{1}{p}-\frac{1}{q})_{+}$, which we call the
basic exponent, is negative.
Our results generalize some of the known estimates for the particular case in
which $\bf M$ is a compact symmetric space of rank one; these estimates were
obtained in papers [5] and [4]. They, in turn generalized and extended results
from [3], [15], [18], [22], [16] and [17].
Our main Theorems could be carried over to Besov spaces on manifolds using
general results about interpolation of compact operators.
Our main Theorems along with some general results in [40] imply similar
results in which balls in Sobolev spaces $B^{r}_{p}({\bf M})$ are replaced by
balls $\mathrm{B}^{r}_{p,t}({\bf M})$ in appropriate Besov spaces (see section
6).
The proofs of all the main results heavily exploit our estimates for the near-
diagonal localization of the kernels of elliptic operators on compact
manifolds (see [12] and section 2 below for the general case and [8]\- [11]
for the case of Laplace-Beltrami operator).
In last section we consider compact homogeneous manifolds and the
corresponding Casimir operator $\mathcal{L}$ (see section 5 for definitions).
For this situation we review our results about approximations by bandlimited
functions. Although we show in Theorem 5.2 that the span of the eigenfunctions
of our operator ${\mathcal{L}}$ is the same as the span of all polynomials
when one equivariantly embeds the manifold, the relation between eigenvalues
and degrees of polynomials is unknown (at least in the general case). However,
it is easy to verify that for compact two-point homogeneous manifolds, the
span of those eigenfunctions whose eigenvalues are not greater than a value
$\ell^{2},\ \ell\in\mathbb{N},$ is the same as the span of all polynomials of
degree at most $\ell$. Thus, on compact two-point homogeneous manifolds, our
Theorem 5.4 about approximations by bandlimited functions can be reformulated
in terms of approximations by polynomials.
## 2\. Kernels elliptic operators on compact Riemannian manifolds
Let $({\bf M},g)$ be a smooth, connected, compact Riemannian manifold without
boundary with Riemannian measure $\mu$. We write $dx$ instead of $d\mu(x)$.
For $x,y\in{\bf M}$, let $d(x,y)$ denote the geodesic distance from $x$ to
$y$. We will frequently need the fact that if $M>s$, $x\in{\bf M}$ and $t>0$,
then
(2.1) $\int_{\bf M}\frac{1}{\left[1+(d(x,y)/t)\right]^{M}}dy\leq
Ct^{s},\>\>\>s=\dim{\bf M},$
with $C$ independent of $x$ or $t$.
Let $L$ be a smooth, positive, second order elliptic differential operator on
${\bf M}$, whose principal symbol $\sigma_{2}(L)(x,\xi)$ is positive on
$\\{(x,\xi)\in T^{*}{\bf M}:\ \xi\neq 0\\}$. In the proof of Theorems 1.1 we
will take $L$ to be the Laplace-Beltrami operator of the metric $g$. We will
use the same notation $L$ for the closure of $L$ from $C^{\infty}({\bf M})$ in
$L_{2}({\bf M})$. In the case $p=2$ this closure is a self-adjoint positive
definite operator on the space $L_{2}({\bf M})$. The spectrum of this
operator, say $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq...$, is discrete
and approaches infinity. Let $u_{0},u_{1},u_{2},...$ be a corresponding
complete system of real-valued orthonormal eigenfunctions, and let
$\textbf{E}_{\omega}(L),\ \omega>0,$ be the span of all eigenfunctions of $L$,
whose corresponding eigenvalues are not greater than $\omega$. Since the
operator $L$ is of order two, the dimension $\mathcal{N}_{\omega}$ of the
space ${\mathbf{E}}_{\omega}(L)$ is given asymptotically by Weyl’s formula,
which says, in sharp form: For some $c>0$,
(2.2)
$\mathcal{N}_{\omega}(L)=c\omega^{s/2}+O(\omega^{(s-1)/2}).\vspace{.3cm}$
where $s=dim{\bf M}$. Since $\mathcal{N}_{\lambda_{l}}=l+1$, we conclude that,
for some constants $c_{1},c_{2}>0$,
(2.3) $c_{1}l^{2/s}\leq\lambda_{l}\leq c_{2}l^{2/s}$
for all $l$. Since $L^{m}u_{l}=\lambda_{l}^{m}u_{l}$, and $L^{m}$ is an
elliptic differential operator of degree $2m$, Sobolev’s lemma, combined with
the last fact, implies that for any integer $k\geq 0$, there exist
$C_{k},\nu_{k}>0$ such that
(2.4) $\|u_{l}\|_{C^{k}({\bf M})}\leq C_{k}(l+1)^{\nu_{k}}.$
Suppose $F\in\mathcal{S}(\bf{R}^{+})$, the space of restrictions to the
nonnegative real axis of Schwartz functions on $\bf{R}$. Using the spectral
theorem, one can define the bounded operator $F(t^{2}L)$ on $L_{2}({\bf M})$.
In fact, for $f\in L_{2}({\bf M})$,
(2.5) $[F(t^{2}L)f](x)=\int K_{t}(x,y)f(y)dy,$
where
(2.6) $K_{t}(x,y)=\sum_{l}F(t^{2}\lambda_{l})u_{l}(x)u_{l}(y)=K_{t}(y,x)$
as one sees easily by checking the case $F=u_{m}$. Using (2.6), (2.2), (2.3)
and (2.4), one easily checks that $K_{t}(x,y)$ is smooth in $(x,y)\in{\bf
M}\times{\bf M}$. We call $K_{t}$ the kernel of $F(t^{2}L)$. $F(t^{2}L)$ maps
$C^{\infty}({\bf M})$ to itself continuously, and may thus be extended to be a
map on distributions. In particular we may apply $F(t^{2}L)$ to any $f\in
L_{p}({\bf M})\subseteq L_{1}({\bf M})$ (where $1\leq p\leq\infty$), and by
Fubini’s theorem $F(t^{2}L)f$ is still given by (2.5).
The following Theorem about $K_{t}$ was proved in [12] for general elliptic
second order differential self-adoint positive operators.
###### Theorem 2.1.
Assume $F\in\mathcal{S}(\bf{R}^{+})$ (the space of restrictions to the
nonnegative real axis of Schwartz functions on $\bf{R}$). For $t>0$, let
$K_{t}(x,y)$ be the kernel of $F(t^{2}L)$. Then:
1. (1)
If $F(0)=0$, then for every pair of $C^{\infty}$ differential operators $X$
$($in $x)$ and $Y$ $($in $y)$ on ${\bf M}$, and for every integer $N\geq 0$,
there exists $C_{N,X,Y}$ such that for $\deg X=j$ and $\deg Y=k$ the following
estimate holds
(2.7) $t^{s+j+k}\left|\left(\frac{d(x,y)}{t}\right)^{N}XYK_{t}(x,y)\right|\leq
C_{N,X,Y},\>\>s=dim\>{\bf M},$
for all $t>0$ and all $x,y\in{\bf M}$.
2. (2)
For general $F\in\mathcal{S}(\bf{R}^{+})$ the estimate (2.7) at least holds
for $0<t\leq 1$.
In this article, we will use the following corollaries of the above result.
###### Corollary 2.1.
Assume $F\in\mathcal{S}(\bf{R}^{+})$. For $t>0$, let $K_{t}(x,y)$ be the
kernel of $F(t^{2}L)$. Suppose that either:
(i) $F(0)=0$, or
(ii) $F$ is general, but we only consider $0<t\leq 1$.
Then for some $C>0$,
(2.8)
$|K_{t}(x,y)|\leq\frac{Ct^{-s}}{\left[1+\frac{d(x,y)}{t}\right]^{s+1}},\>\>\>s=dim\>{\bf
M},$
for all $t$ and all $x,y\in{\bf M}$.
Proof This is immediate from Theorem 2.1, with $X=Y=I$, if one considers the
two cases $N=0$ and $N=s+1$.
###### Corollary 2.2.
Consider $1\leq\alpha\leq\infty$, with conjugate index $\alpha^{\prime}$. In
the situation of Theorem 2.1, there is a constant $C>0$ such that
(2.9) $\left(\int|K_{t}(x,y)|^{\alpha}dy\right)^{1/\alpha}\leq
Ct^{-s/\alpha^{\prime}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{for all
}x,$
and
(2.10) $\left(\int|K_{t}(x,y)|^{\alpha}dx\right)^{1/\alpha}\leq
Ct^{-s/\alpha^{\prime}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{for all
}y,$
Proof We need only prove (2.9), since $K_{t}(y,x)=K_{t}(x,y)$.
If $\alpha<\infty$, (2.9) follows from Corollary 2.1, which tells us that
$\int|K_{t}(x,y)|^{\alpha}dy\leq C\int_{\bf
M}\frac{t^{-s\alpha}}{\left[1+(d(x,y)/t)\right]^{\alpha(s+1)}}dy\leq
Ct^{s(1-\alpha)}$
with $C$ independent of $x$ or $t$, by (2.1).
If $\alpha=\infty$, the left side of (2.9) is as usual to be interpreted as
the $L^{\infty}$ norm of $h_{t,x}(y)=K_{t}(x,y)$. But in this case the
conclusion is immediate from Corollary 2.1.
This completes the proof.
We will use Corollary 2.2 in conjunction with the following fact. We consider
operators of the form $f\to{\mathcal{K}}f$ where
(2.11) $({\mathcal{K}}f)(x)=\int K(x,y)f(y)dy,$
where the integral is over ${\bf M}$, and where we are using Riemannian
measure. In all applications, $K$ will be continuous on ${\bf M}\times{\bf
M}$, and $F$ will be in $L_{1}({\bf M})$, so that ${\mathcal{K}}f$ will be a
bounded continuous function. The following generalization of Young’s
inequality holds:
###### Lemma 2.2.
Suppose $1\leq p,\alpha\leq\infty$, and that $(1/q)+1=(1/p)+(1/\alpha)$.
Suppose that $c>0$, and that
(2.12) $[\int|K(x,y)|^{\alpha}dy]^{1/\alpha}\leq c\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \mbox{for all }x,$
and
(2.13) $[\int|K(x,y)|^{\alpha}dx]^{1/\alpha}\leq c\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \mbox{for all }y,$
Then $\|{\mathcal{K}}f\|_{q}\leq c\|f\|_{p}$ for all $f\in L_{p}$.
###### Proof.
Let $\beta=q/\alpha\geq 1$, so that $\beta^{\prime}=p^{\prime}/\alpha$. For
any $x$, we have
$\displaystyle|({\mathcal{K}}f)(x)|$ $\displaystyle\leq$
$\displaystyle\int|K(x,y)|^{1/\beta^{\prime}}|K(x,y)|^{1/\beta}f(y)|dy$
$\displaystyle\leq$
$\displaystyle\left(\int|K(x,y)|^{p^{\prime}/\beta^{\prime}}dy\right)^{1/p^{\prime}}\left(\int|K(x,y)|^{p/\beta}|f(y)|^{p}dy\right)^{1/p}$
$\displaystyle\leq$ $\displaystyle
c^{1/\beta^{\prime}}\left(\int|K(x,y)|^{p/\beta}|f(y)|^{p}dy\right)^{1/p}$
since $p^{\prime}/\beta^{\prime}=\alpha$,
$\alpha/p^{\prime}=1/\beta^{\prime}$. Thus
$\displaystyle\|{\mathcal{K}}f\|^{p}_{q}$ $\displaystyle\leq$ $\displaystyle
c^{p/\beta^{\prime}}\left(\int\left(\int|K(x,y)|^{p/\beta}|f(y)|^{p}dy\right)^{q/p}dx\right)^{p/q}$
$\displaystyle\leq$ $\displaystyle
c^{p/\beta^{\prime}}\int\left(\int|K(x,y)|^{pq/\beta
p}|f(y)|^{pq/p}dx\right)^{p/q}dy$ $\displaystyle=$ $\displaystyle
c^{p/\beta^{\prime}}\int\left(\int|K(x,y)|^{\alpha}dx\right)^{p/q}|f(y)|^{p}dy$
$\displaystyle\leq$ $\displaystyle
c^{p/\beta^{\prime}}c^{p/\beta}\|f\|^{p}_{p}$
as desired. (In the second line, we have used Minkowski’s inequality for
integrals.)
∎
## 3\. Proof of Theorem 1.1
Now, let $\eta$ be a $C^{\infty}$ function on $[0,\infty)$ which equals $1$ on
$[0,1]$, and which is supported in $[0,4]$. Define, for $x>0$,
$\phi(x)=\eta(x/4)-\eta(x)$
so that $\phi$ is supported in $[1,16]$. For $j\geq 1$, we set
$\phi_{j}(x)=\phi(x/4^{j-1}).$
We also set $\phi_{0}=\eta$, so that $\sum_{j=0}^{\infty}\phi_{j}\equiv 1$. We
claim:
###### Lemma 3.1.
(a) If $r>0$, and $1\leq p\leq q\leq\infty$, then there is a $C>0$ such that
(3.1) $\|\phi_{j}(L)f\|_{q}\leq
C(2^{js})^{-\frac{r}{s}+\frac{1}{p}-\frac{1}{q}}\|f\|_{W_{p}^{r}({\bf M})},$
for all $f\in W_{p}^{r}({\bf M})$. In other words, the norm of $\phi_{j}(L)$,
as an element of ${\bf B}(W_{p}^{r}({\bf M}),L_{q}({\bf M}))$ (the space of
bounded linear operators from $W_{p}^{r}({\bf M})$ to $L_{q}({\bf M})$), is no
more than $C(2^{js})^{-\frac{r}{s}+\frac{1}{p}-\frac{1}{q}}$.
(b) Suppose that
$-\frac{r}{s}+\frac{1}{p}-\frac{1}{q}<0.$
Then $\sum_{j=0}^{\infty}\phi_{j}(L)$ converges absolutely in ${\bf
B}(W_{p}^{r}({\bf M}),L_{q}({\bf M}))$, to the identity operator on
$W_{p}^{r}({\bf M})$.
Proof (a) Define, for $x>0$,
$\psi(x)=\phi(x)/x^{r/2}$
so that $\psi$ is supported in $[1,16]$. For $j\geq 1$, we set
$\psi_{j}(x)=\psi(x/4^{j-1}),$
which implies
$\phi_{j}(x)=2^{-(j-1)r}\psi_{j}(x)x^{r/2}.$
Accordingly, if $f$ is a distribution on ${\bf M}$, for $j\geq 1$,
$\phi_{j}(L)f=2^{-(j-1)r}\psi_{j}(L)(L^{r/2}f),$
in the sense of distributions. If now $f\in W_{p}^{r}({\bf M})$, so that
$L^{r/2}f\in L_{p}({\bf M})$, we see from Lemma 2.2 with $t=2^{-j}$, and from
Lemma 2.2, that if $(1/q)+1=(1/p)+(1/\alpha)$, then
$\|\phi_{j}(L)f\|_{q}\leq C2^{-jr}2^{js/\alpha^{\prime}}\|L^{r/2}f\|_{p}\leq
C(2^{js})^{-\frac{r}{s}+\frac{1}{p}-\frac{1}{q}}\|f\|_{W_{p}^{r}({\bf M})},$
as desired.
For (b), we note that by (a), $\sum_{j=0}^{\infty}\phi_{j}(L)$ converges
absolutely in ${\bf B}(W_{p}^{r}({\bf M}),L_{q}({\bf M}))$. It converges to
the identity on smooth functions, hence in the sense of distributions. Hence
we must have $\sum_{j=0}^{\infty}\phi_{j}(L)=I$ in ${\bf B}(W_{p}^{r}({\bf
M}),L_{q}({\bf M}))$. This completes the proof.
Proof of Theorem 1.1 Since in general $d_{n}\leq\delta_{n}$, it suffices to
prove the upper estimate for $\delta_{n}$. If $q\leq p$, then surely
$\delta_{n}(B^{r}_{p}({\bf M}),L_{q}({\bf M}))\leq C\delta_{n}(B^{r}_{p}({\bf
M}),L_{p}({\bf M}))$. Since the upper estimate is the same for all $q$ with
$q\leq p$, we may as well assume then that $q=p$. In short, we may assume
$q\geq p$.
Let $\eta$ be the same as above and set $\eta_{m}(x)=\eta(x/4^{m-1})$ for
$m\in\mathbb{N}$. Then $\sum_{j=0}^{m-1}\phi_{j}=\eta_{m}$, which is supported
in $[0,4^{m}]$. Examining the kernel of $\eta_{m}(L)$ (see (2.6)), we see that
$\eta_{m}(L):W_{p}^{r}({\bf M})\to{\bf E}_{4^{m}}(L).$
By Weyl’s theorem (2.2), there is a positive integer $c$ such that the
dimension of ${\bf E}_{4^{m}}(L)$ is at most $c2^{ms}$ for every $m$. We see
then by Lemma 3.1 that
$\delta_{c2^{ms}}(B_{p}^{r}({\bf M}),L^{q}({\bf
M}))\leq\|I-\eta_{m}(L)\|\leq\sum_{j=m}^{\infty}\|\phi_{j}(L)\|\leq$
$\sum_{j=m}^{\infty}C(2^{js})^{-\frac{r}{s}+\frac{1}{p}-\frac{1}{q}}\leq
C(2^{ms})^{-\frac{r}{s}+\frac{1}{p}-\frac{1}{q}}\leq
C(c2^{ms})^{-\frac{r}{s}+\frac{1}{p}-\frac{1}{q}},$
where all norms are taken in ${\bf B}(W_{p}^{r}({\bf M}),L_{q}({\bf M}))$.
This proves the basic upper estimate for
$n\in A:=\\{c2^{ms}:m\geq 1\\}$. For any $n\geq c2^{s}$ we may find $m\in A$
with $m\leq n\leq 2^{s}m$, and surely $\delta_{n}\leq\delta_{m}$. This gives
the basic upper estimate for all $n$, and completes the proof.
## 4\. Widths of balls in Besov spaces
The following definitions of Sobolev and Besov spaces are well known [38],
[41]. Let $(U_{i},\chi_{i})$ be a finite atlas on ${\bf M}$ with charts
$\chi_{i}$ mapping $W_{i}$ into the unit ball on ${\bf R}^{n}$, and suppose
$\\{\zeta_{i}\\}$ is a partition of unity subordinate to the $U_{i}$. The
Sobolev space $W_{p}^{r}({\bf M}),\>1\leq p\leq\infty$ and $r$ is natural can
be defined as a space of all distributions $f$ on ${\bf M}$ such that
(4.1) $\sum_{i}\|(\zeta_{i}f)\circ\chi_{i}^{-1}\|_{W_{p}^{r}({\bf
R}^{n})}<\infty.$
The Besov space $\mathcal{B}_{p,t}^{\alpha}({\bf M})$ can be defined as a
space of distributions $f$ on ${\bf M}$ for which
(4.2) $\sum_{i}\|(\zeta_{i}f)\circ\chi_{i}^{-1}\|_{B_{p,t}^{\alpha}({\bf
R}^{n})}<\infty,$
where $\alpha>0$, $1\leq p<\infty$, and $0<t<\infty$ and
$B_{p,t}^{\alpha}({\bf R}^{n})$ is the regular Besov space. This definition
does not depend on the choice of charts or partition of unity ([41]).
An important property of Besov spaces $\mathcal{B}^{\alpha}_{p,t}({\bf
M}),\alpha>0,1\leq p<\infty,1\leq t\leq\infty,$ is that they can be described
using Peetre’s interpolation $K$-functor [1], [20], [42]. Namely,
(4.3) $\mathcal{B}^{\alpha}_{p,t}({\bf M})=\left(L_{p}({\bf M}),W^{r}_{p}({\bf
M})\right)^{K}_{\alpha/r,q},$
where $r$ can be any natural such that $0<\alpha<r,1\leq t<\infty$, or
$0\leq\alpha\leq r,t=\infty$. Since ${\bf M}$ is compact by the Rellich-
Kondrashov theorem the embedding of the ball $B^{r}_{p}({\bf M})$ into
$L_{q}({\bf M})$ is compact as long as the condition
(4.4) $r>s\left(\frac{1}{p}-\frac{1}{q}\right)_{+}$
is satisfied. By an interpolation theorem for compact operators ([40], Theorem
1.16.2) the embedding into $L_{q}({\bf M})$ of the unit ball in the
corresponding Besov space $\mathcal{B}^{\alpha}_{p,t}({\bf M})$ is also
compact.
These facts allow us to use some general results in [40] (Theorem 1.16.3)
about interpolation of compact operators which along with our main results
produce similar theorems about balls $\mathrm{B}^{r}_{p,t}({\bf M})$ in
appropriate Besov spaces.
###### Theorem 4.1.
Let ${\bf M}$ be a compact Riemannian manifold. For every choice of parameters
$\>\>1\leq p<\infty,\>\>1\leq q\leq\infty,\>\>r>0,$ for which the following
relation holds
$d_{n}(B^{r}_{p}({\bf M}),L_{q}({\bf M}))\ll n^{\gamma},$
for the Kolmogorov $n$-width of the unit ball $B^{r}_{p}({\bf M})$ in the
Sobolev space $W_{p}^{r}({\bf M})$ then the similar relation holds for the
Kolmogorov $n$-width of the unit ball $\mathrm{B}^{r}_{p,t}({\bf M})$ in the
Besov space $\mathcal{B}_{p,t}^{r}({\bf M})$ i.e.
$d_{n}(\mathrm{B}^{r}_{p,t}({\bf M}),L_{q}({\bf M}))\ll n^{\gamma}.$
## 5\. Approximation theory on compact homogeneous manifolds
### 5.1. Compact homogeneous manifolds
The most complete results will be obtained for compact homogeneous manifolds.
A homogeneous compact manifold $M$ is a $C^{\infty}$-compact manifold on which
a compact Lie group $G$ acts transitively. In this case $M$ is necessary of
the form $G/K$, where $K$ is a closed subgroup of $G$. The notation
$L_{2}(M),$ is used for the usual Hilbert spaces, with invariant measure $dx$
on $M$.
The Lie algebra g of a compact Lie group $G$ is then a direct sum
$\textbf{g}=\textbf{a}+[\textbf{g},\textbf{g}]$, where a is the center of g,
and $[\textbf{g},\textbf{g}]$ is a semi-simple algebra. Let $Q$ be a positive-
definite quadratic form on g which, on $[\textbf{g},\textbf{g}]$, is opposite
to the Killing form. Let $X_{1},...,X_{d}$ be a basis of g, which is
orthonormal with respect to $Q$. Since the form $Q$ is $Ad(G)$-invariant, the
operator
(5.1) $-X_{1}^{2}-X_{2}^{2}-\ ...-X_{d}^{2},\ d=dim\ G$
is a bi-invariant operator on $G$, which is known as the Casimir operator.
This implies in particular that the corresponding operator on $L_{2}(M)$,
(5.2) $\mathcal{L}=-D_{1}^{2}-D_{2}^{2}-...-D_{d}^{2},\>\>\>D_{j}=D_{X_{j}},\
d=dim\ G,$
commutes with all operators $D_{j}=D_{X_{j}}$. The operator $\mathcal{L}$,
which is usually called the Laplace operator, is the image of the Casimir
operator under differential of quazi-regular representation in $L_{2}(M)$. It
is important to realize that in general, the operator $\mathcal{L}$ is not
necessarily the Laplace-Beltrami operator of the natural invariant metric on
$M$. But it coincides with such operator at least in the following cases: 1)
If $M$ is a $d$-dimensional torus, 2) If the manifold $M$ is itself a compact
semi-simple Lie group group $G$ ([14], Ch. II), 3) If $M=G/K$ is a compact
symmetric space of rank one ([14], Ch. II, Theorem 4.11).
In the case of a compact manifold the norm (4.1) of the Sobolev space
$W_{p}^{r}({\bf M}),\>\>1\leq p\leq\infty,$ $r\in\mathbb{N}$, is equivalent to
one of the following norms [25]
$\|f\|_{p}+\sum_{1\leq k\leq r}\sum_{1\leq i_{1},...,i_{k}\leq
d}\|D_{i_{i}}...D_{i_{k}}f\|_{p}\sim\|f\|_{p}+\sum_{1\leq i_{1},...,i_{r}\leq
d}\|D_{i_{i}}...D_{i_{r}}f\|_{p},$
where $d=dim\>G$.
### 5.2. Bernstein spaces on compact homogeneous manifolds
Returning to the compact homogeneous manifold ${\bf M}=G/K$, let
$\mathbb{D}=\\{D_{1},...,D_{d}\\},\>\>d=\dim G,$ be the same set of operators
as in (5.2). Let us define the Bernstein space
$\textbf{B}_{\omega}^{p}(\mathbb{D})=\\{f\in L_{p}({\bf
M}):\|D_{i_{1}}...D_{i_{k}}f\|_{p}\leq\omega^{k}\|f\|_{p},\>\>1\leq
i_{1},...i_{k}\leq d,\>\omega\geq 0\\}$
where $d=dim\>G$.
As before, the notation ${\bf E}_{\omega}(\mathcal{L}),\>\>\omega\geq 0,$ will
be used for a span of eigenvectors of $\mathcal{L}$ with eigenvalues
$\leq\omega$. For these spaces the next two theorems hold (see [27], [31]):
###### Theorem 5.1.
The following properties hold:
1. (1)
$\textbf{B}_{\omega}^{p}(\mathbb{D})=\textbf{B}_{\omega}^{q}(\mathbb{D}),\>\>\>1\leq
p\leq q\leq\infty,\>\>\omega\geq 0.$
2. (2)
$\textbf{B}^{p}_{\omega}(\mathbb{D})\subset{\bf
E}_{\omega^{2}d}(\mathcal{L})\subset\textbf{B}^{p}_{\omega\sqrt{d}}(\mathbb{D}),\>\>\>d=\dim\>G,\>\>\>\omega\geq
0.$
3. (3)
$\|\mathcal{L}^{k}\varphi\|_{q}\leq C({\bf
M})\omega^{2k+\frac{d}{p}-\frac{d}{q}}\|\varphi\|_{p},\>\>\>\varphi\in{\bf
E}_{\omega}(\mathcal{L}),\>\>\>k\in\mathbb{N},$
where $d=\dim\>G,\>\>1\leq p\leq q\leq\infty$.
Every compact Lie group can be considered to be a closed subgroup of the
orthogonal group $O(\mathbb{R}^{N})$ of some Euclidean space $\mathbb{R}^{N}$.
It means that we can identify ${\bf M}=G/K$ with the orbit of a unit vector
$v\in\mathbb{R}^{N}$ under the action of a subgroup of the orthogonal group
$O(\mathbb{R}^{N})$ in some $\mathbb{R}^{N}$. In this case $K$ will be the
stationary group of $v$. Such an embedding of ${\bf M}$ into $\mathbb{R}^{N}$
is called equivariant.
We choose an orthonormal basis in $\mathbb{R}^{N}$ for which the first vector
is the vector $v$: $e_{1}=v,e_{2},...,e_{N}$. Let $\textbf{P}_{r}({\bf M})$ be
the space of restrictions to ${\bf M}$ of all polynomials in $\mathbb{R}^{N}$
of degree $r$. This space is closed in the norm of $L_{p}({\bf M}),1\leq
p\leq\infty,$ which is constructed with respect to the $G$-invariant
normalized measure on ${\bf M}$ [27], [31].
###### Theorem 5.2.
If ${\bf M}$ is embedded into an $R^{N}$ equivariantly, then
$\textbf{P}_{r}({\bf M})\subset\textbf{B}_{r}(\mathbb{D})\subset{\bf
E}_{r^{2}d}(\mathcal{L})\subset\textbf{B}_{r\sqrt{d}}(\mathbb{D}),\>\>\>d=dim\>G,\>\>\>r\in\mathbb{N},$
and
$span_{r\in\mathbb{N}}\>\textbf{P}_{r}({\bf M})=span_{\omega\geq
0}\>\textbf{B}_{\omega}(\mathbb{D})=span_{j\in\mathbb{N}}\>{\bf
E}_{\lambda_{j}}(\mathcal{L}).$
### 5.3. Besov spaces on compact homogeneous manifolds
For the same operators as above $D_{1},...,D_{d},\ d=dim\ G$, (see section 3)
let $T_{1},...,T_{d}$ be the corresponding one-parameter groups of translation
along integral curves of the corresponding vector fields i.e.
(5.3) $T_{j}(\tau)f(x)=f(\exp\tau X_{j}\cdot
x),\>x\in\mathbb{M}=G/K,\>\tau\in\mathbb{R},\>f\in L_{p}(\mathbb{M}),\>1\leq
p<\infty,$
here $\exp\tau X_{j}\cdot x$ is the integral curve of the vector field $X_{j}$
which passes through the point $x\in\mathbb{M}$. The modulus of continuity is
introduced as
$\Omega_{p}^{r}(s,f)=$ (5.4) $\sum_{1\leq j_{1},...,j_{r}\leq
d}\sup_{0\leq\tau_{j_{1}}\leq s}...\sup_{0\leq\tau_{j_{r}}\leq
s}\|\left(T_{j_{1}}(\tau_{j_{1}})-I\right)...\left(T_{j_{r}}(\tau_{j_{r}})-I\right)f\|_{L_{p}(\mathbb{M})},$
where $d=\dim\>G,\>f\in L_{p}(\mathbb{M}),1\leq p<\infty,\ r\in\mathbb{N},$
and $I$ is the identity operator in $L_{p}(\mathbb{M}).$ We consider the space
of all functions in $L_{p}(\mathbb{M})$ for which the following norm is
finite:
(5.5)
$\|f\|_{L_{p}(\mathbb{M})}+\left(\int_{0}^{\infty}(s^{-\alpha}\Omega_{p}^{r}(s,f))^{t}\frac{ds}{s}\right)^{1/t},1\leq
p,t<\infty,$
with the usual modifications for $t=\infty$. The following theorem is a rather
particular case of general results that can be found in [24], [25].
###### Theorem 5.3.
If ${\bf M}=G/K$ is a compact homogeneous manifold the norm of the Besov space
$\mathcal{B}^{\alpha}_{p,t}({\bf M}),0<\alpha<r\in\mathbb{N},\ 1\leq
p,t<\infty,$ is equivalent to the norm (5.5). Moreover, the norm (5.5) is
equivalent to the norm
(5.6) $\|f\|_{W_{p}^{[\alpha]}(\mathbb{M})}+\sum_{1\leq
j_{1},...,j_{[\alpha]}\leq
d}\left(\int_{0}^{\infty}\left(s^{[\alpha]-\alpha}\Omega_{p}^{1}(s,D_{j_{1}}...D_{j_{[\alpha]}}f)\right)^{t}\frac{ds}{s}\right)^{1/t},d=dim\>G,$
if $\alpha$ is not integer ($[\alpha]$ is its integer part). If
$\alpha=k\in\mathbb{N}$ is an integer then the norm (5.5) is equivalent to the
norm (Zygmund condition)
(5.7) $\|f\|_{W_{p}^{k-1}(\mathbb{M})}+\sum_{1\leq j_{1},...,j_{k-1}\leq
d}\left(\int_{0}^{\infty}\left(s^{-1}\Omega_{p}^{2}(s,D_{j_{1}}...D_{j_{k-1}}f)\right)^{t}\frac{ds}{s}\right)^{1/t},d=dim\>G.$
For $1\leq p\leq\infty$ we define a measure of the best approximation by
functions in ${\bf E}_{\omega}(\mathcal{L})$ as
$\mathcal{E}(f,\omega,p)=\inf_{g\in{\bf
E}_{\omega}(\mathcal{L})}\|f-g\|_{L_{p}({\bf M})}\,\,\mbox{for}\,\,f\in
L_{p}({\bf M}).$
The following theorem was proved in [26], [32], [12].
###### Theorem 5.4.
Suppose that $\alpha>0,1\leq p\leq\infty$, and $0<t<\infty$. Then the norm of
the Besov space $\mathcal{B}^{\alpha}_{p,t}({\bf M})$ is equivalent to the
following one
(5.8) $\|f\|_{{\mathcal{B}^{\alpha}_{p,t}({\bf M})}}:=\|f\|_{L_{p}({\bf
M})}+\left(\sum_{j=0}^{\infty}\left[2^{\alpha
j}{\mathcal{E}}(f,2^{2j},p)\right]^{t}\right)^{1/t}<\infty.$
## References
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* [4] G. Brown and F. Dai (2005), Approximation of smooth functions on compact two-point homogeneous spaces, J. Func. Anal. 220 (2005), 401-423
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* [6] G. Brown, F. Dai, Sun Yongsheng, Kolmogorov width of classes of smooth functions on the sphere , J. Complexity 18 (4) (2002) 1001-1023.
* [7] W. Freeden, T. Gervens, M. Schreiner, Constructive approximation on the spheres. With applications to geomathematics, Numerical Mathematics and Scientific Computation, The Claredon Press, Oxford University Press, New York, 1998.
* [8] D. Geller and A. Mayeli, Continuous wavelets and frames on stratified Lie groups I, Journal of Fourier Analysis and Applications, 12 (2006), 543-579.
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* [40] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978. 528 pp.
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|
arxiv-papers
| 2014-04-20T12:12:17 |
2024-09-04T02:50:01.614960
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Isaac Pesenson, Daryl Geller",
"submitter": "Isaac Pesenson Prof.",
"url": "https://arxiv.org/abs/1404.5035"
}
|
1404.5106
|
# The Hockey Stick Theorems in Pascal and Trinomial Triangles
Sima Mehri
Department of Mathematics
Sharif University of Technology
Tehran, Iran
###### Abstract
There are some theorems in the Pascal’s triangle which their figures resemble
to shoot a ball by hockey stick, so they are called hockey stick theorems. P.
Hilton and J. Pedersen, in the article ”Looking into Pascal Triangle,
Combinatorics, Arithmetic and Geometry”, have stated the little and big hockey
stick and puck theorems in the Pascal’s triangle. The big hockey stick theorem
is a special case of a general theorem which our goal is to introduce it. We
state a hockey stick theorem in the trinomial triangle too.
## 1 Introduction and Description of Results
The big hockey stick and puck theorem, stated in [Hilton1987] is:
###### Theorem 1.1 (Hilton1987).
(The Big Hockey Stick and Puck Theorem)
$\binom{n}{0}+\binom{n+2}{1}+\binom{n+4}{2}+\binom{n+6}{3}=\binom{n+7}{3}-\binom{n+6}{1}$
We have found the general form of above theorem in Pascal triangle as below.
###### Theorem 1.2.
(The Hockey Stick Theorem in Pascal Triangle)
$\sum_{i=0}^{k}\binom{n+2i}{i}=\sum_{j=0}^{\left\lfloor\frac{k}{2}\right\rfloor}\left(-1\right)^{j}\binom{n+2k-j+1}{k-2j}$
(1)
An example of this theorem is illustrated in Figure 1.
11211464116152015611828567056288117213535217115101051133111 Figure 1: Example
of Hocky-Stick:1+3+10+35=56-7
Now we wish to state the hockey stick theorem in trinomial triangle. First
using [3,4], we explain what is the trinomial triangle.
The trinomial triangle is a number triangle of trinomial coefficients. It can
be obtained by starting with a row containing a single ”1” and the next row
containing three 1s and then letting subsequent row elements be computed by
summing the elements above to the left, directly above, and above to the
right. We show the trinomial triangle in Figure 2. The trinomial coefficients
are placed as Figure 3. Following the notation of Andrews (1990) in
[Andrews1990], the trinomial coefficient $\binom{n}{k}_{2}$ with $n\geq 0$ and
$-n\leq k\leq n$, is given by the coefficient of $x^{n+k}$ in the expansion of
$(1+x+x^{2})^{n}$. Therefore,
$\binom{n}{k}_{2}=\binom{n}{-k}_{2}$
Equivalently, the trinomial coefficients are defined by
$(1+x+x^{-1})^{n}=\sum_{k=-n}^{k=n}\binom{n}{k}_{2}x^{k}$
We have proven the following theorem in this triangle:
###### Theorem 1.3.
(The Hockey Stick Theorem in The Trinomial Triangle)
$\sum_{i=0}^{k}\binom{n+i}{n}_{2}=\sum_{s=0}^{\lfloor\frac{k}{2}\rfloor}\left(-1\right)^{s}\binom{n+k+1}{n+2s+1}_{2}.$
(2)
For example see Figures 3 and 2.
1621509012614112690502161151530455145301551141016191610411367631123211111
Figure 2: Hockey Stick in Trinomial Triangle: $1+2+6+16+45=90-21+1$
$\binom{6}{-6}$$\binom{6}{-5}$$\binom{6}{-4}$$\binom{6}{-3}$$\binom{6}{-2}$$\binom{6}{-1}$$\binom{6}{0}$$\binom{6}{1}$$\binom{6}{2}$$\binom{6}{3}$$\binom{6}{4}$$\binom{6}{5}$$\binom{6}{6}$$\binom{5}{-5}$$\binom{5}{-4}$$\binom{5}{-3}$$\binom{5}{-2}$$\binom{5}{-1}$$\binom{5}{0}$$\binom{5}{1}$$\binom{5}{2}$$\binom{5}{3}$$\binom{5}{4}$$\binom{5}{5}$$\binom{4}{-4}$$\binom{4}{-3}$$\binom{4}{-2}$$\binom{4}{-1}$$\binom{4}{0}$$\binom{4}{1}$$\binom{4}{2}$$\binom{4}{3}$$\binom{4}{4}$$\binom{3}{-3}$$\binom{3}{-2}$$\binom{3}{-1}$$\binom{3}{0}$$\binom{3}{1}$$\binom{3}{2}$$\binom{3}{3}$$\binom{2}{-2}$$\binom{2}{-1}$$\binom{2}{0}$$\binom{2}{1}$$\binom{2}{2}$$\binom{1}{-1}$$\binom{1}{0}$$\binom{1}{1}$$\binom{0}{0}$
Figure 3: Trinomial Coefficients
## 2 Proof of Results
In the proof of both theorems, we use induction.
###### Proof.
(Theorem 1.2) We prove this theorem using induction on $k$. By the fact
$\binom{n}{n}=\binom{n+1}{n+1}=1$, statement is obvious for the base of
induction i.e. $k=1$. Now we assume that the statement for $k$ is true, then
the relation (1) would be correct. We wish to verify that it is correct for
the value $k+1$ too. We have
$\displaystyle\sum_{i=0}^{k+1}\binom{n+2i}{i}$
$\displaystyle=\binom{n+2k+2}{k+1}+\sum_{i=0}^{k}\binom{n+2i}{i}=\binom{n+2k+2}{k+1}+\sum_{j=0}^{\left\lfloor\frac{k}{2}\right\rfloor}\left(-1\right)^{j}\binom{n+2k-j+1}{k-2j}$
$\displaystyle=\left[\binom{n+2k+2}{k+1}+\binom{n+2k+1}{k}+\binom{n+2k+1}{k-1}\right]-$
$\displaystyle\quad-\left[\binom{n+2k+1}{k-1}+\binom{n+2k}{k-2}+\binom{n+2k}{k-3}\right]+$
$\displaystyle\quad\quad\vdots$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor-1}\left[\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+3}{k-2\left\lfloor\frac{k}{2}\right\rfloor+3}+\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+2}{k-2\left\lfloor\frac{k}{2}\right\rfloor+2}+\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+2}{k-2\left\lfloor\frac{k}{2}\right\rfloor+1}\right]+$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor}\left[\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+2}{k-2\left\lfloor\frac{k}{2}\right\rfloor+1}+\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+1}{k-2\left\lfloor\frac{k}{2}\right\rfloor}+\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+1}{k-2\left\lfloor\frac{k}{2}\right\rfloor-1}\right]+$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor+1}\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+1}{k-2\left\lfloor\frac{k}{2}\right\rfloor-1}=$
using properties of Pascal triangle, we get
$\displaystyle=\left[\binom{n+2k+2}{k+1}+\binom{n+2k+2}{k}\right]-$
$\displaystyle\quad-\left[\binom{n+2k+1}{k-1}+\binom{n+2k+1}{k-2}\right]+$
$\displaystyle\quad\quad\vdots$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor-1}\left[\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+3}{k-2\left\lfloor\frac{k}{2}\right\rfloor+3}+\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+3}{k-2\left\lfloor\frac{k}{2}\right\rfloor+2}\right]+$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor}\left[\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+2}{k-2\left\lfloor\frac{k}{2}\right\rfloor+1}+\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+2}{k-2\left\lfloor\frac{k}{2}\right\rfloor}\right]+$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor+1}\binom{n+2(k+1)-\left\lfloor\frac{k+1}{2}\right\rfloor+1}{k+1-2\left\lfloor\frac{k+1}{2}\right\rfloor}\mathbf{1}_{\left\\{k+1:even\right\\}}=$
$\displaystyle=\binom{n+2k+3}{k+1}-\binom{n+2k+2}{k-1}+\cdots+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor}\binom{n+2k-\left\lfloor\frac{k}{2}\right\rfloor+3}{k-2\left\lfloor\frac{k}{2}\right\rfloor+1}+$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k+1}{2}\right\rfloor}\binom{n+2(k+1)-\left\lfloor\frac{k+1}{2}\right\rfloor+1}{k+1-2\left\lfloor\frac{k+1}{2}\right\rfloor}\mathbf{1}_{\left\\{k+1:even\right\\}}=$
$\displaystyle=\sum_{j=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}\left(-1\right)^{j}\binom{n+2(k+1)-j+1}{k+1-2j}$
The statement for $k+1$ is also true, and the proof is completed. ∎
###### Proof.
(Theorem 1.3) To prove this theorem similarly we use induction on $k$.
Considering the equation $\binom{n}{n}_{2}=\binom{n+1}{n+1}_{2}$ , the result
being immediate if $k=0$. Assuming that the statement for $k$ is true, then
the relation (2) would be correct. Now we intend to illustrate it is correct
for the value $k+1$ too. We have
$\displaystyle\sum_{i=0}^{k+1}\binom{n+i}{n}_{2}$
$\displaystyle=\binom{n+k+1}{n}_{2}+\sum_{i=0}^{k}\binom{n+i}{n}_{2}=\binom{n+k+1}{n}_{2}+\sum_{s=0}^{\left\lfloor\frac{k}{2}\right\rfloor}(-1)^{s}\binom{n+k+1}{n+2s+1}_{2}=$
$\displaystyle=\left[\binom{n+k+1}{n}_{2}+\binom{n+k+1}{n+1}_{2}+\binom{n+k+1}{n+2}_{2}\right]-$
$\displaystyle\quad-\left[\binom{n+k+1}{n+2}_{2}+\binom{n+k+1}{n+3}_{2}+\binom{n+k+1}{n+4}_{2}\right]+$
$\displaystyle\quad\quad\vdots$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor-1}\left[\binom{n+k+1}{n+2\left\lfloor\frac{k}{2}\right\rfloor-2}_{2}+\binom{n+k+1}{n+2\left\lfloor\frac{k}{2}\right\rfloor-1}_{2}+\binom{n+k+1}{n+2\left\lfloor\frac{k}{2}\right\rfloor}_{2}\right]+$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor}\left[\binom{n+k+1}{n+2\left\lfloor\frac{k}{2}\right\rfloor}_{2}+\binom{n+k+1}{n+2\left\lfloor\frac{k}{2}\right\rfloor+1}_{2}+\binom{n+k+1}{n+2\left\lfloor\frac{k}{2}\right\rfloor+2}_{2}\right]+$
$\displaystyle\quad+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor+1}\binom{n+k+1}{n+2\left\lfloor\frac{k}{2}\right\rfloor+2}_{2}=$
using properties of the trinomial coefficients, we get
$\displaystyle=\sum_{s=0}^{\left\lfloor\frac{k}{2}\right\rfloor}(-1)^{s}\binom{n+k+2}{n+2s+1}_{2}+(-1)^{\left\lfloor\frac{k}{2}\right\rfloor+1}\binom{n+k+2}{n+2\left\lfloor\frac{k}{2}\right\rfloor+3}_{2}$
$\displaystyle=\sum_{s=0}^{\left\lfloor\frac{k+1}{2}\right\rfloor}(-1)^{s}\binom{n+k+2}{n+2s+1}_{2}$
The statement for $k+1$ is also true, and the proof is completed. ∎
The hockey stick theorem in the trinomial triangles has been proved. This
theorem can be translated in Pascal pyramid as follows :
$\sum_{i=0}^{k}\;\sum_{2r+s=2n+i}\binom{n+i}{r,s,r-n}=\sum_{j=0}^{\left\lfloor\frac{k}{2}\right\rfloor}\left(\left(-1\right)^{j}\sum_{2r+s=2n+k+2j+2}\binom{n+k+1}{r,s,r-n-2j-1}\right)$
Other similar theorems might be obtained for Pascal’s four dimensional and
even $n$-dimensional pyramid.
## References
* [Andrews1990] G. Andrews, _Euler’s ’Exemplum Memorabile Inductionis Fallacis’ and Trinomial Coefficients_ J. Amer. Math. Soc. 3 (1990), 653-669.
* [Hilton1987] P. Hilton and J. Pedersen, _Looking into Pascal Triangle, Combinatorics, Arithmetic and Geometry_ Mathematics Magazine, Vol. 60, No. 5 (Dec., 1987), 305-316.
* [3] Eric W. Weisstein, _Trinomial Coefficient_ From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html
* [4] Eric W. Weisstein, _Trinomial Triangle_ From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/TrinomialTriangle.html
|
arxiv-papers
| 2014-04-21T04:10:52 |
2024-09-04T02:50:01.624164
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Sima Mehri",
"submitter": "Sima Mehri",
"url": "https://arxiv.org/abs/1404.5106"
}
|
1404.5110
|
# Generalized Coherent States for the Spherical Harmonics
$Y_{m}^{m}(\theta,\phi)$
H. Fakhri
Department of Theoretical Physics and Astrophysics, Physics Faculty,
University of Tabriz, P O Box 51666-16471, Tabriz, Iran
B. Mojaveri
Department of Physics, Azarbaijan Shahid Madani University,
Tabriz 53741-161, Iran Email: [email protected]:
[email protected]
###### Abstract
The associated Legendre functions $P_{l}^{(m)}(x)$ for a given $l-m$, may be
taken into account as the increasing infinite sequences with respect to both
indices $l$ and $m$. This allows us to construct the exponential generating
functions for them in two different methods by using Rodrigues formula. As an
application then we present a scheme to construct generalized coherent states
corresponding to the spherical harmonics $Y_{m}^{m}(\theta,\phi)$.
PACS Nos: 02.30.Gp; 03.65.-w; 03.65.Fd
MSC Nos: 33C45; 05A15
Keywords: Associated Legendre Functions; Special Functions; Generating
Functions; Coherent States; Spherical Harmonics
## 1 Introduction
Coherent states were first discovered by Schrödinger in the context of quantum
mechanics in order to minimize the uncertainty relation between momentum and
position coordinates [1], and were later generalized successfully to the Lie
group approaches, by Glauber, Klauder, Sudarshan, Barut, Girardello and
Perelomov [2, 3, 4, 5, 6, 7, 8]. Also, for the models with one degree of
freedom either discrete or continuous spectra -with no remark on the existence
of a Lie algebra symmetry- Gazeau and Klauder proposed new coherent states
which are parameterized by two real parameters [9, 10, 11]. The quantum
coherency of states has nowadays pervaded many branches of physics such as
quantum optics, quantum electrodynamics, solid-state physics, nuclear and
atomic physics, from both theoretical and experimental viewpoints [12, 13, 14,
15]. Moreover, some generalized approaches in connection with coherent states
corresponding to shape invariant models have been proposed [16, 17].
Any infinite superposition from the pure states of a quantum mechanical system
cannot form coherent states because they not only must be converged to the
finite-valued functions but also must accept a positive definite measure to
satisfy the resolution of the identity condition on the entire complex plane
or on a unit disc. When both of them are satisfied, then they, as coherent
superpositions, minimize uncertainty relation for some values of the complex
variable. Using the Barut-Girardello eigenvalue equation for the laddering
operators of $su(1,1)$ Lie algebra, the coherent states corresponding to the
spherical harmonics $Y_{m}^{m}(\theta,\phi)$ have been calculated [18]. Also,
in Ref. [19], we have shown that the use of the spatial parity symmetry for
$Y_{m}^{m}(\theta,\phi)$ as angular wave functions of the one-partite systems
can lead to entangled $su(1,1)$-Barut-Girardello coherent states for a
bipartite quantum system. In this paper for the first time, we introduce
exponential and non-exponential generating functions corresponding to the
associated Legendre functions. Then, we construct the generalized coherent
states of $Y_{m}^{m}(\theta,\phi)$ as an application to the new generating
functions.
## 2 New generating functions for the associated Legendre functions
A large number of physical and chemical contexts involve application of the
associated Legendre functions $P_{l}^{(m)}(x)$. They are given by the
Rodrigues formula [20, 21, 22] ($-1<x<+1$)
$\displaystyle
P_{l}^{(m)}(x)=a_{l}^{(m)}(1-x^{2})^{-\frac{m}{2}}\left(\frac{d}{dx}\right)^{l-m}\,\left(1-x^{2}\right)^{l},$
(1)
in which, $l$ is a non-negative integer and $m$, an integer number, is bounded
by $l$: $-l\leq m\leq l$. The normalization coefficients are
$\displaystyle
a_{l}^{(m)}=\left\\{\begin{array}[]{l}\frac{(-1)^{l}}{2^{l}\,l!}\,\frac{(l+m)!}{(l-m)!}\hskip
56.9055pt0\leq m\leq l,\\\ \\\ \frac{(-1)^{l-m}}{2^{l}\,l!}\hskip
71.13188pt-l\leq m\leq 0.\end{array}\right.$ (5)
Also, the functions with positive and negative values of $m$ are proportional
with each other
$\displaystyle P_{l}^{(-m)}(x)=(-1)^{m}\frac{(l-m)!}{(l+m)!}P_{l}^{(m)}(x),$
(6)
whilst the associated Legendre functions with the same $l-m$ ($l+m$) but with
different values of $l$ and $m$ are independent of each other.
Thus, we can introduce two different types of infinite sequences of the
associated Legendre functions depending on whether $l-m$ is even or odd. These
types of sequences are increasing with respect to both indices $l$ and $m$ of
the functions. Due to the fact that whether $l-m$ is even or odd, i.e.
$l-m=2k$ or $l-m=2k+1$ with $k$ as a non-negative integer, the lowest
functions are $P_{k}^{(-k)}(x)$ and $P_{k+1}^{(-k)}(x)$, respectively. It is
obvious that the terminology of lowest functions is devoted to the associated
Legendre functions $P_{l}^{(m)}(x)$ with the lowest value for both indices $l$
and $m$. For a given value of $k$, the generating functions corresponding to
the sequences are calculated as
$\displaystyle G_{k}^{\mbox{{\tiny
even}}}(x,t)=\sum_{m=-k}^{\infty}\frac{t^{k+m}}{(k+m)!}\frac{P_{2k+m}^{(m)}(x)}{a_{2k+m}^{(m)}}$
$\displaystyle\hskip
29.87538pt=\sum_{m=0}^{\infty}\frac{t^{m}}{m!}(1-x^{2})^{\frac{k-m}{2}}\left(\frac{d}{dx}\right)^{2k}\left(1-x^{2}\right)^{k+m}$
$\displaystyle\hskip
29.87538pt=\sum_{m=0}^{\infty}\frac{t^{m}}{m!}(1-x^{2})^{\frac{k-m}{2}}\frac{(2k)!}{2\pi
i}\oint_{C}dz\frac{(1-z^{2})^{k+m}}{(z-x)^{2k+1}}$ $\displaystyle\hskip
29.87538pt=(1-x^{2})^{\frac{k}{2}}\frac{(2k)!}{2\pi
i}\oint_{C}dz\frac{(1-z^{2})^{k}exp\left(\frac{t(1-z^{2})}{\sqrt{1-x^{2}}}\right)}{(z-x)^{2k+1}}$
$\displaystyle\hskip
29.87538pt=(1-x^{2})^{\frac{k}{2}}\left[\left(\frac{d}{dz}\right)^{2k}\left((1-z^{2})^{k}exp\left(\frac{t(1-z^{2})}{\sqrt{1-x^{2}}}\right)\right)\right]_{z=x},$
(7) $\displaystyle G_{k}^{\mbox{{\tiny
odd}}}(x,t)=\sum_{m=-k}^{\infty}\frac{t^{k+m}}{(k+m)!}\frac{P_{2k+m+1}^{(m)}(x)}{a_{2k+m+1}^{(m)}}$
$\displaystyle\hskip
29.87538pt=\sum_{m=0}^{\infty}\frac{t^{m}}{m!}(1-x^{2})^{\frac{k-m}{2}}\left(\frac{d}{dx}\right)^{2k+1}\left(1-x^{2}\right)^{k+m+1}$
$\displaystyle\hskip
29.87538pt=\sum_{m=0}^{\infty}\frac{t^{m}}{m!}(1-x^{2})^{\frac{k-m}{2}}\frac{(2k+1)!}{2\pi
i}\oint_{C}dz\frac{(1-z^{2})^{k+m+1}}{(z-x)^{2k+2}}$ $\displaystyle\hskip
29.87538pt=(1-x^{2})^{\frac{k}{2}}\frac{(2k+1)!}{2\pi
i}\oint_{C}dz\frac{(1-z^{2})^{k+1}exp\left(\frac{t(1-z^{2})}{\sqrt{1-x^{2}}}\right)}{(z-x)^{2k+2}}$
$\displaystyle\hskip
29.87538pt=(1-x^{2})^{\frac{k}{2}}\left[\left(\frac{d}{dz}\right)^{2k+1}\left((1-z^{2})^{k+1}exp\left(\frac{t(1-z^{2})}{\sqrt{1-x^{2}}}\right)\right)\right]_{z=x},$
(8)
for $\left|t\right|<\infty$. $C$ is a closed contour in positive direction on
the complex plane $z$. In order to satisfy Eqs. (4) and (5), it is sufficient
that the arbitrary contour $C$ is chosen so that the point $z=x$ lies inside
of that. The accordance of the above generating functions with the theorem 1
of Ref. [23] can be considered as confirmation for it. Therefore, depending on
whether $l-m$ is even or odd integer, we have obtained two different types of
generating functions as multipliers of $exp(t\sqrt{1-x^{2}})$ for the
associated Legendre functions. Note that Eq. (3) does not allow us to get new
generating functions by means of the sequences with given values of $l+m$.
Furthermore, it should be emphasized that by using the closed contour applied
to Jacobi polynomials, new generating functions corresponding to infinite
sequences of the functions with the same (non-negative) $m$ are derived as
$\displaystyle
G_{m}(x,t)=\sum_{l=m}^{\infty}\frac{t^{l-m}}{(l-m)!}\frac{P_{l}^{(m)}(x)}{2^{l}a_{l}^{(m)}}=\frac{\left(\frac{-1-xt+\sqrt{t^{2}+2xt+1}}{t^{2}\sqrt{1-x^{2}}}\right)^{m}}{\sqrt{t^{2}+2xt+1}},\hskip
48.36967pt\left|t\right|<1.$ (9)
In the special case $m=0$, (6) is converted to the known generating function
of the Legendre polynomials.
## 3 Generalized coherent states for the spherical harmonics
$Y_{m}^{m}(\theta,\phi)$
This section covers the method of making generalized coherent states for the
spherical harmonics $Y_{m}^{m}(\theta,\phi)$ by using the new generating
functions of the associated Legendre functions. The spherical harmonics are
described in terms of the polar (or co-latitude) angle $0\leq\theta\leq\pi$
and the azimuthal (or longitude) angle $0\leq\phi<2\pi$:
$\displaystyle\left|l,m\right\rangle:=Y_{l}^{m}(\theta,\phi)=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}\,e^{im\phi}P_{l}^{(m)}(\cos\theta).$
(10)
They form an orthonormal set with respect to the following inner product
$\displaystyle\left\langle
l,m|l^{\prime},m^{\prime}\right\rangle:=\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}{Y_{l}^{m}}^{*}(\theta,\phi)Y_{l^{\prime}}^{m^{\prime}}(\theta,\phi)d\Omega(\theta,\phi)=\delta_{l\,l^{\prime}}\delta_{m\,m^{\prime}}.$
(11)
For a given $l-m=2k$ with the lower bound $m\geq-k$, the generating function
obtained in (4) can be used to construct coherent states as follows: Let us
define the infinite-dimensional Hilbert space ${\cal
H}_{k}:=\mbox{span}\left\\{\left|2k+m,m\right\rangle\right\\}_{\hskip
2.84526ptm\geq-k}$ equipped with the identity operator
$\sum_{m=-k}^{\infty}\left|2k+m,m\right\rangle\left\langle
2k+m,m\right|=I_{{}_{{\cal H}_{k}}}$. As an infinite superposition of the
spherical harmonics, generalized coherent states together with their explicit
compact forms can be calculated by (4) as
$\displaystyle\left|z\right\rangle_{k}:=N_{k}(|z|)\sum_{m=-k}^{\infty}\frac{(2k+m)!}{(k+m)!}\frac{z^{m}}{\sqrt{(4k+2m+1)(2k+2m)!}}\left|2k+m,m\right\rangle$
$\displaystyle=N_{k}(|z|)\frac{\left(\frac{-\sin\theta}{2e^{i\phi}z}\right)^{k}}{\sqrt{4\pi(2k)!}}\left[\frac{d^{2k}}{du^{2k}}\left(\left(1-u^{2}\right)^{k}e^{-\frac{z\left(1-u^{2}\right)}{2\sin\theta}e^{i\phi}}\right)\right]_{u=\cos\theta},$
(12)
in which $N_{k}(|z|)$ are the the normalization coefficients. Here, $z$ is an
arbitrary complex variable with the polar form $z=re^{i\varphi}$ so that
$0\leq r<\infty$ and $0\leq\varphi<2\pi$. The main ingredient of this work is
the convergence of the infinite expansions $\left|z\right\rangle_{k}$ as
coherent states to explicit compact forms of the known functions.
In what follows it is assumed that $k=0$. If the norm of the coherent states
$\left|z\right\rangle$ is supposed to be normalized to unity with respect to
the inner product (8), i.e., $\left\langle z|z\right\rangle=1$, then we find
the explicit form $N(|z|)=\sqrt{\frac{|z|}{\sinh|z|}}$ for the real
normalization coefficient. Also, we should introduce the appropriate measure
$d\mu(|z|)=rK(r)drd\varphi$ so that the resolution of the identity is realized
for the coherent states
$\displaystyle\left|z\right\rangle=\sqrt{\frac{|z|}{4\pi\sinh|z|}}e^{-\frac{z}{2}e^{i\phi}\sin\theta}$
(13)
in the Hilbert space ${\cal H}_{0}$,
$\displaystyle I_{{}_{{\cal H}_{0}}}=\int_{\mathbb{C}}|z\rangle\langle
z|\,d\mu(|z|)=2\pi\sum_{m=0}^{\infty}\frac{\left|m,m\right\rangle\left\langle
m,m\right|}{(2m+1)!}\int_{0}^{\infty}r^{2m+1}N^{2}(r)K(r)dr.$ (14)
Using the completeness relation
$\sum_{m=0}^{\infty}\left|m,m\right\rangle\left\langle m,m\right|=I_{{}_{{\cal
H}_{0}}}$ it is found that relation (11) is satisfied for the positive
definite measure $K(r)=\frac{\sinh r}{2\pi r}e^{-r}$.
## 4 Conclusions
The parameter $m$ allows us to calculate two different and new types of
exponential generating functions (4) and (5) for the associated Legendre
functions. Also, generating function corresponding to the Legendre polynomials
can be obtained as a special case of the non-exponential generating functions
(6) of the associated Legendre functions. The generating function
corresponding to infinite sequence $\\{P_{m}^{(m)}(x)\\}_{m=0}^{\infty}$ of
the associated Legendre functions is used as an application example to
construct the generalized coherent superposition $\left|z\right\rangle$ of the
spherical harmonics $Y_{m}^{m}(\theta,\phi)$. Its explicit compact form and
also, to realize the resolution of the identity, its corresponding positive
definite measure on the complex plane have been calculated.
## References
* [1] E. Schrödinger, Annalen der Physik 79 (4) (1926) 361.
* [2] R.J. Glauber, Phys. Rev. 130 (6) (1963) 2529.
* [3] R.J. Glauber, Phys. Rev. 131 (6) (1963) 2766.
* [4] J.R. Klauder, J. Math. Phys. 4 (8) (1963) 1055.
* [5] J.R. Klauder, J. Math. Phys. 4 (8) (1963) 1058.
* [6] E.C.G. Sudarshan, Phys. Rev. Lett. 10 (7) (1963) 277.
* [7] A.O. Barut and L. Girardello, Commun. Math. Phys. 21 (1) (1971) 41.
* [8] A.M. Perelomov, Commun. Math. Phys. 26 (3) (1972) 222.
* [9] J.P. Gazeau and J.R. Klauder, J. Phys. A: Math. Gen. 32 (1) (1999) 123.
* [10] J.P. Antoine, J.P. Gazeau, P. Monceau, J.R. Klauder and K.A. Penson, J. Math. Phys. 42 (6) (2001) 2349.
* [11] H. Fakhri, Phys. Lett. A 313 (4) (2003) 243.
* [12] J.R. Klauder and B-S. Skagerstam, “Coherent States. Applications in Physics and Mathematical Physics”, (World Scientific, Singapore, 1985).
* [13] A. Perelomov, “Generalized Coherent States and Their Applications”, (Springer-Verlag, Berlin Heidelberg New York, 1986).
* [14] D.H. Feng, J.R. Klauder and M.R. Strayer, “Coherent States. Past, Present, and Future”, (World Scientific, Singapore, 1994).
* [15] S.T. Ali, J.P. Antoine and J.P. Gazeau, “Coherent States, Wavelets and Their Generalizations”, (Springer-Verlag, Berlin Heidelberg New York, 2000).
* [16] T. Fukui and N. Aizawa, Phys. Lett. A 180 (4-5) (1993) 308.
* [17] A. Chenaghlou and H. Fakhri, Mod. Phys. Lett. A 17 (26) (2002) 1701.
* [18] A. Chenaghlou and H. Fakhri, Mod. Phys. Lett. A 19 (35) (2004) 2619.
* [19] H. Fakhri and A. Dehghani, J. Math. Phys. 50 (5) (2009) 052104.
* [20] H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. I, (McGraw-Hill, New York, 1953).
* [21] Z.E. Wang and D.R. Guo, Special Functions, (World Scientific, Singapore, 1989).
* [22] N. Ja. Vilenkin and A. U. Klimyk, Representations of Lie Groups and Special Functions Vol. II, (Dordrecht, Kluwer, 1993).
* [23] Shy-Der Lin, Shih-Tong Tu and H.M. Srivastava, Rend. Sem. Mat. Univ. Pol. Torino 59 (3) (2001) 199.
|
arxiv-papers
| 2014-04-21T05:11:26 |
2024-09-04T02:50:01.628322
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Fakhri and B. Mojaveri",
"submitter": "Bashir Mojaveri",
"url": "https://arxiv.org/abs/1404.5110"
}
|
1404.5396
|
# Atomic structure and energetics of large vacancies in graphene
J. Kotakoski [email protected] University of Vienna, Department of
Physics, Boltzmanngasse 5, 1090 Vienna, Austria University of Helsinki,
Department of Physics, P.O. Box 43, FI-00014, Finland F. R. Eder University
of Vienna, Department of Physics, Boltzmanngasse 5, 1090 Vienna, Austria J.
C. Meyer University of Vienna, Department of Physics, Boltzmanngasse 5, 1090
Vienna, Austria
(August 27, 2024)
###### Abstract
We present a computational study on the topology, energetics and structural
deformations for a large number of experimentally observed defect
configurations in graphene. We find that both the number of lost hexagonal
carbon rings and introduced non-hexagonal rings increase linearly as a
function of the vacancy order (number of missing atoms). The formation
energies of the defects increase by about 2.2 eV per missing atom after an
initial offset, establishing these defects as the lowest energy vacancy
configurations studied in graphene to date. In addition, we find that even
small point defects, which have been until now assumed flat, cause graphene to
bend out of plane when not restricted into prohibitively confined geometries.
This effect reaches to relative long distances even for some of the smallest
defects, significantly reducing the stress otherwise imposed on the
surrounding lattice.
###### pacs:
61.48.Gh, 61.72.-y, 07.05.Tp
## I Introduction
Two-dimensional materials, such as graphene and single layers of hexagonal
boron nitride and transition metal dichalcogenides, have provided an
unprecedent opportunity to directly image the atomic structure of various
lattice imperfections. For example, high resolution transmission electron
microscopy (HR-TEM) studies have revealed the atomic structure of various
graphene defects hashimoto_direct_2004 ; meyer_direct_2008 ;
warner_dislocation-driven_2012 and grain boundary structures
huang_grains_2011 ; kim_grain_2011 , while scanning tunneling microscopy (STM)
has provided a complementary view ugeda_point_2011 ; ugeda_electronic_2012 ;
tapaszto_mapping_2012 . The energetic electrons, which are used for the
imaging within TEM, have also been utilized to intentionally create defects
Kotakoski_Point_2011 ; Meyer_Accurate_2012 ; Robertson_Spatial_2012 ;
Robertson_Structural_2013 ; Warner_Bond_2013 and to drive their dynamics
Kotakoski_Point_2011 ; kotakoski_stone-walestype_2011 ; Kurasch_Atom-by-
atom_2012 ; Lehtinen_Atomic_2013 . However, with both TEM and STM, one remains
typically agnostic about any small height variations that may occur around the
defects. This is because changes in the sub-nm range are difficult to discern
based on the focus in a TEM experiment, whereas STM measurements for supported
graphene are more sensitive to height variations of the substrate than those
of graphene, and interactions between the STM tip and a freestanding graphene
can significantly alter its shape during the measurement. Eder_Probing_2013
From the theoretical point-of-view, defects in graphene have offered an
intriguing playing field allowing for a multitude of different atomic
configurations via incorporation of various non-hexagonal carbon rings into
the lattice. An overview on research of structural defects in graphene until
2011 is provided in Ref. Banhart_Structural_2011 . Since then, further studies
have been conducted on the atomic structure of point defects Warner_Bond_2013
; Robertson_Spatial_2012 ; Robertson_Structural_2013 , dislocations
warner_dislocation-driven_2012 ; Lehtinen_Atomic_2013 and grain boundaries
Rasool_Measurement_2013 . However, few attempts Jeong_Stability_2008 ;
Wang_Structural_2012 have been made to understand the properties of realistic
larger vacancy complexes in graphene. Further, only rarely in studies of
defects in graphene (with a few notable exceptions Ma_Stone-Wales_2009 ;
Liu_Cones_2010 ; Lehtinen_Atomic_2013 ), have out-of-plane variations been
adequately addressed. Instead, the structures have been typically assumed
flat, and the atomic projections in the TEM images have been interpreted as
corresponding to a flat structure under significant negative strain
Warner_Bond_2013 ; Rasool_Measurement_2013 . However, by bending out from the
flat configuration, a thin membrane can locally reduce the stress around
defects.
In this work, we study a set of defect configurations in graphene, produced
and observed during a TEM experiment under a 100 kV electron beam. The
formation of these defects has been previously described in Refs.
Kotakoski_Point_2011 ; Meyer_Accurate_2012 . TEM images of the defects can be
found in the supplementary video S4 of Ref. Meyer_Accurate_2012 . We find
that, on average, 1.6 hexagons will be transformed into 1.0 non-hexagonal
carbon rings per one missing atom. All of the studied structures are observed
to deform graphene in the out-of-plane direction, which significantly reduces
the local stresses around the defects. The formation energies increase
linearly as a function of the vacancy-order (by $\sim 2.19$ eV per missing
atom after an initial offset), establishing this set of defects as having the
lowest formation energies for any vacancy complexes hitherto studied in
graphene.
## II Methods and results
We start the analysis by looking at the topology of the defects (see Fig. 1a
for four example structures). In Fig. 1b, we plot the number of introduced
non-hexagonal polygons and the number of removed hexagons (with respect to the
pristine lattice) as a function of the vacancy order (number of missing
atoms). Out of all non-hexagonal carbon rings, 0.8% were tetragons, 55.3%
pentagons, 38.2% heptagons and 5.8% octagons. The number of removed hexagons
increases (on average) linearly with the number of removed atoms with a rate
of ca. 1.6 per atom. This is comparable to value of 2 for the V2(5-8-5), but
somewhat lower than those for the V1(5-9) or the energetically more stable
divacancies Banhart_Structural_2011 , namely, the V2(555-777) and the
V2(5555-6-7777), which have values of 3, 3.5 and 4.5 per missing atom,
respectively. The number of non-hexagonal rings, on the other hand, increases
at a rate of about 1.0 polygons/missing atom. Comparing to the average value
of $\sim 0.6$, for all of the above-mentioned divacancies, the difference
between removed hexagons and introduced other polygons is exactly 0.5 per
missing atom (1.0 for the single vacancy). We stress that all of the above
analysis is based on the average properties of all of the defect structures,
which means that for any specific defect, the values can deviate from the ones
listed. All of the analyzed atomic structures are available through Ref. suppl
.
Figure 1: (Color online) (a) Four examples of the 44 vacancy structures used
in this study with varying number of missing atoms ($N$). $x$ and $y$ are the
cartesian coordinates in the in-plane direction. (b) Number of missing
hexagonal carbon rings ($N_{6}^{ideal}-N_{6}$) and number of all other
polygons ($\sum_{i\neq 6}N_{i}$) as a function of $N$. Note that for many $N$,
there are several different configurations. Lines are fits to the data.
The largest defect structure considered in this study has 44 missing atoms and
involves nearly 80 lost hexagons with up to 40 other polygons introduced.
Thus, in order to contain each of the structures within a pristine lattice of
the same size, we had to create a relatively large simulation system into
which the defects were then (separately) introduced. As a consequence, also
the small well-studied defects were in this study relaxed in an atypically
large system. We carried out structural optimization for all of the defect
structures using the conjugate gradient energy minimization scheme employing
two different analytical interaction models to describe the energetics and
inter-atomic forces in the system, namely the PII parametrization by Brenner
from Ref. Brenner_Empirical_1990 and the improved version of the potential
(AIREBO) by Stuart et al. stuart_reactive_2000 . Both potentials reproduced
all of the trends reported within this work. However, since the bond lengths
and formation energies predicted by AIREBO are closer to density functional
theory (DFT) values, as will be to some extent described below, all of the
results shown were obtained with AIREBO. These simulations were carried out
with the LAMMPS code Plimpton_Fast_1995 ; _lammps_???? . The DFT results were
calculated with VASP kresse_efficiency_1996 ; kresse_efficient_1996 using
projector augmented wave potentials blochl_projector_1994 . Plane wave cut-off
was set to 300 eV, and the exchange and correlation were described with the
generalized gradient approximation parametrized by Perdew, Burke and Ernzerhof
perdew_generalized_1996 . Only one $\mathbf{k}$-point was used ($\Gamma$).
These parameters were dictated by the relative large system size (up to 880
atoms for the DFT simulations).
During initial relaxation of the structures with both analytical potentials,
we noticed that the defects caused the lattice to bend out from the flat
configuration. In order to check that this observation was not caused by a
simulation artefact, we selected the V2(5-8-5) divacancy to study this effect.
This defect has been extensively studied in the past (see, f.ex., Ref.
krasheninnikov_bending_2006 ). Curiously, it is known to lead to a local
curvature change in carbon nanotubes krasheninnikov_bending_2006 ;
kotakoski_energetics_2006 , but for graphene, as far as we know, only flat
configurations have hitherto been reported. As can be seen in Fig. 2a, the
formation energy of this defect decreases with increasing system size up to
about 1,000 atoms due to repulsion between the defect and its mirror images
over the periodic boundaries at shorter inter-defect distances. At system
sizes exceeding 100 atoms, the flat configuration becomes less favorable than
the buckled one, and remains so up to the largest studied systems. From this
data, we can conlude that defect-to-defect interactions reach up to at least
15 nm (corresponding to the system size for 20,000 atoms in this study) even
in the case of small point defects in graphene. Interestingly, we find two
possible buckled structures for this defect with an anti-parallel and a
parallel symmetry with respect to the graphene plane [see Fig. 2b]. Both of
these configurations are favored over the flat structure.
Figure 2: (Color online) Effect of system size and out-of-plane corrugations
on the formation energy of a V2(5-8-5) divacancy structure in graphene. (a)
Formation energy $E_{f}$ as a function of the system size ($N_{atoms}$, number
of atoms) for the flat and the two non-flat configurations (with parallel and
anti-parallel symmetries) and the maximum distance of two atoms within the
relaxed non-flat structures in the out-of-plane direction ($\Delta z$). (b)
Out-of-plane corrugations for the two non-flat configurations. The lines are
guides to the eye. $x$ and $y$ are the cartesian coordinates in the in-plane
direction.
We point out that no restriction was required to keep the structure flat
during the relaxation at any system size. According to the analytical
potentials, this configuration presents thus a metastable defect state even
when it is less favored than the nonflat structure. However, one can
reasonably assume that in any practically relevant situation, the graphene
membrane will tend towards the energetically favored configuration due to
external influences (f.ex., thermal vibrations). We assume that these reasons
have until now hidden the corrugated nature of graphene with point defects in
theoretical studies. In contrast to rather large system sizes required to
reveal the out-of-plane bending of the divacancy, earlier quantum monte carlo
and DFT results have shown that the Stone-Wales defect bends the graphene
sheet already at much smaller system sizes (down to just a few tens of atoms)
Ma_Stone-Wales_2009 .
Also in Fig. 2a (second $y$-axis), we plot the maximum distance for any two
atoms in the out-of-plane direction for the optimized locally buckled
configurations ($\Delta z$). The local height of the membrane around the
defect increases continuously indicating that the further decrease in strain,
mediated by the out-of-plane deformations, does not significantly alter the
energetics of the defect and that the defects become sufficiently spatially
isolated at the largest system sizes.
We further looked into how the out-of-plane relaxation correlates with the
release of the local negative strain. The length of the shortest bond in each
relaxed structure was found to increase monotonously with the defect height
for the buckled structures (from 2.4% shrinkage with respect to ideal graphene
up to 1.7% shrinkage over the studied range), whereas for the flat structure
the shortest bond remained at 2.4% shrinkage regardless of the system size.
The number of bonds with more than 0.5% shrinkage saturated at 26 for the non-
flat structures and at 36 for the flat ones at the largest system sizes. These
numbers clearly show that the out-of-plane relaxation is an important way for
the graphene lattice to relieve negative strain caused by the formation of
defects. We also checked how the deformation reacts to strain in the lattice.
It turns out that 0.5% is enough to make the divacancy structure flat for a
20,000-atom system. However, strains up to several % are required to flatten
any of the larger defects. A more in-depth study of these deformations is out
of the scope of the present work.
Based on the above analysis, we established 20,000 atoms as a reasonable
system size for the AIREBO simulations (we tested system sizes up to 180,000
to check that there are no significant changes at formation energies even for
the largest defects). However, structures this large are clearly beyond what
can be typically modeled with any first principles method, $\sim 1,000$ atoms
being more typical. A comparison of our AIREBO results for systems of 20,000
atoms and 880 atoms showed that the formation energies of small defects ($\sim
6$ missing atoms) were already accurate within 0.5 eV for the small system
with a few exceptions being accurate within 0.1 eV, while larger defects
exhibited deviations in the range of $2-3$ eV. Additionally, as was shown
above, the buckled structures appear energetically favored over the flat ones
already for the 880-atom system. (Although, the preferred mode of bending for
an isolated defect may remain hidden.)
In order to qualify our AIREBO simulations, we repeated some of the V2(5-8-5)
simulations with DFT for system sizes of 72, 200 and 880 atoms. For the
880-atomic system, we obtained formation energy of 7.13 eV for the buckled
structure (anti-parallel) and 7.21 eV for the flat one, confirming our AIREBO
results with a similar energy difference (0.04 eV for AIREBO for the same
system size). The value for the flat structure is in line with the results
reported in the literature Banhart_Structural_2011 . (The AIREBO overestimates
$E_{f}$ for this defect by about 1 eV.) The maximum atom-to-atom distance in
the out-of-plane direction ($\Delta z$), as obtained from DFT simulations is
ca. 0.76 Å, indicating that AIREBO slightly overestimates the magnitude of the
deformations. For both of the smaller systems (72 and 200 atoms), DFT
simulations converged into the flat structure, showing that they remain too
small to accompany the out-of-plane relaxation.
Figure 3: Formation energies per number of missing atoms $E_{f}/N$ for vacancy
structures, as calculated with the AIREBO potential for a system of 20,000
atoms. The dashed curve is a fit to the data.
After establishing the reliability of the AIREBO results regarding energetics
and structural deformations, we expanded the formation energy study for all of
the defect structures obtained from TEM images. The results are shown in Fig.
3 (notice that there are several different defects with same number of missing
atoms, $N$). The formation energies scale linearly with the increasing vacancy
order ($E_{f}/N\propto 1/N$), increasing by ca. 2.19 eV per missing atom, with
an initial offset of about 5.35 eV. This energy penalty is associated with the
local deformation of bonds around the locations of removed atoms. The linear
dependency is similar to that found by Jeong et al. Jeong_Stability_2008 for
dislocation lines and local hæckelite structures [composed of merged
V2(555-777) defects] up to 12 missing atoms, although the formation energies
of the vacancy structures considered here are substantially lower (by about
0.5 eV per missing atom). Introduction of the first defect into the pristine
lattice is associated with the highest energy penalty (the initial offset in
$E_{f}$), because the growing defect can easier accumulate the non-ideal bonds
due to effects like overlapping strain fields with opposing signs
kotakoski_energetics_2006 , which leads to lower energy penalty for the
removal of the additional atoms. This behavior is the physical reason why
graphene can be turned into an amorphous 2D carbon glass via introduction of a
growing number of defects Kotakoski_Point_2011 ; eder_journey_2014 .
Figure 4: (Color online) Out-of-plane buckling. (a) Height maps for four
example defects with different number of missing atoms ($N=4,8,16$ and $28$,
from left to right). $x$ and $y$ are the cartesian coordinates in the in-plane
direction. (b) Maximum difference in the out-of-plane direction ($\Delta z$)
for each of the defects as a function of the number of missing atoms, $N$. The
dashed and dotted lines are fits to the data for $N<15$ and $N>15$,
respectively. (c) Relative increase of the in-plane size of the local out-of-
plane deformation as compared to the largest one as a function of $\Delta z$.
Finally, we turn to look at the out-of-plane deformation caused by the larger
vacancies. As an example, four height maps for defects with different $N$ are
presented in Fig. 4a. Also the $\Delta z$ and an estimation of the in-plane
size of the out-of-plane deformation for all defects are plotted in Fig. 4b,c.
This estimation was made by plotting height maps, similar to those in Fig. 4a
but for a larger area, and by drawing a circle around each defect so that all
areas with a height deviation more than an arbitrarily selected $\Delta z_{0}$
were contained inside the circles. The diameter of these circles was then used
as the measure for the size. This approach resulted in estimation of
deformation length in the in-plane direction, which is linearly dependent on
$\Delta z$, showing that each of the buckled structures has approximately the
same local curvature. $\Delta z$ itself appears to increase for defects with
$N<15$ proportional to $\sqrt{N}$, which agrees with the assumption that the
area of the defect grows linearly with $N$. At $N\approx 15$, however, $\Delta
z$ saturates to approximately a constant value of ca. 4.3 Å. This is most
likely due to the fact that at this point the growth of the defect leads to
buckling at the defect itself (increasing the frequency of the out-of-plane
deformations around the defect) without further contribution to the buckling
amplitude of the membrane. This can be seen as the increased number of ripples
for the largest defect in Fig. 4a as compared to the smaller defects. These
trends, and even absolute values, remained almost completely unchanged also
for the largest studied system sizes (up to 180,000 atoms). The actual
interaction distances reach up to tens of nm, as was already discussed above
in the case of V2(5-8-5).
## III Conclusions
As a conclusion, we have presented a computational study on the topology,
energetics and atomic structure of experimentally observed defect structures
in graphene. Unlike what has been previously assumed, all of the defects,
including the smallest ones, cause local buckling of graphene to lower the
negative strain otherwise imposed on the lattice. The number of removed
hexagonal carbon rings and introduced other polygons are shown to
systematically increase as a function of the number of missing atoms. The
ratios of the other polygons remain approximately constant. Formation energies
of the defects increase linearly with the increasing defect size by ca. 2.2 eV
per removed atom after an initial onset of about 5.4 eV. Finally, the height
of the buckling is shown to increase as a square root of the number of removed
atoms until 15 missing atoms before saturating to up to 4 Å for the largest
defects. These deformations result in an interaction length between defects in
graphene in the regime of tens of nanometers.
We acknowledge Austrian Science Fund (FWF): M 1481-N20 and Helsinki University
Funds for funding and the Vienna Scientific Cluster for generous grants of
computational resources.
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|
arxiv-papers
| 2014-04-22T07:10:08 |
2024-09-04T02:50:01.640883
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jani Kotakoski, Franz R. Eder, Jannik C. Meyer",
"submitter": "Jani Kotakoski",
"url": "https://arxiv.org/abs/1404.5396"
}
|
1404.5422
|
# Jacob’s ladders and laws that control chaotic behavior of the measures of
reversely iterated segments
Jan Moser Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
[email protected]
###### Abstract.
The main subject to study in this paper are properties of the sequence of
reversely iterated segments. Especially, we will examine properties of chaotic
behavior of the sequence of measures of corresponding segments. Our results
are not accessible within current methods in the theory of Riemann zeta-
function.
###### Key words and phrases:
Riemann zeta-function
## 1\. Introduction
### 1.1.
Let us start with some notions and formulae to be reminded:
* (A)
the sequence
$\\{\overset{k}{T}\\}_{k=1}^{k_{0}}$
is defined by (see [3], (5.1))
$\varphi_{1}(\overset{k}{T})=\overset{k-1}{T},\ k=1,\dots,k_{0},\
\overset{0}{T}=T,\ T\geq T_{0}[\varphi_{1}],$
where $k_{0}\in\mathbb{N}$ is an arbitrary fixed number and $\varphi_{1}(t)$
is the Jacob’s ladder;
* (B)
next (see [3], (1.3))
(1.1) $\begin{split}&\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm
d}t}=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}}{\omega(t)},\\\
&\omega(t)=\left\\{1+\mathcal{O}\left(\frac{\ln\ln t}{\ln
t}\right)\right\\}\ln t.\end{split}$
where
(1.2)
$\begin{split}&Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\\\
&\vartheta(t)=-\frac{t}{2}\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+i\frac{t}{2}\right),\end{split}$
### 1.2.
We have proved the following theorem (see [3], (2.1) – (2.7)): for every
$L_{2}$-orthogonal system
$\\{f_{n}(t)\\}_{n=1}^{\infty},\ t\in[0,2l],\ l=o\left(\frac{T}{\ln
T}\right),\ T\to\infty$
there is a continuum set of $L_{2}$-orthogonal systems
$\begin{split}&\\{F_{n}(t;T,k,l)\\}_{n=1}^{\infty}=\\\
&=\left\\{f_{n}(\varphi_{1}(t)-T)\prod_{r=0}^{k-1}\left|\tilde{Z}[\varphi_{1}^{r}(t)]\right|\right\\}_{n=1}^{\infty},\
t\in[\overset{k}{T},\overset{k}{\wideparen{T+2l}}],\end{split}$
where
(1.3)
$\begin{split}&\varphi_{1}\\{[\overset{k}{T},\overset{k}{\wideparen{T+2l}}]\\}=[\overset{k-1}{T},\overset{k-1}{\wideparen{T+2l}}],\
k=1,\dots,k_{0},\\\
&[\overset{0}{T},\overset{0}{\wideparen{T+2l}}]=[T,T+2l],\end{split}$
i.e. the following formula is valid
$\begin{split}&\int_{\overset{k}{T}}^{\overset{k}{\wideparen{T+2l}}}f_{m}(\varphi_{1}^{k}(t)-T)f_{n}(\varphi_{1}^{k}(t)-T)\prod_{r=0}^{k-1}\tilde{Z}^{2}[\varphi_{1}^{r}(t)]{\rm
d}t=\\\ &=\left\\{\begin{array}[]{rcl}0&,&m\not=n,\\\
A_{n}&,&m=n,\end{array}\right.\quad A_{n}=\int_{0}^{2l}f_{n}^{2}(t){\rm
d}t.\end{split}$
###### Remark 1.
It is clear that the base of above mentioned result is new notion of reverse
iterations (comp. (1.3)) in the theory of Riemann
$\zeta\left(\frac{1}{2}+it\right)$-function.
In this paper we will study the sequence of reverse iterations
$\left\\{[\overset{r}{T},\overset{r}{\wideparen{T+H}}]\right\\}_{r=0}^{k},\
k=1,\dots,k_{0}$
alone. Namely, we will focus on properties of the sequence of real numbers
(measures of corresponding segments)
$\left\\{|[\overset{r}{T},\overset{r}{\wideparen{T+H}}]|\right\\}_{r=0}^{k}.$
###### Remark 2.
Results of this paper are not accessible by current methods of the theory of
Riemann zeta-function. We mention explicitly that our results are valid also
in the microscopic case
$H\in\left(\left.0,\frac{A}{\ln T}\right]\right.,\quad T\to\infty.$
## 2\. Theorem 1 and motivation behind it
### 2.1.
Let us remind that the segments
$[\overset{r}{T},\overset{r}{\wideparen{T+H}}],\ r=0,1,\dots,k$
are components of disconnected set
(2.1)
$\Delta(T,H,k)=\bigcup_{r=0}^{k}[\overset{r}{T},\overset{r}{\wideparen{T+H}}],\
k=1,\dots,k_{0},$
(comp. [3], (2.9)). Properties of the set (2.1) are listed below (see [3],
(2.5) – (2.7)):
(2.2) $\begin{split}&H=o\left(\frac{T}{\ln T}\right)\ \Rightarrow\\\
&|[\overset{k}{T},\overset{k}{\wideparen{T+H}}]|=\overset{k}{\wideparen{T+H}}-\overset{k}{T}=o\left(\frac{T}{\ln
T}\right),\end{split}$ (2.3)
$|[\overset{k-1}{\wideparen{T+H}},\overset{k}{T}]|=\overset{k}{T}-\overset{k-1}{\wideparen{T+H}}\sim(1-c)\pi(T);\
\pi(T)\sim\frac{T}{\ln T},$ (2.4)
$[T,T+H]\prec[\overset{1}{T},\overset{1}{\wideparen{T+H}}]\prec\dots\prec[\overset{k}{T},\overset{k}{\wideparen{T+H}}]\prec\dots,$
where $c$ is the Euler’s constant and $\pi(T)$ is the prime-counting function.
###### Remark 3.
Consequently, the asymptotic behavior of our disconnected set (2.1) is as
follows (see (2.2), (2.3)): if $T\to\infty$ then the components of the set
(2.1) recede unboundedly each from other and all together are receding to
infinity. Hence, the set (2.1) behaves as a kind of one-dimensional Friedmann-
Hubble expanding universe.
Furthermore, we notice explicitly that the distance $\rho_{l}$ of the two
consecutive segments
$[\overset{l-1}{T},\overset{l-1}{\wideparen{T+H}}],\
[\overset{l}{T},\overset{l}{\wideparen{T+H}}],\quad l=1,2,\dots,k$
is extremely big one, namely (see (2.3))
(2.5) $\rho_{l}\sim(1-c)\frac{T}{\ln T}\to\infty,\quad T\to\infty.$
###### Remark 4.
Since the sequence
$\\{[\overset{r}{T},\overset{r}{\wideparen{T+H}}]\\}_{r=0}^{k}$
is extremely sparse one (see (2.5)) then we may assume that the behavior of
the measures
$\\{|[\overset{r}{T},\overset{r}{\wideparen{T+H}}]|\\}_{r=0}^{k}$
is chaotic one.
Consequently, in correspondence with Remark 3, we wish to obtain some law
controlling this chaotic behavior. In this direction, the following theorem
holds true.
###### Theorem 1.
Let
(2.6) $1\leq n\leq k_{0},\quad\bar{H}=o\left(\frac{T}{\ln T}\right),$
and let the inequality
(2.7)
$|[\overset{n}{T},\overset{n}{\wideparen{T+\bar{H}}}]|=\overset{n}{\wideparen{T+\bar{H}}}-\overset{n}{T}\geq
T^{1/3+\epsilon},\ T\to\infty$
hold true for $\epsilon>0$ \- an arbitrary small fixed number. Then we have
that
(2.8) $\overset{n}{\wideparen{T+\bar{H}}}-\overset{n}{T}\sim\bar{H},\
T\to\infty.$
### 2.2.
Next, let us remind
* (A)
the Hardy-Littlewood-Ingham formula
(2.9)
$\begin{split}&\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=T\ln T+(2c-1-\ln 2\pi)T+R(T),\end{split}$
with the Balasubramanian’s estimate (for example)
(2.10) $R(T)=\mathcal{O}(T^{1/3})$
of the error term in (2.9);
* (B)
the Good’s $\Omega$-theorem that states
(2.11) $R(T)=\Omega(T^{1/4}),\ T\to\infty;$
* (C)
our almost exact formula (see [1], (2.1), (2.2),
$\frac{y}{2}\to\varphi_{1}(t)$)
(2.12)
$\begin{split}&\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t=\\\ &=\varphi_{1}(T)\ln\varphi_{1}(T)+(c-\ln
2\pi)\varphi_{1}(T)+c_{0}+\mathcal{O}\left(\frac{\ln T}{T}\right),\
T\to\infty,\end{split}$
where $c$ is the Euler’s constant and $c_{0}$ is the constant from the
Titchmarsh-Kober-Atkinson formula (see [4], p. 141).
Our discussion concerning formulae (2.10) – (2.12) see in [1], pp. 416, 417.
###### Remark 5.
Consequently, we have proved in [1] that classical Hardy-Littlewood integral
(1918)
$\int_{0}^{T}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm d}t$
has – in addition to the Hardy-Littlewood (and other similar) expressions
possessing unbounded errors (as $T\to\infty$), (comp. (2.10), (2.11)) –
infinite set of almost exact expressions (2.12).
###### Remark 6.
It is clear – in context of (2.10), (2.11) – that our Theorem 1 will be true
for every improvement of the exponent $\frac{1}{3}$:
$\frac{1}{3}\longrightarrow a\in\left(\frac{1}{4},\frac{1}{3}\right).$
## 3\. Proof of Theorem 1
First of all, it follows from (2.9) and (2.10) that
(3.1)
$\begin{split}&\int_{T}^{T+U}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim U\ln T,\ T\to\infty,\\\ &T^{1/3+\epsilon}\leq U=o\left(\frac{T}{\ln
T}\right).\end{split}$
Next, we use, together with (3.1), our formula
(3.2)
$\int_{\overset{k}{T}}^{\overset{k}{\wideparen{T+H}}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim(\overset{k-1}{\wideparen{T+H}}-\overset{k-1}{T})\ln T$
that follows from [3], (1.1) – (1.3), (7.4) with
$[T,T+H]\longrightarrow[\overset{k-1}{T},\overset{k-1}{\wideparen{T+H}}],\
[\overset{1}{T},\overset{1}{\wideparen{T+H}}]\longrightarrow[\overset{k}{T},\overset{k}{\wideparen{T+H}}].$
Of course, we have (see (2.2))
(3.3) $\begin{split}&H=o\left(\frac{T}{\ln T}\right)\ \Rightarrow\\\
&\Rightarrow\ \overset{k}{\wideparen{T+H}}-\overset{k}{T}=o\left(\frac{T}{\ln
T}\right),\quad T\to\infty,\ k=1,\dots,k_{0}.\end{split}$
Further, if $n,\bar{H}$ fulfill the conditions (2.6) and (2.7) then we have
(see (3.1), (3.2)) that
$\int_{\overset{n}{T}}^{\overset{n}{\wideparen{T+\bar{H}}}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim(\overset{n}{\wideparen{T+\bar{H}}}-\overset{n}{T})\ln T,$
and
$\int_{\overset{n}{T}}^{\overset{n}{\wideparen{T+\bar{H}}}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t\sim(\overset{n-1}{\wideparen{T+\bar{H}}}-\overset{n-1}{T})\ln T,$
i.e.
$\overset{n}{\wideparen{T+\bar{H}}}-\overset{n}{T}\sim\overset{n-1}{\wideparen{T+\bar{H}}}-\overset{n-1}{T},\quad
T\to\infty,$
and, consequently,
$\overset{n-1}{\wideparen{T+\bar{H}}}-\overset{n-1}{T}\sim\overset{n-2}{\wideparen{T+\bar{H}}}-\overset{n-2}{T}\sim\dots\sim
T+\bar{H}-T=\bar{H},$
(see also (3.3)). Thus, we have that
$\overset{n}{\wideparen{T+\bar{H}}}-\overset{n}{T}\sim\bar{H},\quad
T\to\infty,$
i.e. the assertion (2.8) is verified.
## 4\. Consequences of Theorem 1
### 4.1.
###### Corollary 1.
Let
(4.1) $H_{1}=A(T)T^{1/3+\epsilon},\quad 0<A(T)<1,$
for example
$H_{1}=\frac{1}{2}T^{1/3+\epsilon},\ \frac{1}{\ln\ln
T}T^{1/3+e\epsilon},\dots$
Then
(4.2) $\overset{k}{\wideparen{T+H_{1}}}-\overset{k}{T}<T^{1/3+e\epsilon},\quad
k=1,\dots,k_{0}.$
###### Remark 7.
Hence, in the case (4.1) we have that all members of the sequence
$\left\\{|[\overset{k}{T},\overset{k}{\wideparen{T+H_{1}}}]|\right\\}_{k=1}^{k_{0}}$
are lying below the level $T^{1/3+\epsilon}$, (see (4.2)).
### 4.2.
Next, as a consequence of Corollary 1, we have
###### Corollary 2.
If
(4.3) $H_{2}=B(T)T^{1/3+\epsilon},\quad B(T)>1,$
for example
$H_{2}=2T^{1/3+\epsilon},\ T^{1/3+\epsilon}\ln T,\dots$
and there is some
$n:\ 1\leq n<k_{0}$
such that
(4.4) $\overset{n}{\wideparen{T+H_{2}}}-\overset{n}{T}<A(T)T^{1/3+\epsilon}$
(see (4.1)), then
(4.5) $\overset{k}{\wideparen{T+H_{2}}}-\overset{k}{T}<T^{1/3+\epsilon},\quad
k=n+1,\dots,k_{0}.$
###### Remark 8.
Consequently, in the case (4.3), (4.4) the second jump of the sequence
$\left\\{|[\overset{k}{T},\overset{k}{\wideparen{T+H_{2}}}]|\right\\}_{k=1}^{k_{0}}$
over the segment
(4.6) $[(1-\epsilon)T^{1/3+\epsilon},(1+\epsilon)T^{1/3+\epsilon}]$
is forbidden. In other words, the oscillations of the sequence of measures
about the measure of the segment (4.6) are forbidden.
## 5\. An estimate from below
### 5.1.
We will use the following in this section:
* (A)
the estimate
$\begin{split}&H=o\left(\frac{T}{\ln T}\right)\ \Rightarrow\\\
&\int_{\overset{k}{T}}^{\overset{k}{\wideparen{T+H}}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t>(1-\epsilon)(\overset{k-1}{\wideparen{T+H}}-\overset{k}{T})\ln T,\
k=1,\dots,k_{0},\end{split}$
that follows from the asymptotic formula (3.2), i.e. we have that
(5.1)
$\begin{split}&\int_{\overset{1}{T}}^{\overset{1}{\wideparen{T+H}}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t>(1-\epsilon)H\ln T,\\\
&\int_{\overset{2}{T}}^{\overset{2}{\wideparen{T+H}}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t>(1-\epsilon)(\overset{1}{\wideparen{T+H}}-\overset{1}{T})\ln T,\\\
&\vdots\\\
&\int_{\overset{k_{0}}{T}}^{\overset{k_{0}}{\wideparen{T+H}}}\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}{\rm
d}t>(1-\epsilon)(\overset{k_{0}-1}{\wideparen{T+H}}-\overset{k_{0}}{T})\ln
T;\end{split}$
* (B)
the property (see (2.1) and [3], sec. 4.1)
(5.2) $\tau\in\Delta(T,H,k)\ \Rightarrow\
\tau\in\left[T,T+\mathcal{O}\left(\frac{T}{\ln T}\right)\right]\subset[T,2T].$
### 5.2.
Since (comp. [4], p. 99)
(5.3) $\begin{split}&\left|\zeta\left(\frac{1}{2}+it\right)\right|<t^{1/6},\
t\to\infty\ \Rightarrow\\\
&\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}<t^{1/3},\ t\to\infty,\
t\to\infty,\end{split}$
then we have (see (5.2), (5.3)), that
(5.4) $\begin{split}&t\in\Delta(T,H,k)\ \Rightarrow\\\
&\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}<\sqrt[3]{2}T^{1/3}<2T^{1/3},\
T\to\infty,\end{split}$
without any hypothesis. Consequently, we have (see (5.1), (5.4)) following
estimates
(5.5)
$\begin{split}&|[\overset{1}{T},\overset{1}{\wideparen{T+H}}]|>\frac{1-\epsilon}{2}HT^{-1/3}\ln
T,\\\
&|[\overset{2}{T},\overset{2}{\wideparen{T+H}}]|>\left(\frac{1-\epsilon}{2}HT^{-1/3}\ln
T\right)^{2}H,\\\ &\vdots\\\
&|[\overset{k_{0}}{T},\overset{k_{0}}{\wideparen{T+H}}]|>\left(\frac{1-\epsilon}{2}HT^{-1/3}\ln
T\right)^{k_{0}}H>\left(\frac{1}{4}T^{-1/3}\ln T\right)^{k_{0}}H=\\\
&=\left(\frac{\ln^{3}T}{64T}\right)^{k_{0}/3}H,\quad\epsilon\in(0,1/2).\end{split}$
Since
$0<\frac{\ln T}{4T^{1/3}}<1,\ T\to\infty,$
then we have the following
###### Theorem 2.
(5.6) $\begin{split}&H=o\left(\frac{T}{\ln T}\right)\ \Rightarrow\\\
&|[\overset{k}{T},\overset{k}{\wideparen{T+H}}]|>\left(\frac{\ln^{3}T}{64T}\right)^{k_{0}/3}H,\
k=1,\dots,k_{0},\ T\to\infty.\end{split}$
###### Example.
If
$H=1,\ k_{0}=3000$
then (see (5.6))
$|[\overset{k}{T},\overset{k}{\wideparen{T+H}}]|>\left(\frac{\ln
T}{64T}\right)^{1000},\ k=1,\dots,3000.$
###### Remark 9.
It appears that only advantage of the estimate (5.6) is, probably, its non-
triviality.
## 6\. Riemann hypothesis and our estimate from below
The following estimate
$\left|\zeta\left(\frac{1}{2}+it\right)\right|<Be^{A\frac{\ln t}{\ln\ln
t}},\quad t\to\infty$
holds true on the Riemann hypothesis (see [4], p. 300). We use this estimate
in the form
$\left|\zeta\left(\frac{1}{2}+it\right)\right|<t^{\frac{C}{\ln\ln t}},\quad
t\to\infty.$
Thus we have (comp. (5.2), (5.4)) that
(6.1) $\begin{split}&t\in\Delta(T,H,k)\ \Rightarrow\
\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}<(2T)^{\frac{2C}{\ln\ln(2T)}}<(2T)^{\frac{2C}{\ln\ln
T}}<\\\ &<2^{\frac{2C}{\ln\ln T}}T^{\frac{2C}{\ln\ln
T}}<(1+\epsilon)T^{\frac{2C}{\ln\ln T}},\quad\epsilon\in(0,1/2),\quad
T\to\infty.\end{split}$
Now, we obtain from (5.1), (6.1), (comp. (5.2), (5.5)) that
$\begin{split}&|[\overset{k_{0}}{T},\overset{k_{0}}{\wideparen{T+H}}]|>\left(\frac{1-\epsilon}{1+\epsilon}\right)^{k_{0}}HT^{-k_{0}\frac{2C}{\ln\ln
T}}\ln^{k_{0}}T>\\\ &>\frac{1}{3^{k_{0}}}HT^{-k_{0}\frac{2C}{\ln\ln
T}}\ln^{k_{0}}T=\\\ &=T^{-k_{0}\frac{\ln 3}{\ln T}}HT^{-k_{0}\frac{2C}{\ln\ln
T}}T^{k_{0}\frac{\ln\ln T}{\ln T}}>HT^{-k_{0}\frac{2D}{\ln\ln T}}.\end{split}$
Hence, the following theorem holds true.
###### Theorem 3.
On Riemann hypothesis we have
(6.2) $\begin{split}&H=o\left(\frac{T}{\ln T}\right)\ \Rightarrow\\\
&|[\overset{k}{T},\overset{k}{\wideparen{T+H}}]|>HT^{-k_{0}\frac{2D}{\ln\ln
T}},\quad k=1,\dots,k_{0},\quad T\to\infty.\end{split}$
###### Remark 10.
The conditional estimate (6.2) is effective particularly in the case
(6.3) $H=T^{\Delta},\ 0<\Delta<1.$
Namely, in this case we obtain from (6.2)
(6.4) $|[\overset{k}{T},\overset{k}{\wideparen{T+H}}]|>T^{\Delta-o(1)},\quad
T\to\infty.$
###### Remark 11.
It was expected that the Riemann hypothesis has essential influence on that
estimate from below. Actually, we have in the case (6.3) that:
* (A)
without any hypothesis (see (5.6))
(6.5)
$|[\overset{k}{T},\overset{k}{\wideparen{T+T^{\Delta}}}]|>\left(\frac{1}{4}\ln
T\right)^{k_{0}}T^{\Delta-\frac{k_{0}}{3}},$
where
(6.6) $k_{0}\geq 3\ \Rightarrow\ \Delta-\frac{k_{0}}{3}<0;$
* (B)
on the Riemann hypothesis (see (6.4))
(6.7)
$|[\overset{k}{T},\overset{k}{\wideparen{T+T^{\Delta}}}]|>T^{\Delta-o(1)},$
where
(6.8) $0<\Delta-o(1)\to\Delta\ \text{as}\ T\to\infty.$
###### Example.
In the case
$\Delta=\frac{1}{3}+\epsilon$
(see Theorem 1) we have, on Riemann hypothesis, that (see (6.4))
$|[\overset{k}{T},\overset{k}{\wideparen{T+T^{1/3+\epsilon}}}]|>T^{1/3+\epsilon-o(1)},\quad
k=1,\dots,k_{0},\quad T\to\infty.$
I would like to thank Michal Demetrian for his help with electronic version of
this paper.
## References
* [1] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral’, Math. Notes 88, (2010) 414-422, arXiv: 0901.3937.
* [2] J. Moser, ‘Jacob’s ladders, the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations‘, Proc. Stek. Inst. 276, (2011), 208-221, arXiv: 1103.0359.
* [3] J. Moser, ‘Jacob’s ladders, reverse iterations and new infinite set of $L_{2}$-orthogonal systems generated by the Riemann $\zeta\left(\frac{1}{2}+it\right)$-function‘, arXiv: 1402.2098, (2014).
* [4] E.C. Titchmarsh, ‘ _The theory of the Riemann zeta-function_ ‘ Clarendon Press, Oxford, 1951.
|
arxiv-papers
| 2014-04-22T08:43:35 |
2024-09-04T02:50:01.647693
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jan Moser",
"submitter": "Jan Moser",
"url": "https://arxiv.org/abs/1404.5422"
}
|
1404.5652
|
# Resolved Multifrequency Radio Observations of GG Tau
Sean M. Andrews11affiliation: Harvard-Smithsonian Center for Astrophysics, 60
Garden Street, Cambridge, MA 02138, USA , Claire J. Chandler22affiliation:
National Radio Astronomy Observatory, P.O. Box O, Socorro, NM 87801, USA ,
Andrea Isella33affiliation: California Institute of Technology, 1200 East
California Blvd., Pasadena, CA 91125, USA , T. Birnstiel11affiliation:
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA , K. A. Rosenfeld11affiliation: Harvard-Smithsonian Center for
Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA , D. J.
Wilner11affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden
Street, Cambridge, MA 02138, USA , L. M. Pérez22affiliation: National Radio
Astronomy Observatory, P.O. Box O, Socorro, NM 87801, USA , L.
Ricci33affiliation: California Institute of Technology, 1200 East California
Blvd., Pasadena, CA 91125, USA , J. M. Carpenter33affiliation: California
Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA ,
N. Calvet44affiliation: Department of Astronomy, University of Michigan, 500
Church Street, Ann Arbor, MI 48109, USA , S. A. Corder55affiliation: Joint
ALMA Observatory, Avenida Alonso de Córdova 3107, Vitacura, Santiago, Chile ,
A. T. Deller66affiliation: The Netherlands Institute for Radio Astronomy
(ASTRON), 7990-AA Dwingeloo, Netherlands , C. P. Dullemond77affiliation:
Heidelberg University, Center for Astronomy, Albert Ueberle Str 2, Heidelberg,
Germany , J. S. Greaves88affiliation: University of St. Andrews, Physics and
Astronomy, North Haugh, St. Andrews KY16 9SS, UK , R. J. Harris99affiliation:
Department of Astronony, University of Illinois, Urbana, IL 61810, USA , Th.
Henning1010affiliation: Max Planck Institut für Astronomie, Königstuhl 17,
69117 Heidelberg, Germany , W. Kwon1111affiliation: SRON Netherlands Institute
for Space Research, Landleven 12, 9747 AD Groningen, The Netherlands ,
J. Lazio1212affiliation: Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91106, USA , H.
Linz1010affiliation: Max Planck Institut für Astronomie, Königstuhl 17, 69117
Heidelberg, Germany , L. G. Mundy1313affiliation: Department of Astronomy,
University of Maryland, College Park, MD 20742, USA , A. I.
Sargent33affiliation: California Institute of Technology, 1200 East California
Blvd., Pasadena, CA 91125, USA , S. Storm1313affiliation: Department of
Astronomy, University of Maryland, College Park, MD 20742, USA , L.
Testi1414affiliation: European Southern Observatory, Karl Schwarzschild Str 2,
85748, Garching, Germany 1515affiliation: INAF-Osservatorio Astrofisico di
Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy [email protected]
###### Abstract
We present sub-arcsecond resolution observations of continuum emission
associated with the GG Tau quadruple star system at wavelengths of 1.3, 2.8,
7.3, and 50 mm. These data confirm that the GG Tau A binary is encircled by a
circumbinary ring at a radius of 235 AU with a FWHM width of $\sim$60 AU. We
find no clear evidence for a radial gradient in the spectral shape of the
ring, suggesting that the particle size distribution is spatially homogeneous
on angular scales $\gtrsim$0$\farcs$1. A central point source, likely
associated with the primary component (GG Tau Aa), exhibits a composite
spectrum from dust and free-free emission. Faint emission at 7.3 mm is
observed toward the low-mass star GG Tau Ba, although its origin remains
uncertain. Using these measurements of the resolved, multifrequency emission
structure of the GG Tau A system, models of the far-infrared to radio spectrum
are developed to place constraints on the grain size distribution and dust
mass in the circumbinary ring. The non-negligible curvature present in the
ring spectrum implies a maximum particle size of 1–10 mm, although we are
unable to place strong constraints on the distribution shape. The
corresponding dust mass is 30–300 $M_{\oplus}$, at a temperature of 20–30 K.
We discuss how this significant concentration of relatively large particles in
a narrow ring at a large radius might be produced in a local region of higher
gas pressures (i.e., a particle “trap”) located near the inner edge of the
circumbinary disk.
protoplanetary disks — radio continuum: planetary systems — stars: individual
(GG Tau) — ISM: dust
## 1 Introduction
The first step of planet formation — the collisional growth of $\mu$m-sized
dust grains into $>$km-sized planetesimals, the building blocks of terrestrial
planets and the cores of giant planets — is fundamental, but physically
complicated and fraught with theoretical uncertainty. A substantial effort
with numerical simulations and laboratory experiments is converging on a basic
model framework for the growth and migration of solids embedded in a
protoplanetary gas disk (see the recent reviews by Testi et al., 2014;
Johansen et al., 2014), but direct astronomical observations of these solids
are required to test and refine it. Thermal continuum emission at mm/radio
wavelengths is well-suited for that task, as a (relatively) bright and
optically thin tracer of solid particles with sizes up to $\sim$10 cm. The
spectral behavior of this emission is diagnostic of the particle size
distribution (e.g., Beckwith & Sargent, 1991; Miyake & Nakagawa, 1993; Henning
& Stognienko, 1996; D’Alessio et al., 2001; Draine, 2006; Ricci et al., 2010b,
a). Therefore, spatially resolved measurements of the mm/radio “colors” can be
used to map out how the particle growth and transport efficiencies vary as a
function of the local physical conditions in the gas disk (Isella et al.,
2010; Banzatti et al., 2011; Guilloteau et al., 2011; Pérez et al., 2012;
Trotta et al., 2013; Menu et al., 2014).
These preliminary studies of the resolved multifrequency continuum emission
from disks indicate that the inward radial transport of mm/cm-sized solids is
a crucial factor for explaining the observed color gradients (cf., Birnstiel
et al., 2012). Birnstiel & Andrews (2014) suggested that this same radial
drift, induced by aerodynamic drag on particles that are partially coupled to
the gas in its sub-Keplerian velocity field (Adachi et al., 1976;
Weidenschilling, 1977), is also responsible for the observed discrepancies
between the sizes of the line and continuum emission in some disks (e.g.,
Panić et al., 2009; Andrews et al., 2012; Rosenfeld et al., 2013b). However,
there is a fundamental issue with the transport timescales: in these idealized
models, radial drift is much too efficient (Takeuchi & Lin, 2002, 2005; Brauer
et al., 2007). Perhaps the most promising option for slowing (or stopping)
this transport mechanism is with a “bump” in the radial gas pressure profile
(e.g., Whipple, 1972), either locally and stochastically in over-densities
generated by turbulence (e.g., Klahr & Henning, 1997; Pinilla et al., 2012b)
or globally and with long duration in density concentrations produced near
sharp ionization boundaries (just outside a “dead” zone; e.g., Dzyurkevich et
al., 2010) or through dynamical interactions with a companion (e.g., Pinilla
et al., 2012a).
The most obvious case in which the latter scenario is relevant is for a
circumbinary disk, where dynamical interactions between the stars and gas
reservoir clear the disk material inside a radius $\sim$3$\times$ larger than
the binary separation (e.g., Artymowicz & Lubow, 1994). The steep gas pressure
gradient created by this clearing will trap particles in a circumbinary
“ring”, which itself might have significantly enhanced pressures due to the
dynamical excitation of density waves (e.g., Artymowicz & Lubow, 1996; Kley &
Dirksen, 2006; Hayasaki & Okazaki, 2009). GG Tau is the canonical example of a
close young stellar pair ($\sim$0$\farcs$25 projected separation; Leinert et
al., 1991; Ghez et al., 1993) with a prominent circumbinary ring, which has
been resolved and extensively studied in mm-wave continuum and molecular line
emission (Dutrey et al., 1994; Guilloteau et al., 1999; Piétu et al., 2011) as
well as scattered light in the optical and near-infrared (Roddier et al.,
1996; Silber et al., 2000; Krist et al., 2002, 2005; McCabe et al., 2002; Itoh
et al., 2002; Duchêne et al., 2004). The (unresolved) radio spectrum indicates
that $\sim$mm/cm-sized particles are present in the GG Tau circumbinary ring
(e.g., Guilloteau et al., 1999; Rodmann et al., 2006; Scaife, 2013), lending
some additional support to the idea that radial drift is halted near the ring
edge.
Here we present resolved measurements of continuum emission from the GG Tau
system at wavelengths of 1.3, 2.8, 7.3, and 50 mm, in an effort to
characterize the dust population in the GG Tau A circumbinary ring. The
observations and data calibration are presented in Section 2. Models of the
resolved emission and broadband spectrum are developed and analyzed in Section
3\. The results are discussed in the context of current ideas about the
evolution of disk solids in Section 4.
## 2 Observations and Data Reduction
GG Tau was observed with the 15-element ($6\times 10.4$ m and $9\times 6.1$ m
antennas) Combined Array for Research in Millimeter Astronomy (CARMA) in its C
and B configurations (30–350 m and 100–1000 m baselines, respectively) with
the 1 mm receivers on 2007 September 17 and November 26, and in the B
configuration with the 3 mm receivers on 2008 January 8, January 17, and
February 15. In the former, the correlator was set up to process two 500 MHz
basebands with coarse spectral resolution in each sideband (2 GHz of total
bandwidth), with a local oscillator frequency of 228 GHz ($\lambda=1.31$ mm).
For the latter, a third 500 MHz baseband was added to each sideband (3 GHz
bandwidth), and the receivers were tuned to 106 GHz ($\lambda=2.83$ mm). The
observations alternated between GG Tau and the nearby quasars J0530+135,
J0510+180, J0431+206, J0449+113, and 3C 111, with a source–calibrator cycle
time of 12–15 minutes. Additional observations of Uranus were made for
calibration purposes. The atmospheric conditions were generally good, with 230
GHz opacities of $\sim$0.15–0.20 and 0.2–0.3 during the 1.3 and 2.8 mm
observations, respectively.
The raw visibilities were calibrated and subsequently imaged using standard
tasks in the MIRIAD software package. The passband shape across the coarse
continuum channels was calibrated using observations of J0530+135, and the
(antenna-based) complex gain response of the array to both instrumental and
atmospheric variations was determined from repeated observations of J0530+135
or 3C 111. Observations of J0510+180, J0431+206, and J0449+113 were used to
assess the quality of the gain calibration; the contribution of decoherence
due to small baseline errors and atmospheric phase noise is found to be small,
representing a “seeing” disk with FWHM $\leq$0$\farcs$1\. The absolute
amplitude scales were set by bootstrapping J0530+135 flux densities from
observations of Uranus, with systematic uncertainties of $\sim$10%. Wideband
continuum visibilities were created by averaging the spectra across the
sampled passbands. These calibrated continuum visibilities were Fourier
inverted assuming natural weighting, deconvolved with the CLEAN algorithm, and
restored with a synthesized beam to produce the emission maps shown in Figure
1. The 1.3 mm visibilities were tapered with a 0$\farcs$1 FWHM Gaussian kernel
before inversion to improve the resulting image quality. The 1.3 mm continuum
map shown in Figure 1 has a $0\farcs 67\times 0\farcs 57$ synthesized beam (at
P.A. = 132°) and an RMS noise level of 2.3 mJy beam-1; the corresponding 2.8
mm map has a $1\farcs 19\times 0\farcs 76$ beam (at P.A. = 118°) and an RMS
noise level of 0.5 mJy beam-1.
Figure 1: Synthesized continuum images of the GG Tau field at (from left to
right) wavelengths of 1.3, 2.8, 7.3, and 50 mm. Contours are drawn at 3
$\sigma$ intervals in each panel (7, 1.5, 0.04, and 0.02 mJy beam-1 from left
to right), and synthesized beam dimensions are marked in the lower left
corners. The primary beam for the CARMA 1.3 mm image is shown as a dotted gray
curve. The key emission components are labeled in the VLA Q-band (7.3 mm)
image, including the circumbinary dust ring around GG Tau Aab, faint radio
emission toward the low-mass binary GG Tau Bab (from component Ba), and the
bright radio emission from the background galaxy GG Tau/N.
New observations of the GG Tau field were also made as part of the
“Disks@EVLA” key project (project code AC982) with the 27-element (25 m
diameter antennas) Karl G. Jansky Very Large Array (VLA), employing the Q-band
receivers in the C configuration (35 m to 3.4 km baselines) on 2010 November
27, and the C-band receivers in the A configuration (0.7–36.4 km baselines) on
2011 July 26. For the Q-band observations, the recently upgraded correlator
was configured to process two contiguous 1 GHz basebands centered around 41.1
GHz ($\lambda=7.29$ mm), each comprising eight 128 MHz-wide spectral windows
with 64 channels. The C-band correlator configuration had a similar setup,
with the 1 GHz basebands centered at 4.5 and 7.5 GHz, for a mean frequency of
6 GHz ($\lambda=5.0$ cm). The observations cycled between GG Tau and the
nearby calibrator J0431+2037 at $\sim$3 and 10 minute intervals for the Q- and
C-bands, respectively. The bright quasars 3C 84 and 3C 147 were also observed
for calibration purposes.
The raw visibilities were calibrated and imaged using the “Disks@EVLA”
pipeline in the CASA software package (now the standard pipeline for high
frequency VLA observations111see https://science.nrao.edu/facilities/vla/data-
processing/pipeline). After flagging the data for radio frequency interference
and other minor issues, the observations of 3C 84 were used to calibrate the
spectral bandpass response of the system after bootstrapping flux densities in
each spectral window from observations of 3C 147 (to properly treat the shape
of the 3C 84 spectrum over the wide relative bandwidth). The Q-band
visibilities were then spectrally averaged into 16 pseudo-continuum sub-bands
(one per 128 MHz spectral window); the C-band data were not averaged, to
minimize bandwidth-smearing. Complex gain variations were calibrated with the
frequent observations of J0431+2037, and the absolute amplitude scale was
determined using a frequency-dependent emission model for the standard flux
density calibrator 3C 147 (Perley & Butler, 2013). The systematic uncertainty
in the amplitude scale is $\sim$10% at Q-band and 5% at C-band. The calibrated
visibilities were Fourier inverted assuming natural weighting, deconvolved
with the multi-frequency synthesis version of CLEAN, and restored with a
synthesized beam to make the composite continuum maps shown in Figure 1. The
Q-band map has a $0\farcs 86\times 0\farcs 61$ synthesized beam (at P.A. =
127°) and an RMS noise level of 13 $\mu$Jy beam-1, and the C-band map has a
$0\farcs 51\times 0\farcs 36$ beam (at P.A. = 127°) with an RMS noise level of
6 $\mu$Jy beam-1. The C-band image reconstruction required substantial extra
care, including imaging of the entire primary beam and faceting on
exceptionally bright background sources, to minimize artifacts at the field
center. The images shown in Figure 1 are not corrected for the response of the
primary beam.
## 3 Analysis and Results
The multifrequency continuum images in Figure 1 show emission from three
distinct components: (1) an extended ring around the close binary GG Tau A at
1.3, 2.8, and 7.3 mm, along with point-like emission at its center detected at
1.3, 7.3, and 50 mm;222It is worth pointing out that we do not detect the
“streamer” identified by Piétu et al. (2011); its estimated surface brightness
is comparable to the RMS noise level in our 1.3 mm map, and would likely lie
well below the noise floor at longer wavelengths if its origin is thermal dust
emission. (2) faint, unresolved emission at 7.3 mm associated with the low-
mass star GG Tau Ba (no emission is found toward its $\sim$1$\farcs$5
companion Bb); and (3) bright, unresolved emission at 7.3 and 50 mm from the
(presumed) extragalactic interloper GG Tau/N. The composite flux densities,
$S_{\nu}$, or upper limits for each of these components are compiled in Table
1 and displayed together in Figure 2 (note that the measurements for the GG
Tau B and N components in this figure and table were determined from images
that were corrected for the response of the primary beam). For completeness,
we also include literature measurements of the radio spectra for each
component in this figure.
Figure 2: The mm/radio spectra for the three emission components in the GG Tau
field: A (left; see Section 3.1 for more details), B (middle), and N (right),
where the emission from the latter two components were estimated from images
that have been corrected for the response of the primary beam. Our
measurements are shown in black, and flux densities from the literature are
marked in green (Table 1 lists $S_{\nu}$ for each component). The error bars
represent the formal statistical uncertainties and the systematic calibration
uncertainties, added in quadrature. Upper limits (at 3 $\sigma$) are marked
with a horizontal line and a downward-pointing arrow.
The radio spectral index of GG Tau/N, $\alpha\approx-0.7$ (defined as
$S_{\nu}\propto\nu^{\,\alpha}$), and its non-detection at mm wavelengths
indicate a non-thermal emission mechanism (e.g., Bieging et al., 1984). Given
this index, GG Tau/N is likely a background extragalactic source (AGN)
emitting an optically thin synchrotron spectrum; however, it is worth noting
that there is no confirmed counterpart at any optical or infrared wavelength.
GG Tau B is a low-mass T Tauri binary (White et al., 1999) with a modest
infrared excess (Luhman et al., 2010); the 7.3 mm emission from the primary
(Ba; spectral type M5) found here is the first detection longward of 24
$\mu$m. The nature of this Q-band emission from Ba is not clear, given the
non-detections at other wavelengths. The derived C-band upper limit allows
$\alpha\gtrsim 0.6$, consistent with origins in a magnetically active corona
(White et al., 1994; Cranmer et al., 2013) or dense wind (Reynolds, 1986).
Likewise, non-detections at 1.3 and 2.8 mm suggest that $\alpha\lesssim 2.3$,
which is also commensurate with thermal emission from a relatively cold disk
(as might be expected around such a low-mass host star; e.g., Andrews et al.,
2013) or optically thin emission from large dust particles (e.g., Ricci et
al., 2010b). Variable non-thermal radio emission is an additional (and not
mutually exclusive) possibility.
We focus here on the multifrequency, resolved emission structure from GG Tau
A, itself composed of a narrow circumbinary ring and a point-like contribution
associated with one or both components of the close binary (Dutrey et al.,
1994; Guilloteau et al., 1999; Piétu et al., 2011). In the following sections,
we describe a simple model to quantify these emission components as a function
of the observing frequency, and use those results to assess the emission
origins. First, we establish the basic spatial structure of the GG Tau A
emission (in Section 3.1), and then we employ that resolved information to
model the spectrum and extract physical constraints on the properties of the
constituent solid particles (e.g., mass, temperature, size distribution; in
Section 3.2).
### 3.1 Models for Resolved Emission from GG Tau A
We adopt a simple model prescription for the brightness distribution of the GG
Tau A emission consisting of an azimuthally-symmetric Gaussian ring and a
central point source. The ring is described by seven parameters: a mean radius
$\mu_{r}$, width $\sigma_{r}$, integrated flux density $S_{\nu,r}$ ($=\int
I_{\nu,r}\,d\Omega$), inclination angle $i$ (0° is face-on), minor axis
position angle $\varphi$ (representing the sky-projected orientation of the
ring rotation axis), and two nuisance parameters to account for the ring
center location {$\Delta\alpha_{r}$, $\Delta\delta_{r}$} (defined as arcsecond
offsets in right ascension and declination from the observed phase center).
The point source component includes three additional parameters: a flux
density $S_{\nu,c}$ and projected offsets {$\Delta\alpha_{c}$,
$\Delta\delta_{c}$} relative to the ring center. We assume a distance of 140
pc (e.g., Torres et al., 2009), and compute models for different frequencies
independently.
For a given set of these 10 parameters, we calculate synthetic complex
visibilities, $V_{\nu}$, sampled at the same spatial frequencies, $(u,v)$, as
the relevant observations. We then evaluate a Cauchy log-likelihood function
(cf., Sivia & Skilling, 2006) to represent the probability of the model,
{$V^{\rm M}_{\nu}(u,v)$}, given the observed complex visibilities, {$V^{\rm
D}_{\nu}(u,v)$}, and their uncertainties, {$\sigma^{\rm D}_{\nu}(u,v)$},
$\mathcal{L}\propto\sum_{k}\ln\left(\frac{1-e^{-R_{k}^{2}/2}}{R_{k}^{2}}\right);\,\,\,\,{\rm
where}\,\,\,\,R_{k}=\left(\frac{V^{\rm D}_{\nu}(u_{k},v_{k})-V^{\rm
M}_{\nu}(u_{k},v_{k})}{\sigma^{\rm D}_{\nu}(u_{k},v_{k})}\right).$ (1)
This log-likelihood function was preferred over its more familiar Gaussian
counterpart (where $\mathcal{L}\propto-\chi^{2}/2=\sum_{k}R_{k}^{2}/2$)
because it is more forgiving of outliers (mitigating parameter bias due to
phase calibration systematics on long baselines) and more conservative (which
is important, since our error estimates are derived solely from the visibility
weights, and therefore do not treat scatter due to issues like pointing
errors, atmospheric phase noise, etc.). The posterior probability distribution
function (PDF) was sampled with Monte Carlo Markov Chain (MCMC) calculations,
using the Goodman & Weare (2010) ensemble sampler as implemented by Foreman-
Mackey et al. (2013).
A set of initial MCMC calculations was conducted with uniform priors on all
parameters. However, given the very weak emission from the central point
source component at 1.3 and 2.8 mm, convergence on the relative offset
parameters {$\Delta\alpha_{c}$, $\Delta\delta_{c}$} was prohibitively slow
(and could therefore lead us to biased inferences of $S_{\nu,c}$). In these
initial calculations, we found that the 1.3 mm relative offsets were entirely
consistent with the well-constrained values at 7.3 mm. Therefore, new MCMC
calculations were made with (independent) Gaussian priors on these offsets,
centered on the 7.3 mm values and with standard deviations corresponding to
the inferred 68% marginal widths of their posterior PDFs. In any case, this
had no impact on the inferences for other parameters.
Figure 3: Summary of the posterior PDFs inferred from MCMC calculations in
reference to resolved interferometer data at 1.3, 2.8, and 7.3 mm, assuming a
model composed of a central point source and a Gaussian ring. The staircase
plot to the left shows marginalized two-parameter posterior PDF surfaces, with
contours drawn at 68 and 95% confidence intervals. Marginalized posterior PDFs
for each parameter are shown along the diagonal. The ring and point source
flux densities, $S_{\nu,r}$ and $S_{\nu,c}$ respectively, are normalized by
their best-fit values for clarity. Note that the 4 directional offset
parameters ($\Delta\alpha_{r}$, $\Delta\delta_{r}$, $\Delta\alpha_{c}$, and
$\Delta\delta_{c}$) are not shown, to simplify the plot. The top right panel
is a visual representation of the (normalized) radial surface brightness
profiles derived from this analysis; the widths of these profiles represent
the 68% confidence intervals. Figure 4: Comparisons of the data and best-fit
models. The left-hand panels show the data and the corresponding images
synthesized from the model and residual visibilities in the same way as the
data. Contour levels are as in Figure 1. A cross marks the ring center and
orientation in the residuals panel. The rightmost panel in each row shows the
azimuthally-averaged (real) visibility profiles, deprojected according to the
derived ring geometry (data in black, models in red).
Figure 3 is a representation of the sampled posterior PDFs in the form of a
staircase diagram, showing both pair-wise parameter covariances (contours are
drawn at the 68 and 95% confidence intervals) and marginalized posterior PDFs
for individual parameters (the four offset parameters are not shown, and the
flux density parameters are normalized to the peaks of their marginal
posterior PDFs, for the sake of clarity). The panel in the upper right corner
is a visualization of the (area-normalized) radial surface brightness profiles
reconstructed from random draws to the joint posterior PDFs, where the shaded
widths of each profile are representative of the 68% (i.e., $\sim$1 $\sigma$)
confidence intervals. Table 2 summarizes the modeling results, listing the
“best-fit” (peaks of the marginal posterior PDFs) parameter values and their
68% confidence intervals at each observing frequency. A direct comparison of
the data and best-fit models is shown in Figure 4, in both the image plane and
with the azimuthally-averaged (and deprojected) visibility profiles.
Within the uncertainties, the GG Tau A circumbinary ring has the same mean
radius and width (as well as inclination, orientation, and center) at all
observed frequencies; $\mu_{r}\approx 230$–240 AU ($\sim$1$\farcs$7) and
$\sigma_{r}\approx 20$–30 AU (corresponding to a FWHM of 0.3–0.5″; i.e., it is
only marginally resolved). There is a hint that the ring is slightly narrower
(at the $\sim$1 $\sigma$ level) at 7.3 mm, although the difference is not
statistically (or practically) significant. Taken together, this indicates
that the spectral behavior of the dust continuum emission does not vary
radially across the ring on the angular scales and at the spectral sensitivity
currently available. If we assume that a Gaussian distribution is an
appropriate spatial model and a power-law with frequency is a reasonable
spectral model, we can very roughly estimate from the fitting results derived
here that $\Delta\alpha\lesssim 0.3$ on (radial) angular scales
$\gtrsim$0$\farcs$1\. Variations at finer scales are possible, and perhaps
likely (see Section 4).
There is some non-negligible curvature in the ring spectrum, with a steeper
spectral index at lower frequencies: $\alpha$(41–106 GHz) $\approx 3.7\pm 0.2$
and $\alpha$(106–228 GHz) $\approx 2.6\pm 0.2$ (the quoted uncertainties
account for the $\sim$10% systematics introduced in the amplitude
calibrations). The peak brightness temperature of the ring at 1.3 mm is only
$\sim$0.2 K, confirming that the emission is optically thin. Therefore, the
observed spectral curvature is a by-product of the intrinsic shape of the dust
opacity spectrum (and perhaps cool temperatures; see Section 3.2 for more
details). The ring is not detected at a wavelength of 5 cm. Assuming it has
the same emission morphology as at shorter wavelengths, a limit on its
integrated flux density can be made based on the measured RMS noise level (6
$\mu$Jy beam-1) in the C-band map: we estimate $S_{\nu,r}<40$ $\mu$Jy (3
$\sigma$).
The central point-like component, originally identified at 1.4 and 1.1 mm by
Guilloteau et al. (1999) and Piétu et al. (2011), is also detected here at
1.3, 7.3, and 50 mm. Models with a relatively faint central source at 2.8 mm
(as listed in Table 2) provide better overall matches to the visibility data,
although formally the inference on $S_{\nu,c}$ is only marginally significant
(greater than zero at the $\sim$2.7 $\sigma$ level): alternatively, we could
quote a 3 $\sigma$ upper limit as $S_{\nu,c}<4.0$ mJy. The (sparse) radio
spectrum of this central source includes contributions from different emission
mechanisms. The spectral index at high frequencies is steep, $\alpha$(41-228
GHz) $\approx 2.1\pm 0.2$, and consistent with optically thick thermal
emission from a warm dust disk. The low-frequency radio spectrum is
considerably more shallow, $\alpha$(6–41 GHz) $\approx 1.3\pm 0.1$; when
combined with the thermal spectrum, the radio emission is best explained with
an intrinsically flat spectrum ($\alpha\approx 0$), suggesting a contribution
from optically thin free-free radiation. We find no evidence that this central
component is resolved: models of the 7.3 mm emission that assume a Gaussian
emission distribution (rather than a point source) indicate a radial width
$<$12 AU (3 $\sigma$; this corresponds to a HWHM $<0\farcs 2$). However, our
models suggest that this component is marginally offset from the ring center,
with a sky-projected separation of $90\pm 30$ mas to the southeast (at P.A.
$\approx 149\pm 1$°) measured from the 7.3 mm data. Assuming the recent orbit
calculations of Köhler (2011), and associating the ring center with the
projected binary center of mass (with the $\sim$0.9:1 mass ratio of White et
al., 1999), these constraints on the scale and orientation of the offsets
suggest that the emission likely originates from the primary component GG Tau
Aa. A similar inference was made for the 1.1 mm emission detected by Piétu et
al. (2011), based on the orbit calculations of Beust & Dutrey (2005).
### 3.2 Models of the GG Tau A Spectrum
Having established an empirical model of the resolved multifrequency emission
from GG Tau A, we shift focus to develop a more physically-motivated model of
the entire far-infrared to radio spectrum. The basic goal is to help
characterize the dust population in the circumbinary ring. The emission
structure of GG Tau A that we derived in the previous section is unusually
simple in the context of protoplanetary disks. The circumbinary ring has a
well-defined mean radius and width, and is spatially isolated from the central
component. The absence of a spatial gradient in the ring spectrum, along with
its apparently low optical depth, suggest that the continuum emission is a
reasonably good tracer of the dust column density distribution. Moreover, the
ring itself is sufficiently narrow and distant from the central heating
sources that we expect it should have a near-constant radial temperature
profile (irradiation heating would produce a variation of $\lesssim$ 15%,
roughly 2–3 K, across the ring; see below). So unlike a typical disk, where
large and uncertain gradients in temperature and density can act as severe
obstacles, we have an interesting opportunity to use the mm/radio spectrum and
our structural constraints from the resolved data to extract some constraints
on the size distribution of the solids in the circumbinary ring.
To that end, we define a spectrum model as the sum of two components: thermal
dust emission in the ring (with flux densities $S_{\nu,r}$) and a composite
emission origin associated with the central point source ($S_{\nu,c}$). For
the latter, we assume a double power-law spectrum,
$S^{\rm M}_{\nu,c}=S_{\nu,0}^{\rm dust}\left(\frac{\nu}{{\rm
10\,GHz}}\right)^{\alpha_{\rm dust}}+S_{\nu,0}^{\rm ff}\left(\frac{\nu}{{\rm
10\,GHz}}\right)^{\alpha_{\rm ff}}.$ (2)
And for the dust emission in the ring, we use the classic and simple one-
dimensional thermal continuum model often adopted for disks (Adams et al.,
1987; Beckwith et al., 1990),
$S^{\rm M}_{\nu,r}=\frac{2\pi\cos{i}}{d^{2}}\int
dr\,r\,B_{\nu}(T_{r})\,(1-e^{-\kappa_{\nu}\Sigma_{r}/\cos{i}}),$ (3)
where $d=140$ pc, $i$ is the ring inclination, $B_{\nu}(T_{r})$ is the Planck
function at a given temperature, $\Sigma_{r}$ is a Gaussian surface density
profile with mean $\mu_{r}$, width $\sigma_{r}$, and peak value $\Sigma_{0}$,
and $\kappa_{\nu}$ is the dust opacity spectrum. Model opacity spectra were
derived assuming a power-law grain size ($a$) distribution, with index $q$
(where $dn/da\propto a^{-q}$) and maximum size $a_{\rm max}$ (the minimum size
was set to 0.1 $\mu$m). For easier comparisons with related work, we employed
the Ricci et al. (2010b) dust mixture333From a material composition
standpoint, this mixture is similar to the one advocated by Pollack et al.
(1994)., with volume fractions of 30% vacuum (porosity), 7% silicates, 21%
carbonaceous materials, and 42% water ice, using optical constants from
Weingartner & Draine (2001), Zubko et al. (1996), and Warren (1984),
respectively. The optical constants for the mixture were determined with the
Bruggeman rule, and the corresponding $\kappa_{\nu}$ for any particle size
were computed with a Mie code. Altogether, the model has 11 parameters,
{$S_{\nu,0}^{\rm dust}$, $\alpha_{\rm dust}$, $S_{\nu,0}^{\rm ff}$,
$\alpha_{\rm ff}$, $i$, $T_{r}$, $\Sigma_{0}$, $\mu_{r}$, $\sigma_{r}$,
$a_{\rm max}$, $q$}.
For any parameter combination, we define two residual terms at each frequency
that record the fit quality relative to the resolved measurements of the ring
and point source flux densities,
$R_{\nu,r}=\left(\frac{S^{\rm D}_{\nu,r}-S^{\rm M}_{\nu,r}}{\sigma^{\rm
D}_{\nu,r}}\right)\,\,{\rm and}\,\,\,\,R_{\nu,c}=\left(\frac{S^{\rm
D}_{\nu,c}-S^{\rm M}_{\nu,c}}{\sigma^{\rm D}_{\nu,c}}\right),$ (4)
respectively; the “observed” flux densities, $S^{\rm D}_{\nu}$, and their
associated uncertainties, $\sigma^{\rm D}_{\nu}$, can be found in Table
2.444Here we also add in quadrature a systematic calibration uncertainty term
at each frequency; see Section 2. Moreover, we made use of the unresolved
photometry in the literature to compare with the sum of the model components,
with an additional residual term
$R_{\nu,{\rm tot}}=\left(\frac{S^{\rm D}_{\nu,{\rm tot}}-[S^{\rm
M}_{\nu,r}+S^{\rm M}_{\nu,c}]}{\sigma^{\rm D}_{\nu,{\rm tot}}}\right),$ (5)
at each frequency, where {$S^{\rm D}_{\nu,{\rm tot}}$, $\sigma^{\rm
D}_{\nu,{\rm tot}}$} are compiled in Table 1. These terms are treated as a
combined residual, $R_{\nu}=\\{R_{\nu,r},R_{\nu,c},R_{\nu,{\rm tot}}\\}$, in
evaluating a log-likelihood function as in Eq. (1) (now the summation is over
$\nu$). The same MCMC algorithm utilized in Section 3.1 was employed to
optimize the model and sample the posterior PDFs for the model parameters.
Because there are more parameters than constraints describing the point
source, we adopted Gaussian priors on {$\log{S_{\nu,0}^{\rm dust}}$,
$\alpha_{\rm dust}$, $\log{S_{\nu,0}^{\rm ff}}$, $\alpha_{\rm ff}$}, with
means {-4.65, 2.1, -4.25, 0.0} and widths {0.2, 0.2, 0.2, 0.1} (the
normalizations are in log Jy units) informed by the examination of the
$S_{\nu,c}$ spectrum described above. To incorporate the measurements of the
resolved emission structure found in Section 3.1, we assumed Gaussian priors
for {$\mu_{r}$, $\sigma_{r}$, $i$} with means {235 AU, 25 AU, 37°} and widths
{5 AU, 5 AU, 1°} (see Table 2). Uniform priors were assumed for the other
parameters, {$T_{r}$, $\Sigma_{0}$, $a_{\rm max}$, $q$}.
Figure 5: The far-infrared to radio spectrum of GG Tau A, decomposed into the
circumbinary ring (blue) and central point source (red), along with their
combination (gray). The model spectrum contributions from each component,
based on random draws from the posterior PDF, are overlaid on the data; their
widths represent the 95% ($\sim$2 $\sigma$) confidence boundaries.
The parameter inferences from these fits are summarized in Table 3. The
corresponding model spectra are compared with the data in Figure 5. In terms
of the physical conditions in the ring, we infer a dust temperature of 20–30
K, comparable to the 35 K determined from the CO spectral line emission by
Guilloteau et al. (1999). The difference is plausibly a manifestation of the
modest temperature inversion that would be expected between the cooler
midplane (where the dust emission is generated) and the warmer atmosphere
traced by the CO (e.g., Dartois et al., 2003; Rosenfeld et al., 2013a). The
dust surface density at the peak of the ring is only 0.01–0.03 g cm-2,
implying a total dust mass of $\sim$30–90 M⊕ (or 1–3$\times 10^{-4}$ M⊙). We
find that relatively top-heavy grain size distributions, $q\approx 1.4$–2.7,
provide substantially better fits than the typical assumptions based on models
of collisional cascades (Dohnanyi, 1969) or measurements in the diffuse
interstellar medium (Mathis et al., 1977), where $q\approx 3.5$. With more
mass concentrated near the maximum particle size, $a_{\rm max}\approx 1$–2 mm,
the corresponding dust opacity spectrum ends up having substantial curvature
at wavelengths near $a_{\rm max}$, reproducing well the observed shape of the
mm-wave ring spectrum. A reconstruction of the inferred $\kappa_{\nu}$ is
shown in Figure 6: the 1.3 mm opacity lies in the range of 5–10 cm2 g-1,
$\sim$2–4$\times$ larger than the standard Beckwith et al. (1990) opacity
prescription for disks. Note that the ring is optically thin at all
frequencies of interest here.
Figure 6: Constraints on the dust opacity spectrum ($\sim$2 $\sigma$
confidence intervals) in the GG Tau A circumbinary ring, reconstructed from
random draws of the joint posterior PDF derived from modeling the far-infrared
to radio spectrum. Overlaid as a dashed curve is the standard opacity
prescription used for protoplanetary disks, originally advocated by Beckwith
et al. (1990).
The best-fit model has a $\chi^{2}\approx 62$, and a reduced
$\tilde{\chi}^{2}\approx 1.8$ (46 datapoints, 11 free parameters).555Note that
we do not double-count the ring+central source photometry measured here in the
“combined” model fit, despite listing those measurements in Table 1 (for the
sake of completeness). Some of these residuals are likely due to under-
estimates of flux density uncertainties, particularly for older single-dish
photometers at challenging submillimeter wavelengths. It is worth explicitly
pointing out that the favored models systematically under-predict the Ku-band
(16 GHz) flux density reported by Scaife (2013) at the $\sim$3 $\sigma$ level
(this datapoint alone drives $\chi^{2}$ up by $\sim$10). Perhaps this is due
to the difficulty of disentangling the GG Tau A and N emission in those
data,666As well as any (relatively) small contribution from GG Tau B. where
the resolution was nearly 3$\times$ the A–N separation and the contrast ratio
is high (in her Figure 1, Scaife indicates that N is roughly an order of
magnitude brighter than A at this frequency). Alternatively, it is possible
that the measurement uncertainties are fine, and instead some of the
assumptions made in the modeling are responsible for the inferred (modest)
residuals. To explore that possibility further, we re-fit the data with
several modifications to our critical assumptions.
In motivating the simplicity of the GG Tau A circumbinary ring structure, we
suggested that its dust temperature is roughly constant. However, a radial
temperature gradient could be present, particularly if the inner part of the
ring intercepts a substantial amount of incident irradiation from the central
stars (analogous to the dust sublimation boundaries of most disks, or the
“wall” features just outside transition disk cavities; e.g., Dullemond et al.,
2001; Calvet et al., 2002). As an extreme counter-example, we re-modeled the
spectrum assuming that the dust temperatures vary inversely with radius in the
ring ($T_{r}\propto 1/r$). No significant differences in the model parameters
were found, as might be expected given the narrowness of the ring. This check
validates the simplified assumption that reasonable temperature gradients have
negligible impact on our results.
The model optimization described above currently requires relatively stringent
priors on the central point source parameters (given the sparse data). But
there is considerable leeway in those prior assumptions that could still offer
reasonable fits to the data. To assess the impact of these priors on the main
physical parameters of the circumbinary ring, we re-modeled the spectrum with
a steeper combination of Gaussian priors on the spectral indices, with means
{$\alpha_{\rm dust}$, $\alpha_{\rm ff}$} = {2.5, 1.0}, and adjustments to
their corresponding normalizations (at 10 GHz), {$\log{S^{\rm dust}_{\nu,0}}$,
$\log{S^{\rm ff}_{\nu,0}}$} = {-5.25, -4.25} (in Jy units; the widths of the
priors were the same as used above). We find that adopting these alternative
priors has no notable effect on the key ring parameters; the point source is
simply too faint to matter in this regard (nor does it improve the fit
quality, even for the Ku-band point noted above). That said, the uncertain
origins of the central point source emission will remain an open question
until resolved measurements at additional frequencies are available.
Perhaps the more relevant assumptions made in the modeling concern the nature
of the grains themselves, particularly their porosities and material
compositions. To investigate these issues further, we first re-modeled the GG
Tau A spectrum assuming different (fixed) porosities. Compared to our nominal
parameters (30% porosity), models with “compact” grains (0% porosity) can
explain the data reasonably well if the circumbinary ring has a similar
$T_{r}$, a slightly ($\sim$30–50%) lower dust mass, and a steeper size
distribution ($q\approx 2.5$–3.7) with $a_{\rm max}$ just under 1 mm. For
higher porosities (60%), we instead infer a (5–8$\times$) higher dust mass and
shallow size distribution ($q\approx 1.0$–2.8) up to a maximum size of a few
mm to $\sim$1 cm. Since $a_{\rm max}$ is comparable to the wavelengths of
interest, resonances preferentially enhance the mm-wave opacities for grains
with higher filling factors (lower porosities; e.g., Miyake & Nakagawa, 1993;
Kataoka et al., 2013, but see Cuzzi et al. 2014): a higher $\kappa_{\nu}$
drives us to find lower $\Sigma_{0}$ (and therefore dust mass) for a fixed
spectrum (and vice versa). These same resonances also naturally introduce an
intrinsic curvature into the mm/cm-wave opacity spectrum that is reflected in
the inferred size distribution index ($q$): since compact grains then do not
require the added $\kappa_{\nu}$ curvature produced by a top-heavy size
distribution, it makes sense that we infer a higher $q$ when the porosity is
low (and vice versa). For a fixed composition, grains with $\sim$20–40%
porosity do produce better fits to the data.
Next, we performed a similar experiment that varied the material composition
of the grains, by re-fitting the data with extreme silicate- or carbon-rich
mixtures.777Since water ice plays only a minor role relative to silicates and
carbonaceous material in setting $\kappa_{\nu}$ in the mm/cm wavelength range,
its volume fraction is left fixed (at 42%; cf., Pollack et al., 1994) in this
experiment. Compared to our adopted mixture (with 7% silicates and 21%
carbonaceous material by volume), models with C-rich grains (28% carbon, no
silicates) in the GG Tau A ring have a comparable dust mass and temperature,
and a modestly steeper grain size distribution ($q\approx 2.2$–3.5) up to
$a_{\rm max}\approx 1$ mm. In the opposite case, models with Si-rich grains
tend toward ($\sim$10$\times$) higher dust masses and shallow size
distributions ($q\approx 1.1$–2.6) with $a_{\rm max}\approx 2$–8 mm. Silicate-
rich grains have mm/cm-wave opacity spectra that exhibit less curvature and
are preferentially reduced compared to C-rich grains, properties that
naturally account for our inferences of top-heavy size distributions and
higher dust masses when C is depleted, respectively. That said, models relying
on these extreme compositions produce poor fits to the data; the experiment is
only intended to convey the sense of the variation.
While there are many alternative assumptions along these lines that could be
made in the modeling, the investigations described above demonstrate that most
of the key parameters related to the dust in the GG Tau A circumbinary ring
are relatively robust. Within a reasonably large range of temperature
gradients, point source emission models, grain porosities and compositional
variations, we find dust temperatures of $\sim$20–30 K, dust masses of
$\sim$1–10$\times$10-4 $M_{\odot}$, and maximum grain sizes of $\sim$1–10 mm
(in all cases with preference near the lower end of the quoted ranges).
However, it should be clear that these data do not provide a strong
quantitative constraint on the power-law index of the grain size distribution,
$q$. Finally, as a point of reference, we find that the dust opacity at 1.3 mm
is restricted to $\sim$0.2–20 cm2 g-1 in all the models explored here.
## 4 Discussion and Conclusions
We have obtained and analyzed high angular resolution observations of
continuum emission associated with GG Tau, a young quadruple star system, at
wavelengths of 1.3, 2.8, 7.3, and 50 mm. These data confirm that GG Tau/N is a
background synchrotron source, and identify faint 7.3 mm emission associated
with the low-mass star GG Tau Ba, although the origin of the latter is
unclear. As had been noted previously (e.g., Guilloteau et al., 1999), we find
a bright emission ring and central peak associated with GG Tau A (although the
ring is undetected at a wavelength of 5 cm). The visibility data were modeled
with a simple surface brightness prescription. We found that the emission at
all frequencies could be described well using a (slightly offset) central
point source and a (projected) circular Gaussian ring with a mean radius
($\mu_{r}$) of $235\pm 5$ AU and width ($\sigma_{r}$) of $25\pm 5$ AU (a FWHM
of $60\pm 10$ AU), reasonably consistent with previous constraints based on
slightly different assumptions (Dutrey et al., 1994; Guilloteau et al., 1999;
Piétu et al., 2011). These morphological constraints indicate that there is
negligible radial variation (on angular scales $\gtrsim 0\farcs 1$) of the
mm/cm-wavelength spectrum ($\alpha\lesssim 0.3$, from 1.3–7.3 mm) across the
circumbinary ring.
The spectrum of the point source flattens considerably at the longest radio
wavelengths, suggesting emission contributions from both dust and free-free
radiation. The integrated spectrum of the ring structure exhibits substantial
curvature, becoming steeper at longer wavelengths, which we suggest is a
manifestation of the intrinsic shape of the dust opacity spectrum. We
developed some simple physical models of the observed GG Tau A spectrum, and
found reasonably good fits for dust temperatures ($T_{r}$) of 20–30 K, dust
masses of 30–60 $M_{\oplus}$ (1–3$\times 10^{-4}$ $M_{\odot}$), and relatively
top-heavy grain size distributions ($dn/da\propto a^{-q}$, with $q\approx
1.4$–2.7) up to maximum particle sizes ($a_{\rm max}$) of $\sim$1–2 mm, for
the grain composition of Ricci et al. (2010b) (effectively the Pollack et al.
1994 mixture with 30% porosity). Alternative assumptions about the
mineralogical makeup and porosities of the grains permit a much wider range of
the size index $q$, but still suggest similar temperatures and a relatively
narrow range of masses ($\sim$1–10$\times 10^{-4}$ $M_{\odot}$) and maximum
particle sizes ($\sim$1–10 mm). In any case, the opacity spectrum inferred in
the ring is not described well with a single power-law in the mm/cm wavelength
range; the standard assumption that $\kappa_{\nu}\propto\nu^{\beta}$ is not
appropriate here. Although we have estimated significantly lower masses (up to
a factor of $\sim$5 compared to, e.g., Guilloteau et al., 1999; Andrews &
Williams, 2005) and maximum grain sizes ($\sim$4–40$\times$ smaller than
inferred by Scaife, 2013) in the ring than in previous studies, these
properties still seem remarkably high given the large separation from the
central stars, narrow radial width of the ring, and age of the system
($\sim$1–3 Myr; or rather time available for particle growth).
A promising way to concentrate and grow dust particles up to $\sim$mm sizes at
such large distances, as well as to halt their normally fast inward migration
due to radial drift, relies on creating a “trap” in a local enhancement of the
gas pressure profile (cf., Whipple, 1972). For GG Tau A, this local pressure
maximum would likely be induced by dynamical interactions between the binary
and disk edge (e.g., Artymowicz & Lubow, 1994). But models of the GG Tau A
stellar orbit highlight a potential issue with this interpretation, since the
ring edge predicted by dynamical simulations lies well inside (at a radius of
roughly $\sim$100 AU) the edge location inferred from dust observations (e.g.,
McCabe et al., 2002; Beust & Dutrey, 2005; Köhler, 2011). Pinilla et al.
(2012a) have suggested that a dust trap could reside substantially beyond the
nominal disk “edge” if the gas pressures decrease (relatively) gradually
toward the inner part of the ring. To schematically illustrate this point in
the case of GG Tau, we used the treatment of Birnstiel et al. (2010) to
simulate the evolution of dust particles — both spatially and in size, subject
to growth, fragmentation, viscous diffusion, and radial drift — in a gas disk
that has a (static) power-law surface density profile with a Gaussian tapered
edge at a radius of 100 AU. Tuning the shape of this taper (i.e., the width of
the Gaussian) can push the gas pressure maximum back to a radius comparable to
the observed dust ring location. For a reasonable turbulent viscosity
parameter ($\alpha\approx 0.002$) and density normalization (a total gas mass
of $\sim$0.1 $M_{\odot}$), we can also reproduce the inferred maximum particle
size and width of the dust ring on appropriate timescales ($\sim$1 Myr). This
simulation is also in qualitative agreement with the infrared scattered light
geometry of the GG Tau A ring (e.g., Duchêne et al., 2004), in that it
predicts a similar distribution of $\mu$m-sized grains produced via
fragmentation in the pressure maximum. Figure 7 summarizes this demonstrative
(although by no means unique) example.
Figure 7: A schematic snapshot for an illustrative model of dust evolution in
a truncated gas disk with a gradually tapered inner edge, tuned to the
relevant parameters of the GG Tau A circumbinary disk. We assumed a static
power-law gas surface density profile ($\Sigma_{\rm gas}\propto 1/r$; red
curve), with an “edge” near the circumbinary disk truncation radius expected
from estimates of the GG Tau A orbit, and a gradual Gaussian taper (with mean
100 AU and width 120 AU). The total gas mass was 0.13 $M_{\odot}$ ($\sim$10%
of the binary mass), consistent with the estimate of Guilloteau et al. (1999).
A population of small (0.1–1 $\mu$m, with $dn/da\propto a^{-3.5}$) grains with
radially constant dust-to-gas mass ratio (0.01) was evolved in size and space
for 1 Myr, following the Birnstiel et al. (2010) prescription for a constant
turbulent viscosity parameter ($\alpha=0.002$) and fragmentation velocity (10
m s-1). The surface density distributions of $\mu$m-sized (0.5–5 $\mu$m;
green) and mm/cm-sized (500 $\mu$m $<a<$ 5 cm; blue) particles predicted by
this model are in reasonably good agreement with both infrared scattered light
observations (e.g., Duchêne et al., 2004) and our estimates of $\Sigma_{\rm
dust}$ inferred from the modeling in Section 3 (the gray shaded region
corresponds to the 2 $\sigma$ uncertainties).
Of course, it is not clear if such “soft” edges are physically realistic for
gas-rich circumbinary disks like the one around GG Tau A; detailed
hydrodynamic simulations (e.g., Bate, 2000; Günther & Kley, 2002; Kley et al.,
2008) tuned to this specific case would be required to assess the feasibility
and internal consistency of the basic scenario proposed above (although see
Beust & Dutrey, 2005). Regardless, the key point is that we should be actively
considering the coupled evolution of gas and solids in this and similar disks
when attempting to reconcile models of their tidal truncation with the (still
uncertain) stellar orbital configurations. In principle, one could test the
proposed explanation with sensitive observations of an optically thin line
tracer that tracks the gas densities near and interior to the dust disk edge
at $\sim$0$\farcs$1 resolution.888It is worth noting that this would also
require a proper accounting of the gas temperatures, which might not be
trivial in such a relatively dust-poor environment (see Bruderer, 2013). In
that context, it is compelling to note that Guilloteau et al. (1999) found
that the 13CO emission extends slightly inside the continuum ring in their
study of the GG Tau A disk; similar data at higher resolution would be
valuable. These dust trap models also predict that larger particles should be
concentrated at the maximum pressure, although the implied spectral gradient
in this particular example would not be resolved with our data. In that sense,
pushing the angular resolution of multifrequency continuum observations in
this and other cases should be considered a high priority.
Although the GG Tau A ring is a particularly spectacular (and useful) example,
the dust in circumbinary disks more generally could serve as important test
cases for studying particle traps induced by dynamical interactions with
companions. The so-called “transition” disks, where the companion is
speculated to be a giant planet, exhibit similar features — narrow dust
continuum rings (e.g., Andrews et al., 2011), often having more extended gas
disks with relatively lower dust masses (e.g., Isella et al., 2012; Rosenfeld
et al., 2013b) and occasionally showing direct (Rosenfeld et al., 2012, 2014;
Casassus et al., 2013; van der Marel et al., 2013; Bruderer et al., 2014) or
indirect (e.g., Dong et al., 2012; Follette et al., 2013) evidence for gas
well inside those dust rings. Given that similarity, we suggest that there is
great value in analyzing resolved multifrequency radio observations of dust
rings (and gas structures) in both circumbinary and transition disks,
analogous to the approach presented here. Ultimately, linking these disk
targets – which are undergoing dynamical interactions with a broad range of
companion masses – could offer important insights on how solids grow inside
gas pressure maxima with a wide diversity of strengths and shapes.
We thank Mark Reid for valuable discussions about data modeling and an
anonymous referee for helpful suggestions. S. M. A. and T. B. acknowledge
support from NASA Origins of Solar Systems grant NNX12AJ04G. A. I., L. M. P.,
and J. M. C. acknowledge support from NSF award AST-1109334. Ongoing CARMA
development and operations are supported by the National Science Foundation
under a cooperative agreement, and by the CARMA partner universities. The VLA
is run by the National Radio Astronomy Observatory, a facility of the National
Science Foundation operated under cooperative agreement by Associated
Universities, Inc.
Table 1: Radio Spectra of GG Tau Components
Component | $\lambda$ (mm) | $S_{\nu}$ (mJy) | Ref.
---|---|---|---
GG Tau A | 0.10 | $5158\pm 1410$ | IRAS
| 0.10 | $6560\pm 1469$ | Howard et al. (2013)
| 0.14 | $8995\pm 2291$ | AKARI
| 0.15 | $7620\pm 3048$ | Howard et al. (2013)
| 0.16 | $8600\pm 3440$ | Howard et al. (2013)
| 0.16 | $6076\pm 1974$ | AKARI
| 0.18 | $8220\pm 3288$ | Howard et al. (2013)
| 0.19 | $7130\pm 2853$ | Howard et al. (2013)
| 0.35 | $6528\pm 1639$ | Andrews & Williams (2005)
| 0.44 | $4540\pm 766$ | Moriarty-Schieven & Butner (1997)
| 0.44 | $4160\pm 665$ | Moriarty-Schieven & Butner (1997)
| 0.44 | $2726\pm 726$ | Andrews & Williams (2005)
| 0.62 | $1370\pm 382$ | Beckwith & Sargent (1991)
| 0.77 | $1250\pm 323$ | Beckwith & Sargent (1991)
| 0.79 | $1110\pm 163$ | Moriarty-Schieven et al. (1994)
| 0.79 | $1710\pm 176$ | Moriarty-Schieven & Butner (1997)
| 0.79 | $1590\pm 170$ | Moriarty-Schieven & Butner (1997)
| 0.79 | $1650\pm 183$ | Moriarty-Schieven & Butner (1997)
| 0.87 | $1255\pm 138$ | Andrews & Williams (2005)
| 1.06 | $800\pm 206$ | Beckwith & Sargent (1991)
| 1.09 | $740\pm 141$ | Moriarty-Schieven et al. (1994)
| 1.09 | $1070\pm 111$ | Moriarty-Schieven & Butner (1997)
| 1.09 | $830\pm 88$ | Moriarty-Schieven & Butner (1997)
| 1.09 | $850\pm 90$ | Moriarty-Schieven & Butner (1997)
| 1.12 | $770\pm 78$ | Piétu et al. (2011)
| 1.25 | $593\pm 130$ | Beckwith et al. (1990)
| 1.26 | $690\pm 75$ | Moriarty-Schieven & Butner (1997)
| 1.26 | $630\pm 66$ | Moriarty-Schieven & Butner (1997)
| 1.26 | $630\pm 70$ | Moriarty-Schieven & Butner (1997)
| 1.31 | $558\pm 58$ | this paper
| 1.33 | $557\pm 56$ | Harris et al. (2012)
| 1.40 | $604\pm 61$ | Guilloteau et al. (1999)
| 1.92 | $320\pm 68$ | Moriarty-Schieven & Butner (1997)
| 2.68 | $85\pm 10$ | Dutrey et al. (1994)
| 2.78 | $73\pm 15$ | Looney et al. (2000)
| 2.83 | $79\pm 9$ | this paper
| 3.06 | $41\pm 9$ | Ohashi et al. (1991)
| 3.40 | $38\pm 4$ | Guilloteau et al. (1999)
| 6.92 | $3.24\pm 0.77$ | Rodmann et al. (2006)
| 7.29 | $2.67\pm 0.29$ | this paper
| 19.09 | $0.25\pm 0.05$ | Scaife (2013)
| 50.00 | $0.036\pm 0.007$ | this paper
GG Tau Ba | 1.31 | $<9$ | this paper
| 1.33 | $<7$ | Harris et al. (2012)
| 2.83 | $<1.7$ | this paper
| 7.29 | $0.058\pm 0.014$ | this paper
| 50.00 | $<0.02$ | this paper
GG Tau Bb | 1.31 | $<9$ | this paper
| 1.33 | $<7$ | Harris et al. (2012)
| 2.83 | $<1.7$ | this paper
| 7.29 | $<0.04$ | this paper
| 50.00 | $<0.02$ | this paper
GG Tau/N | 1.31 | $<14$ | this paper
| 2.83 | $<2.0$ | this paper
| 7.29 | $0.71\pm 0.07$ | this paper
| 19.09 | $2.23\pm 0.12$ | Scaife (2013)
| 20.03 | $3\pm 1$ | Bieging et al. (1984)
| 50.00 | $2.84\pm 0.14$ | this paper
| 61.37 | $3.7\pm 0.4$ | Bieging et al. (1984)
Note. — The uncertainties on the flux densities include both statistical and
systematic calibration terms (added in quadrature). Upper limits are quoted at
the 3 $\sigma$ level, assuming point source emission. The GG Tau B and N flux
densities were determined after correction for the primary beam responses in
each observation.
Table 2: Inferred Visibility Model Parameters
Parameter | 1.3 mm | 2.8 mm | 7.3 mm
---|---|---|---
$S_{\nu,r}$ (mJy) | $543\pm 21$ | $77\pm 4$ | $2.23\,^{+0.08}_{-0.12}$
$\mu_{r}$ (AU) | $232\pm 3$ | $235\pm 3$ | $234\pm 3$
$\sigma_{r}$ (AU) | $26\pm 5$ | $29\,^{+4}_{-5}$ | $17\,^{+4}_{-8}$
$i$ (°) | $37\pm 1$ | $37\pm 1$ | $37\pm 1$
$\varphi$ (°) | $7\pm 2$ | $7\pm 2$ | $7\pm 2$
$S_{\nu,c}$ (mJy) | $15\,^{+3}_{-7}$ | $2.2\,^{+0.6}_{-0.9}$ | $0.44\,^{+0.02}_{-0.04}$
$\Delta\alpha_{c}$ (″) | $+0.03\pm 0.04$ | $+0.06\pm 0.05$ | $+0.05\pm 0.04$
$\Delta\delta_{c}$ (″) | $-0.10\pm 0.05$ | $-0.09\pm 0.06$ | $-0.08\pm 0.03$
Note. — The quoted uncertainties correspond to 68% ($\sim$1 $\sigma$)
confidence intervals; the associated $\sim$10% calibration uncertainty is not
applied to the flux density parameters. The nuisance parameters describing the
offset of the ring center from the observed phase center are not included.
Table 3: Inferred Spectrum Model Parameters
Parameter | Best-Fit Value ($\pm$1 $\sigma$)
---|---
$\Sigma_{0}$ (g cm-2) | $0.02\pm 0.01$
$\mu_{r}$ (AU) | ($235\pm 5$)
$\sigma_{r}$ (AU) | ($25\pm 5$)
$T_{r}$ (K) | $25\pm 5$
$a_{\rm max}$ (mm) | $1.5\,^{+0.3}_{-0.9}$
$q$ | $2.4\,^{+0.3}_{-1.0}$
$\log{S_{\nu,0}^{\rm dust}}$ (Jy) | (-$4.65\pm 0.20$)
$\alpha^{\rm dust}$ | ($2.1\pm 0.2$)
$\log{S_{\nu,0}^{\rm ff}}$ (Jy) | (-$4.50\pm 0.20$)
$\alpha^{\rm ff}$ | ($0.0\pm 0.1$)
Note. — These parameters are valid for the Ricci et al. (2010b) dust mixture
with 30% porosity. See the text for a discussion of alternative assumptions.
The values in parenthesis reflect the Gaussian priors we assumed for these
fits. The formal reduced $\chi^{2}$ statistic for the best-fit model is
$\sim$1.8 (see text).
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|
arxiv-papers
| 2014-04-22T21:07:22 |
2024-09-04T02:50:01.664039
|
{
"license": "Public Domain",
"authors": "Sean M. Andrews, Claire J. Chandler, Andrea Isella, Tilman Birnstiel,\n Katherine A. Rosenfeld, David J. Wilner, Laura M. Perez, Luca Ricci, John M.\n Carpenter, Nuria Calvet, Stuartt A. Corder, Adam T. Deller, Cornelis P.\n Dullemond, Jane S. Greaves, Robert J. Harris, Thomas Henning, Woojin Kwon,\n Joseph Lazio, Hendrik Linz, Lee G. Mundy, Anneila I. Sargent, Shaye Storm,\n and Leonardo Testi",
"submitter": "Sean Andrews",
"url": "https://arxiv.org/abs/1404.5652"
}
|
1404.5673
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2014-069 LHCb-PAPER-2014-012 22 April 2014
Measurement of the resonant and $C\\!P$ components in ${{\kern
3.73305pt\overline{\kern-3.73305ptB}{}}^{0}}\rightarrow J/\psi\pi^{+}\pi^{-}$
decays
The LHCb collaboration†††Authors are listed on the following pages.
The resonant structure of the reaction ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow J/\psi\pi^{+}\pi^{-}$
is studied using data from 3 $\mbox{\,fb}^{-1}$ of integrated luminosity
collected by the LHCb experiment, one-third at
7$\mathrm{\,Te\kern-1.00006ptV}$ center-of-mass energy and the remainder at
8$\mathrm{\,Te\kern-1.00006ptV}$. The invariant mass of the
${{\pi}^{+}}{{\pi}^{-}}$ pair and three decay angular distributions are used
to determine the fractions of the resonant and non-resonant components. Six
interfering ${{\pi}^{+}}{{\pi}^{-}}$ states: $\rho(770)$, $f_{0}(500)$,
$f_{2}(1270)$, $\rho(1450)$, $\omega(782)$ and $\rho(1700)$ are required to
give a good description of invariant mass spectra and decay angular
distributions. The positive and negative $C\\!P$ fractions of each of the
resonant final states are determined. The $f_{0}(980)$ meson is not seen and
the upper limit on its presence, compared with the observed $f_{0}(500)$ rate,
is inconsistent with a model where these scalar mesons are formed from two
quarks and two anti-quarks (tetraquarks) at the eight standard deviation
level. In the $q\overline{q}$ model, the absolute value of the mixing angle
between the $f_{0}(980)$ and the $f_{0}(500)$ scalar mesons is limited to be
less than $17^{\circ}$ at 90% confidence level.
Submitted to Phys. Rev. D
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij41, B. Adeva37, M. Adinolfi46, A. Affolder52, Z. Ajaltouni5, J.
Albrecht9, F. Alessio38, M. Alexander51, S. Ali41, G. Alkhazov30, P. Alvarez
Cartelle37, A.A. Alves Jr25,38, S. Amato2, S. Amerio22, Y. Amhis7, L. An3, L.
Anderlini17,g, J. Anderson40, R. Andreassen57, M. Andreotti16,f, J.E.
Andrews58, R.B. Appleby54, O. Aquines Gutierrez10, F. Archilli38, A.
Artamonov35, M. Artuso59, E. Aslanides6, G. Auriemma25,n, M. Baalouch5, S.
Bachmann11, J.J. Back48, A. Badalov36, V. Balagura31, W. Baldini16, R.J.
Barlow54, C. Barschel38, S. Barsuk7, W. Barter47, V. Batozskaya28, Th.
Bauer41, A. Bay39, L. Beaucourt4, J. Beddow51, F. Bedeschi23, I. Bediaga1, S.
Belogurov31, K. Belous35, I. Belyaev31, E. Ben-Haim8, G. Bencivenni18, S.
Benson50, J. Benton46, A. Berezhnoy32, R. Bernet40, M.-O. Bettler47, M. van
Beuzekom41, A. Bien11, S. Bifani45, T. Bird54, A. Bizzeti17,i, P.M.
Bjørnstad54, T. Blake48, F. Blanc39, J. Blouw10, S. Blusk59, V. Bocci25, A.
Bondar34, N. Bondar30,38, W. Bonivento15,38, S. Borghi54, A. Borgia59, M.
Borsato7, T.J.V. Bowcock52, E. Bowen40, C. Bozzi16, T. Brambach9, J. van den
Brand42, J. Bressieux39, D. Brett54, M. Britsch10, T. Britton59, N.H. Brook46,
H. Brown52, A. Bursche40, G. Busetto22,q, J. Buytaert38, S. Cadeddu15, R.
Calabrese16,f, M. Calvi20,k, M. Calvo Gomez36,o, A. Camboni36, P.
Campana18,38, D. Campora Perez38, A. Carbone14,d, G. Carboni24,l, R.
Cardinale19,38,j, A. Cardini15, H. Carranza-Mejia50, L. Carson50, K. Carvalho
Akiba2, G. Casse52, L. Cassina20, L. Castillo Garcia38, M. Cattaneo38, Ch.
Cauet9, R. Cenci58, M. Charles8, Ph. Charpentier38, S.-F. Cheung55, N.
Chiapolini40, M. Chrzaszcz40,26, K. Ciba38, X. Cid Vidal38, G. Ciezarek53,
P.E.L. Clarke50, M. Clemencic38, H.V. Cliff47, J. Closier38, V. Coco38, J.
Cogan6, E. Cogneras5, P. Collins38, A. Comerma-Montells11, A. Contu15,38, A.
Cook46, M. Coombes46, S. Coquereau8, G. Corti38, M. Corvo16,f, I. Counts56, B.
Couturier38, G.A. Cowan50, D.C. Craik48, M. Cruz Torres60, S. Cunliffe53, R.
Currie50, C. D’Ambrosio38, J. Dalseno46, P. David8, P.N.Y. David41, A.
Davis57, K. De Bruyn41, S. De Capua54, M. De Cian11, J.M. De Miranda1, L. De
Paula2, W. De Silva57, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, N. Déléage4, D. Derkach55, O. Deschamps5, F. Dettori42, A. Di Canto38,
H. Dijkstra38, S. Donleavy52, F. Dordei11, M. Dorigo39, A. Dosil Suárez37, D.
Dossett48, A. Dovbnya43, F. Dupertuis39, P. Durante38, R. Dzhelyadin35, A.
Dziurda26, A. Dzyuba30, S. Easo49,38, U. Egede53, V. Egorychev31, S.
Eidelman34, S. Eisenhardt50, U. Eitschberger9, R. Ekelhof9, L. Eklund51,38, I.
El Rifai5, Ch. Elsasser40, S. Ely59, S. Esen11, T. Evans55, A. Falabella16,f,
C. Färber11, C. Farinelli41, N. Farley45, S. Farry52, D. Ferguson50, V.
Fernandez Albor37, F. Ferreira Rodrigues1, M. Ferro-Luzzi38, S. Filippov33, M.
Fiore16,f, M. Fiorini16,f, M. Firlej27, C. Fitzpatrick38, T. Fiutowski27, M.
Fontana10, F. Fontanelli19,j, R. Forty38, O. Francisco2, M. Frank38, C.
Frei38, M. Frosini17,38,g, J. Fu21,38, E. Furfaro24,l, A. Gallas Torreira37,
D. Galli14,d, S. Gallorini22, S. Gambetta19,j, M. Gandelman2, P. Gandini59, Y.
Gao3, J. Garofoli59, J. Garra Tico47, L. Garrido36, C. Gaspar38, R. Gauld55,
L. Gavardi9, E. Gersabeck11, M. Gersabeck54, T. Gershon48, Ph. Ghez4, A.
Gianelle22, S. Giani’39, V. Gibson47, L. Giubega29, V.V. Gligorov38, C.
Göbel60, D. Golubkov31, A. Golutvin53,31,38, A. Gomes1,a, H. Gordon38, C.
Gotti20, M. Grabalosa Gándara5, R. Graciani Diaz36, L.A. Granado Cardoso38, E.
Graugés36, G. Graziani17, A. Grecu29, E. Greening55, S. Gregson47, P.
Griffith45, L. Grillo11, O. Grünberg62, B. Gui59, E. Gushchin33, Yu. Guz35,38,
T. Gys38, C. Hadjivasiliou59, G. Haefeli39, C. Haen38, S.C. Haines47, S.
Hall53, B. Hamilton58, T. Hampson46, X. Han11, S. Hansmann-Menzemer11, N.
Harnew55, S.T. Harnew46, J. Harrison54, T. Hartmann62, J. He38, T. Head38, V.
Heijne41, K. Hennessy52, P. Henrard5, L. Henry8, J.A. Hernando Morata37, E.
van Herwijnen38, M. Heß62, A. Hicheur1, D. Hill55, M. Hoballah5, C. Hombach54,
W. Hulsbergen41, P. Hunt55, N. Hussain55, D. Hutchcroft52, D. Hynds51, M.
Idzik27, P. Ilten56, R. Jacobsson38, A. Jaeger11, J. Jalocha55, E. Jans41, P.
Jaton39, A. Jawahery58, M. Jezabek26, F. Jing3, M. John55, D. Johnson55, C.R.
Jones47, C. Joram38, B. Jost38, N. Jurik59, M. Kaballo9, S. Kandybei43, W.
Kanso6, M. Karacson38, T.M. Karbach38, M. Kelsey59, I.R. Kenyon45, T. Ketel42,
B. Khanji20, C. Khurewathanakul39, S. Klaver54, O. Kochebina7, M. Kolpin11, I.
Komarov39, R.F. Koopman42, P. Koppenburg41,38, M. Korolev32, A. Kozlinskiy41,
L. Kravchuk33, K. Kreplin11, M. Kreps48, G. Krocker11, P. Krokovny34, F.
Kruse9, M. Kucharczyk20,26,38,k, V. Kudryavtsev34, K. Kurek28, T.
Kvaratskheliya31, V.N. La Thi39, D. Lacarrere38, G. Lafferty54, A. Lai15, D.
Lambert50, R.W. Lambert42, E. Lanciotti38, G. Lanfranchi18, C. Langenbruch38,
B. Langhans38, T. Latham48, C. Lazzeroni45, R. Le Gac6, J. van Leerdam41,
J.-P. Lees4, R. Lefèvre5, A. Leflat32, J. Lefrançois7, S. Leo23, O. Leroy6, T.
Lesiak26, B. Leverington11, Y. Li3, M. Liles52, R. Lindner38, C. Linn38, F.
Lionetto40, B. Liu15, G. Liu38, S. Lohn38, I. Longstaff51, J.H. Lopes2, N.
Lopez-March39, P. Lowdon40, H. Lu3, D. Lucchesi22,q, H. Luo50, A. Lupato22, E.
Luppi16,f, O. Lupton55, F. Machefert7, I.V. Machikhiliyan31, F. Maciuc29, O.
Maev30, S. Malde55, G. Manca15,e, G. Mancinelli6, M. Manzali16,f, J. Maratas5,
J.F. Marchand4, U. Marconi14, C. Marin Benito36, P. Marino23,s, R. Märki39, J.
Marks11, G. Martellotti25, A. Martens8, A. Martín Sánchez7, M. Martinelli41,
D. Martinez Santos42, F. Martinez Vidal64, D. Martins Tostes2, A.
Massafferri1, R. Matev38, Z. Mathe38, C. Matteuzzi20, A. Mazurov16,f, M.
McCann53, J. McCarthy45, A. McNab54, R. McNulty12, B. McSkelly52, B.
Meadows57,55, F. Meier9, M. Meissner11, M. Merk41, D.A. Milanes8, M.-N.
Minard4, N. Moggi14, J. Molina Rodriguez60, S. Monteil5, D. Moran54, M.
Morandin22, P. Morawski26, A. Mordà6, M.J. Morello23,s, J. Moron27, R.
Mountain59, F. Muheim50, K. Müller40, R. Muresan29, M. Mussini14, B. Muster39,
P. Naik46, T. Nakada39, R. Nandakumar49, I. Nasteva2, M. Needham50, N. Neri21,
S. Neubert38, N. Neufeld38, M. Neuner11, A.D. Nguyen39, T.D. Nguyen39, C.
Nguyen-Mau39,p, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin32, T. Nikodem11, A.
Novoselov35, A. Oblakowska-Mucha27, V. Obraztsov35, S. Oggero41, S. Ogilvy51,
O. Okhrimenko44, R. Oldeman15,e, G. Onderwater65, M. Orlandea29, J.M. Otalora
Goicochea2, P. Owen53, A. Oyanguren64, B.K. Pal59, A. Palano13,c, F.
Palombo21,t, M. Palutan18, J. Panman38, A. Papanestis49,38, M. Pappagallo51,
C. Parkes54, C.J. Parkinson9, G. Passaleva17, G.D. Patel52, M. Patel53, C.
Patrignani19,j, A. Pazos Alvarez37, A. Pearce54, A. Pellegrino41, M. Pepe
Altarelli38, S. Perazzini14,d, E. Perez Trigo37, P. Perret5, M. Perrin-
Terrin6, L. Pescatore45, E. Pesen66, K. Petridis53, A. Petrolini19,j, E.
Picatoste Olloqui36, B. Pietrzyk4, T. Pilař48, D. Pinci25, A. Pistone19, S.
Playfer50, M. Plo Casasus37, F. Polci8, A. Poluektov48,34, E. Polycarpo2, A.
Popov35, D. Popov10, B. Popovici29, C. Potterat2, A. Powell55, J.
Prisciandaro39, A. Pritchard52, C. Prouve46, V. Pugatch44, A. Puig Navarro39,
G. Punzi23,r, W. Qian4, B. Rachwal26, J.H. Rademacker46, B.
Rakotomiaramanana39, M. Rama18, M.S. Rangel2, I. Raniuk43, N. Rauschmayr38, G.
Raven42, S. Reichert54, M.M. Reid48, A.C. dos Reis1, S. Ricciardi49, A.
Richards53, M. Rihl38, K. Rinnert52, V. Rives Molina36, D.A. Roa Romero5, P.
Robbe7, A.B. Rodrigues1, E. Rodrigues54, P. Rodriguez Perez54, S. Roiser38, V.
Romanovsky35, A. Romero Vidal37, M. Rotondo22, J. Rouvinet39, T. Ruf38, F.
Ruffini23, H. Ruiz36, P. Ruiz Valls64, G. Sabatino25,l, J.J. Saborido Silva37,
N. Sagidova30, P. Sail51, B. Saitta15,e, V. Salustino Guimaraes2, C. Sanchez
Mayordomo64, B. Sanmartin Sedes37, R. Santacesaria25, C. Santamarina Rios37,
E. Santovetti24,l, M. Sapunov6, A. Sarti18,m, C. Satriano25,n, A. Satta24, M.
Savrie16,f, D. Savrina31,32, M. Schiller42, H. Schindler38, M. Schlupp9, M.
Schmelling10, B. Schmidt38, O. Schneider39, A. Schopper38, M.-H. Schune7, R.
Schwemmer38, B. Sciascia18, A. Sciubba25, M. Seco37, A. Semennikov31, K.
Senderowska27, I. Sepp53, N. Serra40, J. Serrano6, L. Sestini22, P. Seyfert11,
M. Shapkin35, I. Shapoval16,43,f, Y. Shcheglov30, T. Shears52, L. Shekhtman34,
V. Shevchenko63, A. Shires9, R. Silva Coutinho48, G. Simi22, M. Sirendi47, N.
Skidmore46, T. Skwarnicki59, N.A. Smith52, E. Smith55,49, E. Smith53, J.
Smith47, M. Smith54, H. Snoek41, M.D. Sokoloff57, F.J.P. Soler51, F. Soomro39,
D. Souza46, B. Souza De Paula2, B. Spaan9, A. Sparkes50, F. Spinella23, P.
Spradlin51, F. Stagni38, S. Stahl11, O. Steinkamp40, O. Stenyakin35, S.
Stevenson55, S. Stoica29, S. Stone59, B. Storaci40, S. Stracka23,38, M.
Straticiuc29, U. Straumann40, R. Stroili22, V.K. Subbiah38, L. Sun57, W.
Sutcliffe53, K. Swientek27, S. Swientek9, V. Syropoulos42, M. Szczekowski28,
P. Szczypka39,38, D. Szilard2, T. Szumlak27, S. T’Jampens4, M. Teklishyn7, G.
Tellarini16,f, F. Teubert38, C. Thomas55, E. Thomas38, J. van Tilburg41, V.
Tisserand4, M. Tobin39, S. Tolk42, L. Tomassetti16,f, D. Tonelli38, S. Topp-
Joergensen55, N. Torr55, E. Tournefier4, S. Tourneur39, M.T. Tran39, M.
Tresch40, A. Tsaregorodtsev6, P. Tsopelas41, N. Tuning41, M. Ubeda Garcia38,
A. Ukleja28, A. Ustyuzhanin63, U. Uwer11, V. Vagnoni14, G. Valenti14, A.
Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro37, C. Vázquez Sierra37, S.
Vecchi16, J.J. Velthuis46, M. Veltri17,h, G. Veneziano39, M. Vesterinen11, B.
Viaud7, D. Vieira2, M. Vieites Diaz37, X. Vilasis-Cardona36,o, A. Vollhardt40,
D. Volyanskyy10, D. Voong46, A. Vorobyev30, V. Vorobyev34, C. Voß62, H.
Voss10, J.A. de Vries41, R. Waldi62, C. Wallace48, R. Wallace12, J. Walsh23,
S. Wandernoth11, J. Wang59, D.R. Ward47, N.K. Watson45, D. Websdale53, M.
Whitehead48, J. Wicht38, D. Wiedner11, G. Wilkinson55, M.P. Williams45, M.
Williams56, F.F. Wilson49, J. Wimberley58, J. Wishahi9, W. Wislicki28, M.
Witek26, G. Wormser7, S.A. Wotton47, S. Wright47, S. Wu3, K. Wyllie38, Y.
Xie61, Z. Xing59, Z. Xu39, Z. Yang3, X. Yuan3, O. Yushchenko35, M. Zangoli14,
M. Zavertyaev10,b, F. Zhang3, L. Zhang59, W.C. Zhang12, Y. Zhang3, A.
Zhelezov11, A. Zhokhov31, L. Zhong3, A. Zvyagin38.
1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil
2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
3Center for High Energy Physics, Tsinghua University, Beijing, China
4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France
5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont-
Ferrand, France
6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France
7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France
8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot,
CNRS/IN2P3, Paris, France
9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany
10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany
11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg,
Germany
12School of Physics, University College Dublin, Dublin, Ireland
13Sezione INFN di Bari, Bari, Italy
14Sezione INFN di Bologna, Bologna, Italy
15Sezione INFN di Cagliari, Cagliari, Italy
16Sezione INFN di Ferrara, Ferrara, Italy
17Sezione INFN di Firenze, Firenze, Italy
18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy
19Sezione INFN di Genova, Genova, Italy
20Sezione INFN di Milano Bicocca, Milano, Italy
21Sezione INFN di Milano, Milano, Italy
22Sezione INFN di Padova, Padova, Italy
23Sezione INFN di Pisa, Pisa, Italy
24Sezione INFN di Roma Tor Vergata, Roma, Italy
25Sezione INFN di Roma La Sapienza, Roma, Italy
26Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
27AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
28National Center for Nuclear Research (NCBJ), Warsaw, Poland
29Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
30Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
31Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
32Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
33Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
34Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
35Institute for High Energy Physics (IHEP), Protvino, Russia
36Universitat de Barcelona, Barcelona, Spain
37Universidad de Santiago de Compostela, Santiago de Compostela, Spain
38European Organization for Nuclear Research (CERN), Geneva, Switzerland
39Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
40Physik-Institut, Universität Zürich, Zürich, Switzerland
41Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
42Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
43NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
44Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
45University of Birmingham, Birmingham, United Kingdom
46H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
47Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
48Department of Physics, University of Warwick, Coventry, United Kingdom
49STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
50School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
51School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
52Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
53Imperial College London, London, United Kingdom
54School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
55Department of Physics, University of Oxford, Oxford, United Kingdom
56Massachusetts Institute of Technology, Cambridge, MA, United States
57University of Cincinnati, Cincinnati, OH, United States
58University of Maryland, College Park, MD, United States
59Syracuse University, Syracuse, NY, United States
60Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to2
61Institute of Particle Physics, Central China Normal University, Wuhan,
Hubei, China, associated to3
62Institut für Physik, Universität Rostock, Rostock, Germany, associated to11
63National Research Centre Kurchatov Institute, Moscow, Russia, associated
to31
64Instituto de Fisica Corpuscular (IFIC), Universitat de Valencia-CSIC,
Valencia, Spain, associated to36
65KVI - University of Groningen, Groningen, The Netherlands, associated to41
66CBU-Manisa, Manisa, Turkey, associated to38
aUniversidade Federal do Triângulo Mineiro (UFTM), Uberaba-MG, Brazil
bP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
cUniversità di Bari, Bari, Italy
dUniversità di Bologna, Bologna, Italy
eUniversità di Cagliari, Cagliari, Italy
fUniversità di Ferrara, Ferrara, Italy
gUniversità di Firenze, Firenze, Italy
hUniversità di Urbino, Urbino, Italy
iUniversità di Modena e Reggio Emilia, Modena, Italy
jUniversità di Genova, Genova, Italy
kUniversità di Milano Bicocca, Milano, Italy
lUniversità di Roma Tor Vergata, Roma, Italy
mUniversità di Roma La Sapienza, Roma, Italy
nUniversità della Basilicata, Potenza, Italy
oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
pHanoi University of Science, Hanoi, Viet Nam
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
tUniversità degli Studi di Milano, Milano, Italy
## 1 Introduction
The decay mode ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow J/\psi\pi^{+}\pi^{-}$
is of particular interest in the study of $C\\!P$ violation in the $B$
system.111In this paper, mention of a particular decay mode implies the use of
the charge conjugate decay as well, unless stated otherwise. The decay can
proceed either via a tree level process, shown in Fig. 1(a), or via the
penguin mechanisms shown in Fig. 1(b). The ratio of penguin to tree amplitudes
is enhanced in this decay relative to ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\rm\scriptscriptstyle S}}$ [1, *Faller:2008zc]. Thus the
effects of penguin topologies can be investigated by using the
$J/\psi\pi^{+}\pi^{-}$ decay and comparing different measurements of the
$C\\!P$ violating phase, $\beta$, in ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\rm\scriptscriptstyle S}}$, and individual channels such as
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow
J/\psi\rho^{0}$.
Figure 1: (a) Tree level and (b) penguin diagram for ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}}$ decays into
$J/\psi\pi^{+}\pi^{-}$.
The ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow
J/\psi\pi^{+}\pi^{-}$ decay is also useful for the study of the substructure
of light mesons that decay into $\pi^{+}\pi^{-}$. Tests have been proposed to
ascertain if the scalar $f_{0}(500)$ and $f_{0}(980)$ mesons are formed of
$q\overline{q}$ or tetraquarks. In the model of Ref. [3], if these scalar
states are tetraquarks, the ratio of decay widths is predicted to be 1/2. If
instead these are $q\overline{q}$ states, they can be mixtures of two base
states; in this scenario the width ratio can be any value and is determined
principally by the mixing angle between the base states.
The ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow
J/\psi\pi^{+}\pi^{-}$ decay was first observed by the BaBar collaboration
[Aubert:2002vb, *Aubert:2007xw]. It has been previously studied by LHCb using
data from 1 $\mbox{\,fb}^{-1}$ of integrated luminosity [5]. The branching
fraction was measured to be $(3.97\pm 0.22)\times 10^{-5}$. The mass and
angular distributions were used to measure the resonant substructure. That
analysis, however, did not use the angle between the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and $\pi^{+}\pi^{-}$ decay
planes, due to limited statistics. A new theoretical approach [6] now allows
us to include all the angular information and measure the fraction of
$C\\!P$-even and $C\\!P$-odd states. This information is vital to any
subsequent $C\\!P$ violation measurements.
## 2 Data sample and detector
In this paper, we measure the resonant substructure and $C\\!P$ content of the
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow
J/\psi\pi^{+}\pi^{-}$ decay from data corresponding to 3 $\mbox{\,fb}^{-1}$ of
integrated luminosity collected with the LHCb detector [7] using $pp$
collisions. One-third of the data was acquired at a center-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$, and the remainder at
8$\mathrm{\,Te\kern-1.00006ptV}$. The detector is a single-arm forward
spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the
study of particles containing $b$ or $c$ quarks. The detector includes a high-
precision tracking system consisting of a silicon-strip vertex detector
surrounding the $pp$ interaction region, a large-area silicon-strip detector
located upstream of a dipole magnet with a bending power of about
$4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift
tubes [8] placed downstream. The combined tracking system provides a momentum
measurement with relative uncertainty that varies from 0.4% at
5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6% at
100$\mathrm{\,Ge\kern-1.00006ptV}$,222We work in units where $c=1$. and impact
parameter (IP) resolution of 20${\,\upmu\rm m}$ for tracks with large
transverse momentum ($p_{\rm T}$). Different types of charged hadrons are
distinguished by information from two ring-imaging Cherenkov (RICH)
detectors[9]. Photon, electron and hadron candidates are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic calorimeter and a hadronic calorimeter. Muons are identified
by a system composed of alternating layers of iron and multiwire proportional
chambers [10].
The trigger consists of a hardware stage, based on information from the
calorimeter and muon systems, followed by a software stage that applies a full
event reconstruction [11]. Events selected for this analysis are triggered by
a $J/\psi\rightarrow\mu^{+}\mu^{-}$ decay, where the $J/\psi$ meson is
required at the software level to be consistent with coming from the decay of
a ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ meson by use either
of IP requirements or detachment of the $J/\psi$ meson decay vertex from the
primary vertex (PV). In the simulation, $pp$ collisions are generated using
Pythia [12, *Sjostrand:2007gs] with a specific LHCb configuration [14]. Decays
of hadronic particles are described by EvtGen [15], in which final state
radiation is generated using Photos [16]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [17, 18] as described in Ref. [19].
## 3 Decay amplitude formalism
### 3.1 Observables used in the analysis
The ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow
J/\psi\pi^{+}\pi^{-}$ decay with ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow{\mu^{+}}{\mu^{-}}$ can be described by the invariant mass
of the ${{\pi}^{+}}{{\pi}^{-}}$ ($m_{hh}$) pair, and three angles: (i) the
angle between the ${\mu^{+}}$ direction in the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ rest frame with respect to
the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ direction in the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ rest frame,
$\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$; (ii) the angle
between the ${{\pi}^{+}}$ direction and the opposite direction of the ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$ candidate momentum in the
${{\pi}^{+}}{{\pi}^{-}}$ rest frame, $\theta_{hh}$; and (iii) the angle
between the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ and
${{\pi}^{+}}{{\pi}^{-}}$ decay planes in the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ rest frame, $\chi$. The angular
variables are illustrated in Fig. 2.
Figure 2: Illustration of the three angles used in this analysis.
In our previous study [5], we used the “Dalitz-plot” variables: the invariant
mass squared of ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}$,
$s_{12}=m^{2}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}})$, and
the invariant mass squared of the ${{\pi}^{+}}{{\pi}^{-}}$ pair,
$s_{23}=m^{2}({{\pi}^{+}}{{\pi}^{-}})$. Due to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ spin, the event distributions
in the $s_{12}$ and $s_{23}$ plane do not directly show the effect of the
matrix-element squared. Since the probability density functions (PDFs)
expressed as functions of $m_{hh}$ and $\theta_{hh}$ are easier to normalize,
we use them instead. In this paper, the notation $hh$ is equivalent to
$\pi^{+}\pi^{-}$. The Dalitz-plot variables can be translated into ($m_{hh}$,
$\theta_{hh}$), and vice versa. The formalism described below is for the decay
sequence ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}R$, $R\rightarrow\pi^{+}\pi^{-}$.
### 3.2 Amplitude formalism
The decay rate of
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{B}^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ has been described in detail in Ref. [6]. The
differential decay width can be written in terms of the decay time $t$ and the
four other variables $m_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\theta_{hh},$ and $\chi$ as [20, *Bigi:2000yz]
$\displaystyle\frac{d^{5}{\Gamma}}{dt\,dm_{hh}\,d\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\,d\cos\theta_{hh}\,d\chi}=\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\quad\quad\quad\quad\quad\quad{\cal N}e^{-\Gamma
t}\left\\{\frac{|A|^{2}+|(q/p)\overline{A}|^{2}}{2}\cosh\frac{\Delta\Gamma
t}{2}+\frac{|A|^{2}-|(q/p)\overline{A}|^{2}}{2}\cos(\Delta
mt)\right.\quad\quad$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\left.\mathcal{R}e\left((q/p)A^{*}\overline{A}\right)\sinh\frac{\Delta\Gamma
t}{2}-\mathcal{I}m\left((q/p)A^{*}\overline{A}\right)\sin(\Delta
mt)\right\\},$ (1)
$\displaystyle\frac{d^{5}\overline{\Gamma}}{dt\,dm_{hh}\,d\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\,d\cos\theta_{hh}\,d\chi}=\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$
$\displaystyle\quad\quad\quad\quad\left|\frac{p}{q}\right|^{2}{\cal
N}e^{-\Gamma
t}\left\\{\frac{|A|^{2}+|(q/p)\overline{A}|^{2}}{2}\cosh\frac{\Delta\Gamma
t}{2}-\frac{|A|^{2}-|(q/p)\overline{A}|^{2}}{2}\cos(\Delta
mt)\right.\quad\quad$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\left.\mathcal{R}e\left((q/p)A^{*}\overline{A}\right)\sinh\frac{\Delta\Gamma
t}{2}+\mathcal{I}m\left((q/p)A^{*}\overline{A}\right)\sin(\Delta
mt)\right\\},$ (2)
where $\cal N$ is a constant;
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{A}$ is
the amplitude of
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{B}^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$ at the decay time $t=0$, which is itself a
function of $m_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\theta_{hh},$ and $\chi$, summed over all resonant (and possibly non-
resonant) components; $\Delta m$ is the mass difference between the heavy and
light ${B}^{0}$ mass eigenstates, and $\Delta\Gamma$ the width
difference;333We use the conventions that $\Delta m=m_{H}-m_{L}$ and
$\Delta\Gamma=\Gamma_{L}-\Gamma_{H}$, where $L$ and $H$ correspond to the
light and heavy mass eigenstates, respectively. $q$ and $p$ are complex
parameters that describe the relation between mass and flavor eigenstates. In
this analysis we take $|p/q|$ to be equal to unity.
Forming the sum of ${{B}^{0}}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ decay widths and integrating
over decay time, yields the time-integrated and flavor-averaged decay width
$\displaystyle
S(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)=$
$\displaystyle|A(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)|^{2}+|\overline{A}(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)|^{2}$ $\displaystyle-2{\cal
D}\,\mathcal{R}e\left(\frac{q}{p}A^{*}(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)\overline{A}(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)\right)$ $\displaystyle\approx$
$\displaystyle|A(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)|^{2}+|\overline{A}(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)|^{2},$ (3)
where we drop the term arising from quantum interference of the amplitudes in
the last line. This results from the fact that the ${\cal D}$ factor is
negligibly small for ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$
meson decays. Specifically,
${\cal D}=\frac{\int_{0}^{\infty}\alpha(t)e^{-\Gamma t}\sinh\frac{\Delta\Gamma
t}{2}{\rm d}t}{\int_{0}^{\infty}\alpha(t)e^{-\Gamma t}\cosh\frac{\Delta\Gamma
t}{2}{\rm d}t},$ (4)
where $\alpha(t)$ is the decay time dependent detection efficiency.444For
uniform acceptance, ${\cal D}=\Delta\Gamma/(2\Gamma)$. Since
$\Delta\Gamma/\Gamma$ is of the order of $1\%$ for ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ meson decays [22], the ${\cal
D}$ term is about the same size.
We define $A_{R}(m_{hh})$ to be the mass line shape of the resonance $R$,
which in most cases is a Breit-Wigner function. It is combined with the decay
properties of the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ and
resonance to form the expression for the decay amplitude. For each resonance
$R$:
${\mathscr{A}}_{R}(m_{hh})=\sqrt{2J_{R}+1}\sqrt{P_{R}P_{B}}\,F_{B}^{(L_{B})}\left(\frac{P_{B}}{m_{B}}\right)^{L_{B}}F_{R}^{(L_{R})}\left(\frac{P_{R}}{m_{hh}}\right)^{L_{R}}A_{R}(m_{hh}).$
(5)
Here $P_{R}$ ($P_{B}$) is the scalar momentum of one of the two daughters of
the resonance $R$ (or the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ meson) in the $R$ (or ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$) rest frame, $J_{R}$ is the spin
of $R$, $L_{B}$ is the orbital angular momentum between the $J/\psi$ and
$h^{+}h^{-}$ system, and $L_{R}$ the orbital angular momentum in the
$h^{+}h^{-}$ decay, and thus is the same as the spin of the $h^{+}h^{-}$
resonance. $F_{B}^{(L_{B})}$ and $F_{R}^{(L_{R})}$ are the centrifugal barrier
factors for the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ and the
$R$ resonance, respectively [23]. The factor $\sqrt{P_{R}P_{B}}$ results from
converting the phase space of the Dalitz-plot variables $m^{2}_{hh}$ and
$m^{2}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}h^{+}}$ to that of
$m_{hh}$ and $\cos\theta_{hh}$. The function defined in Eq. (5) is based on
previous amplitude analyses [24, 23].
We must sum over all final states, $R$, so for each
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ helicity, denoted by
$\lambda$, equal to $0$, $+1$ and $-1$ we have the overall decay amplitudes:
${\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{\lambda}(m_{hh},\theta_{hh})=\sum_{R}\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\bf
h}_{\lambda}^{R}{\mathscr{A}}_{R}(m_{hh})d_{-\lambda,0}^{J_{R}}(\theta_{hh}),$
(6)
where the Wigner-$d$ functions are defined in Ref. [22] and
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{\lambda}^{R}$
are complex helicity coefficients. We note that the $\lambda$ value of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is equal to that of the $R$
resonance. Finally, the total decay rate of
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{B}^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$ at $t=0$ is given by
$\displaystyle|\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{A}(m_{hh},\theta_{hh},$
$\displaystyle\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)|^{2}=$
$\displaystyle|{\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{0}(m_{hh},\theta_{hh})|^{2}\sin^{2}\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}+\frac{1}{2}\left(|{\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{+}(m_{hh},\theta_{hh})|^{2}+|{\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{-}(m_{hh},\theta_{hh})|^{2}\right)$
$\displaystyle\times(1+\cos^{2}\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}})+\mathcal{R}e\left[{\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{+}(m_{hh},\theta_{hh}){\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{-}^{*}(m_{hh},\theta_{hh})e^{2i\chi}\right]\sin^{2}\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$
$\displaystyle+\sqrt{2}\mathcal{R}e\left[\left({\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{0}(m_{hh},\theta_{hh}){\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{+}^{*}(m_{hh},\theta_{hh})-{\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}^{*}_{0}(m_{hh},\theta_{hh}){\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{-}(m_{hh},\theta_{hh})\right)e^{-i\chi}\right]$
$\displaystyle\times\sin\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}~{}.$ (7)
In order to determine the $C\\!P$ components, it is convenient to replace the
complex helicity coefficients
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{\lambda}^{R}$
by the complex transversity coefficients
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\tau}^{R}$
using the relations
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{0}^{R}$
$\displaystyle=$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{0}^{R},$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{+}^{R}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\parallel}^{R}+\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\perp}^{R}),$
$\displaystyle\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{h}}_{-}^{R}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\parallel}^{R}-\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\perp}^{R}).$
(8)
Here
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{0}^{R}$
corresponds to the longitudinal polarization of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ meson, and the other two
coefficients correspond to polarizations of the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ meson and $h^{+}h^{-}$ system
transverse to the decay axis:
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\parallel}^{R}$
for parallel polarization of the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}$ and $h^{+}h^{-}$, and
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{\textbf{a}}_{\perp}^{R}$
for perpendicular polarization.
Assuming the absence of direct $C\\!P$ violation, the relation between the
$\overline{B}^{0}$ and $B^{0}$ transversity coefficients is
$\bar{\textbf{a}}^{R}_{\tau}=\eta^{R}_{\tau}\textbf{a}^{R}_{\tau}$, where
$\eta^{R}_{\tau}$ is the $C\\!P$ eigenvalue of the $\tau$ transversity
component for the intermediate state $R$, and $\tau$ denotes the
$0,~{}\parallel,$ or $\perp$ components. Note that for the $h^{+}h^{-}$ system
both $C$ and $P$ are given by $(-1)^{L_{R}}$, so the $C\\!P$ of the
$h^{+}h^{-}$ system is always even. The total $C\\!P$ of the final state is
$(-1)^{L_{B}}$, since the $C\\!P$ of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is also even. The final state
$C\\!P$ parities, for S, P, and D-waves, are listed in Table 1.
In this analysis a fit determines the amplitude modulus $a_{\tau}^{R}$ and the
phase $\phi_{\tau}^{R}$ of the amplitude
$\textbf{a}^{R}_{\tau}=a_{\tau}^{R}e^{i\phi_{\tau}^{R}}$ (9)
for each resonance $R$, and each transversity component $\tau$. For the
$\tau=\perp$ amplitude, the $L_{B}$ value of spin-1 (or spin-2) resonances is
1 (or 2). While the other transversity components, $\tau=$ 0 or $\parallel$,
have two possible $L_{B}$ values of 0 and 2 (or 1 and 3) for spin-1 (or -2)
resonances. We use only the smaller values for each. Studies show that our
results for fractions of different interfering components are not sensitive to
these $L_{B}$ choices.
Table 1: $C\\!P$ parity of the full final state for different spin resonances. Note that spin-0 only has the 0 transversity component. Spin | $\eta_{0}$ | $\eta_{\parallel}$ | $\eta_{\perp}$
---|---|---|---
0 | $-1$ | |
1 | 1 | 1 | $-1$
2 | $-1$ | $-1$ | 1
### 3.3 Dalitz fit fractions
A complete description of the decay is given in terms of the fitted complex
amplitudes. Knowledge of the contribution of each component can be summarized
by defining a fit fraction for each transversity $\tau$,
${\cal{F}}_{\tau}^{R}$. To determine ${\cal{F}}_{\tau}^{R}$ one needs to
integrate over all the four variables:
$m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi$. The interference terms between different helicity components
vanish after integrating Eq. (3.2) over the two variables of
$\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$ and $\chi$, i.e.
$\displaystyle\int|\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{A}(m_{hh},\theta_{hh},$
$\displaystyle\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)|^{2}d\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}\,d\chi$
$\displaystyle=\frac{4}{3}\left(|{\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{0}(m_{hh},\theta_{hh})|^{2}+|{\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{+}(m_{hh},\theta_{hh})|^{2}+|{\cal\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{--}\scalebox{0.4}{)}}{H}}_{-}(m_{hh},\theta_{hh})|^{2}\right).$
(10)
The decay rate is the sum of the contributions from the three helicity terms.
To define the transversity fractions, we need to write Eq. (3.3) in terms of
transversity amplitudes. Since $d_{-1,0}^{J_{R}}=-d_{1,0}^{J_{R}}$, the sum of
the three helicity terms is equal to the sum of three transversities, given as
$\displaystyle|{\cal H}_{0}(m_{hh},\theta_{hh})|^{2}+|{\cal
H}_{+}(m_{hh},\theta_{hh})|^{2}+|{\cal H}_{-}(m_{hh},\theta_{hh})|^{2}=$
$\displaystyle\left|\sum_{R}\textbf{a}_{0}^{R}{\mathscr{A}}_{R}(m_{hh})d_{0,0}^{J_{R}}(\theta_{hh})\right|^{2}+\left|\sum_{R}\textbf{a}_{\|}^{R}{\mathscr{A}}_{R}(m_{hh})d_{1,0}^{J_{R}}(\theta_{hh})\right|^{2}+\left|\sum_{R}\textbf{a}_{\perp}^{R}{\mathscr{A}}_{R}(m_{hh})d_{1,0}^{J_{R}}(\theta_{hh})\right|^{2}.$
(11)
Thus, we define the transversity fit fractions as
${\cal{F}}^{R}_{\tau}=\frac{\int\left|a^{R}_{\tau}e^{i\phi^{R}_{\tau}}{\mathscr{A}}_{R}(m_{hh})d_{\lambda,0}^{J_{R}}(\theta_{hh})\right|^{2}dm_{hh}\;d\cos\theta_{hh}}{\int\left(|{\cal{H}}_{0}(m_{hh},\theta_{hh})|^{2}+|{\cal{H}}_{+}(m_{hh},\theta_{hh})|^{2}+|{\cal{H}}_{-}(m_{hh},\theta_{hh})|^{2}\right)dm_{hh}\;d\cos\theta_{hh}},$
(12)
where $\lambda=0$ for $\tau=0$, and $\lambda=1$ for $\tau=\perp$ or
$\parallel$.
The sum of the fit fractions is not necessarily unity due to the potential
presence of interference between two resonances. Interference term fractions
are given by
${\cal{F}}_{\tau}^{RR^{\prime}}=2\mathcal{R}e\left(\frac{\int
a^{R}_{\tau}\;a^{R^{\prime}}_{\tau}e^{i(\phi^{R}_{\tau}-\phi^{R^{\prime}}_{\tau})}{\mathscr{A}}_{R}(m_{hh}){\mathscr{A}}^{*}_{R^{\prime}}(m_{hh})d_{\lambda,0}^{J_{R}}(\theta_{hh})d_{\lambda,0}^{J_{R^{\prime}}}(\theta_{hh})dm_{hh}\;d\cos\theta_{hh}}{\int\left(|{\cal{H}}_{0}(m_{hh},\theta_{hh})|^{2}+|{\cal{H}}_{+}(m_{hh},\theta_{hh})|^{2}+|{\cal{H}}_{-}(m_{hh},\theta_{hh})|^{2}\right)dm_{hh}\;d\cos\theta_{hh}}\right),$
(13)
and
$\sum_{R,\tau}{\cal{F}}_{\tau}^{R}+\sum^{R>R^{\prime}}_{RR^{\prime},\tau}{\cal{F}}_{\tau}^{RR^{\prime}}=1.$
(14)
Interference between different spin-$J$ states vanishes when integrated over
angle, because the $d^{J}_{\lambda 0}$ angular functions are orthogonal.
## 4 Selection requirements
In this analysis we adopt a two step selection. The first step, preselection,
is followed by a multivariate selection based on a boosted decision tree (BDT)
[25]. Preselection criteria are implemented to preserve a large fraction of
the signal events, yet reject easily eliminated backgrounds, and are identical
to those used in Ref. [5]. A ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$ candidate is reconstructed by combining a
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\rightarrow\mu^{+}\mu^{-}$
candidate with two pions of opposite charge. To ensure good track
reconstruction, each of the four particles in the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ candidate is required to have
the track fit $\chi^{2}$/ndf to be less than 4, where ndf is the number of
degrees of freedom of the fit. The ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rightarrow\mu^{+}\mu^{-}$ candidate is formed by two identified muons
of opposite charge having $p_{\rm T}$ greater than 500
$\mathrm{\,Me\kern-1.00006ptV}$, and with a geometrical fit vertex $\chi^{2}$
less than 16. Only candidates with dimuon invariant mass between $-48$
$\mathrm{\,Me\kern-1.00006ptV}$ and $+43$ $\mathrm{\,Me\kern-1.00006ptV}$ from
the observed ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass peak are
selected, and are then constrained to the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ mass [22] for subsequent use.
Each pion candidate is required to have $p_{\rm T}$ greater than 250
$\mathrm{\,Me\kern-1.00006ptV}$, and that the scalar sum, $\mbox{$p_{\rm
T}$}(\pi^{+})+\mbox{$p_{\rm T}$}(\pi^{-})$ is required to be larger than 900
$\mathrm{\,Me\kern-1.00006ptV}$. Both pions must have $\chi^{2}_{\rm IP}$
greater than 9 to reject particles produced from the PV. The $\chi^{2}_{\rm
IP}$ is computed as the difference between the $\chi^{2}$ of the PV
reconstructed with and without the considered track. Both pions must also come
from a common vertex with $\chi^{2}{\rm/ndf}<16$, and form a vertex with the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}$ with a $\chi^{2}$/ndf less
than 10 (here ndf equals five). Pion candidates are identified using the RICH
and muon systems. The particle identification makes use of the logarithm of
the likelihood ratio comparing two particle hypotheses (DLL). For the pion
selection we require DLL$(\pi-K)>-10$ and DLL$(\pi-\mu)>-10$. The ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ candidate must have a flight
distance of more than 1.5 $\rm\,mm$. The angle between the combined momentum
vector of the decay products and the vector formed from the positions of the
PV and the decay vertex (pointing angle) is required to be less than
$2.5^{\circ}$.
The BDT uses eight variables that are chosen to provide separation between
signal and background. These are the minimum of DLL($\mu-\pi$) of the
$\mu^{+}$ and $\mu^{-}$, $\mbox{$p_{\rm T}$}(\pi^{+})+\mbox{$p_{\rm
T}$}(\pi^{-})$, the minimum of $\chi^{2}_{\rm IP}$ of the $\pi^{+}$ and
$\pi^{-}$, and the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$
properties of vertex $\chi^{2}$, pointing angle, flight distance, $p_{\rm T}$
and $\chi^{2}_{\rm IP}$. The BDT is trained on a simulated sample of two
million ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$ signal events generated uniformly in phase space with
unpolarized $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays, and a background data
sample from the sideband $5566<m(J/\psi\pi^{+}\pi^{-})<5616$
$\mathrm{\,Me\kern-1.00006ptV}$. Then the BDT is tested on independent samples
from the same sources. The BDT can take any value from -1 to 1. The
distributions of signal and background are approximately Gaussian shaped with
r.m.s. of about 0.13. Signal peaks at BDT of 0.27 and background at -0.22. To
minimize possible bias on the signal acceptance due to the BDT, we choose a
loose requirement of BDT$>0$, which has about a $95\%$ signal efficiency and a
$90\%$ background rejection rate.
## 5 Fit model
We first select events based on their ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-}$ invariant mass and then perform a full fit to the decay
variables. The invariant mass of the selected $J/\psi\pi^{+}\pi^{-}$
combinations is shown in Fig. 3. There is a large peak at the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}$ mass and a smaller one at
the ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ mass on top of the
background. A double Crystal Ball function with common means models the
radiative tails and is used to fit each of the signals [26]. The known
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}-{{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ mass difference [22] is used to
constrain the difference in mean values. Other components in the fit model
take into account background contributions from $B^{-}\rightarrow J/\psi
K^{-}$ and $B^{-}\rightarrow J/\psi\pi^{-}$ decays combined with a random
$\pi^{+}$, ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}\rightarrow
J/\psi\eta(^{\prime})$ with $\eta(^{\prime})\rightarrow\pi^{+}\pi^{-}\gamma$,
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}\rightarrow
J/\psi\phi$ with $\phi\rightarrow\pi^{+}\pi^{-}\pi^{0}$, ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow J/\psi K^{-}\pi^{+}$
and ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{-}}{p}$ reflections, and combinatorial backgrounds. The
exponential combinatorial background shape is taken from like-sign
combinations, that are the sum of $\pi^{+}\pi^{+}$ and $\pi^{-}\pi^{-}$
candidates, and found to be a good description in previous studies [23, 27].
Figure 3: Invariant mass of $J/\psi\pi^{+}\pi^{-}$ combinations together with
the data fit. The (red) solid curve shows the ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}}$ signal, the (brown) dotted line
shows the combinatorial background, the (green) short-dashed line shows the
$B^{-}$ background, the (purple) dot-dashed curve is ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}_{s}}\rightarrow
J/\psi\pi^{+}\pi^{-}$, the (light blue) long-dashed line is the sum of
${{\kern 1.61993pt\overline{\kern-1.61993ptB}{}}^{0}_{s}}\rightarrow
J/\psi\eta^{(^{\prime})}$, ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}_{s}}\rightarrow J/\psi\phi$ with
$\phi\rightarrow\pi^{+}\pi^{-}\pi^{0}$ backgrounds and the ${\mathchar
28931\relax}_{b}^{0}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}K^{-}p$ reflection, the (black) dot-long dashed curve is the ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}}\rightarrow J/\psi K^{-}\pi^{+}$
reflection and the (blue) solid curve is the total. The points at the bottom
show the difference between the data points and the total fit divided by the
statistical uncertainty on the data.
The shapes of the other components are taken from the Monte Carlo simulation
with their normalizations allowed to vary. The mass fit gives $18\,841\pm 204$
signal and $10\,207\pm 178$ background candidates within $\pm 20$ MeV of the
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ mass peak. Only
candidates within $\pm 20\mathrm{\,Me\kern-1.00006ptV}$ of the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ mass peak are retained for
further analysis. To improve the resolution of the mass and angular variables
used in the amplitude analysis, we perform a kinematic fit constraining the
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ and $J/\psi$ masses to
their PDG mass values [22], and recompute the final state momenta [28].
One of the main challenges in performing a mass and angular analysis is to
construct a realistic probability density function, where both the kinematic
and dynamical properties are modeled accurately. The PDF is given by the sum
of signal, $S$, and background, $B$, functions. The ${\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}$ signal includes events from the
reaction ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{0}_{\rm\scriptscriptstyle S}}$. These are described by a separate
term in the PDF. The total PDF is
$\displaystyle F(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)$ $\displaystyle=$
$\displaystyle f_{\rm
sig}\times\left[\frac{1-f_{{{K}^{0}_{\rm\scriptscriptstyle
S}}}}{{\cal{N}}_{\rm
sig}}\varepsilon(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)S(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)\vphantom{\frac{f_{{{K}^{0}_{\rm\scriptscriptstyle
S}}}}{{\cal N}_{{{K}^{0}_{\rm\scriptscriptstyle S}}}}}\right.$ (15)
$\displaystyle+\left.\frac{f_{{{K}^{0}_{\rm\scriptscriptstyle S}}}}{{\cal
N}_{{{K}^{0}_{\rm\scriptscriptstyle
S}}}}\varepsilon(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)G(m_{hh};m_{{{K}^{0}_{\rm\scriptscriptstyle
S}}},\sigma_{{{K}^{0}_{\rm\scriptscriptstyle
S}}})\sin^{2}\theta_{J/\psi}\right]$ $\displaystyle+(1-f_{\rm
sig})B(m_{hh},\theta_{hh},\theta_{J/\psi},\chi),$
where $f_{\text{sig}}$ is the fraction of the signal in the fitted region
($f_{\text{sig}}=(64.9\pm 1.2)\%$ obtained from the mass fit in Fig. 3),
$\varepsilon$ is the detection efficiency described in Sec. 5.1, and $B$ is
the background PDF described in Sec. 5.2. The ${{K}^{0}_{\rm\scriptscriptstyle
S}}$ component is modeled by a Gaussian function, $G$, with mean
$m_{{{K}^{0}_{\rm\scriptscriptstyle S}}}$ and width
$\sigma_{{{K}^{0}_{\rm\scriptscriptstyle S}}}$. The Gaussian parameters
together with the ${{K}^{0}_{\rm\scriptscriptstyle S}}$ fraction in the
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ peak,
$f_{{{K}^{0}_{\rm\scriptscriptstyle S}}}$, are determined in the fit. The
normalization factors ${\cal N}_{\rm sig}$ for the signal and ${\cal
N}_{{{K}^{0}_{\rm\scriptscriptstyle S}}}$ for the
${{K}^{0}_{\rm\scriptscriptstyle S}}$ candidates are efficiency-multiplied
theoretical functions integrated over the four analysis variables, $m_{hh}$,
$\theta_{hh}$, $\theta_{J/\psi}$, and $\chi$, given by
$\displaystyle{\cal{N}}_{\rm sig}$ $\displaystyle=$
$\displaystyle\int\\!\varepsilon(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)S(m_{hh},\theta_{hh},\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},\chi)\,{\rm d}\,m_{hh}\,{\rm d}\cos\theta_{hh}\,{\rm
d}\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}\,{\rm d}\chi.$
(16)
Examination of the event distribution for $m^{2}({{\pi}^{+}}{{\pi}^{-}})$
versus $m^{2}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}})$ in
Fig. 4 shows obvious structures in $m^{2}({{\pi}^{+}}{{\pi}^{-}})$. To
investigate if there are visible exotic structures in
$m^{2}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}})$, we examine
the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}$ invariant mass
distribution as shown in Fig. 5(a) where we fit the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}})$
distribution to extract the background levels in bins of
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}})$ (red points).
Similarly, Fig. 5(b) shows the ${{\pi}^{+}}{{\pi}^{-}}$ mass distribution.
Apart from a large signal peak due to the $\rho(770)$, there are visible
structures at about 1270 MeV and a ${{K}^{0}_{\rm\scriptscriptstyle S}}$
component at about 500 MeV.
Figure 4: Distribution of $m^{2}({{\pi}^{+}}{{\pi}^{-}})$ versus
$m^{2}({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}})$ for all
events within $\pm 20$ MeV of the ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}}$ mass. Figure 5: Distributions of
(a) $m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}})$ and (b)
$m({{\pi}^{+}}{{\pi}^{-}})$ for ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}}\rightarrow J/\psi\pi^{+}\pi^{-}$
candidates within $\pm 20$ MeV of the ${{\kern
1.61993pt\overline{\kern-1.61993ptB}{}}^{0}}$ mass. The red points with error
bars show the background contributions obtained by fitting the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}})$
distribution in bins of the plotted variables.
### 5.1 Detection efficiency
The detection efficiency is determined from a sample of about four million
simulated ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow
J/\psi\pi^{+}\pi^{-}$ events that are generated uniformly in phase space with
unpolarized $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays. The efficiency model can
be expressed as
$\varepsilon(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)=\varepsilon_{1}(s_{12},s_{13})\times\varepsilon_{2}(\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},m_{hh})\times\varepsilon_{3}(\chi,m_{hh}),$ (17)
where $s_{12}\equiv m^{2}({{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}})$ and $s_{13}\equiv
m^{2}({{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{-}})$ are functions
of $(m_{hh},\theta_{hh})$; such parameter transformations in $\varepsilon_{1}$
are implemented in order to use the Dalitz-plot based efficiency model
developed in previous publications [23, 5].
The efficiency dependence on $\chi$ is modeled by
$\varepsilon_{3}(\chi,m_{hh})=\frac{1}{2\pi}(1+p_{1}\cos\chi+p_{2}\cos
2\chi),$ (18)
where $p_{1}=p_{1}^{0}+p_{1}^{1}\times m^{2}_{hh}$ and
$p_{2}=p_{2}^{0}+p_{2}^{1}\times m^{2}_{hh}+p_{2}^{2}\times m^{4}_{hh}$. The
free parameters are determined by fitting the simulated $\chi$ distributions
using Eq. (18) in bins of $m^{2}_{hh}$. The fit gives $p_{1}^{0}=-0.0065\pm
0.0052$ and $p_{1}^{1}=(0.0011\pm 0.0021)$ GeV-2; $p_{2}^{0}=-0.0006\pm
0.0079$, $p_{2}^{1}=(0.0602\pm 0.0083)$ GeV-2 and $p_{2}^{2}=(-0.0099\pm
0.0018)$ GeV-4.
The acceptance in $\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}$ depends on $m_{hh}$. We disentangle this correlation by fitting the
$\cos\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}}$ distribution in
24 bins of $m^{2}_{hh}$ using the parameterization
$\varepsilon_{2}(\theta_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}},m_{hh})=\frac{1+a\cos^{2}\theta_{J/\psi}}{2+2a/3}.$ (19)
The fitted values of $a$ are modeled by a second order polynomial function
$a(m^{2}_{hh})=a_{0}+a_{1}m^{2}_{hh}+a_{2}m_{hh}^{4},$ (20)
with $a_{0}=0.189\pm 0.021$, $a_{1}=-0.116\pm 0.021$ GeV-2 and $a_{2}=0.017\pm
0.004$ GeV-4.
We model the detection efficiency, $\varepsilon_{1}(s_{12},s_{13})$, by using
the symmetric observables
$x=s_{12}/{\rm GeV}^{2}-18.4~{},~{}~{}~{}~{}{\rm and}~{}~{}~{}~{}y=s_{13}/{\rm
GeV}^{2}-18.4~{}.$ (21)
These variables are related to $s_{23}$ by
$\displaystyle
s_{12}+s_{13}+s_{23}=m^{2}_{B}+m^{2}_{{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}}+m^{2}_{{{\pi}^{+}}}+m^{2}_{{{\pi}^{-}}}.$ (22)
Thus, $\varepsilon_{1}(s_{12},s_{13})$ can be modeled by a two-dimensional
fifth order polynomial function as
$\displaystyle\varepsilon_{1}(s_{12},s_{13})$ $\displaystyle=$ $\displaystyle
1+\epsilon_{1}(x+y)+\epsilon_{2}(x+y)^{2}+\epsilon_{3}xy+\epsilon_{4}(x+y)^{3}+\epsilon_{5}xy(x+y)$
(23)
$\displaystyle+\epsilon_{6}(x+y)^{4}+\epsilon_{7}xy(x+y)^{2}+\epsilon_{8}x^{2}y^{2}$
$\displaystyle+\epsilon_{9}(x+y)^{5}+\epsilon_{10}xy(x+y)^{3}+\epsilon_{11}x^{2}y^{2}(x+y)$
where all the $\epsilon_{i}$ are the fit parameters. The $\chi^{2}/\rm ndf$ is
313/299. The values of the parameters are given in Table 2.
Table 2: Efficiency parameters. There are substantial correlations. $\epsilon_{1}$ | 0.1220$\pm$0.0097
---|---
$\epsilon_{2}$ | 0.1163$\pm$0.0182
$\epsilon_{3}$ | 0.0051$\pm$0.0004
$\epsilon_{4}$ | 0.0399$\pm$0.0101
$\epsilon_{5}$ | -0.0012$\pm$0.0007
$\epsilon_{6}$ | 0.10051$\pm$0.0023
$\epsilon_{7}$ | 0.0002$\pm$0.0005
$\epsilon_{8}$ | -0.000150$\pm$0.000007
$\epsilon_{9}$ | -0.000011$\pm$0.000261
$\epsilon_{10}$ | 0.000350$\pm$0.000146
$\epsilon_{11}$ | -0.000113$\pm$0.000011
The projections of the fit used to measure the efficiency parameters are shown
in Fig. 6. The efficiency shapes are well described by the parametrization.
Figure 6: Projections of invariant mass squared (a) $s_{12}\equiv
m^{2}(J/\psi\pi^{+})$ and (b) $s_{23}\equiv m^{2}(\pi^{+}\pi^{-})$ of the
simulated Dalitz plot used to measure the efficiency parameters. The points
represent the simulated event distributions and the curves the polynomial fit.
The parameterized efficiency as a function of $m(\pi^{+}\pi^{-})$ versus
$\cos\theta_{\pi^{+}\pi^{-}}$ is shown in Fig. 7.
Figure 7: The variation of $\varepsilon_{1}$ is shown as a function of
$m(\pi^{+}\pi^{-})$ and $\cos\theta_{\pi^{+}\pi^{-}}$.
### 5.2 Background composition
The main background source in the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ signal region is combinatorial
and can be taken from the like-sign combinations within $\pm 20$ MeV of the
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ mass peak. In addition,
there is background arising from partially reconstructed ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}$ decays (${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}\rightarrow
J/\psi\eta(^{\prime})$, with $\eta(^{\prime})\rightarrow\pi^{+}\pi^{-}\gamma$
and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\phi$ with $\phi\rightarrow{{\pi}^{+}}{{\pi}^{-}}\pi^{0}$), reflections
from misidentified ${{\mathchar
28931\relax}^{0}_{b}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{K}^{-}}{p}$ and ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow J/\psi K^{-}\pi^{+}$
decays, which cannot be present in the like-sign combinations. We use
simulated samples of these decays to model their contributions. The
${\mathchar 28931\relax}^{0}_{b}$ normalizations are determined from a
previous analysis [29]. The background level in the opposite-sign combination
(${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\pi}^{+}}{{\pi}^{-}}$) is studied by fitting the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}})$
distributions in bins of $m(\pi^{+}\pi^{-})$. The resulting background
distribution in the $\pm 20$ MeV ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ signal region is shown in Fig. 8
by points with error bars. A fit to this distribution gives a partially
reconstructed ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}$
background fraction of 10.7%, the reflection from ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ of 5.3%, and the reflection from
the ${{\mathchar 28931\relax}^{0}_{b}}$ baryon of 15.5% of the total
background. The like-sign combinations summed with the additional backgrounds
modeled by simulation are shown in Fig. 8.
When this data-simulation hybrid sample is used to extract the background
parameters, a further re-weighting procedure is applied based on comparison of
$m(\pi^{+}\pi^{-})$ distributions between the overall fit and the background
data points in Fig. 5(b).
Figure 8: Distributions of $m(\pi^{+}\pi^{-})$ of background components. The
(blue) histogram shows the like-sign combinations added with additional
backgrounds using simulations. The (black) points with error bars show the
background obtained from the fits to the
$m({{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}{{\pi}^{+}}{{\pi}^{-}})$ mass
spectrum in each bin of $\pi^{+}\pi^{-}$ mass.
To better model the angular distributions in the $\rho(770)$ mass region, the
background is separated into the ${{\kern
1.99997pt\overline{\kern-1.99997ptK}{}}^{*0}}$ reflection from ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$, the $\rho$, and other
backgrounds. The total background PDF is sum of these three components:
$\displaystyle B(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)=$
$\displaystyle\frac{f_{{{\kern
1.39998pt\overline{\kern-1.39998ptK}{}}^{*0}}}}{{\cal N}_{{{\kern
1.39998pt\overline{\kern-1.39998ptK}{}}^{*0}}}}B_{{{\kern
1.39998pt\overline{\kern-1.39998ptK}{}}^{*0}}}(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)+\frac{f_{\rho}}{{\cal
N}_{\rho}}B_{\rho}(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)$
$\displaystyle+\frac{1-f_{{{\kern
1.39998pt\overline{\kern-1.39998ptK}{}}^{*0}}}-f_{\rho}}{{\cal N}_{\rm
other}}B_{\rm other}(m_{hh},\theta_{hh},\theta_{J/\psi},\chi),$ (24)
where the $\cal{N}$’s are normalizations, the contributing fractions having
values of $f_{{{\kern 1.39998pt\overline{\kern-1.39998ptK}{}}^{*0}}}=(5.3\pm
0.2)\%$ and $f_{\rho}=(9.5\pm 0.6)\%$; the other background is normalized as
$1-f_{{{\kern 1.39998pt\overline{\kern-1.39998ptK}{}}^{*0}}}-f_{\rho}$.
The ${{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}^{*0}}$ background is
modeled by the function
$\displaystyle B_{{{\kern
1.39998pt\overline{\kern-1.39998ptK}{}}^{*0}}}(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)=$
$\displaystyle\left(\frac{p_{R}}{m_{hh}}\right)^{2}\frac{m_{hh}e^{-a\cdot(1-|\cos\theta_{hh}|)}}{(m^{2}_{0}-m_{hh}^{2})^{2}+m_{0}^{2}\Gamma_{0}^{2}}\times(1-|\cos\theta_{hh}|)^{b}$
$\displaystyle\times\left(1+\alpha_{0}\cos^{2}\theta_{J/\psi}\right)\times(1+p_{b1}\cos\chi+p_{b2}\cos
2\chi),$ (25)
where $m_{0}$, $\Gamma_{0}$, $a$, $b$, $\alpha_{0}$ are free parameters
determined by fitting to the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}{{\kern 1.99997pt\overline{\kern-1.99997ptK}{}}^{*0}}$ simulation. The
last part $(1+p_{b1}\cos\chi+p_{b2}\cos 2\chi)$ is a function of the $\chi$
angle. We have verified that the three backgrounds have consistent $\chi$
distributions, thus the parameters $p_{b1}$ and $p_{b2}$ are determined by
fitting all backgrounds simultaneously.
The $\rho$ background is described by the function
$\displaystyle B_{\rm\rho}(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)=$
$\displaystyle\left(\frac{p_{R}}{m_{hh}}\right)^{2}\frac{m_{hh}}{(m^{2}_{\rho}-m_{hh}^{2})^{2}+m_{\rho}^{2}\Gamma_{\rho}^{2}}\times\sin^{2}\theta_{hh}$
$\displaystyle\times\sin^{2}\theta_{J/\psi}\times(1+p_{b1}\cos\chi+p_{b2}\cos
2\chi),$ (26)
where $m_{\rho}$, $\Gamma_{\rho}$ are free parameters. The parameters are
obtained by fitting to simulated ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\eta^{\prime}(\rightarrow\rho\gamma)$ events.
The model for the remaining backgrounds is
$\displaystyle B_{\rm other}(m_{hh},\theta_{hh},\theta_{J/\psi},\chi)=$
$\displaystyle
m_{hh}B_{1}(m^{2}_{hh},\cos\theta_{hh})\times\left(1+\alpha_{1}\cos^{2}\theta_{J/\psi}\right)$
$\displaystyle\times(1+p_{b1}\cos\chi+p_{b2}\cos 2\chi),$ (27)
with the function
$B_{1}(m_{hh}^{2},\cos\theta_{hh})=\left[B_{2}(\zeta)\frac{p_{B}}{m_{B}}+\frac{b_{0}}{(m^{2}_{1}-m^{2}_{hh})^{2}+m_{1}^{2}\Gamma_{1}^{2}}\right]\times\frac{1+q(\zeta)|\cos\theta_{hh}|+p(\zeta)\cos^{2}\theta_{hh}}{2[1+q(\zeta)/2+p(\zeta)/3]}.$
(28)
Here the variable $\zeta=2(m_{hh}^{2}-m^{2}_{\rm min})/(m^{2}_{\rm
max}-m^{2}_{\rm min})-1$, where $m_{\rm min}$ and $m_{\rm max}$ are the fit
boundaries, $B_{2}(\zeta)$ is a fifth order Chebychev polynomial with
coefficients $b_{i}$ ($i=1$-5), and $q(\zeta)$ and $p(\zeta)$ are two first
order Chebychev polynomials with parameters $c_{j}$ ($j=1$-4).
Figure 9 shows the projections of $\cos\theta_{\pi\pi}$ and
$m({{\pi}^{+}}{{\pi}^{-}})$ from the like-sign data combinations added with
all the additional simulated backgrounds. The other background includes the
${\mathchar 28931\relax}^{0}_{b}$ background and the combinatorial background
which is described by the like-sign combinations. The fitted background
parameters are given in Table 3. The $\cos\theta_{J/\psi}$ background
distribution is shown in Fig. 10. Lastly, the $\chi$ background distribution,
shown in Fig. 11 fit with the function $1+p_{b1}\cos\chi+p_{b2}\cos 2\chi$,
determines the parameters $p_{b1}=-0.004\pm 0.013$ and $p_{b2}=0.070\pm
0.013$.
Figure 9: Projections of (a) $\cos\theta_{\pi\pi}$ and (b) $m({{\pi}^{+}}{{\pi}^{-}})$ of the background. The points with error bars show the like-sign data combinations added with the ${\mathchar 28931\relax}^{0}_{b}$ background and additional simulated backgrounds. The (magenta) dot-dashed line shows the $\eta^{(^{\prime})}\rightarrow\rho\gamma$ background, the (dark-blue) dashed line the ${{\kern 1.79997pt\overline{\kern-1.79997ptK}{}}^{*}}$ reflection background, and the (blue) solid line the total. The points at the bottom show the difference between the data points and the total fit divided by the statistical uncertainty on the data. Figure 10: The $\cos\theta_{J/\psi}$ distribution of the data-simulated hybrid background sample. The points with error bars show the like-sign data combinations added with the ${\mathchar 28931\relax}^{0}_{b}$ background and additional simulated backgrounds. The (magenta) dot-dashed line shows the $\rho$ background, the (dark-blue) dashed the ${{\kern 1.79997pt\overline{\kern-1.79997ptK}{}}^{*}}$ reflection background and the (blue) solid line the total. Figure 11: The $\chi$ distribution of the data-simulated hybrid background sample including the ${\mathchar 28931\relax}^{0}_{b}$ background and the fitted function $1+p_{b1}\cos\chi+p_{b2}\cos 2\chi$. The $p$-value of this fit is $40\%$. Table 3: Parameters for the background model. | $m_{0}$ | $0.7473$ | $\pm$ | $0.0009$$\mathrm{\,Ge\kern-1.00006ptV}$
---|---|---|---|---
| $\Gamma_{0}$ | $0.071$ | $\pm$ | $0.02$$\mathrm{\,Ge\kern-1.00006ptV}$
| $m_{1}$ | $~{}0.45$ | $\pm$ | $0.05$$\mathrm{\,Ge\kern-1.00006ptV}$
| $\Gamma_{1}$ | $~{}0.18$ | $\pm$ | $0.05$$\mathrm{\,Ge\kern-1.00006ptV}$
| $m_{\rho}$ | $~{}0.770$ | $\pm$ | $0.002$$\mathrm{\,Ge\kern-1.00006ptV}$
| $\Gamma_{\rho}$ | $~{}0.110$ | $\pm$ | $0.004$$\mathrm{\,Ge\kern-1.00006ptV}$
| $a$ | $6.94$ | $\pm$ | $0.20$
| $b$ | $0.76$ | $\pm$ | $0.04$
| $b_{0}$ | $0.0019$ | $\pm$ | $0.0004\,$GeV4
| $b_{1}$ | $-0.536$ | $\pm$ | $0.053$
| $b_{2}$ | $~{}0.100$ | $\pm$ | $0.043$
| $b_{3}$ | $-0.100$ | $\pm$ | $0.042$
| $b_{4}$ | $~{}0.080$ | $\pm$ | $0.026$
| $b_{5}$ | $-0.051$ | $\pm$ | $0.025$
| $c_{1}$ | $-0.048$ | $\pm$ | $0.017$
| $c_{2}$ | $-0.172$ | $\pm$ | $0.263$
| $c_{3}$ | $-0.142$ | $\pm$ | $0.170$
| $c_{4}$ | $~{}0.855$ | $\pm$ | $0.259$
| $\alpha_{0}$ | $0.45$ | $\pm$ | $0.04$
| $\alpha_{1}$ | $0.30$ | $\pm$ | $0.03$
### 5.3 Resonance models
Table 4: Possible resonance candidates in the ${{\kern 1.61993pt\overline{\kern-1.61993ptB}{}}^{0}}\rightarrow J/\psi\pi^{+}\pi^{-}$ decay mode. Resonance | Spin | Helicity | Resonance | Mass (MeV) | Width (MeV) | Source
---|---|---|---|---|---|---
| | | formalism | | |
$\rho(770)$ | 1 | $0,\pm 1$ | BW | $775.49\pm 0.34$ | $149.1\pm 0.8$ | PDG [22]
$f_{0}(500)$ | 0 | 0 | BW | $513\pm 32$ | $335\,\pm 67$ | CLEO [30]
$f_{2}(1270)$ | 2 | $0,\pm 1$ | BW | $1275.1\pm 1.2$ | $185.1^{+2.9}_{-2.4}$ | PDG [22]
$\omega(782)$ | 1 | $0,\pm 1$ | BW | $782.65\pm 0.12$ | $8.49\pm 0.08$ | PDG [22]
$f_{0}(980)$ | 0 | 0 | Flatté | $-$ | $-$ | See text
$\rho(1450)$ | 1 | $0,\pm 1$ | BW | $1465\pm 25$ | $400\pm 60$ | PDG [22]
$\rho(1700)$ | 1 | $0,\pm 1$ | BW | $1720\pm 20$ | $250\pm 100$ | PDG [22]
$f_{0}(1500)$ | 0 | 0 | BW | $1461\pm 3$ | $124\pm 7$ | LHCb [31]
$f_{0}(1710)$ | 0 | 0 | BW | $1720\pm 6$ | $135\pm 8$ | PDG [22]
To study the resonant structures of the decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow J/\psi\pi^{+}\pi^{-}$
we use 29 047 event candidates with invariant mass within $\pm 20$ MeV of the
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ mass peak which include
$10\,207\pm 178$ background candidates. The background yield is fixed in the
fit. Apart from non-resonant (NR) decays, the possible resonance candidates in
the decay ${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow
J/\psi\pi^{+}\pi^{-}$ are listed in Table 4. We use Breit-Wigner (BW)
functions for most of the resonances except $f_{0}(980)$. The masses and
widths of the BW resonances are listed in Table 4. When used in the fit, they
are fixed to these values except for the parameters of $f_{0}(500)$ which are
allowed to vary by their uncertainties. For the $f_{0}(980)$ we use a Flatté
shape [32]. Besides the mass, this shape has two additional parameters
$g_{\pi\pi}$ and $g_{KK}$, which are fixed in the fit to the ones obtained
from an amplitude analysis of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}\rightarrow
J/\psi{{\pi}^{+}}{{\pi}^{-}}$ [31], where a large signal is evident. These
parameters are $m_{0}=945.4\pm 2.2$ MeV, $g_{\pi\pi}=167\pm 7$ MeV and
$g_{KK}/g_{\pi\pi}=3.47\pm 0.12$. All background and efficiency parameters are
fixed in the fit.
To determine the complex amplitudes in a specific model, the data are fitted
maximizing the unbinned likelihood given as
$\mathcal{L}=\prod_{i=1}^{N}F(m_{hh}^{i},\theta_{hh}^{i},\theta^{i}_{J/\psi},\chi^{i}),$
(29)
where $N$ is the total number of candidates, and $F$ is the total PDF defined
in Eq. (15).
## 6 Fit results
### 6.1 Final state composition
In order to compare the different models quantitatively, an estimate of the
goodness of fit is calculated from 4D partitions of the fitting variables. To
distinguish between models, we use the Poisson likelihood $\chi^{2}$ [33]
defined as
$\chi^{2}=2\sum_{i=1}^{N_{\rm
bin}}\left[x_{i}-n_{i}+n_{i}\text{ln}\left(\frac{n_{i}}{x_{i}}\right)\right],$
(30)
where $n_{i}$ is the number of events in the four-dimensional bin $i$ and
$x_{i}$ is the expected number of events in that bin according to the fitted
likelihood function. The $\chi^{2}/\text{ndf}$ and the negative of the
logarithm of the likelihood, $\rm-ln\mathcal{L}$, of the fits are given in
Table 5 for various fitting models, where ndf, the number of degrees of
freedom, is equal to $N_{\rm bin}$ minus the number of fit parameters minus
one. Here the five-resonance model (5R) contains the resonances: $\rho(770)$,
$f_{0}(500)$, $f_{2}(1270)$, $\rho(1450)$ and $\omega(782)$, the “Best Model”
adds a $\rho(1700)$ resonance to the 5R model, the 7R model adds a
$f_{0}(980)$ resonance to the Best Model, and the 7R+NR model adds a non-
resonant component. We also give the change of $\rm ln\mathcal{L}$ for various
fits with respect to the 5R model in Table 5.
The 7R model gives a slightly better likelihood compared to the Best Model,
however, the decrease of the $\rm-ln\mathcal{L}$ due to adding $f_{0}(980)$ is
less than the expected $\Delta\rm ln\mathcal{L}$ at 3$\sigma$ significance.
Thus, we use the Best Model, which maintains a significance larger than
3$\sigma$ for each resonance component, as our baseline fit, while the 7R
model is only used to establish an upper limit on the presence of the
$f_{0}(980)$. The Dalitz fit projections on the four observables:
$m(\pi^{+}\pi^{-})$, $\cos(\theta_{{{\pi}^{+}}{{\pi}^{-}}})$,
$\cos\theta_{J/\psi}$ and $\chi$ are shown in Fig. 12 for the Best Model.
Table 5: The $\chi^{2}/\text{ndf}$ and the $\rm-ln\mathcal{L}$ of different resonance models. The decrease of $\rm-ln\mathcal{L}$ is with respect to the 5R model. Resonance model | $\rm-ln\mathcal{L}$ | $\chi^{2}/\text{ndf}$ | Decrease of $\rm-ln\mathcal{L}$
---|---|---|---
5R Model | $-$169271 | 2396/2041 |
5R Model + $\rho(1700)$ (Best Model) | $-$169327 | 2293/2035 | $56$
Best Model + $f_{0}(980)$ (7R Model) | $-$169329 | 2295/2033 | $58$
7R + $f_{0}(1500)$ | $-$169333 | 2293/2031 | $60$
7R + $f_{0}(1710)$ | $-$169329 | 2295/2031 | $56$
7R + NR | $-$169342 | 2292/2031 | $69$
Figure 12: Dalitz fit projections of (a) $m(\pi^{+}\pi^{-})$, (b)
$\cos(\theta_{{{\pi}^{+}}{{\pi}^{-}}})$, (c) $\cos\theta_{J/\psi}$ and (d)
$\chi$ for the 5R Model + $\rho(1700)$ (Best Model). The points with error
bars are data compared with the overall fit, shown by the (blue) solid line.
The individual fit components are signal, shown with a (red) dashed line,
background, shown with a (black) dotted line, and
${{K}^{0}_{\rm\scriptscriptstyle S}}$, shown with a (green) dashed line.
Table 6 shows the summary of fit fractions of different components for various
models. The fit fractions of the interference terms in the Best Model are
computed using Eq. (13) and listed in Table 7. Table 8 shows the resonant
phases from the Best Model. In the Best Model the $C\\!P$-even components sum
to (56.0$\pm$1.4)%, including the interference terms, so that the $C\\!P$-odd
fraction is (44.0$\pm$1.4)%. The structure near the peak of the $\rho(770)$ is
due to $\rho-\omega$ interference. The fit fraction ratio is found to be
$\frac{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\omega(782),~{}\omega\rightarrow\pi^{+}\pi^{-})}{\Gamma({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\rho(770),~{}\rho\rightarrow\pi^{+}\pi^{-})}=(1.07_{-0.22-0.22}^{+0.32+0.29})\times
10^{-2},$
where the uncertainties are statistical and systematic, respectively; wherever
two uncertainties are quoted in this paper, they will be of this form. The
systematic uncertainties will be discussed in detail in Sec. 6.3.
The 7R model fit gives the ratio of observed decays into $\pi^{+}\pi^{-}$ for
$f_{0}(980)/f_{0}(500)$ equal to $(0.6_{-0.4-2.6}^{+0.7+3.3})\times 10^{-2}$.
To determine the statistical uncertainty, the full error matrix and parameter
values from the fit are used to generate 500 data-size sample parameter sets.
For each set, the fit fractions are calculated. The distributions of the
obtained fit fractions are described by bifurcated Gaussian functions. The
widths of the Gaussians are taken as the statistical errors on the
corresponding parameters. We will discuss the implications of this measurement
in Sec. 7.
Table 6: Fit fractions (%) of contributing components for the various models. The uncertainties are statistical only. Sums can differ from 100% due to interference (see Table 7). | 5R | Best Model | 7R | 7R+$f_{0}(1500)$ | 7R+$f_{0}(1710)$ | 7R+NR
---|---|---|---|---|---|---
$\rho(770)_{0}$ | $35.5\pm 1.6$ | $36.2\pm 1.8$ | $36.1\pm 1.8$ | $36.1\pm 1.9$ | $36.1\pm 1.8$ | $36.0\pm 1.9$
$\rho(770)_{\parallel}$ | $13.4\pm 1.1$ | $14.7\pm 1.2$ | $14.8\pm 1.2$ | $14.7\pm 1.2$ | $14.8\pm 1.2$ | $14.9\pm 1.1$
$\rho(770)_{\perp}$ | $11.7\pm 0.9$ | $12.1\pm 1.1$ | $11.9\pm 1.1$ | $12.0\pm 1.1$ | $12.0\pm 1.1$ | $15.0\pm 1.3$
$f_{0}(500)$ | $24.9\pm 1.4$ | $22.2\pm 1.2$ | $21.4\pm 1.7$ | $20.8\pm 1.9$ | $21.1\pm 1.8$ | $18.7\pm 3.1$
$f_{2}(1270)_{0}$ | $4.6\pm 0.4$ | $4.7\pm 0.4$ | $5.0\pm 0.4$ | $4.8\pm 0.4$ | $4.9\pm 0.4$ | $4.5\pm 0.4$
$f_{2}(1270)_{\parallel}$ | $1.0\pm 0.4$ | $0.9\pm 0.4$ | $1.0\pm 0.5$ | $1.0\pm 0.5$ | $1.0\pm 0.4$ | $0.8\pm 0.4$
$f_{2}(1270)_{\perp}$ | $2.1\pm 0.4$ | $2.0\pm 0.4$ | $2.0\pm 0.4$ | $1.9\pm 0.4$ | $2.0\pm 0.5$ | $2.2\pm 0.4$
$\omega(782)_{0}$ | $0.29\pm 0.11$ | $0.26\pm 0.10$ | $0.26\pm 0.11$ | $0.26\pm 0.11$ | $0.26\pm 0.11$ | $0.25\pm 0.11$
$\omega(782)_{\parallel}$ | $0.41\pm 0.15$ | $0.41\pm 0.16$ | $0.41\pm 0.16$ | $0.42\pm 0.16$ | $0.41\pm 0.15$ | $0.39\pm 0.15$
$\omega(782)_{\perp}$ | $0.01_{-0.01}^{+0.06}$ | $0.01_{-0.01}^{+0.06}$ | $0.01_{-0.01}^{+0.05}$ | $0.01_{-0.01}^{+0.05}$ | $0.01_{-0.01}^{+0.05}$ | $0.01_{-0.01}^{+0.05}$
$f_{0}(980)$ | – | – | $0.13\pm 0.11$ | $0.16\pm 0.12$ | $0.14\pm 0.11$ | $0.5\pm 0.3$
$\rho(1450)_{0}$ | $2.5\pm 0.6$ | $6.8\pm 2.0$ | $6.2\pm 2.4$ | $5.3\pm 3.5$ | $6.3\pm 2.3$ | $5.0\pm 1.9$
$\rho(1450)_{\parallel}$ | $1.8\pm 0.8$ | $3.1\pm 1.9$ | $3.2\pm 1.9$ | $2.4\pm 0.8$ | $3.4\pm 2.1$ | $2.7\pm 1.7$
$\rho(1450)_{\perp}$ | $1.6\pm 0.4$ | $1.7\pm 0.7$ | $1.8\pm 0.7$ | $1.5\pm 0.7$ | $1.9\pm 0.8$ | $5.8\pm 2.6$
$f_{0}(1500)$ | – | – | – | $0.33_{-0.18}^{+0.31}$ | – | –
$f_{0}(1710)$ | – | – | – | – | $0.01_{-0.01}^{+0.12}$ | –
$\rho(1700)_{0}$ | – | $2.0\pm 0.9$ | $1.9\pm 1.0$ | $1.4_{-0.8}^{+1.8}$ | $2.0\pm 1.0$ | $1.1\pm 0.7$
$\rho(1700)_{\parallel}$ | – | $1.2_{-0.6}^{+1.2}$ | $1.3_{-0.6}^{+1.1}$ | $1.3_{-0.7}^{+1.3}$ | $1.3\pm 1.0$ | $1.0\pm 0.9$
$\rho(1700)_{\perp}$ | – | $1.8\pm 0.7$ | $1.7\pm 0.6$ | $1.7\pm 0.6$ | $1.8\pm 0.7$ | $3.5\pm 1.2$
NR | – | – | – | – | – | $3.2\pm 1.1$
Sum | 99.8 | 110.2 | 108.8 | 105.9 | 109.3 | 115.5
Table 7: Non-zero interference fractions($\%$) obtained from the fit using the Best Model. The uncertainties are statistical only. Interfering components | Intererence fraction (%)
---|---
$\rho(770)_{0}+\omega(782)_{0}$ | $-0.36$ | $\pm$ | $0.55$
$\rho(770)_{\parallel}+\omega(782)_{\parallel}$ | $0.65$ | $\pm$ | $0.43$
$\rho(770)_{\perp}\\!+\omega(782)_{\perp}$ | $-0.21$ | $\pm$ | $0.37$
$\rho(770)_{0}+\rho(1450)_{0}$ | $-3.34$ | $\pm$ | $2.60$
$\rho(770)_{\parallel}+\rho(1450)_{\parallel}$ | $-4.38$ | $\pm$ | $1.64$
$\rho(770)_{\perp}\\!+\rho(1450)_{\perp}$ | $-0.18$ | $\pm$ | $1.21$
$\rho(770)_{0}+\rho(1700)_{0}$ | $3.34$ | $\pm$ | $0.93$
$\rho(770)_{\parallel}+\rho(1700)_{\parallel}$ | $0.63$ | $\pm$ | $0.88$
$\rho(770)_{\perp}\\!+\rho(1700)_{\perp}$ | $2.10$ | $\pm$ | $0.43$
$\omega(782)_{0}+\rho(1450)_{0}$ | $-0.24$ | $\pm$ | $0.06$
$\omega(782)_{\parallel}+\rho(1450)_{\parallel}$ | $-0.17$ | $\pm$ | $0.06$
$\omega(782)_{\perp}\\!+\rho(1450)_{\perp}$ | $-0.02$ | $\pm$ | $0.03$
$\omega(782)_{0}+\rho(1700)_{0}$ | $0.05$ | $\pm$ | $0.03$
$\omega(782)_{\parallel}+\rho(1700)_{\parallel}$ | $-0.05$ | $\pm$ | $0.03$
$\omega(782)_{\perp}\\!+\rho(1700)_{\perp}$ | $-0.01$ | $\pm$ | $0.02$
$\rho(1450)_{0}+\rho(1700)_{0}$ | $-5.57$ | $\pm$ | $1.98$
$\rho(1450)_{\parallel}+\rho(1700)_{\parallel}$ | $\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-1.31_{-2.89}^{+1.10}$
$\rho(1450)_{\perp}\\!+\rho(1700)_{\perp}$ | $-1.09$ | $\pm$ | $1.02$
Table 8: The fitted resonant phases from the Best Model. The uncertainties are statistical only. Components | phase (∘)
---|---
$\rho(770)_{0}$ | 0 (fixed)
$\rho(770)_{\perp}$ | 0 (fixed)
$\rho(770)_{\parallel}$ | $189.8\pm~{}7.3$
$f_{0}(500)$ | $336.9\pm~{}5.0$
$f_{2}(1270)_{0}$ | $210.1\pm~{}6.9$
$f_{2}(1270)_{\perp}$ | $165.0\pm 13.3$
$f_{2}(1270)_{\parallel}$ | $334.4\pm 21.9$
$\omega(782)_{0}$ | $268.8\pm 11.9$
$\omega(782)_{\perp}$ | $227.4\pm 84.9$
$\omega(782)_{\parallel}$ | $123.5\pm 13.7$
$\rho(1450)_{0}$ | $196.7\pm 12.1$
$\rho(1450)_{\perp}$ | $182.6\pm 22.4$
$\rho(1450)_{\parallel}$ | $74.9\pm 12.6$
$\rho(1700)_{0}$ | $71.1\pm 19.9$
$\rho(1700)_{\perp}$ | $113.4\pm 20.3$
$\rho(1700)_{\parallel}$ | $~{}~{}~{}3.4\pm 24.5$
In Fig. 13 we show the fit fractions of the different resonant components in
the Best Model.
Figure 13: Fit projection of $m(\pi^{+}\pi^{-}$) showing the different
resonant contributions in the Best Model.
Table 9 lists the fit fractions and the transversity fractions of each
contributing resonance. For a $P$\- or $D$-wave resonance, we report its total
fit fraction by summing all the three components.
Table 9: Fit fractions and transversity fractions of contributing resonances in the Best Model. The first uncertainty is statistical and the second the total systematic. | | Transversity fractions (%)
---|---|---
Component | Fit fraction (%) | $\tau=0$ | $\tau=\|$ | $\tau=\perp$
$\rho(770)$ | $63.1\pm 2.2_{-2.2}^{+3.4}$ | $57.4\pm 2.0_{-3.1}^{+1.3}$ | $23.4\pm 1.7_{-1.3}^{+1.0}$ | $19.2\pm 1.7_{-1.2}^{+3.8}$
$f_{0}(500)$ | $22.2\pm 1.2_{-3.5}^{+2.6}$ | 1 | 0 | 0
$f_{2}(1270)$ | $7.5\pm 0.6_{-0.6}^{+0.4}$ | $62\pm 4_{-4}^{+2}$ | $11\pm 5\pm 2$ | $26\pm 5_{-2}^{+4}$
$\omega(782)$ | $0.68_{-0.14-0.13}^{+0.20+0.17}$ | $39_{-13-3}^{+15+4}$ | $60_{-15-4}^{+12+3}$ | $1_{-1}^{+9}\pm 1$
$\rho(1450)$ | $11.6\pm 2.8\pm 4.7$ | $58\pm 10_{-23}^{+14}$ | $27\pm 13_{-11}^{+7}$ | $15\pm 7^{+28}_{-10}$
$\rho(1700)$ | $5.1\pm 1.2\pm 3.0$ | $40\pm 11_{-23}^{+13}$ | $24\pm 14_{-10}^{+7}$ | $36\pm 14_{-9}^{+28}$
Table 10 shows the branching fractions of the resonant modes calculated by
multiplying the fit fraction listed in Table 9 with ${\cal B}({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}\pi^{+}\pi^{-})=(3.97\pm 0.09\pm 0.11\pm 0.16)\times 10^{-5}$, obtained
from our previous study [5], where the uncertainties are statistical,
systematic, and due to normalization, respectively. These branching fractions
are proportional to the squares of the individual resonant amplitudes.
Table 10: Branching fractions for each channel. The first uncertainty is statistical and the second the total systematic. $R$ | ${\cal B}({{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}R,R\rightarrow\pi^{+}\pi^{-})$
---|---
$\rho(770)$ | $(2.50\pm 0.10_{-0.15}^{+0.18})$ | $\times 10^{-5}$
$f_{0}(500)$ | $(8.8\pm 0.5^{+1.1}_{-1.5})$ | $\times 10^{-6}$
$f_{2}(1270)$ | $(3.0\pm 0.3_{-0.3}^{+0.2})$ | $\times 10^{-6}$
$\omega(782)$ | $(2.7_{-0.6-0.5}^{+0.8+0.7})$ | $\times 10^{-7}$
$\rho(1450)$ | $(4.6\pm 1.1\pm 1.9)$ | $\times 10^{-6}$
$\rho(1700)$ | $(2.0\pm 0.5\pm 1.2)$ | $\times 10^{-6}$
### 6.2 Angular moments
Angular moments are defined as an average of the spherical harmonics, $\langle
Y^{0}_{l}(\cos\theta_{\pi\pi})\rangle$, in each efficiency-corrected and
background-subtracted $\pi^{+}\pi^{-}$ invariant mass interval. The moment
distributions provide an additional way of visualizing the effects of
different resonances and their interferences, similar to a partial wave
analysis. Figure 14 shows the distributions of the angular moments for the
Best Model. In general the interpretation of these moments is that $\langle
Y^{0}_{0}\rangle$ is the efficiency corrected and background subtracted event
distribution, $\langle Y^{0}_{1}\rangle$ the sum of the interference between
S-wave and P-wave and between P-wave and D-wave amplitudes, $\langle
Y^{0}_{2}\rangle$ the sum of the P-wave, D-wave and the interference of S-wave
and D-wave amplitudes, $\langle Y^{0}_{3}\rangle$ the interference between
P-wave and D-wave, $\langle Y^{0}_{4}\rangle$ the D-wave, and $\langle
Y^{0}_{5}\rangle$ results from an F-wave [23, 34]. For the moments with
odd-$l$, one will always find that ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ and ${{B}^{0}}$ have opposite
sign; thus the sum of their contributions is expected to be small.
Figure 14: The $\pi^{+}\pi^{-}$ mass dependence of the spherical harmonic
moments of $\cos\theta_{\pi\pi}$ after efficiency corrections and background
subtraction: (a) $\langle Y^{0}_{0}\rangle$, (b) $\langle Y^{0}_{1}\rangle$,
(c) $\langle Y^{0}_{2}\rangle$, (d) $\langle Y^{0}_{3}\rangle$, (e) $\langle
Y^{0}_{4}\rangle$, (f) $\langle Y^{0}_{5}\rangle$. The errors on the black
data points are statistical. The (blue) curves show the fit projections.
### 6.3 Systematic uncertainties
The sources of systematic uncertainties on the results of the amplitude
analysis are summarized in Table 11. Uncertainties due to particle
identification and tracking are taken from Ref. [5] and are taken into account
in the branching fraction results, but do not appear in the fit fractions as
they are independent of pion kinematics. For the uncertainties due to the
acceptance or background modeling, we repeat the data fit 100 times where the
parameters of acceptance or background modeling are varied according to the
corresponding error matrix. For the acceptance function, the error matrix is
obtained by fitting the simulated acceptance as described in Sec. 5.1. For the
background function, the error matrix is obtained by fitting the hybrid data-
simulated sample as described in Sec. 5.2.
There is uncertainty on the fractions of sources in the hybrid MC-data sample
for background modeling. Instead of using the fits to the $\pi^{+}\pi^{-}$
mass distribution to determine the background fractions, we use the fractions
found from the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\pi^{+}\pi^{-}$
mass fit shown in Fig. 3 that finds the ${{\mathchar 28931\relax}^{0}_{b}}$
reflection is $9.6\%$, the ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}$ reflection is $4.2\%$, the
${{\kern 1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}$ background is
$11.5\%$ and the combinatorial part is $74.7\%$, instead of the ones found in
Sec. 5.2. We then fit the new hybrid sample to get the background parameters.
The data fit is repeated with the new background parameters; the changes on
the fit results are added in quadrature with the uncertainties of the
background modeling discussed above. The two background uncertainties have
similar sizes.
We neglect the mass resolution in the fit where the typical resolution is
3$\mathrm{\,Me\kern-1.00006ptV}$. A previous study shows that the resolution
effects are negligible except for the $\omega(782)$ resonance whose total fit
fraction is underestimated by $(0.09\pm 0.08)\%$. We take the quadrature of
0.09% and 0.08%, equal to 0.12%, as the systematic uncertainty of the total
fit fraction of the $\omega(782)$. These uncertainties are included in the
“Acceptance” category.
The uncertainties due to the fit model include adding each resonance that is
listed in Table 4 but not used in the 7R model, varying the centrifugal
barrier factors defined in Eq. (5) substantially, replacing $f_{0}(500)$ model
by a Bugg function [35] and using the alternative Gounaris and Sakurai Model
[36] for the various $\rho$ mesons. The largest variation among those changes
is assigned as the systematic uncertainty for modeling. We also find that
increasing the default angular momentum $L_{B}$ for the P and D-wave cases
gives negligible differences.
Finally, we repeat the amplitude fit by varying the mass and width of all the
resonances except for the $f_{0}(980)$, in the 7R model within their errors
one at a time, and add the changes in quadrature. For the $f_{0}(980)$
resonance, we change the resonance parameters $m_{0}$, $g_{\pi\pi}$ and
$g_{KK}/g_{\pi\pi}$ to the values obtained from Solution II in [31] instead of
using the ones obtained from Solution I.
Table 11: Absolute systematic uncertainties on the results of the amplitude analysis estimated using the Best Model except for the $f_{0}(980)$ where we use the 7R model. Item | Acceptance | Background | Fit model | Resonance | Total
---|---|---|---|---|---
| | model | | parameters |
Fit fractions (%)
$\rho(770)$ | $\pm 0.3$ | $\pm 0.6$ | ${}_{-1.8}^{+3.2}$ | $\pm 1.1$ | ${}_{-2.2}^{+3.4}$
$f_{0}(500)$ | $\pm 0.3$ | $\pm 0.7$ | ${}^{+1.2}_{-2.7}$ | $\pm 2.2$ | ${}^{+2.6}_{-3.5}$
$f_{2}(1270)$ | $\pm 0.1$ | $\pm 0.2$ | ${}_{-0.5}^{+0.1}$ | $\pm 0.3$ | ${}_{-0.6}^{+0.4}$
$\omega(782)$ | $\pm 0.12$ | $\pm 0.02$ | ${}_{-0.03}^{+0.11}$ | $\pm 0.03$ | ${}_{-0.13}^{+0.17}$
$f_{0}(980)$ | $\pm 0.01$ | ${}_{-0.02}^{+0.03}$ | ${}_{-0.04}^{+0.37}$ | $\pm 0.03$ | ${}_{-0.05}^{+0.37}$
$\rho(1450)$ | $\pm 0.15$ | $\pm 1.3$ | ${}_{-1.9}^{+2.3}$ | $\pm 4.0$ | $\pm 4.7$
$\rho(1700)$ | $\pm 0.13$ | $\pm 0.7$ | ${}_{-0.9}^{+0.7}$ | $\pm 2.9$ | $\pm 3.0$
Transversity $0$ fractions (%)
$\rho(770)$ | $\pm 0.5$ | $\pm 0.5$ | ${}_{-3.0}^{+1.0}$ | $\pm 0.5$ | ${}_{-3.1}^{+1.3}$
$f_{2}(1270)$ | $\pm 0.5$ | $\pm 1.7$ | ${}_{-2.9}^{+0.8}$ | $\pm 1.0$ | ${}_{-4}^{+2}$
$\omega(782)$ | $\pm 0.4$ | $\pm 2.1$ | ${}_{-0.6}^{+3.5}$ | $\pm 1.5$ | ${}_{-3}^{+4}$
$\rho(1450)$ | $\pm 0.7$ | $\pm 8.2$ | ${}_{-18.4}^{+~{}2.0}$ | $\pm 11.1$ | ${}_{-23}^{+14}$
$\rho(1700)$ | $\pm 0.6$ | $\pm 9.9$ | ${}_{-18.3}^{+~{}0.4}$ | $\pm 8.7$ | ${}_{-23}^{+13}$
Transversity $\|$ fractions (%)
$\rho(770)$ | $\pm 0.3$ | $\pm 0.5$ | ${}_{-0.8}^{+0.1}$ | $\pm 0.8$ | ${}_{-1.3}^{+1.0}$
$f_{2}(1270)$ | $\pm 0.4$ | $\pm 1.0$ | ${}_{-1.4}^{+1.3}$ | $\pm 1.3$ | ${}_{-2}^{+2}$
$\omega(782)$ | $\pm 0.4$ | $\pm 2.0$ | ${}_{-3.3}^{+0.6}$ | $\pm 1.5$ | ${}_{-4}^{+3}$
$\rho(1450)$ | $\pm 0.6$ | $\pm 5.1$ | ${}_{-8.4}^{+3.0}$ | $\pm 4.4$ | ${}_{-11}^{+~{}7}$
$\rho(1700)$ | $\pm 1.0$ | $\pm 4.2$ | ${}_{-7.9}^{+3.3}$ | $\pm 4.0$ | ${}_{-10}^{+~{}7}$
Transversity $\perp$ fractions (%)
$\rho(770)$ | $\pm 0.3$ | $\pm 0.3$ | ${}_{-0.9}^{+3.8}$ | $\pm 0.6$ | ${}_{-1.2}^{+3.8}$
$f_{2}(1270)$ | $\pm 0.3$ | $\pm 0.9$ | ${}_{-1.9}^{+4.3}$ | $\pm 0.7$ | ${}_{-2}^{+4}$
$\omega(782)$ | $\pm 0.1$ | $\pm 0.2$ | ${}_{-0.2}^{+0.1}$ | $\pm 0.4$ | $\pm 0.5$
$\rho(1450)$ | $\pm 0.6$ | $\pm 3.4$ | ${}_{-~{}0.0}^{+26.8}$ | $\pm 9.3$ | ${}_{-10}^{+29}$
$\rho(1700)$ | $\pm 0.7$ | $\pm 6.2$ | ${}_{-~{}0.0}^{+26.2}$ | $\pm 5.9$ | ${}_{-9}^{+28}$
Ratio of fit fractions (%)
$f_{0}(980)/f_{0}(500)$ | $\pm 0.09$ | $\pm 0.17$ | ${}^{+2.1}_{-0.1}$ | $\pm 2.6$ | ${}^{+3.3}_{-2.6}$
$\omega(782)/\rho(770)$ | $\pm 0.19$ | $\pm 0.04$ | ${}^{+0.21}_{-0.10}$ | $\pm 0.05$ | ${}^{+0.29}_{-0.22}$
## 7 Substructure of the $f_{0}(980)$ and $f_{0}(500)$ mesons
The substructure of mesons belonging to the scalar nonet is controversial.
Most mesons are thought to be formed from a combination of a $q$ and a
$\overline{q}$. Some authors introduce the concept of
$q\overline{q}q\overline{q}$ states or superpositions of the tetraquark state
with the $q\overline{q}$ state [37]. In either case, the $I=0$ $f_{0}(500)$
and the $f_{0}(980)$ are thought to be mixtures of the underlying states whose
mixing angle has been estimated previously. In the $q\overline{q}$ model, the
mixing is parameterized by a normal 2$\times$2 rotation matrix characterized
by the angle $\varphi_{m}$, so that the observed states are given in terms of
the base states as
$\displaystyle|f_{0}(980)\rangle$ $\displaystyle=$
$\displaystyle\;\;\;\cos\varphi_{m}|s\overline{s}\rangle+\sin\varphi_{m}|n\overline{n}\rangle$
$\displaystyle|f_{0}(500)\rangle$ $\displaystyle=$
$\displaystyle-\sin\varphi_{m}|s\overline{s}\rangle+\cos\varphi_{m}|n\overline{n}\rangle,$
$\displaystyle{\rm where~{}}|n\overline{n}\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{1}{\sqrt{2}}\left(|u\overline{u}\rangle+|d\overline{d}\rangle\right).$
(31)
In this case only the $|d\overline{d}\rangle$ part of the
$|n\overline{n}\rangle$ wave function contributes (see Fig. 1). Thus we have
$\tan^{2}\varphi_{m}\equiv r^{f}_{\sigma}=\frac{{\cal{B}}\left({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}f_{0}(980)\right)}{{\cal{B}}\left({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}f_{0}(500)\right)}\frac{\Phi(500)}{\Phi(980)},$ (32)
where the $\Phi$’s are phase space factors [3, 38, 37]. The phase space in
this pseudoscalar to vector-pseudoscalar decay is proportional to the cube of
the $f_{0}$ momenta. Taking the average of the momentum dependent phase space
over the resonant line shapes results in the ratio of phase space factors
$\frac{\Phi(500)}{\Phi(980)}=1.25$.
The 7R model fit gives the ratio of branching fractions
$\frac{{\cal{B}}\left({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}f_{0}(980),~{}f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}{{\cal{B}}\left({{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}f_{0}(500),~{}f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)}=(0.6_{-0.4-2.6}^{+0.7+3.3})\times
10^{-2}.$
We need to correct for the individual branching fractions of the $f_{0}$
resonances decaying into $\pi^{+}\pi^{-}$. BaBar measures the relative
branching ratios of $f_{0}(980)\rightarrow K^{+}K^{-}/\pi^{+}\pi^{-}$ of
$0.69\pm 0.32$ using $B^{\pm}\rightarrow K^{\pm}K^{\pm}K^{\mp}$ and
$B^{\pm}\rightarrow K^{\pm}\pi^{\pm}\pi^{\mp}$ decays [39]. BES has extracted
relative branching ratios using $\psi(2S)\rightarrow\gamma\chi_{c0}$ decays
where the $\chi_{c0}\rightarrow f_{0}(980)f_{0}(980)$, and either both
$f_{0}(980)$’s decay into $\pi^{+}\pi^{-}$ or one into $\pi^{+}\pi^{-}$ and
the other into $K^{+}K^{-}$. Their results [40, *Ablikim:2005kp] are that the
relative branching ratio of $f_{0}(980)\rightarrow
K^{+}K^{-}/{{\pi}^{+}}{{\pi}^{-}}$ is $0.25^{+0.17}_{-0.11}$[42]. Averaging
the two measurements gives
$\frac{{\cal{B}}\left(f_{0}(980)\rightarrow
K^{+}K^{-}\right)}{{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)}=0.35_{-0.14}^{+0.15}.$
(33)
Assuming that the $\pi\pi$ and $KK$ decays are dominant we can also extract
${\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)=\left(0.46\pm
0.06\right),$ (34)
where we have assumed that the only other decays are to $\pi^{0}\pi^{0}$ (one-
half of the $\pi^{+}\pi^{-}$ rate), and to neutral kaons (equal to charged
kaons). We use
${\cal{B}}\left(f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)=\frac{2}{3}$, which
follows from isospin Clebsch-Gordan coefficients, and assuming that the only
decays are into two pions. Since we have only an upper limit on the
${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}f_{0}(980)$ final state, we
will only find an upper limit on the mixing angle, so if any other decay modes
of the $f_{0}(500)$ exist, they would make the limit more stringent.
In order to set an upper limit on $|\varphi_{m}|$, we simulate the final
$\varphi_{m}$ measurement using as input the central value of the measured
ratio, the full statistical error matrix obtained from the 7R model fit, and
asymmetric Gaussian random variables different for the positive, +3.3%, and
negative, ${-2.6}$%, systematic uncertainties (see Table 11). The resulting
rate ratios of $f_{0}(980)$ to $f_{0}(500)$ are then multiplied by a factor of
${\cal{B}}\left(f_{0}(500)\rightarrow\pi^{+}\pi^{-}\right)/{\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)\times\frac{\Phi(500)}{\Phi(980)}$
where a Gaussian random variable is used for
${\cal{B}}\left(f_{0}(980)\rightarrow\pi^{+}\pi^{-}\right)$ to take into
account the uncertainty in the measurement shown in Eq. (34). The upper limit
at 90% confidence level is determined when 10% of the simulations exceed the
limit value. We find
$\tan^{2}\varphi_{m}\equiv
r^{f}_{\sigma}=\left(1.1^{+1.2+6.0}_{-0.7-0.7}\right)\times
10^{-2}<0.098\text{~{} at $90\%$ C.L }$
which translates into a limit of
$|\varphi_{m}|<17^{\circ}\text{~{} at $90\%$ CL },$
where we neglect the effect caused by the small systematic uncertainty on the
ratio of phase space factors.
If the scalar meson substructure is tetraquark, the wave functions are:
$\displaystyle|f_{0}(980)\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(|[su][\overline{s}\,\overline{u}]\rangle+|[sd][\overline{s}\overline{d}]\rangle\right)$
(35) $\displaystyle|f_{0}(500)\rangle$ $\displaystyle=$
$\displaystyle|[ud][\overline{u}\overline{d}]\rangle.$ (36)
The ratio $r^{f}_{\sigma}$ was predicted to be $1/2$ for pure tetraquark
states in Ref. [3]. The measured upper limit on $r^{f}_{\sigma}$ of 0.098 at
90% CL deviates from the tetraquark prediction by 8 standard deviations.
## 8 Conclusions
We have studied the resonance structure of ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}}\rightarrow J/\psi\pi^{+}\pi^{-}$
decays using a modified amplitude analysis. The decay distributions are formed
by a series of final states described by individual $\pi^{+}\pi^{-}$
interfering decay amplitudes. The data are best described by adding coherently
the $\rho(770)$, $f_{2}(1270)$, $f_{0}(500)$, $\omega(782)$, $\rho(1450)$ and
$\rho(1700)$ resonances, with the largest component being the $\rho(770)$. The
final state is $56.0$% $C\\!P$-even, where we have taken into account both the
fit fractions and the interference terms of the different components. Our
understanding of the final state composition allows future measurements of
$C\\!P$ violation in these resonant final states. These results supersede
those obtained in Ref. [5].
There is no evidence for $f_{0}(980)$ resonance production. We limit the
absolute value of the mixing angle between the lightest two scalar states, the
$f_{0}(500)$ and the $f_{0}(980)$, in the $q\overline{q}$ model to be less
than an absolute value of $17^{\circ}$ at 90% confidence level. We find that
$f_{0}(980)$ production is much smaller than predicted for tetraquarks, which
we rule out at the 8 standard deviation level using the model of Ref. [3].
Concern has been expressed [37] that if the $f_{0}(980)$ were a tetraquark
state the measurement of the mixing-dependent $C\\!P$-violating phase in the
decay ${{\kern
1.79993pt\overline{\kern-1.79993ptB}{}}^{0}_{s}}\rightarrow{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}f_{0}(980)$ could be affected due to additional decay mechanisms. Our
result here alleviates this potential source of error.
## Acknowledgements
We express our gratitude to our colleagues in the CERN accelerator departments
for the excellent performance of the LHC. We thank the technical and
administrative staff at the LHCb institutes. We acknowledge support from CERN
and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC
(China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG
(Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR
(Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NASU (Ukraine); STFC and the Royal Society (United Kingdom);
NSF (USA). We also acknowledge the support received from EPLANET, Marie Curie
Actions and the ERC under FP7. The Tier1 computing centres are supported by
IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The
Netherlands), PIC (Spain), GridPP (United Kingdom). We are indebted to the
communities behind the multiple open source software packages on which we
depend. We are also thankful for the computing resources and the access to
software R&D tools provided by Yandex LLC (Russia).
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|
arxiv-papers
| 2014-04-22T23:46:33 |
2024-09-04T02:50:01.674218
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, A. Affolder, Z.\n Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G. Alkhazov, P.\n Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis, L. An, L.\n Anderlini, J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, V.\n Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, V.\n Batozskaya, Th. Bauer, A. Bay, L. Beaucourt, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S.\n Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom,\n A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F.\n Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S.\n Borghi, A. Borgia, M. Borsato, T.J.V. Bowcock, E. Bowen, C. Bozzi, T.\n Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T. Britton,\n N.H. Brook, H. Brown, A. Bursche, G. Busetto, J. Buytaert, S. Cadeddu, R.\n Calabrese, M. Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora\n Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia,\n L. Carson, K. Carvalho Akiba, G. Casse, L. Cassina, L. Castillo Garcia, M.\n Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, S.-F. Cheung, N.\n Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke,\n M. Clemencic, H.V. Cliff, J. Closier, V. Coco, J. Cogan, E. Cogneras, P.\n Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G.\n Corti, M. Corvo, I. Counts, B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz\n Torres, S. Cunliffe, R. Currie, C. D'Ambrosio, J. Dalseno, P. David, P.N.Y.\n David, A. Davis, K. De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De\n Paula, W. De Silva, P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N.\n D\\'el\\'eage, D. Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra,\n S. Donleavy, F. Dordei, M. Dorigo, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya,\n F. Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, S. Eisenhardt, U. Eitschberger, R. Ekelhof,\n L. Eklund, I. El Rifai, Ch. Elsasser, S. Ely, S. Esen, T. Evans, A.\n Falabella, C. F\\\"arber, C. Farinelli, N. Farley, S. Farry, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M.\n Fiore, M. Fiorini, M. Firlej, C. Fitzpatrick, T. Fiutowski, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, J. Fu, E.\n Furfaro, A. Gallas Torreira, D. Galli, S. Gallorini, S. Gambetta, M.\n Gandelman, P. Gandini, Y. Gao, J. Garofoli, J. Garra Tico, L. Garrido, C.\n Gaspar, R. Gauld, L. Gavardi, E. Gersabeck, M. Gersabeck, T. Gershon, Ph.\n Ghez, A. Gianelle, S. Giani', V. Gibson, L. Giubega, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, C. Gotti, M.\n Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G.\n Graziani, A. Grecu, E. Greening, S. Gregson, P. Griffith, L. Grillo, O.\n Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C. Hadjivasiliou, G.\n Haefeli, C. Haen, S.C. Haines, S. Hall, B. Hamilton, T. Hampson, X. Han, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He,\n T. Head, V. Heijne, K. Hennessy, P. Henrard, L. Henry, J.A. Hernando Morata,\n E. van Herwijnen, M. He\\ss, A. Hicheur, D. Hill, M. Hoballah, C. Hombach, W.\n Hulsbergen, P. Hunt, N. Hussain, D. Hutchcroft, D. Hynds, M. Idzik, P. Ilten,\n R. Jacobsson, A. Jaeger, J. Jalocha, E. Jans, P. Jaton, A. Jawahery, M.\n Jezabek, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, N.\n Jurik, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson, T.M. Karbach, M.\n Kelsey, I.R. Kenyon, T. Ketel, B. Khanji, C. Khurewathanakul, S. Klaver, O.\n Kochebina, M. Kolpin, I. Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A.\n Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F.\n Kruse, M. Kucharczyk, V. Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La\n Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E.\n Lanciotti, G. Lanfranchi, C. Langenbruch, B. Langhans, T. Latham, C.\n Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J.\n Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y. Li, M. Liles,\n R. Lindner, C. Linn, F. Lionetto, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H.\n Lopes, N. Lopez-March, P. Lowdon, H. Lu, D. Lucchesi, H. Luo, A. Lupato, E.\n Luppi, O. Lupton, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, M. Manzali, J. Maratas, J.F. Marchand, U.\n Marconi, C. Marin Benito, P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A.\n Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, F.\n Martinez Vidal, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, A. Mazurov, M. McCann, J. McCarthy, A. McNab, R. McNulty, B.\n McSkelly, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, N. Moggi, J. Molina Rodriguez, S. Monteil, D. Moran, M. Morandin, P.\n Morawski, A. Mord\\`a, M.J. Morello, J. Moron, R. Mountain, F. Muheim, K.\n M\\\"uller, R. Muresan, M. Mussini, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neri, S. Neubert, N. Neufeld, M.\n Neuner, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet,\n N. Nikitin, T. Nikodem, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S.\n Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M. Orlandea,\n J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, F.\n Palombo, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J.\n Parkinson, G. Passaleva, G.D. Patel, M. Patel, C. Patrignani, A. Pazos\n Alvarez, A. Pearce, A. Pellegrino, M. Pepe Altarelli, S. Perazzini, E. Perez\n Trigo, P. Perret, M. Perrin-Terrin, L. Pescatore, E. Pesen, K. Petridis, A.\n Petrolini, E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, A.\n Pistone, S. Playfer, M. Plo Casasus, F. Polci, A. Poluektov, E. Polycarpo, A.\n Popov, D. Popov, B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A.\n Pritchard, C. Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B.\n Rachwal, J.H. Rademacker, B. Rakotomiaramanana, M. Rama, M.S. Rangel, I.\n Raniuk, N. Rauschmayr, G. Raven, S. Reichert, M.M. Reid, A.C. dos Reis, S.\n Ricciardi, A. Richards, M. Rihl, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S.\n Roiser, V. Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F.\n Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, C. Sanchez Mayordomo,\n B. Sanmartin Sedes, R. Santacesaria, C. Santamarina Rios, E. Santovetti, M.\n Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D. Savrina, M. Schiller,\n H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A.\n Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A.\n Semennikov, K. Senderowska, I. Sepp, N. Serra, J. Serrano, L. Sestini, P.\n Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L. Shekhtman, V.\n Shevchenko, A. Shires, R. Silva Coutinho, G. Simi, M. Sirendi, N. Skidmore,\n T. Skwarnicki, N.A. Smith, E. Smith, E. Smith, J. Smith, M. Smith, H. Snoek,\n M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B.\n Spaan, A. Sparkes, F. Spinella, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, O. Stenyakin, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S.\n Stracka, M. Straticiuc, U. Straumann, R. Stroili, V.K. Subbiah, L. Sun, W.\n Sutcliffe, K. Swientek, S. Swientek, V. Syropoulos, M. Szczekowski, P.\n Szczypka, D. Szilard, T. Szumlak, S. T'Jampens, M. Teklishyn, G. Tellarini,\n F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Tolk, L. Tomassetti, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier,\n S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning,\n M. Ubeda Garcia, A. Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti,\n A. Vallier, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S.\n Vecchi, J.J. Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D.\n Vieira, M. Vieites Diaz, X. Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D.\n Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H. Voss, J.A. de Vries, R. Waldi,\n C. Wallace, R. Wallace, J. Walsh, S. Wandernoth, J. Wang, D.R. Ward, N.K.\n Watson, D. Websdale, M. Whitehead, J. Wicht, D. Wiedner, G. Wilkinson, M.P.\n Williams, M. Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M.\n Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing,\n Z. Xu, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang,\n L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Sheldon Stone",
"url": "https://arxiv.org/abs/1404.5673"
}
|
1404.5714
|
# The Mystery of the
Cosmic Diffuse Ultraviolet Background Radiation
Richard Conn Henry Henry A. Rowland Department of Physics and Astronomy,
The Johns Hopkins University, Baltimore, MD 21218, USA [email protected] Jayant
Murthy Indian Institute of Astrophysics, Bengaluru, India [email protected]
James Overduin Department of Physics, Astronomy & Geosciences, Towson
University, Towson, MD 21252, USA [email protected] Joshua
Tyler11affiliation: Now at NASA-Goddard Space Flight Center, Greenbelt, MD
20771. Department of Physics, Astronomy & Geosciences, Towson University,
Towson, MD 21252, USA [email protected]
###### Abstract
The diffuse cosmic background radiation in the GALEX far ultraviolet (FUV,
1300 Å - 1700 Å) is deduced to originate only partially in the dust-scattered
radiation of FUV-emitting stars: the source of a substantial fraction of the
FUV background radiation remains a mystery. The radiation is remarkably
uniform at both far northern and far southern Galactic latitudes, and it
increases toward lower Galactic latitudes at all Galactic longitudes. We
examine speculation that it might be due to interaction of the dark matter
with the nuclei of the interstellar medium but we are unable to point to a
plausible mechanism for an effective interaction. We also explore the
possibility that we are seeing radiation from bright FUV-emitting stars
scattering from a “second population” of interstellar grains—grains that are
small compared with FUV wavelengths. Such grains are known to exist (Draine
2011) and they scatter with very high albedo, with an isotropic scattering
pattern. However, comparison with the observed distribution (deduced from
their $100\ \mu$m emission) of grains at high Galactic latitudes shows no
correlation between the grains’ location and the observed FUV emission. Our
modeling of the FUV scattering by small grains also shows that there must be
remarkably few such “smaller” grains at high Galactic latitudes, both North
and South; this likely means simply that there is very little interstellar
dust of any kind at the Galactic poles, in agreement with Perry & Johnston
(1982). We also review our limited knowledge of the cosmic diffuse background
at ultraviolet wavelengths shortward of Lyman $\alpha$—it could be that our
“second component” of the diffuse far-ultraviolet background persists
shortward of the Lyman limit, and is the cause of the re-ionization of the
Universe (Kollmeier et al. 2014).
dust, extinction—ISM: clouds—ultraviolet: ISM—Dark matter
## 1 Introduction
Diffuse celestial background radiation is observed over every wavelength
range. In the microwave the results of observation have been sufficiently
important that they have led to physics Nobel prizes for the observers. What
about the cosmic background in the ultraviolet? Two reviews of the ultraviolet
observations (Bowyer 1991, and Henry 1991) came to apparently quite different
conclusions concerning the origin of the observed (at that time) diffuse
ultraviolet emissions. Bowyer concluded that the diffuse emission includes
dust-scattered starlight; we will see that Bowyer was correct. Henry
discounted dust-scattered starlight and concluded that the observed background
at the highest galactic latitudes must have an exotic origin. Henry, too, was
correct, as we shall demonstrate. The importance of the study of the UV
background is brought out by Murthy (2009).
Today, a powerful new diffuse UV background dataset is available, thanks to
the GALEX mission (Martin et al. 2005)—these new data have been presented by
Murthy, Henry, & Sujatha (2010), and further analysis of that data set (in
particular, the “Total FUV” of column 3 of their Table 1) will form the focus
of the present paper.
The GALEX diffuse background has also been discussed recently by Hamden,
Schiminovich, & Seibert (2013), who are able to fit most of the FUV background
with what might be expected from FUV starlight scattered from dust. We will,
in the present paper, exhibit a simple radiative-transport model of dust
scattering which strongly supports the Hamden et al. fit in many respects.
They also draw attention to “a $\sim 300$ continuum unit FUV isotropic offset
which is likely due to a combination of air glow (likely the dominant
contributor), a small extragalactic background component including continuum
light from unresolved galaxies, and/or a Galactic component not traced by
other indicators.” We will find that it is their last-mentioned possibility
that best accounts for most of that signal. This “exotic” component is
explored in the present paper—we will see that it is not simply an isotropic
offset, it is a strong “second component” of the diffuse FUV background, of
unknown (but Galactic) origin.
Separating these two dominant sources of diffuse far-ultraviolet emission—the
dust scattered starlight (rather mundane); the second component (exotic)—is
particularly difficult because both sources have very similar spectral
distributions. Dust-scattered starlight has a flat spectrum, simply because
the source spectrum (hot stars) is flat, and the albedo and scattering pattern
of the interstellar grains are close to being wavelength-independent. But the
moderate-to-high-Galactic-latitude emission (which we will be showing, below,
to be dominated by something other than dust-scattered starlight) also has a
flat spectrum, as measured by Anderson et al. (1979; $285\pm 32$ photons cm-2
s-1 sr-1 Å-1 over 1250 Å - 1700 Å) and by Tennyson et al. (1988; $300\pm 100$
photons cm-2 s-1 sr-1 Å-1 over 1750 Å - 2800 Å). The spectral resolution, in
both cases, was about 60 Å. This spectral similarity puts the onus on the
present authors to demonstrate that this “second component” of the diffuse FUV
background is truly (mostly at least) not merely dust-scattered starlight—and
attempting to establish that, is the principal aim of the present paper.
## 2 Overview of the data
We focus on analysis and discussion of the diffuse ultraviolet background as
observed with the GALEX mission’s FUV imager (1350 Å - 1750 Å), for which the
images are completely free of zodiacal light—which is not the case for the
GALEX mission’s NUV images (1750 Å - 2800 Å). GALEX is fully described by
Martin et al. (2005).
As indicated above, we will put forward in the present paper the GALEX FUV
observations without allowance for “airglow,” despite certainty that a small
amount of non-astrophysical signal is present. The allowance for that, that
resulted in Column 4 of Murthy, Henry, & Sujatha 2010), is obsolete—to be
replaced by Murthy (2014a)—but airglow is not a problem at the minimum-
brightness locations, and is only a small fraction of the total signal
elsewhere: we can, and we will, ignore it for the present paper.
In Figure 1 we display, in an Aitoff equal-area projection of the entire sky,
the average brightnesses of all of the GALEX FUV imager backgrounds as
determined by Murthy, Henry, & Sujatha (2010). The brightnesses vary from a
low of 285 photon units (which is in good accord with the result of Anderson
et al. 1979), to a high of 8962 photon units close to some very-FUV-bright
stars. The scale above the map in the figure gives the surface brightness of
the observed diffuse radiation in photons cm-2 s-1 sr-1 Å-1, or “photon units”
(which Hamden et al. call “continuum units,” and which we will sometimes call
simply “units”). The virtue of this choice of units is expounded by Henry
(1999). (Regions at the lowest Galactic latitudes—the white areas in Figure
1—most unfortunately, were not observed using the GALEX FUV detector.)
The hypothesis that we will be testing in this paper (and that we will find
that we must reject) is that the only (or, at least the greatly predominant)
source of this GALEX observed diffuse FUV background radiation is ultraviolet
light from stars (their light being scattered by the interstellar dust); so,
to begin the testing of this specific hypothesis, Figure 1 also includes (blue
filled circles and dots) the observed direct FUV emission from each of the
stars in our adaptation (Murthy & Henry 1995) of the TD1 catalog (Thompson et
al. 1978) — these are the putative sources for the diffuse FUV background. The
radius of each circle is proportional to the square root of the flux from that
star.
To obtain a fully-revealing overview of the observations requires a second,
and complementary, plot: Figure 2 is exactly the same as Figure 1, but instead
of being centered on the Galactic center, this plot is centered on the
Galactic anticenter. Comparison of the two figures gives a better impression
of the gross distribution of the cosmic diffuse ultraviolet background
radiation over the sky. A few low-Galactic-latitude stellar constellations are
marked, in both figures, to allow easy orientation.
In looking at each of our figures showing the distribution of the diffuse FUV
background, it is important to keep in mind that interstellar dust strongly
absorbs far-ultraviolet light (whatever the origins of that light). So we can
only see FUV radiation from $\sim 600$ parsecs, at most (Hurwitz et al. 1991),
at low Galactic latitudes. Only at higher Galactic latitudes (above, say, 60∘)
do we see the total diffuse FUV background with no significant absorption.
The ultraviolet-bright stars (our initially-hypothesized source for all of
this light) are strongly concentrated in Gould’s belt, which is tipped
somewhat with respect to the Galactic plane—that inclination is quite apparent
in the two figures, as the brightest diffuse emission (blue, green, in the
figures) adheres closely to Gould’s belt.
There is certainly no doubt that at least some of the diffuse radiation that
is observed (e.g., near Spica) is dust-scattered starlight, as discussed by
Murthy & Henry (2011). The question is, does the radiation that is observed at
other locations have that same origin, or does a significant portion of it
have a separate and independent origin? Let us begin with the question as to
whether the highest Galactic latitude radiation is even astrophysical in its
origin.
## 3 Testing data integrity
The diffuse background that is seen at the highest Galactic latitudes (that
is, the red areas in Figures 1 and 2), although detected with very high
signal-to-noise, is very faint. Is that radiation astrophysical in its origin,
or might it be of solar-system or terrestrial upper-atmospheric origin?
One important but simple test (Figure 3) is to look at the brightness of that
radiation as a function of time over the more than five year history of the
GALEX FUV observations. The figure includes a dashed line at a constant level
of 300 photon units. The observations span a substantial fraction of a solar
cycle. The lower bound of the observations remains steady as a rock,
indicating that solar activity cannot be influencing, directly or indirectly,
what is observed.
The fact that the lowest-level GALEX FUV brightnesses that are observed agree
so precisely with the diffuse background’s spectrum that was reported by
Anderson et al. (1979) is also encouraging, but of course the Anderson et al.
observation, too, was made not far above the Earth’s atmosphere. It would be
good to have observations that were made much farther from Earth; and
fortunately, such observations exist: two scans made from Dynamics Explorer
(Fix, Craven, & Frank 1989). DE was in a polar orbit, with an apogee of 23,250
km. Figure 4 gives these DE observations in an Aitoff projection identical to
that of Figure 1. The color scheme used, however, differs dramatically from
that of Figure 1, for two reasons. First, with regard to the DE data
themselves, we have been at pains to not “adjust” those data in any way: what
is plotted is the brightnesses straight from the Fix et al. paper (the actual
numbers for these important observations were not included in their paper, so
we provide them in Table 1; we thank the authors for supplying them). Now,
what does that have to do with the color scheme? Please note on the
calibration bar (at the top of the figure) that the lowest observed DE
brightness is only 4 units! Well, that value is actually 4 $\pm$ 363 units
(Table 1). The DE field of view was very small, and counting statistics were
significant. Please note the colors of the scale bar, and then note the
complete lack of red or of yellow in the two DE scans of the sky! No, the DE
data agree extremely well with the GALEX data, establishing that the GALEX
data are free of significant upper-atmosphere terrestrial contamination
(although minor contamination is known to be present, because of small
brightness changes observed in the GALEX data as a function of time over each
night-time observing period); see also Murthy (2014b).
The second change in the color scheme, is that in Figure 4 “the blue stars are
red.” We paint them red now, simply so that there is, in the figure, no
overlap of color whatsoever between direct starlight, and diffuse ultraviolet
light—which was not the case in Figures 1 and 2. The present figure allows us
to see Gould’s belt much more clearly than in the previous figures. In
particular, we see that the overwhelming majority of FUV-bright stars are
confined, not only to Gould’s belt, but, in fact to that half of Gould’s belt
that is between Galactic longitudes $180^{\circ}$ and $360^{\circ}$ (Henry
1977). This fact will be extremely helpful to our task of testing (and
ultimately rejecting) our trial hypothesis that essentially all of the diffuse
FUV radiation is simply dust-scattered starlight.
To test the degree of agreement between DE and GALEX quantitatively, we have
identified all of the DE observations that were made at locations that are
within each of the individual GALEX FUV observations, finding that of all of
the GALEX targets, 546 were also observed with DE. The average number of DE
observations at each GALEX location was 3.27—one GALEX location was observed 9
times by DE. We have averaged the DE observations for each of the GALEX
targets to improve the statistics, and we have plotted the result, versus the
GALEX observed value, in Figure 5. The result confirms that Dynamics Explorer
does detect the same background as does GALEX; indeed, the DE background is,
if anything, slightly brighter than the GALEX backgroud.
What does this close agreement tell us about the GALEX observations? Figure 6
is a cartoon showing the Earth, with the GALEX and Dynamics Explorer orbits
drawn as if they were coplanar and located in the page. The great majority of
the DE diffuse background measurements were clearly made from locations that
were much farther from the Earth than were the GALEX observations. The fact
that the DE FUV background and the GALEX FUV background closely agree, as we
have seen in Figure 5 that they do, gives us some confidence that the GALEX
FUV brightnesses can be trusted, at the highest Galactic latitudes, to be
largely astrophysical in their origin.
## 4 Displaying the high-Galactic-latitude cosmic background
Our focus, at least initially, will be on testing our understanding of the
diffuse ultraviolet background at the highest Galactic latitudes, where the
influence of Galactic ultraviolet starlight should be least. So, rather than
an Aitoff projection, we switch to separate polar projections (north Galactic,
and south Galactic). In Figure 7 we display the FUV radiation that is observed
in the northern Galactic hemisphere, on a logarithmic scale similar to that of
Figure 1.
In these polar plots we also include, black open circles (and black dots), the
TD1 FUV stars for that hemisphere, as well as the very brightest FUV stars of
the other hemisphere (dashed open circles). White areas again are regions with
no GALEX FUV observations; many of these contain black circles or black dots
revealing why they were not observed by GALEX: they are the locations of FUV-
bright stars (which might have damaged the GALEX detectors).
Our earlier plots of the observations were for orientation; this plot, and
following plots, will be for analysis and critical discussion. The lack of
color in the star plotting symbols avoids any possible confusion between
direct starlight and the diffuse emission. This plot also brings out the
remarkable confinement of the brightest FUV stars to regions near the galactic
plane; that is, the rim of this figure (though keep in mind that the
projection used emphasizes display of the highest latitudes), and specifically
to the top rim of this figure (and of following figures).
The brightest northern-hemisphere individual stars that were seen in Figures 1
and 2 repay inspection: only Spica provides good evidence for an origin of at
least some of the broader observed diffuse radiation in dust-scattered
starlight.
Note that the brightest stars are, as we have already seen, not only near the
Galactic plane, but also overwhelmingly confined to the longitude range 180∘
to 360∘ (which is the top half of the plot). Notice, very importantly, in
contrast, that there is no asymmetry at all in the diffuse background over
most of this plot. This, by itself, argues that the diffuse background at the
higher Galactic latitudes can hardly originate in dust-scattered starlight.
An important complement to Figure 7 is the corresponding linear intensity
plot, Figure 8. The great virtue of the linear plot is that its lowest
brightness is zero. (On the other extreme of our new intensity scale, the
brightnesses are cut off at 2000 units; regions brighter than that value are
shown as white).
We will be using these plots to test (and, we will see, to reject) the
hypothesis that what we are seeing in the figures is exclusively (or even
predominantly) dust-scattered starlight. We are fortunate that both celestial
Galactic hemispheres are observed by GALEX. Figures 9 and 10 are the same as
the previous two figures, but this time for the opposite, southern, Galactic
hemisphere. Again the Galactic longitude range to which the brightest stars of
Gould’s belt are confined, is over the top half of the figure. One or two
southern Galactic hemisphere stars show evidence for some of the diffuse
background being dust-scattered starlight; these, of course, are among the
stars discussed by Murthy & Henry (2011), that confirm a very strongly forward
scattering property, for FUV radiation, of the interstellar dust.
While forward-scattered light is easy to detect because of its concentration
around the location of the source star, Draine (2011) points out that “the
tendency for the extinction to rise with decreasing $\lambda$, even at the
shortest wavelengths where we can measure it, tells us that grains smaller
than the wavelength must be making an appreciable contribution to the
extinction, down to $\lambda=0.1\ \mu$m.” So we must be sensitive to the fact
that some FUV radiation will be scattered, not forward, but isotropically.
The two methods of display that we have used, linear and logarithmic, are both
of value. The logarithmic brings out clearly the variations in brightness that
occur from place to place over the polar caps, while the linear shows that
these variations are on top of a more uniform base emission that is present
everywhere.
We are testing the hypothesis that the diffuse radiation that is mapped in
Figures 7 and 8, and 9 and 10, originates, at least mostly, in starlight
scattered from interstellar dust, and our tentative rejection of that
hypothesis, so far, rides largely on the uniformity of the faintest diffuse
radiation, in both hemispheres, compared with the strong non-uniformity in the
distribution of the radiation sources (stars). But the radiation is not
completely uniform; it clearly increases in brightness (yellow regions) toward
lower Galactic latitudes, at all Galactic longitudes. It is striking that that
increase is almost entirely independent of Galactic longitude: if we were
seeing starlight scattered from interstellar dust, surely there would be a
very strong top-half-of-figure, bottom-half-of-figure, asymmetry? But no hint
of such an asymmetry can be seen.
This provides us with critical information about our putative exotic “second
component” of the diffuse far-ultraviolet background, namely that at least a
portion of it increases toward lower Galactic latitudes. That argues
conclusively that whatever the second component may be, at least some portion
of it is Galactic, not extragalactic, in its origin; and, equally
conclusively, that at least that portion is celestial, and not terrestrial, in
its origin.
The preceding four figures showed the distribution over both Galactic
hemispheres of the GALEX FUV diffuse background radiation. To avoid confusion,
in these figures direct starlight was only indicated using black circles and
black dots. In Figure 11, we provide a “finder chart” for the FUV-bright stars
of both hemispheres: blue for northern hemisphere stars, and red for southern
hemisphere stars. As in some previous figures, constellation names are
provided around the rim of the figure (the Galactic plane), and, this time,
star names are provided for some of the brightest stars. Once again we note
the extraordinary confinement of the brightest UV-bright stars to the lowest
Galactic latitudes: the three circles are at Galactic latitudes 0∘, 30∘, and
60∘.
## 5 Comparison with the $100\ \mu$m thermal emission
In Figures 7-10 we have displayed the diffuse ultraviolet background, with our
view being centered on the Galactic poles. The hypothesis that we are testing
is that this radiation is (largely, at least) ultraviolet starlight that has
been scattered from interstellar dust. We would of course like to compare the
distribution of the ultraviolet emission, with the distribution of the dust
that is supposedly doing the scattering. Where is that dust located?
Fortunately, we can locate the dust with no ambiguity (that is, as far as its
angular distribution on the sky is concerned; we have no idea of its
distance)—for the dust is not cold, it is somewhat heated by starlight, and
therefore it emits infrared radiation. And so, maps of the $100\ \mu$m thermal
emission will show us where the dust is located on the sky.
Figures 12 and 13 display, on a logarithmic intensity scale, the northern and
southern Galactic hemisphere distributions of the $100\ \mu$m cosmic thermal
emission (Schlegel, Finkbeiner, & Davis 1998). More recent Planck observations
(Abergal et al. 2014) show that the dust density varies on scales smaller than
sampled by Schlegel et al., but are in general agreement. New measurements,
with Pan-Starrs1 (Schafly et al. 2014), are also in good general agreement
with the Schlegel et al. measurements.
We will shortly be making a quantitative comparison of the FUV radiation with
this $100\ \mu$m emission; but even qualitative examination of the
distributions is highly instructive. In particular, a highlight of the polar
distributions of the FUV emission, we saw, was its uniformity over both polar
caps. But now, our two figures displaying the distribution of the thermal
emission (that is, displaying the distribution of the interstellar dust)
reveal that the dust is anything-but-uniformly distributed!
Of course we are assuming that we know with certainty that the distribution of
the $100\ \mu$m emission gives us the distribution of the interstellar dust.
This is not at all a controversial assumption, but it is still interesting to
examine and test that hypothesis too! And the present figures give an
excellent means of scrutinizing that claim. The interstellar dust is heated,
and the interstellar grains then radiate the infrared radiation. But what
exactly is it that heats the interstellar dust? Why it is, largely, the very
same FUV-bright stars that are, in the hypothesis that we are testing,
generating the FUV background that is the subject of the present paper! And we
have already noted that the great majority of those stars are located around
the upper half of the rim of Figures 7-13. But notice, now, that the asymmetry
in the $100\ \mu$m emissions that we have already noted in the two hemispheres
is a quite different asymmetry in each of the two hemispheres! In the Northern
hemisphere, the faintest $100\ \mu$m emission is located farthest from the top
rim of the figure, where the great majority of the FUV originates. Is that
faintness due to angular distance from the heating stars; or is it due simply
to lack of interstellar dust? To answer that question, simply glance at the
southern Galactic hemisphere $100\ \mu$m emission! For this hemisphere, the
asymmetry is almost exactly opposite to what we have just noted in the
northern Galactic hemisphere. The dust that is closest to the heating source
is the faintest! Our test confirms the conventional view that the $100\ \mu$m
emission plots simply show us the (very non-uniform) location of the
interstellar dust: which is exactly what we want to know.
All of this is qualitative, but it does give us confidence that our basic
understanding of the $100\ \mu$m emission’s origin is correct: the infrared
emission is showing us, accurately, the (angular) location of the interstellar
dust. If the $100\ \mu$m emission is faint, that means there is little dust in
that direction: and so, if the FUV emission is relatively strong at such
locations, well, the FUV emission (at least mostly) cannot be FUV starlight
that has been scattered from interstellar dust. And so, that is our
conclusion, from this simple, qualitative, comparison of the FUV and infrared
emissions from the two hemispheres.
We next greatly strengthen our conclusion by a quantitative analysis of these
same data sets.
## 6 Quantitative comparison with the $100\ \mu$m thermal emission
Glancing back at our previous figures, we see that the FUV brightnesses were
plotted in units of photons cm-2 s-1 sr-1 Å-1, while the $100\ \mu$m emission
was in units of MJy sr-1. We now want to plot these observations together, in
a single plot, so as to compare their relative strengths, as a function of
Galactic latitude. We must therefore use the same units for the brightnesses
in the two distinct wavelength ranges. What units should we use? In deciding
what set of units to use, we need to keep in mind what our purpose is. Energy
is conserved. The hypothesis that we are testing is that ultraviolet energy
from the hot stars of (mostly) the Galactic plane, encounters the interstellar
dust at high northern and southern Galactic latitudes, and some of it is
simply scattered and provides the diffuse FUV background that we detect with
GALEX; while some of it is absorbed by the interstellar grains, and then is
re-emitted isotropically, as infrared radiation, which we observe at $100\
\mu$m.
This reexamination of our purposes informs us as to what units we should use:
Henry (1999) has shown that units of photons cm-2 s-1 sr-1 Å-1 are the proper
choice for a spectral plot when comparisons of energy content is the purpose
of that plot—as it is, in the present case.
So! We must convert our infrared brightnesses from MJy sr-1 to photons cm-2
s-1 sr-1 Å-1. That is, given $n$, we need to find $x$, in the equation
$n\mbox{ MJy sr}^{-1}=x\mbox{ photons
cm}^{-2}\mbox{s}^{-1}\mbox{sr}^{-1}\mbox{\AA}^{-1}$ (1)
But 1 Jy = 10-23 erg cm-2 s-1 Hz-1 is the definition of the Jansky (Jy), so
multiplying both sides of this Jansky-definition equation by 106 (as well as
by $n$), and also by sr-1, we write
$n\mbox{ MJy sr}^{-1}=n\mbox{ 10}^{-17}\mbox{ erg
cm}^{-2}\mbox{s}^{-1}\mbox{Hz}^{-1}\mbox{sr}^{-1}=x\mbox{ photons
cm}^{-2}\mbox{s}^{-1}\mbox{\AA}^{-1}\mbox{sr}^{-1}$ (2)
Two conversions remain: from ergs to photons and (somewhat trickier) from Hz-1
to Å-1.
First,
$E=h\nu=\frac{hc}{10^{-8}\lambda_{\AA}}\mbox{ erg photon}^{-1}$ (3)
Dividing the left hand side of Equation (2) by this, we have
$n\mbox{ 10}^{-17}\mbox{
erg}\;\;\frac{1}{\frac{hc}{10^{-8}\lambda_{\AA}}}\;\frac{1}{\mbox{erg
photon}^{-1}}\mbox{ cm}^{-2}\mbox{s}^{-1}\mbox{Hz}^{-1}\mbox{sr}^{-1}=x\mbox{
photons cm}^{-2}\mbox{s}^{-1}\mbox{\AA}^{-1}\mbox{sr}^{-1}$ (4)
or
$n\mbox{ 10}^{-25}\;\,\frac{\lambda_{\AA}}{hc}\,\,\mbox{photons
cm}^{-2}\mbox{s}^{-1}\mbox{Hz}^{-1}\mbox{sr}^{-1}=x\mbox{ photons
cm}^{-2}\mbox{s}^{-1}\mbox{\AA}^{-1}\mbox{sr}^{-1}$ (5)
The bandwidth conversion is less mechanical. First, $\nu=\frac{c}{\lambda}$,
so $\Delta\nu=\frac{-c}{\lambda^{2}}\Delta\lambda\ $ and
$\Delta\nu\;_{Hz}=\frac{-c}{\lambda_{cm}\lambda_{\AA}}\;\;\Delta\lambda\
_{\AA}$ (6)
The minus sign merely recognizes the fact that as wavelength increases,
frequency decreases; it may be omitted. Inserting this, our final conversion,
gives us
$\frac{c}{\lambda_{cm}\lambda_{\AA}}\;\mbox{Hz \AA}^{-1}\>n\mbox{
10}^{-25}\;\,\frac{\lambda_{\AA}}{hc}\,\;\mbox{photons
cm}^{-2}\mbox{s}^{-1}\mbox{Hz}^{-1}\mbox{sr}^{-1}=x\mbox{ photons
cm}^{-2}\mbox{s}^{-1}\mbox{\AA}^{-1}\mbox{sr}^{-1}$ (7)
or
$x=n\times\frac{10^{-25}}{h\lambda_{cm}}$ (8)
which is the transformation that we want. (We hope that this tutorial on the
transformation of units is useful to beginning graduate students; possibly to
some others as well.)
We now apply this transformation to the case at hand: $n=1$ MJy sr-1
corresponds to
$\frac{10^{-25}}{6.625\times 10^{-27}\times 100\ \mu m\times 10^{-4}\mbox{ cm
}\mu m^{-1}}=x=1509.4\mbox{ units}$ (9)
which provides the correspondence that we were seeking. Particularizing to our
lowest observed FUV background,
$300\mbox{ photons
cm}^{-2}\mbox{s}^{-1}\mbox{\AA}^{-1}\mbox{sr}^{-1}=0.2\mbox{ M Jy sr}^{-1}$
(10)
We are now in a position to make our quantitative comparison—which we do in
Figure 14, where we have plotted (black) the diffuse FUV background as a
function of Galactic latitude, both north and south. But our plot includes,
also, the $100\ \mu$m background, which—having learned “what makes sense,”
from examination of Figures 12 and 13—we have here segregated by color: the
$100\ \mu$m background for the Galactic longitude range 30∘ to 210∘ is plotted
in red, while the same, for the Galactic longitude range 210∘ to 0∘ to 30∘, is
plotted in blue. Note the drastic difference in the ratio of red to blue at
northern Galactic latitudes, compared with the ratio of red to blue at
southern Galactic latitudes. (In Figure 15, we verify that no such segregation
occurs for the FUV background radiation.)
Figure 14 is the critical plot that allows us to conclude with confidence that
the diffuse FUV radiation that we see at each polar cap is not starlight
scattered from interstellar dust: for in this plot we see that there is far
too much of it, and it is very wrongly distributed.
Perhaps at one’s first glance at Figure 14, one might feel that (ignoring the
more important longitude-distribution problem for a moment) possibly one can
explain it all as originating in the FUV light of Galactic plane stars: there
is perhaps 2 to 4 times as much thermal radiation as there is FUV radiation,
so we might think: most FUV radiation is absorbed (and then re-radiated as
infrared) while perhaps a third or less is simply scattered. But that will not
do, for while the thermal radiation is emitted isotropically, at least a
substantial fraction of the FUV radiation is very strongly forward-scattered
(see Spica). For us to detect the scattered light at high Galactic latitudes
requires that it be scattered at more than 90∘. But at most only some small
fraction of the scattered light can be other than almost-directly forward
scattered. So, especially given the more important longitude-distribution
problem, this figure seems to rule out any possibility that the diffuse FUV
background at high Galactic latitudes originates significantly in starlight
scattered from forward-scattering interstellar grains.
Thus our conclusion is that we must reject the hypothesis which we posed for
this paper. The diffuse FUV background at high Galactic latitudes (and
increasing in brightness toward lower Galactic latitudes) is, predominantly,
not starlight scattered from interstellar dust. Could more sophisticated dust
models explain the observations? In Section 11 we will consider the
possibility that there is a substantial amount of much smaller, isotropically-
scattering interstellar grains; but, there, we will find that that does not
explain the observations.
## 7 Extragalactic Far-Ultraviolet radiation?
We have just shown that at least a substantial part of the FUV background
originates in an unknown source in our own Galaxy. There is still bound to be
some contribution from other galaxies, and those galaxies emit with a spectrum
that, if we (momentarily) ignore evolution and redshift effects, is somewhat
similar to the observed flat spectrum of the diffuse light that was observed
at high Galactic latitudes as measured by Anderson et al. (1979) and by
Tennyson et al. (1988) using rocket-borne spectrometers.
To estimate the size and shape of this contribution, we plot in Figure 16 two
theoretical models of the spectral intensity of the extragalactic background
light (EBL) at UV wavelengths, due to Finke et al. (2010, “Model C) and
Dominguez et al. (2011, upper and lower limits). Also shown in this figure are
the model predictions from a code that was originally written to compute EBL
intensity at near-optical wavelengths (Overduin & Wesson 2004). This code
takes as inputs the spectral energy distributions of both quiescent and star-
forming galaxies from FUV to sub-mm wavelengths (Devriendt et al. 1999). It
integrates these spectra over redshift and incorporates both galaxy number and
luminosity evolution by normalizing the total luminosity density at each
redshift to a mix of theoretical modeling and observational data, as compiled
by Nagamine et al. (2006). The code includes a model for absorption by dust in
the intergalactic medium, due to Loeb & Haiman (1997). Here we apply it to
calculate the spectral intensity of the EBL at FUV and NUV wavelengths, as
shown in Figure 16. There are four theoretical curves (labeled TVD, SA, Fossil
and H&S in the figure) refering to different galaxy evolution models from
Nagamine et al. (2006), and the parameters of these models have been chosen to
achieve the widest possible spread in model predictions. They can probably be
regarded as firm upper limits on EBL intensity at FUV wavelengths, due to the
limitations of the dust opacty model and galaxy SEDs employed (Overduin,
Prins, & Strobach 2014). All model predictions assume a standard $\Lambda$CDM
cosmology with with $\Omega_{M}=0.3$ and $\Omega_{\Lambda}=0.7$.
In Figure 16, we compare these predictions with forty years of observational
constraints from rocket-borne detectors (Lillie & Witt 1976, Anderson et al.
1979, Jakobsen et al. 1984, Tennyson et al. 1988), Apollo 17 (Henry et al.
1978), Apollo-Soyuz (Paresce et al. 1980), Solrad-11 (Weller 1983), Voyager 2
(Holberg 1986), shuttle-borne spectrometers (Murthy et al. 1990, Martin et al.
1991), Dynamics Explorer-1 (Witt & Petersohn 1994), DUVE (Korpela et al.
1998), EURD/Minisat (Edelstein et al. 2001) and HST/Las Campanas (Bernstein et
al. 2002). Filled symbols indicate mostly spectroscopic data, while open
symbols indicate mostly photometric ones.
The integrated light from distant galaxies is certainly significant, and may
contribute to the excess observed by GALEX at high Galactic latitudes,
particularly for longer wavelengths. However, it is most important to note
that the model predictions for EBL intensity vary strongly with wavelength,
while the observational data—especially those we would regard as most solid
(the spectroscopic measurements denoted by filled symbols in the plot)—show a
brightness that is independent of wavelength. They are also considerably
smaller in magnitude. As we already have presented evidence for a Galactic
source that decreases toward higher Galactic latitudes, we think it reasonable
to conclude that what is observed at the highest Galactic latitudes,
especially at shorter wavelengths, is mostly the same source that is producing
the radiation that is observed at the lower Galactic latitudes.
## 8 Modeling forward-scattered FUV starlight
We have created a simple single-scattering model (Figure 17) for the expected
diffuse FUV background originating in starlight scattering from interstellar
dust. We describe the model below. In Figure 18, we show the result of the
model, for the important case of Henyey-Greenstein (1941) scattering parameter
$g=0.78$ (which is strong forward scattering), and albedo $a=0.62$, those
being the values that are reported by Hamden et al. (2013) from their analysis
of the GALEX data. We have run the model for a variety of values of the albedo
and, most importantly, of the scattering parameter. We only obtain agreement
with the observations if we use strong forward scattering; otherwise the model
predicts far too much scattered light at high Galactic latitudes. (In Section
11 we will find that although a small population of small, isotropically-
scattering grains could easily account for the absolute amount of radiation
that we see at the highest Galactic latitudes, the distribution on the sky is
still quite wrong; and we will conclude from this simply that there is very
little dust indeed—large grains or small—at the highest Galactic latitudes.)
Our model uses our catalog of FUV star brightnesses (Murthy & Henry 1995) as
the sole original source of all FUV radiation. The only additional parameters
in the model are the amount of dust, and the dust scale height above and below
the Galactic plane, which we take to be 100 pc (use of a Gaussian instead of
an exponential produces little difference).
The lowest brightnesses predicted by our model are 148 photon units at high
Galactic latitudes. This is undoubtedly high for these far-northern and far-
southern locations, as our model assumes a uniform distribution of
interstellar dust, with no allowance for the fact (Perry & Johnston 1982) that
there is, in fact, very little dust at our particular location in the Galaxy.
The highest brightness that our model predicts is 22,253 units at
$\ell=299.8,b=-1.0$, which is in the Coalsack nebula: Sujatha, Murthy,
Shalima, & Henry (2007) reported a Voyager 1 observation (at 1159 Å) of 23,700
units at $\ell=301.7,b=-1.7$).
What is shown in Figure 18 is the prediction of our model for the entire sky.
Figure 19 is the same model as in Figure 18, except that now we only include
that portion of the sky that was actually observed by GALEX in the FUV. For
convenience in comparison with the observations, the top line of numbers in
the figure is the actual brightness scale (which is the line just below),
arbitrarily multiplied by 1.706 so as to force agreement with the brightest
FUV background observations (Figures 1 and 19).
In the creation of this model, there are three steps (see Figure 17) to the
calculation (for each distance in a given direction of observation on the sky)
of the contribution to the predicted background of the dust-scattered light of
each star in our catalog.
1) We first calculate the FUV flux arriving at a particular spot on the line
of sight in question, taking into account the attenuation, by the interstellar
dust, of the light from that particular star. For this stage, we assume that
the light that is scattered (as opposed to absorbed) is strongly forward
scattered, so that it, effectively, is simply not removed from the beam. There
are two parts to this calculation:
a) find the total optical depth $\tau$ along the path
$\tau=\Sigma\;\,\kappa\;\,ds\;\,e^{-z/h}$ (11)
The sum is along the path from the star to the particular spot on the line of
sight; $z$ is the distance above the Galactic plane as that path is traversed;
$h$ is the scale height (100 pc) that we have assumed for the interstellar
medium; $\kappa$ is the absorption coefficient, and $ds$ is the one parsec
that we have used in carrying out the calculation. For the absorption
coefficient we begin with an extinction in the visible of 1.0 magnitudes, for
a path length in the Galactic plane of 1 kpc, and we translate to 1500 Å using
the extinction curve of Cardelli et al. (1989) with $R_{V}=3.1$.
b) once we have calculated the total optical depth $\tau$ along the path, we
calculate the flux $F_{\lambda}$ arriving at that particular spot on the GALEX
line of sight:
$F_{\lambda}=f_{\lambda}\times(D/d)^{2}[\,a+(1-a)e^{-\tau}\,]$ (12)
where $f_{\lambda}$ is the brightness of the star in photons cm-2 s-1 Å-1
$(R/D)^{2}$ (from our catalog of star brightnesses), $D$ is the distance from
us to the star, $d$ is the distance from the star to that particular spot
(slice) on our line of sight, $R$ is the radius of the star, and $a$ is our
assumed albedo of the interstellar grains.
2) Next, we calculate how much light is scattered toward us, as the light from
that particular star crosses that particular slice of that particular GALEX
one-degree-field-of-view line of sight.
This is the only point in our calculation where we take into explicit account
the particular value that we have chosen for the Henyey-Greenstein scattering
parameter $g$, instead of just assuming “straight-forward” (i.e., no)
scattering. In particular,
$H=\frac{1-g^{2}}{(1+g^{2}-2g\cos\theta)^{3/2}}$ (13)
(Henyey & Greenstein 1941) gives the fraction scattered in direction $\theta$,
where $\theta$ is the angle of deflection.
The brightness of the slice as seen by us (who are a distance $x$ away) is
given by
$I_{\lambda}=F_{\lambda}\frac{H}{4\pi}\frac{a}{x^{2}}\;(1-e^{w\;\kappa\;e^{-\frac{x}{h}}})$
(14)
where $x$ is in parsecs, $w$ is the thickness of the slice at that distance
from us, and $\kappa$ is the absorption coefficient.
3) Finally, we allow for the absorption between us and that particular slice
by calculating
$I_{\lambda,obs}=I_{\lambda}\times\;[\,a+(1-a)e^{-\tau}\,]$ (15)
where $\tau$ is now the optical depth from us to the slice (at distance $x$).
The optical depth is calculated in the same way that the optical depth $\tau$
from the star to the slice was calculated.
Our predicted brightness is of course the sum of the predicted brightnesses
for each slice (to infinity) on our line of sight, carried out and summed for
all the stars in our catalog.
Note that for two of our steps, we assumed total forward scattering. If we
want to use smaller values of $g$ (and in particular, we are interested in the
case $g=0$, corresponding to isotropic scattering—which is what is expected
for small grains) we will be forced (because of the simplicity of our
calculation scheme) to ignore the interaction of the starlight with the
interstellar medium except at the slice itself. That is not a terribly bad
assumption, and at least gives us some idea of what we might expect to see, if
indeed all that we are seeing is dust-scattered starlight. The result for
small grains will be presented in Section 11.
So, what does our model (Figures 17, 18, and 19) tell us? Our model is to be
compared with the FUV observations, which are shown in Figures 1 and 20. To
assist comparison, in Figure 21 we display a ratio of Observed to Model: (FUV
Observed)/(FUV Model)$\times 0.586$, where $0.586=5253/8962$ to normalize to
highest brightness, and where, for this plot, a small number of high-Galactic-
latitude FUV-bright individual stars (all of the stars in Table 2 except
Spica) have been arbitrarily omitted from the model, as not contributing
significantly to the observed distribution (although they do, if they are
included, to the model). First, note that our model does not do well in
predicting the absolute value of the diffuse FUV background, falling short by
a factor of $\sim 1.7$. Our model for the distribution of the interstellar
dust is crude, but also, of course, we have already concluded that dust-
scattered starlight is only a portion of what is observed. Also, since the
Henyey-Greenstein scattering function has no physical basis, not a great deal
could be hoped for from our model in terms of the detailed distribution of
scattered light on the sky. Indeed, Draine (2003) has warned that, in the
ultraviolet, the use of simple Henyey-Greenstein phases functions is
problematic—he suggests that discrepancies among reports of the values of
these parameters in the ultraviolet may be due to reliance on these functions.
So, in the present paper, we have used the Hamden et al. proposed Henyey-
Greenstein values simply as a “straw man” for discussion.
There are features that are present in the model of Figure 19 that are NOT
present in the observations that we have displayed in the present paper. The
model shows significant asymmetry with Galactic longitude in the predicted
diffuse FUV background at moderate and high Galactic latitudes, particularly
southern. We suggest that this is a robust feature regarding what might be
expected if dust-scattered starlight dominates the diffuse FUV background; the
complete absence of such asymmetry in the observations again confirms that
dust-scattered starlight cannot be a significant source of the diffuse FUV
background at moderate and high Galactic latitudes. Even more importantly, the
model predicts very low brightnesses at low Galactic latitudes at all Galactic
longitudes that are far from where the bulk of the FUV-brightest stars are in
Gould’s belt—yet the observations show a strong brightening toward low
Galactic latitudes at all Galactic longitudes. The observations from Dynamics
Explorer are particularly important in this regard. So we can robustly
conclude that even at the lowest Galactic latitudes starlight forward-
scattered by interstellar dust is not the only source of the diffuse FUV
background. There remains the possibility of a contribution to the FUV
background by isotropic scattering from very small grains; we will deal with
this possibility in Section 11.
## 9 Individual GALEX observations
Sujatha, Murthy, Karnataki, Henry & Bianchi (2009), and Murthy, Henry, &
Sujatha (2010), have begun the work of examining the diffuse FUV background in
individual GALEX observations. They found that the overall level of brightness
at one GALEX target at moderate ($b=+38.6^{\circ}$) Galactic latitude could be
successfully modeled with conventional sources. However, Henry (2010) found
that the model that succeeds for that observation gives incorrect results at
higher Galactic latitudes, and that the model does not predict correctly the
detailed observed distribution across the one-degree field of the observation.
Indeed, this observation caused Henry (2010) to abandon his long-held belief
that the high-Galactic-latitude diffuse ultraviolet background was
extragalactic in its origin, and the revision of that conclusion is of course
strongly reinforced in the present paper.
The result of the GALEX FUV observation of this target is crucial to
supporting the interpretation of the diffuse FUV background that we are
presenting in this paper. The observation was made as part of our Guest
Investigator program with GALEX. The target was specifically proposed because
(we thought at that time) the observation would support our then-held idea
that the high-Galactic-latitude diffuse FUV background was extragalactic in
origin. Instead, the observation provides conclusive proof that the non-
scattered-starlight component of the FUV background is Galactic in origin, and
not extragalactic. The observation is of a high-Galactic-latitude dust cloud
that was discovered by Sandage (1976). Sandage’s observation appears in Figure
22, while our GALEX FUV observation is shown in Figure 23. The GALEX exposure
was 14,821 seconds, so the signal-to-noise is extremely high. We had expected
that the FUV observation would show evidence for the absorption, by the dust
cloud, of our putative extragalactic source. Instead, to our surprise (and
shock, at the time) the glow is bright, and is almost perfectly uniform: see
Henry (2010) for detailed analysis. No, what is detected is clearly forground
diffuse emission of some unknown kind from the interstellar medium,
overwhelming any trace of an effect of the dust cloud behind. This is imaging
of our “second component” of the FUV background!
What can this second component have as its origin? Could it arise from some
kind of interaction of the dark matter with the interstellar medium? We
examine that possibility in the next section, concluding that it is difficult
to support such an attribution. Could the radiation instead be due to
scattering from a population of interstellar grains that are much smaller than
the FUV-wavelengths of the radiation that is being scattered (Draine 2011)? In
our final section before our conclusions, we will examine that possibility as
well—and we will find that we must also reject that possibility—leaving us
with our mystery.
## 10 Hints of new physics?
### 10.1 Background radiation and dark matter
We have presented in this paper evidence for a second component to the diffuse
FUV background, beyond the starlight scattered from conventional dust. Henry
(2010, 2012) has attributed this mysterious second component to unspecified
interaction of the dark matter particles with the nuclei of the interstellar
medium. We have, below, but without success, attempted to find a mechanism or
mechanisms that could justify that attribution. Our lack of success does not,
of course, rule out such an origin; in particular, if the dark matter
particles should turn out to be composite particles, overall electrically
neutral but involving electrically charges components (think of a neutron),
then perhaps our second component could originate in collisions of those dark
matter particles with the nuclei of the interstellar medium. The attraction of
the idea remains that it would account for the fact that our “second
component” is confined to the Galactic plane. Having said that, Dvorkin et al.
(2013) seem to rule out even interactions of dark matter particles possessing
electric dipole moments or tiny electric charges!
In cosmology, and more recently in observations of the Galactic center, there
is a rich tradition of attributing “bumps” and excesses of all kinds in
diffuse background radiation to new physics, usually in the form of decays,
annihilations or other interactions involving dark matter and/or energy
(Overduin & Wesson 2008). Neither dark matter nor dark energy need be
perfectly black; and this way of searching for them is referred to as indirect
detection (as opposed to the direct detection of dark-matter particles
themselves). Features from the one-time “sub-mm excess” in the CMB through the
“MeV bump” and all the way out to the GZK cutoff (or apparent absence of it)
in the high-energy $\gamma$-ray background have been interpreted this way.
Many such features have gone away on closer observation. Will the Galactic UV
excess found by by GALEX suffer the same fate? We have argued above that it is
robust, and that the obstacles to explaining it with conventional astrophysics
(primarily dust scattering) are serious. Before concluding, we therefore
consider whether there are any natural ways to produce such an excess from the
physics of the dark sector.
### 10.2 Neutrinos
One particle considered a particularly plausible dark-matter candidate
throughout much of the 1990s was the massive neutrino (Sciama 1993). Several
independent lines of evidence at that time pointed to a $\tau$ neutrino of
rest energy 27 eV, decaying to a lighter species plus a photon (Figure 24).
Each product carried away half the rest energy of the parent, producing a
signal at $\lambda\sim 900$ Å. One might hope that a lighter version of such a
particle could be associated with a background in the neighborhood of 1200 Å.
This background would, however, be a sharp line and not the broad “shelf” that
we see in the data, from the Lyman limit to the far edge of the GALEX bandpass
(1350-2800 Å). There is no hint of such a signal in observations of other
galaxies and galaxy clusters where dark matter is thought to be even more
concentrated than it is in the Milky Way (Overduin & Wesson 2008). Moreover,
PLANCK measurements of the CMB, in combination with the standard picture of
structure formation by gravitational instability, now imply that the sum of
all three (standard-model) neutrino masses is less than 0.35 eV (Giusarma et
al. 2013). One can evade this constraint with sterile fourth-generation
neutrinos, and indeed these have recently been postulated as warm dark-matter
particles decaying into photons with energies as low as $\gtrsim 0.5$ keV
(Abazajian et al. 2007). However, data on structure formation now rules out
warm dark matter particles in any form with masses below 3.3 keV (Viel et al.
2013). Neutrinos do not seem to be the answer.
### 10.3 Weakly-interacting massive particles (WIMPs)
The leading dark-matter candidates remain weakly-interacting particles or
WIMPs. While still hypothetical, their naturalness stems from the fact that
they automatically produce the right density of cold dark matter, given only
the assumption of weak-scale coupling strength to matter (or vice versa). No
fine-tuning is needed; one property follows from the other, thanks to the
Boltzmann equation. The WIMP mass is also tightly constrained: it cannot be
less than about 2 GeV or WIMPs will overclose the Universe. (This condition,
known as the Lee-Weinberg bound, arises because of a $m^{2}$-dependence in the
cross-section. Low-mass WIMPs interact so weakly that they drop out of thermal
equilibrium too soon after the big bang, producing an unacceptably high relic
density.) As we will see below, this large mass makes it difficult to connect
WIMPs to UV energies, and their small coupling cross-section makes it
difficult to tie them to an excess as bright as that seen in the GALEX data.
WIMPs annihilate “directly” into photons via loop diagrams (Figure 25).
Because it involves loops, the flux of photons from this process is extremely
low. Nevertheless, it is typically considered the most promising way to
discover WIMPs via “indirect detection” (i.e., detection via annihilation
products, rather than WIMPs themselves). This is because, while faint, these
photons are essentially monoenergetic, and therefore distinct from almost any
competing astrophysical background. However, energy conservation dictates that
the energy of the photon products is closely tied to that of the parent WIMPs:
$E_{\gamma}=m_{\tilde{\chi}}$ for
$\tilde{\chi}\tilde{\chi}\rightarrow\gamma\gamma$ (Bergström and Ullio 1997)
and $E_{\gamma}=m_{\tilde{\chi}}(1-m_{Z}^{2}/4m_{\tilde{\chi}})^{2}$ for
$\tilde{\chi}\tilde{\chi}\rightarrow Z\gamma$ (Ullio and Bergström 1998). From
the Lee-Weinberg bound it follows that WIMP annihilation directly to photons
can have nothing to do with the UV background.
More promising might be processes that produce secondary photons via tree-
level WIMP annihilations to quarks or W/Z bosons which then “hadronize” and
decay to charged and neutral pions, as shown schematically in Figure 26.
Example tree-level diagrams for $\tilde{\chi}\tilde{\chi}\rightarrow qq$ are
shown in Figure 27. The flux of photons produced in this way can greatly
exceed that from WIMP annihilations directly into photons. Moreover, these
secondary photons have a broader range of energies, from the WIMP mass down to
perhaps the pion mass scale ($\sim 100$ MeV). This makes them harder to detect
against the many competing astrophysical backgrounds, which is why they are
usually ignored in indirect dark-matter searches (see however Baltz et al.
2008). It might make them more attractive from the point of view of explaining
a relatively flat spectrum (as seen by GALEX). However, the relevant energies
are still orders of magnitude beyond the FUV scale, of order $\sim 10$ eV.
WIMP scattering off nucleons in the interstellar medium is another
possibility. If the scattering is inelastic, it would knock the nucleus into
an excited state, producing de-excitation photons. This is the basis for
direct detection experiments like XENON, whose 129Xe and 131Xe target isotopes
have de-excitation energies in the range 10 -100 keV (Baudis et al. 2013).
Alternatively, Henry (2012) has suggested that WIMP recoils might give rise to
a flux of low-energy photons by bremmstrahlung-type acceleration of charged
quarks inside the nucleus. In high-energy physics, these are called “direct
photons” to distinguish them from the messier photons in jets. However,
fundamental limitations, both experimental and theoretical, mean that it has
only been possible to study this phenomenon in the large transverse momentum
regime; i.e., at energies above about 1 GeV. It is interesting to note that
there are some experimental indications of an unexplained “soft photon” excess
beyond QED expectations at the lower end of this range (Belogianni et al.
2002).
To decide in a model-independent way whether WIMP scattering can be connected
to the FUV excess seen by GALEX, we proceed as follows. The largest WIMP-
nucleon scattering cross-section allowed by current experiment is
$\sigma=2\times 10^{-41}$ cm2 (Aprile et al. 2012). The lightest WIMP mass
allowed by experiment is $m_{\tilde{\chi}}=9$ GeV (Agnese et al. 2013). Hence
the largest possible number density of WIMPs is $n=\rho_{\mbox{\tiny
DM}}/m_{\tilde{\chi}}=0.03$ cm-3, where we have used a canonical figure for
the local dark matter density of $\rho_{\mbox{\tiny DM}}=0.3$ GeV cm-3.
Combining these numbers, we obtain a conservative upper limit on the
scattering rate per WIMP of $n\sigma v=1\times 10^{-35}$ s-1, where $v=220$
km/s is the speed of WIMPs with respect to the interstellar medium. Even if
each WIMP converts its entire rest energy into 10 eV photons, the largest
possible FUV “luminosity per WIMP” is then $1\times 10^{-37}$ erg s-1. Now
consider all the WIMPs scattering off nucleons inside a spherical region whose
radius corresponds to the mean free path of an FUV photon in the local
interstellar medium, $R\approx 600$ pc (Hurwitz, Bowyer & Martin 1991). If
their number density $n$ and luminosity $L$ are uniform throughout this
region, then the total FUV intensity produced cannot exceed $\rho_{\mbox{\tiny
DM}}n\sigma vR<1\times 10^{-17}$ erg cm-2 s-1. By comparison, the intensity of
excess FUV radiation detected by GALEX over its bandpass (1380-2500 Å) is
$1\times 10^{-5}$ erg cm-2 s-1, assuming a flat spectrum with 300 photons cm-2
s-1 sr-1 Å-1. Based on this argument it is hard to see how the excess could be
connected to WIMP scattering in any form. The cross section, which is fixed by
cosmology and by the Boltzmann equation, is simply too small.
### 10.4 Axions
The second leading candidate for dark matter is the axion. Axions are perhaps
not as natural as WIMPs, in that their coupling strength does not
automatically imply the correct cosmological density. On the other hand, they
require less of a leap beyond the standard model, as their existence is
implied within ordinary QCD. Moreover there are now indications that, even in
the leading (supersymmetric) WIMP models, axions arise and may make up more of
the dark matter than the WIMPs themselves (Baer 2013).
Axions come in two main flavors: thermal (meaning they were originally in
equilibrium with standard-model particles in the early universe) and non-
thermal (meaning they arose in some other way, for instance as the result of a
misalignment between the initial value of the axion field and the minimum of
its potential). Most experimental attention has focused on nonthermal axions,
which have masses in the $\mu$eV-meV range. These are numerous enough to make
up the required cosmological density of cold dark matter, and hard to
constrain as they do not decay into standard-model particles. (They are also
known as “invisible axions” for this reason.) They can however convert into
photons inside magnetic fields via the Primakoff effect (Figure 28). This is
the basis for experimental efforts to detect axions from hot stellar cores
using magnetic cavities (Sikivie 1983, van Bibber et al. 1989). The expected
photon flux for a 9.0 T magnetic field inside a 9.26 m cavity aimed at the Sun
(as in the CERN Axion Solar Telescope or CAST) is 0.088 photons day-1 cm-2
keV-1 $(E/\mbox{keV})$ $(L/9.26\mbox{ m})^{2}$ $(B/9.0\mbox{ T})^{2}$
$\exp[-(E/\mbox{keV})/1.305]$ (Andriamonje et al. 2007). On simple dimensional
grounds one might replace such a cavity in the case of the Galactic FUV
background by the “local bubble” of radius $\sim$600 pc, permeated by a
Galactic magnetic field of mean magnitude $B\sim 0.5\mu$G (Mao et al. 2012;
note that the Sun is located about 20 pc above the Galactic plane according to
Humphreys & Larson 1995). A similar mechanism has recently been proposed to
contribute to the diffuse cosmic x-ray background and account for the
unexplained soft x-ray excess in some galaxy clusters (Conlon & Marsh 2013).
However, the corresponding flux per wavelength of $3\times 10^{7}$ erg cm-2
s-1 sr-1 Å-1 $(\lambda/\mbox{10 \AA})^{-4.5}$ $(L/\mbox{600 pc})^{2}$
$(B/0.5~{}\mu\mbox{G})^{2}$ $\exp(-\mbox{10 \AA}/\lambda)$ can have nothing to
do with the GALEX UV excess; it is orders of magnitude too bright, and peaks
at 2 Å (or 5 keV) in the X-ray band, as might be expected since these axions
form in the cores of hot stars.
Thermal axions might be more promising, as they can decay directly into photon
pairs, each with $E_{\gamma}=m_{a}/2$, via a model-dependent axion-photon
coupling constant $g_{a\gamma\gamma}$ (Figure 29). Thus thermal axions with 9
eV$<m_{a}<18$ eV might in principle be associated with a signal like that seen
by GALEX. However, any such mechanism faces considerable challenges. As with
WIMP annihilations, one difficulty would be in reconciling the essentially
monoenergetic nature of these decays with the flat spectrum observed.
Calculations using the Boltzmann equation show that axions this massive would
be able to provide no more than half the observed density of dark matter. They
are also strongly constrained by astrophysical considerations. They would
drain too much energy from the cores of red giant stars, disrupting helium
ignition unless $m_{a}\lesssim 10$ eV in the simplest models (Raffelt 1996).
Upper limits on the intensity of the extragalactic background light impose a
similar bound, $m_{a}<8$ eV (Overduin & Wesson 2008). Observations in the
direction of three rich clusters of galaxies tighten this limit further, to
$m_{a}<3$ eV (Bershady, Ressell & Turner 1991). It might be worth revisiting
these constraints, which depend sensitively on theoretical assumptions
involving the axion coupling strength. More recently, however, an even more
stringent limit has come from data on structure formation. Axions in this mass
range are light enough to act as hot dark matter. Arguments similar to those
mentioned above in connection with massive neutrinos then imply that $m_{a}<2$
eV (Hannestad & Raffelt 2004) or even $m_{a}<0.4$ eV (Melchiorri, Mena &
Slosar 2007). These are gravitational arguments, and do not depend on the
details of axion-photon coupling. Thus axions, too, fall short.
### 10.5 Other candidates
Most of the remaining dark-matter candidates from particle physics are
extremely massive (by definition, more massive than the heaviest standard-
model particle), and hence manifest themselves only in the high-energy
$\gamma$-ray band, if at all. Leading examples include Kaluza-Klein states
(excitations of standard-model particles associated with compact extra
dimensions), branons (similar states in higher-dimensional brane-world
scenarios), cryptons (stable or metastable states in string theory) and
WIMPzillas (heavy non-thermal relic particles). These particles would be
remoter from the FUV band than the WIMPs considered above (whose masses are
tied to the masses of the W and Z bosons). This serves to point up the
essential challenge: in order to explain the GALEX excess, one needs to find
some plausible connection to physics on eV scales. Light neutrinos and axions
come closest to fitting this description, but have now been decisively ruled
out by arguments that are almost purely gravitational (structure formation)
and therefore very hard to evade. The other candidates we have considered here
involve physics at higher energies almost “by design,” and it is hard to see
how they could be connected to the GALEX excess without an unnatural degree of
fine tuning.
Similar remarks apply to one final dark-matter candidate, the primordial black
hole (PBH), though perhaps with more leeway. PBHs are black holes that evade
the upper limit on baryonic mass density because they formed before cosmic
nucleosynthesis. In principle, their cosmological density is unconstrained. In
practice, assuming they formed by gravitational instability with a standard
scale-invariant spectrum of initial masses, it is possible to be quite
specific about the properties they must have. PBHs decay by Hawking
evaporation at a rate $dM/dt=-\alpha/M^{2}$, where $\alpha\approx 7\times
10^{25}$ g3 s-1, so those that are evaporating at the present time $t_{0}$
have $M_{\ast}=(3\alpha t_{0})^{1/3}\sim 10^{15}$ g. The spectrum of
background radiation from PBH evaporation is approximately thermal, peaking at
$\lambda\sim(4\pi/c)^{2}GM$. In the standard scenario described above, this
spectrum is dominated by PBHs with $M\sim M_{\ast}$, giving a sharp peak near
$10^{-4}$ Å (or 100 MeV). No such line is seen in observations of the cosmic
$\gamma$-ray background, leading to the conclusion that PBHs can make up at
most $\sim 10^{-8}$ times the critical density (Page & Hawking 1976, Overduin
& Wesson 2008). It is conceivable that one could evade this conclusion by
imposing a low-mass cutoff on the spectrum of initial PBH masses. They would
then radiate less, and at lower energies. In fact, the same factor of $\sim
10^{-8}$ could push the peak of their contributions to the background close to
the ultraviolet band. However, it is very hard to see how such a cutoff could
arise in a natural way. Many attempts have been made to justify such a
modification for other reasons, generally to connect PBHs to various
phenomena, from microlensing to $\gamma$-ray bursts, but none have gained wide
acceptance.
## 11 Small interstellar grains
The values of the albedo and Henyey-Greenstein scattering parameter $g$ that
were found by Hamden et al. (2013) to account for most of the observed
brightness of the FUV background radiation to their satisfaction, $a=0.62$ and
$g=0.78$, are of course deduced under the assumption that the dust-scattered
FUV light of stars is the predominant source of the observed diffuse FUV
background—which we find in the present paper not to be the case. From his
extensive modeling, Henry (2002) has suggested that the grain albedo might in
fact be as low as $a=0.1$. Mathis (2002) points out that Henry’s result is
very uncertain (and Henry agrees), but the point is made that we do not know
the value of the albedo other than from these diffuse background measurements
and their attribution to scattering by interstellar dust. To see how sensitive
to the adopted albedo value the predicted background is, the reader might look
at Figure 4 of Henry (2012), where a model with $g=0.58$ and $a=0.10$ is
displayed—the brightest predicted diffuse FUV background is only 514 units.
We have already mentioned the very important question raised by Draine’s
(2011) drawing attention to the isotropically-scattered light that must be
produced by interstellar grains that are smaller than the wavelength of the
radiation that is being scattered. In particular, could it be that the 300
photon units background that is observed at the highest Galactic latitudes is
starlight scattered from such very small grains?
Our single-scattering radiative transport model (presented above) does an
excellent job; so good that we even felt comfortable in attributing its
failure (by a factor of 1.7) to account for the observed diffuse FUV
background, to the presence of a second component in addition to that of dust-
scattered starlight. However, the excellence of our model does require that
the grains be strongly forward scattering. Despite that limitation, we will
nevertheless now apply it so as to give us at least some handle on what might
be expected from isotropically-scattering grains.
To cope with isotropically-scattering grains, we now adapt our model to simply
confine scattering to the point of intersection of our line of sight with an
element of the interstellar medium at which the starlight has arrived (Figure
17; step two of Section 8). That is, we will not make any allowance at all for
scattering as the light progressed from the source star to the intersection,
nor as the scattered light subsequently proceeds to the detector. (To see how
serious these omissions are, we experimentally eliminated these same items
from our forward-scattering model that produced Figure 18: the result was not
dramatically different, suggesting that our result in this section can be
trusted—particularly because, as we will now see, our result is dramatic
indeed.)
Our previous model results (Figures 18 and 19) had a non-linear dependence on
the density of the interstellar dust. But that non-linearity came entirely
from the two legs of radiative transport that we have been forced to omit for
the case of isotropic scattering. So for our present test, we can scale the
final brightness, if we wish, by simply increasing or decreasing the density
of the interstellar dust. Since our sole aim is to try to reproduce the
uniform high-Galactic-latitude 300 photon unit background by means of
scattering by small grains, we have simply adjusted the interstellar matter
density as required to force a result of 300 photon units. The factor that we
find to be required is 11.045, that is, we have had to reduce the interstellar
dust density by that large factor if we do wish to not overproduce the
predicted high-Galactic-latitude diffuse background!
Our result is presented in Figure 30, and is (but only at first glance)
extremely surprising. Keep in mind the fact that our maps (Figures 9 and 10)
of the $100\ \mu$m emission give the amount of thermal radiation from the
dust, not the actual amount of dust. The physics that Draine presents is
unexceptionable. And Draine notes that the sharp rise in the extinction curve
that occurs at the shortest FUV wavelengths certifies that small grains do
indeed exist, as expected. Yet at high Galactic latitudes, we clearly do NOT
see the starlight, dust-scattered at large angles, that those facts would lead
us to expect! The answer appears to be that Perry & Johnston (1982) were
correct in asserting that there is negligible reddening within $\sim$ 200 pc
from the Sun on the Galactic plane. We don’t see dust-scattered starlight from
small grains, simply because there is a very low density of interstellar
grains of any kind, in our neck of the Galaxy. The fact that we see little or
no broadly dust-scattered light from all but one of the stars (Spica) in Table
2 is in accord with this interpretation. Also, we do know that we are located
in what is now called the “local bubble” (McClintock, Henry, Linsky, & Moos
1978), a sector of the Galaxy having an exceptionally low density of
interstellar material (perhaps as a result of an ancient supernova explosion).
While we have emphasized the excellence of our model (at least for diffuse
forward-scattered light from stars), there is one element of our model that is
clearly wrong: it assumes a completely locally-homogeneous interstellar
medium. Our model predictions for the highest Galactic latitude observations
should not be trusted at all, in light of the Perry & Johnston (1982) result;
we obtain, with our model, much too high predicted brightnesses for starlight
scattered from dust.
## 12 Conclusions
Very high quality observations of the spatial distribution of the diffuse FUV
background, with excellent signal-to-noise, are available, thanks to the GALEX
mission (Martin et al. 2005) and to the work of Murthy et al. (2010). We have
attempted to account for these observations as originating in the dust-
scattered FUV light of the OB stars of our Galaxy. We have failed in this
attempt. Which leaves us with a mystery: there is an FUV radiation field in
our Galaxy that is of unknown origin, and there seems to be no conventional
source for it that is readily plausible. Henry (2012) has been led to
speculate that the radiation might originate in the particles of the dark
matter of our Galaxy interacting with the nuclei of the interstellar medium.
Such interaction is perhaps possible (Baudis et al. 2013) but production of
the observed flat FUV spectrum has not (or at least, has not yet) been
demonstrated to be possible, and in the present paper we find ourselves unable
to point to a plausible mechanism to produce such radiation. Indeed, Dvorkin,
Blum, & Kamionkowski (2013) seem to rule out such an origin.
If indeed, as we believe to be the case, we have identified a new ultraviolet
radiation field in the Galaxy, then depending on its physical origin, the
possibility exists that its spectrum extends below 912 Å; that is, that it is
a source of ionizing radiation extending well above the Galactic plane. The
critical need for just such an ultraviolet source has been emphasized by
Kollmeier et al. 2014, in their paper “The Photon Underproduction Crisis.”
They speculate that what they call the “missing” ionizing photons could be
coming from decaying or annihlating dark matter particles in the dense cores
of halos and subhalos. In light of their call, we draw attention to both our
own observations (with Voyager) at wavelengths shortward of Lyman $\alpha$
(Murthy, Henry, & Holberg 2012), and also to the unique J-PEX observations of
Kowalski et al. (2006, 2009, 2011) at still shorter wavelengths (Figure 31).
The lack of accord that we find of the Voyager observations nearest the two
J-PEX observations, with those observations, simply means that the spectrum of
the shortest-wavelength background is complex. Our Voyager observations
include remarkably bright patches on the sky—we direct attention to the
possibility that the observed “overproduction” of photons that we report, in
our various papers, might resolve the “underproduction” crisis—and allow us to
understand the reionization of the Universe. While that understanding would be
important indeed, far more important would be the clue to new physics.
If as we suggest there is indeed a substantial component of the diffuse FUV
background that is NOT simply starlight scattered from dust, confirmation
would of course be highly desirable—and would not be terribly difficult to
achieve. What is needed is high-signal-to-noise spectroscopy of the diffuse
FUV background at a few high-Galactic-latitude locations, to probe the
detailed spectral character of the radiation.
We thank Tom Krause for discussions on opacity and Jeff Owens for helpful
comments on direct photons. We are grateful to a host of colleagues with whom
we have worked over the years. We are grateful to Maryland Space Grant
Consortium, to the Johns Hopkins University, to Towson University, and to the
Indian Institute of Astrophysics, for support. Finally, we are grateful to a
thorough and persistent referee.
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Table 1: The Two Dynamics Explorer FUV Scans (Fix, Craven & Frank 1989)aaTable 1 is published in its entirety in the electronic edition of the Astrophysical Journal. A portion is shown here for guidance regarding its form and content. Scan | No. | $\ell$ | _b_ | Photon Units
---|---|---|---|---
1 | 1 | 262.56 | 36.73 | 1352 $\pm$ 533
1 | 2 | 262.73 | 36.54 | 717 $\pm$ 212
1 | 3 | 262.90 | 36.36 | 868 $\pm$ 218
| ……. | | |
1 | 1321 | 261.73 | 37.61 | 838 $\pm$ 216
1 | 1322 | 261.91 | 37.42 | 1231 $\pm$ 241
1 | 1323 | 262.08 | 37.24 | 1020 $\pm$ 241
— | ——— | ——— | ——— | —————
2 | 1 | 244.50 | 45.42 | 510 $\pm$ 397
2 | 2 | 245.47 | 44.81 | 930 $\pm$ 427
2 | 3 | 245.72 | 44.65 | 340 $\pm$ 386
| ……. | | |
2 | 1282 | 242.84 | 46.41 | 1625 $\pm$ 457
2 | 1283 | 243.09 | 46.27 | 567 $\pm$ 393
2 | 1284 | 243.35 | 46.12 | 805 $\pm$ 408
Table 2: Individual Stars _z_ pc | _b_ | Name | Name | HR | HD
---|---|---|---|---|---
-36.8 | $-58^{\circ}$47 | $\alpha$ Eri | Achernar | 472 | 10144
-24.6 | $-52^{\circ}$28 | $\alpha$ Gru | Alnair | 8425 | 209952
-87.3 | $-46^{\circ}$41 | $\gamma$ Peg | Algenib | 39 | 886
-31.7 | $-35^{\circ}$11 | $\alpha$ Pav | Peacock | 7790 | 193924
+18.3 | $+48^{\circ}$56 | $\alpha$ Leo | Regulus | 3982 | 87901
+62.1 | $+50^{\circ}$52 | $\alpha$ Vir | Spica | 5056 | 116658
+29.0 | $+65^{\circ}$19 | $\eta$ UMa | Alcaid | 5191 | 120315
Figure 1: Blue circles and dots mark the locations of 31,215 TD1 far-
ultraviolet-bright stars, while the GALEX FUV diffuse background is also shown
(more than 30,000 one-degree-sized spots). (At the lowest galactic latitudes,
no GALEX FUV images exist.) A few constellations are shown for orientation
purposes. Some FUV-bright stars (e.g., in Crux) are behind significant amounts
of interstellar dust, which forward-scatters their light to us (Murthy, Henry,
& Holberg 1994). Most FUV-bright stars at high galactic latitudes have very
little foreground dust; an important exception being Spica (Murthy & Henry
2011) located at $\ell=316^{\circ},b=50.8^{\circ}$. One crucial question is,
is the ubiquitous red background at high galactic latitudes astrophysical, or
is it largely geocoronal? Attempting to resolve that question is one focus of
the present paper.
Figure 2: This is the same as Figure 1, except that it is centered on the
Galactic anticenter, giving a much better display of the northern-Galactic-
hemisphere distribution of the far ultraviolet background radiation.
Figure 3: All of the GALEX FUV observations from the entire mission have
their diffuse background measurements plotted here as a function of year of
observation. The observations were made over a substantial fraction of a solar
cycle. The minimum value observed, $\sim 300$ photon units, does not change
over that span of time, strongly suggesting that the minimum FUV brightness
measured by GALEX is not significantly affected by terrestrial atmospheric
emission, but rather is astrophysical in its origin.
Figure 4: Dynamics Explorer (“DE”, Fix, Craven, & Frank 1989) was used to
carry out two far-ultraviolet scans around the sky, with the results shown
here. (The UV-bright stars are colored red in this figure, rather than blue,
simply to provide better contrast, with the DE observations, at low galactic
latitudes.) The lowest DE value was $4\pm 363$ units, which (but,
inconsequentially) distorts the color scale. At high galactic latitudes, both
DE scans detect the background at a few hundred photon units, strongly
suggesting, because DE is in a much higher orbit (Fig. 6) than is GALEX, that
the GALEX observations of similar values are not due to geophysical
contamination. Both longitude regions of the galactic plane that were observed
by DE are seen to be very bright: the brightest region that is seen on the
scan that passes closest to Cassiopeia is $6082\pm 455$ photon units, while
the brightest region for the scan that passes slightly farther from Cassiopeia
was $9783\pm 752$ photon units.
Figure 5: That the GALEX observed Galactic Polar glow of $\sim 300$ photon
units is unlikely to be of geophysical origin is supported by this
quantitative comparison with the Dynamics Explorer FUV scans (that were shown
in Figure 4). Each point represents multiple (mostly 3 or 4) observations of a
GALEX target by DE, with the observations averaged to improved the statistics.
Figure 6: The GALEX orbit was nearly circular, but Dynamics Explorer was in
an elliptical orbit having a distant (23,250 km) apogee where terrestrial UV
emissions are expected to be very low indeed. Thus, the DE observations of the
diffuse ultraviolet background are of crucial importance in helping establish
the astrophysical nature of most of the high-galactic-latitude diffuse FUV
backgrounds that were observed by GALEX.
Figure 7: GALEX FUV cosmic background brightness, on a logarithmic scale, for
the Northern Galactic hemisphere. (Regions where no GALEX images are available
are white.) Circles are UV-bright stars (dashed circles are the brightest UV
stars in the Southern Galactic hemisphere; see Figure 11); they are highly
concentrated around the upper rim of the figures. The dust-scattered UV light
of Spica (Murthy & Henry 2011) is apparent at the upper right. The image is
dominated by a very-low-brightness UV glow (red) that shows almost no
dependence on galactic longitude. The dimmest regions (black) are 280 photon
units. This image alone can dispose of the notion that the FUV background at
high galactic latitudes, if astrophysical, is due to dust-scattered starlight:
its origin is therefore a profound mystery. The dashed circle at
$\ell=140^{\circ},b=+40^{\circ}$ indicates the location of the reflection
nebula discovered by Sandage (1976).
Figure 8: The same GALEX FUV brightness as in the previous figure, but this
time on a linear scale, from zero, for the Northern Galactic hemisphere. (Low-
galactic-latitude regions that are brighter than 2000 units in the FUV are
white.) The dust-scattered UV light of Spica (Murthy & Henry 2011) is again
apparent. One virtue of the linear scale is that the brightness and the
relative uniformity of the galactic-cap FUV glow are more clearly seen, and
are striking.
Figure 9: The same logarithmic scale as used in Figure 7, but this time for
the Southern Galactic hemisphere. The UV stars are here shown again , but this
time it is the brightest ones (only) in the Northern Galactic hemisphere that
are shown as dashed circles. Note the feature south of $-60^{\circ}$ between
longitudes $120^{\circ}$ and $150^{\circ}$ ; it is likely dust-scattered
starlight.
Figure 10: The same as Figure 9, but this time a linear plot for the Southern
Galactic Hemisphere. Again the remarkable uniformity of the general glow, with
little or no Galactic longitude dependence, is striking. Note, in this figure
and in the previous one, the structure that appears between $120^{\circ}$ and
$140^{\circ}$ longitude, south of the south $60^{\circ}$ latitude line: this
structure will also be seen in the infrared images, in following figures,
suggesting, for these features, an origin in dust-scattered starlight.
Figure 11: In previous figures, the locations of ultraviolet-bright stars
were shown. The same are shown again here, for both Northern (blue) and
Southern (red) Galactic Latitudes. The sky presents a strikingly different
appearance in the far ultraviolet (compared with the visible) with almost all
bright stars strongly concentrated not only toward the Galactic plane, but
also toward that half of the Galactic plane lying between longitudes
$180^{\circ}$ and $360^{\circ}$ (Henry 1977).
Figure 12: The Northern Galactic hemisphere infrared diffuse background at
$100\ \mu$m, shown here, using a logarithmic intensity scale, provides crucial
assistance in interpeting the observed ultraviolet background. This infrared
background is widely thought to be (and we agree) due to simply thermal
emission from interstellar dust that has been heated by starlight. The plotted
observations are from Schlegel et al. (1998). Unlike the ultraviolet
background (which was displayed in previous figures), in the infrared there is
a strong asymmetry with galactic longitude, and hence a strong asymmetry in
the interstellar dust distribution: the least dust, is present in the
longitude range $30^{\circ}-210^{\circ}$, in the northern galactic hemisphere.
However, a quite different asymmetry (but an asymmetry that is just as strong)
appears in the southern Galactic hemisphere$100\ \mu$m observations (Fig. 13).
Figure 13: The Southern Galactic hemisphere $100\ \mu$m emission is shown
using a logarithmic intensity scale. Here, it is the (very different)
longitude range $210^{\circ}-0^{\circ}-30^{\circ}$ that has the least amount
of interstellar dust. This is in (drastic) contrast with the independence of
Galactic longitude, of the far-ultraviolet background radiation that was
demonstrated in Figures 7-10.
Figure 14: The Galactic Latitude dependence of the far-ultraviolet background
(black), and of the $100\ \mu$m emission (red, for the Galactic Longitude
range $30^{\circ}-210^{\circ}$; blue, for the Galactic Longitude range
$210^{\circ}-0^{\circ}-30^{\circ}$). There is a dramatic difference in the
latitude dependences of the $100\ \mu$m emission for these two longitude
ranges, which is not true for the FUV diffuse emission (see Figure 15), which
therefore must have an independent origin. (The green curve (symmetric about
$-5^{\circ}$ Galactic Latitude) approximates the lower bound of the FUV
emission.) This figure establishes that there is no connection between the
diffuse infrared emission and most of the diffuse FUV emission; the source of
the diffuse FUV emission is unknown—that is the “Mystery” that is referred to
in the title of this paper.
Figure 15: The solid black line is $csc(b-5^{\circ})$, where b is the Galactic
Latitude. The FUV brightnesses have here been subdivided according to Galactic
Longitude (exactly as was done for the $100\ \mu$m emission in Figure 14) in
order to bring out the fact that there is little or no sign of the strong
Galactic Longitude asymmetry that appears in the $100\ \mu$m emission, as
displayed in Figure 14. Also, the fact that the FUV brightness follows the
black line strongly suggests that at least most of the FUV emission originates
in a layer that is centered on the Galactic plane.
Figure 16: Curves give model predictions for the expected diffuse ultraviolet
background, in photon units, on the assumption that the highest Galactic
latitude background is entirely due to the integrated light of distant
galaxies or extragalactic background light (EBL). Shown are predictions by
Finke et al. (2010, “Model C”) and Dominguez et al. (2011, upper and lower
limits). Also shown are upper limits obtained by adapting a code for EBL
intensity at near-optical wavelengths (Overduin, Prins, & Strobach 2014). To
generate the latter curves, we have used templates for both quiescent and
star-forming galaxy spectra over the full spectrum from FUV to sub-mm
wavelengths (Devriendt et al. 1999). Evolution in galaxy luminosity and number
density is incorporated by requiring that the overall luminosity density of
the universe be consistent with theoretical and observational constraints at
every redshift, as compiled by Nagamine et al. (2006). The labels TVD, SA,
Fossil and H&S refer to specific evolution models, and parameters within each
model have been adjusted to give the largest possible spread in predicted EBL
intensities. We have incorporated a model for extinction by dust in the
intergalactic medium due to Loeb & Haiman (1997). All curves assume a standard
$\Lambda$CDM cosmology (with $\Omega_{M}=0.3$ and $\Omega_{\Lambda}=0.7$).
Superimposed on these predicted EBL intensity curves are forty years’ worth of
observational constraints (datapoints and error bars), spectroscopic (filled
symbols) as well as mostly photometric (empty symbols). The GALEX FUV data on
which we focus in this paper gives a minimum background of about 300 photon
units over the range 1350 Å - 1750 Å.
Figure 17: The geometry for our simple single-scattering model giving the
expected flux from the dust-scattered light of stars. The GALEX field of view
is one degree. For each star contributing dust-scattered light to our
observation of the given target, we make three calculations: 1) the diminution
of intensity of the light of each star (one typical star is shown) due to both
its distance from each location on our line of sight, and the attenuation due
to absorption by dust along the path to our observing line of sight, 2) the
amount of light scattered in our direction as the starlight crosses the slice
of our line of sight, and 3) the diminution again, as in the first step, but
now from the scattering location to our observing location at Earth.
Figure 18: Our model for the 1550 Å diffuse background, predicted using the
values of the albedo and scattering parameter g found by Hamden, Schiminovich,
& Siebert (2013). A few constellation patterns are included for orientation.
The brightest spot predicted is at $\ell=299.8,b=-1.0$, which is in the
Coalsack nebula in the constellation Crux. This figure shows our prediction
for the entire sky, whereas the following figure is the same model, but
limited to the regions that were observed in the FUV by GALEX.
Figure 19: Our prediction of the diffuse far-ultraviolet background assuming
that it is entirely due to the light of FUV-bright stars scattered by
interstellar dust having the albedo (0.62) and the Henyey-Greenstein
scattering parameter (0.78) that were found by Hamden et al. (2013) to fit the
GALEX data. The upper scale at the top of the figure is equal to the actual
model scale (given immediately below it), simply multiplied by 1.706 to force
agreement with the data (Figure 17) at the brightest spot. Note the
extraordinarily low predicted FUV brightness between Galactic longitudes 30∘
and 90∘, at the lowest Galactic latitudes. In Figure 20 we again present the
data, for comparison with this model, and in the figure following we give an
adjusted plot of the ratio (observed/model).
Figure 20: This figure is the same as Figure 1, but with the “supposed-source”
star locations no longer marked with blue dots. This permits us to see whether
individual isolated bright stars do give rise to a diffuse background. The
relevant stars are listed in Table 2. We see that only in the case of Spica is
there an extended scattered-light contribution to the observed diffuse FUV
background. For Achernar, there is scattered FUV, but only very close to the
star’s location. In all other cases, it is difficult or impossible to
attribute the diffuse background to a particular source star. Stars that are
detected have been analyzed by Murthy & Henry (2011). (This figure also
includes the Dynamics Explorer scans, on exactly the same intensity scale.
Note, particularily, the two Galactic-plane crossings between Cassiopeia and
Cygnus, showing strong diffuse FUV emission at the lowest Galactic latitudes.)
Figure 21: How well does our dust-scattered FUV model fit the Murthy, Henry, &
Sujatha (2010) observations? This figure shows that (with important caveats)
the fit is remarkably good! What is plotted is the Ratio: (FUV Observed)/(FUV
Model)$\times 0.586$, where $0.586=5253/8962$ to normalize to highest
brightness, and where in this case we have eliminated from our run of the
model six stars (all of those in Table 2 except for Spica) because (Figure 20)
those stars are seen to contribute little or nothing to the diffuse radiation
that is observed. Some of those omitted stars are detectable in this plot as
slight “excesses” of observed/model, meaning of course that they are in fact
detected (if only slightly) by GALEX.
Figure 22: This is the visible-light image of the high-Galactic-latitude
interstellar dust cloud that was discovered by Sandage (1976). Note the
airplane lights! Henry and Murthy obtained GALEX images of part of this nebula
(red circle) in their GALEX Guest Investigator program, to test their
hypothesis (which turned out to be incorrect) that the GALEX FUV radiation was
extragalactic in its origin. Detailed modeling of their observation is given
by Henry (2010).
Figure 23: This is the GALEX FUV image of the Sandage nebula for which a
visible-light image is given in Figure 22. There is no resemblance at all
between the visible and FUV images. Only 6.6% of the counts in this FUV image
are due to stars, the rest is diffuse FUV radiation. This picture is an image
of our “second component” of the FUV diffuse background radiation; its origin,
is the “mystery” referenced in the title of our paper. The same image is
presented in three dimensions in Henry (2010).
Figure 24: Feynman diagram for decay of a massive neutrino into a lighter
neutrino plus a photon. Figure 25: Feynman diagrams for the annihilation of
two WIMPs (here, supersymmetric neutralinos, $\tilde{\chi}$) into a photon
pair (or a photon plus a Z boson) via intermediate fermions and their
supersymmetric partners, the sfermions ($\tilde{f}$). Figure 26: Schematic
depiction of tree-level WIMP annihilations to quarks and bosons, leading to
showers of secondary photons, neutrinos and antimatter. Figure 27: Example
Feynman diagrams for tree-level WIMP annihilations to quark-antiquark pairs
via intermediate Z bosons, neutral Higgs bosons or squarks. These and similar
processes occur inside the circle labeled “??” in Fig. 26. Figure 28: Feynman
diagram for photon-axion interconversion via the Primakoff effect. Figure 29:
Feynman diagram for the decay of light thermal axions to two photons via an
intermediate fermion loop, characterized by a coupling constant
$g_{a\gamma\gamma}$.
Figure 30: This is our model prediction for the case of interstellar grains
that are much smaller than the wavelength of the radiation that is scattered.
Such grains will scatter the radiation isotropically and with unit albedo
(Draine 2011). Our model for forward-scattering grains (Figures 18 and 19),
which is excellent except for its not taking into account local variations
from place to place in the density of the scattering dust, has here been
crudely adapted for the case of isotropic scattering. We have forced the model
to produce a high-Galactic-latitude brightness of 300 photon units (as
observed) by simply reducing the density of dust in the interstellar medium by
a factor of eleven. That drastic reduction seems to be what is necessary to
get the low observed background at high Galactic latitudes. It probably simply
accommodates the fact that we happen to be located in a very low-density
region of the interstellar medium (Perry & Johnston 1982). An important point
to note is that the ratio of the brightest predicted value to the lowest
predicted value is a factor of only 2.95, whereas in Figure 1 the observed
same ratio is a factor of 31.4!
Figure 31: Locations (black dots) are shown for the 1943 Voyager observations
of the diffuse ultraviolet background at wavelengths shorter than Lyman
$\alpha$ that were reported by Murthy, Henry, & Holberg (2012). The red circle
labelled J-PEX shows the location of the J-PEX observation (at 220 - 250 Å) of
Feige 24, reported by Kowalski et al. (2011), who found a mysterious bright
diffuse background. The other red circle is at the location of the J-PEX
observation of Kowalski et al. (2006), which showed no significant background,
despite being at much lower Galactic latitude. The observations by Murthy et
al. show that a similar short-wavelength patchy diffuse background occurs
everywhere.
|
arxiv-papers
| 2014-04-23T06:13:49 |
2024-09-04T02:50:01.687245
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Richard Conn Henry, Jayant Murthy, James Overduin and Joshua Tyler",
"submitter": "Jayant Murthy",
"url": "https://arxiv.org/abs/1404.5714"
}
|
1404.5760
|
# Water isotopic ratios from a continuously melted ice core sample
V. Gkinis Correspondence to: V. [email protected] Centre for Ice and
Climate, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej
30,
2100 Copenhagen, Denmark T. J. Popp Centre for Ice and Climate, Niels Bohr
Institute, University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark T. Blunier Centre for Ice and Climate, Niels Bohr
Institute, University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark M. Bigler Physics Institute, Climate and
Environmental Physics and Oeschger Centre for Climate Change Research
University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland S. Schüpbach
Physics Institute, Climate and Environmental Physics and Oeschger Centre for
Climate Change Research University of Bern, Sidlerstrasse 5, 3012 Bern,
Switzerland E. Kettner Centre for Ice and Climate, Niels Bohr Institute,
University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark S. J. Johnsen Centre for Ice and Climate, Niels
Bohr Institute, University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark Science Institute, University of Iceland, Dunhaga 3,
107, Iceland
###### Abstract
A new technique for on-line high resolution isotopic analysis of liquid water,
tailored for ice core studies is presented. We built an interface between a
Wavelength Scanned Cavity Ring Down Spectrometer (WS-CRDS) purchased from
Picarro Inc. and a Continuous Flow Analysis (CFA) system. The system offers
the possibility to perform simultaneuous water isotopic analysis of
$\delta^{18}$O and $\delta$D on a continuous stream of liquid water as
generated from a continuously melted ice rod. Injection of sub ${\mu}$l
amounts of liquid water is achieved by pumping sample through a fused silica
capillary and instantaneously vaporizing it with 100 % efficiency in a home
made oven at a temperature of 170 ∘C. A calibration procedure allows for
proper reporting of the data on the VSMOW–SLAP scale. We apply the necessary
corrections based on the assessed performance of the system regarding
instrumental drifts and dependance on the water concentration in the optical
cavity. The melt rates are monitored in order to assign a depth scale to the
measured isotopic profiles. Application of spectral methods yields the
combined uncertainty of the system at below 0.1 ‰ and 0.5 ‰ for $\delta^{18}$O
and $\delta$D, respectively. This performance is comparable to that achieved
with mass spectrometry. Dispersion of the sample in the transfer lines limits
the temporal resolution of the technique. In this work we investigate and
assess these dispersion effects. By using an optimal filtering method we show
how the measured profiles can be corrected for the smoothing effects resulting
from the sample dispersion. Considering the significant advantages the
technique offers, i.e. simultaneuous measurement of $\delta^{18}$O and
$\delta$D, potentially in combination with chemical components that are
traditionally measured on CFA systems, notable reduction on analysis time and
power consumption, we consider it as an alternative to traditional isotope
ratio mass spectrometry with the possibility to be deployed for field ice core
studies. We present data acquired in the field during the 2010 season as part
of the NEEM deep ice core drilling project in North Greenland.
## I introduction
Polar ice core records provide some of the most detailed views of past
environmental changes up to 800 000 yr before present, in large part via proxy
data such as the water isotopic composition and embedded chemical impurities.
One of the most important features of ice cores as climate archives, is their
continuity and the potential for high temporal resolution. Greenland ice cores
are particularly well suited for high resolution paleoclimatic studies,
because relatively high snow accumulation rates allow seasonal changes in
proxy data to be identified more than 50 000 yr in the past (Johnsen et al.,
1992; NGRIP members, 2004).
The isotopic signature of polar precipitation, commonly expressed through the
$\delta$ notation111Isotopic abundances are typically reported as deviations
of a sample’s isotopic ratio relative to that of a reference water (e.g.
VSMOW) expressed in per mil (‰) through the $\delta$ notation:
$\delta^{i}=\left(\displaystyle{\frac{{}^{i}{R}_{\textrm{sample}}}{{}^{i}{R}_{\textrm{{SMOW}}}}}-1\right)\times
1000$ where
${}^{2}{R}=\displaystyle{\frac{{}^{2}\textrm{H}}{{}^{1}\textrm{H}}}$ and
${}^{18}{R}=\displaystyle{\frac{{{}^{18}O}}{{{}^{16}O}}}$ (Epstein, 1953;
Mook, 2000) is related to the temperature gradient between the evaporation and
condensation site (Dansgaard, 1964) and has so far been used as a proxy for
the temperature of the cloud at the time of condensation (Jouzel and Merlivat,
1984; Jouzel et al., 1997; Johnsen et al., 2001). One step further, the
combined signal of $\delta\mathrm{D}$ and $\delta{{}^{18}\mathrm{O}}$ commonly
referred to as the deuterium excess (hereafter $\mathrm{D_{xs}}$ ),
constitutes a useful paleothermometer tool. Via its high correlation with the
temperature of the evaporation source (Johnsen et al., 1989), it has been used
to resolve issues related to changes in the location of the evaporation site
(Cuffey and Vimeux, 2001; Kavanaugh and Cuffey, 2002). A relatively recent
advance in the use of water isotope ratios as a direct proxy of firn
temperatures, has been introduced by Johnsen et al. (2000). Assessment of the
diffusivity of the water isotopologues in the porous medium of the firn column
can yield a temperature history, provided a dating model is available.
The measurement of water stable isotopic composition is typically performed
off-line via discrete sampling with traditional isotope ratio mass
spectrometry (hereafter IRMS). While high precision and accuracy can routinely
be achieved with IRMS systems, water isotope analysis remains an elaborate
process, which is demanding in terms of sample preparation, power consumption,
sample size, consumables, isotope standards and carrier gases. The analysis of
a deep ice core at its full length in high resolution (typically 2.5 to 5 cm
per sample) requires the processing of a vast amount of water samples and can
take years to complete. Additionally, these procedures often come at the
expense of not fully exploiting the temporal resolution available in the ice
core.
Laser spectroscopy in the near and mid infrared region has been demonstrated
as a potential alternative for water isotope analysis, presenting numerous
advantages over IRMS (Kerstel et al., 1999; Kerstel, 2004). A major advantage
of the technique is the ability to directly inject the sampled water vapour in
the optical cavity of the spectrometer where both isotopic ratios
${}^{18}\mathrm{O}/^{16}\mathrm{O}$ and ${}^{2}\mathrm{H}/^{1}\mathrm{H}$ are
measured simultaneuously. In contrast, in the most common IRMS techniques
water is not measured as such, but has to be converted to a different gas
prior to measurement. For $\delta{{}^{18}\mathrm{O}}$ analysis, the
$\mathrm{CO}_{2}$ equilibration method (Epstein, 1953) has been widely used,
whereas $\delta\mathrm{D}$ analysis commonly involves the reduction of water
to hydrogen gas over hot uranium (Bigeleisen et al., 1952; Vaughn et al.,
1998; Huber and Leuenberger, 2003), or chromium (Ghere et al., 1996). However,
the combined use of these two methods rules out simultaneous analysis of both
water isotopologues on a given sample. More recently, in combination with the
use of continuous flow mass spectrometers, conversion of water to CO and
$\text{H}_{2}$ is performed in a pyrolysis furnice (Begley and Scrimgeour,
1997) and allows simultaneous $\delta\mathrm{D}$ and
$\delta{{}^{18}\mathrm{O}}$ measurement, but still on a single discrete
sample. One of the drawbacks of this technique is the interference of
$\mathrm{NO}$ , formed at the ion source by the reaction of $\text{N}_{2}$ and
$\text{O}_{2}$ with the CO signal at m/z $=$ 30 (Accoe, 2008). Nowadays,
commercial IR spectrometers are available with a precision comparable to IRMS
systems (Lis et al., 2008; Brand et al., 2009). These units typically receive
a continuous stream of water vapor and offer ease of use and portability.
The analysis of another set of ice core proxies, that of chemical impurities,
has similarly been an elaborate process, traditionally performed with liquid
chromatography techniques. With the advent of Continuous Flow Analysis
(heareafter CFA) from continuously melted ice core segments, the measurement
of chemical impurities has reached the point of largely exploiting the high
resolution available in the core while it is often performed in the field
(Sigg et al., 1994; Röthlisberger et al., 2000; Kaufmann et al., 2008). The
continuous, on-line nature of the technique has resulted in a considerable
reduction in sample preparation and processing times. Recently, Schüpbach et
al. (2009) demonstrated the measurement of $\mathrm{CH}_{4}$ mixing ratios in
an on-line semi continuous mode with the use of a gas chromatograph combined
with a pulsed discharge and a thermal conductivity detector.
Here, we demonstrate the ability to perform continuous measurements of water
isotope ratios from a stream of water vapor derived from a continuously
melting ice rod by coupling a commercial IR spectrometer to a CFA system via a
passive, low volume flash evaporation module. In the following, we assess the
system’s precision, accuracy, and efficient calibration. We then comment on
issues related to sample dispersion in the sample transfer lines, the
evaporation module and the optical cavity of the spectrometer itself in order
to determine the expected smoothing imposed on the acquired data sets.
Finally, isotopic analysis of ice core samples from the NEEM deep ice core are
presented and compared to measurements performed in discrete mode.
## II Experimental
### II-A Continuous flow analysis
In the system described here, (Fig. 1) an ice rod measuring 3.2 $\times$ 3.2
$\times$ 110 cm (hereafter CFA run) is continuously melted on a copper, gold-
nickel coated melter at a regulated temperature of 20 ∘C. The concentric
arrangement of the melter’s surface facilitates the separation of the sample
that originates from the outer and inner part of the core. Approximately 90 %
of the sample from the inner part is transfered to the analytical system by
means of a peristaltic pump with a flow rate of 16 ml min-1. This
configuration provides an overflow of $\approx$10 % from the inner to the
outer part of the melter and ensures that the water sample that is introduced
into the analytical system is not contaminated.
A stainless steel weight sitting on top of the ice rod enhances the stability
and continuity of the melting process. An optical encoder connected to the
stainless steel weight, records the displacement of the rod. This information
is used to accurately define the depth scale of the produced water isotope
data. Breaks in the ice rod are logged prior to the melting process and
accounted for, during the data analysis procedure.
Gases included in the water stream originating from the air bubbles in the ice
core are extracted in a sealed debubbler, with a volume of $\approx$300
$\mu$l. The melt rate of the present system is approximately 3.2 cm min-1,
thus resulting in an analysis time of $\approx$35 min per CFA run. During the
intervals between CFA runs, mQ222filtered and deionized water with a
resistivity more than $18.2~{}\mathrm{M\Omega\,\,\,cm}$ and a total organic
content less than 10 ppb. water is pumped through the system. A 4-port
injection valve (V1 in Fig. 1) allows the selection between the mQ and sample
water. The mQ water is spiked with isotopically enriched water containing 99.8
atom % deuterium, Cortecnet Inc.) in a mixing ratio of $\approx$1 ppm. In this
way a distinction between sample and mQ water is possible, facilitating the
identification of the beginning and end times of a CFA run.
For further details on the analysis of chemical components or the extraction
of gases for greenhouse gas measurements the reader is refered to Kaufmann et
al. (2008) and Schüpbach et al. (2009).
### II-B The water isotope measurement
We follow the same approach as previously presented in Gkinis et al. (2010) by
coupling a commercially available Cavity Ring Down IR spectrometer (hereafter
WS-CRDS) purchasecd from Picarro Inc. (Picarro L1102-i) (Crosson, 2008). The
spectrometer operates with a gas flow rate of 30 standard ml min-1. In the
optical cavity the pressure is regulated at 47 mbar with two proportional
valves in a feedback loop configuration up- and down-stream of the optical
cavity at a temperature of 80 ∘C. The high signal to noise ratio achieved with
the Cavity Ring Down configuration in combination with fine control of the
environmental parameters of the spectrometer, result in a performance
comperable to modern mass spectrometry systems taylored for water stable
isotope analysis.
A 6-port injection valve (V2 in Fig. 1) selects sample from the CFA line or a
set of local water standards. The isotopic composition of the local water
standards is determined with conventional IRMS and reported with respect to
VSMOW standard. A 6-port selection valve (V3 in Fig. 1) is used for the switch
between different water standards. A peristaltic pump (P3 in Fig. 1) in this
line with variable speeds, allows adjustment of the water vapor concentration
in the spectrometer’s optical cavity, by varying the pump speed. In that way,
the system’s sensitivity to levels of different water concentration can be
investigated and a calibration procedure can be implemented. We use high
purity Perfluoroalkoxy (PFA) tubing for all sample transfer lines.
Injection of water sample into the evaporation oven takes place via a
$\varnothing$ 40 $\mu$m fused silica capillary where immediate and 100 %
evaporation takes place avoiding any fractionation effects. The setpoint of
the evaporation temperature is set to 170 ∘C and is regulated with a PID
controller. The amount of the injected water to the oven can be adjusted by
the pressure gradient maintained between the inlet and waste ports of the T1
tee-split (Fig. 1). The latter depends on the ratio of the inner diameters of
the tubes connected to the two ports as well as the length of the waste line.
The total water sample consumption is $\approx$0.1 ml min-1 maintained by the
peristaltic pump P2 (Fig. 1). For a detailed description of the sample
preparation and evaporation module the reader may reffer to Gkinis et al.
(2010). A smooth and undisturbed sample delivery to the spectrometer at the
level of $\approx$20 000 ppm results in optimum performance of the system.
Fluctuations of the sample flow caused by air bubbles or impurities are likely
to result in a deteriorated performance of the measurement and are
occasionally observed as extreme outliers on both $\delta{{}^{18}\mathrm{O}}$
and $\delta\mathrm{D}$ measurements. The processes that control the occurence
of these events are still not well understood.
## III Results and discussion – from raw data to isotope records
In this study we present data collected in the framework of the NEEM ice core
drilling project. Measurements were carried out in the field during the 2010
field season and span 919.05 m of ice core (depth interval 1281.5–2200.55 m).
Here we exemplify the performance of the system over a section of ice from the
depth interval 1382.152–1398.607. The age of this section spans $\approx$411
yr with a mean age of 10.9 ka b2k 333thousand years before 2000 AD (ka b2k).
The reported age is based on a preliminary time scale constructed by
stratigraphic transfer of the GICC05 time scale (Rasmussen et al., 2006) from
the NGRIP to the NEEM ice core.
In Fig. 2 we present an example of raw data as acquired by the system. This
data set covers 7 CFA runs (7.70 m of ice). A clear baseline of the
isotopically heavier mQ water can be seen in between CFA runs. At $t=1.9\times
10{{}^{4}}$ s one can observe a sudden drop in the signal of the water
concentration due to a scheduled change of the mQ water tank. Adjacent to
this, both $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ signals present
a clear spike, characteristic of the sensitivity of the system to the
stability of the sample flow rates.
### III-A VSMOW – water concentration calibrations
Before any further processing we correct the acquired data for fluctuations of
the water concentration in the optical cavity. To a good approximation the
$\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ signals show a linear
response to differences in water concentrations around 20 000 ppmv (Brand et
al., 2009; Gkinis et al., 2010). A correction is performed as:
$\Delta\delta=\alpha(R_{20}-1){}$ (1)
Here $R_{20}=\frac{[\mathrm{H}_{2}\mathrm{O}]}{20\,000}$, $\alpha_{18}=1.94$ ‰
and $\alpha_{\rm D}=3.77$ ‰ as estimated in Gkinis et al. (2010). The
estimation of these values has been performed several times during the period
July 2009–October 2011. Based on compiled data from 6 calibrations we can
report that these values do not appear to be drifting in the course of the two
years. The values reported here are the ones estimated chronologically closer
to the measurements we present here. These values are most likely instrument
specific and should be used with caution for other analyzers. We typically
operate the system in the area of 17 000–22 000 ppmv in which we observe no
impact of the water concentration level on the precision of the isotopic
signal. The mean and standard deviation of the water concentration signal in
the course of aprroximately 7 h as seen in Fig. 2 is $19\;939\pm 306$ ppmv.
The water concentration correction is applied to the raw data, scaling all the
isotopic values to the level of 20 000 ppmv.
Raw data are expressed in per mil values for both $\delta{{}^{18}\mathrm{O}}$
and $\delta\mathrm{D}$ and ppmv for the water vapour concentration. These
values are based on the slope and intercept values of the instrument’s stored
internal calibration line. Due to apparent instrumental drifts though, the
latter are expected to deviate with time. To overcome this problem we perform
frequent VSMOW calibrations by using 3 local water standards with well known
$\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ values measured by
conventional Isotope Ratio Mass Spectrometry combined with a pyrolysis glassy
carbon reactor (Thermo DeltaV–TC/EA). The water standards are transported and
stored in the field using the necessary precautions to avoid evaporation. We
used amber glass bottles with silicon sealed caps. Two of the water standards
are used to calculate the slope and the intercept of the calibration line and
the third is used for a check of the linearity and the accuracy. The frequency
of the VSMOW calibrations depends on the work flow of the overall CFA system.
For the particular section we present here a VSMOW calibration was performed
the same day.
In Fig. 3 we illustrate this calibration. Based on the measured and real
values of the standards “$-$22” and “$-$40” the measurement of the “NEEM”
standard can be used as a test for accuracy and linearity. In Table I we
present the VSMOW calibrated values of the water standards as estimated with
the IRMS system. The slope and intercept of the calibration lines are [1.002,
$-$0.007] for $\delta{{}^{18}\mathrm{O}}$ and [0.963, $-$8.214] for
$\delta\mathrm{D}$ . Based on these values one can calculate the isotopic
composition of the “NEEM” standard. The results we obtain are $-33$.4 ‰ and
$-256$.99 ‰ for $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$
respectively. This results in a difference of $0.04$ ‰ ($0.$31 ‰) for
$\delta{{}^{18}\mathrm{O}}$ ($\delta\mathrm{D}$ ), giving an indication about
the accuracy obtained by the system.
TABLE I: Isotopic composition of the standards used for the VSMOW calibrations
in ‰.
| $-$22 | NEEM | $-$40
---|---|---|---
$\delta{{}^{18}\mathrm{O}}$ | $-$21.9 | $-$33.44 | $-$39.97
$\delta\mathrm{D}$ | $-$168.4 | $-$257.3 | $-$310
### III-B The depth scale
The melting process is recorded by an optical encoder connected to the top of
the stainless steel weight that lies on top of the ice rod. The data acquired
by the optical encoder allow for a conversion of the measurement time scale to
a depth scale. In order to locate the beginning and end of every run we take
advantage of the isotopic step observed during the transition between mQ
baseline and sample water. A smoothed version of the discrete derivative of
the acquired isotope data for both $\delta{{}^{18}\mathrm{O}}$ and
$\delta\mathrm{D}$ reveals a local minimum (maximum) for the beginning (end)
of the measurement (Fig. 4). The logged depth of the top and the bottom of the
CFA run is assigned to these points. Data that lie in the transition interval
between mQ and sample water are manually removed from the series. Additional
breaks within a CFA run that can possibly be created during the drilling or
processing phase of the ice core, are taken into account at the last stage of
the data analysis. If necessary and depending on their size, the gaps can be
filled by means of some interpolation technique. Here, due to the small size
of the gaps we use a linear interpolation scheme. The use of more advanced
methods is also possible but is out of the scope of this work. The processed
profiles presented in Fig. 4 are reported with a nominal resolution of 5 mm.
The interpolated sections are highlighted with gray bars. Their width
indicates the length of the gaps.
### III-C Noise level – comparison with discrete data
An estimate of the noise level of the measurements can be obtained from the
appropriately normalized power spectral density of the time series. Using the
$\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ data of the section under
consideration, we implement an autoregressive spectral estimation method
developed by Burg (1975) by the use of the algorithm introduced by Andersen
(1974). The order of the autoregressive model is $M$ $=$ 300\. The standard
deviation of the time series will be defined as:
$\sigma^{2}=\int_{-f_{c}}^{f_{c}}|\hat{\eta}(f)|^{2}df{}$ (2)
where the Nyquist frequency is $f_{c}=100$ cycles m-1 and
$|\hat{\eta}(f)|^{2}$ can be obtained by a linear fit on the flat high
frequency part of the spectrum (Fig. 5). By performing this analysis we obtain
$\sigma_{18}=0.055$ ‰ and $\sigma_{\rm D}=0.21$ ‰.
In order to validate the quality of the calibrations as well as the estimated
depth scale we compare the CFA data with measurements performed in a discrete
fashion using the same WS-CRDS spectrometer in combination with a sample
preparation evaporator system (Gupta et al., 2009) and an autosampler. The
discrete samples are cut in a resolution of 5 cm. The sample injection
sequence takes into account apparent memory effects and results are reported
on the VSMOW scale by appropriate calibration using local water standards. The
results are illustrated in Fig. 6 for $\delta{{}^{18}\mathrm{O}}$ and
$\delta\mathrm{D}$ . The comparison of the data sets demonstrates validity of
the followed calibration procedures. The benefits of the technique in terms of
achieved resolution can be seen when one compares the two datasets over
isotopic cycles with relatively small amplitude and higher frequency. Such an
example can be seen at the depth of 1390.5 m where a sequence of 4 cycles is
sampled relatively poorly with the discrete method when compared to the on-
line system. This performance can benefit studies that look into the spectral
properties of the signals by providing better statistics for the obtained
measurements.
### III-D Obtained resolution – diffusive sample mixing
One of the advantages of the combined CFA–CRDS technique for water isotopic
analysis of ice cores lies in the potential for higher resolution measurements
relative to discrete sampling. However, diffusion effects in both the liquid
and the vapor phase are expected to attenuate the obtained resolution.
Attenuation of the initial signal of the precipitation occurs also via a
combination of in situ processes that take place after deposition. The porous
medium of the firn column allows for an exchange of water molecules in the gas
phase along the isotopic gradients of the profile. For the case of polar
sites, this process has been studied extensively (Johnsen, 1977; Whillans and
Grootes, 1985) and can be well described and quantified provided that a good
estimate of the diffusivity coefficient and a strain rate history of the ice
core site are available (Johnsen et al., 2000). The process ceases when the
porous medium is closed-off and the diffusivity of air reaches zero at a
density of $\approx$804 kg m-3. Deeper in the ice, diffusion within the ice
crystals takes place via a process that is considerably slower when compared
with the firn diffusion. At a temperature of $-$30 ∘C the diffusivity
coefficients of these two processes differ by 4 orders of magnitude (Johnsen
et al., 2000).
Assuming an isotopic signal ${\delta}_{\rm pr}$ for the precipitation, the
total effect of the diffusive processes, in-situ and experimental, can be seen
as the convolution of ${\delta}_{\rm pr}(z)$ with a smoothing filter
${\mathcal{G}}_{\rm tot}$.
$\delta_{\rm m}(z)=\int_{-\infty}^{\infty}{\delta}_{\rm
pr}(\tau){\mathcal{G}}_{\rm tot}(z-\tau)d\tau=[\delta_{\rm
pr}\ast{\mathcal{G}}_{\rm tot}](z){}$ (3)
where $\delta_{\rm m}(z)$ is the measured signal and ($\ast$) denotes the
convolution operation. Since instrumental and in-situ firn-ice diffusion are
statistically independent, the variance of the total smoothing filter is the
sum of the variances of the in-situ and experimental smoothing filters
(hereafter $\mathcal{G}_{\rm firn}$, $\sigma_{\rm firn}$, $\mathcal{G}_{\rm
cfa}$, $\sigma_{\rm cfa}$).
$\sigma_{\rm tot}^{2}=\sigma_{\rm firn}^{2}+\sigma_{\rm cfa}^{2}{}$ (4)
It can be seen that any attempt to study firn and ice diffusion by means of
ice core data obtained with an on-line method similar to the one we present
here, requires a good assesment of the diffusive properties of the
experimental system. The latter is possible if one is able to estimate the
variance of the smoothing filter $\mathcal{G}_{\rm cfa}$ expressed by the
variance $\sigma_{\rm cfa}^{2}$ (hereafter diffusion length).
One way to approach this problem is to measure the response of the system to a
step function. Ideally, in the case of zero diffusion, a switch between two
isotopic levels would be described by a scaled and shifted version of the the
Heaviside unit step function as:
$\delta_{\rm H}(z)=\left\\{\begin{array}[]{ll}C_{2}&\qquad z<0\\\
C_{1}H(z)+C_{2}&\qquad z\geq 0\end{array}\right.{}$ (5)
where the isotopic shift takes place at $z=0$, $H(z)$ is the Heaviside unit
step function and $C_{1}$ and $C_{2}$ refer to the amplitude and base line
level of the isotopic step. Convolution of the signal of Eq. (5) with
$\mathcal{G}_{\rm cfa}$ and subsequent calculation of the derivative yields,
$\frac{d\delta_{\rm m}}{dz}=\frac{d\delta_{\rm H}}{dz}\ast\mathcal{G}_{\rm
cfa}=C_{1}\frac{dH}{dt}\ast\mathcal{G}_{\rm cfa}=C_{1}\delta_{\rm
Dirac}\ast\mathcal{G}_{\rm cfa}{}$ (6)
Thus the derivative of the measured signal, properly normalized, equals the
impulse respone of the system. Applying the Fourier transform, denoted by the
overhead hat symbol, in Eq. (6), and by using the convolution theorem, we
deduce the transfer function $\hat{\mathcal{G}}_{\rm cfa}$ of the system:
$\widehat{\frac{d\delta_{\rm m}}{dz}}=C_{1}\hat{\delta}_{\rm
Dirac}\cdot\hat{\mathcal{G}}_{\rm cfa}=C_{1}\cdot\hat{\mathcal{G}}_{\rm
cfa}{}$ (7)
In the case of the system presented here, an isotopic transition can be
observed when the main CFA valve (V1 in Fig. 1) switches between mQ water and
sample at the beginning and the end of each CFA run as shown in Fig. 4. By
using these transitions we are able to construct isotopic steps and estimate
the impulse response of the system. Such an isotopic step is illustrated in
Fig. 7a. We fit the data of Fig. 7a with a scaled version of the cumulative
distribution function of a normal distribution described as
$\delta_{\rm model}(z)=\frac{C_{1}^{\prime}}{2}\left[1+{\rm
erf}\left(\frac{z-z_{0}}{\sigma_{\rm
step}\sqrt{2}}\right)\right]+C_{2}^{\prime}{}$ (8)
The values of $C_{1}^{\prime}$, $C_{2}^{\prime}$, $z_{0}$ and $\sigma_{\rm
step}$ are estimated by means of a least square optimization and used
accordingly to normalize the length scale and the isotopic values of the step.
A nominal melt rate of 3.2 cm min-1 is used for all the calculations presented
here. We focus our analysis on the $\delta\mathrm{D}$ signal. The same
approach can be followed for $\delta{{}^{18}\mathrm{O}}$ . In Fig. 7b we
present the calculated impulse response of the system. The latter can be well
approximated by a Gaussian type filter described as:
$\mathcal{G}_{\rm cfa}(z)=\frac{1}{\sigma_{\rm
cfa}\sqrt{2\pi}}{e}^{-\frac{z^{2}}{2\sigma_{\rm cfa}^{2}}}{}$ (9)
The diffusion length term $\sigma_{\rm cfa}$ is equal to $13.4\pm 0.17$ mm
($1\sigma$) as calculated with the least squares optimization. The transfer
function for this filter will be given by its Fourier transform, which is
itself a Gaussian and is equal to (Abramowitz and Stegun, 1964):
$\displaystyle\mathfrak{F}[\mathcal{G}_{\rm cfa}(z)]=\hat{\mathcal{G}}_{\rm
cfa}$ $\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{1}{\sigma_{\rm
cfa}\sqrt{2\pi}}{e}^{-\frac{z^{2}}{2\sigma_{\rm
cfa}^{2}}}{e}^{-ikz}dk={e}^{\frac{-k^{2}\sigma_{\rm cfa}^{2}}{2}}$ (10)
where $k=\frac{2\pi}{\lambda}$ and $\lambda$ is the wavelength444Here the term
wavelength refers to the isotopic signal in the ice and should not be confused
with the wavelength of the light emitted by the laser diode of the
spectrometer. of a harmonic of the isotopic signal. Harmonics with an initial
amplitude $A_{0}$ and wavenumber $k$ will be attenuated to a final amplitude
equal to:
$A=A_{0}{e}^{\frac{-k^{2}\sigma_{\rm cfa}^{2}}{2}}{}$ (11)
An estimate of the transfer function based on the data and the cumulative
distribution model is presented in Fig. 8 (blue and pink curve, respectively).
As seen in this plot, cycles with wavelengths longer than 25 cm experience
negligible attenuation, whereas cycles with a wavelength of 7 cm are
attenuated by $\approx 50$ %.
The step response approach has been followed in the past for on-line chemistry
data. In some studies such as Sigg et al. (1994) and Rasmussen et al. (2005),
the resolution of the experimental system was assessed via the estimation of
the transfer function. In other studies (Röthlisberger et al., 2000; Kaufmann
et al., 2008), the characteristic time in which a step reaches a certain level
(typically $1/e$) with respect to its final value, is used as a measure of the
obtained resolution of the system. A common weakness of this approach as
applied in the current, as well as previous studies, is that it is based on
the analysis of a step that is introduced in the analytical system by
switching a valve that is typically situated downstream of the melting and the
debubbling system. Consequently, the impact of these last two elements on the
smoothing of the obtained signals is neglected. In this study, this is the
valve V1 in Fig. 1.
To overcome this problem we will present here an alternative way, based on the
comparison of the spectral properties of the on-line CFA data and the off-line
discrete data in 5 cm sampling resolution, presented in Sect. 3.2. In this
approach the diffusion length of the total smothing filter for the off-line
discrete analysis will be:
$\sigma_{\rm off}^{2}=\sigma_{\rm firn}^{2}+\sigma_{\rm 5\,cm}^{2}{}$ (12)
where $\sigma_{\rm 5\,cm}^{2}$ is the diffusion length of the smoothing
imposed by the sample cutting scheme at a 5 cm resolution. If one averages the
on-line CFA data at a 5 cm resolution by means of a running mean filter, the
diffusion length of the total smoothing filter for the on-line CFA
measurements averaged on a 5 cm resolution will be:
$\sigma_{\rm on}^{2}=\sigma_{\rm firn}^{2}+\sigma_{\rm 5\,cm}^{2}+\sigma_{\rm
cfa}^{2}{}$ (13)
From Eqs. (12) and (13) we get:
$\sigma_{\rm cfa}^{2}=\sigma_{\rm on}^{2}-\sigma_{\rm off}^{2}{}$ (14)
As a result, the term $\sigma_{\rm on}^{2}-\sigma_{\rm off}^{2}$ is directly
related to the diffusion length of the smoothing filter of the whole CFA-water
isotope system including the melting and debubbling sections. Based on Eq.
(11), the power spectral density of the signals will be:
$P=P_{0}{e}^{-k^{2}\sigma^{2}}{}$ (15)
where $\sigma^{2}$ refers in this case to $\sigma_{\rm on}^{2}$ or
$\sigma_{\rm off}^{2}$. Combining the power spectral densities of the on-line
and off-line time series we finally get:
$\ln\left(\frac{P_{\rm off}}{P_{\rm on}}\right)=\ln\left(\frac{P_{\rm
0off}}{P_{\rm 0on}}\right)+\sigma_{\rm cfa}^{2}k^{2}{}$ (16)
Hence, the logarithm of the ratio $P_{\rm off}/P_{\rm on}$ is linearly related
to $k^{2}$ with a slope equal to $\sigma_{\rm cfa}^{2}$. In Fig. 9 we perform
this analysis for $\delta\mathrm{D}$ and by applying a linear fit we calculate
the $\sigma_{\rm cfa}[D]$ to be equal to $16.4\pm 2.4$ mm. In a similar manner
$\sigma_{\rm cfa}[O18]$ is found to be equal to $16.8\pm 2.3$ mm.
The higher value calculated with the spectral method points to the additional
diffusion of the sample at the melter and debubbler system that could not be
considered in the analysis based on the step response. The impulse response of
the system based on the updated value of $\sigma_{\rm cfa}^{2}$ is presented
in Fig. 6.
### III-E Optimal filtering
In the ideal case of a noise-free measured signal ${\delta^{\prime}_{\rm
m}}(z)$ and provided that the transfer function $\hat{\mathcal{G}}_{\rm cfa}$
is known, one can reconstruct the initial isotopic signal $\delta_{\rm i}(z)$
from Eq. (3) as:
$\delta_{\rm
i}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\hat{\delta}^{\prime}_{\rm
m}(k)}{\hat{\mathcal{G}}(k)}e^{ikz}dk{}$ (17)
where the integral operation denotes the inverse Fourier transform and
$k=\frac{2\pi}{\lambda}$ with $\lambda$ being the wavelength of the isotopic
signals. In the presence of measurement noise $\eta(z)$, this approach will
fail due to excess amplification of the high frequency noise channels in the
spectrum of the signal.
Hereby we use the Wiener approach in deconvoluting the acquired isotopic
signals for the diffusion that takes place during the measurement. Considering
a measured isotopic signal
$\delta_{\rm m}(z)={\delta^{\prime}_{\rm m}}(z)+\eta(z){}$ (18)
an optimal filter $\varphi(z)$ can be constructed that when used at the
deconvolution step, it results in an estimate of the initial isotopic signal
described as:
$\tilde{\delta}_{i}(z)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\hat{\delta}_{\rm
m}(k)}{\hat{\mathcal{G}}(k)}\hat{\varphi}(k)e^{ikz}dk{}$ (19)
Assuming that ${\delta^{\prime}_{\rm m}}(z)$ and $\eta(z)$ are uncorrelated
signals, the optimal filter is given by:
$\hat{\varphi}(k)=\frac{|\hat{\delta}_{\rm
m}^{\prime}(k)|^{2}}{|\hat{\delta}_{\rm
m}^{\prime}(k)|^{2}+|\hat{\eta}(k)|^{2}}{}$ (20)
(Wiener, 1949); where $|\hat{\delta}_{\rm m}^{\prime}(k)|^{2}$ and
$|\hat{\eta}(k)|^{2}$ are the power spectral densities of the signals
$\delta_{\rm m}^{\prime}(z)$ and $\eta(z)$.
In the same fashion as in the previous section we assume that the spectrum of
the noise free measured signal $|\hat{\delta}_{\rm m}^{\prime}(k)|^{2}$, is
described by Eq. (15) where $\sigma^{2}=\sigma_{\rm tot}^{2}$. Regarding the
noise, we assume red noise described by an AR1 process. The spectrum of the
noise signal will then be described by (Kay and Marple, 1981):
$|\hat{\eta}(k)|^{2}=\frac{\sigma_{\eta}^{2}\Delta
z}{\left|1+a_{1}\exp{\left(-ik\Delta z\right)}\right|^{2}}{}$ (21)
where $\sigma_{\eta}^{2}$ is the variance of the noise and $a_{1}$ is the
coefficient of the AR1 process. We vary the parameters $\sigma_{\rm tot}^{2}$,
$P_{0}$, $\sigma_{\eta}^{2}$ and $a_{1}$ so that the sum $|\hat{\delta}_{\rm
m}(k)|^{2}=|\hat{\delta}_{\rm m}^{\prime}(k)|^{2}+|\hat{\eta}(k)|^{2}$ fits
the spectrum of the measured signal. The set of parameters that results in the
optimum fit is used to calculate the optimal filter.
The constructed filters together with the transfer functions that were
calculated based on the two different techniques outlined in Sect. III-D are
illustrated in Fig. 8. One can observe how the restoration filters work by
amplifying cycles with wavelengths as low as 7 mm. Beyond that point, the
shape of the optimal filter attenuates cycles with higher frequency, which lie
in the area of noise. An example of deconvoluted $\delta\mathrm{D}$ data
section is given in Fig. 10. It can be seen that the effect of the optimal
filtering results in both the amplification of the signals that are damped due
to the instrumental diffusion, as well as in the filtering of the measurement
noise.
### III-F Information on deuterium excess
Combining $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ gives the
deuterium excess as $\mathrm{D_{xs}}$ $=$ $\delta\mathrm{D}$ $-$
8$\delta{{}^{18}\mathrm{O}}$ (Craig et al., 1963; Mook, 2000). The noise level
of the $\mathrm{D_{xs}}$ signal can be calculated by the estimated noise
levels of $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ as:
${\sigma}_{\mathrm{D_{xs}}}=\sqrt{\sigma_{\rm
D}^{2}+64\cdot{\sigma}_{18}^{2}}=0.48\,\,\,\mbox{\permil}$ (22)
As seen in Fig. 11, the $\mathrm{D_{xs}}$ signal presents a low signal to
noise ratio. In this case, the technique of optimal filtering can effectively
attenuate unwanted high frequency noise components, thus reveiling a “clean”
$\mathrm{D_{xs}}$ signal.
The latter offers the possibility for the study of abrubt transitions as they
have previously been investigated in $\delta{{}^{18}\mathrm{O}}$ ,
$\delta\mathrm{D}$ and $\mathrm{D_{xs}}$ time series from discrete high
resolution samples (Steffensen et al., 2008). The on-line fashion in which
these measurements are performed has the potential to yield not only higher
temporal resolution but also better statistics for those climatic transitions.
## IV Conclusions
[Summary and conclusions]
We have succesfully demonstrated the possibility for on-line water isotopic
analysis on a continuously melted ice core sample. We used an infrared laser
spectrometer in a cavity ring down configuration in combination with a
continuous flow melter system. A custom made continuous stream flash
evaporator served as the sample preparation unit, interfacing the laser
spectrometer to the melter system.
Local water standards have been used in order to calibrate the measurements to
the VSMOW scale. Additionally, dependencies related to the sample size in the
optical cavity have been accounted for. The melting procedure is recorded by
an optical encoder that provides the necessary information for assigning a
depth scale to the isotope measurements. We verified the validity of the
applied calibrations and the calculated depth scale by comparing the CFA
measurements with measurements performed on discrete samples in 5 cm
resolution.
By means of spectral methods we provide an estimate of the noise level of the
measurements. The combined uncertainty of the measurement is estimated at
$\approx$0.06, 0.2, and 0.5 ‰ for $\delta{{}^{18}\mathrm{O}}$ ,
$\delta\mathrm{D}$ and $\mathrm{D_{xs}}$ , respectively. This performance is
comparable to, or better than the performance typically achieved with
conventional IRMS systems in a discrete mode.
Based on the isotopic step at the beginning of each CFA run, the impulse
response, as well as the transfer function of the system can be estimated. We
show how this method does not take into account the whole CFA system, thus
underestimating the sample diffusion that takes place from the melter until
the optical cavity of the spectrometer. We proposed a different method that
considers the power spectrum of the CFA data in combination with the spectrum
of a data set over the same depth interval measured in a discrete off-line
fashion. The use of the optimal filtering deconvolution technique, provides a
way to deconvolute the measured isotopic profiles for apparent sample
dispersion effects.
The combination of infrared spectroscopy on gaseuous samples with continuous
flow melter systems provides new possibilities for ice core science.The non
destructive, continuous, and on-line technique offers the possibility for
analysis of multiple species on the same sample in high resolution and
precision and can potentially be performed in the field.
## Acknowledgements
We would like to thank Dorthe Dahl Jensen for supporting our research.
Numerous drillers, core processors and general field assistants have
contributed to the NEEM ice core drilling project with weeks of intensive
field work. Withought this collective effort, the measurements we present here
would not be possible. Bruce Vaughn and James White have contributed to this
project with valuable comments and ideas. This project was partly funded by
the Marie Curie Research Training Network for Ice Sheet and Climate Evolution
(MRTN-CT-2006-036127).
Edited by: P. Werle
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Figure 1: Block diagram of the CFA-CRDS system. Figure 2: Raw signals spanning
7 CFA runs on 29 May 2010. Figure 3: VSMOW calibrations with three water
standards “$-$22”, “$-$40” and “NEEM”; Left – $\delta\mathrm{D}$ , right –
$\delta{{}^{18}\mathrm{O}}$ . Figure 4: The beginning and end of each CFA run
is determined by the extrema of the 1st derivative of the isotopic signal,
presented on the top graph. With gray bars we indicate the position and width
of sections with data that are missing due to breaks in the ice, or removed in
order to account for the transition from mQ water to sample and vice cersa.
Figure 5: $\delta{{}^{18}\mathrm{O}}$ and $\delta\mathrm{D}$ power spectral
density based on the data of the interval 1382.152–1398.607 m. Figure 6:
Comparison CRDS-CFA with 5 cm discrete samples for $\delta\mathrm{D}$ (top)
and $\delta{{}^{18}\mathrm{O}}$ (bottom). Bars indicate the position and width
of sections with missing/removed data. Figure 7: (a) Isotopic
$\delta\mathrm{D}$ step. The length scale is normalized so normalized length
$=$ 0 when the normalized $\delta\mathrm{D}$ value equals 0.5 ‰. (b) Impulse
Response of the system for $\delta\mathrm{D}$ based on the step response (red)
and the spectral analysis (blue) with $\sigma_{\rm cfa}=13.4$ and 16.4 mm,
respectively. Figure 8: (a) Power spectral density of $\delta\mathrm{D}$ . (b)
Transfer function calculated based on the step response with $\sigma_{\rm
cfa}=13.4$ mm (pink – squares) and the comparison between discrete and CFA
analysis with $\sigma_{\rm cfa}=16.4$ mm (green – triangles). Restoration
filters built considering the two different transfer functions are illustrated
with orange circles ($\sigma_{\rm cfa}=16.4$ mm) and black diamonds
($\sigma_{\rm cfa}=13.4$ mm). Figure 9: Calculation of the diffusion length
for the transfer function of the CFA system. The dashed and solid lines
represent, respectively the power spectral density of the offline discrete
data and the CFA data averaged on a 5 cm resolution. Figure 10:
$\delta\mathrm{D}$ signal before and after optimal filtering. Figure 11:
$\delta{{}^{18}\mathrm{O}}$ , $\delta\mathrm{D}$ and $\mathrm{D_{xs}}$ signals
after the optimal filtering. For the $\mathrm{D_{xs}}$ we present the signal
before (light green) and after (black) the filtering. Gray bars indicate the
position and width of sections with missing/removed data.
|
arxiv-papers
| 2014-04-23T09:30:41 |
2024-09-04T02:50:01.699703
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. Gkinis, T. J. Popp, T. Blunier, M. Bigler, S. Sch\\\"upbach, E.\n Kettner, S. J. Johnsen",
"submitter": "Vasileios Gkinis",
"url": "https://arxiv.org/abs/1404.5760"
}
|
1404.5805
|
# A note on Liouville type theorem of elliptic inequality $\Delta
u+u^{\sigma}\ls 0$ on Riemannian manifolds
Hui-Chun Zhang Department of Mathematics
Sun Yat-sen University
Guangzhou 510275
E-mail address: [email protected]
###### Abstract.
Let $\sigma>1$ and let $M$ be a complete Riemannian manifold. In a very recent
work [10], Grigor′yan and Sun proved that a Liouville type theorem for
nonnegative solutions of elliptic inequality
$(*)\quad\qquad\qquad\qquad\Delta u(x)+u^{\sigma}(x)\ls 0,\qquad x\in
M.\quad\qquad\qquad\qquad$
via a pointwise condition of volume growth of geodesic balls. In this note, we
improve their result to that an _integral condition_ on volume growth implies
the same uniqueness of ($*$). It is inspired by the well-known Varopoulos-
Grigor′yan’s criterion for parabolicity of $M$.
## 1\. Introduction
Let $\sigma>1$ and let $M$ be a complete noncompact Riemannian manifold
without boundary. Consider the semilinear elliptic inequality
(1.1) $\Delta u(x)+u^{\sigma}(x)\ls 0,\quad x\in M,$
where $\Delta$ is the Laplace-Beltrami opertor on $M$. A function $u\in
W^{1,2}_{\rm loc}(M)$ is called a _weak solution_ of the inequality (1.1) if
$-\int_{M}\left<{\nabla u},{\nabla\psi}\right>d\mu+\int_{M}u^{\sigma}\psi
d\mu\ls 0$
holds for any nonnegative function $\psi\in W^{1,2}(M)$ with compact support.
In Euclidean setting, i.e. $M=\mathbb{R}^{n}$, it has a long history to study
the uniqueness of nonnegative solutions for (1.1) (or more general elliptic
inequations and equalities). There are many beautiful results have been
obtained in this subject. We refer the readers to, for instance, [1, 2, 3, 4,
11, 12, 13] and references therein for them. Many of these results are based
on comparison principle and careful choices of test functions for (1.1). To
use this method on a manifold $M$, one have to estimate the second order
derivative of distance functions, which needs some assumptions on curvature of
$M$.
Surprisingly, in recent works Grigor′yan-Kondratiev [9] and Grigor′yan-Sun
[10] proved a _curvature-free_ Liouville type theorem for nonnegative weak
solution of (1.1) in terms of volume growth of geodesic balls in $M$ as
follows.
###### Theorem 1.1 (Grigor′yan-Sun [10]).
Let $M$ be a complete Riemannian manifold without boundary. Fix a point
$x_{0}\in M$ and set $V(r):=\mu\big{(}B(x_{0},r)\big{)}$ the volume of
geodesic ball of radius $r$ centered at $x_{0}$.
Assume that, for some $C>0$, the inequality
(1.2) $V(r)\ls Cr^{\frac{2\sigma}{\sigma-1}}(\ln r)^{\frac{1}{\sigma-1}}$
holds for all large enough $r$. Then any nonnegative weak solution of (1.1) is
identically equal to 0.
They also showed that the exponents $\frac{2\sigma}{\sigma-1}$ and
$\frac{1}{\sigma-1}$ are sharp.
On the other hand, let us recall that a manifold $M$ is said to be _parabolic_
if a Liouville type theorem holds for nonnegative solution of inequality
$\Delta u(x)\ls 0,\qquad x\in M,$
i.e., any nonnegative weak solution of $\Delta u\ls 0$ on $M$ must be
constant. Cheng and Yau [5] proved that $V(r)\ls Cr^{2}$, for some $C>0$, is a
sufficient condition for parabolicity of $M$. Nowdays, a well-known sharp
sufficient condition for parabolicity is the following _integral_ condition,
which was proved independly by Varopoulos [14] and Grigor′yan [7, 8]:
$\int^{\infty}\frac{r}{V(r)}dr=\infty.$
Inspired by Varopoulos-Grigor′yan’s condition for the parabolicity of $M$, we
ask a natural question: what is a sufficient condition for Liouville type
theorem of inequlity (1.1) via an _integral_ estimate of $V(r)$? Of course,
such a condition should cover the above pointwise condition (1.2).
In this remark, we solve this question. Our main result states as follows:
###### Theorem 1.2.
Let $M$ be a complete Riemannian manifold without boundary. Assume that
(1.3) $\liminf_{t\to
0^{+}}t^{\frac{\sigma}{\sigma-1}}\int_{1}^{\infty}\frac{V(r)}{r^{\frac{3\sigma-1}{\sigma-1}+t}}dr<\infty.$
Then any nonnegative weak solution of (1.1) is identically equal to 0.
###### Remark 1.3.
Condition (1.2) in Theorem 1.1 implies the condition (1.3). In fact,
$\eqref{eq1.2}\Longrightarrow
t^{\frac{\sigma}{\sigma-1}}\int_{1}^{\infty}\frac{V(r)dr}{r^{\frac{3\sigma-1}{\sigma-1}+t}}\ls
t^{\frac{\sigma}{\sigma-1}}\int_{1}^{\infty}\frac{(\ln
r)^{\frac{1}{\sigma-1}}dr}{r^{1+t}}=\Gamma(\frac{\sigma}{\sigma-1}),$
where $\Gamma(\cdot)$ is Gamma function.
## 2\. Proof of Theorem 1.2
###### Proof of Theorem 1.2.
Let $u\in W^{1,2}_{\rm loc}(M)$ be a nontrivial nonnegative solution to the
inequality (1.1).
The proof of Theorem 1.1 in [10] contains two main parts. Firstly, the authors
derived a useful priori estimate in terms of a test function and positive
parameters (which will be recalled in Lemma 2.1 below). Secondly, they chose
specific test functions to conclude $\int_{M}u^{\sigma}d\mu=0.$ Our proof of
Theorem 1.2 is basically along the same line in [10]. The different from
Grigor′yan-Sin’s proof will appear in the second part. We will choose a
variation of their test functions to conclude $\int_{M}u^{\sigma}d\mu=0.$
Firstly, let us recall the useful priori estimate given in [10]. We summarize
it as the following lemma:
###### Lemma 2.1 (Grigor′yan-Sun, [10]).
Set $s=8\sigma/(\sigma-1)$. Then there exists a constant $C_{0}>0$ such that
the following property holds:
For any
$t\in\big{(}0,\min\\{1,\frac{\sigma-1}{2}\\}\big{)},$
any nonempty compact set $K\subset M$, and any Lipschitz function $\phi$ on
$M$ with conpact support such that $0\ls\phi\ls 1$ on $M$ and $\phi\equiv 1$
in a neighborhood of $K$, we have
(2.1) $\int_{M}\phi^{s}u^{\sigma}d\mu\ls C_{0}\Big{(}\int_{M\backslash
K}\phi^{s}u^{\sigma}d\mu\Big{)}^{\frac{t+1}{2\sigma}}\cdot J(t,\phi)$
and
(2.2) $\Big{(}\int_{M}\phi^{s}u^{\sigma}d\mu\Big{)}^{1-\frac{t+1}{2\sigma}}\ls
C_{0}\cdot J(t,\phi),$
where
$J(t,\phi):=t^{-\frac{1}{2}-\frac{\sigma}{2(\sigma-1)}}\Big{(}\int_{M}|\nabla\phi|^{2\frac{\sigma-t}{\sigma-1}}d\mu\Big{)}^{\frac{1}{2}}\cdot\Big{(}\int_{M}|\nabla\phi|^{\frac{2\sigma}{\sigma-t-1}}d\mu\Big{)}^{\frac{\sigma-t-1}{2\sigma}}.$
###### Proof.
Inequality (2.1) is Eq.(2.10) in [10], and inequality (2.2) is Eq.(2.11) in
[10]. ∎
In the following, we will consider a family of specific test functions
$\phi_{n}$, which are modifications from original structures in [10].
Fix any $t\in\big{(}0,\min\\{1,\frac{\sigma-1}{2}\\}\big{)}$. We set
$R=R(t):=\exp(1/t)$. We consider the function
$\phi_{t}(x)=\begin{cases}1,&r(x)<R,\\\
\Big{(}\frac{r(x)}{R}\Big{)}^{-t},&r(x)\geqslant R,\end{cases}$
and a family of functions, for any $n=1,2,3,\cdots,$
$\xi_{t,n}(x)=\begin{cases}1,&0\ls r(x)\ls 2^{n}R,\\\
2-\frac{r(x)}{2^{n}R},&2^{n}R\ls r(x)\ls 2^{n+1}R,\\\ 0,&r(x)\geqslant
2^{n+1}R.\end{cases}$
Consider the functions
(2.3) $\phi_{t,n}(x):=\phi_{t}(x)\cdot\xi_{t,n}(x).$
Then, for each $n=1,2,\cdots$, function $\phi_{t,n}(x)$ is Lipschitz
continuous on $M$ and has compact support, and $\phi_{t,n}\equiv 1$ on
$B_{R(t)}:=B(x_{0},R(t))$.
Claim: _There exists a constant $C_{1}>0$ such that, for any
$t\in\\!\big{(}0,\min\\{1,\frac{\sigma-1}{2}\\}\big{)}$ with_
$A(t):=\int_{1}^{\infty}\frac{V(r)}{r^{\frac{3\sigma-1}{\sigma-1}+t}}dr<\infty,$
_we have_
(2.4) $\limsup_{n\to\infty}[J(t,\phi_{t,n})]^{\frac{2\sigma}{2\sigma-t-1}}\ls
C_{1}\cdot t^{\frac{\sigma}{\sigma-1}}\cdot A(t).$
###### Proof of Claim:.
In the proof, the parameter $t$ is fixed. To simplify the notations, we denote
by
$\phi:=\phi_{t},\quad\xi_{n}:=\xi_{t,n}\quad{\rm
and}\quad\phi_{n}:=\phi_{t,n}.$
Notice that
$\nabla\phi_{n}=\xi_{n}\cdot\nabla\phi+\phi\cdot\nabla\xi_{n}.$
We have
$|\nabla\phi_{n}|\ls\xi_{n}\cdot|\nabla\phi|+\phi\cdot|\nabla\xi_{n}|;$
and, by the inequality $(A+B)^{a}\ls 2^{a-1}(A^{a}+B^{a})$ for all $A,B>0$ and
$a\geqslant 1$,
$|\nabla\phi_{n}|^{a}\ls
2^{\frac{4\sigma}{\sigma-1}-1}\big{[}\xi_{n}^{a}\cdot|\nabla\phi|^{a}+\phi^{a}\cdot|\nabla\xi_{n}|^{a}\big{]}$
for any $a\in[1,\frac{4\sigma}{\sigma-1}]$. In the following, we denote by
$\sigma_{0}:=\frac{4\sigma}{\sigma-1}.$
Similar as in [10], we need to estimate the integral
$\int_{M}|\nabla\phi_{n}|^{a}d\mu.$ For any $a\in[1,\sigma_{0}]$, we have
(2.5) $\begin{split}\int_{M}\\!|\nabla\phi_{n}|^{a}d\mu&\ls
2^{\sigma_{0}-1}\cdot\Big{(}\int_{M\backslash
B_{R}}\\!\\!|\nabla\phi|^{a}d\mu+\\!\int_{B_{2^{n+1}R}\backslash
B_{2^{n}R}}\\!\\!\phi^{a}|\nabla\xi_{n}|^{a}d\mu\Big{)}\\\
&:=2^{\sigma_{0}-1}\cdot\big{(}I(a)+I\\!I(a,n)\big{)},\end{split}$
where $B_{R}:=B(x_{0},R)$, and we have used that $\nabla\phi=0$ in $B_{R}$ and
that $|\nabla\xi_{n}|$ supported in $\overline{B_{2^{n+1}R}}\backslash
B_{2^{n}R}$.
Before we estimate the above integrals $I(a)$ and $I\\!I(a,n)$, we need the
following simple (but important) observation:
_If the parameter $a\in[1,\sigma_{0}]$ satisfies_
(2.6) $a(t+1)\geqslant t+\frac{2\sigma}{\sigma-1}.$
_Then we have_
(2.7) $\sum_{n=1}^{\infty}\frac{V(2^{n}R)}{\big{(}2^{n-1}R\big{)}^{a(t+1)}}\ls
2\cdot 16^{\sigma_{0}}\cdot A(t):=C_{2}\cdot A(t).$
_In particular, it implies that_
(2.8) $\lim_{n\to\infty}\frac{V(2^{n}R)}{\big{(}2^{n-1}R\big{)}^{a(t+1)}}=0.$
Indeed, we calculate directly to conclude
(2.9)
$\begin{split}\sum_{n=1}^{\infty}&\frac{V(2^{n}R)}{\big{(}2^{n-1}R\big{)}^{a(t+1)}}\\\
&\quad=4^{a(t+1)}\cdot
2\cdot\sum_{n=1}^{\infty}\frac{V(2^{n}R)}{\big{(}2^{n+1}R\big{)}^{a(t+1)}}\cdot\frac{2^{n+1}R-2^{n}R}{2^{n+1}R}\\\
&\quad\ls 4^{a(t+1)}\cdot
2\cdot\sum_{n=1}^{\infty}\int_{2^{n}R}^{2^{n+1}R}\frac{V(r)dr}{r^{a(t+1)+1}}\\\
&\quad\ls 2\cdot
16^{\sigma_{0}}\cdot\int_{1}^{\infty}\frac{V(r)dr}{r^{a(t+1)+1}},\end{split}$
we we have used that $t<1$, $a\ls\sigma_{0}$ and that $R=\exp(1/t)>1$.
Combining with (2.6) and (2.9), we can obtain
$\sum_{n=1}^{\infty}\frac{V(2^{n}R)}{\big{(}2^{n-1}R\big{)}^{a(t+1)}}\ls
2\cdot
16^{\sigma_{0}}\int_{1}^{\infty}\frac{V(r)dr}{r^{t+\frac{2\sigma}{\sigma-1}+1}}=2\cdot
16^{\sigma_{0}}\cdot A(t).$
Ths is the desired estimate (2.7).
Now let us estimate $I(a)$. Assume that the parameter $a$ satisfies (2.6), we
have
(2.10) $\begin{split}I(a)&=\int_{M\backslash
B_{R}}\\!\\!\\!|\nabla\phi|^{a}d\mu\ls\int_{M\backslash
B_{R}}\Big{[}\frac{t}{R}\cdot\Big{(}\frac{r}{R}\Big{)}^{-t-1}\Big{]}^{a}d\mu\\\
&=e^{a}\cdot t^{a}\int_{M\backslash
B_{R}}\\!\frac{1}{r^{a(t+1)}}d\mu\qquad\quad({\rm since}\ R^{t}=e)\\\
&=e^{a}\cdot t^{a}\cdot\sum_{n=1}^{\infty}\int_{B_{2^{n}R}\backslash
B_{2^{n-1}R}}\\!\frac{1}{r^{a(t+1)}}d\mu\\\ &\ls e^{a}\cdot
t^{a}\cdot\sum_{n=1}^{\infty}\frac{V(2^{n}R)}{\big{(}2^{n-1}R\big{)}^{a(t+1)}}\\\
&\ls e^{\sigma_{0}}\cdot C_{2}\cdot t^{a}A(t)\qquad\big{(}{\rm by}\ \
a\ls\sigma_{0}\ {\rm and}\ \ \eqref{eq2.7}\big{)}.\end{split}$
Let us estimate $I\\!I(a,n)$. Assume that the parameter $a$ satisfies (2.6),
we have
(2.11) $\begin{split}I\\!I(a,n)&=\int_{B_{2^{n+1}R}\backslash
B_{2^{n}R}}\\!\\!\phi^{a}|\nabla\xi_{n}|^{a}d\mu\\\
&\ls\Big{(}\frac{2^{n}R}{R})^{-at}\Big{(}\frac{1}{2^{n}R}\Big{)}^{a}\cdot
V(2^{n+1}R)\\\
&=R^{at}\cdot\frac{V(2^{n+1}R)}{(2^{n}R)^{a(t+1))}}\overset{R^{t}=e}{=}e^{a}\cdot\frac{V(2^{n+1}R)}{(2^{n}R)^{a(t+1))}}.\end{split}$
Combining with (2.8), (2.11) and that $a\ls\sigma_{0}$, we have
(2.12) $\lim_{n\to\infty}I\\!I(a,n)=0.$
Therefore, according to (2.5),(2.10) and (2.12), we obtain, for any
$a\in[1,\sigma_{0}]$ satisfying (2.6),
(2.13) $\limsup_{n\to\infty}\int_{M}|\nabla\phi_{n}|^{a}d\mu\ls
2^{\sigma_{0}-1}\cdot e^{\sigma_{0}}\cdot C_{2}\cdot t^{a}A(t):=C_{3}\cdot
t^{a}A(t).$
We take
$a_{1}=2\frac{\sigma-t}{\sigma-1}\qquad{\rm and}\quad
a_{2}=\frac{2\sigma}{\sigma-t-1}.$
Then it is easy to check that $a_{1},a_{2}$ satisfy (2.6). Indeed,
$a_{1}(t+1)=\frac{2\sigma}{\sigma-1}+2t-\frac{2t^{2}}{\sigma-1}\geqslant\frac{2\sigma}{\sigma-1}+t\qquad({\rm
since}\ \ t\ls\frac{\sigma-1}{2})$
and
$a_{2}(t+1)=\frac{2\sigma}{\sigma-1}\cdot\frac{\sigma-1}{\sigma-t-1}\cdot(t+1)\geqslant\frac{2\sigma}{\sigma-1}\cdot(t+1)\geqslant\frac{2\sigma}{\sigma-1}+t.$
Now, by using
$J(t,\phi_{n})=t^{-\frac{1}{2}-\frac{\sigma}{2(\sigma-1)}}\Big{(}\int_{M}|\nabla\phi_{n}|^{a_{1}}d\mu\Big{)}^{\frac{1}{2}}\cdot\Big{(}\int_{M}|\nabla\phi_{n}|^{a_{2}}d\mu\Big{)}^{\frac{1}{a_{2}}}$
and (2.13), we can conclude that
$\begin{split}\limsup_{n\to\infty}J(t,\phi_{n})&\ls
t^{-\frac{1}{2}-\frac{\sigma}{2(\sigma-1)}}\cdot
C^{\frac{1}{2}+\frac{1}{a_{2}}}_{3}\cdot
t^{\frac{a_{1}}{2}+1}\cdot[A(t)]^{\frac{1}{2}+\frac{1}{a_{2}}}\\\
&=C_{3}^{\frac{2\sigma-t-1}{2\sigma}}\cdot
t^{\frac{1}{2}+\frac{\sigma}{2(\sigma-1)}-\frac{t}{\sigma-1}}\cdot[A(t)]^{\frac{2\sigma-t-1}{2\sigma}}\end{split}$
Then
(2.14)
$\begin{split}\limsup_{n\to\infty}[J(t,\phi_{n})]^{\frac{2\sigma}{2\sigma-t-1}}&\ls
C_{3}\cdot
t^{(\frac{1}{2}+\frac{\sigma}{2(\sigma-1)}-\frac{t}{\sigma-1})\cdot\frac{2\sigma}{2\sigma-t-1}}\cdot
A(t)\\\ &=C_{3}\cdot
t^{\frac{\sigma}{\sigma-1}\cdot(1-\frac{t}{2\sigma-t-1})}\cdot
A(t).\end{split}$
Noticing that
$\lim_{t\to 0^{+}}t^{-\frac{\sigma}{\sigma-1}\cdot\frac{t}{2\sigma-t-1}}=1,$
we have that the function $t\mapsto
t^{-\frac{\sigma}{\sigma-1}\cdot\frac{t}{2\sigma-t-1}}$ is bounded on $(0,1)$
uniformly.
Set the constant
$C_{1}:=C_{3}\cdot\sup_{0<t<1}t^{-\frac{\sigma}{\sigma-1}\cdot\frac{t}{2\sigma-t-1}}.$
Then the desired estimate (2.4) follows from (2.14), and hence the proof of
Claim is completed. ∎
Now let us continue the proof of Theorem 1.2.
According to (1.3), there is a sequence of numbers
$\\{t_{\alpha}\\}_{\alpha=1}^{\infty}$, going to 0, such that
(2.15) $t_{\alpha}^{\frac{\sigma}{\sigma-1}}\cdot
A(t_{\alpha})=t_{\alpha}^{\frac{\sigma}{\sigma-1}}\int_{1}^{\infty}\frac{V(r)}{r^{\frac{3\sigma-1}{\sigma-1}+t_{\alpha}}}dr\ls
C_{4},\qquad\forall\ \alpha=1,2,\cdots$
for some constant $C_{4}$, independent of $\alpha.$ Without loss the
generality, we can also assume that
$t_{\alpha}\in(0,\min\\{1,\frac{\sigma-1}{2}\\}),$ for all
$\alpha=1,2,3,\cdots.$
By using the above Claim, we have, for each $\alpha=1,2,\cdots$,
(2.16) $\limsup_{n\to\infty}\
J(t_{\alpha},\phi_{t_{\alpha},n})\\!\ls\\!(C_{1}\cdot C_{4})^{\frac{2\sigma-
t_{\alpha}-1}{2\sigma}}\\!\ls\\!\max\\{(C_{1}C_{4})^{\frac{2\sigma-1}{2\sigma}},1\\}\\!:=C_{5}.$
In the following is similar as in [10]. We want to show $u\in L^{\sigma}(M)$,
and moreover $\int_{M}u^{\sigma}d\mu=0.$ Fix arbitrary a nonempty compact set
$K\subset M$.
Notice that $R(t_{\alpha})=\exp(1/t_{\alpha})\to\infty$ as $\alpha\to\infty$.
So, we have
$K\subset B_{R(t_{\alpha})}$
for all large enough $\alpha$. Hence, for any sufficient large $\alpha$,
$\phi_{t_{\alpha},n}\equiv 1$ on $K$ holds for any $n=1,2,\cdots.$. For such
$\alpha$, we can apply Lemma 2.1 to $t_{\alpha},$ $K$ and function
$\phi_{t_{\alpha},n}$; and we conclude that
(2.17)
$\int_{K}u^{\sigma}d\mu\\!\ls\int_{M}\phi_{t_{\alpha},n}^{s}u^{\sigma}d\mu\\!\ls
C_{0}\Big{(}\int_{M\backslash
K}\\!\phi_{t_{\alpha},n}^{s}u^{\sigma}d\mu\Big{)}^{\frac{t_{\alpha}+1}{2\sigma}}\cdot
J(t_{\alpha},\phi_{t_{\alpha},n})$
and
(2.18)
$\int_{K}u^{\sigma}d\mu\ls\int_{M}\phi_{t_{\alpha},n}^{s}u^{\sigma}d\mu\ls\Big{(}C_{0}\cdot
J(t_{\alpha},\phi_{t_{\alpha},n})\Big{)}^{\frac{2\sigma}{2\sigma-
t_{\alpha}-1}},$
for all $n=1,2,\cdots$, where we have used that $\phi_{t_{\alpha},n}\equiv 1$
on $K$.
By combining (2.16) and (2.18), we obtain
$\int_{K}u^{\sigma}d\mu\ls\Big{(}C_{0}\cdot
C_{5}\Big{)}^{\frac{2\sigma}{2\sigma-t_{\alpha}-1}}$
for all large enough $\alpha$. Letting $\alpha\to\infty$, we have
(2.19) $\int_{K}u^{\sigma}d\mu\ls\Big{(}C_{0}\cdot
C_{5}\Big{)}^{\frac{2\sigma}{2\sigma-1}}:=C_{6}.$
By combining with(2.17),(2.19),(2.16) and that $\phi_{t_{\alpha},n}\ls 1$ on
$M$, we have
$\int_{K}u^{\sigma}d\mu\\!\ls C_{0}\Big{(}\int_{M\backslash
K}\\!u^{\sigma}d\mu\Big{)}^{\frac{t_{\alpha}+1}{2\sigma}}\cdot C_{5},$
for all large enough $\alpha$. Letting $\alpha\to\infty$, we have
(2.20) $\int_{K}u^{\sigma}d\mu\\!\ls C_{0}\cdot
C_{5}\cdot\Big{(}\int_{M\backslash
K}\\!u^{\sigma}d\mu\Big{)}^{\frac{1}{2\sigma}}.$
By using the arbitrariness of $K$, we can take $K=\overline{B_{r}}$ for any
$r>0$. Combining with (2.19) and (2.20) and letting $r\to\infty$, we have
$\int_{M}u^{\sigma}d\mu=0,$
which implies $u\equiv 0$ on $M$, and the proof of Theorem 1.2 is completed. ∎
Acknowledgements. We would like to thank Dr. Yuhua Sun for his interesting in
the paper. The author is partially supported by Guangdong Natural Science
Foundation S2012040007550 and by China Postdoctoral Science Foundation
2012T50736, 2012M521639.
## References
* [1] S. Alarcón; J. Garciá-Melián & A. Quaas, _Nonexistence of positive supersolutions to some nonlinear elliptic porblems,_ J. Math. Pures Appl. 99(2013) 618–634.
* [2] Biduat-Veron, M.-F., & Pohozaev, S., _Nonexistence results and estimates for some nonlinear elliptic problems_ , J. Anal. Math., 84 (2001), 1–49.
* [3] Caffarelli, L.; Garofalo, N. & Segala, F., _A gradient bound for entire solutions of quasi-linear equatios and its consequences_ , Comm. Pure Appl. Math., 47 (1994), 1457–1473.
* [4] Caristi, G.; Mitidieri, E.& Pohozaev, S. I., _Some Liouville theorems for quasilinear elliptic inequalities_ , Doklady Math. 79 (2009), no. 1, 118–124.
* [5] S. Y. Cheng & S. T. Yau, _Differential equations on Riemannian manifolds and their geometric applications_ , Comm. Pure Appl. Math., 28(1975), no. 3, 333–354.
* [6] Gidas, B.& Spruck, J., _Global and local behavior of positive solutions of nonlinear elliptic equations_ , Comm. Pure Appl. Math. 34 (1981), no. 4, 525–598.
* [7] A. Grigor′yan, _On the existence of a Green function on a manifold_ (in Russian), Uspekhi Math. Nauk 38(1)(1983) 161–162, Engl. transl.: Russian Math. Surveys 38(1)(1983) 190–191.
* [8] A. Grigor′yan, _On the existence of positive fundamental solution of the Laplace equation on Riemannian manifolds_ (in Russian), Maat. Sb. 128(3)(1985) 354–363, Engl. transl.: Math. USSR Sb. 56 (1987) 349–358.
* [9] A. Grigor′yan & V. A. Kondratiev, _On the existence of positive solutions of semilinear elliptic inequalities on Riemannian manifolds_. Around the research of Vladimir Maz ya II, 203–218. International Mathematical Series (New York), 12. Springer, New York, 2010.
* [10] A. Grigor′yan & Y. Sun, _On Nonnegative Solutions of the Inequality $\Delta u+u^{\sigma}\ls 0$ on Riemnannian Manifolds_, to appear in Comm. Pure Appl. Math., (2013).
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|
arxiv-papers
| 2014-04-23T12:44:42 |
2024-09-04T02:50:01.708355
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Hui-Chun Zhang",
"submitter": "Hui-Chun Zhang",
"url": "https://arxiv.org/abs/1404.5805"
}
|
1404.5893
|
# The Solar Meridional Circulation and Sunspot Cycle Variability
###### Abstract
We have measured the meridional motions of the magnetic elements in the Sun’s
surface layers since 1996 and find systematic and substantial variations. In
general the meridional flow speed is fast at cycle minima and slow at cycle
maxima. We find that these systematic variations are characterized by a
weakening of the meridional flow on the poleward sides of the active (sunspot)
latitudes. This can be interpreted as a inflow toward the sunspot zones
superimposed on a more general poleward meridional flow profile. We also find
variations in the meridional flow which vary from cycle-to-cycle. The
meridional flow was slower at both the minimum and maximum of cycle 23
compared to similar phases of cycles 21, 22, and 24. Models of the magnetic
flux transport by a variable meridional flow suggest that it can significantly
modulate the size and timing of the following sunspot cycle through its impact
on the Sun’s polar magnetic fields. We suggest that the meridional flow
variations observed in cycle 23 contributed to the weak polar fields at the
end of the cycle which then produced a weak cycle 24 and the extraordinary
cycle 23/24 minimum.
HATHAWAY & UPTON MERIDIONAL FLOW AND THE SUNSPOT CYCLE Corresponding author:
D. H. Hathaway, Mail Code ZP13, NASA/Marshall Space Flight Center, Huntsville,
AL 35812 USA. ([email protected])
11affiliationtext: Space Science Office, NASA Marshall Space Flight Center,
Huntsville, Alabama, USA.22affiliationtext: Department of Physics and
Astronomy, Vanderbilt University, Nashville, Tennessee, USA.33affiliationtext:
Center for Space Physics and Aeronomy Research, University of Alabama in
Huntsville, Huntsville, Alabama, USA.
## 1 Introduction
The sunspot cycle minimum between cycles 23 and 24 (cycle 23/24 minimum) was
exceptional compared to others in modern times. In December of 2008 the
13-month smoothed sunspot number reached its lowest level since July of 1913
and the smoothed number of spotless days in a month reached its highest level
since August of 1913. In September of 2009 geomagnetic activity, as measured
by the aa index, reached record lows (since measurements began in 1868) while
galactic cosmic rays, as measured by neutron monitors, reached record highs
(since measurements began in 1953).
Since cycle 23/24 minimum in late-2008 we have seen the rise of cycle 24 as
the smallest sunspot cycle in 100 years. This provides the simple answer to
the question: What caused this extraordinary sunspot cycle minimum? This deep
and extended minimum was caused by the typically delayed start of a small
cycle. Statistically, small cycles start late and leave behind a long cycle
and a deep minimum (Hathaway et al., 1999) (c.f. minima preceding cycles
12-15). Two effects contribute to this: the actual start times for small
cycles are delayed (Hathaway et al., 1994) and the Waldmeier Effect
(Waldmeier, 1935) (in which small cycles rise more slowly) moves the date of
cycle minimum between overlapping cycles.
This explanation raises an obvious follow-on question: What caused cycle 24 to
be so small? This can be attributed to the weak polar fields built up during
cycle 23. Models of the Sun’s magnetic dynamo suggest that the Sun’s largely
dipole magnetic field at cycle minimum is the seed for the magnetic field that
erupts in the form of sunspots after amplification by the Sun’s differential
rotation (Babcock, 1961; Leighton, 1969).
Based on these models, Schatten et al. (1978) proposed that the strength of
the polar fields at sunspot cycle minimum should be a good predictor of the
amplitude of the following sunspot cycle maximum. Recently, Muñoz-Jaramillo et
al. (2013) confirmed (by using counts of polar faculae as a proxy for the
polar fields prior to 1976) that the polar fields themselves are indeed well
correlated with the amplitude of the following sunspot cycle maximum. Sunspot
cycle predictions based on the polar fields at minimum have successfully
predicted the last three cycles (Schatten et al., 1978; Schatten and Sofia,
1987; Schatten et al., 1996) as well as cycle 24 (Svalgaard et al., 2005).
Wang and Sheeley (2009) have noted that the strength of Sun’s axial dipole is
more closely attuned to dynamo theory and may be measured more accurately than
the polar fields for previous cycles. The axial dipole largely determines the
interplanetary magnetic field near cycle minima and this field can be derived
from historical geomagnetic measurements (Svalgaard and Cliver, 2005;
Rouillard et al., 2007). In fact, Wang and Sheeley (2009) suggest that it is
this connection that makes the geomagnetic aa index at its minimum such a good
predictor for the amplitude of following cycle (Ohl, 1966; Hathaway et al.,
1999).
This explanation for the cause of a weak cycle 24 minimum raises yet another
follow-on question: Why did cycle 23 produce such weak polar fields and axial
dipole? Dynamo models (Babcock, 1961; Leighton, 1969) and models for the
transport of magnetic flux in the solar photosphere (DeVore et al., 1984; Wang
et al., 1989; van Ballegooijen et al., 1998; Schrijver and Title, 2001;
Baumann et al., 2004) produce the polar fields through the emergence of tilted
bipolar active regions followed by the poleward transport of the magnetic
flux. The strength of the polar fields depends on the details of both the
sources (the total magnetic flux and tilt of the active regions) and the
transport processes (the meridional flow and the non-axisymmetric cellular
convective flows). Either the sources or the transport processes, or both,
must vary to give variable polar fields and the consequent solar cycle
variability.
## 2 Producing the Sun’s Axial Dipole
Both the active region sources and the transport processes that produce the
axial dipole moment are evident in magnetic butterfly diagrams (Fig. 1). These
diagrams are produced by averaging the Sun’s photospheric magnetic field over
longitude during each solar rotation to show the magnetic field as a function
of latitude and time. The butterfly wings in these diagrams show the emergence
of magnetic flux in the active latitudes with a predominance of one polarity
at higher latitudes and the opposite polarity at lower latitudes due to the
Joy’s Law tilt of active regions (Hale et al., 1919). This polarity pattern
alternates from hemisphere-to-hemisphere and cycle-to-cycle reflecting Hale’s
Law (Hale et al., 1919).
As each cycle progresses, new tilted dipoles emerge at progressively lower
latitudes so that the higher latitude following polarity flux progressively
cancels the preceding polarity flux that previously occupied those latitudes.
The poleward transport of the high latitude polarity is evident in Fig. 1 in
the form of unipolar streams that move poleward with time. As each cycle
reaches its maximum phase the sunspot zones extend to their lowest latitudes
with leading polarity in close proximity to the equator. This allows for the
diffusive transport of leading polarity flux across the equator and
cancellation with the (opposite polarity) leading polarity flux in the other
hemisphere.
Figure 1: A magnetic butterfly diagram produced from data acquired by the
National Solar Observatory. This shows the emergence of magnetic flux with the
systematic separation of polarities in the active latitude bands as well as
the poleward transport of this flux.
Figure 2: The Sun’s axial dipole moment (black line) as measured at the Wilcox
Solar Observatory for the last 40 years. The smoothed sunspot number is shown
in red. This shows that the dipole reverses at about the time of cycle maximum
and reaches its maximum at about the time of cycle minimum. The amplitudes of
the following cycles decrease as the amplitudes of the dipole moment decrease.
The polarity of the high latitude flux is opposite to that of the polar fields
at the start of the cycle so the emergence and poleward transport cancels the
polar fields from the previous cycle and then builds up opposite polarity
polar fields over the remainder of the sunspot cycle. This is seen in the
axial dipole moment (Fig. 2) calculated from the latitudinal distribution of
the surface magnetic field. The dipole reversals occur at about the time of
cycle maximum and the axial dipole moment that is built up at the end of the
cycle is well correlated with the strength of the following cycle (Wang and
Sheeley, 2009; Muñoz-Jaramillo et al., 2013).
This process could potentially lead to catastrophic behavior for solar cycle
amplitudes. Big sunspot cycles have more magnetic flux emerging in active
regions which would produce stronger axial dipoles along with increasing cycle
amplitudes. Likewise, smaller sunspot cycles should produce a string of cycles
with diminishing amplitudes. While the sunspot record does show periods of
increasing and decreasing cycle amplitudes, these trends are always limited to
4-5 cycles as part of the 100-year Gleissberg Cycle (Gleissberg, 1939). This
suggests that either the transport processes or the Joy’s Law tilt of active
regions (or both) must somehow change systematically with sunspot cycle
amplitude to stem the tide of increasing or decreasing cycle amplitudes.
A study of active region tilt by Dasi-Espuig et al. (2010) indicates that the
tilt may indeed vary with sunspot cycle amplitude. They found a tendency for a
smaller proportionality between active region tilt and latitude in bigger
cycles. This effect would help to modulate the polar field production by
leaving less of an excess of the higher latitude polarity during big cycles as
shown by Cameron et al. (2010) and Jiang et al. (2011).
Another mechanism for modulating the polar fields is to modulate the
latitudinal transport. This transport is facilitated by two processes – a
random walk by the rapidly evolving convection pattern and the direct
transport by the poleward meridional flow. The convection pattern includes
granules with velocities of $\sim 3000$ m s-1, lifetimes of $\sim 10$ minutes,
and diameters of $\sim 1$ Mm and supergranules with velocities of $\sim 300$ m
s-1, lifetimes of a day, and diameters of $\sim 30$ Mm. In one year, a random
walk by supergranules should have $\sim 400$ “steps” of 15 Mm giving a
displacement of $\sim 300$ Mm The meridional flow velocity is only $\sim 10$ m
s-1 but this direct flow also gives a displacement of $\sim 300$ Mm over the
course of a year. While these simple calculations suggest that these two
processes have similar impact, it should be noted that the diffusive effect of
the supergranules gives both poleward and equatorward motions and the
magnitude of the diffusivity still remains uncertain (Hagenaar et al., 1999).
In this paper we report on our measurements of the meridional motions of the
magnetic elements. We find variations in the strength and structure of the
meridional flow that may help to explain in general how the polar fields are
modulated and in particular how they were modulated during cycle 23. These
variations may have contributed to the production of the weak axial dipole
moment during cycle 23 which, in turn, caused the small amplitude for cycle
24.
## 3 Measuring the Meridional Flow
Measurements of the meridional flow can be made by a variety of techniques
including direct Doppler, local helioseismology, and feature tracking.
Advantages and disadvantages of the different techniques are discussed in the
Appendix. We choose to use magnetic feature tracking because is not masked by
large systematic signals and it measures the motions of the features of
interest for magnetic flux transport – magnetic field elements.
Our measurement technique is described in more detail in Hathaway and
Rightmire (2010), Hathaway and Rightmire (2011), and Rightmire-Upton et al.
(2012). We acquired full-disk magnetograms from the ESA/NASA Solar and
Heliospheric Observatory Michelson Doppler Investigation (SOHO/MDI) from May
1996 through March 2011 and from the NASA Solar Dynamics Observatory
Helioseismic and Magnetic Imager (SDO/HMI) from April 2010 through July 2013.
The SOHO/MDI magnetograms (Scherrer et al., 1995) were individual or 5-minutes
averages obtained every 96 minutes except for a data gap (the SOHO summer
vacation) from June to October 1998. The SDO/HMI magnetograms (Scherrer et
al., 2012) were averaged over 12-minutes and obtained every 60 minutes.
We projected each full-disk magnetogram onto a Mercator projection grid in
heliographic longitude and latitude. We extracted long, thin strips of data
with longitudinal widths of $105^{\circ}$ and latitudinal heights of
$2^{\circ}$ centered on the central meridian at a series of latitudes. These
strips were cross-correlated with similar strips from magnetograms obtained 8
hours later and 8 hours earlier. The offset in longitude and latitude giving
the highest correlation gives the differential rotation and meridional flow.
The average meridional flow profile obtained with the SDO/HMI data is shown in
Fig. 3.
Figure 3: The average meridional flow profile obtained with data from the
SDO/HMI instrument acquired between April 2010 and July 2013. Meridional flow
profiles used in previous surface flux transport models are shown with the
dotted (Wang et al., 1989), dashed (van Ballegooijen et al., 1998) and dashed-
dotted (Schrijver and Title, 2001) lines.
This meridional flow profile is problematic for most surface flux transport
models – it indicates that the poleward motions of the magnetic elements peak
in mid-latitudes and extend right to the poles. The early surface flux
transport models (Wang et al., 1989) employed a meridional flow that had
maximum poleward velocities immediately adjacent to the equator with a slow
fall-off to higher latitudes in each hemisphere (dotted line in Fig. 3.).
Later models (van Ballegooijen et al., 1998; Schrijver and Title, 2001) used
profiles that do match ours up to latitudes of $40-50^{\circ}$ but then have
little or no poleward flow at latitudes above $75^{\circ}$ (dashed and dashed-
dotted lines in Fig. 3).
The latitudinal structure of the meridional flow can significantly alter the
strength and structure of the polar fields it produces in these models. The
continued poleward flow at high latitudes tends to make the fields more
tightly concentrated at the poles. A peak flow velocity at very low latitudes
keeps opposite polarity elements from canceling across the equator.
## 4 Meridional Flow Speed Variations
We calculated the average profile for each 27-day rotation of the Sun and fit
each of them to orthogonal polynomials in $\sin B$ where $B$ is the
heliographic latitude. The fits to the meridional flow are dominated by the
terms of order 1 and 3.
Fig. 4 shows the history of the fit coefficients for the meridional flow
profiles. Here the coefficients found with SOHO/MDI for cycle 23 (Hathaway and
Rightmire, 2010) are now augmented by those we find with SDO/HMI data for
cycle 24 and by those found with NSO/Kitt Peak data for cycles 21 and 22 by
Komm et al. (1993a) using a similar method. (Komm et al. (1993a) correlated
square areas at $\sim 24$ hour time lags. We both excluded areas of strong
field associated with sunspots.) The variations in the meridional flow speed
over the course of each cycle are substantial. The meridional flow is fast at
cycle minima and slow at cycle maxima as was previously noted by Komm et al.
(1993a).
Figure 4: The meridional flow speed history from 1980 through mid-2013 is
represented in terms of the coefficients (S1 and S3) of the polynomials fit to
the profiles. Results for individual Carrington rotations are shown in black
for data from SOHO/MDI and in blue for the new results from SDO/HMI. Both sets
of data are shown with $2\sigma$ error bars. The data points (with $1\sigma$
error bars) for the 2-year intervals from 1980 to 1990 from Komm et al.
(1993a) are also shown in black. The smoothed sunspot number is shown in red
as a reference for comparing the meridional flow speed variations with the
phases of the sunspot cycles.
Fig. 4 also shows variations in the meridional flow speed from cycle-to-cycle.
The meridional flow was slower at both the preceding minimum (1996) and the
maximum (2000) of cycle 23 when compared to the minima and maxima of both the
earlier cycles (21 and 22) and the later cycle (24).
## 5 Meridional Flow Structure Variations
The fit coefficients shown in Fig. 4 help to characterize the amplitude of the
variability but do not fully describe the meridional flow variations. Cameron
and Schüssler (2010) noted that the variations in the S1 term shown in Fig. 4
could be explained in terms of the previously reported changes to the
meridional flow profiles derived from local helioseismology (Zhao and
Kosovichev, 2004; Gizon, 2004; González Hernández et al., 2008, 2010) and
interpreted as inflows toward the active latitudes. This behavior is seen in
Fig. 5 by showing each individual profile in a color-coded image.
The nature of the slow-down of the meridional flow at cycle maxima is clearly
evident in Fig 5. The slow-down is seen as a weakening of the poleward flow on
the poleward sides of the active latitudes (lighter shades of red in the north
and blue in the south. This can indeed be interpreted as an inflow toward the
active latitudes that is superimposed on the average poleward flow profile as
suggested by Cameron and Schüssler (2010), noted earlier from helioseismology
(Zhao and Kosovichev, 2004; Gizon, 2004; González Hernández et al., 2008,
2010), and from one of our earlier studies of magnetic element motion
(Hathaway and Rightmire, 2011).
Figure 5: The meridional flow profile history from May 1996 to July 2013 is
represented by showing the flow profiles from each 27-day solar rotation as
color-coded vertical strips. Shades of blue represent southward flow while
shades of red represent northward flow. Black dots indicate the location of
the active latitudes as given by the latitudinal centroid of the sunspot group
area in each hemisphere.
Fig. 5 also shows substantial differences between cycle 23 and our new results
for cycle 24. The weakening of the meridional flow on the poleward sides of
the active latitudes is hardly discernible in cycle 24. We also find polar
counter-cells (equatorward flow at high latitudes) with the MDI data during
cycle 23 that are not yet seen in HMI data during cycle 24. Cycle 23 appeared
to have a counter-cell in the south that extended down to $60^{\circ}$ in
mid-1996. By 2001 the counter-cell boundary moved poleward of our $75^{\circ}$
observation limit. At about this time a counter-cell is seen to dip below this
limit in the north and then proceed to grow to its maximum extent in 2006-2007
after which it shrinks but is still apparent in MDI data in 2010. It is worth
noting that during the 1-year overlap between MDI and HMI from April of 2010
to March of 2011 this northern counter-cell was visible in the MDI data but
not seen in the HMI data even at $85^{\circ}$ north (Rightmire-Upton et al.,
2012).
## 6 Possible Effects on the Polar Fields
The effects of these meridional flow variations on the polar fields and the
axial dipole need to be determined. Several previous studies have
characterized the effects of simply changing the amplitude of the meridional
flow. In addition, Jiang et al. (2010) and Cameron and Schüssler (2012) have
recently examined the effects associated with variations in the shape of the
meridional flow profile that are very similar to those found here (inflows
toward the active latitudes).
The parametric study of Baumann et al. (2004) used the meridional flow profile
of van Ballegooijen et al. (1998) (dashed line in Fig. 3) and varied the peak
velocity from 0 to 30 m s-1 with a fixed diffusivity of 600 km2 s-1. They
found that the polar fields increased in strength as the meridional flow
amplitude increased from 0 to 8 m s-1 — more high latitude, following polarity
flux is carried to the poles and the meridional flow itself counters the
diffusion away from the poles. As the meridional flow amplitude increases
above 8 m s-1 the polar fields become weaker — the higher velocities prevent
opposite polarities from canceling across the equator so that the net flux
carried by the meridional flow decreases. (This result is similar to what was
found earlier by Wang et al. (1989)).
The switchover in the sensitivity of polar field strength to meridional flow
speed variations at 8 m s-1 makes any conclusions about the effects of the
observed variations difficult based on this study. The average flow speed of
11 m s-1 is very close to this switchover point and the switchover point
itself must depend on the diffusivity and meridional flow profile used in the
calculations.
The structural changes to the meridional flow profile (the slowdown above the
active latitudes) should have a direct and negative effect on the poleward
transport of the high latitude following polarity but have little effect on
the flux cancellation across the equator. In general, this should modulate the
sunspot cycle amplitudes – big cycles would have stronger inflows that would
produce weaker polar fields than expected from the increase of active region
sources. In fact, we find (Fig. 5) that this inflow was stronger in cycle 23
than it is in cycle 24. Jiang et al. (2010) and Cameron and Schüssler (2012)
show that variations like this may indeed provide a nonlinear feedback on the
amplitudes of the solar cycles and may even be a key ingredient in determining
the amplitudes of solar cycles. Although the variations they employ are
significantly stronger than those we find in cycles 23 and 24 (neither cycle
shows any actual equatorward flow within the active latitudes) their results
do suggest that the meridional flow slowdown poleward of the active latitudes
in cycle 23 contributed to producing the weak polar fields.
## 7 Conclusions
The exceptional depth of cycle 23/24 minimum and the exceptional length of
cycle 23 can both be attributed to the small amplitude of cycle 24 – small
cycles typically start late and leave behind low minima (Hathaway et al.,
1999). The small amplitude of cycle 24 can be attributed to the weak polar
fields produced prior to cycle 23/24 minimum by the activity and flux
transport of cycle 23 – weak polar fields produce weak cycles (Muñoz-Jaramillo
et al., 2013). The strength of the polar fields (or the axial dipole moment)
is largely determined by a combination of three processes: the total magnetic
flux emerging in active regions, the characteristic tilt of those active
regions, and the transport (meridional flow and diffusion) of that emerging
flux.
Cycle 23 was significantly smaller than cycles 21 and 22 that preceded it.
This change in active region sources alone can be a significant source of the
change in polar fields. Dasi-Espuig et al. (2010) found that there are changes
to the characteristic tilt of active regions as a function of cycle amplitude.
However, they did not examine cycle 23 for comparison with other cycles. Here
we examined changes in the meridional flow measured from the motions of the
small magnetic elements that populate the Sun’s surface. We found variations
in the meridional flow that may have contributed to producing the weak polar
fields in cycle 23 and may play a more general role in modulating the
amplitudes of the solar cycle.
We found that cycle 23 was characterized by a slower meridional flow and that
this variation in the meridional flow was primarily due to a slowdown of the
meridional flow at the latitudes poleward of the sunspot zones. This slowdown
is not seen in cycle 24 – a much weaker solar cycle. This suggests that this
is a characteristic feature of solar cycles: the weakening of the meridional
flow is greater in bigger cycles. Cameron and Schüssler (2012) suggest that
this may be a key nonlinear process needed to modulate the amplitudes of the
solar cycles.
We suggest that this change in the meridional flow contributed to the
production of the weak polar fields at cycle 23/24 minimum. While this
suggestion is supported by the modeling work of Cameron and Schüssler (2012),
more detailed modeling work with the actual meridional flow profiles is needed
for confirmation.
## Appendix A Flow Measurement Methods
Measurements of the meridional flow can be made by a variety of techniques
including direct Doppler, time-distance helioseismology, and magnetic feature
tracking. The different techniques give information about flows at different
depths and about motions of different features. Both the direct Doppler and
time-distance helioseismology methods must first characterize and remove
systematic signals larger than the meridional flow signal itself. Magnetic
feature tracking is not subject to such large systematic errors and has the
distinct advantage of measuring the motions of the features of interest for
magnetic flux transport – magnetic field elements.
The Doppler signal due to the axisymmetric meridional flow is masked by the
much larger signal due to the convective blue shift and by
instrumental/scanning artifacts. The convective blue shift is produced by the
correlation between brightness and radial flow velocities in granules. It
gives a blue shift at disk center that falls off toward the limb and,
depending on spectral line, becomes a red shift near the limb itself. The
convective blue shift signal typically varies by $\sim 500$ m s-1 from disk
center to limb and the signal is known to vary locally in the presence of
magnetic field elements. Various methods have been devised to separated the
convective blue shift signal from the meridional flow signal (Snodgrass, 1984;
Ulrich et al., 1988; Hathaway, 1996) but all require an accurate measurement
of the convective blue shift and all assume that this signal does not vary
with latitude. Imaging artifacts, which (like the convective blue shift and
meridional flow signals) remain relatively fixed on the image can also
introduce systematic errors to the measurement of the meridional flow. Fig. 6
shows examples of these Doppler signals to illustrate the difficulties
entailed in making direct Doppler measurements of the meridional flow. While
this method is sensitive to the meridional flow at high latitudes, it is
insensitive to the meridional flow near the equator.
Figure 6: The direct Doppler signals associated with measuring the meridional
flow are shown here with the same scaling. The convective Blue Shift (left)
has a dynamic range of 500 m s-1 and is show here without any local variations
due to the presence of magnetic field. Imaging artifacts (center, from HMI)
can have a dynamic range of 200 m s-1. The meridional flow (right) has a
dynamic range of only 10-15 m s-1.
Local helioseismology measures the meridional flow by determining small
differences in the sound travel time between north-to-south and south-to-north
moving acoustic waves. Time-Distance helioseismology does this by correlating
signals observed at points separated in latitude. Duvall and Hanasoge (2009)
recently reported on a previously unnoticed systematic center-to-limb
variation in acoustic wave travel times. The source of this signal is still
uncertain but it is clearly seen as an apparent flow away from disk center
that has affected all previous measurements of the meridional flow using this
method. The signal itself can be characterized by measuring the signal away
from disk center along the equator as was done by Zhao et al. (2012). Their
characterization of this signal indicates that it can be several times larger
than the meridional flow signal itself (see their Fig. 2.). As with the direct
Doppler method, this method requires an accurate measurement of a systematic
signal and assumes that this signal does not vary with latitude. This method
is sensitive to meridional flow near the equator and it provides important
depth information but it becomes more uncertain at high latitudes. Even with
high resolution data these measurements rarely extend poleward of
$60^{\circ}$. Furthermore, when using meridional flow profiles from
helioseismology one must chose the appropriate depth for the flux transport
profile.
The feature tracking method using the small magnetic elements has significant
advantages over either direct Doppler or local helioseismology. It can be used
to measure the meridional flow from the equator up to at least $85^{\circ}$
latitude (Rightmire-Upton et al., 2012). It does not require characterizing
and removing any substantial systematic signals and it directly measures the
motions of the features of interest for magnetic flux transport – the magnetic
elements. Conversely, the motions of the magnetic elements are not given by
the motions of the surface plasma. The rotation rate of the small magnetic
elements (Snodgrass, 1983; Komm et al., 1993b) is significantly faster than
the surface Doppler rate – indicating that the magnetic elements are anchored
further down in the surface shear layer.
We should note that Dikpati et al. (2010) have suggested that magnetic element
feature tracking is subject to a large systematic signal associated with the
“diffusion” produced by supergranules. They argue that the diffusive transport
of magnetic elements away from the active latitudes should give a fictitious
flow velocity away from the active latitudes with this method and they
estimated the magnitude of the effect using a 1D (latitudinal) transport model
in which there were no magnetic elements per se. We (Hathaway and Rightmire,
2011) investigated this possibility by transporting magnetic elements in both
latitude and longitude using only supergranular flows and then attempted to
measure this associated fictitious meridional flow. We found that it must be
less than the 1-2 m s-1 noise level in our measurements (Fig. 10 of that
paper). Furthermore, the meridional flow variations we find using magnetic
feature tracking are in the form of an inflow toward the active latitudes -
not the outflow that would be produced according to Dikpati et al. (2010).
###### Acknowledgements.
The authors were supported by a grant from the NASA Living with a Star Program
to Marshall Space Flight Center. The HMI data used are courtesy of the
NASA/SDO and the HMI science team. The SOHO/MDI project was supported by NASA
grant NAG5-10483 to Stanford University. SOHO is a project of international
cooperation between ESA and NASA.
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|
arxiv-papers
| 2014-04-23T17:08:13 |
2024-09-04T02:50:01.718360
|
{
"license": "Public Domain",
"authors": "David H. Hathaway and Lisa Upton",
"submitter": "David Hathaway",
"url": "https://arxiv.org/abs/1404.5893"
}
|
1404.6010
|
# Stanley depth of monomial ideals
Dorin Popescu Dorin Popescu, Simion Stoilow Institute of Mathematics of
Romanian Academy, Research unit 5, P.O.Box 1-764, Bucharest 014700, Romania
[email protected]
###### Abstract.
Let $I\supsetneq J$ be two monomial ideals of a polynomial algebra over a
field generated in degree $\geq d$, resp. $\geq d+1$ . We study when the
Stanley Conjecture holds for $I/J$ using the recent result of [6] concerning
the polarization.
Key words : Monomial Ideals, Depth, Stanley depth.
2010 Mathematics Subject Classification: Primary 13C15, Secondary 13F20,
13F55, 13P10.
The support from grant PN-II-RU-TE-2012-3-0161 of Romanian Ministry of
Education, Research and Innovation are gratefully acknowledged.
## Introduction
Let $K$ be a field and $S=K[x_{1},\ldots,x_{n}]$ be the polynomial $K$-algebra
in $n$ variables. Let $I\supsetneq J$ be two monomial ideals of $S$ and
suppose that $I$ is generated by monomials of degrees $\geq d$ for some
positive integer $d$. Using a multigraded isomorphism we may assume either
that $J=0$, or $J$ is generated in degrees $\geq d+1$.
If $I,J$ are squarefree monomial ideals then $d$ is a lower bound of
$\operatorname{depth}_{S}I/J$ by [3, Proposition 3.1] (see also [15, Lemma
1.1]). Proposition 2 gives a lower bound of $\operatorname{depth}_{S}I/J$ in
terms of degrees also in the case when $I,J$ are not squarefree using the
polarization and the so called the canonical form of $I/J$ (see [10]).
A Stanley decomposition of a multigraded $S$-module $M$ is a finite family
$\mathcal{D}=(S_{l},u_{l})_{l\in L}$ in which $u_{l}$ are homogeneous elements
of $M$ and $S_{l}$ are multigraded $K-$algebra retract of $S$ for all $l\in L$
such that $S_{l}\cap\operatorname{Ann}_{S}u_{l}=0$ and $M=\sum_{l\in
L}S_{l}u_{l}$ as a multigraded $K-$vector space. The Stanley depth of
$\mathcal{D}$, denoted by $\operatorname{sdepth}(\mathcal{D})$, is the depth
of the $S-$module $\sum_{l\in L}S_{l}u_{l}$. The Stanley depth of $M$ is
defined as
$\operatorname{sdepth}\ (M):=\operatorname{max}\\{\operatorname{sdepth}\
({\mathcal{D}})\ |\ {\mathcal{D}}\;\text{is a Stanley decomposition
of}\;M\\}.$
Depth and Stanley depth behave in a different way for instance
$\operatorname{depth}_{S}(M\oplus
M^{\prime})=\operatorname{min}\\{\operatorname{depth}_{S}M,\operatorname{depth}_{S}M^{\prime}\\}$
while for sdepth it can happen $\operatorname{sdepth}_{S}(M\oplus
M^{\prime})>\operatorname{min}\\{\operatorname{sdepth}_{S}M,\operatorname{sdepth}_{S}M^{\prime}\\}$
sometimes as seen in [5, Examples 14, 16] with the help of [9]. These results
were obtained using the so called the Hilbert depth (see [1], [23]). The same
notion is important also in other properties of depth and Stanley depth (see
[21, Proposition 2.4]).
An upper bound for $\operatorname{depth}_{S}M$ could be given by the following
conjecture.
###### Conjecture 1.
(Stanley [22]) $\operatorname{depth}_{S}M\leq\operatorname{sdepth}_{S}M$.
It will be very nice if this conjecture holds for $M=I/J$. Recently Ichim,
Katthän and Moyano-Fernández proved that Stanley’s Conjecture holds for all
factors $I/J$ as above if and only if it holds for their polarizations [6,
Theorem 4.3]. Thus we may restrict to the case when $I,J$ are squarefree
monomial ideals. Unfortunately, there are few results in this case in spite of
the many papers appeared on this subject (see [3], [12], [13], [7], [14], [2],
[15]). It is the purpose of our paper to study what these few results say in
the non squarefree case using [6, Theorem 4.3]. We use here the lower bound
given by Proposition 2 (see Theorems 5, 6 and Proposition 4).
A particular case of this conjecture is the following one.
###### Conjecture 2.
Suppose that $I\subset S$ is minimally generated by some squarefree monomials
$f_{1},\ldots,f_{r}$ of degrees $d$, and a set $E$ of squarefree monomials of
degree $\geq d+1$. If $\operatorname{sdepth}_{S}I/J=d+1$ then
$\operatorname{depth}_{S}I/J\leq d+1$.
This conjecture is studied in [17], [18], [19], [11], [16] when $r\leq 4$ and
some cases when $r=5$ (see Theorems 3, 4). Proposition 3 proves Conjecture 2
also when $r=6$ but $d=1$ and $E=\emptyset$.
## 1\. A lower bound of depth and Stanley depth
Let $S=K[x_{1},\ldots,x_{n-1}]$ be the polynomial $K$-algebra over a field $K$
and $J\subsetneq I\subset R$ two monomial ideals. Denote by $G(I)$,
respectively $G(J)$, the minimal monomial system of generators of $I$,
respectively $J$.
A very important result concerning the Stanley depth is given by [6, Corollary
4.4] and we recall it below.
###### Theorem 1.
(Ichim, Katthän, Moyano-Fernández) Let $J\subsetneq I$ be monomial ideals of
$S$, and let $I^{p}\subset J^{p}$ be their (complete) polarizations. Then
$\operatorname{sdepth}_{S}I/J-\operatorname{depth}_{S}I/J=\operatorname{sdepth}I^{p}/J^{p}-\operatorname{depth}I^{p}/J^{p}.$
For $i\in[n]$ let $e_{i}=\operatorname{max}_{u\in G(I)\cup
G(J)}\operatorname{deg}_{x_{i}}u$ and set
$e_{I/J}=\sum_{i\in[n],e_{i}>0}(e_{i}-1)$. We have
$e_{I/J}=\operatorname{depth}I^{p}/J^{p}-\operatorname{depth}_{S}I/J$, that is
$e_{I/J}$ is the number of the new variables necessary for polarization.
Suppose that $I$ is generated by some monomials $f_{1},\ldots,f_{r}$ of
degrees $d_{I/J}$ and a set of monomials $E$ of degrees $\geq d_{I/J}+1$. Then
###### Proposition 1.
$\operatorname{depth}_{S}I/J\geq d_{I/J}-e_{I/J}$ and
$\operatorname{sdepth}_{S}I/J\geq d_{I/J}-e_{I/J}$.
###### Proof.
By [3, Proposition 3.1] (see also [15, Lemma 1.1]) we have
$\operatorname{depth}I^{p}/J^{p}\geq d_{I/J}$ because $I^{p},J^{p}$ are
squarefree monomial ideals. Note that by polarization the degrees of monomials
are preserved. It follows that
$\operatorname{depth}_{S}I/J=\operatorname{depth}I^{p}/J^{p}-e_{I/J}\geq
d_{I/J}-e_{I/J}$. The inequality concerning sdepth is similar but easier since
obviously the sdepth is $\geq d_{I/J}$ in the case of a factor of some
squarefree monomial ideals. ∎
###### Example 1.
Let $n=3$, $d=12$, $I=(x_{1}^{3}x_{2}^{4}x_{3}^{5},x_{1}^{10}x_{2}^{2})$. Note
that $e_{1}=10$, $e_{2}=4$, $e_{3}=5$ and $e_{I}=16$.
###### Remark 1.
In the above example Proposition 1 gives $\operatorname{depth}_{S}I\geq-4$
which is obvious. This situation will be next improved considering the so
called the canonical form of $I$ given by [10].
We recall some definitions and results from [10].
###### Definition 1.
The power $x_{n}^{r}$ enters in a monomial $u$ if $x_{n}^{r}|u$ but
$x_{n}^{r+1}\nmid u$. We say that $I/J$ is of type $(k_{1},\ldots,k_{s})$ with
respect to $x_{n}$ if $x_{n}^{k_{i}}$ are all the powers of $x_{n}$ which
enter in a monomial of $G(I)\cup G(J)$ for $i\in[s]$ and $1\leq
k_{1}<\ldots<k_{s}$. $I/J$ is in the canonical form with respect to $x_{n}$ if
$I/J$ is of type $(1,\ldots,s)$ for some $s\in{\mathbb{N}}$ and we say that
$I/J$ is the canonical form if it is in the canonical form with respect to all
variables $x_{1},\ldots,x_{n}$.
###### Remark 2.
Suppose that $I/J$ is of type $(k_{1},\ldots,k_{s})$ with respect to $x_{n}$.
It is easy to get the canonical form $I^{\prime}/J^{\prime}$ of $I/J$ with
respect to $x_{n}$ replacing $x_{n}^{k_{i}}$ by $x_{n}^{i}$ whenever
$x_{n}^{k_{i}}$ enters in a generators of $G(I)\cup G(J)$. Applying by
recurrence this procedure for other variables we get the canonical form of
$I/J$, that is with respect to all variables.
###### Theorem 2.
(A. Popescu [10, Theorems 1, 2]) Let $I^{\prime}/J^{\prime}$ be the canonical
form of $I/J$. Then
$\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}=\operatorname{sdepth}_{S}I/J$
and
$\operatorname{depth}_{S}I^{\prime}/J^{\prime}=\operatorname{depth}_{S}I/J$.
###### Definition 2.
Let $I^{\prime}/J^{\prime}$ be the canonical form of $I/J$ and set
$t_{I/J}=\operatorname{max}\\{d_{I^{\prime}/J^{\prime}}-e_{I^{\prime}/J^{\prime}},0\\}$
(we may have $d_{I^{\prime}/J^{\prime}}<e_{I^{\prime}/J^{\prime}}$ as shows
Example 3). We call $t_{I/J}$ the index of $I/J$. When $J=0$ we write $t_{I}$
instead $t_{I/J}$ for simplicity. If $I,J$ are squarefree monomial ideals then
$t_{I/J}=d_{I/J}$.
Using the terminology defined above we get a better lower bound for sdepth and
depth as in Proposition 1.
###### Proposition 2.
$\operatorname{depth}_{S}I/J\geq t_{I/J}$ and
$\operatorname{sdepth}_{S}I/J\geq t_{I/J}$.
###### Proof.
By Theorem 2 we have
$\operatorname{depth}_{S}I/J=\operatorname{depth}_{S}I^{\prime}/J^{\prime}\geq\operatorname{max}\\{d_{I^{\prime}/J^{\prime}}-e_{I^{\prime}/J^{\prime}},0\\}=t_{I/J}$.
The second inequality holds similarly. ∎
###### Remark 3.
This lower bound is easy to describe but it is not the best known lower bound.
For example, when $J=0$ then a better lower bound is given by
$1+\operatorname{size}(I)$ in the terminology of [8], [4]. More precisely, if
$n=3$, $d_{I}=1$, $I=(x_{1},x_{2}x_{3})=(x_{1},x_{2})\cap(x_{1},x_{3})$ then
$\operatorname{size}(I)=1$ and $t_{I}=d_{I}$ since $I$ is squarefree. Thus
$1+\operatorname{size}(I)>t_{I}$.
###### Remark 4.
In Example 1 note that $I$ is of type $(3,10)$ with respect to $x_{1}$, of
type $(2,4)$ with respect to $x_{2}$ and of type $(5)$ with respect to
$x_{3}$. Then the canonical form of $I$ is
$I^{\prime}=(x_{1}x_{2}^{2}x_{3},x_{1}^{2}x_{2})$. Note that $I$ is generated
by monomials of degrees $12$ but in $I^{\prime}$ one generator has degree $4$
and the other $3$. Clearly, $e_{I^{\prime}}=2$, $d_{I^{\prime}}=3$ and so the
index $t_{I}$ of $I$ is $1$. Thus Proposition 2 says that
$\operatorname{depth}_{S}I\geq 1$, which is also trivial but anyway better
than what follows from Proposition 1 (see Remark 1).
## 2\. Stanley depth of monomial ideals which are not necessarily squarefree
Suppose that $I$ is minimally generated by some squarefree monomials
$f_{1},\ldots,f_{r}$ of degrees $d$ for some $d\in{\mathbb{N}}$ and a set of
squarefree monomials $E$ of degree $\geq d+1$. Let $B$ be the set of the
squarefree monomials of degrees $d+1$ of $I\setminus J$.
We start recalling two results of [16] (see also [19] and [11]).
###### Theorem 3.
Conjecture 2 holds for $r\leq 4$.
###### Theorem 4.
Conjecture 2 holds for $r=5$ if there exists
$j\not\in\cup_{i\in[5]}\operatorname{supp}f_{i}$, $j\in[n]$ such that
$(B\setminus E)\cap(x_{j})\not=\emptyset$ and $E\subset(x_{j})$.
For simplicity we denote $t=t_{I/J}$, that is the index of $I/J$.
###### Theorem 5.
Let $J\subsetneq I$ be monomial ideals of $S$ not necessarily squarefree and
$I^{\prime}/J^{\prime}$ the canonical form of $I/J$. Suppose that $I^{\prime}$
is generated by $r^{\prime}$ monomials $f_{1},\ldots,f_{r^{\prime}}$ of degree
$d_{I^{\prime}/J^{\prime}}$ and a set $E^{\prime}$ of monomials of degree
$\geq d_{I^{\prime}/J^{\prime}}+1$. Let $B^{\prime}$ be the set of monomials
of degree $d_{I^{\prime}/J^{\prime}}+1$ from $I^{\prime}\setminus J^{\prime}$.
Assume that $\operatorname{sdepth}_{S}I/J=t+1$. Then the following statements
hold:
1. (1)
If $r^{\prime}\leq 4$ then $\operatorname{depth}_{S}I/J\leq t+1$,
2. (2)
If $r^{\prime}=5$ and there exists
$j\not\in\cup_{i\in[5]}\operatorname{supp}f_{i}$, $j\in[n]$ such that
$(B^{\prime}\setminus E^{\prime})\cap(x_{j})\not=\emptyset$ and
$E^{\prime}\subset(x_{j})$, then $\operatorname{depth}_{S}I/J\leq t+1$.
###### Proof.
Let $I^{\prime}/J^{\prime}$ be the canonical form of $I/J$. By Theorem 1 we
have
$\operatorname{sdepth}I^{\prime p}/J^{\prime
p}=\operatorname{sdepth}_{S}I^{\prime}/J^{\prime}-\operatorname{depth}I^{\prime
p}/J^{\prime
p}+\operatorname{depth}_{S}I^{\prime}/J^{\prime}=d_{I^{\prime}/J^{\prime}}+1.$
Since $I^{\prime p}$ is generated by $r^{\prime}$ squarefree monomials of
degree $d_{I^{\prime}/J^{\prime}}$ and a set $E^{\prime p}$ of squarefree
monomials of degree $d_{I^{\prime}/J^{\prime}}+1$ we get by Theorem 3 that
$\operatorname{depth}I^{\prime p}/J^{\prime p}\leq
d_{I^{\prime}/J^{\prime}}+1$. Hence
$\operatorname{depth}_{S}I/J=\operatorname{depth}_{S}I^{\prime}/J^{\prime}\leq
t+1$ by Theorem 2, that is (1) holds. The proof of (2) is the same using
Theorem 4 instead Theorem 3. ∎
###### Remark 5.
Let $I$ be generated by some monomials $h_{1},\ldots,h_{r}$ of degree $d$ and
a set of monomials $E$ of monomials of degree $\geq d+1$. It is possible that
$I^{\prime}$ is generated by $f_{1},\ldots,f_{r^{\prime}}$ of degrees
$d_{I^{\prime}/J^{\prime}}$ with $r^{\prime}>r$ and a set $E^{\prime}$ of
monomials of degree $\geq d_{I^{\prime}/J^{\prime}}+1$. For example when $n=2$
and $I=(x_{1}^{3}x_{2}^{4},x_{1}^{11}x_{2})$ we have $r=1$ and we see that
$I^{\prime}=(x_{1}x_{2}^{2},x_{1}^{2}x_{2})$ has $r^{\prime}=2$.
###### Example 2.
Let $n=2$, $d=1$, $I=(x_{1})$, $J=(x_{1}x_{2}^{2})$. Then $e_{1}=1$,
$e_{2}=2$, $e_{I/J}=1$, $t=0$ and $I^{p}/J^{p}=(x_{1})/(x_{1}x_{2}y_{2})$,
where $y_{2}$ is the new variable from polarization. We have $I/J\cong
x_{1}K[x_{1}]\oplus x_{1}x_{2}K[x_{1}]$ as graded $K$-vector spaces. Thus
$\operatorname{sdepth}_{S}I/J=1=t+1$. By (1) of the above theorem we get
$\operatorname{depth}_{S}I/J\leq 1$, the inequality being in fact an equality.
###### Theorem 6.
Let $J\subsetneq I$ be monomial ideals of $S$ not necessarily squarefree.
Assume that $\operatorname{sdepth}_{S}I/J=t$. Then
$\operatorname{depth}_{S}I/J=t$
The proof is similar to the proof of Theorem 5 using now [15, Theorem 4.3]
instead Theorem 3.
###### Example 3.
Let $n=2$, $d=1$, $I=(x_{2})$, $J=(x_{1}^{2}x_{2},x_{1}x_{2}^{2})$. Then
$e_{1}=e_{2}=e_{I/J}=2$, $t=\operatorname{max}\\{-1,0\\}=0$ and
$I^{p}/J^{p}=(x_{2})/(x_{1}y_{1}x_{2},x_{1}x_{2}y_{2})$, where $y_{1},y_{2}$
are the new variables from polarization. Since $x_{2}$ induces a nonzero
element of the socle of $I/J$ we see that $\operatorname{sdepth}_{S}I/J=0$.
Thus $\operatorname{sdepth}_{S}I/J=t=0$. By the above theorem we get
$\operatorname{depth}_{S}I/J=0$.
## 3\. Stanley depth of a factor of squarefree monomial ideals
The above theorem implies the following corollary.
###### Proposition 3.
Suppose that $I\subset S$ is minimally generated by $6$ variables
$\\{x_{1},\ldots,x_{6}\\}$ and $J\subsetneq I$ a squarefree monomial ideal. If
$\operatorname{sdepth}_{S}I/J=2$ then $\operatorname{depth}_{S}I/J\leq 2$.
###### Proof.
By [17, Proposition 1.3] we see that there exists $c=x_{6}x_{k}x_{q}\not\in J$
for $6<k<q\leq n$. Let $B$ be the set of all squarefree monomials from
$I\setminus J$ and $\tilde{I}$ be the ideal generated by $x_{1},\ldots,x_{5}$
and ${\tilde{E}}=B\setminus((x_{1},\ldots,x_{5})\cup[x_{6},c])$. Set
${\tilde{J}}=J\cap{\tilde{I}}$. Then for $j=6$ we have
${\tilde{E}}\subset(x_{j})$. In the following exact sequence
$0\rightarrow{\tilde{I}}/{\tilde{J}}\rightarrow I/J\rightarrow
I/J+{\tilde{I}}\rightarrow 0$
the last term is isomorphic with $(x_{6})/(x_{6})\cap(J+{\tilde{I}})$ and has
depth $\geq 2$ and sdepth $3$ because it has just the interval $[x_{6},c]$.
Suppose that $\operatorname{sdepth}_{S}I/J=2$. By [20, Lemma 2.2] we get
$\operatorname{sdepth}_{S}{\tilde{I}}/{\tilde{J}}\leq 2$. When
$\operatorname{sdepth}_{S}{\tilde{I}}/{\tilde{J}}=1$ then it is enough to
apply [15, Theorem 4.3]. If
$\operatorname{sdepth}_{S}{\tilde{I}}/{\tilde{J}}=2$ and
$(B\setminus{\tilde{E}})\cap(x_{j})\not=\emptyset$ then it is enough to apply
Theorem 4.
Now suppose that $(B\setminus{\tilde{E}})\cap(x_{j})=\emptyset$, that is
$B\cap(x_{6})\cap(x_{1},\ldots,x_{5})=\emptyset$. In the following exact
sequence
$0\rightarrow(x_{6})/(x_{6})\cap J\rightarrow I/J\rightarrow
I/(J,x_{6})\rightarrow 0$
if the last term has sdepth $\geq 3$ then the first term has sdepth $\leq 2$
as above and so also depth $\leq 2$. Otherwise, the last term has sdepth $\leq
2$. But the last term is isomorphic with
$(x_{1},\ldots,x_{5})/(x_{1},\ldots,x_{5})\cap J$ because
$B\cap(x_{6})\cap(x_{1},\ldots,x_{5})=\emptyset$. Thus in the exact sequence
$0\rightarrow(x_{1},\ldots,x_{5})/(x_{1},\ldots,x_{5})\cap J\rightarrow
I/J\rightarrow I/(J,x_{1},\ldots,x_{5})\rightarrow 0$
the first term has sdepth $\leq 2$ and so its depth $\leq 2$ by Theorem 4 when
there exists $k>6$ such that
$B\cap(x_{1},\ldots,x_{5})\cap(x_{k})\not=\emptyset$. Otherwise,
$J\geq(x_{1},\ldots,x_{5})(x_{6},\ldots,x_{n})$ and we get
$\operatorname{depth}_{S}(x_{1},\ldots,x_{5})/(x_{1},\ldots,x_{5})\cap
J=\operatorname{depth}_{\tilde{S}}(x_{1},\ldots,x_{5}){\tilde{S}}/(x_{1},\ldots,x_{5})\cap
J\cap{\tilde{S}}\leq 1$
for ${\tilde{S}}=K[x_{1},\ldots,x_{5}]$. Since the last term is isomorphic
with $(x_{6})/J\cap(x_{6})$ it has depth $\geq 2$ and the Depth Lemma ends the
proof. ∎
###### Proposition 4.
Suppose that $I\subset S$ is minimally generated by $6$ variables
$\\{x_{1},\ldots,x_{6}\\}$ and $J\subsetneq I$ is a monomial ideal not
necessarily squarefree. Suppose that $\operatorname{sdepth}_{S}I/J=t+1$. Then
$\operatorname{depth}_{S}I/J\leq t+1$.
The proof is similar to the proof of Theorem 5 using now Proposition 3 instead
Theorem 3.
###### Example 4.
Let $n=7$, $I=(x_{1},\ldots,x_{6}),$
$J=(x_{1}^{2},x_{1}x_{2},\ldots,x_{1}x_{5},x_{1}x_{7})$. Then $t=0$. The
element ${\hat{x}}_{1}\in I/J$ induced by $x_{1}$ is annihilated by all
variables but $x_{6}$. It follows that $\operatorname{sdepth}_{S}I/J\leq 1$.
Thus $\operatorname{sdepth}_{S}I/J\leq t+1$ and so
$\operatorname{depth}_{S}I/J\leq 1$ by Proposition 4. Note that
$I^{p}/J^{p}=(x_{1},\ldots,x_{6})/(x_{1}y,x_{1}x_{2},\ldots,x_{1}x_{5},x_{1}x_{7})$
has sdepth $\leq 2$ because now the element of $I^{p}/J^{p}$ induced by
$x_{1}$ is annihilated by all variables but $x_{6},y$.
## References
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|
arxiv-papers
| 2014-04-24T01:41:56 |
2024-09-04T02:50:01.726706
|
{
"license": "Public Domain",
"authors": "Dorin Popescu",
"submitter": "Dorin Popescu",
"url": "https://arxiv.org/abs/1404.6010"
}
|
1404.6025
|
# Randomized Benchmarking with Confidence
Joel J. Wallman Institute for Quantum Computing, University of Waterloo,
Waterloo, Canada Centre for Engineered Quantum Systems, School of Physics,
The University of Sydney, Sydney, NSW 2006, Australia Steven T. Flammia
Centre for Engineered Quantum Systems, School of Physics, The University of
Sydney, Sydney, NSW 2006, Australia
###### Abstract
Randomized benchmarking is a promising tool for characterizing the noise in
experimental implementations of quantum systems. In this paper, we prove that
the estimates produced by randomized benchmarking (both standard and
interleaved) for arbitrary Markovian noise sources are remarkably precise by
showing that the variance due to sampling random gate sequences is small. We
discuss how to choose experimental parameters, in particular the number and
lengths of random sequences, in order to characterize average gate errors with
rigorous confidence bounds. We also show that randomized benchmarking can be
used to reliably characterize time-dependent Markovian noise (e.g., when noise
is due to a magnetic field with fluctuating strength). Moreover, we identify a
necessary property for time-dependent noise that is violated by some sources
of non-Markovian noise, which provides a test for non-Markovianity.
## I Introduction
One of the key obstacles to realizing large-scale quantum computation is the
need for error correction and fault tolerance Gottesman (2009), which require
the coherent implementation of unitary operations to high precision.
Characterizing the accuracy of an experimental implementation of a unitary
operation is therefore an important prerequisite for constructing a large-
scale quantum computer.
It is possible to completely characterize an experimental implementation of a
unitary using full quantum process tomography Chuang and Nielsen (1997);
Poyatos _et al._ (1997). However, this approach has several major
deficiencies when applied to large quantum systems. Firstly, it is provably
exponential in the number of qubits of the system for any procedure that can
identify general noise sources and hence it cannot be performed practically
for even intermediate numbers of qubits, despite improvements such as
compressed sensing Flammia _et al._ (2012); Gross _et al._ (2010). Secondly,
it is sensitive to state preparation and measurement (SPAM) errors, which
create a noise floor below which an accurate process estimation becomes
impossible Merkel _et al._ (2013). Finally, it does not capture any notion of
systematic, time-dependent errors that can arise from applying many unitaries
in sequence.
One can avoid the exponential scaling by accepting a partial characterization
of an experimental implementation. A partial characterization of, for example,
the average error rate and/or the worst-case error rate compared to a perfect
implementation of a target unitary is typically enough to determine whether an
experimental implementation of a unitary is sufficient for achieving fault-
tolerance in a specific scheme for fault-tolerant quantum computation. Such
partial characterizations can be obtained efficiently (in the number of
quantum systems) using either randomized benchmarking Emerson _et al._
(2005); Knill _et al._ (2008); Magesan _et al._ (2011, 2012a); Gaebler _et
al._ (2012); Magesan _et al._ (2012b) or direct fidelity estimation da Silva
_et al._ (2011); Flammia and Liu (2011).
While direct fidelity estimation gives an unconditional and assumption-free
estimate of the average gate fidelity, it is prone to state preparation and
measurement (SPAM) errors, which leads to conflation of noise sources. Thus, a
key advantage of randomized benchmarking is that it is not sensitive to SPAM
errors. Unfortunately, however, current proposals for randomized benchmarking
assume that the noise is time-independent, although time-dependence can be
partially characterized by a deviation from the expected fidelity decay curve
Magesan _et al._ (2011, 2012a). Furthermore, experimental implementations of
randomized benchmarking typically use on the order of 100 random sequences of
Clifford gates, which is three orders of magnitude smaller than the number of
sequences suggested by the rigorous bounds in Ref. Magesan _et al._ (2012a)
to obtain an accuracy comparable to the claimed experimental accuracies
Magesan _et al._ (2012b); Brown _et al._ (2011). Numerical investigations of
a variety of noise models have shown that between 10–100 random sequences for
each length are sufficient to provide a tight estimate of the average gate
fidelity Epstein _et al._ (2014). Ideally, one would like to combine the
advantages of both randomized benchmarking and direct fidelity estimation to
achieve a method that is insensitive to SPAM, requires few measurements, is
nearly assumption-free (i.e., does not assume a specific noise model), and
comes with rigorous guarantees on the errors involved.
In this paper, we provide a new analysis of randomized benchmarking which
brings it closer in line with this ideal. We first show that the standard
protocol can be modified to provide a means of estimating the time-dependent
average gate fidelity (which characterizes the average error rate), provided
that the gate-dependent fluctuations at each time step are sufficiently small.
Under the assumption that the noise is Markovian (that is, that the noise can
be written as a sequence of noisy channels acting on the system of interest),
all the time-dependent parameters that are estimated by our procedure are
upper-bounded by 1, so if some of the parameters are observed to be greater
than 1, the experimental noise must be non-Markovian.
We then provide a rigorous justification for taking a small number of random
sequences at each length that is on the same order as used in practice by
obtaining bounds on the variance due to sampling gate sequences. Our work
complements the approach of Ref. Epstein _et al._ (2014), where it was shown
that the width of the confidence interval for the parameters extracted from
randomized benchmarking is on the order of the square root of the variance.
Our work therefore proves that this confidence interval is generally very
narrow, that is, the parameters extracted from randomized benchmarking are
determined with high precision.
Numerically, we observe that our bounds (at least for qubits) are saturated
and so cannot be improved without further assumptions on the noise (e.g., that
the noise is diagonal in the Pauli basis). Therefore any experiments using
fewer random sequences than justified by our analysis (unless there is solid
evidence that the noise has a specific structure) will potentially
underestimate the error due to sampling random sequences.
As a particular example, our results provide a rigorous proof that for single-
qubit noise with an average error rate of $10^{-4}$, the error for randomized
benchmarking with 100 random sequences of 100 random gates will be less than
$0.9\%$ with $99\%$ confidence. If we use the parameters estimated in the
experiment of Ref. Brown _et al._ (2011), with 100 random sequences of length
987 at an average error rate of $2\times 10^{-5}$, we find the error is less
than $.8\%$ with $99\%$ confidence.
We emphasize that our results are solely in terms of the number of random gate
_sequences_ , and a given sequence must still be repeated many times to gather
statistics about expectation values of an observable. This is of course an
unavoidable consequence of quantum mechanics. However, these statistical
fluctuations in the estimates of expectation values can be analyzed separately
with standard statistical tools for binomial distributions or with the recent
Bayesian methods introduced in Granade _et al._ (2014) and combined
seamlessly with our results.
In order to give a rigorous statement of results, we will first review the
randomized benchmarking protocol.
## II The randomized benchmarking protocol
The goal of randomized benchmarking is to efficiently but partially
characterize the average noise in an experimental implementation of a group
$\mathcal{G}=\\{g_{1},\ldots,g_{\left\lvert\mathcal{G}\right\rvert}\\}\subset\mathsf{U}(d)$
of operations acting on a $d$-dimensional quantum system. In order to
characterize the average noise in an implementation of $\mathcal{G}$ using
randomized benchmarking, we require $\mathcal{G}$ to be a unitary 2-design
(e.g., the Clifford group on $n$ qubits for $d=2^{n}$), meaning that sampling
over $\mathcal{G}$ reproduces the second moments of the Haar measure Dankert
_et al._ (2009); Gross _et al._ (2007). To accomplish this, the following
protocol is implemented.
* •
Choose a random sequence $s=s_{1}\ldots s_{m}\in\mathbb{N}_{\left\lvert
G\right\rvert}^{m}$ of $m$ integers chosen uniformly at random from
$\mathbb{N}_{\left\lvert
G\right\rvert}=\left\\{1,\ldots,\left\lvert\mathcal{G}\right\rvert\right\\}$.
* •
Prepare a $d$-dimensional system in some state $\rho$ (usually taken to be the
pure state $|0\rangle$).
* •
At each time step $t=0,\ldots,m$, apply $g_{t}$ where $g_{t}=g_{s_{t}}$ and
$g_{0}:=\prod_{t=1}^{m}g_{t}^{-1}$. Alternatively, to perform interleaved
randomized benchmarking for the gate $g_{\rm int}\in\mathcal{G}$, apply
$g_{t,{\rm int}}$ where $g_{t,{\rm int}}=g_{\rm int}g_{t}$ for $t\neq 0$ and,
as before, $g_{0,{\rm int}}=\prod_{t=1}^{m}g_{t,{\rm int}}^{-1}$. (In general,
each gate must be compiled into a sequence of elementary gates as well.)
* •
Perform a POVM $\left\\{E,\mathbbm{1}-E\right\\}$ for some $E$ (usually taken
to be $|0\rangle\\!\langle 0|$) and repeat with the sequence $s$ sufficiently
many times to obtain an estimate of the probability $F_{m,s}=p(E|s,\rho)$ to a
suitable precision.
We can regard the probability $F_{m,s}$ as a realization of a random variable
$F_{m}$. We will denote the variance of the distribution
$\\{F_{m,s}:s\in\mathbb{N}_{\left\lvert G\right\rvert}\\}$ for a fixed $m$ by
$\sigma_{m}^{2}$. Averaging $F_{m,s}$ over a number of random sequences will
give an estimate $\hat{F}_{m}$ of $\bar{F}_{m}$, the average of $F_{m,s}$ over
all sequences $s$ of fixed length $m$ (that is, $\bar{F}_{m}$ is the
expectation of the random variable $F_{m}$). The accuracy of this estimate
will be a function of the number of random sequences and $\sigma_{m}^{2}$.
Obtaining estimates $\hat{F}_{m}$ for multiple $m$ and fitting to the model
$\displaystyle\bar{F}_{m}=A+Bf^{m}$ (1)
will give an estimate of $f$ provided that the noise does not depend too
strongly on the target gate Magesan _et al._ (2012a), where Nielsen (2002)
$\displaystyle f=\frac{d\mathcal{F}_{\rm avg}(\mathcal{E})-1}{d-1}$ (2)
and
$\displaystyle\mathcal{F}_{\rm avg}(\mathcal{E})$
$\displaystyle=\int\mathrm{d}\psi\mathrm{Tr}\bigl{[}\psi\mathcal{E}(\psi)\bigr{]}$
(3)
is the average gate fidelity of a noise channel $\mathcal{E}$ with respect to
the identity channel and $\mathrm{d}\psi$ is the uniform Haar measure over all
pure states. The average gate fidelity of $\mathcal{E}$ gives the average
probability that preparing a state $\psi$, applying $\mathcal{E}$ and then
measuring $\\{\psi,\mathbbm{1}-\psi\\}$ will give the outcome $\psi$, averaged
over all pure states $\psi$.
For standard randomized benchmarking, $\mathcal{E}$ is the error channel per
operation, averaged over all operations in $\mathcal{G}$. For interleaved
benchmarking, $\mathcal{E}$ is the error channel on a composite channel,
namely, the interleaved channel composed with an element of $\mathcal{G}$,
averaged over all $\mathcal{G}$. We note in passing that separating the error
in the interleaved channel from the error in the composite channel is one of
the key difficulties in obtaining meaningful results from interleaved
benchmarking Kimmel _et al._ (2013), though we do not address this issue
here.
## III Statement of Results and Paper Outline
The first principal contribution of this paper is to show that the number of
random sequences that need to be averaged is comparable to the number actually
used in contemporary experiments (compared to previous best estimates, which
require 3 orders of magnitude more random sequences than currently used). The
second principal contribution is to show that randomized benchmarking can be
used to characterize time-dependent fluctuations in the noise strength.
In more detail, and in order of appearance, we show the following.
* •
We use the results derived later in the paper to obtain explicit confidence
intervals for the estimates $\hat{F}_{m}$ when $mr\ll 1$, where
$r=1-\mathcal{F}_{\rm avg}(\mathcal{E})$ is the average gate infidelity (Sec.
IV.1).
* •
Again, using results derived later, we show that a more thorough analysis of
randomized benchmarking data can be used to characterize time-dependent
Markovian noise, and consequently as a sufficient condition for the presence
of non-Markovian noise in a system (Sec. IV.2).
* •
We review representation theory and the Liouville representation of quantum
channels and prove some elementary results (Sec. V). We give an explicit proof
of bounds on the diamond norm (which characterizes the worst-case error rate)
in terms of the average gate fidelity (which characterizes the average error
rate). These give slight improvements over previously stated (but unproven)
bounds (Sec. V.4).
* •
We derive an expression for the mean of the randomized benchmarking
distribution with time-dependent noise (Sec. VI.1).
* •
We show that the variance for randomized benchmarking $d$-level systems with
average gate infidelity and sequences of length $m$ satisfies
$\displaystyle\sigma_{m}^{2}\leq 4d(d+1)mr+O(m^{2}r^{2}d^{4})\,.$ (4)
Furthermore, we provide an argument that suggests that this bound can be
improved to
$\displaystyle\sigma_{m}^{2}\leq mr+O(m^{2}r^{2}d^{4})\,.$ (5)
* •
For qubits, we improve the upper bound to
$\displaystyle\sigma_{m}^{2}\leq m^{2}r^{2}+\frac{7mr^{2}}{4}+6\delta
mr+O(m^{2}r^{3})+O(\delta m^{2}r^{2})\,,$ (6)
where $\delta$ quantifies the deviation from preparations and measurements in
a Pauli eigenstate. We use this improved bound to derive confidence intervals
that rigorously justify the use of a small number of random sequences for
qubits in the regime $mr\ll 1$.
* •
For the special case of single-qubit noise that is diagonal in the Pauli
basis, we further improve the upper bound to
$\displaystyle\sigma_{m}^{2}\leq\frac{11mr^{2}}{4}+O(m^{2}r^{3})\,,$ (7)
which is independent of preparations and measurements.
* •
We show that the variance for unital (but nonunitary) channels decays
exponentially to zero asymptotically, while the variance for nonunital noise
converges exponentially to a positive constant proportional to the degree of
nonunitality (as suitably quantified).
* •
We prove that our results are robust under gate-dependent noise, which is one
of the key assumptions under which randomized benchmarking produces a
meaningful result. Furthermore, since our results apply to interleaved
randomized benchmarking, gate dependence can be experimentally tested and used
to bound the contribution from gate-dependent terms.
## IV Analyzing data from randomized benchmarking with finite sampling
In this section, we summarize the implications of our results for analyzing
the data obtained from randomized benchmarking experiments. In particular, we
derive confidence intervals for the estimates $\hat{F}_{m}$ of $\bar{F}_{m}$
and show how randomized benchmarking can be used to characterize time-
dependent noise.
### IV.1 Confidence interval for randomized benchmarking
For a fixed sequence length $m$, randomized benchmarking provides an estimate
$\hat{F}_{m}$ of $\bar{F}_{m}$, which is exact in the limit when all random
sequences are sampled. We will only consider the variance $\sigma_{m}^{2}$ due
to sampling a finite number $K_{m}$ of random sequences of length $m$, and we
ignore the random fluctuations resulting from the use of a finite number of
measurements to estimate a probability.
In Ref. Magesan _et al._ (2012a), the variance-independent form of
Hoeffding’s inequality was used to estimate the number of sequences $K_{m}$
required to obtain a given level of accuracy. The estimate in Ref. Magesan
_et al._ (2012a) erroneously restricted the range of the random variable in
Hoeffding’s inequality. That is, they assumed that all the probabilities
$F_{m,s}$ lay in a strict subset of $[0,1]$. This assumption, while valid for
depolarizing noise, is not valid in general. A simple counterexample is where
the noise is a single-qubit preparation channel into the $|0\rangle\langle 0|$
state and $\rho=E=|0\rangle\langle 0|$. Then any sequence of $m$ gates ending
in an identity gate or a $z$-axis rotation has $F_{m,s}=1$, while any sequence
ending in an $X$ gate gives $F_{m,s}=0$. Correcting for this (which does not
change any of the conclusions of Ref. Magesan _et al._ (2012a)), the
variance-independent form of Hoeffding’s inequality requires $10^{5}$ samples
to ensure that the estimate $\bar{F}_{m}$ is within $5\times 10^{-3}$ of the
true mean $\bar{F}_{m}$ with $99\%$ probability. However, many experimental
implementations of randomized benchmarking only use 30–100 sequences for each
value of $m$ Brown _et al._ (2011); Gambetta _et al._ (2012); Magesan _et
al._ (2012b).
One of the principal contributions of this paper is to provide a theoretical
justification for choosing a relatively small number of sequences by showing
that the variance is small for the short sequences that are of practical
relevance. For the special case of qubits, we show that even for small $m$
(e.g. $m\approx 100$) the variance is at most $4\times 10^{-4}$ for currently
achievable gate infidelities $r\approx 10^{-4}$, which is comparable to the
numerical estimates presented in Fig. 1. Utilizing this very small variance
gives substantial improvements over the previous rigorous bounds obtained in
Refs. Magesan _et al._ (2012a); Kimmel _et al._ (2013). However, our bound
on the variance (which is numerically almost optimal for qubits) implies that
$K_{m}$ should scale _quadratically_ with $m$ to make the variance is
independent of $m$.
Our upper bound $\sigma_{m}^{2}\leq
m^{2}r^{2}+\tfrac{7}{4}mr^{2}+O(m^{2}r^{3})$ (for qubits, neglecting the
negligible $\delta r$ terms) can be used together with a stronger version of
Hoeffding’s inequality Hoeffding (1963) to obtain a rigorous confidence
interval comparable to the standard errors of the mean reported in current
experiments Brown _et al._ (2011). The stronger version of Hoeffding’s
inequality implies that
$\displaystyle\Pr\biggl{(}\left\lvert\hat{F}_{m}-\bar{F}_{m}\right\rvert>\epsilon\biggr{)}\leq
2\Bigl{[}H(\epsilon,\sigma_{m}^{2})\Bigr{]}^{K}\,,$ (8)
where $K$ is the number of randomly sampled sequences of length $m$ and
$\displaystyle
H(\epsilon,v)=\Bigl{(}\frac{1}{1-\epsilon}\Bigr{)}^{\frac{1-\epsilon}{v+1}}\Bigl{(}\frac{v}{v+\epsilon}\Bigr{)}^{\frac{v+\epsilon}{v+1}}\,.$
(9)
Consequently, sampling
$\displaystyle
K=-\frac{\log\bigl{(}2/\delta\bigr{)}}{\log\bigl{(}H(\epsilon,\sigma_{m}^{2})\bigr{)}}$
(10)
random sequences is sufficient to obtain an absolute precision of $\epsilon$
with probability $1-\delta$. Since $r$ is determined by the fitting procedure,
which in turn depends on the uncertainties, this procedure would be applied
recursively with an initial upper bound on $r$. Similarly, for qudits, a
straightforward generalization of the above argument can be used, but with
$\sigma_{m}^{2}\leq 4d(d+1)mr+O(d^{4}m^{2}r^{2})$. (There are various
inefficiencies in this estimate which mean that it does not reduce to the same
answer as above for $d=2$; see Theorem 10 for more details.)
To get a feel for the sort of estimates that this bound provides, consider the
following parameters for a single-qubit benchmarking experiment:
$m=100,r=10^{-4},\epsilon=1\%,\delta=1\%$, and use our upper bound of
$\sigma^{2}_{m}=m^{2}r^{2}+\tfrac{7}{4}mr^{2}$ (ignoring the higher-order
terms). Then our bound shows that $K=145$ random sequences suffices. This is
an improvement by orders of magnitude over the previous best rigorously
justifiable upper bound of $10^{5}$ using the variance-independent Hoeffding
inequality Magesan _et al._ (2012a).
Importantly, however, we note that the quadratic scaling with $m$ in the
regime $mr\ll 1$ seems to be necessary (see Fig. 1). Even in the optimal case
of noise that is diagonal in the Pauli basis, $K_{m}$ would still need to
scale linearly with $m$ to make the variance independent of $m$ (where having
the variance depend on $m$ would generally cause less weight to be assigned to
larger $m$ when fitting). The linear scaling can be understood intuitively as
following from the fact that there are $m$ places for an error in a sequence
of length $m$, and the errors could add up in the worst case. Therefore, a
corollary of our result is that longer sequence lengths should be averaged
over more random sequences in this regime.
Furthermore, we prove in Sec. VII that there are noise sources such that the
variance due to sampling random sequences is constant (or decays on an
arbitrarily long timescale). If such noise sources (including nonunital noise
and any unitary noise, such as over- and under-rotations) are believed to be
present, substantially more random sequences need to be sampled. As such,
randomized benchmarking is most reliable in the regime $mr\ll 1$, although,
since the next lowest order terms in our bound are $\delta m^{2}r^{2}$ and
$m^{2}r^{3}$, the lowest order bounds on the variance should be approximately
valid for $mr\approx 0.1$.
Figure 1: Plot of a random sampling of the exact variance $\sigma_{m}^{2}$ as
a function of the sequence length $m$ for randomized benchmarking with 100
randomly generated, time-independent noisy qubit channels. The noise was
sampled from the set of extremal qubit channels, characterized in Ref. Ruskai
_et al._ (2002), with average gate infidelity $r\lesssim 2.69\times 10^{-4}$.
Each channel was evolved for increasing sequence lengths to track the behavior
of the variance as a function of $m$, which is why the data points track
parabolic curves (furthermore, the spread in the parabolic curves is generated
by the spread in the infidelity of the samples). The green curve is the upper
bound $\sigma_{m}^{2}=m^{2}r^{2}+\tfrac{7mr^{2}}{4}$, where we neglect the
corrections to the bound at order $O(m^{2}r^{3})$ and corrections due to
measurement imprecision. Note that our bound is almost optimal. Our analytic
results show that the variance $\sigma_{m}^{2}$ will _increase_ with $m$, at
least until some threshold sequence length where the exponential decay for
generic channels proven in Theorem 17 begins to dominate.
### IV.2 Characterizing time-dependent noise
The original presentation of randomized benchmarking assumed that the noise
was approximately time independent (i.e., independent of the time step at
which the gate is applied), with any Markovian time-dependence being partially
characterized by deviations from the time-independent case Magesan _et al._
(2012a). However, in many practical applications there may be a nonnegligible
time dependence, which it would be desirable to characterize more fully.
We show that randomized benchmarking can also be used to characterize time-
dependent noise, provided the gate-dependence is negligible (in the sense
established in Theorem 18) and that the time-dependent noise is identically
distributed between different experiments. However, the number of random
sequences of length $m$ will typically need to be increased relative to the
number required for time-independent noise. In particular, we will show in
Theorem 8 that
$\displaystyle\bar{F}_{m}=A+B\prod_{t=1}^{m}f_{t}\,,$ (11)
where $A$ and $B$ are constants that depend only upon the preparation and
measurement procedures (and so account for SPAM) and the average gate fidelity
at time $t$ is $F_{t}=f_{t}+(1-f_{t})/d$, where $d$ is the dimensionality of
the system being benchmarked. In the case of time-independent noise, $f_{t}$
is a constant and Eq. (11) reduces to the standard equation for the fidelity
decay curve.
By performing randomized benchmarking for a set of sequence lengths $m_{1}$
and $m_{2}$, we can estimate $\hat{F}_{m_{j}}$ with associated uncertainties
$\delta_{j}$. Combining these estimates with a procedure for obtaining an
estimate $\hat{A}$ of $A$ with associated uncertainty $\delta_{A}$ Johnson
_et al._ (2014), we can estimate the ratio
$\displaystyle\frac{\bar{F}_{m_{2}}-\bar{A}}{\bar{F}_{m_{1}}-\bar{A}}=\prod_{t=m_{1}+1}^{m_{2}}f_{t}$
(12)
with uncertainty on the order of
$\displaystyle\delta_{1,2,A}\approx\sqrt{\left(\delta_{1}+\delta_{A}\right)^{2}+\left(\delta_{2}+\delta_{A}\right)^{2}}\,.$
(13)
Therefore we can estimate the average gate infidelity $r$ over the time
interval $[m_{1}+1,m_{2}]$. From our rigorous analysis, we can infer that
$\delta_{1}$ and $\delta_{2}$ will be small for small $m_{j}r$, while
$\delta_{A}$ will be determined only by finite measurement statistics.
Furthermore, when $m_{2}\approx m_{1}$, $\prod_{t=m_{1}+1}^{m_{2}}f_{t}\approx
1-r(m_{2}-m_{1})$ and so the above method gives a reliable method of
characterizing the time-dependent gate fidelity.
We also note that if there are no temporal correlations in the noise, than all
of the parameters $r_{t}$ (where $r_{t}$ is the average gate infidelity at
time $r$) are lower-bounded by zero. Therefore any negative values (or average
values) of $r_{t}$ are an indicator of temporal correlations in the noise,
that is, of non-Markovian behavior.
## V Mathematical preliminaries
Randomized benchmarking involves composing random sequences of quantum
channels that are sampled in a way which approximates a group average. For
this reason, it is natural to consider both the representation theory of
groups and the structure of quantum channels, especially the composition of
channels. In this section we collect several mathematical results in this vein
that we will need to prove our main results. We begin by considering group
representation theory, and in particular prove a proposition showing how the
tensor product of certain representations couple together. Most of this
material is standard and can be found in any textbook on the subject, e.g.
Goodman and Wallach (2009).
### V.1 Representation Theory and Some Useful Lemmas
A _representation_ (rep) of a group $\mathcal{G}$ is a pair $(\phi,V)$, where
$V$ is vector space known as the representation space (which we always take to
be $\mathbb{R}^{d}$ or $\mathbb{C}^{d}$ for different values of $d$) and
$\phi:\mathcal{G}\to\mathsf{GL}(V)$—where $\mathsf{GL}(V)$ is the general
linear group over $V$—is a homomorphism. A rep is _faithful_ if $\phi$ is
injective and _unitary_ (resp. _orthogonal_) if $\phi(g)$ is a unitary (resp.
orthogonal) operator for all $g\in\mathcal{G}$. The dimension of a rep is the
dimension of $V$. A _subrepresentation_ (subrep) is a pair $(\phi_{W},W)$ such
that $\phi(g)W\subseteq W$ for all $g\in\mathcal{G}$ and $\phi_{W}$ denotes
the restriction of $\phi$ to the subspace $W$. We sometimes refer to a space,
subspace, or homomorphism as being a rep or subrep, with the complementary
ingredients understood from the context.
A rep is called _irreducible_ or an _irrep_ if the only subreps are
$\emptyset$ and $V$. Since the reps we consider are unitary reps of compact
groups, if $W$ is a subrep of $V$ then the orthogonal complement $W^{\bot}$ is
a subrep as well. Therefore any rep can be decomposed into a direct sum of
irreps, which may occur with some multiplicity. Any basis that decomposes a
rep into a direct sum of irreps is called a Schur basis.
The simplest rep is the trivial rep $(1,\mathbb{C})$, which is also an irrep.
The trivial rep is defined for any group $\mathcal{G}$ and take any element of
$\mathcal{G}$ to 1\. While the trivial rep deserves its name, it frequently
appears as a subrep of tensor powers of other reps and so will appear
throughout this paper.
The randomized benchmarking protocol is designed so that the sequence of
operators applied to a system correspond to noise channels conjugated by
uniformly random elements of a group $\mathcal{G}$. Given a rep $(\phi,V)$ of
a group $\mathcal{G}$, a matrix $A\in\mathsf{GL}(V)$ and an element
$g\in\mathcal{G}$, we define $A^{g}=\phi(g)A\phi(g^{-1})$. The uniform average
of this action on $A$ is called the _$\mathcal{G}$ -twirl_ of $A$, and is
given by
$A^{\mathcal{G}}=\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}A^{g}$.
Note that, for notational convenience, the map $\phi$ is left implicit but
will always be obvious given the dimensionality of the matrix being twirled.
An important property of $A^{\mathcal{G}}$ is that it commutes with the action
of $\mathcal{G}$ for any rep $(\phi,V)$ (reducible or not) since $\phi$ is a
homomorphism and $\mathcal{G}$ is a group. That is,
$A^{\mathcal{G}}=(A^{g})^{\mathcal{G}}=(A^{\mathcal{G}})^{g}$ for all
$g\in\mathcal{G}$.
Expressions for the expected value $\bar{F}_{m}$ and variance $\sigma_{m}^{2}$
for the randomized benchmarking protocol for a fixed value of $m$ will be
obtained using the following propositions.
###### Proposition 1.
Let $(\phi,\mathbb{C}^{d})$ be a nontrivial $d$-dimensional irreducible
representation of a group $\mathcal{G}$ and
$A\in\mathsf{GL}(\mathbb{C}^{d}),B\in\mathsf{GL}(\mathbb{C}^{d+1})$. Then
* •
$A^{\mathcal{G}}=a\mathbbm{1}_{d}$;
* •
$B^{\mathcal{G}}=B_{11}\oplus b\mathbbm{1}_{d}$ [where the representation of
$\mathcal{G}$ is $(1\oplus\phi,\mathbb{C}^{d+1})$]; and
* •
$\sum_{g\in\mathcal{G}}\phi(g)=0$,
where $a=\mathrm{Tr}A/d$ and $b=(\mathrm{Tr}B-B_{11})/d$.
Proof. All three statements follow directly from Schur’s Lemma Goodman and
Wallach (2009). $\Box$
###### Proposition 2.
If $(\phi,V)$ is an irreducible representation of a finite group $\mathcal{G}$
with a real-valued character $\chi_{\phi}$, then the trivial representation is
a subrepresentation of $(\phi,V)^{\otimes 2}$ with multiplicity 1.
Proof. As the rep is irreducible, Schur’s orthogonality relations Goodman and
Wallach (2009) give
$\displaystyle\left\lvert\mathcal{G}\right\rvert=\sum_{g\in\mathcal{G}}\chi_{\phi}(g)^{*}\chi_{\phi}(g)=\sum_{g\in\mathcal{G}}\chi_{\phi}(g)^{2}=\sum_{g\in\mathcal{G}}\chi_{\phi^{\otimes
2}}(g)\chi_{1}(g)\,,$ (14)
where we have used $\chi_{\phi^{\otimes
2}}(g)=\left[\chi_{\phi}(g)\right]^{2}$ and that the character for the trivial
representation is $\chi_{1}(g)=1$ for all $g\in\mathcal{G}$. $\Box$
### V.2 The Liouville Representation of Quantum Channels
A quantum channel is a linear map
$\mathcal{E}:\mathcal{D}_{d_{1}}\to\mathcal{D}_{d_{2}}$, where
$\mathcal{D}_{d}$ is the set of $d$-dimensional density operators. Quantum
channels can be represented in a variety of equivalent ways, with different
representations naturally suited to particular applications.
In this paper, we will primarily use the Liouville representation because it
is defined so that quantum channels compose under matrix multiplication. We
occasionally also use the Choi representation in order to apply results from
the literature, but we will introduce it only as required.
#### V.2.1 States and measurements
We begin by introducing the Liouville representation (also called the transfer
matrix representation) of quantum states and measurements. States and
measurement effects (i.e., elements of a positive-operator valued measure, or
POVM) can be viewed as channels from
$\mathcal{E}:\mathbb{R}\to\mathcal{D}_{d}$ and
$\mathcal{E}:\mathcal{D}_{d}\to\mathbb{R}$ respectively, hence they can be
treated on the same footing as any other quantum channel. However, we
introduce them separately for pedagogical and notational clarity.
In the standard formulation of quantum mechanics in terms of density operators
and POVMs, a quantum state $\rho\in\mathcal{D}_{d}$ is any Hermitian, positive
semi-definite operator such that $\mathrm{Tr}\rho=1$. In addition, we always
have $\mathrm{Tr}\rho^{2}\in\left[0,1\right]$. We can always choose a basis
$\mathbb{A}=\left\\{A_{0},A_{1},\ldots,A_{d^{2}-1}\right\\}$ of orthonormal
operators for $\mathsf{GL}(\mathbb{C}^{d})$, where orthonormality is according
to the Hilbert-Schmidt inner product $\langle
A,B\rangle=\mathrm{Tr}\left(A^{\dagger}B\right)$. We can expand any density
operator relative to such a basis as $\rho=\sum_{j}\rho_{j}A_{j}$, where
$\rho_{j}=\langle A_{j},\rho\rangle$. Throughout this paper we set
$A_{0}=\mathbbm{1}/\sqrt{d}$, which fixes $\rho_{0}=\mathrm{Tr}\rho/\sqrt{d}$,
and makes all other $A_{j}$ for $j\not=0$ traceless.
We can then identify a density operator $\rho$ with a corresponding column
vector
$\displaystyle|\rho)=\left(\begin{array}[]{c}\rho_{0}\\\
\vec{\rho}\end{array}\right)\in\mathbb{C}^{d^{2}}$ (17)
such that $\vec{\rho}_{j}=\rho_{j}$ for $j=1,\ldots,d^{2}-1$. Here we make the
important distinction between the density operator itself, $\rho$, and the
representation of $\rho$ in terms of the column vector $|\rho)$. Note that
$|\rho)$ is just a generalized version of a Bloch vector (with a different
normalization) for $d\geq 2$.
The conditions for $\rho$ to correspond to a density operator now translate
into geometric conditions on $|\rho)$. In particular, we will use the fact
that
$\left\lVert|\rho)\right\rVert_{2}^{2}=\mathrm{Tr}\rho^{2}\in\left[0,1\right]$,
where $\left\lVert v\right\rVert_{2}$ for $v\in\mathbb{C}^{d^{2}}$ is the
standard isotropic Euclidean norm.
Measurements in the standard formulation correspond to POVMs, that is, to sets
of Hermitian, positive semidefinite operators $\left\\{E_{j}\right\\}$ such
that $\sum_{j}E_{j}=\mathbbm{1}$. As with quantum states, we can expand an
element $E$ of a POVM (an effect) as $E=\sum E_{j}A_{j}^{\dagger}$, where
$E_{j}=\langle E,A_{j}\rangle$. We then identify an effect $E$ with a row
vector
$\displaystyle(E|=\bigl{(}\begin{array}[]{cc}E_{0}&\vec{E}\end{array}\bigr{)}\in{\mathbb{C}^{*}}^{d^{2}}\,,$
which must satisfy similar conditions to $|\rho)$.
In this formalism, the probability of observing an effect $E$ given that the
quantum state $\rho$ was prepared is $p(E|\rho)=\mathrm{Tr}E\rho=(E|\rho)$.
#### V.2.2 Transformations
For simplicity, we will only consider quantum channels that are either states,
measurements or completely positive and trace-preserving (CPTP) maps
$\mathcal{E}:\mathcal{D}_{d}\to\mathcal{D}_{d}$. We do not consider channels
that reduce the trace or change the dimension because, while conceptually no
more difficult, they require cumbersome additional notation and we do not use
any such channels.
A quantum channel $\mathcal{E}$ maps a density operator $\rho$ to another
density operator $\mathcal{E}(\rho)$. We want to determine the map
$\mathcal{E}$ between the corresponding vectors $|\rho)$ and
$(\mathcal{E}(\rho)|$. Since quantum channels are linear,
$\displaystyle\mathcal{E}(\rho)=\sum_{j}\mathcal{E}(A_{j})\rho_{j}\,,$
which implies that
$\displaystyle\bigl{\lvert}\mathcal{E}(\rho)\bigr{)}_{k}=\sum_{j}\langle
A_{k},\mathcal{E}(A_{j})\rangle\bigl{\lvert}\rho_{j}\bigr{)}\,.$
That is, $\bigl{\lvert}\mathcal{E}(\rho)\bigr{)}=\mathcal{E}|\rho)$ where we
abuse notation slightly and define $\mathcal{E}$ as the matrix such that
$\mathcal{E}_{j,k}=\langle A_{k},\mathcal{E}(A_{j})\rangle$. That is, we use
$\mathcal{E}$ to denote both the abstract operator as well as its
representation as a matrix acting on vectors $|\rho)$. In this representation,
the identity channel is represented by $\mathbbm{1}_{d^{2}}$, the composition
of two channels is given by matrix multiplication, and furthermore, the
conjugate channel of a unitary channel $\mathcal{E}$ is given by
$\mathcal{E}^{\dagger}$. In particular, these properties imply that the
Liouville representation of the unitary channels is a faithful and unitary
representation of $\mathsf{U}(d)$ (though technically, it is a projective
representation since a global phase is lost). Given our choice of $\mathbb{A}$
(recall that we have fixed $A_{0}$) and the fact that we consider only trace-
preserving channels, we will always write the matrix representation of a
quantum channel as
$\displaystyle\mathcal{E}=\left(\begin{array}[]{cc}1&0\\\
\alpha(\mathcal{E})&\varphi(\mathcal{E})\end{array}\right)\,.$ (20)
A channel $\mathcal{E}$ is unital (i.e., the identity is a fixed point of the
channel) iff $\alpha(\mathcal{E})=0$. Therefore we can regard
$\left\lVert\alpha(\mathcal{E})\right\rVert$ as quantifying the nonunitality
of $\mathcal{E}$.
The representation $\bigl{(}\varphi,\mathbb{C}^{d^{2}-1}\bigr{)}$ of
$\mathsf{U}(d)$ is irreducible Gross _et al._ (2007), which will play a
crucial role in our analysis of randomized benchmarking since it allows us to
use tools from representation theory such as Schur’s Lemma. Note also that the
representation $(\varphi,\mathbb{C}^{d^{2}-1})$ of any subgroup
$\mathcal{G}\subseteq\mathsf{U}(d)$ that is a unitary 2-design is also
irreducible by the same argument (which can be regarded as a defining property
of a unitary 2-design Gross _et al._ (2007)). Therefore we can also use tools
from representation theory when considering channels twirled over a unitary
2-design. This fact allows randomized benchmarking to be performed efficiently
because unitary 2-designs can be efficiently sampled while the full unitary
group cannot Emerson _et al._ (2005); Dankert _et al._ (2009).
The representation $\varphi(g)$ of $\mathcal{G}$ will be one of the basic
tools we use in this paper. As such, whenever $g$ appears in a matrix
multiplication, it will refer to $\varphi(g)$.
Randomized benchmarking will allow for the estimation of
$\displaystyle
f(\mathcal{E}):=\tfrac{1}{d^{2}-1}\mathrm{Tr}\varphi(\mathcal{E})\,,$ (21)
which corresponds to the average gate fidelity of $\mathcal{E}$ with the
identity channel. We will sometimes omit the argument of $\alpha$, $\varphi$
and $f$, or indicate the argument via a subscript. However, in all cases the
argument will be clear from the context.
#### V.2.3 Properties of channels in the Liouville representation
Since the Liouville representation associates a unique matrix to each channel,
we can characterize properties of quantum channels by properties of the
corresponding matrix. In particular, we will consider the spectral radius,
$\displaystyle\varrho(M)=\max_{j}\left\lvert\eta_{j}(M)\right\rvert\,,$ (22)
and the spectral norm, $\left\lVert M\right\rVert_{\infty}=\max\sigma_{j}(M)$,
where $\left\\{\eta_{j}(M)\right\\}$ and $\left\\{\sigma_{j}(M)\right\\}$ are
the eigenvalues and singular values of a matrix $M$ respectively. These norms
satisfy $\varrho(M)\leq\left\lVert\mathcal{M}\right\rVert_{\infty}$, which we
will use to obtain bounds on valid quantum channels.
###### Proposition 3.
Let $\mathcal{E}$ be a completely positive map. Then the adjoint channel
$\mathcal{E}^{\dagger}$ is also completely positive.
Proof. Any map can be written as
$\displaystyle\mathcal{E}=\sum K_{j}\otimes L_{j}^{T}\,,$ (23)
where the superscript $T$ denotes the transpose and the $K_{j}$ and $L_{j}$
are Kraus operators for $\mathcal{E}$. We then have
$\displaystyle\mathcal{E}^{\dagger}=\sum K_{j}^{\dagger}\otimes L_{j}^{*}\,,$
(24)
where the $*$ denotes complex conjugation.
By Choi’s theorem on completely positive maps, $\mathcal{E}$ is completely
positive if and only if Kraus operators can be chosen so that
$L_{j}=K_{j}^{\dagger}$. Therefore Kraus operators $K_{j}^{\dagger}$ and
$L_{j}^{\dagger}$ for $\mathcal{E}^{\dagger}$ can be chosen so that
$L_{j}^{\dagger}=(K_{j}^{\dagger})^{\dagger}$. $\Box$
###### Corollary 4.
The adjoint channel of a unital, completely-positive and trace-preserving
channel is also a unital, completely-positive and trace-preserving channel.
###### Proposition 5.
Any completely positive and trace-preserving channel
$\mathcal{E}:\mathcal{D}_{d}\to\mathcal{D}_{d}$ satisfies the following
relations:
(i) $\displaystyle\det\mathcal{E}$ $\displaystyle\leq 1\,,$ (ii)
$\displaystyle\left\lVert\mathcal{E}\right\rVert_{\infty}$
$\displaystyle\leq\sqrt{d}\,,$ (iii) $\displaystyle\varrho(\mathcal{E})$
$\displaystyle\leq 1\,,$ (iv)
$\displaystyle\left\lVert\alpha(\mathcal{E})\right\rVert_{2}$
$\displaystyle\leq\sqrt{d-1}\,.$ (25)
Inequality (i) is saturated if and only if $\mathcal{E}$ is unitary.
Furthermore, if $\mathcal{E}$ is unital, then (ii) can be improved to
$\left\lVert\mathcal{E}\right\rVert_{\infty}=1$.
Proof. (i): See Ref. (Wolf and Cirac, 2008, Thm 2).
(ii): See (Pérez-García _et al._ , 2006, Thm. II.1), noting that
$\|\mathcal{E}\|_{\infty}=\|\mathcal{E}\|_{2\to 2}$.
(iii): See Evans and Hoegh-Krohn (1978).
(iv): For any density operator $\rho$, we have
$\mathrm{Tr}\rho^{2}\in\left[0,1\right]$. In particular consider
$\mathcal{E}(\mathbbm{1}/d)$, which must be a density operator since
$\mathbbm{1}/d$ is a density operator and $\mathcal{E}$ is a quantum channel.
Then
$\displaystyle\mathrm{Tr}\,\mathcal{E}(\mathbbm{1}/d)^{2}=\left\lVert\mathcal{E}B(\mathbbm{1}/d)\right\rVert_{2}^{2}=\frac{1+\left\lVert\alpha(\mathcal{E})\right\rVert_{2}^{2}}{d}\leq
1$ (26)
gives the desired result.
Now let $\mathcal{E}$ be a unital channel. Then $\mathcal{E}^{\dagger}$ and
thus $\mathcal{E}^{\dagger}\mathcal{E}$ are also channels by Corollary 4.
Substituting (i) into the equality
$\varrho(\mathcal{E}^{\dagger}\mathcal{E})=\left\lVert\mathcal{E}\right\rVert_{\infty}$
gives the improved bound. $\Box$
### V.3 Representing noisy channels
An attempt to physically implement a quantum channel $\mathcal{E}$ will
generally result in some other channel $\mathcal{E}^{\prime}$, with the aim
being, loosely speaking, to make $\mathcal{E}^{\prime}$ as close to
$\mathcal{E}$ as possible. We will now outline how noisy channels can be
related to the intended channel in the linear representation.
Consider an attempt to implement a target unitary channel $\mathcal{U}$ that
results in some (noisy) channel $\mathcal{E}$. Then since $\mathcal{E}$ is a
real square matrix, it can be written as $\mathcal{E}=LQ$ where $L$ is a lower
triangular matrix and $Q$ is an orthogonal matrix. Since $\mathcal{U}$ is an
orthogonal matrix, we can always write $\mathcal{E}=\Lambda^{\rm
post}(\mathcal{U})\mathcal{U}$, where $\Lambda^{\rm
post}(\mathcal{U})=LQ\mathcal{U}^{T}$. Similarly, we can always write
$\mathcal{E}=\mathcal{U}\Lambda^{\rm pre}(\mathcal{U})$.
While the difference between these expressions is trivial for any _single_
channel, it can cause confusion when comparing channels. Since $\Lambda^{\rm
pre}(\mathcal{U})=\mathcal{U}^{T}\Lambda^{\rm post}(\mathcal{U})\mathcal{U}$,
the notion of “the” noise in an implementation of $\mathcal{U}$ depends on
which expression is used. (We will fix a representation below to avoid this
ambiguity.)
The convergence of randomized benchmarking depends crucially upon the
assumption that the noise is approximately independent of the target. However,
in a general scenario, at most one of $\Lambda^{\rm post}$ or $\Lambda^{\rm
pre}$ will be approximately independent of the target, with the specific
choice depending upon the physical implementation. As a specific example,
consider amplitude damping for a single qubit, which can be written as
$\displaystyle\Delta=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&\sqrt{g}&0&0\\\
0&0&\sqrt{g}&0\\\ 1-g&0&0&g\\\ \end{array}\right)$ (31)
in the Pauli basis $\tfrac{1}{\sqrt{2}}\left(\mathbbm{1},X,Y,Z\right)$, where
$g\in\left[0,1\right]$ determines the strength of the damping. Assume that
this noise is applied independently from the left (i.e., $\Lambda^{\rm
post}=\Delta$), independently of the target. Then for $X$ and $Z$,
$\displaystyle\Lambda^{\rm pre}(Z)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\sqrt{g}&0&0\\\ 0&0&\sqrt{g}&0\\\ 1-g&0&0&g\\\ \end{array}\right)\quad$
$\displaystyle\quad\Lambda^{\rm pre}(X)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\sqrt{g}&0&0\\\ 0&0&\sqrt{g}&0\\\ -(1-g)&0&0&g\\\ \end{array}\right)$ (40)
which is only independent of the target when $g=1$ (i.e., when there is no
noise).
In this work, we write noise operators as _pre-multiplying the target_ rather
than post-multiplying the target as in Ref. Magesan _et al._ (2011). The
reason for this change is so that the residual noise term that is not averaged
is in the first time step rather than the last and so is independent of the
sequence length. While this simplifies the analysis, all the results of this
paper can be derived for the other form with small modifications.
### V.4 Measures of noise
The fidelity and trace distance between two quantum states are defined as
$\displaystyle
F(\rho,\sigma)=\bigl{\lVert}\sqrt{\rho}\sqrt{\sigma}\bigr{\rVert}_{1}^{2}\,,$
$\displaystyle D(\rho,\sigma)=\frac{1}{2}\lVert\rho-\sigma\rVert_{1}\,,$ (41)
respectively111Note that some authors define fidelity to be the square root of
the fidelity defined here., where the 1-norm (or trace norm) is given by
$\lVert X\rVert_{1}=\mathrm{Tr}\sqrt{X^{\dagger}X}$. These two quantities are
related by the Fuchs-van de Graaf inequalities Fuchs and Graaf (1999),
$\displaystyle 1-\sqrt{F(\rho,\sigma)}\leq
D(\rho,\sigma)\leq\sqrt{1-F(\rho,\sigma)}\,,$ (42)
where the right-hand inequality is always saturated when both states are pure.
When one of the states is a pure state $\psi$, the left-hand inequality in Eq.
(42) can be sharpened to
$\displaystyle 1-F(\psi,\sigma)\leq D(\psi,\sigma)\,.$ (43)
Both of these quantities for quantum states can be promoted to distance
measures for quantum channels Gilchrist _et al._ (2005). Two such measures
are the average gate fidelity and the diamond distance.
The average gate fidelity between a channel $\mathcal{E}$ and a unitary
$\mathcal{U}$ is defined to be
$\displaystyle F_{\rm avg}(\mathcal{E},\mathcal{U})$
$\displaystyle=\int\mathrm{d}\psi
F\left[\mathcal{E}(\psi),\mathcal{U}(\psi)\right]\,,$ (44)
where $\mathrm{d}\psi$ is the unitarily invariant Haar measure. For
convenience, it is typical to define a single argument version,
$\displaystyle F_{\rm avg}(\mathcal{U}^{\dagger}\mathcal{E})=F_{\rm
avg}(\mathcal{E},\mathcal{U})\,,$ (45)
which is technically the average gate fidelity between
$\mathcal{U}^{\dagger}\mathcal{E}$ and $\mathbbm{1}$.
The _diamond distance_ between two quantum channels $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$ with $\mathcal{E}_{j}:\mathcal{D}_{d}\to\mathcal{D}_{d}$ is
defined in terms of a norm of their difference
$\Delta=\mathcal{E}_{1}-\mathcal{E}_{2}$ as follows:
$\displaystyle\frac{1}{2}\left\lVert\Delta\right\rVert_{\diamond}=\frac{1}{2}\sup_{\psi}\left\lVert\mathbbm{1}_{d}\otimes\Delta(\psi)\right\rVert_{1}\,.$
(46)
The norm in the above definition is indeed a valid norm, called the diamond
norm, and it extends naturally to any Hermiticity-preserving linear map
between operators. The factor of $1/2$ is to ensure that the diamond distance
between two channels is bounded between $0$ and $1$.
The diamond distance is useful for several reasons. Firstly, allowing for
larger entangled inputs does not change the value of the diamond distance,
hence it is _stable_. Secondly, it has an operational meaning as determining
the optimal success probability for distinguishing two unknown quantum
channels $\mathcal{E}_{1}$ and $\mathcal{E}_{2}$ Kitaev (1997). Equivalently,
the diamond distance gives the worst-case error rate between the pair of
channels. Although we will not be able to measure the diamond distance
directly, we will be able to bound it in terms of measurable quantities
obtainable via randomized benchmarking.
To obtain upper and lower bounds on the diamond norm, we will use the
following two lemmas to relate the average gate fidelity to the trace norm of
the corresponding Choi matrix and then to the diamond norm. Recall that the
Choi matrix of a linear map $\Delta$ is given by
$J(\Delta)=\Delta\otimes\mathbbm{1}_{d}(\Phi)$, where
$\Phi=\sum_{j\in\mathbb{Z}_{d}}|jj\rangle/\sqrt{d}$ is the maximally entangled
state. The first of these lemmas was proven in Refs. Horodecki _et al._
(1999); Nielsen (2002).
###### Lemma 6 (Horodecki _et al._ (1999); Nielsen (2002)).
The average fidelity of a CPTP map $\mathcal{E}$ is related to its Choi matrix
$J(\mathcal{E})$ by
$\displaystyle(d+1)F_{\mathrm{avg}}(\mathcal{E})=dF\bigl{[}\Phi,J(\mathcal{E})\bigr{]}+1\,.$
(47)
###### Lemma 7.
Let $\Delta$ be a Hermiticity-preserving linear map between $d$-dimensional
operators. Then the following inequalities bound the diamond norm and are
saturated:
$\displaystyle\left\lVert
J(\Delta)\right\rVert_{1}\leq\left\lVert\Delta\right\rVert_{\diamond}\leq
d\left\lVert J(\Delta)\right\rVert_{1}\,.$ (48)
Proof. We first prove the lower bound and show that it is saturated. We have
$\displaystyle\left\lVert\Delta\right\rVert_{\diamond}=\sup_{\psi}\left\lVert\Delta\otimes\mathbbm{1}_{d}(\psi)\right\rVert_{1}\geq\left\lVert\Delta\otimes\mathbbm{1}_{d}(\Phi)\right\rVert_{1}=\left\lVert
J(\Delta)\right\rVert_{1}\,.$ (49)
To see that the above inequality is saturated, simply let
$\Delta=\mathbbm{1}_{d}$.
To prove the upper bound, we write
$\displaystyle\left\lVert\Delta\right\rVert_{\diamond}=\sup\Bigl{\\{}\left\lVert\left(\mathbbm{1}_{d}\otimes\sqrt{\rho_{0}}\right)J(\Delta)\left(\mathbbm{1}_{d}\otimes\sqrt{\rho_{1}}\right)\right\rVert_{1}:\rho_{0},\rho_{1}\in\mathcal{D}_{d}\Bigr{\\}}$
(50)
which follows from Theorem 6 of Ref. Watrous (2012). Using (Bhatia, 1997, Thm.
IX.4.5) and the triangle inequality, then the submultiplicitivity of the norm,
we find that
$\displaystyle\left\lVert\Delta\right\rVert_{\diamond}$
$\displaystyle\leq\sup\bigl{\\{}\tfrac{1}{2}\left\lVert\left(\mathbbm{1}_{d}\otimes\rho_{0}\right)J(\Delta)\right\rVert_{1}+\tfrac{1}{2}\left\lVert
J(\Delta)\left(\mathbbm{1}_{d}\otimes\rho_{1}\right)\right\rVert_{1}:\rho_{0},\rho_{1}\in\mathcal{D}_{d}\bigr{\\}}$
$\displaystyle\leq\sup\bigl{\\{}\tfrac{1}{2}\left\lVert\left(\mathbbm{1}_{d}\otimes\rho_{0}\right)\right\rVert_{1}\left\lVert
J(\Delta)\right\rVert_{1}+\tfrac{1}{2}\left\lVert
J(\Delta)\right\rVert_{1}\left\lVert\left(\mathbbm{1}_{d}\otimes\rho_{1}\right)\right\rVert_{1}:\rho_{0},\rho_{1}\in\mathcal{D}_{d}\bigr{\\}}$
$\displaystyle=\tfrac{1}{2}d\left\lVert
J(\Delta)\right\rVert_{1}\sup\bigl{\\{}\left\lVert\rho_{0}\right\rVert_{1}+\left\lVert\rho_{1}\right\rVert_{1}:\rho_{0},\rho_{1}\in\mathcal{D}_{d}\bigr{\\}}$
$\displaystyle=d\left\lVert J(\Delta)\right\rVert_{1}\,.$ (51)
To see that this bound is saturated, let $\Delta$ be the projector onto
$|0\rangle\langle 0|$. $\Box$
We note that it would be interesting to see if the previous bounds are still
saturated when restricting the input $\Delta$ to be a difference of channels.
## VI Time-dependent gate-independent errors in randomized benchmarking
We consider the ideal case in which the noise depends only upon the time step.
For such types of noise, we derive expressions for the mean $\bar{F}_{m}$ and
variance $\sigma_{m}^{2}$ of the randomized benchmarking distribution
$\left\\{F_{m,k}\right\\}$ for fixed $m$. In particular, we will show that for
unital but nonunitary noise, $\sigma_{m}^{2}$ decreases exponentially with
$m$, while for non-unital noise, $\sigma_{m}^{2}$ converges to a constant
dependent on the strength of the non-unitality. We will also upper-bound the
variance for small $m$, which enables the derivation of rigorous confidence
intervals for the estimate of the average gate infidelity in Sec. IV.1. We
will also show that our results are stable under gate-dependent perturbations
in the noise in Sec. VIII.
In order to present our results in as clear a form as possible, we will only
explicitly consider the original proposal for randomized benchmarking.
Interleaved benchmarking can also be treated in an almost identical manner,
except that the noise is conjugated by the interleaved gate and is redefined
to absorb the noise term for the interleaved gate.
Denoting the noise at time step $t$ by $\Lambda_{t}$, the sequence of
operations applied to the system in the randomized benchmarking experiment
with sequence $s\in\mathbb{N}_{\left\lvert\mathcal{G}\right\rvert}^{m}$ is
$\displaystyle\mathcal{S}_{s}=\prod_{t=m}^{0}g_{t}\Lambda_{t}\,.$ (52)
Here $g_{t}$ are the ideal unitary gates which are sampled from any unitary
2-design $\mathcal{G}$.
To make it easier to analyze the above expression, we define
$h_{t}=\prod_{b=m}^{t}g_{b}$, so that $h_{0}=\mathbbm{1}$, $h_{m}=g_{m}$ and
$g_{t}=h_{t+1}^{\dagger}h_{t}$ for all $t\in(0,m)$. Uniformly sampling the
$g_{t}$ is equivalent to uniformly sampling the $h_{t}$ since $\mathcal{G}$ is
a group; the exception is $h_{0}$ and $g_{0}$, which are chosen so that the
product of all the gates is the identity (c.f. Sec. II). We can then rewrite
Eq. (52) as
$\displaystyle\mathcal{S}_{s}=h_{m}\Lambda_{m}\ldots
h_{2}^{\dagger}h_{1}\Lambda_{1}h_{1}^{\dagger}\Lambda_{0}=\prod_{t=m}^{1}\Lambda_{t}^{h_{t}}\,,$
(53)
where we incorporate the first noise term into the preparation by setting
$\rho\leftarrow\Lambda_{0}\rho$. This redefinition of $\rho$ is independent of
the sequence length because we write the noise as pre- rather than post-
multiplying the target. (Note that if the noise post-multiplied the target,
then incorporating the final noise term in $E$ would make $E$ depend on the
sequence length $m$.)
The probability of observing the outcome $E$ for the sequence $S_{s}$ is
$F_{m,k}=(E|\mathcal{S}_{s}|\rho)$. We regard the set
$\left\\{F_{m,k}\right\\}$ as the realizations of a random variable with mean
$\bar{F}_{m}$ and variance $\sigma_{m}^{2}$. Randomized benchmarking then
corresponds to randomly sampling from the distribution
$\left\\{F_{m,s}\right\\}$ (which we henceforth refer to as the randomized
benchmarking distribution) to approximate the mean $\bar{F}_{m}$.
### VI.1 Mean of the benchmarking distribution
We now derive an expression for $\bar{F}_{m}$ for general CPTP maps with time-
dependent noise. A similar expression was derived for time-_independent_ noise
in Ref. Magesan _et al._ (2011). We will then show how $\bar{F}_{m}$ can be
used to approximate quantities of experimental interest, namely, the SPAM
error, average time-dependent gate fidelity and the worst-case error due to
the noise.
###### Theorem 8.
The mean of the distribution $\left\\{F_{m,s}\right\\}$ for fixed $m$ is
$\displaystyle\bar{F}_{m}=E_{0}\rho_{0}+\vec{E}\cdot\vec{\rho}\prod_{t=1}^{m}f_{t}\,.$
(54)
Proof. By definition, the mean is
$\displaystyle\bar{F}_{m}=\left\lvert\mathcal{G}\right\rvert^{-m}\sum_{s\in\mathbb{N}_{\left\lvert\mathcal{G}\right\rvert}^{m}}(E|\mathcal{S}_{s}|\rho)=(E|\prod_{t=m}^{1}\Lambda^{\mathcal{G}}|\rho)\,.$
(55)
Using Proposition 1 gives
$\displaystyle\bar{F}_{m}=\left(\begin{array}[]{c
c}E_{0}&\vec{E}\end{array}\right)\left(\begin{array}[]{c c}1&0\\\
0&\prod_{t=1}^{m}f_{t}\mathbbm{1}_{d^{2}-1}\end{array}\right)\left(\begin{array}[]{c}\rho_{0}\\\
\vec{\rho}\end{array}\right)\,.$ (61)
$\Box$
The parameters $E_{0}\rho_{0}$ and $\vec{E}\vec{\rho}$ directly characterize
the quality of the state and measurement procedure (with the caveat that
$\rho$ has been redefined to include a noise term), since
$\mathrm{Tr}E\rho=E_{0}\rho_{0}+\vec{E}\vec{\rho}$. This can be viewed as an
instance of gate set tomography using a limited number of combinations of
gates Blume-Kohout _et al._ (2013).
The parameters $f_{t}$ that give the mean of a randomized benchmarking
distribution are closely related to an operational characterization of the
amount of noise, namely, the average gate infidelity Magesan _et al._ (2012b,
a) (which gives the average error rate), as
$\displaystyle
r_{t}=1-F_{\mathrm{avg}}(\Lambda_{t})=\frac{d-1}{d}(1-f_{t})\,.$ (62)
The randomized benchmarking protocol will enable the estimation of
$\prod_{t}f_{t}$, which can then be used to estimate the average gate
infidelity averaged over arbitrary time intervals (as shown in Sec. IV.2).
We now show that the average gate infidelity provides an upper and a lower
bound on $\tfrac{1}{2}\left\lVert\Lambda-\mathbbm{1}\right\rVert_{\diamond}$,
which gives the worst-case error introduced by using $\Lambda$ instead of
$\mathbbm{1}$. An upper bound of the same form was stated without proof in
Ref. Gambetta _et al._ (2012), however, the bound here is a factor of two
better. The following relation between the diamond distance and the average
gate fidelity can also be applied at each time step to relate the time-
averaged average gate fidelity to the average diamond distance from the
identity channel. Note that the following bound is very loose in the regime
$mr\ll 1$ (since in that regime, $r\ll\sqrt{r}$), which is also the regime in
which we will typically use it.
###### Proposition 9.
Let $r=1-F_{\rm avg}(\Lambda)$ be the average error rate for $\Lambda$. Then
$\displaystyle
r(d+1)/d\leq\tfrac{1}{2}\left\lVert\Lambda-\mathbbm{1}\right\rVert_{\diamond}\leq\sqrt{d(d+1)r}\,.$
(63)
Proof. Applying Lemma 7 to $\Delta=\Lambda-\mathbbm{1}$ gives
$\displaystyle D\left[\Phi,J(\Lambda)\right]=\tfrac{1}{2}\left\lVert
J(\Lambda)-\Phi\right\rVert_{1}\leq\tfrac{1}{2}\left\lVert\Lambda-\mathbbm{1}\right\rVert_{\diamond}\leq\tfrac{d}{2}\left\lVert
J(\Lambda)-\Phi\right\rVert_{1}=dD\left[\Phi,J(\Lambda)\right]\,.$ (64)
Recalling that $\Phi$, the maximally entangled state, is a pure state and
using Eq. (42) and (43) gives
$\displaystyle 1-F\left[\Phi,J(\Lambda)\right]\leq
D\left[\Phi,J(\Lambda)\right]\leq\sqrt{1-F\left[\Phi,J(\Lambda)\right]}\,.$
(65)
From Lemma 6, $1-F\left[\Phi,J(\Lambda)\right]=d^{-1}(d+1)r$. Substituting
this into the above expression and combining the inequalities completes the
proof. $\Box$
### VI.2 Upper bounds on the variance
We now consider the variance $\sigma_{m}^{2}$ of the distribution
$\left\\{F_{m,k}\right\\}$ for fixed $m$. It has been observed that the
standard error of the mean (and hence the sample variance) can be remarkably
small in experimental applications of randomized benchmarking using relatively
few random sequences Gambetta _et al._ (2012); Magesan _et al._ (2012b). In
this section, we will prove that the variance due to sampling random sequences
is indeed small in scenarios of practical interest (i.e., $mr\ll 1$) by
obtaining an upper bound on $\sigma_{m}^{2}$ in terms of $mr$. For the special
case of a qubit, we will also obtain a significantly improved upper bound in
terms of $m$ and $r$.
We begin by obtaining a general bound on $\sigma_{m}^{2}$ that depends only on
$m$, $r$ and the dimension $d$ of the system being benchmarked. In order to
present results in a simple form, we assume that the noise is time- and gate-
independent, however, the results in this section can readily be generalized
to time-dependent noise.
As a first attempt at obtaining a good bound on the variance, we use only the
fact that when $mr\ll 1$, we have $\bar{F}_{m}\approx A+B$, where
$A=E_{0}\rho_{0}$ and $B=\vec{E}\cdot\vec{\rho}$. Expanding the expression
from Theorem 8 to first order in $r$ using Eq. (62) gives
$\displaystyle\bar{F}_{m}=A+B-\frac{Bmdr}{d-1}\,.$ (66)
The value of all realizations of $\bar{F}_{m}$ (i.e., the probabilities
$F_{m,s}$) are all in the unit interval. Since the distribution with the
largest variance that has mean $\bar{F}_{m}$ and takes values in the unit
interval is the binomial distribution with that mean, we then have
$\displaystyle\sigma_{m}^{2}\leq\bar{F}_{m}(1-\bar{F}_{m})=(A+B)(1-A-B)+\frac{mdBr}{d-1}\,.$
(67)
While simple to obtain, this bound has a constant off-set term that depends
upon the SPAM which seems to be unavoidable. This term would be zero in the
absence of SPAM, and could even be eliminated if the probabilities $F_{m,s}$
could be restricted to the interval $[1-A-B,A+B]$. However, as illustrated in
Sec. IV.1, this cannot be done in general. Moreover, we expect that the above
argument substantially overestimates the variance because it ignores the
possibility that many sequences may have $F_{m,s}$ closer to $\bar{F}_{m}$.
We now obtain an alternative bound that has a larger coefficient for $r$, but
no constant term. We note from the outset that the following bound is not
tight in general (and the previous bound suggests that the dimensional factor
is an artifact of the proof technique), though by improving one of the steps
we will be able to obtain a tight bound for qubits. To facilitate our
analysis, we use the identity $(E|\mathcal{E}|\rho)^{2}=(E^{\otimes
2}|\mathcal{E}^{\otimes 2}|\rho^{\otimes 2})$ to write the variance as
$\displaystyle\sigma_{m}^{2}=\left\lvert\mathcal{G}\right\rvert^{-m}\sum_{k}F_{m,k}^{2}-\bar{F}_{m}^{2}=(E^{\otimes
2}|\Bigl{(}\left[(\Lambda^{\otimes
2})^{\mathcal{G}}\right]^{m}-\left[\left(\Lambda^{\mathcal{G}}\right)^{\otimes
2}\right]^{m}\Bigr{)}|\rho^{\otimes 2})\,.$ (68)
###### Theorem 10.
The variance for time- and gate-independent randomized benchmarking of
$d$-level systems with time- and gate-independent noise satisfies
$\displaystyle\sigma_{m}^{2}\leq 4d(d+1)mr+O(m^{2}r^{2}d^{4})\,.$ (69)
Proof. We write $\Lambda=\mathbbm{1}-r\Delta$, where the first row of $\Delta$
is zero since $\Lambda$ is CPTP. Since
$(d^{2}-1)f=\mathrm{Tr}\varphi=\mathrm{Tr}\Lambda-1$, we can use Eq. (62) to
obtain $\mathrm{Tr}\Delta=d(d+1)$
We then expand the expression
$\displaystyle\sigma_{m}^{2}=(E^{\otimes 2}|\Bigl{(}\left[(\Lambda^{\otimes
2})^{\mathcal{G}}\right]^{m}-\left[\left(\Lambda^{\mathcal{G}}\right)^{\otimes
2}\right]^{m}\Bigr{)}|\rho^{\otimes 2})$ (70)
to second order in $r\Delta$. Note that
$(\Delta\otimes\mathbbm{1})^{g}=\Delta^{g}\otimes\mathbbm{1}$ and so all the
first-order terms and the second-order terms where the $\Delta$ act at
different times will cancel. Therefore the only second-order terms are the $m$
terms with $\Delta^{\otimes 2}$ and so the variance is
$\displaystyle\sigma_{m}^{2}=mr^{2}(E^{\otimes 2}|\left[(\Delta^{\otimes
2})^{\mathcal{G}}-\left(\Delta^{\mathcal{G}}\right)^{\otimes
2}\right]|\rho^{\otimes 2})+O(r^{3}\Delta^{3})\,.$ (71)
Noting that $\Delta^{\mathcal{G}}=\frac{d(d+1)}{d^{2}-1}\mathbbm{1}$, the
variance satisfies
$\displaystyle\sigma_{m}^{2}$ $\displaystyle\leq
mr^{2}\left\lvert\mathcal{G}\right\rvert^{-1}\left\lVert\sum_{g\in\mathcal{G}}(\Delta^{\otimes
2})^{g}\right\rVert_{\diamond}+O(r^{3}\Delta^{3})+O(mr^{2})$
$\displaystyle\leq mr^{2}\left\lVert\Delta^{\otimes
2}\right\rVert_{\diamond}+O(r^{3}\Delta^{3})+O(mr^{2})$ $\displaystyle\leq
mr^{2}\left\lVert\Delta\right\rVert_{\diamond}^{2}+O(r^{3}\Delta^{3})$
$\displaystyle\leq 4d(d+1)mr+O(r^{3}\Delta^{3})+O(mr^{2})\,,$ (72)
where we have used the triangle inequality, the invariance of the diamond norm
under unitary conjugation, the submultiplicativity of the diamond norm [with
$\Delta\otimes\Delta=(\Delta\otimes\mathbbm{1})(\mathbbm{1}\otimes\Delta)$]
and Proposition 9.
Finally, consider terms of $O(r^{k}\Delta^{k})$ for $k>2$. For $k\geq 3$, all
$O(m^{k})$ such terms are upper-bounded by
$r^{k}\left\lVert\Delta^{k}\right\rVert_{\diamond}$ and so are
$O(r^{k/2}d^{k})$. Therefore the only contributions of $O(r^{2})$ or greater
are from $k=3$ and $k=4$.
For $k=3$, the only terms that will not cancel are products whose only
nontrivial terms are a $(\Delta\otimes\Delta)^{\mathcal{G}}$ and a
$\Delta^{\mathcal{G}}\otimes\mathbbm{1}=\frac{d(d+1)}{d^{2}-1}\mathbbm{1}$.
There are only $O(m^{2})$ such terms, and applying the diamond norm bound to
$\frac{d}{d-1}(\Delta\otimes\Delta)^{\mathcal{G}}$ shows that such terms
contribute at most $O(m^{2}r^{2}d^{2})$.
For $k=4$, the only terms that will be of $O(r^{2})$ are those that are
products with two $(\Delta\otimes\Delta)^{\mathcal{G}}$ terms. Again, there
are only $O(m^{2})$ such terms and so such terms also contribute at most
$O(m^{2}r^{2}d^{4})$. $\Box$
While the bound in Theorem 10 is promising, it is not sufficiently small to
justify the sequence lengths chosen in many experimental implementations of
randomized benchmarking for a single qubit, since $mr\approx 10^{-2}$ in many
such experiments and so the contribution to standard error of the mean due to
sampling random gate sequences is expected to be on the order of
$0.1K^{-1/2}$, where $K$ is the number of random sequences of length $m$ that
are sampled.
One of the loosest approximations in Theorem 10 is the use of the triangle
inequality to upper-bound the contribution from terms of the form
$(\Delta^{\otimes 2})^{\mathcal{G}}$. Avoiding this is difficult in general,
however, for the case of a single qubit, we can significantly improve the
following bound by understanding the irrep structure of the representation
$g\otimes g$. This irrep structure will depend on the choice of 2-design, so
we now fix the 2-design to be the single qubit Clifford group,
$\mathcal{C}_{2}$ and work in the Pauli basis
$\mathbb{A}=\\{\mathbbm{1},X,Y,Z\\}/\sqrt{2}$ (where the factor of $\sqrt{2}$
makes the basis trace-orthonormal). In particular, we will work in the block
basis
$\displaystyle\left(\begin{array}[]{c}\mathbbm{1}\\\
\mathbbm{1}\otimes\vec{\sigma}\\\ \vec{\sigma}\otimes\mathbbm{1}\\\
\vec{\sigma}\otimes\vec{\sigma}\end{array}\right)$ (77)
where $\vec{\sigma}=\\{X,Y,Z\\}/\sqrt{2}$. Restricting the Liouville
representation to each of the blocks in the above basis will give a rep of
$\mathcal{C_{2}}$, where the first three reps have already been characterized.
We now characterize the final subrep, $(\phi^{\otimes 2},\mathbb{C}^{9})$.
###### Proposition 11.
The representation $(\phi^{\otimes 2},\mathbb{C}^{9})$ of $\mathcal{C}_{2}$ is
the direct sum of four inequivalent irreps.
Proof. The proof follows from a direct application of Schur’s orthogonality
relations, which imply
$\displaystyle\left\lvert\mathcal{C}_{2}\right\rvert^{-1}\sum_{g\in\mathcal{C}_{2}}\chi_{\phi^{\otimes
2}}(g)^{*}\chi_{\phi^{\otimes 2}}(g)=\sum_{\lambda}n_{\lambda}^{2}\,,$ (78)
where $n_{\lambda}$ is the multiplicity of the irrep $\lambda$ in the rep
$\phi^{\otimes 2}$.
The character is given by
$\displaystyle\chi_{\phi^{\otimes 2}}(g)=\mathrm{Tr}g^{\otimes
2}=(\mathrm{Tr}g)^{2}\,.$ (79)
Since the elements of $\mathcal{C}_{2}$ permute Paulis (up to signs), the
diagonal elements of $\mathcal{G}$ in the Pauli basis are either $1$ or $-1$
and there are 0, 1 or 3 diagonal elements that can contribute to
$\mathrm{Tr}g$.
There are eight elements of $\mathcal{C}_{2}$ with no diagonal elements,
namely, the eight permutations $X\to\pm Y\to\pm Z$ and $X\to\pm Z\to\pm Z$.
There is 1 element with all diagonal elements equal, namely, the identity
(note that $-\mathbbm{1}$ is antiunitary so is not in the Clifford group). All
other 15 elements of the Clifford group have $\chi_{\phi}(g)=\pm 1$ since the
diagonal elements cannot sum to any values in $\left\\{0,\pm 2,\pm
3\right\\}$.
Plugging these character values into Eq. (78) gives
$\displaystyle\left\lvert\mathcal{C}_{2}\right\rvert^{-1}\sum_{g\in\mathcal{C}_{2}}\chi_{\phi^{\otimes
2}}(g)^{*}\chi_{\phi^{\otimes
2}}(g)=\frac{1}{24}\sum_{g\in\mathcal{C}_{2}}\left\lvert\chi_{\phi}(g)\right\rvert^{4}=\frac{1}{24}(3^{4}+15)=4\,.$
(80)
Given that the multiplicity of an irrep must be a nonnegative integer, there
are two possibilities. Either there are 4 inequivalent irreps or the rep
$\phi^{\otimes 2}$ contains two equivalent irreps. By Proposition 2,
$\phi^{\otimes 2}$ contains a trivial irrep with multiplicity 1 and so cannot
contain two equivalent irreps.
The following bases of operators:
$\displaystyle\mathbb{A}_{1}$
$\displaystyle=\frac{1}{2\sqrt{3}}\left(XX+YY+ZZ\right)\,,$
$\displaystyle\mathbb{A}_{2}$
$\displaystyle=\left\\{\frac{1}{2\sqrt{2}}\left(XX-
YY\right),\frac{1}{2\sqrt{6}}\left(XX+YY-2ZZ\right)\right\\}\,,$
$\displaystyle\mathbb{A}_{S}$ $\displaystyle=\frac{1}{2\sqrt{2}}\left\\{XY-
YX,XZ-ZX,YZ-ZY\right\\}\,,$ $\displaystyle\mathbb{A}_{T}$
$\displaystyle=\frac{1}{2\sqrt{2}}\left\\{XY+YX,XZ+ZX,YZ+ZY\right\\}\,,$ (81)
span the four irreps. $\Box$
The fact that $g\otimes g$ is a direct sum of four inequivalent irreps will
allow us to use Schur’s lemma on the unital block of $\Delta$. To account for
the nonunital component, we use the following bound.
###### Proposition 12.
For any completely positive and trace-preserving qubit channel
$\Lambda:\mathcal{D}_{2}\to\mathcal{D}_{2}$ with average gate infidelity
$r<1/3$, the nonunital part $\alpha$ obeys the inequality
$\displaystyle\left\lVert\alpha\right\rVert_{2}^{2}\leq 9r^{2}\,.$ (82)
Proof. Any trace-preserving qubit channel as
$\displaystyle\Lambda$ $\displaystyle=(1\oplus
U)\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&w_{1}&0&0\\\ 0&0&w_{2}&0\\\
t&0&0&w_{3}\\\ \end{array}\right)(1\oplus U^{\dagger})(1\oplus V)\,.$ (87)
for some $U,V\in O(3)$ [corresponding to unitaries $u,v\in U(2)$], where
$\left\lvert w_{j}\right\rvert$ are the singular values of $\varphi$ with
$\left\lvert w_{j}\right\rvert\in[0,1]$ for all $j$ Ruskai _et al._ (2002)
and we have added the $(1\oplus U^{\dagger})$ term for convenience. By Von
Neumann’s trace inequality Mirsky (1975),
$\displaystyle
3-6r=\mathrm{Tr}\varphi=\left\lvert\mathrm{Tr}UWU^{\dagger}V\right\rvert\leq\sum_{j}\left\lvert
w_{j}\right\rvert$ (88)
where $W={\rm diag}(w_{1},w_{2},w_{3})$ and we have used the fact that the
singular values of $V$ are all one. For notational convenience, we define
perturbations $\delta_{j}$ by $\left\lvert w_{j}\right\rvert=1-\delta_{j}r$
which then satisfy $\sum_{j}\delta_{j}\leq 6$ and $\delta_{j}\geq 0$ for all
$j$.
The conditions for $\Lambda$ to be completely positive are
$\displaystyle\left\lvert t\right\rvert+\left\lvert w_{3}\right\rvert$
$\displaystyle\leq 1$ $\displaystyle(w_{j}\pm w_{k})^{2}$
$\displaystyle\leq(1\pm w_{l})^{2}$ (89)
for any permutation $\left\\{j,k,l\right\\}$ of $\left\\{1,2,3\right\\}$.
Therefore
$\displaystyle\left\lvert t\right\rvert\leq 1-\left\lvert
w_{3}\right\rvert=\delta_{3}r\,,$ (90)
and, since $\left\lVert\alpha\right\rVert_{2}$ is invariant under the unitary
transformations in Eq. (87), $\left\lVert\alpha\right\rVert_{2}=\left\lvert
t\right\rvert$. Therefore the only remaining problem is to bound $\delta_{3}$
(note that at this point, we could accept the trivial bound $\delta_{3}\leq
6$).
If $w_{3}<0$, then complete positivity implies
$\displaystyle(2-\delta_{1}r-\delta_{2}r)^{2}\leq\delta_{3}^{2}r^{2}\,,$ (91)
which cannot be satisfied subject to $\sum_{j}\delta_{j}\leq 6$ and
$\delta_{j}\geq 0$ for $r<1/3$. Therefore for all $r<1/3$,
$w_{3}=1-\delta_{3}r>0$.
Considering the conditions
$\displaystyle(\delta_{j}-\delta_{k})^{2}$ $\displaystyle\leq\delta_{l}^{2}$
(92)
for all permutations $\left\\{j,k,l\right\\}$ of $\left\\{1,2,3\right\\}$, we
see that $\delta_{3}\leq\max_{j}\delta_{j}\leq 3$ and so
$\left\lVert\alpha\right\rVert_{2}\leq 3r$. $\Box$
Combining the irrep structure of $g\otimes g$ and the bound on the nonunital
component allows us to improve the bound in Theorem 10 for the special case of
one qubit. As discussed in Sec. IV, the following bound provides a rigorous
justification of current experiments and allows values of $K_{m}$ to be chosen
that are substantially smaller then previously justified rigorously, that is,
$K_{m}\approx 145$ as opposed to $K_{m}\approx 7\times 10^{4}$ as estimated in
Ref. Magesan _et al._ (2012a).
###### Theorem 13.
The variance for arbitrary time- and gate-independent noise satisfies
$\displaystyle\sigma_{m}^{2}\leq m^{2}r^{2}+\frac{7}{4}mr^{2}+6\delta
mr+O(m^{2}r^{3})+O(\delta m^{2}r^{2})\,,$ (93)
where $\delta=\lvert\vec{\delta}_{E}\cdot\vec{\delta}_{\rho}\rvert\leq 1/2$
for any choice of $\vec{w}\in\left\\{\vec{x},\vec{y},\vec{z}\right\\}$ and
$\vec{\delta}_{\rho},\vec{\delta}_{E}\perp\vec{w}$ such that
$\displaystyle\vec{E}^{T}=a\vec{w}+\vec{\delta}_{E}$
$\displaystyle\vec{\rho}=b\vec{w}+\vec{\delta}_{\rho}\,.$ (94)
Proof. To prove the theorem, we will derive an exact expression for the
variance and then approximate it in the relevant regimes.
We begin by noting that in the basis $\left\\{\mathbbm{1}^{\otimes
2},\mathbbm{1}\otimes\mathbb{A},\mathbb{A}\otimes\mathbbm{1},\mathbb{A}\otimes\mathbb{A}\right\\}$
we have
$\displaystyle(\Lambda^{\otimes
2})^{\mathcal{C}_{2}}=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\varphi^{\mathcal{C}_{2}}&0&0\\\ 0&0&\varphi^{\mathcal{C}_{2}}&0\\\
P_{1}\alpha^{\otimes
2}&\left\lvert\mathcal{C}_{2}\right\rvert^{-1}\sum_{g\in\mathcal{C}_{2}}g\alpha\otimes\varphi^{(g)}&\left\lvert\mathcal{C}_{2}\right\rvert^{-1}\sum_{g\in\mathcal{C}_{2}}\varphi^{(g)}\otimes
g\alpha&(\varphi^{\otimes 2})^{\mathcal{C}_{2}}\end{array}\right)$ (99)
where
$P_{1}=\left\lvert\mathcal{C}_{2}\right\rvert^{-1}\sum_{g\in\mathcal{C}_{2}}g^{\otimes
2}$. It can be verified that $\vec{E}^{\otimes 2}$ is in the null space of
$\sum_{g\in\mathcal{C}_{2}}\varphi^{(g)}\otimes g\alpha$ and
$\sum_{g\in\mathcal{C}_{2}}g\alpha\otimes\varphi^{(g)}$ for any $\vec{E}$ by,
for example, considering a basis for the space of $\varphi$’s. We note in
passing that this property is not a general property of 2-designs, in that it
does not hold for the single-qutrit Clifford group.
By Propositions 11 and 1 $(\varphi^{\otimes
2})^{\mathcal{C}_{2}}=\sum_{R}\lambda_{R}P_{R}$ where the $P_{R}$ are the
projectors onto the irreps from Proposition 1 and
$\lambda_{R}=\mathrm{Tr}P_{R}\varphi^{\otimes 2}/\mathrm{Tr}P_{R}$. From Eq.
(68), together with the orthogonality of the projectors $P_{R}$, we have
$\displaystyle\sigma_{m}^{2}=\rho_{\mathbbm{1}}^{2}\vec{E}^{\otimes
2}P_{1}\vec{\alpha}^{\otimes
2}\sum_{t=0}^{m-1}\lambda_{1}^{t}+\sum_{R}\lambda_{R}^{m}\vec{E}^{\otimes
2}P_{R}\vec{\rho}^{\otimes 2}-(1-2r)^{2m}(\vec{E}\vec{\rho})^{2}\,.$ (100)
We can bound the first term using
$\displaystyle\rho_{0}^{2}\vec{E}^{\otimes
2}P_{1}\vec{\alpha}\sum_{t=0}^{m-1}\lambda_{1}^{t}$
$\displaystyle=\frac{1}{3}\rho_{0}^{2}\left\lVert\vec{E}\right\rVert_{2}^{2}\left\lVert\alpha\right\rVert_{2}^{2}\sum_{t=0}^{m-1}1\leq\frac{3mr^{2}}{4}$
(101)
where we have used and the trivial bound $\lambda_{1}\leq 1$ (for the first
term only) and Proposition 12 to obtain the final inequality.
Similarly, the eigenvalues can be calculated to be
$\displaystyle\lambda_{1}$
$\displaystyle=\frac{1}{3}\sum_{j,k}\varphi_{j,k}^{2}=\frac{1}{3}\mathrm{Tr}\left(\varphi^{\dagger}\varphi\right)$
$\displaystyle\lambda_{2}$
$\displaystyle=\frac{1}{3}\sum_{j}\varphi_{j,j}^{2}-\frac{1}{6}\sum_{j\neq
k}\varphi_{j,k}^{2}=\frac{1}{2}\sum_{j}\varphi_{j,j}^{2}-\frac{1}{6}\mathrm{Tr}\varphi^{\dagger}\varphi$
$\displaystyle\lambda_{T}$ $\displaystyle=\frac{1}{6}\sum_{j\neq
k}\left(\varphi_{j,k}\varphi_{k,j}+\varphi_{j,j}\varphi_{k,k}\right)=\frac{1}{6}\mathrm{Tr}\left(\varphi^{2}\right)+\frac{1}{6}\left(\mathrm{Tr}\varphi\right)^{2}-\frac{1}{3}\sum_{j}\varphi_{j,j}^{2}\,,$
(102)
where we have omitted $\lambda_{S}$ since it will not contribute to the
variance since any symmetric vector (such as $\vec{E}^{\otimes 2}$) will be
orthogonal to $P_{S}$.
We now consider general noise with $\vec{E}$ and $\vec{\rho}$ as in Eq. (13),
where, without loss of generality, we set $\vec{w}=\vec{z}$. We begin by
considering the case $\delta=0$, for which $\vec{E}^{\otimes
2}P_{T}=\vec{E}^{\otimes 2}P_{S}=0$. Then a simple calculation using
Proposition 12 gives $\vec{E}^{\otimes 2}P_{1}\vec{\rho}^{\otimes
2}=a^{2}b^{2}/3$, $(\vec{E}\vec{\rho})^{2}=a^{2}b^{2}$ and $\vec{E}^{\otimes
2}P_{2}\vec{\rho}^{\otimes 2}=\tfrac{2}{3}a^{2}b^{2}$.
The eigenvalues $\lambda_{1}$ and $\lambda_{2}$ can be written as $x+2y$ and
$x-y$ respectively, where $x=\frac{1}{3}\sum_{j}\varphi_{j,j}^{2}$ and
$y=\frac{1}{6}\sum_{j\neq k}\varphi_{j,k}^{2}$. Writing
$\varphi=\mathbbm{1}-\Delta r$, where $\mathrm{Tr}\Delta=6$ (cf. the
discussion in the proof of Theorem 10), we have
$\displaystyle 1-4r+4r^{2}\leq
x:=\frac{1}{3}\sum_{j}\varphi_{jj}^{2}=1-4r+\frac{r^{2}}{3}\sum_{j}\Delta_{jj}^{2}\leq
1-4r+12r^{2}\,,$ (103)
where the maximum and the minimum are obtained by maximizing and minimizing
$\sum_{j}\Delta_{jj}^{2}$ subject to $\sum_{j}\Delta_{jj}=6$ for real matrices
$\Delta$ with nonnegative diagonal entries respectively. The diagonal entries
of $\Delta$ must be nonnegative since all entries of $\varphi$ have modulus
upper-bounded by 1 [which can be easily verified from the form of extremal
channels in Eq. (87)]. Therefore the variance satisfies
$\displaystyle\sigma_{m}^{2}\leq\frac{3mr^{2}}{4}+\frac{a^{2}b^{2}}{3}\left[(x+2y)^{m}+2(x-y)^{m}-3(1-2r)^{2m}\right]\,.$
(104)
Since $1\geq\lambda_{1}-2y=x\geq 1-4r$ by Eq. (103), we have $y\leq 2r$ and
so, using a binomial expansion to $O(r^{3})$ gives
$\displaystyle(x+2y)^{m}+2(x-y)^{m}-3(1-2r)^{2m}$ $\displaystyle\leq
12mr^{2}+12m^{2}r^{2}+O(m^{3}r^{3})\,.$ (105)
Noting that $a^{2}b^{2}\leq 1/4$ gives
$\displaystyle\sigma_{m}^{2}\leq
m^{2}r^{2}+\frac{7}{4}mr^{2}+O(m^{2}r^{3})\,.$ (106)
We now consider the correction when $\delta>0$ in Eq. (13), which will
realistically always be the case since $\rho$ incorporates a residual noise
term. Then we define functions $h_{R}(\vec{\delta}_{1},\vec{\delta}_{2})$ by
$\displaystyle\vec{E}^{\otimes 2}P_{R}\vec{\rho}^{\otimes
2}=a^{2}b^{2}\vec{z}^{\otimes 2}P_{R}\vec{z}^{\otimes
2}+h_{R}(\vec{\delta}_{1},\vec{\delta}_{2})\,,$ (107)
where we will henceforth omit the arguments of $h_{R}$. Since
$\sum_{R}\vec{E}^{\otimes 2}P_{R}\vec{\rho}^{\otimes
2}=(\vec{E}\vec{\rho})^{2}$, we can write the variance as
$\displaystyle\sigma_{m}^{2}\leq
a^{2}b^{2}\sigma_{m,z}^{2}+\sum_{R}h_{R}\left[\lambda_{R}^{m}-(1-2r)^{2m}\right]\,.$
(108)
To $O(r^{2})$, the smallest eigenvalue is $\lambda_{2}$, since
$\displaystyle\frac{1}{6}\sum_{j\neq k}\varphi_{j,j}\varphi_{k,k}$
$\displaystyle=\frac{1}{3}\sum_{j}\varphi_{j,j}^{2}+O(r^{2})$
$\displaystyle\frac{1}{6}\left\lvert\sum_{j\neq
k}\varphi_{j,k}\varphi_{k,j}\right\rvert$
$\displaystyle\leq\frac{1}{6}\sum_{j\neq k}\varphi_{j,k}^{2}$ (109)
where the first line follows by writing $\varphi_{j,j}=1-r\Delta_{j,j}$ and
the second from the inequality $\varphi_{j,k}^{2}+\varphi_{k,j}^{2}\geq
2\left\lvert\varphi_{j,k}\varphi_{k,j}\right\rvert$ and the triangle
inequality. Therefore, to $O(r^{2})$, $1-8r\leq
x-y=\lambda_{2}\leq\lambda_{R}\leq 1$ for all $R$ [where the bounds on $x$ and
$y$ are as in Eq. (103)] and so
$\left\lvert\lambda_{R}^{m}-(1-2r)^{2m}\right\rvert\leq 4mr+O(m^{2}r^{2})$ for
all $R$. Therefore
$\displaystyle\sigma_{m}^{2}$ $\displaystyle\leq
a^{2}b^{2}\sigma_{m,z}^{2}+\left[4mr+O(m^{2}r^{2})\right]\sum_{R}h_{R}$
$\displaystyle\leq
m^{2}r^{2}+\frac{7}{4}mr^{2}+\left[4mr+O(m^{2}r^{2})\right]\sum_{R}h_{R}+O(m^{2}r^{3})$
$\displaystyle\leq m^{2}r^{2}+\frac{7}{4}mr^{2}+6\delta
mr+O(m^{2}r^{3})+O(\delta m^{2}r^{2})$ (110)
where we have obtained the final inequality using
$\displaystyle
a^{2}b^{2}+\sum_{R}h_{R}=(\vec{E}\vec{\rho})^{2}=a^{2}b^{2}+2ab(\vec{\delta}_{1}\cdot\vec{\delta}_{2})+(\vec{\delta}_{1}\cdot\vec{\delta}_{2})^{2}\leq
a^{2}b^{2}+\frac{3\delta}{2}\,.$ (111)
where the final inequality follows since
$\delta=\lvert\vec{\delta}_{1}\cdot\vec{\delta}_{2}\rvert,\left\lvert
ab\right\rvert\leq 1/2$. $\Box$
It is worth noting that one could in principle fill in the implicit constants
given in the big-$O$ notation by following the previous argument with
sufficient care. To have a truly rigorous confidence region, one would need to
take this into account, but for current parameter regimes of interest, the
terms really are negligible, so it hardly seems worth optimizing this concern.
We also note that $\delta_{\rho}$ will typically have entries of order
$\sqrt{r}$ even without SPAM, since the off-diagonal terms for generic noise
are of order $\sqrt{r}$ and there is a residual noise term that has been
incorporated into $\rho$. However, the corresponding entries in $\delta_{E}$
will generally be smaller (or at least, are determined only by SPAM).
We now show that the variance can be even further improved (by a factor of $m$
and with no dependence on the state and measurement) for noise that is
diagonal in the Pauli basis.
###### Corollary 14.
If the unital block of the noise is diagonal in the Pauli basis, this bound
can be improved to
$\displaystyle\sigma_{m}^{2}\leq\frac{11mr^{2}}{4}+O(m^{2}r^{3})\,.$ (112)
Proof. For noise such that $\varphi$ is diagonal in the Pauli basis,
$\lambda_{1}=\lambda_{2}=x$ and $\lambda_{T}\leq\lambda_{1}$, which can be
shown using the inequality $2ab\leq a^{2}+b^{2}$ for $a,b\in\mathbb{R}$.
Therefore, for noise that is diagonal in the Pauli basis, we have
$\displaystyle\sigma_{m}^{2}\leq\frac{3mr^{2}}{4}+\frac{1}{4}\left[\left(1-4r+12r^{2}\right)^{m}-(1-2r)^{2m}\right]\leq\frac{11mr^{2}}{4}+O(m^{2}r^{3})$
(113)
by Eq. (103). $\Box$
One consequence of the above corollary is that the variance of the randomized
benchmarking distribution will typically depend strongly upon the choice of
2-design even for gate independent noise. This observation follows from the
above theorem by noting that the unital block can be perturbed by an
arbitrarily small amount to allow it to be unitarily diagonalized. Performing
randomized benchmarking in the basis where the unital block is diagonalized
(i.e., setting $\mathcal{G}=\mathcal{C}_{2}^{U}$) will give variances of order
$mr^{2}$, while randomized benchmarking in other bases will give variances of
order $m^{2}r^{2}$.
## VII Asymptotic variance of randomized benchmarking
We now consider the variance $\sigma_{m}^{2}$ of the distribution
$\left\\{F_{m,s}\right\\}$ as $m\to\infty$. While not directly relevant to
current experiments, the asymptotic behavior is nevertheless interesting in
that it may provide a method of estimating the amount of nonunitality.
We will prove that, for the class of channels defined below called
$n$-contractive channels (which are generic in the space of CPTP channels),
$\sigma_{m}^{2}$ decays exponentially in $m$ to a constant that quantifies the
amount of nonunitality. Unfortunately, we will not be able to provide a bound
on the decay rate. In fact, no such bound is possible without further
assumptions since the channel
$[(1-\epsilon)U+\epsilon\mathcal{E}]^{\mathcal{G}}$ for any unitary $U$ and
$2$-contractive channel $\mathcal{E}$ will have an eigenvalue
$1-\epsilon+O(\epsilon)<1$ corresponding to the trivial subrep (this can be
seen by following the proof of Proposition 16). This eigenvalue will result in
a variance that decays as $(1-\epsilon)^{m}$ for arbitrary $\epsilon>0$.
###### Definition 15.
A channel $\Lambda:\mathcal{D}_{d}\to\mathcal{D}_{d}$ is _$n$ -contractive_
with respect to a group $\mathcal{G}\subseteq\mathsf{U}(d)$ if
$(\Lambda^{\otimes n})^{\mathcal{G}}$ has at most one eigenvalue of modulus 1.
We now prove that all unital but nonunitary channels are $2$-contractive with
respect to any finite 2-design. We conjecture that all nonunitary channels are
in fact $2$-contractive with respect to any unitary 2-design. An equivalent
statement for trace-preserving channels $\Lambda$ is that $(\Lambda^{\otimes
2})^{\mathcal{G}}$ is strongly irreducible whenever $\Lambda$ is not unitary
Sanz _et al._ (2010). As a corollary of the following proposition, this
conjecture holds for qubits, since, for qubits, the projection onto the unital
part of a CPTP map is also a CPTP map Kimmel _et al._ (2013). However,
proving it for higher dimensions remains an open problem.
###### Proposition 16.
Let $\Lambda$ be a completely positive, trace-preserving and unital channel
and $\mathcal{G}$ a unitary 2-design. Then $\Lambda$ is $2$-contractive with
respect to $\mathcal{G}$ if and only if it is nonunitary.
Proof. First assume $\Lambda$ is unitary. Since
$(\varphi,\mathbb{R}^{d^{2}-1})$ is an orthogonal irrep of $\mathsf{U}(d)$,
$(\varphi,\mathbb{R}^{d^{2}-1})^{\otimes 2}$ contains the trivial rep as a
subrep with multiplicity 1 by Proposition 2. Therefore for any
$U\in\mathsf{U}(d)$ and in a fixed Schur basis (i.e., independent of $U$),
$\varphi(U)^{\otimes 2}=1\oplus\mathcal{T}(U)$ for some homomorphism
$\mathcal{T}$. Therefore any vector $v$ in the (one-dimensional) trivial
representation is a +1-eigenvector of $\varphi(U)^{\otimes 2}$ for any $U$ and
consequently is a +1-eigenvector of $\bigl{[}\varphi^{\otimes
2}(\Lambda)\bigr{]}^{\mathcal{G}}$.
We now show that for all completely positive, trace-preserving and unital
$\Lambda$,
$\displaystyle\left\lVert\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right\rVert_{\infty}^{2}$ $\displaystyle\leq
1-\left\lvert\mathcal{G}\right\rvert^{-1}\left[1-\frac{\mathrm{Tr}\varphi^{\dagger}\varphi}{d^{2}-1}\right]\,.$
(114)
Recall that one of the equivalent definitions of the spectral norm is
$\displaystyle\left\lVert\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right\rVert_{\infty}^{2}=\max_{u:\left\lVert
u\right\rVert_{2}=1}u^{\dagger}\left[\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right]^{\dagger}\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}u\,.$ (115)
Expanding the averages over $\mathcal{G}$ gives
$\displaystyle\left\lVert\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right\rVert_{\infty}^{2}$
$\displaystyle=\max_{u:\left\lVert
u\right\rVert_{2}=1}\left\lvert\mathcal{G}\right\rvert^{-2}\sum_{g,h\in\mathcal{G}}u^{\dagger}\left[\left(\varphi^{\otimes
2}\right)^{g}\right]^{\dagger}\left(\varphi^{\otimes 2}\right)^{h}u$
$\displaystyle\leq
1-\left\lvert\mathcal{G}\right\rvert^{-1}+\max_{u:\left\lVert
u\right\rVert_{2}=1}\left\lvert\mathcal{G}\right\rvert^{-2}\sum_{g\in\mathcal{G}}u^{\dagger}\left[\left(\varphi^{\dagger}\varphi\right)^{\otimes
2}\right]^{g}u\,.$ (116)
where in the second line we have used the improved bound for unital channels
in Proposition 5 to bound the contribution from the
$\left\lvert\mathcal{G}\right\rvert^{2}-\left\lvert\mathcal{G}\right\rvert$
terms with $g\neq h$.
Now let $u$ be an arbitrary unit vector and write $u=\sum u_{j,k}v_{j}\otimes
v_{k}$, where $\left\\{v_{j}\right\\}$ is an orthonormal basis of
$\mathbb{C}^{d}$. Then, since $\left(\varphi^{\dagger}\varphi\right)^{g}$ is
positive semidefinite with eigenvalues upper-bounded by 1 by Proposition 5, we
have
$\displaystyle
u^{\dagger}\left[\left(\varphi^{\dagger}\varphi\right)^{g}\right]^{\otimes
2}u$ $\displaystyle=\sum_{j,k}\left\lvert
u_{j,k}\right\rvert^{2}\left[v_{j}^{\dagger}\left(\varphi^{\dagger}\varphi\right)^{g}v_{j}\right]\times\left[v_{k}^{\dagger}\left(\varphi^{\dagger}\varphi\right)^{g}v_{k}\right]$
$\displaystyle\leq\sum_{j,k}\left\lvert
u_{j,k}\right\rvert^{2}v_{j}^{\dagger}\left(\varphi^{\dagger}\varphi\right)^{g}v_{j}\,,$
(117)
where we have used $0\leq
v_{k}^{\dagger}\left(\varphi^{\dagger}\varphi\right)^{g}v_{k}\leq 1$ for all
$k$ and $g$ to obtain the second line. By Proposition 1,
$\displaystyle\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}\left(\varphi^{\dagger}\varphi\right)^{g}=\frac{\mathrm{Tr}\varphi^{\dagger}\varphi}{d^{2}-1}\mathbbm{1}\,,$
(118)
so
$\displaystyle\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}u^{\dagger}\left[\left(\varphi^{\dagger}\varphi\right)^{g}\right]^{\otimes
2}u$
$\displaystyle\leq\frac{\mathrm{Tr}\varphi^{\dagger}\varphi}{d^{2}-1}\sum_{j,k}\left\lvert
u_{j,k}\right\rvert^{2}$
$\displaystyle\leq\frac{\mathrm{Tr}\varphi^{\dagger}\varphi}{d^{2}-1}$ (119)
for all $u$ such that $\left\lVert u\right\rVert_{2}=1$. Therefore
$\displaystyle\left\lVert\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right\rVert_{\infty}^{2}$ $\displaystyle\leq
1-\left\lvert\mathcal{G}\right\rvert^{-1}\left[1-\frac{\mathrm{Tr}\varphi^{\dagger}\varphi}{d^{2}-1}\right]\,.$
(120)
$\Box$
###### Theorem 17.
Let $\Lambda$ be a $2$-contractive channel with respect to a group
$\mathcal{G}$ that is also a 2-design. Then the variance due to sampling
random gate sequences of elements from $\mathcal{G}$ decays exponentially to
$\displaystyle\frac{\rho_{0}^{2}\vec{E}^{\otimes 2}P_{1}\alpha^{\otimes
2}}{1-\lambda_{1}}\,,$ (121)
where
$P_{1}=\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}g^{\otimes
2}$ is a rank-1 projector and $\lambda_{1}=\mathrm{Tr}P_{1}\varphi^{\otimes
2}$.
Proof. For convenience, we use the block basis $\left\\{\mathbbm{1}^{\otimes
2},\mathbbm{1}\otimes\mathbb{A},\mathbb{A}\otimes\mathbbm{1},\mathbb{A}\otimes\mathbb{A}\right\\}$
for the matrix representation. In this basis, we can write
$\displaystyle(\Lambda^{\mathcal{G}})^{\otimes 2}$
$\displaystyle=\left(\begin{array}[]{ccccc}1&0&0&0\\\ 0&f\mathbbm{1}&0&0\\\
0&0&f\mathbbm{1}&0\\\ 0&0&0&f^{2}\mathbbm{1}\end{array}\right)$ (126)
$\displaystyle(\Lambda^{\otimes 2})^{\mathcal{G}}$
$\displaystyle=\left(\begin{array}[]{ccccc}1&0&0&0\\\ 0&f\mathbbm{1}&0&0\\\
0&0&f\mathbbm{1}&0\\\ P_{1}\alpha^{\otimes 2}&b&c&\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\end{array}\right)\,,$ (131)
where we have used $\sum_{g\in\mathcal{G}}g=0$ by Proposition 1,
$P_{1}=\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}g^{\otimes
2}$, and
$\displaystyle b$
$\displaystyle=\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}\varphi^{(g)}\otimes\left[g\alpha\right]$
$\displaystyle c$
$\displaystyle=\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}\left[g\alpha\right]\otimes\varphi^{(g)}\,.$
(132)
By Propositions 1 and 2, $P_{1}$ is a rank-1 projector onto the trivial
subrep, which occurs with multiplicity 1.
It can easily be shown using an inductive step that
$\displaystyle\left[(\Lambda^{\otimes
2})^{\mathcal{G}}\right]^{m}=\left(\begin{array}[]{ccccc}1&0&0&0\\\
0&f^{m}\mathbbm{1}&0&0\\\ 0&0&f^{m}\mathbbm{1}&0\\\
A_{m}&B_{m}&C_{m}&\left[\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right]^{m}\end{array}\right)\,,$ (137)
where
$\displaystyle A_{m}$
$\displaystyle=\sum_{t=0}^{m-1}\biggl{[}(\varphi^{\otimes
2})^{\mathcal{G}}\biggr{]}^{t}P_{1}\alpha^{\otimes 2}$ $\displaystyle B_{m}$
$\displaystyle=\sum_{t=0}^{m-1}f^{m-1-t}\left[(\varphi^{\otimes
2})^{\mathcal{G}}\right]^{t}b$ $\displaystyle C_{m}$
$\displaystyle=\sum_{t=0}^{m-1}f^{m-1-t}\left[(\varphi^{\otimes
2})^{\mathcal{G}}\right]^{t}c\,.$ (138)
Since the trivial subrep occurs with multiplicity 1 and $(\varphi^{\otimes
2})^{\mathcal{G}}$ commutes with $g^{\otimes 2}$ for all $g$, by Proposition 1
we can write $(\varphi^{\otimes 2})^{\mathcal{G}}=\lambda_{1}P_{1}+M$ for some
matrix $M$ orthogonal to $P_{1}$, where
$\lambda_{1}=\mathrm{Tr}P_{1}\left(\varphi^{\otimes 2}\right)^{\mathcal{G}}$.
Therefore we have
$\displaystyle A_{m}$ $\displaystyle=P_{1}\alpha^{\otimes
2}\sum_{t=0}^{m-1}\lambda_{1}^{t}\,.$ (139)
Substituting these expressions into Eq. (68) gives
$\displaystyle\sigma_{m}^{2}=\rho_{0}^{2}\vec{E}^{\otimes
2}P_{1}\alpha^{\otimes
2}\sum_{t=1}^{m}\lambda_{1}^{t}+\rho_{0}\vec{E}^{\otimes
2}(B_{m}+C_{m})\vec{\rho}+\vec{E}^{\otimes 2}\left[\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right]^{m}\vec{\rho}^{\otimes
2}-f^{2m}\left(\vec{E}\vec{\rho}\right)^{2}\,.$ (140)
We now prove that all but the first term decay exponentially for any
$2$-contractive channel with respect to $\mathcal{G}$. Let $SJS^{-1}$ be the
Jordan decomposition of $\left(\varphi^{\otimes 2}\right)^{\mathcal{G}}$.
Then, by the submultiplicativity of the spectral norm and a standard identity,
$\displaystyle\left\lVert\left[\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right]^{m}\right\rVert_{\infty}$
$\displaystyle=\left\lVert\left(SJS^{-1}\right)^{m}\right\rVert_{\infty}$
$\displaystyle\leq\left\lVert J^{m}\right\rVert_{\infty}\left\lVert
S\right\rVert_{\infty}\left\lVert S^{-1}\right\rVert_{\infty}$
$\displaystyle\leq(d^{2}-1)^{2}\left\lVert J^{m}\right\rVert_{\max}\left\lVert
S\right\rVert_{\infty}\left\lVert S^{-1}\right\rVert_{\infty}$ (141)
where $\left\lVert M\right\rVert_{\max}=\max_{j,k}\left\lvert
M_{j,k}\right\rvert$ and $J$ is a $(d^{2}-1)^{2}\times(d^{2}-1)^{2}$ matrix.
Note that since $S$ is invertible, both $\left\lVert S\right\rVert_{\infty}$
and $\left\lVert S^{-1}\right\rVert_{\infty}$ are finite.
By explicit calculation,
$\displaystyle\left\lVert
J^{m}\right\rVert_{\max}=\max_{k,j=1,\ldots,d_{k}}\left\lvert\eta_{k}\right\rvert^{m-j}\binom{m}{j}$
(142)
where $J_{k}$ is the $k$th Jordan block of $J$ with eigenvalue $\eta_{k}$ and
dimension $d_{k}$. By Proposition 16, $(\Lambda^{\otimes 2})^{\mathcal{G}}$
has at most one eigenvalue of modulus 1, which can be identified as the top
left entry in the expression in Eq. (126). Consequently, all the $\eta_{k}$
have modulus strictly less than 1 and so $\left\lVert
J^{m}\right\rVert_{\max}$ and consequently
$\left\lVert\left[\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}\right]^{m}\right\rVert_{\infty}$ decay exponentially
to zero with $m$.
Therefore the only term in Eq. (68) that does not decay exponentially to zero
in $m$ is the first term, namely,
$\displaystyle\rho_{0}^{2}\vec{E}^{\otimes 2}P_{1}\alpha^{\otimes
2}\sum_{t=1}^{m}\lambda_{1}^{t}\,,$ (143)
which converges exponentially in $m$ to
$\displaystyle\frac{\rho_{0}^{2}\vec{E}^{\otimes 2}P_{1}\alpha^{\otimes
2}}{1-\lambda_{1}}\,,$ (144)
provided $\left\lvert\lambda_{1}\right\rvert<1$, otherwise it diverges. Since
$P_{1}=uu^{\dagger}$ for some unit vector $u$ and the trivial rep occurs with
multiplicity 1, $u$ is an eigenvector of $\left(\varphi^{\otimes
2}\right)^{\mathcal{G}}$ with eigenvalue $\lambda_{1}$, which must be strictly
less than 1 by Proposition 16. $\Box$
## VIII Stability under gate-dependent perturbations
In our treatment of randomized benchmarking, we have assumed that the noise is
independent of the target gate (although the noise may depend on time). In a
physical implementation, the noise will depend on the target. We can account
for gate-dependent noise perturbatively by writing
$\displaystyle\Lambda_{t,g}=\Lambda_{t}+\epsilon\Delta_{t,g}\,,$ (145)
where
$\Lambda_{t}=\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}\Lambda_{t,g}$
is a valid quantum channel and $\epsilon$ is scaled such that
$\lVert\Delta_{t,g}\rVert_{\infty}\leq 1$ for all $t$ and $g$.
In Ref. Magesan _et al._ (2012a) it was shown that the mean of the
benchmarking distribution is robust under gate-dependent perturbations. We now
show that the variance is also stable under gate-dependent perturbations. Let
us write $\sigma_{m,0}^{2}$ for the gate-averaged variance, and
$\delta(\sigma_{m}^{2})$ as the correction due gate-dependent perturbations.
Then we have the following theorem.
###### Theorem 18.
The gate-dependent correction to the variance satisfies
$\delta(\sigma_{m}^{2})\leq\delta_{0}$ whenever the gate-dependent noise in
Eq. (145) satisfies
$\displaystyle\epsilon\leq\frac{\delta_{0}}{9dm}\,.$ (146)
Proof. The variance can be written in terms of an average over all gate
sequences of length $m$ as
$\displaystyle\sigma_{m}^{2}$
$\displaystyle=-\bar{F}_{m}^{2}+\left\lvert\mathcal{G}\right\rvert^{-m}\sum_{k}F_{m,k}^{2}$
$\displaystyle=-\bar{F}_{m,0}^{2}-\delta(\bar{F}_{m}^{2})+\left\lvert\mathcal{G}\right\rvert^{-m}\sum_{k}\left[F_{m,k,0}^{2}+\delta(F_{m,k}^{2})\right]$
$\displaystyle\leq\sigma_{m,0}^{2}+2\left\lvert\delta(\bar{F}_{m})\right\rvert+2\left\lvert\mathcal{G}\right\rvert^{-m}\sum_{k}\left\lvert\delta(F_{m,k})\right\rvert$
$\displaystyle\leq\sigma_{m,0}^{2}+4\left\lvert\mathcal{G}\right\rvert^{-m}\sum_{k}\left\lvert\delta(F_{m,k})\right\rvert\,.$
(147)
where $M_{0}$ and $\delta(M)$ denote the average term and the perturbation
from the average of $M$ respectively and we have used
$\displaystyle\left\lvert\delta(M^{2})\right\rvert$
$\displaystyle=\left\lvert[M_{0}+\delta(M)]^{2}-M_{0}^{2}\right\rvert=\left\lvert
2M_{0}+\delta(M)\right\rvert\left\lvert\delta(M)\right\rvert\leq
2\left\lvert\delta(M)\right\rvert\,.$ (148)
The second-to-last inequality follows since $M_{0}$ and $M_{0}+\delta(M)$ are
in the unit interval. We have also used
$\left\lvert\delta(\bar{F}_{m})\right\rvert\leq\left\lvert\mathcal{G}\right\rvert^{-1}\sum_{g\in\mathcal{G}}\left\lvert
F_{m,k}\right\rvert$, which follows from the triangle inequality.
With noise written in the form of Eq. (145), the sequence of operators applied
in the randomized benchmarking experiment with sequence $S_{m,k}$ is
$\displaystyle\mathcal{S}_{m,k}=\prod_{t=m}^{0}g_{t}\Lambda_{t,g}=\sum_{a=0}^{m+1}\epsilon^{a}\sum_{b\in\mathbb{Z}_{2}^{m+1}:H(b)=a}\prod_{t=m}^{0}g_{t}M_{t,g,b_{t}}\,,$
(149)
where $H(b)$ is the Hamming weight of the bit string $b$ and
$\displaystyle M_{t,g,b_{t}}=\begin{cases}\Lambda_{t}&\mbox{if }b_{t}=0\\\
\Delta_{t,g}&\mbox{if }b_{t}=1\,.\end{cases}$ (150)
Substituting Eq. (149) into the expression for the probability
$F_{m,k}=(E|\mathcal{S}_{m,k}|\rho)$ and using the triangle inequality gives
$\displaystyle\left\lvert\delta(F_{m,k})\right\rvert$
$\displaystyle\leq\sum_{a=1}^{m+1}\epsilon^{a}\sum_{b}\left\lvert(E|\left(\prod_{t=m}^{0}g_{t}M_{t,g,b_{t}}\right)|\rho)\right\rvert$
$\displaystyle\leq\sum_{a=1}^{m+1}\epsilon^{a}\sum_{b}\left\lVert\prod_{t=m}^{0}g_{t}M_{t,g,b_{t}}\right\rVert_{\infty}$
$\displaystyle\leq\sum_{a=1}^{m+1}\epsilon^{a}\binom{m+1}{a}d^{(a+1)/2}$
$\displaystyle=\sqrt{d}\bigl{[}(1+\epsilon\sqrt{d})^{m+1}-1\bigr{]}$
$\displaystyle\leq\sqrt{d}\bigl{[}\textrm{e}^{\epsilon\sqrt{d}(m+1)}-1\bigr{]}\,,$
(151)
where, to get from the second to the third line, we note that for a fixed
order $a$ there are at most $a+1$ quantum channels (i.e., products of
$\Lambda_{t}$’s and the elements of $\mathcal{G}$, which are channels)
interleaved by the $a$ different $\Delta_{t,g}$ and we have also used the
submultiplicativity of the spectral norm and Proposition 5.
We can then substitute the above sequence-independent upper bound into Eq.
(VIII), setting
$\delta_{0}=\left\lvert\delta(\sigma_{m}^{2})-\sigma_{m,0}^{2}\right\rvert$
and solving for $\epsilon$ gives the sufficient condition
$\displaystyle\epsilon\leq\frac{\ln(1+\delta_{0}/(4\sqrt{d}))}{\sqrt{d}(m+1)}\,.$
(152)
To extract the slightly weaker but more transparent bound stated in the
theorem, we use the simple bounds $m+1\leq 2m$ (for $m\geq 1$),
$\delta_{0}\leq 1/4$ (because the fidelity is contained in the unit interval),
the inequality $x/(1+x)\leq\log(1+x)$, and the loose bound
$4\sqrt{d}+\delta_{0}\leq 9/2\sqrt{d}$ for $d\geq 2$. $\Box$
We remark that this result can surely be improved, though we have not
attempted to do so. In particular, there should certainly be a factor of at
least $r$, the average infidelity, bounding the change in the variance.
## IX Conclusion
We have proven that the randomized benchmarking protocol can be applied to
experimental scenarios in which the noise is time dependent in an efficient
and reliable manner. Moreover, the ability to estimate time-dependent average
gate fidelities using randomized benchmarking provides an indicator for non-
Markovianity over long timescales.
In particular, we have proven that the variance is small for short sequences
and asymptotically decays exponentially to a (small) constant, that, in the
case of unital noise, is zero. The fact that the variance is remarkably small
(e.g., on the order of $4\times 10^{-4}$ for currently achievable noise
levels) enables experimental realizations of randomized benchmarking to be
accurate even when using a small number of random sequences (e.g., 145
sequences compared to the $10^{5}$ proposed in Ref. Magesan _et al._
(2012a)).
Our results show rigorously that randomized benchmarking with arbitrary
Markovian noise is generically almost as accurate as has previously been
estimated in experiments Gaebler _et al._ (2012); Magesan _et al._ (2012b);
Brown _et al._ (2011) and numerics Epstein _et al._ (2014). However, we find
that $K_{m}$ should scale with $m$ so that the variance is independent of $m$.
We also find that if near-unitary noise (such as under- and over-rotations)
are a predominant noise source, then randomized benchmarking should be
conducted in the regime $mr\ll 1$, since the variance due to sampling random
sequences with such noise sources will remain large as $m$ increases.
It has recently been suggested that the unexpectedly good accuracy of
randomized benchmarking arises because data is simultaneously fit to
$\hat{F}_{m}$ for all sequence lengths Epstein _et al._ (2014). However, this
suggestion presupposed a more fundamental fact, which we have now proven,
namely, that the variance due to sampling random gate sequences is remarkably
small.
Our results also apply directly to interleaved benchmarking Magesan _et al._
(2012b) since the interleaved gate sequence can be rewritten as a standard
randomized benchmarking gate sequence with the interleaved gate and its
inversion incorporated into the noise, with a consequent (but small) increase
in the error rate. As such, interleaved randomized benchmarking is essentially
as accurate as randomized benchmarking, provided the noise is Markovian and
approximately gate-independent.
Another possible application of our results is in estimating the nonunitality
of a channel by estimating the constant to which the variance asymptotically
converges. While it is not immediately apparent how to guarantee that the
variance has (approximately) converged (given that the decay rate of the
variance can be arbitrarily small), it may be possible to artificially boost
the decay rate using, for example, the technique introduced in Ref. Kimmel
_et al._ (2013).
While our results prove that randomized benchmarking can reliably be performed
using the number of sequences currently used in practice for qubits, the
weaker bound on the variance for qudits implies that, to obtain results that
are currently rigorously justified to a given confidence level, many more
sequences are required when benchmarking higher-dimensional systems (or
multiple qubits). Consequently, a major open problem is to improve the bound
for qudits. Since the primary source of the improvement for qubits arose by
considering the irrep structure of the tensor product representation, one
route to obtaining an improved bound for qudits is to analyze the general
representation structure of tensor product representations.
Acknowledgements—JJW acknowledges helpful discussions with Joseph Emerson. The
authors were supported by the IARPA MQCO program, by the ARC via EQuS project
number CE11001013, and by the US Army Research Office grant numbers
W911NF-14-1-0098 and W911NF-14-1-0103. STF also acknowledges support from an
ARC Future Fellowship FT130101744.
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|
arxiv-papers
| 2014-04-24T04:52:31 |
2024-09-04T02:50:01.733971
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Joel J. Wallman and Steven T. Flammia",
"submitter": "Joel Wallman",
"url": "https://arxiv.org/abs/1404.6025"
}
|
1404.6041
|
# Leader-Contention-Based User Matching for 802.11 Multiuser MIMO Networks
†‡Tung-Wei Kuo, †Kuang-Che Lee, †Kate Ching-Ju Lin and ‡Ming-Jer Tsai
†Research Center for Information Technology Innovation, Academia Sinica,
Taiwan
‡Department of Computer Science, National Tsing Hua University, Taiwan
###### Abstract
In multiuser MIMO (MU-MIMO) LANs, the achievable throughput of a client
depends on who are transmitting concurrently with it. Existing MU-MIMO MAC
protocols however enable clients to use the traditional 802.11 contention to
contend for concurrent transmission opportunities on the uplink. Such a
contention-based protocol not only wastes lots of channel time on multiple
rounds of contention, but also fails to maximally deliver the gain of MU-MIMO
because users randomly join concurrent transmissions without considering their
channel characteristics. To address such inefficiency, this paper introduces
MIMOMate, a leader-contention-based MU-MIMO MAC protocol that matches clients
as concurrent transmitters according to their channel characteristics to
maximally deliver the MU-MIMO gain, while ensuring all users to fairly share
concurrent transmission opportunities. Furthermore, MIMOMate elects the leader
of the matched users to contend for transmission opportunities using
traditional 802.11 CSMA/CA. It hence requires only a single contention
overhead for concurrent streams, and can be compatible with legacy 802.11
devices. A prototype implementation in USRP-N200 shows that MIMOMate achieves
an average throughput gain of 1.42x and 1.52x over the traditional contention-
based protocol for 2-antenna and 3-antenna AP scenarios, respectively, and
also provides fairness for clients.
###### Index Terms:
Multiuser MIMO, User matching, Channel orthogonality
## I Introduction
With the growing technique of multiple antenna systems, the number of antennas
on an access point (AP) is increasing steadily. Most of mobile devices, such
as smartphones or tablets, are however limited by their size and power
constraints, and hence have a fewer number of antennas as compared to the AP.
Traditional 802.11 protocols, which enable only a single client to communicate
with the AP, hence cannot fully utilize concurrent transmission opportunities
supported by a multi-antenna AP. To address this problem, recent work has
advocated developing multiuser MIMO (MU-MIMO) LANs [1, 2, 3, 4, 5, 6, 7] to
enable multiple clients to communicate concurrently with an AP and fully
utilize all the available degrees of freedom [8].
|
---|---
(a) client C3 sends concurrently with C1 | (b) SNR after projection changes
|
(c) client C3 sends concurrently with C2 | (d) SNR after projection is larger than in (b)
Figure 1: SNR after projection changes
Though some MU-MIMO MAC protocols [1, 2, 3, 4, 5, 6, 7] have been proposed to
realize concurrent transmissions across different nodes, in either the uplink
or downlink scenarios, they simply select a random subset of users to
communicate concurrently with an AP. However, in a MU-MIMO LAN, the achievable
throughput of a client highly depends on who are transmitting concurrently
with it. Consider the example in Fig. 1(a), where three single-antenna clients
contend for communicating with a 2-antenna AP. Say clients C1 and C3 win the
first and second contentions, respectively, and transmit concurrently to the
AP. The 2-antenna AP receives the signals in a 2-dimensional antenna space, as
shown in Fig. 1(b). The basic approach for decoding the concurrent packets is
called zero-forcing with successive interference cancellation (ZF-SIC) [5][8].
The AP first zero-forces (ZF) the interference from client C1 by projecting
the signal of the second client C3 on a direction orthogonal to C1, and hence
can decode C3. The AP then uses interference cancellation (IC) to subtract
C3’s signal and decode the first client C1. We note that the second client
decoded by ZF however experiences SNR reduction after projection, as shown in
Fig. 1(b), while the first client decoded by IC obtains the same SNR as it
transmits alone. The amount of SNR reduction for the second client depends on
channel orthogonality between two concurrent clients, i.e., the angle $\theta$
between C1 and C3. Consider another example in Fig. 1(c), where clients C2 and
C3 transmit the first and second streams, respectively. Client C3 in this case
gets a higher SNR after projection, as in Fig. 1(d), and thus achieves a
higher throughput, as compared to sending concurrently with C1, because its
channel is more orthogonal to C2’s channel. This example illustrates that
random user selection cannot efficiently deliver the gain of a MU-MIMO LAN.
To better deliver MU-MIMO gains, some theoretical works on downlink MU-MIMO
LANs have focused on allowing the single transmitter, i.e., AP, to select a
proper subset of users as concurrent receivers. User selection in the uplink
scenario is however much more challenging because multiple contending clients
compete for transmission opportunities without coordination. As a result,
existing uplink MU-MIMO LANs [4][5] still adopt traditional 802.11 contention
to ask all the clients to contend for concurrent transmissions without
sophisticated user selection. Such a contention-based scheme not only fails to
select concurrent transmitters according to their channel characteristics, but
also wastes a lot of channel time for multiple rounds of contention for
concurrent transmission opportunities. For example, if multiple single-antenna
clients contend for transmitting to a 3-antenna AP, they need to contend for
three transmission opportunities sequentially. If a client fails to win the
first contention, it needs another round of contention to contend for the
opportunity of sending the second stream. If it fails again, it needs to
contend for transmitting the third stream. The channel time occupied by such
sequential contention could significantly offset the gain of MU-MIMO LANs.
Though some prior studies work on uplink MU-MIMO user scheduling, they either
maximize the sum rate of concurrent transmissions without considering fairness
[9][10], or cannot be compatible with legacy 802.11 contention-based MAC [11,
12, 13, 14, 15, 16, 17]. In addition, all the above proposals do not give any
formal model to formulate the relationship between two conflict design goals,
i.e., throughput and fairness. Hence, in this paper, we propose MIMOMate, a
leader-contention-based MU-MIMO MAC protocol that matches concurrent
transmitters according to their channel characteristics to maximally deliver
the MU-MIMO gain under the fairness requirement. Our contributions are as
follows:
* •
We first formulate, in Section IV, a rigorous model to formally define the
user selection problem of maximizing throughput under the fairness constraint,
and propose a matching algorithm to solve the problem. We further show that
MIMOMate’s user matching algorithm can achieve the optimal solution for the
2-antenna AP scenario.
* •
We propose a MU-MIMO MAC design, in Section V, that integrates our proposed
user matching algorithm with a leader-based contention scheme. MIMOMate’s
leader-contention-based MAC is hence compatible with traditional 802.11, and,
more importantly, requires only a single contention overhead not scaling up
with the number of concurrent streams.
* •
Unlike prior theoretical work that only mathematically analyzes the effect of
MU-MIMO user selection, we build a prototype of MIMOMate using the USRP-N200
radio platform [18], and use testbed measurements to understand the
inefficiency of random user selection in real channels in Section III.
* •
We finally experimentally evaluate, in Section VI, the performance of
MIMOMate. The results show that MIMOMate achieves an average throughput gain
of 1.42x and 1.52x over the sequential-contention-based protocol for 2-antenna
and 3-antenna AP scenarios, respectively, and also provides users fair
transmission opportunities.
The remainder of this paper is organized as follows. We review related works
in Section II. Section III measures how existing schemes fail to deliver MU-
MIMO gains and provide fairness in real channels. Sections IV and V describe
our MIMOMate algorithm and how to realize it as a MAC protocol, respectively.
In Sections VI and VII, we evaluate the performance of MIMOMate via
experiments and simulations, respectively. Finally, Section VIII concludes
this paper.
## II Related Work
In the last few years, the advantage of MU-MIMO LANs has been verified
theoretically [19, 20, 21] and demonstrated empirically [1, 2, 3, 4, 5, 22,
23]. In Beamforming [1, 2, 3], a multi-antenna AP uses the precoding technique
to transmit multiple streams to multiple single-antenna clients. SAM [4]
focuses on the uplink scenario and allows multiple single-antenna clients to
communicate concurrently with a multi-antenna AP. TurboRate [5] proposes a
rate adaptation protocol for uplink MU-MIMO LANs. IAC [22] connects multiple
APs through the Ethernet to form a virtual MIMO node that communicates
concurrently with multiple clients. 802.11n+ [23] enables concurrent
transmissions across different links. All the above practical MU-MIMO systems
leverage the traditional 802.11 content mechanism to share concurrent
transmission opportunities. In contrast, MIMOMate enables clients with a
better channel orthogonality to transmit concurrently.
Prior theoretical work on user selection in downlink MU-MIMO LANs [24, 25, 26,
27, 28, 29] selects the optimal subset of clients from those who have packets
queued in the AP to maximize the sum rate of concurrent transmissions. The
works [26][28] further address the issue of fairness. However, their solutions
are designed for downlink MU-MIMO, and cannot be easily applied in the uplink
scenarios due to the lack of a coordinator.
User selection in uplink MU-MIMO LANs [9][10] requires the AP to explicitly
coordinate between the clients for every packet and select the optimal subset
of clients to transmit at the rate specified by the AP. Enabling coordination
among clients for uplink traffic however requires a significant signaling
overhead. Our work differs from those user selection algorithms in that it
matches multiple potential subsets of concurrent transmitters to improve the
system throughput, but elects a leader from the matched users to perform
traditional 802.11 contention without coordination among clients. Some
previous works [4, 11, 12, 13] propose to use multi-round contention to enable
as many concurrent transmissions as possible, while giving clients a fair
opportunity to transmit concurrent streams. SAM [4] proposes a preamble-
counting protocol, which allows each client to count the number of existing
streams and determine whether it can contend for sending a concurrent stream
in a distributed way. Multi-round contention [11][12] is proposed to let
clients send RTSs in multiple rounds of contention. Multiple clients might
send RTSs concurrently in each single round of contention, and the AP then
feedbacks a CTS to notify those clients that can transmit concurrently. An
asynchronous MAC protocol [13] is proposed to enable clients to independently
start their concurrent transmissions, i.e., without the need of starting
concurrent transmissions at the same time. It however relies on a control
channel to feedback who can join concurrent transmissions. There are also some
papers that have considered channel orthogonality and fairness jointly in
uplink MU-MIMO [16, 17, 14, 15]. However, these solutions are heuristics
without any formal performance analysis, and also not compatible with the
existing 802.11 standard. Our work is the first that maximizes the throughput
under the fairness constraint, and is able to coexist with legacy 802.11
devices.
## III MU-MIMO Background and Motivations
Before describing our proposed protocol, we first use testbed measurements in
real channels to understand the limitation of the existing MU-MIMO MAC
protocols. The measurement results also give us an insight to the motivation
of enabling user selection in a MU-MIMO MAC. We consider again the network in
Fig. 1(a) where two single-antenna clients communicate with a 2-antenna AP.
The measurements are empirically performed using USRP-N200 [18] on a 10 MHz
OFDM channel with 802.11 modulations and coding rates. The available bit-rates
hence range from 3–27 Mb/s. The measurements are designed to answer the
following questions:
a) How often is a client unable to transmit concurrently? Recall that the
second client’s SNR reduces after projection, and the amount of SNR reduction
depends on the angle between its channel and the channel of its concurrent
transmitter, i.e., the first client. This means that the throughput of the
second client in a MU-MIMO network depends on not only its own SNR but also
channel orthogonality between concurrent transmitters. To illustrate this
point, in Fig. 2, we analytically compute the throughput of the second client
decoded by ZF for the whole range of the inter-client angle
$\theta\in[0,\pi/2]$ when its original SNR at the AP is 5, 10, 15, 20 and 25
dB, respectively.111We empirically measure an SNR-throughput mapping table and
map the SNR after projection to the corresponding throughput.
Figure 2: Throughput after projection
The figure shows that a small inter-client angle significantly reduces the SNR
after projection and hence the throughput. If the client has a high original
SNR, e.g., 25 dB, by selecting the best bit rate according to the SNR after
projection [5], it can still get a relatively high throughput even after
projection. In contrast, if the original SNR of the client is low, the client
would be very likely to get zero throughput even if it already uses perfect
bit rate adaptation. This is because a small SNR reduction could make its SNR
after projection become lower than the 802.11 operational SNR region, i.e., 4
dB. In particular, if the client’s original SNR is 10 dB, then for any angle
smaller than 31 degree, its SNR after projection drops below 4 dB, leading to
zero throughput. The situation is even worse if the client’s original SNR is
only 5 dB.
One important thing worth noting is that the success of decoding the first
client using ZF-SIC relies on the AP being able to decode the second client
correctly. Otherwise, the AP cannot remove the interfering signal of the
second client, and hence also fails to decode the first client. Also, after
removing the interfering signal from the second client, the first client can
select its best bit-rate according to its original SNR without considering who
later joins the concurrent transmission. The above constraint hence requires
the second client to give up its transmission opportunity if its SNR after
projection is lower than the 802.11 operational SNR region. This also
motivates why selecting a suitable subset of clients to transmit concurrently
is important for delivering the gain of MU-MIMO.
b) How different is the throughput of a client when it transmits concurrently
with different clients? We have shown in Fig. 2 that the throughput of a
client could change significantly with the inter-client angle. We next check
whether the angle between the channels of two clients is actually randomly
distributed between $[0,\pi/2]$. To validate this point, we empirically
measure how much throughput a client can achieve if it transmits concurrently
with different clients in a real testbed where 6 single-antenna clients, named
Tx1-Tx6, contend for communicating concurrently with a 2-antenna AP. We repeat
the experiment twice with different random locations of the clients.
Experiment 1 locates Tx1 close to the AP, while Experiment 2 locates Tx1 far
from the AP. Two experiments represent the scenarios when Tx1 has a high and
low original SNR, respectively.
|
---|---
(a) Experiment 1 | (b) Experiment 2
Figure 3: Heterogeneous throughput in a MU-MIMO LAN
Fig. 3 plots the throughput of one client (denoted by Tx1) when it transmits
alone or when it transmits concurrently with any of other five clients
(denoted by Tx2–Tx6). The figure shows that, in both experiments, as compared
to transmitting alone, Tx1 usually gets a lower throughput when it joins the
concurrent transmission and is decoded by projection. In addition, the
client’s throughput, as transmitting concurrently with different users, could
be very different. For example, in experiment 1, Tx1 obtains a high throughput
when it transmits concurrently with Tx4, while suffering a low throughput as
joining Tx6’s transmission. The situation becomes worse when Tx1 has a low
original SNR (as in experiment 2); in many cases, it gets zero throughput as
transmitting concurrently with another client. These results are consistent
with the analysis shown in Fig. 2. Thus, to get a high throughput, Tx1 would
like to transmit with a client whose channel is more orthogonal to its
channel, as a result experiencing less SNR reduction. The current random
access protocols however do not consider this effect, and hence cannot
efficiently deliver the MU-MIMO gain.
|
---|---
(a) Experiment 1 | (b) Experiment 2
Figure 4: Fairness of transmission opportunities based on the throughput
c) Is selecting concurrent clients to maximize the throughput fair enough? A
naïve solution to improving the throughput is to deterministically assign the
client that achieves the highest throughput to join the transmission of the
first contention winner. This simple solution however might not be fair for
all the clients. To see why this is a problem, we measure the throughput of
each client in the above scheme. Specifically, in each experiment, we let each
client have an equal probability to transmit the first stream; given the first
contention winner, we then select the client that can achieve the highest
throughput after projection to transmit the second stream. Each experiment
includes 1,000 rounds of contention, i.e., 1,000 concurrent transmissions.
Because every client has an equal probability to win the first contention, we
do not consider the throughput of a client transmitting using the first
stream. Instead, in Fig. 4, we only plot the average throughput of a client
transmitting in the second stream. The results of two experiments show that,
by applying such a naïve solution, some clients, e.g., Tx1, Tx4, Tx5 and Tx6
in experiment 1, do not have any opportunities to join concurrent
transmissions. Even worse, the clients with a low original SNR are very likely
to starve because they cannot compete with those clients in the high SNR
regime. One might think, alternatively, we can assign the client that has the
largest inter-client angle with the first contention winner to transmit
concurrently. We repeat the same experiment by applying the above assignment.
The results in Fig. 5 show that this solution again fails to provide fairness
because some clients happen to have a small angle with all the other clients
and hence do not get any transmission opportunities.
|
---|---
(a) Experiment 1 | (b) Experiment 2
Figure 5: Fairness of transmission opportunities based on the angle
## IV MIMOMate Matching
Motivated by the above measurements, we aim at designing a matching protocol,
called MIMOMate, that pairs concurrent clients to deliver the maximum
throughput gain enabled by concurrent transmissions, while, at the same time,
providing clients fair concurrent transmission opportunities. For simplicity,
we describe our MIMOMate protocol assuming that multiple single-antenna
clients communicate concurrently with a multi-antenna AP in an uplink MU-MIMO
LAN. Our design however can be generalized to clients with multiple antennas
and downlink MU-MIMO LANs. We will describe in the next section how to realize
MIMOMate as a leader-contention-based MAC protocol.
### IV-A Overview
The goal of MIMOMate is to build chain relation in concurrent transmissions.
When a client wins contention, the following concurrent transmissions are
determined a priori. Hence, all the concurrent transmitters only require to
precede their streams with one contending process. In particular, MIMOMate
matches clients whose channels are more orthogonal to each other as a group of
concurrent transmitters with a precedence relation, which is called MIMO-
mates. To see how it works, let us consider an example where two clients are
allowed to communicate concurrently with a 2-antenna AP. We match two clients
as MIMO-Mates, in which one is the lead and the other is the follower. When
the lead of MIMO-Mates wins contention and transmits the first stream, its
follower transmits the second stream concurrently immediately after it detects
the transmission from the lead.
The protocol can be generalized to an $N$-antenna AP scenario where $N$
clients can transmit concurrently. We match $N$ clients as a MIMO-Mate
(precedence) relation $(u_{1},u_{2},\cdots,u_{N})$ such that clients
$u_{1},u_{2},\cdots,u_{N}$ join the concurrent transmissions one after another
in order of precedence. In particular, after the lead $u_{1}$ wins contention,
any of the following clients $u_{i}$ can count the number of preambles to
figure out the time that it should transmit. The above protocol benefits
throughput gains from two factors: 1) it matches clients with a higher channel
orthogonality to transmit concurrently and minimizes throughput reduction
caused by projection; 2) it requires only one contending process for
concurrent transmissions, as a result reducing the overhead significantly.
However, the benefit of MIMOMate might not be able to be fully delivered when
any of the matched MIMO-Mates does not have traffic to send. In this case, the
unused transmission opportunities should be exploited by other clients to
avoid waste. We thus further integrate MIMOMate with an angle-based contention
mechanism, which will be discussed in Section V.
### IV-B Problem Formulation
Our objective is to match clients as MIMO-mates in order to maximize the
throughput subject to the fairness constraint. We first define our problem in
a 2-antenna AP scenario, and next extend it to a 3-antenna AP scenario and
even a more general $N$-antenna AP scenario.
Let us first consider the 2-antenna AP scenario. Say $u$ is the client that
wins the first contention, and $v$ is the follower of $u$, who joins $v$’s
transmission. We define $(u,v)$ as the MIMO-mate relation of clients $u$ and
$v$. Let $r^{(u,v)}_{v}$ denote the throughput of $v$ as it transmits
concurrently with $u$ and is decoded by using ZF to project orthogonal to
client $u$. We note that client $v$ might get a different throughput if it is
assigned to follow a different predecessor, i.e., $r^{(u,v)}_{v}$ could be
different from $r^{(u^{\prime},v)}_{v}$ if $u\neq u^{\prime}$. The MIMO-mate
matching problem in a 2-antenna AP scenario can be defined as follows:
###### Problem 1.
(2-MIMOMate) Given a set of clients $V$ and the throughput $r^{(u,v)}_{v}$ for
all $u,v\in V$, the matching problem is to find a set $M\subseteq V\times V$
such that
1. 1.
$r^{(u,v)}_{v}>0,\forall(u,v)\in M$,
2. 2.
$u\neq v,\forall(u,v)\in M$,
3. 3.
$u_{1}\neq u_{2}$ and $v_{1}\neq v_{2}$ for any two distinct elements
$(u_{1},v_{1}),(u_{2},v_{2})\in M$,
4. 4.
$|M|$ is maximized,
5. 5.
$\sum_{(u,v)\in M}r^{(u,v)}_{v}$ is maximized among those $M$ satisfying
Constraint 4.
To ensure the success of ZF-SIC decoding, Constraint 1 allows a client to join
the concurrent transmission only if it can be successively decoded, i.e.,
getting a positive throughput. Constraints 2–3 force each client to follow at
most one of other clients, and hence guarantee fairness. The rationale of
Constraint 4 is that, since each client in traditional 802.11 has an equal
probability to win the first contention and transmit the first stream, then,
by finding the maximum set $M$, we allow as many clients as possible to join
the concurrent transmission. This ensures clients to also have a fair
probability to transmit the second stream. Under such a fairness constraint,
our goal is to find a feasible solution that maximizes the total throughput of
the followers, i.e., the second streams. Note that we are only interested in
the throughput of the followers because, by using ZF-SIC, a client who
transmits the first stream can get about the same throughput no matter who its
follower is [5].
|
---|---
(a) 2-MIMOMate matching | (b) 3-MIMOMate matching
Figure 6: MIMOMate matching for a network with 4 clients
The above 2-MIMOmate problem can actually be illustrated as a bipartite graph,
as shown in Figure 6(a). Each edge $(u,v)$ is associated with a weight, which
is set to the throughput of $v$ when it follows $u$, i.e., $r^{(u,v)}_{v}$. We
observe that the 2-MIMOmate problem is exactly equivalent to the Bipartite
Maximum Weighted Maximum Cardinality Matching problem, which finds a maximum
cardinality matching with maximum weight in a bipartite graph and can be
solved in polynomial time by the algorithm proposed in [30] (see Chapter 7.8).
Note that, since MIMOMate always assigns any leader a specific follower, any
feasible solution of Problem 1 is therefore deterministic. That is, the
relationship between the first contention winner and its follower is fixed
until we solve Problem 1 again when the channels change. This is very
different from the probabilistic nature of uniformly random contention used in
most of existing protocols [4, 11, 12], where every client has the same
probability to win the contention of sending concurrently with a given first
winner. A natural question then arises: Would choosing the follower uniformly
randomly, e.g., via uniformly random contention, results in a solution better
than the output of Problem 1? We give the following positive result on Problem
1:
###### Theorem 1.
If the throughput $r^{(u,v)}_{v}$ is greater than $0$ for all $u,v\in V,u\neq
v$, then the average throughput of Problem 1’s output is higher than or equal
to the average throughput achieved by using uniformly random contention to
choose the follower, even if the contention overhead is ignored.
In fact, we can prove the following stronger theorem, which replaces the
uniformly random contention by any probabilistic assignment that obeys the
fairness constraint. More specifically, in any fair probabilistic assignment,
every client has the same probability to transmit the second stream; however,
any first winner might choose its follower with a non-uniform probability, and
different leaders might have different probability distributions.
###### Theorem 2.
If the throughput $r^{(u,v)}_{v}$ is greater than $0$ for all $u,v\in V,u\neq
v$, then the average throughput of Problem 1’s output is higher than or equal
to the average throughput achieved by any fair probabilistic assignment.
We prove the two theorems in the Appendix.B. Note that the two theorems only
holds when the throughput $r^{(u,v)}_{v}$ is greater than $0$ for all $u,v\in
V,u\neq v$. If this condition does not hold, i.e., some $r^{(u,v)}_{v}=0$,
then using contention to choose the followers, e.g., the method used in [4,
11, 12], would fail ZF-SIC decoding. On the other hand, to guarantee the
success of ZF-SIC decoding, such a pair of clients would not be chosen in
Problem 1. In addition, if we further consider the overhead of using
contention to choose followers, the throughputs of the methods proposed in [4,
11, 12] would further decrease.
We next consider the 3-antenna AP scenario. Say clients $u,v$ and $w$
communicate with a 3-antenna AP concurrently and join the concurrent
transmissions one after another. We define $(u,v,w)$ as the MIMO-mate relation
of clients $u,v$ and $w$. The AP can use ZF-SIC to decode client $w$ by
projecting along the direction orthogonal to both clients $u$ and $v$. It re-
encodes client $w$’s stream and subtracts it from the received signals. The AP
then decodes client $v$ by projecting the resulting signal along the direction
orthogonal to client $u$, and decodes client $u$ after removing the signals of
clients $v$ and $w$. Let $r^{(u,v,w)}_{v}$ and $r^{(u,v,w)}_{w}$ denote the
throughput of clients $v$ and $w$, respectively. The MIMOMate matching problem
in a 3-antenna AP scenario can be defined as follows:
###### Problem 2.
(3-MIMOMate) Given a set of clients $V$ and the throughput $r^{(u,v,w)}_{u}$
and $r^{(u,v,w)}_{w}$ for all $u,v,w\in V$, the matching problem is to find a
set $M\subseteq V\times V\times V$ such that
1. 1.
$r^{(u,v,w)}_{v}>0$ and $r^{(u,v,w)}_{w}>0,\forall(u,v,w)\in M$,
2. 2.
$u\neq v\neq w,\forall(u,v,w)\in M$,
3. 3.
$u_{1}\neq u_{2},v_{1}\neq v_{2}$, and $w_{1}\neq w_{2}$ for any two distinct
elements $(u_{1},v_{1},w_{1}),(u_{2},v_{2},w_{2})\in M$,
4. 4.
$|M|$ is maximized,
5. 5.
$\sum_{(u,v,w)\in M}(r^{(u,v,w)}_{v}+r^{(u,v,w)}_{w})$ is maximized among
those $M$ satisfying Constraint 4.
input : a set of clients $V$ and the channel state information from each
client to AP’s $N$ antennas
1
2Duplicate $V$ to $V_{1},V_{2},\cdots,V_{N}$
3 Remove legacy 802.11 nodes from $V_{2},\cdots,V_{N}$
4 Initialize $M\leftarrow\\{\\}$
5for _$k:=1$ to $N-1$_ do
6 For each edge $(u_{i},v_{j}){\in}V_{k}\times V_{k+1}$, if $u_{i}$ has a
predecessor or $u_{i}{\in}V_{1}$, set the weight of edge $(u_{i},v_{j})$ to
the throughput of $v_{j}$ as it transmits concurrently with $u_{i}$ and all
its predecessors; otherwise, set the weight of $(u_{i},v_{j})$ to 0
7 $M^{\prime}\leftarrow$ the solution of the 2-MIMOMate matching problem for
$V_{k}\times V_{k+1}$ solved by [30]
8 if _$k=1$_ then
9 Add each $(u_{i},v_{j}){\in}M^{\prime}$ to $M$
10
11 else
12 For each $(u_{i},v_{j}){\in}M^{\prime}$, find the MIMO-Mate relation
(element) $m\in M$ that includes $u_{i}\in V_{k}$ and add client $v_{j}\in
V_{k+1}$ to the element $m$
13
14
15return $M$
Algorithm 1 $N$-MIMOMate Matching Algorithm
Similarly, in the 3-MIMOMate problem, we are only interested in maximizing the
throughput of the followers, i.e., clients $v$ and $w$. We observe that the
3-MIMOMate problem, as illustrated in Figure 6(b), is actually a variation of
the Maximum 3-Dimensional Matching problem [31], which is defined as follows:
Let $X$, $Y$, and $Z$ be disjoint sets, and let $T$ be a subset of
$X{\times}Y{\times}Z$ that includes all feasible matching combinations. The
problem finds the maximum matching $M\subseteq T$ such that
$u_{1}{\neq}u_{2},v_{1}{\neq}v_{2}$ and $w_{1}{\neq}w_{2}$ for any two
distinct elements $(u_{1},v_{1},w_{1})$ and $(u_{2},v_{2},w_{2})$ in $M$.
Hence, the differences between our 3-MIMOMate problem and the 3-dimensional
matching problem are 1) our problem further considers the total weight of a
matching (i.e., Constraint 5), and 2) Constraint 2 in our problem restricts
each client to be included in an element at most once. For example, $(u,u,v)$
is not a feasible combination in our problem because client $u$ cannot
transmit two streams from its single antenna at the same time.
We note that a general $N$-MIMOMate matching problem can be formulated in a
similar manner, and it is a variation of the $N$-dimensional matching problem
[31]. On the other hand, although the 2-MIMOMate problem is polynomial time
solvable, the $N$-MIMOMate problem for any $N\geq 3$ is however NP-hard. We
will prove the NP-hardness of the 3-MIMOMate problem in the Appendix.A by
deriving a reduction from the 3-dimensional matching problem (which is also
NP-hard) to our problem. The NP-hardness of the $N$-MIMOMate problem can be
proved in a similar way.
### IV-C Heuristic Matching Algorithm
There is an approximation algorithm [32] proposed to solve the $N$-dimensional
matching problem. We can use the algorithm to solve our $N$-MIMOMate matching
problem and achieve an approximation ratio, 3/2+$\epsilon$, for any
$\epsilon>0$, in terms of the size of matching. It however does not ensure to
find the one achieving the maximal throughput (i.e., Constraint 5) among all
maximum matchings. In addition, our problem requires an additional cost to
compute the weights (throughputs) of all possible MIMO-Mates, which is an
$O(|V|^{N})$ computational cost. We hence propose an algorithm, as shown in
Algorithm 1, to solve our MIMOMate matching problem with a reduced cost of
weight computation. The basic idea of Algorithm 1 is to decompose the
$N$-MIMOMate matching problem into $(N-1)$ 2-MIMOMate matching problems, each
of which can be solved by the bipartite maximum weighted maximum cardinality
matching algorithm [30] in polynomial time. The advantage of such
decomposition is that it reduces the cost of weight computation from
$O(|V|^{N})$ to $O(N|V|^{2})$.
|
---|---
(a) step 1 | (b) step 2
Figure 7: Example of solving 3-MIMOMate Matching
For simplicity, we use the 3-antenna AP scenario to describe our algorithm,
and next explain how to generalize it to an $N$-antenna AP scenario. We first
duplicate the client set $V$ to $V_{1},V_{2}$ and $V_{3}$ (line 1 in Algorithm
1). Our algorithm solves the 3-MIMOMate matching problem in two steps, as
illustrated in Fig. 7. In the first step, as in Fig. 7(a), we set the weight
of edge $(u_{i},v_{j}){\in}V_{1}{\times}V_{2}$ by computing the throughput of
$v_{j}{\in}V_{2}$ as it transmits concurrently with $u_{i}{\in}V_{1}$, and try
to optimally match any second client in $V_{2}$ to a MIMO-Mate lead in
$V_{1}$, which is actually the 2-MIMOMate matching problem for $V_{1}\times
V_{2}$. After the first step, we determine the MIMO-Mate relation for the
first and second transmitters. Then, in the second step, our goal is to add
any third transmitter in $V_{3}$ into each MIMO-Mate relation. To do so, we
first assign edge $(v_{j},w_{k}){\in}V_{2}{\times}V_{3}$ a weight that equals
the throughput of $w_{k}$ when it transmits concurrently with $v_{j}$ and
$v_{j}$’s predecessor, which is solved in the first step. For example, in Fig.
7(b), because $(u_{4},v_{1})$ is matched as MIMO-Mates in the first step, the
weight of edge $(v_{1},w_{2})$ equals the throughput of $w_{2}$ when it joins
concurrent transmissions of $u_{4}$ and $v_{1}$. We note that, for any
$v_{j}{\in}V_{2}$, if it is not assigned a predecessor in the first step, we
set the weight of all its outgoing edges to 0 because it is not allowed to
match with any clients in $V_{3}$ in the second step. After weight assignment,
we can solve another 2-MIMOMate matching problem to match clients in $V_{3}$
to clients in $V_{2}$.
Our algorithm can be generalized to the $N$-MIMOMate matching problem.
Specifically, it iteratively solves a 2-MIMOMate matching problem to match
clients in $V_{k+1}$ to clients in $V_{k}$, where $k=1,2,\cdots,N-1$. Hence,
each iteration includes a client in a MIMO-Mate relation. In particular, the
$k^{th}$ iteration adds the $(k+1)^{th}$ concurrent client to MIMO-Mates. In
addition, in the $k^{th}$ iteration, because we already know the first $k$
clients in MIMO-Mates, we thus only need to compute the weight of edges
$(u,v)\in V_{k}\times V_{k+1}$ according to the given MIMO-Mate relation. This
is why our algorithm can reduce the cost of weight computation to
$O(N|V|^{2})$.
So far we describe our algorithm by assuming that all the clients are MIMOMate
nodes. Our algorithm can be slightly adjusted to allow the coexistence of
MIMOMate nodes and legacy 802.11 nodes. Recall that we duplicate the client
set $V$ to $V_{1},V_{2},\cdots,V_{N}$, where $N$ is the number of antennas
equipped on the AP. Each node in the duplicated set $V_{i}$ is a candidate of
sending the $i^{th}$ stream. Note that legacy nodes follow the traditional
802.11 operation and can only contend for sending the first stream. In other
words, legacy nodes do not leverage the concurrent transmission opportunities
and will not follow any ongoing transmissions. We can hence simply remove
those legacy nodes from $V_{2},\cdots,V_{N}$, and only keep them in $V_{1}$.
By doing this, legacy nodes can still use conventional 802.11 contention to
occupy the first dimension, and can further be followed by some other MIMOMate
nodes. Consider Fig. 7 as an example. Assume that node $u_{4}$ is a legacy
node. We only put it in $V_{1}$, but not in $V_{2}$ and $V_{3}$. It hence can
be followed by some other MIMOMate nodes, but cannot join concurrent
transmissions.
## V MIMOMate’s Medium Access Protocol
We consider a MU-MIMO MAC protocol similar to SAM [4], where clients join the
concurrent transmissions one after another. Like SAM [4], clients join
concurrent transmissions one after another. Each client counts the number of
concurrent streams by cross-correlating with the known preamble in the
presence of ongoing transmissions. Clients can join the concurrent
transmissions until they detect that the number of existing streams equals the
number of antennas at the AP.222The preamble-counting based protocol, like SAM
[4], could suffer from collisions when hidden nodes interrupt the preamble-
counting process. We apply a multi-round light-weight handshaking mechanism
proposed in [33] to address the hidden terminal problem with minimum overhead.
Each client determines its best bit-rate based on TurboRate, the MU-MIMO rate
adaptation scheme proposed in [5]. TurboRate allows each client to announce
training symbols before data transmission. All the clients who contend for the
later transmissions can hence learn the channels of the ongoing streams from
those training symbols, and adapt the bit-rates based on their channels.
Moreover, TurboRate asks clients to give up contention opportunities if their
SNR after ZF-SIC decoding is too low to be decodable. To increase the gain of
MU-MIMO, the protocol forces concurrent clients to end their transmissions at
about the same time. To do so, concurrent clients overhear the information
about the frame duration of the first stream, which is embedded in the MAC
header, and fragment or aggregate their packets accordingly [5][23]. MIMOMate
differs from the existing MAC protocols in that it only allows clients to use
802.11’s CSMA/CA to contend for the first stream, but lets the remaining
clients join the concurrent transmission of its predecessor in the MIMO-Mate
relation scheduled by Algorithm 1. In particular, say a client is scheduled to
transmit the $k^{th}$ stream in the MIMO-Mates; it can start transmitting once
it detects $k-1$ preambles from all its predecessors after its leader wins the
contention. Hence, all clients in the MIMO-Mates only require one contending
process.
To realize such a user matching protocol, MIMOMate’s MAC needs tow major
modifications: 1) the AP needs to learn the uplink channel information of all
its clients, and 2) the AP needs to announce the matching result to its
clients. To learn the channel information, one possible solution is to let all
the clients learn their uplink channels and report this information to the AP.
To do so, the clients leverage channel reciprocity [34], which refers to the
property that the channels in the forward and reverse directions are the same.
Using reciprocity, every client can exploit the beacons to learn the downlink
channel and use it to estimate the uplink channel. It is however an expensive
overhead to ask all the clients to report their channels for every packet
transmission. On the other hand, legacy nodes, which follow the traditional
802.11 operation, do not feedback this information. We hence perform the
following optimizations to reduce the overhead of channel feedback: The AP
learns the uplink channel of a client from its uplink frames, including the
association frames when that client joins the network, the data frame of its
uplink packets, and the ACK of its downlink packets. The AP hence only needs
to re-schedule MIMO-Mates when it detects that the channels of certain clients
change due to channel variation or user mobility. Once the AP reschedules
MIMO-Mates, it announces the updated matching result to the MIMOMate nodes. A
simple solution is to annotate the periodical beacon messages with the
announcement. However, legacy nodes might not be able to identify the modified
beacon format. To enable the coexistence of MIMOMate nodes and legacy nodes,
we let the AP send the matching result in another control frame using the
subtype not used in conventional 802.11 [35].
Figure 8: Angle-based contention
We notice that the above protocol only operates properly if the scheduled
MIMO-Mates always have traffic to transmit. However, in practice, a client
might have a bursty traffic pattern. Hence, to enable users to fully utilize
concurrent transmission opportunities, we can further integrate MIMOMate with
any contention-based MAC protocol. Specifically, clients can contend for the
unused degrees of freedom if the scheduled MIMO-Mates do not have traffic to
transmit. However, to better exploit the gain of concurrent transmissions, we
propose to integrate MIMOMate with an angle-based contention scheme. In
particular, when any of the scheduled MIMO-Mates does not have traffic to
transmit, we allow other clients to contend for the concurrent transmission
opportunity, e.g., the second stream in the example shown in Fig. 8, until the
number of concurrent streams equals the number of antennas supported by the
AP, $N$. However, we modify the contention mechanism to assign different users
a different probability of winning a concurrent transmission opportunity,
according to their channel orthogonality with the ongoing streams.
Specifically, we tend to let a client with a larger angle between its channel
and the channels of the ongoing streams have a higher probability to win the
concurrent transmission opportunity such that SNR reduction due to projection
can be minimized.
input : the initial contention window of the $k^{th}$ stream $CW^{k}\leftarrow
CW_{min}$; the initial update of the $k^{th}$ stream
$\delta^{k}_{cur}\leftarrow 0$; $N$ antennas at the AP
1
2for _$1\leq k\leq N$_ do
3 //contention for the $k^{th}$ stream in each packet transmission
4 Learn the angle $\theta\in[0,\pi/2]$ between the client’s channel and the
channels of the $(k-1)$ ongoing streams
5 if _SNR after projection $\leq$ 802.11 SNR regime_ then
6 return; //give up this contention
7 Update $CW^{k}$ using traditional 802.11 backoff
8 if _$k >1$_ then
9 $\delta^{k}_{last}\leftarrow\delta^{k}_{cur}$
10 $CW_{orig}^{k}\leftarrow CW^{k}-\delta^{k}_{last}$
11 $CW^{k}\leftarrow CW_{orig}^{k}-\frac{\theta-\pi/4}{\pi/4}*CW_{orig}^{k}$
12 $CW^{k}\leftarrow\max(CW_{min},\min(CW_{max},CW^{k}))$
13 $\delta^{k}_{cur}\leftarrow CW^{k}-CW^{k}_{orig}$
14
15
Algorithm 2 Angle-based Contention Scheme
To achieve this goal, we apply Algorithm 2 to adjust the contention window for
each concurrent stream according to the channels of the concurrent
transmitters. Specifically, when a node contends for sending the $k^{th}$
stream in the presence of $(k-1)$ ongoing transmissions, it will adjust its
contention windows based on the angle between its own channel and the channels
of the $(k-1)$ ongoing transmitting clients. We assume that clients can learn
the angle between its own channel and the channels of the ongoing transmitters
using the distributed method proposed in [5]. The high-level idea of the
angle-based contention is that, if this angle is large, we let the client
decrease its contention window and hence earn a higher probability to win the
concurrent transmission opportunity. Otherwise, the client gives other clients
a higher priority to transmit concurrently by increasing its contention
window.
To realize the above design, we let each client maintain a distinct contention
window $CW^{k}$ for the contention of the $k^{th}$ stream. The contention
windows are adjusted according to the channels of the ongoing clients. The
amount of increment (or decrement) is proportional to the inter-client angle,
i.e. $\frac{\theta-\pi/4}{\pi/4}$ in line 10. To ensure fairness, we ask a
client assigned a higher priority in the current packet to pay back its
opportunity in the next packet. To this end, if a client decreases (increases)
the contention window by $\delta$ for the current packet, it pays (earns) the
priority back by increasing (decreasing) $\delta$ to its contention window for
the next packet, i.e., $\delta_{last}$ in line 8\. The above contention scheme
can be applied for the contention of each concurrent stream until the number
of streams reaches the number of antennas supported by the AP, i.e., $1\leq
k\leq N$.
Overhead and complexity: Recall that implementing MIMOMate as a MAC protocol
relies on two modifications: 1) the AP needs to learn the uplink channels, and
2) the AP needs to announce the matching results. Note that we let the AP
measure the channel information from historical uplink frames without any
additional message overhead. The only additional overhead required by our
design is the matching announcement. We will show in Section VI that such a
small overhead does not offset the gain of our matching algorithm. On the
other hand, since our matching algorithm is performed in the access point, and
the complexity of clients should not change much. Therefore, the only supports
we need from MIMOMate clients are that 1) they need to receive the matching
announcement, and 2) they need to adapt the contention window size based on a
simple operation defined in our angle-based contention scheme, as in Algorithm
2. We believe additional power consumption in the clients due to our design
should be negligible.
## VI Experimental Results
We build a prototype of MIMOMate using the USRP-N200 radio platform, which is
equipped with an RFX2400 daughterboard. A multi-antenna AP is built by
combining multiple USRP-N200 boards using an external clock. We implement an
OFDM PHY layer with standard 802.11 modulations (BPSK, 4-64QAM) and code
rates. Since USRP-N200 operates on a 10MHz channel, the possible bit rates
range from 3 to 27 Mb/s. We evaluate the performance of MIMOMate in both
2-antenna and 3-antenna AP scenarios. Limited by the number of USRPs we have,
we set 6 clients to contend for transmitting two packets concurrently to the
2-antenna AP, while setting 5 clients to contend for transmitting three
packets concurrently to the 3-antenna AP. To allow multiple clients to
transmit concurrently, we leverage the synchronization method used in [5][23].
Specifically, for each experiment, the AP broadcasts a trigger signal. Each
client records the timestamp of detecting the trigger, $t_{trigger}$, waits a
pre-defined period of time, $t_{\Delta}$, and sets the timestamp of the
beginning of its transmission to $t_{start}=t_{trigger}+t_{\Delta}$. In our
testbed, $t_{\Delta}$ is set to 0.1s, which is long enough to tackle the
delays introduced by software.
We compare the following schemes: 1) MIMOMate, which is our proposed protocol,
2) max-throughput first, which always allows the client that achieves the
maximal throughput after projection to join the concurrent transmissions, 3)
max-angle first, which always allows the client that has the maximum angle
with the ongoing transmissions to transmit concurrently, 4) SAM [4], i.e.,
contention-based protocol without RTS/CTS, which assigns all users an equal
probability to sequentially contend for each concurrent transmission
opportunity, and 5) MRC [11], i.e., multi-round contention, which also assigns
each client an equal probability of winning contentions, but precedes
concurrent transmissions with multiple rounds of RTS and a single CTS. For all
the comparison schemes, we apply TurboRate [5], a MU-MIMO rate adaptation
scheme, to allow concurrent clients to select their best bit rates.
Due to the timing constraints limited by software radio, we do not implement
contention, random backoff and ACK in USRPs. Instead, for each experiment, we
offline create a packet trace of 1,000 1500-byte packets for each client. The
traces of different clients are generated based on the above four comparison
schemes, and ensure that there are at most 2 and 3 clients assigned to
transmit concurrently in a particular time-slot in 2- and 3- antenna AP
scenarios, respectively. In particular, in the beginning of each experiment,
we let each client transmit training symbols, one after another, for the AP to
estimate its uplink channel. The AP then performs offline contention to
generate 1,000 rounds of concurrent transmissions. For all the comparison
schemes, in each round of concurrent transmissions, the AP assigns each client
a randomly-selected backoff value between 1 and its contention window, and
picks the client with the smallest backoff value to send the first stream. The
contention window of each client is updated according to the 802.11 standard
if collisions occur. The AP then assigns the remaining concurrent transmission
opportunities to other clients based on design of various comparison
protocols. For example, in SAM, the AP uses the same contention scheme to
assign the remaining transmission opportunities; in MIMOMate the AP assigns
MIMO-Mates of the first contention winner to transmit concurrently; in max-
throughput first and max-angle first, the AP assigns the remaining
transmission opportunities based on the throughput and the angle between
channels, respectively. Based on the contention results, the AP generates the
packet trace for each client, and immediately sends the trace to each client
through Ethernet connection. Each USRP client can hence read its offline-
generated packet trace and determine the time that it should transmit packets
accordingly. Clients are asked to send null symbols, i.e., 0, if they are not
selected to transmit in a round of packet transmissions. Since offline
contention performed in the AP does not take too much time, we expect that the
channels do not change significantly, i.e., the channels during data
transmissions would be similar to those learned in the training phase. In
addition, since USRPs cannot implement real-time ACK, we disable
retransmissions in the experiments. That is, the AP simply drops a packet if
the packet cannot be received or decoded correctly.
We first evaluate the performance of MIMOMate when clients have a continuous
traffic pattern, and next evaluate the performance of integrating the angle-
based contention mechanism (Algorithm 2) with MIMOMate when clients have a
bursty traffic pattern.
### VI-A Performance Comparison for Continuous Traffic
We evaluate the performance of the comparison schemes in terms of 1)
throughput gain, 2) fairness, and 3) overhead.
---
(a) Total throughput in the 2-antenna AP scenario
(b) Total throughput in the 3-antenna AP scenario
Figure 9: Throughput for continuous traffic
Throughput gain: We first check the throughput gain delivered by MIMOMate when
users have a continuous traffic pattern, i.e., always have packets to send.
Hence, in MIMOMate, the scheduled MIMO-Mates can always transmit concurrently
if their lead wins the first contention. We repeat the experiment with random
assignment of client locations in our testbed.
Figs. 9(a) and 9(b) plot the CDFs of the total throughput in 2-antenna and
3-antenna scenarios, respectively. The figures show that traditional
contention-based protocols, i.e., SAM and MRC, assign each user an equal
probability to win the contention, without considering the channel
characteristics, and produce a low throughput. MRC requires additional RTS-CTS
overhead, and hence performs a little bit worse than SAM. Compared to SAM
(MRC), the average throughput gain from enabling concurrent transmissions with
MIMOMate’s user selection is about 42% (45%) and 52% (57%) in 2- and 3-antenna
AP scenarios, respectively. The gain mainly comes from two design principles
in MIMOMate: 1) minimizing SNR reduction due to MIMO decoding, and 2) reducing
the channel time wasted for contending for concurrent transmissions. Note that
the gain in the 3-antenna AP scenario is higher than that in the 2-antenna AP
scenario. It implies that user matching plays an important role to deliver the
MU-MIMO gain especially when the number of concurrent transmissions supported
by the system increases. The figures also show that max-angle first and max-
throughput first produce a throughput comparable to (or even slightly higher
than) our MIMOMate because they greedily select the users with the best
channel characteristics or with the highest throughput to join the concurrent
transmissions. In addition, similar to MIMOMate, they also require only one
contending process. We will show later that these two schemes however result
in unfair resource sharing.
---
(a) Fairness of the second stream
(b) Fairness of the third stream
Figure 10: Fairness comparison
Fairness: We next examine fairness of sharing concurrent transmission
opportunities among clients in a 3-antenna AP scenario. We plot in Figs. 10(a)
and 10(b) the number of the second transmission opportunities and the third
transmission opportunities obtained by each client over the total number of
transmissions, which is the metric used to evaluate fairness in our
experiments. The figures show that both the contention-based schemes, i.e.,
SAM and MRC, and our MIMOMate enable all clients to get almost an equal
probability to transmit the second stream and the third stream, respectively.
This implies that our matching algorithm enables users to achieve the same
level of fairness as if they use a fair contention mechanism. The probability
of sending the third stream in SAM is however slightly lower than that in
MIMOMate and MRC. This is because, if the transmission time of the first
stream is too short due to a high data rate, then there might be no enough
time for SAM to hold the third stream and its contention. On the other hand,
in max-throughput first and max-angle first, users cannot have a fair
opportunity to transmit concurrently because these two schemes always favor
certain users to achieve a high throughput. Based on the results in Figs. 9
and 10, we conclude that MIMOMate achieves a throughput comparable to the
greedy algorithms, while providing users a fairness level similar to the
contention mechanism.
Figure 11: Airtime occupied by the data streams
Overhead: We now compare the overhead of different systems. Recall that we do
not implement ACK and contention in USRPs. Thus, we offline compute the
channel time occupied by the protocol overhead. To do so, we feed the
throughput outputted by the experiments (i.e., the rate of the data frames
without considering the 802.11 overhead) to the offline computation, and add
the overhead of each protocol, including contention, interframe timing
(SIFS/DIFS), the PLCP/MAC headers, and the ACK, to each packet transmission.
Fig. 11 plots the percentage of airtime occupied by the data frames, which is
computed by the ratio of airtime for data transmission to the overall airtime
occupied by a packet (i.e., including the overhead). The figure shows that,
since clients in our MIMOMate and SAM use 802.11’s contention to compete for
sending the first stream, their overhead for the first stream is the same with
that of the conventional 802.11. By eliminating the contending process for the
second stream, MIMOMate and the greedy algorithms can utilize about 63% of
airtime to transmit the second streams. In contrast, both SAM and MRC require
multiple rounds of contention, which significantly offset the MIMO gain when
the number of antennas supported by the AP keeps increasing. SAM allows each
client to transmit immediately once it wins the contention. Hence, the airtime
of data sent in a higher dimension becomes shorter and shorter. For MRC, all
concurrent clients start their transmissions after receiving the CTS, their
available airtime is hence shorter yet the same. However, since multiple
clients might send RTS concurrently in MRC, the number of contention rounds
required in MRC might be fewer than that in SAM. This explains why the airtime
of the third stream in MRC is longer than that in SAM.
---
(a) Total throughput in the 2-antenna AP scenario
(b) Total throughput in the 3-antenna AP scenario
Figure 12: Throughput for bursty traffic
### VI-B Throughput Gain for Bursty Traffic
We next evaluate the performance of MIMOMate when clients have a bursty
traffic pattern. The packet traces are generated using the following model.
Each user transmits several files to the AP, and the size of each file is
randomly selected from 500 to 550 KB. The arrival of file transmission follows
a Poisson process with an arrival rate $\lambda=2$ files per second. Thus,
when any of the scheduled MIMO-Mates does not have traffic to send, other
clients contend for transmitting concurrently using the contention window
computed based on Algorithm 2. Fig. 12 plots the CDFs of the total throughput.
The figure shows that, compared to SAM (MRC), the average throughput gain
achieved by combining MIMOMate with angle-based contention is about 22% (33%)
and 19% (36%) for 2- and 3-antenna scenarios, respectively. The throughput
gain in this case is lower than that in the continuous traffic scenario
because angle-based contention introduces additional contention overhead.
Also, for some periods, there are only a few clients with traffic demand and
hence concurrent transmission opportunities cannot be fully utilized. The gap
between SAM and MRC for bursty traffic is larger than that for continuous
traffic. This is because, in MRC, when there are always at least $N$
contending clients, where $N$ is the degrees of freedom, the AP might be able
to detect $N$ clients within fewer than $N$ contention rounds, if some clients
send RTS at the same time in one contention round. However, when the number of
contending clients is less than $N$, the AP always needs to wait for the
duration of $N$ rounds of RTS and then responds the CTS.
---
(a) Total throughput in the 2-antenna AP scenario
(b) Total throughput in the 3-antenna AP scenario
Figure 13: Impact of number of users on throughput
## VII Simulation Results
We further perform simulations to evaluate the performance of MIMOMate in
large-scale scenarios. The simulations are designed to answer the following
questions.
* •
How does MIMOMate perform in different scales of networks?
* •
How does the packet size affect the throughput performance of comparison
protocols?
* •
Can legacy 802.11 devices operate normally in the presence of MIMOMate nodes?
In each simulation, we uniformly randomly distribute the users in a disk with
center the AP and radius 100 m. Furthermore, the antennas on the AP are
collinear with a gap of 0.05 m between two neighboring antennas. The channels
are generated according to the Rayleigh fading channel model, and we assume
the available transmission bit-rates are 6, 9, 12, 18, 24, 36, 48, and 54
Mb/s, which are identical with 802.11a. The default number of users is set to
15, and the default packet size is set to 1500 bytes. The detailed settings
will be specified in each simulation.
---
(a) Total throughput in the 2-antenna AP scenario
(b) Total throughput in the 3-antenna AP scenario
Figure 14: Impact of various packet sizes on throughput
### VII-A Impact of Number of Clients
We evaluate the performance of MIMOMate when the number of clients varies from
3 to 30. Figs. 13(a) and 13(b) plot the total throughput for 2-antenna and
3-antenna AP scenarios, respectively. The trend of the simulation results are
similar to that of small-scale experimental results. The effect of increasing
the number of clients on MIMOMate, max-angle first, and max-throughput first
is relatively small, showing that MIMOMate operates well even when the network
scales up. One thing worth noting is that SAM outperforms MRC when the number
of clients is small, e.g., less than twelve, because its clients do not need
to wait for transmitting concurrently after multiple rounds of contention. The
throughput of SAM however decreases when the number of clients increases. The
reason is that, although both SAM and MRC require multiple rounds of
contention, contention failure can only occur in the last round of contention
in MRC, yet could happen in any round of contention in SAM. The more clients
exist, the higher probability that contention fails is. When contention fails,
the number of concurrent streams would exceed the degrees of freedom, which
makes ZF-SIC decoding fail. Hence the result.
### VII-B Impact of Dynamic Packet Sizes
We next evaluate how MIMOMate performs when the packet size varies
dynamically. In this simulation, we uniformly randomly pick a size between 200
bytes and 1500 bytes for the first client of each transmission. The clients
joining later end their transmissions at the same time with the first stream.
Again, Figs. 14(a) and 14(b) plot the total throughput for 2-antenna and
3-antenna AP scenarios, respectively. Due to a smaller average packet size,
i.e., around $(200+1500)/2$ bytes, and, as a result, a higher proportion of
airtime occupied by the overhead, the throughput in this simulation is less
than those found in Fig. 13; However, the advantage of MIMOMate does not get
affected when the packet size changes. Observer that, compared to Figs. 13(a)
and 13(b), the throughput of SAM decreases slower here. This is because, in
SAM, when the packet size is small, then there might be no remaining airtime
for clients to exploit the transmission opportunities of high dimensions. In
other words, the number of contentions for SAM in this simulation is less than
that in the previous simulation. As a result, less contention failures occur
here. Therefore, compared with the previous simulation, the effect of
increasing the number of users on SAM is less here.
---
(a) Total throughput in the 2-antenna AP scenario
(b) Total throughput in the 3-antenna AP scenario
Figure 15: Impact of number of legacy nodes on throughput
### VII-C Impact of Existence of Legacy Devices
We finally check the performance of MIMOMate in the presence of legacy nodes.
Since MRC is not compatible with the traditional 802.11 standard, we exclude
it from this simulation. In this simulation, we set the total number of
clients at 15, and let some of nodes be legacy 802.11 clients. Figs. 15(a) and
15(b) show the total throughput when the number of legacy nodes varies from 1
to 15. The results show that, in MIMOMate max-angle first and max-throughput
first, the total throughput decreases as the number of legacy nodes increases.
The first reason is that, since legacy nodes can only send the first stream,
i.e., occupying the first dimension, the probability of picking concurrent
clients with a good channel decreases when the number of non-legacy nodes
decreases. Second, when the number of non-legacy clients is less than the
degrees of freedom, the concurrent transmission opportunities might not be
able to be fully utilized, as a result reducing the throughput significantly.
When all the users are legacy users, i.e., the number of legacy users is
fifteen, all the methods degenerate to the traditional 802.11 protocol and
hence perform the same. The performance of SAM however does not change much
with various numbers of legacy users because it simply randomly picks clients
to fully utilize the available degrees of freedom, without considering channel
orthogonality between concurrent clients. On the contrary, the performance of
SAM increases slightly when the number of non-legacy nodes decreases, because
the probability of collisions due to contention failure decreases when more
nodes join contention.
---
(a) Throughput of legacy nodes in the 2-antenna AP scenario
(b) Throughput of legacy nodes in the 3-antenna AP scenario
Figure 16: Average throughput of legacy devices
We further check whether the throughput performance of legacy nodes gets
affected by our matching design. Figs. 16(a) and 16(b) compare the total
throughput of legacy nodes in comparison schemes with that in the traditional
802.11 protocol. The results show that their throughput in SAM is
significantly worse than that in traditional 802.11. This is because SAM
suffers from contention failures that might happen in the second and third
streams. The figure also shows that legacy nodes can coexist with MIMOMate
nodes well and achieve a similar performance, as compared to that in
traditional 802.11. A small gap between MIMOMate and traditional 802.11 is due
to occasional contention failures that might happen when contention is
required for some first winners who are not assigned any follower.
## VIII Conclusion
This paper introduces MIMOMate, a user matching protocol that maximally
delivers the gain of a MU-MIMO LAN, while, at the same time, ensuring all the
clients to have a fair opportunity to transmit concurrent packets. The clients
scheduled as the MIMO-Mates can join concurrent transmissions one after
another with only one contending process, as a result reducing the MAC
overhead significantly. We also integrate MIMOMate with an angle-based
contention mechanism to best utilize concurrent transmission opportunities
when any of the scheduled MIMO-Mates does not have traffic to transmit. Our
prototype implementation shows that MIMOMate increases the throughput by 42%
and 52% over the contention-based protocol for 2- and 3-antenna AP scenarios,
respectively, and also provides fairness for clients. Our theoretical analysis
also proved that MIMOMate can always achieve a higher or equal throughput as
compared to the traditional contention scheme in the 2-antenna AP scenario.
Analytic performance evaluation for any general scenarios is considered as our
future work.
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### -A Proof of the NP-hardness of the 3-MIMOMate Problem
###### Theorem 3.
The 3-MIMOMate problem is NP-hard.
###### Proof.
We start our proof by relaxing the 3-MIMOMate problem to a simpler problem,
denoted by the maximum fairness (3MF) problem, which finds a set $M$ such that
Constraints 1–4 in problem 2 are satisfied. Any optimal solution of the 3MF
problem is hence a feasible solution of our problem. We then proceed the proof
by showing that even the decision version of the relaxed 3MF problem is NP-
hard.
Let 3MF$(V,r,k)$ denote an instance of the decision version of the 3MF
problem, which seeks a matching $M\subseteq V\times V$ with $|M|\geq k$. We
prove its NP-hardness by polynomial-time reduction from the 3-dimensional
matching problem, which is NP-hard. Let 3DM$(X,Y,Z,T,k)$ denote the decision
version of the 3-dimensional matching problem, which finds a matching
$M\subseteq T$, where $T{\subseteq}X{\times}Y{\times}Z$, such that
$u_{1}{\neq}u_{2},v_{1}{\neq}v_{2}$, and $w_{1}{\neq}w_{2}$ for any
$(u_{1},v_{1},w_{1}),(u_{2},v_{2},w_{2})\in M$ and $|M|\geq k$. For every
instance $3DM(X,Y,Z,T,k)$, we can construct an instance 3MF$(V,r,k)$ in
polynomial time as follows:
1. 1.
$V=\\{u_{i}|1\leq i\leq|X|+|Y|+|Z|\\}$.
2. 2.
Set
$r^{(u_{i},u_{j},u_{k})}_{u_{j}}+r^{(u_{i},u_{j},u_{k})}_{u_{k}}=1,\forall(u_{i},u_{j},u_{k})$,
if $(x_{i},y_{j-|X|},z_{k-|X|-|Y|})\in T$; otherwise, set it to 0.
We show that 3DM$(X,Y,Z,T,k)$ has a feasible solution $M$ if, and only if,
3MF$(V,r,k)$ has a feasible solution $M^{\prime}$. For the “only if”
direction, for all the elements $(x_{i},y_{j},z_{k})$ in $M$, we add
$(u_{i},u_{j+|X|},u_{k+|X|+|Y|})$ to the solution of 3MF$(V,r,k)$,
$M^{\prime}$. This solution is feasible because, by the construction of the
instance,
$r^{(u_{i},u_{j+|X|},u_{k+|X|+|Y|})}_{u_{j+|X|}}+r^{(u_{i},u_{j+|X|},u_{k+|X|+|Y|})}_{u_{k+|X|+|Y|}}=1$,
and thus Constraints 1 and 3 of the 3MF problem hold. Constraint 2 is also
satisfied, i.e., $u_{i},u_{j+|X|}$ and $u_{k+|X|+|Y|}$ are distinct clients,
because $i\leq|X|<j{+}|X|\leq|X|{+}|Y|<k{+}|{X}|{+}|Y|$. Finally, since
different elements $(x_{i},y_{j},z_{k})$ map to different
$(u_{i},u_{j+|X|},u_{k+|X|+|Y|})$, then $|M^{\prime}|=|M|\geq k$.
For the “if” direction, for every $(u_{i},u_{j},u_{k})\in M^{\prime}$, we add
$(x_{i},y_{j-|X|},z_{k-|X|-|Y|})$ to the solution of 3MF, $M$. Since
$(u_{i},u_{j},u_{k})$ in $M^{\prime}$, we have
$r^{(u_{i},u_{j},u_{k})}_{u_{j}}+r^{(u_{i},u_{j},u_{k})}_{u_{k}}=1$. Thus, by
the construction of the instance, $(x_{i},y_{j-|X|},z_{k-|X|-|Y|})\in T$. In
addition, by Constraint 3, the constraint of the 3DM problem that restricts
$u_{1}{\neq}u_{2},v_{1}{\neq}v_{2}$, and $w_{1}{\neq}w_{2}$, for any two
distinct elements $(u_{1},v_{1},w_{1}),(u_{2},v_{2},w_{2})\in T$, holds.
Finally, again, since $|M|{=}|M^{\prime}|$, $M\geq k$ holds as well. Hence, we
conclude that the 3FM problem is also NP-hard. ∎
### -B Proofs of Theorem 1 and Theorem 2
Since Theorem 2 implies Theorem 1, we only need to show Theorem 2.
Furthermore, it is sufficient to show that the average throughput of the
output of Problem 1, denoted as $T_{M}$, is greater than or equal to that of
the optimal fair probabilistic assignment selection (the fair probabilistic
assignment selection that achieves the highest average throughput), denoted as
$T_{R}$. For ease of presentation, we give an order to the clients, so that
the input $V$ in Problem 1, i.e., the set of all clients, is
$\\{1,2,...,|V|\\}$. Note that since the throughput $r^{(i,j)}_{j}$ is greater
than 0 for all $i,j\in V,i\neq j$, i.e., ZF-SIC decoding is successful, a
client who transmits the first stream can get about the same throughput no
matter who its follower is [5]. In addition, every client has an equal
probability of winning the first contention. Therefore, we assume that the
throughputs contributed by the first stream are the same in both the output of
Problem 1 and the optimal fair probabilistic assignment. Thus, we can ignore
the average throughput contributed by the first stream in the following proof,
since we only need to show $T_{M}\geq T_{R}$.
To derive the average throughput, we introduce a variable $p^{i,j}$ for all
$i,j\in V,i\neq j$. $p^{i,j}$ represents the probability that client $j$ is
chosen as client $i$’s follower. Therefore, given $p^{i,j}$s, the average
throughput can be expressed as follows.
$\displaystyle\sum_{i\in V}{\\{Pr\\{\text{client }i\text{ wins the first
contention}\\}\sum_{j\in V\setminus\\{i\\}}{p^{i,j}r^{(i,j)}_{j}}\\}}$
$\displaystyle=\frac{1}{|V|}\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{p^{i,j}r^{(i,j)}_{j}}}.$
Now, we are ready to derive $T_{M}$ and $T_{R}$. Denote $p_{M}^{i,j}$s and
$p_{R}^{i,j}$s as the $p^{i,j}$s used in the output of Problem 1 and the
optimal fair probabilistic assignment, respectively. It is then sufficient to
show that
$\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{p_{M}^{i,j}r^{(i,j)}_{j}}}\geq\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{p_{R}^{i,j}r^{(i,j)}_{j}}}.$ (1)
The following proof is done by three steps: 1) show that $p_{M}^{i,j}$s is an
optimal solution of the integer programming of the bipartite maximum weighted
matching problem; 2) show that any $p_{R}^{i,j}$s is a feasible solution of
the relaxed integer programming of the bipartite maximum weighted matching
problem, which allows the integer variables to be relaxed as any real number
between $[0,1]$; 3) Start from the fact that the achieved objective value of
$p_{M}^{i,j}$s in the integer programming is greater than or equal to that of
$p_{R}^{i,j}$s in the relaxed integer programming for the maximum weight
matching problem [36] and show that Eq. (1) holds.
Step 1: Note that $p_{M}^{i,j}$s are either 0 or 1, indicating whether to use
client $j$ as client $i$’s follower or not. First observe that since
$r^{(i,j)}_{j}$ is greater than $0$ for all $i,j\in V,i\neq j$, the size of
the output of Problem 1, $|M|$, must be $|V|$. In other words, every client
has a different follower, which implies
$\sum_{j\in V\setminus\\{i\\}}{p_{M}^{i,j}}=1,\forall i\in V.$ (2)
Then, recall that Problem 1 is actually a bipartite maximum weighted maximum
cardinality matching problem. The book [30] shows that, to solve the bipartite
maximum weighted maximum cardinality matching problem, we can add a
sufficiently large number, $C$, to the weight of each edge, and solve the
bipartite maximum weighted matching problem on the new graph instead. We can
now use the integer programming of the bipartite maximum weighted matching
problem to find $p_{M}^{i,j}$s. We first give a detailed construction of the
bipartite graph. Given the set $V$ and throughputs $r^{(i,j)}_{j}$s in Problem
1, we construct a bipartite graph $G=(V_{1}\cup V_{2},E)$, where
$V_{1}=\\{v_{1}^{1},v_{1}^{2},...,v_{1}^{|V|}\\}$ (the set of winners of the
first contention),
$V_{2}=\\{v_{2}^{1},v_{2}^{2},...,v_{2}^{|V|}\\}$ (the set of followers),
$E=\\{(v_{1}^{i},v_{2}^{j})|r^{(i,j)}_{j}>0\\}$, and
$w(v_{1}^{i},v_{2}^{j})=r^{(i,j)}_{j}+C,\forall(v_{1}^{i},v_{2}^{j})\in E$
(the increased throughput).
The integer programming of the bipartite maximum weighted matching problem on
$G$ can then be formulated as
maximize $\displaystyle\sum_{(v_{1}^{i},v_{2}^{j})\in
E}{w(v_{1}^{i},v_{2}^{j})p^{i,j}}\quad$ subject to $\displaystyle\sum_{j\in
V\setminus\\{i\\}}{p^{i,j}}\leq 1,\forall i\in V,$ $\displaystyle\text{(the
maximum number of edges incident to }v_{1}^{i}\text{ is 1)}$
$\displaystyle\sum_{i\in V\setminus\\{j\\}}{p^{i,j}}\leq 1,\forall j\in V,$
$\displaystyle\text{ (the maximum number of edges incident to }v_{2}^{j}\text{
is 1)}$ $\displaystyle p^{i,j}=0\text{ or }1,\forall i,j\in V,i\neq j.$
The optimal $p^{i,j}$s of the above integer programming are then
$p_{M}^{i,j}$s.
Step 2: To show that $p_{R}^{i,j}$s correspond to a feasible solution of the
relaxed integer programming of the bipartite maximum weighted matching
problem, we need to derive some properties of $p_{R}^{i,j}$s. The fairness
constraint requires that every client has the same probability to transmit the
second stream. We hence have the following equation.
$\sum_{i\in V\setminus\\{j\\}}p_{R}^{i,j}=\sum_{i\in
V\setminus\\{j^{\prime}\\}}p_{R}^{i,j^{\prime}}\leq 1,\forall j,j^{\prime}\in
V.$ (3)
The inequality must follows. Otherwise, $\sum_{i\in V}\sum_{j\in
V\setminus\\{i\\}}p_{R}^{i,j}=\sum_{j\in V}\sum_{i\in
V\setminus\\{j\\}}p_{R}^{i,j}>|V|$, which implies $\sum_{j\in
V\setminus\\{i\\}}p_{R}^{i,j}>1$ for some $i$ and contradicts to the fact that
any fair probabilistic assignment chooses at most one follower at each time,
i.e.,
$\sum_{j\in V\setminus\\{i\\}}p_{R}^{i,j}\leq 1,\forall i\in V.$ (4)
Then, by Eq. (3) and Eq. (4), we get that $p_{R}^{i,j}$s correspond to a
feasible solution of the relaxed integer programming.
Step 3: It has been shown in [36] that, for the relaxed integer programming of
the bipartite maximum weighted matching problem, the objective value achieved
by an optimal integral solution is no less than that achieved by any feasible
solution of the relaxed integer programming. Hence, we have
$\sum_{(v_{1}^{i},v_{2}^{j})\in
E}{w(v_{1}^{i},v_{2}^{j})p_{M}^{i,j}}\geq\sum_{(v_{1}^{i},v_{2}^{j})\in
E}{w(v_{1}^{i},v_{2}^{j})p_{R}^{i,j}}$. Therefore, it is sufficient to show
that
$\sum_{(v_{1}^{i},v_{2}^{j})\in
E}{w(v_{1}^{i},v_{2}^{j})p_{M}^{i,j}}\geq\sum_{(v_{1}^{i},v_{2}^{j})\in
E}{w(v_{1}^{i},v_{2}^{j})p_{R}^{i,j}}\text{ implies Eq.~{}\eqref{eq: goal}}.$
Before showing this, we must further refine Eq. (4) and get the following
equation.
$\sum_{j\in V\setminus\\{i\\}}p_{R}^{i,j}=1,\forall i\in V,$ (5)
which can be proved by contradiction. If there exists some $i^{\prime}\in V$,
such that $\sum_{j\in V\setminus\\{i^{\prime}\\}}p_{R}^{i^{\prime},j}<1$, then
we must have $\sum_{i\in V\setminus\\{j\\}}p_{R}^{i,j}=\sum_{i\in
V\setminus\\{j^{\prime}\\}}p_{R}^{i,j^{\prime}}<1,\forall j,j^{\prime}\in V$;
otherwise, if $\sum_{i\in V\setminus\\{j\\}}p_{R}^{i,j}=\sum_{i\in
V\setminus\\{j^{\prime}\\}}p_{R}^{i,j^{\prime}}=1,\forall j,j^{\prime}\in V$,
then $\sum_{j\in V\setminus\\{i\\}}p_{R}^{i,j}=1,\forall i\in V$, which
contradicts to $\sum_{j\in V\setminus\\{i^{\prime}\\}}p_{R}^{i^{\prime},j}<1$.
Therefore, we can add a small value to all $p_{R}^{i^{\prime},j}$s, $j\in
V\setminus\\{i^{\prime}\\}$, such that Eq. (3) and Eq. (4) still hold, i.e.,
after addition, $p_{R}^{i,j}$s still correspond to a fair probabilistic
assignment. Obviously, the average throughput is higher after we increase
$p_{R}^{i^{\prime},j}$s, which contradicts to the fact that $p_{R}^{i,j}$s
correspond to an optimal fair probabilistic assignment. Hence, Eq. (5) holds.
We are now ready to accomplish the final part of the proof. Observe that
$\displaystyle\sum_{(v_{1}^{i},v_{2}^{j})\in
E}{w(v_{1}^{i},v_{2}^{j})p_{M}^{i,j}}=\sum_{v_{1}^{i}\in
V_{1}}{\sum_{v_{2}^{j}\in V_{2},j\neq i}{w(v_{1}^{i},v_{2}^{j})p_{M}^{i,j}}}$
$\displaystyle=$ $\displaystyle\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{(r^{(i,j)}_{j}+C)p_{M}^{i,j}}}$ $\displaystyle=$
$\displaystyle\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{r^{(i,j)}_{j}p_{M}^{i,j}}}+C\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{p_{M}^{i,j}}},$
and
$\sum_{(v_{1}^{i},v_{2}^{j})\in
E}{w(v_{1}^{i},v_{2}^{j})p_{R}^{i,j}}=\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{r^{(i,j)}_{j}p_{R}^{i,j}}}+C\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{p_{R}^{i,j}}}$
by a similar reasoning. Therefore, it is sufficient to show that $\sum_{i\in
V}{\sum_{j\in V\setminus\\{i\\}}{p_{M}^{i,j}}}=\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{p_{R}^{i,j}}}$. By Eq. (2) and Eq. (5), we have $\sum_{i\in
V}{\sum_{j\in V\setminus\\{i\\}}{p_{M}^{i,j}}}=\sum_{i\in V}{\sum_{j\in
V\setminus\\{i\\}}{p_{R}^{i,j}}}=|V|$. The proof is then completed.
|
arxiv-papers
| 2014-04-24T06:43:52 |
2024-09-04T02:50:01.754548
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tung-Wei Kuo, Kuang-Che Lee, Kate Ching-Ju Lin and Ming-Jer Tsai",
"submitter": "Kate Ching-Ju Lin",
"url": "https://arxiv.org/abs/1404.6041"
}
|
1404.6043
|
# Exothermic isospin-violating dark matter after SuperCDMS and CDEX
Nan Chen1 Qing Wang1,2,3 [email protected] 1Department of Physics,
Tsinghua University, Beijing 100084, P. R. China
2Center for High Energy Physics, Tsinghua University, Beijing 100084, P. R.
China
3Collaborative Innovation Center of Quantum Matter, Beijing 100084, P. R.
China Wei Zhao4 Shin-Ted Lin 5 Qian Yue4 Jin Li4 4Key Laboratory of Particle
and Radiation Imaging (Ministry of Education) and Department of Engineering
Physics, Tsinghua University, Beijing 100084, P. R. China
5College of Physical Science and Technology, Sichuan University, Chengdu
610064, P. R. China
###### Abstract
We show that exothermic isospin-violating dark matter (IVDM) can make the
results of the latest CDMS-Si experiment consistent with recent null
experiments, such as XENON10, XENON100, LUX, CDEX, and SuperCDMS, whereas for
the CoGeNT experiment, a strong tension still persists. For CDMS-Si, separate
exothermic dark matter or isospin-violating dark matter cannot fully
ameliorate the tensions among these experiments; the tension disappears only
if exothermic scattering is combined with an isospin-violating effect of
$f_{n}/f_{p}=-0.7$. For such exothermic IVDM to exist, at least a new vector
gauge boson (dark photon or dark Z’) that connects SM quarks to Majorana-type
DM particles is required.
###### pacs:
95.35.+d, 95.30.Cq
Low-mass dark matter (DM) in the GeV energy region is currently the main topic
of DM searches. On the one hand, some direct-detection experiments have
claimed to have observed low-energy recoil events in excess of known
backgrounds; these include DAMA DAMA1 ; DAMA2 ; DAMA3 ; DAMA4 , CoGeNT Cogent1
; Cogent2 ; Cogent3 ; Cogent4 , and CRESST-II CRESSTII 111A possible excess
over the background reported for the previous run (from 2009 to 2011) has not
been confirmed in the upgraded CRESST-II detector, with an exposure of 29.35
kg live days collected in 2013CRESSTIInew ., and the latest such result is the
positive CDMS-Si signal CDMSSi . These excesses, if interpreted in terms of DM
particles elastically scattering off target nuclei, may imply the existence of
light DM particles with a mass of $<10$ GeV and a scattering cross section of
approximately $10^{-41}\sim 10^{-40}\mathrm{cm}^{2}$. However, many other
experiments, such as CDMS-II CDMSII1 ; CDMSII2 ; CDMSII3 ; CDMSlite ,
XENON10/100 XENON10 ; XENON1001 ; XENON1002 ; XENON1003 , SIMPLE SIMPLE ,
TEXONO TEXONO1 ; TEXONO2 , CDEX CDEX1 , LUX LUX , the latest SuperCDMS
SuperCDMS , and CDEX CDEX2 , have reported null results in the same DM mass
range. The serious conflict between these two completely different sets of
results contrasts sharply with the situation in particle physics collider
experiments, where all data appear to be in harmony with Standard Model (SM)
predictions and, to date, no evidence of new physics beyond the SM has been
observed. These tensions in the direct detection of DM are a strong motivation
driving further investigations, which seek a deeper understanding either of
the direct-detection experiments or of present theoretical interpretations of
the experimental results. We may also treat the reconciliation of these
contradictory experimental results as a guide for the identification of
certain properties of the DM particle.
From the theoretical viewpoint, we need to establish whether there exists some
mechanism to reconcile the present tension. If so, then we are closer to the
discovery of the DM particle in particle physics experiments; if not, then
experimentalists must conduct more complex background analysis to extract
additional events from the observed signals. Note that most of the
experimentally detected signals have been recorded using target materials that
are different from those used in the experiments in which the null results
have been obtained; the exception is CoGeNT, for which the target material is
Ge, which is also used in CDMS-II, TEXONO, CDEX, and SuperCDMS. Although a
contingent of CoGeNT researchers has recently released an improved analysis of
3.4 years of CoGeNT data Cogent5 , exhibiting a close similarity to previously
reported results Cogent3 ; Cogent4 on the annual modulation, though different
and weaker, questions remain regarding their surface event analysis
QuestionCogent . By contrast, J.H. Davis has announced that the DAMA result
can be fitted using neutrons from muons and neutrinos instead of DM
DAMAalternativeExplanation . Shortly following this announcement, comments
appeared claiming that the effect from muons and neutrinos is
negligibleDAMAcomment1 ; DAMAcomment2 . R. Foot has attempted to use MeV-order
DM scattering off electrons to explain the DAMA
resultDAMAalternativeExplanation1 . Recently, some researchers have used MeV-
range axion-like particles to explain the DAMA
signalDAMAalternativeExplanation2 ; soon after, others claimed that this model
has already been ruled out by many orders of magnitude based on existing
experimental resultsDAMAcomment3 . Considering that CRESST-II’s signal has
already disappeared CRESSTIInew , if we ignore these debatable CoGeNT and DAMA
results, then, in our analysis, there are two popular theoretical scenarios
that are able to ameliorate the tensions between the remaining CDMS-Si results
and the other null experiments in regard to the details of the different
structures of their target nuclei. One is isospin-violating DM (IVDM)IVDM1 ;
IVDM2 ; IVDM3 ; IVDM4 ; IVDM5 ; IVDM6 ; IVDM7 ; IVDM8 , wherein the DM
particle couples to protons and neutrons with different strengths; possible
destructive interference resulting from these two couplings can weaken the
bounds of XENON10/100 and move the signal regions of DAMA and CoGeNT closer to
each other IVDM5 ; IVDM6 . To reconcile the data from DAMA, CoGeNT, and
XENON10/100, a large destructive interference is required; this interference
is dependent on the ratio of the spin-independent scatterings of the couplings
of the DM particle to the neutron ($f_{n}$) and to the proton ($f_{p}$), which
must be of order $f_{n}/f_{p}\approx-0.7$ IVDM5 . However, from indirect DM
searches, such as the antiproton flux measured by BESS-Polar II, the relevant
couplings for IVDM have been found to be severely constrained IVDM9 ; IVDM10 .
Furthermore, after the appearance of the LUX data LUX , it was observed
tension1 ; tension2 ; tension3 that LUX and CDMS-Si are now in tension even
for IVDM. The other possible scenario is to go beyond conventional elastic
scattering and consider whether DM scatters inelastically to a lower mass
state; such DM is termed exothermic ExcitingDM1 ; ExcitingDM2 ; ExcitingDM3 ;
ExcitingDM4 ; ExcitingDM5 ; ExcitingDM6 ; ExcitingDM7 . For a sufficiently
long-lived heavier state, there must be sufficient numbers of such DM
particles in the vicinity of the Earth to produce a signal in direct-detection
experiments, and the splitting cannot be too large. We consider $\delta\leq
200$ keV for appropriately small splittings; in that case, the only available
decays are to neutrinos or photons. If couplings in the SM occur through the
kinetic mixing of a dark-sector gauge boson with the SM gauge bosons, then the
lifetimes are longer than the age of the universe ExcitingDM2 ; Lifetime1 . By
choosing a mass splitting between the DM excited and de-excited states of
approximately $\delta\sim-200$ keV, ExcitingDM8 ; ExcitingDM9 one can
accommodate both the LUX and CDMS-Si results and simultaneously account for
the high- and low-energy events.
Although the exothermic DM model succeeds in relaxing the tension between LUX
and CDMS-Si ExcitingDM8 ; ExcitingDM9 , this model has not been considered in
the aftermath of the latest results from SuperCDMS SuperCDMS and CDEX CDEX2 .
Such an analysis is the topic of the present paper. Our objective in the study
is to examine the consistency of the SuperCDMS and CDEX null results with the
excess CDMS-Si result by implementing the two scenarios mentioned above. The
CoGeNT and DAMA results will also be considered as references in our
discussion, although the interpretations thereof are still a subject of
debate. Because exothermic DM, unlike endothermic DM, for which the DM
scatters inelastically to a higher mass state, can reduce the relative
modulation amplitude, the tension with the CoGeNT result is not expected to be
reduced, and we shall see later that the DAMA result even shrinks to zero. Our
strategy is first to apply exothermic DM to the SuperCDMS and CDEX results to
confirm whether these more stringent experiments are consistent with the CDMS-
Si result. If so, then exothermic DM becomes a unique type of DM that is
consistent with all existing (except CoGeNT and DAMA) direct-detection
experiments; if not, we will add in the IVDM effect and evaluate the results.
As mentioned above, the IVDM model is already in tension with the results of
LUX and CDMS-Si, and therefore, relying solely on IVDM to ameliorate the
tensions among the different experiments is impossible; however, we can
combine this mechanism with the exothermic DM model to further reduce these
tensions.
For SuperCDMS, its latest result has recently been reported SuperCDMS , in
which the data obtained during 577 kg-days of exposure were analyzed for WIMPs
of mass $<30~{}\mathrm{GeV}/c^{2}$ with a blinded signal region. Eleven events
were observed once the analysis was complete. The authors set an upper limit
on the spin-independent WIMP-nucleon cross section of $1.2\times
10^{-42}\mathrm{cm}^{2}$ at $8~{}\mathrm{GeV}/c^{2}$. In the meantime, CDEX
already published its latest null results CDEX2 for 53.9 kg-days of data.
To interpret the above experimental results in terms of inelastic scattering,
we note that exothermic DM particles are those DM particles $\chi_{1}$ of mass
$m_{1}$ that inelastically down-scatter to DM particles $\chi_{2}$ of mass
$m_{2}$ from a nucleus $N$ as follows: $\chi_{1}+N\rightarrow\chi_{2}+N$. The
requisite velocity to produce a nuclear recoil of energy $E_{R}$ is
$\displaystyle
v_{\mathrm{min}}=\frac{1}{\sqrt{2E_{R}m_{N}}}|\delta+\frac{m_{N}E_{R}}{\mu}|,\hskip
56.9055pt\delta\equiv m_{2}-m_{1}<0,$ (1)
where $\mu$ is the reduced mass of the DM-nucleon system. Up-scattering
($\delta>0$) is more prevalent from heavy nuclei, whereas down-scattering
($\delta<0$) is more prevalent from light nuclei, where the energy of the
recoiling nucleus is peaked near a scale that is proportional to the splitting
between the dark matter states and is inversely proportional to the nuclear
mass. Consequently, nuclear recoils caused by exothermic DM ($\delta<0$) are
more visible in experiments with light nuclei and low thresholds. Given the
lightness of Si with respect to Xe and Ge, down-scattering is one avenue for
explaining the CDMS-Si data while remaining consistent with the null XENON,
LUX, SuperCDMS, and CDEX searches. Figure 1 shows a plot of the elastic-
scattering (corresponding to $\delta=0$) results from the CoGeNT, DAMA and
CDMS-Si signal regions alongside the null results of XENON100, XENON10, LUX,
SuperCDMS, and CDEX. For the DAMA experiment, it has been noted DAMA6 that
nuclei recoiling along the characteristic axes or planes of the crystal
structure may travel large distances without colliding with other nuclei. This
means that recoils that undergo such ion channeling have quenching factors of
$Q_{T}\approx 1$. We consider the cases both with and without this ion-
channeling effect. The null experiments of XENON100, LUX and SuperCDMS are in
strong tension with the CoGeNT, DAMA (both with and without the ion-channeling
effect), and CDMS-Si results.
Figure 1: Elastic-scattering results without isospin violation. The exclusion
lines for LUX (solid blue), CDEX (dashed brown), XENON10 (solid red), XENON100
(solid purple), SuperCDMS (dash-dotted magenta) are all at the 90$\%$ CL and
are superimposed over the 68$\%$ (dark yellow) and 90$\%$ (light yellow) CL
CDMS-Si best-fit regions, the 95$\%$ (dark cyan) CL CoGeNT best-fit region and
the 95$\%$ CL DAMA best-fit regions without ion channeling (dark yellow) and
with ion channeling (magenta).
In Fig. 2, we consider the results for the inelastic scattering of exothermic
DM (corresponding to $\delta<0$) from the CoGeNT and CDMS-Si signal regions
along with the null results from XENON100, XENON10, LUX, SuperCDMS, and CDEX.
The left panel corresponds to $\delta=-50$ keV, and the right panel
corresponds to $\delta=-200$ keV.
Figure 2: Inelastic-scattering results without isospin violation. The left
panel corresponds to $\delta=-50$ keV, and the right panel corresponds to
$\delta=-200$ keV. For an explanation of the legend, see Fig. 1.
We see that for $\delta=-50$ keV, the situation is slightly improved, whereas
for $\delta=-200$ keV, the situation is much improved. The exception is
CoGeNT, which is, as expected, still in tension with all null experiments;
XENON10, XENON100 (lying outside the plot area to the right) and CDEX are
already consistent with CDMS-Si. LUX covers almost the entire CDMS-Si contour
for $\delta=-50$ keV but covers only a small portion for $\delta=-200$ keV.
Only SuperCDMS still fully covers the CDMS-Si contour and is strongly in
tension with the CDMS-Si result. One observation is that the signal region of
CoGeNT becomes significantly larger than that for CDMS-Si at $\delta=-200$
keV. This behavior arises from the difficulty in fitting the data from the
multi-events to a relatively large DM mass splitting ExcitingDM9 . The
$\chi^{2}_{\mathrm{min}}$ of this fitting is significantly larger than that
for the elastic fitting. Because $\delta=-200$ keV is already approaching the
lower limit on the allowed mass difference for exothermic DM ExcitingDM7 , the
results of Fig. 2 indicate that exothermic DM alone, even when an extreme
$\delta$ value is used and the CoGeNT result is ignored, is still not
sufficient to accommodate both the SuperCDMS and CDMS-Si results. For DAMA,
note that when the inelastic scattering of DM is considered, the area of the
low-mass signal region from the DAMA experiment shown in Fig. 1 reduces as the
mass splitting $|\delta|$ grows. This effect can also be observed in Fig. 1 of
ExcitingDM5 . In our analysis with $\delta=-50$ keV and $\delta=-200$ keV, the
signal region of DAMA for low masses (masses comparable to the signal regions
of CDMS-Si and CoGeNT) completely disappears, or the DAMA result shrinks to
zero. For this reason, in the following, as long as we are discussing
exothermic DM with $\delta=-50$ keV or $\delta=-200$ keV, we shall no longer
consider the DAMA experiment. Furthermore, the inelastic-scattering DM does
not fit the DAMA data well even for larger masses ($m_{\chi}>30$GeV); the
$\chi^{2}_{\mathrm{min}}/$d.o.f is approximately 35/34 for $\delta=-200$ keV,
whereas $\chi^{2}_{\mathrm{min}}/$d.o.f =27.8/34 for $\delta=0$.
Next, we include an isospin-violating effect. The general low-energy
differential cross section is ExcitingDM3
$\displaystyle\frac{d\sigma}{dE_{R}}=\frac{m_{N}}{2\mu^{2}v^{2}}\sigma_{\mathrm{el}}[Zf_{p}+(A-Z)f_{n}]^{2}F(q^{2}),$
(2)
where $Z$ is the atomic number of the target nucleus; $A$ is its mass number;
$f_{p}$ and $f_{n}$ are constants that represent the relative coupling
strengths to protons and neutrons, respectively; and $F(q^{2})$ a form factor
that depends on the momentum transfer to the nucleus, $q^{2}=2m_{N}E_{R}$.
$\sigma_{\mathrm{el}}$ is the elastic limit of the above cross section, which
is reached when the splitting is much less than the kinetic energy of the
collision. The differential scattering rate of dark matter per unit recoil
energy $E_{R}$ is given by
$\displaystyle\frac{dR}{dE_{R}}=N_{T}n_{\chi}\int_{v_{\mathrm{min}}}\frac{d\sigma}{dE_{R}}vf(v)dv,$
(3)
where $v_{\mathrm{min}}$, which is determined using Eq. (1), is the minimum
velocity required to produce a recoil of energy $E_{R}$; $N_{T}$ is the number
of target nuclei; $n_{\chi}$ is the local number density of the dark matter;
and $f(v)$ is the distribution of DM velocities relative to the target. With
$N_{T}m_{N}=m_{\mathrm{detector}}$ and $\rho_{\chi}=n_{\chi}m_{\chi}$, the
differential recoil rate per unit detector mass can be written as
$\displaystyle\frac{dR}{dE_{R}}=\frac{\rho_{\chi}}{2m_{\chi}\mu^{2}}\sigma_{\mathrm{el}}[Zf_{p}+(A-Z)f_{n}]^{2}F_{A}(q^{2})\eta(E_{R},t),$
(4)
where $\rho_{\chi}=0.3~{}\mathrm{GeV}/\mathrm{c}^{3}$ is the local DM density.
Details of the DM velocity distribution are included via the mean inverse
speed $\eta(E,t)$,
$\displaystyle\eta(E_{R},t)=\int_{v_{min}(E_{R})}\frac{f(v)}{v}d^{3}v,$ (5)
where $f(v)$ at any given time of the year is determined by the velocity of
the Earth through the halo and by the distribution of DM velocities within the
halo itself, here assumed to be of the form
$\displaystyle f(v)=\frac{N_{0}}{(\pi
v_{0}^{2})^{3/2}}e^{-v^{2}/v_{0}^{2}}\Theta(v_{esc}-v).$ (6)
We have assumed a Maxwell-Boltzmann distribution for the DM halo velocities
with a mean of $v_{0}=220\mathrm{km}/\mathrm{s}$ and a sharp cutoff (i.e., the
galactic escape velocity) at $v_{esc}=544\mathrm{km}/\mathrm{s}$. $N_{0}$ is
chosen to normalize the probability distribution to one. Because the Earth is
moving with a velocity $v_{E}=220\mathrm{km}/\mathrm{s}$, $\eta(E,t)$ can be
written as DAMA2
$\displaystyle\eta(E,t)=\left\\{\begin{array}[]{lr}\frac{1}{v_{0}y},&\mbox{for}\quad
z<y,x<|y-z|,\\\
\frac{1}{2N_{esc}v_{0}y}[\mbox{erf}(x+y)-\mbox{erf}(x-y)-\frac{4}{\sqrt{pi}}ye^{-z^{2}}],&\mbox{for}\quad
z>y,x<|y-z|,\\\
\frac{1}{2N_{esc}v_{0}y}[\mbox{erf}(z)-\mbox{erf}(x-y)-\frac{2}{\sqrt{pi}}(y+z-x)e^{-z^{2}}],&\mbox{for}\quad|y-z|<x<y+z,\\\
0,&\mbox{for}\quad y+z<x,\\\ \end{array}\right.$ (11)
where
$\displaystyle x=v_{min}/v_{0},\quad y=v_{E}/v_{0},\quad z=v_{esc}/v_{0}\hskip
56.9055ptN_{esc}=\mbox{erf}(z)-\frac{2z}{\sqrt{\pi}}e^{-z^{2}}.$ (12)
For the annual modulation, the count rate generally has an approximate time
dependence as follows:
$\displaystyle\frac{dR}{dE_{R}}(E_{R},t)\approx
S_{0}(E_{R})+S_{m}(E_{R})\cos\omega(t-t_{c})$ (13)
where $t_{c}$ is the time of year at which $v_{obs}(t)$ is at its maximum,
$S_{0}(E_{R})$ is the average differential recoil rate over a year, and
$S_{m}(E_{R})$ is referred to as the modulation amplitude. For the standard
halo model,
$\displaystyle
S_{m}(E_{R})=\frac{1}{2}\Big{[}\frac{dR}{dE_{R}}\bigg{|}_{v_{E}=v_{sun}+v_{orb}\cos\gamma}-\frac{dR}{dE_{R}}\bigg{|}_{v_{E}=v_{sun}-v_{orb}\cos\gamma}\Big{]},$
(14)
where $v_{orb}=30\mathrm{km/s}$ and $\cos\gamma=0.51$.
Finally, to consider isospin-violating scattering from dark matter, different
mass numbers will yield different differential recoil rates. The event rate is
given by
$\displaystyle R=\sum_{i}r_{i}N_{T}m_{A_{i}}\int
dE_{R}\frac{\rho_{\chi}}{2m_{\chi}\mu^{2}}F_{A_{i}}(q^{2})\eta(E_{R},t),$ (15)
where the sum is over the isotopes $A_{i}$ with fractional number abundances
$r_{i}$ IVDM5 .
Using these formulae, and with a ratio of $f_{n}/f_{p}\approx-0.7$, we
performed the relevant calculations, and in Fig. 3, we plot the elastic-
scattering (corresponding to $\delta=0$) IVDM results of the CoGeNT, DAMA
(both with and without the ion-channeling effect) and CDMS-Si signal regions,
alongside the null results of XENON100, XENON10, LUX, SuperCDMS, and CDEX.
Through comparison with Fig. 1, we find that the IVDM model does slightly
reduce the tensions, but the null experiments LUX and SuperCDMS are
essentially still in tension with the CoGeNT, DAMA and CDMS-Si result. In
particular, we recover the previously mentioned result that LUX and CDMS-Si
are in tension for IVDM tension1 ; tension2 ; tension3 .
Figure 3: Elastic-scattering IVDM results for $f_{n}/f_{p}=-0.7$. For an
explanation of the legend, see Fig. 1.
We continue by considering the inelastic-scattering effects. The underlying
model for inelastic scattering is typically constructed with a vector
particle–dark photon (or dark Z’) mixing kinetically with an SM U(1) gauge
boson and coupling to the two different DM particles, $\chi_{1}$ and
$\chi_{2}$ ExcitingDM2 ; ExcitingDM9 ; here, to ensure that the coupling of
the DM particles to the dark photon is strictly off-diagonal in the mass
basis, the DM particles must be Majorana states because there then exists no
vector current for a single Majorana particle. In this scenario, elastic
scattering between DM and nucleons can occur happen at second order (right
panel of Fig. 4) and is thus suppressed, whereas inelastic scattering can
occur at first order (left panel of Fig. 4) and thus plays the leading role in
direct-detection experiments. If, furthermore, the kinetic energy is smaller
than the mass splitting of the DM, then up-scattering on nucleons is
kinetically prohibited, and we are left with the exothermic scattering of the
DM.
Figure 4: First- and second-order Born amplitudes for DM-nucleus scattering
ExcitingDM2 .
To further account for the large isospin-violating effect, the conventional
Higgs portal scheme of a scalar field mixing with the SM Higgs to communicate
between the SM and DM sectors causes no significant isospin violationZ'IVDM1
because only a very small percentage of the nucleon constituents are related
to the current quark mass and thus connected to the Higgs field. We then must
exploit vector instead of scalar particles to connect the dark world with SM
particles 222Indeed, we have investigated the possibility that instead of
treating the extra vector boson as a messenger that connects the DM world with
SM particles, we may treat it merely as a single DM candidate Z'SingleDM . The
result reveals that DM of this type must have a mass larger than the weak W
boson mass and therefore is unrelated to the present GeV DM, which is a
possibility that supports the present choice of a messenger role for the
vector boson in our low-energy-region search for DM.. For such a model with a
single messenger, the isospin-violating effect depends on the choice of SM
U(1) with which the new vector boson mixes. For example, if, as usual, we take
U(1) to be the SM hypercharge $\mathrm{U(1)_{Y}}$ ExcitingDM9 , because the
proton and neutron have the same hypercharge, we then expect the plot for the
left diagram of Fig. 4 to be the same for both neutrons and protons, leading
to $f_{n}=f_{p}$, i.e., there is no isospin violation. If, instead, we
consider that in the low-energy region, the $Z$-boson component of
$\mathrm{U(1)_{Y}}$ decouples, then effectively, only the electromagnetic part
will contribute, and we can then take U(1) to be the SM electromagnetic
$\mathrm{U(1)_{em}}$ ExcitingDM2 ; ExcitingDM7 ; because the neutron is
neutral and the proton is charged, we then expect the same plot to be zero for
neutrons, resulting in $f_{n}=0$ and $f_{p}\neq 0$, i.e., we have isospin
violation. In Fig. 5, we plot the $f_{n}=0$ IVDM exothermic DM result for the
CoGeNT and CDMS-Si signal regions along with the null results of XENON10, LUX,
SuperCDMS, and CDEX (XENON100 lies to the right, outside the plot area).
Figure 5: Inelastic IVDM scattering result for $f_{n}=0$ and $\delta=-200$
keV. For an explanation of the legend, see Fig. 1.
Comparison of the plots of Fig. 5 and the right panel of Fig. 2 reveals only a
very few changes. In particular, the SuperCDMS result is only marginally in
tension with the CDMS-Si result. This is because the maximum suppression
values of $f_{n}/f_{p}$ are $-0.785$ (for Ge), $-0.697$ (for Xe), and $-0.992$
(for Si), and a detailed computation shows that if we take $f_{n}/f_{p}=-0.7$,
then the energy spectra of Ge and Xe relative to Si are suppressed by
approximately 90$\%$ and 95$\%$, respectively; if we set $f_{n}=0$, then the
suppression of Ge and Xe relative to the Si energy spectra is reduced by
20$\%$. Hence, $f_{n}=0$ offers an insufficient isospin-violating effect, and
we must increase its strength by setting $f_{n}/f_{p}=-0.7$. In the
literature, the first discussion of vector boson exchange leading to
$f_{n}/f_{p}=-0.7$ was presented in Ref. Z'IVDM2 , and in that discussion, the
key roles were played by three factors: the conventional kinetic mixing and
the mass mixing between SM $U(1)$ and the dark photon or Z’ as well as the
coupling of the dark photon to SM quarks. Although the original model
presented in Ref. Z'IVDM2 does not include the inelastic-scattering effect,
we can modify the model by adding a Majorana mass term to the DM fields, which
will yield exactly an off-diagonal dark-photon coupling to the DM fermions,
and this improvement does not change the value of $f_{n}/f_{p}$. To be more
explicit, we write a Lagrangian for our proposed schematic model as follows:
$\displaystyle\mathcal{L}=\mathcal{L}_{\mathrm{SM}}-\frac{1}{4}X^{\mu\nu}X_{\mu\nu}+\frac{1}{2}m_{X}^{2}X_{\mu}{X}^{\mu}-m{{}_{\chi}}\bar{\chi}\chi-\frac{1}{2}\sin\epsilon\;{B}_{\mu\nu}X^{\mu\nu}+\delta
m^{2}Z_{\mu}{X}^{\mu}~{}~{}~{}~{}$ (16)
$\displaystyle+\bar{\chi}(i\not{\partial}-f_{\chi}^{V}\not{X})\chi-{\displaystyle\sum_{f}}f_{f}^{V}\bar{f}\not{X}f-\frac{\delta}{2}(\bar{\chi}^{c}\chi+\bar{\chi}\chi^{c}),$
where the extra $U(1)_{\mathrm{X}}$ is assumed to be broken and the
corresponding vector boson mass is $m_{Z^{\prime}}$. We denote the fields in
the interaction basis by $(B,W^{3},X)$ and in the mass-eigenstate basis by
$(A,Z,Z^{\prime})$, and we define $Z\equiv{c}_{W}{W}^{3}-s_{W}{B}$, where
$s_{W}$($c_{W}$) is the sine (cosine) of the Weinberg angle. $\chi$ is the
fermionic DM field with Dirac mass $M$ and Majorana mass $\delta\ll M$.
For this Lagrangian, the discussions of Ref. Z'IVDM2 demonstrate that there
exist suitable parameter spaces $(\epsilon,\delta m^{2},f_{f}^{V})$ to account
for $f_{n}/f_{p}=-0.7$, as described in greater detail below.
* •
For the dark Z’ scenario, in which the SM fields are uncharged under the extra
$U(1)_{\mathrm{X}}$ group and, thus, $f_{f}^{V}=0$, Fig. 2 of Ref.Z'IVDM2
shows that the ratio $f_{n}/f_{p}\sim 0.7$ with $m_{Z^{\prime}}=4$ GeV can be
achieved by adjusting the remaining two parameters $\epsilon$ and $\delta
m^{2}$ appropriately. The figure shows that for $\epsilon\approx\delta
m^{2}/m_{Z}^{2}$ and $\epsilon\ll 1$, we have $f_{n}/f_{p}\approx
1/3s_{W}\approx-0.7$.
* •
For the baryonic Z’ scenario, the SM is charged under the $U(1)_{\mathrm{X}}$
group, whereas the leptons are uncharged under $U(1)_{\mathrm{X}}$ and
$U(1)_{\mathrm{X}}\equiv U(1)_{\mathrm{B}}$. In this case,
$f_{u}^{V}=f_{d}^{V}\equiv f_{q}^{V}$. Now, there are three parameters,
$(\epsilon,\delta m^{2},f_{f}^{V})$. Figure. 3 of Ref.Z'IVDM2 shows that the
ratio $f_{n}/f_{p}\sim 0.7$ can be achieved by adjusting two of the three
parameters; the left panel illustrates the variation of $\epsilon$ and
$f_{q}^{V}$ with $\delta m^{2}=0$, and the right panel illustrates the
variation of $\epsilon$ and $\delta m^{2}$ with $f_{q}^{V}\approx 10^{-5}$.
The figure shows that to obtain $f_{n}/f_{p}\approx-0.7$, $f_{q}^{V}$ must be
more than an order of magnitude smaller than $\epsilon$. Suppose that
$\epsilon$ in Ref. Z'IVDM2 is constrained to be on the order of $10^{-2}$ or
smaller, such that $f_{q}^{V}\leq 10^{-3}$. The requisite smallness of
$f_{q}^{V}$ may be achieved by coupling $Z^{\prime}$ only to the second- and
third-generation quarks, and this relaxes the restriction that the additional
$U(1)_{\mathrm{X}}$ must be baryonic, thereby allowing for couplings to
leptons to facilitate the construction of an anomaly-free model.
By contrast, diagonalizing the DM mass matrix leads to mass eigenstates
$\chi_{1,2}$ of masses $M_{1,2}=m_{\chi}\mp\delta$ and an off-diagonal gauge
interaction, which leads to the DM scattering picture previously considered in
Fig. 4.
$\displaystyle\bar{\chi}(i\not{\partial}-f_{\chi}^{V}\not{X})\chi=\bar{\chi}_{1}i\not{\partial}\chi_{1}+\bar{\chi}_{2}i\not{\partial}\chi_{2}-f_{\chi}^{V}(\bar{\chi}_{1}\not{X}\chi_{2}+\bar{\chi}_{2}\not{X}\chi_{1}).$
(17)
Ref. Z'IVDM3 presents similar discussions with two additional important
extensions: first, noting the possibility of applying the model to inelastic
scattering, and second, proving that a combination with the conventional Higgs
mediator will help to achieve the desired isospin violation. These extensions
are also investigated in Ref. Z'IVDM1 , and the combination of the dark photon
and the conventional Higgs mediator is further generalized to the combination
of the dark photon and another new light vector gauge boson. Using our
schematic model (16), especially the parameter range represented in Fig. 2 and
Fig. 3 of Ref.Z'IVDM2 , in addition to these other possible underlying
exothermic IVDM models that give rise to an expected value of
$f_{n}/f_{p}=-0.7$, we plot the IVDM exothermic DM result for the CoGeNT and
CDMS-Si signal regions along with the null results of XENON100, XENON10, LUX,
SuperCDMS, and CDEX (see Fig. 6). The left plot corresponds to $\delta=-50$
keV, and the right plot corresponds to $\delta=-200$ keV.
Figure 6: Inelastic IVDM scattering results for $\delta=-50$ keV (left panel)
and $\delta=-200$ keV (right panel). For an explanation of the legend, see
Fig. 1.
Apart from the strong tension remaining between CoGeNT and the null
experiments, we see that even for $\delta=-50$ keV, CDMS-Si is already
consistent with most of the null experiments, although LUX cuts through half
of the contour. For $\delta=-200$ keV, the tensions between CDMS-Si and the
null experiments are over-relaxed. Therefore, with the assistance of isospin-
violating effects from the dark photon or Z’, we can readily make CDMS-Si
consistent with all current null experiments, even without invoking the
extreme case of exothermic DM with $\delta=-200$ keV. This leaves open a
region in the parameter space for exothermic DM to fit other current and
future DM detection experiments.
It should be noted that there are several other possible methods of
reconciling the tensions among various direct-detection experiments. The first
is to interpret the possible signals appearing in DAMA, CoGeNT, and CDMS-Si
not as DM signals but as some atmospherically produced neutral particle with a
relatively large magnetic dipole moment MdipoleMoment , as such particles can
mimic DM signals. A very definite flux could explain the signals observed in
DAMA/LIBRA, CDMS-Si, and CoGeNT that are consistent with the bounds from
XENON100 and CDMS-II. In this scenario, the key is that the recoil energy of
the assumed particle must lie some specific energy range that is above the
thresholds of DAMA/LIBRA, CDMS-Si, and CoGeNT but below those of XENON100 and
CDMS-II. If we further consider the latest results of SuperCDMS and CDEX, then
this recoil energy must lie above the thresholds of these two experiments and
therefore is expected to produce signals in these detectors. This has is not
occurred, hence implying that this alternative interpretation is not favored
by the latest SuperCDMS and CDEX null results.
The second possibility is to invoke composite DM, wherein stable particles of
charge 2 bind with primordial helium to form O-helium ”atoms” (OHe),
representing a specific warmer-than-cold nuclear-interacting form of dark
matter OHe . Because it slows down in terrestrial matter, OHe is elusive in
direct methods of underground DM detection such as those used in the CDMS
experiment, but its reactions with nuclei can lead to annual variations in the
energy released in the energy range of $2-6$ keV such as those observed in the
DAMA/NaI and DAMA/LIBRA experiments. However, this class of solution cannot
explain the unmodulated signals in experiments such as CoGeNT and CDMS-Si and
therefore is not favored by these experiments.
Finally, for completeness, we will list for each experiment some of the
details of the computations used to obtain all the above plots (except Figs.
4):
1. (i)
CDMS-Si: We used the acceptance from CDMSSi and a total exposure of 140.2 kg-
days, assuming zero background. We considered an energy interval of [7,100]
keV and binned the data in 2 keV intervals as in tension3 . The three
candidate events appeared in the first three bins. To find the best-fit
regions, we obtained the extended log-likelihood function and simply plotted
constant values of the likelihood that it would correspond to the 68$\%$ and
95$\%$ CL regions under the assumption that the likelihood distribution is
Gaussian.
2. (ii)
CoGeNT: We used the data and flat background shown in Fig. 23 of Cogent3 ,
which has been corrected for efficiency (i.e., bin counts have been scaled to
reflect the numbers of events expected based on those observed and the deduced
efficiency). We performed a $\chi^{2}$ scan over a cross section using the DM
mass and background from the data of Ref. Cogent3 . The curves for the region
of interest correspond to the 90$\%$ C.L. regions. The energy resolution below
10 keV was taken to be that reported by CoGeNT, namely,
$\sigma^{2}=\sigma_{n}^{2}+2.35^{2}E\eta F$, where $\sigma_{n}=69.4$ eV is the
intrinsic electronic noise, $E$ is the energy in eV, $\eta=2.96$ eV is the
average energy required to create an electron-hole pair in Ge at approximately
80 K, and $F=0.29$ is the Fano factor. The number of expected events in a
given range was taken to be tension1
$\displaystyle
N_{[E_{1},E_{2}]}=Ex.\int_{0}^{\infty}\frac{dR}{dE_{R}}\mbox{res}(E_{1},E_{2},E_{R})dE_{R}+b_{[E_{1},E_{2}]},$
(18)
where $b$ is the flat, floating background and
$2\mbox{res}(E_{1},E_{2};E_{R})=\mbox{erf}((E_{1}-E_{R})/(\sqrt{2}\sigma))-\mbox{erf}((E_{2}-E_{R})/(\sqrt{2}\sigma))$.
3. (iii)
DAMA: The average amplitude over the energy interval $[E_{1},E_{2}]$ is
$\displaystyle
S_{m,[E_{1},E_{2}]}=\frac{1}{E_{2}-E_{1}}\sum_{T=Na,I}c_{T}\int_{E_{1}/Q_{T}}^{E_{2}/Q_{T}}S(E_{R})dE_{R},$
(19)
where $c_{T}$ is the mass fraction of the target and $Q_{T}$ is the quenching
factor for the target, which we take to be $Q_{Na}=0.3$ and $Q_{I}=0.09$. To
account for the ion-channeling effect, we take the channeling fraction to be
$\displaystyle f_{Na}=10^{-\sqrt{E/(6.9\mathrm{keV})}},\qquad
f_{I}=10^{-\sqrt{E/(11.5\mathrm{keV})}},$ (20)
as in Ref. DAMA6 . The measured energy will be normally distributed with a
standard deviation of
$\displaystyle\sigma(E)=(0.448\mathrm{keV})\sqrt{E/\mathrm{keV}}+0.0091E.$
(21)
We used the data presented in Fig. 6 of DAMA3 . We calculated $\chi^{2}$ using
all 36 bins corresponding to energies from 2 keV to 20 keV. The 95$\%$ C.L.
contours of the region of interest satisfy
$\chi^{2}=\chi^{2}_{min}+\mathrm{CDF}^{-1}(\mathrm{ChiSq[2],C.L.}).$
4. (iv)
XENON10: We simply adopted the collaboration’s parameterization from Fig. 1 of
XENON10 , assuming a sharp cutoff to zero at a nuclear recoil energy of 1.4
keV. The signal region is from 5 to 35 electrons, corresponding to nuclear
recoils of 1.4 keV to 10 keV. A limit of 90$\%$ C.L. was obtained using the
$p_{\mbox{max}}$ method Pmax and the 23 highlighted S2 event signals from
Fig. 2 of XENON10 .
5. (v)
XENON100: We used the mean $\nu(E)$ characterized in XENON1004 . For the
scintillation efficiency $\mathcal{L}_{eff}$, we used the efficiency used in
XENON100’s 225-live-day analysis XENON1003 obtained from Fig. 1 of XENON1001
, which included a linear extrapolation to 0 for $E$ below 3 keV. The response
of the detector was modeled as a Gaussian distribution with a mean of $n$ and
a variance of $\sqrt{n}\sigma_{PMT}$, where $\sigma_{PMT}=0.5\mbox{PE}$
XENON1004 . The Gaussian smearing also included a photoelectron-dependent
acceptance, which we parameterized based on Fig. 1 of XENON1003 . To obtain
the total rate, we summed the differential rate over the signal region, which
corresponds to $S1\in(3,30)\mbox{PE}$ for the analysis presented in XENON1003
, and used a total exposure of 225$\times$34 kg-days XENON1003 . We then used
Poisson statistics to obtain a 90$\%$ C.L. upper limit, where two events were
observed, as shown in Fig. 2 of XENON1003 .
6. (vi)
LUX: The experimental design of LUX is quite similar to that of XENON100
ExcitingDM9 . Both experiments use a combination of scintillation (S1) and
ionization signals (S2) to effectively reject background. Following XENON1004
, we computed the number of signal events as follows:
$\displaystyle
N_{DM}=\int_{S1_{lower}}^{S1_{upper}}dS1\sum_{n=1}^{\infty}\mbox{Gauss}(S1|n,\sqrt{n}\sigma_{PMT})\int_{0}^{\infty}dE_{R}\epsilon(E_{R})\mbox{Poisson}(n|\nu(E_{R}))\frac{dR}{dE_{R}}\times\mbox{Ex.},$
(22)
where Ex. denotes the experimental exposure, $\epsilon(E_{R})$ is the S1
efficiency, and $\sigma_{PMT}=0.37~{}\mbox{PE}$ accounts for the PMT
resolution. For the LUX analysis, $S1_{lower}=2$ and $S1_{upper}=30$. The
expected number of photoelectrons $\nu(E_{R})$ is
$\displaystyle\nu(E_{R})=E_{R}\times\mathcal{L}_{eff}(E_{R})\times\frac{S_{n}}{S_{e}}\times
L_{y},$ (23)
where $\mathcal{L}_{eff}$ is the energy-dependent scintillation efficiency of
liquid xenon, $L_{y}$ is the light yield, and $S_{n}$ and $S_{e}$ are the
nuclear- and electron-recoil quenching factors, respectively, that arise from
the applied electric field. We used the energy-dependent absolute light yield,
$\mathcal{L}_{eff}(E_{R})\frac{S_{n}}{S_{e}}L_{y}$, from slide 25 of LUX1 ,
with a hard cutoff below 3 keV. Finally, for the DM detection efficiency, we
used the efficiency calculated after threshold cuts from Fig. 9 of LUX . We
computed 90$\%$ CL limits using Poisson statistics with no events detected.
7. (vii)
SuperCDMS: For the efficiency reported in Fig. 1 of SuperCDMS , we used the
577-kg-day data from SuperCDMS . To obtain the 90$\%$ C.L. limits, we used
Poisson statistics with 11 candidate events detected, which are listed in
Table 1 of SuperCDMS , and zero background was assumed.
8. (viii)
CDEX: We assumed perfect efficiency and used the 53.9-kg-day data from the
residual spectrum presented in Fig. 3(b) of CDEX2 . A flat background was
assumed, as given by the minimum $\chi^{2}$ method. The quenching factor of a
recoiling Ge nucleus was obtained from the TRIM program as in CDEX3 . To
obtain the 90$\%$ C.L. limits, the binned Poisson method DAMA2 with bins of
$0.1$ keVee was used.
To summarize, we find that exothermic DM alone is not sufficient to fully
resolve the tensions between CDMS-Si and the null experiments. However, if
some underlying interaction allows isospin-violating effects to be
incorporated into exothermic DM models, then, with the aid of the strongest
setting $f_{n}/f_{p}=-0.7$, exothermic IVDM can make the CDMS-Si result
consistent with the results of all the latest null experiments, except the
CoGeNT experiment. Meanwhile, for exothermic IVDM to exist, at least a new
vector gauge boson (dark photon or dark Z’) that connects SM quarks with
Majorana-type DM particles is required.
## Acknowledgments
This work was supported by the National Basic Research Program of China (973
Program) under Grant No. 2010CB833000, the National Science Foundation of
China (NSFC) under Grant No. 11475092, the Specialized Research Fund for the
Doctoral Program of High Education of China No. 20110002110010, and the
Tsinghua University Initiative Scientific Research Program No. 20121088494.
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|
arxiv-papers
| 2014-04-24T07:35:03 |
2024-09-04T02:50:01.767916
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nan Chen, Qing Wang, Wei Zhao, Shin-Ted Lin, Qian Yue, Jin Li",
"submitter": "Wang Qing",
"url": "https://arxiv.org/abs/1404.6043"
}
|
1404.6119
|
arxiv-papers
| 2014-04-24T13:35:50 |
2024-09-04T02:50:01.777135
|
{
"license": "Public Domain",
"authors": "V. V. Vien and H.N.Long",
"submitter": "Vien Vo Van",
"url": "https://arxiv.org/abs/1404.6119"
}
|
|
1404.6202
|
# Mixed Hessian inequalities and uniqueness in the class
$\mathcal{E}(X,\omega,m)$
Sławomir Dinew Faculty of Mathematics and Computer Science
Jagiellonian University 30-348 Krakow
Lojasiewicza 6, Poland [email protected] and Chinh H. Lu Chalmers
University of Technology
Mathematical Sciences
412 96 Gothenburg
Sweden [email protected]
(Date:
The first-named author is partially supported by NCN grant
2013/08/A/ST1/00312.
The second-named author is partially supported by the french ANR project MACK)
###### Abstract.
We prove a general inequality for mixed Hessian measures by global arguments.
Our method also yields a simplification for the case of complex Monge-Ampère
equation. Exploiting this and using Kołodziej’s mass concentration technique
we also prove the uniqueness of the solutions to the complex Hessian equation
on compact Kähler manifolds in the case of probability measures vanishing on
$m$-polar sets.
## 1\. Introduction
Nonlinear equations of Monge-Ampère and more generally of Hessian type have
proven to be a very fruitful branch of research. In the set-up of compact
Kähler manifolds the solution of the Calabi conjecture by Yau ([25]) has
literally opened the door for PDE methods in complex geometry. In fact Yau’s
theorem is still a subject of numerous generalizations (see, in particular
[15, 4] and references therein) and new powerful tools coming from
pluripotential theory allowed applications that were previously unreachable.
The complex Hessian equation on compact Kähler manifolds is not that geometric
because the solutions do not yield Kähler metrics (see hovewer [1] for some
geometric applications). Nevertheless the PDE theory is interesting on its own
right, and a strong motivation for considering it is its real counterpart that
has been developed some time ago thanks to the works of Trudinger, Wang, Chou
and others (see [8, 9, 20, 21, 24] and references therein). Interestingly
while many estimates known from the Monge-Ampère theory apply verbatim, some
like the a priori gradient estimate for the Hessian equation turned out to be
unexpectedly difficult (this estimate was finally proven in [12]). This
finally terminated the program for proving an analogue of the Calabi-Yau
theorem in the Hessian setting. Afterwards a fruitful potential theory was
established culminating in the solution of the smoothing problem for
m-subharmonic functions (see [19, 18]).
In this note we continue the investigation of the pluripotential theory for
complex Hessian equations on compact Kähler manifolds. We establish some
technical results that allow us to prove a general inequality for mixed
Hessian measures (an analogue of the main result in [10]). It is worth
mentioning that while the methods from [10] are local in spirit, here our
approach is global which allows an essential simplification. In particular we
avoid the delicate approximation by Dirichlet solutions with potentially
discontinuous boundary values from [10]. Note that the local inequalities can
be deduced from the global ones (see Theorem 3.9) thus our method yields a
simplification even for the Monge-Ampère equation.
###### Theorem 1.
[Theorem 3.7] Let $u_{1},...,u_{m}$ be functions in $\mathcal{E}(X,\omega,m)$
and $\mu$ be a positive Radon measure vanishing on $m$-polar sets such that
$H_{m}(u_{j})\geq f_{j}\mu\ ,\forall j=1,...,m,$
where $f_{j}$ are non-negative integrable (with respect to $\mu$) functions.
Then one has
$\omega_{u_{1}}\wedge\cdots\wedge\omega_{u_{m}}\wedge\omega^{n-m}\geq(f_{1}\cdots
f_{m})^{1/m}\mu.$
Having these inequalities in hand it is straightforward to generalize the
uniqueness theorem from [11] thus establishing a very general uniqueness
result for the solutions to the complex Hessian equation living in the class
$\mathcal{E}(X,\omega,m)$.
###### Theorem 2.
[Theorem 4.1] Let $u,v$ be functions in $\mathcal{E}(X,\omega,m)$ such that
$H_{m}(u)=H_{m}(v).$
Then $u-v$ is a constant.
The note is organized as follows. First we briefly recall the notions and
tools that we shall use later on. Then in Section 3 we establish the
inequality for mixed Hessian measures. In Section 4 we prove the uniqueness
theorem. Finally in Section 5 we give an explicit example that the mixed
Hessian inequality fails outside the class $\mathcal{E}(X,\omega,m)$.
## 2\. Preliminaries
We recall basic facts on $m$-subharmonic functions which will be used later.
We first consider the local setting.
### 2.1. $m$-subharmonic functions
Let $M$ be a non-compact Kähler manifold of dimension $n$ and $\omega$ be a
Kähler form. Fix an integer $m$ such that $1\leq m\leq n$.
###### Definition 2.1.
A smooth function $u$ is called $m$-subharmonic ($m$-sh for short) on $M$ if
the following holds in the classical sense
$(dd^{c}u)^{k}\wedge\omega^{n-k}\geq 0,\ \forall k=1,\cdots,m.$
Equivalently, $u$ is $m$-sh if the vector of eigenvalues
$\lambda(x)\in\mathbb{R}^{n}$ of $dd^{c}u$ with respect to $\omega$ satisfies
$S_{k}(\lambda(x))\geq 0,\ \forall x\in M,\ \forall k=1, \cdots,m.$
Yet another characterization of $m$-subharmonicty can be obtained via the
inequalities
$\forall\varepsilon>0\ \ (dd^{c}u+\varepsilon\omega)^{m}\wedge\omega^{n-m}\geq
0.$
From Gårding’s inequality [13] we easily get the smooth version of the mixed
Hessian inequalities:
###### Lemma 2.2.
Let $u_{1},...,u_{m}$ be smooth $m$-sh functions on $M$ and $f_{1},...,f_{m}$
be smooth nonnegative functions such that
$(dd^{c}u_{j})^{m}\wedge\omega^{n-m}=f_{j}\omega^{n},\ \forall j=1,...,m.$
Then the following mixed Hessian inequality holds
$dd^{c}u_{1}\wedge...\wedge
dd^{c}u_{m}\wedge\omega^{n-m}\geq(f_{1}...f_{m})^{1/m}\omega^{n}.$
###### Definition 2.3.
Let $u$ be a locally integrable, upper semicontinuous function on $M$. Then
$u$ is called $m$-sh on $M$ if the following two conditions are satisfied:
1. i)
in the weak sense of currents
$dd^{c}u\wedge dd^{c}\varphi_{1}\wedge\cdots\wedge
dd^{c}\varphi_{m-1}\wedge\omega^{n-m}\geq 0,$
for any collection $\varphi_{1}\cdots\varphi_{m-1}$ of smooth $m$-sh
functions;
2. ii)
if $v$ is another function satisfying the above inequalities and $v=u$ almost
everywhere on $M$ then $u\leq v$.
Note that two $m$-subharmonic functions are the same if they are equal almost
everywhere on $M$. Observe also that by Gårding’s inequality Definition 2.1
and Definition 2.3 are equivalent for smooth functions. The class of $m$-sh
functions on $M$ (with respect to $\omega$) is denoted by
$\mathcal{SH}_{m}(M)$. Note that this class depends heavily on the metric
$\omega$ which makes the regularization process of $m$-sh functions very
complicated. More precisely, it is not clear whether the convolution of a
$m$-sh functions (when $\omega$ is not flat) with a smooth kernel is $m$-sh.
Nevertheless any $m$-sh function can be approximated by smooth $m$-sh
functions (see [19], [18]).
The smoothing property allows us to define the domain of definition of the
complex Hessian operator:
###### Definition 2.4.
A $m$-sh function $u$ belongs to the domain of definition of the Hessian
operator $\mathcal{D}_{m}(M)$ if there exists a regular Borel measure $\mu$
such that for any sequence $(u_{j})$ of smooth $m$-sh functions decreasing to
$u$ then $(dd^{c}u_{j})^{m}\wedge\omega^{n-m}$ converges weakly to $\mu$.
We shall need the following result of Błocki [6]:
###### Lemma 2.5.
[6] Assume that $M$ is an open subset of $\mathbb{C}^{n}$ and
$\omega=dd^{c}|z|^{2}$ is the standard Kähler form in $\mathbb{C}^{n}$. If
$m=2$ and $u$ is $m$-sh on $M$ then $u\in\mathcal{D}_{m}(M)$ if and only if
$u\in W^{1,2}_{\rm loc}(M)$.
### 2.2. $\omega$-$m$-subharmonic functions
Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$ and $m$ be an
integer such that $1\leq m\leq n$.
###### Definition 2.6.
A function $u:X\rightarrow\mathbb{R}\cup\\{-\infty\\}$ is called
$\omega$-$m$-subharmonic ($\omega$-$m$-sh for short) if in any local chart
$\Omega$ of $X$, the function $\rho+u$ is $m$-sh, where $\rho$ is a local
potential of $\omega$.
Observe that a smooth function $u$ is $\omega$-$m$-sh if and only if
$(\omega+dd^{c}u)^{k}\wedge\omega^{n-k}\geq 0,\ \forall k=1,...,m,$
or equivalently iff
$\ \forall\varepsilon>0\ \
((1+\varepsilon)\omega+dd^{c}u)^{m}\wedge\omega^{n-m}\geq 0.$
We let $\mathcal{SH}_{m}(X,\omega)$ denote the class of $\omega$-$m$-sh
functions on $X$. It follows from [18] that for any
$u\in\mathcal{SH}_{m}(X,\omega)$ there exists a decreasing sequence of smooth
$\omega$-$m$-sh functions on $X$ which converges to $u$. Following the
classical pluripotential method of Bedford and Taylor [2] one can then define
the complex Hessian operator for any bounded $\omega$-$m$-sh function:
$H_{m}(u):=(\omega+dd^{c}u)^{m}\wedge\omega^{n-m},$
which is a non-negative (regular) Borel measure on $X$. The complex Hessian
operator is local in the plurifine topology which in particular implies that
the sequence
${\bf 1}_{\\{u>-j\\}}H_{m}(\max(u,-j))$
is non-decreasing, for any $u\in\mathcal{SH}_{m}(X,\omega)$. Moreover given
any $\omega$-$m$-sh function $u$
$\forall j\in\mathbb{N},\ \int_{X}{\bf
1}_{\\{u>-j\\}}H_{m}(\max(u,-j))\leq\int_{X}\omega^{n}.$
We then define the class $\mathcal{E}(X,\omega,m)$ as the set of these of
$\omega$-$m$-sh functions for which
$\lim_{j\rightarrow\infty}\int_{X}{\bf
1}_{\\{u>-j\\}}H_{m}(\max(u,-j))=\int_{X}\omega^{n},$
(see [18], [14]). For any $u\in\mathcal{E}(X,\omega,m)$ we define
$H_{m}(u):=\lim_{j\rightarrow\infty}{\bf 1}_{\\{u>-j\\}}H_{m}(\max(u,-j)).$
We recall the following result:
###### Theorem 2.7.
[18] Let $\mu$ be a non-negative Radon measure on $X$ vanishing on all
$m$-polar sets. Assume that $\mu(X)=\int_{X}\omega^{n}$. Then there exists
$u\in\mathcal{E}(X,\omega,m)$ such that
$H_{m}(u)=\mu.$
Finally by $\mathcal{E}^{1}(X,\omega,m)$ we denote the subset of
$\mathcal{E}(X,\omega,m)$ of all $u$ integrable with respect to their own
Hessian measure $H_{m}(u)$.
Note that when $\mu=f\omega^{n}$ for some smooth positive function $f$ then
$u$ is also smooth (see [12]). The variational method used in [18] (which
originated from [4]) can be applied in the same way to get the following
result:
###### Theorem 2.8.
Let $\mu$ be a non-negative Radon measure on $X$ vanishing on all $m$-polar
sets. Then for any $\varepsilon>0$ there exists
$u\in\mathcal{E}^{1}(X,\omega,m)$ such that
$H_{m}(u)=e^{\varepsilon u}\mu.$
The proof of Theorem 2.8 is left to the reader as an easy exercise. We will
need this result in the next section. We remark that the solution $u$ is
unique- this follows from the domination principle in
$\mathcal{E}(X,\omega,m)$ which is an easy consequence of Theorem 2 as shown
in [7] (when $m=n$).
Next we shall need the notion of capacity associated to the Hessian operator
and convergence with respect to it:
###### Definition 2.9.
The $m$-capacity of a Borel subset of $X$ is defined as
${\rm Cap}_{\omega,m}(E):=\sup\left\\{\int_{E}H_{m}(u)\ \big{|}\
u\in\mathcal{SH}_{m}(X,\omega),\ -1\leq u\leq 0\right\\}.$
###### Definition 2.10.
A sequence $(u_{j})$ of converges in ${\rm Cap}_{\omega,m}$ to $u$ if for any
$\varepsilon>0$
$\lim_{j\to+\infty}{\rm Cap}_{\omega,m}(|u_{j}-u|>\varepsilon)=0.$
Exactly as it the plurisubharmonic setting we have convergence in capacity for
monotonely decreasing sequences and quasi-continuity of $\omega$-$m$-sh
functions:
###### Proposition 2.11.
If $(u_{j})\subset\mathcal{SH}_{m}(X,\omega)$ decreases to
$u\not\equiv-\infty$ then $u_{j}$ converges to $u$ in ${\rm Cap}_{\omega,m}$.
###### Lemma 2.12.
Any $\omega$-$m$-sh function $u$ is quasi-continuous, i.e. for any
$\varepsilon>0$ there exists an open subset $U$ such that ${\rm
Cap}_{\omega,m}(U)<\varepsilon$ and $u$ restricted on $X\setminus U$ is
continuous.
The following convergence result is an easy adaptation of [16], Corollary
1.14:
###### Lemma 2.13.
Let $(\varphi^{1}_{j}),...,(\varphi^{m}_{j})$ be uniformly bounded sequence of
functions in $\mathcal{SH}_{m}(X,\omega)\cap L^{\infty}(X)$ converging in
${\rm Cap}_{\omega,m}$ to $\varphi^{1},...,\varphi^{m}$ respectively. Assume
that $(f_{j})$ is a uniformly bounded sequence of quasi continuous functions
converging in ${\rm Cap}_{\omega,m}$ to $f$. Then we have the weak convergence
of measures
$f_{j}\omega_{\varphi_{j}^{1}}\wedge...\wedge\omega_{\varphi_{j}^{m}}\wedge\omega^{n-m}\rightharpoonup
f\omega_{\varphi^{1}}\wedge...\wedge\omega_{\varphi^{m}}\wedge\omega^{n-m}.$
## 3\. An inequality for mixed Hessian measures
Our main goal in this section is to prove an inequality for mixed Hessian
measures (Theorem 1). We first prove some technical results which we need
later on. Our approach, which makes use of the ”$\beta$-convergence” method of
Berman [3], is global in nature and therefore avoids some difficulties arising
from rough boundary data present in the local Dirichlet problem (compare with
[10]).
Let $\mu$ be a positive Radon measure on $X$. Let $L_{\mu}$ denote the mapping
$\mathcal{SH}_{m}(X,\omega)\ni\varphi\mapsto L_{\mu}(\varphi):=\int_{X}\varphi
d\mu.$
The following result has been proven in [18] (it is in fact an an easy
adaptation of a theorem from [4]):
###### Lemma 3.1.
Let $\mu$ be a positive Radon measure on $X$ which is dominated by ${\rm
Cap}_{\omega,m}$. Then $L_{\mu}$ is continuous on
$\mathcal{E}^{1}(X,\omega,m)$ with respect to the $L^{1}$ topology.
###### Lemma 3.2.
Let $(u_{j})$ and $(\varphi_{j})$ be sequences of bounded $\omega$-$m$-sh
functions on $X$. Assume that $u_{j}$ converges in ${\rm Cap}_{\omega,m}$ to
$u\in\mathcal{SH}_{m}(X,\omega)\cap L^{\infty}(X)$ and $\varphi_{j}$ converges
in $L^{1}(X)$ to $\varphi\in\mathcal{SH}_{m}(X,\omega)\cap L^{\infty}(X)$. If
there exists $A>0$ such that
$H_{m}(\varphi_{j})\leq A\,H_{m}(u_{j}),\ \forall j,$
then $\varphi_{j}\rightarrow\varphi$ in ${\rm Cap}_{\omega,m}$. In particular,
$H_{m}(\varphi_{j})$ converges weakly to $H_{m}(\varphi)$.
###### Proof.
Without loss of generality we can assume that all the functions are negative.
Fix $\varepsilon>0$ and set $E_{j}:=\\{\varphi<\varphi_{j}-3\varepsilon\\}$.
We are going to prove that ${\rm Cap}_{\omega,m}(E_{j})\to 0$ as
$j\to+\infty$. Let $C>1$ be a constant such that
$\sup_{X}(|u_{j}|+|\varphi_{j}|+|u|+|\varphi|)\leq C\ ,\ \forall j.$
Fix $1>\delta>0$ such that $\delta C<\varepsilon$ and let
$v\in\mathcal{SH}_{m}(X,\omega)$ such that $-1\leq v\leq 0$. The following
inclusions are obvious:
$\\{\varphi<\varphi_{j}-3\varepsilon\\}\subset\\{\varphi<(1-\delta)\varphi_{j}+\delta
v-2\varepsilon\\}\subset\\{\varphi<\varphi_{j}-\varepsilon\\}.$
By the comparison principle we thus get
$\delta^{m}\int_{\\{\varphi<\varphi_{j}-3\varepsilon\\}}H_{m}(v)\leq\int_{\\{\varphi<\varphi_{j}-\varepsilon\\}}H_{m}(\varphi)\leq\varepsilon^{-1}\int_{X}[\max(\varphi,\varphi_{j})-\varphi]H_{m}(\varphi).$
Taking the supremum over all $v$ and using Lemma 3.1 we obtain
$\lim_{j\to+\infty}{\rm Cap}_{\omega,m}(E_{j})=0.$
Now, set $F_{\varepsilon}:=\\{\varphi_{j}<\varphi-3\varepsilon\\}$. It remains
to prove that ${\rm Cap}_{\omega,m}(F_{j})$ also converges to $0$ as
$j\to+\infty$. Arguing as in the first step we get
$\displaystyle\delta^{m}{\rm Cap}_{\omega,m}(F_{j})$ $\displaystyle\leq$
$\displaystyle\varepsilon^{-1}\int_{X}\left[\max(\varphi_{j},\varphi)-\varphi_{j}\right]H_{m}(\varphi_{j})$
$\displaystyle\leq$ $\displaystyle
A\varepsilon^{-1}\int_{X}\left[\max(\varphi_{j},\varphi)-\varphi_{j}\right]H_{m}(u_{j}).$
On the other hand, since
$\int_{X}\left[\max(\varphi_{j},\varphi)-\varphi_{j}\right]H_{m}(u)\longrightarrow
0$
as follows from the first step, it suffices to prove that
$\int_{X}\left[\max(\varphi_{j},\varphi)-\varphi_{j}\right]\left[H_{m}(u_{j})-H_{m}(u)\right]\longrightarrow
0.$
By an integration by parts the above term is dominated by
$C_{1}\int_{X}|u_{j}-u|H_{m}(\psi_{j}),$
where $\psi_{j}$ is a sequence of uniformly bounded $\omega$-$m$-sh functions
and $C_{1}$ depends only on $C$. Now, fix a small $t>0$. Since $u_{j}$
converges to $u$ in ${\rm Cap}_{\omega,m}$ we get for some $C_{2}$ also
dependent only on $C$ that
$\displaystyle\limsup_{j\to+\infty}\int_{X}|u_{j}-u|H_{m}(\psi_{j})$
$\displaystyle\leq$ $\displaystyle
t\int_{X}\omega^{n}+\limsup_{j\to+\infty}C_{2}{\rm
Cap}_{\omega,m}(\\{|u_{j}-u|>t\\}$ $\displaystyle=$ $\displaystyle
t\int_{X}\omega^{n},$
from which the result follows. ∎
The uniqueness result for bounded $\omega$-$m$-sh functions can be proven by
repeating the arguments in [5]:
###### Lemma 3.3.
Let $u,v$ be bounded $\omega$-$m$-sh functions. If $H_{m}(u)=H_{m}(v)$ then
$u-v$ is constant.
We shall also need the following domination principle:
###### Lemma 3.4.
Let $u,v$ be bounded $\omega$-$m$-sh functions. Assume that $H_{m}(u)$
vanishes on the set $\\{u<v\\}$. Then $u\geq v$ on $X$.
###### Proof.
We can assume that $v\leq 0$. Fix $\varepsilon>0$ and $1>s>0$ such that
$s\max(1,\sup_{X}|v|)<\varepsilon$. Let $\varphi$ be a $\omega$-$m$-sh
function on $X$ such that $-1\leq\varphi\leq 0$. Applying the comparison
principle we obtain
$\displaystyle s^{m}\int_{\\{u<v-2\varepsilon\\}}H_{m}(\varphi)$
$\displaystyle\leq$
$\displaystyle\int_{\\{u<v-2\varepsilon\\}}H_{m}((1-s)v+s\varphi)$
$\displaystyle\leq$
$\displaystyle\int_{\\{u<(1-s)v+s\varphi-\varepsilon\\}}H_{m}(s\varphi+(1-s)v)$
$\displaystyle\leq$
$\displaystyle\int_{\\{u<(1-s)v+s\varphi-\varepsilon\\}}H_{m}(u)$
$\displaystyle\leq$ $\displaystyle\int_{\\{u<v\\}}H_{m}(u)=0.$
We then get ${\rm Cap}_{\omega,m}(\\{u<v-2\varepsilon\\})=0$ which implies the
result. ∎
From the domination principle we immediately get the following:
###### Lemma 3.5.
Let $u,v$ be bounded $\omega$-$m$-sh functions. Assume that $\mu$ is a
positive Radon measure such that
$H_{m}(u)\leq e^{u}\mu\ {\rm and}\ H_{m}(v)\geq e^{v}\mu.$
Then $v\leq u$.
###### Proof.
By the comparison principle we have
$\int_{\\{u<v\\}}H_{m}(u)\leq\int_{\\{u<v\\}}e^{u}d\mu\leq\int_{\\{u<v\\}}e^{v}d\mu\leq\int_{\\{u<v\\}}H_{m}(v)\leq\int_{\\{u<v\\}}H_{m}(u)$
Thus all inequalities above become equalities and we infer that
$H_{m}(u)(u<v)=\mu(u<v)=0$. Now, it suffices to apply the domination
principle. ∎
###### Lemma 3.6.
Let $u,v,\varphi,\psi_{1},\psi_{2}$ be bounded $\omega$-$m$-sh functions such
that
$H_{m}(u)=fH_{m}(\varphi)+h_{1}H_{m}(\psi_{1})\ ,\
H_{m}(v)=gH_{m}(\varphi)+h_{2}H_{m}(\psi_{2}),$
where $f,g,h_{1},h_{2}$ are non-negative bounded functions. Then for any
$1\leq k\leq m-1$,
$\omega_{u}^{k}\wedge\omega_{v}^{m-k}\wedge\omega^{n-m}\geq
f^{k/m}g^{(m-k)/m}H_{m}(\varphi).$
###### Proof.
Step 1: Assume that $\varphi,\psi_{1},\psi_{2}$ are smooth on $X$.
Let $f^{j},g^{j},h_{1}^{j},h_{2}^{j}$ be uniformly bounded sequences of smooth
non-negative functions which converge in $L^{1}(X)$ to $f,g,h_{1},h_{2}$
respectively. We can also assume that these sequences are normalized so that
the corresponding measures have the same mass as $\int_{X}\omega^{n}$. We now
use the main result of [12] to solve the following equations
$H_{m}(u_{j})=f_{j}H_{m}(\varphi)+h_{1}^{j}H_{m}(\psi_{1}),\
H_{m}(v_{j})=g_{j}H_{m}(\varphi)+h_{2}^{j}H_{m}(\psi_{2}),$
where $u_{j},v_{j}$ are smooth $\omega$-$m$-sh functions. We also normalize
$u_{j},v_{j}$ so that $\sup_{X}u_{j}=\sup_{X}u$ and $\sup_{X}v_{j}=\sup_{X}v$.
It follows from Lemma 3.2 and Lemma 3.3 that $u_{j}$ and $v_{j}$ converge in
${\rm Cap}_{\omega,m}$ to $u$ and $v$ respectively. Since $u_{j}$ and $v_{j}$
are smooth by Gårding’s inequality ([13]) we get
$\omega_{u_{j}}^{k}\wedge\omega_{v_{j}}^{m-k}\wedge\omega^{n-m}\geq(f_{j})^{k/m}(g_{j})^{(m-k)/m}H_{m}(\varphi).$
Now, the result follows by letting $j\to+\infty$.
Step 2: Assume that $f,g$ are quasi-continuous on $X$ and
$\min(f,g)\geq\delta>0$ for some positive constant $\delta$.
It follows from [18] that we can approximate $\varphi,\psi_{1},\psi_{2}$ from
above by decreasing sequences of smooth $\omega$-$m$-sh functions, say
$(\varphi^{j}),(\psi_{1}^{j}),(\psi_{2}^{j})$. Let
$u_{j}\in\mathcal{SH}_{m}(X,\omega)$ solve the equation
$H_{m}(u_{j})=e^{u_{j}-u}\left[fH_{m}(\varphi^{j})+h_{1}H_{m}(\psi_{1}^{j})\right].$
Since $\varphi_{j},\psi_{1}^{j}$ are uniformly bounded and $u,f,h_{1}$ are
bounded (in fact $\delta\leq f\leq C$) we deduce from the comparison principle
that $u_{j}$ is also uniformly bounded. Assume that $u_{j}$ converges in
$L^{1}(X)$ to some bounded $\omega$-$m$-sh function $u_{\infty}$. It follows
from Lemma 3.2 that $u_{j}$ converges in ${\rm Cap}_{\omega,m}$ to
$u_{\infty}$. By letting $j\to+\infty$ we get
$H_{m}(u_{\infty})=e^{u_{\infty}-u}H_{m}(u).$
Applying Lemma 3.5 we get $u=u_{\infty}$. Now, we do the same thing for $v$ to
get a sequence $v_{j}$ converging in ${\rm Cap}_{\omega,m}$ to $v$. Applying
Step 1 for $u_{j}$ and $v_{j}$ we get
$\omega_{u_{j}}^{k}\wedge\omega_{v_{j}}^{m-k}\wedge\omega^{n-m}\geq
e^{k(u_{j}-u)/m+(m-k)(v_{j}-v)/m}f^{k/m}g^{(m-k)/m}H_{m}(\varphi_{j}).$
Since $u_{j},v_{j}$ converge in ${\rm Cap}_{\omega,m}$ to $u$ and $v$
respectively and $f,g$ are quasi-continuous we can argue as in [16] (see Lemma
2.13) to see that the right-hand side of the above inequality converges to
$f^{k/m}g^{(m-k)/m}H_{m}(\varphi),$
while the left-hand side converges to
$\omega_{u}^{k}\wedge\omega_{v}^{m-k}\wedge\omega^{n-m}$.
Step 3: Assume that $\min(f,g)\geq\delta>0$ for some positive constant
$\delta$.
Let $f_{j},g_{j}$ be uniformly bounded sequences of continuous non-negative
functions which converge in $L^{1}(X,H_{m}(\varphi))$ to $f,g$ respectively.
We can assume also that $\min(f_{j},g_{j})\geq\delta$ for all $j$. Let $u_{j}$
solve the equation
$H_{m}(u_{j})=e^{u_{j}-u}\left[f_{j}H_{m}(\varphi)+h_{1}H_{m}(\psi_{1})\right].$
It follows from Lemma 3.5 that $u_{j}$ is uniformly bounded. We can argue as
in Step 2 to see that $u_{j}$ converges in ${\rm Cap}_{\omega,m}$ to $u$. Now,
do the same thing for $v$ and use Step 2 to get
$\omega_{u_{j}}^{k}\wedge\omega_{v_{j}}^{m-k}\wedge\omega^{n-m}\geq
e^{k(u_{j}-u)/m+(m-k)(v_{j}-v)/m}f_{j}^{k/m}g_{j}^{(m-k)/m}H_{m}(\varphi).$
The result follows by letting $j\to+\infty$.
Step 4: We now prove the general statement. Consider
$f_{\varepsilon}:=\max(f,\varepsilon)$ and solve
$H_{m}(u_{\varepsilon})=e^{u_{\varepsilon}-u}\left[f_{\varepsilon}H_{m}(\varphi)+h_{1}H_{m}(\psi_{1})\right].$
Then $u_{\varepsilon}\leq u$ and $u_{\varepsilon}\geq\varphi/2+\psi_{1}/2-C$
as follows from the comparison principle (see Lemma 3.5). Do the same thing
for $v$ and apply Step 3 to conclude. ∎
Now, we are ready to prove Theorem 1. For the sake of simplicity we only treat
the case when there are only two functions instead of a collection of $m$
functions. The general case follows in an obvious way by changing the
notation.
###### Theorem 3.7.
Let $u,v$ be functions in $\mathcal{E}(X,\omega,m)$ and $\mu$ be a positive
Radon measure vanishing on $m$-polar sets such that
$H_{m}(u)\geq f\mu\ ,\ H_{m}(v)\geq g\mu,$
where $f,g$ are non-negative integrable (with respect to $\mu$) functions.
Then for any $1\leq k\leq m-1$, one has
$\omega_{u}^{k}\wedge\omega_{v}^{m-k}\wedge\omega^{n-m}\geq
f^{k/m}g^{(m-k)/m}\mu.$
###### Proof.
We first treat the case when $u,v,f,g$ are bounded and $\mu=H_{m}(\varphi)$
for some bounded $\omega$-$m$-sh function $\varphi$. Fix $\delta>0$. For each
$j>1$ let $u_{j}$ solve the following equation
$H_{m}(u_{j})=e^{j(u_{j}-u)}\left[(1-\delta)f\mu+\delta H_{m}(u)\right].$
One can solve the above equation by using the variational method exactly the
same way as in [18] (see Theorem 2.8). By the comparison principle one gets
$u\leq u_{j}\leq u+j^{-1}\log(\delta^{-1})$. Thus $u_{j}$ converges uniformly
on $X$ to $u$. Now, do the same thing for $v$ and get a sequence $v_{j}$ which
converges uniformly to $v$. Applying Lemma 3.6 and letting $j\to+\infty$ we
get
$\omega_{u}^{k}\wedge\omega_{v}^{m-k}\wedge\omega^{n-m}\geq(1-\delta)f^{k/m}g^{(m-k)/m}\mu.$
Now, the result follows by letting $\delta\to 0$.
If $f,g$ and $\varphi$ are bounded but $u,v$ are not bounded we can argue as
follows. Consider $u_{j}=\max(u,-j)$ and $v_{j}=\max(v,-j)$. Then
$H_{m}(u_{j})\geq{\bf 1}_{\\{u>-j\\}}fH_{m}(\varphi)\ {\rm and}\
H_{m}(v_{j})\geq{\bf 1}_{\\{v>-j\\}}fH_{m}(\varphi).$
We then apply the previous step for $u_{j},v_{j}$ and let $j$ go to $+\infty$,
noting that $H_{m}(\varphi)$ does not charge $m$-polar sets.
For the general case observe that since $\mu$ does not charge $m$-polar sets,
it follows from [18] and the generalized Radon-Nikodym theorem [22] that we
can write $\mu=hH_{m}(\varphi)$ for some
$\varphi\in\mathcal{SH}_{m}(X,\omega)\cap L^{\infty}(X)$ and $h\in
L^{1}(H_{m}(\varphi))$. We thus can assume that $\mu=H_{m}(\phi)$ for some
bounded $\omega$-$m$-sh function $\phi$. Now, consider $f_{j}:=\min(f,j)$ and
$g_{j}:=\min(g,j)$. Applying the first step and letting $j\to+\infty$ we get
the result. ∎
###### Remark 3.8.
In the first step of the proof, instead of solving a family of equations with
parameter $j$ we can argue as follows. Observe first that $H_{m}(u)-f\mu$ is a
positive Radon measure dominated by $H_{m}(u)$ with $u\in L^{\infty}(X)$. Then
one can find $\psi\in\mathcal{SH}_{m}(X,\omega)\cap L^{\infty}(X)$ such that
$H_{m}(\psi)=e^{\psi}[H_{m}(u)-f\mu].$
Indeed, the existence of a solution follows from the variational approach and
it is bounded since there is a bounded subsolution (see also [26] for a more
detailed discussion about the envelop method for the complex Monge-Ampère
equation).
We finish this section by presenting how local inequalities for Hessian
measures stem from their global counterpart. More precisely we prove the
following theorem:
###### Theorem 3.9.
Let $\omega$ be a germ of a Kähler metric near $0\in\mathbb{C}^{n}$. Let also
$u_{1},u_{2},\cdots,u_{m}$ be bounded $m$-subharmonic functions satisfying
$(dd^{c}u_{j})^{m}\wedge\omega^{n-m}\geq f_{j}\mu$ for some non negative Borel
measure vanishing on all $m$-polar sets. Then
(3.1) $dd^{c}u_{1}\wedge\cdots\wedge
dd^{c}u_{m}\wedge\omega^{n-m}\geq(f_{1}\cdots f_{m})^{1/m}\mu.$
Before we start the proof we recall the following standard fact, based on the
patching of local potentials (see, for example [23], Lemma 3.8 for a
discussion):
###### Lemma 3.10.
Let $M$ be the unit ball in $\mathbb{C}^{n}$ equipped with the Kähler metric
$\omega=dd^{c}\phi$, with $\phi$ bounded. Fix a smaller ball $B$ centered at
$0$. Then $(M,\omega)$ admits an isometric embedding into a compact complex
torus $(X,\tilde{\omega})$ such that $\omega|_{B}=\tilde{\omega}_{Im(B)}$ with
$Im(B)$ being the image of $B$ under the isometry. Moreover $\tilde{\omega}$
can be taken to be the standard flat metric on a neighborhood of $X\setminus
Im(M)$.
Now we can prove the local mixed Hessian inequalities:
###### Proof of theorem 3.9.
Suppose first that all the functions $u_{1},\cdots u_{n}$ are bounded near
$0\in C^{n}$. It is enough to establish the inequality in a small neighborhood
of $0$. Shrinking that neighborhood if necessary we may assume that
$\omega=dd^{c}\phi$ for some bounded plurisubharmonic function $\phi$.
Exploiting the previous lemma we may assume that (with the obvious
identifications) $\omega$ is a Käler form on a compact Kähler manifold $X$.
Shrinking the domain further if necessary, we can patch each of the functions
$u_{j}-\phi$ with a suitable global $\omega$-$m$-subharmonic function with an
isolated pole at $0\in X$ (strictly speaking we have to use the regularized
maximum technique instead of ordinary maximum- see [23] for a discussion) we
get global $\omega$-$m$-subharmonic functions $\tilde{u}_{j}$ such that
$\tilde{u}_{j}=u_{j}-\phi$ in a neighborhood of $0\in X$. Now by the global
inequality for mixed Hessian measures we get
$\omega_{\tilde{u}_{1}}\wedge\cdots\wedge\omega_{\tilde{u}_{m}}\wedge\omega^{n-m}\geq(f_{1}\cdots
f_{m})^{1/m}\mu$
in a neighborhood of $0\in X$ which is exactly what we seek.
Now the passage from $u_{j}$ bounded to general $u_{j}$ is done exactly like
in the global argument. ∎
## 4\. uniqueness
The main result of this section is the following uniqueness result for the
normalized solutions to the complex hessian equations:
###### Theorem 4.1.
$u,v$ be functions in $\mathcal{E}(X,\omega,m)$ and $\mu$ be a positive Radon
measure vanishing on $m$-polar sets such that
$H_{m}(u)=\mu=H_{m}(v).$
Then $u-v$ is a constant.
###### Proof.
Given the inequality for mixed hessian measures the argument pretty much
follows the one from [11]. We present the details for the sake of
completeness.
Suppose on contrary that $u-v\neq$ const. Note that $\forall
t\in\mathbb{R}\cup\\{\infty\\}\cup\\{-\infty\\}$ the sets $\\{u-v=t\\}$ are
Borel and hence at most countably many of these are charged by $\mu$. Observe
that by assumption the sets with $t=+\infty$ or $-\infty$ are massless for
they are $m$-polar . Just like in [11] we prove that actually precisely one of
the remaining sets has positive $\mu$-mass. For if not then there must exist
$t\in\mathbb{R}$ such that $\\{u-v=t\\}$ is massless, while
$\mu(\\{u-v<t\\}),\mu(\\{u-v>t\\})>c>0$ for some constant $c\leq 1/2$. Note
that after adding a constant we can and will assume $t$ to be zero.
Consider the new measure
$\widehat{\mu}:=\begin{cases}C\mu,&on\ \\{u<v\\}\\\ 0,&on\ \\{u\geq
v\\},\end{cases}$
where $C>1$ is a nonnegative normalization constant so that $\widehat{\mu}$ is
a nonnegative probability measure (note that this is possible, since, by
assumption, $\mu$ charges the set $\\{u\geq v\\}$).
Of course $\widehat{\mu}$ does not charge pluripolar sets either (and is also
a Borel measure since the set $\\{u\geq v\\}$ is Borel). By [18] we can solve
the Hessian equation
$H_{m}(w)=\widehat{\mu},\ \ w\in\mathcal{E}(X,\omega,m),\ sup_{X}w=0.$
Consider the set inclusion
$U_{t}:=\\{(1-t)u<(1-t)v+tw\\}\subset\\{u<v\\}$
for every $t\in(0,1)$. Hence on $U_{t}$ we have
$\omega_{(1-t)v+tw}\wedge\omega_{u}^{m-1}\wedge\omega^{n-m}\geq(1-t)\mu+tC^{1/m}\mu=(1+t(C^{1/m}-1))H_{m}(u),$
where we have made use of Theorem 3.7.
Exploiting the partial comparison principle (see [18]) we get
$\displaystyle(1+((C)^{1/m}-1)t)\int_{U_{t}}H_{m}(u)\leq\int_{U_{t}}\omega_{(1-t)v+tw}\wedge\omega_{u}^{m-1}\wedge\omega^{n-m}$
$\displaystyle\leq\int_{U_{t}}\omega_{(1-t)u+t0}\wedge\omega_{u}^{m-1}\wedge\omega^{n-m}=(1-t)\int_{U_{t}}H_{m}(u)+t\int_{U_{t}}\omega_{u}^{m-1}\wedge\omega^{n-m+1}.$
In other words we get
(4.1)
$C^{1/m}\int_{U_{t}}\mu\leq\int_{U_{t}}\omega_{u}^{m-1}\wedge\omega^{n-m+1}.$
Note that the inequality for Hessian measures coupled with total volume
considerations yields
$\omega_{u}^{k}\wedge\omega_{v}^{m-k}\wedge\omega^{n-m}=\mu$
for any $k\in\\{0,1,\cdots,m\\}$, see Corollary 2.2. in [11]. But then the
same argument with $u$ exchanged by $v$ leads to the inequality
$C^{1/m}\int_{U_{t}}\mu\leq\int_{U_{t}}\omega_{v}^{m-1}\wedge\omega^{n-m+1}.$
If we let now $t$ to zero we observe that $U_{t}$ converges to
$\\{u<v\\}\setminus\\{w=-\infty\\}$, but $\\{w=-\infty\\}$ is an $m$-polar set
and hence is negligible with respect to $\mu$. Thus we get
$C^{1/m}\int_{\\{u<v\\}}\mu\leq\int_{\\{u<v\\}}\omega_{v}^{m-1}\wedge\omega^{n-m+1},$
as well as
$C^{1/m}\int_{\\{u<v\\}}\mu\leq\int_{\\{u<v\\}}\omega_{u}^{m-1}\wedge\omega^{n-m+1}.$
Playing the same game on $\\{u>v\\}$ (namely we define a measure like
$\hat{\mu}$ with respect to the set $\\{u>v\\}$) we also get for a different
constant $D>1$ the inequality
$D^{1/m}\int_{\\{u>v\\}}\mu\leq\int_{\\{u>v\\}}\omega_{v}^{m-1}\wedge\omega^{n-m+1}.$
Coupling these with the fact that $\\{u=v\\}$ is massless by construction we
end up with the inequalities
$\min\\{C^{1/m},D^{1/m}\\}\int_{X}\mu\leq\int_{\\{u<v\\}}\omega_{v}^{m-1}\wedge\omega^{n-m+1}+\int_{\\{u>v\\}}\omega_{v}^{m-1}\wedge\omega^{n-m+1}\leq\int_{X}\mu,$
so $1<\min\\{C^{1/m},D^{1/m}\\}\leq 1$, a contradiction.
Thus we know that $\mu(\\{u\neq v\\})=0$. Just like in [11] our next and final
task will be to prove analogous mass vanishing for $H_{j}(u),H_{j}(v),\
j=0,1,\cdots,m-1$. Indeed, consider the sets
$V_{t,j}:=\\{u+(t/j)u_{j}+(3/2)t<v\\}\subset\\{u<v\\},$
where $u_{j}:=\max\\{u,-j\\}$.
The partial comparison principle results in
$\int_{V_{t,j}}\omega_{u}^{k}\wedge\omega_{v}^{m-1-k}\wedge(\omega_{v}+(t/j)\omega)\wedge\omega^{n-m}\leq\int_{V_{t,j}}\omega_{u}^{k}\wedge\omega_{v}^{m-1-k}\wedge(\omega_{u}+(t/j)\omega_{u_{j}})\wedge\omega^{n-m}.$
Now, (recall $\omega_{u}^{k}\wedge\omega_{v}^{m-k}\wedge\omega^{n-m}=\mu,\
\forall k\in\\{0,\cdots m\\}$) the equation above reduces to
$\int_{V_{t,j}}\omega_{u}^{k}\wedge\omega_{v}^{m-1-k}\wedge\omega^{n-m+1}\leq\int_{V_{t,j}}\omega_{u}^{k}\wedge\omega_{v}^{m-1-k}\wedge\omega_{u_{j}}\wedge\omega^{n-m}.$
Note that $V_{t,j}$ is a decreasing sequence of sets in terms of $j$. Letting
$j\rightarrow\infty$, using vanishing on pluripolar sets and then letting $t$
to zero we obtain
$\int_{\\{u<v\\}}\omega_{u}^{k}\wedge\omega_{v}^{m-1-k}\wedge\omega^{n-m+1}\leq\int_{\\{u<v\\}}H_{m}(u)=0.$
In the same vein the measures
$\omega_{u}^{k}\wedge\omega_{v}^{m-1-k}\wedge\omega^{n-m+1}$ put no mass on
$\\{u>v\\}$. Finally exchanging
$\omega_{u}^{k}\wedge\omega_{v}^{m-1-k}\wedge\omega^{n-m}$ in the argument
above with $\omega_{u}^{k}\wedge\omega_{v}^{m-2-k}\wedge\omega^{n-m+1}$ we
obtain mass vanishing on $\\{u<v\\}$ for the measures
$\omega_{u}^{k}\wedge\omega_{v}^{m-2-k}\wedge\omega^{n-m+2},\
k=0,1,\cdots,m-2$. An easy induction finally yields that $\omega^{n}$ has its
mass supported on $\\{u=v\\}$ which is impossible unless $u$ equals $v$. ∎
## 5\. An example
In this section we give an example which shows that vanishing on $m$-polar
sets cannot be removed from the assumptions. The example (adapted from [10])
actually shows that our mixed Hessian inequality fails outside the class
$\mathcal{E}(X,\omega,m)$.
###### Example 5.1.
Let $n\geq 3,m=2$ and $\omega=dd^{c}|z|^{2}$ be the flat Kähler metric in
$\mathbb{C}^{n}$. Consider the following functions
$u_{k}(z):=\max\left(\frac{1}{k}\log|z_{1}|,k^{2}\log|z_{2}|\right),\ {\rm
and}\ v_{k}(z):=\max\left(\frac{1}{k}\log|z_{2}|,k^{2}\log|z_{1}|\right).$
Then
$(dd^{c}u_{k})^{2}\wedge\omega^{n-2}(dd^{c}v_{k})^{2}\wedge\omega^{n-2}=\frac{(2\pi)^{2}k}{2}[z_{1}=z_{2}=0],$
where $[Z]$ means the current of integration along $Z$. But
$dd^{c}u_{k}\wedge
dd^{c}v_{k}\wedge\omega^{n-2}=\frac{(2\pi)^{2}}{2k^{2}}[z_{1}=z_{2}=0],$
which violates (3.1) when $k>1$.
Note that in this example $u_{k},v_{k}$ (which are plurisubharmonic functions)
belong to the domain of definition of $H_{m}$ since they belong to
$W^{1,2}(\mathbb{C}^{n})$ (see Lemma 2.5). But their Hessian measures charge
the $m$-polar set $\\{z_{1}=z_{2}=0\\}$.
###### Proof.
From [10, Example 4.1] we know that $(dd^{c}u_{k})^{2}=(dd^{c}v_{k})^{2}$,
when considered as measure in $\mathbb{C}^{2}$, is the Dirac measure of the
origin with coefficient $a(k)=\frac{(2\pi)^{2}k}{2}$. Thus the Hessian measure
of $u_{k}$ and $v_{k}$ is the integration along the line $\\{z_{1}=z_{2}=0\\}$
with coefficient $a(k)$, while $dd^{c}u_{k}\wedge
dd^{c}v_{k}\wedge\omega^{n-2}$ is the integration along the same line with
coefficient $b(k)=\frac{(2\pi)^{2}}{2k^{2}}$. When $k>1$ this violates
inequality (3.1). ∎
###### Remark 5.2.
The example above has Hessian measure charging some non-dicrete analytic
$m$-polar set. It is interesting to note that in the plurisubharmonic case it
is a deep open problem whether such a function actually exists.
## References
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* [2] E. Bedford, B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), no. 1, 1-44.
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* [4] R. J. Berman, S. Boucksom, V. Guedj, A. Zeriahi, A variational approach to complex Monge-Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179-245.
* [5] Z. Błocki, Uniqueness and stability for the complex Monge-Ampère equation on compact Kähler manifold, Indiana Univ. Math. J. 52 (2003), no. 6, 1697-1701.
* [6] Z. Błocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble) 55 (2005), no. 5, 1735-1756.
* [7] T. Bloom, N. Levenberg, Pluripotential energy, Potential Analysis, Volume 36, Issue 1 (2012), 155-176.
* [8] L. Caffarelli, L. Nirenberg, J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261-301.
* [9] K.-S. Chou and X.-J. Wang, Variational theory for Hessian equations, Comm. Pure Appl. Math., 54 (2001), 1029-1064.
* [10] S. Dinew, An inequality for mixed Monge-Ampère measures, Math. Z. 262 (2009), 1-15.
* [11] S. Dinew, Uniqueness in $\mathcal{E}(X,\omega)$, J. Funct. Anal. 256 (2009), no. 7, 2113-2122.
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* [13] L. Gårding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959) 957-965.
* [14] V. Guedj, A. Zeriahi, The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), no. 2, 442-482.
* [15] S. Kołodziej, The complex Monge-Ampère equation, Acta Math. 180 (1998) 69-117.
* [16] S. Kołodziej, The complex Monge-Ampère equation and theory, Memoirs Amer. Math. Soc. 178 (2005) 64p.
* [17] S.-Y. Li, On the Dirichlet problems for symmetric function equations of the eigenvalues of the complex Hessian, Asian J. Math. 8 (2004),no. 1, 87-106.
* [18] H. C. Lu, V. D. Nguyen, Degenerate complex Hessian equations on compact Kähler manifolds, arXiv:1402.5147.
* [19] S. Pliś, The smoothing of $m$-subharmonic functions, arXiv:1312.1906.
* [20] N. S. Trudinger, On the Dirichlet problem for Hessian equations, Acta Math. 175 (1995), 151-164.
* [21] N. S. Trudinger, X.-J. Wang, Hessian measures II, Ann. of Math. 150 (1999), 579-604.
* [22] J. Rainwater, A note on the preceding paper, Duke Math. J. 36 (1969) 799-800.
* [23] M. Verbitsky, Plurisubharmonic functions in calibrated geometry and $q$-convexity, Math. Z. 264 (2010), 939-957.
* [24] X.-J. Wang, The k-Hessian equation, Lecture Notes in Math., 1977, Springer, Dordrecht, 2009.
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|
arxiv-papers
| 2014-04-24T17:58:15 |
2024-09-04T02:50:01.782951
|
{
"license": "Public Domain",
"authors": "S{\\l}awomir Dinew and Chinh H. Lu",
"submitter": "Chinh Lu Hoang",
"url": "https://arxiv.org/abs/1404.6202"
}
|
1404.6247
|
# Career on the Move: Geography, Stratification, and Scientific Impact
Pierre Deville Department of Applied Mathematics, Université catholique de
Louvain, Belgium CCNR and Physics Department, Northeastern University, Boston,
MA 02115, USA Dashun Wang IBM Thomas J. Watson Research Center, Yorktown
Heights, NY, 10598, USA CCNR and Physics Department, Northeastern University,
Boston, MA 02115, USA Roberta Sinatra CCNR and Physics Department,
Northeastern University, Boston, MA 02115, USA Center for Cancer Systems
Biology, Dana-Farber Cancer Institute, Boston, MA 02115, USA Chaoming Song
Department of Physics, University of Miami, Coral Gables, FL 33124,USA CCNR
and Physics Department, Northeastern University, Boston, MA 02115, USA Vincent
D. Blondel Department of Applied Mathematics, Université catholique de
Louvain, Belgium Albert-László Barabási ∗, CCNR and Physics Department,
Northeastern University, Boston, MA 02115, USA Department of Medicine, Brigham
and Women’s Hospital, Harvard Medical School, Boston, MA 02115 Center for
Network Science, Central European University, Budapest, Hungary,
∗ [email protected]
###### Abstract
Changing institutions is an integral part of an academic life. Yet little is
known about the mobility patterns of scientists at an institutional level and
how these career choices affect scientific outcomes. Here, we examine over
420,000 papers, to track the affiliation information of individual scientists,
allowing us to reconstruct their career trajectories over decades. We find
that career movements are not only temporally and spatially localized, but
also characterized by a high degree of stratification in institutional
ranking. When cross-group movement occurs, we find that while going from elite
to lower-rank institutions on average associates with modest decrease in
scientific performance, transitioning into elite institutions does not result
in subsequent performance gain. These results offer empirical evidence on
institutional level career choices and movements and have potential
implications for science policy.
Despite their importance for education, scientific productivity, reward and
hiring procedures, our quantitative understandings of how individuals make
career moves and relocate to new institutions, and how such moves shape and
affect performance, remains limited. Indeed, previous research on migration
patterns of scientists auriol2007labour ; auriol2010careers tended to focus
on large-scale surveys on country-level movements, revealing long-term
cultural and economical priorities vannoorden2004 ; schiermeier2011career ;
jans2010study ; solimano2008international . At a much finer scale, research on
human dynamics and mobility has emerged as an active line of enquiry
brockmann/nature/2006 ; simini2012universal ; PhysRevLett.86.3200 ;
gonzalez/nature/2008 ; bagrow2011collective ; lu2012predictability ;
szell2012understanding , owing to new and increasingly available massive
datasets providing time resolved individual trajectories blondel2012data .
While these studies cover a much shorter time scale than a typical career,
they uncover a set of regularities and reproducible patterns behind human
movements brockmann/nature/2006 ; gonzalez/nature/2008 ; song/naturePhys/2010
. Less is known about patterns behind career moves at an institutional level
and how these moves affect individual performance.
Here we take advantage of the fact that scientists publish somewhat regularly
along their career petersen2012persistence ; petersen2013reputation , and for
each publication, the institution in which the work was performed is listed as
an affiliation in the paper, documenting career trajectories at a fine scale
and in great detail. These digital traces, offering data on not only
individual scientific output at each institution but also career moves from
one institution to another, can provide insights for science policy, helping
us understand how institutions shape knowledge, the typical moves of
individual career development and help us evaluate scientific outcomes
associated with professional mobility.
We use the Physical Review dataset to extract mobility information,
publication record, and citations for individual scientists. The data consists
of 237,038 physicists and 425,369 scientific papers, out of which 4,052
different institutions are extracted after the disambiguation process for
authors and affiliations (see SM for disambiguation process). To reconstruct
the career trajectory of a scientist, we use the affiliation given in each of
his/her publications (Fig 1). For authors with multiple affiliations listed on
a paper we consider the first affiliation as primary institution. We compute
the impact of each paper by counting its cumulative citations collected 5
years after its publication jones2008multi ; radicchi2008universality ;
barabasi2012publishing ; wang2013quantifying .
Figure 1: Illustrative example of career trajectory reconstruction for
hypothetical authors. Given the paper N°1 and N°2, we know that the author
John J. Smith was affiliated to Northeastern University in 1963 and Harvard
University in 1988. Extracting information from all his other publications
allows us to reconstruct his career trajectory and discover that he was
affiliated to Northeastern University for 8 years where he published 5 papers
and then moved to Harvard University for 23 years where he published 16
papers. The cumulative number of citations of a paper obtained within 5 years
after the publication is also known.
## I Results
Three characteristics are computed for each institution $i$ (Fig. 2): the
institution size ($A_{i}$), representing the total number of distinct authors
that published at least one paper at institution $i$; the number of papers
($P_{i}$) published under affiliation $i$; the cumulative number of citations
$C_{i}$ collected by all papers $P_{i}$. We find that $P(A)$ follows a fat
tailed distribution, indicating significant population heterogeneity among
different institutions (Fig. 2a). While most institutions are small, a few
have a large number of scientists, often corresponding to large institutes or
universities. We observe similar disparity in $P(C)$ (Fig. 2b): few
institutions acquire a large number of citations, while most research labs or
universities receive few citations.
Figures 2c-d show the correlation between the institution size $A$ and both
the average publications impact ${C}/{P}$ and the average productivity
${P}/{A}$ of institutions. The average productivity and impact of an
institution are different but complementary measures of scientific
performance. We find the institution size has little influence on productivity
($R^{2}=0.43$) (Fig. 2d), yet it positively correlates with the impact of
publications ($R^{2}=0.85$), indicating that large institutions offer a more
innovative/higher impact environment than smaller ones as captured by
citations per paper (Fig. 2c). Also, as larger institutions have more internal
collaborations, the number of co-authors in publications from large
institutions might be larger and, as a consequence, attracts more citations
jones2008multi .
Figure 2: Basic features of research institutions. (a) The probability density
function of institution size, $A$, follows a fat tailed distribution,
indicating a significant heterogeneity. While most institutions size are
small, a few have a large population, often representing large institutes or
universities with a long history. (b) The probability density function of
citations of institutions, $C$, is also very heterogeneous. Few institutions
acquired a large number of citations, while most research labs or universities
received few citations. Only the first thousand locations are taken into
account in further analyses (shaded area). (c) The correlation between
institution size and average publication impact is reported. Institution size
positively correlates with the impact of publications ($R^{2}=0.9$) ,
indicating that large institutions offer a more innovative/higher impact
environment than smaller ones as captured by citations per paper. The dashed
line indicates a power-law behaviour with exponent $\alpha=0.204\pm 0.006$ (d)
The correlation between institution size and institution average productivity
is also reported, indicating institution size has little influence on
productivity ($R^{2}=0.43$). The dashed line indicates a power-law behaviour
with exponent $\alpha=0.037\pm 0.003$.
Many institutions are small with few citations, hence they account for very
small portion of the data. For the rest of the paper, we will focus on the
thousand most cited institutions, accounting for more than $99\%$ of papers.
They correspond to institutions with at least 698 citations within the APS
data over the 120-year period (shaded area in Fig. 2).
Mobility is often important in furthering a professional career
schiermeier2011career . In science, the best lab for the type of research you
are doing is usually not where you are zhang2013characterizing ;
borner2005spatio ; mazloumian2013global . Nowadays changing countries is a
rite of passage for many young researchers who follow the resources and
facilities vannoorden2004 ; petersen2012persistence . As the patterns and
characteristics of these migrations are blurry, we need to systematically
study the mobility of scientists. Thanks to the large disambiguated data
spanning the last 120 years that we have compiled, a systematic study of
scientific mobility is now possible.
The strong correlations between the three quantities ($A,P,C$) indicate any of
the three could characterise an institution, serving as a proxy of its ranking
against others. Here, we choose $C$ (the total number of citations) as our
parameter to approximate the ranking by reputation. Other parameters such as
the h-index of an institution or the number of papers $P$ could also be used
hirsch2005index ; hirsch2007does ; lehmann2006measures . But the results
should be insensitive to this choice owing to good correlations between these
quantities ($R^{2}=0.96$ and $R^{2}=0.92$ respectively). The top-ranked
institutions all correspond to well-known universities or research labs with
long tradition of excellence in physics (Fig. 3), corroborating our hypothesis
that $C$ is a reasonable proxy for ranking. We can also observe the similarity
and stability of other rankings when comparing with other metrics.
Figure 3: Ten most cited institutions in physics. Comparison between different
rankings. The H-index is closely related to the number of citations as we can
observe. Top-ranked institutions all correspond to well-known universities or
research labs with long tradition of excellence in physics, corroborating our
hypothesis that $C$ is a reasonable proxy for ranking
We focus on authors with similar career longevity, restricting our corpus to
those who began their career between 1950 and 1980 and published for at least
20 years without any interruption exceeding 5 years. Following these criteria,
we arrived at a subset of 2,725 scientists to study the mobility patterns and
their impact on their careers. A total of 5,915 career movements are recorded
for this corpus.
In Figure 4a we select three individuals as exemplary career histories. Each
line represents one individual, with circles denoting his/her publications,
allowing us to observe his/her location. The size of the circle is
proportional to citations the paper acquires in five years, approximating the
impact of the work. By studying the whole corpus, we compute $P(m)$, the
probability for a scientist to have visited $m$ different institutions along
his career (Fig. 4c), finding that career movements are common but infrequent:
Only $14\%$ of them never moved at all ($m=1$). For the ones that move, they
mostly move once or twice, $P(m)$ decaying quickly as $m$ increases. We also
compute $P(t)$, the probability to observe a movement at time $t$, where $t=0$
corresponds to the date of the scientist’s first publication. We find that
most movements occurred in the early stage of the career (Fig. 4b), supporting
the hypothesis that changing affiliations is a rite of passage for young
researchers schiermeier2011career . This likely corresponds to the postdoc
period where graduates broaden their horizons through mobility. This may also
reflect the increasing cost of relocation and family constraints as family
developed vannoorden2004 ; jans2010study . A third characteristic is the
geographical distance of movements, $\Delta d$. Existing literature hints for
somewhat competing hypothesis in the role geography plays in career movements.
Indeed, research on human mobility suggests that regular human movements
mostly cover short distances with occasional longer trips, characterized by a
power law distance distribution erlander1990gravity ; simini2012universal ;
brockmann/nature/2006 ; gonzalez/nature/2008 ; in contrast, country-level
surveys find increasing cross-country movements mostly due to cultural
exposure and life quality concerns, indicating potential dominance in long
distance moves in career choices comparing with typical human travels
auriol2010careers ; auriol2007labour ; levin1999foreign ; zucker2007star ;
vannoorden2004 ; franzoni2012foreign ; jans2010study . We measure the distance
distribution over all moves observed in our dataset, finding that our result
is supported by a combination of both hypothesis. We find the probability to
move to further locations decays as a power law newman2005power ;
milojevic2010power , whereas the null model predicts this probability to be
flat (Fig. 4d). This observation is consistent with studies on human mobility,
that short distance moves dominate career choices. Yet, when comparing the
power law exponents, we find the exponent characterizing career moves
($\gamma=0.65\pm 0.053$) is much smaller than those observed in human travel
($\gamma\approx 2$), corresponding to higher likelihood of observing long
range movements. This observation might be explained by the influence that
scientific collaborations can have on career movements as similar low
exponents are observed for collaboration network between cities pan2012world .
Figure 4: Basic features of scientists career. (a) Illustration of three
scientific trajectories based on publications where each line corresponds to
one scientist and each publication is represented by a circle whose size is
proportional to its number of citations cumulated within $5$ years after its
publication. The institutions are ranked according to the total number of
citations they obtained (see Methods), $1$ being the most cited institution.
(b) The probability density function of movement according to time, $P(t)$,
shows that most movements occurred in the early stage of the career. This
likely corresponds to the postdoc period where graduates broaden their
horizons through mobility. (c) The probability density function of number of
visited institutions for a scientist along his career, $P(m)$, indicates that
career movements are common but infrequent. Scientists mostly move once or
twice, $P(m)$ decaying quickly as $m$ increases. (d) The probability density
function of distance of movements, $P(\Delta d)$, has a fat-tail that can be
fitted by a power law with an exponent $\gamma=0.65\pm 0.053$, whereas the
null model predicts this probability to be roughly flat.
Taken together, the preceding results indicate that career moves mostly happen
during the early stage of a career and are more likely to cover short
distances. The observed location in both time and space raises the question of
how individual moves as a function of institutional rankings. To this end,
denoting with $T_{i,j}$ the number of transitions from the institution of rank
$i$ to the one of rank $j$, we measure $P(i,j)$, the probability to have a
transition from rank $i$ to rank $j$ as
$P(i,j)=\frac{T_{i,j}}{\sum\limits_{i,j}T_{i,j}}.$ (1)
Interestingly, we find that most movements involve elite institutions (rank is
small), and transitions between bottom institutions are rare (Fig. 5a). This
is due to the fact that elite institutions are characterised by larger
populations, hence translating into more events.
To account for the population based heterogeneity, we compare the observed
$P(i,j)$ with the probability $P^{null}(i,j)$ expected in a random model where
we randomly shuffle the transitions from institution $i$ to $j$ while
preserving the total number of transitions from and to each institution.
Formally, in this null model, we have
$P^{null}(i,j)={\sum\limits_{k}P(k,j)\cdot\sum\limits_{l}P(i,l)},$ (2)
and we compare $P(i,j)$ with the null model by computing the matrix
$M(i,j)=\frac{P(i,j)}{\sum\limits_{k}P(k,j)\sum\limits_{l}P(i,l)}.$ (3)
$M(i,j)$ is the ratio between the probability $P(i,j)$ to have a transition
from rank $i$ to $j$ divided by the probability $P^{null}(i,j)$ when the
movements are shuffled, measuring the likelihood for a move to take place by
accounting for the size of the institutions. Hence, $M(i,j)=1$ indicates the
amount of observed movements is about what one would expect if movements were
random. Similarly, $M(i,j)>1$ indicates that we observe more transitions from
$i$ to $j$ than we expected, whereas $M(i,j)<1$ corresponds to transitions
that are underrepresented. We find that career moves are characterized by a
high degree of stratification in institutional rankings (Fig. 5b). Indeed, we
observe two distinct clubs (red spots in Fig. 5b), indicating that the
overrepresented movements are the ones within elite institutions (lower-left
corner) or within lower-rank institutions (upper-right corner), and scientists
belonging to one of the two groups tend to move to institutions within the
same group. On the other hand, both upper-left and lower-right corners are
colored blue, indicating cross group movements (transitions from elite to
lower-rank institutions and vice-versa) are significantly underrepresented.
Also, scientists from medium-ranked institutions move to the next institution
with a probability that is indistinguishable from the random case. In other
words, their movements indicate no bias towards middle, elite or lower-ranked
institutions.
The high intensity of stratification in career movements raises an interesting
question: how does individual performance in science relate to their moves
across different institutional rankings ?
To answer this question, we need to quantify the performance change for each
individual before and after the move. Imagine that a scientist moves from $i$
to $j$, and published $n$ papers at location $i$ and $m$ papers at $j$. The
impact of a paper $k$ can be approximated by $c_{k}$, the number of citations
cumulated within 5 years after its publication jones2008multi ;
radicchi2008universality ; barabasi2012publishing ; wang2013quantifying . Let
$c^{-}=\\{c^{-}_{1},c^{-}_{2},...,c^{-}_{n}\\}$ and
$c^{+}=\\{c^{+}_{1},c^{+}_{2},...,c^{+}_{m}\\}$ be the lists of number of
citations for papers published before ($c^{-}$) and after ($c^{+}$) the
transition from $i$ to $j$ ($T_{i,j}$). To quantify the change in performance,
we introduce
$\Delta c^{*}=\frac{\overline{c^{+}}-\overline{c^{-}}}{\sigma_{c}}$ (4)
where $\overline{c^{+}}$ and $\overline{c^{-}}$ are the average of $c^{+}$ and
$c^{-}$, respectively, and $\sigma_{c}$ corresponds to the standard deviation
of the concatenation of both $c^{+}$ and $c^{-}$ while preserving the moment
when the movement took place (see SM for more information about $\sigma_{c}$).
Therefore, $\Delta c^{*}$ captures the statistical difference in the average
citations between papers published before and after the movement normalized by
the random expectation when the same author’s publications were shuffled. A
positive $\Delta c^{*}$ indicates papers following the move on average result
in higher citation impact, hence representing an improvement in scientific
performance. A negative value corresponds to a decline in performance.
To quantify the influence of movements on individual performance, we divide
all movements into two categories based on the performance change: movements
associated with positive and negative $\Delta c^{*}$, and measure
$M(i,j|\Delta c^{*}>0)$ and $M(i,j|\Delta c^{*}<0)$. We find the observed
stratification in career moves is robust against individual performance (Fig.
5c-d). That is, the two clubs emerge for both categories in a similar fashion
as in Figure 5b, indicating the pattern of moving within elite or lower-rank
institutions is nearly universal for people whose performance is improved or
decreased following the move. Comparing Figure 5c and Figure 5d, we find the
red spot in lower-left corner is more concentrated in Figure 5d than in Figure
5c, hinting that being more mobile in the space of rankings may lead to
variable performance. To test this hypothesis, for each transition $T_{i,j}$
we calculate the rank difference between the origin and destination
($\Delta{r_{ij}}=i-j$).
Figure 5: Stratification of career movement. (a) The matrix of probability to
have a transition from rank $i$ to rank $j$, ($1$ being the top institution)
indicates that most movements involve elite institutions (rank is small) while
transitions between bottom institutions are rather rare. (b) The likelihood
$M(i,j)$ for a move to take place by accounting for the size of the
institutions is characterized by a high degree of stratification in
institutional rankings. Indeed, we observe two distinct clubs (red regions),
indicating that the overrepresented movements are the ones within elite
institutions (lower-left corner) or within lower-rank institutions (upper-
right corner), and scientists belonging to one of the two groups tend to move
to institutions within the same group. (c)-(d) The Likelihood $M(i,j)|\Delta
c^{*}<0$ and $M(i,j)|\Delta c^{*}>0$ for transitions resulting in higher and
lower scientific impact, respectively, indicates that the stratification in
career moves is robust against individual performance. We find the red region
in lower-left corner is more concentrated in Fig. 5d than in c, hinting that
being more mobile in the space of rankings may lead to variable performance.
A positive value of $\Delta{r_{ij}}$ indicates $i>j$, hence a movement to a
lower-rank institution, whereas $\Delta{r_{ij}}<0$ corresponds to transitions
into institutions with a higher rank. In Figure 6 we measure the relation
between $\Delta c^{*}$ and $\Delta{r}$. When scientists move to institutions
with a lower rank ($\Delta{r}>0$), we find that their average change in
performance is negative, corresponding to a decline in the impact of their
work. Yet, what is particularly interesting lies in the $\Delta{r}<0$ regime.
Indeed, when people move from lower rank location to elite institutions, we
observe no performance change on average. This is rather unexpected, as
transitioning from lower-rank institutions to elite institutions is thought to
provide better access to ideas and lab resources, which in turn should fuel
scientific productivity. A possible explanation may be that scientist who have
the opportunity to make big jumps in the ranking space may have already had an
excellent performance in their previous institutions. A move therefore will
not affect their impact.
Figure 6: Impact of movements on career performance. The relation between the
statistical difference of citations ($\Delta c^{*}$) and the ranking
difference ($\Delta{r}$) associated to a transition shows that, when people
move to institutions with a lower rank ($\Delta{r}>0$), their average change
in performance is negative, corresponding to a decline in the impact of their
work. Yet, what is particularly interesting lies in the $\Delta{r}<0$ regime.
Indeed, when people move from lower rank location to elite institutions, we
observe no performance change on average.
## II Discussion
In summary, we extracted affiliation information from the publications of each
scientist, allowing us to reconstruct their career moves between different
institutions as well as the body of work published at each location. We find
career movements are common yet infrequent. Most people move only once or
twice, and usually in the early stage of their career. Career movements are
affected by geography. The distance covered by the move can be approximated
with a power law distribution, indicating that most movements are local and
moving to faraway locations is less probable. We also observe a high degree of
stratification in career movements. People from elite institutions are more
likely to move to other elite institutions, whereas people from lower rank
institutions are more likely to move to places with similar ranks. We further
confirm that the observed stratification is robust against the change in
individual performance before and after the move. When cross-group movement
occurs, we find that while going from elite to lower-rank institutions on
average results in a modest decrease in scientific impact, transitioning into
elite institutions, does not result in gain in impact.
The nature of our dataset restricted our study on a sample of scientists. As a
result of this selection process, our results are biased towards physicists
from 1960s to 1980s with high career longevity. Yet, these limitations also
suggest new avenues for further investigations. Indeed, as datasets become
more comprehensive and of higher resolution, newly available data sources like
Web of Science or Google Scholar can provide new and deeper insights towards
generalization of the results across different disciplines, temporal trends,
and more. Further investigations regarding the influence of career longevity
on scientific mobility should also be considered as it could reveal as well
results of importance. Taken together our results offer the first systematic
empirical evidence on how career moves affect scientific performance and
impact.
## III Method
Dataset. The data provided by the American Physical Society (APS) contains
over $450,000$ publications, each identified with a unique number,
corresponding to all papers published in 9 different journals, namely Physical
Review A, B, C, D, E, I, L, ST and Review of Modern Physics, spanning a period
of 117 years from 1893 to 2010. For each paper the dataset includes title,
date of publication (day,month,year), author names and affiliations of each of
the authors. A separate dataset also provides list of citations within the APS
data only, using unique paper identifiers. About 5% of publications with
ambiguous author-affiliation links or massively authored were removed from
this dataset (see SM for more details).
Author Name Disambiguation. To derive individual information, one has to
reconnect papers belonging to a single scientist. Since no unique author
identifier is present in the data, author names must be disambiguated. The
dataset contains about $1,2$ millions of author-paper pairs. To overcome the
ambiguities present in the data, we design a procedure that uses information
about the author but also metadata about the paper such as coauthors and
citations. By computing similarities between authors, our procedure can
successfully detect single authors as well as homonymies (see SM for more
details about the disambiguation method). A total of $237,038$ distinct
scientists are detected by our method.
Affiliation Disambiguation. A major disadvantage when dealing with publication
data is the inconsistencies and errors associated with affiliation names on
papers. A total of $319,829$ different affiliation names are identified in the
dataset. The disambiguation procedure for affiliations uses geocoded
information as well as a similarity measure between affiliation names in order
to disambiguate institutions. The disambiguated set of authors also plays a
crucial role in the procedure (see SM for more details about the
disambiguation method). A total of $4,052$ distinct institutions are
identified by our algorithm.
Resolving individual career trajectory Based on the information present in the
publications of a scientist, we can reconstruct his/her career trajectory. In
order to detect career movements, i.e. changes in a scientist’s institution,
one has to remove artificial movements induced by short-term stays and by
errors and typos in the affiliation names on the papers. To do so, only
institutions reported in at least two consecutive papers are considered in a
career trajectory.
Ranking the institutions Three variables are considered to rank an
institution: (i) the total number of papers, $P_{i}$, published with
institution $i$, (ii) the cumulated number of citations, $C_{i}$,
corresponding to institution $i$, (iii) the h-index, $H_{i}$, of institution
$i$. The variable $C_{i}$ is defined as $C_{i}=\sum\limits_{k=1}^{P_{i}}c_{k}$
where $c_{k}$ is the number of citations within the APS data of paper $k$
cumulated within $5$ years after its publications. An institution has an
h-index $H$ if $H$ of its $P$ papers have at least $H$ citations each, and the
other $(P-H)$ papers have no more than $H$ citations each. $H$ for papers
indicates the cumulative number of citations obtained within $5$ years after
the publication.
Binning the institutions. About $6,000$ transitions between $1,000$
institutions are detected for our subset of scientists. In order to have a
statistically significant number of transitions to derive the values of
$P(i,j)$ and $M(i,j)$ ( Fig. 5), institutions are binned logarithmically
according to their rank ($r$) into five groups.
## IV Acknowledgments
We thank Nicolas Boumal and colleagues from the Center for Complex Network
Research (CCNR) for the valuable discussions and comments. DW, CS, and ALB are
supported by Lockheed Martin Corporation (SRA 11.18.11), the Network Science
Collaborative Technology Alliance is sponsored by the U.S. Army Research
Laboratory under agreement W911NF-09-2-0053, Defense Advanced Research
Projects Agency under agreement 11645021, and the Future and Emerging
Technologies Project 317 532 ”Multiplex” financed by the European Commission.
PD is supported by the National Fund for Scientific Research (FNRS) and by the
Research Department of the Communauté française de Belgique (Large Graph
Concerted Research Action). RS acknowledges support from the James S.
McDonnell Foundation.
## V Author contributions
PD designed research, analysed the data and wrote the paper. DW, RS, CS, VB
and ALB, designed research and wrote the paper.
## VI Additional Information
Competing financial interests: The authors declare no competing financial
interests.
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|
arxiv-papers
| 2014-04-24T19:59:38 |
2024-09-04T02:50:01.792256
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pierre Deville, Dashun Wang, Roberta Sinatra, Chaoming Song, Vincent\n D. Blondel and Albert-Laszlo Barabasi",
"submitter": "Pierre Deville Pierre",
"url": "https://arxiv.org/abs/1404.6247"
}
|
1404.6375
|
Remarks on remnants by fermions’ tunnelling from black strings
Deyou Chen 111 E-mail: [email protected] and Zhonghua Li 222 E-mail:
[email protected]
Institute of Theoretical Physics, China West Normal University,
Nanchong 637009, China
Abstract: Hawking’s calculation is unable to predict the final stage of the
black hole evaporation. When effects of quantum gravity are taken into
account, there is a minimal observable length. In this paper, we investigate
fermions’ tunnelling from the charged and rotating black strings. With the
influence of the generalized uncertainty principle, the Hawking temperatures
are not only determined by the rings, but also affected by the quantum numbers
of the emitted fermions. Quantum gravity corrections slow down the increases
of the temperatures, which naturally leads to remnants left in the
evaporation.
## 1 Introduction
Hawking radiation is a quantum tunnelling phenomenon of particles across black
holes’ horizons. To describe this phenomenon, the semi-classical tunnelling
method, which relies on calculating the imaginary part of a emission
particle’s action, was put forward [2]. Adopting the WKB approximation, one
can get the relationship between the tunnelling rate and the action of the
classically forbidden trajectory of the particle. Here we adopt the
canonically invariant expression [3, 4, 5]
$\displaystyle\Gamma\propto exp\left[-Im\oint pdr\right].$ (1)
This canonically invariant relation was first derived in [6, 7, 8].
The null geodesic method and the Hamilton-Jacobi method are usual methods
employed to derive the imaginary part [9, 10, 11, 12, 13]. In the null
geodesic method [9], we should first perform the Painleve coordinate
transformation on a metric. Then use canonical momenta and Hamilton canonical
equations to get the imaginary part. When the variable background spacetime is
taken into account, the corrected Hawking temperature is higher than the
standard one. Therefore, the variable background spacetime implies the
accelerated evaporation. The equation of motion of a massive particle is
different from that of the massless one. The former obeys de Broglie wave
function relation. Therefore, the phase velocity of the particle was adopted
to research the tunnelling radiation of massive particles in the subsequent
investigations [14, 15]. The embryonic form of the the Hamilton-Jacobi method
[12] was first found in [10, 11]. In this method, the action satisfies the
Hamilton-Jacobi equation. Taking into account the property of the spacetime,
one carries out separation of variables on the action. Then inserting the
separated variables into the Hamilton-Jacobi equation and solving it, one gets
the imaginary part. Extending this work to the tunnelling radiation of
fermions, the standard Hawking temperatures of the spherically symmetric and
charged black holes were recovered [16]. Other work about fermions’ tunnelling
radiation is referred to [17, 18, 19, 20, 21, 22, 23, 24, 25]. The standard
temperatures were also recovered by anomaly cancellations [26, 27, 28].
Various theories of quantum gravity predict the existence of a minimal
observable length [29, 30, 31, 32, 33]. This length can be implemented in the
model of the generalized uncertainty principle (GUP)
$\displaystyle\Delta x\Delta p\geq\frac{\hbar}{2}\left[1+\beta\Delta
p^{2}\right],$ (2)
where $\beta=\beta_{0}\frac{l^{2}_{p}}{\hbar^{2}}$, $\beta_{0}$ is a
dimensionless parameter and $l_{p}$ is the Planck length. The derivation of
the GUP is relied on the modified fundamental commutation relations. Kempf et.
al. first modified commutation relations [34] and got
$\left[x_{i},p_{j}\right]=i\hbar\delta_{ij}\left[1+\beta p^{2}\right]$, where
$x_{i}$ and $p_{i}$ are operators of position and momentum defined by
$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle x_{0i},$ $\displaystyle
p_{i}$ $\displaystyle=$ $\displaystyle p_{0i}(1+\beta p^{2}),$ (3)
and $x_{0i}$ and $p_{0i}$ satisfy the canonical commutation relations
$\left[x_{0i},p_{0j}\right]=i\hbar\delta_{ij}$.
This modification plays an important role in quantum gravity. With
considerations of modifications, the cosmological constant problem was
discussed and the finiteness of the constant was derived in [35]. Using a new
form of GUP, the Unruh effect has been analyzed in [36]. The quantum dynamics
of the Friedmann-Robertson-Walker universe was gotten in [37]. The related
predictions on post inflation preheating in the cosmology were derived in
[38]. Using the modifications, the thermodynamics of the black holes were
researched in [39, 40, 41] and the tunnelling radiation of scalar particles
was investigated in [42]. In recent work [43], taking into account effects of
quantum gravity, the authors modified the Dirac equation in curved spacetime
and investigated fermions’ tunnelling from the Schwarzschild black hole. They
derived that the quantum correction slows down the increase of the Hawking
temperature, which leads to the remnant.
In this paper, we extend this work to anti de Sitter spacetimes and
investigate the tunnelling radiation of fermions from black strings, where
effects of quantum gravity are taken into account. Black strings are
cylindrically symmetric solutions of the Einstein-Maxwell equations with a
negative cosmological constant. The solutions are asymptotically anti de
Sitter in transverse direction and along the axis. There are three Killing
vectors, $\partial_{t}$, $\partial_{\theta}$, $\partial_{z}$ as the minimal
symmetry. The AdS/CFT correspondence is an important topic in modern physics.
Researches of anti de Sitter spacetimes are helpful to understand this
correspondence. To incorporate effects of quantum gravity, we first modify the
Dirac equation in curved spacetime by operators of position and momentum
defined in [34]. Then adopt the Hamilton-Jacobi method to get the imaginary
parts of the actions. The corrected Hawking temperatures are not only
determined by the mass, charge and angular momentum of the strings, but also
affected by the quantum numbers (charge, angular momentum, mass and energy) of
the emitted fermions. Quantum gravity corrections slow down the increases of
the Hawking temperatures. It is natural to lead to the remnants left in the
evaporation.
The rest is organized as follows. In the next section, using operators of
position and momentum defined in [34], we modify the Dirac equation in curved
spacetime. In Sect. 3, we investigate the tunnelling radiation of charged
fermions from the charged black string. The remnant is observed in the
evaporation. In Sect. 4, the radiation of uncharged fermions in the rotating
black string is discussed. Sect. 5 is devoted to our conclusion.
## 2 Generalized Dirac equation
Here we adopt the modified fundamental commutation relation put forward in
[34] to modify the Dirac equation in curved spacetime. Using Eq. (3), the
square of momentum operators is gotten as
$\displaystyle p^{2}$ $\displaystyle=$ $\displaystyle
p_{i}p^{i}=-\hbar^{2}\left[{1-\beta\hbar^{2}\left({\partial_{j}\partial^{j}}\right)}\right]\partial_{i}\cdot\left[{1-\beta\hbar^{2}\left({\partial^{j}\partial_{j}}\right)}\right]\partial^{i}$
(4) $\displaystyle\simeq$
$\displaystyle-\hbar^{2}\left[{\partial_{i}\partial^{i}-2\beta\hbar^{2}\left({\partial^{j}\partial_{j}}\right)\left({\partial^{i}\partial_{i}}\right)}\right].$
The higher order terms of $\beta$ are neglected in the last step. In the
theory of quantum gravity, the generalized frequency is found as [44]
$\displaystyle\tilde{\omega}=E(1-\beta E^{2}),$ (5)
where $E$ is the energy operator and defined as $E=i\hbar\partial_{t}$. From
the energy mass shell condition $p^{2}+m^{2}=E^{2}$, the generalized
expression of the energy was derived [42, 44, 45, 46]. It is
$\displaystyle\tilde{E}=E[1-\beta(p^{2}+m^{2})].$ (6)
The generalized Dirac equation without considerations of electromagnetic
effects in the flat spacetime has been derived in [45] by the consequence of
the GUP. In curved spacetime, the Dirac equation with an electromagnetic field
takes on the form
$\displaystyle
i\gamma^{\mu}\left(\partial_{\mu}+\Omega_{\mu}+\frac{i}{\hbar}eA_{\mu}\right)\Psi+\frac{m}{\hbar}\Psi=0,$
(7)
where $\Omega_{\mu}\equiv\frac{i}{2}\omega_{\mu}\,^{ab}\Sigma_{ab}$,
$\omega_{\mu}\,^{ab}$ is the spin connection defined by the tetrad
$e^{\lambda}\,_{b}$ and ordinary connection
$\displaystyle\omega_{\mu}\,^{a}\,{}_{b}=e_{\nu}\,^{a}e^{\lambda}\,_{b}\Gamma^{\nu}_{\mu\lambda}-e^{\lambda}\,_{b}\partial_{\mu}e_{\lambda}\,^{a}.$
(8)
The Latin indices live in the flat metric $\eta_{ab}$ while Greek indices are
raised and lowered by the curved metric $g_{\mu\nu}$. The tetrad can be
constructed from
$\displaystyle g_{\mu\nu}=e_{\mu}\,^{a}e_{\nu}\,^{b}\eta_{ab},\hskip
14.22636pt\eta_{ab}=g_{\mu\nu}e^{\mu}\,_{a}e^{\nu}\,_{b},\hskip
14.22636pte^{\mu}\,_{a}e_{\nu}\,^{a}=\delta^{\mu}_{\nu},\hskip
14.22636pte^{\mu}\,_{a}e_{\mu}\,^{b}=\delta_{a}^{b}.$ (9)
In equation (7), $\Sigma_{ab}$ is the Lorentz spinor generators defined by
$\Sigma_{ab}=\frac{i}{4}\left[{\gamma^{a},\gamma^{b}}\right],\hskip
14.22636pt\\{\gamma^{a},\gamma^{b}\\}=2\eta^{ab}.$ (10)
Then one can construct $\gamma^{\mu}$’s in the curved spacetime as
$\gamma^{\mu}=e^{\mu}\,_{a}\gamma^{a},\hskip
19.91692pt\left\\{{\gamma^{\mu},\gamma^{\nu}}\right\\}=2g^{\mu\nu}.$ (11)
To get the generalized Dirac equation in the curved spacetime, we rewrite Eq.
(7) as
$\displaystyle-i\gamma^{0}\partial_{0}\Psi=\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\Psi.$
(12)
Using Eqs. (4), (6) and (12) and neglecting the higher order terms of $\beta$,
we get [42, 44, 45, 46]
$\displaystyle-i\gamma^{0}\partial_{0}\Psi=\left(i\gamma^{i}\partial_{i}+i\gamma^{\mu}\Omega_{\mu}+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}+\frac{m}{\hbar}\right)\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\Psi,$ (13)
which is rewritten as
$\displaystyle\left[i\gamma^{0}\partial_{0}+i\gamma^{i}\partial_{i}\left(1-\beta
m^{2}\right)+i\gamma^{i}\beta\hbar^{2}\left(\partial_{j}\partial^{j}\right)\partial_{i}+\frac{m}{\hbar}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right.$
$\displaystyle\left.+i\gamma^{\mu}\frac{i}{\hbar}eA_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)+i\gamma^{\mu}\Omega_{\mu}\left(1+\beta\hbar^{2}\partial_{j}\partial^{j}-\beta
m^{2}\right)\right]\Psi=0.$ (14)
Thus the generalized Dirac equation is derived. When $A_{\mu}=0$, it describes
a equation without electromagnetic fields. In the following sections, we adopt
Eq. (14) to describe fermions tunnelling from the charged and rotating black
strings.
## 3 Fermions’ tunnelling from a charged black string
The 4-dimensional neutral black string solutions to Einstein-Maxwell equations
with a negative cosmological constant were derived in [47]. Subsequently, the
general solutions with electric charges were gotten [48]. In this section, we
investigate charged fermions’ tunnelling from a cylindrically symmetric black
string. The black string solution is given by [48]
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(r)dt^{2}+\frac{1}{g(r)}dr^{2}+r^{2}d{\theta}^{2}+\alpha^{2}r^{2}dz^{2},$
(15)
with the electromagnetic potential
$\displaystyle A_{\mu}=\left(A_{t},0,0,0\right)=\left(\frac{2Q}{\alpha
r},0,0,0\right),$ (16)
where $f(r)=g(r)=\alpha^{2}r^{2}-\frac{4M}{\alpha
r}+\frac{4Q^{2}}{\alpha^{2}r^{2}}$, $0\leq\theta\leq 2\pi$,
$\alpha^{2}=-\frac{\Lambda}{3}$, $\Lambda$ is the negative cosmology constant.
$M$ and $Q$ are the ADM mass and charge per unit length in the $z$ direction,
respectively. The above spacetime is asymptotically anti-de Sitter in the
transverse directions and string directions. The singularity at $r=0$ is
enclosed by the horizon $r_{+}$ if the condition
$Q^{2}\leq\frac{3}{4}M^{\frac{4}{3}}$ holds. The event horizon $r_{+}$ is
located at
$\displaystyle
r_{+}=\frac{1}{2}\left[\sqrt{2R}+\left(-2R+\frac{8M}{\alpha^{3}\sqrt{2R}}\right)\right],$
(17)
where
$\displaystyle
R=\left[\frac{M^{2}}{\alpha^{6}}+\left(\left(\frac{M^{2}}{\alpha^{6}}\right)^{2}-\left(\frac{4Q^{2}}{3\alpha^{4}}\right)^{3}\right)^{\frac{1}{2}}\right]^{\frac{1}{3}}+\left[\frac{M^{2}}{\alpha^{6}}-\left(\left(\frac{M^{2}}{\alpha^{6}}\right)^{2}-\left(\frac{4Q^{2}}{3\alpha^{4}}\right)^{3}\right)^{\frac{1}{2}}\right]^{\frac{1}{3}}.$
(18)
The metric (15) describes a neutral black string solution when $Q=0$.
For a spin-1/2 fermion, there are two states corresponding to spin up and spin
down. Here we only investigate the state with spin up. The investigation of
the state with spin down is parallel and the same result can be obtained. To
describe the motion of a charge fermion, we suppose that the wave function
takes on the form
$\displaystyle\Psi=\left(\begin{array}[]{c}A\\\ 0\\\ B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I\left(t,r,\theta,z\right)\right),$
(23)
where $A$ and $B$ are functions of $t,r,\theta,z$, and $I$ is the action of
the fermion with spin up state. To find gamma matrices, we should first
construct a tetrad. It is straightforward to construct a tetrad from the
metric (15). The tetrad is
$\displaystyle e_{\mu}\,^{a}=\rm{diag}\left(\sqrt{f},1/\sqrt{g},r,\alpha
r\right).$ (24)
Then gamma matrices are gotten as
$\displaystyle\gamma^{t}=\frac{1}{\sqrt{f\left(r\right)}}\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right),$
$\displaystyle\gamma^{\theta}=\sqrt{g^{\theta\theta}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (29)
$\displaystyle\gamma^{r}=\sqrt{g\left(r\right)}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),$
$\displaystyle\gamma^{z}=\sqrt{g^{zz}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right).$ (34)
In the above equations, $\sqrt{g^{\theta\theta}}=\frac{1}{r}$ and
$\sqrt{g^{zz}}=\frac{1}{\alpha r}$. To apply the WKB approximation, we insert
the wave function and the gamma matrices into the generalized Dirac equation.
Then divide by the exponential term and multiply by $\hbar$. The resulting
equation to leading order in $\hbar$ is derived and decoupled into four
equations
$\displaystyle-iA\frac{1}{\sqrt{f}}\partial_{t}I-B\left(1-\beta
m^{2}\right)\sqrt{g}\partial_{r}I-Am\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]$
$\displaystyle+B\beta\sqrt{g}\partial_{r}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]+Am\left(1-\beta
m^{2}\right)$ $\displaystyle-iA\frac{eA_{t}}{\sqrt{f}}\left[1-\beta
m^{2}-\left(g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right)\right]=0,$
(35)
$\displaystyle iB\frac{1}{\sqrt{f}}\partial_{t}I-A\left(1-\beta
m^{2}\right)\sqrt{g}\partial_{r}I-Bm\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]$
$\displaystyle+A\beta\sqrt{g}\partial_{r}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]+Bm\left(1-\beta
m^{2}\right)$ $\displaystyle+iB\frac{eA_{t}}{\sqrt{f}}\left[1-\beta
m^{2}-\left(g^{rr}\left(\partial_{r}I\right)^{2}+g^{\theta\theta}\left(\partial_{\theta}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right)\right]=0,$
(36)
$\displaystyle A\left\\{-\left(1-\beta
m^{2}\right)\sqrt{g^{\theta\theta}}\partial_{\theta}I+\beta\sqrt{g^{\theta\theta}}\partial_{\theta}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{zz}(\partial_{z}I)^{2}\right]\right.$
$\displaystyle\left.-i\left(1-\beta
m^{2}\right)\sqrt{g^{zz}}\partial_{z}I+i\beta\sqrt{g^{zz}}\partial_{z}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{zz}(\partial_{z}I)^{2}\right]\right\\}=0.$
(37)
$\displaystyle B\left\\{-\left(1-\beta
m^{2}\right)\sqrt{g^{\theta\theta}}\partial_{\theta}I+\beta\sqrt{g^{\theta\theta}}\partial_{\theta}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{zz}(\partial_{z}I)^{2}\right]\right.$
$\displaystyle\left.-i\left(1-\beta
m^{2}\right)\sqrt{g^{zz}}\partial_{z}I+i\beta\sqrt{g^{zz}}\partial_{z}I\left[g^{rr}(\partial_{r}I)^{2}+g^{\theta\theta}(\partial_{\theta}I)^{2}+g^{zz}(\partial_{z}I)^{2}\right]\right\\}=0.$
(38)
Obviously, it is difficult to get the solution of the action $I$ from the
above equations. However, the action can be separated by the property of the
black string. Considering the Killing vectors of the spacetime, the author
separated the action as $I=-\omega t+W(r)+l\theta+Jz$ [18], where $\omega$ is
the energy of the emitted fermion. From the above four equations, we carry out
separation of variables as
$\displaystyle I=-\omega t+W(r)+\Theta(\theta,z).$ (39)
We first observe Eqs. (37) and (38) and find that they are irrelevant to $A$
and $B$ and can be reduced to the same equation. Inserting Eq. (39) into Eqs.
(37) and (38) yields
$\displaystyle\left(\sqrt{g^{\theta\theta}}\partial_{\theta}\Theta+i\sqrt{g^{zz}}\partial_{z}\Theta\right)\left[1-\beta
m^{2}-\beta g^{rr}(\partial_{r}W)^{2}-\beta
g^{\theta\theta}(\partial_{\theta}\Theta)^{2}-\beta
g^{zz}(\partial_{z}\Theta)^{2}\right]=0.$ (40)
In the above equation, the summation of factors in the square brackets can not
be zero. Therefore, it should be
$\displaystyle\sqrt{g^{\theta\theta}}\partial_{\theta}\Theta+i\sqrt{g^{zz}}\partial_{z}\Theta=0,$
(41)
which yields a complex function solution (other than the trivial constant
solution) of $\Theta$. However, this solution has no contribution to the
tunnelling rate. Therefore, we will not consider its contribution in the
calculation. Another important relation predicted by Eq. (41) is
$g^{\theta\theta}(\partial_{\theta}\Theta)^{2}+g^{zz}(\partial_{z}\Theta)^{2}=0$.
Now we focus our attention on the first two equations. Inserting Eq. (39) into
Eqs. (35) and (36) and canceling $A$ and $B$ yield
$\displaystyle
A_{6}\left({\partial_{r}W}\right)^{6}+A_{4}\left({\partial_{r}W}\right)^{4}+A_{2}\left({\partial_{r}W}\right)^{2}+A_{0}=0,$
(42)
where
$\displaystyle A_{6}$ $\displaystyle=$ $\displaystyle\beta^{2}g^{3}f,$
$\displaystyle A_{4}$ $\displaystyle=$ $\displaystyle\beta
g^{2}f\left(m^{2}\beta-2\right)-\beta^{2}g^{2}e^{2}A_{t}^{2},$ $\displaystyle
A_{2}$ $\displaystyle=$ $\displaystyle gf\left(1-\beta
m^{2}\right)\left(1+\beta m^{2}\right)-2\beta geA_{t}[\omega-eA_{t}(1-\beta
m^{2})],$ $\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle-m^{2}f\left(1-\beta m^{2}\right)^{2}-\left[\omega-
eA_{t}\left(1-\beta m^{2}\right)\right]^{2}.$ (43)
Neglect the higher order terms of $\beta$ and solve Eq. (42) at the event
horizon. Thus the imaginary part of the radial action is
$\displaystyle ImW_{\pm}$ $\displaystyle=$
$\displaystyle\pm\int\frac{dr}{\sqrt{gf}}\sqrt{\left[\omega-eA_{t}(1-\beta
m^{2})\right]^{2}+m^{2}f}\left(1+\beta
m^{2}+\beta\frac{\tilde{\omega}_{0}^{2}-eA_{t}\tilde{\omega}_{0}}{f}\right)$
(44) $\displaystyle=$ $\displaystyle\pm\pi\frac{\omega-
eA_{t+}}{f^{\prime}}\left(1+\beta\xi\right),$
where $+(-)$ denote the outgoing (ingoing) solutions,
$f^{\prime}=2\alpha^{2}r_{+}+\frac{4M}{\alpha
r_{+}^{2}}-\frac{8Q^{2}}{\alpha^{2}r_{+}^{3}}$,
$\xi=\frac{3}{2}m^{2}+\frac{2m^{2}+1}{2\omega_{0}}+\frac{2eA_{t+}}{f^{\prime}r_{+}}-\frac{2}{3}\frac{\omega_{0}}{f^{\prime}r_{+}}$,
$\tilde{\omega}_{0}=\omega-eA_{t}$, $\omega_{0}=\omega-eA_{t+}$,
$A_{t+}=\frac{2Q}{\alpha r_{+}}$ is the electromagnetic potential at the event
horizon. Using the relation of the roots of $f=0$, it is easily proved that
$\xi>0$. The tunnelling rate of the charged fermion at the event horizon is
$\displaystyle\Gamma$ $\displaystyle\propto$ $\displaystyle exp[-Im\oint
p_{r}dr]=exp\left[-Im\left(\int p_{r}^{out}dr-\int p_{r}^{in}dr\right)\right]$
(45) $\displaystyle=$ $\displaystyle exp\left[\mp 2Im\int
p_{r}^{out,in}dr\right].$
Here $p_{r}=\partial_{r}W$, and $out(in)$ correspond to $+(-)$. Thus the
tunnelling rate is gotten as
$\displaystyle\Gamma\propto\exp\left[-2\pi\frac{\omega-
eA_{t+}}{f^{\prime}}\left(1+\beta\xi\right)\right].$ (46)
However, the temporal contribution to the tunneling amplitude was missed in
the above calculation [3, 6, 7, 8]. We use Kruskal coordinates $(T,R)$ to find
this temporal contribution. The region exterior is described by
$\displaystyle T$ $\displaystyle=$ $\displaystyle e^{\kappa r_{*}}sinh(\kappa
t),$ $\displaystyle R$ $\displaystyle=$ $\displaystyle e^{\kappa
r_{*}}cosh(\kappa t),$ (47)
where $r_{*}=r+\frac{1}{2\kappa}ln\frac{r-r_{+}}{r_{+}}$ is the tortoise
coordinate, and $\kappa$ is the surface gravity. The interior region is given
by
$\displaystyle T$ $\displaystyle=$ $\displaystyle e^{\kappa r_{*}}cosh(\kappa
t),$ $\displaystyle R$ $\displaystyle=$ $\displaystyle e^{\kappa
r_{*}}sinh(\kappa t).$ (48)
To find the temporal contribution, we connect these two patches across the
horizon. Rotate the time $t$ as $t\rightarrow t-\frac{\pi}{2}i\kappa$. As
pointed in [3, 4, 5], this rotation would lead to an additional imaginary
contribution coming from the temporal part, namely, $Im[(\omega-eA_{t+})\Delta
t^{out,in}]=\frac{1}{2}\pi(\omega-eA_{t+})\kappa$. So the total temporal
contribution is $Im[(\omega-eA_{t+})\Delta t]=\pi(\omega-eA_{t+})\kappa$.
Therefore, the tunnelling rate with the consideration of the temporal
contribution is
$\displaystyle\Gamma$ $\displaystyle\propto$ $\displaystyle
exp\left[-\frac{1}{\hbar}\left(Im((\omega-eA_{t+})\Delta t)+Im\oint
p_{r}dr\right)\right]$ (49) $\displaystyle=$
$\displaystyle\exp\left[-4\pi\frac{\omega-
eA_{t+}}{f^{\prime}}\left(1+\frac{1}{2}\beta\xi\right)\right].$
This is the Boltzmann factor with the Hawking temperature at the event horizon
taking
$\displaystyle
T=\frac{f^{\prime}}{4\pi\left(1+\beta\xi\right)}=T_{0}\left(1-\frac{1}{2}\beta\xi\right),$
(50)
where $T_{0}=\frac{1}{2\pi}\left(\alpha^{2}r_{+}+\frac{2M}{\alpha
r_{+}^{2}}-\frac{4Q^{2}}{\alpha^{2}r_{+}^{3}}\right)$ is the standard Hawking
temperature of the black string.
It is shown that the corrected temperature appears and is lower than the
standard one. The correction is not only determined by the mass and charge of
the black string, but also affected by the quantum number (mass, charge and
energy) of the emitted fermion. Quantum gravity correction slows down the
increase of the Hawking temperature caused by the evaporation. Finally, the
black string is in a balance state. At this state, the evaporation stops and
the remnant is produced.
It is of interest to discuss the corrected area entropy. The entropy can be
derived by the first law of thermodynamics with the corrected temperature
(50). However, the expression is complicated, so we don’t write it here. The
corrected temperature were also gotten in[49, 50, 51, 52, 53, 54]. When
$\beta=0$, the standard Hawking temperature is recovered [18, 48].
## 4 Fermions’ tunnelling from a rotating black string
In this section, we investigate uncharged fermions’ tunnelling from the event
horizon of a rotating black string. Therefore, effects of the electromagnetic
field in the generalized Dirac equation are not taken into account here. The
rotating black string solution in a spacetime asymptotically anti de Sitter in
the radial direction was derived by Lemos [55]. The solution is
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(\alpha^{2}r^{2}-\frac{4M\left(1-\frac{a^{2}\alpha^{2}}{2}\right)}{\alpha
r}\right)dt^{2}+\left(\alpha^{2}r^{2}-\frac{4M\left(1-\frac{3}{2}a^{2}\alpha^{2}\right)}{\alpha
r}\right)^{-1}dr^{2}$ (51)
$\displaystyle-\frac{8Ma\sqrt{1-\frac{a^{2}\alpha^{2}}{2}}}{\alpha
r}dtd\varphi+\left(r^{2}-\frac{4Ma^{2}}{\alpha
r}\right)d\varphi^{2}+\alpha^{2}r^{2}dz^{2},$
where $\alpha^{2}=-\frac{\Lambda}{3}$, $\Lambda$ is the negative cosmological
constant, $a$ is the angular momentum per unit mass. It is defined that
$a^{2}\alpha^{2}=1-\frac{\epsilon}{M}$ and
$\epsilon=\sqrt{M^{2}-\frac{8J^{2}\alpha^{2}}{9}}$. $M$ and $J$ are the mass
and angular momentum line densities of the spacetime, respectively. The
relation between $J$ and $a$ is given by
$J=\frac{3}{2}Ma\sqrt{1-\frac{a^{2}\alpha^{2}}{2}}$. For convenience of the
investigation, the metric (51) is rewritten as
$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-\Delta\left(\gamma
dt-\frac{\delta}{\alpha^{2}}d\varphi\right)^{2}+r^{2}\left(\gamma
d\varphi-\delta dt\right)^{2}+\frac{dr^{2}}{\Delta}+\alpha^{2}r^{2}dz^{2},$
(52)
where
$\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\alpha^{2}r^{2}-\frac{b}{\alpha r},\quad
b=4M\left(1-\frac{3a^{2}\alpha^{2}}{2}\right),$ $\displaystyle\gamma$
$\displaystyle=$
$\displaystyle\sqrt{\frac{2-a^{2}\alpha^{2}}{2-3a^{2}\alpha^{2}}},\quad\delta=\frac{a\alpha^{2}}{\sqrt{1-\frac{3}{2}a^{2}\alpha^{2}}}.$
(53)
The event horizon is located at $r_{+}=\alpha^{-1}b^{\frac{1}{3}}$ which is
given for $\Delta=0$. To describe the fermion’s tunnelling from the event
horizon, one can directly construct a tetrad and gamma matrices from the
metric (52). For simplicity to construct the tetrad and gamma matrices, we
perform the dragging coordinate transformation
$\displaystyle\varphi=\phi+\Omega t,\hskip
14.22636pt\Omega=\frac{-\Delta\gamma\delta\alpha^{2}+r^{2}\gamma\delta\alpha^{4}}{-\Delta\delta^{2}+r^{2}\gamma^{2}\alpha^{4}},$
(54)
on the metric (52) and get
$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-F(r)dt^{2}+\frac{1}{G(r)}dr^{2}+g_{\phi\phi}d\phi^{2}+g_{zz}dz^{2}$
(55) $\displaystyle=$ $\displaystyle-\frac{\Delta
r^{2}\left(\alpha^{2}\gamma^{2}-\delta^{2}\right)^{2}}{-\Delta\delta^{2}+\alpha^{4}r^{2}\gamma^{2}}dt^{2}+\frac{1}{\Delta}dr^{2}+\left(-\frac{\Delta\delta^{4}}{\alpha^{4}}+r^{2}\gamma^{2}\right)d\phi^{2}+\alpha^{2}r^{2}dz^{2}.$
Here we still only investigate the state with spin up. Assume that the wave
function of the fermion with spin up state shares the similar expression as Eq
. (23), namely,
$\displaystyle\Psi=\left(\begin{array}[]{c}A\\\ 0\\\ B\\\
0\end{array}\right)\exp\left(\frac{i}{\hbar}I\left(t,r,\phi,z\right)\right).$
(60)
The tetrad is easily constructed as
$\displaystyle
e_{\mu}\,^{a}=\rm{diag}\left(\sqrt{F},1/\sqrt{G},\sqrt{g_{\phi\phi}},\sqrt{g_{zz}}\right).$
(61)
Now gamma matrices take on the form as
$\displaystyle\gamma^{t}=\frac{1}{\sqrt{F\left(r\right)}}\left(\begin{array}[]{cc}i&0\\\
0&-i\end{array}\right),$
$\displaystyle\gamma^{\phi}=\sqrt{g^{\phi\phi}}\left(\begin{array}[]{cc}0&\sigma^{1}\\\
\sigma^{1}&0\end{array}\right),$ (66)
$\displaystyle\gamma^{r}=\sqrt{G\left(r\right)}\left(\begin{array}[]{cc}0&\sigma^{3}\\\
\sigma^{3}&0\end{array}\right),$
$\displaystyle\gamma^{z}=\sqrt{g^{zz}}\left(\begin{array}[]{cc}0&\sigma^{2}\\\
\sigma^{2}&0\end{array}\right).$ (71)
In the above equations,
$g^{\phi\phi}=\frac{\alpha^{4}}{-\Delta\delta^{4}+\alpha^{4}r^{2}\gamma^{2}}$,
$g^{zz}=\frac{1}{\alpha^{2}r^{2}}$. Inserting the wave function and the gamma
matrices into the generalized Dirac equation and adopting the same process as
the above section, we get
$\displaystyle-iA\frac{1}{\sqrt{F}}\partial_{t}I-B\left(1-\beta
m^{2}\right)\sqrt{G}\partial_{r}I-Am\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]$
$\displaystyle+B\beta\sqrt{G}\partial_{r}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]+Am\left(1-\beta
m^{2}\right)=0,$ (72)
$\displaystyle iB\frac{1}{\sqrt{F}}\partial_{t}I-A\left(1-\beta
m^{2}\right)\sqrt{G}\partial_{r}I-Bm\beta\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]$
$\displaystyle+A\beta\sqrt{G}\partial_{r}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]+Bm\left(1-\beta
m^{2}\right)=0,$ (73)
$\displaystyle A\left\\{-\left(1-\beta
m^{2}\right)\sqrt{g^{\phi\phi}}\partial_{\phi}I+\beta\sqrt{g^{\phi\phi}}\partial_{\phi}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]\right.$
$\displaystyle\left.-i\left(1-\beta
m^{2}\right)\sqrt{g^{zz}}\partial_{z}I+i\beta\sqrt{g^{zz}}\partial_{z}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]\right\\}=0,$
(74)
$\displaystyle B\left\\{-\left(1-\beta
m^{2}\right)\sqrt{g^{\phi\phi}}\partial_{\phi}I+\beta\sqrt{g^{\phi\phi}}\partial_{\phi}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}\right]\right.$
$\displaystyle\left.-i\left(1-\beta
m^{2}\right)\sqrt{g^{zz}}\partial_{z}I+i\beta\sqrt{g^{zz}}\partial_{z}I\left[g^{rr}\left(\partial_{r}I\right)^{2}+g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}\right]+g^{zz}\left(\partial_{z}I\right)^{2}\right\\}=0.$
(75)
It is also difficult to solve the action $I$ from the above equations. We
first observe the last two equations. They can be reduced into the same
equation and yields
$\sqrt{g^{\phi\phi}}\partial_{\phi}I+i\sqrt{g^{zz}}\partial_{z}I=0$. This
implies
$\displaystyle
g^{\phi\phi}\left(\partial_{\phi}I\right)^{2}+g^{zz}\left(\partial_{z}I\right)^{2}=0.$
(76)
Now our interest is the first two equations which determine the Hawking
temperature of the black string. Considering the properties of the metrics
(51) and (55), we carry out separation of variables as
$\displaystyle I=-\left(\omega-j\Omega\right)t+W\left(r,z\right)+j\phi,$ (77)
where $\omega$ and $j$ are the energy and angular momentum of the emitted
fermion, respectively. Inserting Eq. (77) into Eqs. (72) and (73) and
canceling $A$ and $B$ yield
$\displaystyle
B_{6}\left({\partial_{r}W}\right)^{6}+B_{4}\left({\partial_{r}W}\right)^{4}+B_{2}\left({\partial_{r}W}\right)^{2}+B_{0}=0,$
(78)
where
$\displaystyle B_{6}$ $\displaystyle=$ $\displaystyle\beta^{2}G^{3}F,$
$\displaystyle B_{4}$ $\displaystyle=$ $\displaystyle\beta
G^{2}F\left(m^{2}\beta-2\right),$ $\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle GF\left[\left(1-\beta m^{2}\right)^{2}+2\beta
m^{2}\left(1-m^{2}\beta\right)\right],$ $\displaystyle B_{0}$ $\displaystyle=$
$\displaystyle-m^{2}\left(1-\beta
m^{2}\right)^{2}F-\left(\omega-j\Omega\right)^{2}.$ (79)
Neglecting the higher order terms of $\beta$ and solving Eq. (78) at the event
horizon, we get the solution of $W$. Thus the imaginary part of $W$ is
$\displaystyle ImW_{\pm}$ $\displaystyle=$ $\displaystyle\pm\int
dr\sqrt{\frac{m^{2}F+\left(\omega-j\Omega\right)^{2}}{GF}}\left(1+\beta
m^{2}+\beta\frac{\left(\omega-j\Omega\right)^{2}}{F}\right)$ (80)
$\displaystyle=$
$\displaystyle\pm\pi\frac{\left(\omega-j\Omega_{+}\right)\alpha^{4}r_{+}^{2}\gamma}{\left(2\alpha^{4}r_{+}^{3}+b\right)\left(\alpha^{2}\gamma^{2}-\delta^{2}\right)}\left(1+\beta\chi\right),$
where $+(-)$ are the outgoing (ingoing) solutions,
$\chi=\frac{3}{2}m^{2}+\frac{3j\left(\omega-j\Omega_{+}\right)\delta}{\left(\alpha^{2}\gamma^{2}-\delta^{2}\right)\gamma
r_{+}^{2}}-\frac{3}{2}\frac{\left(\omega-j\Omega_{+}\right)^{2}\delta^{2}}{\left(\alpha^{2}\gamma^{2}-\delta^{2}\right)r_{+}^{2}}$,
$\Omega_{+}=\delta/\gamma$ is the angular velocity at the event horizon. It is
not difficult to prove that $\chi>0$.
To find the temporal contribution, we use the Kruskal coordinates $(T,R)$. The
region exterior to the string $(r>r_{+})$ is described by
$\displaystyle T$ $\displaystyle=$ $\displaystyle
e^{\kappa_{+}r_{*}}sinh(\kappa_{+}t),$ $\displaystyle R$ $\displaystyle=$
$\displaystyle e^{\kappa_{+}r_{*}}cosh(\kappa_{+}t),$ (81)
where $r_{*}=r+\frac{1}{2\kappa_{+}}ln\frac{r-r_{+}}{r_{+}}$, and $\kappa_{+}$
denote the surface gravity. The interior region is
$\displaystyle T$ $\displaystyle=$ $\displaystyle
e^{\kappa_{+}r_{*}}cosh(\kappa_{+}t),$ $\displaystyle R$ $\displaystyle=$
$\displaystyle e^{\kappa_{+}r_{*}}sinh(\kappa_{+}t).$ (82)
Adopt the same process as the above section, we get the total temporal
contribution is $Im[(\omega-j\Omega_{+})\Delta
t]=\pi(\omega-j\Omega_{+})\kappa_{+}$. Therefore, the tunnelling rate is
$\displaystyle\Gamma$ $\displaystyle\propto$ $\displaystyle
exp\left[-\frac{1}{\hbar}\left(Im((\omega-j\Omega_{+})\Delta t)+Im\oint
p_{r}dr\right)\right]$ (83) $\displaystyle=$
$\displaystyle\exp\left[-\frac{4\pi\left(\omega-j\Omega_{+}\right)\alpha^{4}r_{+}^{2}\gamma}{\left(2\alpha^{4}r_{+}^{3}+b\right)\left(\alpha^{2}\gamma^{2}-\delta^{2}\right)}\left(1+\frac{1}{2}\beta\chi\right)\right].$
Eq. (83) is the Boltzmann factor of the Hawking temperature at the event
horizon taking
$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{\left(2\alpha^{4}r_{+}^{3}+b\right)\left(\alpha^{2}\gamma^{2}-\delta^{2}\right)}{4\pi\alpha^{4}r_{+}^{2}\gamma\left(1+\frac{1}{2}\beta\chi\right)}=T_{0}\left(1-\frac{1}{2}\beta\chi\right),$
(84)
where
$T_{0}=\frac{\left(2\alpha^{4}r_{+}^{3}+b\right)\left(\alpha^{2}\gamma^{2}-\delta^{2}\right)}{4\pi\alpha^{4}r_{+}^{2}\gamma}$
is the standard Hawking temperature. Obviously, the corrected Hawking
temperature is lower than the standard one. The correction is related not only
to the mass and angular momentum of the black string but also to the quantum
number (mass, angular momentum and energy) of the emitted fermion. Due to
$\chi>0$, there is a balance point. At this point, the evaporation stops and
the remnant is left.
## 5 Conclusion
In this paper, taking into account the influence of quantum gravity, we
modified the Dirac equation in curved spacetime by the modified fundamental
commutation relations put forward in [34]. Then the tunnelling radiation of
fermions from the event horizons of the charged and rotating black strings was
investigated. The corrected Hawking temperatures were gotten. In the charged
spacetime, the correction is related not only to the mass and charge of the
black string but also to the quantum number (mass, charge and energy) of the
emitted fermion. In the rotating spacetime, the quantum number (mass, angular
momentum and energy) of the emitted fermion and the mass and angular momentum
of the black string affect the Hawking temperature at the same time. Due to
the quantum gravity corrections, the evaporation of the black strings slows
down. Finally, the evaporation stops and the remnants are left. The remnants
in the final state were also discussed in [56, 57, 58, 59]. It is of interest
to discuss remnants. A review of this topic can be found in [60].
When $\beta=0$, the standard Hawking temperatures are recovered.
Acknowledgments
This work is supported by the National Natural Science Foundation of China
with Grant No. 11205125.
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|
arxiv-papers
| 2014-04-25T10:12:01 |
2024-09-04T02:50:01.808775
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Deyou Chen and Zhonghua Li",
"submitter": "Deyou Chen",
"url": "https://arxiv.org/abs/1404.6375"
}
|
1404.6383
|
# Bloscpack: a compressed lightweight serialization format for numerical data
Valentin Haenel∗† * Corresponding author: [email protected]†
IndependentCopyright © 2014 Valentin Haenel. This is an open-access article
distributed under the terms of the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium,
provided the original author and source are credited.
http://creativecommons.org/licenses/by/3.0/
###### Abstract
This paper introduces the Bloscpack file format and the accompanying Python
reference implementation. Bloscpack is a lightweight, compressed binary file-
format based on the Blosc codec and is designed for lightweight, fast
serialization of numerical data. This article presents the features of the
file-format and some some API aspects of the reference implementation, in
particular the ability to handle Numpy ndarrays. Furthermore, in order to
demonstrate its utility, the format is compared both feature- and performance-
wise to a few alternative lightweight serialization solutions for Numpy
ndarrays. The performance comparisons take the form of some comprehensive
benchmarks over a range of different artificial datasets with varying size and
complexity, the results of which are presented as the last section of this
article.
###### Index Terms:
applied information theory, compression/decompression, python, numpy, file
format, serialization, blosc
## 1 Introduction
When using compression during storage of numerical data there are two
potential improvements one can make. First, by using compression, naturally
one can save storage space. Secondly—and this is often overlooked—one can save
time. When using compression during serialization, the total compression time
is the sum of the time taken to perform the compression and the time taken to
write the compressed data to the storage medium. Depending on the compression
speed and the compression ratio, this sum maybe less than the time taken to
serialize the data in uncompressed format i.e.
$write_{uncompressed}>write_{compressed}+time_{compress}$
The Bloscpack file format and Python reference implementation aims to achieve
exactly this by leveraging the fast, multithreaded, blocking and shuffling
Blosc codec.
## 2 Blosc
Blosc [Blosc] is a fast, multitreaded, blocking and shuffling compressor
designed initially for in-memory compression. Contrary to many other available
compressors which operate sequentially on a data buffer, Blosc uses the
blocking technique [Alted2009, Alted2010] to split the dataset into individual
blocks. It can then operate on each block using a different thread which
effectively leads to a multithreaded compressor. The block size is chosen such
that it either fits into a typical L1 cache (for compression levels up to 6)
or L2 cache (for compression levels larger than 6). In modern CPUs L1 and L2
are typically non-shared between other cores, and so this choice of block size
leads to an optimal performance during multi-thread operation.
Also, Blosc features a shuffle filter [Alted2009] (p.71) which may reshuffle
multi-byte elements, e.g. 8 byte doubles, by significance. The net result for
series of numerical elements with little difference between elements that are
close, is that similar bytes are placed closer together and can thus be better
compressed (this is specially true on time series datasets). Internally, Blosc
uses its own codec, _blosclz_ , which is a derivative of FastLZ [FastLZ] and
implements the LZ77 [LZ77] scheme. The reason for Blosc to introduce its own
codec is mainly the desire for simplicity (blosclz is a highly streamlined
version of FastLZ), as well as providing a better interaction with Blosc
infrastructure.
Moreover, Blosc is designed to be extensible, and allows other codecs than
blosclz to be used in it. In other words, one can consider Blosc as a meta-
compressor, in that it handles the splitting of the data into blocks,
optionally applying the shuffle filter (or other future filters), while being
responsible of coordinating the individual threads during operation. Blosc
then relies on a "real" codec to perform that actual compression of the data
blocks. As such, one can think of Blosc as a way to parallelize existing
codecs, while allowing to apply filters (also called pre-conditioners). In
fact, at the time when the research presented in this paper was conducted
(Summer 2013), a proof-of-concept implementation existed to integrate the well
known Snappy codec [Snappy] as well as LZ4 [LZ4] into the Blosc framework. As
of January 2014 this proof of concept has matured and as of version 1.3.0
Blosc comes equipped with support for Snappy [Snappy], LZ4 [LZ4] and even Zlib
[zlib].
Blosc was initially developed to support in-memory compression in order to
mitigate the effects of the memory hierarchy [Jacob2009]. More specifically,
to mitigate the effects of memory latency, i.e. the ever growing divide
between the CPU speed and the memory access speed–which is also known as the
problem of the _starving CPUs_ [Alted2009].
The goal of in-memory compression techniques is to have a numerical container
which keeps all data as in-memory compressed blocks. If the data needs to be
operated on, it is decompressed only in the caches of the CPU. Hence, data can
be moved faster from memory to CPU and the net result is faster computation,
since less CPU cycles are wasted while waiting for data. Similar techniques
are applied successfully in other settings. Imagine for example, one wishes to
transfer binary files over the internet. In this case the transfer time can be
significantly improved by compressing the data before transferring it and
decompressing it after having received it. As a result the total compressed
transfer time, which is taken to be the sum of the compression and
decompression process and the time taken to transfer the compressed file, is
less than the time taken to transfer the plain file. For example the well
known UNIX tool rsync [rsync] implements a -z switch which performs
compression of the data before sending it and decompression after receiving
it. The same basic principle applies to in-memory compression, except that we
are transferring data from memory to CPU. Initial implementations based on
Blosc exist, c.f. Blaze [Blaze] and carray [CArray], and have been shown to
yield favourable results [Personal communication with Francesc Alted].
## 3 Numpy
The Numpy [VanDerWalt2011, Numpy] ndarray is the de-facto multidimensional
numerical container for scientific python applications. It is probably the
most fundamental package of the scientific python ecosystem and widely used
and relied upon by third-party libraries and applications. It consists of the
N-dimensional array class, various different initialization routines and many
different ways to operate on the data efficiently.
## 4 Existing Lightweight Solutions
There are a number of other plain (uncompressed) and compressed lightweight
serialization formats for Numpy arrays that we can compare Bloscpack to. We
specifically ignore more heavyweight solutions, such as HDF5, in this
comparison.
* •
NPY
* •
NPZ
* •
ZFile
### 4.1 NPY
_NPY_ [NPY] is a simple plain serialization format for numpy. It is considered
somewhat of a gold standard for the serialization. One of its advantages is
that it is very, very lightweight. The format specification is simple and can
easily be digested within an hour. In essence it simply contains the ndarray
metadata and the serialized data block. The metadata amounts to the dtype, the
order and the shape or the array. The main drawback is that it is a plain
serialization format and does not support compression.
### 4.2 NPZ
_NPZ_ is, simply put, a Zip file which contains multiple NPY files. Since this
is a Zip file it may be optionally compressed, however the main uses case is
to store multiple ndarrays in a single file. Zip is an implementation of the
DEFLATE [DEFLATE] algorithm. Unlike the other evaluated compressed formats,
NPZ does not support a compression level setting.
### 4.3 ZFile
_ZFile_ is the native serialization format that ships with the Joblib [Joblib]
framework. Joblib is equipped with a caching mechanism that supports caching
input and output arguments to functions and can thus avoid running heavy
computations if the input has not changed. When serializing ndarrays with
Joblib, a special subclass of the Pickler is used to store the metadata
whereas the datablock is serialized as a ZFile. ZFile uses zlib [zlib]
internally and simply runs zlib on the entire data buffer. zlib is also an
implementation of the DEFLATE algorithm. One drawback of the current ZFile
implementation is that no chunking scheme is employed. This means that the
memory requirements might be twice that of the original input. Imagine trying
to compress an incompressible buffer of 1GB: in this case the memory
requirement would be 2GB, since the entire buffer must be copied in memory as
part of the compression process before it can be written out to disk.
## 5 Bloscpack Format
The Bloscpack format and reference implementation builds a serialization
format around the Blosc codec. It is a simple chunked file-format well suited
for the storage of numerical data. As described in the Bloscpack format
description, the big-picture of the file-format is as follows:
>
> |-header-|-meta-|-offsets-|
>
> |-chunk-|-checksum-|-chunk-|-checksum-|...|
>
The format contains a 32 byte header which contains various options and
settings for the file, for example a magic string, the format version number
and the total number of chunks. The meta section is of variable size and can
contain any metadata that needs to be saved alongside the data. An optional
offsets section is provided to allow for partial decompression of the file in
the future. This is followed by a series of chunks, each of which is a blosc
compressed buffer. Each chunk can be optionally followed by a checksum of the
compressed data which can help to protect against silent data corruption.
The chunked format was initially chosen to circumvent a 2GB limitation of the
Blosc codec. In fact, the ZFile format suffers from this exact limitation
since zlib—at least the Python bindings—is also limited to buffers of 2GB in
size. The limitation stems from the fact that int32 are used internally by the
algorithms to store the size of the buffer and the maximum value of an int32
is indeed 2GB. In any case, using a chunked scheme turned out to be useful in
its own right. Using a modest chunk-size of e.g. 1MB (the current default)
causes less stress on the memory subsystem. This also means that in contrast
to ZFile, only a small fixed overhead equal to the chunk-size is required
during the compression and decompression process, for example when compressing
or decompression from/to an external storage medium.
With version 3 the format was enhanced to allow appending data to an existing
Bloscpack compressed file. This is achieved by over-allocating the offsets and
metadata section with dummy values to allow chunks to be appended later and
metadata to be enlarged. One caveat of this is that we can not pre-allocate an
infinite amount of space and so only a limited amount of data can potentially
be appended. However, to provide potential consumers of the format with as
much flexibility as possible, the amount of space to be pre-allocated is
configurable.
For an in-depth discussion of the technical details of the Bloscpack format
the interested reader is advised to consult the official documentation
[Bloscpack]. This contains a full description of the header layout, the sizes
of the entries and their permissible values.
## 6 Command Line Interface
Initially, Bloscpack was conceived as a command-line compression tool. At the
time of writing, a Python API is in development and, in fact, the command-line
interface is being used to drive and dog-food the Python API. Contrary to
existing tools such as gzip [gzip], bloscpack doesn’t use command-line options
to control its mode of operation, but instead uses the _subcommand_ style.
Here is a simple example:
$ ./blpk compress data.dat$ ./blpk decompress data.dat.blp data.dcmpAnother
interesting subcommand is info which can be used to inspect the header and
metadata of an existing file:
$ ./blpk info data.dat.blp[...]The Bloscpack documentation contains extensive
descriptions of the various options and many examples of how to use the
command line API.
## 7 Packing Numpy Arrays
As of version 0.4.0 Bloscpack comes with support for serializing Numpy
ndarrays. The approach is simple and lightweight: the data buffer is saved in
Blosc compressed chunks as defined by the Bloscpack format. The shape, dtype
and order attributes—the same ones saved in the NPY format—are saved in the
metadata section. Upon de-serialization, first an empty ndarray is allocated
from the information in the three metadata attributes. Then, the Bloscpack
chunks are decompressed directly into the pre-allocated array.
The Bloscpack Python API for Numpy ndarray is very similar to the simple NPY
interface; arrays can be serialized/de-serialized using single function
invocations.
Here is an example of serializing a Numpy array to file:
>>> import numpy as np>>> import bloscpack as bp>>> a = np.linspace(0, 100,
2e8)>>> bp.pack_ndarray_file(a, ’a.blp’)>>> b =
bp.unpack_ndarray_file(’a.blp’)>>> assert (a == b).all()And here is an example
of serializing it to a string:
>>> import numpy as np>>> import bloscpack as bp>>> a = np.linspace(0, 100,
2e8)>>> b = bp.pack_ndarray_str(a)>>> c = bp.unpack_ndarray_str(b)>>> assert
(a == c).all()The compression parameters can be configured as keyword
arguments to the pack functions (see the documentation for detail).
## 8 Comparison to NPY
The [NPY] specification lists a number of requirements for the NPY format. To
compare NPY and Bloscpack feature-wise, let us look at the extent to which
Bloscpack satisfies these requirements when dealing with Numpy ndarrays.
* 1.
_Represent all NumPy arrays including nested record arrays and object arrays._
Since the support for Numpy ndarrays is very fresh only some empirical results
using toy arrays have been tested. Simple integer, floating point types and
string arrays seem to work fine. Structured arrays are also supported (as of
0.4.1), even those with nested data types. Finally, object arrays also seem to
survive the round-trip tests.
* 2.
_Represent the data in its native binary form._
Since Bloscpack will compress the data it is impossible to represent the data
in its native binary form.
* 3.
_Be contained in a single file._
Using the metadata section of the Bloscpack format all required metadata for
decompressing a Numpy ndarray can be included alongside the compressed data.
* 4.
_Support Fortran-contiguous arrays directly._
If an array has Fortran ordering we can save it in Fortran ordering in
Bloscpack. The order is saved as part of the metadata and the contiguous
memory block is saved as is. The order is set during decompression and hence
the array is deserialized correctly.
* 5.
_Store all of the necessary information to reconstruct the array including
shape and dtype on a machine of a different architecture […] Endianness […]
Type._
As mentioned above all integer types as well as string and object arrays are
handled correctly and their shape is preserved. As for endianness, initial toy
examples with large-endian dtypes pass the roundtrip test
* 6.
_Be reverse engineered._
In this case _reverse engineering_ refers to the ability to decode a Bloscpack
compressed file after both the Bloscpack code and file-format specification
have been lost. For NPY this can be achieved if one roughly knows what to look
for, namely three metadata attributes and one plain data block. In the
Bloscpack case, things are more difficult. First of all, the header does have
a larger number of entries which must first be deciphered. Secondly the data
is compressed and without knowledge of the compression scheme and
implementation this will be very difficult to reverse engineer.
* 7.
_Allow memory-mapping of the data._
Since the data is compressed it is not possible to use the mmap primitive to
map the file into memory in a meaningful way. However, due to the chunk-wise
nature of the storage, it is theoretically possible to implement a quasi-mem-
mapping scheme. Using the chunk offsets and the typesize and shape from the
Numpy ndarray metadata, it will be possible to determine which chunk or chunks
contain a single element or a range and thus load and decompress only those
chunks from disk.
* 8.
_Be read from a file-like stream object instead of an actual file._
This has been part of the Bloscpack code base since very early versions since
it is essential for unit testing w/o touching the file system, e.g. by using a
file-like StringIO object. In fact this is how the Numpy ndarray
serialization/de-serialization to/from strings is implemented.
* 9.
_Be read and written using APIs provided in the numpy package._
Bloscpack does not explicitly aspire to being part of Numpy.
## 9 Benchmarks
The benchmarks were designed to compare the following three alternative
serialization formats for Numpy ndarrays: NPY, NPZ and ZFile with Bloscpack.
To this end, we measured compression speed, decompression speed, both with and
without the Linux file system cache and compression ratio for a number of
different experimental parameters.
### 9.1 Parameters
Three different array sizes were chosen:
* •
small 1e4 8 = 80000 Bytes = 80KB
* •
mid 1e7 8 = 80000000 Bytes = 80MB
* •
large 2e8 * 8 = 1600000000 Bytes = 1.4 GB
Three different dataset complexities were chosen:
* •
low arange (very low Kolmogorov complexity*)
* •
medium sin \+ noise
* •
high random numbers
And lastly two different storage mediums were chosen:
* •
ssd encrypted (LUKS) SSD
* •
sd SD card
The SD card was chosen to represent a class of very slow storage, not because
we actually expect to serialize anything to an SD card in practice.
To cut down on the number of data points we choose only to evaluate the
compression levels 1, 3 and 7 for ZFile and 1, 3, 7 and 9 for Bloscpack.
Although NPZ is a compressed format it does not support modifying the
compression level. This results in using 1 + 1 + 3 + 4 = 9 different codec
values.
This configuration leads to 3 * 3 * 2 * 9 = 160 data points. Additionally to
account for fluctuations, each datapoint was run multiple times depending on
the size of the dataset. In each case of number of sets each with a number of
runs were performed. Then, the mean across runs for each set and then the
minimum across all sets was taken as the final value for the datapoint. For
the small size, 10 sets with 10 runs were performed. For the mid size, 5 sets
with 5 runs were performed. And finally, for the large size, 3 sets with 3
runs each were
performed.\raisebox{10.00002pt}{\hypertarget{id26}{}}\hyperlink{id25}{*}\raisebox{10.00002pt}{\hypertarget{id26}{}}\hyperlink{id25}{*}footnotetext:
The inquisitive reader will note the following caveat at this stage. Perhaps
Kolmogorov complexity is not the correct choice of complexity measure to
define low entropy data for a Lempel-Ziv style dictionary encoder. In fact,
any sequence of consecutive integers by definition has high Lempel-Ziv
complexity and is not compressible. However, as will be shown during the
benchmarks later on, Bloscpack is actually very good at compressing these
kinds of sequences, whereas ZFile and NPZ are not. This is a result of the
fact that arange generated muti-byte type integer data and the shuffle filter
for Bloscpack can optimize this very well. At this stage we simply state that
the proposed low entropy dataset has been sufficient for the benchmarks. An
in-depth treatment of the effects the shuffle filter has on the Lempel-Ziv
complexity is beyond the scope of this paper and will perhaps be the subject
of a future publication.
### 9.2 Timing
The timing algorithm used was a modified version of the timeit tool which
included in the Python standard library. This supports deactivation of the
Python interpreters garbage collector during the run and executing code before
and after each run. For example, when measuring decompression speed without
the Linux file system cache, one needs to clear this cache before each run and
it is imperative that this operation does not enter into the timing. Also,
when measuring compression speed, one needs to make sure sync is executed
after the run, to ensure the data is actually written out to the storage
medium. Contrary to clearing the file system cache, the time required by the
sync operation must enter the timing to not contaminate the results.
### 9.3 Hardware
The machine used was a Lenovo Carbon X1 ultrabook with an Intel Core i7-3667U
Processor [CPU]. This processor has 2 physical cores with active
hyperthreading resulting in 4 threads. The CPU scaling governor was set to
performance which resulted in a CPU frequency of 2.0Ghz per core. The CPU has
three levels of cache at: 32K, 256K and 4096k as reported by Linux sysfs. The
memory bandwidth was reported to be 10G/s write and 6G/s read by the Blosc
benchmarking tool. Interestingly this is in stark contrast to the reported
maximum memory bandwidth of 25G/s which is advertised on the manufacturers
data sheet. The operating system used was Debian Stable 7.1 with the following
64bit kernel installed from Debian Backports: 3.9-0.bpo.1-amd64 #1 SMP Debian
3.9.6-1~bpo70+1 x86_64 GNU/Linux.
The IO bandwidth of the two storage media was benchmarked using dd:
$ dd if=/dev/zero of=outputfile bs=512 count=32M$ dd if=outputfile
of=/dev/null
* •
SSD: 230 MB/s write / 350 MB/sd read
* •
SD: 20 MB/sd read/write
### 9.4 Disabled OS Defaults
Additionally certain features of the operating system were disabled explicitly
while running the benchmarks. These optimizations were chosen based on
empirical observations while running initial benchmarks, observing suspicious
behaviour and investigating possible causes. While there may be other
operating system effects, the precautions listed next were found to have
observably detrimental effects and disabling them lead to increased
reliability of the results.
First, the daily cronjobs were disabled by commenting out the corresponding
line in /etc/crontab. This is important because when running the benchmarks
over night, certain IO intensive cronjobs might contaminate the benchmarks.
Secondly, the Laptop Mode Tools were disabled via a setting in /etc/laptop-
mode/laptop-mode.conf. These tools will regulate certain resource settings, in
particular disk write-back latency and CPU frequency scaling governor, when
certain system aspects—e.g. the connectivity to AC power—change and again this
might contaminate the benchmarks.
## 10 Versions Used
The following versions and git-hashes—where available—were used to acquire the
data reported in this article:
* •
benchmark-script: NA / 7562c6d
* •
bloscpack: 0.4.0 / 6a984cc
* •
joblib: 0.7.1 / 0cfdb88
* •
numpy: 1.7.1 / NA
* •
conda: 1.8.1 / NA
* •
python: ’Python 2.7.5 :: Anaconda 1.6.1 (64-bit)’
The benchmark-script and results files are available from the repository of
the EuroScipy2013 talk about Bloscpack [Haenel2013]. The results file analysed
are contained in the csv file results_1379809287.csv.
### 10.1 Bloscpack Settings
In order to reduce the overhead when running Bloscpack some optional features
have not be enabled during the benchmarks. In particular, no checksum is used
on the compressed chunks and no offsets to the chunks are stored.
## 11 Results
The results of the benchmark are presented in the figures 1, 2, 3, 4 and 5.
Figures 1 to 4 show timing results and are each a collection of subplots where
each subplot shows the timing results for a given combination of dataset size
and entropy. The dataset size increases horizontally across subplots whereas
the dataset entropy increases vertically across subplots. Figures 1 and 2 show
results for the SSD storage type and figures 3 and four show results for the
SD storage type. Figures 1 and 3 compare Bloscpack with NPY whereas figures 2
and 4 compare Bloscpack with NPZ and ZFile. NPY is shown separately from NPZ
and ZFile since their performance characteristics are so different that they
can not be adequately compared visually on the same plot. For all timing plots
black bars indicate compression time, white is used to denote decompression
time w/o the file system cache and gray identifies decompression time with a
hot file system cache. For all timing plots, larger values indicate worse
performance. Lastly, figure 5 shows the compression ratios for all examined
formats.
Figure 1: Compare Bloscpack and NPY on the SSD storage type.
Figure 2: Compare Bloscpack, NPZ and ZFile on the SSD storage type.
Figure 3: Compare Bloscpack and NPY on the SD storage type.
Figure 4: Compare Bloscpack, NPZ and ZFile on the SD storage type.
Figure 5: Compression ratios for all examined formats
In Fig. 1 we can see how Bloscpack compares to NPY on the SSD storage type.
The first thing to note, is that for small datasets (first column of
subplots), Bloscpack does not lag behind much compared to NPY for compression
and is actually slightly faster for decompression. However the absolute
differences here are in the millisecond range, so one might perhaps argue that
Bloscpack and NPY are on par for small datasets. As soon as we move to the
medium size datasets first gains can be seen. Especially for the low entropy
case where Bloscpack beats NPY for both compression and decompression w/o file
system cache. For the medium entropy case, Bloscpack is slightly faster for a
few settings, at least for the compression and decompression cases.
Surprisingly, for the decompression with a hot file system cache, Bloscpack is
actually 2 times slower under the compression levels 7 and 9. One possibility
for this might be that, even though the file contents are in memory, reading
from the file necessitates an initial memory-to-memory copy, before the data
can actually be decompressed. For the high entropy case, Bloscpack is mostly
slightly slower than NPY. For the large dataset the results are simply a
scaled version of the medium dataset size results and yield no additional
insights.
Fig. 2 shows the comparison between Bloscpack, NPZ and ZFile on the SSD
storage type. In this comparison, the speed of the Blosc compressor really
shines. For every combination of dataset size and entropy the is a compression
level for Bloscpack that can compress faster than any of the competitors. In
the extreme case of the large size and the low entropy, Bloscpack is over 300
times faster during compression than NPZ (302 seconds for NPZ vs. 0.446
seconds for Bloscpack). Even for the high entropy case, where very very little
compression is possible due to the statistics of the dataset, Bloscpack is
significantly faster during compression. This is presumably because Blosc will
try to compress a buffer, finish very quickly because there is no work to be
done and then it simply copies the input verbatim.
One very surprising result here is that both NPZ and ZFile with level 7 take
extraordinary amounts of time to compress the low entropy dataset. In fact
they take the longest on the low entropy dataset compared to the medium and
high entropies. Potentially this is related to the high Lempel-Ziv complexity
of that dataset, as mentioned before. Recall that both NPZ and ZFile use the
DEFLATE algorithm which belongs to the class of LZ77 dictionary encoders, so
it may suffer since it no shuffle filter as in the case of Blosc is employed.
Figures 3. and 4. show the same results as figures 1. and 2. respectively but
but for the SD storage class. Since the SD card is much slower than the SSD
card the task is strongly IO bound and therefore benefits of compression can
be reaped earlier. For example, Bloscpack level 7 is twice as fast as NPY
during compression on the medium size medium entropy dataset. For the low
entropy dataset at medium and large sizes, Bloscpack is about an order of
magnitude faster. For the high entropy dataset Bloscpack is on par with NPY
because the overhead of trying to compress but not succeeding is negligible
due to the IO boundedness resulting from the speed of the SD card. When
comparing Bloscpack to NPZ and ZFile on the SD card, the IO boundedness means
that any algorithm that can achieve a high compression ratio in a reasonable
amount of time will perform better. For example for medium size and medium
entropy, NPZ is actually 1.6 times faster than Bloscpack during compression.
As in the SSD case, we observe that NPZ and ZFile perform very slowly on low
entropy data.
Lastly in Figure 5. we can see the compression ratios for each codec, size and
entropy. This is mostly just a sanity check. NPY is always at 1, since it is a
plain serialization format. Bloscpack gives better compression ratios for low
entropy data. NPZ and ZFile give better compression ratios for the medium
entropy data. And all serializers give a ratio close to zero for the high
entropy dataset.
## 12 Conclusion
This article introduced the Bloscpack file-format and python reference
implementation. The features of the file format were presented and compared to
other serialization formats in the context of Numpy ndarrays. Benchmarking
results are presented that show how Bloscpack can yield performance
improvements for serializing Numpy arrays when compared to existing solutions
under a variety of different circumstances.
## 13 Future Work
As for the results obtained so far, some open questions remain unsolved. First
of all, it is not clear why Bloscpack at level 7 and 9 gives comparatively bad
results when decompressing with a hot file system cache. Also the bad
performance of ZFile and NPY on the so-called low entropy dataset must be
investigated and perhaps an alternative can be found that is not biased
towards Bloscpack. Additionally, some mathematical insights into the
complexity reduction properties of Blosc’s shuffle filter would be most
valuable.
Lastly, more comprehensive benchmarks need to be run. This means, first
finding non-artificial benchmark datasets and establishing a corpus to run
Bloscpack and the other solutions on. Furthermore, It would be nice to run
benchmarks on other architectures for machines with more than 2 physical
cores, non-uniform memory access and an NFS file-system as commonly found in
compute clusters.
## 14 Gratitude
The author would like to thank the following people for advice, helpful
comments and discussions: Pauli Virtanen, Gaël Varoquaux, Robert Kern and
Philippe Gervais. Also, the author would like to specially thank Stéfan van
der Walt and Francecs Alted for reviewing drafts of this paper.
## References
* [Alted2009] Francesc Alted. _The Data Access Problem_ EuroScipy 2009 Keynote Presentation http://www.blosc.org/docs/StarvingCPUs.pdf
* [Alted2010] Francesc Alted. _Why modern CPUs are starving and what can be done about it_ , Computing in Science & Engineering, Vol. 12, No. 2. (March 2010), pp. 68-71 http://www.blosc.org/docs/StarvingCPUs-CISE-2010.pdf
* [DEFLATE] Peter. Deutsch _DEFLATE Compressed Data Format Specification version 1.3_ RFC1951 1996 http://tools.ietf.org/html/rfc1951
* [Haenel2013] Valentin Haenel. _Introducing Bloscpack_ EuroScipy 2013 Presentation https://github.com/esc/euroscipy2013-talk-bloscpack
* [Jacob2009] Bruce Jacob. _The Memory System: You Can’t Avoid It, You Can’t Ignore It, You Can’t Fake It_ Synthesis Lectures on Computer Architecture 2009, 77 pages,
* [VanDerWalt2011] Stefan Van Der Walt, S. Chris Colbert, Gaël Varoquaux _The NumPy array: a structure for efficient numerical computation_ Computing in Science and Engineering 13, 2 (2011) 22-30
* [LZ77] Ziv, Jacob; Lempel, Abraham (May 1977). _A Universal Algorithm for Sequential Data Compression_. IEEE Transactions on Information Theory 23 (3): 337–343.
* [NPY] Robert Kern. _The NPY format_ https://github.com/numpy/numpy/blob/master/doc/neps/npy-format.txt
* [Joblib] Joblib http://pythonhosted.org/joblib/
* [zlib] Zlib http://www.zlib.net/
* [gzip] Gzip http://www.gzip.org/
* [rsync] Rsync http://rsync.samba.org/
* [Blaze] Blaze http://blaze.pydata.org/
* [CArray] CArray http://carray.pytables.org/docs/manual/
* [Numpy] Numpy http://www.numpy.org/
* [FastLZ] FastLZ http://fastlz.org/
* [Snappy] Snappy http://code.google.com/p/snappy/
* [LZ4] LZ4 http://code.google.com/p/lz4/
* [Blosc] Blosc http://blosc.org
* [Bloscpack] Bloscpack https://github.com/Blosc/bloscpack
* [CPU] Intel® Core™ i7-3667U Processor
|
arxiv-papers
| 2014-04-25T10:53:23 |
2024-09-04T02:50:01.816590
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Valentin Haenel",
"submitter": "Pierre de Buyl",
"url": "https://arxiv.org/abs/1404.6383"
}
|
1404.6384
|
# CATOS: Computer Aided Training/Observing System
Jinook Oh∗† * Corresponding author: [email protected]† Cognitive Biology
Dept., University of ViennaCopyright © 2014 Jinook Oh. This is an open-access
article distributed under the terms of the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are credited.
http://creativecommons.org/licenses/by/3.0/
###### Abstract
In animal behavioral biology, there are several cases in which an autonomous
observing/training system would be useful. 1) Observation of certain species
continuously, or for documenting specific events, which happen irregularly; 2)
Longterm intensive training of animals in preparation for behavioral
experiments; and 3) Training and testing of animals without human
interference, to eliminate potential cues and biases induced by humans. The
primary goal of this study is to build a system named CATOS (Computer Aided
Training/Observing System) that could be used in the above situations. As a
proof of concept, the system was built and tested in a pilot experiment, in
which cats were trained to press three buttons differently in response to
three different sounds (human speech) to receive food rewards. The system was
built in use for about 6 months, successfully training two cats. One cat
learned to press a particular button, out of three buttons, to obtain the food
reward with over 70 percent correctness.
###### Index Terms:
animal training, animal observing, automatic device
## 1 Introduction
It is often the case in animal behavioral biology that a large amount of human
resources, time, and data storage (such as video recordings) are required in
animal observation and training. Some representative examples of these cases
are:
* •
Observation of certain species continuously or monitoring for specific events,
which occur irregularly, when behavior of certain species during any time
period or specific time period, such as nocturnal behaviors, are investigated.
* •
Certain experiments require a prolonged training period, sometimes over a
year. This type of experiment requires reliable responses, which may not
correspond to usual behavior patterns, from animals in tasks. Therefore,
training may require a long period of time until the subject is ready to be
tested. Additionally, long periods of human supervised training can introduce
unintended cues and biases for animals.
In the first case, an autonomous system for observing animals can save human
resources and reduce the amount of data storage. The reduced amount of data
can also conserve other types of human resources such as investigation and
maintenance of large-scale data. There have been attempts to build autonomous
observing or surveillance systems in the fields of biology, such as Kritzler
et al. [Kri08]’s work, and security systems, such as Belloto et al. [Bel09],
Vallejo et al. [Val09], for instance. There are also commercial products for
surveillance systems with various degrees of automation, or incorporating
artificial intelligence. However, the intelligence of each system is case-
specific and it is difficult to apply these specific systems to novel
situations without considerable adjustments. In the second case, an autonomous
system for prolonged, intensive training can also save human resources and
eliminate potential cues and biases caused by humans. Training with an
autonomous system is an extension of traditional operant conditioning chambers
and many modern and elaborated versions have been developed and used, such as
in Markham et al. [Mar96], Takemoto et al. [Tak11], Kangas et al. [Kan12],
Steurer et al. [Ste12], and Fagot & Bonte [Fag09]. However, many of the
previous devices use commercial software. Also, they do not possess the
observational features developed in the current project. It would be useful to
have an open-source, relatively low-budget, and modularized system which could
be customized for the observation, training and the experimentation on animal
subjects of various species. CATOS, the system built in the present study,
fulfills these necessities. The difference between the previous systems and
CATOS (Computer Aided Training/Observing System) in the present work is that
the animals do not have to be captured or transported to a separated space at
a specific time in order to be trained. The disadvantages of separating
animals (e.g., primates) are well-known, and include stress on animals
separated from their group or moved from their usual confines, the risky
catching procedure for both animal and human (cf. Fagot & Bonte [Fag09]).
Similar arguments apply to most animal species, especially when they are
social. The automatic learning device for monkeys (ALDM) described in Fagot &
Bonte [Fag09] is very similar to the trainer aspect of CATOS described in the
present work, but CATOS is different in following features. First of all, it
aimed to be open-source based and more modular so that it can be more easily
adjusted and adopted to different species and experiments. Another feature is
that CATOS is equipped with various observational features, including visual
and auditory recording and recognition through video camera and microphone,
which make the system able to interact with the subjects, such as reacting
immediately to a subject with a motion detection from a camera or a sound
recognition from a microphone. CATOS should offer the following advantages.
* •
The system should be flexible in terms of its adjustability and the
extendibility to various projects and species. The software should be open-
source, and both software and hardware components should be modularized as
much as possible, thus the system reassembly for researchers in animal
behavioral biology is practical.
* •
The system should have various observational features applicable to a broad
range of animal species and observational purposes.
* •
The system should perform continuous monitoring, and it should record video
and/or sound only when a set of particular conditions is fulfilled. This would
reduce the amount of data produced during the procedure.
* •
The system should have actuators to react in certain situations, which allows
it to act as a trainer/experimenter. The human trainer/experimenter designs
the procedure by adjusting parameters and modules, but the actual performance
should be done by the system. In this way, the system could help reducing the
amount of time required for training, and eliminating cues/biases which might
be induced by the human interferences.
* •
With this system, the animal should not have to be transported to a certain
space, or separated from its group, for training. The animals should be able
to choose when to start a trial on their own.
Two CATOS prototypes have been built during this study. The first build of
CATOS has 3 pushbuttons as a main input device for cats and the second build
has a touch-screen as a main input device. The first build was an initial
attempt to build and test such a system. The second build is the final product
of the study. The basic structures of these two builds are more or less the
same. The differences are that the second version has improved functions and
it uses the touch-screen instead of pushbuttons. The first build of CATOS was
tested with domestic cats (Felis catus) to train them to press three different
buttons differently depending on the auditory stimuli (three different human
speech sounds). The final goal of this training is to investigate human speech
perception in cats. There is no doubt in that many animal species can
recognize some words in human speech. The examples of speech perception in
dogs and chimpanzees can be found in the work of Kaminski et al. [Kam04] and
Heimbauer et al. [Hei11] respectively. In some cases, animals can even
properly produce words with specific purposes. An example of speech perception
and production in a parrot can be found in the work of Pepperberg [Pep87].
Despite these findings, there is ongoing debate about whether the same
perceptual mechanisms are used in speech recognition by humans and animals
(Fitch [Fit11]). To investigate this issue, animals have to be trained to show
different and reliable responses to different human speech sounds. Then, we
can test which features of human speech are necessary for different animal
species to understand it. Thus, the final aim of the training in this study
would be to obtain cats showing different responses to different human speech
sounds with statistical significance (over 75 percent). Before reaching this
final goal, several smaller steps and goals are required.
## 2 Brief description of CATOS (Computer Aided Training/Observing System)
The overall system is composed of a combination of software and hardware
components. The software components are mainly composed of the Python script
named as ’AA.<version>.py’ and the program for the microcontroller. The ’AA’
runs all of the necessary processes and communicates with the microcontroller
program. The microcontroller program operates sensors and actuators as it
communicates with the ’AA’ program. The hardware components are composed of
various devices, some of which are directly connected to the computer via USB
cables. Some other devices only have GPIO (General Purpose Input Output) pins;
therefore they are connected to the microcontroller. The microcontroller
itself is connected to the computer via a USB cable. The hardware devices,
which are directly connected via USB cables, can be accessed using various
software modules, which are imported into the ’AA’ program. The access to
other devices only using GPIO pins is performed in the microcontroller and the
’AA’ program simply communicates with the microcontroller program via a serial
connection for sending commands to actuators and receiving values from
sensors.
Figure 1: Overall system diagram.
The software for this system is called AA (Agent for Animals). This software
was build with helps of many external libraries such as OpenCV [Bra00], and
NumPy/SciPy [Jon01]. Once it starts, seven processes were launched using
multiprocessing package of Python and it runs until the user terminates the
program. The multiprocessing was used because the heavy calculation for image
processing from multiple webcams were concerned. The number of processes can
be changed as some of them can be turned on or off. These processes include a
video-in process for each camera, a video-out process, an audio-in process, an
audio-out process, a schema process, and a message-board process. Figure 1.
Even though some of these processes have quite simple tasks, they were
separated in order to prevent them from interfering with each other and/or
becoming the bottleneck. The system has to process the visual, auditory, and
other sensory and motor information simultaneously to recognize the change of
the environment and respond to it properly. The output data such as captured
video input images, recorded WAV files, movement-records, CSV files for trial
results, and the log file are temporarily stored in the ’output’ folder. After
the daily session is finished, all of these output files go through an
archiving process which can include, but is not restricted to, generating
movies, generating images with the movement analysis, labeling sound files,
and moving different types of files into the categorized subfolders of an
archiving folder named with a timestamp.
Figure 2: AA_DataViewer
Besides combining all the above modules and implementing some common
functions, one more Python program was implemented to facilitate the process
of analyzing the recorded data. The program is called ”AA DataViewer”, which
is based on wxPython GUI toolkit and Matplotlib [Hun07] for drawing graphs.
Figure 2. It loads the log file, the result CSV (comma separated values) file
containing the results of the trial, the movement-record CSV files, the MP4
movie files, and the WAV files from one folder containing all data collected
for one session (day). For each video clip, there is a JPEG image showing the
movements of the blobs. The circles in the image represent the positions of
the blobs and their color represents the time-flow, with the black
corresponding to the beginning of the movie, and the white to the end of the
movie. A line connecting multiple circles means that those blobs occurred at
the same time. Another feature of this program is its ability to generate a
graph with selected sessions. In the ’archive’ folder, there are sub-folders,
each of which contains all the data for a session. When the ’select sessions’
button is clicked, a pop-up window appears for selecting multiple folders. The
result data from these selected sub-folders of ’archive’ folder is drawn as a
graph using Matplotlib [Hun07]. By visualizing the data for certain period, it
helps the trainer or experimenter quickly assess the current status of the
training procedure.
Figure 3: Automatic feeder
The two feeders used in this study is a device mainly comprising the Arduino
microcontroller; (refer http://www.arduino.cc/), a motor-shield for the
microcontroller, a servomotor, and a frame encasing the whole feeder. Both
Feeder variants work in a similar way, by rotating the servomotor by a certain
number of degrees, although the second feeder shows better performance in
terms of consistent amount of food released, due to the usage of an
Archimedes’ screw. Initially, an estimate of the amount of food left in the
food container was obtained using an IR distance sensor, but this feature was
discarded in the second build since the distance information from the IR
sensor was not accurate enough for this application. The second feeder
confirms the emission of a food reward via the piezoelectric sensor, which is
positioned right below the Archimedes’ screw. Figure 3.
Figure 4: Circuit with a microcontroller
Communication between the Arduino chip and the main computer was accomplished
by using the Arduino module of the ’AA’ program. In the circuit, Figure 4,
* •
The temperature sensor measures the temperature inside of the protective
wooden platform.
* •
The photocell sensor measures the ambient light level.
* •
The light bulb can be turned on when the photocell sensor indicates the
ambient light level is below a user-defined threshold.
* •
Two fans are turned on when the temperature sensor indicates the temperature
is too high in the platform.
* •
The piezoelectric sensor is read while the servomotor is actuating, in order
to confirm the occurrence of the food reward. This sensor reading is required
because occasionally the food dispensing fails due to the combination of the
short motor activation time (<0.5 seconds) and the shape of the dry food
pieces (which can fit into other pieces easily and then fail to emerge).
* •
The servomotor is responsible for the food dispense by turning the Archimedes’
screw back and forth.
## 3 Results of building CATOS and its testing on 2 domesticated cats
The hardware and software were built and tested. The software is available at
https://github.com/jinook0707/CATOS_alpha with GNU General Public License,
version 3. Both hardware and software are curretnly in its alpha stage.
Although its potential to be used to train and test animal cognition was
tested and its usage seemed promising to save human resources in certain
situations, both hardware and software should be developed further to be
practically used for experimenting animal cognition.
The two web-cams observed the experimental area for 8 to 12 hours per day for
about 5 months (from the middle of October 2012 to the middle of March 2013).
The movement records, MP4 movie files, JPEG image files, and WAV sound files
generated during this period took 37.35 Giga bytes of storage. To obtain a
rough idea of the degree of reduction in data storage that was achieved using
the system, the number of recorded frames in the video recording was assessed.
Data for 15 days were taken to calculate it. The total observation period was
406138 seconds, corresponding to 112.8 hours. The number of frames recorded
was 206024 and the average FPS(Frame Per Second) was 7.5, therefore,
approximately, the video recordings were stored for 27470 seconds (=7.6
hours), which is about 6.7 percent of entire observation period. These
specific numbers are not very meaningful since they can fluctuate with the
increase or decrease of the subject’s movements, but the point is that the
most of the meaningless recordings were successfully filtered out by CATOS.
Human presence during session is not necessary. Data transfer from one
computer to another, maintenance, or modification of the system requires human
interaction, but no time and effort is required concerning the training and
testing sessions. Because no one attends the sessions, a periodic analysis of
the animal’s performance with the system is required. A simple assessment of
how much food the animals took, or more specifically, how many correct and
incorrect trials occurred, can be done quickly since this information is
already stored in result CSV file displaying the number of correct and
incorrect trials generated with timestamps at the end of each session. Also,
the data-viewer utility program displays all the timestamps and its JPEG
image, which presents a brief report on the movement detected in the recorded
video-clip. Thus, simply browsing the JPEG images is often enough to assess
the session. If it is not enough, then one can obtain a more detailed
assessment by playing the video-clips recorded around the trial times.
Figure 5: Recent performance of the trained cat on three human speech
discrimination task.
Two domesticated cats were trained for testing the system. Both cats learned
that approaching the feeder on a playback sound could lead to a food reward.
Then one cat further learned that pressing one out of three buttons could lead
to a food reward. The training of the association between three different
sound stimuli and three different buttons is an ongoing process. The most
recent performance data Figure 5 shows over 70 percent of overall performance
and also the performance on each button is significantly higher than 33.3
percent of chance level.
## References
* [Bel09]
* N.
Bellotto, E. Sommerlade, B. Benfold, C. Bibby, I. Reid, D. Roth, C. Fernandez,
L.V. Gool and J. Gonzalez. _A distributed camera system for multi-resolution
surveillance_ , Proc. of the third ACM/IEEE Int. Conf. on Distributed Smart
Cameras (ICDSC), 2009.
* [Bra00]
* G.
Bradski. _The OpenCV Library, Dr. Dobb’s Journal of Software Tools_ ,
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* E.
Jones, T. Oliphant, P. Peterson, and others, _SciPy: Open source scientific
tools for Python_ , 2001.
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Fagot and D. Paleressompoulle. _Automatic testing of cognitive performance in
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* [Fag10]
* J.
Fagot and E. Bonte. _Automated testing of cognitive performance in monkeys:
Use of a battery of computerized test systems by a troop of semi-free-ranging
baboons (Papio papio)_ , Behavior Research Methods, 42(2):507-516, May 2010.
* [Fit11]
* T.
Fitch. (2011). _Speech perception: a language-trained chimpanzee weighs in_ ,
Current Biology, 21(14):R543-R546, July 2011.
* [Hei11] L.A. Heimbauer, M.J. Beran and M.J. Owren. _A chimpanzee recognizes synthetic speech with significantly reduced acoustic cues to phonetic content_ , Current Biology, 21(14):1210-1214, June 2011.
* [Hun07]
* J.
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|
arxiv-papers
| 2014-04-25T10:53:33 |
2024-09-04T02:50:01.823569
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jinook Oh",
"submitter": "Pierre de Buyl",
"url": "https://arxiv.org/abs/1404.6384"
}
|
1404.6385
|
# High-Content Digital Microscopy with Python
Fabrice Salvaire∗† * Corresponding author: [email protected]† Genomic
Vision SACopyright © 2014 Fabrice Salvaire. This is an open-access article
distributed under the terms of the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium,
provided the original author and source are credited.
http://creativecommons.org/licenses/by/3.0/
###### Abstract
High-Content Digital Microscopy enhances user comfort, data storage and
analysis throughput, paving the way to new researches and medical diagnostics.
A digital microscopy platform aims at capturing an image of a cover slip, at
storing information on a file server and a database, at visualising the image
and analysing its content. We will discuss how the Python ecosystem can
provide such software framework efficiently. Moreover this paper will give an
illustration of the data chunking approach to manage the huge amount of data.
###### Index Terms:
high-content microscopy, digital microscopy, high-throughput scanner, virtual
slide, slide viewer, multi-processing, HDF5, ZeroMQ, OpenGL, data chunking
## 1 Introduction
Since early times optical microscopy plays an important role in biology
research and medical diagnostic. Nowadays digital microscopy is a natural
evolution of the technology that provides many enhancements on user comfort,
data storage and analysis throughput. First, in comparison to binocular
microscopy where the low light emission intensity of the specimens causes
sever stress to the eyes, the digital microscopy monitor display offers
greater comfort to the users. Second, the digitization of the output allows to
freeze and store information for short to long term storage, to compress the
data, and to easily duplicate it, to protect its integrity (by checksum) and
its confidentiality (by cryptography). On the other hand, optical microscopy
implies conservation of the specimens themselves at low temperature and in the
dark. Last, the automation of a high content application provides a
considerable scale-up of the data processing throughput, thus paving the way
to new researches and medical diagnostics.
We will discuss in this paper how the Python ecosystem can provide efficiently
a software framework for the digital microscopy. Our discussion will first
present the data acquisition method, then we will describe the data storage
and finally the image viewer.
## 2 Data Acquisition
The first challenge of high-content digital microscopy is the quantity of
data. Let us first evaluate how large the data is, and enlighten our reader of
the reasons of such quantity of data. To reach the required resolution to see
the details of a specimen, optical microscopes use objectives magnifying up to
the diffraction limit which is about $100\times$. Nowadays the pixel size for
a CCD and sCMOS camera is about $6.5\,\text{um}$, thus we reach a resolution
of $162.5\,\text{nm}$ at a magnification of $40\times$. Now consider a
specimen put on a cover slip: a glass square surface of 18 mm wide (we will
later relate the support and the specimen by the more generic word _slide_ ,
which corresponds to a larger glass surface). Consequently to cover this
surface at such magnification we have to acquire an area larger than
$100\,000$ px wide, thus of the order of 10 billion of pixels. This is roughly
300 times larger than the actual largest professional digital camera
($36\,\text{Mpx}$). In light of foregoing digital microscopy are big data
similar to spatial images and involve a software framework similar to the
well-known Google Map.
For scientific application, we preferably use monochrome camera so as to avoid
the interpolation of a Bayer mosaic. Instead, to capture the entire colour
spectrum at the same time, colours are captured sequentially by placing a
filter of the colour’s corresponding wave length transmission in front of the
camera. These shots are called _colour fields of view_ here. Figure 1 shows
the schematic of an application of this acquisition method called an
epifluorescence microscope.
Figure 1: Schematic of an epifluorescence microscope. Specimens are labelled
with fluorescent molecules so called fluorophores. In this example we are
capturing an image for a fluorophore having an excitation wave length in the
blue and an emission wave length in the green. The filters are used to
restrict the excitation and filter the emission, respectively.
A camera like the Andor Neo sCMOS features a sensor of resolution $2560\times
2160\,\text{px}$ and a surface of $416\times 351\,\text{um}$. Thus to cover
the whole specimen surface we have to capture a mosaic of fields of view of
size $43\times 51$ (2193 tiles) using an automated stagger. We will also refer
to the fields of view as _tiles_ or _images_ according to the context.
To observe the specimen in several colours, two strategies can be used to
acquire the mosaic: one is to acquire a mosaic per colour and the other is to
acquire several colours per field of view. Both methods have advantages and
drawbacks. One of the differences is the uncertainty that occurs on the
registration of the colour fields of view. When capturing several colours per
field of view at the same staging position, the relative positioning error is
due to the optical path. When capturing a mosaic per colour, the error is also
due to the reproducibility of the stagger. On the other hand the accuracy of
the tile positions is always due to the stagger precision. So as to perform a
field of view registration without black zone in the reconstructed image, we
drive the stagger with a sufficient overlapping zone on both directions.
Another irregularity on the mosaic is due to the camera alignment error
according to the stagger axes that draw a sheared mosaic pattern (figure 2).
The shearing doesn’t have any serious effect on the reconstructed image since
it only displaces systematically the fields of view in the mosaic frame.
Figure 2: Field of View Mosaic showing a sheared effect due to the camera
misalignment. The tiles are rotated in the stagger frame but not in the mosaic
frame.
All these uncertainties can be studied using fluorescent beads with an
appropriate density on the cover slip and an image registration algorithm.
The third dimension of a specimen can be observed using the vertical focus
axis of the microscope so as to perform a so called _z-stack_ of images that
enlarge the depth of field virtually and thus improve the focus accuracy.
The Neo camera features a standard amplifier-DAC stage with a 12-bit
resolution and another stage with a combination of two amplifier-DACs to
achieve a 16-bit resolution for high dynamic image. Thus image pixels must be
encoded using an unsigned 16-bit integer data type. It means a colour field of
view weights $10.5\,\text{MB}$ and our mosaic weights $23\,\text{GB}$ per
colour.
Depending of the intensity dynamic of the specimen and the zero-padding
arising from the DAC, most of the pixels can have many zeros on the most
significant bits. Therefore, the amount of data can be efficiently reduced
using a lossless compression algorithm in conjunction with a bit shuffling to
group the zeros together and form long zero sequences in the byte stream.
## 3 Virtual Slide Format and Storage
We can now define the data structure of an acquisition so called later a
_virtual slide_. A virtual slide is made of a mosaic of fields of view and a
set of attributes that constitute the so called _slide header_. Examples of
attributes are a slide identifier, a date of acquisition or a type of assay.
The mosaic is a set of colour fields of view made of a mosaic index $(r,c)$, a
stagger position $(x,y,z)$, a colour index $w$ and an image array of unsigned
16-bit integers.
From this mosaic of field of views, we can imagine to reconstruct the slide
image once and for all and produce a giant image, where we could use for this
purpose the BigTIFF [BigTIFF] extension to the TIFF format. But if we want to
keep raw data without information loss we have to imagine a way to store the
original fields of view and process them on-line. This case is particularly
important when the registration matters for the interpretation of the
reconstructed image.
The HDF5 [HDF5] library and its h5py [h5py] Python binding are perfectly
suited for this purpose. The content of an HDF5 file is self-defined and the
library is open source which guaranties a long term access to the data. The
structure of an HDF5 file is similar to a file system having folder objects so
called _groups_ and N-dimensional array objects so called _dataset_ that
corresponds here to files. Each of these objects can have attached attributes.
This virtual file system provides the same flexibility than a real file system
similar to a UNIX loop device. Figure 3 shows an example.
Figure 3: HDF5 Virtual File System. Attributes can be attached to each node.
The h5py module provides a Pythonic API and map Numpy [Numpy] arrays to
datasets and reciprocally. The Numpy library is well appropriate to store
images in memory since it maps efficiently a C linear array data structure on
Python. The following code snippet gives an overview of its usage:
import numpy as npimport h5pyslide_file = h5py.File(’slide.hdf5’,
’w’)slide_file.attrs[’slide_name’] = u’John Doe’root_group =
slide_file[’/’]image_group = root_group.create_group(’images’)n =
1000image_dataset = image_group.create_dataset( ’image1’, shape=(100*n,
100*n), dtype=np.uint16)data = np.arange(n*n,
dtype=np.uint16).reshape((n,n))image_dataset[n:2*n,n:2*n] = dataAs usual when
large data sets are involved, the HDF5 library implements a data blocking
concept so called _chunk_ which is an application of the divide-and-conquer
paradigm. Indeed the data compression as well as the efficiency of the data
transfer requires datasets to be splitted in chunks. This feature is a
cornerstone for many features. It permits to read and write only a subset of
the dataset (a _hyperslab_), providing means for Python to map concepts such
view and broadcasting. Moreover it permits to implement a read-ahead and cache
mechanism to speed up the data transfer from storage to memory.
Another cornerstone of the HDF5 library is the implementation of a modular and
powerful data transfer pipeline shown on figure 4 whose aim is to decompress
the data from chunks stored on disk, scatter-gather the data and transform
them, for example to apply a scale-offset filter. The h5py module provides the
classic GZIP compression as well its faster counterpart LZF [LZF] and other
compression algorithms can be added easily as plugins.
Figure 4: HDF5 Data Transfer Pipeline.
The flexibility of HDF5 permits to use different strategies to store our
fields of view according to our application. The guideline is to think how
images will be retrieved and used. For example if we want to get the fields of
view as a planar image then we should use the same shape for the dataset, i.e.
if the image shape is $(H,W)$ then the dataset shape should be $(N_{w}\,H,W)$
where $N_{w}$ is the number of colour planes. Like this we can map directly
the data from storage to memory. The planar format is usually more suited for
analysis purpose, but if we want to privilege the display then we should
choose an interleaved format. However we cannot use an interleaved format in
practice if we consider there is an offset between the colour fields of view.
To store the mosaic we can use a dataset per field of view or pack everything
in only one dataset. This second approach would be the natural choice if we
had reconstructed the slide image. For example if the mosaic shape is $(R,C)$
then we can create a dataset of shape $(R\,N_{w}\,H,C\,W)$ with a chunk size
of $(h,w)$ where $(H,W)=(n\,h,n\,w)$ and $n\in\mathbb{Z}^{*+}$. Figure 5 shows
an example of a packed mosaic. The induced overhead will be smoothed by the
fact the images are stored on disk as chunks.
Figure 5: A dataset for a $2\times 2$ mosaic, chunks are represented by dotted
squares.
However if we want to load at the same time a set of consecutive tiles, then
we can use this linear dataset shape $(R\,C\,N_{w}\,H,W)$ and index the image
using the linearised index $r\,C+c$. Figure 6 shows an example of a linearised
mosaic. For example the code to get the fields of view in the slice
$[10,20:30]$ would be:
lower_index = 10*C + 20upper_index = 10*C + 30field_of_view_step = NW *
Hlower_r = lower_index * field_of_view_stepupper_r = upper_index *
field_of_view_stepmemory_map = image_dataset[lower_r:upper_r,:]And to get from
here the wth colour plane of the ith field of view, the code would be:
row_offset = i * field_of_view_step + w * Hcolour_image =
memory[row_offset:row_offset +H,:]If the mosaic is sparse we can still pack
the mosaic and use a bisection algorithm to perform a binary search to get the
corresponding linear index used for the storage.
Figure 6: A linear dataset for an acquisition having 3 colours where the
pointer to a tile and a plane are shown.
One can argue this approach is not natural, but encapsulating the slice
computation in a virtual slide API allows for efficient ways to store and
retrieve the data. A better approach would be to have a direct access to the
chunks, but actually the HDF5 API does not provide such facility (it only
provides direct chunk write up to now). Thus if we do not want to rewrite or
extend the library, the hyperslab mechanism is a nice alternative. However if
we dislike this packing method, we can still use the following dataset layout
$(R,C,N_{w},H,W)$ with this chunk layout $(1,1,1,H,W)$, where the slicing is
more natural. Anyway the right approach is to test several dataset layouts and
measure the I/O performance, using the tool _h5perf_ provided with the HDF5
SDK. More details about chunking can be found in the reference
[HDF5-Chunking].
This storage method can be easily extended to a more complicated acquisition
scheme having z-stacks or a time dimension.
### 3.1 Remote Virtual Slide
We have now defined a framework to store our virtual slide based on top of the
stack HDF5/h5py that relies on an HDF5 file stored on a local system or a
network file system to work in a client-server manner. This framework works
perfectly, but a network file system has some limitations in comparison to a
real client-server framework. In particular a network file system is complex
and has side effects on an IT infrastructure, for example the need to setup an
authentication mechanism for security. Moreover we cannot build a complex
network topology made of a virtual slide broadcast server and clients.
We will now introduce the concept of remote virtual slide so as to add a real
client-server feature to our framework. We have two types of data to send over
the network, the slide header and the images. Since images are a flow of
bytes, it is easy to send them over the network and use the Blosc [Blosc]
real-time compression algorithm to reduce the payload. For the slide header,
we can serialise the set of attributes to a JSON [JSON] string, since the
attributes data types are numbers, strings and tuples of them.
For the networking layer, we use the ZeroMQ [ZMQ] library and its Python
binding PyZMQ [PyZMQ]. ZeroMQ is a socket library that acts as a concurrency
framework, carries message across several types of socket and provides several
connection patterns. ZeroMQ is also an elegant solution to the global
interpreter lock [GIL] of the CPython interpreter that prevent real multi-
threading. Indeed the connection patterns and the message queues offer a
simple way to exchange data between processes and synchronise them. This
library is notably used by IPython [IPython] for messaging.
The remote virtual slide framework is build on the request-reply pattern to
provide a client-server model. This pattern can be used to build a complex
network topology with data dealer, router and consumer.
## 4 Microscope Interconnection
As a first illustration of the remote virtual slide concept, we will look at
the data flow between the automated microscope so called _scanner_ and the
software component, so called _slide writer_ , whose aim is to write the HDF5
file on the file server. This client-server or producer-consumer framework is
shown on figure 7. To understand the design of this framework, we have to
consider these constrains. The first one is due to the fact that the producer
does not run at the same speed than the consumer. Indeed we want to maximise
the scanner throughput and at the same time maximise the data compression
which is a time consuming task. Thus there is a contradiction in our
requirements. Moreover the GIL prevents real time multi-threading. Thus we
must add a FIFO buffer between the producer and the consumer so as to handle
the speed difference between them. This FIFO is called _slide proxy_ and acts
as an image cache. The second constraint is due to the fact that the slide
writer can complete its job after the end of scan. It means the slide writer
will not be ready to process another slide immediately, which is a drawback if
we want to scan a batch of slides. Thus we need a third process called _slide
manager_ whose aim is to fork a slide writer for each scan that will itself
fork the slide proxy. Due to this fork mechanism, these three processes, slide
manager, slide writer and slide proxy must run on same host so called _slide
server_. For the other component, all the configurations can be envisaged.
The last component of this framework is the slide database whose aim is to
store the path of the HDF5 file on the slide server so as to retrieve the
virtual slide easily.
Figure 7: Virtual Slide Writer Architecture.
## 5 Slide Viewer Graphic Engine
The slide viewer graphic engine works as Google Map using image tiles and
follows our concept to reconstruct the slide image online. We can imagine
several strategies to reconstruct the slide image. The first one would be to
perform all the computation on CPU. But nowadays we have GPU that offer a
higher level of parallelism for such a task. GPUs can be programmed using
several API like CUDA, OpenCL and OpenGL [OpenGL]. The first ones are more
suited for an exact computation and the last one for image rendering. In the
followings we are talking about modern OpenGL where the fixed pipeline is
deprecated in favour of a programmable pipeline.
The main features of the slide viewer are to manage the viewport, the zoom
level and to provide an image processing to render a patchwork of 16-bit
images. All these requirements are fulfilled by OpenGL. The API provides a way
to perform a mapping of a 2D texture to a triangle and by extension to a
quadrilateral which is a particular form of a triangle strip. This feature is
perfectly suited to render a tile patchwork.
The OpenGL programmable pipeline is made of several stages. For our topic, the
most important ones are the vertex shader, the rasterizer and the fragment
shader, where a fragment corresponds to a pixel. The vertex shader is mainly
used to map the scene referential to the OpenGL window viewport. Then the
rasterizer generates the fragments of the triangles using a scanline algorithm
and discards fragments which are outside the viewport. Finally a fragment
shader provides a way to perform an image processing and to manage the zoom
level using a texture sampler. Figure 8 shows an illustration of the texture
painting on the viewport.
Figure 8: OpenGL viewport and texture painting. The overlapped black
rectangles represent the mosaic of tiles. The red rectangle shows the viewport
area. And the blue rectangle illustrates the rendering of a texture for a tile
which is partially out of the viewport area. The horizontal line represents
the sampling of the triangle defined by the vertexes (1, 2, 3) using a
scanline algorithm. Pixels out of the viewport are discarded.
A texture can have from one to four colour components (RGBA), which make easy
to render a slide acquisition with up to four colours. To render more colours,
we just need more than one texture by tile and a more complicated fragment
shader. If the tiles are stored in a planar format then we have to convert
them to an interleaved format, we call this task texture preparation. However
we can also use a texture per colour but in this case we have to take care to
the maximal number of texture slots provided by the OpenGL implementation,
else we have to perform a framebuffer blending. The main advantage of using a
multi-colour texture is for efficiency since the colour processing is
vectorised in the fragment shader. However if we want to register the colour
on-line, then the texture lookup is any more efficient.
To render the viewport, the slide viewer must perform several tasks. First it
must find the list of tiles that compose the viewport and load these tiles
from the HDF5 file. Then it must prepare the data for the corresponding
textures and load them to OpenGL. The time consuming tasks are the last three
ones. In order to accelerate the rendering, it would be judicious to perform
these tasks in parallel, which is not simple using Python.
For the tile loading, we can build on our remote virtual slide framework in
order to perform an intelligent read-ahead and to eventually prepare the data
for the texture.
The parallelisation of the texture loading is the most difficult part and it
relies of the OpenGL implementation. Modern OpenGL Extension to the X Window
server (GLX) supports texture loading within a thread, but this approach
cannot be used efficiently in Python due to the GIL. Moreover we cannot use a
separate process to do that since it requires processes could share an OpenGL
context, which is only available for indirect rendering (glXImportContextExt).
Also we cannot be sure the multi-threading would be efficient in our case due
to the fact we are rendering a subset of the mosaic at a time and thus
textures have a short life time. And the added complexity could prove to be a
drawback.
Since our mosaic can be irregular, we cannot found by a simple computation
which tiles are in the viewport. Instead we use an R-tree for this purpose.
All the tiles boundaries are filled in the R-tree. And to get the list of
tiles within the viewport, we perform an intersection query of the R-tree with
the viewport boundary.
### 5.1 Slide Viewer Architecture
Figure 9: Slide Viewer Architecture.
Figure 9 shows the architecture of our slide viewer where the virtual slide
API can access the data through the HDF5 file or the remote framework. In our
IT infrastructure, HDF5 files are stored on a file server that can provide a
network file system to access files remotely. The remote virtual slide can be
used in two different ways according to the machine where the process of the
server side, called _tile dealer_ , is executed. If this process runs on the
same host as the slide viewer, then we can use it to implement a read-ahead
mechanism to parallelise the tile loading. And if it runs on the file server,
then we can use it as an alternative to the network file system in a similar
way as a virtual slide broadcast service. This second example demonstrates the
remote virtual slide is a fundamental software component in our framework that
open the way to many things.
Another way to access efficiently the data, it to use a local cache to store
temporally the virtual slide. Nowadays we can build on a very fast locale
cache using a PCI-e SSD card, which commonly reach a read/write bandwidth of
$1000\,\text{MB/s}$ and thus outperforms most of the hardware RAID.
The slide viewer implements two Least Recently Used caches to store the tiles
and the textures. These caches are a cornerstone for the fluidity of the
navigation within the slide, since it helps to reduce the viewer latency.
Nowadays we can have on a workstation $64\,\text{GB}$ of RAM for a decent
cost, which open the way to a large in memory cache in complement to a PCI-e
SSD cache. In this way we can build a 3-tier system made of a file server to
store tera bytes of data, a PCI-e SSD cache to store temporally slides and an
in memory cache to store a subset of the virtual slide.
### 5.2 Vertex and Fragment Shader
In modern OpenGL all the computations must be performed by hand from the
viewport modelling to the fragment processing, excepted the texture sampling
which is provided by the OpenGL Shading Language.
Since we are doing a two dimensional rendering, it simplifies considerably the
viewport model and the coordinate transformation. OpenGL discards all the
fragment that are outside the $[-1,1]\times[-1,1]$ interval. Thus to manage
the viewport, we have to transform the slide frame coordinate using the
following model matrix:
$\left(\begin{array}[]{c}x\\\ y\\\ z\\\ w\\\
\end{array}\right)=\left(\begin{array}[]{cccc}\frac{2}{x_{sup}-x_{inf}}&0&0&-\frac{x_{inf}+x_{sup}}{x_{sup}-x_{inf}}\\\
0&\frac{2}{y_{sup}-y_{inf}}&0&-\frac{y_{inf}+y_{sup}}{y_{sup}-y_{inf}}\\\
0&0&1&0\\\ 0&0&0&1\\\ \end{array}\right)\left(\begin{array}[]{c}x_{s}\\\
y_{s}\\\ 0\\\ 1\\\ \end{array}\right)$ (1)
where $[x_{inf},x_{sup}]\times[y_{inf},y_{sup}]$ is the viewport interval and
$(x_{s},y_{s})$ is a coordinate in the slide frame.
OpenGL represents fragment colour by a normalised float in the range $[0,1]$
and values which are outside this range are clamped. Thus to transform our
16-bit pixel intensity we have to use this formula:
$\hat{l}=\frac{l-I_{inf}}{I_{sup}-I_{inf}}$ (2)
where $0\leq I_{inf}<I_{sup}<2^{16}$. This normalisation can be used to
perform an image contrast by adjusting the values of $I_{inf}$ and $I_{sup}$.
The fact OpenGL supports the unsigned 16-bit data type for texture permits to
load the raw data directly in the fragment shader without information loss.
According to the configuration of OpenGL, the RAMDAC of the video adapter will
convert the normalised floats to an unsigned 8-bit intensity for a standard
monitor or to 10-bit for high-end monitor like DICOM compliant models.
As soon as we have converted our pixel intensities to float, we can apply some
image processing treatments like a gamma correction for example.
In the previous paragraphs, we told we can load in a texture up to four colour
components using RGBA textures. Since monitors can only render three colour
components (RGB), we have to transform a four components colour space to a
three components colour space using a _mixer matrix_. This computation can be
easily extended to any number of colours using more than one texture. The
mixer matrix coefficients should be choose so as to respect the normalised
float range.
Another important feature of the slide viewer is to permit to the user to
select which colours will be displayed on the screen. This feature is easily
implemented using a diagonal matrix so called _status matrix_ with its
coefficients set to zero or one depending of the colour status.
We can now write the matrix computation for the rendering of up to four
colours:
$\left(\begin{array}[]{c}r\\\ g\\\ b\\\
\end{array}\right)=\underbrace{\left(\begin{array}[]{ccc}m_{r0}&\ldots&m_{r3}\\\
m_{g0}&\ldots&m_{g3}\\\ m_{b0}&\ldots&m_{b3}\\\
\end{array}\right)}_{\text{mixer
matrix}}\underbrace{\left(\begin{array}[]{ccc}s_{0}&&\\\ &\ddots&\\\
&&s_{3}\\\ \end{array}\right)}_{\text{status
matrix}}\left(\begin{array}[]{c}\hat{l}_{0}\\\ \vdots\\\ \hat{l}_{3}\\\
\end{array}\right)$ (3)
If we consider a GPU with more than 1024 cores, then most of the rows of our
display will be processed in parallel which is nowadays impossible to perform
with a multi-core CPU. It is why our approach to render a mosaic of tiles is
so efficient and the rendering is nearly done in real time.
### 5.3 Zoom Layer
When the texture must be magnified, it is important to enlarge the pixel
without interpolation. In OpenGL it is achieved by using the _GL_NEAREST_ mode
for the texture magnification filter.
Despite GPU are very powerful, there is a maximal number of tiles in the
viewport that can be reasonably processed. The amount of memory of the GPU is
an indicator of this limitation. If we consider a GPU with $2048\,\text{MB}$,
then we can load 66 textures having a layout of $2560\times 2160\,\text{px}$
and a 16-bit RGB format. It means we can display a mosaic of $8\times 8$ at
the same time. If we want to display more tiles at the same time, then we have
to compute a so called _mipmaps_ which is a pyramidal collection of mignified
textures. Usually we perform a geometric series that corresponds to divide by
two the size of the texture recursively. Due to the power of the GPU, it is
not necessary to compute the entire pyramid, but just some levels. In our case
we can compute the levels 8 and 16. For higher levels according to the size of
the mosaic, it could be more efficient to compute a reconstructed image. These
mignified textures can be computed online using CUDA or stored in the HDF5
files.
Our slide viewer implements a zoom manager in order to control according to
the current zoom which zoom layer is active and to limit the zoom amplitude to
an appropriate range. Moreover we can implement some excluded zoom ranges and
force the zoom to the nearest authorised zoom according to the zoom direction.
Figure 10: Cell displayed in the slide viewer. The slide was acquired with an
epifluorescence-microscope at magnification $40\times$ with a camera of
resolution $1392\times 1040\,\text{px}$ and with four colours. The size of the
part of the mosaic shown on the viewport is $19\times 22$ , which corresponds
to 418 tiles and thus around $595\,\text{Mpx}$. The dimension of the visible
surface is around $4.9\times 3.1\,\text{mm}$. Here the slide image is rendered
at magnification $2.5\times$ and the zoom layer corresponds to a mignification
of level $2^{4}=16$ and thus to a texture of dimension $87\times
65\,\text{px}$. So there is around $2\,\text{Mpx}$ to process.
### 5.4 Detection Layer
Our slide viewer is not limited to display raw images, but can also display
tiles from an image processing pipeline. When the viewer render a viewport, it
first looks which tiles compose the viewport, then for each tile, it looks if
the OpenGL LRU cache has a texture for the corresponding tile and image
processing pipeline, if the texture does not exists yet then it cascades the
request to the tile LRU cache and finally it will asks the image processing
pipeline to generate the image. The tile loading from the virtual slide
corresponds to the so called raw image pipeline and each zoom layer owns its
image pipeline. Moreover each pipeline can have its own fragment shader to
customise the rendering.
### 5.5 Benchmark
Figure 10 show a reconstructed image made of 418 tiles. For a tile dimension
of $1392\times 1040\,\text{px}$ and a four colours acquisition, our slide
viewer needs around $2\,\text{s}$ to render the zoom layer 16 and
$6\,\text{s}$ for the layer 8 (100 raw tiles) on a workstation with a Xeon
E5-1620 CPU, a GeForce GTX-660 GPU and the HDF5 file stored on a local SATA
hard disk. The required time to load a tile form the HDF5 file is around
$50\,\text{ms}$, thus the tile loading account for $80\,\%$ of the full
rendering time.
## 6 Conclusion
This paper gives an overview how the Python ecosystem can be used to build a
software platform for high-content digital microscopy. Our achievement
demonstrates Python is well suited to build a framework for big data. Despite
Python is a high level language, we can handle a large amount of data
efficiently by using powerful C libraries and GPU processing.
First we gave an overview how to store and handle virtual slides using Python,
Numpy and the HDF5 library. Different methods to store the images of the
fields of view within a dataset was discussed. In particular the case where we
do not reconstruct an image of slide once and for all, but rather perform an
on-line reconstruction from the raw images. Despite our method to store the
images works well, it would be interesting to look deeper in the HDF5 library
to see if we could do something still better.
We described the concept of remote virtual slide which is a client-server
model build on top of our virtual slide framework. We gave two examples of
utilisation of this client-server model, the scanner interconnection with the
slide writer and the tile dealer. Also we shown how this architecture solve
the GIL problem and enhance the performance.
Finally we described our slide viewer architecture based on the OpenGL
programmable pipeline and a texture patchwork rendering. We gave an overview
on the vertex and the fragment shader. Thanks to the power of GPU, this method
can render more than three colours in quasi real time. Moreover we explained
how to manage the zoom level efficiently so as to overcome the limited amount
of RAM of the GPU.
In a near future, it would be interesting to see how the JIT Python
interpreter PyPy will enhance the performance of this framework. Up to now the
lake of support of C library like Numpy and Qt prevents to run the code with
it.
The Git repository https://github.com/FabriceSalvaire/PyOpenGLV4 provides an
oriented object API on top of PyOpenGL to work with the OpenGL programmable
pipeline. This module is used in our slide viewer.
## References
* [BigTIFF] Ole Eichhorn of Aperio, http://bigtiff.org
* [Blosc] Francesc Alted, http://blosc.org, https://github.com/FrancescAlted/python-blosc
* [GIL] http://www.dabeaz.com/python/UnderstandingGIL.pdf
* [HDF5] HDF Group, http://www.hdfgroup.org/HDF5
* [h5py] Andrew Collette and contributers, http://www.h5py.org
* [HDF5-Chunking] http://www.hdfgroup.org/HDF5/doc/Advanced/Chunking/index.html, http://www.hdfgroup.org/HDF5/doc/Advanced/Chunking/Chunking_Tutorial_EOS13_2009.pdf, http://www.hdfgroup.org/HDF5/doc/Advanced/DirectChunkWrite/UsingDirectChunkWrite.pdf
* [IPython] http://ipython.org/ipython-doc/stable/development/messaging.html
* [JSON] http://www.json.org
* [LZF] Andrew Collette http://www.h5py.org/lzf, Marc Lehmann http://oldhome.schmorp.de/marc/liblzf.html
* [Numpy] Travis Oliphant and Numpy developers, http://www.numpy.org
* [OpenGL] Khronos Group, http://www.opengl.org
* [PyOpenGL] http://pyopengl.sourceforge.net
* [PyZMQ] https://github.com/zeromq/pyzmq
* [ZMQ] iMatix Corporation, http://zeromq.org
|
arxiv-papers
| 2014-04-25T10:54:26 |
2024-09-04T02:50:01.830002
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fabrice Salvaire",
"submitter": "Pierre de Buyl",
"url": "https://arxiv.org/abs/1404.6385"
}
|
1404.6387
|
# PySTEMM: Executable Concept Modeling for K-12 STEM Learning
Kelsey D’Souza∗† * Corresponding author: [email protected]† Senior at
Westwood High SchoolCopyright © 2014 Kelsey D’Souza. This is an open-access
article distributed under the terms of the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are credited.
http://creativecommons.org/licenses/by/3.0/
###### Abstract
Modeling should play a central role in K-12 STEM education, where it could
make classes much more engaging. A model underlies every scientific theory,
and models are central to all the STEM disciplines (Science, Technology,
Engineering, Math). This paper describes executable concept modeling of STEM
concepts using immutable objects and pure functions in Python. I present
examples in math, physics, chemistry, and engineering, built using a proof-of-
concept tool called PySTEMM . The approach applies to all STEM areas and
supports learning with pictures, narrative, animation, and graph plots. Models
can extend each other, simplifying getting started. The functional-programming
style reduces incidental complexity and code debugging.
###### Index Terms:
STEM education, STEM models, immutable objects, pure functions
## 1 Introduction
A _model_ is a simplified representation of part of some world, focused on
selected aspects. A model underlies every scientific theory, and models are
central to all STEM areas — science, technology, engineering, and mathematics
— helping us conceptualize, understand, explain, and predict phenomena
objectively. Children form mental models and physical models during play to
understand their world. Scientists use bio-engineered tissue as a model of
human organs. Computational modeling is revolutionizing science and
engineering, as recognized by the 2013 Nobel Price in Chemistry going for
computational modeling of biochemical systems.
Previous research [Whi93], [Orn08] has shown significant learning benefits
from model-building and exploring in STEM education. Students should create,
validate, refute, and use models to better understand deep connections across
subject areas, rather than mechanically drilling through problems. In this
paper I demonstrate that executable concept modeling, based on using immutable
objects and pure functions in Python:
* •
applies across multiple STEM areas,
* •
supports different representations and learning modes,
* •
is feasible and approachable,
* •
encourages bottom-up exploration and assembly, and
* •
builds deep understanding of underlying concepts.
### 1.1 Executable Concept Models
A _concept model_ describes something by capturing relevant concepts,
attributes, and rules. A _concept instance_ is a specific individual of a
_concept type_ e.g. NO2 is a concept instance of the general concept type
Molecule. The concept type Molecule might define a formula attribute for any
molecule instance to list how many atoms of each element it contains. The
concept instance NO2 has one Nitrogen and two Oxygen atoms. This is similar to
the idea from object-oriented programming of an object that is an instance of
a class.
Concepts and attributes are chosen to suit a purpose. A different model of
Molecule might describe atoms, functional groups, bonds, sites at which other
molecules can interact, site geometry, and forces that govern geometry and
interactions.
An _executable concept model_ is represented on a computer, so concept
instances and concept types can be manipulated and checked by the machine,
increasing confidence in the model.
### 1.2 PySTEMM Models
“Executable” typically entails programming language complexity, debugging
headaches, and distractions from the actual concepts under study. Much of this
complexity stems from _imperative programming_ , where variables and object
attributes are modified as the program executes its procedures.
_Functional programming_ is a good alternative. It uses (a) _immutable
objects_ , whose attribute values do not get modified by program code; and (b)
_pure functions_ , producing a result that depends solely on inputs, without
modifying any other attributes or variables.
PySTEMM, by using immutable objects and pure functions, and providing multiple
model representations, reduces needless complexity and debugging. It uses the
_Python_ programming language to define executable concept models that have
three parts:
* 1.
Structure: A concept type is defined by a Python _class_ that describes
attributes together with their types (which can reference other concept
types). A concept instance is a Python _object_ instantiated from that class,
with values for its attributes.
* 2.
Functions: The pure functions that represent additional properties or rules on
concept instances are defined as Python _methods_ on the class1.
* 3.
Visualization: The visualization of concept types and instances are defined
with Python _dictionaries_ of visual properties, used as _templates_.
PySTEMM models focus on defining _what terms and concepts mean_ , rather than
step-by-step instructions about _how to compute_. PySTEMM functions manipulate
not just numbers, but molecules, rigid bodies, planets, visualizations, and
even concept types and
functions.\raisebox{10.00002pt}{\hypertarget{id4}{}}\hyperlink{id3}{1}\raisebox{10.00002pt}{\hypertarget{id4}{}}\hyperlink{id3}{1}footnotetext:
Since we use methods on a class for functions, in "a.f(x)" the inputs to f
include argument x, and the object a on which the method is invoked.
In the rest of this paper I present example models from math, chemistry,
physics, and engineering, introduce key aspects of PySTEMM, and show Python
model source code as well as multiple model representations generated by
PySTEMM. The last section describes the implementation of PySTEMM.
## 2 Mathematics
We begin with models of math functions, because math forms the basis of all
other models. Next we move on to _high-order_ functions i.e. functions that
accept functions as inputs, or whose results are functions. Since our focus in
this section is modeling math concepts, we will model math functions as
objects. In subsequent sections on physics, chemistry, etc., we will directly
use normal Python code for math computations.
### 2.1 Basic Numeric Functions
Figure 1: Three Function concept types.
The Python model of _concept types_ for basic functions is:
1 # file: function_types.py23 class Function(Concept):4 domain =
Property(List(Int))5 def eval(self, x): pass6 class_template =
{K.gradient_color: ’Green’}78 class RuleFunction(Function):9 rule = Callable10
domain = List(Int)1112 def eval(self, x):13 return self.rule(x)1415
class_template = {K.gradient_color: ’Yellow’}1617 class
TableFunction(Function):18 points = List(Tuple(Int, Int))19 domain =
Property(List(Int))2021 def _get_domain(self):22 return [x for x, y in
self.points]2324 def eval(self, x):25 return find(y1 for x1,y1 in
self.points26 if x1==x)2728 class_template = {K.gradient_color: ’Maroon’}29
instance_template = {K.name: ’Circle’}The concept type Function is defined as
a class (line 3), with an attribute domain which is a list of integers (line
4). "Property" allows domain to be represented differently for different
subclasses of Function. Function evaluation is modeled by method eval (line 5)
whose specifics are deferred to subclasses. The visualization of functions is
defined by class_template (line 6).
We define two subclasses of Function, each with different representations.
RuleFunctions (line 8-15) are defined by an attribute rule that is a Python
_callable_ expression, an explicit domain, and eval that simply invokes rule.
TableFunctions (line 17-29) are defined by a list of (x,y) pairs in an
attribute points, a domain computed from points by _get_domain, and eval that
finds the matching pair in points. The class_template (lines 15, 28) is a
dictionary of visualization properties for the concept type, and
instance_template (line 29) is for visualizing instances. PySTEMM generates
the visual and English narrative in Figure 1 for these concept types.
Figure 2: TableFunction concept instance.
Below, we _extend_ this model with a TableFunction instance tf with its list
of points (line 4), and customize what the model should visualize:
1 # file function_instances.py2 from function_types.py import *34 tf =
TableFunction(points=[(1, 10), (2, 15)])56 M = Model()7 M.addInstances(tf)8
M.showMethod(tf, ’eval’)9 M.showEval(tf,’eval’,[1])
PySTEMM generates the visualization in Figure 2. The domain of tf was
calculated from its points, its value at x=1 is 10, and the code for eval() is
shown in the context of the instance. Since eval is a _pure function_ ,
tf.eval(1) depends solely on the input 1 and the definition of tf itself, so
it is easy to understand the source code: it returns the y1 from the x1,y1
pair that matches the input x.
Note that tf is drawn as a circle of the same color as the TableFunction
class: the instance_template for TableFunction is merged with the
class_template before being applied to tf.
### 2.2 Inverse Functions
Figure 3: InverseFunction type and instance.
An InverseFunction inverts another: $g=f^{-1}(x)$. The model below extends the
function_instances model with a class and an instance. On line 5, the
InverseFunction(...) constructor is a _high-order function_ corresponding to
the $f^{-1}$ operator, since it receives a function tf to invert, and produces
the new inverted function inv.
1from function_instances import *23class InverseFunction(Concept): ...45inv =
InverseFunction(inverts=tf)67M.addClasses(InverseFunction)8M.addInstances(inv)9M.showEval(inv,
’eval’,[15])The instance visualization generated by PySTEMM in Figure 3 shows
the inverse function as a blue square, its eval() effectively flips the (x,y)
pairs of the function it inverts, and its domain is computed as the set of y
values of the function it inverts.
### 2.3 Graph Transforms and High-Order Functions
Figure 4: Function Transforms: A Bump of a Shift of $x^{2}$.
A graph transformation as taught in middle school — translation, scaling,
rotation — is modeled as a function that operates on a source function,
producing the transformed function. In Figure 4, PySTEMM generates a graph
plot of the original function, a shifted version, and a “bumped” version of
the shifted function. The instances are defined as:
Bump(source = ShiftX(source = RuleFunc(rule=square), by=3), start=0, end=5,
val=100)Similarly, the _limit_ of a function is a high-order function: it
operates on another function and a target point, and evaluates to a single
numeric value. Calculus operators, such as _differentiation_ and _integration_
, can be modeled as high-order functions as well: they operate on a function
and produce a new function.
## 3 Chemistry: Reaction
Figure 5: Reaction concept type.
Figure 6: An instance of Reaction.
1class Element(Concept):2 name = String34class Molecule(Concept):5 formula =
List(Tuple(Element, Int))6 instance_template = {7 K.text: lambda m:
computed_label(m)}89class Reaction(Concept):10 products = List(Tuple(Int,
Molecule))11 reactants = List(Tuple(Int, Molecule))An Element is modeled as
just a name, since we ignore electron and nuclear structure. A Molecule has an
attribute formula with a list of pairs of element with a number indicating the
number of atoms of that element. A Reaction has reactants and products, each
some quantity of a certain molecule. This Python model is visualized by
PySTEMM in Figure 5.
Note that convenient Python constructs, like _lists_ of _tuples_ , are
visualized in a similarly convenient manner. Also, the instance_template for
molecule (lines 6-7), specifying the visualization properties for a molecule
instance, contains a _function_ which takes a molecule instance and computes
its label. Visualization templates are parameterized by the objects they will
be applied to.
Figure 6 shows an instance of a reaction, showing reaction structure and
computed labels for reactions and molecules, while hiding the formula
structure within molecules.
### 3.1 Reaction Balancing
Figure 7: Reaction balance matrix and solved coefficients.
Our next model computes reaction balancing for reactions. An unbalanced
reaction has lists ins and outs of molecules without coefficients. Figure 7
shows how PySTEMM visualizes a reaction with the balance computation,
coefficients, and intermediate values, as explained below.
We formulate reaction-balancing as an _integer-linear programming_ problem
[Sen06], which we solve for molecule coefficients. The formula of the
molecules constrain the coefficients, since atoms of every element must
balance. The function elem_balance_matrix computes a matrix of _molecule_ vs.
_element_ , with the number of atoms of each element in each molecule, with +
for ins and - for outs. This matrix multiplied by the vector of coefficients
must result in all 0. All coefficients have to be positive integers
(diagonal_matrix), and the objective_function seeks the smallest coefficients
satisfying these constraints.
Once we have balanced reactions, we can add attributes and functions to model
reaction stoichiometry and thermodynamics. For example:
class Element(Concept): name = String atomic_mass = Floatclass
Molecule(Concept): formula = List(Tuple(Element, Int)) molar_mass =
Property(Float) def _get_molar_mass(self): return sum([n * el.atomic_mass for
el, n in self.formula])Fe = Element(name=’Fe’, atomic_mass=56)Cl =
Element(name=’Cl’, atomic_mass=35.5)FeCl2 = Molecule(formula=[(Fe,1),
(Cl,2)])FeCl2.molar_mass # = 127
### 3.2 Reaction Network
class Network(Concept): reactions = List(Reaction)R1 = Reaction(reactants=[(2,
NO2)], products=[(1, NO3), (1, NO)])R2 = Reaction(reactants=[(1, NO3), (1,
CO)], products=[(1, NO2), (1, CO2)])Net = Network(reactions=[R1, R2])
Figure 8: A reaction Network with two reactions.
A Network of coupled chemical reactions has a list of reactions. Given this
Python model, and a narrative template for Reaction, PySTEMM generates Figure
8, including the _instance-level_ English narrative. Just as there are element
balance constraints on an individual reaction, we could model network-level
constraints on the reaction rates and concentrations of chemical species, but
have not shown this here.
### 3.3 Layered Models
Figure 9: Layered concept models and generated math.
The reaction examples illustrate an important advantage of PySTEMM modeling;
instead of directly modeling the mathematics of reaction, we focus on the
structure of the concept instances; in this case, what constitutes a molecule,
or a reaction?
From this model, we compute the math model. The math version of a molecule is
a single column with the number of atoms of each element type in that
molecule. The math for a reaction collects this column from each molecule and
combines them into an element_balance_matrix. Pure functions thus easily
traverse the concept instances to build corresponding math models such as
matrices of numbers.
## 4 Physics
Figure 10: Ball in motion: functions of time as code, graphs, animation
Below is a model of the motion of a ball under constant force. The ball has
vector-valued attributes for initial position, velocity, and forces (lines
2,3). The functions acceleration, velocity, and position are pure functions of
time and use numerical integration. We visualize ball b via showGraph and
animate (lines 18-19). Like all visualizations, the animation is specified by
a _template_ (line 21): a dictionary of visual properties, except that these
properties can be _functions_ of the _object_ being animated, and the _time_
at which its attributes values are computed.
1class Ball(Concept):2 mass, p0, v0 = Float, Instance(vector), ...3 forces =
List(vector)4 def net_force(self):5 return v_sum(self.forces)6 def
acceleration(self, time):7 return self.net_force() / self.mass8 def
velocity(self, time):9 return self.v0 + v_integrate(self.acceleration, time)10
def position(self, time):11 return self.p0 + v_integrate(self.velocity,
time)1213 def p_x(self, time): ....14 def p_y(self, time): ....1516b =
Ball(p0=..., v0=..., mass=..., forces=...)17m = Model(b)18m.showGraph(b,
(’a_y’,’v_y’,’p_y’), (0,10))19m.animate(b,20 (0,10),21 [{K.new: K.shape,22
K.origin: lambda b,t: [b.p_x(t), b.p_y(t)]]},23 {K.new: K.line,
point_list=lambda b,t: ...},24 {K.new: K.line, point_list=lambda b,t: ...}]
)PySTEMM generates graphs of the time-varying functions, and a 2-D animation
of the position and velocity vectors of the ball over time (Figure 10).
## 5 Engineering
Figure 11: ROV made of PVCPipes.
In Summer 2012 I attended the OEX program at MIT, where we designed and built
a marine remote-operated vehicle (ROV) with sensors to monitor water
conditions. I later used PySTEMM to recreate models of the ROV, and generate
engineering attributes and 3-D visualizations like Figure 11.
The ROV is built from PVCPipes in a functional style. To create several
PVCPipes positioned and sized relative to each other, the model uses pure
functions like shift and rotate that take a PVCPipe and some geometry, and
produce a transformed PVCPipe. This makes it simple to define parametric
models and rapidly try different ROV structures. The model shown excludes
motors, micro-controller, and computed drag, net force, and torque.
class PVCPipe(Concept): length, radius, density = Float, Float, Float def
shift(self, v): return PVCPipe(self.p0 + v, self.r, self.axis) def
rotate(self, a): return PVCPipe(self.p0, self.r, self.axis + a)class
ROV(Concept): body = List(PVCPipe) def mass(self): ... def
center_of_mass(self): ... def moment_of_inertia(self): ...p1 = PVCPipe(....)p2
= p1.shift((0,0,3), ...)c1, c2 = p1.rotate((0,0,90))...rov = ROV(body=p1, p2,
c1, c2)
## 6 Implementation
### 6.1 Architecture
The overall architecture of PySTEMM, illustrated in Figure 12, has two main
parts: _Tool_ and _Model Library_. The _tool_ manipulates _models_ ,
traversing them at the type and instance level and generating visualizations.
The _model library_ includes the models presented in this paper and any
additional models any PySTEMM user would create. The _tool_ is implemented
with 3 classes:
* •
Concept: a superclass that triggers special handling of the concept type to
process attribute-type definitions.
* •
Model: a collection of concepts classes and concept instances, configured with
some visualization.
* •
View: an interface to a drawing application scripted via AppleScript.
Figure 12 explains the architecture in more detail, and lists external modules
that were used for specific purposes. PySTEMM uses the Enthought traits module
[Tra14] to define attribute types for a concept. Traits provides a class
HasTraits with a custom meta-class, and pre-defined traits such as List,
Tuple, String, and Int. The Concept class derives from HasTraits, which
triggers traits to capture concept attribute type definitions and generate
constructor and attribute logic for checked attribute assignment.
Figure 12: Architecture of PySTEMM.
We gain several benefits by building models with immutable objects and pure
functions:
* •
The _user models_ can be manipulated by the _tool_ more easily to provide tool
capabilities like animation and graph-plotting, based on evaluating pure
functions at different points in time.
* •
The values of computed attributes and other intermediate values can be
visualized as easily and unambiguously as any stored attributes.
* •
Debugging becomes much less of an issue since values do not change while
executing a model, and the definitions parallel the math taught in school
science.
The source code for PySTEMM is available at https://github.com/kdz/pystemm.
### 6.2 Python
Python provides many advantages to this project:
* •
adequate support for high-order functions and functional programming;
* •
lightweight and flexible syntax, with convenient modeling constructs like
lists, tuples, and dictionaries;
* •
good facilities to manipulate classes, methods, and source code;
* •
vast ecosystem of open-source libraries, including excellent ones for
scientific computing.
### 6.3 Templates
All visualization is defined by _templates_ containing visual property values,
or functions to compute those values:
Concept_Template = { K.text: lambda concept: computeClassLabel(concept),
K.name: ’Rectangle’, K.corner_radius: 6, ... K.gradient_color: "Snow"}The
primary operation on a template is to _apply_ it to some modeling object,
typically a concept class or instance:
def apply_template(t, obj, time=None): # t.values are drawing-app values, or
functions # obj: any object, passed into template functions # returns copy of
t, F(obj) replaces functions F if isinstance(t, dict): return {k:
apply_template(v, obj, time) for k, v in t.items()} if isinstance(t, list):
return [apply_template(x, obj, time) for x in t] if callable(t): return t(obj)
if arity(t)==1 else t(obj, time) return tAnimation templates have special case
handling, since their functions take two parameters: the _instance_ to be
rendered, and the _time_ at which to render its attributes.
Templates can also be _merged_. Figure 2 shows an instance of TableFunction as
a circle in the same color as the TableFunction class, by merging an
instance_template with a class_template.
## 7 Summary
I have described PySTEMM as a tool, model library, and approach for building
executable concept models for a variety of STEM subjects. The PySTEMM
approach, using immutable objects and pure functions in Python, can apply to
all STEM areas. It supports learning through pictures, narrative, animation,
and graph plots, all generated from a single model definition, with minimal
incidental complexity and code debugging issues. Such modeling, if given a
more central role in K-12 STEM education, could make STEM learning much more
deeply engaging.
## References
* [Whi93] White, Barbara Y. “ThinkerTools: Causal Models, Conceptual Change, and Science Education”, Cognition and Instruction, Vol. 10, No. 1.
* [Orn08] Ornek, Funda. “Models in Science Education: Applications of Models in Learning and Teaching Science”, International Journal of Environmental & Science Education, 2008.
* [Edw04] Edwards, Jonathan. “Example Centric Programming”, The College of Information Sciences and Technology (Pennsylvania State University: 2004), http://www.subtext-lang.org/OOPSLA04.pdf
* [Fun13] "9.8. Functools — Higher-order Functions and Operations on Callable Objects.", Python Software Foundation, 2013. http://docs.python.org/2/library/functools.html.
* [Bla07] Blais, Martin. “True Lieberman-style Delegation in Python." Active State Software, 2007, http://code.activestate.com/recipes/519639-true-lieberman-style-delegation-in-python/.
* [Sen06] Sen, S. K., Hans Agarwal, and Sagar Sen. “Chemical Equation Balancing: An Integer Programming Approach”, Mathematical and Computer Modeling, Vol. 44, No.7-8, 2006.
* [Tra14] Enthought Traits Library, http://code.enthought.com/projects/traits/
|
arxiv-papers
| 2014-04-25T10:55:44 |
2024-09-04T02:50:01.837812
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kelsey D'Souza",
"submitter": "Pierre de Buyl",
"url": "https://arxiv.org/abs/1404.6387"
}
|
1404.6388
|
# Performance of Python runtimes on a non-numeric scientific code
Riccardo Murri∗† * Corresponding author: [email protected]† University
of Zurich, Grid Computing Competence CenterCopyright © 2014 Riccardo Murri.
This is an open-access article distributed under the terms of the Creative
Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original author and source are
credited. http://creativecommons.org/licenses/by/3.0/
###### Abstract
The Python library FatGHol [FatGHoL] used in [Murri2012] to reckon the
rational homology of the moduli space of Riemann surfaces is an example of a
non-numeric scientific code: most of the processing it does is generating
graphs (represented by complex Python objects) and computing their
isomorphisms (a triple of Python lists; again a nested data structure). These
operations are repeated many times over: for example, the spaces $M_{0,6}$ and
$M_{1,4}$ are triangulated by 4’583’322 and 747’664 graphs, respectively.
This is an opportunity for every Python runtime to prove its strength in
optimization. The purpose of this experiment was to assess the maturity of
alternative Python runtimes, in terms of: compatibility with the language as
implemented in CPython 2.7, and performance speedup.
This paper compares the results and experiences from running FatGHol with
different Python runtimes: CPython 2.7.5, PyPy 2.1, Cython 0.19, Numba 0.11,
Nuitka 0.4.4 and Falcon.
###### Index Terms:
python runtime, non-numeric, homology, fatgraphs
## 1 Introduction
The moduli space $M_{g,n}$ of smooth Riemann surfaces is a topological space
which has been subject of much research both in algebraic geometry and in
string theory. It is known since the ’90s that this space has a triangulation
indexed by a special kind of graphs [Penner1988, Kontsevich1992,
ConantVogtmann2003], nicknamed "fat graphs".
Since graphs are combinatorial and discrete objects, a computational approach
to the problem of computing topological invariants of $M_{g,n}$ is now
feasible; algorithms to enumerate fatgraphs and compute their graph homology
have been devised in [Murri2012] and implemented in Python.
We propose an experiment whose purpose is to assess the maturity of
alternative Python runtimes, in terms of:
* (a)
compatibility with the language as implemented in CPython 2.7, and
* (b)
performance speedup.
In particular, we were interested in the possible speedups of a large non-
numeric code.
## 2 Experiment setup
The FatGHoL <http://fatghol.googlecode.com/> [FatGHoL] program was used as a
test code. FatGHoL computes homology of the moduli spaces of Riemann surfaces
$M_{g,n}$ via Penner-Kontsevich’s fatgraph simplicial complex [Penner1988,
Kontsevich1992]. Homology is one of the most important invariants of
topological spaces. There are several homology theories but they all share
this computational procedure outline: given a vector space of (generalized)
_simplex chains_ and a _boundary operator_ , which is by definition a linear
operator $D$ such that $D^{2}=0$, the homology space is by definition
$\mathop{\textrm{Ker }}D/\mathop{\textrm{Im }}D$. In graph homology, however,
it is the computation of these simplices and boundary that takes up the
largest fraction of compute time: the simplex chains are defined as formal
linear combinations of graphs, and the boundary operator maps a graph into a
linear combination of graphs obtained by contracting its edges. Thus, explicit
construction of the simplices requires enumerating all distinct isomorphism
classes of fatgraphs, and then computing their mutual relationships upon
contraction of edges.
The FatGHoL program runs in three stages:
* 1.
generate fatgraphs,
* 2.
explicitly compute the boundary operator in matrix form,
* 3.
actually solve the homology linear system.
The last step has been disabled in the test code as it is implemented in C++
for speed. What remains is 100% pure Python code that runs on Python 2.6+ (but
could run on 2.5 with minimal modifications).
FatGHoL involves a large number of graph isomorphism computations: especially
during fatgraph generation, each candidate fatgraph needs to be compared to
all fatgraphs already discovered, in order to avoid duplicates. In later
stages, the isomorphism computations are cached in memory, but in step 2
additional data is created for each graph, in order to pass from fatgraphs to
simplices.
It is worth noting that the FatGHoL code exercises many of Python’s advanced
data manipulation features, like list and dictionary comprehensions, slicing,
etc. but does not use any kind of tight nested loops of the kind normally
featured in numeric codes.
Profiling data show more precisely how much work is done at the Python level
in the simplest case $M_{0,4}$. The following listing shows profiling data
extracted from a complete run of FatGHoL on CPython 2.7.5; 15787953 function
calls (15728052 primitive calls) were effected in 39.572 seconds; the top 10
most called functions, ordered by call count are:
>
> ncalls tottime filename:lineno(function)
> 2216088 2.175 rg.py:227(<genexpr>)
> 966575 0.819 rg.py:143(is_loop)
> 775362 0.839 cyclicseq.py:88(__getitem__)
> 775362 0.634 rg.py:170(other_end)
> 722308 3.438 combinatorics.py:368(__init__)
> 539039 1.689 cyclicseq.py:112(__getslice__)
> 506075/447917 0.745 cache.py:181(wrapper)
> 476134 1.122 combinatorics.py:441(rearranged)
> 385725 0.355 rg.py:137(__init__)
> 345740 0.849 rg.py:568(_first_unused_corner)
>
The FatGHoL code was run with seven different alternative Python runtimes:
* •
CPython 2.7.5;
* •
Cython 0.19.1;
* •
Cython 0.19.1 in "pure Python mode";
* •
Falcon 0.05;
* •
Nuitka 0.4.4;
* •
PyPy 2.1;
* •
Numba 0.10.0 and 0.11.0 with @autojit.
A detailed description of each of these is given in a later section; Table I
provides an overview of the installation and usage features of the different
runtimes. The code used to install the software and run the experiments is
available on GitHub at https://github.com/riccardomurri/python-runtimes-
shootout.
Runtime | _Cython 0.19.1_ | _Falcon 0.05_ | _Nuitka 0.4.4_ | _Numba 0.11.0_ | _PyPy 2.1_
---|---|---|---|---|---
_Installed size_ (MB) | 30 a | 14 a | 25 a | 97 a (+ 518MB of LLVM 3.2) | 162 b
_Install script length_ (SLOC) | 6 | 8 | 10 | 24 | 19
_Usage documentation_ | extensive | minimal | short how-to to explain the different compilation options available | minimal, mostly examples | none
_Porting/optimization documentation_ | extensive | none | list of optimizations that the runtime does (or will) support | none | provides only a list of compatibility issues; I could find no list of _Do_ -s and _Don’t_ -s for better speed in PyPy
_Porting/optimization effort_ | none ("Pure Python" mode) to very heavy (.pxd hinting) | none: runs unmodified Python code | none: runs unmodified Python code | light (w/ @autojit) to heavy (@jit with types) | none: runs unmodified Python code
TABLE II: Comparison of installation features of the Python runtimes. a Plus
123MB for the CPython interpreter, which is anyway needed. b Does not need the
CPython interpreter in addition, as all others do.
Except for Cython in "pure Python mode" and Numba, all runtimes run the
unmodified Python code of FatGHoL. Cython in "pure Python mode" requires the
addition of decorators to the Python code that specify the types of function
arguments and local variables to increase speedup of selected portions of the
code. Similarly, Numba uses the decorators @jit or @autojit to mark functions
that should be compiled to native code (the difference between the two
decorators is that that @autojit infers types at runtime, whereas @jit
requires the programmer to specify them1); we only used the @autojit decorator
to mark the same functions that were marked as optimization candidates in the
Cython experiment.\raisebox{10.00002pt}{\hypertarget{no-more-
autojit}{}}\hyperlink{id10}{1}\raisebox{10.00002pt}{\hypertarget{no-more-
autojit}{}}\hyperlink{id10}{1}footnotetext: Note that in more recent versions
of Numba, the two decorators have been fused into one: @jit uses the supplied
types, or infers them if none are given.
Each Python runtime was run on 4 test cases: computing the homology of the
$M_{0,4}$, $M_{0,5}$, $M_{1,3}$, and $M_{2,1}$ moduli spaces. The test cases
take from 0.20s to more than 2 minutes of runtime with CPython 2.7. Each test
case was run 10 times and the best time and lowest RAM occupation are reported
in the summary tables below.
## 3 Results
Falcon and Numba could not run the code (see details in a later section) and
thus do not appear in the report below.
For each runtime, the total used CPU time and memory were measured: results
and summary graphs are given in figures 1 and 2. Detailed comparisons are
given in the other figures.
Figure 1: Comparison of the total CPU time used by each runtime on the
different test cases. The $x$-axis is sorted so that the runtimes for CPython
2.7.5 are ascending. The $y$-axis shows values in seconds (smaller is better).
Note that the $y$-axis is drawn on a logarithmic scale!
The CPU time data prompt a few observations:
* •
PyPy gives the best results, provided the code runs long enough to discount
for the startup time of the JIT compiler. Given enough time, the JIT compiler
gives extremely good results, with speedups of 100% to 400% relative to
CPython in the $M_{0,5}$ and $M_{1,3}$ cases. In other words, for the JIT
approach to pay off, the code needs to perform many iterations of the same
code path (this is certainly the case for FatGHoL), because compiling a single
function to native code takes a non-negligible amount of time. The break-even
point for the FatGHoL code seems to be around 5 seconds of runtime: on
$M_{2,1}$, the CPU time taken by CPython and PyPy are almost equal.
* •
Cython gives consistently about a 30% speedup on unmodified Python code.
However, the "pure Python mode", in which Cython takes variable typing hints
embedded in the code does not seem to give any advantage: results of the two
runs are not significantly different. This might be related to a bug in the
current version of Cython, see details in a later section.
Figure 2: Comparison of the total RAM used by each runtime on the different
test cases. The $x$-axis is sorted so that the RAM usage for CPython 2.7.5 are
ascending. The $y$-axis shows values in MBs (smaller is better). Note that the
$y$-axis is drawn on a logarithmic scale!
The large memory consumption from PyPy and Nuitka stands out in the memory
data of figure 2. On the other hand, there is no significant increase in
memory usage between CPython and Cython.
The large memory usage of PyPy can be explained by the fact that the JIT
infrastructure must keep in memory the profile and traces for all the code
paths taken. In any long-running program, the memory should eventually reach a
steady state and not increase any further; it should be noted however, that in
these benchmarks the memory used by the PyPy JIT framework dwarfs the memory
used by the program itself.
We have no explanation for the large memory consumption of Nuitka.
Figure 3: Comparison of the total CPU time used by each runtime on the
$M_{0,4}$ test case. The $x$-axis shows values in seconds.
Figure 4: Comparison of the total CPU time used by each runtime on the
$M_{0,5}$ test case. The $x$-axis shows values in seconds.
Figure 5: Comparison of the total CPU time used by each runtime on the
$M_{1,3}$ test case. The $x$-axis shows values in seconds.
Figure 6: Comparison of the total CPU time used by each runtime on the
$M_{2,1}$ test case. The $x$-axis shows values in seconds.
Figure 7: Comparison of the total RAM usage by each runtime on the $M_{0,4}$
test case. The $x$-axis shows values in MBs.
Figure 8: Comparison of the total RAM usage by each runtime on the $M_{0,5}$
test case. The $x$-axis shows values in MBs.
Figure 9: Comparison of the total RAM usage by each runtime on the $M_{1,3}$
test case. The $x$-axis shows values in MBs.
Figure 10: Comparison of the total RAM usage by each runtime on the $M_{2,1}$
test case. The $x$-axis shows values in MBs.
## 4 Runtime systems details
### 4.1 Cython 0.19.1 <http://cython.org/>
Cython is a compiler for a superset of the Python language. It translates
Python modules to a C or C++ source that is then compiled to a native code
library that CPython can load and use. Cython optimizes best when users
decorate the source code with hints at the types of variables and functions;
it can also translate unmodified Python code, but then no type inference is
performed. Cython allows a variety of ways for giving these type hints; its
so-called "pure Python" mode requires users to insert functions and variable
decorators in the code: the Cython compiler can act on these directives, but
the CPython interpreter will instead load a cython module which turns them
into no-ops.
We tested Cython twice: on the unmodified Python sources, and with hinting in
the "pure Python" mode. The graphs show however very little difference between
the two modes; this could be a consequence of Cython defect ticket #477.
Cython does its best when the source code is annotated with its extended
keywords, which allow specifying the types of variables (which allows
optimizations, e.g., in loops), or marking certain functions as C-only (which
saves time when dereferencing variables). This extended markup can be provided
either in the sources, or in additional .pxd files. We have not done this
exercise, however, as the amount of coding time required to properly mark all
functions and variables is quite substantial.
### 4.2 Falcon 0.05 <https://github.com/rjpower/falcon>
Falcon is a Python extension module that hacks into a CPython interpreter and
changes the execution loop, implementing several optimizations (for instance,
using a register-based VM instead of a stack-based one) that the Falcon
authors think should be used in the upstream CPython interpreter too. However,
Falcon is still in early stages of development and crashes on FatGHoL code
with a segmentation fault.
### 4.3 Numba <http://numba.pydata.org/>
As its website states:
> > Numba is an optimizing compiler for Python; it uses the LLVM compiler
> infrastructure to compile Python syntax to machine code. It is NumPy-aware
> and can speed up code using NumPy arrays. Other, less well-typed code will
> be translated to Python C-API calls effectively removing the "interpreter"
> but not removing the dynamic indirection. Numba is also not a Just-In-Time
> compiler.
Numba requires the code developer to use either the @autojit (use run-time
type info) or the @jit (explicitly provide type information) decorators to
mark those functions that should be compiled. For our experiment, we used the
decorator @autojit on all functions that were decorated also in the Cython
test.
Versions 0.10.0 and 0.11.0 of Numba were tested; we could not get either
version to work.
Numba version 0.10.0 dies with an internal error ("TypeError: type_container()
takes exactly 1 argument (3 given)", reported as Issue #295 on Numba’s GitHub
issue tracker), that has been fixed in version 0.11.
However, Numba 0.11.0 with a "NotImplementedError: Unable to cast from { i64,
i8* }* to { i64, i8* }" message. This has been reported as Issue #350 on the
issue tracker and is waiting for a fix.
### 4.4 Nuitka 0.4.4 <http://www.nuitka.net/>
Nuitka translates Python (2.6+) into a C++ program that then uses libpython to
execute in the same way as CPython does, in a very compatible way. Although
still in development, Nuitka claims that it already:
> > create[s] the most efficient native code from this. This means to be fast
> with the basic Python object handling.
Results of this experiment do not seem to corroborate this claim.
### 4.5 PyPy 2.1 <http://pypy.org/>
PyPy is a Python language interpreter with a Just-In-Time compiler (and many
other features!). It can thus translate repetitive Python code into native
code on the fly. PyPy must first be bootstrapped by compiling itself, which
takes a lot of time and RAM, but then it is a drop-in replacement for the
python command and just works.
## 5 Acknowledgements
The author acknowledges support of the Informatik Dienste of the University of
Zurich, particularly for the usage of the new SGI UV 2000 machine for running
the tests. I would also like to thank Kay Hayen, Marc Florisson, Russel Power
and Alex Rubynstein for their readiness to discuss and fix the bugs I reported
on Nuitka, Numba, and Falcon. Finally, I would like to express my gratitude to
all those who made remarks and inquiries at the EuroSciPy poster session, and
particularly Ronan Lamy and Denis Engemann for their insightful comments.
Finally, I would like to thank Mike Mueller and Pierre de Buyl for reviewing
the initial draft paper and making many useful suggestions for improving it.
## References
* [Murri2012] R. Murri. _Fatgraph Algorithms and the Homology of the Kontsevich Complex_ , arXiv preprint arXiv:1202.1820, February 2012.
* [FatGHoL] R. Murri. _The FatGHoL software website_ , http://fatghol.googlecode.com/
* [Penner1988] R. C. Penner. _Perturbative series and the moduli space of Riemann surfaces_ , J. Differential Geom, 1988.
* [Kontsevich1992] M. Kontsevich. _Formal (non)-commutative symplectic geometry_ , The Gelfand Mathematical Seminars, 1990–1992.
* [ConantVogtmann2003] J. Conant, K. Vogtmann. _On a theorem of Kontsevich_ , Algebr. Geom. Topol., 2003.
|
arxiv-papers
| 2014-04-25T10:55:48 |
2024-09-04T02:50:01.843710
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Riccardo Murri",
"submitter": "Pierre de Buyl",
"url": "https://arxiv.org/abs/1404.6388"
}
|
1404.6389
|
# Computing an Optimal Control Policy for an Energy Storage
Pierre Haessig∗†, Thibaut Kovaltchouk†, Bernard Multon†, Hamid Ben Ahmed†,
Stéphane Lascaud‡ * Corresponding author: [email protected]† SATIE
CNRS laboratory - ENS Rennes, Bruz, France‡ LME department - EDF R&D,
Écuelles, FranceCopyright © 2014 Pierre Haessig et al. This is an open-access
article distributed under the terms of the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are credited.
http://creativecommons.org/licenses/by/3.0/
###### Abstract
We introduce StoDynProg, a small library created to solve Optimal Control
problems arising in the management of Renewable Power Sources, in particular
when coupled with an Energy Storage System. The library implements generic
Stochastic Dynamic Programming (SDP) numerical methods which can solve a large
class of Dynamic Optimization problems.
We demonstrate the library capabilities with a prototype problem: smoothing
the power of an Ocean Wave Energy Converter. First we use time series analysis
to derive a stochastic Markovian model of this system since it is required by
Dynamic Programming. Then, we briefly describe the “policy iteration”
algorithm we have implemented and the numerical tools being used. We show how
the API design of the library is generic enough to address Dynamic
Optimization problems outside the field of Energy Management. Finally, we
solve the power smoothing problem and compare the optimal control with a
simpler heuristic control.
###### Index Terms:
Stochastic Dynamic Programming, Policy Iteration Algorithm, Autoregressive
Models, Ocean Wave Energy, Power Smoothing.
## 1 Introduction to Power Production Smoothing
Electric power generated by renewable sources like wind, sun or ocean waves
can exhibit a strong _variability_ along time. Because on an electricity grid
the energy production must match the consumption, this variability can be an
issue for the grid stability. Yet most of the time, fluctuations of renewable
power sources are absorbed without trouble thanks to regulation mechanisms
which make flexible generation units adjust their production in real-time.
Therefore, the production-consumption equilibrium can be maintained.
However, there are cases where fluctuations may be considered too strong to be
fed directly to the grid so that an _energy storage system_ , acting as a
_buffer_ , may be required to smooth out the production. The schematic of the
system considered in this article is given on figure 1.
Figure 1: Power smoothing with an Energy Storage: an example of an Optimal
Control problem.
### 1.1 Smoothing with an Energy Storage
Electricity generation from _ocean waves_ (with machines called Wave Energy
Converters) is an example where the output power can be _strongly
fluctuating_. This is illustrated on figure 2 where the output power
$P_{prod}(t)$ from a particular wave energy converter called SEAREV is
represented during 100 seconds.
We just mention that this production time series does not come from
measurements but from an hydro-mechanical simulation from colleagues since the
SEAREV is a big 1 MW - 30 meters long machine which is yet to be built
[Ruellan-2010].
The oscillations of $P_{prod}(t)$ at a period of about 1.5 s comes from the
construction of the SEAREV: in short, it is a floating _double-pendulum_ that
oscillates with the waves. Also, because _ocean waves have a stochastic
behavior_ , the amplitude of these oscillations is irregular.
Figure 2: Smoothing the Ocean Power injected to the grid using an Energy
Storage controlled by the simple linear law (3). The storage buffers the
difference between the two powers.
Therefore an energy storage absorbing a power $P_{sto}$ can be used to smooth
out the power $P_{grid}$ injected to the electricity network:
$P_{grid}(t)=P_{prod}(t)-P_{sto}(t)$ (1)
The energy of the storage then evolves as:
$E_{sto}(k+1)=E_{sto}(k)+P_{sto}(k)\Delta t$ (2)
expressed here in discrete time ($\Delta t=0.1\text{ s}$ throughout this
article), without accounting for losses. The storage energy is bounded: $0\leq
E_{sto}\leq E_{rated}$, where $E_{rated}$ denotes the storage capacity which
is set to 10 MJ in this article (i.e. about 10 seconds of reserve at full
power)
It is a control problem to choose a power smoothing law. We present the
example of a linear feedback control:
$P_{grid}(t)=\frac{P_{max}}{E_{rated}}E_{sto}(t)$ (3)
where $P_{max}$ is the rated power of the SEAREV (1.1 MW). This law gives
“good enough” smoothing results as it can be seen on figure 2.
The performance of the smoothing is greatly influenced by the _storage sizing_
(i.e. the choice of the capacity $E_{rated}$). This question is not addressed
in this article but was discussed by colleagues [Aubry-2010]. We also don’t
discuss the choice of the storage _technology_ , but it is believed that
super-capacitors would be the most suitable choice. Because energy storage is
very expensive (~20 k€/kWh or ~5 k€/MJ for supercaps), there is an interest in
studying how to make the best use of a given capacity to avoid a costly over-
sizing.
### 1.2 Finding an Optimal Smoothing Policy
Control law (3) is an example of heuristic choice of policy and we now try to
go further by finding an _optimal_ policy.
Optimality will be measured against a _cost function_ $J$ that penalizes the
average variability of the power injected to the grid:
$J=\frac{1}{N}\mathbb{E}\left\\{\sum_{k=0}^{N-1}c(P_{grid}(k))\right\\}\quad\text{with
$N\rightarrow\infty$}$ (4)
where $c$ is the _instantaneous cost_ (or penalty) function which can be
$c(P_{grid})=P_{grid}^{2}$ for example. Expectation $\mathbb{E}$ is needed
because the production $P_{prod}$ is a stochastic input, so that the output
power $P_{grid}$ is also a random variable.
This minimization problem falls in the class of _stochastic dynamic
optimization_. It is _dynamic_ because decisions at each time-step cannot be
taken independently due to the coupling along time introduced by the evolution
of the stored energy (2). To describe the dynamics of the system, we use the
generic notation
$x_{k+1}=f(x_{k},u_{k},\varepsilon_{k})$ (5)
where $x,u,\varepsilon$ are respectively _state_ variables, _control_
variables and _perturbations_. State variables are the “memory” of the system.
The stored energy $E_{sto}$ is here the only state variable, but more will
appear in section 2.2. Control variables are the ones which values must be
chosen at each instant to optimize the cost $J$. The injected power $P_{grid}$
is here the single control variable.
Dynamic optimization (also called _optimal control_) is addressed by the
Dynamic Programming method [Bertsekas-2005] which yields a theoretical
analysis of the solution structure. Indeed, once all state variables (i.e.
“memories”) of the system are identified, the optimum of the cost $J$ is
attained by a “state feedback” policy, that is a policy where the control is
chosen as _a function of the state:_
$P_{grid}(t)=\mu(x(t))$ (6)
The goal is then to find the _optimal_ feedback function $\mu$. Since
$E_{sto}$ is a state variable, policy (3) is in fact a special case of (6).
Since $\mu$ has no special structure in the general case1, it will be
_numerically computed on a grid_ over the state space. We cover the algorithm
for this computation in section 3.
#### 1.2.1 Prerequisite
Dynamic Programming does require that stochastic perturbations are
_independent_ random variables (i.e. the overall dynamical model must be
Markovian) and this is not true for the $P_{prod}(k)$ time series. Therefore
we devote section 2 to the problem of expressing $P_{prod}$ as a discrete-time
Markov process, using _time series analysis_. This will yield new state
variables accounting for the dynamics of $P_{prod}$.
## 2 Stochastic Model of a Wave Energy Production
We now take a closer look at the $P_{prod}$ time series. A 1000 s long
simulation is presented on figure 4, along with a zoom to better see the
structure at short time scales. An histogram is also provided which shows that
$P_{prod}$ is clearly _non-gaussian_. This precludes the direct use of
“standard” time series models based on Autoregressive Moving Average (ARMA)
models [Brockwell-1991].
However, we can leverage the knowledge of the inner working of the SEAREV.
Indeed, by calling $\Omega$ the rotational speed of the inner pendulum with
respect to the hull, we know that the output power is:
$P_{prod}=T_{PTO}(\Omega).\Omega$ (7)
where $T_{PTO}$ is the torque applied to the pendulum by the electric machine
which harvests the energy (PTO stands for “Power Take Off”). Finding the best
$T_{PTO}$ command is actually another optimal control problem which is still
an active area of research in the Wave Energy Conversion community
[Kovaltchouk-2013]. We use here a “viscous damping law, with power leveling”,
that is $T_{PTO}(\Omega)=\beta.\Omega$. This law is applied as long as it
yields a power below $P_{max}$. Otherwise the torque is reduced to level the
power at 1.1 MW as can be seen on figure 4 whenever the speed is more than 0.5
rad/s.
Thanks to equation (7), we can thus model the speed $\Omega$ and then deduce
$P_{prod}$. Modeling the speed is much easier because it is quite Gaussian
(see fig. 4) and has a much more regular behavior which can be captured by an
ARMA process.
### 2.1 Autoregressive Model of the Speed
Within the ARMA family, we restrict ourselves to the autoregressive (AR)
processes because we need a Markovian model. The equation of an AR(p) model
for the speed is:
$\Omega(k)=\phi_{1}\Omega(k-1)+\dots+\phi_{p}\Omega(k-p)+\varepsilon(k)$ (8)
where $p$ is the order of the model and $\varepsilon(k)$ is a series
independent random variables. Equation (8) indeed yields a Markovian process,
using the lagged observations of the speed $\Omega(k-1),\dots,\Omega(k-p)$ as
state variables.
AR(p) model fitting consists in _selecting_ the order $p$ and _estimating_ the
unknown coefficients $\phi_{1},\dots,\phi_{p}$ as well as the unknown variance
of $\varepsilon$ which we denote $\sigma_{\varepsilon}^{2}$.
Figure 3: Autocorrelation function (acf) of the speed data, compared with the
acf from two AR(2) models, fitted with two different methods.
Figure 4: Speed & Power time series from a 1000 seconds SEAREV simulation
(sample Em_1.txt). The gray rectangle time interval is enlarged in the middle
panel. Distribution histogram on the right.
#### 2.1.1 Order selection
is generally done using _information criterions_ such as AIC or BIC
[Brockwell-1991], but for this modeling problem, we restrict ourselves to the
smallest order which can reproduce the _decaying oscillations_ of the
autocorrelation function. Autocorrelation (acf) of the speed is plotted on
figure 3 where we can see that a model of order $p=2$ can indeed reproduce the
autocorrelation up to about 15 s of time lags (a 1st order model would only
yield an exponential decay without oscillations). 15 s is thought to be the
time horizon of interest when using a 10 MJ/1.1 MW energy storage.
Keeping the model order low is required to maintain the dimension of the
overall state vector under 3 or 4. The underlying issue of an exponentially
growing complexity will appear in section 3 when solving the Dynamic
Programming equation.
#### 2.1.2 Parameter estimation
Once the order is selected, we have to estimate coefficients $\phi_{1}$,
$\phi_{2}$ and $\sigma_{\varepsilon}^{2}$. “Classical” fitting methodology
[Brockwell-1991] is based on Conditional Maximum Likelihood Estimators (CMLE).
This method is readily available in GNU R with the arima routine or in Python
with statsmodel.tsa.ar_model.
However, we have plotted the autocorrelation of the estimated AR(2) model on
figure 3 to show that CMLE is _not appropriate:_ oscillations of the acf
clearly decay too slowly compared to the data acf.
The poor adequacy of this fit is actually a consequence of our choice of a low
order model which implies that the AR(2) process can only be an _approximation
of the true process_. Statistically speaking, our model is _misspecified_ ,
whereas CMLE is efficient for correctly specified models only. This problem
has been discussed in the literature [McElroy-2013] and has yielded the
“Multi-step ahead fitting procedure”.
Being unfamiliar with the latter approach, we compute instead
$\phi_{1},\phi_{2}$ estimates which _minimize the difference_ between the
theoretical AR(2) acf and the data acf. The minimization criterion is the sum
of the squared acf differences over a range of lag times which can be chosen.
We name this approach the “multi-lags acf fitting” method. Minimization is
conducted with fmin from scipy.optimize.
method | $\hat{\phi}_{1}$ | $\hat{\phi}_{2}$ | $\hat{\sigma}_{\varepsilon}$
---|---|---|---
CMLE | 1.9883 (.0007) | -0.9975 (.0007) | 0.00172
fit on 15 s | 1.9799 | -0.9879 | 0.00347
TABLE II: AR(2) fitting results from the two methods (along with standard
error when available).
The result of this acf fitting over time lags up to 15 s (i.e. 150 lags) is
shown on figure 3 while numerical estimation results are given in table I.
With the model obtained from this multi-lags method, we can simulate speed and
power trajectories and check that they have a “realistic behavior”. We can
thus infer that the dynamic optimization algorithm should make appropriate
control decisions out of it. This will be discussed in section 3.3. Going
further, it would be interesting to study the influence of the AR parameters
(including order $p$) on the dynamic optimization to see if the “multi-lags
acf fitting” indeed brings an improvement of the final cost function $J$.
### 2.2 Reformulation as a state-space model
The AR(2) model is a state-space model with state variables being the lagged
observations of the speed $\Omega(k-1)$ and $\Omega(k-2)$. In order to get a
model with a better “physical interpretation” we introduce the variable
$A_{k}=(\Omega_{k}-\Omega_{k-1})/\Delta t$ which is the backward discrete
derivative of $\Omega$. As the timestep gets smaller $A_{k}$ comes close to
the acceleration (in rad/s2) of the pendulum. Using $(\Omega,A)$ as the state
vector, we obtain the following state-space model:
$\begin{split}\begin{pmatrix}\Omega_{k}\\\
A_{k}\end{pmatrix}=&\begin{bmatrix}\phi_{1}+\phi_{2}&-\phi_{2}\Delta t\\\
(\phi_{1}+\phi_{2}-1)/\Delta
t&-\phi_{2}\end{bmatrix}\begin{pmatrix}\Omega_{k-1}\\\ A_{k-1}\end{pmatrix}\\\
+&\begin{bmatrix}1\\\ 1/\Delta t\end{bmatrix}\varepsilon_{k}\end{split}$ (9)
We now have a stochastic Markovian model for the power production of the
SEAREV. Taken together with state equation of the storage (2) and algebraic
relations (1) and (7), we have a Markovian model of the overall system. The
state vector $x=(E_{sto},\Omega,A)$ is of dimension 3 which is just small
enough to apply the Stochastic Dynamic Programming method.
## 3 Optimal storage control with Dynamic Programming
### 3.1 The Policy Iteration Algorithm
We now give an overview of the _policy iteration_ algorithm that we
implemented to solve the power smoothing problem described in the
introduction. Among the different types of dynamic optimization problems, it
is an “infinite horizon, average cost per stage problem” (as seen in (4)).
While at first this cost equation involves a summation over an infinite number
of instants, the Dynamic Programming approach cuts this into two terms: the
present and the whole future. In the end, the optimization falls back to
solving this equilibrium equation:
$\begin{split}J+\tilde{J}(x)=\min_{u\in
U(x)}\underset{w}{\mathbb{E}}\Big{\\{}\underbrace{c(x,u,w)}_{\text{instant
cost}}\\\ +\underbrace{\tilde{J}(f(x,u,w))}_{\text{cost of the
future}}\Big{\\}}\end{split}$ (10)
where $J$ is the minimized average cost and $\tilde{J}$ is the transient (or
differential) cost function, also called _value function_.
Note that eq. (10) is a functional equation for $\tilde{J}$ which should be
solved for _any value_ of the state $x$ in the state space. In practice, it is
solved in a _discrete grid_ that must be chosen so that the variations of
$\tilde{J}$ are represented with enough accuracy. Also, the optimal policy
$\mu$ appears implicitly as the _argmin_ of this equation, that is the optimal
control $u$ for each $x$ value of the state grid.
#### 3.1.1 Equation solving
The simplest way to solve eq. (10) is to iterate the right-hand side, starting
with a zero value function. This is called _value iteration_.
A more efficient approach is _policy iteration_. It starts with an initial
policy (like the heuristic linear (3)) and gradually improves it with a two
steps procedure:
* 1.
policy evaluation: the current policy is evaluated, which includes computing
the average cost (4) and the so-called _value function_
* 2.
policy improvement: a single step of optimization with policy iteration yields
a improved policy. Then this policy must be again evaluated (step 1).
The policy evaluation involves solving the equilibrium equation without the
minimization step but with a fixed policy $\mu$ instead:
$\begin{split}J_{\mu}+\tilde{J}_{\mu}(x)=\underset{w}{\mathbb{E}}\Big{\\{}\underbrace{c(x,\mu(x),w)}_{\text{instant
cost}}\\\ +\underbrace{\tilde{J}_{\mu}(f(x,\mu(x),w))}_{\text{cost of the
future}}\Big{\\}}\end{split}$
It can be solved by iterating the right-hand side like for policy iteration
but much faster due to the absence of minimization. In the end, a few policy
improvement iterations are needed to reach convergence. More details about the
value and policy iteration algorithms can be found in [Bertsekas-2005]
textbook for example. The conditions for the convergence, omitted here, are
also discussed.
### 3.2 StoDynProg library description
We have created a small library to _describe_ and _solve_ optimal control
problems (in discrete time) using the Stochastic Dynamic Programming (SDP)
method. It implements the value iteration and policy iteration algorithms
introduced above. Source code is available on GitHub
https://github.com/pierre-haessig/stodynprog under a BSD 2-Clause license.
#### 3.2.1 Rationale for a library, benefits of Python
Because the SDP algorithms are in fact quite simple (they can be written with
one set of nested for loops) we were once told that they should be written
from scratch for each new problem. However we face in our research in energy
management several optimization problems with slight structural differences so
that code duplication would be unacceptably high. Thus the motivation to write
a unified code that can handle all our use cases, and hopefully some others’.
StoDynProg is pure Python code built with numpy for multi-dimensional array
computations. We also notably use an external multidimensional interpolation
routine by Pablo Winant (see 3.2.5 below).
The key aspect of the flexibility of the code is its ability to handle
problems of _arbitrary dimensions_ (in particular the state space and the
control space). This impacts particularly the way to iterate over those
variables. Our code makes thus a heavy use of Python tuple packing/unpacking
machinery and itertools.product to iterate on rectangular grids of arbitrary
dimension.
#### 3.2.2 API description
StoDynProg provides two main classes: SysDescription and DPSolver.
#### 3.2.3 SysDescription
holds the description of the discrete-time dynamic optimization problem.
Typically, a user writes its dynamics function (the Python implementation of
$f$ in (5)) and handles it to a SysDescription instance:
from stodynprog import SysDescription# SysDescription object with proper
dimensions# of state (2), control (1) and perturbation (1)mysys =
SysDescription((2, 1, 1))def my_dyn(x1, x2, u, w): ’dummy dynamics’ x1_next =
x1 + u + w x2_next = x2 + x1 return (x1_next, x2_next)# assign the dynamics
function:mysys.dyn = my_dynWe use here a setter/getter approach for the dyn
property. The same is used to describe the cost function ($c$ in (4)). We
believe the property approach enables simplified user code compared to a class
inheritance mechanism. With some inspiration of Enthought traits, the setter
has a basic validation mechanism that checks the signature of the function
being assigned (with getargspec from the inspect module).
#### 3.2.4 DPSolver
holds parameters that tunes the optimization process, in particular the
discretized grid of the state. Also, it holds the code of the optimization
algorithm in its methods. We illustrate here the creation of the solver
instance attached to the previous system:
from stodynprog import DPSolver# Create the solver for ‘mysys‘ system:dpsolv =
DPSolver(mysys)# state discretizationx1_min, x1_max, N1 = (0, 2.5, 100)x2_min,
x2_max, N2 = (-15, 15, 100)x_grid = dpsolv.discretize_state(x1_min, x1_max,
N1, x2_min, x1_max, N2)Once the problem is fully described, the optimization
can be launched by calling dpsolv.policy_iteration with proper arguments about
the number of iterations.
For more details on StoDynProg API usage, an example problem of _Inventory
Control_ is treated step-by-step in the documentation (created with Sphinx).
#### 3.2.5 Multidimensional Interpolation Routine
StoDynProg makes an intensive use of a multidimensional interpolation routine
that is not available in the “standard scientific Python stack”. Interpolation
is needed because the algorithm manipulates two scalar functions which are
discretized on a grid over the state space: the value function $\tilde{J}$ and
feedback policy $\mu$. Thus, functions are stored as $n$-d arrays, where $n$
is the dimension of the state vector ($n=3$ for ocean power smoothing
example). In the course of the algorithm, the value function needs to be
evaluated between grid points, thus the need for interpolation.
#### 3.2.6 Requirements and Algorithm Selection
No “fancy” interpolation method is required so linear interpolation is a good
candidate. Speed is very important because it is called many times. Also, it
should accept vectorized inputs, so that interpolation of multiple points can
be done efficiently in one call. We assert that the functions will be stored
on a _rectangular grid_ which should simplify interpolation computations. The
most stringent requirement is _multidimensionality_ (for $0\leq n\leq 4$)
which rules out most available tools.
We have evaluated 4 routines (details available in an IPython Notebook within
StoDynProg source tree):
* •
LinearNDInterpolator class from scipy.interpolate
* •
RectBivariateSpline class from scipy.interpolate
* •
map_coordinates routine from scipy.ndimage
* •
and MultilinearInterpolator class written by Pablo Winant within its Dolo
project [Winant-2010] for Economic modelling (available on
https://github.com/albop/dolo).
The most interesting in terms of performance and off-the-shelf availability is
RectBivariateSpline which exactly meets our needs expect for
multidimensionality because it’s limited to $n=2$. LinearNDInterpolator has no
dimensionality limitations but works with unstructured data and so does not
leverage the rectangular structure. Interpolation time was found 4 times
longer in 2D, and unacceptably long in 3D. Then map_coordinates and
MultilinearInterpolator were found to both satisfy all our criterions but the
latter being consistently 4 times faster (both 2D and 3D case). Finally we
also selected MultilinearInterpolator because it can be instantiated to retain
the data once and then called several time. We find the usage of this object-
oriented interface more convenient than functional interface of
map_coordinates.
### 3.3 Results for Searev power smoothing
Figure 5: Storage control policy: Power injected to the grid as a function of
speed and acceleration, for 7 levels of stored energy between empty and full.
Figure 6: Comparison of the power smoothing behavior between the _heuristic_
(dark blue) and _optimized_ (light blue) storage management policies (storage
capacity of 10 MJ). Stored energy on the bottom panel.
We have applied the policy iteration algorithm to the SEAREV power smoothing
problem introduced in section 1. The algorithm is initialized with the linear
storage control policy (3). This heuristic choice is then gradually improved
by each policy improvement step.
#### 3.3.1 Algorithm parameters
About 5 policy iterations only are needed to converge to an optimal strategy.
In each policy iteration, there is a policy evaluation step which requires
1000 iterations to converge. This latter number is dictated by the time
constant of the system (1000 steps $\leftrightarrow$ 100 seconds) and 100
seconds is the time it takes for the system to “decorrelate”, that is loose
memory of its state (both speed and stored energy).
We also need to decide how to discretize and bound the state space of the
{SEAREV + storage} system:
* •
for the stored energy $E_{sto}$, bounds are the natural limits of the storage:
$E_{sto}\in[0,E_{rated}]$. A grid of 30 points yields precise enough results.
* •
for the speed $\Omega$ and the acceleration $A$, there are no natural bounds
so we have chosen to limit the values to $\pm 4$ _standard deviations_. This
seems wide enough to include most observations but not too wide to keep a good
enough resolution. We use grids of 60 points to keep the grid step small
enough.
This makes a state space grid of $30\times 60\times 60\approx 110\text{k}$
points. Although this number of points can be handled well by a present
desktop computer, this simple grid size computation illustrates the commonly
known weakness of Dynamic Programming which is the “Curse of Dimensionality”.
Indeed, this size grows exponentially with the number of dimensions of the
state so that for practical purpose state dimension is limited to 3 or 4. This
explains the motivation to search a low order model for the power production
time series in section 2.
#### 3.3.2 Algorithm execution time
With the aforementioned discretization parameters, policy evaluation takes
about 10 s (for the 1000 iterations) while policy improvement takes 20 s (for
one single value iteration step). This makes 30 s in total for one policy
iteration step, which is repeated 5 times. Therefore, the optimization
converges in about 3 minutes. This duration would grow steeply should the grid
be refined.
As a comparison of algorithm efficiency, the use of _value iteration_ would
takes much longer than _policy iteration_. Indeed, it needs 1000 iterations,
just like policy evaluation (since it is dictated by the system’s
“decorrelation time”) but each iteration involves a costly optimization of the
policy so that it takes 20 s. This makes altogether 5 hours of execution time,
i.e. 100 times more than policy iteration!
As possible paths to improve the execution time, we see, at the _code level_ ,
the use of more/different vectorization patterns although vectorized
computation is already used a lot. Maybe the use of Cython may speed up
unavoidable loops but this may not be worth the loss of flexibility and the
decrease in coding speed. Optimization at the _algorithm level_ , just as
demonstrated with “policy vs. value iteration”, is also worth investigating
further. In the end, more use of Robert Kern’s line_profiler will be needed to
decide the next step.
#### 3.3.3 Output of the computation
The policy iteration algorithm solves equation (10) and outputs the minimized
cost $J$ and two arrays: function $\tilde{J}$ (transient cost) and function
$\mu$ (optimal policy (6)), both expressed on the discrete state grid (3d
grid).
We focus on $\mu$ which yields the power $P_{grid}$ that should be injected to
the grid for any state of the system. Figure 5 is a Mayavi surface plot which
shows $P_{grid}(\Omega,A)$ for various levels of $E_{sto}$. Observations of
the result are in agreement with what can be expected from a reasonable
storage control:
* •
the more energy there is in the storage, the more power should be injected to
the grid (similar to the heuristic control (3)).
* •
the speed and acceleration of the SEAREV also modulates the injected power,
but to a lesser extent. We may view speed and acceleration as approximate
measurements of the _mechanical energy_ of the SEAREV. This energy could be a
hidden influential state variable, in parallel with the stored energy.
* •
the injected power is often set between 0.2 and 0.3 MW, that is _close to the
average_ power production.
Such observations show that the algorithm has _learned_ from the SEAREV
behavior to take sharper decisions compared to the heuristic policy it was
initialized with.
#### 3.3.4 Qualitative analysis of the trajectory
To evaluate the storage control policy, we simulate its effect on the sample
SEAREV data we have (instead of using the state space model used for the
optimization). The only adaptation required for this trajectory simulation is
to transform the _policy array_ ($\mu$ known on the state grid only) into a
_policy function_ ($\mu$ evaluable on the whole state space). This is achieved
using the same n-dimensional interpolation routine used in the algorithm.
A simulated trajectory is provided on figure 6 to compare the effect of the
optimized policy with the heuristic linear policy (3). As previously said, the
storage capacity is fixed at $E_{rated}=10\text{ MJ}$ or about 9 seconds of
charge/discharge at the rated power.
Positive aspect, the optimized policy yields an output power that is generally
closer to the average (thin gray line) than the linear policy. This better
smoothing of the “peaks and valleys” of the production is achieved by a better
usage of the available storage capacity. Indeed, the linear policy generally
under-uses the higher levels of energy.
As a slight negative aspect, the optimized policy yields a “spiky” output
power in the situations of high production (200–220 s). In this situation, the
output seems worse that the linear policy. We connect this underperformance to
the linear model (9) used to represent the SEAREV dynamics. The linearity
holds well for small movements but not when the speed is high and the pendulum
motion gets very abrupt (acceleration high above 4 standard deviations which
contradicts the Gaussian distribution assumption). Since the control
optimization is based on the linear model, the resulting control law cannot
appropriately manage these non-linear situations. Only an upgraded model would
genuinely solve this problem but we don’t have yet an appropriate low-order
non-linear model of the SEAREV. One quick workaround to reduce the power peaks
is to shave the acceleration measurements (not demonstrated here).
#### 3.3.5 Quantitative assessment
We now numerically check that the optimized policy brings a true enhancement
over the linear policy. We simulate the storage with the three 1000 s long
samples we have and compute the power variability criterion2 for each.
Figure 7 shows the standard deviation for each sample in three situations:
without storage (which yields the natural standard deviation of the SEAREV
production), with a storage controlled by the linear policy and finally the
same storage controlled by the optimized policy. Sample Em_1.txt was used to
fit the state space model (9) but we don’t think this should introduce a too
big bias because of the low model order.
Beyond the intersample variability, we can see the consistent improvement
brought by the optimized law. Compared to the linear policy, the standard
deviation of the injected power is reduced by about 20 % (27 %, 16 %, 22 % for
each sample respectively). We can conclude that the variability of the
injected power is indeed reduced by using the Dynamic Programming.
Figure 7: Effect of optimizing the storage control on three SEAREV production
time series. Standard deviation compared to the heuristic linear control case
is reduced by about 20 %.
Because the RMS deviation criterion used in this article is not directly
limited or penalized in current grid codes, there is no financial criterion to
decide whether the observed deviations are acceptable or not. Therefore we
cannot conclude if the ~20% reduction of the variability brought by optimal
control is valuable. Nevertheless, there exists criterions like the "flicker"
which are used in grid codes to set standards of power quality. Flicker, which
is way more complicated to than an additive criterion like (4) could be used
to put an economic value on a control strategy. This is the subject of ongoing
research.
## 4 Conclusion
With the use of standard Python modules for scientific computing, we have
created StoDynProg, a small library to solve Dynamic Optimization problems
using Stochastic Dynamic Programming.
We have described the mathematical and coding steps required to apply the SDP
method on an example problem of realistic complexity: smoothing the output
power of the SEAREV Wave Energy Converter. With its generic interface,
StoDynProg should be applicable to other Optimal Control problems arising in
Electrical Engineering, Mechanical Engineering or even Life Sciences. The only
requirement is an appropriate mathematical structure (Markovian model), with
the “Curse of Dimensionality” requiring a state space of low dimension.
Further improvements on this library should include a better source tree
organization (make a proper package) and an improved test coverage.
\raisebox{10.00002pt}{\hypertarget{id13}{}}\hyperlink{id4}{1}\raisebox{10.00002pt}{\hypertarget{id13}{}}\hyperlink{id4}{1}footnotetext:
In the special case of a linear dynamics and a quadratic cost (“LQ control” ),
the optimal feedback is actually a _linear_ function. Because of the state
constraint $0\leq E_{sto}\leq E_{rated}$, the storage control problem falls
outside this classical case.
\raisebox{10.00002pt}{\hypertarget{id14}{}}\hyperlink{id12}{2}\raisebox{10.00002pt}{\hypertarget{id14}{}}\hyperlink{id12}{2}footnotetext:
Instead of using the exact optimization cost (4) (average quadratic power in
MW2), we actually compute the standard deviation (in MW). It is mathematically
related to the quadratic power and we find it more readable.
## References
* [Aubry-2010] J. Aubry, P. Bydlowski, B. Multon, H. Ben Ahmed, and B. Borgarino. _Energy Storage System Sizing for Smoothing Power Generation of Direct Wave Energy Converters_ , 3rd International Conference on Ocean Energy, 2010.
* [Bertsekas-2005] D. P. Bertsekas, _Dynamic Programming and Optimal Control_ , Athena Scientific, 2005.
* [Brockwell-1991] P. J. Brockwell, and R. A. Davis. _Time Series: Theory and Methods_ , Springer Series in Statistics, Springer, 1991.
* [Kovaltchouk-2013] T. Kovaltchouk, B. Multon, H. Ben Ahmed, F. Rongère, J. Aubry, and A. Glumineau. _Influence of control strategy on the global efficiency of a Direct Wave Energy Converter with electric Power Take-Off_ , EVER 2013 conference, 2013.
* [McElroy-2013] T. McElroy, and M. Wildi. _Multi-step-ahead estimation of time series models_ , International Journal of Forecasting, 29: 378–394, 2013.
* [Ruellan-2010] M. Ruellan, H. Ben Ahmed, B. Multon, C. Josset, A. Babarit, and A. Clément. _Design Methodology for a SEAREV Wave Energy Converter_ , IEEE Trans. Energy Convers, 25: 760–767, 2010.
* [Winant-2010] P. Winant. _Dolo, a python library to solve global economic models_ , http://albop.github.io/dolo, 2010.
|
arxiv-papers
| 2014-04-25T10:55:51 |
2024-09-04T02:50:01.849408
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Pierre Haessig, Thibaut Kovaltchouk, Bernard Multon, Hamid Ben Ahmed,\n St\\'ephane Lascaud",
"submitter": "Pierre de Buyl",
"url": "https://arxiv.org/abs/1404.6389"
}
|
1404.6390
|
# JyNI – Using native CPython-Extensions in Jython
Stefan Richthofer∗† * Corresponding author: [email protected]†
Institute for Neural Computation, Ruhr-Universität BochumCopyright © 2014
Stefan Richthofer. This is an open-access article distributed under the terms
of the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and
source are credited. http://creativecommons.org/licenses/by/3.0/
###### Abstract
Jython is a Java based Python implementation and the most seamless way to
integrate Python and Java. However, it does not support native extensions
written for CPython like NumPy or SciPy. Since most scientific Python code
fundamentally depends on exactly such native extensions directly or
indirectly, it usually cannot be run with Jython. JyNI (Jython Native
Interface) aims to close this gap. It is a layer that enables Jython users to
load native CPython extensions and access them from Jython the same way as
they would do in CPython. In order to leverage the JyNI functionality, you
just have to put it on the Java classpath when Jython is launched. It neither
requires you to recompile the extension code, nor to build a customized Jython
fork. That means, it is binary compatible with existing extension builds.
At the time of writing, JyNI does not fully implement the Python C-API and it
is only capable of loading simple examples that only involve most basic built-
in types. The concept is rather complete though and our goal is to provide the
C-API needed to load NumPy as soon as possible. After that we will focus on
SciPy and others.
We expect that our work will also enable Java developers to use CPython
extensions like NumPy in their Java code.
###### Index Terms:
Jython, Java, Python, CPython, extensions, integration, JNI, native, NumPy,
C-API, SciPy
## 1 Introduction
[JyNI] is a compatibility layer with the goal to enable [JYTHON] to use native
CPython extensions like NumPy or SciPy. This way we aim to enable scientific
Python code to run on Jython.
Figure 1: Extensions
Since Java is rather present in industry, while Python is more present in
science, JyNI is an important step to lower the cost of using scientific code
in industrial environments.
Figure 2: JyNI
Because of the complexity of the Python C-API, the task of developing JyNI
revealed to be a true challenge. Especially the frequent occurrence of
preprocessor macros in the public C-API allows for no naive approaches like
directly delegating every C-API call to Jython. It turned out, that most
built-in types need individual fixes, considerations and adjustments to allow
seamless integration with Jython.
There have been similar approaches for other Python implementations, namely
[IRONCLAD] for IronPython and [CPYEXT] for PyPy. As far as we know, these
suffer from equal difficulties as JyNI and also did not yet reach a satisfying
compatibility level for current Python versions.
Another interesting work is NumPy4J [NP4J], which provides Java interfaces for
NumPy by embedding the CPython interpreter. It features automatic conversion
to Java suitable types and thus allows easy integration with other Java
frameworks. A more general approach is Jepp [JEPP], which also works via
embedding the CPython interpreter. It also includes conversion methods between
basic Python and Java types, but is not specifically NumPy-aware.
However, none of these approaches aims for integration with Jython. In
contrast to that, JyNI is entirely based on Jython. Though large parts are
derived from CPython, the main Python runtime is provided by Jython and JyNI
delegates most C-API calls to Jython directly or indirectly (i.e. some objects
are mirrored natively, so calls to these can be processed entirely on native
side, syncing the results with Jython afterwards; see implementation section
for details).
## 2 Usage
Thanks to Jython’s hooking capabilities, it is sufficient to place JyNI.jar on
the classpath (and some native libraries on the library path) when Jython is
launched. Then Jython should “magically” be able to load native extensions, as
far as the needed Python C-API is already implemented by JyNI. No
recompilation, no forking – it just works with original Jython and original
extensions (up to version compatibility; see the versioning notes at the end
of this section).
Note that the naive way does not actually set the classpath for Jython:
>
> java -cp "[...]:JyNI.jar:[JyNI binaries folder]"
> -jar jython.jar
>
The correct way is:
>
> java -cp "[...]:JyNI.jar:[JyNI binaries folder]:
> jython.jar" org.python.util.jython
>
Alternatively, one can use Jython’s start script:
>
> jython -J-cp "[...]:JyNI.jar:[JyNI binaries folder]"
>
Usually the JVM does not look for native library files on the classpath. To
ease the configuration, we built into JyNI’s initializer code that it also
searches for native libraries on the classpath. Alternatively you can place
libJyNI.so and libJyNI-loader.so anywhere the JVM finds them, i.e. on the java
library path (java.library.path) or the system’s library path
(LD_LIBRARY_PATH).
To get an impression of JyNI, proceed as described in the following
subsection.
### 2.1 Instructions to run JyNIDemo.py
* •
Go to [JyNI], select the newest release in the download section and get the
sources and binaries appropriate for your system (32 or 64 bit).
* •
Extract JyNI-Demo/src/JyNIDemo.py from the sources.
* •
To launch it with CPython, extract DemoExtension.so from the bin archive.
* •
JyNIDemo.py adds the extension folder via sys.path.append([path]). You can
modify that line so it finds your extracted DemoExtension.so or delete the
line and put DemoExtension.so on the pythonpath.
* •
If you launch JyNIDemo.py with Jython, it won’t work. Put JyNI.jar, libJyNI-
Loader.so and libJyNI.so on Jython’s classpath. libJyNI-Loader.so and
libJyNI.so can alternatively be placed somewhere on the Java library path.
Jython should now be able to run JyNIDemo.py via
>
> java -cp "[...]:JyNI.jar:[JyNI binaries folder]:
> jython.jar" org.python.util.jython JyNIDemo.py
>
Be sure to use Jython 2.7 (beta) or newer. If you are not using the Jython
stand-alone version, make sure that Jython’s Lib-folder is on the Python path.
### 2.2 Versioning note
JyNI’s version consists of two parts. The first part (currently 2.7) indicates
the targeted API version. Your Jython should meet this version if you intend
to use it with JyNI. For extensions, the API version means that a production
release of JyNI would be able to load any native extension that a CPython
distribution of the same version (and platform) can load. Of course, this is
an idealistic goal – there will always remain some edgy, maybe exotic API-
aspects JyNI won’t be able to support.
The second part of the JyNI version (currently alpha.2.1) indicates the
development status. As long as it contains “alpha” or “beta”, one can’t expect
that the targeted API version is already met. Once out of beta, we will
maintain this version part as a third index of the targeted API version (i.e.
JyNI 2.7.x).
## 3 Capabilities
JyNI is currently available for Linux only. Once it is sufficiently complete
and stable, we will work out a cross platform version compilable on Windows,
Mac OS X and others. The following built-in types are already supported:
* •
Number types PyInt, PyLong, PyFloat, PyComplex
* •
Sequence types PyTuple, PyList, PySlice, PyString, PyUnicode
* •
Data structure types PyDict, PySet, PyFrozenSet
* •
Operational types PyModule, PyClass, PyMethod, PyInstance, PyFunction, PyCode,
PyCell
* •
Singleton types PyBool, PyNone, PyEllipsis, PyNotImplemented
* •
Native types PyCFunction, PyCapsule, PyCObject
* •
Natively defined custom types
* •
Exception types
* •
PyType as static type or heap type
The function families PyArg_ParseTuple and Py_BuildValue are also supported.
## 4 Implementation
To create JyNI we took the source code of CPython 2.7 and stripped away all
functionality that can be provided by Jython and is not needed for mirroring
objects (see below). We kept the interface unchanged and implemented it to
delegate calls to Jython via JNI and vice versa. The most difficult thing is
to present JNI jobject s from Jython to extensions such that they look like
PyObject* from Python (C-API). For this task, we use the three different
approaches explained below, depending on the way a native type is implemented.
In this section, we assume that the reader is familiar with the Python [C-API]
and has some knowledge about the C programming language, especially about the
meaning of pointers and memory allocation.
### 4.1 Python wraps Java
The best integration with Jython is obtained, if PyObject* is only a stub that
delegates all its calls to Jython (figure 3). This is only possible, if Jython
features a suitable counterpart of the PyObject (i.e. some subclass of
org.python.core.PyObject with similar name, methods and functionality).
Further, there must not exist macros in the official C-API that directly
access the PyObject’s memory. Consequently, one cannot use tp_dictoffset to
obtain the object’s dictionary or offset from PyMemberDef to access the
object’s members.
Since members are usually only accessed via generic getter or setter methods
that also look for a PyGetSetDef with the right name, we usually re-implement
the members as get-sets. Also the dictionary access is usually performed in
methods we can safely rewrite to versions that get the dictionary from Jython.
Figure 3: Python wraps Java
Examples for this method are PyDict, PySlice and PyModule.
The cases where this approach fails are
* •
if Jython features no corresponding type
* •
if the Python C-API features macros to access the Object’s memory directly
We deal with these cases in the following.
### 4.2 Mirroring objects
If the Python C-API provides macros to access an object’s data, we cannot
setup the object as a stub, because the stub would not provide the memory
positions needed by the macros. To overcome this issue, we mirror the object
if its C-API features such direct access macros (figure 4).
Figure 4: Objects are mirrored
Examples, where this approach is successfully applied are PyTuple, PyList,
PyString, PyInt, PyLong, PyFloat and PyComplex.
The difficulty here is to provide a suitable synchronization between the
counterparts. If the CPython object is modified by C code, these changes must
be reflected immediately on Jython side. The problem here is, that such
changes are not reported; they must be detected. Performing the
synchronization when the C call returns to Jython is only suitable, if no
multiple threads exist. However, most of the affected objects are immutable
anyway, so an initial data synchronization is sufficient.
PyList is an example for an affected object that is mutable via a macro. For
PyList, we perform an individual solution. The Jython class
org.python.core.PyList uses a variable of type java.util.List (which is an
interface) to store its backend. We wrote a wrapper, that provides access to
the memory of the C struct of PyListObject and implements the java.util.List
interface on Java side. If a PyList is mirrored, we replace its backend by our
wrapper. If it was initially created on the Jython side, we insert all its
elements into the C counterpart on initialization.
PyCell and PyByteArray are other examples that need mirror mode, but are
mutable. However, we have rough ideas how to deal with them, but since they
are not used by NumPy, we don’t put priority on implementing them.
### 4.3 Java wraps Python
If Jython provides no counterpart of an object type, the two approaches
described above are not feasible. Typically, this occurs, if an extension
natively defines its own PyType objects, but there are also examples for this
in the original Python C-API. If the types were previously known, we could
simply implement Jython counterparts for them and apply one of the two
approaches above. However, we decided to avoid implementing new Jython objects
if possible and solve this case with a general approach. PyCPeer extends
org.python.core.PyObject and redirects the basic methods to a native PyObject*
(figure 5). The corresponding PyObject* pointer is tracked as a java long in
PyCPeer. Currently PyCPeer supports attribute access by delegating
__findattr_ex__, which is the backend method for all attribute accessing
methods in Jython (i.e. __findattr__ and __getattr__ in all variants).
Further, PyCPeer delegates the methods __str__, __repr__ and __call__. A more
exhaustive support is planned. PyCPeerType is a special version of PyCPeer
that is suited to wrap a natively defined PyType.
Let’s go through an example. If you execute the Python code "x = foo.bar",
Jython compiles it equivalently to the Java call "x =
foo.__getattr__("bar");". If foo is a PyCPeer wrapping a native PyObject*,
Java’s late binding would call __findattr_ex__("bar") implemented in PyCPeer.
Via the native method JyNI.getAttrString(long peerHandle, String name) the
call is delegated to JyNI_getAttrString in JyNI.c and then finally to
PyObject_GetAttrString in object.c. To convert arguments and return values
between Java jobject and CPython PyObject*, we use the conversion methods
JyNI_JythonPyObject_FromPyObject and JyNI_PyObject_FromJythonPyObject (see
next section). Our version of PyObject_GetAttrString falls back to the
original CPython implementation, if it is called with a PyCPeer or a mirrored
object. A flag in the corresponding JyObject (see next section) allows to
detect these cases.
Figure 5: Java wraps Python
An example from the C-API that needs the approach from this section is
PyCFunction.
### 4.4 Object lookup
Every mentioned approach involves tying a jobject to a PyObject*. To resolve
this connection as efficiently as possible, we prepend an additional header
before each PyObject in memory. If a PyGC_Head is present, we prepend our
header even before that, as illustrated in figure 6.
Figure 6: Memory layout
In the source, this additional header is called JyObject and defined as
follows:
typedef struct{ jobject jy; unsigned short flags; JyAttribute* attr;}
JyObject;jy is the corresponding jobject, flags indicates which of the above
mentioned approaches is used, whether a PyGC_Head is present, initialization
state and synchronization behavior. attr is a linked list containing void
pointers for various purpose. However, it is intended for rare use, so a
linked list is a sufficient data structure with minimal overhead. A JyObject
can use it to save pointers to data that must be deallocated along with the
JyObject. Such pointers typically arise when formats from Jython must be
converted to a version that the original PyObject would have contained
natively.
To reserve the additional memory, allocation is adjusted wherever it occurs,
e.g. when allocations inline as is the case for number types. The adjustment
also occurs in PyObject_Malloc. Though this method might not only be used for
PyObject allocation, we always prepend space for a JyObject. We regard this
slight overhead in non-PyObject cases as preferable over potential
segmentation fault if a PyObject is created via PyObject_NEW or
PyObject_NEW_VAR. For these adjustments to apply, an extension must be
compiled with the WITH_PYMALLOC flag activated. Otherwise several macros would
direct to the raw C methods malloc, free, etc., where the neccessary extra
memory would not be reserved. So an active WITH_PYMALLOC flag is crucial for
JyNI to work. However, it should be not much effort to recompile affected
extensions with an appropriate WITH_PYMALLOC flag value.
Statically defined PyType objects are treated as a special case, as their
memory is not dynamically allocated. We resolve them simply via a lookup table
when converting from jobject to PyObject* and via a name lookup by Java
reflection if converting the other way. PyType objects dynamically allocated
on the heap are of course not subject of this special case and are treated
like usual PyObject s (the Py_TPFLAGS_HEAPTYPE flag indicates this case).
The macros AS_JY(o) and FROM_JY(o), defined in JyNI.h, perform the necessary
pointer arithmetics to get the JyObject header from a PyObject* and vice
versa. They are not intended for direct use, but are used internally by the
high-level conversion functions described below, as these also consider
special cases like singletons or PyType objects.
The other lookup direction is done via a hash map on the Java side. JyNI
stores the PyObject* pointers as Java Long objects and looks them up before
doing native calls. It then directly passes the pointer to the native method.
The high-level conversion functions
jobject JyNI_JythonPyObject_FromPyObject (PyObject* op);PyObject*
JyNI_PyObject_FromJythonPyObject (jobject jythonPyObject);take care of all
this, do a lookup and automatically perform initialization if the lookup
fails. Of course the jobject mentioned in these declarations must not be an
arbitrary jobject, but one that extends org.python.core.PyObject. Singleton
cases are also tested and processed appropriately. NULL converts to NULL.
Though we currently see no use case for it, one can use the declarations in
JyNI.h as JyNI C-API. With the conversion methods one could write hybrid
extensions that do C, JNI and Python calls natively.
### 4.5 Global interpreter lock (GIL)
The global interpreter lock is a construction in CPython that prevents
multiple threads from running Python code in the same process. It is usually
acquired when the execution of a Python script begins and released when it
ends. However, a native extension and some parts of native CPython code can
release and re-acquire it by inserting the Py_BEGIN_ALLOW_THREADS and
Py_END_ALLOW_THREADS macros. This way, an extension can deal with multiple
threads and related things like input events (f.i. Tkinter needs this).
In contrast to that, Jython does not have a GIL and allows multiple threads at
any time, using Java’s threading architecture. Since native extensions were
usually developed for CPython, some of them might rely on the existence of a
GIL and might produce strange behaviour if it was missing. So JyNI features a
GIL to provide most familiar behaviour to loaded extensions. To keep the Java
parts of Jython GIL-free and have no regression to existing multithreading
features, the JyNI GIL is only acquired when a thread enters native code and
released when it enters Java code again – either by returning from the native
call or by performing a Java call to Jython code. Strictly speaking, it is not
really global (thus calling it “GIL” is a bit misleading), since it only
affects threads in native code. While there can always be multiple threads in
Java, there can only be one thread in native code at the same time (unless the
above mentioned macros are used).
## 5 A real-world example: Tkinter
To present a non-trivial example, we refere to Tkinter, one of the most
popular GUI frameworks for Python. There has already been an approach to make
Tkinter available in Jython, namely jTkinter – see [JTK]. However, the last
update to the project was in 2000, so it is rather outdated by now and must be
considered inactive.
Since release alpha.2.1, JyNI has been tested successfully on basic Tkinter
code. We load Tkinter from the place where it is usually installed on Linux:
import sys#Include native Tkinter:sys.path.append(’/usr/lib/python2.7/lib-
dynload’)sys.path.append(’/usr/lib/python2.7/lib-tk’)from Tkinter import *root
= Tk()txt = StringVar()txt.set("Hello World!")def print_text(): print
txt.get()def print_time_stamp(): from java.lang import System print
"System.currentTimeMillis: " +str(System.currentTimeMillis())Label(root,
text="Welcome to JyNI Tkinter-Demo!").pack()Entry(root,
textvariable=txt).pack()Button(root, text="print text",
command=print_text).pack()Button(root, text="print timestamp",
command=print_time_stamp).pack()Button(root, text="Quit",
command=root.destroy).pack()root.mainloop()
Figure 7: Tkinter demonstration
Note that the demonstration also runs with CPython in principle. To make this
possible, we perform from java.lang import System inside the method body of
print_time_stamp rather than at the beginning of the file. Thus, only the
button “print timestamp” would produce an error, since it performs Java calls.
In a Jython/JyNI environment, the button prints the current output of
java.lang.System.currentTimeMillis() to the console (see figure 7).
## 6 Another example: The datetime module
As a second example, we refere to the datetime module. Jython features a Java-
based version of that module, so this does not yet pay off in new
functionality. However, supporting the original native datetime module is a
step toward NumPy, because it features a public C-API that is needed by NumPy.
The following code demonstrates how JyNI can load the original datetime
module. Note that we load it from the place where it is usually installed on
Linux. To overwrite the Jython version, we put the new path to the beginning
of the list in sys.path:
import syssys.path.insert(0, ’/usr/lib/python2.7/lib-dynload’)import
datetimeprint datetime.__doc__print "-" * 22printprint datetime.__name__now =
datetime.datetime(2013, 11, 3, 20, 30, 45)print nowprint repr(now)print
type(now)To verify that the original module is loaded, we print out the
__doc__ string. It must read "Fast implementation of the datetime type.". If
JyNI works as excpected, the output is:
>
> Fast implementation of the datetime type.
> ----------------------
>
> datetime
> 2013-11-03 20:30:45
> datetime.datetime(2013, 11, 3, 20, 30, 45)
> <type ’datetime.datetime’>
>
## 7 Roadmap
The main goal of JyNI is compatibility with NumPy and SciPy, since these
extensions are of most scientific importance. Since NumPy has dependencies on
several other extensions, we will have to ensure compatibility with these
extensions first. Among these are ctypes and datetime (see previous section).
In order to support ctypes, we will have to support the PyWeakRef object.
### 7.1 Garbage Collection
Our first idea to provide garbage collection for native extensions, was to
adopt the original CPython garbage collector source and use it in parallel
with the Java garbage collector. The CPython garbage collector would be
responsible to collect mirrored objects, native stubs and objects created by
native extensions. The stubs would keep the corresponding objects alive by
maintaining a global reference. However, this approach does not offer a clean
way to trace reference cycles through Java/Jython code (even pure Java Jython
objects can be part of reference cycles keeping native objects alive forever).
To obtain a cleaner solution, we plan to setup an architecture that makes the
native objects subject to Java’s garbage collector. The difficulty here is
that Java’s mark and sweep algorithm only traces Java objects. When a Jython
object is collected, we can use its finalizer to clean up the corresponding
C-PyObject (mirrored or stub), if any. To deal with native PyObject s that
don’t have a corresponding Java object, we utilize JyGCHead s (some
minimalistic Java objects) to track them and clean them up on finalization. We
use the visitproc mechanism of original CPython’s garbage collection to obtain
the reference connectivity graph of all relevant native PyObject s. We mirror
this connectivity in the corresponding JyGCHead s, so that the Java garbage
collector marks and sweeps them according to native connectivity.
A lot of care must be taken in the implementation details of this idea. For
instance, it is not obvious, when to update the connectivity graph. So a
design goal of the implementation is to make sure that an outdated
connectivity graph can never lead to the deletion of still referenced objects.
Instead, it would only delay the deletion of unreachable objects. Another
issue is that the use of Java finalizers is discouraged for various reasons.
An alternative to finalizers are the classes from the package java.lang.ref.
We would have JyGCHead extend PhantomReference and register all of them to a
ReferenceQueue. A deamon thread would be used to poll references from the
queue as soon as the garbage collector enqueues them. For more details on Java
reference classes see [JREF].
### 7.2 Cross-Platform support
We will address cross-platform support when JyNI has reached a sufficiently
stable state on our development platform. At least we require rough solutions
for the remaining gaps. Ideally, we focus on cross-platform support when JyNI
is capable of running NumPy.
## 8 License
JyNI is released under the GNU [GPL] version 3. To allow for commercial use,
we add the classpath exception [GPL_EXC] like known from GNU Classpath to it.
We were frequently asked, why not LGPL, respectively what the difference to
LGPL is. In fact, the GPL with classpath exception is less restrictive than
LGPL. [GPL_EXC] states this as follows: The LGPL would additionally require
you to "allow modification of the portions of the library you use". For C/C++
libraries this especially requires distribution of the compiled .o-files from
the pre-linking stage. Further you would have to allow "reverse engineering
(of your program and the library) for debugging such modifications".
## References
* [JyNI] Stefan Richthofer, Jython Native Interface (JyNI) Homepage, http://www.JyNI.org, 6 Apr. 2014, Web. 7 Apr. 2014
* [JYTHON] Python Software Foundation, Corporation for National Research Initiatives, Jython: Python for the Java Platform, http://www.jython.org, Mar. 2014, Web. 7 Apr. 2014
* [IRONCLAD] Resolver Systems, Ironclad, http://code.google.com/p/ironclad, 26 Aug. 2010, Web. 7 Apr. 2014
* [CPYEXT] PyPy team, PyPy/Python compatibility, http://pypy.org/compat.html, Web. 7 Apr. 2014
* [NP4J] Joseph Cottam, NumPy4J, https://github.com/JosephCottam/Numpy4J, 11. Dec. 2013, Web. 7 Apr. 2014
* [JEPP] Mike Johnson, Java embedded Python (JEPP), http://jepp.sourceforge.net, 14 May 2013, Web. 7 Apr. 2014
* [JTK] Finn Bock, jTkinter, http://jtkinter.sourceforge.net, 30 Jan. 2000, Web. 7 Apr. 2014
* [C-API] Python Software Foundation, Python/C API Reference Manual, http://docs.python.org/2/c-api, Web. 7 Apr. 2014
* [JREF] Peter Haggar, IBM Corporation, http://www.ibm.com/developerworks/library/j-refs, 1 Oct. 2002, Web. 7 Apr. 2014
* [GPL] Free Software Foundation, GNU General Public License v3, http://www.gnu.org/licenses/gpl.html, 29 June 2007, Web. 7 Apr. 2014
* [GPL_EXC] Wikipedia, GPL linking exception, http://en.wikipedia.org/wiki/GPL_linking_exception#The_classpath_exception, 5 Mar 2014, Web. 7 Apr. 2014
|
arxiv-papers
| 2014-04-25T10:56:33 |
2024-09-04T02:50:01.857492
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Stefan Richthofer",
"submitter": "Pierre de Buyl",
"url": "https://arxiv.org/abs/1404.6390"
}
|
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