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1305.6192
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arxiv-papers
| 2013-05-27T12:25:33 |
2024-09-04T02:49:45.773446
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{
"license": "Public Domain",
"authors": "M.J. Nikmehr and F. Heydari",
"submitter": "Mohammad Nikmehr",
"url": "https://arxiv.org/abs/1305.6192"
}
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1305.6193
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arxiv-papers
| 2013-05-27T12:27:18 |
2024-09-04T02:49:45.777077
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{
"license": "Public Domain",
"authors": "M.J. Nikmehr, A. Nejati and M. Deldar",
"submitter": "Mohammad Nikmehr",
"url": "https://arxiv.org/abs/1305.6193"
}
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1305.6199
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arxiv-papers
| 2013-05-27T12:45:22 |
2024-09-04T02:49:45.780571
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{
"license": "Public Domain",
"authors": "M.J. Nikmehr and F. Heydari",
"submitter": "Mohammad Nikmehr",
"url": "https://arxiv.org/abs/1305.6199"
}
|
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1305.6287
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# The intersection graph of ideals of $\mathbb{Z}_{n}$ is
weakly perfect ††thanks: Key Words: Intersection graph of ideals of
$\mathbb{Z}_{n}$, Clique number, Chromatic number. ††thanks: 2010 Mathematics
Subject Classification: 05C15, 05C69.
R. Nikandish and M.J. Nikmehr
Department of Mathematics, K. N. Toosi University of Technology, Tehran, Iran
$\mathsf{r\\[email protected]}$ $\mathsf{[email protected]}$
###### Abstract
A graph is called weakly perfect if its vertex chromatic number equals its
clique number. Let $R$ be a ring and $I(R)^{*}$ be the set of all left proper
non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by
$G(R)$, is a graph with the vertex set $I(R)^{*}$ and two distinct vertices
$I$ and $J$ are adjacent if and only if $I\cap J\neq 0$. In this paper, it is
shown that $G(\mathbb{Z}_{n})$, for every positive integer $n$, is a weakly
perfect graph. Also, for some values of $n$, we give an explicit formula for
the vertex chromatic number of $G(\mathbb{Z}_{n})$. Furthermore, it is proved
that the edge chromatic number of $G(\mathbb{Z}_{n})$ is equal to the maximum
degree of $G(\mathbb{Z}_{n})$ unless either $G(\mathbb{Z}_{n})$ is a null
graph with two vertices or a complete graph of odd order.
1\. Introduction
The study of algebraic structures, using the properties of graphs, becomes an
exciting research topic in the last twenty years, leading to many fascinating
results and questions. There are many papers on assigning a graph to a ring,
for instance see [1] and [4]. Let $R$ be a ring with unity. By $I(R)$ and
$I(R)^{*}$, we mean the set of all left ideals of $R$ and the set of all left
proper non-trivial ideals of $R$, respectively.
Let $G$ be a graph with the vertex set $V(G)$. For any $x\in V(G)$, $d(x)$
represents the number of edges incident to $x$, called the degree of the
vertex $x$ in $G$. The maximum degree of vertices of $G$ is denoted by
$\Delta(G)$. A graph $G$ is connected if there is a path between every two
distinct vertices. The complete graph of order $n$, denoted by $K_{n}$, is a
graph in which any two distinct vertices are adjacent. A clique of $G$ is a
maximal complete subgraph of $G$ and the number of vertices in the largest
clique of $G$, denoted by $\omega(G)$, is called the clique number of $G$. For
a graph $G$, let $\chi(G)$ denote the vertex chromatic number of $G$, i.e.,
the minimal number of colors which can be assigned to the vertices of $G$ in
such a way that every two adjacent vertices have different colors. Clearly,
for every graph $G$, $\omega(G)\leq\chi(G)$. A graph $G$ is said to be weakly
perfect if $\omega(G)=\chi(G)$. Recall that a $k$-edge coloring of a graph $G$
is an assignment of $k$ colors $\\{1,\ldots,k\\}$ to the edges of $G$ such
that no two adjacent edges have the same color, and the edge chromatic number
$\chi^{\prime}(G)$ of a graph $G$ is the smallest integer $k$ such that $G$
has a $k$-edge coloring. To find some graph coloring methods, we refer the
reader to [5].
Let $F=\\{S_{i}|i\in I\\}$ be an arbitrary family of sets. The intersection
graph, $G(F)$, is a graph with $V(G(F))=F$ and two distinct vertices are
adjacent if and only if they have non-empty intersection. The following
theorem is an interesting fact about intersection graphs due to Marczewski
([6]).
###### Theorem 1
. Every simple graph is an intersection graph.
This result shows that intersection graphs are not weakly perfect in general
(for example, the clique number of a cycle with five vertices is 2, but its
vertex chromatic number is 3). The intersection graph of ideals of a ring $R$,
denoted by $G(R)$, is a graph with the vertex set $I(R)^{*}$ and two distinct
vertices $I$ and $J$ are adjacent if and only if $I\cap J\neq 0$. This graph
was first defined in [4]. While the authors were mainly interested in the
study of intersection graph of ideals of $\mathbb{Z}_{n}$, where
$\mathbb{Z}_{n}$ is the ring of integers modulo $n$. For instance, they
determined the values of $n$ for which the graph of $\mathbb{Z}_{n}$ is
complete, Eulerian or Hamiltonian. The present article is a natural
continuation of the study in this direction. The main aim of this paper is to
show that $G(\mathbb{Z}_{n})$ is a weakly perfect graph, for every positive
integer $n$. Moreover, we determine all integers $n$ when
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$.
2\. Vertex Chromatic Number and Clique Number of $G(\mathbb{Z}_{n})$
In this section, we prove that
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))$, for every positive
integer $n$. Also, for some values of $n$, we give an explicit formula for
$\chi(G(\mathbb{Z}_{n}))$.
Let $n$ be a natural number. Throughout the paper, without loss of generality,
we assume that $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\ldots p_{m}^{n_{m}}$, where
$p_{i}$’s are distinct primes and $n_{i}$’s are natural numbers and $n_{1}\leq
n_{2}\leq\cdots\leq n_{m}$.
We begin with the following remarks.
###### Remark 2
. Consider the ring $\mathbb{Z}_{n}$. It follows from Chinese Remainder
Theorem that
$\mathbb{Z}_{n}\cong\mathbb{Z}_{p_{1}^{n_{1}}}\times\cdots\times\mathbb{Z}_{p_{m}^{n_{m}}}$.
###### Remark 3
. $I\in I(\mathbb{Z}_{n})$ if and only if $I=I_{1}\times\cdots\times I_{m}$,
where $I_{i}\in I(\mathbb{Z}_{p_{i}^{n_{i}}})$.
By Remarks 2 and 3, one can easily see that
$|I(\mathbb{Z}_{n})^{*}|=\prod_{i=1}^{m}(n_{i}+1)-2$.
Let $I=I_{1}\times\cdots\times I_{m}\in I(\mathbb{Z}_{n})$. Each $I_{i}$,
$1\leq i\leq m$, is called a component of $I$. Suppose that $k$, $1\leq k\leq
m$, is an integer and $F$ is the family of all ideals of $\mathbb{Z}_{n}$ with
exactly $k$ non-zero components in which $I_{i_{j}}\neq 0$, $1\leq j\leq k$.
Obviously, one can consider $F$ as the set of ideals with no zero component of
the subring
$\mathbb{Z}_{p_{i_{1}}^{n_{i_{1}}}}\times\cdots\times\mathbb{Z}_{p_{i_{k}}^{n_{i_{k}}}}$.
$\mathbf{Definition.}$ (i) The number of elements of $F$ is denoted by
$\mathcal{W}(F)$.
(ii) Let $G$ be the family of ideals with no zero component of subring
$\mathbb{Z}_{p_{j_{1}}^{n_{j_{1}}}}\times\cdots\times\mathbb{Z}_{p_{j_{l}}^{n_{j_{l}}}}$.
We write $G\subseteq F$, if
$\\{p_{j_{1}}^{n_{j_{1}}},\ldots,p_{j_{l}}^{n_{j_{l}}}\\}\subseteq\\{p_{i_{1}}^{n_{i_{1}}},\ldots,p_{i_{k}}^{n_{i_{k}}}\\}$.
(iii) We write $F\cap G=\varnothing$, if
$\\{p_{j_{1}}^{n_{j_{1}}},\ldots,p_{j_{l}}^{n_{j_{l}}}\\}\cap\\{p_{i_{1}}^{n_{i_{1}}},\ldots,p_{i_{k}}^{n_{i_{k}}}\\}=\varnothing$.
(iv) The family $G$ is said to be complement to $F$, if $F\cap G=\varnothing$
and
$\\{p_{j_{1}}^{n_{j_{1}}},\ldots,p_{j_{l}}^{n_{j_{l}}}\\}\cup\\{p_{i_{1}}^{n_{i_{1}}},\ldots,p_{i_{k}}^{n_{i_{k}}}\\}=\\{p_{1}^{n_{1}},\ldots,p_{m}^{n_{m}}\\}$
and we write $F^{c}=G$. Clearly, $(F^{c})^{c}=F$.
The following example investigate families of ideals of $\mathbb{Z}_{n}$, when
$n=p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$.
###### Example 4
. Let $n=p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}$. Then there exists one
family of ideals with no zero component. Suppose that $F_{0}$ is the family of
ideals of the form $I_{1}\times I_{2}\times I_{3}$, where each $I_{i}$ is a
non-zero ideal of $\mathbb{Z}_{p_{i}^{n_{i}}}$. Let $F_{1},F_{2},F_{3}$ be
families of ideals of the form $0\times I_{2}\times I_{3},I_{1}\times 0\times
I_{3},I_{1}\times I_{2}\times 0$, respectively, where each $I_{i}$ is a non-
zero ideal of $\mathbb{Z}_{p_{i}^{n_{i}}}$. Also, assume that
$F_{4},F_{5},F_{6}$ are families of ideals of the form $0\times 0\times
I_{3},0\times I_{2}\times 0,I_{1}\times 0\times 0$, respectively, where each
$I_{i}$ is a non-zero ideal of $\mathbb{Z}_{p_{i}^{n_{i}}}$. Then
$\mathcal{W}(F_{0})=n_{1}n_{2}n_{3},\mathcal{W}(F_{1})=n_{2}n_{3},\mathcal{W}(F_{2})=n_{1}n_{3}$
and $\mathcal{W}(F_{3})=n_{2}n_{3}$. Also, $F_{i}\subset F_{0}$, for
$i=1,\ldots,6$. Moreover, $F_{1}^{c}=F_{6},F_{2}^{c}=F_{5}$ and
$F_{4}^{c}=F_{3}$. If we let $F_{7}=0\times 0\times 0$, then
$\\{F_{0},\ldots,F_{7}\\}$ is a partition of ideals of
$\mathbb{Z}_{p_{1}^{n_{1}}}\times\mathbb{Z}_{p_{2}^{n_{2}}}\times\mathbb{Z}_{p_{3}^{n_{3}}}$.
###### Theorem 5
. Let $n=p_{i}^{n_{i}}$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=n_{i}-1$.
###### Proof.
It is clear. $\Box$
###### Theorem 6
. The graph $G(\mathbb{Z}_{n})$ is weakly perfect, for every $n>0$.
###### Proof.
Let $C=\\{G\,|\,\mathcal{W}(G)>\mathcal{W}(G^{c})\\}\cup A$, where $G\in A$ if
and only if $\mathcal{W}(G)=\mathcal{W}(G^{c})$ and $|A\cap\\{G,G^{c}\\}|=1$.
We show that $C_{1}=\bigcup_{G\in C}G$ is a clique of $G(\mathbb{Z}_{n})$.
Assume to the contrary that there exist ideals $I$ and $J$ in $C_{1}$ and
$I\cap J=0$. Then there exist two families $G_{1}$ and $G_{2}$ such that
$G_{1}\cap G_{2}=\varnothing$. This implies that $G_{1}\subseteq G_{2}^{c}$
and $G_{2}\subseteq G_{1}^{c}$. Consequently,
$\mathcal{W}(G_{2})\geq\mathcal{W}(G_{2}^{c})\geq\mathcal{W}(G_{1})\geq\mathcal{W}(G_{1}^{c})\geq\mathcal{W}(G_{2})$
and so
$\mathcal{W}(G_{2})=\mathcal{W}(G_{2}^{c})=\mathcal{W}(G_{1})=\mathcal{W}(G_{1}^{c})$.
Hence, $G_{1}=G_{2}^{c}$ and $G_{1},G_{2}\in C$, a contradiction. Therefore,
$|C_{1}|\leq\omega(G(\mathbb{Z}_{n}))$. To complete the proof, we show that
$\chi(G(\mathbb{Z}_{n}))\leq|C_{1}|$. We color all ideals in $C_{1}$ with
different colors and color each family $H$ of ideals out of $C$ with colors of
ideals of $H^{c}$. Now, we show that this is a proper vertex coloring of
$G(\mathbb{Z}_{n})$. Suppose that $I,J$ are adjacent vertices in
$G(\mathbb{Z}_{n})$. Without loss of generality, one can assume that there are
different families $F_{i}$ and $F_{j}$ such that $I\in F_{i}$ and $J\in F_{j}$
and at least one of $I,J$ is not contained in $C_{1}$. Since $F_{i}\neq
F_{j}$, we deduce that $F_{i}\neq F_{j}^{c},F_{i}^{c}\neq F_{j},F_{i}^{c}\neq
F_{j}^{c}$. Therefore, $I$ and $J$ have different colors. Thus we obtain a
proper vertex coloring for $G(\mathbb{Z}_{n})$, as desired. $\Box$
###### Theorem 7
. Let $n_{m}\geq\prod_{i=1}^{n_{m-1}}n_{i}$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=n_{m}\prod_{i=1}^{n_{m-1}}(n_{i}+1)-1.$
###### Proof.
It is enough to see that $F\in C$ if for every $J\in F$ the ideal $J$ contains
$I=0\times\cdots\times 0\times(p_{m}^{n_{m}-1})$. $\Box$
From the previous theorem we have the following immediate corollaries.
###### Corollary 8
. Let $n=p_{1}^{n_{1}}p_{2}^{n_{2}}$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=n_{2}(n_{1}+1)-1$.
###### Proof.
By the assumption, $n_{2}\geq n_{1}$. So the result follows from Theorem 7.
$\Box$
###### Corollary 9
. Let $n_{1}=n_{2}=\cdots=n_{m}=1$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=2^{m-1}-1$.
Let $R$ be a ring and $R\cong F_{1}\times\cdots\times F_{m}$, where each
$F_{i}$ is a field. By a similar argument in the proof of Theorem 7 and
Corollary 9, one can show that $\omega(G(R))=\chi(G(R))=2^{m-1}-1$.
###### Theorem 10
. Let $m>1$ be an odd number such that
$\prod_{i=0}^{\lfloor\frac{m}{2}\rfloor-1}n_{m-i}\leq\prod_{i=1}^{m-\lfloor\frac{m}{2}\rfloor}n_{i}$.
Then $\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=$
$|\\{I\in
I(\mathbb{Z}_{n})^{*}\,|\,I\rm{\,has\,at\,most\lfloor\frac{m}{2}\rfloor\,zero\,components}\\}|.$
###### Proof.
It is enough to see that $F\in C$ if for every $J\in F$ the ideal $J$ has at
most $\lfloor\frac{m}{2}\rfloor$ zero components. $\Box$
###### Corollary 11
. Let $m$ be an odd number and $n_{1}=n_{2}=\cdots=n_{m}=\alpha$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=\sum_{i=0}^{\lfloor\frac{m}{2}\rfloor}\big{(}\,^{m}_{i}\,\big{)}\alpha^{m-i}-1.$
###### Proof.
By Theorem 10, $\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=$
$|\\{I\in
I(\mathbb{Z}_{n})^{*}|I\rm{\,has\,at\,most\lfloor\frac{m}{2}\rfloor\,zero\,components}\\}|.$
It is clear that there exist $\big{(}\,^{m}_{i}\,\big{)}\alpha^{m-i}$ ideals
with $i$ non-zero components. Note that the ideal
$\mathbb{Z}_{p_{1}^{\alpha}}\times\cdots\times\mathbb{Z}_{p_{m}^{\alpha}}$ is
not a vertex of $G(\mathbb{Z}_{n})$. $\Box$
In the sequel, similar results in case $m$ is an even number are given.
###### Theorem 12
. Let $m>2$ be an even number such that
$\prod_{i=0}^{\frac{m}{2}-2}{n_{m-i}}\leq\prod_{i=1}^{\frac{m}{2}+1}{n_{i}}$.
Then $\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=$
$|\\{I\in
I(\mathbb{Z}_{n})^{*}\,|\,I\rm{\,has\,at\,most\frac{m}{2}-1\,zero\,components}\\}|+\sum_{F\in
A}\mathcal{W}(F).$
###### Proof.
It is enough to see that $F\in C$ if for every $J\in F$ either $J$ has at most
$\frac{m}{2}-1$ zero components or $J$ has exactly $\frac{m}{2}$ zero
components and $\mathcal{W}(F)=\mathcal{W}(F^{c})$. $\Box$
We close this section with the following corollary.
###### Corollary 13
. Let $m$ be an even number and $n_{1}=n_{2}=\cdots=n_{m}=\alpha$. Then
$\omega(G(\mathbb{Z}_{n}))=\chi(G(\mathbb{Z}_{n}))=\sum_{i=0}^{\frac{m}{2}-1}\big{(}\,^{m}_{i}\,\big{)}\alpha^{m-i}+\frac{\big{(}\,^{m}_{\frac{m}{2}}\,\big{)}\alpha^{\frac{m}{2}}}{2}-1.$
###### Proof.
Since $n_{1}=n_{2}=\cdots=n_{m}=\alpha$, it is easily seen that
$\prod_{i=0}^{\frac{m}{2}-2}{n_{m-i}}\leq\prod_{i=1}^{\frac{m}{2}+1}{n_{i}}$.
Moreover, there are $\big{(}\,^{m}_{i}\,\big{)}\alpha^{m-i}$ ideals with $i$
non-zero components. Note that, if $i=\frac{m}{2}$, exactly
$\frac{\big{(}\,^{m}_{\frac{m}{2}}\,\big{)}\alpha^{\frac{m}{2}}}{2}$ of proper
ideals are adjacent together and
$\mathbb{Z}_{p_{1}^{\alpha}}\times\cdots\times\mathbb{Z}_{p_{m}^{\alpha}}$ is
not a vertex of $G(\mathbb{Z}_{n})$. The result, now, follows from Theorem 12.
$\Box$
It follows from [2], for every ring $R$, if $\omega(G(R))<\infty$, then
$\chi(G(R))<\infty$. Also, we have not found any example of a ring $R$ such
that $G(R)$ is not weakly perfect. These positive results motivate the
following conjecture.
$\mathbf{Conjecture.}$ For every ring $R$, $G(R)$ is a weakly perfect graph.
3\. Edge Chromatic Number of $G(\mathbb{Z}_{n})$
In this section we study the edge chromatic number of $G(\mathbb{Z}_{n})$ and
prove that, for every positive integer $n$,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$, unless $n=pq$,
where $p,q$ are distinct primes, or $n=p^{m}$, where $m$ is an even number. If
$n=pq$, then $G(\mathbb{Z}_{n})$ is a null graph with two vertices and if
$n=p^{m}$, where $m$ is an even number, then $G(\mathbb{Z}_{n})$ is a complete
graph of order $m-1$. First, we need the following lemmas:
###### Lemma 14
. [8, p. 16] If $G$ is a simple graph, then either
$\chi^{\prime}(G)=\Delta(G)$ or $\chi^{\prime}(G)=\Delta(G)+1$.
###### Lemma 15
. [3, Corollary 5.4] Let $G$ be a simple graph. Suppose that for every vertex
$u$ of maximum degree, there exists an edge $u--v$ such that
$\Delta(G)-d(v)+2$ is more than the number of vertices with maximum degree in
$G$. Then $\chi^{\prime}(G)=\Delta(G)$.
###### Lemma 16
. [7, Theorem D] If $G$ has order $2s$ and maximum degree $2s-1$, then
$\chi^{\prime}(G)=\Delta(G)$. If $G$ has order $2s+1$ and maximum degree $2s$,
then $\chi^{\prime}(G)=\Delta(G)+1$ if and only if the size of $G$ is at least
$2s^{2}+1$.
Now, we are ready to state our main result in this section.
###### Theorem 17
. Let $n$ be a positive integer. Then
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$, unless
$n=p_{1}p_{2}$, where $p_{1},p_{2}$ are distinct primes, or $n=p^{m}$, where
$m$ is an even number.
###### Proof.
Suppose that $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\ldots p_{m}^{n_{m}}$, where
$p_{i}$’s are all distinct primes and $n_{i}$’s are all natural numbers. If
$m=1$, then $G(\mathbb{Z}_{n})$ is a complete graph of order $n_{1}-1$. If
$n_{1}$ is an even number, then
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))+1$, otherwise
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$. Thus assume that
$m\geq 2$. We continue the proof in the following cases:
Case 1. $n_{1}=n_{2}=\cdots=n_{m}=1$. In this case, $\mathbb{Z}_{n}\cong
F_{1}\times\cdots\times F_{m}$, where each $F_{i}$ is a field. If $m=2$, then
$\mathbb{Z}_{n}\cong F_{1}\times F_{2}$ and so $G(\mathbb{Z}_{n})$ is a null
graph with two vertices. Therefore,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))+1$. Then suppose
that $m\geq 3$. For each $j=1,\ldots,m$, define the ideal
$I_{j}=F_{1}\times\cdots\times F_{j-1}\times 0\times F_{j+1}\times\cdots\times
F_{m}$. Then $d(I_{j})=\Delta(G(\mathbb{Z}_{n}))=2^{m}-4$, for each $1\leq
j\leq m$. Also, $I_{1},\ldots,I_{m}$ are all vertices with maximum degree in
$G(\mathbb{Z}_{n})$. Let $u=I_{i}$ ($1\leq i\leq m$) be a vertex of maximum
degree, say $I_{1}$. Then $v=0\times F_{2}\times 0\times\cdots\times 0$ is an
adjacent vertex to $u$. Thus
$\Delta(G(\mathbb{Z}_{n}))-d(v)+2=2^{m}-4-(2^{m-1}-2)+2=2^{m}-2^{m-1}$. Since
$m\geq 3$, we conclude that $2^{m}-2^{m-1}>m$, for each $m$. By Lemma 15,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$.
Case 2. Every $n_{i}$ is an even number. Then $|V(G(\mathbb{Z}_{n}))|$ is an
odd number of the form $2s+1$. Note that every vertex of the form
$I_{1}\times\cdots\times I_{m}$, where $I_{i}$ is a non-zero ideal of
$\mathbb{Z}_{p_{i}^{n_{i}}}$ is adjacent to all vertices of
$G(\mathbb{Z}_{n})$. Since the size of a complete graph of order $2s+1$ is
$2s^{2}+s$, if we prove that the intersection graph $G(\mathbb{Z}_{n})$, in
this case, losses at least $s$ edges, then by Lemma 16,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$. One can consider
the ring $\mathbb{Z}_{n}$ of the form $R_{1}\times R_{2}$, where $R_{1},R_{2}$
are two rings with $|I(R_{1})|=\alpha$ and $|I(R_{2})|=\beta$. Therefore,
$\alpha\beta=2s+3$. Clearly, every vertex of the form $(I,0)$ is not adjacent
to every vertex of the form $(0,J)$. Then $G(R_{1}\times R_{2})$ losses at
least $\alpha\beta-(\alpha+\beta)+1$ edges. It is easily checked that
$\alpha\beta-(\alpha+\beta)+1>s=\frac{\alpha\beta-3}{2}$, as desired.
Case 3. There exists at least one $i$ such that $n_{i}$ is an odd number and
$n_{j}>1$ for some $j$. Therefore $\mathbb{Z}_{p_{j}^{n_{j}}}$ has at least
one non-trivial proper ideal and then $\mathbb{Z}_{n}$ has an ideal with no
zero complement which is adjacent to every other vertex. Since
$|V(G(\mathbb{Z}_{n}))|$ is an even number, by Lemma 16,
$\chi^{\prime}(G(\mathbb{Z}_{n}))=\Delta(G(\mathbb{Z}_{n}))$. $\Box$
Acknowledgements. The authors would like to express their deep gratitude to
the professor Saieed Akbari for his fruitful comments. Also, we thank to the
referee for his/her careful reading and his/her valuable suggestions.
## References
* [1] S. Akbari, D. Kiani, F. Mohammadi, S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra 213 (12) (2009), 2224–2228.
* [2] S. Akbari, R. Nikandish, M.J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl., to appear.
* [3] L.W. Beineke, B.J. Wilson, Selected Topics in Graph Theory, Academic Press Inc., London, 1978.
* [4] I. Chakrabarty, S. Ghosh, T.K. Mukherjee, M.K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (17) (2009), 5381–5392.
* [5] M. Kubale, Graph Colorings, American Mathematical Society, 2004.
* [6] E. Marczewski, Sur deux propriétés des class d’ensembles, Fund. Math. 33 (1945), 303–307.
* [7] M.J. Plantholt, The chromatic index of graphs with large maximum degree, Discrete Math. 47 (1981), 91–96.
* [8] H.P. Yap, Some Topics in Graph Theory, in: London Math. Soc. Lecture Note Ser., vol. 108, 1986.
|
arxiv-papers
| 2013-05-27T18:29:12 |
2024-09-04T02:49:45.786025
|
{
"license": "Public Domain",
"authors": "R.Nikandish and M.J. Nikmehr",
"submitter": "Mohammad Nikmehr",
"url": "https://arxiv.org/abs/1305.6287"
}
|
1305.6350
|
# An efficient dynamic ID based remote user authentication scheme using self-
certified public keys for multi-server environment
Dawei Zhaoa,b Haipeng Penga,b Shudong Lic Yixian Yanga,b
aInformation Security Center, Beijing University of Posts and
Telecommunications,
Beijing 100876, China.
bNational Engineering Laboratory for Disaster Backup and Recovery,
Beijing University of Posts and Telecommunications, Beijing 100876, China.
c School of Mathematics, Shandong Institute of Business and Technology,
Shandong Yantai, 264005 China.
††E-mail address: [email protected] (Dawei Zhao); [email protected]
(Haipeng Peng).
Abstract. Recently, Li et al. analyzed Lee et al.’s multi-server
authentication scheme and proposed a novel smart card and dynamic ID based
remote user authentication scheme for multi-server environments. They claimed
that their scheme can resist several kinds of attacks. However, through
careful analysis, we find that Li et al.’s scheme is vulnerable to stolen
smart card and offline dictionary attack, replay attack, impersonation attack
and server spoofing attack. By analyzing other similar schemes, we find that
the certain type of dynamic ID based multi-server authentication scheme in
which only hash functions are used and no registration center participates in
the authentication and session key agreement phase is hard to provide perfect
efficient and secure authentication. To compensate for these shortcomings, we
improve the recently proposed Liao et al.’s multi-server authentication scheme
which is based on pairing and self-certified public keys, and propose a novel
dynamic ID based remote user authentication scheme for multi-server
environments. Liao et al.’s scheme is found vulnerable to offline dictionary
attack and denial of service attack, and cannot provide user’s anonymity and
local password verification. However, our proposed scheme overcomes the
shortcomings of Liao et al.’s scheme. Security and performance analyses show
the proposed scheme is secure against various attacks and has many excellent
features.
Keyword. Authentication, Multi-server, Pairing-based, Hash function, Self-
certified public keys.
## §1 Introduction
With the rapid development of network technologies, more and more people begin
using the network to acquire various services such as on-line financial, on-
line medical, on-line shopping, on-line bill payment, on-line documentation
and data exchange, etc. And the architecture of server providing services to
be accessed over the network often consists of many different servers around
the world instead of just one. While enjoying the comfort and convenience of
the internet, people are facing with the emerging challenges from the network
security.
Identity authentication is the key security issue of various types of on-line
applications and service systems. Before an user accessing the services
provided by a service provider server, mutual identity authentication between
the user and the server is needed to prevent the unauthorized personnel from
accessing services provided by the server and avoid the illegal system
cheating the user by masquerading as legal server. In the single server
environment, password based authentication scheme [1] and its enhanced version
which additionally uses smart cards [2-9] are widely used to provide mutual
authentication between the users and servers. However, the conventional
password based authentication methods are not suitable for the multi-servers
environment since each user does not only need to log into different remote
servers repetitively but also need to remember many various sets of identities
and passwords if he/she wants to access these service providing servers. In
order to resolve this problem, in 2000, based on the difficulty of
factorization and hash function, Lee and Chang [10] proposed a user
identification and key distribution scheme which agrees with the multi-server
environment. Since then, authentication schemes for the multi-server
environment have been widely investigated and designed by many researchers
[11-28].
Based on the used of the basic cryptographic algorithms, the existing multi-
server authentication schemes can be divided into two types, namely the hash
based authentication schemes and the public-key based authentication schemes.
At the same time, among these existing multi-server authentication schemes,
some of them need the registration center (RC) to participate in the
authentication and session key agreement phase, while others don’t. Therefore,
according to the participation or not of the RC in the authentication and
session key agreement phase, we divide the multi-server authentication schemes
into RC dependented authentication schemes and non-RC dependented
authentication schemes.
In this paper, we analyze a novel multi-server authentication scheme, Li et
al.’s scheme [20] which is only based on hash function and a non-RC
dependented authentication scheme. We find that this scheme is vulnerable to
stolen smart card and offline dictionary attack, replay attack, impersonation
attack and server spoofing attack. By analyzing some other similar schemes
[15,17-19], we find that the type of dynamic ID based multi-server
authentication scheme which is only using hash functions and non-RC
dependented is hard to provide perfect efficient and secure authentication. To
compensate for these shortcomings, we improve the recently proposed Liao et
al.’s multi-server authentication scheme [27] which is based on pairing and
self-certified public keys, and propose a novel dynamic ID based remote user
authentication scheme for multi-server environments. Liao et al.’s scheme is
found vulnerable to offline dictionary attack [28] and denial of service
attack, and cannot provide user’s anonymity and local password verification.
However, our proposed scheme overcomes the shortcomings of Liao et al.’s
scheme. Security and performance analyses show the proposed scheme is secure
against various attacks and has many excellent features.
## §2 Related works
A large number of authentication schemes have been proposed for the multi-
server environment. Hash function is one of the key technologies in the
construction of multi-server authentication scheme. In 2004, Juang et al. [11]
proposed an efficient multi-server password authenticated key agreement scheme
based on a hash function and symmetric key cryptosystem. In 2009, Hsiang and
Shih [12] proposed a dynamic ID based remote user authentication scheme for
multi-server environment in which only hash function is used. However, Sood et
al. [13] found that Hsiang and Shih’s scheme is susceptible to replay attack,
impersonation attack and stolen smart card attack. Moreover, the password
change phase of Hsiang and Shih’s scheme is incorrect. Then Sood et al.
presented a novel dynamic identity based authentication protocol for multi-
server architecture to resolve the security flaws of Hsiang and Shih’s scheme
[13]. After that, Li et al. [14] pointed out that Sood et al.’s protocol is
still vulnerable to leak-of-verifier attack, stolen smart card attack and
impersonation attack. At the same time, Li et al. [14] proposed another
dynamic identity based authentication protocol for multi-server architecture.
However, the above mentioned scheme are all RC dependented multi-server
authentication scheme. In 2009, Liao and Wang [15] proposed a dynamic ID based
multi-server authentication scheme which is based on hash function and non-RC
dependented. But, Liao and Wang’s scheme is vulnerable to insider’s attack,
masquerade attack, server spoofing attack, registration center spoofing attack
and is not reparable [16]. After that, Shao et al. [17] and Lee et al. [18,19]
proposed some similar types of multi-server authentication schemes. In 2012,
Li et al.[20] pointed out that Lee et al.’s scheme [18] cannot withstand
forgery attack, server spoofing attack and cannot provide proper
authentication, and then proposed a novel dynamic ID based multi-server
authentication schemes which is only using hash function and non-RC
dependented. However, with careful analysis, we find that Li et al.’s scheme
[20] is still vulnerable to stolen smart card and offline dictionary attack,
replay attack, impersonation attack and server spoofing attack. We also
analyzed Shao et al.’s scheme [17] and Lee et al.’s scheme [19], they are all
vulnerable to stolen smart card and offline dictionary attack, replay attack,
impersonation attack and server spoofing attack. In general, it is difficult
to construct a secure dynamic ID based and non-RC dependented multi-server
authentication scheme if only hash functions are used.
Public-key cryptograph is another useful technique which is widely used in the
construction of multi-server authentication scheme. In 2000, Lee and Chang
[21] proposed a user identification and key distribution scheme in which the
difficulty of factorization on public key cryptography is used. In 2001, Tsaur
[22] proposed a remote user authentication scheme based on RSA cryptosystem
and Lagrange interpolating polynomial for multi-server environments. Then Lin
et al. [23] proposed a multi-server authentication protocol based on the
simple geometric properties of the Euclidean and discrete logarithm problem
concept. Since the traditionally public key cryptographic algorithms require
many expensive computations and consume a lot of energy, Geng and Zhang [24]
proposed a dynamic ID-based user authentication and key agreement scheme for
multi-server environment using bilinear pairings. But Geng and Zhang’s scheme
cannot withstand user spoofing attack [25]. After that, Tseng et al. [26]
proposed an efficient pairing-based user authentication scheme with smart
cards. However, in 2013, Liao and Hsiao [27] pointed out that Tseng et al.’s
scheme is vulnerable to insider attack, offline dictionary attack and
malicious server attack, and cannot provide proper mutual authentication and
session key agreement. At the same time, Liao and Hsiao proposed a novel non-
RC dependented multi-server remote user authentication scheme using self-
certified public keys for mobile clients [27]. Recently, Chou et al. [28]
found Liao and Hsiao’s scheme cannot withstand password guessing attack.
Furthermore, with careful analysis, we find that Liao and Hsiao’s scheme is
still vulnerable to denial of service attack, and cannot provide user’s
anonymity and local password verification. In this paper, based on the Liao
and Hsiao’s scheme, we propose a secure dynamic ID based and non-RC
dependented multi-server authentication scheme using the pairing and self-
certified public keys.
## §3 Review and cryptanalysis of Li et al.’s authentication scheme
### 3.1 Review of Li et al.’s scheme
Li et al.’s contains three participants, the user $U_{i}$, the server $S_{j}$,
and the registration center $RC$. $RC$ chooses the master secret key $x$ and a
secret number $y$ to compute $h(x\|y)$ and $h(SID_{j}\|h(y))$, and then shares
them with $S_{j}$ via a secure channel. $SID_{j}$ is the identity of server
$S_{j}$. There are four phases in the scheme: registration phase, login phase,
verification phase, and password change phase.
#### 3.1.1 Registration phase
When the remote user authentication scheme starts, the user $U_{i}$ and the
registration center $RC$ need to perform the following steps to finish the
registration phase:
(1) $U_{i}$ freely chooses his/her identity $ID_{i}$, the password $PW_{i}$,
and computes $A_{i}=h(b\oplus PW_{i})$, where $b$ is a random number generated
by $U_{i}$. Then $U_{i}$ sends $ID_{i}$ and $A_{i}$ to the registration center
$RC$ for registration through a secure channel.
(2) $RC$ computes $B_{i}=h(ID_{i}\|x)$, $C_{i}=h(ID_{i}\|h(y)\|A_{i})$,
$D_{i}=h(B_{i}\|h(x\|y))$ and $E_{i}=B_{i}\oplus h(x\|y)$. $RC$ stores
$\\{C_{i},D_{i},E_{i},h(\cdot),h(y)\\}$ on the user’s smart card and sends it
to user $U_{i}$ via a secure channel.
(3) $U_{i}$ keys $b$ into the smart card, and finally the smart card contains
$\\{C_{i},D_{i},E_{i},b,h(\cdot),h(y)\\}$.
#### 3.1.2 Login phase
Whenever $U_{i}$ wants to login $S_{j}$, he/she must perform the following
steps to generate a login request message:
(1) $U_{i}$ inserts his/her smart card into the card reader and inputs
$ID_{i}$ and $PW_{i}$. Then the smart card computes $A_{i}=h(b\oplus PW_{i})$,
$C_{i}^{*}=h(ID_{i}\|h(y)\|A_{i})$, and checks whether the computed
$C_{i}^{*}$ is equal to $C_{i}$. If they are equal, $U_{i}$ proceeds the
following steps. Otherwise the smart card aborts the session.
(2) The smart card generates a random number $N_{i}$ and computes
$P_{ij}=E_{i}\oplus h(h(SID_{j}\|h(y))\|N_{i})$, $CID_{i}=A_{i}\oplus
h(D_{i}\|SID_{j}\|N_{i})$, $M_{1}=h(P_{ij}\|CID_{i}\|D_{i}\|N_{i})$ and
$M_{2}=h(SID_{j}\|h(y))\oplus N_{i}$.
(3) $U_{i}$ submits $\\{P_{ij},CID_{i},M_{1},M_{2}\\}$ to $S_{j}$ as a login
request message.
#### 3.1.3 Verification phase
Wher $S_{j}$ receiving the login message $\\{P_{ij},CID_{i},M_{1},M_{2}\\}$,
$S_{j}$ and $U_{i}$ perform the following steps to finish the mutual
authentication and session key agreement.
(1) $S_{j}$ computes $N_{i}=M_{2}\oplus h(SID_{j}\|h(y))$, $E_{i}=P_{ij}\oplus
h(h(SID_{j}\|h(y))\|N_{i})$, $B_{i}=E_{i}\oplus h(x\|y)$,
$D_{i}=h(B_{i}\|h(x\|y))$ and $A_{i}=CID_{i}\oplus h(D_{i}\|SID_{j}\|N_{i})$
by using $\\{P_{ij},CID_{i},M_{1},M_{2}\\}$, $h(SID_{j}\|h(y))$ and $h(x\|y)$.
(2) $S_{j}$ computes $h(P_{ij}\|CID_{i}\|D_{i}\|N_{i})$ and checks whether it
is equal to $M_{1}$. If they are not equal, $S_{j}$ rejects the login request
and terminates this session. Otherwise, $S_{j}$ accepts the login request
message. Then $S_{j}$ generates a random number $N_{j}$ and computes
$M_{3}=h(D_{i}\|A_{i}\|N_{j}\|SID_{j})$, $M_{4}=A_{i}\oplus N_{i}\oplus
N_{j}$. Finally, $S_{j}$ sends the message $\\{M_{3},M_{4}\\}$ to $U_{i}$.
(3) After receiving the response message $\\{M_{3},M_{4}\\}$ sent from
$S_{j}$, $U_{i}$ computes $N_{j}=A_{i}\oplus N_{i}\oplus M_{4}$,
$M_{3}^{*}=h(D_{i}\|A_{i}\|N_{j}\|SID_{j})$ and checks $M_{3}^{*}$ with the
received message $M_{3}$. If they are not equal, $U_{i}$ rejects these
messages and terminates this session. Otherwise, $U_{i}$ successfully
authenticates $S_{j}$. Then, the user $U_{i}$ computes the mutual
authentication message $M_{5}=h(D_{i}\|A_{i}\|N_{i}\|SID_{j})$ and sends
$\\{M_{5}\\}$ to the server $S_{j}$.
(4) Upon receiving the message $\\{M_{5}\\}$ from $U_{i}$, $S_{j}$ computes
$h(D_{i}\|A_{i}\|N_{i}\|SID_{j})$ and checks it with the received message
$\\{M_{5}\\}$. If they are equal, $S_{j}$ successfully authenticates $U_{i}$
and the mutual authentication is completed. After the mutual authentication
phase, the user $U_{i}$ and the server $S_{j}$ compute
$SK=h(D_{i}\|A_{i}\|N_{i}\|N_{j}\|SID_{j})$, which is taken as their session
key for future secure communication.
#### 3.1.4 Password change phase
This phase is invoked whenever $U_{i}$ wants to change his password $PW_{i}$
to a new password $PW^{new}_{i}$. There is no need for a secure channel for
password change, and it can be finished without communicating with the
registration center $RC$.
(1) $U_{i}$ inserts his/her smart card into the card reader and inputs
$ID_{i}$ and $PW_{i}$.
(2) The smart card computes $A_{i}=h(b\oplus PW_{i})$,
$C^{*}_{i}=h(ID_{i}\|h(y)\|A_{i})$, and checks whether the computed
$C^{*}_{i}$ is equal to $C_{i}$. If they are not equal, the smart card rejects
the password change request. Otherwise, the user $U_{i}$ inputs a new password
$PW^{new}_{i}$ and a new random number $b^{new}$.
(3) The smart card computes $A^{new}_{i}=h(b^{new}\oplus PW^{new}_{i})$ and
$C^{new}_{i}=h(ID_{i}\|h(y)\|A^{new}_{i})$.
(4) Finally, the smart card replaces $C_{i}$ and $b$ with $C^{new}_{i}$ and
$b^{new}$ to finish the password change phase.
### 3.2 Cryptanalysis of Li et al.’s scheme
Li et al. claimed that their scheme can resist many types of attacks and
satisfy all the essential requirements for multi-server architecture
authentication. However, if we assume that $A$ is an adversary who has broken
a user $U_{m}$ and a server $S_{n}$, or a combination of a malicious user
$U_{m}$ and a dishonest server $S_{n}$. Then $A$ could get the secret number
$h(x\|y)$ and $h(y)$, and can perform the stolen smart card and offline
dictionary attack, replay attack, impersonation attack and server spoofing
attack to Li et al.’s scheme. The concrete cryptanalysis of the Li et al.’s
scheme is shown as follows.
#### 3.2.1 Stolen smart card and offline dictionary attack
If a user $U_{i}$’s smart card is stolen by an adversary $A$, $A$ can extract
the information $\\{C_{i},D_{i},E_{i},b,$ $h(\cdot),h(y)\\}$ from the memory
of the stolen smart card. Furthermore, in case $A$ intercepts a valid login
request message $\\{P_{ij},CID_{i},M_{1},M_{2}\\}$ sent from user $U_{i}$ to
server $S_{j}$ in the public communication channel, $A$ can compute
$N_{i}=h(SID_{j}\|h(y))\oplus M_{2}$, $E_{i}=P_{ij}\oplus
h(h(SID_{j}\|h(y))\|N_{i})$, $B_{i}=E_{i}\oplus h(x\|y)$,
$D_{i}=h(B_{i}\|h(x\|y))$ and $A_{i}=CID_{i}\oplus h(D_{i}\|SID_{j}\|N_{i})$
by using $h(y)$ and $h(x\|y)$. Then $A$ can launch offline dictionary attack
on $C_{i}=h(ID_{i}\|h(y)\|A_{i})$ to know the identity $ID_{i}$ of the user
$U_{i}$ because $A$ knows the values of $A_{i}$ corresponding to the user
$U_{i}$. Besides $A$ can launch offline dictionary attack on $A_{i}=h(b\oplus
PW_{i})$ to know the password $PW_{i}$ of $U_{i}$ because $A$ knows the value
of $b$ from the stolen smart card of the user $U_{i}$. Now $A$ possesses the
valid smart card of user $U_{i}$, knows the identity $ID_{i}$, password
$PW_{i}$ corresponding to the user $U_{i}$ and hence can login on to any
service server.
#### 3.2.2 Replay attack
The replay attack is replaying the same message of the receiver or the sender
again. If adversary $A$ has intercepted a valid login request message
$\\{P_{ij},CID_{i},M_{1},M_{2}\\}$ sent from user $U_{i}$ to server $S_{j}$ in
the public communication channel. Then $A$ can compute
$N_{i}=h(SID_{j}\|h(y))\oplus M_{2}$, $E_{i}=P_{ij}\oplus
h(h(SID_{j}\|h(y))\|N_{i})$, $B_{i}=E_{i}\oplus h(x\|y)$,
$D_{i}=h(B_{i}\|h(x\|y))$ and $A_{i}=CID_{i}\oplus h(D_{i}\|SID_{j}\|N_{i})$
by using $h(y)$ and $h(x\|y)$. Then adversary $A$ can replay this login
request message $\\{P_{ij},CID_{i},M_{1},M_{2}\\}$ to $S_{j}$ by masquerading
as the user $U_{i}$ at some time latter. After verification of the login
request message, $S_{j}$ computes $M_{3}=h(D_{i}\|A_{i}\|N_{j}\|SID_{j})$ and
$M_{4}=A_{i}\oplus N_{i}\oplus N_{j}$, and sends the message
$\\{M_{3},M_{4}\\}$ to $A$ who is masquerading as the user $U_{i}$. The
adversary $A$ can verify the received value of $\\{M_{3},M_{4}\\}$ and compute
$M_{5}^{\prime}=h(D_{i}\|A_{i}\|N_{i}\|SID_{j})$ since he knows the values of
$N_{i},E_{i},B_{i},D_{i}$ and $A_{i}$. Then $A$ sends $\\{M^{\prime}_{5}\\}$
to the server $S_{j}$. The $S_{j}$ computes $h(D_{i}\|A_{i}\|N_{i}\|SID_{j})$
and checks it with the received message $\\{M_{5}^{\prime}\\}$. This
equivalency authenticates the legitimacy of the user $U_{i}$, the service
provider server $S_{j}$ and the login request is accepted. Finally after
mutual authentication, adversary $A$ masquerading as the user $U_{i}$ and the
server $S_{j}$ agree on the common session key as
$SK=h(D_{i}\|A_{i}\|N_{i}\|N_{j}\|SID_{j})$. Therefore, the adversary $A$ can
masquerade as user $U_{i}$ to login on to server $S_{j}$ by replaying the same
login request message which had been sent from $U_{i}$ to $S_{j}$.
#### 3.2.3 Impersonation attack
In this subsection, we show that the adversary $A$ who possesses $h(y)$ and
$h(x\|y)$ can masquerade as any user $U_{i}$ to login any server $S_{j}$ as
follows.
Adversary $A$ chooses two random numbers $a_{i}$ and $b_{i}$, and computes
$A_{i}=h(a_{i})$ and $B_{i}=h(b_{i})$. Then $A$ can compute
$D_{i}=h(B_{i}\|h(x\|y))$, $E_{i}=B_{i}\oplus h(x\|y)$, $P_{ij}=E_{i}\oplus
h(h(SID_{j}\|h(y))\|N_{i})$, $CID_{i}=A_{i}\oplus h(D_{i}\|SID_{j}\|N_{i})$,
$M_{1}=h(P_{ij}\|CID_{i}\|D_{i}\|N_{i})$ and $M_{2}=h(SID_{j}\|h(y))\oplus
N_{i}$ by using $h(y)$ and $h(x\|y)$. Now $A$ sends the login request message
$\\{P_{ij},CID_{i},M_{1},M_{2}\\}$ by masquerading as the user $U_{i}$ to
server $S_{j}$. After receiving the login request message, $S_{j}$ computes
$N_{i}=h(SID_{j}\|h(y))\oplus M_{2}$, $E_{i}=P_{ij}\oplus
h(h(SID_{j}\|h(y))\|N_{i})$, $B_{i}=E_{i}\oplus h(x\|y)$,
$D_{i}=h(B_{i}\|h(x\|y))$ and $A_{i}=CID_{i}\oplus h(D_{i}\|SID_{j}\|N_{i})$
by using $\\{P_{ij},CID_{i},M_{1},M_{2}\\}$, $h(x\|y)$ and $h(SID_{j}\|h(y))$.
Then $S_{j}$ computes $M_{3}=h(D_{i}\|A_{i}\|N_{j}\|SID_{j})$ and
$M_{4}=A_{i}\oplus N_{i}\oplus N_{j}$, and sends the message
$\\{M_{3},M_{4}\\}$ to $A$ who is masquerading as the user $U_{i}$. Then
adversary $A$ computes $N_{j}=A_{i}\oplus N_{i}\oplus M_{4}$ and verifies
$M_{3}$ by computing $h(D_{i}\|A_{i}\|N_{j}\|SID_{j})$. Then $A$ computes
$M_{5}=h(D_{i}\|A_{i}\|N_{i}\|SID_{j})$ and sends $\\{M_{5}\\}$ back to the
server $S_{j}$. The $S_{j}$ computes $h(D_{i}\|A_{i}\|N_{i}\|SID_{j})$ and
checks it with the received message $\\{M_{5}\\}$. This equivalency
authenticates the legitimacy of the user $U_{i}$, the service provider server
$S_{j}$ and the login request is accepted. Finally after mutual
authentication, adversary $A$ masquerading as the user $U_{i}$ and the server
$S_{j}$ agree on the common session key as
$SK=h(D_{i}\|A_{i}\|N_{i}\|N_{j}\|SID_{j})$.
#### 3.2.4 Server spoofing attack
In this subsection, we show that the adversary $A$ who possesses $h(y)$ and
$h(x\|y)$ can masquerade as the server $S_{j}$ to spoof user $U_{i}$, if $A$
has intercepted a valid login request message
$\\{P_{ij},CID_{i},M_{1},M_{2}\\}$ sent from user $U_{i}$ to server $S_{j}$ in
the public communication channel.
After intercepting a valid login request message
$\\{P_{ij},CID_{i},M_{1},M_{2}\\}$ sent from user $U_{i}$ to server $S_{j}$ in
the public communication channel, $A$ can compute
$N_{i}=h(SID_{j}\|h(y))\oplus M_{2}$, $E_{i}=P_{ij}\oplus
h(h(SID_{j}\|h(y))\|N_{i})$, $B_{i}=E_{i}\oplus h(x\|y)$,
$D_{i}=h(B_{i}\|h(x\|y))$ and $A_{i}=CID_{i}\oplus h(D_{i}\|SID_{j}\|N_{i})$
corresponding to $U_{i}$. Then $A$ can choose a random number
$N^{\prime}_{j}$, and compute $M_{3}=h(D_{i}\|A_{i}\|N^{\prime}_{j}\|SID_{j})$
and $M_{4}=A_{i}\oplus N_{i}\oplus N^{\prime}_{j}$. $A$ then sends the message
$\\{M_{3},M_{4}\\}$ by masquerading as server $S_{j}$ to the user $U_{i}$.
After receiving the message $\\{M_{3},M_{4}\\}$, $U_{i}$ computes
$N^{\prime}_{j}=A_{i}\oplus N_{i}\oplus M_{4}$ and verifies $M_{3}$ by
computing $h(D_{i}\|A_{i}\|N^{\prime}_{j}\|SID_{j})$. Then $U_{i}$ computes
$M_{5}=h(D_{i}\|A_{i}\|N_{i}\|SID_{j})$ and sends it to the $S_{j}$ who is
masquerading as the adversary $A$. Then $A$ computes
$h(D_{i}\|A_{i}\|N_{i}\|SID_{j})$ and checks it with the received message
$\\{M_{5}\\}$. Finally after mutual authentication, adversary $A$ masquerading
as the server $S_{j}$ and the user $U_{i}$ agree on the common session key as
$SK=h(D_{i}\|A_{i}\|N_{i}\|N^{\prime}_{j}\|SID_{j})$.
### 3.3 Discussion
Except the Li et al.’s scheme, we also analyzed other four dynamic ID based
authentication schemes for multi-server environment [15,17-19]. These schemes
are all based on hash functions and non-RC dependented. We found that such
type of multi-server remote user authentication scheme are almost vulnerable
to stolen smart card and offline dictionary attacks, impersonation attack and
server spoofing attack etc. The cryptanalysis methods of these schemes are
similar to that of Li et al.’s scheme shown in section 3.2. We think that
under the assumptions that no registration center participates in the
authentication and session key agreement phase, the dynamic ID and hash
function based user authentication schemes for multi-server environment is
hard to provide perfect efficient and secure authentication. Fortunately,
there is another technique, public-key cryptograph which is widely used in the
construction of authentication scheme. Therefore, in order to construct a
secure, low power consumption and non-RC dependented authentication scheme, we
adopt the elliptic curve cryptographic technology of public-key techniques,
and propose a novel dynamic ID based and non-RC dependented remote user
authentication scheme using pairing and self-certified public keys for multi-
server environment.
## §4 Preliminaries
Before presenting our scheme, we introduce the concepts of bilinear pairings,
self-certified public keys, as well as some related mathematical assumptions.
### 4.1 Bilinear pairings
Let $G_{1}$ be an additive cyclic group with a large prime order $q$ and
$G_{2}$ be a multiplicative cyclic group with the same order $q$.
Particularly, $G_{1}$ is a subgroup of the group of points on an elliptic
curve over a finite field $E(F_{p})$ and $G_{2}$ is a subgroup of the
multiplicative group over a finite field. $P$ is a generator of $G_{1}$.
A bilinear pairing is a map $e:G_{1}\times G_{1}\rightarrow G_{2}$ and
satisfies the following properties:
(1) Bilinear: $e(aP,bQ)=e(P,Q)^{ab}$ for all $P,Q\in G_{1}$ and $a,b\in
Z^{*}_{q}$.
(2) Non-degenerate: There exists $P,Q\in G_{1}$ such that $e(P,Q)\neq 1$.
(3) Computability: There is an efficient algorithm to compute $e(P,Q)$ for all
$P,Q\in G_{1}$.
### 4.2 self-certified public keys
In [27], Liao et al. first proposes a key distribution based on self-certified
public keys (SCPKs) [29,30] among the service servers. By using the SCPK, a
user’s public key can be computed directly from the signature of the third
trust party (TTP) on the user’s identity instead of verifying the public key
using an explicit signature on a user’s public key. The SCPK scheme is
described as follows.
(1) Initialization: The third trust party (TTP) first generates all the needed
parameters of the scheme. TTP chooses a non-singular high elliptic curve
$E(F_{p})$ defined over a finite field, which is used with a based point
generator $P$ of prime order $q$. Then TTP freely chooses his/her secret key
$s_{T}$ and computes his/her public key $pub_{T}=s_{T}\cdot P$. The related
parameters and $pub_{T}$ are publicly and authentically available.
(2) Private key generation: An user $A$ chooses a random number $k_{A}$,
computes $K_{A}=k_{A}\cdot P$ and sends his/her identity $ID_{A}$ and $K_{A}$
to the TTP. TTP chooses a random number $r_{A}$, computes
$W_{A}=K_{A}+r_{A}\cdot P$ and $\bar{s}_{A}=h(ID_{A}\parallel W_{A})+r_{A}$,
and sends $W_{A}$ and $\bar{s}_{A}$ to user $A$. Then $A$ obtains his/her
secret key by calculating $s_{A}=\bar{s}_{A}+k_{A}$.
(3) Public key extraction: Anyone can calculate $A$’s public key
$pub_{A}=h(ID_{A}\parallel W_{A})pub_{T}+W_{A}$ when he/she receives $W_{A}$.
### 4.3 Related mathematical assumptions
To prove the security of our proposed protocol, we present some important
mathematical problems and assumptions for bilinear pairings defined on
elliptic curves. The related concrete description can be found in [31,32].
(1) Computational discrete logarithm (CDL) problem: Given $R=x\cdot P$, where
$P,R\in G_{1}$. It is easy to calculate $R$ given $x$ and $P$, but it is hard
to determine $x$ given $P$ and $R$.
(2) Elliptic curve factorization (ECF) problem: Given two points $P$ and
$R=x\cdot P+y\cdot P$ for $x,y\in Z^{*}_{q}$ , it is hard to find $x\cdot P$
and $y\cdot P$.
(3) Computational Diffie-Hellman (CDH) problem: Given $P,xP,yP\in G_{1}$, it
is hard to compute $xyP\in G_{1}$.
## §5 The proposed scheme
In this section, by improving the recently proposed Liao et al.’s multi-server
authentication scheme [27] which is found vulnerable to offline dictionary
attack and denial of service attack [28], and cannot provide user’s anonymity
and local password verification, we propose a novel dynamic ID based remote
user authentication scheme for multi-server environment using pairing and
self-certified public keys. Our scheme contains three participants: the user
$U_{i}$, the service provider server $S_{j}$, and the registration center
$RC$. The legitimate user $U_{i}$ can easily login on to the service provider
server using his smart card, identity and password. There are six phases in
the proposed scheme: system initialization phase, the user registration phase,
the server registration phase, the login phase, the authentication and session
key agreement phase, and the password change phase. The notations used in our
proposed scheme are summarized in Table 1.
Table 1: Notations used in the proposed scheme. $e$ | | A bilinear map, $e:G_{1}\times G_{1}\longrightarrow G_{2}$.
---|---|---
$U_{i}$ | | The $i$th user.
$ID_{i}$ | | The identity of the user $U_{i}$.
$S_{j}$ | | The $j$th service provider server.
$SID_{j}$ | | The identity of the service provider server $S_{j}$.
$RC$ | | The registration center.
$s_{RC}$ | | The master secret key of the registration center $RC$ in $Z_{q}^{*}$.
$pub_{RC}$ | | The public key of $RC$, $pub_{RC}=s_{RC}\cdot P$.
$P$ | | A generator of group $G_{1}$.
$H()$ | | A map-to-point function, $H:{0,1}^{*}\longrightarrow G_{1}$.
$h()$ | | A one way hash function, $h:{0,1}^{*}\longrightarrow{0,1}^{k}$, where $k$ is the
| | output length. $h()$ allows the concatenation of some integer
| | values and points on an elliptic curve.
$\oplus$ | | A simple XOR operation in $G_{1}$. If $P_{1},P_{2}\in G_{1}$, $P_{1}$ and $P_{2}$ are
| | points on an elliptic curve over a finite field, the operation
| | $P_{1}\oplus P_{2}$ means that it performs the XOR operations of the
| | x-coordinates and y-coordinates of $P_{1}$ and $P_{2}$, respectively.
$\parallel$ | | The concatenation operation.
### 5.1 System initialization phase
In the proposed scheme, registration center $RC$ is assumed a third trust
party. In the system initialization phase, $RC$ generates all the needed
parameters of the scheme.
(1) $RC$ selects a cyclic additive group $G_{1}$ of prime order $q$, a cyclic
multiplicative group $G_{2}$ of the same order $q$, a generator $P$ of
$G_{1}$, and a bilinear map $e:G_{1}\times G_{1}\longrightarrow G_{2}$.
(2) $RC$ freely chooses a number $s_{RC}\in Z_{q}^{*}$ keeping as the system
private key and computes $pub_{RC}=s_{RC}\cdot P$ as the system public key.
(3) $RC$ selects two cryptographic hash functions $H(\cdot)$ and $h(\cdot)$.
Finally, all the related parameters
$\\{e,G_{1},G_{2},q,P,Pub_{RC},H(\cdot),h(\cdot)\\}$ are publicly and
authentically available.
### 5.2 User registration phase
When the user $U_{i}$ wants to access the services, he/she has to submit
his/her some related information to the registration center $RC$ for
registration. The steps of the user registration phase are as follows:
(1) The user $U_{i}$ freely chooses his/her identity $ID_{i}$ and password
$pw_{i}$, and chooses a random number $b_{i}$. Then $U_{i}$ computes
$HPW_{i}=h(ID_{i}\parallel pw_{i}\parallel b_{i})\cdot P$, and submits
$ID_{i}$ and $HPW_{i}$ to $RC$ for registration via a secure channel.
(2) When receiving the message $ID_{i}$ and $HPW_{i}$, $RC$ computes
$QID_{i}=H(ID_{i})$, $CID_{i}=s_{RC}\cdot QID_{i}$,
$Reg_{ID_{i}}=CID_{i}\oplus s_{RC}\cdot HPW_{i}$ and $H_{i}=h(QID_{i}\parallel
CID_{i})$. Then $RC$ stores the message $\\{Reg_{ID_{i}},H_{i}\\}$ in
$U_{i}$’s smart card and submits the smart card to $U_{i}$ through a secure
channel.
(3) After receiving the smart card, $U_{i}$ enters $b_{i}$ into the smart
card. Finally, the smart card contains parameters
$\\{Reg_{ID_{i}},H_{i},b_{i}\\}$.
### 5.3 Server registration phase
If a service provider server $S_{j}$ wants to provides services for the users,
he/she must perform the registration to the registration center $RC$ to become
a legal service provider server. The process of server registration phase of
the proposed scheme is based on SCPK mentioned in section 4.2.
(1) $S_{j}$ chooses a random number $v_{j}$ and computes $V_{j}=v_{j}\cdot P$.
Then $S_{j}$ submits $SID_{j}$ and $V_{j}$ to $RC$ for registration via a
secure channel.
(2) After receiving the message $\\{SID_{j},V_{j}\\}$, $RC$ chooses a random
number $w_{j}$, and computes $W_{j}=w_{j}\cdot P+V_{j}$ and
$s^{\prime}_{j}=(s_{RC}\cdot h(SID_{j}\parallel W_{j})+w_{j})$ mod $q$. Then
$RC$ submits the message $\\{W_{j},s^{\prime}_{j}\\}$ to $S_{j}$ through a
secure channel.
(3) After receiving $\\{W_{j},s^{\prime}_{j}\\}$, $S_{j}$ computes the private
key $s_{j}=(s^{\prime}_{j}+v_{j})$ mod $q$, and checks the validity of the
values issued to him/her by checking the following equation:
$pub_{j}=s_{j}\cdot P=h(SID_{j}\parallel W_{j})\cdot pub_{RC}+W_{j}$. At last,
$S_{j}$’s personal information contains $\\{SID_{j},pub_{j},s_{j},W_{j}\\}$
The details of user registration phase and server registration phase are shown
in Fig.1.
Figure 1: User and server registration phase of the proposed scheme.
### 5.4 Login phase
If user $U_{i}$ wants to access the services provided by server $S_{j}$,
$U_{i}$ needs to login on to $S_{j}$, the process of the login phase are as
following:
(1) $U_{i}$ inserts his/her smart card into the smart card reader, and inputs
identity $ID_{i}$ and password $pw_{i}$. Then the smart card computes
$QID_{i}=H(ID_{i})$, $CID_{i}=Reg_{ID_{i}}\oplus h(ID_{i}\parallel
pw_{i}\parallel b_{i})\cdot pub_{RC}$, $H^{*}_{i}=h(QID_{i}\parallel
CID_{i})$, and checks whether $H^{*}_{i}=H_{i}$. If they are equal, it means
$U_{i}$ is a legal user. Otherwise the smart card aborts the session.
(2) The smart card generates two random numbers $u_{i}$ and $r_{i}$, and
computes $DID_{i}=u_{i}\cdot QID_{i}$ and $R_{i}=r_{i}\cdot P$. Then the smart
card sends the login request message $\\{DID_{i},R_{i}\\}$ to server $S_{j}$
over a public channel.
### 5.5 Authentication and session key agreement phase
(1) After receiving the login request $\\{DID_{i},R_{i}\\}$ sent from $U_{i}$,
$S_{j}$ chooses a random number $r_{j}$, and computes $R_{j}=r_{j}\cdot P$,
$T_{ji}=r_{j}\cdot R_{i}$, $K_{ji}=s_{j}\cdot R_{i}$ and
$Auth_{ji}=h(DID_{i}\parallel SID_{j}\parallel K_{ji}\parallel R_{j})$. Then
$S_{j}$ sends the message $\\{W_{j},R_{j},Auth_{ji}\\}$ to $U_{i}$.
(2) When receiving $\\{W_{j},R_{j},Auth_{ji}\\}$, $U_{i}$ computes
$T_{ij}=r_{i}\cdot R_{j}$, $pub_{j}=h(SID_{j}\parallel W_{j})\cdot
pub_{RC}+W_{j}$, $K_{ij}=r_{i}\cdot pub_{j}$ and $Auth_{ij}=h(DID_{i}\parallel
SID_{j}\parallel K_{ij}\parallel R_{j})$. Then $U_{i}$ checks $Auth_{ij}$ with
the received $Auth_{ji}$. If they are not equal, $U_{i}$ terminates this
session. Otherwise, $S_{j}$ is authenticated, and $U_{i}$ continues to compute
$M_{i}=r_{i}\cdot DID_{i}$, $N_{i}=u_{i}\cdot CID_{i}$,
$d_{ij}=h(DID_{i}\parallel SID_{j}\parallel K_{ij}\parallel M_{i})$ and
$B_{i}=(r_{i}+d_{ij})\cdot N_{i}$. Finally, $U_{i}$ sends the message
$\\{M_{i},B_{i}\\}$ to $S_{j}$.
(3) After receiving the message $\\{M_{i},B_{i}\\}$ sent from $U_{i}$, $S_{j}$
computes $d_{ji}=h(DID_{i}\parallel SID_{j}\parallel K_{ji}\parallel M_{i})$
and checks whether $e(M_{i}+d_{ji}\cdot DID_{i},pub_{RC})=e(B_{i},P)$. If they
are not equal, $S_{j}$ terminates this session. Otherwise, $U_{i}$ is
authenticated.
Finally, the user $U_{i}$ and the server $S_{j}$ agree on a common session key
as $U_{i}:SK=h(DID_{i}\parallel SID_{j}\parallel K_{ij}\parallel T_{ij})$,
$S_{j}:SK=h(DID_{i}\parallel SID_{j}\parallel K_{ji}\parallel T_{ji})$.
The login phase and authentication and session key agreement phase are
depicted in Fig.2.
Figure 2: Login and verification phase of the proposed scheme.
### 5.6 Password change phase
The following steps show the process of the password change phase of a user
$U_{i}$.
(1) The user $U_{i}$ inserts his/her smart card into the smart card reader,
and inputs identity $ID_{i}$ and password $pw_{i}$. Then the smart card
computes $QID_{i}=H(ID_{i})$, $CID_{i}=Reg_{ID_{i}}\oplus h(ID_{i}\parallel
pw_{i}\parallel b_{i})\cdot pub_{RC}$, $H^{*}_{i}=h(QID_{i}\parallel
CID_{i})$, and checks whether $H^{*}_{i}=H_{i}$. If they are equal, it means
$U_{i}$ is a legal user. Otherwise the smart card aborts the session.
(2) The smart card generates a random number $z_{i}$, and computes
$Z_{i}=z_{i}\cdot P$ and $AID_{i}=CID_{i}\oplus z_{i}\cdot pub_{RC}$. Then the
smart card sends the message $\\{ID_{i},AID_{i},Z_{i}\\}$ to the registration
center $RC$.
(3) After receiving the message $\\{ID_{i},AID_{i},Z_{i}\\}$, $RC$ computes
$CID_{i}=AID_{i}\oplus s_{RC}\cdot Z_{i}$, $QID_{i}=H(ID_{i})$, and checks
whether $e(CID_{i},P)=e(QID_{i},pub_{RC})$. If they are equal, user $U_{i}$ is
authenticated. Then $RC$ computes $V_{1}=h(CID_{i}\parallel s_{RC}\cdot
Z_{i})$ and sends $\\{V_{1}\\}$ to $U_{i}$.
(4) When receiving $\\{V_{1}\\}$, user computes $h(CID_{i}\parallel z_{i}\cdot
pub_{RC})$ and checks it with the received $V_{1}$. If they are equal, the
registration center $RC$ is authenticated. Then $U_{i}$ chooses his/her new
password $pw_{i}^{new}$ and the new random number $b_{i}^{new}$, and computes
$HPW_{i}^{new}=h(ID_{i}\parallel pw_{i}^{new}\parallel b_{i}^{new})\cdot P$,
$V_{2}=HPW_{i}^{new}\oplus z_{i}\cdot pub_{RC}$ and $V_{3}=h(CID_{i}\parallel
z_{i}\cdot pub_{RC}\parallel HPW_{i}^{new})$. Then $U_{i}$ submits
$\\{V_{2},V_{3}\\}$ to $RC$.
(5) Upon receiving the response $\\{V_{2},V_{3}\\}$, the registration server
$RC$ computes $HPW_{i}^{new}=V_{2}\oplus s_{RC}\cdot Z_{i}$ and
$V_{3}^{*}=h(CID_{i}\parallel s_{RC}\cdot Z_{i}\parallel HPW_{i}^{new})$. Then
$RC$ compares $V_{3}^{*}$ with the received $V_{3}$. If they are equal, $RC$
continues to compute $Reg_{ID_{i}}^{new}=CID_{i}\oplus s_{RC}\cdot
HPW_{i}^{new}$, $V_{4}=Reg_{ID_{i}}^{new}\oplus s_{RC}\cdot Z_{i}$ and
$V_{5}=h(s_{RC}\cdot Z_{i}\parallel Reg_{ID_{i}}^{new})$. After that, $RC$
sends $\\{V_{4},V_{5}\\}$ to $U_{i}$.
(6) After receiving $\\{V_{4},V_{5}\\}$, $U_{i}$ computes
$Reg_{ID_{i}}^{new}=V_{4}\oplus z_{i}\cdot pub_{RC}$ and
$V_{5}^{*}=h(z_{i}\cdot pub_{RC}\parallel Reg_{ID_{i}}^{new})$. Then $U_{i}$
checks whether $V_{5}^{*}=V_{5}$. If they are equal, user $U_{i}$ replaces the
original $Reg_{ID_{i}}$ and $b_{i}$ with $Reg_{ID_{i}}^{new}$ and
$b_{i}^{new}$.
The details of a password change phase of the proposed scheme are shown in
Fig.3.
Figure 3: Password change phase of the proposed scheme.
## §6 Security analysis
### 6.1 Stolen smart card and offline dictionary attacks
In the proposed scheme, we assume that if a smart card is stolen, physical
protection methods cannot prevent malicious attackers to get the stored secure
elements. At the same time, adversary $A$ can access to a big dictionary of
words that likely includes user’s password and intercept the communications
between the user and server.
In the proposed scheme, in case a user $U_{i}$’s smart card is stolen by an
adversary $A$, he can extract $\\{Reg_{ID_{i}},H_{i}\\}$ from the memory of
the stolen smart card. At the same time, it is assumed that adversary $A$ has
intercepted a previous full session messages
$\\{DID_{i},R_{i},W_{j},R_{j},Auth_{ji},M_{i},B_{i}\\}$ between the user
$U_{i}$ and server $S_{j}$. However, the adversary still cannot obtain the
$U_{i}$’s identity $ID_{i}$ and password $pw_{i}$ except guessing $ID_{i}$ and
$pw_{i}$ at the same time. Therefore, it is impossible to get the $U_{i}$’s
identity $ID_{i}$ and password $pw_{i}$ from stolen smart card and offline
dictionary attack in our proposed scheme.
### 6.2 Replay attack
Replaying a message of previous session into a new session is useless in our
proposed scheme because user’s smart card and the server choose different rand
numbers $r_{i}$ and $r_{j}$, and the user’identity is different in each new
session, which make all messages dynamic and valid for that session only. If
we assume that an adversary $A$ replies an intercepted previous login request
$\\{DID_{i},R_{i}\\}$ to $S_{j}$, after receiving the response message
$\\{W_{j},R_{j},Auth_{ji}\\}$ sent from $S_{j}$, $A$ cannot compute the
correct response message $\\{M_{i},B_{i}\\}$ to pass the $S_{j}$’s
authentication since he does not know the values of $ID_{i}$, $pw_{i}$,
$u_{i}$ and $r_{i}$. Therefore, the proposed scheme is robust for the replay
attack.
### 6.3 Impersonation attack
If an adversary $A$ wants to masquerade as a legal user $U_{i}$ to pass the
authentication of a server $S_{j}$, he must have the values of both $QID_{i}$
and $CID_{i}$. However, $QID_{i}$ and $CID_{i}$ are protected by $U_{i}$’s
smart card, $ID_{i}$ and $pw_{i}$ since $QID_{i}=H(ID_{i})$ and
$CID_{i}=Reg_{ID_{i}}\oplus h(ID_{i}\parallel pw_{i}\parallel b_{i})\cdot
pub_{RC}$. Therefore, unless the adversary $A$ can obtain the $U_{i}$’s smart
card, $ID_{i}$ and $pw_{i}$ at the same time, the proposed scheme is secure to
the impersonation attack.
### 6.4 Server spoofing attack
If an adversary $A$ wants to masquerade as a legal server $S_{j}$ to cheat a
user $U_{i}$, he must calculate a valid $Auth_{ji}$ which is embedded with the
shared secret key $K_{ji}=s_{j}\cdot R_{i}$ to pass the authentication of
$U_{i}$. However, adversary $A$ cannot derive the shared secret key $K_{ji}$
without knowing the private key $s_{j}$ of the server $S_{j}$. Therefore, our
scheme is secure against the server spoofing attack.
### 6.5 Insider attack
In the proposed scheme, the registration center $RC$ cannot obtain the
$U_{i}$’s password $pw_{i}$. Since in the registration phase, $U_{i}$ chooses
a random number $b_{i}$ and sends $ID_{i}$ and $HPW_{i}=h(ID_{i}\parallel
pw_{i}\parallel b_{i})\cdot P$ to $RC$, $RC$ can not derive $pw_{i}$ from
$HPW_{i}$ based on CDL problem. Therefore, the proposed scheme is robust for
insider attack.
### 6.6 Denial of service attack
In denial of service attack, an adversary $A$ updates identity and password
verification information on smart card to some arbitrary value and hence
legitimate user cannot login successfully in subsequent login request to the
server. In the proposed scheme, smart card checks the validity of user
$U_{i}$’s identity $ID_{i}$ and password $pw_{i}$ before password update
procedure. An adversary can insert the stolen smart card of the user $U_{i}$
into smart card reader and has to guess the identity $ID_{i}$ and password
$pw_{i}$ correctly corresponding to the user $U_{i}$. Since the smart card
computes $H^{*}_{i}=h(QID_{i}\parallel CID_{i})$, and compares it with the
stored value of $H_{i}$ in its memory to verify the legitimacy of the user
$U_{i}$ before smart card accepts password update request. It is not possible
to guess identity $ID_{i}$ and password $pw_{i}$ correctly at the same time in
real polynomial time even after getting the smart card of the user $U_{i}$.
Therefore, the proposed scheme is secure against the denial of service attack.
### 6.7 Perfect forward secrecy
Perfect forward secrecy means that even if an adversary compromises all the
passwords of the users, it still cannot compromise the session key. In the
proposed scheme, the session key $SK=h(DID_{i}\parallel SID_{j}\parallel
K_{ij}\parallel T_{ij})$ ($SK=h(DID_{i}\parallel SID_{j}\parallel
K_{ji}\parallel T_{ji})$) is generated by three one-time random numbers
$u_{i}$, $r_{i}$ and $r_{j}$ in each session. These one-time random numbers
are only held by the user $U_{i}$ and the server $S_{j}$, and cannot be
retrieved from $SK$ based on the security of CDH problem. Thus, even if an
adversary obtains previous session keys, it cannot compromise other session
key. Hence, the proposed scheme achieves perfect forward secrecy.
### 6.8 User’s anonymity
In our proposed scheme, the user $U_{i}$’s login message is different in each
login phase. Among each login message, $DID_{i}=u_{i}\cdot H(ID_{i})$ is
associated with a random number $u_{i}$ which is known by $U_{i}$ only.
Therefore, any adversary cannot identity the real identity of the logon user
and our scheme can provide the user’s anonymity.
### 6.9 No verification table
In our proposed scheme, it is obvious that the user, the server and the
registration center do not maintain any verification table.
### 6.10 Local password verification
In the proposed scheme, smart card checks the validity of user $U_{i}$’s
identity $ID_{i}$ and password $pw_{i}$ before logging into server $S_{j}$.
Since the adversary cannot compute the correct $CID_{i}$ without the knowledge
of $ID_{i}$ and $pw_{i}$ to pass the verification equation $H^{*}_{i}=H_{i}$,
thus our scheme can avoid the unauthorized accessing by the local password
verification.
### 6.11 Proper mutual authentication
In our scheme, the user first authenticates the server. $U_{i}$ sends the
message $\\{DID_{i},R_{i}\\}$ to the server $S_{j}$ to build an connection.
After receiving the response message $\\{W_{j},R_{j},Auth_{ji}\\}$ sent from
$S_{j}$, $U_{i}$ computes $T_{ij}$, $pub_{j}$, $K_{ij}$, $Auth_{ij}$, and
checks whether $Auth_{ij}=Auth_{ji}$. If they are equal, $S_{j}$ is
authenticated by $U_{i}$. Otherwise, $U_{i}$ stops to login onto this server.
Since $Auth_{ji}=h(DID_{i}\parallel SID_{j}\parallel K_{ji}\parallel R_{j})$
and $K_{ji}=s_{j}\cdot R_{i}$, an adversary $A$ cannot compute the correct
$K_{ji}$ without the knowledge of value of $s_{j}$. Any fabricated message
$\\{W^{\prime}_{j},R^{\prime}_{j},Auth^{\prime}_{ji}\\}$ cannot pass the
verification. Then $U_{i}$ computes $M_{i}$, $N_{i}$, $d_{ij}$, $B_{i}$, and
sends the message $\\{M_{i},B_{i}\\}$ to $S_{j}$. After receiving the message
$\\{M_{i},B_{i}\\}$ sent from $U_{i}$, $S_{j}$ computes $d_{ji}$ and checks
whether $e(M_{i}+d_{ji}\cdot DID_{i},pub_{RC})=e(B_{i},P)$. If they are not
equal, $S_{j}$ terminates this session. Otherwise, $U_{i}$ is authenticated.
Since $B_{i}=(r_{i}+d_{ij})\cdot N_{i}$, an adversary $A$ cannot compute the
correct $B_{i}$ without the knowledge of values of $u_{i}$ and $r_{i}$ etc.
Any fabricated message $\\{M^{\prime}_{i},B^{\prime}_{i}\\}$ cannot pass the
verification. Therefore, our proposed scheme can provide proper mutual
authentication.
## §7 Performance comparison and functionality analysis
In this section, we compares the performance and functionality of our proposed
scheme with some previously schemes. To analyze the computation cost, some
notations are defined as follows.
$TG_{e}$: The time of executing a bilinear map operation, $e:G_{1}\times
G_{1}\longrightarrow G_{2}$.
$TG_{mul}$: The time of executing point scalar multiplication on the group
$G_{1}$.
$TG_{H}$: The time of executing a map-to-point hash function H(.).
$TG_{add}$: The time of executing point addition on the group $G_{1}$.
$T_{h}$: The time of executing a one-way hash function $h(.)$.
Since the XOR operation and the modular multiplication operation require very
few computations, it is usually negligible considering their computation cost.
Table 2 shows the performance comparisons of our proposed scheme and some
other related protocols. We mainly focus on three computation costs including:
C1, the total time of all operations executed in the user registration phase;
C2, the total time spent by the user during the process of login phase and
verification phase; C3, the total time spent by the server during the process
of verification phase. As shown in Table 2, Tseng et al.’s scheme are more
efficient in terms of computation cost. However, Tseng et al.’s scheme is
vulnerable to stolen smart card and offline dictionary attacks, server
spoofing attack and insider attack, and cannot provide perfect forward
secrecy, user’s anonymity, proper mutual authentication and session key
agreement. In our proposed scheme, the total computation cost of the user (C2)
is 9$TG_{mul}$+$TG_{H}$+$TG_{add}$+5$T_{h}$. But similar to that in Liao et
al.’s scheme, the user $U_{i}$ can pre-compute $R_{i}=r_{i}\cdot P$ in the
client, and then the computation cost of the user (C2) requires
8$TG_{mul}$+$TG_{H}$+$TG_{add}$+5$T_{h}$ on-line computation. It can be found
that our proposed scheme spends a little more computation cost than Liao et
al.’s scheme in C2, and the others are almost equal. However, Liao et al.’s
scheme is vulnerable to stolen smart card and offline dictionary attacks and
denial of service attack, and cannot provide user’s anonymity and local
password verification.
Table 2: Computational cost comparison of our scheme and other schemes. | Proposed scheme | Liao et al.’scheme [27] | Tseng et al.’scheme [26] |
---|---|---|---|---
C1 | 3$TG_{mul}$+$TG_{H}$+2$T_{h}$ | 3$TG_{mul}$+$TG_{H}$+$T_{h}$ | 2$TG_{mul}$+$TG_{H}$+$T_{h}$ |
C2 | 8$TG_{mul}$+$TG_{H}$+$TG_{add}$+5$T_{h}$ | 5$TG_{mul}$+$TG_{H}$+$TG_{add}$+5$T_{h}$ | 3$TG_{mul}$+2$T_{h}$ |
C3 | 2$TG_{e}$+4$TG_{mul}$+$TG_{add}$+2$T_{h}$ | 2$TG_{e}$+5$TG_{mul}$+$TG_{add}$+2$T_{h}$ | 2$TG_{e}$+$TG_{mul}$+$TG_{H}$+$TG_{add}$+$T_{h}$ |
Table 3 lists the functionality comparisons among our proposed scheme and
other related schemes. It is obviously that our scheme has many excellent
features and is more secure than other related schemes.
Table 3: Functionality comparisons among related multi-server authentication protocols. | Proposed | Liao | Tseng | Li | Lee | Shao | Lee
---|---|---|---|---|---|---|---
| scheme | et al. | et al. | et al. | et al. | et al. | et al.
| | [27] | [26] | [20] | [18] | [17] | [19]
Resist stolen smart card and | Yes | No | No | No | No | No | No
offline dictionary attacks | | | | | | |
Resist replay attack | Yes | Yes | Yes | No | No | No | No
Resist impersonation attack | Yes | Yes | Yes | No | No | No | No
Resist server spoofing attack | Yes | Yes | No | No | No | No | No
Resist insider attack | Yes | Yes | No | Yes | Yes | No | Yes
Resist denial of service attack | Yes | No | Yes | Yes | Yes | Yes | No
Perfect forward secrecy | Yes | Yes | No | Yes | Yes | No | No
User’s anonymity | Yes | No | No | Yes | Yes | No | Yes
No verification table | Yes | Yes | Yes | Yes | Yes | Yes | Yes
Local password verification | Yes | No | Yes | Yes | Yes | Yes | No
Proper mutual authentication | Yes | Yes | No | Yes | No | Yes | Yes
## §8 Conclusion
In this paper, we point out that Li et al.’s scheme is vulnerable to stolen
smart card and offline dictionary attack, replay attack, impersonation attack
and server spoofing attack. Furthermore, by analyzing some other similar
schemes, we find the certain type of dynamic ID based and non-RC dependented
multi-server authentication scheme in which only hash functions are used is
hard to provide perfect efficient and secure authentication. To compensate for
these shortcomings, we improve the Liao et al.’s multi-server authentication
scheme which is based on pairing and self-certified public keys, and propose a
novel dynamic ID based and non-RC dependented remote user authentication
scheme for multi-server environments. The security and performance analyses
show the proposed scheme is secure against various attacks and has many
excellent features.
## §9 Acknowledgment
This paper was supported by the National Natural Science Foundation of China
(Grant Nos. 61070209, 61202362, 61121061), and the Asia Foresight Program
under NSFC Grant (Grant No. 61161140320).
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|
arxiv-papers
| 2013-05-28T02:18:02 |
2024-09-04T02:49:45.793384
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dawei Zhao, Haipeng Peng, Shudong Li, Yixian Yang",
"submitter": "Dawei Zhao",
"url": "https://arxiv.org/abs/1305.6350"
}
|
1305.6506
|
11institutetext: MapR Technologies EMEA, Galway, Ireland
11email: [email protected]
# Notes on Physical & Logical Data Layouts
Michael Hausenblas 11
###### Abstract
In this short note I review and discuss fundamental options for physical and
logical data layouts as well as the impact of the choices on data processing.
I should say in advance that these notes offer no new insights, that is,
everything stated here has already been published elsewhere. In fact, it has
been published in so many different places, such as blog posts, in the
literature, etc. that the main contribution is to bring it all together in one
place.
## 1 Motivation
Data processing and management systems such as databases, datastores [Cat11]
or query engines usually have to answer to two kinds of entities: _humans_ and
_hardware_.
Towards humans, they provide means to query, manipulate or manage111The
management aspect can span a wide range of activities including but not
limited to snapshots, mirroring, etc. data. Towards the hardware, they issue
store and retrieve commands. They depend directly or indirectly on the very
nature of the hardware. Almost all systems— for example, Hadoop’s distributed
file system—are designed with strong though not necessarily explicit
assumptions about the underlying hardware such as hard disk drives (HDD)
[Ele09], their spindles, heads, etc.
Conceptually, there are three levels present in data processing and management
systems (Fig. 1):
Figure 1: The three levels of data representation and interaction in data
management systems, including examples for each of the levels.
* •
The _User Interface_ level. Any database or datastore needs to provide a way
to interact with the data under management. This can be something elaborate,
standardised and mature as the Structured Query Language (SQL) found in
relational database management systems (RDBMS), such as Oracle DB, PostgreSQL,
or MySQL. This can be a RESTful interface, found in many NoSQL datastores,
like, for example, CouchDB’s API222See documentation at
http://wiki.apache.org/couchdb/HTTP_Document_API. Of course, this can also be
a programming-language-level API such as the case with
Hadoop333http://hadoop.apache.org/docs/current/api/org/apache/hadoop/mapreduce/package-
summary.html.
* •
The _Logical Data Layout_ level. Addresses how the user conceptually thinks
about and deals with the data. In case of a RDBMS the data units might be
tables and records, in a key-value store like Redis it may be an entry
identified via a key and in a wide-column store the data unit might be a row
containing different columns, and last but not least in an RDF store a single
triple might be the unit one logically manipulates.
* •
The _Physical Data Layout_ level. On this level, we’re concerned with the
question how the data is laid out once serialised. The serialisation takes
place from main memory (RAM) either to send the data in question over the
wire, or, to be stored on a durable medium such as a hard disk drive or a
solid-state drive (SSD) [Cor12]. Concrete serialisations may be textual based,
such as CSV and JSON or of binary nature, like the RCFile format [HLH+11].
## 2 Manifestations of Data Layouts
In [HVTC12] we introduced the three fundamental data shapes _tabular_ , _tree_
—henceforth _nested_ , and _graph_. It turns out that it is useful to further
differentiate the shapes, distinguishing between logical and physical layouts,
as hinted above.
In the following, I propose a non-exhaustive, lightweight taxonomy for logical
and physical data layouts and serialisation formats as depicted in Fig. 2.
The main point of this taxonomy is to decouple the logical from the physical
level. While for the human user the logical level is of importance, from a
software and systems perspective the physical level dominates. There are
cases, however, where the abstraction is leaking and the user is forced to
accommodate.
Figure 2: A non-exhaustive, lightweight taxonomy for logical and physical data
layouts and serialisation formats commonly used in the data processing
community.
Take, for example, best practices concerning NoSQL data modeling444As found in
the blog post “NoSQL Data Modeling Techniques” via
http://highlyscalable.wordpress.com/2012/03/01/nosql-data-modeling-
techniques/: with a wide-column store, such as HBase, one can easily get into
a situation where one must take into account the physical location of the data
in order to avoid performance
penalties555http://stackoverflow.com/questions/10806955/hbase-schema-key-for-
real-time-analytics-solution.
Also, the choice of the serialisation format (for example, textual vs. binary) can have severe implications, both in terms of performance and maintenance. Look at a case where one decides to use JSON as the wire format in contrast to, say, Avro. In the former case, one can debug any document simply by issuing a command on the shell like cat datafile.json | more while with Avro more specialised tooling is necessary. On the other hand, one can probably expect a better I/O performance and disk utilisation with a binary format such as Avro, compared to JSON.
Now we’re already entering the discussion of the impact of choices we make
concerning how the data is laid out. Let’s jump right into it.
## 3 Impact on Data Processing at Scale
There are two schools of thought concerning the organisation of data units:
data _normalisation_ , and data _denormalisation_. The former wants to
minimise redundancy, the latter aims to minimise assembly. Both have their own
built-in assumptions, characteristics and use cases:
### Normalised data …
* •
As data items are not redundant, data consistency is relatively easy to
achieve compared to denormalised data.
* •
When updating data in place one only has to deal with it once and not in
multiple locations.
* •
Storage is efficiently used, that is, it takes up less disk space.
### Denormalised data …
* •
The access to data units is fast as no joins are necessary; the data can be
considered to be pre-joined.
* •
As it provides an entity-centric view, it is in general more straight-forward
to employ automated sharding of the data.
* •
Due to keeping multiple copies of data items or fragments thereof around, it
requires typically a multitude more space on disk than normalised data.
In Table 1 I’m providing a comparison and summary of the two different ways to
handle data including typical examples of workloads and technologies
concerning use cases.
Table 1: A comparison of normalised vs. denormalised handling of data on the logical and physical level across SQL and NoSQL data management systems. | NORMALISED | DENORMALISED
---|---|---
_characteristics_ | Each data item is stored exactly in one place. | The data items are repeated as needed.
_advantages_ | Built-in data consistency and storage-efficiency. | Fast, entity-centric data access without the need for joins.
_disadvantages_ | Joints are costly and hard to implement (especially distributed). | Inflexible and storage-hungry.
_workloads_ | OLTP, write-intensive | OLAP, read-intensive
_examples_ | Classical, textbook relational database modelling | Wide-column datastores (HBase, Cassandra), document-oriented datastores (MongoDB, CouchDB), key-value datastores (Redis, Memcached, etc.), graph databases (Neo4j, RDF stores), large-scale relational databases
Allow me a side remark relative to the ongoing and tiring debate SQL vs.
NoSQL: it turns out that the focus on SQL as the representative of the evil is
really a rather backward view. As stated in many places all over the Web666For
example, see the blog post http://gigaom.com/2013/02/21/sql-is-whats-next-for-
hadoop-heres-whos-doing-it/ many open source projects and commercial entities
are introducing SQL bindings or interfaces on top of Hadoop and NoSQL
datastores.
This is quite understandable, given the huge number of deployed (business
intelligence) tools that natively speak SQL and of course the many people out
there trained in this language.
Joining the dots. We are now in a position to wrap up on the impact of choices
we make concerning how the data is laid out: one dimension of freedom is the
choice how we organise the data: normalised vs. denormalised. The second
choice we have is concerning the physical data representation. Interestingly,
some systems are more rigid and upfront with what they support, expect or
allow. While, for example, in the Hadoop ecosystem it is entirely up to you
how you serialise your data—and depending on your requirements and the
workload you might end up with a different result—traditional RDBMS are much
more restrictive. Seldom you get to choose the physical data layout and the
logical layout is hard-coded anyways.
Coming back full circle to the initial Fig. 1 one should, however, not
underestimate the _User Interface_ level. At the end of the day the usability,
integrability and user familiarity of this level can be the reason why some
data management systems may have a better chance to survive than others. Last
but not least, one should take into account the emerging _Polyglot
Persistence_ 777http://martinfowler.com/bliki/PolyglotPersistence.html meme
that essentially states that one size does not fit it all concerning data
storage and manipulation. I suggest embracing this meme together with Pat
Helland’s advice [Hel11]: “In today’s humongous database systems, clarity may
be relaxed, but business needs can still be met.”
## 4 Acknowledgements
I’d like to thank Eric Brewer, whose RICON2012 keynote motivated me to write
up this short note. His keynote is available via https://vimeo.com/52446728
and more than certainly worth watching it in its entirety.
## References
* [Cat11] Rick Cattell. Scalable SQL and NoSQL data stores. SIGMOD Rec., 39:12–27, 2011.
* [Cor12] Michael Cornwell. Anatomy of a solid-state drive. Commun. ACM, 55(12):59–63, 2012.
* [Ele09] Jon Elerath. Hard Disk Drives: The Good, the Bad and the Ugly! Commun. ACM, 52(6):38–45, 2009.
* [Hel11] Pat Helland. If You Have Too Much Data, then “Good Enough” Is Good Enough. ACM Queue, 9:40:40–40:50, 2011.
* [HLH+11] Yongqiang He, Rubao Lee, Yin Huai, Zheng Shao, Namit Jain, Xiaodong Zhang, and Zhiwei Xu. RCFile: A fast and space-efficient data placement structure in MapReduce-based warehouse systems. In Serge Abiteboul, Klemens Böhm, Christoph Koch, and Kian-Lee Tan, editors, ICDE, pages 1199–1208. IEEE Computer Society, 2011.
* [HVTC12] Michael Hausenblas, Boris Villazon-Terrazas, and Richard Cyganiak. Data Shapes and Data Transformations. arXiv, 1211.1565, 2012.
|
arxiv-papers
| 2013-05-28T14:16:11 |
2024-09-04T02:49:45.804931
|
{
"license": "Public Domain",
"authors": "Michael Hausenblas",
"submitter": "Michael Hausenblas",
"url": "https://arxiv.org/abs/1305.6506"
}
|
1305.6511
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-088 LHCb-PAPER-2013-024
Observation of
$\boldmath{\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi}$
decay and study of
$\boldmath{\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1,2}\mathrm{K}^{*0}}$
decays
The LHCb collaboration†††Authors are listed on the following pages.
The first observation of the decay
${\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi}$ and a
study of ${\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1,2}\mathrm{K}^{*0}}$
decays are presented. The analysis is performed using a dataset, corresponding
to an integrated luminosity of 1.0 $\mbox{\,fb}^{-1}$, collected by the LHCb
experiment in $\mathrm{p}\mathrm{p}$ collisions at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$. The following ratios of branching fractions
are measured:
$\begin{array}[]{lll}\dfrac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi)}{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi)}&=&(18.9~{}\pm 1.8\,\mathrm{(stat)}\pm 1.3\,\mathrm{(syst)}\pm
0.8\,({\cal B}))\times 10^{-2},\\\ \vskip 3.0pt\cr\dfrac{{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0})}&=&(19.8~{}\pm 1.1\,\mathrm{(stat)}\pm
1.2\,\mathrm{(syst)}\pm 0.9\,({\cal B}))\times 10^{-2},\\\ \vskip
3.0pt\cr\dfrac{{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0})}{{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}&=&(17.1~{}\pm
5.0\,\mathrm{(stat)}\pm 1.7\,\mathrm{(syst)}\pm 1.1\,({\cal B}))\times
10^{-2},\\\ \vskip 3.0pt\cr\end{array}$
where the third uncertainty is due to the limited knowledge of the branching
fractions of
${\upchi_{\mathrm{c}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma}$ modes.
Submitted to Nucl. Phys. B
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
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Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
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Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
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Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V.
Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P.
Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A. Hicheur1, E.
Hicks51, D. Hill54, M. Hoballah5, M. Holtrop40, C. Hombach53, P. Hopchev4, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
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March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
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Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, M. Palutan18, J. Panman37,
A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i,
A. Phan58, E. Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S.
Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, I.
Polyakov30, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C.
Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C. Prouve7, V.
Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H. Rademacker45, B.
Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41,
S. Redford54, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K.
Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35, G.
Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d,
V. Salustino Guimaraes2, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i,
R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, I. Sepp52, N. Serra39,
J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P. Shatalov30, Y.
Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30,
A. Shires52, R. Silva Coutinho47, M. Sirendi46, T. Skwarnicki58, N.A. Smith51,
E. Smith54,48, J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D.
Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V.
Vagnoni14, G. Valenti14, A. Vallier7, M. Van Dijk45, R. Vazquez Gomez18, P.
Vazquez Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M.
Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X.
Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A.
Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R.
Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D.
Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D.
Wiedner11, L. Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55,
F.F. Wilson48, J. Wimberley57, J. Wishahi9, M. Witek25, S.A. Wotton46, S.
Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, R. Young49,
X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A.
Zvyagin37.
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 Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
Two-body $\mathrm{B}$-meson decays into a final states containing charmonium
meson have played a crucial role in the observation of $C\\!P$ violation in
the $\mathrm{B}$-meson system. These decay modes also provide a sensitive
laboratory for studying the effects of the strong interaction. Such decays are
expected to proceed predominantly via the colour-suppressed tree diagram
involving
$\overline{}\mathrm{b}\rightarrow\overline{}\mathrm{c}\mathrm{c}\overline{}\mathrm{s}$
transition shown in Fig. 1. Under the factorization hypothesis the branching
ratios of the
${\mathrm{B}^{0}_{(\mathrm{s})}\rightarrow\upchi_{\mathrm{c}0,2}\mathrm{X}}$
decays, where $\mathrm{X}$ denotes a $\mathrm{K}^{*0}$ or a $\upphi$ meson,
are expected to be small in comparison to
$\mathrm{B}^{0}_{(\mathrm{s})}\rightarrow\upchi_{\mathrm{c}1}\mathrm{X}$
decays [1]. However, non-factorizable contributions may be large [1]; the
branching fraction for the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}0}\mathrm{K}^{*0}$ decay was
measured by the BaBar collaboration to be $(1.7\pm 0.3\pm 0.2)\times 10^{-4}$
[2] while the branching fraction for the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ decay was
measured by the BaBar and Belle collaborations to be $(2.5\pm 0.2\pm
0.2)\times 10^{-4}$ [3] and $(1.73^{+0.15+0.34}_{-0.12-0.22})\times 10^{-4}$
[4], respectively. The branching fraction for the decay
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}$ has been
measured by the BaBar collaboration to be $(6.6\pm 1.8\pm 0.5)\times 10^{-5}$
[3] and, unlike the branching fraction for the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}0}\mathrm{K}^{*0}$ decay, can
still be explained in the factorization approach [5]. Therefore, future
measurements of the branching fractions of both
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}$ decays can
provide valuable information for the understanding of the production of
$\upchi_{\mathrm{c}}$ states in $\mathrm{B}$ meson decays, where
$\upchi_{\mathrm{c}}$ denotes $\upchi_{\mathrm{c}1}$ and
$\upchi_{\mathrm{c}2}$ states. The decay modes
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}}\upphi$ have not
been observed previously.
$\mathrm{W}^{+}$$\mathrm{B}^{0}_{(\mathrm{s})}\left\\{\begin{tabular}[]{r}$\overline{}\mathrm{b}$\\\
\\\ \\\ \vspace{0.5em}\hfil\\\
$\mathrm{d}${($\mathrm{s}$)}\end{tabular}\right.$$\left.\begin{tabular}[]{l}$\overline{}\mathrm{c}${}{}{}\\\
$\mathrm{c}$\end{tabular}\right\\}\upchi_{\mathrm{c}}$$\left.\begin{tabular}[]{l}$\overline{}\mathrm{s}$\\\
$\mathrm{d}${($\mathrm{s}$)}\end{tabular}\right\\}\mathrm{K}^{*0}(\upphi)$
Figure 1: Leading-order tree level diagram for the
$\mathrm{B}^{0}_{(\mathrm{s})}\rightarrow\upchi_{\mathrm{c}}\mathrm{X}$
decays.
In this paper, the first observation of the decay
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ and a study
of the $\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1,2}\mathrm{K}^{*0}$ decays
are presented. The analysis is based on a data sample, corresponding to an
integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, collected with the LHCb
detector in $\mathrm{p}\mathrm{p}$ collisions at a centre-of-mass energy of
7$\mathrm{\,Te\kern-1.00006ptV}$.
## 2 LHCb detector
The LHCb detector [6] is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta<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 placed downstream. The combined tracking system has momentum
resolution $\Delta p/p$ 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 ($p_{\rm T}$).
Charged hadrons are identified using two ring-imaging Cherenkov detectors [7].
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
[8].
The trigger [9] 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. Candidate events are first required to pass a
hardware trigger which selects muons with $\mbox{$p_{\rm
T}$}>1.48{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the subsequent software
trigger, at least one of the muons is required to have both $\mbox{$p_{\rm
T}$}>0.8{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter larger than
$100\,\upmu\rm m$ with respect to all of the primary $\mathrm{p}\mathrm{p}$
interaction vertices (PVs) in the event. Finally, the two final state muons
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 6.4
[10] with a specific LHCb configuration [11]. Decays of hadronic particles are
described by EvtGen [12] in which final state radiation is generated using
Photos [13]. The interaction of the generated particles with the detector and
its response are implemented using the Geant4 toolkit [14, 15] as described in
Ref. [16].
## 3 Event selection
The decays $\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}}\upphi$ (the
inclusion of charged conjugate processes is implied throughout) are
reconstructed using the
$\upchi_{\mathrm{c}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma$ decay mode. The decays
${\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi$ are used as normalization channels. The intermediate resonances
are reconstructed in the ${{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\rightarrow\upmu^{+}\upmu^{-}}$,
${\mathrm{K}^{*0}\rightarrow\mathrm{K}^{+}\uppi^{-}}$ and
${\upphi\rightarrow\mathrm{K}^{+}\mathrm{K}^{-}}$ final states.
As in Refs. [17, 18, 19], pairs of oppositely-charged tracks identified as
muons, each having $\mbox{$p_{\rm
T}$}>0.55{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and 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. Track quality is ensured by
requiring the $\chi^{2}$ per number of degrees of freedom ($\chi^{2}$/ndf)
provided by the track fit to be less than 5. Well identified muons are
selected by requiring that the difference in logarithms of the likelihood of
the muon hypothesis with respect to the hadron hypothesis is larger than zero
[8]. The fit of the common two-prong vertex is required to satisfy
$\chi^{2}/{\rm ndf}<20$. The vertex is required to be well separated from the
reconstructed primary vertex of any of the $\mathrm{p}\mathrm{p}$ interactions
by requiring the decay length to be at least three times its uncertainty.
Finally, the invariant mass of the dimuon combination is required to be
between 3.020 and 3.135 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
To create $\upchi_{\mathrm{c}}$ candidates, the selected
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ candidates are
combined with a photon that has been reconstructed using clusters in the
electromagnetic calorimeter that have transverse energy greater than
0.7$\mathrm{\,Ge\kern-1.00006ptV}$. To suppress the large combinatorial
background from $\uppi^{0}\rightarrow\upgamma\upgamma$ decays, photons that
can form part of a $\uppi^{0}\rightarrow\upgamma\upgamma$ candidate with
invariant mass within $10{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
known $\uppi^{0}$ mass [20] are not used for reconstruction of
$\upchi_{\mathrm{c}}$ candidates. To be considered as a $\upchi_{\mathrm{c}}$,
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upgamma$
combination needs to have a transverse momentum larger than
3${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and an invariant mass in the range 3.4
– 3.7${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
The selected $\upchi_{\mathrm{c}}$ and
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ candidates are then
combined with $\mathrm{K}^{+}\uppi^{-}$ or $\mathrm{K}^{+}\mathrm{K}^{-}$
pairs to create $\mathrm{B}^{0}_{(\mathrm{s})}$ meson candidates. To identify
kaons (pions), the difference in logarithm of the likelihood of the kaon and
pion hypotheses [7] is required to be greater than (less than) zero. The track
$\chi^{2}/\rm{ndf}$ provided by the track fit is required to be less than 5.
The kaons and pions are required to have transverse momentum larger than
0.8${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and to have an impact parameter
$\chi^{2}$, defined as the difference between the $\chi^{2}$ of the
reconstructed $\mathrm{p}\mathrm{p}$ collision vertex formed with and without
the considered track, larger than 4. The invariant mass of the kaon and pion
system, $M_{\mathrm{K}^{+}\uppi^{-}}$, is required to be
${0.675<M_{\mathrm{K}^{+}\uppi^{-}}<1.215{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$
and the invariant mass of the kaon pair, $M_{\mathrm{K}^{+}\mathrm{K}^{-}}$,
is required to be
${0.999<M_{\mathrm{K}^{+}\mathrm{K}^{-}}<1.051{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}}$.
In the reconstruction of $\mathrm{K}^{*0}$ candidates, a possible background
arises from $\upphi\rightarrow\mathrm{K}^{+}\mathrm{K}^{-}$ decays when a kaon
is misidentified as a pion. To suppress this contribution, the invariant mass
of the kaon and pion system, calculated under the kaon mass hypothesis for the
pion track, is required to be outside the range from 1.01 to
1.03${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
In addition, the decay time of $\mathrm{B}$ candidates is required to be
larger than 150$\,\upmu\rm m$/$c$ to reduce the large combinatorial background
from particles produced in the primary $\mathrm{p}\mathrm{p}$ interaction. To
improve the invariant mass resolution of the $\mathrm{B}^{0}_{(s)}$ meson
candidate a kinematic fit [21] is performed. In this fit, constraints are
applied to the masses of the intermediate
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\upchi_{\mathrm{c}}$ resonances [20] and it is also required that the
$\mathrm{B}^{0}_{(\mathrm{s})}$ meson candidate momentum vector points to the
primary vertex. The $\chi^{2}$/ndf for this fit is required to be less than 5.
## 4 $\mathrm{B}^{0}$ $\rightarrow$$\upchi_{\mathrm{c}}$$\mathrm{K}^{*0}$
and $\mathrm{B}^{0}_{\mathrm{s}}$
$\rightarrow$$\upchi_{\mathrm{c}1}$$\upphi$ decays
LHCbLHCbLHCbLHCb$\begin{array}[]{l}(\mathrm{a})\\\
\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}\\\
\rm{\upchi_{\mathrm{c}1}~{}constraint}\end{array}$$\begin{array}[]{l}(\mathrm{b})\\\
\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi\\\
\rm{\upchi_{\mathrm{c}1}~{}constraint}\end{array}$$\begin{array}[]{l}(\mathrm{c})\\\
\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}\\\
\rm{\upchi_{\mathrm{c}2}~{}constraint}\end{array}$$\begin{array}[]{l}(\mathrm{d})\\\
\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi\\\
\rm{\upchi_{\mathrm{c}2}~{}constraint}\end{array}$$\mathrm{M_{\upchi_{\mathrm{c}}\mathrm{K}^{*0}}}$$\mathrm{M_{\upchi_{\mathrm{c}}\mathrm{K}^{*0}}}$$\mathrm{M_{\upchi_{\mathrm{c}1}\upphi}}$$\mathrm{M_{\upchi_{\mathrm{c}1}\upphi}}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$
Candidates / (5${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)Candidates /
(5${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)Candidates /
(5${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)Candidates /
(5${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)
Figure 2: Invariant mass distributions for: (a)
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}$ and (b)
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ candidates
with $\upchi_{\mathrm{c}1}$ mass constraint; (c)
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}$ and (d)
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ candidates
with $\upchi_{\mathrm{c}2}$ mass constraint. The total fitted function (thick
solid blue), signal for the $\upchi_{\mathrm{c}1}$ and $\upchi_{\mathrm{c}2}$
modes (thin green solid and dotted, respectively) and the combinatorial
background (dashed blue) are shown.
The invariant mass distributions after selecting
${\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}}$ and
${\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi}$
candidates, separately with a $\upchi_{\mathrm{c}1}$ and
$\upchi_{\mathrm{c}2}$ mass constraints, are shown in Fig. 2. The signal is
modelled by a single Gaussian function and the combinatorial background is
modelled by an exponential function. In the $\mathrm{B}^{0}$ channel (Figs.
2(a) and (c)), the right peak in the mass distributions corresponds to the
$\upchi_{\mathrm{c}1}$ mode and the left one to the $\upchi_{\mathrm{c}2}$
mode. Owing to the small
$\upchi_{\mathrm{c}0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma$ branching fraction [20] the contribution from the
$\upchi_{\mathrm{c}0}$ mode is negligible. As the $\mathrm{B}^{0}$ candidate
mass is calculated with the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma$ invariant mass constrained to the $\upchi_{\mathrm{c}1}$
($\upchi_{\mathrm{c}2}$) known mass, the signal peak corresponding to the
$\upchi_{\mathrm{c}2}$ ($\upchi_{\mathrm{c}1}$) mode is shifted to a lower
(higher) value with respect to the $\mathrm{B}^{0}$ mass. The same effect is
observed in simulation. The ratio of the mass resolutions of these two signal
peaks is fixed to the value obtained from simulation. In the
$\mathrm{B}^{0}_{\mathrm{s}}$ channel no significant contribution from the
$\upchi_{\mathrm{c}2}$ decay mode is expected and therefore it is not
considered in the fit. The statistical significance for the observed signal is
determined as $S=\sqrt{-2\ln\frac{\mathcal{L}_{\rm B}}{\mathcal{L}_{\rm
S+B}}}$, where $\mathcal{L}_{\rm S+B}$ and $\mathcal{L}_{\rm B}$ denote the
likelihood of the signal plus background hypothesis and the background only
hypothesis, respectively. The statistical significance of the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ signal is
found to be larger than 9 standard deviations.
The positions and resolutions of the signal peaks are consistent with the
expectations from simulation. To investigate the different signal yields
obtained with the $\upchi_{\mathrm{c}1}$ and $\upchi_{\mathrm{c}2}$ mass
constraints, a simplified simulation study was performed, which accounts for
correlations, differences in selection efficiencies and background
fluctuations. This study demonstrates that the yields are in agreement within
the statistical uncertainty.
To examine the resonance structure of the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ decays, the
sPlot technique [22] was used with weights determined from the
$\mathrm{B}^{0}_{(\mathrm{s})}$ candidate invariant mass fits described above.
The invariant mass distributions for each signal component are obtained. For
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upgamma$
invariant mass distributions the requirement on the invariant mass of the
$\mathrm{K}^{+}\uppi^{-}$($\mathrm{K}^{+}\mathrm{K}^{-}$) system is tightened
to be within $50(10){\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the known
$\mathrm{K}^{*0}$($\upphi$) mass to reduce background.
The resulting invariant mass distributions for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upgamma$,
$\mathrm{K}^{+}\uppi^{-}$ and $\mathrm{K}^{+}\mathrm{K}^{-}$ from
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ candidates
are shown in Fig. 3. The ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma$ invariant mass distributions are modelled with the sum of a
constant and a Crystal Ball function [23] with tail parameters fixed to
simulation. In the $\upchi_{\mathrm{c}2}$ mode the signal peak position is
fixed to the sum of the $\upchi_{\mathrm{c}1}$ peak position and the known
difference between $\upchi_{\mathrm{c}1}$ and $\upchi_{\mathrm{c}2}$ masses
[20]. The $\upchi_{\mathrm{c}2}$ mass resolution is fixed to the
$\upchi_{\mathrm{c}1}$ mass resolution multiplied by a scale factor determined
using simulation. The $\mathrm{K}^{+}\uppi^{-}$ and
$\mathrm{K}^{+}\mathrm{K}^{-}$ invariant mass distributions are modelled with
the sum of a relativistic P-wave Breit-Wigner function with the natural width
fixed to the known value [20] and a non-resonant component modelled with the
LASS parametrization [24]. For the $\mathrm{K}^{+}\mathrm{K}^{-}$ case the
relativistic P-wave Breit-Wigner function is convolved with a Gaussian
function for the detector resolution.
LHCbLHCbLHCbLHCbLHCbLHCb$\begin{array}[]{l}(\mathrm{e})\\\
\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi\end{array}$$\begin{array}[]{l}(\mathrm{f})\\\
\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi\end{array}$$\begin{array}[]{l}(\mathrm{c})\\\
\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}\end{array}$$\begin{array}[]{l}(\mathrm{d})\\\
\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}\end{array}$$\begin{array}[]{l}(\mathrm{a})\\\
\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}\end{array}$$\begin{array}[]{l}(\mathrm{b})\\\
\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}\end{array}$$\mathrm{M_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma}}$$\mathrm{M_{\mathrm{K}^{+}\mathrm{K}^{-}}}$$\mathrm{M_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma}}$$\mathrm{M_{\mathrm{K}^{+}\uppi^{-}}}$$\mathrm{M_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma}}$$\mathrm{M_{\mathrm{K}^{+}\uppi^{-}}}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$
Candidates / (15${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)Candidates /
(4${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)
Candidates / (30${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)Candidates /
(30${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)
Candidates / (15${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)Candidates /
(15${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)
Figure 3: Background-subtracted invariant mass distributions for: (a)
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upgamma$ and (b)
$\mathrm{K}^{+}\uppi^{-}$ final states from
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ decays obtained
with the $\upchi_{\mathrm{c}1}$ mass constraint applied to the
$\mathrm{B}^{0}_{(\mathrm{s})}$ candidate invariant mass; (c)
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upgamma$ and (d)
$\mathrm{K}^{+}\uppi^{-}$ final states from
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}$ decays obtained
with the $\upchi_{\mathrm{c}2}$ mass constraint applied to the
$\mathrm{B}^{0}_{(\mathrm{s})}$ candidate invariant mass; (e)
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upgamma$ and (f)
$\mathrm{K}^{+}\mathrm{K}^{-}$ final states from
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ decays
obtained with the $\upchi_{\mathrm{c}1}$ mass constraint applied to the
$\mathrm{B}^{0}_{(\mathrm{s})}$ candidate invariant mass. The total fitted
function (solid) and the non-resonant contribution (dotted) are shown.
The signal peak positions are consistent with the known masses of the mesons
while the invariant mass resolutions are consistent with the expectation from
simulation. In the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma$ invariant mass distributions, the non-resonant contribution is
consistent with zero. The resonant contributions for the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ decays are
determined with the $\upchi_{\mathrm{c}1}$ mass constraint while the resonant
contribution for the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}$ decay is
determined with the $\upchi_{\mathrm{c}2}$ mass constraint. The resulting
resonant yields, obtained from the fits to the background-subtracted
$\mathrm{K}^{+}\uppi^{-}$ and $\mathrm{K}^{+}\mathrm{K}^{-}$ distributions,
are shown in Table 1.
Table 1: Signal yields for the $\mathrm{B}$ decays. Decay | Yield
---|---
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ | $\;\,\\!\;\,566\pm 31$
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}$ | $\;\,\\!\;\,\;\;66\pm 19$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ | $\;\,\\!\;\,146\pm 14$
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\mathrm{K}^{*0}$ | $56,\\!707\pm 279$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upphi$ | $15,\\!027\pm 139$
## 5 $\mathrm{B}^{0}$ $\rightarrow$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
$\mathrm{K}^{*0}$ and $\mathrm{B}^{0}_{\mathrm{s}}$ $\rightarrow$
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ $\upphi$ decays
The $\mathrm{B}^{0}\rightarrow\upchi_{c}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ branching
fractions are measured with respect to the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi$ decays to reduce the systematic uncertainties. The invariant
mass distributions for the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi$ candidates after selection requirements are shown in Fig. 4. The
signal and the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ invariant mass distributions are modelled by a double-
sided Crystal Ball function and the combinatorial background is modelled by an
exponential function. The parameters of the $\mathrm{B}^{0}_{\mathrm{s}}$ peak
are fixed to be the same as those of the $\mathrm{B}^{0}$ peak except the
position and yield. The difference between the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ peak positions is fixed to the world average [20]. The
positions of the signal peaks are consistent with the known masses of the
$\mathrm{B}^{0}_{(\mathrm{s})}$ mesons [20] and the mass resolutions are
consistent with expectations from simulation.
LHCbLHCb$\begin{array}[]{l}(\mathrm{a})\\\
\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}\end{array}$$\begin{array}[]{l}(\mathrm{b})\\\
\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi\end{array}$$\mathrm{M_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}}}$$\mathrm{M_{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi}}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$
Candidates / (5${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)Candidates /
(5${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)
Figure 4: Invariant mass distributions for (a)
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ and (b)
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi$. The total fitted function (thick solid blue), signal (thin
solid green), the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ (green dotted) and the combinatorial background (dashed
blue) are shown.
The resonant contributions in the
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi$ decays are determined using the sPlot technique with the same
method as that used for the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}}\upphi$ decays. The
resulting $\mathrm{K}^{+}\uppi^{-}$ and $\mathrm{K}^{+}\mathrm{K}^{-}$
invariant mass distributions from
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi$ candidates are shown in Fig. 5. The resulting resonant yields
are summarized in Table 1. The S-wave contributions are consistent with those
considered in other analyses [17, 25, 26].
LHCbLHCb$\begin{array}[]{l}(\mathrm{a})\\\
\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}\end{array}$$\begin{array}[]{l}(\mathrm{b})\\\
\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi\end{array}$$\mathrm{M_{\mathrm{K}^{+}\uppi^{-}}}$$\mathrm{M_{\mathrm{K}^{+}\mathrm{K}^{-}}}$$\mathrm{[GeV/c^{2}]}$$\mathrm{[GeV/c^{2}]}$
Candidates / (1${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)Candidates /
(15${\mathrm{\,Me\kern-0.70004ptV\\!/}c^{2}}$)
Figure 5: Background-subtracted invariant mass distributions for (a)
$\mathrm{K}^{+}\uppi^{-}$ combinations from
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ decays and (b) $\mathrm{K}^{+}\mathrm{K}^{-}$
combination from
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi$ decays. The total fitted function (solid) and the non-resonant
contribution (dotted) are shown.
## 6 Efficiencies and systematic uncertainties
The branching fraction ratios are calculated using the formulas
$\displaystyle{\begin{array}[]{lll}\frac{\displaystyle{\cal
B}(\mathrm{B}\rightarrow\upchi_{\mathrm{c}1}\mathrm{X})}{\displaystyle{\cal
B}(\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X})}&=&\frac{\displaystyle
N_{\mathrm{B}\rightarrow\upchi_{\mathrm{c}1}\mathrm{X}}}{\displaystyle
N_{\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X}}}\times\frac{\displaystyle\upvarepsilon_{\mathrm{B}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{X}}}{\displaystyle\upvarepsilon_{\mathrm{B}\rightarrow\upchi_{\mathrm{c}1}\mathrm{X}}}\times\frac{\displaystyle
1}{\displaystyle{\cal
B}(\upchi_{\mathrm{c}1}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}~{},\\\\[13.00005pt] \frac{\displaystyle{\cal
B}(\mathrm{B}\rightarrow\upchi_{\mathrm{c}2}\mathrm{X})}{\displaystyle{\cal
B}(\mathrm{B}\rightarrow\upchi_{\mathrm{c}1}\mathrm{X})}&=&\frac{\displaystyle
N_{\mathrm{B}\rightarrow\upchi_{\mathrm{c}2}\mathrm{X}}}{\displaystyle
N_{\mathrm{B}\rightarrow\upchi_{\mathrm{c}1}\mathrm{X}}}\times\frac{\displaystyle\upvarepsilon_{\mathrm{B}\rightarrow\upchi_{\mathrm{c}1}\mathrm{X}}}{\displaystyle\upvarepsilon_{\mathrm{B}\rightarrow\upchi_{\mathrm{c}2}\mathrm{X}}}\times\frac{\displaystyle{\cal
B}(\upchi_{\mathrm{c}1}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}{\displaystyle{\cal
B}(\upchi_{\mathrm{c}2}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}~{},\par\end{array}}$ (1)
where $N$ represents the measured yield and $\upvarepsilon$ represents the
total efficiency. The total efficiency is the product of the geometrical
acceptance, the detection, reconstruction, selection and trigger efficiencies.
The efficiencies are derived using simulation and are presented in Table 2.
Table 2: Total efficiencies for all decay modes. Uncertainties are statistical only and reflect the size of the simulation sample. Decay | Efficiency $[10^{-4}]$
---|---
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ | $\;\,7.89\pm 0.12$
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}$ | $\;\,9.45\pm 0.13$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ | $12.7\;\,\pm 0.2\;\,$
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\mathrm{K}^{*0}$ | $53.9\;\,\pm 0.3\;\,$
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upphi$ | $85.1\;\,\pm 0.4\;\,$
Most potential 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. The remaining systematic uncertainties are summarized in Table
3 and each is now discussed in turn.
Systematic uncertainties related to the signal determination procedure are
estimated using a number of alternative options. For each of the alternatives
the ratio of event yields is calculated and the systematic uncertainty is then
determined as the maximum deviation of this ratio from the ratio obtained with
the baseline model. For the $\mathrm{B}^{0}_{(\mathrm{s})}$ meson decays a fit
with a second-order polynomial for the combinatorial background description, a
fit with a Crystal Ball [23] function for the signal peaks and fit over
different ranges of invariant mass are used. In the
$\mathrm{B}^{0}_{\mathrm{s}}$ channel a fit including the
$\upchi_{\mathrm{c}2}$ decay mode is also performed. For the
$\mathrm{K}^{+}\uppi^{-}$ and $\mathrm{K}^{+}\mathrm{K}^{-}$ combinations the
fits are repeated, modelling the background with an S-wave two-body phase-
space function or an S-wave two-body phase-space function multiplied by a
linear function. The $\mathrm{K}^{+}\uppi^{-}$ and
$\mathrm{K}^{+}\mathrm{K}^{-}$ invariant mass ranges and the bin size are also
varied. The resulting uncertainties are 3% on ${{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})/{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0})}$, $5\%$ on ${{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi)/{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi)}$, and $9\%$ on ${{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0})/{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}$.
Another important source of systematic uncertainty arises from the potential
disagreement between data and simulation in the estimation of efficiencies. To
study this source of uncertainty, the selection criteria are varied in ranges
corresponding to as much as $30\%$ change in the signal yields and the ratios
of the selection and reconstruction efficiencies are compared between data and
simulation. The largest difference ($3\%$) is assigned as a systematic
uncertainty in each mode.
A further source of possible disagreement between data and simulation is the
photon reconstruction efficiency. As in Ref. [18], the photon reconstruction
efficiency has been studied using
$\mathrm{B}^{+}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*+}$, followed by
$\mathrm{K}^{*+}\rightarrow\mathrm{K}^{+}\uppi^{0}$ and
$\uppi^{0}\rightarrow\upgamma\upgamma$ decays. For photons with transverse
momentum greater than 0.7${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ the agreement
between data and simulation is at the level of 4%, which is assigned as a
systematic uncertainty to the ratios ${{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})/{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0})}$ and ${{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi)/{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi)}$. As the transverse momentum spectra of photons are similar in
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}$ decays, this
systematic uncertainty cancels in the ratio ${{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0})/{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}$.
The systematic uncertainty related to the trigger efficiency has been 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(2\mathrm{S})\mathrm{K}^{+}$ which have
similar kinematics and the same trigger requirements as the channels under
study in this analysis [17]. An agreement within 1% is found, which is
assigned as systematic uncertainty.
The uncertainty due to the finite simulation sample size is included in the
statistical uncertainty of the result by adding it in quadrature to the
statistical uncertainty on the ratio of yields.
Table 3: Relative systematic uncertainties (in $\%$) on the ratio of branching fractions. Source | $\frac{{\cal B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi)}{{\cal B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\upphi)}$ | $\frac{{\cal B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}{{\cal B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\mathrm{K}^{*0})}$ | $\frac{{\cal B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0})}{{\cal B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}$
---|---|---|---
Signal determination | $5$ | $3$ | $9$
Efficiencies from simulation | $3$ | $3$ | $3$
Photon reconstruction | $4$ | $4$ | $-$
Trigger | $1$ | $1$ | $1$
Sum in quadrature | $7$ | $6$ | $10$
## 7 Results and summary
The first observation of the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi$ decay has
been made with a data sample, corresponding to an integrated luminosity of
1.0$\mbox{\,fb}^{-1}$ of $\mathrm{p}\mathrm{p}$ collisions at a centre-of-mass
energy of 7$\mathrm{\,Te\kern-1.00006ptV}$, collected with the LHCb detector.
Its branching fraction, normalized to that of the
$\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi$ decay and using the known value ${\cal
B}(\upchi_{\mathrm{c}1}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)=(34.4\pm 1.5)\%$ [20], is measured to be
$\begin{array}[]{llll}\dfrac{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow\upchi_{\mathrm{c}1}\upphi)}{{\cal
B}(\mathrm{B}^{0}_{\mathrm{s}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upphi)}&=&(6.51~{}\pm 0.64\,\mathrm{(stat)}\pm
0.46\,\mathrm{(syst)})\times 10^{-2}\times\dfrac{1}{{\cal
B}(\upchi_{\mathrm{c}1}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}&=\\\ \vskip 3.0pt\cr&=&(18.9~{}\pm 1.8\,\mathrm{(stat)}\pm
1.3\,\mathrm{(syst)}\pm 0.8\,({\cal B}))\times 10^{-2},\par\end{array}$
where the third uncertainty corresponds to the uncertainty on the branching
fraction of the
$\upchi_{\mathrm{c}1}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma$ decay. Using the same dataset, the ratio of the branching
fractions of the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0}$ modes and the ratio of the branching fractions of the
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0}$ and
$\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0}$ modes have been
measured. The ratios are determined using Eq. 1 and the known value
$\frac{{\cal
B}(\upchi_{\mathrm{c}1}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}{{\cal
B}(\upchi_{\mathrm{c}2}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}=\frac{(34.4\pm 1.5)\%}{(19.5\pm 0.8)\%}=1.76\pm 0.11$ [20]
and are
$\begin{array}[]{llll}\dfrac{{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}{{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0})}&=&(6.82~{}\pm 0.39\,\mathrm{(stat)}\pm
0.41\,\mathrm{(syst)})\times 10^{-2}\times\dfrac{1}{{\cal
B}(\upchi_{\mathrm{c}1}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}&=\\\ \vskip 3.0pt\cr&=&(19.8~{}\pm 1.1\,\mathrm{(stat)}\pm
1.2\,\mathrm{(syst)}\pm 0.9\,({\cal B}))\times 10^{-2},\\\ \vskip
5.0pt\cr\dfrac{{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0})}{{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}&=&(9.74~{}\pm
2.86\,\mathrm{(stat)}\pm 0.97\,\mathrm{(syst)})\times
10^{-2}\times\dfrac{{\cal
B}(\upchi_{\mathrm{c}1}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}{{\cal
B}(\upchi_{\mathrm{c}2}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma)}&=\\\ \vskip 3.0pt\cr&=&(17.1~{}\pm 5.0\,\mathrm{(stat)}\pm
1.7\,\mathrm{(syst)}\pm 1.1\,({\cal B}))\times 10^{-2},\par\end{array}$
where the third uncertainty is due to the uncertainty on the branching
fractions of the
${\upchi_{\mathrm{c}}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\upgamma}$ modes.
The ratio ${\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})/{\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0})$ obtained in this paper is compatible with, but more
precise than, the previous best value of $(17.2^{+3.6}_{-3.0}){\times
10^{-2}}$ determined from the world average value ${\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})=(2.22^{+0.40}_{-0.31})\times
10^{-4}$ [20] and the branching fraction ${\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0})=(1.29\pm 0.05\pm 0.13)\times 10^{-3}$ measured by the
Belle collaboration [27]. Other measurements of ${\cal
B}(\mathrm{B}^{0}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\mathrm{K}^{*0})$ are not considered as they do not take into account
the $\mathrm{K}^{+}\uppi^{-}$ S-wave component. The ratio ${{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}2}\mathrm{K}^{*0})/{\cal
B}(\mathrm{B}^{0}\rightarrow\upchi_{\mathrm{c}1}\mathrm{K}^{*0})}$ obtained in
this paper is compatible with the value derived from BaBar measurements,
$(26\pm 7\mathrm{(stat)})\times 10^{-2}$ [3], taking only the statistical
uncertainties into account.
## 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); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from 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 thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-05-28T14:26:05 |
2024-09-04T02:49:45.811866
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, 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, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R.\n Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M.\n Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M.\n Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M.\n Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra\n Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, P. Griffith, 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, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, M. Holtrop, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten,\n R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M.\n Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, 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, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O.\n Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C.\n Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki,\n J. Marks, G. Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli,\n D. Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, E. Maurice, A. Mazurov, B. Mc Skelly, J. McCarthy, A. McNab, R.\n McNulty, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P.\n Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo\n Casasus, F. Polci, G. Polok, A. Poluektov, I. Polyakov, 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, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, E. Rodrigues, P.\n Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet, T.\n Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, C. Salzmann, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, M.\n Sirendi, T. Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D.\n Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek,\n V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M.\n Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin, U.\n Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez, P.\n Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri,\n G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, M. Witek, S.A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Ivan Belyaev",
"url": "https://arxiv.org/abs/1305.6511"
}
|
1305.6677
|
# Technicolor with Scalar Doublet After the Discovery of Higgs Boson
Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, P.R. China
Abstract
The SM-like Higgs boson with mass of 125 GeV discovered at the LHC is subject
to a natural interpretation of electroweak symmetry breaking. As a successful
theory in offering this naturalness, technicolor with a scalar doublet and two
both $SU(3)_{c}$ and $SU(N_{TC})$ colored scalars, which is considered as a
low-energy effective theory, is proposed after the discovery of SM-like Higgs
boson. At present status, the model can be consistent with both the direct and
indirect experimental limits. In particular, the consistency with precision
electroweak measurements is realized by the colored scalars, which give rise
to a large negative contribution to $S$ parameter. It is also promising to
detect techni-pions and these colored scalars at the LHC.
## 1 Introduction
Since the discovery of a standard model (SM)-like Higgs boson with mass around
125 GeV [1] reported by the ATLAS and CMS collaboration, extensive efforts
have been devoted to explore its implication to electroweak symmetry breaking
(EWSB) in the context of new physics. It is now believed that a part of most
favored parameter space of natural supersymmetry (SUSY) fails to achieve this,
due to the absence of SUSY signals at the large hadron collider (LHC) with
$\sqrt{s}=8$ TeV.
In parallel to SUSY as candidate of new physics which provides natural EWSB,
technicolor (TC) was also considered as an interesting scenario decades ago.
TC model differs from SUSY in many ways. In particular, it provides EWSB
through condensation of techni-fermions. Therefore, unlike in SUSY models
where the electroweak mass parameters are tightly related to the SUSY-breaking
scale due to EWSB, naturalness doesn’t concern us in TC-model. However, it
also suffers from its own problems, such as too large $S$ parameter in
precision electroweak measurements and explanation of mass hierarchy of SM
flavors (for a review, see [2]).
In this paper, we consider a variety of TC model based on previous works in
[3, 4]. These authors proposed scalar doublet(s) to original TC, which is
known as TC with scalar. This variety of TC-model should be considered as a
low-energy effective theory. Otherwise, the hierarchy problem appears as in
SM. It can be either embedded into walking TC or supersymmetric TC. In the
former case, the scalar is composite [5]. In other words, this variety of TC
model imitates the low-energy behavior of a set of extended TC. In the later
case, the scalar can be either fundamental or composite. The superpartners
receive their masses from supersymmetry breaking, and below SUSY-breaking
scale we obtain a TC with massive scalar doublet and SM111For recent works on
other variants of TC model, see, e.g, [6].. In contrast to the earliest TC,
the SM fermions obtain their mass similarly to SM in this type of TC model, in
which the vacuum expectation value (vev) of Higgs is-induced by the
condensation of techni-fermions-through the Yukawa couplings of Higgs scalar
to techni-fermions.
In particular, we study TC-model with one scalar doublet $\phi$ and two both
$SU(3)_{c}$ and $SU(N_{TC})$ colored scalars, in which there are two mass
scales $f$ and $f^{\prime}$, respectively, referring to decay constant of
techni-pions and vev of $\phi$. They are supposed to satisfy $f^{2}+f^{\prime
2}=\upsilon^{2}=(246{\it GeV})^{2}$ from consideration of EWSB. The coupling
of neutral scalar of $\phi$ to SM fermions are the same as those of SM except
an additional factor $f^{\prime}/\upsilon$. Therefore this factor determines
the deviation of our model from SM. Using the LHC data about Higgs immediately
leads to,
$\displaystyle{}0<\theta\lesssim 0.2,~{}~{}~{}~{}~{}~{}\theta\equiv
f/f^{\prime}.$ (1.1)
The couplings of techni-pions to SM fermions are similar to those of charged
Higgs boson in type $\mathbf{I}$ Higgs doublet model, except a common $\theta$
factor also.
The physical states below the scale $4\pi f$ are $\sigma$ scalar identified as
Higgs boson, techni-pions and colored techni-scalars. Distinctive features in
this model include $(1)$ the small $\theta\sim 0.2$ in (1.1) is sufficient to
provide a Higgs scalar of 125 GeV and techni-pion of $210-280$ GeV
simultaneously; $(2)$ the suppression from $\theta$ factor is sufficient for
the techni-pions to evade both the direct and indirect experimental limits;
$(3)$ the colored techni-scalars receive their masses of order $\upsilon$ via
their $\phi^{4}$ couplings to scalar doublet $\phi$; $(4)$ finally the colored
techni-scalars provide a large negative contribution to $S$, which eliminates
the large positive contribution arising from condensation, and therefore
reconciles the model with precision electroweak measurements.
The paper is organized as follows. In section 2, we present our model in
details. Then we discuss direct experimental limits on Higgs scalar, techni-
pions and colored techni-scalars in subsection 3.1. We address indirect
experimental limits on techni-pions in subsection 3.2 and colored techni-
scalars in subsection 3.3, respectively. We discuss masses of colored techni-
scalars and corrections to parameters of precision electroweak measurements
due to these scalars. We finally conclude in section 4.
## 2 The Model
The content of TC model which we will explore in this paper is as follows,
$\displaystyle{}Y_{L}=\left(\begin{array}[]{c}p\\\ m\\\
\end{array}\right)_{L}$ $\displaystyle:$
$\displaystyle\left(\mathbf{N}_{TC},~{}\mathbf{1},~{}\mathbf{2}\right)_{0}$
(2.3) $\displaystyle p_{R}$ $\displaystyle:$
$\displaystyle\left(\mathbf{N}_{TC},~{}\mathbf{1},~{}\mathbf{1}\right)_{1/2}$
$\displaystyle m_{R}$ $\displaystyle:$
$\displaystyle\left(\mathbf{N}_{TC},~{}\mathbf{1},~{}\mathbf{1}\right)_{-1/2}$
$\displaystyle\omega_{t}$ $\displaystyle:$
$\displaystyle\left(\mathbf{N}_{TC},~{}\bar{\mathbf{3}},~{}\mathbf{1}\right)_{-1/6}$
(2.4) $\displaystyle\omega_{b}$ $\displaystyle:$
$\displaystyle\left(\mathbf{N}_{TC},~{}\bar{\mathbf{3}},~{}\mathbf{1}\right)_{+1/6}$
with addition of a fundamental scalar
$\phi:\left(\mathbf{1},~{}\mathbf{1},~{}\mathbf{2}\right)_{1/2}$. The
assignments of representations are under the notation of
$SU(N_{TC})\times~{}SU(3)_{c}\times~{}SU(2)_{L}\times~{}U(1)_{Y}$. In
comparison with the simplest model of TC with scalar, two additional colored
scalars $\omega_{t,b}$ are added. The choices of hypercharge follow from the
requirement of anomaly free.
The Lagrangian for the model reads,
$\displaystyle{}\mathcal{L}=\mathcal{L}_{SM}+\mathcal{L}_{TC}+\mathcal{L}_{\phi}+\mathcal{L}_{\omega}-V(\phi,\omega)$
(2.5)
The hidden TC and SM matters communicate either via $\phi$ or $\omega$
scalars. For the former case, doublet $\phi$ couples to techni-fermions and SM
flavors in $\mathcal{L}_{\phi}=\mathcal{L}(\phi,T)+\mathcal{L}(\phi,f_{SM})$
respectively, which read
$\displaystyle{}\mathcal{L}(\phi,T)=h_{+}\bar{Y}_{L}\tilde{\phi}p_{R}+h_{-}\bar{Y}_{L}\phi~{}m_{R}+H.C$
(2.6)
and
$\displaystyle{}\mathcal{L}(\phi,f_{SM})=h_{l}\bar{L}\tilde{\phi}l_{R}+h_{U}\bar{Q}_{L}\tilde{\phi}~{}U_{R}+h_{D}\bar{Q}_{L}\phi~{}D_{R}+H.C$
(2.7)
Here, $Q_{L}$ and $L$ being the quark and lepton doublets of SM, respectively,
$U_{R}$, $D_{R}$ being right-hand top and bottom quark respectively, and
$l_{R}$ the right hand lepton. $h$s in (2.7) are the ordinary SM Yukawa
couplings.
There also exists strong communication between the quarks of third family and
techni-fermions through $\omega$ scalars 222In dynamical models of EWSB,
similarly to the magnitudes of Yukawa couplings of Higgs boson to SM fermions
, we assume that the largest effect is in the Yukawa couplings of top-bottom
doublet. ,
$\displaystyle{}\mathcal{L}_{\omega}=\lambda_{\omega_{t}}\bar{t}_{R}p_{R}\omega_{t}^{{\dagger}}+\lambda_{\omega_{b}}\bar{b}_{R}m_{R}\omega_{b}^{{\dagger}}+H.C$
(2.8)
An advantage of adding $\omega$ scalars is that the four-fermions operators
involving top and bottom quark induced by Yukawa interaction in (2.8)
contribute to significantly negative $S$, which cancels the large positive
tree-level contribution due to condensation of techni-fermions. There is also
a disadvantage of introducing colored scalars. Because carrying colors implies
that $\omega$ scalars can be directly produced at hadron colliders, making the
model more constrained. This will be discussed in the next section.
The potential $V(\phi,\omega)$ in (2.5) can be determined from the symmetries
in (2.3). Below the scale of $\Lambda_{TC}=4\pi~{}f$ we will work in effective
field analysis, it is more convenient to express $\phi$ doublet and its
conjugate in the form of unitary matrix $\Phi$ [4], which can be defined as,
$\displaystyle{}\Phi=\frac{\sigma+f^{\prime}}{\sqrt{2}}\Sigma^{\prime},~{}~{}~{}~{}~{}~{}\Sigma^{\prime}=\exp(2i\Pi^{\prime}/f^{\prime})$
(2.9)
Using $\Phi$, we can write the self-couplings as,
$\displaystyle{}V(\phi,\omega)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}}{8}[Tr(\Phi^{{\dagger}}\Phi)]^{2}+\lambda_{5}[\omega_{t}^{{\dagger}}\omega_{t}]^{2}+\lambda_{6}[\omega_{b}^{{\dagger}}\omega_{b}]^{2}$
$\displaystyle+$
$\displaystyle\lambda_{2}Tr(\Phi^{{\dagger}}\Phi)\omega_{t}^{{\dagger}}\omega_{t}+\lambda_{3}Tr(\Phi^{{\dagger}}\Phi)\omega_{b}^{{\dagger}}\omega_{b}+\lambda_{4}\omega_{t}^{{\dagger}}\omega_{t}\omega_{b}^{{\dagger}}\omega_{b}$
## 3 Experimental Limits
As well known we can use the effective chiral Lagrangian to describe the TC
model below the TC scale $\Lambda_{TC}$. In this approach, the pseudoscalars
that result from the chiral symmetry breaking are the isotriplet of technipion
$\Sigma$. Guided by non-linear realization of $\pi$ mesons in QCD, $\Sigma$
can also be similarly treated as,
$\displaystyle{}\Sigma=\exp(2i\Pi/f)$ (3.1)
which transforms as $\Sigma\rightarrow~{}L\Sigma~{}R^{{\dagger}}$ under the
chiral symmetries. It is then straightforward to write the kinetic terms of
our model,
$\displaystyle{}\mathcal{L}=\frac{f^{2}}{4}Tr\left(D_{\mu}\Sigma^{{\dagger}}D^{\mu}\Sigma\right)+\frac{1}{2}Tr\left(D_{\mu}\Phi^{{\dagger}}D^{\mu}\Phi\right)$
(3.2)
with the derivative $D_{\mu}\Sigma=\partial^{\mu}\Sigma-
igW^{\mu}_{a}\frac{\tau^{a}}{2}\Sigma+ig^{\prime}B^{\mu}\Sigma\frac{\tau^{3}}{2}$.
From (3.2) one observes that the linear combination $\pi_{a}\sim
f\Pi+f^{\prime}\Pi^{\prime}$ become the longitudinal components of the EW
gauge bosons, leaving its orthogonal combination
$\pi_{p}=(-f^{\prime}\Pi+f\Pi^{\prime})/\sqrt{f^{2}+f^{\prime 2}}$ as the
physical states of low energy region. One finds that $f^{2}+f^{\prime
2}=\upsilon^{2}$.
### 3.1 Direct Searches
Now we understand $\sigma$ and $\pi_{p}$ are the freedoms below
$\Lambda_{TC}$. The mass of $\sigma$ can be directly determined from (2) as in
[4],
$\displaystyle{}m^{2}_{\sigma}=\frac{3}{2}\tilde{\lambda}f^{\prime 2},$ (3.3)
where
$\displaystyle{}\tilde{\lambda}=\lambda_{1}+\frac{11}{24}\left[3h^{4}_{t}+N_{TF}(h^{4}_{+}+h^{4}_{-})\right].$
(3.4)
As for mass of $\pi_{p}$, it follows from the effective potential 333Here
$H=\left(\begin{array}[]{cc}h_{+}&0\\\ 0&h_{-}\\\ \end{array}\right)$. As
manifested in (2.6), it combines with $\Phi$ to transform as $\Phi
H\rightarrow L\Phi HR^{{\dagger}}$.,
$\displaystyle{}V_{eff}(\sigma)=c_{1}4\pi
f^{3}Tr\left(\Phi~{}H\Sigma^{{\dagger}}\right)+H.C$ (3.5)
with the coefficient $c_{1}\sim\mathcal{O}(1)$. Substituting $\pi_{p}$ into
(3.5) gives rise to444In what follows, we set $c_{1}=1$ for discussion.
$\displaystyle{}m_{\pi_{p}}=2c_{1}\sqrt{2}\frac{4\pi
f}{f^{\prime}}h\upsilon^{2}=8\sqrt{2}\pi
h\theta\upsilon^{2},~{}~{}~{}h=(h_{+}+h_{-})/2.$ (3.6)
Direct search on $\sigma$
The couplings of $\sigma$ to SM fermions and EW gauge bosons are suppressed by
a factor $f^{\prime}/\upsilon$.Identifying $\sigma$ as the Higgs boson
discovered at the LHC implies that the ratio $\mu_{\gamma}$ of signal strength
of $h\rightarrow\gamma\gamma$ over its SM prediction, and ratio $\mu_{V}$ of
signal strength of Higgs decaying into four-leptons via $WW^{*}$ and $ZZ^{*}$
both equal to
$\displaystyle{}\mu_{\gamma}=\mu_{VV}=(f^{\prime}/\upsilon)^{2},$ (3.7)
Global fit to the LHC data [7] suggests that (1.1) and
$\displaystyle{}\tilde{\lambda}=0.15-0.18,~{}~{}~{}~{}h=1.75\tilde{\lambda}.$
(3.8)
The requirement $4\pi f>\upsilon$ from consistency further constrains $\theta$
being in the range $(0.08,0.2)$. As for the decays of Higgs boson to $bb$ and
$\tau\tau$, the uncertainty is still large at present status.
Direct search on $\pi_{p}$
The Yukawa couplings of charged technipion to SM fermions can be extracted
from (2.7) [4] ,
$\displaystyle{}i\left(\frac{f}{\upsilon}\right)[\bar{D}_{L}V^{{\dagger}}\pi^{-}_{p}h_{U}U_{R}+\bar{U}_{L}\pi^{+}_{p}Vh_{D}D_{R}+H.c~{}]$
(3.9)
where $V$ denotes the CKM matrix of SM. Eq.(3.9) implies that couplings of
$\pi_{p}$ to SM fermions are similar to those of charged Higgs boson in type
$\mathbf{I}$ Higgs doublet model, except a suppression by $f/\upsilon\simeq
f/f^{\prime}=\theta$. From (3.6) and (3.8) $m_{\pi_{p}}$ is determined to be
in the range $(210.3,334.3)$ (GeV) . Searches on this range of mass for
charged Higgs boson are mainly from the channel $H^{+}\rightarrow t\bar{b}$.
We find that the ratio of signal strength for $\pi_{p}\rightarrow t\bar{b}$
over SM background can be expressed in terms of that for $H^{+}$,
$\displaystyle{}\mu^{(\pi_{p})}(\pi_{p}^{+}\rightarrow
t\bar{b})=\theta^{2}\mu^{(H^{+})}(H^{+}\rightarrow t\bar{b})$ (3.10)
Charged Higgs boson mass below 78.6 GeV has been excluded by direct searches
at LEP [8] (for searches at the LHC, see, e.g., [10] ). This bound on
$m_{\pi_{p}}$ however can be significantly relaxed due to the $\theta^{2}$
suppression on event rate.
Direct search on $\omega_{t,b}$
As we will discuss in the next section, the fit to precision electroweak
measurements typically suggests that,
$\displaystyle{}\lambda_{\omega_{t}}$ $\displaystyle\simeq$ $\displaystyle
0.3-1.0,~{}~{}~{}~{}m_{\omega_{t}}\simeq 580-1500{\it~{}GeV},$
$\displaystyle\lambda_{\omega_{b}}$ $\displaystyle\simeq$ $\displaystyle
1.3-3.0,~{}~{}~{}~{}m_{\omega_{b}}\simeq 100-250{\it~{}GeV}$ (3.11)
Note that this spectra are constrained to be no excess at 1$\sigma$ level.
Allowing no excess at 3$\sigma$ level further decreases the values of
$\lambda_{\omega_{t,b}}$ $\lambda_{\omega_{b}}$ required, which helps evading
the direct experimental limits. The spectra of (3.1) can easily evade the
direct detection at the $e^{+}e^{-}$ collider. The dominant channel for
searching $\omega_{t,b}$ scalars is through
$e^{+}e^{-}\rightarrow\omega_{t/b}\omega^{*}_{t/b}\rightarrow
t\bar{t}/b\bar{b}$. The ratio of cross section of
$\sigma(e^{+}e^{-}\rightarrow\omega_{t/b}\omega^{*}_{t/b})$ over its SM
background $\sigma_{SM}(e^{+}e^{-}\rightarrow t\bar{t}/b\bar{b})$ is very
small for each of them. The reason is due to severe suppression by
$\beta=\sqrt{1-4m^{2}_{\omega}/s}$ even if light $\omega$ scalars near 100 GeV
can be produced. At a hadron collider such as LHC $\omega_{t/b}$ scalar is
mainly produced from gluon fusion (GF), and its decay is dominated by
$\omega_{t/b}\rightarrow t/b+p_{R}/m_{R}$. The SM background for this is
$gg\rightarrow m$-jets ( with either 2t-jets for $\omega_{t}$ or 2b-jets for
$\omega_{b}$ included) plus missing energy. Their mass bounds can be estimated
in terms of their analogies in supersymmetric models, i.e, stop and sbottom,
$\displaystyle{}\mu^{(GF)}_{\omega_{b}}$ $\displaystyle=$
$\displaystyle\frac{\sigma(gg\rightarrow\omega_{b}\omega^{*}_{b})Br(\omega_{b}\omega^{*}_{b}\rightarrow
b\bar{b}+E_{T})}{\sigma(gg\rightarrow\tilde{b}_{1}\tilde{b}^{*}_{1})Br(\tilde{b}_{1}\tilde{b}^{*}_{1}\rightarrow
b\bar{b}+E_{T})}\mu^{(GF)}_{\tilde{t}_{1}}(gg\rightarrow\tilde{t}_{1}\tilde{t}^{*}_{1}\rightarrow{\it
2b-jets+other~{}jets+E_{T}}),$ $\displaystyle\mu^{(GF)}_{\omega_{t}}$
$\displaystyle=$
$\displaystyle\frac{\sigma(gg\rightarrow\omega_{t}\omega^{*}_{t})Br(\tilde{t}_{1}\tilde{t}^{*}_{1}\rightarrow
t\bar{t}+E_{T})}{\sigma(gg\rightarrow\tilde{t}_{1}\tilde{t}^{*}_{1})Br(\tilde{t}_{1}\tilde{t}^{*}_{1}\rightarrow
t\bar{t}+E_{T})}\mu^{(GF)}_{\tilde{t}_{1}}(gg\rightarrow\tilde{t}_{1}\tilde{t}^{*}_{1}\rightarrow
2t-jets+{\it other~{}jets+E_{T}})$
where $\mu^{(GF)}$s refer to the ratio of signal strength over the SM
prediction via production of gluon fusion. The small ratio between couplings
$\lambda_{\omega_{t}}/h^{SM}_{t}\simeq 0.5$ indicates that mass bound on
$m_{\omega_{t}}$ can be relaxed in comparison with that on
$m_{\tilde{t}_{1}}$. The bound on $m_{\omega_{b}}$ is heavily dependent on the
mass of techni-fermion $m_{R}$ [9], a large part of mass range in (3.1) can
still survive in specific situation.
### 3.2 Indirect Searches
In what follows, we consider the indirect experimental limits on $\pi_{p}$ in
the case of $\theta\sim 0.08-0.2$.
Correction to Br$(Z\rightarrow b\bar{b})$
The radiative correction to Br$(Z\rightarrow b\bar{b})$ coming from technipion
is mostly through the exchange of technipion and top quark. There are three
kinds of Feynman diagrams, the calculation of which can be similarly
considered as for that of charged Higgs bosons in the minimal supersymmetric
standard model in [11]. In addition, there are higher-order corrections due to
$\omega_{t}$ scalar, which are smaller effects and will be neglected. We
summarize the experimental and theoretical results in Table one. One observes
that small $\theta$ factor severely suppresses the correction for technipion,
and forbids it from exposition through the measurement of $R_{b}$.
| Exp value | SM prediction | Exp-SM | TC-correction
---|---|---|---|---
$R_{b}$ | $0.21629\pm 0.00066$ [12] | 0.21581 | $(4.8\pm 6.6)\times 10^{-4}$ | $-5.0\times 10^{-4}\cdot\theta^{2}$
Table 1: The correction to Br$(Z\rightarrow b\bar{b})$ and its experimental
limit.
Correction to $B_{s}^{0}-\bar{B}_{s}^{0}$
The measurement of $B_{s}^{0}-\bar{B}_{s}^{0}$ mixing is another experiment
which can be useful to expose the technipion. Because $\pi^{\pm}_{p}$ gives
rise to two additional one-loop diagrams to this process, which involve
one-$\pi^{\pm}_{p}$-W and two-$\pi^{\pm}_{p}$ exchange, respectively.
Following the results in [4, 13](for an earlier work, see [14]), the
correction is derived to be,
$\displaystyle{}\Delta~{}M^{(\pi_{p})}_{s}\simeq\Delta~{}M_{s}^{SM}\left(-0.18\cdot\theta^{2}-0.63\cdot\theta^{4}\right)$
(3.13)
when $\theta$ closes to $\theta_{max}$. The updated analysis of
$\Delta~{}M^{SM}$ in SM is discussed in [16], whereas its latest experimental
value is given in [12]. We collect these results in Table 2. Similar to the
correction to Br$(Z\rightarrow b\bar{b})$, as a result of $\theta$ suppression
technipion doesn’t produce obviously effects in this experiment.
| Exp value | SM prediction | (Exp-SM) | TC-correction
---|---|---|---|---
$\Delta~{}M_{s}$ | $17.719\pm 0.036$(stat) | $17.3\pm 2.6$ | $0.42\pm 2.6$ | $-3.11\cdot\theta^{2}-10.90\cdot\theta^{4}$
Table 2: Correction to $B_{s}^{0}-\bar{B}_{s}^{0}$ mixing in our model and its
experimental limit. Here $\Delta~{}M_{s}$ is in unite of ps-1.
Correction to $b\rightarrow s\gamma$
The partial width for $b\rightarrow s\gamma$ in our model is similar to that
of type $\mathbf{I}$ two Higgs doublet model (see [15] for the calculation),
with the replacement of $H^{\pm}$ by $\pi_{p}^{\pm}$ in the one-loop Feynman
diagrams. However, our model differs from the type $\mathbf{I}$ two Higgs
doublet model in the way that the couplings of $\pi_{p}$ to SM fermions are
suppressed by $\theta$ factor. In Table 3 we show the experimental and
theoretic results. In Fig 1, we plot $\delta\Gamma/\Gamma_{SM}$ as function of
$\theta$. At present status, no excess beyond $3\sigma$ level is expected in
the range $\theta=0.08-0.15$, or equivalently in the range of mass $210-287$
GeV.
Exp value | SM prediction | (Exp-SM)$(\lesssim 2\sigma)$ | (TC-correction)
---|---|---|---
$3.55\pm 0.26$ [17] | $3.15\pm 0.23$ [18] | $-0.3-1.1$ | Fig.1
Table 3: Correction to $\mathcal{B}(b\rightarrow~{}s\gamma)$ and its
experimental limit. $\mathcal{B}$ is in unite of $10^{-4}$.
Figure 1: Contour of $\delta~{}\Gamma/\Gamma_{SM}$ for
$b\rightarrow~{}s\gamma$. At present status, no excess beyond $3\sigma$ level
is expected in the range of $0.08-0.16$. Note that the choice of $\theta>0.08$
is required by $hf^{\prime}<4\pi f$ and $\upsilon<4\pi f$ as explained above.
One may wonder the implications of direct search on the charged Higgs boson to
technipion. For $\pi_{p}$ with mass of about 90 GeV which is the lower bound
found at colliders, it corresponds to $\theta=0.015$ in our model. It easily
evades indirect experiments such as $b\rightarrow~{}s\gamma$.
### 3.3 Precision Measurement
As well known a severe problem that plagues TC is the precision electroweak
measurement since the report of Peskin and Takeuchi [19]. Because the
condensation of techni-fermions gives rise to a positive and large tree-level
contribution to the oblique parameter
$\displaystyle{}S_{0}$ $\displaystyle\simeq$ $\displaystyle
0.1~{}\frac{N_{TF}}{2}N_{TC}\simeq 0.1~{}N_{TC}$ $\displaystyle T_{0}$
$\displaystyle\simeq$ $\displaystyle
0.01\left(\frac{\Lambda^{\prime}}{1{\it~{}TeV}}\right)^{4}$ (3.14)
for two flavors $N_{TF}=2$. $\Lambda^{\prime}$ is a mass scale close to
$\Lambda_{TC}$, which is hard to be estimated precisely. As noted from (3.3),
$S_{0}$ is too large. In what follows, we consider the case $N_{TC}=4$. The
introduction of colored $\omega$ scalars also produces significant
contributions to $S$. What is of interest is these new contributions always
cancel out the tree-level part of $S$, and help evading the precision
measurements when the masses of $\omega_{t,b}$ are of order $\sim\upsilon$.
Following the definition of $S$ and $T$ parameters, we derive the total
contribution in our model 555To calculate $S$ and $T$ we follow the notation
in [20]. In this reference, the four fermions interactions below
$\Lambda_{TC}$ induced by the $\omega$ scalars are carefully considered. The
effects on oblique parameters from these operators can be extracted in terms
of the effective field theory analysis.,
$\displaystyle{}S$ $\displaystyle=$ $\displaystyle
S_{0}+\frac{2}{3\pi}(2\delta g^{t}_{R}-\delta g^{b}_{R})$ $\displaystyle T$
$\displaystyle=$ $\displaystyle T_{0}+\delta
g^{t}_{R}\frac{3m^{2}_{t}}{\pi^{2}\alpha\upsilon^{2}}\ln\left(\frac{\Lambda_{TC}}{m_{t}}\right)$
(3.15)
where
$\displaystyle{}\delta
g^{t}_{R}=-\frac{\lambda^{2}_{\omega_{t}}\upsilon^{2}}{8m^{2}_{\omega_{t}}},~{}~{}~{}~{}~{}~{}\delta
g^{b}_{R}=\frac{\lambda^{2}_{\omega_{b}}\upsilon^{2}}{8m^{2}_{\omega_{b}}}$
(3.16)
The experimental limits on $S$ and $T$ of (3.3) have been updated from global
fit. Following the results in the second reference of [21],
$\displaystyle{}S=0.07\pm 0.10,~{}~{}~{}~{}~{}~{}~{}T=0.05\pm 0.12.$ (3.17)
in Table 4 we show four benchmark points involving parameters of
$\omega$-scalar mass and their Yukawa couplings.
As shown in Table 4 , it is sufficient for $m_{\omega_{t,b}}$ of order $\sim$
EW scale to cancel the tree-level contribution $S_{0}$. This requirement can
be naturally realized in our model. Note that the vev of $\Phi$ induced by the
condensation gives rise to masses of techni-scalar in terms of potential
$V(\phi,\omega)$ in (2), which read,
$\displaystyle{}m^{2}_{\omega_{t}}=\frac{1}{2}\lambda_{2}f^{\prime
2},~{}~{}~{}~{}m^{2}_{\omega_{b}}=\frac{1}{2}\lambda_{3}f^{\prime 2}.$ (3.18)
In this sense, bounds on $m_{\omega_{t,b}}$ due to precise electroweak
measurements can be used to constrain Yukawa couplings in (2).
$\lambda_{\omega_{t}}$ | $\lambda_{\omega_{b}}$ | $m_{\omega_{t}}$ | $m_{\omega_{b}}$ | $\Lambda^{\prime}$
---|---|---|---|---
0.3 | 1.3 | 583.4 | 100 | 800
0.5 | 3.0 | 972.4 | 229 | 800
0.3 | 1.3 | 476.3 | 100 | 1500
0.5 | 3.0 | 794 | 229 | 1500
Table 4: Benchmark points hinted by the precision measurements. Here mass is
in unite of GeV. $\Lambda^{\prime}=0.8~{}(1.5)$ TeV corresponds to central
value $\delta g^{t}_{R}=-0.002$ ($-0.003$), respectively, and $\delta
g^{b}_{R}$=1.3.
## 4 Conclusions
In this paper we consider TC-model with one scalar doublet and two extra
colored techni-scalars. After the condensation of techni-fermion at scale
$\Lambda=4\pi f\sim 480$ GeV which is above the EW scale, the scalar doublet
receives its vev $f^{\prime}$ through its coupling to techni-fermions, gives
rises to a SM-like scalar $\sigma$ discovered at the LHC when
$\theta=f/f^{\prime}\lesssim 0.2$. With $m_{\sigma}=125$ GeV, experimental
limits suggest that $m_{\pi_{p}}$ in the range $210-280$ GeV. Because of
$\theta$ suppression on Yukawa couplings of techni-pions to SM fermions, they
can evade the present experimental limits from both direct and indirect
searches.
On the other hand, the colored techni-scalars obtain their masses of order
$\mathcal{O}(0.1-1)$ TeV, through $\phi^{4}$ coupling with scalar doublet.
They can provide a large negative contribution to $S$, which eliminates the
large positive contribution arising from condensation, therefore reconcile our
model with precision electroweak measurements.
A detailed analysis on bounds of $\omega$ scalars masses is needed in the
further. It is also of interest to consider TC models with two fundamental
scalar doublets instead.
$\bf{Acknowledgement}$
The author thanks C. D. Carone for correspondence and D. Shih for discussions.
This work is supported in part by the Natural Science Foundation of China
under Grant No. 11247031.
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|
arxiv-papers
| 2013-05-29T02:42:35 |
2024-09-04T02:49:45.825419
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sibo Zheng",
"submitter": "Sibo Zheng",
"url": "https://arxiv.org/abs/1305.6677"
}
|
1305.6681
|
# Large eddy simulation of a muffler with the high-order spectral difference
method
Matteo Parsani Current Institution: Computational Aerosciences Branch, NASA
Langley Research Center, Hampton, VA 23681, USA
([email protected])Division of Computer, Electrical and Mathematical
Sciences & Engineering, King Abdullah University of Science and Technology,
Thuwal, 23955-6900, KSA Michael Bilka University of Notre Dame, Department of
Aerospace and Mechanical Engineering, 365 Fitzpatrick Hall, Notre Dame, IN
46556-5637, USA ([email protected]) Chris Lacor Department of Mechanical
Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
([email protected])
###### Abstract
The combination of the high-order accurate spectral difference discretization
on unstructured grids with subgrid-scale modelling is investigated for large
eddy simulation of a muffler at $Re=4.64\cdot 10^{4}$ and low Mach number. The
subgrid-scale stress tensor is modelled by the wall-adapting local eddy-
viscosity model with a cut-off length which is a decreasing function of the
order of accuracy of the scheme. Numerical results indicate that although the
high-order solver without subgrid-scale modelling is already able to capture
well the features of the flow, the coupling with the wall-adapting local eddy-
viscosity model improves the quality of the solution.
## 1 Introduction
Throughout the past two decades, the development of high-order accurate
spatial discretization has been one of the major fields of research in
computational fluid dynamics (CFD), computational aeroacoustics (CAA),
computational electromagnetism (CEM) and in general computational physics
characterized by linear and nonlinear wave propagation phenomena. High-order
accurate discretizations have the potential to improve the computational
efficiency required to achieve a desired error level. In fact, compared with
low order schemes, high order methods offer better wave propagation properties
and increased accuracy for a comparable number of degrees of freedom (DOFs).
Therefore, it may be advantageous to use such schemes for problems that
require very low numerical dissipation and small error levels [1]. Moreover,
since computational science is increasingly used as an industrial design and
analysis tool, high accuracy must be achieved on unstructured grids which are
required for efficient meshing. These needs have been the driving force for
the development of a variety of higher order schemes for unstructured meshes
such as the Discontinuous Galerkin (DG) method [2, 3], the Spectral Volume
(SV) method [4], the Spectral Difference (SD) method [5, 6], the Energy Stable
Flux Reconstruction [7] and many others.
In this study we focus on a SD solver for unstructured hexahedral grids
(tensorial cells). The SD method has been proposed as an alternative high
order collocation-based method using local interpolation of the strong form of
the equations. Therefore, the SD scheme has an important advantage over
classical DG and SV methods, that no integrals have to be evaluated to compute
the residuals, thus avoiding the need for costly high-order accurate
quadrature formulas.
Although the formulation of high-order accurate spatial discretization is now
fairly mature, their application for the simulation of general turbulent flows
implies that particular attention has still to be paid to subgrid-scale (SGS)
models. So far, the combination of the SD method with SGS models for LES has
not been widely investigated. In 2010, Parsani et al. [8] reported the first
implementation in study of a two-dimensional (2D) third-order accurate SD
solver coupled with the Wall-Adapting Local Eddy-viscosity (WALE) model [14]
and a cut-off length which is a decreasing function of the order of accuracy.
A successful extension of that approach to a three-dimensional (3D) second-
order accurate SD solver has been reported in [12]. Very recently, Lodato and
Jameson [13] have presented an alternative technique to model the unresolved
scales in the flow field: A structural SGS approach with the WALE Similarity
Mixed model (WSM), where constrained explicit filtering represents a key
element to approximate subgrid-scale interactions. The performance of such an
algorithm has been also satisfactory.
In this study, we couple for the first time the approach proposed in [8] with
a 3D fourth-order accurate SD solver, for the simulation of the turbulent flow
in an industrial-type muffler at $Re=4.64\cdot 10^{4}$. The goal is to
investigate if the coupling of a high-order SD scheme with a sub-grid closure
model improves the quality of the results when the grid resolution is
relatively low. The latter requirement is often desirable when a high-order
accurate spatial discretization is used.
## 2 Physical model and numerical algorithm
In this study the system of the Navier-Stokes equations for a compressible
flow are discretized in space using the SD method and the subgrid-scale stress
tensor is modelled by the WALE approach.
### 2.1 Filtered Navier-Stokes equations
The three physical conservation laws for a general Newtonian fluid, i.e., the
continuity, the momentum and energy equations, are introduced using the
following notation: $\rho$ for the mass density, $\vec{u}\in\mathbb{R}^{dim}$
for the velocity vector in a physical space with $dim$ dimensions, $P$ for the
static pressure and $E$ for the specific total energy which is related to the
pressure and the velocity vector field by
$E=\frac{1}{\gamma-1}\frac{P}{\rho}+\frac{|\vec{u}|^{2}}{2}$, where $\gamma$
is the constant ratio of specific heats and it is $1.4$ for air in standard
conditions.
The system, written in divergence form and equipped with suitable initial-
boundary conditions, is
$\frac{\partial\textbf{w}}{\partial
t}+\vec{\nabla}\cdot\left(\vec{\textbf{f}}_{C}\left(\textbf{w}\right)-\vec{\textbf{f}}_{D}\left(\textbf{w},\vec{\nabla}\textbf{w}\right)\right)=\frac{\partial\textbf{w}}{\partial
t}+\vec{\nabla}\cdot\vec{\textbf{f}}=0,$ (1)
where
$\textbf{w}=\left(\overline{\rho},\overline{\rho}\tilde{\vec{u}},\overline{\rho}\tilde{E}\right)^{T}$
is the vector of the filtered conservative variables and
$\vec{\textbf{f}}_{C}=\vec{\textbf{f}}_{C}\left(\textbf{w}\right)$ and
$\vec{\textbf{f}}_{D}=\vec{\textbf{f}}_{D}\left(\textbf{w},\vec{\nabla}\textbf{w}\right)$
represent the convective and the diffusive fluxes, respectively. Here the
symbols $(\overline{\cdot})$ and $(\tilde{\cdot})$ represent the spatially
filtered field and the Favre filtered field defined as
$\tilde{\vec{u}}=\overline{\rho\vec{u}}/\overline{\rho}$.
In a general 3D ($dim=3$) Cartesian space,
$\vec{x}=\left[x_{1},x_{2},x_{3}\right]^{T}$, the components of the flux
vector
$\vec{\textbf{f}}\left(\textbf{w},\vec{\nabla}\textbf{w}\right)=\left[\textbf{f}_{1},\textbf{f}_{2},\textbf{f}_{3}\right]^{T}$
are given by
$\textbf{f}_{1}=\left(\begin{array}[]{c}\overline{\rho}\tilde{u}_{1}\\\
\overline{\rho}\tilde{u}_{1}^{2}+\overline{P}-\tilde{\sigma}_{11}+\tau_{11}^{sgs}\\\
\overline{\rho}\tilde{u}_{1}\tilde{u}_{2}-\tilde{\sigma}_{21}+\tau_{21}^{sgs}\\\
\overline{\rho}\tilde{u}_{1}\tilde{u}_{3}-\tilde{\sigma}_{31}+\tau_{31}^{sgs}\\\
\tilde{u}_{1}\left(\overline{\rho}\tilde{E}+\overline{P}\right)-\tilde{u}_{1}\left(\tilde{\sigma}_{11}-\tau_{11}^{sgs}\right)-\tilde{u}_{2}\left(\tilde{\sigma}_{21}-\tau_{21}^{sgs}\right)-\tilde{u}_{3}\left(\tilde{\sigma}_{31}-\tau_{31}^{sgs}\right)-c_{P}\frac{\mu}{Pr}\frac{\partial\tilde{T}}{\partial
x_{1}}+q_{1}^{sgs}\end{array}\right),$
$\textbf{f}_{2}=\left(\begin{array}[]{c}\overline{\rho}\tilde{u}_{2}\\\
\overline{\rho}\tilde{u}_{1}\tilde{u}_{2}-\tilde{\sigma}_{12}+\tau_{12}^{sgs}\\\
\overline{\rho}\tilde{u}_{2}^{2}+\overline{P}-\tilde{\sigma}_{22}+\tau_{22}^{sgs}\\\
\overline{\rho}\tilde{u}_{2}\tilde{u}_{3}-\tilde{\sigma}_{32}+\tau_{32}^{sgs}\\\
\tilde{u}_{2}\left(\overline{\rho}\tilde{E}+\overline{P}\right)-\tilde{u}_{1}\left(\tilde{\sigma}_{12}-\tau_{12}^{sgs}\right)-\tilde{u}_{2}\left(\tilde{\sigma}_{22}-\tau_{22}^{sgs}\right)-\tilde{u}_{3}\left(\tilde{\sigma}_{32}-\tau_{32}^{sgs}\right)-c_{P}\frac{\mu}{Pr}\frac{\partial\tilde{T}}{\partial
x_{2}}+q_{2}^{sgs}\end{array}\right),$
$\textbf{f}_{3}=\left(\begin{array}[]{c}\overline{\rho}\tilde{u}_{3}\\\
\overline{\rho}\tilde{u}_{1}\tilde{u}_{3}-\tilde{\sigma}_{13}+\tau_{13}^{sgs}\\\
\overline{\rho}\tilde{u}_{2}\tilde{u}_{3}-\tilde{\sigma}_{23}+\tau_{23}^{sgs}\\\
\overline{\rho}\tilde{u}_{3}^{2}+\overline{P}-\tilde{\sigma}_{33}+\tau_{33}^{sgs}\\\
\tilde{u}_{3}\left(\overline{\rho}\tilde{E}+\overline{P}\right)-\tilde{u}_{1}\left(\tilde{\sigma}_{13}-\tau_{13}^{sgs}\right)-\tilde{u}_{2}\left(\tilde{\sigma}_{23}-\tau_{23}^{sgs}\right)-\tilde{u}_{3}\left(\tilde{\sigma}_{33}-\tau_{33}^{sgs}\right)-c_{P}\frac{\mu}{Pr}\frac{\partial\tilde{T}}{\partial
x_{3}}+q_{3}^{sgs}\end{array}\right),$
where $c_{P}$, $\mu$, $Pr$ and $T$ represent respectively the specific heat
capacity at constant pressure, the dynamic viscosity, the Prandtl number and
the temperature of the fluid. Moreover, $\sigma_{ij}$ represents the
$ij-$component of the resolved viscous stress tensor [15].
Both momentum and energy equations differ from the classical fluid dynamic
equations only for two terms which take into account the contributions from
the unresolved scales. These contributions, represented by the specific
subgrid-scale stress tensor $\tau_{ij}^{sgs}$ and by the subgrid heat flux
vector defined $q_{i}^{sgs}$, appear when the spatial filter is applied to the
convective terms [15]. The interactions of $\tau_{ij}^{sgs}$ and $q_{i}^{sgs}$
with the resolved scales have to be modeled through a subgrid-scale closure
model because they cannot be determined using only the resolved flow field w.
#### 2.1.1 The wall-adapted local eddy-viscosity closure model
The smallest scales present in the flow field and their interaction with the
resolved scales have to be modeled through the subgrid-scale term
$\tau_{ij}^{sgs}$. The most common approach to model such a tensor is based on
the eddy-viscosity concept in which one assumes that the residual stress is
proportional to a measure of the filtered local strain rate [15], which is
defined as follows:
$\tau_{ij}^{sgs}-\tau_{kk}^{sgs}\delta_{ij}=-2\,\overline{\rho}\,\nu_{t}\left(\tilde{S}_{ij}-\frac{\delta_{ij}}{3}\tilde{S}_{kk}\right).$
(2)
In the WALE model, it is assumed that the eddy-viscosity $\nu_{t}$ is
proportional to the square of the length scale of the cut-off length (or width
of the grid filter) and the filtered local rate of strain. Although the model
was originally developed for incompressible flows, it can also be used for
variable density flows by giving the formulation as follows
$\nu_{t}=\left(C\Delta\right)^{2}\left|\tilde{S}\right|.$ (3)
Here $\left|\tilde{S}\right|$ is defined as
$\left|\tilde{S}\right|=\frac{\left[\tilde{S}_{ij}^{d}\,\tilde{S}_{ij}^{d}\right]^{3/2}}{\left[\tilde{S}_{ij}\,\tilde{S}_{ij}\right]^{5/2}+\left[\tilde{S}_{ij}^{d}\,\tilde{S}_{ij}^{d}\right]^{5/4}},$
(4)
where $\tilde{S}_{ij}^{d}$ is the traceless symmetric part of the square of
the resolved velocity gradient tensor
$\tilde{g}_{ij}=\frac{\partial\tilde{u}_{i}}{\partial x_{j}}$. Note that in
Equation (3) $\Delta$, i.e., the cut-off length, is an unknown function. Often
the cut-off length is taken proportional to the smallest resolvable length
scale of the discretization. In the present work, the definition of the grid
filter function is given in Section 2.2, where the SD method is discussed.
### 2.2 Spectral difference method
Consider a problem governed by a general system of conservation laws given by
Equation (1) and valid on a domain $\Omega\subset\mathbb{R}^{dim}$ with
boundary $\partial\Omega$ and completed with consistent initial and boundary
conditions. The domain is divided into $N$ non-overlapping cells, with cell
index $i$.
In order to achieve an efficient implementation of the SD method, all
hexahedral cells in the physical domain are mapped into cubic elements using
local coordinates $\vec{\xi}=\left[\xi_{1},\xi_{2},\xi_{3}\right]^{T}$. Such a
transformation is characterized by the Jacobian matrix
$\vec{\vec{\left.\mathrm{J}\right.}}_{i}$ with determinant
$det(\vec{\vec{\left.\mathrm{J}\right.}}_{i})$. Therefore, system (1) can be
written in the mapped coordinate system as
$\frac{\partial\mathbf{w}_{i}^{\hskip 1.13791pt\vec{\xi}}}{\partial
t}=-\frac{\partial\mathbf{f}_{1,i}^{\hskip
1.13791pt\vec{\xi}}}{\partial\xi_{1}}-\frac{\partial\mathbf{f}_{2,i}^{\hskip
1.13791pt\vec{\xi}}}{\partial\xi_{2}}-\frac{\partial\mathbf{f}_{3,i}^{\hskip
1.13791pt\vec{\xi}}}{\partial\xi_{3}}=-\vec{\nabla}^{\hskip
1.13791pt\vec{\xi}}\cdot\vec{\textbf{f}}^{\hskip 1.13791pt\vec{\xi}}_{i},$ (5)
where $\mathbf{w}^{\hskip 1.13791pt\vec{\xi}}_{i}\equiv
det(\vec{\vec{\left.\mathrm{J}\right.}}_{i})\,\mathbf{w}$ and
$\vec{\nabla}^{\hskip 1.13791pt\vec{\xi}}$ are the conserved variables and the
generalized divergence differential operator in the mapped coordinate system,
respectively.
For a $\left(p+1\right)$-th-order accurate $dim$-dimensional scheme, $N^{s}$
_solution collocation points_ with index $j$ are introduced at positions
$\vec{\xi}_{j}^{s}$ in each cell $i$, with $N^{s}$ given by
$N^{s}=\left(p+1\right)^{dim}$. Given the values at these points, a polynomial
approximation of degree $p$ of the solution in cell $i$ can be constructed.
This polynomial is called the _solution polynomial_ and is usually composed of
a set of Lagrangian basis polynomial $L_{j}^{s}\left(\vec{\xi}\right)$ of
degree $p$:
$\mathbf{W}_{i}\left(\vec{\xi}\right)=\sum_{j=1}^{N^{s}}\mathbf{W}_{i,j}\,L_{j}^{s}\left(\vec{\xi}\right).$
(6)
Therefore, the unknowns of the SD method are the interpolation coefficients
$\mathbf{W}_{i,j}=\mathbf{W}_{i}\left(\vec{\xi}_{j}^{s}\right)$ which are the
approximated values of the conserved variables $\mathbf{w}_{i}$ at the
solution points.
The divergence of the mapped fluxes $\vec{\nabla}^{\hskip
1.13791pt\vec{\xi}}\cdot\vec{\textbf{f}}^{\hskip 1.13791pt\vec{\xi}}$ at the
solution points is computed by introducing a set of $N^{f}$ flux collocation
points with index $l$ and at positions $\vec{\xi}_{l}^{f}$, supporting a
polynomial of degree $p+1$. The evolution of the mapped flux vector
$\vec{\textbf{f}}^{\hskip 1.13791pt\vec{\xi}}$ in cell $i$ is then
approximated by a flux polynomial $\vec{\textbf{F}}^{\hskip
1.13791pt\vec{\xi}}_{i}$, which is obtained by reconstructing the solution
variables at the flux points and evaluating the fluxes
$\vec{\textbf{F}}^{\hskip 1.13791pt\vec{\xi}}_{i,l}$ at these points. The flux
is also represented by a Lagrange polynomial:
$\vec{\textbf{F}}^{\hskip
1.13791pt\vec{\xi}}_{i}\left(\vec{\xi}\right)=\sum_{l=1}^{N^{f}}\vec{\textbf{F}}^{\hskip
1.13791pt\vec{\xi}}_{i,l}\,L_{l}^{f}\left(\vec{\xi}\right),$ (7)
where the coefficients of the flux interpolation are defined as
$\vec{\textbf{F}}^{\hskip
1.13791pt\vec{\xi}}_{i,l}=\begin{cases}\vec{\textbf{F}}^{\hskip
1.13791pt\vec{\xi}}_{i}\left(\vec{\xi}_{l}^{f}\right),&\quad\vec{\xi}_{l}^{f}\in\Omega_{i},\\\
\vec{\textbf{F}}_{\mathrm{num}}^{\hskip
1.13791pt\vec{\xi}}\left(\vec{\xi}_{l}^{f}\right),&\quad\vec{\xi}_{l}^{f}\in\partial\Omega_{i}.\end{cases}$
(8)
Here $\vec{\textbf{F}}_{\mathrm{num}}^{\hskip 1.13791pt\vec{\xi}}$ is the
numerical flux vector at the cell interface. In fact, the solution at a face
is in general not continuous and requires the solution of a Riemann problem to
maintain conservation at a cell level (i.e., the flux component normal to a
face $\vec{\textbf{F}}_{\mathrm{num}}^{\hskip
1.13791pt\vec{\xi}}\cdot\vec{n}^{\hskip 1.13791pt\vec{\xi}}$ must be
continuous between two neighboring cells). Approximate Riemann solvers are
typically used (e.g. Rusanov Riemann solver). The tangential component of
$\vec{\textbf{F}}_{\mathrm{num}}^{\hskip 1.13791pt\vec{\xi}}$ is usually taken
from the interior cell.
Taking the divergence of the flux polynomial $\vec{\nabla}^{\hskip
1.13791pt\vec{\xi}}\cdot\vec{\textbf{F}}^{\hskip 1.13791pt\vec{\xi}}_{i}$ in
the solution points results in the following modified form of (5), describing
the evolution of the conservative variables at the solution points:
$\frac{d\mathbf{W}_{i,j}}{dt}=-\left.\vec{\nabla}\cdot\vec{\textbf{F}}_{i}\right|_{j}=-\frac{1}{J_{i,j}}\left.\vec{\nabla}^{\hskip
1.13791pt\vec{\xi}}\cdot\vec{\textbf{F}}^{\hskip
1.13791pt\vec{\xi}}_{i}\right|_{j}=\mathbf{R}_{i,j},$ (9)
where $\vec{\textbf{F}}_{i}$ is the flux polynomial vector in the physical
space whereas $\mathbf{R}_{i,j}$ is the SD residual associated with
$\mathbf{W}_{i,j}$. This is a system of ODEs, in time, for the unknowns
$\mathbf{W}_{i,j}$. In this work, the optimized explicit eighteen-stages
fourth-order Runge-Kutta schemes presented in [16] is used to solve such a
system at each time step.
#### 2.2.1 Solution and flux points distributions
In 2007, Huynh [9] showed that for quadrilateral and hexahedral cells, tensor
product flux point distributions based on a one-dimensional flux point
distribution consisting of the end points and the Legendre-Gauss quadrature
points lead to stable schemes for arbitrary order of accuracy. In 2008, Van
den Abeele et al. [10] showed an interesting property of the SD method, namely
that it is independent of the positions of its solution points in most general
circumstances. This property implies an important improvement in efficiency,
since the solution points can be placed at flux point positions and thus a
significant number of solution reconstructions can be avoided. Recently, this
property has been proved by Jameson [11].
#### 2.2.2 Cut-off length $\Delta$
In Section 2.1.1 we have seen that in the WALE model the cut-off length
$\Delta$ is used to compute the turbulent eddy-viscosity $\nu_{t}$, i.e.,
Equation (3). Following the approach presented in [8], for each cell with
index $i$ and each flux points with index $l$ and positions
$\boldsymbol{\xi}_{l}^{f}$, we use the following definition of filter width
$\Delta_{i,l}=\left[\frac{1}{N^{s}}det\left(\left.\vec{\vec{\left.\mathrm{J}\right.}}_{i}\right|_{\boldsymbol{\xi}_{l}^{f}}\right)\right]^{1/dim}=\left(\frac{det(J_{i,l})}{N^{s}}\right)^{1/dim}.$
(10)
Notice that the cell filter width is not constant in one cell, but it varies
because the Jacobian matrix is a function of the positions of the flux points.
Moreover, for a given mesh, the number of solution points depends on the order
of the SD scheme, so that the grid filter width decreases by increasing the
polynomial order of the approximation.
## 3 Numerical results
The main purpose of this section is to evaluate the accuracy and the
reliability of the fourth-order SD-LES solver for simulating a 3D turbulent
flow in an industrial-type muffler. The results are compared with the particle
image velocimetry (PIV) measurement performed at the Department of
Environmental and Applied Fluid Dynamics of the von Karman Institute for Fluid
Dynamics [17]. In Figure 1, the geometry of the muffler and its characteristic
dimensions are illustrated, where the flow is from left to right.
$5\,d$$7.5\,d$$5\,d$$4.75\,d$
$d$
$0.625\,d$
$0.625\,d$
$\chi_{3}$$\chi_{2}$
Figure 1: Configuration of the 3D muffler test case.
At the inlet, mass density and velocity profiles are imposed. The inlet
velocity profile in the $x_{3}$ direction is given by
$u_{3}=u_{max}\left\\{\frac{1}{2}-\frac{1}{2}\tanh{\left[2.2\left(\frac{r}{d/2}-\frac{d/2}{r}\right)\right]}\right\\}.$
At the outlet only the pressure is prescribed. In accordance to the
experiments, the inlet Mach number and the Reynolds number, based on maximum
velocity at the inlet $u_{max}$ and the diameter of the inlet/outlet $d$
($d=4\,cm$), are set respectively to $M_{inlet}=0.05$ and $Re=4.64\cdot
10^{4}$.
The flow is computed using fourth-order ($p=3$) SD scheme on a grid with
$36,612$ hexahedral elements which was generated with the open source software
Gmsh [18]. Second-order boundary elements are used to approximate the curved
geometry. The total number of DOFs is approximately $2.3\cdot 10^{6}$ (i.e.,
$36,612\cdot(p+1)^{3}$). The maximum CFL number used for the computations
started from $0.1$ and increased up to $0.65$. After the flow field was fully
developed, time averaging is performed for a period corresponding to about 25
flow-through times.
The computation is validated on the center plane of the expansion coinciding
with the center planes of the inlet and outlet pipes using the PIV results
from [17]. All of the measurements are taken on the symmetrical center plane
of the muffler. The reference cross section corresponds to the entrance of the
expansion chamber. It should be noted that the circular nature of the geometry
acts as a lens causing a change in magnification in the radial direction
($x_{2}$) which prevents from capturing images close to the wall. It is found
that outside $~{}1\,cm$ from the wall the magnification effect is negligible
and as the mean stream-wise direction is in the direction of constant
magnification and has only little effect on the particle correlations no
corrections are deemed necessary.
In Figure 2, the non-dimensional mean velocity profile in the axial direction
$\langle\tilde{u}_{3}\rangle/u_{max}$ is shown for four different cross
sections in the expansion chamber, where the PIV measurements were done. In
this figure, the PIV data are also plotted for comparison. Figure 3 shows the
non-dimensional Reynolds stress $\langle
u_{2}^{\prime}u_{3}^{\prime}\rangle/{u_{max}^{2}}$ at the same cross sections.
Although the high-order implicit LES is already able to capture well the
features of the flow field, the use of the WALE model improves the results. In
particular, when the SGS model is active, the local extrema of the time-
averaged velocity profiles and the second-order statistical moment (which get
fairly oscillatory by moving far away from the inlet pipe) are better
captured.
## 4 Conclusions
The fourth-order SD method in combination with the WALE model and the variable
filter width performed well. The numerical results confirm that the model is
correctly accounting for the unresolved shear stress computed from the
resolved field, for the present internal flow. However, it should be noted
that the SD discretization without subgrid-scale modelling also worked rather
well, at least for the grid resolution used in this study.
Work is currently under way to test both approaches for different orders of
accuracy, grid resolutions and other realistic turbulent flows. We believe
that the flexibility of the high-order SD scheme on unstructured grids
together with the development of robust sub-grid closure models for highly
separated flows and efficient grid generators for high-order accurate schemes
will allow to perform LES of industrial-type flows in the near future.
(a) $1d$ downstream. (b) $4d$ downstream. (c) $6d$ downstream. (d) $7d$
downstream.
Figure 2: Time-averaged velocity profile in the axial direction
$\langle\tilde{u}_{3}\rangle/u_{max}$ at four cross sections in the expansion
chamber, obtained with fourth-order ($p=3$) SD-LES method. Comparison with
experimental measurements (PIV) [17].
(a) $1d$ downstream. (b) $4d$ downstream. (c) $6d$ downstream. (d) $7d$
downstream.
Figure 3: Reynolds stress $\langle
u_{2}^{\prime}u_{3}^{\prime}\rangle/u_{max}^{2}$ in the axial direction at
four cross sections in the expansion chamber, obtained with fourth-order
($p=3$) SD-LES method. Comparison with experimental measurements (PIV) [17].
## Acknowledgement
The authors would like to thank Professor David I. Ketcheson and Professor
Mark H. Carpenter for their support. This research used partially the
resources of the KAUST Supercomputing Laboratory and was supported in part by
an appointment to the NASA Postdoctoral Program at Langley Research Center,
administered by Oak Ridge Associates Universities. These supports are
gratefully acknowledged.
## References
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* [2] Busch, K., König, M., and Niegemann, J.: Discontinuous Galerkin methods in nanophotonics. Laser Photonics Rev. 5(6), 773–809 (2011).
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|
arxiv-papers
| 2013-05-29T03:22:15 |
2024-09-04T02:49:45.832243
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Matteo Parsani and Michael Bilka and Chris Lacor",
"submitter": "Matteo Parsani",
"url": "https://arxiv.org/abs/1305.6681"
}
|
1305.6700
|
# Early warning signals: The charted and uncharted territories
Carl Boettiger [email protected] Noam Ross Alan Hastings Center for Stock
Assessment Research, Department of Applied Math and Statistics, University of
California, Mail Stop SOE-2, Santa Cruz, CA 95064, USA Department of
Environmental Science and Policy, University of California Davis, 1 Shields
Avenue, Davis, CA 95616 USA
###### Abstract
The realization that complex systems such as ecological communities can
collapse or shift regimes suddenly and without rapid external forcing poses a
serious challenge to our understanding and management of the natural world.
The potential to identify early warning signals that would allow researchers
and managers to predict such events before they happen has therefore been an
invaluable discovery that offers a way forward in spite of such seemingly
unpredictable behavior. Research into early warning signals has demonstrated
that it is possible to define and detect such early warning signals in advance
of a transition in certain contexts. Here we describe the pattern emerging as
research continues to explore just how far we can generalize these results. A
core of examples emerges that shares three properties: the phenomenon of rapid
regime shifts, a pattern of ’critical slowing down’ that can be used to detect
the approaching shift, and a mechanism of bifurcation driving the sudden
change. As research has expanded beyond these core examples, it is becoming
clear that not all systems that show regime shifts exhibit critical slowing
down, or vice versa. Even when systems exhibit critical slowing down,
statistical detection is a challenge. We review the literature that explores
these edge cases and highlight the need for (a) new early warning behaviors
that can be used in cases where rapid shifts do not exhibit critical slowing
down, (b) the development of methods to identify which behavior might be an
appropriate signal when encountering a novel system; bearing in mind that a
positive indication for some systems is a negative indication in others, and
(c) statistical methods that can distinguish between signatures of early
warning behaviors and noise.
###### keywords:
early warning signals , regime shifts , bifurcation , critical slowing down
## Introduction
Many natural systems exhibit regime shifts - rapid changes in the state and
conditions of system behavior. Examples of such shifts include lake
eutrophication (Carpenter et al. 1999), algal overgrowth of coral systems
(Mumby et al. 2007), fishery collapse (Jackson et al. 2001), desertification
of grasslands (Kéfi et al. 2007), and rapid changes in climate (Dakos et al.
2008, Lenton et al. 2009). Such dramatic shifts have the potential to impact
ecosystem health and human well-being. Thus, it is important to develop
strategies for adaptation, mitigation, and avoidance of such shifts.
The idea that complex systems such as ecosystems could change suddenly and
without warning goes back to the 1960s (Lewontin 1969, Holling 1973, May
1977). Such early work revealed that even simple models with the appropriate
nonlinearities were capable of unpredictable behavior. The only way to predict
the transition was to have the right model – and that meant having already had
the chance to observe the transition. One cogent early example (Ludwig et al.
1978) demonstrated how knowledge of the forms and time scales of interactions
among insects, birds, and trees could lead to a qualitative model that
essentially predicted the possibility of regime shifts.
Management of systems that could potentially undergo shifts requires balancing
the costs of adaptation, mitigation, or avoidance against the costs of the
shift itself. Avoidance depends on an ability to predict regime shifts in
advance, or depending on the time scale of response and response of the
system, on the ability to recognize a shift as it is occurring. Adaptation and
mitigation might require an ability to predict a shift in advance if the time
scale of implementation is long relative to the rate at which damages occur.
An important component of this management challenge is the development of
early warning signals (EWS) of impending rapid regime shifts (Scheffer et al.
2009). Since regime shifts occur in a variety of systems, and underlying
mechanisms for the shifts are not always known, the development of generic
signals applicable to a variety of systems would be particularly valuable.
This naturally leads to the questions of when such generic signals would be
valuable tools versus the need to develop system-specific approaches in all
cases.
Foundational research in EWS identified certain patterns that may forecast a
sudden transition in a wide variety of systems (Scheffer et al. 2009). Most
extensively researched is the phenomenon of critical slowing down (CSD), which
is manifested as a pattern of increasing variance or autocorrelation of a
system. Subsequent work has begun to identify a growing library of cases in
which these indicators are not present before a transition (Schreiber and
Rittenhouse 2004, Schreiber and Rudolf 2008, Hastings and Wysham 2010, Bel et
al. 2012), or are observed in the absence of any transition (Kéfi et al.
2012). These examples are distinct from the more well-known case of
statistical error – such as a signal that is present but too weak to detect
due to insufficient available data (see Dakos et al. (2008); Scheffer et al.
(2009) and Perretti and Munch (2012)). Instead, such work moves into new
territory where different underlying mechanisms have lead to starkly different
patterns. Determining which underlying mechanisms are present is a substantial
empirical and theoretical challenge. When does critical slowing down
correspond to the assumptions made?
Here we review a variety of mechanisms that may lead to rapid (or
“catastrophic”) regime shifts in ecological systems, as well as mechanisms
that generate early warning signals. We focus on CSD and its manifestations as
they are the most commonly studied warning signals. We illustrate that not all
rapid shifts exhibit CSD, and not all observations of CSD involve rapid
shifts. Thus the issue of determining EWS is really two-fold: first, to
identify classes of systems where the warning signal is expected and
conversely systems that may undergo shifts without such signals, and second,
to determine appropriate statistical tools to detect the warning signal. In
this paper we review both aspects of the overall question.
Critical slowing down (CSD) | A system’s slowing response to perturbations as it’s dominant eigenvalue approaches zero, often expressed in greater variance, autocorrelation, and return time. CSD is one possible EWS.
---|---
Early warning signals (EWS) | A general term for dynamic patterns in system behavior that precede regime shifts. Though CSD phenomena are among the best studied EWS, some shifts will require alternative signals; Figure 1.
Definitions: In this paper we refer to two closely related but different
phenomena
## Relationships between Critical Slowing Down, Bifurcations, and Regime
Shifts
CSD has been studied extensively in theoretical (Wissel 1984, Gandhi et al.
1998, Carpenter and Brock 2006, Hastings and Wysham 2010, Dakos et al. 2011a,
Lade and Gross 2012, Boettiger and Hastings 2012a) and empirical contexts
(Drake and Griffen 2010, Carpenter et al. 2011, Veraart et al. 2012, Dai et
al. 2012, Wang et al. 2012) as a potential EWS for regime shifts. CSD occurs
as a system’s dominant eigenvalue approaches zero due to a changing (possibly
deteriorating) environment. As the eigenvalue approaches zero, the system’s
response to small perturbations slows. This change in dynamic properties of a
system can be expressed in greater variance, autocorrelation, and return time
of observed state variables.
In Figure 1, we illustrate the domains of overlap between three distinct
phenomena. The first, _rapid regime shifts_ , are abrupt changes in system
behavior. The second, _bifurcations_ , are qualitative changes in system
behavior due to the passing of a threshold in underlying parameters or
conditions. Where these two overlap, we sometimes call the phenomenon a
“catastrophic bifurcation.” Finally, _critical slowing down_ is the observed
behavior of slow system response to perturbation. The labels in italics
describe examples of phenomena that fall into these various domains. Below we
describe cases that fall into each of these regions.
Figure 1: Venn diagram representing the intersecting domains of rapid regime
shifts, bifurcations, and critical slowing down. Labels in italic are example
phenomena that occur in each domain. Roman numerals and indicate example
literature (right) exploring each domain, and also refer to sections below
describing those domains. Each dot represents a study in the domain. Studies
and dots in grey represent literature not explicitly testing EWS, but which
demonstrate phenomena related to EWS. The center domain (I) where all three
phenomena intersect, is the most extensively researched domain of the EWS
field. Literature outside this charted region does not yet provide the needed
EWS, but hints where existing signals based on CSD may be insufficient or
misleading.
### Catastrophic Bifurcations Preceded by CSD (I)
Much of the (most visible) recent research in EWS has focused on the center of
the diagram, where all three concepts intersect. The warning signal patterns
postulated, such as increasing variance and coefficient of variation,
(Carpenter and Brock 2006), increasing autocorrelation (Dakos et al. 2008),
increasing skewness (Guttal and Jayaprakash 2008a) can all be directly derived
from the changing eigenvalue in a saddle node (also called fold) bifurcation.
Consequently, experimental evaluations of warning signals have largely focused
on this situation as well. CSD has frequently been studied in the context of
models exhibiting saddle-node bifurcations.
Dai et al. (2012) studied yeast cell growth in a microcosm and demonstrated
that an Allee effect created a saddle-node bifurcation in the system. When the
cell density was reduced to levels near the bifurcation point, a decrease in
recovery time (increase in variance and autocorrelation over time) was
observed. Veraart et al. (2012) studied a system of cyanobacteria where models
suggest a saddle-node bifurcation driven by light inhibition. They also found
increases in autocorrelation and decreased recovery rates as the system
approached the bifurcation. These important experiments are among the best
demonstrations that saddle-node bifurcation dynamics really occur in natural
systems, and can be accompanied by reliable detection of EWS, at least when
sufficient data sampling, replicates, and controls are available.
Carpenter et al. (2011) provide a larger-scale example in which a lake
ecosystem is manipulated towards a sudden transition through the introduction
of a predator, while a neighboring experimental lake provides a control. In
this and similar lake systems, bifurcation is thought to be driven in part by
trophic interactions where adult fish prey on the competitors of their
juveniles (Carpenter and Kitchell 1996, Walters and Kitchell 2001, Carpenter
et al. 2008) which leads to a saddle-node bifurcation. While the underlying
dynamics of a whole lake ecosystem are less tractable than the laboratory
controlled chemoststats of microorganisms, the system is understood well
enough to anticipate that a sudden transition can be induced under the
intended manipulation. Like the laboratory examples, this helps eliminate the
options outside the circle “bifurcations,” in Figure 1. The observed warning
signals then place it in the center of the diagram.
These studies have provided valuable demonstrations of the potential to find
early warning signals of sudden transitions. However, this literature has
begun to enumerate examples of similar transitions in which no such signal is
present.
### Catastrophic Bifurcations _not_ Preceded by CSD (II)
Saddle nodes are only one of a variety of bifurcations, which can cause rapid
changes in system dynamics. Other bifurcations can cause long-term changes in
system dynamics without a gradual pass through a state with zero eigenvalue,
and therefore, not exhibit CSD. Many of these examples can in fact show
patterns in typical early warning indicator variables such as variance or
autocorrelation that are completely opposite to the patterns seen in the
saddle-node case. Several of these examples are found outside the EWS
literature, indicating a need to expand the range of systems studied for EWS.
These are some of the most problematic cases. They represent disruptive but
potentially avoidable events, but would not be detected by using CSD as an
EWS. These cases include bifurcations in continuous time (Schreiber and Rudolf
2008) and discrete time (Schreiber 2003), explicitly spatial (Bel et al. 2012)
and non-spatial, chaotic (Schreiber 2003, Hastings and Wysham 2010) and non-
chaotic (Schreiber and Rudolf 2008, Hastings and Wysham 2010, Bel et al. 2012)
examples. Before warning signals can be reliably applied to novel systems,
research must provide a way to discern if the dynamics correspond to the
better understood warning signals of the saddle-node case or the more complex
patterns such as the examples discussed here.
One class of bifurcations in which we would not expect to see CSD prior to
regime shift are sometimes known as _crises_. Crises are sudden changes in the
dynamics of chaotic attractors that occur in response to small changes in
parameters (Grebogi et al. 1983). Chaotic attractors are features of many
ecological models (Hastings et al. 1993), and chaotic behavior has been shown
in some ecological systems (Costantino et al. 1997).
Hastings and Wysham (2010) examined a continuous model of a stochastic three-
species food chain where all species migrate between six patches. When
environmental stochasticity (represented as random variation in the carrying
capacity) is low, all species coexist in a chaotic but stable attractor. A
small increase in environmental stochasticity, though, causes extinction of
the top predator and rapid shift to a non-chaotic cycle. Despite an increase
in environmental variability, neither the variance nor skew of the populations
of any species change as the system approaches this bifurcation.
Another example of a chaotic crisis can be found in a simple discrete-time
model where a population is subject to strong density dependence (an Allee
effect) and harvested by predators with a Type II (saturating) functional
response (Schreiber 2003). This case is illustrated in Figure 2. When prey
have high growth rates, the system has chaotic dynamics. Small increases in
the predation intensity cause a bifurcation with chaotic but persistent prey
populations to prey extinction. As predation intensity increases towards this
threshold, the population exhibits _decreasing_ variance.
Figure 2: A system where variance decreases prior to a population collapse;
adapted from Schreiber (2003). In this model, prey species with high growth
rates exhibit chaotic dynamics under predation, but populations collapse when
predation increases beyond a threshold value. Left: The population level as a
function of predation rate. Mean dynamics shown as black line, realizations
with varying initial conditions shown as grey dots; see Schreiber (2003).
Middle: Variance of the prey population level. Note that it _decreases_ as
predation rate approaches the threshold. Right: Lag-1 Autocorrelation in prey
population dynamics increases as the threshold is approached
Examples are not restricted to chaotic dynamics. An example is found in
Schreiber and Rudolf (2008), in which variance is observed to decrease before
a sudden transition that results in the extinction of the population.
Another non-chaotic example is found in some spatially extended systems that
exhibit a type of bifurcation not accompanied by CSD. In this class of models,
individual locations are subject to saddle node-type regime shifts and
influence adjacent locations via short-range facilitation and long-range
competition. Such models are used represent transitions between vegetation
types in response to changing water availability, and reproduce naturally
occurring vegetation patterns (Rietkerk and van de Koppel 2008). In such
systems, a regime shift in one location can propagate spatially and transition
the whole system from one regime to another. Such a transition occurs if the
control parameter (e.g., rainfall), exceeds the _Maxwell point_ \- the value
at which a local disturbance propagates outwards (Bel et al. 2012). The
Maxwell point may be far from the level at which an individual location would
undergo a saddle-node bifurcation, and thus the system’s global dynamics would
not exhibit CSD prior to such a transition. This case illustrates the
importance of distinguishing between _local_ and _global_ system dynamics and
identifying the appropriate scale of observation.
Finally, Boerlijst et al. (2013) found that indicators of CSD do not appear
prior to saddle-node bifurcations when perturbations are not in the direction
of a system’s dominant eigenvalue, and even then may only appear in one
variable of the system. In their example case, increased variance and
autocorrelation only occurred when noise was applied to the juvenile
population of a model with juveniles, adults, and predators, and it did not
appear when identical noise was applied to all three. When CSD indicators did
appear, they only did so in the juvenile population variables. This represents
another under-explored area - selecting appropriate variables for early-
warning detection in multivariate systems. Even where CSD is present, it may
not be expressed in all system components.
### Non-Catastrophic Bifurcations Preceded by CSD (III)
Not all regime shifts are rapid. Some systems undergo bifurcations between
qualitatively different, but quantitatively similar regimes. These transitions
may be reversible. In a management setting, such qualitative changes may be
gradual, so warning signals that detect such transitions may be effective
“false positives.”
CSD precedes several types of these non-catastrophic bifurcations. In the
subcritical form of a Hopf bifurcation, a system transitions from a stable
equilibrium to a stable cycle. As a control parameter approaches the critical
threshold, the system’s dominant eigenvalue approaches zero and thus exhibits
CSD (Chisholm and Filotas 2009, Kéfi et al. 2012). However, the mean value of
the equilibrium does not change dramatically, and the transition from stable
equilibrium to cycles is gradual as the cycle sizes grow from zero at the
threshold value. To appreciate how this bifurcation is gradual rather than
catastrophic, note that in the presence of stochasticity, the system behavior
observed on either side of the threshold may be indistinguishable: on one side
stochasticity bounces the system around a stable node, while on the other it
bounces the system around a very small limit cycle in the same region of state
space. Even when oscillations grow quickly, returning the environmental
conditions (bifurcation parameter) to the previous conditions restores the
stable node – the bifurcation does not exhibit the hysteresis of the saddle
node bifurcation. Contrast this to a critical transition in which any
stochastic fluctuation across the threshold could lead to a qualitatively
different state.
The system’s eigenvalue also passes through zero in the case of the
transcritical bifurcation. The transcritical is a degenerate case of the
saddle-node, and occurs in many of the same systems. However, when a system
passes through a transcritical bifurcation, the stable equilibrium transitions
smoothly from positive to zero, or the reverse. In population systems, this
corresponds to a transition from an equilibrium of a very small population
size to extinction - an important but non-catastrophic, and probably directly
observable, event. CSD is observed prior to the transcritical bifurcations
(Chisholm and Filotas 2009, Kéfi et al. 2012).
An experimental example of a transcritical bifurcation is found in Drake and
Griffen (2010), where a population of _Daphnia_ was forced through a
transcritical bifurcation by reducing food supplies and driving population
growth rates below zero. Indicators of CSD (variation, skewness,
autocorrelation, and spatial correlation) increased prior to collapse of the
population.
### CSD in the absence of bifurcations or regime shifts. (IV)
Critical slowing down may appear in systems without any bifurcations. Kéfi et
al. (2012) showed that smooth transitions that modify a system’s potential and
decrease the value of its dominant eigenvalue would result in longer return
times and greater variance and autocorrelation in system behavior (See Figure
3). When the transition between states is smooth, these measures will exhibit
a smooth increase to a maximum and then a decrease, unlike the sharp peaks
found in systems with bifurcations. Nonetheless, both exhibit increasing
measures of CSD that may be indistinguishable.
Figure 3: A system where critical slowing down is observed without a critical
threshold, from Kéfi et al. (2012). In this model, prey have logistic growth
and are subject to predation with a Type III functional response, but there is
no bifurcation. Instead, average prey population exhibits a smooth response to
increased predation (grazing). Left: The population level as a function of
predation rate. Middle: Variance of the prey population level. Right: Lag-1
Autocorrelation in prey population dynamics as grazing rate increases. Note
that both indicators increase despite the lack of a bifurcation.
### Catastrophic Regime Shifts without Bifurcations or CSD (V)
Some rapid regime shifts are not due to bifurcations at all. A large external
forcing (as illustrated in Figure 4) may change the behavior of a system
without any warning. This mechanism is commonly recognized, (Scheffer et al.
2001, 2009, Barnosky et al. 2012, Scheffer et al. 2012), but others are
possible. An internal stochastic event may switch a system between dynamic
regimes, or a change in system behavior may be the manifestation of a long-
term transient. In none of these cases would CSD be expected to precede such
changes. Nonetheless, it may be difficult to distinguish such cases from
bifurcations.
Figure 4: Difference between different types of perturbations. On the
horizontal axis is the bifurcation parameter, representing the state of the
environment (e.g. annual mean temperature) whose slow change could lead to a
sudden shift. A direct disturbance to the system state (e.g. population size,
vertical axis) could also cause a transition if it is large enough to cross
the stability threshold (dashed line). Such a perturbation can come from
exogenous factors such as anthropogenic pressures or occur by chance from
intrinsic stochasticity. These distinct mechanisms of disturbance and
environmental change are coupled – as the environment deteriorates, moving the
system right on the diagram, the probability that a disturbance crosses the
threshold increases. From Bel et al. (2012).
Large, rapid changes in external conditions will result in rapid changes in
ecological system dynamics. For instance, rapid changes in North American
vegetation at the start of the Bølling-Allerød and end of the Younger Dryas
period are thought to be responses to similarly large, rapid changes in
climate (Williams et al. 2011). Doney and Sailley (2013) interpret a recent
analysis by Di Lorenzo and Ohman (2013) as demonstrating that what were
previously thought of as regime shifts in krill dynamics in the Pacific ocean
(Hare and Mantua 2000) could actually be explained by a close coupling to the
external forcing of El Nino environmental dynamics through the Pacific Decadal
Oscillation (PDO). Schooler et al. (2011) found that lakes with the invasive
plant _Salvaniai molesta_ and herbivorous weevils alternated between low- and
high-_Salvnia_ states driven by disturbances from regular external flooding
events. These examples highlight cases that involve critical transitions
between regimes under circumstances that do not permit the discovery of early
warning signals, as CSD is not anticipated under these mechanisms.
Internally-driven stochastic perturbations may shift systems from one state to
another even if underlying environmental conditions remain the same. In such
conditions EWS would not be expected. Hastings and Wysham (2010) showed that
in a model where one species with stochastic Ricker dynamics disperses among
eight patches, model behavior can switch stochastically between wildly
oscillatory behavior and regularly cycling regimes even while parameters
(including stochastic variability) remain the same. Ditlevsen and Johnsen
(2010) examined 25 abrupt climate changes that occurred during the last
glacial period (Dansgaard-Oeschger events) and found no evidence for CSD in
high-resolution climate data from ice cores, and concluded that the events
were driven by endogenous climate stochasticity rather than regime shifts
(though see Cimatoribus et al. 2013 for an alternative conclusion).
Some events that appear to be regime shifts may actually be transients in some
systems. Sudden changes in dynamics can occur in simple ecological models with
strong density dependence that take long times to reach equilibrium. Hastings
(1998) showed such dynamics in model of dispersal of inter- or sub-tidal
organisms whose larvae disperse along a coastline. Over the thousands of years
it takes the model to reach equilibrium, it may alternate between temporary
regimes of regular cycles and chaos that switch in only a few years. While on
long time scales these are technically not regime shifts, such changes would
effectively appear to be regime shifts on shorter ones. We would not expect
such regime shifts to be preceded with CSD.
Of course, stochastically-driven regime shifts may occur in systems where
bifurcations are also possible, and it may be difficult to distinguish between
the two. Renne et al. (2013), for example suggest that ecosystems were under
near-critical stress due to climate changes just prior to the Chicxulub meteor
impact, which resulted in mass extinction. In such a case, EWS may precede the
regime shift even if it is ultimately triggered by a stochastic event.
## Statistical problems in detecting early warning signals
The above cases show that behavior providing EWS before regime shifts may only
be present in certain types of ecological systems (e.g. see the conditions
outlined in Scheffer et al. 2009). An additional important consideration is
whether these behaviors will be _detectable_. To be usable as EWS, system
behavior must be detectable well enough in advance of a regime shift to serve
in decision-making, and be reliably distinguishable from other patterns.
Ecological data is often sparse, noisy, autocorrelated and subject to
confounding driving variables, in contrast to much of the experimental or
simulated data used to test EWS. Under common levels of noise found in field
data, CSD-based EWS often fail (Perretti and Munch 2012).
A wide variety of statistical summary indicators have been examined as
potential detectors of CSD. The most common are variance and autocorrelation.
Others include skewness (Guttal and Jayaprakash 2008a) and conditional
heteroscedasticity (Seekell et al. 2011). These statistics are typically
calculated on sliding windows of time-series data and tested formally or
informally for trends. The relative power of these tests varies considerably
with context; no indicator has consistently outperformed others (Dakos et al.
2011b, 2012, Lindegren et al. 2012, Perretti and Munch 2012). Also, measuring
these indicators requires making sometimes arbitrary calculations. For
instance, the power of lag-1 autocorrelation to detect a regime shift may be
modified by changing methods of data aggregation, de-trending, changing
sliding window length, filtering signal bandwidth (Lenton et al. 2012). These
choices may be optimized when enough calibration data is available, as Lenton
et al. (2012) were able to do with several sets of paleoclimate data. However,
such calibration may not be possible with many ecological datasets. Multiple-
method (Lindegren et al. 2012) and composite indices (Drake and Griffen 2010)
have been proposed, but their power relative to other indicators is unknown.
Another approach to detecting CSD has been fitting time series data to models.
Two approaches have been used for these model-based methods. First, models may
be used to calculate summary statistics related to CSD, such as eigenvalues
(Lade and Gross 2012) or diffusion terms in jump-diffusion models (Carpenter
2011, Brock and Carpenter 2012). These statistics are then examined for trends
in the same fashion as the summary statistics above. Alternatively, models
representing both deteriorating and stable conditions may be fit to the data
and in order to determine which is more likely (Dakos et al. 2012, ).
Boettiger and Hastings (2012a) found that likelihood ratio tests were more
powerful than trend-based summary statistic tests across several real and
simulated ecological data sets. This approach is also more robust than
summary-statistic methods to spurious correlations that arise when collapses
are driven by purely stochastic events (Boettiger and Hastings 2012a).
Care is required in the criteria used to judge the power of warning signal
methods. The trade-off between false negatives and false positives is a matter
of not just statistical but economic efficiency. For instance, a large number
of false positives may be acceptable if they reduce the probability of a false
warning that would result in an otherwise avoidable catastrophic regime shift,
and the costs of failing to detect such a shift exceed that of the false
positives. Boettiger and Hastings (2012a) suggest the use of receiver-
operating characteristic (ROC) curves to describe the performance of various
EWS. ROC curves (Figure 5) represent the false positive rate at any true
positive rate. The area under the curve (AUC) is a useful metric of overall
performance. AUC will be one if the signal is perfect and 0.5 if the signal
performs no better than random. The complete shape of the curve provides more
information on the possible trade-offs under different sensitivities. This
information, combined with a decision-theoretic framework, has the potential
to illuminate the cases in which EWS can be useful.
Figure 5: Receiver-operating characteristic (ROC) curves illustrate the trade-
off between false positive and true positive detection rates of an EWS.
Perfect warning signals (solid curve) would identify all thresholds while
generating no false positives, while very poor signals would have no ability
to distinguish false from true signals (dotted line). In reality, warning
signals’ have a trade-off between the two which is described by a curve
(dotted line) or summarized by the area under the ROC curve
## Discussion
Recognizing the potential for early warning signals of critical transitions
represents a substantial leap forward in addressing one of the most
challenging questions in ecology and ecosystem management today. In the
decades prior, the prospect that ecosystems could make sudden transitions into
an undesirable state due to gradual, slow changes in their environment hung
like a specter over both our understanding and management of natural systems.
Research that points to the possibility of detecting these transitions holds
the promise of meeting this challenge and has attracted justifiably widespread
attention among both theoretical and empirical communities. Nonetheless, our
understanding of early warning signals is still in its infancy. Thus far, our
best understanding and empirical experience lies in transitions that are
driven by saddle-node bifurcations.
While saddle-node bifurcations may be common, they represent only part of the
potential mechanisms for rapid regime shift. Occupying the center of our
diagram, Figure 1, such transitions represent our best-understood cases.
Researchers have relied on existing expertise and prior research to identify
empirical systems most likely to experience critical transitions through the
saddle-node-like mechanism (e.g. Carpenter et al. 2011, Dai et al. 2012), and
have achieved a close match to theoretical predictions of early warning
signals. While these examples provide a much needed proof-of-principle that
these signals can be detected in the real world, it is too early to apply the
same methods to novel systems where the saddle-node is only one of many
possible mechanisms. We are not yet able to determine if a natural system is
likely to have a saddle-node bifurcation without detailed study, despite the
popularity of saddle-node models.
Thus, establishing the saddle node mechanism is a necessary condition of using
CSD as a warning signal. This can be done via manipulation in simple
experimental systems (Veraart et al. 2012, Dai et al. 2012), but this is
impractical in most natural systems. Another approach is to assume the saddle-
node mechanism applies to a limited set of systems that have well-studied
examples, such as lakes undergoing eutrophication (Scheffer et al. 2001),
lakes with ‘trophic-triangle’ cascade mechanisms (Carpenter and Kitchell 1996,
Walters and Kitchell 2001, Carpenter et al. 2008), forest/savannah transitions
(Staver et al. 2011, Hirota et al. 2011), and rangeland transitions
(Walker1993; Anderies et al. 2002). Fitting simplified saddle-node models to
past regime shifts (Boettiger and Hastings 2012a) in less well-understood
systems may provide evidence for the mechanism. However, care must be taken to
specify sufficient alternative models.
CSD alone cannot be used as evidence of regime shifts. In some cases, it will
be present when no transition is approaching. In other cases, regime shifts
occur without CSD. Though false alarms and missed events can occur in any
statistical procedure, the cases discussed here demonstrate that these errors
will also arise when the underlying dynamics do not correspond to our
assumptions. These situations fall in the uncharted area beyond the center of
Figure 1, where research has just begun to illuminate their existence and
properties. A better theoretical and empirical understanding of these cases
will allow us to construct novel warning signals, that may be opposite the
patterns observed in the familiar saddle-node bifurcations. Before early
warning signals can be applied in novel systems, additional information is
needed in order to determine the best signal to use.
One area that requires further exploration is the effect of different forms of
stochasticity on the existence of EWS and signal detectability. Many processes
contribute to stochastic behavior in ecological systems, and different forms
of stochasticity have different effects on system behavior far beyond greater
variance (Melbourne and Hastings 2008). Hastings and Wysham (2010) argued that
most examples of detectable CSD indicators were found in models with additive
stochasticity and smooth potentials. Boerlijst et al. (2013), however, found
that stochasticity had the same effects whether it was additive or included in
the population growth rate. Instead, they found the _direction_ of stochastic
perturbations relative to the system’s eigenvalue determined whether CSD
indicators were detectable. The form of stochasticity may be important in the
detectability of CSD indicators even where CSD is present, because stochastic
perturbations are needed to explore system state-space, while at the same time
can reduce the statistical power. More work such as Perretti and Munch (2012),
which examined the role of noise color in detecting CSD, will be useful.
Another area that has is understanding how the relationship between the scale
of observation and the scale of ecological processes affect the efficacy of
EWS. As shown by the Maxwell Point example in Bel et al. (2012), EWS which
detect local bifurcations may not detect global bifurcations in system
behavior. The scale of observation likely also will affect the statistical
power of EWS. Similarly, as illustrated in Boerlijst et al. (2013), the choice
of variables to observe in multivariate systems is important, but little is
known about how to select the appropriate variable for detecting EWS.
The future of early warning signals lies in the uncharted territory. For
certain classes of transitions, such as stochastically-driven regime shifts,
prediction may not be possible. In such cases, management options include
optimizing outcomes despite the possibility of regime shifts, or possibly
taking actions to reduce the long-term probability of regime shifts, despite
short-term unpredictability. Likewise, regime shifts driven by external
perturbation or strong forcing are not predictable _if_ the scope of
management does not include the external causes. Proper scoping of the
management problem can avoid this situation (Fischer et al. 2009, Alliance
2010, Polasky et al. 2011). More research is needed in methods of
distinguishing such cases from those in which early detection may be possible.
For other classes of transitions, prediction may be possible but other EWS
must be explored. Flickering (Brock and Carpenter 2010, Wang et al. 2012), or
rapid transitions between states prior to a more permanent transition, is one
signal that may apply across many types of systems. It manifests in bi-
modality and high variance in times series. Spatial pattern development may be
a warning signal in systems with short-distance positive feedbacks but long-
distance negative feedbacks, such as grassland-desert transitions (Rietkerk et
al. 2004). Other spatial signals may apply where systems include both saddle
nodes and positive feedbacks across space (Litzow et al. 2008, Guttal and
Jayaprakash 2008a, Dakos et al. 2009, Bailey 2010, Dakos et al. 2011b, Bel et
al. 2012). A critical task for EWS research is to map these signals to their
domains of applicability, and create methods to establish if ecosystems fall
into these domains.
## Acknowledgments
This work was partially supported by the Center for Stock Assessment Research,
a partnership between the University of California Santa Cruz and the
Fisheries Ecology Division, Southwest Fisheries Science Center, Santa Cruz,
CA, to CB; the NSF Integrative Graduate Education and Research Traineeship
Program to NR and by funding from NSF Grant EF 0742674 to AH.
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|
arxiv-papers
| 2013-05-29T06:00:23 |
2024-09-04T02:49:45.839422
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Carl Boettiger, Noam Ross, Alan Hastings",
"submitter": "Carl Boettiger",
"url": "https://arxiv.org/abs/1305.6700"
}
|
1305.6866
|
R.R. KamalianExamples of cyclically-interval non-colorable bipartite graphs
cyclically-interval edge coloring, bipartite graph 05C15 For an undirected,
simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets
of its vertices and edges, respectively. A function
$\varphi:E(G)\rightarrow\\{1,2,\ldots,t\\}$ is called a proper edge
$t$-coloring of a graph $G$ if adjacent edges are colored differently and each
of $t$ colors is used. An arbitrary nonempty subset of consecutive integers is
called an interval. If $\varphi$ is a proper edge $t$-coloring of a graph $G$
and $x\in V(G)$, then $S_{G}(x,\varphi)$ denotes the set of colors of edges of
$G$ which are incident with $x$. A proper edge $t$-coloring $\varphi$ of a
graph $G$ is called a cyclically-interval $t$-coloring if for any $x\in V(G)$
at least one of the following two conditions holds: a) $S_{G}(x,\varphi)$ is
an interval, b) $\\{1,2,\ldots,t\\}\setminus S_{G}(x,\varphi)$ is an interval.
For any $t\in\mathbb{N}$, let $\mathfrak{M}_{t}$ be the set of graphs for
which there exists a cyclically-interval $t$-coloring, and let
$\mathfrak{M}\equiv\bigcup_{t\geq 1}\mathfrak{M}_{t}.$
Examples of bipartite graphs that do not belong to the class $\mathfrak{M}$
are constructed. R.R. Kamalian 2013 R.R. Kamalian
Institute for Informatics and Automation Problems National Academy of Sciences
of RA, 0014 Yerevan, Republic of Armenia
E-mail: [email protected]_
# Examples of cyclically-interval non-colorable bipartite graphs
R.R. Kamalian
## 1 Introduction
We consider undirected, simple, finite, and connected graphs. For a graph $G$
we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges,
respectively. For a graph $G$, we denote by $\Delta(G)$ and $\chi^{\prime}(G)$
the maximum degree of a vertex of $G$ and the chromatic index of $G$ [14],
respectively. The terms and concepts which are not defined can be found in
[17].
For an arbitrary finite set $A$, we denote by $|A|$ the number of elements of
$A$. The set of positive integers is denoted by $\mathbb{N}$. An arbitrary
nonempty subset of consecutive integers is called an interval. An interval
with the minimum element $p$ and the maximum element $q$ is denoted by
$[p,q]$.
For any $t\in\mathbb{N}$ and arbitrary integers $i_{1},i_{2}$ satisfying the
conditions $i_{1}\in[1,t]$, $i_{2}\in[1,t]$, we define [8, 9] the sets
$intcyc_{1}[(i_{1},i_{2}),t],$ $intcyc_{1}((i_{1},i_{2}),t),$
$intcyc_{2}((i_{1},i_{2}),t),$ $intcyc_{2}[(i_{1},i_{2}),t]$ as follows:
$intcyc_{1}[(i_{1},i_{2}),t]\equiv[\min\\{i_{1},i_{2}\\},\max\\{i_{1},i_{2}\\}],$
$intcyc_{1}((i_{1},i_{2}),t)\equiv
intcyc_{1}[(i_{1},i_{2}),t]\backslash(\\{i_{1}\\}\cup\\{i_{2}\\}),$
$intcyc_{2}((i_{1},i_{2}),t)\equiv[1,t]\backslash
intcyc_{1}[(i_{1},i_{2}),t],$
$intcyc_{2}[(i_{1},i_{2}),t]\equiv[1,t]\backslash
intcyc_{1}((i_{1},i_{2}),t).$
If $t\in\mathbb{N}$ and $Q$ is a non-empty subset of the set $\mathbb{N}$,
then $Q$ is called a $t$-cyclic interval if there exist integers
$i_{1},i_{2},j_{0}$ satisfying the conditions $i_{1}\in[1,t]$,
$i_{2}\in[1,t]$, $j_{0}\in\\{1,2\\}$, $Q=intcyc_{j_{0}}[(i_{1},i_{2}),t]$.
A function $\varphi:E(G)\rightarrow[1,t]$ is called a proper edge $t$-coloring
of a graph $G$ if adjacent edges are colored differently and each of $t$
colors is used.
For a graph $G$ and a positive integer $t$, where $\chi^{\prime}(G)\leq
t\leq|E(G)|$, we denote by $\alpha(G,t)$ the set of all proper edge
$t$-colorings of $G$. Let us set
$\alpha(G)\equiv\bigcup_{t=\chi^{\prime}(G)}^{|E(G)|}\alpha(G,t).$
If $G$ is a graph, $\varphi\in\alpha(G)$, and $x\in V(G)$, then the set
$\\{\varphi(e)/e\in E(G),e\textrm{ is incident with }x\\}$ is denoted by
$S_{G}(x,\varphi)$.
A proper edge $t$-coloring $\varphi$ of a graph $G$ is called a cyclically-
interval $t$-coloring of $G$, if for any $x\in V(G)$ at least one of the
following two conditions holds: a) $S_{G}(x,\varphi)$ is an interval, b)
$[1,t]\setminus S_{G}(x,\varphi)$ is an interval.
For any $t\in\mathbb{N}$, we denote by $\mathfrak{M}_{t}$ the set of graphs
for which there exists a cyclically-interval $t$-coloring. Let
$\mathfrak{M}\equiv\bigcup_{t\geq 1}\mathfrak{M}_{t}.$
For an arbitrary tree $D$, it was shown in [8, 9] that $D\in\mathfrak{M}$,
and, moreover, all possible values of $t$ were found for which
$D\in\mathfrak{M}_{t}$. For an arbitrary simple cycle $C$, it was shown in [7,
10] that $C\in\mathfrak{M}$, and, moreover, all possible values of $t$ were
found for which $C\in\mathfrak{M}_{t}$. Some interesting results on this and
related topics were obtained in [1, 3, 4, 15, 16, 11, 12, 13, 2, 5, 6].
In this paper, the examples of bipartite graphs that do not belong to the
class $\mathfrak{M}$ are constructed.
For any integer $m\geq 2$, set:
$V_{0,m}\equiv\\{x_{0}\\},\qquad V_{1,m}\equiv\\{x_{i,j}/\;1\leq i<j\leq
m\\},$ $V_{2,m}\equiv\\{y_{p,q}/\;1\leq p\leq m,1\leq q\leq m\\},$
$E^{\prime}_{m}\equiv\\{(x_{0},y_{p,q})/\;1\leq p\leq m,1\leq q\leq m\\}.$
For any integers $i,j,m$ satisfying the inequalities $m\geq 2$, $1\leq i<j\leq
m$, set:
$E^{\prime\prime}_{i,j,m}\equiv\\{(x_{i,j},y_{i,q})/\;1\leq q\leq
m\\}\cup\\{(x_{i,j},y_{j,q})/\;1\leq q\leq m\\}.$
For any integer $m\geq 2$, let us define a graph $G(m)$ by the following way:
$G(m)\equiv\Bigg{(}\bigcup_{k=0}^{2}V_{k,m},E^{\prime}_{m}\cup\Bigg{(}\bigcup_{1\leq
i<j\leq m}E^{\prime\prime}_{i,j,m}\Bigg{)}\Bigg{)}.$
It is not difficult to see that for any integer $m\geq 2$, $G(m)$ is a
bipartite graph with $\Delta(G(m))=\chi^{\prime}(G(m))=m^{2}$,
$|V(G(m))|=\frac{3m^{2}-m}{2}+1$, $|E(G(m))|=m^{3}$.
###### Theorem 1.
For any integer $m\geq 8$, $G(m)\not\in\mathfrak{M}$.
###### Proof 1.1.
Assume the contrary. It means that there exist integers $m_{0},t_{0},k_{0}$,
satisfying the conditions $m_{0}\geq 8$, $m_{0}^{2}\leq t_{0}\leq m_{0}^{3}$,
$t_{0}=m_{0}^{2}+k_{0}$, $0\leq k_{0}\leq m_{0}^{3}-m_{0}^{2}$,
$G(m_{0})\in\mathfrak{M}_{t_{0}}$.
Let $\varphi_{0}$ be a cyclically-interval $t_{0}$-coloring of the graph
$G(m_{0})$. Without loss of generality, we can suppose that
$S_{G(m_{0})}(x_{0},\varphi_{0})=[1,m_{0}^{2}]$. Let us consider the edges
$e^{\prime}$ and $e^{\prime\prime}$ of the graph $G(m_{0})$, which are
incident with the vertex $x_{0}$ and satisfy the equalities
$\varphi_{0}(e^{\prime})=1$,
$\varphi_{0}(e^{\prime\prime})=\lfloor\frac{m_{0}^{2}}{2}\rfloor$.
Suppose that $e^{\prime}=(x_{0},y^{\prime})$,
$e^{\prime\prime}=(x_{0},y^{\prime\prime})$. Clearly, there exists a vertex
$\widetilde{x}\in V_{1,m_{0}}$ in the graph $G(m_{0})$ which is adjacent to
the vertices $y^{\prime}$ and $y^{\prime\prime}$. It is not difficult to see
that $S_{G(m_{0})}(y^{\prime},\varphi_{0})\cup
S_{G(m_{0})}(\widetilde{x},\varphi_{0})\cup
S_{G(m_{0})}(y^{\prime\prime},\varphi_{0})$ is a $t_{0}$-cyclic interval.
Clearly, the inequalities $m_{0}^{2}+k_{0}-4m_{0}+4>4m_{0}-2$ and
$4m_{0}-1\leq\lfloor\frac{m_{0}^{2}}{2}\rfloor\leq m_{0}^{2}+k_{0}-4m_{0}+3$
are true. Consequently, $\lfloor\frac{m_{0}^{2}}{2}\rfloor\in
intcyc_{1}((4m_{0}-2,m_{0}^{2}+k_{0}-4m_{0}+4),m_{0}^{2}+k_{0})$. But it is
incompatible with the evident relations $\lfloor\frac{m_{0}^{2}}{2}\rfloor\in
S_{G(m_{0})}(y^{\prime\prime},\varphi_{0})$ and
$S_{G(m_{0})}(y^{\prime},\varphi_{0})\cup
S_{G(m_{0})}(\widetilde{x},\varphi_{0})\cup
S_{G(m_{0})}(y^{\prime\prime},\varphi_{0})\subseteq
intcyc_{2}[(4m_{0}-2,m_{0}^{2}+k_{0}-4m_{0}+4),m_{0}^{2}+k_{0}]$.
Contradiction.
The author thanks P.A. Petrosyan for his attention to this work.
## References
* [1] Altinakar S., Caporossi G., Hertz A. On compact $k$-edge-colorings: A polynomial time reduction from linear to cyclic. Discrete Optimization 8(2011), 502–512.
* [2] Asratian A.S. Investigation of some mathematical model of scheduling theory. Doctoral Thesis, Moscow, 1980 (in Russian).
* [3] Bartholdi J.J., Orlin J.B., Ratliff H.D. Cyclic scheduling via integer programs with circular ones. Operations Research 28, 1980, 1074–1085.
* [4] Dauscha W., Modrow H.D., Neumann A. On cyclic sequence type for constructing cyclic schedules. Zeitschrift für Operations Research 29, 1985, 1–30.
* [5] Jensen T.R., Toft B. Graph Coloring Problems. Wiley Interscience Series in Discrete Mathematics and Optimization, 1995.
* [6] Kamalian R.R. Interval Edge Colorings of Graphs. Doctoral dissertation. The Institute of Mathematics of the Siberian Branch of the Academy of Sciences of USSR, Novosibirsk, 1990 (in Russian).
* [7] Kamalian R.R. On cyclically continuous edge colorings of simple cycles. Proceedings of the CSIT Conference, Yerevan, 2007, 79–80 (in Russian).
* [8] Kamalian R.R. On a number of colors in cyclically interval edge colorings of trees. Research report LiTH-MAT-R-2010/09-SE, Linköping University, 2010.
* [9] Kamalian R.R. On cyclically-interval edge colorings of trees. Buletinul of Academy of Sciences of the Republic of Moldova, Matematica 1(68), 2012, 50–58.
* [10] Kamalian R.R. On a number of colors in cyclically-interval edge-colorings of simple cycles. Open Journal of Discrete Mathematics, 3(2013), 43–48.
* [11] Kubale M. Graph Colorings. American Mathematical Society, 2004.
* [12] Kubale M., Nadolski A. Chromatic scheduling in a cyclic open shop. European Journal of Operational Research 164(2005), 585–591.
* [13] Nadolski A. Compact cyclic edge-colorings of graphs. Discrete Mathematics 308(2008), 2407–2417.
* [14] Vizing V.G. The chromatic index of a multigraph. Kibernetika 3 (1965), 29–39.
* [15] de Werra D., Mahadev N.V.R., Solot Ph. Periodic compact scheduling. ORWP 89/18, Ecole Polytechnique Fédérale de Lausanne, 1989.
* [16] de Werra D., Solot Ph. Compact cylindrical chromatic scheduling. ORWP 89/10, Ecole Polytechnique Fédérale de Lausanne, 1989.
* [17] West D.B. Introduction to Graph Theory. Prentice-Hall, New Jersey, 1996.
|
arxiv-papers
| 2013-05-29T16:53:38 |
2024-09-04T02:49:45.852444
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R.R. Kamalian",
"submitter": "Rafayel Kamalian",
"url": "https://arxiv.org/abs/1305.6866"
}
|
1305.6962
|
# Mechanical resonators for storage and transfer of electrical and optical
quantum states
S. A. McGee1, D. Meiser1,2, C. A. Regal1, K. W. Lehnert1,3, and M. J. Holland1
1JILA and Department of Physics, University of Colorado, Boulder CO
80309-0440, USA.
2Tech-X Corp., 5621 Arapahoe Ave. Ste. A, Boulder, CO 80303, USA.
3National Institute of Standards and Technology, Boulder CO 80309, USA.
###### Abstract
We study an optomechanical system in which a microwave field and an optical
field are coupled to a common mechanical resonator. We explore methods that
use these mechanical resonators to store quantum mechanical states and to
transduce states between the electromagnetic resonators from the perspective
of the effect of mechanical decoherence. Besides being of fundamental
interest, this coherent quantum state transfer could have important practical
implications in the field of quantum information science, as it potentially
allows one to overcome intrinsic limitations of both microwave and optical
platforms. We discuss several state transfer protocols and study their
transfer fidelity using a fully quantum mechanical model that utilizes quantum
state-diffusion techniques. This work demonstrates that mechanical decoherence
should not be an insurmountable obstacle in realizing high fidelity storage
and transduction.
## I Introduction
Recent experiments have demonstrated the ability to control mesoscopic
mechanical resonators near the quantum limit Teufel et al. (2011a); Chan et
al. (2011); Verhagen et al. (2012); O Connell et al. (2010). This achievement
provides novel opportunities for fundamental physics Kleckner et al. (2008)
and a technology for engineering quantum systems Stannigel et al. (2012); Hong
et al. (2012). The mechanical resonators are formally equivalent to
electromagnetic resonators, which form basic elements of quantum optics, but
offer many new and unique opportunities. Mechanical resonators are massive
objects that can be coaxed into interacting strongly with many different
systems. For instance, in experiments to date, mesoscopic mechanical objects
have been coupled to electrical and optical photons in cavities, although not
to both simultaneously. As proposed in Ref. Regal and Lehnert (2011), it may
be possible in the near future to couple a mechanical resonator to both
electrical and optical cavities at the same time (Fig. 1). Such an interface
would provide a way to connect quantum resources that are more suited for
creating and manipulating quantum states (i.e. electrical circuits) Hofheinz
et al. (2009) to resources that are more suited for transmitting quantum
states (i.e. optical platforms).
Figure 1: (Color online) A schematic diagram for an optical cavity coupled to
a thin dielectric membrane mechanical resonator, which in turn is coupled to a
resonant electrical LC-circuit. The system can be pumped or read out by both
microwave and optical drives.
Given this emerging possibility, an important question is how to harness
mechanical resonators within electromagnetic cavities to transduce and store
quantum states. By transduction, we mean the transfer of energy between
distinct degrees of freedom; in this case, between electromagnetic oscillators
whose frequencies are separated by many orders of magnitude. The ability to
strongly couple single photons in optomechanical systems would open up a wide
variety of quantum protocols Rabl (2011). However, in order to achieve
sufficient coupling, experiments mainly operate the electromagnetic-mechanical
interface in an analogous way to three-wave mixing in nonlinear optics Zhang
et al. (2003); Akram et al. (2010). A strong pump-tone red-detuned from the
cavity is introduced to bridge most of the energy gap between the
electromagnetic and mechanical oscillators. This produces an effective beam-
splitter interaction that can conveniently be turned on and off by varying the
pump-tone intensity. Single-photon states detuned from the pump can then be
transduced between mechanical excitations and any number of electromagnetic
modes.
The quantum optomechanics experiments envisioned here are thus rooted in the
well developed toolbox associated with two-mode quantum optics. However, when
we introduce low-frequency mechanical resonators, the presence of a thermal
bath damping and exciting the phonon resonances must be accounted for in the
theoretical analysis. To create a versatile interface between microwave and
optical photons, we consider use of a megahertz membrane microresonator Teufel
et al. (2011a). Despite recent progress toward bringing such mechanical
systems to the quantum regime, the mechanical decoherence rate, which is
proportional to the product of the mechanical resonator line strength and
occupation number of thermal bath phonons, remains a significant decay
pathway.
In this paper, we consider the effect of decoherence on the application of
mechanical resonators to store quantum mechanical states and to transduce
states between electromagnetic resonators. Our analysis is based on a quantum
state diffusion (QSD) method Wiseman and Milburn (1993) for solving the
evolution of open quantum systems. This method provides an exact unraveling of
the quantum master equation into parallel pure-state quantum trajectories.
Using the QSD method we are able to calculate the fidelity of this system for
quantum state memory applications and quantum state transduction.
It should be emphasized that this numerical method is not restricted to
solving the dynamical evolution of only Gaussian or classical states. The QSD
method allows for any quantum input state to be tested. In this paper, we
compare the memory and transduction fidelity for coherent states, squeezed-
states, Schrödinger-cat states, and nonclassical superpositions of Fock
states. Such numerical approaches also allow for the analysis of many types of
coupling schemes. The simplest scheme is the coherent swapping of the quantum
state of two oscillators at a transfer rate determined by the strength of the
coupling. This is the optomechanics analogue of Rabi flopping between internal
states of a two-level atom. In addition to this scheme Tian and Wang (2010),
we explore several more diverse swapping schemes and find them to be more
robust against the omnipresent mechanical decoherence Wang and Clerk (2012a,
b); Tian (2012).
As we will show, the basic system we consider formally corresponds precisely
to a set of adjustable beam-splitters and cavities as illustrated in Fig. 2.
The optomechanical and electromechanical coupling strength is represented by
the reflectivity of the effective beam splitters and can be adjusted by
variation of external parameters to be anywhere from 0 to $100\%$. We first
look at the system from the perspective of utilizing this beam-splitter
interaction to facilitate the storage and retrieval of an electromagnetic
quantum state in a mechanical resonator. Second, we look at the system from
the perspective of transduction of a quantum state from a microwave to optical
resonator, or vice versa. We investigate the effect of different protocols on
the population of the mechanical state and, hence, the susceptibility to
mechanical decoherence.
Figure 2: (Color online) An equivalent system of two coupled adjustable beam
splitters which can be a formal analogue to the two-mode optomechanical system
in the linearized approximation. In this open quantum system, each oscillator
is also coupled to its respective reservoir (not shown).
In all cases, we set the decay of the optical and microwave cavities equal to
zero and consider the state preparation of the optical and microwave modes as
an initial condition. This simplifies the analysis and allows us to focus on
the role of mechanical decoherence. Nonetheless, the internal and external $Q$
of the optical and microwave cavities is also an important topic, and has
recently been treated in both the context of swapping Tian and Wang (2010) and
itinerant photon schemes Wang and Clerk (2012a); Safavi-Naeini and Painter
(2011); Wang and Clerk (2012b).
## II Theory
The coupled electro-opto-mechanical system Zhang et al. (2003); Akram et al.
(2010) is described by the Hamiltonian in the Schrödinger picture,
$\hat{H}=\hat{H}_{\text{self}}+\hat{H}_{\text{coupling}}+\hat{H}_{\text{pump}}$,
where:
$\displaystyle\hat{H}_{\text{self}}$ $\displaystyle=$
$\displaystyle\hbar\omega_{o,c}\
\hat{a}^{{\dagger}}\hat{a}+\hbar\omega_{\mu,c}\
\hat{b}^{{\dagger}}\hat{b}+\hbar\omega_{m}\hat{d}^{{\dagger}}\hat{d}\;,$
$\displaystyle\hat{H}_{\text{coupling}}$ $\displaystyle=$
$\displaystyle-\frac{\hbar
X_{\text{ZP}}}{2}\left(\hat{d}+\hat{d}^{{\dagger}}\right)\left(g_{o}\hat{a}^{{\dagger}}\hat{a}+g_{\mu}\hat{b}^{{\dagger}}\hat{b}\right)\;,$
$\displaystyle\hat{H}_{\text{pump}}$ $\displaystyle=$
$\displaystyle\hbar\left(\hat{a}A_{o}^{*}e^{i\omega_{o}t}+\hat{a}^{{\dagger}}A_{o}e^{-i\omega_{o}t}\right)$
(1)
$\displaystyle{}+\hbar\left(\hat{b}A_{\mu}^{*}e^{i\omega_{\mu}t}+\hat{b}^{{\dagger}}A_{\mu}e^{-i\omega_{\mu}t}\right)\;.$
Here the operators $\hat{a}$, $\hat{b}$, and $\hat{d}$ are the annihilation
operators for a photon in the optical cavity, a photon in the microwave
cavity, and a phonon in the mechanical resonator, respectively. The system is
driven by classical pump fields with frequencies $\omega_{(o,\mu)}$ and the
bare (i.e. uncoupled) cavity resonance frequencies are $\omega_{(o,\mu),c}$,
where $o$ stands for optical and $\mu$ stands for microwave. For the
mechanical resonator, the resonance frequency is $\omega_{m}$ and the harmonic
oscillator length is $X_{\text{ZP}}$. The coupling constants $g_{o}$ and
$g_{\mu}$ are physically determined by the amount of the shift of the resonant
frequency of each cavity with respect to changes in the mechanical resonator
position.
The strong coherent pump amplitudes, $A_{o}$ and $A_{\mu}$, lead to a buildup
of large steady-state fields in the optical and microwave resonators, whose
purpose is to increase the optomechanical coupling (a fact which we
demonstrate below). The resulting steady-state intracavity field amplitudes in
turn shift the equilibrium position of the mechanical resonator through the
radiation pressure force. We begin by finding these steady-state semiclassical
solutions. We remove the time-dependence of the Hamiltonian in Eq. (1) by
transforming to an interaction picture rotating at the drive frequency, i.e.
with $e^{i\hat{H_{0}}t}$ where
$\hat{H_{0}}=\hbar\omega_{o}\ \hat{a}^{{\dagger}}\hat{a}+\hbar\omega_{\mu}\
\hat{b}^{{\dagger}}\hat{b}\,.$ (2)
We define detunings as the difference of the drive frequencies from their
respective bare-cavity resonances
$\Delta_{(o,\mu)}=\omega_{(o,\mu),c}-\omega_{(o,\mu)}\;.$ (3)
The interaction Hamiltonian then becomes,
$\displaystyle\hat{V_{0}}$ $\displaystyle=$
$\displaystyle\hbar\Delta_{o}\hat{a}^{{\dagger}}\hat{a}+\hbar\Delta_{\mu}\hat{b}^{{\dagger}}\hat{b}+\hbar\omega_{m}\hat{d}^{{\dagger}}\hat{d}$
(4) $\displaystyle{}-\frac{\hbar
X_{\text{ZP}}}{2}\left(\hat{d}+\hat{d}^{{\dagger}}\right)\left(g_{o}\hat{a}^{{\dagger}}\hat{a}+g_{\mu}\hat{b}^{{\dagger}}\hat{b}\right)$
$\displaystyle{}+\hbar\left(\hat{a}A_{o}^{*}+\hat{a}^{{\dagger}}A_{o}\right)+\hbar\left(\hat{b}A_{\mu}^{*}+\hat{b}^{{\dagger}}A_{\mu}\right)\,.$
From this we derive coupled equations of motion for the macroscopic fields,
$\alpha=\bigl{<}\hat{a}\bigr{>}$, $\beta=\bigl{<}\hat{b}\bigr{>}$, and
$\delta=\bigl{<}\hat{d}\bigr{>}$. These equations are found by writing the
Heisenberg operator equations and directly substituting for each operator its
corresponding classical amplitude:
$\displaystyle i\frac{d\alpha}{dt}$ $\displaystyle=$
$\displaystyle\left(\Delta_{o}-i\frac{\kappa_{o}}{2}\right)\alpha-\frac{g_{o}X_{\text{ZP}}}{2}\left(\delta+\delta^{*}\right)\alpha+A_{o}\,,$
$\displaystyle i\frac{d\beta}{dt}$ $\displaystyle=$
$\displaystyle\left(\Delta_{\mu}-i\frac{\kappa_{\mu}}{2}\right)\beta-\frac{g_{\mu}X_{\text{ZP}}}{2}\left(\delta+\delta^{*}\right)\beta+A_{\mu}\,,$
$\displaystyle i\frac{d\delta}{dt}$ $\displaystyle=$
$\displaystyle\omega_{m}\delta-\frac{X_{\text{ZP}}}{2}\left(g_{o}|\alpha|^{2}+g_{\mu}|\beta|^{2}\right)\,.$
(5)
where $\kappa_{o,\mu}$ is the damping of each electromagnetic oscillator. We
introduce damping to insure the system relaxes to steady-state. We find the
steady-state solution by setting the time derivatives on the left-hand side of
each equation to zero. Solving the last equation shows that the equilibrium
position of the mechanical oscillator is shifted due the force exerted on it
by the radiation pressure in the cavities, with steady-state value
$\delta_{\text{s}}=X_{\text{ZP}}\left(g_{o}|\alpha|^{2}+g_{\mu}|\beta|^{2}\right)/(2\omega_{m})$.
Note this is purely real indicating no shift in the equilibrium momentum of
the mechanical oscillator.
Substituting the steady-state result for $\delta$ into the remaining two
equations for $\alpha$ and $\beta$ produces two coupled algebraic equations
that contain cubic terms in the field amplitudes. These equations have
multiple roots for small $\kappa_{o,\mu}$, corresponding to the well-known
steady-state solutions for two-mode optical bistability. For both the optical
and microwave cavities, we are interested in the stable solutions on the high-
intensity branch, which we denote by $\alpha_{\text{s}}$ and
$\beta_{\text{s}}$ respectively.
Having determined the semiclassical solutions, we now proceed to consider the
fluctuations about these solutions. We linearize all three fields:
$\displaystyle\hat{a}$ $\displaystyle\rightarrow\alpha_{\text{s}}+\hat{a}$
$\displaystyle\hat{b}$ $\displaystyle\rightarrow\beta_{\text{s}}+\hat{b}$
$\displaystyle\hat{d}$ $\displaystyle\rightarrow\delta_{\text{s}}+\hat{d}$ (6)
and substitute these into Eq. (4). We may then identify the mean-field energy
shift that contains no operators
$\displaystyle E$ $\displaystyle=$
$\displaystyle\hbar\Delta_{o}|\alpha_{\text{s}}|^{2}+\hbar\Delta_{\mu}|\beta_{\text{s}}|^{2}+\hbar\omega_{m}|\delta_{\text{s}}|^{2}$
(7) $\displaystyle{}-\hbar
X_{\text{ZP}}\delta_{\text{s}}\left(g_{o}|\alpha_{\text{s}}|^{2}+g_{\mu}|\beta_{\text{s}}|^{2}\right)$
$\displaystyle{}+\hbar\left(\alpha_{\text{s}}A_{o}^{*}+\alpha_{\text{s}}^{*}A_{o}\right)+\hbar\left(\beta_{\text{s}}A_{\mu}^{*}+\beta_{\text{s}}^{*}A_{\mu}\right)$
Using this, the final step in the derivation is to subtract the energy offset
$E$ in Eq. (7) from the Hamiltonian in Eq. (4) and transform the resulting
interaction into an appropriate rotating frame. Since at this point we are
working in the interaction picture rotating at the drive frequency, we now
need to make a second interaction picture transformation, that is, on top of
the previous one, to transform into a rotating frame at the cavity frequencies
for all oscillator modes; optical, microwave, and mechanical. In doing this we
must take careful account of the radiation pressure shift that we have just
derived for the electromagnetic resonant frequencies. We thus perform a second
transformation into an interaction picture rotating with $e^{i\hat{H_{1}}t}$
where
$\hat{H_{1}}\equiv\hbar\tilde{\Delta}_{o}\
\hat{a}^{{\dagger}}\hat{a}+\hbar\tilde{\Delta}_{\mu}\
\hat{b}^{{\dagger}}\hat{b}+\hbar\omega_{m}\hat{d}^{{\dagger}}\hat{d}\,.$ (8)
and we have defined $\tilde{\Delta}_{o}=\Delta_{o}-X_{\text{ZP}}\delta g_{o}$
and $\tilde{\Delta}_{\mu}=\Delta_{\mu}-X_{\text{ZP}}\delta g_{\mu}$ in order
to account for the aforementioned shift in the cavity frequency due to
radiation pressure. This leads to the interaction Hamiltonian
$\displaystyle\hat{V_{1}}$ $\displaystyle=$ $\displaystyle-\frac{\hbar
X_{\text{ZP}}g_{0}}{2}\Bigl{(}\hat{d}\hat{a}^{{\dagger}}\alpha_{\text{s}}e^{i(\tilde{\Delta}_{o}-\omega_{m})t}+\hat{d}\hat{a}\alpha_{\text{s}}^{*}e^{-i(\tilde{\Delta}_{o}+\omega_{m})t}$
(9)
$\displaystyle{}\quad+\hat{d}^{{\dagger}}\hat{a}^{{\dagger}}\alpha_{\text{s}}e^{i(\tilde{\Delta}_{o}+\omega_{m})t}+\hat{d}^{{\dagger}}\hat{a}\alpha_{\text{s}}^{*}e^{-i(\tilde{\Delta}_{o}-\omega_{m})t}\Bigr{)}$
$\displaystyle{}-\frac{\hbar
X_{\text{ZP}}g_{\mu}}{2}\Bigl{(}\hat{d}\hat{b}^{{\dagger}}\beta_{\text{s}}e^{i(\tilde{\Delta}_{\mu}-\omega_{m})t}+\hat{d}\hat{b}\beta_{\text{s}}^{*}e^{-i(\tilde{\Delta}_{\mu}+\omega_{m})t}$
$\displaystyle{}\quad+\hat{d}^{{\dagger}}\hat{b}^{{\dagger}}\beta_{\text{s}}e^{i(\tilde{\Delta}_{\mu}+\omega_{m})t}+\hat{d}^{{\dagger}}\hat{b}\beta_{\text{s}}^{*}e^{-i(\tilde{\Delta}_{\mu}-\omega_{m})t}\Bigr{)}$
From this result, we can see that in order to maximize the beam-splitter
couplings, the strong pump fields should be detuned to the red of their
respective microwave or optical cavity by
$\tilde{\Delta}_{o}=\tilde{\Delta}_{\mu}=\omega_{m}\;.$ (10)
In the resolved sideband limit where the frequency, $\omega_{m}$, of the
mechanical resonator is much larger than the mechanical decay rate as well as
the effective coupling constants, we can then employ the rotating wave
approximation where all the rapidly oscillating time dependent terms that
contain $e^{2i\omega_{m}}$ average to zero. Putting all this together leads to
the following effective Hamiltonian
$\displaystyle\hat{H}_{\text{eff}}=$
$\displaystyle\hbar\omega_{m}\left(\hat{a}^{{\dagger}}\hat{a}+\hat{b}^{{\dagger}}\hat{b}+\hat{d}^{{\dagger}}\hat{d}\right)$
$\displaystyle-\hbar\frac{\Omega_{o}}{2}\left(\hat{a}^{{\dagger}}\hat{d}+\hat{a}\
\hat{d}^{{\dagger}}\right)-\hbar\frac{\Omega_{\mu}}{2}\left(\hat{b}^{{\dagger}}\hat{d}+\hat{b}\
\hat{d}^{{\dagger}}\right)\;.$ (11)
where the modified coupling constants are now
$\displaystyle\Omega_{o}$ $\displaystyle=$ $\displaystyle g_{o}X_{\rm
ZP}\alpha_{\text{s}}\,,$ $\displaystyle\Omega_{\mu}$ $\displaystyle=$
$\displaystyle g_{\mu}X_{\rm ZP}\beta_{\text{s}}\,.$ (12)
Without loss of generality, we have taken both $\alpha_{\text{s}}$ and
$\beta_{\text{s}}$ to be real. The largest values achieved in experiment are
about $\Omega_{o}\sim 0.1\omega_{m}$ and $\Omega_{\mu}\sim 0.1\omega_{m}$.
Teufel et al. (2011b); Verhagen et al. (2012)
As noted earlier, this bilinear Hamiltonian is analogous to three quantized
single mode fields coupled to each other by beam splitters. The beam splitters
are adjustable by adjusting the $\Omega$’s. A coupling of $\pi/2$ will be like
a 50/50 beam splitter. A coupling of $\pi$ will be like a mirror, swapping the
states perfectly. If we turn the coupling off, the oscillators will propagate
freely as if there is no beam splitter. Thus, by varying the coupling
constants, we can change from 0 to $100\%$ reflection and transmission. It is
important to note that the quantum mechanical systems described by the field
operators $\hat{a}$, $\hat{b}$, and $\hat{d}$ are really fluctuations of the
bare fields around their stationary values at each of their respective
resonator frequencies.
In addition to the Hamiltonian in Eq. 11, the resonators are subject to
dissipative processes that stem from their coupling to the environment. The
mechanical resonator is coupled to a thermal bath. Thermal phonons are
absorbed into the mechanical resonator at a rate $\gamma_{m}\bar{n}$ where
$\bar{n}$ is the average Bose occupancy of the resonator and phonons are lost
at a rate $\gamma_{m}(\bar{n}+1)$. The optical and microwave cavities are
taken to be at zero temperature, and we set the loss rates given by
$\gamma_{o}$ and $\gamma_{\mu}$ respectively to zero.
We use the quantum state diffusion (QSD) method to simulate the fully quantum
evolution of this open system Gisin and Percival (1992); Persival (1999);
Belavkin and Staszewski (1992); Wiseman and Milburn (1993). The QSD method is
well suited for this problem because it yields the conditional evolution of an
open quantum system subject to homodyne measurements of the output fields. In
this way, we may obtain amplitude and phase information about the decay
channel. In the QSD approach, we stochastically evolve each resonator
subsystem as if we were performing a continuous fictitious homodyne
measurement of the photons or phonons coming out of each resonator. Even
though an actual experiment can not measure the phonons in the mechanical
resonator, the numerical simulation gives us access to this information in the
spirit of a ‘Gedanken’ measurement.
For the scenarios we explore, the decay rates of the optical and microwave
resonators are set to zero, so that only the mechanical resonator is coupled
to the environment. Thus, in those cases, the stochastic evolution only
applies to the mechanical resonator subsystem, while the optical and microwave
resonators evolve according to the Schrödinger equation. In actual
experiments, the optical and microwave resonators are also coupled to the
environment and actual homodyne measurements can be performed on those output
fields. The QSD method works for the fully open system as well. But here, we
are reducing the fully open system to focus only on the mechanical
decoherence.
In the QSD method, the evolution of the total density matrix of the system is
unraveled into an ensemble of stochastic parallel pure state trajectories.
Each trajectory is evolved according to a stochastic differential equation.
The trajectories are then averaged in the ensemble sense to recreate the total
density matrix. In the limit of a large number of trajectories, the ensemble
average of the stochastic trajectories goes to a state diffusion evolution.
Each trajectory evolves according to the stochastic differential equation
Wiseman and Milburn (1993)
$\displaystyle{|\tilde{\Psi}(t+dt)\rangle}=$
$\displaystyle\Bigg{\\{}1-\bigg{[}\frac{i}{\hbar}\hat{H}_{\text{eff}}+\frac{\gamma_{m}(2\bar{n}+1)}{2}\hat{d}^{{\dagger}}\hat{d}$
$\displaystyle\quad\qquad+2\gamma_{m}(2\bar{n}+1)\langle\hat{d}^{{\dagger}}+\hat{d}\rangle\hat{d}\bigg{]}dt$
$\displaystyle\quad+\hat{d}\sqrt{\gamma_{m}\bar{n}}\ dW_{\hat{d}}(t)$
$\displaystyle\quad+\hat{d}^{{\dagger}}\sqrt{\gamma_{m}(\bar{n}+1)}\
dW_{\hat{d}^{{\dagger}}}(t)\Bigg{\\}}{|\Psi(t)\rangle}$ (13)
where, $dW(t)$ is the continuum limit of a Wiener increment, $\Delta W$, which
satisfies the ensemble average $\langle(\Delta W)^{2}\rangle=\Delta t$ of a
Gaussian random distribution with a width $\Delta t$. There are two Wiener
increments, one for each noise process in the mechanical oscillator where
there are two types of decay channels, one for phonons entering the system,
$dW_{\hat{d}^{{\dagger}}}(t)$, and one for phonons leaving the system,
$dW_{\hat{d}}(t)$. We numerically integrate these stochastic differential
equations using a second order scheme Kloeden and Platen (1992).
## III Quantum State Memory
One of the possible applications of this system is to store a quantum state in
the mechanical resonator. One could prepare the microwave resonator in any of
a variety of quantum states. This can be done with superconducting quantum
circuits or other experimental setups such as those described in Lvovsky and
Mlynek (2002); Kakuyanagi et al. (2010); Wallraff et al. (2004); Hofheinz et
al. (2009). Then, the states of the microwave and mechanical resonators could
be swapped using a “$\pi$-pulse”, effectively storing the quantum state in the
mechanical resonator. At some later time, another swap could be done to put
the state back into the microwave resonator where it can be retrieved Zhang et
al. (2003); Tian and Wang (2010). The objective would be to maintain high
fidelity of the quantum state involved. These swaps are achieved by varying
the coupling constant, $\Omega_{(o,\mu)}$ which behaves like a Rabi frequency
in the beam splitter Hamiltonian in Eq. 11. The coupling constants can be
changed by modulating the bare coupling constants $g_{(o,\mu)}$ or the complex
pump amplitude $A_{(o,\mu)}$, or the detuning $\Delta_{(o,\mu)}$. For this
problem, it is sufficient to consider just a pair of resonators. We reduce the
system to one electromagnetic resonator and one mechanical resonator by
setting one of the coupling constants to zero.
A “$\pi$-pulse” in this context is a pulse where the time integral over the
coupling constant in frequency units is $\pi$. For the Gaussian coupling
pulses, $\Omega(t)=\Omega e^{-(t-t_{c})^{2}/(w^{2}\pi)}$, that we employ, the
pulse area is $w\pi\Omega$ where $w\pi$ is the width of the Gaussian and the
peak amplitude is $\Omega$. The pulse sequence is schematically shown in Fig.
3.
Figure 3: A schematic diagram of the coupling $\pi$ pulse sequence for Quantum
State Memory tests. The sign of the coupling constant for retrieval must be
the opposite of the sign for storage in order to cancel the phase accumulation
during the pulses.
To see how well the system is able to store classical and quantum states, we
study a variety of initial states with the same pulse sequence for storage and
retrieval. Recall that these states in fact represent the phonon or photon
fluctuations around the stationary value of each resonator, as we have
previously described in the derivation of the linearized effective Hamiltonian
in Eq. 11.
We use coherent states ${|\alpha\rangle}{|n\rangle}_{m}$, squeezed coherent
states ${|\alpha,\xi\rangle}{|n\rangle}_{m}$, cat states
${|\psi_{cat}\rangle}$, and a superposition of Fock states
${|\psi_{SF}\rangle}$ as inputs to the microwave resonator. The squeezed
coherent state Walls and Milburn (1994) is a displaced squeezed vacuum state,
${|\alpha,\xi\rangle}=\hat{D}(\alpha)\hat{S}(\xi){|0\rangle}$, where $\alpha$
is the mean value and $\xi$ is the squeezing parameter. The displacement
operator is $\hat{D}(\alpha)=e^{\alpha\hat{a}^{{\dagger}}-\alpha^{*}\hat{a}}$
and the squeezing operator is
$\hat{S}(\xi)=e^{\frac{1}{2}(\xi^{*}\hat{a}^{2}-\xi(\hat{a}^{{\dagger}})^{2})}$.
The cat state is
${|\psi_{cat}\rangle}=N({|\alpha\rangle}+{|-\alpha\rangle}){|n\rangle}_{m}$
where $N$ is a normalization constant. The input state for the superposition
of Fock states that we will study is
${|\psi_{SF}\rangle}=\frac{1}{\sqrt{2}}({|0\rangle}_{\mu}+{|1\rangle}_{\mu}){|n\rangle}_{m}$,
where ${|i\rangle}_{\mu}$ indicates the photon fluctuations around the
stationary value inside the microwave cavity and ${|n\rangle}_{m}$ indicates
the phonon fluctuations around the stationary value inside the mechanical
resonator.
The initial Fock state for the mechanical resonator for all input states,
${|n\rangle}_{m}$, is randomly chosen according to the probability
distribution for a thermal state,
$P_{n}=\frac{\bar{n}^{n}}{(\bar{n}+1)^{n+1}}\;,$ (14)
in order to sample the thermal density matrix at a temperature corresponding
to $\bar{n}$.
To measure the success of this scheme, we look at the fidelity Tian and Wang
(2010) of retrieving the input state as a function of the mechanical quality
factor, $Q_{m}=\omega_{m}/\gamma_{m}$. The fidelity Uhlmann (1976) is defined
as
$F(\rho_{i},\rho_{f})=\left[Tr\left(\sqrt{\sqrt{\rho_{i}}\rho_{f}\sqrt{\rho_{i}}}\right)\right]^{2}$
(15)
where $\rho_{i}$ and $\rho_{f}$ are the reduced density matrices for the input
and output states respectively. The fidelity for pure states reduces to the
overlap between the initial and final states.
At zero temperature, in the case where the pulse duration is short compared to
the decay time of the mechanical resonator, we can neglect the decay during
the swapping pulses and assume that the swaps happened perfectly. In that
case, the final state will only have decayed exponentially, at the mechanical
decay rate $\gamma_{m}$, during the time, $\Delta T$, between the two $\pi$
pulses to become ${|\psi_{f}\rangle}={|e^{-\gamma_{m}\Delta
T/2}\alpha\rangle}$. Thus we can analytically find the fidelity for a pure
final state to be
$F=|\langle e^{-\frac{\omega_{m}\Delta
T}{2Q_{m}}}\alpha|\alpha\rangle|^{2}=e^{-|\alpha|^{2}(1-e^{-\omega_{m}\Delta
T/(2Q_{m})})^{2}}\;.$ (16)
However, for non-zero temperatures or large decay rates the final state will
thermalize quickly and this formula will no longer be valid. The state does
not decay to the vacuum as Eq. 16 suggests. Rather it decays to a thermalized
value. Thus, the final fidelity for pure states will saturate at low-Q to the
overlap between a thermalized state and the initial state. Thus, the coherent
state fidelity takes on the form,
$F(Q)=F(0)+\left(1-F(0)\right)e^{-|\alpha|^{2}(1-e^{-\omega_{m}\Delta
T/(2Q_{m})})^{2}}\;.$ (17)
We take the overlap between the thermal state given by Eq. 14 and the coherent
state in the number basis to obtain the thermal saturation overlap value,
$F(0)=\frac{e^{-|\alpha|^{2}/(1+\bar{n})}}{1+\bar{n}}\;.$ (18)
Further complicating matters, the full density matrix, which is pure, must be
reduced to the resonator subsystem that is being read out before taking the
fidelity overlap with the reduced input density matrix. This makes the reduced
density matrices impure. Consequently, we need to employ the more general
formula to find the fidelity that is valid for states that are not pure.
Fig. 4A shows the memory fidelity at zero temperature for the input states
${|\alpha\rangle}$, ${|\psi_{SF}\rangle}$, and ${|\psi_{cat}\rangle}$ for
$\Delta T=64(unitsof1/\omega_{m})$. The coherent state fidelity agrees well
with our analytical result in Eq. 17 except at very low Q. At low Q values,
the input state thermalizes quickly to a thermal state. For these low Q
values, the decay time is smaller than the pulse width. Even though there is
almost no time between the pulses for these cases, the state still thermalizes
during the pulse width. A thermal state has a constant fidelity overlap with
the initial state. Thus, the fidelity for low Q mechanical resonators levels
off to a constant value. This causes the actual fidelity to deviate from the
analytic formula. Across the range of $Q_{m}$ values considered in Fig. 4, the
non-classical states have lower fidelities than the coherent state.
Figure 4: (Color online) Memory fidelity as a function of mechanical resonator
quality. (a): For $\bar{n}=0$. Input states ${|\alpha\rangle}$ (red circles)
with $\alpha=1$, ${|\psi_{SF}\rangle}$ (black diamonds),
${|\psi_{cat}\rangle}$ (blue squares). The solid line is the analytic formula
for the coherent fidelity from Eq. 17. (b): For $\bar{n}=3$. (c): For squeezed
states ${|\alpha,\xi\rangle}$ with $\alpha=1$ and $\bar{n}=0$ for various
squeezing parameters $\xi$. In all cases,
${\Delta}T=64(unitsof\frac{1}{\omega_{m}})$ and the peak coupling is
$\Omega_{\mu}=0.1\omega_{m}$. The horizontal dotted black line indicates 95%
fidelity for reference.
The fidelities for a finite temperature corresponding to $\bar{n}=3$ are shown
in Fig. 4B. As expected, the fidelities are consistently reduced by the
thermal noise. At the current experimental values for the mechanical quality
Teufel et al. (2011b) (far right hand side of the plots), all the fidelities
are above $99\%$ and there is little difference between the various input
states.
Figure 5: (Color online) (a): Memory fidelity of ${|\psi_{SF}\rangle}$ as a
function of mechanical resonator quality for various wait times for
$\bar{n}=3$. The fidelity decreases exponentially with decreasing Q and with
increased wait times. At low-Q, the fidelity saturates to the thermalized
value. (b): The memory fidelity vs. scaled wait time for $\bar{n}=0$. (c): The
memory fidelity vs. scaled wait time for different temperatures showing how
all the curves collapse onto one universal curve. In all cases, the peak
coupling is $\Omega_{\mu}=0.1\omega_{m}$.
For the squeezed states ${|\alpha,\xi\rangle}$, shown in Fig. 4C, we varied
the squeezing parameter $\xi$, keeping $\alpha$ constant at $\alpha=1$. For
$\xi=0$ the input state is just the coherent state, so the data is similar to
the coherent state data. The solid red line is the analytic formula from Eq.
17 showing good agreement with the no squeezing, $\xi=0$, input state. As the
squeezing increases, the input state becomes more and more nonclassical. For
low-Q cavities, the mechanical resonator quickly decays to a thermal state
that is farther and farther away from the initial squeezed state. This causes
the fidelity for these highly squeezed states to go to zero.
Next, we look at how long the mechanical oscillator can store a quantum state
before significant degradation occurs. Fig. 5A shows the memory fidelity of
${|\psi_{SF}\rangle}$ for increasing wait times as a function of $Q_{m}$ for
$\bar{n}=0$. As expected, the fidelity decreases with increasing wait time,
but we can still achieve above a 95% fidelity for the higher $Q_{m}$ values.
At the current experimental Q values of about $Q_{m}=360,000$ Teufel et al.
(2011b) the quantum state can be stored in the mechanical resonator for longer
than ${\Delta}T=160(unitsof1/\omega_{m})$ at low temperatures.
For the coherent case, we have a universal curve for the zero temperature
fidelity,
$F=e^{-|\beta(0)|^{2}(1-e^{-\zeta_{o}/2})^{2}}\;.$ (19)
where, we have rescaled the fidelity data by a dimensionless variable scaled
by the thermal average occupation number $\bar{n}$,
$\displaystyle\zeta_{o}=$ $\displaystyle\omega_{m}(\Delta
T-\frac{\pi}{\Omega_{\mu}})/Q_{m}$ (20) $\displaystyle\zeta=$
$\displaystyle\omega_{m}(\Delta T-\frac{\pi}{\Omega_{\mu}})\bar{n}/Q_{m}\;,$
(21)
For the Fock states, the exponential dependence on $\zeta$ is similar,
although we no longer have an analytic formula. We subtract the width of the
$\pi$ pulses to more accurately reflect the actual storage time. Fig. 5B shows
the same memory fidelity as for subplot A versus $\zeta_{o}$ for $\bar{n}=0$.
Fig. 5C shows the memory fidelity versus $\zeta$ for finite $\bar{n}$ in order
to remove the dependence on the temperature. As we increase the wait time
before the second $\pi$ pulse, the fidelity decreases exponentially as
expected. By removing the dependence on the temperature, the fidelity curves
all collapse onto one universal curve. Even though we are running our
simulations at low $\bar{n}$ values, these results can be scaled up to more
experimentally practical $\bar{n}$ values while keeping $\zeta$ constant.
All the input states in Figure 4 reach above a 95% fidelity for $\zeta_{o}$
values below about 0.26 for coherent states, 0.11 for Fock states, and 0.04
for Cat states for $\bar{n}=0$. The fidelity is above 95% for $\zeta$ values
below 0.03 for coherent states, 0.02 for Fock states, and 0.01 for Cat states
for $\bar{n}=3$ with the peak coupling set to $\Omega_{\mu}=0.1\omega_{m}$.
Figure 6: (Color online) Distribution of fidelities for ${|\psi_{SF}\rangle}$
for $Q_{m}=10,000$ (a): $Q_{m}=1000$, (b): and $Q_{m}=500$, (c): for 1000
independent QSD trajectories. In all cases,
${\Delta}T=64(unitsof\frac{1}{\omega_{m}})$, $\bar{n}=3$, and the peak
coupling is $\Omega_{\mu}=0.1\omega_{m}$. Note the different horizontal axis
in each plot.
The distribution of the fidelities over the individual stochastic QSD
trajectories is not Gaussian and thus the mean and variance are not
representative of the outcome of a single run. In an experiment, many runs
must be performed to get good statistics about the memory transfer fidelity.
However, as the mean fidelity increases, the distribution becomes more and
more narrow and thus more repeatable. Fig. 6 shows the fidelity distribution
for ${|\psi_{SF}\rangle}$ for $Q_{m}=10,000$, $Q_{m}=1000$, $Q_{m}=500$, where
the wait time is ${\Delta}T=64(unitsof\frac{1}{\omega_{m}})$, $\bar{n}=3$, and
the peak coupling is $\Omega_{\mu}=0.1\omega_{m}$ for 1000 trajectories.
## IV Quantum State Transduction
We turn now to the problem of transducing the quantum state from the microwave
domain to the optical domain via the mechanical resonator or vice versa. This
situation is formally equivalent to the quantum memory case if the
electromechanical coupling pulse and the optomechanical coupling pulse do not
overlap. This is because transduction can be realized by two sequential steps
from one resonator to the mechanical resonator and then to the second
resonator, playing the same roles as storage and retrieval in the previously
studied case of quantum memory. However, with three resonators, more varied
protocols are possible. Although we specialize our analysis to the case where
the initial state is in a superposition of Fock states, which now also
contains the optical vacuum state,
${|\psi_{t}\rangle}=\frac{1}{\sqrt{2}}({|0\rangle}_{\mu}+{|1\rangle}_{\mu}){|n\rangle}_{m}{|0\rangle}_{o}$,
these procedures could be similarly applied to consider other quantum states.
The coupling parameters can be of similar orders of magnitude for optical and
microwave cavities. So, to simplify the analysis, we set the respective
optical and microwave coupling parameters equal to each other,
$\Omega_{o}=\Omega_{\mu}=0.1\omega_{m}$. In addition, we set both optical and
microwave detunings equal to the mechanical resonator frequency.
For transduction, we want to minimize the decay and thermal effects caused by
leaving the state in the mechanical resonator for any appreciable time period.
The simplest method to accomplish this is to move the quantum state through
the mechanical resonator as quickly as possible. This method would be good for
applications such as quantum information processing where speed is desirable.
Another method is to adiabatically move the state from the microwave to the
optical resonator or vice versa without fully populating the mechanical
resonator. Naturally, adiabaticity requires longer times, but it is also less
susceptible to variations in the pulse profiles. We will discuss this method
in more detail in the next section.
We begin by using a similar protocol to the quantum memory scheme. We set the
first $\pi$ pulse to swap the state from the microwave cavity to the
mechanical resonator. Then, we set the second $\pi$ pulse to occur right after
the first one to swap the state from the mechanical resonator to the optical
cavity. This pulse sequence is shown schematically in Fig. 7C.
The resulting fidelity is shown by the red dots in Figure 7A as a function of
the Q of the mechanical resonator. As this is formally equivalent to the
memory scheme, we see similar behavior. The separation between the peaks of
the $\pi$ pulses used here is the same as the wait time we used for the
quantum memory scheme. Thus, the numerical data are similar. Just like in the
quantum memory case, as the mechanical quality decreases, the thermal noise
and decay processes become more significant and the fidelity exponentially
decays down to the fully thermalized value.
A natural extension of this scheme is to move the two $\pi$ pulses closer
together, which could allow for faster transfer. Taking this to its logical
extreme, we study a scheme where both coupling pulses occur simultaneously.
This allows for the state to move from the microwave to optical cavity, or
vice versa, without fully occupying the mechanical resonator, but note, that
this in not in the adiabatic regime.
The overlap in the coupling modifies the optimal pulse area of both couplings.
The effective Rabi frequency for the beam splitter Hamiltonian in Eq. 11 is
$\tilde{\Omega}=\sqrt{\Omega_{o}^{2}+\Omega_{\mu}^{2}}\;.$ (22)
When the optical coupling is turned off, as in the Quantum Memory case of the
last section, the effective Rabi frequency reduces to
$\tilde{\Omega}=\Omega_{\mu}$ and the pulse area for each swapping pulse is
$\pi$. However, when both couplings are on and equal in magnitude, the
effective Rabi frequency becomes,
$\tilde{\Omega}=\sqrt{2}\Omega_{o}=\sqrt{2}\Omega_{\mu}$. Then the pulse area
of both pulses increase to $\sqrt{2}\pi$. This is shown schematically in Fig.
7E.
This scheme achieves significantly higher fidelities than the separated pulse
scheme for all Q values. The fidelity is shown in Fig. 7A by the blue squares.
In the low-Q regime, the decay and thermal noise processes become increasingly
more significant. However, the population going through the mechanical
resonator is smaller, so the effect is lessened. Also, there is no waiting
time between the pulses for decay and thermal noise processes to occur. The
only decay happens during the width of the pulse. Thus, the simultaneous pulse
scheme is more robust against these processes.
Figure 7: (Color online) (a): Transfer fidelity vs. mechanical resonator
quality for ${|\psi_{t}\rangle}$ for the separated pulse scheme (red dots),
the simultaneous pulse scheme (blue squares), intuitive pulse scheme (green
diamonds), and counter-intuitive pulse scheme (black stars) with $\bar{n}=3$.
(b): Number of photons or phonons in each resonator for intuitive, separated
coupling pulses for zero temperature where
${\Delta}T=64(unitsof\frac{1}{\omega_{m}})$. (c): The pulse profile for
separated pulses. (d): Number of photons or phonons in each resonator for
simultaneous coupling pulses for zero temperature. (e): The pulse profile for
simultaneous pulses. In all cases, the peak couplings are
$\Omega_{o}=\Omega_{\mu}=0.1\omega_{m}$.
To illustrate the differences between the separated and simultaneous pulse
schemes, we examine the populations in each of the three resonators throughout
the swapping process. Fig. 7B shows how the populations change for the
separated $\pi$ pulse scheme. As expected, the state in the electrical
resonator moves into the mechanical resonator and then into the optical
resonator. In contrast, Fig. 7D shows populations for the simultaneous $\pi$
pulse scheme. Here, the state transfers from the electrical resonator to the
optical resonator without ever fully populating the mechanical resonator.
As the pulses move closer together, the pulse area needed to make the swap
smoothly changes from $\pi$ to $\sqrt{2}\pi$. Also, in real experiments, there
may be slight imperfections in the pulse preparation that would result in
varying pulse areas and peak separations. We have run our simulations over a
range of varying pulse areas and separations to investigate the effect this
had on the fidelity.
Fig. 8 shows the transduction fidelity versus the pulse area and the peak
separation between the two coupling pulses for $\bar{n}=0$ and $Q_{m}=100,000$
for the superposition of Fock states, ${|\psi_{t}\rangle}$. The horizontal
axis represents the separation between the peaks of the two coupling pulses as
a percentage of the total transduction sequence time as shown in Figure 7C.
The zero on the horizontal axis is the point where the two coupling pulses
occur simultaneously. The positive horizontal axis represents peak separations
that are “intuitive”, i.e., the electromechanical coupling pulse occurs before
the optomechanical coupling pulse.
As the intuitive Gaussian pulses move farther apart, they eventually become
effectively separated and the distance between them no longer matters, since
this is at zero temperature with very low decay rates. This is the area on the
far right-hand side of Fig. 8 where the regions of high fidelity level off at
odd integer multiples of $\pi$. Pulses that are $\pi$ pulses or an odd integer
multiple of a $\pi$ pulse will swap the state completely from the microwave
resonator to the optical resonator. Any other pulse area will not perfectly
swap the states. Thus, we see the oscillatory behavior we expect in that part
of the plot.
At zero pulse separation, the pulses are identical and the Rabi swapping pulse
area is $\sqrt{2}\pi$. Thus, the peak fidelity oscillations are at odd integer
multiples of $\sqrt{2}\pi$. For partially overlapping intuitive pulses, the
peak fidelity oscillations smoothly drop from the simultaneous values to the
separated values.
The fidelity for a representative partially overlapping intuitive coupling
pulse configuration is shown in Fig. 7A by the green diamonds. For this case,
the pulse area is $1.2\pi$, the peak separation is $10\%$, and $\bar{n}=3$. In
the next section, we will discuss the counter-intuitive half of the plot.
Figure 8: (Color online) The fidelity vs. pulse area and pulse separation for
$Q_{m}=100,000$, $\bar{n}=0$, and $\Omega_{o}=\Omega_{\mu}=0.1\omega_{m}$. The
negative horizontal axis values represent the peak separation for counter-
intuitively ordered pulses while the positive horizontal axis values represent
the peak separation for intuitively ordered pulses.
### IV.1 Adiabatic State Transfer
The left half of Fig. 8 shows the transduction fidelity for overlapping
coupling pulses that are in the counter-intuitive order, which means, that the
optomechanical coupling pulse occurs before the electromechanical coupling
pulse. As the peak separation increases, at some point, the pulses become
effectively separated and the fidelity goes to zero. However, when the
coupling pulses are significantly overlapping, there is a large area of high
fidelity for any pulse area. As we increase the pulse area and thus the
adiabaticity, the zone of high fidelity transduction increases.
This counter-intuitive coupling scheme closely resembles the Stimulated Raman
Adiabatic Passage (STIRAP) process in a three level atom. Our system of three
coupled harmonic resonators can be formally mapped onto such a three state
system Wang and Clerk (2012a). The energy levels for a three state system are
shown in Fig. 9. If we identify the microwave cavity with state ${|1\rangle}$,
the optical cavity with state ${|2\rangle}$, and the mechanical cavity with
state ${|3\rangle}$, then, we can get population transfer from state
${|1\rangle}$ to ${|2\rangle}$ via the normal STIRAP process Bogolubov et al.
(1985), which leaves virtually no population in state ${|3\rangle}$. In the
STIRAP process, the Stokes coupling (the coherent coupling between states
${|2\rangle}$ and ${|3\rangle}$) is turned on first which splits the energy
levels for state ${|3\rangle}$. Then, the pump coupling (the coherent coupling
between states ${|1\rangle}$ and ${|3\rangle}$) is turned on and the
population in state ${|1\rangle}$ is seen to be transferred to state
${|2\rangle}$ without ever having any significant population in state
${|3\rangle}$ because of the interference between the pathways corresponding
to transversing each of the two split energy levels.
Figure 9: Schematic of a three state system.
For illustration purposes, we examine the populations of the three resonators
through out this scheme shown in Fig. 10. Just as we would expect for a
STIRAP-like process, the state adiabatically transfers from the microwave to
the optical resonator while minimally populating the mechanical resonator.
Thus, this scheme is much more robust against decay and thermal noise as well
as imperfections in pulse area. However, adiabaticity generally requires
longer times. The simultaneous pulse scheme will transduce the state much
quicker, but the pulses must be precisely generated. So, there is a trade-off
between the transduction time and the strigentness of pulse preparation and
also decay during the simultaneous pulse width.
Figure 10: (Color online) The number of photons or phonons in each resonator
for the STIRAP-like coupling pulses. The pulse area is $10\pi$ and the peak
separation is -14%.
A representative counter-intuitive partially overlapping coupling pulse
configuration is shown in Fig. 7A by the black stars. For this case, the pulse
area is $2.4\pi$, the peak separation is $-15\%$, and $\bar{n}=3$.
## V Conclusion
Quantum state memory and transduction is possible for a broad range of
experimentally achievable parameters in driven cavity optomechanics. Many
different types of states can be stored in the mechanical resonator and
retrieved with a high fidelity in the range of experimentally achievable
mechanical quality factors. At the current experimental Q values of about
$Q_{m}=360,000$ Teufel et al. (2011b) the quantum state can be stored in the
mechanical resonator for longer than ${\Delta}T=160(unitsof1/\omega_{m})$ at
low temperatures. As the experiments improve the coupling strength, the time
the state can be stored without significant degradation will increase. If the
mechanical mode is cooled before the swapping pulses are applied, then the
storage time will also increase.
We have shown several procedures to accomplish transduction of quantum states.
High fidelity transfer is possible for Rabi type pulses of varying widths and
separations even for very low Q mechanics. We have shown that over 95%
transduction fidelity can be achieved for $Q_{m}>4525$ for
$\Omega_{m}=0.1\omega_{m}$ and $\bar{n}=3$ for the simultaneous pulse scheme.
This scheme is quicker and more robust against thermal noise and decay than
the more common separated pulse scheme. Higher bath temperatures will require
a shorter simultaneous pulse with stronger coupling to achieve high
fidelities.
Counter-intuitively ordered adiabatic pulses can also be used to transfer the
quantum state through a STIRAP like scheme with high fidelity. This scheme is
robust against thermal noise and decay and imperfections in pulse preparation
and overlap, but requires longer transduction times. Higher bath temperatures
will also require longer transduction times to maintain the adiabaticity.
###### Acknowledgements.
This work was supported by the National Science Foundation.
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* Uhlmann (1976) A. Uhlmann, Rep. Math. Phys. 9, 273 (1976).
* Bogolubov et al. (1985) N. N. Bogolubov, F. L. Kien, and A. S. Shumovsky, Physics Letters 107A, 173 (1985).
|
arxiv-papers
| 2013-05-29T21:49:39 |
2024-09-04T02:49:45.859709
|
{
"license": "Public Domain",
"authors": "S. A. McGee and D. Meiser and C. A. Regal and K. W. Lehnert and M. J.\n Holland",
"submitter": "Murray Holland",
"url": "https://arxiv.org/abs/1305.6962"
}
|
1305.6992
|
# Coexistence and Interference Mitigation for Wireless Body Area Networks:
Improvements using On-Body Opportunistic Relaying
Jie Dong, David Smith National ICT Australia (NICTA)† and The Australian
National University
(e-mail: [email protected])
###### Abstract
Coexistence, and hence interference mitigation, across multiple wireless body
area networks (WBANs) is an important problem as WBANs become more pervasive.
Here, two-hop relay-assisted cooperative communications using opportunistic
relaying (OR) is proposed for enhancement of coexistence for WBANs. Suitable
time division multiple access (TDMA) schemes are employed for both intra-WBAN
and inter-WBANs access protocols. To emulate actual conditions of WBAN use,
extensive on-body and inter-body “everyday” channel measurements are employed.
In addition, a realistic inter-WBAN channel model is simulated to investigate
the effect of body shadowing and hub location on WBAN performance in enabling
coexistence. When compared with single-link communications, it is found that
opportunistic relaying can provide significant improvement, in terms of
signal-to-interference+noise ratio (SINR), to outage probability and level
crossing rate (LCR) into outages. However, average outage duration (AOD) is
not affected. In addition a lognormal distribution shows a better fit to
received SINR when the channel coherence time is small, and a Nakagami-m
distribution is a more common fit when the channel is more stable. According
to the estimated SINR distributions, theoretical outage probability, LCR and
AOD are shown to match the empirical results well.
###### Index Terms:
Cooperative communications, interference management, opportunistic relaying,
wireless body area networks.
## I Introduction
The rapid development of semiconductor technology and wireless communications
has led to a new generation of wireless sensor networks, wireless body area
networks (WBANs), which allow inexpensive and continuous monitoring of human
physiological condition and activity, with real-time updates [1]. Sensors can
be implanted inside or placed on the human body without causing inconvenience
and impairing daily activities. Compared with traditional monitoring, WBAN can
provide flexibility and location independency [2]. In 2009, there were
approximately 11 million active WBAN units around the world, and this number
was predicted to reach 420 million by 2014 [3].
A traditional WBAN has a single-link star topology, consisting of several
sensors and a hub (i.e., gateway device) [4]. According to the WBAN’s
application requirements, it is essential to maintain high communications
reliability at the same time as minimizing power consumption. Therefore, in
order to overcome large path losses typically experienced in single-link, star
topology, WBAN communications, two-hop cooperative communications is an
alternative implementation in the IEEE 802.15.6 WBAN standard [4]. In both
narrow-band [5, 6, 7, 8, 9] and ultra-wideband WBAN systems [10], some two-hop
cooperative communication schemes have been investigated that provide
significant performance improvement. In [8] idle sensors are used as relays
and selection combining (SC) is used to provide diversity gain at the hub.
When using either maximal ratio combining (MRC) or SC with decode-and-forward
relay communications, it shows an up-to-14 dB increase in bit error
probability, when employing an IEEE 802.15.6 compliant BCH coding and GFSK
modulation scheme [5].
Importantly, the improvement in [5] was determined for an isolated WBAN.
However, in a lot of circumstances, the WBAN will need to perform reliably in
the vicinity of other WBANs such as in the foyer of a medical center. In
addition, as a WBAN may be highly mobile, it is generally not feasible to
globally coordinate multiple WBANs[11]. Therefore, inter-WBAN interference may
lead to unreliable communications. Furthermore, according to specifications
for the IEEE 802.15.6 standard, the performance of a WBAN should remain
reliable when up-to 10 WBANs are co-located in a 6-by-6-by-6m space [12]. In
light of all these considerations for WBANs this paper focuses on interference
mitigation and hence coexistence enhancement by using simple two-hop, relay-
assisted cooperative communications.
The contributions of this paper are as follows:
1. 1.
Multiple WBANs are simulated employing extensive “everyday” on-body and inter-
body channel gain measurements[13], with the WBAN-of-interest consisting of
one hub, two possible relays/sensors and eight other sensors. Within each
WBAN, the central hub coordinates associated sensors using time division
multiple access (TDMA). And, in accordance with an option for physical layer
transmission in the IEEE 802.15.6 BAN standard [4], sensors use BCH coded GFSK
modulation to transmit packets at 2.4 GHz carrier frequency.
2. 2.
In this context, two non-coordinated neighboring WBANs are modeled to
investigate the WBANs coexistence. Modified TDMA is employed as an inter-WBAN
access scheme, due to good performance in interference mitigation with respect
to channel quality and low power consumption [14];
3. 3.
Two-hop cooperative communications is employed for the WBAN-of-interest. Two
different implementations are investigated. In the first implementation, the
choice of relay node changes according to an activation of any of the sensors
as relays[8]. Whereas, two fixed relays attached to the left hip and right hip
respectively are introduced in the second implementation [9]. In this paper,
we will primarily focus on the second implementation.
4. 4.
Three-branch opportunistic relaying (OR), similar to using selection
combining, is used as it has low complexity and low power consumption. In OR,
only a single relay with the most reliable signal path to the destination
forwards a packet per hop. This OR is subsequently shown to have significant
performance advantages over single-hop communications for both first and
second-order outage statistics in the presence of interference, both
theoretically and empirically in terms of signal-to-interference-plus-noise-
ratio (SINR)
The performance of closely-located WBANs will be interference limited.
Therefore, the effectiveness of two-hop communications scheme using relays on
interference mitigation is studied based on signal-to-interference-plus-noise-
ratio (SINR) of each received packet. Both the first order statistics of
outage probability and second order statistics of level crossing rate (LCR)
and average outage duration (AOD) for SINR are derived in this paper. The
results are compared with respect to the coexistence of two traditional star-
topology, single-hop WBANs with the same channel data and the same TDMA
schemes used. Here the work published in [8] and [9] is combined and extended,
the extension including analysis of AOD. In other important extensions of
[8],[9], the best, simple distribution of SINR values is found according to
maximum-likelihood parameter estimation; and based on the SINR distributions
derived, we generate theoretical outage probability, level crossing rates and
average outage duration and compare them with experimental results.
The rest of the paper is organized as follows. In Section II, details of the
system model used are given, including two different implementations of
cooperative communications. Then Section III provides the analysis of
experimental results for WBANs where relay positions are varied over the
entire body. In Section IV, analysis of cooperative communications in the
WBAN-of-interest using two set relay positions are presented. First order
statistics, outage probability and second order statistics, level crossing
rate and average outage duration, are derived from contiguous empirical SINR
values and are compared with theory according to fitted distributions . In
Section V, coexistence performance of WBANs employing two-hop opportunistic
relaying cooperative communications with fixed relays is compared with star
topology single-link communications. It’s effectiveness is shown based on a
comparison of outage probability, level crossing rate and average outage
duration of the two schemes. Section VI provides some concluding remarks on
coexistence between non-coordinated WBANs being better facilitated by
cooperative communications on any given WBAN.
## II System Model
As mentioned in the introduction, there are two different dual-hop relay-
assisted cooperative communications schemes implemented. In this section, we
first present common configurations shared by both implementations, which
includes intra- and inter-WBAN access schemes, coding and modulation, and
opportunistic relaying. Then their differences in relay selection, and intra-
and inter-WBAN channel model, together with the techniques for overlaying the
models, are explained.
### II-A Common Setup
#### II-A1 Intra-WBAN Access Scheme
The basic model of a WBAN used in the simulation consists of one hub (gateway)
and three active sensor nodes. In this star topology system, the hub
coordinates the sensors with a time division multiple access (TDMA) scheme.
Therefore, as soon as the hub broadcasts a beacon signal, nodes respond by
transmitting the collected information back according to a pre-defined
sequence for their allocated time slots. This process is shown in Fig. 1, in
which the labels indicate node numbers. In Fig. 1, the beacon signal is
neglected since it is so short compared to the time-length of transmissions by
nodes. In terms of two-hop cooperative communications, two relays are chosen
from at the hub. On-body hub and relay locations and the way to select active
relays will be explained in Section II-B.
Here we denote the time period starting from beacon signal to all nodes
finishing transmission as a superframe, and this concept will be used in
latter sections. In analysis and simulation, it is assumed that each sensor
only transmits one packet of information in a single superframe. After
receiving information packets from all sensors, the WBAN goes into idle mode
and waits until the next beacon period starts. The length of the idle mode
depends on the inter-WBAN access scheme.
Figure 1: Intra-WBAN and Inter-WBANs TDMA scheme
#### II-A2 Inter-WBAN Access Scheme
In consideration of co-channel interference mitigation and power consumption
reduction, TDMA is employed as a co-channel access scheme across all WBANs.
Assume there are $N_{c}$ WBANs located in close proximity and the number is
fixed during the period of simulation, then the channel is evenly divided into
$N_{c}$ time slots. Each time slot has a superframe length $T_{d}$, which
allows every WBAN to collect all information packets from sensors in its own
network. Therefore, under this TDMA scheme, a WBAN is required to wait for
time $(N_{c}-1)\times T_{d}$ for other co-located systems to complete their
operations before its next transmission cycle starts. This waiting period is
also called an idle period, which is denoted as $T_{idle}$. However, because
global inter-network coordination is generally not feasible, the execution of
this scheme is different from a traditional implementation of TDMA. As shown
in Fig. 1, a WBAN chooses the start time of every superframe randomly,
following a uniform distribution over $[0,T_{d}+T_{idle}]$. In this paper,
inter-WBAN TDMA is used across two WBANs employing this configuration, but
this can be generalized to any number of closely located WBANs.
#### II-A3 Coding and Modulation Scheme
In the IEEE 802.15.6 BAN standard [4], BCH coding together with GFSK
modulation at 2.4 GHz carrier frequency is specified as an option for physical
layer transmission. For this option sensors encode their messages with
BCH(31,19), which is a shortened version of BCH(63,51). These encoded messages
are then modulated prior to transmission using Gaussian-minimum-shift-keying
(GMSK), which is a special type of Gaussian-FSK (GFSK) with a modulation index
of 0.5.
#### II-A4 Opportunistic Relaying
As mentioned in the introduction, two relays are potentially chosen from while
each sensor transmits. Therefore, there are three paths that the signal can
take from sensor to hub. One is the direct link from the active sensor to the
hub, while the other two are decode-and-forward links via either relay. Here,
a three-branch opportunistic relaying (OR) scheme is implemented similar to
using selection combining (SC) at the hub. The difference between OR and SC is
that in OR, the best path is chosen prior to the sensor transmission and only
the best path is activated. However, in SC, signals are sent over all three
links, and the hub receives three copies of the signal, and the one with the
best quality is subsequently chosen. In this way, OR reduces power consumption
and mitigates interference with other co-located WBANs. In this paper, the
channel quality of different paths is estimated using SINR of packets received
at relay and hub nodes. SINR $\nu$ at every node is defined as:
$\nu=\frac{a_{sig}|h_{TxRx}|^{2}}{|\varepsilon_{noise}|^{2}+\sum(a_{int,i}|h_{int,i}|^{2})}~{},$
(1)
where $a_{sig}$ and $a_{int}$ represent the transmit power of a packet from
WBAN-of-interest and interfering WBAN respectively; $|h_{TxRx}|$ and
$|h_{int}|$ represent the average channel gain for the source to destination
and interfering source to destination links across the time duration of the
transmitted signal packet. The subscript $i$ indicates the $i$th interfering
source. $|\varepsilon_{noise}|$ is the average noise power experienced at the
node-of-interest across the time duration of the transmitted signal packet.
Throughout this paper, normally distributed additive white Gaussian noise is
assumed, with $|\varepsilon_{noise}|=-95$ dBm in all analysis111Where we note
that $-95$ dBm corresponds to a receiver sensitivity specified for 2.4 GHz
carrier frequency in IEEE 802.15.6 [4]..
In terms of two relay links, the channel quality is evaluated according to the
minimal SINR among two hops at the beginning of a superframe. The selected
path $\mathrm{j}_{OR}$ is then determined by the following algorithm:
$\displaystyle\mathrm{j}_{OR}$
$\displaystyle=\arg\max[\nu_{1},\nu_{2},\nu_{sh}],$ $\displaystyle\nu_{k}$
$\displaystyle=\min[\nu_{r_{k}h},\nu_{sr_{k}}],\>\textrm{with}\>k=1\>\textrm{or}\>2.$
(2)
The subscripts $s,~{}r_{k}~{}and~{}h$ indicate sensor, $k$th relay and the hub
respectively; $\nu_{ab}$ represents the SINR of received packet transmitted
from node $a$ to node $b$.
### II-B Types of Relay Selection and WBAN Channel Models
#### II-B1 Relay Selection
Two different relay selection schemes are investigated. The first
implementation does not require additional hardware, but uses inactive sensors
as relays, as shown in Fig. 2(a). As mentioned before, the WBAN consists of
one hub and three active sensors. While one sensor is transmitting, the other
two act as possible relays and decode-and-forward received packets to the hub
as required. In contrast, for the second implementation in Fig. 2(b), besides
the existing sensors and hub, two fixed relays are added to the overall
configuration of the WBAN system. The relays are placed at the left and right
hips respectively. When active, they listen to the channel and decode-and-
forward packets transmitted by sensors.
(a) Varying Relays Implementation
(b) Fixed Relays Implementation
Figure 2: Two different relay selection implementations
#### II-B2 intra- and inter-WBAN channel model
Extensive on-body and inter-body channel data was measured with small wearable
channel sounders operating at 2.36 GHz (close to the 2.4 GHz ISM band) over
several hours of normal everyday activity [13]. In terms of on-body channel
data, experiments were repeated on different individual subjects. In each
experiment, subjects wore the same measuring system, which consisted of 3
transceivers and 7 receivers [13][8]. Throughout the entire experiment,
transceivers worked in a round-robin fashion broadcasting every 5 ms at 0 dBm.
During one radio transmission, the remaining channel sounders, including the
remaining inactive transceivers, recorded the received signal strength
indicator (RSSI) upon successful detection of a packet.
While observing different subjects on-body channel profiles, Subject 1’s data
indicates that these is a sudden change in experiment environment. Fig. 3
shows a typical channel gain plot for Subject 1 over a period of approximately
3 hours. As seen in the figure, the sudden change occurs roughly at the
40000th sample (i.e., after approximately 80 minutes). The channel before that
moment shows a slow fading characteristic with larger coherence time, and it
is more stable. In contrast, after the 40000th sample, the coherence time
decreases significantly and the channel starts to vary more rapidly. In the
simulation of two-hop cooperative communication scheme with fixed relays, two
distinct parts of Subject 1’s channel are investigated and the impact of this
difference is then compared.
Figure 3: A typical on-body channel of Subject 1
To capture the inter-body channel data, more subjects were involved in one
experiment [13, 9]. For the data set used in our simulation, there were 8
people walking together to a cafe, sitting there for a while and then walking
back to office. Each subject wore a measuring system, which consisted of only
1 transceiver at left hip and 2 receivers at right upper arm and left wrist
respectively. Similar to the on-body channel experiment, each transceiver
broadcasted every 5 ms in round-robin sequence at 0 dBm. However, since the
total number of transceivers was different, on-body and inter-body channel
data sets have unmatched sampling rates, 15 ms per sample and 40 ms per sample
respectively.
For the analysis with _variation in relays used_ , on-body channel data and a
simulated inter-body channel model are used, _and hence interference used in
this analysis is based on simulation_. It simulates a realistic scenario which
may happen in an open indoor or outdoor environment. As shown in Fig. 4, two
subjects walk approach each other, passing by each other and then move apart
in a space with a dimension of $6\times 0.5~{}\mathrm{m^{2}}$. [8] provides a
detailed description of the motion model for this simulation. It also explains
how the inter-body channel model is simulated. Briefly, the inter-body channel
model incorporates both large scale and small scale fading. The large scale
fading model is simulated by combining the effect of free space path loss and
shadowing. For WBAN applications, a mean path loss with an exponent of 2 is
used since it corresponds to a relatively ‘open’ environment with few objects
nearby causing reflection and diffraction. In terms of shadowing, there will
not always exist a line-of-sight path between two WBANs. It will be obstructed
by the human body and influenced by body movement in most cases. According to
the on-body channel data, it can be observed that the average attenuation
caused by shadowing is approximately 40 dB. Thus, the large scale fading model
is simulated by adding a $-40$ dB offset to the mean path loss. Different
levels of shadowing are investigated; no shadowing, partial shadowing and full
shadowing, among which full shadowing is the most common. Around shadowed mean
path loss, small scale fading is added. This is achieved by using Jakes model
with a Doppler frequency of 2Hz and Rayleigh fading with fading coefficients
that are $\mathcal{CN}(0,1)$ distributed around the shadowed path loss.
_Please note that Rayleigh small-scale fading is a poor model for the on-body
channel over the WBAN-of-interest, e.g.,[15, 16], but this model is applied
here specifically for an open environment body-to-body._
Figure 4: Simulation of two subjects’ motion for the scenario with variation
in relays used
In the case of analysis with _a fixed relay implementation_ , both on-body and
inter-body channel measurements are employed, _and hence interference used in
this analysis is based on measurement_. Recall that on-body and inter-body
sampling rates are different, 15 ms and 40 ms respectively. Therefore, it is
essential to synchronize them to the same sampling rate. As opposed to the
experiment described in [17], in which the experiment involves continuous
strenuous activity, the inter-body channel data used in this paper was
measured differently. The orientation and distance between subjects were more
stable, and there were less scatterers in the surrounding environment. As
defined in [18], channel coherence time is typically the time lag until the
autocorrelation coefficient reduces to 0.7. Based on that, calculations show
that on-body and inter-body channel data used in this simulation have an
average coherence time of 2087 ms and 892 ms respectively. Therefore, it is
possible to down-sample both types of channel data to 120 ms per sample
without losing accuracy. In addition, since the transmission time of a packet
is shorter than the 120 ms sampling rate, a block fading model is used whereby
the Tx-Rx channel gain is constant over each packet. Besides temporal
synchronization, spatial overlaying is also required. Since opportunistic
relaying used in this simulation is based on the SINR value at the relays and
the hub, it needs channel data for all three interfering links. Data for
channels 1, 2 and 3 is shown in Fig. 5. However, due to the limitation of the
inter-body experiment, there are no direct measurements for Channels 2 and 3.
Therefore, they are modeled by overlaying the shadowed components of second
hop on-body channels, which are Channels 4 and 5 in Fig. 5, to the existing
inter-body channel data. A detailed description of this overlaying process can
be found in [9]. Here, different combinations of subjects are used in the
overall analysis, as shown in Table I.
Figure 5: Overlaying of inter-body and on-body channels TABLE I: Choice of WBAN-of-interest and Interfering WBAN: each X indicates an independent analysis set. | Interfering WBAN
---|---
Subject-of-interest | #1 | #2 | #3 | #4 | #5 | #6
Subject #1 | | x | x | x | x | x
Subject #2 | x | | x | x | x | x
## III Simulation Results with Varied Relay Positions
In this section, the performance of two-hop opportunistic relaying, using (2),
with varied relay positions is compared with a system using single-link
communications. We also investigate the impact of different hub locations and
levels of shadowing on the overall WBAN-of-interest performance. The analysis
is based on received SINR outage probability, where the received SINR is
derived from recorded signal strength, interference is simulated as described
in the last section and noise is AWGN with
$|\varepsilon_{\mathrm{noise}}|=-95$ dBm, see (1). In this paper, the outage
probability at a given SINR threshold $\nu_{\mathrm{th}}$ is defined as the
probability of the SINR value of a random received packet being smaller than
$\nu_{\mathrm{th}}$, i.e. $Pr(\nu\textless\nu_{\mathrm{th}})$. Performance is
compared with respect to a SINR threshold value at outage probabilities of 1%
and 10%, where 10% corresponds to the guideline for 10% maximum packet error
rates in the IEEE 802.15.6 BAN standard [12].
In these simulations, the hub and two sensors are placed at one of the three
locations (chest, left and right hips) separately. The last sensor is placed
at one of the locations among left and right ankle, left and right wrist, left
upper arm, head and back, which correspond to the channel sounder locations in
the on-body channel data capturing experiment. In Fig. 6, it is obvious that
shadowing influences the system performance significantly. In the case of the
hub at chest, Fig. 6(a), partial shadowing raises the SINR threshold value by
about 29 dB at an outage probability of 10% over the case where no shadowing
is employed. Full shadowing gives a further 6 dB increment. Similarly, when
the hub is placed either at left or right hips, the same observation can be
made. Therefore, shadowing is able to mitigate interference from other co-
located WBANs, and hence improve the SINR performance for the WBAN-of-
interest.
Considering the orientation of interfering hubs or sensors to the WBAN-of-
interest, full shadowing is, commonly, a more realistic assumption. Therefore,
we compare the performance of two-hop cooperative communications assuming full
shadowing. In Fig. 6, a significant and consistent improvement in outage
probability performance is demonstrated using cooperative communications.
Figs. 6(a), 6(b) and 6(c) show that the proposed cooperative scheme can
provide approximately 7 dB improvement at an outage probability of 10%
regardless of the hub location. However, comparing different hub placements,
the case where the hub is at the chest shows a better overall performance in
terms of outage probability compared with the cases of the hub placed at
either the left or right hip.
(a) Outage Probability for hub at chest
(b) Outage Probability for hub at left hip
(c) Outage Probability for hub at right hip
Figure 6: SINR Outage Probability for Subjects 1 and 2, varied relay
positions.
## IV Theoretical Analysis for Fixed-Relay Scheme
In this section, analysis is once again based on two-hop relay-assisted
communications. For each analysis set marked in Table I, results are collected
for each packet. For each packet, received signal and interference power are
both recorded, and AWGN with power $|\varepsilon_{\mathrm{noise}}|=-95$ dBm is
added. An opportunistic relaying decision using (2) is made according to all
calculated SINR values for each packet (1). Here, the first and second order
statistics according to fitted distributions of the data, including SINR
distribution, level crossing rate and average outage duration, are obtained.
### IV-A Correlation between Signal and Interference
Firstly, the cross correlation between signal and interference received at the
same destinations is investigated, for all destination links. Received signal
and interference powers, $s$ and $int$ respectively, are normalised, and their
cross correlation, $\forall i$ packets, is calculated as
$\varrho_{s,int}=\frac{E\\{\left(s_{i}-E(s)\right).\left(int_{i}-E(int)\right)\\}}{\sqrt{\mathrm{var}(s)}\sqrt{\mathrm{var}(int)}}~{}.$
(3)
According to the analysis of the fixed-relay data used here the cross
correlation, $\varrho_{s,int}$ from (3), between signal and interference for
all destination links is averaged and found to be 0.27. As described for
autocorrelation in the Section II, if the cross-correlation coefficients were
consistently above 0.7, then signal and interference would be significantly
correlated.Thus, an average cross-correlation of 0.27 indicates signal and
interference are, generally, independent.
Moreover, based on the signal and interference powers recorded, we calculate
the best envelope distribution for received signal and interference powers
separately, and the best fit is found among normal, lognormal, gamma, Weibull,
Nakagami-m and Rayleigh distributions, according to the minimum negative log-
likelihood of maximum-likelihood (ML) estimation of distribution parameters.
Based on these distribution estimations, we form separate probability density
functions, $f_{S}(s)$ and $f_{I}(int)$, for signal and interference
respectively. We also calculate the empirical joint PDF of both signal and
interference for each packet, $f_{S,I}(s,int)$. We confirm the independence of
signal and interference as we find that,
$f_{S,I}(s,int)\simeq f_{S}(s).f_{I}(int)~{}.$ (4)
### IV-B SINR Distribution
The distribution of SINR for each simulation using two-hop communications with
fixed relays is found according to the ML-Estimation, the same as used for
signal and interference powers, described in Section IV.A. A lognormal best
fit for both Subjects 1 and 2 is found when the channel is less stable, while
a Nakagami-m distribution is found to best describe the channel with a larger
coherence time. The parameters of best-fitted distribution for each analysis
set are shown in Table II. To see how well the distribution approximation fits
the empirical data, the cumulative distribution functions for each results set
are derived based on the parameters found using ML-Estimation and then
compared with the corresponding outage probability $Pr(\nu<\nu_{th})$. In Fig.
7, two examples, Subject 1 with interfering Subject 4 and Subject 2 with
interfering Subject 5, are shown. It can be observed that there is a very good
distribution fit to experimental results.
TABLE II: Distribution of the SINR values, Fixed Relay Scheme, across analysis
sets.
Subject-of-interest | Interfering Subject | Distribution | Parameters | Comments
---|---|---|---|---
Subject #1 | Subject #2 | normal | $\mu_{n}=16.3322,\sigma_{n}=5.1008$ | On-body channel gain data of Subject 1, from first sample to $\sim$40000 th sample (i.e., first 80 minutes), is very stable, as shown in Fig. 3
Subject #3 | Nakagami-m | $m=1.3618,w=254.5702$
Subject #4 | Nakagami-m | $m=1.6476,w=268.4434$
Subject #5 | Nakagami-m | $m=1.0556,w=212.0279$
Subject #6 | Weibull | $k=2.2253,\lambda=15.0594$
Subject #1 | Subject #2 | lognormal | $\mu=2.1292,\sigma=0.6879$ | Channel gain data starting from $\sim$40000th sample is far less stable than start of dataset, as shown in Fig. 3
Subject #3 | gamma | $a=2.5504,b=2.7798$
Subject #4 | lognormal | $\mu=2.1374,\sigma=0.7004$
Subject #5 | lognormal | $\mu=1.5759,\sigma=0.7354$
Subject #6 | lognormal | $\mu=1.5933,\sigma=0.8602$
Subject #2 | Subject #1 | lognormal | $\mu=2.4652,\sigma=0.9402$ |
Subject #3 | lognormal | $\mu=2.3622,\sigma=1.0179$
Subject #4 | gamma | $a=1.2619,b=12.9156$
Subject #5 | lognormal | $\mu=2.2883,\sigma=1.0579$
Subject #6 | lognormal | $\mu=2.5453,\sigma=0.9327$
$\\{\mu_{n},\sigma_{n}\\}$, mean and standard deviation of normal
distribution; $\\{m,w\\}$, shape and spread parameters of Nakagami-m
distribution;$\\{k,\lambda\\}$, shape and scale parameters of Weibull
distribution; $\\{\mu,\sigma\\}$, log-mean and log-standard deviation of
lognormal distribution; $\\{a,b\\}$, shape and scale parameters of gamma
distribution.
Figure 7: Experimental and derived outage probability comparison for WBANs-of-
interest on Subjects 1 and 2
### IV-C Level Crossing Rate
In this paper, level crossing rate (LCR) is defined as the average rate at
which a received packet’s SINR value going below a particular threshold. In
the simulation, the LCR value at threshold $\nu_{th}$ is calculated as
$LCR_{\nu_{\mathrm{th}}}=\frac{n}{\sum\limits_{i=1}^{n-1}{t_{i,i+1}}}~{},$ (5)
where $n$ is the total number of crossings at threshold $\nu_{\mathrm{th}}$,
and $t_{\mathrm{i,i+1}}$ indicates the time between two consecutive crossings.
Here, we also derive the theoretical level crossing rate for every simulation
based on the distribution parameters shown in Table II, and compare with the
corresponding experimental LCR. Analysis is performed on simulation for the
less stable on-body channel, i.e. Rows 6 – 15 in Table II. The theoretical
level crossing rate for lognormal and gamma distributions are,
$\displaystyle LCR_{ln}(\nu_{th})$
$\displaystyle=f_{D}\exp\left(\frac{-(\ln(\nu_{th})-\mu)^{2}}{2\sigma^{2}}\right)~{},$
(6) $\displaystyle LCR_{gamma}(\nu_{th})$
$\displaystyle=\frac{f_{D}\sqrt{2\pi}\nu_{th}^{a-0.5}}{\Gamma(a)b^{a-0.5}}\exp(-\nu_{th}/b)~{},$
(7)
where $\nu_{th}$ is the linear value of the threshold; $\ln(\cdot)$ is the
natural logarithm and $\Gamma(\cdot)$ is the standard gamma function and
$f_{D}$ indicates the Doppler spread which is found to be 1.0 Hz. It is found
that the theoretical LCR curves match the experimental results for all
simulations with different subjects-of-interest and interfering subjects.
Examples are demonstrated as shown in Fig. 8. Therefore, the fitted
distribution and their ML-estimated parameters provide very good estimates for
the first and second-order statistics of SINR.
Figure 8: Experimental and derived level crossing rate comparison for WBANs-
of-interest on Subjects 1 and 2
### IV-D Average Outage Duration
Average outage duration (AOD), analogous to average fading duration for
channel gain profiles, is defined as the average period of continuous time
that SINR values stay below a threshold value $\nu_{\mathrm{th}}$ across one
or more packets. Assume in a given duration, there are a total of $n\geq 1$
periods when received packets have SINR values lower than $\nu_{\mathrm{th}}$
and the length of each period is denoted as $t_{i}$, then the corresponding
AOD is calculated as
$AOD_{\nu_{th}}=\frac{\sum\limits_{i=1}^{n}{t_{i}}}{n}~{}.$ (8)
The experimental AOD result is then compared with corresponding theoretical
AOD, which is derived as
$AOD_{theo}(\nu_{th})=\frac{F(\nu_{th})}{LCR_{\nu_{th}}}~{},$ (9)
where $F(\nu_{th})$ is the cumulative distribution function of the SINR
values, which is also known as the outage probability at given SINR threshold;
$LCR_{\nu_{\mathrm{th}}}$ is the theoretical level crossing rate for the same
simulation. From the examples shown in Fig. 9, we can see that theoretical AOD
matches experimental results very well.
Figure 9: Experimental and derived average outage duration comparison for
WBANs-of-interest on Subjects 1 and 2
## V Simulation Results for the Fixed Relay Scheme
In this section, the experimental performance of two-hop opportunistic
relaying cooperative communications using relays at left and right hips is
compared with the system using single-link communications. The analysis is
presented, with respect to the SINR values across packets, for the system’s
first order statistics — in terms of outage probability; and second order
statistics — in terms of level crossing rate and average outage duration.
Simulation results for using Subjects 1 and 2 as subject-of-interest are shown
separately. In addition, on-body channels of Subject 1 shows two distinct
characteristics after a sudden change of experiment environment, as shown in
Fig. 3. Channels are very stable at an earlier stage, i.e., first 80 minutes,
and then they become less stable. Therefore, the performance of the proposed
scheme in these two circumstances is also presented.
### V-A Experimental Outage Probability
Fig. 10(a) \- 10(c) shows the resultant outage probability for Subjects 1 and
2. In the figures, the overall outage probability is obtained by averaging
across the entire set of outage probability for each subject-of-interest. For
Subject 1 when the channel is less stable, it is shown in Fig. 10(a) that
cooperative communications provides an average of 3 dB and 7 dB improvement
over single link communications at outage probabilities of 10% and 1%
respectively. Looking at individual analysis sets, the improvement can reach
up-to 11 dB increase in threshold value at an outage probability of 1% when
Subject 2 is taken as an interferer. In addition, simulation on Subject 2
shows similar improvements, with 3 dB and more than 10 dB increase at 10% and
1% respectively (As shown in Fig. 10(b)).
In contrast, simulations on Subject 1 with a very stable channel leads to a
different result. In Fig. 10(c), it is shown that the cooperative
communications doesn’t provide significant performance improvement in terms of
outage probability. However, while comparing with Fig. 10(a) where there is a
less stable channel, the system in a very stable channel has a 10 dB
performance gain, for both two hop cooperative and single-link communication
schemes, over the less stable channel.
(a) Outage Probability for Subject 1, less stable channel
(b) Outage Probability for Subject 2
(c) Outage Probability for Subject 1, very stable channel
Figure 10: SINR Outage Probability for Subjects 1 and 2, Fixed Relay Scheme
### V-B Experimental Level Crossing Rate
As defined in Section IV.C, level crossing rate (LCR) is the average rate of a
received packet’s SINR value going below a given threshold
$\nu_{\mathrm{th}}$. It is calculated as (5). In Fig. 11(a) and Fig. 11(b), it
is shown that cooperative communications reduces LCR at low SINR threshold
values significantly. For Subject 1 with less stable channels, the proposed
scheme raises the threshold value by an average of 4 dB at an LCR of 1 Hz and
an average of 7.5 dB at an LCR of 0.1 Hz. In terms of Subject 2, the
improvement, as shown in Fig. 11(b), is about 2.5 dB and 3 dB at an LCR of 1
Hz and 0.1 Hz respectively. Furthermore, the same observation as that made for
outage probability can be made from Fig. 11(c) when on-body channels are more
stable. In this case of larger channel coherence time, no real performance
advantage in terms of level crossing rate is shown for using cooperative
communication scheme over single-link communication schemes.
(a) Level Crossing Rate for Subject 1, less stable channel
(b) Level Crossing Rate for Subject 2
(c) Level Crossing Rate for Subject 1, very stable channel
Figure 11: Level Crossing Rate of SINR for Subjects 1 and 2, Fixed Relay
Scheme
### V-C Experimental Average Outage Duration
Average outage duration (AOD) of time varying SINR, is calculated as (8). Fig.
12(a) and Fig. 12(b) presents the simulation results of AOD of both
corresponding subjects-of-interest and interfering subjects. For both of them,
the curves for single-link and two-hop cooperative communication schemes
overlap at most of the SINR threshold values. In other words, there is no real
performance advantage of using the proposed scheme for WBANs coexistence in
terms of average outage duration.
(a) Average Outage Duration for Subject 1
(b) Average Outage Duration for Subject 2
Figure 12: Average Outage Duration of SINR for Subjects 1 and 2, Fixed Relay
Scheme
## VI Conclusion
This paper has studied the performance of two-hop opportunistic relaying
cooperative communications with respect to interference mitigation and
coexistence enhancement. TDMA was employed as the intra-network access scheme,
as well as the access scheme across multiple WBANs. Three-branch opportunistic
relaying was used and the best path was selected between an active sensor and
the hub. Two different relay implementations were investigated to provide
diversity gain at the hub. The first one uses two inactive sensors as relays,
while the other implementation adds two additional fixed relays, not providing
sensor functions, at the left and right hips. Empirical on-body channel
measurements were used as a reliable and realistic intra-WBAN channel model
for both implementations. However, there were two implementations of inter-
WBAN channel models. For the implementation of varied relay positions, it was
simulated by the superposition of free space path loss, shadowing and small
scale fading. In terms of simulation using fixed relays, empirical inter-body
channel measurements were used in the inter-WBAN channel model of a practical
WBAN working environment. The performance of both schemes were compared with
traditional single-link communications where no relays are used. The analysis
was performed with respect to first and second order statistics of received
packets’ SINR values. In addition, distributions of SINR values were found
using maximum-likelihood (ML) estimation. Theoretical outage probability,
level crossing rate and average outage duration were derived based on the
distribution parameters, and shown to match empirical results well.
When using the varied relay positions implementation, it has been found that
the use of cooperative communications can mitigate interference by providing
an average of 7 dB improvement at SINR outage probability of 10% over the use
of single-link communications. This improvement is consistent no matter where
the location of the hub is, but better overall performance is obtained when
the hub is placed at the chest. In addition, body shadowing can assist WBAN
coexistence significantly. In terms of using the implementation with fixed
relay positions, similar conclusions on the the effectiveness of cooperative
communications can be made. The average improvements in outage probability at
10% and 1% are 3 dB and 8 dB respectively. Level crossing rate reduces
significantly at low SINR threshold values when using the proposed scheme.
However, it does not provide improvement for coexistence in terms of average
outage duration of SINR. Based on ML estimation, it is found that a lognormal
distribution best describes the probability distribution of the received
packets’ SINR when the channel coherence time is smaller. In contrast, a
Nakagami-m distribution fits SINR better when the channel coherence time is
very large, i.e. the channel is highly stable.
## Acknowledgment
NICTA is funded by the Australian Government as represented by the Department
of Broadband, Communications and the Digital Economy and the Australian
Research Council through the ICT Centre of Excellence program.
## References
* [1] A. Astrin, “The promise of body area networks IEEE 802.15.6,” _ISMICT 2007_ , Dec. 2007.
* [2] D. Lewis, “IEEE 802.15.6 call for applications - summary ID: 802.15-05-0407-05,” IEEE submission, July 2008.
* [3] ABIresearch, “Body Area Networks for Sports and Healthcare,” [Online] https://www.telematicsresearch.net/research/product/1005246-body-area-networks-for-sports-and-healthca/, 2011\.
* [4] A. Astrin _et al._ , “IEEE Standard for Local and metropolitan area networks part 15.6: Wireless Body Area Networks: IEEE Std 802.15.6-2012,” Feb. 2012.
* [5] J. Dong and D. Smith, “Cooperative receive diversity for coded gfsk body-area communications,” _Electronics Letters_ , vol. 47, no. 19, pp. 1098 –1100, Sep. 2011.
* [6] D. Smith and D. Miniutti, “Cooperative body-area-communications: First and second-order statistics with decode-and-forward,” in _Wireless Communications and Networking Conference (WCNC), 2012 IEEE_ , Paris, France, Apr. 2012.
* [7] P. Ferrand, M. Maman, C. Goursaud, J.-M. Gorce, and L. Ouvry, “Performance evaluation of direct and cooperative transmissions in body area networks,” _Annals of Telecommunications_ , vol. 66, pp. 213–228, 2011.
* [8] J. Dong and D. Smith, “Cooperative body-area-communications: Enhancing coexistence without coordination between networks,” in _Personal, Indoor and Mobile Radio Communication (PIMRC), 2012 IEEE_ , Sydney, Australia, September. 2012.
* [9] ——, “Opportunistic relaying in wireless body area networks: Coexistence performance,” _CoRR_ , vol. abs/1212.4489, 2012.
* [10] Y. Chen _et al._ , “Cooperative communications in ultra-wideband wireless body area networks: Channel modeling and system diversity analysis,” _Selected Areas in Communications, IEEE Journal on_ , vol. 27, no. 1, pp. 5 –16, Jan. 2009.
* [11] E.-J. Kim, S. Youm, T. Shon, and C.-H. Kang, “Asynchronous inter-network interference avoidance for wireless body area networks,” _The Journal of Supercomputing_ , pp. 1–18, 2012. [Online]. Available: http://dx.doi.org/10.1007/s11227-012-0840-4
* [12] B. Zhen, M. Patel, S. Lee, and E. Won, “Body Area Network (BAN) Technical Requirements,” July 2008, IEEE802.15.6 technical contribution, document ID:15-08-0037-01-0006-ieee-802-15-6-technical-requirements-document-v-4-0.
* [13] D. Smith, L. Hanlen, D. Rodda, B. Gilbert, J. Dong, and V. Chaganti, “Body Area Network Radio Channel Measurement Set,” [Online] http://opennicta.com/datasets, 2012.
* [14] A. Zhang, D. Smith, D. Miniutti, L. Hanlen, D. Rodda, and B. Gilbert, “Performance of piconet co-existence schemes in wireless body area networks,” in _Wireless Communications and Networking Conference (WCNC), 2010 IEEE_ , Sydney, Australia, Apr. 2010, pp. 1 –6.
* [15] D. Smith _et al._ , “Statistical characterization of the dynamic narrowband body area channel,” _Applied Sciences on Biomedical and Communication Technologies, 2008. ISABEL ’08. First International Symposium on_ , pp. 1–5, Oct. 2008.
* [16] D. Smith, L. Hanlen, J. Zhang, D. Miniutti, D. Rodda, and B. Gilbert, “First- and second-order statistical characterizations of the dynamic body area propagation channel of various bandwidths,” _Annals of Telecommunications_ , vol. 66, pp. 187–203, 2011.
* [17] D. Smith, J. Zhang, L. Hanlen, D. Miniutti, D. Rodda, and B. Gilbert, “Temporal correlation of dynamic on-body area radio channel,” _Electronics Letters_ , vol. 45, no. 24, pp. 1212–1213, 2009.
* [18] A. Paulraj, R. Nabar, and D. Gore, _Introduction to Space-Time Wireless Communications_ , 1st ed. New York, NY, USA: Cambridge University Press, 2008.
|
arxiv-papers
| 2013-05-30T02:58:34 |
2024-09-04T02:49:45.870086
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jie Dong and David Smith",
"submitter": "Jie Dong",
"url": "https://arxiv.org/abs/1305.6992"
}
|
1305.7055
|
# Bare Higgs mass and potential at ultraviolet cutoff
Yuta Hamada Hikaru Kawai Department of Physics, Kyoto University, Kyoto
606-8502, Japan Kin-ya Oda Department of Physics, Osaka University, Osaka
560-0043, Japan
###### Abstract
We first review the current status of the top mass determination paying
attention to the difference between the $\overline{\text{MS}}$ and pole
masses. Then we present our recent result on the bare Higgs mass at a very
high ultraviolet cutoff scale.
## I Introduction
It is more and more likely that the 125 GeV particle discovered at the Large
Hadron Collider (LHC) Aad:2012tfa ; Chatrchyan:2012ufa is the Standard Model
(SM) Higgs. Its couplings to the $W$ and $Z$ gauge bosons, to the top and
bottom quarks, and to the tau lepton are all consistent to those in the SM
within one standard deviation even though their values vary two orders of
magnitude, see e.g. Ref. Giardino:2013bma . No hint of new physics beyond the
SM has been found so far at the LHC up to 1 TeV. It is important to examine to
what scale the SM can be a valid effective description of nature.
In Ref. Froggatt:1995rt , Froggatt and Nielsen have predicted the top and
Higgs masses to be $173\pm 5\,\text{GeV}$ and $135\pm 9\,\text{GeV}$,
respectively, based on the assumption that the SM Higgs potential must have
another minimum at the Planck scale and that its height is order-of-magnitude-
wise degenerate to the SM one. (This assumption is equivalent to the vanishing
Higgs quartic coupling and its beta function at the Planck scale.) The success
of this prediction indicates that at least the top-Higgs sector of the SM is
not much altered up to a very high ultraviolet (UV) cutoff scale.
As all the parameters in the SM are fixed by the Higgs mass determination, we
can now obtain the _bare_ parameters at the UV cutoff scale, which then become
important inputs for a given UV completion of the SM. If the UV theory fails
to fit them, it is killed.
The parameters in the SM are dimensionless except for the Higgs mass (or
equivalently its vacuum expectation value (VEV)). The dimensionless bare
coupling constants can be approximated by the running ones at the UV cutoff
scale, see e.g. Appendix of Ref. Hamada:2012bp . The latter can be evaluated
through the standard renormalization group equations (RGEs) once the low
energy inputs are given. After fixing all the dimensionless bare couplings,
the last remaining one is the bare Higgs mass which is the main subject of
this work.
## II Top quark Yukawa coupling at top mass scale
The largest ambiguity for the Higgs mass parameter is coming from the low
energy input of the top Yukawa coupling at the top mass scale. Let us review
the present status of its determination. Important point is the distinction
between the modified minimal subtraction ($\overline{\text{MS}}$) mass and the
pole one, which are utilized, respectively, in the $\overline{\text{MS}}$ and
on-shell schemes.
Currently the most accurate value of the top quark mass is obtained from
combination of the Tevatron data, basically reconstructed as an invariant mass
of its decay products CDF:2013jga :
$\displaystyle M_{t}^{\text{Tevatron}}$ $\displaystyle=173.20\pm
0.87\,\text{GeV},$ (1)
whereas the similar analysis of the LHC data gives LHC top mass ;
Deliot:2013hc : $M_{t}^{\text{LHC}}=173.3\pm 1.4\,\text{GeV}$. (If we naively
combine these two results, we get $M_{t}^{\text{inv}}=173.2\pm
0.7\,\text{GeV}$, which is of 0.4% accuracy.) However, the authors of Ref.
Alekhin:2012py criticize that the top quark mass, measured at the Tevatron
and LHC via kinematical reconstruction from the top quark decay products and
comparison to Monte Carlo simulations, is not necessarily the pole mass
$M_{t}$ but is merely the mass parameter in the Monte Carlo program which does
not resort to any given renormalization scheme. The point is that the mass of
the colored top quark is reconstructed from the color singlet final states.111
Note however that the pole mass of the colored quark is well defined to all
orders in perturbation theory and that its infrared renormalon ambiguity
appears only at the non-perturbative level of order $\Lambda_{\text{QCD}}$.
To circumvent this problem, they propose to determine the
$\overline{\text{MS}}$ top quark mass directly from the dependence of the
inclusive $t\bar{t}$ cross section on it. In Ref. Alekhin:2012py , the
observed values at Tevatron:
$\displaystyle\sigma(p\bar{p}\to t\bar{t}+X)$
$\displaystyle=7.56^{+0.63}_{-0.56}\,\text{pb}\quad\text{(D0)}\qquad\text{and}\qquad
7.50^{+0.48}_{-0.48}\,\text{pb}\quad\text{(CDF)}$ (2)
are combined and fit by the theoretical prediction, which is obtained by using
four different parton distribution functions (PDFs) at the NNLO and by
including the NNLO QCD contributions to $\sigma(p\bar{p}\to t\bar{t}+X)$. The
resultant value of the $\overline{\text{MS}}$ running top mass at the top mass
scale becomes Alekhin:2012py :
$\displaystyle m_{t}^{\text{QCD}}(M_{t})$ $\displaystyle=163.3\pm
2.7\,\text{GeV}.$ (3)
In the above computation, the NLO electroweak (EW) radiative corrections
($\propto\alpha\alpha_{s}$) to $\sigma(p\bar{p}\to t\bar{t}+X)$ are neglected.
The ratio of the sum of such EW corrections to the $t\bar{t}$ total cross
section at the Tevatron in the on-shell scheme is shown to be less than 0.2%
for the Higgs mass 120–200 GeV and the top pole mass 165–180 GeV Kuhn:2006vh .
Theoretically the $\overline{\text{MS}}$ mass in Eq. (3) is related to the
pole mass $M_{t}$ by
$\displaystyle m_{t}^{\text{QCD}}(M_{t})$
$\displaystyle=M_{t}\left(1+\delta_{t}^{\text{QCD}}(M_{t})\right),$ (4)
where up to the NNLO QCD corrections of $O(\alpha_{s}^{3})$, see e.g. Ref.
Jegerlehner:2012kn ,
$\displaystyle\delta_{t}^{\text{QCD}}(M_{t})$ $\displaystyle=-{4\over
3}{\alpha_{s}(M_{t})\over\pi}-9.125\left(\alpha_{s}(M_{t})\over\pi\right)^{2}-80.405\left(\alpha_{s}(M_{t})\over\pi\right)^{3},$
(5)
with $\alpha_{s}(\mu)=g_{s}^{2}(\mu)/4\pi$ being the strong coupling in the
six flavor $\overline{\text{MS}}$ scheme. This relation result in the pole
mass Alekhin:2012py
$\displaystyle M_{t}$ $\displaystyle=173.3\pm 2.8\,\text{GeV}.$ (6)
In Ref. Alekhin:2012py the pole mass of the top quark $M_{t}$ is also
directly extracted from the NNLO theory prediction using the on-shell scheme.
The resultant central value varies $169.9$–$172.7$ GeV, depending on the PDF,
with the error less than 2.4 GeV for each. These values are consistent to Eq.
(6). To summarize, the derived pole mass is close to the experimentally
obtained invariant mass (1).
It is customary to define the $\overline{\text{MS}}$ running VEV $v(\mu)$ in
such a way that
$\displaystyle-m^{2}(\mu)=\lambda(\mu)\,v^{2}(\mu)$ (7)
holds for the potential
$\displaystyle\mathcal{V}=m^{2}\phi^{\dagger}\phi+\lambda\left(\phi^{\dagger}\phi\right)^{2},$
(8)
with $\left\langle\phi\right\rangle=v/\sqrt{2}$. Then the
$\overline{\text{MS}}$ top mass is commonly defined as, see e.g. Ref.
Jegerlehner:2012kn ,
$\displaystyle m_{t}(\mu)$ $\displaystyle={y_{t}(\mu)\,v(\mu)\over\sqrt{2}}.$
(9)
This definition of the $\overline{\text{MS}}$ mass leads to Jegerlehner:2012kn
$\displaystyle m_{t}(M_{t})$
$\displaystyle=m_{t}^{\text{QCD}}(M_{t})+M_{t}\,\Delta_{t}^{\text{EW}}(M_{t}),$
(10)
where, taking into account up to NLO EW contributions of
$O(\alpha\alpha_{s})$,
$\displaystyle\Delta_{t}^{\text{EW}}(M_{t})$
$\displaystyle=0.0664-0.00115\left({M_{H}\over\text{GeV}}-125\right).$ (11)
This discrepancy (11) is due to the definition of the $\overline{\text{MS}}$
top mass via Eq. (9), and is dominantly coming from the tadpole contribution
to the shift of $v(\mu)$. If we instead use the definition of the
$\overline{\text{MS}}$ mass $m_{t}^{\text{QCD}}(\mu)=y_{t}(\mu)V/\sqrt{2}$
with $V=\left(\sqrt{2}G_{\mu}\right)^{-1/2}=246.22$ GeV, where
$G_{\mu}=1.1663787(6)\times 10^{-5}\,\text{GeV}^{-2}$ is the Fermi constant
determined from the muon life time, then we would get the one given in Eq.
(4).
Plugging Degrassi:2012ry
$\displaystyle g_{s}(M_{t})$
$\displaystyle=1.1645+0.0031\left({\alpha_{s}(M_{Z})-0.1184\over
0.0007}\right)-0.00046\left({M_{t}\over\text{GeV}}-173.15\right)$ (12)
into Eq. (10), we get
$\displaystyle m_{t}(M_{t})$
$\displaystyle=M_{t}\left[1.00658-0.00041\left({\alpha_{s}(M_{Z})-0.1184\over
0.0007}\right)+0.00006\left({M_{t}\over\text{GeV}}-173.15\right)-0.00115\left({M_{H}\over\text{GeV}}-125\right)\right].$
(13)
The $\overline{\text{MS}}$ mass (10) becomes _larger_ than the top quark pole
mass $M_{t}$ Jegerlehner:2012kn .
Let us review the derivation of $\Delta_{t}^{\text{EW}}(M_{t})$ more in
detail. First the $\overline{\text{MS}}$ running VEV $v(\mu)$ is obtained and
then it is multiplied to the $\overline{\text{MS}}$ running Yukawa
$y_{t}(\mu)$ in order to obtain the running mass $m_{t}(\mu)$ in Eq. (10).
$v(\mu)$ can be read from the Fermi constant
$G_{F}(\mu)=1/\sqrt{2}v^{2}(\mu)$:
$\displaystyle G_{\mu}$
$\displaystyle=G_{F}(\mu)\left(1+\Delta_{G_{F},\alpha}+\Delta_{G_{F},\alpha\alpha_{s}}+\cdots\right)=G_{F}(\mu)\left[1+{\alpha_{2}(\mu)\over
4\pi}{m_{t}^{4}(\mu)\over
m_{W}^{2}(\mu)\,m_{H}^{2}(\mu)}\left(6-12\ln{m_{t}(\mu)\over\mu}\right)+\cdots\right].$
(14)
The $O(\alpha)$ and $O(\alpha\alpha_{s})$ contributions
$\Delta_{G_{F},\alpha}$ and $\Delta_{G_{F},\alpha\alpha_{s}}$ are given in
Eqs. (A.3) and (A.6) in Ref. Bezrukov:2012sa . The dominant tadpole
contribution is picked up in the last step in Eq. (14) for explicitness. The
resultant $\overline{\text{MS}}$ VEV is $v(M_{t})\sim 260$ GeV at the top mass
scale. On the other hand, the $\overline{\text{MS}}$ Yukawa coupling is given
by Hempfling:1994ar
$\displaystyle y_{t}(\mu)$
$\displaystyle={M_{t}\over\sqrt{2}V}\left(1+\delta_{t}^{\text{QCD}}(\mu)+\delta_{t,\alpha}^{\text{QED}}(\mu)+\delta_{t,\alpha}^{W}(\mu)+\delta_{t,\alpha\alpha_{s}}^{\text{EW}}(\mu)+\cdots\right),$
(15)
where the $O(\alpha)$ corrections are
$\displaystyle\delta_{t,\alpha}^{\text{QED}}(\mu)$
$\displaystyle={Q_{t}^{2}\alpha(\mu)\over
4\pi}\left(6\log{M_{t}\over\mu}-4\right),$ (16)
$\displaystyle\delta_{t,\alpha}^{W}(\mu)$
$\displaystyle={G_{\mu}\,m_{t}^{2}(\mu)\over
8\pi^{2}\sqrt{2}}\left[-\left(2N_{c}+3\right)\ln{M_{t}\over\mu}+{N_{c}\over
2}+4-r+2r\left(2r-3\right)\ln(4r)-8r^{2}\left({1\over
r}-1\right)^{3/2}\arccos\sqrt{r}\right],$ (17)
with $Q_{t}=2/3$, $N_{c}=3$, and $r=M_{H}^{2}/4M_{t}^{2}$. The resultant
explicit analytic formula of $\Delta_{t}^{\text{EW}}(M_{t})$ is given in Ref.
Jegerlehner:2003py that takes into account up to the NLO EW corrections of
$O(\alpha\alpha_{s})$. The tiny $O(\alpha\alpha_{s})$ correction to the Yukawa
coupling, $\delta_{t,\alpha\alpha_{s}}^{\text{EW}}(\mu)$ in Eq. (15), can be
read off from $\Delta_{t,\alpha\alpha_{s}}^{\text{EW}}$ by subtracting the
tadpole contribution to $v(\mu)$. In Ref. Degrassi:2012ry , numerical value of
Eq. (15) is evaluated as
$\displaystyle y_{t}(M_{t})$
$\displaystyle=0.93587+0.00557\left({M_{t}\over\text{GeV}}-173.15\right)-0.00003\left({M_{H}\over\text{GeV}}-125\right)-0.00041\left(\alpha_{s}(M_{Z})-0.1184\over
0.0007\right)\pm 0.00200_{\text{th}}.$ (18)
Multiplying Eq. (18) by the running VEV $v(\mu)$ that is read from Eq. (14),
we obtain $\Delta_{t}^{\text{EW}}$, and hence the running mass (13).
## III Bare parameters at high scale
In this proceedings we show our result Hamada:2012bp based on Eq. (18) with
the pole mass (6). In Figure 1, we show a plot with the two loop RGEs,
summarized in Ref. Hamada:2012bp , for the dimensionless SM couplings, with
the low energy boundary condition for the top Yukawa as explained above,
namely with $y_{t}(M_{t})=0.93587$. $\beta_{\lambda}$ is the beta function for
the Higgs quartic coupling: $\beta_{\lambda}=d\lambda/d\ln\mu$.
$m_{B}^{2}/I_{1}$ is explained in the following.
Figure 1: RGE running of the SM couplings and of the beta function for the
quartic coupling $\beta_{\lambda}$, except for $m_{B}^{2}$ which is _not_ a
running mass but is the (quadratically divergent part of) bare Higgs mass
parameter, obtained by taking each scale on the horizontal axis to be the
cutoff $\Lambda$. $I_{1}=\Lambda^{2}/16\pi^{2}$ is the one loop integral.
In the bare perturbation theory, the renormalized Higgs mass squared parameter
is given at the one loop level by
$\displaystyle m_{R}^{2}$
$\displaystyle=m_{B,\text{1-loop}}^{2}+\left(6\lambda_{B}+\frac{3}{4}g_{YB}^{2}+\frac{9}{4}g_{2B}^{2}-6y_{tB}^{2}\right)I_{1}+\delta
m^{2},$ (19)
where $I_{1}=\int^{\Lambda}d^{4}p/(2\pi)^{4}p^{2}=\Lambda^{2}/16\pi^{2}$ is
the quadratically divergent one loop integral over the Euclidean momentum. In
obtaining the expression (19), we assume existence of an underlying gauge
invariant regularization, such as string theory, and freely shift the
integrated momenta. The requirement that the sum in the parentheses in Eq.
(19) to be zero is the celebrated Veltman condition. In a mass independent
renormalization scheme, including the dimensional regularization, the bare
mass squared $m_{B}^{2}$ is chosen in such a way that the renormalized mass
parameter becomes zero, $m_{R}^{2}=0$, when $\delta m^{2}=0$; and then non-
zero $\delta m^{2}$ is introduced as a perturbation. (This choice of the bare
mass $m_{B}^{2}$ to cancel the quadratic divergence is automatic in the
dimensional regularization scheme.) Consequently the bare mass $m_{B}^{2}$
contains a quadratic divergence: $\Lambda^{2}$, whereas the running mass
$\delta m^{2}$ only logarithmic one: $\log\Lambda$. Note that this
cancellation of $\Lambda^{2}$ by $m_{B}^{2}$ is done once and for all, and
then we never see $\Lambda^{2}$. Bardeen has argued that therefore the
quadratic divergence is not a real problem, see e.g. Ref. Iso:2013aqa for a
recent review. However, e.g. in obtaining a low energy effective field theory
from string theory, this procedure of matching $\Lambda^{2}$ by $m_{B}^{2}$ is
not a fake, and we focus on this largest part $m_{B}^{2}$ in the bare
Lagrangian, neglecting the subleasing $\delta m^{2}\propto v^{2}\log\Lambda$.
In Ref. Hamada:2012bp we have obtained the bare Higgs mass at two loop orders
in the bare perturbation theory:
$\displaystyle m_{B,\,\text{2-loop}}^{2}$
$\displaystyle=-\bigg{\\{}9y_{tB}^{4}+y_{tB}^{2}\left(-\frac{7}{12}g_{YB}^{2}+\frac{9}{4}g_{2B}^{2}-16g_{3B}^{2}\right)+\frac{77}{16}g_{YB}^{4}+\frac{243}{16}g_{2B}^{4}+\lambda_{B}\left(-18y_{tB}^{2}+3g_{YB}^{2}+9g_{2B}^{2}\right)-10\lambda_{B}^{2}\bigg{\\}}\,I_{2},$
(20)
which realizes $m_{R}^{2}=0$ for $\delta m^{2}=0$ at this order, where
$\displaystyle I_{2}$
$\displaystyle=\int{d^{4}p\over(2\pi)^{4}}\int{d^{4}q\over(2\pi)^{4}}{1\over
p^{2}q^{2}(p+q)^{2}}$ (21)
is the quadratically divergent two loop integral over the Euclidean momenta.
As a non-trivial check of the consistency of our treatment, we have confirmed
that the coefficients of the infrared divergent integral
$\int{d^{4}p\over(2\pi)^{4}p^{4}}\int{d^{4}q\over(2\pi)^{4}q^{2}}$ cancel out
in each gauge invariant set of diagrams. We note that our two loop computation
is for the quadratically divergent part $m_{B}^{2}$ and is irrelevant to the
higher loop result $\propto\left(\log\Lambda\right)^{n}\Lambda^{2}$, obtained
in Ref. Einhorn:1992um and used in Ref. Kolda:2000wi ; Casas:2004gh , which
would correspond to the effects of the RGE running of the dimensionless
couplings in our language.
In obtaining the two-loop corrected bare mass
$m_{B}^{2}=m_{B,\text{1-loop}}^{2}+m_{B,\text{2-loop}}^{2}$, one needs a
relation between the one- and two-loop integrals $I_{1}$ and $I_{2}$, which is
necessarily regularization scheme dependent. When we employ
$\displaystyle\int{d^{4}p\over p^{2}}=\int_{\epsilon}^{\infty}d\alpha\int
d^{4}p\,e^{-\alpha p^{2}},$ (22)
we get
$\displaystyle I_{2}={I_{1}\over 16\pi^{2}}\ln{2^{6}\over 3^{3}}\simeq
0.005I_{1}.$ (23)
With the blue solid line in Fig. (2), we plot $m_{B}^{2}/I_{1}$ when the UV
cutoff is at the Planck scale, that is, when $I_{1}=M_{P}^{2}/16\pi^{2}$. The
blue dashed line is the same with the 1-loop only bare mass
$m_{B,\text{1-loop}}^{2}/I_{1}$. We see that the two-loop correction is small,
so is the regularization dependence. For comparison, we also plot the quartic
coupling $\lambda$ at the Planck scale with the red dotted line. We see that
both become small for $M_{t}\simeq 170$ GeV.
Figure 2: The blue solid (dashed) line corresponds to the one-plus-two-loop
(one-loop) bare mass $m_{B}^{2}$ ($m_{B,\,\text{1-loop}}^{2}$) in units of
${M_{\text{Pl}}^{2}/16\pi^{2}}$ for $\Lambda=M_{\text{Pl}}$. For comparison,
we also plot the quartic coupling $\lambda$ at the Planck scale with the red
dotted line.
## IV Summary
We have computed two loop correction to the quadratically divergent part of
the bare SM Higgs mass in the bare perturbation theory. We have found that in
generic regularizations the two loop correction to the bare mass is small.
Therefore the regularization dependence is not large. Both the resultant bare
Higgs mass and quartic coupling can become small if the top quark pole mass is
around 170 GeV. Possible consequences of this result will be presented
elsewhere.
###### Acknowledgements.
We are grateful to Abdelhak Djouadi, Mikhail Yu. Kalmykov, and Bernd A. Kniehl
for valuable comments.
## References
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* (3) P. P. Giardino, K. Kannike, I. Masina, M. Raidal and A. Strumia, “The Universal Higgs Fit,” arXiv:1303.3570 [hep-ph].
* (4) C. D. Froggatt and H. B. Nielsen, “Standard Model Criticality Prediction: Top Mass 173 +- 5-Gev and Higgs Mass 135 +- 9-Gev,” Phys. Lett. B 368 (1996) 96 [hep-ph/9511371].
* (5) Y. Hamada, H. Kawai and K. Oda, “Bare Higgs Mass at Planck Scale,” arXiv:1210.2538 [hep-ph].
* (6) Tevatron Electroweak Working Group for the CDF and D0 Collaborations, “Combination of CDF and D0 results on the mass of the top quark using up to 8.7 fb-1 at the Tevatron,” arXiv:1305.3929 [hep-ex].
* (7) The ATLAS and CMS Collaborations, “Combination of ATLAS and CMS results on the mass of the top quark using up to 4.9 fb-1 of data,” ATLAS-CONF-2012-095, CMS PAS TOP-12-001 (2012), http://cds.cern.ch/record/1460441.
* (8) F. Deliot et al. [ATLAS and D0 Collaborations], “Combination of the top-quark mass measurements from the Tevatron and from the LHC colliders,” arXiv:1302.0830 [hep-ex].
* (9) S. Alekhin, A. Djouadi and S. Moch, “The Top Quark and Higgs Boson Masses and the Stability of the Electroweak Vacuum,” Phys. Lett. B 716 (2012) 214 [arXiv:1207.0980 [hep-ph]].
* (10) J. H. Kuhn, A. Scharf and P. Uwer, “Electroweak Effects in Top-Quark Pair Production at Hadron Colliders,” Eur. Phys. J. C 51 (2007) 37 [hep-ph/0610335].
* (11) F. Jegerlehner, M. Y. .Kalmykov and B. A. Kniehl, “On the Difference Between the Pole and the Msbar Masses of the Top Quark at the Electroweak Scale,” Phys. Lett. B 722 (2013) 123 [arXiv:1212.4319 [hep-ph]].
* (12) F. Bezrukov, M. Y. .Kalmykov, B. A. Kniehl and M. Shaposhnikov, “Higgs Boson Mass and New Physics,” JHEP 1210 (2012) 140 [arXiv:1205.2893 [hep-ph]].
* (13) R. Hempfling and B. A. Kniehl, “On the Relation Between the Fermion Pole Mass and Ms Yukawa Coupling in the Standard Model,” Phys. Rev. D 51 (1995) 1386 [hep-ph/9408313].
* (14) F. Jegerlehner and M. Y. .Kalmykov, “$O(\alpha\alpha_{s})$ correction to the pole mass of the t-Quark within the Standard Model,” Nucl. Phys. B 676 (2004) 365 [hep-ph/0308216].
* (15) G. Degrassi, S. Di Vita, J. Elias-Miro, J. R. Espinosa, G. F. Giudice, G. Isidori and A. Strumia, “Higgs Mass and Vacuum Stability in the Standard Model at NNLO,” JHEP 1208 (2012) 098 [arXiv:1205.6497 [hep-ph]].
* (16) S. Iso, “What Can We Learn from the 126 GeV Higgs Boson for the Planck Scale Physics?—Hierarchy Problem and the Stability of the Vacuum—,” arXiv:1304.0293 [hep-ph].
* (17) M. B. Einhorn and D. R. T. Jones, “The Effective Potential and Quadratic Divergences,” Phys. Rev. D 46 (1992) 5206.
* (18) C. F. Kolda and H. Murayama, “The Higgs Mass and New Physics Scales in the Minimal Standard Model,” JHEP 0007 (2000) 035 [hep-ph/0003170].
* (19) J. A. Casas, J. R. Espinosa and I. Hidalgo, “Implications for New Physics from Fine-Tuning Arguments. 1. Application to SUSY and Seesaw Cases,” JHEP 0411 (2004) 057 [hep-ph/0410298].
|
arxiv-papers
| 2013-05-30T10:24:22 |
2024-09-04T02:49:45.882176
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yuta Hamada, Hikaru Kawai, and Kin-ya Oda",
"submitter": "Kin-ya Oda",
"url": "https://arxiv.org/abs/1305.7055"
}
|
1305.7147
|
# Cell Growth and Size Homeostasis in Silico
Yucheng Hu,2
Zhou Pei-yuan Center for Applied Mathematics,
Tsinghua University, Beijing, China, 100008 Corresponding author. E-mail:
[email protected] Tianqi Zhu
Beijing Institute of Genomics,
Chinese Academy of Sciences, Beijing, China, 100101 The two authors have
contributed equally to this work.
###### Abstract
Cell growth in size is a complex process coordinated by intrinsic and
environmental signals. In a recent work [Tzur et al., Science, 2009,
325:167-171], size distributions in an exponentially growing population of
mammalian cells were used to infer the growth rate in size. The results
suggest that cell growth is neither linear nor exponential, but subject to
size-dependent regulation. To explain their data, we build a model in which
the cell growth rate is controlled by the relative amount of mRNA and
ribosomes in a cell. Plus a stochastic division rule, the evolutionary process
of a population of cells can be simulated and the statistics of the _in-
silico_ population agree well with the experimental data. To further explore
the model space, alternative growth models and division rules are studied.
This work may serve as a starting point for us to understand the rational
behind cell growth and size regulation using predictive models.
_Key words:_ cell growth rate; Collins-Richmond method; cell cycle
progression; size regulation; cell size distribution; protein synthesis
## Introduction
Understanding the dynamical process of cell growth in size between divisions
is a classic problem in biology. Over the decades there has been extensive
research on this subject and yet much is still unknown about it (1, 2, 3, 4,
5). Earlier attempts to measure growth rate at single-cell level suffer from
technical limitations (6, 7, 8). Now the state-of-art method can monitor cell
size with much greater accuracy (9, 10), but the intrinsic noise in cells and
limited sample size that can be measured by experiment hindered the
interpretation of the single-cell measurement data. Alternatively, the
statistics of a population of synchronized or asynchronized cells can be
accurately measured, from which the cell growth dynamics can be inferred (11,
8, 12). Together, these two types of measurements provide complementary data,
shedding light on the mechanisms that regulate cell growth.
In a recent work, Tzur et al. (13) estimated the mean growth rate in size of a
mouse lymphoblasts cell line (L1210) based on a population level approach. The
rational behind this method is that, an asynchronous population growing
exponentially (in number) has a steady size distribution, and the zero-flux
condition of this steady distribution establishes a functional relation
between the growth rate and size distributions of the asynchronous, newborn
and dividing cell populations. This relation is known as the Collins-Richmond
equation (11) and has been used to estimate growth rate of bacteria and animal
cells before (11, 14, 15). Compared with earlier work using similar approach,
Tzur et al. managed to remove all the unproven assumptions and obtain the cell
size distributions in greater accuracy, thus significantly improve the
fidelity of the results.
The estimated growth rate as a function of cell size from (13) is replotted
here in Fig. 1A (Fig. 2A of the original paper). It appears that cells exhibit
an exponential-like growth before their size reaching a certain threshold,
after that the growth rate begin to drop (although a majority of cells have
already divided before reaching this critical size). This “$\Lambda$”-shaped
growth pattern is consistent with previous results in (11, 15) and more recent
results in (12) for different cell lines. Although the reduction of growth
rate for large cells has been noticed before (11, 15), a mathematical model
that explicitly explain this growth pattern is still missing. In fact,
previous work tend to treat this part of data as outlier, probably because (i)
the data was not accurate enough for a quantitative analysis and (ii) only a
small proportion of cells in the population are found in this rate reduction
region (about 10 as estimated in (15) and 35% in (13)). However, as we will
show, albeit only affecting a small proportion of cells, this growth rate
reduction can play an important role in maintaining cell size homeostasis. We
suspect it might function as a regulatory mechanism for size control and is
worth to be re-examined more carefully with the newly available experimental
data.
In this paper we present a simple model to explain the experimental data in
(13). The model assumes that the growth rate of a cell is determined by both
ribosome number and mRNA level. Their relative abundance changes as cell-cycle
progresses, coordinating the dynamics of cell growth. Plus an empirical
division rule that tells a cell when to divide, the _in-silico_ cell
population generated under our growth model can reproduce the observed
experimental results, i. e., the “$\Lambda$”-shaped growth curve and cell size
distributions of the asynchronous and newborn populations.
We emphasis that even though we build this model with some biological rational
in our mind, it is still semi-phenomenological and serves mainly for the
purpose of explaining the data. Mathematically, finding a model that can
regenerate the observed data is an inverse problem and the solution (the
model) is usually not unique. To this end, we also explore other
phenomenological models as well as alternative division rules that may or may
not give rise to results consistent with the experiment and explain why it is
so.
## A Simple Cell Growth Model
Denote $v(s)$ as the cell growth rate as a function of cell size $s$ in an
asynchronous population of cells in which the frequency density of any
observable (size, age, growth rate, etc) is time-invariant. Since cells with
the same size may have different growth rate, $v(s)$ should be understood as
an ensemble average conditioning on the given size $\langle ds/dt\rangle_{s}$
(see Eq. (3) for its mathematical definition). The estimated $v(s)$ from
experimental measurements obtained by Tzur et al. (13) is shown in Fig. 1A. It
shows that $v(s)$ is neither a constant, which corresponding to linear growth,
nor proportional to $s$, which corresponding to exponential growth. Instead,
$v(s)$ is “$\Lambda$”-shaped: it increases linearly with respect to $s$ when
$s$ is small and then decreases after $s$ exceeds a certain threshold. We
suspect this growth pattern is caused by some form of size-dependent
regulation.
Figure 1: Cell growth rate as a function of cell size. (A) Experimental result
obtained using Collins-Richmond method in (13) (permission from AAAS to reuse
this figure, different curves correspond to different detailed
implementations). (B) Averaged growth rate obtained from the _in-silico_
population simulated using our cell growth model. The dashed-red curve
represents pure exponential growth ($v(s)=\lambda_{2}s$) for $0\leq s<2000$
and linear decay in growth rate ($v(s)=200-\gamma_{2}s$) for $s\geq 2000$.
We propose a model sketched in Fig. 2A. It is assumed that the size (volume)
of a cell is proportional to its protein mass, and the later is further
assumed to be proportional to the total number of ribosomes in the cell
(thinking ribosome as a representative of proteome). Under the above
assumptions, cell size, protein mass and ribosome number can be represented by
one variable, $s$, after proper rescaling. The degradation rate per unit of
protein mass is a constant $\gamma_{2}$. The protein synthesis rate is
proportional to the total number of working ribosomes in the cell, i. e.,
ribosomes that can allocate mRNA to initiate translation. The amount of mRNA
is denoted by $m$, and its units is rescaled so that one unit of ribosome need
one unit of mRNA. So the total working ribosomes in a cell equals to
$\min\\{m,s\\}$ (for simplicity $m$ and $s$ are treated as continuous
variable), and the protein synthesis rate is $\lambda_{2}\min\\{m,s\\}$. The
dynamics of mRNA is assumed to be age-dependent, with degradation rate
$\gamma_{1}$ and production rate $\lambda_{1}(\kappa t)^{q}/(1+(\kappa
t)^{q})$. Here $t$ is the cell age, $q$ and $\kappa$ are two parameters. So we
have
$\displaystyle\frac{dm}{dt}$ $\displaystyle=\frac{\lambda_{1}(\kappa
t)^{q}}{1+(\kappa t)^{q}}-\gamma_{1}m,$ (1a) $\displaystyle\frac{ds}{dt}$
$\displaystyle=\left[\lambda_{2}\min\\{m,s\\}-\gamma_{2}s\right]^{+},$ (1b)
where $[x]^{+}=\max\\{0,x\\}$ keeps $ds/dt$ to be non-negative (even if
protein degradation is faster than synthesis, the constituting amino acids
remain in the cell, so the cell size will not shrink). The Hill’s function
term allows mRNA level to saturate quickly at a plateau. A typical solution of
this system is shown in Fig. 2B (see Methods for parameter values).
Figure 2: (A) A two-variable cell growth model. Cell size is proportional to
the number of ribosomes it contains. The degradation rate per cell mass is
$\gamma_{2}$ and the production rate is proportional to the number of working
ribosomes, $\lambda_{2}\min\\{m,s\\}$. (B) Trajectories of mRNA and cell size
according to Eqs. (1a) and (1b). Initially the mRNA level is set to zero.
According to the relative abundance of mRNA and ribosomes, three growth stages
can be identified in which mRNA and ribosomes play different roles in
regulating cell growth (see main article).
Initially mRNA level is low in the newborn (old mRNA degraded during mitosis
and chromosomes need time to unfold). This is growth stage I in which
insufficient mRNA supply limits protein synthesis. In stage II mRNA level
builds up quickly, allowing all ribosomes to work full time, and the cell will
grow exponentially. But the maximum mRNA level a cell can possibly support is
limited. If the cell reaches a critical size without dividing, it will take up
all the mRNA and its growth rate decreases in stage III.
To simulate an evolving population we also need to know how and when a cell
divides. Following (13) we assume the size difference between two sibling
daughter cells obeys a Gaussian distribution $N(0,\sigma^{2})$, with a
standard deviation $\sigma\approx 68.8$ (fl) that is independent with the size
of the mother cell. For convenience we set the mRNA level of the newborns to
be 0, although a small non-zero value will give similar results. The detailed
rules specifying when cell divides will be postponed in later sections as they
have little effect in determining the shape of the growth rate curve.
From the _in-silico_ cell population we obtain the mean growth rate $v(s)$
(Fig. 1B. See Methods for detailed implementation). By tuning model
parameters, quantitative agreement with the experiment result can be made.
## Fitting the Growth Curve
Deducing a model that fits observed data is an inverse problem and the
solution is never unique (as long as the model allows unlimited complexity).
Nevertheless, giving the richness of information contained in our data,
finding a simple model that fits all the data (growth rate and size
distributions) is non-trivial. Next we analysis several models and explain why
they can or cannot reproduce the growth curve.
In general, dynamics of a cell can be described by
$\frac{d\mathbf{X}}{dt}=\mathbf{A}(\mathbf{X},t),$
where the cell state $\mathbf{X}$ usually lives in a high dimension.
$\mathbf{A}$ is some deterministic or stochastic operator. In an asynchronous
population, the frequency density of $\mathbf{X}$, denoted by $p(\mathbf{x})$,
is in steady state. Suppose in some experiment one can measure a component in
$\mathbf{X}$, say $S$, while the rest components $\mathbf{M}$ are hidden
variables. $S$ can be cell size, DNA mass, or surface marker intensity, etc.
The Collins-Richmond equation can be applied on the frequency distribution of
$S$ to get the growth rate of $S$, which is essentially in the following
marginal form,
$\bar{v}(s)\propto\frac{2\bar{F}_{0}(s)-\bar{F}_{mi}(s)-\bar{F}_{a}(s)}{\bar{f}_{a}(s)}.$
Here $\bar{f}_{a}(s),\bar{f}_{mi}(s)$ and $\bar{f}_{0}(s)$ (with
$\bar{F}_{a}(s),\bar{F}_{mi}(s)$ and $\bar{F}_{0}(s)$ being their accumulative
distributions) are the marginal of $p(\mathbf{x})$ (asynchronous population),
$p_{mi}(\mathbf{x})$ (dividing cells) and $p_{0}(\mathbf{x})$ (newborns),
respectively, i. e.,
$\displaystyle\bar{f}_{a}(s)$
$\displaystyle=\int_{\mathbf{m}}p(s,\mathbf{m})d\mathbf{m},$
$\displaystyle\bar{f}_{mi}(s)$
$\displaystyle=\int_{\mathbf{m}}p_{mi}(s,\mathbf{m})d\mathbf{m},$
$\displaystyle\bar{f}_{0}(s)$
$\displaystyle=\int_{\mathbf{m}}p_{0}(s,\mathbf{m})d\mathbf{m}.$
Intuitively $\bar{v}(s)$ is the average of growth rate of $S$, $dS/dt\equiv
v(s,\mathbf{m})$, in the one-dimensional marginal space (12). To put it in a
more rigorous mathematical form, since the flux of $S$ across the point $S=s$
in the marginal space equals to the flux across the hyperplane $S=s$ in the
full space where $\mathbf{X}=(S,\mathbf{M})$ lives, we have
$\bar{v}(s)\bar{f}_{a}(s)=\int_{\mathbf{m}}v(s,\mathbf{m})p(s,\mathbf{m})d\mathbf{m},\\\
$
which gives
$\bar{v}(s)=\frac{\int_{\mathbf{m}}v(s,\mathbf{m})p(s,\mathbf{m})d\mathbf{m}}{\bar{f}_{a}(s)}=\int_{\mathbf{m}}v(s,\mathbf{m})p(\mathbf{m}|s)d\mathbf{m}\equiv\left\langle\frac{ds}{dt}\right\rangle_{s},$
(3)
where $p(\mathbf{m}|s)$ is the conditional probability density of $\mathbf{M}$
given $S=s$. In other words, $\bar{v}(s)$ is the expectation of the growth
rate $v(s,\mathbf{m})$ conditioning on a given $s$.
Next we consider a class of models with one hidden variable:
$\displaystyle\frac{dm}{dt}$ $\displaystyle=h(s,m,t),$
$\displaystyle\frac{ds}{dt}$ $\displaystyle=v(s,m).$
Here $t$ denotes the cell age, $s$ is the cell size and the hidden variable
$m$ may have different meaning in different models. The model given by Eqs.
(1a) and (1b), which will be referred as Model 1, belongs to this class, with
$m$ represents the mRNA level in a cell. Other examples includes:
Model 2:
$\frac{ds}{dt}=\lambda s-\gamma s,$ (4)
where cell size $s$ is decoupled from any hidden variable.
Model 3
$\displaystyle m$ $\displaystyle=\begin{cases}\lambda+k_{1}(s-s_{1}),&\text{
if $s<s_{1}$},\\\ \lambda,&\text{ if $s_{1}\leq s<s_{2}$},\\\
\lambda+k_{2}(s-s_{2}),&\text{ if $s\geq s_{2}$}.\end{cases}$ (5a)
$\displaystyle\frac{ds}{dt}$ $\displaystyle=[ms]^{+}.$ (5b)
Here $m$ is the effective growth rate per unit of cell size and is chosen as a
piecewise linear function of $s$ based on results from direct observation of
single cell growth in (9). For $\lambda=0.1$, $k_{1}=0.0001$, $k_{2}=-0.0001$,
$s_{1}=1500$, $s_{2}=2000$, the $m\sim s$ relation is plotted in the insertion
of Fig. 3A. The shape of this curve is partially consistent with Fig. S4 in
(9), but we exaggerated over the part that $m$ decreases for large $s$,
otherwise it cannot fit the data here. This inconsistence may be caused by
different cell culture condition in the single-cell experiment compared with
the population-level experiment (e. g., loss of cell-cell interaction).
Another possibility is that more than 65% cells have already divided before
reaching the critical size (13), so if the sample size in the single-cell
measurement is small, this growth reduction may be missed.
Model 4
$\displaystyle m$
$\displaystyle\sim\mathcal{U}\left(-\frac{\lambda}{2},\frac{\lambda}{2}\right)\text{,
heritable}$ (6a) $\displaystyle\frac{ds}{dt}$
$\displaystyle=\begin{cases}0,&\text{ if $t<C-\ln 2/(\lambda+m)$},\\\
(\lambda+m)s,&\text{otherwise}.\end{cases}$ (6b)
It is briefly mentioned in (12) that the observed growth rate reduction for
large cells might be caused by the factor that fast growing cells divide
earlier at relative small size, leaving slow growing cells with relative large
size behind. In the meanwhile all cells grow exponentially. To test this idea
we build the above model in which cells have different intrinsic growth rate
as indicated by $m$, a random variable uniformly distributed in
$(-\lambda/2,\lambda/2),\lambda=0.1$. When dividing, the same $m$ is passed
from a mother cell to the daughter cells. We have to use a special division
rule (see Methods) to make the fastest growing cells divide at $s=2000$ and
the slowest-growing cells divide at $s=2500$ so as to fit the growth curve.
One problem caused by different growth rate in this model is that the time
needed for a cell to double its size via exponential growth is different
(which is $\ln 2/(\lambda+m)$). To give no selective bias in the heterogeneous
population, we add an idle time with length $C-\ln 2/(\ lambda+m),C=14$
(hour), to each cell before it starts to grow, so that cells would have equal
cell cycle length.
For each model, we check if the model-predicted growth rate
$\bar{v}(s)=\int_{m}v(s,m)p(m|s)dm,$ (7)
agrees with the experimental result (Fig. 1A). For model 2, replacing $v(s,m)$
in the above equation by Eq. (4) immediately gives
$\bar{v}(s)=(\lambda-\gamma)s$. It gives a straight line so is inconsistent
with the experiment.
For model 3, since $m$ depends on $s$ only, $p(m|s)=\delta(m-m(s))$, and Eq.
(7) can be computed explicitly as
$\bar{v}(s)=\begin{cases}[\lambda_{1}s+k_{1}(s-s_{1})s]^{+},&\text{ if
$s<s_{1}$},\\\ \lambda_{1}s,&\text{ if $s_{1}\leq s<s_{2}$},\\\
\lambda_{1}s+k_{2}(s-s_{2})s,&\text{ if $s\geq s_{2}$}.\end{cases}$ (8)
This growth curve agrees with the experiment reasonably well (Fig. 3A).
Figure 3: (A) Growth rate of model 3 as given by Eq. (8). The insertion is the
corresponding growth rate per volume ($m$ in Eq. (5a)). (B) Mean growth rate
in model 4 (black curve).
For model 4 computer simulation shows that the averaged growth rate
$\bar{v}(s)$ is indeed “$\Lambda$”-shaped (Fig. 3B). In fact, within the range
$1300<s<2000$, most cells are in exponential growth stage and no cell divide,
so $m$ is uniformly distributed and $\bar{v}(s)=\lambda s$. As $s$ increases,
the mean value of $m$ in the population shifts from 0 to $-\lambda/2$ because
cells with larger $m$ start to divide, so the mean growth rate decreases.
However, inconsistent with the experiment, for $s$ smaller than 1300 the mean
growth rate is close to zero because many newborns are idle. In addition, the
cell size distributions under this model differ from the measured
distributions (result not shown). Thus the model in its current form only
provides partial explanation to the experiment.
For model 1, explicit expression for $\bar{v}(s)$ is not available, but some
qualitative analysis can still be made. On one hand, for small $s$
($500<s<1500$), replace the $v(s,m)$ in Eq. (7) with Eq. (1b) and split the
integral, we have
$\bar{v}(s)=\int_{0}^{s}(\lambda_{2}m-\gamma_{2}s)^{+}p(m|s)dm+\int_{s}^{m_{max}}(\lambda_{2}-\gamma_{2})sp(m|s)dm,$
where $m_{max}\approx 2000$ is the maximum mRNA level. For small cells
($s<1500$) the mRNA level tend to be low (if they are in growth stage I), so
$p(m|s)>0$ and $(\lambda_{2}m-\gamma_{2}s)^{+}<(\lambda_{2}-\gamma_{2})s$ for
$0<m<s$. Overall $\bar{v}(s)<(\lambda_{2}-\gamma_{2})s$, which explains why
for small $s$ the growth curve (black-solid line in Fig. 1B) lies below the
exponential curve (red-dashed line in Fig. 1B). On the other hand, for large
cells ($s\geq 1500$) the mRNA level tend to be saturated around $m_{max}$,
approximately we have $p(m|s)\approx\delta(m-m_{max})$ and
$\bar{v}(s)=\lambda_{2}\min(s,m_{max})-\gamma_{2}s$. So for $1500\leq s<2000$
$\bar{v}(s)$ increases linearly with $s$ (an exponential growth with rate
$\lambda_{2}-\gamma_{2}$) and for $s\geq 2000$ $\bar{v}(s)$ decreases linearly
with $s$ (with a slope equals to $\gamma_{2}$). This is also the way how the
parameters $\lambda_{1},\gamma_{1}$, which controls $m_{max}$, and
$\lambda_{2},\gamma_{2}$ are chosen by comparing with the experimental growth
curve in Fig. 1A.
## Division Rules and Size Homeostasis
A division rule tells a cell when to divide. It effects the size homeostasis
in a cell population. Here we ask, for cell growth model 1, what kind of
division rule can reproduce the asynchronous and newborn size distributions
that have been directly measured by experiment in (13).
We assume there exists a division rate function $p(\mathbf{x},t)$ that depends
on cell state $\mathbf{x}$ and cell age $t$. The probability that a cell
divides during an infinitesimal time interval $dt$ is $p(\mathbf{x},t)dt$. In
particular, we consider the following division rules.
1. 1.
Age-gate:
$p(t)=\begin{cases}0,&\text{ if $t<t_{0}$},\\\ p_{0},&\text{ if $t\geq
t_{0}$}.\end{cases}$
2. 2.
Age-gate plus size-gate:
$p(s,t)=p_{1}(t)+p_{2}(s),$
with
$p_{1}(t)=\begin{cases}0,&\text{ if $t<t_{0}$},\\\ p_{0},&\text{ if $t\geq
t_{0}$},\end{cases}\ \ \ \ \ \ \ \ \ \ p_{2}(s)=\begin{cases}0,&\text{ if
$s<s_{0}$},\\\ p_{0},&\text{ if $s\geq s_{0}$}.\end{cases}$
3. 3.
Signal integration:
$p(t)=\begin{cases}0,&\text{ if $A(t)<A_{0}$},\\\ p_{0},&\text{ if $A(t)\geq
A_{0}$},\end{cases}$
where
$A(t)=\int_{t_{II}}^{t}\min\\{m,s\\}dt^{\prime}$
is the area of the part of the shaded region in Fig. 2B up to $t$.
For each division rule, we search for the parameters ($p_{0}$ and $t_{0}$ for
rule 1, $p_{0},s_{0}$ and $t_{0}$ for rule 2, $p_{0}$ and $A_{0}$ for rule 3)
that minimize the $L_{1}$-distance between the _in-slico_ and experimental
distributions. The best-fit results are shown in Fig. 4. See Methods for the
optimization procedure.
Figure 4: Asynchronous (left) and newborn (right) cell size distributions from
direct measurement (black thick line) and _in-silico_ population simulated
using division rule 1 (green dashed line), rule 2 (red dot-dashed line) and
rule 3 (blue solid line). Sample size is $N=10^{5}$. See Methods for the
parameter values.
Under division rule 1, a homeostasis population can be established.
Interestingly, if we apply the same division rule to an exponential growth
population (model 2), there is no stable size distribution. This is consistent
with the result in (15) saying that, if cells grow exponentially in size and
the division rule depends on age only, the variance of cell size in a
population will diverge. In this sense the “$\Lambda$”-shaped growth pattern
can be understood as a regulation mechanism bounding the cell size from above,
which leads to size homeostasis even for a division rule that depends only on
cell age. However, by itself this mechanism is not good enough as the _in-
silico_ size distributions (Fig. 4 green dashed curves) cannot match the
measured ones very well. In particular, the size distribution of the newborns
is much wider and some newborns are relatively smaller in size. These cells
born to be too small are likely to find themselves in a disadvantageous place
to start with compared with other cells. In other words, the quality of the
population produced by this division rule is compromised. The reason is that
there is no quality-checking (in terms of cell size) in this scheme.
Under division rule 2, the _in-silico_ distributions agree with the
experimental data reasonably well (Fig. 4 red dot-dashed curves). It
degenerates to rule 1 when $s_{0}\rightarrow\infty$. So by taking extra size
information into account, this division scheme achieves better agreement with
experimental data than the one that only uses age information.
Division rule 3 also leads to a good fitting (Fig. 4 blue solid curves). It is
assumed in this scheme that after a cell leaves region I (Fig. 2B), it begins
to measure a mitosis-signal in an integral way. Here we take the signal to be
proportional to the protein synthesis rate, $\min\\{s,m\\}$. Once the time
integral of this signal reaches a critical value, the cell begins to divide
with a constant rate $p_{0}$. (If the area within region I is also taken into
the integration, similar results still hold.)
For each division rule (using the optimal parameters we found), the
$L_{1}$-distance between the _in-silico_ and experiment distributions as a
function of simulation time is plotted in Fig. 5. Initially all cells are
identical in size and synchronized at age zero. As the population evolves
under the growth model and division rule, the size distributions gradually
reach homeostasis. It appears division rule 3 gives the best fit to the
experimental data. For all division rules, the time taken for the _in-silico_
population to reach size homeostasis is around 10 days. Given that the average
cell cycle length is roughly 10 hours in our model (see Fig. 1B), more than 20
rounds of divisions are needed for a synchronous population to reach
homeostasis.
Figure 5: $L_{1}$-distance between the _in-silico_ and experimental size
distributions (see Methods). Initially the population is synchronized at age
zero and all cells have identical cell size. Different curves correspond to
different division rules.
Overall, for growth model 1, both division rules 2 and 3 explain the
experimental data reasonably well and they both predict that, for cells with
the same size, older cells are more likely to divide than younger cells, and
for cells with the same age, larger cells are more likely to divide than
smaller cells, which is consistent with the observation made in (13). Again,
we note that there could be many other division rules which can match the
data.
## Discussions
In this work we proposed a simple cell growth model aiming to explain the
“$\Lambda$”-shaped growth rate curve observed in experiments. It makes an
excellent demonstration that complex experimental results can sometimes be
captured by very simple intuition. In our model the growth rate of a cell is
regulated by its mRNA content: when there is enough mRNA, cell enjoys an
exponential growth, otherwise its growth rate is compromised. The rate-
limiting effect of mRNA occurs when a cell is newly born or its size exceeds a
certain threshold. The statistics of the _in-silico_ population generated
using our growth model matches with the experimental results very well.
While our motivation in building the model is to explain the data, two central
features of the model may resemble the actual biological mechanisms regulating
cell growth. The first one is a transit slow-growing period right after a cell
is born, a behavior which is observed in several cell lines (12). There could
be many explanations for this and that the mRNA is playing a rate-limiting
role is one of them. After all, during mitosis the production of mRNA should
be minimum as the chromosomes are tightly packed. In addition, chromosome-
unfolding and transcription initialization also delay the mRNA production in
the newborns. The other crucial feature of the model is the existence of a
maximum mRNA level that a cell can reach. Recent experiments show that the
mRNA output reaches a plateau in division arrested yeast because of limited
DNA copy number (16). Together, the interplay between mRNA and protein
synthesis provide a potential mechanism for growth control, consistent with
the idea that gene expression and cell size are tightly correlated (17).
Cell size homeostasis is coordinated by growth and division. We studied
several empirical division rules to see if they can regenerate the
experimental data under our growth model. It turns out that the observed size
distributions cannot be explained simply by an age-gate division rule and
certain quality control mechanism, or “sizer”, in mitosis-decision is needed.
The division rule based on the time integral of a growth signal fits the
experiment very well. Interestingly, there is evidence showing budding yeast
takes a similar approach in its division control (18).
The Collins-Richmond relation provides an elegant way of extracting dynamic
properties of some observable from its frequency distribution in an
asynchronous population. Examples of data on which this method could be
applied include DNA content (19) and RNA Polymerase II distribution from ChIP-
seq experiments (20). We described a general framework for consistency
checking between a growth model and results based on the Collins-Richmond
method. It can be useful in providing some heuristic insight or guidance in
building and tuning more complex models.
Finally we note that, the biggest limitation of the Collins-Richmond method in
its current form is that it can only handle one variable at a time and is thus
unable to directly capture the dynamical interaction between different
variables. Recently Kafri et al. (12) developed a new framework called ERA
(ergodic rate analysis) that extends the density balance law, based on which
the Collins-Richmond equation holds, to high-dimensional data. They were able
to apply this new method on asynchronous population of Hela cells which
revealed more insights on cell cycle dynamics.
## Methods
### Simulation of cell population
We want to simulate the evolution of a population of cells and collect its
statistical information when cell size reaches homeostasis. It is practically
impossible to simulate the entire population because the number of cells grows
exponentially. Instead, we keep track of only a limited number of cells. In
particular, we maintain a population of $N=10^{5}$ cells, and randomly delete
one cell among them with equal chance whenever there is a cell division.
Essentially this is the Moran population process (21) with fixed population
size. It mimics drawing random samples from an exponentially growing
population provided that there is no inheritable fitness difference between
cell lineages, so the statistics thus obtained will be the same as that of the
total population (apart from a sampling error).
To simulate the growth of each cell, Eqs. (1a) and (1b) is solved using fourth
order Rugger-Kutta method with constant step-size $\Delta t=0.05$ (hour). The
parameters used are: $\lambda_{1}=2000$, $\gamma_{1}=1$, $\lambda_{2}=0.25$,
$\gamma_{2}=0.15$, $\kappa=0.5$, $q=4$. These values are chosen empirically to
fit the experiment. In principle the number of mRNA molecules and ribosomes in
a cell are integers and driven by stochastic processes, which can be described
by a stochastic chemical reaction system. However, since their copy numbers
are huge in a cell, a continuous and deterministic approach makes a very good
approximation here. Additionally, since we are mainly interested about the
statistics of the population, the random fluctuations in the individual cells
are averaged out and have almost no effect on the growth rate and size
distributions.
While the growth of individual cells can be approximated by deterministic
ordinary differential equations, cell divisions need to be treated as
stochastic events that arrive random in time. From the division rule we can
compute the division rate $p(\mathbf{x},t)$ for a cell with state $\mathbf{x}$
at age $t$, and the probability that the cell divides during a small time
interval $dt$ is $p(\mathbf{x},t)dt$. So a straight-forward but less efficient
implementation is to generate a random variable $u$ uniformly distributed in
$(0,1)$, and compare it with the mitosis probability $p(\mathbf{x},t)dt$. Only
if $u<p(\mathbf{x},t)dt$ then the cell divides. Doing so we need to generate
one random variable for each cell at every time step thus is very
computational expensive for large population. A more efficient way is to
assign an uniformly distributed random variable $u$ to each newborn cell and
monitor the value
$\int_{0}^{\tau}p(\mathbf{x},t)dt+\ln u,$
for this cell. This term is negative when $\tau$ is small and increases as
$\tau$ marching forward. At the time when its value reaches zero the cell
divides. It can be shown that the two procedures generate statistically
equivalent waiting time for mitosis as time step-size approaches zero (22).
Now for one cell we only need to generate one random variable in its whole
cell cycle thus the simulation speed is much faster.
### Collecting statistics from the _in-silico_ population
Growth rate curve: For each cell we record its size $s(t)$ at each time-step
$t_{n}$. The growth rate of an individual cell at age $t_{n}$ is estimated by
$(s(t_{n})-s(t_{n-1}))/\Delta t$. To compute the average growth rate $v(s)$ as
a function of cell size $s$, we sort cells by size, partition them into small
intervals of $s$, and average the growth rate of cells within each interval.
Asynchronous and newborn size distributions: The asynchronous size
distribution can be directly sampled from the homeostasis _in-silico_
population. For the size distribution of the newborns, we collect and record
the size information of the newborns during the simulation, and when needed,
the most recent $N=10^{5}$ newborns in the database are used to sample the
size distribution.
### Fitting the parameters in division rules
Each division rule contains several parameters. For a given set of parameters
of a division rule, we simulate the population and obtain its asynchronous
size distribution $f^{sim}_{a}$ and newborn distribution $f^{sim}_{0}$ as
described above. Define the error as the sum of $L_{1}$-distance between
$f^{sim}_{a,0}$ and the experimental results $f^{exp}_{a,0}$,
$Err\equiv\left\|f^{sim}_{a}-f^{exp}_{a}\right\|_{1}+\left\|f^{sim}_{0}-f^{exp}_{0}\right\|_{1}.$
Minimizing the error within the parameter space is a nonlinear optimization
problem and we use the “optim” routine in R for this task. The parameters we
found are: division rule 1, $t_{0}=6.4,p_{0}=0.8$; division rule 2,
$t_{0}=8,s_{0}=1440,p_{0}=0.18$, division rule 3, $A_{0}=6400,p_{0}=0.5$.
We also tested the $L_{\infty}$, $L_{2}$ and Kullback-Leibler distances. The
first two have similar result as the $L_{1}$-distance, but the last one seems
to be a little less sensitive for our case.
### Division rule for model 4
$p(t)=\begin{cases}0,&\text{ if $s(\lambda+m+0.35)\leq 1000$},\\\
p_{0},&\text{ if $s(\lambda+m+0.35)>1000$}.\end{cases}$
$\lambda=0.1$, $m\sim\mathcal{U}(-0.05,0.05)$ and $p_{0}=10$ (see Eq.(6)). It
is easy to verify that, the fastest growing cell ($m=0.05$) divides around
size $s=2000$, and the slowest growing cell ($m=-0.05$) divides around size
$s=2500$.
## Acknowledgment
YH are thankful to A. Lander for introducing this topic and to S. Wang for her
initiative programing work. We thank R. Kafri, J. Lei, C. Tang and X. Wang for
helpful discussions. We also thank the anonymous reviewers for their helpful
comments. YH is supported by the NSFC (National Science Foundation of China)
under Grant No. 11301294 and TZ is supported by NSFC under Grant No. 31301093.
## References
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|
arxiv-papers
| 2013-05-30T15:52:42 |
2024-09-04T02:49:45.890193
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yucheng Hu and Tianqi Zhu",
"submitter": "Yucheng Hu",
"url": "https://arxiv.org/abs/1305.7147"
}
|
1305.7198
|
# Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms
are minimal
Yong Luo111The author is supported by the DFG Collaborative Research Center
SFB/Transregio 71.
###### Abstract
A submanifold $M^{m}$ of a Euclidean space $R^{m+p}$ is said to have harmonic
mean curvature vector field if $\Delta\vec{H}=0$, where $\vec{H}$ is the mean
curvature vector field of $M\hookrightarrow R^{m+p}$ and $\Delta$ is the rough
Laplacian on $M$. There is a conjecture named after Bangyen Chen which states
that submanifolds of Euclidean spaces with harmonic mean curvature vector
fields are minimal. In this paper we prove that weakly convex hypersurfaces
(i.e. hypersurfaces whose principle curvatures are nonnegative) with harmonic
mean curvature vector fields in Euclidean spaces are minimal. Furthermore we
prove that weakly convex biharmonic hypersurfaces in nonpositive curved space
forms are minimal.
## 1 Introduction
Let $\vec{x}:M^{m}\to R^{m+p}$ be an immersion from a Riemmanian manifold $M$
of dimension $m$ to a Euclidean space of dimension $m+p$, $p\geq 1$. Denote by
$\vec{x},\vec{H},\Delta$ respectively the position vector of $M$, the mean
curvature vector field of $M$ and the Laplacian operator with respect to the
induced metric $g$ on $M$. Then it is well known that (see for example [5])
$\Delta\vec{x}=-n\vec{H}.$
This shows that $M$ is a minimal submanifold if and only if its coordinates
functions are harmonic functions. According to this equation, a submanifold in
a Euclidean space with harmonic mean curvature vector field, i.e.
$\displaystyle\Delta\vec{H}=0,$ (1.1)
if and only if
$\displaystyle\Delta^{2}\vec{x}=0.$ (1.2)
Therefore a submanifold with harmonic mean curvature vector field is called a
biharmonic submanifold.
There is also a variational description of biharmonic submanifolds as follows.
Assume that $\vec{x}:M\to N^{m+p}$ is an immersion to a Riemnnian manifold,
then the biharmonic energy of $\vec{x}$ is defined by
$\displaystyle E_{2}(\vec{x})=\int_{M}|\tau(\vec{x})|^{2}dM,$ (1.3)
where $\tau(\vec{x})$ is the tension field of $\vec{x}$. The critical points
of the functional $E_{2}$ satisfy the following E-L equation (see [10])
$\displaystyle-\Delta\vec{H}=\sum_{i=1}^{m}R^{N}(e_{i},\vec{H})e_{i},$ (1.4)
where $\\{e_{i}\\}$ is a local orthonormal frame on $M$. In particular, when
$N$ is a Euclidean space, this equation is just $\Delta\vec{H}=0$.
###### Definition 1.1.
A Submanifold satisfying equation (1.4) is called a biharmonic submanifold.
As is easy to see, minimal submanifolds are biharmonic submanifolds. It is
nature to arise the question whether the space of biharmonic submanifolds is
strictly larger than the space of minimal submanifolds. Concerning this
problem, when the ambient manifold is a Euclidean space Chen conjectured that
the answer is no.
Chen’s Conjecture. Suppose that $\vec{x}:M^{m}\to R^{m+p}$, $p\geq 1$,
satisfies
$\displaystyle\Delta\vec{H}=0.$ (1.5)
Then $\vec{H}=0$.
Since Chen’s conjecture arose, it remains to be open with very little progress
even for hypersurfaces with dimensions grater that 4. But in recent years it
attracts many attentions and there are some partial answers to this
conjecture. Now we give an overview of them, as to the author’s knowledge.
Chen’s conjecture is proved:
$\bullet$ For Surfaces in $R^{3}$ in [6] and [10] and in an unpublished work
by Chen himself, as reported in [4].
$\bullet$ For hypersurfaces in $R^{4}$ in [9] and a different proof in [7].
$\bullet$ For hypersurfaces which admit at most 2 distinct principle
curvatures in [8].
$\bullet$ For curves [8].
$\bullet$ For submanifolds $M^{m}$ which are pseudo-umbilic and $m\neq 4$ [8].
$\bullet$ For submanifolds of finite type [8].
$\bullet$ For submanifolds which are proper, i.e. any preimage of compact
subsets are compact in [1].
In this paper we prove Chen’s conjecture for weakly convex biharmonic
hypersurfaces in Euclidean spaces.
###### Theorem 1.2.
Assume that $\vec{x}:M^{m}\to R^{m+1}$ is a weakly convex biharmonic
submanifold in $R^{m+1}$, i.e. $\Delta\vec{H}=0.$ Then $\vec{H}=0$.
Chen’s conjecture is generalized to the so called generalized Chen’s
conjecture (see [2] [4] [5] [16] [17] and [15] etc.) allowing the ambient
manifolds to be nonpositive curved.
Generalized Chen’s Conjecture. Suppose that $f:M\to(N,h)$ is an isometric
immersion and the section curvature of $(N,h)$ nonpositive. Then if $f$ is
biharmonic, it is minimal.
The generalized Chen’s curvature turned out to be false by a counter example
constructed by Y-L. Ou and L. Tang (see [17]). But it remains interesting to
find out sufficient conditions which make it be true. It is easy to see from
the proof of theorem 1.2 that our argument is also applicable to find out that
for hypersurfaces weakly convex is a sufficient condition to guarantee the
generalized Chen’s conjecture to be true when the ambient manifold is a space
form $N(c)$ of constant curvature $c\leq 0$. Thus we have
###### Theorem 1.3.
Assume that $\vec{x}:M^{m}\to N^{m+1}(c)$ is a weakly convex biharmonic
hypersurfaces in a spcae form $N^{m+1}(c)$ with $c\leq 0$, then $\vec{H}=0$.
Other sufficient conditions have also been found out to guarantee the
generalized Chen’s conjecture to be true. For recent development in this
direction, we refer to [13] [11] etc..
We would like to point out that for closed (compact without boundary) bi-
harmonic submanifolds the Chen’s conjecture and the generalized Chen’s
conjecture is easily proved to be true by an argument due to Jiang (see [10])
or by directly using an integration by parts argument.
The Chen’s and generalized Chen’s conjectures belong to the research field of
classification of biharmonic submanifolds in Riemannian or pseudo-Riemannian
manifolds. Nowadays it is an active research field and for readers who have
interest with it we refer to a survey paper [12] by Montaldo and Oniciuc and
references therein.
Organization. In section 2 we give a brief sketch of submanifolds geometry.
Theorem 1.2 and theorem 1.3 are proved in section 3.
## 2 Preliminaries
Assume that $\vec{x}:M^{m}\to N^{m+p}$ is an immersion to a Riemannian
manifold $N$ with Riemannian metric $\langle,\rangle$, which is a bilinear
form on $TN\otimes TN$, the tensor of the tangent vector space of $N$. Then
$M$ inherits a Riemannian metric from $N$ by
$g_{ij}:=\langle\partial_{i}\vec{x},\partial_{j}\vec{x}\rangle$ and a volume
form by $\sqrt{\det g_{ij}}dx$. The second fundamental form of
$M\hookrightarrow N$, $h:TM\otimes TM\to NM$, is defined by
$h(X,Y):=D_{X}Y-\nabla_{X}Y,$
for any $X,Y\in TM$, where $D$ is the covariant derivative with respect to the
Levi-Civita connection on $N$, $\nabla$ is the Levi-Civita connection on $M$
with respect to the induced metric and $NM$ is the normal bundle of $M$. For
any normal vector field $\eta$ the Weingarten map $A_{\eta}:TM\to TM$ is
defined by
$\displaystyle D_{X}\eta=-A_{\eta}X+\nabla^{\bot}_{X}\eta,$ (2.1)
where $\nabla^{\bot}$ is the normal connection and as is well known that $h$
and $A$ are related by
$\displaystyle\langle h(X,Y),\eta\rangle=\langle A_{\eta}X,Y\rangle.$ (2.2)
For any $p\in M$, let $\\{e_{1},e_{2},...,e_{m},e_{m+1},...,e_{m+p}\\}$ be a
local orthonormal basis of $N$ such that $\\{e_{1},...,e_{m}\\}$ is an
orthonormal basis of $T_{p}M$. Then $h$ is decomposed at $p$ as
$h(X,Y)=\sum_{\alpha=m+1}^{m+p}h_{\alpha}(X,Y)e_{\alpha}.$
The mean curvature vector field is defined as
$\displaystyle\vec{H}:=\frac{1}{m}\sum_{i=1}^{m}h(e_{i},e_{i})=\sum_{\alpha=m+1}^{m+p}H_{\alpha}e_{\alpha},$
(2.3)
where
$\displaystyle H_{\alpha}:=\frac{1}{m}\sum_{i=1}^{m}h_{\alpha}(e_{i},e_{i}).$
(2.4)
## 3 Proof of theorem 1.2
It is well known that for a submanifold $M^{m}$ in a Euclidean space to be
biharmonic, i.e. $\Delta\vec{H}=0$ if and only if ([4])
$\displaystyle\Delta^{\bot}\vec{H}-\sum_{i=1}^{m}h(A_{\vec{H}}e_{i},e_{i})$
$\displaystyle=$ $\displaystyle 0,$ (3.1) $\displaystyle
m\nabla|\vec{H}|^{2}+4\sum_{i=1}^{m}A_{\nabla^{\bot}_{e_{i}}\vec{H}}e_{i}$
$\displaystyle=$ $\displaystyle 0,$ (3.2)
where $\Delta^{\bot}$ is the (nonpositive) Laplace operator associated with
the normal connection $\nabla^{\bot}$.
Assume that $\vec{H}=H\nu$, where $\nu$ is the unit normal vector field on
$M$. Note that by the assumption that $M$ is weakly convex, we have $H\geq 0$.
Define
$\displaystyle B:=\\{p\in M:H(p)>0\\}$ (3.3)
We will prove that $B$ is an empty set by a contradiction argument, and so $M$
is minimal and we are done.
If $B$ is not empty, we see that $B$ is an open subset of $M$. We assume that
$B_{1}$ is a nonempty connect component of $B$. We will prove that $H\equiv 0$
in $B_{1}$, thus a contradiction.
We prove it in two steps.
Step 1. $H$ is a constant in $B_{1}$.
Let $p\in B_{1}$ be a point. Around $p$ we choose a local orthonormal frame
$\\{e_{k},k=1,...,m\\}$ such that $\langle h,\nu\rangle$ is a diagonal matrix
at $p$, where $\nu$ is the unit normal vector field of $M$.
For any $1\leq k\leq m$ we have at $p$
$\displaystyle\langle\sum_{i=1}^{m}A_{\nabla^{\bot}_{e_{i}}\vec{H}}e_{i},e_{k}\rangle$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{m}\langle
h(e_{i},e_{k}),\nabla^{\bot}_{e_{i}}\vec{H}\rangle$ $\displaystyle=$
$\displaystyle\langle h(e_{k},e_{k}),\nabla^{\bot}_{e_{k}}\vec{H}\rangle.$
By equation (3.2) we have
$\displaystyle 0$ $\displaystyle=$ $\displaystyle
m\nabla_{e_{k}}|\vec{H}|^{2}+4\langle\sum_{i=1}^{m}A_{\nabla^{\bot}_{e_{i}}\vec{H}}e_{i},e_{k}\rangle$
$\displaystyle=$ $\displaystyle m\nabla_{e_{k}}|\vec{H}|^{2}+4\langle
h(e_{k},e_{k}),\nabla^{\bot}_{e_{k}}\vec{H}\rangle$ $\displaystyle=$
$\displaystyle
2mH\nabla_{e_{k}}H+4\lambda_{k}\langle\nu,\nabla^{\bot}_{e_{k}}\vec{H}\rangle$
$\displaystyle=$ $\displaystyle(2mH+4\lambda_{k})\nabla_{e_{k}}H,$
where $\lambda_{k}:=h(e_{k},e_{k})$ is the $k$th principle curvature of $M$ at
$p$, which is nonnegative by the assumption that $M$ is weakly convex.
By $2mH+4\lambda_{k}>0$ at $p$, one gets
$\displaystyle\nabla_{e_{k}}H=0\,\,\,\,at\,\,\,\,p,$ (3.4)
for any $k=1,...,m$, which implies that
$\displaystyle\nabla H=0\,\,\,\,at\,\,\,\,p.$ (3.5)
Because $p$ is an arbitrary point in $B_{1}$, we see that
$\displaystyle\nabla H=0\,\,\,\,in\,\,\,\,B_{1}.$ (3.6)
Therefore we get that $H$ is a constant in $B_{1}$.
Step 2. $H$ is zero in $B_{1}$.
Let $p\in B_{1}$, by step 1, we see that
$\displaystyle\Delta|\vec{H}|^{2}(p)=0.$ (3.7)
On the other hand, by equation (3.1), we have
$\displaystyle\Delta|\vec{H}|^{2}$ $\displaystyle=$ $\displaystyle
2|\nabla^{\bot}\vec{H}|^{2}+2\langle\vec{H},\Delta^{\bot}\vec{H}\rangle$ (3.8)
$\displaystyle\geq$ $\displaystyle 2\sum_{i=1}^{m}\langle
h(A_{\vec{H}}e_{i},e_{i}),\vec{H}\rangle$ $\displaystyle=$ $\displaystyle
2\sum_{i=1}^{m}\langle A_{\vec{H}}e_{i},A_{\vec{H}}e_{i}\rangle$
$\displaystyle=$ $\displaystyle 2\sum_{i=1}^{m}H^{2}\langle
A_{\nu}e_{i},A_{\nu}e_{i}\rangle$ $\displaystyle\geq$ $\displaystyle 2nH^{4}.$
From (3.7)-(3.8), we get $H(p)=0$. Because $p$ is an arbitrary point in
$B_{1}$, we see that $H\equiv 0$ in $B_{1}$, a contradiction.
This completes the proof of theorem 1.2 $\hfill\Box$
A sketch of proof of theorem 1.3. If $M$ is a submanifolds in a space form
$N(c)$, then it is biharmonic if and only if
$\displaystyle\Delta^{\bot}\vec{H}-\sum_{i=1}^{m}h(A_{\vec{H}}e_{i},e_{i})+cm\vec{H}$
$\displaystyle=$ $\displaystyle 0,$ (3.9) $\displaystyle
m\nabla|\vec{H}|^{2}+4\sum_{i=1}^{m}A_{\nabla^{\bot}_{e_{i}}\vec{H}}e_{i}$
$\displaystyle=$ $\displaystyle 0.$ (3.10)
The same as the step 1 in the proof of theorem 1.2, by using equation (3.10),
we can prove that if $H(p)\neq 0$ at a point $p\in M$, then it is a constant
around a neighborhood of the point $p$. Thus we have at $p$,
$\Delta|\vec{H}|^{2}=0.$
On the other hand at $p$
$\displaystyle\Delta|\vec{H}|^{2}$ $\displaystyle=$ $\displaystyle
2|\nabla\vec{H}|^{2}+2\langle\vec{H},\Delta\vec{H}\rangle$ $\displaystyle=$
$\displaystyle
2|\nabla\vec{H}|^{2}-2\sum_{i=1}^{m}R^{N}(e_{i},\vec{H},e_{i},\vec{H})$
$\displaystyle=$ $\displaystyle 2|\nabla\vec{H}|^{2}-2cm|\vec{H}|^{2}$
$\displaystyle\geq$ $\displaystyle 2|\nabla\vec{H}|^{2}.$
Therefore $\nabla\vec{H}(p)=0$. Now we choose an orthogonal basis
$\\{e_{i},i=1,...,m\\}$ of $T_{p}M$. Computing directly one gets at $p$
$\displaystyle
0=\langle\nabla_{e_{i}}\vec{H},e_{j}\rangle=H\langle\nabla_{e_{i}}\nu,e_{j}\rangle=-H\langle\nu,\nabla_{e_{i}}e_{j}\rangle=H\langle
h(e_{i},e_{j}),\nu\rangle,$ (3.11)
for any $1\leq i,j\leq m$. Taking trace over this equality we get
$H^{2}(p)=0.$
Therefore $H(p)=0$. This is a contradiction to $H(p)\neq 0$.
This completes the proof of theorem 1.3. $\hfill\Box$
Acknowledgement. Special thanks due to my supervisor professor Guofang Wang
for his constant encouragement and support.
## References
* [1] K. Akutagawa S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164(1)(2013), 351-355.
* [2] R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of $S^{3}$, Internat. J. Math.12(8)(2001), 867–876.
* [3] R. Caddeo, S. Montaldo and C. Oniciuc, On biharmonic maps, Contemp. Math. 288(2001), 286–290.
* [4] B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17(2)(1991), 169–188.
* [5] B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapore, 1984.
* [6] B. Y. Chen and S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo- Euclidean spaces, Kyushu J. Math. 52(1)(1998), 167–185.
* [7] F. Defever, Hypersurfaces of $E^{4}$ with harmonic mean curvature vector, Math. Nachr.196(1998), 61–69.
* [8] I. Dimitric, Submanifolds of $E^{m}$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica 20(1)(1992), 53-65.
* [9] T. Hasanis and T. Vlachos, Hypersurfaces in $E^{4}$ with harmonic mean curvature vector field, Math. Nachr. 172(1)(1995), 145–169.
* [10] G. Y. Jiang, 2-harmonic maps and their first and second variational formulas, Chinese Ann. Math. Ser. A7(1986), 389–402.
* [11] Y. Luo, Biharmonic submanifolds in a nonpositive curved manifold and $\varepsilon$-superbiharmonic submanifolds, preprint(2013).
* [12] S. Montaldo and C. Oniciuc, A short survey on biharmonic maps between Riemannian manifolds, Rev. Un. Mat. Argentina 47(2)(2006), 1–22.
* [13] N. Nakauchi and H. Urakawa, Biharmonic Submanifolds in a Riemannian Manifold with Non-Positive Curvature, Results. Math. 63(2013), 467–474.
* [14] N. Nakauchi, H. Urakawa and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. dedicata(2013), Doi:10.1007/s10711-013-9854-1, to appear.
* [15] C. Oniciuc, Biharmonic maps between Riemannian manifolds, An. St. Al. Univ. Al. I. Cuza , Iasi, Vol. 68(2002), 237–248.
* [16] Y.-L. Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. Math. 248(2010), 217–232.
* [17] The generalized chen’s conjecture on biharmonic submanifolds is false, Michigan Math. J. 61(3)(2012), 531–542.
Yong Luo
Mathematisches Institut,
Albert-Ludwigs-Universität Freiburg,
Eckerstr. 1, 79104 Freiburg, Germany.
[email protected]
|
arxiv-papers
| 2013-05-30T18:39:09 |
2024-09-04T02:49:45.899460
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yong Luo",
"submitter": "Yong Luo",
"url": "https://arxiv.org/abs/1305.7198"
}
|
1305.7272
|
# Accuracy of Range-Based Cooperative Localization in Wireless Sensor
Networks:
A Lower Bound Analysis
Liang Heng, and Grace Xingxin Gao The authors are with the Department of
Aerospace Engineering and the Coordinated Science Laboratory, University of
Illinois at Urbana-Champaign, Urbana, IL 61801, USA. E-mail:
[email protected], [email protected].
###### Abstract
Accurate location information is essential for many wireless sensor network
(WSN) applications. A location-aware WSN generally includes two types of
nodes: _sensors_ whose locations to be determined and _anchors_ whose
locations are known a priori. For range-based localization, sensors’ locations
are deduced from anchor-to-sensor and sensor-to-sensor range measurements.
Localization accuracy depends on the network parameters such as network
connectivity and size. This paper provides a generalized theory that
quantitatively characterizes such relation between network parameters and
localization accuracy. We use the _average degree_ as a connectivity metric
and use geometric _dilution of precision_ (DOP), equivalent to the Cramér-Rao
bound, to quantify localization accuracy. We prove a novel lower bound on
expectation of average geometric DOP (LB-E-AGDOP) and derives a closed-form
formula that relates LB-E-AGDOP to only three parameters: average anchor
degree, average sensor degree, and number of sensor nodes. The formula shows
that localization accuracy is approximately inversely proportional to the
average degree, and a higher ratio of average anchor degree to average sensor
degree yields better localization accuracy. Furthermore, the paper
demonstrates a strong connection between LB-E-AGDOP and the best achievable
accuracy. Finally, we validate the theory via numerical simulations with three
different random graph models.
###### Index Terms:
Wireless sensor networks, range-based localization, cooperative localization,
accuracy, network connectivity, dilution of precision (DOP), Cramér-Rao bound,
Laplacian matrix
## I Introduction
Wireless sensor networks (WSNs) hold considerable promise for large-scale,
flexible, robust, cost-effective data collection and information processing in
complex environments[1, 2, 3]. Location awareness is a fundamental feature in
many WSN applications because “sensing data without knowing the sensor’s
location is meaningless” [4]. Location information can also help a node
interact with its neighbors and surroundings, improving networking operations
such as geographic routing and topology control [5].
To enable location awareness in WSNs, a wide variety of localization schemes
have been explored over the past decade. According to the measurements used to
estimate locations, these schemes can be generally classified as range-based
[6, 7], angle-based [8, 5], proximity-based [9, 10], and event-driven [11,
12]. Besides, the localization schemes can be categorized as either
noncooperative or cooperative [13, 14]. In a noncooperative scheme, the
unknown-location nodes (hereinafter referred to as _sensor nodes_ or simply
_sensors_) make measurements with known-location references (hereinafter
referred to as _anchor nodes_ or _anchors_), without any communication between
sensor nodes. In a cooperative scheme, in addition to anchor-to-sensor
measurements, each sensor also makes measurements with neighboring sensors;
the additional information gained from sensor-to-sensor measurements enhances
localization accuracy, availability, and robustness. In the literature,
cooperative localization has also been named as “relative,” “GPS-free,”
“multi-hop,” or “network” localization.
This paper focuses on range-based cooperative localization schemes.
Figure 1: A scenario of range-based cooperative localization in 2 dimensions.
Black squares denote anchor nodes whose locations are known, red circles
denote sensor nodes whose locations are to be estimated, and blue lines
represent ranging links which provide inter-node distance information. Sensors
are randomly distributed and ranging links are randomly established according
to a certain random graph model. This paper aims for a generalized theory that
characterizes the connection between system parameters (namely network
connectivity and network size) and localization accuracy.
As illustrated in Fig. 1, anchor nodes (denoted by black squares) are aware of
their locations, and sensor nodes (denoted by red circles) determine their
locations using inter-node distance information. Range-based cooperative
localization is essentially a graph embedding problem [15, 16, 17].
Connectivity of the graph exerts significant influence on many performance
metrics, such as accuracy, energy efficiency, localizability, robustness, and
scalability. Although localizability has been extensively studied with respect
to connectivity [15, 16], the relationship between accuracy and connectivity
has not yet been theoretically treated. The objective of this paper is a
generalized theory that quantitatively characterizes the connection between
network parameters (namely, network connectivity and network size) and
localization accuracy. The theory provides compendious guidelines on the
design and deployment of location-aware WSNs.
### I-A Related work
As previously mentioned, range-based cooperative localization is a graph
embedding problem. Saxe [18] has shown that testing the embeddability of
weighted graphs (equivalently, testing localizability) is strongly NP-hard.
Aspnes et al.[19] have further proven that localization in sparse networks is
NP-hard. However, when the network is densely connected such that $O(N^{2})$
pairs of nodes know their relative distances, where $N$ is the number of
nodes, there are efficient algorithms such as multidimensional scaling (MDS)
[20] and semidefinite programming (SDP) [21] for solving the localization
problem.
Cooperative localization can also be seen as a high-dimension optimization
problem that finds a vector of node locations such that inter-node distances
are as close to range measurements as possible. In general, this optimization
problem may have many local optimums. The MDS and SDP algorithms [20, 21], as
well as some stochastic optimization algorithms [22], are able to find a
solution close to the global optimum under certain conditions (e.g., dense
connectivity). The solution is not necessarily very accurate but can be
treated as an initial guess. Then, the location solution can be improved using
an iterative algorithm such as _lateration_ (also referred to as
_trilateration_ , _multilateration_ , and even mistakenly _triangulation_)
[23, 6, 7, 24, 25].
It should be noted that even when the localization problem is overdetermined,
noisy range measurements can lead to flip ambiguity in a bad geometry, as
discussed in [26, 27]. The flip ambiguity can cause incorrect initial guess or
unconvergence of some iterative lateration algorithms. The localization
accuracy discussed in this paper (as well as many previous papers) is about
the lateration errors under the assumption that the initial guess is correct
and the lateration converges.
The accuracy of lateration has been widely studied using the Cramér-Rao (CR)
bound [28, 29, 30, 31, 32, 33, 34, 35], which is the reciprocal of the Fisher
information matrix [36]. Many of these studies ended up with a complicated
Fisher information matrix (or an equivalent form) involving node locations,
and did not give an explicit closed-form expression that can characterize
localization accuracy with respect to network connectivity.
There have been some papers (e.g., [31, 37]) using Monte-Carlo simulations to
reveal that (1) for one sensor node and $m$ randomly-distributed anchor nodes,
the CR bound is inversely proportional to $m$; (2) higher percentage of
anchors results in better accuracy. However, there is still a dearth of theory
to describe these relationships precisely for a more generalized setting.
Two papers [38, 39] deserve special mentions here because they contain a
similar flavor to this paper. Shen et al. [38] presented scaling laws of
localization accuracy for randomly-deployed nodes. The scaling laws indicate
that sensors and anchors “contribute equally” to localization accuracy.
However, this statement is correct only for dense network (a larger number of
nodes); in this paper, we shall show that anchors generally contribute more
than sensors do, and the sensors and anchors tend to contribute equally as the
number of sensors increases. In [39], Javanmard and Montanari offered neat
upper and lower bounds of localization accuracy for random geometric graphs.
However, the bounds are only applicable to random geometric graphs, and
require range errors to be uniformly bounded.
### I-B Our contributions
In this paper, we use the _average degree_ as a connectivity metric and use
geometric _dilution of precision_ (DOP), equivalent to the CR bound, to
quantify localization accuracy. We have proved a lower bound on the
expectation of average geometric DOP (E-AGDOP) under the assumption that nodes
are randomly distributed, and nodes are randomly connected such that the graph
of network can reach a certain average degree. We have further derived a
closed-form formula that relates the lower bound to only three parameters:
average anchor degree, average sensor degree, and number of sensor nodes. The
formula shows that (1) localization accuracy is approximately inversely
proportional to the average degree, and (2) average anchor degree contributes
more to localization accuracy than average sensor degree does. Our numerical
examples and simulations have validated the formula and further shown that (1)
the lower bound is strongly connected to the best achievable localization
accuracy, and (2) the lower bound is applicable to many random graph models.
### I-C Outline of the paper
The remainder of this paper is organized as follows. Section II formulates the
cooperative localization problem and introduces the assumptions, definitions,
and notations used throughout this paper. Section III analyzes localization
accuracy and connects it to DOP. Section IV derives a closed-form expression
LB-E-AGDOP, which is a function of network connectivity and size. Section V
shows the strong connection between LB-E-AGDOP and the best achievable
accuracy. Numerical simulation results are presented in Section VI to validate
the theory. Finally, Section VII concludes the paper. Proofs of key theorems
and equations are provided in Appendices A and B.
## II Preliminaries
### II-A Problem formulation
In this paper, a sensor network is modeled as a _simple_ graph111A simple
graph, also known as a strict graph, is an unweighted, undirected graph
containing no self-loops or multiple edges [40]. $\mathcal{G}=(V,E)$, where
$V=\\{1,2,\ldots,N\\}$ is a set of $N$ nodes (or vertices), and
$E=\\{e_{1},e_{2},\ldots,e_{K}\\}\subseteq V\times V$ is a set of $K$ links
(or edges) that connect the nodes [16].
All nodes are in a $d$-dimensional Euclidean space ($d\geq 1$), with the
locations denoted by $p_{n}\in\mathbb{R}^{d}$, $n=1$, …, $N$. The first
$N_{S}$ nodes, labeled 1 through $N_{S}$, are _sensor_ nodes (or mobile
nodes), whose locations are unknown; the rest $N_{A}=N-N_{S}$ nodes, labeled
$N_{S}+1$ through $N$, are _anchor_ nodes (or beacon nodes). Anchors are aware
of their exact locations through built-in GPS receivers or manual pre-
programming during deployment.
An unordered pair $e_{k}=(i_{k},j_{k})\in E$ if and only if there exists a
direct ranging link between nodes $i_{k}$ and $j_{k}$. The link provides
inter-node distance information $\rho_{k}=r_{k}+\epsilon_{k}$, where
$r_{k}=\|p_{i_{k}}-p_{j_{k}}\|$ is the actual Euclidean distance between nodes
$i$ and $j$, and $\epsilon_{k}$ is the range measurement error.
The cooperative localization problem is to determine the locations of sensor
nodes $p_{n}$, $n=1$, …, $N_{S}$, given a fixed network graph $G$, known
locations of anchors $p_{n}$, $n=N_{S}+1$, …, $N$, and range measurements
$\rho_{k}$, $k=1$, …, $K$.
### II-B Assumptions
#### II-B1 Range measurement errors
The range measurements $\rho_{k}$ can be obtained by a variety of methods,
such as one-way time of arrival (ToA), two-way ToA, or received signal
strength indication (RSSI) [41]. One-way ToA usually result in biased range
measurements due to unsynchronized clocks [24], while two-way ToA and RSSI do
not depend on clocks. In this paper, we assume zero clock biases in range
measurements. Our assumption holds for the cases of two-way ToA, RSSI, and
one-way ToA with perfect clock synchronization.
Range measurement errors in RSSI-based methods are usually treated as having a
log-normal distribution [42]. For most ToA-based methods, line-of-sight range
measurement errors can be modeled as zero-mean Gaussian random variables [6,
24]. In this paper, we adopt the Gaussian assumption.
#### II-B2 Coordinate symmetry
For any link $e_{k}=(i_{k},j_{k})\in E$, we assume that the direction vector,
defined as
$v_{k}=r_{k}^{-1}(p_{i_{k}}-p_{j_{k}})=[v_{k,1},\ldots,v_{k,d}]^{\mathsf{T}},$
(1)
satisfies the following condition:
$\operatorname{E}(v_{k,1}^{2})=\operatorname{E}(v_{k,2}^{2})=\cdots=\operatorname{E}(v_{k,d}^{2}).$
(2)
This assumption holds for all sensor-to-sensor links if all sensor nodes are
uniformly distributed in a space that is symmetrical in all coordinates. This
assumption holds for anchor-to-sensor links if anchors are uniformly
distributed or anchors are fixed at certain special locations such as the
scenario shown in Fig. 1.
The list below shows three well-studied models of random graphs.
* •
Erdős–Rényi random graph (ERG) $\mathcal{G}(N,p)$: Nodes are connected
randomly regardless of the distance. Each link is included in the graph with
probability $p$ independent from every other link [43].
* •
Random geometric graph (RGG) $\mathcal{G}(N,r)$: Two nodes are connected if
and only if the distance between them is at most a threshold $r$ [44, 45].
* •
Random proximity graph (RPG) $\mathcal{G}(N,k)$: Each node connects to its $k$
nearest neighbors. These graphs are also denoted $k$-NNG [45].
With properly chosen locations of anchor nodes, all of them satisfies the
coordinate symmetry condition. Therefore, the theory developed in this paper
is applicable to, but not limited to, the above models.
### II-C Metrics of connectivity
For all nodes $n=1$, …, $N$, we define the following _degrees_ :
* •
Anchor degree: $\operatorname{deg}_{A}(n)$, the number of anchor nodes
incident to node $n$;
* •
Sensor degree: $\operatorname{deg}_{S}(n)$, the number of sensor nodes
incident to node $n$;
* •
Degree:
$\operatorname{deg}(n)=\operatorname{deg}_{A}(n)+\operatorname{deg}_{S}(n)$,
the number of nodes incident to node $n$;
We assume that there are no anchor-to-anchor links, i.e.,
$\operatorname{deg}_{A}(n)=0$ for $n=N_{S}+1$, …, $N$, because anchor-to-
anchor links are helpless when locations of anchors are perfectly known.
In graph theory, connectivity is usually described by vertex connectivity or
edge connectivity: a graph is $\kappa$-vertex/edge-connected if it remains
connected whenever fewer than $\kappa$ vertices/edges are removed [46].
Unfortunately, vertex/edge connectivity mainly reflects some “minimum”
properties of connectivity, such as
$\min_{n\in\\{1,\ldots,N\\}}\operatorname{deg}(n)$ [46], and does not
distinguish between sensor and anchor nodes. This paper uses average degrees
to characterize the overall connectivity of the network. Average degrees are
defined as
$\delta_{*}=\frac{1}{N_{S}}\sum_{n=1}^{N_{S}}\operatorname{deg}_{*}(n),$ (3)
where the subscript ∗ can be blank, A, or S, for the average degree, average
anchor degree, or average sensor degree, respectively.
Let $K_{S}$ and $K_{A}$ denote the number of sensor-to-sensor and anchor-to-
sensor links in the network, respectively. It is easy to verify the equalities
$K=K_{S}+K_{A}$, $N_{S}\delta_{S}=2K_{S}$, $N_{S}\delta_{A}=K_{A}$, and
$\delta=\delta_{S}+\delta_{A}$.
### II-D List of notations
$d$
dimensionality
$\operatorname{deg}(n)$
degree of node $n$
$\operatorname{deg}_{A}(n)$
anchor degree of node $n$
$\operatorname{deg}_{S}(n)$
sensor degree of node $n$
$\delta$
average degree
$\delta_{A}$
average anchor degree
$\delta_{S}$
average sensor degree
$E$
set of all links, $\\{e_{1},\ldots,e_{K}\\}$
$\epsilon_{k}$
range error of link $e_{k}$
$\bm{\varepsilon}$
localization errors, $\bm{\varepsilon}=\bm{p}^{(\infty)}-\bm{p}$
$F$
inverse of DOP matrix, $F=G^{\mathsf{T}}G$
$\mathcal{G}$
graph $(V,E)$
$G$
geometry matrix
$H$
DOP matrix, $H=(G^{\mathsf{T}}G)^{-1}$
$I$
identity matrix
$i_{k}$
head of link $e_{k}=(i_{k},j_{k})$
$j_{k}$
tail of link $e_{k}=(i_{k},j_{k})$
$K$
number of all links
$K_{A}$
number of all anchor-to-sensor links
$K_{S}$
number of all sensor-to-sensor links
$L$
Laplacian matrix of graph $\mathcal{G}$,
$d\check{\Xi}=[L_{ij}]_{i,j\in\\{1,2,\ldots,N_{S}\\}}$
$m$
index of dimensions, $m\in\\{1,\ldots,d\\}$.
$N$
number of nodes, $N=N_{A}+N_{S}$
$N_{A}$
number of anchor nodes
$N_{S}$
number of sensor nodes
$\mathcal{N}(\mu,\sigma^{2})$
Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$
$p_{i}$
location of node $i$
$r_{k}$
actual distance of link $e_{k}$
$\rho_{k}$
distance measurement of link $e_{k}$
$\Sigma$
covariance of range errors
$\sigma_{k}$
standard deviation of range errors of link $e_{k}$
$V$
set of all nodes, $\\{1,\ldots,N\\}$
$v_{k}$
unit vector denoting the direction of link $e_{k}$,
$v_{k}=r_{k}^{-1}(p_{i_{k}}-p_{j_{k}})$
$\Xi$
conditional expectation of $F$ given certain links,
$\Xi=\operatorname{E}_{\text{locations}}(F|\text{links})$
$\check{\Xi}$
submatrix of $\Xi$, representting one coordinate
$\|z\|$
Euclidean norm of vector $z$
$A\succeq B$
$A-B$ is positive semidefinite
## III Localization Accuracy
Localization is essentially an optimization problem that finds coordinate
vectors $p_{n}\in\mathbb{R}^{d}$, $n=1$, …, $N_{S}$, such that for each
ranging link $e_{k}=(i_{k},j_{k})\in E$, the distance
$r_{k}=\|p_{i_{k}}-p_{j_{k}}\|$ is as close to the range measurement
$\rho_{k}$ as possible.
Assume that range errors follow a zero-mean Gaussian distribution:
$\epsilon_{k}=\rho_{k}-r_{k}\sim\mathcal{N}(0,\sigma_{k}^{2}),\quad\forall
k=1,\ldots,K.$ (4)
The maximum-likelihood estimation of $\\{p_{n}\\}_{n=1}^{N_{S}}$ is equivalent
to the weighted least squares (LS) problem
$\begin{split}&\operatorname*{arg\,max}_{\\{p_{n}\\}_{n=1}^{N_{S}}}\operatorname{P}\bigl{(}\\{\rho_{k}\\}_{k=1}^{K}\bigm{|}\\{p_{n}\\}_{n=1}^{N_{S}}\bigr{)}\\\
&=\operatorname*{arg\,max}_{\\{p_{n}\\}_{n=1}^{N_{S}}}\prod_{k=1}^{K}\frac{1}{2\pi\sigma_{k}^{2}}\exp\Bigl{(}-\frac{(\|p_{i_{k}}-p_{j_{k}}\|-\rho_{k})^{2}}{2\sigma_{k}^{2}}\Bigr{)}\\\
&=\operatorname*{arg\,min}_{\\{p_{n}\\}_{n=1}^{N_{S}}}\sum_{k=1}^{K}\frac{(\|p_{i_{k}}-p_{j_{k}}\|-\rho_{k})^{2}}{\sigma_{k}^{2}}.\end{split}$
(5)
The LS problem cannot be directly solved because the distance
$r_{k}=\|p_{i_{k}}-p_{j_{k}}\|$ is a nonlinear function of the coordinate
vectors $p_{i_{k}}$ and $p_{j_{k}}$. Let
$\bm{r}=(r_{1},r_{2},\ldots,r_{K})^{\mathsf{T}}\in\mathbb{R}^{K}$ and
$\bm{p}=\operatorname{column}\\{p_{1},p_{2},\ldots,p_{N_{S}}\\}\allowbreak\in\mathbb{R}^{dN_{S}}$.
The first-order linear approximation of the distance function
$\bm{r}(\bm{p}^{(0)}+\Delta\bm{p})$ with respect to an initial guess
$\bm{p}^{(0)}$ can be written as
$\bm{r}(\bm{p}^{(0)}+\Delta\bm{p})=\bm{r}(\bm{p}^{(0)})+G\Delta\bm{p},$ (6)
where the _geometry matrix_ $G\in\mathbb{R}^{K\times dN_{S}}$ is given by
$\begin{split}G&=\frac{\partial\bm{r}}{\partial\bm{p}}\\\
&=\begin{bmatrix}\frac{\partial r_{1}}{\partial p_{1,1}}&\ldots&\frac{\partial
r_{1}}{\partial p_{1,d}}&\ldots&\frac{\partial r_{1}}{\partial
p_{N_{S},1}}&\ldots&\frac{\partial r_{1}}{\partial p_{N_{S},d}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\ \frac{\partial r_{K}}{\partial
p_{1,1}}&\ldots&\frac{\partial r_{K}}{\partial p_{1,d}}&\ldots&\frac{\partial
r_{K}}{\partial p_{N_{S},1}}&\ldots&\frac{\partial r_{K}}{\partial
p_{N_{S},d}}\\\ \end{bmatrix},\end{split}$ (7)
where $p_{i,m}$, $m=1$, …, $d$, is the $m$th element of the coordinate vector
$p_{i}$. Each element of the geometry matrix $G$ is given by
$\begin{split}G_{k,(n-1)d+m}&=\frac{\partial r_{k}}{\partial
p_{n,m}}=\frac{\partial\|p_{i_{k}}-p_{j_{k}}\|}{\partial p_{n,m}}\\\
&=\begin{cases}\frac{p_{i_{k},m}-p_{j_{k},m}}{\|p_{i_{k}}-p_{j_{k}}\|}=v_{k,m}&\text{if
}n=i_{k},\\\
\frac{p_{j_{k},m}-p_{i_{k},m}}{\|p_{i_{k}}-p_{j_{k}}\|}=-v_{k,m}&\text{if
}n=j_{k},\\\ 0&\text{otherwise}.\\\ \end{cases}\end{split}$ (8)
Each row of $G$ represents a link. There are only $d$ nonzero elements in a
row for an anchor-to-sensor link, and there are $2d$ nonzero elements for an
sensor-to-sensor link. Given that each row of $G$ has $dN_{S}$ elements, $G$
is highly sparse when the network contains many sensor nodes.
When the network is localizable, $G$ must be a tall matrix (i.e., $K\geq
dN_{S}$ [15, 16, 47]) with full column rank. Then, the weighted LS problem (5)
can be solved by the following iterative algorithm based on the Newton–Raphson
method [24]:
$\bm{p}^{(n+1)}=\bm{p}^{(n)}+(G^{\mathsf{T}}\Sigma^{-1}G)^{-1}G^{\mathsf{T}}\Sigma^{-1}[\bm{\rho}-\bm{r}(\bm{p}^{(n)})],$
(9)
where $\bm{\rho}=(\rho_{1},\rho_{2},\ldots,\rho_{K})^{\mathsf{T}}$,
$\Sigma=\operatorname{Cov}(\bm{\epsilon},\bm{\epsilon})$ is the covariance of
range errors, where
$\bm{\epsilon}=(\epsilon_{1},\ldots,\epsilon_{K})^{\mathsf{T}}$.
When the initial guess $\bm{p}^{(0)}$ is accurate enough and the iteration
converges, the localization errors $\bm{\varepsilon}$ have the following
relationship to the range errors
$\bm{\epsilon}=(\epsilon_{1},\ldots,\epsilon_{K})^{\mathsf{T}}$:
$\begin{split}\bm{\varepsilon}&=\bm{p}^{(\infty)}-\bm{p}=(G^{\mathsf{T}}\Sigma^{-1}G)^{-1}G^{\mathsf{T}}\Sigma^{-1}(\bm{\rho}-\bm{r})\\\
&=(G^{\mathsf{T}}\Sigma^{-1}G)^{-1}G^{\mathsf{T}}\Sigma^{-1}\bm{\epsilon}.\end{split}$
(10)
The covariance of localization errors is thus given by
$\begin{split}\operatorname{Cov}(\bm{\varepsilon},\bm{\varepsilon})&=(G^{\mathsf{T}}\Sigma^{-1}G)^{-1}G^{\mathsf{T}}\Sigma^{-1}\operatorname{Cov}(\bm{\epsilon},\bm{\epsilon})\\\
&\qquad\Sigma^{-1}G^{\mathsf{T}}(G^{\mathsf{T}}\Sigma^{-1}G)^{-1}\\\
&=(G^{\mathsf{T}}\Sigma^{-1}G)^{-1}.\end{split}$ (11)
This has achieved the CR bound [28, 31, 32, 33, 29, 34].
If range measurement errors are independent and identically distributed (iid),
i.e., $\Sigma=\operatorname{diag}(\sigma^{2},\ldots,\sigma^{2})$, we have
$\operatorname{Cov}(\bm{\varepsilon},\bm{\varepsilon})=(G^{\mathsf{T}}\Sigma^{-1}G)^{-1}=\sigma^{2}(G^{\mathsf{T}}G)^{-1}.$
(12)
The matrix $H=(G^{\mathsf{T}}G)^{-1}\in\mathbb{R}^{dN_{S}\times dN_{S}}$ is
referred to as dilution of precision (DOP) matrix. DOP is a term widely used
in satellite navigation specifying the multiplicative effect on positioning
accuracy due to satellite geometry222The DOP is usually defined in the form of
$\sqrt{\operatorname{tr}[(G^{\mathsf{T}}G)^{-1}]}$ [37, 24]. In this paper, we
define DOP in the form of $\operatorname{tr}[(G^{\mathsf{T}}G)^{-1}]$ for
simplicity in calculation and analysis. [24]. For cooperative localization,
DOP specifies the multiplicative effect due to not only geometry of the nodes
but also connectivity of the network. DOP decouples localization accuracy from
range accuracy. The smaller DOP is, the better localization accuracy one would
expect.
A diagonal element $H_{(n-1)d+m,(n-1)d+m}$ is the DOP of coordinate $m$ for
node $n$. The sum of all the diagonal elements, $\operatorname{tr}(H)$, is the
geometric DOP (GDOP) of the whole network. In this paper, we define average
GDOP (AGDOP) as GDOP divided by the number of sensor nodes,
$\operatorname{tr}(H)/N_{S}$. AGDOP is a performance indicator of localization
accuracy due to network geometry and connectivity.
For a network where nodes are deployed and connected randomly, AGDOP is a
random variable. The expectation of AGDOP (E-AGDOP) indicates the expected
localization accuracy because the root-mean-square localization error is
proportional to $\sqrt{\text{E-AGDOP}}$. We shall use E-AGDOP and its lower
bound to study the relationship between localization accuracy and network
connectivity in the rest of the paper.
## IV Lower Bound on E-AGDOP
In this section, we shall prove that
$[\operatorname{E}(G^{\mathsf{T}}G)]^{-1}$ is a lower bound on
$\operatorname{E}[(G^{\mathsf{T}}G)^{-1}]$. Furthermore, we shall show that it
is possible to evaluate this lower bound analytically for a random network
(randomly-deployed nodes and randomly-established links) that achieves a
certain level of connectivity.
###### Theorem 1 (Lower bound on DOP matrix)
For a random network with a non-singular geometry matrix $G$ defined in (7),
$\operatorname{E}[(G^{\mathsf{T}}G)^{-1}]\succeq[\operatorname{E}(G^{\mathsf{T}}G)]^{-1},$
(13)
where the operator $X\succeq Y$ denotes that $X-Y$ is positive semidefinite.
###### Proof:
Detailed in Appendix A. ∎
The matrix $F=G^{\mathsf{T}}G$ is a function of node locations and links, both
of which have been assumed to be random. Let us calculate $\operatorname{E}F$
by the following two steps:
1. 1.
$\Xi=\operatorname{E}_{\text{nodes}}(F|\text{links})$, conditional expectation
of $F$ for randomly-deployed nodes given certain links;
2. 2.
$\operatorname{E}F=\operatorname{E}_{\text{links}}(\Xi)$, expectation of $\Xi$
for randomly-established links.
### IV-A Step 1: randomly-deployed nodes
Recall (8) which describes the elements in $G$. Note that when link $e_{k}$
connects to node $n$, i.e., $n\in\\{i_{k},j_{k}\\}$,
$\sum_{m=1}^{d}\Bigl{(}\frac{\partial r_{k}}{\partial
p_{n,m}}\Bigr{)}^{2}=\frac{\sum_{m=1}^{d}(p_{i_{k},m}-p_{j_{k},m})^{2}}{\|p_{i_{k}}-p_{j_{k}}\|^{2}}=1.$
(14)
By the coordinate symmetry assumption (Section II-B2), we have
$\operatorname{E}\Bigl{(}\frac{\partial r_{k}}{\partial
p_{n,1}}\Bigr{)}^{2}=\operatorname{E}\Bigl{(}\frac{\partial r_{k}}{\partial
p_{n,2}}\Bigr{)}^{2}=\cdots=\operatorname{E}\Bigl{(}\frac{\partial
r_{k}}{\partial p_{n,m}}\Bigr{)}^{2}.$ (15)
To satisfy (14), we must have
$\operatorname{E}\Bigl{(}\frac{\partial r_{k}}{\partial
p_{n,m}}\Bigr{)}^{2}=\frac{1}{d},\quad\forall m=1,\ldots,d.$ (16)
Therefore, the elements of matrix
$F=\\{F_{\tilde{i}\tilde{j}}\\}\in\mathbb{R}^{dN_{S}\times dN_{S}}$ have the
conditional expectation
$\begin{split}\Xi_{\tilde{i}\tilde{j}}&=\operatorname{E}_{\text{nodes}}(F_{\tilde{i}\tilde{j}}|\text{links})=\operatorname{E}\sum_{k=1}^{K}\frac{\partial
r_{k}}{\partial p_{i,m_{1}}}\frac{\partial r_{k}}{{\partial p_{j,m_{2}}}}\\\
&=\begin{cases}\frac{1}{d}\operatorname{deg}(i)&\text{if }i=j\text{ and
}m_{1}=m_{2},\\\ -\frac{1}{d}&\text{if }(i,j)\in E\text{ and }m_{1}=m_{2},\\\
0&\text{otherwise},\\\ \end{cases}\end{split}$ (17)
where $\tilde{i}=(i-1)d+m_{1}$, $\tilde{j}=(j-1)d+m_{2}$, $1\leq
m_{1},m_{2}\leq d$. For instance, let us consider a very simple sensor network
shown in Fig. 2. The matrix $\Xi$ for this network is given by
$\Xi_{\text{Fig.~{}\ref{fig:simpleSN}}}=\begin{bmatrix}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\frac{1}{2}}&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-\frac{1}{2}}&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0\\\
0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{1}{2}}&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\frac{1}{2}}&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-\frac{1}{2}}&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-\frac{1}{2}}&0\\\
0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\frac{1}{2}}&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\frac{1}{2}}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}0}&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}-\frac{1}{2}}&0&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}&0\\\
0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}0}&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}-\frac{1}{2}}&0&{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}1}\\\
\end{bmatrix}.$ (18)
1234 Figure 2: A simple sensor network comprised of 4 nodes and 3 links in 2
dimensions. Nodes 1 to 3 are sensors; node 4 is an anchor ($N_{S}=3$,
$N_{A}=1$, $K_{S}=2$, $K_{A}=1$). Eq. (18) shows the matrix $\Xi$ for this
network.
As shown by the red- and blue-colored elements in (18), we have
$\Xi=\check{\Xi}\otimes I$, where $\otimes$ denotes the Kronecker product, and
$I$ is the identity matrix of size $d$. The elements of the matrix
$\check{\Xi}\in\mathbb{R}^{N_{S}\times N_{S}}$ are given by
$\check{\Xi}_{ij}=\begin{cases}\frac{1}{d}\operatorname{deg}(i)&\text{if
}i=j,\\\ -\frac{1}{d}&\text{if }(i,j)\in E,\\\ 0&\text{otherwise}.\\\
\end{cases}$ (19)
For the sensor network shown by Fig. 2, the matrix $\check{\Xi}$ is given by
$\check{\Xi}_{\text{Fig.~{}\ref{fig:simpleSN}}}=\begin{bmatrix}\frac{1}{2}&-\frac{1}{2}&0\\\
-\frac{1}{2}&1&-\frac{1}{2}\\\ 0&-\frac{1}{2}&1\\\ \end{bmatrix}.$ (20)
The matrix $\check{\Xi}$ (as well as $\Xi$) indicates a relationship between
localization accuracy and graph Laplacians [48]. Let $L$ denote the Laplacian
matrix of the graph $\mathcal{G}$. It can be seen that
$\check{\Xi}=d^{-1}[L_{ij}]_{i,j\in\\{1,2,\ldots,N_{S}\\}}$, i.e.,
$d\check{\Xi}$ is the matrix of $L$ obtained by deleting its last $N_{A}$ rows
and columns that are related to the anchor nodes.
The lower bound on E-AGDOP (LB-E-AGDOP) can be calculated by inverting
$F=\operatorname{E}\Xi$ or, equivalently, inverting
$\check{F}=\operatorname{E}\check{\Xi}$, because
$\operatorname{tr}[(\operatorname{E}\Xi)^{-1}]=d\operatorname{tr}[(\operatorname{E}\check{\Xi})^{-1}]$.
### IV-B Step 2: randomly-established links
Given an average degree $\delta$, the trace of $\check{\Xi}$ is given by
$\operatorname{tr}(\check{\Xi})=\sum_{i=1}^{N_{S}}\check{\Xi}_{ii}=\sum_{i=1}^{N_{S}}\operatorname{deg}(i)/d=N_{S}\delta/d.$
(21)
Given an average sensor degree $\delta_{S}$, there are
$K_{S}=N_{S}\delta_{S}/2$ sensor-to-sensor links in the network, and thus
$\check{\Xi}$ includes $N_{S}\delta_{S}$ off-diagonal elements with a non-zero
value of $-1/d$. Assume that the sensor-to-sensor links are chosen uniformly
at random from the set $\\{(i,j)|1\leq i<j\leq N_{S},i,j\in\mathbb{Z}\\}$.
Then, each off-diagonal element $\check{\Xi}_{ij}$, $i\neq j$ satisfies the
Bernoulli distribution
$\check{\Xi}_{ij}=\begin{cases}-1/d&\text{with probability
}\frac{\delta_{S}}{N_{S}-1},\\\ 0&\text{with probability
}1-\frac{\delta_{S}}{N_{S}-1}.\end{cases}$ (22)
Then, the expectation of $\check{F}$ is given by
$\operatorname{E}\check{F}_{ij}=\operatorname{E}_{\text{links}}(\Xi_{ij})=\begin{cases}\frac{\delta}{d}&\text{if
}i=j,\\\ -\frac{\delta_{S}}{d(N_{S}-1)}&\text{otherwise}.\\\ \end{cases}$ (23)
Appendix B shows that
$\operatorname{tr}[(\operatorname{E}\check{F})^{-1}]=\frac{N_{S}}{\eta}\Bigl{(}1+\frac{\zeta}{1-N_{S}\zeta}\Bigr{)},$
(24)
where $\eta=d^{-1}[\delta+\delta_{S}/(N_{S}-1)]$ and
$\zeta=\delta_{S}/[\delta(N_{S}-1)+\delta_{S}]$. Therefore, LB-E-AGDOP is
given by
$\begin{split}\text{LB-E-
AGDOP}&=\frac{\operatorname{tr}[(\operatorname{E}F)^{-1}]}{N_{S}}=\frac{d\operatorname{tr}[(\operatorname{E}\check{F})^{-1}]}{N_{S}}\\\
&=\frac{d}{\eta}\Bigl{(}1+\frac{1}{\zeta^{-1}-N_{S}}\Bigr{)}\\\
&=\frac{d^{2}}{\delta+\delta_{S}/(N_{S}-1)}\Bigl{(}1+\frac{\delta_{S}}{\delta_{A}(N_{S}-1)}\Bigr{)}\\\
&=\frac{d^{2}}{\delta}\frac{N_{S}-1+\delta_{S}/\delta_{A}}{N_{S}-1+\delta_{S}/\delta}.\end{split}$
(25)
Thus far, we have obtained a closed-form expression for LB-E-AGDOP. It depends
on two parameters of network connectivity, $\delta_{S}$ and $\delta_{A}$ (note
$\delta=\delta_{S}+\delta_{A}$), and one parameter of network size, $N_{S}$.
In (25), the first term ${d^{2}}/{\delta}$ shows that LB-E-AGDOP is
approximately inversely proportional to the average degree, and grows
quadratically with dimensionality. The second term
$(N_{S}-1+\delta_{S}/\delta_{A})\big{/}(N_{S}-1+\delta_{S}/\delta)$ shows that
a higher ratio of average anchor degree to average sensor degree leads to
better localization accuracy. However, this effect diminishes when number of
sensor nodes increase. As $N_{S}\rightarrow\infty$, the LB-E-AGDOP approaches
${d^{2}}/{\delta}$ regardless of the ratio of average anchor degree to average
sensor degree.
## V LB-E-AGDOP and the Best Achievable Accuracy
The previous section has proven that the LB-E-AGDOP given by (25) is a lower
bound on E-AGDOP. In this section, we shall show that LB-E-AGDOP describes the
best achievable accuracy with certain network connectivity. We first prove
that LB-E-AGDOP is equal to the minimum AGDOP for one sensor node. For
multiple sensor nodes, we use several numerical examples to show that LB-E-
AGDOP is less than and very close to the minimum AGDOP.
### V-A Minimum AGDOP for one sensor node
Let us first consider the simplest case that there is only one sensor node,
i.e., $N_{S}=1$. By (25), the lower bound becomes
$\text{LB-E-
AGDOP}=\frac{d^{2}}{\delta}\frac{\delta_{S}/\delta_{A}}{\delta_{S}/\delta}=\frac{d^{2}}{\delta_{A}}=\frac{d^{2}}{N_{A}}.$
(26)
In this subsection, we shall show that the lower bound ${d^{2}}/{N_{A}}$
represents the best achievable performance.
When $N_{S}=1$, the geometry matrix can be written as
$G=\begin{bmatrix}\cos\theta_{1}&\sin\theta_{1}\\\
\cos\theta_{2}&\sin\theta_{2}\\\ \vdots&\vdots\\\
\cos\theta_{N_{A}}&\sin\theta_{N_{A}}\\\ \end{bmatrix},$ (27)
where $\theta_{i}$, $i=1$, …, $N_{A}$ is the angular coordinate of anchor node
$i$ in the polar coordinate system poled at the sensor node. Fig. 3 shows a
scenario of $N_{A}=3$.
$\theta_{1}$$\theta_{2}$$\theta_{3}$ Figure 3: Geometry of one sensor node and
three anchor nodes.
Noting that
$G^{\mathsf{T}}G=\begin{bmatrix}\sum_{i=1}^{N_{A}}\cos^{2}\theta_{i}&\sum_{i=1}^{N_{A}}\cos\theta_{i}\sin\theta_{i}\\\
\sum_{i=1}^{N_{A}}\cos\theta_{i}\sin\theta_{i}&\sum_{i=1}^{N_{A}}\sin^{2}\theta_{i}\\\
\end{bmatrix},$ (28)
we obtain a closed-form expression of $(G^{\mathsf{T}}G)^{-1}$ as
$\begin{split}&(G^{\mathsf{T}}G)^{-1}=\frac{1}{\det(G^{\mathsf{T}}G)}\cdot\\\
&\quad\begin{bmatrix}\sum_{i=1}^{N_{A}}\sin^{2}\theta_{i}&-\sum_{i=1}^{N_{A}}\cos\theta_{i}\sin\theta_{i}\\\
-\sum_{i=1}^{N_{A}}\cos\theta_{i}\sin\theta_{i}&\sum_{i=1}^{N_{A}}\cos^{2}\theta_{i}\\\
\end{bmatrix},\end{split}$ (29)
where the determinant of $G^{\mathsf{T}}G$ is given by
$\begin{split}\det(G^{\mathsf{T}}G)&=\biggl{(}\sum_{i=1}^{N_{A}}\cos^{2}\theta_{i}\biggr{)}\biggl{(}\sum_{i=1}^{N_{A}}\sin^{2}\theta_{i}\biggr{)}\\\
&\qquad{}-\biggl{(}\sum_{i=1}^{N_{A}}\cos\theta_{i}\sin\theta_{i}\biggr{)}^{2}\\\
&=\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{A}}(\cos^{2}\theta_{i}\sin^{2}\theta_{j}\\\
&\qquad{}-\cos\theta_{i}\sin\theta_{i}\cos\theta_{j}\sin\theta_{j})\\\
&=\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{A}}\cos\theta_{i}\sin\theta_{j}\sin(\theta_{j}-\theta_{i})\\\
&=\sum_{1\leq i<j\leq N_{A}}\sin^{2}(\theta_{j}-\theta_{i}).\end{split}$ (30)
Therefore, the GDOP is given by
$\operatorname{tr}[(G^{\mathsf{T}}G)^{-1}]=\frac{N_{A}}{\sum_{1\leq i<j\leq
N_{A}}\sin^{2}(\theta_{j}-\theta_{i})},$ (31)
which agrees with the result obtained in [49].
From (31) we can find the minimum AGDOP for one sensor node and $N_{A}$ anchor
nodes. Since the sum $S=\sum_{1\leq i<j\leq
N_{A}}\sin^{2}(\theta_{j}-\theta_{i})$ is continuous differentiable and
bounded, a maximum or a minimum is attained if and only if
$\begin{split}\frac{\partial
S}{\partial\theta_{i}}&=\sum_{j=1}^{N_{A}}\sin[2(\theta_{i}-\theta_{j})]\\\
&=0,\qquad\forall i\in\\{1,\ldots,N_{A}\\}.\end{split}$ (32)
It is easy to verify that the values $\theta_{i}=2\pi i/N_{A}$ satisfy the
above condition. The corresponding maximum of $S$ is given by
$\begin{split}\sum_{1\leq i<j\leq
N_{A}}\sin^{2}(\theta_{j}-\theta_{i})&=\sum_{1\leq i<j\leq
N_{A}}\sin^{2}\frac{(j-i)2\pi}{N_{A}}\\\
&=\frac{1}{2}\sum_{i=1}^{N_{A}}\sum_{j=1}^{N_{A}}\sin^{2}\frac{(j-i)2\pi}{N_{A}}\\\
&=\frac{1}{2}\sum_{i=1}^{N_{A}}N_{A}/2=\frac{N_{A}^{2}}{4}.\\\ \end{split}$
(33)
Therefore, the minimum AGDOP is $4/N_{A}$, equal to the lower bound given by
(26).
Fig. 4 compares the LB-E-AGDOP calculated from (25) and the AGDOP obtained
from simulations with the parameters $N_{S}=1$, $N_{A}=3,4,\ldots,9$. The
green solid curve shows the LB-E-AGDOP. The box-and-whisker plot [50] shows
the sample minimum, lower quartile, median, upper quartile, and sample maximum
of the AGDOP. The red plus marks denote statistical outliers. It can be seen
that our LB-E-AGDOP is equal to the sample minimum, and is closer to the
median when $N_{A}$ is greater.
Figure 4: Comparison between LB-E-AGDOP from (25) and AGDOP from simulations
with the parameters $N_{S}=1$, $N_{A}=3,4,\ldots,9$. The green solid curve
shows the LB-E-AGDOP. The box-and-whisker plot shows the sample minimum (lower
whisker, black), lower quartile (lower edge of the box, blue), median (central
mark, red), upper quartile (upper edge of the box, blue), and sample maximum
(upper whisker, black; outliers excluded). Our LB-E-AGDOP is equal to the
sample minimum, and is closer to the median when $N_{A}$ is greater.
### V-B Minimum AGDOP for multiple sensor nodes
For multiple sensor nodes, the minimum AGDOP can be hardly derived from a
theoretical analysis. Instead, we present several results obtained from
numerical optimization. From the following results it can be seen that for
multiple sensor nodes, our LB-E-AGDOP is less than the minimum achievable
AGDOP. The gap between LB-E-AGDOP and the minimum AGDOP is smaller when the
LB-E-AGDOP is smaller.
#### Case 1: $N_{S}=2$, $\delta_{S}=1$, $\delta_{A}=2$
One of the best geometries is given by
98.68∘
The geometry has up-down and left-right reflection symmetry. The minimum AGDOP
is 1.633, greater than the LB-E-AGDOP 1.500.
#### Case 2: $N_{S}=2$, $\delta_{S}=1$, $\delta_{A}=3$
One of the best geometries is given by
112.59∘
The geometry has up-down and left-right reflection symmetry. The minimum AGDOP
is 1.124, slightly greater than the LB-E-AGDOP 1.067.
#### Case 3: $N_{S}=3$, $\delta_{S}=2$, $\delta_{A}=1$
One of the best geometries is given by
60.00∘
The geometry has $\pm 120^{\circ}$ rotational symmetry. The minimum AGDOP is
2.667, greater than the LB-E-AGDOP 2.000. It should be noted that the
connectivity of this case does not ensure unique localizability [16, 7].
#### Case 4: $N_{S}=3$, $\delta_{S}=2$, $\delta_{A}=2$
One of the best geometries is given by
104.15∘
The geometry has left-right reflection symmetry and $\pm 120^{\circ}$
rotational symmetry. The minimum AGDOP is 1.313, slightly greater than the LB-
E-AGDOP 1.200.
TABLE I: LB-E-AGDOP establishes a lower bound on AGDOP $N_{S}$ | $N_{A}$ | $\delta_{S}$ | $\delta_{A}$ | minimum AGDOP | LB-E-AGDOP
---|---|---|---|---|---
1 | $n$ | $0$ | $n$ | $4/n$ | $4/n$
2 | 4 | 1 | 2 | 1.633 | 1.500
2 | 6 | 1 | 3 | 1.124 | 1.067
3 | 3 | 2 | 1 | 2.667 | 2.000
3 | 6 | 2 | 2 | 1.313 | 1.200
Table I summaries the minimum AGDOP and LB-E-AGDOP calculated for the cases
mentioned in this section. Although the LB-E-AGDOP is derived from the
expectation of AGDOP, we observe that LB-E-AGDOP may also establish a lower
bound on AGDOP.
Figure 5: Comparison between the LB-E-AGDOP from (25) and the sample mean of
AGDOP from simulations with the parameters $N_{S}=8,16,24,32$. The black line
shows $y=x$. Figure 6: Comparison between LB-E-AGDOP from (25) and the sample
minimum of AGDOP from simulations with the parameters $N_{S}=8,16,24,32$. The
black line shows $y=x$.
## VI Simulation Results
In this section, we conduct numerical simulations to validate the theoretical
results obtained in Sections IV and V.
### VI-A Simulation settings
All simulation results presented in this section are based on the following
settings.
* •
Two dimensions ($d=2$);
* •
Sensor nodes are uniformly distributed in the unit square $[0,1]\times[0,1]$;
* •
Four anchors ($N_{A}=4$) located at the corners of the unit square, i.e.,
$(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$;
* •
Random graph models: ERG, RGG, and RPG.
Fig. 1 has shown a snapshot excerpted from the simulation of the RGG model
with the parameters $r=0.495$ and $N_{S}=16$.
### VI-B Comparison between the LB-E-AGDOP and E-AGDOP
Fig. 5 compares the LB-E-AGDOP calculated using (25) and the sample mean of
AGDOP obtained from simulations. The simulations are based on the parameters
$N_{S}=8,16,24,32$ and the three random graph models. Each marker in Fig. 5
represents a network configuration with certain $N_{S}$, $\delta_{S}$ and
$\delta_{A}$.
Our theoretical lower bound is validated by the simulation results as no
markers are below the black line $y=x$. Although the relationship between LB-
E-AGDOP and E-AGDOP is nonlinear, if two different network configurations
result in the same LB-E-AGDOP value, they also lead to very close E-AGDOP
values. Therefore, our derived lower bound, LB-E-AGDOP, can be used as a
performance indicator of the expected accuracy of range-based cooperative
localization in random sensor networks.
Furthermore, it can be seen that our lower bound is validated for all the
three random graph models, with resulting similar gaps between LB-E-AGDOP and
E-AGDOP. This demonstrates that our lower bound is applicable to various
random graph models, as long as the coordinate symmetry assumption holds.
Fig. 5 also shows that the lower bound is tighter when LB-E-AGDOP is smaller.
When LB-E-AGDOP is greater than 1, E-AGDOP grows dramatically. In the last
part of this section, we shall see that this is because E-AGDOP is likely to
be infinite when $\delta<4$. Therefore, if E-AGDOP is finite, LB-E-AGDOP is
usually less than 1, and the lower bound is considerably tight.
### VI-C Comparison between the LB-E-AGDOP and minimum AGDOP
Fig. 6 compares the LB-E-AGDOP calculated using (25) and the sample minimum of
AGDOP obtained from simulations. The simulations are based on the parameters
$N_{S}=8,16,24,32$ and the three random graph models. Each marker in Fig. 5
represents a network configuration with certain $N_{S}$, $\delta_{S}$ and
$\delta_{A}$. Comparing Fig. 6 to Fig. 5, we can see that the theoretical
lower bound matches the minimum AGDOP better than matches the E-AGDOP.
Similar to our discussion about Fig. 5, it can be seen that (1) the
theoretical lower bound is validated by the simulation results; (2) the lower
bound can be used as a performance indicator of the best accuracy of range-
based cooperative localization in random sensor networks; (3) the lower bound
works for all the three random graph models as well as other models that
satisfy the conditions in Section II-B; and (4) the lower bound is tighter
when LB-E-AGDOP is smaller.
### VI-D Additional discussion
When we compare LB-E-AGDOP to E-AGDOP, there is an implicit assumption
E-AGDOP${}<\infty$. To many people’s surprise, localizability, i.e.,
uniqueness of the solutions to the localization problem (as treated in [16,
21, 7]), does not necessarily guarantee E-AGDOP${}<\infty$. More specifically,
for one sensor node, three anchors almost surely achieve unique
localizability. However, by (31) it can be verified that
$\operatorname{E}\mathinner{\bigl{(}\operatorname{tr}[(G^{\mathsf{T}}G)^{-1}]\bigr{)}}<\infty$
if and only if $N_{A}\geq 4$. Our ongoing work [51] has proven that for $d$
dimensions, at least $d+2$ anchors are required to locate one sensor node with
finite accuracy.
In cooperative localization, if $\delta<d+2$, there must be one node that has
$d+1$ neighbors or fewer. Thus, expectation of GDOP of this node is infinite,
so is the E-AGDOP. When $N_{S}$ is large, LB-E-AGDOP${}\rightarrow
d^{2}/\delta$. Therefore, E-AGDOP is likely to be infinite if
$\text{LB-E-AGDOP}>d^{2}/(d+2).$ (34)
This explains why in Fig. 5, E-AGDOP grows dramatically when LB-E-AGDOP is
greater than 1. This also suggests that in practice, the network connectivity
should meet the requirement $\text{LB-E-AGDOP}\leq d^{2}/(d+2)$ so that the
nodes can be accurately located. Furthermore, if this requirement is met, from
the simulation results we can see that E-AGDOP and minimum AGDOP are very
close, and both of them can be approximated by our lower bound.
## VII Conclusion
This paper has presented a generalized theory that characterizes the
connection between system parameters (network connectivity and size) and the
accuracy of range-based localization schemes in random WSNs. We have proven a
novel lower bound on expectation of AGDOP and derived a closed-form formula
(25) that relates LB-E-AGDOP and E-AGDOP to only three parameters: average
sensor degree $\delta_{S}$, average anchor degree $\delta_{A}$, and number of
sensor nodes $N$. The formula shows that LB-E-AGDOP is approximately inversely
proportional to the average degree, and a higher ratio of average anchor
degree to average sensor degree leads to better localization accuracy.
The simulation results have validated the theoretical results, and shown that
(1) the lower bound are applicable to various random graph models that satisfy
our coordinate symmetry assumption; (2) E-AGDOP and minimum AGDOP are very
close, and both of them can be approximated by LB-E-AGDOP when LB-E-AGDOP is
small. The theory and simulation results presented in this paper provide
guidelines on the design of range-based localization schemes and the
deployment of sensor networks.
## Appendix A Proof of Theorem 1
There are a few approaches to proving Theorem 1. One of the simplest proofs is
based on a recent result about the Cauchy–Schwarz inequality for the
expectation of random matrices [52, 53]:
###### Lemma 1 (Cauchy–Schwarz inequality [52, 53])
Let $A\in\mathbb{R}^{n\times p}$ and $B\in\mathbb{R}^{n\times p}$ be random
matrices such that $\operatorname{E}\|A\|^{2}<\infty$,
$\operatorname{E}\|B\|^{2}<\infty$, and $\operatorname{E}(A^{\mathsf{T}}A)$ is
non-singular. Then
$\operatorname{E}(B^{\mathsf{T}}B)\succeq\operatorname{E}(B^{\mathsf{T}}A)[\operatorname{E}(A^{\mathsf{T}}A)]^{-1}\operatorname{E}(A^{\mathsf{T}}B).$
(35)
With the substitutions $A=G$ and $B=G(G^{\mathsf{T}}G)^{-1}$ into the above
inequality, we have
$U=\operatorname{E}[(G^{\mathsf{T}}G)^{-1}]\succeq
V=[\operatorname{E}(G^{\mathsf{T}}G)]^{-1},$ (36)
which already proves Theorem 1.
Since the diagonal elements of a positive semidefinite matrix must be non-
negative, we have
$U_{ii}\geq V_{ii},\quad\forall i=1,\ldots,dN_{S},$ (37)
where $U=[U_{ij}]$ and $V=[V_{ij}]$. In particular, the expectation of GDOP,
$\operatorname{tr}(U)$, has a lower bound $\operatorname{tr}(V)$.
## Appendix B Proof of Eq. (24)
###### Lemma 2 (Sherman–Morrison formula [54])
Suppose $A$ is an invertible square matrix, and $u$ and $v$ are vectors.
Suppose furthermore that $1+v^{\mathsf{T}}A^{-1}u\neq 0$. Then the
Sherman–Morrison formula states that
$(A+uv^{\mathsf{T}})^{-1}=A^{-1}-\frac{A^{-1}uv^{\mathsf{T}}A^{-1}}{1+v^{\mathsf{T}}A^{-1}u}.$
(38)
With $\eta=d^{-1}[\delta+\delta_{S}/(N_{S}-1)]$, (23) can be written as
$\eta^{-1}\operatorname{E}\check{F}=I-uu^{\mathsf{T}},$ (39)
where $u=\sqrt{\zeta}(1,1,\ldots,1)^{\mathsf{T}}$, and
$\zeta=\delta_{S}/[\delta(N_{S}-1)+\delta_{S}]$.
Letting $u=-v=\sqrt{\zeta}(1,1,\ldots,1)^{\mathsf{T}}$, by the
Sherman–Morrison formula we have
$(I-uu^{\mathsf{T}})^{-1}=I+uu^{\mathsf{T}}/(1-u^{\mathsf{T}}u),$ (40)
and thus
$\begin{split}\eta\operatorname{tr}\bigl{[}(\operatorname{E}\check{F})^{-1}\bigr{]}&=\operatorname{tr}[(I-uu^{\mathsf{T}})^{-1}]\\\
&=N_{S}+N_{S}\zeta/(1-N_{S}\zeta).\end{split}$ (41)
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| Liang Heng received the B.S. and M.S. degrees in electrical engineering
from Tsinghua University, Beijing, China in 2006 and 2008. He received the PhD
degree in electrical engineering from Stanford University under the direction
of Per Enge in 2012. He is currently a postdoctoral research associate in the
Department of Aerospace Engineering, University of Illinois at Urbana-
Champaign. His research interests are cooperative navigation and satellite
navigation. He is a member of the IEEE and the Institute of Navigation (ION).
---|---
| Grace Xingxin Gao received the B.S. degree in mechanical engineering and
the M.S. degree in electrical engineering from Tsinghua University, Beijing,
China in 2001 and 2003. She received the PhD degree in electrical engineering
from Stanford University in 2008. From 2008 to 2012, she was a research
associate at Stanford University. Since 2012, she has been with University of
Illinois at Urbana-Champaign, where she is presently an assistant professor in
the Aerospace Engineering Department. Her research interests are systems,
signals, control, and robotics. She is a member of the IEEE and the Institute
of Navigation (ION).
---|---
|
arxiv-papers
| 2013-05-30T23:55:26 |
2024-09-04T02:49:45.909577
|
{
"license": "Public Domain",
"authors": "Liang Heng and Grace Xingxin Gao",
"submitter": "Liang Heng",
"url": "https://arxiv.org/abs/1305.7272"
}
|
1305.7284
|
11institutetext: M.Stat. final semester project
Indian Statistical Institute, 203 B.T.Road, Kolkata - 700 108, India
# Adjusting for Treatment Effects in Studies of Quantitative Traits
Guide: Prof. Saurabh Ghosh Human Genetics Unit
Project done by: Subhabrata Majumdar M.Stat. 2nd year
###### Abstract
A population-based study of a quantitative trait, e.g. Blood Pressure(BP) may
be seriously compromised when the trait is subject to the effects of a
treatment. Without appropriate corrections this can lead to considerable
reduction of statistical power. Here we demonestrate this in the scenario of
QTL mapping through Single-Marker Analysis. The data are simulated from a
normal mixtrure for different values of allele frequencies, separation between
normal populations and Linkage Disequilibrium, and several methods of
correction are compared to check which can best compensate for the loss of
power if treatment effects are ignored. In one of these methods, underlying
BPs are approximated by subtracting an estimate of mean value of medicine
effect from obsereved BPs in treated subjects. We domonestrate the efficacy of
this method throughout different choices of parameters. Finally to account for
quantitative traits that follow non-normal distributions, data are simulated
from lognormal mixtures similarly and Kruskal-Wallis test is used to obtain
estimates of powers for different methods of analysis.
Keywords : Quantitative traits; Imputation; QTL mapping; Single-marker
analysis; Mixed models; Kruskal-Wallis test
## 1 Introduction
Quantitative traits are traits that take continuous values, like body height,
body weight, Blood Pressure (BP) etc., and can be attributed to polygenic
effects i.e. product of two or more genes. Population-based studies of a
quantitative trait help to determine the genetic determinants of the trait.
But in many such studies selective individuals receive treatments, like
antihypertensive medicines for high BP. As a result, In treated subjects, the
outcome of primary interest, the underlying BP, which is the BP that an
individual would have if he=she was not treated, cannot be measured and
analysis must therefore be based on the observed BP. Without any correction,
this leads to a reduction in statistical power and shrinkage in the estimated
effects of determinants[1][2].
Some common and often-used ways to tackle this situation are as follows. One
can completely ignore the information on treatment status and perform the
analysis assuming that the observed values are same as the underlying values
of the BP of the subjects[3]. Treated or affected subjects are sometimes
excluded from the analysis[4]. Another common method is to adjust for BP
treatment by incorporating it as a covariate [5]. One can also convert BP into
a binary trait by defining as hypertensive the subjects who are treated or
have an observed BP in excess of a stated threshold[6][7].
Apart from these, a general approach is Imputation. Here the observed values
are replaced by some plausible approximation of the actual BPs. The methods
under this include addition of some statistic estimated from the data or some
constant to the treated BP values [8], replacing the treated values by some
random or constant quantity [9], adding residuals modified by a non-parametric
algorithm [10], and censored normal regression [11].
Here we first consider a simulation model for BP where observations are drawn
from a normal mixture and are treated with medicine with a high probability if
they cross a certain threshold. These are taken as observed BPs. Since in a
practical scenario the genotype at the Quantitative Trait Locus (QTL) is not
observed, genotypes of marker loci with different degrees of linkage with the
QTL are considered and ANOVAs are calculated with respect to them. With this
basic premise, different methods of analysis were applied on the simulated
datasets and the resulting powers were compared. Lastly, we consider deviation
from normality for underlying QT values. We now simulate from a lognormal
mixture instead and then compare different methods of analysis through powers
obtained through Kruskal-Wallis test, instead of ANOVA as before.
## 2 Methods
### 2.1 The simulation model
We consider our parameter of interest the systolic blood presure (SBP) of a
subject. An individual is defined hypertensive if the observed SBP exceeds a
certain fixed threshold, which we take to be 140 mm Hg. In our case this
ensures that the proportion of hypertensive individuals to the full population
is not more than 0.2.
Suppose that the alleles of a gene affecting BP are $A$ and $a$, their
frequencies being $1-p$ and $p$. Then the possible genotypes are $AA,Aa$ and
$aa$. From Hardy Weinberg Equilibrium we know that these genotypes occur in
the population with probabilities $(1-p)^{2},2p(1-p)$ and $p^{2}$,
respectively. For the populations under these genotypes, we assume three
equidistant normal distributions with means at $120-d,120$ and $120+d$ mm Hg:
$d$ being the distance between two adjacent means. With a given pair of values
for $p$ and $d$, 1000 datasets each with 100 datapoints are simulated from the
this normal mixture. Furthermore, a subject was declared hypertensive if the
simulated BP exceeds 140 mm Hg. If deemed hypertensive, a subject was assigned
to the treatment group with probability 0.8, and also a random treatment
effect from a $N(-10,3^{2})$ distribution is added to the underlying BP to
obtain the observed BP. These simulations were performed with three choices of
$p$ (0.1, 0.3, 0.5) and five choices of $d$ (10, 15, 20, 25, 30).
### 2.2 Methods of analysis
Seven different methods of analysis were compared on the simulated data sets.
Except one method all others make use of the ANOVA model for calculating
powers, with the effects of genotypes taken as fixed effects. The methods are
described below. Consider in general $X_{i},Y_{i}$ as the underlying and
observed SBPs of the $i^{th}$ subject, respectively. We also define $M_{i}$ to
be the indicator of the $i^{th}$ subject taking medicine.
#### (a) Taking underlying BPs as observed:
Although this is not feasible in practice, this is done to estimate the actual
power of ANOVA on a sample from the full population.
#### (b) No adjustment from treatment:
We ignore the information on treatment status and perform the ANOVA assuming
that observed BPs are same as underlying BPs, i.e. here $X_{i}=Y_{i}$ for all
$i$. This gives an idea about the reduction of power due to treatment.
#### (c) Omitting all affected individuals:
Here we take $X_{i}=Y_{i}$ if $Y_{i}<140$ and $M_{i}=0$.
#### (d) Omitting all treated individuals:
We take $X_{i}=Y_{i}$ if $M_{i}=0$.
#### (e) Treatment effect modeled as a covariate:
Here the underlying model is
$X_{i}=\mu+\alpha_{i}+\beta M_{i}+\epsilon_{i}$
and $X_{i}=Y_{i}$ for all $i$, with $\alpha_{i}$ being the fixed effect of
genotype and $\beta$ being the random treatment effect. This is a mixed model,
and the power is calculated considering the p-value of the F-statistic
corresponding to the fixed effect.
#### (f) Correction by a fixed quantity:
The difference of means in treated and untreated but affected subjects, say
$m$ is an unbiased and consistent estimate of the medicine effect (Proof in
Appendix). We subtract this from each treated observation to get an estimate
of the underlying BP and then perform the ANOVA, i.e. here we have
$X_{i}=Y_{i}-mM_{i}$.
#### (g) Correction by a non-parametric algorithm:
Here we use the non-parametric adjustment algorithm by Levy et al[10] which
has already been shown to give good approximations of actual powers in similar
situation[11].
In this method, first observations are centered around the mean and raw
residuals are obtained:
$r_{i}=Y_{i}-\bar{Y}$
Then the residuals are ordered and modified residuals are obtained as follows
(assume now that $\\{r_{i}\\}$ is the set of ordered residuals):
$r_{k}^{*}=(1-M_{i})r_{k}+M_{i}\left(\frac{r_{k}+\sum_{j=1}^{k-1}r_{j}^{*}}{k}\right)$
The residuals are then sorted back to their original order and added to the
mean observed BP to get estimates of the underlying BPs:
$X_{i}=\bar{Y}+r_{i}^{*}=Y_{i}-r_{i}+r_{i}^{*}$
### 2.3 Modifications for QTL mapping
A caveat in the above approach is that in a practical situation, we do not
exactly know the location of the Quantitative Trait Locus (QTL). Instead one
can obtain genotypes at several marker loci near to the approximate position
of the QTL within the genome. In that case, we perform the ANOVA with respect
to each of these genotypes ignoring the effect of others and infer the QTL to
be closest to the most significant marker within a given chromosomal region.
This is called Single-Marker Analysis. We now attempt to integrate this
scenario into our approach.
Proximity of the marker loci to the QTL means a high degree of linkage among
them, i.e. alleles at these two loci in an organism tend to pass on
simultaneously to the next generation while reproducing. Now the effect of
linkage between two loci in the genome is measured by a quantity called
Linkage Disequilibrium (LD). It measures the of non-random association of
alleles at two or more loci. Given the allele-pairs $A/a$ and $B/b$ the LD
between two loci is defined as:
$\delta=P(AB)-P(A)P(B)$
Where $P(AB)$ is the probability of the alleles $A$ and $B$ co-segregating,
and $P(A),P(B)$ the respective individual allele frequencies. To make the
value independent of the allele frequencies, $\delta$ is divided by its
theoretical maximum to obtain a scale free quantity:
$\delta^{\prime}=\begin{cases}\frac{\delta}{\min\\{P(A)P(B),P(a)P(b)\\}}&\text{if
}\delta<0\\\ \frac{\delta}{\max\\{P(A)P(b),P(a)P(B)\\}}&\text{if }\delta>0\\\
\end{cases}$
Note that $\delta^{\prime}$ = 1 means complete linkage and $\delta^{\prime}$ =
0 means no linkage at all i.e. independent assortment of alleles.
Our previous simulation model is now changed to incorporate this situation.
Say we denote the alleles of the marker locus by $B/b$. Then under the above
model, the frequencies of the four haplotypes are:
$\displaystyle P(AB)$ $\displaystyle=$ $\displaystyle(1-p)^{2}+\delta$
$\displaystyle P(Ab)$ $\displaystyle=$ $\displaystyle p(1-p)-\delta$
$\displaystyle P(aB)$ $\displaystyle=$ $\displaystyle p(1-p)-\delta$
$\displaystyle P(ab)$ $\displaystyle=$ $\displaystyle p^{2}+\delta$
Now two biallelic haplotypes, each with a QTL and a marker allele, are
generated and they are fused to obtain a biallelic genotype. Then the
underlying BP value is generated from the distribution corresponding QTL
genotype, but the observed genotype is taken as that of the marker locus. For
example, if the haplotypes generated are $AB$, $aB$ and thus the genotype of a
subject is $AaBB$, then the observation is taken from a $N(120,20^{2})$
population which is the distribution corresponding to the QTL genotype $Aa$,
but the observed genotype is taken as $BB$ and ANOVA is done based on the
genotypes $BB,Bb$ and $bb$.
As before, 1000 datasets with 100 points each were simulated for
$\delta^{\prime}$ = 1/3 and 2/3. Note that our previous simulation corresponds
to $\delta^{\prime}$ = 1.
## 3 Results and discussion
All analyses were done on MATLAB version R2008a[12]. For all the methods
except (e), single-factor ANOVA was used to calculate the p-values, while for
method (e), the p-value was calculated from the F-statistic of fixed effect in
the mixed model. All significance levels were set at $\alpha=0.05$. For a
choice of $(p,d,\delta^{\prime})$ the power to detect the effect of QTL, was
therefore estimated as the proportion of datasets which were found to have the
F-statistic with p-value $<0.05$. The following three tables contain the
powers for the three values of $\delta$ considered.
| | Method of analysis (powers in percentage)
---|---|---
$p$ | $d$ | All | All | Omit affected | Omit treated | Treatment | Constant | Non-parametric
| (mm Hg) | underlying | observed | subjects | subjects | as covariate | adjustment | adjustment
| 10 | 44.7 | 42.4 | 29.9 | 31.7 | 29.0 | 44.3 | 41.7
| 15 | 81.2 | 79.1 | 60.8 | 68.6 | 59.5 | 80.9 | 79.8
0.1 | 20 | 97.3 | 97.0 | 86.1 | 89.9 | 85.9 | 97.3 | 97.1
| 25 | 99.7 | 99.7 | 97.5 | 98.1 | 97.0 | 99.7 | 99.7
| 30 | 100.0 | 100.0 | 99.7 | 99.7 | 99.5 | 100.0 | 100.0
| 10 | 80.3 | 79.1 | 53.3 | 65.9 | 56.6 | 80.0 | 78.0
| 15 | 98.9 | 97.8 | 90.2 | 93.9 | 91.6 | 98.6 | 98.8
0.3 | 20 | 100.0 | 100.0 | 99.3 | 99.8 | 99.7 | 100.0 | 100.0
| 25 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
| 30 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
| 10 | 86.0 | 84.3 | 65.5 | 74.0 | 70.4 | 86.0 | 83.2
| 15 | 99.7 | 98.8 | 95.2 | 97.6 | 96.9 | 99.7 | 99.5
0.5 | 20 | 100.0 | 100.0 | 99.9 | 100.0 | 100.0 | 100.0 | 100.0
| 25 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
| 30 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
Table 1: Powers obtained by different methods for $\delta^{\prime}=1$ | | Method of analysis (powers in percentage)
---|---|---
$p$ | $d$ | All | All | Omit affected | Omit treated | Treatment | Constant | Non-parametric
| (mm Hg) | underlying | observed | subjects | subjects | as covariate | adjustment | adjustment
| 10 | 24.8 | 23.1 | 12.1 | 15.3 | 16.9 | 24.4 | 22.6
| 15 | 45.5 | 43.1 | 24.3 | 32.0 | 27.9 | 45.3 | 41.7
0.1 | 20 | 68.3 | 64.8 | 42.1 | 49.5 | 46.5 | 67.2 | 64.5
| 25 | 84.5 | 83.0 | 59.0 | 68.4 | 65.2 | 84.5 | 82.9
| 30 | 92.7 | 92.4 | 74.7 | 78.4 | 80.7 | 92.5 | 92.0
| 10 | 43.4 | 42.5 | 25.8 | 29.9 | 29.2 | 43.0 | 41.2
| 15 | 75.1 | 74.9 | 47.7 | 58.1 | 56.0 | 75.1 | 73.8
0.3 | 20 | 92.9 | 92.8 | 71.7 | 81.1 | 76.9 | 92.6 | 91.8
| 25 | 98.6 | 98.3 | 84.3 | 91.2 | 90.6 | 98.5 | 98.2
| 30 | 99.5 | 99.5 | 93.3 | 96.3 | 94.9 | 99.5 | 99.5
| 10 | 50.2 | 49.6 | 31.8 | 39.4 | 35.1 | 50.0 | 49.5
| 15 | 82.1 | 80.9 | 58.3 | 68.7 | 61.4 | 82.1 | 79.2
0.5 | 20 | 95.9 | 95.6 | 79.4 | 89.0 | 82.8 | 95.7 | 94.3
| 25 | 98.6 | 98.6 | 90.1 | 96.7 | 90.8 | 98.6 | 98.2
| 30 | 99.8 | 99.8 | 93.6 | 98.1 | 96.1 | 99.9 | 99.6
Table 2: Powers obtained by different methods for $\delta^{\prime}=2/3$ | | Method of analysis (powers in percentage)
---|---|---
$p$ | $d$ | All | All | Omit affected | Omit treated | Treatment | Constant | Non-parametric
| (mm Hg) | underlying | observed | subjects | subjects | as covariate | adjustment | adjustment
| 10 | 8.9 | 7.9 | 6.8 | 7.7 | 6.5 | 8.5 | 7.3
| 15 | 15.5 | 14.6 | 8.4 | 10.5 | 9.8 | 15.4 | 14.5
0.1 | 20 | 24.5 | 22.5 | 12.5 | 14.2 | 16.2 | 24.4 | 22.2
| 25 | 32.1 | 30.4 | 17.2 | 20.5 | 19.6 | 32.0 | 30.0
| 30 | 39.4 | 36.9 | 22.5 | 28.1 | 26.0 | 39.2 | 36.0
| 10 | 12.9 | 12.5 | 9.9 | 10.3 | 8.5 | 12.2 | 12.5
| 15 | 27.0 | 25.6 | 14.0 | 18.4 | 16.9 | 25.5 | 25.4
0.3 | 20 | 36.8 | 34.9 | 17.5 | 24.6 | 21.3 | 36.0 | 34.5
| 25 | 46.8 | 45.7 | 24.3 | 29.8 | 28.9 | 46.3 | 43.9
| 30 | 55.8 | 55.6 | 32.2 | 37.6 | 34.9 | 56.2 | 53.3
| 10 | 14.7 | 15.0 | 11.1 | 13.5 | 12.7 | 14.4 | 14.8
| 15 | 24.1 | 23.3 | 17.6 | 18.9 | 15.1 | 23.9 | 22.7
0.5 | 20 | 40.2 | 38.9 | 27.6 | 29.6 | 26.0 | 40.2 | 37.0
| 25 | 50.6 | 50.2 | 29.2 | 40.3 | 31.6 | 49.8 | 47.9
| 30 | 59.7 | 56.6 | 32.8 | 41.8 | 36.3 | 58.5 | 54.3
Table 3: Powers obtained by different methods for $\delta^{\prime}=1/3$
First of all, as shown before[13], it is found that as the value of LD and
thus degree of linkage between the QTL and marker locus decreases, the
estimated power also falls for a given allele frequency and distance between
marker genotype means. More importantly, from the above tables it is clear
that there is reduction of power when single-marker ANOVA is performed
considering the observed BPs instead of the true underlying BP values. Since a
large proportion of hypertensive subjects, i.e. whose underlying BPs exceed
the threshold of 140 mm Hg, are subjected to treatments that reduce the BP, a
negative bias comes to the measurement of BP from these subjects. Without any
adjustments this leads to shrinkage in the estimates of the effects of the
genetic determinant[11].
Among the adjustment methods considered, the first two methods (coulmns 5 and
6 in the tables) are found to be very inefficient, understandably so because
they selectively remove all or most of the affected individuals, who have high
underlying BP, thus discarding a lot of useful information.
The method of including treatment status as covariates also results in marked
reduction of statistical powers. Although taking the information that whether
a subject is being treated or not as a random effect in the assumed model
seems to be a valid approach, it is actually flawed. This is so because a
subject can receive treatment only if the underlying BP crosses a certain
threshold, i.e. treatments are not assigned in a completely random way for all
values of underlying BP. To be precise, if the underlying BP is $>$ 140 mm Hg,
the probability of the subject receiving treatment is 0.8, but if it is less
than the threshold, the probability is 0, i.e. none of these subjects are
treated. Thus, rather than being an explanation to the underlying BP,
treatment status is an outcome of it. Since much of the variability of the
observed BPs is explained by underlying BPs, including treatment status as a
covariate in a model with underlying BPs as the response variable is not a
good idea.
Finally, correction by a fixed estimate of the effect of antihypertensive
medicine gives the closest estimates of the true powers of the procedure in
almost all the cases. The non-parametric adjustment algorithm by Levy et
al[10] also gives good estimates of the powers, but it is not much reliable
since in most of the cases they are less than the powers when analysis is done
ignoring treatment status.
## 4 Deviation from normality
Many quantitative traits follow non-normal distributions, like lognormal[14],
skew-normal[15] etc. For analyzing data on these traits from a population we
cannot directly use ANOVA to obtain statistical power. To tackle this, one
either has to go for ANOVA after some data transformation. In case a suitable
transformation is not found or there are issues regarding interpretation, non-
parametric methods can be used to obtain estimates of powers.
For this situation, we simulate the datasets with the same choices of values
for the parameters $(p,d,\delta^{\prime})$, from a lognormal mixture
population with the same values of means and variance as before. Kruskal-
Wallis test is used to obtain estimates of powers. Among the seven methods of
analysis described before, all except the one considering treatment effect as
covariate are used in this case as well. In addition, in the method of
correcting by a fixed value, we obtain the estimate of medicine effect in a
slightly different way. Instead of using the difference of mthe eans of
observed BPs of treated and affected but untreated subjects, the difference of
the corresponding medians is used. The results obtained are as follows.
| | Method of analysis (powers in percentage)
---|---|---
$p$ | $d$ | All | All | Omit affected | Omit treated | Constant | Non-parametric
| (mm Hg) | underlying | observed | subjects | subjects | adjustment | adjustment
| 10 | 43.3 | 42.2 | 35.8 | 36.6 | 43.3 | 43.3
| 15 | 79.3 | 78.6 | 72.7 | 73.1 | 79.3 | 77.5
0.1 | 20 | 96.8 | 95.9 | 91.5 | 93.5 | 96.8 | 96.8
| 25 | 99.6 | 99.1 | 98.9 | 99.8 | 99.6 | 99.6
| 30 | 100.0 | 100.0 | 99.9 | 100.0 | 100.0 | 100.0
| 10 | 79.8 | 79.0 | 69.6 | 72.5 | 79.8 | 78.5
| 15 | 98.8 | 98.2 | 96.0 | 97.4 | 98.8 | 98.8
0.3 | 20 | 100.0 | 100.0 | 100.0 | 99.8 | 100.0 | 100.0
| 25 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
| 30 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
| 10 | 85.5 | 85.4 | 76.8 | 82.3 | 85.5 | 83.3
| 15 | 99.8 | 99.6 | 98.7 | 99.3 | 99.8 | 99.8
0.5 | 20 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
| 25 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
| 30 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0
Table 4: Non-parametric case: powers obtained by different methods for $\delta^{\prime}=1$ | | Method of analysis (powers in percentage)
---|---|---
$p$ | $d$ | All | All | Omit affected | Omit treated | Constant | Non-parametric
| (mm Hg) | underlying | observed | subjects | subjects | adjustment | adjustment
| 10 | 21.1 | 20.5 | 15.3 | 19.8 | 21.1 | 20.6
| 15 | 44.8 | 44.5 | 33.8 | 35.2 | 44.5 | 44.1
0.1 | 20 | 63.5 | 63.0 | 52.5 | 58.4 | 63.5 | 63.2
| 25 | 77.9 | 77.7 | 70.1 | 71.4 | 78.0 | 76.8
| 30 | 90.0 | 89.9 | 79.2 | 79.4 | 89.7 | 89.6
| 10 | 45.3 | 45.2 | 34.7 | 40.0 | 45.1 | 43.6
| 15 | 75.7 | 75.0 | 63.0 | 67.0 | 75.0 | 72.8
0.3 | 20 | 94.3 | 94.2 | 82.1 | 85.1 | 94.3 | 93.5
| 25 | 98.6 | 98.3 | 92.3 | 94.2 | 98.6 | 98.1
| 30 | 99.9 | 99.7 | 96.8 | 97.7 | 99.8 | 99.5
| 10 | 52.7 | 52.1 | 40.6 | 45.9 | 52.4 | 50.7
| 15 | 84.3 | 83.0 | 69.6 | 77.4 | 84.3 | 81.2
0.5 | 20 | 95.5 | 95.1 | 88.0 | 91.0 | 95.3 | 94.3
| 25 | 99.0 | 98.8 | 94.8 | 98.0 | 98.8 | 98.6
| 30 | 99.9 | 99.8 | 97.0 | 98.8 | 99.8 | 99.6
Table 5: Non-parametric case: powers obtained by different methods for
$\delta^{\prime}=2/3$
| | Method of analysis (powers in percentage)
---|---|---
$p$ | $d$ | All | All | Omit affected | Omit treated | Constant | Non-parametric
| (mm Hg) | underlying | observed | subjects | subjects | adjustment | adjustment
| 10 | 8.2 | 7.7 | 6.4 | 6.3 | 8.3 | 7.2
| 15 | 12.4 | 12.1 | 9.8 | 12.6 | 12.2 | 11.7
0.1 | 20 | 18.2 | 18.1 | 15.9 | 14.3 | 18.1 | 18.3
| 25 | 27.8 | 27.1 | 16.9 | 18.0 | 27.6 | 27.0
| 30 | 32.8 | 32.6 | 22.2 | 27.8 | 32.8 | 33.3
| 10 | 12.7 | 12.1 | 9.8 | 11.8 | 12.7 | 12.1
| 15 | 22.3 | 22.0 | 17.2 | 19.5 | 22.1 | 22.4
0.3 | 20 | 36.6 | 36.3 | 25.0 | 27.3 | 36.6 | 35.6
| 25 | 39.7 | 39.7 | 33.0 | 35.0 | 39.7 | 38.4
| 30 | 52.8 | 52.2 | 34.6 | 37.9 | 52.7 | 51.7
| 10 | 13.5 | 12.8 | 13.1 | 13.7 | 13.5 | 13.4
| 15 | 26.2 | 25.7 | 20.0 | 20.7 | 26.2 | 25.7
0.5 | 20 | 38.9 | 38.0 | 27.6 | 32.9 | 38.3 | 36.7
| 25 | 50.1 | 49.5 | 36.8 | 40.5 | 49.7 | 47.8
| 30 | 59.3 | 58.4 | 39.4 | 46.2 | 59.1 | 55.0
Table 6: Non-parametric case: powers obtained by different methods for
$\delta^{\prime}=1/3$
The powers obtained by different methods are comparable with the same powers
when simulated from the normal mixture. In this case the reduction of power
due to treatment effects seems to be less severe than the normal mixture
simulation. The constant adjustment gives the closest powers to the estimated
true powers, as before.
## 5 Conclusion
We have demonestrated through simulation that the distorting effect of
antihypertensive therapy in studies of quantitatively measured blood pressure
can lead to loss of statistical power in the Single-marker analysis approach
of QTL mapping. Among the adjustment methods considered, ignoring the problem
altogether and analysing observed BP in treated subjects as if it was the
underlying BP, excluding affected or treated subjects from analysis, or
fitting a mixed model with treatment as a binary covariate perform very poorly
and thus should not be used. Finally we have concluded that adding an estimate
of the reduced BP due to medicine can reasonably nullify the reduction of
powers.
## Acknowledgement
I thank my mentor, Prof. Saurabh Ghosh of Human Genetics Unit, Indian
Statistical Institute, Kolkata for his unfailing help in guiding me through my
doubts and providing ideas while doing this project.
## References
* [1] White IR, Chaturvedi N, McKeigue PM. Median analysis of blood pressure for a sample including treated hypertensives. Statistics in Medicine 1994; 13(16):1635-1641.
* [2] Cui JS, Hopper JL, Harrap SB. Antihypertensive treatments obscure familial contributions to blood pressure variation. Hypertension 2003; 41(2):207-210.
* [3] Brand E, Wang JG, Herrmann SM, Staessen JA. An epidemiological study of blood pressure and metabolic phenotypes in relation to the Gbeta3 C825T polymorphism. Journal of Hypertension 2003; 21(4):729-737.
* [4] Rice T, Rankinen T, Province MA, Chagnon YC, Perusse L, Borecki IB, Bouchard C, Rao DC. Genome-wide linkage analysis of systolic and diastolic blood pressure: the Quebec family study. Circulation 2000; 102(16):1956-1963.
* [5] O’Donnell CJ, Lindpaintner K, Larson MG, Rao VS, Ordovas JM, Schaefer EJ, Myers RH, Levy D. Evidence for association and genetic linkage of the angiotensin-converting enzyme locus with hypertension and blood pressure in men but not women in the Framingham heart study. Circulation 1998; 97(18):1766-1772.
* [6] Sethi AA, Nordestgaard BG, Agerholm-Larsen B, Frandsen E, Jensen G, Tybjaerg-Hansen A. Angiotensinogen polymorphisms and elevated blood pressure in the general population: the Copenhagen city heart study. Hypertension 2001; 37(3):875-881.
* [7] Zhu X, Chang YP, Yan D, Weder A, Cooper R, Luke A, Kan D, Chakravarti A. Associations between hypertension and genes in the renin-angiotensin system. Hypertension 2003; 41(5):1027-1034.
* [8] Cui J, Hopper JL, Harrap SB. Genes and family environment explain correlations between blood pressure and body mass index. Hypertension 2002; 40(1):7-12.
* [9] Hunt SC, Ellison RC, Atwood LD, Pankow JS, Province MA, Leppert MF. Genome scans for blood pressure and hypertension: the National Heart, Lung, and Blood Institute family heart study. Hypertension 2002; 40(1):1-6.
* [10] Levy D, DeStefano AL, Larson MG, O’Donnell CJ, Lifton RP, Gavras H, Cupples LA, Myers RH. Evidence for a gene influencing blood pressure on chromosome 17. Genome scan linkage results for longitudinal blood pressure phenotypes in subjects from the Framingham Heart Study. Hypertension 2000; 36(4):477-483.
* [11] Tobin MD, Sheehan NA, Scurrah KJ, Burton PR. Adjusting for treatment effects in studies of quantitative traits: antihypertensive therapy and systolic blood pressure. Statistics in Medicine 2005; 24:2911-2935.
* [12] Mathworks Inc. MATLAB Version 7.6 (R2008a), 2008.
* [13] Collard, BCY, Jaufer MZZ, Brouwer JB, Pang ECK. An introduction to markers, quantitative trait locus (QTL) mapping and marker-assisted selection for crop improvement: The basic concepts. Euphytica 2005; 142:169-196.
* [14] Sillanpää MJ, Hoti F. Mapping Quantitative Trait Loci From a Single-Tail Sample of the Phenotype Distribution Including Survival Data. Genetics 2007; 177(4):2361-2377.
* [15] Fernandes E, Pacheco A, Penha-Gonçalves C. Journal of Zhejiang University SCIENCE B 2007; 8(11): 792-801.
## Appendix: Properties of the estimate obtained in method (f) of section 2
Suppose in a dataset there are total $n$ subjects, among whom $m$ are
hypertensive and $k$ are treated with medicine. Given affected, a subject is
treated with probability $p$, i.e. $k\thicksim Bin(m,p)$. Now, without loss of
generality, say the subjects $1,...,m$ are affected, and among them $1,...,k$
are treated. $X_{i},Y_{i}$ are the underlying and observed BPs of the $i^{th}$
subject, respectively.
For the sake of simplicity, we assume that underlying BP of all subjects come
from a $N(\mu,\sigma^{2})$ population, instead of the normal mixture
considered in the main work. Subjects with underlying BP above a threshold $c$
are considered affected, and the medicine effects (say $B_{i}$, for the
$i^{th}$ subject) are assumed to follow a $N(\nu,\tau^{2})$ distribution. With
this setting, our estimate under consideration will be:
$\hat{\nu}=\frac{1}{k}\sum_{i=1}^{k}X_{i}-\frac{1}{m-k}\sum_{i=k+1}^{m}X_{i}$
#### Unbiasedness:
For $i=1,...,k$ we have $X_{i}=Y_{i}+B_{i}$, and for $i=k+1,...,m$, we have
$X_{i}=Y_{i}$. Thus
$E(\hat{\nu})=\frac{1}{k}\sum_{i=1}^{k}E(Y_{i}+B_{i})-\frac{1}{m-k}\sum_{i=k+1}^{m}EY_{i}=\mu_{c}+\nu-\mu_{c}=\nu$
with $\mu_{c}$ being the mean of the $N(\mu,\sigma^{2})$ distribution
truncated at $c$.
#### Consistency:
$\displaystyle Var(\hat{\nu})$ $\displaystyle=$
$\displaystyle\frac{1}{k}\left(Var(Y_{1})+Var(B_{1})\right)+\frac{1}{m-k}Var(Y_{1})$
$\displaystyle=$
$\displaystyle\left(\frac{1}{k}+\frac{1}{m-k}\right)Var(Y_{1})+\frac{1}{k}Var(B_{1})$
$\displaystyle=$ $\displaystyle\frac{m\sigma^{2}}{k(m-k)}+\frac{\tau^{2}}{k}$
$\displaystyle\Rightarrow\lim_{n\rightarrow\infty}Var(\hat{\nu})$
$\displaystyle=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{m}\left[\frac{\sigma^{2}}{\frac{k}{m}\left(1-\frac{k}{m}\right)}+\frac{\tau^{2}}{\frac{k}{m}}\right]$
because as $n\rightarrow\infty,m\rightarrow\infty$.
Now $k\thicksim Bin(m,p)\Rightarrow\lim_{m\rightarrow\infty}\dfrac{k}{m}=p$,
hence the quantity inside brackets will tend to
$\sigma^{2}/p(1-p)+\tau^{2}/p$, i.e. a fixed quantity as $m\rightarrow\infty$.
It follows that $\lim_{n\rightarrow\infty}Var(\hat{\nu})=0$.
|
arxiv-papers
| 2013-05-31T02:30:56 |
2024-09-04T02:49:45.919584
|
{
"license": "Public Domain",
"authors": "Saurabh Ghosh, Subhabrata Majumdar",
"submitter": "Subhabrata Majumdar",
"url": "https://arxiv.org/abs/1305.7284"
}
|
1305.7316
|
11institutetext: ${}^{1~{}}$The Graduate University for Advanced Studies,
${}^{2~{}}$The University of Tokyo,
${}^{3~{}}$National Institute of Informatics,
Tokyo, Japan
{nqminh,giovanni,goran_topic,aizawa}@nii.ac.jp
Authors’ Instructions
# A hybrid approach for semantic enrichment of MathML mathematical expressions
Minh-Quoc Nghiem${}^{~{}1}$ Giovanni Yoko Kristianto${}^{~{}2}$
Goran Topić${}^{~{}3}$ Akiko Aizawa${}^{~{}2,3}$
###### Abstract
In this paper, we present a new approach to the semantic enrichment of
mathematical expression problem. Our approach is a combination of statistical
machine translation and disambiguation which makes use of surrounding text of
the mathematical expressions. We first use Support Vector Machine classifier
to disambiguate mathematical terms using both their presentation form and
surrounding text. We then use the disambiguation result to enhance the
semantic enrichment of a statistical-machine-translation-based system.
Experimental results show that our system archives improvements over prior
systems.
###### Keywords:
MathML, Semantic Enrichment, Disambiguation, Statistical Machine Translation
## 1 Introduction
The semantic enrichment of mathematical documents is among the most
significant areas of math-aware technologies. It is the process of associating
semantic tags, usually concepts, with mathematical expressions. We use MathML
[10] Presentation and Content Markup to represent mathematical expressions and
their meaning. The semantic enrichment task then becomes the task of
generating Content MathML outputs from Presentation MathML expressions. It is
an important technology towards fulfilling the dream of global digital
mathematical library (DML).
The semantic enrichment of mathematical expression is a challenging task.
Mathematical notations are ambiguous, context-dependent, and vary from
community to community. Given a Presentation MathML element, there are many
potential mappings to its Content MathML element. For example, the token
$\delta$ can be mapped to $KroneckerDelta$, $DiracDelta$, $DiscreteDelta$, or
$\delta$. By correctly disambiguating these token elements, we can get a more
accurate semantic enrichment system.
Disambiguation of mathematical elements is an important component in the
semantic enrichment system. Basic methods for dealing with ambiguities so far
were either rule-based [1] or statistics-based [7]. The rule-based approach is
of course generally not able to derive meaning from arbitrary Presentation
MathML expressions. The statistics-based approach resolves ambiguities based
on the probabilities, and thus gets better results than the rule-based system.
In this paper, we enhance the statistics-based approach by combining it with a
disambiguation component.
So far, there has been limited discussion about the contribution of
surrounding text to mathematical element disambiguation problem. It is
becoming increasingly difficult to ignore the surrounding text of mathematical
expressions. For example, the token $\delta$ can be mapped to $KroneckerDelta$
if its surrounding text contains the word ‘Kronecker delta’. It is difficult
to disambiguate using only the presentation of mathematical expression. The
combination of mathematical expression itself and its surrounding text can
lead to improvements in disambiguation process.
The aim of this paper is to examine and solve the ambiguity when mapping
Presentation MathML elements to their Content elements. This paper also
attempts to find the contribution of surrounding text to mathematical element
disambiguation problem. We use a Support Vector Machine (SVM) learning model
for MathML Presentation token element (mi) disambiguation. Both presentation
of mathematical expression and its surrounding text are encoded in a feature
vector used in SVM. We evaluate the efficacy of the system by incorporating it
into an SMT-based semantic enrichment system.
We formulate the problem as follows: given a Presentation MathML expression
and its surrounding text, can we interpret its Content MathML expression? This
paper provides contributions in three main areas of mathematical semantic
enrichment problem. First, we show that combination of a disambiguation
component and the SMT-based system improves the system’s performance. Second,
we show that the text surrounding the mathematical expressions contributes to
the disambiguation process. Third, we show that the name of the category that
a mathematical expression belongs to is the most important text feature for
disambiguation.
The remainder of this paper is organized as follows. Sections 2 provides a
brief overview of the background and related work on semantic enrichment of
mathematical expressions. Section 3 presents our method. Section 4 describes
the experimental setup and results. Section 5 concludes the paper and points
to avenues for future work.
## 2 Related Work
MathML [10] is the best-known open markup format for representing mathematical
formulas. It is recommended by the W3C Math Working Group as a standard to
represent mathematical expressions. MathML is an application of XML for
describing mathematical notations and encoding mathematical content within a
text format. MathML has two types of encoding: Content MathML addresses the
meaning of formulas; and Presentation MathML addresses the display of
formulas. We use MathML Presentation Markup to display mathematical
expressions and MathML Content Markup to convey mathematical meaning.
Most major computer algebra systems, such as Mathematica [5] and Maple [6],
are capable of importing and exporting MathML of both formats. These importing
and exporting functions enable the conversion from Presentation to Content
MathML. Importing, of course, depends on the interpretation of each computer
algebra systems engine.
There is a project called SnuggleTeX [1], which addresses the semantic
interpretation of mathematical expressions. The project provides a direct way
to generate Content MathML from Presentation MathML based on manually encoded
rules. The current version at the time of writing this paper supports
operators that are the same as ASCIIMathML [2]. For example, it uses the ASCII
string “\$in$” instead of the symbol “$\in$”. One major drawback of this
approach is that it always makes the same interpretation for the same
Presentation MathML element.
A recent study by Nghiem et al. [7] also addressed the semantic interpretation
of mathematical expressions. This study applied a method based on statistical
machine translation to extract translation rules automatically. This approach
contrasted with previous research, which tended to rely on manually encoded
rules. This study also introduced segmentation rules used to segment
mathematical expressions. Combining segmentation rules and translation rules
strengthened the translation system and the best system achieved 20.89% error
rate. The shortcoming of this approach is that it did not make use of text
information surrounding mathematical expressions.
Wolska et al. [8, 9] presented a knowledge-poor method of finding a denotation
of simple symbolic expressions in mathematical discourse. The system used
statistical co-occurrence measures to classify a simple symbolic expression
into one of seven predefined concepts. They showed that the lexical
information from the linguistic context immediately surrounding the expression
improved the results. The lexical information from the larger document context
also contributed to the best interpretation results. This approach had been
evaluated on a gold standard manually annotated by experts, achieving 66%
precision.
## 3 Our Approach
The system has two phases, a training phase and a running phase, and consists
of three main modules.
* •
Statistical-based rule extraction: Extracts rules for translation, given the
training data. We establish two types of rules: segmentation rules and
translation rules. Each rule is associated with its probability.
* •
SVM-based disambiguation: An SVM training algorithm builds a model that
assigns to identifiers ($mi$) their correct content. Features are extracted
from both the presentation of mathematical expressions and their surrounding
text.
* •
Translation: The input of this module includes a Presentation MathML
expression, a set of rules for translation, and the output from the
disambiguation module. This module translates Presentation into Content MathML
expression.
Figure 1 shows the system framework.
Figure 1: System Framework
### 3.1 Statistical-based rule extraction
The rules for translation were extracted according to the procedure used by
Nghiem et al. [7]. Given a set of training mathematical expressions in MathML
parallel markup, we extracted two types of rules: segmentation rules and
translation rules. Translation rules are used to translate (sub)trees of
Presentation MathML markup to (sub)trees of Content MathML markup.
Segmentation rules are used to combine and reorder the (sub)trees to form a
complete tree. The output of this module is a set of segmentation and
translation rules, each rule is associated with its probability.
### 3.2 SVM disambiguation
An $mi$ token element in MathML presentation markup can be translated into
many different elements in MathML content markup. In this paper, we assumed
that one $mi$ element can be translated into one of a limited predefined set
of Content elements. Given an $mi$ element, we use an SVM training algorithm
to build a model that assigns to its correct Content element. When
translating, each of the Presentation $mi$ elements will be disambiguated
before generating Content MathML expressions. The accuracy of the SVM
disambiguation is a crucial preprocessing step for a high-quality MathML
Presentation to Content translation.
We used the alignment output of GIZA++111https://code.google.com/p/giza-pp/
[11] to generate training and testing data for the disambiguation problem.
Given a training data consists of several parallel markup expressions, we used
GIZA++ to align the Presentation terms to the Content terms. From this
alignment results, we extract pairs of Presentation $mi$ elements and their
associated Content elements. Only $mi$ elements that have ambiguities in their
translation are kept to generate training and testing data. Table shows 1 the
examples of Presentation $mi$ elements and their associated Content elements.
Table 1: Presentation $mi$ elements and their associated Content elements Presentation elements | Content elements
---|---
$<$mi$>\sigma<$/mi$>$ | $<$ci$>$Weierstrass Sigma$<$/ci$>$
| $<$ci$>$Divisor Sigma$<$/ci$>$
| $<$ci$>\sigma<$/ci$>$
$<$mi$>\mu<$/mi$>$ | $<$ci$>$MoebiusMu$<$/ci$>$
| $<$ci$>\mu<$/ci$>$
$<$mi$>$H$<$/mi$>$ | $<$ci$>$StruveH$<$/ci$>$
| $<$ci$>$Harmonic Number$<$/ci$>$
| $<$ci$>$Hankel H1$<$/ci$>$
| $<$ci$>$Hankel H2$<$/ci$>$
| $<$ci$>$Hermite H2$<$/ci$>$
| $<$ci$>$H$<$/ci$>$
$<$mi$>$y$<$/mi$>$ | $<$ci$>$Bessel Y Zero$<$/ci$>$
| $<$ci$>$Spherical Bessel Y$<$/ci$>$
| $<$ci$>$y$<$/ci$>$
For each mathematical expression, an $mi$ element has only one correct
translation. In other mathematical expressions, the same $mi$ element might
have another correct translation. Assume that an $mi$ element $e$ has $n$ ways
of translating from Presentation into Content MathML. For each mathematical
expression, we create one positive instance by combining $e$ and its correct
translation. We also create $n-1$ negative instances by combining $e$ and its
incorrect translations.
The features used in the SVM disambiguation may be divided into two main
groups: Presentation MathML features and surrounding text features.
Presentation MathML features are extracted from the Presentation MathML markup
of the mathematical expression. Surrounding text features are extracted from
the text surrounding the mathematical expression. The category which the
mathematical expression belongs to is also used. Table 2 shows the features we
used for classification.
Table 2: Features used for classification Feature | Description
---|---
Presentation MathML | Only child | Is it the only child of its parent node
feature | Preceded by mo | Is it preceded by an $<$mo$>$ node
| Followed by mo | Is it followed by an $<$mo$>$ node
| $\&\\#8289;$ | Is it followed by a Function Application
| Parent’s name | The name of its parent node
| Name | The name of the identifier
Text feature | Category | Relation between category name and candidate translation
| Unigram | Vector represents unigram feature
| Bigram | Vector represents bigram feature
| Trigram | Vector represents trigram feature
Candidate translation | One of $n$ candidate translations of the $mi$ element
There were six Presentation MathML features in our experiment. The first one
determines whether the $mi$ element is the only child of its parent. The
relation between the $mi$ element and its surrounding $mo$ elements is encoded
in the following three features. The last two features represent the name of
the $mi$ element and its parent. Among these features, the name of the $mi$
element is the most important feature.
Among the text features, the first one is the category that mathematical
expression belongs to. In mathematical resource websites, such as the Wolfram
Functions Site, mathematical expressions belong to different categories. But
usually we do not have the text surrounding these mathematical expressions. We
then can calculate the relation between the category name and the Content
translation of each $mi$ element. The relation has one of three values: the
same as the Content translation, contains the Content translation, or does not
contain the Content translation.
In case we have the text surrounding or the description of the mathematical
expressions, we can use n-gram features [12]. In this paper, we use unigram,
bigram and trigram features. These features are implemented as the vectors
containing the n-grams which appear in the training data. We will assign each
instance into one of two classes, depending on the candidate translation. The
class is ‘true’ if the candidate translation is the correct Content
translation of the $mi$ element, and ‘false’ otherwise.
### 3.3 Translation
After disambiguation, we use the result to enhance the semantic enrichment of
a statistical-machine-translation-based system. The input of this module
includes a Presentation MathML expression, a set of rules for translation, and
the output from the disambiguation module. The output of this module is the
Content MathML expression which represents the meaning of the Presentation
MathML expression. If there is only one mapping from a Presentation element,
that Content element is chosen. If the disambiguation module accepts more than
two mappings from a Presentation element, the Content element with higher
probability is chosen.
## 4 Evaluation
The first dataset for the experiments is the Wolfram Functions site [3]. This
site was created as a resource for educational, mathematical, and scientific
communities. All formulas on this site are available in both Presentation
MathML and Content MathML format. The only text information on this dataset is
the function category of each mathematical expression. In our experiments, we
used 136,685 mathematical expressions divided into seven categories.
The second dataset for the experiments is the Archives of the Association for
Computational Linguistics Corpus [4] (ACL-ARC). It contains mathematical
expressions extracted from scientific papers in the area of Computational
Linguistics and Language Technology. Currently, we use mathematical
expressions drawn from 20 papers which were selected from this dataset. We
have manually annotated all mathematical expressions with MathML parallel
Markup and their textual descriptions. Out of 2,065 mathematical expressions
in the dataset, only 648 expressions have their own description. Table 3 shows
examples of mathematical expressions and their description in ACL-ARC dataset.
Table 3: Examples of mathematical expressions and their description in ACL-ARC dataset Textual description | MathML Presentation expression | MathML Content expressions
---|---|---
a word to be translated | $<$mrow$>$ $<$mi$>$w$<$/mi$>$ $<$/mrow$>$ | $<$ci$>$w$<$/ci$>$
a word in a dependency relationship | $<$mrow$>$ $<$mi$>$w$<$/mi$>$ $<$/mrow$>$ | $<$ci$>$w$<$/ci$>$
a matrix | $<$mrow$>$ $<$mi$>$t$<$/mi$>$ $<$/mrow$>$ | $<$ci$>$t$<$/ci$>$
a similarity matrix which specifies the similarity between individual elements | $<$mrow$>$ $<$mi$>$sim$<$/mi$>$ $<$/mrow$>$ | $<$ci$>$sim$<$/ci$>$
argument | $<$mrow$>$ $<$msub$>$ $<$mi$>$S$<$/mi$>$ $<$msub$>$ $<$mi$>$j$<$/mi$>$ $<$mi$>$i$<$/mi$>$ $<$/msub$>$ $<$/msub$>$ $<$/mrow$>$ | $<$apply$>$ $<$selector /$>$ $<$ci$>$S$<$/ci$>$ $<$apply$>$ $<$selector /$>$ $<$ci$>$j$<$/ci$>$ $<$ci$>$i$<$/ci$>$ $<$/apply$>$ $<$/apply$>$
The LM probabilities | $<$mrow$>$ $<$mi$>$P$<$/mi$>$ $<$mo$>$$<$/mo$>$ $<$mrow$>$ $<$mo$>$($<$/mo$>$ $<$mrow$>$ $<$mi$>$v$<$/mi$>$ $<$mo$>|<$/mo$>$ $<$mrow$>$ $<$mi$>$Parent$<$/mi$>$ $<$mo$>$$<$/mo$>$ $<$mrow$>$ $<$mo$>$($<$/mo$>$ $<$mi$>$v$<$/mi$>$ $<$mo$>$)$<$/mo$>$ $<$/mrow$>$ $<$/mrow$>$ $<$/mrow$>$ $<$mo$>$)$<$/mo$>$ $<$/mrow$>$ $<$/mrow$>$ | $<$apply$>$ $<$ci$>$P$<$/ci$>$ $<$apply$>$ $<$ci$>|<$/ci$>$ $<$ci$>$v$<$/ci$>$ $<$apply$>$ $<$ci$>$Parent$<$/ci$>$ $<$ci$>$v$<$/ci$>$ $<$/apply$>$ $<$/apply$>$ $<$/apply$>$
The evaluation was done using two metrics: accuracy score for disambiguation
and tree edit distance rate score for semantic enrichment. The accuracy score
of disambiguation is the ratio of correctly classified instances to total
instances. The tree edit distance rate (TEDR) score [7] is defined as the
ratio of (1) the minimal cost of transforming the generated into the reference
Content MathML tree using edit operations and (2) the maximum number of nodes
of the generated and the reference Content MathML tree. We also compare our
semantic enrichment results to the results of Nghiem et al.
First, we set up an experiment to examine the disambiguation result on each
Presentation MathML $mi$ element. In this experiment, we compare three
systems. The first system uses both Presentation MathML and text features. The
second system uses only Presentation MathML features. The last system chooses
the interpretation with highest probability.
Training and testing were performed using ten-fold cross-validation. For each
category, we partitioned the original corpus into ten subsets. Of the ten
subsets, we retained a single subset as validation data for testing the model,
remaining subsets are used as training data. The cross-validation process was
repeated ten times, and the ten results from the folds then averaged to
produce a single estimate. Table 4 shows the results of the disambiguation
component.
Table 4: Disambiguation accuracy Category | Number of instances | With text features | Without text features | Most frequent
---|---|---|---|---
ACL-ARC | 2,996 | 92.9573 | 93.7583 | 93.4246
Bessel-TypeFunctions | 1,352 | 92.8254 | 92.3077 | 86.0947
Constants | 714 | 91.1765 | 90.3361 | 83.7535
ElementaryFunctions | 6,073 | 96.1963 | 96.3774 | 89.6427
GammaBetaErf | 3,816 | 95.2830 | 94.4706 | 78.0136
HypergeometricFunctions | 72,006 | 97.5571 | 97.0697 | 88.0746
IntegerFunctions | 11,955 | 95.8009 | 95.1652 | 90.0711
Polynomials | 5,905 | 98.2388 | 95.3091 | 87.3328
All WFS Data | 320,726 | 98.9243 | 98.4398 | 92.7025
The results in Table 4 show that disambiguation result using SVM outperformed
the ‘most frequent’ method. The reason ‘most frequent’ method got high scores
is because mathematical elements often have a preferred meaning. The systems
that used only Presentation MathML features achieved even better scores,
because they use surrounding mathematical elements. It is interesting to note
that on the ACL-ARC data, the ‘most frequent’ system get higher score than the
system with text features. Overall, on WFS data, we gained 5 to 16 percent
accuracy improvements.
The systems that also used text features outperform the systems that used only
Presentation MathML features in most of WFS categories. This result may be
explained by the fact that the category of a mathematical expression is
closely related to that expression. Contrary to expectations, this study did
not find any improvement in ACL-ARC data. It seems possible that these results
are due to the lack of training data and the sparseness of n-gram features.
This finding was unexpected and suggests that in order to use n-gram text
features, we need more data.
Second, we set up an experiment to examine the semantic enrichment result. The
results from disambiguation component are used in the semantic enrichment
system. We compare three systems: with text feature, without text feature, and
the system of Nghiem et al. which used ‘most frequent’ method. In this
experiment, we use 90 percent of expressions for training both SVM-based
disambiguation and translation components. We use the other 10 percent of
expressions for testing. Table 5 shows the translation result.
Table 5: Semantic enrichment TEDR Category | Number of expression | With text feature | Without text feature | Most frequent
---|---|---|---|---
Bessel-TypeFunctions | 701 | 18.0604 | 18.0604 | 18.4118
Constants | 555 | 33.9016 | 34.0328 | 34.6230
ElementaryFunctions | 9,537 | 7.4879 | 7.4809 | 7.7343
GammaBetaErf | 1,558 | 17.2308 | 17.2851 | 18.4796
HypergeometricFunctions | 9,347 | 49.4678 | 49.4797 | 49.6902
IntegerFunctions | 1,175 | 20.5292 | 20.5874 | 20.9945
Polynomials | 727 | 19.6309 | 19.7987 | 20.2685
All WFS Data | 23,600 | 29.0707 | 29.0869 | 29.2769
The results in Table 5 show that combining disambiguation and statistical
machine translation improved the system. Expressions in ‘Gamma Beta Erf’
category benefit from the disambiguation module the most with 1.2 percent
error rate reduction. Less ambiguity in elementary functions might lead to
lower performance in ‘Elementary Functions’ category. We did not evaluate on
ACL-ARC data because the disambiguation result was almost the same as the
‘most frequent’ method. Overall, on WFS data, we achieved 0.2 to 1.2 percent
error rate reduction.
## 5 Conclusion
In this paper, we have presented a new approach to the semantic enrichment for
mathematical expression problem. Our approach, which combines statistical
machine translation and disambiguation component, shows promise. This study
has shown that the disambiguation component using presentation features
improved the system performance. The use of text features, especially the
category of each expression, also played an important role in the
disambiguation of mathematical elements. Experimental results of this study
showed that our system achieves improvements over prior systems.
This research has raised many questions in need of further investigation. One
question is finding and combining new features, such as the style of the font,
for the disambiguation task. Another possible improvement is making use of co-
occurrence of mathematical elements in the same document. In the scope of this
paper, we only disambiguated lexical ambiguities of mathematical expressions.
Structural ambiguities should also be considered to achieve better results.
The evidence from this study suggests that in a small dataset, descriptions of
mathematical expressions did not improve the system performance. Further work
needs to be done to establish whether descriptions of mathematical expressions
contribute to the the task in a larger dataset.
## References
* [1] D. McKain: SnuggleTeX version 1.2.2. http://www2.ph.ed.ac.uk/snuggletex/
* [2] ASCII MathML. http://www1.chapman.edu/jipsen/mathml/asciimath.html
* [3] The Wolfram Functions Site. http://functions.wolfram.com/
* [4] The archives of the Association for Computational Linguistics. http://acl-arc.comp.nus.edu.sg/
* [5] Wolfram Research, Inc.: Mathematica Edition: Version 8.0. Wolfram Research, Inc. (2010)
* [6] Maplesoft: Maple. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario. (2013)
* [7] M.Q. Nghiem, G.K. Yoko, Y. Matsubayashi, and A. Aizawa: Automatic Approach to Understanding Mathematical Expressions Using MathML Parallel Markup Corpora. The 26th Annual Conference of the Japanese Society for Artificial Intelligence (2012)
* [8] M. Wolska, M. Grigore, and M. Kohlhase: Using Discourse Context to Interpret Object-Denoting Mathematical Expressions. Towards a Digital Mathematics Library, pages 85-101 (2011)
* [9] M. Wolska, and M. Grigore: Symbol Declarations in Mathematical Writing. Towards a Digital Mathematics Library, pages 119-127 (2010)
* [10] R. Ausbrooks et al.: Mathematical Markup Language (MathML) version 3.0. W3C Recommendation, World Wide Web Consortium (2010)
* [11] F. J. Och and H. Ney: A systematic comparison of various statistical alignment models. Computational Linguistics Volume 29 Issue 1, pages 19-51 (2003)
* [12] W. B. Cavnar , J. M. Trenkle: N-Gram-Based Text Categorization. In Proceedings of SDAIR-94, 3rd Annual Symposium on Document Analysis and Information Retrieval, pages 161-175 (1994)
|
arxiv-papers
| 2013-05-31T07:34:07 |
2024-09-04T02:49:45.926566
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Minh-Quoc Nghiem, Giovanni Yoko Kristianto, Goran Topic, Akiko Aizawa",
"submitter": "Minh-Quoc Nghiem",
"url": "https://arxiv.org/abs/1305.7316"
}
|
1305.7404
|
# Ice Cube Observed PeV Neutrinos from AGN Cores
Floyd W. Stecker NASA Goddard Space Flight Center
Greenbelt, MD 20771, USA
###### Abstract
I show that the high energy neutrino flux predicted to arise from AGN cores
can explain the PeV neutrinos detected by Ice Cube without conflicting with
the constraints from the observed extragalactic cosmic ray and $\gamma$-ray
backgrounds.
Recently, the Ice Cube collaboration has reported the first observation of
cosmic 2 PeV energy neutrinos giving a signal $\sim 3\sigma$ above the
atmospheric background aa13 . More recently 18 events $\sim$4$\sigma$ above
the expected atmospheric background were reported with energies above 100 TeV.
These neutrinos are likely to be of cosmic origin; their angular distribution
is consistent with isotropy. The average spectral index for these neutrinos
was approximately -2 over the energy range between $\sim$ 100 TeV and 2 PeV
ha13 .
In 1991 we proposed a model suggesting that very high energy neutrinos could
be produced in the cores of active galaxies (AGN) such as Seyfert galaxies
sdss91 . Using that model, we gave estimates of the flux and spectrum of high
energy neutrinos to be expected. In light of subsequent AGN observations and
the discovery of neutrino oscillations the flux estimates for this model were
revised downward s05 , although the shape of the predicted neutrino spectrum
remained unchanged.
The new estimate was obtained by lowering the flux shown in the Figure in Ref.
sdss91E by a factor of 20. This rescaling gives a value for the $\nu_{\mu}$
flux at 100 TeV of $E_{\nu}^{2}\Phi(E_{\nu})\sim 10^{-8}$ GeV cm-2s-1sr-1 and
a flux of $\sim 5.6\times 10^{-8}$ GeV cm-2s-1sr-1 at $\sim 1$ PeV. The peak
flux in these units occurs at an energy $\sim$1 PeV
In our model protons are accelerated by shocks in the cores of AGN in the
vicinity of the black hole accretion disk (see, e.g., Ref. ke86 ). Being
trapped by the magnetic field, they lose energy dramatically by interactions
with the dense photon field of the ”big blue bump” of thermal emission from
the accretion disk (see, e.g., Ref. laor90 ) which is optically thick to
protons sdss91 . The primary interactions are those from photomeson
production. The primary neutrino producing channel, which occurs near
threshold, is
$\gamma+p\rightarrow\Delta^{+}\rightarrow n+\pi^{+}.$ (1)
giving the pion an energy roughly $\approx 0.2$ of that of the emitting
protons st68 . Since the secondary $\gamma$-rays from $\Delta^{+}\rightarrow
p+\pi^{0}$ followed by $\pi^{0}\rightarrow 2\gamma$ lose energy in the source
from electron-positron pair production and the protons in our model do not
reach ultrahigh energies, losing energy in the source, there is no conflict
with ultrahigh energy cosmic ray (UHECR) or extragalactic $\gamma$-ray
background constraints ro13 . Full details of the model may be found in Ref.
sdss91 , Ref. s05 and references therein.
The important decays leading to $\nu$ production are
$\pi^{+}\rightarrow\mu^{+}+\nu_{\mu}$ followed by
$\mu^{+}\rightarrow\bar{\nu_{\mu}}+\nu_{e}+e^{+}$, with all leptons carrying
off about 1/4 of the pion energy. Thus in our model we expect that with an
assumed power-law proton spectrum from shock acceleration, followed by
photomeson production, there will be both more and higher energy $\nu_{e}$’s
produced than $\bar{\nu_{e}}$’s, reducing the effect of Glashow resonance
production at 6.3 PeV sg60 in the detector (see also Ref. ro13 ).
Given the effective area of Ice Cube for an isotropic flux of $\nu_{e}$’s at
$\sim$1 PeV of 5 m2 and an exposure time of 615.9 days aa13 , and giving a
total predicted $\nu$ flux at $\sim$1 PeV of $\sim 6\times 10^{-14}$ (cm2 s
sr)-1, we would predict that Ice Cube should see a total of $\sim$6 neutrino
induced events.
The main uncertainty in the magnitude of the modeled flux is from the
uncertainty in the number of AGN per Mpc3. The neutrino spectrum was predicted
to be proportional to $E^{-2.1}$ between 1 and 10 PeV under the assumption
that the proton spectrum has an $E^{-2}$ dependence from shock acceleration.
The average slope of the neutrino spectrum was predicted to be $\sim 3.1$
between 1 and 100 PeV. Protons were assumed to be accelerated up to a maximum
energy of $2.5\times 10^{4}$ PeV. It is, of course, possible that the average
proton spectrum can be steeper than $E^{-2}$ and that the maximum energy to
which protons are accelerated is less than $2.5\times 10^{4}$ PeV. Since the
sources are extragalactic, we expect that the observed neutrinos will have a
roughly isotropic angular distribution on the sky.
Given the uncertainties in the model parameters, the general agreement with
the AGN core model is significant, particularly the prediction of a peak
neutrino energy flux at $\sim$ 1 PeV. I conclude that, given the present Ice
Cube results, AGN cores may naturally account for the implied $\nu$ flux and
angular distribution without violating constraints from $\gamma$-ray
background and UHECR fluxes. More Ice Cube data should soon be forthcoming.
Acknowledgment: I thank Francis Halzen for helpful comments.
## References
* (1) M. G. Aartsen et al., arXiv:1304.5356.
* (2) F. Halzen, Proc. 33rd Intl. Cosmic Ray Conf., Rio de Janiero, 2013.
* (3) F. W. Stecker, C. Done, M.H. Salamon and P. Sommers, Phys. Rev. Lett. 66, 2697 (1991),
* (4) F. W. Stecker, Phys. Rev. Lett. 69, 2738 (1992).
* (5) F. W. Stecker, Phys. Rev. D 72, 107301 (2005).
* (6) D. Kazanas and D. C. Ellison, Astrophys. J. 304, 178 (1986).
* (7) A. Laor, Mon. Not. Royal Astr. Soc. 246, 369 (1990).
* (8) F. W. Stecker, Phys. Rev. Lett. 21, 1016 (1968).
* (9) E. Roulet et al., J. Cosmol. Astropart. Phys. 2013/01/028 (2013).
* (10) S. L. Glashow, Phys. Rev. 118, 316 (1960).
|
arxiv-papers
| 2013-05-31T14:15:59 |
2024-09-04T02:49:45.936750
|
{
"license": "Public Domain",
"authors": "Floyd W. Stecker",
"submitter": "Floyd Stecker",
"url": "https://arxiv.org/abs/1305.7404"
}
|
1305.7456
|
# Angle-dependent Gap state in Asymmetric Nuclear Matter
Xin-le Shang [email protected] Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China Wei Zuo Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China State Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing 100190, China
###### Abstract
We propose an axi-symmetric angle-dependent gap (ADG) state with the broken
rotational symmetry in isospin-asymmetric nuclear matter. In this state, the
deformed Fermi spheres of neutron and proton increase the pairing
probabilities along the axis of symmetry breaking near the average Fermi
surface. We find the state possesses lower free energy and larger gap value
than the angle-averaged gap state at large isospin asymmetries. These
properties are mainly caused by the coupling of different $m_{j}$ components
of the pairing gap. Furthermore, we find the transition from the ADG state to
normal state is of second order and the ADG state vanishes at the critical
isospin asymmetry $\alpha_{c}$ where the angle-averaged gap vanishes.
###### pacs:
21.65.Cd, 26.60.-c, 74.20.Fg, 74.25.-q
## I Introduction
The neutron-proton (n-p) pairing properties play an important role in the
description of superfluidity of finite nuclei with $N\simeq Z$ Fn ; Fn2 and
symmetric nuclear matterSnm1 ; Snm2 ; Snm3 . In general, the n-p pair
correlations are considered in different dominant partial-wave channels,
depending on the relevant density and temperature. For weakly isospin-
asymmetric systems, the isospin singlet attractive ${}^{3}S_{1}-^{3}D_{1}$
(${}^{3}SD_{1}$) channel dominates the pairing interaction at relatively low
densities around the nuclear saturation density due to the tensor component of
the nuclear forceSnm1 ; Sdd ; Sdd2 ; Sdd3 ; Sdd4 ; Sdd5 , and the
${}^{3}D_{{}_{2}}$ channel dominates at high densities well above the
saturation densityD1 ; D2 . In neutron star matter, the n-p pair correlations
are strongly suppressed by the isospin-asymmetry. However, the dilute nuclear
matter at sub-saturation densities in supernovas and hot proto-neutron stars
can support ${}^{3}SD_{1}$ channel pairingSdd3 ; Sh1 ; Sh2 ; Sh3 .
Since n-p pair correlations depend crucially on the overlap between the
neutron and proton Fermi surfaces, the pairing gap is suppressed rapidly as
the system is driven out of the isospin-symmetric state. At zero temperature,
a small isospin-asymmetry is enough to prevent the formation of the Cooper
pairs between neutrons and protons with momenta $\overrightarrow{k}$ and
$-\overrightarrow{k}$ around their average Fermi surface where the
contribution to superfluidity is dominant. Near zero temperature, thermal
excitations can reduce the suppression by smearing out the two Fermi surfaces,
however, it is ineffective when the separation between the two Fermi surfaces
is large compared to the temperature. In isospin-asymmetric nuclear matter,
the FFLO ff ; lo state and the DFS (deformed Fermi surfaces) dfs state have
been studied in Refs.ffn ; dfsn . In a FFLO state, the shift of the two Fermi
spheres with respect to each other, resulting form the collective motion of
the Cooper pairs with a finite momentum, enhances the overlap between the
neutron and proton Fermi surfaces. The overlap regions then provide the
kinematical phase space for n-p pairing phenomena to occur. And in a DFS
state, the deformation of the neutron and proton Fermi surfaces may increase
the phase-space overlap between the two Fermi surfaces. Both in these two
kinds of possible superfluid states, the quasiparticle excitation spectra are
no longer isotropic, since the anisotropic overlapping configurations could
increase the pairing energy. On the other hand, the usually adopted angle-
averaging procedure in the previous calculationsSdd2 ; ffn , which has been
proved to be a quite good approximation in symmetry nuclear matter aap ,
considers the gap as an isotopic gap by ignoring the angle dependence. As the
true ground state corresponds to the anisotropic overlapping configuration,
the angle-averaging procedure may be an insufficient approximation in isospin-
asymmetric nuclear matter.
In this paper we consider an axi-symmetric angle dependent gap (ADG) state,
and give a general and systematic comparison between the ADG state and the
angle-averaged gap (AAG) state in isospin-asymmetric nuclear matter. The paper
is organized as follows: In Sec. II we briefly review the formalism for the
isotropic AAG state, and derive the angle dependent gap equations from the
Gorkov equations. The numerical solutions of these equations are shown and
discussed in Sec. III, where we compare the AAG state with the ADG state at
finite temperature. Finally, a summary and a conclusion are given in Sec. IV.
## II Formalism
For isospin-asymmetric nuclear matter, the isospin singlet ${}^{3}SD_{1}$
pairing channel dominates the attractive pairing force at low densities. In
this case we can consider ${}^{3}SD_{1}$ channel only, the gap function is
thus expanded according to
$\displaystyle\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})=\sum_{l,m_{j}}\Delta_{l}^{m_{j}}(k)[G_{l}^{m_{j}}(\textbf{\^{k}})]_{\sigma_{1},\sigma_{2}},$
(1)
with the elements of the spin-angle matrices
$\displaystyle[G_{l}^{m_{j}}(\textbf{\^{k}})]_{\sigma_{1},\sigma_{2}}\equiv\langle\frac{1}{2}\sigma_{1},\frac{1}{2}\sigma_{2}\mid
1\sigma_{1}+\sigma_{2}\rangle\langle 1\sigma_{1}+\sigma_{2},lm_{l}\mid
1m_{j}\rangle Y_{l}^{m_{l}}(\textbf{\^{k}}),$
where $m_{j}$ and $m_{l}$ are the projections of the total angular momentum
$j=1$ and the orbit angular momentum $l=0,2$ of the pair, respectively. The
$Y_{l}^{m_{l}}(\textbf{\^{k}})$ denotes the spherical harmonic with
$\textbf{\^{k}}\equiv\textbf{k}/k$. The anomalous density matrix follows the
same expansion. Moreover the time-reversal invariance implies that
$\displaystyle\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})=(-1)^{1+\sigma_{1}+\sigma_{2}}\Delta_{-\sigma_{1},-\sigma_{2}}^{*}(\textbf{k}).$
(3)
Namely, the pairing gap matrix $\Delta(\textbf{k})$ in spin space possesses
the property
$\displaystyle\Delta(\textbf{k})\Delta^{{\dagger}}(\textbf{k})=ID^{2}(\textbf{k}),$
(4)
i.e., the gap function has the structure of a ``unitary triplet" state aap .
$I$ is the identity matrix and $D(\textbf{k})$ is a scalar quantity in spin
space.
Once the the isospin singlet ${}^{3}SD_{1}$ channel has been selected, the
pairing gap is an isoscalar and the isospin indices can be dropped off. The
proton/neutron propagators follow from the solution of the Gorkov equations,
and can be present in the form ($\hbar=1$)
$\displaystyle\textbf{G}_{\sigma,\sigma^{{}^{\prime}}}^{(p/n)}(\textbf{k},\omega_{m})=-\delta_{\sigma,\sigma^{{}^{\prime}}}\frac{i\omega_{m}+\xi_{\textbf{k}}\mp\delta\varepsilon_{\textbf{k}}}{(i\omega_{m}+E_{\textbf{k}}^{+})(i\omega_{m}-E_{\textbf{k}}^{-})}.$
(5)
The neutron-proton anomalous propagator matrix in spin space has the form
$\displaystyle\textbf{F}^{{\dagger}}(\textbf{k},\omega_{m})=-\frac{\Delta^{{\dagger}}(\textbf{k})}{(i\omega_{m}+E_{\textbf{k}}^{+})(i\omega_{m}-E_{\textbf{k}}^{-})},$
(6)
where $\omega_{m}$ are the Matsubara frequencies, the uper sign in
$\textbf{G}_{\sigma,\sigma^{{}^{\prime}}}^{(p/n)}$ corresponds to protons, and
the lower to neutrons. The quasiparticle excitation spectra are determined by
finding the poles of the propagators in Gorkov equations,
$\displaystyle
E_{\textbf{k}}^{\pm}=\sqrt{\xi_{\textbf{k}}^{2}+\frac{1}{2}Tr(\Delta\Delta^{{\dagger}})\pm\frac{1}{2}\sqrt{[Tr(\Delta\Delta^{{\dagger}})]^{2}-4\det(\Delta\Delta^{{\dagger}})}}\pm\delta\varepsilon_{\textbf{k}},$
(7)
where
$\displaystyle\xi_{\textbf{k}}\equiv\frac{1}{2}(\varepsilon_{\textbf{k}}^{p}+\varepsilon_{\textbf{k}}^{n}),\delta\varepsilon_{\textbf{k}}\equiv\frac{1}{2}(\varepsilon_{\textbf{k}}^{p}-\varepsilon_{\textbf{k}}^{n}),$
and $\varepsilon_{\textbf{k}}^{(n,p)}$ are the single particle energies of
neutrons and protons. Using the ``unitary" property in Eq. (4), the
quasiparticle spectra are simplified to
$\displaystyle
E_{\textbf{k}}^{\pm}=\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}\pm\delta\varepsilon_{\textbf{k}},$
(8)
which are separated into two branches due to the isospin-asymmetry.
In the present ``unitary triplet" case, the gap equation at finite temperature
can be written in the standard form
$\displaystyle\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})=-\sum_{\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>$
$\displaystyle\times\frac{\Delta_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})}{2\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(\textbf{k}^{{}^{\prime}})}}[1-f(E_{\textbf{k}^{{}^{\prime}}}^{+})-f(E_{\textbf{k}^{{}^{\prime}}}^{-})],$
(9)
where $f(E)=[1+\exp(\beta E)]^{-1}$ is the Fermi distribution at finite
temperature and $V$ is the interaction in the ${}^{3}SD_{1}$ channel.
$\beta^{-1}=k_{B}T$, where $k_{B}$ is the Boltzmann constant and $T$ is the
temperature. Substituting the expansion Eq.(1) into Eqs.(9) and (4), one gets
a set of coupled equations for the quantities $\Delta_{l}^{m_{j}}(k)$
$\displaystyle\Delta_{l}^{m_{j}}(k)=\frac{-1}{\pi}\int_{0}^{\infty}dk^{{}^{\prime}}k^{{}^{\prime}2}\sum_{l^{{}^{\prime}}=0,2}i^{l^{{}^{\prime}}-l}V^{l^{{}^{\prime}}l}_{\lambda}(k^{{}^{\prime}},k)\sum_{l^{{}^{\prime\prime}}\mu}\Delta_{l^{{}^{\prime\prime}}}^{\mu}(k^{{}^{\prime}})$
$\displaystyle\times\int
d\Omega_{\textbf{k}^{{}^{\prime}}}Tr[G_{l^{{}^{\prime}}}^{m_{j}*}(\textbf{\^{k}}^{{}^{\prime}})G_{l^{{}^{\prime\prime}}}^{\mu}(\textbf{\^{k}}^{{}^{\prime}})]\frac{1-f(E_{\textbf{k}^{{}^{\prime}}}^{+})-f(E_{\textbf{k}^{{}^{\prime}}}^{-})}{\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(\textbf{k}^{{}^{\prime}})}},$
(10)
with
$\displaystyle
D^{2}(\textbf{k})=\frac{1}{2}Tr(\Delta\Delta^{{\dagger}})=\sum_{ll^{{}^{\prime}}=0,2}\sum_{m_{j}m_{j^{{}^{\prime}}}}\Delta_{l}^{m_{j}*}(k)\Delta_{l^{{}^{\prime}}}^{m_{j^{{}^{\prime}}}}(k)Tr[G_{l}^{m_{j}{\dagger}}(\textbf{\^{k}})G_{l^{{}^{\prime}}}^{m_{j^{{}^{\prime}}}}(\textbf{\^{k}})],$
where
$\displaystyle
V^{l^{{}^{\prime}}l}_{\lambda}(k^{{}^{\prime}},k)\equiv<k^{{}^{\prime}}\mid
V^{l^{{}^{\prime}}l}_{\lambda}\mid
k>=\int_{0}^{\infty}r^{2}drj_{l^{{}^{\prime}}}(k^{{}^{\prime}}r)V^{l^{{}^{\prime}}l}_{\lambda}(r)j_{l}(kr),$
(12)
is the matrix elements of the NN interaction in different partial wave
($\lambda=T,S,l,l^{{}^{\prime}}$) channels. Here $\lambda$ corresponds to the
coupled ${}^{3}SD_{1}$ channel. Following from Eq.(5), we can get the
densities of neutrons and protons
$\displaystyle\rho^{(p/n)}=\sum_{\textbf{k},\sigma}n_{\sigma}^{(p/n)}(\textbf{k}),$
(13)
with the distributions
$\displaystyle
n_{\sigma}^{(p/n)}(\textbf{k})=\\{\frac{1}{2}(1+\frac{\xi_{\textbf{k}}}{\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}})f(E_{\textbf{k}}^{\pm})$
$\displaystyle+\frac{1}{2}(1-\frac{\xi_{\textbf{k}}}{\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}})[1-f(E_{\textbf{k}}^{\mp})]\\}.$
(14)
Summation over frequencies in Eq.(6) leads to the density matrix of the
particles in the condensate,
$\displaystyle\nu(\textbf{k})=\frac{\Delta(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$
(15)
It is essential that the coupled Eqs.(10) and (13) should be solved self-
consistently.
The six components $\Delta_{l}^{m_{j}}(k)$ of $\Delta(\textbf{k})$ are
strongly coupled due to the angle dependent energy denominator
$\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}$ in Eqs.(10) and (13). The
equations are thus complicated to be solved accurately, and approximation has
been employed. Before introducing the angle-averaging procedure and ADG, we
need to substitute $\Delta_{l}^{m_{j}}(k)$ with real variables. From Eq.(3) we
can find the relation
$\displaystyle\Delta_{l}^{m_{j}*}(k)=-(-1)^{m_{j}}\Delta_{l}^{-m_{j}}(k).$
(16)
Therefore, we have four independent components $\Delta_{0}^{0}(k)$,
$\Delta_{0}^{1}(k)$, $\Delta_{2}^{0}(k)$ and $\Delta_{2}^{1}(k)$ for
${}^{3}SD_{1}$ channel , and we can describe $\Delta_{l}^{m_{j}}(k)$ as
$\displaystyle\Delta_{0}^{0}(k)$ $\displaystyle=i\delta_{0}(k),$
$\displaystyle\Delta_{0}^{1}(k)$ $\displaystyle=\delta_{1}(k)+in_{1}(k),$
$\displaystyle\Delta_{2}^{0}(k)$ $\displaystyle=i\delta_{2}(k),$
$\displaystyle\Delta_{2}^{1}(k)$ $\displaystyle=\delta_{3}(k)+in_{3}(k),$ (17)
where the six independent variables $\delta_{0}(k)$, $\delta_{1}(k)$,
$n_{1}(k)$, $\delta_{2}(k)$, $\delta_{3}(k)$ and $n_{3}(k)$ are real
quantities. Inserting Eq.(17) into Eq.(11), we get
$\displaystyle D^{2}(\textbf{k})=$
$\displaystyle\frac{1}{32\pi}\Big{\\{}4\delta_{0}^{2}(k)-4\sqrt{2}\delta_{0}(k)\delta_{2}(k)[3\cos^{2}\theta-1]+2\delta_{2}^{2}(k)[3\cos^{2}\theta-1]$
$\displaystyle+8[\delta_{1}^{2}(k)+n_{1}^{2}(k)]+8[\delta_{3}^{2}(k)+n_{3}^{2}(k)]+6[\delta_{3}^{2}(k)+n_{3}^{2}(k)]\sin^{2}\theta$
$\displaystyle+4\sqrt{2}n_{1}(k)n_{3}(k)[3\cos^{2}\theta-1]+4\sqrt{2}\delta_{1}(k)\delta_{3}(k)[3\cos^{2}\theta-1]$
$\displaystyle+12[2\delta_{0}(k)n_{3}(k)+2\delta_{2}(k)n_{1}(k)-\sqrt{2}\delta_{2}(k)n_{3}(k)]\cos\theta\sin\theta\cos\varphi$
$\displaystyle+12[2\delta_{1}(k)\delta_{2}(k)+2\delta_{0}(k)\delta_{3}(k)-\sqrt{2}\delta_{2}(k)\delta_{3}(k)]\cos\theta\sin\theta\sin\varphi$
$\displaystyle+6[n_{3}^{2}(k)-\delta_{3}^{2}(k)+2\sqrt{2}\delta_{1}(k)\delta_{3}(k)-2\sqrt{2}n_{1}(k)n_{3}(k)]\sin^{2}\theta\cos
2\varphi$
$\displaystyle+12[\delta_{3}(k)n_{3}(k)-\sqrt{2}\delta_{1}(k)n_{3}(k)-\sqrt{2}\delta_{3}(k)n_{1}(k)]\sin^{2}\theta\sin
2\varphi\Big{\\}}.$
### II.1 The angle-averaging procedure
Supposing the angle dependence of the energy denominator
$\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}$ can be neglected, the gap
equations are simplified by substituting $D^{2}(\textbf{k})$ with its angular
average value,
$\displaystyle D^{2}(\textbf{k})\rightarrow$ $\displaystyle
d^{2}(k)=\frac{1}{4\pi}\int d\Omega_{\textbf{k}}D^{2}(\textbf{k})$ (19)
$\displaystyle=\frac{1}{8\pi}\Big{[}2\delta_{1}^{2}(k)+\delta_{0}^{2}(k)+2n_{1}^{2}(k)+2\delta_{3}^{2}(k)+\delta_{2}^{2}(k)+2n_{3}^{2}(k)\Big{]}.$
Thereby, the energy denominator and the quasiparticle spectra are isotropic.
Noting the properties of $G_{l}^{m_{j}}(\textbf{\^{k}})$
$\displaystyle\int
d\Omega_{\textbf{k}}Tr[G_{l}^{m_{j}*}(\textbf{\^{k}})G_{l^{{}^{\prime}}}^{m_{j^{{}^{\prime}}}*}(\textbf{\^{k}})]=\delta_{ll^{{}^{\prime}}}\delta_{m_{j}m_{j^{{}^{\prime}}}},$
(20)
the different $m_{j}$ components $\Delta_{l}^{m_{j}}(k)$ with the same $l$
become uncoupled and all equal to each other. It follows that
$\displaystyle\delta_{1}(k)=n_{1}(k)=\sqrt{\frac{1}{2}}\delta_{0}(k),\delta_{3}(k)=n_{3}(k)=\sqrt{\frac{1}{2}}\delta_{2}(k),$
(21)
and
$\displaystyle d^{2}(k)=\frac{3}{8\pi}[\delta_{0}^{2}(k)+\delta_{2}^{2}(k)].$
(22)
Taking the normalization
$\displaystyle\Delta_{0}(k)=\sqrt{\frac{3}{8\pi}}\delta_{0}(k),\Delta_{2}(k)=-\sqrt{\frac{3}{8\pi}}\delta_{2}(k),$
(23)
the set of equations in Eq.(10) reduces to two coupled equations for the
${}^{3}S_{1}$ and ${}^{3}D_{1}$ gap components $\Delta_{0}(k)$ and
$\Delta_{2}(k)$, respectively. They read
$\displaystyle\left(\begin{array}[]{l}\Delta_{0}\\\
\Delta_{2}\end{array}\right)(k)=\frac{-1}{\pi}\int
dk^{{}^{\prime}}k^{{}^{\prime}2}\left(\begin{array}[]{ll}V^{00}&V^{02}\\\
V^{20}&V^{22}\end{array}\right)(k,k^{{}^{\prime}})\frac{1-f(E_{k^{{}^{\prime}}}^{+})-f(E_{k^{{}^{\prime}}}^{-})}{\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(k^{{}^{\prime}})}}\left(\begin{array}[]{l}\Delta_{0}\\\
\Delta_{2}\end{array}\right)(k^{{}^{\prime}})$ (30) , (31)
where $V^{00}$, $V^{02}$, $V^{20}$, $V^{22}$ are given in Eq.(12) with
$l,l^{{}^{\prime}}=0,2$ and
$\displaystyle
E^{\pm}_{k}=\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(k)}\pm\delta\varepsilon_{\textbf{k}},$
$\displaystyle D^{2}(k)\equiv d^{2}(k)=\Delta_{0}^{2}(k)+\Delta_{2}^{2}(k).$
(32)
Eqs.(13), (24) and (25) compose the angle-averaged gap equations and should be
solved simultaneously for isospin-asymmetric nuclear matter. The quasiparticle
spectra here are isotropic and the gapless excitation exists at large
asymmetry ($|\delta\varepsilon_{\textbf{k}_{F}}|\geq D(k_{F})$) near zero
temperature.
### II.2 The angle dependent gap
As pointed out in the Sec.I, the angle dependence of quasiparticle spectra due
to $D^{2}(\textbf{k})$ may increase the phase-space overlap of neutron and
proton near their average Fermi surface. We consider an axi-symmetric
$D^{2}(\textbf{k})$ solution which corresponds to an axi-symmetric deformation
of the neutron and proton Fermi spheres. From the expression in Eq.(18), the
axi-symmetric solutions are restricted by
$\displaystyle
2\delta_{0}(k)n_{3}(k)+2\delta_{2}(k)n_{1}(k)-\sqrt{2}\delta_{2}(k)n_{3}(k)=0,$
$\displaystyle
2\delta_{1}(k)\delta_{2}(k)+2\delta_{0}(k)\delta_{3}(k)-\sqrt{2}\delta_{2}(k)\delta_{3}(k)=0,$
$\displaystyle
n_{3}^{2}(k)-\delta_{3}^{2}(k)+2\sqrt{2}\delta_{1}(k)\delta_{3}(k)-2\sqrt{2}n_{1}(k)n_{3}(k)=0,$
$\displaystyle\delta_{3}(k)n_{3}(k)-\sqrt{2}\delta_{1}(k)n_{3}(k)-\sqrt{2}\delta_{3}(k)n_{1}(k)=0.$
(33)
There exists only one nontrivial solution
$\displaystyle\delta_{1}(k)=n_{1}(k)=\delta_{3}(k)=n_{3}(k)=0,$ (34)
which corresponds to the $m_{j}=0$ gap components of $\Delta_{l}^{m_{j}}(k)$.
In this case
$\displaystyle D^{2}(\textbf{k})\rightarrow D^{2}(k,\theta)=$
$\displaystyle\frac{1}{8\pi}\Big{[}\delta_{0}^{2}(k)-\sqrt{2}\delta_{0}(k)\delta_{2}(k)(3\cos^{2}\theta-1)$
(35) $\displaystyle+\delta_{2}^{2}(k)\frac{3\cos^{2}\theta+1}{2}\Big{]}.$
Using the normalization
$\displaystyle\Delta_{0}(k)=\sqrt{\frac{1}{8\pi}}\delta_{0}(k),\Delta_{2}(k)=-\sqrt{\frac{1}{8\pi}}\delta_{2}(k),$
(36)
one gets the angle dependent gap equations
$\displaystyle\left(\begin{array}[]{l}\Delta_{0}\\\
\Delta_{2}\end{array}\right)(k)$ $\displaystyle=\frac{-1}{\pi}\int
dk^{{}^{\prime}}k^{{}^{\prime}2}\left(\begin{array}[]{ll}V^{00}&V^{02}\\\
V^{20}&V^{22}\end{array}\right)(k,k^{{}^{\prime}})$ (46)
$\displaystyle\times\int
d\Omega_{\textbf{k}^{{}^{\prime}}}\frac{1-f(E_{k^{{}^{\prime}}}^{+})-f(E_{k^{{}^{\prime}}}^{-})}{\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(k^{{}^{\prime}},\theta)}}\left(\begin{array}[]{ll}\emph{f}(\theta)&\emph{g}(\theta)\\\
\emph{g}(\theta)&\emph{h}(\theta)\end{array}\right)\left(\begin{array}[]{l}\Delta_{0}\\\
\Delta_{2}\end{array}\right)(k^{{}^{\prime}}),$
with the following axi-symmetric quantities,
$\displaystyle D^{2}(k,\theta)$
$\displaystyle=\Delta_{0}^{2}(k)+\sqrt{2}\Delta_{0}(k)\Delta_{2}(k)[3\cos^{2}\theta-1]+\Delta_{2}^{2}(k)[\frac{3\cos^{2}\theta+1}{2}],$
$\displaystyle E^{\pm}_{k}$
$\displaystyle=\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(k,\theta)}\pm\delta\varepsilon_{\textbf{k}}.$
(47)
The angle matrix
$\left(\begin{array}[]{ll}\emph{f}(\theta)&\emph{g}(\theta)\\\
\emph{g}(\theta)&\emph{h}(\theta)\end{array}\right)$ comes from the coupling
among the different $m_{j}$ components of $\Delta(\textbf{k})$. The matrix
elements are
$\displaystyle\emph{f}(\theta)$
$\displaystyle=Tr[G_{0}^{0*}(\textbf{\^{k}}^{{}^{\prime}})G_{0}^{0}(\textbf{\^{k}}^{{}^{\prime}})]=\frac{1}{4\pi},$
$\displaystyle\emph{g}(\theta)$
$\displaystyle=-Tr[G_{0}^{0*}(\textbf{\^{k}}^{{}^{\prime}})G_{2}^{0}(\textbf{\^{k}}^{{}^{\prime}})]=\frac{\sqrt{2}}{8\pi}(3\cos^{2}\theta-1),$
$\displaystyle\emph{h}(\theta)$
$\displaystyle=Tr[G_{2}^{0*}(\textbf{\^{k}}^{{}^{\prime}})G_{2}^{0}(\textbf{\^{k}}^{{}^{\prime}})]=\frac{1}{8\pi}(3\cos^{2}\theta+1).$
(48)
As a first inspection, when applying following the substitution [both in the
gap equations (30) and the expression of $E_{k}^{\pm}$ in Eq.(31)]
$\displaystyle\frac{3\cos^{2}\theta}{8\pi}\rightarrow\frac{1}{8\pi},$ (49)
which has been used as the angle-averaging procedure for ${}^{3}PF_{2}$
superfluidity in Ref.3pf2 , Eq.(30) reduces to the form of angle-averaged gap
Eq.(24). At zero temperature, the pairing is suppressed by the gapless
excitation near the average Fermi surface in the AAG state. However, pairing
can exist in the interval $(0,\theta_{1})\bigcup(\pi,\pi-\theta_{1})$ of
$\theta$ near the average Fermi surface in the ADG state, where
$\displaystyle\cos^{2}\theta_{1}=\frac{\delta\mu^{2}-\Delta_{0}^{2}(k_{F})+\sqrt{2}\Delta_{0}(k_{F})\Delta_{2}(k_{F})-\Delta_{2}^{2}(k_{F})/2}{3\sqrt{2}\Delta_{0}(k_{F})\Delta_{2}(k_{F})+3\Delta_{2}^{2}(k_{F})/2}$
and $\delta\mu$ is the difference between the neutron and proton chemical
potentials. This mechanism is consistent with that of the FFLO state.
Furthermore, the influences from the coupling of different $m_{j}$ components
are partially taken into account via the angle matrix
$\left(\begin{array}[]{ll}\emph{f}(\theta)&\emph{g}(\theta)\\\
\emph{g}(\theta)&\emph{h}(\theta)\end{array}\right)$ in the ADG state.
### II.3 Thermodynamics
For isospin-asymmetric nuclear matter at a fixed temperature and given neutron
and proton densities, the essential quantity to describe the thermodynamics of
the system is the free energy defined as
$\displaystyle\emph{F}|_{\rho,\beta}=\emph{U}-\beta^{-1}\emph{S},$ (50)
where U is the internal energy and S is the entropy. In the mean-field
approximation, the entropy of the superfluid state is
$\displaystyle\emph{S}=-2k_{B}\sum_{\textbf{k}}$
$\displaystyle\\{f(E_{\textbf{k}}^{+})\ln
f(E_{\textbf{k}}^{+})+\bar{f}(E_{\textbf{k}}^{+})\ln\bar{f}(E_{\textbf{k}}^{+})$
(51) $\displaystyle+f(E_{\textbf{k}}^{-})\ln
f(E_{\textbf{k}}^{-})+\bar{f}(E_{\textbf{k}}^{-})\ln\bar{f}(E_{\textbf{k}}^{-})\\},$
where $\bar{f}(E_{\textbf{k}}^{\pm})=1-f(E_{\textbf{k}}^{\pm})$. The internal
energy of the superfluid state reads
U
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]$
(52)
$\displaystyle+\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\nu^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\nu_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}}).$
The first term in Eq.(36) includes the kinetic energy of the quasiparticles
which is a functional of the pairing gap. In the normal state it reduces to
the kinetic energy of the neutrons and protons. The second term includes the
BCS mean-field interaction among the particles in the condensate and can be
eliminated in terms of the gap equation (9) (shown in Appendix). Finally, the
internal energy is written as
U
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]$
(53)
$\displaystyle-\sum_{\textbf{k}}\frac{D^{2}(\textbf{k})}{\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$
A thermodynamically stable state minimizes the difference of the free energies
between the superconducting and normal states,
$\delta\emph{F}=\emph{F}_{S}-\emph{F}_{N}$ [the free energy in the normal
state follows from Eqs.(35) and (37) when $\Delta\rightarrow 0$].
## III Results
The numerical calculations here focus on the effects of the angle dependence
of the quasiparticle spectra and the emergence of the ADG phase in isospin-
asymmetric nuclear matter. To simplify the calculations, several assumptions
have been adopted. Firstly, the pairing interaction is approximated by the
bare interaction; i.e., the effects of the screening of the pairing
interaction are ignored. Secondly, we adopt the free single particle (s.p.)
spectrum, which may affect the density of the states at the Fermi surface.
Previous calculations bhf1 ; bhf2 show that using a more realistic s.p.
spectrum obtained from the BHF approach (the BHF spectrum) may reduce the
${}^{3}SD_{1}$ channel pairing gap as compared with the free spectrum. As for
the pairing interaction, the screening potential (i.e., the higher-order
contribution in the pairing interaction) for the ${}^{3}S_{1}$ pairing channel
in nuclear matter under different approximations has been discussed in
Refs.scr1 . It has been shown the screening potential is repulsive at low
densities in the one-bubble approximation, whereas it is slightly attractive
in the full RPA (suitably renormalized to cure the low density mechanical
instability of nuclear matter scr1 ; scr2 ). Up to now, the screening effect
on the pairing gap remains an open problem. Finally, we ignore the isospin
triplet states, which is valid when the pairing in the isospin singlet channel
is much larger than that in the isospin triplet channel. However, the argument
could be questionable when the first two approximations are abandoned. In the
present calculations, the net density is fixed at the empirical saturation
density of nuclear matter $\rho=\rho_{0}=0.17fm^{-3}$ except for Fig.7, and
the Argonne $V_{18}$ potential is adopted as the pairing interaction.
Figure 1: (Color online). The upper and lower curves in the figures are
related to the values of $\Delta_{0}(k_{F})$ and $\Delta_{2}(k_{F})$ vs
isospin-asymmetry $\alpha$. The blue solid and red dashed lines correspond to
the ADG and angle-averaged gap, respectively.
Fig.1 shows the angle-averaged and angle dependent gaps $\Delta_{0}(k_{F})$
and $\Delta_{2}(k_{F})$ in the ${}^{3}SD_{1}$ partial-wave channel as a
function of isospin-asymmetry $\alpha$, defined as
$\alpha=(\rho_{n}-\rho_{p})/\rho$. The temperatures are set at low-temperature
regime $\beta^{-1}=$ $0.5$ MeV, $1.0$ MeV, $2.0$ MeV, $3.0$ MeV (the critical
temperature $\beta^{-1}_{c}$ where the superfluid vanishes is about $7.5$ MeV
for isospin-symmetric case). At temperature $\beta^{-1}=0.5$ MeV, the value of
$\Delta_{0}(k_{F})$ in the ADG state becomes larger than that of the angle-
averaged gap state for $\alpha\geq 0.07$, and the difference of
$\Delta_{0}(k_{F})$ between the ADG and angle-averaged gap states reaches 22
percent at $\alpha=0.23$. With increasing temperature, the difference of
$\Delta_{0}(k_{F})$ between the two kinds of states decreases rapidly. The
critical isospin-asymmetries $\alpha_{c}$ at which the gaps vanish are the
same in the two states, and their values are $0.267$, $0.275$, $0.30$ and
$0.315$ for the temperatures $0.5$ MeV, $1.0$ MeV, $2.0$ MeV and $3.0$ MeV,
respectively. It implies that the thermal excitation can promote pairing in
large isospin-asymmetry nuclear matter at low temperature regime.
In order to have an entire inspection of the difference between the pairing
gaps of the ADG state and the angle-averaged gap state, we exhibit the gap
functions in Fig.2.
Figure 2: (Color online). The curves marked with symbols ${}^{3}S_{1}$ and
${}^{3}D_{1}$ are related to the gap functions $\Delta_{0}(k)$ and
$\Delta_{2}(k)$ in Eqs.(24) and (30). The blue solid and red dashed lines
correspond to the ADG and angle-averaged gap, respectively.
At temperature $\beta^{-1}=0.5$ MeV, the gap functions of the two different
kinds of states are almost the same except a little difference of
$\Delta_{0}(k)$ near the zero momentum for the asymmetry $\alpha=0.02$ [in
Fig.2.(a)]. When the system becomes more asymmetric, the difference gets
larger [in Fig.2.(b)]. However, the curves of the ADG coincide with these of
the angle-averaged gap for $\beta^{-1}=3.0$ MeV with $\alpha=0.16$ [in
Fig.2.(d)]. That implies the angle-averaging procedure is a satisfactory
approximation for asymmetric nuclear matter at high temperatures.
Figure 3: (Color online). The difference of the free energy between the
superconducting and normal states as a function of the isospin-symmetry
$\alpha$ for diffderent temperatures. The blue solid and red dashed lines
correspond to the ADG and angle-averaged gap, respectively.
A larger gap value in ADG state may result in a larger pairing energy in the
condensate [second term in Eqs.(36) and (37)], which has important influence
on the free energy of the superconducting state. Thus we calculate the free-
energy difference $\delta\emph{F}$ between the normal and superconducting
states. The results are shown in Fig.3, where the parameters are set as the
same as those in Fig.1. At temperature $\beta^{-1}=0.5$ MeV, $\delta F$ in the
ADG state gets smaller than that of the angle-averaged gap state when
$\alpha\geq 0.06$, especially, the former is about 35 percent lower than the
latter in the regime $\alpha>0.17$. We can conclude that the ADG state is more
favored than the angle-averaged gap state for large asymmetry at low
temperature, since the angle dependence of the pairing gap enhances the
pairing energy and has little effect on the kinetic energy. However, the
thermal excitation can reduce the effects of angle dependence of the pairing
gap [comparing the Fig.3.(a) with Fig.3.(d)]. It is also shown in Fig.3 that
the values of $\delta F$ tend to zero gently when
$\alpha\rightarrow\alpha_{c}$ at different temperatures.
Figure 4: (Color online). The higher and lower curves in the upper two figures
are related to the neutron and proton occupation probabilities, respectively.
The curves in the lower two figures are the pairing probabilities. The blue
solid and red dashed lines correspond to the ADG and angle-averaged gap,
respectively.
One straightforward way to understand the effects of angle dependence of the
pairing gap is to investigate the normal and superconducting occupation
probabilities [obtained from Eqs.(14) and (15)] near the average Fermi surface
(related to the average chemical potential of neutron and proton). The results
are depicted in Fig.4, where the spin summation has been carried out. In this
figure, the neutron/proton and pairing particle occupation probabilities at
the average Fermi surface for a fixed asymmetry $\alpha=0.16$ at temperature
$\beta^{-1}=$ $0.5$ MeV has been compared with those at $3.0$ MeV. In isospin-
asymmetric nuclear matter, the large splitting between the neutron and proton
occupation probabilities prevents the pairing around the average Fermi surface
in the angle-averaging procedure. However, in the ADG state, the splitting is
reduced by the angle dependence of the pairing gap in partial area around the
average Fermi surface, i.e., in the regime
$\theta\subset(0,\frac{\pi}{5})\cup(\frac{4\pi}{5},\pi)$ as shown in
Fig.4.(a). In Fig.4.(c), as compared with the angle-averaged gap, although the
pairing in the ADG state is almost fully suppressed in the regime
$\theta\subset(\frac{\pi}{5},\frac{4\pi}{5})$ in ADG state, it is obviously
enhanced at $\theta$ smaller than $\frac{\pi}{5}$ and greater than
$\frac{4\pi}{5}$.
Substituting the expression of $D^{2}(\textbf{k})$ in Eq.(31) into Eq.(14), we
can find that the Fermi spheres of neutron and proton are no longer isotropic
in the ADG state. Since we assume an axi-symmetric quasiparticle spectrum in
the ADG state, the rotational symmetry is spontaneously broken [in terms of
group theory the $O(3)$ symmetry breaks down to $O(2)$] and there exists one
favored direction. The neutron Fermi sphere possesses an oblate deformation
perpendicular to the favored direction, whereas the proton Fermi sphere has a
prolate deformation along the favored direction. The two different
deformations enhance the correlation between neutrons and protons near their
average Fermi surface. However, at high temperature the neutron/proton
occupation probability in the ADG becomes almost isotropic as shown in
Fig.4.(b), namely, the thermal excitation reduces the angle dependence of
quasiparticle spectra. In this case, the deformation of the neutron/proton
Fermi sphere fails to increase the phase-space overlap of neutron and proton
near their average Fermi surface effectively. Thus the results of the ADG
state are nearly the same as that of the angle-averaged gap state, i.e., the
angle-averaging procedure becomes an adequate approximation at high
temperatures $\beta^{-1}\geq 3$ MeV.
Figure 5: (Color online). The entropy (scaled by $\beta^{-1}$) as a function
of the isospin-symmetry $\alpha$ for different temperature. The blue solid,
red dashed and black dash-dotted lines correspond to the ADG, angle-averaged
gap and normal state, respectively.
Fig.5 displays the entropy ($\beta^{-1}S$) as a function of isospin-asymmetry
$\alpha$ for different temperatures $\beta^{-1}=$ $0.5$ MeV, $1.0$ MeV, $2.0$
MeV and $3.0$ MeV. The entropy in the superconducting state is smaller than
that in the normal state near $\alpha=0$, and gets larger than that in the
normal state at sufficiently large asymmetry. However, around the transition
point $\alpha_{c}$ from the superconducting state to the normal state, the
entropies of the superconducting states (both of the ADG state and the angle-
averaged gap state) approach to the value of the normal state, i.e., the
latent heats $Q=\beta^{-1}(S_{s}-S_{n})\rightarrow 0$ when
$\alpha\rightarrow\alpha_{c}$. Hence the transitions are of second order. At
temperature $\beta^{-1}=$ $0.5$ MeV, the entropy in the ADG state is nearly a
linear function of the isospin-asymmetry when $0.02<\alpha<0.22$. With
increasing temperature, the linear property of the entropy curve disappears
and the difference between the ADG state and the angle-averaged gap state gets
smaller.
Figure 6: (Color online). $\Delta_{0}(k_{F})$ and $\Delta_{2}(k_{F})$ as a
function of isospin-asymmetry $\alpha$ for the ADG, angle-averaged gap and
`approximation in ADG' are shown in Fig.(a). Fig.(b) exhibits the gap
functions for the three case. The normal and superconducting occupation
probabilities at the average Fermi surface for the three case are shown in
Figs.(c) and (d), respectively. The blue solid, red dashed and green dash-
dotted lines correspond to the ADG, angle-averaged gap and the `approximation
in ADG', respectively. The temperature is set to be $\beta^{-1}=0.5$ MeV, and
the isospin-asymmetry $\alpha=0.16$ in (b), (c), (d).
Comparing the gap equations (30) for the ADG state with Eq.(24) for the angle-
averaged gap state, two differences appear in the ADG state, i.e., the angle
dependent quasiparticle spectrum and the angle matrix
$\left(\begin{array}[]{ll}\emph{f}(\theta)&\emph{g}(\theta)\\\
\emph{g}(\theta)&\emph{h}(\theta)\end{array}\right)$. The first leads to the
deformation of neutron/proton Fermi sphere, and the second corresponds to the
coupling among different $m_{j}$ gap components. Actually, the angle matrix
modifies the strength of $V^{l^{{}^{\prime}}l}_{\lambda}(k^{{}^{\prime}},k)$
in different directions in momentum space. We replace the angle matrix by
$\frac{1}{4\pi}\left(\begin{array}[]{ll}1&0\\\ 0&1\end{array}\right)$ to
inspect the influence of the angle matrix. The results are shown in Fig.6 for
asymmetry $\alpha=0.16$ in (b), (c), (d) and the temperature is set to be
$\beta^{-1}=0.5$ MeV. The dash-doted lines denoted by `approximation in ADG'
are obtained by replacing the angle matrix with
$\frac{1}{4\pi}\left(\begin{array}[]{ll}1&0\\\ 0&1\end{array}\right)$.
Figs.6.(c) and (d) exhibit the neutron/proton and pairing particle occupation
probabilities at the average Fermi surface, respectively. The curves of ADG
and `approximation in ADG' are nearly the same in Figs.6.(c) and (d). Whereas
the gap functions in Fig.6.(b) show that the curves of `approximation in ADG'
behave closer to those of the angle-averaged gap state than those of the ADG
state. Fig.6.(a) displays the $\Delta_{0}(k_{F})$ and $\Delta_{2}(k_{F})$ vs
isospin-asymmetry $\alpha$. The gaps of `approximation in ADG' turn out to be
smaller than both the gaps in the ADG state and angle-averaged gap state when
$\alpha>0.07$. Moreover, the curves of `approximation in ADG' are much closer
to that of the angle-averaged gap state. All these results indicate that the
influence of the angle matrix is much more important than that of the angle
dependence of quasiparticle spectrum. Furthermore, the coupling from different
$m_{j}$ gap components may strengthen the pairing interaction for large
isospin-asymmetry at low temperatures.
Figure 7: (Color online). The difference of the free energy between the
superconducting and normal states as a function of the isospin-symmetry
$\alpha$ for different densities at a fixed temperature $\beta^{-1}=0.5$ MeV.
The blue solid and red dashed lines correspond to the ADG and angle-averaged
gap, respectively.
In order to discuss the effect of angle dependence of the pairing gap for
different densities, we show the free energy difference $\delta\emph{F}$
between the superconducting and normal states at temperature $\beta^{-1}=0.5$
MeV vs isospin-asymmetry $\alpha$ in Fig.7. The densities are set to be
$\rho=0.05$, $0.1$, $1.5\rho_{0}$ and $2\rho_{0}$ for (a), (b), (c) and (d),
respectively. At the density $\rho=0.05$ [in Fig.7.(a)], the two curves of
$\delta\emph{F}$ for the ADG and angle-averaged gap states are very close to
each other, indicating the effect of angle dependence of the pairing gap is
quite small at low densities. When the density increases, the difference of
$\delta\emph{F}$ for the ADG and angle-averaged gap states increases rapidly,
implying that the angle dependence of the pairing gap is more important at
higher densities. As the Fermi energy $\emph{E}_{F}\propto\rho^{\frac{2}{3}}$,
the value of $\frac{\Delta}{\emph{E}_{F}}$ is thus small at high densities. In
this case, the summations over ${k}^{{}^{\prime}}$ in the gap equation (9)
concentrate near the average Fermi surface (i.e., the contribution to
superfluidity from the Cooper pairs around the average Fermi surface is
dominant). A little separation of the neutron and proton Fermi surfaces
$\delta\mu$ may suppress the superfluidity strongly. In the ADG configuration,
the angle dependence can reduce the suppression. However, at low densities,
the value of $\frac{\Delta}{\emph{E}_{F}}$ gets large. Thus the contribution
to superfluidity from the Cooper pairs near the average Fermi surface is no
longer as important as that at high densities. Since the angle dependence
mainly increases the pairing probability around the average Fermi surface, the
effect of the angle dependence becomes weak at low densities.
## IV Summary and Outlook
The fermionic condensation in asymmetric nuclear matter leads to
superconducting states which spontaneously break the spatial symmetries (such
as FFLO and DFS states). The quasiparticle spectrum behaves as an isotropic
one and the angle dependence of the pairing gap should be reconsidered. In
this work we propose an axi-symmetric angle dependent gap state in which the
isotropic symmetry is broken in isospin-asymmetric nuclear matter, and compare
with the angle-averaged gap state. It is shown the ADG state is more favored
than the angle-averaged gap state for large asymmetry at low temperature, and
the differences of both the gap values and the free energies between the two
kinds of states get small with increasing temperature. At temperature
$\beta^{-1}=0.5$ MeV with density $\rho_{0}$, the maximal differences of
$\Delta_{0}(k_{F})$ and $\delta\emph{F}$ between the ADG state and angle-
averaged gap state are about 22 and 35 percent, respectively. The differences
get larger at higher densities for $\beta^{-1}=0.5$ MeV. In the ADG state, the
neutron and proton deformed Fermi spheres increase the pairing probability
along the axis of symmetry breaking near their average Fermi surface. The
effect of the coupling among different $m_{j}$ gap components is also
investigated in this work and we find the coupling dominates the main
contribution to the mechanism of the ADG state.
The ADG state vanishes at the critical value $\alpha_{c}$, where the angle-
averaged gap vanishes. And the phase transition from the ADG state to the
normal state is of the second order. When temperature goes up, $\alpha_{c}$
rises and the effect of angle dependence of pairing gap becomes weak. In a
certain region of $\alpha$ the latent heat has an anomalous negative sign,
which is consistent with the result if Ref.Sdd2 . However, this does not
affect the stability of the ADG state, since its energy budget is dominated by
the pair-condensation energy.
In the ADG state, the symmetry is broken spontaneously. It is different from
that in the FFLO state, where the symmetry is broken by the collective motion
of the cooper pairs (the translation and rotational symmetries are both
broken). The translation symmetry is maintained in the ADG state. The
deformation of the neutron/proton Fermi sphere in the ADG state is similar to
the DFS configuration, however, the mechanisms are different. In the DFS state
the symmetry breaking corresponds to the deformed Fermi surface, while in the
ADG state the symmetry breaking results from the angle dependence of the
pairing gap. As is well known, the continuous symmetry breaking leads to
collective excitations with vanishing minimal frequency (Goldstone's theorem).
The breaking of rotational symmetry, which corresponds to the anisotropic
$D^{2}(\textbf{k})$ in the ADG state, may imply new collective bosonic modes
in asymmetric nuclear matter. However, the true ground state could be a
combination of the ADG state and the FFLO state, we should consider the ADG
state with the cooper pair momentum together which is in progress.
## Acknowledgments
The work is supported by the 973 Program of China under No. 2013CB834405, the
National Natural Science Foundation of China (No. 11175219), and the Knowledge
Innovation Project(No. KJCX2-EW-N01) of the Chinese Academy of Sciences.
## Appendix
We present here the main steps of the elimination of the second term in
Eq.(36) by using the gap equation (9). The elements of the density matrix of
the particles in condensate are,
$\displaystyle\nu_{\sigma_{1},\sigma_{2}}(\textbf{k})=\frac{\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$
(54)
The second term of Eq.(36) is written as
$\displaystyle\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\nu^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\nu_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})$
$\displaystyle=\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\frac{\Delta^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})]$
$\displaystyle\times\frac{\Delta_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})}{2\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(\textbf{k}^{{}^{\prime}})}}[1-f(E_{\textbf{k}^{{}^{\prime}}}^{+})-f(E_{\textbf{k}^{{}^{\prime}}}^{-})]$
$\displaystyle=\sum_{\textbf{k},\sigma_{1},\sigma_{2}}\frac{\Delta^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})]$
$\displaystyle\times\sum_{\textbf{k}^{{}^{\prime}},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\frac{\Delta_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})}{2\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(\textbf{k}^{{}^{\prime}})}}[1-f(E_{\textbf{k}^{{}^{\prime}}}^{+})-f(E_{\textbf{k}^{{}^{\prime}}}^{-})].$
(55)
Noting that the second summation over
${k}^{{}^{\prime}},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}$ is
$-\Delta_{\sigma_{2},\sigma_{1}}$ (using the gap equation (9)), thus
$\displaystyle\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\nu^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\nu_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})$
$\displaystyle=-\sum_{\textbf{k},\sigma_{1},\sigma_{2}}\frac{\Delta^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})]$
$\displaystyle=-\sum_{\textbf{k}}\frac{Tr[\Delta(\textbf{k})\Delta^{{\dagger}}(\textbf{k})]}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$
(56)
Using the ``unitary" property Eq.(4),
U
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]$
(57)
$\displaystyle+\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\nu^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\nu_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})$
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]-\sum_{\textbf{k}}\frac{2D^{2}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})]$
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]-\sum_{\textbf{k}}\frac{D^{2}(\textbf{k})}{\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$
## References
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|
arxiv-papers
| 2013-05-31T15:34:52 |
2024-09-04T02:49:45.942493
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xinle Shang and Wei Zuo",
"submitter": "Xinle Shang",
"url": "https://arxiv.org/abs/1305.7456"
}
|
1305.7489
|
[by]Jianxin Chen, Joel Klassen, and Bei ZengSimone Severini (chair) & Fernando
Brandao (co-chair)2The 8th Conference on the Theory of Quantum Computation,
Communication and Cryptography111 10.4230/LIPIcs.xxx.yyy.p
# Universal Entanglers for Bosonic and Fermionic Systems111This work was
partially supported by NSERC and CIFAR
Joel Klassen Department of Physics, University of Guelph
50 Stone Road East, Guelph, Ontario, Canada
[email protected] Institute for Quantum Computing
200 University Avenue West, Waterloo, Ontario, Canada Jianxin Chen
Department of Mathematics & Statistics, University of Guelph
50 Stone Road East, Guelph, Ontario, Canada
{jianxinc,zengb}@uoguelph.ca Institute for Quantum Computing
200 University Avenue West, Waterloo, Ontario, Canada UTS-AMSS Joint Research
Laboratory for Quantum Computation and Quantum Information Processing
Academy of Mathematics and Systems Science, Chinese Academy of Sciences,
Beijing, China Bei Zeng Department of Mathematics & Statistics, University
of Guelph
50 Stone Road East, Guelph, Ontario, Canada
{jianxinc,zengb}@uoguelph.ca Institute for Quantum Computing
200 University Avenue West, Waterloo, Ontario, Canada
###### Abstract.
A universal entangler (UE) is a unitary operation which maps all pure product
states to entangled states. It is known that for a bipartite system of
particles $1,2$ with a Hilbert space
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$, a UE exists when
$\min{(d_{1},d_{2})}\geq 3$ and $(d_{1},d_{2})\neq(3,3)$. It is also known
that whenever a UE exists, almost all unitaries are UEs; however to verify
whether a given unitary is a UE is very difficult since solving a quadratic
system of equations is NP-hard in general. This work examines the existence
and construction of UEs of bipartite bosonic/fermionic systems whose wave
functions sit in the symmetric/antisymmetric subspace of
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. The development of a theory of UEs for
these types of systems needs considerably different approaches from that used
for UEs of distinguishable systems. This is because the general entanglement
of identical particle systems cannot be discussed in the usual way due to the
effect of (anti)-symmetrization which introduces “pseudo entanglement” that is
inaccessible in practice. We show that, unlike the distinguishable particle
case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are
symmetric (resp. antisymmetric) subspaces of
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ if and only if $d\geq 3$ (resp. $d\geq
8$). To prove this we employ algebraic geometry to reason about the different
algebraic structures of the bosonic/fermionic systems. Additionally, due to
the relatively simple coherent state form of unentangled bosonic states, we
are able to give the explicit constructions of two bosonic UEs. Our
investigation provides insight into the entanglement properties of systems of
indisitinguishable particles, and in particular underscores the difference
between the entanglement structures of bosonic, fermionic and distinguishable
particle systems.
###### Key words and phrases:
Universal Entangler, Bosonic States, Fermionic States
###### 1991 Mathematics Subject Classification:
J.2 Physics
## 1\. Introduction
Entanglement sits at the core of the counterintuitive and useful properties of
quantum mechanics. At its inception Schrödinger labeled entanglement “the
characteristic trait of quantum mechanics, the one that enforces its entire
departure from classical lines of thought.” [19] This observation remains true
today, and with the advent of quantum computing, its practical consequences
have never before been more real. However after decades of effort,
entanglement remains poorly understood [13, 1, 11]. A promising avenue for
furthering our understanding of entanglement is cataloguing and analyzing the
various means of generating it. There is a sense that those mechanisms which
generate maximal amounts of entanglement, or most consistently generate
entanglement, are especially enlightening because they serve as bounds on what
can and can not be done, thus restricting our domain of inquiry.
One outcome of this line of thought is the concept of a universal entangler
(UE). A UE is a unitary operator which maps any non-entangled state to an
entangled state [3]. A UE can act as a useful tool, both theoretically and
experimentally, due to its generality. This generality is derived from the
fact that a UE admits any non-entangled quantum states. However this
generality also makes demonstrating the properties of UEs very difficult. For
instance, while it has been shown that UEs do exist for a system with Hilbert
space $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ when
$\min{(d_{1},d_{2})}\geq 3$ and $(d_{1},d_{2})\neq(3,3)$, proving this fact
has been nontrivial, requiring techniques from algebraic geometry [3]. To date
no elementary method is known which can achieve the same results.
Additionally, although it has been shown that whenever UEs exist almost all
unitaries are UEs [4], explicit constructions of UEs remain ellusive. This is
due to the fact that the problem of verifying whether a given unitary is a UE
is in general intractable since the verification is equivalent to solving a
quadratic system of equations which is hard in general [6]. So far the only
explicitly known UE is an example for the $(d_{1},d_{2})=(3,4)$, from an order
$12$ Hadamard matrix [4]. In general more advanced methods may be needed in
order to construct UEs, as well as to verify their universality.
The theory of entanglement of systems of indistinguishable particles has
garnered much attention during the past decade [17, 16, 12, 14, 5, 1]. The
entanglement of systems of indistinguishable particles cannot necessarily be
approached in the same way as the distinguishable particle case because the
symmetry requirement of the wave functions (i.e. symmetrization for bosonic
system and antisymmetrization for fermionic system) may introduce ‘pseudo
entanglement’ which is not accessible in practice [5, 16, 17, 12, 14]. It is
now widely agreed that non-entangled states correspond to the coherent states
$|v\rangle^{\otimes N}$ [15] for indistinguishable bosonic systems and to
Slater determinants for indistinguishable fermionic systems [5, 1]. A natural
line of inquiry is to identify the existence and construction of UEs for
systems of indistinguishable particles. Indistinguishable bipartite
bosonic/fermionic states are symmetric/antisymmetric states of the Hilbert
space $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. This does not necessarily mean
that the theory of UEs of distinguishable particles is readily generalizable
to UEs for indistinguishable particles. Some obvious reasons for this are: 1.
although almost all unitaries are UEs when $d>3$, the lack of understanding of
explicit constructions prevents us from directly verifying whether there exist
any UEs which are symmetric under particle permutation; 2. the definition of a
non-entangled state for systems of indistinguishable fermions is dramatically
different from that of systems of distinguishable particles (in fact, a single
Slater determinant, when viewed as an antisymmetric distinguishable particle
state, is indeed entangled).
This paper discusses the existence and construction of UEs for both
indistinguishable bipartite bosonic (BUE) and fermionic (FUE) systems.
Employing techniques in algebraic geometry, considering the different
algebraic structures of the bosonic and fermionic systems, we show that, in
contrast to the distinguishable particle case, BUEs exist for bosonic systems
if and only if the single particle Hilbert space has dimension $d\geq 3$, and
FUEs exist for fermionic systems if and only if the single particle Hilbert
space has dimension $d\geq 8$. We also show, similarly to the distinguishable
particle case, that for dimensions where BUEs/FUEs exist, almost all unitaries
are BUEs/FUEs. Finally, because the unentangled states of indistinguishable
bosonic systems are of a relatively simple coherent state form
$|v\rangle\otimes|v\rangle$, which implies a hidden linear structure for the
product states (i.e. the set of all single particle states $|v\rangle$ form a
vector space), the construction of BUEs becomes significantly simpler. We have
found a simple explicit construction of a BUE based on permutation matrices
which holds for all $d\geq 3$, and another one based on Householder-type gates
[10] which holds for all $d\geq 5$. Unfortunately the explicit construction
and verification of FUEs, like distinguishable particle UEs, remains a
significantly more intractable problem.
We believe that our investigation provides insight into the entanglement
properties of identical particle systems, and in particular the different
entanglement structures between bosonic, fermionic and distinguishable
particle systems.
We organize our paper as follows. In section 2 we review some previously
established results about UEs and provide some preliminaries about bosonic and
fermionic systems to help establish our main results. In section 3 we give a
proof for the existence and prevalence of BUEs, and give two explicit examples
of their construction. In Section 4 we give a proof for the existence and
prevalence of FUEs. Finally, in section 5, we provide a brief summary of our
results and a discussion of future directions.
## 2\. Preliminaries
This section provides preliminaries to help establish our main results for
BUEs and FUEs. We first briefly review UEs for distinguishable particle
systems established in [3]. We then further briefly review basic entanglement
theory for bosonic and fermionic systems.
### 2.1. Universal entanglers
For the case of distinguishable particles, it is known that any given quantum
system is identified with some finite (or infinite) Hilbert space
$\mathcal{H}$. Moreover, two unit vectors are indistinguishable if they differ
only by a global phase factor. Hence, distinct pure states can be put in
correspondence with “rays” in $\mathcal{H}$, or equivalently, points in the
projective Hilbert space $\mathbb{P}(\mathcal{H})$.
We consider pure states for bipartite systems, whose Hilbert space is
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$. A bipartite quantum state is a
product state if $|\psi\rangle=|\psi_{1}\rangle\otimes|\psi_{2}\rangle$ for
some $|\psi_{1}\rangle\in\mathbb{C}^{d_{1}}$ and
$|\psi_{2}\rangle\in\mathbb{C}^{d_{2}}$. Otherwise, it is an entangled state.
It is straightforward to see that the set of all the product states do not
form a linear vector space, so one does not expect that the UE problem can be
examined using basic tools from linear algebra.
Instead, it is observed that the set of normalized product states in a
composite system associated with $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$
is isomorphic to a projective variety in $\mathbb{P}^{d_{1}d_{2}-1}$, a well
studied object in algebraic geometry. Before continuing, we need some basic
notations and necessary background materials from algebraic geometry [8].
For any positive integer $n$, the set of all $n$-tuples from $\mathbb{C}$ is
called an $n$-dimensional affine space over $\mathbb{C}$. An element of
$\mathbb{C}^{n}$ is called a point, and if point
$P=(a_{1},a_{2},\cdots,a_{n})$ with $a_{i}\in\mathbb{C}$, then the $a_{i}$’s
are called the coordinates of $P$. Informally, an affine space is what is left
of a vector space after forgetting its origin.
We define projective $n$-space, denoted by $\mathbb{P}^{n}$, to be the set of
equivalence classes of $(n+1)-$tuples $(a_{0},\cdots,a_{n})$ from
$\mathbb{C}$, not all zero, under the equivalence relation given by
$(a_{0},\cdots,a_{n})\sim(\lambda a_{0},\cdots,\lambda a_{n})$ for all
$\lambda\in\mathbb{C}$, $\lambda\neq 0$. We use $[a_{0}:\cdots:a_{n}]$ to
denote the projective coordinates of this point.
The polynomial ring in $n$ variables, denoted by
$\mathbb{C}[x_{1},x_{2},\cdots,x_{n}]$, is the set of polynomials in $n$
variables with coefficients in field $\mathbb{C}$.
A subset $Y$ of $\mathbb{C}^{n}$ is an algebraic set if it is the common zeros
of a finite set of polynomials $f_{1},f_{2},\cdots,f_{r}$ with
$f_{i}\in\mathbb{C}[x_{1},x_{2},\cdots,x_{n}]$ for $1\leq i\leq r$, which is
also denoted by $Z(f_{1},f_{2},\cdots,f_{r})$.
One may observe that the union of a finite number of algebraic sets is an
algebraic set, and the intersection of any family of algebraic sets is again
an algebraic set. Therefore, by taking the open subsets to be the complements
of algebraic sets, we can define a topology, called the Zariski topology on
$\mathbb{C}^{n}$.
A nonempty subset $Y$ of a topological space $X$ is called irreducible if it
cannot be expressed as the union of two proper closed subsets. The empty set
is not considered to be irreducible.
An affine algebraic variety is an irreducible closed subset of
$\mathbb{C}^{n}$, with respect to the induced topology.
A notion of algebraic variety may also be introduced in projective spaces,
called projective algebraic variety: a subset $Y$ of $\mathbb{P}^{n}$ is an
algebraic set if it is the common zeros of a finite set of homogeneous
polynomials $f_{1},f_{2},\cdots,f_{r}$ with
$f_{i}\in\mathbb{C}[x_{0},x_{1},\cdots,x_{n}]$ for $1\leq i\leq r$. We call
open subsets of irreducible projective varieties quasi-projective varieties.
Observe that a product state in $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$
can be written as the Kronecker product of a vector
$v_{1}\in\mathbb{C}^{d_{1}}$ and another vector $v_{2}\in\mathbb{C}^{d_{2}}$.
Let’s further write these vectors in the computational basis, say
$v_{1}=\left(\begin{array}[]{cccc}x_{1},&x_{2},&\cdots&,x_{d_{1}}\end{array}\right)$
and
$v_{2}=\left(\begin{array}[]{cccc}y_{1},&y_{2},&\cdots&,y_{d_{2}}\end{array}\right)$.
Their product state is a $d_{1}d_{2}$-dimensional vector
$\displaystyle\left(\begin{array}[]{ccccccc}z_{1},&z_{2},&\cdots&z_{d_{2}},&z_{d_{2}+1},&\cdots&,z_{d_{1}d_{2}}\end{array}\right)$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccccc}x_{1}y_{1},&x_{1}y_{2},&\cdots&x_{1}y_{d_{2}},&x_{2}y_{1},&\cdots&,x_{d_{1}}y_{d_{2}}\end{array}\right)$
Hence $z_{(i-1)d_{2}+j}=x_{i}y_{j}$ for any $1\leq i\leq d_{1},1\leq j\leq
d_{2}$. It follows that
$\displaystyle z_{(i_{1}-1)d_{2}+j_{1}}z_{(i_{2}-1)d_{2}+j_{2}}$
$\displaystyle=$ $\displaystyle
z_{(i_{1}-1)d_{2}+j_{2}}z_{(i_{2}-1)d_{2}+j_{1}}$
for any $1\leq i_{1},i_{2}\leq d_{1},1\leq j_{1},j_{2}\leq d_{2}$. On the
other hand, any $d_{1}d_{2}$-dimensional vector $(z_{k})_{k=1}^{d_{1}d_{2}}$
satisfying the above polynomials can be written as the tensor product of
$v_{1}\in\mathbb{C}^{d_{1}}$ and $v_{2}\in\mathbb{C}^{d_{2}}$ [8]. This
implies that the set of normalized product states in
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ is isomorphic to a projective
variety in $\mathbb{P}^{d_{1}d_{2}-1}$ which is called a “Segre variety” and
denoted as $\Sigma_{d_{1},d_{2}}$. This simple observation provides an
algebraic geometric description of product states and entangled states.
Therefore, a unitary operator $U$ acting on
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ is a UE if and only
$U(\Sigma_{d_{1},d_{2}})\bigcap\Sigma_{d_{1},d_{2}}=\emptyset.$
From the geometric point of view, a UE will rotate the set of product states
to another set which is completely void of product states.
In [3], it is proved that UEs exist if and only if $\min\\{d_{1},d_{2}\\}\geq
3$ and $(d_{1},d_{2})\neq(3,3)$. Surprisingly, it is further illustrated that
a random unitary operator acting on such a bipartite system will even rotate
the set of product states to another set which contains nothing but nearly
maximally entangled states [4].
Although it has been shown that a random unitary gate will almost surely be a
UE of a bipartite quantum system $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$
if $\min\\{d_{1},d_{2}\\}\geq 3$ and $(d_{1},d_{2})\neq(3,3)$, constructing an
explicit UE for any bipartite quantum system is not that easy. One simple
strategy is to randomly pick a unitary gate acting on
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$, and then verify whether it is a
UE by solving a family of polynomial equations. Unfortunately, there is no
known efficient way to solve quadratic polynomial systems [6]. So far,
explicit UEs are only known for $(d_{1},d_{2})=(3,4)$ [4].
### 2.2. Bosonic systems
It is known that bosonic states lie in the $2$nd symmetric tensor power of
$\mathbb{C}^{d}$, denoted by $\vee^{2}\mathbb{C}^{d}$. A state in
$\vee^{2}\mathbb{C}^{d}$ is a product state if it can be written as some
$|\alpha\rangle\otimes|\alpha\rangle$, i.e. it is a coherent state [14, 5].
Any state which cannot be written as such a symmetric product form does
demonstrate correlation which can be potentially used in quantum information
processing [14], and hence is considered entangled.
Any bipartite bosonic pure state is local unitarily equivalent to
$\sum_{\alpha}\lambda_{\alpha}|\alpha\rangle\otimes|\alpha\rangle$ [12, 14].
This then indicates a hidden linear structure for bipartite bosonic pure
states because the single particles states $|\alpha\rangle$ form a vector
space.
From the algebraic geometric point of view, any bosonic product state
$|\alpha\rangle\otimes|\alpha\rangle$ can be written as a vector with
projective coordinates
$[a_{1}a_{1}:a_{1}a_{2}:\cdots:a_{1}a_{d}:a_{2}a_{1}:a_{2}a_{2}:\cdots:a_{2}a_{d}:a_{3}a_{1}:\cdots:a_{d}a_{d}]$
where $[a_{1}:\cdots:a_{d}]$ are the projective coordinates of
$|\alpha\rangle$.
Such points can be characterized by a family of polynomials again. In fact,
the set of projective points with coordinates
$[a_{1}a_{1}:a_{1}a_{2}:\cdots:a_{1}a_{d}:a_{2}a_{1}:a_{2}a_{2}:\cdots:a_{2}a_{d}:a_{3}a_{1}:\cdots:a_{d}a_{d}]$
is obviously isomorphic to the set of the following points
$[a_{1}^{2}:a_{2}^{2}:\cdots:a_{d}^{2}:a_{1}a_{2}:a_{1}a_{3}:\cdots:a_{1}a_{d}:a_{2}a_{3}:\cdots:a_{d-1}a_{d}]$
which is known as the Veronese variety in algebraic geometry [7].
Hence the set of bosonic product states corresponds to a special case of
Veronese variety whose dimension is $d-1$. This fact will be used in our
further investigation.
### 2.3. Fermionic systems
Consider the pure states of a bipartite fermionic system whose Hilbert space
is the antisymmetric subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$. The
Pauli exclusion principle requires that $d\geq 2$. We denote the $2$nd
exterior power of $\mathbb{C}^{d}$, i.e. the antisymmetric subspace of
$\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ by $\wedge^{2}\mathbb{C}^{d}$. For any
$|\alpha\rangle,|\beta\rangle\in\mathbb{C}^{d}$, we use the notation
$|\alpha\rangle\wedge|\beta\rangle=\frac{1}{\sqrt{2}}(|\alpha\rangle\otimes|\beta\rangle-|\beta\rangle\otimes|\alpha\rangle),$
(3)
to denote a single Slater determinant.
A quantum state $|\psi\rangle$ in $\wedge^{2}\mathbb{C}^{d}$ is said to be
decomposable if it can be written as an exterior product of individual vectors
from $\mathbb{C}^{d}$, i.e. there exists
$|\alpha\rangle,|\beta\rangle\in\mathbb{C}^{d}$ such that
$|\psi\rangle=|\alpha\rangle\wedge|\beta\rangle$. Decomposable states are
considered unentangled, as any correlation results purely from the fermionic
statistics, and so is not useful for quantum information processing [17, 16].
Any state which cannot be written in such a decomposable form does demonstrate
correlation which can be potentially used in quantum information processing
[17, 16] , and hence is considered to be entangled.
Any bipartite fermionic pure state is local unitarily equivalent to
$\sum_{\alpha}\lambda_{i}|\alpha_{i}\rangle\wedge|\beta_{i}\rangle$ [17, 16],
where $|\alpha_{i}\rangle,|\beta_{i}\rangle\in\mathbb{C}^{d}$,
$\langle{\alpha_{i}}|\beta_{j}\rangle=0$,
$\langle\alpha_{i}|\alpha_{j}\rangle=\delta_{ij}$, and
$\langle\beta_{i}|\beta_{j}\rangle=\delta_{ij}$. This is an analogue of the
Schmidt decomposition of a distinguishable particle system and hence is called
the Slater decomposition. Similarly to the distinguishable particle case, the
set of all decomposable states do not form a linear vector space, so one does
not expect that the FUE problem can be examined using basic tools from linear
algebra.
Again, let’s look over the decomposable (or fermionic product) states from the
algebraic geometric point of view. As we showed before, a decomposable state
can be written as $|\psi\rangle=|\alpha\rangle\wedge|\beta\rangle$ where
$|\alpha\rangle$ and $|\beta\rangle$ are two vectors in $\mathbb{C}^{d}$. Let
$S_{\psi}$ be the $2$-dimensional subspace spanned by $|\alpha\rangle$ and
$|\beta\rangle$. A different basis for $S_{\psi}$ will give a different
exterior product, but the two exterior products will differ only by a nonzero
scale. Ignoring the nonzero scale, any decomposable state corresponds to a
$2$-dimensional subspace in $\mathbb{C}^{d}$ and vice versa. Hence the set of
decomposable states is isomorphic to the set of $2$-dimensional subspaces
which is known as a Grassmannian $G(2,d)$ [7]. It is not that obvious that
$G(2,d)$ can be characterized by a set of polynomials, but it can be. The
correspondence we have just shown is known as the Plücker embedding of a
Grassmannian into a projective space:
$\tau:G(2,d)\rightarrow\mathbb{P}(\wedge^{2}\mathbb{C}^{d}).$
This embedding satisfies certain simple quadratic polynomials and is called
the Grassmann-Plücker relations (see e.g. p. A III.172 Eq. (84-(J,H)) in [2],
Prop 11-32 in [9], and [5]). This implies the Grassmannian embeds as an
algebraic variety of $\mathbb{P}(\wedge^{2}\mathbb{C}^{d})$.
## 3\. Bosonic Universal Entanglers
### 3.1. Existence and Prevalence
Recall that a bosonic state in $\vee^{2}\mathbb{C}^{d}$ is a product state if
it can be written as $|\alpha\rangle\otimes|\alpha\rangle$ for some
$|\alpha\rangle$. A quantum gate acting on $\vee^{2}\mathbb{C}^{d}$ is said to
be a bosonic universal entangler (BUE) if it will map every product state to
some entangled state.
Note that the set of product states of a bosonic system can also be
characterized by a set of polynomials. Indeed, let
$\Lambda=\\{|\alpha\rangle\otimes|\alpha\rangle:|\alpha\rangle\in\mathbb{C}^{d}\\}$,
this is a precisely the Veronese variety [7]. Furthermore, $\Lambda$ is
isomorphic to $\mathbb{C}^{d}$. For any
$|\psi\rangle\in\vee^{2}\mathbb{C}^{2}$, let us denote
$\text{rank}|\psi\rangle\equiv\min\\{r:|\psi\rangle=\sum\limits_{i=1}^{r}|a_{i}\rangle|a_{i}\rangle\\}$.
###### Theorem 3.1.
There is a BUE acting on $\vee^{2}\mathbb{C}^{d}$ if and only if $d\geq 3$.
Furthermore, when $d\geq 3$, almost every quantum gate acting on
$\vee^{2}\mathbb{C}^{d}$ is a BUE.
###### Proof 3.2.
For $d\leq 2$, we have
$\displaystyle\dim U(\Lambda)+\dim\Lambda=2\dim\Lambda=2(d-1)\geq{d+1\choose
2}-1=\dim\mathbb{P}(\vee^{2}\mathbb{C}^{d}).$ (4)
This implies there is no BUE for $\vee^{2}\mathcal{C}^{d}$. This assertion
follows from the dimension counting theorem which states that the intersection
of any two projective varieties $\mathcal{A}$ and
$\mathcal{B}\subseteq\mathbb{P}^{m}$ is nonempty if
$\dim\mathcal{A}+\dim\mathcal{B}\geq m$. More specifically, we have
$U(\Lambda)\bigcap\Lambda\neq\emptyset$.
On the other hand, consider the set of quantum gates acting on a system of two
indistinguishable bosons. Any quantum gate acting on this system should be a
symmetric gate, i.e., $SUS=U$, where $S$ is the swap operator. Equivalently,
$U$ is a quantum gate acting on $\vee^{2}\mathbb{C}^{d}$.
Let
$\mathcal{X}=\\{\Phi|\Phi\in\mathcal{U}(\vee^{2}\mathbb{C}^{d}),\Phi(\Lambda)\cap\Lambda\neq\emptyset\\}$.
Our aim is to show that $\mathcal{X}$ is a proper subset of
$\mathcal{U}(\vee^{2}\mathbb{C}^{d})$. If this is so, then the existence of
BUEs will be automatically guaranteed.
Let’s consider the Zariski topology on the projective space. In this setting,
the unitary group $\mathcal{U}(\vee^{2}\mathbb{C}^{d})$ is Zariski dense in
the general linear group $GL(\vee^{2}\mathbb{C}^{d})$ [18]. We further define
$\mathcal{X}^{\prime}=\\{\Phi|\Phi\in
GL(\vee^{2}\mathbb{C}^{d}),\Phi(\Lambda)\cap\Lambda\neq\emptyset\\}$. It is
easy to see that $\mathcal{X}\subseteq\mathcal{X}^{\prime}$.
The dimension of its Zariski closure $\dim\overline{\mathcal{X}^{\prime}}$ is
bounded by ${d+1\choose 2}^{2}-({d+1\choose 2}-1)+2(d-1)$. See Lemma C.1 in
Appendix C for details.
Now we prove the existence of a BUE as follows. If $U$ is not a BUE,
$\mathcal{U}(\vee^{2}\mathbb{C}^{d})\subset\mathcal{X}^{\prime}$, then
$GL(\vee^{2}\mathbb{C}^{d})=\overline{\mathcal{U}(\vee^{2}\mathbb{C}^{d})}\subset{\overline{\mathcal{X}^{\prime}}}$.
However, $\dim(\overline{\mathcal{X}^{\prime}})\leq{d+1\choose
2}^{2}-({d+1\choose 2}-1)+2(d-1)<{d+1\choose 2}^{2}=\dim
GL(\vee^{2}\mathbb{C}^{d})$. This is a contradiction. So
$\mathcal{U}(\vee^{2}\mathbb{C}^{d})\not\subset\mathcal{X}^{\prime}$, i.e. a
unitary operator $\Phi\in\mathcal{U}(\vee^{2}\mathbb{C}^{d})$ with universal
entangling power exists.
We will now show that $\mathcal{X}$ is not only a proper subset, but also a
negligible subset of $\mathcal{U}(\vee^{2}\mathbb{C}^{d})$.
$\mathcal{U}(\vee^{2}\mathbb{C}^{d})$ is a locally compact Lie group of
dimension ${d+1\choose 2}^{2}$. Recall that
$\dim(\overline{\mathcal{X}^{\prime}})$ is at most ${d+1\choose
2}^{2}-({d+1\choose 2}-1)+2(d-1)<{d+1\choose
2}^{2}=\dim(\mathcal{U}(\vee^{2}\mathbb{C}^{d}))$.
We have shown $\dim(\overline{\mathcal{X}^{\prime}})<{d+1\choose
2}^{2}=\dim(\mathcal{U}(\vee^{2}\mathbb{C}^{d}))$.
$\overline{\mathcal{X}^{\prime}}$ is Noetherian (i.e. any descending sequence
of its closed subvarieties is stationary), then
$\overline{\mathcal{X}^{\prime}}$ is a union of finitely many smooth
subvarieties of $GL(\vee^{2}\mathbb{C}^{d})$ with lower dimensions. Hence
$\overline{\mathcal{X}^{\prime}}\cap\mathcal{U}(\vee^{2}\mathbb{C}^{d})$
(which contains $\mathcal{X}^{\prime}\cap\mathcal{U}(\vee^{2}\mathbb{C}^{d})$,
the set of our main interest) is a union of finite many submanifolds of
$\mathcal{U}(\vee^{2}\mathbb{C}^{d})$ with lower dimensions. Therefore,
$\mathcal{X}^{\prime}\cap\mathcal{U}(\vee^{2}\mathbb{C}^{d})$ is measure zero
in $\mathcal{U}(\vee^{2}\mathbb{C}^{d})$ which implies that a random unitary
operator $U$ is almost surely a BUE.
### 3.2. Explicit Construction
As we have shown in Theorem 3.1, a random unitary acting on
$\vee^{2}\mathbb{C}^{d}$ will almost surely be a BUE. Hence we can pick an
arbitrary unitary acting on $\vee^{2}\mathbb{C}^{d}$ and verify whether it
will map some product state to another product state. Recall that the set of
product states in a bosonic system is isomorphic to $\mathbb{C}^{d}$. This
will make it easier to verify whether a unitary is a BUE. Here we provide
verifications of two different classes of BUEs.
#### 3.2.1. Householder-type Bosonic Universal Entanglers
For $d\geq 5$ and any subspace $S\subset\vee^{2}\mathbb{C}^{d}$, let’s
consider the following gate $U=\mathbb{I}_{\vee^{2}\mathbb{C}^{d}}-2P_{S}$
where $P_{S}$ is a projection to some subspace $S$. These gates are known as
Householder matrices in linear algebra [10] and they are widely used to
perform QR decomposition.
A gate $U$ constructed in this way will be a BUE if the subspace $S$ is chosen
properly to satisfy the following two constraints:
1. 1.
There is no product state in $S^{\perp}$.
2. 2.
$\text{rank}|\psi\rangle\geq 3$ for any $|\psi\rangle\in S$.
This claim can be proved by contradiction. Assume there are two product states
$|\psi\rangle|\psi\rangle$ and $|\phi\rangle|\phi\rangle$ such that
$(\mathbb{I}_{\vee^{2}\mathbb{C}^{d}}-2P_{S})|\psi\rangle|\psi\rangle=|\phi\rangle|\phi\rangle$,
we have
$2P_{S}|\psi\rangle|\psi\rangle=|\psi\rangle|\psi\rangle-|\phi\rangle|\phi\rangle$.
$P_{S}|\psi\rangle|\psi\rangle\neq 0$ since there is no product state in
$S^{\perp}$. On the other hand, $P_{S}|\psi\rangle|\psi\rangle$ is a vector in
$S$ which is a subspace completely void of states with rank no more than $2$.
This contradicts our assumption.
In this subsection, we will construct a subspace $S$ to satisfy the above two
constraints for any $d\geq 5$. A family of BUEs will follow immediately.
Let $S$ be the span of the following vectors.
$\displaystyle|11\rangle+|23\rangle+|32\rangle,$
$\displaystyle|22\rangle+|34\rangle+|43\rangle,$ $\displaystyle\textrm{\ \ \ \
\ \ \ \ \ }\cdots,$
$\displaystyle|d-2,d-2\rangle+|d-1,d\rangle+|d,d-1\rangle,$
$\displaystyle|d-1,d-1\rangle+|d,1\rangle+|1,d\rangle,$
$\displaystyle|d,d\rangle+|12\rangle+|21\rangle.$
We first show there is no product state in $S^{\perp}$. Assume
$|\psi\rangle|\psi\rangle\perp S$ where
$|\psi\rangle=\sum\limits_{i=1}^{d}a_{i}|i\rangle$. The orthogonality implies
the following equations.
$\displaystyle(E1)\left\\{\begin{array}[]{lll}a_{1}^{2}+2a_{2}a_{3}&=&0,\\\
a_{2}^{2}+2a_{3}a_{4}&=&0,\\\ &\vdots&\\\
a_{d}^{2}+2a_{1}a_{2}&=&0.\end{array}\right.$
The only common solution to the above equations is
$(a_{1},a_{2},\cdots,a_{d})=(0,0,\cdots,0)$ when $d\geq 3$. See Appendix A for
details.
Hence, there is no product state in $S^{\perp}$.
Next, we will verify that $\text{rank}|\psi\rangle\geq 3$ for any
$|\psi\rangle\in S$.
Assume there is some state $|\psi\rangle\in S$ with rank no more than $2$.
Let’s say
$\displaystyle c_{1}(|11\rangle+|23\rangle+|32\rangle),$ (6) $\displaystyle+$
$\displaystyle c_{2}(|22\rangle+|34\rangle+|43\rangle),$ (7) $\displaystyle+$
$\displaystyle\textrm{\ \ \ \ \ \ \ \ \ }\cdots,$ (8) $\displaystyle+$
$\displaystyle c_{d}(|d,d\rangle+|12\rangle+|21\rangle),$ (9) $\displaystyle=$
$\displaystyle(x_{1}|1\rangle+x_{2}|2\rangle+\cdots+x_{d}|d\rangle)(x_{1}|1\rangle+x_{2}|2\rangle+\cdots+x_{d}|d\rangle),$
(10) $\displaystyle+$
$\displaystyle(y_{1}|1\rangle+y_{2}|2\rangle+\cdots+y_{d}|d\rangle)(y_{1}|1\rangle+y_{2}|2\rangle+\cdots+y_{d}|d\rangle).$
(11)
Then we have the following equations.
$\displaystyle(E2)\left\\{\begin{array}[]{lll}x_{1}^{2}+y_{1}^{2}&=&c_{1},\\\
x_{2}^{2}+y_{2}^{2}&=&c_{2},\\\ &\vdots&\\\ x_{d}^{2}+y_{d}^{2}&=&c_{d},\\\
x_{1}x_{2}+y_{1}y_{2}&=&c_{d},\\\ x_{2}x_{3}+y_{2}y_{3}&=&c_{1},\\\
&\vdots&\\\ x_{d}x_{1}+y_{d}y_{1}&=&c_{d-1},\\\
x_{i}x_{j}+y_{i}y_{j}&=&0\forall|i-j|\geq 2.\end{array}\right.$ (21)
There is no nonzero $(c_{0},c_{2},\cdots,c_{d})$ satisfying the above
equations when $d\geq 5$. See Appendix B for details. Hence
$\text{rank}|\psi\rangle\geq 3$ for any $|\psi\rangle\in S$.
This implies that $U=I-2P_{S}$ is a bosonic universal entangler for any $d\geq
5$.
#### 3.2.2. Permutation Universal Entanglers
Any product state can be written as the following.
$\displaystyle|\phi\rangle|\phi\rangle$ $\displaystyle=$
$\displaystyle(\sum\limits_{i=1}^{d}a_{i}|i\rangle)(\sum\limits_{j=1}^{d}a_{j}|j\rangle)$
(22) $\displaystyle=$
$\displaystyle\sum\limits_{i,j=1}^{d}a_{i}a_{j}|ij\rangle$ (23)
$\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{d}a_{i}^{2}|ii\rangle+\sum\limits_{1\leq
i<j\leq d}\sqrt{2}a_{i}a_{j}(\frac{|ij\rangle+|ji\rangle}{\sqrt{2}}).$ (24)
Any bosonic state $|\psi\rangle\in\vee^{2}\mathbb{C}^{d}$ can be denoted as a
${d+1\choose 2}$-dimensional vector
$(x_{11},x_{22},\cdots,x_{dd},x_{12},\cdots,x_{1d},x_{21},\cdots,x_{d-1d})$
since we can always write $|\psi\rangle$ as a linear combination of bosonic
basis states
$x_{11}|11\rangle+x_{22}|22\rangle+\cdots+x_{dd}|dd\rangle+x_{12}\frac{|12\rangle+|21\rangle}{\sqrt{2}}+x_{13}\frac{|13\rangle+|31\rangle}{\sqrt{2}}+\cdots+x_{d-1,d}\frac{|d-1,d\rangle+|d,d-1\rangle}{\sqrt{2}}.$
$|\psi\rangle$ is a product state if and only if there exists some nonzero
vector $(a_{1},a_{2},\cdots,a_{d})$ such that
$\displaystyle(x_{11},\cdots,x_{dd},x_{12},x_{13},\cdots,x_{1d},x_{23},\cdots,x_{d-1d})$
(25) $\displaystyle=$
$\displaystyle(a_{1}^{2},\cdots,a_{d}^{2},\sqrt{2}a_{1}a_{2},\sqrt{2}a_{1}a_{3},\cdots,\sqrt{2}a_{1}a_{d},\sqrt{2}a_{2}a_{3},\cdots,\sqrt{2}a_{d-1}a_{d}).$
(26)
A permutation matrix $U$ acting on the ${d+1\choose 2}$-dimensional vector
space is certainly a bosonic quantum gate.
For any $d\geq 3$, let’s define a permutation matrix $U$ as the following:
$\displaystyle U$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{d}(\frac{|i,i+1\rangle+|i+1,i\rangle}{\sqrt{2}})\langle
ii|+\sum\limits_{i=1}^{d}|ii\rangle(\frac{\langle i,i+1|+\langle
i+1,i|}{\sqrt{2}})$ $\displaystyle+\sum\limits_{1\leq i<i+1<j\leq
d}\frac{(|ij\rangle+|ji\rangle)(\langle ij|+\langle ji|)}{2}.$
Here the addition and subtraction are all modulo $d$, but the results range
from $1$ to $d$.
$U$ is a unitary matrix since it is simply a rotation of the ${d+1\choose
2}$-dimensional vector space. Let’s assume $U$ will map some (bosonic) product
state to another (bosonic) product state. Without loss of generality, let’s
assume
$\displaystyle U(\sum\limits_{i=1}^{d}a_{i}^{2}|ii\rangle+\sum\limits_{1\leq
i<j\leq
d}\sqrt{2}a_{i}a_{j}(\frac{|ij\rangle+|ji\rangle}{\sqrt{2}}))=\sum\limits_{i=1}^{d}b_{i}^{2}|ii\rangle+\sum\limits_{1\leq
i<j\leq d}\sqrt{2}b_{i}b_{j}(\frac{|ij\rangle+|ji\rangle}{\sqrt{2}}).$
It follows that
$\displaystyle a_{1}^{2}$ $\displaystyle=$ $\displaystyle\sqrt{2}b_{1}b_{2},$
$\displaystyle a_{2}^{2}$ $\displaystyle=$ $\displaystyle\sqrt{2}b_{2}b_{3},$
$\displaystyle\vdots$ , $\displaystyle a_{d}^{2}$ $\displaystyle=$
$\displaystyle\sqrt{2}b_{d}b_{1},$ $\displaystyle\sqrt{2}a_{1}a_{2}$
$\displaystyle=$ $\displaystyle b_{1}^{2},$ $\displaystyle\sqrt{2}a_{2}a_{3}$
$\displaystyle=$ $\displaystyle b_{2}^{2},$ $\displaystyle\vdots$ ,
$\displaystyle\sqrt{2}a_{d}a_{1}$ $\displaystyle=$ $\displaystyle b_{d}^{2}.$
Hence we have
$\prod\limits_{i=1}^{d}a_{i}^{2}=(\sqrt{2})^{d}\prod\limits_{i=1}^{d}b_{i}b_{i+1}=\sqrt{2}^{d}\prod\limits_{i=1}^{d}b_{i}^{2}$.
Similarly,
$\prod\limits_{i=1}^{d}b_{i}^{2}=\sqrt{2}^{d}\prod\limits_{i=1}^{d}a_{i}^{2}$.
The above two equations imply that there exists some $1\leq t\leq d$ such that
$a_{t}=0$.
The equation $b_{t}^{2}=\sqrt{2}a_{t}a_{t+1}$ implies $b_{t}=0$. Then
$a_{t-1}^{2}=\sqrt{2}b_{t-1}b_{t}=0$ will implies $a_{t-1}=0$. By repeating
the above procedure, we will eventually have $a_{i}=0$ for any $1\leq i\leq
d$. This contradicts our assumption that $U$ will map some (bosonic) product
state to another (bosonic) product state. Hence $U$ is a bosonic universal
entangler.
## 4\. Fermionic Universal Entanglers
Given a bipartite system of indisitinguishable fermions
$\wedge^{2}\mathbb{C}^{d}$, a $2$-vector in $\wedge^{2}\mathbb{C}^{d}$ is said
to be decomposable if it can be written as an exterior product of individual
vectors from $\mathbb{C}^{d}$. Decomposable $2$-vectors are also considered to
be unentangled states in this fermionic system.
We say a quantum gate $U$ is a fermionic universal entangler (FUE) if $U$ will
transform every product state to some entangled state.
###### Theorem 4.1.
There is some FUE acting on a bipartite system of indisitinguishable fermions
$\wedge^{2}\mathbb{C}^{d}$ if and only if $d\geq 8$. Furthermore, almost every
quantum gate acting on $\wedge^{2}\mathbb{C}^{d}$is an FUE when $d\geq 8$.
###### Proof 4.2.
Let
$\Gamma_{d}=\\{|\phi\rangle\in\wedge^{2}\mathbb{C}^{d}:|\phi\rangle=|\psi_{1}\rangle\wedge|\psi_{2}\rangle\textrm{\
\ for some\ }|\psi_{1}\rangle,|\psi_{2}\rangle\in\mathbb{C}^{d}\\}$. A quantum
gate $U$ is an FUE if and only if
$\displaystyle U(\Gamma_{d})\bigcap\Gamma_{d}=\emptyset.$ (27)
Observe that decomposable $2$-vectors in $\wedge^{2}\mathbb{C}^{d}$ correspond
to weighted $2$-dimensional linear subspaces of $\mathbb{C}^{d}$. If we ignore
the phase factor, decomposable $2$-vectors can be characterized by the
Grassmannian of $2$-dimensional subspaces of $\mathbb{C}^{d}$, an algebraic
subvariety of the projective space $\mathbb{P}(\wedge^{2}\mathbb{C}^{d})$[8].
We will denote the Grassmannian of $r$-dimensional subspaces of
$\mathbb{C}^{d}$ as $G(r,d)$.
First, we examine the necessary condition.
According to the intersection theorem, if $\dim
U(\Gamma_{d})+\dim\Gamma_{d}\geq\dim\mathbb{P}(\wedge^{2}\mathbb{C}^{d})$, or
equivalently, $2\times 2(d-2)=2\dim G(2,d)\geq{d\choose 2}-1$, then for any
$U$, $U(\Gamma_{d})\bigcap\Gamma_{d}\neq\emptyset$. This inequality holds only
for $2\leq d\leq 7$ which implies the fermionic universal entangling device
does not exist for $d\leq 7$.
Now, let’s look into the sufficient condition.
The set of quantum gates acting on a bipartite system of indisitinguishable
fermions $\wedge^{2}\mathbb{C}^{d}$ is the unitary group acting on
$\wedge^{2}\mathbb{C}^{d}$, denoted as
$\mathcal{U}(\wedge^{2}\mathbb{C}^{d})$.
Similarly, let
$\mathcal{Y}=\\{\Phi|\Phi\in\mathcal{U}(\wedge^{2}\mathbb{C}^{d}),\Phi(\Gamma_{d})\cap\Gamma_{d}\neq\emptyset\\}$.
We will show that $\mathcal{Y}$ is a proper subset in
$\mathcal{U}(\wedge^{2}\mathbb{C}^{d})$.
Again, let’s consider the Zariski topology on the projective space. In this
setting, the unitary group $\mathcal{U}(\wedge^{2}\mathbb{C}^{d})$ is Zariski
dense in the general linear group $GL(\wedge^{2}\mathbb{C}^{d})$[18]. We
further define $\mathcal{Y}^{\prime}=\\{\Phi|\Phi\in
GL(\wedge^{2}\mathbb{C}^{d}),\Phi(\Gamma_{d})\cap\Gamma_{d}\neq\emptyset\\}$.
It is easy to see $\mathcal{X}\subseteq\mathcal{Y}^{\prime}$.
Similar to the proof of Lemma C.1 in Appendix C, the dimension of
$\mathcal{Y}^{\prime}$’s Zariski closure $\dim\overline{\mathcal{Y}^{\prime}}$
is bounded by ${d\choose 2}^{2}-({d\choose 2}-1)+2\times 2(d-2)$.
Now we prove the existence of an FUE $U$ as follows. If it does not exist,
$\mathcal{U}(\wedge^{2}\mathbb{C}^{d})\subset\mathcal{Y}^{\prime}$, then
$GL(\wedge^{2}\mathbb{C}^{d})=\overline{\mathcal{U}(\wedge^{2}\mathbb{C}^{d})}\subset{\overline{\mathcal{Y}^{\prime}}}$.
However, $\dim(\overline{\mathcal{Y}^{\prime}})\leq{d\choose 2}^{2}-({d\choose
2}-1)+4(d-2)<{d\choose 2}^{2}=\dim GL(\wedge^{2}\mathbb{C}^{d})$. This is a
contradiction. So
$\mathcal{U}(\wedge^{2}\mathbb{C}^{d})\not\subset\mathcal{Y}^{\prime}$, i.e.
an FUE $\Phi\in\mathcal{U}(\wedge^{2}\mathbb{C}^{d})$ exists.
Following the lines of the proof of Theorem 3.1, we can prove that
$\mathcal{Y}$ is not only a proper subset, but also a neglectable subset in
$\mathcal{U}(\wedge^{2}\mathbb{C}^{d})$.
## 5\. Summary and Discussion
Employing properties of algebraic geometry, we have shown that for bipartite
systems of indistinguishable bosons with a Hilbert space that is the symmetric
subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$, bosonic universal
entanglers (BUEs) exist if and only if $d\geq 3$. Similarly, we have shown
that for bipartite systems of indistinguishable fermions with a Hilbert space
that is the antisymmetric subspace of $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$,
fermionic universal entanglers (FUEs) exist if and only if $d\geq 8$. These
two results are in contrast to previous results regarding bipartite systems of
distinguishable particles with a Hilbert space
$\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$, for which universal entanglers
exist if and only if min $(d_{1},d_{2})\geq 3$ and $(d_{1},d_{2})\neq(3,3)$.
This illustrates some of the important differences between the entanglement of
systems of indistinguishable particles to the entanglement of systems of
distinguishable particles.
In contrast, we have illustrated one feature which holds for both systems of
distinguishable and indistinguishable particles. Previous work has shown that,
for systems of distinguishable particles, if a universal entangler exists for
some Hilbert space, then almost all unitaries operating on that space are
universal entanglers. We have shown that this result also holds for systems of
indistinguishable bosons and fermions. However to verify whether or not a
given bipartite unitary is a universal entangler is in general an intractable
problem for both distinguishable particle systems and fermionic systems. This
intractability arises from the fact that solving a system of quadratic
equations is, in general, NP-hard.
Bosonic systems turn out to be special though. Because the set of all product
states is isomorphic to a linear vector space, it is possible to use
elementary methods to verify bosonic universal entanglers. We have given
explicit constructions of two types of BUE, one is of the Householder type
which is valid for $d\geq 5$ and the other is of a permutation type which is
valid for $d\geq 3$. Both are very simple constructions.
It is our hope that our success in finding explicit constructions of BUEs will
help inform the search for explicit constructions of both FUEs and UEs,
problems which remain intractable in general. We can not rule out the
possibiliy that there might be some other structure, beyond just the
corresponding general algebraic varieties, which would provide some special
family of explicitly verifiable UEs or FUEs. In fact, the explicit
construction for the $(3,4)$ system from an order $12$ Hadamard matrix
demonstrated in [4] provides a hint of the possibility of such families.
Another natural direction of inquiry is to explore the entangling power of
these BUEs and FUEs. As demonstrated in [4], a random unitary is not only
almost surely a UE, but it also almost surely maps the set of product states
to another set which contains nothing but nearly maximally entangled states,
with respect to almost any kind of entanglement measure. One would expect
similar properties for BUEs and FUEs. However to go further in that direction
one would need to first establish reasonable entanglement measures for bosonic
and fermionic systems (see, e.g. entanglement measures discussed in [5]).
Finally, it would be useful to generalize these results to multipartite
bosonic and fermionc systems. Our guess is that the bosonic systems might
remain easy to solve since they retain the nice property that the set of all
product states is isomorphic to a linear vector space. The fermionic case is
expected to be much more complicated given that in the multipartite case even
the Grassmann-Plücker relations themselves are harder to describe [2, 9, 5].
We would leave these cases for future investigation.
##### Acknowledgements
JK is supported by NSERC. JC is supported by NSERC, UTS-AMSS Joint Research
Laboratory for Quantum Computation and Quantum Information Processing and NSF
of China (Grant No. 61179030). BZ is supported by NSERC and CIFAR.
## Appendix A There Is No Nonzero Solution For Polynomial System (E1)
Here we will show there is no nonzero solution $(a_{1},\cdots,a_{d})$
satisfying the following equations.
$\displaystyle(E1)\left\\{\begin{array}[]{lll}a_{1}^{2}+2a_{2}a_{3}&=&0,\\\
a_{2}^{2}+2a_{3}a_{4}&=&0,\\\ &\vdots&\\\
a_{d}^{2}+2a_{1}a_{2}&=&0.\end{array}\right.$
Assume $a_{i}\neq 0$, then $a_{i+1},a_{i+2}$ are nonzero. This follows all
$a_{i}$’s are nonzero.
$\displaystyle\mathop{\Pi}\limits_{i=1}^{d}a_{i}^{2}=\mathop{\Pi}\limits_{i=1}^{d}(-2a_{i+1}a_{i+2})=(-2)^{d}\mathop{\Pi}\limits_{i=1}^{d}a_{i}^{2}.$
(29)
This implies $\mathop{\Pi}\limits_{i=1}^{d}a_{i}^{2}=0$. Hence it is a
contradiction.
Therefore, the only solution to this polynomial system is
$(a_{1},\cdots,a_{d})=(0,\cdots,0)$.
## Appendix B There Is No Nonzero Solution For Polynomial System (E2)
Here we will show there is no nonzero solution $(c_{0},c_{1},\cdots,c_{d})$
satisfying the following equations.
$\displaystyle(E2)\left\\{\begin{array}[]{lll}x_{1}^{2}+y_{1}^{2}&=&c_{1},\\\
x_{2}^{2}+y_{2}^{2}&=&c_{2},\\\ &\vdots&\\\ x_{d}^{2}+y_{d}^{2}&=&c_{d},\\\
x_{1}x_{2}+y_{1}y_{2}&=&c_{d},\\\ x_{2}x_{3}+y_{2}y_{3}&=&c_{1},\\\
&\vdots&\\\ x_{d}x_{1}+y_{d}y_{1}&=&c_{d-1},\\\
x_{i}x_{j}+y_{i}y_{j}&=&0\forall|i-j|\geq 2.\end{array}\right.$ (39)
For $d\geq 5$, let’s assume there is some $1\leq i\leq d$ such that $x_{i}=0$
and $y_{i}\neq 0$.
It follows from $x_{i}x_{j}+y_{i}y_{j}=0$ for any $|j-i|\geq 2$ that $y_{j}=0$
for any $|j-i|\geq 2$. So, $y_{i+2}=y_{i+3}=0$.
Then, $0\neq
x_{i}^{2}+y_{i}^{2}=x_{i+1}x_{i+2}+y_{i+1}y_{i+2}=x_{i+1}x_{i+2}$. This
implies $x_{i+1},x_{i+2}\neq 0$.
From $x_{i+1}x_{i+3}+y_{i+1}y_{i+3}=0$, we have $x_{i+3}=0$. So, $0\neq
x_{i+2}^{2}+y_{i+2}^{2}=x_{i+3}x_{i+4}+y_{i+3}y_{i+4}=0$. This is a
contradiction.
So, for any $1\leq i\leq d$, we have $x_{i}=y_{i}=0$ or $x_{i}y_{i}\neq 0$.
Let’s look into the various situations.
1. 1.
There is some $i$ such that $x_{i}=y_{i}=0$. Then we have
$x_{i-1}^{2}+y_{i-1}^{2}=0$. If $x_{i-1}=y_{i-1}=0$, we consider
$x_{i-2}^{2}+y_{i-2}^{2}=0$. By repeating this procedure, if all
$x_{j}$’s,$y_{j}$’s are not all zero, we will find some $i^{\prime}$ such that
$y_{i^{\prime}}=ix_{i^{\prime}}$ or $y_{i^{\prime}}=-ix_{i^{\prime}}$ and
$x_{i^{\prime}+1}=y_{i^{\prime}+1}=0$. We will further have
$y_{i^{\prime}+k}=\pm x_{i^{\prime}+k}$ for any $k=2,\cdots,d-1$. This implies
that $(c_{1},\cdots,c_{d})=0$.
2. 2.
All $x_{i}$’s, $y_{i}$’s are nonzero. For any fixed $i$,
$\frac{y_{j}}{x_{j}}=-\frac{x_{i}}{y_{i}}$ for any $j=i+2,\cdots,i+d-2$. This
implies $\frac{y_{k}}{x_{k}}$ is a constant $i$ or $-i$. This also implies
$(c_{1},\cdots,c_{d})=0$.
## Appendix C Proof of Lemma C.1
###### Lemma C.1.
$\dim(\overline{\mathcal{X}^{\prime}})\leq{d+1\choose 2}^{2}-({d+1\choose
2}-1)+2(d-1)$, where $\overline{\mathcal{X}^{\prime}}$ is the Zariski closure
of $\mathcal{X}^{\prime}$.
The following technical lemmas will be needed.
###### Lemma C.2 ([20]).
If $Z_{1}$ and $Z_{2}$ are both irreducible varieties over $\mathbb{C}$, and
$\phi:Z_{1}\rightarrow Z_{2}$ is a dominant morphism, then
$\dim(Z_{2})\leq\dim(Z_{1})$. Here, dominant means $\Phi(Z_{1})$ is dense in
$Z_{2}$.
###### Lemma C.3 ([20]).
If $Z_{1}$ and $Z_{2}$ are both varieties over $\mathbb{C}$, and
$\phi:Z_{1}\rightarrow Z_{2}$ is a morphism, then
$\dim(Z_{1})\leq\dim(Z_{2})+\max\limits_{z\in Z_{2}}{\dim(\phi^{-1}(z))}$.
Lemma C.2 and Lemma C.3 establish a connection between the dimensions of
domain and codomain of a variety morphism.
###### Proof C.4.
We have a morphism $F:GL(\vee^{2}\mathbb{C}^{d})\times\mathbb{P}^{{d+1\choose
2}-1}\rightarrow\mathbb{P}^{{d+1\choose 2}-1}$ which is just the left action
of $GL(\vee^{2}\mathbb{C}^{d})$ on $\mathbb{P}^{{d+1\choose 2}-1}$, defined by
$F(g,[w])=[g\cdot w]$.
We let $y_{0}=(1,0,\cdots,0)$ be a row vector with $d+1\choose 2$ entries, and
for any given $y_{1}$, $y_{2}\in\mathbb{P}^{{d+1\choose 2}-1}$, we choose
proper $g_{1}$ and $g_{2}\in GL(\vee^{2}\mathbb{C}^{d})$, such that
$[g_{1}\cdot y_{0}]=[y_{1}]$ and $[g_{2}\cdot y_{0}]=[y_{2}]$. Then we have
$\displaystyle[g\cdot y_{2}]=[y_{1}]\iff[gg_{2}\cdot y_{0}]=[gg_{1}\cdot
y_{0}]\iff$ $\displaystyle[g_{1}^{-1}gg_{2}\cdot y_{0}]=[y_{0}].$ (40)
From the above observations, F has the following property: for any $y_{1}$,
$y_{2}\in\mathbb{P}^{{d+1\choose 2}-1}$,
$F^{-1}(y_{2})\cap\\{GL(\vee^{2}\mathbb{C}^{d})\times\\{y_{1}\\}\\}\cong\\{\left(\begin{array}[]{cc}z_{1}&\alpha\\\
0&g^{\prime}\end{array}\right):z_{1}\in\mathbb{C}\backslash\\{0\\},g^{\prime}\in
GL({d+1\choose 2}-1),\alpha\in\mathbb{C}^{{d+1\choose 2}-1}\ is\ a\ row\
vector.\\}$. Hence
$\dim(F^{-1}(y_{2})\cap{GL(\vee^{2}\mathbb{C}^{d})\times\\{y_{1}\\}})={d+1\choose
2}^{2}-({d+1\choose 2}-1)$.
Let $P_{1}$, $P_{2}$ be projections of
$GL(\vee^{2}\mathbb{C}^{d})\times\mathbb{P}^{{d+1\choose 2}-1}$ to
$GL(\vee^{2}\mathbb{C}^{d})$, $\mathbb{P}^{{d+1\choose 2}-1}$ respectively.
Now we only look at $GL(\vee^{2}\mathbb{C}^{d})\times\Lambda\subseteq
GL(\vee^{2}\mathbb{C}^{d})\times\mathbb{P}^{{d+1\choose 2}-1}$, to get
$F:GL(\vee^{2}\mathbb{C}^{d})\times\Lambda\rightarrow\mathbb{P}^{{d+1\choose
2}-1}$. Then we have a characterization of $\mathcal{X}^{\prime}$:
$\mathcal{X}^{\prime}=P_{1}F^{-1}(\Lambda)$. In fact
$\displaystyle g\in\mathcal{X}^{\prime}$ $\displaystyle\iff$ $\displaystyle
g(\Lambda)\cap\Lambda\neq\emptyset$ $\displaystyle\iff$ $\displaystyle\exists
z_{1},z_{2}\in\Lambda,s.t.g(z_{1})=z_{2}$ $\displaystyle\iff$
$\displaystyle\exists z_{1},z_{2}\in\Lambda,s.t.(g,z_{1})\in F^{-1}(z_{2})$
$\displaystyle\iff$ $\displaystyle\exists z_{2}\in\Lambda,s.t.g\in
P_{1}F^{-1}(z_{2})$ $\displaystyle\iff$ $\displaystyle g\in
P_{1}F^{-1}(\Lambda).$
So $\overline{\mathcal{X}^{\prime}}\subseteq GL(\vee^{2}\mathbb{C}^{d})$ is
the Zariski closure of $\mathcal{X}^{\prime}$, which is also an algebraic
variety.
Next, we assert that
$P_{1}:F^{-1}(\Lambda)\rightarrow\overline{\mathcal{X}^{\prime}}$ is a
dominant morphism.
Furthermore, consider $\Psi:F^{-1}(\Lambda)\rightarrow\Lambda\times\Lambda$
given by $\Psi(g,[z])=([z],[g\cdot z])$.
For $\forall z_{1}\in\Lambda$, $z_{2}\in\Lambda$, we have
$\Psi^{-1}(z_{1},z_{2})=(g_{2}Tg_{1}^{-1},z_{1})$, where
$T=\\{\left(\begin{array}[]{cc}z_{0}&\alpha\\\
0&g^{\prime}\end{array}\right):z_{0}\in\mathbb{C}\backslash\\{0\\},g^{\prime}\in
GL({d+1\choose 2}-1),\alpha\in\mathbb{C}^{{d+1\choose 2}-1}\ is\ a\ row\
vector\\}$, and $g_{1},g_{2}\in GL(\vee^{2}\mathbb{C}^{d})$, s.t.
$g_{1}(y_{0})=z_{1}$, $g_{2}(y_{0})=z_{2}$. So this is a dominant morphism.
Then we obtain
$\displaystyle\dim(F^{-1}(\Lambda))\leq$
$\displaystyle\dim(T)+\dim(\Lambda\times\Lambda)$ $\displaystyle=$
$\displaystyle{d+1\choose 2}^{2}-({d+1\choose
2}-1)+\dim(\Lambda)+\dim(\Lambda).$
It is required in Lemma C.2 that varieties $Z_{1}$ and $Z_{2}$ be irreducible.
Actually, this condition can be weakened. Lemma C.2 is still true for the more
general case that $Z_{1}$ and $Z_{2}$ are closed subsets of irreducible
varieties[8]. Through this approach, we can fill out the gap and apply this
lemma without danger of confusion. Indeed, the irreduciblity of $Z_{1}$ and
$Z_{2}$ really holds, but verification of this is not easy.
Then from Lemma C.2 and Lemma C.3, we will have
$\displaystyle\dim(\overline{\mathcal{X}^{\prime}})\leq$
$\displaystyle\dim(F^{-1}(\Lambda))$ $\displaystyle\leq$
$\displaystyle{d+1\choose 2}^{2}-({d+1\choose
2}-1)+\dim(\Lambda)+\dim(\Lambda)$ $\displaystyle=$ $\displaystyle{d+1\choose
2}^{2}-({d+1\choose 2}-1)+2(d-1).$
## References
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* [7] J. Harris. Algebraic geometry: a first course, volume 133. Springer, 1992.
* [8] Robin Hartshorne. Algebraic geometry. Springer-Verlag, 1983.
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* [10] R.A. Horn and C.R. Johnson. Matrix analysis. Cambridge university press, 1990.
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* [12] Y. S. Li, B. Zeng, X. S. Liu, and G. L. Long. Entanglement in a two-identical-particle system. Physical Review A, 64(5):054302, November 2001.
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| 2013-05-31T17:03:03 |
2024-09-04T02:49:45.951554
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Joel Klassen, Jianxin Chen, Bei Zeng",
"submitter": "Joel Klassen",
"url": "https://arxiv.org/abs/1305.7489"
}
|
1306.0044
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arxiv-papers
| 2013-05-31T22:39:31 |
2024-09-04T02:49:45.964172
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Nick Thomas",
"submitter": "Nicholas Thomas",
"url": "https://arxiv.org/abs/1306.0044"
}
|
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1306.0066
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# A further simplification of Tarski’s axioms of geometry
T. J. M. Makarios111mailto:[email protected]@gmail.com
###### Abstract
A slight modification to one of Tarski’s axioms of plane Euclidean geometry is
proposed. This modification allows another of the axioms to be omitted from
the set of axioms and proven as a theorem. This change to the system of axioms
simplifies the system as a whole, without sacrificing the useful modularity of
some of its axioms. The new system is shown to possess all of the known
independence properties of the system on which it was based; in addition,
another of the axioms is shown to be independent in the new system.
## 1 Background
Alfred Tarski’s axioms of geometry were first described in a course he gave at
the University of Warsaw in 1926–1927. Since then, they have undergone
numerous improvements, with some axioms modified, and other superfluous axioms
removed; for a history of the changes, see [TG99] (especially Section 2), or
for a summary, see Figure 2 in [Nar07].
The axioms rely on only one primitive notion — that of points — and two
primitive relations: betweenness and congruence. Congruence is denoted
$ab\equiv cd$, and can be interpreted as asserting that the line segment from
$a$ to $b$ is congruent to the line segment from $c$ to $d$. Betweenness is
denoted $\mathrm{B}abc$, and can be interpreted as asserting that $b$ lies on
the segment from $a$ to $c$ (and may be equal to $a$ or $c$).
The version of the axioms used in [SST83] (see pages 10–14) consists of ten
first-order axioms, together with either a first-order axiom schema, or a
single higher-order axiom. This version has been adopted in later
publications, such as [Mak12] (see Sections 2.3 and 2.4) and [Nar07] (see
Figure 3), and Victor Pambuccian has called it the “most polished form” of
Tarski’s axioms (see [Pam06], page 122).
This semi-canonical version of the system is the result of many
simplifications to the original system of twenty axioms plus one axiom schema.
At least one of these simplifications appears to have taken the form of a
slight alteration to one axiom in order to allow another axiom to be dropped
and subsequently proven as a theorem.
Specifically, in [Tar57] (see note 18), axiom (ix) is a version of what is
called the _axiom of Pasch_ , which states (modulo notational differences):
$\forall a,b,c,p,q.\;\mathrm{B}apc\wedge\mathrm{B}qcb\longrightarrow\exists
x.\;\mathrm{B}axq\wedge\mathrm{B}xpb$
In [Tar59], this has been replaced by axiom A9, which states (again, modulo
notational changes):
$\forall a,b,c,p,q.\;\exists
x.\;\mathrm{B}apc\wedge\mathrm{B}qcb\longrightarrow\mathrm{B}axq\wedge\mathrm{B}bpx$
The significant change between the two is the reversal of the order of points
in the final betweenness relation. They are easily shown to be equivalent
using the symmetry of betweenness, which states:
$\forall a,b,c.\;\mathrm{B}abc\longrightarrow\mathrm{B}cba$
The interesting thing is that in [Tar57], (1) is an axiom (axiom (iii)), but
in [Tar59], it has been removed from the list of axioms. Haragauri Gupta’s
proof of (1) (see [Gup65], Theorem 2.18) relies on the precise ordering of
points in the final betweenness relation in (1).
Also, Wolfram Schwabhäuser, Wanda Szmielew, and Alfred Tarski, on page 12 of
[SST83], draw attention to the ordering of points in their version of the
axiom of Pasch (labelled (• ‣ 2) in this paper), noting that it is important
until after the proof of (1) (which they call Satz 3.2). Again, their proof of
(1) (which is essentially the same as this paper’s proof of Lemma 4) relies on
the precise ordering of points in (• ‣ 2).
Therefore, although it does not appear to be explicitly acknowledged in the
published literature, it seems likely that the change from (1) to (1) was
necessary to allow the removal of (1) from the set of axioms. It may be the
case that (1) is a theorem even with (1), and that this was not known when it
was replaced by (1), or that it was known, but (1) allows a simpler proof.
In any case, it appears that Tarski was willing to reorder points in his
axioms to allow the simplification of the axiom system as a whole, either by
removing axioms or merely by simplifying proofs of theorems.
In the tradition of such simplifications, this paper presents one further
simplification of the axiom system; one of the axioms is slightly modified,
allowing another of the traditional axioms to be proven as a theorem, rather
than assumed as an axiom.
## 2 The axioms
Tarski’s axioms, as stated in [SST83], pages 10–14, are as follows. The names
are adopted from [Mak12] (Section 2.4), which provides some diagrams and
intuitive explanations for the axioms.
* •
Reflexivity axiom for equidistance
$\forall a,b.\;ab\equiv ba$
* •
Transitivity axiom for equidistance
$\forall a,b,p,q,r,s.\;ab\equiv pq\wedge ab\equiv rs\longrightarrow pq\equiv
rs$
* •
Identity axiom for equidistance
$\forall a,b,c.\;ab\equiv cc\longrightarrow a=b$
* •
Axiom of segment construction
$\forall a,b,c,q.\;\exists x.\;\mathrm{B}qax\wedge ax\equiv bc$
* •
Five-segments axiom
$\displaystyle\forall a,b,c,d,a^{\prime},b^{\prime},c^{\prime},d^{\prime}.\;$
$\displaystyle{a\neq
b\wedge\mathrm{B}abc\wedge\mathrm{B}a^{\prime}b^{\prime}c^{\prime}}$
$\displaystyle\qquad\wedge ab\equiv a^{\prime}b^{\prime}\wedge bc\equiv
b^{\prime}c^{\prime}\wedge ad\equiv a^{\prime}d^{\prime}\wedge bd\equiv
b^{\prime}d^{\prime}$ $\displaystyle\longrightarrow cd\equiv
c^{\prime}d^{\prime}$ (FS)
* •
Identity axiom for betweenness
$\forall a,b.\;\mathrm{B}aba\longrightarrow a=b$
* •
Axiom of Pasch
$\forall a,b,c,p,q.\;\mathrm{B}apc\wedge\mathrm{B}bqc\longrightarrow\exists
x.\;\mathrm{B}pxb\wedge\mathrm{B}qxa$
* •
Lower $2$-dimensional axiom
$\exists
a,b,c.\;\neg\mathrm{B}abc\wedge\neg\mathrm{B}bca\wedge\neg\mathrm{B}cab$
* •
Upper $2$-dimensional axiom
$\forall a,b,c,p,q.\;p\neq q\wedge ap\equiv aq\wedge bp\equiv bq\wedge
cp\equiv cq\longrightarrow\mathrm{B}abc\vee\mathrm{B}bca\vee\mathrm{B}cab$
* •
Euclidean axiom
$\forall a,b,c,d,t.\;\mathrm{B}adt\wedge\mathrm{B}bdc\wedge a\neq
d\longrightarrow\exists
x,y.\;\mathrm{B}abx\wedge\mathrm{B}acy\wedge\mathrm{B}xty$
* •
Axiom of continuity
$\displaystyle\forall X,Y.\;$ $\displaystyle{(\exists a.\;\forall x,y.\;x\in
X\wedge y\in Y\longrightarrow\mathrm{B}axy)}$
$\displaystyle\longrightarrow(\exists b.\;\forall x,y.\;x\in X\wedge y\in
Y\longrightarrow\mathrm{B}xby)$ (Co)
This collection of eleven axioms, Tarski’s axioms of the continuous Euclidean
plane, will be denoted CE2. A similar collection of axioms, denoted
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$, is obtained by removing (• ‣ 2)
from CE2 and replacing (FS) with
$\displaystyle\forall a,b,c,d,a^{\prime},b^{\prime},c^{\prime},d^{\prime}.\;$
$\displaystyle{a\neq
b\wedge\mathrm{B}abc\wedge\mathrm{B}a^{\prime}b^{\prime}c^{\prime}}$
$\displaystyle\qquad\wedge ab\equiv a^{\prime}b^{\prime}\wedge bc\equiv
b^{\prime}c^{\prime}\wedge ad\equiv a^{\prime}d^{\prime}\wedge bd\equiv
b^{\prime}d^{\prime}$ $\displaystyle\longrightarrow dc\equiv
c^{\prime}d^{\prime}$ (FS′)
The only difference between (FS) and (FS′) is the reversal of the first two
points in the last congruence relation.
This paper will show that CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ is
equivalent to CE2, and will, in fact, show a stronger result — Theorem 7 —
about a smaller set of axioms.
One of the features of Tarski’s axiom system is its modularity: some texts
omit or delay the introduction of (Co) (see, for example, [Ehr97], page 61);
(• ‣ 2) can be replaced by another axiom in order to investigate hyperbolic
geometry, or omitted entirely, for absolute geometry (see [Pam02], pages
331–333); (• ‣ 2) and (• ‣ 2) can be replaced by other axioms that
characterize other dimensions (see [Tar59], footnote 5). For this reason, let
A denote the collection of axioms (• ‣ 2), (• ‣ 2), (• ‣ 2), (• ‣ 2), (FS), (•
‣ 2), and (• ‣ 2). These are Tarski’s axioms of absolute dimension-free
geometry without the axiom of continuity. Let A′ denote the collection of
axioms (• ‣ 2), (• ‣ 2), (• ‣ 2), (FS′), (• ‣ 2), and (• ‣ 2).
The stronger result that this paper shows is that A′ is equivalent to A. Thus,
the modularity of axioms (• ‣ 2), (• ‣ 2), (• ‣ 2), and (Co) is unaffected by
the proposed change to Tarski’s axiom system.
## 3 Proof of equivalence
###### Lemma 1.
If (• ‣ 2) and (• ‣ 2) hold, then given any points $a$ and $b$, we have
$ab\equiv ab$.
###### Proof.
Given $a$ and $b$, (• ‣ 2) lets us obtain a point $x$ such that $ax\equiv ab$.
Using this twice in (• ‣ 2) gives us $ab\equiv ab$. ∎
###### Lemma 2.
If (• ‣ 2) and (• ‣ 2) hold and $a$, $b$, $c$, and $d$ are points such that
$ab\equiv cd$, then $cd\equiv ab$.
###### Proof.
By Lemma 1, we have $ab\equiv ab$. Using $ab\equiv cd$ and $ab\equiv ab$, (• ‣
2) tells us that $cd\equiv ab$. ∎
###### Lemma 3.
If (• ‣ 2) and (• ‣ 2) hold, then given any points $a$ and $b$, we have
$\mathrm{B}abb$.
###### Proof.
Given $a$ and $b$, (• ‣ 2) lets us obtain a point $x$ such that
$\mathrm{B}abx$ and $bx\equiv bb$. Then (• ‣ 2) tells us that $b=x$, so
$\mathrm{B}abb$. ∎
###### Lemma 4.
(• ‣ 2), (• ‣ 2), (• ‣ 2), and (• ‣ 2) together imply (1).
###### Proof.
Suppose we are given points $a$, $b$, and $c$ such that $\mathrm{B}abc$. We
also have $\mathrm{B}bcc$, by Lemma 3. Then (• ‣ 2) lets us obtain a point $x$
such that $\mathrm{B}bxb$ and $\mathrm{B}cxa$. According to (• ‣ 2), the
former implies $b=x$, so the latter tells us that $\mathrm{B}cba$. ∎
###### Lemma 5.
A′ implies (• ‣ 2).
###### Proof.
Given arbitrary points $a$ and $b$, (• ‣ 2) lets us obtain a point $x$ such
that $\mathrm{B}bax$ and $ax\equiv ba$. We consider two cases: $x=a$ and
$x\neq a$.
If $x=a$, then $aa\equiv ba$. By Lemma 2, we have $ba\equiv aa$, so by (• ‣
2), we have $b=a$. Substituting this back into the congruence as necessary
gives us $ab\equiv ba$, as desired.
Suppose, on the other hand, that $x\neq a$. Lemma 4 and $\mathrm{B}bax$ tell
us that $\mathrm{B}xab$. Lemma 1 tells us that $xa\equiv xa$, $ab\equiv ab$,
and $aa\equiv aa$. We make the following substitutions in (FS′):
$a,a^{\prime}\mapsto x$; $b,b^{\prime},d,d^{\prime}\mapsto a$; and
$c,c^{\prime}\mapsto b$. Then all of the hypotheses of (FS′) are satisfied,
and its conclusion is that $ab\equiv ba$. ∎
###### Lemma 6.
If (• ‣ 2) and (• ‣ 2) hold, then (FS′) is equivalent to (FS).
###### Proof.
Because the hypotheses of (FS) and (FS′) are identical, we need only show that
their conclusions are equivalent.
(• ‣ 2) tells us that $cd\equiv dc$ and $dc\equiv cd$.
If $cd\equiv c^{\prime}d^{\prime}$, then $cd\equiv dc$ together with this fact
and (• ‣ 2) let us conclude that $dc\equiv c^{\prime}d^{\prime}$.
Similarly, if $dc\equiv c^{\prime}d^{\prime}$, then $dc\equiv cd$ together
with this fact and $\eqref{te}$ let us conclude that $cd\equiv
c^{\prime}d^{\prime}$.
Therefore (FS′) is equivalent to (FS). ∎
###### Theorem 7.
A′ is equivalent to A.
###### Proof.
By Lemmas 5 and 6, A′ implies (• ‣ 2) and (FS). A′ contains all of the other
axioms of A, so A′ implies A.
By Lemma 6, A implies (FS′), and it contains all of the other axioms of A′, so
A implies A′.
Therefore A′ is equivalent to A. ∎
As an immediate corollary, we have the following:
###### Corollary 8.
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ is equivalent to CE2. ∎
## 4 Independence results
The first part of Section 5 of [TG99] (see pages 199 and 200) concerns the
independence of Tarski’s axioms. One problem seen there is that the various
historical changes to Tarski’s axioms often force a reconsideration of
previously established independence results. This paper’s suggested
simplification of Tarski’s axioms is no exception, so this section aims to
establish which of the known independence results apply to the specific set of
axioms CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$.
Because CE2 and CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ differ only in their
subsets A and A′, Theorem 7 tells us that the axioms in
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ but not in A′ are independent if and
only if they are independent in CE2. In fact, we can go further than this.
###### Theorem 9.
Suppose that (Ax) is an axiom of CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$
other than (• ‣ 2) or (FS′). If (Ax) is independent in CE2 then (Ax) is also
independent in CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$.
###### Proof.
Note that (Ax) is not (• ‣ 2), because this was explicitly excluded; nor is it
(• ‣ 2) or (FS), because these are not axioms of
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$, from which (Ax) was chosen.
Therefore
$\textnormal{{{CE${}_{\mathnormal{\mathsf{2}}}$}}}\setminus\left\\{\textup{(Ax)}\right\\}$
contains (• ‣ 2), (• ‣ 2), and (FS), so by Lemma 6,
$\textnormal{{{CE${}_{\mathnormal{\mathsf{2}}}$}}}\setminus\left\\{\textup{(Ax)}\right\\}\vdash\eqref{fsp}$.
All of the axioms of CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ other than
(FS′) are also axioms of CE2, so
$\textnormal{{{CE${}_{\mathnormal{\mathsf{2}}}$}}}\setminus\left\\{\textup{(Ax)}\right\\}\vdash\textnormal{{{CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$}}}\setminus\left\\{\textup{(Ax)}\right\\}$.
Therefore, if
$\textnormal{{{CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$}}}\setminus\left\\{\textup{(Ax)}\right\\}\vdash\textup{(Ax)}$,
then
$\textnormal{{{CE${}_{\mathnormal{\mathsf{2}}}$}}}\setminus\left\\{\textup{(Ax)}\right\\}\vdash\textup{(Ax)}$.
Taking the contrapositive of this statement, we see that if (Ax) is
independent in CE2 then it is independent in
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$. ∎
This allows us to adopt almost all of the independence results known for CE2.
###### Corollary 10.
(• ‣ 2), (• ‣ 2), (• ‣ 2), (• ‣ 2), (• ‣ 2), and (Co) are each individually
independent in CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$.
###### Proof.
Schwabhäuser and his colleagues note the independence of each of these axioms
in CE2; see [SST83], page 26. ∎
The remaining independence result noted in [SST83] can also be adapted to
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$:
###### Theorem 11.
(FS′) is independent in CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$.
###### Proof.
Schwabhäuser and his co-authors note the existence of a model $\mathcal{M}$
demonstrating the independence of (FS) in CE2; see [SST83], page 26. Because
$\mathcal{M}$ satisfies (• ‣ 2) and (• ‣ 2), but violates (FS), we can
conclude, using Lemma 6, that $\mathcal{M}$ also violates (FS′). $\mathcal{M}$
satisfies every other axiom of CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$,
because all such axioms are also axioms of CE2 (and are not equal to (FS));
therefore, $\mathcal{M}$ demonstrates the independence of (FS′) in
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$. ∎
Because of the absence of (• ‣ 2) from
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$, we can easily show the independence
of (• ‣ 2) in that axiom system:
###### Theorem 12.
(• ‣ 2) is independent in CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$.
###### Proof.
We proceed as is usual in independence proofs, by defining a model
$\mathcal{M}$ of every axiom of CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$
except (• ‣ 2). As in the real Cartesian plane (the standard model of CE2, and
hence of CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$), we take $\mathbb{R}^{2}$
to be the set of points, and define betweenness so that $\mathrm{B}abc$ if and
only if $b$ is a convex combination of $a$ and $c$. Departing from the
standard model, we define congruence so that $ab\equiv cd$ if and only if
$a=b$.
Because the real Cartesian plane is a model of
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$, and $\mathcal{M}$ differs from the
real Cartesian plane only in its definition of congruence, we can conclude
that $\mathcal{M}$ is a model of all of the axioms of
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ that make no mention of congruence.
Those axioms are (• ‣ 2), (• ‣ 2), (• ‣ 2), (• ‣ 2), and (Co).
The definition of congruence ensures that $\mathcal{M}$ trivially satisfies (•
‣ 2).
Choosing $x=a$ ensures that (• ‣ 2) is satisfied.
The hypotheses of (FS′) include $a\neq b$ and $ab\equiv a^{\prime}b^{\prime}$,
which implies $a=b$; this contradiction in the hypotheses means that (FS′) is
vacuously true.
The hypotheses of (• ‣ 2) imply that $a=b=c=p$; it is always the case that
$\mathrm{B}ppp$, so (• ‣ 2) is satisfied.
Finally, in $\mathcal{M}$, it is the case that $(0,0)(0,0)\equiv(0,0)(0,1)$,
but not the case that $(0,0)(0,1)\equiv(0,0)(0,1)$, so $\mathcal{M}$ does not
satisfy (• ‣ 2).
Because $\mathcal{M}$ satisfies every axiom of
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ except (• ‣ 2), we can conclude that
(• ‣ 2) is independent in CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$. ∎
Notice that $\mathcal{M}$ in the proof of Theorem 12 also violates (• ‣ 2)
(because it is not the case that $(0,0)(0,1)\equiv(0,1)(0,0)$), so this model
would not demonstrate the independence of (• ‣ 2) in CE2, which is, as far as
the author is aware, still an open question.
We now have independence results for all of the axioms of
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ except (• ‣ 2) and (• ‣ 2). As far
as the author knows, the independence of these in CE2 is still an open
question (see also [TG99], pages 199 and 200), although there are independence
results relating to these axioms in other versions of Tarski’s axiom system.
Gupta shows the independence of (• ‣ 2) (his axiom A5; see [Gup65], pages 41
and 41a), but only in the context of a particular variant of Tarski’s axiom
system. This variant uses a weaker but more complicated form of the upper
$2$-dimensional axiom, Gupta’s A′12;222It seems that A′12 in Gupta’s thesis
ought to include $u\neq v$ among the hypotheses, as A12 does. Taken exactly as
it is printed, A′12 is violated by the real Cartesian plane whenever $x$, $y$,
and $z$ are any non-collinear points and $u=v$. his independence model for (•
‣ 2) also violates (• ‣ 2), which is trivially equivalent to his original
axiom A12.
Lesław Szczerba established the independence of a version of the axiom of
Pasch within a certain variant of Tarski’s axiom system (see [Szc70]). This
variant used, instead of (• ‣ 2), an axiom essentially asserting that any
three non-collinear points have a circumcentre.333There appears to be a
typographical error in the statement of this axiom in [Szc70] (page 492). His
axiom A8′ states (in our notation): $\forall a,b,c.\;\exists
p.\;\neg(\mathrm{B}abc\vee\mathrm{B}bca\vee\mathrm{B}cab)\longrightarrow
ap\equiv bp\wedge bp\equiv cb$ It seems that the second congruence relation in
the consequent should state $bp\equiv cp$; see also [TG99], pages 199 and 184.
## 5 Conclusions
Although this paper has not answered the question of whether (• ‣ 2) is
independent in CE2, it has demonstrated that (• ‣ 2) is superfluous to
Tarski’s axioms of geometry. A slight modification to (FS) allows (• ‣ 2) to
be proven as a theorem, and therefore removed from the set of axioms. This
simplification of the axiom system (to
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$) does not diminish its deductive
power or the important ways in which it exhibits modularity.
Besides removing one of the axioms not known to be independent in CE2,
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ allows an easy proof of the
independence of (• ‣ 2), which was also not known to be independent in CE2.
The two remaining independence questions for
CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$ are, as far as the author knows,
still open questions for CE2; furthermore, if either axiom is shown to be
independent in CE2, then Theorem 9 would immediately establish its
independence in CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$.
As well as trying to resolve these remaining independence questions, future
work might seek other slight modifications to the axioms that may allow even
known independent axioms to be dropped. That this may be possible can be seen
by considering Gupta’s fully independent version of Tarski’s axioms for plane
Euclidean geometry (see [Gup65], pages 41–41c).
His version had eleven axioms,444Strictly speaking, as he presented it, it had
infinitely many axioms, but if his axiom schema A′13 is replaced by a
comparable second-order axiom, then there are eleven axioms in total, and the
second-order axiom is shown to be independent by his example on page 41c. some
of which were deliberately made more complicated in order to allow easy proofs
of the independence of other axioms (see, for example, his note on the
independence of his axiom A7 on page 41b). CE2 already has simpler axioms than
Gupta’s system (which he used only for the demonstration that a fully
independent system is possible), but CE$\mathnormal{\mathsf{{}_{2}^{\prime}}}$
now shows that a reduction in the number of axioms is also possible, without
making any of them more complex, despite Gupta’s system being fully
independent.
## References
* [Ehr97] Philip Ehrlich. From completeness to Archimedean completeness: An essay in the foundations of Euclidean geometry. Synthese, 110:57–76, 1997. http://www.ohio.edu/people/ehrlich/FromCompletenessToArch.pdf.
* [Gup65] Haragauri Narayan Gupta. Contributions to the Axiomatic Foundations of Geometry. PhD thesis, University of California, Berkeley, 1965.
* [Mak12] Timothy James McKenzie Makarios. A mechanical verification of the independence of Tarski’s Euclidean axiom. Master’s thesis, Victoria University of Wellington, New Zealand, 2012\. http://researcharchive.vuw.ac.nz/handle/10063/2315.
* [Nar07] Julien Narboux. Mechanical theorem proving in Tarski’s geometry. In Francisco Botana and Tomas Recio, editors, Automated Deduction in Geometry, volume 4869 of Lecture Notes in Computer Science, pages 139–156. Springer Berlin / Heidelberg, 2007.
* [Pam02] Victor Pambuccian. Axiomatizations of hyperbolic geometry: A comparison based on language and quantifier type complexity. Synthese, 133:331–341, 2002.
* [Pam06] Victor Pambuccian. Axiomatizations of hyperbolic and absolute geometries. In András Prékopa and Emil Molnár, editors, Non-Euclidean Geometries, volume 581 of Mathematics and Its Applications, pages 119–153. Springer US, 2006.
* [SST83] Wolfram Schwabhäuser, Wanda Szmielew, and Alfred Tarski. Metamathematische Methoden in der Geometrie. Springer-Verlag, 1983.
* [Szc70] Lesław W. Szczerba. Independence of Pasch’s axiom. Bulletin de l’Academie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, 18(8):491–498, 1970.
* [Tar57] Alfred Tarski. A decision method for elementary algebra and geometry. Technical report, The RAND Corporation, 1957. First published August 1948.
* [Tar59] Alfred Tarski. What is elementary geometry? In Leon Henkin, Patrick Suppes, and Alfred Tarski, editors, The Axiomatic Method: with Special Reference to Geometry and Physics, pages 16–29. North-Holland Publishing Company, 1959.
* [TG99] Alfred Tarski and Steven Givant. Tarski’s system of geometry. The Bulletin of Symbolic Logic, 5(2):175–214, June 1999.
|
arxiv-papers
| 2013-06-01T02:45:46 |
2024-09-04T02:49:45.968789
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Timothy Makarios",
"submitter": "Timothy Makarios",
"url": "https://arxiv.org/abs/1306.0066"
}
|
1306.0249
|
# Performance of the Muon Identification at LHCb
F. Archillia W. Baldinib G. Bencivennia N. Bondarc W. Boniventod S. Cadeddud
P. Campanaa A. Cardinid P. Ciambronea X. Cid Vidale C. Deplanod P. De Simonea
A. Falabellaf,r M. Frosinig,15 S. Furcasa Now at Sezione INFN di Milano,
Milano, Italy. E. Furfaroh M. Gandelmani J.A. Hernando Morataj G. Grazianig A.
Laid G. Lanfranchia J.H. Lopesi O. Maevc G. Mancad G. Martellottih A.
Massafferrik D. Milanesl Now at LPNHE, Université Pierre et Marie Curie,
Université Paris Diderot, CNRS/IN2P3, Paris, France. R. Oldemand,m M. Palutana
G. Passalevag D. Pincih E. Polycarpoi Corresponding author. R. Santacesariah
E. Santovettin,p A. Sartia,q A. Sattan B. Schmidte B. Sciasciaa F. Soomroa A.
Sciubbah,q and S. Vecchib
aINFN - Laboratori Nazionali di Frascati
Frascati Italy
bSezione INFN di Ferrara
Ferrara Italy
cPetersburg Nuclear Physics Institute
Gatchina St-Petersburg Russia
dSezione INFN di Cagliari
Cagliari Italy
eEuropean Organisation for Nuclear Research (CERN)
Geneva Switzerland
fSezione INFN di Bologna
Bologna Italy
gSezione INFN di Firenze
Firenze Italy
hSezione INFN di Roma
Roma Italy
iUniversidade Federal do Rio de Janeiro (UFRJ)
Rio de Janeiro Brasil
jUniversidade de Santiago de Compostela
Santiago de Compostela Spain
kCentro Brasileiro de Pesquisas Físicas (CBPF)
Rio de Janeiro Brasil
lSezione INFN di Bari
Bari Italy
mUniversità di Cagliari
Cagliari Italy
nSezione INFN di Roma Tor Vergata
Roma Italy
oUniversità di Firenze
Firenze Italy
pUniversità di Roma Tor Vergata
Roma Italy
qSapienza Universita di Roma
Roma Italy
rUniversità di Ferrara
Ferrara Italy
E-mail [email protected]
###### Abstract
The performance of the muon identification in LHCb is extracted from data
using muons and hadrons produced in $J/\psi\rightarrow\mu^{+}\mu^{-}$ ,
$\Lambda^{0}\rightarrow p\pi^{-}$ and
$D^{\star+}\rightarrow\pi^{+}D^{0}(K^{-}\pi^{+})$ decays. The muon
identification procedure is based on the pattern of hits in the muon chambers.
A momentum dependent binary requirement is used to reduce the probability of
hadrons to be misidentified as muons to the level of 1%, keeping the muon
efficiency in the range of 95-98%. As further refinement, a likelihood is
built for the muon and non-muon hypotheses. Adding a requirement on this
likelihood that provides a total muon efficiency at the level of 93%, the
hadron ${\rm misidentification}\,\,{\rm probabilities}$ are below 0.6%.
###### keywords:
Particle identification methods;Performance of High Energy Physics Detectors
## 1 Introduction
LHCb [1] is a dedicated heavy flavour experiment, designed to exploit the high
$pp\rightarrow c\bar{c}$ and $pp\rightarrow b\bar{b}$ cross-sections at the
LHC in order to perform precision measurements of CP violation and rare
decays. Muons are present in the final state of many of the key decays,
sensitive to new physics, as shown, for example, in [2, 3, 4, 5, 6], among
others. Moreover, they play a crucial role in the determination of the flavor
tagging of the neutral $B$ mesons and are also present in the signatures of
interesting electroweak and strong processes. The muon identification
procedure must provide high muon efficiency while keeping the incorrect
identification probability of hadrons as muons (misidentification
probabilities) at the lowest possible level. The pion misidentification is one
of the major sources of combinatoric background for decays with muons in the
final state. It is also important to keep the other hadron misidentification
probabilities at low levels so that rare decays can be separated from more
abundant hadronic decays with similar or identical topology.
This paper presents the performance of the muon identification in LHCb,
obtained from the data recorded in 2011, corresponding to approximately
1$\mbox{\,fb}^{-1}$. In Section 2, a brief description of the LHCb
spectrometer and the muon detection system is given. The muon identification
algorithm is discussed in Section 3. The method used to extract the muon
efficiency and the misidentification probability from data is explained in
Section 4. Finally, the performance results are presented in Section 5,
followed by the conclusions in Section 6.
## 2 The LHCb experiment and the muon system
The LHCb detector [1] is a single-arm forward spectrometer. A vertex locator
(VELO) determines with high precision the positions of the vertices of $pp$
collisions (PVs) and the decay vertices of long-lived particles. The tracking
system includes a silicon strip detector located in front of a dipole magnet
with an integrated field of about 4 Tm, and a combination of silicon strip
detectors and straw drift chambers placed behind the magnet. The momentum of
charged particles is determined with a resolution of
$\sigma_{p}/p\sim$0.4(0.6)% at a momentum scale of
3(100)${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Charged hadron identification is achieved with two ring-imaging Cherenkov
(RICH) detectors. The calorimeter system consists of a scintillator pad
detector, a preshower, an electromagnetic calorimeter and a hadronic
calorimeter. It identifies high transverse energy111The transverse energy of a
2$\times$2 cells cluster is defined as
$E_{T}=\displaystyle\sum_{i=1}^{4}E_{i}\sin{\theta_{i}}$, where $E_{i}$ is the
energy deposited in cell $i$ and $\theta_{i}$ is the angle between the z-axis
and a neutral particle assumed to be coming from the mean position of the
interaction envelope hitting the centre of the cell [11]. hadron, electron and
photon candidates and provides information for the trigger.
The muon system [7] is composed of five stations (M1-M5) of rectangular shape,
placed along the beam axis, as shown in Fig. 1. Station M1 is located in front
of the calorimeters and is used to improve the transverse momentum measurement
in the first level hardware trigger. Stations M2 to M5 are placed downstream
the calorimeters and are interleaved with iron absorbers 80 cm thick to select
penetrating muons. The total absorber thickness in front of station M2,
including the calorimeters, is approximately 6.6 interaction lengths. More
than 99% of the total area of the system is equipped with multi-wire
proportional chambers (MWPC) with Ar/CO2/CF4(40:55:5) as gas mixture. Only the
inner part of the first station is instrumented with triple-GEM detectors
filled with Ar/CO2/CF4(45:15:40).
The chambers are positioned to provide with their sensitive area a hermetic
geometric acceptance to high momentum particles coming from the interaction
point. In addition, the chambers of different stations form projective towers
pointing to the interaction point. The detectors provide digital space point
measurements on the particle trajectories, supplying information to the
trigger processor and to the data acquisition (DAQ). The information is
obtained by partitioning the detector into rectangular logical pads whose
dimensions define the x, y resolution in the plane perpendicular to the beam
axis. Each station is divided into four regions, R1 to R4 with increasing
distance from the beam axis, as shown in Fig. 2. The linear dimensions of the
regions R1, R2, R3, R4, and their segmentation scale in the ratio 1:2:4:8.
Each muon station is designed to perform with an efficiency above 99% in a 20
ns time window with a noise rate below 1 kHz per physical channel, which was
achieved during operation, as described in [7].
Figure 1: Schematic view of the LHCb experiment. The muon stations are seen as
the five green vertical bars, the second one placed just after the
calorimeters, shown as the blue rectangles. Figure 2: Schematic view of one
muon system station (reproduced from [7]).
The muon system provides information for the selection of high transverse
momentum muons at the trigger level and for the offline muon identification.
This document refers to the latter procedure, which uses only the information
from the 4 stations located after the calorimeters. The muon identification in
the trigger system is described in [8].
## 3 The muon identification procedure
The muon identification strategy can be divided in three steps:
* •
A loose binary selection of muon candidates based on the penetration of the
muons through the calorimeters and iron filters, which provides high
efficiency while reducing the misidentification probability of hadrons to the
percent level (called IsMuon);
* •
Computation of a likelihood for the muon and non-muon hypotheses, based on the
pattern of hits around the extrapolation to the different muon stations of the
charged particles trajectories reconstructed with high precision in the
tracking system. The logarithm of the ratio between the muon and non-muon
hypotheses is used as discriminating variable and called muDLL.
* •
Computation of a combined likelihood for the different particle hypotheses,
including information from the calorimeter and RICH systems. The logarithm of
the ratio between the muon and pion hypotheses is used as discriminating
variable and called DLL.
Additionally the number of tracks identified as muons that share a hit with a
given muon candidate (called NShared) can be used to further reject false
candidates.
### 3.1 IsMuon binary selection
The binary selection is defined according to the number of stations where a
hit is found within a field of interest (FOI) defined around the track
extrapolation. The number of stations required to have a muon signal is a
function of track momentum ($p$), as shown in Table 1. The sizes of the fields
of interest also depend on the particle momentum and are defined according to
the expected multiple scattering suffered by a muon when traversing the
material. The FOI are parameterized separately for the 4 regions of the 4
different stations downstream the calorimeter in both $x$ and $y$ directions
according to:
$\mathrm{FOI}=a+b\times\exp(-c\times p).$ (1)
The parameters $a$, $b$ and $c$ have been determined using muons from a full
detector Monte Carlo simulation [12].
Momentum range | Muon stations
---|---
3 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ $<p<$ 6 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ | M2 and M3
6 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ $<p<$ 10 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ | M2 and M3 and (M4 or M5)
$p>$ 10 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ | M2 and M3 and M4 and M5
Table 1: Muon stations required to trigger the IsMuon decision as a function
of momentum range.
For tracks passing the IsMuon requirement, the muon identification can be
further improved by a selection based on the logarithm of the ratio between
the likelihoods for the muon and non-muon hypotheses (muDLL).
### 3.2 Muon and non-muon likelihoods
The likelihoods are computed as the cumulative probability distributions of
the average squared distance significance $D^{2}$ of the hits in the muon
chambers with respect to the linear extrapolation of the tracks from the
tracking system. True muons tend to have a much narrower $D^{2}$ distribution,
close to zero, than the other particles that are incorrectly selected by the
IsMuon requirement.
The average squared distance significance is defined as:
$D^{2}={1\over N}\sum_{i}\left\\{\left({x^{i}_{closest}-x^{i}_{track}\over
pad^{i}_{x}}\right)^{2}+\left({y^{i}_{closest}-y^{i}_{track}\over
pad^{i}_{y}}\right)^{2}\right\\}$ (2)
where the index $i$ runs over the stations containing hits within the FOI,
($x^{i}_{closest,i},y^{i}_{closest}$) are the coordinates of the closest hit
to the track extrapolation point for each station
($x^{i}_{track},y^{i}_{track}$) and $pad^{i}_{x,y}$ correspond to one half of
the pad sizes in the x,y directions. The total number of stations containing
hits within their FOI is denoted by $N$.
The $D^{2}$ distribution for muons depends on the multiple scattering and,
therefore, on the momentum ($p$) and polar angle ($\theta$) distributions of
the analyzed sample. In order to avoid a dependence of the muon likelihood on
the calibration sample (with particular $p$ and $\theta$), the tuning of the
muon likelihood is performed separately in momentum bins and muon detector
regions (which correspond to 4 intervals in $\theta$).
The likelihood for the non-muon hypothesis is calibrated with the $D^{2}$
distribution for protons, since the other charged hadrons (pions or kaons)
selected by IsMuon will present a $D^{2}$ distribution with a component
identical to the protons and a component very similar to the true muons, due
to decays in flight before the calorimeter. For protons, the hits in the muon
system found around the track extrapolation are essentially due to three
sources: hits from punch-though [9] protons, hits from true muons pointing to
the same direction of the proton or random hits. The last two are at first
order uncorrelated to the proton momentum while the first one can present some
momentum dependence, less important however than the dependence expected for
muons.
Hence, the tuning of the non-muon likelihood is merely performed separately
for the 4 muon system regions, due to their different granularity.
The likelihood for the muon (or non-muon) hypothesis is then defined, for each
candidate, as the integral of the calibrated muon (or proton) $D^{2}$
probability density function from 0 to the measured value, $D^{2}_{0}$.
The results presented in this document are obtained with a muon likelihood
calibrated with muons from $J/\psi\rightarrow\mu^{+}\mu^{-}$ decays selected
from the data taken in 2010, as described in Section 4. The non-muon
likelihood has been calibrated with a simulated sample of decays
$\Lambda^{0}\rightarrow p\pi^{-}$ .
The $D^{2}$ distributions for muons, protons, pions and kaons obtained from
data are shown in Fig. 3(a). The distributions of the logarithm of the ratio
between the muon and non-muon hypotheses (muDLL) are shown in Fig. 3(b). More
details about the selection of the particles used to make these plots and to
extract the performance are given in Section 4.
### 3.3 Combined likelihoods
The muon and non-muon likelihoods presented in Section 3.2 can be combined
with the likelihoods provided by the RICH systems and the calorimeters to
improve the muon identification performance.
The Cherenkov angles measured in the two RICH detectors are combined with the
track momentum using an overall event log-likelihood algorithm. For each track
in the event, a likelihood is assigned to each of the different mass
hypotheses (electron, muon, pion, kaon and proton). The RICH likelihood can
differentiate between muon and other particles in particular at low momentum,
below 5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ [10].
The energy deposition in the calorimeters also allows the evaluation of
likelihoods for the muon (minimum ionizing particle), electron and hadron
hypotheses.
A combined log-likelihood is then obtained for each track and for each of the
different mass hypotheses by summing the logarithms of the likelihoods
obtained using the muon system, the RICH and the calorimeters. In this
computation, the non-muon likelihood obtained in the muon system is assigned
to the electron, pion, kaon and proton hypotheses. The difference of the
combined log-likelihoods for the muon and pion hypotheses (DLL) is then used
to identify the muons.
### 3.4 Discriminating variable based on hits sharing
Different tracks can be associated to the same muon hits when the matching of
tracks to muon chamber hits is performed. Reducing the number of tracks that
share hits can help to improve the misidentification probability. To use this
information, a discriminant variable named NShared is built for tracks
satisfying the IsMuon criteria and a score of 1 is added to a given track if
it shares any hits with another one. The score is given to the track to which
the hit is more distant. With this definition, a track having NShared=3, for
example, shares at least one hit with 3 other tracks in the event, all of them
with $D^{2}$ values smaller than the track own $D^{2}$. Selecting muons with
NShared=0 is the usual way to reduce the probability of incorrectly
identifying hadrons as muons due to nearby true muons in high multiplicity
events, but looser requirements can also be applied as shown in Fig. 4.
---
Figure 3: Average square distance significance distributions for muons,
protons, pions and kaons (a) and the corresponding muDLL distributions (b).
Figure 4: Normalized NShared distributions for muons, protons, kaons and
pions.
## 4 Method for the extraction of efficiencies
In order to extract the performance of the muon identification from data,
muon, proton, pion, and kaon candidates are selected with high purity from two
body decays using kinematical requirements only. When necessary, the purity is
improved by using a tag and probe technique where particle identification
requirements are applied to one of the tracks (tag) while the other (probe) is
used for the computation of the muon efficiency or of the hadron ${\rm
misidentification}\,\,{\rm probability}$.
### 4.1 Selection of control samples
An abundant source of muons is provided in the experiment by the
$J/\psi\rightarrow\mu^{+}\mu^{-}$ decay. By requiring the muons to have a high
impact parameter with respect to the primary vertex and the reconstructed
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ to have a large flight distance
significance and good decay vertex quality, most of the combinatorial
background originating from the tracks coming from the primary vertex is
removed and the sample gets enriched by $B\rightarrow J/\psi X$ candidates. In
order to reduce further the combinatorial background, one of the muons is
required to be identified as a muon. This is defined as the tag muon, while
the one being probed is only required to have $p_{\rm T}$
$>0.8\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
Protons are selected from the $\Lambda^{0}\rightarrow p\pi^{-}$ decays
reconstructed using decay vertex quality criteria and detachment of the decay
vertex from the primary one. Besides, the invariant mass obtained by assigning
the $\pi$ mass to the two daughters is required to be out of a window of 20
${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the nominal $K^{0}_{s}$
mass.
The $D^{*+}$ $\rightarrow$$\pi^{+}$ $D^{0}$ ($\rightarrow$$K^{-}$ $\pi^{+}$)
decays are the source of pions and kaons. Once again relatively high impact
parameter is required for the daughters of the $D^{0}$ while the $D^{0}$
flight direction is required to point to the primary vertex. To evaluate the
pion misidentification probability, the tag kaon is selected using a suitable
cut on the $\pi$-$K$ log-likelihoods difference, based on the RICH
information. To evaluate the kaon misidentification probability, the RICH
particle identification is used to identify the pion. Quality criteria are
used for the $D^{*+}$ and $D^{0}$ decay vertices. A window of
$25\,$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the nominal $D^{0}$
mass is used to exclude the doubly Cabibbo suppressed mode and the
$K^{+}K^{-}$ and $\pi^{+}\pi^{-}$ decay channels.
To avoid potential biases from the trigger requirements, in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\Lambda^{0}$ samples only
events triggered independently on the probe track are used; this condition has
to be satisfied at both hardware and software level, as explained in [11]. For
the $D^{0}\\!\rightarrow K^{-}\pi^{+}$ sample, a substantial fraction of the
events would be lost by such requirement. Therefore the hardware trigger is
required to be activated independently on the probe track (kaon or pion) and
the software trigger decision is based on impact parameter and detachment from
the primary vertex only, with no particle identification requirement.
After the background subtraction of selected two-body decays, the number of
muon, proton, pion and kaon candidates in the 2011 data samples are 2.4, 16.1,
11.7 and 12.3 millions, respectively.
### 4.2 Efficiency evaluation
As a baseline method to evaluate the efficiency $\epsilon_{muonID}$ of a
generic muon identification requirement denoted in this section by $muonID$
(e.g. IsMuon true or DLL greater than a given cut), is used :
$\epsilon_{muonID}=\frac{S_{true}}{S_{true}+S_{false}},$ (3)
where $S_{true}$ and $S_{false}$ are the numbers of signal events satisfying
and not satisfying $muonID$, extracted from data using
$S_{true,false}=N_{true,false}-B_{true,false}.$ (4)
$N_{true,false}$ are obtained by counting the number of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates with invariant mass
lying within a signal mass window around the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass; the number of background
events within the same mass window, $B_{true,false}$, is computed by
extrapolating to the signal window the mass fit done in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ sidebands.
For the proton ${\rm misidentification}\,\,{\rm probability}$, the same method
is used. The kaon and pion ${\rm misidentification}\,\,{\rm probabilities}$
are also obtained with Eq. 3, but $S_{true,false}$ and $B_{true,false}$ are
extracted directly from a full fit of the signal and background shapes to the
invariant mass distribution of the $D^{0}$ candidates.
## 5 Results
The muon identification performance is presented in terms of the muon
efficiency and hadron ${\rm misidentification}\,\,{\rm probabilities}$ for the
different requirements. In all cases, the performance is evaluated for tracks
extrapolated within the geometrical acceptance of the muon detector.
### 5.1 Performance of the IsMuon binary selection
The efficiency of the IsMuon requirement, $\varepsilon_{IM}\,\,$, is the
efficiency of finding hits within the fields of interest in the muon chambers
for tracks extrapolated to the muon system. In Fig. 5,
$\varepsilon_{IM}\,\,$is shown as a function of the muon momentum, for
different transverse momentum ranges. A weak dependency with transverse
momentum is observed and in particular a drop of $\sim$2% is measured for the
lowest $p_{\rm T}$ interval. This efficiency drop is essentially due to tracks
close to the inner edges of region R1 which in principle have their
extrapolation points within M1 and M5 acceptance, but are in fact scattered
outside the detector. For particles with $p_{\rm T}$ above 1.7
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, the efficiency is above 97% in the
whole momentum range, from 3 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 100
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The average efficiency obtained for
the $\,\mu_{\rm probe}$ in the $\rm J\\!/\\!\psi$ calibration sample is
$\varepsilon_{IM}\,\,$=($98.13\pm 0.04)$%, for particles with
$p>3\,$$\mathrm{\,Ge\kern-1.00006ptV}$ and $p_{\rm T}$
$>0.8\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
|
---|---
|
Figure 5: IsMuon efficiency and misidentification probabilities, as a function
of momentum, in ranges of transverse momentum: $\varepsilon_{IM}\,\,$(a),
$\wp_{IM}(p\rightarrow\mu)\,\,$(b), $\wp_{IM}(\pi\rightarrow\mu)\,\,$(c) and
$\wp_{IM}(K\rightarrow\mu)\,\,$(d).
The ${\rm misidentification}\,\,{\rm probabilities}$
$\wp_{IM}(p\rightarrow\mu)\,\,$, $\wp_{IM}(\pi\rightarrow\mu)\,\,$and
$\wp_{IM}(K\rightarrow\mu)\,\,$are also shown in Fig. 5. The observed decrease
of $\wp_{IM}\,\,$with increasing transverse momentum is expected, since tracks
with higher transverse momentum traverse the detector at higher polar angles,
in the lower occupancy regions. The proton ${\rm misidentification}\,\,{\rm
probability}$ is smaller than 0.5% for all $p_{T}$ ranges and momentum above
30${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. It drops quickly with momentum for
the lowest $p_{\rm T}$ ranges, reaching a plateau at about 30-40
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The pion and kaon ${\rm
misidentification}\,\,{\rm probabilities}$ have a similar behavior, increasing
with decreasing $p_{T}$. Above 40 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, the
pion ${\rm misidentification}\,\,{\rm probability}$ is almost at the level of
the proton ${\rm misidentification}\,\,{\rm probability}$. At low momentum,
decays in flight are the dominant source of incorrect identification, as can
be seen from the difference between the pion/kaon and proton curves. While the
proton ${\rm misidentification}\,\,{\rm probability}$, within the $p_{\rm T}$
intervals chosen, lies within 0.1-1.3%, the pion and kaon ${\rm
misidentification}\,\,{\rm probabilities}$ are within 0.2-5.6% and 0.6-4.5%,
respectively. For momentum above 30 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
$\wp_{IM}(\pi\rightarrow\mu)\,\,$and $\wp_{IM}(K\rightarrow\mu)\,\,$have a
small dependence on $p_{\rm T}$. At the lowest $p_{T}$ range, the kaon ${\rm
misidentification}\,\,{\rm probability}$ is lower than the pion for the lowest
momentum interval, in spite of the larger decay width of kaons to muons. Since
the muon is produced with a larger opening angle with respect to the original
track trajectory in kaon decays than in pion decays, and on average low
momentum particles tend to decay further upstream in the detector, then the
hits in the muon chambers have a higher probability to lie outside the fields
of interest.
When integrated over $p>3\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and the
whole $p_{\rm T}$ spectra of our calibration samples, the average values for
the ${\rm misidentification}\,\,{\rm probabilities}$ are
$\wp_{IM}(p\rightarrow\mu)\,\,$=(1.033 $\pm$ 0.003)%,
$\wp_{IM}(\pi\rightarrow\mu)\,\,$=(1.025$\pm$0.003)% and
$\wp_{IM}(K\rightarrow\mu)\,\,$=(1.111$\pm$0.003)%. For pions and kaons, about
60% of the ${\rm misidentification}\,\,{\rm probability}$ is due to decays in
flight, for these particular samples. The average efficiency and ${\rm
misidentification}\,\,{\rm probabilities}$, integrated over momentum
($p>3\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$), are also given in Table 2,
for 5 different $p_{\rm T}$ intervals. There are not enough candidates in the
muon, pion and kaon samples for a measurement dependent on momentum in the
lowest $p_{\rm T}$ bin. Similarly for the protons, in the highest $p_{\rm T}$
interval.
Table 2: Average IsMuon efficiency and ${\rm misidentification}\,\,{\rm probabilities}$ in different transverse momentum intervals (%). Uncertainties are statistical. $p_{T}$ interval (${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$) | muon | proton | pion | kaon
---|---|---|---|---
$p_{T}<0.8$ | | 1.393$\pm$ 0.005 | 6.2$\pm$ 0.1 | 4.3$\pm$0.1
$0.8<p_{T}<1.7$ | 96.94$\pm$ 0.07 | 0.737$\pm$ 0.003 | 2.19$\pm$ 0.01 | 1.93$\pm$0.1
$1.7<p_{T}<3.0$ | 98.53$\pm$ 0.05 | 0.149$\pm$ 0.004 | 0.61$\pm$ 0.01 | 0.93$\pm$0.01
$3.0<p_{T}<5.0$ | 98.51$\pm$ 0.06 | 0.12$\pm$ 0.02 | 0.40$\pm$ 0.01 | 0.72$\pm$0.01
$5.0<p_{T}$ | 98.51$\pm$ 0.07 | | 0.33$\pm$ 0.02 | 0.69$\pm$0.01
The LHCb detector has been designed to operate at the luminosity of ${\cal
L}=2\times 10^{32}\,$cm-2s-1 and with a probability of having one interaction
per beam crossing maximal with respect to higher numbers. However, in the 2011
run the experiment operated with an average number of interactions per beam
crossing about 2.5 times the nominal average, with a corresponding increase of
the overall detector occupancies. The behavior of $\varepsilon_{IM}\,\,$and
$\wp_{IM}\,\,$was then evaluated as a function of the number of tracks which
contain hits in the tracking subsystems, from the VELO to the tracking
stations. No significant decrease of $\varepsilon_{IM}\,\,$is observed, while
an increase of the ${\rm misidentification}\,\,{\rm probabilities}$ is seen
with higher track multiplicities, as expected. The detailed behaviour of both
the efficiency and the ${\rm misidentification}\,\,{\rm probabilities}$ as a
function of momentum is shown in Fig. 6. The probability
$\wp_{IM}(p\rightarrow\mu)\,\,$increases by a factor 2.7 for particles with
momentum in the range 3 to 5 ${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, when
comparing events with track multiplicity smaller than 40 and events with track
multiplicity between 150 and 250, which is the highest interval of
multiplicity analysed. At high momentum, the difference is much less
pronounced. For pions and kaons, the increase at low momentum is a factor of
two, approximately, and drops quickly to a plateau value starting at 20
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Since the FOI are smaller at high
momentum, the ${\rm misidentification}\,\,{\rm probability}$ becomes less
sensitive to the multiplicity of the underlying event.
|
---|---
|
Figure 6: IsMuon efficiency $\varepsilon_{IM}\,\,$(a) and $\wp_{IM}\,\,$for
protons (b), pions (c) and kaons (d) as a function of momentum for different
ranges of the number of trajectories reconstructed in the event (ntracks).
The charge dependence of the efficiency $\varepsilon_{IM}\,\,$is also
analysed. No difference between the efficiencies is seen up to the level of
the statistical fluctuations. When integrating over the whole momentum range,
the relative difference is 0.09$\pm$0.08%, compatible with zero within the
statistical uncertainty.
### 5.2 Performance of muon likelihoods
The muon identification efficiency ($\varepsilon_{\rm muDLL}\,\,$) is measured
as a function of a selection cut in the variable muDLL, for different momentum
ranges, as shown in Fig. 7(a). The ${\rm misidentification}\,\,{\rm
probabilities}$ are also shown in Fig. 7(b) to Fig. 7(d), for the same
momentum ranges. The black solid line shows the average fractions, when
integrated over $p>3\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ (and $p_{\rm T}$
$>0.8\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ for the muons). All curves
start at the efficiency or ${\rm misidentification}\,\,{\rm probability}$
corresponding to the IsMuon requirement. For tracks with
$p>10\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, the muon efficiency is
independent of momentum up to muDLL$\sim$2\. To achieve a misidentification
probability independent from the momentum, the value of the muDLL cut must
depend on particle momentum. By applying a muDLL cut irrespective of the
momentum, the ${\rm misidentification}\,\,{\rm probabilities}$ show a strong
momentum dependence.
|
---|---
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Figure 7: The efficiency $\varepsilon_{\rm muDLL}\,\,$as a function of muon
DLL cut for muons (a) and ${\rm misidentification}\,\,{\rm probabilities}$ for
protons (b), pions (c) and kaons (d). The black solid line shows the average
values integrated over $p>3\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The blue
dotted line correspond to particles in the range
$3<p<10\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The red dashed lines show
results for $10<p<20\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and the green
dashed-dotted for $p>20\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.
As an example, when requiring muDLL$\geq$1.74, a cut that provides a final
muon efficiency of 93.2%, the final misidentification probabilities are 0.21%,
0.78% and 0.52% for protons, kaons and pions respectively. This cut, which
provides a sharp decrease of 5% of the efficiency with respect to the IsMuon
efficiency, is used here as an example only for a clear comparison between the
muon DLL and the DLL. Since the average efficiency and ${\rm
misidentification}\,\,{\rm probabilities}$ values are given for our
calibration samples, which have their particular momentum and $p_{T}$
spectrum, they can be different for samples with different kinematic
distributions.
The momentum dependence of $\varepsilon_{\rm muDLL}\,\,$and of $\wp_{\rm
muDLL}\,\,$for particles satisfying this particular cut, muDLL$\geq$1.74, are
shown in Fig. 8, compared to the IsMuon requirement alone and a tighter
selection, muDLL$\geq$2.25. Again, this second cut was chosen for providing a
sharp reduction of the muon efficiency of 10% with respect to the IsMuon
efficiency. Once more, since the performance is integrated over $p_{T}$, small
variations from these values are expected for different samples, in particular
for the ${\rm misidentification}\,\,{\rm probabilities}$, which present a
stronger dependence with transverse momentum.
|
---|---
|
Figure 8: Muon efficiency (a) and ${\rm misidentification}\,\,{\rm
probabilities}$ for protons (b), pions (c) and kaons (d) as a function of the
particle momentum for the IsMuon requirement alone (black solid circles) and
with the additional cuts muDLL$\geq$1.74 (red triangles) and muDLL$\geq$2.25
(blue open circles).
### 5.3 Performance of combined likelihoods
The DLL efficiency is shown as a function of the pion and kaon ${\rm
misidentification}\,\,{\rm probabilities}$ in Fig. 9, together with the
results obtained using the muDLL alone, allowing for a direct comparison of
their performances.
The DLL benefits from RICH and calorimeter information, being more effective
than the muon DLL alone in separating pions and kaons from muons. After
IsMuon, this is the most used particle identification requirement used to
select muons in LHCb and the actual cut value is usually chosen according to
the compromise between purity and efficiency needed for that specific study.
The average misidentification rates corresponding to a cut which provides an
average decrease of 5% (equivalent to the one obtained with muDLL$\geq$1.74,
as previously shown) are around 0.65% and 0.38% for the kaons and pions,
respectively.
|
---|---
Figure 9: Average efficiency $\varepsilon_{\rm DLL}\,\,$as a function of the
pion (a) and kaon (b) ${\rm misidentification}\,\,{\rm probabilities}$ for
particles with momentum in the range $p>3$
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The dotted lines show the DLL
performance, while the muon DLL performance is shown with a solid line.
### 5.4 Performance of selections based on hits sharing
As mentioned in Section 3, after requiring IsMuon, an additional way of
reducing the incorrect identification probability of hadrons as muons, in
particular at high occupancy, is the use of a cut on NShared.
The muon efficiency is shown as a function of the pion ${\rm
misidentification}\,\,{\rm probability}$ for corresponding NShared cut in Fig.
10(a); protons are shown in Fig. 10(b). Due to similar decay-in-flight
pollution at low momentum, kaons behave as pions. The black solid line shows
the average values integrated over
$p>3\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The NShared selection is
particularly effective at low momenta, with increasing the FOI size.
|
---|---
Figure 10: Muon efficiency $\varepsilon_{\rm NShared}\,\,$as a function of the
pion and proton ${\rm misidentification}\,\,{\rm probabilities}$. The average
values, for all particles with $p>3\,$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$,
are shown with a black line, compared to the three momentum ranges separately,
as for Fig. 7.
### 5.5 Systematic checks
The effect of the trigger and of the method chosen to evaluate the efficiency
and misidentificatin probabilities are investigated. Alternatively to the
requirement of the $J/\psi\rightarrow\mu^{+}\mu^{-}$ sample being triggered
independently of the probe muon, a muon trigger decision based on the tag muon
was used to evaluate the IsMuon efficiency. The systematic uncertainty due to
the choice of trigger strategy is taken as the difference between the two
determinations, which is 0.2%.
When performing a full fit to the signal and background components of the mass
distributions used to extract the yields of signal events satisfying or not
the muon identification requirements, the resulting efficiencies and proton
${\rm misidentification}\,\,{\rm probability}$ rates agree within the
statistical uncertainties with the results shown in Section 5.
For the pion and kaon misidentification probabilities, the effect of the
trigger is studied and found to be negligible within the uncertainties,
independently of momentum and transverse momentum. Also the systematic
uncertainty related to the method used for the evaluation of the efficiency is
found to be negligible as a function of momentum, apart from a few intervals
where it is comparable with the statistical accuracy.
## 6 Conclusions
The performance of the muon identification procedure used in the LHCb
experiment has been evaluated, using a dataset corresponding to
1$\mbox{\,fb}^{-1}$ recorded in 2011 at
$\sqrt{s}=7\,$$\mathrm{\,Te\kern-1.00006ptV}$.
A loose binary criterium that can be used to select muons is based on the
matching of muon hits with the particle trajectory. For candidates satisfying
this requirement, likelihoods for muon and non-muon hypotheses are built with
the pattern of hits around the trajectories, which can be used to refine the
selection. An additional way of rejecting fake muon candidates is provided by
a variable sensitive to hit sharing by nearby particles.
The muon identification efficiency was observed to be robust against the
variation of detector occupancies and presents a weak dependence on momentum
and transverse momentum. Hadron ${\rm misidentification}\,\,{\rm
probabilities}$ present a stronger dependence on hit or track multiplicity,
however the highest increase factors are observed only for low momentum
particles.
Average muon identification efficiencies at the 98% level are attainable for
pion and kaon misidentification below the 1% level at high transverse
momentum, using the loosest identification criterium. The performance of
additional requirements based on likelihoods or on hits sharing can be tuned
according to the needs of each analysis and reduce the ${\rm
misidentification}\,\,{\rm probabilities}$ dependence on track multiplicity.
Adding a requirement on the difference of the log-likelihoods that provides a
total muon efficiency at the level of 93%, the hadron ${\rm
misidentification}\,\,{\rm probabilities}$ are below 0.6%.
###### 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 CERN and at the LHCb institutes, and acknowledge
support from the National Agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil);
CERN; NSFC (China); CNRS/IN2P3 (France); BMBF, DFG, HGF and MPG (Germany); SFI
(Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); ANCS
(Romania); MinES of Russia and Rosatom (Russia); MICINN, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from the ERC
under FP7 and the Region Auvergne.
## References
* [1] The LHCb Collaboration, _The LHCb Detector at the LHC_ , 32008S08005.
* [2] The LHCb Collaboration, _First evidence for the decay $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$_, _Phys. Rev. Lett._ 110 (2013) 021801.
* [3] The LHCb collaboration, _Measurement of the isospin asymmetry in ${B}\rightarrow{K}^{(*)}\mu^{+}\mu^{-}$ decays_, _J. High Energy Phys._ 07 (2012) 133.
* [4] The LHCb Collaboration, _Differential branching fraction and angular analysis of the decay ${B}^{0}\rightarrow{K}^{*0}\mu^{+}\mu^{-}$_, _Phys. Rev. Lett._ (2011) 108 181806\.
* [5] The LHCb Collaboration, _Measurement of the CP-violating phase $\phi_{s}$ in the decay${B}^{0}_{s}\rightarrow{J}/\psi\phi$_, _Phys. Rev. Lett._ 108 (2011) 101803.
* [6] The LHCb Collaboration, _Measurement of the CP violating phase $\phi_{s}$ in $\overline{B}^{0}_{s}\rightarrow{J}/\psi f_{0}(980)$_, _Phys. Lett. B_ 707 (2011) 497.
* [7] Alves Jr. et al, _Performance of the LHCb muon system_ , 82013P02022.
* [8] R. Aaij and J. Albrecht, _Muon triggers in the High Level Trigger of LHCb_ , LHCb-PUB-2011-017.
* [9] C. Grupen, _Particle Detectors_ , Cambridge university press, Cambridge, England 1996.
* [10] M. Adinolfi et al, _Performance of the LHCb RICH detector at the LHC_ , Eur. Phys. J. C (2013) 73:2431.
* [11] R. Aaij et al, _The LHCb Trigger and its Performance_ , 82013P04022.
* [12] M. Clemencic et al, _The LHCb simulation application, Gauss: design, evolution and experience_, _J. Phys.: Conf. Ser._ 331 (2011) 032023.
|
arxiv-papers
| 2013-06-02T20:52:19 |
2024-09-04T02:49:45.981236
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "F. Archilli, W. Baldini, G. Bencivenni, N. Bondar, W. Bonivento, S.\n Cadeddu, P. Campana, A. Cardini, P. Ciambrone, X. Cid Vidal, C. Deplano, P.\n De Simone, A. Falabella, M. Frosini, S. Furcas, E. Furfaro, M. Gandelman,\n J.A. Hernando Morata, G. Graziani, A. Lai, G. Lanfranchi, J.H. Lopes, O.\n Maev, G. Manca, G. Martellotti, A. Massafferri, D. Milanes, R. Oldeman, M.\n Palutan, G. Passaleva, D. Pinci, E. Polycarpo, R. Santacesaria, E.\n Santovetti, A. Sarti, A. Satta, B. Schmidt, B. Sciascia, F. Soomro, A.\n Sciubba and S. Vecchi",
"submitter": "\\'Erica Polycarpo",
"url": "https://arxiv.org/abs/1306.0249"
}
|
1306.0266
|
# Predicted alternative structure for tantalum metal under high pressure and
high temperature
Zhong-Li Liu [email protected] College of Physics and Electric Information,
Luoyang Normal University, Luoyang 471022, China Laboratory for Shock Wave
and Detonation Physics Research, Institute of Fluid Physics, P.O. Box 919-102,
621900 Mianyang, Sichuan, China Ling-Cang Cai Laboratory for Shock Wave and
Detonation Physics Research, Institute of Fluid Physics, P.O. Box 919-102,
621900 Mianyang, Sichuan, China Xiu-Lu Zhang Laboratory for Shock Wave and
Detonation Physics Research, Institute of Fluid Physics, P.O. Box 919-102,
621900 Mianyang, Sichuan, China Laboratory for Extreme Conditions Matter
Properties, Southwest University of Science and Technology, 621010 Mianyang,
Sichuan, China Feng Xi Laboratory for Shock Wave and Detonation Physics
Research, Institute of Fluid Physics, P.O. Box 919-102, 621900 Mianyang,
Sichuan, China
###### Abstract
First-principles simulations have been performed to investigate the phase
stability of tantalum metal under high pressure and high temperature (HPHT).
We searched its low-energy structures globally using our developed multi-
algorithm collaborative (MAC) crystal structure prediction technique. The
body-centred cubic (bcc) was found to be stable at pressure up to 300 GPa. The
previously reported $\omega$ and A15 structures were also reproduced
successfully. More interestingly, we observed another phase (space group:
Pnma, 62) that is more stable than $\omega$ and A15\. Its stability is
confirmed by its phonon spectra and elastic constants. For $\omega$-Ta, the
calculated elastic constants and high-temperature phonon spectra both imply
that it is neither mechanically nor dynamically stable. Thus, $\omega$ is not
the structure to which bcc-Ta transits before melting. On the contrary, the
good agreement of Pnma-Ta shear sound velocities with experiment suggests Pnma
is the new structure of Ta implied by the discontinuation of shear sound
velocities in recent shock experiment [J. Appl. Phys. 111, 033511 (2012)].
## I Introduction
Tantalum (Ta) metal is used frequently as the pressure scale in the diamond-
anvil cell (DAC) and shock wave (SW) experiments, thanks to its high
stability, chemical inertness, and very high melting temperature. It has
attracted tremendous interests on its very wide range of properties in recent
years. Among these properties, melting is the most intriguing one because
there are very large discrepancies in its high-pressure melting temperature
between different experiments. In detail, there exist several thousand degrees
of discrepancies in the melting temperature of Ta when extrapolating from the
diamond-anvil cell (DAC) Errandonea _et al._ (2001, 2003) pressures of
$\sim$100 GPa to the shock wave (SW) Brown and Shaner (1984) pressure of
$\sim$300 GPa. Extensive investigations have been performed to understand the
underlying causes both experimentally and theoretically. Foata-Prestavoine
_et al._ (2007); Taioli _et al._ (2007); Liu _et al._ (2008); Errandonea
(2005); Luo and Swift (2007); Ross _et al._ (2007); Verma _et al._ (2004);
Wang _et al._ (2001); Moriarty _et al._ (2002); Wu _et al._ (2009);
Burakovsky _et al._ (2010); Dewaele _et al._ (2010); Klug (2010); Haskins
_et al._ (2012); Ruiz-Fuertesa _et al._ (2010) Nevertheless, the melting
discrepancies of Ta still remain inconclusive up to now.
Most recently, Dewaele et al. Dewaele _et al._ (2010) revisited the melting
curve of Ta via DAC experiment and obtained a much higher result compared to
previously reported flat DAC melting curve Errandonea _et al._ (2001, 2003).
In their DAC experiment, they excluded the effects of chemical reactivity of
Ta samples with pressure medium and pressure medium melting. And they believe
that the previous tantalum melting curve Errandonea _et al._ (2001, 2003) has
been underestimated because of the undetected chemical reactions or errors of
pyrometry. Furthermore, the measured melting data are in general agreement
with our previous molecular dynamics simulations Liu _et al._ (2008) and ab
initio results by Taioli et al. Taioli _et al._ (2007) with the difference of
around 1000 K. While, almost at the same time, in another new DAC experiment
Ruiz-Fuertesa et al. still obtained the same flat melting curve, Ruiz-Fuertesa
_et al._ (2010) in contrast with the DAC results of Dewaele et al. Dewaele
_et al._ (2010)
Whether or not a solid-solid (SS) phase transition occurs before melting is
key to understand the large discrepancies between DAC and SW experiments, and
those between different DAC experiments. So, the SS transition in Ta has been
another attractive subject in recent years. Theoretically, Burakovsky et al.
reported a phase transition from bcc to hexagonal omega (hex-$\omega$) phase
above 70 GPa based on first-principles simulations. Burakovsky _et al._
(2010) They also believe that the previous DAC melting data Errandonea _et
al._ (2001, 2003) are problematic, because the observed sample motion in DAC
experiments is very likely not due to melting but internal stresses
accompanying a SS transition such as bcc-$\omega$ transition. Burakovsky _et
al._ (2010) In addition, in highly undercooled tantalum liquid Jakse et al.
observed another metastable structure, A15, in the molecular dynamics
simulations. Jakse _et al._ (2004) While, no SS phase transition was observed
in the new DAC experiment by Dewaele et al. Dewaele _et al._ (2010)
On one hand, there is at least about 1000 K regime between the new DAC data
Dewaele _et al._ (2010) and the theoretical predictions, Liu _et al._
(2008); Taioli _et al._ (2007) where it is possible to occur SS phase
transitions. On the other hand, the discontinuity in recent experimental shear
sound velocities $C_{l}$ at $\sim$60 GPa Hu _et al._ (2012) and the softening
of experimental and theoretical shear sound velocities Antonangeli _et al._
(2010) have the obvious characteristics of SS phase transitions. Furthermore,
Hsiung observed the displacive $\omega$ phase transition within the shock-
recovered polycrystalline tantalum after being shocked to relatively lower
pressure, 45 GPa. Hsiung and Lassila (1998, 2000); Hsiung (2010) It was
recently reported that the observed $\omega$ phase in the shock-recovered Ta
is not ideal hexagonal but pseudo-hexagonal. Hsiung (2010) But what Burakovsky
et al. Burakovsky _et al._ (2010) reported is the ideal $\omega$ phase. So,
it is necessary to further explore the phase stability of Ta at high pressure
and high temperature (HPHT) to understand the high-pressure melting
discrepancies between different experiments. This motivates us to further
investigate whether or not bcc-Ta transit to other phase before melting.
First, we globally searched the low-energy structures of Ta using our recently
developed ab initio MAC crystal structure prediction technique. Then, the
stability of the produced metastable structures was carefully checked.
Finally, the sound velocities of the stable and metastable structures were
deduced from the calculated elastic constants and compared with experiments.
The rest of this paper is organized as follows. Section II is the
computational details. We present the results and discussion in Section III.
The conclusions are drawn in Section IV.
## II Computational details
In order to determine the stable structures and alternative metastable
structures for Ta, we searched its low-energy structures from 0 to 300 GPa
with the interval of 10 GPa using the MAC crystal structure prediction
technique as implemented in our Muse code. The MAC algorithm combines
organically the multi algorithms including the evolutionary algorithm, the
simulated annealing algorithm and the basin hopping algorithm to search
collaboratively the global energy minima of materials with the fixed
stoichiometry. Liu After introduced the competition in all the evolutionary
and variation operators, the evolution of the crystal population and the
choice of the operators are self-adaptive automatically. That is to say the
crystal population undergoes a self-adaptive evolution process. So, it can
effectively find materials’ stable and metastable structures under certain
conditions only provided the chemical information of the materials. Liu More
importantly, Muse generates the random structures of the first generation with
symmetry constraints, and then largely shortens the optimization time of the
first generation and increase the diversity of crystal population. The random
structures can be created according to the randomly chosen space group numbers
from 2 to 230 and Wyckoff positions must be fit to the atom-number ratio of
different kinds. Liu Tests on systems including metallic, covalent and ionic
systems all show that Muse has very high efficiency and almost 100% success
rate.
The structures generated by the Muse code were optimized by vasp package.
Kresse and Hafner (1994); Kresse and Furthmüller (1996) We applied the
generalized gradient approximation (GGA) parametrized by PBE Perdew _et al._
(1996) and the electron-ion interaction described by the PAW scheme. Blöchl
(1994); Kresse and Joubert (1999) The pseudopotential for Ta has the valence
electrons’ configuration of $\mathrm{5p^{6}6s^{2}5d^{3}}$. To achieve good
convergences the kinetic energy cutoff and the k-point grids spacing were
chosen to be 500 eV and 0.03 $\mathrm{\AA}^{-1}$ in the calculations,
respectively. The accuracies of the target pressure and the energy convergence
for all optimizations are better than 0.1 GPa and $10^{-5}$ eV, respectively.
The systems containing 4, 8 and 16 atoms in the primitive cell were used in
all the structure searches.
## III Results and discussion
### III.1 Structure search for Ta under high pressure
The structures were generated with symmetry constraints and optimized at fixed
pressure in our MAC crystal structure searches. Liu The enthalpies of these
structures were calculated and compared to find the proper path towards the
lowest-enthalpy structure. Bcc has the lowest enthalpy among all the searched
structures. It is interesting that we found an energetically competitive
structure with the orthorhombic Pnma symmetry (space group: 62). Pnma-Ta has
four atoms in its primitive cell (see Fig. 1). The four atoms are at Wickoff’s
4c positions (0.132, 0.250, 0.366) with the lattice constants 4.98, 4.30 and
2.56 $\mathrm{\AA}$ at $\sim$100 GPa. This structure was then checked and
confirmed by Burakovsky. Burakovsky It is noted that we also reproduced the
previous reported Pm$\bar{3}$n (A15) Streitz and Moriarty ; Jakse _et al._
(2004) and P6/mmm (hex-$\omega$) Burakovsky _et al._ (2010); Hsiung and
Lassila (1998, 2000); Hsiung (2010) structures whose energy are much higher
than Pnma structure. Meanwhile, other produced structures of Ta with space
groups I4/mmm, C2/c, Fddd and so on, were also found to have much higher
energies than Pnma-Ta.
Figure 1: (Color online) The crystal structure of Pnma-Ta Figure 2: (Color
online) Enthalpy differences vs pressure with respective to bcc
In detail, we found that bcc is stable at pressure up to 300 GPa. The enthalpy
differences of some low-enthalpy structures relative to bcc are plotted in
Fig. 2. We note that Pnma-Ta has much lower enthalpies than the previously
reported hex-$\omega$ Burakovsky _et al._ (2010); Hsiung and Lassila (1998,
2000); Hsiung (2010) and A15 Streitz and Moriarty ; Jakse _et al._ (2004)
phases. But it has lower symmetry than hex-$\omega$ (P6/mmm, 191) and A15
(Pm$\bar{3}$n, 223). Their enthalpy differences with respective to bcc are all
positive and increase with pressure. This indicates that, at 0 K, bcc is the
most stable phase and becomes more and more stable as pressure increases.
While, it is probably not the case at elevated temperature and the Pnma
structure is expected to be more stable than bcc at HPHT. After all, Pnma-Ta
has much lower energy that hex-$\omega$ and A15 structures at high pressure.
Before we treat Pnma structure as an alternative metastable phase of Ta, we
should check its stability at high pressure.
### III.2 Stability of Pnma-Ta
To test the dynamic stability of this candidate metastable structure, we
calculated its phonon dispersion curve. We determined the vibrational
frequencies of Pnma-Ta using the density functional perturbation theory
(DFPT), Baroni _et al._ (1987, 2001) as implemented in the quantum-espresso
package. Giannozzi _et al._ (2009) We used the generalized gradient
approximation (GGA) proposed by Perdew, Burke and Ernzerhof (PBE) Perdew _et
al._ (1996) as the exchange-correlation functional. A nonlinear core
correction to the exchange-correlation energy function was introduced to
generate a Vanderbilt ultrasoft pseudopotential for Ta with the valence
electrons’ configuration 4s25p65d36s2, and the ultrasoft pseudopotentials were
generated with a scalar-relativistic calculation. As an example, the
calculated phonon dispersion curve of Pnma-Ta at 96 GPa was shown in Fig. 3.
There are no imaginary frequencies in any position of the first Brillouin zone
at pressure up to $\sim$250 GPa. This indicates the predicted Pnma-Ta is
dynamically stable under compression.
Figure 3: Phonon dispersion curve of Pnma-Ta at 96 GPa
The mechanical stability of material is reflected by the elastic constants. We
calculated the high-pressure elastic constants of Pnma-Ta through stress-
strain relationship, similar to Ref. Jochym and Parlinski, 2000. All the
resulting elastic constants of Pnma-Ta are positive and increase with pressure
(Table. 1). To check its mechanical stability under compression, we used the
stability criteria of orthorhombic crystal, Wu _et al._ (2007)
$C_{11}>0,\quad C_{22}>0,\quad C_{33}>0,$
$C_{44}>0,\quad C_{55}>0,\quad C_{66}>0,$
$C_{11}+C_{22}+C_{33}+2(C_{12}+C_{13}+C_{23})>0,$
$C_{11}+C_{22}-2C_{12}>0,$
$C_{11}+C_{33}-2C_{13}>0,$
$C_{22}+C_{33}-2C_{23}>0.$
After careful check, we found the calculated elastic constants conform to
these stability criteria at all pressures, suggesting Pnma-Ta is also
mechanically stable in the pressure range of interest.
Table 1: Elastic constants of Pnma-Ta at different pressures $P$ and atomic volumes $V$. The elastic constants are all in GPa. $P$(GPa) | $V$($\mathrm{\AA^{3}}$) | $C_{11}$ | $C_{22}$ | $C_{33}$ | $C_{12}$ | $C_{13}$ | $C_{23}$ | $C_{44}$ | $C_{55}$ | $C_{66}$
---|---|---|---|---|---|---|---|---|---|---
41.1 | 15.75 | 513 | 535 | 492 | 249 | 190 | 235 | 99 | 86 | 132
52.5 | 15.25 | 575 | 600 | 568 | 269 | 206 | 248 | 118 | 98 | 151
65.5 | 14.75 | 646 | 674 | 650 | 290 | 227 | 261 | 140 | 112 | 172
80.5 | 14.25 | 727 | 759 | 746 | 316 | 250 | 276 | 165 | 128 | 196
118.3 | 13.25 | 936 | 965 | 997 | 384 | 301 | 307 | 227 | 171 | 252
142.1 | 12.75 | 1066 | 1087 | 1147 | 431 | 333 | 327 | 264 | 196 | 286
170.0 | 12.25 | 1210 | 1226 | 1315 | 483 | 371 | 356 | 304 | 224 | 326
203.0 | 11.75 | 1372 | 1385 | 1502 | 551 | 414 | 390 | 348 | 254 | 371
242.2 | 11.25 | 1569 | 1584 | 1727 | 630 | 467 | 429 | 395 | 298 | 423
### III.3 Sound velocities of Pnma-Ta
From the elastic constants, we deduced its high-pressure sound velocities in
the way similar to our previous work. Liu _et al._ (2011) To compare with
experiments, we also calculated the elastic constants of bcc and deduced its
high-pressure velocities with the same method. The comparison of the
calculated sound velocities with experiments are plotted in Fig. 4. It is
noted that our calculated sound velocities are at 0 K. So, one should be
careful in the comparison. Below $\sim$100 GPa, our obtained bcc shear sound
velocities are in excellent agreement with the results from the inelastic
x-ray scattering (IXS) experiment at room temperature. Antonangeli _et al._
(2010) While at 100 GPa, the shear sound velocity begins the softening
behavior, consistent with the IXS experiment, Antonangeli _et al._ (2010) and
it recovers at about 250 GPa.
Figure 4: (Color online) High pressure sound velocities of bcc and Pnma Ta
compared with experiments. The experimental data are from Refs. Brown and
Shaner, 1984; Antonangeli _et al._ , 2010; Hu _et al._ , 2012 and Yu _et
al._ , 2006.
Recently, Hu et al. observed a discontinuation of Hugoniot shear sound
velocity $C_{l}$ at $\sim$60 GPa, implying a SS phase transition under shock
compression. Hu _et al._ (2012) Our calculated 0 K bcc-Ta sound velocities
are in good agreement with their experimental data below 50 GPa. While, their
experimental shear sound velocities have an abrupt decrease at about 60 GPa,
implying a SS phase transition. Our bcc-Ta shear sound velocities have no such
discontinuation at 60 GPa. But our shear sound velocities of Pnma-Ta deviate
from those of bcc-Ta. It is interesting that our calculated shear sound
velocities of Pnma-Ta reproduced the experimental discontinuation of shear
sound velocities above 60 GPa. For bulk sound velocity $C_{b}$, our calculated
values of both bcc and Pnma are in good agreement with experiments,
respectively, and they have no discontinuation in the pressure range of
interest. Our shear sound velocities of Pnma-Ta are also in good agreement
with those of Hu et al. This suggests that solid Ta transits from bcc to Pnma
phase under shock compression. It is noted that both our calculated 0 K shear
sound velocities of bcc-Ta and the root temperature IXS data Antonangeli _et
al._ (2010) show the elastic softening at 100 GPa, but the Hugoniot shear
sound velocities Hu _et al._ (2012) have softening at 60 GPa. This is
attributed to the effects of the high temperature and the shock shear stress,
which lower the phase transition pressure.
### III.4 Stability of hex-$\omega$ Ta
In order to determine the mechanical stability of hex-$\omega$-Ta, we also
calculated its high-pressure elastic constants after full relaxation. For
hexagonal lattices, there are five independent elastic constants, i.e.
$C_{11}$, $C_{12}$, $C_{13}$, $C_{33}$ and $C_{44}$. We calculated the elastic
constants as the second derivatives of the internal energy with respect to the
strain tensor which leads to volume conserving lattice. The elastic constants
were calculated at the equilibrium relaxed structure at fixed volume $V$ by
keeping the strains in the lattice and relaxing the symmetry allowed internal
degrees of freedom, and by finding the quadratic coefficients after fitting
the forth-order polynomial curves to the total energies and strains. The
method for reducing the five elastic constants is the same as that in Ref.
Steinle-Neumann _et al._ , 1999.
Figure 5: The elastic constants of hex-$\omega$ Ta vs pressure.
The elastic constants of hex-$\omega$-Ta as the function of pressure are
plotted in Fig. 5. As one can see, all the elastic constants except $C_{44}$
are positive and increase with pressure. Shear modulus $C_{44}$ is negative
and decreases with increasing pressure. This indicates that hex-$\omega$ phase
is mechanically unstable at high pressure and 0 K. However, one can not
conclude that it is unstable at high-temperature. So, we should check the
high-temperature stability of hex-$\omega$ phase from the high-temperature
phonon dispersion.
It was pointed that the mechanically unstable phase at 0 K can be stabilized
by vibrational entropy of lattice at elevated temperature, i.e. the
interactions between phonons. Souvatzis _et al._ (2008) The harmonic phonon
calculation methods, typically the frozen phonon method and the density
functional perturbation theory (DFPT), Baroni _et al._ (1987, 2001) only
treat with the 0 K lattice, despite that they can include the electronic
temperature through the finite-temperature DFT. In order to include the high-
temperature lattice vibration, we applied the self-consistent ab initio
lattice dynamics (SCAILD) method recently developed by Souvatzis et al.
Souvatzis _et al._ (2008) It has been successfully applied to many metals and
other systems. Souvatzis _et al._ (2008); Souvatzis and Rudin (2008); Luo
_et al._ (2010); Souvatzis _et al._ (2009); Božin _et al._ (2010); Souvatzis
_et al._ (2010, 2011); Souvatzis (2011) The SCAILD method which is based on
the calculation of Hellman-Feynman forces on atoms in a supercell is the
robust extension of the frozen phonon method. It takes into account the
anharmonic effects induced by the interactions between phonons.
In the SCAILD method, all phonons are excited together in the same cell by
displacing atoms situated at the undistorted positions
$\mathbf{R+b_{\sigma}}$, and the distorted positions are changed to
$\mathbf{R+b_{\sigma}+U_{R\sigma}}$, where the displacements are written as
Souvatzis _et al._ (2008, 2010)
$\mathbf{U_{R\sigma}}=\frac{1}{\sqrt{N}}\sum_{\mathbf{q,s}}\mathcal{A}_{\mathbf{qs}}^{\sigma}\epsilon_{\mathbf{qs}}^{\sigma}e^{i\mathbf{q(R+b_{\sigma})}}.$
(1)
In Eq.(1), $\mathbf{R}$ is the $N$ Bravais lattice sites of the supercell,
$\mathbf{b_{\sigma}}$ the position of atom $\sigma$ with respect to this site,
$\epsilon_{\mathbf{qs}}^{\sigma}$ are the phonon eigenvectors corresponding to
the phonon mode, $s$. The mode amplitude $\mathcal{A}_{\mathbf{qs}}^{\sigma}$
can be evaluated from the different phonon frequencies $\omega_{\mathbf{q,s}}$
through Souvatzis _et al._ (2008, 2010)
$\mathcal{A}_{\mathbf{qs}}^{\sigma}=\pm\sqrt{\frac{\hbar}{2M_{\sigma}\omega_{\mathbf{ks}}}coth\left(\frac{\hbar\omega_{\mathbf{qs}}}{2k_{B}T}\right)},$
(2)
where $T$ is the temperature of the system. The phonon frequencies Souvatzis
_et al._ (2008, 2010)
$\omega_{\mathbf{qs}}=\left[\sum_{\sigma}\frac{\epsilon_{\mathbf{qs}}^{\sigma}\mathbf{F_{q}^{\sigma}}}{\mathcal{A}_{\mathbf{qs}}^{\sigma
M_{\sigma}}}\right]^{1/2},$ (3)
are obtained from the Fourier transform $\mathbf{F_{q}^{\sigma}}$ of the
forces acting on the atoms in the supercell. The phonon frequencies are
calculated in a self-consistent manner according to Eqs.1-3, through which we
alternate the forces on the displaced atoms and calculate new phonon
frequencies and new displacements. The interaction between different lattice
vibrations are included in SCAILD as the simultaneous presence of all the
commensurate phonons in the same force calculation. The phonon frequencies are
thus renormalized by the very same interaction.Souvatzis _et al._ (2010)
Figure 6: (Color online) The 0 K and high-temperature phonon dispersion curves
of hex-$\omega$-Ta at different pressures.
The force calculations were performed with vasp, Kresse and Hafner (1994);
Kresse and Furthmüller (1996) within GGA parametrized by PBE. Perdew _et al._
(1996) The electron-ion interactions were also described by the PAW scheme
Blöchl (1994); Kresse and Joubert (1999) with the energy cutoff of 350 eV.
Fermi-Dirac temperature smearing was applied to the Kohn-Sham occupational
number together with a $6\times 6\times 6$ Monkhorst-Pack k-point grid. The
electronic temperature was set as the same of lattice through Fermi smearing
width. After tests, the supercell was chosen to contain 54 atoms resulted from
the $3\times 3\times 2$ replication of hex-$\omega$ unit cell.
In the SCAILD calculations, the forces calculations are very demanding, and
the high temperature phonon dispersion spectra were converged after more than
100 SCAILD loops. We calculated the finite temperature phonon dispersion from
0 to 300 GPa. As examples, the phonon dispersion spectra at 70, 100, 150 and
300 GPa are plotted in Fig. 6. At 0 K, all the dispersion curves contain
considerable imaginary frequencies in the acoustic branches, in agreement with
the calculations of Burakovsky et al. Burakovsky _et al._ (2010) And at high
temperature the converged phonon dispersion curves still have considerable
imaginary frequencies at pressures from 0 to 300 GPa and temperatures from
4000 to 7000 K. So according to the SCAILD theory, hex-$\omega$ phase is
unstable at high temperature under compression. This does not support the
previous theoretical predictions that bcc-$\omega$ transition occurs before
melting above 70 GPa. Burakovsky _et al._ (2010)
### III.5 Discussion
From our elastic constants and high-temperature phonon dispersion curves, we
see that the ideal hex-$\omega$ phase is not stable at 0 K and elevated
temperature under high pressure. Furthermore, the hex-$\omega$ phase is not
energetically competitive compared with the Pnma structure according to our
structure searches. However, what Hsiung observed in the recent experiment is
the pseudo-hex-$\omega$ with fractional coordinates: (0,0,0),
(2.056/3,0.944/3,1/2), and (0.944/3,2.056/3,1/2). Hsiung (2010) We started
from these initial coordinates in our optimizations, but obtained the ideal
hex-$\omega$ with coordinates: (0,0,0), (2/3,1/3,1/2), and (1/3,2/3,1/2). And
the high-temperature phonon dispersion curves of hex-$\omega$ Ta also show
considerable imaginary frequencies, suggesting it is not dynamically stable.
The negative shear moduli $C_{44}$ of hex-$\omega$ Ta reflected its mechanical
instability. Why the pseudo-hex-$\omega$ structure is stable and observed in
experiment is probably because of the existence of impurities such as W atoms
Hsiung (2010) in the samples. W atoms prevent Ta atoms from moving to their
ideal equilibrium positions in hexagonal lattice.
From the ab initio molecular dynamics simulations of Ta, Wu et al. Wu _et
al._ (2009) found that shear induces bcc to transit to a viscous plastic flow
(partially disordered, partially crystalline structure) before melting. So
they argued that this transition was misinterpreted as melting in previous DAC
Errandonea _et al._ (2001, 2003) experiments due to the similarity of the
plastic flow to liquid. The substantial elastic softening and the
discontinuities of the Hugoniot shear sound velocities Hu _et al._ (2012) and
the shear sound speed at room temperature Burakovsky _et al._ (2010) of the
bcc phase imply the SS phase transition in solid Ta. Just as Klug reviewed,
possible additional phases except $\omega$ may complicate the measurement of
the melting point and the pyrometric techniques also have space to improve.
Klug (2010) The additional phases with very low symmetries resemble the local
structure in the liquid phase. So, the transformations from bcc to such low
symmetry phases bring the difficulties to identify the occurrence of melting
in experiments. Our predicted new phase with Pnma low symmetry is expected to
be the phase to which bcc transits before melting.
## IV Conclusions
In conclusion, in order to find low-energy structures of Ta at high pressure,
we globally searched the structures for pure Ta from 0 to 300 GPa using the ab
initio MAC crystal structure prediction technique. We found that bcc is the
unique stable phase in this pressure range. It is interesting that we also
found the energetically competitive Pnma-Ta. Pnma-Ta was tested to be more
stable than hex-$\omega$ and A15 structures. The calculated elastic constants
of Pnma-Ta satisfy the mechanical stability criteria of orthorhombic crystal.
The absence of imaginary frequencies in Pnma-Ta at high pressure indicates
that it is also dynamically stable. To check the stability of the previously
reported hex-$\omega$ Ta, we calculated its elastic constants and found it is
not mechanically stable at high pressure and 0 K. To further explore the high
temperature stability of hex-$\omega$ Ta, we calculated its high-temperature
phonon spectra within the framework of the SCAILD approximation. The resulting
high temperature phonon spectra contain considerable imaginary frequencies in
the acoustic branches, also indicating the instability of hex-$\omega$ phase
at HPHT. So we believe that Ta transits from bcc to Pnma at HPHT before
melting.
## V acknowledgments
The research was supported by the National Natural Science Foundation of China
(11104127, 11104227 and U1230201/A06), the Project 2010A0101001 funded by
CAEP, and the Science Research Scheme of Henan Education Department under
Grand No. 2011A140019.
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|
arxiv-papers
| 2013-06-03T01:06:06 |
2024-09-04T02:49:45.990107
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhong-Li Liu, Ling-Cang Cai, Xiu-Lu Zhang, Feng Xi",
"submitter": "Z. L. Liu",
"url": "https://arxiv.org/abs/1306.0266"
}
|
1306.0272
|
# One can hear the area and curvature of boundary of a domain by hearing the
Steklov eigenvalues
Genqian Liu
###### Abstract.
For a given bounded domain $\Omega$ with smooth boundary in a smooth
Riemannian manifold $(\mathcal{M},g)$, we show that the Poisson type upper-
estimate of the heat kernel associated to the Dirichlet-to-Neumann operator,
the Sobolev trace inequality, the Log-Sobolev trace inequality, the Nash trace
inequality, and the Rozenblum-Lieb-Cwikel type inequality are all equivalent.
Upon decomposing the Dirichlet-to-Neumann operator into a sum of the square
root of the Laplacian and a pseudodifferntial operator and by applying Grubb’s
method of symbolic calculus for the corresponding pseudodifferential heat
kernel operators, we establish a procedure to calculate all the coefficients
of the asymptotic expansion of the trace of the heat kernel associated to
Dirichlet-to-Neumann operator as $t\to 0^{+}$. In particular, we explicitly
give the first four coefficients of this asymptotic expansion. These
coefficients give precise information regarding the area and curvatures of the
boundary of the domain in terms of the spectrum of the Steklov problem.
###### 1991 Mathematics Subject Classification:
35P20, 53C44, 58J35, 58J50
Key words and phrases. Sobolev trace inequality; Dirichlet-to-Neumann
operator; Steklov eigenvalue; heat kernel associated to Dirichlet-to-Neumann
operator; Poisson upper bound; Asymptotic expansion; Curvature
Department of Mathematics, Beijing Institute of Technology, Beijing 100081,
the People’s Republic of China. E-mail address: [email protected]
## 1\. Introduction
An interesting question in Analysis is the following: what is the relationship
between the geometrical quantitative characteristics of a bounded domain and
the spectrum of the Steklov problem on the boundary of the domain? This
problem originates from inverse spectral problems, where the known data is the
spectrum of a differential or a pseudodifferential operator and one wishes to
recover the geometry of a manifold. A rather efficient method to deal with
this question is the asymptotic expansion of the trace of the heat kernel of
the Dirichlet-to-Neumann (also called the Steklov-Poincaré) operator (the so-
called “heat kernel method”). The coefficients of the asymptotic expansion of
the trace of such a heat kernel not only provides spectral invariants but also
gives some geometric and topological information. The heat kernel method is
closely related to the Calderón problem and also has many applications in
shape recognition, detection of distant physical objects (such as stars or
atoms, or moving objects) from the light or sound they emit, identification of
inhomogeneities (for instance in medical imaging) through the determination of
different conductivities (see, [56] and [59], [105], [58]).
Let $\mathcal{M}$ be an $(n+1)$-dimensional, smooth, complete Riemannian
manifold with metric tensor $g=(g_{jk})$, and let $\Omega$ be a bounded domain
in $\mathcal{M}$ with smooth boundary $\partial\Omega$. The Laplace-Beltrami
operator associated with the metric $g$ is given in local coordinates by
(1.1)
$\displaystyle\triangle_{g}u=\frac{1}{\sqrt{|g|}}\sum_{j,k=1}^{n+1}\frac{\partial}{\partial
x_{j}}\left(\sqrt{|g|}\,g^{jk}\frac{\partial u}{\partial x_{k}}\right),$
where $(g^{jk})$ is the inverse of the metric tensor $(g_{jk})$ and
$|g|=\mbox{det}\,g$. For the Dirichlet problem associated with (1.1),
(1.4)
$\displaystyle\left\\{\begin{array}[]{ll}\Delta_{g}u=0&\mbox{in}\;\;\Omega,\\\
u=\phi&\mbox{on}\;\;\partial\Omega,\end{array}\right.$
we denote the Dirichlet-to-Neumann operator in this case by
(1.5) $\displaystyle{\mathcal{N}}_{g}\phi=\frac{\partial u}{\partial\nu},$
where $\nu=(\nu_{1},\cdots,\nu_{n+1})$ denotes the unit inward normal to
$\partial\Omega$.
The Dirichlet-to-Neumann operator is a self-adjoint, first order elliptic
pseudodifferential operator (see p. 37-38 of [107]). Prototypical in inverse
problems, the Dirichlet-to-Neumann operator is related to the Calderón problem
[16] of determining the anisotropic conductivity of a body from current and
voltage measurements at its boundary. The Calderón problem has been solved
affirmatively for the isotropic conductivity of a body by J. Sylvester and G.
Uhlmann in higher dimensional case ($\mbox{dim}\,M\geq 3$) [104], and by A.
Nachman for two dimensional case [83]. Generalizations to less regular
conductivities has been obtained by a number of authors (see, [2], [10], [19],
[49], [83], [85], [89], [14], [87], [84], [15], [62], [105], [6]).
Unfortunately, ${\mathcal{N}}_{g}$ doesn’t determine $g$ uniquely for general
$n$-dimensional Riemannian manifold (This observation is due to L. Tartar, see
[65] for an account). Because $\partial\Omega$ is compact, the spectrum of
${\mathcal{N}}_{g}$ is nonnegative, discrete and unbounded (see p. 95 of [9],
[40]). The spectrum $\\{\lambda_{k}\\}_{k=1}^{\infty}$ of this operator is
just the Steklov spectrum of the domain $\Omega$. More precisely,
$\displaystyle\left\\{\begin{array}[]{ll}\Delta_{g}u_{k}=0\quad\;\quad\mbox{in}\;\;\Omega,\\\
\frac{\partial
u_{k}}{\partial\nu}=-\lambda_{k}u_{k}\quad\;\mbox{on}\;\;\partial\Omega,\end{array}\right.$
where $u_{k}$ is the eigenfunction corresponding to the $k$-th Steklov
eigenvalue $\lambda_{k}$. The study of the spectrum of ${\mathcal{N}}_{g}$ was
initiated by Steklov in 1902 (see [100]). Eigenvalues and eigenfunctions of
this operator are used in fluid mechanics, heat transmission and vibration
problems (see [39] and [63]). Denote by $\omega_{n}$ the volume of the unit
ball in ${\mathbb{R}}^{n}$. A famous asymptotic formula of L. Sandgren [96]
(This formula was first established by Sandgren, and a sharp form was given by
the author in [77]) states that
(1.7) $\displaystyle
N(\tau)=\\#\\{k\big{|}\lambda_{k}\leq\tau\\}=\frac{\omega_{n}\big{(}\mbox{vol}(\partial\Omega)\big{)}\tau^{n}}{(2\pi)^{n}}+o(\tau^{n})\quad\;\mbox{as}\;\;\tau\to+\infty,$
or, what is the same,
(1.8)
$\displaystyle\quad\quad\quad\;Z=\mbox{Tr}\;e^{t{\mathcal{N}}_{g}}=\sum_{k=1}^{\infty}e^{-\lambda_{k}t}\sim\frac{\Gamma(n+1)\,\omega_{n}(\mbox{vol}(\partial\Omega))}{(2\pi)^{n}\,t^{n}}\,=\frac{\Gamma(\frac{n+1}{2})\,\mbox{vol}(\partial\Omega)}{\pi^{\frac{n+1}{2}}t^{n}}\;\quad\;\mbox{as}\;\;t\to
0^{+}.$
Asymptotic formula (1.8) shows that you can hear the area of the boundary
$\partial\Omega$ by the first term of asymptotic expansion for
Tr$\,e^{t{\mathcal{N}}_{g}}$. Therefore our problem just asks: how can one
obtain more terms in the asymptotic expansion for the
Tr$\,e^{t{\mathcal{N}}_{g}}$ ? This problem is quite similar to the well-known
Kac problem for the Laplacian on a domain (The Kac question asks: is it
possible to hear the shape of a domain just by hearing all of the eigenvalues
of the Dirichlet Laplacian? see [8], [12], [28], [57], [61], [67], [5], [47],
[46], [79], [3], [97], [27], [21], [24] and the references therein).
In order to explain the main method of this paper, we briefly review the
historical background for the case of Dirichlet Laplacian on domains. In 1910,
H. A. Lorentz conjectured that for a two-dimensional domain
$\Omega\subset{\mathbb{R}}^{2}$, the asymptotics of the counting function of
the Dirichlet eigenvalues $\\{\mu_{k}\\}$ are given by:
(1.9) $\displaystyle
N_{D}(\tau)=\\#\\{k\big{|}\mu_{k}\leq\tau\\}=\frac{\mbox{vol}(\Omega)}{2\pi}\tau+o(\tau)\quad\,\,\mbox{as}\;\;\tau\to\infty.$
This asymptotic in particular implies that $\mbox{vol}(\Omega)$ is a spectral
invariant. Lorentz’s conjecture was proved in 1913 by Hermann Weyl (see [111]
and [112]). With Weyl’s formula as a starting point, Pleijel [90] in 1954
showed that
(1.10)
$\displaystyle\sum_{k=1}^{\infty}e^{-\mu_{k}t}\sim\frac{\mbox{vol}(\Omega)}{2\pi
t}-\frac{\mbox{vol}(\partial\Omega)}{4}\,\frac{1}{\sqrt{2\pi
t}}\quad\;\mbox{as}\;\;t\to 0^{+},$
where $\mbox{vol}\,(\partial\Omega)$ is the length of boundary
$\partial\Omega$. By a Tauberian theorem the asymptotic formula for the first
term on the right side of (1.10) is equivalent to Weyl’s formula (1.9). For
simply connected domains Pleijel established the formula
(1.11)
$\displaystyle\sum_{k=1}^{\infty}e^{-\mu_{k}t}\sim\frac{\mbox{vol}(\Omega)}{2\pi
t}+\frac{\mbox{vol}(\partial\Omega)}{4}\,\frac{1}{\sqrt{2\pi
t}}+\frac{1}{6}\quad\;\mbox{as}\;\;t\to 0^{+}.$
Kac [61] used a combination of probability techniques and heat equation
methods to establish (1.10) for convex domains, and he obtained (1.11) as a
limiting case of convex polygonal domains. Kac also conjectures that for
multiply connected domains in ${\mathbb{R}}^{2}$ with $r$ holes, the number
$\frac{1}{6}$ in (1.11) should be replaced by $\frac{1}{6}(1-r)$. McKean and
Singer in a celebrated paper [80] gave an affirmative answer to the conjecture
of Kac with respect to the third term for multiply connected domains in
$n$-dimensional Riemannian manifold (with or without boundary). McKean and
Singer [80] also obtained information about the curvature of the boundary of
$\Omega$, which showed that the Euler characteristic $\chi(\Omega)$ is also a
spectral invariant. Gilkey [44] explicitly calculated the first four
coefficients of the expansion of the trace of the heat kernel. In 1991,
Gordon, Webb and Wolpert [46], found examples of pairs of distinct plane
domains with the same spectrum (there are higher dimensional examples for a
similar problem by Gordon-Webb [47]). In [80], the heat kernel estimates of
the Laplacian play an important role in the asymptotic expansion. The two-
sided Gaussian estimates for the heat kernel associated with a uniformly
elliptic operator in ${\mathbb{R}}^{n}$ was proved by Aronson [4] (see also
[92], [32], [22], [50], [88], [38]):
(1.12) $\displaystyle\frac{c}{t^{n/2}}e^{-|x-y|^{2}/ct}\leq
G(t,x,y)\leq\frac{C}{t^{n/2}}e^{-|x-y|^{2}/Ct},$
where $c$ and $C$ are two positive constants. Varadhan [108], [109] first
realized that the Riemannian distance should be used instead. His result
implies that, on any manifold,
$\displaystyle\lim_{t\to 0^{+}}t\,\ln G(t,x,y)=-\frac{d^{2}(x,y)}{4},$
where $d(x,y)$ is the Riemannian distance between $x$ and $y$ (see also,
[86]). For a complete Riemannian manifold with Ricci curvature bounded below,
Li-Yau [73] and Sturm [102] obtained upper and lower Gaussian estimates on the
heat kernel, from which Varadhan’s asymptotic result follows immediately.
Varopoulos [110] proved that the Sobolev inequality is not only sufficient but
also necessary for the on-diagonal upper bound of heat kernel of Laplacian.
Carlen, Kusuoka and Stroock [17] proved that the upper bound of heat kernel is
equivalent to the Nash inequality (see also [48]). Davies [30], [31], [32]
proved that the on-diagonal upper bound of the heat kernel is also equivalent
to the log-Sobolev inequality. In addition, the Sobolev inequality is
equivalent to the Rozenblum-Lieb-Cwikel inequality (see [93], [72], [29],
[74], [26], [70] and [68]).
Let us come back to the Steklov spectrum. Because ${\mathcal{N}}_{g}$ is a
pseudodifferntial operator, the corresponding problems become much more
difficult than those of the Laplacian. To get more geometric information from
the Steklov eigenvalues on $\partial\Omega$, we consider the heat kernel
associated to the Dirichlet-to-Neumann operator on $\partial\Omega$:
(1.15) $\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial
u(t,x)}{\partial
t}={\mathcal{N}}_{g}u(t,x)&\mbox{in}\;\;[0,+\infty)\times\partial\Omega,\\\
u(0,x)=\phi(x)&\mbox{on}\;\;\partial\Omega,\end{array}\right.$
where ${\mathcal{N}}_{g}$ is the Dirichlet-to-Neumann operator on
$\partial\Omega$. By perturbation of a fractional Laplacian, Gimperlein and
Grubb in [45] have recently proved the following important Poisson upper and
lower bounds for the heat kernel ${\mathcal{K}}(t,x,y)$ of the Dirichlet-to-
Neumann operator:
$\displaystyle{\mathcal{K}}(t,x,y)\leq\frac{Ct}{(t^{2}+d^{2}(x,y))^{1/2}}\left[\frac{1}{(t^{2}+d^{2}(x,y))^{n/2}}+1\right]\quad\mbox{for
all}\;\;t>0\;\;\mbox{and}\;\;x,y\in\partial\Omega,$
$\displaystyle{\mathcal{K}}(t,x,y)\geq\frac{ct}{(t^{2}+d^{2}(x,y))^{(n+1)/2}}\quad\;\,\mbox{for}\;\;t+d(x,y)<r,$
where $r>0$ is some constant (The Poisson upper bound estimate
${\mathcal{K}}(t,x,y)\leq C(t\wedge
1)^{-n}\big{(}1+\frac{d(x,y)}{t}\big{)}^{-n-1}\;\;\mbox{for
all}\;\;x,y\in\partial\Omega\;\;\mbox{and}\,\,t>0,$
of the kernel was also obtained by Elst and Ouhabaz in [35]).
In the first part of this paper, combining classical methods and some new
techniques we show that the Poisson type upper-estimate of the heat kernel
associated to the Dirichlet-to-Neumann operator is equivalent to the Sobolev
trace inequality, to the Log-Sobolev trace inequality, to the Nash trace
inequality, and to the Rozenblum-Lieb-Cwikel type inequality.
For the coefficients of asymptotic expansion of the trace of the heat kernel
associated to the Dirichlet-to-Neumann operator, the first coefficient
$a_{0}(n,x)$ had been known by (1.8); $a_{1}(2,x)$ had been obtain in [37];
the coefficients $a_{1}(n,x)$ and $a_{2}(n,x)$ were explicitly calculated in
[91] and [78] in different ways. In the second part of the paper, by
decomposing the Ditrichlet-to-Neumann operator and by applying Grubb’s method
(see [51]), we establish a procedure, from which all coefficients of the
asymptotic expansion of the trace of the heat kernel associated to the
Dirichlet-to-Neumann operator can be calculated as $t\to 0^{+}$. In
particular, we explicitly give the first four coefficients of the asymptotic
expansion. This provides the information for the area and curvature of the
boundary.
The main ideas are as follows: we first decompose the Dirichlet-to-Neumann
operator ${\mathcal{N}}_{g}$ into a sum of the negative square root
$-\sqrt{-\Delta_{h}}$ of the Laplacian and a pseudodifferential operator $B$
on $\partial\Omega$, where $h$ is the induced metric on $\partial\Omega$ by
$g$, then we calculate their principal, second, third and $(M-1)$-th symbols
for $-\sqrt{-\Delta_{h}}$ and $B$. Because the heat kernels of $\Delta_{h}$
and $-\sqrt{-\Delta_{h}}$ have an analytical formula, we can explicitly
calculate the heat kernel ${\mathcal{K}}_{V}(t,x,y)$ of $-\sqrt{-\Delta_{h}}$.
Therefore the heat kernel $\mathcal{K}(t,x,y)$ of ${\mathcal{N}}_{g}$ can be
expanded into the form
${\mathcal{K}}_{V}(t,x,y)+{\mathcal{K}}_{V_{-2}}(t,x,y)+\cdots+{\mathcal{K}}_{V_{-M}}(t,x,y)+{\mathcal{K}}_{V^{\prime}_{M}}(t,x,y)$
as $t\to 0^{+}$, where ${\mathcal{K}}_{V_{-2}}(t,x,y)$,
${\mathcal{K}}_{V_{-3}}(t,x,y)$ and ${\mathcal{K}}_{V_{M}}(t,x,y)$ corresponds
to the principal, second and $(M-1)$-th symbols of $e^{tB}$. By considering
the trace of the heat kernel of the Dirichlet-to-Neumann operator,
(1.16)
$\displaystyle\mbox{Tr}\,e^{t{\mathcal{N}}_{g}}=\int_{\partial\Omega}\mathcal{K}(t,x,x)dS(x)=\sum_{k=1}^{\infty}e^{-\lambda_{k}t},$
and by a remainder estimate we eventually obtain the following asymptotic
expansion:
$\displaystyle\int_{\partial\Omega}{\mathcal{K}}(t,x,x)dS(x)=t^{-n}\int_{\partial\Omega}\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\,dS(x)+t^{1-n}\int_{\partial\Omega}a_{1}(n,x)\,dS(x)$
$\displaystyle\quad\;+\cdots+t^{M-1-n}\int_{\partial\Omega}a_{M-1}(n,x)\,dS(x)+\left\\{\begin{array}[]{ll}O(t^{M-n})\\\
O(t\log t),\end{array}\right.\quad\;\mbox{as}\;\;t\to 0^{+},$
where $M$ can be taken as $2,3,4,\cdots$ with $n\geq M-1$, and the notation
$O(t^{m})$ (respectively, $O(t\log t)$) denotes a function which satisfies
$|O(t^{m})|\leq c_{0}t^{m}$ (respectively, $|O(t\log t)|\leq c_{0}t\log t$)
for some constant $c_{0}>0$ and all $t>0$. The first coefficient
$a_{0}(n,x)=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}$ is independent
of $n$ and $x$. The second coefficient $a_{1}(n,x)$ depends only on the “area”
$\mbox{vol}(\partial\Omega)$ and the mean curvature of the boundary
$\partial\Omega$. The coefficient $a_{2}(n,x)$ depends not only on the “area ’
$\mbox{vol}(\partial\Omega)$ and the principal curvatures
$\kappa_{1},\cdots,\kappa_{n}$, but also on the scalar curvature
${\tilde{R}}_{\Omega}$ (respectively $R_{\partial\Omega}$) of $\Omega$
(respectively $\partial\Omega$). Finally the fourth coefficient $a_{3}(n,x)$
depends on $\mbox{vol}(\partial\Omega)$, the principal curvatures, the Ricci
tensor ${\tilde{R}}_{jj}$ (respectively $R_{jj}$) and the scalar curvature
${\tilde{R}}_{\Omega}$ (respectively $R_{\partial\Omega}$ of $\Omega$
(respectively $\partial\Omega$) as well as the covariant derivative
$\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1),(n+1)}$ of the Ricci curvature with
respect to $(\bar{\Omega},g)$ (see Theorem 6.1). Generally, $a_{M-1}(n,x)$ can
be represented in terms of the geometical quantities (see, (ii) of Remark
6.2). This asymptotic expansion shows that one can hear the “area” of
$\partial\Omega$ and $\int_{\partial\Omega}a_{M-1}(n,x)dS(x)$
($M=1,2,3,\cdots$) by “hearing” all of the Steklov eigenvalues. It also
implies that $\int_{\partial\Omega}a_{M-1}(n,x)dS(x)$ ($M\geq 1$) are all
spectral invariants, from which we know that two domains with different
spectral invariants just mentioned above, can never have the same Steklov
spectrum. As a by-product, we explicitly give the first four coefficients of
the asymptotic expansion of the trace of the corresponding heat kernel
associated to the negative square root $-\sqrt{-\Delta_{h}}$ of the Laplacian,
which also tells us that the first four terms of asymptotic expansion of the
heat kernel can be obtained not only for any smooth elliptic partial
differential operator of degree $2,3,4,5$ (see, p. 45 of [80] and p. 613 of
[44]) but also for the elliptic pseudodifferential operator of fractional-
order $\frac{1}{2}$.
## 2\. Preliminaries
Let $\Omega$ be a bounded domain with smooth boundary in a smooth Riemannian
manifold $(\mathcal{M},g)$. $\mathcal{K}$ is said to be a fundamental solution
of $\frac{\partial u}{\partial t}={\mathcal{N}}_{g}u$, if for any fixed
$y\in\partial\Omega$,
(2.3)
$\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial\mathcal{K}(t,x,y)}{\partial
t}={\mathcal{N}}_{g}\mathcal{K}(t,x,y),\quad t>0,\,\,x\in\partial\Omega,\\\
\mathcal{K}(0,x,y)=\delta(x-y),\end{array}\right.$
where ${\mathcal{N}}_{g}$ acts on the $x$ variable and $\delta(x-y)$ is the
delta function concentrated at $y$. $\mathcal{K}$ is also called the heat
kernel associated to the Dirichlet-to-Neumann operator (or the kernel of the
semigroup $e^{t{\mathcal{N}}_{g}}$). The initial condition in (2.3) means that
for every continuous function $\phi(x)$ on $\partial\Omega$, if
$x\in\partial\Omega$ then
(2.4) $\displaystyle\lim_{t\to
0^{+}}\int_{\partial\Omega}\mathcal{K}(t,x,y)\phi(y)\,dS(y)=\phi(x).$
For the special manifold $({\mathbb{R}}^{n+1}_{+},g)$, where
${\mathbb{R}}^{n+1}_{+}$ is the upper half-space
$\\{(x_{1},\cdots,x_{n+1})\in{\mathbb{R}}^{n+1}\big{|}x_{n+1}\geq 0\\}$, and
(2.5)
$\displaystyle\left(g_{jk}\right)_{(n+1)\times(n+1)}=\begin{pmatrix}h_{11}&h_{12}&\cdots&h_{1n}&0\\\
h_{21}&h_{22}&\cdots&h_{2n}&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
h_{n1}&h_{n2}&\cdots&h_{nn}&0\\\ 0&0&\cdots&0&1\end{pmatrix}$
is a positive-definite, real symmetric $(n+1)\times(n+1)$ constant matrix. It
is easy to verify that the function
(2.6) $\displaystyle
u(x,x_{n+1})=\int_{{\mathbb{R}}^{n}}P(x-z,x_{n+1})\phi(z)\,\sqrt{|h|}\,dz$
is a solution of the problem:
(2.9)
$\displaystyle\left\\{\begin{array}[]{ll}\sum_{j,k=1}^{n+1}h^{jk}\frac{\partial^{2}u}{\partial
x_{j}\partial x_{k}}=0,&\mbox{in}\;\;R^{n+1}_{+},\\\
u=\phi,&\mbox{on}\;\;\partial R^{n+1}_{+},\end{array}\right.$
where $(h^{jk})$ is the inverse of $\left(h_{jk}\right)$, and the Poisson
kernel is
(2.10) $\displaystyle
P(x,x_{n+1})=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\,\frac{x_{n+1}}{(\sum_{i,j=1}^{n}h_{jk}x_{j}x_{k}+x_{n+1}^{2})^{\frac{n+1}{2}}}.$
This implies
$\displaystyle\frac{\partial u}{\partial\nu}=\frac{\partial u}{\partial
x_{n+1}}\bigg{|}_{x_{n+1}=0}=\int_{{\mathbb{R}}^{n}}\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\,\left(\sum_{j,k=1}^{n}h_{jk}(x_{j}-z_{j})(x_{k}-z_{k})\right)^{-\frac{n+1}{2}}\phi(z)\,\sqrt{|h|}\,dz.$
In other words, the Dirichlet-to-Neumann operator
${\mathcal{N}}_{g}:H^{\frac{1}{2}}(\partial{\mathbb{R}}^{n+1}_{+})\rightarrow
H^{-\frac{1}{2}}(\partial{\mathbb{R}}^{n+1}_{+})$ is given by
(2.11)
$\displaystyle\qquad\quad\quad{\mathcal{N}}_{g}\phi(x)=\int_{{\mathbb{R}}^{n}}\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\,\left(\sum_{j,k=1}^{n}h_{jk}(x_{j}-z_{j})(x_{k}-z_{k})\right)^{-\frac{n+1}{2}}\phi(z)\,\sqrt{|h|}\,dz,$
from which we can see
(2.12)
$\displaystyle{\mathcal{N}}_{g}\phi=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\,\sqrt{-\sum_{j,k=1}^{n}h_{jk}\frac{\partial^{2}}{\partial
x_{j}\partial x_{k}}}\;\;\phi.$
Furthermore by applying Fourier transform, with the aid of the basic method of
the linear transform and the following formula
$\displaystyle\left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\int_{{\mathbb{R}}^{n}}e^{-t|\xi|}e^{\pm
ix\cdot\xi}d\xi=\frac{\sqrt{2^{n}}\,\Gamma\big{(}\frac{n+1}{2}\big{)}}{\sqrt{\pi}}\,\frac{t}{\left(t^{2}+|x|^{2}\right)^{\frac{n+1}{2}}},$
we can get that for a positive definite and symmetric constant matrix
$(h_{jk})_{n\times n}$,
(2.13) $\displaystyle
u(t,x)=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\,\frac{t}{\left(t^{2}+\sum_{j,k=1}^{n}h_{jk}x_{j}x_{k}\right)^{\frac{n+1}{2}}}$
is a fundamental solution of
(2.16) $\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial
u(t,x)}{\partial t}={\mathcal{N}}_{g}u(t,x),\quad
x\in\partial{\mathbb{R}}^{n+1}_{+},\;t>0,\\\
u(0,x)=\delta(x).\end{array}\right.$
For the Laplacian $-\Delta_{h}$ on $\partial\Omega$, we can define the square
root operator $\sqrt{-\Delta_{h}}$ (see [107]). Stinga and Torrea (see [101])
showed that if $u$ satisfies
(2.19)
$\displaystyle\left\\{\begin{array}[]{ll}\Delta_{h}u+\frac{\partial^{2}u}{\partial
x_{n+1}^{2}}=0&\mbox{in}\;\;\partial\Omega\times(0,\infty)\\\
u(x,0)=\phi(x)&\mbox{for}\;\;x\in\partial\Omega,\end{array}\right.$
then
(2.20) $\displaystyle\frac{\partial u}{\partial
x_{n+1}}=-\sqrt{-\Delta_{h}}\,\phi(x)$
and
(2.21)
$\displaystyle-\sqrt{-\Delta_{h}}\,\phi(x)=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\left(e^{\mu\Delta_{h}}\phi(x)-\phi(x)\right)\frac{d\mu}{\mu^{3/2}},$
where $e^{t\Delta_{h}}$ is the heat semigroup generated by $\Delta_{h}$ on
$\partial\Omega$.
If $U$ is an open subset of ${\mathbb{R}}^{n}$, we denote by
$S^{m}_{1,0}=S^{m}_{1,0}(U,{\mathbb{R}}^{n})$ the set of all $p\in
C^{\infty}(U,{\mathbb{R}}^{n})$ such that for every compact set $K\subset U$
we have
(2.22) $\displaystyle|D^{\beta}_{x}D^{\alpha}_{\xi}p(x,\xi)|\leq
C_{K,\alpha,\beta}(1+|\xi|)^{m-|\alpha|},\quad\;x\in
K,\,\,\xi\in{\mathbb{R}}^{n}$
for all $\alpha,\beta\in{\mathbb{N}}^{n}_{+}$. The elements of $S^{m}_{1,0}$
are called symbols (or total symbols) of order $m$. It is clear that
$S^{m}_{1,0}$ is a Fréchet space with semi-norms given by the smallest
constants which can be used in (2.22) (i.e.,
$\displaystyle\|p\|_{K,\alpha,\beta}=\,\sup_{x\in
K}\bigg{|}\left(D_{x}^{\beta}D_{\xi}^{\alpha}p(x,\xi)\right)(1+|\xi|)^{|\alpha|-m}\bigg{|}).$
Let $p(x,\xi)\in S^{m}_{1,0}$. A pseudo-differential operator in an open set
$U\subset{\mathbb{R}}^{n}$ is essentially defined by a Fourier integral
operator (cf. [55], [98]):
(2.23) $\displaystyle
P(x,D)u(x)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{{\mathbb{R}}^{n}}p(x,\xi)e^{ix\cdot\xi}\hat{u}(\xi)d\xi,$
and denoted by $OPS^{m}$. Here $u\in C_{0}^{\infty}(U)$ and
$\hat{u}(\xi)=\left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\int_{{\mathbb{R}}^{n}}e^{-i\langle
y,\xi\rangle}u(y)dy$ is the Fourier transform of $u$. If there are smooth
$p_{m-j}(x,\xi)$, homogeneous in $\xi$ of degree $m-j$ for $|\xi|\geq 1$, that
is, $p_{m-j}(x,r\xi)=r^{m-j}p_{m-j}(x,\xi)$ for $r,|\xi|\geq 1$, and if
(2.24) $\displaystyle p(x,\xi)\sim\sum_{j\geq 0}p_{m-j}(x,\xi)$
in the sense that
(2.25) $\displaystyle p(x,\xi)-\sum_{j=0}^{N}p_{m-j}(x,\xi)\in
S^{m-N-1}_{1,0},$
for all $N$, then we say $p(x,\xi)\in S_{cl}^{m}$, or just $p(x,\xi)\in
S^{m}$. We call $p_{m}(x,\xi)$, $p_{m-1}(x,\xi)$ and $p_{m-2}(x,\xi)$ the
principal, second and third symbols of $P(x,D)$, respectively. Let
$\mathcal{M}$ be a smooth $n$-dimensional Riemannian manifold (of class
$C^{\infty}$). We will denote by $C^{\infty}(\mathcal{M})$ and
$C_{0}^{\infty}(\mathcal{M})$ the space of all smooth complex-valued functions
on $\mathcal{M}$ and the subspace of all functions with compact support,
respectively. Assume that we are given a linear operator
$\displaystyle P:C^{\infty}_{0}(\mathcal{M})\to C^{\infty}(\mathcal{M}).$
If $G$ is some chart in $\mathcal{M}$ (not necessarily connected) and
$\kappa:G\to U$ its diffeomorphism onto an open set
$U\subset{\mathbb{R}}^{n}$, then let ${\tilde{P}}$ be defined by the diagram
$\displaystyle\begin{CD}C_{0}^{\infty}(G)@>{P}>{}>C^{\infty}(G)\\\
@A{\kappa^{*}}A{}A@A{}A{\kappa^{*}}A\\\
C_{0}^{\infty}(U)@>{\tilde{P}}>{}>C^{\infty}(U)\end{CD}$
where $\kappa^{*}$ is the induced transformation from $C^{\infty}(U)$ into
$C^{\infty}(G)$, taking a function $u$ to the function $u\circ\kappa$. (note,
in the upper row is the operator $r_{G}\circ P\circ i_{G}$, where $i_{G}$ is
the natural embedding $i_{G}:C_{0}^{\infty}(G)\to C^{\infty}_{0}(M)$ and
$r_{G}$ is the natural restriction $r_{G}:\,C^{\infty}(M)\to C^{\infty}(G)$;
for brevity we denote this operator by the same letter $P$ as the original
operator). An operator $P:C_{0}^{\infty}(\mathcal{M})\to
C^{\infty}(\mathcal{M})$ is called a pseudodifferential operator on
$\mathcal{M}$ if for any chart diffeomorphism $\kappa:G\to U$, the operator
$\tilde{P}$ defined above is a pseudodifferential operator on $U$. An operator
$P$ is said to be an elliptic pseudodifferential operator of order $m$ if for
every compact $K\subset\Omega$ there exists a positive constant $c=c(K)$ such
that
$\displaystyle|p(x,\xi)\geq c|\xi|^{m},\,x\in K,\,|\xi|\geq 1$
for any compact set $K\subset\Omega$. If $P$ is a non-negative elliptic
pseudodifferential operator of order $m$, then the spectrum of $P$ lies in a
right half-plane and has a finite lower bound
$\gamma(P)=\inf\\{\mbox{Re}\,\lambda\big{|}\lambda\in\sigma(P)\\}$. We can
modify $p_{m}(x,\xi)$ for small $\xi$ such that $p_{m}(x,\xi)$ has a positive
lower bound throughout and lies in
$\\{\lambda=re^{i\theta}\big{|}r>0,|\theta|\leq\theta_{0}\\}$, where
$\theta_{0}\in(0,\frac{\pi}{2})$. According to [51], the resolvent
$(P-\lambda)^{-1}$ exists and is holomorphic in $\lambda$ on a neighborhood of
a set
$\displaystyle
W_{r_{0},\epsilon}=\\{\lambda\in{\mathbb{C}}\big{|}|\lambda|\geq
r_{0},\mbox{arg}\,\lambda\in[\theta_{0}+\epsilon,2\pi-\theta_{0}-\epsilon],\,\mbox{Re}\,\lambda\leq\gamma(P)-\epsilon\\}$
(with $\epsilon>0$). There exists a parametrix $Q^{\prime}_{\lambda}$ on a
neighborhood of a possibly larger set (with $\delta>0,\epsilon>0$)
$\displaystyle
V_{\delta,\epsilon}=\\{\lambda\in{\mathbb{C}}\big{|}|\lambda|\geq\delta\;\;\mbox{or
arg}\,\lambda\in[\theta_{0}+\epsilon,2\pi-\theta_{0}-\epsilon]\\}$
such that this parametrix coincides with $(P-\lambda)^{-1}$ on the
intersection. Its symbol $q(x,\xi,\lambda)$ in local coordinates is
holomorphic in $\lambda$ there and has the form (cf. Section 3.3 of [51])
(2.26) $\displaystyle q(x,\xi,\lambda)\sim\sum_{l\geq
0}q_{-m-l}(x,\xi,\lambda)$
where
(2.27) $\displaystyle
q_{-m}=(p_{m}(x,\xi)-\lambda)^{-1},\quad\;q_{-m-1}=b_{1,1}(x,\xi)q^{2}_{-m},$
$\displaystyle\,\cdots,\,q_{-m-l}=\sum_{k=1}^{2l}b_{l,k}(x,\xi)q^{k+1}_{-m},\cdots,$
with symbols $b_{l,k}$ independent of $\lambda$ and homogeneous of degree
$mk-l$ in $\xi$ for $|\xi|\geq 1$. The semigroup $e^{-tP}$ can be defined from
$P$ by the Cauchy integral formula (see p. 4 of [45]):
$\displaystyle
e^{-tP}=\frac{i}{2\pi}\int_{\mathcal{C}}e^{-t\lambda}(P-\lambda)^{-1}d\lambda,$
where $\mathcal{C}$ is a suitable curvature in the positive direction around
the spectrum of $P$.
## 3\. Upper estimates of the heat kernel, Sobolev trace inequality, Log-
Sobolev trace inequality, Nash trace inequality and Rozenblum-Lieb-Cwikel type
inequality
We now can establish the equivalence for the Poisson type upper-estimate of
the heat kernel associated to the Dirichlet-to-Neumann operator with a number
of other inequalities.
Theorem 3.1. Let $(\mathcal{M},g)$ be an $n+1$ dimensional Riemannian manifold
($n\geq 2$), and let $\Omega$ be a bounded domain with smooth boundary. Then
the following statements are equivalent:
(i) Sobolev trace inequality: there exist positive constants $A$ and $B$ such
that for all $v\in W^{1,2}(\Omega)$,
(3.1)
$\displaystyle\left(\int_{\partial\Omega}|v|^{\frac{2n}{n-1}}dS\right)^{\frac{n-1}{n}}\leq
A\int_{\Omega}|\nabla_{g}v|^{2}dV+B\int_{\partial\Omega}|v|^{2}dS,$
where $dV$ and $dS$ are the Riemannian elements of volume and area on $\Omega$
and $\partial\Omega$, respectively;
(ii) Log-Sobolev trace inequality: for all $v\in W^{1,2}(\Omega)$ and all
$\epsilon>0$,
(3.2)
$\displaystyle\int_{\partial\Omega}v^{2}\ln|v|\,dS\leq\epsilon\int_{\Omega}|\nabla_{g}v|^{2}dV+\beta(\epsilon)\|v\|_{L^{2}(\partial\Omega)}^{2}+\|v\|_{L^{2}(\partial\Omega)}^{2}\ln\|v\|_{L^{2}(\partial\Omega)},$
where
$\beta(\epsilon)=\frac{n}{2}\ln\frac{nA}{2e}+BA^{-1}\epsilon-\frac{n}{2}\ln\epsilon$;
(iii) Nash trace inequality: for all $v\in W^{1,2}(\Omega)$,
(3.3)
$\displaystyle\|v\|_{L^{2}(\partial\Omega)}^{2+\frac{2}{n}}\leq\left(A\|\nabla_{g}v\|_{L^{2}(\Omega)}^{2}+B\|v\|_{L^{2}(\partial\Omega)}^{2}\right)\|v\|_{L^{1}(\partial\Omega)}^{2/n};$
(iv) On-diagonal Dirichlet-to-Neumann heat kernel upper bound:
(3.4)
$\displaystyle{\mathcal{K}}(t,x,y)\leq\left(nAe/4\right)^{n}\,\frac{e^{BA^{-1}t}}{t^{n}}\,\,\mbox{for
all}\;\;x,y\in\partial\Omega\;\;\mbox{and}\;\;t>0;$
(v) Off-diagonal Dirichlet-to-Neumann heat kernel upper bound:
(3.5)
$\displaystyle{\mathcal{K}}(t,x,y)\leq\frac{Ct}{(t^{2}+d^{2}(x,y))^{\frac{n+1}{2}}}\quad\,\mbox{for
all}\;\;x,y\in\partial\Omega\,\,\mbox{and}\,\,0\leq t\leq 1.$
(vi) The Rozenblum-Lieb-Cwikel type inequality: let $q\in
L^{1}_{loc}(\partial\Omega)$ and $q_{-}\in L^{n}(\partial\Omega)$, where
$q_{-}(x)=-\min\\{q(x),0\\}$. For $\alpha\geq 0$, let $I_{q}(\alpha)$ be the
number of eigenvalues $\lambda$ satisfying
(3.6)
$\displaystyle(A{\mathcal{N}}_{g}-B-q)\phi(x)=-\lambda\phi(x)\quad\,x\in\partial\Omega$
with $\lambda\leq-\alpha$, where $A$ and $B$ are the positive constants in the
Sobolev trace inequality (3.1). Then there exists a positive constant $C(n)$
depending only on $n$ such that
(3.7) $\displaystyle I_{q}(0)\leq C(n)\int_{\partial\Omega}q_{-}^{n}\,dS.$
Remark 3.2. (a) The sharp Sobolev trace inequality was proved by Y.-Y. Li and
M. Zhu (see Theorem 0.1 of [75]). Li and Zhu proved that the constant
$A=2(n-1)^{-1}\big{(}(n+1)\omega_{n+1}\big{)}^{-\frac{1}{n}}$ in the front of
$\int_{\partial\Omega}|\nabla_{g}v|^{2}dV$ is optimal. It cannot be replaced
by any smaller number.
(b) In the Rozenblum-Lieb-Cwikel type inequality, we can take $C(n)=e^{n}$
(see the proof for (i)$\Rightarrow$(vi)). If we replace the Sobolev constants
$A$ and $B$ in (3.6) by any two positive real numbers ${\tilde{A}}$ and
${\tilde{B}}$, then the corresponding inequality (3.7) still holds; however,
the constant $C(n)$ should be replaced by a new constant
$C(n,{\tilde{A}},{\tilde{B}})$ which depends only on $n$, ${\tilde{A}}$ and
${\tilde{B}}$.
(c) The estimate (3.7) is sharp, in the following sense. Replacing $q$ by
$\beta q$ in (3.6), where $\beta>0$ is a large parameter, we derive from (3.7)
$\displaystyle I_{\beta q}(0)\leq
C(n)\beta^{n}\int_{\partial\Omega}q^{n}_{-}dS.$
Similar to Rozenblum’s method [93] (see also, [68]), it is easy to prove that
for any $q\in C^{\infty}(\partial\Omega)$, $I_{\beta q}(0)$ has the asymptotic
formula:
(3.8) $\displaystyle\lim_{\beta\to+\infty}\beta^{-n}I_{\beta
q}(0)=\frac{\omega_{n}}{(2\pi)^{n}}\int_{\partial\Omega}\frac{q^{n}_{-}}{A^{n}}dS.$
Here the asymptotic constant $\frac{\omega_{n}}{(2\pi)^{n}}$ is just the
corresponding coefficient in Sandgren’s asymptotic formula (1.7).
Proof of Theorem 3.1. (i) $\Longrightarrow$ (ii): Given $f\in W^{1,2}(\Omega)$
such that $\|f\|_{L^{2}(\partial\Omega)}=1$, we introduce the measure
$\displaystyle dW(x)=f^{2}(x)dS(x).$
It follows from $\int_{\partial\Omega}f^{2}dS=1$ that
$\int_{\partial\Omega}dW=1$. Since $\ln\phi$ is a concave function of $\phi$,
by applying Jensen’s inequality we have
$\int_{\partial\Omega}\ln\phi\,dW\leq\ln\int_{\partial\Omega}\phi\,dW$.
Putting $\phi=|f|^{q-2}$ with $q=\frac{2n}{n-1}$, we get
$\displaystyle\int_{\partial\Omega}(\ln|f|^{q-2})f^{2}dS\leq\ln\int_{\partial\Omega}|f|^{q-2}f^{2}dS=\ln\|f\|_{L^{q}(\partial\Omega)}^{q},$
i.e.,
$\displaystyle\int_{\partial\Omega}f^{2}\ln|f|\,dS\leq\frac{q}{q-2}\ln\|f\|_{L^{q}(\partial\Omega)}.$
Since $\frac{q}{q-2}=n$, by applying the Sobolev trace inequality (3.1) we
have
$\displaystyle\int_{\partial\Omega}f^{2}\ln f^{2}\,dS$ $\displaystyle\leq$
$\displaystyle n\ln\|f\|_{L^{q}(\partial\Omega)}^{2}\leq
n\ln\left(A\int_{\Omega}|\nabla_{g}f|^{2}dV+B\int_{\partial\Omega}f^{2}dS\right)$
$\displaystyle=$ $\displaystyle
n\ln\left(A\int_{\Omega}|\nabla_{g}f|^{2}dV+B\right).$
Note that for any fixed $a>0$, the following elementary inequality holds:
$\displaystyle\ln x\leq ax-1-\ln a\;\quad\mbox{for all}\;\;x>0.$
Therefore
$\displaystyle\int_{\partial\Omega}f^{2}\ln f^{2}dS$ $\displaystyle\leq$
$\displaystyle n\ln\big{(}A\int_{\Omega}|\nabla_{g}f|^{2}dV+B)$
$\displaystyle\leq$ $\displaystyle
na\big{(}A\int_{\Omega}|\nabla_{g}f|^{2}dV+B\big{)}-n(1+\ln a).$
Taking $\epsilon=(naA)/2$, we deduce
$\displaystyle\int_{\partial\Omega}f^{2}\ln f^{2}dS\leq
2\epsilon\int_{\Omega}|\nabla_{g}f|^{2}dV-n\ln\epsilon+2BA^{-1}\epsilon+n\ln\frac{nA}{2e}.$
Finally, by putting $f=\frac{v}{\|v\|_{L^{2}(\partial\Omega)}}$ we get
$\displaystyle\int_{\partial\Omega}\frac{v^{2}}{\|v\|_{L^{2}(\partial\Omega)}^{2}}\,\ln\frac{v^{2}}{\|v\|_{L^{2}(\partial\Omega)}^{2}}\,dS\leq
2\epsilon\int_{\Omega}\frac{|\nabla_{g}v|^{2}}{\|v\|_{L^{2}(\partial\Omega)}^{2}}dV-n\ln\epsilon+2BA^{-1}\epsilon+n\ln\frac{nA}{2e},$
so that
$\displaystyle\int_{\partial\Omega}v^{2}\ln\frac{|v|}{\|v\|_{L^{2}(\partial\Omega)}}dS\leq\epsilon\int_{\Omega}|\nabla_{g}v|^{2}dV+\left(\frac{n}{2}\,\ln\frac{nA}{2e}+BA^{-1}\epsilon-\frac{n}{2}\ln\,\epsilon\right)\|v\|^{2}_{L^{2}(\partial\Omega)}.$
This is just the desired Log-Sobolev trace inequality (3.2).
(i) $\Longrightarrow$ (iii): Using Hölder interpolation inequality, we have
$\displaystyle\int_{\partial\Omega}v^{2}dS$ $\displaystyle=$
$\displaystyle\int_{\partial\Omega}v^{2-\frac{2}{n+1}}v^{\frac{2}{n+1}}dS=\int_{\partial\Omega}v^{\frac{2n}{n+1}}v^{\frac{2}{n+1}}dS$
$\displaystyle\leq$
$\displaystyle\left(\int_{\partial\Omega}v^{\frac{2np}{n+1}}dS\right)^{\frac{1}{p}}\left(\int_{\partial\Omega}v^{\frac{2p^{\prime}}{n+1}}dS\right)^{\frac{1}{p^{\prime}}}.$
By choosing $p=\frac{n+1}{n-1}$, $\,p^{\prime}=\frac{n+1}{2}$, we obtain
$\displaystyle\int_{\partial\Omega}v^{2}dS\leq\left(\int_{\partial\Omega}v^{\frac{2n}{n-1}}dS\right)^{\frac{n-1}{n+1}}\left(\int_{\partial\Omega}|v|dS\right)^{\frac{2}{n+1}},$
so that
$\displaystyle\|v\|_{L^{2}(\partial\Omega)}^{2+\frac{2}{n}}\leq\left(\int_{\partial\Omega}v^{\frac{2n}{n-1}}dS\right)^{\frac{n-1}{n}}\left(\int_{\partial\Omega}|v|dS\right)^{\frac{2}{n}}.$
Applying the Sobolev trace inequality (3.1), we have
$\displaystyle\|v\|_{L^{2}(\partial\Omega)}^{2+\frac{2}{n}}\leq\left(A\|\nabla_{g}v\|_{L^{2}(\Omega)}^{2}+B\|v\|_{L^{2}(\partial\Omega)}^{2}\right)\|v\|_{L^{1}(\partial\Omega)}^{\frac{2}{n}},$
which is the Nash trace inequality (3.3).
(ii) $\Longrightarrow$ (iv): Let $u(t,x)$ be a smooth solution of the heat
equation for the Dirichlet-to-Neumann operator. Clearly, $u$ can be written as
$\displaystyle u(t,x)=\int_{\partial\Omega}{\mathcal{K}}(t,x,y)u(0,y)dS(y),$
where ${\mathcal{K}}$ is the fundamental solution of (2.3). Note that
$\displaystyle\sup_{u\neq
0}\frac{\|u(t,\cdot)\|_{L^{\infty}(\partial\Omega)}}{\|u(0,\cdot)\|_{L^{1}(\partial\Omega)}}=\sup_{x,y}{\mathcal{K}}(t,x,y).$
For any fixed $t>0$, we can take $p(s)=\frac{t}{t-s}$. It is obvious that
$p(0)=1$ and $p(t)=\infty$. Let $u=u(t,x)$ be a positive solution to the heat
equation (1.15) of the Dirichlet-to-Neumann operator. By a direct calculation,
we have
(3.9) $\displaystyle\frac{\partial\|u\|_{L^{p(s)}(\partial\Omega)}}{\partial
s}=-\|u\|_{L^{p(s)}(\partial\Omega)}\cdot\frac{p^{\prime}(s)}{p^{2}(s)}\cdot\ln\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}$
$\displaystyle\quad\quad\;+\frac{\|u\|_{L^{p(s)}(\partial\Omega)}^{1-p(s)}}{p(s)}\left(p^{\prime}(s)\int_{\partial\Omega}u^{p(s)}\ln
u\,dS+p(s)\int_{\partial\Omega}u^{p(s)-1}({\mathcal{N}}_{g}u)dS\right).$
Multiplying by $p^{2}(s)\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}$ on both
sides, and applying the property of ${\mathcal{N}}_{g}$, we obtain
$\displaystyle
p(s)^{2}\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}\,\frac{\partial\|u\|_{L^{p(s)}(\partial\Omega)}}{\partial
s}$ $\displaystyle=$
$\displaystyle-p^{\prime}(s)\|u\|_{L^{p(s)}(\partial\Omega)}^{1+p(s)}\ln\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}$
$\displaystyle+p(s)p^{\prime}(s)\|u\|_{L^{p(s)}(\partial\Omega)}\int_{\partial\Omega}u^{p(s)}\ln
u\,dS$
$\displaystyle-p^{2}(s)(p(s)-1)\|u\|_{L^{p(s)}(\partial\Omega)}\int_{\Omega}u^{p(s)-2}|\nabla_{g}u|^{2}dV.$
Dividing both sides by $\|u\|_{L^{p(s)}(\partial\Omega)}$, we find that
(3.10) $\displaystyle
p^{2}(s)\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}\frac{\partial\left(\ln\|u\|_{L^{p(s)}(\partial\Omega)}\right)}{\partial
s}=-p^{\prime}(s)\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}\ln\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}$
$\displaystyle\qquad\quad\quad\;\;+p(s)p^{\prime}(s)\int_{\partial\Omega}u^{p(s)}\ln
u\,dS-4(p(s)-1)\int_{\Omega}|\nabla_{g}(u^{p(s)/2})|^{2}dV.$
Putting
$\displaystyle
v=\frac{u^{p(s)/2}(s,x)}{\|u^{p(s)/2}\|_{L^{2}(\partial\Omega)}},$
we get
$\displaystyle\|v\|_{L^{2}(\partial\Omega)}=1\;\;\mbox{and}\;\;\ln v^{2}=\ln
u^{p(s)}-\ln\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)},$
so that
$\displaystyle p^{\prime}(s)\int_{\partial\Omega}v^{2}\ln v^{2}\,dS$
$\displaystyle=$ $\displaystyle
p^{\prime}(s)\int_{\partial\Omega}\frac{u^{p(s)}}{\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}}\left(\ln
u^{p(s)}-\ln\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}\right)dS$ $\displaystyle=$
$\displaystyle\frac{p(s)p^{\prime}(s)}{\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}}\int_{\partial\Omega}u^{p(s)}\ln
u\,dS-p^{\prime}(s)\ln\|u\|_{L^{p(s)}(\partial\Omega)}^{p(s)}.$
We substitute $v$ into the right-hand side of (3.10) to obtain
$\displaystyle
p^{2}(s)\,\frac{\partial\big{(}\ln\|u\|_{L^{p(s)}(\partial\Omega)}\big{)}}{\partial
s}=p^{\prime}(s)\left(\int_{\partial\Omega}v^{2}\ln
v^{2}dS-\frac{4(p(s)-1)}{p^{\prime}(s)}\int_{\Omega}|\nabla_{g}v|^{2}dV\right).$
Choose
$\displaystyle 2\epsilon=\frac{4(p(s)-1)}{p^{\prime}(s)}=\frac{4s(t-s)}{t}\leq
t.$
By the Log-Sobolev trace inequality, we get
$\displaystyle
p^{2}(s)\frac{\partial\big{(}\ln\|u\|_{L^{p(s)}(\partial\Omega)}\big{)}}{\partial
s}\leq
p^{\prime}(s)\left[-n\ln\big{(}\frac{2s(t-s)}{t}\big{)}+A^{-1}Bt+n\ln\frac{nA}{2e}\right].$
Since $\frac{p^{\prime}(s)}{p^{2}(s)}=\frac{1}{t}$, we deduce
$\displaystyle\frac{\partial\big{(}\ln\|u\|_{L^{p(s)}(\partial\Omega)}\big{)}}{\partial
s}\leq\frac{1}{t}\left[-n\ln\frac{s(t-s)}{t}+A^{-1}Bt+n\ln\frac{nA}{4e}\right].$
Integrating the above inequality from $0$ to $t$, we have
$\displaystyle\ln\frac{\|u(t,x)\|_{L^{p(t)}(\partial\Omega)}}{\|u(0,x)\|_{L^{p(0)}(\partial\Omega)}}\leq-n\ln
t+2n+A^{-1}Bt+n\ln\frac{nA}{4e}.$
Noting that $p(0)=1,p(t)=\infty$, we get
$\displaystyle\|u(t,x)\|_{L^{\infty}(\partial\Omega)}\leq\left(\frac{\exp(A^{-1}Bt+2n+n\ln\frac{nA}{4e})}{t^{n}}\right)\|u(0,x)\|_{L^{1}(\partial\Omega)},$
which implies
$\displaystyle{\mathcal{K}}(t,x,y)\leq\left(\frac{nAe}{4}\right)^{n}\frac{e^{A^{-1}Bt}}{t^{n}}.$
(iii) $\Longrightarrow$ (iv): Let ${\mathcal{K}}={\mathcal{K}}(t,x,y)$ be the
heat kernel associated to the Dirichlet-to-Neumann operator. For any fixed
$y\in\partial\Omega$, we define $v=v(t,x)={\mathcal{K}}(t,x,y)$. It is clear
that $\int_{\partial\Omega}v(t,x)dS(x)=1$, i.e.,
$\|v\|_{L^{2}(\partial\Omega)}=1$. Since
$\displaystyle\frac{\partial}{\partial
t}\left(\int_{\partial\Omega}v^{2}(t,x)dS\right)$ $\displaystyle=$
$\displaystyle\int_{\partial\Omega}2v\,\frac{\partial v}{\partial
t}dS=\int_{\partial\Omega}2v({\mathcal{N}}_{g}v)dS$ $\displaystyle=$
$\displaystyle\int_{\partial\Omega}2v\,\frac{\partial
v}{\partial\nu}\,dS=-2\int_{\Omega}|\nabla_{g}v|^{2}dV.$
From the Nash trace inequality with $\|v\|_{L^{2}(\partial\Omega)}=1$, we have
$\displaystyle\|v\|_{L^{2}(\partial\Omega)}^{2+\frac{2}{n}}\leq\big{(}A\|\nabla_{g}v\|_{L^{2}(\Omega)}^{2}+B\|v\|_{L^{2}(\partial\Omega)}^{2}\big{)}$
so that
$\displaystyle-\|\nabla_{g}v\|_{L^{2}(\Omega)}^{2}\leq-\frac{1}{A}\|v\|_{L^{2}(\partial\Omega)}^{2+\frac{2}{n}}+\frac{B}{A}\|v\|_{L^{2}(\partial\Omega)}^{2}.$
Therefore
$\displaystyle\frac{\partial}{\partial
t}\left(\int_{\partial\Omega}v^{2}(t,x)dS\right)\leq-\alpha\|v\|_{L^{2}(\partial\Omega)}^{2+\frac{2}{n}}+\gamma\|v\|_{L^{2}(\partial\Omega)}^{2},$
where $\alpha=\frac{2}{A},\gamma=\frac{2B}{A}$. Write
$\displaystyle f(t)=\int_{\partial\Omega}v^{2}(t,x)dS,\quad\,g(t)=e^{-\gamma
t}f(t),$
we have
$\displaystyle\frac{\partial f(t)}{\partial
t}\leq-\alpha[f(t)]^{1+\frac{1}{n}}+\gamma f(t)$
and
$\displaystyle\frac{\partial g(t)}{\partial t}$ $\displaystyle=$
$\displaystyle-\gamma e^{-\gamma t}f(t)+e^{-\gamma t}\,\frac{\partial
f(t)}{\partial t}$ $\displaystyle\leq$ $\displaystyle-\gamma e^{-\gamma
t}f(t)+e^{-\gamma t}\gamma f(t)-\alpha e^{-\gamma
t}\left[f(t)\right]^{1+\frac{1}{n}}.$
Thus
$\displaystyle\frac{\partial g(s)}{\partial s}\leq-\alpha e^{\frac{\gamma
s}{n}}[g(s)]^{1+\frac{1}{n}},\quad\,s\in(0,t].$
Integrating the above inequality from $\frac{t}{2}$ to $t$, we get
$\displaystyle-n\left([g(s)]^{-\frac{1}{n}}\right)\bigg{|}_{s=\frac{t}{2}}^{s=t}\leq-\frac{n\alpha}{\gamma}\left[e^{\frac{\gamma
t}{n}}-e^{\frac{\gamma t}{2n}}\right],$
so that
$\displaystyle ne^{\frac{\gamma
t}{n}}[f(t)]^{-\frac{1}{n}}\geq\frac{n\alpha}{\gamma}\left(e^{\frac{\gamma
t}{n}}-e^{\frac{\gamma t}{2n}}\right).$
Therefore
(3.11) $\displaystyle e^{\frac{\gamma
t}{2n}}[f(t)]^{-\frac{1}{n}}\geq\frac{\alpha}{\gamma}\left(e^{\frac{\gamma
t}{2n}}-1\right)\geq\frac{\alpha}{\gamma}\cdot\frac{\gamma t}{2n}=\frac{\alpha
t}{2n},$
i.e.,
$\displaystyle
f(t)\leq\frac{1}{t^{n}}\left(\frac{2n}{\alpha}\right)^{n}e^{\frac{\gamma
t}{2}}.$
This implies
$\displaystyle\int_{\partial\Omega}{\mathcal{K}}(t,x,y){\mathcal{K}}(t,x,y)dS(x)\leq\frac{\big{(}2n/\alpha)^{n}}{t^{n}}e^{\gamma
t/2}.$
Noticing that ${\mathcal{K}}(t,x,y)$ is symmetric with respect to $x$ and $y$,
we find by the reproducing property that for any $x\in\partial\Omega$,
$\displaystyle{\mathcal{K}}(2t,x,x)=\int_{\partial\Omega}{\mathcal{K}}(t,x,y){\mathcal{K}}(t,y,x)dS(y)\leq\frac{(2n/\alpha)^{n}}{t^{n}}e^{\gamma
t/2}.$
It follows that
$\displaystyle{\mathcal{K}}(t,x,y)$ $\displaystyle=$
$\displaystyle\int_{\partial\Omega}{\mathcal{K}}(t/2,x,z){\mathcal{K}}(t/2,z,y)dS(z)$
$\displaystyle\leq$
$\displaystyle\left(\int_{\partial\Omega}{\mathcal{K}}^{2}(t/2,x,z)dS(z)\right)^{\frac{1}{2}}\left(\int_{\partial\Omega}{\mathcal{K}}^{2}(t/2,z,y)dS(z)\right)^{\frac{1}{2}}$
$\displaystyle=$
$\displaystyle\big{(}{\mathcal{K}}(t,x,x)\big{)}^{1/2}\big{(}{\mathcal{K}}(t,y,y)\big{)}^{1/2}\leq\frac{(4n/\alpha)^{n}}{t^{n}}e^{\gamma
t/4},$
i.e.,
$\displaystyle{\mathcal{K}}(t,x,y)\leq\frac{(2nA)^{n}}{t^{n}}e^{A^{-1}Bt/2}.$
When $t$ is large, the above bound can be improved. Indeed, we find by the
first inequality of (3.11) that for $t\geq 1$,
$\displaystyle f(t)={\mathcal{K}}(2t,x,x)\leq\frac{e^{\gamma
t/2}\left(\frac{\gamma}{\alpha}\right)^{n}}{(e^{\frac{\gamma
t}{2n}}-1)^{n}}\leq\frac{B^{n}e^{A^{-1}Bt}}{(e^{B/(nA)}-1)^{n}}.$
Therefore, there exists a positive constant $C$ depending only on $A$ and $B$
such that
$\displaystyle{\mathcal{K}}(t,x,y)\leq\min\left\\{\frac{(2nA)^{n}}{t^{n}}e^{A^{-1}Bt/2},C\right\\},\;\;t>0,\;\;x,y\in\partial\Omega.$
(iv) $\Longrightarrow$ (i): Let $H=H(t,x,y)$ be the shift heat kernel of the
Dirichlet-to-Neumann operator:
$\displaystyle{\mathcal{N}}_{g}u-c_{2}u-\frac{\partial u}{\partial
t}=0\quad\;\mbox{on}\;\;\partial\Omega.$
Since
$\displaystyle{\mathcal{K}}(t,x,y)\leq\frac{c_{1}}{t^{n}}e^{c_{2}t},$
we have
$\displaystyle
H=H(t,x,y)=e^{-c_{2}t}{\mathcal{K}}(t,x,y)\leq\frac{c_{1}}{t^{n}}.$
Note that
$\displaystyle\int_{\partial\Omega}H(t,x,y)dS(y)\leq\int_{\partial\Omega}{\mathcal{K}}(t,x,y)dS(y)=1.$
It follows that for all $f\in L^{2}(\partial\Omega)$,
$\displaystyle\|H*f\|_{L^{\infty}(\partial\Omega)}$ $\displaystyle=$
$\displaystyle\sup_{x\in\partial\Omega}\bigg{|}\int_{\partial\Omega}H(t,x,y)f(y)dS(y)\bigg{|}\leq\sup_{x\in\partial\Omega}\left(\int_{\partial\Omega}H^{2}(t,x,y)dS(y)\right)^{1/2}\|f\|_{L^{2}(\partial\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{c_{1}}}{t^{n/2}}\left(\int_{\partial\Omega}H(t,x,y)dS(y)\right)^{1/2}\|f\|_{L^{2}(\partial\Omega)}\leq\frac{\sqrt{c_{1}}}{t^{n/2}}\|f\|_{L^{2}(\partial\Omega)}.$
Also, by Hölder’s inequality we find that for all $q\in[1,n)$ and
$q^{\prime}=\frac{q}{q-1}$,
$\displaystyle\|H*f\|_{L^{\infty}(\partial\Omega)}$ $\displaystyle\leq$
$\displaystyle\sup_{x\in\partial\Omega}\left(\int_{\partial\Omega}H^{q^{\prime}}(t,x,y)dS(y)\right)^{1/{q^{\prime}}}\|f\|_{L^{q}(\partial\Omega)}$
$\displaystyle\leq$
$\displaystyle\sup_{x\in\partial\Omega}\left(\int_{\partial\Omega}H^{q^{\prime}/q}(t,x,y)\cdot
H(t,x,y)dS(y)\right)^{1/q^{\prime}}\|f\|_{L^{q}(\partial\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{c_{1}^{1/q}}{t^{n/q}}\|f\|_{L^{q}(\partial\Omega)}.$
For the operator $L=\left(\sqrt{(-{\mathcal{N}}_{g}+c_{2})}\right)^{-1}$, by
eigenfunction expansion (Laplace transform) we get that for $f\in
C^{\infty}(\partial\Omega)$,
$\displaystyle(Lf)(x)$ $\displaystyle=$
$\displaystyle\big{(}\Gamma(1/2)\big{)}^{-1}\int_{0}^{\infty}t^{-1/2}\big{[}e^{({\mathcal{N}}_{g}-c_{2})t}f\big{]}(t,x)dt$
$\displaystyle=$
$\displaystyle\big{(}\Gamma(1/2)\big{)}^{-1}\int_{0}^{\infty}t^{-\frac{1}{2}}(H*f)(t,x)dt,$
where $e^{({\mathcal{N}}_{g}-c_{2})t}f$ is the semigroup notation for $H*f$.
For fixed $t>0$, we write $Lf=L_{1}f+L_{2}f,$ where
$\displaystyle
L_{1}f(x)=\Gamma(1/2)^{-1}\int_{0}^{t}s^{-\frac{1}{2}}[H*f](s,x)ds,$
$\displaystyle
L_{2}f(x)=\Gamma(1/2)^{-1}\int_{t}^{\infty}s^{-\frac{1}{2}}[H*f](s,x)ds.$
Note that for any $\lambda>0$,
$\displaystyle|\\{x\big{|}|Lf(x)|\geq\lambda\\}|\leq|\\{x\big{|}|L_{1}f(x)|\geq\frac{\lambda}{2}\\}|+|\\{x\big{|}|L_{2}f(x)|>\frac{\lambda}{2}\\}|.$
Form (3) and the definition of $L_{2}f$,
$\displaystyle\|L_{2}f\|_{\infty}\leq\frac{1}{\sqrt{\pi}}\,c_{1}^{1/q}\int_{t}^{\infty}s^{-\frac{1}{2}-\frac{n}{q}}\|f\|_{L^{q}(\partial\Omega)}ds=\frac{2q}{(2n-q)\sqrt{\pi}}c_{1}^{\frac{1}{q}}t^{\frac{1}{2}-\frac{n}{q}}\|f\|_{L^{q}(\partial\Omega)}.$
We choose $t$ such that
(3.13)
$\displaystyle\frac{\lambda}{2}=\frac{2q}{(2n-q)\sqrt{\pi}}\,c_{1}^{\frac{1}{q}}t^{\frac{1}{2}-\frac{n}{q}}\|f\|_{L^{q}(\partial\Omega)}.$
It follows that
$\displaystyle|\\{x\big{|}|Lf(x)|\geq\lambda\\}|\leq|\\{x\big{|}|L_{1}f(x)|\geq\frac{\lambda}{2}\\}|,$
so that
$\displaystyle|\\{x\big{|}|Lf(x)|\geq\lambda\\}|\leq|\\{x\big{|}|L_{1}f(x)|\geq\frac{\lambda}{2}\\}|\leq\left(\frac{\lambda}{2}\right)^{-q}\int_{\partial\Omega}|L_{1}f(x)|^{q}dS(x).$
From Minkowski’s inequality and Young’s inequality, we get
$\displaystyle\|L_{1}f\|_{L^{q}(\partial\Omega)}$ $\displaystyle\leq$
$\displaystyle\big{(}\Gamma(1/2)\big{)}^{-1}\int_{0}^{t}s^{-\frac{1}{2}}\|H*f(s,\cdot)\|_{L^{q}(\partial\Omega)}ds$
$\displaystyle\leq$
$\displaystyle\big{(}\Gamma(1/2)\big{)}^{-1}\int_{0}^{t}s^{-\frac{1}{2}}\left(\sup_{x}\|H(s,x,\cdot)\|_{L^{1}(\partial\Omega)}\right)\|f\|_{L^{q}(\partial\Omega)}ds$
$\displaystyle\leq$
$\displaystyle\frac{2}{\sqrt{\pi}}\,t^{\frac{1}{2}}\|f\|_{L^{q}(\partial\Omega)},$
which shows
$\displaystyle|\\{x\big{|}Lf(x)|\geq\lambda\\}|\leq\left(\frac{2}{\sqrt{\pi}}\right)^{q}\left(\frac{\lambda}{2}\right)^{-q}t^{\frac{q}{2}}\|f\|_{L^{q}(\partial\Omega)}^{q}.$
According to the choice of $t$ in (3.13), this is equivalent to
$\displaystyle\\{x\big{|}|Lf(x)|\geq\lambda\\}|\leq\left(\frac{16}{\pi}\right)^{nq/(2n-q)}\left(\frac{q}{2n-q}\right)^{q^{2}/(2n-q)}(c_{1})^{\frac{q}{2n-q}}\lambda^{-r}\|f\|_{L^{q}(\partial\Omega)}^{r},$
where $r=(2qn)/(2n-q)$. By the Marcinkiewicz interpolation lemma, we know that
$L$ is a bounded operator from $L^{2}(\partial\Omega)$ to
$L^{p}(\partial\Omega)$ with $p=\frac{2n}{n-1}$ (taking $q=2$), i.e.,
(3.14) $\displaystyle\|Lu\|_{L^{p}(\partial\Omega)}\leq
c(c_{1})^{\frac{1}{n-1}}\|u\|_{L^{2}(\partial\Omega)}\quad\,\mbox{for
all}\;\;u\in C^{\infty}(\partial\Omega).$
Write $v=Lu$. Then $u=L^{-1}v$ and
$\displaystyle\|u\|_{L^{2}(\partial\Omega)}^{2}$ $\displaystyle=$
$\displaystyle\langle L^{-1}v,L^{-1}v\rangle=\langle L^{-2}v,v\rangle$
$\displaystyle=$
$\displaystyle\langle-{\mathcal{N}}_{g}v+c_{2}v,v\rangle=\int_{\Omega}|\nabla_{g}v|^{2}dV+c_{2}\int_{\partial\Omega}v^{2}ds.$
Substitute this into (3.14) we arrive at the Sobolev trace inequality:
$\displaystyle\|v\|_{L^{2}(\partial\Omega)}^{p}\leq
const.(c_{1})^{\frac{1}{n-1}}\left(\|\nabla_{g}v\|_{L^{2}(\Omega)}^{2}+c_{2}\|v\|_{L^{2}(\partial\Omega)}^{2}\right).$
If we take $c_{1}=\left(\frac{nAe}{4}\right)^{n-1}$ and $c_{2}=A^{-1}B$ as in
(3.4), then we get a Sobolev trace inequality with the claimed constants:
$\displaystyle\|v\|_{L^{p}(\partial\Omega)}^{2}\leq
const.A\|\nabla_{g}v\|_{L^{2}(\Omega)}^{2}+const.B\|v\|_{L^{2}(\partial\Omega)}^{2}.$
(v) $\Longrightarrow$ (iv): It is obvious because $t\in[0,1]$ and $d(x,y)$ is
bounded for all $x,y\in\partial\Omega$.
(iv) $\Longrightarrow$ (v): By (iv) there exists a constant $c_{1}>0$ such
that
(3.15) $\displaystyle{\mathcal{K}}(t,x,y)\leq c_{1}t^{-n}\quad\,\mbox{for
all}\;\;t\in[0,1]\;\;\mbox{and}\;\;x,y\in\partial\Omega.$
Clearly, $\partial\Omega$ has at most a finite number of connected components,
say $\Lambda_{1},\cdots,\Lambda_{N}$, with $\Lambda_{i}\neq\Lambda_{j}$ if
$i\neq j$. Let
$\displaystyle\mathfrak{T}=\\{\psi\in
C^{\infty}(\partial\Omega)\big{|}\max_{\underset{x_{i}\in\Lambda_{i},x_{j}\in\Lambda_{j}}{i,j\in\\{1,\cdots,N\\}}}|\psi(x_{i})-\psi(x_{j})|+D\|\nabla\psi\|_{\infty}\leq
D\\},$
where $D=1+\sum_{i=1}^{N}\mbox{diam}\,\Lambda_{i}$. We define the derivative
$\delta_{\psi}$ on the space $\mathcal{L}(L^{p}(\partial\Omega))$ of all the
linear operators on $L^{p}(\partial\Omega)$ by
$\delta_{\psi}(E)=[M_{\psi},E]$, where $M_{\psi}$ denotes the multiplication
operator with the function $\psi$. Furthermore, we can define
$\delta_{\psi}^{m}(E)=\delta_{\psi}(\delta_{\psi}^{m-1}(E))$ for all $m>1$ by
induction. It follows from (3.15) and Proposition 4.1 of [35] that there
exists a constants $c_{2}>0$ such that
(3.16) $\displaystyle\|\delta_{\psi}^{n+1}(T_{t})\|_{1\to\infty}\leq
c_{2}t\quad\;\mbox{for
all}\;\;\psi\in\mathfrak{T}\,\,\mbox{and}\;\;t\in[0,1].$
Note that
$\displaystyle\delta_{\psi}^{n+1}(T_{t})=[M_{\psi},[\cdots,[M_{\psi},T_{t}]\cdots]]=\sum_{k=0}^{n+1}(-1)^{k}\begin{pmatrix}k\\\
n+1\end{pmatrix}\psi^{n+1-k}T_{t}\psi^{k}.$
It follows from (3.16) that for any fixed
$\psi\in\mathfrak{T}\,\,\mbox{and}\;\;t\in[0,1]$,
$\displaystyle-
c_{2}t\int_{\partial\Omega}v(y)\,dS(y)\leq\int_{\partial\Omega}\sum_{k=0}^{n+1}(-1)^{k}\begin{pmatrix}k\\\
n+1\end{pmatrix}\psi^{n+1-k}(x)\,\mathcal{K}(t,x,y)\,\psi^{k}(y)\,v(y)\,dS(y)$
$\displaystyle\qquad\qquad\leq
c_{2}t\int_{\partial\Omega}v(y)\,dS(y)\quad\,\mbox{for all}\;\;0\leq v\in
C^{\infty}(\partial\Omega).$
This implies
$\displaystyle|(\psi(x)-\psi(y))^{n+1}{\mathcal{K}}(t,x,y)|\leq
c_{2}t\quad\,\mbox{for
all}\;\;t\in[0,1],x,y\in\partial\Omega\,\,\mbox{and}\;\;\psi\in\mathfrak{T}.$
Since $d(x,y)=\sup\\{|\psi(x)-\psi(y)|\big{|}\psi\in\mathfrak{T}\\}$, we get
$\displaystyle(d(x,y))^{n+1}{\mathcal{K}}(t,x,y)\leq c_{2}t,$
i.e.,
$\displaystyle\left(\frac{d(x,y)}{t}\right)^{n+1}{\mathcal{K}}(t,x,y)\leq
c_{2}t^{-n}\quad\,\mbox{for
all}\;\;t\in[0,1],x,y\in\partial\Omega\,\,\mbox{and}\;\;\psi\in\mathfrak{T}.$
Combining this and (3.15) we have
$\displaystyle\left(1+\frac{d(x,y)}{t}\right)^{n+1}{\mathcal{K}}(t,x,y)\leq
2^{n+1}(c_{1}+c_{2})t^{-n}\quad\,\mbox{for
all}\;\;t\in[0,1],x,y\in\partial\Omega\,\,\mbox{and}\;\;\psi\in\mathfrak{T}.$
This is equivalent to
$\displaystyle{\mathcal{K}}(t,x,y)\leq
ct^{-n}\left(1+\frac{d^{2}(x,y)}{t^{2}}\right)^{-(n+1)/2}\quad\,\mbox{for
all}\;\;t\in[0,1],x,y\in\partial\Omega\,\,\mbox{and}\;\;\psi\in\mathfrak{T},\quad\;\,$
which just is the inequality (3.5).
(vi)$\Longrightarrow$(i): If the inequality (3.7) holds for all $q\in
C^{\infty}(\partial\Omega)$, then $I_{q}(0)\leq(1-\epsilon)$ for all $q\in
C^{\infty}(\partial\Omega)$ with
$\|q\|_{L^{n}(\partial\Omega)}^{n}\leq\frac{1-\epsilon}{C(n)}$, where
$0<\epsilon<1$. This implies $I_{q}(0)=0$ for all such $q$, and consequently
$(-A{\mathcal{N}}_{g}+B+q)$ is a nonnegative operator for all $q\in
C^{\infty}(\partial\Omega)$ with
$\|q\|_{L^{n}(\partial\Omega)}^{n}\leq\frac{1-\epsilon}{C(n)}$. It follows
that
$\displaystyle
0\leq\langle(-A{\mathcal{N}}_{g}+B+q)f,f\rangle=A\int_{\Omega}|\nabla_{g}u|^{2}dV+B\int_{\partial\Omega}f^{2}dS+\int_{\partial\Omega}q|f|^{2}dS$
for all such $q$ and all $0\leq f\in C^{\infty}(\partial\Omega)$, where $u\in
C^{\infty}(\bar{\Omega})$ satisfies $\Delta_{g}u=0$ in $\Omega$ and $u=f$ on
$\partial\Omega$, i.e.,
$\displaystyle\sup_{\underset{-q\in
C^{\infty}(\partial\Omega)}{\|q\|_{L^{n}(\partial\Omega)\leq((1-\epsilon)/C(n))^{1/n}}}}\bigg{\\{}\int_{\partial\Omega}-q|f|^{2}dS\bigg{\\}}\leq
A\int_{\Omega}|\nabla_{g}u|^{2}dV+B\int_{\partial\Omega}f^{2}dS.$
As the dual of $L^{n}(\partial\Omega)$ is $L^{n/(n-1)}(\partial\Omega)$, this
yields
$\displaystyle\bigg{(}\int_{\partial\Omega}|f|^{2n/(n-1)}dx\bigg{)}^{(n-1)/n}\leq\left(\frac{C(n)}{1-\epsilon}\right)^{1/n}\left(A\int_{\Omega}|\nabla_{g}u|^{2}dV+B\int_{\partial\Omega}f^{2}dS\right)\quad\;\mbox{for
all}\;\;f\in C^{\infty}(\partial\Omega).$
By virtue of
$\displaystyle\inf_{\underset{v=f\;\;on\;\;\partial\Omega}{v\in
W^{1,2}(\Omega)}}\int_{\partial\Omega}|\nabla_{g}v|^{2}dV=\int_{\Omega}|\nabla_{g}u|^{2}dV,$
we immediately get that for all $v\in W^{1,2}(\Omega)$, the Sobolev trace
inequality (3.1) holds with new constant constants
$A^{\prime}=A\left(\frac{C(n)}{1-\epsilon}\right)^{1/n},\;B^{\prime}=B\left(\frac{C(n)}{1-\epsilon}\right)^{1/n}$.
(i)$\Longrightarrow$(vi): By monotonicity of $I_{q}(0)$ with respect to the
$q(x)$, we may assume $q(x)<0$ for all $x\in\partial\Omega$. Then the number
$I_{q}(0)$ of non-positive eigenvalues for (3.6) is equal to the number of
eigenvalues less than or equal to $1$ for the problem
(3.17) $\displaystyle(A{\mathcal{N}}_{g}-B)\psi(x)=\mu
q(x)\psi(x)\quad\,\mbox{for all}\;\;x\in\partial\Omega,$
where $A$ and $B$ are positive constants in the Sobolev trace inequality
(3.1). In fact, since
$\displaystyle\frac{\int_{\Omega}A|\nabla_{g}{\tilde{\psi}}|^{2}dV+(B+q)\int_{\partial\Omega}\psi^{2}dS}{\int_{\partial\Omega}\psi^{2}dS}=\frac{\int_{\partial\Omega}|q|\psi^{2}dS}{\int_{\partial\Omega}\psi^{2}dS}\bigg{[}\frac{A\int_{\Omega}|\nabla_{g}{\tilde{\psi}}|^{2}dV+B\int_{\partial\Omega}\psi^{2}dS}{\int_{\partial\Omega}|q|\psi^{2}dS}-1\bigg{]},$
we see that the dimension of the subspace on which the left-hand side is non-
positive is equal to the dimension on which the quadratic form
$\big{(}A\int_{\Omega}|\nabla_{g}{\tilde{\psi}}|^{2}dV+B\int_{\partial\Omega}\psi^{2}dS\big{)}/\big{(}\int_{\partial\Omega}|q|\psi^{2}dS\big{)}$
is less than or equal to $1$, where $\tilde{\psi}$ is the harmonic extension
of $\psi$ to $\bar{\Omega}$. Let $\\{\psi_{i}(x)\\}_{i=1}^{\infty}$ be a set
of orthonormal eigenfunctions satisfying
$\displaystyle(A{\mathcal{N}}_{g}-B)\psi_{i}=\mu_{i}q\psi_{i}\quad\mbox{on}\;\;\partial\Omega$
with the eigenvalues $\\{\mu_{i}\\}$. It follows from [96] (or [77]) that
$\mu_{i}>0$ for all $i\geq 1$. Then the kernel of the “heat” equation
$\left(\frac{-A{\mathcal{N}}_{g}+B}{q}-\frac{\partial}{\partial
t}\right)u=0\quad\,\mbox{in}\;\;[0,\infty)\times(\partial\Omega)$
must take the form
(3.18) $\displaystyle
0<H(t,x,y)=\sum_{i=1}^{\infty}e^{-\mu_{i}t}\psi_{i}(x)\psi_{i}(y)\quad\;\mbox{on}\;\;(0,\infty)\times(\partial\Omega)\times(\partial\Omega),$
where the $L^{2}$-norm is given by the volume form $-q(x)dS(x)$ instead of
$dS(x)$. Consider the function
(3.19) $\displaystyle
h(t)=\sum_{i=1}^{\infty}e^{-2\mu_{i}t}=\int_{\partial\Omega}\int_{\partial\Omega}H^{2}(t,x,y)q(x)q(y)dS(x)\,dS(y).$
In view of
$\bigg{(}\frac{A{\mathcal{N}}_{g}-B}{(-q(y))}-\frac{\partial}{\partial
t}\bigg{)}H(t,x,y)\equiv 0$, we have
$\displaystyle\quad\;\quad\quad\;\frac{\partial h}{\partial t}$
$\displaystyle=$ $\displaystyle
2\int_{\partial\Omega}\int_{\partial\Omega}H(t,x,y)q(x)q(y)\frac{\partial
H(t,x,y)}{\partial t}dS(x)\,dS(y)$ $\displaystyle=$ $\displaystyle
2\int_{\partial\Omega}\int_{\partial\Omega}H(t,x,y)\big{[}\big{(}-A{\mathcal{N}}_{g}+B\big{)}H(t,x,y)\big{]}q(x)dS(x)\,dS(y).$
$\displaystyle=$
$\displaystyle-2\int_{\partial\Omega}(-q(x))\bigg{(}A\int_{\Omega}|\nabla_{y}{\tilde{H}}(t,x,y)|^{2}dV(y)+B\int_{\partial\Omega}H^{2}(t,x,y)dS(y)\bigg{)}dS(x).$
Here, for any fixed $t>0$ and $x\in\partial\Omega$, ${\tilde{H}}(t,x,y)$
satisfies
$\displaystyle\left\\{\begin{array}[]{ll}\Delta_{y}{\tilde{H}}(t,x,y)=0\quad\;\mbox{for}\;\;y\in\Omega,\\\
{\tilde{H}}(t,x,y)=H(t,x,y)\quad\;\mbox{for}\;\;y\in\partial\Omega.\end{array}\right.$
Similar to the method of [74], we get
$\displaystyle\quad\;h(t)=\int_{\partial\Omega}(-q(x))\int_{\partial\Omega}H^{2}(t,x,y)(-q(y))dS(y)\,dS(x)$
$\displaystyle\leq$
$\displaystyle\int_{\partial\Omega}(-q(x))\left[\left(\int_{\partial\Omega}H^{\frac{2n}{n-1}}(t,x,y)dS(y)\right)^{\frac{n-1}{n+1}}\left(\int_{\partial\Omega}H(t,x,y)(-q(y))^{\frac{n+1}{2}}dS(y)\right)^{\frac{2}{n+1}}\right]dS(x)$
$\displaystyle\leq$
$\displaystyle\bigg{[}\int_{\partial\Omega}(-q(x))\bigg{(}\int_{\partial\Omega}H^{\frac{2n}{n-1}}(t,x,y)dS(y)\bigg{)}^{\frac{n-1}{n}}dS(x)\bigg{]}^{\frac{n}{n+1}}$
$\displaystyle\times\bigg{[}\int_{\partial\Omega}(-q(x))\bigg{(}\int_{\partial\Omega}H(t,x,y)(-q(y))^{\frac{n+1}{2}}dS(y)\bigg{)}^{2}dS(x)\bigg{]}^{\frac{1}{n+1}}.$
Set
$\displaystyle
Q(t,x)=\int_{\partial\Omega}H(t,x,y)(-q(y))^{\frac{n+1}{2}}dS(y)$
It is obvious that
$\displaystyle\bigg{(}\frac{-A{\mathcal{N}}_{g}+B}{q(x)}-\frac{\partial}{\partial
t}\bigg{)}Q(t,x)\equiv 0$
and
$Q(0,x)=\int_{\partial\Omega}\delta(x-y)(-q(y))^{\frac{n-1}{2}}(-q(y))dS(y)=(-q(x))^{\frac{n-1}{2}}.$
Similar to the argument of (3), we have $\frac{\partial}{\partial
t}\int_{\partial\Omega}Q^{2}(t,x)(-q(x))dS(x)\leq 0$, so that
$\displaystyle\int_{\partial\Omega}Q^{2}(t,x)(-q(x))dS(x)\leq\int_{\partial\Omega}Q^{2}(0,x)(-q(x))dS(x)=\int_{\partial\Omega}(-q(x))^{n}dS(x).$
By this and (3), we obtain
(3.23) $\displaystyle
h^{\frac{n+1}{n}}(t)\bigg{(}\int_{\partial\Omega}(-q(x))^{n}dS(x)\bigg{)}^{-1/n}$
$\displaystyle\quad\quad\;\;\leq\int_{\partial\Omega}(-q(x))\bigg{(}\int_{\partial\Omega}H^{\frac{2n}{n-1}}(t,x,y)dS(y)\bigg{)}^{\frac{n-1}{n}}dS(x).$
Combining (3), (3.23) and the Sobolev trace inequality (3.1), we get
$\displaystyle\frac{\partial h}{\partial
t}\leq-2\bigg{(}\int_{\partial\Omega}(-q(x))^{n}dS(x)\bigg{)}^{-1/n}h^{\frac{n+1}{n}}(t).$
Thus
(3.24) $\displaystyle
h(t)\leq\left(\frac{n}{2}\right)^{n}\bigg{(}\int_{\partial\Omega}(-q(x))^{n}dS(x)\bigg{)}t^{-n}.$
From (3.19) we obtain
$\displaystyle\left(\frac{n}{2}\right)^{n}\bigg{(}\int_{\partial\Omega}(-q(x))^{n}dS(x)\bigg{)}t^{-n}\geq\sum_{i=1}^{\infty}e^{-2\mu_{i}t}.$
Finally, we set $\mu_{k}$ to be the greatest eigenvalue less than or equal to
$1$. By setting $t=\frac{n}{2\mu_{k}}$, we get
$\displaystyle\left(\int_{\partial\Omega}|q(x)|^{n}dS(x)\right)\mu_{k}^{n}$
$\displaystyle=$
$\displaystyle\left(\frac{n}{2}\right)^{n}\bigg{(}\int_{\partial\Omega}(-q(x))^{n}dS(x)\bigg{)}\left(\frac{n}{2}\right)^{-n}\mu_{k}^{n}$
$\displaystyle\geq$ $\displaystyle\sum_{i=1}^{\infty}e^{-n\mu_{i}/\mu_{k}}\geq
ke^{-n},$
which leads to
$\displaystyle\int_{\partial\Omega}|q(x)|^{n}dS(x)$ $\displaystyle\geq$
$\displaystyle\mu_{k}^{n}\int_{\partial\Omega}|q(x)|^{n}dS(x)$
$\displaystyle\geq$ $\displaystyle ke^{-n}=I_{q}(0)e^{-n}.$
Therefore, the Rozenblum-Lieb-Cwikel type inequality holds with $C(n)=e^{n}$.
$\quad\quad\square$
## 4\. Symbol expression of the Dirichlet-to-Neumann operator
Let $\Omega$ be a bounded domain $\Omega\subset(\mathcal{M},g)$ with smooth
boundary $\partial\Omega$. Then $\partial\Omega$ has an induced Riemannian
metric $h$, and $\partial\Omega\hookrightarrow\bar{\Omega}$ has a second
fundamental form, with associated Weingarten map
$\displaystyle A_{\nu}:T_{x}(\partial\Omega)\to T_{x}(\partial\Omega).$
Let $A^{*}_{\nu}:T^{*}_{x}(\partial\Omega)\to T_{x}^{*}(\partial\Omega)$ be
the adjoint of $A_{\nu}$, and let $\langle\cdot,\cdot\rangle$ be the inner
product on $T^{*}_{x}(\partial\Omega)$ arising from the given Riemannian
metric. As shown in [71] and [20], ${\mathcal{N}}_{g}$ is a negative-
semidefinite, self-adjoint, elliptic pseudodifferential operator in
$OPS^{0}(\partial\Omega)$ with the principal symbol
$-\left(\sum_{j,k=1}^{n}h^{jk}\xi_{j}\xi_{k}\right)^{1/2}$, where $(h^{jk})$
is the inverse of $h$. It was proved (see, Proposition C.1 of [107] ) that
$\displaystyle{\mathcal{N}}_{g}=-\sqrt{-\Delta_{h}}+B\quad\;\mbox{mod}\;\,OPS^{-1}(\partial\Omega),$
where $B\in OPS^{0}(\partial\Omega)$ has principal symbol
$\displaystyle p_{0}(x,\xi)=\frac{1}{2}\left(Tr\,A_{\nu}-\frac{\langle
A^{*}_{\nu}\xi,\xi\rangle}{\langle\xi,\xi\rangle}\right).$
The following two Lemmas provide more information for the operators
$-\sqrt{-\Delta_{h}}$ and $B$.
Lemma 4.1. Let $p(x,\xi)$ be the symbol of the operator $-\sqrt{-\Delta_{h}}$
on $\partial\Omega$, and let $p(x,\xi)\sim\sum_{j\geq 0}p_{1-j}(x,\xi)$. Then
(4.1) $\displaystyle
p_{1}(x,\xi)=-\left(\sum_{j,k=1}^{n}h^{jk}(x)\xi_{j}\xi_{k}\right)^{1/2},$
(4.2) $\displaystyle
p_{0}(x,\xi)=\frac{i}{2}\left(\sum_{j,k=1}^{n}h^{jk}(x)\xi_{j}\xi_{k}\right)^{-1/2}\left[\sum_{j,k=1}^{n}\frac{1}{\sqrt{|h(x)|}}\,\frac{\partial(\sqrt{|h(x)|}\,h^{jk}(x))}{\partial
x_{k}}\,\xi_{j}\right.$
$\displaystyle\left.\quad\qquad\quad\;\;-\frac{1}{2}\left(\sum_{j,k=1}^{n}h^{jk}(x)\xi_{j}\xi_{k}\right)^{-1}\left(\sum_{j,k,l,m=1}^{n}h^{lm}(x)\,\frac{\partial
h^{jk}(x)}{\partial x_{l}}\xi_{j}\xi_{k}\xi_{m}\right)\right]$
and
(4.3) $\displaystyle
p_{-1}(x,\xi)=\frac{1}{2p_{1}}\bigg{[}-p_{0}^{2}+i\sum_{l=1}^{n}\bigg{(}\frac{\partial
p_{1}}{\partial\xi_{l}}\,\frac{\partial p_{0}}{\partial x_{l}}+\frac{\partial
p_{0}}{\partial\xi_{l}}\,\frac{\partial p_{1}}{\partial
x_{l}}\bigg{)}+\frac{1}{2}\sum_{j,k=1}^{n}\frac{\partial^{2}p_{1}}{\partial\xi_{j}\partial\xi_{k}}\,\frac{\partial^{2}p_{1}}{\partial
x_{j}\partial x_{k}}\bigg{]}.$
Proof. Denote by $c(x,\xi)$ the symbol of $-\Delta_{h}$ on $\partial\Omega$.
It follows from the composition formula (see, for example, (3.17) of p. 13 of
[107]) that
$\displaystyle c(x,\xi)$ $\displaystyle\sim$ $\displaystyle\sum_{\alpha\geq
0}\frac{i^{|\alpha|}}{\alpha!}D^{\alpha}_{\xi}p(x,\xi)D_{x}^{\alpha}p(x,\xi)$
$\displaystyle=$
$\displaystyle\left(p_{1}+p_{0}+p_{-1}+r\right)^{2}-i\sum_{l=1}^{n}\frac{\partial(p_{1}+p_{0}+p_{-1}+r)}{\partial\xi_{l}}\,\frac{\partial(p_{1}+p_{0}+p_{-1}+r)}{\partial
x_{l}}$
$\displaystyle-\frac{1}{2}\,\sum_{j,k=1}^{n}\frac{\partial^{2}(p_{1}+p_{0}+p_{-1}+r)}{\partial\xi_{j}\partial\xi_{k}}\,\frac{\partial^{2}(p_{1}+p_{0}+p_{-1}+r)}{\partial
x_{j}\partial x_{k}}+\cdots,$
where $r=p-p_{1}-p_{0}-p_{-1}\in S_{1,0}^{-2}$,
$D^{\alpha}=D^{\alpha_{1}}_{1}\cdots D^{\alpha_{n}}_{n}$ and
$D_{l}=\frac{1}{i}\,\frac{\partial}{\partial x_{l}}$. On the other hand, it is
well known that
$\displaystyle
c(x,\xi)=\sum_{j,k=1}^{n}h^{jk}(x)\xi_{j}\xi_{k}-i\sum_{j,k=1}^{n}\frac{1}{\sqrt{|h(x)|}}\,\frac{\partial(\sqrt{|h(x)|}\,h^{jk}(x))}{\partial
x_{j}}\,\xi_{k}.$
Then the symbol equations in $\xi$ of degree $2$, $1$ and $0$, respectively,
give
$\displaystyle p_{1}^{2}(x,\xi)=\sum_{j,k=1}^{n}h^{jk}(x)\xi_{j}\xi_{k},$
$\displaystyle 2p_{0}(x,\xi)p_{1}(x,\xi)-i\sum_{l=1}^{n}\frac{\partial
p_{1}(x,\xi)}{\partial\xi_{l}}\,\frac{\partial p_{1}(x,\xi)}{\partial
x_{l}}=-i\sum_{j,k=1}^{n}\frac{1}{\sqrt{|h(x)|}}\,\frac{\partial(\sqrt{|h(x)|}\,h^{jk}(x))}{\partial
x_{j}}\,\xi_{k},$ $\displaystyle
p_{0}^{2}+2p_{1}p_{-1}-i\sum_{l=1}^{n}\bigg{(}\frac{\partial
p_{1}}{\partial\xi_{l}}\,\frac{\partial p_{0}}{\partial x_{l}}+\frac{\partial
p_{0}}{\partial\xi_{l}}\,\frac{\partial p_{1}}{\partial
x_{l}}\bigg{)}-\frac{1}{2}\sum_{j,k=1}^{n}\frac{\partial^{2}p_{1}}{\partial\xi_{j}\partial\xi_{k}}\,\frac{\partial^{2}p_{1}}{\partial
x_{j}\partial x_{k}}=0,$
which implies
$\displaystyle
p_{1}(x,\xi)=\pm\bigg{(}\sum_{j,k=1}^{n}h^{jk}(x)\xi_{j}\xi_{k}\bigg{)}^{1/2},$
$\displaystyle
p_{0}(x,\xi)=\frac{i}{2p_{1}}\bigg{[}\sum_{l=1}^{n}\frac{\partial
p_{1}(x,\xi)}{\partial\xi_{l}}\,\frac{\partial p_{1}(x,\xi)}{\partial
x_{l}}-\sum_{j,k=1}^{n}\frac{1}{\sqrt{|h(x)|}}\,\frac{\partial(\sqrt{|h(x)|}\,h^{jk}(x))}{\partial
x_{j}}\,\xi_{k}\bigg{]},$ $\displaystyle
p_{-1}(x,\xi)=\frac{1}{2p_{1}}\bigg{[}-p_{0}^{2}+i\sum_{l=1}^{n}\bigg{(}\frac{\partial
p_{1}}{\partial\xi_{l}}\,\frac{\partial p_{0}}{\partial x_{l}}+\frac{\partial
p_{0}}{\partial\xi_{l}}\,\frac{\partial p_{1}}{\partial
x_{l}}\bigg{)}+\frac{1}{2}\sum_{j,k=1}^{n}\frac{\partial^{2}p_{1}}{\partial\xi_{j}\partial\xi_{k}}\,\frac{\partial^{2}p_{1}}{\partial
x_{j}\partial x_{k}}\bigg{]}.$
Noticing that the principal symbol of $-\sqrt{-\Delta_{h}}$ is non-positive,
we immediately obtain (4.1), (4.2) and (4.3). $\quad\quad\square$
Denote by $(A_{\nu})_{jk}$ (respectively, $(A^{*}_{\nu})_{jk}$) the entry in
the $j$-th row and $k$-th column of $A_{\nu}$ (respectively, $A_{\nu}^{*}$)
under a basis of $T_{x}(\partial\Omega)$ (respectively,
$T^{*}_{x}(\partial\Omega)$), and let $C$ be an $n\times n$ matrix satisfying
$C^{T}C=(h^{jk})$. Let $2Q_{1}$ and $6Q_{2}$ be the principal minor
determinants of orders $2$ and $3$ for the matrix
$\big{(}(-2A_{\nu})_{jk}\big{)}$, respectively. We denote
$Q_{3}=\sum_{j\neq
k}\left[(-2A_{\nu})_{jj}\big{(}-2{\tilde{R}}_{k(n+1)k(n+1)}+2(A_{\nu}^{2})_{kk}\big{)}-(-2A_{\nu})_{jk}\big{(}-2{\tilde{R}}_{k(n+1)j(n+1)}+2(A_{\nu}^{2})_{kj}\big{)}\right],$
where ${\tilde{R}}_{j(n+1)k(n+1)}$ is the curvature tensor with respect to
$g$, and $e_{n+1}=\nu$.
Lemma 4.2. Let $\Omega$ be a bounded domain with smooth boundary
$\partial\Omega$ in an $(n+1)$-dimensional Riemannian manifold
$(\mathcal{M},g)$. Then the Dirichlet-to-Neumann operator ${\mathcal{N}}_{g}$
is given by
(4.4)
$\displaystyle{\mathcal{N}}_{g}=-\sqrt{-\Delta_{h}}+B=-\sqrt{-\Delta_{h}}-B_{0}-B_{-1}-B_{-2}\quad\;\mbox{mod}\;\,OPS^{-3}(\partial\Omega),$
where
(4.5) $\displaystyle
B_{0}=-\frac{1}{2}\left[(\mbox{Tr}\,A_{\nu})\,Id+\frac{\Gamma\big{(}\frac{n-2}{2}\big{)}}{4\pi^{\frac{n}{2}}}\,\frac{1}{|x|^{n-2}}*\sum_{j,k=1}^{n}\big{(}(C^{-1})^{T}A_{\nu}^{*}C^{-1}\big{)}_{jk}\,\frac{\partial^{2}}{\partial
x_{j}\partial x_{k}}\right]$ (4.6) $\displaystyle
B_{-1}=-\frac{1}{8}\left[\bigg{(}-2Q_{1}+2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}-2\,\mbox{Tr}\,A_{\nu}^{2}+3(\mbox{Tr}\,A_{\nu})^{2}\bigg{)}2^{-1+\frac{n}{2}}\frac{\Gamma\big{(}\frac{n-1}{2}\big{)}}{\Gamma(\frac{1}{2})}\,\frac{1}{|x|^{n-1}}\right.$
$\displaystyle\quad\quad\quad\left.+\frac{5\Gamma\big{(}\frac{n-5}{2}\big{)}}{2^{5}\pi^{\frac{n}{2}}\Gamma\big{(}\frac{5}{2}\big{)}}\,\frac{1}{|x|^{n-5}}*\bigg{(}\sum_{j,k=1}^{n}\big{(}(C^{-1})^{T}A_{\nu}^{*}C^{-1}\big{)}_{jk}\frac{\partial^{2}}{\partial
x_{j}\partial x_{k}}\bigg{)}^{2}\right.$
$\displaystyle\quad\quad\quad\left.+\frac{\Gamma\big{(}\frac{n-3}{2}\big{)}}{4\pi^{\frac{n}{2}}\Gamma\big{(}\frac{3}{2}\big{)}}\,\frac{1}{|x|^{n-3}}*\sum_{j,k=1}^{n}\bigg{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6(A_{\nu}^{2})_{jk}\bigg{)}\,\frac{\partial^{2}}{\partial
x_{j}\partial x_{k}}\right],$ $\displaystyle
B_{-2}\in{\mbox{OPS}}^{-2}(\partial\Omega),$
where $f*g$ is the convolution of two functions $f$ and $g$. Furthermore, the
principal, second and third symbols of $B$ are
(4.7) $\displaystyle
p_{0}^{B}=\frac{1}{2}\left(\mbox{Tr}\,A_{\nu}-\frac{\langle
A^{*}_{\nu}\xi,\xi\rangle}{\langle\xi,\xi\rangle}\right),$
$\displaystyle\quad\;\qquad p_{-1}^{B}(x,\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left[-2Q_{1}+2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}-2\,\mbox{Tr}\,A_{\nu}^{2}+3(\mbox{Tr}\,A_{\nu})^{2}+\frac{5\langle
A_{\nu}^{*}\xi,\xi\rangle^{2}}{\langle\xi,\xi\rangle^{2}}\right.$
$\displaystyle\left.-\frac{1}{\langle\xi,\xi\rangle}\sum_{j,k=1}^{n}\left(2{\tilde{R}}_{j(n+1)k(n+1)}+6(A_{\nu}^{2})_{jk}\right)\xi_{j}\xi_{k}\right]\frac{1}{\sqrt{\langle\xi,\xi\rangle}}$
and
(4.9) $\displaystyle p_{-2}^{B}(x,\xi)=\frac{\langle
A_{\nu}^{*}\xi,\xi\rangle}{8\langle\xi,\xi\rangle^{2}}\bigg{(}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\mbox{Tr}\,A_{\nu}^{2}-3(\mbox{Tr}\,A_{\nu})^{2}$
$\,\,-\frac{5\langle
A_{\nu}^{*}\xi,\xi\rangle^{2}}{\langle\xi,\xi\rangle^{2}}+\frac{1}{\langle\xi,\xi\rangle}\,\sum_{j,k=1}^{n}\big{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6(A_{\nu}^{2})_{jk}\big{)}\xi_{j}\xi_{k}\bigg{)}\qquad\qquad\qquad\quad\\\
$
$\displaystyle\quad\;\;-\frac{1}{4\langle\xi,\xi\rangle}\,\left\\{\frac{3}{2}(\mbox{Tr}\,A_{\nu})\bigg{(}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\mbox{Tr}\,A_{\nu}^{2}\bigg{)}-4(\mbox{Tr}\,A_{\nu})^{3}\right.$
$\displaystyle\left.\quad\quad+\frac{1}{4}\bigg{[}6Q_{2}+3Q_{3}+\sum_{j=1}^{n}\bigg{(}-2{\tilde{R}}_{j(n+1)j(n+1),(n+1)}+4\sum_{l=1}^{n}{\tilde{R}}_{l(n+1)j(n+1)}(A_{\nu})_{lj}\bigg{)}\bigg{]}\right.$
$\displaystyle\left.-\frac{1}{2}\bigg{[}\frac{\langle
A_{\nu}^{*}\xi,\xi\rangle}{\langle\xi,\xi\rangle}+\mbox{Tr}\,A_{\nu}\bigg{]}\bigg{[}\frac{1}{2\langle\xi,\xi\rangle}\bigg{(}\sum_{j,k=1}^{n}\big{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6(A_{\nu}^{2})_{jk}\big{)}\xi_{j}\xi_{k}\bigg{)}\right.\quad\quad\;$
$\displaystyle\ \left.-\frac{2\langle
A_{\nu}^{*}\xi,\xi\rangle^{2}}{\langle\xi,\xi\rangle^{2}}+2(\mbox{Tr}\,A_{\nu})^{2}-\frac{1}{2}\bigg{(}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\mbox{Tr}\,A_{\nu}^{2}\bigg{)}\bigg{]}\right.\quad\quad\;$
$\displaystyle\quad\qquad\left.+\frac{1}{4\langle\xi,\xi\rangle}\bigg{[}\sum_{j,k=1}^{n}\bigg{(}2{\tilde{R}}_{j(n+1)k(n+1),(n+1)}+20\sum_{l=1}^{n}{\tilde{R}}_{j(n+1)l(n+1)}(A_{\nu})_{lk}+24(A_{\nu}^{3})_{jk}\bigg{)}\xi_{j}\xi_{k}\bigg{]}\right.\quad\;\;$
$\displaystyle\quad\qquad\left.-\frac{3\langle
A_{\nu}^{*}\xi,\xi\rangle}{2\langle\xi,\xi\rangle^{2}}\bigg{[}\sum_{j,k=1}^{n}\bigg{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6(A_{\nu}^{2})_{jk}\bigg{)}\xi_{j}\xi_{k}\bigg{]}\right.\quad\;\;$
$\displaystyle\quad\qquad\left.+\frac{4\langle
A_{\nu}^{*}\xi,\xi\rangle^{3}}{\langle\xi,\xi\rangle^{3}}+\frac{\mbox{Tr}\,A_{\nu}}{2}\bigg{[}\frac{1}{2\langle\xi,\xi\rangle}\bigg{(}\sum_{j,k=1}^{n}\big{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6(A_{\nu}^{2})_{jk}\big{)}\xi_{j}\xi_{k}\bigg{)}\right.$
$\displaystyle\quad\qquad\left.-\frac{2\langle
A_{\nu}^{*}\xi,\xi\rangle^{2}}{\langle\xi,\xi\rangle^{2}}+2(\mbox{Tr}\,A_{\nu})^{2}-\frac{1}{2}\bigg{(}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\mbox{Tr}\,A_{\nu}^{2}\bigg{)}\bigg{]}\right.$
$\displaystyle\quad\qquad\left.-\frac{1}{4}\bigg{[}\frac{\langle
A_{\nu}^{*}\xi,\xi\rangle}{\langle\xi,\xi\rangle}+\mbox{Tr}\,A_{\nu}\bigg{]}\bigg{[}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\mbox{Tr}\,A_{\nu}^{2}-4(\mbox{Tr}\;A_{\nu})^{2}\bigg{]}\right\\},$
respectively.
Proof. We choose coordinates $x=(x_{1},\cdots,x_{n})$ on an open set in
$\partial\Omega$ and then coordinates $(x,x_{n+1})$ on a neighborhood in
$\bar{\Omega}$ such that $x_{n+1}=0$ on $\partial\Omega$ and $|\nabla
x_{n+1}|=1$ near $\partial\Omega$ while $x_{n+1}>0$ on $\Omega$ and such that
$x$ is constant on each geodesic segment in $\bar{\Omega}$ normal to
$\partial\Omega$. Then the metric tensor on $\bar{\Omega}$ has the form (see,
p. 532 of [107])
(4.10)
$\displaystyle\big{(}g_{jk}(x,x_{n+1})\big{)}_{(n+1)\times(n+1)}=\begin{pmatrix}(h_{jk}(x,x_{n+1}))_{n\times
n}&0\\\ 0&1\end{pmatrix}.$
The Laplace-Beltrami operator $\Delta_{g}$ on $\Omega$ is given in local
coordinates by
$\displaystyle\Delta_{g}u$ $\displaystyle=$
$\displaystyle\sum_{j,k=1}^{n+1}\frac{1}{\sqrt{|g|}}\,\frac{\partial}{\partial
x_{j}}\left(\sqrt{|g|}\,g^{jk}\frac{\partial u}{\partial x_{k}}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{|h|}}\,\frac{\partial}{\partial
x_{n+1}}\left(\sqrt{|h|}\,\frac{\partial u}{\partial
x_{n+1}}\right)+\sum_{j,k=1}^{n}\frac{1}{\sqrt{|h|}}\frac{\partial}{\partial
x_{j}}\left(\sqrt{|h|}h^{jk}\,\frac{\partial u}{\partial x_{k}}\right)$
$\displaystyle=$ $\displaystyle\frac{\partial^{2}u}{\partial
x_{n+1}^{2}}+a(x_{n+1})\frac{\partial u}{\partial
x_{n+1}}+L(x,x_{n+1},D_{x})u,\quad\;\;a(x_{n+1})=\frac{1}{2|h|}\,\frac{\partial|h|}{\partial
x_{n+1}},$
where, as usual, $|g|=\mbox{det}(g_{jk}),\,\,|h|=\mbox{det}(h_{jk})$, and
$L(x_{n+1})=L(x,x_{n+1},D_{x})$ is a family of Laplace-Beltrami operators on
$\partial\Omega$, associated to the family of metric $\left(h_{jk}\right)$ on
$\partial\Omega$, so $L(0)=\Delta_{h}$. Similar to the proof of Proposition
C.1 of [107], we will construct smooth families of operators
$A_{j}(x_{n+1})\in OPS^{1}(\partial\Omega)$ such that
(4.11) $\displaystyle\frac{\partial^{2}}{\partial
x_{n+1}^{2}}+a(x_{n+1})\frac{\partial}{\partial x_{n+1}}+L(x_{n+1})$
$\displaystyle\quad\;=\left(\frac{\partial}{\partial
x_{n+1}}-A_{1}(x_{n+1})\right)\left(\frac{\partial}{\partial
x_{n+1}}+A_{2}(x_{n+1})\right),$
modulo a smooth operator. It will follow that the principal parts of
$A_{1}(x_{n+1})$ and $A_{2}(x_{n+1})$ is $\sqrt{-L(x_{n+1})}$, and
(4.12)
$\displaystyle{\mathcal{N}}_{g}=-A_{2}(0)\quad\,\mbox{mod}\;\;OPS^{-\infty}(\partial\Omega).$
In view of $[\frac{\partial}{\partial x_{n+1}},A_{2}(x_{n+1})]=\frac{\partial
A_{2}(x_{n+1})}{\partial x_{n+1}}$ (see (4.9) of [82]), it follows that the
right-hand side of (4.11) is equal to
$\displaystyle\frac{\partial^{2}}{\partial
x_{n+1}^{2}}-A_{1}(x_{n+1})\frac{\partial}{\partial
x_{n+1}}+A_{2}(x_{n+1})\frac{\partial}{\partial x_{n+1}}+\frac{\partial
A_{2}(x_{n+1})}{\partial x_{n+1}}-A_{1}(x_{n+1})A_{2}(x_{n+1}).$
Thus we get
$\displaystyle\left\\{\begin{array}[]{ll}A_{2}(x_{n+1})-A_{1}(x_{n+1})=a(x_{n+1}),\\\
-A_{1}(x_{n+1})A_{2}(x_{n+1})+\frac{\partial A_{2}(x_{n+1})}{\partial
x_{n+1}}=L(x_{n+1}),\end{array}\right.$
from which we have an equation for $A_{1}(x_{n+1})$:
$\displaystyle A_{1}(x_{n+1})^{2}+A_{1}(x_{n+1})a(x_{n+1})-\frac{\partial
A_{1}(x_{n+1})}{\partial x_{n+1}}=-L(x_{n+1})+\frac{\partial
a(x_{n+1})}{\partial x_{n+1}}.$
Setting
(4.14) $\displaystyle
A_{1}(x_{n+1})=\Upsilon(x_{n+1})+B(x_{n+1}),\quad\,\Upsilon(x_{n+1})=\sqrt{-L(x_{n+1})},$
we obtain an equation for $B(x_{n+1})$:
(4.15) $\displaystyle
2B(x_{n+1})\Upsilon(x_{n+1})+[\Upsilon(x_{n+1}),B(x_{n+1})]+B(x_{n+1})^{2}-\frac{\partial
B(x_{n+1})}{\partial x_{n+1}}$
$\displaystyle\quad\quad\;\;+B(x_{n+1})a(x_{n+1})=\frac{\partial\Upsilon(x_{n+1})}{\partial
x_{n+1}}-\Upsilon(x_{n+1})a(x_{n+1})+\frac{\partial a(x_{n+1})}{\partial
x_{n+1}}.$
We will inductively obtain terms $B_{j}(x_{n+1})\in OPS^{-j}(\partial\Omega)$
and establish that, with $B(x_{n+1})\sim\sum_{j\geq 0}B_{j}(x_{n+1})$, the
operators
$\displaystyle
A_{1}(x_{n+1})=\sqrt{-L(x_{n+1})}+B(x_{n+1}),\quad\;A_{2}(x_{n+1})=\sqrt{-L(x_{n+1})}+B(x_{n+1})+a(x_{n+1})$
do yield (4.11) modulo a smooth operator. Since $B(x_{n+1})$ is a smooth
family in $OPS^{0}(\partial\Omega)$, the principal part $B_{0}(x_{n+1})$ must
satisfy
$2B_{0}(x_{n+1})\Upsilon(x_{n+1})=\frac{\partial\Upsilon(x_{n+1})}{\partial
x_{n+1}}-a(x_{n+1})\Upsilon(x_{n+1})$, i.e.,
(4.16) $\displaystyle
B_{0}(x_{n+1})=\frac{1}{2}\,\frac{\partial\Upsilon(x_{n+1})}{\partial
x_{n+1}}\Upsilon(x_{n+1})^{-1}-\frac{1}{2}a(x_{n+1})\quad\;\,OPS^{-1}(\partial\Omega).$
By the second identity of (4.14) we derive
(4.17) $\displaystyle\left(\frac{\partial\Upsilon(x_{n+1})}{\partial
x_{n+1}}(0)\right)\Upsilon(0)^{-1}=\frac{1}{2}\left(\frac{\partial
L(x_{n+1})}{\partial
x_{n+1}}(0)\right)L(0)^{-1}\quad\;\mbox{mod}\;\;OPS^{-1}(\partial\Omega).$
Therefore,
(4.18) $\displaystyle B_{0}(0)=\frac{1}{4}\left[\left(\frac{\partial
L(x_{n+1})}{\partial
x_{n+1}}(0)\right)L(0)^{-1}-|h|^{-1}\frac{\partial|h|}{\partial
x_{n+1}}(x,0)\right].$
Next, noting that $B(x_{n+1})$ is also a smooth family of
$OPS^{-1}(\partial\Omega)$, by compare the second symbol of $B(x_{n+1})$ we
have
$\displaystyle
2B_{-1}(x_{n+1})\Upsilon(x_{n+1})+B_{0}(x_{n+1})^{2}-\frac{\partial
B_{0}(x_{n+1})}{\partial x_{n+1}}+B_{0}(x_{n+1})a(x_{n+1})=\frac{\partial
a(x_{n+1})}{\partial x_{n+1}},$
i.e.,
$\displaystyle B_{-1}(x_{n+1})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\frac{\partial a(x_{n+1})}{\partial
x_{n+1}}-B_{0}(x_{n+1})^{2}+\frac{\partial B_{0}(x_{n+1})}{\partial
x_{n+1}}\right.$
$\displaystyle\left.-B_{0}(x_{n+1})\frac{}{}a(x_{n+1})\right)\Upsilon(x_{n+1})^{-1}.$
From (4.16) we get
$\displaystyle\frac{\partial B_{0}(x_{n+1})}{\partial
x_{n+1}}=\frac{1}{2}\left[\frac{\partial^{2}\Upsilon(x_{n+1})}{\partial
x_{n+1}^{2}}\Upsilon(x_{n+1})^{-1}-\left(\frac{\partial\Upsilon(x_{n+1})}{\partial
x_{n+1}}\right)^{2}\Upsilon(x_{n+1})^{-2}-\frac{\partial a(x_{n+1})}{\partial
x_{n+1}}\right],$
so that
$\displaystyle B_{-1}(x_{n+1})=\frac{1}{2}\bigg{[}\frac{1}{2}\,\frac{\partial
a(x_{n+1})}{\partial
x_{n+1}}-B_{0}(x_{n+1})^{2}+\frac{1}{2}\frac{\partial^{2}\Upsilon(x_{n+1})}{\partial
x_{n+1}^{2}}\Upsilon(x_{n+1})^{-1}$
$\displaystyle\quad\quad\;-\frac{1}{2}\,\bigg{(}\frac{\partial\Upsilon(x_{n+1})}{\partial
x_{n+1}}\bigg{)}^{2}\Upsilon(x_{n+1})^{-2}-B_{0}(x_{n+1})a(x_{n+1})\bigg{]}\Upsilon(x_{n+1})^{-1}.$
Again, by using the second identity of (4.14), we have
$\displaystyle
2\Upsilon(x_{n+1})\,\frac{\partial^{2}\Upsilon(x_{n+1})}{\partial
x_{n+1}^{2}}+2\left(\frac{\partial\Upsilon(x_{n+1})}{\partial
x_{n+1}}\right)^{2}=-\frac{\partial^{2}L(x_{n+1})}{\partial x_{n+1}^{2}},$
$\displaystyle\quad\,\,\left(\frac{\partial\Upsilon(x_{n+1})}{\partial
x_{n+1}}\right)^{2}=\frac{1}{4}\left(\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}\right)^{2}\Upsilon(x_{n+1})^{-2}.$
Combining these identities we obtain
$\displaystyle B_{-1}(x_{n+1})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{2}\,\frac{\partial
a(x_{n+1})}{\partial
x_{n+1}}-B_{0}(x_{n+1})^{2}-\frac{1}{4}\,\frac{\partial^{2}L(x_{n+1})}{\partial
x_{n+1}^{2}}(-L(x_{n+1}))^{-1}\right.$
$\displaystyle\left.-\frac{1}{4}\left(\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}\right)^{2}L(x_{n+1})^{-2}-B_{0}(x_{n+1})a(x_{n+1})\right]{\sqrt{-L(x_{n+1})}\,}^{-1},$
so that
$\displaystyle B_{-1}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{2}\,\frac{\partial
a(x_{n+1})}{\partial
x_{n+1}}(x,0)-B_{0}(0)^{2}-\frac{1}{4}\left(\frac{\partial^{2}L(x_{n+1})}{\partial
x_{n+1}^{2}}(0)\right)(-L(0))^{-1}\right.$
$\displaystyle\left.-\frac{1}{4}\left(\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}(0)\right)^{2}L(0)^{-2}-B_{0}(0)a(0)\right]{\sqrt{-L(0)}\,}^{-1}.$
Inserting (4.18) into the above identity, we get
$\displaystyle B_{-1}(0)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{1}{4|h|}\,\frac{\partial^{2}|h|}{\partial
x_{n+1}^{2}}(x,0)-\frac{3}{16|h|^{2}}\left(\frac{\partial|h|}{\partial
x_{n+1}}(x,0)\right)^{2}\right.$
$\displaystyle\left.-\frac{5}{16}\left(\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}(0)\right)^{2}L(0)^{-2}-\frac{1}{4}\left(\frac{\partial^{2}L(x_{n+1})}{\partial
x_{n+1}^{2}}(0)\right)\big{(}-L(0)\big{)}^{-1}\right]{\sqrt{-L(0)}\,}^{-1}.$
Furthermore, by compare the third symbol of $B(x_{n+1})$ we get
$\displaystyle
2B_{-2}(x_{n+1})\Psi(x_{n+1})+2B_{0}(x_{n+1})B_{-1}(x_{n+1})-\frac{\partial
B_{-1}(x_{n+1})}{\partial x_{n+1}}+a(x_{n+1})B_{-1}(x_{n+1})=0,$
i.e.,
$\displaystyle B_{-2}(x_{n+1})$ $\displaystyle=$
$\displaystyle\bigg{(}-B_{0}(x_{n+1})B_{-1}(x_{n+1})+\frac{1}{2}\,\frac{\partial
B_{-1}(x_{n+1})}{\partial x_{n+1}}$
$\displaystyle-\frac{a(x_{n+1})}{2}\,B_{-1}(x_{n+1})\bigg{)}\Psi^{-1}(x_{n+1}),$
where
(4.21) $\displaystyle\frac{\partial B_{-1}(x_{n+1})}{\partial
x_{n+1}}=\frac{1}{4}\bigg{[}\frac{1}{2}\,\frac{\partial a(x_{n+1})}{\partial
x_{n+1}}-B_{0}(x_{n+1})^{2}-\frac{1}{4}\,\frac{\partial^{2}L(x_{n+1})}{\partial
x_{n+1}^{2}}(-L(x_{n+1}))^{-1}$ $\displaystyle-\frac{1}{4}\left(\frac{\partial
L(x_{n+1})}{\partial
x_{n+1}}\right)^{2}L(x_{n+1})^{-2}-a(x_{n+1})B_{0}(x_{n+1})\bigg{]}\frac{\partial
L(x_{n+1})}{\partial x_{n+1}}(-L(x_{n+1}))^{-\frac{3}{2}}$
$\displaystyle+\frac{1}{2}\bigg{[}\frac{1}{2}\,\frac{\partial^{2}a(x_{n+1})}{\partial
x_{n+1}^{2}}-2B_{0}(x_{n+1})\,\frac{\partial B_{0}(x_{n+1})}{\partial
x_{n+1}}+\frac{1}{4}\,\frac{\partial^{3}L(x_{n+1})}{\partial
x_{n+1}^{3}}L(x_{n+1})^{-1}$
$\displaystyle-\frac{3}{4}\,\frac{\partial^{2}L(x_{n+1})}{\partial
x_{n+1}^{2}}\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}L(x_{n+1})^{-2}+\frac{1}{2}\big{(}\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}\big{)}^{2}\,\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}L(x_{n+1})^{-3}$ $\displaystyle-a(x_{n+1})\,\frac{\partial
B(x_{n+1})}{\partial x_{n+1}}-B_{0}(x_{n+1})\frac{\partial
a(x_{n+1})}{\partial x_{n+1}}\bigg{]}\sqrt{-L(x_{n+1})}^{\;-1}.$
Hence
$\displaystyle{\mathcal{N}}_{g}$ $\displaystyle=$ $\displaystyle-
A_{2}(0)=-\sqrt{-L(0)}-\big{(}B_{0}(0)+a(0)\big{)}-B_{-1}(0)-B_{-2}(0)$
$\displaystyle=$
$\displaystyle-\sqrt{-\Delta_{g}}-\frac{1}{4}\left[\left(\frac{L(x_{n+1})}{\partial
x_{n+1}}(0)\right)L(0)^{-1}+|h|^{-1}\frac{\partial|h|}{\partial
x_{n+1}}(x,0)\right]$
$\displaystyle-\frac{1}{2}\left[\frac{1}{4|h|}\,\frac{\partial^{2}|h|}{\partial
x_{n+1}^{2}}(x,0)-\frac{3}{16|h|^{2}}\left(\frac{\partial|h|}{\partial
x_{n+1}}(x,0)\right)^{2}\right.$
$\displaystyle\left.-\frac{5}{16}\left(\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}(0)\right)^{2}L(0)^{-2}-\frac{1}{4}\left(\frac{\partial^{2}L(x_{n+1})}{\partial
x_{n+1}^{2}}(0)\right)\big{(}-L(0)\big{)}^{-1}\right]{\sqrt{-L(0)}\,}^{-1}-B_{-2}(0)$
$\displaystyle\quad\;\;\mbox{mod}\;\;OPS^{-3}(\partial\Omega).$
We will compute the first, second and third symbols of $B$. It is obvious that
the principal symbols of $L(0)$ and $\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}(0)$ are
$-\sum_{j,k=1}^{n}h^{jk}(x,0)\xi_{j}\xi_{k}=-\langle\xi,\xi\rangle$ and
$-\sum\frac{\partial h^{jk}}{\partial x_{n+1}}(x,0)\xi_{j}\xi_{k}$,
respectively. Furthermore, we choose a normal coordinate system on
$(\partial\Omega,h)$, centered at $x_{0}\in\partial\Omega$. From (C.24) of
[107] (see also, (4.68)–(4.70) of Appendix C of [107]), one has
(4.22) $\displaystyle\sum_{j,k=1}^{n}\frac{\partial h^{jk}}{\partial
x_{n+1}}(x_{0},0)\xi_{j}\xi_{k}=2\langle A_{\nu}^{*}\xi,\xi\rangle,$
so that the principal symbol of $\frac{\partial L(x_{n+1})}{\partial
x_{n+1}}(0)\,L(0)^{-1}$ is $\frac{2\langle
A^{*}_{\nu}\xi,\xi\rangle}{\langle\xi,\xi\rangle}$. It follows from (C.26) of
[107] that
(4.23) $\displaystyle|h|^{-1}\frac{\partial|h|}{\partial
x_{n+1}}(x_{0},0)=\sum_{j=1}^{n}\frac{\partial h_{jj}}{\partial
x_{n+1}}(x_{0},0)=-2\,\mbox{Tr}\,A_{\nu}.$
Consequently
$\displaystyle p_{0}^{B}=\frac{1}{2}\left(\mbox{Tr}\,A_{\nu}-\frac{\langle
A^{*}_{\nu}\xi,\xi\rangle}{\langle\xi,\xi\rangle}\right)$
and
$\displaystyle p_{-1}^{B}(x_{0},\xi)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\frac{1}{4|h|}\,\frac{\partial^{2}|h|}{\partial
x_{n+1}^{2}}(x_{0},0)-\frac{3}{16}\left(\frac{1}{|h|}\,\frac{\partial|h|}{\partial
x_{n+1}}(x_{0},0)\right)^{2}\right.$
$\displaystyle\left.-\frac{5}{16}\left(\sum_{j,k=1}^{n}\frac{\partial
h^{jk}}{\partial
x_{n+1}}(x_{0},0)\,\xi_{j}\xi_{k}\right)^{2}\left(\sum_{j,k=1}^{n}h^{jk}(x_{0},0)\,\xi_{j}\xi_{k}\right)^{-2}\right.$
$\displaystyle\left.+\frac{1}{4}\left(\sum_{j,k=1}^{n}\frac{\partial^{2}h^{jk}}{\partial
x_{n+1}^{2}}(x_{0},0)\,\xi_{j}\xi_{k}\right)\left(\sum_{j,k=1}^{n}h^{jk}(x_{0},0)\,\xi_{j}\xi_{k}\right)^{-1}\right]$
$\displaystyle\quad\,\times\left(\sum_{j,k=1}^{n}h^{jk}(x_{0},0)\,\xi_{j}\xi_{k}\right)^{-1/2}$
$\displaystyle=-\frac{1}{8}\left[\frac{\partial^{2}|h|}{\partial
x_{n+1}^{2}}(x_{0},0)-3\left(\mbox{Tr}\,A_{\nu}\right)^{2}-\frac{5\langle
A^{*}_{\nu}\xi,\xi\rangle^{2}}{\langle\xi,\xi\rangle^{2}}\right.$
$\displaystyle\quad\;\left.+\frac{1}{\langle\xi,\xi\rangle}\left(\sum_{j,k=1}^{n}\frac{\partial^{2}h^{jk}}{\partial
x_{n+1}^{2}}(x_{0},0)\,\xi_{j}\xi_{k}\right)\right]\langle\xi,\xi\rangle^{-1/2}.\quad\;$
Note that $\left(-\frac{1}{2}\,\frac{\partial h_{jk}}{\partial
x_{n+1}}(x_{0},0)\right)_{n\times n}$ is the matrix of Weingarten’s map under
a basis of $T_{0}(\partial\Omega)$ at $x_{0}$, and its eigenvalues are just
the principal curvatures $\kappa_{1},\cdots,\kappa_{n}$ of $\partial\Omega$ in
the direction $\nu$. We can write
$\displaystyle g_{jk}(x)$ $\displaystyle=$
$\displaystyle\delta_{jk}+\sum_{l=1}^{n+1}\frac{\partial g_{jk}}{\partial
x_{l}}(0)\,x_{l}+\frac{1}{2}\sum_{l,m=1}^{n+1}\frac{\partial^{2}g_{jk}}{\partial
x_{l}\partial x_{m}}(0)\,x_{l}x_{m}$
$\displaystyle+\frac{1}{6}\sum_{l,m,v=1}^{n+1}\frac{\partial^{3}g_{jk}}{\partial
x_{l}\partial x_{m}\partial
x_{v}}(0)\,x_{l}x_{m}x_{v}+O(|x|^{4})\;\;\mbox{near}\;\,0.$
By taking $x=(0,\cdots,0,x_{n+1}):=x_{n+1}e_{n+1}$, we find that
$\displaystyle|h(0+x_{n+1}e_{n+1})|=|g(0+x_{n+1}e_{n+1})|$
$\displaystyle\quad\;=\mbox{det}\,\left(\delta_{jk}+\frac{\partial
g_{jk}}{\partial
x_{n+1}}(0)x_{n+1}+\frac{1}{2}\,\frac{\partial^{2}g_{jk}}{\partial
x_{n+1}^{2}}(0)x_{n+1}^{2}+\frac{1}{6}\,\frac{\partial^{3}g_{jk}}{\partial
x_{n+1}^{3}}(0)\,x_{n+1}^{3}+O(|x_{n+1}|^{4})\right),$
so that
$\displaystyle\quad\quad\quad\;\frac{\partial^{2}|h|}{\partial
x_{n+1}^{2}}(0)$ $\displaystyle=$ $\displaystyle 2\sum_{1\leq j<k\leq
n}\bigg{(}\frac{\partial h_{jj}}{\partial x_{n+1}}(0)\,\frac{\partial
h_{kk}}{\partial x_{n+1}}(0)-\frac{\partial h_{jk}}{\partial
x_{n+1}}(0)\,\frac{\partial h_{kj}}{\partial
x_{n+1}}(0)\bigg{)}+\sum_{j=1}^{n}\frac{\partial^{2}h_{jj}}{\partial
x_{n+1}^{2}}(0)$ $\displaystyle:=$ $\displaystyle
2Q_{1}+\sum_{j=1}^{n}\frac{\partial^{2}h_{jj}}{\partial x_{n+1}^{2}}(0),$
$\displaystyle\frac{\partial^{3}|h|}{\partial x_{n+1}^{3}}(0)$
$\displaystyle=$ $\displaystyle 6\sum_{1\leq j<k<l\leq
n}\bigg{(}\frac{\partial h_{jj}}{\partial x_{n+1}}(0)\,\frac{\partial
h_{kk}}{\partial x_{n+1}}(0)\frac{\partial h_{ll}}{\partial
x_{n+1}}(0)+\frac{\partial h_{jk}}{\partial x_{n+1}}(0)\,\frac{\partial
h_{kl}}{\partial x_{n+1}}(0)\frac{\partial h_{lj}}{\partial x_{n+1}}(0)$
$\displaystyle+\frac{\partial h_{jl}}{\partial x_{n+1}}(0)\,\frac{\partial
h_{kj}}{\partial x_{n+1}}(0)\frac{\partial h_{lk}}{\partial
x_{n+1}}(0)-\frac{\partial h_{jl}}{\partial x_{n+1}}(0)\,\frac{\partial
h_{kk}}{\partial x_{n+1}}(0)\frac{\partial h_{lj}}{\partial x_{n+1}}(0)$
$\displaystyle-\frac{\partial h_{jk}}{\partial x_{n+1}}(0)\,\frac{\partial
h_{kj}}{\partial x_{n+1}}(0)\frac{\partial h_{ll}}{\partial
x_{n+1}}(0)-\frac{\partial h_{jj}}{\partial x_{n+1}}(0)\,\frac{\partial
h_{kl}}{\partial x_{n+1}}(0)\frac{\partial h_{lk}}{\partial
x_{n+1}}(0)\bigg{)}$ $\displaystyle+3\sum_{j\neq k}\bigg{(}\frac{\partial
h_{jj}}{\partial x_{n+1}}(0)\,\frac{\partial^{2}h_{kk}}{\partial
x_{n+1}^{2}}(0)-\frac{\partial h_{jk}}{\partial
x_{n+1}}(0)\,\frac{\partial^{2}h_{kj}}{\partial
x_{n+1}^{2}}(0)\bigg{)}+\sum_{j=1}^{n}\frac{\partial^{3}h_{jj}}{\partial
x_{n+1}^{3}}(0)$ $\displaystyle:=$ $\displaystyle
6Q_{2}+3Q_{3}+\sum_{j=1}^{n}\frac{\partial^{3}h_{jj}}{\partial
x_{n+1}^{3}}(0).$
It is well-known (see, for example, [60], [99], [24], [52]) that
$Q_{1}=4\sum_{1\leq j<k\leq
n}\kappa_{j}\kappa_{k}=2\big{(}R_{\partial\Omega}-{\tilde{R}}+2{\tilde{R}}_{\nu\nu}\big{)}$
and $Q_{2}=-8\sum_{1\leq j<k<l\leq n}\kappa_{j}\kappa_{k}\kappa_{l}$. A simple
calculation shows that, at $x_{0}=0$,
$\displaystyle\quad\,\Gamma_{(n+1)(n+1)}^{s}=0,\quad\,\Gamma_{(n+1)k}^{s}=\frac{1}{2}\sum_{l=1}^{n+1}g^{sl}\frac{\partial
g_{lk}}{\partial
x_{n+1}},\quad\Gamma_{j(n+1)}^{t}=\frac{1}{2}\sum_{l=1}^{n+1}g^{tl}\,\frac{\partial
g_{lj}}{\partial x_{n+1}},$
where $\Gamma_{\beta\gamma}^{\alpha}$ are the Christoffel symbols. It follows
from p. 188 of [99] (or [53]) that
$\displaystyle{\tilde{R}}_{j(n+1)k(n+1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\frac{\partial^{2}g_{jk}}{\partial
x_{n+1}^{2}}(0)+\sum_{s,t=1}^{n+1}g_{st}\left(\Gamma_{(n+1)k}^{s}\Gamma_{j(n+1)}^{t}-\Gamma_{(n+1)(n+1)}^{s}\Gamma_{jk}^{t}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\,\frac{\partial^{2}h_{jk}}{\partial
x_{n+1}^{2}}(0)+\frac{1}{4}\sum_{t,l=1}^{n}\frac{\partial h_{tk}}{\partial
x_{n+1}}(0)h^{tl}(0)\,\frac{\partial h_{lj}}{\partial x_{n+1}}(0)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\,\frac{\partial^{2}h_{jk}}{\partial
x_{n+1}^{2}}(0)+(A_{\nu}^{2})_{jk},$
$\displaystyle{\tilde{R}}_{j(n+1)k(n+1),(n+1)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\frac{\partial^{3}h_{jk}}{\partial
x_{n+1}^{3}}(0)-\frac{1}{4}\,\sum_{m,l=1}^{n+1}\big{(}\frac{\partial
h_{km}}{\partial x_{n+1}}(0)\,\frac{\partial h_{ml}}{\partial
x_{n+1}}(0)\,\frac{\partial h_{lj}}{\partial x_{n+1}}(0)\big{)}$
$\displaystyle+\frac{1}{4}\sum_{l=1}^{n+1}\big{(}\frac{\partial^{2}h_{lk}}{\partial
x_{n+1}^{2}}(0)\,\frac{\partial h_{lj}}{\partial x_{n+1}}(0)+\frac{\partial
h_{lk}}{\partial x_{n+1}}(0)\,\frac{\partial^{2}h_{lj}}{\partial
x_{n+1}^{2}}(0)\big{)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\frac{\partial^{3}h_{jk}}{\partial
x_{n+1}^{3}}(0)+2\sum_{l=1}^{n+1}{\tilde{R}}_{l(n+1)k(n+1)}(A_{\nu})_{lj},$
so that
(4.30)
$\displaystyle\left.\begin{array}[]{l}\frac{\partial^{2}h_{jk}}{\partial
x_{n+1}^{2}}(x_{0},0)=-2{\tilde{R}}_{j(n+1)k(n+1)}+2\,(A_{\nu}^{2})_{jk}\\\
\frac{\partial^{3}h_{jk}}{\partial
x_{n+1}^{3}}(x_{0},0)=-2{\tilde{R}}_{j(n+1)k(n+1),(n+1)}+4\sum_{l=1}^{n}{\tilde{R}}_{l(n+1)k(n+1)}(A_{\nu})_{lj}.\end{array}\right.$
Combining these and (4)–(4) we have
(4.31) (4.34)
$\displaystyle\left.\begin{array}[]{l}\frac{\partial^{2}|h|}{\partial
x_{n+1}^{2}}(x_{0},0)=2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\,\mbox{Tr}\,A_{\nu}^{2},\\\
\frac{\partial^{3}|h|}{\partial
x_{n+1}^{3}}(x_{0},0)=6Q_{2}+3Q_{3}+\sum_{j=1}^{n}\big{(}-2{\tilde{R}}_{j(n+1)j(n+1),(n+1)}+4\sum_{l=1}^{n}{\tilde{R}}_{l(n+1)j(n+1)}(A_{\nu})_{lj}\big{)}\end{array}\right.$
with
$Q_{3}=\sum_{j\neq
k}\left[(-2A_{\nu})_{jj}\big{(}-2{\tilde{R}}_{k(n+1)k(n+1)}+2(A_{\nu}^{2})_{kk}\big{)}-(-2A_{\nu})_{jk}\big{(}-2{\tilde{R}}_{k(n+1)j(n+1)}+2(A_{\nu}^{2})_{kj}\big{)}\right].$
From $hh^{-1}=I$, we get
$\displaystyle\frac{\partial h}{\partial x_{n+1}}h^{-1}+h\frac{\partial
h^{-1}}{\partial x_{n+1}}=0,\quad\quad\,\,\frac{\partial^{2}h}{\partial
x_{n+1}^{2}}h^{-1}+2\frac{\partial h}{\partial x_{n+1}}\frac{\partial
h^{-1}}{\partial x_{n+1}}+h\,\frac{\partial^{2}h^{-1}}{\partial
x_{n+1}^{2}}=0,$ $\displaystyle\frac{\partial^{3}h}{\partial
x_{n+1}^{3}}h^{-1}+3\frac{\partial^{2}h}{\partial x_{n+1}^{2}}\,\frac{\partial
h^{-1}}{\partial x_{n+1}}+3\frac{\partial h}{\partial
x_{n+1}}\,\frac{\partial^{2}h^{-1}}{\partial
x_{n+1}^{2}}+h\frac{\partial^{3}h^{-1}}{\partial x_{n+1}^{3}}=0,$
which imply
$\displaystyle\frac{\partial^{2}h^{jk}}{\partial x_{n+1}^{2}}(x_{0},0)$
$\displaystyle=$ $\displaystyle-\frac{\partial^{2}h_{jk}}{\partial
x_{n+1}^{2}}(x_{0},0)-2\sum_{l=1}^{n}\frac{\partial h_{jl}}{\partial
x_{n+1}}(x_{0},0)\,\frac{\partial h^{lk}}{\partial x_{n+1}}(x_{0},0)$
$\displaystyle=$ $\displaystyle-\frac{\partial^{2}h_{jk}}{\partial
x_{n+1}^{2}}(x_{0},0)+8(A_{\nu}^{2})_{jk},$
$\displaystyle\frac{\partial^{3}h^{jk}}{\partial x_{n+1}^{3}}(x_{0},0)$
$\displaystyle=$ $\displaystyle-\frac{\partial^{3}h_{jk}}{\partial
x_{n+1}^{3}}(x_{0},0)-3\sum_{l=1}^{n}\frac{\partial^{2}h_{jl}}{\partial
x_{n+1}^{2}}(x_{0},0)\,\frac{\partial h^{lk}}{\partial x_{n+1}}(x_{0},0)$
$\displaystyle-3\sum_{l=1}^{n}\frac{\partial h_{jl}}{\partial
x_{n+1}}(x_{0},0)\,\frac{\partial^{2}h^{lk}}{\partial x_{n+1}^{2}}(x_{0},0).$
Thus
(4.37)
$\displaystyle\quad\quad\qquad\left.\begin{array}[]{l}\frac{\partial^{2}h^{jk}}{\partial
x_{n+1}^{2}}(x_{0},0)=2{\tilde{R}}_{j(n+1)k(n+1)}+6(A_{\nu}^{2})_{jk},\\\
\frac{\partial^{3}h^{jk}}{\partial
x_{n+1}^{3}}(x_{0},0)=2{\tilde{R}}_{j(n+1)k(n+1),(n+1)}+20\sum_{l=1}^{n}{\tilde{R}}_{j(n+1)l(n+1)}(A_{\nu})_{lk}+24(A_{\nu}^{3})_{jk}.\end{array}\right.$
Combining (4), (4), (4.30), (4.31) and (4.37), we obtain (4) and (4.9).
Finally, it follows from p. 363 of [43] that the Fourier transform of
$2^{\lambda+\frac{n}{2}}\,\frac{\Gamma\big{(}\frac{\lambda+n}{2}\big{)}}{\Gamma\big{(}-\frac{\lambda}{2}\big{)}}|x|^{-\lambda-n}$
is $|\eta|^{\lambda}$. Since $(h^{jk})$ is positive-definite, there exists a
matrix $C$ such that $C^{T}C=(h^{jk})$. By setting $\eta=C\xi$ and by using
the fact that $\widehat{f*g}=(2\pi)^{\frac{n}{2}}\hat{f}\cdot\hat{g}$, we
obtain the expressions (4.4)–(4.6) for ${\mathcal{N}}_{g}$.
Remark 4.3. The proofs of Theorems 4.1 and 4.2 also provide a general method
to calculate all $p_{m}(x,\xi)$ ($m=1,0,-1,-2,\cdots$) and $p^{B}_{m}(x,\xi)$
($m=0,-1,-2,\cdots$) for the pseudodifferential operators
$-\sqrt{-\Delta_{h}}$ and $B$, respectively.
## 5\. Series representation of the heat kernel associated to the Dirichlet-
to-Neumann operator
Let $\Omega$ is a bounded domain with smooth boundary $\partial\Omega$ in
$(n+1)$-dimensional Riemannian manifold $(\mathcal{M},g)$. Define
$-\Delta^{0}_{h}$ to be the Laplacian $-\Delta_{h}$ on $\partial\Omega$ with
its coefficients frozen at $y\in\partial\Omega$, and let
$\sqrt{-\Delta_{h}^{0}}$ be the square root operator of $-\Delta_{h}^{0}$,
where $h$ is the induced metric on $\partial\Omega$ by $g$. Denote by
$G_{0}(t,x,y)$ and $F_{0}(t,x,y)$ the fundamental solutions of $\frac{\partial
u}{\partial t}=\Delta_{g}^{0}u$ and $\frac{\partial v}{\partial
t}=-\sqrt{-\Delta_{h}^{0}}v$ evaluated $t>0$, $x\in\partial\Omega$, and the
same point $y\in\partial\Omega$ at which the coefficients of $-\Delta_{g}^{0}$
and $\sqrt{-\Delta_{h}^{0}}$ are computed, respectively. Leu us denote
$\displaystyle\left(G_{0}\\#\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)\right)(t,x,y)=\int_{0}^{t}\int_{\partial\Omega}G_{0}(s,x,z)\left((\Delta_{h}-\Delta_{h}^{0})G_{0}(t-s,z,y)\right)ds\,dz.$
Furthermore
$\displaystyle
G_{0}\\#\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)\\#\cdots\\#\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)\quad(m\mbox{-fold})$
can be constructed inductively.
Theorem 5.1. Suppose that $F(t,x,y)$ satisfies
$\displaystyle\left\\{\begin{array}[]{ll}\frac{\partial F(t,x,y)}{\partial
t}=-\sqrt{-\Delta_{h}}\,F(t,x,y)&\mbox{for}\;\;(t,x,y)\in[0,\infty)\times\partial\Omega\times\partial\Omega,\\\
F(0,x,y)=\delta(x-y)&\mbox{for}\;\;x,y\in\partial\Omega.\end{array}\right.$
Then $F(t,x,y)$ has the following series representation:
(5.2) $\displaystyle
F(t,x,y)=\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}G_{0}(\mu,x,y)d\mu$
$\displaystyle\quad\quad+\sum_{m\geq
1}\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left[G_{0}\\#\underset{m}{\underbrace{\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)\\#\cdots\\#\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)}}(\mu,x,y)\right]d\mu.$
Proof. It is well-known (see, for example, (5.22) of p 247 of [106], [48] or
[51]) that for $\lambda\geq 0$,
$\displaystyle
e^{-t\sqrt{\lambda}}=\int_{0}^{\infty}\frac{t}{\sqrt{4\pi\mu^{3}}}e^{-t^{2}/4\mu}e^{-\mu\lambda}d\mu,$
i.e., the Laplace transform of $\frac{t}{\sqrt{4\pi\mu^{3}}}e^{-t^{2}/4\mu}$
is $e^{-t\sqrt{\lambda}}$. By applying the spectral theorem, we get that for
all $t>0$,
(5.3) $\displaystyle
e^{-t\sqrt{-\Delta_{h}}}\phi(x)=\int_{0}^{\infty}\frac{t}{\sqrt{4\pi\mu^{3}}}e^{-t^{2}/4\mu}e^{\mu\Delta_{h}}\phi(x)d\mu,\quad\;\forall\phi\in
H^{\frac{1}{2}}(\partial\Omega).$
Therefore, we have
$\displaystyle
e^{-t\sqrt{-\Delta_{h}}}\delta(x-y)=\int_{0}^{\infty}\frac{t}{\sqrt{4\pi\mu^{3}}}e^{-t^{2}/4\mu}\left(e^{\mu\Delta_{h}}\delta(x-y)\right)d\mu,$
i.e.,
(5.4) $\displaystyle
F(t,x,y)=\int_{0}^{\infty}\frac{t}{\sqrt{4\pi\mu^{3}}}e^{-t^{2}/4\mu}G(\mu,x,y)d\mu,$
where $G(t,x,y)$ is the fundamental solution of heat equation $\frac{\partial
u}{\partial t}=\Delta_{h}u$ on $[0,\infty)\times\partial\Omega$. It is obvious
(see, for example, [80] or p. 4 of [41]) that in the normal coordinates
(5.5) $\displaystyle G_{0}(t,x,y)=\left(\frac{1}{4\pi
t}\right)^{n/2}e^{-\sum_{j,k=1}^{n}(h_{jk}(y))(x_{j}-y_{j}-b_{j}(y)t)(x_{k}-y_{k}-b_{k}(y)t)/4t},$
where
$\displaystyle
b_{k}(y)=\sum_{j=1}^{n}\frac{1}{\sqrt{|h(y)|}}\,\frac{\partial(\sqrt{|}h|\,h^{jk})}{\partial
x_{j}}(y),\;\;k=1,2,\cdots,n.$
From the well-known estimate (see (3.5b) of [80]):
(5.6)
$\displaystyle\;\;\big{|}G_{0}\\#\underset{m}{\underbrace{\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)\\#\cdots\\#\left((\Delta_{g}-\Delta_{h}^{0})G_{0}\right)(t,x,y)}}\big{|}\leq\frac{c_{2}^{m}}{(m/2)!}\,t^{(m-n)/2}e^{-c_{3}(d(x,y))^{2}/4t},$
one immediately get that
(5.7) $\displaystyle\,\,\quad\quad G=G_{0}+\sum_{m\geq
1}G_{0}\\#\underset{m}{\underbrace{\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)\\#\cdots\\#\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)}}.$
Note that the sum of right-hand side converges rapidly to the unique
fundamental solution $G$. Simple calculation shows that
(5.8)
$\displaystyle\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}G_{0}(\mu,x,y)d\mu=\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\frac{1}{(4\pi\mu)^{n/2}}e^{-\sum_{j,k=1}^{n}h_{jk}(x_{j}-y_{j})(x_{k}-y_{k})/4\mu}d\mu$
$\displaystyle\quad\qquad\quad\;\;+\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\frac{1}{(4\pi\mu)^{n/2}}e^{\sum_{j,k=1}^{n}h_{jk}[(b_{k}(y))(x_{j}-y_{j})+(b_{j}(y))(x_{k}-y_{k})]/4}d\mu$
$\displaystyle\quad\qquad\quad\;\;+\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\frac{1}{(4\pi\mu)^{n/2}}e^{-\sum_{j,k=1}^{n}h_{jk}(b_{j}(y))(b_{k}(y))\mu/4}d\mu$
$\displaystyle\quad\qquad\quad\leq\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\left[\frac{t}{\big{(}t^{2}+\sum_{j,k=1}^{n}h_{jk}(x_{j}-y_{j})(x_{k}-y_{k})\big{)}^{(n+1)/2}}\right.$
$\displaystyle\left.\quad\quad\qquad\;\;+t^{-n}e^{\sum_{j,k=1}^{n}h_{jk}[(b_{k}(y))(x_{j}-x_{k})+(b_{j}(y))(x_{k}-y_{k})]}+t^{-n}\frac{}{{}}\right]$
and, for $m\geq 1$,
(5.9)
$\displaystyle\bigg{|}\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left[G_{0}\\#\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)\\#\cdots\\#\left((\Delta_{h}-\Delta_{h}^{0})G_{0}\right)(\mu,x,y)\right]d\mu\bigg{|}$
$\displaystyle\quad\qquad\leq\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left[\frac{c_{2}^{m}}{(m/2)!}\mu^{(m-n)/2}e^{-c_{3}(d(x,y))^{2}/4\mu}\right]d\mu$
$\displaystyle\quad\qquad=\frac{c_{2}^{m}\Gamma(\frac{n-m+1}{2})}{2^{m-n}[(m/2)!]\sqrt{\pi}}\,\frac{t}{\big{(}t^{2}+c_{3}(d(x,y))^{2}\big{)}^{(n-m+1)/2}}.$
Therefore, (5.2) holds in the strong sense. $\quad\quad\square$
Remark 5.2. We can also give another series representation for the heat kernel
$F(t,x,y)$ of the square root $-\sqrt{-\Delta_{h}}$ of the Laplacian on
$\partial\Omega$. In fact, from Proposition 13.3 of p. 62 of [107], one has
that
$\displaystyle G(t,x,y)\sim\sum_{j\geq
0}t^{(j-n)/2}p_{j}\big{(}x,t^{-1/2}(x-y)\big{)}e^{-\sum_{l,k=1}^{n}h_{lk}(x)(x_{l}-y_{l})(x_{k}-y_{k})/4t},$
where $p_{j}(x,z)$ is a polynomial in $z$ which is even or odd in $z$
according to the parity of $j$ (here $p_{\beta}(x,z)$, a polynomial of degree
$|\beta|$ in $z$, is determined by the following formula:
$\left[\mbox{det}\big{(}4\pi\big{(}h^{jk}(x)\big{)}\big{)}\right]^{-1/2}D_{z}^{\beta}e^{-\sum_{j,k=1}^{n}h_{jk}(x)\,z_{j}z_{k}/4}=p_{\beta}(x,z)e^{-\sum_{j,k=1}^{n}h_{jk}(x)\,z_{j}z_{k}/4}).$
Therefore
$\displaystyle
F(t,x,y)\sim\sum_{j=0}^{\infty}\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left[\mu^{(j-n)/2}p_{j}\big{(}x,\mu^{-1/2}(x-y)\big{)}e^{-\sum_{l,k=1}^{n}h_{lk}(x)\,(x_{l}-y_{l})(x_{k}-y_{k})/4\mu}\right]d\mu.$
Theorem 5.3. Let $\Omega$ be a bounded domain with smooth boundary
$\partial\Omega$ in $(n+1)$-dimensional Riemannian manifold $(\mathcal{M},g)$.
Then the fundamental solution ${\mathcal{K}}(t,x,y)$ of the heat equation for
the Dirichlet-to-Neumann operator on $\partial\Omega$, defined by (2.3), has
the following series representation:
$\displaystyle{\mathcal{K}}(t,x,y)$ $\displaystyle=$
$\displaystyle{\mathcal{K}}_{V}(t,x,y)+\sum_{0\leq
l<M}{\mathcal{K}}_{V_{-1-l}}(t,x,y)+{\mathcal{K}}_{V^{\prime}_{M}}(t,x,y)$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left[\frac{}{}G_{0}(\mu,x,y)\right.$
$\displaystyle\left.+\sum_{m=1}^{\infty}G_{0}\\#\underset{m}{\underbrace{\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}\\#\cdots\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}}}(\mu,x,y)\right]d\mu$
$\displaystyle+\sum_{0\leq
l<M}{\mathcal{K}}_{V_{-1-l}}(t,x,y)+{\mathcal{K}}_{V^{\prime}_{M}}(t,x,y),$
where
(5.11) (5.13)
$\displaystyle\left.\begin{array}[]{ll}{\mathcal{K}}_{V}(t,x,y)=\left(\frac{1}{2\pi}\right)^{n}\int_{{\mathbb{R}}^{n}}e^{i(x-y)\cdot\xi}e^{tp(x,\xi)}d\xi=F(t,x,y),\\\
{\mathcal{K}}_{V_{-2}}(t,x,y)=\left(\frac{1}{2\pi}\right)^{n}\int_{{\mathbb{R}}^{n}}e^{i(x-y)\cdot\xi}t(p_{0}^{B}(x,\xi))e^{tp_{1}(x,\xi)}d\xi,\\\
{\mathcal{K}}_{V_{-3}}(t,x,y)=\left(\frac{1}{2\pi}\right)^{n}\int_{{\mathbb{R}}^{n}}e^{i(x-y)\cdot\xi}\left[t(p_{-1}^{B}(x,\xi))\right.\\\
\;\,\quad\qquad\quad\qquad\left.+\frac{t^{2}}{2}\left((p_{0}^{B}(x,\xi))^{2}+2p_{0}(x,\xi)p_{0}^{B}(x,\xi)\right)\right]e^{tp_{1}(x,\xi)}d\xi,\\\
{\mathcal{K}}_{V_{-4}}(t,x,y)=\left(\frac{1}{2\pi}\right)^{n}\int_{{\mathbb{R}}^{n}}e^{i(x-y)\cdot\xi}\left[t(p_{-2}^{B}(x,\xi))\right.\\\
\left.\qquad\qquad\qquad\;\,+t^{2}(p_{0}p_{-1}^{B}+p_{-1}p_{0}^{B}+p_{0}^{B}p_{-1}^{B})+\frac{t^{3}}{6}\big{(}(p_{0}^{B})^{3}+3p_{0}^{2}p_{0}^{B}+3p_{0}(p_{0}^{B})^{2}\big{)}\right]e^{tp_{1}(x,\xi)}d\xi,\\\
{\mathcal{K}}_{V^{\prime}_{M}}(t,x,y)=\left(\frac{1}{2\pi}\right)^{n}\int_{{\mathbb{R}}^{n}}e^{i(x-y)\cdot\xi}v^{\prime}_{M}(t,x,\xi)d\xi,\end{array}\right.$
and
(5.17) $\displaystyle\big{|}{\mathcal{K}}_{V^{\prime}_{M}}(t,x,y)\big{|}\leq
c_{0}e^{-c_{1}t}\left\\{\begin{array}[]{ll}t\big{[}t+d(x,y)\big{]}^{M-1-n}&\mbox{if}\;\;1-M>-n,\\\
t(|\log(d(x,y)+t)|+1)&\mbox{if}\;\;1-M=-n,\\\
t&\mbox{if}\;\;1-M<-n,\end{array}\right.$
for some $c_{0}>0$ and any $c_{1}<0$, where $M=2,3,4$.
Proof. Let $p(x,\xi)$ and $p^{B}(x,\xi)$ are the symbols of
$-\sqrt{-\Delta_{h}}$ and $B$, respectively. Then
$\displaystyle p(x,\xi)\sim\sum_{k\geq
0}p_{1-k}(x,\xi),\quad\;\;p^{B}(x,\xi)\sim\sum_{k\geq 0}p^{B}_{-k}(x,\xi).$
Let us write
$\displaystyle r(x,\xi)$ $\displaystyle=$ $\displaystyle
p(x,\xi)-p_{1}(x,\xi)-p_{0}(x,\xi)-p_{-1}(x,\xi),$ $\displaystyle
r^{B}(x,\xi)$ $\displaystyle=$ $\displaystyle
p^{B}(x,\xi)-p^{B}_{0}(x,\xi)-p^{B}_{-1}(x,\xi)-p^{B}_{-2}(x,\xi).$
It is clear that $r(x,\xi)\in S^{-2}_{1,0}$ and $r^{B}(x,\xi)\in
S^{-3}_{1,0}$. Let $\mathcal{C}$ is a suitable curve in the complex plane
going in the positive direction around the spectrum of $-{\mathcal{N}}_{g}$;
it can be taken as the boundary $W_{r_{0},\epsilon}$ for a small $\epsilon$
(see, Section 2) (here we can use ${\mathcal{C}}_{\theta,R}$ consisting of the
two rays $re^{i\theta}$ and $re^{-i\theta}$, $\theta=\theta_{0}+\epsilon$
($0<\theta<\frac{\pi}{2}$) by a circular piece in the right-plane with radius
$R\geq 2t|p_{1}(x,\xi)|$). We may choose $R$ large enough. Then for
$\lambda\in{\mathcal{C}}_{\theta,R}$, we can write the symbol
$q(x,\xi,\lambda)$ of the resolvent operator
$(-{\mathcal{N}}_{g}-\lambda)^{-1}$ in local coordinates as
$\displaystyle\left[-p_{1}(x,\xi)-p_{0}(x,\xi)-p_{-1}(x,\xi)-r(x,\xi)-p^{B}_{0}(x,\xi)-p^{B}_{-1}(x,\xi)-p^{B}_{-2}(x,\xi)-r^{B}(x,\xi)-\lambda\right]^{-1}$
$\displaystyle=\left(-p_{1}-\lambda\right)^{-1}\left[1+\frac{-p_{0}-p_{-1}-r-p^{B}_{0}-p^{B}_{-1}-p_{-2}^{B}-r^{B}}{-p_{1}-\lambda}\right]^{-1}$
$\displaystyle=\left(-p_{1}-\lambda\right)^{-1}\left[1-\frac{-p_{0}-p_{-1}-r-p^{B}_{0}-p^{B}_{-1}-p_{-2}^{B}-r^{B}}{-p_{1}-\lambda}\right.$
$\displaystyle\left.\;+\left(\frac{-p_{0}-p_{-1}-r-p^{B}_{0}-p^{B}_{-1}-p_{-2}^{B}-r^{B}}{-p_{1}-\lambda}\right)^{2}\right.$
$\displaystyle\left.\;-\left(\frac{-p_{0}-p_{-1}-r-p^{B}_{0}-p^{B}_{-1}-p_{-2}^{B}-r^{B}}{-p_{1}-\lambda}\right)^{3}\right.$
$\displaystyle\left.\;+\left(\frac{p_{0}-p_{-1}-r-p_{0}^{B}-p_{-1}^{B}-p_{-2}^{B}-r^{B}}{-p_{1}-\lambda}\right)^{4}-\cdots\right]$
$\displaystyle=\left[\frac{1}{-p_{1}-\lambda}-\frac{-p_{0}-p_{-1}-r}{(-p_{1}-\lambda)^{2}}+\frac{(-p_{0}-p_{-1}-r)^{2}}{(-p_{1}-\lambda)^{3}}-\frac{(-p_{0}-p_{-1}-r)^{3}}{(-p_{1}-\lambda)^{4}}+\cdots\right]$
$\displaystyle\;+\left[-\frac{-p^{B}_{0}}{(-p_{1}-\lambda)^{2}}+\left(-\frac{-p^{B}_{-1}}{(-p_{1}-\lambda)^{2}}+\frac{(p_{0}^{B})^{2}+2p_{0}p_{0}^{B}}{(-p_{1}-\lambda)^{3}}\right)\right.$
$\displaystyle\left.\;+\left(-\frac{-p_{-2}^{B}}{(-p_{1}-\lambda)^{2}}+\frac{2(p_{0}p_{-1}^{B}+p_{-1}p_{0}^{B}+p_{0}^{B}p_{-1}^{B})}{(-p_{1}-\lambda)^{3}}\right.\right.$
$\displaystyle\left.\left.\;+\frac{3p_{0}^{2}p_{0}^{B}+3p_{0}(p_{0}^{B})^{2}+(p_{0}^{B})^{3}}{(-p_{1}-\lambda)^{4}}\right)+\cdots\right].$
In the above asymptotic expansion of symbol, the “$\cdots$” denotes all the
terms which belong to $S_{1,0}^{-5}$. Noting that
$\displaystyle\frac{1}{-p-\lambda}=\frac{1}{-p_{1}-\lambda}-\frac{-p_{0}-p_{-1}-r}{(-p_{1}-\lambda)^{2}}+\frac{(-p_{0}-p_{-1}-r)^{2}}{(-p_{1}-\lambda)^{3}}-\frac{(-p_{0}-p_{-1}-r)^{3}}{(-p_{1}-\lambda)^{4}}+\cdots,$
we get
(5.18) $\displaystyle
q(x,\xi,\lambda)=\left(-p-\lambda\right)^{-1}+\left[\frac{p_{0}^{B}}{(-p_{1}-\lambda)^{2}}\right]+\left[\frac{p_{-1}^{B}}{(-p_{1}-\lambda)^{2}}+\frac{(p_{0}^{B})^{2}+2p_{0}p_{0}^{B}}{(-p_{1}-\lambda)^{3}}\right]$
$\displaystyle\quad\;\;+\left(\frac{p_{-2}^{B}}{(-p_{1}-\lambda)^{2}}+\frac{2(p_{0}p_{-1}^{B}+p_{-1}p_{0}^{B}+p_{0}^{B}p_{-1}^{B})}{(-p_{1}-\lambda)^{3}}\right.$
$\displaystyle\quad\;\;\left.\;+\frac{3p_{0}^{2}p_{0}^{B}+3p_{0}(p_{0}^{B})^{2}+(p_{0}^{B})^{3}}{(-p_{1}-\lambda)^{4}}\right)+r^{\prime}$
$\displaystyle\quad=(-p(x,\xi)-\lambda)^{-1}+q_{-2}(x,\xi,\lambda)+q_{-3}(x,\xi,\lambda)+q_{-4}(x,\xi,\lambda)+r^{\prime}(x,\xi,\lambda),$
with
$\displaystyle
q_{-2}(x,\xi,\lambda)=\frac{p_{0}^{B}}{(-p_{1}-\lambda)^{2}},\quad\,q_{-3}(x,\xi,\lambda)=\frac{p_{-1}^{B}}{(-p_{1}-\lambda)^{2}}+\frac{(p_{0}^{B})^{2}+2p_{0}p_{0}^{B}}{(-p_{1}-\lambda)^{3}},$
$\displaystyle
q_{-4}(x,\xi,\lambda)=\frac{p_{-2}^{B}}{(-p_{1}-\lambda)^{2}}+\frac{2(p_{0}p_{-1}^{B}+p_{-1}p_{0}^{B}+p_{0}^{B}p_{-1}^{B})}{(-p_{1}-\lambda)^{3}}$
$\displaystyle\quad\qquad\qquad\quad+\frac{3p_{0}^{2}p_{0}^{B}+3p_{0}(p_{0}^{B})^{2}+(p_{0}^{B})^{3}}{(-p_{1}-\lambda)^{4}},\quad\quad\;r^{\prime}\in
S^{-4}_{1,0}.$
The semigroup $e^{-t(-{\mathcal{N}}_{g})}$ can be defined from
$-{\mathcal{N}}_{g}$ by the Cauchy integral formula
(5.19) $\displaystyle
e^{t{\mathcal{N}}_{g}}=e^{-t(-{\mathcal{N}}_{g})}=\frac{i}{2\pi}\int_{{\mathcal{C}}_{\theta,R}}e^{-t\lambda}(-{\mathcal{N}}_{g}-\lambda)^{-1}d\lambda,$
where ${\mathcal{C}}_{\theta,R}$ is the curve in complex plane defined as
before. By virtue of (5.18) and (5.19), we get that in the local coordinate
patch the symbol of $e^{t{\mathcal{N}}_{g}}$ is
$\displaystyle v(t,x,\xi)\sim e^{tp(x,\xi)}+\sum_{1\leq
l<M}v_{-1-l}(t,x,\xi)+v^{\prime}_{M}(t,x,\xi),$
where
$\displaystyle
v_{-1-l}(t,x,\xi)=\frac{i}{2\pi}\int_{{\mathcal{C}}_{\theta,R}}e^{-t\lambda}q_{-1-l}(x,\xi,\lambda)d\lambda,\quad\,(l=1,2,\cdots,M-1),$
$\displaystyle
v^{\prime}_{M}(t,x,\xi)=\frac{i}{2\pi}\int_{{\mathcal{C}}_{\theta,R}}e^{-t\lambda}\,r^{\prime}(x,\xi,\lambda)d\lambda,\quad\,M=2,3,4,\cdots.$
In order to give an explicit expansion of the heat kernel, we rewrite
$q_{-2}$, $q_{-3}$ and $q_{-4}$ as
$\displaystyle
q_{-2}(x,\xi,\lambda)=\frac{b_{1,1}(x,\xi)}{(-p_{1}-\lambda)^{2}},\quad\,q_{-1-l}(x,\xi,\lambda)=\sum_{k=1}^{2l}\frac{b_{l,k}(x,\xi)}{(-p_{1}-\lambda)^{k+1}},\,(l=2,3)$
where
$\displaystyle
b_{1.1}(x,\xi)=p_{0}^{B}(x,\xi),\quad\,b_{1,2}(x,\xi)=0,\quad\;b_{2,1}(x,\xi)=p_{-1}^{B}(x,\xi),$
$\displaystyle
b_{2,2}(x,\xi)=\left(p_{0}^{B}(x,\xi)\right)^{2}+2p_{0}(x,\xi)\,p_{0}^{B}(x,\xi),\quad\;b_{2,3}(x,\xi)=b_{2,4}(x,\xi)=0,$
$\displaystyle
b_{3,1}(x,\xi)=\frac{p_{-2}^{B}}{(-p_{1}-\lambda)^{2}},\quad\;b_{3,2}(x,\xi)=\frac{2(p_{0}p_{-1}^{B}+p_{-1}p_{0}^{B}+p_{0}^{B}p_{-1}^{B})}{(-p_{1}-\lambda)^{3}},$
$\displaystyle
b_{3,3}(x,\xi)=\frac{3p_{0}^{2}p_{0}^{B}+3p_{0}(p_{0}^{B})^{2}+(p_{0}^{B})^{3}}{(-p_{1}-\lambda)^{4}},\;\;b_{3,4}(x,\xi)=b_{3,5}(x,\xi)=b_{3,6}(x,\xi)=0.$
Then, for $t>0$, we find by setting $\varrho=t\lambda$ that
$\displaystyle\frac{i}{2\pi}\int_{{\mathcal{C}}_{\theta,R}}e^{-t\lambda}\frac{b_{l,k}(x,\xi)}{(-p_{1}(x,\xi)-\lambda)^{k+1}}d\lambda$
$\displaystyle=$
$\displaystyle\frac{i}{2\pi}\int_{{\mathcal{C}}_{\theta,R}}e^{-\varrho}\frac{t^{k}b_{l,k}(x,\xi)}{(-tp_{1}(x,\xi)-\varrho)^{k+1}}d\varrho$
$\displaystyle=$
$\displaystyle\frac{1}{k!}t^{k}b_{l,k}(x,\xi)\,e^{tp_{1}(x,\xi)},$
Thus,
$\displaystyle v(t,x,\xi)$ $\displaystyle=$ $\displaystyle
e^{tp(x,\xi)},\quad\;\;v_{-1-l}(t,x,\xi)=\sum_{k=1}^{2l}\frac{1}{k!}t^{k}b_{l,k}(x,\xi)\,e^{tp_{1}(x,\xi)}$
$\displaystyle\qquad\;\mbox{for}\;\;l=1,2,\cdots,M-1.$
We define $V(t)$, $V_{-1-l}(t)$ and $V^{\prime}_{M}(t)$ in local coordinates
to be the pseudodifferential operators with symbols $v(t,x,\xi)$,
$v_{-1-l}(t,x,\xi)$ and $v^{\prime}_{M}(t,x,\xi)$, respectively. It follows
that the heat kernel ${\mathcal{K}}(t,x,\xi)$ of $e^{t{\mathcal{N}}_{g}}$ is
in local coordinates expanded according to the symbol expansion:
$\displaystyle{\mathcal{K}}(t,x,y)=F(t,x,y)+\sum_{1\leq
l<M}{\mathcal{K}}_{V_{-1-l}}(t,x,y)+{\mathcal{K}}_{V^{\prime}_{M}}(t,x,y),$
where ${\mathcal{K}}_{V}(t,x,y)$, ${\mathcal{K}}_{V_{-1-l}}(t,x,y)$ and
${\mathcal{K}}_{V^{\prime}_{M}}(t,x,y)$ are as in (5.11) because of (5), and
$M=2,3,4$. Finally, in view of $\gamma(-{\mathcal{N}}_{g})=0$, we find by
(2.24) of Theorem 2.5 of [45] that
$\displaystyle\big{|}{\mathcal{K}}_{V^{\prime}_{M}}(t,x,y)\big{|}\leq
c_{0}e^{-c_{1}t}\left\\{\begin{array}[]{ll}t\big{[}t+d(x,y)\big{]}^{M-1-n}&\mbox{if}\;\;1-M>-n,\\\
t(|\log(d(x,y)+t)|+1)&\mbox{if}\;\;1-M=-n,\\\
t&\mbox{if}\;\;1-M<-n,\end{array}\right.$
for some $c_{0}>0$ and any $c_{1}<0$. $\qquad\quad\;\;\square$
## 6\. Estimates of the pole
Let $u_{k}$ be the eigenfunction corresponding to the $k$-th Steklov
eigenvalue $\lambda_{k}$ (i.e., ${\mathcal{N}}_{g}u_{k}=-\lambda_{k}u_{k}$),
and denote by $dS(x)=\sqrt{|h|}dx$ the area element of $\partial\Omega$. Since
$Z=\int_{\partial\Omega}{\mathcal{K}}(t,x,x)dS(x)$ converges,
$e^{t{\mathcal{N}}_{g}}:\phi\to\int_{\partial\Omega}{\mathcal{K}}\phi$ is a
compact mapping of the (real) Hilbert space
$H=L^{2}(\partial\Omega,\sqrt{|h|}\,dx)$. This implies
(6.1) $\displaystyle{\mathcal{K}}(t,x,y)=\sum_{k\geq
1}e^{-\lambda_{k}t}u_{k}(x)u_{k}(y)$
with uniform convergence on compact figures of
$(0,\infty)\times\partial\Omega\times\partial\Omega$, and the spur $Z$ is
easily evaluated as (see, for example, [81] or Chapter 4 of [51])
(6.2) $\displaystyle Z=\int_{\partial\Omega}\sum_{k\geq
1}e^{-\lambda_{k}t}u_{k}^{2}(x)dS(x)=\sum_{k\geq 1}e^{-\lambda_{k}t}.$
Formula (5) can now be used to estimate the pole ${\mathcal{K}}(t,x,x)$ for
$t\downarrow 0$, up to terms of magnitude $t^{3-n}$, then integrate the result
over $\partial\Omega$ to get an estimate of
$Z=\mbox{Tr}\;e^{t{\mathcal{N}}_{g}}=\int_{\partial\Omega}{\mathcal{K}}(t,x,x)dS(x)$.
In order to state our main result regarding the relationship between the
spectrum of the Dircichlet-to-Neumann (or Steklov Poincar ) operator and
various informations on $\partial\Omega$, we introduce the following
notations: We denote by $\,{\tilde{R}}_{jkjk}(x)$ (respectively
$R_{jkjk}(x)$), $\,{\tilde{R}}_{jj}(x)=\sum_{k=1}^{n+1}{\tilde{R}}_{jkjk}(x)$
(respectively ${R}_{jj}(x)=\sum_{k=1}^{n}R_{jkjk}(x)$),
$\,{\tilde{R}}_{\Omega}$ (respectively $R_{\partial\Omega}(x)$) the the
curvature tensor, the Ricci curvature tensor, the scalar curvature with
respect to $\Omega$ (respectively $\partial\Omega$) at $x\in\partial\Omega$.
Denote by $\,\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1),(n+1)}(x)$ the covariant
derivative of the curvature tensor with respect to $(\bar{\Omega},g)$. In
addition, we denote by $\kappa_{1}(x),\cdots,\kappa_{n}(x)$ the principal
curvatures of $\partial\Omega$ at $x\in\partial\Omega$.
Theorem 6.1. Suppose that $(\mathcal{M},g)$ is an $(n+1)$-dimensional, smooth
Riemannian manifold, and assume that $\Omega\subset\mathcal{M}$ is a bounded
domain with smooth boundary $\partial\Omega$. Let $\mathcal{K}(t,x,y)$ be the
heat kernel associated to the Dirichlet-to-Neumann operator.
(a) If $n\geq 1$, then
(6.3)
$\displaystyle\int_{\partial\Omega}\mathcal{K}(t,x,x)dx=t^{-n}\int_{\partial\Omega}a_{0}(n,x)\,dS(x)+t^{1-n}\int_{\partial\Omega}a_{1}(n,x)\,dS(x)$
(6.6)
$\displaystyle\quad\quad\quad\quad+\left\\{\begin{array}[]{ll}O(t^{2-n})\quad\;\,\mbox{when}\;\;n>1,\\\
O(t\log
t)\quad\,\mbox{when}\;\;n=1,\end{array}\right.\quad\;\;\mbox{as}\;\;t\to
0^{+};$
(b) If $n\geq 2$, then
(6.7)
$\displaystyle\int_{\partial\Omega}\mathcal{K}(t,x,x)dx=t^{-n}\int_{\partial\Omega}a_{0}(n,x)\,dS(x)+t^{1-n}\int_{\partial\Omega}a_{1}(n,x)\,dS(x)$
(6.10)
$\displaystyle+t^{2-n}\int_{\partial\Omega}a_{2}(n,x)\,dS(x)+\left\\{\begin{array}[]{ll}O(t^{3-n})\quad\,\mbox{when}\;\;n>2,\\\
O(t\log
t)\;\;\;\mbox{when}\;\;n=2,\end{array}\right.\quad\;\;\mbox{as}\;\;t\to
0^{+};$
(c) If $n\geq 3$, then
(6.11)
$\displaystyle\int_{\partial\Omega}\mathcal{K}(t,x,x)dx=t^{-n}\int_{\partial\Omega}a_{0}(n,x)\,dS(x)+t^{1-n}\int_{\partial\Omega}a_{1}(n,x)\,dS(x)$
$\displaystyle\qquad\qquad+t^{2-n}\int_{\partial\Omega}a_{2}(n,x)\,dS(x)+t^{3-n}\int_{\partial\Omega}a_{3}(n,x)\,dS(x)$
(6.14)
$\displaystyle\qquad\qquad+\left\\{\begin{array}[]{ll}O(t^{4-n})\quad\,\,\mbox{when}\;\;n>3,\\\
O(t\log t)\quad\mbox{when}\;\;n=3,\end{array}\right.\quad\;\;\mbox{as}\;\;t\to
0^{+}.$
Here
(6.15) $\displaystyle
a_{0}(n,x)=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}},$ (6.16)
$\displaystyle\quad
a_{1}(n,x)=\left(\frac{1}{2\pi}\right)^{n}\frac{(n-1)\Gamma(n)\,\mbox{vol}({\mathbb{S}}^{n-1})}{2n}\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)},$
(6.17) $\displaystyle
a_{2}(n,x)=\frac{\Gamma(n-1)\mbox{vol}({\mathbb{S}}^{n-1})}{8(2\pi)^{n}}\left[\frac{3-n}{3n}R_{\partial\Omega}+\frac{n-1}{n}{\tilde{R}}_{\Omega}\right.$
$\displaystyle\left.+\frac{n^{3}-n^{2}-4n+6}{n(n+2)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{2}+\frac{n^{2}-n-2}{n(n+2)}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\right]$
and
(6.18) $\displaystyle
a_{3}(n,x)=\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\frac{\Gamma(n-2)\,\mbox{vol}(S^{n-1})}{8n}\bigg{[}\frac{n^{3}-2n^{2}-7n+7}{2(n+2)}{\tilde{R}}_{\Omega}(x)\,\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}$
$\displaystyle\quad\,+\frac{-3n^{4}-4n^{3}+59n^{2}+75n-180}{6(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}R_{\partial\Omega}$
$\displaystyle\quad\,+\frac{n^{5}-20n^{3}+2n^{2}+61n-74}{6(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{3}$
$\displaystyle\quad\,+\frac{n^{4}+8n^{3}+15n^{2}+3n-32}{2(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\big{)}$
$\displaystyle\quad\,+\frac{-6n^{3}-34n^{2}+40}{3(n+2)(n+4)}\sum_{j=1}^{n}\kappa_{j}^{3}(x)+\frac{4n^{2}-6}{n+2}\sum_{j=1}^{n}\kappa_{j}(x){\tilde{R}}_{jj}(x)$
$\displaystyle\quad\,-\frac{12n^{3}+50n^{2}-6n-104}{3(n+2)(n+4)}\sum_{j=1}^{n}\kappa_{j}(x)R_{jj}(x)+(n-1)\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1),(n+1)}(x)$
$\displaystyle\quad\,-\frac{n-2}{2}{\tilde{R}}_{\Omega}(x)+\frac{n-2}{2}R_{\partial\Omega}-\frac{n-2}{2}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\bigg{]}.$
Proof. Put $x_{0}=0$ for simplicity, and (as in proof of Lemma 4.2) choose
coordinates $x=(x_{1},\cdots,x_{n})$ on an open set in $\partial\Omega$
(centered at $x_{0}=0$) and then coordinates $(x,x_{n+1})$ on a neighborhood
in $\bar{\Omega}$ such that $x_{n+1}=0$ on $\partial\Omega$ and $|\nabla
x_{n+1}|=1$ near $\partial\Omega$ while $x_{n+1}>0$ on $\Omega$ and such that
$x$ is constant on each geodesic segment in $\bar{\Omega}$ normal to
$\partial\Omega$. Clearly, the metric tensor on $\bar{\Omega}$ has the form
$\displaystyle\big{(}g_{jk}(x,x_{n+1})\big{)}_{(n+1)\times(n+1)}=\begin{pmatrix}(h_{jk}(x,x_{n+1}))_{n\times
n}&0\\\ 0&1\end{pmatrix}.$
Recall that $\left(-\frac{1}{2}\,\frac{\partial h_{jk}}{\partial
x_{n+1}}(x_{0})\right)_{n\times n}$ is the matrix of the Weingarten map under
a basis of $T_{0}(\partial\Omega)$ at $x_{0}=0$, its eigenvalues are just the
principal curvatures $\kappa_{1},\cdots,\kappa_{n}$ of $\partial\Omega$ at
$x_{0}=0$ in the direction $\nu$. Therefore we can choose the principal
curvature vectors $e_{1},\cdots,e_{n}$ as an orthonormal basis of
$T_{0}(\partial\Omega)$ at $x_{0}=0$ such that the symmetric matrix
$\left(-\frac{1}{2}\,\frac{\partial h_{jk}}{\partial
x_{n+1}}(x_{0})\right)_{n\times n}$ becomes the diagonal matrix
$\displaystyle\begin{pmatrix}\kappa_{1}&0&\cdots&0\\\ 0&\kappa_{2}&\cdots&0\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&\kappa_{n}\end{pmatrix}.$
Furthermore, we take a geodesic normal coordinate system for
$(\partial\Omega,h)$ centered at $x_{0}=0$, with respect to
$e_{1},\cdots,e_{n}$. As Riemann showed, one has (see p. 555 of [107], or
[99])
(6.19) $\displaystyle h_{jk}(x_{0})=\delta_{jk},\;\;\frac{\partial
h_{jk}}{\partial x_{l}}(x_{0})=0,\;\;\frac{\partial^{2}h_{jk}}{\partial
x_{l}\partial x_{m}}(x_{0})=-\frac{1}{3}R_{jlkm}-\frac{1}{3}R_{jmkl}$
$\displaystyle\qquad\qquad\mbox{for all}\;\;1\leq j,k,l,m\leq n,$
so that
(6.20) $\displaystyle\frac{\partial^{2}|h|}{\partial x_{k}\partial
x_{l}}(x_{0})=\sum_{m=1}^{n}\frac{\partial^{2}h_{mm}}{\partial x_{k}\partial
x_{l}}(x_{0})=-\sum_{m=1}^{n}\frac{R_{mkml}+R_{mlmk}}{3}=-\sum_{m=1}^{n}\frac{2R_{mkml}}{3}.$
Also, from $hh^{-1}=I$ we get
(6.21) $\displaystyle\frac{\partial h^{jk}}{\partial
x_{l}}(x_{0})=0,\quad\frac{\partial^{2}h^{jk}}{\partial x_{l}\partial
x_{m}}(x_{0})=-\frac{\partial^{2}h_{jk}}{\partial x_{l}\partial
x_{m}}(x_{0})\quad\;\mbox{for all}\,\,1\leq j,k,l,m\leq n.$
In view of (5.9) and (5.17), we find by taking $x=y$ that
(6.25) $\displaystyle|{\mathcal{K}}_{V^{\prime}_{M}}(t,x,x)|\leq
c_{0}\left\\{\begin{array}[]{ll}t^{M-n},&\mbox{if}\;\;1-M>-n,\\\ t(1+\log
t),&\mbox{if}\;\;1-M=-n,\\\ t,&\mbox{if}\;\;1-M<-n\end{array}\right.$
and, for $m\geq 1$,
(6.26)
$\displaystyle\bigg{|}\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\bigg{[}G_{0}\\#\underset{m}{\underbrace{\left((\Delta_{g}-\Delta_{g}^{0})G_{0}\right)\\#\cdots\\#\left((\Delta_{g}-\Delta_{g}^{0})G_{0}\right)}}(\mu,x,x)\bigg{]}d\mu\bigg{|}$
$\displaystyle\quad\qquad\quad\;\;\leq\frac{c_{2}^{m}\Gamma(\frac{n-m+1}{2})\,t^{m-n}}{2^{m-n}[(m/2)!]\sqrt{\pi}},$
where $M=2,3,4$ will be taken later (according to the term number of
asymptotic expansion of the heat kernel). Consequently, by (5), the trace of
the operator $e^{t{\mathcal{N}}_{g}}$ has the following asymptotic expansion
(6.27)
$\displaystyle\mbox{Tr}\,e^{t{\mathcal{N}}_{g}}=\int_{\partial\Omega}\left[\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left(\frac{}{{}}G_{0}(\mu,x,x)+\big{(}G_{0}\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}\big{)}(\mu,x,x)\right)d\mu\right.$
$\displaystyle\qquad\qquad\quad\;\left.+\sum_{1\leq
l<M}{\mathcal{K}}_{V_{-1-l}}(t,x,x)\right]dS(x)+{\mathcal{K}}_{V^{\prime}_{M}}(t,x,x)\quad\mbox{as}\;\;t\to
0^{+},$
i.e., up to terms of magnitude $\leq$ constant$\times t^{M-n}$, one is left
with
(6.28)
$\displaystyle\int_{\partial\Omega}{\mathcal{K}}(t,x,x)dS(x)=\int_{\partial\Omega}\left[\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left(\frac{}{}G_{0}(\mu,x,x)\right.\right.$
$\displaystyle\left.\left.\quad\quad+\big{(}G_{0}\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}\big{(}\mu,x,x)\frac{}{}\right)d\mu+\sum_{1\leq
l<M}{\mathcal{K}}_{V_{-1-l}}(t,x,x)\right]dS(x).$
According to the definition of $\Delta_{h}$, we get that at the fixed point
$x_{0}=0$,
$\displaystyle
b_{k}(x_{0})=\sum_{j=1}^{n}\frac{1}{\sqrt{|h|}}\,\frac{\partial(\sqrt{|h|}\,h^{jk})}{\partial
x_{j}}(x_{0})=0,$
so that
(6.29)
$\displaystyle\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\,G_{0}(\mu,x,x)d\mu=\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left(\frac{1}{4\pi\mu}\right)^{\frac{n}{2}}d\mu=\frac{\Gamma\big{(}\frac{n+1}{2}\big{)}}{\pi^{\frac{n+1}{2}}}\,t^{-n}.$
It follows from [80] that up to the desired precision, the integrand
$G_{0}(t-s,0,x)\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}(s,x,0)\big{)}\sqrt{\mbox{det}\;h}$
can be replaced by the product of a factor $1+$ a line function $f$ of
$x+o(|x|^{2})$ and expression
(6.30)
$\displaystyle\frac{e^{-|x|^{2}/4(t-s)}}{[4\pi(t-s)]^{n/2}}\left[\frac{1}{2}\frac{\partial^{2}h^{ij}}{\partial
x_{k}\partial x_{l}}(0)\,x_{k}x_{l}\frac{\partial^{2}}{\partial x_{i}\partial
x_{j}}\right.$
$\displaystyle\left.\quad\quad\quad\;\;+\left(\frac{\partial}{\partial
x_{k}}\,\frac{1}{\sqrt{|h|}}\frac{\partial}{\partial
x_{i}}\sqrt{|h|}\,h^{ij}\right)(0)x_{k}\frac{\partial}{\partial
x_{j}}\right]\frac{e^{-|x|^{2}/4s}}{(4\pi s)^{n/2}}$
$\displaystyle\quad\quad\;\;=(4\pi t)^{-n/2}\frac{e^{-|x|^{2}/4r}}{(4\pi
r)^{n/2}}\left[\frac{1}{2}\,\frac{\partial^{2}h^{ij}}{\partial x_{k}\partial
x_{l}}(0)x_{k}x_{l}\left(\frac{x_{i}x_{j}}{4s^{2}}-\frac{\delta_{ij}}{2s}\right)\right.$
$\displaystyle\left.\quad\quad\quad\;\;-\left(\frac{\partial}{\partial
x_{k}}\,\frac{1}{\sqrt{|h|}}\,\frac{\partial}{\partial
x_{i}}\sqrt{|h|}\,h^{ij}\right)(0)\frac{x_{k}x_{j}}{2s}\right],$
where $r=s(t-s)/t$. Now the factor alluded to above (6.30) can be replaced by
$1$, since $f\times$ (6.30) integrates to $0$ while the last $2$ terms
contribute $\leq ct^{3-n}$. Consequently, up to the desired precision, we find
by p. 52–53 of [80] and (6.19)–(6.21) that at $x_{0}=0$,
$\displaystyle
G_{0}\\#\big{(}(\Delta_{h}-\Delta^{0}_{h})G_{0}\big{)}=\int_{0}^{t}ds\int_{{\mathbb{R}}^{n}}(\ref{j1})\,dx$
$\displaystyle\quad\;\;=\frac{t}{(4\pi
t)^{n/2}}\left[-\frac{1}{6}\sum_{i,j=1}^{n}\frac{\partial^{2}h^{ij}}{\partial
x_{i}\partial
x_{j}}-\frac{1}{2}\sum_{i=1}^{n}\frac{\partial^{2}\sqrt{|h(x)|}}{\partial
x_{i}^{2}}-\frac{1}{12}\sum_{i,j=1}^{n}\frac{\partial^{2}h^{ii}}{\partial
x_{j}^{2}}\right]$ $\displaystyle\quad\;\;=\frac{t}{(4\pi
t)^{n/2}}\left[\frac{1}{6}\sum_{i,j=1}^{n}\frac{\partial^{2}h_{ij}}{\partial
x_{i}\partial
x_{j}}-\frac{1}{4}\sum_{i=1}^{n}\frac{\partial^{2}|h(x)|}{\partial
x_{i}^{2}}+\frac{1}{12}\sum_{i,j=1}^{n}\frac{\partial^{2}h_{ii}}{\partial
x_{j}^{2}}\right]$ $\displaystyle\quad\;\;=\frac{t}{(4\pi
t)^{n/2}}\sum_{i,j=1}^{n}\bigg{[}-\frac{1}{18}R_{ijji}+\frac{1}{6}R_{jiji}-\frac{1}{18}\,R_{ijij}\bigg{]}$
$\displaystyle\quad\;\;=\frac{t}{(4\pi
t)^{n/2}}\left(\frac{1}{6}\sum_{i,j=1}^{n}R_{jiji}\right)=\frac{t}{(4\pi
t)^{n/2}}\,\frac{R_{\partial\Omega}}{6},$
so that
$\displaystyle\quad\quad\quad\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\left[\big{(}G_{0}\\#\big{(}(\Delta_{g}-\Delta_{g}^{0})G_{0}\big{)}(\mu,x,x)\big{)}\right]d\mu$
$\displaystyle=$
$\displaystyle\frac{R_{\partial\Omega}(x)}{6}\int_{0}^{\infty}\frac{te^{-t^{2}/4\mu}}{\sqrt{4\pi\mu^{3}}}\,\frac{\mu}{(4\pi\mu)^{n/2}}d\mu$
$\displaystyle=$
$\displaystyle\frac{\Gamma(\frac{n-1}{2})\,R_{\partial\Omega}(x)}{24\pi^{\frac{n+1}{2}}}\,t^{2-n},$
where $R_{\partial\Omega}(x)$ is the scalar curvature of $(\partial\Omega,h)$.
It is easy to calculate that
$R_{\partial\Omega}={\tilde{R}}-2{\tilde{R}}_{\nu\nu}+\frac{1}{2}Q_{1}$, where
$Q_{1}=4\sum_{1\leq j<k\leq n}\kappa_{j}\kappa_{k}$ (see (4)). In the normal
coordinate system on $\partial\Omega$ with respect to the principal curvature
vectors $e_{1},\cdots,e_{n}$ centered at $x_{0}=0$, the Weingarten map
$A_{\nu}$ is self-adjoint, and (see p. 159 of [60])
(6.32)
$\displaystyle\mbox{Tr}\,A_{\nu}=\sum_{j=1}^{n}\kappa_{j},\quad\;\,\frac{\langle
A^{*}_{\nu}\xi,\xi\rangle}{\langle\xi,\xi\rangle}=\frac{\sum_{j=1}^{n}\kappa_{j}\xi_{j}^{2}}{\sum_{j=1}^{n}\xi_{j}^{2}}.$
By (4.7), (4) and (4.9), we can easily see that
$\displaystyle
p_{0}^{B}(x,\xi)=\frac{1}{2}\left(\sum_{j=1}^{n}\kappa_{j}(x)-\frac{\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}}{\sum_{j=1}^{n}\xi_{j}^{2}}\right),$
$\displaystyle p_{-1}^{B}(x,\xi)$ $\displaystyle=$
$\displaystyle\frac{1}{8\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}\left[-2Q_{1}+2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}-2\sum_{j=1}^{n}\kappa_{j}^{2}(x)+3\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}^{2}\right.$
$\displaystyle\left.+\frac{5\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}\big{)}^{2}}{\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{2}}-\frac{1}{\sum_{j=1}^{n}\xi_{j}^{2}}\,\sum_{j,k=1}^{n}\bigg{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6\kappa_{j}^{2}(x)\delta_{jk}\bigg{)}\xi_{j}\xi_{k}\right],$
$\displaystyle
p_{-2}^{B}(x,\xi)=\frac{\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}}{8(\sum_{j=1}^{n}\xi_{j}^{2})^{2}}\bigg{(}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\sum_{j=1}^{n}\kappa_{j}^{2}(x)-3\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{2}$
$\quad\quad\;\;-\frac{5\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}\big{)}^{2}}{\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{2}}+\frac{1}{\sum_{j=1}^{n}\xi_{j}^{2}}\,\sum_{j,k=1}^{n}\big{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6\kappa_{j}^{2}(x)\delta_{jk}\big{)}\xi_{j}\xi_{k}\bigg{)}\qquad\qquad\qquad\quad\\\
$
$\displaystyle-\frac{1}{4\sum_{j=1}^{n}\xi_{j}^{2}}\,\left\\{\frac{3}{2}(\sum_{j=1}^{n}\kappa_{j}(x))\big{(}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\sum_{j=1}^{n}\kappa_{j}^{2}(x)\big{)}-4(\sum_{j=1}^{n}\kappa_{j}(x))^{3}\right.$
$\displaystyle\left.+\frac{1}{4}\bigg{(}-48\sum_{1\leq j<k<l\leq
n}\kappa_{j}\kappa_{k}\kappa_{l}\,-6\sum_{j\neq
k}\kappa_{j}\big{(}-2{\tilde{R}}_{k(n+1)k(n+1)}\right.$
$\displaystyle\left.+2\kappa_{k}^{2}\big{)}-\sum_{j=1}^{n}\big{(}2{\tilde{R}}_{j(n+1)j(n+1),(n+1)}-4\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}\kappa_{j}\big{)}\bigg{)}\right.$
$\displaystyle\left.-\frac{1}{2}\bigg{[}\frac{\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}}{\sum_{j=1}^{n}\xi_{j}^{2}}+\sum_{j=1}^{n}\kappa_{j}(x)\bigg{]}\bigg{[}\frac{1}{2\sum_{j=1}^{n}\xi_{j}^{2}}\bigg{(}\sum_{j,k=1}^{n}\big{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6\kappa_{j}^{2}\delta_{jk}\big{)}\xi_{j}\xi_{k}\bigg{)}\right.\;\;$
$\displaystyle\left.-\frac{2\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}\big{)}^{2}}{\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{2}}+2\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}^{2}-\frac{1}{2}\bigg{(}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\sum_{j=1}^{n}\kappa_{j}^{2}(x)\bigg{)}\bigg{]}\right.$
$\displaystyle\left.+\frac{1}{4\sum_{j=1}^{n}\xi_{j}^{2}}\bigg{[}\sum_{j,k=1}^{n}\bigg{(}2{\tilde{R}}_{j(n+1)k(n+1),(n+1)}+20{\tilde{R}}_{j(n+1)k(n+1)}\kappa_{{}_{k}}+24\kappa_{j}^{3}\delta_{jk}\bigg{)}\xi_{j}\xi_{k}\bigg{]}\right.$
$\displaystyle\left.-\frac{3\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}}{2\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{2}}\bigg{[}\sum_{j,k=1}^{n}\bigg{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6\kappa_{j}^{2}\delta_{jk}\bigg{)}\xi_{j}\xi_{k}\bigg{]}\right.$
$\displaystyle\left.+\frac{4\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}\big{)}^{3}}{\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{3}}+\frac{\sum_{j=1}^{n}\kappa_{j}(x)}{2}\bigg{[}\frac{1}{2\sum_{j=1}^{n}\xi_{j}^{2}}\bigg{(}\sum_{j,k=1}^{n}\big{(}2{\tilde{R}}_{j(n+1)k(n+1)}+6\kappa_{j}^{2}\delta_{jk}\big{)}\xi_{j}\xi_{k}\bigg{)}\right.$
$\displaystyle\left.-\frac{2\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}\big{)}^{2}}{\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{2}}+2\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}^{2}-\frac{1}{2}\bigg{(}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\sum_{j=1}^{n}\kappa_{j}^{2}(x)\bigg{)}\bigg{]}\right.$
$\displaystyle\left.-\frac{1}{4}\bigg{[}\frac{\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}}{\sum_{j=1}^{n}\xi_{j}^{2}}+\sum_{j=1}^{n}\kappa_{j}(x)\bigg{]}\bigg{[}2Q_{1}-2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}+2\sum_{j=1}^{n}\kappa_{j}^{2}(x)-4(\sum_{j=1}^{n}\kappa_{j}(x))^{2}\bigg{]}\right\\}$
Also, it follows from (4.2), (4.3), (6.19)–(6.21) that $p_{0}(x,\xi)$ and
$p_{-1}(x,\xi)$ have the following simple expressions in the normal
coordinates chosen before:
(6.33) $\displaystyle p_{0}(x,\xi)=0,$ (6.34) $\displaystyle
p_{-1}(x,\xi)=\frac{1}{4\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{3}{2}}}\bigg{[}-\frac{1}{2}\sum_{k,l=1}^{n}\frac{\partial^{2}|h(x)|}{\partial
x_{k}\partial
x_{l}}\,\xi_{k}\xi_{l}-\sum_{j,k,l=1}^{n}\frac{\partial^{2}h^{jk}}{\partial
x_{k}\partial x_{l}}\,\xi_{j}\xi_{l}\quad\quad$
$\displaystyle\qquad\quad+\frac{1}{\sum_{j=1}^{n}\xi_{j}^{2}}\sum_{j,k,l,m=1}^{n}\frac{\partial^{2}h^{jk}}{\partial
x_{m}\partial
x_{l}}\,\xi_{j}\xi_{k}\xi_{l}\xi_{m}-\frac{1}{2}\sum_{j,k,m=1}^{n}\frac{\partial^{2}h^{jk}}{\partial
x_{m}^{2}}\,\xi_{j}\xi_{k}\bigg{]}\quad\qquad$
$\displaystyle\quad\;\;=\frac{1}{4\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{3}{2}}}\bigg{[}\sum_{k,l,m=1}^{n}\frac{R_{mkml}}{3}\xi_{k}\xi_{l}+\sum_{j,k,l=1}^{n}\frac{R_{jklk}}{3}\xi_{j}\xi_{l}$
$\displaystyle\;\;\quad\quad+\frac{1}{\sum_{j=1}^{n}\xi_{j}^{2}}\sum_{j,k,l,m=1}^{n}\frac{R_{jmkl}+R_{jlkm}}{3}\xi_{j}\xi_{k}\xi_{l}\xi_{m}-\sum_{j,k,m=1}^{n}\frac{R_{jmkm}}{3}\xi_{j}\xi_{k}\bigg{]}\;$
$\displaystyle\;\,\quad=\frac{1}{4\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{3}{2}}}\bigg{[}\sum_{j,k,l=1}^{n}\frac{R_{jklk}}{3}\xi_{j}\xi_{l}$
$\displaystyle\;\,\quad\quad+\frac{1}{\sum_{j=1}^{n}\xi_{j}^{2}}\sum_{j,k,l,m=1}^{n}\frac{R_{jmkl}+R_{jlkm}}{3}\xi_{j}\xi_{k}\xi_{l}\xi_{m}\bigg{]}.$
Therefore
(6.35)
$\displaystyle\quad\;\quad\quad\;{\mathcal{K}}_{V_{-2}}(t,x,x)=\left(\frac{1}{2\pi}\right)^{n}\int_{{\mathbb{R}}^{n}}\frac{t}{2}\left[\sum_{j=1}^{n}\kappa_{j}(x)-\frac{\sum_{j=1}^{n}\kappa_{j}(x)\xi_{j}^{2}}{\sum_{j=1}^{n}\xi_{j}^{2}}\right]e^{-t\,\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}\,d\xi.$
Next, from (4.2) and (4.7), we see that
$p_{0}^{B}(x,-\xi)=p_{0}^{B}(x,\xi),\quad\;p^{B}_{-1}(x,-\xi)=p_{-1}^{B}(x,\xi),\;\;\mbox{for
all}\;\;\xi\in{\mathbb{R}}^{n},$
so that
(6.36)
$\displaystyle{\mathcal{K}}_{V_{-3}}(t,x,x)=\big{(}\frac{1}{2\pi}\big{)}^{n}\int_{{\mathbb{R}}^{n}}\big{[}tp^{B}_{-1}(x,\xi)+\frac{t^{2}}{2}\big{(}\big{(}p_{0}^{B}(x,\xi)\big{)}^{2}+2p_{0}(x,\xi)p_{0}^{B}(x,\xi)\big{)}\big{]}e^{tp_{1}(x,\xi)}d\xi$
$\displaystyle\quad\;\;=\left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\int_{{\mathbb{R}}^{n}}\left[tp^{B}_{-1}(x,\xi)+\frac{t^{2}}{2}\big{(}p_{0}^{B}(x,\xi)\big{)}^{2}\right]e^{-t\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi$
$\displaystyle\quad\;\;=\left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\int_{{\mathbb{R}}^{n}}\left\\{\frac{t}{8\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{1}{2}}}\left[-2Q_{1}+2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)\right.\right.$
$\displaystyle\quad\;\;\left.\left.-2\sum_{j=1}^{n}\kappa_{j}^{2}(x)+3\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}^{2}+\frac{5\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\,\xi_{j}^{2}\big{)}^{2}}{\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{2}}\right.\right.$
$\displaystyle\quad\;\;\left.\left.-\,\frac{1}{\sum_{j=1}^{n}\xi_{j}^{2}}\,\sum_{j=1}^{n}\bigg{(}2{\tilde{R}}_{j(n+1)j(n+1)}+6\kappa_{j}^{2}(x)\bigg{)}\xi_{j}^{2}\right]\right.$
$\displaystyle\quad\;\;\left.\,+\,\frac{t^{2}}{8}\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)-\frac{\sum_{j=1}^{n}\kappa_{j}(x)\,\xi_{j}^{2}}{\sum_{j=1}^{n}\xi_{j}^{2}}\bigg{)}^{2}\right\\}e^{-t\,\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}\,d\xi,$
and
(6.37)
$\displaystyle{\mathcal{K}}_{V_{-4}}(t,x,x)=\left(\frac{1}{2\pi}\right)^{n}\int_{{\mathbb{R}}^{n}}\bigg{[}t\,p_{-2}^{B}(x,\xi)+t^{2}\bigg{(}p_{0}(x,\xi)p_{-1}^{B}(x,\xi)$
$\displaystyle\quad\;\;\quad+p_{-1}(x,\xi)p_{0}^{B}(x,\xi)+p_{0}^{B}(x,\xi)p_{-1}^{B}(x,\xi)\bigg{)}+\frac{t^{3}}{6}\bigg{(}(p_{0}^{B}(x,\xi))^{3}$
$\displaystyle\quad\;\;\quad+3(p_{0}(x,\xi))^{2}p_{0}^{B}(x,\xi)+3p_{0}(x,\xi)(p_{0}^{B}(x,\xi))^{2}\bigg{)}\bigg{]}e^{-t\,\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}\,d\xi.$
$\displaystyle\quad\quad=\left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\int_{{\mathbb{R}}^{n}}\bigg{[}t\,p_{-2}^{B}(x,\xi)+t^{2}\bigg{(}p_{-1}(x,\xi)p_{0}^{B}(x,\xi)+p_{0}^{B}(x,\xi)p_{-1}^{B}(x,\xi)\bigg{)}$
$\displaystyle\quad\;\;\quad+\frac{t^{3}}{6}\big{(}p_{0}^{B}(x,\xi)\big{)}^{3}\bigg{]}e^{-t\,\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}\,d\xi.$
By applying the spherical coordinates transform
$\displaystyle\left\\{\begin{array}[]{ll}\xi_{1}=r\cos\phi_{1}\\\
\xi_{2}=r\sin\phi_{1}\cos\phi_{2}\\\
\xi_{3}=r\sin\phi_{1}\sin\phi_{2}\cos\phi_{3}\\\ \cdots\cdots\cdots\cdots\\\
\xi_{n-1}=r\sin\phi_{1}\sin\phi_{2}\cdots\sin\phi_{n-2}\cos\phi_{n-1}\\\
\xi_{n}=r\sin\phi_{1}\sin\phi_{2}\cdots\sin\phi_{n-2}\sin\phi_{n-1},\end{array}\right.$
where $0\leq
r<+\infty,\,0\leq\phi_{1}\leq\pi,\,\cdots,\,0\leq\phi_{n-2}\leq\pi,\;0\leq\phi_{n-1}\leq
2\pi$, we then have
$d\xi=r^{n-1}dS=r^{n-1}\sin^{n-2}\phi_{1}\sin^{n-3}\phi_{2}\cdots\sin\phi_{n-2}.$
Since
$\displaystyle\int_{0}^{\pi}\sin^{k}\phi\;d\phi=\frac{\Gamma(\frac{n+1}{2})}{\Gamma(\frac{n+2}{2})}\,\sqrt{\pi}=\left\\{\begin{array}[]{ll}\frac{(2m-1)!!}{(2m)!!}\,\pi,\quad\,k=2m,\\\
2\frac{(2m)!!}{(2m+1)!!},\quad\;\,k=2m+1,\end{array}\right.\quad\mbox{vol}\,({\mathbb{S}}^{n-1})=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})},$
and since the space ${\mathbb{R}}^{n}$ is symmetric about the coordinate axes
$\xi_{1},\cdots,\xi_{n}$, we immediately get
(6.42)
$\displaystyle\int_{{\mathbb{R}}^{n}}\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{m}{2}}\xi_{k}^{\alpha}\,e^{-\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=\left\\{\begin{array}[]{ll}\Gamma(n+m)\,vol({\mathbb{S}}^{n-1})&\mbox{for}\;\;\alpha=0\\\
0&\mbox{for}\;\;\alpha=1,\end{array}\right.\;\;n\geq 1,$ (6.45)
$\displaystyle\int_{{\mathbb{R}}^{n}}\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{m-2}{2}}\xi_{k}\xi_{l}\,e^{-\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=\left\\{\begin{array}[]{ll}\frac{\Gamma(n+m)\,vol({\mathbb{S}}^{n-1})}{n}&\mbox{for}\;\;k=l\\\
0&\mbox{for}\;\;k\neq l,\end{array}\right.\;\;n\geq 2,$ (6.48)
$\displaystyle\int_{{\mathbb{R}}^{n}}\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{m-4}{2}}\xi_{k}^{2}\xi_{l}^{2}\,e^{-\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=\left\\{\begin{array}[]{ll}\frac{3\,\Gamma(n+m)\,vol({\mathbb{S}}^{n-1})}{n(n+2)}&\mbox{for}\;\,k=l\\\
\frac{\Gamma(n+m)\,vol({\mathbb{S}}^{n-1})}{n(n+2)}&\mbox{for}\;\;k\neq
l,\end{array}\right.\;\;n\geq 2,$ (6.49)
$\displaystyle\int_{{\mathbb{R}}^{n}}\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{m-4}{2}}\xi_{k}\xi_{l}\xi_{m}\xi_{p}\,e^{-\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=0\quad\;klmp\;\;\mbox{comprises}\;\,\leq
1\;\;\mbox{pairs},\;\;n\geq 3,$
$\displaystyle\int_{{\mathbb{R}}^{n}}\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{m-6}{2}}\xi_{k}^{6}\,e^{-\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=\frac{15\,\Gamma(n+m)\,vol({\mathbb{S}}^{n-1})}{n(n+2)(n+4)},\;\;n\geq
3$
$\displaystyle\int_{{\mathbb{R}}^{n}}\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{m-6}{2}}\xi_{k}^{4}\xi_{l}^{2}\,e^{-\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=\frac{3\,\Gamma(n+m)\,vol({\mathbb{S}}^{n-1})}{n(n+2)(n+4)},\quad\;k\neq
l,\;\;n\geq 3,$
$\displaystyle\int_{{\mathbb{R}}^{n}}\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{m-6}{2}}\xi_{k}^{2}\xi_{l}^{2}\xi_{m}^{2}\,e^{-\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=\frac{\Gamma(n+m)\,vol({\mathbb{S}}^{n-1})}{n(n+2)(n+4)},\;\quad
k\neq l\neq m,\;\;n\geq 3,$
$\displaystyle\int_{{\mathbb{R}}^{n}}\big{(}\sum_{j=1}^{n}\xi_{j}^{2}\big{)}^{\frac{m-6}{2}}\xi_{k}\xi_{l}\xi_{m}\xi_{p}\xi_{q}\xi_{v}\,e^{-\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=0\quad\;klmpqv\,\;\mbox{comprises}\,\leq
2\,\;\mbox{pairs},\,n\geq 3.$
It follows from (6.35) and (6.36) that
$\displaystyle{\mathcal{K}}_{V_{-2}}(t,x,x)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{2\pi}\right)^{n}\frac{t^{1-n}}{2}\int_{0}^{\infty}\frac{(tr)^{n-1}d(tr)}{e^{tr}}$
$\displaystyle\quad\times\int_{{\mathbb{S}}^{n}}\bigg{[}\sum_{j=1}^{n}\kappa_{j}(x)-\sum_{j=1}^{n-1}\kappa_{j}(x)\sin^{2}\phi_{1}\cdots\sin^{2}\phi_{j-1}\cos^{2}\phi_{j}$
$\displaystyle\quad\;-\kappa_{n}(x)\sin^{2}\phi_{1}\cdots\sin^{2}\phi_{n-1}\bigg{]}dS$
$\displaystyle=t^{1-n}\left(\frac{1}{2\pi}\right)^{\frac{n}{2}}\frac{(n-1)\Gamma(n)\,\mbox{vol}({\mathbb{S}}^{n-1})}{2n}\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}$
$\displaystyle:=t^{1-n}a_{1}(n,x)$
and
$\displaystyle{\mathcal{K}}_{V_{-3}}(t,x,x)=\left(\frac{1}{2\pi}\right)^{n}\int_{{\mathbb{S}}^{n}}dS\left\\{\frac{t^{2-n}}{8}\int_{0}^{\infty}\frac{(tr)^{n-2}d(tr)}{e^{tr}}\left[-2Q_{1}(x)\frac{}{}\right.\right.$
$\displaystyle\left.\left.+2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)-2\sum_{j=1}^{n}\kappa_{j}^{2}(x)+3\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}^{2}\right.\right.$
$\displaystyle\left.\left.+5\bigg{(}\sum_{j=1}^{n-1}\kappa_{j}(x)\sin^{2}\phi_{1}\cdots\sin^{2}\phi_{j-1}\cos^{2}\phi_{j}+\kappa_{n}(x)\sin^{2}\phi_{1}\cdots\sin^{2}\phi_{n-1}\bigg{)}^{2}\right.\right.\;\;$
$\displaystyle\left.\left.-\sum_{j=1}^{n-1}\bigg{(}2{\tilde{R}}_{j(n+1)j(n+1)}(x)+6\kappa_{j}^{2}(x)\bigg{)}\sin^{2}\phi_{1}\cdots\sin^{2}\phi_{j-1}\cos^{2}\phi_{j}\right.\right.\qquad\quad\quad\;\qquad\qquad$
$\displaystyle\left.\left.-6\kappa_{n}^{2}(x)\sin^{2}\phi_{1}\cdots\sin^{2}\phi_{n-1}\right]+\frac{t^{2-n}}{8}\int_{0}^{\infty}\frac{(tr)^{n-1}d(tr)}{e^{tr}}\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\right.$
$\displaystyle\left.-\sum_{j=1}^{n-1}\kappa_{j}(x)\sin^{2}\phi_{1}\cdots\sin^{2}\phi_{j-1}\cos^{2}\phi_{j}-\kappa_{n}(x)\sin^{2}\phi_{1}\cdots\sin^{2}\phi_{n-1}\bigg{)}^{2}\right\\}\qquad\quad\qquad\quad$
$\displaystyle=$ $\displaystyle
t^{2-n}\left(\frac{1}{2\pi}\right)^{n}\frac{\Gamma(n-1)\,\mbox{vol}({\mathbb{S}}^{n-1})}{8}\left[-8\sum_{1\leq
j<k\leq
n}\kappa_{j}\kappa_{k}+\frac{2(n-1)}{n}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)\right.$
$\displaystyle\left.+\,\frac{-2n^{2}-8n-4}{n(n+2)}\sum_{j=1}^{n}\kappa_{j}^{2}(x)+\frac{n^{3}+2n^{2}+3n+8}{n(n+2)}\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}^{2}\right]t^{2-n}$
$\displaystyle=$ $\displaystyle
t^{2-n}\left(\frac{1}{2\pi}\right)^{n}\frac{\Gamma(n-1)\,\mbox{vol}({\mathbb{S}}^{n-1})}{8}\left[\frac{n^{3}-2n^{2}-5n+8}{n(n+2)}\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}^{2}\right.$
$\displaystyle\left.+\frac{2(n-1)}{n}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)+\,\frac{2n^{2}-4}{n(n+2)}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\right]$
$\displaystyle:=$ $\displaystyle t^{2-n}\tilde{a}_{2}(n,x).$
Furthermore,
(6.50)
$\displaystyle\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\int_{{\mathbb{R}}^{n}}tp_{-2}^{B}(x,\xi)\,e^{-t\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi=t^{3-n}\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\Gamma(n-2)\,\mbox{vol}({\mathbb{S}}^{n-1})\qquad\quad\qquad\;$
$\displaystyle\times\bigg{\\{}\frac{-n^{2}-4n+5}{8n(n+2)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}\big{)}+\frac{-n^{2}+6n-7}{8n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{3}$
$\displaystyle+\bigg{(}\frac{1}{4}+\frac{-12n^{2}-18n}{8n(n+2)(n+4)}\bigg{)}\sum_{j=1}^{n}\kappa_{j}^{3}(x)-\frac{n^{3}+8n^{2}-7n-2}{8n(n+2)(n+4)}\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}\bigg{(}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\bigg{)}$
$\displaystyle+\bigg{(}\frac{1}{2}+\frac{-5n-1}{4n(n+2)}\bigg{)}\sum_{j=1}^{n}\kappa_{j}(x)\,{\tilde{R}}_{j(n+1)j(n+1)}+\frac{n-1}{8n}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1),(n+1)}\bigg{\\}},$
(6.51)
$\displaystyle\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\int_{{\mathbb{R}}^{n}}\frac{t^{3}}{6}\big{[}(p_{0}^{B}(x,\xi))^{3}+3p_{0}^{2}(x,\xi)\,p_{0}^{B}(x,\xi)+3p_{0}(x,\xi)(p_{0}^{B}(x,\xi))^{2}\big{]}e^{-t\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}\,d\xi\qquad\quad\qquad$
$\displaystyle\qquad\qquad=\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\int_{{\mathbb{R}}^{n}}\frac{t^{3}}{6}\big{(}p_{0}^{B}(x,\xi)\big{)}^{3}e^{-t\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}\,d\xi\qquad\qquad$
$\displaystyle=\frac{t^{3-n}}{48}\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\Gamma(n)\,\mbox{vol}({\mathbb{S}}^{n-1})\bigg{[}\frac{n^{3}+3n^{2}-7n-13}{n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{3}$
$\displaystyle\quad+\frac{6n+18}{n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\big{)}-\frac{8}{n(n+2)(n+4)}\sum_{j=1}^{n}\kappa_{j}^{3}(x)\bigg{]}$
and
(6.52)
$\displaystyle\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\int_{{\mathbb{R}}^{n}}t^{2}p_{0}^{B}(x,\xi)\big{[}p_{-1}(x,\xi)+p_{-1}^{B}(x,\xi)\big{]}e^{-t\sqrt{\sum_{j=1}^{n}\xi_{j}^{2}}}d\xi\qquad\quad\quad\qquad\quad\quad\;\,$
$\displaystyle=$
$\displaystyle\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\frac{\Gamma(n-1)\,\mbox{vol}(S^{n-1})t^{3-n}}{8}\bigg{[}\frac{n^{2}+9n+24}{3n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j,k=1}^{n}R_{jkjk}(x)\big{)}$
$\displaystyle+\frac{n^{2}+n-1}{n(n+2)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)\big{)}+\frac{-n^{3}-6n^{2}-3n+15}{2n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{3}$
$\displaystyle+\frac{n^{3}+3n^{2}-7n-7}{n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\big{)}+\frac{6n+4}{n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}^{3}(x)\big{)}$
$\displaystyle+\frac{2}{n(n+2)}\sum_{j=1}^{n}\kappa_{j}(x)\,{\tilde{R}}_{j(n+1)j(n+1)}(x)-\frac{2n+16}{3n(n+2)(n+4)}\sum_{j,k=1}^{n}\kappa_{j}(x)\,{R}_{jkjk}(x)$
$\displaystyle-\frac{1}{n}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)+\frac{1}{2n}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{2}-\frac{1}{n}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\bigg{]}.$
Note that for any $x\in\partial\Omega$
(6.53) $\displaystyle
R_{jkjk}(x)={\tilde{R}}_{jkjk}(x)+\kappa_{j}(x)\kappa_{k}(x),\quad\;1\leq
j,k\leq n.$
Since
$\displaystyle{\tilde{R}}_{\Omega}(x)=\sum_{j=1}^{n+1}{\tilde{R}}_{jj}(x)=\sum_{j=1}^{n+1}\big{(}\sum_{k=1}^{n}{\tilde{R}}_{jkjk}(x)+{\tilde{R}}_{j(n+1)j(n+1)}\big{)}$
$\displaystyle\quad\quad\quad=2\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)+\sum_{j,k=1}^{n}{\tilde{R}}_{jkjk},$
we obtain
(6.54)
$\displaystyle\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)=\frac{1}{2}\left({\tilde{R}}_{\Omega}(x)-R_{\partial\Omega}+\sum_{1\leq
j\neq k\leq n}^{n}\kappa_{j}(x)\kappa_{k}(x)\right).$
Inserting (6.50)—(6.52) into (6.37) and then using (6.54) we obtain that
(6.55)
$\displaystyle{\mathcal{K}}_{V_{-4}}(t,x,x)=t^{3-n}\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\Gamma(n-2)\,\mbox{vol}(S^{n-1})\bigg{[}\frac{n^{3}-2n^{2}-7n+7}{8n(n+2)}\qquad\quad$
$\displaystyle\times\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)\big{)}+\frac{n^{5}-3n^{4}-26n^{3}+47n^{2}+124n-158}{48n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{3}$
$\displaystyle+\frac{3n^{3}+7n^{2}-9n-16}{12n(n+2)(n+4)}\sum_{j=1}^{n}\kappa_{j}^{3}(x)+\frac{n^{4}+n^{3}-16n^{2}-3n+22}{8n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\big{)}$
$\displaystyle+\frac{2n^{2}-3}{4n(n+2)}\sum_{j=1}^{n}\kappa_{j}(x)\,{\tilde{R}}_{j(n+1)j(n+1)}(x)+\frac{n-1}{8n}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1),(n+1)}(x)$
$\displaystyle+\frac{n^{3}+7n^{2}+6n-48}{24n(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j,k=1}^{n}R_{jkjk}(x)\big{)}+\frac{-n^{2}-6n+16}{12n(n+2)(n+4)}\sum_{j,k=1}^{n}\kappa_{j}(x)\,R_{jkjk}(x)$
$\displaystyle-\frac{n-2}{8n}\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1)}(x)+\frac{n-2}{16n}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{2}-\frac{n-2}{8n}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\bigg{]}$
$\displaystyle=t^{3-n}\bigg{(}\frac{1}{2\pi}\bigg{)}^{n}\frac{\Gamma(n-2)\,\mbox{vol}(S^{n-1})}{8n}\bigg{[}\frac{n^{3}-2n^{2}-7n+7}{2(n+2)}{\tilde{R}}(x)\,\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}$
$\displaystyle\quad\,+\frac{-3n^{4}-4n^{3}+59n^{2}+75n-180}{6(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}R_{\partial\Omega}$
$\displaystyle\quad\,+\frac{n^{5}-20n^{3}+2n^{2}+61n-74}{6(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{3}$
$\displaystyle\quad\,+\frac{n^{4}+8n^{3}+15n^{2}+3n-32}{2(n+2)(n+4)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}\big{(}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\big{)}$
$\displaystyle\quad\,+\frac{-6n^{3}-34n^{2}+40}{3(n+2)(n+4)}\sum_{j=1}^{n}\kappa_{j}^{3}(x)+\frac{4n^{2}-6}{n+2}\sum_{j=1}^{n}\kappa_{j}(x)\big{(}\sum_{k=1}^{n+1}{\tilde{R}}_{jkjk}(x)\big{)}$
$\displaystyle\quad\,-\frac{12n^{3}+50n^{2}-6n-104}{3(n+2)(n+4)}\sum_{j=1}^{n}\kappa_{j}(x)\big{(}\sum_{k=1}^{n}R_{jkjk}(x)\big{)}+(n-1)\sum_{j=1}^{n}{\tilde{R}}_{j(n+1)j(n+1),(n+1)}(x)$
$\displaystyle\quad\,-\frac{n-2}{2}{\tilde{R}}_{\Omega}(x)+\frac{n-2}{2}R_{\partial\Omega}-\frac{n-2}{2}\sum_{j=1}^{n}\kappa_{j}^{2}(x)$
$\displaystyle:=t^{3-n}a_{3}(n,x).$
Put
$a_{2}(n,x)=\frac{\Gamma(\frac{n-1}{2})\,R_{\partial\Omega}(x)}{24\pi^{\frac{n+1}{2}}}+{\tilde{a}}_{2}(n,x)$.
In view of (6.53) we get that
$\displaystyle a_{2}(n,x)$ $\displaystyle=$
$\displaystyle\frac{R_{\partial\Omega}}{24\pi^{\frac{n+1}{2}}}\,\frac{2^{1-n}\Gamma(n-1)vol({\mathbb{S}}^{n-1})}{\pi^{\frac{n-1}{2}}}$
$\displaystyle+\left(\frac{1}{2\pi}\right)^{n}\frac{\Gamma(n-1)\,\mbox{vol}({\mathbb{S}}^{n-1})}{8}\left[\frac{n^{3}-2n^{2}-5n+8}{n(n+2)}\bigg{(}\sum_{j=1}^{n}\kappa_{j}(x)\bigg{)}^{2}\right.$
$\displaystyle\left.+\frac{n-1}{n}\left({\tilde{R}}_{\Omega}(x)-R_{\partial\Omega}(x)+\sum_{1\leq
j\neq k\leq
n}^{n}\kappa_{j}(x)\kappa_{k}(x)\right)\sum_{j=1}^{n}+\,\frac{2n^{2}-4}{n(n+2)}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\right]$
$\displaystyle=$
$\displaystyle\frac{\Gamma(n-1)vol({\mathbb{S}}^{n-1})}{8(2\pi)^{n}}\left[\frac{3-n}{3n}R_{\partial\Omega}+\frac{n-1}{n}{\tilde{R}}_{\Omega}\right.$
$\displaystyle\left.+\frac{n^{3}-n^{2}-4n+6}{n(n+2)}\big{(}\sum_{j=1}^{n}\kappa_{j}(x)\big{)}^{2}+\frac{n^{2}-n-2}{n(n+2)}\sum_{j=1}^{n}\kappa_{j}^{2}(x)\right].\qquad\qquad\qquad\quad$
From the above arguments, (6.28), (6.29), (6) and (6.25), we find that
$\displaystyle\int_{\partial\Omega}{\mathcal{K}}(t,x,x)dS(x)=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}t^{-n}\int_{\partial\Omega}1\,dS(x)+t^{1-n}\int_{\partial\Omega}a_{1}(n,x)dS(x)$
$\displaystyle\quad\quad\quad+\cdots+t^{M-1-n}\int_{\partial\Omega}a_{M-1}(n,x)dS(x)+{\mathcal{K}}_{V^{\prime}_{M}}(t,x,x)\quad\;\mbox{as}\;\;t\to
0^{+}.$
Finally, if we choose $M=2$, $M=3$ and $M=4$ then (6.3), (6.7), (6.11) follow,
respectively. $\square$.
Remark 6.2. (i) Our asymptotic expansion is sharp because the following
asymptotic expansion holds for $t\to 0^{+}$ (see, (4.2.62) of [51]):
(6.56)
$\displaystyle\quad\quad{\mathcal{K}}(t,x,x)\sim\sum_{j\in{\mathbb{N}},\;j-n\notin{\mathbb{N}}_{+}}c_{j-n}(x)t^{j-n}+\sum_{j-n\in{\mathbb{N}}_{+}}c_{j-n}(x)t^{j-n}\log\,t+\sum_{l\in{\mathbb{N}}_{+}}r_{l}(x)t^{l},$
in the usual sense: the difference between ${\mathcal{K}}(t,x,x)$ and the sum
of terms up to $j-n=N\in{\mathbb{N}}_{+}$, $l=N$, is $O(t^{N+1})$.
Consequently (see (4.2.64) of [51]),
(6.57) $\displaystyle\mbox{Tr}\,e^{t{\mathcal{N}}_{g}}\sim
t^{j-n}\sum_{j\in{\mathbb{N}},\;j-n\notin{\mathbb{N}}_{+}}\int_{\partial\Omega}c_{j-n}(x)dS(x)\,$
$\displaystyle\qquad\qquad+t^{j-n}\log\,t\sum_{j-n\in{\mathbb{N}}_{+}}\int_{\partial\Omega}c_{j-n}(x)dS(x)+t^{l}\sum_{l\in{\mathbb{N}}_{+}}\int_{\partial\Omega}r_{l}(x)dS(x).$
(ii) By our method, we can also get the asymptotic expansion for any integer
$M>4$:
(6.58)
$\displaystyle\int_{\partial\Omega}{\mathcal{K}}(t,x,x)dS(x)=t^{-n}\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}\int_{\partial\Omega}1\,dS(x)+t^{1-n}\int_{\partial\Omega}a_{1}(n,x)\,dS(x)$
$\displaystyle\qquad\quad\quad+t^{2-n}\int_{\partial\Omega}a_{2}(n,x)\,dS(x)+\cdots+t^{M-1-n}\int_{\partial\Omega}a_{M-1}(n,x)\,dS(x)$
(6.60)
$\displaystyle\qquad\quad\quad+\left\\{\begin{array}[]{ll}O(t^{M-n})\quad\,\mbox{when}\;\;n>M-1,\\\
O(t\log
t)\;\quad\;\mbox{when}\;\;n=M-1,\end{array}\right.\quad\;\;\,\mbox{as}\;\;t\to
0^{+},$
if we further calculate the lower-order symbol equations for the operators
$-\sqrt{-\Delta_{h}}$ and $B$, respectively (note that
${\mathcal{N}}_{g}=-\sqrt{-\Delta_{h}}+B$).
(iii) The above asymptotic expansion shows that one can “hear”
$\int_{\partial\Omega}a_{M-1}(n,x)dS(x)$ ($M=1,2,3,\cdots$) by “hearing” all
of the Steklov eigenvalues.
(iv) We can obtain the same result when we directly calculate the first three
symbols of $-\sqrt{-\Delta_{h}}$ and $B$ instead of using
${\mathcal{K}}_{V}(t,x,y)$.
(v) By applying the Tauberian theorem (see, for example, Theorem 15.3 of p. 30
of [64]) for the first term on the right side of (6.3), we immediately get
Sandgren’s asymptotic formula (1.7) for the Steklov eigenvalues.
In particular, we have the following:
Corollary 6.3. Let $(\mathcal{M},g)$ be an $(n+1)$-dimensional, smooth
Riemannian manifold, and let $\Omega\subset\mathcal{M}$ be a bounded domain
with smooth boundary $\partial\Omega$. Let $h$ be the induced metric on
$\partial\Omega$ by $g$. If $F(t,x,y)$ is a fundamental solution of
$\frac{\partial u}{\partial t}=-\sqrt{-\Delta_{h}}\,u$ on
$[0,+\infty)\times(\partial\Omega)$, then
$\displaystyle\int_{\partial\Omega}F(t,x,x)dS(x)=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}t^{-n}\int_{\partial\Omega}1\,dS(x)+\frac{\Gamma(\frac{n-1}{2})}{12\pi^{\frac{n+1}{2}}}t^{2-n}\int_{\partial\Omega}R_{\partial\Omega}\,dS(x)$
$\displaystyle\quad\quad\quad\;+\frac{\Gamma(\frac{n-3}{2})}{720\pi^{\frac{n+1}{2}}}t^{4-n}\int_{\partial\Omega}(10A-B+2C)\,dS(x)+\frac{\Gamma(\frac{n-5}{2})}{4^{3}\pi^{\frac{n+1}{2}}}t^{6-n}D+o(t^{8-n})\quad\;\mbox{as}\;\;t\to
0^{+}$
with
$\displaystyle A=\big{(}\sum_{1\leq i<j\leq
n}R_{ijij}\big{)}^{2}=(R_{\partial\Omega})^{2},\,\quad B=\sum_{1\leq j,k\leq
n}\big{(}\sum_{1\leq i\leq n}R_{ijik}\big{)}^{2},\,\quad C=\sum_{1\leq
i,j,k,l\leq n}\big{(}R_{ijkl}\big{)}^{2},$ $\displaystyle
D=\int_{\partial\Omega}D(n,x)dS(x)=\frac{(4\pi)^{-n/2}}{7!}\int_{\partial\Omega}\big{(}-\frac{142}{9}R_{ijij,k}R_{mlml,k}-\frac{26}{9}R_{ijik,l}R_{mjmk,l}\qquad\qquad\qquad\qquad\quad\qquad\quad\qquad\qquad$
$\displaystyle\;\;-\frac{7}{9}R_{ijkm,l}R_{ijkm,l}-\frac{35}{9}R_{ijij}R_{mlml}R_{pqpq}+\frac{14}{3}R_{ijij}R_{mlmp}R_{qlqp}-\frac{14}{3}R_{ijij}R_{mlpq}R_{mlpq}$
$\displaystyle\;\;+4R_{ijik}R_{jlml}R_{kpmp}-\frac{20}{9}R_{ijik}R_{lpmp}R_{jlkm}+\frac{8}{9}R_{ijik}R_{jlmp}R_{klmp}-\frac{8}{3}R_{ijkl}R_{ijmp}R_{klmp}\big{)}dS(x),$
where $R_{ijkl}$ and $R_{ijkl,m}$ denote the curvature tensor and its
covariant derivative on $(\partial\Omega,h)$,respectively.
Proof. It follows from (1.5a) and (7.2) of [80] and p.613 of [44] that, as
$t\to 0^{+}$,
$\displaystyle\int_{\partial\Omega}G(t,x,x)dS(x)=\int_{\partial\Omega}\left[\left(G_{0}(t,x,x)+\big{(}G_{0}\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}\big{)}(t,x,x)\frac{}{}\right.\right.$
$\displaystyle\,\left.\left.+G_{0}\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}\right)(t,x,x)\right.$
$\displaystyle\,\left.+G_{0}\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}\\#\big{(}(\Delta_{h}-\Delta_{h}^{0})G_{0}\big{)}(t,x,x)\right]dS(x)+o(t^{4-\frac{n}{2}})$
$\displaystyle=\frac{1}{(4\pi
t)^{n/2}}\left[\int_{\partial\Omega}1\,dS(x)+\frac{t}{3}\int_{\partial\Omega}R_{\partial\Omega}\,dS(x)+\frac{t^{2}}{180}\int_{\partial\Omega}(10A-B+2C)dS(x)\right.$
$\displaystyle\left.\,+Dt^{3}+o(t^{4})\right].$
Combing this and (5.2), we immediately get an asymptotic expansion:
$\displaystyle\int_{\partial\Omega}F(t,x,x)dS(x)=\int_{0}^{\infty}\frac{te^{-\frac{t^{2}}{4\mu}}}{\sqrt{4\pi\mu^{3}}}\left\\{\frac{1}{(4\pi\mu)^{n/2}}\left[\int_{\partial\Omega}1\,dS(x)+\frac{\mu}{3}\int_{\partial\Omega}R_{\partial\Omega}\,dS(x)\right.\right.$
$\displaystyle\quad\left.\left.+\frac{\mu^{2}}{180}\int_{\partial\Omega}(10A-B+2C)dS(x)+D\mu^{3}+o(\mu^{4})\right]\right\\}d\mu$
$\displaystyle=\frac{\Gamma(\frac{n+1}{2})}{\pi^{\frac{n+1}{2}}}t^{-n}\int_{\partial\Omega}1\,dS(x)+\frac{\Gamma(\frac{n-1}{2})}{12\pi^{\frac{n+1}{2}}}t^{2-n}\int_{\partial\Omega}R_{\partial\Omega}\,dS(x)$
$\displaystyle\quad+\frac{\Gamma(\frac{n-3}{2})}{2880\,\pi^{\frac{n+1}{2}}}t^{4-n}\int_{\partial\Omega}(10A-B+2C)\,dS(x)+\frac{\Gamma(\frac{n-5}{2})}{4^{3}\pi^{\frac{n+1}{2}}}t^{6-n}D+o(t^{8-n})\quad\;\mbox{as}\;\;t\to
0^{+}.$
Remark 6.4. For $n=2$, it follows from [80] that
$10A-B+2C=12(R_{\partial\Omega})^{2}$. Therefore by applying the classical
Gauss-Bonnet formula for the Euler characteristic $E$ of the (two-dimensional)
boundary $\partial\Omega$ (i.e., $\int_{\partial\Omega}R_{\partial\Omega}=2\pi
E$), we get
$\displaystyle\int_{\partial\Omega}F(t,x,x)dS(x)=\frac{\Gamma(\frac{3}{2})}{\pi^{3/2}t^{2}}\,\mbox{vol}\,(\partial\Omega)+\frac{\Gamma(\frac{1}{2})}{6\pi^{1/2}}E+\frac{\Gamma(-\frac{1}{2})\,t^{2}}{240\pi^{3/2}}\int_{\partial\Omega}(R_{\partial\Omega})^{2}$
$\displaystyle\quad\quad\qquad+\frac{\Gamma(-\frac{3}{2})D}{4^{3}\pi^{3/2}}t^{4}+o(t^{6})\quad\mbox{as}\;\;t\to
0^{+},$
in particular, the Euler characteristic of boundary $\partial\Omega$ is
audible by hearing all the eigenvalues of $-\sqrt{-\Delta_{h}}$ on
$\partial\Omega$.
Acknowledgments
I wish to express my sincere gratitude to Professor L. Nirenberg and Professor
Fang-Hua Lin for their support and help. This research was supported by SRF
for ROCS, SEM (No. 2004307D01) and NNSF of China (11171023/A010801).
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|
arxiv-papers
| 2013-06-03T01:53:49 |
2024-09-04T02:49:46.002930
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Genqian Liu",
"submitter": "Genqian Liu",
"url": "https://arxiv.org/abs/1306.0272"
}
|
1306.0310
|
# pre-saddle neutron multiplicity for fission reactions induced by heavy ions
and light particles
S. Soheyli111Corresponding author: [email protected], M. K. Khalili Bu-Ali
Sina University, Department of Physics, Hamedan, Iran
###### Abstract
Pre-saddle neutron multiplicity has been calculated for several fission
reactions induced by heavy ions and light particles. Experimentally, it is
impossible to determine the contribution of neutrons being emitted before the
saddle point and those emitted between the saddle and the scission points.
Determination of the pre-saddle neutron multiplicity in our research is based
on the comparison between the experimental anisotropies and those predicted by
the standard saddle-point statistical model. Analysis of the results shows
that the pre-saddle neutron multiplicity depends on the fission barrier height
and stability of the compound nucleus. In heavy ion induced fission, the
number of pre-saddle neutrons decreases with increasing the excitation energy
of the compound nucleus. A main cause of this behavior is due to a reduction
in the ground state-to-saddle point transition time with increasing the
excitation energy of the compound nucleus. Whereas in induced fission by light
particles, the number of pre-saddle neutrons increases with increasing the
excitation energy of the compound nucleus.
###### pacs:
25.70.Jj, 25.85.Ge
## I Introduction
In the last two decades, much theoretical attention has directed towards
understanding the dynamics of fission. According to reports, measuring the
number of neutrons emitted during fission most likely gives information on the
timescale of fission as well as on the nuclear dynamics. The transition state
model of fission, based on appropriate level densities, predicts the widths
(and thus lifetimes) of fission and neutron emission. This model is also
suitable for determining the pre-fission neutron multiplicity if the
calculated lifetimes are long compared to the dynamically constrained fission
lifetime. Several groups have invested an extensive effort in measuring the
number of emitted neutrons associated with fission reactions induced by heavy
ions [1-17]. The measurement of emitted neutrons is usually limited to the
measurement of pre-scission neutron multiplicity, post-scission neutron
multiplicity, and therefore total neutron multiplicity. These measurements
show that the transition state model of fission leads to an underestimation of
the number of measured pre-scission neutrons emitted in heavy ion induced
fission at high excitation energies. This discrepancy can be related to the
viscosity of the hot nucleus [18]. Hence, the fission lifetime of a hot
nucleus is substantially longer than that determined by statistical model of
Bohr and Wheeler [19]. As a result, it is natural to expect that a dissipative
dynamical model would provide an appropriate description of nuclear fission at
high excitation energies [20].
Pre-scission neutrons $\nu_{pre}$, can be emitted between the ground state of
the compound nucleus and the saddle point (pre-saddle neutrons) $\nu_{gs}$, or
between the saddle and the scission points (saddle-to-scission neutrons)
$\nu_{ss}$. The number of pre-saddle neutrons as well as the number of saddle-
to-scission neutrons can be determined by a combined dynamical statistical
model ( CDSM ) [21-23]. The contributions $\nu_{gs}$ and $\nu_{ss}$ to the
pre-scission neutron multiplicity are also estimated by a stochastic approach
based on three-dimensional Langevin equation [24]. Recently, a more accurate
four-dimensional Langevin model as an extension of the three-dimensional
Langevin model by adding the fourth collective coordinate ( the projection of
the total spin about the symmetry axis of the fissioning nucleus ) is used to
calculate the pre-scission neutron multiplicity [25].
A common assumption in the calculation of the angular anisotropy of fission
fragments using by the transition state model is that all pre-scission
neutrons are emitted prior to reaching the saddle-point, since it is not
straightforward to separate experimentally the contribution of neutrons being
emitted before the saddle-point and those emitted between the saddle and the
scission points [26-35]. It is well known that the standard saddle-point
statistical model (SSPSM) has become the standard theory of fission fragment
angular distributions and received great success since it was proposed. The
effect of neutron evaporation prior to reaching the saddle-point is to reduce
the temperature of the fissioning nucleus which in turn increases the fission
fragment anisotropy prediction by using this model. Only $\nu_{gs}$ has an
influence over the prediction of angular anisotropy by using SSPSM. The upper
limit of the angular anisotropy of fission fragments, based on the prediction
of SSPSM is determined by assuming that all the pre-scission neutrons are
emitted before the saddle-point.
The pre-saddle neutrons as a crucial quantity in determining the angular
anisotropy of fission fragments by using SSPSM plays a main role, although any
precise method to determine it has not been introduced. In this article, we
calculate the number of pre-saddle neutrons by a novel method. In this method,
the values of $\nu_{gs}$ for several induced fission reactions by light
particles and heavy ions are determined by the fission fragments angular
distribution method. This method is based on comparison between the
experimental anisotropies and those predicted by the standard saddle-point
statistical model. This method is limited to the calculation of pre-saddle
neutrons in induced fission in which the angular anisotropy of fission
fragments has a normal behavior, i.e., it is observed a good agreement between
the angular anisotropy of fission fragments and that predicted by the SSPSM.
In order to make the present paper self-contained, we present in Sec. II, an
brief description of the standard saddle-point statistical model as well as
the calculating method of the pre-saddle neutron multiplicity on the basis of
the SSPSM in detail. Section III is devoted to the results obtained in this
study. Finally, the concluding remarks are given in Sec. IV.
## II Method of Calculations
### II.1 Standard saddle-point Statistical Model
The standard transition-state model has been used to analyze the angular
anisotropy of fission fragments in fission. In the transition-state model, the
equilibrium distribution over the K degree of freedom (the projection of total
angular momentum of the compound nucleus (I) on the symmetry axis of the
fissioning nucleus) is assumed to be established at the transition state. Two
versions of the transition-state models based on assumptions on the position
of the transition state: standard saddle-point statistical model (SSPSM), and
scission-point statistical model (SPSM) can be used for the prediction of
fission fragment angular distributions. The basic assumption of the SSPSM is
that fission proceeds along the symmetry axis of a deformed compound nucleus,
and that the distribution of K is frozen from the saddle point to the scission
point. In this model, the fission fragment angular distribution $W(\theta)$
for the fission of spin zero nuclei is given by the following expression
Vandenbosch1:1973
$W(\theta)\propto\sum_{I=0}^{\infty}\frac{(2I+1)^{2}T_{I}\exp[-p\sin^{2}{\theta}]J_{0}[-\emph{i}p\sin^{2}{\theta}]}{\textrm{erf}[\sqrt{2p}]}.$
(1)
Where ${T_{I}}$, and ${J_{0}}$ are the transmission coefficient for fission,
and the zeroth-order Bessel function,
$p=(I+\frac{1}{2})^{2}/(4K_{\circ}^{2})$, and the variance of the equilibrium
K distribution ($K_{\circ}$) is
$K_{\circ}^{2}=\frac{\Im_{eff}T}{\hbar^{2}},$ (2)
here ${\Im_{eff}}$ and $T$ are the effective moment of inertia and the nuclear
temperature of the compound nucleus at the saddle point, respectively.
The angular anisotropy of fission fragments is defined as
$A=\frac{W(0^{\circ})}{W(90^{\circ})}.$ (3)
The nuclear temperature of the compound nucleus at the saddle point is given
by
$T=\sqrt{\frac{E_{ex}}{\emph{a}}},$ (4)
where $E_{ex}$ is the excitation energy of the fissioning system and a is the
nuclear level density parameter at the saddle point. $E_{ex}$ can be expressed
by the following relation
$E_{ex}=E_{c.m.}+Q-B_{f}(I)-E_{R}(I)-{\nu_{gs}}E_{n}.$ (5)
In this equation, $E_{c.m.}$, $Q$, $B_{f}(I)$, $E_{R}(I)$, $\nu_{gs}$, and
$E_{n}$ represent the center-of-mass energy of the projectile, the $Q$ value,
the spin dependent fission barrier height, the spin dependent rotational
energy of the compound nucleus, the number of pre-saddle neutrons, and the
average excitation energy lost due to evaporation of one neutron from the
compound nucleus prior to the system reaching to the saddle point,
respectively. In the case of $p\gg 1$, the angular anisotropy of fission
fragments by using Eq. (1) is given by the following approximate relation
$A\approx 1+\frac{<I^{2}>}{4K_{\circ}^{2}}.$ (6)
The prediction of angular anisotropy of fission fragments by using the SSPSM
is valid only under restrictive assumptions. At high angular momentum, or at
high fissility, the rotating liquid drop model (RLDM) predicts that the
fission barrier height($B_{f}(I)$) vanishes even for a spherical nucleus,
which leads to $K_{\circ}^{2}\rightarrow\infty$. Subsequently, the
distribution of K is uniform and hence the prediction of the SSPSM for the
fission fragments angular anisotropy is nearly uniform by using Eq. (1). This
predicted tendency toward isotropy for fission fragments at high angular
momentum is not seen in the experiments. This discrepancy is taken as a clear
indication that the width of the K distribution is not determined at the
predicted spherical saddle point shape, but at a point where nucleus is more
deformed. Therefore, it has been proposed that the standard saddle-point
statistical model breaks down at high spin and/or large values of
$\frac{Z^{2}}{A}$ of the compound nucleus (CN), and the angular distribution
of fission fragments is governed by an effective transition state different
from saddle point transition state.
### II.2 Pre-saddle Neutron multiplicity
It is clear that because of the hindrance to fission, a large number of
particles more that those predicted by the statistical model are emitted from
the fissioning system. In heavy ion fusion reactions, due to the formation of
a heavy compound nucleus, the competition between neutron emission and fission
describes the decay possibilities rather well. During the collective motion to
the scission point, neutrons will be evaporated if energetically possible, and
would be experimentally as pre-fission, or more correctly, pre-scission
neutrons. A longer saddle to scission time due to the viscosity effect, will
result in a higher pre-scission neutron multiplicity Hinde3:1989 . The
calculation of pre-saddle neutrons in heavy ion induced reactions based on the
comparison between the experimental data of angular anisotropy and those
predicted by the SSPSM depends on the kinetic energy and the binding energy of
evaporated neutron from the compound nucleus prior to the system reaching to
the saddle point. The energy spectrum of evaporated neutrons is usually given
by the following form (an evaporation spectrum) Tsang:1983
$\frac{dN}{dE}=CE\exp(-\frac{E}{T}).$ (7)
Hence, the average kinetic energy of the emitted neutron, $\overline{E}_{K}$
is given by
$\overline{E}_{K}=2T.$ (8)
The average excitation energy lost due to evaporation of one neutron from the
compound nucleus prior to the system reaching to the saddle point is given by
$E_{n}=B_{n}+2T,$ (9)
where, $B_{n}$ denotes the average neutron separation energy.
In this work, the average energy lost by an emitted neutron over the energy
range of the projectile is calculated by Eq. (9), for heavy ion induced
fission reactions, as well as for induced fissions by light projectiles. The
level density parameter, a is taken $\frac{A_{C.N.}}{8}$ ( Considering the
level density parameter as $\frac{A_{C.N.}}{10}$, rather than
$\frac{A_{C.N.}}{8}$, the number of pre-saddle neutrons varies at most by
10$\%$). Hence, number of pre-saddle neutrons is not sensitive to the level
density parameter selected in the calculation. $\Im_{eff}$, $B_{f}(I)$, and
$E_{R}(I)$ are accounted by the use of rotating finite range model (RFRM)
Sierk:1986 , while $<I^{2}>$ quantities are calculated by several models
[39-44]. In the following sections, the determination of the number of pre-
saddle neutrons, $\nu_{gs}$ for these systems is based on the comparison
between the experimental data of angular anisotropies and those predicted by
the SSPSM. In the present work, it is determined pre-saddle neutron
multiplicities for several systems undergoing heavy ion induced fission in
which fission fragments angular anisotropies have a normal behavior as well as
those systems undergoing light particle induced fission. In order to determine
number of pre-saddle neutrons in heavy ion reactions with anomalous angular
anisotropies, it is necessary to predict the average contribution of non
compound nucleus fission events Soheyli:2012 .
## III Results and discussion
The calculated multiplicities of pre-saddle neutrons as a function of $E_{ex}$
for the two ${}^{16}\textrm{O}+^{209}\textrm{Bi}\rightarrow^{225}$ Pa and
${}^{19}\textrm{F}+^{208}\textrm{Pb}\rightarrow^{227}\textrm{Pa}$ reaction
systems leading to Protactinium isotopes, are shown in Fig. 1(a). For above
studied systems, the experimental data of angular anisotropy are taken from
literature [46, 47]. As illustrated in the figure, the number of pre-saddle
neutrons decreases with increasing the excitation energy of the compound
nucleus. This behavior is due to the fact that the fission barrier height (and
thus the ground state-to-saddle point transition time ) decreases with
increasing the excitation energy of the compound nucleus, which can be lead to
that $\nu_{gs}$ decreases with $E_{ex}$. In this figure, the general trend of
the number of pre-saddle neutrons as a function of the excitation energy of
the compound nucleus is represented by a line using the method of least
squares. Fig. 1(b), shows a similar case for the
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system. For this system, the experimental data of angular anisotropy are taken
from literature [8]. Multiplicities of pre-saddle neutrons for the two
${}^{11}\textrm{B}+^{237}\textrm{Np}$ and
${}^{16}\textrm{O}+^{232}\textrm{Th}$ reaction systems, both populating the
same compound nucleus ${}^{248}\textrm{Cf}$ are also shown in Fig. 1(c). For
these two systems, the experimental data of $<\textrm{A}>$ are taken from
literature [56-58]. It is interesting to note that for these two systems, as
well as for the two ${}^{16}\textrm{O}+^{209}\textrm{Bi}\rightarrow^{225}$ Pa
and ${}^{19}\textrm{F}+^{208}\textrm{Pb}\rightarrow^{227}\textrm{Pa}$ reaction
systems as shown in Fig. 1(a), the number of pre-saddle neutrons at any given
excitation energy appears to be nearly equal. As a result, the multiplicities
of pre-saddle neutrons for heavy ion fusion reactions populating the same
compound nucleus are nearly independent of the entrance channel asymmetry and
depend on the mass number of the compound nucleus.
Figure 1: Calculated multiplicities of pre-saddle neutrons (a) for the two
${}^{16}\textrm{O}+^{209}\textrm{Bi}\rightarrow^{225}$ Pa and
${}^{19}\textrm{F}+^{208}\textrm{Pb}\rightarrow^{227}\textrm{Pa}$ reaction
systems. Thick and dotted lines represent the general trends of $\nu_{gs}$
against the excitation energy of the compound nucleus for the two
${}^{16}\textrm{O}+^{209}\textrm{Bi}\rightarrow^{225}$ Pa and
${}^{19}\textrm{F}+^{208}\textrm{Pb}\rightarrow^{227}\textrm{Pa}$ reaction
systems, respectively. (b) For the
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}{\textrm{Th}}$ reaction
system. Thick line represents the general trend of $\nu_{gs}$ against the
excitation energy of the compound nucleus, and (c) for the two
${}^{11}\textrm{B}+^{237}\textrm{Np}$ and
${}^{16}\textrm{O}+^{232}\textrm{Th}$ reaction systems, both populating the
same compound nucleus ${}^{248}\textrm{Cf}$. Thick and thin lines represent
the general trends of $\nu_{gs}$ against the excitation energy of the compound
nucleus for the two ${}^{11}\textrm{B}+^{237}\textrm{Np}$ and
${}^{16}\textrm{O}+^{232}\textrm{Th}$ reaction systems, respectively.
The ratio of the calculated pre-saddle neutron multiplicity , $\nu^{gs}_{cal}$
to experimental pre-scission neutron multiplicity $\nu^{pre}_{exp}$
Rossner:1992 , and also the ratio of theoretical pre-saddle neutron
multiplicity to theoretical pre-scission neutron multiplicity Frobrich1: 1994
for the ${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$
reaction system, are given in Table I.
$E_{c.m.}(MeV)$ | $E_{ex}(MeV)$ | $\nu^{pre}_{exp}$ | $\nu^{gs}_{cal}$ | $\nu^{gs}_{cal}/{\nu^{pre}_{exp}}$ | $\nu^{gs}_{th}/{\nu^{pre}_{th}}$
---|---|---|---|---|---
76.9 | 22.7 | 1.50 | 1.81 | 1.21 | 0.96
82.6 | 27.6 | 1.90 | 1.60 | 0.84 | 0.91
92.0 | 32.0 | 2.40 | 1.30 | 0.54 | 0.78
105.9 | 42.5 | 2.80 | 0.52 | 0.18 | 0.64
119.0 | 55.0 | 3.40 | 0.00 | 0.00 | 0.56
Table 1: Comparison between the calculated $\nu^{gs}_{cal}$,
$\nu^{gs}_{cal}/{\nu^{pre}_{exp}}$ and $\nu^{gs}_{th}/\nu^{pre}_{th}$
Frobrich1: 1994 for the
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system.
As can be seen from Table I, the calculated number of pre-saddle neutrons for
the ${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system is greater than ${\nu^{pre}_{exp}}$ at
$\textrm{E}~{}_{\textrm{c.m.}}=76.9~{}\textrm{MeV}$. This unexpected result
can be related to the measured value of fission fragment angular anisotropy at
low energy . It seems that the measured value of the angular anisotropy at
$\textrm{E}~{}_{\textrm{c.m.}}=76.9~{}\textrm{MeV}$ is reported more than its
actual value.
As the nucleus is heated, the excitation energy of the compound nucleus,
$E_{ex}$ exceeds the fission barrier height, $B_{f}$. Hence, it becomes
possible for the nucleus to fission after passing through excited states above
the fission barrier ( transient state ) Hinde5: 1993 . In this transient state
picture, the fission width, $\Gamma_{f}$ depends on the level density above
the fission barrier. The fission width and the neutron width can be shown to
be approximately given by $\Gamma_{f}\propto\exp(-\frac{B_{f}}{T})$ and
$\Gamma_{n}\propto\exp(-\frac{B_{n}}{T})$ ( $B_{n}$ is the neutron binding
energy ), respectively. Therefore, the energy dependence of the ratio
$\frac{\Gamma_{n}}{\Gamma_{f}}$ is expected to be dominated by the ratio of
appropriate Boltzmann factors, i.e., $\frac{\Gamma_{n}}{\Gamma_{f}}\approx
exp[(B_{f}-B_{n})/T]$.
In general, in heavy ion induced fission, $B_{f}$ will be relatively high at
low excitation energy or at low angular momentum, $I$, however as $I$, as well
as $E_{ex}$ is increased, the larger moment of inertia of the elongated saddle
point configuration causes its energy to increase less rapidly than that of
the compact equilibrium deformation, so the barrier height falls to zero at
some $I$. The ratio $\frac{\Gamma_{n}}{\Gamma_{f}}$ is known to decrease
sharply as $E_{ex}$ increases in nuclei of the Lead-Bismuth region, and it is
expected to do just the opposite for nuclei with the largest known atomic
numbers Bishop: 1972 . For the lighter group of fissioning elements
$B_{f}\gg{B_{n}}$, and for the very heavy ones, it is expected that
$B_{n}\gg{B_{f}}$. For nuclei of intermediate mass like the Neptunium, $B_{n}$
and $B_{f}$ are nearly equal and one expects only a slow variation of
$\frac{\Gamma_{n}}{\Gamma_{f}}$ with $E_{ex}$. In a heavy ion reaction, there
is sufficient excitation energy to emit several neutrons, and fission can
compete at each stage ( if the excitation energy is greater than the fission
barrier height ), thus the fission probability and neutron evaporation
probability at stage i, are given by
$p_{{}_{f,i}}=(\frac{\Gamma_{{}_{f}}}{\Gamma_{{}_{tot}}})_{{}_{i}}$ and
$p_{{}_{n,i}}=(\frac{\Gamma_{{}_{n}}}{\Gamma_{{}_{tot}}})_{{}_{i}}=1-(\frac{\Gamma_{{}_{f}}}{\Gamma_{{}_{tot}}})_{{}_{i}}$,
respectively. As a result, the total fission probability, $P_{{}_{f}}$ is
given by
$P_{{}_{f}}=\sum_{k=1}^{\nu}\prod_{i=1}^{k}(p_{{}_{f,i}})(p_{{}_{n,i-1}}),$
(10)
where $\Gamma_{tot}=\Gamma_{f}+\Gamma_{n}$. The mean number of neutrons
emitted before fission, $\nu_{pre}$ can be derived by the following expression
$\nu_{pre}=(\frac{1}{P_{f}})\sum_{k=1}^{\nu}(k-1)\prod_{i=1}^{k}(p_{{}_{f,i}})(p_{{}_{n,i-1}}).$
(11)
As $I$ increases, the fission barrier height decreases, then $p_{{}_{f,1}}$
along the decay chain approaches unity, and steps with $k>1$ become
insignificant, and $\nu_{pre}\longrightarrow 0$; thus fission is predicted to
occur at the first step in the decay chain. It is obvious that as the
projectile energy rises, $\nu_{pre}$ will initially rises, due to more chances
for fission, but should subsequently falls as the angular momentum reaches the
value at which $P_{f}$ nears unity. It is shown that, the transient time at
the scission point, $\tau_{sci}$ by using a diffusion model for the fission
process is given by Hassani: 1984
$\tau_{sci}\simeq\tau_{sad}+\overline{\tau}=\beta^{-1}\ln(10B_{f}/T)+\overline{\tau},$
(12)
where, $\tau_{sad}$, $\overline{\tau}$ and $\beta$ are the transient time at
the saddle point, the average traveling time between the saddle and scission
points, and the nuclear friction, respectively. The time $\overline{\tau}$ is
a function of the value of the nuclear friction, of the shape of potential and
of the excitation energy. The above equation shows that $\tau_{sad}$ depends
sensibily on the nuclear friction $\beta$ and on the excitation energy of the
compound nucleus.
Earlier calculations of fission fragment anisotropies based on SSPSM have been
corrected to include the effect of pre scission neutron emission. The
calculation of fission fragment anisotropies with taking into account the
effect of pre-scission neutron emission better compares with the SSPSM
predictions with the experimental results. However, there is a small
discrepancy between model predictions and the data at high excitation
energies. A fraction of pre-scission neutrons is expected to be emitted
between saddle to scission. These latter neutrons do not longer influence the
prediction of angular anisotropy by SSPSM, since it is assumed that the SSPSM
is decided at the saddle point. In Fig. 2, the effect of pre-saddle neutrons
in the prediction of angular anisotropy by SSPSM is demonstrated for the
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system Frobrich1: 1994 . As it is shown in the figure, the discrepancy between
the experimental data of angular anisotropies and the prediction of the SSPSM
can be removed to a large extent by taking into account the pre-saddle neutron
emission correction.
Figure 2: Experimental and calculated anisotropies in
${}^{16}\textrm{O}+^{208}\textrm{Pb}\rightarrow^{224}\textrm{Th}$ reaction
system [8, 54]. Thick, dashed, dotted curves are theoretical analysis in the
framework of the SSPSM without neutron emission correction, with pre-saddle
neutrons [${\nu^{pre}_{exp.}}(\nu^{gs}_{th.}/\nu^{pre}_{th.})$] correction,
and pre-scission neutrons correction [$\nu^{pre}_{exp.}$], respectively.
We observe that for the above studied system, the ratio of the calculated pre-
saddle neutron multiplicity to experimental pre-scission neutron multiplicity,
$\nu^{gs}_{cal}/{\nu^{pre}_{exp}}\approx\frac{1}{4.1}$ at $\frac{B_{f}}{T}=1$
is in agreement with
$\frac{\tau_{gs}}{\tau_{gs}+\tau_{ss}}\approx\frac{1}{3.7}$( where,
$\tau_{gs}$ and $\tau_{ss}$ are ground-to-saddle and saddle-to-scission
transition times, respectively) Saxena1:1994 . Hence, the neutron emission
rate by the compound nucleus in the transition from the ground state to the
saddle point and then in the transition from saddle to the scission points are
approximately uniform.
The calculated multiplicities of pre-saddle neutrons as a function of $E_{ex}$
for the
${}^{11}\textrm{B}+^{197}\textrm{Au},^{209}\textrm{Bi},^{235}\textrm{U},^{237}\textrm{Np}$
reaction systems are shown in Fig. 3(a). For these studied systems, the
experimental data of angular anisotropies are taken from literature [46,
48-50, 54-56]. The values of $\nu_{gs}$ as a function of $E_{ex}$ for the
${}^{14}\textrm{N},^{16}\textrm{O}+^{197}\textrm{Au}$ and
${}^{14}\textrm{N},^{16}\textrm{O}+^{209}\textrm{Bi}$ reaction systems are
also shown in Fig. 3(b). For these systems, the experimental data of angular
anisotropies are taken from literature [46, 47]. The calculated multiplicities
of pre-saddle neutrons as a function of the excitation energy of the compound
nucleus for induced fission of the ${}^{209}\textrm{Bi}$ target by using
different projectiles are shown in Fig. 3(c).
Figure 3: Calculated pre-saddle neutron multiplicities, (a) for the
${}^{11}\textrm{B}+^{197}\textrm{Au},^{209}\textrm{Bi},^{235}\textrm{U},^{237}\textrm{Np}$
reaction systems. Thin, dashed, thick, and dashed-dotted lines represent the
general trends of the number of pre-saddle neutrons against the excitation
energy of the compound nucleus, respectively. (b) for the
${}^{14}\textrm{N},^{16}\textrm{O}+^{197}\textrm{Au}$ and
${}^{14}\textrm{N},^{16}\textrm{O}+^{209}\textrm{Bi}$ reaction systems. Thin,
thick, dashed, and dashed-dotted lines represent the general trends of the
number of pre-saddle neutrons against the excitation energy of the compound
nucleus for these systems, respectively., and (c) for the
${}^{11}\textrm{B},^{12}\textrm{C},^{14}\textrm{N},^{16}\textrm{O}+^{209}\textrm{Bi}$
reaction systems. Thick, dashed-dotted, thin, and dashed lines represent the
general trends of the number of pre-saddle neutrons against the excitation
energy of the compound nucleus for these systems, respectively.
The average values of $\nu_{gs}$, as well as ranges of pre-saddle neutron
multiplicities for the fission reactions of different targets induced by the
same projectile over the same projectile energy range are shown in Table II.
In this Table, the quantity $V_{b}$ denotes the Coulomb barrier height. It can
be observed that $\overline{\nu}_{gs}$ decreases with increasing the mass
number of the target.
reaction systems | Projectile energy (in $\frac{E_{c.m.}}{V_{b}}$) | $\nu_{gs}$ | $\overline{\nu}_{gs}$
---|---|---|---
${}^{11}\textrm{B}+^{197}\textrm{Au}$ | 1.4-1.9 | 3.1-1.4 | 2.0
${}^{11}\textrm{B}+^{209}\textrm{Bi}$ | 1.4-1.9 | 1.8-0.8 | 1.6
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{12}\textrm{C}+^{197}\textrm{Au}$ | 1.3-1.8 | 2.4-1.6 | 2.1
${}^{12}\textrm{C}+^{209}\textrm{Bi}$ | 1.3-1.8 | 1.5-0.4 | 1.0
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{14}\textrm{N}+^{197}\textrm{Au}$ | 1.2-1.7 | 3.0-0.5 | 1.9
${}^{14}\textrm{N}+^{209}\textrm{Bi}$ | 1.2-1.7 | 1.6-0.1 | 0.9
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{16}\textrm{O}+^{197}\textrm{Au}$ | 1.0-1.6 | 3.3-0.7 | 2.0
${}^{16}\textrm{O}+^{208}\textrm{Pb}$ | 1.0-1.6 | 1.9-0.1 | 1.5
${}^{16}\textrm{O}+^{209}\textrm{Bi}$ | 1.0-1.6 | 1.7-0.9 | 1.4
Table 2: Comparison between the calculated pre-saddle neutron multiplicity in
the form of a range, as well as $\overline{\nu}_{gs}$ for fission reactions of
the different targets induced by the same projectile.
The average values of $\nu_{gs}$, as well as the pre-saddle neutron
multiplicity in the form of a range for the induced fission of the same target
by different projectiles over the same projectile energy are also given in
Table III. As can be seen in Table III, the quantity $\overline{\nu}_{gs}$
decreases with increasing the mass number of projectile. All heavy ion induced
reactions show that $\nu_{gs}$ falls quite rapidly with increasing the mass
asymmetry, since it is partly due to a reduction of the dynamical fission time
scale with the mass asymmetry.
reaction systems | Projectile energy (in $\frac{E_{c.m.}}{V_{b}}$) | $\nu_{gs}$ | $\overline{\nu}_{gs}$
---|---|---|---
${}^{11}\textrm{B}+^{209}\textrm{Bi}$ | 1.2-1.7 | 2.2-1.2 | 1.9
${}^{12}\textrm{C}+^{209}\textrm{Bi}$ | 1.2-1.7 | 1.8-0.8 | 1.2
${}^{14}\textrm{N}+^{209}\textrm{Bi}$ | 1.2-1.7 | 1.6-0.2 | 0.9
${}^{16}\textrm{O}+^{209}\textrm{Bi}$ | 1.2-1.7 | 1.4-0.6 | 0.8
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{12}\textrm{C}+^{197}\textrm{Au}$ | 1.2-1.6 | 2.8-1.8 | 2.4
${}^{14}\textrm{N}+^{197}\textrm{Au}$ | 1.2-1.6 | 3.0-1.0 | 2.3
${}^{16}\textrm{O}+^{197}\textrm{Au}$ | 1.2-1.6 | 2.5-0.7 | 2.0
$------$ | $------$ | $------$ | $~{}~{}-------$
${}^{16}\textrm{O}+^{208}\textrm{Pb}$ | 1.1-1.6 | 1.9-1.0 | 1.9
${}^{19}\textrm{F}+^{208}\textrm{Pb}$ | 1.1-1.6 | 1.4-0.4 | 1.4
Table 3: Comparison between the calculated pre-saddle neutron multiplicity in
the form of a range, as well as $\overline{\nu}_{gs}$ for fission reactions of
the same target induced by different heavy ions.
We now attempt to estimate the pre-saddle neutron multiplicities in several
fission reactions induced by light projectiles. We must pay attention to some
important points expressing the difference between fission induced by light
projectiles and heavy ions. In the fission induced by light projectiles, the
energy in the center-of-mass framework, $E_{c.m.}$ is roughly the same as that
in the laboratory framework, as well as due to the low weight of projectile,
rotational energy, $E_{R}$ can be neglected. Fig. 4, shows calculated pre-
saddle neutron multiplicities for the two $\alpha+^{182}\textrm{W}$, and
$\textrm{p}+^{185}\textrm{Re}$ reaction systems which are leading to similar
${}^{186}\textrm{Os}$ compound nucleus, as well as for the two
$\textrm{p}+^{209}\textrm{Bi}$, and $\alpha+^{206}\textrm{Pb}$ that formed the
same ${}^{210}\textrm{Po}$ compound nucleus. For these systems, the
experimental data of angular anisotropies are taken from literature [57-59].
The values of $<I^{2}>$ for these systems are given by Ignatyuk1: 1984 :
$<I^{2}>=\frac{\sum(2I+1)T_{I}I(I+1)}{\sum(2I+1)T_{I}}$ (13)
where $T_{I}$ is the entrance channel transmission coefficients and satisfy
$T_{I}=1$ for $I\leq{I_{max}}$ and $T_{I}=0$ for $I>I_{max}$. If the maximum
angular momentum is determined by the relation $<I^{2}>=1/2I_{max}^{2}$, the
following relations give the values of the mean square angular momentum of the
compound nucleus for the fission of pre-actinide nuclei induced by proton and
$\alpha$ particle, respectively:
$<I^{2}>=2.08E_{p}(MeV)-15,$ (14) $<I^{2}>=10.2E_{\alpha}(MeV)-199.$ (15)
In heavy ion induced fission at low bombarding energies, several neutrons are
evaporated prior to the reaching to the saddle point, and at the highest
bombarding energy essentially all the neutrons are evaporated by the fission
fragments, i.e., the fission process is rapid compared to the time scale for
neutron evaporation. However, the number of pre-saddle neutrons, $\nu_{gs}$
increases with increasing the excitation energy of the compound nucleus in
fission induced by light projectiles. This behavior is mainly due to that in
the induced fission by light projectile, the fission barrier height is higher
than the neutron binding energy, as well as $B_{f}$ is approximately
independent of the excitation energy of the compound nucleus. Therefore, the
fission probability, $P_{f}=\frac{\Gamma_{f}}{\Gamma_{tot}}$ is negligible at
low energies. When $E_{ex}<{B_{f}}$, it is impossible that the compound
nucleus undergoes fission, but there is sufficient excitation energy to emit
several neutrons. It is clear that the fission becomes significant if
$E_{ex}>{B_{f}}$.
Figure 4: The values of $\nu_{gs}$ for the two $\alpha+^{206}\textrm{Pb}$ ,
and $\textrm{p}+^{209}\textrm{Bi}$ reaction systems which are leading to the
similar ${}^{210}\textrm{Po}$ compound nucleus, as well as for the two
$\alpha+^{182}\textrm{W}$, and $\textrm{p}+^{185}\textrm{Re}$ reaction systems
that formed the same ${}^{186}\textrm{Os}$ compound nucleus. Thick, thin,
dashed, and dashed-dotted lines represent the general trends of the pre-saddle
neutrons against the excitation energy of the compound nucleus for these
systems, respectively.
## IV Summary and Conclusions
We have presented in this paper the calculated pre-saddle neutron
multiplicities for several heavy ion induced fission reactions, as well as for
several fission reactions induced by light projectiles. The calculation by
using the experimental data of fission fragment angular anisotropies, as well
as the prediction of the SSPSM is a novel method, which has been carried out
in this work for the first time. We have also considered the behavior of pre-
saddle neutron multiplicities in fission reactions induced by heavy ions and
light projectiles. In heavy ion induced fission, the number of pre-saddle
neutrons decreases with increasing the excitation energy of the compound
nucleus. Whereas in fission induced by light particles, the number of pre-
saddle neutrons increases with increasing the excitation energy of the
compound nucleus. The fission barrier height in heavy ion fission reaction
depends on the excitation energy of the compound nucleus. On the other hand,
the fission barrier height ( and thus ground-to-saddle transition time ;
$\tau_{gs}\propto\ln(10B_{f}/T)$ ) decreases with increasing the excitation
energy of the compound nucleus. As a result, in heavy ion induced fission the
number of pre-saddle neutrons decreases with increasing the excitation energy
of the compound nucleus. Our results also shows that the emission rate of
neutrons is approximately constant in transition from the ground state to the
saddle point and then from the saddle to the scission points. On the contrary,
in fission induced by light projectiles, the fission barrier height is greater
than the neutron binding energy, and the fission barrier is approximately
independent of the excitation energy. Hence, the compound nucleus does not
undergo fission, unless the excitation energy of the compound nucleus exceeds
the fission barrier. As a result, in fission induced by light projectiles, the
number of pre-saddle neutrons exhibits an increasing function against the
excitation energy of the compound nucleus as shown our calculations. The
number of pre-saddle neutrons for reactions lead to the same compound nucleus
at any given excitation energy appears to be nearly equal, since the number of
pre-saddle neutrons depends only on the mass of the compound nucleus and it is
independent of the entrance mass asymmetry parameter. This behavior of the
number of the pre-saddle neutrons as a function of the projectile mass and/or
of the target mass may also be related to the size of compound nucleus. We
observe that the average number of pre-saddle neutrons decreases with
increasing the mass number of projectile in fission reactions of the same
target induced by different projectiles. A similar behavior in the
multiplicities of pre-saddle neutrons is also observed in fission reactions of
different targets induced by the same projectile. At the end, our results may
provide useful information on the ground state-to-saddle and saddle-to-
scission transition times.
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|
arxiv-papers
| 2013-06-03T07:15:15 |
2024-09-04T02:49:46.023805
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Soheyli, M. K. Khalili",
"submitter": "Saeid Soheyli",
"url": "https://arxiv.org/abs/1306.0310"
}
|
1306.0319
|
# On characterization of Poisson integrals of Schrödinger operators with BMO
traces
Xuan Thinh Duong, Lixin Yan and Chao Zhang Department of Mathematics,
Macquarie University, NSW 2109, Australia [email protected] Department of
Mathematics
Zhongshan University
Guangzhou 510275, PR China [email protected] School of Statistics and
Mathematics
Zhejiang Gongshang University
Hangzhou 310018, PR China [email protected]
###### Abstract.
Let $\mathcal{L}$ be a Schrödinger operator of the form
$\mathcal{L}=-\Delta+V$ acting on $L^{2}(\mathbb{R}^{n})$ where the
nonnegative potential $V$ belongs to the reverse Hölder class $B_{q}$ for some
$q\geq n.$ Let ${\rm BMO}_{{\mathcal{L}}}(\mathbb{R}^{n})$ denote the BMO
space on $\mathbb{R}^{n}$ associated to the Schrödinger operator
$\mathcal{L}$. In this article we will show that a function $f\in{\rm
BMO}_{{\mathcal{L}}}(\mathbb{R}^{n})$ is the trace of the solution of
${\mathbb{L}}u=-u_{tt}+\mathcal{L}u=0,u(x,0)=f(x),$ where $u$ satisfies a
Carleson condition
$\displaystyle\sup_{x_{B},r_{B}}r_{B}^{-n}\int_{0}^{r_{B}}\int_{B(x_{B},r_{B})}t|\nabla
u(x,t)|^{2}{dxdt}\leq C<\infty.$
Conversely, this Carleson condition characterizes all the
${\mathbb{L}}$-harmonic functions whose traces belong to the space ${\rm
BMO}_{{\mathcal{L}}}(\mathbb{R}^{n})$. This result extends the analogous
characterization founded by Fabes, Johnson and Neri in [11] for the classical
BMO space of John and Nirenberg.
###### Key words and phrases:
Poisson integrals, Schrödinger operators, BMO space, Lipschitz space, Carleson
measure, reverse Hölder inequality, Dirichlet problem.
###### 2010 Mathematics Subject Classification:
42B35, 42B37, 35J10, 47F05
###### Contents
1. 1 Introduction and statement of the main result
2. 2 Basic properties of the heat and Poisson semigroups of Schrödinger operators
3. 3 Proof of the Main Theorem
1. 3.1 Existence of boundary values of ${\mathbb{L}}$-harmonic functions
2. 3.2 The characterization of ${\rm BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ in terms of Carleson measure
4. 4 The spaces ${\rm HMO}^{\alpha}_{\mathcal{L}}({\mathbb{R}}^{n+1}_{+})$ and their characterizations
## 1\. Introduction and statement of the main result
Consider the Laplace operator $\Delta=\sum_{i=1}^{n}\partial_{x_{i}}^{2}$ on
the Euclidean space $\mathbb{R}^{n}$. A basic tool in harmonic analysis to
study a (suitable) function $f(x)$ on $\mathbb{R}^{n}$ is to consider a
harmonic function on $\mathbb{R}^{n+1}_{+}$ which has the boundary value as
$f(x)$. A standard choice for such a harmonic function is the Poisson integral
$e^{-t\sqrt{-\Delta}}f(x)$ and one recovers $f(x)$ when letting $t\rightarrow
0^{+}$. In other words, one obtains $u(x,t)=e^{-t\sqrt{-\Delta}}f(x)$ as the
solution of the equation $u_{tt}+\Delta u=0,u(x,0)=f(x)$. This approach is
intimately related to the study of singular integrals. For the classical case
$f\in L^{p}(\mathbb{R}^{n})$, $1\leq p\leq\infty$, we refer the reader to
Chapter 2 of the standard textbook [24].
At the end-point space $L^{\infty}(\mathbb{R}^{n})$, the study of singular
integrals has a natural substitution, the BMO space, i.e. the space of
functions of bounded mean oscillation. A celebrated theorem of Fefferman and
Stein [14] states that a BMO function is the trace of the solution of
$\partial_{tt}u+\Delta u=0,u(x,0)=f(x),$ whenever $u$ satisfies
(1.1)
$\displaystyle\sup_{x_{B},r_{B}}r_{B}^{-n}\int_{0}^{r_{B}}\int_{B(x_{B},r_{B})}t|\nabla
u(x,t)|^{2}{dxdt}\leq C<\infty,$
where $\nabla=(\nabla_{x},\partial_{t}).$ Expanding on this result, Fabes,
Johnson and Neri [11] showed that condition (1.1) characterizes all the
harmonic functions whose traces are in ${\rm BMO}(\mathbb{R}^{n})$. The study
of this topic has been widely extended to more general operators such as
elliptic operators (instead of the Laplacian) and for domains other than
$\mathbb{R}^{n}$ such as Lipschitz domains. See for examples [4, 12, 13, 18].
The main aim of this article is to study a similar characterization to (1.1)
for the Schrödinger operators with appropriate conditions on its potentials.
Let us consider the Schrödinger operator
(1.2) $\mathcal{L}=-\Delta+V(x)\ \ \ {\rm on}\ \ L^{2}(\mathbb{R}^{n}),\ \ \ \
n\geq 3.$
As to the nonnegative potential $V$, we assume that it is not identically zero
and that $V\in B_{q}$ for some $q\geq n/2$, which by definition means that
$V\in L^{q}_{\rm loc}(\mathbb{R}^{n}),V\geq 0$, and there exists a constant
$C>0$ such that the reverse Hölder inequality
(1.3)
$\left(\frac{1}{\left|B\right|}\int_{B}V(y)^{q}~{}dy\right)^{1/q}\leq\frac{C}{\left|B\right|}\int_{B}V(y)~{}dy,$
holds for all balls $B$ in $\mathbb{R}^{n}.$
The operator $\mathcal{L}$ is a self-adjoint operator on
$L^{2}(\mathbb{R}^{n})$. Hence $\mathcal{L}$ generates the
$\mathcal{L}$-Poisson semigroup $\\{e^{-t\sqrt{\mathcal{L}}}\\}_{t>0}$ on
$L^{2}(\mathbb{R}^{n})$. Since the potential $V$ is nonnegative, the semigroup
kernels ${\mathcal{P}}_{t}(x,y)$ of the operators $e^{-t\sqrt{\mathcal{L}}}$
satisfy
$\displaystyle 0\leq{\mathcal{P}}_{t}(x,y)\leq p_{t}(x-y)$
for all $x,y\in\mathbb{R}^{n}$ and $t>0$, where
$p_{t}(x-y)=c_{n}{t\over(t^{2}+|x|^{2})^{{n+1\over 2}}},\ \ \ \
c_{n}={\Gamma\Big{(}{n+1\over 2}\Big{)}\over\pi^{(n+1)/2}}$
is the kernel of the classical Poisson semigroup
$\left\\{{P}_{t}\right\\}_{t>0}=\\{e^{-t\sqrt{-\Delta}}\\}_{t>0}$ on
$\mathbb{R}^{n}$. For $f\in L^{p}(\mathbb{R}^{n})$, $1\leq p<\infty,$ it is
well known that the Poisson extension
$u(x,t)=e^{-t\sqrt{\mathcal{L}}}f(x),t>0,x\in\mathbb{R}^{n}$, is a solution to
the equation
(1.4) $\displaystyle{\mathbb{L}}u=-u_{tt}+{\mathcal{L}}u=0\ \ \ {\rm in}\
{\mathbb{R}}^{n+1}_{+}$
with the boundary data $f$ on $\mathbb{R}^{n}$ (see Remark 3.3 for $p=\infty$
below). The equation ${\mathbb{L}}u=0$ is understood in the weak sense, that
is, $u\in{W}^{1,2}_{{\rm loc}}({\mathbb{R}}^{n+1}_{+})$ is a weak solution of
${\mathbb{L}}u=0$ if it satisfies
$\int_{{\mathbb{R}}^{n+1}_{+}}{\nabla}u\cdot{\nabla}\psi\,dY+\int_{{\mathbb{R}}^{n+1}_{+}}Vu\psi\,dY=0,\
\ \ \ \forall\psi\in C_{0}^{1}({\mathbb{R}}^{n+1}_{+}).$
In the sequel, we call such a function $u$ an ${\mathbb{L}}$-harmonic function
associated to the operator ${\mathbb{L}}$.
As mentioned above, we are interested in deriving the characterization of the
solution to the equation ${\mathbb{L}}u=0$ in ${\mathbb{R}}^{n+1}_{+}$ having
boundary values with BMO data. Following [10], a locally integrable function
$f$ belongs to BMO${}_{{\mathcal{L}}}({\mathbb{R}}^{n})$ whenever there is
constant $C\geq 0$ so that
(1.5) ${1\over|B|}\int_{B}|f(y)-f_{B}|dy\leq C$
for every ball $B=B(x,r)$, and
(1.6) ${1\over|B|}\int_{B}|f(y)|dy\leq C$
for every ball $B=B(x,r)$ with $r\geq\rho(x)$. Here
$f_{B}=|B|^{-1}\int_{B}f(x)dx$ and the critical radii above are determined by
the function $\rho(x;V)=\rho(x)$ which takes the explicit form
(1.7) $\rho(x)=\sup\Big{\\{}r>0:\ {1\over r^{n-2}}\int_{B(x,r)}V(y)dy\leq
1\Big{\\}}.$
We define $\|f\|_{{\rm BMO}_{\mathcal{L}}(\mathbb{R}^{n})}$ to be the smallest
$C$ in the right hand sides of (1.5) and (1.6). Because of (1.6), this ${\rm
BMO}_{{\mathcal{L}}}(\mathbb{R}^{n})$ space is in fact a proper subspace of
the classical BMO space of John and Nirenberg, and it turns out to be a
suitable space in studying the case of the end-point estimates for $p=\infty$
concerning the boundedness of some classical operators associated to
$\mathcal{L}$ such as the Littlewood-Paley square functions, fractional
integrals and Riesz transforms (see [1, 2, 5, 6, 10, 17, 19]).
The following theorem is the main result of this article.
###### Theorem 1.1.
Suppose $V\in B_{q}$ for some $q\geq n.$ We denote by ${\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$ the class of all $C^{1}$-functions
$u(x,t)$ of the solution of ${\mathbb{L}}u=0$ in $\mathbb{R}_{+}^{n+1}$ such
that
(1.8) $\displaystyle\|u\|^{2}_{{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}$
$\displaystyle=$
$\displaystyle\sup_{x_{B},r_{B}}r_{B}^{-n}\int_{0}^{r_{B}}\int_{B(x_{B},r_{B})}t|\nabla
u(x,t)|^{2}{dxdt}<\infty,$
where $\nabla=(\nabla_{x},\partial_{t}).$ Then we have
* (1)
If $u\in{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$, then there exists some
$f\in{\rm BMO_{\mathcal{L}}}(\mathbb{R}^{n})$ such that
$u=e^{-t\sqrt{\mathcal{L}}}f$, and
$\|f\|_{{\rm BMO_{\mathcal{L}}}(\mathbb{R}^{n})}\leq C\|u\|_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}$
with some constant $C>0$ independent of $u$ and $f$.
* (2)
If $f\in{\rm BMO_{\mathcal{L}}}(\mathbb{R}^{n})$, then the function
$u=e^{-t\sqrt{\mathcal{L}}}f$ satisfies estimates (1.8) with $\|u\|_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}\approx\|f\|_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}.$
We should mention that for the Schrödinger operator $\mathcal{L}$ in (1.2), an
important property of the $B_{q}$ class, proved in [15, Lemma 3], assures that
the condition $V\in B_{q}$ also implies $V\in B_{q+\epsilon}$ for some
$\epsilon>0$ and that the $B_{q+\epsilon}$ constant of $V$ is controlled in
terms of the one of $B_{q}$ membership. This in particular implies $V\in
L^{q}_{\rm loc}(\mathbb{R}^{n})$ for some $q$ strictly greater than $n/2.$
However, in general the potential $V$ can be unbounded and does not belong to
$L^{p}(\mathbb{R}^{n})$ for any $1\leq p\leq\infty.$ As a model example, we
could take $V(x)=|x|^{2}$. Moreover, as noted in [21], if $V$ is any
nonnegative polynomial, then $V$ satisfies the stronger condition
$\displaystyle\max_{x\in B}V(x)\leq\frac{C}{\left|B\right|}\int_{B}V(y)~{}dy,$
which implies $V\in B_{q}$ for every $q\in(1,\infty)$ with a uniform constant.
This article is organized as follows. In Section 2, we recall some preliminary
results including the kernel estimates of the heat and Poisson semigroups of
$\mathcal{L}$, the $H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$ and ${\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ spaces associated to the Schrödinger
operators and certain properties of ${\mathbb{L}}$-harmonic functions. In
Section 3, we will prove our main result, Theorem 1.1. The proof of part (1)
follows a similar method to that of [11], which depends heavily on three non-
trivial results: Alaoglu’s Theorem on the weak-$\ast$ compactness of the unit
sphere in the dual of a Banach space, the duality theorem asserting that the
space ${\rm BMO}_{{\mathcal{L}}}(\mathbb{R}^{n})$ is the dual space of
$H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$, and some specific properties of the
Hardy space $H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$ and the Carleson measure.
For part (2), we will prove it by making use of the full gradient estimates on
the kernel of the Poisson semigroup in the $(x,t)$ variables under the
assumption on $V\in B_{q}$ for some $q\geq n$. This improves previously known
results (see [6, 10, 17, 19]) which characterize the space ${\rm
BMO}_{{\mathcal{L}}}(\mathbb{R}^{n})$ in terms of Carleson measure which are
only related to the gradient in the $t$ variable. In Section 4, we will extend
the method for the space ${\rm BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ in Section
3 to obtain some generalizations to Lipschitz-type spaces
${\Lambda}_{\mathcal{L}}^{\alpha}(\mathbb{R}^{n})$ for $\alpha\in(0,1)$.
Throughout the article, the letters “$c$ ” and “$C$ ” will denote (possibly
different) constants that are independent of the essential variables.
## 2\. Basic properties of the heat and Poisson semigroups of Schrödinger
operators
In this section, we begin by recalling some basic properties of the critical
radii function $\rho(x)$ under the assumption (1.3) on $V$ (see Section 2,
[10]).
###### Lemma 2.1.
Suppose $V\in B_{q}$ for some $q>n/2.$ There exist $C>0$ and $k_{0}\geq 1$
such that for all $x,y\in\mathbb{R}^{n}$
(2.1)
$C^{-1}\rho(x)\left(1+\frac{\left|x-y\right|}{\rho(x)}\right)^{-k_{0}}\leq\rho(y)\leq
C\rho(x)\left(1+\frac{\left|x-y\right|}{\rho(x)}\right)^{\frac{k_{0}}{k_{0}+1}}.$
In particular, $\rho(x)\sim\rho(y)$ when $y\in B(x,r)$ and $r\leq c\rho(x)$.
It follows from Lemmas 1.2 and 1.8 in [21] that there is a constant $C_{0}$
such that for a nonnegative Schwartz class function $\varphi$ there exists a
constant $C$ such that
(2.2)
$\int_{\mathbb{R}^{n}}\varphi_{t}(x-y)V(y)dy\leq\left\\{\begin{array}[]{lll}Ct^{-1}\big{(}{\sqrt{t}\over\rho(x)}\big{)}^{\delta}&{\rm
for}\ t\leq\rho(x)^{2},\\\\[6.0pt]
C\big{(}{\sqrt{t}\over\rho(x)}\big{)}^{C_{0}+2-n}&{\rm for}\
t>\rho(x)^{2},\end{array}\right.$
where $\varphi_{t}(x)=t^{-n/2}\varphi(x/\sqrt{t}),$ and
$\delta=2-\frac{n}{q}>0.$
Let $\\{e^{-t\mathcal{L}}\\}_{t>0}$ be the heat semigroup associated to
$\mathcal{L}$:
(2.3) $\displaystyle
e^{-t\mathcal{L}}f(x)=\int_{\mathbb{R}^{n}}{\mathcal{K}}_{t}(x,y)f(y)~{}dy,\qquad
f\in L^{2}(\mathbb{R}^{n}),~{}x\in\mathbb{R}^{n},~{}t>0.$
From the Feynman-Kac formula, it is well-known that the kernel
${\mathcal{K}}_{t}(x,y)$ of the semigroup $e^{-t\mathcal{L}}$ satisfies the
estimate
(2.4) $\displaystyle 0\leq{\mathcal{K}}_{t}(x,y)\leq h_{t}(x-y)$
for all $x,y\in\mathbb{R}^{n}$ and $t>0$, where
(2.5) $h_{t}(x-y)=\frac{1}{(4\pi
t)^{n/2}}~{}e^{-\frac{\left|x-y\right|^{2}}{4t}}$
is the kernel of the classical heat semigroup
$\left\\{T_{t}\right\\}_{t>0}=\\{e^{t\Delta}\\}_{t>0}$ on $\mathbb{R}^{n}$.
This estimate can be improved in time when $V\not\equiv 0$ satisfies the
reverse Hölder condition $B_{q}$ for some $q>n/2$. The function $\rho(x)$
arises naturally in this context.
###### Lemma 2.2 (see [10]).
Suppose $V\in B_{q}$ for some $q>n/2.$ For every $N>0$ there exist the
constants $C_{N}$ and $c$ such that for $x,y\in\mathbb{R}^{n},t>0$,
* (i)
$0\leq{\mathcal{K}}_{t}(x,y)\leq
C_{N}t^{-n/2}e^{-\frac{\left|x-y\right|^{2}}{ct}}\left(1+\frac{\sqrt{t}}{\rho(x)}+\frac{\sqrt{t}}{\rho(y)}\right)^{-N}\
{\rm and}$
* (ii)
$\left|\partial_{t}{\mathcal{K}}_{t}(x,y)\right|\leq
C_{N}t^{-\frac{n+2}{2}}e^{-\frac{\left|x-y\right|^{2}}{ct}}\left(1+\frac{\sqrt{t}}{\rho(x)}+\frac{\sqrt{t}}{\rho(y)}\right)^{-N}.$
Kato-Trotter formula (see for instance [9]) asserts that
$\displaystyle
h_{t}(x-y)-{\mathcal{K}}_{t}(x,y)=\int_{0}^{t}\int_{\mathbb{R}^{n}}h_{s}(x-z)V(z){\mathcal{K}}_{t-s}(z,y)dzds.$
Then we have the following result.
###### Lemma 2.3 (see [9]).
Suppose $V\in B_{q}$ for some $q>n/2.$ There exists a nonnegative Schwartz
function $\varphi$ on $\mathbb{R}^{n}$ such that
$\left|h_{t}(x-y)-{\mathcal{K}}_{t}(x,y)\right|\leq\left(\frac{\sqrt{t}}{\rho(x)}\right)^{\delta}\varphi_{t}(x-y),\quad
x,y\in\mathbb{R}^{n},~{}t>0,$
where $\varphi_{t}(x)=t^{-n/2}\varphi\left(x/\sqrt{t}\right)$ and
$\delta=2-\frac{n}{q}>0.$
The Poisson semigroup associated to $\mathcal{L}$ can be obtained from the
heat semigroup (2.2) through Bochner’s subordination formula (see [23]):
$\displaystyle e^{-t\sqrt{\mathcal{L}}}f(x)$
$\displaystyle=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-u}}{\sqrt{u}}~{}e^{-{t^{2}\over
4u}\mathcal{L}}f(x)~{}du$ (2.6)
$\displaystyle=\frac{t}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{~{}e^{-{t^{2}\over
4s}}}{s^{3/2}}e^{-s\mathcal{L}}f(x)~{}ds.$
From (2), the semigroup kernels ${\mathcal{P}}_{t}(x,y)$, associated to
$e^{-t\sqrt{\mathcal{L}}}$, satisfy the following estimates. For its proof, we
refer to [19, Proposition 3.6].
###### Lemma 2.4.
Suppose $V\in B_{q}$ for some $q>n/2.$ For any
$0<\beta<\min\\{1,2-\frac{n}{q}\\}$ and every $N>0$, there exists a constant
$C=C_{N}$ such that
* (i)
${\displaystyle|{\mathcal{P}}_{t}(x,y)|\leq
C{t\over(t^{2}+|x-y|^{2})^{n+1\over
2}}\left(1+{(t^{2}+|x-y|^{2})^{1/2}\over\rho(x)}+{(t^{2}+|x-y|^{2})^{1/2}\over\rho(y)}\right)^{-N};}$
* (ii)
For every $m\in{\mathbb{N}}$,
$\displaystyle|t^{m}\partial^{m}_{t}{\mathcal{P}}_{t}(x,y)|\leq
C{t^{m}\over(t^{2}+|x-y|^{2})^{n+m\over
2}}\left(1+{(t^{2}+|x-y|^{2})^{1/2}\over\rho(x)}+{(t^{2}+|x-y|^{2})^{1/2}\over\rho(y)}\right)^{-N};$
* (iii)
${\displaystyle\big{|}t\partial_{t}e^{-t\sqrt{\mathcal{L}}}(1)(x)\big{|}\leq
C\left({t\over\rho(x)}\right)^{\beta}\left(1+{t\over\rho(x)}\right)^{-N}.}$
Recall that a Hardy-type space associated to $\mathcal{L}$ was introduced by
J. Dziubański et al. in [8, 9, 10], defined by
(2.7) $H^{1}_{\mathcal{L}}(\mathbb{R}^{n})=\big{\\{}f\in
L^{1}(\mathbb{R}^{n}):{\mathcal{P}}^{\ast}f(x)=\sup_{t>0}|e^{-t\sqrt{\mathcal{L}}}f(x)|\in
L^{1}(\mathbb{R}^{n})\big{\\}}$
with
$\|f\|_{H^{1}_{\mathcal{L}}(\mathbb{R}^{n})}=\|{\mathcal{P}}^{\ast}f\|_{L^{1}(\mathbb{R}^{n})}.$
For the above class of potentials, $H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$
admits an atomic characterization, where cancellation conditions are only
required for atoms with small supports. It can be verified that for every
$m\in{\mathbb{N}}$, for fixed $t>0$ and $x\in\mathbb{R}^{n},$
$\partial_{t}^{m}{\mathcal{P}}_{t}(x,\cdot)\in
H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$ with
(2.8)
$\|\partial_{t}^{m}{\mathcal{P}}_{t}(x,\cdot)\|_{H^{1}_{\mathcal{L}}(\mathbb{R}^{n})}\leq
Ct^{-m}.$
Indeed, from (ii) of Lemma 2.4 we have that for a fixed $y\in\mathbb{R}^{n},$
$\displaystyle\sup_{s>0}|e^{-s\sqrt{\mathcal{L}}}\big{(}t^{m}\partial_{t}^{m}{\mathcal{P}}_{t}(\cdot,y)\big{)}(x)|$
$\displaystyle\leq$ $\displaystyle
C\sup_{s>0}\left({t^{m}\over(t+s)^{m}}{(t+s)^{m}\over(t+s+|x-y|)^{n+m}}\right)$
$\displaystyle\leq$ $\displaystyle C{t\over(t+|x-y|)^{n+1}}\in
L^{1}(\mathbb{R}^{n},dx),$
which, in combination with the fact that
$\partial_{t}^{m}{\mathcal{P}}_{t}(x,\cdot)=\partial_{t}^{m}{\mathcal{P}}_{t}(\cdot,x)$,
shows estimate (2.8).
###### Lemma 2.5.
Suppose $V\in B_{q}$ for some $q>n/2.$ Then the dual space of
$H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$ is ${{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})}$, i.e.,
$(H^{1}_{\mathcal{L}}(\mathbb{R}^{n}))^{\ast}={{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})}.$
###### Proof.
For the proof, we refer to [10, Theorem 4]. See also [6, 17]. ∎
In the sequel, we may sometimes use capital letters to denote points in
${\mathbb{R}}^{n+1}_{+}$, e.g., $Y=(y,t),$ and set
$\displaystyle{\nabla}_{Y}u(Y)=(\nabla_{y}u,\partial_{t}u)\ \ \ {\rm and}\ \ \
|{\nabla}_{Y}u|^{2}=|\nabla_{y}u|^{2}+|\partial_{t}u|^{2}.$
For simplicity we will denote by $\nabla$ the full gradient $\nabla_{Y}$ in
$\mathbb{R}^{n+1}$. We now recall a Moser type local boundedness estimate (see
for instance, [17]) and include a proof here for the sake of self-containment.
###### Lemma 2.6.
Suppose $0\leq V\in L^{q}_{\rm loc}(\mathbb{R}^{n})$ for some $q>n/2.$ Let $u$
be a weak solution of ${\mathbb{L}}u=0$ in the ball
$B(Y_{0},2r)\subset\mathbb{R}^{n+1}$. Then for any $p\geq 1$, there exists a
constant $C=C(n,p)>0$ such that
$\displaystyle\sup_{B(Y_{0},r)}|u(Y)|\leq C\Big{(}{1\over
r^{n+1}}\int_{B(Y_{0},2r)}|u(Y)|^{p}dY\Big{)}^{1/p}.$
###### Proof.
It is enough to show that $u^{2}$ is a subharmonic function. Since for any
$\varphi\in C_{0}^{1}(B(Y_{0},2r))$ with $\varphi\geq 0$, we have that $\int
V\varphi u^{2}dY\leq C_{r}\|u\|_{W^{1,2}(B(Y_{0},2r))}<\infty$ (see for
instance, [16, Lemma 3.3]). This gives
$\displaystyle\int$ $\displaystyle{\nabla}u^{2}\cdot{\nabla}\varphi\,dY$
$\displaystyle=$ $\displaystyle
2\int{\nabla}u\cdot{\nabla}(u\varphi)\,dY-2\int\varphi|{\nabla}u|^{2}\,dY$
$\displaystyle=$ $\displaystyle-2\int V\varphi
u^{2}\,dY-2\int\varphi|{\nabla}u|^{2}\,dY$ $\displaystyle\leq$ $\displaystyle
0.$
The desired result follows readily. ∎
###### Lemma 2.7.
Suppose $V\in B_{q}$ for some $q\geq n/2.$ Assume that $u\in W_{\rm
loc}^{1,2}(\mathbb{R}^{n})$ is a weak solution of $(-\Delta+V)u=0$ in
$\mathbb{R}^{n}$. Also assume that there is a $d>0$ such that
(2.9) $\displaystyle\int_{\mathbb{R}^{n}}{|u(x)|\over 1+|x|^{n+d}}dx\leq
C_{d}<\infty.$
Then $u=0$ in $\mathbb{R}^{n}$.
###### Proof.
Fix an $R\geq 10$, we let $\varphi\in C_{0}^{\infty}(B(0,3R/4))$ such that
$0\leq\varphi\leq 1,\varphi=1$ on $B(0,5R/8)$, and $|\nabla\varphi|\leq
C/R,|\nabla^{2}\varphi|\leq C/R^{2}.$ Following [22, Lemma 6.1], one writes
$(-\Delta+V)(u\varphi)=-2\nabla u\cdot\nabla\varphi-u\Delta\varphi\ \ \ {\rm
in}\ \mathbb{R}^{n}.$
Then we have
$u(x)\varphi(x)=\int_{\mathbb{R}^{n}}\Gamma_{V}(x,y)\\{-2\nabla
u\cdot\nabla\varphi-u\Delta\varphi\\}dy,$
where $\Gamma_{V}(x,y)$ denotes the fundamental solution of $-\Delta+V$ in
$\mathbb{R}^{n}$. Hence, for any $x\in B(0,R/2)$,
(2.10) $\displaystyle|u(x)|$ $\displaystyle\leq$ $\displaystyle{C\over
R}\int_{5R/8\leq|y|\leq 3R/4}|\Gamma_{V}(x,y)|\left(|\nabla u(y)|+{|u(y)|\over
R}\right)dy$ $\displaystyle\leq$ $\displaystyle{C\over
R^{2}}\left\\{\int_{5R/8\leq|y|\leq
3R/4}|\Gamma_{V}(x,y)|^{2}dy\right\\}^{1/2}\left\\{\int_{B(0,R)}|u(y)|^{2}dy\right\\}^{1/2},$
where we have used the Hölder inequality and Caccioppoli’s inequality.
From the upper bound of $\Gamma_{V}(x,y)$ in [21, Theorem 2.7], we have that
for every $k>d$ and every $x\in B(0,R/2)$,
(2.11) $\displaystyle\int_{5R/8\leq|y|\leq 3R/4}|\Gamma_{V}(x,y)|^{2}dy$
$\displaystyle\leq$ $\displaystyle C_{k}\int_{5R/8\leq|y|\leq
3R/4}\bigg{|}{1\over\left(1+{|x-y|\over\rho(x)}\right)^{k}}{1\over|x-y|^{n-2}}\bigg{|}^{2}dy$
$\displaystyle\leq$ $\displaystyle C_{k}\rho(x)^{2k}R^{-2k-n+4}.$
Recall that the condition $B_{n/2}$ implies $V\in B_{q_{0}}$ for some
$q_{0}>n/2$. It follows from Lemma 2.9 of [22] that
$|u(y)|\leq|B|^{-1}\int_{B}|u(z)|dz$ if $B=B(y,R)\subset\mathbb{R}^{n}$ (see
also Lemma 2.6). Then we have
$\displaystyle\int_{B(0,R)}|u(y)|^{2}dy$ $\displaystyle\leq$ $\displaystyle
CR^{n}\left({C\over R^{n}}\int_{B(0,2R)}|u(y)|dy\right)^{2}\leq
C{R^{n+2d}}\left(\int_{\mathbb{R}^{n}}{|u(y)|\over
1+|y|^{n+d}}dy\right)^{2}\leq CC_{d}^{2}{R^{n+2d}}.$
This, in combination with (2.10) and (2.11), yields that for every $x\in
B(0,R/2)$,
$\displaystyle|u(x)|$ $\displaystyle\leq$ $\displaystyle
C^{\prime}_{k}C_{d}\,\rho(x)^{k}R^{d-k}.$
Letting $R\to+\infty$, we obtain that $u(x)=0$ and therefore, $u=0$ in the
whole $\mathbb{R}^{n}.$ The proof is complete. ∎
###### Remark 2.8.
For any $d\geq 0$, one writes
$\displaystyle{{\mathcal{H}}_{d}(\mathcal{L})}=\Big{\\{}f\in W^{1,2}_{\rm
loc}({\mathbb{R}}^{n}):\mathcal{L}f=0\ {\rm and}\ \ |f(x)|=O(|x|^{d})\ \ {\rm
as}\ |x|\rightarrow\infty\Big{\\}}$
and
$\displaystyle{{\mathcal{H}}_{\mathcal{L}}}=\bigcup_{d:\ 0\leq
d<\infty}{{\mathcal{H}}_{d}(\mathcal{L})}.$
By Lemma 2.7, it follows that for any $d\geq 0$,
${{\mathcal{H}}_{\mathcal{L}}}={{\mathcal{H}}_{d}(\mathcal{L})}=\big{\\{}0\big{\\}}.$
See also Proposition 6.5 of [6].
At the end of this section, we establish the following characterization of
Poisson integrals of Schrödinger operators with functions in
$L^{p}(\mathbb{R}^{n}),1\leq p<\infty.$
###### Proposition 2.9.
Suppose $V\in B_{q}$ for some $q\geq(n+1)/2.$ If $u$ is a continuous weak
solution of ${\mathbb{L}}u=0$ in ${\mathbb{R}}^{n+1}_{+}$ and there exist a
constant $C>0$ and a $1\leq p<\infty$, such that
$\|u(\cdot,t)\|_{L^{p}(\mathbb{R}^{n})}=\left(\int_{\mathbb{R}^{n}}|u(x,t)|^{p}dx\right)^{1/p}\leq
C<\infty$
for all $t>0$, then
* (1)
when $1<p<\infty$, $u(x,t)$ is the Poisson integral of a function $f$ in
$L^{p}(\mathbb{R}^{n});$
* (2)
if $p=1$, $u(x,t)$ is the Poisson-Stieltjes integral of a finite Borel
measure; if, in addition, $u(\cdot,t)$ is Cauchy in the $L^{1}$ norm as $t>0$
then $u(x,t)$ is the Poisson integral of a function $f$ in
$L^{1}(\mathbb{R}^{n}).$
###### Proof.
The proof of Proposition 2.9 is standard (see for instance, Theorem 2.5,
Chapter 2 in [24]). We give a brief argument of this proof for completeness
and the convenience of the reader.
If $1<p<\infty$ and $\|u(\cdot,t)\|_{L^{p}(\mathbb{R}^{n})}\leq C<\infty$ for
all $t>0$, then there exists a sequence $\\{t_{k}\\}$ such that
$\lim\limits_{k\rightarrow\infty}t_{k}=0$, and a function $f\in
L^{p}(\mathbb{R}^{n})$ such that $u(\cdot,t_{k})$ converges weakly to $f$ as
$k\rightarrow\infty.$ That is, for each $g\in
L^{p^{\prime}}(\mathbb{R}^{n}),1/p+1/p^{\prime}=1,$
$\lim\limits_{k\rightarrow\infty}\int_{\mathbb{R}^{n}}u(y,t_{k})g(y)dy=\int_{\mathbb{R}^{n}}f(y)g(y)dy.$
If $p=1$ there exists a finite Borel measure $\mu$ that is the weak-$\ast$
limit of a sequence $\\{u(\cdot,t_{k})\\}$. That is, for each $g$ in
$C_{0}(\mathbb{R}^{n})$,
$\lim\limits_{k\rightarrow\infty}\int_{\mathbb{R}^{n}}u(y,t_{k})g(y)dy=\int_{\mathbb{R}^{n}}g(y)d\mu(y).$
For any $x\in\mathbb{R}^{n},t>0$, we take $g(y)=p_{t}(x,y)\in
L^{p^{\prime}}(\mathbb{R}^{n})$ for $1\leq p^{\prime}\leq\infty$ and also
belongs to $C_{0}(\mathbb{R}^{n})$ we have, in particular,
$\lim\limits_{k\rightarrow\infty}\int_{\mathbb{R}^{n}}p_{t}(x,y)u(y,t_{k})dy=\int_{\mathbb{R}^{n}}p_{t}(x,y)f(y)dy$
when $1<p<\infty$, and
$\lim\limits_{k\rightarrow\infty}\int_{\mathbb{R}^{n}}p_{t}(x,y)u(y,t_{k})dy=\int_{\mathbb{R}^{n}}p_{t}(x,y)d\mu(y)$
when $p=1.$
Since $u$ is continuous, $\lim\limits_{t\rightarrow
0^{+}}u(x,t+t_{k})=u(x,t_{k})$. It is well known that if $u(\cdot,t_{k})\in
L^{p}(\mathbb{R}^{n}),1\leq p<\infty$, then $\lim\limits_{t\rightarrow
0^{+}}e^{-t\sqrt{\mathcal{L}}}(u(\cdot,t_{k}))=u(x,t_{k})$ for almost every
$x\in\mathbb{R}^{n}.$ Set
$w(x,t)=e^{-t\sqrt{\mathcal{L}}}(u(\cdot,t_{k}))(x)-u(x,t+t_{k})$. The
function $w$ satisfies ${\mathbb{L}}w=0,w(x,0)=0$. Define,
$\displaystyle{\overline{w}}(x,t)=\left\\{\begin{array}[]{rrl}w(x,t),&&t\geq
0,\\\\[6.0pt] -w(x,-t),&&t<0.\end{array}\right.$
Then ${\overline{w}}$ satisfies
${\overline{\mathbb{L}}}{\overline{w}}(x,t)=0,\ \ \ \ \ \
(x,t)\in{\mathbb{R}}^{n+1},$
where ${\overline{\mathbb{L}}}$ is an extension operator of ${{\mathbb{L}}}$
on $\mathbb{R}^{n+1}$. Observe that if $V(x)\in B_{q}(\mathbb{R}^{n})$ with
$q\geq(n+1)/2,$ then it can be verified that $V(x,t)=V(x)\in B_{q}$ on
$\mathbb{R}^{n+1}$. To apply Lemma 2.7, we need to verify (2.9). Indeed,
$\displaystyle\int_{{\mathbb{R}}^{n+1}}{|{\overline{w}}(x,t)|\over
1+|(x,t)|^{n+3}}dxdt$ $\displaystyle\leq$ $\displaystyle
C\int_{0}^{\infty}{1\over 1+t^{2}}\left(\int_{\mathbb{R}^{n}}{|w(x,t)|\over
1+|x|^{n+{1}}}dx\right)dt$ $\displaystyle\leq$ $\displaystyle
C\int_{0}^{\infty}{1\over 1+t^{2}}\left(\sup_{t>0}\|w(\cdot,t)\|_{p}\right)dt$
$\displaystyle\leq$ $\displaystyle
C\sup_{t>0}\left(\int_{\mathbb{R}^{n}}|u(x,t)|^{p}dx\right)^{1/p}<\infty,$
and so (2.9) holds. By Lemma 2.7, we have that $\overline{w}\equiv 0$, and
then $w=0$, that is,
$u(x,t+t_{k})=e^{-t\sqrt{\mathcal{L}}}(u(\cdot,t_{k}))(x),\ \ \
x\in\mathbb{R}^{n},\ t>0.$
Therefore,
$u(x,t)=\lim\limits_{k\rightarrow\infty}u(x,t+t_{k})=\lim\limits_{k\rightarrow\infty}e^{-t\sqrt{\mathcal{L}}}(u(\cdot,t_{k}))(x)=e^{-t\sqrt{\mathcal{L}}}f(x)$
when $1<p<\infty$, and $u(x,t)=\int_{\mathbb{R}^{n}}p_{t}(x,y)d\mu(y)$ when
$p=1.$ The proof is complete. ∎
## 3\. Proof of the Main Theorem
### 3.1. Existence of boundary values of ${\mathbb{L}}$-harmonic functions
###### Lemma 3.1.
For every $u\in{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$ and for every
$k\in{\mathbb{N}}$, there exists a constant $C_{k}>0$ such that
$\int_{\mathbb{R}^{n}}{|u(x,{1/k})|^{2}\over(1+|x|)^{2n}}dx\leq C_{k}<\infty,$
hence $u(x,1/k)\in L^{2}((1+|x|)^{-2n}dx)$. Therefore for all
$k\in{\mathbb{N}}$, $e^{-t\sqrt{\mathcal{L}}}(u(\cdot,{1/k}))(x)$ exists
everywhere in ${\mathbb{R}}^{n+1}_{+}$.
###### Proof.
Since $u\in C^{1}({\mathbb{R}}^{n+1}_{+})$, it reduces to show that for every
$k\in{\mathbb{N}},$
(3.1) $\displaystyle\int_{|x|\geq
1}{|u(x,{1/k})-u(x/|x|,1/k)|^{2}\over(1+|x|)^{2n}}dx\leq C_{k}\|u\|^{2}_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}<\infty.$
To do this, we write
$\displaystyle u(x,1/k)-u(x/|x|,1/k)$
$\displaystyle=\big{[}u(x,1/k)-u(x,|x|)\big{]}+\big{[}u(x,|x|)-u(x/|x|,|x|)\big{]}+\big{[}u(x/|x|,|x|)-u(x/|x|,1/k)\big{]}.$
Let
$\displaystyle I=\int_{|x|\geq
1}{|u(x,1/k)-u(x,|x|)|^{2}\over(1+|x|)^{2n}}dx,$ $\displaystyle
II=\int_{|x|\geq 1}{|u(x,|x|)-u(x/|x|,|x|)|^{2}\over(1+|x|)^{2n}}dx,$
and
$\displaystyle III=\int_{|x|\geq
1}{|u(x/|x|,|x|)-u(x/|x|,1/k)|^{2}\over(1+|x|)^{2n}}dx.$
Set $Y_{0}=(x,t)$ and $r=t/4$. We use Lemma 2.6 for $\partial_{t}u$ and
Schwarz’s inequality to obtain
(3.2) $\displaystyle\big{|}\partial_{t}u(x,t)\big{|}$ $\displaystyle\leq$
$\displaystyle C\Big{(}{1\over
r^{n+1}}\int_{B(Y_{0},2r)}|\partial_{s}u(y,s)|^{2}{dY}\Big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle C\Big{(}{1\over
t^{n+1}}\int_{B(x,t/2)}\int_{t/2}^{3t/2}|\partial_{s}u(y,s)|^{2}{dsdy}\Big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle
Ct^{-1}\Big{(}{1\over|B(x,2t)|}\int_{0}^{2t}\int_{B(x,2t)}s|\partial_{s}u(y,t)|^{2}{dyds}\Big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle Ct^{-1}\|u\|_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})},$
which gives
(3.3)
$\displaystyle|u(x,|x|)-u(x,1/k)|=\Big{|}\int_{1/k}^{|x|}\partial_{t}u(x,t)dt\
\Big{|}\leq C\log(k|x|)\|u\|_{{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}.$
It follows that
$\displaystyle I+III$ $\displaystyle\leq$ $\displaystyle C\|u\|^{2}_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}\int_{|x|\geq
1}{1\over(1+|x|)^{2n}}\log^{2}(k|x|)dx$ $\displaystyle\leq$ $\displaystyle
C_{k}\|u\|^{2}_{{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}.$
For the term $II,$ we have that for any $x\in\mathbb{R}^{n},$
$u(x,|x|)-u(x/|x|,|x|)=\int_{1}^{|x|}D_{r}u(r\omega,|x|)dr,\ \ \ \
x=|x|\omega.$
Let $B=B(0,1)$ and $2^{m}B=B(0,2^{m})$. Note that for every
$m\in{\mathbb{N}}$, we have
$\displaystyle\int_{2^{m}B\backslash
2^{m-1}B}\left|\int_{1}^{|x|}\left|D_{r}u(r\omega,|x|)\right|dr\right|^{2}dx$
$\displaystyle=\int_{2^{m-1}}^{2^{m}}\int_{|\omega|=1}\left|\int_{1}^{\rho}D_{r}u(r\omega,\rho)dr\right|^{2}\rho^{n-1}d\omega
d\rho$ $\displaystyle\leq
2^{mn}\int_{2^{m-1}}^{2^{m}}\int_{|\omega|=1}\int_{1}^{2^{m}}|D_{r}u(r\omega,\rho)|^{2}drd\omega
d\rho$ $\displaystyle\leq 2^{mn}\int_{2^{m-1}}^{2^{m}}\int_{2^{m}B\backslash
B}|\nabla_{y}u(y,t)|^{2}|y|^{1-n}dydt$ $\displaystyle\leq
2^{mn}\int_{2^{m-1}}^{2^{m}}\int_{2^{m}B}|\nabla_{y}u(y,t)|^{2}dydt,$
which gives
$\displaystyle\int_{2^{m}B\backslash 2^{m-1}B}|u(x,|x|)-u(x/|x|,|x|)|^{2}dx$
$\displaystyle\leq
C2^{m(2n-1)}\left({1\over|2^{m}B|}\int_{0}^{2^{m}}\int_{2^{m}B}|t\nabla_{y}u(y,t)|^{2}{dydt\over
t}\right)$ $\displaystyle\leq C2^{m(2n-1)}\|u\|^{2}_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}.$
Therefore,
$\displaystyle II$ $\displaystyle\leq C\sum_{m=1}^{\infty}{1\over
2^{2mn}}\int_{2^{m}B\backslash 2^{m-1}B}|u(x,|x|)-u(x/|x|,|x|)|^{2}dx\leq
C\|u\|^{2}_{{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}.$
Combining estimates of $I,II$ and $III$, we have obtained (3.1).
Note that by Lemma 2.4, if $V\in B_{q}$ for some $q\geq n/2$, then the
semigroup kernels ${\mathcal{P}}_{t}(x,y)$, associated to
$e^{-t\sqrt{\mathcal{L}}}$, decay faster than any power of $1/|x-y|$. Hence,
for all $k\in{\mathbb{N}}$, $e^{-t\sqrt{\mathcal{L}}}(u(\cdot,{1/k}))(x)$
exists everywhere in ${\mathbb{R}}^{n+1}_{+}$. This completes the proof. ∎
###### Lemma 3.2.
For every $u\in{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$, we have that
for every $k\in{\mathbb{N}}$,
$u(x,t+{1/k})=e^{-t\sqrt{\mathcal{L}}}\big{(}u(\cdot,{1/k})\big{)}(x),\ \ \ \
x\in\mathbb{R}^{n},\ t>0.$
###### Proof.
Since $u(x,\cdot)$ is continuous on $\mathbb{R}_{+}$, we have that $\lim_{t\to
0^{+}}u(x,t+{1}/{k})=u(x,{1}/{k}).$ Let us first show that for every
$k\in{\mathbb{N}}$,
(3.4) $\lim_{t\to 0^{+}}e^{-t\sqrt{\mathcal{L}}}(u(\cdot,1/k))(x)=u(x,1/k),\ \
x\in\mathbb{R}^{n}.$
One writes
$\displaystyle e^{-t\sqrt{\mathcal{L}}}(u(\cdot,1/k))(x)$ $\displaystyle=$
$\displaystyle
e^{-t\sqrt{\mathcal{L}}}(u(\cdot,1/k)1\\!\\!1_{|x-\cdot|>1})(x)$
$\displaystyle+\big{(}e^{-t\sqrt{\mathcal{L}}}-e^{-t\sqrt{-\Delta}}\big{)}(u(\cdot,1/k)1\\!\\!1_{|x-\cdot|\leq
1})(x)+e^{-t\sqrt{-\Delta}}(u(\cdot,1/k)1\\!\\!1_{|x-\cdot|\leq 1})(x)$
$\displaystyle=$ $\displaystyle I(x,t)+II(x,t)+III(x,t).$
By Lemma 3.1, we have that $u(x,1/k)\in L^{1}((1+|x|)^{-2n}dx)$. From Lemma
2.4, estimates of the semigroup kernels ${\mathcal{P}}_{t}(x,y)$, associated
to $e^{-t\sqrt{\mathcal{L}}}$, show that
$\displaystyle I(x,t)$ $\displaystyle\leq$ $\displaystyle
Ct\rho(x)^{n}\int_{|x-y|>1}{1\over 1+|x-y|^{2n+1}}|u(y,1/k)|dy\leq
Ct\rho(x)^{n}(1+|x|^{2n})\int_{\mathbb{R}^{n}}{1\over
1+|y|^{2n}}|u(y,1/k)|dy,$
hence $\lim_{t\to 0^{+}}I(x,t)=0.$
For the term $II(x,t)$, it follows from Lemma 2.3 that there exists a
nonnegative Schwartz function $\varphi$ such that
$\displaystyle II(x,t)$ $\displaystyle\leq$ $\displaystyle
C\int_{0}^{\infty}\int_{|x-y|\leq 1}{t\over s^{3/2}}\exp\big{(}-{t^{2}\over
4s}\big{)}|{\mathcal{K}}_{s}(x,y)-h_{s}(x-y)||u(y,1/k)|dyds$
$\displaystyle\leq$ $\displaystyle
C\|u(\cdot,1/k)\|_{L^{\infty}(B(x,1))}\bigg{[}\int_{0}^{\rho(x)^{2}}\int_{\mathbb{R}^{n}}{t\over
s^{3/2}}\exp\big{(}-{t^{2}\over
4s}\big{)}\Big{(}{\sqrt{s}\over\rho(x)}\Big{)}^{\delta}\varphi_{s}(x-y)dyds$
$\displaystyle\hskip
122.34692pt+\int_{\rho(x)^{2}}^{\infty}\int_{\mathbb{R}^{n}}{t\over
s^{3/2}}\exp\big{(}-{t^{2}\over 4s}\big{)}h_{s}(x-y)dyds\bigg{]}$
$\displaystyle\leq$ $\displaystyle
Ct\rho(x)^{-1}\|u(\cdot,1/k)\|_{L^{\infty}(B(x,1))},$
which implies that $\lim_{t\to 0^{+}}II(x,t)=0.$ Finally, we can follow a
standard argument as in the proof of Theorem 1.25, Chapter 1 of [24] to show
that for every $x\in\mathbb{R}^{n},$ $\lim_{t\to
0^{+}}e^{-t\sqrt{-\Delta}}(u(\cdot,1/k)1\\!\\!1_{|x-\cdot|\leq
1})(x)=u(x,{1}/{k}),$ and so (3.4) holds.
Next, we follow an argument as in Proposition 2.9 to set
$w(x,t)=e^{-t\sqrt{\mathcal{L}}}(u(\cdot,1/k))(x)-u(x,t+1/k)$. The function
$w$ satisfies ${\mathbb{L}}w=0,w(x,0)=0$. Define,
$\displaystyle{\overline{w}}(x,t)=\left\\{\begin{array}[]{rrl}w(x,t),&&t\geq
0,\\\\[6.0pt] -w(x,-t),&&t<0.\end{array}\right.$
Then ${\overline{w}}$ satisfies
${\overline{\mathbb{L}}}{\overline{w}}(x,t)=0,\ \ \ \ \ \
(x,t)\in{\mathbb{R}}^{n+1},$
where ${\overline{\mathbb{L}}}$ is an extension operator of ${{\mathbb{L}}}$
on $\mathbb{R}^{n+1}$. Observe that if $V(x)\in B_{q}(\mathbb{R}^{n})$ with
$q\geq n,$ then it can be verified that $V(x,t)=V(x)\in B_{q}$ on
$\mathbb{R}^{n+1}$. Next, let us verify (2.9). One writes
$\displaystyle\int_{{\mathbb{R}}^{n+1}}{|{\overline{w}}(x,t)|\over
1+|(x,t)|^{4(n+1)}}dxdt$ $\displaystyle\leq$ $\displaystyle
2\int_{{\mathbb{R}}^{n+1}_{+}}{|e^{-t\sqrt{\mathcal{L}}}(u(\cdot,1/k))(x)|\over
1+(|x|^{2}+t^{2})^{2(n+1)}}dxdt$
$\displaystyle+2\int_{{\mathbb{R}}^{n+1}_{+}}{|u(x,1/k)|\over
1+(|x|^{2}+t^{2})^{2(n+1)}}dxdt$
$\displaystyle+2\int_{{\mathbb{R}}^{n+1}_{+}}{|u(x,t+1/k)-u(x,1/k)|\over
1+(|x|^{2}+t^{2})^{2(n+1)}}dxdt=IV+V+VI.$
Observe that if $t\geq 1,$ then by Lemmas 2.1 and 2.4,
$\displaystyle\int_{\mathbb{R}^{n}}{|e^{-t\sqrt{\mathcal{L}}}(u(\cdot,1/k))(x)|\over(1+|x|^{2})^{2n}}dx$
$\displaystyle\leq$ $\displaystyle
C\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}^{n}}{1\over(1+|x|)^{4n}}{t\,\rho(x)^{2n}\over(t+|x-y|)^{3n+1}}dx\right)|u(y,1/k)|dy$
$\displaystyle\leq$ $\displaystyle
C\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}^{n}}{1\over(1+|x|)^{2n}}{t\over(t+|x-y|)^{3n+1}}dx\right)|u(y,1/k)|dy$
$\displaystyle\leq$ $\displaystyle C(t+1)\int_{\mathbb{R}^{n}}{|u(y,1/k)|\over
1+|y|^{2n}}dy\leq C_{k}(t+1),$
which gives
$\displaystyle IV\leq
C\int_{0}^{\infty}{1\over(1+t)^{4}}\left(\int_{\mathbb{R}^{n}}{|e^{-t\sqrt{\mathcal{L}}}(u(\cdot,1/k))(x)|\over(1+|x|^{2})^{4n}}dx\right)dt\leq
C_{k}\int_{0}^{\infty}{1\over(1+t)^{2}}dt\leq C^{\prime}_{k}.$
If $t\leq 1,$ then $IV\leq C_{k}$ can be verified by using condition $u\in
C^{1}({\mathbb{R}}^{n+1}_{+})$ and Lemmas 2.1 and 2.4. By Lemma 3.1, we have
that $V\leq C_{k}$. For term $VI$, we use (3.2) to obtain that
$|\partial_{t}u(x,t)|\leq C/t,$ and then
$\displaystyle|u(x,t+1/k)-u(x,1/k)|=\Big{|}\int_{1/k}^{t+1/k}\partial_{s}u(x,s)ds\
\Big{|}\leq C\log(1+kt),$
which gives
$\displaystyle VI$ $\displaystyle\leq$ $\displaystyle
C\int_{{\mathbb{R}}^{n+1}_{+}}{\log(1+kt)\over
1+(|x|^{2}+t^{2})^{2(n+1)}}dxdt\leq C_{k}.$
Estimate (2.9) then follows readily.
By Lemma 2.7, we have that $\overline{w}\equiv 0$, and then $w=0$, that is,
$u(x,t+1/k)=e^{-t\sqrt{\mathcal{L}}}(u(\cdot,1/k))(x),\ \ \
x\in\mathbb{R}^{n},\ t>0.$
The proof is complete. ∎
###### Remark 3.3.
Suppose $V\in B_{q}$ for some $q\geq n/2$ and let $\mathcal{L}=-\Delta+V.$
Using a similar argument as in (3.4), we have that for every $f\in
L^{\infty}(\mathbb{R}^{n})$,
* (i)
$\lim\limits_{t\rightarrow 0^{+}}e^{-t\mathcal{L}}f(x)=f(x),$ a.e.
$x\in{\mathbb{R}^{n}};$
* (ii)
$\lim\limits_{t\rightarrow 0^{+}}e^{-t\sqrt{\mathcal{L}}}f(x)=f(x),\ $ a.e.
${x\in\mathbb{R}^{n}.}$
For the heat and Poisson integrals of the harmonic oscillator
$\mathcal{L}=-\Delta+|x|^{2}$ on $\mathbb{R}^{n}$, we refer to Remarks
2.9-2.11, [25]. See also [20].
From (ii), it follows from an argument as in Proposition 2.9 that for $V\in
B_{q}$ for some $q\geq(n+1)/2$, if $u$ is a continuous weak solution of
${\mathbb{L}}u=0$ in ${\mathbb{R}}^{n+1}_{+}$ with
$\sup_{t>0}\|u(\cdot,t)\|_{L^{\infty}(\mathbb{R}^{n})}\leq C<\infty$, then
$u(x,t)$ is the Poisson integral of a function $f$ in
$L^{\infty}(\mathbb{R}^{n}).$
From now on, for any $k\in{\mathbb{N}}$, we set
$u_{k}(x,t)=u(x,t+{1/k}).$
Following an argument as in [11, Lemma 1.4], we have
###### Lemma 3.4.
For every $u\in{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$, there exists a
constant $C>0$ (depending only on $n$) such that for all $k\in{\mathbb{N}},$
(3.5)
$\sup_{x_{B},r_{B}}r_{B}^{-n}\int_{0}^{r_{B}}\int_{B(x_{B},r_{B})}t|\partial_{t}u_{k}(x,t)|^{2}{dxdt}\leq
C\|u\|^{2}_{{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}<\infty.$
###### Proof.
Let $B=B(x_{B},r_{B})$. If $r_{B}\geq 1/k$, then letting $s=t+1/k$, it follows
that
$\displaystyle\int_{0}^{r_{B}}\int_{B}t|\partial_{t}u(x,t+1/k)|^{2}{dxdt}$
$\displaystyle\leq C\int_{0}^{2r_{B}}\int_{2B}s|\partial_{s}u(x,s)|^{2}{dxds}$
$\displaystyle\leq C|B|\|u\|^{2}_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}<\infty.$
If $r_{B}<1/k$, then it follows from Lemma 2.6 for $\partial_{t}u(x,t+{1/k})$
and a similar argument as in (3.2) that
$\big{|}\partial_{t}u(x,t+{1/k})\big{|}\leq
C\big{(}t+k^{-1}\big{)}^{-1}\|u\|_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}.$
Therefore,
$\displaystyle|B|^{-1}\int_{0}^{r_{B}}\int_{B}t|\partial_{t}u(x,t+1/k)|^{2}{dxdt}$
$\displaystyle\leq C|B|^{-1}\|u\|^{2}_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}\int_{0}^{r_{B}}\int_{B}t\big{(}t+k^{-1}\big{)}^{-2}{dxdt}$
$\displaystyle\leq C\|u\|^{2}_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}\,\big{(}k^{2}\int_{0}^{r_{B}}t{dt}\big{)}$
$\displaystyle\leq C\|u\|^{2}_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}<\infty$
since $r_{B}<1/k.$
By taking the supremum over all balls $B\subset\mathbb{R}^{n},$ we complete
the proof of (3.5). ∎
Letting $f_{k}(x)=u(x,1/k),k\in{\mathbb{N}}$, it follows from Lemma 3.2 that
$u_{k}(x,t)=e^{-t\sqrt{\mathcal{L}}}(f_{k})(x),\ \ \ x\in\mathbb{R}^{n},\
t>0.$
Recall that a measure $\mu$ defined on ${\mathbb{R}}^{n+1}_{+}$ is said to be
a Carleson measure if there is a positive constant $c$ such that for each ball
$B$ on ${\mathbb{R}}^{n}$,
(3.6) $\mu({\widehat{B}})\leq c|B|,$
where ${\widehat{B}}$ is the tent over $B$. The smallest bound $c$ in (3.6) is
defined to be the norm of $\mu$, and is denoted by $|||\mu|||_{car}$.
It follows from Lemma 3.4 that
$\mu_{\nabla_{t},f_{k}}(x,t)=|t\partial_{t}e^{-t\sqrt{\mathcal{L}}}(f_{k})(x)|^{2}{dxdt\over
t}$
is a Carleson measure with $|||\mu_{\nabla_{t},f_{k}}|||_{car}\leq
C\|u\|^{2}_{{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}$.
###### Lemma 3.5.
For every $u\in{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$, there exists a
constant $C>0$ independent of $k$ such that
$\|f_{k}\|_{\rm BMO_{\mathcal{L}}(\mathbb{R}^{n})}\leq C<\infty,\ \ \hbox{ for
any }k\in\mathbb{N}.$
Hence for all $k\in{\mathbb{N}}$, $f_{k}$ is uniformly bounded in ${\rm
BMO_{\mathcal{L}}(\mathbb{R}^{n})}$.
The proof of Lemma 3.5 was given in [10, Theorem 2]; see also [6, 17, 19].
These arguments depend on three non-trivial results: the duality theorem that
${\rm
BMO}_{{\mathcal{L}}}(\mathbb{R}^{n})=\big{(}H^{1}_{\mathcal{L}}(\mathbb{R}^{n})\big{)}^{\ast}$,
Carleson inequality on tent spaces (see [3, Theorem 1]) and some special
properties of the space $H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$. In the sequel,
we are going to give a direct proof of Lemma 3.5 which is independent of these
results such as the duality of $H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$ and ${\rm
BMO}_{{\mathcal{L}}}(\mathbb{R}^{n})$ mentioned above (we thank Jie Xiao for
this observation).
To prove Lemma 3.5, we need to establish the following Lemmas 3.6 and 3.7.
Given a function $f\in L^{2}((1+|x|)^{-2n}dx)$ and an $L^{2}$ function $g$
supported on a ball $B=B(x_{B},r_{B})$, for any
$(x,t)\in{\mathbb{R}}^{n+1}_{+}$, set
(3.7) $\displaystyle F(x,t)=t\partial_{t}e^{-t\sqrt{\mathcal{L}}}f(x)\ \ \
{\rm and}\ \ \
G(x,t)=t\partial_{t}e^{-t\sqrt{\mathcal{L}}}(I-e^{-r_{B}\sqrt{\mathcal{L}}})g(x).$
###### Lemma 3.6.
Suppose $f,g,F,G$ are as in (3.7). If $f$ satisfies
$|||\mu_{\nabla_{t},f}|||^{2}_{car}=\sup_{x_{B},r_{B}}r_{B}^{-n}\int_{0}^{r_{B}}\int_{B(x_{B},r_{B})}t|\partial_{t}e^{-t\sqrt{\mathcal{L}}}f(x)|^{2}{dxdt}<\infty,$
then there exists a constant $C>0$ such that
(3.8) $\displaystyle\int_{{\mathbb{R}}^{n+1}_{+}}|F(x,t)G(x,t)|{dxdt\over
t}\leq C|B|^{1/2}|||\mu_{\nabla_{t},f}|||_{car}\|g\|_{{L}^{2}(B)}.$
###### Proof.
To prove (3.8), let us consider the square function ${\mathcal{G}}f$ given by
${\mathcal{G}}(f)(x)=\Big{(}\int_{0}^{\infty}|t\partial_{t}e^{-t\sqrt{\mathcal{L}}}f(x)|^{2}{dt\over
t}\Big{)}^{1/2}.$
By the spectral theory, the function ${\mathcal{G}}(f)$ is bounded on
$L^{2}(\mathbb{R}^{n})$. Now, given a ball
$B=B(x_{B},r_{B})\subset{\mathbb{R}}^{n}$ with radius $r_{B}$, we put
$T(B)=\\{(x,t)\in{\mathbb{R}}^{n+1}_{+}:x\in B,\ 0<t<r_{B}\\}.$
We then write
$\displaystyle\int_{{\mathbb{R}}^{n+1}_{+}}$
$\displaystyle|F(x,t)G(x,t)|{dxdt\over t}$
$\displaystyle=\int_{T(2B)}\big{|}F(x,t)G(x,t)\big{|}{dxdt\over
t}+\sum_{k=2}^{\infty}\int_{T(2^{k}B)\backslash
T(2^{k-1}B)}\big{|}F(x,t)G(x,t)\big{|}{dxdt\over t}$ $\displaystyle={\rm
A_{1}}+\sum_{k=2}^{\infty}{\rm A_{k}}.$
Using the Hölder inequality and $L^{2}$-boundedness of ${\mathcal{G}}$, we
obtain
$\displaystyle{\rm A_{1}}$
$\displaystyle\leq\Big{\|}\Big{\\{}\int_{0}^{2r_{B}}|t\partial_{t}e^{-t\sqrt{\mathcal{L}}}f(x)|^{2}{dt\over
t}\Big{\\}}^{1/2}\Big{\|}_{L^{2}(2B)}\|{\mathcal{G}}({{I}}-e^{-r_{B}\sqrt{\mathcal{L}}})g\|_{{L}^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq Cr_{B}^{n\over
2}|||\mu_{\nabla_{t},f}|||_{car}\|({{I}}-e^{-r_{B}\sqrt{\mathcal{L}}})g\|_{{L}^{2}(\mathbb{R}^{n})}$
$\displaystyle\leq Cr_{B}^{n\over
2}|||\mu_{\nabla_{t},f}|||_{car}\|g\|_{{L}^{2}(B)}.$
Let us estimate ${\rm A_{k}}$ for $k=2,3,\cdots.$ Observe that
$\displaystyle{\rm A}_{k}$
$\displaystyle\leq\Big{\|}\Big{\\{}\int_{0}^{2^{k}r_{B}}\big{|}t\partial_{t}e^{-t\sqrt{\mathcal{L}}}f(x)\big{|}^{2}{dt\over
t}\Big{\\}}^{1/2}\Big{\|}_{L^{2}(2^{k}B)}$
$\displaystyle\quad\quad\times\Big{\|}\Big{\\{}\int_{0}^{2^{k}r_{B}}\big{|}t\partial_{t}e^{-t\sqrt{\mathcal{L}}}(I-e^{-r_{B}\sqrt{\mathcal{L}}})g(x)\chi_{T(2^{k+1}B)\backslash
T(2^{k}B)}\big{|}^{2}{dt\over t}\Big{\\}}^{1/2}\Big{\|}_{{L}^{2}(2^{k}B)}$
$\displaystyle\leq C(2^{k}r_{B})^{n\over
2}|||\mu_{\nabla_{t},f}|||_{car}\times{\rm B}_{k},$
where
$\displaystyle{\rm
B}_{k}=\Big{\|}\Big{\\{}\int_{0}^{2^{k}r_{B}}\big{|}t\partial_{t}e^{-t\sqrt{\mathcal{L}}}(I-e^{-r_{B}\sqrt{\mathcal{L}}})g(x)\chi_{T(2^{k+1}B)\backslash
T(2^{k}B)}\big{|}^{2}{dt\over t}\Big{\\}}^{1/2}\Big{\|}_{{L}^{2}(2^{k}B)}.$
To estimate ${\rm B}_{k},$ we set
$\Psi_{t,s}(\mathcal{L})h(y)=(t+s)^{2}\Big{(}{d^{2}{e^{-r\sqrt{\mathcal{L}}}}\over
dr^{2}}\Big{|}_{r=t+s}h\Big{)}(y).$
Note that
$({{I}}-e^{-r_{B}\sqrt{\mathcal{L}}})g=\int_{0}^{r_{B}}s\sqrt{\mathcal{L}}e^{-s\sqrt{\mathcal{L}}}g{\frac{ds}{s}}.$
Let $\epsilon\in(0,1/4)$. By Lemma 2.4, we have
$\displaystyle{\rm B}_{k}$ $\displaystyle\leq
C\Big{\|}\Big{\\{}\int_{0}^{2^{k}r_{B}}\Big{|}\int_{0}^{r_{B}}{ts\over(t+s)^{2}}\Psi_{t,s}(\mathcal{L})g(x)\chi_{T(2^{k+1}B)\backslash
T(2^{k}B)}{ds\over s}\Big{|}^{2}{dt\over
t}\Big{\\}}^{1/2}\Big{\|}_{{L}^{2}(2^{k}B)}$ $\displaystyle\leq
C\Big{\|}\Big{\\{}\int_{0}^{2^{k}r_{B}}\Big{|}\int_{0}^{r_{B}}\int_{B(x_{B},r_{B})}{ts\over(t+s)^{2}}{(t+s)^{\epsilon}\over(t+s+|x-y|)^{n+\epsilon}}$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\times|g(y)|\chi_{T(2^{k+1}B)\backslash
T(2^{k}B)}(x){dyds\over s}\Big{|}^{2}{dt\over
t}\Big{\\}}^{1/2}\Big{\|}_{{L}^{2}(2^{k}B)}.$
Note that for $x\in T(2^{k+1}B)\backslash T(2^{k}B)$ and $y\in B$, we have
that $|x-y|\geq 2^{k}r_{B}$. The inequality
${ts(t+s)^{\epsilon}\over(t+s)^{2}}\leq C\ \\!{\rm
min}\big{(}(ts)^{\epsilon/2},t^{-\epsilon/2}s^{3\epsilon/2}\big{)},$
together with Hölder’s inequality and elementary integration, produces a
positive constant $C$ such that
$\displaystyle{\rm B}_{k}$ $\displaystyle\leq
C(2^{k}r_{B})^{-(n+\epsilon)+{n\over
2}}\|g\|_{L^{1}(B)}\Big{\\{}\int_{0}^{2^{k}r_{B}}\Big{|}\int_{0}^{r_{B}}{\rm
min}\big{(}(ts)^{\epsilon/2},t^{-\epsilon/2}s^{3\epsilon/2}\big{)}{ds\over
s}\Big{|}^{2}{dt\over t}\Big{\\}}^{1/2}$ $\displaystyle\leq
C(2^{k}r_{B})^{-(n+\epsilon)+{n\over 2}}r_{B}^{{n\over
2}+\epsilon}\|g\|_{{L}^{2}(B)},$
where we used the fact that
$\Big{\\{}\int_{0}^{2^{k}r_{B}}\big{|}\int_{0}^{r_{B}}{\rm
min}\big{(}(ts)^{\epsilon/2},t^{-\epsilon/2}s^{3\epsilon/2}\big{)}{ds\over
s}\big{|}^{2}{dt\over t}\Big{\\}}^{1/2}\\\
\leq\Big{\\{}\int_{0}^{r_{B}}\big{|}\int_{0}^{r_{B}}(ts)^{\epsilon/2}{ds\over
s}\big{|}^{2}{dt\over
t}\Big{\\}}^{1/2}+\Big{\\{}\int_{r_{B}}^{\infty}\big{|}\int_{0}^{r_{B}}t^{-\epsilon/2}s^{3\epsilon/2}{ds\over
s}\big{|}^{2}{dt\over t}\Big{\\}}^{1/2}\leq Cr_{B}^{\epsilon}.$
Consequently,
${\rm A}_{k}\leq C2^{-k\epsilon}r_{B}^{n\over
2}|||\mu_{\nabla_{t},f}|||_{car}\left\|g\right\|_{L^{2}(B)},$
which implies
$\displaystyle\int_{{\mathbb{R}}^{n+1}_{+}}|F(x,t)G(x,t)|{dxdt\over t}$
$\displaystyle\leq Cr_{B}^{{n\over
2}}|||\mu_{\nabla_{t},f}|||_{car}\|g\|_{{L}^{2}(B)}+C\sum_{k=2}^{\infty}2^{-k\epsilon}r_{B}^{{n\over
2}}|||\mu_{f}|||_{car}\|g\|_{{L}^{2}(B)}$ $\displaystyle\leq Cr_{B}^{{n\over
2}}|||\mu_{\nabla_{t},f}|||_{car}\|g\|_{{L}^{2}(B)}$
as desired. ∎
###### Lemma 3.7.
Suppose $B,f,g,F,G$ are defined as in Lemma 3.6. If
$|||\mu_{\nabla_{t},f}|||_{car}<\infty$, then we have the equality:
$\displaystyle\int_{{\mathbb{R}}^{n}}f(x)({\mathcal{I}}-e^{-r_{B}\sqrt{\mathcal{L}}})g(x)dx={1\over
4}\int_{{\mathbb{R}}^{n+1}_{+}}F(x,t)G(x,t){dxdt\over t}.$
###### Proof.
The proof follows by making minor modifications to that of [7, Proposition 4],
and so we skip it here. See also [6, 10, 19]. ∎
###### Proof of Lemma 3.5.
First, we have an equivalent characterization of ${\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ that $f\in{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ if and only if $f\in
L^{2}((1+|x|)^{-(n+\epsilon)}dx)$ and
(3.9)
$\sup_{B}\Big{(}|B|^{-1}\int_{B}|f(x)-e^{-r_{B}\sqrt{\mathcal{L}}}f(x)|^{2}dx\Big{)}^{1/2}\leq
C<\infty.$
This can be obtained by making minor modifications to that of [6, Proposition
6.11] (see also [5, 17]) corresponding to the case in which the function
$e^{-r_{B}\sqrt{\mathcal{L}}}f$ is replaced by $e^{-r_{B}^{2}\mathcal{L}}f$,
and we omit the detail here.
Now if $\|u\|_{{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}<\infty$, then it
follows from Lemma 3.1 that
$\int_{\mathbb{R}^{n}}{|f_{k}(x)|^{2}\over 1+|x|^{2n}}dx\leq C_{k}<\infty.$
Given an $L^{2}$ function $g$ supported on a ball $B=B(x_{B},r_{B})$, it
follows by Lemma 3.7 that we have
$\displaystyle\int_{{\mathbb{R}}^{n}}f_{k}(x)(I-e^{-r_{B}\sqrt{\mathcal{L}}})g(x)dx={1\over
4}\int_{{\mathbb{R}}^{n+1}_{+}}t\partial_{t}e^{-t\sqrt{\mathcal{L}}}f_{k}(x)\
t\partial_{t}e^{-t\sqrt{\mathcal{L}}}(I-e^{-r_{B}\sqrt{\mathcal{L}}})g(x){dxdt\over
t}.$
By Lemmas 3.6 and 3.4,
$\displaystyle|\int_{{\mathbb{R}}^{n}}f_{k}(x)(I-e^{-r_{B}\sqrt{\mathcal{L}}})g(x)dx|$
$\displaystyle\leq
C|B|^{1/2}|||\mu_{\nabla_{t},f_{k}}|||_{car}\|g\|_{{L}^{2}(B)}$
$\displaystyle\leq C|B|^{1/2}\|u\|_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}\|g\|_{{L}^{2}(B)}.$
Then the duality argument for ${L}^{2}$ shows that
$\displaystyle\Big{(}|B|^{-1}\int_{B}|f_{k}(x)-e^{-r_{B}\sqrt{\mathcal{L}}}f_{k}(x)|^{2}dx\Big{)}^{1/2}$
$\displaystyle=|B|^{-1/2}\sup\limits_{\|g\|_{{L}^{2}(B)\leq
1}}\Big{|}\int_{{\mathbb{R}}^{n}}(I-e^{-r_{B}\sqrt{\mathcal{L}}})f_{k}(x)g(x)dx\Big{|}$
$\displaystyle=|B|^{-1/2}\sup\limits_{\|g\|_{{L}^{2}(B)\leq
1}}\Big{|}\int_{{\mathbb{R}}^{n}}f_{k}(x)\big{(}I-e^{-r_{B}\sqrt{\mathcal{L}}}\big{)}g(x)dx\Big{|}$
$\displaystyle\leq C\|u\|_{{\rm HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}$
for some $C>0$ independent of $k.$
It then follows that for all $k\in{\mathbb{N}}$, $f_{k}$ is uniformly bounded
in ${\rm BMO}_{\rm\mathcal{L}}({\mathbb{R}}^{n})$. ∎
###### Proof of part (1) of Theorem 1.1.
Letting $f_{k}(x)=u(x,1/k)$, it follows by Lemma 3.2 that
$u(x,t+{1/k})=e^{-t\sqrt{\mathcal{L}}}(f_{k})(x)$ and so
(3.10)
$\displaystyle\mathcal{L}u(x,t+{1/k})=\mathcal{L}e^{-t\sqrt{\mathcal{L}}}(f_{k})(x).$
Then we have the following facts:
* (i)
$H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$ is a Banach space;
* (ii)
For each $t>0$ and $x\in\mathbb{R}^{n}$,
$\partial^{2}_{t}{\mathcal{P}}_{t}(x,\cdot)\in
H^{1}_{\mathcal{L}}(\mathbb{R}^{n})$ with
$\|\partial^{2}_{t}{\mathcal{P}}_{t}(x,\cdot)\|_{H^{1}_{\mathcal{L}}(\mathbb{R}^{n})}\leq
C/t^{2}$ (see (2.8));
* (iii)
The duality theorem that $(H^{1}_{\mathcal{L}}(\mathbb{R}^{n}))^{\ast}={\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})$(see Lemma 2.5).
From (i), we use Lemma 3.5 and pass to a subsequence, we have that
$f_{k}\rightarrow f$ (in weak-${\ast}$ convergence) for some $f\in{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ such that $\|f\|_{{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})}\leq C\|u\|_{{\rm
HMO_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}$. Then by (ii) and (iii), we
conclude that, for each $(x,t)\in{\mathbb{R}}_{+}^{n+1}$, the right-hand side
of (3.10) converges to $\mathcal{L}e^{-t\sqrt{\mathcal{L}}}(f)(x)$ when
$k\to\infty$. On the other hand, as $k\rightarrow\infty$, the left-hand side
of (3.10) converges pointwisely to $\mathcal{L}u(x,t)$. Hence, for a fixed
$t>0,$
$\mathcal{L}w(x,t)=0\ \ {\rm in}\ \ {\mathbb{R}}^{n},$
where $w(x,t)=u(x,t)-e^{-t\sqrt{\mathcal{L}}}(f)(x)$. Next, let us verify
(2.9). Letting $k$ small enough such that $t>2/k$, one writes
$\displaystyle\int_{\mathbb{R}^{n}}{|w(x,t)|\over 1+|x|^{2n}}dx$
$\displaystyle\leq$ $\displaystyle\int_{\mathbb{R}^{n}}{|u(x,1/k)|\over
1+|x|^{2n}}dx$ $\displaystyle+\int_{\mathbb{R}^{n}}{|u(x,t)-u(x,1/k)|\over
1+|x|^{2n}}dx$
$\displaystyle+\int_{\mathbb{R}^{n}}{|e^{-t\sqrt{\mathcal{L}}}(f)(x)|\over
1+|x|^{2n}}dx\leq I+II+III.$
By Lemma 3.1, we have that $I\leq C_{k}$. For term $II$, we use (3.2) to
obtain that $|\partial_{t}u(x,t)|\leq C/t,$ and then
$\displaystyle|u(x,t)-u(x,1/k)|=\Big{|}\int_{1/k}^{t}\partial_{s}u(x,s)ds\
\Big{|}\leq C\log(kt),$
which gives
$\displaystyle II$ $\displaystyle\leq$ $\displaystyle
C\int_{\mathbb{R}^{n}}{\log(kt)\over 1+|x|^{2n}}dx\leq C\log(kt).$
Consider the term $III$. For a fixed $t>0$ and $x\in R^{n}$, we set
$f_{B}=t^{-n}\int_{B(x,t)}f(y)dy.$ It can be verified by a standard argument
(see [10, Theorem 2]) that $|e^{-t\sqrt{\mathcal{L}}}f(x)|\leq C\|f\|_{{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})}+|f_{B}|.$ By Lemma 2 of [10], we have that
$|f_{B}|\leq C(1+{\rm log}[{\rho(x)/t}])\|f\|_{{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})}$ if $t<\rho(x)$; $|f_{B}|\leq C\|f\|_{{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})}$ if $t\geq\rho(x)$. It then follows by
Lemma 2.1 that there is a constant $k_{0}\geq 1$ such that
$\displaystyle|e^{-t\sqrt{\mathcal{L}}}f(x)|\leq C\|f\|_{{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})}\left(1+{\rm log}\left[1+{C\rho(0)\over
t}\left(1+{|x|\over\rho(0)}\right)^{k_{0}\over k_{0}+1}\right]\right)$
for every $t>0$ and $x\in\mathbb{R}^{n},$ which yields
$III\leq C\|f\|_{{\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})}\int_{\mathbb{R}^{n}}{1\over
1+|x|^{2n}}\left(1+{\rm log}\left[1+{C\rho(0)\over
t}\left(1+{|x|\over\rho(0)}\right)^{k_{0}\over k_{0}+1}\right]\right)dx\leq
C_{t}<\infty.$
Estimate (2.9) then follows readily.
By Lemma 2.7, we have that $u(x,t)=e^{-t\sqrt{\mathcal{L}}}(f)(x)$ with
$f\in{\rm BMO}_{\mathcal{L}}(\mathbb{R}^{n}).$ The proof of part (1) of
Theorem 1.1 is complete. ∎
### 3.2. The characterization of ${\rm BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ in
terms of Carleson measure
To prove part (2) of Theorem 1.1, we need the following Lemmas 3.8 and 3.9.
###### Lemma 3.8.
Suppose $V\in B_{q}$ for some $q>n.$ Let $\beta=1-{n\over q}$. For every
$N>0$, there exist constants $C=C_{N}>0$ and $c>0$ such that for all
$x,y\in\mathbb{R}^{n}$ and $t>0,$ the semigroup kernels
${\mathcal{K}}_{t}(x,y)$, associated to $e^{-t{\mathcal{L}}}$, satisfy the
following estimates:
(3.11)
$\displaystyle|\nabla_{x}{\mathcal{K}}_{t}(x,y)|+|\sqrt{t}\nabla_{x}\partial_{t}{\mathcal{K}}_{t}(x,y)|\leq
Ct^{-(n+1)/2}e^{-\frac{\left|x-y\right|^{2}}{ct}}\left(1+\frac{\sqrt{t}}{\rho(x)}+\frac{\sqrt{t}}{\rho(y)}\right)^{-N},$
and for $|h|<|x-y|/4,$
(3.12)
$\displaystyle|\nabla_{x}{\mathcal{K}}_{t}(x+h,y)-\nabla_{x}{\mathcal{K}}_{t}(x,y)|\leq
C\Big{(}{|h|\over\sqrt{t}}\Big{)}^{\beta}t^{-(n+1)/2}e^{-\frac{\left|x-y\right|^{2}}{ct}}.$
###### Proof.
The proof of Lemma 3.8 is inspired by ideas developed in [21]. Let
$\Gamma_{0}(x,y)$ denote the fundamental solution for the operator $-\Delta$
in $\mathbb{R}^{n}.$ It is well-known that
$\Gamma_{0}(x,y)=-{1\over n(n-2)\omega(n)}{1\over|x-y|^{n-2}},\ \ \ \ n\geq
3,$
where $\omega(n)$ is the area of the unit sphere in $\mathbb{R}^{n}$. Fix
$t>0$ and $x_{0},y_{0}\in\mathbb{R}^{n}$. Assume that
$\partial_{t}u+\mathcal{L}u=0$. Let $\eta\in C_{0}^{\infty}(B(x_{0},2R))$ such
that $\eta=1$ on $B(x_{0},3R/2)$, $|\nabla\eta|\leq C/R$ and
$|\nabla^{2}\eta|\leq C/R^{2}.$ Following an argument as in Lemma 4.6 of [21],
we have
$\displaystyle u(x)\eta(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}\Gamma_{0}(x,y)\\{-\Delta\\}(u\eta)(y)dy$
$\displaystyle=\int_{\mathbb{R}^{n}}\Gamma_{0}(x,y)\\{-Vu\eta-\eta\partial_{t}u-2\nabla
u\cdot\nabla\eta-u\Delta\eta\\}dy$
$\displaystyle=\int_{\mathbb{R}^{n}}\Gamma_{0}(x,y)\\{-Vu\eta-\eta\partial_{t}u-u\Delta\eta\\}dy$
(3.13)
$\displaystyle\quad+2\int_{\mathbb{R}^{n}}\nabla\Gamma_{0}(x,y)\cdot(\nabla\eta)udy.$
Thus, for $x\in B(x_{0},R),$
$\displaystyle|\nabla_{x}u(x)|$ $\displaystyle\leq
C\int_{B(x_{0},2R)}{V(y)|u(y)||\eta(y)|\over|x-y|^{n-1}}dy$
$\displaystyle\hskip
56.9055pt+C\int_{B(x_{0},2R)}{|\partial_{t}u(y)||\eta(y)|\over|x-y|^{n-1}}dy+{C\over
R^{n+1}}\int_{B(x_{0},2R)}|u(y)|dy.$
It follows from Lemmas 1.2 and 1.8 in [21] that there exist $C$ and $m_{0}>0$
such that for all $R>0,$
$\int_{B(x_{0},2R)}{V(y)\over|x-y|^{n-1}}dy\leq{C\over
R^{n-1}}\int_{B(x_{0},2R)}V(y)dy\leq{C\over
R}\Big{(}{R\over\rho(x_{0})}\Big{)}^{m_{0}}.$
Therefore,
$\displaystyle|\nabla_{x}u(x)|$ $\displaystyle\leq{C\over
R}\sup_{B(x_{0},2R)}|u(y)|\Big{\\{}\Big{(}{R\over\rho(x_{0})}\Big{)}^{m_{0}}+1\Big{\\}}+CR\sup_{B(x_{0},2R)}|\partial_{t}u(y)|,\
\ \ \ \forall x\in B(x_{0},R).$
Let us prove (3.11). We take $u={\mathcal{K}}_{t}(x,y_{0})$ and
$R=|x_{0}-y_{0}|/8$. If $x\in B(x_{0},2R)$, then
$\left|x-y_{0}\right|\sim\left|x_{0}-y_{0}\right|$. Using Lemma 2.2, we have
that for any $N>m_{0}+1,$
$\displaystyle|\nabla_{x}{\mathcal{K}}_{t}(x_{0},y_{0})|$
$\displaystyle\leq{C\over
R}\sup_{B(x_{0},2R)}|{\mathcal{K}}_{t}(x,y_{0})|\Big{\\{}\Big{(}{R\over\rho(x_{0})}\Big{)}^{m_{0}}+1\Big{\\}}+R\sup_{B(x_{0},2R)}|\partial_{t}{\mathcal{K}}_{t}(x,y_{0})|$
$\displaystyle\leq{C_{N}\over
R}t^{-n/2}e^{-\frac{\left|x_{0}-y_{0}\right|^{2}}{ct}}\Big{(}1+\frac{\sqrt{t}}{\rho(x_{0})}+\frac{\sqrt{t}}{\rho(y_{0})}\Big{)}^{-N}\Big{\\{}\Big{(}{R\over\rho(x_{0})}\Big{)}^{m_{0}}+1+{R^{2}\over
t}\Big{\\}}$ (3.14) $\displaystyle\leq
C^{\prime}_{N}t^{-(n+1)/2}e^{-\frac{\left|x_{0}-y_{0}\right|^{2}}{c_{1}t}}\left(1+\frac{\sqrt{t}}{\rho(x_{0})}+\frac{\sqrt{t}}{\rho(y_{0})}\right)^{-N+m_{0}},$
which implies the desired result. Now, letting
$u=\sqrt{t}\partial_{t}{\mathcal{K}}_{t}(x,y_{0})$, we use a similar argument
as in (3.2) to obtain estimate for the term
$|\sqrt{t}\nabla_{x}\partial_{t}{\mathcal{K}}_{t}(x_{0},y_{0})|$ in (3.11).
To prove (3.12), we follow an argument as in Remark 4.10 and (6.6) of [21]. It
follows from (3.2) that
$\displaystyle\|\nabla^{2}_{x}(u\eta)\|_{q}\leq
CR^{(n/q)-2}\Big{\\{}\Big{(}{R\over\rho(x_{0})}\Big{)}^{m_{0}}+1\Big{\\}}\sup_{B(x_{0},2R)}|u|+CR^{(n/q)}\sup_{B(x_{0},2R)}|\partial_{t}u|.$
To see (3.12), we fix $x_{0},y_{0}\in\mathbb{R}^{n}$ and
$h\in\mathbb{R}^{n},|h|<|x_{0}-y_{0}|/4$. Let $u={\mathcal{K}}_{t}(x,y_{0})$
and $R=|x_{0}-y_{0}|/8$. It then follows from the imbedding theorem of Morrey
that
$\displaystyle|\nabla_{x}{\mathcal{K}}_{t}(x_{0}+h,y_{0})-\nabla_{x}{\mathcal{K}}_{t}(x_{0},y_{0})|\leq
C|h|^{1-(n/q)}\Big{(}\int_{B(x_{0},R)}|\nabla^{2}_{x}{\mathcal{K}}_{t}(x,y_{0})|^{q}dx\Big{)}^{1/q}$
$\displaystyle\leq$ $\displaystyle C\Big{(}{|h|\over
R}\Big{)}^{1-(n/q)}{1\over
R}\Big{\\{}\Big{(}{R\over\rho(x_{0})}\Big{)}^{m}+1\Big{\\}}\left(\sup_{B(x_{0},2R)}|{\mathcal{K}}_{t}(x,y_{0})|+{R^{2}\over
t}\sup_{B(x_{0},2R)}|t\partial_{t}{\mathcal{K}}_{t}(x,y_{0})|\right)$
$\displaystyle\leq$ $\displaystyle C\Big{(}{|h|\over
R}\Big{)}^{1-(n/q)}t^{-(n+1)/2}e^{-\frac{\left|x_{0}-y_{0}\right|^{2}}{ct}}$
$\displaystyle\leq$ $\displaystyle
C\Big{(}{|h|\over\sqrt{t}}\Big{)}^{1-(n/q)}t^{-(n+1)/2}e^{-\frac{\left|x_{0}-y_{0}\right|^{2}}{c^{\prime}t}}.$
Estimate (3.12) follows readily. ∎
Turning to the Poisson semigroup $\\{e^{-t\sqrt{\mathcal{L}}}\\}_{t>0}$, we
have
###### Lemma 3.9.
Suppose $V\in B_{q}$ for some $q>n.$ Let $\beta=1-{n\over q}$. For every
$N>0$, there exist constants $C=C_{N}>0$ such that for all
$x,y\in\mathbb{R}^{n}$ and $t>0$,
* (i)
${\displaystyle|t\nabla_{x}{\mathcal{P}}_{t}(x,y)|\leq
C\frac{t}{(t^{2}+\left|x-y\right|^{2})^{\frac{n+1}{2}}}\left(1+\frac{(t^{2}+\left|x-y\right|^{2})^{1/2}}{\rho(x)}+\frac{(t^{2}+\left|x-y\right|^{2})^{1/2}}{\rho(y)}\right)^{-N};}$
* (ii)
For all $|h|\leq|x-y|/4$,
$\displaystyle|t\nabla_{x}{\mathcal{P}}_{t}(x+h,y)-t\nabla_{x}{\mathcal{P}}_{t}(x,y)|\leq
C\left({|h|\over t}\right)^{\beta}{t\over(t^{2}+|x-y|^{2})^{{n+1\over 2}}};$
* (iii)
There is some $\delta>1$ such that
${\displaystyle\big{|}t\nabla_{x}e^{-t\sqrt{\mathcal{L}}}(1)(x)\big{|}\leq
C\min\Big{\\{}\Big{(}\frac{{t}}{\rho(x)}\Big{)}^{\delta},\Big{(}{t\over\rho(x)}\Big{)}^{-N}\Big{\\}}.}$
###### Proof.
The proofs of (i) and (ii) follow from the subordination formula (2) and Lemma
3.8. To prove (iii), we consider two cases.
Case 1: $t>\rho(x).$ We use (i) to obtain
$\displaystyle|t\nabla_{x}e^{-t\sqrt{\mathcal{L}}}(1)(x)|\leq
C\left({t\over\rho(x)}\right)^{-N}.$
Case 2: $t\leq\rho(x).$ In this case, it follows from the subordination
formula (2) that
$\displaystyle\big{|}t\nabla_{x}e^{-t\sqrt{\mathcal{L}}}(1)(x)\big{|}$
$\displaystyle\leq C\int_{0}^{\infty}\Big{(}{t\over
u}\Big{)}^{2}\exp\big{(}-{t^{2}\over
4u}\big{)}\left|\sqrt{u}\nabla_{x}e^{-u{\mathcal{L}}}(1)(x)\right|~{}du$
$\displaystyle\leq C\int_{0}^{\rho(x)^{2}}\Big{(}{t\over
u}\Big{)}^{2}\exp\big{(}-{t^{2}\over
4u}\big{)}\left|\sqrt{u}\nabla_{x}e^{-u{\mathcal{L}}}(1)(x)\right|~{}du$
$\displaystyle\ +\ C\int_{\rho(x)^{2}}^{\infty}\Big{(}{t\over
u}\Big{)}^{2}\exp\big{(}-{t^{2}\over
4u}\big{)}\left|\sqrt{u}\nabla_{x}e^{-u{\mathcal{L}}}(1)(x)\right|~{}du$
$\displaystyle=I_{1}(x)+I_{2}(x).$
By Lemma 3.8, we have
$\displaystyle I_{2}(x)\leq C\int_{\rho(x)^{2}}^{\infty}\Big{(}{t\over
u}\Big{)}^{2}~{}du\leq C\left(\frac{t}{\rho(x)}\right)^{2}.$
Consider the term $I_{1}(x).$ We apply Kato-Trotter formula to obtain
$\displaystyle
h_{u}(x-y)-{\mathcal{K}}_{u}(x,y)=\int_{0}^{u}\int_{\mathbb{R}^{n}}h_{s}(x-z)V(z){\mathcal{K}}_{u-s}(z,y)dzds,$
which gives
$\displaystyle|{\sqrt{u}}\nabla_{x}e^{-u\mathcal{L}}(1)(x)\big{|}$
$\displaystyle=\Big{|}\int_{\mathbb{R}^{n}}\int_{0}^{u}\int_{\mathbb{R}^{n}}\sqrt{u}\nabla_{x}h_{s}(x-z)V(z){\mathcal{K}}_{u-s}(z,y)dzdsdy\Big{|}$
$\displaystyle\leq C\int_{0}^{u}\Big{(}{u\over
s}\Big{)}^{1/2}\int_{\mathbb{R}^{n}}s^{-n/2}\exp\Big{(}-{|x-z|^{2}\over
cs}\Big{)}V(z)dzds.$
It follows from (2.2) that for $s\leq\rho(x)^{2},$
$\displaystyle\int_{\mathbb{R}^{n}}s^{-n/2}\exp\Big{(}-{|x-y|^{2}\over
cs}\Big{)}V(y)dy\leq\frac{C}{s}\left(\frac{\sqrt{s}}{\rho(x)}\right)^{\delta},$
where $\delta=2-\frac{n}{q}>1$. This implies that
$\displaystyle|{\sqrt{u}}\nabla_{x}e^{-u\mathcal{L}}(1)(x)|$
$\displaystyle\leq C\int_{0}^{u}\Big{(}{u\over
s}\Big{)}^{1/2}s^{-1}\Big{(}{\sqrt{s}\over\rho(x)}\Big{)}^{\delta}ds\leq
C\Big{(}{\sqrt{u}\over\rho(x)}\Big{)}^{\delta}.$
Therefore,
$\displaystyle I_{1}(x)$ $\displaystyle\leq
C\int_{0}^{\rho(x)^{2}}\Big{(}{t\over u}\Big{)}^{2}\exp\big{(}-{t^{2}\over
4u}\big{)}\left(\frac{\sqrt{u}}{\rho(x)}\right)^{\delta}~{}du\leq
C\left(\frac{t}{\rho(x)}\right)^{\delta}.$
Combining the estimates of Case 1 and Case 2, we obtain (iii). ∎
To prove part (2) of Theorem 1.1, we recall that the Carleson measure is
closely related to the space ${\rm BMO}_{\mathcal{L}}(\mathbb{R}^{n})$. We
note that for every $f\in{\rm BMO}_{\mathcal{L}}(\mathbb{R}^{n})$,
(3.15)
$\mu_{\nabla_{t},f}(x,t)=t|\partial_{t}e^{-t\sqrt{\mathcal{L}}}(f)(x)|^{2}{dxdt}$
is a Carleson measure with $|||\mu_{\nabla_{t},f_{k}}|||_{car}\leq
C\|f\|^{2}_{{\rm BMO_{\mathcal{L}}}(\mathbb{R}^{n})}$(see [6, 19]).
###### Proof of part (2) of Theorem 1.1.
Recall that the condition $V\in B_{n}$ implies $V\in B_{q_{0}}$ for some
$q_{0}>n$. From Lemmas 2.4 and 3.9, we see that
$u(x,t)=e^{-t\sqrt{\mathcal{L}}}f(x)\in C^{1}({\mathbb{R}}^{n+1}_{+})$. Let us
fix a ball $B=B(x_{B},r_{B})$. From (3.15), it suffices to show that there
exists a constant $C>0$ such that
$\displaystyle\int_{0}^{r_{B}}\int_{B}|t\nabla_{x}e^{-t\sqrt{\mathcal{L}}}f(x)|^{2}\frac{dxdt}{t}$
$\displaystyle\leq C|B|\left\|f\right\|_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}^{2}.$
To do this, we split the function $f$ into local, global, and constant parts
as follows
$f=(f-f_{2B})\chi_{2B}+(f-f_{2B})\chi_{(2B)^{c}}+f_{2B}=f_{1}+f_{2}+f_{3},$
where $2B=B(x_{B},2r_{B})$.
Since the Riesz transform $\nabla\mathcal{L}^{-1/2}$ is bounded on
$L^{2}(\mathbb{R}^{n})$, we have
$\displaystyle\int_{0}^{r_{B}}\int_{B}|t\nabla_{x}e^{-t\sqrt{\mathcal{L}}}f_{1}(x)|^{2}\frac{dxdt}{t}$
$\displaystyle=\int_{0}^{r_{B}}\int_{\mathbb{R}^{n}}|\nabla_{x}\mathcal{L}^{-{1/2}}t\mathcal{L}^{{1/2}}e^{-t\sqrt{\mathcal{L}}}f_{1}(x)|^{2}\frac{dxdt}{t}$
$\displaystyle\leq
C\int_{0}^{\infty}\int_{\mathbb{R}^{n}}|t\mathcal{L}^{1/2}e^{-t\sqrt{\mathcal{L}}}f_{1}(x)|^{2}\frac{dxdt}{t}$
$\displaystyle\leq C\left\|f_{1}\right\|_{L^{2}(\mathbb{R}^{n})}^{2}$
$\displaystyle=C\int_{2B}\left|f(x)-f_{2B}\right|^{2}dx$ $\displaystyle\leq
C|B|\left\|f\right\|_{{\rm BMO_{\mathcal{L}}}(\mathbb{R}^{n})}^{2}.$
To estimate the global term, we use (i) of Lemma 3.9 and then the standard
argument as in Theorem 2 of [10] shows that for $x\in B$ and $t<r_{B}$,
$\displaystyle|t\nabla_{x}e^{-t\sqrt{\mathcal{L}}}f_{2}(x)|\leq
C\int_{(2B)^{c}}\left|f(y)-f_{2B}\right|\frac{t}{\left|x_{B}-y\right|^{n+1}}dy$
$\displaystyle\leq
C\sum_{k=2}^{\infty}\frac{t}{(2^{k}r_{B})^{n+1}}\left[\int_{2^{k}B\backslash
2^{k-1}B}\left|f(y)-f_{2^{k}B}\right|dy+(2^{k}r_{B})^{n}\left|f_{2^{k}B}-f_{2B}\right|\right]$
$\displaystyle\leq C\big{(}{t\over
r_{B}}\big{)}\sum_{k=2}^{\infty}2^{-k}\left[\left\|f\right\|_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}+k\left\|f\right\|_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}\right]\leq C\big{(}{t\over
r_{B}}\big{)}\left\|f\right\|_{{\rm BMO_{\mathcal{L}}}(\mathbb{R}^{n})},$
which yields
$\displaystyle\int_{0}^{r_{B}}\int_{B}|t\nabla_{x}e^{-t\sqrt{\mathcal{L}}}f_{2}(x)|^{2}\frac{dxdt}{t}\leq
C|B|r_{B}^{-2}\int_{0}^{r_{B}}t{dt}\left\|f\right\|^{2}_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}\leq C|B|\left\|f\right\|^{2}_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}.$
It remains to estimate the constant term $f_{3}=f_{2B}$, for which we make use
of (i) and (iii) of Lemma 3.9. Assume first that $r_{B}\leq\rho(x_{B})$. By
Lemma 3.1, $\rho(x)\sim\rho(x_{B})$ for $x\in B$, we have
(3.16)
$\displaystyle\int_{0}^{r_{B}}\int_{B}|t\nabla_{x}e^{-t\sqrt{\mathcal{L}}}f_{3}(x)|^{2}\frac{dxdt}{t}$
$\displaystyle\leq$
$\displaystyle{\left|f_{2B}\right|^{2}}\int_{0}^{r_{B}}\int_{B}\big{(}t/\rho(x)\big{)}^{2\delta}\frac{dxdt}{t}$
$\displaystyle\leq$ $\displaystyle
C|B|\left|f_{2B}\right|^{2}\big{(}r_{B}/\rho(x_{B})\big{)}^{2\delta}$
$\displaystyle\leq$ $\displaystyle C|B|\|f\|_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}^{2}\Big{(}1+\log{\rho(x_{B})\over
r_{B}}\Big{)}^{2}\big{(}r_{B}/\rho(x_{B})\big{)}^{2\delta}$
$\displaystyle\leq$ $\displaystyle C|B|\|f\|_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}^{2}.$
Suppose finally that $r_{B}>\rho(x_{B})$, we use an argument as in Theorem 2
of [10] to select a finite family of critical balls $\left\\{Q_{k}\right\\}$
such that $B\subset\cup Q_{k}$ and $\sum\left|Q_{k}\right|\leq\left|B\right|$.
Then, using the fact that $\left|f_{2B}\right|\leq\left\|f\right\|_{\rm
BMO_{\mathcal{L}}(\mathbb{R}^{n})},$ we can bound the left hand side of (3.16)
by
$\displaystyle C{\left\|f\right\|_{\rm
BMO_{\mathcal{L}}}^{2}}\sum_{k}\left(\int_{0}^{\rho(x_{k})}\int_{Q_{k}}\left({t\over{\rho(x_{k})}}\right)^{2\delta}{dxdt\over
t}+\int_{\rho(x_{k})}^{\infty}\int_{Q_{k}}\left(\frac{t}{\rho(x_{k})}\right)^{-2N}{dxdt\over
t}\right)$ $\displaystyle\leq C{\left\|f\right\|_{{\rm
BMO_{\mathcal{L}}}(\mathbb{R}^{n})}^{2}}\sum_{k}\left|Q_{k}\right|\leq
C|B|\left\|f\right\|_{{\rm BMO_{\mathcal{L}}}(\mathbb{R}^{n})}^{2},$
which establishes the proof of part (2) of Theorem 1.1. ∎
## 4\. The spaces ${\rm HMO}^{\alpha}_{\mathcal{L}}({\mathbb{R}}^{n+1}_{+})$
and their characterizations
In this section we will extend the method for the space ${\rm
BMO}_{\mathcal{L}}(\mathbb{R}^{n})$ in Section 3 to obtain some
generalizations to Lipschitz-type spaces
$\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$ with $0<\alpha<1$ (see [1]).
Let us recall that a locally integrable function $f$ in
$\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n}),0<\alpha<1,$ if there exists a
constant $C>0$ such that
(4.1) $\displaystyle|f(x)-f(y)|\leq C|x-y|^{\alpha}$
and
(4.2) $\displaystyle|f(x)|\leq C\rho(x)^{\alpha}$
for all $x,y\in\mathbb{R}^{n}.$ The norm of $f$ in
$\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$ is given by
$\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}=\sup_{\begin{subarray}{c}x,y\in\mathbb{R}^{n}\\\
x\not=y\end{subarray}}{|f(x)-f(y)|\over|x-y|^{\alpha}}+\sup_{x\in\mathbb{R}^{n}}|\rho(x)^{-\alpha}f(x)|.$
Because of (4.2), this space $\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$
is in fact a proper subspace of the classical Lipschitz space
$\Lambda^{\alpha}(\mathbb{R}^{n})$ (see [1, 11, 19]). Following [1], a locally
integrable function $f$ in $\mathbb{R}^{n}$ is in ${\rm
BMO}_{\mathcal{L}}^{\alpha}(\mathbb{R}^{n})$, $\alpha>0,$ if there is a
constant $C>0$ such that
(4.3) $\displaystyle\int_{B}|f(y)-f_{B}|dy\leq Cr^{n+\alpha}$
for every ball $B=B(x,r)$, and
(4.4) $\displaystyle\int_{B}|f(y)|~{}dy\leq Cr^{n+\alpha}$
for every ball $B=B(x,r)$ with $r\geq\rho(x)$. Define
$\displaystyle\|f\|_{{\rm
BMO}_{\mathcal{L}}^{\alpha}(\mathbb{R}^{n})}=\inf\big{\\{}C:C\ {\rm
satisfies}\ \eqref{e4.3}\ {\rm and}\ \eqref{e4.4}\big{\\}}.$
It is proved in [1, Proposition 4] that for $0<\alpha<1$, ${\rm
BMO}_{\mathcal{L}}^{\alpha}(\mathbb{R}^{n})=\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n}).$
###### Theorem 4.1.
Suppose $V\in B_{q}$ for some $q\geq n,$ and $\alpha\in(0,1).$ We denote by
${\rm HMO^{\alpha}_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$ the class of all
$C^{1}$-functions $u(x,t)$ of the solution of
${\mathbb{L}}u=-u_{tt}+\mathcal{L}u=0$ such that
(4.5) $\displaystyle\|u\|^{2}_{{\rm
HMO^{\alpha}_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}$ $\displaystyle=$
$\displaystyle\sup_{x_{B},r_{B}}r_{B}^{-(n+2\alpha)}\int_{0}^{r_{B}}\int_{B(x_{B},r_{B})}t|\nabla
u(x,t)|^{2}{dxdt}<\infty.$
Then we have
* (1)
If $u\in{\rm HMO^{\alpha}_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$, then there
exist some $f\in\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$ and a constant
$C>0$ such that $u(x,t)=e^{-t\sqrt{\mathcal{L}}}f(x)$, and
$\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\leq C\|u\|_{{\rm
HMO^{\alpha}_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}.$
* (2)
If $f\in\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$, then
$u(x,t)=e^{-t\sqrt{\mathcal{L}}}f(x)\in{\rm
HMO^{\alpha}_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$ with $\|u\|_{{\rm
HMO^{\alpha}_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}\approx\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}.$
Part (2) of Theorem 4.1 is a straightforward result from the following
proposition.
###### Proposition 4.2.
Suppose $V\in B_{q}$ for some $q\geq n,$ and $\alpha\in(0,1)$ and $f$ is a
function such that
$\int_{\mathbb{R}^{n}}\frac{\left|f(x)\right|}{(1+\left|x\right|)^{n+\alpha+\varepsilon}}~{}dx<\infty$
for some $\varepsilon>0$. Then the following statements are equivalent:
* (1)
$f\in\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$;
* (2)
There exists a constant $C>0$ such that
$\|t\nabla e^{-t\sqrt{\mathcal{L}}}f(x)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
Ct^{\alpha};$
* (3)
$u(x,t)=e^{-t\sqrt{\mathcal{L}}}f(x)\in{\rm
HMO^{\alpha}_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})$ with $\|u\|_{{\rm
HMO^{\alpha}_{\mathcal{L}}}(\mathbb{R}_{+}^{n+1})}\approx\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}.$
###### Proof.
$\rm(1)\Rightarrow(2)$. Let
$f\in\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$. One writes
$\displaystyle t\nabla e^{-t\sqrt{\mathcal{L}}}f(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}t\nabla{\mathcal{P}}_{t}(x,z)\left(f(z)-f(x)\right)~{}dz+f(x)t\nabla
e^{-t\sqrt{\mathcal{L}}}(1)(x)$ $\displaystyle=I(x)+II(x).$
From Lemma 3.9, we have
$\displaystyle|I(x)|$ $\displaystyle\leq$ $\displaystyle
C\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\int_{\mathbb{R}^{n}}|t\nabla{\mathcal{P}}_{t}(x,z)|\left|x-z\right|^{\alpha}dz$
$\displaystyle\leq$ $\displaystyle
C\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\int_{\mathbb{R}^{n}}\frac{t\left|x-z\right|^{\alpha}}{\left(t+\left|x-z\right|\right)^{n+1}}~{}dz$
$\displaystyle\leq$ $\displaystyle
Ct^{\alpha}\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}.$
To estimate the term $II(x)$, we consider two cases.
Case 1: $\rho(x)\leq t$. In this case we use Lemma 3.9 to obtain
$\displaystyle|II(x)|$ $\displaystyle\leq$
$\displaystyle\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\rho(x)^{\alpha}\big{|}t\nabla
e^{-t\sqrt{\mathcal{L}}}(1)(x)\big{|}$ $\displaystyle\leq$ $\displaystyle
Ct^{\alpha}\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}.$
Case 2: $\rho(x)>t$. By Lemma 3.9, there exists a $\delta>1$ such that
$\displaystyle|II(x)|$ $\displaystyle\leq$ $\displaystyle
C\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\rho(x)^{\alpha}\Big{(}{t\over\rho(x)}\Big{)}^{\delta}$
$\displaystyle\leq$ $\displaystyle
C\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\rho(x)^{\alpha}\Big{(}{t\over\rho(x)}\Big{)}^{\alpha}$
$\displaystyle=$ $\displaystyle
Ct^{\alpha}\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})},$
which, together with estimate of $I(x)$, yields $\|t\nabla
e^{-t\sqrt{\mathcal{L}}}f(x)\|_{L^{\infty}(\mathbb{R}^{n})}\leq
Ct^{\alpha}\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}.$
$\rm(2)\Rightarrow(3)$. For every ball $B=B(x_{B},r_{B})$, we have
$\displaystyle\int_{0}^{r_{B}}\int_{B}t|\nabla
e^{-t\sqrt{\mathcal{L}}}f(x)|^{2}{dx~{}dt}$ $\displaystyle\leq
C\|f\|^{2}_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\int_{0}^{r_{B}}\int_{B}t^{2\alpha-1}~{}{dt~{}dx}$
$\displaystyle\leq
Cr_{B}^{n+2\alpha}\|f\|^{2}_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})},$
and hence $u(x,t)=e^{-t\sqrt{\mathcal{L}}}f(x)\in{\rm
HMO_{\mathcal{L}}^{\alpha}}(\mathbb{R}_{+}^{n+1}).$
The proof of $\rm(3)\Rightarrow(1)$ is a direct consequence of [19, Theorem
1.3]. This completes the proof. ∎
To prove part (1) of Theorem 4.1, we need some preliminary results.
###### Lemma 4.3.
Let $\alpha\in(0,1).$ For every $u\in{\rm
HMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}_{+}^{n+1})$ and every
$k\in{\mathbb{N}}$, there exists a constant $C_{k,\alpha}>0$ such that
$\int_{\mathbb{R}^{n}}{|u(x,{1/k})|^{2}\over 1+|x|^{2n+1}}dx\leq
C_{k,\alpha}<\infty,$
and so $u(x,1/k)\in L^{2}((1+|x|)^{-(2n+1)}dx)$. Hence for all
$k\in{\mathbb{N}}$, $e^{-t\sqrt{\mathcal{L}}}(u(\cdot,{1/k}))(x)$ exists
everywhere in ${\mathbb{R}}^{n+1}_{+}$.
###### Lemma 4.4.
Let $\alpha\in(0,1).$ For every $u\in{\rm
HMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}_{+}^{n+1})$ and for every
$k\in{\mathbb{N}}$, there exists a constant $C>0$ independent of $k$ such that
$\displaystyle\|u(\cdot,1/k)\|_{{\rm
BMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}$ $\displaystyle\leq$
$\displaystyle C\|u\|_{{\rm
HMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}_{+}^{n+1})}.$
Hence for all $k\in{\mathbb{N}}$, $u(x,1/k)$ is uniformly bounded in ${\rm
BMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$.
The proof of Lemma 4.3 is analogous to that of Lemma 3.1. For Lemma 4.4,
similar arguments as in Lemmas 3.2 and 3.4 show that
$u(x,t+1/k)=e^{-t\sqrt{\mathcal{L}}}\big{(}u(\cdot,1/k)\big{)}(x)$ satisfies
(4.6) $\displaystyle\hskip
28.45274pt\sup_{x_{B},r_{B}}r_{B}^{-(n+2\alpha)}\int_{0}^{r_{B}}\int_{B(x_{B},r_{B})}t|\partial_{t}e^{-t\sqrt{\mathcal{L}}}\big{(}u(\cdot,1/k)\big{)}(x)|^{2}{dxdt}\leq
C\|u\|^{2}_{{\rm HMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}_{+}^{n+1})}$
for all $k\in{\mathbb{N}}.$ This, together with [19, Theorem 1.3], shows that
$u(x,1/k)\in{\rm BMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$, and then the
norm of $u(x,1/k)$ in the space ${\rm
BMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})$ is less than the left hand side
of (4.6). We omit the detail here.
###### Proof of part (1) Theorem 4.1.
By using Lemma 2.6 for $\partial_{t}u(x,t+{1/k})$ and a similar argument as in
(3.2) we show that
$|\partial_{t}u(x,t)|\leq Ct^{\alpha-1}\|u\|_{{\rm
HMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}_{+}^{n+1})},$
and hence if $t_{1},t_{2}>0$,
$\displaystyle|u(x,t_{1})-u(x,t_{2})|$
$\displaystyle=\big{|}\int_{t_{2}}^{t_{1}}\partial_{s}u(x,s)ds\big{|}\leq
C\|u\|_{{\rm
HMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}_{+}^{n+1})}\big{|}\int_{t_{2}}^{t_{1}}s^{\alpha-1}ds\big{|}$
$\displaystyle\leq C|t_{1}^{\alpha}-t_{2}^{\alpha}|\leq
C|t_{1}-t_{2}|^{\alpha}$
since $0<\alpha<1.$ The family $\\{u(x,t)\\}$ is a Cauchy sequence as $t$
tends to zero and hence converges to some function $f(x)$ everywhere.
Now we apply Lemma 4.4, and note that for all $k\in{\mathbb{N}}$,
$\|u(\cdot,1/k)\|_{{\rm BMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\leq
C\|u\|_{{\rm HMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}_{+}^{n+1})}<\infty.$
Letting $k$ tend to $\infty$, we conclude
$\|f\|_{\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\leq C\|f\|_{{\rm
BMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}^{n})}\leq C\|u\|_{{\rm
HMO}^{\alpha}_{\mathcal{L}}(\mathbb{R}_{+}^{n+1})},$
and hence $u(x,t)=e^{-t\sqrt{\mathcal{L}}}f(x).$ This completes the proof of
part (1) of Theorem 4.1. ∎
Acknowledgments. X.T. Duong was supported by Australia Research Council (ARC).
L. Yan was supported by NNSF of China (Grant No. 10925106 and 11371378),
Guangdong Province Key Laboratory of Computational Science and Grant for
Senior Scholars from the Association of Colleges and Universities of
Guangdong. C. Zhang was supported by the General Financial Grant from the
China Postdoctoral Science Foundation (Grant No. 2013M531883). L.X. Yan would
like to thank S. Hofmann, Z.W. Shen and J. Xiao for helpful discussions.
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|
arxiv-papers
| 2013-06-03T08:18:06 |
2024-09-04T02:49:46.032258
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xuan Thinh Duong, Lixin Yan and Chao Zhang",
"submitter": "Chao Zhang",
"url": "https://arxiv.org/abs/1306.0319"
}
|
1306.0361
|
# An Analysis of issues against the adoption of Dynamic Carpooling
Daniel Graziotin, Free University of Bozen-Bolzano, [email protected]
(December 2010)
###### Abstract
Using a private car is a transportation system very common in industrialized
countries. However, it causes different problems such as overuse of oil,
traffic jams causing earth pollution, health problems and an inefficient use
of personal time.
One possible solution to these problems is carpooling, i.e. sharing a trip on
a private car of a driver with one or more passengers. Carpooling would reduce
the number of cars on streets hence providing worldwide environmental,
economical and social benefits. The matching of drivers and passengers can be
facilitated by information and communication technologies. Typically, a driver
inserts on a web-site the availability of empty seats on his/her car for a
planned trip and potential passengers can search for trips and contact the
drivers. This process is slow and can be appropriate for long trips planned
days in advance. We call this static carpooling and we note it is not used
frequently by people even if there are already many web-sites offering this
service and in fact the only real open challenge is widespread adoption.
Dynamic carpooling, on the other hand, takes advantage of the recent and
increasing adoption of Internet-connected geo-aware mobile devices for
enabling impromptu trip opportunities. Passengers request trips directly on
the street and can find a suitable ride in just few minutes. Currently there
are no dynamic carpooling systems widely used. Every attempt to create and
organize such systems failed.
This paper reviews the state of the art of dynamic carpooling. It identifies
the most important issues against the adoption of dynamic carpooling systems
and the proposed solutions for such issues. It proposes a first input on
solving the problem of mass-adopting dynamic carpooling systems.
## 1 Introduction
Using a private car is a transportation system very common in industrialized
countries. Between 2004 and 2009, the worldwide production of private vehicles
has been of 295 millions of new units 111(Accessed Dec 13 2010)
http://oica.net/ and, as of 2004, there were 199 millions registered drivers
in the U.S.A.222 (Accessed Dec. 13 2010) http://www.fhwa.dot.gov/ [U.S.
Department of Transportation - Federal Highway Administration]. Road transport
is responsible for about 16% of man-made CO2 emissions333(Accessed Dec. 13
2010) http://oica.net/ [Organisation Internationale des Constructeurs
d’Automobile].
Private car travelling is a common but wasteful transportation system. Most
cars are occupied by just one or two people. Average car occupancy in the U.K.
is reported to be 1.59 persons/car, in Germany only 1.05 [4]. Private car
travelling creates a number of different problems and societal costs
worldwide. Environmentally, it is responsible for a wasteful use of a scarce
and finite resource, i.e. oil, and causes unnecessary earth pollution. The
traffic caused by single occupancy vehicles causes traffic jams and hugely
increases the amount of time spent by people in queues on streets. This is a
unsavvy use of another scarce resource: time. Moreover, the additional
pollution creates health problems to millions of individuals. Lastly, lone
drivers in separate cars miss opportunities to meet and talk, incurring in a
loss of potential social capital.
One possible solution to all these problems is carpooling, i.e. the act of
sharing a trip on a private vehicle between one or more other passengers. The
shared use of a single car by two or more people would reduce the number of
cars on streets. Carpooling helps the environment by allowing to use oil
wisely, to reduce earth pollution and consequent health problems. It reduces
traffic and - consequently - time that people spend in their cars. Carpooling
has also the potential of increasing social capital by letting people meet and
know each other.
Carpooling is not a widespread practice. There are already many systems
facilitating the match between drivers and passengers, most of them in form of
bulletin board-like web-sites. The intention of offering empty seats of a
vehicle is usually announced by a driver many days before the start of the
trip. The coordination between a driver and the passengers who are candidating
for sharing the trip with him/her is usually carried out by e-mails or private
messages in the web-site. Therefore, we may see carpooling as a _static_ way
of sharing a trip.
The availability of geo-aware, mobile devices connected to the Internet opens
up possibilities for the formation of carpools in short notice, directly on
streets. This phenomenon is called dyamic carpooling (also known as dynamic
ridesharing, instant ridesharing and agile ridesharing). Dan Kirshner,
researcher in this field and maintainer of http://dynamicridesharing.org
website defines it as follows: “A system that facilitates the ability of
drivers and passengers to make one-time ride matches close to their departure
time, with sufficient convenience and flexibility to be used on a daily
basis.”444Kirshner, D. (Accessed Dec 10${}^{\text{th}}$ 2010) -
http://dynamicridesharing.org .
Currently there are no dynamic carpooling systems widely used. In fact, there
are many problematic issues related to the implementation and the adoption of
dynamic carpooling systems. In Section 2 we present our analysis of the state
of the art on dynamic carpooling. We collected all research papers about
dynamic carpooling and issues against its mass adoption. We critically analyze
the issues and the solution proposals in Section 3 of this paper. While we
acknowledge all aspects are critical, we claim that the basic technological
infrastructure is an important required and key building block. In Section 4
we summarize our thoughts and outcomes about this analysis.
### 1.1 Terminology
In Table 1 we introduce some key concepts used in this paper.
Term | Definition
---|---
Person | A user registered in the system, with login and password
Trip | The Driver can create Trips in the system. A Trip is the information about the availability of seats in a car going from a certain location to a certain destination, driven by a Driver on a certain date
Driver | The role of a Person when he/she offers to share some seats on his/her car for a specific trip
Passenger | The role of a Person when he/she accepts to occupy a seat on a car of a Driver
Participation | The act of taking part into a Trip. The Driver participates by default in a Trip he/she created. A Passenger can participate in Trips created by a Driver. Participation can be just requested by the passenger or already confirmed by the driver.
Location | A geographical location.
Table 1: Key Terms of this paper
## 2 State of the art
This section contains a summary of previously published papers, in order of
publication. In the next section we present the outcomes of the analysis of
the whole state of the art and how we decided to move in order to provide a
significant contribute in solving the problem of adopting dynamic ridesharing
services.
#### Sociotechnical support for Ride Sharing[9]
This paper lists barriers to dynamic carpooling adoption and possible actions
to reduce them. It reports about High Occupancy Vehicles (HOV) lane - which
are lanes dedicated for people doing carpooling - on streets of San Francisco
and Oakland and complains that there should be no fees on bridges for HOVs.
The author suggests conventions developed between drivers and passengers (e.g.
pickup points near public transportation stops). Regarding security, the paper
suggests to give priority to female passengers, to not leave them alone
waiting for a ride. The paper reports that there are no stories about rape,
kidnapping or murder and the most common reported problem is bad driving.
There are suggestions on needed research:
* •
Need of location-aware devices, because dynamic carpooling is actually limited
to fixed pickups and drop-off locations.
* •
Simple user interfaces for passengers and drivers.
* •
Routing matching algorithms: short window of opportunity to match passenger
and driver.
* •
Time-to-pickup algorithms: to help passenger decide whether to use carpooling
or Public Transportation System.
* •
Safety and reputation system design: authenticate passenger and driver before
making the match, monitor arrival at destination, feedback system.
The paper discusses about social capital impacts: there is the potential for
creating new social connections and also matching drivers and passengers
according to their profiles creates bridging across class, race and religious
views.
#### Pilot Tests of Dynamic Ridesharing[6]
The author presents three pilot tests done in the USA, all of them failed. The
reasons of failure are the following:
* •
Too complicated rules and user interface
* •
Too weak marketing effort
* •
Too few users. After 1 month, 1000 flyers distributed to the public and a
proposed discount on parking, only 12 users were using the system.
The paper adds the idea of saving money when parking. It also enforces the
idea of using social networks to allow car pooling on the fly. The author
envisions using a web – and mobile service, also introducing some interesting
user stories.
#### The smart Jitney: Rapid, Realistic Transport [8]
The paper focuses on environmental benefits of dynamic carpooling. It asserts
that dynamic carpooling would lower greenhouse gas emissions in a better way
than electric/hydrogen/hybrid cars would do. It introduces the idea of Smart
Jitney: an unlicensed car driving on a defined route according to a schedule.
The author suggests the installation of Auto Event Recorders on cars,
enforcing security. It complains that challenges are all focused in convincing
the population to use the service, proposing a cooperative public development
of the system.
#### Auction negotiation for mobile Rideshare service[1]
The paper proposes the use of agent-based systems powered auction mechanisms
for driver-passenger matching.
#### Casual Carpooling - enhanced[5]
The author considers areas without HOV lanes and proposes the use of Radio
Frequency IDentification (RFID) chips to quickly identify passengers and
drivers. Readers should be installed at common pick-up points. The paper
complains that it would cost less to pay passengers and drivers for using the
service than to build a HOV lane.
#### Empty seats travelling[4]
This white paper by Nokia suggests to use the phone as a mean of
transportation, creating a value in terms of a transport opportunity. It
points out some factors limiting static carpooling, arranged via websites:
* •
Trip arrangements are not ad hoc
* •
It is impossible to arrange trips to head home from work or to drive shopping.
The paper notices that people are not widely encouraged to practice carpooling
by local governments. It collects obstacles and success factors in terms of
human sentences, and their solution. The authors say that the challenge is in
the definition of a path leading from existing ride share services to a fully
automated system.
#### Interactive systems for real time dynamic multi hop carpooling[3]
The author proposes a dynamic multi-hop system, by dividing a passenger route
into smaller segments being part of other trips. The author claims that the
problems of static carpooling are that matching drivers and passengers based
on their destinations limits the number of possible rides, and with high
waiting times. Carpooling is static and does not adapt itself well to ad hoc
traveling. The paper asks governments to integrate carpooling in laws and to
push for its use. The author complains that the perceived quality of service
is increased even driving the passenger away from destination: a driver and a
passenger should not be matched only if they share the same or similar
destination because perfect matching would require high waiting times.
The paper also addresses social aspects: in a single trip with 3 hops a
passenger might meet 3 to 10 people, therefore passengers may be socially
matched. It suggests to link the application with some social networks like
Facebook, MySpace and use profile information to match drivers and passengers.
As security improvement, the paper suggests: the use of finger-prints, RFID,
voice signature, display the location of vehicles on a map, using user
pictures, assigning random numbers to be used as passwords.
#### Instant Social Ride Sharing[2]
The paper proposes matching methodologies based on both a minimization of
detours and the maximization of social connections. It assumes the existence
of a social network data source in which users are connected by means of
groups, interests, etc. In such a network, the number of relatively short
paths between a driver and a passenger indicates the strength of their social
connection.
It provides algorithms and SQL queries. The authors assume that there is
already a large scale of users, and no barriers to adoption are taken into
account.
#### Combining Ridesharing & Social Networks[10]
The author envisions a mobile and web system that interacts with social
networks profiles that should improve security and trust by users. Users can
register to the system in a traditional way (e.g., by giving email, username,
password), then complete their profiles by linking their accounts to multiple
existing social networks account, to fill the remaining fields. Otherwise,
they have to fill the fields manually and verify their identity in more
classical ways. The paper proposes Opensocial555Google, MySpace et al.
(Accessed Dec. 19${}^{\text{th}}$ 2010) - http://www.opensocial.org/ as
connection interface. An own rating system is also complained, which keeps
scores of persons. Amongst the criteria are factors like reliability, safety
and friendliness.
It suggests the use of mobile systems, that should make use of GPS and
creation of a match on the fly (real-time algorithms). The paper provides some
results of surveys: people are willing to loose 23% more time to pickup a
friend of their social network rather than a stranger (6%). It also provides a
high-level description of the system and implementation details.
The author asks for extra research on psychological factors that increase
trust and perceived safety.
#### SafeRide: Reducing Single Occupancy Vehicles [7]
The publication is about a project in the U.S.A. It reports that there is a
market-formation problem: to achieve the system that attracts passengers,
there will have to be many drivers available. But the drivers will emerge only
when it appears profitable or otherwise desiderable, and that depends on there
being many passengers, etc. The author complains that someone must discover a
winning formula before anyone will invest.
The paper lists some interesting user stories, as well as algorithms and
requirements.
## 3 Comparative Analysis of Dynamic Carpooling Issues
The analysis of the state of the art brought some issues related to adopting
dynamic carpooling systems. We categorized the issues gathered from the state
of the art and their proposals in the following categories:
* •
Interface Design - all issues related to graphical implementation of clients
and ease of use
* •
Algorithms - the instructions regarding driver/passengers matching problems
* •
Coordination - the aspects related on how to let people meet, authenticate and
coordinate.
* •
Trustiness - the problems related on raising user confidence on dynamic
carpooling systems
* •
Safety - the issues regarding ensuring protection of users
* •
Social Aspects - all the issues related to create social connections and
raising social capital in dynamic carpooling systems
* •
Reaching Critical Mass - the problems on reaching a sufficient amount of
persons using the system that would attract more other people
* •
Incentives - all the political, motivational and economical issues related to
dynamic ridesharing systems
* •
System Suggestions - everything else that we consider relevant for building
dynamic carpooling systems
We attach the comparative analysis summarized in tables in Appendix, Table 2
up to table 5. For each category (columns), we list the suggestions and
interesting points made in the different research papers (rows), in form of
imperative sentences. Tables 2 to 5 in Appendix is our contribution
rationalizing the many problematic issues involved in the creation and
deployment of dynamic carpooling systems and in summarizing best practices and
suggestions in how to deal with them.
Our rationalization of dynamic carpooling issues and possible solutions shows
how dynamic carpooling systems still have many important open issues to be
addressed and solved. This fact explains the current absence of any dynamic
carpooling system deployed and used for real. The most addressed issue is
“Reaching the critical mass”. This problem has been faced in several ways but
noone of them worked. This issue seems very dependant to the issues named
“Incentives”, “Safety” and “Trustiness”. In order to receive incentives from
the government, there must be a system providing safety. Incentives and safety
would produce positive feedbacks and provide trustiness among the general
public, therefore providing at least a palliative for the critical mass issue.
We decided to address the overall challenge from a very core point of view and
to focus on technical aspects. Among the projects, we observed that their
source code and the prototypes produced were not freely accessible by the
general public. There are no information regarding the servers, that are all
proprietary and obscured. Another issue seems related to a missing
standardization of the protocols used. Therefore, every project started from
zero, “reinventing the wheel”. In order to overcome the “reaching critical
mass” issue, we believe that it is important that providers of dynamic
carpooling services can exchange their data easily so that cross provider
matching are possible.
That is, the open problem of dynamic carpooling still needs the basic building
blocks of research on which future work should be performed in order to
overcome all the other issues.
## 4 Conclusions and future work
In this paper we presented the outcomes of a research aimed at providing a
better understanding to the open problem of dynamic carpooling. Through an
analysis of the state of the art (reported in Section 2), we identified the
key issues in the domain of dynamic carpooling, presented in Section 3. Based
on the comparative analysis of the open issues, we notice that the basic
building block is missing: an open and extendable technological
infrastructure. A future field of research will be to create an opensource
framework providing basic dynamic carpooling functionalities and an open,
discussable protocol. This framework should be as much clear and extendable as
possible, to let other researchers work on different technical aspects. The
other, non technical issues should be solved afterwards.
Concluding, we believe the topic of this paper is a recent and challenging
one, still waiting for at least initial solutions and steps forwards. Common
goal of solving an important problem for our world: too many cars on our
streets with just one passenger in them.
## Appendix
Paper | Interface Design | Algorithms | Coordination
---|---|---|---
Sociotechnical support for Ride Sharing[9] | Give start, ending points and clear indications. Filter what information to reveal | |
Pilot Tests of Dynamic Ridesharing[6] | Provide lots of flexible settings to satisfy users. | | Provide a static/dynamic approach, let users insert entries days before the start
The smart Jitney: Rapid, Realistic Transport [8] | Provide different levels of services: \- simple: just destination and pickup \- groups preferences (only women etc.) \- scheduling of rides | |
Auction negotiation for mobile Rideshare service[1] | | |
Casual Carpooling - enhanced[5] | | | Implement one-time registration process, simple. Provide RFID devices for drivers and passengers
Empty seats travelling[4] | | |
Interactive systems for real time dynamic multi hop carpooling[3] | Focus on simplicity. Provide voice, speech recognition. Allow users to communicate each other. | | Driving passenger away from the destination but near transportation locations (e.g. a bus station) increases quality of service and enhances coordination.
Instant Social Ride Sharing[2] | | Given, built around social connections. Social network needed. | Built around social connection between users
Combining Ridesharing & Social Networks[10] | Implement a simple registration system from mobile phone. In a second phase link social networks profiles, or manual fill. Develop a very simple UI | |
SafeRide: Reducing Single Occupancy Vehicles[7] | | Both data structures and Algorithms for matching are given |
Table 2: Paper Analysis: Interface Design, Algorithms, Coordination Paper | Trustiness | Safety | Social Aspects
---|---|---|---
Sociotechnical support for Ride Sharing[9] | | Authenticate before the match: password / PIN monitor arrival at destination Provide a feedback system a la EBay | Announce matching items in profiles before the ride Do research in social capital aspects
Pilot Tests of Dynamic Ridesharing[6] | | Create a PIN at registration phase to be used by the client | Add social networking support to help finding neighbours
The smart Jitney: Rapid, Realistic Transport [8] | Brand the idea: apply stickers on every car that participates. Give limitations to drivers: age limits, extra driving tests, check on criminal records etc. | Provide Auto Event Recorders on cars. Implement an emergency button on mobile phone, record GPS data. Provide a feedback system a la EBay |
Auction negotiation for mobile Rideshare service[1] | | |
Casual Carpooling - enhanced[5] | Record carpooling activity when cars pass through RFID readers | Build it around RFID, record lots of data and positions |
Empty seats travelling[4] | Involve community and governments in planning and implementation phases | Let the service be available only to registered users; Provide a Feedback system | Give the possibility to create social connections
Interactive systems for real time dynamic multi hop carpooling[3] | | Use RFID, GPS. Implement a complete rating system. Display vehicle and driver information before entering a vehicle. Display participants pictures. Assign random numbers for passenger pickups to confirm the ride. Provide voice and video features. | Match passengers socially. Link the application to social networks.
Instant Social Ride Sharing[2] | Use social networks to enhance it. | |
Combining Ridesharing & Social Networks[10] | People are ready to spend 17% more time to pickup a friend of the social network rather than a stranger. Implement it. | Implement a rating system. Use and record GPS data. Do extra research in this field. |
SafeRide: Reducing Single Occupancy Vehicles[7] | Use social networks to enhance it. | Implement a GPS Help button. Record time, place, and sound. Develop a Feedback system |
Table 3: Paper Analysis: Trustiness, Safety, Social Aspects Paper | Critical Mass | Incentives | Suggestions
---|---|---|---
Sociotechnical support for Ride Sharing[9] | | | Provide a location-aware system Make use of mobile phones
Pilot Tests of Dynamic Ridesharing[6] | Provide mass marketing before, during and after deployment. Search for start-up incentives | Search an institutional sponsor. Make the government provide parking spaces to participants | Implement both Web and mobile clients. Implement a static and a dynamic approach. Start with a many-to-one system: all at a single destination
The smart Jitney: Rapid, Realistic Transport [8] | | Use a cooperative, public development of the system | Implement a Web interface and mobile clients (using phone calls)
Auction negotiation for mobile Rideshare service[1] | | |
Casual Carpooling - enhanced[5] | | Make employers incentive employees. Involve Regional Transportation Boards |
Empty seats travelling[4] | Create an incremental service, starting from a thread of backwards compatible services (bus, taxi). Don’t introduce new devices for the service, use mobile phones | Find a way to make the service a business case. Search for public incentives | Implement the system mobile only. Record GPS data. Provide a non-obtrusive system for authentication Research on quality of service measures
Interactive systems for real time dynamic multi hop carpooling[3] | A multi-hop system will solve the problem, as more rides will be available, waiting times will decrease and quality will rise. | Convince governments to change laws to enforce carpooling | Use a dynamic, multi-hop, real- time mobile system to minimize waiting times, one hop at a time
Instant Social Ride Sharing[2] | | | Use mobile phones and sms. Use GPS. Use a provided high-level description of the system
Table 4: Paper Analysis: Critical Mass, Incentives, Suggestions Pt. 1 Paper | Critical Mass | Incentives | Suggestions
---|---|---|---
Combining Ridesharing & Social Networks[10] | Involve users in some parts of development process. Research further on this topic. | | Implement a mobile and a web system that interacts with social networks profiles. Use Opensocial and other social networks. Use our high level description of the whole system
SafeRide: Reducing Single Occupancy Vehicles[7] | Market-formation problem: discover a new, winning formula. Start with an existing service, like taxis. Find large employers. Serve events (i.e.. concerts) | Find money. Search for incentives from governments | Implement our Use Cases Provide our functional requirements. Provide our non-functional requirements
Table 5: Paper Analysis: Critical Mass, Incentives, Suggestions Pt.2
## References
* [1] Abdel-Naby, S., Fante, S.: Auctions negotiation for mobile rideshare service. In Proc. IEEE Second International Conference on Pervasive Computing and Applications (2007)
* [2] Gidófalvi, G. et al. Instant Social Ride-Sharing. In Proc. 15th World Congress on Intelligent Transport Systems, p 8, Intelligent Transportation Society of America (2008)
* [3] Gruebele, P.A., Interactive System for Real Time Dynamic Multi-hop Carpooling. Global Transport Knowledge Partnership (2008).
* [4] Hartwig, S., Buchmann, M.: Empty seats traveling: Next-generation ridesharing and its potential to mitigate traffic and emission problems in the 21st century. Technical report, Nokia (2007), http://research.nokia.com/files/tr/NRC-TR-2007-003.pdf
* [5] Kelley, K. L. Casual Carpooling-Enhanced. Journal of Public Transportation 10, 4 (2007), 119.
* [6] Kirshner, D. Pilot Tests of Dynamic Ridesharing. Technical report (2006), http://www.ridenow.org/ridenow_summary.html
* [7] Morris, J. et. al. SafeRide: Reducing Single Occupancy Vehicles. Technical report (2008), http://www.cs.cmu.edu/~jhm/SafeRide.pdf
* [8] Murphy, P.: The smart jitney: Rapid, realistic transport. New Solutions Journal (4) (2007)
* [9] Resnick, P. SocioTechnical support for ride sharing. In Working Notes of the Symposium on Crossing Disciplinary Boundaries in the Urban and Regional Contex (UTEP-03) (2006).
* [10] Wessels, R. Combining Ridesharing & Social Networks. Technical report (2009), http://www.aida.utwente.nl/education/ITS2-RW-Pooll.pdf
|
arxiv-papers
| 2013-06-03T11:32:04 |
2024-09-04T02:49:46.044916
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniel Graziotin (Free University of Bozen-Bolzano)",
"submitter": "Daniel Graziotin",
"url": "https://arxiv.org/abs/1306.0361"
}
|
1306.0557
|
# Five lectures on DPG methods
Jay Gopalakrishnan Address: PO Box 751, Portland State University, Portland,
OR 97207-0751. [email protected]
Lecture Notes Jay Gopalakrishnan
FIVE LECTURES ON DPG METHODS Spring 2013, Portland
These lectures present a relatively recent introduction into the class of
discontinuos Galerkin (DG) methods, named Discontinuous Petrov-Galerkin (DPG)
methods. DPG methods, in which DG spaces form a critical ingredient, can be
thought of as least-square methods in nonstandard norms, or as Petrov-Galerkin
methods with special test spaces, or as a nonstandard mixed method. We will
pursue all these points of view in this lecture.
By way of preliminaries, let us recall two results from the Babuška-Brezzi
theory. Throughout, $b(\cdot,\cdot):X\times Y\to\mathbb{C}$ is a continuous
sesquilinear form where $X$ and $Y$ are (generally not equal) normed linear
spaces over the complex field $\mathbb{C}$, $Y^{*}$ denotes the space of
continuous conjugate-linear functionals on $Y$, and $\ell\in Y^{*}$.
###### Theorem 1.
Suppose $X$ is a Banach space and $Y$ is a reflexive Banach space. The
following three statements are equivalent:
1. a)
For any $\ell\in Y^{*}$, there is a unique $x\in X$ satisfying
$b(x,y)=\ell(y)\qquad\forall y\in Y.$ (1)
2. b)
$\\{y\in Y:\;b(z,y)=0,\;\forall z\in X\\}=\\{0\\}$ and there is a $C_{1}>0$
such that
$\displaystyle\inf_{0\neq z\in X}\sup_{0\neq y\in
Y}\frac{|b(z,y)|}{\|z\|_{X}\|y\|_{Y}}\geq C_{1},$ (2)
3. c)
$\\{z\in X:\;b(z,y)=0,\;\forall y\in Y\\}=\\{0\\}$ and there is a $C_{1}>0$
such that
$\displaystyle\inf_{0\neq y\in Y}\sup_{0\neq z\in
X}\frac{|b(z,y)|}{\|y\|_{Y}\|z\|_{X}}\geq C_{1}.$ (3)
###### Theorem 2.
Suppose $X$ and $Y$ are Hilbert spaces, $X_{h}\subset X$ and $Y_{h}\subset Y$
are finite dimensional subspaces, $\dim(X_{h})=\dim(Y_{h})$, and suppose one
of (a), (b) or (c) of Theorem 1 hold. If, in addition, there exists a
$C_{3}>0$ such that
$\inf_{0\neq z_{h}\in X_{h}}\sup_{0\neq y_{h}\in
Y_{h}}\frac{|b(z_{h},y_{h})|}{\|y_{h}\|_{Y}}\geq C_{3},$ (4)
then there is a unique $x_{h}\in X_{h}$ satisfying
$b(x_{h},y_{h})=\ell(y_{h})\qquad\forall y_{h}\in Y_{h},$ (5)
and
$\|x-x_{h}\|_{X}\leq\frac{C_{2}}{C_{3}}\inf_{z_{h}\in X_{h}}\|x-z_{h}\|_{X},$
where $C_{2}>0$ is any constant for which the inequality $|b(x,y)|\leq
C_{2}\|x\|_{X}\|y\|_{Y}$ holds for all $x\in X$ and $y\in Y$.
We studied these well-known theorems in earlier lectures – see your earlier
class notes for their full proofs and examples. Methods of the form (5) with
$X_{h}\neq Y_{h}$ are called Petrov-Galerkin (PG) methods with trial space
$X_{h}$ and test space $Y_{h}$. Note the standard difficulty: The inf-sup
condition (2) does not in general imply the discrete inf-sup condition (4).
## 1\. Optimal test spaces
Although (2)$\centernot\implies$ (4) in general, we now ask: Is it possible to
find a test space $Y_{h}$ for which (2)$\implies$ (4)? We now show that the
answer is simple and affirmative. From now on, $X$ and $Y$ are Hilbert spaces,
$(\cdot,\cdot)_{Y}$ denotes the inner product on $Y$, and $X_{h}$ is a finite-
dimensional subspace of $X$ (where $h$ is some parameter related to the
dimension).
###### Definition 3.
Given any trial space $X_{h}$, we define its optimal test space for the
continuous sesquilinear form $b(\cdot,\cdot):X\times Y\to\mathbb{C}$ by
$Y_{h}^{\mathrm{opt}}=T(X_{h})$
where $T:X\to Y$ (the trial-to-test operator) is defined by
$(Tz,y)_{Y}=b(z,y)\qquad\forall y\in Y,\;z\in X.$ (6)
Equation (6) uniquely defines a $Tz$ for any given $z\in X$, by Riesz
representation theorem. We call $Tz$ the “optimal” test function of $z$,
because it solves an optimization problem, as we see next.
###### Proposition 4 (Optimizer).
For any $z\in X$, the maximum of
$f_{z}(y)=\frac{|b(z,y)|}{\|y\|_{Y}}$
over all nonzero $y\in Y$ is attained at $y=Tz$.
###### Proof.
By duality in Hilbert spaces,
$\displaystyle\sup_{0\neq y\in Y}f_{z}(y)$ $\displaystyle=\sup_{0\neq y\in
Y}\frac{|(Tz,y)_{Y}|}{\|y\|_{Y}}=\|Tz\|_{Y},$
and $f_{z}(Tz)=\|Tz\|_{Y}.$ ∎
###### Proposition 5 (Exact inf-sup condition $\implies$ Discrete inf-sup
condition).
If (2) holds, then (4) holds with $C_{3}=C_{1}$ when we set
$Y_{h}=Y_{h}^{\mathrm{opt}}$.
###### Proof.
For any $z_{h}\in X_{h}$, letting
$s_{1}=\sup_{0\neq y\in Y}\frac{|b(z_{h},y)|}{\|y\|_{Y}},\qquad
s_{2}=\sup_{0\neq y_{h}\in
Y_{h}^{\mathrm{opt}}}\frac{|b(z_{h},y_{h})|}{\|y_{h}\|_{Y}}$
it is obvious that $s_{1}\geq s_{2}$. To prove that $s_{1}\leq s_{2}$, since
$s_{1}=\|Tz_{h}\|_{Y}$ by Proposition 4,
$s_{1}=\|Tz_{h}\|_{Y}=\frac{|(Tz_{h},Tz_{h})_{Y}|}{\|Tz_{h}\|_{Y}}\leq\sup_{y_{h}\in
Y_{h}^{\mathrm{opt}}}\frac{|(Tz_{h},y_{h})_{Y}|}{\|y_{h}\|_{Y}}=\sup_{y_{h}\in
Y_{h}^{\mathrm{opt}}}\frac{|b(z_{h},y_{h})|}{\|y_{h}\|_{Y}}=s_{2},$
so $s_{1}=s_{2}$. Hence the discrete inf-sup constant equals the exact inf-sup
constant. ∎
###### Definition 6.
For any trial subspace $X_{h}\subset X$, the ideal PG method finds
$x_{h}\in{X_{h}}$ solving
$b({x_{h}},{y_{h}})=\ell({y_{h}}),\qquad\forall{y_{h}}\in
Y_{h}^{\mathrm{opt}}.$ (7)
###### Assumption 7.
Suppose $\\{z\in X:\;b(z,y)=0,\;\forall y\in Y\\}=\\{0\\}$ and suppose there
exist $C_{1},C_{2}>0$ such that
$C_{1}\|y\|_{Y}\leq\sup_{0\neq z\in X}\frac{|b(z,y)|}{\|z\|_{X}}\leq
C_{2}\|y\|_{Y}\qquad\forall y\in Y.$
The lower and upper inequalities are the exact inf-sup and continuity bounds,
respectively.
###### Theorem 8 (Quasioptimality).
Assumption 7 $\implies$ the ideal PG method (7) is uniquely solvable for
$x_{h}$ and
$\|x-x_{h}\|_{X}\leq\frac{C_{2}}{C_{1}}\inf_{z_{h}\in X_{h}}\|x-z_{h}\|_{X}$
where $x$ is the unique exact solution of (1).
###### Proof.
We want to apply Theorem 2. To this end, first observe that $T$ is injective:
Indeed, if $Tz=0$, then by (6), we have $b(z,y)=0$ for all $y\in Y$, so
Assumption 7 implies that $z=0$. Thus
$\dim(X_{h})=\dim(Y_{h}^{\mathrm{opt}})$.
Furthermore, if the inf-sup condition of Assumption 7 holds, then it is an
exercise (see Exercise 10 below) to show that the other inf-sup condition,
$C_{1}\|z\|_{Y}\leq\sup_{0\neq y\in Y}\frac{|b(z,y)|}{\|y\|_{Y}}\qquad\forall
z\in X,$ (8)
holds with the same constant $C_{1}$. This, together with Proposition 5 shows
that the discrete inf-sup condition (4) holds with the same constant. Hence
Theorem 2 gives the result. ∎
###### Exercise 9.
Suppose $Z_{1},Z_{2}$ are Banach spaces and $A:Z_{1}\to Z_{2}$ is a linear
continuous bijection. Then prove that the inverse of its dual
$(A^{\prime})^{-1}$ exists, is continuous, and
$(A^{\prime})^{-1}=(A^{-1})^{\prime}.$
###### Exercise 10.
Prove that, under the assumptions of Theorem 1, if Statement (c) of Theorem 1
holds for some $C_{1}$, then Statement (b) also holds with the same constant
$C_{1}$. (Hint: Use Exercise 9 but do not forget that our spaces are over
$\mathbb{C}$.)
###### Definition 11.
Let $R_{Y}:Y\to Y^{*}$ denote the Riesz map defined by
$(R_{Y}y)(v)=(y,v)_{Y},$ for all $y$ and $v$ in $Y$. It is well known to be
invertible and isometric:
$\|R_{Y}y\|_{Y^{*}}=\|y\|_{Y}.$ (9)
Let $B:X\to Y^{*}$ be the operator generated by the form $b(\cdot,\cdot)$,
i.e., $Bx(y)=b(x,y)$ for all $x\in X$ and $y\in Y$. By the definition of $T$
in (6), it is obvious that
$T=R_{Y}^{-1}\circ B.$ (10)
Finally, for any $z\in X$, we define the energy norm of $z$ by
$\vvvert{z}\vvvert_{X}\;\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\;\|Tz\|_{Y}$.
Clearly, by Proposition 4,
$\vvvert{z}\vvvert_{X}=\|Tz\|_{Y}=\sup_{0\neq y\in
Y}\frac{|b(z,y)|}{\|y\|_{Y}}.$
###### Exercise 12.
Prove that if Assumption 7 holds, then $\vvvert{\cdot}\vvvert_{X}$ and
$\|\cdot\|_{X}$ are equivalent norms:
$C_{1}\|z\|_{X}\leq\vvvert{z}\vvvert_{X}\leq C_{2}\|z\|_{X}\qquad\forall z\in
X.$
###### Theorem 13 (Residual minimization).
Suppose Assumption 7 holds and $x$ solves (1). Then, the following are
equivalent statements:
1. i)
$x_{h}\in X_{h}$ is the unique solution of the ideal PG method (7).
2. ii)
$x_{h}$ is the best approximation to $x$ from $X_{h}$ in the following sense:
$\vvvert{x-x_{h}}\vvvert_{X}=\inf_{z_{h}\in X_{h}}\vvvert{x-z_{h}}\vvvert_{X}$
3. iii)
$x_{h}$ minimizes residual in the following sense:
$x_{h}=\operatorname{arg}\min_{z_{h}\in X_{h}}\|\ell-Bz_{h}\|_{Y^{*}}.$
###### Proof.
$(\ref{item:dpgls-1})\iff~{}(\ref{item:dpgls-2}):$
$\displaystyle x_{h}\text{ solves~{}\eqref{eq:ipg}}$ $\displaystyle\iff
b(x-x_{h},y_{h})=0$ $\displaystyle\forall y_{h}\in Y_{h}^{\mathrm{opt}}$
$\displaystyle\iff b(x-x_{h},Tz_{h})=0$ $\displaystyle\forall z_{h}\in X_{h}$
$\displaystyle\iff(T(x-x_{h}),Tz_{h})_{Y}=0$ $\displaystyle\forall z_{h}\in
X_{h},$
and the result follows since $(T\cdot,T\cdot)_{Y}$ is the inner product
generating the $\vvvert{\cdot}\vvvert_{X}$ norm.
$(\ref{item:dpgls-2})\iff~{}(\ref{item:dpgls-3}):$
$\displaystyle\vvvert{x-x_{h}}\vvvert_{X}=\inf_{z_{h}\in
X_{h}}\vvvert{x-z_{h}}\vvvert_{X}$
$\displaystyle\iff\|T(x-x_{h})\|_{Y}=\inf_{z_{h}\in X_{h}}\|T(x-z_{h})\|_{Y}$
$\displaystyle\iff\|R_{Y}^{-1}B(x-x_{h})\|_{Y}=\inf_{z_{h}\in
X_{h}}\|R_{Y}^{-1}B(x-z_{h})\|_{Y},$ by (10),
$\displaystyle\iff\|B(x-x_{h})\|_{Y^{*}}=\inf_{z_{h}\in
X_{h}}\|B(x-z_{h})\|_{Y^{*}},$ by (9), $\displaystyle\iff\|\ell-
Bx_{h}\|_{Y^{*}}=\inf_{z_{h}\in X_{h}}\|\ell-Bz_{h}\|_{Y^{*}}$
since $\ell=Bx$. This proves the result. ∎
###### Definition 14.
Let $x$ solve (1) and $x_{h}$ solve (7). We call $\varepsilon=R_{Y}^{-1}(\ell-
Bx_{h})$ the error representation function. Clearly,
$\|\varepsilon\|_{Y}=\|R_{Y}^{-1}B(x-x_{h})\|_{Y}=\|T(x-x_{h})\|_{Y}=\vvvert{x-x_{h}}\vvvert_{X},$
i.e., the $Y$-norm of $\varepsilon$ measures the error in the energy norm.
Note that $\varepsilon$ is the unique element of $Y$ satisfying
$(\varepsilon,y)_{Y}=\ell(y)-b(x_{h},y),\qquad\forall y\in Y.$
###### Theorem 15 (Mixed Galerkin formulation).
The following are equivalent statements:
1. i)
$x_{h}\in X_{h}$ solves the ideal PG method (7).
2. ii)
$x_{h}$ and $\varepsilon$ solves the mixed formulation
$\displaystyle(\varepsilon,y)_{Y}+b(x_{h},y)$ $\displaystyle=\ell(y)$
$\displaystyle\forall y\in Y,$ (11a) $\displaystyle b(z_{h},\varepsilon)$
$\displaystyle=0$ $\displaystyle\forall z_{h}\in X_{h}.$ (11b)
###### Proof.
$(\ref{item:dpgmixed-1})\implies~{}(\ref{item:dpgmixed-2}):$ Since (11a) holds
by the definition of the error representation function, we only need to prove
(11b). To this end,
$b(z_{h},\varepsilon)=(Tz_{h},\varepsilon)_{Y}=(Tz_{h},R_{Y}^{-1}(\ell-
Bx_{h}))_{Y}=(Tz_{h},T(x-x_{h}))_{Y}$, which being the conjugate of
$b(x-x_{h},Tz_{h})$, vanishes.
$(\ref{item:dpgmixed-2})\implies~{}(\ref{item:dpgmixed-1}):$ Since (11a)
implies $b(x_{h},y_{h})=\ell(y_{h})-(\varepsilon,y_{h})_{Y}$ for all $y_{h}\in
Y_{h}^{\mathrm{opt}}$, it suffices to prove that $(\varepsilon,y_{h})_{Y}=0$
for all $y_{h}\in Y_{h}^{\mathrm{opt}}$: But this is obvious from
$\displaystyle(Tz_{h},\varepsilon)_{Y}$ $\displaystyle=b(z_{h},\varepsilon)=0$
$\displaystyle\forall z_{h}\in X_{h},$
which holds by virtue of (11b). ∎
To summarize the theory so far, we have shown that there are test spaces that
can pair with any given trial space to generate an ideal Petrov-Galerkin
method that is guaranteed to be stable. Moreover, the discrete inf-sup
constant of the method is the same as the exact inf-sup constant. We then
showed, in Theorem 13, that the resulting methods are least square methods
that minimize the residual in a dual norm. Finally, we also showed that the
method can be interpreted as a (standard Galerkin rather than a Petrov-
Galerkin) mixed formulation on $Y\times X_{h}$, after introducing an error
representation function.
###### Definition 16.
Suppose an open $\varOmega\subset\mathbb{R}^{N}$ is partitioned into disjoint
open subsets $K$ (called elements), forming the collection ${\varOmega_{h}}$
(called mesh), such that the union of $\bar{K}$ for all $K\in{\varOmega_{h}}$
is $\bar{\varOmega}.$ Let $Y(K)$ denote a Hilbert space of some functions on
$K$ with inner product $(\cdot,\cdot)_{Y(K)}$. An ideal DPG method is an ideal
PG method with
$Y=\prod_{K\in{\varOmega_{h}}}Y(K),$ (12)
endowed with the inner product
$(y,v)_{Y}=\sum_{K\in{\varOmega_{h}}}(y|_{K},v|_{K})_{Y(K)}\qquad\forall
y,v\in K,$
where $y|_{K}$ denotes the $Y(K)$-component of any $y$ in the product space
$Y$. For example, the space $Y=H^{1}(\varOmega)$ is not of the form (12),
while
$Y=H^{1}({\varOmega_{h}})\;\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\;\\{v\in
L^{2}(\varOmega):v|_{K}\in H^{1}(K)$ for all $K\in{\varOmega_{h}}\\}$ is of
the form (12).
Such DPG methods are interesting due to the resulting locality of $T$. Note
that to compute a basis for the optimal test space, we must solve (6) to
compute $Tz$ for each $z$ in a basis of $X_{h}$. That equation, namely
$(Tz,y)_{Y}=b(z,y)$, decouples into independent equations on each element, if
$Y$ has the form (12). Indeed, the component of $Tz$ on an element $K$, say
$t_{K}=Tz|_{K}$ can be computed (independently of other elements) by solving
$(t_{K},y_{K})_{Y(K)}=b(z,y_{K})$ for all $y_{K}\in Y(K).$ The adjective
discontinuous in the name “DPG” should no longer be a surprise since test
spaces $Y$ of the form (12) admit (discontinuous) functions with no continuity
constraints across element interfaces.
## 2\. Examples and Connections
Although we presented the theory over $\mathbb{C}$ for generality (e.g., to
cover harmonic wave propagation), it continues to apply for real-valued
bilinear and linear forms (in place of sesquilinear and conjugate-linear
forms). All the examples in this section are over $\mathbb{R}$.
###### Example 17 (Standard FEM).
Set
$\displaystyle(v,w)_{Y}$
$\displaystyle=\int_{\varOmega}\mathop{\mathrm{grad}}v\cdot\mathop{\mathrm{grad}}w\;,\quad
X=Y=H_{0}^{1}(\varOmega),\quad\|u\|_{X}^{2}=\|u\|_{Y}^{2}=(u,u)_{Y},$
and consider the standard weak formulation of the Dirichlet problem: For any
given $F\in H^{-1}(\varOmega)$, find $u\in H_{0}^{1}(\varOmega)$ solving
$b(u,v)=F(v),\qquad\forall v\in H_{0}^{1}(\varOmega),$
where $b(u,v)=(u,v)_{Y}$. Clearly, in this case, the trial-to-test operator
$T$ is the identity map $I$ on $X$. Hence, if we set $X_{h}$ to the standard
Lagrange finite element subspace of $H_{0}^{1}(\varOmega)$ based on a finite
element mesh, we get $Y_{h}^{\mathrm{opt}}=X_{h}$. Thus the standard finite
element method uses an optimal test space. Since the form is coercive, it
obviously satisfies Assumption 7, so the previously discussed theorems apply
for this method.
###### Example 18 ($L^{2}$-based least-squares method).
Suppose $X$ is a Hilbert space and $A:X\to L^{2}(\varOmega)$ is a continuous
bijective linear operator. Then setting $Y=L^{2}(\varOmega)$, the problem of
finding a $u\in X$ such that $Au=f$, for any given $f\in Y$, can be put into a
variational formulation by setting
$b(u,v)=(Au,v)_{Y},\qquad\ell(v)=(f,v)_{Y}.$
Then it is obvious that $Tu=Au,$ so $Y_{h}^{\mathrm{opt}}=AX_{h}$. It is also
easy to verify that Assumption 7 holds: By the bijectivity of $A$,
uniqueness: $\displaystyle z\in X,b(z,y)=0\;\forall\;y\in
L^{2}(\varOmega)\implies Az=0\implies z=0,$ inf-sup: $\displaystyle\sup_{0\neq
z\in
X}\frac{|(Az,y)_{Y}|}{\|z\|_{X}}\geq\frac{|(y,y)_{Y}|}{\|A^{-1}y\|_{X}}\geq
C_{1}\|y\|_{Y},$
with $C_{1}=\|A^{-1}\|^{-1}$. Hence, Theorems 8 and 13 hold for this method.
Finally, since $B=R_{L^{2}(\varOmega)}A$ and $\ell=R_{L^{2}(\varOmega)}f$, by
the isometry of the Riesz map, the residual minimization property of Theorem
13(iii) implies that for any trial subspace $X_{h}\subset X$, we have
$x_{h}=\operatorname{arg}\min_{z_{h}\in
X_{h}}\|f-Az_{h}\|_{L^{2}(\varOmega)},$
i.e., in this example, the ideal PG method coincides with the standard
$L^{2}(\varOmega)$-based least-squares method.
###### Example 19 (1D o.d.e. without integration by parts).
Let $\varOmega=(0,1)$, $f\in L^{2}(\varOmega)$. Consider the boundary value
problem (where primes denote differentiation) to find $u(x)$ solving
$\displaystyle u^{\prime}$ $\displaystyle=f$ $\displaystyle\text{ on }(0,1),$
(13a) $\displaystyle u(0)$ $\displaystyle=0$ $\displaystyle\text{(boundary
condition at $x=0$)}.$ (13b)
A Petrov-Galerkin variational formulation is immediately obtained by
multiplying (13a) with an $L^{2}$ test function $v$ and integrating. The
resulting forms are
$b(u,v)=\int_{0}^{1}u^{\prime}v,\quad\ell(v)=\int_{0}^{1}fv,$
and the spaces are
$\displaystyle X=\\{u\in H^{1}(0,1):\;u(0)=0\\},\quad Y=L^{2}(0,1).$
This now fits into the framework of Example 18. (Indeed, the operator
$Au=u^{\prime},\quad A:X\to Y\text{ is a bijection,}$ (14)
because for any $f\in Y$, the function $u(x)=\int_{0}^{x}f(s)\;ds$ is in $X$
and satisfies $Au=f$.) Hence, as already discussed in Example 18, this method
reduces to a standard $L^{2}$-based least-squares method.
###### Example 20 (1D o.d.e. with integration by parts).
We consider the same boundary value problem as above, namely (13), but now
develop a different variational formulation for it. Multiply (13a) by a test
function $v\in C^{1}(\bar{\varOmega})$ and integrate by parts to get
$-\int_{0}^{1}uv^{\prime}+u(1)v(1)-u(0)v(0)=\int_{0}^{1}fv$
Using (13b) and letting the unknown value $u(1)$ to be a separate variable
$\hat{u}_{1}$, to be determined, we have derived the variational equation
$-\int_{0}^{1}uv^{\prime}+\hat{u}_{1}v(1)=\int_{0}^{1}fv.$
We let the pair $(u,\hat{u}_{1})$ to be a group variable $z$, and fix an
appropriate functional setting. Set the forms by
$b(z,v)\equiv
b(\,(u,\hat{u}_{1}),v)=\hat{u}_{1}v(1)-\int_{0}^{1}uv^{\prime},\qquad\ell(v)=\int_{0}^{1}fv,$
the spaces by
$X=L^{2}(\varOmega)\times\mathbb{R},\quad Y=H^{1}(\varOmega),\quad\text{where
}\varOmega=(0,1),$
and the norms by
$\displaystyle\|z\|_{X}^{2}$
$\displaystyle\equiv\|(u,\hat{u}_{1})\|_{X}^{2}=\|u\|_{L^{2}(\varOmega)}^{2}+|\hat{u}_{1}|^{2}$
$\displaystyle\|v\|_{Y}^{2}$
$\displaystyle=\|v^{\prime}\|_{L^{2}(\varOmega)}^{2}+|v(1)|^{2}.$ (15)
By Sobolev inequality, $v(1)$ makes sense for $v\in Y$, so the above set
$b(\cdot,\cdot)$ and $\|\cdot\|_{Y}$ are well-defined. In fact (by Exercise
21), the above set norm $\|v\|_{Y}$ is equivalent to the standard
$H^{1}(\varOmega)$ norm.
Next, let us verify Assumption 7. First, suppose $(u,\hat{u}_{1})$ satisfies
$b(\,(u,\hat{u}_{1}),v)=0\quad\forall v\in Y.$ (16)
Then, choosing $v\in\mathcal{D}(\varOmega)$, the set of infinitely
differentiable compactly supported functions in $\varOmega$, we find that the
distributional derivative $u^{\prime}$ vanishes. Hence $u\in
H^{1}(\varOmega)$. Going back to a general $v\in Y$, we may now integrate
equation (16) by parts to obtain
$-u(1)v(1)+u(0)v(0)+\hat{u}_{1}v(1)=0.$ (17)
Choosing $v(x)=1-x$, we obtain $u(0)=0$. Thus, $u$ solves (13) with zero data,
so $u\equiv 0$ by (14). From (17) we also have $u(1)=\hat{u}_{1}$, which
together with $u\equiv 0$ implies
$\hat{u}_{1}=0,\qquad u=0.$ (18)
Thus the uniqueness part of Assumption 7, $\\{z\in X:\;b(z,y)=0,\;\forall y\in
Y\\}=\\{0\\}$ holds.
For the remaining parts, we begin by noting that by Cauchy-Schwarz inequality,
$\displaystyle\sup_{z\in X}\frac{|b(z,v)|^{2}}{\|z\|_{X}^{2}}$
$\displaystyle=\sup_{(u,\hat{u}_{1})\in
X}\frac{\displaystyle{\left|\hat{u}_{1}v(1)-\int_{0}^{1}uv^{\prime}\right|^{2}}}{\|u\|_{L^{2}(\varOmega)}^{2}+|\hat{u}_{1}|^{2}}\leq\sup_{(u,\hat{u}_{1})\in
X}\frac{\displaystyle{\left(|\hat{u}_{1}|^{2}+\|u\|_{L^{2}(\varOmega)}^{2}\right)\|v\|_{Y}^{2}}}{\|u\|_{L^{2}(\varOmega)}^{2}+|\hat{u}_{1}|^{2}}=\|v\|_{Y}^{2}$
while on the other hand, given any $v\in Y$, choosing $u=-v^{\prime}$ and
$\hat{u}_{1}=v(1)$, we get
$\displaystyle\sup_{z\in X}\frac{|b(z,v)|^{2}}{\|z\|_{X}^{2}}$
$\displaystyle=\sup_{(u,\hat{u}_{1})\in
X}\frac{\displaystyle{\left|\hat{u}_{1}v(1)-\int_{0}^{1}uv^{\prime}\right|^{2}}}{\|u\|_{L^{2}(\varOmega)}^{2}+|\hat{u}_{1}|^{2}}\geq\frac{\displaystyle{\left||v(1)|^{2}+\int_{0}^{1}|v^{\prime}|^{2}\right|^{2}}}{\|-v^{\prime}\|_{L^{2}(\varOmega)}^{2}+|v(1)|^{2}}=\|v\|_{Y}^{2}$
Thus Assumption 7 holds with $C_{1}=C_{2}=1$.
We can now calculate the optimal test space (see Exercise 22 below) for any
given trial space. Let $P_{p}(\varOmega)$ denote the space of polynomials of
degree at most $p$ on $\varOmega$. We experiment with
$X_{h}=P_{p}(\varOmega)\times\mathbb{R},$
i.e., the discrete solution $x_{h}=(u_{h},\hat{u}_{1,h})$ has $u_{h}$ in
$P_{p}(\varOmega)\subset L^{2}(\varOmega)$ and the point flux value
approximation $\hat{u}_{1,h}$ in $\mathbb{R}$. The resulting method was
implemented in FEniCS (code can be downloaded from here). Collecting the
results obtained with an $f$ corresponding to an exact solution with a sharp
layer,
$u=\frac{e^{M(x-1)}-e^{-M}}{1-e^{-M}},$
we obtain Figure 1.
The first graph in Figure 1 plots the exact solution $u$ and the computed
$u_{h}$ for three values of $p$. We also implemented the method of Example 19
and plotted the corresponding solutions in the next graph in Figure 1.
Comparing, we find that the ideal PG method of the current example performs
better than that of Example 19. Finally, we also plotted the
$L^{2}(\varOmega)$-projections of the exact solution on $P_{p}(\varOmega)$ in
the last graph in Figure 1. Comparing the plots, the first and the third
figures appear identical. Exercise 23 asks you to show that this is indeed the
case.
Figure 1. Solutions from one-element one-dimensional computations
###### Exercise 21.
Prove that the norm defined in (15) is equivalent to the standard Sobolev norm
defined by
$\|v\|^{2}_{H^{1}(\varOmega)}=\|v\|_{L^{2}(\varOmega)}^{2}+\|v^{\prime}\|_{L^{2}(\varOmega)}^{2}$.
Hint: Use a Sobolev inequality and a Poincaré-type inequality.
###### Exercise 22.
Prove that in the setting of Example 20, an explicit formula for
$T(u,\hat{u}_{1})$ can be given for any $(u,\hat{u}_{1})\in X$:
$T(u,\hat{u}_{1})=\hat{u}_{1}+\int_{x}^{1}u(s)\,ds.$ (19)
Next, use (41) to prove that if $X_{h}=P_{p}(\varOmega)\times\mathbb{R}$, then
$Y_{h}^{\mathrm{opt}}=P_{p+1}(\varOmega)$.
###### Exercise 23.
Prove that the $u_{h}$ resulting from the ideal Petrov-Galerkin method of
Example 20 equals the $L^{2}(0,1)$ projection of $u$ and that
$\hat{u}_{1,h}=\hat{u}_{1}$. Hint: Apply Theorem 8.
###### Exercise 24.
Suppose $\varOmega=(0,1)$ is partitioned by the mesh
$0=x_{0}<x_{1}<\cdots<x_{m}=1$. Consider the method of Example 20, modified to
use the different trial subspace $X_{h}=\\{u:u|_{(x_{i-1},x_{i})}\in
P_{p}(x_{i-1},x_{i}),$ for $i=1,\ldots,m\\}\times\mathbb{R}$. Show that $T$
does not, in general, map locally supported trial functions to locally
supported test functions, by exhibiting a $(u,\hat{u}_{1})\in X_{h}$ such that
$\mathop{\mathrm{missing}}{supp}(u)\subseteq[x_{i-1},x_{i}]$ for some $i$ but
$\mathop{\mathrm{missing}}{supp}(T(u,\hat{u}_{1}))\not\subseteq[x_{i-1},x_{i}]$.
###### Example 25 (An ideal DPG method).
Continuing to consider (13), we now sketch how to extend the ideal PG scheme
of Example 20 to an ideal DPG scheme. Following the setting of Definition 16,
we assume $\varOmega=(0,1)$ is partitioned into ${\varOmega_{h}}$ consisting
of $m$ intervals $(x_{i-1},x_{i})$ for all $i=1,\ldots,m$, with $x_{0}=0$ and
$x_{m}=1$. Let $v^{\pm}(x)$ denote the limiting value of $v$ at $x$ from the
right and left, respectively. Set
$\displaystyle Y$ $\displaystyle=H^{1}({\varOmega_{h}})=\\{z\in
L^{2}(\varOmega):z|_{K}\in H^{1}(K),\;\forall K\in{\varOmega_{h}}\\}$
$\displaystyle\|y\|_{Y}^{2}$
$\displaystyle=\sum_{i=1}^{m}\left(|y^{-}(x_{i})|^{2}+\int_{x_{i-1}}^{x_{i}}|y^{\prime}|^{2}\right)$
$\displaystyle X$ $\displaystyle=L^{2}(\varOmega)\times\mathbb{R}^{m}$
$\displaystyle\|(u,{\hat{u}}_{1},{\hat{u}}_{2},\ldots,{\hat{u}}_{m})\|_{X}^{2}$
$\displaystyle=\|u\|_{L^{2}(\varOmega)}^{2}+|\hat{u}_{1}|^{2}+|\hat{u}_{2}|^{2}+\cdots+|\hat{u}_{m}|^{2}$
$\displaystyle\ell(y)$ $\displaystyle=(f,y)_{L^{2}(\varOmega)}$ $\displaystyle
b(\,(u,{\hat{u}}_{1},{\hat{u}}_{2},\ldots,{\hat{u}}_{m}),\,y)$
$\displaystyle=\sum_{i=1}^{m}\left({\hat{u}}_{i}y^{-}(x_{i})-{\hat{u}}_{i-1}y^{+}(x_{i-1})-\int_{x_{i-1}}^{x_{i}}uy^{\prime}\right),$
with the understanding that ${\hat{u}}_{0}=0$. Note that if $m=1$, then this
reduces to the method of Example 20. For general $m$, the action of the trial-
to-test operator $T$ is local and can be computed element by element (see
Exercise 26). The method for general $m$ can be analyzed as in Example 20 (see
Exercise 27).
###### Exercise 26.
Prove that, in the setting of Example 25,
$T(0,\cdots,0,\hat{u}_{i},0,\cdots,0)=\hat{u}_{i}\times\left\\{\begin{aligned}
&1&&\text{ if }x\in(x_{i-1},x_{i})\\\ &x-x_{i+1}-1&&\text{ if
}x\in(x_{i},x_{i+1})\\\ &0&&\text{ elsewhere.}\end{aligned}\right.$
###### Exercise 27.
Verify Assumption 7 for the formulation of Example 25.
The process by which we extended the formulation of Example 20 to that in
Example 25 is an instance of “hybridization”. Variables like $\hat{u}_{i}$ in
Example 25 are referred to by various names such as facet, or inter-element,
or interface unknowns, and in the DG community, by names like numerical fluxes
or numerical traces. To put the hybrid method in a more general PG context, we
use the abstract setting stated next.
###### Assumption 28.
Suppose $X$ takes the form $X_{0}\times\hat{X}$ where $X_{0}$ and $\hat{X}$
are two Hilbert spaces and let the finite-dimensional subspace $X_{h}$ have
the form $X_{h,0}\times\hat{X}_{h}$ with subspaces $X_{h,0}\subset X_{0}$ and
$\hat{X}_{h}\subset\hat{X}$. Suppose there are continuous sesquilinear forms
$\hat{b}(\cdot,\cdot):\hat{X}\times Y\to\mathbb{C}$ and
$b_{0}(\cdot,\cdot):X_{0}\times Y\to\mathbb{C}$, in terms of which
$b(\cdot,\cdot)$ is set by
$b(\,(u,\hat{u}),y\,)=b_{0}(u,y)+\hat{b}(\hat{u},y),$
for all $(u,\hat{u})\in X$ and $y\in Y$, and suppose
$Y_{0}=\\{y\in
Y:\hat{b}(\hat{u}_{h},y)=0,\;\forall\hat{u}_{h}\in\hat{X}_{h}\\}$ (20)
is a closed subspace of $Y$. In addition to the already defined $T:X\to Y$,
define $T_{0}:X_{0}\to Y_{0}$ by $(T_{0}u,y)_{Y}=b_{0}(u,y)$, for all $y\in
Y_{0}.$
Under this setting, we consider two ideal PG methods: $\displaystyle\text{Find
}(x_{h},\hat{x}_{h})\in X_{h}:$ $\displaystyle\quad
b(\,(x_{h},\hat{x}_{h}),y\,)=\ell(y)$ $\displaystyle\forall y\in
Y_{h}^{\mathrm{opt}}\equiv T(X_{h}).$ (21a) $\displaystyle\text{Find }x_{h}\in
X_{h,0}:$ $\displaystyle\quad b_{0}(x_{h},y)=\ell(y)$ $\displaystyle\forall
y\in Y_{h,0}^{\mathrm{opt}}\equiv T_{0}(X_{h,0}).$ (21b)
The interest in the “hybridized” form (21a) arises because, when moving from
$Y_{0}$ to $Y$, one can often obtain test spaces of the form in Definition 16,
which make $T$ local. This will become clearer in Example 30, discussed after
the next theorem, and later in Example 55.
###### Theorem 29 (Hybrid method).
Suppose Assumption 28 holds. Then, the test spaces in (21) satisfy
$Y_{h,0}^{\mathrm{opt}}\subset Y_{h}^{\mathrm{opt}}$. Hence,
$(x_{h},\hat{x}_{h})\in X_{h}\text{ solves~{}\eqref{eq:15}}\implies
x_{h}\text{ solves~{}\eqref{eq:16}.}$
###### Proof.
Since $Y_{h}^{\mathrm{opt}}$ is a closed subspace of $Y$, we have the
orthogonal decomposition
$Y=Y_{h}^{\mathrm{opt}}+Y_{\perp}$ (22)
where $Y_{\perp}$ is the $Y$-orthogonal complement of $Y_{h}^{\mathrm{opt}}$.
Let $y_{0}\in Y_{h,0}^{\mathrm{opt}}$. Apply (22) to decompose
$y_{0}=y_{h}+y_{\perp}$, with $y_{h}\in Y_{h}^{\mathrm{opt}}$ and
$y_{\perp}\in Y_{\perp}$.
First, we claim that $y_{\perp}\in Y_{0}$. This is because
$\displaystyle\hat{b}(\hat{u}_{h},y_{\perp})=(T(0,\hat{u}_{h}),y_{\perp})_{Y}=0\qquad\forall\hat{u}_{h}\in\hat{X}_{h}.$
The last identity followed from the orthogonality of $y_{\perp}$ to
$T(X_{h})$.
Next, we claim that $y_{\perp}=0$. It suffices to prove that
$(y_{0},y_{\perp})_{Y}=0$ since $\|y_{\perp}\|_{Y}^{2}=(y_{0},y_{\perp})_{Y}$.
Since $y_{0}\in Y_{h,0}^{\mathrm{opt}}$, there is a $u_{h}\in X_{h}$ such that
$y_{0}=T_{0}u_{h}$. Then,
$\displaystyle(y_{0},y_{\perp})_{Y}$
$\displaystyle=(T_{0}u_{h},y_{\perp})_{Y}=b_{0}(u_{h},y_{\perp})$
$\displaystyle\text{as }y_{\perp}\in Y_{0}$
$\displaystyle=(T(u_{h},0),y_{\perp})_{Y}=0$ $\displaystyle\text{as
}T(X_{h})\perp y_{\perp}.$
Finally, since $y_{\perp}=0$, we have $y_{0}=y_{h}+0\in Y_{h}^{\mathrm{opt}}.$
Thus $Y_{h,0}^{\mathrm{opt}}\subset Y_{h}^{\mathrm{opt}}$. The second
statement of the theorem is now obvious by choosing $y\in
Y_{h,0}^{\mathrm{opt}}$ in (21a). ∎
###### Example 30.
Set $\varOmega=(0,1)$,
$X_{0}=L^{2}(\varOmega)\times\mathbb{R},\hat{X}=\hat{X}_{h}=\mathbb{R}^{m-1},Y=H^{1}({\varOmega_{h}}),$
$\displaystyle b_{0}(\,(u,\hat{u}_{m}),y)$
$\displaystyle={\hat{u}}_{m}y^{-}(1)-\sum_{i=1}^{m}\int_{x_{i-1}}^{x_{i}}uy^{\prime},$
$\displaystyle\hat{b}(\,(\hat{u}_{1},\ldots,\hat{u}_{m-1}),y)$
$\displaystyle=\hat{u}_{1}y^{-}(x_{1})-\hat{u}_{m-1}y^{+}(x_{m-1})+\sum_{i=2}^{m-1}\left({\hat{u}}_{i}y^{-}(x_{i})-{\hat{u}}_{i-1}y^{+}(x_{i-1})\right),$
Then, the method (21a) yields the method of Example 25. It is easy to see that
$Y_{0}=H^{1}(\varOmega)$. Hence the method (21b) yields the method of Example
20. By Theorem 29, the (global basis of) optimal test functions of Example 20
can be expressed as a linear combination of the (local basis of) optimal test
functions of Example 25.
## 3\. Inexact test spaces
To compute the optimal test spaces, we need to apply $T$, which requires
solving (6), typically an infinite-dimensional problem. Although we have seen
some examples where the action of $T$ can be computed in closed form, for the
vast majority of interesting boundary value problems, this is not feasible.
Hence we are motivated to substitute the optimal test functions by inexact
(approximations of) optimal test functions.
Let $Y^{r}$ denote a finite-dimensional subspace of $Y$ (with the index $r$
related to its dimension.) Let $T^{r}:X\to Y^{r}$ be defined by
$(T^{r}w,y)_{Y}=b(w,y)$ for all $y\in Y^{r}$. In general, $T^{r}\neq T$.
###### Definition 31.
A DPG method for (1) uses a space $Y$ as in the ideal DPG method of Definition
16, finite-dimensional subspaces $X_{h}\subset X$ and $Y^{r}\subset Y$, and
computes $x_{h}$ in $X_{h}$ satisfying
$b({x_{h}},{y})=\ell({y}),\qquad\forall{y}\in
Y_{h}^{r}\;\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\;T^{r}(X_{h}).$ (23)
The DPG method is sometimes also called the “practical” DPG method, because it
uses the inexact, but practically computable, test space $Y_{h}^{r}$ (in
contrast to the ideal DPG method, which uses the exact optimal test space
$Y_{h}^{\mathrm{opt}}$).
###### Assumption 32.
There is a linear operator $\varPi:Y\to Y^{r}$ and a $C_{\varPi}>0$ such that
for all ${w_{h}}\in{X_{h}}$ and all $v\in Y$,
$b(w_{h},v-{\varPi}v)=0,\qquad\text{and}\qquad\|{\varPi}v\|_{Y}\leq
C_{{\varPi}}\|v\|_{Y}.$
###### Theorem 33.
Suppose Assumptions 7 and 32 hold. Then the DPG method (23) is uniquely
solvable for $x_{h}$ and
$\|x-x_{h}\|_{X}\leq\frac{C_{2}C_{\varPi}}{C_{1}}\inf_{z_{h}\in
X_{h}}\|x-z_{h}\|_{X}$
where $x$ is the unique exact solution of (1).
###### Proof.
First, note that by Assumption 32, $T^{r}:X_{h}\to Y^{r}$ is injective:
$T^{r}w_{h}=0$ for some $w_{h}\in X_{h}\implies b(w_{h},y^{r})=0$ for all
$y^{r}\in Y^{r}\implies b(w_{h},\varPi y)=0$ for all $y\in Y\implies
b(w_{h},y)=0$ for all $y\in Y$, which by Assumption 7 implies that $w_{h}=0$.
Thus,
$\dim(Y_{h}^{r})=\dim(X_{h}).$
Next, for any $z_{h}\in X_{h}$, let
$s_{0}=\sup_{0\neq y\in Y}\frac{|b(z_{h},y)|}{\|y\|_{Y}},\qquad
s_{1}=\sup_{0\neq y\in Y^{r}}\frac{|b(z_{h},y^{r})|}{\|y^{r}\|_{Y}},\qquad
s_{2}=\sup_{0\neq y\in
Y_{h}^{r}}\frac{|b(z_{h},y_{h}^{r})|}{\|y_{h}^{r}\|_{Y}}.$
The result will follow from Theorem 2 once we prove the discrete inf-sup
condition
$C_{1}C_{\varPi}^{-1}\|z_{h}\|_{X}\leq s_{2}.$ (24)
We proceed to bound $\|z_{h}\|_{X}$ using $s_{0}$, then $s_{1}$, and finally
$s_{2}$. Assumption 7 implies (by Exercise 10) that the inf-sup condition
$C_{1}\|z_{h}\|_{X}\leq s_{0}$
holds. Hence Assumption 32 implies
$\displaystyle C_{1}\|z_{h}\|_{X}$ $\displaystyle\leq\sup_{0\neq y\in
Y}\frac{|b(z_{h},y)|}{\|y\|_{Y}}=\sup_{0\neq y\in Y}\frac{|b(z_{h},\varPi
y)|}{\|y\|_{Y}}$ $\displaystyle\forall z_{h}\in X_{h}$
$\displaystyle\leq\sup_{0\neq y\in Y}\frac{|b(z_{h},\varPi
y)|}{C_{\varPi}^{-1}\|\varPi y\|_{Y}}\leq\sup_{0\neq y\in
Y^{r}}\frac{|b(z_{h},y^{r})|}{C_{\varPi}^{-1}\|y^{r}\|_{Y}}$
$\displaystyle\forall z_{h}\in X_{h},$
i.e., we have proven the tighter inf-sup condition
$C_{1}C_{\varPi}^{-1}\|z_{h}\|_{X}\leq s_{1}$. To finish the proof of (24), it
only remains to tighten it further by proving that $s_{1}\leq s_{2}$.
Analogous to Proposition 4, $s_{1}$ is attained at $T^{r}z_{h}$, so
$\displaystyle
s_{1}=\frac{(T^{r}z_{h},T^{r}z_{h})_{Y}}{\|T^{r}z_{h}\|_{Y}}\leq\sup_{0\neq
y_{h}^{r}\in
Y_{h}^{r}}\frac{(T^{r}z_{h},y_{h}^{r})_{Y}}{\|y_{h}^{r}\|_{Y}}=s_{2}.$
This shows (24) and finishes the proof. ∎
###### Remark 34.
Although Theorem 33 has more hypotheses than Theorem 8,
$\text{Theorem~{}\ref{thm:inex}}\implies\text{Theorem~{}\ref{thm:quasiopt}.}$
Indeed, the ideal PG method is obtained by simply setting $Y^{r}=Y$, and in
that case, the trivial operator $\varPi=I$ satisfies Assumption 32 with
$C_{\varPi}=1$. (Note that Theorem 33 holds if we use any closed subspace
$Y^{r}\subset Y$ in (23), not only finite-dimensional $Y^{r}$.)
###### Exercise 35 (Necessity & Sufficiency of Assumption 32).
Suppose Assumption 7 holds. If there is a $C_{0}>0$ such that for all
$\ell\in(Y_{h}^{r})^{*}$ there exists a unique $x_{h}\in X_{h}$ satisfying
(23) and moreover
$\|x_{h}\|_{X}\leq C_{0}\|\ell\|_{(Y_{h}^{r})^{*}},$
then the method (23) is called stable and $C_{0}$ is the stability constant of
the method. Show that
$\text{the method~{}\eqref{eq:pdpg} is
stable}\iff\text{Assumption~{}\ref{asm:Pi} holds},$
and relate the stability constant to the other constants.
###### Definition 36 (cf. Definition 11).
Let
$\vvvert{x}\vvvert_{r}\;\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\;\|T^{r}x\|_{Y}$.
Let $R_{Y^{r}}:{Y^{r}}\to{(Y^{r})}^{*}$ be the Riesz map defined by
$(R_{Y^{r}}y)(v)=(y,v)_{Y},$ for all $y$ and $v$ in ${Y^{r}}$. By the
definition of $T^{r}$, it is easy to see that
$T^{r}w=R_{Y^{r}}^{-1}Bw.$ (25)
###### Theorem 37 (cf. Theorem 13).
Suppose Assumptions 7 and 32 hold and let $x$ solve (1). Then, the following
are equivalent statements:
1. i)
$x_{h}\in X_{h}$ is the unique solution of the DPG method (23).
2. ii)
$x_{h}$ is the best approximation to $x$ from $X_{h}$ in the following sense:
$\vvvert{x-x_{h}}\vvvert_{r}=\inf_{z_{h}\in X_{h}}\vvvert{x-z_{h}}\vvvert_{r}$
3. iii)
$x_{h}$ minimizes residual in the following sense:
$x_{h}=\operatorname{arg}\min_{z_{h}\in X_{h}}\|\ell-
Bz_{h}\|_{{(Y^{r})^{*}}}.$
###### Proof.
Follow along the lines of proof of Theorem 13 but use (25) instead of (10). ∎
###### Definition 38 (cf. Definition 14).
Let $x$ solve (1). We call $\varepsilon^{r}=R_{Y^{r}}^{-1}(\ell-Bx_{h})$ the
error estimator of an $x_{h}$ in $X_{h}$. It is easy to see that it is the
unique element of $Y^{r}$ satisfying
$(\varepsilon^{r},y)_{Y}=\ell(y)-b(x_{h},y),$ for all $y\in Y^{r}$.
###### Theorem 39 (cf. Theorem 15).
The following are equivalent statements:
1. i)
$x_{h}\in X_{h}$ solves the DPG method (23).
2. ii)
$x_{h}\in X_{h}$ and $\varepsilon^{r}\in Y^{r}$ solve the mixed formulation
$\displaystyle(\varepsilon^{r},y)_{Y}+b(x_{h},y)$ $\displaystyle=\ell(y)$
$\displaystyle\forall y\in Y^{r},$ (26a) $\displaystyle
b(z_{h},\varepsilon^{r})$ $\displaystyle=0$ $\displaystyle\forall z_{h}\in
X_{h}.$ (26b)
###### Proof.
Follow along the lines of the proof of Theorem 15. ∎
###### Exercise 40.
Prove that $\varepsilon^{r}$ is $Y$-orthogonal to $Y_{h}^{r}$.
Next, recall the setting of Assumption (28) with
$b(\,(u,\hat{u}),y\,)=b_{0}(u,y)+\hat{b}(\hat{u},y).$ Analogous to (20),
define
$Y_{0}^{r}=\\{y\in
Y^{r}:\;\hat{b}(\hat{u}_{h},y)=0,\;\forall\hat{u}_{h}\in\hat{X}_{h}\\}$ (27)
and let $T_{0}^{r}:X_{0}\to Y_{0}^{r}$ be defined by
$(T_{0}^{r}u,y)_{Y}=b_{0}(u,y)$ for all $y\in Y_{0}^{r}.$ Then consider the
corresponding DPG methods:
$\displaystyle\text{Find }(x_{h},\hat{x}_{h})\in X_{h}:$ $\displaystyle\quad
b(\,(x_{h},\hat{x}_{h}),y\,)=\ell(y)$ $\displaystyle\forall y\in
Y_{h}^{r}\equiv T^{r}(X_{h}).$ (28a) $\displaystyle\text{Find }x_{h}\in
X_{h,0}:$ $\displaystyle\quad b_{0}(x_{h},y)=\ell(y)$ $\displaystyle\forall
y\in Y_{h,0}^{r}\equiv T_{0}^{r}(X_{h,0}).$ (28b)
###### Theorem 41 (cf. Theorem 29).
Suppose Assumption 28 holds. Then, the test spaces satisfy $Y_{h,0}^{r}\subset
Y_{h}^{r}$. Hence,
$(x_{h},\hat{x}_{h})\in X_{h}\text{ solves~{}\eqref{eq:26}}\implies
x_{h}\text{ solves~{}\eqref{eq:27}.}$
###### Proof.
Proceed as in the proof of Theorem 29, after replacing $Y$ by $Y^{r}$, and
$Y_{\perp}$ by the orthogonal complement of $Y_{h}^{r}$ in $Y^{r}$. ∎
###### Remark 42 (Some ways to implement DPG methods).
1. (1)
Choose a local basis for $X_{h}$, say $e_{j}$. Compute $v_{i}=T^{r}e_{i}$
(usually precomputed on a fixed reference element and mapped to physical
elements). Then assemble the square matrix
$A_{ij}=b(e_{j},v_{i})$ (29)
by usual finite element techniques and solve.
2. (2)
Let $e_{j}$ be as in item (1) and additionally select a local basis for
$Y^{r}$, say $y_{i}$. Assemble the rectangular (since $\dim X_{h}\leq\dim
Y^{r}$ typically) matrix
$B_{ij}=b(e_{j},y_{i})$
and the (block-diagonal) Gram matrix $M_{lm}=(y_{l},y_{m})_{Y}$. (Again, their
assembly can be done by precomputing element matrices on a fixed reference
element and mapping to physical elements.) Then form the square matrix
$A=B^{t}M^{-1}B.$ It is easy to see that this matrix equals (29), so we
proceed as in item (1).
3. (3)
Let $e_{j}$ and $y_{i}$ be as in item (2). Assemble the matrices of (26) and
solve. Since (26) is a standard Galerkin formulation, not a Petrov-Galerkin
formulation, this technique requires no further explanation. We will opt for
this method in the code in the next section.
###### Exercise 43.
Suppose the basis $e_{j}$ and the matrix $A$ are as in Remark 42(1). Prove
that Assumptions 7 and 32 imply the spectral condition number of $A$ satisfies
$\kappa(A)\leq\frac{\lambda_{1}}{\lambda_{0}}\frac{C_{2}^{2}C_{\varPi}^{2}}{C_{1}^{2}},$
where $\lambda_{0},\lambda_{1}$ are positive numbers such that
$\lambda_{0}\|\vec{\chi}\|_{2}^{2}\leq\|x\|_{X}^{2}\leq\lambda_{1}\|\vec{\chi}\|_{2}^{2}$
holds for all $x=\sum_{j}\chi_{j}e_{j}$ in $X_{h}$.
## 4\. The Laplacian
Let $\varOmega$ be a bounded connected open subset of $\mathbb{R}^{N}$ for any
$N\geq 2$ with Lipschitz boundary $\partial\varOmega$. We focus on the simple
boundary value problem
$\displaystyle-\Delta u$ $\displaystyle=f$ $\displaystyle\text{ on
}\varOmega,$ (30a) $\displaystyle u$ $\displaystyle=0$ $\displaystyle\text{ on
}\partial\varOmega.$ (30b)
All functions are real-valued in this section. We assume we have a mesh
${\varOmega_{h}}$ as in Definition 16 and additionally assume that $\partial
K$ is Lipschitz for all $K\in{\varOmega_{h}}$ (so that we may use trace
theorems on each element), but the shape of the elements is unimportant for
now.
To develop our PG formulation for (30), we set the test space by
$\displaystyle Y=H^{1}({\varOmega_{h}})=\\{v:\;v|_{K}\in H^{1}(K),\;\forall
K\in{\varOmega_{h}}\\}\equiv\prod_{K\in{\varOmega_{h}}}H^{1}(K),$
$\displaystyle(v,y)_{Y}=(v,y)_{\varOmega_{h}}+(\mathop{\mathrm{grad}}v,\mathop{\mathrm{grad}}y)_{\varOmega_{h}}.$
Multiplying (30) by a $v\in Y$ and integrating by parts on any element
$K\in{\varOmega_{h}}$, we obtain
$\int_{K}\mathop{\mathrm{grad}}u\cdot\mathop{\mathrm{grad}}v-\int_{\partial
K}(n\cdot\mathop{\mathrm{grad}}u)v=\int_{K}fv.$ (31)
As usual, the integral over $\partial K$ must be interpreted as a duality
pairing in $H^{1/2}(\partial K)$ if $u$ is not sufficiently regular. Summing
up (31) over all $K\in{\varOmega_{h}}$ and letting
$n\cdot\mathop{\mathrm{grad}}u$ be an independent unknown, denoted by
$\hat{q}_{n}$, we derive the PG formulation. To state it precisely, we use
these notations: Let
$(r,s)_{\varOmega_{h}}=\sum_{K\in{\varOmega_{h}}}(r,s)_{K}$ where
$(\cdot,\cdot)_{D}$, for any domain $D$, denotes the $L^{2}(D)$-inner product
and
$\langle{\ell,w}\rangle_{\partial{\varOmega_{h}}}=\sum_{K\in{\varOmega_{h}}}\langle{\ell,w}\rangle_{1/2,\partial
K}$ where $\langle{\ell,\cdot}\rangle_{1/2,\partial K}$ denotes the action of
a functional $\ell$ in $H^{-1/2}(\partial K)$. The PG weak formulation finds
$(u,\hat{q}_{n})\in X$ satisfying
$(\mathop{\mathrm{grad}}u,\mathop{\mathrm{grad}}v)_{\varOmega_{h}}-\langle{\hat{q}_{n},v}\rangle_{\partial{\varOmega_{h}}}=(f,v)_{\varOmega},\qquad\forall
v\in Y,$ (32)
where the trial space $X$ is defined as follows: First, define the element-by-
element trace operator $\mathop{\mathrm{tr}_{n}}$ by
$\mathop{\mathrm{tr}_{n}}:H(\mathop{\mathrm{missing}}{div},{\varOmega})\to\prod_{K\in{\varOmega_{h}}}H^{-1/2}(\partial
K),\qquad\mathop{\mathrm{tr}_{n}}r|_{\partial K}=r\cdot n|_{\partial K}.$
Here and throughout, $n$ generically denotes the unit outward normal of any
domain under consideration. Now set
$\displaystyle H^{-1/2}(\partial\varOmega_{h})$
$\displaystyle=\mathop{\mathrm{missing}}{ran}(\mathop{\mathrm{tr}_{n}}),$ (33)
$\displaystyle\|\hat{r}_{n}\|_{H^{-1/2}(\partial{\varOmega_{h}})}$
$\displaystyle=\inf\bigg{\\{}\|q\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}:\;\forall
q\in H(\mathop{\mathrm{missing}}{div},{\varOmega})\text{ such that
}\mathop{\mathrm{tr}_{n}}(q)=\hat{r}_{n}\bigg{\\}}.$
The trial space is then given by
$\displaystyle X$ $\displaystyle=H_{0}^{1}(\varOmega)\times
H^{-1/2}(\partial{\varOmega_{h}}),$ $\displaystyle\|(w,\hat{r}_{n})\|_{X}^{2}$
$\displaystyle=\|\mathop{\mathrm{grad}}w\|_{L^{2}(\varOmega)}^{2}+\|\hat{r}_{n}\|_{H^{-1/2}(\partial{\varOmega_{h}})}^{2}.$
In (33), the norm is a quotient norm (see Exercises 44–46). With this quotient
norm, we will not need to explicitly use the subspace topology inherited from
$\prod_{K}H^{-1/2}(\partial K)$.
###### Exercise 44.
Suppose $X_{1}$ and $X_{2}$ are linear spaces, $A:X_{1}\to X_{2}$ is a linear
onto map, and let $\pi:X_{1}\to X_{1}/\ker A$ be the quotient map.
1. (1)
Prove that there is a unique linear one-to-one and onto map
$\hat{A}:X_{1}/\ker A\to X_{2}$ such that $A=\hat{A}\circ\pi$.
2. (2)
If in addition, $X_{1}$ is a normed linear space and $\ker A$ is closed, then
using the quotient norm $\|\pi(u)\|_{X_{1}/\ker A}=\inf_{w\in\ker
A}\|u+w\|_{X_{1}}$, prove that
$\|y\|_{X_{2}}=\|\hat{A}^{-1}y\|_{X_{1}/\ker A}$ (34)
makes $X_{2}$ into a normed linear space and $\hat{A}$ establishes an
isometric isomorphism between $X_{1}/\ker A$ and $X_{2}$.
###### Exercise 45.
For all $r\in H(\mathop{\mathrm{missing}}{div},{\varOmega})$, define
$\mathop{\mathrm{tr}_{n}}r\in H^{-1/2}(\partial{\varOmega_{h}})$ by
$\mathop{\mathrm{tr}_{n}}r|_{\partial K}=r\cdot n|_{\partial K}$. What is
$\ker(\mathop{\mathrm{tr}_{n}})$? Verify that $\ker(\mathop{\mathrm{tr}_{n}})$
is a closed subspace of $H(\mathop{\mathrm{missing}}{div},{\varOmega})$. Apply
Exercise 44 with $X_{1}=H(\mathop{\mathrm{missing}}{div},{\varOmega})$,
$X_{2}=H^{-1/2}(\partial{\varOmega_{h}})$, and $A=\mathop{\mathrm{tr}_{n}}$,
to conclude that the norm in (33) is the same as (34), and that
$H^{-1/2}(\partial{\varOmega_{h}})$ is complete under that norm.
###### Exercise 46.
Prove that there is a continuous linear map
$E:H^{-1/2}(\partial{\varOmega_{h}})\to
H(\mathop{\mathrm{missing}}{div},{\varOmega})$ such that
$\|\hat{r}_{n}\|_{H^{-1/2}(\partial{\varOmega_{h}})}=\|E\hat{r}_{n}\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}$
and $\mathop{\mathrm{tr}_{n}}E\hat{r}_{n}=\hat{r}_{n}$. (Hint: Consider
$\hat{A}^{-1}$ from Exercise 44 and find a minimizer over a coset.)
We now set
$b(\,(w,\hat{r}_{n}),\,v)=(\mathop{\mathrm{grad}}w,\mathop{\mathrm{grad}}v)_{\varOmega_{h}}-\langle{\hat{r}_{n},v}\rangle_{\partial{\varOmega_{h}}},\qquad\ell(v)=(f,v)_{\varOmega}$
and proceed to analyze the formulation (32). Let
$H(\mathop{\mathrm{missing}}{div},{{\varOmega_{h}}})=\\{v\in
L^{2}(\varOmega)^{N}:v|_{K}\in H(\mathop{\mathrm{missing}}{div},{K})$ for all
$K\in{\varOmega_{h}}\\}$. Define
$\displaystyle\big{|}[\tau\cdot n]\big{|}_{\partial{\varOmega_{h}}}$
$\displaystyle\;\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\;\sup_{0\neq\phi\in
H_{0}^{1}(\varOmega)}\frac{|\langle{\tau\cdot
n,\phi}\rangle_{\partial{\varOmega_{h}}}|}{\;\|\phi\|_{H^{1}(\varOmega)}},$
$\displaystyle\forall\tau\in
H(\mathop{\mathrm{missing}}{div},{{\varOmega_{h}}}),$ (35)
$\displaystyle\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}$
$\displaystyle\;\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\;\sup_{0\neq r\,\in
H(\mathop{\mathrm{missing}}{div},{\varOmega})}\frac{|\langle{r\cdot
n,v}\rangle_{\partial{\varOmega_{h}}}|}{\;\|r\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}},$
$\displaystyle\forall v\in H^{1}({\varOmega_{h}}).$ (36)
###### Exercise 47.
Prove that any $v\in H^{1}({\varOmega_{h}})$ has
$\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}=0$ if and only if $v\in
H_{0}^{1}(\varOmega)$.
###### Exercise 48.
Prove that
$\displaystyle{\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}=\sup_{0\neq\hat{r}_{n}\in
H^{-1/2}(\partial{\varOmega_{h}})}\frac{\\!\\!|\langle{\hat{r}_{n},v}\rangle_{\partial{\varOmega_{h}}}|}{\quad\|\hat{r}_{n}\|_{H^{-1/2}(\partial{\varOmega_{h}})}}.}$
Next, define an orthogonal projection
$P:L^{2}(\varOmega)^{N}\to\mathop{\mathrm{grad}}H_{0}^{1}(\varOmega)$, by
$(Pq,\mathop{\mathrm{grad}}\phi)_{\varOmega}=(q,\mathop{\mathrm{grad}}\phi)_{\varOmega}\qquad\forall\phi\in
H_{0}^{1}(\varOmega).$ (37)
###### Exercise 49.
Prove that $\mathop{\mathrm{grad}}H_{0}^{1}(\varOmega)$ is a closed subspace
of $L^{2}(\varOmega)^{N}$ (under the current assumptions on $\varOmega$).
###### Lemma 50 (A Poincaré-type inequality).
There is a positive constant $C$ independent of ${\varOmega_{h}}$ such that
for all $v$ in $H^{1}({\varOmega_{h}})$,
$C\|v\|_{\varOmega_{h}}\leq\|P\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}+\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}.$
###### Proof.
Let $\phi$ in $H_{0}^{1}(\varOmega)$ solve the Dirichlet problem
$-\Delta\phi=v$. Then,
$\displaystyle\|v\|_{{L^{2}(\varOmega)}}^{2}$
$\displaystyle=(-\Delta\phi,v)_{\varOmega}=(\mathop{\mathrm{grad}}\phi,\mathop{\mathrm{grad}}v)_{\varOmega_{h}}-\langle{\frac{\partial\phi}{\partial
n},v}\rangle_{\partial{\varOmega_{h}}}$
$\displaystyle=(\mathop{\mathrm{grad}}\phi,P\mathop{\mathrm{grad}}v)_{\varOmega_{h}}-\langle{\frac{\partial\phi}{\partial
n},v}\rangle_{\partial{\varOmega_{h}}}$
$\displaystyle\leq\|P\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}\|\mathop{\mathrm{grad}}\phi\|_{\varOmega_{h}}+\bigg{(}\frac{\langle{\mathop{\mathrm{grad}}\phi\cdot
n,v}\rangle_{\partial{\varOmega_{h}}}}{\quad\|\mathop{\mathrm{grad}}\phi\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}}\bigg{)}\|\mathop{\mathrm{grad}}\phi\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}$
$\displaystyle\leq\left(\|P\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}+\sup_{q\in
H(\mathop{\mathrm{missing}}{div},{\varOmega})}\\!\\!\frac{\langle{v,q\cdot
n}\rangle_{\partial{\varOmega_{h}}}}{\quad\|q\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}}\right)\|\mathop{\mathrm{grad}}\phi\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}.$
Since $\varOmega$ is bounded and connected, the standard Poincaré inequality
holds, so $\|\mathop{\mathrm{grad}}\phi\|^{2}_{{L^{2}(\varOmega)}}\leq
C\|v\|^{2}_{L^{2}(\varOmega)}$. Moreover,
$\|\mathop{\mathrm{missing}}{div}(\mathop{\mathrm{grad}}\phi)\|_{L^{2}(\varOmega)}=\|v\|_{L^{2}(\varOmega)}$,
so the result follows. ∎
###### Lemma 51 (Piecewise harmonic functions).
There is a $C>0$ independent of ${\varOmega_{h}}$ such that for all $v$ in
$H^{1}({\varOmega_{h}})$ satisfying $\Delta(v|_{K})=0$ for all
$K\in{\varOmega_{h}}$, we have
$C\|\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}\leq\big{|}[\mathop{\mathrm{grad}}v\cdot
n]\big{|}_{\partial{\varOmega_{h}}}+\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}.$
###### Proof.
Let $\tau=\mathop{\mathrm{grad}}v$. We construct the Helmholtz-Hodge
decomposition of $\tau$, namely $\tau=\mathop{\mathrm{grad}}\psi+z$ with
$\psi$ in $H_{0}^{1}(\varOmega)$ and $z$ in
$H(\mathop{\mathrm{missing}}{div},{\varOmega})$, as follows: First define
$\psi$ by
$(\mathop{\mathrm{grad}}\psi,\mathop{\mathrm{grad}}\varphi)_{\varOmega}=(\tau,\mathop{\mathrm{grad}}\varphi)_{\varOmega_{h}},\qquad\forall\varphi\in
H_{0}^{1}(\varOmega).$ (38)
Then, set $z=\tau-\mathop{\mathrm{grad}}\psi$.
By (38),
$(\tau-\mathop{\mathrm{grad}}\psi,\mathop{\mathrm{grad}}\varphi)_{\varOmega}=0$,
so $\mathop{\mathrm{missing}}{div}z=0$. Hence the two components,
$\mathop{\mathrm{grad}}\psi$ and $z,$ are $L^{2}(\varOmega)$-orthogonal and
$\|z\|_{\varOmega}^{2}+\|\mathop{\mathrm{grad}}\psi\|^{2}_{\varOmega}=\|\tau\|_{\varOmega}^{2}.$
(39)
Thus,
$\displaystyle\|\tau\|_{\varOmega}^{2}$
$\displaystyle=(\tau,\tau)=(\tau,\mathop{\mathrm{grad}}\psi+z)_{\varOmega_{h}}=(\tau,\mathop{\mathrm{grad}}\psi)_{\varOmega_{h}}+(\mathop{\mathrm{grad}}v,z)_{\varOmega_{h}}$
$\displaystyle=-(\mathop{\mathrm{missing}}{div}\tau,\psi)_{\varOmega_{h}}+\langle{\tau\cdot
n,\psi}\rangle_{\partial{\varOmega_{h}}}+\langle{n\cdot
z,v}\rangle_{\partial{\varOmega_{h}}},$
Since $v$ is harmonic on each element, the first term vanishes. Hence
$\displaystyle\|\tau\|_{\varOmega}^{2}$ $\displaystyle=\frac{\langle{\tau\cdot
n,\psi}\rangle_{\partial{\varOmega_{h}}}}{\|\psi\|_{H^{1}(\varOmega)}}\|\psi\|_{H^{1}(\varOmega)}+\frac{\langle{n\cdot
z,v}\rangle_{\partial{\varOmega_{h}}}}{\|z\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}}\|z\|_{L^{2}(\varOmega)}$
$\displaystyle\leq\bigg{(}\sup_{w\in
H_{0}^{1}(\varOmega)}\frac{\langle{\tau\cdot
n,w}\rangle_{\partial{\varOmega_{h}}}}{\|w\|_{H^{1}(\varOmega)}}\bigg{)}\|\psi\|_{H^{1}(\varOmega)}+\bigg{(}\sup_{q\in
H(\mathop{\mathrm{missing}}{div},{\varOmega})}\frac{\\!\langle{n\cdot
q,v}\rangle_{\partial{\varOmega_{h}}}}{\quad\|q\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}}\;\bigg{)}\|z\|_{L^{2}(\varOmega)}.$
The result now follows from (39) and the standard Poincaré inequality applied
to $\psi$. ∎
###### Lemma 52.
There is a positive constant $C$ independent of ${\varOmega_{h}}$ such that
for all $v$ in $H^{1}({\varOmega_{h}})$,
$\|\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}\leq\|P\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}+C\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}.$
###### Proof.
Let $z\in H_{0}^{1}(\varOmega)$ be such that
$P\mathop{\mathrm{grad}}v=\mathop{\mathrm{grad}}z$, let $\varepsilon=v-z$, and
let $r|_{K}=-\mathop{\mathrm{grad}}(\varepsilon|_{K})$ on all
$K\in{\varOmega_{h}}$. Then, (37) implies
$(P\mathop{\mathrm{grad}}v-\mathop{\mathrm{grad}}v,\mathop{\mathrm{grad}}\phi)=(r,\mathop{\mathrm{grad}}\phi)_{K}=0\qquad\forall\phi\in
H_{0}^{1}(\varOmega).$ (40)
Choosing $\phi\in\mathcal{D}(K)$, we immediately find that
$\mathop{\mathrm{missing}}{div}(r|_{K})=0$, i.e., $\varepsilon$ is harmonic on
each $K\in{\varOmega_{h}}$. Applying Lemma 51, we thus obtain
$C\|\mathop{\mathrm{grad}}\varepsilon\|_{\varOmega_{h}}\leq\big{|}[r\cdot
n]\big{|}_{\partial{\varOmega_{h}}}+\big{|}[\varepsilon
n]\big{|}_{\partial{\varOmega_{h}}}.$ (41)
But $\big{|}[r\cdot n]\big{|}_{\partial{\varOmega_{h}}}=0$. This is because we
may integrate by parts element by element to conclude from (40) that
$0=(r,\mathop{\mathrm{grad}}\phi)_{\varOmega_{h}}=-(\mathop{\mathrm{missing}}{div}r,\phi)_{\varOmega_{h}}+\langle{r\cdot
n,\phi}\rangle_{\partial{\varOmega_{h}}}=\langle{r\cdot
n,\phi}\rangle_{\partial{\varOmega_{h}}},$
for all $\phi\in H_{0}^{1}(\varOmega)$, so definition (35) implies
$\big{|}[r\cdot n]\big{|}_{\partial{\varOmega_{h}}}=0$. Moreover, definition
(36) shows that $\big{|}[\varepsilon
n]\big{|}_{\partial{\varOmega_{h}}}=\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}$.
Therefore, returning to (41), we conclude that
$\displaystyle\|\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}\leq\|P\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}+\|\mathop{\mathrm{grad}}\varepsilon\|_{\varOmega_{h}}\leq\|P\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}+C\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}},$
which proves the lemma. ∎
###### Theorem 53.
Assumption 7 holds for the formulation (32).
###### Proof.
The uniqueness part of Assumption 7, namely $\\{(w,\hat{s}_{n})\in
X:\;b(\,(w,\hat{s}_{n}),y)=0,\;\forall y\in Y\\}=\\{0\\}$ can be proved by an
argument analogous to what we have seen previously (see between (16) and
(18)), so is left as an exercise.
To prove the continuity estimate, we use
$|\langle{\hat{s}_{n},v}\rangle_{\partial{\varOmega_{h}}}|\leq\|\hat{s}_{h}\|_{H^{-1/2}(\partial{\varOmega_{h}})}\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}$,
a consequence of Exercise 48, to get
$|b(\,(w,\hat{s}_{n}),y)|\leq\|(w,\hat{s}_{n})\|_{X}\left(\|\mathop{\mathrm{grad}}v\|_{\varOmega_{h}}^{2}+\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}\right)^{1/2}$
Now, since (36) implies
$\displaystyle\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}}$
$\displaystyle=\sup_{r\,\in
H(\mathop{\mathrm{missing}}{div},{\varOmega})}\frac{(r,\mathop{\mathrm{grad}}v)_{\varOmega_{h}}+(\mathop{\mathrm{missing}}{div}r,v)_{\varOmega_{h}}}{\;\|r\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}}\leq\|v\|_{Y},$
the continuity estimate is proved.
It only remains to prove the inf-sup condition. But
$\displaystyle\sup_{\,(w,\hat{s}_{n})\in{{X}}}\frac{|b(\,(w,\hat{s}_{n}),v)|}{\|(w,\hat{s}_{n})\|_{{X}}}$
$\displaystyle\geq\sup_{w\in
H_{0}^{1}(\varOmega)}\frac{(\mathop{\mathrm{grad}}w,\mathop{\mathrm{grad}}v)_{\varOmega_{h}}}{\|\mathop{\mathrm{grad}}w\|_{L^{2}(\varOmega)}}=\|P\mathop{\mathrm{grad}}v\|_{L^{2}(\varOmega)},$
$\displaystyle\sup_{\,(w,\hat{s}_{n})\in{{X}}}\frac{|b(\,(w,\hat{s}_{n}),v)|}{\|(w,\hat{s}_{n})\|_{{X}}}$
$\displaystyle\geq\sup_{\hat{s}_{n}\in
H^{-1/2}(\partial{\varOmega_{h}})}\frac{|\langle{\hat{s}_{n},v}\rangle_{\partial{\varOmega_{h}}}|}{\|\hat{s}_{n}\|_{H^{-1/2}(\partial{\varOmega_{h}})}}=\big{|}[vn]\big{|}_{\partial{\varOmega_{h}}},$
so the required inf-sup condition follows by adding and using Lemmas 52 and
50. ∎
To consider a particular instance of the DPG method, we now fix element shapes
to be triangles. For any integer $p\geq 0$ let $P_{p}(K)$ denote the space of
polynomials of degree at most $p$ restricted to $K$. For any triangle $K$, let
$P_{p}(\partial K)$ denote the set of functions on $\partial K$ whose
restrictions to each edge of $K$ is a polynomial of degree at most $p$. We now
set
$\displaystyle X_{h}$ $\displaystyle=\\{(w,\hat{s}_{n})\in X:\;w|_{K}\in
P_{p+1}(K),\;\hat{s}_{n}|_{\partial K}\in P_{p}(\partial K)\;\;\forall
K\in{\varOmega_{h}}\\},$ (42a) $\displaystyle Y^{r}$ $\displaystyle=\\{v\in
Y:\;v|_{K}\in P_{r}(K),\;\;\forall K\in{\varOmega_{h}}\\},$ (42b)
compute the inexact test space $Y_{h}^{r}$, and consider the DPG method that
finds $(u_{h},\hat{q}_{n,h})\in X_{h}$ solving
$(\mathop{\mathrm{grad}}u_{h},\mathop{\mathrm{grad}}v)_{\varOmega_{h}}-\langle{\hat{q}_{n,h},v}\rangle_{\partial{\varOmega_{h}}}=(f,v)_{\varOmega},\qquad\forall
v\in Y_{h}^{r}.$ (43)
###### Theorem 54.
Suppose $N=2$, ${\varOmega_{h}}$ is a shape regular finite element mesh of
triangles and $X_{h}$ and $Y^{r}$ are set as in (42). Then, whenever $r\geq
p+N$, Assumption 32 holds. Consequently, by Theorem 33, the DPG method (43) is
quasioptimal.
###### Proof.
Let $r=p+N$. It is easy to see that for every $v\in H^{1}(K)$, there is a
unique $\varPi_{r}^{0}v\in P_{r}(K)$ satisfying
$\displaystyle\varPi_{r}^{0}v$ $\displaystyle=0$ $\displaystyle\text{at all 3
vertices of }K,$ $\displaystyle(\varPi_{r}^{0}v-v,q_{p-1})_{K}$
$\displaystyle=0,$ $\displaystyle\forall q_{p-1}\in P_{p-1}(K),$
$\displaystyle\langle\varPi_{r}^{0}v-v,\mu_{p}\rangle_{\partial K}$
$\displaystyle=0,$ $\displaystyle\forall\mu_{p}\in P_{p}(\partial K).$
Setting $\varPi v=\varPi_{r}^{0}(v-\bar{v})+\bar{v}$, where $\bar{v}$ denotes
the mean value of $v$ on $K$, it is an exercise to show that there is a $C$
independent of the size of the triangle $K$ (but dependent on the shape
regularity of $K$) such that
$\displaystyle(\varPi v-v,q_{p-1})_{K}$ $\displaystyle=0,$
$\displaystyle\forall q_{p-1}\in P_{p-1}(K),$ (44a)
$\displaystyle\langle\varPi v-v,\mu_{p}\rangle_{\partial K}$
$\displaystyle=0,$ $\displaystyle\forall\mu_{p}\in P_{p}(\partial K),$ (44b)
$\displaystyle\|\varPi v\|_{H^{1}(K)}$ $\displaystyle\leq C\|v\|_{H^{1}(K)}$
$\displaystyle\forall v\in H^{1}(K).$ (44c)
Then,
$\displaystyle b(\,(w_{h},\hat{s}_{n,h}),v-\varPi v)$
$\displaystyle=(\mathop{\mathrm{grad}}w_{h},\mathop{\mathrm{grad}}(v-\varPi
v))_{\varOmega_{h}}-\langle{\hat{s}_{n,h},v-\varPi
v}\rangle_{\partial{\varOmega_{h}}}$ $\displaystyle=-(\Delta w_{h},v-\varPi
v)_{\varOmega_{h}}-\langle{\hat{s}_{n,h}-n\cdot\mathop{\mathrm{grad}}w_{h},v-\varPi
v}\rangle_{\partial{\varOmega_{h}}}=0,$
by (44a) and (44b). ∎
###### Example 55.
To put this method into the framework of Assumption 28, set
$\displaystyle
X_{0}=H_{0}^{1}(\varOmega),\quad\hat{X}=H^{-1/2}(\partial{\varOmega_{h}}),$
$\displaystyle
b_{0}(u,y)=(\mathop{\mathrm{grad}}u,\mathop{\mathrm{grad}}y)_{\varOmega_{h}},\quad\hat{b}(\hat{q}_{n},y)=-\langle{\hat{q}_{n},y}\rangle_{\partial{\varOmega_{h}}}.$
Furthermore, the $X_{h}$ in (42a) can be split into $X_{h,0}\times\hat{X}_{h}$
with
$\displaystyle X_{h,0}$ $\displaystyle=\\{w\in
H_{0}^{1}(\varOmega):\;w|_{K}\in P_{p+1}(K),\;\;\forall
K\in{\varOmega_{h}}\\},$ $\displaystyle\hat{X}_{h}$
$\displaystyle=\\{\hat{s}_{n}\in
H^{-1/2}(\partial{\varOmega_{h}}):\;\hat{s}_{n}|_{\partial K}\in
P_{p}(\partial K)\;\;\forall K\in{\varOmega_{h}}\\},$
so that $Y_{0}$ in (20) becomes
$Y_{0}=\\{y\in
H^{1}({\varOmega_{h}}):\langle{\hat{s}_{n,h},y}\rangle_{\partial{\varOmega_{h}}}=0,\;\forall\hat{s}_{n,h}\in\hat{X}_{h}\\},$
a weakly conforming subspace of $H^{1}(\varOmega)$. Its subspace, defined in
(27) becomes $Y_{0}^{r}=\\{y\in
Y^{r}:\langle{\hat{s}_{n,h},y}\rangle_{\partial{\varOmega_{h}}}=0,\;\forall\hat{s}_{n,h}\in\hat{X}_{h}\\}$.
The non-hybrid form of the DPG method, namely (28b) uses this $Y_{0}^{r}$ and
finds $u_{h}\in X_{h,0}$ satisfying
$(\mathop{\mathrm{grad}}u_{h},\mathop{\mathrm{grad}}y_{0})_{\varOmega_{h}}=(f,y_{0})\qquad\forall
y_{0}\in Y_{h,0}^{r}.$ (45)
Recall that $y_{0}\in Y_{h,0}^{r}$ if and only if it is in $Y_{0}^{r}$ and
solves
$(\mathop{\mathrm{grad}}y_{0},\mathop{\mathrm{grad}}v)_{\varOmega_{h}}+(y_{0},v)_{\varOmega_{h}}=(\mathop{\mathrm{grad}}w,\mathop{\mathrm{grad}}v)_{\varOmega_{h}}\qquad\forall
v\in Y^{r}$ (46)
for some $w\in X_{h,0}$. By Theorem 41, the $u_{h}$ in (45) coincides with the
first solution component of the hybrid DPG method (43). The difficulty with
implementing (45) is that the computation of $Y_{h,0}^{r}$, requiring multiple
solves of the global weakly conforming problem (46), is too expensive. In
contrast the hybrid form (43) is easily implementable as the computation of
$Y_{h}^{r}$ amounts to inverting a block diagonal matrix.
Before concluding, let us consider convergence rates. The error estimate of
Theorem 33 (which holds by virtue of Theorems 53 and 54) gives
$\|u-u_{h}\|_{H^{1}(\varOmega)}+\|\hat{q}_{n}-\hat{q}_{n,h}\|_{H^{-1/2}({\varOmega_{h}})}\leq
C\inf_{(w_{h},\hat{s}_{n,h})\in
X_{h}}\left(\|u-w_{h}\|_{H^{1}(\varOmega)}+\|\hat{q}_{n}-\hat{s}_{n,h}\|_{H^{-1/2}({\varOmega_{h}})}\right).$
Henceforth $C>0$ denotes a generic constant independent of
$h=\max_{K\in{\varOmega_{h}}}\mathop{\mathrm{missing}}{diam}(K)$ but dependent
on the mesh’s shape regularity. To obtain convergence rates in terms of $h$,
we must bound the infimum above. Suppose $u$ is smooth. By the Bramble-Hilbert
lemma,
$\inf_{w_{h}\in X_{h,0}}\|u-w_{h}\|_{H^{1}(\varOmega)}\leq
Ch^{p+1}|u|_{H^{p+2}(\varOmega)}.$ (47)
For the error in $\hat{q}_{n}$, let $q=\mathop{\mathrm{grad}}u$ and let
$\mathop{\varPi_{\scriptscriptstyle{\mathrm{RT}}}}q$ denote the Raviart-Thomas
projection of $q$ into $\\{r\in
H(\mathop{\mathrm{missing}}{div},{\varOmega}):r|_{K}\in
P_{p}(K)+xP_{p}(K),\;\forall K\in{\varOmega_{h}}\\}$. Then
$\mathop{\varPi_{\scriptscriptstyle{\mathrm{RT}}}}q\cdot n\in\hat{X}_{h}$, so
$\inf_{\hat{s}_{n,h}\in\hat{X}_{h}}\|\hat{q}_{n}-\hat{s}_{n,h}\|_{H^{-1/2}({\varOmega_{h}})}\leq\|(q-\mathop{\varPi_{\scriptscriptstyle{\mathrm{RT}}}}q)\cdot
n\|_{H^{-1/2}({\varOmega_{h}})}\leq\|q-\mathop{\varPi_{\scriptscriptstyle{\mathrm{RT}}}}q\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}$
where we have used (33), by which, the $H^{-1/2}({\varOmega_{h}})$-norm of a
function can be bounded by the
$H(\mathop{\mathrm{missing}}{div},{\varOmega})$-norm of any of its extensions.
Estimating
$\|q-\mathop{\varPi_{\scriptscriptstyle{\mathrm{RT}}}}q\|_{H(\mathop{\mathrm{missing}}{div},{\varOmega})}$
as usual,
$\inf_{\hat{s}_{n,h}\in\hat{X}_{h}}\|\hat{q}_{n}-\hat{s}_{n,h}\|_{H^{-1/2}({\varOmega_{h}})}\leq
Ch^{p+1}\left(|q|_{H^{p+1}(\varOmega)}+|\mathop{\mathrm{missing}}{div}q|_{H^{p+1}(\varOmega)}\right).$
(48)
From (47) and (48), we obtain $O(h^{p+1})$ convergence for $u_{h}$ and
$\hat{q}_{n,h}$.
$h/\sqrt{2}$ | $\|u-u_{h}\|_{H^{1}(\varOmega)}$ | $\|\varepsilon^{r}\|_{H^{1}({\varOmega_{h}})}$
---|---|---
1/4 | 0.008277 | 0.008987
1/8 | 0.002111 | 0.002297
1/16 | 0.000531 | 0.000579
1/32 | 0.000133 | 0.000145
1/64 | 0.000033 | 0.000036
Let us now check if we see this convergence rate in practice. We use a FEniCS
code (download code from here) which implements the mixed reformulation of the
DPG method given in Theorem 39 (see also Remark 42). Solving a simple problem
with a smooth solution (see the code for details) on the unit square, using
$p=1$ and uniform meshes with various $h$, we collect the results in the table
aside. Clearly, $\|u-u_{h}\|_{H^{1}(\varOmega)}$ appears to converge at
$O(h^{2})$, in accordance with the theory. Also, the error estimator
$\varepsilon^{r}$ (see Definition 38) appears to converge to zero at the same
rate as the error.
Figure 2. Initial, midway, and final iterates in an adaptive scheme using the
DPG error estimator $\varepsilon^{r}$.
It is possible to prove that the error estimator $\varepsilon^{r}$ is an
efficient and reliable indicator of the actual error, but to keep these
lectures introductory, we omit the details. Instead, let us consider a FEniCS
implementation of a typical adaptive algorithm using the element-wise norms of
$\varepsilon^{r}$ as the error indicators (download the code from here). In
the code, we compute the element error indicator
$\|\varepsilon^{r}\|_{H^{1}(K)}$ on each $K\in{\varOmega_{h}}$, and sort the
elements in decreasing order of the indicators. The elements falling in the
top half are marked for refinement. In the next iteration of the adaptive
algorithm, those elements (and possibly other adjacent elements) are refined
by bisection, the DPG problem is solved on the new mesh, and the newly
obtained $\varepsilon^{r}$ is used to mark elements as before. We use this
process to approximate the solution of the Dirichlet problem (30) on the unit
square with $f=e^{-100(x_{0}^{2}+x_{1}^{2})}.$ We expect the solution to have
interesting variations only near the origin. As seen in Figure 2, the error
estimator automatically identifies the right region for refinement even though
we started with a very coarse mesh.
## Appendix
### Codes
The programs are in the python FEniCS environment. You will need to download
and install FEniCS from fenicsproject.org for them to run. (I am not an expert
in FEniCS and suggestions to improve the codes are very welcome.) Here are the
available downloads on DPG methods:
* •
The FEniCS code for implementing the Petrov Galerkin method of Example 20 and
generating Figure 1 can be downloaded from here. The code also implements a
comparable least square method and the computation of $L^{2}$ projections.
* •
You can download download a FEniCS implementation of the DPG method for the
Dirichlet problem.
* •
A code implementing an adaptive algorithm using the DPG error estimator is
also available. This code is modeled after a FEniCS demo for standard finite
elements.
### Acknowledgments
These are (unpublished) notes from a few of my lectures in a Spring 2013
graduate class at PSU (MTH 610). The DPG research is in close collaboration
with Leszek Demkowicz (see references below). I am grateful to my students for
their feedback on the notes and to Kristian Ølgaard for clarifying FEniCS
syntax. I am also grateful to NSF and AFOSR for supporting my research into DG
and mixed methods and for encouraging the integration of such research into
graduate education.
### Bibliographic remarks
The presentation in Section 1, including the terminology of ‘optimal test
spaces’, Theorem 8, etc. is based on [7]. The DPG methods were developed in a
series of papers, beginning with [5, 7]. The name “DPG” was previously used by
others [1], but without the concept of optimal test functions. The
interpretation as a mixed formulation (Theorem 15) is motivated by [3].
Theorem 33 is from [9]. Operators such as $\varPi$, in the standard mixed
Galerkin context, are sometimes known as Fortin operators. Theorems 29 and 41
have not appeared in this form previously. Lemmas 50 and 51 are from [6], but
the method of Section 4 was developed later, independently in [8] and [2]. A
more comprehensive bibliography is available in [4].
##
* [1] C. L. Bottasso, S. Micheletti, and R. Sacco. The discontinuous Petrov-Galerkin method for elliptic problems. Comput. Methods Appl. Mech. Engrg., 191(31):3391–3409, 2002.
* [2] D. Broersen and R. Stevenson, A Petrov-Galerkin discretization with optimal test space of a mild-weak formulation of convection-diffusion equations in mixed form, Preprint, 2013.
* [3] W. Dahmen, C. Huang, C. Schwab, and G. Welper. Adaptive Petrov-Galerkin methods for first order transport equations. SIAM J Numer. Anal., 50(5):2420–2445, 2012.
* [4] L. Demkowicz and J. Gopalakrishnan. An overview of the discontinuous Petrov Galerkin method. In X. Feng, O. Karakashian, and Y. Xing, editors, Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations: 2012 John H Barret Memorial Lectures, volume 157 of The IMA Volumes in Mathematics and its Applications, pages 149–180. Institute for Mathematics and its Applications, Minneapolis, Springer, 2013.
* [5] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods. Part I: The transport equation. Comput. Methods Appl. Mech. Engrg., 199:1558–1572, 2010.
* [6] L. Demkowicz and J. Gopalakrishnan. Analysis of the DPG method for the Poisson equation. SIAM J Numer. Anal., 49(5):1788–1809, 2011.
* [7] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions. Numerical Methods for Partial Differential Equations, 27(1):70–105, 2011.
* [8] L. Demkowicz and J. Gopalakrishnan. A primal DPG method without a first-order reformulation. Computers and Mathematics with Applications, 66(6):1058–1064, 2013\.
* [9] J. Gopalakrishnan and W. Qiu. An analysis of the practical DPG method. Math. Comp, in press, doi:10.1090/S0025-5718-2013-02721-4, electronically appeared, 2013.
|
arxiv-papers
| 2013-06-03T17:43:01 |
2024-09-04T02:49:46.060156
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jay Gopalakrishnan",
"submitter": "Jay Gopalakrishnan",
"url": "https://arxiv.org/abs/1306.0557"
}
|
1306.0714
|
# $L^{p}$ mean estimates for an operator preserving inequalities between
polynomials
###### Abstract.
If $P(z)$ be a polynomial of degree at most $n$ which does not vanish in
$|z|<1$, it was recently formulated by Shah and Liman [20, Integral estimates
for the family of $B$-operators, Operators and Matrices, 5(2011), 79 - 87]
that for every $R\geq 1$, $p\geq 1$,
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq\frac{R^{n}|\Lambda_{n}|+|\lambda_{0}|}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p},$
where $B$ is a $\mathcal{B}_{n}$-operator with parameters
$\lambda_{0},\lambda_{1},\lambda_{2}$ in the sense of Rahman [14],
$\sigma(z)=Rz$ and
$\Lambda_{n}=\lambda_{0}+\lambda_{1}\frac{n^{2}}{2}+\lambda_{2}\frac{n^{3}(n-1)}{8}$.
Unfortunately the proof of this result is not correct. In this paper, we
present a more general sharp $L_{p}$-inequalities for
$\mathcal{B}_{n}$-operators which not only provide a correct proof of the
above inequality as a special case but also extend them for $0\leq p<1$ as
well.
††footnotetext: AMS Mathematics Subject Classification (2000): 26D10,
41A17.††footnotetext: Keywords and Phrases: $L^{p}$-inequalities;
$\mathcal{B}_{n}$-operators; polynomials.
N. A. Rather1 and Suhail Gulzar2,***The work is supported by Council of
Scientific and Industrial Research, New Delhi, under grant F.No.
09/251(0047)/2012-EMR-I.
## 1\. Introduction and statement of results
Let $\mathscr{P}_{n}$ denote the space of all complex polynomials
$P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}$ of degree at most $n$. For
$P\in\mathscr{P}_{n}$, define
$\left\|P(z)\right\|_{0}:=\exp\left\\{\frac{1}{2\pi}\int_{0}^{2\pi}\log\left|P(e^{i\theta})\right|d\theta\right\\},$
$\left\|P(z)\right\|_{p}:=\left\\{\frac{1}{2\pi}\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}\right\\}^{1/p},\,\,0<p<\infty$
$\left\|P(z)\right\|_{\infty}:=\underset{\left|z\right|=1}{Max}\left|P(z)\right|,$
and denote for any complex function $\sigma:\mathbb{C}\rightarrow\mathbb{C}$
the composite function of $P$ and $\sigma$, defined by
$\left(P\circ\sigma\right)(z):=P\left(\sigma(z)\right)\,\,\,\,(z\in\mathbb{C})$,
as $P\circ\sigma$.
A famous result known as Bernstein’s inequality (for reference, see [13,
p.531], [15, p.508] or [19] states that if $P\in\mathscr{P}_{n}$, then
$\left|P^{\prime}(z)\right|_{\infty}\leq n\left\|P(z)\right\|_{\infty},$ (1.1)
whereas concerning the maximum modulus of $P(z)$ on the circle
$\left|z\right|=R>1$, we have
$\left\|P(Rz)\right\|_{\infty}\leq
R^{n}\left\|P(z)\right\|_{\infty},\,\,\,R\geq 1,$ (1.2)
(for reference, see [12, p.442] or [13, vol.I, p.137] ).
Inequalities (1.1) and (1.2) can be obtained by letting $p\rightarrow\infty$
in the inequalities
$\left\|P^{\prime}(z)\right\|_{p}\leq n\left\|P(z)\right\|_{p},p\geq 1$ (1.3)
and
$\left\|P(Rz)\right\|_{p}\leq
R^{n}\left\|P(z)\right\|_{p},\,\,\,\,R>1,\,\,\,\,\,p>0,$ (1.4)
respectively. Inequality (1.3) was found by Zygmund [21] whereas inequality
(1.4) is a simple consequence of a result of Hardy [9] (see also [16, Th.
5.5]). Since inequality (1.3) was deduced from M. Riesz’s interpolation
formula [18] by means of Minkowski’s inequality, it was not clear, whether the
restriction on $p$ was indeed essential. This question was open for a long
time. Finally Arestov [2] proved that (1.3) remains true for $0<p<1$ as well.
If we restrict ourselves to the class of polynomials $P\in\mathscr{P}_{n}$
having no zero in $|z|<1$, then inequalities (1.1) and (1.2) can be
respectively replaced by
$\left\|P^{\prime}(z)\right\|_{\infty}\leq\dfrac{n}{2}\left\|P(z)\right\|_{\infty},$
(1.5)
and
$\left\|P(Rz)\right\|_{\infty}\leq\frac{R^{n}+1}{2}\left\|P(z)\right\|_{\infty}\,\,\,\,\,\,R>1.$
(1.6)
Inequality (1.5) was conjectured by Erdös and later verified by Lax [10],
whereas inequality (1.6) is due to Ankey and Ravilin [1].
Both the inequalities (1.5) and (1.6) can be obtain by letting
$p\rightarrow\infty$ in the inequalities
$\left\|P^{\prime}(z)\right\|_{p}\leq
n\frac{\left\|P(z)\right\|_{p}}{\left\|1+z\right\|_{p}},\,\,\,\,\,p\geq 0$
(1.7)
and
$\left\|P(Rz)\right\|_{p}\leq\frac{\left\|R^{n}z+1\right\|_{p}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p}\,\,\,\,\,\,R>1,\,\,\,\,\,p>0.$
(1.8)
Inequality (1.7) is due to De-Bruijn [7] for $p\geq 1$. Rahman and Schmeisser
[17] extended it for $0\leq p<1$ whereas the inequality (1.8) was proved by
Boas and Rahman [6] for $p\geq 1$ and later it was extended for $0\leq p<1$ by
Rahman and Schmeisser [17].
Q. I. Rahman [14] (see also Rahman and Schmeisser [15, p. 538]) introduced a
class $\mathcal{B}_{n}$ of operators $B$ that carries a polynomial
$P\in\mathscr{P}_{n}$ into
$B[P](z):=\lambda_{0}P(z)+\lambda_{1}\left(\dfrac{nz}{2}\right)\dfrac{P^{\prime}(z)}{1!}+\lambda_{2}\left(\dfrac{nz}{2}\right)^{2}\dfrac{P^{\prime\prime}(z)}{2!},$
(1.9)
where $\lambda_{0},\lambda_{1}$ and $\lambda_{2}$ are such that all the zeros
of
$U(z):=\lambda_{0}+\lambda_{1}C(n,1)z+\lambda_{2}C(n,2)z^{2}\,\,\,\,\,$ (1.10)
where $C(n,r)=\dfrac{n!}{r!(n-r)!}\,\,\,\,\,0\leq r\leq n,$ lie in half plane
$|z|\leq\left|z-n/2\right|.$
As a generalization of inequality (1.1) and (1.5), Q. I. Rahman [14,
inequality 5.2 and 5.3] proved that if $P\in\mathscr{P}_{n},$ and
$B\in\mathcal{B}_{n}$ then
$|B[P](z)|\leq|\Lambda_{n}|\|P(z)\|_{\infty},\,\,\,\,\,\,\textnormal{for}\,\,\,\,\,\,\,|z|\geq
1,$ (1.11)
and if $P\in\mathscr{P}_{n},$ $P(z)\neq 0$ in $|z|<1,$ then
$|B[P](z)|\leq\dfrac{1}{2}\left\\{|\Lambda_{n}|+|\lambda_{0}|\right\\}\|P(z)\|_{\infty},\,\,\,\,\,\,\textnormal{for}\,\,\,\,\,\,\,|z|\geq
1,$ (1.12)
where
$\Lambda_{n}=\lambda_{0}+\lambda_{1}\frac{n^{2}}{2}+\lambda_{2}\frac{n^{3}(n-1)}{8}.$
(1.13)
As a corresponding generalization of inequalities (1.2) and (1.4), Rahman and
Schmeisser [15, p. 538] proved that if $P\in\mathscr{P}_{n},$ then
$\left|B[P\circ\sigma](z)\right|\leq
R^{n}|\Lambda_{n}|\left\|P(z)\right\|_{\infty}\,\,\,\mbox{}\textrm{for}\,\mbox{}\,\,\,|z|=1.$
(1.14)
and if $P\in\mathscr{P}_{n},$ $P(z)\neq 0$ in $|z|<1,$ then as a special case
of Corollary 14.5.6 in [15, p. 539], we have
$\left|B[P\circ\sigma](z)\right|\leq\frac{1}{2}\left\\{R^{n}|\Lambda_{n}|+|\lambda_{0}|\right\\}\left\|P(z)\right\|_{\infty}\,\,\,\mbox{}\textrm{for}\,\mbox{}\,\,\,|z|=1,$
(1.15)
where $\sigma(z):=Rz,\,\,R\geq 1$ and $\Lambda_{n}$ is defined by (1.13).
Inequality (1.15) also follows by combining the inequalities (5.2) and (5.3)
due to Rahman [14].
As an extension of inequality (1.14) to $L_{p}$-norm, recently Shah and Liman
[20, Theorem 1] proved:
###### Theorem A.
If $P\in\mathscr{P}_{n}$, then for every $R\geq 1$ and $p\geq 1$,
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq
R^{n}|\Lambda_{n}|\left\|P(z)\right\|_{p},$ (1.16)
where $B\in\mathcal{B}_{n}$, $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by
(1.13).
While seeking the analogue of (1.15) in $L_{p}$ norm, they [20, Theorem 2]
have made an incomplete attempt by claiming to have proved the following
result:
###### Theorem B.
If $P\in\mathscr{P}_{n}$, and $P(z)$ does not vanish for $|z|\leq 1,$ then for
each $p\geq 1$, $R\geq 1$,
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq\frac{R^{n}|\Lambda_{n}|+|\lambda_{0}|}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p},$
(1.17)
where $B\in\mathcal{B}_{n}$, $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by
(1.13).
Unfortunately the proof of inequality (1.17) and other related results
including the key lemma [20, Lemma 4] given by Shah and Liman is not correct.
The reason being that the authors in [20] deduce [20, line 10 from line 7 on
page 84, line 19 on page 85 from Lemma 3, line 16 from line 14 on page 86] by
using the argument that if $P^{\star}(z):=z^{n}\overline{P(1/\overline{z})}$,
then for $\sigma(z)=Rz$, $R\geq 1$ and $|z|=1,$
$|B[P^{\star}\circ\sigma](z)|=|B[(P^{\star}\circ\sigma)^{\star}](z)|,$
which is not true, in general, for every $R\geq 1$ and $|z|=1$. To see this,
let
$P(z)=a_{n}z^{n}+\cdots+a_{k}z^{k}+\cdots+a_{1}z+a_{0}$
be an arbitrary polynomial of degree $n$, then
$P^{\star}(z):=z^{n}\overline{P(1/\overline{z})}=\bar{a_{0}}z^{n}+\bar{a_{1}}z^{n-1}+\cdots+\bar{a_{k}}z^{n-k}+\cdots+\bar{a_{n}}.$
Now with $\omega_{1}:=\lambda_{1}n/2$ and $\omega_{2}:=\lambda_{2}n^{2}/8$, we
have
$B[P^{\star}\circ\sigma](z)=\sum_{k=0}^{n}\left(\lambda_{0}+\omega_{1}(n-k)+\omega_{2}(n-k)(n-k-1)\right)\bar{a_{k}}z^{n-k}R^{n-k},$
and in particular for $|z|=1$, we get
$B[P^{\star}\circ\sigma](z)=R^{n}z^{n}\sum_{k=0}^{n}\left(\lambda_{0}+\omega_{1}(n-k)+\omega_{2}(n-k)(n-k-1)\right)\overline{{a_{k}}\left(\frac{z}{R}\right)^{k}},$
whence
$|B[P^{\star}\circ\sigma](z)|=R^{n}\left|\sum_{k=0}^{n}\overline{\left(\lambda_{0}+\omega_{1}(n-k)+\omega_{2}(n-k)(n-k-1)\right)}a_{k}\left(\frac{z}{R}\right)^{k}\right|.$
But
$|B[(P^{\star}\circ\sigma)^{\star}](z)|=R^{n}\left|\sum_{k=0}^{n}\left(\lambda_{0}+\omega_{1}k+\omega_{2}k(k-1)\right)a_{k}\left(\frac{z}{R}\right)^{k}\right|,$
so the asserted identity does not hold in general for every $R\geq 1$ and
$|z|=1$ as e.g. the immediate counterexample of $P(z):=z^{n}$ demonstrates in
view of $P^{\star}(z)=1$, $|B[P^{\star}\circ\sigma](z)|=|\lambda_{0}|$ and
$|B[(P^{\star}\circ\sigma)^{\star}](z)|=|\lambda_{0}+\lambda_{1}(n^{2}/2)+\lambda_{2}n^{3}(n-1)/8|\,\,\,\,(|z|=1).$
As claimed by [20], Theorem B is sharp has remained to be verified. In fact,
this claim is also wrong.
The main aim of this paper is to establish $L_{p}$-mean extensions of the
inequality (1.15) for $0\leq p<\infty$ and present correct proofs of the
results mentioned in [20]. In this direction, we present the following
interesting compact generalization of Theorem B which yields $L_{p}$ mean
extension of the inequality (1.12) for $0\leq p<\infty$ which among other
things includes a correct proof of inequality (1.17) for $1\leq p<\infty$ as a
special case.
###### Theorem 1.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$\alpha,\delta\in\mathbb{C}$ with $|\alpha|\leq 1,|\delta|\leq 1,$ $0\leq
p<\infty$ and $R>1,$
$\displaystyle\Big{\|}B[P\circ\sigma](e^{i\theta})$ $\displaystyle-\alpha
B[P](e^{i\theta})+\delta\Big{\\{}\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}\Big{\|}_{p}$
$\displaystyle\leq\dfrac{\left\|(R^{n}-\alpha)\Lambda_{n}z+(1-\alpha)\lambda_{0}\right\|_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$
(1.18)
where $m=Min_{|z|=1}|P(z)|,$ $B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and
$\Lambda_{n}$ is defined by (1.13). The result is best possible and equality
in (1) holds for $P(z)=az^{n}+b,$ $|a|=|b|=1.$
Setting $\delta=0$ in (1), we get the following result.
###### Corollary 1.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$\alpha,\delta\in\mathbb{C}$ with $|\alpha|\leq 1,|\delta|\leq 1,$ $0\leq
p<\infty$ and $R>1,$
$\displaystyle\Big{\|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\Big{\|}_{p}\leq\dfrac{\left\|(R^{n}-\alpha)\Lambda_{n}z+(1-\alpha)\lambda_{0}\right\|_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$
(1.19)
$B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by (1.13).
The result is best possible and equality in (1) holds for $P(z)=az^{n}+b,$
$|a|=|b|=1.$
If we take $\alpha=0$ in (1.19), we get the following result which is the
generalization of Theorem B for $p\geq 1$ and also extends it for $0\leq p<1.$
###### Corollary 2.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$0\leq p<\infty$ and $R>1,$
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq\dfrac{\left\|R^{n}\Lambda_{n}z+\lambda_{0}\right\|_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$
(1.20)
$B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by (1.13).
The result is sharp as shown by $P(z)=az^{n}+b,$ $|a|=|b|=1.$
By triangle inequality, the following result follows immediately from
Corollary 2.
###### Corollary 3.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$0\leq p<\infty$ and $R>1,$
$\left\|B[P\circ\sigma](z)\right\|_{p}\leq\dfrac{R^{n}\left|\Lambda_{n}|+|\lambda_{0}\right|}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$
(1.21)
$B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and $\Lambda_{n}$ is defined by (1.13).
###### Remark 1.
Corollary 3 establishes a correct proof of a result due to Shah and Liman [20,
Theorem 3] for $p\geq 1$ and also extends it for $0\leq p<1$ as well.
###### Remark 2.
If we choose $\lambda_{0}=0=\lambda_{2}$ in (1.20), we get for $R>1$ and
$0\leq p<\infty,$
$\left\|P^{\prime}(Rz)\right\|_{p}\leq\frac{nR^{n-1}}{\left\|1+z\right\|_{p}}\left\|P(z)\right\|_{p},$
which, in particular, yields inequality (1.7). Next if we take
$\lambda_{1}=0=\lambda_{2}$ in (1.20), we get inequality (1.8). Inequality
(1.12) can be obtained from corollary 2 by letting $p\rightarrow\infty$ in
(1).
Taking $\alpha=0$ in (1), we get the following result.
###### Corollary 4.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$\delta\in\mathbb{C}$ with $|\delta|\leq 1,$ $0\leq p<\infty$ and $R>1,$
$\displaystyle\Big{\|}B[P\circ\sigma](z)+\delta\Big{\\{}\dfrac{(R^{n}|\Lambda_{n}|-|\lambda_{0}|)m}{2}\Big{\\}}\Big{\|}_{p}\leq\dfrac{\left\|R^{n}\Lambda_{n}z+\lambda_{0}\right\|_{p}}{\|1+z\|_{p}}\left\|P(z)\right\|_{p},$
(1.22)
where $m=Min_{|z|=1}|P(z)|,$ $B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and
$\Lambda_{n}$ is defined by (1.13). The result is best possible and equality
in (1.22) holds for $P(z)=az^{n}+b,$ $|a|=|b|=1.$
The following corollary immediately follows from Theorem 1 by letting
$p\rightarrow\infty$ in (1) and choosing the argument of $\delta$ suitably
with $|\delta|=1.$
###### Corollary 5.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish for $|z|<1,$ then for
$\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$ $R>1,$
$\displaystyle\Big{\|}B[P\circ\sigma](z)-\alpha B[P](z)\Big{\|}_{\infty}\leq$
$\displaystyle\dfrac{\left|R^{n}-\alpha\right||\Lambda_{n}|+\left|(1-\alpha)\lambda_{0}\right|}{2}\left\|P(z)\right\|_{\infty}$
$\displaystyle-\dfrac{\left|R^{n}-\alpha\right||\Lambda_{n}|-\left|(1-\alpha)\lambda_{0}\right|}{2}m,$
(1.23)
where $m=Min_{|z|=1}|P(z)|,$ $B\in\mathcal{B}_{n},$ $\sigma(z)=Rz$ and
$\Lambda_{n}$ is defined by (1.13).
## 2\. Lemma
For the proofs of this theorem, we need the following lemmas. The first lemma
follows from Corollary $18.3$ of [11, p. 86].
###### Lemma 1.
If $P\in\mathscr{P}_{n}$ and $P(z)$ has all zeros in $|z|\leq 1,$ then all the
zeros of $B[P](z)$ also lie in $|z|\leq 1.$
###### Lemma 2.
If $P\in\mathscr{P}_{n}$ and $P(z)$ have all its zeros in $\left|z\right|\leq
1$ then for every $R>1,$ and $\left|z\right|=1$,
$\left|P(Rz)\right|\geq\left(\frac{R+1}{2}\right)^{n}\left|P(z)\right|.$
###### Proof.
Since all the zeros of $P(z)$ lie in $\left|z\right|\leq 1$, we write
$P(z)=C\prod_{j=1}^{n}\left(z-r_{j}e^{i\theta_{j}}\right),$
where $r_{j}\leq 1$. Now for $0\leq\theta<2\pi$, $R>1$, we have
$\displaystyle\left|\frac{Re^{i\theta}-r_{j}e^{i\theta_{j}}}{e^{i\theta}-r_{j}e^{i\theta_{j}}}\right|$
$\displaystyle=\left\\{\frac{R^{2}+r_{j}^{2}-2Rr_{j}\cos(\theta-\theta_{j})}{1+r_{j}^{2}-2r_{j}\cos(\theta-\theta_{j})}\right\\}^{1/2},$
$\displaystyle\geq\left\\{\frac{R+r_{j}}{1+r_{j}}\right\\},$
$\displaystyle\geq\left\\{\frac{R+1}{2}\right\\},\textrm{for}\,\,\,j=1,2,\cdots,n.$
Hence
$\displaystyle\left|\frac{P(Re^{i\theta})}{P(e^{i\theta})}\right|$
$\displaystyle=\prod_{j=1}^{n}\left|\frac{Re^{i\theta}-r_{j}e^{i\theta_{j}}}{e^{i\theta}-r_{j}e^{i\theta_{j}}}\right|,$
$\displaystyle\geq\prod_{j=1}^{n}\left(\frac{R+1}{2}\right),$
$\displaystyle=\left(\frac{R+1}{2}\right)^{n},$
for $0\leq\theta<2\pi$. This implies for $|z|=1$ and $R>1$,
$\left|P(Rz)\right|\geq\left(\frac{R+1}{2}\right)^{n}\left|P(z)\right|,$
which completes the proof of Lemma 2. ∎
###### Lemma 3.
If $P\in\mathscr{P}_{n}$ and $P(z)$ has all its zeros in $|z|\leq 1,$ then for
every real or complex number $\alpha$ with $|\alpha|\leq 1,$ $R>1$ and
$|z|\geq 1,$
$\displaystyle|B[P\circ\sigma](z)-\alpha
B[P](z)|\geq|R^{n}-\alpha||\Lambda_{n}||z|^{n}m,$ (2.1)
where $m=\underset{|z|=1}{Min}|P(z)|,$ $\sigma(z)=Rz$ and $\Lambda_{n}$ is
given by (1.13).
###### Proof.
By hypothesis, all the zeros of $P(z)$ lie in $|z|\leq 1$ and
$m|z|^{n}\leq|P(z)|\,\,\,\,\textnormal{for}\,\,\,\,|z|=1.$
We first show that the polynomial $g(z)=P(z)-\beta mz^{n}$ has all its zeros
in $|z|\leq 1$ for every real or complex number $\beta$ with $|\beta|<1.$ This
is obvious if $m=0,$ that is if $P(z)$ has a zero on $|z|=1.$ Henceforth, we
assume $P(z)$ has all its zeros in $|z|<1,$ then $m>0$ and it follows by
Rouche’s theorem that the polynomial $g(z)$ has all its zeros in $|z|<1$ for
every real or complex number $\beta$ with $|\beta|<1.$ Applying Lemma 2 to the
polynomial $g(z),$ we deduce
$\displaystyle|g(Rz)|\geq\left(\dfrac{R+1}{2}\right)^{n}|g(z)|\,\,\,\,\textnormal{for}\,\,\,\,|z|=1\,\,\,\,R>1.$
Since $R>1,$ therefore $\frac{R+1}{2}>1,$ this gives
$\displaystyle|g(Rz)|>|g(z)|\,\,\,\,\textnormal{for}\,\,\,\,|z|=1\,\,\,\,R>1.$
(2.2)
Since all the zeros of $G(Rz)$ lie in $|z|<1/R<1,$ by Rouche’s theorem again
it follows from (2.2) that all the zeros of polynomial
$\displaystyle H(z)=g(Rz)-\alpha g(z)=P(Rz)-\alpha
P(z)-\beta(R^{n}-\alpha)z^{n}m$
lie in $|z|<1,$ for every $\alpha,\beta$ with $|\alpha|\leq 1,|\beta|<1$ and
$R>1.$ Applying Lemma 1 to $H(z)$ and noting that $B$ is a linear operator, it
follows that all the zeros of polynomial
$\displaystyle B[H](z)$ $\displaystyle=B[g\circ\sigma](z)-\alpha B[g](z)$
$\displaystyle=\left\\{B[P\circ\sigma](z)-\alpha
B[P](z)\right\\}-\beta(R^{n}-\alpha)mB[z^{n}]$ (2.3)
lie in $|z|<1.$ This gives
$\displaystyle\left|B[P\circ\sigma](z)-\alpha
B[P](z)\right|\geq|R^{n}-\alpha||\Lambda_{n}||z|^{n}m\,\,\,\,\,\textnormal{for}\,\,\,\,\,|z|\geq
1.$ (2.4)
If (2.4) is not true, then there is point $w$ with $|w|\geq 1$ such that
$\displaystyle\left|B[P\circ\sigma](w)-\alpha
B[P](w)\right|<|R^{n}-\alpha||\Lambda_{n}||w|^{n}m.$ (2.5)
We choose
$\beta=\dfrac{B[P\circ\sigma](w)-\alpha
B[P](w)}{(R^{n}-\alpha)\Lambda_{n}w^{n}m},$
then clearly $|\beta|<1$ and with this choice of $\beta,$ from (Proof.), we
get $B[H](w)=0$ with $|w|\geq 1.$ This is clearly a contradiction to the fact
that all the zeros of $H(z)$ lie in $|z|<1.$ Thus for every real or complex
$\alpha$ with $|\alpha|\leq$ 1,
$\left|B[P\circ\sigma](z)-\alpha
B[P](z)\right|\geq|R^{n}-\alpha||\Lambda_{n}||z|^{n}m$
for $|z|\geq 1$ and $R>1.$ ∎
###### Lemma 4.
If $P\in\mathscr{P}_{n}$ and $P(z)$ has no zero in $\left|z\right|<1,$ then
for every $\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$ $R>1$ and $|z|\geq 1$,
$\left|B[P\circ\sigma](z)-\alpha
B[P](z)\right|\leq\left|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)\right|,$ (2.6)
where $P^{\star}(z)=z^{n}\overline{P(1/\overline{z})}$ and $\sigma(z)=Rz.$
###### Proof.
Since the polynomial $P(z)$ has all its zeros in $|z|\geq 1,$ therefore, for
every real or complex number $\lambda$ with $|\lambda|>1,$ the polynomial
$f(z)=P(z)-\lambda P^{\star}(z),$ where
$P^{\star}(z)=z^{n}\overline{P(1/\overline{z})},$ has all zeros in $|z|\leq
1.$ Applying Lemma 2 to the polynomial $f(z),$ we obtain for every $R>1$ and
$0\leq\theta<2\pi,$
$|f(Re^{i\theta})|\geq\left(\dfrac{R+1}{2}\right)^{n}|f(e^{i\theta})|.$ (2.7)
Since $f(Re^{i\theta})\neq 0$ for every $R>1,$ $0\leq\theta<2\pi$ and $R+1>2,$
it follows from (2.7) that
$|f(Re^{i\theta})|>\left(\dfrac{R+1}{2}\right)^{n}|f(Re^{i\theta})|\geq|f(e^{i\theta})|,$
for every $R>1$ and $0\leq\theta<2\pi.$ This gives
$|f(z)|<|f(Rz)|\,\,\,\textnormal{for}\,\,\,\,|z|=1,\,\,\,\,\textnormal{and}\,\,\,R>1.$
Using Rouche’s theorem and noting that all the zeros of $f(Rz)$ lie in
$|z|\leq 1/R<1,$ we conclude that the polynomial
$T(z)=f(Rz)-\alpha f(z)=\left\\{P(Rz)-\alpha
P(z)\right\\}-\lambda\left\\{P^{\star}(Rz)-\alpha P^{\star}(z)\right\\}$
has all its zeros in $|z|<1$ for every real or complex $\alpha$ with
$|\alpha|\geq 1$ and $R>1.$
Applying Lemma 1 to polynomial $T(z)$ and noting that $B$ is a linear
operator, it follows that all the zeros of polynomial
$\displaystyle B[T](z)$ $\displaystyle=B[f\circ\sigma](z)-\alpha B[f](z)$
$\displaystyle=\left\\{B[P\circ\sigma](z)-\alpha
B[P](z)\right\\}-\lambda\left\\{B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)\right\\}$
lie in $|z|<1$ where $\sigma(z)=Rz.$ This implies
$|B[P\circ\sigma](z)-\alpha B[P](z)|\leq|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)|$ (2.8)
for $|z|\geq 1$ and $R>1.$ If inequality (2.8) is not true, then there exits a
point $z=z_{0}$ with $|z_{0}|\geq 1$ such that
$|B[P\circ\sigma](z_{0})-\alpha
B[P](z_{0})|>|B[P^{\star}\circ\sigma](z_{0})-\alpha B[P^{\star}](z_{0})|.$
(2.9)
But all the zeros of $P^{\star}(Rz)$ lie in $|z|<1/R<1,$ therefore, it follows
(as in case of $f(z)$) that all the zeros of $P^{\star}(Rz)-\alpha
P^{\star}(z)$ lie in $|z|<1.$ Hence, by Lemma 1, we have
$B[P^{\star}\circ\sigma](z_{0})-\alpha B[P^{\star}](z_{0})\neq 0.$
We take
$\lambda=\dfrac{B[P\circ\sigma](z_{0})-\alpha
B[P](z_{0})}{B[P^{\star}\circ\sigma](z_{0})-\alpha B[P^{\star}](z_{0})},$
then $\lambda$ is well defined real or complex number with $|\lambda|>1$ and
with this choice of $\lambda,$ we obtain $B[T](z_{0})=0$ where $|z_{0}|\geq
1.$ This contradicts the fact that all the zeros of $B[T](z)$ lie in $|z|<1.$
Thus (2.8) holds true for $|\alpha|\leq 1$ and $R>1.$ ∎
###### Lemma 5.
If $P\in\mathscr{P}_{n}$ and $P(z)$ has no zero in $\left|z\right|<1,$ then
for every $\alpha\in\mathbb{C}$ with $|\alpha|\leq 1,$ $R>1$ and $|z|\geq 1$,
$\displaystyle\big{|}B[P\circ\sigma]$ $\displaystyle(z)-\alpha B[P](z)\big{|}$
$\displaystyle\leq$ $\displaystyle\left|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)\right|-(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m,$
(2.10)
where $P^{\star}(z)=z^{n}\overline{P(1/\overline{z})},$
$m=\underset{|z|=1}{Min}|P(z)|$ and $\sigma(z)=Rz.$
###### Proof.
By hypothesis $P(z)$ has all its zeros in $|z|\geq 1$ and
$\displaystyle m\leq|P(z)|\,\,\,\textnormal{for}\,\,\,\,|z|=1.$ (2.11)
We show $F(z)=P(z)+\lambda m$ does not vanish in $|z|<1$ for every $\lambda$
with $|\lambda|<1.$ This is obvious if $m=0$ that is, if $P(z)$ has a zero on
$|z|=1.$ So we assume all the zeros of $P(z)$ lie in $|z|>1,$ then $m>0$ and
by the maximum modulus principle, it follows from (2.11),
$\displaystyle m<|P(z)|\,\,\,\textnormal{for}\,\,\,|z|<1.$ (2.12)
Now if $F(z)=P(z)+\lambda m=0$ for some $z_{0}$ with $|z_{0}|<1,$ then
$\displaystyle P(z_{0})+\lambda m=0$
This implies
$\displaystyle|P(z_{0})|=|\lambda|m\leq
m,\,\,\,\textnormal{for}\,\,\,|z_{0}|<1$ (2.13)
which is clearly contradiction to (2.12). Thus the polynomial $F(z)$ does not
vanish in $|z|<1$ for every $\lambda$ with $|\lambda|<1.$ applying Lemma 4 to
the polynomial $F(z),$ we get
$\displaystyle|B[F\circ\sigma](z)-\alpha
B[F](z)|\leq|B[F^{\star}\circ\sigma](z)-\alpha B[F^{\star}](z)$
for $|z|=1$ and $R>1.$ Replacing $F(z)$ by $P(z)+\lambda m,$ we obtain
$\displaystyle|B[P\circ\sigma](z)$ $\displaystyle-\alpha
B[P](z)+\lambda(1-\alpha)\lambda_{0}m|$
$\displaystyle\leq|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)+\bar{\lambda}(R^{n}-\alpha)\Lambda_{n}z^{n}m|$ (2.14)
Now choosing the argument of $\lambda$ in the right hand side of (Proof.) such
that
$\displaystyle|B[P^{\star}\circ\sigma](z)$ $\displaystyle-\alpha
B[P^{\star}](z)+\bar{\lambda}(R^{n}-\alpha)\Lambda_{n}z^{n}m|$
$\displaystyle=|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)|-|\lambda||R^{n}-\alpha||\Lambda_{n}|m$
for $|z|=1,$which is possible by Lemma 3,we get
$\displaystyle|B[P^{\star}\circ\sigma](z)-\alpha B[P^{\star}](z)|$
$\displaystyle-|\lambda||1-\alpha||\lambda_{0}|m$ $\displaystyle\leq$
$\displaystyle|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)|-|\lambda||R^{n}-\alpha||\Lambda_{n}|m$
Equivalently,
$\displaystyle\big{|}B[P\circ\sigma]$ $\displaystyle(z)-\alpha B[P](z)\big{|}$
$\displaystyle\leq$ $\displaystyle\left|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)\right|-(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m,$
∎
Next we describe a result of Arestov [2].
For $\delta=(\delta_{0},\delta_{1},\cdots,\delta_{n})\in\mathbb{C}^{n+1}$ and
$P(z)=\sum_{j=0}^{n}a_{j}{z}^{j}\in\mathscr{P}_{n}$, we define
$\psi_{\delta}P(z)=\sum_{j=0}^{n}\delta_{j}a_{j}{z}^{j}.$
The operator $\psi_{\delta}$ is said to be admissible if it preserves one of
the following properties:
(i) $P(z)$ has all its zeros in $\left\\{z\in\mathbb{C}:|z|\leq 1\right\\},$
(ii) $P(z)$ has all its zeros in$\left\\{z\in\mathbb{C}:|z|\geq 1\right\\}.$
The result of Arestov [2] may now be stated as follows.
###### Lemma 6.
[2, Theorem 4] Let $\phi(x)=\rho(logx)$ where $\rho$ is a convex non
decreasing function on $\mathbb{R}.$ Then for all $P\in\mathscr{P}_{n}$ and
each admissible operator $\psi_{\delta}$,
$\int_{0}^{2\pi}\phi(|\psi_{\delta}P(e^{i\theta})|)d\theta\leq\int_{0}^{2\pi}\phi(C(\delta,n)|P(e^{i\theta})|)d\theta,$
where $C(\delta,n)=max(|\delta_{0}|,|\delta_{n}|).$
In particular, Lemma 6 applies with $\phi:x\rightarrow x^{p}$ for every
$p\in(0,\infty)$. Therefore, we have
$\left\\{\int_{0}^{2\pi}(|\psi_{\delta}P(e^{i\theta})|^{p})d\theta\right\\}^{1/p}\leq
C(\delta,n)\left\\{\int_{0}^{2\pi}|P(e^{i\theta})|^{p}d\theta\right\\}^{1/p}.$
(2.15)
We use (2.15) to prove the following interesting result.
###### Lemma 7.
If $P\in\mathscr{P}_{n}$ and $P(z)$ does not vanish in $|z|<1,$ then for every
$p>0$, $R>1$ and for $\gamma$ real, $0\leq\gamma<2\pi$,
$\displaystyle\int_{0}^{2\pi}\Big{|}\Big{\\{}B[P\circ\sigma](e^{i\theta})$
$\displaystyle-\alpha B[P](e^{i\theta})\Big{\\}}\mbox{}\,e^{i\gamma}$
$\displaystyle+\left\\{B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\right\\}\Big{|}^{p}d\theta$
$\displaystyle\leq\Big{|}(R^{n}-\alpha)$
$\displaystyle\Lambda_{n}e^{i\gamma}+(1-\bar{\alpha})\bar{\lambda_{0}}\Big{|}^{p}\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta,$
(2.16)
where $B\in\mathcal{B}_{n}$, $\sigma(z):=Rz$,
$B[P^{\star}\circ\sigma]^{\star}(z):=(B[P^{\star}\circ\sigma](z))^{\star}$ and
$\Lambda_{n}$ is defined by (1.13).
###### Proof.
Since $P\in\mathscr{P}_{n}$ and $P^{\star}(z)=z^{n}\overline{P(1/\bar{z})}$,
by Lemma 4 , we have for $|z|\geq 1,$
$\left|B[P\circ\sigma](z)-\alpha
B[P](z)\right|\leq\left|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)\right|,$ (2.17)
Also, since $P^{\star}(Rz)-\alpha
P^{\star}(z)=R^{n}z^{n}\overline{P(1/R\bar{z})}-\alpha
z^{n}\overline{P(1/\bar{z})}$,
$\displaystyle B[P^{\star}\circ\sigma](z)$ $\displaystyle-\alpha
B[P^{\star}](z)=\lambda_{0}\Big{\\{}R^{n}z^{n}\overline{P(1/R\bar{z})}-\alpha
z^{n}\overline{P(1/\bar{z})}\Big{\\}}$ $\displaystyle+\lambda_{1}$
$\displaystyle\left(\frac{nz}{2}\right)\Big{\\{}\left(nR^{n}z^{n-1}\overline{P(1/R\bar{z})}-R^{n-1}z^{n-2}\overline{P^{\prime}(1/R\bar{z})}\right)$
$\displaystyle-\alpha\left(nz^{n-1}\overline{P(1/\bar{z})}-z^{n-2}\overline{P^{\prime}(1/\bar{z})}\right)\Big{\\}}$
$\displaystyle+\frac{\lambda_{2}}{2!}$
$\displaystyle\left(\frac{nz}{2}\right)^{2}\Big{\\{}\Big{(}n(n-1)R^{n}z^{n-2}\overline{P(1/R\bar{z})}$
$\displaystyle-2(n-1)R^{n-1}z^{n-3}\overline{P^{\prime}(1/R\bar{z})}+R^{n-2}z^{n-4}\overline{P^{\prime\prime}(1/R\bar{z})}\Big{)}$
$\displaystyle-\alpha\Big{(}n$
$\displaystyle(n-1)z^{n-2}\overline{P(1/\bar{z})}-2(n-1)z^{n-3}\overline{P^{\prime}(1/\bar{z})}$
$\displaystyle+r^{n-2}z^{n-4}\overline{P^{\prime\prime}(1/\bar{z})}\Big{)}\Big{\\}}$
and therefore,
$\displaystyle B[P^{\star}\circ\sigma]^{\star}(z)$
$\displaystyle-\bar{\alpha}B[P^{\star}]^{\star}(z)=\Big{(}B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)\Big{)}^{\star}$ $\displaystyle=\Big{(}\bar{\lambda_{0}}+$
$\displaystyle\bar{\lambda_{1}}\frac{n^{2}}{2}+\bar{\lambda_{2}}\frac{n^{3}(n-1)}{8}\Big{)}\Big{\\{}R^{n}P(z/R)-\bar{\alpha}P(z)\Big{\\}}$
$\displaystyle-\Big{(}\bar{\lambda_{1}}$
$\displaystyle\frac{n}{2}+\bar{\lambda_{2}}\frac{n^{2}(n-1)}{4}\Big{)}\Big{\\{}R^{n-1}zP^{\prime}(z/R)-\bar{\alpha}zP^{\prime}(z)\Big{\\}}$
$\displaystyle+\bar{\lambda_{2}}$
$\displaystyle\frac{n^{2}}{8}\Big{\\{}R^{n-2}z^{2}P^{\prime\prime}(z/R)-\bar{\alpha}z^{2}P^{\prime\prime}(z)\Big{\\}}.$
(2.18)
Also,
$|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)|=|B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z)|\,\,\,\hbox{}\,\,\,\textrm{for}\,\,\,|z|=1.$
Using this in (2.17), we get
$|B[P\circ\sigma](z)-\alpha
B[P](z)|\leq|B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z)|\,\,\,\hbox{}\,\,\,\textrm{for}\,\,\,|z|=1.$
As in Lemma 4, the polynomial $P^{\star}\circ\sigma(z)-\alpha P^{\star}(z),$
has all its zeros in $|z|<1$ and by Lemma 1,
$B[P^{\star}\circ\sigma](z)-\alpha B[P^{\star}](z),$ also has all its zero in
$|z|<1,$ therefore,
$B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z)$ has
all its zeros in $|z|\geq 1.$ Hence by the maximum modulus principle,
$|B[P\circ\sigma](z)-\alpha
B[P](z)|<|B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z)|\,\,\,\hbox{}\,\,\,\textrm{for}\,\,\,|z|<1$
(2.19)
A direct application of Rouche’s theorem shows that with
$P(z)=a_{n}z^{n}+\cdots+a_{0},$
$\displaystyle\psi_{\delta}P(z)=$
$\displaystyle\Big{\\{}B[P\circ\sigma](z)-\alpha
B[P](z)\Big{\\}}e^{i\gamma}+B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z),$
$\displaystyle=$
$\displaystyle\left\\{(R^{n}-\alpha)\left(\lambda_{0}+\lambda_{1}\frac{n^{2}}{2}+\lambda_{2}\frac{n^{3}(n-1)}{8}\right)e^{i\gamma}+(1-\bar{\alpha})\bar{\lambda_{0}}\right\\}a_{n}z^{n}$
$\displaystyle+\cdots+\left\\{(R^{n}-\bar{\alpha})\left(\bar{\lambda_{0}}+\bar{\lambda_{1}}\frac{n^{2}}{2}+\bar{\lambda_{2}}\frac{n^{3}(n-1)}{8}\right)+e^{i\gamma}(1-\alpha)\lambda_{0}\right\\}a_{0},$
has all its zeros in $|z|\geq 1$, for every real $\gamma,$ $0\leq\gamma\leq
2\pi.$ Therefore, $\psi_{\delta}$ is an admissible operator. Applying (2.15)
of Lemma 6, the desired result follows immediately for each $p>0$.
∎
We also need the following lemma [4].
###### Lemma 8.
If $A,B,C$ are non-negative real numbers such that $B+C\leq A,$ then for each
real number $\gamma,$
$|(A-C)e^{i\gamma}+(B+C)|\leq|Ae^{i\gamma}+B|.$
## 3\. Proof of Theorem
###### Proof of Theorem 1.
By hypothesis $P(z)$ does not vanish in $|z|<1,$ therefore by Lemma 5, we have
$\displaystyle\big{|}B[P\circ\sigma]$ $\displaystyle(z)-\alpha B[P](z)\big{|}$
$\displaystyle\leq$ $\displaystyle\left|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)\right|-(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m,$
(3.1)
for $|z|=1,$ $|\alpha|\leq 1$ and $R>1$ where
$P^{\star}(z)=z^{n}\overline{P(1/\overline{z})}.$ Since
$B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z)$ is
the conjugate of $B[P^{\star}\circ\sigma](z)-\alpha B[P^{\star}](z)$ and
$|B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z)|=|B[P^{\star}\circ\sigma](z)-\alpha
B[P^{\star}](z)|\,\,\,\,\textnormal{for}\,\,\,\,\,|z|=1.$
Thus (3) can be written as
$\displaystyle|B[P\circ\sigma](z)$ $\displaystyle-\alpha
B[P](z)|+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}$
$\displaystyle\leq$
$\displaystyle\left|B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z)\right|-\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\,\,\,\,\,\mbox{}\textrm{for}\,\,\,\,\,|z|=1$
(3.2)
Taking
$A=\left|B[P^{\star}\circ\sigma]^{\star}(z)-\bar{\alpha}B[P^{\star}]^{\star}(z)\right|,\,\,\,B=\left|B[P\circ\sigma](z)-\alpha
B[P](z)\right|$
and
$C=\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}$
in Lemma 8 and noting by (3) that
$B+C\leq A-C\leq A,$
we get for every real $\gamma$,
$\displaystyle\Big{|}\Big{\\{}\big{|}B[P^{\star}$
$\displaystyle\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}-\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}e^{i\gamma}$
$\displaystyle+\Big{\\{}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}\Big{|}$
$\displaystyle\leq\Big{|}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}e^{i\gamma}+\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}\Big{|}.$
This implies for each $p>0,$
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{|}\Big{\\{}\big{|}B[P^{\star}$
$\displaystyle\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}-\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}e^{i\gamma}$
$\displaystyle+\Big{\\{}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}\Bigg{|}^{p}d\theta$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Big{|}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}e^{i\gamma}+\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}\Big{|}^{p}d\theta.$ (3.3)
Integrating both sides of (3) with respect to $\gamma$ from $0$ to $2\pi,$ we
get with the help of Lemma 7 for each $p>0,$
$\displaystyle\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}\Bigg{|}$
$\displaystyle\Big{\\{}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}-\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}e^{i\gamma}$
$\displaystyle+\Big{\\{}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}\Bigg{|}^{p}d\theta
d\gamma$ $\displaystyle\leq$
$\displaystyle\int\limits_{0}^{2\pi}\int\limits_{0}^{2\pi}\Bigg{|}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}e^{i\gamma}+\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}\Bigg{|}^{p}d\theta d\gamma.$ $\displaystyle\leq$
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}\Bigg{|}\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}e^{i\gamma}+\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}\Bigg{|}^{p}d\gamma\Bigg{\\}}d\theta$
$\displaystyle\leq$
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}\Bigg{|}\big{\\{}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{\\}}e^{i\gamma}+\big{\\{}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{\\}}\Bigg{|}^{p}d\gamma\Bigg{\\}}d\theta$
$\displaystyle\leq$
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{\\{}\int\limits_{0}^{2\pi}\Bigg{|}\big{\\{}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{\\}}e^{i\gamma}+\big{\\{}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{\\}}\Bigg{|}^{p}d\theta\Bigg{\\}}d\gamma$
$\displaystyle\leq$
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{|}(R^{n}-\alpha)\Lambda_{n}e^{i\gamma}+(1-\bar{\alpha})\bar{\lambda_{0}}\Bigg{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
(3.4)
Now it can be easily verified that for every real number $\gamma$ and $s\geq
1$,
$\left|s+e^{i\alpha}\right|\geq\left|1+e^{i\alpha}\right|.$
This implies for each $p>0$,
$\int_{0}^{2\pi}\left|s+e^{i\gamma}\right|^{p}d\gamma\geq\int_{0}^{2\pi}\left|1+e^{i\gamma}\right|^{p}d\gamma.$
(3.5)
If $\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha|-|1-\alpha||\lambda_{0}|)m}{2}\neq
0,$ we take
$s=\dfrac{\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}-\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}}{\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}}$
, then by (3), $s\geq 1$ and we get with the help of (3.5),
$\displaystyle\int\limits_{0}^{2\pi}\Big{|}\Big{\\{}$
$\displaystyle\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}-\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}e^{i\gamma}$
$\displaystyle+\Big{\\{}$
$\displaystyle\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}\Big{|}^{p}d\gamma$
$\displaystyle=$
$\displaystyle\Bigg{|}\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Bigg{|}^{p}$
$\displaystyle\times\int\limits_{0}^{2\pi}\left|e^{i\gamma}+\dfrac{\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}-\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}}{\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}}\,\right|^{p}d\gamma$
$\displaystyle=$
$\displaystyle\Bigg{|}\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Bigg{|}^{p}$
$\displaystyle\times\int\limits_{0}^{2\pi}\left|e^{i\gamma}+\Bigg{|}\dfrac{\big{|}B[P^{\star}\circ\sigma]^{\star}(e^{i\theta})-\bar{\alpha}B[P^{\star}]^{\star}(e^{i\theta})\big{|}-\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}}{\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}}\Bigg{|}^{p}\,\right|d\gamma$
$\displaystyle\geq$
$\displaystyle\Bigg{|}\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Bigg{|}^{p}\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma$
(3.6)
For $\big{|}B[P\circ\sigma](e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}=0$,
then (3) is trivially true. Using this in (3), we conclude for every real or
complex number $\alpha$ with $|\alpha|\leq 1,$ $R>1$ and $p>0$,
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{|}\big{|}B[P\circ\sigma]$
$\displaystyle(e^{i\theta})-\alpha
B[P](e^{i\theta})\big{|}+\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Bigg{|}^{p}d\theta\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Big{|}(R^{n}-\alpha)\Lambda_{n}e^{i\gamma}+(1-\bar{\alpha})\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
This gives for every real or complex number $\delta,\alpha$ with $|\delta|\leq
1,$ $|\alpha|\leq 1,$ $R>1$ and $\gamma$ real
$\displaystyle\int\limits_{0}^{2\pi}\Bigg{|}B[P\circ\sigma]$
$\displaystyle(e^{i\theta})-\alpha
B[P](e^{i\theta})+\delta\Big{\\{}\dfrac{(|R^{n}-\alpha||\Lambda_{n}|-|1-\alpha||\lambda_{0}|)m}{2}\Big{\\}}\Bigg{|}^{p}d\theta\int\limits_{0}^{2\pi}|1+e^{i\gamma}|^{p}d\gamma$
$\displaystyle\leq\int\limits_{0}^{2\pi}\Big{|}(R^{n}-\alpha)\Lambda_{n}e^{i\gamma}+(1-\bar{\alpha})\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta.$
(3.7)
Since
$\displaystyle\int\limits_{0}^{2\pi}\Big{|}(R^{n}$
$\displaystyle-\alpha)\Lambda_{n}e^{i\gamma}+(1-\bar{\alpha})\bar{\lambda_{0}}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}|(R^{n}-\alpha)\Lambda_{n}|e^{i\gamma}+|(1-\bar{\alpha})\bar{\lambda_{0}}|\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta$
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}|(R^{n}-\alpha)\Lambda_{n}|e^{i\gamma}+|(1-\alpha)\lambda_{0}|\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta,$
$\displaystyle=\int\limits_{0}^{2\pi}\Big{|}(R^{n}-\alpha)\Lambda_{n}e^{i\gamma}+(1-\alpha)\lambda_{0}\Big{|}^{p}d\gamma\int_{0}^{2\pi}\left|P(e^{i\theta})\right|^{p}d\theta,$
(3.8)
the desired result follows immediately by combining (3) and (3). This
completes the proof of Theorem 1 for $p>0$. To establish this result for
$p=0$, we simply let $p\rightarrow 0+$.
∎
## References
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* [9] G.H.HARDY, The mean value of the modulus of an analytic function, Proc. London Math. Soc., 14(1915), 269-277.
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* [11] M. MARDEN, Geometry of Polynomials, Math. Surves No. 3, Amer. Math. Soc., Providence, R I (1966).
* [12] G. V. MILOVANOVIC, D. S. MITRINOVIC and TH. M. RASSIAS, Topics in Polynomials: Extremal Properties, Inequalities, Zeros, World scientific Publishing Co., Singapore, (1994).
* [13] G. PÓLYA an G. SZEGÖ, Aufgaben und lehrsätze aus der Analysis, Springer-Verlag, Berlin (1925).
* [14] Q. I. RAHMAN, Functions of exponential type, Trans. Amer. Soc., 135(1969), 295 – 309
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|
arxiv-papers
| 2013-06-04T09:59:35 |
2024-09-04T02:49:46.074042
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "N. A. Rather and Suhail Gulzar",
"submitter": "Suhail Gulzar Mattoo Suhail Gulzar",
"url": "https://arxiv.org/abs/1306.0714"
}
|
1306.0834
|
defipropdefi defiprop defilemmadefi defilemma thmdefi thm lemmadefi lemma
propdefi prop cordefi cor remarkdefi remark exmdefi exm
# Sets of lengths in maximal orders in central simple algebras
Daniel Smertnig Institut für Mathematik und Wissenschaftliches Rechnen
Karl-Franzens-Universität Graz
Heinrichstraße 36
8010 Graz, Austria [email protected]
###### Abstract.
Let $\mathcal{O}$ be a holomorphy ring in a global field $K$, and $R$ a
classical maximal $\mathcal{O}$-order in a central simple algebra over $K$. We
study sets of lengths of factorizations of cancellative elements of $R$ into
atoms (irreducibles). In a large majority of cases there exists a transfer
homomorphism to a monoid of zero-sum sequences over a ray class group of
$\mathcal{O}$, which implies that all the structural finiteness results for
sets of lengths—valid for commutative Krull monoids with finite class
group—hold also true for $R$. If $\mathcal{O}$ is the ring of algebraic
integers of a number field $K$, we prove that in the remaining cases no such
transfer homomorphism can exist and that several invariants dealing with sets
of lengths are infinite.
###### Key words and phrases:
sets of lengths, maximal orders, global fields, Brandt groupoid, divisorial
ideals, Krull monoids
###### 2010 Mathematics Subject Classification:
16H10, 16U30, 20M12, 20M13, 11R54
The author is supported by the Austrian Science Fund (FWF): W1230, Doctoral
Program “Discrete Mathematics”
## 1\. Introduction
Let $H$ be a (left- and right-) cancellative semigroup and $H^{\times}$ its
group of units. An element $u\in H\setminus H^{\times}$ is called
_irreducible_ (or an _atom_) if $u=ab$ with $a,b\in H$ implies that $a\in
H^{\times}$ or $b\in H^{\times}$. If $a\in H\setminus H^{\times}$, then
$l\in\mathbb{N}$ is a _length of $a$_ if there exist atoms
$u_{1},\ldots,u_{l}\in H$ with $a=u_{1}\cdot\ldots\cdot u_{l}$, and the _set
of lengths_ of $a$, written as $\mathsf{L}(a)$, consists of all such lengths.
If there is a non-unit $a\in H$ with $\lvert\mathsf{L}(a)\rvert>1$, say
$1<k<l\in\mathsf{L}(a)$, then for every $n\in\mathbb{N}$, we have
$\mathsf{L}(a^{n})\supset\\{\,kn+\nu(l-k)\mid\nu\in[0,n]\,\\}$, which shows
that sets of lengths become arbitrarily large. If $H$ is commutative and
satisfies the ACC on divisorial ideals, then all sets of lengths are finite
and non-empty.
Sets of lengths (and all invariants derived from them, such as the set of
distances) are among the most investigated invariants in factorization theory.
So far research has almost been entirely devoted to the commutative setting,
and it has focused on commutative noetherian domains, commutative Krull
monoids, numerical monoids, and others (cf. [1, 12, 27, 28, 26, 20, 7]).
Recall that a commutative noetherian domain is a Krull domain if and only if
the monoid of non-zero elements is a Krull monoid and this is the case if and
only if the domain is integrally closed. Suppose that $H$ is a Krull monoid
(so completely integrally closed and the ACC on divisorial two-sided ideals
holds true). Then the monoid of divisorial two-sided ideals is a free abelian
monoid. If $H$ is commutative (or at least normalizing), this gives rise to
the construction of a transfer homomorphism $\theta\colon
H\to\mathcal{B}(G_{P})$, where $\mathcal{B}(G_{P})$ is the monoid of zero-sum
sequences over a subset $G_{P}$ of the class group $G$ of $H$. Transfer
homomorphisms preserve sets of lengths, and if $G_{P}$ is finite, then
$\mathcal{B}(G_{P})$ is a finitely generated commutative Krull monoid, whose
sets of lengths can be studied with methods from combinatorial number theory.
This approach has lead to a large variety of structural results for sets of
lengths in commutative Krull monoids (see [27, 24] for an overview).
Only first hesitant steps were taken so far to study factorization properties
in a non-commutative setting (for example, quaternion orders are investigated
in [19, 18, 16]), semifirs in ([14, 15], semigroup algebras in [37]). The
present paper provides an in-depth study of sets of lengths in classical
maximal orders over holomorphy rings in global fields.
Let $\mathcal{O}$ be a commutative Krull domain with quotient field $K$, $A$ a
central simple algebra over $K$, $R$ a maximal order in $A$, and $R^{\bullet}$
the semigroup of cancellative elements (equivalently, $R$ is a PI Krull ring).
Any approach to study sets of lengths, which runs as described above and
involves divisorial two-sided ideals, is restricted to normalizing Krull
monoids ([25, Theorem 4.13]). For this reason we develop the theory of
divisorial one-sided ideals. In Section 3 we fix our terminology in the
setting of cancellative small categories. Following ideas of Asano and Murata
[5] and partly of Rehm [45, 46], we provide in Section 4 a factorization
theory of integral elements in arithmetical groupoids, and introduce an
abstract transfer homomorphism for a subcategory of such a groupoid (Section
4). In Section 5 the divisorial one-sided ideal theory of maximal orders in
quotient semigroups is given, and Section 5 establishes the relationship with
arithmetical groupoids. Section 5 is a main result in the abstract setting of
arithmetical maximal orders (Remarks 5.2 and 5.1 reveal how the well-known
transfer homomorphisms for normalizing Krull monoids fit into our abstract
theory). For maximal orders over commutative Krull domains, we see that all
sets of lengths are finite and non-empty (Section 5.2). In Section 6 we
demonstrate that classical maximal orders over holomorphy rings in global
fields fulfill the abstract assumptions of Section 5, which implies the
following structural finiteness results on sets of lengths.
###### Theorem .
Let $\mathcal{O}$ be a holomorphy ring in a global field $K$, $A$ a central
simple algebra over $K$, and $R$ a classical maximal $\mathcal{O}$-order of
$A$. Suppose that every stably free left $R$-ideal is free. Then there exists
a transfer homomorphism $\theta\colon
R^{\bullet}\to\mathcal{B}(\operatorname{\mathcal{C}}_{A}(\mathcal{O}))$, where
$\operatorname{\mathcal{C}}_{A}(\mathcal{O})=\mathcal{F}^{\times}(\mathcal{O})\,/\,\\{\,a\mathcal{O}\mid
a\in K^{\times},\,a_{v}>0\text{ for all archimedean places $v$ of $K$ where
$A$ is ramified.}\,\\}$
is a ray class group of $\mathcal{O}$, and
$\mathcal{B}(\operatorname{\mathcal{C}}_{A}(\mathcal{O}))$ is the monoid of
zero-sum sequences over $\operatorname{\mathcal{C}}_{A}(\mathcal{O})$. In
particular,
1. 1.
The set of distances $\Delta(R^{\bullet})$ is a finite interval, and if it is
non-empty, then $\min\Delta(R^{\bullet})=1$.
2. 2.
For every $k\in\mathbb{N}$, the union of sets of lengths containing $k$,
denoted by $\mathcal{U}_{k}(R^{\bullet})$, is a finite interval.
3. 3.
There is an $M\in\mathbb{N}_{0}$ such that for every $a\in R^{\bullet}$ the
set of lengths $\mathsf{L}(a)$ is an AAMP with difference
$d\in\Delta(R^{\bullet})$ and bound $M$.
Thus, under the additional hypothesis that every stably free left $R$-ideal is
free, we obtain a transfer homomorphism to a monoid of zero-sum sequences over
a finite abelian group. Therefore, sets of lengths in $R$ are the same as sets
of lengths in a commutative Krull monoid with finite class group.
If $A$ satisfies the Eichler condition relative to $\mathcal{O}$, then every
stably free left $R$-ideal is free by Eichler’s Theorem. In particular, if $K$
is a number field and $\mathcal{O}$ is its ring of algebraic integers, then
$A$ satisfies the Eichler condition relative to $\mathcal{O}$ unless $A$ is a
totally definite quaternion algebra. Thus in this setting Section 1 covers the
large majority of cases, and the following complementary theorem shows that
the condition that every stably free left $R$-ideal is free is indeed
necessary.
###### Theorem .
Let $\mathcal{O}$ be the ring of algebraic integers in a number field $K$, $A$
a central simple algebra over $K$, and $R$ a classical maximal
$\mathcal{O}$-order of $A$. If there exists a stably free left $R$-ideal that
is not free, then there exists no transfer homomorphism $\theta\colon
R^{\bullet}\to\mathcal{B}(G_{P})$, where $G_{P}$ is any subset of an abelian
group. Moreover,
1. 1.
$\Delta(R^{\bullet})=\mathbb{N}$.
2. 2.
For every $k\geq 3$, we have $\mathbb{N}_{\geq
3}\subset\mathcal{U}_{k}(R^{\bullet})\subset\mathbb{N}_{\geq_{2}}$.
The proof of Section 1 is based on recent work of Kirschmer and Voight ([39,
40]), and will be given in Section 7. If $H$ is a commutative Krull monoid
with an infinite class group such that every class contains a prime divisor,
then Kainrath showed that every finite subset of $\mathbb{N}_{\geq 2}$ can be
realized as a set of lengths ([38], or [27, Section 7.4]), whence
$\Delta(H)=\mathbb{N}$ and $\mathcal{U}_{k}(H)=\mathbb{N}_{\geq 2}$ for all
$k\geq 2$. However, we explicitly show that in the above situation no transfer
homomorphism is possible, implying that the factorization of $R^{\bullet}$
cannot be modeled by a monoid of zero-sum sequences. A similar statement about
sets of lengths in the integer-valued polynomials, as well as the
impossibility of a transfer homomorphism to a monoid of zero-sum sequences,
was recently shown by Frisch [21].
## 2\. Preliminaries
Let $\mathbb{N}$ denote the set of positive integers and put
$\mathbb{N}_{0}=\\{\,0\,\\}\cup\mathbb{N}$. For integers $a,b\in\mathbb{Z}$,
let $[a,b]=\\{\,x\in\mathbb{Z}\mid a\leq x\leq b\,\\}$ denote the discrete
interval. All semigroups and rings are assumed to have an identity element,
and all homomorphisms respect the identity. By a factorization we always mean
a factorization of a cancellative element into irreducible elements (a formal
definition follows in Section 3). In order to study factorizations in
semigroups we will have to investigate their divisorial one-sided ideal
theory, in which the multiplication of ideals only gains sufficiently nice
properties if one considers it as a partial operation that is only defined for
certain pairs of ideals. This is the reason why we introduce our concepts in
the setting of groupoids and consider subcategories of these groupoids.
Throughout the paper there will be many statements that can be either
formulated “from the left” or “from the right”, and most of the time it is
obvious how the symmetric statement should look like. Therefore often just one
variant is formulated and it is left to the reader to fill in the symmetric
definition or statement if required.
### 2.1. Small categories as generalizations of semigroups
Let $H$ be a small category. In the sequel the objects of $H$ play no role,
and therefore we shall identify $H$ with the set of morphisms of $H$. We
denote by $H_{0}$ the set of identity morphisms (representing the objects of
the category). There are two maps $s,t\colon H\to H_{0}$ such that two
elements $a,b\in H$ are composable to a (uniquely determined) element $ab\in
H$ if and only if $t(a)=s(b)$. 111This choice of $t$ and $s$ is compatible
with the usual convention for groupoids, but unfortunately opposite to the
usual convention for categories. For $e,f\in H_{0}$ we set $H(e,f)=\\{\,a\in
H\mid s(a)=e,\,t(a)=f\,\\}$, $H(e)=H(e,e)$, $H(e,\cdot)=\bigcup_{f^{\prime}\in
H_{0}}H(e,f^{\prime})$ and $H(\cdot,f)=\bigcup_{e^{\prime}\in
H_{0}}H(e^{\prime},f)$. Note that an element $e\in H$ lies in $H_{0}$ if and
only if $s(e)=t(e)=e$, $ea=a$ for all $a\in H(e,\cdot)$ and $ae=a$ for all
$a\in H(\cdot,e)$.
A semigroup may be viewed as a category with a single object (corresponding to
its identity element), and elements of the semigroup as morphisms with source
and target this unique object. In this way the notion of a small category
generalizes the usual notion of a semigroup ($H$ is a semigroup if and only if
$\lvert H_{0}\rvert=1$). We will consider a semigroup to be a small category
in this sense whenever this is convenient, without explicitly stating this
anymore. For $A,B\subset H$ we write $AB=\\{\,ab\in H\mid a\in A,b\in B\text{
and }t(a)=s(b)\,\\}$ for the set of all possible products, and if $b\in H$,
then $Ab=A\\{b\\}$ and $bA=\\{b\\}A$.
An element $a\in H$ is called _left-cancellative_ if it is an epimorphism
($ab=ac$ implies $b=c$ for all $b,c\in H(t(a),\cdot)$), and it is called
_right-cancellative_ if it is a monomorphism ($ba=ca$ implies $b=c$ for all
$b,c\in H(\cdot,s(a))$), and _cancellative_ if it is both. The set of all
cancellative elements is denoted by $H^{\bullet}$, and $H$ is called
_cancellative_ if $H=H^{\bullet}$. The set of isomorphisms of $H$ will also be
called the _set of units_ , and we denote it by $H^{\times}$. A subcategory
$D\subset H$ is _wide_ if $D_{0}=H_{0}$.
In line with the multiplicative notation, if $H$ and $D$ are two small
categories, we call a functor $f\colon H\to D$ a homomorphism (of small
categories). Explicitly, a map $f\colon H\to D$ is a homomorphism if
$f(H_{0})\subset D_{0}$ and whenever $a,b\in H$ with $t(a)=s(b)$ then also
$f(a)\cdot f(b)$ is defined (i.e., $t(f(a))=s(f(b))$) and $f(ab)=f(a)f(b)$.
If $H$ is a commutative semigroup, and $D\subset H$ is a subsemigroup, then a
localization $D^{-1}H$ with an embedding $H\hookrightarrow D^{-1}H$ exists
whenever all elements of $D$ are cancellative, and in particular $H$ has a
group of fractions if and only if $H$ is cancellative. If $H$ is a non-
commutative semigroup and $D\subset H$, then a semigroup of right fractions
with respect to $D$, $HD^{-1}$, in which every element can be represented as a
fraction $ad^{-1}$ with $a\in H$, $d\in D$, together with an embedding
$H\hookrightarrow HD^{-1}$, exists if and only if $D$ is cancellative and $D$
satisfies the _right Ore condition_ , meaning $aD\cap dH\neq\emptyset$ for all
$a\in H$ and $d\in D$. For a semigroup of left fractions, $D^{-1}H$, one gets
the analogous _left Ore condition_ , and if $D$ satisfies both, the left and
the right Ore condition, then every semigroup of right fractions is a
semigroup of left fractions and conversely. In this case we write
$D^{-1}H=HD^{-1}$. If $H^{\bullet}$ satisfies the left and right Ore
condition, we also write
$\mathbf{q}(H)=H(H^{\bullet})^{-1}=({H^{\bullet}}^{-1})H$ for the
corresponding semigroup of fractions.
The notion of semigroups of fractions generalizes to categories of fractions
with analogous conditions ([23]). Let $H$ be a small category, and $D\subset
H^{\bullet}$ a subset of the cancellative elements. Then $D$ _admits a
calculus of right fractions_ if $D$ is a wide subcategory of $H$ and it
satisfies the right Ore condition, i.e., $aD\cap dH\neq\emptyset$ for all
$a\in H$ and $d\in D$ with $s(a)=s(d)$. In that case there exists a small
category $HD^{-1}$ with $(HD^{-1})_{0}=H_{0}$ and an embedding $j\colon H\to
HD^{-1}$ (i.e., $j$ is a faithful functor) with $j\mid
H_{0}=\operatorname{id}$ and such that every element of $HD^{-1}$ can be
represented in the form $j(a)j(d)^{-1}$ with $a\in H$, $d\in D$ and
$t(a)=t(d)$, $j(D)\subset H^{\times}$ and it is universal with respect to that
property, i.e., if $f\colon H\to S$ is any homomorphism with $f(D)\subset
S^{\times}$, then there exists a unique $D^{-1}f\colon HD^{-1}\to S$ such that
$D^{-1}f\circ j=f$. We can assume $H\subset HD^{-1}$ and take $j$ to be the
inclusion map, and we call $HD^{-1}$ the _category of right fractions_ of $H$
with respect to $D$. If $D$ also admits a _left calculus of fractions_ , then
$HD^{-1}$ is also a category of left fractions, and we write
$HD^{-1}=D^{-1}H$.
A _monoid_ is a cancellative semigroup satisfying the left and right Ore
condition (following the convention of [25]). Every monoid has a (left and
right) group of fractions which is unique up to unique isomorphism. A
semigroup $H$ is called _normalizing_ if $aH=Ha$ for all $a\in H$. It is
easily checked that a normalizing cancellative semigroup is already a
normalizing monoid.
Let $\mathcal{M}$ be a directed multigraph (i.e., a quiver). For every edge
$a$ of $\mathcal{M}$ we write $s(a)$ for the vertex that is its source and
$t(a)$ for the vertex that is its target. The _path category_ on
$\mathcal{M}$, denoted by $\mathcal{F}(\mathcal{M})$, is defined as follows:
It consists of all tuples $y=(e,a_{1},\ldots,a_{k},f)$ with
$k\in\mathbb{N}_{0}$, $e,f$ vertices of $\mathcal{M}$ and $a_{1},\ldots,a_{k}$
edges of $\mathcal{M}$ with either $k=0$ and $e=f$ or $k>0$, $s(a_{1})=e$,
$t(a_{i})=s(a_{i+1})$ for all $i\in[1,k-1]$ and $t(a_{k})=f$. The set of
identities $\mathcal{F}(\mathcal{M})_{0}$ is the set of all tuples with $k=0$,
and given any tuple $y$ as above, $s(y)=(e,e)$ and $t(y)=(f,f)$. Composition
is defined in the obvious manner by concatenating tuples and removing the two
vertices in the middle. We identify the set of vertices of $\mathcal{M}$ with
$\mathcal{F}(M)_{0}$ so that $(e,e)=e$. Every subset $M$ of a small category
$H$ will be viewed as a quiver, with vertices $\\{\,s(a)\mid a\in
M\,\\}\cup\\{\,t(a)\mid a\in M\,\\}$ and for each $a\in M$ a directed edge
(again called $a$) from $s(a)$ to $t(a)$.
### 2.2. Groupoids
A groupoid $G$ is a small category in which every element is a unit (i.e.,
every morphism is an isomorphism). If $e,f,e^{\prime},f^{\prime}\in G_{0}$ and
there exist $a\in G(e,f)$ and $b\in G(e^{\prime},f^{\prime})$, then
$\begin{cases}G(e,e^{\prime})&\to G(f,f^{\prime})\\\ x&\mapsto
a^{-1}xb\end{cases}$ (1)
is a bijection.
For all $e\in G_{0}$ the set $G(e)$ is a group, called the _vertex group_ or
_isotropy group_ of $G$ at $e$. If $f\in G_{0}$ and $a\in G(e,f)$, then,
taking $b=a$, the map in (1) is a group isomorphism from $G(e)$ to $G(f)$. If
$G(e)$ is abelian, it can be easily checked that this isomorphism does not
depend on the choice of $a$: If $a,a^{\prime}\in G(e,f)$, then
$a^{\prime}(a^{-1}xa)a^{\prime-1}=(a^{\prime}a^{-1})x(aa^{\prime-1})=(a^{\prime}a^{-1})(aa^{\prime-1})x=x.$
In particular, if $G$ is connected (meaning $G(e,e^{\prime})\neq\emptyset$ for
all $e,e^{\prime}\in G_{0}$) and one vertex group is abelian, then all vertex
groups are abelian, and they are canonically isomorphic.
In this case we define for $e\in G_{0}$ and $x\in G(e)$ the set
$(x)=\\{\,a^{-1}xa\mid a\in G(e,\cdot)\,\\}$, and the _universal vertex group_
as
$\mathbb{G}=\\{\,(x)\mid x\in G(e),\,e\in G_{0}\,\\}.$
$\mathbb{G}$ indeed has a natural abelian group structure: For every $e\in
G_{0}$ there is a bijection $j_{e}\colon G(e)\to\mathbb{G},x\mapsto(x)$
inducing the structure of an abelian group on $\mathbb{G}$, and because the
diagrams
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commute for every choice of $e,f\in G_{0}$ and $a\in G(e,f)$, this group
structure is independent of the choice of $e$, yielding a canonical group
isomorphism $j_{e}\colon G(e)\to\mathbb{G}$ for every $e\in G_{0}$. We will
use calligraphic letters to denote elements of $\mathbb{G}$. If
$\mathcal{X}\in\mathbb{G}$, then the unique representative of $\mathcal{X}$ in
$G(e)$, $j_{e}^{-1}(\mathcal{X})$, will be denoted by $\mathcal{X}_{e}$.
If $G$ is a groupoid, and $H\subset G$ is a subcategory, then $HH^{-1}$
denotes the set of all right fractions of elements of $H$. Furthermore,
$HH^{-1}\subset G$ is a subgroupoid if and only if $H$ satisfies the right Ore
condition.
### 2.3. Krull monoids and Krull rings
A monoid $H$ is called a _Krull monoid_ if it is completely integrally closed
(in other words, a maximal order) and satisfies the ACC on divisorial two-
sided ideals. A prime Goldie ring $R$ is a _Krull ring_ if it is completely
integrally closed and satisfies the ACC on divisorial two-sided ideals
(equivalently, its monoid $R^{\bullet}$ of cancellative elements is a Krull
monoid; see [25]). The theory of commutative Krull monoids is presented in
[32, 27]. The simplest examples of non-commutative Krull rings are classical
maximal orders in central simple algebras over Dedekind domains (see Section
5.2). We discuss monoids of zero-sum sequences.
Let $G=(G,0_{G},+)$ be an additively written abelian group, $G_{P}\subset G$ a
subset and let $\mathcal{F}_{ab}(G_{P})$ be the (multiplicatively written)
free abelian monoid with basis $G_{P}$. Elements $S\in\mathcal{F}_{ab}(G_{P})$
are called _sequences over $G_{P}$_, and are written in the form
$S=g_{1}\cdot\ldots\cdot g_{l}$ where $l\in\mathbb{N}_{0}$ and
$g_{1},\ldots,g_{l}\in G_{P}$. We denote by $\lvert S\rvert=l$ the _length_ of
$S$. Such a sequence $S$ is said to be a _zero-sum sequence_ if
$\sigma(S)=g_{1}+\ldots+g_{l}=0_{G}$. The submonoid
$\mathcal{B}(G_{P})=\\{\,S\in\mathcal{F}_{ab}(G_{P})\mid\sigma(S)=0\,\\}\;\subset\;\mathcal{F}_{ab}(G_{P})$
is called the _monoid of zero-sum sequences_ over $G_{P}$. It is a reduced
commutative Krull monoid, which is finitely generated whenever $G_{P}$ is
finite ([27, Theorem 3.4.2]). Moreover, every commutative Krull monoid
possesses a transfer homomorphism onto a monoid of zero-sum sequences, and
thus $\mathcal{B}(G_{P})$ provides a model for the factorization behavior of
commutative Krull monoids ([27, Section 3.4]).
## 3\. Arithmetical Invariants
In this section we introduce our main arithmetical invariants (rigid
factorizations, sets of lengths, sets of distances) and transfer homomorphisms
in the setting of cancellative small categories.
_Throughout this section, let $H$ be a cancellative small category._
$H$ is _reduced_ if $H^{\times}=H_{0}$. An element $u\in H\setminus
H^{\times}$ is an _atom_ (or _irreducible_) if $u=bc$ with $b,c\in H$ implies
$b\in H^{\times}$ or $c\in H^{\times}$. By $\mathcal{A}(H)$ we denote the set
of all atoms of $H$, and call $H$ _atomic_ if every $a\in H\setminus
H^{\times}$ can be written as a (finite) product of atoms. A left ideal of $H$
is a subset $I\subset H$ with $HI\subset I$, and a right ideal of $H$ is
defined similarly. A _principal left (right) ideal of $H$_ is a set of the
form $Ha$ ($aH$) for some $a\in H$. If $H$ is a commutative monoid, then $p\in
H\setminus H^{\times}$ is a _prime element_ if $p\mid ab$ implies $p\mid a$ or
$p\mid b$ for all $a,b\in H$.
###### Proposition .
If $H$ satisfies the ACC on principal left and right ideals, then $H$ is
atomic.
###### Proof.
We first note that if $a,b\in H$ then $aH=bH$ if and only if $a=b\varepsilon$
with $\varepsilon\in H^{\times}$, and similarly $Ha=Hb$ if and only if
$a=\varepsilon b$ with $\varepsilon\in H^{\times}$. [We only show the
statement for the right ideals. The non-trivial direction is showing that
$aH=bH$ implies $a=b\varepsilon$. Since $aH=bH$ implies $a=bx$ and $b=ay$ with
$x,y\in H$, we get $a=a(yx)$ and $b=b(xy)$. Since $H$ is cancellative, this
implies $xy=t(b)=s(x)$ and $yx=t(a)=s(y)$, hence $y=x^{-1}$ and therefore
$x,y\in H^{\times}$.]
1. Claim A.
If $a\in H\setminus H^{\times}$, then there exist $u\in\mathcal{A}(H)$ and
$a_{0}\in H$ such that $a=ua_{0}$.
###### Proof of Claim A.
Assume the contrary. Then the set
$\Omega=\\{\,a^{\prime}H\mid\text{$a^{\prime}\in H\setminus H^{\times}$ such
that there are no $u\in\mathcal{A}(H)$, $a_{0}\in H$ with
$a^{\prime}=ua_{0}$}\,\\}$
is non-empty, and hence, using the ascending chain condition on the principal
right ideals, possesses a maximal element $aH$ with $a\in H\setminus
H^{\times}$. Then $a\not\in\mathcal{A}(H)$, and therefore $a=bc$ with $b,c\in
H\setminus H^{\times}$. But $aH\subsetneq bH$ since $c\not\in H^{\times}$, and
thus maximality of $aH$ in $\Omega$ implies $b=ub_{0}$ with
$u\in\mathcal{A}(H)$ and $b_{0}\in H$. But then $a=u(b_{0}c)$, a
contradiction. ∎
We proceed to show that every $a\in H\setminus H^{\times}$ is a product of
atoms. Again, assume that this is not the case. Then
$\Omega^{\prime}=\\{\,Ha^{\prime}\mid\text{$a^{\prime}\in H\setminus
H^{\times}$ such that $a^{\prime}$ is not a product of atoms}\,\\}$
is non-empty, and hence possesses a maximal element $Ha$ with $a\in H\setminus
H^{\times}$ (this time using the ascending chain condition on principal left
ideals). Again $a\not\in\mathcal{A}(H)$ as otherwise it would be a product of
atoms. By Claim A, $a=ua_{0}$ with $u\in\mathcal{A}(H)$ and $a_{0}\in H$.
Since $a\not\in\mathcal{A}(H)$, $a_{0}\not\in H^{\times}$. Moreover,
$Ha\subsetneq Ha_{0}$ since $u\not\in H^{\times}$ and therefore
$a_{0}=u_{1}\cdot\ldots\cdot u_{l}$ with $l\in\mathbb{N}$ and
$u_{1},\ldots,u_{l}\in\mathcal{A}(H)$. Thus $a=uu_{1}\cdot\ldots\cdot u_{l}$
is a product of atoms, a contradiction. ∎
The following definition provides a natural notion of an ordered factorization
(called a _rigid factorization_) in a cancellative small category. It is
modeled after a terminology by Cohn [14, 15].
Let $\mathcal{F}(\mathcal{A}(H))$ denote the path category on atoms of $H$. We
define
$H^{\times}\times_{r}\mathcal{F}(\mathcal{A}(H))=\\{\,(\varepsilon,y)\in
H^{\times}\times\mathcal{F}(\mathcal{A}(H))\mid t(\varepsilon)=s(y)\,\\},$
and define an associative partial operation on
$H^{\times}\times_{r}\mathcal{F}(\mathcal{A}(H))$ as follows: If
$(\varepsilon,y),(\varepsilon^{\prime},y^{\prime})\in
H^{\times}\times_{r}\mathcal{F}(\mathcal{A}(H))$ with
$\varepsilon,\varepsilon^{\prime}\in H^{\times}$,
$y=(e,u_{1},u_{2},\ldots,u_{k},f)\in\mathcal{F}(\mathcal{A}(H))\;\text{ and
}\;y^{\prime}=(e^{\prime},v_{1},v_{2},\ldots,v_{l},f^{\prime})\in\mathcal{F}(\mathcal{A}(H)),$
then the operation is defined if $t(y)=s(\varepsilon^{\prime})$, and
$(\varepsilon,y)\cdot(\varepsilon^{\prime},y^{\prime})=(\varepsilon,(e,u_{1},\ldots,u_{k}\varepsilon^{\prime},v_{1},v_{2},\ldots,v_{l},f^{\prime}))\quad\text{if
$k>0$, }$
while
$(\varepsilon,y)\cdot(\varepsilon^{\prime},y^{\prime})=(\varepsilon\varepsilon^{\prime},y^{\prime})$
if $k=0$. In this way $H^{\times}\times_{r}\mathcal{F}(\mathcal{A}(H))$ is
again a cancellative small category (with identities $\\{\,(e,(e,e))\mid e\in
H_{0}\\}$ that we identify with $H_{0}$ again,
$s(\varepsilon,y)=s(\varepsilon)$ and $t(\varepsilon,y)=t(y)$). We define a
congruence relation $\sim$ on it as follows: If
$(\varepsilon,y),(\varepsilon^{\prime},y^{\prime})\in
H^{\times}\times_{r}\mathcal{F}(\mathcal{A}(H))$ with $y,y^{\prime}$ as
before, then $(\varepsilon,y)\sim(\varepsilon^{\prime},y^{\prime})$ if $k=l$,
$\varepsilon u_{1}\cdot\ldots\cdot
u_{k}=\varepsilon^{\prime}v_{1}\cdot\ldots\cdot v_{l}\in H$ and either $k=0$
or there exist $\delta_{2},\ldots,\delta_{k}\in H^{\times}$ and
$\delta_{k+1}=t(u_{k})$ such that
$\varepsilon^{\prime}v_{1}=\varepsilon
u_{1}\delta_{2}^{-1}\quad\text{and}\quad
v_{i}=\delta_{i}u_{i}\delta_{i+1}^{-1}\text{ for all $i\in[2,k]$}.$
###### Definition 3.1.
The _category of rigid factorizations of $H$_ is defined as
$\mathsf{Z}^{*}(H)=(H^{\times}\times_{r}\mathcal{F}(\mathcal{A}(H)))/\sim,$
For $z\in\mathsf{Z}^{*}(H)$ with
$z=[(\varepsilon,(e,u_{1},u_{2},\ldots,u_{k},f))]_{\sim}$ we write
$z=\varepsilon u_{1}*\ldots*u_{k}$ and the operation on $\mathsf{Z}^{*}(H)$ is
also denoted by $*$. The _length_ of $z$ is $\lvert z\rvert=k$. There is a
surjective homomorphism $\pi\colon\mathsf{Z}^{*}(H)\to H$, induced by
multiplying out the elements of the factorization in $H$, explicitly
$\pi(z)=\varepsilon u_{1}u_{2}\cdot\ldots\cdot u_{k}\in H$. For $a\in H$, we
define $\mathsf{Z}^{*}(a)=\mathsf{Z}_{H}^{*}(a)=\pi^{-1}(\\{a\\})$ to be the
_set of rigid factorization of $a$_.
To simplify the notation, we make the following conventions:
* •
If, for a rigid factorization $z=\varepsilon
u_{1}*\ldots*u_{k}\in\mathsf{Z}^{*}(H)$, we have $k>0$ (i.e., $\pi(z)\not\in
H^{\times}$), then the unit $\varepsilon$ can be absorbed into the first
factor $u_{1}$ (replacing it by $\varepsilon u_{1}$), and we can essentially
just work in $\mathcal{F}(\mathcal{A}(H))/\sim$, with $\sim$ defined to match
the equivalence relation on $H^{\times}\times_{r}\mathcal{F}(\mathcal{A}(H))$.
* •
If $H$ is reduced but $\lvert H_{0}\rvert>1$, we often still write
$s(u_{1})u_{1}*\ldots*u_{k}$ instead of the shorter $u_{1}*\ldots*u_{k}$, as
$k=0$ is allowed and in the path category there is a different empty path for
every $e\in H_{0}$.
###### Remark .
1. 1.
If $H$ is reduced, then $\mathsf{Z}^{*}(H)=\mathcal{F}(\mathcal{A}(H))$.
If $H$ is not reduced, the $H^{\times}$ factor allows us to represent trivial
factorizations of units, and the equivalence relation $\sim$ allows us to deal
with trivial insertion of units. In the commutative setting these
technicalities can easily be avoided by identifying associated elements and
passing to the reduced monoid $H_{\text{red}}=\\{\,aH^{\times}\mid a\in
H\,\\}$. Unfortunately, associativity (left, right or two-sided) is in general
no congruence relation in the non-commutative case.
2. 2.
If $H$ is a commutative monoid, then $\mathsf{Z}^{*}(H)\cong
H^{\times}\times\mathcal{F}(\mathcal{A}(H_{\text{red}}))$, where
$\mathcal{F}(\mathcal{A}(H_{\text{red}}))$ is the free monoid on
$\mathcal{A}(H_{\text{red}})$, while a _factorization_ in this setting is
usually defined as an element of the free abelian monoid
$\mathsf{Z}(H)=\mathcal{F}_{\text{ab}}(\mathcal{A}(H_{\text{red}}))$, implying
in particular that factorizations are unordered while rigid factorizations are
ordered. The homomorphism $\pi:\mathsf{Z}^{*}(H_{\text{red}})\to
H_{\text{red}}$ obviously factors through the multiplication homomorphism
$\mathsf{Z}(H_{\text{red}})\to H_{\text{red}}$, and the fibers consist of the
different permutations of a factorization.
In the following we will only be concerned with invariants related to the
lengths of factorizations, which may as well be defined using rigid
factorizations.
###### Definition 3.2.
Let $a\in H$.
1. 1.
We call
$\mathsf{L}(a)=\mathsf{L}_{H}(a)=\\{\,\lvert z\rvert\in\mathbb{N}_{0}\mid
z\in\mathsf{Z}^{*}(a)\,\\}$
the _set of lengths of $a$_.
2. 2.
The _system of sets of lengths of $H$_ is defined as
$\mathcal{L}(H)=\\{\,\mathsf{L}(a)\subset\mathbb{N}_{0}\mid a\in H\,\\}$.
3. 3.
A positive integer $d\in\mathbb{N}$ is a _distance of $a$_ if there exists an
$l\in\mathsf{L}(a)$ such that $\\{\,l,l+d\,\\}\in\mathsf{L}(a)$ and
$\mathsf{L}(a)\cap[l+1,l+d-1]=\emptyset$. The _set of distances of $a$_ is the
set consisting of all such distances and is denoted by
$\Delta(a)=\Delta_{H}(a)$. The _set of distances of $H$_ is defined as
$\Delta(H)=\bigcup_{a\in H}\Delta(a).$
4. 4.
We define
$\mathcal{U}_{k}(H)=\bigcup_{\begin{subarray}{c}L\in\mathcal{L}(H)\\\ k\in
L\end{subarray}}L$ for $k\in\mathbb{N}_{0}$.
5. 5.
$H$ is _half-factorial_ if $\lvert\mathsf{L}(a)\rvert=1$ for all $a\in H$
(equivalently, $H$ is atomic and $\Delta(H)=\emptyset$).
We write $b\mid_{H}^{r}a$ if $a\in Hb$ and similarly $b\mid_{H}^{l}a$ if $a\in
bH$.
###### Definition & Lemma .
Let $H\subset D$ be subcategories of a groupoid. The following are equivalent:
1. (a)
For all $a,b\in H$, $b\mid_{D}^{r}a$ implies $b\mid_{H}^{r}a$,
2. (b)
$HH^{-1}\cap D=H$.
$H\subset D$ is called _right-saturated_ if these equivalent conditions are
fulfilled.
###### Proof.
(a) $\Rightarrow$ (b): Let $c=ab^{-1}$ with $a,b\in H$, $t(a)=t(b)$ and $c\in
D$. Then $cb=a$, i.e., $b\mid_{D}^{r}a$ and hence also $b\mid_{H}^{r}a$. Since
the left factor is uniquely determined as $c=ab^{-1}$, it follows that $c\in
H$.
(b) $\Rightarrow$ (a): Let $b\mid_{D}^{r}a$. There exists $c\in D$ with
$cb=a$, and thus $c=ab^{-1}$. Therefore $c\in HH^{-1}\cap D=H$, hence
$b\mid_{H}^{r}a$. ∎
###### Definition 3.3.
Let $B$ be a reduced cancellative small category. A homomorphism $\theta\colon
H\to B$ is called a _transfer homomorphism_ if it has the following
properties:
1. (T1)
$B=\theta(H)$ and $\theta^{-1}(B_{0})=H^{\times}$.
2. (T2)
If $a\in H$, $b_{1},b_{2}\in B$ and $\theta(a)=b_{1}b_{2}$, then there exist
$a_{1},a_{2}\in H$ such that $a=a_{1}a_{2}$, $\theta(a_{1})=b_{1}$ and
$\theta(a_{2})=b_{2}$.
The notion of a transfer homomorphism plays a central role in studying sets of
lengths. It is easily checked that the following still holds in our
generalized setting (cf. [27, §3.2] for the commutative case, [25, Proposition
6.4] for the non-commutative monoid case).
###### Proposition .
If $\theta\colon H\to B$ is a transfer homomorphism, then
$\mathsf{L}_{H}(a)=\mathsf{L}_{B}(\theta(a))$ for all $a\in H$ and hence all
invariants defined in terms of lengths coincide for $H$ and $B$. In
particular,
* •
$\mathcal{L}(H)=\mathcal{L}(B)$,
* •
$\mathcal{U}_{k}(H)=\mathcal{U}_{k}(B)$ for all $k\in\mathbb{N}_{0}$,
* •
$\Delta_{H}(a)=\Delta_{B}(\theta(a))$ for all $a\in H$, and
$\Delta(H)=\Delta(B)$.
###### Proposition .
Let $H$ be a cancellative small category, $G$ a finite abelian group and
$\theta\colon H\to\mathcal{B}(G)$ a transfer homomorphism. Then $H$ is half-
factorial if and only if $\lvert G\rvert\leq 2$. If $\lvert G\rvert\geq 3$,
then we have
1. 1.
$\Delta(H)$ is a finite interval, and if it is non-empty, then
$\min\Delta(H)=1$,
2. 2.
for every $k\geq 2$, the set $\mathcal{U}_{k}(H)$ is a finite interval,
3. 3.
there exists an $M\in\mathbb{N}_{0}$ such that for every $a\in H$ the set of
lengths $\mathsf{L}(a)$ is an almost arithmetical multiprogression (AAMP) with
difference $d\in\Delta(H)$ and bound $M$.
###### Proof.
By the previous lemma it is sufficient to show these statements for the monoid
of zero-sum sequences $\mathcal{B}(G)$ over a finite abelian group $G$.
$\mathcal{B}(G)$ is half-factorial if and only if $\lvert G\rvert\leq 2$ by
[27, Proposition 2.5.6]. The first statement is proven in [29], the second can
be found in [24, Theorem 3.1.3]. For the definition of AAMPs and a proof of 3
see [27, Chapter 4]. ∎
The description in 3. is sharp by a realization theorem of W.A. Schmid [49].
## 4\. Factorization of integral elements in arithmetical groupoids
In this section we introduce arithmetical groupoids and study the
factorization behavior of integral elements. In Section 5 we will see that the
divisorial fractional one-sided ideals of suitable semigroups form such
groupoids. Thus in non-commutative semigroups arithmetical groupoids
generalize the free abelian group of divisorial fractional two-sided ideals
familiar from the commutative setting (see Section 4 and Section 4). This
abstract approach to factorizations was first used by Asano and Murata in [5].
We follow their ideas and also those of Rehm in [45, 46], who studies
factorizations of ideals in rings in a different abstract framework. The
notation and terminology for lattices follows [30], a reference for l-groups
is [51]. Section 4 is the main result on factorizations of integral elements
in a lattice-ordered groupoid (due to Asano and Murata). We introduce an
abstract norm homomorphism $\eta$, and as the main result in this section, we
present a transfer homomorphism to a monoid of zero-sum sequences in Section
4.
###### Definition 4.1.
A lattice-ordered groupoid $(G,\leq)$ is a groupoid $G$ together with a
relation $\leq$ on $G$ such that for all $e,f\in G_{0}$
1. 1.
$(G(e,\cdot),\,\leq\mid_{G(e,\cdot)})$ is a lattice (we write
$\wedge^{\prime}_{e}$ and $\vee^{\prime}_{e}$ for the meet and join),
2. 2.
$(G(\cdot,f),\,\leq\mid_{G(\cdot,f)})$ is a lattice (we write
$\wedge^{\prime\prime}_{f}$ and $\vee^{\prime\prime}_{f}$ for the meet and
join),
3. 3.
$(G(e,f),\,\leq\mid_{G(e,f)})$ is a sublattice of both $G(e,\cdot)$ and
$G(\cdot,f)$. Explicitly: For all $a,b\in G(e,f)$ it holds that
$a\wedge^{\prime}_{e}b=a\wedge^{\prime\prime}_{f}b\in G(e,f)$ and
$a\vee^{\prime}_{e}b=a\vee^{\prime\prime}_{f}b\in G(e,f)$.
If $a,b\in G$ and $s(a)=s(b)$ we write $a\wedge b=a\wedge^{\prime}_{s(a)}b$
and $a\vee b=a\vee^{\prime}_{s(a)}b$. If $t(a)=t(b)$ we write $a\wedge
b=a\wedge^{\prime\prime}_{t(a)}b$ and $a\vee b=a\vee^{\prime\prime}_{t(a)}b$.
By 3 this is unambiguous if $s(a)=s(b)$ and $t(a)=t(b)$ both hold. The
restriction of $\leq$ to any of $G(e,\cdot)$, $G(\cdot,f)$ or $G(e,f)$ will in
the following simply be denoted by $\leq$ again. (Keep in mind however that
$\leq$ need not be a partial order on the entire set $G$, and $\wedge$ and
$\vee$ do not represent meet and join operations on the entire set $G$ in the
order-theoretic sense.)
An element $a$ of a lattice-ordered groupoid is called _integral_ if $a\leq
s(a)$ and $a\leq t(a)$, and we write $G_{+}$ for the subset of all integral
elements of $G$.
###### Definition 4.2.
A lattice-ordered groupoid $G$ is called an _arithmetical groupoid_ if it has
the following properties for all $e,f\in G_{0}$:
1. (P1)
For $a\in G$, $a\leq s(a)$ if and only if $a\leq t(a)$.
2. (P2)
$G(e,\cdot)$ and $G(\cdot,f)$ are modular lattices.
3. (P3)
If $a\leq b$ for $a,b\in G(e,\cdot)$ and $c\in G(\cdot,e)$, then $ca\leq cb$.
Analogously, if $a,b\in G(\cdot,f)$ and $c\in G(f,\cdot)$, then $ac\leq bc$.
4. (P4)
For every non-empty subset $M\subset G(e,\cdot)\cap G_{+}$, $\sup(M)\in
G(e,\cdot)$ exists, and similarly for $M\subset G(\cdot,f)\cap G_{+}$. If
moreover $M\subset G(e,f)$ then $\sup_{G(e,\cdot)}(M)=\sup_{G(\cdot,f)}(M)$.
5. (P5)
$G(e,f)$ contains an integral element.
6. (P6)
$G(e,\cdot)$ and $G(\cdot,f)$ satisfy the ACC on integral elements.
_For the remainder of this section, let $G$ be an arithmetical groupoid._
P5 implies in particular $G(e,f)\neq\emptyset$ for all $e,f\in G_{0}$, i.e.,
$G$ is connected. If $e,e^{\prime}\in G_{0}$ and $c\in G(e^{\prime},e)$, then
$G(e,\cdot)\to G(e^{\prime},\cdot),x\mapsto cx$ is an order isomorphism by P3,
and similarly every $d\in G(f,f^{\prime})$ induces an order isomorphism from
$G(\cdot,f)$ to $G(\cdot,f^{\prime})$. P2 could therefore equivalently be
required for a single $e$ and a single $f\in G_{0}$. Moreover, since the map
$(G(e,\cdot),\leq)\to(G(\cdot,e),\geq),x\mapsto x^{-1}$ is also an order
isomorphism (Lemma 4.1) and the property of being modular is self-dual, it is
in fact sufficient that one of $G(e,\cdot)$ and $G(\cdot,e)$ is modular for
one $e\in G_{0}$.
Using P5 we also observe that it is sufficient to have the ACC on integral
elements on one $G(e,\cdot)$ and one $G(\cdot,f)$: If, say, $a_{1}\leq
a_{2}\leq a_{3}\leq\ldots$ is an ascending chain of integral elements in
$G(e^{\prime},\cdot)$ and $c\in G(e,e^{\prime})$ is integral, then $ca_{1}\leq
ca_{2}\leq ca_{3}\leq\ldots$ is an ascending chain of integral elements in
$G(e,\cdot)$ (Lemma 4.2), hence becomes stationary, and multiplying by
$c^{-1}$ from the left again shows that the original chain also becomes
stationary.
We summarize some basic properties that follow immediately from the
definitions.
###### Lemma .
Let $e,f\in G_{0}$.
1. 1.
$a\leq x\Leftrightarrow a^{-1}\geq x^{-1}$ holds if either $a,x\in G(e,\cdot)$
or $a,x\in G(\cdot,f)$. In particular, for $a\in G$ the following are
equivalent: (a) $a\leq s(a)$; (b) $a\leq t(a)$; (c) $a^{-1}\geq s(a)$; (d)
$a^{-1}\geq t(a)$.
2. 2.
Let $a\in G(e,f)$. If $x\in G(\cdot,e)$ and $y\in G(f,\cdot)$ are integral,
then $xa\leq a$ and $ay\leq a$.
3. 3.
If $a\in G(e,f)$, $x\in G(\cdot,e)$ and $y\in G(f,\cdot)$, then
1. (i)
$x(a\vee b)=xa\vee xb$ and $x(a\wedge b)=xa\wedge xb$ if $b\in G(e,\cdot)$,
2. (ii)
$(a\vee b)y=ay\vee by$ and $(a\wedge b)y=ay\wedge by$ if $b\in G(\cdot,f)$.
4. 4.
Let $\emptyset\neq M\subset G(e,\cdot)$ and $x\in G(\cdot,e)$. If
$\sup_{G(e,\cdot)}(M)$ exists, then also $\sup_{G(s(x),\cdot)}(xM)$ exists,
and $\sup(xM)=x\sup(M)$. Moreover, then also $\inf_{G(\cdot,e)}(M^{-1})$
exists and $\inf(M^{-1})=\sup(M)^{-1}$. Analogous statements hold for
$\emptyset\neq M\subset G(\cdot,f)$ and $x\in G(f,\cdot)$.
5. 5.
$G(e,\cdot)$, $G(\cdot,f)$, $G(e,f)$ and in particular $G(e)$ are
conditionally complete as lattices.
6. 6.
The set $G_{+}$ of all integral elements forms a reduced wide subcategory of
$G$, and $G=\mathbf{q}(G_{+})$ is the groupoid of (left and right) fractions
of this subcategory.
7. 7.
For every $a\in G(e,f)$, there exist $b\in G(e)$ and $c\in G(f)$ with $b\leq
a$ and $c\leq a$.
###### Proof.
1. 1.
Assume first $s(x)=s(a)$. By P3, $a\leq x$ if and only if $x^{-1}a\leq t(x)$.
By P1 this is equivalent to $x^{-1}a\leq t(a)$. Again by P3 this is equivalent
to $x^{-1}\leq a^{-1}$. The case $t(x)=t(a)$ is proven similarly.
(a) $\Leftrightarrow$ (b) and (c) $\Leftrightarrow$ (d) by P1. For (a)
$\Leftrightarrow$ (c) set $x=s(a)$.
2. 2.
Since $x\leq t(x)=s(a)$, we have $xa\leq s(a)a=a$ by P3. Similarly, $by\leq
y$.
3. 3.
We show (i), (ii) is similar. Since $a\leq a\vee b$ and $b\leq a\vee b$, P3
implies $xa\leq x(a\vee b)$ and $xb\leq x(a\vee b)$, thus $xa\vee xb\leq
x(a\vee b)$. Therefore
$a\vee b=(x^{-1}xa)\,\vee\,(x^{-1}xb)\leq x^{-1}(xa\vee xb),$
and multiplying by $x$ from the left gives $x(a\vee b)\leq xa\vee xb$. Dually,
$x(a\wedge b)=xa\wedge xb$.
4. 4.
Let $c=\sup(M)$. Then for all $m\in M$, $xm\leq xc$, hence $xc$ is an upper
bound for $xM$. If $d\in G(s(x),\cdot)$ is another upper bound for $xM$, then
$m\leq x^{-1}d$ for all $m\in M$, hence $c\leq x^{-1}d$ and thus $xc\leq d$.
Therefore $xc=\sup(xM)$.
For $d\in G(e,\cdot)$ we have $m\leq d$ for all $m\in M$ if and only if
$m^{-1}\geq d^{-1}$ (in $G(\cdot,e)$), and $\inf(M^{-1})=\sup(M)^{-1}$
follows.
5. 5.
We show the claim for $G(e,\cdot)$, for $G(\cdot,f)$ the proof is similar. Let
$\emptyset\neq M\subset G(e,\cdot)$ be bounded, say $x\leq m\leq y$ for some
$x,y\in G(e,\cdot)$ and all $m\in M$. Then $y^{-1}M\subset G(t(y),\cdot)$ is
integral, hence $\sup(y^{-1}M)$ exists by P4, and $\sup(M)=y\sup(y^{-1}M)$ by
4. Similarly, $M^{-1}x\subset G(\cdot,t(x))$ is integral, and therefore
$\sup(M^{-1}x)$ exists, implying
$\inf(M)=\sup(M^{-1})^{-1}=x\sup(M^{-1}x)^{-1}$.
The proof for $G(e,f)$ is similar but uses in addition
$\sup_{G(t(y),\cdot)}(y^{-1}M)=\sup_{G(\cdot,f)}(y^{-1}M)$ (from P4), to
ensure that the supremum lies in $G(e,f)$ again.
6. 6.
By 2 and the fact that every $e\in G_{0}$ is integral by definition, $G_{+}$
forms a wide subcategory of $G$. If $a\in G_{+}\setminus G_{0}$, then
$a<s(a)$, thus $a^{-1}>s(a)$ and therefore $a^{-1}$ is not integral. Hence the
subcategory of integral elements is reduced. Let $x\in G$ and $e=s(x)$. Then
$a=x\wedge e\leq e$, hence $a$ is integral. Since $a\leq x$, also $x^{-1}a\leq
t(x)$ is integral. Set $b=x^{-1}a$. Then $x=ab^{-1}$ with $a,b\in G_{+}$.
Similarly one can find $c,d\in G_{+}$ with $x=d^{-1}c$.
7. 7.
By P5 there exist integral $b^{\prime}\in G(f,e)$ and $c^{\prime}\in G(e,f)$.
Set $b=ab^{\prime}$ and $c=c^{\prime}a$. Then $b\leq a$, $c\leq a$ and $b\in
G(e)$, $c\in G(f)$. ∎
For $e,f\in G$ it is immediate from the definitions that
$G_{+}(e,\cdot)=G(e,\cdot)\cap G_{+}$, $G_{+}(\cdot,f)=G(\cdot,f)\cap G_{+}$
and $G_{+}(e,f)=G(e,f)\cap G_{+}$. Moreover, $G_{+}(e,\cdot)$ is a sublattice
of $G(e,\cdot)$, $G_{+}(\cdot,f)$ is a sublattice of $G(\cdot,f)$, and
$G_{+}(e,f)$ is a sublattice of $G(e,f)$.
If $a,b\in G_{+}(e,\cdot)$, then $a\leq b$ if and only if $b\mid_{G_{+}}^{l}a$
as $a=b(b^{-1}a)$, and $b^{-1}a$ is integral if and only if $a\leq b$.
Similarly, if $a,b\in G_{+}(\cdot,f)$, then $a\leq b$ if and only if
$b\mid_{G_{+}}^{r}a$. Correspondingly, for integral elements with the same
left (right) identity, we may view the join and meet operations as left
(right) gcd and lcm.
###### Definition & Lemma .
For $u\in G$ the following are equivalent:
1. (a)
$u$ is maximal in $G_{+}(s(u),\cdot)\setminus\\{\,s(u)\,\\}$,
2. (b)
$u$ is maximal in $G_{+}(\cdot,t(u))\setminus\\{\,t(u)\,\\}$,
3. (c)
$u\in\mathcal{A}(G_{+})$.
An element $u\in G$ satisfying these equivalent conditions is called _maximal
integral_.
###### Proof.
(a) $\Rightarrow$ (b): By definition, $u$ is maximal in $G(s(u),\,\cdot)$ with
$u<s(u)$. If $u\leq y<t(u)$ with $y\in G(\cdot,\,t(u))$, then $uy^{-1}\leq
yy^{-1}=s(y)$, hence $uy^{-1}\in G(s(u),\,\cdot)$ is integral, and therefore
$u<uy^{-1}\leq s(u)$. By maximality of $u$ in the first set, therefore
$uy^{-1}=s(u)$, whence $y=u$ and $u$ is maximal in the second set.
(b) $\Rightarrow$ (c): Assume $u=vw$ with $v,w\in G_{+}\setminus G_{0}$. Then
$u<w<t(u)$, contradicting the maximality of $u$ in $G_{+}(\cdot,t(u))$.
(c) $\Rightarrow$ (a): Let $v\in G_{+}(s(u),\cdot)$ with $u\leq v<s(u)$. Then
$u=v(v^{-1}u)$ with $v$ and $v^{-1}u$ integral, and since $v\not\in G_{0}$
necessarily $v^{-1}u\in G_{0}$, i.e., $u=v$. ∎
###### Lemma .
Let $U$ be an l-group. For $p\in U$ the following are equivalent:
1. (a)
$p$ is maximal integral,
2. (b)
$p$ is a prime element in $U_{+}$,
3. (c)
$p\in\mathcal{A}(U_{+})$.
###### Proof.
(a) $\Leftrightarrow$ (c) is shown as in Section 4. It suffices to show (a)
$\Rightarrow$ (b) and (b) $\Rightarrow$ (c). Let $e$ be the identity of $U$.
(a) $\Rightarrow$ (b): Let $p$ be maximal in $U_{+}$ with $p\neq e$. Assume
$p\mid ab$ for $a,b\leq e$. That means $ab\leq p$. Assume $a\not\leq p$. Then
$b\,=\,(a\vee p)b\,=\,ab\,\vee\,pb\,\leq\,p\,\vee\,pb\,=\,p$, i.e., $p\mid b$.
(b) $\Rightarrow$ (c): Let $p$ be a prime, $p=ab$ with $a,b\leq e$. Say $p\mid
a$, i.e., $a\leq p$. Then $a\leq p\leq a$ implies $p=a$ and therefore $b=e$. ∎
###### Proposition .
1. 1.
If $G$ is a group (i.e., $\lvert G_{0}\rvert=1$), then $G$ is the free abelian
group with basis $\mathcal{A}(G_{+})$, and $G_{+}$ is the free abelian monoid
with basis $\mathcal{A}(G_{+})$. Moreover $\gcd(a,b)=a\vee b$ and
$\operatorname{lcm}(a,b)=a\wedge b$.
2. 2.
Let $\mathcal{M}$ be a set, $F$ the free abelian group with basis
$\mathcal{M}$, and $H\subset F$ the free abelian monoid with the same basis. A
lattice order is defined on $F$ by $a\leq b$ if $a=cb$ with $c\in H$, and
$(F,\leq)$ is an arithmetical groupoid with $F_{+}=H$ and
$\mathcal{M}=\mathcal{A}(F_{+})$.
3. 3.
For every $e\in G_{0}$, the group isomorphism $j_{e}\colon G(e)\to\mathbb{G}$
induces the structure of an arithmetical groupoid on $\mathbb{G}$, and the
induced structure on $\mathbb{G}$ is independent of the choice of $e$.
$\mathbb{G}$ ($G(e)$) is a free abelian group and $\mathbb{G}_{+}$
($G(e)_{+}$) is a free abelian monoid with basis $\mathcal{A}(\mathbb{G}_{+})$
($\mathcal{A}(G(e)_{+})$). Moreover, $j_{e}(G(e)_{+})=\mathbb{G}_{+}$ and
$j_{e}(\mathcal{A}(G(e)_{+}))=\mathcal{A}(\mathbb{G}_{+})$.
###### Proof.
1. 1.
$G$ is an l-group, and by Lemma 4.5 it is conditionally complete. Therefore
$G$ is commutative ([51, Theorems 2.3.1(d) and 2.3.9]). Since it satisfies the
ACC on integral elements, $G_{+}$ is atomic (Section 3). By the previous
lemma, every atom of $G_{+}$ is a prime element, and therefore $G_{+}$ is
factorial. Because it is also reduced, $G_{+}$ is the free abelian monoid with
basis $\mathcal{A}(G_{+})$. Now $G=\mathbf{q}(G_{+})$ implies that $G$ is the
free abelian group with basis $\mathcal{A}(G_{+})$. Finally, $\gcd(a,b)=a\vee
b$ and $\gcd(a,b)=a\wedge b$ for $a,b\in G_{+}$ follow because $c\leq d$ if
and only if $d\mid c$ for all $c,d\in G_{+}$.
2. 2.
Clearly $\leq$ defines a lattice order on $F$, and the properties of an
arithmetical groupoid are, except for P2, either trivial, or easily checked.
For P2 recall that every l-group is distributive, hence modular, as a lattice
([51, Theorem 2.1.3(a)]). Now $F_{+}=H$ and $\mathcal{A}(F_{+})=\mathcal{M}$
are immediate from the definitions.
3. 3.
For every $e\in G_{0}$ the vertex group $G(e)$ is an arithmetical groupoid, as
is easily checked. Via the group isomorphism $j_{e}:G(e)\to\mathbb{G}$
therefore $\mathbb{G}$ gains the structure of an arithmetical groupoid. If
$f\in G_{0}$ and $c\in G(e,f)$, then for all $x,y\in G(e)$ we have $x\leq
y\Leftrightarrow c^{-1}xc\leq c^{-1}yc$, and since $j_{f}^{-1}\circ
j_{e}(x)=c^{-1}xc$, the induced order on $\mathbb{G}$ is independent of the
choice of $e$.
By 1., applied to $\mathbb{G}$, respectively $G(e)$, the remaining claims
follow (for $j_{e}(\mathcal{A}(\mathbb{G}_{+}))=\mathcal{A}(G(e)_{+})$ use the
characterization of atoms as maximal integral elements from Section 4). ∎
Let $[a,b]$, $[c,d]$ be intervals in a lattice. Recall that $[a,b]$ is _down-
perspective_ to $[c,d]$ if $c=a\wedge d$ and $b=a\vee d$. Moreover, $[a,b]$ is
_perspective_ to $[c,d]$ if either $[a,b]$ is down-perspective to $[c,d]$, or
$[c,d]$ is down-perspective to $[a,b]$. The intervals $[a,b]$, $[c,d]$ are
_projective_ if there exists a finite sequence of intervals
$[a,b]=[a_{0},b_{0}],[a_{1},b_{1}],\ldots,[a_{k},b_{k}]=[c,d]$ such that
$[a_{i-1},b_{i-1}]$ is perspective to $[a_{i},b_{i}]$ for all $i\in[1,k]$.
(See [30, Chapter I.3.5].)
###### Definition 4.3.
1. 1.
An element $a\in G_{+}(e,f)$ is _transposable_ to an element $b\in
G_{+}(e^{\prime},f^{\prime})$ if there exists an element $c\in
G_{+}(e,e^{\prime})$ such that $[a,e]$ is down-perspective to $[cb,c]$.
Explicitly: $b=c^{-1}(c\wedge a)$ and $c\vee a=e$.
2. 2.
An element $a\in G_{+}$ is _projective_ to an element $b\in G_{+}$ if there
exists a sequence of integral elements $a=c_{0},c_{1},\ldots,c_{n},c_{n+1}=b$,
such that for any pair of successive elements $(c_{i},c_{i+1})$ either $c_{i}$
is transposable to $c_{i+1}$ or $c_{i+1}$ is transposable to $c_{i}$.
It is easily checked that being transposable is a transitive and reflexive
relation (but not symmetric), and projectivity is an equivalence relation.
Note that in a modular lattice perspective intervals are isomorphic ([30,
p.308, Theorem 348]), and therefore in particular the lengths of $[a,s(a)]$
and $[b,s(b)]$ coincide if $a$ is projective to $b$.
###### Lemma .
If $a,b,c$ are as in the definition of transposability, then $cb=c\wedge a=ad$
for some $d\in G_{+}$ and $c$ is transposable to $d$.
###### Proof.
Since $c\wedge a\leq a$, there exists an integral $d$ with $ad=c\wedge a$, and
$cb=c\wedge a$ by definition of transposability. The claim follows from
$d=a^{-1}(c\wedge a)$ and $c\vee a=e$. ∎
###### Lemma .
If two lattice intervals $[a,b]$ and $[c,d]$ of $G(e,\cdot)$ are projective,
then the integral elements $b^{-1}a$ and $d^{-1}c$ are projective to each
other in the sense of the previous definition.
###### Proof.
It suffices to show that if the lattice interval $[a,b]$ is down-perspective
to $[c,d]$, then $b^{-1}a$ is transposable to $d^{-1}c$. Then $a=bx$, $d=by$
and $c=dz=byz$ with $x,y,z\in G_{+}$.
$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{d=by}$$\textstyle{bx=a}$$\textstyle{c=byz}$
Since $[a,b]$ is down-perspective to $[c,d]$, we get
$\displaystyle b$ $\displaystyle=a\vee d=bx\vee by=b(x\vee y)$ and therefore
$\displaystyle x\vee y$ $\displaystyle=s(x)=t(b),\text{ and }$ $\displaystyle
c$ $\displaystyle=a\wedge d=bx\wedge by=b(x\wedge y)$ and therefore
$\displaystyle x\wedge y$ $\displaystyle=b^{-1}c=yz.$
Thus $d^{-1}c=z=y^{-1}(x\wedge y)$ with $y\in G(t(b),t(d))$, and hence
$x=b^{-1}a$ is transposable to $d^{-1}c$. ∎
###### Definition & Lemma .
For every $a\in G$,
$\\{\,\mathcal{X}\in\mathbb{G}\mid\mathcal{X}_{s(a)}\leq
a\,\\}=\\{\,\mathcal{X}\in\mathbb{G}\mid\mathcal{X}_{t(a)}\leq a\,\\},$
and we write $\mathbb{G}_{\leq a}$ for this set. The _lower bound_ $\Phi\colon
G\to\mathbb{G}$ is defined by $\Phi(a)=\sup(\mathbb{G}_{\leq a})$.
###### Proof.
Let $\mathcal{X}\in\mathbb{G}$. Recall from Section 2.2 that
$\mathcal{X}_{s(a)}=j_{s(a)}^{-1}(\mathcal{X})$ denotes the unique
representative of $\mathcal{X}$ in $G(s(a))$, and that
$\mathcal{X}_{t(a)}=a^{-1}\mathcal{X}_{s(a)}a$. We have to show that
$\mathcal{X}_{s(a)}\leq a$ if and only if $\mathcal{X}_{t(a)}\leq a$, but this
follows from $\mathcal{X}_{s(a)}\leq
a\;\;\Leftrightarrow\;\;a^{-1}\mathcal{X}_{s(a)}a\leq a^{-1}aa=a$. ∎
With the definition of $\Phi$ and the notation of Section 2.2 we have: If
$a\in G(e,f)$, then $\Phi(a)_{e}=\sup\\{\,x\in G(e)\mid x\leq a\,\\}\in G(e)$,
$\Phi(a)_{f}=\sup\\{\,x\in G(f)\mid x\leq a\,\\}\in G(f)$,
$\Phi(a)_{f}=a^{-1}\Phi(a)_{e}a=b^{-1}\Phi(a)_{e}b$ for all $b\in G(e,f)$ and
$\Phi(a)=j_{e}(\Phi(a)_{e})=j_{f}(\Phi(a)_{f})$.
###### Lemma .
Let $e,f\in G_{0}$.
1. 1.
If $a,b\in G(e,\cdot)$ or $a,b\in G(\cdot,f)$ with $a\leq b$, then
$\Phi(a)\leq\Phi(b)$. In particular, if $a\in G_{+}$, then
$\Phi(a)\in\mathbb{G}_{+}$.
2. 2.
If $a\in G(e,f)$, $b\in G(f,\cdot)$ then $\Phi(a)\Phi(b)\leq\Phi(ab)$. If
moreover $a,b\in G_{+}$, then $\Phi(ab)\leq\Phi(a)\wedge\Phi(b)$, and if
furthermore $\Phi(a)$ and $\Phi(b)$ are coprime, then
$\Phi(ab)=\Phi(a)\Phi(b)$.
3. 3.
If $u\in\mathcal{A}(G_{+})$, then $\Phi(u)\in\mathbb{G}_{+}$ is prime.
4. 4.
Let $u,v\in G_{+}$ be projective elements. If $u\in\mathcal{A}(G_{+})$, then
$v\in\mathcal{A}(G_{+})$, and $\Phi(u)=\Phi(v)$.
###### Proof.
1. 1.
Immediate from the definition of $\Phi$.
2. 2.
Observe that $c=a^{-1}\Phi(a)_{e}\in G(f,e)$ is integral. Therefore
$\Phi(a)_{e}\Phi(b)_{e}=ac\Phi(b)_{e}=acc^{-1}\Phi(b)_{f}c=a\Phi(b)_{f}c\leq
a\Phi(b)_{f}\leq ab,$
and hence $\Phi(a)\Phi(b)\leq\Phi(ab)$.
Let now $a,b$ be integral. Then 1. implies $\Phi(ab)\leq\Phi(a)$ and
$\Phi(ab)\leq\Phi(b)$, so $\Phi(ab)\leq\Phi(a)\wedge\Phi(b)$. The last
statement follows because
$\Phi(a)\wedge\Phi(b)=\operatorname{lcm}(\Phi(a),\Phi(b))$ in
$\mathbb{G}_{+}$.
3. 3.
By Section 4 it suffices to show $\Phi(u)\in\mathcal{A}(\mathbb{G}_{+})$. If
$e=s(u)$, then it suffices to prove $\Phi(u)_{e}\in\mathcal{A}(G(e)_{+})$ (by
Proposition 4.3). Assume that $\Phi(u)_{e}=ab$ with $a,b\in G(e)$ such that
$a<e$ and $b<e$. Then $b\vee u=e$, since $b>ab=\Phi(u)_{e}$, and therefore
$u\,\geq\,ab\vee au\,=\,a(b\vee u)\,=\,a,$
a contradiction to $a>ab=\Phi(u)_{e}$.
4. 4.
We first show that $v$ is maximal integral, and may assume that either $u$ is
transposable to $v$ or $v$ is transposable to $u$. Let $e=s(u)$ and $f=s(v)$.
Assume first that $u$ is transposable to $v$ via $c\in G_{+}(e,f)$. Then
$[u,e]$ is down-perspective to $[cv,c]$, and since $G(e,\cdot)$ is modular,
the intervals are isomorphic, hence have the same length (namely $1$).
Multiplying from the left by $c^{-1}$ therefore also $[v,f]$ has length $1$,
and thus $v$ is maximal integral. If $v$ is transposable to $u$, one argues
along similar lines.
For the remainder of the claim we may now assume that $u$ is transposable to
$v$ (since we already know that $v$ is also maximal integral). Let again $c\in
G_{+}(e,f)$ be such that $cv=c\wedge u$ and $e=c\vee u$. If $p=\Phi(u)_{e}$,
then $c^{-1}pc=\Phi(u)_{f}$. Since $pc\leq c\wedge p\leq c\wedge u=cv$, we get
$c^{-1}pc\leq v$ and therefore
$\Phi(v)\geq\Phi(c^{-1}pc)=\Phi(\Phi(u)_{f})=\Phi(u)$. By 3., $\Phi(u)$ is
prime and thus maximal integral in $\mathbb{G}_{+}$, which implies
$\Phi(u)=\Phi(v)$. ∎
The converse of Lemma 4.3 is false in general: A non-maximal integral element
can have a prime lower bound.
###### Proposition .
1. 1.
The category $G_{+}$ is half-factorial. Explicitly: Every $a\in G_{+}$
possesses a rigid factorization
$s(a)u_{1}*\ldots*u_{k}\in\mathsf{Z}^{*}(a)$
with $k\in\mathbb{N}_{0}$ and $u_{1},\ldots,u_{k}\in\mathcal{A}(G_{+})$ and
the number of factors, $k\in\mathbb{N}_{0}$, is uniquely determined by $a$.
Moreover, if $s(a)v_{1}*\ldots*v_{k}\in\mathsf{Z}^{*}(a)$ is another rigid
factorization with $v_{1},\ldots,v_{k}\in\mathcal{A}(G_{+})$, then there
exists a permutation $\tau\in\mathfrak{S}_{k}$ such that $u_{\tau(i)}$ is
projective to $v_{i}$ for all $i\in[1,k]$. In particular,
$\Phi(u_{\tau(i)})=\Phi(v_{i})$ for all $i\in[1,k]$.
2. 2.
Any two rigid factorizations of $a\in G_{+}$ can be transformed into each
other by a number of steps, each of which only involves replacing two
successive elements by two new ones.
3. 3.
(Transposition.) If $a=uv$ with $u,v\in\mathcal{A}(G_{+})$ and
$\Phi(u)=\mathcal{P}$, $\Phi(v)=\mathcal{Q}$, $\mathcal{Q}\neq\mathcal{P}$,
then there exist uniquely determined
$v^{\prime},u^{\prime}\in\mathcal{A}(G_{+})$ such that
$\Phi(v^{\prime})=\mathcal{Q}$, $\Phi(u^{\prime})=\mathcal{P}$ and
$uv=v^{\prime}u^{\prime}$.
Explicitly,
$\displaystyle u^{\prime}$ $\displaystyle=a\vee\mathcal{P}_{t(a)},$
$\displaystyle u^{\prime}\wedge v$ $\displaystyle=a,$ $\displaystyle
u^{\prime}\vee v$ $\displaystyle=t(a),$ $\displaystyle v^{\prime}$
$\displaystyle=a\vee\mathcal{Q}_{s(a)},$ $\displaystyle u\wedge v^{\prime}$
$\displaystyle=a,$ $\displaystyle u\vee v^{\prime}$ $\displaystyle=s(a).$
So $u$ is transposable to $u^{\prime}$ and $v^{\prime}$ is transposable to
$v$.
4. 4.
Given any permutation $\tau^{\prime}\in\mathfrak{S}_{k}$, there exist
$w_{1},\ldots,w_{k}\in\mathcal{A}(G_{+})$, such that
$s(a)w_{1}*\ldots*w_{k}\in\mathsf{Z}^{*}(a)$
and $\Phi(w_{i})=\Phi(u_{\tau^{\prime}(i)})$ for all $i\in[1,k]$.
###### Proof.
1. 1, 2.
We observe that rigid factorizations of $a$ correspond bijectively to maximal
chains of the sublattice $[a,s(a)]$ of $G(s(a),\cdot)$: If
$s(a)u_{1}*\ldots*u_{k}\in\mathsf{Z}^{*}(a)$, then
$s(a)>u_{1}>u_{1}u_{2}>\ldots>u_{1}\cdot\ldots\cdot u_{k}=a$ is a chain in
$[a,s(a)]$ and since $u_{1},\ldots,u_{k}$ are maximal integral, it is in fact
a maximal chain of $[a,s(a)]$. Conversely, if
$s(a)=x_{0}>x_{1}>x_{2}>\ldots>x_{k}=a$ is a maximal chain of $[a,s(a)]$ then
we set $u_{i}=x_{i-1}^{-1}x_{i}$ for all $i\in[1,k]$. These elements are
maximal integral, i.e., atoms of $G_{+}$, and
$a=x_{k}=s(a)x_{1}(x_{1}^{-1}x_{2})\cdot\ldots\cdot(x_{k-2}^{-1}x_{k-1})(x_{k-1}^{-1}x_{k})=s(a)u_{1}\cdot\ldots\cdot
u_{k}$.
By P6, $[a,s(a)]$ satisfies the ACC, but also the DCC because if
$s(a)=x_{0}\geq x_{1}\geq\ldots\geq a$ is a descending chain in $[a,s(a)]$,
then $x_{0}^{-1}a\leq x_{1}^{-1}a\leq\ldots\leq a^{-1}a=t(a)$ is an ascending
chain in $G_{+}(\cdot,t(a))$ and therefore becomes stationary again by P6.
Being a modular lattice, $[a,s(a)]$ is therefore of finite length.
The claims now follow from the Jordan-Hölder Theorem for modular lattices (see
e.g., [30, p.333, Theorem 377]). The existence of maximal chains implies that
$G_{+}$ is atomic (alternatively, use Section 3 together with the ACC on
integral elements). For half-factoriality, and projectivity of the factors,
assume that $s(a)=x_{0}>x_{1}>x_{2}>\ldots>x_{k}=a$ and
$s(a)=y_{0}>y_{1}>y_{2}>\ldots>y_{l}=a$ are two maximal chains from which
rigid factorizations with factors $u_{i}=x_{i-1}^{-1}x_{i}$ for $i\in[1,k]$
and $v_{i}=y_{i-1}^{-1}y_{i}$ for $i\in[1,l]$ are derived. Then the uniqueness
part of the Jordan-Hölder Theorem implies $k=l$ and that there exists a
permutation $\tau\in\mathfrak{S}_{k}$ such that $[x_{\tau(i)},x_{\tau(i)-1}]$
is projective to $[y_{i},y_{i-1}]$ for all $i\in[1,k]$. By Section 4, this
implies that $u_{\tau(i)}$ is projective to $v_{i}$ for all $i\in[1,k]$.
Finally, 2. follows in a similar manner by induction on the length of $a$. Fix
a composition series of $[u_{1}\wedge v_{1},a]$. This gives rise to
refinements of $s(a)>u_{1}>u_{1}\wedge v_{1}>a$ and $s(a)>v_{1}>u_{1}\wedge
v_{1}>a$ to composition series of $[a,s(a)]$. Applying the induction
hypothesis to $u_{2}*\ldots*u_{k}$ (respectively $v_{2}*\ldots*v_{k}$), and
the rigid factorization derived from the refined chain
$t(u_{1})>u_{1}^{-1}(u_{1}\wedge v_{1})>\ldots>u_{1}^{-1}a$ (respectively
$t(v_{1})>v_{1}^{-1}(u_{1}\wedge v_{1})>\ldots>v_{1}^{-1}a$) one proves the
claim.
2. 3.
Let $e=s(u)$, $q=\mathcal{Q}_{e}$ and set $v^{\prime}=uv\vee q$. We first
show:
1. Claim A.
$q\not\leq u$,
2. Claim B.
$v^{\prime}\wedge u=uv$,
3. Claim C.
$v^{\prime}$ is maximal integral.
_Proof of Claim A._ Suppose $q\leq u$. Then
$\mathcal{Q}\leq\Phi(u)=\mathcal{P}$, a contradiction to $\mathcal{P}$ and
$\mathcal{Q}$ being distinct prime elements of $\mathbb{G}_{+}$ (Lemma 4.3).
_Proof of Claim B._ Since $G(e,\cdot)$ is modular and $uv\leq u$,
$v^{\prime}\wedge u=(uv\vee q)\wedge u=uv\vee(q\wedge u).$
Because $q\not\leq u$ (Claim A), we have $q>q\wedge u\geq qu$ and thus, by
maximality of $u$, $qu=q\wedge u$. Therefore
$uv\vee(q\wedge u)=uv\vee qu=uv\vee
u\mathcal{Q}_{t(u)}=u(v\vee\mathcal{Q}_{t(u)})=uv.$
_Proof of Claim C._ Since $uv$ is a product of two atoms and $G_{+}$ is half-
factorial, it suffices to show $uv<v^{\prime}<e$. Suppose first
$uv=v^{\prime}$. Then $q\leq uv\leq u$, contradicting Claim A. Assume now
$v^{\prime}=e$. Then $u=e\wedge u=v^{\prime}\wedge u$, and by Claim B
therefore $u=uv$, contradicting $v<s(v)$.
_Existence._ We have $uv=v^{\prime}u^{\prime}$ with
$v^{\prime}\in\mathcal{A}(G_{+})$ (by Claim C) and
$u^{\prime}=v^{\prime-1}uv\in G_{+}$. Since $G_{+}$ is half-factorial, this
necessarily implies $u^{\prime}\in\mathcal{A}(G_{+})$. By definition of
$v^{\prime}$, $\mathcal{Q}\leq\Phi(v^{\prime})<1_{\mathbb{G}}$, where the
latter inequality is strict because $v^{\prime}<e$. Thus
$\Phi(v^{\prime})=\mathcal{Q}$ and 1. implies $\Phi(u^{\prime})=\mathcal{P}$.
_Uniqueness._ If $v^{\prime\prime}u^{\prime\prime}=uv$ with
$\Phi(v^{\prime\prime})=\mathcal{Q}$, then $v^{\prime\prime}<e$ and
$v^{\prime\prime}\geq uv\vee q=v^{\prime}$. By Claim C, $v^{\prime}$ is
maximal integral and thus $v^{\prime\prime}=v^{\prime}$, and then also
$u^{\prime\prime}=u^{\prime}$.
_Explicit formulas._ Since $e\geq u\vee
v^{\prime}\geq\Phi(u)_{e}\vee\Phi(v^{\prime})_{e}=\mathcal{P}_{e}\vee\mathcal{Q}_{e}=(\mathcal{P}\vee\mathcal{Q})_{e}=(1_{\mathbb{G}})_{e}=e$,
it follows that $u\vee v^{\prime}=e$. By Claim B, $u\wedge v^{\prime}=uv=a$.
The equalities $u^{\prime}=a\vee\mathcal{P}_{t(a)}$, $u^{\prime}\wedge v=a$
and $u^{\prime}\vee v=t(a)$ are shown similarly.
3. 4.
Write $\tau^{\prime}$ as a product of transpositions and use 3. ∎
Proposition 4.3 gives an explicit and complete description of the possible
relations between two maximal integral elements with coprime lower bound. The
case where the lower bounds coincide is more complicated (there can be no
relations, or many), but in the case where we will need it, it is quite simple
(see Section 7.2).
###### Corollary .
If $H\subset G_{+}$ is a subcategory, then $\mathsf{L}_{H}(a)$ is finite and
non-empty for all $a\in H$. If for every prime $\mathcal{P}\in\mathbb{G}_{+}$
and all (equivalently, one) $e\in G_{0}$ the set
$\\{\,u\in\mathcal{A}(G_{+})\mid\Phi(u)=\mathcal{P}\text{ and }s(u)=e\,\\}$ is
finite, then $\mathsf{Z}^{*}_{H}(a)$ is finite for all $a\in H$.
###### Proof.
Using that $G_{+}$ is reduced, it follows from P6 that $H$ satisfies the ACC
on principal left and right ideals, and hence
$\mathsf{Z}_{H}^{*}(a)\neq\emptyset$. If
$s(a)u_{1}*\ldots*u_{k}\in\mathsf{Z}_{H}^{*}(a)$ with $k\in\mathbb{N}_{0}$ and
$u_{1},\ldots,u_{k}\in\mathcal{A}(H)$, then in particular $u_{i}<s(u_{i})$ for
all $i\in[1,k]$, and therefore $k$ is bounded by the length of the
factorization of $a$ in $G_{+}$. A similar argument shows the second claim. ∎
The properties that all sets of lengths, respectively that all sets of
factorizations, are finite have been studied a lot in the commutative setting.
Note, if $H$ is a commutative monoid and $a\in H$, then $\mathsf{Z}_{H}(a)$ is
finite if and only if $\mathsf{Z}^{*}_{H}(a)$ is finite.
###### Definition & Lemma .
There exists a unique groupoid epimorphism $\eta\colon G\to\mathbb{G}$ such
that $\eta(u)=\Phi(u)$ for all $u\in\mathcal{A}(G_{+})$. We call $\eta$ the
_abstract norm_ of $G$.
###### Proof.
We need to show existence and uniqueness of such a homomorphism. Let $a\in
G_{+}$, and let $s(a)u_{1}*\ldots*u_{k}\in\mathsf{Z}^{*}(a)$ with
$u_{1},\ldots,u_{k}\in\mathcal{A}(G_{+})$. Since the sequence of
$\Phi(u_{1}),\ldots,\Phi(u_{k})$ is, up to order, uniquely determined by $a$
(Section 4), it follows that we can define
$\eta(a)=\Phi(u_{1})\cdot\ldots\cdot\Phi(u_{k})$, and this is a homomorphism
$G_{+}\to\mathbb{G}_{+}$ with $\eta(u)=\Phi(u)$ for all
$u\in\mathcal{A}(G_{+})$. $G$ is the category of (left and right) fractions of
$G_{+}$, and hence $\eta$ extends to a unique groupoid homomorphism
$\eta\colon G\to\mathbb{G}$.
To verify that $\eta$ is surjective, let first $\mathcal{P}\in\mathbb{G}$ be a
prime element of $\mathbb{G}_{+}$, and let $e\in G_{0}$. Let $u\in
G_{+}(e,\cdot)$ be a maximal integral element with $\mathcal{P}_{u}\leq u$.
Then $\Phi(u)=\mathcal{P}$, and therefore $\eta(u)=\mathcal{P}$. The claim
follows since $\mathbb{G}$ is the free abelian group with basis
$\mathcal{A}(\mathbb{G}_{+})$. ∎
In general $\eta\neq\Phi$, since $\Phi$ need not be a homomorphism, but from
Lemma 4.2 it follows that for integral $a$ the prime factorizations of
$\Phi(a)$ and $\eta(a)$ have the same support and
$\mathsf{v}_{\mathcal{P}}(\eta(a))\geq\mathsf{v}_{\mathcal{P}}(\Phi(a))$ for
all primes $\mathcal{P}$ of $\mathbb{G}_{+}$.
###### Theorem .
Let $G$ be an arithmetical groupoid, $\eta\colon G\to\mathbb{G}$ the abstract
norm, $H$ a right-saturated subcategory of $G_{+}$, and
$C=\mathbb{G}/\mathbf{q}(\eta(H))$. For $\mathcal{G}\in\mathbb{G}$ set
$[\mathcal{G}]=\mathcal{G}\mathbf{q}(\eta(H))\in C$, and
$C_{M}=\\{\,[\eta(u)]\in C\mid\text{$u\in\mathcal{A}(G_{+})$}\,\\}$. Assume
that
1. 1.
for $a\in G$ with $s(a)\in H_{0}$, $a\in HH^{-1}$ if and only if
$\eta(a)\in\mathbf{q}(\eta(H))$.
2. 2.
for every $e\in G_{0}$ and $g\in C_{M}$, there exists an element
$u\in\mathcal{A}(G_{+})$ such that $s(u)=e$ and $[\eta(u)]=g$.
Then there exists a transfer homomorphism $\theta\colon
H\to\mathcal{B}(C_{M})$.
###### Proof.
Let $\theta\colon H\to\mathcal{B}(C_{M})$ be defined as follows: For $a\in H$
and $s(a)u_{1}*\ldots*u_{k}\in\mathsf{Z}^{*}_{G_{+}}(a)$ with
$u_{1},\ldots,u_{k}\in\mathcal{A}(G_{+})$, set
$\theta(a)=[\eta(u_{1})]\cdot\ldots\cdot[\eta(u_{k})]\in\mathcal{B}(C_{M})$
(in particular, identities are mapped to the empty sequence). We have to show
that this definition depends only on $a$, and not on the particular rigid
factorization into maximal integral elements chosen. Let
$s(a)v_{1}*\ldots*v_{k}\in\mathsf{Z}^{*}_{G_{+}}(a)$ be another such rigid
factorization. Then there exists a permutation $\tau\in\mathfrak{S}_{k}$ with
$\eta(u_{i})=\Phi(u_{i})=\Phi(v_{\tau(i)})=\eta(v_{\tau(i)})$ (due to
Proposition 4.1 and by definition of $\eta$). Therefore
$[\eta(u_{1})]\cdot\ldots\cdot[\eta(u_{k})]=[\eta(v_{1})]\cdot\ldots\cdot[\eta(v_{k})]$.
With this definition $\theta$ is a homomorphism: Obviously
$\theta(e)=\mathbf{1}_{\mathcal{B}(C_{M})}$ for all $e\in H_{0}$, and if $b\in
H$ with $t(a)=s(b)$ and $s(b)w_{1}*\ldots*w_{l}\in\mathsf{Z}^{*}(b)$ is a
rigid factorization of $b$ into maximal integral elements, then
$s(a)u_{1}*\ldots*u_{k}*w_{1}*\ldots*w_{l}$ is a rigid factorization of $ab$.
Thus
$\theta(ab)=[\eta(u_{1})]\cdot\ldots\cdot[\eta(u_{k})][\eta(w_{1})]\cdot\ldots\cdot[\eta(w_{l})]=\theta(a)\theta(b).$
We still have to check that $\theta$ has properties T1 and T2.
If $\theta(a)=\mathbf{1}_{\mathcal{B}(C_{M})}$, then $a$ possesses an empty
factorization into maximal elements, hence $a\in G_{0}\cap
H=H_{0}=H^{\times}$.
$\theta$ is surjective: $\theta(e)=\mathbf{1}_{\mathcal{B}(C_{M})}$ for any
$e\in H_{0}$. Let $k\in\mathbb{N}$ and $g_{1}\cdot\ldots\cdot
g_{k}\in\mathcal{B}(C_{M})$. By definition of $C_{M}$ and our second
assumption, there exists an element $u_{1}\in\mathcal{A}(G_{+})$ with
$[\eta(u_{1})]=g_{1}$ and $s(u_{1})\in H_{0}$. Again by our second assumption,
for all $i\in[2,k]$, there exist $u_{i}\in\mathcal{A}(G_{+})$ with
$s(u_{i})=t(u_{i-1})$ and $[\eta(u_{i})]=g_{i}$. With $a=u_{1}\cdot\ldots\cdot
u_{k}\in G$ we get
$[\eta(a)]=[\eta(u_{1})\cdot\ldots\cdot\eta(u_{k})]=[\eta(u_{1})]+\ldots+[\eta(u_{k})]=\mathbf{0}\in
C$ and $s(a)\in H_{0}$, and hence $\eta(a)\in\mathbf{q}(\eta(H))$. By our
first assumption, therefore $a\in HH^{-1}$, and since moreover $a$ is integral
in $G$ and $H$ is right-saturated in $G_{+}$, we get $a\in H$ and
$\theta(a)=g_{1}\cdot\ldots\cdot g_{k}$.
$\theta$ satisfies T2: Let $a\in H$, $\theta(a)=ST$ with
$S,T\in\mathcal{B}(C_{M})$ and $S=g_{1}\cdot\ldots\cdot g_{k}$,
$T=g_{k+1}\cdot\ldots\cdot g_{l}$, where $k\in\mathbb{N}_{0}$ and
$l\in\mathbb{N}_{\geq k}$. By Proposition 4.4, we can find a rigid
factorization $s(a)u_{1}*\ldots*u_{l}\in\mathsf{Z}^{*}(a)$ with
$u_{i}\in\mathcal{A}(G_{+})$ and $[\eta(u_{i})]=g_{i}$ for all $i\in[1,l]$.
Let $b=s(a)u_{1}\cdot\ldots\cdot u_{k}$ and $c=t(b)u_{k+1}\cdot\ldots\cdot
u_{l}$. Then $a=bc$. Since $s(b)\in H_{0}$ and
$[\eta(b)]=\sigma(S)=\mathbf{0}$, the first assumption implies $b\in
HH^{-1}\cap G_{+}=H$. Then $s(c)\in H_{0}$ and $c\in H$ follows similarly.
Finally, $\theta(b)=S$ and $\theta(c)=T$. ∎
The theorem remains true if $H$ is a left-saturated subcategory of $G_{+}$,
and in the first condition the set $HH^{-1}$ is replaced by $H^{-1}H$, and the
condition $s(a)\in H_{0}$ is replaced by $t(a)\in H_{0}$. Similarly, one can
replace the second condition by a symmetrical one, requiring $t(u)=e$ instead
of $s(u)=e$ (in the proof of the surjectivity of $\theta$ one then first
chooses $u_{k}$, followed by $u_{k-1}$ and so on).
###### Remark .
If $G$ is a group, then $G_{+}$ is the free abelian monoid with basis
$\mathcal{A}(G_{+})$ (Section 4). As a saturated submonoid of this free
abelian monoid, $H$ is therefore a reduced commutative Krull monoid [27,
Theorem 2.4.8]. Since $\eta=\operatorname{id}_{G}$ and
$HH^{-1}=\eta(H)\eta(H)^{-1}$ the first condition is trivially satisfied, and
because of $G_{0}=\\{\,1\,\\}$, the second condition is also trivially
satisfied.
Conversely, let $H$ be a normalizing Krull monoid. Then
$H_{\text{red}}=\\{\,aH^{\times}\mid a\in H\,\\}$ is a reduced commutative
Krull monoid, isomorphic to the monoid of its non-zero principal ideals ([25,
Corollary 4.14]). The latter is a submonoid of the divisorial fractional
ideals of $H$, which form the free abelian monoid of integral elements in the
free abelian group of divisorial ideals of $H$. In this way we recover the
well-known transfer homomorphism for Krull monoids as given for example in
[27, Proposition 3.4.8] for commutative Krull monoids, and in [25, Theorem
6.5] for normalizing Krull monoids.
We continue the discussion of normalizing Krull monoids in Remarks 5.2 and
5.1, where the divisorial two-sided ideal theory appears as a special case of
the divisorial one-sided ideal theory.
## 5\. Divisorial ideal theory in semigroups
In this section we develop a divisorial one-sided ideal theory in semigroups.
This follows again original ideas of Asano and Murata and generalizes the
corresponding theory in rings and the theory of divisorial two-sided ideals in
cancellative semigroups (see [2, 34, 3, 4, 6, 5, 17] for classical treatments,
and [42, 32, 33, 25, 36] for more modern treatments in this area). In
particular, the one-sided ideal theory of classical maximal orders over
Dedekind domains is a special case of the theory presented here.
The divisorial fractional one-sided ideals with left- and right-orders maximal
in a fixed equivalence class of orders form a groupoid as studied in the
previous section (this was in fact the motivation for Brandt to introduce the
notion of a groupoid, see [9, 10]). We connect the factorization theory of
elements of a maximal order $H$ with the one for the cancellative small
category of integral principal ideals with left- and right-order conjugate to
$H$, and apply results from the previous section to the factorization of
elements in $H^{\bullet}$. The main result in this section is Section 5. After
having derived it we discuss in detail the case of rings, and of classical
maximal orders (Section 5.1 and Section 5.2).
A semigroup $Q$ is called a _quotient semigroup_ if every cancellative element
is invertible in $Q$, in short $Q^{\bullet}=Q^{\times}$. A subsemigroup
$H\subset Q$ is a _right order_ in $Q$ if $H(H\cap Q^{\bullet})^{-1}=Q$, and
$H$ is a _left order_ in $Q$ if $(H\cap Q^{\bullet})^{-1}H=Q$. $H$ is an
_order_ in $Q$ if it is a left and a right order. We summarize the connection
between a subsemigroup $H\subset Q$ being an order, and $Q$ being a semigroup
of (left and right) fractions of $H$.
###### Lemma .
Let $Q$ be a quotient semigroup, and $H\subset Q$ a subsemigroup.
1. 1.
If $H$ is an order in $Q$, then $H^{\bullet}=H\cap Q^{\bullet}$ and
$Q=\mathbf{q}(H)$.
2. 2.
If $\mathbf{q}(H)=Q$, then $H^{\bullet}=H\cap Q^{\bullet}$ and $H$ is an order
in $Q$.
3. 3.
If $H$ is an order in $Q$, $H^{\prime}$ is a subsemigroup of $Q$ and there
exist $a,b\in Q^{\bullet}$ with $aHb\subset H^{\prime}$, then $H^{\prime}$ is
an order in $Q$.
###### Proof.
1. 1.
It suffices to show $H^{\bullet}=H\cap Q^{\bullet}$, and the inclusion $H\cap
Q^{\bullet}\subset H^{\bullet}$ is clear. Let $a\in H^{\bullet}$, and
$q,q^{\prime}\in Q$ with $aq=aq^{\prime}$. Since $H$ is a right order in $Q$,
there exist $c,d\in H$ and $s\in H\cap Q^{\bullet}$ with $q=cs^{-1}$ and
$q^{\prime}=ds^{-1}$, where we can choose a common denominator because $H\cap
Q^{\bullet}$ satisfies the right Ore condition in $H$. Then $ac=ad$, and,
because $a\in H^{\bullet}$, also $c=d$, showing $q=q^{\prime}$. Since $H$ is a
left order it follows in the same way that $a$ is right-cancellative in
$Q^{\bullet}$, and hence $a\in H\cap Q^{\bullet}$.
2. 2.
It again suffices to show $H^{\bullet}=H\cap Q^{\bullet}$, and this follows in
the same way as in 1.
3. 3.
It suffices to show that every $q\in Q$ has representations of the form
$q=cs^{-1}=t^{-1}d$ with $c,d\in H^{\prime}$ and $s,t\in H^{\prime}\cap
Q^{\bullet}$. Since $H$ is an order in $Q$, there exist
$c^{\prime},d^{\prime}\in H$ and $s^{\prime},t^{\prime}\in H\cap Q^{\bullet}$
with $a^{-1}qa=c^{\prime}s^{\prime-1}$ and $bqb^{-1}=t^{\prime-1}d^{\prime}$.
Setting $c=ac^{\prime}b$, $d=ad^{\prime}b$, $s=as^{\prime}b$ and
$t=at^{\prime}b$, the claim follows. ∎
_For the remainder of this section, let $Q$ be a quotient semigroup._
If $H$ and $H^{\prime}$ are orders in $Q$, then $H$ is (Asano-)equivalent to
$H^{\prime}$, written $H\sim H^{\prime}$, if there exist $a,b,c,d\in
Q^{\bullet}$ with $aHb\subset H^{\prime}$ and $cH^{\prime}d\subset H$. This is
an equivalence relation on the set of orders in $Q$. An order $H$ is _maximal_
if it is maximal within its equivalence class with respect to set inclusion.
A feature of the non-commutative theory is that often there is no unique
maximal order in a given equivalence class, and in fact in the most important
cases we study there are usually infinitely many, but only finitely many
conjugacy classes of them. In studying the divisorial one-sided ideal theory
of a maximal order $H$, one has to study the ideal theory of all maximal
orders in its equivalence class at the same time.
Let $H,H^{\prime}\subset Q$ be subsemigroups (not necessarily orders), and let
$X,Y\subset Q$. As in the previous sections $XY=\\{\,xy\mid x\in X,y\in
Y\,\\}$. $X$ is a _left $H$-module_ if $HX\subset X$, and a _right
$H^{\prime}$-module_ if $XH^{\prime}\subset X$. It is an _$(H,H^{\prime})$
-module_ if it is a left $H$\- and a right $H^{\prime}$-module, i.e.,
$HXH^{\prime}\subset X$. We define
${({Y}\\!\\!:_{r}\\!{X})}=\\{\,q\in Q\mid Xq\subset
Y\,\\}\quad\text{and}\quad{({Y}\\!\\!:_{l}\\!{X})}=\\{\,q\in Q\mid qX\subset
Y\,\\}.$
Every left $H$-module is an $(H,\\{1\\})$-module, and similarly every right
$H^{\prime}$-module is a $(\\{1\\},H^{\prime})$-module. We set
$\mathcal{O}_{l}(X)={({X}\\!\\!:_{l}\\!{X})}$ and
$\mathcal{O}_{r}(X)={({X}\\!\\!:_{r}\\!{X})}$.
###### Lemma .
Let $H,H^{\prime}$ be subsemigroups of $Q$ and let $X$ be an
$(H,H^{\prime})$-module.
1. 1.
${({H}\\!\\!:_{r}\\!{X})}$ and ${({H^{\prime}}\\!\\!:_{l}\\!{X})}$ are
$(H^{\prime},H)$-modules.
2. 2.
$X\subset{({H}\\!\\!:_{l}\\!{{({H}\\!\\!:_{r}\\!{X})}})}$ and
$X\subset{({H^{\prime}}\\!\\!:_{r}\\!{{({H^{\prime}}\\!\\!:_{l}\\!{X})}})}$.
3. 3.
$\mathcal{O}_{l}(X)$ and $\mathcal{O}_{r}(X)$ are subsemigroups of $Q$.
4. 4.
${({\mathcal{O}_{l}(X)}\\!\\!:_{r}\\!{X})}={({\mathcal{O}_{r}(X)}\\!\\!:_{l}\\!{X})}=\\{\,q\in
Q\mid XqX\subset X\,\\}$.
5. 5.
$X\subset\mathcal{O}_{l}(X)$ if and only if $X\subset\mathcal{O}_{r}(X)$ if
and only if $X^{2}\subset X$.
###### Proof.
1. 1.
$XH^{\prime}{({H}\\!\\!:_{r}\\!{X})}H\,\subset\,X{({H}\\!\\!:_{r}\\!{X})}H\,\subset
HH=H$ and similarly for ${({H^{\prime}}\\!\\!:_{l}\\!{X})}$.
2. 2.
$X{({H}\\!\\!:_{r}\\!{X})}\subset H$ by definition of
${({H}\\!\\!:_{r}\\!{X})}$ and thus
$X\subset{({H}\\!\\!:_{l}\\!{{({H}\\!\\!:_{r}\\!{X})}})}$. The other identity
is proven analogously.
3. 3.
Clearly $1\in\mathcal{O}_{l}(X)$ and
$\mathcal{O}_{l}(X)\mathcal{O}_{l}(X)X\subset\mathcal{O}_{l}(X)X\subset X$
implies $\mathcal{O}_{l}(X)\mathcal{O}_{l}(X)\subset\mathcal{O}_{l}(X)$. The
claim for $\mathcal{O}_{r}(X)$ is shown similarly.
4. 4.
We show ${({\mathcal{O}_{l}(X)}\\!\\!:_{r}\\!{X})}=\\{\,q\in Q\mid XqX\subset
X\,\\}$. Let $q\in Q$. Then $XqX\subset X$ is equivalent to
$Xq\subset\mathcal{O}_{l}(X)$, which in turn is equivalent to
$q\in{({\mathcal{O}_{l}(X)}\\!\\!:_{r}\\!{X})}$.
5. 5.
Immediate from the definitions of $\mathcal{O}_{l}(X)$ and
$\mathcal{O}_{r}(X)$. ∎
###### Definition 5.1.
For $X\subset Q$ as in Section 5, we define
$X^{-1}={({\mathcal{O}_{l}(X)}\\!\\!:_{r}\\!{X})}={({\mathcal{O}_{r}(X)}\\!\\!:_{l}\\!{X})}=\\{\,q\in
Q\mid XqX\subset X\,\\}\quad\text{and}\quad X_{v}=(X^{-1})^{-1}.$
###### Definition 5.2.
Let $H$ and $H^{\prime}$ be orders in $Q$.
1. 1.
A _fractional left $H$-ideal_ is a left $H$-module $I$ such that $I\cap
Q^{\bullet}\neq\emptyset$ and ${({H}\\!\\!:_{r}\\!{I})}\cap
Q^{\bullet}\neq\emptyset$.
2. 2.
A _fractional right $H^{\prime}$-ideal_ is a right $H^{\prime}$-module $I$
such that $I\cap Q^{\bullet}\neq\emptyset$ and
${({H^{\prime}}\\!\\!:_{l}\\!{I})}\cap Q^{\bullet}\neq\emptyset$.
3. 3.
If $I$ is a fractional left $H$-ideal and a fractional right
$H^{\prime}$-ideal, then $I$ is a _fractional $(H,H^{\prime})$-ideal_.
4. 4.
A _fractional $H$-ideal_ is a fractional $(H,H)$-ideal.
5. 5.
$I$ is a _left $H$-ideal_ if it is a fractional left $H$-ideal and $I\subset
H$. A _right $H^{\prime}$-ideal_ is defined analogously.
6. 6.
If $I$ is a left $H$-ideal and a right $H^{\prime}$-ideal, then $I$ is an
_$(H,H^{\prime})$ -ideal_.
7. 7.
An _$H$ -ideal_ is an $(H,H)$-ideal.
8. 8.
A fractional left $H$-ideal $I$ is _integral_ if $I\subset\mathcal{O}_{l}(I)$
(equivalently, $I\subset\mathcal{O}_{r}(I)$). The same definition is made for
fractional right $H^{\prime}$-ideals.
If $H$ is a maximal order, then the notions of a left $H$-ideal and that of an
integral fractional left $H$-ideal coincide (this will follow from Lemma 5.1
and 5.2). We will sometimes call a fractional left (right) $H$-ideal _one-
sided_ to emphasize that it need not be a fractional right (left) $H$-ideal,
or _two-sided_ to emphasize that it is indeed a fractional $H$-ideal.
We recall some properties of fractional left $H$-ideals and first observe the
following.
###### Lemma .
If $H$ is an order in $Q$ and $I$ is a fractional left $H$-ideal, then
$\mathcal{O}_{l}(I)$ and $\mathcal{O}_{r}(I)$ are orders. $I$ is a fractional
$(\mathcal{O}_{l}(I),\mathcal{O}_{r}(I))$-ideal.
###### Proof.
Let $a\in I\cap Q^{\bullet}$, and let $b\in{({H}\\!\\!:_{r}\\!{I})}\cap
Q^{\bullet}$. By definition, $H\subset\mathcal{O}_{l}(I)$ and since $H$ is an
order and $\mathcal{O}_{l}(I)$ a semigroup, it is also an order.
$\mathcal{O}_{l}(I)I\subset I$ and
$b\in{({\mathcal{O}_{l}(I)}\\!\\!:_{r}\\!{I})}$ imply that $I$ is a fractional
left $\mathcal{O}_{l}(I)$-ideal. Since
$b\in{({\mathcal{O}_{l}(I)}\\!\\!:_{r}\\!{I})}={({\mathcal{O}_{r}(I)}\\!\\!:_{l}\\!{I})}$,
it holds that $b\mathcal{O}_{l}(I)a\subset bI\subset\mathcal{O}_{r}(I)$, and
since $\mathcal{O}_{r}(I)$ is a semigroup and $\mathcal{O}_{l}(I)$ an order,
$\mathcal{O}_{r}(I)$ is also an order. Therefore $I$ is also a fractional
right $\mathcal{O}_{r}(I)$-ideal. ∎
The previous lemma implies that it is no restriction to require $I$ to be a
fractional $(H,H^{\prime})$-ideal over it, say, being a fractional left
$H$-ideal (set $H^{\prime}=\mathcal{O}_{r}(I)$).
###### Lemma .
Let $H$ and $H^{\prime}$ be orders in $Q$, and let $I$ be a fractional
$(H,H^{\prime})$-ideal.
1. 1.
The orders $H$, $H^{\prime}$, $\mathcal{O}_{l}(I)$ and $\mathcal{O}_{r}(I)$
are all equivalent.
2. 2.
If $H$ is maximal, then $\mathcal{O}_{l}(I)=H$, and similarly, if $H^{\prime}$
is maximal, then $\mathcal{O}_{r}(I)=H^{\prime}$.
3. 3.
${({H}\\!\\!:_{r}\\!{I})}$ is a fractional right $H$-ideal, and
${({H^{\prime}}\\!\\!:_{l}\\!{I})}$ is a fractional left $H^{\prime}$-ideal.
4. 4.
If $J$ is a fractional left $H$-ideal, then $I\cap J$ and $I\cup J$ are
fractional left $H$-ideals.
5. 5.
If $(I_{m})_{m\in M}$ is a non-empty family of left $H$-ideals for some index
set $M$, then $\bigcup_{m\in M}I_{m}$ is a left $H$-ideal.
6. 6.
If $H^{\prime\prime}$ is an order, and $K$ is a fractional
$(H^{\prime},H^{\prime\prime})$-ideal, then $IK$ is a fractional
$(H,H^{\prime\prime})$-ideal.
###### Proof.
1. 1.
By definition of the right and left order, $H\subset\mathcal{O}_{l}(I)$ and
$H^{\prime}\subset\mathcal{O}_{r}(I)$. Let $a\in I\cap Q^{\bullet}$,
$b\in{({H}\\!\\!:_{r}\\!{I})}\cap Q^{\bullet}$ and
$c\in{({H^{\prime}}\\!\\!:_{l}\\!{I})}\cap Q^{\bullet}$. Then
$\mathcal{O}_{l}(I)ab\subset Ib\subset H$ and $ca\mathcal{O}_{r}(I)\subset
H^{\prime}$, so $H\sim\mathcal{O}_{l}(I)$ and
$H^{\prime}\sim\mathcal{O}_{r}(I)$. Finally, $cHa\subset cI\subset H^{\prime}$
and $aH^{\prime}b\subset H$ imply $H\sim H^{\prime}$.
2. 2.
By 1., $\mathcal{O}_{l}(I)\sim H$ and by definition of the left order
$H\subset\mathcal{O}_{l}(I)$. Maximality of $H$ implies
$H=\mathcal{O}_{l}(I)$. Analogously, $H^{\prime}=\mathcal{O}_{r}(I)$ if
$H^{\prime}$ is maximal.
3. 3.
${({H}\\!\\!:_{r}\\!{I})}$ is an $(H^{\prime},H)$-module, and
${({H}\\!\\!:_{r}\\!{I})}\cap Q^{\bullet}\neq\emptyset$ because $I$ is a
fractional left $H$-ideal. Since
$I\subset{({H}\\!\\!:_{l}\\!{{({H}\\!\\!:_{r}\\!{I})}})}$, also
${({H}\\!\\!:_{l}\\!{{({H}\\!\\!:_{r}\\!{I})}})}\cap
Q^{\bullet}\neq\emptyset$, and thus ${({H}\\!\\!:_{r}\\!{I})}$ is a fractional
right $H$-ideal. Similarly one shows that ${({H^{\prime}}\\!\\!:_{l}\\!{I})}$
is a fractional left $H^{\prime}$-ideal.
4. 4.
Clearly $H(I\cap J)\subset I\cap J$ and if $c\in{({H}\\!\\!:_{r}\\!{I})}\cap
Q^{\bullet}$, then $(I\cap J)c\subset Ic\subset H$. It remains to show $I\cap
J\cap Q^{\bullet}\neq\emptyset$. Let $a\in I\cap Q^{\bullet}$ and $b\in J\cap
Q^{\bullet}$. Then $a=a^{\prime}s^{-1}$ and $b=b^{\prime}s^{-1}$ with
$a^{\prime},b^{\prime},s\in H\cap Q^{\bullet}$ (we can choose a common
denominator using the right Ore condition). By the left Ore condition there
exist $a^{\prime\prime}\in H\cap Q^{\bullet}$ and $b^{\prime\prime}\in H$ with
$a^{\prime\prime}a^{\prime}=b^{\prime\prime}b^{\prime}$. Then
$a^{\prime\prime}a^{\prime}s^{-1}=b^{\prime\prime}b^{\prime}s^{-1}\in I\cap
J\cap Q^{\bullet}$.
For the union, again $H(I\cup J)\subset I\cup J$, and there exists $a\in(I\cup
J)\cap Q^{\bullet}$. It remains to show ${({H}\\!\\!:_{r}\\!{(I\cup J)})}\cap
Q^{\bullet}\neq\emptyset$. But ${({H}\\!\\!:_{r}\\!{(I\cup
J)})}={({H}\\!\\!:_{r}\\!{I})}\cap{({H}\\!\\!:_{r}\\!{J})}$, and we are done
by applying our previous statement about the intersection to the fractional
right $H$-ideals ${({H}\\!\\!:_{r}\\!{I})}$ and ${({H}\\!\\!:_{r}\\!{J})}$.
5. 5.
Set $I=\bigcup_{m\in M}I_{m}$. Then $HI\subset I$ and $I\cap
Q^{\bullet}\neq\emptyset$ are clear, and $I_{m}\subset H$ for all $m\in M$
implies $1\in{({H}\\!\\!:_{r}\\!{I})}$.
6. 6.
Certainly $HIKH^{\prime\prime}\subset IK$. If $a\in I\cap Q^{\bullet}$ and
$b\in K\cap Q^{\bullet}$, then $ab\in IK\cap Q^{\bullet}$. Let
$c\in{({H}\\!\\!:_{r}\\!{I})}\cap Q^{\bullet}$ and
$d\in{({H^{\prime}}\\!\\!:_{r}\\!{K})}\cap Q^{\bullet}$. Then $IKdc\subset
IH^{\prime}c\subset Ic\subset H$, i.e., $dc\in{({H}\\!\\!:_{r}\\!{IK})}\cap
Q^{\bullet}$. If $c^{\prime}\in{({H^{\prime}}\\!\\!:_{l}\\!{I})}\cap
Q^{\bullet}$ and $d^{\prime}\in{({H^{\prime\prime}}\\!\\!:_{l}\\!{K})}\cap
Q^{\bullet}$, then $d^{\prime}c^{\prime}IK\subset d^{\prime}H^{\prime}K\subset
d^{\prime}K\subset H^{\prime\prime}$, i.e.,
$d^{\prime}c^{\prime}\in{({H^{\prime\prime}}\\!\\!:_{l}\\!{IK})}\cap
Q^{\bullet}$. ∎
###### Lemma .
Let $H$ and $H^{\prime}$ be orders in $Q$.
1. 1.
If $H\sim H^{\prime}$, then there exist $a,b\in H^{\prime\bullet}$ with
$aH^{\prime}b\subset H$. If moreover $H\subset H^{\prime}$, then we can even
take $a,b\in H^{\bullet}$.
2. 2.
If $H\sim H^{\prime}$ and $H\subset H^{\prime}$, then there exists an order
$H^{\prime\prime}$ and $a,b\in H^{\bullet}$ such that $H\subset
H^{\prime\prime}\subset H^{\prime}$ and $H^{\prime\prime}b\subset H$ and
$aH^{\prime}\subset H^{\prime\prime}$.
3. 3.
The following statements are equivalent:
1. (a)
$H\sim H^{\prime}$.
2. (b)
There exists a fractional $(H,H^{\prime})$-ideal.
3. (c)
There exists a fractional $(H^{\prime},H)$-ideal.
###### Proof.
1. 1.
There exist $x,y\in Q^{\bullet}$ with $xH^{\prime}y\subset H$. Since
$H^{\prime}$ is an order in $Q$, $x=ac^{-1}$ and $y=d^{-1}b$ with $a,b\in
H^{\prime}$ and $c,d\in H^{\prime}\cap Q^{\bullet}$. If $H\subset H^{\prime}$
we even take $a,b,c,d\in H$. Then $aH^{\prime}b\subset
ac^{-1}H^{\prime}d^{-1}b\subset H$.
2. 2.
Using 1. choose $a,b\in H^{\bullet}$ with $aH^{\prime}b\subset H$. Let
$H^{\prime\prime}=H\cup aH^{\prime}\cup HaH^{\prime}$. Then it is easily
checked that $H^{\prime\prime}H^{\prime\prime}\subset H^{\prime\prime}$ and
obviously $H\subset H^{\prime\prime}$, thus $H^{\prime\prime}$ is an order.
Moreover, $H^{\prime\prime}\subset H^{\prime}$, $H^{\prime\prime}b\subset H$
and $aH^{\prime}\subset H^{\prime\prime}$, as claimed.
3. 3.
(a) $\Rightarrow$ (b): By 1. there exist $a,b\in H^{\bullet}$ with $aHb\subset
H^{\prime}$ and $c,d\in H^{\prime\bullet}$ with $cH^{\prime}d\subset H$.
Define $I=HbcH^{\prime}$. Clearly $I$ is an $(H,H^{\prime})$-module with
$bc\in I\cap Q^{\bullet}$. Since $aI=aHbcH^{\prime}\subset
H^{\prime}cH^{\prime}\subset H^{\prime}$ and $Id=HbcH^{\prime}d\subset
HbH\subset H$, $I$ is a fractional $(H,H^{\prime})$-ideal.
(b) $\Rightarrow$ (a): By Lemma 5.1.
(a) $\Leftrightarrow$ (c) follows by symmetry, swapping the roles of $H$ and
$H^{\prime}$. ∎
###### Lemma .
Let $H$ be an order in $Q$. The following statements are equivalent.
1. (a)
$H$ is a maximal order.
2. (b)
If $I$ is a fractional left $H$-ideal, then $\mathcal{O}_{l}(I)=H$ and if $J$
is a fractional right $H$-ideal, then $\mathcal{O}_{r}(I)=H$.
3. (c)
If $I$ is a fractional $H$-ideal, then
$\mathcal{O}_{r}(I)=\mathcal{O}_{l}(I)=H$.
4. (d)
If $I$ is an $H$-ideal, then $\mathcal{O}_{r}(I)=\mathcal{O}_{l}(I)=H$.
###### Proof.
(a) $\Rightarrow$ (b): By Lemma 5.2, $\mathcal{O}_{l}(I)=H$ and
$\mathcal{O}_{r}(J)=H$.
(b) $\Rightarrow$ (c) $\Rightarrow$ (d): Trivial.
(d) $\Rightarrow$ (a): Assume $H^{\prime}$ is an order equivalent to $H$ and
$H\subset H^{\prime}$. Applying Lemma 5.2 we find an equivalent order
$H^{\prime\prime}$ with $H\subset H^{\prime\prime}\subset H^{\prime}$ and
$a,b\in Q^{\bullet}$ with $aH^{\prime}\subset H^{\prime\prime}$ and
$H^{\prime\prime}b\subset H$. Let $I=\\{\,x\in Q\mid H^{\prime\prime}x\subset
H\,\\}$. Then $I$ is an $H$-ideal, and
$H^{\prime\prime}\subset\mathcal{O}_{l}(I)$, implying $H^{\prime\prime}=H$ by
(d). Set $J=\\{\,x\in Q\mid xH^{\prime}\subset H\,\\}$. Then $J$ is again an
$H$-ideal (we use $a\in J$ since $H^{\prime\prime}=H$), and
$H^{\prime}\subset\mathcal{O}_{r}(J)$ implies $H^{\prime}=H$. ∎
###### Lemma .
Let $H$ be a maximal order in $Q$, let $I$ and $J$ be fractional left
$H$-ideals, and let $K$ be a fractional left $\mathcal{O}_{r}(I)$-ideal.
1. 1.
$\mathcal{O}_{l}(I)=H$, $I^{-1}$ is a fractional right $H$-ideal with
$\mathcal{O}_{r}(I^{-1})=H$ and $I_{v}$ is a fractional left $H$-ideal with
$\mathcal{O}_{l}(I_{v})=H$ and $I\subset I_{v}$. Moreover,
$\mathcal{O}_{l}(I)_{v}=H_{v}=H$.
2. 2.
If $a\in Q^{\bullet}$, then $Ha$ is a fractional left $H$-ideal with
$\mathcal{O}_{r}(Ha)=a^{-1}Ha$, $(Ha)^{-1}=a^{-1}H$ and $(Ha)_{v}=Ha$. $Ha$ is
integral (equivalently, a left $H$-ideal), if and only if $a\in H^{\bullet}$.
3. 3.
If $I\subset J$ then $J^{-1}\subset I^{-1}$ and $I_{v}\subset J_{v}$.
4. 4.
$I\subset I_{v}=(I_{v})_{v}$, $I_{v}^{-1}=(I^{-1})_{v}=I^{-1}$ and
$\mathcal{O}_{l}(I_{v})=\mathcal{O}_{l}(I)=\mathcal{O}_{l}(I)_{v}=H$.
5. 5.
$(I_{v}\cap J_{v})_{v}=I_{v}\cap J_{v}$.
6. 6.
$(I\cup J)_{v}=(I_{v}\cup J)_{v}=(I\cup J_{v})_{v}=(I_{v}\cup J_{v})_{v}$.
7. 7.
If $\mathcal{O}_{r}(I)$ and $\mathcal{O}_{r}(K)$ are also maximal, then
$(IK)_{v}=(I_{v}K)_{v}=(IK_{v})_{v}=(I_{v}K_{v})_{v}$.
###### Proof.
1. 1.
By Lemma 5.2, $\mathcal{O}_{l}(I)=H$, and thus Lemma 5.3 implies that
$I^{-1}={({H}\\!\\!:_{r}\\!{I})}$ is a right $H$-ideal. By the symmetric
statement of what we just showed for fractional right $H$-ideals, therefore
$\mathcal{O}_{r}(I^{-1})=H$ and $I_{v}$ is a fractional left $H$-ideal.
Applying the first part of the statement to $I_{v}$ yields
$\mathcal{O}_{l}(I_{v})=H$. Now $I\subset I_{v}$ follows from Lemma 5.2, and
$H_{v}=H$ from $H^{-1}={({H}\\!\\!:_{r}\\!{H})}=H$.
2. 2.
Since $a\in Ha\cap Q^{\bullet}$ and $a^{-1}\in{({H}\\!\\!:_{r}\\!{Ha})}$, $Ha$
is a fractional left $H$-ideal. Certainly
$a^{-1}Ha\subset\mathcal{O}_{r}(Ha)$. Conversely, if
$x\in\mathcal{O}_{r}(Ha)$, then $ax\in Ha$ and thus $x\in a^{-1}Ha$, so that
altogether $\mathcal{O}_{r}(Ha)=a^{-1}Ha$. Moreover, $(Ha)a^{-1}H\subset H$
and if $Hax\subset H$ for $x\in Q^{\bullet}$, then $ax\in H$ and hence $x\in
a^{-1}H$, implying $(Ha)^{-1}=a^{-1}H$. Finally, $(Ha)_{v}=Ha$ because
$(Ha)_{v}=((Ha)^{-1})^{-1}=Ha$. $Ha$ is a left $H$-ideal if and only if it is
integral due to maximality of $H$, and $Ha\subset H$ if and only if $a\in
H\cap Q^{\bullet}=H^{\bullet}$.
3. 3.
If $x\in Q$ with $Jx\subset H$, then $Ix\subset Jx\subset H$, and hence
$J^{-1}\subset I^{-1}$. By 1., $J^{-1}$ and $I^{-1}$ are fractional right
$H$-ideals with $\mathcal{O}_{r}(I^{-1})=\mathcal{O}_{r}(J^{-1})=H$, and we
apply the symmetric statement for right fractional $H$-ideals to obtain
$I_{v}\subset J_{v}$.
4. 4.
By 1., $I^{-1}$ is a fractional right $H$-ideal with
$\mathcal{O}_{r}(I^{-1})=H$, and $I_{v}$ is a fractional left $H$-ideal with
$\mathcal{O}_{l}(I_{v})=H$. Moreover, also by 1, $I\subset I_{v}$ and
$I^{-1}\subset(I^{-1})_{v}=[(I^{-1})^{-1}]^{-1}=I_{v}^{-1}$. It follows from
3., that $I_{v}\subset(I_{v})_{v}$ and $I^{-1}=I_{v}^{-1}$. Therefore
$(I_{v})_{v}=(I_{v}^{-1})^{-1}\subset(I^{-1})^{-1}=I_{v}$, whence
$I_{v}=(I_{v})_{v}$.
5. 5.
$I_{v}\cap J_{v}\subset(I_{v}\cap
J_{v})_{v}\subset(I_{v})_{v}\cap(J_{v})_{v}=I_{v}\cap J_{v}$.
6. 6.
$I\cup J\subset I_{v}\cup J\subset I_{v}\cup J_{v}\subset(I\cup J)_{v}$ and by
taking divisorial closures, and 4., the claim follows.
7. 7.
We use $\mathcal{O}_{r}(I_{v})=\mathcal{O}_{r}(I)$ and
$\mathcal{O}_{l}(K)=\mathcal{O}_{l}(K_{v})$ (from 1.). We have $IK\subset
I_{v}K\subset I_{v}K_{v}$, and similarly $IK\subset IK_{v}\subset I_{v}K_{v}$
(by 1.). By 3., this implies
$(IK)_{v}\subset(IK_{v})_{v}\subset(I_{v}K_{v})_{v}$ and
$(IK)_{v}\subset(I_{v}K)_{v}\subset(I_{v}K_{v})_{v}$.
To prove the claim it suffices to show $(I_{v}K_{v})_{v}\subset(IK)_{v}$,
which will follow from 3. and 4. if we show $I_{v}K_{v}\subset(IK)_{v}$. Since
$IK(IK)^{-1}\subset\mathcal{O}_{l}(I)$, we have
$K(IK)^{-1}\subset{({\mathcal{O}_{l}(I)}\\!\\!:_{r}\\!{I})}=I^{-1}=I_{v}^{-1}$,
where the last equality is due to 4. Multiplying by $I_{v}$ from the right
gives $K(IK)^{-1}I_{v}\subset I_{v}^{-1}I_{v}$. By definition,
$I_{v}^{-1}I_{v}\subset\mathcal{O}_{r}(I_{v})$. Since
$\mathcal{O}_{r}(I_{v})=\mathcal{O}_{r}(I)=\mathcal{O}_{l}(K)$, therefore
$K(IK)^{-1}I_{v}\subset\mathcal{O}_{l}(K)$. Now $(IK)^{-1}I_{v}\subset
K^{-1}=K_{v}^{-1}$ (using 4. again). Multiplying by $K_{v}$ from the right and
using $\mathcal{O}_{r}(K_{v})=\mathcal{O}_{r}(K)$, we obtain
$(IK)^{-1}I_{v}K_{v}\subset\mathcal{O}_{r}(K)$. Since
$\mathcal{O}_{l}((IK)^{-1})=\mathcal{O}_{r}(K)$, this implies
$I_{v}K_{v}\subset((IK)^{-1})^{-1}=(IK)_{v}$. ∎
###### Definition 5.3.
Let $H$ be an order in $Q$. A fractional left or right $H$-ideal $I$ is called
_divisorial_ if $I=I_{v}$.
If $I$ is a fractional left $H$-ideal for a maximal order $H$, then it is not
necessarily true that $\mathcal{O}_{r}(I)$ is again a maximal order. The next
proposition shows that for divisorial fractional left or right $H$-ideals with
$H$ maximal, already both, $\mathcal{O}_{l}(I)$ and $\mathcal{O}_{r}(I)$, are
maximal. We can define an associative partial operation, the _$v$ -product_,
by $I\cdot_{v}J=(IJ)_{v}$ when $J$ is a divisorial fractional left
$\mathcal{O}_{r}(I)$-ideal. Moreover it shows that every divisorial fractional
left or right ideal is _$v$ -invertible_, i.e., invertible with respect to
this operation.
###### Proposition .
Let $H$ be a maximal order in $Q$. Let $I$ be a fractional left $H$-ideal.
Then
1. 1.
$\mathcal{O}_{l}(I^{-1})$ is a maximal order. In particular,
$\mathcal{O}_{r}(I_{v})$ is a maximal order.
2. 2.
$(II^{-1})_{v}=\mathcal{O}_{l}(I)$ and if $\mathcal{O}_{r}(I)$ is also
maximal, then $(I^{-1}I)_{v}=\mathcal{O}_{r}(I)$.
3. 3.
If $I$ is a divisorial fractional left $H$-ideal, $J$ a divisorial fractional
left $\mathcal{O}_{r}(I)$-ideal and $K$ a divisorial fractional left
$\mathcal{O}_{r}(J)$-ideal, then
$(I\cdot_{v}J)\cdot_{v}K=I\cdot_{v}(J\cdot_{v}K).$
###### Proof.
1. 1.
Because $H$ is maximal, $\mathcal{O}_{l}(I)=H$. Trivially,
$\mathcal{O}_{r}(I)\subset\mathcal{O}_{l}(I^{-1})$. Let
$H^{\prime}\supset\mathcal{O}_{l}(I^{-1})$ be an order with
$H^{\prime}\sim\mathcal{O}_{l}(I^{-1})$. Then $J=IH^{\prime}I^{-1}$ is an
$(H,H)$-module and if $a\in I\cap Q^{\bullet}$ and $b\in I^{-1}\cap
Q^{\bullet}$, then $ab\in J$ and moreover
$J^{2}=IH^{\prime}I^{-1}IH^{\prime}I^{-1}\subset
IH^{\prime}\mathcal{O}_{l}(I^{-1})H^{\prime}I^{-1}=IH^{\prime}I^{-1}=J,$
showing that $J$ is an integral left $\mathcal{O}_{l}(J)$-ideal.
We claim $H=\mathcal{O}_{l}(J)$. Since
$H=\mathcal{O}_{l}(I)\subset\mathcal{O}_{l}(J)$ and $H$ is maximal, it
suffices to show $\mathcal{O}_{l}(J)\sim H$. To this end we first show
$bJa\subset H^{\prime}$:
$bJa=bIH^{\prime}I^{-1}a\subset
I^{-1}IH^{\prime}I^{-1}I\subset\mathcal{O}_{r}(I)H^{\prime}\mathcal{O}_{r}(I)=H^{\prime}.$
Since $H^{\prime}\sim H$, there exist $c,d\in Q^{\bullet}$ with
$cH^{\prime}d\subset H$. Since $ab\in J\cap Q^{\bullet}$ therefore
$cb(\mathcal{O}_{l}(J)ab)ad\subset cbJad\subset H$, proving the claim.
Therefore, from the definition of $J$,
$H^{\prime}I^{-1}\subset{({J}\\!\\!:_{r}\\!{I})}\subset{({\mathcal{O}_{l}(J)}\\!\\!:_{r}\\!{I})}={({H}\\!\\!:_{r}\\!{I})}=I^{-1}$
and thus $H^{\prime}\subset\mathcal{O}_{l}(I^{-1})$, and, because we started
out with the converse inclusion, also $H^{\prime}=\mathcal{O}_{l}(I^{-1})$.
Now $\mathcal{O}_{r}(I_{v})=\mathcal{O}_{l}(I^{-1})$ implies the “in
particular” statement.
2. 2.
We have to show $(II^{-1})_{v}=\mathcal{O}_{l}(I)$ and
$(I^{-1}I)_{v}=\mathcal{O}_{r}(I)$, and we check the first equality as the
second one then follows analogously. The inclusion
$II^{-1}\subset\mathcal{O}_{l}(I)$ implies
$(II^{-1})_{v}\subset\mathcal{O}_{l}(I)_{v}=\mathcal{O}_{l}(I)$. It remains to
prove $\mathcal{O}_{l}(I)\subset(II^{-1})_{v}$. Due to maximality of
$\mathcal{O}_{l}(I)$, it holds that
$\mathcal{O}_{l}(II^{-1})=\mathcal{O}_{l}(I)$, and therefore
$II^{-1}(II^{-1})^{-1}\subset\mathcal{O}_{l}(I)$. Thus
$I^{-1}(II^{-1})^{-1}\subset I^{-1}$, and
$(II^{-1})^{-1}\subset\mathcal{O}_{r}(I^{-1})=\mathcal{O}_{l}(I)$. By Lemma
5.3, therefore $\mathcal{O}_{l}(I)\subset(II^{-1})_{v}$.
3. 3.
Using Lemma 5.7, which can be applied due to 1.,
$((IJ)_{v}K)_{v}=(IJK)_{v}=(I(JK)_{v})_{v}$. ∎
###### Corollary .
If $H$ is a maximal order, then every order $H^{\prime}$ with $H^{\prime}\sim
H$ is contained in a maximal order equivalent to $H$.
###### Proof.
By Lemma 5.3, there exists a fractional $(H,H^{\prime})$-ideal $I$. Then
$I_{v}$ is divisorial, $\mathcal{O}_{r}(I_{v})\sim\mathcal{O}_{l}(I_{v})=H$,
$H^{\prime}\subset\mathcal{O}_{r}(I_{v})$ and by Proposition 5.1
$\mathcal{O}_{r}(I_{v})$ is maximal. ∎
###### Corollary .
Let $\alpha$ denote an equivalence class of maximal orders of $Q$. Let
$\mathcal{F}_{v}(\alpha)=\\{\,I\mid\text{$I$ is a divisorial fractional left
(or right) $H$-ideal with $H\in\alpha$}\,\\}$
and
$\mathcal{I}_{v}(\alpha)=\\{\,I\mid\text{$I$ is a divisorial left (or right)
$H$-ideal with $H\in\alpha$}\,\\}.$
Then $(\mathcal{F}_{v}(\alpha),\cdot_{v},\subset)$ is a lattice-ordered
groupoid, with identity elements the maximal orders in $\alpha$. If $I,J$ are
in $\mathcal{F}_{v}(\alpha)$ with $\mathcal{O}_{l}(I)=\mathcal{O}_{l}(J)$ or
$\mathcal{O}_{r}(I)=\mathcal{O}_{r}(J)$, then $I\wedge J=I\cap J$ and $I\vee
J=(I\cup J)_{v}$. Moreover, $\mathcal{I}_{v}(\alpha)$ is the subcategory of
integral elements.
###### Proof.
For $I\in\mathcal{F}_{v}(\alpha)$ we have
$\mathcal{O}_{l}(I)\cdot_{v}I=I=I\cdot_{v}\mathcal{O}_{r}(I)$. If
$J\in\mathcal{F}_{v}(\alpha)$, then the $v$-product $I\cdot_{v}J$ is defined
whenever $\mathcal{O}_{r}(I)=\mathcal{O}_{l}(J)$, and then $I\cdot_{v}J$ is a
divisorial fractional $(\mathcal{O}_{l}(I),\mathcal{O}_{r}(J))$-ideal. The
$v$-product is associative when it is defined (Proposition 5.3). Therefore
$\mathcal{F}_{v}(\alpha)$ with $\cdot_{v}$ as composition is a category where
the set of identities is the set of maximal orders, $\alpha$, and for
$I\in\mathcal{F}_{v}(\alpha)$ we have $s(I)=\mathcal{O}_{l}(I)$ and
$t(I)=\mathcal{O}_{r}(I)$. This category is a groupoid due to Proposition 5.2.
On $\mathcal{F}_{v}(\alpha)$ set inclusion defines a partial order, and
obviously also the restrictions to
$\\{\,I\in\mathcal{F}_{v}(\alpha)\mid\mathcal{O}_{l}(I)=H\,\\}$ and
$\\{\,I\in\mathcal{F}_{v}(\alpha)\mid\mathcal{O}_{r}(I)=H\,\\}$ for
$H\in\alpha$, given by set inclusion in these subsets, are partial orders. Let
$I,J\in\mathcal{F}_{v}(\alpha)$ with $\mathcal{O}_{l}(I)=\mathcal{O}_{l}(J)$.
Then $I\cap J\in\mathcal{F}_{v}(\alpha)$ and $(I\cup
J)_{v}\in\mathcal{F}_{v}(\alpha)$ (by Lemmas 5 and 5), and clearly they are
the infimum respectively supremum of $\\{\,I,J\,\\}$ in
$\\{\,I\in\mathcal{F}_{v}(\alpha)\mid\mathcal{O}_{l}(I)=H\,\\}$, making this
set lattice-ordered. Symmetric statements hold if
$\mathcal{O}_{r}(I)=\mathcal{O}_{r}(J)$. If
$\mathcal{O}_{l}(I)=\mathcal{O}_{l}(J)$ and
$\mathcal{O}_{r}(I)=\mathcal{O}_{r}(J)$ both hold, then also
$\mathcal{O}_{l}(I\cap J)=\mathcal{O}_{l}((I\cup J)_{v})=\mathcal{O}_{l}(I)$
and $\mathcal{O}_{r}(I\cap J)=\mathcal{O}_{r}((I\cup
J)_{v})=\mathcal{O}_{r}(I)$ both hold. Therefore
$(\mathcal{F}_{v}(\alpha),\subset)$ is a lattice-ordered groupoid with the
claimed meet and join. It is immediate from the definitions that
$\mathcal{I}_{v}(\alpha)$ is the subcategory of integral elements of this
lattice-ordered groupoid. ∎
###### Definition & Lemma .
An order $H$ is _bounded_ if it satisfies the following equivalent conditions:
1. (a)
Every fractional left $H$-ideal and every fractional right $H$-ideal contains
a fractional (two-sided) $H$-ideal.
2. (b)
Every left $H$-ideal and every right $H$-ideal contains a (two-sided)
$H$-ideal.
3. (c)
For all $a\in Q^{\bullet}$ there exist $b,c\in Q^{\bullet}$ such that
$bH\subset Ha$ and $Hc\subset aH$.
4. (d)
For all $a\in Q^{\bullet}$ there exist $b,c\in Q^{\bullet}$ such that
$Ha\subset bH$ and $aH\subset Hc$.
5. (e)
For all $a\in Q^{\bullet}$, $HaH$ is a fractional (two-sided) $H$-ideal.
6. (f)
For all $a\in Q^{\bullet}$ there exists a fractional (two-sided) $H$-ideal $I$
with $a\in I$.
7. (g)
If $M\subset Q$ and $a,b\in Q^{\bullet}$ with $aMb\subset H$, then there exist
$c,d\in Q^{\bullet}$ with $cM\subset H$ and $Md\subset H$.
###### Proof.
(a) $\Rightarrow$ (b): Trivial.
(b) $\Rightarrow$ (c): Let $a\in Q^{\bullet}$. Then $a=d^{-1}c$ with $c,d\in
H\cap Q^{\bullet}$, and $Ha=Hd^{-1}c\supset Hc$. By (b), $Hc$ contains an
$H$-ideal $J$. If $b\in J\cap Q^{\bullet}$, then $bH\subset J\subset Ha$. The
symmetric claim follows similarly.
(c) $\Rightarrow$ (d): By (c) applied to $a^{-1}$, there exist $b,c\in
Q^{\bullet}$ with $b^{-1}H\subset Ha^{-1}$ and $Hc^{-1}\subset a^{-1}H$. Then
$Ha\subset bH$ and $aH\subset Hc$.
(d) $\Rightarrow$ (e): $HaH$ is an $(H,H)$-module and contains the element
$a\in Q^{\bullet}$. Let $b,c\in Q^{\bullet}$ with $Ha\subset bH$ and
$aH\subset Hc$. Then $HaH\subset bH$ and $HaH\subset Hc$, hence
$b^{-1}\in{({H}\\!\\!:_{l}\\!{HaH})}$ and
$c^{-1}\in{({H}\\!\\!:_{r}\\!{HaH})}$.
(e) $\Rightarrow$ (f): Trivial.
(f) $\Rightarrow$ (g): $aM\subset Hb^{-1}\subset Hb^{-1}H$, and the latter
being contained in a fractional $H$-ideal, there exists $a^{\prime}\in
Q^{\bullet}\cap{({H}\\!\\!:_{l}\\!{Hb^{-1}H})}$ and thus $a^{\prime}aM\subset
H$. Similarly, $Mb\subset a^{-1}H\subset Ha^{-1}H$, and there exists a
$b^{\prime}\in Q^{\bullet}\cap{({H}\\!\\!:_{r}\\!{Ha^{-1}H})}$. Thus
$Mbb^{\prime}\subset H$.
(g) $\Rightarrow$ (a): Let $I$ be a fractional left $H$-ideal and $a\in I\cap
Q^{\bullet}$. Then $(Ha^{-1})a\subset H$ and so there exists a $b\in
Q^{\bullet}$ such that $b(Ha^{-1})\subset H$, and thus $bH\subset Ha$.
Therefore $HbH\subset Ha\subset I$ and $HbH$ is a fractional $H$-ideal
contained in $I$ (as $b\in HbH$, $a^{-1}\in{({H}\\!\\!:_{r}\\!{HbH})}$ and, by
(g) again, there exists a $c\in Q^{\bullet}$ with $c(Hb)\subset H$, whence
$c\in{({H}\\!\\!:_{l}\\!{HbH})}$). The case where $I$ is a fractional right
$H$-ideal is similar. ∎
###### Lemma .
1. 1.
Let $H$ and $H^{\prime}$ be orders in $Q$. If $H$ is bounded and $H\sim
H^{\prime}$, then $H^{\prime}$ is also bounded.
2. 2.
Let $H$ and $H^{\prime}$ be bounded equivalent maximal orders of $Q$. Then
there exists an $(H,H^{\prime})$-ideal $I$.
###### Proof.
1. 1.
Because $H\sim H^{\prime}$ and $H$ is bounded, there exist $c,d\in
Q^{\bullet}$ with $cH\subset H^{\prime}\subset dH$ (using (c) and (d) of the
equivalent characterizations of boundedness). We verify condition (c) for
$H^{\prime}$. Let $a\in Q^{\bullet}$. Then $cHa\subset H^{\prime}a$, and there
exists an $x\in Q^{\bullet}$ with $xH\subset Ha$. Then $cxH\subset
H^{\prime}a$ and finally $cxd^{-1}H^{\prime}\subset cxH\subset H^{\prime}a$.
Similarly, one finds $z\in Q^{\bullet}$ with $H^{\prime}z\subset aH^{\prime}$.
2. 2.
We show: If $H$ and $H^{\prime}$ are bounded equivalent maximal orders, then
$H^{\prime}H$ is a fractional $(H^{\prime},H)$-ideal. Then
$(H^{\prime}H)^{-1}$ is an $(H,H^{\prime})$-ideal (since
$\mathcal{O}_{r}(H^{\prime}H)=H$ by maximality of $H$ and $H\subset
H^{\prime}H$, Lemma 5.3 implies $(H^{\prime}H)^{-1}\subset H$; similarly, one
shows $(H^{\prime}H)^{-1}\subset H^{\prime}$).
Clearly $H^{\prime}H$ is an $(H^{\prime},H)$-module and $1\in H^{\prime}H$. We
need to show that there exist $a,b\in Q^{\bullet}$ with $H^{\prime}Ha\subset
H^{\prime}$ and $bH^{\prime}H\subset H$. Since $H$ and $H^{\prime}$ are
bounded and equivalent there exist $a,b\in Q^{\bullet}$ with $Ha\subset
H^{\prime}$ and $bH^{\prime}\subset H$, and the claim follows. ∎
###### Proposition .
Let $\alpha$ be an equivalence class of maximal orders in $Q$.
$(\mathcal{F}_{v}(\alpha),\cdot_{v},\subset)$ is an arithmetical groupoid if
and only if all $H\in\alpha$ (equivalently, one $H\in\alpha$) satisfy the
following three conditions:
1. (A1)
$H$ satisfies the ACC on divisorial left $H$-ideals and the ACC on divisorial
right $H$-ideals,
2. (A2)
$H$ is bounded,
3. (A3)
the lattice of divisorial fractional left $H$-ideals is modular, and the
lattice of divisorial right $H$-ideals is modular.
###### Proof.
From Section 5 we already know that $\mathcal{F}_{v}(\alpha)$ is a lattice-
ordered groupoid. As in the discussion after Definition 4.2 and from Section
5, we see that if one representative $H\in\alpha$ satisfies A1–A3, then the
same is true for all $H^{\prime}\in\alpha$.
Assume first that A1–A3 hold. Then P1 holds due to Lemma 5.5, property P2 is
just A3 in the present setting. P3 follows easily: If $I\subset J$ are
divisorial fractional left $H$-ideals, and $K$ is a divisorial fractional
right $H$-ideal, then $KI\subset KJ$ and therefore
$K\cdot_{v}I=(KI)_{v}\subset(KJ)_{v}=K\cdot_{v}J$, and similarly for the
symmetric statement. P4 also holds: Let $(I_{m})_{m\in M}$ be a non-empty
family of divisorial left $H$-ideals. Then $(\bigcup_{m\in M}I_{m})_{v}\subset
H$ is also a divisorial left $H$-ideal, and if $(I_{m})_{m\in M}$ is a family
of $(H,H^{\prime})$-ideals with $H^{\prime}\in\alpha$, then the divisorial
closure of the union is again an $(H,H^{\prime})$-ideal. P6 is just A1. A2
implies P5: If $H,H^{\prime}\in\alpha$, then there exists an
$(H,H^{\prime})$-ideal $I$ by Lemma 5.2. Then $I_{v}$ is as required in P5.
Assume now that $(\mathcal{F}_{v}(\alpha),\cdot_{v},\subset)$ is an
arithmetical groupoid, and $H\in\alpha$. Then P2 implies A3, and P6 implies
A3. From P5 we can derive A2: Let $I$ be a fractional left $H$-ideal, and
$x\in I\cap Q^{\bullet}$. Then $Hx\in\mathcal{F}_{v}(\alpha)$. By P5, there
exists $J\in\mathcal{I}_{v}(\alpha)$ with
$\mathcal{O}_{l}(J)=\mathcal{O}_{r}(Hx)$ and $\mathcal{O}_{r}(J)=H$. Thus
$HxJ$ is a fractional $H$-ideal, and $HxJ\subset I$. We proceed similarly if
$I$ is a fractional right $H$-ideal. ∎
###### Remark .
1. 1.
From the discussion after Definition 4.2, we also see that we can equivalently
formulate A3 as “the lattice of divisorial fractional left (right) $H$-ideals
is modular”, as the property for the other side then holds automatically.
2. 2.
Let $H$ be a normalizing monoid. By definition of a monoid, $H$ satisfies the
left and right Ore condition, hence it is an order in its quotient group.
Lemma 5.2 shows that every fractional left or right $H$-ideal is in fact
already a two-sided $H$-ideal, and thus $H$ is bounded.
Assume that $H$ is a normalizing Krull monoid. Then $\alpha=\\{\,H\,\\}$, and
the lattice-ordered groupoid $\mathcal{F}_{v}(\alpha)$ is in fact a group. The
lattice of divisorial fractional $H$-ideals is then modular, even distributive
[51, Theorem 2.1.3(a)], and hence by the previous theorem an arithmetical
groupoid.
###### Definition 5.4.
We call a maximal order $H$ satisfying A1–A3 an _arithmetical maximal order_.
If $\alpha$ is its equivalence class of arithmetical maximal orders, then we
denote by $\mathcal{M}_{v}(\alpha)\subset\mathcal{I}_{v}(\alpha)$ the (quiver
of) maximal integral elements.
_Let from here on $H$ be an arithmetical maximal order in $Q$, and let
$\alpha$ be its equivalence class of arithmetical maximal orders._
By Lemma 5.2, every principal left ideal $Ha$ with $a\in H^{\bullet}$ is a
divisorial left $H$-ideal with inverse $a^{-1}H\in\mathcal{F}_{v}(\alpha)$.
Let
$\mathcal{H}(\alpha)=\\{\,H^{\prime}a\in\mathcal{I}_{v}(\alpha)\mid
H^{\prime}\in\alpha,\,a\in H^{\prime\bullet}\,\\}.$
The $v$-product coincides with the usual proper product on
$\mathcal{H}(\alpha)$. Thus
$(\mathcal{H}(\alpha),\cdot)\subset(\mathcal{I}_{v}(\alpha),\cdot_{v})$ is a
wide subcategory, with the product $IJ=I\cdot J=I\cdot_{v}J$ for
$I,J\in\mathcal{H}(\alpha)$ defined whenever
$\mathcal{O}_{r}(I)=\mathcal{O}_{l}(J)$, and then
$\mathcal{O}_{l}(IJ)=\mathcal{O}_{l}(I)$ and
$\mathcal{O}_{r}(IJ)=\mathcal{O}_{r}(J)$. The inclusion
$(\mathcal{H}(\alpha),\cdot)\subset(\mathcal{I}_{v}(\alpha),\cdot_{v})$ is
left- and right-saturated. By $\mathcal{H}_{H}(\alpha)$ (or shorter,
$\mathcal{H}_{H}$, since $H$ determines $\alpha$) we denote the subcategory of
$\mathcal{H}(\alpha)$ where the left and right orders of every element are not
only equivalent but in fact conjugate to $H$. Explicitly,
$\mathcal{H}_{H}=\mathcal{H}_{H}(\alpha)=\\{\,d(Ha)d^{-1}\in\mathcal{I}_{v}(\alpha)\mid
a\in H^{\bullet},\,d\in Q^{\bullet}\,\\}.$
If $H^{\prime}\in\alpha$, then $\mathcal{H}_{H}=\mathcal{H}_{H^{\prime}}$ if
and only if $H^{\prime}$ and $H$ are conjugate. Again the inclusion
$(\mathcal{H}_{H},\cdot)\subset(\mathcal{H}(\alpha),\cdot)$ is left- and
right-saturated, and thus so is the inclusion
$(\mathcal{H}_{H},\cdot)\subset(\mathcal{I}_{v}(\alpha),\cdot_{v})$.
The following simple lemma gives a correspondence between $H$ and
$\mathcal{H}_{H}$.
###### Lemma .
Let $d\in Q^{\bullet}$.
1. 1.
If $a,a_{1},a_{2}\in H^{\bullet}$ with $a=a_{1}a_{2}$, then
$d^{-1}(Ha)d=d^{-1}(Ha_{2})d\cdot
d^{-1}a_{2}^{-1}(Ha_{1})a_{2}d\in\mathcal{H}_{H}$ with
$d^{-1}(Ha_{2})d,\,d^{-1}a_{2}^{-1}(Ha_{1})a_{2}d\in\mathcal{H}_{H}$.
2. 2.
If $a\in H^{\bullet}$ and $d^{-1}(Ha)d=I_{2}\cdot I_{1}$ with
$I_{1},I_{2}\in\mathcal{H}_{H}$, then there exist $a_{1},a_{2}\in H^{\bullet}$
with $I_{2}=d^{-1}(Ha_{2})d$, $I_{1}=d^{-1}a_{2}^{-1}(Ha_{1})a_{2}d$ and
$a=a_{1}a_{2}$.
3. 3.
If $a_{1},a_{2},b_{1},b_{2}\in H^{\bullet}$ with $Ha_{2}=Hb_{2}$ and
$a_{2}^{-1}(Ha_{1})a_{2}=b_{2}^{-1}(Hb_{1})b_{2}$, then there exist
$\varepsilon_{1},\varepsilon_{2}\in H^{\times}$ with
$b_{1}=\varepsilon_{1}a_{1}\varepsilon_{2}^{-1}$ and
$b_{2}=\varepsilon_{2}a_{2}$.
In particular, for $a\in H^{\bullet}$ we have $a\in\mathcal{A}(H^{\bullet})$
if and only if $d^{-1}(Ha)d\in\mathcal{A}(\mathcal{H}_{H})$.
###### Proof.
1. 1.
The multiplication is defined because
$\mathcal{O}_{r}(d^{-1}(Ha_{2})d)=d^{-1}a_{2}^{-1}Ha_{2}d=\mathcal{O}_{l}(d^{-1}a_{2}^{-1}(Ha_{1})a_{2}d)$.
The remaining statements are then clear.
2. 2.
Since $\mathcal{O}_{l}(I_{2})=d^{-1}Hd$ we have $I_{2}=d^{-1}Hda_{2}^{\prime}$
with $a_{2}^{\prime}\in(d^{-1}Hd)^{\bullet}$, and hence, with
$a_{2}=da_{2}^{\prime}d^{-1}\in H^{\bullet}$, $I_{2}=d^{-1}(Ha_{2})d$. Then
$\mathcal{O}_{l}(I_{1})=\mathcal{O}_{r}(I_{2})=d^{-1}a_{2}^{-1}Ha_{2}d$, and
therefore similarly $I_{1}=d^{-1}a_{2}^{-1}(Ha_{1}^{\prime})a_{2}d$ with
$a_{1}^{\prime}\in H^{\bullet}$. Hence
$d^{-1}(Ha)d=d^{-1}(Ha_{1}^{\prime}a_{2})d$, and thus $a=\varepsilon
a_{1}^{\prime}a_{2}$ with $\varepsilon\in H^{\times}$. Taking
$a_{1}=\varepsilon a_{1}^{\prime}$ the claim follows.
3. 3.
Since $Ha_{2}=Hb_{2}$, there exists an $\varepsilon_{2}\in H^{\times}$ with
$b_{2}=\varepsilon_{2}a_{2}$. Then
$a_{2}^{-1}(Ha_{1})a_{2}=b_{2}^{-1}(Hb_{1})b_{2}=a_{2}^{-1}\varepsilon_{2}^{-1}(Hb_{1})\varepsilon_{2}a_{2}=a_{2}^{-1}(Hb_{1}\varepsilon_{2})a_{2},$
and thus there exists $\varepsilon_{1}\in H^{\times}$ with
$\varepsilon_{1}a_{1}=b_{1}\varepsilon_{2}$, i.e.,
$b_{1}=\varepsilon_{1}a_{1}\varepsilon_{2}^{-1}$. ∎
Observe that we may view a rigid factorization
$Ha_{2}*a_{2}^{-1}(Ha_{1})a_{2}\in\mathsf{Z}^{*}(\mathcal{H}_{H})$ as a
multiplicative way of writing the chain $H\supset Ha_{2}\supset Ha_{1}a_{2}$.
###### Proposition .
Let $a\in H^{\bullet}$. For every $d\in Q^{\bullet}$ there is a bijection
$\mathsf{Z}_{H}^{*}(a)\to\mathsf{Z}_{\mathcal{H}_{H}}^{*}(d^{-1}(Ha)d)$, given
by
$u_{1}*u_{2}*\ldots*u_{k}\,\mapsto\,d^{-1}(Hu_{k})d*(d^{-1}u_{k}^{-1}(Hu_{k-1})u_{k}d)*\ldots*(d^{-1}u_{k}^{-1}\cdot\ldots\cdot
u_{2}^{-1}(Hu_{1})u_{2}\cdot\ldots\cdot u_{k}d).$
If $\overline{\theta}\colon\mathcal{H}_{H}\to B$ is a transfer homomorphism to
a reduced cancellative small category $B$ and having the additional property
that $\overline{\theta}(d^{-1}(Ha)d)=\overline{\theta}(Ha)$ for all $a\in
H^{\bullet}$ and $d\in Q^{\bullet}$, then it induces a transfer homomorphism
$\theta\colon H^{\bullet}\to B^{\text{op}}$ given by
$\theta(a)=\overline{\theta}(Ha)$.
###### Proof.
The claimed bijection follows by iterating the previous lemma.
We need to verify that $\theta$ is a transfer homomorphism and first check
that $\theta$ is a homomorphism: For $a,b\in H^{\bullet}$
$\theta(ab)=\overline{\theta}(Hab)=\overline{\theta}(Hb\cdot
b^{-1}(Ha)b)=\overline{\theta}(Hb)\cdot\overline{\theta}(b^{-1}(Ha)b)=\overline{\theta}(Hb)\cdot\overline{\theta}(Ha)=\theta(a)\cdot^{\text{op}}\theta(b),$
and if $a\in H^{\times}$ then $Ha=H$, whence
$\theta(a)=\overline{\theta}(H)\in B_{0}$. We verify T1: Let $b\in B$. Then
there exist $d\in Q^{\bullet}$ and $a\in H^{\bullet}$ with
$\overline{\theta}(d^{-1}(Ha)d)=b$, hence $\theta(a)=b$. If $a\in H^{\bullet}$
with $\theta(a)\in B_{0}$, then $\overline{\theta}(Ha)\in B_{0}$, hence
$Ha\in(\mathcal{H}_{H})_{0}$, i.e., $Ha=H$ and $a\in H^{\times}$. It remains
to check T2: Let $a\in H^{\bullet}$ and $b_{1},b_{2}\in B$ with
$\theta(a)=b_{1}\cdot^{\text{op}}b_{2}$. Then
$\overline{\theta}(Ha)=b_{2}b_{1}$, hence there exist $a_{1},a_{2}\in
H^{\bullet}$ with $Ha=Ha_{2}\cdot a_{2}^{-1}(Ha_{1})a_{2}$ and
$\overline{\theta}(Ha_{2})=b_{2}$,
$\overline{\theta}(a_{2}^{-1}(Ha_{1})a_{2})=b_{1}$. This implies
$a=\varepsilon a_{1}a_{2}$ with $\varepsilon\in H^{\times}$, and
$\theta(\varepsilon a_{1})=b_{1}$, $\theta(a_{2})=b_{2}$. ∎
###### Remark .
The condition $\overline{\theta}(d^{-1}(Ha)d)=\overline{\theta}(Ha)$ implies
in particular $\lvert\overline{\theta}(\mathcal{H}_{H})_{0}\rvert=1$. Thus in
fact $B$ is necessarily a semigroup.
Let $\mathbb{G}$ be the universal vertex group of $\mathcal{F}_{v}(\alpha)$,
and let $\eta\colon\mathcal{F}_{v}(\alpha)\to\mathbb{G}$ be the abstract norm,
as defined in the previous section.
###### Lemma .
1. 1.
If $I\in\mathcal{F}_{v}(\alpha)$ and $d\in Q^{\bullet}$, then
$\eta(d^{-1}Id)=\eta(I)$.
2. 2.
$\mathbf{q}(\eta(\mathcal{H}_{H}))=\mathbf{q}(\\{\,\eta(Ha)\mid a\in
H^{\bullet}\,\\})=\\{\,\eta(Hq)\mid q\in Q^{\bullet}\,\\}$.
###### Proof.
1. 1.
It suffices to verify the claim for maximal integral
$I\in\mathcal{I}_{v}(\alpha)$. If $P\in\mathcal{I}_{v}(\alpha)$ is the maximal
divisorial two-sided $\mathcal{O}_{l}(I)$-ideal contained in $I$, then
$d^{-1}Pd$ is the maximal divisorial two-sided ideal contained in $d^{-1}Id$,
and since
$d^{-1}Pd=(\mathcal{O}_{l}(d^{-1}Id)d^{-1}\mathcal{O}_{l}(I))\cdot_{v}P\cdot_{v}(\mathcal{O}_{l}(I)d\mathcal{O}_{l}(d^{-1}Id))$
we have $\eta(I)=(P)=(d^{-1}Pd)=\eta(d^{-1}Id)\in\mathbb{G}$.
2. 2.
The first equality is immediate from 1. For the second equality, note that if
$q=ab^{-1}$ with $a,b\in H^{\bullet}$, then (using 1. multiple times and the
fact that $\eta$ is a homomorphism)
$\begin{split}\eta(Hq)&=\eta(Hab^{-1})=\eta(Hb^{-1}\cdot
b(Ha)b^{-1})=\eta(Hb^{-1})\eta(b(Ha)b^{-1})=\eta(bH)^{-1}\eta(Ha)\\\
&=\eta(b^{-1}(bH)b)^{-1}\eta(Ha)=\eta(Ha)\eta(Hb)^{-1}.\qed\end{split}$
Applying Section 4 to the present situation, we obtain a transfer homomorphism
$\mathcal{H}_{H}\to\mathcal{B}(C_{M})$ if we impose some additional crucial
conditions on $H$.
###### Theorem .
Let $Q$ be a quotient semigroup, $H$ an arithmetical maximal order in $Q$, and
$\alpha$ its equivalence class of arithmetical maximal orders.
1. 1.
For all $a\in H^{\bullet}$, $\mathsf{L}_{H^{\bullet}}(a)$ is finite and non-
empty. If, for every maximal divisorial $H$-ideal $P$, the number of maximal
divisorial left $H$-ideals $I$ with $P\subset I$ is finite, then
$\mathsf{Z}^{*}_{H^{\bullet}}(a)$ is finite for all $a\in H^{\bullet}$.
2. 2.
Let $P_{H^{\bullet}}=\\{\,\eta(Hq)\mid q\in
Q^{\bullet}\,\\}\subset\mathbb{G}$, $C=\mathbb{G}/P_{H^{\bullet}}$, and
$C_{M}=\\{\,[\eta(I)]\in C\mid I\in\mathcal{I}_{v}(\alpha)\text{ maximal
integral}\,\\}$. Assume:
1. (i)
A divisorial fractional left $H$-ideal $I$ is principal if and only if
$\eta(I)\in P_{H^{\bullet}}$.
2. (ii)
For all $H^{\prime}\in\alpha$ and all $g\in C_{M}$ there exists a maximal
divisorial left $H^{\prime}$-ideal with $[\eta(I)]=g$.
Then there exists a transfer homomorphism $\theta\colon
H^{\bullet}\to\mathcal{B}(C_{M})$.
###### Proof.
By Section 5, $(\mathcal{F}_{v}(\alpha),\cdot_{v},\subset)$ is an arithmetical
groupoid, and $\mathcal{I}_{v}(\alpha)$ is its subcategory of integral
elements. $(\mathcal{H}_{H},\cdot)$ is a left- and right-saturated subcategory
of $(\mathcal{I}_{v},\cdot_{v})$.
1. 1.
This follows immediately from Section 4 and Section 5.
2. 2.
Let $I$ be a fractional left $H^{\prime}$-ideal with $H^{\prime}=dHd^{-1}$.
Then $I$ is principal if and only if the fractional left $H$-ideal $d^{-1}Id$
is, and this is the case if and only if $\eta(I)=\eta(d^{-1}Id)\in
P_{H^{\bullet}}=\mathbf{q}(\eta(\mathcal{H}_{H}))$ (where the last equality is
due to the previous lemma). Therefore the first condition of Section 4 is
satisfied. Condition (ii) of the present theorem is equivalent to the second
condition of Section 4. Thus there exists a transfer homomorphism
$\overline{\theta}\colon\mathcal{H}_{H}\to\mathcal{B}(C_{M})$ as in Section 4.
By Section 5, there exists a transfer homomorphism $\theta\colon
H^{\bullet}\to\mathcal{B}(C_{M})$. ∎
###### Remark .
1. 1.
We continue our discussion from Section 5. Let $H$ be a normalizing Krull
monoid. Then $\alpha=\\{\,H\,\\}$, $Ha=HaH=aH$ for all $a\in Q^{\bullet}$ and
associativity is a congruence relation [25, Lemma 4.4.1], thus
$H_{\text{red}}=\\{\,H^{\times}a\mid a\in H\,\\}$ with the induced operation
is also a monoid. Therefore $\mathcal{H}=\mathcal{H}_{H}=\\{\,HaH\mid a\in
H\,\\}\cong H_{\text{red}}$ and $G=\mathcal{F}_{v}(\alpha)$ is the free
abelian group on the maximal divisorial (two-sided) $H$-ideals, while
$\mathcal{I}_{v}(\alpha)$ is the free abelian monoid on the same basis.
In the previous theorem we therefore have $\mathbb{G}=G$,
$\eta=\operatorname{id}$, $P_{H^{\bullet}}=\\{\,Hq\mid q\in Q^{\bullet}\,\\}$,
and hence $C$ is the divisorial class group of $H$, and $C_{M}$ is the set of
divisorial ideal classes that contain a maximal divisorial $H$-ideal. The
second condition of the theorem is trivially true by virtue of $\lvert
G_{0}\rvert=1$ and the definition of $C_{M}$, and the first condition is
trivially true because $\eta=\operatorname{id}$. We thus get a transfer
homomorphism $H\to\mathcal{B}(C_{M})$ (induced from the transfer homomorphism
$H_{\text{red}}\cong\mathcal{H}_{H}\to\mathcal{B}(C_{M})$), which is the same
one as in [25, Theorem 6.5].
2. 2.
If $H$ is a maximal order satisfying only A1 and A3, then
$\mathsf{L}_{H^{\bullet}}(a)$ is finite and non-empty for all $a\in
H^{\bullet}$. In Section 4 one may drop P5 and P6, and still obtain
Proposition 4.1 in the weaker form that, for each $a\in G_{+}$, either
$\mathsf{Z}^{*}_{G_{+}}(a)=\emptyset$ or $\lvert\mathsf{L}_{G_{+}}(a)\rvert=1$
(and of course without any statement about $\Phi$, which can only be defined
in the presence of P5). This is possible because P6 is only used to show
existence of a rigid factorization of $a$. A sufficient condition for
$\mathsf{Z}^{*}_{G_{+}}(a)\neq\emptyset$ is that $G_{+}(s(a),\cdot)$ and
$G_{+}(\cdot,t(a))$ satisfy the ACC. If $H$ satisfies A1, then
$G_{+}(e,\cdot)$ and $G_{+}(\cdot,e)$ with $e\in(\mathcal{H}_{H})_{0}$
(corresponding to conjugate orders of $H$) satisfy the ACC, and as in Section
4 one shows that $\mathsf{L}_{\mathcal{H}_{H}}(a)$ is finite and non-empty for
all $a\in\mathcal{H}_{H}$. Hence the same is true for $H^{\bullet}$.
### 5.1. Rings
Suppose that $Q=(Q,+,\cdot)$ is a quotient ring in the sense of [42, Chapter
3] (but recall that we in addition require it to be unital, as we do for all
rings). Then $(Q,\cdot)$ is a quotient semigroup. In the remainder of this
section we show that the ring-theoretic divisorial one-sided ideal theory for
maximal orders in $(Q,+,\cdot)$ coincides with the semigroup-theoretic one.
222In [42, Chapter 5] the terminology “reflexive” is used in place of
“divisorial”. If $R$ is a ring-theoretic order in $Q$, then a fractional left
$R$-ideal $I$ in the semigroup-theoretic sense is a fractional left $R$-ideal
in the ring-theoretic sense if and only if $I-I\subset I$ (see [42, §3.1.11]
for the usual definition).
_Let for the remainder of this subsection $Q=(Q,+,\cdot)$ be a quotient ring._
###### Lemma .
Let $H$ be an order in the multiplicative semigroup $(Q,\cdot)$ and $I$ a
fractional left $H$-ideal (in the semigroup-theoretic sense). Consider the
following statements:
1. (a)
$I-I\subset I$.
2. (b)
$\mathcal{O}_{l}(I)$ is a subring of $Q$.
3. (c)
$\mathcal{O}_{r}(I)$ is a subring of $Q$.
Then (a) $\Rightarrow$ (b) and (a) $\Rightarrow$ (c). If $H$ is a maximal
order and $I$ is divisorial, then (a) $\Leftrightarrow$ (b) $\Leftrightarrow$
(c).
###### Proof.
Assume that (a) holds. We show (b): Let $a,b\in\mathcal{O}_{l}(I)$. Then
$aI\subset I$ and $bI\subset I$ and hence $(a-b)I\subset aI-bI\subset
I-I\subset I$, thus $a-b\in\mathcal{O}_{l}(I)$.
Assume now that $H$ is maximal, $I=I_{v}$ and (b) holds. We show (a). Let
$a,b\in I=I_{v}=(I^{-1})^{-1}$. Then $aI^{-1}\subset\mathcal{O}_{l}(I)$, and
$bI^{-1}\subset\mathcal{O}_{l}(I)$, whence $(a-b)I^{-1}\subset
aI^{-1}-bI^{-1}\subset\mathcal{O}_{l}(I)-\mathcal{O}_{l}(I)=\mathcal{O}_{l}(I)$
and thus $a-b\in(I^{-1})^{-1}=I$. ∎
###### Lemma .
A ring-theoretic order $R$ in $Q$ is maximal in the ring-theoretic sense if
and only if it is maximal in the semigroup-theoretic sense.
###### Proof.
We show that if $R$ is maximal in the ring-theoretic sense, then it is maximal
in the semigroup-theoretic sense, as the other direction is trivial. Let $I$
be a fractional left $R$-ideal in the semigroup-theoretic sense. Then
${}_{R}\langle I\rangle$ is a fractional left $R$-ideal in the ring-theoretic
sense, and using $R\subset\mathcal{O}_{l}(I)$, it follows that
$\mathcal{O}_{l}(I)\subset\mathcal{O}_{l}({}_{R}\langle I\rangle)$. Maximality
of $R$ in the ring-theoretic sense implies $R=\mathcal{O}_{l}({}_{R}\langle
I\rangle)$, hence also $R=\mathcal{O}_{l}(I)$. Similarly, if $J$ is a
fractional right $R$-ideal in the ring-theoretic sense then
$\mathcal{O}_{r}(J)=R$. Therefore Section 5 implies that $R$ is maximal in the
semigroup-theoretic sense. ∎
As before let $\alpha$ be an equivalence class of maximal orders of
$(Q,\cdot)$ in the semigroup-theoretic sense.
###### Lemma .
Let $H\in\alpha$ and assume that $H$ is a subring of $Q$ (i.e., an order in
$Q$ in the ring-theoretic sense).
1. 1.
Every $H^{\prime}\in\alpha$ is a subring of $Q$ (and therefore an order in $Q$
in the ring-theoretic sense).
2. 2.
If $I$ is a divisorial fractional left $H$-ideal and $J$ is a divisorial
fractional left $\mathcal{O}_{r}(I)$-ideal, then
$\\{\,ab\mid a\in I,b\in J\,\\}_{v}=\big{(}{}_{H}\langle\\{\,ab\mid a\in
I,b\in J\,\\}\rangle\big{)}_{v},$
i.e., the semigroup-theoretic $v$-product coincides with the ring-theoretic
one.
3. 3.
If $I$ and $J$ are divisorial fractional left $H$-ideals, then $(I\cup
J)_{v}=(I+J)_{v}$.
###### Proof.
1. 1.
By Lemma 5.3 there exists a fractional $(H,H^{\prime})$-ideal $I$. By
maximality of $H$ and $H^{\prime}$, also $\mathcal{O}_{l}(I_{v})=H$ and
$\mathcal{O}_{r}(I_{v})=H^{\prime}$ and the claim follows from Section 5.1
applied to $I_{v}$.
2. 2.
Write $I\cdot_{S}J=\\{\,ab\mid a\in I,b\in J\,\\}$ for the semigroup-theoretic
ideal product and $I\cdot_{R}J={}_{H}\langle\\{\,ab\mid a\in I,b\in
J\,\\}\rangle$ for the ring-theoretic one. Then $I\cdot_{S}J\subset
I\cdot_{R}J$, and both of these sets are fractional left $H$-ideals (in the
semigroup-theoretic sense). Therefore
$(I\cdot_{S}J)_{v}\subset(I\cdot_{R}J)_{v}$. For the converse inclusion, it
suffices to show $I\cdot_{R}J\subset(I\cdot_{S}J)_{v}$, but this is true
because by Section 5.1 $(I\cdot_{S}J)_{v}$ is additively closed.
3. 3.
Clearly $I\cup J\subset I+J$ and both sets are fractional left $H$-ideals (for
$I+J$ proceed as in the proof of Lemma 5.4; in particular observe
${({H}\\!\\!:_{r}\\!{I\cup J})}\subset{({H}\\!\\!:_{r}\\!{I+J})}$). As before
it therefore suffices to show $I+J\subset(I\cup J)_{v}$. This again holds due
to Section 5.1. ∎
Altogether, if $R$ is a maximal order in $Q$ in the ring-theoretic sense, then
it does not matter whether we form $\mathcal{F}_{v}(\alpha)$ by using the
ring-theoretic or the semigroup-theoretic notions. We use the same notion of
boundedness for ring-theoretic orders as in Section 5; for semiprime Goldie
rings this coincides with the notion in [42].
###### Theorem .
Let $R$ be a maximal order in a quotient ring $Q$, $\alpha$ its equivalence
class of maximal orders in the semigroup-theoretic sense, and $\beta$ its
equivalence class of maximal orders in the ring-theoretic sense. Then
$\alpha=\beta$ and $\mathcal{F}_{v}(\alpha)=\mathcal{F}_{v}(\beta)$, where the
latter is the ring-theoretic analogue of $\mathcal{F}_{v}(\alpha)$.
If $R$ is bounded, satisfies the ACC on divisorial left $R$-ideals and on
divisorial right $R$-ideals, and the lattice of divisorial fractional left
(right) $R$-modules is modular, then $(R,\cdot)$ is an arithmetical maximal
order in $(Q,\cdot)$ in the semigroup-theoretic sense. In particular, the
conclusions of Section 5 hold for $R$.
###### Proof.
By Lemma 5.1.1, $\alpha=\beta$, and by Section 5.1,
$\mathcal{F}_{v}(\alpha)=\mathcal{F}_{v}(\beta)$ as sets. By Section 5.1, the
$v$-product, meet and join coincide, and hence
$\mathcal{F}_{v}(\alpha)=\mathcal{F}_{v}(\beta)$ as lattice-ordered groupoids.
The remaining claims follow from this. ∎
In [46, §5(d)], Rehm gives examples for bounded maximal orders $E$, that are
prime and satisfy the ACC on divisorial two-sided $E$-ideals, but do not
satisfy the ACC on divisorial left $E$-ideals or the ACC on divisorial right
$E$-ideals. In fact (unless one takes the special case where $E$ itself is a
quotient ring), the orders $E$ are not even atomic. However, these orders are
not Goldie, as they are not of finite left or right uniform dimension, and do
not satisfy the ACC on left or right annihilator ideals.
Before going to maximal orders in central simple algebras, we discuss
principal ideal rings.
###### Example (Principal ideal rings).
Let $R$ be a bounded order in a quotient ring $Q$. Assume that every left
$R$-ideal and every right $R$-ideal is principal. By the characterization in
Section 5, $R$ is then already a maximal order, and it satisfies A1–A3. Thus
$\mathcal{H}_{R}=\mathcal{I}_{v}(\alpha)$, and facts about the rigid
factorizations in $\mathcal{I}_{v}(\alpha)$ trivially descend to facts about
rigid factorizations of $R^{\bullet}$. Examples we have in mind include
bounded skew polynomial rings $D[X,\sigma]$, where $D$ is a division ring and
$\sigma\colon D\to D$ is an automorphism, and the Hurwitz quaternions
$\mathbb{Z}[1,i,j,\frac{1+i+j+ij}{2}]$ with $i^{2}=-1$, $j^{2}=-1$ and
$ij=-ji$. Both of these examples are left- and right-euclidean domains, and
hence principal ideal rings. In this way we can for example rediscover Theorem
2 in [16, §5].
Let $Q$ be a quaternion algebra over a field $K$ with
$\operatorname{char}(K)\neq 2$, and $a\in Q^{\bullet}\setminus K^{\times}$.
Then $\operatorname{nr}(a)=a\overline{a}\in K^{\times}$ and
$\operatorname{tr}(a)=a+\overline{a}\in K$. For the polynomial ring $Q[X]$ in
the central variable $X$, therefore
$f=X^{2}-\operatorname{tr}(a)X+\operatorname{nr}(a)=(X-cac^{-1})(X-c\overline{a}c^{-1})\quad\text{for
all $c\in Q^{\bullet}$},$
and thus $\lvert\mathsf{Z}^{*}_{Q[X]}(f)\rvert=\infty$ if $K$ is infinite.
(But these rigid factorizations are usually considered to be identical
factorizations, and $Q[X]$, being left- and right-euclidean, is even a UFD
with suitable definitions, see for example [8, Chapter 3.2] and [14, Chapter
3].) In terms of ideal theory, every element $X-cac^{-1}$ with $c\in
Q^{\bullet}$ generates a maximal left $Q[X]$-ideal lying above the maximal
two-sided $Q[X]$-ideal $Q[X]f$. If also $d\in Q^{\bullet}$, then
$Q[X](X-cac^{-1})=Q[X](X-dad^{-1})$ if and only if $cac^{-1}=dad^{-1}$, i.e.,
$d^{-1}c\in K(a)$.
### 5.2. Classical maximal orders over Dedekind domains in CSAs
Let $\mathcal{O}$ be a commutative domain with quotient field $K$. By a
central simple algebra $A$ over $K$, we mean a $K$-algebra with
$\dim_{K}(A)<\infty$, which is simple as a ring, and has center $K$. Then $A$
is artinian because it is a finite-dimensional $K$-algebra, and hence it is a
quotient ring (in an artinian ring, every non-zero-divisor is invertible [42,
§3.1.1], hence it is a quotient ring and an element is left-cancellative if
and only if it is right-cancellative if and only if it is cancellative). By
Posner’s Theorem ([42, §13.6.6]), a ring $R$ is a prime PI ring if and only if
it is an order in a central simple algebra, and hence in particular, prime PI
rings are bounded Goldie rings. Furthermore, PI Krull rings are characterized
as those maximal orders in central simple algebras whose center is a
commutative Krull domain ([35, Theorem 2.4]). We start with a simple corollary
of Section 5.1.
###### Corollary .
If $R$ is a PI Krull ring, then $\mathsf{L}_{R^{\bullet}}(a)$ is finite and
non-empty for all $a\in R^{\bullet}$.
###### Proof.
We only have to verify the conditions of Section 5.1. By [35, Theorem 2.4] the
various notions of Krull rings coincide for prime PI rings. Thus $R$ is a
bounded Chamarie-Krull ring. The ACC on divisorial left $R$-ideals and
divisorial right $R$-ideals follows from [11] (or [31, Corollary 3.11]).
Moreover, for every divisorial prime $R$-ideal $P$, the set of regular
elements modulo $P$, denoted $\mathcal{C}(P)$, is cancellative, satisfies the
left and right Ore condition, and for the localization
$R_{\mathcal{C}(P)}={}_{\mathcal{C}(P)}R\subset Q$ every left (right)
$R_{\mathcal{C}(P)}$-ideal is principal ([11, Proposition 2.5]). The lattice
of divisorial fractional left (right) $R_{\mathcal{C}(P)}$-ideals is hence
modular. Using the ACC on divisorial left and right $R$-ideals, one checks as
in the commutative case that
$I_{v}R_{\mathcal{C}(P)}=(IR_{\mathcal{C}(P)})_{v}$ for a fractional right
$R$-ideal $I$.
Suppose now $I,J,K$ are divisorial fractional right $R$-ideals, and $K\subset
I$. We have to check $I\cap(J+K)_{v}=((I\cap J)+K)_{v}$. But
$(I\cap(J+K)_{v})R_{\mathcal{C}(P)}=IR_{\mathcal{C}(P)}\cap(JR_{\mathcal{C}(P)}+KR_{\mathcal{C}(P)})_{v}$
and $((I\cap J)+K)_{v}R_{\mathcal{C}(P)}=((IR_{\mathcal{C}(P)}\cap
JR_{\mathcal{C}(P)})+KR_{\mathcal{C}(P)})_{v}$, and thus, by modularity in the
localizations, they are equal for every divisorial prime $R$-ideal $P$. The
claim now follows from [11, Lemme 2.7], by which the global divisorial
fractional right $R$-ideals can be recovered as intersections from the local
ones. ∎
Using Remark 5.2, we get the above result even for more general classes of
rings, namely for Dedekind prime rings and bounded Chamarie-Krull rings (cf.
[11]).
But the aim of this subsection is to restrict to the situation where the base
ring $\mathcal{O}$ is a Dedekind domain, as a preparation for the structural
results on sets of lengths in the setting of holomorphy rings. Suppose that
$\mathcal{O}$ is a Dedekind domain. A ring $R$ is a _classical
$\mathcal{O}$-order_ of $A$ if $\mathcal{O}\subset R$, $R$ is finitely
generated as $\mathcal{O}$-module and $KR=A$. $R$ is a _classical maximal
$\mathcal{O}$-order_ if it is maximal with respect to set inclusion within the
set of all classical $\mathcal{O}$-orders. Such classical maximal
$\mathcal{O}$-orders as well as their ideal theory are well-studied, in
particular Reiner’s book [47] provides a thorough description of them. If $R$
is a classical $\mathcal{O}$-order, then it is a ring-theoretic order in $A$
in the sense we discussed, and it is a maximal order if and only if it is a
classical maximal $\mathcal{O}$-order (for this see [42, §5.3]). The set of
all classical maximal $\mathcal{O}$-orders forms an equivalence class of
(ring-theoretic) maximal orders, call it $\beta$ for a moment. If we write
$\alpha$ for the same semigroup-theoretic equivalence class of maximal orders
(i.e., $\alpha=\beta$ as sets, but we view the elements of $\beta$ as rings
and those of $\alpha$ just as semigroups), then
$\mathcal{F}_{v}(\alpha)=\mathcal{F}_{v}(\beta)$ by Section 5.1. Next, we
recall that our notion of ideals coincides with that of [47] and [53] in the
case of maximal orders, thereby seeing how the one-sided ideal theory of
classical maximal $\mathcal{O}$-orders is a special case of the semigroup-
theoretic divisorial one-sided ideal theory developed in this section. We also
recognize the abstract norm homomorphism $\eta$ of Section 4 as a
generalization of the reduced norm of ideals (in the sense of [47, §24]).
###### Lemma .
Let $I\subset A$ and let $T$ be a classical $\mathcal{O}$-order in $A$. The
following are equivalent:
1. (a)
$I$ is a fractional left $T$-ideal in the ring-theoretic sense (i.e., as in
[42, §3.1.11]).
2. (b)
$I$ is a finitely generated $\mathcal{O}$-module with $KI=A$ and $TI\subset I$
. 333Here $KI=\\{\,\lambda a\mid\lambda\in K,a\in
I\,\\}=\\{\,\sum_{i=1}^{n}\lambda_{i}a_{i}\mid\text{$\lambda_{i}\in K,a_{i}\in
I$}\,\\}=K\otimes_{\mathcal{O}}I$.
If $T$ is maximal, then in addition the following statements are equivalent to
the previous ones:
1. (c)
$I$ is a divisorial fractional left $T$-ideal in the semigroup-theoretic sense
(Definitions 5.2 and 5.3).
2. (d)
$I$ is a divisorial fractional left $T$-ideal in the ring-theoretic sense
(i.e., a reflexive fractional left $T$-ideal as in [42, §5.1]).
###### Proof.
(a) $\Rightarrow$ (b): Recall that $I$ is a fractional left $T$-ideal in the
ring-theoretic sense if $TI\subset I$, $I+I\subset I$ and there exist $x,y\in
A^{\times}$ with $x\in I$ and $Iy\subset T$. $\mathcal{O}$ is the center of
$T$, and $T$ is finitely generated over the noetherian ring $\mathcal{O}$.
Since $Iy\subset T$, therefore also $I$ is a finitely generated
$\mathcal{O}$-module. Writing $x^{-1}=rc^{-1}$ with $r\in T^{\bullet}$ and
$c\in\mathcal{O}^{\bullet}$ we see that $c=rx\in I\cap\mathcal{O}^{\bullet}$.
If $a\in A$ is arbitrary, then $a=r^{\prime}d^{-1}$ with $r^{\prime}\in T$,
$d\in\mathcal{O}^{\bullet}$ and therefore $a=(r^{\prime}c)(c^{-1}d^{-1})\in
KI$.
(b) $\Rightarrow$ (a): Certainly $TI\subset I$ and $I+I\subset I$. We have to
find $x,y\in A^{\times}$ with $x\in I$ and $Iy\subset T$. Since $KI=A$, there
exist $\lambda\in K^{\times}$ and $x\in I\cap A^{\times}$ with $1=\lambda x$
(in fact even $x\in K^{\times}$). If $I={}_{\mathcal{O}}\langle
y_{1},\ldots,y_{l}\rangle$ with $y_{1},\ldots,y_{l}\in I$, then due to $KT=A$
there exists a common denominator $y\in\mathcal{O}^{\bullet}$ with $y_{i}y\in
T$, hence $Iy\subset T$.
Let now $T$ be maximal. (d) $\Rightarrow$ (a) is trivial, and (a)
$\Rightarrow$ (d) follows because $T$ is a Dedekind prime ring, and hence
every fractional left $T$-ideal (in the ring-theoretic sense) is invertible
(see [42, §5.2.14] or [47, §22,§23] for the more specific case where $R$ is a
maximal order in a CSA), and therefore divisorial.
(c) $\Leftrightarrow$ (d) follows from Section 5.1. ∎
A subset $I\subset A$ satisfying the second condition of the previous lemma
and additionally $\mathcal{O}_{l}(I)=T$ is considered to be a left $T$-ideal
in [47] and [53]. Thus, a left $T$-ideal in the sense of [47, 53] is (in our
terms) a fractional left $T$-ideal in the ring-theoretic sense with
$\mathcal{O}_{l}(I)=T$. If $T$ is maximal, then the extra condition
$\mathcal{O}_{l}(I)=T$ is trivially satisfied, and the definitions are
equivalent, but for a non-maximal order the definitions do not entirely agree
(we will only need to work with ideals of maximal orders).
Since all $I\in\mathcal{F}_{v}(\alpha)$ are invertible (i.e.,
$II^{-1}=\mathcal{O}_{l}(I)$ and $I^{-1}I=\mathcal{O}_{r}(I)$ for the ring-
theoretic products), the $v$-product coincides with the usual proper product
of ideals: $I\cdot_{v}J=I\cdot J$ whenever $I,J\in\mathcal{F}_{v}(\alpha)$
with $\mathcal{O}_{r}(I)=\mathcal{O}_{l}(J)$. Therefore,
$\mathcal{F}_{v}(\alpha)$ is the groupoid of all normal ideals of $A$ in
Reiner’s terminology ($\mathcal{O}$ is fixed implicitly).
To be able to apply our abstract results we still have to check that A1
through A3 are true for $\alpha$: A1 follows because every $R\in\alpha$ is
noetherian, while A2 is true because every fractional left $R$-ideal with
$R\in\alpha$ in fact even contains a non-zero element of the center (cf. [42,
Prop. 5.3.8(i) and (ii)] or see “(b) $\Rightarrow$ (a)” of the last proof).
Since every fractional left (right) $R$-ideal is divisorial, A3 follows from
the modularity of the lattice of left (right) $R$-modules.
Writing $\mathcal{F}^{\times}(\mathcal{O})$ for the non-zero fractional ideals
of the commutative Dedekind domain $\mathcal{O}$, and $\mathbb{G}$ for the
universal vertex group of $\mathcal{F}_{v}(\alpha)$, we have the following.
###### Lemma .
If $R,R^{\prime}\in\alpha$ and $\mathcal{P}\in\mathbb{G}$, then
$\mathcal{P}_{R}\cap\mathcal{O}=\mathcal{P}_{R^{\prime}}\cap\mathcal{O}\in\max(\mathcal{O})$
and there is a canonical bijection
$\\{\,\mathcal{P}\mid\mathcal{P}\in\mathbb{G}\text{ maximal integral
\\}}\to\max(\mathcal{O}),$
inducing an isomorphism of free abelian groups
$r\colon\mathbb{G}\overset{\sim}{\rightarrow}\mathcal{F}^{\times}(\mathcal{O})$.
The inverse map is given by $\mathfrak{p}\mapsto(\mathfrak{P})$ where
$\mathfrak{P}$ is the unique maximal (two-sided) $R$-ideal lying over
$\mathfrak{p}$. If $R$ is unramified at $\mathfrak{p}$, then
$\mathfrak{P}=R\mathfrak{p}$.
If all residue fields of $\mathcal{O}$ are finite, and
$\eta\colon\mathcal{F}_{v}(\alpha)\to\mathbb{G}$ is the abstract norm
homomorphism, then $r\circ\eta=\operatorname{nr}_{A/K}$.
###### Proof.
All but the last statement follow from [47, Theorem 22.4]. Since $r\circ\eta$
and $\operatorname{nr}_{A/K}$ are both homomorphisms
$\mathcal{F}_{v}(\alpha)\to\mathcal{F}^{\times}(\mathcal{O})$, it suffices to
verify equality for $M$ a maximal integral left $R^{\prime}$-ideal with
$R^{\prime}\in\alpha$, where it holds due to [47, Theorem 24.13]. ∎
## 6\. Proof of Section 1
_Throughout this section, let $K$ be a global field and $\mathcal{O}$ be a
holomorphy ring in $K$. 444For us, $\mathcal{O}$ is a holomorphy ring if it is
an intersection of all but finitely many of the valuation domains associated
to valuations of $K$. Furthermore, let $A$ be a central simple algebra over
$K$, and $R$ a classical maximal $\mathcal{O}$-order. _
Setting $P_{A}=\\{\,aO\mid a\in K^{\times},\text{ $a_{v}>0$ for all
archimedean places $v$ of $K$ where $A$ is ramified}\,\\}$, and denoting by
$\operatorname{\mathcal{C}}_{A}(\mathcal{O})=\mathcal{F}^{\times}(\mathcal{O})/P_{A}$
the corresponding ray class group, we have the following.
###### Lemma .
Let $r$ be as in Section 5.2. Then $r$ induces an isomorphism
$\mathbb{G}/P_{R^{\bullet}}\,\cong\,\operatorname{\mathcal{C}}_{A}(\mathcal{O}),$
where $P_{R^{\bullet}}=\\{\,\eta(Rx)\mid x\in
A^{\times}\,\\}\subset\mathbb{G}$.
###### Proof.
By Section 5.2, $r\circ\eta=\operatorname{nr}_{A/K}$. The isomorphism follows
because $\operatorname{nr}(Rx)=\mathcal{O}\operatorname{nr}(x)$ for all $x\in
A^{\times}$, and $\operatorname{nr}(A^{\times})=\\{\,a\in K^{\times}\mid
a_{v}>0\text{ for all archimedean places $v$ of $K$ where $A$ is ramified
}\,\\}$ by the Hasse-Schilling-Mass theorem on norms ([47, Theorem 33.15]). ∎
###### Lemma .
For all classical maximal $\mathcal{O}$-orders $R^{\prime}$, and all
$g\in\operatorname{\mathcal{C}}_{A}(\mathcal{O})$, there exist infinitely many
maximal left $R^{\prime}$-ideals $I$ with $[\operatorname{nr}(I)]=g$.
###### Proof.
Let $g\in\operatorname{\mathcal{C}}_{A}(\mathcal{O})$. Then there exist
infinitely many distinct maximal ideals $\mathfrak{p}$ of $\mathcal{O}$ with
$[\mathfrak{p}]=g$: The number field case for $\mathcal{O}=\mathcal{O}_{K}$,
the ring of algebraic integers, can be found in [27, Corollary 2.11.16] or
[43, Corollary 7 to Proposition 7.9]. The general case then follows because
$\mathcal{O}$ is obtained from $\mathcal{O}_{K}$ by localizing at finitely
many maximal ideals, hence the induced epimorphism
$\operatorname{\mathcal{C}}_{A}(\mathcal{O}_{K})\to\operatorname{\mathcal{C}}_{A}(\mathcal{O})$
yields the statement. For the function field case see [27, Proposition 8.9.7].
For each $\mathfrak{p}\in\max(\mathcal{O})$ with $[\mathfrak{p}]=g$ and every
maximal left $R^{\prime}$-ideal $M$ with $\mathfrak{p}\subset M$, we have
$[\operatorname{nr}(M)]=[\mathfrak{p}]=g$ ([47, Theorem 24.13], or use Section
5.2). ∎
In the following equivalent characterizations of the first condition of
Section 1, “left” may be replaced by “right” in each statement; this follows
easily from the first statement. We write $\operatorname{\mathcal{LC}}(R)$ for
the finite set of isomorphism classes of fractional left $R$-ideals, i.e.,
$[I]=[J]$ in $\operatorname{\mathcal{LC}}(R)$ if and only if $J=Ix$ with $x\in
A^{\times}$. The reduced norm induces a surjective map of finite sets
$\mu_{R}\colon\operatorname{\mathcal{LC}}(R)\to\operatorname{\mathcal{C}}_{A}(\mathcal{O})$,
given by $[I]\mapsto[\operatorname{nr}(I)]$.
###### Lemma .
The following are equivalent.
1. (a)
A fractional left $R^{\prime}$-ideal with $R^{\prime}$ conjugate to $R$ is
principal if and only if $\operatorname{nr}(I)\in P_{A}$.
2. (b)
A fractional left $R$-ideal is principal if and only if
$\operatorname{nr}(I)\in P_{A}$.
3. (c)
Every fractional left $R$-ideal $I$ with $[\operatorname{nr}(I)]=\mathbf{0}$
is principal.
4. (d)
For the map of finite sets
$\mu_{R}\colon\operatorname{\mathcal{LC}}(R)\to\operatorname{\mathcal{C}}_{A}(\mathcal{O})$
it holds that $\lvert\mu_{R}^{-1}(\mathbf{0})\rvert=1$.
5. (e)
Every stably free left $R$-ideal is free.
6. (f)
Every finitely generated projective $R$-module that is stably free is free.
###### Proof.
The equivalence of (a), (b), (c) and (d) is trivial. The remaining
equivalences follow from standard literature: (f) $\Rightarrow$ (e) is true
because $R$ is hereditary noetherian.
(e) $\Rightarrow$ (f): Let $M\neq\mathbf{0}$ be a stably free finitely
generated projective $R$-module. Then $M\cong R^{n}\oplus I$ for some left
$R$-ideal $I$ and $n\in\mathbb{N}_{0}$ ([47, Theorem 27.8] or [42, §5.7.8]).
$I$ is stably free and hence free by (e), but then so is $M$.
To see (d) $\Leftrightarrow$ (e) it suffices to recall that
$\operatorname{\mathcal{C}}_{A}(\mathcal{O})$ is isomorphic to the projective
class group $\operatorname{\mathcal{C}}(R)$ (see e.g. [52, Corollary 9.5]) and
that $\operatorname{\mathcal{LC}}(R)$ is just the set of isomorphism classes
of locally free $R$-modules of rank one, i.e., the map $\mu_{R}$ corresponds
to $\mathsf{LF}_{1}\to\operatorname{\mathcal{C}}(R),[I]\mapsto[I]-[R]$ in the
notation of [52]. (See also [47, Theorem 35.14] or [22] for the number field
case.) ∎
###### Proof of Section 1.
By Section 5.1, $R$ is an arithmetical maximal order in $Q$. We verify
conditions (i) and (ii) of Section 5. Let $I$ be a fractional left $R$-ideal.
By Section 6, $\eta(I)\in P_{R^{\bullet}}$ if and only if
$\operatorname{nr}(I)\in P_{A}$. By Section 6, and the fact that every stably
free left $R$-ideal is free, this is the case if and only if $I$ is principal,
thus condition (i) holds. Condition (ii) holds due to Section 6. By Section 6,
$C\cong\operatorname{\mathcal{C}}_{A}(\mathcal{O})$, and by Section 6,
therefore $C=C_{M}$. Hence there exists a transfer homomorphism $\theta\colon
R^{\bullet}\to\mathcal{B}(\operatorname{\mathcal{C}}_{A}(\mathcal{O}))$. The
remaining claims in the theorem follow from this by Section 3. ∎
###### Remark .
If more generally $\mathcal{O}^{\prime}$ is an arbitrary Dedekind domain with
quotient field the global field $K$, then there is a transfer homomorphism to
either $\mathcal{B}(\operatorname{\mathcal{C}}_{A}(\mathcal{O}^{\prime}))$ or
$\mathcal{B}(\operatorname{\mathcal{C}}_{A}(\mathcal{O}^{\prime})\setminus\\{\mathbf{0}\\})$,
depending on whether or not $\mathcal{O}^{\prime}$ contains prime elements.
Only Section 6 has to be adapted: $\mathcal{O}^{\prime}$ is a localization of
a holomorphy ring $\mathcal{O}$, and hence there is an epimorphism
$\operatorname{\mathcal{C}}_{A}(\mathcal{O})\to\operatorname{\mathcal{C}}_{A}(\mathcal{O}^{\prime})$.
This implies that every class
$g\in\operatorname{\mathcal{C}}_{A}(\mathcal{O}^{\prime})\setminus\\{\mathbf{0}\\}$
contains a maximal ideal (see [13] for details). Therefore, for all classical
maximal $\mathcal{O}^{\prime}$-orders $R^{\prime}$ and all
$g\in\operatorname{\mathcal{C}}_{A}(\mathcal{O}^{\prime})\setminus\\{\mathbf{0}\\}$,
there exists a maximal left $R^{\prime}$-ideal $I$ with
$[\operatorname{nr}(I)]=g$. The trivial class however may or may not contain a
maximal ideal. In either case, the statements 1–3 of Section 1 hold true.
Thanks to Kainrath for pointing this out.
## 7\. Proof of Section 1
_Throughout this section, let $\mathcal{O}_{K}$ be the ring of algebraic
integers in a number field $K$, $A$ a central simple algebra over $K$, and $R$
a classical maximal $\mathcal{O}_{K}$-order in $A$ having a stably free left
$R$-ideal that is not free. Furthermore, the discriminant of $A$ is denoted
by_
$\mathfrak{D}=\prod_{\begin{subarray}{c}\mathfrak{p}\in\max(\mathcal{O}_{K})\\\
\textnormal{$A$ is ramified at
$\mathfrak{p}$}\end{subarray}}\mathfrak{p}\,\triangleleft\,\mathcal{O}_{K}.$
The aim of this section is to prove Theorem 1. The existence of a stably free
left $R$-ideal that is not free implies that $A$ is a totally definite
quaternion algebra and that $K$ is totally real (Note, that conversely, for
all but finitely many isomorphism classes of such classical maximal
$\mathcal{O}_{K}$-orders in totally definite quaternion algebras there exist
stably free left $R$-ideals that are non-free). We proceed in three
subsections.
### 7.1. Reduction.
We state two propositions and show how they imply Theorem 1. The proofs of
these two propositions will then be given in Section 7.3.
###### Proposition .
There exists a totally positive prime element $p\in\mathcal{O}_{K}$, a non-
empty subset $E\subset\\{\,2,3,4\,\\}$ and for every $l\in\mathbb{N}_{0}$ an
atom $y_{l}\in\mathcal{A}(R^{\bullet})$ such that
$\mathsf{L}_{R^{\bullet}}(y_{l}p)=\\{\,3\,\\}\,\cup\,(l+E).$
(We emphasize that $E$ does not depend on $l$.)
###### Proposition .
If $L\in\mathcal{L}(R^{\bullet})$ and $n\in\mathbb{N}$, then
$n+L=\\{\,n+l\,\mid\,l\in L\,\\}\in\mathcal{L}(R^{\bullet})$.
###### Proof of Section 1 (based on Section 7.1 and Section 7.1).
We first show that there is no transfer homomorphism
$R^{\bullet}\to\mathcal{B}(G_{P})$ for any subset $G_{P}$ of an abelian group.
Assume to the contrary that $\theta\colon R^{\bullet}\to\mathcal{B}(G_{P})$ is
such a transfer homomorphism.
1. Claim A.
If $S\in\mathcal{B}(G_{P})$ and $U\in\mathcal{A}(\mathcal{B}(G_{P}))$, then
$\max\mathsf{L}_{\mathcal{B}(G_{P})}(SU)\leq\lvert S\rvert+1.$
###### Proof of A.
Let $S=g_{1}\cdot\ldots\cdot g_{l}$, with $l=\lvert S\rvert$ and
$g_{1},\ldots,g_{l}\in G_{P}$, and suppose that $SU=T_{1}\cdot\ldots\cdot
T_{k}$ with $k\in\mathbb{N}$ and
$T_{1},\ldots,T_{k}\in\mathcal{A}(\mathcal{B}(G_{P}))$. Then for every
$i\in[1,k]$ either $T_{i}\mid U$, but then already $T_{i}=U$, or $g_{j}\mid
T_{i}$ for some $j\in[1,l]$. This shows $k\leq\lvert S\rvert+1$. ∎
By Section 7.1, there exists a totally positive prime element
$p\in\mathcal{O}_{K}$, and for every $l\in\mathbb{N}_{0}$ an atom
$y_{l}\in\mathcal{A}(R^{\bullet})$ with
$\max\mathsf{L}_{R^{\bullet}}(y_{l}p)\geq l+2$. But, if
$l\geq\lvert\theta(p)\rvert$, then
$l+2\,\leq\,\max\mathsf{L}_{R^{\bullet}}(y_{l}p)=\max\mathsf{L}_{\mathcal{B}(G_{P})}(\theta(y_{l})\theta(p))\,\leq\,\lvert\theta(p)\rvert+1\,\leq\,l+1,$
a contradiction.
In order to show $\Delta(R^{\bullet})=\mathbb{N}$, we choose $d\in\mathbb{N}$.
Let $p$ and $E$ be as in Section 7.1 and set $\epsilon=\min E$. If
$l=d+3-\epsilon$ and $y_{l}$ as in Section 7.1, then we find
$d=(l+\epsilon)-3\in\Delta_{R^{\bullet}}(y_{l}p)$.
Let $k\in\mathbb{N}_{\geq 3}$. By definition, we have
$\mathcal{U}_{k}(R^{\bullet})\subset\mathbb{N}_{\geq 2}$. Thus it remains to
show that for every $k^{\prime}\geq 3$ there exists an element $a\in
R^{\bullet}$ with $\\{\,k,k^{\prime}\,\\}\subset\mathsf{L}(a)$. Assume without
restriction that $k\leq k^{\prime}$ and let $k=3+n$ with $n\in\mathbb{N}_{0}$.
Using Section 7.1, we find an element $a^{\prime}\in R^{\bullet}$ with
$\\{\,3=k-n,\,k^{\prime}-n\,\\}\in\mathsf{L}(a^{\prime})$, and hence by
Section 7.1 there exists an element $a\in R^{\bullet}$ with
$\\{\,k,k^{\prime}\,\\}\in\mathsf{L}(a)$. ∎
### 7.2. Preliminaries.
_Algebraic number theory:_ Our notation mainly follows Narkiewicz [43]. Let
$L/K$ be an extension of number fields. Then $D_{L/K}$ is the relative
different, $\operatorname{N}_{L/K}$ the relative field norm,
$d_{L/K}=\operatorname{N}_{L/K}(D_{L/K})$ is the relative discriminant and
$d_{K}=d_{K/\mathbb{Q}}$ the absolute discriminant (we tacitly identify ideals
of $\mathbb{Z}$ with their positive generators for the absolute discriminant
and norm). If $\mathcal{O}\subset\mathcal{O}_{K}$ is an order, then
$\mathfrak{f}_{\mathcal{O}}$ is the conductor of $\mathcal{O}$ in
$\mathcal{O}_{K}$ and
$h(\mathcal{O})=\lvert\operatorname{Pic}(\mathcal{O})\rvert$ is the class
number of $\mathcal{O}$. Given $a\in L$ with minimal polynomial $f\in K[X]$
over $K$, $\delta_{L/K}(a)=f^{\prime}(a)$ is the different of $a$. Completion
at a prime $\mathfrak{p}\in\max(\mathcal{O}_{K})$ is denoted by a subscript
$\mathfrak{p}$, e.g., $\mathcal{O}_{K,\mathfrak{p}}$, $K_{\mathfrak{p}}$, and
so on. If $\mathfrak{m}\,\triangleleft\,\mathcal{O}_{K}$ is a squarefree
ideal, then
$\operatorname{\mathcal{C}}^{+}_{\mathfrak{m}}(\mathcal{O}_{K})=\\{\,\mathfrak{a}\in\mathcal{F}^{\times}(\mathcal{O}_{K})\mid(\mathfrak{a},\mathfrak{m})=\mathcal{O}_{K}\,\\}\,\big{/}\,\\{\,a\mathcal{O}_{K}\mid\text{$a\in
K^{\times}$ is totally positive, $a\equiv 1\mod\mathfrak{m}$}\,\\}$
denotes the corresponding ray class group. We will repeatedly make use of the
fact that every class in
$\operatorname{\mathcal{C}}^{+}_{\mathfrak{m}}(\mathcal{O}_{K})$ contains
infinitely many maximal ideals of $\mathcal{O}_{K}$ ([43, Corollary 7 to
Proposition 7.9]).
_Quaternion algebras:_ We follow [53, 41], and [39, 40] for computational
aspects. Denote by
$\overline{\,\cdot\,}:A\overset{\sim}{\rightarrow}A^{\text{op}}$ the anti-
involution given by conjugation of elements. Then
$\operatorname{nr}_{A/K}(x)=\operatorname{nr}(x)=x\overline{x}=\overline{x}x\qquad\text{and}\qquad\operatorname{tr}_{A/K}(x)=\operatorname{tr}(x)=x+\overline{x}\qquad\text{for
all $x\in A$.}$
Every element $x\in A$ satisfies an equation of the form
$x^{2}-\operatorname{tr}(x)x+\operatorname{nr}(x)=0,$
and if $x\in A\setminus K$, then $K(x)/K$ is a quadratic field extension. From
the equation above we see that
$\operatorname{N}_{K(x)/K}=\operatorname{nr}_{A/K}|K(x)$ and
$\operatorname{Tr}_{K(x)/K}=\operatorname{tr}_{A/K}|K(x)$.
A classical $\mathcal{O}_{K}$-order $T$ of $A$ is called a _classical Eichler
( $\mathcal{O}_{K}$)-order_ if it is the intersection of two classical maximal
$\mathcal{O}_{K}$-orders. 555Though unconventional, we keep the qualifier
“classical” for consistency with the earlier sections. The reduced
discriminant of a classical $\mathcal{O}_{K}$-order $T$ takes the form
$\mathfrak{D}\mathfrak{N}$ where
$\mathfrak{N}\,\triangleleft\,\mathcal{O}_{K}$ is the level of $T$.
Furthermore, because $A$ is totally definite, we have
$[T^{\times}:\mathcal{O}_{K}^{\times}]<\infty$ for the unit group, and
$\operatorname{\mathcal{C}}_{A}(\mathcal{O}_{K})=\operatorname{\mathcal{C}}^{+}(\mathcal{O}_{K})$
is the narrow class group of $\mathcal{O}_{K}$.
As in the previous section, $\operatorname{\mathcal{LC}}(R)$ is the set of
isomorphism classes of left $R$-ideals, and
$\mu_{R}\colon\operatorname{\mathcal{LC}}(R)\to\operatorname{\mathcal{C}}^{+}(\mathcal{O}_{K}),[I]\mapsto[\operatorname{nr}(I)]$.
###### Proposition .
Let $C\in\mathbb{N}$. Let $\mathfrak{p}\in\max(\mathcal{O}_{K})$ with
$\mathfrak{p}\nmid d_{K}\mathfrak{D}$, and such that
$\frac{h^{+}}{Mw^{2}}(\operatorname{N}_{K/\mathbb{Q}}(\mathfrak{p})+1)-\frac{2}{w}\sqrt{\operatorname{N}_{K/\mathbb{Q}}(\mathfrak{p})}\geq
C.$
Then, for every $c\in\operatorname{\mathcal{LC}}(R)$ with
$\mu_{R}(c)=[\mathfrak{p}]$, there exist at least $C$ maximal left (right)
$R$-ideals of reduced norm $\mathfrak{p}$ and class $c$. Here
$h^{+}=\lvert\operatorname{\mathcal{C}}^{+}(\mathcal{O}_{K})\rvert$ is the
narrow class number, and $w$ and $M$ are constants depending on $\mathfrak{D}$
(see [39, 40]).
###### Proof.
Although not explicitly stated in this way, this is proved by Kirschmer and
Voight in [39, 40]: In the proof of [39, Proposition 7.7], a lower bound on
the entries of a matrix $T^{\prime}$ (in their notation) is derived, which
immediately gives a lower bound on the entries of a matrix $T(\mathfrak{p})$
(in their notation). This is exactly what we need, as is clear from their
definition of $T(\mathfrak{p})$. ∎
_Optimal embeddings:_ Let $L/K$ be a quadratic field extension, and $T$ a
classical Eichler $\mathcal{O}_{K}$-order in $A$ of squarefree level
$\mathfrak{N}\,\triangleleft\,\mathcal{O}_{K}$. If $\mathcal{O}$ is an order
in $L$, every embedding $\iota\colon\mathcal{O}\to T$ gives rise to a unique
embedding $\iota\colon L\to A$, and $\iota$ is an _optimal embedding_ if
$\iota(L)\cap T=\mathcal{O}$. For $a\in T^{\times}$, $\mathcal{O}\to
T,x\mapsto a\iota(x)a^{-1}$ is then another such embedding. The number of
optimal embeddings up to conjugation by units is bounded above by a constant
(depending only on $\mathfrak{D}$ and $\mathfrak{N}$) times $h(\mathcal{O})$
(see [53, Corollaire III.5.12]). Since
$[T^{\times}\colon\mathcal{O}_{K}^{\times}]$ is finite, the total number of
optimal embeddings of $\mathcal{O}$ into $T$ is still bounded by a constant
times $h(\mathcal{O})$.
_Quadratic forms:_ We use a theorem about representation numbers of totally
positive definite quadratic forms over totally real fields. Let $V$ be an
$n$-dimensional $K$-vector space. An $\mathcal{O}_{K}$-lattice $L$ of rank $n$
is a finitely generated $\mathcal{O}_{K}$-submodule of $V$ that generates $V$
(over $K$). Together with a quadratic form $q\colon V\to K$ with
$q(L)\subset\mathcal{O}_{K}$, $(L,q)$ it is a quadratic lattice. For
$a\in\mathcal{O}_{K}$ we set
$r(L,a)=\lvert\\{x\in L\mid q(x)=a\\}\rvert.$
An element $a\in\mathcal{O}_{K}$ is locally represented everywhere by $(L,q)$
if it is represented by the completion
$L_{v}=L\otimes_{\mathcal{O}_{K}}\mathcal{O}_{K,v}$ for all places $v$ of $K$.
The following result is a special case of Theorem 5.1 in [50].
###### Proposition .
Let $(L,q)$ be a quadratic $\mathcal{O}_{K}$-lattice of rank four and suppose
that $q$ is totally positive definite. Then, for every $\eta>0$ and
$s\in\mathbb{N}_{0}$, there exists a constant $C_{\eta,s}>0$, such that for
all $a\in\mathcal{O}_{K}$ that are locally represented everywhere by $L$, with
$\lvert\operatorname{N}_{K/\mathbb{Q}}(a)\rvert$ sufficiently large and
$\mathfrak{p}^{s}\nmid a\mathcal{O}_{K}$ if $L_{\mathfrak{p}}$ is anisotropic,
the asymptotic formula
$r(L,a)\,=\,r(\operatorname{gen}L,a)+O(\lvert\operatorname{N}_{K/\mathbb{Q}}(a)\rvert^{\frac{11}{18}+\varepsilon})$
holds with
$r(\operatorname{gen}L,a)\,\geq\,C_{\eta,s}\cdot\operatorname{N}_{K/\mathbb{Q}}\big{(}a\,({}_{\mathcal{O}_{K}}\langle
q(L)\rangle)^{-1}\big{)}^{1-\eta}.$
In particular, $r(L,a)$ is of order of magnitude
$\lvert\operatorname{N}_{K/\mathbb{Q}}(a)\rvert^{1-\eta}$. If $T$ is any
classical $\mathcal{O}_{K}$-order and $I$ any left $T$-ideal (in particular if
$I=T$), then the restriction of the reduced norm to $I$ makes
$(I,\operatorname{nr}\mid I)$ into a quadratic $\mathcal{O}_{K}$-lattice of
rank four and this is the situation that we will apply this result to.
_Ideal theory in $R$:_ Let $\alpha$ be the set of all classical maximal
$\mathcal{O}_{K}$-orders in $A$ (i.e., the equivalence class of the maximal
order $R$). Conjugation extends to ideals: For $I\in\mathcal{F}_{v}(\alpha)$
define $\overline{I}=\\{\,\overline{x}\mid x\in I\,\\}$. Then $\overline{I}$
is a fractional $(\mathcal{O}_{r}(I),\mathcal{O}_{l}(I))$-ideal,
$I\cdot\overline{I}=\mathcal{O}_{l}(I)\operatorname{nr}(I)$, and
$\overline{I}\cdot I=\mathcal{O}_{r}(I)\operatorname{nr}(I)$, and hence
$I^{-1}=\overline{I}\cdot(\mathcal{O}_{l}(I)\operatorname{nr}(I))^{-1}=(\operatorname{nr}(I)\mathcal{O}_{r}(I))^{-1}\cdot\overline{I}$.
$\mathcal{F}_{v}(\alpha)$ takes a particularly simple form: If
$\mathfrak{p}\mid\mathfrak{D}$ then there exists a maximal two-sided $R$-ideal
$\mathfrak{P}$ with $\mathfrak{P}^{2}=\mathfrak{p}$,
$\operatorname{nr}(\mathfrak{P})=\mathfrak{p}$ and if $I$ is a left or right
$R$-ideal with $\operatorname{nr}(I)=\mathfrak{p}^{k}$, then
$I=\mathfrak{P}^{k}$.
If $\mathfrak{p}\nmid\mathfrak{D}$, then $\mathfrak{P}=\mathfrak{p}R$ is the
maximal two-sided $R$-ideal lying above $\mathfrak{p}$, and
$R_{\mathfrak{p}}/\mathfrak{P}_{\mathfrak{p}}\cong
M_{2}(\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}_{\mathfrak{p}})\cong
M_{2}(\mathbb{F}_{\operatorname{N}_{K/\mathbb{Q}}(\mathfrak{p})})$. In
particular there are $\operatorname{N}_{K/\mathbb{Q}}(\mathfrak{p})+1$ maximal
left $R$-ideals (respectively maximal right $R$-ideals) with reduced norm
$\mathfrak{p}$. If $M,N$ are two distinct maximal left $R$-ideals with
$\operatorname{nr}(M)=\operatorname{nr}(N)=\mathfrak{p}$, then $M\cap
N=\mathfrak{P}$ (since the composition length of
$M_{2}(\mathbb{F}_{\operatorname{N}_{K/\mathbb{Q}}(\mathfrak{p})})$ is two).
This implies that if $M\cdot M^{\prime}=N\cdot N^{\prime}$ with maximal
integral $M^{\prime},N^{\prime}\in\mathcal{F}_{v}(\alpha)$, then $M\cdot
M^{\prime}=N\cdot N^{\prime}=\mathfrak{P}$, and thus necessarily
$M^{\prime}=\overline{M}$, $N^{\prime}=\overline{N}$.
We therefore explicitly know all relations between maximal integral elements
of $\mathcal{F}_{v}(\alpha)$: From Section 4 we know that all relations are
generated from those between pairs of products of two elements and it also
characterizes the only relation between maximal integral elements of coprime
reduced norm. With the discussion above we now also know the relations between
two maximal integral elements of the same reduced norm: Either there are none,
or the product is $\mathfrak{P}$.
A left $R$-ideal $I$ is _primitive_ if it is not contained in an ideal of the
form $R\mathfrak{a}$ with $\mathfrak{a}\,\triangleleft\,\mathcal{O}_{K}$. If
$\operatorname{nr}(I)=\mathfrak{p}^{k}$ with
$\mathfrak{p}\in\max(\mathcal{O}_{K})$ and $I$ is primitive, then it has a
unique rigid factorization in $\mathcal{I}_{v}(\alpha)$.
### 7.3. Proofs of Section 7.1 and Section 7.1
We start with some lemmas.
###### Lemma .
Let $T$ be a classical $\mathcal{O}_{K}$-order in $A$. For all but finitely
many associativity classes of totally positive prime elements
$q\in\mathcal{O}_{K}$ we have: If $x\in T$ with $\operatorname{nr}(x)=q$ and
$x^{2}=\varepsilon q$ for some $\varepsilon\in T^{\times}$, then
$\varepsilon=-1$.
###### Proof.
$x$ satisfies the polynomial equation
$x^{2}-\operatorname{tr}(x)x+\operatorname{nr}(x)=0$. Substituting
$x^{2}=\varepsilon q$ and $\operatorname{nr}(x)=q$ yields
$\operatorname{tr}(x)x=(1+\varepsilon)q.$ (2)
It will thus suffice to show that for all but finitely many
$\mathcal{O}_{K}q$, we have $\operatorname{tr}(x)=0$.
Assume that $q\in\mathcal{O}_{K}$ is a totally positive prime element, $x\in
T$ with $x^{2}=\varepsilon q$ and $\operatorname{tr}(x)\neq 0$. Then
$K(x)=K(\varepsilon)$ by (2). Let $L=K(\varepsilon)$. Since $\varepsilon\in
L\cap T^{\times}\subset\mathcal{O}_{L}^{\times}$,
$\mathcal{O}_{L}q=\mathcal{O}_{L}\varepsilon q=(\mathcal{O}_{L}x)^{2},$
and therefore $q$ ramifies in $\mathcal{O}_{L}$, implying
$\mathcal{O}_{K}q\mid d_{L/K}$. Hence, for fixed $L$ there are only finitely
many possibilities for $\mathcal{O}_{K}q$, and moreover there are only
finitely many possibilities for $L=K(\varepsilon)$ because
$[T^{\times}:\mathcal{O}_{K}^{\times}]$ is finite since $A$ is totally
definite. 666It should be pointed out that an even stronger statement is true.
For any fixed totally real field $K$, there are only finitely many totally
imaginary quadratic extensions that have larger unit group (i.e., weak unit
defect), while all other totally imaginary quadratic extensions $L/K$ have
$\mathcal{O}_{L}^{\times}=\mathcal{O}_{K}^{\times}$ (i.e., strong unit
defect). This follows from
$[\mathcal{O}_{L}^{\times}:\mu(L)\mathcal{O}_{K}^{\times}]\in\\{1,2\\}$ ([54,
Theorem 4.12]): For $\varepsilon\in\mathcal{O}_{L}^{\times}$ we have
$\varepsilon^{2}=\eta\zeta$ with $\zeta\in\mu(L)$ and
$\eta\in\mathcal{O}_{K}^{\times}$. But
$\operatorname{ord}(\zeta)\mid[L:\mathbb{Q}]=2[K:\mathbb{Q}]$ and if
$\gamma\in(\mathcal{O}_{K}^{\times})^{2}$, then $\eta\gamma\zeta$ yields the
same extension. Since
$[\mathcal{O}_{K}^{\times}:(\mathcal{O}_{K}^{\times})^{2}]<\infty$ there are
therefore only finitely many such extensions. This argument is due to Remak in
[48, §3]. Thus there are, up to associativity, only finitely many such $q$. ∎
###### Lemma .
Let $T$ be a classical $\mathcal{O}_{K}$-order in $A$. For every
$M\in\mathbb{N}$ there exists a $C\in\mathbb{N}$ such that for all totally
positive prime elements $q\in\mathcal{O}_{K}$ with
$q\in\operatorname{nr}_{A_{\mathfrak{p}}/K_{\mathfrak{p}}}(T_{\mathfrak{p}})$
for all $\mathfrak{p}\in\max(\mathcal{O}_{K})$ and
$\operatorname{N}_{K/\mathbb{Q}}(q)\geq C$
$\lvert\\{a\in T\mid\operatorname{nr}(a)=q\text{ and }a^{2}\neq-q\\}\rvert\geq
M.$
###### Proof.
Let $q$ be a totally positive prime element of $\mathcal{O}_{K}$. We derive an
upper bound with order of magnitude
$\sqrt{\operatorname{N}_{K/\mathbb{Q}}(q)}\log(\operatorname{N}_{K/\mathbb{Q}}(q))^{2[K:\mathbb{Q}]-1}$
on the number of elements $a\in T$ with $a^{2}=-q$ (based on counting optimal
embeddings). Comparing this to the lower bound of order of magnitude
$\operatorname{N}_{K/\mathbb{Q}}(q)^{1-\eta}$ for the number of elements $a\in
T$ with $\operatorname{nr}(a)=q$ obtained from Section 7.2 will give the
result.
If $a\in T$ with $\operatorname{nr}(a)=q$ and $a^{2}=-q$, then
$\mathcal{O}_{K}[a]\subset T$ is isomorphic to the order
$\mathcal{O}_{K}[\sqrt{-q}]$ in the relative quadratic extension
$K(\sqrt{-q})$. We determine an upper bound the number of embeddings of
$\mathcal{O}_{K}[\sqrt{-q}]$ into $T$. For this we may without loss of
generality assume that $T$ is a classical Eichler order of squarefree level,
for otherwise we may replace it by a classical Eichler order of squarefree
level in which it is contained (e.g., a classical maximal order), and bound
the number of embeddings there.
Let $L=K(\sqrt{-q})$. Since $f=X^{2}+q$ is the minimal polynomial of
$\sqrt{-q}$ over $K$, we get for the different
$\delta_{L/K}(\sqrt{-q})=f^{\prime}(\sqrt{-q})=2\sqrt{-q}$. Therefore, we find
for the conductor of $\mathcal{O}_{K}[\sqrt{-q}]$ in $\mathcal{O}_{L}$,
$\mathfrak{f}_{\mathcal{O}_{K}[\sqrt{-q}]}=\delta_{L/K}(\sqrt{-q})D_{L/K}^{-1}\;\mid\;2\mathcal{O}_{L}.$
(cf. [43, Proposition 4.12 and Theorem 4.8]). Since
$2\mathcal{O}_{L}\subset\mathcal{O}_{K}[\sqrt{-q}]\subset\mathcal{O}_{L}$ and
$\lvert\mathcal{O}_{L}/2\mathcal{O}_{L}\rvert=2^{[L:\mathbb{Q}]}$, there are
at most $2^{2^{[L:\mathbb{Q}]}}$ orders in $\mathcal{O}_{L}$ that contain
$\mathcal{O}_{K}[\sqrt{-q}]$. For any such order $\mathcal{O}$ with
$\mathcal{O}_{K}[\sqrt{-q}]\subset\mathcal{O}\subset\mathcal{O}_{L}$, we have
$h(\mathcal{O})=h(\mathcal{O}_{L})\frac{\lvert(\mathcal{O}_{L}/\mathfrak{f}_{\mathcal{O}})^{\times}\rvert}{\lvert(\mathcal{O}/\mathfrak{f}_{\mathcal{O}})^{\times}\rvert}\,\leq\,h(\mathcal{O}_{L})2^{[L:\mathbb{Q}]}$
(cf. [44, §I.12.9 and §I.12.11]). The number of optimal embeddings of
$\mathcal{O}$ into $T$ is bounded by a constant times $h(\mathcal{O})$, and
hence the total number of embeddings of $\mathcal{O}_{K}[\sqrt{-q}]$ into $T$
is bounded by a constant times $h(\mathcal{O}_{L})$, where the constant does
not depend on $q$. Combining the upper bound
$h(\mathcal{O}_{L})\,\ll\,\sqrt{\lvert d_{L}\rvert}\,\log(\lvert
d_{L}\rvert)^{[L:\mathbb{Q}]-1}$
(cf. [43, Theorem 4.4]), with
$d_{L}=\operatorname{N}_{L/\mathbb{Q}}(D_{L/\mathbb{Q}})=\operatorname{N}_{L/\mathbb{Q}}(D_{K/\mathbb{Q}})\operatorname{N}_{L/\mathbb{Q}}(D_{L/K})\leq
d_{K}^{2}2^{[L:\mathbb{Q}]}\lvert\operatorname{N}_{L/\mathbb{Q}}(\sqrt{-q})\rvert=d_{K}^{2}2^{[L:\mathbb{Q}]}\operatorname{N}_{K/\mathbb{Q}}(q).$
(here $D_{L/K}\mid 2\sqrt{-q}\mathcal{O}_{L}$ was used), we obtain
$h(\mathcal{O}_{L})\,\ll\,\sqrt{\operatorname{N}_{K/\mathbb{Q}}(q)}\,\log{(\operatorname{N}_{K/\mathbb{Q}}(q))}^{[L:\mathbb{Q}]-1},$
and thus an upper bound of the same order for $\lvert\\{a\in
T\mid\operatorname{nr}(a)=q\text{ and }a^{2}=-q\\}\rvert$.
By Section 7.2, for every $\eta>0$ and sufficiently large (in norm) $q$ with
$q$ being locally represented everywhere by the norm form, $\lvert\\{a\in
T\mid\operatorname{nr}(a)=q\\}\rvert$ grows with order of magnitude
$\operatorname{N}_{K/\mathbb{Q}}(q)^{1-\eta}$, and the claim follows, by
choosing $\eta$ small enough, say $\eta<\frac{1}{4}$. ∎
###### Remark .
For any classical $\mathcal{O}_{K}$-order $T$ of $A$ there are infinitely many
pairwise non-associated totally positive primes $q\in\mathcal{O}_{K}$ that are
locally represented everywhere by $\operatorname{nr}_{A/K}$ on $T$. This can
easily be seen as follows: Let
$\mathfrak{D}\mathfrak{N}\,\triangleleft\,\mathcal{O}_{K}$ be the discriminant
of $T$. If $\mathfrak{p}\in\max(\mathcal{O}_{K})$ with
$\mathfrak{p}\nmid\mathfrak{D}\mathfrak{N}$, then $T_{\mathfrak{p}}\cong
M_{2}(\mathcal{O}_{K,\mathfrak{p}})$ and thus
$\operatorname{nr}_{A_{\mathfrak{p}}/K_{\mathfrak{p}}}(T_{\mathfrak{p}})=\mathcal{O}_{K,\mathfrak{p}}$.
If $\mathfrak{p}\mid\mathfrak{D}\mathfrak{N}$, since
$\operatorname{center}(T_{\mathfrak{p}})=\mathcal{O}_{K,\mathfrak{p}}$,
certainly still every square of $\mathcal{O}_{K,\mathfrak{p}}$ is represented
by $\operatorname{nr}_{A_{\mathfrak{p}}/K_{\mathfrak{p}}}$ on
$T_{\mathfrak{p}}$ (in fact, if $\mathfrak{p}\mid\mathfrak{D}$ but
$\mathfrak{p}\nmid\mathfrak{N}$ then $T_{\mathfrak{p}}$ is isomorphic to the
unique classical maximal $\mathcal{O}_{K,\mathfrak{p}}$-order in the unique
quaternion division algebra over $K_{\mathfrak{p}}$, for which
$\operatorname{nr}(T_{\mathfrak{p}})=\mathcal{O}_{K,\mathfrak{p}}$ also
holds). By Hensel’s Lemma therefore every totally positive prime element
$q\in\mathcal{O}_{K}$ with $q\equiv 1\mod 4\mathfrak{D}\mathfrak{N}$ is
locally represented everywhere by $\operatorname{nr}_{A/K}$ on $T$. But there
are infinitely many pairwise non-associated such primes, because every class
of the ray class group
$\operatorname{\mathcal{C}}_{4\mathfrak{D}\mathfrak{N}}^{+}(\mathcal{O}_{K})$
contains infinitely many pairwise distinct maximal ideals, and primes $q$ of
the required form correspond exactly to the trivial class in
$\operatorname{\mathcal{C}}_{4\mathfrak{D}\mathfrak{N}}^{+}(\mathcal{O}_{K})$.
###### Lemma .
Let $q$ be a totally positive prime element of $\mathcal{O}_{K}$. Let $I$ be a
non-principal right $R$-ideal with $\operatorname{nr}(I)=q^{m}\mathcal{O}_{K}$
for some $m\in\mathbb{N}$, and $J$ be a left $S=\mathcal{O}_{l}(I)$-ideal with
$\operatorname{nr}(J)=q^{n}\mathcal{O}_{K}$ for some $n\in\mathbb{N}$ such
that: $I\cong J$ (as left $S$-ideals) and $I$ (respectively $J$) is not
contained in any principal left $S$-ideal except $S$ itself, and not contained
in any principal right $\mathcal{O}_{r}(I)$-ideal (respectively right
$\mathcal{O}_{r}(J)$-ideal) except $\mathcal{O}_{r}(I)$ (respectively
$\mathcal{O}_{r}(J)$) itself.
Assume further that $a\in S$ with $\operatorname{nr}(a)=q$ and $a^{2}S\neq
qS$.
1. 1.
For all $l\in\mathbb{N}$, $(a^{l}\overline{J}a^{-l})a^{l}I$ is a principal
right $R$-ideal and an atom of $\mathcal{H}_{R^{\bullet}}$. In particular,
$a^{l}q^{m}\in\mathcal{A}(R^{\bullet})$ for all $l\in\mathbb{N}$.
2. 2.
$\overline{J}I\in\mathcal{A}(\mathcal{H}_{R^{\bullet}})$ if it is primitive.
In particular if $m=n=1$ and $I\neq J$, then
$\overline{J}I\in\mathcal{A}(\mathcal{H}_{R^{\bullet}})$.
###### Proof.
Since $I$ is not contained in any principal right $R$-ideal, it is in
particular not contained in $qR$, hence primitive. Similarly, $J$ is
primitive. Let
$M_{1}*\ldots*M_{m}\in\mathsf{Z}^{*}_{\mathcal{I}_{v}(\alpha)}(I)$ and
$N_{1}*\ldots*N_{n}\in\mathsf{Z}^{*}_{\mathcal{I}_{v}(\alpha)}(J)$, with
$M_{1},\ldots,M_{m},N_{1},\ldots,N_{n}\in\mathcal{M}_{v}(\alpha)$, be the
unique rigid factorizations of $I$ and $J$.
1. 1.
Since $I\cong J$ as left $S$-ideals,
$(a^{l}\overline{J}a^{-l})a^{l}I=a^{l}\overline{J}I$ is principal. A rigid
factorization of it is given by
$(a^{l}\overline{N_{n}}a^{-l})*\ldots*(a^{l}\overline{N_{1}}a^{-l})*(a^{l}Sa^{-l})a*(a^{l-1}Sa^{-(l-1)})a*\ldots*(aSa^{-1})a*M_{1}*\ldots*M_{m}$
with
$M_{i},\,a^{l}\overline{N_{j}}a^{-l},\,(a^{l-k}Sa^{-(l-k)})a\in\mathcal{M}_{v}(\alpha)$
for $i\in[1,m]$, $j\in[1,n]$ and $k\in[0,l-1]$. By the restrictions imposed on
$I$, $J$ and $a$, this is the only rigid factorization of
$a^{l}\overline{J}I$. Since any non-empty proper subproduct starting from the
left (or the right) is non-principal, it is an atom in
$\mathcal{H}_{R^{\bullet}}$. The “in particular” statement follows by setting
$J=I$, as then
$a^{l}\overline{J}I=a^{l}q^{m}R\in\mathcal{A}(\mathcal{H}_{R^{\bullet}})$ and
because of Section 5, therefore $a^{l}q^{m}\in\mathcal{A}(R^{\bullet})$.
2. 2.
By primitivity,
$\overline{N_{n}}*\ldots*\overline{N_{1}}*M_{1}*\ldots*M_{m}\in\mathsf{Z}^{*}_{\mathcal{I}_{v}(\alpha)}(\overline{J}I)$
is the unique rigid factorization of $\overline{J}I$, and since as before no
non-empty proper subproduct from the left (or the right) is principal, it is
an atom in $\mathcal{H}_{R^{\bullet}}$. For the “in particular” statement,
note that if $m=n=1$ (i.e., $I$ and $J$ are both maximal left $S$-ideals),
then $\overline{J}I=qR$ if and only if $I=J$, and otherwise $\overline{J}I$ is
necessarily primitive.
∎
###### Lemma .
Let $I$ be a left $R$-ideal, $S=\mathcal{O}_{r}(I)$, and $a\in R\cap S$. Then
$\prod_{i=1}^{l}(a^{l-i+1}Ra^{-(l-i+1)})a\cdot
I=a^{l}I=(a^{l}Ia^{-l})a^{l}=(a^{l}Ia^{-l})\cdot\prod_{i=1}^{l}(a^{l-i+1}Sa^{-(l-i+1)})a$
with the left-most and the right-most expressions being proper products of
$(a^{l-i+1}Ra^{-(l-i+1)})a,\;I,\;(a^{l-i+1}Sa^{-(l-i+1)})a,\;a^{l}Ia^{-l}\,\in\,\mathcal{I}_{v}(\alpha).$
(The products have to be read in ascending order with “$i=1$” to the very
left.)
###### Proof.
The formulas are clear, and so is that the products are proper ones. The key
point is that these one-sided ideals are indeed integral. But this is so
because $a\in S$, hence $a\in a^{k}Sa^{-k}$ for all $k\in\mathbb{N}_{0}$,
implying that $(a^{k}Sa^{-k})a\in\mathcal{I}_{v}(\alpha)$, and similarly $a\in
R$, thus $(a^{k}Ra^{-k})a\in\mathcal{I}_{v}(\alpha)$. ∎
###### Lemma .
Let $M$ be a maximal left $R$-ideal, and $N$ a maximal left
$\mathcal{O}_{r}(M)$-ideal. If $M\cdot N=N^{\prime}\cdot M^{\prime}$, then
$\overline{M}\cdot N^{\prime}=N\cdot\overline{M^{\prime}}$.
###### Proof.
Since
$\mathcal{O}_{r}(\overline{M})=\mathcal{O}_{l}(M)=\mathcal{O}_{l}(N^{\prime})$
and
$\mathcal{O}_{r}(N)=\mathcal{O}_{r}(M^{\prime})=\mathcal{O}_{l}(\overline{M^{\prime}})$
the product is proper. We have
$\overline{M}\cdot N^{\prime}\cdot M^{\prime}=\overline{M}\cdot M\cdot
N=\operatorname{nr}(M)\mathcal{O}_{l}(N)\cdot
N=N\cdot\mathcal{O}_{r}(N)\operatorname{nr}(M)=N\cdot\overline{M^{\prime}}\cdot
M^{\prime},$
and thus $\overline{M}\cdot N^{\prime}=N\cdot\overline{M^{\prime}}$. ∎
###### Proof of Section 7.1.
Let $p\in\mathcal{O}_{K}$ be a totally positive prime element with
$p\mathcal{O}_{K}\nmid d_{K}\mathfrak{D}\mathfrak{N}$ and with
$\operatorname{nr}(p)$ satisfying the bound of Section 7.2 for the classical
maximal order $R$ (with $C=1$). Then there exists a maximal right $R$-ideal
$U$ with $\operatorname{nr}(U)=p\mathcal{O}_{K}$ that is non-free (i.e., non-
principal) but is stably free (i.e., $[\operatorname{nr}(U)]=\mathbf{0}$ in
$\operatorname{\mathcal{C}}^{+}(\mathcal{O}_{K})$). Let $U=U_{0},\ldots,U_{r}$
be the maximal left $\mathcal{O}_{l}(U)$-ideals of reduced norm
$p\mathcal{O}_{K}$ (of which there are
$r+1=\operatorname{N}_{K/\mathbb{Q}}(p)+1$). By Section 7.3, there exists a
totally positive prime element $q\in\mathcal{O}_{K}$, $q\mathcal{O}_{K}\nmid
pd_{K}\mathfrak{D}\mathfrak{N}$, and an element
$a\,\in\,\mathcal{O}_{l}(U)\,\cap\,\bigcap_{j=0}^{r}\mathcal{O}_{r}(U_{j})\quad\text{with}\quad\operatorname{nr}(a)=q\text{
and }a^{2}\neq-q,$
and in fact, by Section 7.3, we may make this choice such that
$a^{2}\neq\varepsilon q$ for any $\varepsilon\in R^{\times}$. In addition, we
may take $\operatorname{N}_{K/\mathbb{Q}}(q)$ to be sufficiently large to
satisfy the bound of Section 7.2 for $\mathcal{O}_{l}(U)$ (with $C=2$). Then
there exist distinct left $\mathcal{O}_{l}(U)$-ideals $I$ and $J$ such that
$I\cong J\cong U$ and
$\operatorname{nr}(I)=\operatorname{nr}(J)=q\mathcal{O}_{K}$.
Set $S=\mathcal{O}_{r}(I)$, and observe that $S\cong R$, because $U\cong I$.
By Section 7.3,
$(a^{l}\overline{J}a^{-l})a^{l}I\in\mathcal{A}(\mathcal{H}_{S^{\bullet}})$ for
all $l\in\mathbb{N}_{0}$, say $(a^{l}\overline{J}a^{-l})a^{l}I=y_{l}S$ with
$y_{l}\in\mathcal{A}(S^{\bullet})$. We consider the principal right $S$-ideal
$X_{l}=(a^{l}\overline{J}a^{-l})a^{l}Ip\subset S$, say $X_{l}=x_{l}S$ with
$x_{l}\in S^{\bullet}$. We will first determine all possible rigid
factorizations of $X_{l}$ in $\mathcal{I}_{v}(\alpha)$. As in Section 7.3, the
right $S$-ideal $(a^{l}\overline{J}a^{-l})a^{l}I$ has reduced norm
$q^{l+2}\mathcal{O}_{K}$, is primitive, and thus possesses a unique rigid
factorization,
$(a^{l}\overline{J}a^{-l})*(a^{l}\mathcal{O}_{l}(I)a^{-l})a*(a^{l-1}\mathcal{O}_{l}(I)a^{-(l-1)})a*\ldots*(a\mathcal{O}_{l}(I)a^{-1})a*I\;\in\;\mathsf{Z}^{*}_{\mathcal{I}_{v}(\alpha)}((a^{l}\overline{J}a^{-l})a^{l}I),$
with the $l+2$ factors
$a^{l}\overline{J}a^{-l},\,(a^{l-k}\mathcal{O}_{l}(I)a^{-(l-k)})a$ for
$k\in[0,l-1]$ and $I$ all in $\mathcal{M}_{v}(\alpha)$.
For an element with a unique rigid factorization we make the convention of
identifying the element and its factorization when this is notationally
convenient. For principal ideals we omit the order and only write the
generator if it is clear from the neighboring elements in the factorization
what the order must be. For example, we can write the previous rigid
factorization as $a^{l}\overline{J}a^{-l}*a^{l}*I$.
$X_{l}$ has $(r+1)\binom{l+4}{2}$ rigid factorizations: They arise from the
different rigid factorizations
$U_{i}*\overline{U_{i}}\in\mathsf{Z}^{*}_{\mathcal{I}_{v}(\alpha)}(\mathcal{O}_{l}(U)p)$
for $i\in[0,r]$ and the possible transpositions of $U_{i}$ and
$\overline{U_{i}}$. We denote the rigid factorization of $X_{l}$ that arises
from $a^{l}Ja^{-l}*a^{l}*U_{i}*\overline{U_{i}}*I$ by transposing the one-
sided ideals of norm $p$ to the positions $m\in[-1,l+1]$ and $n\in[m,l+1]$ in
the factorization by $F_{i,m,n}$: Here, the left-most position in the rigid
factorization is denoted by $-1$, the right-most by $l+1$. So, by “the rigid
factorization obtained by transposing $U_{i}$ to the position $-1$ and
$\overline{U_{i}}$ to $l+1$” we mean the unique rigid factorization of $X_{l}$
that has a factor of norm $p\mathcal{O}_{K}$ as the first factor and as the
last factor, and that can be transformed into
$a^{l}\overline{J}a^{-l}*a^{l}*U_{i}*\overline{U_{i}}*I$ by transposition of
maximal integral elements with coprime norm.
For $i\in[0,r]$ let $V_{i}\in\mathcal{M}_{v}(\alpha)$ and
$M_{i}\in\mathcal{M}_{v}(\alpha)$ be defined by
$\overline{U_{i}}I=M_{i}\overline{V_{i}}$ under transposition, and let
$W_{i}\in\mathcal{M}_{v}(\alpha)$ and $N_{i}\in\mathcal{M}_{v}(\alpha)$ be
defined by $W_{i}\overline{N_{i}}=\overline{J}U_{i}$ under transposition.
($\\{V_{i}\mid i\in[0,r]\\}$ is then the set of all $r+1$ left
$S=\mathcal{O}_{r}(I)$-ideals of reduced norm $p$. Similarly $\\{W_{i}\mid
i\in[0,r]\\}$ is then the set of all $r+1$ left $\mathcal{O}_{r}(J)$-ideals of
reduced norm $p$, and since $\mathcal{O}_{r}(I)\cong\mathcal{O}_{r}(J)$ the
sets are actually conjugate under conjugation by an element of $A^{\times}$.)
By Section 7.3 we then also have $U_{i}M_{i}=IV_{i}$ and
$\overline{W_{i}}\,\overline{J}=\overline{N_{i}}\,\overline{U_{i}}$ under
transposition. Using Section 7.3 to see that $a$ transposes “nicely” with
$U_{i}$ and $\overline{U_{i}}$, we can explicitly describe all $F_{i,m,n}$ as
follows:
1. (Case 1)
If $m=n=-1$:
$F_{i,m,n}=a^{l}W_{i}a^{-l}*a^{l}\overline{W_{i}}a^{-l}*a^{l}\overline{J}a^{-l}*a^{l}*I.$
2. (Case 2)
If $m=-1$ and $0\leq n\leq l$:
$F_{i,m,n}=a^{l}W_{i}a^{-l}*a^{l}\overline{N}_{i}a^{-l}*a^{n}*a^{l-n}\overline{U_{i}}a^{-(l-n)}*a^{l-n}*I.$
3. (Case 3)
If $0\leq m\leq n\leq l$:
$F_{i,m,n}=a^{l}\overline{J}a^{-l}*a^{m}*a^{l-m}U_{i}a^{-(l-m)}*a^{n-m}*a^{l-n}\overline{U_{i}}a^{-(l-n)}*a^{l-n}*I.$
4. (Case 4)
If $m=-1$ and $n=l+1$:
$F_{i,m,n}=a^{l}W_{i}a^{-l}*a^{l}\overline{N_{i}}a^{-l}*a^{l}*M_{i}*\overline{V_{i}}.$
5. (Case 5)
If $0\leq m\leq l$ and $n=l+1$:
$F_{i,m,n}=a^{l}\overline{J}a^{-l}*a^{m}*a^{l-m}U_{i}a^{-(l-m)}*a^{l-m}*M_{i}*\overline{V_{i}}.$
6. (Case 6)
If $m=n=l+1$:
$F_{i,m,n}=a^{l}\overline{J}a^{-l}*a^{l}*I*V_{i}*\overline{V_{i}}.$
For each of these rigid factorizations of the ideal $X_{l}$ in
$\mathcal{I}_{v}(\alpha)$ we can form minimal subproducts of principal one-
sided ideals (starting from the left or the right) to obtain a representation
of $X_{l}$ as a product in $\mathcal{H}_{S^{\bullet}}$ (and hence a
representation of $x_{l}$ as a product of elements of $S^{\bullet}$). But only
when each of these minimal principal subproducts is an atom of
$\mathcal{H}_{S^{\bullet}}$ this gives rise to an actual rigid factorization
of $x_{l}$ into atoms. We discuss the individual cases one-by-one:
1. Case 1.
If $m=n=-1$: If $W_{i}$ is non-principal, then this does not give rise to a
rigid factorization into atoms, as the first principal factor is
$a^{l}(W_{i}\overline{W_{i}})a^{-l}=a^{l}(p\mathcal{O}_{r}(J))a^{-l}$, and
this is not an atom (since there is at least one element in $\\{W_{i}\mid
i\in[0,r]\\}$ that is principal by Section 7.2). If on the other hand $W_{i}$
is principal, then this gives rise to a rigid factorization of $X_{l}$ in
$\mathcal{H}_{S^{\bullet}}$ of length $3$, with atomic factors
$a^{l}W_{i}a^{-l}$, $a^{l}\overline{W_{i}}a^{-l}$ and
$(a^{l}\overline{J}a^{-l})a^{l}I$, which in turn gives rise to a rigid
factorization of length $3$ of $x_{l}\in S$.
2. Case 2.
If $m=-1$ and $0\leq n\leq l$:
1. Case 2a
If $U_{i}\cong I$: Then the last principal factor is necessarily
$(a^{l-n}\overline{U_{i}}a^{-(l-n)})a^{l-n}I$. If $n<l$, then transposing
$\overline{U_{i}}$ to the right shows that this is not an atom in
$\mathcal{H}_{S^{\bullet}}$. If $n=l$ then
$\overline{U_{i}}I=M_{i}\overline{V_{i}}$ is an atom if and only if $V_{i}$ is
non-principal. Since also $U_{i}\cong J$, the factor
$W_{i}\overline{N_{i}}=\overline{J}U_{i}$ is principal, and, because
$\overline{J}$ and $\overline{U_{i}}$ are non-principal, this is either an
atom (if $W_{i}$ is non-principal), or a product of two atoms (if $W_{i}$ is
principal). So if $V_{i}$ is non-principal we get a rigid factorization of
length either $l+2$ or $l+3$, and if $V_{i}$ is principal we get no rigid
factorization into atoms.
2. Case 2b
If $U_{i}\not\cong I$: Then either there are no non-trivial principal factors
(if $W_{i}$ is non-principal), or the first factor is $W_{i}$ and the
remaining product does not factor into non-trivial principal factors. But then
this second factor is not an atom, because after transposition of
$\overline{U_{i}}$ to the very left of the second factor (i.e, position $0$),
we have a principal factor $\overline{W_{i}}$. So in any case, this does not
give rise to a rigid factorization into atoms.
3. Case 3.
If $0\leq m\leq n\leq l$: If $I\not\cong U_{i}$, then there are no non-trivial
principal factors, and hence no rigid factorization into atoms is obtained.
If $I\cong U_{i}$, then the first principal factor is
$(a^{l}\overline{J}a^{-l})a^{m}(a^{l-m}U_{i}a^{-(l-m)})$, and the last one is
$(a^{l-n}\overline{U_{i}}a^{-(l-n)})a^{l-n}I$. If $m>0$ (or $n<l$), then by
transposing $U_{i}$ to the left in the first factor (or $\overline{U_{i}}$ to
the right in the second factor) once, we see that this does not give rise to a
rigid factorization into atoms. Consider now $m=0$ and $n=l$. If $V_{i}$ is
principal, then $\overline{J}U_{i}=V_{i}\overline{K_{i}}$ implies that the
first factor $a^{l}\overline{J}U_{i}a^{-l}$ is no atom, and hence again we get
no rigid factorization into atoms. Analogously we get no rigid factorization
into atoms if $W_{i}$ is principal. If on the other hand $V_{i}$ and $W_{i}$
are both non-principal then $\overline{J}U_{i}$ is an atom, and so is
$\overline{U_{i}}I$. Thus we obtain a rigid factorization of $X_{l}$ (and
hence of $x_{l}$) of length $l+2$. (It is then in fact the same one as the one
obtained from Case 2 in the same situation.)
4. Case 4.
If $m=-1$ and $n=l+1$:
1. Case 4a
If $U_{i}\cong I$: Then $W_{i}\overline{N_{i}}$ and $M_{i}\overline{V_{i}}$
are both principal. If $W_{i}$ is non-principal, then
$W_{i}\overline{N_{i}}=\overline{J}U_{i}$ is an atom since $J$ is non-
principal. If $W_{i}$ is principal, then $W_{i}\overline{N_{i}}$ is a product
of two atoms. Similarly, $M_{i}\overline{V_{i}}$ is either an atom or a
product of two atoms. So in this case we get a rigid factorization of length
$l+2$, $l+3$ or $l+4$. (In the case that $V_{i}$ is non-principal, it is the
same one as in Case 2. In the case that $V_{i}$ and $W_{i}$ are both non-
principal it is the same as in Case 3 in the same situation.)
2. Case 4b
If $U_{i}\not\cong I$: Then $W_{i}\overline{N_{i}}$ and
$M_{i}\overline{V_{i}}$ are both non-principal. If $W_{i}$ is principal, but
$\overline{V_{i}}$ is not, then the second principal factor is necessarily
$(a^{l}\overline{N_{i}}a^{-l})a^{l}M_{i}\overline{V_{i}}$, and this cannot be
split as a non-trivial product of principal factors. But transposing
$\overline{V_{i}}$ to the very left in this factor gives a principal factor
$\overline{W_{i}}$, hence
$(a^{l}\overline{N_{i}}a^{-l})a^{l}M_{i}\overline{V_{i}}$ is not an atom.
Arguing analogously, if $\overline{V_{i}}$ is principal but $W_{i}$ is not, no
rigid factorization into atoms is obtained.
Finally, if $W_{i}$ and $V_{i}$ are both principal, we get a rigid
factorization into $3$ atoms.
5. Case 5.
If $0\leq m\leq l$ and $n=l+1$: This is analogous to Case 2.
6. Case 6.
If $m=n=l+1$: This is analogous to Case 1.
Since there is at least one $i\in[0,r]$ for which $W_{i}$ is principal, we get
at least one rigid factorization of $x_{l}$ of length $3$ from Case 1. For
$i=0$, $U_{i}\cong I$, so Case 4 gives at least one factorization with length
in $[l+2,l+4]$. Note that which of the lengths in $[l+2,l+4]$ occur in Case 2,
Case 3, and Case 4 depends only on the principality of certain one-sided
ideals, and not on $l$. Thus we have shown that there exists a set
$\emptyset\neq E\subset\\{\,2,3,4\,\\}$ such that, for any choice of
$l\in\mathbb{N}_{0}$,
$\mathsf{L}_{S^{\bullet}}(x_{l})=\\{\,3\,\\}\;\cup\;(l+E),$
and $x_{l}$ has the claimed form $x_{l}=y_{l}p$ with
$y_{l}\in\mathcal{A}(S^{\bullet})$ and $p$ a totally positive prime element of
$\mathcal{O}_{K}$. Since $S\cong R$, the same is true for $R$. ∎
###### Remark .
1. 1.
In the proof, the classical $\mathcal{O}_{K}$-order
$T=\mathcal{O}_{l}(U)\,\cap\,\bigcap_{j=0}^{r}\mathcal{O}_{r}(U_{j})$
is maximal at every prime $\mathfrak{r}\in\max(\mathcal{O}_{K})$ with
$\mathfrak{r}\neq p\mathcal{O}_{K}$ (thus $T_{\mathfrak{r}}\cong
M_{2}(\mathcal{O}_{K,\mathfrak{r}})$ if $\mathfrak{r}\nmid p\mathfrak{D}$ and
$T_{\mathfrak{r}}$ is isomorphic to the unique classical maximal
$\mathcal{O}_{K,\mathfrak{r}}$-order in the unique quaternion division algebra
over $K_{\mathfrak{r}}$ if $\mathfrak{r}\mid\mathfrak{D}$). At
$\mathfrak{p}=p\mathcal{O}_{K}$, it is not hard too see by local calculations
that
$T_{\mathfrak{p}}\cong\left\\{\begin{pmatrix}a&b\\\
p^{2}c&a+pd\end{pmatrix}\;\bigg{\lvert}\;a,b,c,d\in\mathcal{O}_{K,\mathfrak{p}}\right\\}.$
But $T_{\mathfrak{p}}$ is not a classical Eichler order, and so neither is
$T$.
2. 2.
If $\mathfrak{r}\in\max(\mathcal{O}_{K})$ we can find infinitely many pairwise
non-associated totally positive prime elements $q\in\mathcal{O}_{K}$ such that
$\mathfrak{r}$ splits in $K(\sqrt{-q})$, and infinitely many pairwise non-
associated totally positive prime elements $q\in\mathcal{O}_{K}$ such that
$\mathfrak{r}$ is inert in $K(\sqrt{-q})$: We may restrict ourselves to $q$
with $\operatorname{N}_{K/\mathbb{Q}}(q)$ odd, and
$q\mathcal{O}_{K}\neq\mathfrak{r}$. Let
$\mathfrak{r}^{\prime}=\mathfrak{r}^{1+\mathsf{v}_{\mathfrak{r}}(4)}$. If
$-q\equiv 1\mod\mathfrak{r}^{\prime}$, then $-q$ is a square in
$\mathcal{O}_{K}/\mathfrak{r}^{\prime}$ and hence $\mathfrak{r}$ splits in
$K(\sqrt{-q})$. If $-q\equiv a\mod\mathfrak{r}^{\prime}$, with $a$ a non-
square in $\mathcal{O}_{K}/\mathfrak{r}^{\prime}$, then $\mathfrak{r}$ is
inert in $K(\sqrt{-q})$. It therefore suffices to show that in every class of
$(\mathcal{O}_{K}/\mathfrak{r}^{\prime})^{\times}$ there are infinitely many
pairwise non-associated totally positive prime elements of $\mathcal{O}_{K}$.
Since we have the exact sequence
$\mathbf{1}\to\mathcal{O}_{K}^{\times,+}/\\{x\in\mathcal{O}_{K}^{\times,+}\mid
x\equiv_{\mathfrak{r}^{\prime}}1\\}\to(\mathcal{O}_{K}/\mathfrak{r}^{\prime})^{\times}\to\operatorname{\mathcal{C}}_{\mathfrak{r}^{\prime}}^{+}(\mathcal{O}_{K})\to\operatorname{\mathcal{C}}^{+}(\mathcal{O}_{K})\to\mathbf{1},$
(cf. [43, Lemma 3.2], [44, Exercises VI.1.12, VI.1.13]), it suffices that
every class in the kernel of
$\operatorname{\mathcal{C}}_{\mathfrak{r}^{\prime}}^{+}(\mathcal{O}_{K})\to\operatorname{\mathcal{C}}^{+}(\mathcal{O}_{K})$
contains infinitely many pairwise non-associated prime elements. But this is
so, because in fact every class of
$\operatorname{\mathcal{C}}_{\mathfrak{r}^{\prime}}^{+}(\mathcal{O}_{K})$
contains infinitely many pairwise distinct maximal ideals (cf. [43, Corollary
7 to Proposition 7.9]).
3. 3.
Using the previous observation to find a suitable element $a$ in the proof, we
can replace Section 7.3 by a simpler one if $\mathfrak{D}\neq\mathcal{O}_{K}$:
Choosing the totally positive prime element $q\in\mathcal{O}_{K}$ such that a
prime divisor $\mathfrak{r}\mid\mathfrak{D}$ splits in $K(\sqrt{-q})$, the
field $K(\sqrt{-q})$ does not embed into $A$ at all (see e.g. [53, Theoreme
III.3.8] or [41, Theorem 7.3.3]).
If $\mathfrak{D}=\mathcal{O}_{K}$, we may make use of the fact that the
particular classical order $T$ in the proof is contained in a classical
Eichler order of squarefree level $\mathfrak{p}$. Taking $q$ such that
$\mathfrak{p}$ is inert in $K(\sqrt{-q})$, the formulas for counting optimal
embeddings ([53, Corollaire III.5.12]) show that no order of $K(\sqrt{-q})$
embeds into $T$.
For this approach we only need the qualitative statement of Section 7.2, but
not the order of magnitude.
###### Proof of Section 7.1.
It suffices to prove the claim for $n=1$. Let $a$ in $R^{\bullet}$, and let
$\\{I_{1}^{(1)}*\ldots*I_{k}^{(1)},\;\ldots,\;I_{1}^{(l)}*\ldots*I_{k}^{(l)}\\}=\mathsf{Z}^{*}_{\mathcal{I}_{v}(\alpha)}(Ra)\subset\mathcal{F}(\mathcal{M}_{v}(\alpha))$
be the set of all rigid factorizations of $Ra$ in $\mathcal{I}_{v}(\alpha)$.
Using Section 7.2, we can choose a totally positive prime element
$q\in\mathcal{O}_{K}$ with $q\nmid\operatorname{nr}(a)$ and such that there
exists an $x\in A^{\times}$ with $\operatorname{nr}(x)=q$ and
$x\;\in\;T=\bigcap_{i=1}^{l}\bigcap_{j=1}^{k}\mathcal{O}_{l}(I_{j}^{(i)})\cap\mathcal{O}_{r}(I_{j}^{(i)}).$
We claim $\mathsf{L}(xa)=1+\mathsf{L}(a)$. The rigid factorizations of $Rxa$
in $\mathcal{I}_{v}(\alpha)$ are given by all possible transpositions of $x$
to any position in
$I_{1}^{(i)}*\ldots*I_{m}^{(i)}*x,$
for all $i\in[1,l]$. But, since $x\in T$, it follows from Section 7.3 that any
such rigid factorization is of the form
$I_{1}^{(i)}*\ldots*I_{m}^{(i)}*x*x^{-1}I_{m+1}^{(i)}x*\ldots*x^{-1}I_{k}^{(i)}x.$
for $m\in[0,k]$. We see that for the principal subproducts this does not
change anything except insert one additional factor (corresponding to $x$) at
some position. Thus, for each $i\in[1,l]$,
$I_{1}^{(i)}*\ldots*I_{m}^{(i)}*x*x^{-1}I_{m+1}^{(i)}x*\ldots*x^{-1}I_{k}^{(i)}x$
gives rise to a rigid factorization of $ax$ in $R^{\bullet}$ of length $l+1$
if and only if $I_{1}^{(i)}*\ldots*I_{k}^{(i)}$ gives rise to a rigid
factorization of $a$ of length $l$. ∎
## 8\. Acknowledgments
I thank Alfred Geroldinger for suggesting the topic, and him and Franz Halter-
Koch for reading preliminary versions of this manuscript and providing many
valuable comments.
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|
arxiv-papers
| 2013-06-04T15:41:51 |
2024-09-04T02:49:46.089759
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Daniel Smertnig",
"submitter": "Daniel Smertnig",
"url": "https://arxiv.org/abs/1306.0834"
}
|
1306.0951
|
# Velocity bunching in travelling wave accelerator with low acceleration
gradient††thanks: Supported by National Natural Science Foundation of China
(11205152)
HUANG Rui-Xuan HE Zhi-Gang1) LI Wei-Wei JIA Qi-Ka [email protected] National
Synchrotron Radiation Laboratory, University of Science and Technology of
China, Hefei, 230029, Anhui, China
###### Abstract
We present the analytical and simulated results concerning the influences of
the acceleration gradient in the velocity bunching process, which is a bunch
compression scheme that uses a traveling wave accelerating structure as a
compressor. Our study shows that the bunch compression application with low
acceleration gradient is more tolerant to phase jitter and more successful to
obtain compressed electron beam with symmetrical longitudinal distribution and
low energy spread. We also present a transverse emittance compensation scheme
to compensate the emittance growth caused by the increasing of the space
charge force in the compressing process that is easy to be adjusted for
different compressing factors.
###### keywords:
traveling wave accelerating structure, velocity bunching, acceleration
gradient, emittance compensation
###### pacs:
4
1.85.Ew, 41.85.Ct, 41.60.Cr
## 1 Introduction
In recent years, the demand for applications of high brightness - low
emittance, high current, with sub-picosecond pulse length electron beams has
increased dramatically. In the fourth generation synchrotron light source
community, high-brightness beams are needed for application to short
wavelength free electron lasers (FEL), as well as for inverse-Compton-
scattering (ICS) generation of short x-ray pulses. For studying novel
accelerating techniques such as plasma-based accelerators and generation of
coherent THz radiation, short electron bunches are also required.
Short bunches are commonly obtained by magnetic compression. In this scheme,
the bunch is compressed when drifting through a series of dipoles arranged in
a chicane configuration which can introduce an energy-dependent path length.
Therefore an electron bunch with the proper time-energy correlation can be
shortened in the chicane. The time-energy correlation along the bunch can be
tuned by means of an accelerating section upstream from the chicane. Great
progress has been made in this this field, but magnetic compression may
introduce momentum spread and transverse emittance dilution due to the bunch
self-interaction via coherent synchrotron radiation[2]. To obtain a smaller
and more symmetrical electron beam, a linear energy-time correlation is
required along the bunch, which can be realized by a accelerating structure at
a higher harmonic[3], with respect to the main accelerating linac RF.
Velocity bunching relies on the phase slippage between the electrons and the
rf wave that occurs during the acceleration of nonultrarelativistic electrons.
It was experimentally observed in photocathode rf gun[4] and proposed to
integrate the velocity bunching scheme in the next photoinjector designs using
a dedicated rf structure downstream of the rf electron source[5]. Previous
experimental works showed the compression ability of the velocity bunching
method[6, 7]. Furthermore, the emittance growth in the compressing process was
completely compensated by long solenoids[8] when the compression factor is 3.
This paper mainly focuses on the beam bunching in the traveling wave structure
with low acceleration gradient (4 MV/m) instead of high acceleration gradient
(normally 20 MV/m) in previous work. A brief analysis of the velocity bunching
mechanism is presented in the Section 2 firstly. In Section 3, the analytical
and simulated results of bunch compression within high and low acceleration
gradient accelerators are described, and the conclusion is figured out that a
traveling wave accelerating structure with low acceleration gradient is more
tolerant to phase jitter and easier to obtain compressed electron beam with
symmetrical longitudinal distribution and low energy spread in the velocity
bunching process. A transverse emittance compensation scheme is shown in
Section 4, which is easy to be adjusted for different compressing factors.
Section 5 presents a summary of this paper.
## 2 Velocity bunching mechanism
In the velocity bunching process, the longitudinal phase space rotation is
based on a correlated time-velocity chirp in the electron bunch, so that
electrons on the tail of the bunch are faster than electrons in the bunch
head. This rotation occurs inside a traveling rf wave of a long multicell rf
structure which applies an off crest energy chirp to the injected beam as well
as accelerates it.. This is possible if the injected beam is slightly slower
than the phase velocity of the rf wave so that when injected at the zero
crossing field phase it slips back to phases where the field is accelerating,
but is simultaneously chirped and compressed.
An electron in an rf traveling wave accelerating structure experiences the
longitudinal electric field:
$\displaystyle{E_{z}}={E_{0}}\sin(\phi)$ (1)
where $E_{0}$ is the peak field, $\phi=kz-\omega t+{\phi_{0}}$ is the phase of
the electron with respect to the wave and $\phi_{0}$ is the injection phase of
the electron with respect to the rf wave. The evolution of $\phi$ can be
expressed as a function of z:
$\displaystyle\frac{{d\phi}}{{dz}}=k-\omega\frac{{dt}}{{dz}}=k-\frac{\omega}{{\beta
c}}=k(1-\frac{\gamma}{{\sqrt{{\gamma^{2}}-1}}})$ (2)
The energy gradient can be written as[9]:
$\displaystyle\frac{{d\gamma}}{{dz}}=\alpha k\sin(\phi)$ (3)
where $\alpha\equiv e{E_{0}}/m{c^{2}}k$ is defined as dimensionless vector
potential amplitude of the wave. The equations (2) and (3) with the initial
conditions ${\gamma_{z=0}}={\gamma_{0}}$ and ${\phi_{z=0}}={\phi_{0}}$
describe the longitudinal motion of an electron in the rf structure. Using a
separation of variables approach, one can get
$\displaystyle\alpha\cos\phi+\gamma-\sqrt{{\gamma^{2}}-1}=C$ (4)
the $\phi$ can be expressed as a function of $\gamma$ :
$\displaystyle\phi(\gamma)=\arccos(\frac{{C-\gamma+\sqrt{{\gamma^{2}}-1}}}{\alpha})$
(5)
where the constant C is set by the initial conditions of the problem:
$C=\alpha\cos\phi_{0}+\gamma_{0}-\sqrt{{\gamma_{0}^{2}}-1}$ . The final phase
of electron at the exit of the accelerator is
$\displaystyle\phi_{e}(\gamma_{e})=\arccos(\frac{{C-\gamma_{e}+\sqrt{{\gamma_{e}^{2}}-1}}}{\alpha})$
(6)
If the $\gamma_{e}$ is high enough,
$\sqrt{{\gamma_{e}^{2}}-1}-\gamma_{e}\simeq 0$ . With the approximation
$\gamma_{0}-\sqrt{{\gamma_{0}^{2}}-1}\simeq 1/(2\gamma_{0})$ . Then, the final
phase becomes:
$\displaystyle\phi_{e}=\arccos(\cos\phi_{0}+1/(2\alpha\gamma_{0}))$ (7)
Expanding Eq. (7) to first order in the initial energy spread and initial
phase spread gives
$\displaystyle\Delta{\phi_{e}}=\frac{{\sin{\phi_{0}}}}{{\sin{\phi_{e}}}}\Delta{\phi_{0}}+\frac{1}{{2\alpha\gamma_{0}^{2}\sin{\phi_{e}}}}\Delta{\gamma_{0}}$
(8)
Hence depending upon the incoming energy and phase extents (initial bunch
length), the phase of injection in the rf structure $\phi_{0}$ can be tuned to
minimize the phase extent of extraction (final bunch length).
## 3 Bunch compression in low and high gradient accelerators
Based on equations (6) and (8), one can find that the compression factor is
less sensitive to the injection phase $\phi_{0}$ and more tolerant to the
phase jitter when the peak field $E_{0}$ is lower. Fig. 1 shows the simulated
phase scanning results at different acceleration gradients by using the code
ASTRA[10].
The rms bunch length as a function of relative phase (There is about 28.7
degrees phase shift between the two results. In the figure, we do a
translation for contrast).
The simulation shows a coincident result to the analytical conclusion. It
indicates that higher precision phase jitter control and power supply for the
solenoids to compensate the transverse emittance growth (The emittance
compensation scheme will be presented in next section) are needed in
experimental practice when the acceleration gradient is high. A lower
acceleration gradient in the velocity bunching process has been confirmed to
be more tolerant to phase jitter and power precision. Furthermore, with a
lower gradient, the compressed electron beam tends to develop more symmetrical
longitudinal distribution. Analytical and simulated results will be presented
in following paragraph.
The velocity bunching process can be shown as Fig. 2. To compress the electron
bunch, the injection phase should be set at the phase that is tens degrees off
the crest. The velocity of the initial beam is smaller than the rf wave (
$\beta_{\phi}=1$ ), there is phase slippage between the initial and compressed
beam. For a high gradient accelerator, the phase slippage is smaller, but to
obtain the same compression factor as in the low gradient accelerator, the
injection phase should be closer to the crest; when the injection phase is
farther from the crest, one can get a compressed beam with a single spike and
most of electrons are in the spike, but the compression factor is too large,
the emittance compensation is difficult at present. In this paper, the
accelerators are the normal S-band ( $2856~{}MHz$ ), $2/3\pi$ mode, $3~{}m$
long travelling wave accelerators. One also can set an appropriate injection
phase to make the phase slippage in the range of linearity within a short
accelerator, then the electron bunch is imposed a more linear negative energy
chirp (the speed of electrons in the head is slower than the speed of
electrons in the tail), and the electron bunch can be compressed enormously in
a drift space with appropriate length, which has been presented in
Reference[7] and named by the authors as “ballistic bunching”.
Velocity bunching process.
Eq. (1) can be expanded as:
$\displaystyle{E_{z}}={E_{0}}(\phi-\frac{{{\phi^{3}}}}{{3!}}+\frac{{{\phi^{5}}}}{{5!}}-\frac{{{\phi^{7}}}}{{7!}}+\frac{{{\phi^{9}}}}{{9!}}\cdots)$
(9)
If there is no higher order terms in the equation, the longitudinal
distribution state of the initial beam can be preserved during the velocity
bunching process when the initial energy spread is very small or it can be
treat as a linear one. Unfortunately, there are always higher order terms in
the equation. So reducing the $E_{0}$ may be an available way to prevent the
longitudinal distortion. By using the equations (2) and (3), the numerical
calculation results show the rationality of the analysis. However, the space
charge force and magnetic force are not considered. In the following content,
the simulation results will be presented.
Longitudinal distribution after compression by different $E_{0}$ should be
compared. The current profiles of the initial bunch and the compressed bunches
(with almost identical bunch length) are shown in Fig. 3.
The current profiles of the initial bunch (blue line) and the compressed
bunches (black line: compressed bunch in $4~{}MV/m$ gradient accelerator; red
line: compressed bunch in $20~{}MV/m$ gradient accelerator).
From the above figure it appears that a traveling wave accelerating structure
with low acceleration gradient is more successful to obtain compressed
electron with symmetrical longitudinal distribution. Fig. 4 shows the bunch
length evolutions along the beam direction when the acceleration gradients are
$E_{0}=4~{}MV/m$ and $E_{0}=20~{}MV/m$ .
RMS bunch length evolution along the longitudinal position.
As we know that to avoid the longitudinal distortion in magnetic compression,
an X-band accelerator system is normally used to linearize the energy chirp of
the electron beam[3], which is cost almost one million dollar. About this
problem in the velocity bunching process, theoretical and experimental works
are worth doing in detail.
When the $E_{0}$ is higher, the energy spread in the bunch is greater, because
the bunch compression is achieved by the velocity difference within the bunch.
Fig. 5 shows a contrast of the energy spread evolution when the gradient are
$E_{0}=4~{}MV/m$ and $E_{0}=20~{}MV/m$ .
RMS energy spread evolution at different acceleration gradients.
## 4 Emittance compensation
In this section, an emittance compensation scheme will be suggested during the
velocity bunching process.
The electric field of accelerators along the beam line.
The electric and magnetic fields along the beam line are shown in Fig. 6 and
Fig. 7. The peak acceleration gradients of the photocathode rf gun,
accelerator 1 and accelerator 2 are $80~{}MV/m$ , $4~{}MV/m$ and $20~{}MV/m$
respectively. Four solenoids are used for emittance compensation, $B_{0}$ is
located at $10~{}cm$ downstream from the photocathode, and the other three are
around the accelerators.
The magnetic field of solenoids along the beam line.
To prevent irreversible emittance growth during bunch compression, the crux is
to preserve the laminarity of the electron beam with an envelope propagated as
close as possible to a Brillouin-like flow, represented by an invariant
envelope[11] as generalized to the context of beam compression and thus
increasing $I$ during acceleration. Mismatches between the space charge forces
and the external focusing gradient produce slice envelope oscillations that
cause normalized emittance oscillations. It has been shown that in order to
keep such oscillations under control during the velocity bunching, the beam
has to be injected into the accelerator with a laminar envelope waist (
$\sigma^{{}^{\prime}}=0$ ) and the envelope has to be matched to the
accelerating and focusing gradients so that it can stay close to an
equilibrium mode[11]. Long solenoids around the accelerator are used to
provide the required focusing.
The transverse emittance and beam size evolutions along the beam direction are
shown in Fig. 8. For electron bunches with different compression factors at
different injection phases, one just need to scan the strength of the
solenoids to compensate the emittance growth as shown in Fig. 8.
The transverse emittance and beam size evolutions along the beam direction at
different injection phases, the optimal magnetic strengths of solenoids are
also shown in the figure.
Fig. 9 shows the current profiles for different compressed beam at different
injection phases. We should point out that the magnetic force of long
solenoids (around the accelerator) can affect the compression, which can be
found by contrasting the results in Fig. 9 and phase scanning result at
$E_{0}=4~{}MV/m$ without the long solenoids in Fig. 1.
The current profiles for different compressed beam at different injection
phases.
## 5 Summary and conclusion
Based on the analyses, we contrast the ASTRA simulated results of velocity
bunching process in travelling wave structure with low ( $4~{}MV/m$ ) and high
( $20~{}MV/m$ ) acceleration gradient. According to the results, the
conclusion is figured out that the bunch compression application with low
acceleration gradient is more tolerant to phase jitter and should be useful
for obtaining better performance beams with symmetrical longitudinal
distribution and low energy spread. Furthermore a successful improvement of
transverse emittance during compression is possible with optimized long
solenoids which is easily satisfied for different compressing factor.
###### Acknowledgements.
The authors thank Prof. PEI, Yuan-Ji for insightful discussions.
## References
* [1]
* [2] Bane K L F, et al. Phys. Rev. ST Accel. Beams, 2009, 12: 030704.
* [3] Emma P, LCLS-TN-01-1, November 14, 2001.
* [4] Wang X J, Qiu X, and Ben-Zvi I. Phys. Rev. E, 1996, 54: R3121.
* [5] Serafini L and Ferrario M. AIP Conf. Proc, 2001, 581: 87.
* [6] Piot P, Carr L, Graves W S and Loos H. Phys. Rev. ST Accel. Beams, 2003, 6: 033503.
* [7] Anderson S G, et al. Phys. Rev. ST Accel. Beams, 2005, 8: 014401.
* [8] Ferrarion M, et al. Phys. Rev. Lett, 2010, 104: 054801
* [9] Kim K J, Nucl. Instr. and Meth. A, 1989, 275: 201.
* [10] Flottmann K. ASTRA user manual, http://www.desy.de/~mpyflo/
* [11] Serafini L and Rosenzweig J B, Phys. Rev. E, 1997, 55: 7565.
|
arxiv-papers
| 2013-06-05T01:11:00 |
2024-09-04T02:49:46.119054
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Rui-Xuan Huang, Zhi-Gang He, Wei-Wei Li, Qi-Ka Jia",
"submitter": "Zhigang He",
"url": "https://arxiv.org/abs/1306.0951"
}
|
1306.1047
|
∎
11institutetext: College of Mathematics and Yantze Center of Mathematics,
Sichuan University, Chengdu 610064,P.R.China
11email: [email protected]
11email: [email protected]
# Saari’s Conjecture and Variational Minimal Solutions for $N$-Body
Problems††thanks: Supported partially by NSF of China
Yu Xiang Zhang Shiqing
(Received: date / Accepted: date)
###### Abstract
In this paper, we will prove Saari’s conjecture in a particular case by using
a arithmetic fact, and then, apply it to prove that for any given positive
masses, the variational minimal solutions of the N-body problem in
${\mathbb{R}}^{2}$ are precisely a relative equilibrium solution whose
configuration minimizes the function $IU^{2}$ in ${{\mathbb{R}}}^{2}$.
###### Keywords:
N-body problems Central configurations Saari’s conjecture Variational
minimization Homographic solutions
###### MSC:
11J17 11J71 34C25 42A05 70F10 70F15 70G75
## 1 Introduction
In 1970, Donald Saari saari1970bounded proposed the following conjecture : In
the Newtonian $N$-body problem, if the moment of inertia,
$I=\Sigma^{n}_{k=1}m_{k}|q_{k}|^{2}$, is constant, where
$q_{1},q_{2},\cdots,q_{n}$ represent the position vectors of the bodies of
masses $m_{1},\cdots,m_{n}$, then the corresponding solution is a relative
equilibrium. In other words: Newtonian particle systems of constant moment of
inertia rotate like rigid bodies.
A lot of energies have been spent to understand Saari’s conjecture, but most
of those works palmore1979relative ; palmore1981saari failed to achieve
crucial results. But there have been a few successes in the struggle to
understand Saari’s conjecture. McCord mccord2004saari proved that the
conjecture is true for three bodies of equal masses. Llibre and Pina
llibre2002saari gave an alternative proof of this case, but they never
published it.In particular, Moeckel moeckel2005computer ; moeckel2005proof
obtained a computer-assisted proof for the Newtonian three-body problem with
positive masses when physical space is $\mathbb{R}^{d}$ for all positive
integer $d\geq 2$. Diacu, P$\acute{\rm e}$rez-Chavela, and Santoprete
diacu2005saari showed that the conjectre is true for any $n$ in the collinear
case for potentials that depend only on the mutual distances between point
masses. There have been results, such as santoprete2004counterexample ;
roberts2006some ; schmah2007saari , which studied the conjecture in other
contexts than the Newtonian one.
Recently the interest in this conjecture has grown considerably due to the
discovery of the figure eight solution chenciner2000remarkable , which, as
numerical arguments show, has an approximately constant moment of inertia but
is not a relative equilibrium.
The variational minimal solutions of the N-body problem are attractive, since
they are nature from the viewpoint of the principle of least action.
Unfortunately, there were very few works about the variational minimal
solutions before 2000. It’s worth noticing that a lot of results have been got
by the action minimization methods in recent years, please see
barutello2004action ; chen2001action ; chen2003binary ; chen2008existence ;
Chenciner2002 ; chenciner2000remarkable ; chenciner2000minima ;
ferrario2004existence ; long2000geometric ; zhang2002variational ;
zhang2004nonplanar ; zhang2004new ; zhang2001minimizing and the references
there.
In this paper, we will first prove a arithmetic fact, then use it to prove
Saari’s conjecture in a particular case: the position $q_{i}(t)$ of $i$-th
point particle has the form
$q_{i}(t)=a_{i}\cos(\frac{2\pi}{T}t)+b_{i}\sin(\frac{2\pi}{T}t),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (1)
and $a_{i},b_{i}\in\mathbb{R}^{d}$ for all $i=1,\ldots,N$. In the last part,
we describe the shapes of the variational minimal solution of the N-body
problem in some constraints.
Let $\mathcal{X}_{d}$ denote the space of configurations of $N\geq 2$ point
particles with masses $m_{1},\ldots,m_{N}$ in Euclidean space $\mathbb{R}^{d}$
of dimension $d$, whose center of masses is at the origin, that is,
$\mathcal{X}_{d}=\\{q=(q_{1},\cdots,q_{N})\in(\mathbb{R}^{d})^{N}:\sum_{i=1}^{N}{m_{i}q_{i}}=0\\}$.
Let $\mathbb{T}=\mathbb{R}/T\mathbb{Z}$ denote the circle of length
$T=|\mathbb{T}|$, embedded as $\mathbb{T}\subset\mathbb{R}^{2}$.By the loop
space $\Lambda$, we mean the Sobolev space
$\Lambda=H^{1}(\mathbb{T},\mathcal{X}_{d})$. We consider the opposite of the
potential energy defined by
$U(q)=\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}.$ (2)
The kinetic energy is defined (on the tangent bundle of $\mathcal{X}_{d}$) by
$K=\sum_{i=1}^{N}{\frac{1}{2}{m_{i}|\dot{q}_{i}|^{2}}}$, the total energy is
$E=K-U$ and the Lagrangian is
$L(q,\dot{q})=L=K+U=\sum_{i}\frac{1}{2}m_{i}|\dot{q}|^{2}+\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}$.
Given the Lagrangian L, the positive definite functional
$\mathcal{{A}}:\Lambda\rightarrow\mathbb{R}\cup\\{+\infty\\}$ defined by
$\mathcal{{A}}(q)=\int_{\mathbb{T}}{L(q(t),\dot{q}(t))dt}.$ (3)
is termed action functional (or the Lagrangian action).
The action functional $\mathcal{{A}}$ is of class $C^{1}$ on the subspace
$\hat{\Lambda}\subset\Lambda$, which is collision-free space. Hence critical
point of $\mathcal{{A}}$ in $\hat{\Lambda}$ are T-periodic classical solutions
(of class $C^{2}$) of Newton’s equations
$m_{i}\ddot{q}_{i}=\frac{\partial U}{\partial q_{i}}.$ (4)
Definition chenciner2000remarkable . A configuration
$q=(q_{1},\cdots,q_{N})\in{\mathcal{X}}_{d}\setminus\Delta_{d}$ is called a
central configuration if there exists a constant $\lambda\in{\mathbb{R}}$ such
that
$\sum_{j=1,j\neq
k}^{N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}(q_{j}-q_{k})=-\lambda
m_{k}q_{k},1\leq k\leq N$ (5)
The value of $\lambda$ in (1.1) is uniquely determined by
$\lambda=\frac{U(q)}{I(q)}$ (6)
Where
$\Delta_{d}=\left\\{q=(q_{1},\cdots,q_{N})\in(\mathbb{R}^{d})^{N}:q_{j}=q_{k}~{}\mbox{for~{}some}~{}j\neq
k\right\\}$ (7) $I(q)=\sum_{1\leq j\leq
N}m_{j}|q_{j}|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(8)
It’s well known that the central configurations are the critical points of the
function $IU^{2}$, and $IU^{2}$ attains its infimum on
${\mathcal{X}}_{d}\setminus\Delta_{d}$. Furthermore, we know
moeckel1990central that
$inf_{{\mathcal{X}}_{2}\setminus\Delta_{2}}{IU^{2}}<inf_{{\mathcal{X}}_{1}\setminus\Delta_{1}}{{IU^{2}}}$and
$inf_{{\mathcal{X}}_{3}\setminus\Delta_{3}}{{IU^{2}}}<inf_{{\mathcal{X}}_{2}\setminus\Delta_{2}}{IU^{2}}$
when$N\geq 4$. It is well known that the homographic solutions derived by the
central configurations which minimize the function $IU^{2}$ when $N\geq 4$ and
${\mathbb{R}^{d}}={\mathbb{R}^{3}}$ are homothetic, furthermore, a homographic
motion in ${\mathbb{R}^{3}}$ which is not homothetic takes place in a fixed
planealbouy1997probleme ; arnold2006dynamical ; Chenciner2002 ;
wintner1941analytical .This is an important reason for us only to consider
$d=2$. In fact, A. Chenciner Chenciner2002 and Zhang-Zhou zhang2004nonplanar
had proved that the minimizer of Lagrangian action among (anti)symmetric loops
for the spatial $N$-body($N\geq 4$) problem is a collision-free non-planar
solution.
Notations. Let $\mathcal{S}={\\{q\in
H^{1}(\mathbb{T},(\mathbb{R}^{2})^{N}):\int_{\mathbb{T}}{q(t)dt}=0}\\}$. Let
$[x]$ denote the unique integer such that $x-1<[x]\leq x$ for any real $x$.
The difference $x-[x]$ is written as $\\{x\\}$ and satisfies $0\leq\\{x\\}<1$.
First of all, we need a famous arithmetic fact which belongs to Kronecker:
Lemma 1. If 1,$\theta_{1}$, …, $\theta_{n}$ are linearly independent over the
rational field, then the set {($\\{k\theta_{1}\\}$, …, $\\{k\theta_{n}\\}$):
$k\in\mathbb{N}\\}$ are dense in the $n$-dim unite cube
$\\{(\varphi_{1},\ldots,\varphi_{n}):0\leq\varphi_{i}\leq 1,i=1,\ldots,n\\}$.
The main results in this paper are the following theorems:
Theorem 1. Given $\theta_{1}$, …, $\theta_{n}$ and any $\epsilon>0$, there are
infinitely many integers $k\in\mathbb{N}$ such that
$\\{k\theta_{i}\\}<\epsilon$ or $\\{k\theta_{i}\\}>1-\epsilon$ for every
$i=1,\ldots,n$.
Theorem 2. If $U(q)\equiv const$, where $q=(q_{1},\cdots,q_{N})$,
$q_{i}(t)=a_{i}\cos(\frac{2\pi}{T}t)+b_{i}\sin(\frac{2\pi}{T}t),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (9)
and $a_{i},b_{i}\in\mathbb{R}^{d}$ for all $i=1,\ldots,N$. Then
$q_{i}(t)(i=1,\ldots,N)$ is is a rigid motion.
Corollary 1. Saari’s Conjecture is true if $i$-th point particle has mode of
motion
$q_{i}(t)=a_{i}\cos(\frac{2\pi}{T}t)+b_{i}\sin(\frac{2\pi}{T}t),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (10)
and $a_{i},b_{i}\in\mathbb{R}^{d}$, for all $i=1,\ldots,N$.
Corollary 2. Saari’s Conjecture is true if in a barycentric reference frame
the configurations formed by the bodies remain the central configurations all
the time.
Remark. If Finiteness of Central Configurations is true hampton2006finiteness
; smale1998mathematical ; wintner1941analytical , the proposition is obvious.
But we don’t need this hypothesis here.
Theorem 3. The regular solutions of the N-body problem which minimize the
functional ${\mathcal{A}}$ in $\mathcal{S}$ are precisely a relative
equilibrium solution whose configuration minimizes the function $IU^{2}$ in
${\mathbb{R}^{2}}$.
Remark. Compared with the result of A.Chenciner Chenciner2002 : For the planar
$N$-body problem, a relative equilibrium solution whose configuration
minimizes $I^{\frac{1}{2}}U$ is always a minimizer of the action among
(anti)symmetric loops; moreover, all minimizers are of this form provided
there exists only a finite number of similitude classes of $N$-body central
configurations. For the second part, he could only prove rigorously for 3-body
and 4-body problems, since we know that the Finiteness of Central
Configurations have only been proved for 3-body and 4-body problems until now.
## 2 The Proof
Proof of Theorem 1:
If all of $\theta_{1}$, …, $\theta_{n}$ are rational, the proposition is
obviously right. Hence, without loss of generality, we will suppose that
1,$\theta_{1}$, …, $\theta_{l}$($1\leq l\leq n$) are linearly independent over
the rational field and $\theta_{l+1}$, …, $\theta_{n}$ can be spanned by
rational linear combination. That is, we have
$\theta_{i}=x_{i}^{0}+\sum_{1\leq j\leq l}x_{i}^{j}\theta_{j}$, where $l<i\leq
n$ and $x_{i}^{j}$ are rational numbers for $0\leq j\leq l$. Let integer $p$
satisfy that all of $px_{i}^{0}$ are integers for $l<i$. It is easy to know
that 1,$p\theta_{1}$, …, $p\theta_{l}$ are still linearly independent over the
rational field. Then for any $\delta>0$, there are infinitely many integers
$k\in\mathbb{N}$ such that $\\{kp\theta_{i}\\}<\delta$ or
$\\{kp\theta_{i}\\}>1-\delta$ for every $i=1,\ldots,l$ by the
${\mathbf{Lemma~{}1}}$ in Section 1, and it is easy to know that
$\\{kp\theta_{i}\\}<C\delta$ or $\\{kp\theta_{i}\\}>1-C\delta$ for some
constant $C$ which only depends on $x_{i}^{j}$. So for any $\epsilon>0$, there
are infinitely many integers $k\in\mathbb{N}$ such that
$\\{k\theta_{i}\\}<\epsilon$ or $\\{k\theta_{i}\\}>1-\epsilon$ for every
$i=1,\ldots,n$.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Proof of Theorem 2:
Firstly, we represent $U(q(t))$ as Fourier series.
$\displaystyle U$ $\displaystyle=$ $\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{|q_{j}-q_{k}|}$ $\displaystyle=$ $\displaystyle\sum_{1\leq
j<k\leq
N}\frac{m_{j}m_{k}}{[|a_{j}-a_{k}|^{2}\cos^{2}(\frac{2\pi}{T}t)+|b_{j}-b_{k}|^{2}\sin^{2}(\frac{2\pi}{T}t)+2(a_{j}-a_{k})\cdot(b_{j}-b_{k})\sin(\frac{2\pi}{T}t)\cos(\frac{2\pi}{T}t)]^{\frac{1}{2}}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{[\frac{|a_{j}-a_{k}|^{2}+|b_{j}-b_{k}|^{2}}{2}+(\frac{|a_{j}-a_{k}|^{2}-|b_{j}-b_{k}|^{2}}{2})\cos(\frac{4\pi}{T}t)+(a_{j}-a_{k})\cdot(b_{j}-b_{k})\sin(\frac{4\pi}{T}t)]^{\frac{1}{2}}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{[\frac{|a_{j}-a_{k}|^{2}+|b_{j}-b_{k}|^{2}}{2}+(\frac{|a_{j}-a_{k}|^{2}-|b_{j}-b_{k}|^{2}}{2})\cos(\frac{4\pi}{T}t)+(a_{j}-a_{k})\cdot(b_{j}-b_{k})\sin(\frac{4\pi}{T}t)]^{\frac{1}{2}}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{[A_{jk}+B_{jk}\cos(\frac{4\pi}{T}t+\theta_{jk})]^{\frac{1}{2}}}$
where
$A_{jk}=\frac{|a_{j}-a_{k}|^{2}+|b_{j}-b_{k}|^{2}}{2}$ (11)
$B_{jk}=[(\frac{|a_{j}-a_{k}|^{2}-|b_{j}-b_{k}|^{2}}{2})^{2}+((a_{j}-a_{k})\cdot(b_{j}-b_{k}))^{2}]^{\frac{1}{2}}$
(12)
and $\theta_{jk}$ can be determined when $B_{jk}>0$.In the following, we will
prove $B_{jk}=0$ for any $j,k\in\\{{1,\ldots,N}\\}$. It is easy to know that
$A_{jk}\geq B_{jk}$, let $C_{jk}=\frac{B_{jk}}{A_{jk}}$, then we have
$\displaystyle U$ $\displaystyle=\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}[1+(-\frac{1}{2})C_{jk}\cos(\frac{4\pi}{T}t+\theta_{jk})+\ldots+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}(C_{jk})^{n}\cos^{n}(\frac{4\pi}{T}t+\theta_{jk})+\ldots]$
$\displaystyle=\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}\\{1+(-\frac{1}{2})C_{jk}\frac{\exp\sqrt{-1}(\frac{4\pi}{T}t+\theta_{jk})+\exp-\sqrt{-1}(\frac{4\pi}{T}t+\theta_{jk})}{2}+\ldots+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}(C_{jk})^{n}[\frac{\exp\sqrt{-1}(\frac{4\pi}{T}t+\theta_{jk})+\exp-\sqrt{-1}(\frac{4\pi}{T}t+\theta_{jk})}{2}]^{n}+\ldots\\}$
$\displaystyle=\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}[1+(-\frac{1}{2})C_{jk}\frac{\exp\sqrt{-1}(\frac{4\pi}{T}t+\theta_{jk})+\exp-\sqrt{-1}(\frac{4\pi}{T}t+\theta_{jk})}{2}+\ldots+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}(C_{jk})^{n}\frac{\sum_{0\leq
l\leq n}\left(\begin{array}[]{c}n\\\ l\\\
\end{array}\right)\exp\sqrt{-1}((\frac{4\pi}{T}t+\theta_{jk})(2l-n))}{2^{n}}+\ldots]$
$\displaystyle=\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}\\{1+\sum_{1\leq
l}\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-2l+1)}{(2l)!}(C_{jk})^{2l}\frac{\left(\begin{array}[]{c}2l\\\
l\\\ \end{array}\right)}{2^{2l}}+$ $\displaystyle\sum_{1\leq
n}\exp\sqrt{-1}(\frac{4n\pi}{T}t)[\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}\frac{(C_{jk})^{n}\exp\sqrt{-1}(n\theta_{jk})}{2^{n}}+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n-1)}{(n+2)!}\frac{(C_{jk})^{n+2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)\exp\sqrt{-1}(n\theta_{jk})}{2^{n+2}}+\ldots]+$
$\displaystyle\sum_{1\leq
n}\exp\sqrt{-1}(\frac{-4n\pi}{T}t)[\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}\frac{(C_{jk})^{n}\exp\sqrt{-1}(-n\theta_{jk})}{2^{n}}+$
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n-1)}{(n+2)!}\frac{(C_{jk})^{n+2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)\exp\sqrt{-1}(-n\theta_{jk})}{2^{n+2}}+\ldots]\\}$
Since $U\equiv const$, we have
$\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}[\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}\frac{(C_{jk})^{n}\exp\sqrt{-1}(n\theta_{jk})}{2^{n}}+$
(13)
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n-1)}{(n+2)!}\frac{(C_{jk})^{n+2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)\exp\sqrt{-1}(n\theta_{jk})}{2^{n+2}}+\ldots]=0$
$\displaystyle\sum_{1\leq j<k\leq
N}\frac{m_{j}m_{k}}{A_{jk}^{\frac{1}{2}}}[\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n+1)}{n!}\frac{(C_{jk})^{n}\exp-\sqrt{-1}(n\theta_{jk})}{2^{n}}+$
(14)
$\displaystyle\frac{(-\frac{1}{2})(-\frac{1}{2}-1)\ldots(-\frac{1}{2}-n-1)}{(n+2)!}\frac{(C_{jk})^{n+2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)\exp-\sqrt{-1}(n\theta_{jk})}{2^{n+2}}+\ldots]=0$
for any $n\geq 1$. Hence we have
$\sum_{1\leq j<k\leq N}D^{(n)}_{jk}\exp
2\pi\sqrt{-1}(n\frac{\theta_{jk}}{2\pi})=0$ (15)
for any $n\geq 1$, where
$D^{(n)}_{jk}=\frac{m_{j}m_{k}C_{jk}^{n}}{A_{jk}^{\frac{1}{2}}}[1+\frac{(\frac{1}{2}+n)(\frac{1}{2}+n+1)}{(n+1)(n+2)}\frac{(C_{jk})^{2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)}{2^{2}}+\ldots]$ (16)
We claim that the right side of (16) is convergent. In fact, let
$\displaystyle f_{jk}$
$\displaystyle=1+\frac{(\frac{1}{2}+n)(\frac{1}{2}+n+1)}{(n+1)(n+2)}\frac{(C_{jk})^{2}\left(\begin{array}[]{c}n+2\\\
n+1\\\ \end{array}\right)}{2^{2}}+\ldots$
$\displaystyle=1+c_{1}(C_{jk})^{2}+c_{2}(C_{jk})^{4}+\ldots+c_{l}(C_{jk})^{2l}+\ldots$
where
$c_{l}=\frac{(\frac{1}{2}+n)(\frac{1}{2}+n+1)\ldots(2l-1-\frac{1}{2}+n)(2l-\frac{1}{2}+n)}{(n+1)(n+2)\ldots(n+2l-1)(n+2l)}\frac{\left(\begin{array}[]{c}n+2l\\\
n+l\\\ \end{array}\right)}{2^{2l}}$ (17)
Then we have
$\frac{c_{l+1}}{c_{l}}=\frac{(2l+\frac{1}{2}+n)(2l+1+\frac{1}{2}+n)}{4(l+1)(l+1+n)}$
(18) $\lim_{l\rightarrow\infty}\frac{c_{l+1}}{c_{l}}=1$ (19)
Hence the series (16) is convergent when $(C_{jk})^{2}<1$. Furthermore, we can
prove the convergence of the series (16) by using Gauss’ text when
$(C_{jk})^{2}=1$. In fact, we have
$\frac{c_{l}}{c_{l+1}}=1+\frac{\frac{n+2}{2}}{l}+\beta_{l}$ (20)
where
$\beta_{l}=-\frac{2n^{2}+2n+\frac{3}{4}+\frac{(n+\frac{1}{2})(n+\frac{3}{2})(n+2)}{2l}}{4l^{2}+2l(n+2)+(n+\frac{1}{2})(n+\frac{3}{2})}$
(21)
Since $\frac{n+2}{2}>1$ and $|\beta_{l}|\sim\frac{c}{l^{2}}$, where $c$ is a
constant. Then it is easy to know that the series (16) is convergent when
$C_{jk}^{2}=1$.
From Theorem 1, we know there exists some $n$ such that
$n\frac{\theta_{jk}}{2\pi}=k_{n}+\varphi_{jk}$, where $k_{n}$ is an integer
and $-\frac{1}{4}<\varphi_{jk}<\frac{1}{4}$. Since $D^{(n)}_{jk}\geq 0$, there
must be $D^{(n)}_{jk}=0$ for any $j,k$ by (13). So we have $C_{jk}=0$,
$|q_{j}-q_{k}|\equiv\sqrt{A_{jk}}$. Hence $q_{i}(t)(i=1,\ldots,N)$ is a rigid
motion.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Remark. It is easy to know that the same result is still true when the
potential function is defined by
$U(q)=\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|^{\alpha}}}$ for any
$\alpha>0$ and if $U(q(t))$ is a trigonometric polynomial when $i$-th point
particle has mode of motion
$q_{i}(t)=a_{i}\cos(\frac{2\pi}{T}t)+b_{i}\sin(\frac{2\pi}{T}t),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (22)
and $a_{i},b_{i}\in\mathbb{R}^{d}$, for all $i=1,\ldots,N$.
Proof of Corollary 1:
It is well known that Newtonian particle systems of constant moment of inertia
must satisfy that $U$ is constant.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Proof of Corollary 2:
From the conditions of $\mathbf{Corollary~{}2}$, we have
$m_{i}\ddot{q}_{i}=-\lambda m_{i}q_{i}.$ (23)
where $\lambda=\frac{U(q)}{I(q)}$ is a constant. Then it is easy to know that
$q_{i}(t)=a_{i}\cos(\sqrt{\lambda}t)+b_{i}\sin(\sqrt{\lambda}t),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (24)
for some $a_{i},b_{i}\in\mathbb{R}^{d}$, $i=1,\ldots,N$.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Proof of Theorem 3:
We have
$\displaystyle{\mathcal{A}}(q)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{T}}{[\sum_{i}\frac{1}{2}m_{i}|\dot{q_{i}}|^{2}+\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}]dt}$
$\displaystyle\geq$
$\displaystyle\int_{\mathbb{T}}{[(\frac{2\pi}{T})^{2}\sum_{i}\frac{1}{2}m_{i}|{q_{i}}|^{2}+\sum_{i<j}{\frac{m_{i}m_{j}}{|q_{i}-q_{j}|}}]dt}$
$\displaystyle=$
$\displaystyle\int_{\mathbb{T}}{[\frac{1}{2}(\frac{2\pi}{T})^{2}I(q)+\frac{1}{2}U(q)+\frac{1}{2}U(q)]dt}$
$\displaystyle\geq$ $\displaystyle
3\int_{\mathbb{T}}{[(\frac{1}{2})^{3}(\frac{2\pi}{T})^{2}I(q)U^{2}(q)]^{\frac{1}{3}}dt}$
$\displaystyle\geq$ $\displaystyle
3[\frac{(inf_{\mathcal{X}_{2}\setminus\Delta_{2}}{IU^{2}})\pi^{2}}{2}]^{\frac{1}{3}}T^{\frac{1}{3}}$
then,
${\mathcal{A}}(q)=3[\frac{(inf_{\mathcal{X}_{2}\setminus\Delta_{2}}{IU^{2}})\pi^{2}}{2}]^{\frac{1}{3}}T^{\frac{1}{3}}$
if and only if:
${(\textit{i})}.$ there exist $a_{i},b_{i}\in\mathbb{R}^{2}$, for all
$i=1,\ldots,N$, such that
$q_{i}(t)=a_{i}\cos(\frac{2\pi}{T}t)+b_{i}\sin(\frac{2\pi}{T}t),~{}~{}~{}~{}~{}~{}\forall
t\in\mathbb{T}.$ (25)
${(\textit{ii})}.$ $(\frac{2\pi}{T})^{2}I(q)=U(q).$
${(\textit{iii})}.$ $q$ minimizes the function $IU^{2}$.
By ${(\textit{ii})}$ and ${(\textit{iii})}$ we know $I(q)\equiv
const,U(q)\equiv const$, and $q(t)$ is always a central configuration. Then
$q$ is a relative equilibrium solution whose configuration minimizes the
function $IU^{2}$ by Theorem 2.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Box$
Remark. If the Finiteness of Central Configurations is true, ${(\textit{ii})}$
and ${(\textit{iii})}$ are sufficient to prove Theorem 3. But this problem
don’t need so strong Conjecture, it just need the weaker assumption: the
minimum points of the function $IU^{2}$ are finite. However, as far as we know
there doesn’t exist rigorous proof under the weaker assumption. So we prove
that Saari’s conjecture in the particular case (9) to get over the obstacle.
## References
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* (8) Chenciner, A., Montgomery, R.: A remarkable periodic solution of the three-body problem in the case of equal masses. Annals of Mathematics-Second Series 152(3), 881–902 (2000)
* (9) Chenciner, A., Venturelli, A.: Minima de l’intégrale d’action du problème newtoniende 4 corps de masses égales dans r3: Orbites’ hip-hop’. Celestial Mechanics and Dynamical Astronomy 77(2), 139–151 (2000)
* (10) Diacu, F., Perez-Chavela, E., Santopetre, M.: Saari’s conjecture of the n-body problem in the collinear case. Trans. Amer. Math. Soc 357, 4215–4223 (2005)
* (11) Ferrario, D.L., Terracini, S.: On the existence of collisionless equivariant minimizers for the classical n-body problem. Inventiones mathematicae 155(2), 305–362 (2004)
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* (22) Saari, D.G.: On bounded solutions of the n-body problem. In: Periodic Orbits, Stability and Resonances, pp. 76–81. Springer (1970)
* (23) Santoprete, M.: A counterexample to a generalized saari’s conjecture with a continuum of central configurations. Celestial Mechanics and Dynamical Astronomy 89(4), 357–364 (2004)
* (24) Schmah, T., Stoica, C.: Saari’s conjecture is true for generic vector fields. Transactions of the American Mathematical Society 359(9), 4429–4448 (2007)
* (25) Smale, S.: Mathematical problems for the next century. The Mathematical Intelligencer 20(2), 7–15 (1998)
* (26) Wintner, A.: The analytical foundations of celestial mechanics. Princeton, NJ, Princeton university press; London, H. Milford, Oxford university press, 1941. 1 (1941)
* (27) Zhang, S., Zhou, Q.: Variational methods for the choreography solution to the three-body problem. Science in China Series A: Mathematics 45(5), 594–597 (2002)
* (28) Zhang, S., Zhou, Q.: Nonplanar and noncollision periodic solutions for n-body problems. Discrete and Continuous Dynamical Systems-A 10(3), 679–686 (2004)
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|
arxiv-papers
| 2013-06-05T10:26:14 |
2024-09-04T02:49:46.126730
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yu Xiang and Zhang Shiqing",
"submitter": "Shiqing Zhang",
"url": "https://arxiv.org/abs/1306.1047"
}
|
1306.1053
|
###### Abstract
Halo models of the large scale structure of the Universe are critically
examined, focusing on the definition of halos as smooth distributions of cold
dark matter. This definition is essentially based on the results of
cosmological $N$-body simulations. By a careful analysis of the standard
assumptions of halo models and $N$-body simulations and by taking into account
previous studies of self-similarity of the cosmic web structure, we conclude
that $N$-body cosmological simulations are not fully reliable in the range of
scales where halos appear. Therefore, to have a consistent definition of halos
is necessary either to define them as entities of arbitrary size with a grainy
rather than smooth structure or to define their size in terms of small-scale
baryonic physics.
###### keywords:
large-scale structure; dark matter halo; $N$-body simulation
10.3390/galaxies1010031 1 Received: 15 April 2013; in revised form: 22 May
2013 / Accepted: 23 May 2013 /
Published: 29 May 2013 Halo Models of Large Scale Structure and Reliability of
Cosmological $N$-Body Simulations José Gaite .
## 1 Introduction
Halo models of the large scale structure of matter are now very popular, as
simple searches on the Internet show: for example, a Google search with the
three words “halo model cosmology” produces 4,680,000 results, and an ArXiv
search for “halo model” yields “too many hits” and recommends a more specific
search. Naturally, the halos to which halo models refer are dark matter halos,
initially introduced to model the invisible matter surrounding galaxies.
However, present halo models are concerned with the large scale distribution
of halos in space as well as with the distribution of matter within a single
halo. In this respect, the modern report on halo models by Cooray and Sheth
CooSh traces the appearance of these models to 1952, in a paper about the
spatial distribution of galaxies written by Neyman and Scott NeySc , where
they argue that it is “useful to think of the galaxy distribution as being
made up of distinct clusters with a range of sizes.” Thus, Neyman and Scott
propose that the statistical theory of the galaxy distribution is simplified
by separating the full distribution into one part corresponding to the
distribution of galaxies within clusters and another corresponding to the
distribution of cluster centers in space. In particular, they favor “quasi-
uniform” distributions of clusters and mention the Poisson distribution. Of
course, this hypothesis is not in accord with modern ideas, in which the
strong clustering of clusters plays a fundamental role in the large scale
structure of matter. When galaxy clusters are replaced with dark matter halos
and we consider the distribution of halo centers in space, we have the basic
halo model. This distribution is indeed assumed to be non-uniform and the
study of halo correlation functions is an important part of halo models CooSh
.
At any rate, halo models are not sufficiently supported by observations of the
large scale structure of matter, inasmuch as dark matter has not been observed
directly and the indirect observations of it, through gravity, are strongly
model dependent. Actually, our knowledge of the dark matter distribution is
mainly due to the results of cosmological $N$-body simulations. As
collisionless cold dark matter is assumed to be the main component of the
cosmic fluid and its dynamics is very simple to simulate, many $N$-body
simulations with large $N$ have been carried out and the type of structure to
which they give rise is well studied. Halo models seem to adapt well to this
type of structure, since the particles (or bodies) tend to form smooth
distributions on small scales that one can associate with halos, and these
halos are, on larger scales, clustered in irregular distributions with
definite features, such as filaments. Therefore, there appear distinct halos
with a range of sizes, which make up the large scale structure of matter.
As the cold dark matter (CDM) dynamics is purely gravitational and does not
introduce any scale, one may ask what determines the range of sizes of halos.
This is one of the points we intend to unveil. In fact, the absence of scales
immediately suggests that the CDM distribution should be scale invariant,
namely, a fractal distribution. In fact, fractal models and halo models of the
large scale structure can be merged in a model of fractal distributions of
halos EPL . However, the resulting model is actually a multifractal model in
which halos are characterized by point-like singularities and, if the full
matter distribution is statistically homogeneous, halos consist of grainy
rather than smooth mass distributions, of arbitrary size. Point-like
singularities can also be present at the centers of smooth halos, but halos of
this type have, in addition, smooth components and definite sizes, in contrast
with multifractal halos, in which both definite sizes and smoothness are
precluded by scale invariance and statistical homogeneity.
Of course, multifractal singularities only appear in continuous matter
distributions, and $N$-body simulations amount to a discretization of matter
that breaks the scale invariance of CDM dynamics. The discreteness limitations
of cosmological $N$-body simulations have been studied by Splinter et al.
Splinter . Their conclusions are very relevant to our problem and are
reproduced at the end, after examining the definition of halos in cosmological
$N$-body simulations.
In summary, our main concern is to find out if halo models of the large scale
structure are well justified, specifically, if smooth halos with a range of
sizes are well justified by cosmological $N$-body simulations. Regarding this
problem, we have to assess the spherical collapse and virialization model that
is supposed to lead to the formation of halos. The large scale distribution of
matter, whether made up by halos or not, displays definite features, namely,
filaments and walls, which constitute the famous “cosmic web” structure. This
structure is reproduced by the adhesion model, which is worth studying with
regard to the formation of halos. Taking account of the conclusions from our
study of the spherical collapse and adhesion models, the results of $N$-body
simulations are reconsidered, to establish the role of the breaking of scale
invariance in them and its consequences for halo models. At the end, some
critical conclusions are presented and discussed.
## 2 The Spherical Collapse Model and Virialization
The spherical collapse model is a simplified model of gravitational collapse
that is supposed to give rise to the simplest type of halo germs. Its main
interest is that the spherical collapse, namely, the collapse of an initial
matter distribution that is spherically symmetric, is soluble, in the sense
that it consists of the one-dimensional gravitational dynamics of spherical
shells. In particular, let us consider, in a spatially flat
Friedmann–Lemaitre–Robertson–Walker universe, the collapse of an initial “top-
hat” overdensity, namely, a sphere with constant density slightly larger than
the background density. Its evolution has a straightforward solution in
Lagrangian coordinates, and it undergoes several stages: (i) an initial
expansion that follows the Hubble expansion at a slower rate; (ii) as the
expansion decelerates, the “top-hat” overdensity reaches a maximum size and
begins to contract; (iii) then, it collapses and, if it stays spherically
symmetric, its size tends to zero, but in practice it is supposed to virialize
and stabilize at some non-zero size.
Thus, the spherical collapse model leads to the formation of what one may call
a spherical halo, but the model per se does not prescribe its size. It just
assumes that a different process, namely, “virialization”, takes over at the
end and produces an object of a definite size. On the other hand, there is no
way to predict this size, so the virialized object is supposed to be a
spherical object with a radius that is precisely one half of the turn-around
radius. This has the advantage of linking the sizes of the final virialized
halos to the initial spectrum of linear overdensities. On the other hand, this
link may look suspicious, because virialization embodies the nonlinear and
chaotic nature of gravitational dynamics, and chaos implies erasure of initial
conditions. Therefore, it is necessary to look into the meaning of
virialization in some detail.
### 2.1 Virialization
Naturally, the stage of contraction in the spherical collapse of a uniform
sphere produces homologous spheres of decreasing size and increasing density
until reaching zero size and infinite density. This is as true for CDM as for
a gas, assuming that the process is adiabatic, namely, that the entropy does
not increase in it. A point of infinite density is a singularity, but one can
predict, under the assumption of reversibility, that it is followed by a
rebound and an expansion, until the sphere’s radius gets back to the turn-
around radius. Therefore, the motion is oscillatory. However, this
reversibility is not realistic and one must expect irreversibility and entropy
growth, and, plausibly, the formation of a stable state of smaller size. This
stable, collapsed state must fulfill the (scalar) virial theorem, namely,
$E=-K+3PV$, where $E$ is its total energy, $K$ its (average) kinetic energy,
$V$ its (average) volume and $P$ the external pressure, which vanishes in the
“top-hat” collapse model. On the other hand, the reversible oscillatory motion
also fulfills the virial theorem, so “virialization” is actually a misnomer.
In fact, there is no way in which the virial theorem can select a preferred
size for the stable state. This stable, collapsed state is rather the
consequence of the type of processes known as “violent relaxation”
(redistribution in a rapidly varying gravitational potential) or “chaotic
mixing” (exponential spreading of trajectories in phase space). We must
consider these processes to unveil how the collapse proceeds.
Of course, relaxation and mixing take place because the initial “top-hat”
overdensity cannot be taken totally uniform and must contain density
perturbations inside. The inner overdensities (or underdensities) evolve and
grow just as the total overdensity does. Therefore, the spherical symmetry is
lost and the infalling particles do not converge to a point. Anyway, some
effects of non-uniformity can be studied within the spherical collapse model,
so that solubility in Lagrangian coordinates is maintained until shell-
crossing takes place. Sánchez-Conde _et al_. Sanchez-Conde undertake this
study, after criticizing the standard assumptions of the spherical collapse
model, in particular the stabilization radius at one half of the turn-around
radius (“the justification …is poor and lack a solid theoretical background”).
Their results do not support the common assumptions, namely, the collapse
factor $1/2$ and the time of “virialization”. Presumably, the breaking of
spherical symmetry makes the common assumptions even less justifiable.
As a matter of principle, the characteristics of a stable state that has
undergone a relaxation process in which the thermodynamical entropy grows
cannot be determined by the initial conditions. Actually, entropy growth is
equivalent to loss of information, and the more entropy, the less information
about the process that has led to the stable state. Indeed, as we know from
thermodynamics, the most stable state is the one with the maximum entropy
allowed by the boundary conditions. In the gravitational case, the maximum-
entropy spherically-symmetric states are called isothermal spheres. However,
these are only local maxima of entropy, and there is no global maximum. This
is a consequence of the “gravothermal catastrophe”: a sufficient large central
density tends to keep growing indefinitely (such an isothermal sphere has
negative specific heat). This shows, on the one hand, that there can be
temporary stable states of various sizes and, on the other hand, that one must
inevitably deal with singularities in the end.
At any rate, since the spherical collapse is unstable against non-radial
perturbations and, furthermore, cosmological $N$-body simulations show that
gravitational collapse is usually anisotropic and involves tidal interactions
with the surrounding matter, we consider next a more advanced model of
structure formation that includes these aspects.
## 3 The Zeldovich Approximation and the Adhesion Model
The Zeldovich approximation somewhat resembles the spherical collapse model,
insofar as it is soluble and indeed consists of a very simple dynamics in
Lagrangian coordinates, which holds until (non-spherical) shells cross.
However, the Zeldovich approximation, complemented by the adhesion model,
which gives a simple prescription for the dynamics after shell-crossing,
constitutes a more powerful and successful approach to the formation of the
large scale structure of the Universe GSS . Interestingly, the Zeldovich
approximation implies that “spherical collapse is specifically forbidden” GSS
, because its probability vanishes.
The Zeldovich approximation can be understood as the first order perturbative
approximation to the gravitational motion in Lagrangian coordinates Lagrangian
; namely, the motion is given by
${\mbox{\boldmath{$x$}}}={\mbox{\boldmath{$x$}}}_{0}+D(t)\,{\mbox{\boldmath{$g$}}}({\mbox{\boldmath{$x$}}}_{0})$,
where $x$ is the comoving coordinate, $g$ the peculiar gravitational field,
and $D(t)$ the growth rate of linear density fluctuations. Redefining time as
$\tau=D(t)$, the motion is simply uniform linear motion, with a constant
velocity given by the initial peculiar gravitational field. Naturally, nearby
particles have different velocities, and, as the linear solution is prolonged
into the nonlinear regime, trajectories cross at caustic surfaces, called
“Zeldovich pancakes” in this context GSS . On the other hand, the formation of
caustics is a general feature of irrotational dust models, in Newtonian
dynamics or in general relativity, so it is reasonable to assume that caustics
are indeed the first cosmological structures.
After a set of particles merge at a caustic, their subsequent evolution is
undefined. If no kinetic energy is dissipated (adiabatic collapse), the
particles cross (or rebound), like in a spherical collapse. Hence, if there
was no dissipation in caustics, there would be no real structure formation.
Therefore, the linear motion in the Zeldovich approximation is supplemented
with a viscosity term, resulting in the equation
$\frac{d\widetilde{\mbox{\boldmath{$u$}}}}{d\tau}\equiv\frac{\partial\widetilde{\mbox{\boldmath{$u$}}}}{\partial\tau}+\widetilde{\mbox{\boldmath{$u$}}}\cdot\nabla\widetilde{\mbox{\boldmath{$u$}}}=\nu\nabla^{2}\widetilde{\mbox{\boldmath{$u$}}}$
(1)
where $\widetilde{\mbox{\boldmath{$u$}}}$ is the peculiar velocity in
$\tau$-time. Let us remark that dissipation and viscosity in CDM dynamics may
not have the same origin as in normal baryonic fluids GZ ; BD . To Equation
(1), it must be added the no-vorticity (potential flow) condition,
$\nabla\times\widetilde{\mbox{\boldmath{$u$}}}=0$, implied by
$\nabla\times{\mbox{\boldmath{$g$}}}({\mbox{\boldmath{$x$}}}_{0})=0$. Thus,
Equation (1) is the three-dimensional form of the Burgers equation for very
compressible (pressureless) fluids GSS . The limit ${\nu\rightarrow 0}$ might
seem to recover the caustic-crossing solutions but actually is the high
Reynolds-number limit and gives rise to Burgers turbulence. Whereas
incompressible turbulence is associated with the development of vorticity,
Burgers turbulence is associated with the development of shock fronts, namely,
discontinuities of the velocity. These discontinuities arise at caustics and
give rise to matter accumulation by inelastic collision of particles. The
viscosity $\nu$ measures the thickness of shock fronts, which become true
singularities in the limit ${\nu\rightarrow 0}$. This is the adhesion model,
which produces, with the appropriate random initial conditions, a
characteristic network of sheets, filaments and nodes, called “the cosmic web”
GSS . This distribution of caustics is actually self-similar, with
multifractal features V-Frisch . A simulation of the Burgers equation in the
limit ${\nu\rightarrow 0}$ is shown in Figure 1.
Figure 1: Cosmic web produced by the Burgers equation with random initial
conditions.
One might think of identifying the nodes of the cosmic web with halos, but the
nodes produced by the adhesion model are just Dirac-delta singularities of
vanishing size. If ${\nu}$ is not zero, nodes have a size proportional to
${\nu}$. This size will be negligible if ${\nu}$ is identified with molecular
(baryon) viscosity, but it may be the right size if “viscosity” is due to the
mechanism proposed by Buchert and Domínguez BD . At any rate, nodes are just
one of the three types of singularities predicted by the adhesion model, and
the other types, namely, filaments and sheets, cannot be identified with
halos.
## 4 $N$-Body Simulations
$N$-body simulations of gravitational dynamics DR have been very helpful in
the study of large scale structure formation and, in a way, have been
complementary to observations, since observations are biased towards the
baryonic matter, while $N$-body simulations take full account of the dark
matter, in particular, of non-baryonic matter. Collisionless non-baryonic CDM
is only subjected to gravitational forces, so it is fairly simple to simulate
its dynamics. Moreover, due to the advances in both hardware and software, now
it is possible to simulate the combined dynamics of CDM and baryon gas with
relatively good resolution. At any rate, the large scale dynamics is ruled by
the dominant component, namely, CDM. We employ the data from a large
simulation of CDM and gas carried out by the Mare Nostrum supercomputer in
Barcelona Gott1 . This simulation contains $1024^{3}$ dark matter particles
and the same number of gas particles in a comoving cube of 500 $h^{-1}$ Mpc
edges. Later, we also employ, for a comparison, the CDM-only Virgo Consortium
GIF2 simulation, with $400^{3}$ particles in a 110 $h^{-1}$ Mpc cube, as
described by Gao _et al_. GIF2 . Both simulations have already been the object
of multifractal analyses, by means of counts-in-cells I1 ; I2 , and we can
take advantage of the methods and results of these analyses.
Naturally, we use the zero-redshift (present time) snapshots of either
simulation. A representative image of the matter distribution is given by the
distribution in a slice, see Figure 2. This slice is prepared as follows.
First of all, we focus on the dominant CDM component of the Mare Nostrum
simulation. Since the number of particles is very large, it is useful to
coarse-grain the particle distribution to obtain a density representation I1 ;
I2 . The coarse-graining is carried out by using a mesh of cubes with length
such that the average density is one particle per cube I2 , so the mesh-cube’s
edge is $1/1024$ of the simulation cube’s edge. Furthermore, given that the
homogeneity scale is about 3% of the simulation cube’s edge I2 , a quarter of
a full slice is adequate to perceive the features of the matter distribution
(the lower left quarter is taken). In summary, our slice consists of
$512\times 512$ mesh-cubes, and the density is given by the number of
particles in each one. To each mesh-cube corresponds a pixel in the image,
with an intensity proportional to the density in the mesh-cube. In addition,
in the slice represented in Figure 2, the density field has been cut off at
$\varrho=4$ ($\varrho=1$ is the average density), so that the contrast does
not render invisible the pixels corresponding to low-density cubes and one can
appreciate the full cosmic-web structure. Indeed, Figure 2 shows a self-
similar structure that looks like the structure in Figure 1.
Figure 2: Dark matter slice of Mare Nostrum $N$-body simulation (cut off at
$\varrho=4$).
Nevertheless, one can wonder what is the appearance of the full density,
namely, including $\varrho>4$. To see this, let us raise the cutoff to
$\varrho=256$, a value that is only exceeded by a few mesh-cubes and that, at
the same time, preserves some contrast in the picture. Amazingly, this change
makes the cosmic-web structure of Figure 2 vanish and the new image, Figure 3,
resembles what one can see in a starry night, namely, distinct bright spots
with a (small) range of sizes. Naturally, these bright spots must be
identified with dark matter halos rather than with stars. To understand the
transformation from a cosmic web structure to a distribution of halos of
similar size, we must spell out the various scales that play a role in
cosmological $N$-body simulations.
Figure 3: The same slice of Figure 2 but cut off at $\varrho=256$. Notice the
halos.
Of course, the first scale is the simulation cube’s edge, but we can refer the
remaining scales to it and, hence, assign it the value of unity. The next
scale is the discretization length $N^{-1/3}$, namely, the length of the edge
of the mesh cube such that there is one particle per cube on average.
Naturally, the mesh of these cubes is used for the counts-in-cells and
$N^{-1/3}=1/1024$ is the coarse-graining length (in physical units,
$0.5\;h^{-1}$ Mpc, in the Mare Nostrum simulation). There is another scale:
the gravity cutoff or softening length, which is necessary to avoid numerical
problems when two particles get close and the force between them gets too
large. The softening length is of the order of some Kpc, in particular, it is
$15\;h^{-1}$ Kpc in the Mare Nostrum simulation. (In the GIF2 simulation, the
discretization length is $0.25\;h^{-1}$ Mpc and the softening length is
$=7\;h^{-1}$ Kpc, so their ratio is almost the same.) Besides, there are other
scales in the initial conditions, but we are only concerned with the scales in
the dynamics. In summary, while CDM dynamics is scale free, we see that
$N$-body simulations of it introduce two scales. Therefore, the appearance of
halos in Figure 3 must be due to the presence of these scales, which prevent
the formation of a truly self-similar cosmic web. Indeed, scale invariance can
certainly be measured for scales between the homogeneity scale and the
discretization scale I1 ; I2 . In addition, the halos in Figure 3 have sizes
of the order of the discretization length, which is the larger of the two
scales.
While Figure 2 or Figure 3 show the matter distribution between the
homogeneity scale and the discretization scale, they do not show the
distribution on smaller scales, that is, they do not show what one might call
the mass distribution inside halos. The most populated mesh-cube is located at
the position $(380/512,159/512)$ in the slice and contains $2466$ particles
(to be compared with one per mesh-cube, on average). Nearby cubes are
overpopulated as well, so we define the heaviest halo as the one formed by all
of them together. To be precise, we choose $4\times 4$ adjacent cubes of the
slice and we display the (projected) particle positions in them in Figure 4.
Other halos in the slice have a similar aspect. Patently, the matter
distribution on these small scales is very different from the cosmic-web
distribution between the homogeneity scale and the discretization scale: now
we perceive a nearly smooth distribution (this also happens for the GIF2
simulation, naturally). The smoothness of the distribution inside halos is
presumably due to the gravity softening. However, the scale that seems to mark
the transition from an irregular cosmic-web distribution to a smooth
distribution is the discretization scale. The transition over this scale has
an even more definite and sharp effect on the statistics of halo masses, as we
show next.
Figure 4: Zooming in on the largest halo: $4\times 4$ pixels at position
$(0.74,0.31)$ in Figure 3.
### 4.1 Halo Mass Statistics
Since we have seen that the halo sizes in $N$-body simulations are about the
discretization scale, it is appropriate to define halos so that they have
precisely this size, for the sake of simplification. Then, one can easily
measure by counts-in-cells the mass function of halos, namely, the number of
halos of a given mass. The halo mass functions of the Mare Nostrum or GIF2
zero-redshift snapshots follow very definite power laws precisely when the
halos have the size of the discretization scale I1 ; I2 . This is shown in
Figure 5, where the respective constant size halo mass functions for variable
size are displayed: power laws for size $N^{-1/3}$ are fulfilled by all halos
except the most massive ones. As the halo size increases, the straight line
bends, becoming convex from above, as expected in a multifractal distribution
I1 ; I2 . On the contrary, as the halo size decreases, the straight line
becomes concave from above, because the number of mesh-cubes with few
particles must then increase. Notice that the sizes chosen in the GIF2
simulation are not exact multiples of $N^{-1/3}$, in consonance with the
characteristics of this simulation and our numerical methods: the GIF2
simulation contains $400^{3}$ dark matter particles and we use powers of 2 I1
.
Figure 5: Constant size halo mass functions for variable size $\lambda$, for
the Mare Nostrum, above, and GIF2, below, $z=0$ snapshots. Abscissas: halo
mass; ordinates: number of halos.
The power-law mass function of halos at the discretization scale is found in
every cosmological $N$-body simulation that we have analyzed, besides the Mare
Nostrum and GIF2 simulations, but we have no simple explanation of it. It can
be connected with the Press–Schechter theory of structure formation by
spherical collapse of overdensities in a Gaussian distribution, but the power-
law exponent is just beyond the allowed range I1 ; I2 . In contrast, the
parabola like shape (in a log-log plot) seen on larger scales is explained by
a lognormal like model that, in turn, corresponds to a multifractal model of
the matter distribution on those scales. This model has been described in
detail before (see I1 ; I2 and references therein), so we now restrict
ourselves to properties that are relevant with regard to halos.
## 5 Scale Invariance and Halos
Let us review very briefly the multifractal model of the large-scale structure
in cosmological $N$-body simulations, focusing on the CDM component of the
Mare Nostrum simulation. In the multifractal model, the coarse-grained density
$\varrho({\mbox{\boldmath{$x$}}},r)$ (defined by counts-in-cells or any
suitable method), at the point $x$ and for coarse-graining length $r$,
fulfills $\varrho({\mbox{\boldmath{$x$}}},r)\sim
r^{\alpha(}\mbox{\boldmath{${}^{x}$}}^{)}{}^{-3}$, where $\alpha\geq 0$ (see
Falcon for a precise definition). Consequently, the point density
$\lim_{r\rightarrow 0}\varrho({\mbox{\boldmath{$x$}}},r)$ is finite and non-
vanishing only if $\alpha=3$, while it is infinite for $\alpha<3$, and zero
for $\alpha>3$. Therefore, it is natural to associate points $x$ such that
$0\leq\alpha({\mbox{\boldmath{$x$}}})<3$, namely, density singularities, with
halos and points such that $\alpha({\mbox{\boldmath{$x$}}})>3$ with cosmic
voids. At any rate, multifractality is ensured by the power-law behavior of
the density with respect to the coarse-graining length or, equivalently, by
the power-law behavior of the statistical moments $M_{q}(r)$ of the
distribution Falcon . A multifractal can be characterized by its multifractal
spectrum, namely, the fractal dimension $f(\alpha)$ of the set of points $x$
with exponent $\alpha$. Notice that the multifractal spectrum can be defined
for any distribution with singularities, not just for self-similar
distributions. However, multifractal spectra of self-similar distributions
have typical parabola like shapes Falcon ; Halsey .
One proof of multifractality consists in computing the coarse multifractal
spectrum for several coarse-graining lengths $r$ and showing that it does not
depend on $r$. We reproduce in Figure 6 the coarse multifractal spectrum of
the dark matter in the Mare Nostrum simulation for $r=\\{1,2,4,8\\}\times
N^{-1/3}$, which cover most of the scaling range I2 . The extent of the
scaling range and the transition to homogeneity are better perceived in the
scaling of moments $M_{q}(r)$ I2 . The scaling range, which goes from the
discretization length (or somewhat below) to the homogeneity scale, extends
over two decades, at the most.
Figure 6: Multifractal spectra of the dark matter in the Mare Nostrum
simulation, for $r=\\{1,2,4,8\\}\times N^{-1/3}$ (blue, red, brown and green,
respectively).
Unfortunately, the scaling range in three-dimensional $N$-body simulations
cannot be very large, for the time being. In contrast, one-dimensional
cosmological $N$-body simulations with moderate $N$ can reach truly compelling
scaling ranges: the simulations of Miller et al. Miller , with $N\leq
2^{18}\simeq 2.6\times 10^{5}$, and of Joyce and Sicard Joyce , with
$N=10^{5}$, reach almost 4 decades! Moreover, the analysis by Joyce and Sicard
of the “halos” formed in their simulation has led them to state, regarding
three-dimensional halos, that CDM halos are “not well modeled as smooth
objects” and that “the supposed ‘universality’ of [halo] profiles is, like
apparent smoothness, an artifact of poor numerical resolution”. These
conclusions agree with the conclusions from our own analyses in three
dimensions EPL ; I1 ; I2 : if we are to preserve the concept of CDM halos,
they are to be defined as grainy structures in a self-similar distribution
rather than smooth structures with a range of sizes.
A different but very interesting demonstration of, on the one hand, scale
invariance and of, on the other hand, the discreteness limitations of
cosmological $N$-body simulations is provided by the work of Gottlöber et al.
Gott . The purpose of their work is to assess the problem of the emptiness of
cosmic voids by means of $N$-body simulations. To do this, Gottlöber et al.
Gott performed a low resolution simulation and then resimulate voids with
high resolution, namely, with the resolution corresponding to replacing each
particle with 512 particles. Naturally, they find that the voids are no longer
empty and, furthermore, the high-resolution dark matter distribution inside
large voids is such that “haloes are arranged in a pattern, which looks like a
miniature universe.” In other words, Gottlöber et al. find that a higher
resolution brings out in a void the invisible structure below the
discretization scale, demonstrating self-similarity of the full structure. One
can infer that a resimulation of halos with higher resolution must bring out
as well their grainy, self-similar structure.
## 6 Discussion and Conclusions
It seems inevitable to conclude that the presence of an intrinsic scale in
cosmological $N$-body simulations, namely, the discreteness scale $N^{-1/3}$,
severely affects the type of mass distribution that is produced below that
scale, to the extent that the smooth halos with a range of sizes about that
scale that are commonly seen in these simulations are probably an artifact of
insufficient resolution.
The problems of cosmological $N$-body simulations below the discreteness scale
have already been noticed by Splinter et al. Splinter : their comparison of
results of various $N$-body simulations reveals that “codes never agree well
below the mean comoving interparticle separation” (which is another name for
the discreteness scale). Therefore, one might think that the smoothness of
halos that is seen on these small scales should have been questioned before.
Probably, this has not occurred (or has had no consequences) because of the
popularity of the spherical collapse model. However, now it appears that this
model does not necessarily predict smooth spherical halos and, in addition,
its range of application is far more restricted than usually assumed.
In fact, the adhesion model is more adequate than the spherical collapse model
to provide a general description of large scale structure, namely, to produce
the typical cosmic web structure perceived in both CDM simulations and
observations of the galaxy distribution. The cosmic web is self-similar, so
the adhesion model suggests that the size of halos or, in general, the size of
cosmic-web structures is determined by small-scale processes that can be
lumped into an effective viscosity that breaks the scale invariance. The
question is, of course, how such small-scale processes determine the scale at
which scale invariance is broken and how this scale compares with the
discreteness scale $N^{-1/3}~{}>~{}0.2\,h^{-1}$ Mpc (generally).
First of all, let us remark that the real CDM is probably discrete. Indeed,
current models of CDM favor a WIMP composition. Neutralinos, for example, may
have a mass $<1$ TeV. Therefore, comparing with the mass resolution of
cosmological $N$-body simulations (e.g., $8.24\times 10^{9}\,h^{-1}$ M⊙ in the
Mare Nostrum simulation), there is such a huge factor ($10^{64}$ in the Mare
Nostrum simulation) that the CDM distribution on astrophysical scales is
continuous, in practice, and, hence, $N$-body simulations can in no way
reproduce the real CDM dynamics on small scales.
We could also consider that the scale invariance of CDM dynamics is broken on
small scales by an aspect of it that is not taken into account by $N$-body
simulations: the collapse of CDM overdensities eventually produces densities
and velocities that make Newtonian physics invalid and require relativistic
physics. In general relativity, a mass has an associated length scale, namely,
its Schwarschild radius. Consequently, in the “top-hat” collapse model, for
example, there is an intrinsic scale, which, in contrast with the usually
assumed scale, is not arbitrary and, furthermore, is independent of the
initial conditions. Naturally, this new scale arises in connection with
supermassive black hole formation and, arguably, the size of these black holes
is not relevant on cosmological scales.
In conclusion, the CDM dynamics does not seem to generate any small scale that
is cosmologically relevant. Thus, it seems natural to either define a sort of
scale invariant halos EPL ; I1 ; I2 or to turn to the baryon physics.
However, it is not easy to think of a definite scale in the baryonic physics
that marks the end of scale invariance. As a matter of fact, the gas in the
Mare Nostrum simulation follows the same scaling laws as the CDM does, despite
the presence of biasing I2 . At any rate, the modeling of baryonic physics in
cosmological $N$-body simulations is still in its infancy DR . What seems
clear is that the standard conclusions about smooth halos with a range of
sizes drawn from state-of-the-art $N$-body simulations, especially, CDM-only
simulations, must be reassessed.
## References
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* (2) Neyman, J.; Scott, E.L. A theory of the spatial distribution of galaxies. Astrophys. J. 1952, 116, 144–163.
* (3) Gaite, J. The fractal distribution of haloes. Europhys. Lett. 2005, 71, 332–338.
* (4) Splinter, R.J.; Melott, A.L.; Shandarin, S.F.; Suto, Y. Fundamental discreteness limitations of cosmological $N$-body clustering simulations. Astrophys. J. 1998, 497, 38–61.
* (5) Sánchez-Conde, M.A.; Betancort-Rijo, J.; Prada, F. The spherical collapse model with shell crossing. Mon. Not. R. Astron. Soc. 2007, 378, 339–352.
* (6) Gurbatov, S.N.; Saichev, A.I.; Shandarin, S.F. Large-scale structure of the Universe. The Zeldovich approximation and the adhesion model. _Phys. Usp._ 2012, 55, 223–249.
* (7) Buchert, T. Lagrangian theory of gravitational instability of Friedman-Lemaitre cosmologies and the ‘Zel’dovich approximation’. Mon. Not. R. Astron. Soc. 1992, 254, 729–737.
* (8) Gurevich, A.V.; Zybin, K.P. Large-scale structure of the Universe: Analytic theory. _Phys. Usp._ 1995, 38, 687–722.
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* (13) Gao, L.; White, S.D.M.; Jenkins, A.; Stoehr, F.; Springel, V. The subhalo populations of LCDM dark haloes. Mon. Not. R. Astron. Soc. 2004, 355, 819–834.
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* (17) Halsey, T.C.; Jensen, M.H.; Kadanoff, L.P.; Procaccia, I.; Shraiman, B.I. Fractal measures and their singularities-The characterization of strange sets. Phys. Rev. A 1986, 33, 1141–1151.
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* (19) Joyce, M.; Sicard, F. Non-linear gravitational clustering of cold matter in an expanding universe: Indications from 1D toy models. Mon. Not. R. Astron. Soc. 2011, 413, 1439–1446.
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|
arxiv-papers
| 2013-06-05T10:52:03 |
2024-09-04T02:49:46.133639
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jose Gaite",
"submitter": "Jos\\'e Gaite",
"url": "https://arxiv.org/abs/1306.1053"
}
|
1306.1095
|
DETERMINATION OF THE STRONG COUPLING FROM HADRONIC TAU DECAYS USING
RENORMALIZATION GROUP SUMMED
PERTURBATION THEORY111Contribution to the proceedings of the workshop
”Determination of the Fundamental Parameters of QCD”, Nanyang Technological
University, Singapore, 18-22 March 2013, to be published in Mod. Phys. Lett.
A.
GAUHAR ABBAS222Speaker
The Institute of Mathematical Sciences, C.I.T.Campus, Taramani, Chennai 600
113, India
B. ANANTHANARAYAN
Centre for High Energy Physics, Indian Institute of Science, Bangalore 560
012, India
I. CAPRINI
Horia Hulubei National Institute for Physics and Nuclear Engineering,
P.O.B. MG-6, 077125 Bucharest-Magurele, Romania
###### Abstract
We determine the strong coupling constant $\alpha_{s}$ from the $\tau$
hadronic width using a renormalization group summed (RGS) expansion of the QCD
Adler function. The main theoretical uncertainty in the extraction of
$\alpha_{s}$ is due to the manner in which renormalization group invariance is
implemented, and the as yet uncalculated higher order terms in the QCD
perturbative series. We show that new expansion exhibits good renormalization
group improvement and the behaviour of the series is similar to that of the
standard CIPT expansion. The value of the strong coupling in ${\overline{\rm
MS}}$ scheme obtained with the RGS expansion is
$\alpha_{s}(M_{\tau}^{2})=0.338\pm 0.010$. The convergence properties of the
new expansion can be improved by Borel transformation and analytic
continuation in the Borel plane. This is discussed elsewhere in these
proceedings.
## 1 Introduction
The inclusive hadronic decay width of the $\tau$ lepton provides a clean
source for the determination of $\alpha_{s}$ at low energies [1, 2, 3, 4]. The
perturbative QCD contribution is known up to $O(\alpha_{s}^{4})$ [5] and the
nonperturbative corrections are found to be small [6, 7, 8, 9, 10, 11]. The
main uncertainties arise from the treatment of higher-order corrections and
improvement of the perturbative series through renormalization group (RG)
methods. The leading methods for the treatment of the perturbative series are
fixed-order perturbation theory (FOPT) and contour-improved perturbation
theory (CIPT) [12, 13]. The predictions from these methods are not in
agreement and the discrepancy between them is one of the main sources of
ambiguity in the extraction of $\alpha_{s}$ [9, 11, 14, 15, 16].
The above theoretical discrepancy has been the motivation for proposing an
alternative approach. We consider the method developed in [17, 18], using a
procedure originally suggested in [19, 20, 21] which we refer to as
renormalization-group summation (RGS). This is a framework involving leading
logarithms summation, in which terms in powers of the coupling constant and
logarithms are regrouped, so that for a given order, the new expansion
includes every term in the perturbative series that can be calculated using
the RG invariance. The results, which are summarized in this talk, are similar
to those in CIPT and predicts $\alpha_{s}$ close to the CIPT prediction [22].
Note that this method was used for the study of the inclusive decays of the
b-quark and the hadronic cross section in $e^{+}e^{-}$ annihilation [17, 18].
Our work demonstrates that the method can also be applied with success to the
determination of $\alpha_{s}$ from $\tau$ hadronic decays.
It must be noted that the QCD perturbative corrections are also sensitive to
as yet uncalculated higher order terms in the series. The coefficients of the
perturbative series grow as $n!$ and the series have zero radius of
convergence[23, 24, 25, 26]. Consequently one can study the Borel transform of
the QCD Adler function which has ultraviolet (UV) and infrared (IR) renormalon
singularities in the Borel plane. The divergent behaviour can be considerably
tamed by using techniques of series acceleration based on conformal mappings
and ‘singularity softening’ [27, 28, 29, 30, 31, 32]. The method is not
applicable to the perturbative series in powers of $\alpha_{s}$ since the
expanded correlators are singular at $\alpha_{s}=0$, but can be applied in the
Borel plane. The large order behaviour of the RGS expansion can be improved
using this method [33]. This provides a new class of expansions, which
simultaneously implement the RG invariance and the large-order summation by
the analytic continuation in the Borel plane. The details may be found
elsewhere in these proceedings [34].
## 2 The QCD Adler function
Our treatment begins with the QCD Adler function, which enters the expression
for the total inclusive hadronic width of the $\tau$ lepton. The total
inclusive hadronic width of the $\tau$ provides an accurate calculation of the
ratio
$R_{\tau}\,\equiv\,\frac{\Gamma[\tau^{-}\to{\rm
hadrons}\,\nu_{\tau}]}{\Gamma[\tau^{-}\to
e^{-}\overline{\nu}_{e}\nu_{\tau}]}.$ (1)
The Cabibbo-allowed combination $R_{\tau,V+A}$ which proceeds through a vector
and an axial vector current can be written as
$R_{\tau,V+A}=S_{\rm EW}|V_{ud}|^{2}\bigg{(}1+\delta^{(0)}+\delta_{\rm
EW}+\delta^{(2,m_{q})}_{ud,V/A}+\sum_{D=4,6,\dots}\delta_{ud,V/A}^{(D)}\bigg{)}\,,$
(2)
where $S_{\rm EW}$ and $\delta_{\rm EW}$ are electroweak corrections,
$\delta^{(0)}$ is the dominant universal perturbative QCD correction, and
$\delta_{ud}^{(D)}$ denote quark mass and higher $D$-dimensional operator
corrections (condensate contributions) arising in the operator product
expansion (OPE). Our main interest is in the perturbative correction
$\delta^{(0)}$ which can be written as
$\delta^{{(0)}}=\frac{1}{2\pi
i}\\!\\!\oint\limits_{|s|=M_{\tau}^{2}}\\!\\!\frac{ds}{s}\left(1-\frac{s}{M_{\tau}^{2}}\right)^{3}\left(1+\frac{s}{M_{\tau}^{2}}\right)\hat{D}_{\rm
pert}(s),$ (3)
where $\hat{D}_{\rm pert}(s)\equiv D^{(1+0)}(s)-1$ is the perturbative part of
the reduced function Adler function. The perturbative expansion of
$\widehat{D}(s)$ in the “fixed-order perturbation theory” at
$\mu^{2}=M_{\tau}^{2}$ reads [9]
$\widehat{D}_{\rm
FOPT}(a_{s},L)=\sum\limits_{n=1}^{\infty}a_{s}^{n}\sum\limits_{k=1}^{n}k\,c_{n,k}\,L^{k-1}\,.$
(4)
where
$a_{s}\equiv\alpha_{s}(\mu^{2})/\pi,\quad\quad L\equiv\,\ln(-s/\mu^{2}).$ (5)
The coefficients $c_{n,1}$ in the ${\overline{\rm MS}}$ scheme for $n_{f}=3$
flavours are (cf. [5] and references therein):
$c_{1,1}=1,\,c_{2,1}=1.640,\,c_{3,1}=6.371,\,c_{4,1}=49.076.$ (6)
Several estimates for the next coefficient $c_{5,1}$ are available [8, 9, 11,
16]. The remaining coefficients $c_{n,k}$ for $k>1$ are determined from
renormalization-group invariance. The function $\widehat{D}_{\rm pert}$ is
scale independent and therefore satisfies the RG equation
$\mu^{2}\frac{\mathrm{d}}{\mathrm{d}\mu^{2}}\left[\widehat{D}_{\rm
pert}(s)\right]=0,$ (7)
which takes the form
$\beta(a_{s})\frac{\partial\widehat{D}_{\rm pert}(s)}{\partial
a_{s}}-\frac{\partial\widehat{D}_{\rm pert}(s)}{\partial\ln(-s/\mu^{2})}=0,$
(8)
where
$\beta(a_{s})\equiv\mu^{2}\frac{\mathrm{d}a_{s}(\mu^{2})}{\mathrm{d}\mu^{2}}=-(a_{s}(\mu^{2}))^{2}\sum_{k=0}^{\infty}\beta_{k}(a_{s}(\mu^{2}))^{k}$
(9)
is the $\beta$ function. The known $\beta_{j}$ coefficients are [35, 36, 37]:
$\beta_{0}=9/4,\,\,\beta_{1}=4,\,\,\beta_{2}=10.0599,\,\,\beta_{3}=47.228.$
(10)
The series (4) is badly behaved especially near the time-like axis due to the
large imaginary part of the logarithm $\ln(-s/M_{\tau}^{2})$ along the circle
$|s|=M_{\tau}^{2}$ [12, 13]. The choice $\mu^{2}=-s$ provides the CIPT
expansion [12, 13]
$\widehat{D}_{\rm
CIPT}(s)=\sum\limits_{n=1}^{\infty}c_{n,1}(a_{s}(-s))^{n}\,,$ (11)
where the running coupling $a_{s}(-s)$ is determined by solving the
renormalization-group equation (9) numerically in an iterative way along the
circle, starting with the input value $a_{s}(M_{\tau}^{2})$ at
$s=-M_{\tau}^{2}$. This expansion avoids the appearance of large logarithms
along the circle $|s|=M_{\tau}^{2}$.
## 3 Renormalization-Group Summation
The expansion of the Adler function (4) can be written in the RGS form [22,
33]
$\widehat{D}_{\rm FOPT}(a_{s},L)=\sum_{n=1}^{\infty}a_{s}^{n}D_{n}(a_{s}L),$
(12)
where the functions $D_{n}(a_{s}L)$, depending on a single variable
$u=a_{s}L$, are defined as
$D_{n}(u)\equiv\sum_{k=n}^{\infty}(k-n+1)c_{k,k-n+1}u^{k-n}.$ (13)
The function $D_{1}$ sums all the leading logarithms, the second function
$D_{2}$ sums the next-to-leading logarithms, and so on. These function can be
obtained in a closed analytical form using RG invariance through the equation
(8) and the expansion (12). This gives the following RGE equation
$\displaystyle 0$
$\displaystyle=-\sum_{n=1}^{\infty}\sum_{k=2}^{n}k(k-1)c_{n,k}\,a_{s}^{n}L^{k-2}$
$\displaystyle-\left(\beta_{0}a_{s}^{2}+\beta_{1}a_{s}^{3}+\beta_{2}a_{s}^{4}+\ldots+\beta_{l}a_{s}^{l+2}+\ldots\right)\times\sum_{n=1}^{\infty}\sum_{k=1}^{n}nkc_{n,k}a_{s}^{n-1}L^{k-1}.$
(14)
By extracting the aggregate coefficient of $a_{s}^{n}L^{n-p}$ one obtains the
recursion formula $(n\geq p)$
$0=(n-p+2)c_{n,n-p+2}+\sum_{\ell=0}^{p-2}(n-\ell-1)\beta_{\ell}c_{n-\ell-1,n-p+1}.$
(15)
Multiplying both sides of (15) by $(n-p+1)u^{n-p}$ and summing from $n=p$ to
$\infty$, we obtain a set of first-order linear differential equation for the
functions defined in (13) for $n\geq 1$:
$\frac{\mathrm{d}D_{n}}{\mathrm{d}u}+\sum_{\ell=0}^{n-1}\beta_{\ell}\left(u\frac{\mathrm{d}}{\mathrm{d}u}+n-\ell\right)D_{n-\ell}=0,$
(16)
with the initial conditions $D_{n}(0)=c_{n,1}$ which follow from (13). The
solution of the system (16) can be found iteratively in an analytical form.
The solutions $D_{n}(u)$ depend on the coupling constant and logarithms. The
expressions of $D_{n}(u)$ for $n\leq 4$, written in terms of the coefficients
$c_{k,1}$ with $k\leq n$ and $\beta_{j}$ with $0\leq j\leq n-1$, are:
$\displaystyle D_{1}(u)$ $\displaystyle=\frac{c_{1,1}}{w},\quad\quad\quad
w=1+\beta_{0}u,$ $\displaystyle D_{2}(u)$
$\displaystyle=\frac{c_{2,1}}{w^{2}}-\frac{\beta_{1}c_{1,1}\ln
w}{\beta_{0}w^{2}},$ (17) $\displaystyle D_{3}(u)$
$\displaystyle=\frac{(\beta_{1}^{2}-\beta_{0}\beta_{2})c_{1,1}}{\beta_{0}^{2}w^{2}}+\left[\frac{(-\beta_{1}^{2}+\beta_{0}\beta_{2})c_{1,1}}{\beta_{0}^{2}}+c_{3,1}\right]w^{-3}$
(18)
$\displaystyle+\left[-\frac{\beta_{1}(\beta_{1}c_{1,1}+2\beta_{0}c_{2,1})\ln
w}{\beta_{0}^{2}}+\frac{\beta_{1}^{2}c_{1,1}\ln^{2}w}{\beta_{0}^{2}}\right]w^{-3}.$
$\displaystyle
D_{4}(u)=-\frac{(\beta_{1}^{3}-2\beta_{0}\beta_{1}\beta_{2}+\beta_{0}^{2}\beta_{3})c_{1,1}}{2\beta_{0}^{3}}w^{-2}-\left[\frac{\beta_{1}(-\beta_{1}^{2}+\beta_{0}\beta_{2})c_{1,1}}{\beta_{0}^{3}}+\frac{2(-\beta_{1}^{2}+\beta_{0}\beta_{2})c_{2,1}}{\beta_{0}^{2}}\right]w^{-3}$
$\displaystyle+\frac{2\beta_{1}(-\beta_{1}^{2}+\beta_{0}\beta_{2})c_{1,1}\ln
w}{\beta_{0}^{3}}w^{-3}+\left[\frac{(-\beta_{1}^{3}+\beta_{0}^{2}\beta_{3})c_{1,1}}{2\beta_{0}^{3}}+\frac{2(-\beta_{1}^{2}+\beta_{0}\
\beta_{2})c_{2,1}}{\beta_{0}^{2}}+c_{4,1}\right]w^{-4}$
$\displaystyle-\frac{\beta_{1}(-2\beta_{1}^{2}c_{1,1}+3\beta_{0}\beta_{2}c_{1,1}+2\beta_{0}\beta_{1}c_{2,1}+3\beta_{0}^{2}c_{3,1})\ln
w}{\beta_{0}^{3}}w^{-4}$
$\displaystyle+\frac{\beta_{1}^{2}(5\beta_{1}c_{1,1}+6\beta_{0}c_{2,1})\ln^{2}w}{2\beta_{0}^{3}}w^{-4}-\frac{\beta_{1}^{3}c_{1,1}\ln^{3}w}{\beta_{0}^{3}}w^{-4}.$
The higher order solutions can be found in Ref. [22]. These expressions are
used for computing the perturbative contribution to the hadronic width of the
$\tau$ lepton and the subsequent extraction of $\alpha_{s}(M_{\tau}^{2})$.
## 4 The properties of the RGS expansion
We now discuss the properties of the RGS expansion in the complex momentum
plane, along the circle $s=M_{\tau}^{2}\exp(i\theta)$. In Fig. 1, we show the
behaviour of the modulus of the Adler function along the circle given by the
first $N=5$ terms in the expansions (4), (11) and (12), respectively. In this
calculation and below we used the standard value
$\alpha_{s}(M_{\tau}^{2})=0.34$, adopted also in previous studies [9, 31]. One
may see that the behaviour of the new RGS expansion is similar to that of the
CIPT expansion.
Figure 1: Modulus of the Adler function expansions (4), (11) and (12), summed
up to the order $N=5$, along the circle $s=M_{\tau}^{2}\exp(i\theta)$.
In Table 1, we show the values of the quantity $\delta^{(0)}$ defined in (3),
calculated with FOPT, CIPT and RGSPT, as a function of the perturbative order
up to which the series was summed. We observe that CIPT shows a faster
convergence compared to FOPT. To order $N=4$, the difference between FOPT and
CIPT is $0.0215$, which is the main source of the theoretical uncertainty in
the extraction of $\alpha_{s}$ from the hadronic $\tau$ decay rate. The new
RGS expansion, as remarked earlier, shows the convergence which is similar to
the CIPT expansion. We note that for $N=4$, the difference between the results
of the RGS and the standard FOPT is $0.01754$, and the difference from the RGS
and CIPT is $0.0039$, which confirms that the new expansion gives results
close to those of the CIPT. For $N=5$, using the estimate $c_{5,1}=283$ from
[9], we find that the RGS differs from FOPT by $0.0232$, and from CIPT by
$0.0035$.
Table 1: Predictions of $\delta^{(0)}$ by the standard FOPT, CIPT and the RGS, for various truncation orders $N$. | $\delta^{(0)}_{\rm FOPT}$ | $\delta^{(0)}_{\rm CIPT}$ | $\delta^{(0)}_{\rm RGS}$
---|---|---|---
$N=1$ | 0.1082 | 0.1479 | 0.1455
$N=2$ | 0.1691 | 0.1776 | 0.1797
$N=3$ | 0.2025 | 0.1898 | 0.1931
$N=4$ | 0.2199 | 0.1984 | 0.2024
$N=5$ | 0.2287 | 0.2022 | 0.2056
It would be of interest to study the behaviour of RGS expansion at higher
orders which are not shown in the Table 1. This is the subject of the next
section, in a model for higher order coefficients of the Adler function.
## 5 Higher order behaviour of the RGS expansion
As discussed earlier, the extraction of $\alpha_{s}$ from the hadronic $\tau$
decays width is also sensitive to the large order behaviour of the QCD
perturbative series. It is of interest to check if the low order behaviour of
the new RGS expansion persists at higher orders. This investigation was
carried out in [22, 33] in a model proposed in [9]. In this model, the RGS
expansion of the QCD Adler function has a behaviour which is similar to that
of CIPT and eventually exhibits big oscillations, showing the divergent
character of the QCD perturbative series at higher orders. In Fig. 2, we show
the behaviour of FO, CI and RGS expansions in the so-called “reference model”
defined in [9, 38]. The RGS results are close to those of CIPT at every order
up to $N=10$. In this model, FOPT expansion approaches better the ‘true
value’.
Figure 2: Dependence of $\delta^{(0)}$ in FOPT, CIPT and RGS on the truncation
order $N$ in the Beneke and Jamin model [9]. The gray band is the exact value.
A method for taming the divergent behaviour of the QCD perturbative expansions
was proposed in [27, 28, 29, 30, 31, 32], using the series acceleration by the
conformal mappings of the Borel plane and the implementation of the known
nature of the leading singularities in this plane.
In Ref. [33], we applied these techniques to the RGS expansion and defined a
novel, RGS non-power (RGSNP) expansion of the QCD Adler function, in which the
powers of the coupling of the standard expansion are replaced by suitable
functions that resemble the expanded amplitude as concerns their singularities
and the expansions in powers of $\alpha_{s}$. The non-power expansions show
remarkable convergence properties for the Adler function, which is the crucial
input in the determination of $\alpha_{s}$ from hadronic $\tau$ decays. This
is reviewed elsewhere in these proceedings [34].
## 6 Determination of $\alpha_{s}(M_{\tau}^{2})$ from RGS expansion
In this section, we present the derivation of $\alpha_{s}(M_{\tau}^{2})$
following Ref. [22]. We use as input the coefficients $c_{n,1}$ calculated
from Feynman diagrams, given in (6), and the estimate $c_{5,1}=283\pm 283$ [9,
16]. We need also the phenomenological value of the pure perturbative
correction to the hadronic $\tau$ width, for which we adopt the recent
estimate [16],
$\delta^{(0)}_{\rm phen}=0.2037\pm 0.0040_{exp}\pm 0.0037_{\rm PC},$ (19)
where the first error is experimental and the second shows the uncertainty due
to the power corrections. This value has been used in several recent
determinations [30, 16, 31]. With this input we obtain
$\displaystyle\alpha_{s}(M_{\tau}^{2})$ $\displaystyle=$ $\displaystyle
0.3378\pm 0.0046_{\rm exp}\pm 0.0042_{\rm PC}~{}^{+0.0062}_{-0.0072}(c_{5,1})$
(20) ${}^{+0.0005}_{-0.0004}{(\rm
scale)}\pm^{+0.000085}_{-0.000082}{(\rm\beta_{4})}.$
In the above, the first two errors are due to the corresponding uncertainties
of $\delta^{(0)}_{\rm phen}$ given in (19), while the third is due to the
uncertainty of the coefficient $c_{5,1}$ with the conservative range adopted
above, the fourth is due to scale variation, and the last one is due to the
effect of the truncation of the $\beta$-function expansion. The details may be
found in [22]. We observe that $\alpha_{s}(M_{\tau}^{2})$ is not sensitive to
the variation of the scale. The largest error comes from the uncertainty of
the five loop coefficient $c_{5,1}$. This was also observed in the standard
CIPT analysis [11, 14] and in the analysis based on the CI expansions improved
by the conformal mappings of the Borel plane [30, 31].
Combining in quadrature the errors given in (20), the prediction based on RGS
expansion reads [22]
$\alpha_{s}(M_{\tau}^{2})=0.338\pm 0.010.$ (21)
We mention that for the same input (19) the standard FOPT and CIPT give,
respectively,
$\displaystyle\alpha_{s}(M_{\tau}^{2})$ $\displaystyle=$ $\displaystyle
0.320^{+0.012}_{-0.007},\quad\quad\quad{\rm FOPT},$
$\displaystyle\alpha_{s}(M_{\tau}^{2})$ $\displaystyle=$ $\displaystyle
0.342\pm 0.012,\quad\quad{\rm CIPT}.$ (22)
For comparison we mention also the value
$\alpha_{s}(M_{\tau}^{2})=0.320^{+0.019}_{-0.014}$, obtained recently in Ref.
[31] with the same input (19) and the CI non-power (CINP) expansion and the
value $\alpha_{s}(M_{\tau}^{2})=0.319~{}^{+0.015}_{-0.012}$ determined by the
RGS non-power (RGSNP) expansion in the Ref. [33] based on the analytic
continuation in the Borel plane.333The question of the uncertainties due to
the nonperturbative corrections has been recently addressed in detail in the
Ref. [39], where an error larger than that quoted for power corrections in the
Eq. (19) has been obtained. Using this more conservative input will slightly
increase the error in the predictions above. For instance, as discussed in
[33], for the RGSNP prediction the upper and lower errors change to $0.017$
and $0.015$ respectively. After evolving to the scale of $M_{Z}$, the RGSNP
prediction of $\alpha_{s}$ reads
$\alpha_{s}(M_{Z}^{2})=0.1184~{}^{+0.0018}_{-0.0015}$. The special features of
this latter expansion will be discussed elsewhere in these proceedings [34].
## 7 Conclusion
In this talk, we have presented our recent results on the determination of the
strong coupling constant $\alpha_{s}$ from the $\tau$ hadronic width, based on
the formalism which we denote as RGS expansion of the Adler function. Due to
the present discrepancy in the determination of the $\alpha_{s}$ from hadronic
$\tau$ decays, any alternative approach besides FOPT and CIPT, must be
pursued. It must, however, be noted that the RGS framework is an important
framework, which was simply not explored in detail earlier in the context of
the hadronic decay of the $\tau$ lepton. Our work fills this gap. The method
discussed in this talk exploits RG invariance in a complete way, providing
analytical closed form solutions to a definite order. The truncated summation
of the perturbative series differ among each other by terms of the order
$\alpha_{s}^{N+1}$. This difference turns out to be quite important for the
low scale relevant in $\tau$ decays.
We have discussed in detail the properties of the RGS expansion, including its
properties in the complex energy plane and concluded that these properties are
similar to those of CIPT expansion. We also provide the value of the strong
coupling $\alpha_{s}$ from this expansion, which is closer to CIPT prediction.
We conclude that the the summation of leading logarithms provides a systematic
expansion with good convergence properties in the complex plane, including the
critical region near the time-like region.
The determination of $\alpha_{s}$ is also ambiguous due to the effect of the
as yet uncalculated higher order terms in the perturbative expansion of the
hadronic width. This ambiguity is amplified by the fact that the perturbative
series is divergent, the coefficients displaying a factorial increase. In QCD,
due to the presence of the UV and IR renormalon singularities situated on the
real axis in the Borel plane, the Borel-Laplace integral giving the expanded
correlators in terms of their Borel transform requires a prescription. Using
the Principal Value prescription and the technique of series acceleration by
conformal mappings and singularity softening in the Borel plane developed in
[27] \- [32], we have defined in [33] a new kind of expansion, referred to as
RGS non-power (RGSNP) expansion. The divergent behaviour of the standard
perturbative series is considerably tamed in the non-power expansions, which
show good convergence in the complex energy plane. More details may be found
in Refs. [33, 34].
## Acknowledgments
We thank Prof. K. K. Phua and the workshop organizers for the kind hospitality
at the Institute for Advanced Studies, Nanyang Technological University,
Singapore. IC acknowledges support from the Ministry of National Education
under Contracts PN 09370102/2009 and Idei-PCE No 121/2011.
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|
arxiv-papers
| 2013-06-05T13:07:33 |
2024-09-04T02:49:46.142325
|
{
"license": "Public Domain",
"authors": "Gauhar Abbas, B. Ananthanarayan, I. Caprini",
"submitter": "Gauhar Abbas",
"url": "https://arxiv.org/abs/1306.1095"
}
|
1306.1108
|
# Inertial longitudinal magnetization reversal for non-Heisenberg ferromagnets
E. G. Galkina Institute of Physics, 03028 Kiev, Ukraine CEMS, RIKEN,
Saitama, 351-0198, Japan V. I. Butrim Vernadsky Taurida National University,
Simferopol, 95007 Ukraine Yu. A. Fridman Vernadsky Taurida National
University, Simferopol, 95007 Ukraine B. A. Ivanov [email protected]
Institute of Magnetism, 03142, Kiev, Ukraine Taras Shevchenko National
University of Kiev, 01601, Ukraine CEMS, RIKEN, Saitama, 351-0198, Japan
Franco Nori CEMS, RIKEN, Saitama, 351-0198, Japan Department of Physics, The
University of Michigan, Ann Arbor, MI 48109-1040, USA.
###### Abstract
We analyze theoretically the novel pathway of ultrafast spin dynamics for
ferromagnets with high enough single-ion anisotropy (non-Heisenberg
ferromagnets). This longitudinal spin dynamics includes the coupled
oscillations of the modulus of the magnetization together with the quadrupolar
spin variables, which are expressed through quantum expectation values of
operators bilinear on the spin components. Even for a simple single-element
ferromagnet, such a dynamics can lead to an inertial magnetization reversal
under the action of an ultrashort laser pulse.
###### pacs:
75.10.Jm, 75.10.Hk, 78.47.J-, 05.45.-a
## I Introduction
Which is the fastest way to reverse the magnetization of either a magnetic
particle or a small region of a magnetic film? This question has attracted
significant interest, both fundamental and practical, for magnetic information
storage. [1] Intense laser pulses, with durations less than a hundred
femtoseconds, are able to excite the ultrafast evolution of the total spin of
a magnetically-ordered system on a picosecond time scale, see e.g. the
reviews. [2] ; [3] ; [4] The limitations for the time of magnetization
reversal come from the characteristic features of the spin evolution for a
magnet with a concrete type of magnetic order. The dynamical time cannot be
shorter than the characteristic period of spin oscillations $T$,
$T=2\pi/\omega_{0}$, where $\omega_{0}$ is the magnetic resonance frequency.
For ferromagnets, the frequency of standard spin oscillations (precession) is
$\omega_{0,\mathrm{FM}}=\gamma H_{\mathrm{r}}$, where $\gamma$ is the
gyromagnetic ratio, and $H_{\mathrm{r}}$ is an effective field of relativistic
origin, like the anisotropy field, which is usually less than a few Tesla.
Thus, the dynamical time for Heisenberg ferromagnets cannot be much shorter
than one nanosecond. For antiferromagnets, all the dynamical characteristics
are exchange enhanced, and
$\omega_{0,\mathrm{AFM}}=\gamma\sqrt{H_{\mathrm{ex}}H_{\mathrm{r}}}$, where
$H_{\mathrm{ex}}$ is the exchange field,
$H_{\mathrm{ex}}=J/2\mu_{\mathrm{B}}$, $J$ is the exchange integral, and
$\mu_{\mathrm{B}}$ is the Bohr magneton, see Ref. [5], . The excitation of
terahertz spin oscillations has been experimentally demonstrated for
transparent antiferromagnets using the inverse Faraday effect or the inverse
Cotton-Mouton effect. [6] ; [7] ; [8] ; [9] ; [10] ; [11] The non-linear
regimes of such dynamics include the inertia-driven dynamical reorientation of
spins on a picosecond time scale, which was observed in orthoferrites. [10] ;
[11]
The exchange interaction is the strongest force in magnetism, and the exchange
field $H_{\mathrm{ex}}$ can be as strong as $10^{3}$ Tesla. The modulus of the
magnetization is determined by the exchange interaction, and the direction of
the magnetization is governed by relativistic interactions. It would be very
tempting to produce a magnetization reversal by changing the modulus of the
magnetization vector, i.e., via the longitudinal dynamics of ${\rm{\bf M}}$.
For such a process, dictated by the exchange interaction, the characteristic
times could be of the order of the exchange time $\tau_{\mathrm{ex}}=1/\gamma
H_{\mathrm{ex}}$, which is shorter than one picosecond. However, within the
standard approach such dynamics is impossible. The evolution of the modulus of
the magnetization, $M=|{\rm{\bf M}}|$, within the closed Landau-Lifshitz
equation for the magnetization only (or the set of such equations for the
sublattice magnetizations, ${\rm{\bf M}}_{\alpha})$, is purely dissipative.
[12] This feature could be explained as follows: two angular variables,
$\theta$ and $\varphi$, describing the direction of the vector ${\rm{\bf M}}$,
within the Landau-Lifshitz equation determine the pair of conjugated Hamilton
variables $(\cos\theta$ and $\varphi$ are the momentum and coordinate,
respectively). Also, the evolution of the single remaining variable
$M=|{\rm{\bf M}}|$, governed by a first-order equation can be only
dissipative; see a more detailed discussion below. Moreover, the exchange
interaction conserves the total spin of the system, and the relaxation of the
total magnetic moment of any magnet can be present only when accounting for
relativistic effects. Thus the relaxation time for the total magnetic moment
is relativistic but it is exchange-enhanced, as was demonstrated within the
irreversible thermodynamics of the magnon gas. [12] Note here that the
relaxation of the magnetization of a single sublattice for multi-sublattice
magnets can be of purely exchange origin. [13] Recently, magnetization
reversal on a picosecond time scale has been experimentally demonstrated for
the ferrimagnetic alloy GdFeCo, see Refs. [13], ; [14], . These results can be
explained within the concept of exchange relaxation, developed by Baryakhtar,
[15] accounting for the purely exchange evolution of the sublattice
magnetization. [16] Such an exchange relaxation can be quite fast, but its
characteristic time is again longer than the expected “exchange time”
$\tau_{\mathrm{ex}}=1/\gamma H_{\mathrm{ex}}$.
Thus, the ultrafast mechanisms of magnetization reversal implemented so far
are: the dynamical (inertial) switching possible for antiferromagnets, [10] ;
[11] and the exchange longitudinal evolution for ferrimagnets. [13] ; [14] ;
[16] These are both quite fast, with a characteristic time of the order of
picoseconds; but their characteristic times are longer than the “ideal
estimate”: the exchange time $\tau_{\mathrm{ex}}$.
In this work, we present a theoretical study of the possibility of the
dynamical evolution of the modulus of the magnetization for non-Heisenberg
ferromagnets with high enough single-ion anisotropy that can be called
longitudinal spin dynamics. For such a dynamics, an inertial magnetization
reversal is possible even for a simple single-element ferromagnet.
Longitudinal dynamics does not exist in Heisenberg magnets, and this dynamics
cannot be described in terms of the Landau-Lifshitz equation, or using the
Heisenberg Hamiltonian, which is bilinear over the components of spin
operators for different spins, see more details below in Sec. II. The key
ingredient of our theory is the inclusion of higher-order spin quadrupole
variables. It is known that for magnets with atomic spin $S>1/2$, allowing the
presence of single-ion anisotropy, the spin dynamics is not described by a
closed equation for spin dipolar variable $\left\langle{\rm{\bf
S}}\right\rangle$ alone (or magnetization ${\rm{\bf
M}}=-2\mu_{\mathrm{B}}\left\langle{\rm{\bf S}}\right\rangle$). [17] ; [18] ;
[19] ; [20] ; [21] ; Chubuk90 ; Matveev ; AndrGrishchuk Here and below
$\left\langle{...}\right\rangle$ means quantum and (at finite temperature)
thermal averaging. To be specific, we choose the spin-one ferromagnet with
single-ion anisotropy, the simplest system allowing this effect. The full
description of these magnets requires taking into account the dynamics of
quadrupolar variables,
$S_{ik}=(1/2)\left\langle{S_{i}S_{k}+S_{k}S_{i}}\right\rangle$, that represent
the quantum averages of the operators, bilinear in the spin components. Our
theory is based on the consistent semiclassical description of a full set of
spin quantum expectation values (dipolar and quadrupolar) for the spin-one
system, which was investigated by many authors from different viewpoints. [17]
; [18] ; [19] ; [20] ; [21] ; Chubuk90 ; Matveev ; AndrGrishchuk As we will
show, the longitudinal dynamics of spin, including nonlinear regimes, can be
excited by a femtosecond laser pulse. With natural accounting for the
dissipation, the longitudinal spin dynamics can lead to changing the sign of
the total spin of the system (longitudinal magnetization reversal).
## II MODEL DESCRIPTION
The Landau-Lifshitz equation was proposed many years ago as a phenomenological
equation, and it is widely used for the description of various properties of
ferromagnets. Concerning its quantum and microscopic basis, it is worth noting
that this equation naturally arises using the so-called spin coherent states.
[22] ; [23] These states can be introduced for any spin $S$ as the state with
the maximum value of the spin projection on an arbitrary axis ${\rm{\bf n}}$.
Such states can be parameterized by a unit vector ${\rm{\bf n}}$; the
direction of the latter coincides with the quantum mean values for the spin
operator $\left\langle{\rm{\bf S}}\right\rangle=S{\rm{\bf n}}$ (dipolar
variables). This property is quite convenient for linking the quantum physics
of spins to a phenomenological Landau-Lifshitz equation. The use of spin
coherent states is most efficient when the Hamiltonian of the system is linear
with respect to the operators of the spin components. If an initial state is
described by a certain spin coherent state, its quantum evolution will reduce
to a variation of the parameters of the state (namely, the direction of the
unit vector ${\rm{\bf n}})$, which are described by the classical Landau-
Lifshitz equation. Thus, spin coherent states are a convenient tool for the
analysis of spin Hamiltonians containing only operators linear on the spin
components or their products on different sites. An important example is the
bilinear Heisenberg exchange interaction, described by the first term in Eq.
(1) below.
In contrast to the cases above, for the full description of spin-$S$ states,
one needs to introduce $SU(2S+1)$ generalized coherent states. [17] ; [19] ;
[20] ; [21] The analysis shows that spin coherent states are less natural for
the description of magnets whose Hamiltonian contains products of the spin
component operators at a single site. Such terms are present for magnets with
single-ion anisotropy or a biquadratic exchange interaction. Magnets with non-
small interaction of this type are often called non-Heisenberg. For such
magnets, some non-trivial features, absent for Heisenberg magnets, are known.
Among them we note the possibility of so-called quantum spin reduction; namely
the possibility to have the value of $|\left\langle{\rm{\bf S}}\right\rangle|$
less than its nominal value, $|\left\langle{\rm{\bf S}}\right\rangle|<S$, even
for pure states at zero temperature. This was first mentioned by Moriya, [24]
as early as 1960. As an extreme realization of the effect of quantum spin
reduction, we note the existence of the so-called spin nematic phases with a
zero mean value of the spin in the ground state at zero temperature. In the
last two decades, the interest on such states has been considerable, motivated
by studies of multicomponent Bose-Einstein condensates of atoms with non-zero
spin. [25] ; [26] ; [27] ; [28]
A significant manifestation of quantum spin reduction is the appearance of an
additional branch of the spin oscillations, which is characterized by the
dynamics (oscillations) of the length of the mean value of spin without spin
precession. [17] ; [18] ; [19] ; [20] ; [21] ; Chubuk90 ; Matveev The
characteristic frequency of this mode can be quite high (of the order of the
exchange integral). For this reason, for a description of resonance properties
or thermodynamic behavior of magnets, this mode is usually neglected, and the
common impression is that the dynamics of magnetic materials with constant
single-ion anisotropy $K<$(0.2-0.3)$J$ is fully described by the standard
phenomenological theory. However, for an ultrafast evolution of the spin
system under a femtosecond laser pulse, one can expect a lively demonstration
of this longitudinal high-frequency mode. Thus, it is important to explore the
possible manifestations of the effects of quantum spin reduction in the
dynamic properties of ferromagnets.
The simplest model allowing spin dynamics with effects of quantum spin
reduction is described by the Hamiltonian
$H=-\frac{1}{2}\sum\limits_{\mathbf{n},\boldsymbol{\ell}}{\bar{J}{\rm{\bf
S}}_{\mathbf{n}}{\rm{\bf
S}}_{\mathbf{n}+\boldsymbol{\ell}}}+\frac{K}{2}\sum\limits_{\mathbf{n}}{\left({S_{\mathbf{n},x}}\right)^{2}},$
(1)
where ${\rm{\bf S}}_{\mathbf{n}}$ is the spin-one operator at the site
$\mathbf{n}$; $\bar{J}>0$ is the exchange constant for nearest-neighbors
$\boldsymbol{\ell}$, and $K>0$ is the constant of the easy-plane anisotropy
with the plane $yz$ as the easy plane. The quantization axis can be chosen
parallel to the $z$-axis and $\left\langle{\rm{\bf
S}}\right\rangle=\left\langle{S_{z}}\right\rangle{\rm{\bf e}}_{z}$. For the
full description of spin $S=1$ states, let us introduce $SU(3)$ coherent
states [17] ; [19] ; [20] ; [21]
$\left|{\mathbf{u},\mathbf{v}}\right\rangle=\sum\limits_{j=x,y,z}{\left({u_{j}+iv_{j}}\right)\left|{\psi_{j}}\right\rangle},$
(2)
where the states $\left|{\psi_{j}}\right\rangle$ determine the Cartesian
states for $S=1$ and are expressed in terms of the ordinary states
{$\left|{\pm 1}\right\rangle,\;\left|0\right\rangle$} with given projections
$\pm 1,\;0$ of the operator $S_{z}$ by means of the relations
$\left|{\psi_{x}}\right\rangle=\left({\left|{-1}\right\rangle-\left|{+1}\right\rangle}\right)/\sqrt{2}$,
$\left|{\psi_{y}}\right\rangle=i\left({\left|{-1}\right\rangle+\left|{+1}\right\rangle}\right)/\sqrt{2}$,
$\left|{\psi_{z}}\right\rangle=\left|0\right\rangle$, with the real vectors
$\mathbf{u}$ and $\mathbf{v}$ subject to the constraints
$\mathbf{u}^{2}+\mathbf{v}^{2}=1$, $\mathbf{u}\cdot\mathbf{v}=0$. All
irreducible spin averages, which include the dipolar variable
$\left\langle\mathbf{S}\right\rangle$ (average value of the spin) and
quadrupole averages $S_{ik}$, bilinear over the spin components, can be
written through $\mathbf{u}$ and $\mathbf{v}$ as follows
$\displaystyle\left\langle\mathbf{S}\right\rangle$ $\displaystyle=$
$\displaystyle 2(\mathbf{u}\times\mathbf{v}),$ $\displaystyle
S_{ik}=\frac{1}{2}\left\langle{S_{i}S_{k}+S_{k}S_{i}}\right\rangle$
$\displaystyle=$ $\displaystyle\delta_{ik}-u_{i}u_{k}-v_{i}v_{k}.$ (3)
At zero temperature and within the mean-field approximation, the spin dynamics
is described by the Lagrangian [21]
$L=-2\hbar\sum\limits_{n}{\mathbf{v}_{n}(\partial\mathbf{u}_{n}/\partial
t})-W\left({\mathbf{u},\mathbf{v}}\right),$ (4)
where
$W\left({\mathbf{u},\mathbf{v}}\right)=\left\langle{\mathbf{u},\mathbf{v}}\right|H\left|{\mathbf{u},\mathbf{v}}\right\rangle$
is the energy of the system.
We are interested in spin oscillations which are uniform in space, and hence
we assume that the discrete variables $\mathbf{u}$ and $\mathbf{v}$ have the
same values for all spins and are only dependent on time. The frequency
spectrum of linear excitations, which consists of two branches, can be easily
obtained on the basis of the linearized version of the Lagrangian (4). In the
general case, the system of independent equations for $\mathbf{u}$ and
$\mathbf{v}$, taking into account the aforementioned constrains
$\mathbf{u}^{2}+\mathbf{v}^{2}=1$, $\mathbf{u}\cdot\mathbf{v}=0$, consists of
four nonlinear equations, describing two different regimes of spin dynamics.
One regime is similar to that for an ordinary spin dynamics treated on the
basis of the Landau-Lifshitz equation; it corresponds to oscillations of the
spin direction. The second regime corresponds to oscillations of the modulus
of the magnetization $\left\langle{\rm{\bf S}}\right\rangle=S(t){\rm{\bf
e}}_{z}$, with the vectors $\mathbf{u}$ and $\mathbf{v}$ rotating in the
$xy$-plane perpendicular to $\left\langle{\rm{\bf S}}\right\rangle$. This mode
of the spin oscillations corresponds to the longitudinal spin dynamics. It is
convenient to consider these two types of dynamics separately. Particular non-
linear longitudinal solutions, with $\left\langle{\rm{\bf
S}}\right\rangle=s(t){\rm{\bf e}}_{z}$ and $u_{z}=0,\;v_{z}=0$, were found in
Refs. [29], ; [30], . Note here that the longitudinal dynamics is much faster
than the standard transversal one, and the standard spin precession (described
by the Landau-Lifshitz equation) at a picosecond time scale just cannot
develop. Therefore, these two regimes, longitudinal and transverse, can be
treated independently, and we limit ourselves only to the longitudinal
dynamics with $\left\langle{\rm{\bf S}}\right\rangle=s(t){\rm{\bf e}}_{z}$ and
$u_{z}=0,\;v_{z}=0$.
## III LONGITUDINAL SPIN DYNAMICS
To describe the longitudinal spin dynamics, it is convenient to introduce new
variables: the spin modulus $s=2|{\rm{\bf u}}||{\rm{\bf v}}|=2uv$ and angular
variable $\gamma$, with
${\rm{\bf u}}=u\left({{\rm{\bf e}}_{x}\cos\gamma-{\rm{\bf
e}}_{y}\sin\gamma}\right),\;{\rm{\bf v}}=v\left({{\rm{\bf
e}}_{x}\sin\gamma+{\rm{\bf e}}_{y}\cos\gamma}\right),$ (5)
In this representation $\left\langle{S_{z}}\right\rangle=s$, and the non-
trivial quadrupolar variables are
$\left\langle{S_{x}S_{y}+S_{y}S_{x}}\right\rangle=\sqrt{1-s^{2}}\sin 2\gamma$
and $\left\langle{S_{y}^{2}-S_{x}^{2}}\right\rangle=\sqrt{1-s^{2}}\cos
2\gamma$, with all other quantum averages being either zero (as the transverse
spin components $\left\langle{S_{x,y}}\right\rangle$ or $S_{xz}$, $S_{yz}$) or
trivial, independent on $s$ and $\gamma$, as
$\left\langle{S_{z}^{2}}\right\rangle=1$. The mean-field energy, written per
one spin through the variables $s,\;\gamma$, takes the form
$W(s,\gamma)=-\frac{J}{2}s^{2}-\frac{K}{4}\sqrt{1-s^{2}}\cos 2\gamma,$ (6)
where $J=\bar{J}Z$, $Z$ is the number of nearest neighbors. The ground state
at $\sqrt{1-s^{2}}>0$ corresponds to $\cos 2\gamma=1$, with the mean value of
the spin $s=\pm\bar{s}$, $\bar{s}=\sqrt{1-\kappa^{2}}<1$, that is a
manifestation of quantum spin reduction at non-zero anisotropy. Here we
introduce the dimensionless parameter $\kappa=K/4J$. For these variables, the
Lagrangian can be written as
$L=\hbar s\frac{\partial\gamma}{\partial t}-W(s,\gamma),$ (7)
and $\hbar s$ and $\gamma$ play the role of canonical momentum and coordinate,
respectively, with the Hamilton function $W(s,\gamma)$. The physical meaning
of the above formal definitions is quite clear: the angular variable $\gamma$
describes the transformation of quadrupolar variables under rotation around
the $z$-axis, with $\hbar s$ as the projection of the angular momentum on this
axis.
### III.1 Small oscillations
Let us now start with the description of the dynamics of small-amplitude
oscillations. After linearization around the ground state, the equation leads
to a simple formula for the frequency of longitudinal oscillations
$\hbar\omega_{l}=2J\bar{s}=2J\sqrt{1-\kappa^{2}},$ (8)
which are in fact coupled oscillations of the projection of the spin and
quadrupolar variables, see Fig. 1.
One can see that, for a wide range of values of the anisotropy constant, like
$\kappa<0.2$-0.8, this frequency $\omega_{l}$ is of the order of
(1.8-1.2)$J/\hbar$, i.e., $\omega_{l}$ is comparable to the exchange frequency
$J/\hbar$. Thus the longitudinal spin dynamics is expected to be quite fast.
In contrast, standard transversal oscillations for a purely easy-plane model
(1) are gapless (they acquire a finite gap when accounting for a magnetic
anisotropy in the easy plane, which is usually small). Thus the essential
difference in the frequencies of these two dynamical regimes is clearly seen.
Figure 1: (color online) Graphic presentation of the variables $s$ and
$\gamma$ and their evolution. The thick red arrow represents the mean value of
the spin. The quadrupolar variables are shown by the blue three-axial
ellipsoid with the directions of the main axis (chosen to have
$\left\langle{S_{1}S_{2}}\right\rangle=0)$: ${\rm{\bf e}}_{3}={\rm{\bf
e}}_{z}$ and ${\rm{\bf e}}_{1}$, ${\rm{\bf e}}_{2}$. The half-axes of the
ellipsoid are equal to $\left\langle{S_{1}^{2}}\right\rangle$,
$\left\langle{S_{2}^{2}}\right\rangle$ and
$\left\langle{S_{3}^{2}}\right\rangle=\left\langle{S_{z}^{2}}\right\rangle=1$.
(a) the ground state, (b) the standard transverse dynamics, i.e. the spin
precession. The other frames (c)-(e) present the transient values of the
variables in longitudinal oscillations. (c) and (e) correspond to the longest
and shortest length of the spin, and at the moment depicted in (d) the spin
length equals to its equilibrium value, but the quadrupolar ellipsoid is
turned on the angle $\gamma$ with respect to the $x$-axis. On the frames
(c)-(e), the shape of the unperturbed ellipsoid is shown by light grey.
At a first glance, there is a contradiction between the concept of
longitudinal dynamics caused by single-ion anisotropy and the result present
in equation (8): the value of $\omega_{l}$ is still finite for vanishing
anisotropy constant $K$; and it is even growing to the value $2J/\hbar$ when
$\kappa\to 0$. This can be explained as follows: for a given energy, the ratio
of amplitudes for the oscillations of the spin variable $s$ and quadrupolar
variable $\gamma$ vanish at $\kappa\to 0$ as $\kappa$. In fact, for extremely
low anisotropy the spin oscillations are not present in this mode, which
becomes just a free rotation of the quadrupole ellipsoid of the form
$\gamma=2Jt/\hbar$, with $s=\bar{s}=\mathrm{const}$. On a phase plane with
coordinates $(s,\gamma)$ this dynamics is depicted by vertical straight lines
parallel to the $\gamma$-axis, see Fig. 2(c) below. We will discuss this
feature in more detail with the analysis of non-linear oscillations.
### III.2 Nonlinear dynamics and phase plane analysis.
Before considering damped oscillations, it is instructive to discuss
dissipationless non-linear longitudinal oscillations. It is convenient to
present an image of the dynamics as a “phase portrait” on the plane momentum-
coordinate $(s,\;\gamma)$, which shows the behavior of the system for
arbitrary initial conditions. The phase trajectories in the plane without
dissipation can be found from the condition $W(s,\gamma)=\mathrm{const}$.
The energy (6) has an infinite set of minima, with $s=\pm\bar{s}$ and
$\gamma=\pi n,$ with equal energies (green ellipses on the Fig.2), and an
infinite set of maxima at $s=0$ and $\gamma=\pi/2+\pi n$ (red ellipses on the
Fig.2), here $n$ is an integer. Only the minima with $s=\bar{s}$ and
$s=-\bar{s}$ are physically different; equivalent extremes with different
values of $n$ correspond to the equal values of the observables and are
completely equivalent. The minima on the phase plane correspond to foci with
two physically different equilibrium states with antiparallel orientation of
spin $s=\pm\bar{s}$, and
$\left\langle{S_{y}^{2}-S_{x}^{2}}\right\rangle=\sqrt{1-\bar{s}^{2}}$,
$\left\langle S_{z}^{2}\right\rangle=1$,
$\left\langle{S_{x}S_{y}+S_{y}S_{x}}\right\rangle=0$. The saddle points are
located at the values $s=0$ and $\gamma=\pi n$. The lines with $s=\pm 1$ are
singular; these correspond to degenerate motion with $\gamma$ linear in time
$\gamma=\pm tJ/\hbar$; the points at these lines where $d\gamma/dt$ change
sign, can be treated as some non-standard saddle points.
The shape of the phase trajectories, i.e., the characteristic features of
oscillations, varies with the change of the anisotropy parameter $\kappa$.
Note first the general trend, the relative amplitude of the changes of the
spin and $\gamma$ depends on $\kappa$: the bigger $\kappa$ is, the larger
values of the change of spin are observed. The topology of the phase
trajectories change at the critical value of the anisotropy parameter
$\kappa$. At small $\kappa<1/2$, the trajectories with infinite growing
$\gamma$ are present, and the standard separatrix trajectories connect
together different saddle points, see Fig. 2 (c). As mentioned above; only
such trajectories are present at the limit $\kappa\to 0$, but, in fact, even
for the small value of $\kappa=0.2$ used in Fig.2 (c), the main part of the
plane is occupied by the trajectories that change the spin. At the critical
value $\kappa=1/2,$ the separatrix trajectories connect the saddle points at
$s=0$, $\gamma=0$ and the degenerated saddle points at the singular lines with
$s=\pm 1$. At larger anisotropy $\kappa>1/2$, as in Fig. 2(a), the only
separatrix loops entering the same saddle point are present.
The minima, saddle point and the separatrix are the key ingredients for the
switching between the ground states with $s=\bar{s}$ and $s=-\bar{s}$. The
case of large anisotropy $\kappa>1/2$ looks like the standard one for the
switching phenomena, whereas for small anisotropy the situation is more
complicated.
(a) $\ \kappa=$0.6 (b) $\ \kappa=$0.5
(c) $\ \kappa=$0.2
Figure 2: (color online) Phase plane representations for dissipation-free non-
linear longitudinal spin oscillations for different values of the parameter
$\kappa$; $\kappa=$0.6, 0.5 and 0.2 for panels (a), (b), and (c),
respectively. The green and red ellipses present the minima and maxima,
respectively; the standard saddle points are depicted by red rectangles, while
the standard separatrix trajectories are drawn by red lines. The singular
trajectories with $s=\pm 1$ and the separatrix trajectories entering the non-
standard saddle points on these lines $s=\pm 1$ are shown by blue lines on the
frames (a) and (c). For the critical value $\kappa=0.5$, all the separatrix
trajectories and the singular trajectories with $s=\pm 1$ organize a common
net; and on the corresponding frame (b) all of them are presented by red
lines.
### III.3 Damped longitudinal motion
Damping is a crucial ingredient for the dynamical switching between different,
but equivalent in energy, states. The high-frequency mode of longitudinal
oscillations have high-enough relative damping; as was found from microscopic
calculations, [29] the decrement of longitudinal mode
$\Gamma=\lambda\omega_{l}$, where $\lambda\sim 0.2$. To account for the
damping in the dynamic equations for $s$ and $\gamma$, it is useful to
consider a different parametrization of the longitudinal dynamics. Let us now
introduce a unit vector, $\boldsymbol{\sigma}=\sigma_{1}{\rm{\bf
e}}_{1}+\sigma_{2}{\rm{\bf e}}_{2}+\sigma_{3}{\rm{\bf e}}_{3}$, with
components $\sigma_{3}=s$, $\sigma_{1}=\langle S_{y}^{2}-S_{x}^{2}\rangle,$
and $\sigma_{2}=\langle S_{1}S_{2}+S_{2}S_{1}\rangle$. Being written through
$\boldsymbol{\sigma}$, the equation of motion takes the form of the familiar
Landau-Lifshitz equation
$\hbar\frac{\partial\boldsymbol{\sigma}}{\partial
t}=[\boldsymbol{\sigma}\times{\rm{\bf h}}_{\mathrm{eff}}]+{\rm{\bf
R}},\quad{\rm{\bf h}}_{\mathrm{eff}}=-\frac{\partial
W}{\partial\boldsymbol{\sigma}},$ (9)
where ${\rm{\bf h}}_{\mathrm{eff}}$ can be treated as an effective field for
longitudinal dynamics, and the relaxation term ${\rm{\bf R}}$ is added. The
equation of motion with ${\rm{\bf R}}=0$ is fully equivalent to the Hamilton
form of the equation found from (7), but the form of the dissipation is more
straightforward in unit-vector presentation. The choice of the damping term in
a standard equation for the motion of the transverse spin is still under
debate, see. [15] ; [31] But here the damping term can be written in the
simplest form, as in the original paper of Landau and Lifshitz, ${\rm{\bf
R}}=\lambda[{\rm{\bf h}}_{\mathrm{eff}}-\boldsymbol{\sigma}({\rm{\bf
h}}_{\mathrm{eff}}\boldsymbol{\sigma})]$. The arguments are as follows: (i)
this form gives the correct value of the decrement of linear oscillations,
$\Gamma=\lambda\omega_{l}$; (ii) it is convenient for analysis, because it
keeps the condition $\boldsymbol{\sigma}^{2}=1$. Finally, the equations of
motion with the dissipation term of the aforementioned form are:
$\hbar\frac{ds}{dt}=-\frac{\partial
W}{\partial\gamma}-\lambda(1-s^{2})\frac{\partial W}{\partial
s},\;\hbar\frac{d\gamma}{dt}=\frac{\partial W}{\partial
s}-\frac{\lambda}{(1-s^{2})}\frac{\partial W}{\partial\gamma}.$ (10)
These equations describe the damped counterpart of the non-linear longitudinal
oscillations discussed in the previous subsection and present as phase
portraits on Fig. 2. The character of the motion at not-too-large $\lambda$
can be qualitatively understood from energy arguments. The trajectories of
damped oscillations in any point of the phase plane approximately follow the
non-damped (described by equation $W(s,\gamma)=\mathrm{const})$ ones, but
cross them passing from larger to smaller values of $W$, see Figs. 3 and 4. It
happens that for the case of interest, the dynamics is caused by the time-
dependent stimulus. An action of the stimulus on the system can be described
by adding the corresponding time-dependent interaction energy $\Delta W$ to
the system Hamiltonian, $W\to W(s,\gamma)+\Delta W(s,\gamma,t)$. Within this
dynamical picture, $\Delta W$ produces an “external force” driving the system
far from equilibrium.
The analysis is essentially simplified for a pulse-like stimulus of a short
duration $\Delta t$ (much shorter than the period of motion, $\omega_{l}\Delta
t\ll 1)$. In this case, the role of the pulse is reduced to the creation of
some non-equilibrium state, which then evolves as some damped nonlinear
oscillations described by the “free” equations (10) with $\Delta W=0$. The
phase plane method, which shows the behavior of the system for arbitrary
initial conditions, is the best tool for the description of such an evolution.
Figure 3: (color online) Phase plane representation of damped longitudinal
spin oscillations for $\kappa=$0.6. Here and in Fig. 4, the dashed lines
(obtained analytically before) represent the phase trajectories without
dissipation, while the full lines are trajectories for dissipation constant
$\lambda=0.2$, found numerically. The separatrix lines are drawn by red
curves.
It is worth noting that the asymptotic behavior of the separatrix trajectories
at $\gamma,s\to 0$, is important for this analysis, and can be easily found
analytically as
$\left({\frac{\gamma}{s}}\right)_{\mathrm{separ}}=R_{\mathrm{separ}}=\\\
=\frac{1}{8\kappa}\left[{\lambda(1+3\kappa)+\sqrt{\lambda^{2}(1+3\kappa)^{2}+16\kappa(1-\kappa)}}\right]\quad.$
(11)
First let us start with the analysis for high-enough anisotropy. The
corresponding phase portrait is present in Fig. 3. The general property of the
phase plane is that the phase trajectories cannot cross each other; they can
only merge at the saddle points. Thus the trajectories coming to different
minima are stretched between two separatrix lines entering the same saddle
point from different directions, as shown in Fig. 3. From this it follows that
any initial state with arbitrary non-equilibrium values of spin $s(+0)$, but
without deviation of $\gamma$ from its equilibrium value, evolve to the state
with the same sign of the spin as for $s(+0)$, and no switching occurs. On the
other hand, if the initial condition is above the separatrix trajectory,
entering the saddle point, the evolution will move the system to the
equivalent minimum with the sign of the spin opposite to the initial one,
$s(+0)$, realizing the switching.
Figure 4 shows the phase plane for equations (10) for the more complicated
case of low anisotropy, demonstrating possible scenarios of the switching of
the sign of the spin value during such dynamics. Here the separatrix
trajectories for the damped motion can be monitored from their maxima, and the
full picture of the behavior can be understood only when including a few
equivalent foci with $\gamma=0,\,\pm\pi,\,\pm 2\pi$, ect., with different, but
equivalent in energy, values of the spin, $s=\pm\bar{s}$,
$\bar{s}=\sqrt{1-\kappa^{2}}$, and different saddle points, located at
$\gamma=0,\,\pm\pi,\,\pm 2\pi$. As for small anisotropy, the trajectories
coming to different minima are located between two branches of the separatrix
lines, but now this “separatrix corridor” is organized by separatrix lines
entering different saddle points. The switching phenomena is also possible,
but the process involves a few full turns of the variable $\gamma$.
The general regulation for any anisotropy can be formulated as follows: for
realizing spin switching, one needs to have the initial deviation (reduction)
of the spin value, and, simultaneously, a non-zero deviation of the
quadrupolar variable $\gamma$. To switch the positive spin value to negative,
one needs to start from the states just above the separatrix line entering the
saddle point from positive values of $s$. The smaller the initial value of the
spin, the smaller value of $\gamma(0)$ would realize the switching. From the
asymptotic equation (11), the corresponding ratio
$R_{\mathrm{separ}}=\gamma(0)/m(0)$ is smaller for small values of $\lambda$;
but even when $\lambda\to 0,$ it exceeds the value
$R_{\mathrm{separ}}(\lambda=0)=0.5\sqrt{(1-\kappa)/\kappa}$. Thus, the
switching could occur for non-zero values of $\kappa$.
Figure 4: (color online) Phase plane for the damped spin evolutions for low
anisotropy, $\kappa=0.2$. The central frame shows the full diagram; left and
right panels demonstrate the details of the behavior near the equilibrium
values $s=-\bar{s}$ and $s=\bar{s}$, respectively. On this frame, the regions
colored by green and yellow correspond to different basins of attraction with
initial values leading to the equilibrium states with $s=\bar{s}$ and
$s=-\bar{s}$, respectively.
## IV INTERACTION OF THE LIGHT PULSE ON THE SPIN SYSTEM: CREATION OF THE
INITIAL STATE FOR SWITCHING.
Let us now consider the longitudinal spin evolution caused by a specific
stimulus: a femtosecond laser pulse. The reduction of the spin to small values
was observed in many experiments, and the only non-trivial remaining question
is: how can we create a deviation of the quadrupolar variable $\gamma$ from
its equilibrium value $\gamma=0$. To find this, we now consider possible
mechanisms of light interaction with quadrupolar variables of non-Heisenberg
magnets.
The interaction of the spin system of magnetically-ordered media and light is
described by the Hamiltonian (as above, written per spin) $\Delta
W=\bar{\varepsilon}_{ij}v_{0}E_{i}(t)E_{j}^{\ast}(t)/16\pi$, where $v_{0}$ is
the volume per spin, $E_{i}(t)$ is the time-dependent amplitude of the light
in the pulse,
$\bar{\varepsilon}_{ij}=d(\omega\varepsilon_{ij}^{(\mathrm{spin})})/d\omega$,
$\varepsilon_{ij}^{(\mathrm{spin})}$ is the spin-dependent part of the
dielectric permittivity tensor, and $\omega$ is the frequency of light. For
the longitudinal dynamics considered here, circularly-polarized light
propagating along the $z$-axis acts on the $z$-component of the spin via the
standard inverse Faraday effect, with the antisymmetric part of
$\bar{\varepsilon}_{ij}^{(a)}$ as,
$\bar{\varepsilon}_{xy}^{(a)}=-\bar{\varepsilon}_{xy}^{(a)}=s\alpha_{\mathrm{F}}$,
giving an interaction of the form
$\Delta
W_{\mathrm{circular}}=s\frac{\alpha_{\mathrm{F}}v_{0}}{16\pi}|E_{\mathrm{circ}}|^{2}\sigma,$
(12)
where $E_{\mathrm{circ}}$ is the (complex) amplitude, $\sigma$ describes the
pulse helicity: $\sigma=\pm 1$ for right-handed and left-handed circularly
polarized laser pulses. To describe qualitatively the result of the action of
the light pulse, let us now assume that the pulse duration
$\tau_{\mathrm{pulse}}$ is the shortest time of the problem. If the pulse
duration is shorter than the period of spin oscillations, the real pulse shape
can be replaced by the Dirac delta function, $|E_{\mathrm{circ}}|^{2}\to
E_{\mathrm{p}}^{2}\tau_{\mathrm{pulse}}\delta(t),$ where
$E_{\mathrm{p}}^{2}=\int{|E_{\mathrm{circ}}|^{2}dt}/\tau_{\mathrm{pulse}}$
characterizes the pulse intensity. (Note that this approximation is still
qualitatively valid even for any comparable values of $\tau_{\mathrm{pulse}}$
and $2\pi/\omega)$ Then, using equations (10) one can find the effect produced
by the pulse. Within this approximation, the action of a pulse leads to an
instantaneous deviation of the variable $\gamma$ from its equilibrium value,
which then evolves following the non-perturbed equations of motion (10).
Keeping in mind that before the pulse action the system is in equilibrium,
$s(-0)=\bar{s}$ and $\gamma(-0)=0$, it is straightforward to find the values
of these variables $(s$ and $\gamma)$ after the action of the pulse, $s(+0)$
and $\gamma(+0)$. For our purposes, the non-equilibrium value of the
quadrupolar variable $\gamma$ is important:
$\gamma(+0)=-\frac{\alpha_{\mathrm{F}}}{16\pi\hbar}E_{\mathrm{p}}^{2}v_{0}\tau_{\mathrm{pulse}}\sigma.$
(13)
The cumulative action of the circularly-polarized pulse, including an
essential reduction of the spin value (caused either by thermal or non-thermal
mechanisms) and the deviation of $\gamma$ described by (13) could lead to the
evolution we are interested here, switching the spin of the system. Note here
that for standard spin reduction the polarization of the light pulse is not
essential, [32] ; [33] ; [34] whereas the values of $\gamma(+0)$ are opposite
for right- and left-handed circularly polarized pulses. These features are
characteristic of the effect described here. Note the recent experiment where
the role of circular polarization in spin switching for GdFeCo alloy was
mentioned, but the authors have attributed it to magnetic circular dichroism.
[35]
## V CONCLUDING REMARKS.
Let us now compare the approach developed in this article with previous
results on subpicosecond spin evolution. The first experimental observation of
demagnetization for ferromagnetic metals under femtosecond laser pulses shows
that the magnetic moment can be quenched very fast to small values, much
faster than one picosecond. [32] ; [33] ; [34] These effects are associated
with a new domain of the physics of magnets, femtomagnetism, [36] and its
analysis is based on the microscopic consideration of spins of atomic
electrons, [37] ; [38] or itinerant electrons. [39] Not discussing this
fairly promising and fruitful domain of magnetism, note that, to the best of
our knowledge, no effects of magnetization reversal during this “femtomagnetic
stage” has been reported in the literature. For example, the subpicosecond
quenching processes for the ferromagnetic alloy GdFeCo are responsible for the
creation of a far-from-equilibrium state, but the evolution of this state,
giving the spin reversal, can be described within the standard set of
equations for the sublattice magnetizations. [16]
In contrast, here we propose some pathway to switch the sign of the magnetic
moment during extremely short times, of order of the exchange time. It is
shown here that the spin dynamics for magnets with non-small single-ion
anisotropy can lead to the switching of the sign of the magnetic moment via
the longitudinal evolution of the spin modulus together with quadrupolar
variables, i.e., quantum expectation values of operators bilinear over the
spin components $S_{x}$ and $S_{y}$. It is worth to stress here that the
“restoring force” for this dynamics is the _exchange interaction_ , and the
characteristic time is the exchange time. On the other hand, to realize this
scenario, one needs to a have non-Heisenberg interaction, e.g., single-ion
anisotropy, which couples the spin dipole and quadrupole variables.
Obviously, this effect is beyond the standard picture of spin dynamics based
on any closed set of equations for the spin dipolar variables (i.e., the
quantum expectation values linear on the spin components) alone. Note that our
approach based on the full set of variables for the atomic spin is “more
macroscopic” than the “femtomagnetic” approach, [37] ; [38] ; [39] dealing
with electronic states. To realize this type of switching, it is necessary to
have a significant coupling between dipolar and quadrupolar spin variables,
which is present in magnets with strong single-ion anisotropy. Such anisotropy
is known for the numerous magnets based on anisotropic ions of transition
elements such as Ni2+, Cr2+, Fe2+. As the classic example, note nickel
fluosilicate hexahydrate NiSiF${}_{6}\cdot$6H2O, with spin-one Ni2+ ions,
coupled by isotropic ferromagnetic exchange interaction and subject to high
single-ion anisotropy. For this compound, the strong effect of quantum spin
reduction is known, with its strength dependent on the pressure: the value of
$K/J$ is growing with the pressure $P$ resulting in the value $\langle
S\rangle=0.6$ at $P=6$ kbar and leading to the transition to the non-magnetic
state with $\langle S\rangle=0$ at $P\sim 10$ kbar. Bar+NiSiF ; Dyakonov+jetp
A number of recent experiments were done with rare-earth transition-metals
compounds. [13] ; [14] ; [35] ; [40] Note here a rich variety of non-linear
spin dynamics observed for thin films of the FeTb alloy under the action of
femtosecond laser pulses. [40] However, the theory developed here for simple
one-sublattice ferromagnet cannot be directly applied for the description of
such compounds. Ferromagnetic order with high easy-plane anisotropy is present
for many heavy rare-earth elements, such as Tb and Dy at low temperatures.
REbook This feature is known both for bulk monocrystals, REbook and in thin
layers and superlattices, see Ref. Dy_Ysuperlattices and references wherein.
Strictly speaking, in our article only spin-one ions were considered. Rare-
earth metals have non-zero values of both spin and orbital momentum, forming
the total angular momentum of the ion, and for their description the theory
needs some modifications. However, we believe that the effects of spin
switching caused by quadrupolar spin dynamics will be present as well for such
magnets with high values of atomic angular momentum.
The scenario proposed here includes inertial features, with the evolution of
an initial deviation from one equilibrium state to the other, located far from
the initial one. The initial deviations should include both deviation of the
magnetization and of the quadrupolar variable, $\gamma$. Thus the effect is
based on standard magnetization reduction, but it is helicity dependent as
well. The necessary initial deviations can be created by a light pulse of
circular polarization and the possibility of switching depends on the
connection of the initial spin direction and the pulse helicity. The possible
materials should satisfy a number of general conditions: they should be very
susceptible to magnetization quenching, which is typical for many materials,
as well as a sizeable Faraday effect, and they should also have spin one and a
high enough easy-plane anisotropy.
This work is partly supported by the Presidium of the National Academy of
Sciences of Ukraine via projects no.VTs/157 (EGG) and No.0113U001823 (BAI) and
by the grants from State Foundation of Fundamental Research of Ukraine No.
F33.2/002 (VIB and YuAF) and No. F53.2/045 (BAI). FN acknowledges partial
support from the ARO, RIKEN iTHES Project, MURI Center for Dynamic Magneto-
Optics, JSPS-RFBR Contract No. 12-02-92100, Grant-in-Aid for Scientific
Research (S), MEXT Kakenhi on Quantum Cybernetics, and Funding Program for
Innovative R&D on S&T (FIRST).
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|
arxiv-papers
| 2013-06-05T14:07:13 |
2024-09-04T02:49:46.149535
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E.G. Galkina, V.I. Butrim, Yu.A. Fridman, B.A. Ivanov, F. Nori",
"submitter": "Boris Ivanov A",
"url": "https://arxiv.org/abs/1306.1108"
}
|
1306.1154
|
# Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-
rank Matrices111The research was supported in part by NSF FRG Grant
DMS-0854973 and NIH Grant R01 CA127334-05.
T. Tony Cai and Anru Zhang
Department of Statistics
The Wharton School
University of Pennsylvania
###### Abstract
This paper considers compressed sensing and affine rank minimization in both
noiseless and noisy cases and establishes sharp restricted isometry conditions
for sparse signal and low-rank matrix recovery. The analysis relies on a key
technical tool which represents points in a polytope by convex combinations of
sparse vectors. The technique is elementary while leads to sharp results.
It is shown that for any given constant $t\geq{4/3}$, in compressed sensing
$\delta_{tk}^{A}<\sqrt{(t-1)/t}$ guarantees the exact recovery of all $k$
sparse signals in the noiseless case through the constrained $\ell_{1}$
minimization, and similarly in affine rank minimization
$\delta_{tr}^{\mathcal{M}}<\sqrt{(t-1)/t}$ ensures the exact reconstruction of
all matrices with rank at most $r$ in the noiseless case via the constrained
nuclear norm minimization. Moreover, for any $\epsilon>0$,
$\delta_{tk}^{A}<\sqrt{\frac{t-1}{t}}+\epsilon$ is not sufficient to guarantee
the exact recovery of all $k$-sparse signals for large $k$. Similar result
also holds for matrix recovery. In addition, the conditions
$\delta_{tk}^{A}<\sqrt{(t-1)/t}$ and
$\delta_{tr}^{\mathcal{M}}<\sqrt{(t-1)/t}$ are also shown to be sufficient
respectively for stable recovery of approximately sparse signals and low-rank
matrices in the noisy case.
Keywords: Affine rank minimization, compressed sensing, constrained $\ell_{1}$
minimization, low-rank matrix recovery, constrained nuclear norm minimization,
restricted isometry, sparse signal recovery.
## 1 Introduction
Efficient recovery of sparse signals and low-rank matrices has been a very
active area of recent research in applied mathematics, statistics, and machine
learning, with many important applications, ranging from signal processing
[28, 16] to medical imaging [22] to radar systems [3, 21]. A central goal is
to develop fast algorithms that can recover sparse signals and low-rank
matrices from a relatively small number of linear measurements. Constrained
$\ell_{1}$-norm minimization and nuclear norm minimization are among the most
well-known algorithms for the recovery of sparse signals and low-rank matrices
respectively.
In compressed sensing, one observes
$y=A\beta+z,$ (1)
where $y\in\mathbb{R}^{n}$, $A\in\mathbb{R}^{n\times p}$ with $n\ll p$,
$\beta\in\mathbb{R}^{p}$ is an unknown sparse signal, and $z\in\mathbb{R}^{n}$
is a vector of measurement errors. The goal is to recover the unknown signal
$\beta\in\mathbb{R}^{p}$ based on the measurement matrix $A$ and the observed
signal $y$. The constrained $\ell_{1}$ minimization method proposed by Candés
and Tao [11] estimates the signal $\beta$ by
$\hat{\beta}=\mathop{\rm
arg\min}_{\beta\in\mathbb{R}^{p}}\\{\|\beta\|_{1}:\;\mbox{ subject to
}\;A\beta-y\in\mathcal{B}\\},$ (2)
where $\mathcal{B}$ is a set determined by the noise structure. In particular,
$\mathcal{B}$ is taken to be $\\{0\\}$ in the noiseless case. This constrained
$\ell_{1}$ minimization method has now been well studied and it is understood
that the procedure provides an efficient method for sparse signal recovery.
A closely related problem to compressed sensing is the affine rank
minimization problem (ARMP) (Recht et al. [26]), which aims to recover an
unknown low-rank matrix based on its affine transformation. In ARMP, one
observes
$b=\mathcal{M}(X)+z,$ (3)
where $\mathcal{M}:\mathbb{R}^{m\times n}\to\mathbb{R}^{q}$ is a known linear
map, $X\in\mathbb{R}^{m\times n}$ is an unknown low-rank matrix of interest,
and $z\in\mathbb{R}^{q}$ is measurement error. The goal is to recover the low-
rank matrix $X$ based on the linear map $\mathcal{M}$ and the observation
$b\in\mathbb{R}^{q}$. Constrained nuclear norm minimization [26], which is
analogous to $\ell_{1}$ minimization in compressed sensing, estimates $X$ by
$X_{\ast}=\mathop{\rm arg\min}_{B\in\mathbb{R}^{m\times
n}}\\{\|B\|_{\ast}:\;\mbox{ subject to }\;\mathcal{M}(B)-b\in\mathcal{B}\\},$
(4)
where $\|B\|_{\ast}$ is the nuclear norm of $B$, which is defined as the sum
of all singular values of $B$.
One of the most widely used frameworks in compressed sensing is the _restrict
isometry property_ (RIP) introduced in Candés and Tao [11]. A vector
$\beta\in\mathbb{R}^{p}$ is called $s$-sparse if $|$supp$(\beta)|\leq s$,
where supp$(\beta)=\\{i:\beta_{i}\neq 0\\}$ is the support of $\beta$.
###### Definition 1.1
Suppose $A\in\mathbb{R}^{n\times p}$ is a measurement matrix and $1\leq s\leq
p$ is an integer. The restricted isometry constant (RIC) of order $s$ is
defined as the smallest number $\delta_{k}^{A}$ such that for all $s$-sparse
vectors $\beta\in\mathbb{R}^{p}$,
$(1-\delta_{s}^{A})\|\beta\|_{2}^{2}\leq\|A\beta\|_{2}^{2}\leq(1+\delta_{s}^{A})\|\beta\|_{2}^{2}.$
(5)
When $s$ is not an integer, we define $\delta_{s}^{A}$ as $\delta_{\lceil
s\rceil}^{A}$.
Different conditions on the RIC for sparse signal recovery have been
introduced and studied in the literature. For example, sufficient conditions
for the exact recovery in the noiseless case include $\delta_{2k}<\sqrt{2}-1$
in [14], $\delta_{2k}<0.472$ in [6], $\delta_{2k}<0.497$ in [23],
$\delta_{k}<0.307$ in [8], $\delta_{k}<1/3$ and $\delta_{2k}\leq 1/2$ in [9].
There are also other sufficient conditions that involve the RIC of different
orders, e.g. $\delta_{3k}^{A}+3\delta_{4k}^{A}<2$ in [12],
$\delta_{k}^{A}+\delta_{2k}^{A}<1$ in [10], $\delta_{2k}^{A}<0.5746$ jointly
with $\delta_{8k}^{A}<1$, $\delta_{3k}^{A}<0.7731$ jointly with
$\delta_{16k}^{A}<1$ in [30] and $\delta_{2k}^{A}<4/\sqrt{41}$ in [1].
Similar to the RIP for the measurement matrix $A$ in compressed sensing given
in Definition 1.1, a restricted isometry property for a linear map
$\mathcal{M}$ in ARMP can be given. For two matrices $X$ and $Y$ in
$\mathbb{R}^{m\times n}$, define their inner product as $\langle
X,Y\rangle=\sum_{i,j}X_{ij}Y_{ij}$ and the Frobenius norm as
$\|X\|_{F}=\sqrt{\langle X,X\rangle}=\sqrt{\sum_{i,j}X_{ij}^{2}}$.
###### Definition 1.2
Suppose $\mathcal{M}:\mathbb{R}^{n\times m}\to\mathbb{R}^{q}$ is a linear map
and $1\leq r\leq\min(m,n)$ is an integer. The restricted isometry constant
(RIC) of order $r$ for $\mathcal{M}$ is defined as the smallest number
$\delta_{r}^{\mathcal{M}}$ such that for all matrices $X$ with rank at most
$r$,
$(1-\delta_{r}^{\mathcal{M}})\|X\|_{F}^{2}\leq\|\mathcal{M}(X)\|_{2}^{2}\leq(1+\delta_{r}^{\mathcal{M}})\|X\|_{F}^{2}.$
(6)
When $r$ is not an integer, we define $\delta_{r}^{\mathcal{M}}$ as
$\delta_{\lceil r\rceil}^{\mathcal{M}}$.
As in compressed sensing, there are many sufficient conditions based on the
RIC to guarantee the exact recovery of matrices of rank at most $r$ through
the constrained nuclear norm minimization (4). These include
$\delta_{4r}^{\mathcal{M}}<\sqrt{2}-1$ [15],
$\delta_{5r}^{\mathcal{M}}<0.607$, $\delta_{4r}^{\mathcal{M}}<0.558$, and
$\delta_{3r}^{\mathcal{M}}<0.4721$ [24], $\delta_{2r}^{\mathcal{M}}<0.4931$
[29], $\delta_{r}^{\mathcal{M}}<0.307$ [29], $\delta_{r}^{\mathcal{M}}<1/3$
[9], and $\delta_{2r}^{\mathcal{M}}<1/2$ [9].
Among these sufficient RIP conditions, $\delta_{k}^{A}<1/3$ and
$\delta_{r}^{\mathcal{M}}<1/3$ have been verified in [9] to be sharp for both
sparse signal recovery and low-rank matrix recovery problems. Sharp conditions
on the higher order RICs are however still unknown. As pointed out by
Blanchard and Thompson [4], higher-order RIC conditions can be satisfied by a
significantly larger set of Gaussian random matrices in some settings. It is
therefore of both theoretical and practical interests to obtain sharp
sufficient conditions on the high order RICs.
In this paper, we develop a new elementary technique for the analysis of the
constrained $\ell_{1}$-norm minimization and nuclear norm minimization
procedures and establish sharp RIP conditions on the high order RICs for
sparse signal and low-rank matrix recovery. The analysis is surprisingly
simple, while leads to sharp results. The key technical tool we develop states
an elementary geometric fact: Any point in a polytope can be represented as a
convex combination of sparse vectors. The following lemma may be of
independent interest.
###### Lemma 1.1 (Sparse Representation of a Polytope)
For a positive number $\alpha$ and a positive integer $s$, define the polytope
$T(\alpha,s)\subset\mathbb{R}^{p}$ by
$T(\alpha,s)=\\{v\in\mathbb{R}^{p}:\|v\|_{\infty}\leq\alpha,\;\|v\|_{1}\leq
s\alpha\\}.$
For any $v\in\mathbb{R}^{p}$, define the set of sparse vectors
$U(\alpha,s,v)\subset\mathbb{R}^{p}$ by
$U(\alpha,s,v)=\\{u\in\mathbb{R}^{p}:\;{\rm supp}(u)\subseteq{\rm
supp}(v),\;\|u\|_{0}\leq
s,\;\|u\|_{1}=\|v\|_{1},\;\|u\|_{\infty}\leq\alpha\\}.$ (7)
Then $v\in T(\alpha,s)$ if and only if $v$ is in the convex hull of
$U(\alpha,s,v)$. In particular, any $v\in T(\alpha,s)$ can be expressed as
$v=\sum_{i=1}^{N}\lambda_{i}u_{i},\quad\mbox{and }\;0\leq\lambda_{i}\leq
1,\quad\sum_{i=1}^{N}\lambda_{i}=1,\quad\mbox{and }\;u_{i}\in U(\alpha,s,v).$
Lemma 1.1 shows that any point $v\in\mathbb{R}^{p}$ with
$\|v\|_{\infty}\leq\alpha$ and $\|v\|_{1}\leq s\alpha$ must lie in a convex
polytope whose extremal points are $s$-sparse vectors $u$ with
$\|u\|_{1}=\|v\|_{1}$ and $\|u\|_{\infty}\leq\alpha$, and vice versa. This
geometric fact turns out to be a powerful tool in analyzing constrained
$\ell_{1}$-norm minimization for compressed sensing and nuclear norm
minimization for ARMP, since it represents a non-sparse vector by the sparse
ones, which provides a bridge between general vectors and the RIP conditions.
A graphical illustration of Lemma 1.1 is given in Figure 1.
Figure 1: A graphical illustration of sparse representation of a polytope in
one orthant with $p=3$ and $s=2$. All the points in the colored area can be
expressed as convex combinations of the sparse vectors represented by the
three pointed black line segments on the edges.
Combining the results developed in Sections 2 and 3, we establish the
following sharp sufficient RIP conditions for the exact recovery of all
$k$-sparse signals and low-rank matrices in the noiseless case. We focus here
on the exact sparse and noiseless case; the general approximately sparse (low-
rank) and noisy case is considered in Sections 2 and 3.
###### Theorem 1.1
Let $y=A\beta$ where $\beta\in\mathbb{R}^{p}$ is a $k$-sparse vector. If
$\delta_{tk}^{A}<\sqrt{\frac{t-1}{t}}$ (8)
for some $t\geq 4/3$, then the $\ell_{1}$ norm minimizer $\hat{\beta}$ of (2)
with $\mathcal{B}=\\{0\\}$ recovers $\beta$ exactly.
Similarly, suppose $b=\mathcal{M}(X)$ where the matrix
$X\in\mathbb{R}^{m\times n}$is of rank at most $r$. If
$\delta_{tr}^{\mathcal{M}}<\sqrt{\frac{t-1}{t}}$ (9)
for some $t\geq 4/3$, then the nuclear norm minimizer $X_{*}$ of (4) with
$\mathcal{B}=\\{0\\}$ recovers $X$ exactly.
Moreover, it will be shown that for any $\epsilon>0$,
$\delta_{tk}^{A}<\sqrt{\frac{t-1}{t}}+\epsilon$ is not sufficient to guarantee
the exact recovery of all $k$-sparse signals for large $k$. Similar result
also holds for matrix recovery. For the more general approximately sparse
(low-rank) and noisy cases considered in Sections 2 and 3, it is shown that
Conditions (8) and (9) are also sufficient respectively for stable recovery of
(approximately) $k$-sparse signals and (approximately) rank-$r$ matrices in
the noisy case. An oracle inequality is also given in the case of compressed
sensing with Gaussian noise under the condition
$\delta_{tk}^{A}<\sqrt{(t-1)/t}$ when $t\geq 4/3$.
The rest of the paper is organized as follows. Section 2 considers sparse
signal recovery and Section 3 focuses on low-rank matrix recovery. Discussions
on the case $t<4/3$ and some related issues are given in Section 4. The proofs
of the key technical result Lemma 1.1 and the main theorems are contained in
Section 5.
## 2 Compressed Sensing
We consider compressed sensing in this section and establish the sufficient
RIP condition $\delta_{tk}^{A}<\sqrt{(t-1)/t}$ in the noisy case which implies
immediately the results in the noiseless case given in Theorem 1.1. For
$v\in\mathbb{R}^{p}$, we denote $v_{\max(k)}$ as $v$ with all but the largest
$k$ entries in absolute value set to zero, and $v_{-\max(k)}=v-v_{\max(k)}$.
Let us consider the signal recovery model (1) in the setting where the
observations contain noise and the signal is not exactly $k$-sparse. This is
of significant interest for many applications. Two types of bounded noise
settings,
$z\in\mathcal{B}^{\ell_{2}}(\varepsilon)\triangleq\\{z:\|z\|_{2}\leq\varepsilon\\}\quad\mbox{and}\quad
z\in\mathcal{B}^{DS}(\varepsilon)\triangleq\\{z:\|Az\|_{\infty}\leq\varepsilon\\},$
are of particular interest. The first bounded noise case was considered for
example in [18]. The second case is motivated by the _Dantzig Selector_
procedure proposed in [13]. Results on the Gaussian noise case, which is
commonly studied in statistics, follow immediately. For notational
convenience, we write $\delta$ for $\delta^{A}_{tk}$.
###### Theorem 2.1
Consider the signal recovery model (1) with $\|z\|_{2}\leq\varepsilon$.
Suppose $\hat{\beta}^{\ell_{2}}$ is the minimizer of (2) with
$\mathcal{B}=\mathcal{B}^{\ell_{2}}(\eta)=\\{z:\|z\|_{2}\leq\eta\\}$ for some
$\eta\geq\varepsilon$. If $\delta=\delta_{tk}^{A}<\sqrt{(t-1)/t}$ for some
$t\geq 4/3$, then
$\|\hat{\beta}^{\ell_{2}}-\beta\|_{2}\leq\frac{\sqrt{2(1+\delta)}}{1-\sqrt{t/(t-1)}\delta}(\varepsilon+\eta)+\left(\frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1\right)\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k}}.$
(10)
Now consider the signal recovery model (1) with
$\|A^{T}z\|_{\infty}\leq\varepsilon$. Suppose $\hat{\beta}^{DS}$ is the
minimizer of (2) with
$\mathcal{B}=\mathcal{B}^{DS}(\eta)=\\{z:\|A^{T}z\|_{\infty}\leq\eta\\}$ for
some $\eta\geq\varepsilon$. If $\delta=\delta_{tk}^{A}<\sqrt{(t-1)/t}$ for
some $t\geq 4/3$, then
$\|\hat{\beta}^{DS}-\beta\|_{2}\leq\frac{\sqrt{2tk}}{1-\sqrt{t/(t-1)}\delta}(\varepsilon+\eta)+\left(\frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1\right)\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k}}.$
(11)
###### Remark 2.1
The result for the noiseless case follows directly from Theorem 2.1. When
$\beta$ is exactly $k$-sparse and there is no noise, by setting
$\eta=\epsilon=0$ and by noting $\beta_{-\max(k)}=0$, we have
$\hat{\beta}=\beta$ from (10), where $\hat{\beta}$ is the minimizer of (2)
with $\mathcal{B}=\\{0\\}$.
###### Remark 2.2
It should be noted that Theorems 1.1 and 2.1 also hold for $1<t<4/3$ with
exactly the same proof. However the bound $\sqrt{(t-1)/t}$ is not sharp for
$1<t<4/3$. See Section 4 for further discussions. The condition $t\geq 4/3$ is
crucial for the “sharpness” results given in Theorem 2.2 at the end of this
section.
The signal recovery model (1) with Gaussian noise is of particular interest in
statistics and signal processing. The following results on the i.i.d. Gaussian
noise case are immediate consequences of the above results on the bounded
noise cases using the same argument as that in [5, 6], since the Gaussian
random variables are essentially bounded.
###### Proposition 2.1
Suppose the error vector $z\sim N_{n}(0,\sigma^{2}I)$ in (1).
$\delta_{tk}^{A}<\sqrt{(t-1)/t}$ for some $t\geq 4/3$. Let
$\hat{\beta}^{\ell_{2}}$ be the minimizer of (2) with
$\mathcal{B}=\\{z:\|z\|_{2}\leq\sigma\sqrt{n+2\sqrt{n\log n}}\\}$ and let
$\hat{\beta}^{DS}$ be the minimizer of (2) with
$\mathcal{B}=\\{z:\|A^{T}z\|_{\infty}\leq 2\sigma\sqrt{\log p}\\}$. Then with
probability at least $1-1/n$,
$\begin{split}\|\beta^{\ell_{2}}-\beta\|_{2}\leq&\frac{2\sqrt{2(1+\delta)}}{1-\sqrt{t/(t-1)}\delta}\sigma\sqrt{n+2\sqrt{n\log
n}}\\\
&+\left(\frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1\right)\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k}},\end{split}$
and with probability at least $1-1/\sqrt{\pi\log p}$,
$\begin{split}\|\hat{\beta}^{DS}-\beta\|_{2}\leq&\frac{4\sqrt{2t}}{1-\sqrt{t/(t-1)}\delta}\sigma\sqrt{k\log
p}\\\
&+\left(\frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1\right)\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k}}.\end{split}$
The oracle inequality approach was introduced by Donoho and Johnstone [20] in
the context of wavelet thresholding for signal denoising. It provides an
effective way to study the performance of an estimation procedure by comparing
it to that of an ideal estimator. In the context of compressed sensing, oracle
inequalities have been given in [7, 9, 13, 15] under various settings.
Proposition 2.2 below provides an oracle inequality for compressed sensing
with Gaussian noise under the condition $\delta_{tk}^{A}<\sqrt{(t-1)/t}$ when
$t\geq 4/3$.
###### Proposition 2.2
Given (1), suppose the error vector $z\sim N_{n}(0,\sigma^{2}I)$, $\beta$ is
$k$-sparse. Let $\hat{\beta}^{DS}$ be the minimizer of (2) with
$\mathcal{B}=\\{z:\|A^{T}z\|_{\infty}\leq 4\sigma\sqrt{\log p}\\}$. If
$\delta_{tk}^{A}<\sqrt{(t-1)/t}$ for some $t\geq 4/3$, then with probability
at least $1-1/\sqrt{\pi\log p}$,
$\|\hat{\beta}^{DS}-\beta\|_{2}^{2}\leq\frac{256t}{(1-\sqrt{t/(t-1)}\delta_{tk}^{A})^{2}}\log
p\sum_{i}\min(\beta_{i}^{2},\sigma^{2}).$ (12)
We now turn to show the sharpness of the condition
$\delta_{tk}^{A}<\sqrt{(t-1)/t}$ for the exact recovery in the noiseless case
and stable recovery in the noisy case. It should be noted tha tthe result in
the special case $t=2$ was shown in [17].
###### Theorem 2.2
Let $t\geq 4/3$. For all $\varepsilon>0$ and $k\geq 5/\varepsilon$, there
exists a matrix $A$ satisfying $\delta_{tk}<\sqrt{\frac{t-1}{t}}+\varepsilon$
and some $k$-sparse vector $\beta_{0}$ such that
* •
in the noiseless case, i.e. $y=A\beta_{0}$, the $\ell_{1}$ minimization method
(2) with $\mathcal{B}=\\{0\\}$ fail to exactly recover the $k$-sparse vector
$\beta_{0}$, i.e. $\hat{\beta}\neq\beta_{0}$, where $\hat{\beta}$ is the
solution to (2).
* •
in the noisy case, i.e. $y=A\beta_{0}+z$, for all constraints
$\mathcal{B}_{z}$ (may depends on $z$), the $\ell_{1}$ minimization method (2)
fails to stably recover the $k$-sparse vector $\beta_{0}$, i.e.
$\hat{\beta}\nrightarrow\beta$ as $z\to 0$, where $\hat{\beta}$ is the
solution to (2).
## 3 Affine Rank Minimization
We consider the affine rank minimization problem (3) in this section. As
mentioned in the introduction, this problem is closely related to compressed
sensing. The close connections between compressed sensing and ARMP have been
studied in Oymak, et al. [25]. We shall present here the analogous results on
affine rank minimization without detailed proofs.
For a matrix $X\in\mathbb{R}^{m\times n}$ (without loss of generality, assume
that $m\leq n$) with the singular value decomposition
$X=\sum_{i=1}^{m}a_{i}u_{i}v_{i}^{T}$ where the singular values $a_{i}$ are in
descending order, we define $X_{\max(r)}=\sum_{i=1}^{r}a_{i}u_{i}v_{i}^{T}$
and $X_{-\max(r)}=\sum_{i=r+1}^{m}a_{i}u_{i}v_{i}^{T}$. We should also note
that the nuclear norm $\|\cdot\|_{\ast}$ of a matrix equals the sum of the
singular values, and the spectral norm $\|\cdot\|$ of a matrix equals its
largest singular value. Their roles are similar to those of $\ell_{1}$ norm
and $\ell_{\infty}$ norm in the vector case, respectively. For a linear
operator $\mathcal{M}:\mathbb{R}^{m\times n}\to\mathbb{R}^{q}$, its dual
operator is denoted by
$\mathcal{M}^{\ast}:\mathbb{R}^{q}\to\mathbb{R}^{m\times n}$.
Similarly as in compressed sensing, we first consider the matrix recovery
model (3) in the case where the error vector $z$ is in bounded sets:
$\|z\|_{2}\leq\epsilon$ and $\|\mathcal{M}^{\ast}(z)\|\leq\varepsilon$. The
corresponding nuclear norm minimization methods are given by (4) with
$\mathcal{B}=\mathcal{B}^{\ell_{2}}(\eta)$ and
$\mathcal{B}=\mathcal{B}^{DS}(\eta)$ respectively, where
$\displaystyle\mathcal{B}^{\ell_{2}}(\eta)$ $\displaystyle=$
$\displaystyle\\{z:\|z\|_{2}\leq\eta\\},$ (13)
$\displaystyle\mathcal{B}^{DS}(\eta)$ $\displaystyle=$
$\displaystyle\\{z:\|\mathcal{M}^{*}(z)\|\leq\eta\\}.$ (14)
###### Proposition 3.1
Consider ARMP (3) with $\|z\|_{2}\leq\varepsilon$. Let $X_{\ast}^{\ell_{2}}$
be the minimizer of (4) with $\mathcal{B}=\mathcal{B}^{\ell_{2}}(\eta)$
defined in (13) for some $\eta\geq\epsilon$. If
$\delta_{r}^{\mathcal{M}}<\sqrt{(t-1)/t}$ with $t\geq 4/3$, then
$\|X_{\ast}^{\ell_{2}}-X\|_{F}\leq\frac{\sqrt{2(1+\delta)}}{1-\sqrt{t/(t-1)}\delta}(\varepsilon+\eta)+\left(\frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1\right)\frac{2\|X_{-\max(r)}\|_{1}}{\sqrt{r}}.$
(15)
Similarly, consider ARMP (3) with $z$ satisfying
$\|\mathcal{M}^{\ast}(z)\|\leq\varepsilon$. Let $X_{\ast}^{DS}$ be the
minimizer of (4) with $\mathcal{M}=\mathcal{B}^{DS}(\eta)$ defined in (14),
then
$\|X_{\ast}^{DS}-X\|_{F}\leq\frac{\sqrt{2tr}}{1-\sqrt{t/(t-1)}\delta}(\varepsilon+\eta)+\left(\frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1\right)\frac{2\|X_{-\max(r)}\|_{1}}{\sqrt{r}}.$
(16)
In the special noiseless case where $z=0$, it can be seen from either of these
two inequalities above that all matrices $X$ with rank at most $r$ can be
exactly recovered provided that $\delta_{tr}^{\mathcal{M}}<\sqrt{(t-1)/t}$,
for some $t\geq 4/3$.
The following result shows that the condition
$\delta_{tr}^{\mathcal{M}}<\sqrt{(t-1)/t}$ with $t\geq 4/3$ is sharp. These
results together establish the optimal bound on $\delta_{tr}^{\mathcal{M}}$
$(t\geq 4/3)$ for the exact recovery in the noiseless case.
###### Proposition 3.2
Suppose $t\geq 4/3$. For all $\varepsilon>0$ and $r\geq 5/\varepsilon$, there
exists a linear map $\mathcal{M}$ with
$\delta_{tr}^{\mathcal{M}}<\sqrt{(t-1)/t}+\varepsilon$ and some matrix $X_{0}$
of rank at most $r$ such that
* •
in the noiseless case, i.e. $b=\mathcal{M}(X_{0})$, the nuclear norm
minimization method (4) with $\mathcal{B}=\\{0\\}$ fails to exactly recover
$X_{0}$, i.e. $X_{\ast}\neq X_{0}$, where $X_{\ast}$ is the solution to (4).
* •
in the noisy case, i.e. $b=\mathcal{M}(X_{0})+z$, for all constraints
$\mathcal{B}_{z}$ (may depends on $z$), the nuclear norm minimization method
(4) fails to stably recover $X_{0}$, i.e. $X_{\ast}\nrightarrow X_{0}$ as
$z\to 0$, where $X_{\ast}$ is the solution to (4) with
$\mathcal{B}=\mathcal{B}_{z}$.
## 4 Discussion
We shall focus the discussions in this section exclusively on compressed
sensing as the results on affine rank minimization is analogous. In Section 2,
we have established the sharp RIP condition on the high-order RICs,
$\delta_{tk}^{A}<\sqrt{\frac{t-1}{t}}\quad\mbox{for some $t\geq\frac{4}{3}$,}$
for the recovery of $k$-sparse signals in compressed sensing. In addition, it
is known from [9] that $\delta_{k}^{A}<1/3$ is also a sharp RIP condition. For
a general $t>0$, denote the sharp bound for $\delta_{tk}^{A}$ as
$\delta_{\ast}(t)$. Then
$\delta_{\ast}(1)=1/3\quad\mbox{and}\quad\delta_{\ast}(t)=\sqrt{(t-1)/t},\quad
t\geq 4/3.$
A natural question is: What is the value of $\delta_{\ast}(t)$ for $t<4/3$ and
$t\neq 1$? That is, what is the sharp bound for $\delta_{tk}^{A}$ when $t<4/3$
and $t\neq 1$? We have the following partial answer to the question.
###### Proposition 4.1
Let $y=A\beta$ where $\beta\in\mathbb{R}^{p}$ is $k$-sparse. Suppose $0<t<1$
and $tk\geq 0$ to be an integer
* •
When $tk$ is even and $\delta_{tk}^{A}<\frac{t}{4-t}$, the $\ell_{1}$
minimization (2) with $\mathcal{B}=\\{0\\}$ recovers $\beta$ exactly.
* •
When $tk$ is odd and
$\delta_{tk}^{A}<\frac{\sqrt{t^{2}-1/k^{2}}}{4-2t+\sqrt{t^{2}-1/k^{2}}}$, the
$\ell_{1}$ minimization $\eqref{eq:signalmini}$ with $\mathcal{B}=\\{0\\}$
recovers $\beta$ exactly.
In addition, the following result shows that $\delta_{*}(t)\leq\frac{t}{4-t}$
for all $0<t<4/3$. In particular, when $t=1$, the upper bound $t/(4-t)$
coincides with the true sharp bound $1/3$.
###### Proposition 4.2
For $0<t<4/3$, $\varepsilon>0$ and any integer $k\geq 1$,
$\delta_{tk}^{A}<\frac{t}{4-t}+\varepsilon$ is not suffient for the exact
recovery. Specifically, there exists a matrix $A$ with
$\delta_{tk}^{A}=\frac{t}{4-t}$ and a $k$-sparse vector $\beta_{0}$ such that
$\hat{\beta}\neq\beta_{0}$, where $\hat{\beta}$ is the minimizer of (2) with
$\mathcal{B}=\\{0\\}$.
Propositions 4.1 and 4.2 together show that $\delta_{*}(t)=\frac{t}{4-t}$ when
$tk$ is even and $0<t<1$. We are not able to provide a complete answer for
$\delta_{*}(t)$ when $0<t<4/3$. We conjecture that
$\delta_{*}(t)=\frac{t}{4-t}$ for all $0<t<4/3$. The following figure plots
$\delta_{*}(t)$ as a function of $t$ based on this conjecture for the interval
$(0,4/3)$.
Figure 2: Plot of $\delta_{\ast}$ as a function of $t$. The dotted line is
$t=4/3$.
Our results show that exact recovery of $k$-sparse signals in the noiseless
case is guaranteed if $\delta_{tk}^{A}<\sqrt{(t-1)/t}$ for some $t\geq 4/3$.
It is then natural to ask the question: Among all these RIP conditions
$\delta_{tk}^{A}<\delta_{\ast}(t)$, which one is easiest to be satisfied?
There is no general answer to this question as no condition is strictly weaker
or stronger than the others. It is however interesting to consider special
random measurement matrices $A=(A_{ij})_{n\times p}$ where
$A_{ij}\sim\mathcal{N}(0,1/n),\quad
A_{ij}\sim\left\\{\begin{array}[]{ll}1/\sqrt{n}&\text{w.p.}1/2\\\
-1/\sqrt{n}&\text{w.p.}1/2\end{array}\right.,\;\text{ or }\quad
A_{ij}\sim\left\\{\begin{array}[]{ll}\sqrt{3/n}&\text{w.p.}1/6\\\
0&\text{w.p.}1/2\\\ -\sqrt{3/n}&\text{w.p.}1/6\end{array}\right..$
Baraniuk et al [2] provides a bound on RICs for a set of random matrices from
concentration of measure. For these random measurement matrices, Theorem 5.2
of [2] shows that for positive integer $m<n$ and $0<\lambda<1$,
$P(\delta_{m}^{A}<\lambda)\geq
1-2\left(\frac{12ep}{m\lambda}\right)^{m}\exp\left(-n(\lambda^{2}/16-\lambda^{3}/48)\right).$
(17)
Hence, for $t\geq 4/3$,
$P(\delta_{tk}^{A}<\sqrt{(t-1)/t})\geq
1-2\exp\left(tk\left(\log(12e/\sqrt{t(t-1)})+\log(p/k)\right)-n\left(\frac{t-1}{16t}-\frac{(t-1)^{3/2}}{48t^{3/2}}\right)\right).$
For $0<t<4/3$, using the conjectured value $\delta_{*}(t)=\frac{t}{4-t}$, we
have
$P(\delta_{tk}^{A}<t/(4-t))\geq
1-2\exp\left(tk(\log(12(4-t)e/t^{2})+\log(p/k))-n\left(\frac{t^{2}}{16(4-t)^{2}}-\frac{t^{3}}{48(4-t)^{3}}\right)\right).$
It is easy to see when $p,k,$ and $p/k\to\infty$, the lower bound of $n$ to
ensure $\delta_{tk}^{A}<t/(4-t)$ or $\delta_{tk}^{A}<\sqrt{(t-1)/t}$ to hold
in high probability is $n\geq k\log(p/k)n^{\ast}(t)$, where
$n^{\ast}\triangleq\left\\{\begin{array}[]{ll}t/\left(\frac{t^{2}}{16(4-t)^{2}}-\frac{t^{3}}{48(4-t)^{3}}\right)&t<4/3;\\\
t/\left(\frac{t-1}{16t}-\frac{(t-1)^{3/2}}{48t^{3/2}}\right),&t\geq
4/3.\end{array}\right.$
For the plot of $n^{\ast}(t)$, see Figure 1. $n^{\ast}(t)$ has minimum $83.2$
when $t=1.85$. Moreover, among integer $t$, $t=2$ can also provide a near-
optimal minimum: $n^{\ast}(2)=83.7$.
Figure 3: Plot of $n_{\ast}$ as a function of $t$.
We should note that the above analysis is based on the bound given in (17)
which itself can be possibly improved.
## 5 Proofs
We shall first establish the technical result, Lemma 1.1, and then prove the
main results.
Proof of Lemma 1.1. First, suppose $v\in T(\alpha,s)$. We can prove $v$ is in
the convex hull of $U(\alpha,s,v)$ by induction. If $v$ is $s$-sparse, $v$
itself is in $U(\alpha,s,v)$.
Suppose the statement is true for all $(l-1)$-sparse vectors $v$ ($l-1\geq
s$). Then for any $l$-sparse vector $v$ such that $\|v\|_{\infty}\leq\alpha$,
$\|v\|_{1}\leq s\alpha$, without loss of generality we assume that $v$ is not
$(l-1)$-sparse (otherwise the result holds by assumption of $l-1$). Hence we
can express $v$ as $v=\sum_{i=1}^{l}a_{i}e_{i}$, where $e_{i}$’s are different
unit vectors with one entry of $\pm 1$ and other entries of zeros; $a_{1}\geq
a_{2}\geq\cdots\geq a_{l}>0$. Since $\sum_{i=1}^{l}a_{i}=\|v\|_{1}\leq
s\alpha$, so
$1\in D\triangleq\\{1\leq j\leq
l-1:a_{j}+a_{j+1}+\cdots+a_{l}\leq(l-j)\alpha\\},$
which means $D$ is not empty. Take the largest element in $D$ as $j$, which
implies
$a_{j}+a_{j+1}+\cdots+a_{l}\leq(l-j)\alpha,\quad
a_{j+1}+a_{j+2}+\cdots+a_{l}>(l-j-1)\alpha.$ (18)
(It is noteworthy that even if the largest $j$ in $D$ is $l-1$, (18) still
holds). Define
$b_{w}\triangleq\frac{\sum_{i=j}^{l}a_{i}}{l-j}-a_{w},\quad j\leq w\leq l,$
(19)
which satisfies $\sum_{i=j}^{l}a_{i}=(l-j)\sum_{i=j}^{l}b_{i}$. By (18), for
all $j\leq w\leq l$,
$b_{w}\geq
b_{j}=\frac{\sum_{i=j+1}^{l}a_{i}}{l-j}-\frac{l-j-1}{l-j}a_{j}\geq\frac{\sum_{i=j+1}^{l}a_{i}-(l-j-1)\alpha}{l-j}>0.$
In addition, we define
$v_{w}\triangleq\sum_{i=1}^{j-1}a_{i}e_{i}+(\sum_{i=j}^{l}b_{i})\sum_{i=j,i\neq
w}^{l}e_{i}\in
R^{p},\quad\lambda_{w}\triangleq\frac{b_{w}}{\sum_{i=j}^{l}b_{i}},\quad j\leq
w\leq l,$ (20)
then $0\leq\lambda_{w}\leq 1$, $\sum_{w=j}^{l}\lambda_{w}=1$,
$\sum_{w=j}^{l}\lambda_{w}v_{w}=v$,
$\text{supp}(v_{w})\subseteq\text{supp}(v)$. We also have
$\|v_{w}\|_{1}=\sum_{i=1}^{j-1}a_{i}+(l-j)\sum_{w=j}^{l}b_{w}=\sum_{i=1}^{j-1}a_{i}+\sum_{i=j}^{l}a_{i}=\|v\|_{1},$
$\|v_{w}\|_{\infty}=\max\\{a_{1},\cdots,a_{j-1},\sum_{i=j}^{l}b_{i}\\}\leq\max\\{\alpha,\frac{\sum_{i=j}^{l}a_{i}}{l-j}\\}\leq\alpha.$
The last inequality is due to the first part of (18). Finally, note that
$v_{w}$ is $(l-1)$-sparse, we can use the induction assumption to find
$\\{u_{i,w}\in\mathbb{R}^{p},\lambda_{i,w}\in\mathbb{R}:1\leq i\leq
N_{w},j\leq w\leq l\\}$ such that
$u_{i,w}\text{ is
}s\text{-sparse},\quad\text{supp}(u_{i,w})\subseteq\text{supp}(v_{i})\subseteq\text{supp}(v),\quad\|u_{i,w}\|_{1}=\|v_{i}\|_{1}=\|v\|_{1},\quad\|u_{i,w}\|_{\infty}\leq\alpha;$
In addition, $v_{i}=\sum_{i=1}^{N_{w}}\lambda_{i,w}u_{i,w}$, so
$v=\sum_{w=j}^{l}\sum_{i=1}^{N_{w}}\lambda_{w}\lambda_{i,w}u_{i,w}$, which
proves the result for $l$.
The proof of the other part of the lemma is easier. When $v$ is in the convex
hull of $U(\alpha,s,v)$, then we have
$\|v\|_{\infty}=\|\sum_{i=1}^{N}\lambda_{i}u_{i}\|_{\infty}\leq\sum_{i=1}^{N}\lambda_{i}\|u_{i}\|_{\infty}\leq\alpha,$
$\|v\|_{1}=\|\sum_{i=1}^{N}\lambda_{i}u_{i}\|_{1}\leq\sum_{i=1}^{N}\lambda_{i}\|u_{i}\|_{1}\leq\sum_{i=1}^{N}\lambda_{i}\|u_{i}\|_{0}\|u_{i}\|_{\infty}\leq
s\alpha,$
which finished the proof of the lemma. $\square$
Proof of Theorem 1.1 First, we assume that $tk$ is an integer. By the well-
known Null Space Property (Theorem 1 in [27]), we only need to check for all
$h\in\mathcal{N}(A)\setminus\\{0\\}$,
$\|h_{\max(k)}\|_{1}<\|h_{-\max(k)}\|_{1}$. Suppose there exists
$h\in\mathcal{N}(A)\setminus\\{0\\}$, such that
$\|h_{\max(k)}\|_{1}\geq\|h_{-\max(k)}\|_{1}$. Set
$\alpha=\|h_{\max(k)}\|_{1}/k$. We divide $h_{-\max(k)}$ into two parts,
$h_{-\max(k)}=h^{(1)}+h^{(2)}$, where
$h^{(1)}=h_{-\max(k)}\cdot 1_{\\{i||h_{-\max(k)}(i)|>\alpha/(t-1)\\}},\quad
h^{(2)}=h_{-\max(k)}\cdot 1_{\\{i||h_{-\max(k)}(i)|\leq\alpha/(t-1)\\}}.$
Then $\|h^{(1)}\|_{1}\leq\|h_{-\max(k)}\|_{1}\leq\alpha k$. Denote
$|\text{supp}(h^{(1)})|=\|h^{(1)}\|_{0}=m$. Since all non-zero entries of
$h^{(1)}$ have magnitude larger than $\alpha/(t-1)$, we have
$\alpha
k\geq\|h^{(1)}\|_{1}=\sum_{i\in\text{supp}(h^{(1)})}|h^{(1)}(i)|\geq\sum_{i\in\text{supp}(h^{(1)})}\alpha/(t-1)=m\alpha/(t-1).$
Namely $m\leq k(t-1)$. In addition we have
$\begin{split}\|h^{(2)}\|_{1}&=\|h_{-\max(k)}\|_{1}-\|h^{(1)}\|_{1}\leq
k\alpha-\frac{m\alpha}{t-1}=(k(t-1)-m)\cdot\frac{\alpha}{t-1},\\\
\|h^{(2)}\|_{\infty}&\leq\frac{\alpha}{t-1}.\end{split}$ (21)
We now apply Lemma 1.1 with $s=k(t-1)-m$. Then $h^{(2)}$ can be expressed as a
convex combination of sparse vectors:
$h^{(2)}=\sum_{i=1}^{N}\lambda_{i}u_{i}$, where $u_{i}$ is $(k(t-1)-m)$-sparse
and
$\|u_{i}\|_{1}=\|h^{(2)}\|_{1},\quad\|u_{i}\|_{\infty}\leq\frac{\alpha}{(t-1)},\quad\text{supp}(u_{i})\subseteq\text{supp}(h^{(2)}).$
(22)
Hence,
$\|u_{i}\|_{2}\leq\sqrt{\|u_{i}\|_{0}}\|u_{i}\|_{\infty}\leq\sqrt{k(t-1)-m}\|u_{i}\|_{\infty}\leq\sqrt{k(t-1)}\|u_{i}\|_{\infty}\leq\sqrt{k/(t-1)}\alpha.$
(23)
Now we suppose $\mu\geq 0,c\geq 0$ are to be determined. Denote
$\beta_{i}=h_{\max(k)}+h^{(1)}+\mu u_{i}$, then
$\sum_{j=1}^{N}\lambda_{j}\beta_{j}-c\beta_{i}=h_{max{(k)}}+h^{(1)}+\mu
h^{(2)}-c\beta_{i}=(1-\mu-c)(h_{\max(k)}+h^{(1)})-c\mu u_{i}+\mu h.$ (24)
Since $h_{\max(k)}$, $h^{(1)}$, $u_{i}$ are $k$-, $m$-, $(k(t-1)-m)$-sparse
respectively, $\beta_{i}=h_{\max(k)}+h^{(1)}+\mu u_{i}$,
$\sum_{j=1}^{N}\lambda_{j}\beta_{j}-c\beta_{i}-\mu
h=(1-\mu-c)(h_{\max(k)}+h^{(1)})-c\mu u_{i}$ are all $tk$-sparse vectors.
We can check the following identity in $\ell_{2}$ norm,
$\begin{split}\sum_{i=1}^{N}\lambda_{i}\|A(\sum_{j=1}^{N}\lambda_{j}\beta_{j}-c\beta_{i})\|_{2}^{2}+(1-2c)\sum_{1\leq
i<j\leq
N}\lambda_{i}\lambda_{j}\|A(\beta_{i}-\beta_{j})\|_{2}^{2}=\sum_{i=1}^{N}\lambda_{i}(1-c)^{2}\|A\beta_{i}\|_{2}^{2}.\end{split}$
(25)
Since $Ah=0$ and (24), we have
$A(\sum_{j=1}^{N}\lambda_{j}\beta_{j}-c\beta_{i})=A((1-\mu-c)(h_{\max(k)}+h^{(1)})-c\mu
u_{i})$. Set $c=1/2$, $\mu=\sqrt{t(t-1)}-(t-1)$, let the left hand side of
(25) minus the right hand side, we get
$\displaystyle 0$ $\displaystyle\leq$
$\displaystyle(1+\delta_{tk}^{A})\sum_{i=1}^{N}\lambda_{i}\left((1-\mu-c)^{2}\|h_{\max(k)}+h^{(1)}\|_{2}^{2}+c^{2}\mu^{2}\|u_{i}\|_{2}^{2}\right)$
$\displaystyle-(1-\delta_{tk}^{A})\sum_{i=1}^{N}\lambda_{i}(1-c)^{2}\left(\|h_{\max(k)}+h^{(1)}\|_{2}^{2}+\mu^{2}\|u_{i}\|_{2}^{2}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\lambda_{i}\left[\left((1+\delta_{tk}^{A})(\frac{1}{2}-\mu)^{2}-(1-\delta_{tk}^{A})\cdot\frac{1}{4}\right)\|h_{\max(k)}+h^{(1)}\|_{2}^{2}+\frac{1}{2}\delta_{tk}^{A}\mu^{2}\|u_{i}\|_{2}^{2}\right]$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{N}\lambda_{i}\|h_{\max(k)}+h^{(1)}\|_{2}^{2}\left[(\mu^{2}-\mu)+\delta_{tk}^{A}\left(\frac{1}{2}-\mu+(1+\frac{1}{2(t-1)})\mu^{2}\right)\right]$
$\displaystyle=$
$\displaystyle\|h_{\max(k)}+h^{(1)}\|_{2}^{2}\left[\delta_{tk}^{A}\left((2t-1)t-2t\sqrt{t(t-1)}\right)-\left((2t-1)\sqrt{t(t-1)}-2t(t-1)\right)\right]$
$\displaystyle<$ $\displaystyle 0.$
We used the fact that
$\delta_{tk}^{A}<\sqrt{(t-1)/t},$
$\quad\|u_{i}\|_{2}\leq\sqrt{k/(t-1)}\alpha\leq\frac{\|h_{\max(k)}\|_{2}}{\sqrt{(t-1)}}\leq\frac{\|h_{\max(k)}+h^{(1)}\|_{2}}{\sqrt{t-1}}$
above. This is a contradiction.
When $tk$ is not an integer, note $t^{\prime}=\lceil tk\rceil/k$, then
$t^{\prime}>t$, $t^{\prime}k$ is an integer,
$\delta_{t^{\prime}k}=\delta_{tk}<\sqrt{\frac{t-1}{t}}<\sqrt{\frac{t^{\prime}k-1}{t^{\prime}k}},$
which can be deduced to the former case. Hence we finished the proof.
$\square$
Proof of Theorem 2.1. We first prove the inequality on
$\hat{\beta}^{\ell_{2}}$ (10). Again, we assume that $tk$ is an integer at
first. Suppose $h=\hat{\beta}^{\ell_{2}}-\beta$, we shall use a widely known
result (see, e.g., [5], [13], [12], [19]),
$\|h_{-\max(k)}\|_{1}\leq\|h_{\max(k)}\|_{1}+2\|\beta_{-\max(k)}\|_{1}.$
Besides,
$\|Ah\|_{2}\leq\|y-A\beta\|_{2}+\|A\hat{\beta}^{\ell_{2}}-y\|_{2}\leq\varepsilon+\eta.$
(26)
Define $\alpha=(\|h_{\max(k)}\|_{1}+2\|\beta_{-\max(k)}\|_{1})/k$. Similarly
as the proof of Theorem 1.1, we divide $h_{-\max(k)}$ into two parts,
$h_{-\max(k)}=h^{(1)}+h^{(2)}$, where
$h^{(1)}=h_{-\max(k)}\cdot 1_{\\{i||h_{-\max(k)}(i)|>\alpha/(t-1)\\}},\quad
h^{(2)}=h_{-\max(k)}\cdot 1_{\\{i||h_{-\max(k)}(i)|\leq\alpha/(t-1)\\}}.$
Then $\|h^{(1)}\|_{1}\leq\|h_{-\max(k)}\|_{1}\leq\alpha k$. Denote
$|\text{supp}(h^{(1)})|=\|h^{(1)}\|_{0}=m$. Since all non-zero entries of
$h^{(1)}$ have magnitude larger than $\alpha/(t-1)$, we have
$\alpha
k\geq\|h^{(1)}\|_{1}=\sum_{i\in\text{supp}(h^{(1)})}|h^{(1)}(i)|\geq\sum_{i\in\text{supp}(h^{(1)})}\alpha/(t-1)=m\alpha/(t-1).$
Namely $m\leq k(t-1)$. Hence, (21) still holds. Besides,
$\|h_{\max(k)}+h^{(1)}\|_{0}=k+m\leq tk$, we have
$\langle
A(h_{\max(k)}+h^{(1)}),Ah\rangle\leq\|A(h_{\max(k)}+h^{(1)})\|_{2}\|Ah\|_{2}\leq\sqrt{1+\delta}\|h_{\max(k)}+h^{(1)}\|_{2}(\varepsilon+\eta).$
(27)
Again by (21), we apply Lemma 1.1 by setting $s=k(t-1)-m$, we can express
$h^{(2)}$ as a weighted mean: $h^{(2)}=\sum_{i=1}^{N}\lambda_{i}u_{i}$, where
$u_{i}$ is $(k(t-1)-m)$-sparse and (22) still holds. Hence,
$\|u_{i}\|_{2}\leq\sqrt{\|u_{i}\|_{0}}\|u_{i}\|_{\infty}\leq\sqrt{k(t-1)-m}\|u_{i}\|_{\infty}\leq\sqrt{k(t-1)}\|u_{i}\|_{\infty}\leq\sqrt{k/(t-1)}\alpha.$
Now we suppose $1\geq\mu\geq 0,c\geq 0$ are to be determined. Denote
$\beta_{i}=h_{\max(k)}+h^{(1)}+\mu u_{i}$, then we still have (24). Similarly
to the proof of Theorem 1.1, since $h_{\max(k)},h^{(1)},u_{i}$ are $k$-, $m$-,
$(k(t-1)-m)$-sparse vectors, respectively, we know
$\beta_{i}=h_{\max(k)}+h^{(1)}+\mu u_{i}$,
$\sum_{j=1}^{N}\lambda_{j}\beta_{j}-c\beta_{i}-\mu
h=(1-\mu-c)(h_{\max(k)}+h^{(1)})-c\mu u_{i}$ are all $tk$ sparse vectors.
Suppose $x=\|h_{\max(k)}+h^{(1)}\|_{2}$,
$P=\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k}}$, then
$\quad\|u_{i}\|_{2}\leq\sqrt{k/(t-1)}\alpha\leq\frac{\|h_{\max(k)}\|_{2}}{\sqrt{(t-1)}}+\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k(t-1)}}\leq\frac{\|h_{\max(k)}+h^{(1)}\|_{2}}{\sqrt{t-1}}+\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k(t-1)}}=\frac{x+P}{\sqrt{t-1}}.$
We still use the $\ell_{2}$ identity (25). Set $c=1/2$,
$\mu=\sqrt{t(t-1)}-(t-1)$ and take the difference of the left- and right-hand
sides of (25), we get
$\begin{split}0=&\sum_{i=1}^{N}\lambda_{i}\left\|A\left((h_{\max(k)}+h^{(1)}+\mu
h^{(2)})-\frac{1}{2}(h_{\max(k)}+h^{(1)}+\mu
u_{i})\right)\right\|_{2}^{2}-\sum_{i=1}^{N}\frac{\lambda_{i}}{4}\|A\beta_{i}\|_{2}^{2}\\\
=&\sum_{i=1}^{N}\lambda_{i}\left\|A\left((\frac{1}{2}-\mu)(h_{\max(k)}+h^{(1)})-\frac{\mu}{2}u_{i}+\mu
h\right)\right\|_{2}^{2}-\sum_{i=1}^{N}\frac{\lambda_{i}}{4}\|A\beta_{i}\|_{2}^{2}\\\
=&\sum_{i=1}^{N}\lambda_{i}\left\|A\left((\frac{1}{2}-\mu)(h_{\max(k)}+h^{(1)})-\frac{\mu}{2}u_{i}\right)\right\|_{2}^{2}+\mu^{2}\|Ah\|_{2}^{2}\\\
&+2\left\langle
A\left((\frac{1}{2}-\mu)(h_{\max(k)}+h^{(1)})-\frac{\mu}{2}h^{(2)}\right),\mu
Ah\right\rangle-\sum_{i=1}^{N}\frac{\lambda_{i}}{4}\|A\beta_{i}\|_{2}^{2}\\\
=&\sum_{i=1}^{N}\lambda_{i}\left\|A\left((\frac{1}{2}-\mu)(h_{\max(k)}+h^{(1)})-\frac{\mu}{2}u_{i}\right)\right\|_{2}^{2}\\\
&+\mu(1-\mu)\left\langle
A(h_{\max(k)}+h^{(1)}),Ah\right\rangle-\sum_{i=1}^{N}\frac{\lambda_{i}}{4}\|A\beta_{i}\|_{2}^{2}.\end{split}$
Now since $\beta_{i}$,
$(\frac{1}{2}-\mu)(h_{\max(k)}+h^{(1)})-\frac{\mu}{2}u_{i}$ are all
$tk$-sparse vectors, we apply the definition of $\delta_{tk}^{A}$ and also
(27) to get
$\begin{split}0\leq&(1+\delta)\sum_{i=1}^{N}\lambda_{i}\left((\frac{1}{2}-\mu)^{2}\|h_{\max(k)}+h^{(1)}\|_{2}^{2}+\frac{\mu^{2}}{4}\|u_{i}\|_{2}^{2}\right)+\mu(1-\mu)\sqrt{1+\delta}\|h_{\max(k)}+h^{(1)}\|_{2}(\varepsilon+\eta)\\\
&-(1-\delta)\sum_{i=1}^{N}\frac{\lambda_{i}}{4}\left(\|h_{\max(k)}+h^{(1)}\|_{2}^{2}+\mu^{2}\|u_{i}\|_{2}^{2}\right)\\\
=&\sum_{i=1}^{N}\lambda_{i}\left[\left((1+\delta)(\frac{1}{2}-\mu)^{2}-(1-\delta)\cdot\frac{1}{4}\right)\left\|h_{\max(k)}+h^{(1)}\right\|_{2}^{2}+\frac{1}{2}\delta\mu^{2}\|u_{i}\|_{2}^{2}\right]\\\
&+\mu(1-\mu)\sqrt{1+\delta}\left\|h_{\max(k)}+h^{(1)}\right\|_{2}(\varepsilon+\eta)\\\
\leq&\left[(\mu^{2}-\mu)+\delta\left(\frac{1}{2}-\mu+(1+\frac{1}{2(t-1)})\mu^{2}\right)\right]x^{2}+\left[\mu(1-\mu)\sqrt{1+\delta}(\varepsilon+\eta)+\frac{\delta\mu^{2}P}{t-1}\right]x+\frac{\delta\mu^{2}P^{2}}{2(t-1)}\\\
=&-t\left((2t-1)-2\sqrt{t(t-1)}\right)\left(\sqrt{\frac{t-1}{t}}-\delta\right)x^{2}+\left[\mu^{2}\sqrt{\frac{t}{t-1}}\cdot\sqrt{1+\delta}(\varepsilon+\eta)+\frac{\delta\mu^{2}P}{t-1}\right]x+\frac{\delta\mu^{2}P^{2}}{2(t-1)}\\\
=&\frac{\mu^{2}}{t-1}\left[-t\left(\sqrt{\frac{t-1}{t}}-\delta\right)x^{2}+\left(\sqrt{t(t-1)(1+\delta)}(\varepsilon+\eta)+\delta
P\right)x+\frac{\delta P^{2}}{2}\right],\end{split}$ (28)
which is an second-order inequality for $x$. By solving this inequality we get
$\displaystyle x$ $\displaystyle\leq$
$\displaystyle\frac{\left(\sqrt{t(t-1)(1+\delta)}(\varepsilon+\eta)+\delta
P\right)+\sqrt{\left(\sqrt{t(t-1)(1+\delta)}(\varepsilon+\eta)+\delta
P\right)^{2}+2t(\sqrt{(t-1)/t}-\delta)\delta
P^{2}}}{2t(\sqrt{(t-1)/t}-\delta)}$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{t(t-1)(1+\delta)}}{t(\sqrt{(t-1)/t}-\delta)}(\varepsilon+\eta)+\frac{2\delta+\sqrt{2t(\sqrt{(t-1)/t}-\delta)\delta}}{2t(\sqrt{(t-1)/t}-\delta)}P.$
Finally, note that $\|h_{-\max(k)}\|_{1}\leq\|h_{\max(k)}\|_{1}+P\sqrt{k}$, by
Lemma 5.3 in [9], we obtain $\|h_{-\max(k)}\|_{2}\leq\|h_{\max(k)}\|_{2}+P$,
so
$\displaystyle\|h\|_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\|h_{\max(k)}\|_{2}^{2}+\|h_{-\max(k)}\|_{2}^{2}}$
$\displaystyle\leq$
$\displaystyle\sqrt{\|h_{\max(k)}\|_{2}^{2}+(\|h_{\max(k)}\|_{2}+P)^{2}}$
$\displaystyle\leq$ $\displaystyle\sqrt{2\|h_{\max(k)}\|_{2}^{2}}+P$
$\displaystyle\leq$ $\displaystyle\sqrt{2}x+P$ $\displaystyle\leq$
$\displaystyle\frac{\sqrt{2t(t-1)(1+\delta)}}{t(\sqrt{(t-1)/t}-\delta)}(\varepsilon+\eta)+\left(\frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1\right)\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k}}$
$\displaystyle=$
$\displaystyle\frac{\sqrt{2(1+\delta)}}{1-\sqrt{t/(t-1)}\delta}(\varepsilon+\eta)+\left(\frac{\sqrt{2}\delta+\sqrt{t(\sqrt{(t-1)/t}-\delta)\delta}}{t(\sqrt{(t-1)/t}-\delta)}+1\right)\frac{2\|\beta_{-\max(k)}\|_{1}}{\sqrt{k}},$
which finished the proof.
When $tk$ is not an integer, again we define $t^{\prime}=\lceil tk\rceil/k$,
then $t^{\prime}>t$ and
$\delta_{t^{\prime}k}^{A}=\delta_{tk}^{A}<\sqrt{\frac{t-1}{t}}<\sqrt{\frac{t^{\prime}-1}{t^{\prime}}}$.
We can prove the result by working on $\delta_{t^{\prime}k}^{A}$.
For the inequality on $\hat{\beta}^{DS}$ (11), the proof is similar. Define
$h=\hat{\beta}^{DS}-\beta$. We have the following inequalities
$\|A^{T}Ah\|_{\infty}\leq\|A^{T}(A\hat{\beta}^{\ell_{2}}-y)\|_{\infty}+\|A^{T}(y-A\beta)\|_{\infty}\leq\eta+\varepsilon,$
$\langle A(h_{\max(k)}+h^{(1)}),Ah\rangle=\langle
h_{\max(k)}+h^{(1)},A^{T}Ah\rangle\leq\|h_{\max(k)}+h^{(1)}\|_{1}(\varepsilon+\eta)\leq\sqrt{tk}(\varepsilon+\eta)\|h_{\max(k)+h^{(1)}}\|_{2},$
(29)
instead of (26) and (27). We can prove (11) basically the same as the proof
above except that we use (29) instead of (27) when we go from the third term
to the fourth term in (28). $\square$
Proof of Proposition 2.1. By a small extension of Lemma 5.1 in [5], we have
$\|z\|_{2}\leq\sigma\sqrt{n+2\sqrt{n\log n}}$ with probability at least
$1-1/n$; $\|A^{T}z\|_{\infty}\leq\sigma\sqrt{2(1+\delta_{1}^{A})\log p}\leq
2\sigma\sqrt{\log p}$ with probability at least $1-1/\sqrt{\pi\log p}$. Then
the Proposition is immediately implied by Theorem 2.1. $\square$
Proof of Proposition 2.2. The proof of Proposition (2.2) is similar to that of
Theorem 4.1 in [9] and Theorem 2.7 in [15].
First, as in the proof of Proposition 2.1, we have
$\|A^{T}z\|_{\infty}\leq\lambda/2$ with probability at least $1/\sqrt{\pi\log
n}$. In the rest proof, we will prove (12) in the event that
$\|A^{T}z\|_{\infty}\leq\lambda/2$. Define
$K(\xi,\beta)=\gamma\|\xi\|_{0}+\|A\beta-A\xi\|_{2}^{2},\quad\gamma=\frac{\lambda^{2}}{8}=2\sigma^{2}\log
p.$
Let $\bar{\beta}=\arg\min_{\xi}K(\xi,\beta)$. Since $K(\bar{\beta},\beta)\leq
K(\beta,\beta)$, we have $\gamma\|\bar{\beta}\|_{0}\leq\gamma\|\beta\|_{0}$,
which means $\bar{\beta}$ is $k$-sparse.
Now we introduce the following lemma which can be regarded as an extension of
Lemma 4.1 in [9].
###### Lemma 5.1
Suppose $A\in\mathbb{R}^{n\times p}$, $k\geq 2$ is an integer, $s>1$ is real
and $sk$ is integer. Then we have $\delta_{sk}^{A}\leq(2s-1)\delta_{k}^{A}$.
Similarly, suppose $\mathcal{M}:\mathbb{R}^{m\times n}\to\mathbb{R}^{q}$ is a
linear map, $r\geq 2$ is an integer, $s>1$ is real and $sr$ is integer. Then
we have $\delta_{sr}^{\mathcal{M}}\leq(2s-1)\delta_{r}^{\mathcal{M}}$.
We omit the proof here as the proof of Lemma 4.1 in [9] can still apply to
this lemma.
By Lemma 5.1, we can see when $1<t<2$,
$\delta_{2k}^{A}\leq(2\frac{2k}{\lceil tk\rceil}-1)\delta_{\lceil
tk\rceil}^{A}\leq(4/t-1)\delta_{tk}^{A}\leq\sqrt{t/(t-1)}\delta_{tk}^{A}.$
When $t\geq 2$, $\delta_{2k}^{A}\leq\delta_{tk}^{A}$, which means
$\delta_{2k}^{A}\leq\sqrt{t/(t-1)}\delta_{tk}^{A},$ (30)
whenever $t\geq 4/3$.
Next, we have
$\|\bar{\beta}-\beta\|_{2}^{2}\leq\frac{1}{1-\delta_{2k}^{A}}\|A\bar{\beta}-A\beta\|_{2}^{2}\leq\frac{1}{1-\sqrt{t/(t-1)}\delta_{tk}^{A}}\|A\bar{\beta}-A\beta\|_{2}^{2}.$
With a small edition on Lemma 5.4 in [9] and Lemma 3.5 in [15], we have
$\|A^{T}(y-A\bar{\beta})\|_{\infty}\leq\|A^{T}(y-A\beta)\|_{\infty}+\|A^{T}A(\beta-\bar{\beta})\|_{\infty}\leq\lambda.$
Since $\bar{\beta}$ is $k$-sparse, we can apply Theorem 2.1 by plugging
$\beta$ by $\bar{\beta}$ and get
$\|\hat{\beta}-\bar{\beta}\|_{2}\leq\frac{\sqrt{2t\|\bar{\beta}\|_{0}}}{{1-\sqrt{t/(t-1)}\delta_{tk}^{A}}}2\lambda.$
Hence,
$\begin{split}\|\hat{\beta}-\beta\|_{2}^{2}&\leq
2\|\hat{\beta}-\bar{\beta}\|_{2}^{2}+2\|\bar{\beta}-\beta\|_{2}^{2}\leq\frac{16t\|\bar{\beta}\|_{0}\lambda^{2}}{(1-\sqrt{t/(t-1)}\delta_{tk}^{A})^{2}}+\frac{2}{1-\sqrt{t/(t-1)}\delta_{tk}^{A}}\|A\bar{\beta}-A\beta\|_{2}^{2}\\\
&\leq\frac{128t}{(1-\sqrt{t/(t-1)}\delta_{tk}^{A})^{2}}K(\bar{\beta},\beta).\end{split}$
Suppose $\beta^{\prime}=\sum_{i=1}^{p}\beta\cdot 1_{\\{|\beta_{i}|>\mu\\}}$,
where $\mu=\sqrt{\frac{\gamma}{1+\delta_{k}^{A}}}$. Then
$\begin{split}K(\bar{\beta},\beta)&\leq
K(\beta^{\prime},\beta)\leq\gamma\sum_{i=1}^{p}1_{\\{|\beta_{i}|>\mu\\}}+\|A\beta^{\prime}-A\beta\|_{2}^{2}\\\
&\leq\gamma\sum_{i=1}^{p}1_{\\{|\beta_{i}|>\mu\\}}+(1+\delta_{k}^{A})\sum_{i=1}^{p}1_{\\{|\beta_{i}|\leq\mu\\}}|\beta_{i}|^{2}\leq\sum_{i=1}^{p}\min(\gamma,(1+\delta_{k}^{A})|\beta_{i}|^{2})\\\
&\leq 2\log p\sum_{i=1}^{p}\min(\sigma^{2},|\beta_{i}|^{2}).\end{split}$
Therefore, we have proved (12) in the event that
$\|A^{T}z\|_{\infty}\leq\lambda/2$. $\square$
Proof of Theorem 2.2. For any $\varepsilon>0$ and $k\geq 5/\varepsilon$,
suppose $p\geq 2tk$, $m^{\prime}=((t-1)+\sqrt{t(t-1)})k$, $m$ is the largest
integer strictly smaller than $m^{\prime}$. Then $m<m^{\prime}$ and
$m^{\prime}-m\leq 1$. Since $t\geq 4/3$, we have $m^{\prime}\geq k$. Define
$\beta_{1}=\sqrt{k+\frac{mk^{2}}{m^{\prime
2}}}^{-1}(\overbrace{1,\cdots,1}^{k},\overbrace{-\frac{k}{m^{\prime}},\cdots,-\frac{k}{m^{\prime}}}^{m},0,\cdots,0)\in\mathbb{R}^{p},$
then $\|\beta_{1}\|_{2}=1$. We define linear map
$A:\mathbb{R}^{p}\to\mathbb{R}^{p}$, such that for all
$\beta\in\mathbb{R}^{p}$,
$A\beta=\sqrt{1+\sqrt{\frac{t-1}{t}}}\left(\beta-\langle\beta_{1},\beta\rangle\beta_{1}\right).$
Now for all $\lceil tk\rceil$-sparse vector $\beta$,
$\|A\beta\|_{2}^{2}=\left(1+\sqrt{\frac{t-1}{t}}\right)(\beta-\langle\beta_{1},\beta\rangle\beta_{1})^{T}(\beta-\langle\beta_{1},\beta\rangle\beta_{1})=\left(1+\sqrt{\frac{t-1}{t}}\right)\left(\|\beta\|_{2}^{2}-|\langle\beta_{1},\beta\rangle|^{2}\right).$
Since $\beta$ is $\lceil tk\rceil$-sparse, by Cauchy-Schwarz Inequality,
$\displaystyle 0\leq|\langle\beta_{1},\beta\rangle|^{2}$ $\displaystyle\leq$
$\displaystyle\|\beta\|_{2}^{2}\cdot\|\beta_{1}\cdot
1_{\text{supp}(\beta)}\|_{2}^{2}$ $\displaystyle\leq$
$\displaystyle\|\beta\|_{2}^{2}\|\beta_{1,\max(\lceil
tk\rceil)}\|_{2}^{2}=\|\beta\|_{2}^{2}\cdot\frac{m^{\prime 2}+k(\lceil
tk\rceil-k)}{m^{\prime 2}+mk}$ $\displaystyle\leq$
$\displaystyle\frac{m^{\prime 2}+k^{2}(t-1)+k}{m^{\prime
2}+m^{\prime}k}\cdot\frac{1}{1-\frac{k(m^{\prime}-m)}{m^{\prime
2}+m^{\prime}k}}\|\beta\|_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{m^{\prime 2}+k^{2}(t-1)}{m^{\prime
2}+m^{\prime}k}\cdot\frac{m^{\prime 2}+k^{2}(t-1)+k}{m^{\prime
2}+k^{2}(t-1)}\cdot\frac{1}{1-\frac{k(m^{\prime}-m)}{m^{\prime
2}+m^{\prime}k}}\|\beta\|_{2}^{2}$ $\displaystyle=$ $\displaystyle
2\sqrt{t-1}(\sqrt{t}-\sqrt{t-1})\cdot(1+\frac{1}{tk})\cdot\frac{1}{1-\frac{1}{2k}}\|\beta\|_{2}^{2}$
$\displaystyle\leq$
$\displaystyle\left(2\sqrt{t(t-1)}-2(t-1)\right)\cdot(1+\frac{5}{2k})\|\beta\|_{2}^{2}$
$\displaystyle\leq$
$\displaystyle\left(2\sqrt{t(t-1)}-2(t-1)+\frac{5}{2k}\right)\|\beta\|_{2}^{2}.$
We used the fact that $m^{\prime}\geq k$, $0<m^{\prime}-m\leq 1$ and
$\begin{split}\frac{m^{\prime 2}+k^{2}(t-1)}{m^{\prime
2}+m^{\prime}k}&=\frac{\left((t-1)+\sqrt{t(t-1)}\right)^{2}+t-1}{\left((t-1)+\sqrt{t(t-1)}\right)^{2}+\left((t-1)+\sqrt{t(t-1)}\right)}\\\
&=\frac{(t-1)\left(t-1+t+2\sqrt{t(t-1)}+1\right)}{\sqrt{t(t-1)}\left(\sqrt{t}+\sqrt{(t-1)}\right)^{2}}\\\
&=\frac{2\sqrt{t-1}}{\sqrt{t}+\sqrt{t-1}}=2\sqrt{t-1}\left(\sqrt{t}-\sqrt{t-1}\right)\end{split}$
above. Hence,
$\left(1+\sqrt{\frac{t-1}{t}}\right)\|\beta\|_{2}^{2}\geq\|A\beta\|_{2}^{2}\geq\left(1-\sqrt{\frac{t-1}{t}}-\left(1+\sqrt{\frac{t-1}{t}}\right)\frac{5}{2k}\right)\|\beta\|_{2}^{2}\geq\left(1-\sqrt{\frac{t-1}{t}}-\varepsilon\right)\|\beta\|_{2}^{2},$
which implies $\delta_{tk}^{A}\leq\sqrt{(t-1)/t}+\varepsilon$.
Now we consider
$\beta_{0}=(\overbrace{1,\cdots,1}^{k},0,\cdots,0)\in\mathbb{R}^{p},$
$\gamma_{0}=(\overbrace{0,\cdots,0}^{k},\overbrace{\frac{k}{m^{\prime}},\cdots,\frac{k}{m^{\prime}}}^{m},0,\cdots,0).$
Note that $A\beta_{1}=0$, so $A\beta_{0}=A\gamma_{0}$. Besides, $\beta_{0}$ is
$k$-sparse and $\|\gamma_{0}\|_{1}<\|\beta_{0}\|_{1}$.
* •
In the noiseless case, i.e. $y=A\beta_{0}$, the $\ell_{1}$ minimization method
(2) fails to exactly recover $\beta_{0}$ through $y$ since $y=A\gamma_{0}$,
but $\|\gamma_{0}\|_{1}<\|\beta_{0}\|_{1}$.
* •
In the noisy case, i.e. $y=A\beta_{0}+z$, assume that $\ell_{1}$ minimization
method (2) can stably recover $\beta_{0}$ with constraint $\mathcal{B}_{z}$.
Suppose $\hat{\beta}_{z}$ is the solution of $\ell_{1}$ minimization, then
$\lim_{z\to 0}\hat{\beta}_{z}=\beta_{0}$. Note that
$y-A(\hat{\beta}_{z}-\beta_{0}+\gamma_{0})=y-A\hat{\beta}_{z}\in\mathcal{B}_{z}$,
by the definition of $\hat{\beta}_{z}$, we have
$\|\hat{\beta}_{z}-\beta_{0}+\gamma_{0}\|_{1}\geq\|\hat{\beta}_{z}\|_{1}$. Let
$z\to 0$, it contradicts that $\|\gamma_{0}\|_{1}<\|\beta_{0}\|_{1}$.
Therefore, $\ell_{1}$ minimization method (2) fails to stably recover
$\beta_{0}$. $\square$
Proof of Proposition 4.1. We use the technical tools developed in Cai and
Zhang [10] to prove this result. We begin by introducing another important
concept in the RIP framework - restricted orthogonal constants (ROC) proposed
in [11].
###### Definition 5.1
Suppose $A\in\mathbb{R}^{n\times p}$, define the restricted orthogonal
constants (ROC) of order $k_{1},k_{2}$ as the smallest non-negative number
$\theta_{k_{1},k_{2}}^{A}$ such that
$|\langle
A\beta_{1},A\beta_{2}\rangle|\leq\theta_{k_{1},k_{2}}^{A}\|\beta_{1}\|_{2}\|\beta_{2}\|_{2},$
for all $k_{1}$-sparse vector $\beta_{1}\in\mathbb{R}^{p}$ and $k_{2}$-sparse
vector $\beta_{2}\in\mathbb{R}^{p}$ with disjoint supports.
Based on Theorem 2.5 in [10],
$\delta_{tk}^{A}+\frac{2k-tk}{tk}\theta_{tk,tk}^{A}<1$ (31)
is a sufficient condition for exact recovery of all $k$-sparse vectors. By
Lemma 3.1 in [10], $\theta_{tk,tk}^{A}\leq 2\delta_{tk}^{A}$ when $tk$ is
even; $\theta_{tk,tk}^{A}\leq\frac{2tk}{\sqrt{(tk)^{2}-1}}\delta_{tk}^{A}$
when $tk$ is odd. Hence,
$\delta_{tk}^{A}+\frac{2k-tk}{tk}\theta_{tk,tk}^{A}\leq\frac{4-t}{t}\delta_{tk}^{A},\quad\text{
when $tk$ is even;}$
$\delta_{tk}^{A}+\frac{2k-tk}{tk}\theta_{tk,tk}^{A}\leq\left(1+\frac{4k-2tk}{\sqrt{(tk)^{2}-1}}\right)\delta_{tk}^{A},\quad\text{
when $tk$ is odd.}$
The proposition is implied by the inequalities above and (31). $\square$
Proof of Proposition 4.2. The idea of the proof is quite similar to Theorem
3.2 by Cai and Zhang [9]. Define
$\gamma=\frac{1}{\sqrt{2k}}(\overbrace{1,\cdots,1}^{2k},0,\cdots,0),$
$\begin{split}A:\mathbb{R}^{p}&\to\mathbb{R}^{p}\\\
\beta&\mapsto\frac{2}{\sqrt{4-t}}\left(\beta-\langle\beta,\gamma\rangle\gamma\right).\end{split}$
Now for all non-zero $\lceil tk\rceil$-sparse vector $\beta\in\mathbb{R}^{p}$,
$\|A\beta\|_{2}^{2}=\frac{4}{4-t}\langle\beta-\langle\beta,\gamma\rangle\gamma,\beta-\langle\beta,\gamma\rangle\gamma\rangle=\frac{4}{4-t}(\|\beta\|_{2}^{2}-\langle\beta,\gamma\rangle^{2}).$
We can immediately see $\|A\beta\|_{2}^{2}\leq(1+t/(4-t))\|\beta\|_{2}^{2}$.
On the other hand by Cauchy-Schwarz’s inequality,
$\langle\beta,\gamma\rangle^{2}=\langle\beta,\gamma\cdot
1_{\\{supp(\beta)\\}}\rangle^{2}\leq\|\beta\|_{2}^{2}(\sum_{i\in
supp(\beta)}\gamma_{i}^{2})\leq\|\beta\|_{2}^{2}\cdot\frac{\lceil
tk\rceil}{2k}.$
For $k>1/\varepsilon$, we have
$\|A\beta\|_{2}^{2}\geq\frac{4}{4-t}(1-\frac{\lceil
tk\rceil}{2k})\|\beta\|_{2}^{2}\geq\frac{4}{4-t}(1-\frac{tk}{2k}-\varepsilon/2)\|\beta\|_{2}^{2}>(1-\frac{t}{4-t}-\varepsilon)\|\beta\|_{2}^{2}.$
Therefore, we must have $\delta_{tk}^{A}=\delta_{\lceil
tk\rceil}^{A}<t/(4-t)+\varepsilon$.
Finally, we define
$\beta_{0}=(\overbrace{1,\cdots,1}^{k},0,\cdots,0),\quad\beta_{0}^{\prime}=(\overbrace{0,\cdots,0}^{k},\overbrace{-1,\cdots,-1}^{k},0,\cdots,0).$
Then $\beta_{0},\beta_{0}^{\prime}$ are both $k$-sparse, and
$y=A\beta_{0}=A\beta_{0}^{\prime}$. There’s no way to recover both
$\beta_{0},\beta_{0}^{\prime}$ only from $(y,A)$. $\square$
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|
arxiv-papers
| 2013-06-05T15:50:28 |
2024-09-04T02:49:46.160231
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "T. Tony Cai and Anru Zhang",
"submitter": "Anru Zhang",
"url": "https://arxiv.org/abs/1306.1154"
}
|
1306.1246
|
The LHCb collaboration
# Measurement of $C\\!P$ violation in the phase space of
$B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}$ and $B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}$ decays
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48,37, U.
Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U.
Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, D.
Elsby44, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry51, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V.
Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P.
Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A. Hicheur1, E.
Hicks51, D. Hill54, M. Hoballah5, M. Holtrop40, C. Hombach53, P. Hopchev4, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-
March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E.
Maurice6, A. Mazurov16,32,37,e, B. Mc Skelly51, J. McCarthy44, A. McNab53, R.
McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, P.
Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40, F.
Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T.
Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N.
Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V.
Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34,
A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O.
Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora Goicochea2, P.
Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, M. Palutan18, J. Panman37,
A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G.
Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C. Patrignani19,i, C.
Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G. Penso24,l, M. Pepe
Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35,
P. Perret5, M. Perrin-Terrin6, G. Pessina20, K. Petridis52, A. Petrolini19,i,
A. Phan58, E. Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S.
Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E.
Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54,
J. Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig
Navarro38, G. Punzi22,r, W. Qian4, J.H. Rademacker45, B. Rakotomiaramanana38,
M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, M.M.
Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives
Molina35, D.A. Roa Romero5, P. Robbe7, D.A. Roberts57, E. Rodrigues53, P.
Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, J.
Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35, G.
Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d,
V. Salustino Guimaraes2, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i,
R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, I. Sepp52, N. Serra39,
J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P. Shatalov30, Y.
Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30,
A. Shires52, R. Silva Coutinho47, M. Sirendi46, T. Skwarnicki58, N.A. Smith51,
E. Smith54,48, J. Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F.
Soomro18, D. Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P.
Spradlin50, F. Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58,
B. Storaci39, M. Straticiuc28, U. Straumann39, V.K. Subbiah37, L. Sun56, S.
Swientek9, V. Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26,
S. T’Jampens4, M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E.
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Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38,
M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M.
Ubeda Garcia37, A. Ukleja27, D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V.
Vagnoni14, G. Valenti14, A. Vallier7, M. Van Dijk45, R. Vazquez Gomez18, P.
Vazquez Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M.
Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X.
Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A.
Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R.
Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D.
Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D.
Wiedner11, L. Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55,
F.F. Wilson48, J. Wimberley57, J. Wishahi9, M. Witek25, S.A. Wotton46, S.
Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, R. Young49,
X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L.
Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A.
Zvyagin37.
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 Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
###### Abstract
The charmless decays $B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}$ and
$B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}$ are reconstructed using data,
corresponding to an integrated luminosity of 1.0 fb-1, collected by LHCb in
2011. The inclusive charge asymmetries of these modes are measured as
$A_{C\\!P}(B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-})=0.032\pm
0.008\mathrm{\,(stat)}\pm 0.004\mathrm{\,(syst)}\pm
0.007(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0muK^{\pm})$ and
$A_{C\\!P}(B^{\pm}\rightarrow K^{\pm}K^{+}K^{-})=-0.043\pm
0.009\mathrm{\,(stat)}\pm 0.003\mathrm{\,(syst)}\pm
0.007(J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0muK^{\pm})$, where the third
uncertainty is due to the $C\\!P$ asymmetry of the $B^{\pm}\rightarrow
J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0muK^{\pm}$ reference mode. The
significance of $A_{C\\!P}(B^{\pm}\rightarrow K^{\pm}K^{+}K^{-})$ exceeds
three standard deviations and is the first evidence of an inclusive $C\\!P$
asymmetry in charmless three-body $B$ decays. In addition to the inclusive
$C\\!P$ asymmetries, larger asymmetries are observed in localised regions of
phase space.
###### pacs:
Valid PACS appear here
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
| |
---|---|---
| | CERN-PH-EP-2013-090
| | LHCb-PAPER-2013-027
| | 5 June 2013
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
Submitted to Phys. Rev. Lett.
Decays of $B$ mesons to three-body hadronic charmless final states provide an
interesting environment to search for $C\\!P$ violation through the study of
its signatures in the Dalitz plot Miranda1 ; *Miranda2. Theoretical
predictions are mostly based on quasi-two-body decays to intermediate states,
e.g. $\rho^{0}K^{\pm}$ and $K^{*0}(892)\pi^{\pm}$ for ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ decays and $\phi K^{\pm}$ for ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ decays (see, e.g. Ref. Neubert ). These intermediate
states are accessible through amplitude analyses of data, such as those
performed by the Belle and the BaBar collaborations, who reported evidence of
$C\\!P$ violation in the intermediate channel $\rho^{0}K^{\pm}$ bellek2pi ;
BaBark2pi in ${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ decays and more
recently in the channel $\phi K^{\pm}$ BaBarkkk in ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ decays. However, the inclusive $C\\!P$ asymmetry of
${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ decays was found to be consistent with zero.
For direct $C\\!P$ violation to occur, two interfering amplitudes with
different weak and strong phases must be involved in the decay process BSS1979
. Large $C\\!P$ violation effects have been observed in charmless two-body $B$
meson decays such as $B^{0}\rightarrow K^{\pm}\pi^{\mp}$ and
$B^{0}_{s}\rightarrow K^{\mp}\pi^{\pm}$ LHCb-PAPER-2013-018 . However, the
source of the strong phase difference in these processes is not well
understood, which limits the potential to use these measurements to search for
physics beyond the Standard Model. One possible source of the required strong
phase is from final-state hadron rescattering, which can occur between two or
more decay channels with the same flavour quantum numbers, such as
${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ Marshak ; Wolfenstein ; Branco ; livro_Bigi . This effect,
referred to as “compound $C\\!P$ violation” Soni2005 is constrained by
$C\\!PT$ conservation so that the sum of the partial decay widths, for all
channels with the same final-state quantum numbers related by the S-matrix,
must be equal for charge-conjugated decays.
In this Letter we report measurements of the inclusive $C\\!P$-violating
asymmetries in ${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ and
${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ decays with unprecedented precision.
The inclusion of charge-conjugate decay modes is implied except in the
asymmetry definitions. The $C\\!P$ asymmetry in $B^{\pm}$ decays to a final
state $f^{\pm}$ is defined as
$A_{C\\!P}(B^{\pm}\rightarrow f^{\pm})=\Phi[\Gamma(B^{-}\rightarrow
f^{-}),\Gamma(B^{+}\rightarrow f^{+})],$ (1)
where $\Phi[X,Y]\equiv(X-Y)/(X+Y)$ is the asymmetry operator, $\Gamma$ is the
decay width, and the final states are $f^{\pm}=K^{\pm}\pi^{+}\pi^{-}$ or
$f^{\pm}=K^{\pm}K^{+}K^{-}$. We also study their asymmetry distributions
across the phase space.
The LHCb detector Alves:2008zz 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 analysis is based on $pp$ collision data,
corresponding to an integrated luminosity of 1.0 fb-1, collected in 2011 at a
centre-of-mass energy of 7 TeV.
Events are selected by a trigger LHCb-DP-2012-004 that 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. Candidate events
are first required to pass a hardware trigger, which selects particles with a
large transverse energy. The software trigger requires a two-, three- or four-
track secondary vertex with a high sum of the transverse momenta, $p_{\rm T}$,
of the tracks and a significant displacement from the primary $pp$ interaction
vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with
respect to any primary interaction greater than 16, where $\chi^{2}_{\rm IP}$
is defined as the difference between the $\chi^{2}$ of a given PV
reconstructed with and without the considered track. A multivariate algorithm
is used for the identification of secondary vertices consistent with the decay
of a $b$ hadron.
A set of selection criteria is applied to reconstruct $B$ mesons and suppress
the combinatorial backgrounds. The $B^{\pm}$ decay products are required to
satisfy a set of selection criteria on their momenta, transverse momenta, the
$\chi^{2}_{\rm IP}$ of the final-state tracks, and the distance of closest
approach between any two tracks. The $B$ candidates are required to have
$\mbox{$p_{\rm T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $\chi^{2}_{\rm
IP}<10$ and displacement from any PV greater than 3 mm. Additional
requirements are applied to variables related to the $B$ meson production and
decay, such as quality of the track fits for the decay products, and the angle
between the $B$ candidate momentum and the direction of flight from the
primary vertex to the decay vertex. Final-state kaons and pions are further
selected using particle identification information, provided by two ring-
imaging Cherenkov detectors LHCb-DP-2012-003 . The kinematic selection is
common to both decay channels, while the particle identification selection is
specific to each final state. Charm contributions are removed by excluding the
regions of $\pm 30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ around the $D^{0}$
mass in the two-body invariant masses $m_{\pi\pi}$, $m_{K\pi}$ and $m_{KK}$.
The contribution of the
${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$
decay is also excluded from the ${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$
sample by removing the mass region
$3.05<m_{\pi\pi}<3.15{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
Figure 1: Invariant mass spectra of (a) ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ decays and (b) ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ decays. The left panel in each figure shows the $B^{-}$
modes and the right panel shows the $B^{+}$ modes. The results of the unbinned
maximum likelihood fits are overlaid. The main components of the fit are also
shown.
The simulated events used in this analysis are generated using Pythia 6.4
Sjostrand:2006za with a specific LHCb configuration LHCb-PROC-2010-056 .
Decays of hadronic particles are produced by EvtGen Lange:2001uf , in which
final-state radiation is generated using Photos Golonka:2005pn . The
interaction of the generated particles with the detector and its response are
implemented using the Geant4 toolkit Allison:2006ve ; *Agostinelli:2002hh as
described in Ref. LHCb-PROC-2011-006 .
Unbinned extended maximum likelihood fits to the mass spectra of the selected
$B^{\pm}$ candidates are performed. The ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ signal
components are parameterised by so-called Cruijff functions Cruijff to
account for the asymmetric effect of final-state radiation on the signal
shape. The combinatorial background is described by an exponential function,
and the background due to partially reconstructed four-body $B$ decays is
parameterised by an ARGUS function Argus convolved with a Gaussian resolution
function. Peaking backgrounds occur due to decay modes with one misidentified
particle, and consist of the channels ${B^{\pm}\rightarrow
K^{+}K^{-}\pi^{\pm}}$, ${B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}}$ and
$B^{\pm}\rightarrow\eta^{\prime}(\rho^{0}\gamma)K^{\pm}$ for the
${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ mode, and ${B^{\pm}\rightarrow
K^{+}K^{-}\pi^{\pm}}$ for the ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ mode.
The shapes and yields of the peaking backgrounds are obtained from simulation.
The invariant mass spectra of the ${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$
and ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ candidates are shown in Fig. 1.
The mass fits of the two samples are used to obtain the signal yields,
$N(K\pi\pi)=35\,901\pm 327$ and $N(K\\!K\\!K)=22\,119\pm 164$, and the raw
asymmetries, $A_{\rm raw}(K\pi\pi)=0.020\pm 0.007$ and $A_{\rm
raw}(K\\!K\\!K)=-0.060\pm 0.007$, where the uncertainties are statistical. In
order to determine the $C\\!P$ asymmetries, the measured raw asymmetries are
corrected for effects induced by the detector acceptance and interactions of
final-state particles with matter, as well as for a possible $B$-meson
production asymmetry. The $C\\!P$ asymmetry is expressed in terms of the raw
asymmetry and a correction $A_{\Delta}$,
$\\!\\!\\!A_{C\\!P}\\!=\\!A_{\rm raw}\\!-\\!A_{\Delta},\qquad
A_{\Delta}\\!=\\!A_{\rm D}(K^{\pm})\\!+\\!A_{\rm P}(B^{\pm}).$ (2)
Here $A_{\rm D}(K^{\pm})$ is the kaon detection asymmetry, given in terms of
the charge-conjugate kaon detection efficiencies $\varepsilon_{D}$ by $A_{\rm
D}(K^{\pm})=\Phi[\varepsilon_{D}(K^{-}),\varepsilon_{D}(K^{+})]$, and $A_{\rm
P}(B^{\pm})$ is the production asymmetry, defined from the $B^{\pm}$
production rates, $R(B^{\pm})$, as $A_{\rm
P}(B^{\pm})=\Phi[R(B^{-}),R(B^{+})]$. The decay products are regarded as a
pair of charge-conjugate hadrons $h^{+}h^{-}=\pi^{+}\pi^{-},K^{+}K^{-}$, and a
kaon with the same charge as the $B^{\pm}$ meson, whose detection asymmetry is
given by $A_{\rm D}(K^{\pm})$.
The correction term $A_{\Delta}$ is measured from data using a sample of
approximately $6.3\times 10^{4}$
$B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}(\mu^{+}\mu^{-})K^{\pm}$ decays. The
${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$
sample passes the same trigger, kinematic, and kaon particle identification
selection as the signal samples, and it has a similar event topology. The
kaons from ${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm}}$ decay also have similar kinematics in the laboratory frame to
those from the ${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ and
${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ modes. The correction is obtained
from the raw asymmetry of the
${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$ mode
as
$A_{\Delta}=A_{\rm raw}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K)-A_{C\\!P}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K),$ (3)
using the world average of the $C\\!P$ asymmetry
$A_{C\\!P}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K)=(0.1\pm 0.7)\%$
PDG2012 . The $C\\!P$ asymmetries of the ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ channels
are then determined using Eqs. (2) and (3).
Since the detector efficiencies for the signal modes are not flat in the
Dalitz plot, and the raw asymmetries are also not uniformly distributed, an
acceptance correction is applied to the integrated raw asymmetries.
Furthermore, the detector acceptance and reconstruction depend on the trigger
selection. The efficiency of the hadronic hardware trigger is found to have a
small charge asymmetry for final-state kaons. Therefore, the data are divided
into two samples with respect to the hadronic hardware trigger decision:
events with candidates selected by the hadronic trigger, and events selected
by other triggers independently of the signal candidate. In order to apply Eq.
(3) to ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ events selected by the
hadronic hardware trigger, the difference in trigger efficiencies caused by
the presence of three kaons compared to one kaon is taken into account. An
acceptance correction is applied to each trigger sample of the
${B^{-}\rightarrow K^{-}\pi^{+}\pi^{-}}$ and ${B^{-}\rightarrow
K^{-}K^{+}K^{-}}$ modes. It is determined by the ratio between the $B^{-}$ and
$B^{+}$ average efficiencies in simulated events, reweighted to reproduce the
population in the Dalitz plot of signal data. The subtraction of $A_{\Delta}$
is performed separately for each trigger configuration. The integrated $C\\!P$
asymmetries are then the weighted averages of the $C\\!P$ asymmetries for the
two trigger samples.
The systematic uncertainties on the asymmetries are related to the mass fit
models, possible trigger asymmetry, and phase-space acceptance correction. In
order to estimate the uncertainty due to the choice of the signal mass shape,
the initial model is replaced with the sum of a Gaussian and a Crystal Ball
function Skwarnicki:1986xj . The uncertainty associated with the combinatorial
background model is estimated by repeating the fit with a first-order
polynomial. We evaluate three uncertainties related to the peaking
backgrounds: one due to the uncertainty on their yields, another due to the
difference in mass resolution between simulation and data, and a third due to
their possible non-zero asymmetries. The largest deviations from the nominal
results are accounted for as systematic uncertainties. The systematic
uncertainties related to the possible asymmetry induced by the trigger
selection are of two kinds: one due to an asymmetric response of the hadronic
hardware trigger to kaons, and a second due to the choice of sample division
by trigger decision. The former is evaluated by reweighting the
${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$ mode
with the charge-separated kaon efficiencies from calibration data. The latter
is determined by varying the trigger composition of the samples in order to
estimate the systematic differences in trigger admixture between the signal
channels and the ${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{\pm}}$ mode. Two distinct uncertainties are attributed to the phase-
space acceptance corrections: one is obtained from the uncertainty on the
detection efficiency given by the simulation, and the other is evaluated by
varying the binning of the acceptance map. The systematic uncertainties for
the measurements of $A_{C\\!P}({B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}})$
and $A_{C\\!P}({B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}})$ are summarised in
Table 1.
Table 1: Systematic uncertainties on $A_{C\\!P}({K^{\pm}\pi^{+}\pi^{-}})$ and $A_{C\\!P}({K^{\pm}K^{+}K^{-}})$. The total systematic uncertainties are the sum in quadrature of the individual contributions. Systematic uncertainty | $A_{C\\!P}(K\pi\pi)$ | $A_{C\\!P}(K\\!K\\!K)$
---|---|---
Signal model | 0.0010 | 0.0002
Combinatorial background | 0.0006 | $<0.0001$
Peaking background | 0.0007 | 0.0001
Trigger asymmetry | 0.0036 | 0.0019
Acceptance correction | 0.0012 | 0.0019
Total | 0.0040 | 0.0027
The results obtained for the inclusive $C\\!P$ asymmetries of the
${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ decays are
$\displaystyle A_{C\\!P}({B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}})$
$\displaystyle=$ $\displaystyle 0.032\pm 0.008\pm 0.004\pm 0.007,$
$\displaystyle A_{C\\!P}({B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}})$
$\displaystyle=$ $\displaystyle-0.043\pm 0.009\pm 0.003\pm 0.007,$
where the first uncertainty is statistical, the second is the experimental
systematic, and the third is due to the $C\\!P$ asymmetry of the
${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$
reference mode PDG2012 . The significances of the inclusive charge
asymmetries, calculated by dividing the central values by the sum in
quadrature of the statistical and both systematic uncertainties, are 2.8
standard deviations ($\sigma$) for ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ and $3.7\sigma$ for ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ decays.
Figure 2: Asymmetries of the number of signal events in bins of the Dalitz
plot, $A_{C\\!P}^{N}$, for (a) ${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$
and (b) ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ decays. The inset figures
show the projections of the number of background-subtracted events in bins of
(left) the $m^{2}_{\pi^{+}\pi^{-}}$ variable for
$m^{2}_{K^{\pm}\pi^{\mp}}<15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
(right) the $m^{2}_{K^{+}K^{-}\,{\rm low}}$ variable for
$m^{2}_{K^{+}K^{-}\,{\rm
high}}<15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The distributions are
not corrected for acceptance.
Figure 3: Invariant mass spectra of (a) ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ decays in the region
$0.08<m^{2}_{\pi^{+}\pi^{-}}<0.66{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$
and $m^{2}_{K^{\pm}\pi^{\mp}}<15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$,
and (b) ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ decays in the region
$1.2<m^{2}_{K^{+}K^{-}\,{\rm
low}}<2.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$m^{2}_{K^{+}K^{-}\,{\rm
high}}<15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. The left panel in each
figure shows the $B^{-}$ modes and the right panel shows the $B^{+}$ modes.
The results of the unbinned maximum likelihood fits are overlaid.
In addition to the inclusive charge asymmetries, we also study the asymmetry
distributions in the two-dimensional phase space of two-body invariant masses.
The background-subtracted Dalitz plot distributions of the signal region,
defined as the region within three Gaussian widths from the signal peak, are
divided into bins with equal numbers of events in the combined $B^{-}$ and
$B^{+}$ samples. An asymmetry variable,
$A_{C\\!P}^{N}=\Phi[N(B^{-}),N(B^{+})]$, is computed from the number
$N(B^{\pm})$ of negative and positive entries in each bin of the background-
subtracted Dalitz plots.
The distributions of the $A_{C\\!P}^{N}$ variable in the Dalitz plots of
${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ are shown in Fig. 2, where the ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ Dalitz plot is symmetrised and its two-body invariant mass
squared variables are defined as $m^{2}_{K^{+}K^{-}\,{\rm
low}}<m^{2}_{K^{+}K^{-}\,{\rm high}}$. For ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ we identify a positive asymmetry located in the low
$\pi^{+}\pi^{-}$ invariant mass region, around the $\rho(770)^{0}$ resonance,
as seen by Belle bellek2pi and BaBar BaBark2pi , and above the $f_{0}(980)$
resonance. This can be seen also in the inset figure of the $\pi^{+}\pi^{-}$
invariant mass projection, where there is an excess of $B^{-}$ candidates. No
significant asymmetry is present in the low-mass region of the
${K^{\pm}\pi^{\mp}}$ invariant mass projection. The $A_{C\\!P}^{N}$
distribution of the ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ mode reveals an
asymmetry concentrated at low values of $m^{2}_{K^{+}K^{-}\,{\rm low}}$ and
$m^{2}_{K^{+}K^{-}\,{\rm high}}$ in the Dalitz plot. The distribution of the
projection of the number of events onto the $m^{2}_{K^{+}K^{-}\,{\rm low}}$
invariant mass (inset in the right plot of Fig. 2) shows that this asymmetry
is not related to the $\phi(1020)$ resonance, but is instead located in the
region $1.2<m^{2}_{K^{+}K^{-}\,{\rm
low}}<2.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$.
The $C\\!P$ asymmetries are measured in two regions of phase space with large
asymmetry. The ${B^{\pm}\rightarrow K^{\pm}K^{+}K^{-}}$ region,
$m^{2}_{K^{+}K^{-}\,{\rm
high}}<15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$1.2<m^{2}_{K^{+}K^{-}\,{\rm
low}}<2.0{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, is defined such that
the $\phi(1020)$ resonance is excluded. For the ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ mode we measure the $C\\!P$ asymmetry of the region
$m^{2}_{K^{\pm}\pi^{\mp}}<15{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
$0.08<m^{2}_{\pi^{+}\pi^{-}}<0.66{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$,
which spans the lowest $\pi^{+}\pi^{-}$ masses including the $\rho(770)^{0}$
resonance. Unbinned extended maximum likelihood fits are performed to the mass
spectra of the candidates in the two regions, using the same models as the
global fits. The spectra are shown in Fig. 3. The resulting signal yields and
raw asymmetries for the two regions are ${N^{\mathrm{reg}}(K\pi\pi)=552\pm
47}$ and ${A_{\rm raw}^{\mathrm{reg}}(K\pi\pi)=0.687\pm 0.078}$ for the
${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ mode, and
${N^{\mathrm{reg}}(K\\!K\\!K)=2581\pm 55}$ and ${A_{\rm
raw}^{\mathrm{reg}}(K\\!K\\!K)=-0.239\pm 0.020}$ for the ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ mode. The $C\\!P$ asymmetries are obtained from the raw
asymmetries by applying an acceptance correction and subtracting the detection
and production asymmetry correction $A_{\Delta}$ from
${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$
decays. The validity of the global $A_{\Delta}$ from
${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$
decays for the results in the regions was tested by comparing the kinematic
distributions of their decay products. Systematic uncertainties are estimated
due to the signal models, trigger asymmetry, acceptance correction for the
region and due to the limited validity of Eq. (2) for large asymmetries. The
local charge asymmetries for the two regions are measured to be
$\displaystyle A_{C\\!P}^{\mathrm{reg}}(K\pi\pi)$ $\displaystyle=$
$\displaystyle 0.678\pm 0.078\pm 0.032\pm 0.007,$ $\displaystyle
A_{C\\!P}^{\mathrm{reg}}(K\\!K\\!K)$ $\displaystyle=$ $\displaystyle-0.226\pm
0.020\pm 0.004\pm 0.007,$
where the first uncertainty is statistical, the second is the experimental
systematic, and the third is due to the $C\\!P$ asymmetry of the
${B^{\pm}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{\pm}}$
reference mode.
In conclusion, we have measured the inclusive $C\\!P$ asymmetries of the
${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ modes with significances of $2.8\sigma$ and $3.7\sigma$,
respectively. The latter represents the first evidence of an inclusive $C\\!P$
asymmetry in charmless three-body $B$ decays. These charge asymmetries are not
uniformly distributed in the phase space. For ${B^{\pm}\rightarrow
K^{\pm}\pi^{+}\pi^{-}}$ decays, we observe positive asymmetries at low
$\pi^{+}\pi^{-}$ masses, around the $\rho(770)^{0}$ resonance as indicated by
Belle bellek2pi and BaBar BaBark2pi , and also above the $f_{0}(980)$
resonance, where it is not clearly associated to resonances. Although it is
possible to identify the signature of the $\rho(770)^{0}$ resonance for any
value of $m^{2}_{K^{\pm}\pi^{\mp}}$, the asymmetry appears only at low
$K^{\pm}\pi^{\mp}$ mass around the $\rho(770)^{0}$ invariant mass. A signature
of $C\\!P$ violation is present in the ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ Dalitz plot, mostly concentrated in the region of low
$m^{2}_{K^{+}K^{-}\,{\rm low}}$ and low $m^{2}_{K^{+}K^{-}\,{\rm high}}$. A
similar pattern of the $C\\!P$ asymmetry was shown in the preliminary results
of the ${B^{\pm}\rightarrow K^{+}K^{-}\pi^{\pm}}$ and
${B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}}$ decay modes by LHCb LHCb-
CONF-2012-028 , in which the positive asymmetries are at low $\pi^{+}\pi^{-}$
masses and the negative at low $K^{+}K^{-}$ masses, both not clearly
associated to intermediate resonant states.
Moreover, the excess of events in the ${B^{-}\rightarrow K^{-}\pi^{+}\pi^{-}}$
with respect to the ${B^{+}\rightarrow K^{+}\pi^{+}\pi^{-}}$ sample is
comparable to the excess of ${B^{+}\rightarrow K^{+}K^{+}K^{-}}$ with respect
to the ${B^{-}\rightarrow K^{-}K^{+}K^{-}}$ mode. This apparent correlation,
together with the inhomogeneous $C\\!P$ asymmetry distribution in the Dalitz
plot, could be related to compound $C\\!P$ violation. Since the
${B^{\pm}\rightarrow K^{\pm}\pi^{+}\pi^{-}}$ and ${B^{\pm}\rightarrow
K^{\pm}K^{+}K^{-}}$ modes have the same flavour quantum numbers (as do the
pair ${B^{\pm}\rightarrow K^{+}K^{-}\pi^{\pm}}$ and
${B^{\pm}\rightarrow\pi^{+}\pi^{-}\pi^{\pm}}$), $C\\!P$ violation induced by
hadron rescattering could play an important role in these charmless three-body
$B$ decays. In order to quantify a possible compound $C\\!P$ asymmetry, the
introduction of new amplitude analysis techniques, which would take into
account the presence of hadron rescattering in three-body $B$ decays, is
necessary.
## 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); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from 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 thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-06-05T21:10:15 |
2024-09-04T02:49:46.171552
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, 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, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R.\n Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M.\n Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A. Falabella, C.\n F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M.\n Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M.\n Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra\n Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, P. Griffith, 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, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, M. Holtrop, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten,\n R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M.\n Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, 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, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O.\n Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C.\n Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki,\n J. Marks, G. Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli,\n D. Martinez Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C.\n Matteuzzi, E. Maurice, A. Mazurov, B. Mc Skelly, J. McCarthy, A. McNab, R.\n McNulty, B. Meadows, F. Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N.\n Minard, J. Molina Rodriguez, S. Monteil, D. Moran, P. Morawski, A. Mord\\`a,\n M.J. Morello, R. Mountain, I. Mous, F. Muheim, K. M\\\"uller, R. Muresan, B.\n Muryn, B. Muster, P. Naik, T. Nakada, R. Nandakumar, I. Nasteva, M. Needham,\n S. Neubert, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V.\n Niess, R. Niet, N. Nikitin, T. Nikodem, A. Nomerotski, A. Novoselov, A.\n Oblakowska-Mucha, V. Obraztsov, S. Oggero, S. Ogilvy, O. Okhrimenko, R.\n Oldeman, M. Orlandea, J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K.\n Pal, A. Palano, M. Palutan, J. Panman, A. Papanestis, M. Pappagallo, C.\n Parkes, C.J. Parkinson, G. Passaleva, G.D. Patel, M. Patel, G.N. Patrick, C.\n Patrignani, C. Pavel-Nicorescu, A. Pazos Alvarez, A. Pellegrino, G. Penso, M.\n Pepe Altarelli, S. Perazzini, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P.\n Perret, M. Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E.\n Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo\n Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov,\n B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V.\n Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P.\n Rodriguez Perez, S. Roiser, V. Romanovsky, A. Romero Vidal, J. Rouvinet, T.\n Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N.\n Sagidova, P. Sail, B. Saitta, V. Salustino Guimaraes, C. Salzmann, B.\n Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, M.\n Sirendi, T. Skwarnicki, N.A. Smith, E. Smith, J. Smith, M. Smith, M.D.\n Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza De Paula, B. Spaan, A.\n Sparkes, P. Spradlin, F. Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone,\n B. Storaci, M. Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, S. Swientek,\n V. Syropoulos, M. Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M.\n Teklishyn, E. Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg,\n V. Tisserand, M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E.\n Tournefier, S. Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P.\n Tsopelas, N. Tuning, M. Ubeda Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin, U.\n Uwer, V. Vagnoni, G. Valenti, A. Vallier, M. Van Dijk, R. Vazquez Gomez, P.\n Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J. Velthuis, M. Veltri,\n G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward,\n N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J.\n Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M.\n Williams, F.F. Wilson, J. Wimberley, J. Wishahi, M. Witek, S.A. Wotton, S.\n Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang, R. Young, X. Yuan, O.\n Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C. Zhang, Y.\n Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin",
"submitter": "Irina Nasteva",
"url": "https://arxiv.org/abs/1306.1246"
}
|
1306.1298
|
11institutetext: Claremont Graduate University, Institute of Mathematical
Sciences,
Claremont, CA 91711, USA,
[email protected], [email protected] 22institutetext: Naval Air
Warfare Center, Physics and Computational Sciences,
China Lake, CA 93555, USA
# Multiclass Semi-Supervised Learning on Graphs
using Ginzburg-Landau Functional Minimization
Cristina Garcia-Cardona 11 Arjuna Flenner 22 Allon G. Percus 11
###### Abstract
We present a graph-based variational algorithm for classification of high-
dimensional data, generalizing the binary diffuse interface model to the case
of multiple classes. Motivated by total variation techniques, the method
involves minimizing an energy functional made up of three terms. The first two
terms promote a stepwise continuous classification function with sharp
transitions between classes, while preserving symmetry among the class labels.
The third term is a data fidelity term, allowing us to incorporate prior
information into the model in a semi-supervised framework. The performance of
the algorithm on synthetic data, as well as on the COIL and MNIST benchmark
datasets, is competitive with state-of-the-art graph-based multiclass
segmentation methods.
###### Keywords:
diffuse interfaces, learning on graphs, semi-supervised methods
## 1 Introduction
Many tasks in pattern recognition and machine learning rely on the ability to
quantify local similarities in data, and to infer meaningful global structure
from such local characteristics [8]. In the classification framework, the
desired global structure is a descriptive partition of the data into
categories or classes. Many studies have been devoted to the binary
classification problems. The multiple-class case, where data are partitioned
into more than two clusters, is more challenging. One approach is to treat the
problem as a series of binary classification problems [1]. In this paper, we
develop an alternative method, involving a multiple-class extension of the
diffuse interface model introduced in [4].
The diffuse interface model by Bertozzi and Flenner combines methods for
diffusion on graphs with efficient partial differential equation techniques to
solve binary segmentation problems. As with other methods inspired by physical
phenomena [3, 17, 21], it requires the minimization of an energy expression,
specifically the Ginzburg-Landau (GL) energy functional. The formulation
generalizes the GL functional to the case of functions defined on graphs, and
its minimization is related to the minimization of weighted graph cuts [4]. In
this sense, it parallels other techniques based on inference on graphs via
diffusion operators or function estimation [8, 7, 31, 26, 28, 5, 25, 15].
Multiclass segmentation methods that cast the problem as a series of binary
classification problems use a number of different strategies: (i) deal
directly with some binary coding or indicator for the labels [9, 28], (ii)
build a hierarchy or combination of classifiers based on the one-vs-all
approach or on class rankings [14, 13] or (iii) apply a recursive partitioning
scheme consisting of successively subdividing clusters, until the desired
number of classes is reached [25, 15]. While there are advantages to these
approaches, such as possible robustness to mislabeled data, there can be a
considerable number of classifiers to compute, and performance is affected by
the number of classes to partition.
In contrast, we propose an extension of the diffuse interface model that
obtains a simultaneous segmentation into multiple classes. The multiclass
extension is built by modifying the GL energy functional to remove the
prejudicial effect that the order of the labelings, given by integer values,
has in the smoothing term of the original binary diffuse interface model. A
new term that promotes homogenization in a multiclass setup is introduced. The
expression penalizes data points that are located close in the graph but are
not assigned to the same class. This penalty is applied independently of how
different the integer values are, representing the class labels. In this way,
the characteristics of the multiclass classification task are incorporated
directly into the energy functional, with a measure of smoothness independent
of label order, allowing us to obtain high-quality results. Alternative
multiclass methods minimize a Kullback-Leibler divergence function [23] or
expressions involving the discrete Laplace operator on graphs [30, 28].
This paper is organized as follows. Section 2 reviews the diffuse interface
model for binary classification, and describes its application to semi-
supervised learning. Section 3 discusses our proposed multiclass extension and
the corresponding computational algorithm. Section 4 presents results obtained
with our method. Finally, section 5 draws conclusions and delineates future
work.
## 2 Data Segmentation with the Ginzburg-Landau Model
The diffuse interface model [4] is based on a continuous approach, using the
Ginzburg-Landau (GL) energy functional to measure the quality of data
segmentation. A good segmentation is characterized by a state with small
energy. Let $u(\boldsymbol{x})$ be a scalar field defined over a space of
arbitrary dimensionality, and representing the state of the system. The GL
energy is written as the functional
$\mathrm{GL}(u)=\frac{\epsilon}{2}\int\\!|\nabla
u|^{2}\;d\boldsymbol{x}+\frac{1}{\epsilon}\int\\!\Phi(u)\;d\boldsymbol{x},$
(1)
with $\nabla$ denoting the spatial gradient operator, $\epsilon>0$ a real
constant value, and $\Phi$ a double well potential with minima at $\pm 1$:
$\Phi(u)=\frac{1}{4}\left(u^{2}-1\right)^{2}.$ (2)
Segmentation requires minimizing the GL functional. The norm of the gradient
is a smoothing term that penalizes variations in the field $u$. The potential
term, on the other hand, compels $u$ to adopt the discrete labels of $+1$ or
$-1$, clustering the state of the system around two classes. Jointly
minimizing these two terms pushes the system domain towards homogeneous
regions with values close to the minima of the double well potential, making
the model appropriate for binary segmentation.
The smoothing term and potential term are in conflict at the interface between
the two regions, with the first term favoring a gradual transition, and the
second term penalizing deviations from the discrete labels. A compromise
between these conflicting goals is established via the constant $\epsilon$. A
small value of $\epsilon$ denotes a small length transition and a sharper
interface, while a large $\epsilon$ weights the gradient norm more, leading to
a slower transition. The result is a diffuse interface between regions, with
sharpness regulated by $\epsilon$.
It can be shown that in the limit $\epsilon\to 0$ this function approximates
the total variation (TV) formulation in the sense of functional ($\Gamma$)
convergence [18], producing piecewise constant solutions but with greater
computational efficiency than conventional TV minimization methods. Thus, the
diffuse interface model provides a framework to compute piecewise constant
functions with diffuse transitions, approaching the ideal of the TV
formulation, but with the advantage that the smooth energy functional is more
tractable numerically and can be minimized by simple numerical methods such as
gradient descent.
The GL energy has been used to approximate the TV norm for image segmentation
[4] and image inpainting [3, 10]. Furthermore, a calculus on graphs equivalent
to TV has been introduced in [12, 25].
### Application of Diffuse Interface Models to Graphs
An undirected, weighted neighborhood graph is used to represent the local
relationships in the data set. This is a common technique to segment classes
that are not linearly separable. In the $N$-neighborhood graph model, each
vertex $v_{i}\in V$ of the graph corresponds to a data point with feature
vector $\boldsymbol{x}_{i}$, while the weight $w_{ij}$ is a measure of
similarity between $v_{i}$ and $v_{j}$. Moreover, it satisfies the symmetry
property $w_{ij}=w_{ji}$. The neighborhood is defined as the set of $N$
closest points in the feature space. Accordingly, edges exist between each
vertex and the vertices of its $N$-nearest neighbors. Following the approach
of [4], we calculate weights using the local scaling of Zelnik-Manor and
Perona [29],
$w_{ij}=\exp\left(-\frac{||\boldsymbol{x}_{i}-\boldsymbol{x}_{j}||^{2}}{\tau(\boldsymbol{x}_{i})\;\tau(\boldsymbol{x}_{j})}\right).$
(3)
Here, $\tau(\boldsymbol{x}_{i})=||\boldsymbol{x}_{i}-\boldsymbol{x}^{M}_{i}||$
defines a local value for each $\boldsymbol{x}_{i}$, where
$\boldsymbol{x}^{M}_{i}$ is the position of the $M$th closest data point to
$\boldsymbol{x}_{i}$, and $M$ is a global parameter.
It is convenient to express calculations on graphs via the graph Laplacian
matrix, denoted by $\mathbf{L}$. The procedure we use to build the graph
Laplacian is as follows.
1. 1.
Compute the similarity matrix $\mathbf{W}$ with components $w_{ij}$ defined in
(3). As the neighborhood relationship is not symmetric, the resulting matrix
$\mathbf{W}$ is also not symmetric. Make it a symmetric matrix by connecting
vertices $v_{i}$ and $v_{j}$ if $v_{i}$ is among the $N$-nearest neighbors of
$v_{j}$ or if $v_{j}$ is among the $N$-nearest neighbors of $v_{i}$ [27].
2. 2.
Define $\mathbf{D}$ as a diagonal matrix whose $i$th diagonal element
represents the degree of the vertex $v_{i}$, evaluated as
$d_{i}=\sum_{j}w_{ij}.$ (4)
3. 3.
Calculate the graph Laplacian: $\mathbf{L}=\mathbf{D}-\mathbf{W}$.
Generally, the graph Laplacian is normalized to guarantee spectral convergence
in the limit of large sample size [27]. The symmetric normalized graph
Laplacian ${\mathbf{L}\boldsymbol{{}_{s}}}$ is defined as
${\mathbf{L}\boldsymbol{{}_{s}}}=\mathbf{D}^{-1/2}\;\mathbf{L}\;\mathbf{D}^{-1/2}=\mathbf{I}-\mathbf{D}^{-1/2}\;\mathbf{W}\;\mathbf{D}^{-1/2}.$
(5)
Data segmentation can now be carried out through a graph-based formulation of
the GL energy. To implement this task, a fidelity term is added to the
functional as initially suggested in [11]. This enables the specification of a
priori information in the system, for example the known labels of certain
points in the data set. This kind of setup is called semi-supervised learning
(SSL). The discrete GL energy for SSL on graphs can be written as [4]:
$\displaystyle\mathrm{GL}_{\mathrm{SSL}}(\boldsymbol{u})$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\langle\boldsymbol{u},{\mathbf{L}\boldsymbol{{}_{s}}}\boldsymbol{u}\rangle+\frac{1}{\epsilon}\sum_{v_{i}\in
V}\Phi(u(v_{i}))+\sum_{v_{i}\in
V}\frac{\mu(v_{i})}{2}\;\left(u(v_{i})-\hat{u}(v_{i})\right)^{2}$ (7)
$\displaystyle=$ $\displaystyle\frac{\epsilon}{4}\sum_{v_{i},v_{j}\in
V}w_{ij}\left(\frac{u(v_{i})}{\sqrt{d_{i}}}-\frac{u(v_{j})}{\sqrt{d_{j}}}\right)^{2}+\frac{1}{\epsilon}\sum_{v_{i}\in
V}\Phi(u(v_{i}))+$ $\displaystyle\sum_{v_{i}\in
V}\frac{\mu(v_{i})}{2}\;\left(u(v_{i})-\hat{u}(v_{i})\right)^{2}.$
In the discrete formulation, $\boldsymbol{u}$ is a vector whose component
$u(v_{i})$ represents the state of the vertex $v_{i}$, $\epsilon>0$ is a real
constant characterizing the smoothness of the transition between classes, and
$\mu(v_{i})$ is a fidelity weight taking value $\mu>0$ if the label
$\hat{u}(v_{i})$ (i.e. class) of the data point associated with vertex $v_{i}$
is known beforehand, or $\mu(v_{i})=0$ if it is not known (semi-supervised).
Minimizing the functional simulates a diffusion process on the graph. The
information of the few labels known is propagated through the discrete
structure by means of the smoothing term, while the potential term clusters
the vertices around the states $\pm 1$ and the fidelity term enforces the
known labels. The energy minimization process itself attempts to reduce the
interface regions. Note that in the absence of the fidelity term, the process
could lead to a trivial steady-state solution of the diffusion equation, with
all data points assigned the same label.
The final state $u(v_{i})$ of each vertex is obtained by thresholding, and the
resulting homogeneous regions with labels of $+1$ and $-1$ constitute the two-
class data segmentation.
## 3 Multiclass Extension
The double-well potential in the diffuse interface model for SSL drives the
state of the system towards two definite labels. Multiple-class segmentation
requires a more general potential function $\Phi_{M}(u)$ that allows clusters
around more than two labels. For this purpose, we use the periodic-well
potential suggested by Li and Kim [21],
$\Phi_{M}(u)=\frac{1}{2}\,\\{u\\}^{2}\,(\\{u\\}-1)^{2},$ (8)
where $\\{u\\}$ denotes the fractional part of $u$,
$\\{u\\}=u-\lfloor u\rfloor,$ (9)
and $\lfloor u\rfloor$ is the largest integer not greater than $u$.
This periodic potential well promotes a multiclass solution, but the graph
Laplacian term in Equation (7) also requires modification for effective
calculations due to the fixed ordering of class labels in the multiple class
setting. The graph Laplacian term penalizes large changes in the spatial
distribution of the system state more than smaller gradual changes. In a
multiclass framework, this implies that the penalty for two spatially
contiguous classes with different labels may vary according to the (arbitrary)
ordering of the labels.
This phenomenon is shown in Fig. 1. Suppose that the goal is to segment the
image into three classes: class 0 composed by the black region, class 1
composed by the gray region and class 2 composed by the white region. It is
clear that the horizontal interfaces comprise a jump of size 1 (analogous to a
two class segmentation) while the vertical interface implies a jump of size 2.
Accordingly, the smoothing term will assign a higher cost to the vertical
interface, even though from the point of view of the classification, there is
no specific reason for this. In this example, the problem cannot be solved
with a different label assignment. There will always be an interface with
higher costs than others independent of the integer values used.
Thus, the multiclass approach breaks the symmetry among classes, influencing
the diffuse interface evolution in an undesirable manner. Eliminating this
inconvenience requires restoring the symmetry, so that the difference between
two classes is always the same, regardless of their labels. This objective is
achieved by introducing a new class difference measure.
Figure 1: Three-class segmentation. Black: class 0. Gray: class 1. White:
class 2.
### 3.1 Generalized Difference Function
The final class labels are determined by thresholding each vertex $u(v_{i})$,
with the label $y_{i}$ set to the nearest integer:
$y_{i}=\left\lfloor u(v_{i})+\frac{1}{2}\right\rfloor.$ (10)
The boundaries between classes then occur at half-integer values corresponding
to the unstable equilibrium states of the potential well. Define the function
$\hat{r}(x)$ to represent the distance to the nearest half-integer:
$\hat{r}(x)=\left|\frac{1}{2}-\\{x\\}\right|.$ (11)
A schematic of $\hat{r}(x)$ is depicted in Fig. 2. The $\hat{r}(x)$ function
is used to define a generalized difference function between classes that
restores symmetry in the energy functional. Define the generalized difference
function $\rho$ as:
$\rho(u(v_{i}),u(v_{j}))=\left\\{\begin{array}[]{lll}\hat{r}(u(v_{i}))+\hat{r}(u(v_{j}))&&y_{i}\neq
y_{j}\\\ &&\\\
\left|\hat{r}(u(v_{i}))-\hat{r}(u(v_{j}))\right|&&y_{i}=y_{j}\end{array}\right.$
(12)
Thus, if the vertices are in different classes, the difference $\hat{r}(x)$
between each state’s value and the nearest half-integer is added, whereas if
they are in the same class, these differences are subtracted. The function
$\rho(x,y)$ corresponds to the tree distance (see Fig. 2). Strictly speaking,
$\rho$ is not a metric since it does not satisfy $\rho(x,y)=0\Rightarrow x=y$.
Nevertheless, the cost of interfaces between classes becomes the same
regardless of class labeling when this generalized distance function is
implemented.
Half-integerInteger$\hat{r}(x)$ Figure 2: Schematic interpretation of
generalized difference: $\hat{r}(x)$ measures distance to nearest half-
integer, and $\rho$ is a tree distance measure.
The GL energy functional for SSL, using the new generalized difference
function $\rho$ and the periodic potential, is expressed as
$\displaystyle\mathrm{MGL}_{\mathrm{SSL}}(\boldsymbol{u})$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{2}\sum_{v_{i}\in V}\sum_{v_{j}\in
V}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\,\left[\rho(u(v_{i}),u(v_{j}))\,\right]^{2}+$
(13) $\displaystyle\frac{1}{2\epsilon}\sum_{v_{i}\in
V}\\{u(v_{i})\\}^{2}\,(\\{u(v_{i})\\}-1)^{2}+$ $\displaystyle\sum_{v_{i}\in
V}\frac{\mu(v_{i})}{2}\;\left(u(v_{i})-\hat{u}(v_{i})\right)^{2}.$
Note that the smoothing term in this functional is composed of an operator
that is not just a generalization of the normalized symmetric Laplacian
${\mathbf{L}\boldsymbol{{}_{s}}}$. The new smoothing operation, written in
terms of the generalized distance function $\rho$, constitutes a non-linear
operator that is a symmetrization of a different normalized Laplacian, the
random walk Laplacian
$\mathbf{L}\boldsymbol{{}_{w}}=\mathbf{D}^{-1}\mathbf{L}=\mathbf{I}-\mathbf{D}^{-1}\mathbf{W}$
[27]. The reason is as follows. The Laplacian $\mathbf{L}$ satisfies
$(\mathbf{L}\boldsymbol{u})_{i}=\sum_{j}w_{ij}\left(u_{i}-u_{j}\right)$
and $\mathbf{L}\boldsymbol{{}_{w}}$ satisfies
$(\mathbf{L}\boldsymbol{{}_{w}}\boldsymbol{u})_{i}=\sum_{j}\frac{w_{ij}}{d_{i}}\left(u_{i}-u_{j}\right).$
Now replace $w_{ij}/{d_{i}}$ in the latter expression with the symmetric form
${w_{ij}}/{\sqrt{d_{i}d_{j}}}$. This is equivalent to constructing a
reweighted graph with weights $\widehat{w}_{ij}$ given by:
$\widehat{w}_{ij}=\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}.$
The corresponding reweighted Laplacian $\mathbf{\widehat{L}}$ satisfies:
$(\mathbf{\widehat{L}}\boldsymbol{u})_{i}=\sum_{j}\widehat{w}_{ij}\left(u_{i}-u_{j}\right)=\sum_{j}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\left(u_{i}-u_{j}\right),$
(14)
and
$\langle\boldsymbol{u},\mathbf{\widehat{L}}\boldsymbol{u}\rangle=\frac{1}{2}\sum_{i,j}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\left(u_{i}-u_{j}\right)^{2}.$
(15)
While $\mathbf{\widehat{L}}=\mathbf{\widehat{D}}-\mathbf{\widehat{W}}$ is not
a standard normalized Laplacian, it does have the desirable properties of
stability and consistency with increasing sample size of the data set, and of
satisfying the conditions for $\Gamma$-convergence to TV in the $\epsilon\to
0$ limit [2]. It also generalizes to the tree distance more easily than does
${\mathbf{L}\boldsymbol{{}_{s}}}$. Replacing the difference
$\left(u_{i}-u_{j}\right)^{2}$ with the generalized difference
$\left[\rho(u_{i},u_{j})\right]^{2}$ then gives the new smoothing multiclass
term of equation (13). Empirically, this new term seems to perform well even
though the normalization procedure differs from the binary case.
By implementing the generalized difference function on a tree, the cost of
interfaces between classes becomes the same regardless of class labeling.
### 3.2 Computational Algorithm
The GL energy functional given by (13) may be minimized iteratively, using
gradient descent:
$u_{i}^{n+1}=u_{i}^{n}-dt\,\left[\frac{\delta\mathrm{MGL}_{\mathrm{SSL}}}{\delta
u_{i}}\right],$ (16)
where $u_{i}$ is a shorthand for $u(v_{i})$, $dt$ represents the time step and
the gradient direction is given by:
$\frac{\delta\mathrm{MGL}_{\mathrm{SSL}}}{\delta
u_{i}}=\epsilon\;\hat{R}(u_{i}^{n})+\frac{1}{\epsilon}\Phi_{M}^{\prime}(u_{i}^{n})+\mu_{i}\left(u_{i}^{n}-\hat{u}_{i}\right)$
(17)
$\hat{R}(u_{i}^{n})=\sum_{j}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\left[\hat{r}(u_{i}^{n})\pm\hat{r}(u_{j}^{n})\right]\hat{r}^{\prime}(u_{i}^{n})$
(18)
$\Phi_{M}^{\prime}(u_{i}^{n})=2\;\\{u_{i}^{n}\\}^{3}-3\;\\{u_{i}^{n}\\}^{2}+\\{u_{i}^{n}\\}$
(19)
The gradient of the generalized difference function $\rho$ is not defined at
half integer values. Hence, we modify the method using a greedy strategy:
after detecting that a vertex changes class, the new class that minimizes the
smoothing term is selected, and the fractional part of the state computed by
the gradient descent update is preserved. Consequently, the new state of
vertex $i$ is the result of gradient descent, but if this causes a change in
class, then a new state is determined.
Algorithm 1 Calculate $\boldsymbol{u}$
0: $\epsilon,dt,N_{D},n_{\mathrm{max}},K$
0: $\mathrm{out}=\boldsymbol{u^{\mathrm{end}}}$
for $i=1\to N_{D}$ do
$u_{i}^{~{}0}\leftarrow
rand((0,K))-\frac{1}{2}.\quad\mathrm{If~{}}\mu_{i}>0,~{}u_{i}^{~{}0}\leftarrow\hat{u}_{i}$
end for
for $n=1\to n_{\mathrm{max}}$ do
for $i=1\to N_{D}$ do
$u_{i}^{n+1}\leftarrow
u_{i}^{n}-dt\left(\epsilon\>\hat{R}(u_{i}^{n})+\frac{1}{\epsilon}\>\Phi_{M}^{\prime}(u_{i}^{n})+\mu_{i}\left(u_{i}^{n}-\hat{u}_{i}\right)\right)$
if $\mathrm{Label}(u_{i}^{n+1})\neq\mathrm{Label}(u_{i}^{n})$ then
$\hat{k}=\arg\min_{\;0\leq
k<K}\;\sum_{j}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\,\left[\rho(k+\\{u_{i}^{n+1}\\},u_{j}^{n+1})\,\right]^{2}$
$u_{i}^{n+1}\leftarrow\hat{k}+\\{u_{i}^{n+1}\\}$
end if
end for
end for
Specifically, let $k$ represent an integer in the range of the problem, i.e.
$k\in[0,K-1]$, where $K$ is the number of classes in the problem. Given the
fractional part $\\{u\\}$ resulting from the gradient descent update, find the
integer $k$ that minimizes
$\sum_{j}\frac{w_{ij}}{\sqrt{d_{i}d_{j}}}\,\left[\rho(k+\\{u_{i}\\},u_{j})\,\right]^{2}$,
the smoothing term in the energy functional, and use $k+\\{u_{i}\\}$ as the
new vertex state. A summary of the procedure is shown in Algorithm 1 with
$N_{D}$ representing the number of points in the data set and
$n_{\mathrm{max}}$ denoting the maximum number of iterations.
## 4 Results
The performance of the multiclass diffuse interface model is evaluated using a
number of data sets from the literature, with differing characteristics. Data
and image segmentation problems are considered on synthetic and real data
sets.
### 4.1 Synthetic Data
#### 4.1.1 Three Moons.
A synthetic three-class segmentation problem is constructed following an
analogous procedure to the one used in [5] for “two moon” binary
classification. Three half circles (“three moons”) are generated in
$\mathbb{R}^{2}$. The two top circles have radius $1$ and are centered at
$(0,0)$ and $(3,0)$. The bottom half circle has radius $1.5$ and is centered
at $(1.5,0.4)$. 1,500 data points (500 from each of these half circles) are
sampled and embedded in $\mathbb{R}^{100}$. The embedding is completed by
adding Gaussian noise with $\sigma^{2}=0.02$ to each of the 100 components for
each data point. The dimensionality of the data set, together with the noise,
make this a nontrivial problem.
The symmetric normalized graph Laplacian is computed for a local scaling graph
using $N=10$ nearest neighbors and local scaling based on the $M=10^{th}$
closest point. The fidelity term is constructed by labeling 25 points per
class, 75 points in total, corresponding to only 5% of the points in the data
set. The multiclass GL method was further refined by geometrically decreasing
$\epsilon$ over the course of the minimization process, from $\epsilon_{0}$ to
$\epsilon_{f}$ by factors of $1-\Delta_{\epsilon}$ ($n_{\mathrm{max}}$
iterations per value of $\epsilon$), to allow sharper transitions between
states as in [4]. Table 1 specifies the parameters used. Average accuracies
and computation times are reported over 100 runs. Results for $k$-means and
spectral clustering (obtained by applying $k$-means to the first 3
eigenvectors of $\mathbf{L}\boldsymbol{{}_{s}}$) are included as reference.
Table 1: Three-moons results Method | Parameters | Correct % (stddev %) | Time [s]
---|---|---|---
$k$-means | – | 72.1 (0.35) | 0.66
Spectral clustering | 3 eigenvectors | 80.0 (0.59) | 0.02
Multiclass GL | ${\mu=30}$, ${\epsilon=1}$, ${dt=0.01}$, | 95.1 (2.33) | 0.89
${n_{\mathrm{max}}=1,000}$
Multiclass GL (adaptive $\epsilon$) | ${\mu=30},{\epsilon_{0}=2},{\epsilon_{f}=0.01},$ | 96.2 (1.59) | 1.61
${\Delta_{\epsilon}=0.1},{dt=0.01},$
${n_{\mathrm{max}}=40}$
Segmentations obtained for spectral clustering and for multiclass GL with
adaptive $\epsilon$ methods are shown in Fig. 3. The figure displays the
_best_ result obtained over 100 runs, corresponding to accuracies of $81.3\%$
(spectral clustering) and 97.9% (multiclass GL with adaptive $\epsilon$). The
same graph structure is used for the spectral clustering decomposition and the
multiclass GL method.
Figure 3: Three-moons segmentation. Left: spectral clustering. Right:
multiclass GL with adaptive $\epsilon$.
(a) 100 iterations
(b) 300 iterations
(c) 1,000 iterations
(d) Energy evolution
Figure 4: Evolution of label values in three moons, using multiclass GL (fixed
$\epsilon$): $\mathbb{R}^{2}$ projections at 100, 300 and 1,000 iterations,
and energy evolution.
For comparison, we note the results from the literature for the simpler two-
moon problem (also $\mathbb{R}^{100}$, $\sigma^{2}=0.02$ noise). The best
results reported include: 94% for $p$-Laplacian [5], 95.4% for ratio-
minimization relaxed Cheeger cut [25], and 97.7% for binary GL [4]. While
these are not SSL methods, the last of these does involve other prior
information in the form of a mass balance constraint. It can be seen that our
procedures produce similarly high-quality results even for the more complex
three-class segmentation problem.
It is instructive to observe the evolution of label values in the multiclass
method. Fig. 4 displays $\mathbb{R}^{2}$ projections of the results of
multiclass GL (with fixed $\epsilon$), at 100, 300 and 1,000 iterations. The
system starts from a random configuration. Notice that after 100 iterations,
the structure is still fairly inhomogeneous, but small uniform regions begin
to form. These correspond to islands around fidelity points and become seeds
for further homogenization. The system progresses fast, and by 300 iterations
the configuration is close to the final result: some points are still
incorrectly labeled, mostly on the boundaries, but the classes form nearly
uniform clusters. By 1,000 iterations the procedure converges to a steady
state and a high-quality multiclass segmentation (95% accuracy) is obtained.
In addition, the energy evolution for one typical run is shown in Fig. 4(d)
for the case with fixed $\epsilon$. The figure includes plots of the total
energy (red) as well as the partial contributions of each of the three terms,
namely smoothing (green), potential (blue) and fidelity (purple). Observe that
at the initial iterations, the principal contribution to the energy comes from
the smoothing term, but it has a fast decay due to the homogenization taking
place. At the same time, the potential term increases, as $\rho$ pushes the
label values toward half-integers. Eventually, the minimization process is
driven by the potential term, while small local adjustments are made. The
fidelity term is satisfied quickly and has almost negligible influence after
the first few iterations. This picture of the “typical” energy evolution can
serve as a useful guide in evaluating the performance of the method when no
ground truth is available.
#### 4.1.2 Swiss Roll.
Table 2: Swiss roll results Method | Parameters | Correct % (stddev %) | Time [s]
---|---|---|---
$k$-means | – | 37.9 (0.91) | 0.05
Spectral Clustering | 4 eigenvectors | 49.7 (0.96) | 0.05
Multiclass GL | ${\mu=50}$, ${\epsilon=1}$, ${dt=0.01}$ | 91.0 (2.72) | 0.75
${n_{\mathrm{max}}=1,000}$
A synthetic four-class segmentation problem is constructed using the Swiss
roll mapping, following the procedure in [24]. The data are created in
$\mathbb{R}^{2}$ by randomly sampling from a Gaussian mixture model of four
components with means at $(7.5,7.5)$, $(7.5,12.5)$, $(12.5,7.5)$ and
$(12.5,12.5)$, and all covariances given by the $2\times 2$ identity matrix.
1,600 points are sampled (400 from each of the Gaussians).The data are then
converted from 2 to 3 dimensions, with the following Swiss roll mapping:
$(x,y)\rightarrow(x\cos(x),y,x\sin(x))$.
As before, we construct the weight matrix for a local scaling graph, with
$N=10$ and scaling based on the $M=10^{th}$ closest neighbor. The fidelity set
is formed by labeling $5$% of the points selected randomly.
Table 2 gives a description of the parameters used, as well as average results
over 100 runs for $k$-means, spectral clustering and multiclass GL. The _best_
results achieved over these 100 runs are shown in Fig. 5. These correspond to
accuracies of $50.1\%$ (spectral clustering) and $96.4\%$ (multiclass GL).
Notice that spectral clustering produces results composed of compact classes,
but with a configuration that does not follow the manifold structure. In
contrast, the multiclass GL method is capable of segmenting the manifold
structure correctly, achieving higher accuracies.
(a) Spectral clustering
(b) Multiclass GL
Figure 5: Swiss roll results.
### 4.2 Image Segmentation
We apply our algorithm to the color image of cows shown in Fig. 6(a). This is
a $213\times 320$ color image, to be divided into four classes: sky, grass,
black cow and red cow. To construct the weight matrix, we use feature vectors
defined as the set of intensity values in the neighborhood of a pixel. The
neighborhood is a patch of size $5\times 5$. Red, green and blue channels are
appended, resulting in a feature vector of dimension 75. A local scaling graph
with $N=30$ and $M=30$ is constructed. For the fidelity term, 2.6% of labeled
pixels are used (Fig. 6(b)).
The multiclass GL method used the following parameters: $\mu=30$,
$\epsilon=1$, $dt=0.01$ and $n_{\mathrm{max}}=800$. The average time for
segmentation using different fidelity sets was $19.9$ s. Results are depicted
in Figs. 6(c)-6(f). Each class image shows in white the pixels identified as
belonging to the class, and in black the pixels of the other classes. It can
be seen that all the classes are clearly segmented. The few mistakes made are
in identifying some borders of the black cow as part of the red cow, and vice-
versa.
(a) Original
(b) Sampled
(c) Black cow
(d) Red cow
(e) Grass
(f) Sky
Figure 6: Color (multi-channel) image. Original image, sampled fidelity and
results.
### 4.3 Benchmark Sets
#### 4.3.1 COIL-100.
The Columbia object image library (COIL-100) is a set of 7,200 color images of
100 different objects taken from different angles (in steps of 5 degrees) at a
resolution of $128\times 128$ pixels [22]. This image database has been
preprocessed and made available by [6] as a benchmark for SSL algorithms. In
summary, the red channel of each image is downsampled to $16\times 16$ pixels
by averaging over blocks of $8\times 8$ pixels. Then $24$ of the objects are
randomly selected and partitioned into six arbitrary classes: $38$ images are
discarded from each class, leaving $250$ per class or 1,500 images in all. The
downsampled $16\times 16$ images are further processed to hide the image
structure by rescaling, adding noise and masking 15 of the 256 components. The
result is a data set of 1,500 data points, of dimension 241.
We build a local scaling graph, with $N=4$ nearest neighbors and scaling based
on the $M=4^{th}$ closest neighbor. The fidelity term is constructed by
labeling $10$% of the points, selected at random. The multiclass GL method
used the following parameters: $\mu=100$, $\epsilon=4$, $dt=0.02$ and
$n_{\mathrm{max}}=$ 1,000. An average accuracy of 93.2%, with standard
deviation of 1.27%, is obtained over 100 runs, with an average time for
segmentation of $0.29$s.
For comparison, we note the results reported in [23]: 83.5% ($k$-nearest
neighbors), 87.8% (LapRLS), 89.9% (sGT), 90.9% (SQ-Loss-I) and 91.1% (MP). All
these are SSL methods (with the exception of $k$-nearest neighbors which is
supervised), using 10% fidelity just as we do. As can be seen, our results are
of greater accuracy.
#### 4.3.2 MNIST Data.
The MNIST data set [20] is composed of 70,000 $28\times 28$ images of
handwritten digits $0$ through $9$. The task is to classify each of the images
into the corresponding digit. Hence, this is a $10$-class segmentation
problem.
The weight matrix constructed corresponds to a local scaling graph with $N=8$
nearest neighbors and scaling based on the $M=8^{th}$ closest neighbor. We
perform no preprocessing, so the graph directly uses the $28\times 28$ images.
This yields a data set of 70,000 points of dimension 784. For the fidelity
term, 250 images per class (2,500 images, corresponding to $3.6\%$ of the
data) are chosen randomly. The multiclass GL method used the following
parameters: $\mu=50$, $\epsilon=1$, $dt=0.01$ and $n_{\mathrm{max}}=$ 1,500.
An average accuracy of 96.9%, with standard deviation of 0.04%, is obtained
over 50 runs. The average time for segmentation using different fidelity sets
was $60.89$ s.
Comparative results from other methods reported in the literature include:
87.1% (p-Laplacian [5]), 87.64% (multicut normalized 1-cut [15]), 88.2%
(Cheeger cuts [25]), 92.6% (transductive classification [26]). As with the
three-moon problem, some of these are based on unsupervised methods but
incorporate enough prior information that they can fairly be compared with SSL
methods. Comparative results from supervised methods are: 88% (linear
classifiers [19, 20]), 92.3-98.74% (boosted stumps [20]), 95.0-97.17%
($k$-nearest neighbors [19, 20]), 95.3-99.65% (neural/convolutional nets [19,
20]), 96.4-96.7% (nonlinear classifiers [19, 20]), 98.75-98.82% (deep belief
nets [16]) and 98.6-99.32% (SVM [19]). Note that all of these take 60,000 of
the digits as a training set and 10,000 digits as a testing set [20], in
comparison to our approach where we take only $3.6\%$ of the points for the
fidelity term. Our SSL method is nevertheless competitive with these
supervised methods. Moreover, we perform no preprocessing or initial feature
extraction on the image data, unlike most of the other methods we compare with
(we have excluded from the comparison, however, methods that explicitly deskew
the image). While there is a computational price to be paid in forming the
graph when data points use all 784 pixels as features, this is a simple one-
time operation.
## 5 Conclusions
We have proposed a new multiclass segmentation procedure, based on the diffuse
interface model. The method obtains segmentations of several classes
simultaneously without using one-vs-all or alternative sequences of binary
segmentations required by other multiclass methods. The local scaling method
of Zelnik-Manor and Perona, used to construct the graph, constitutes a useful
representation of the characteristics of the data set and is adequate to deal
with high-dimensional data.
Our modified diffusion method, represented by the non-linear smoothing term
introduced in the Ginzburg-Landau functional, exploits the structure of the
multiclass model and is not affected by the ordering of class labels. It
efficiently propagates class information that is known beforehand, as
evidenced by the small proportion of fidelity points (2% – 10% of dataset)
needed to perform accurate segmentations. Moreover, the method is robust to
initial conditions. As long as the initialization represents all classes
uniformly, different initial random configurations produce very similar
results. The main limitation of the method appears to be that fidelity points
must be representative of class distribution. As long as this holds, such as
in the examples discussed, the long-time behavior of the solution relies less
on choosing the “right” initial conditions than do other learning techniques
on graphs.
State-of-the-art results with small classification errors were obtained for
all classification tasks. Furthermore, the results do not depend on the
particular class label assignments. Future work includes investigating the
diffuse interface parameter $\epsilon$. We conjecture that the proposed
functional converges (in the $\Gamma$-convergence sense) to a total
variational type functional on graphs as $\epsilon$ approaches zero, but the
exact nature of the limiting functional is unknown.
#### Acknowledgements.
This research has been supported by the Air Force Office of Scientific
Research MURI grant FA9550-10-1-0569 and by ONR grant N0001411AF00002.
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|
arxiv-papers
| 2013-06-06T05:32:00 |
2024-09-04T02:49:46.181805
|
{
"license": "Public Domain",
"authors": "Cristina Garcia-Cardona, Arjuna Flenner, Allon G. Percus",
"submitter": "Allon G. Percus",
"url": "https://arxiv.org/abs/1306.1298"
}
|
1306.1509
|
# Characterisation of the muon beams for the Muon Ionisation Cooling
Experiment
The MICE Collaboration D. Adams STFC Rutherford Appleton Laboratory, Harwell
Oxford, Didcot, UK D. Adey Department of Physics, University of Warwick,
Coventry, UK Now at Fermilab, Batavia, IL, USA A. Alekou Department of
Physics, Blackett Laboratory, Imperial College London, London, UK Now at
CERN, Geneva, Switzerland M. Apollonio Department of Physics, Blackett
Laboratory, Imperial College London, London, UK Now at Diamond Light Source,
Harwell Science and Innovation Campus, Didcot, Oxfordshire, UK R.
Asfandiyarov DPNC, Section de Physique, Université de Genève, Geneva,
Switzerland J. Back Department of Physics, University of Warwick, Coventry,
UK G. Barber Department of Physics, Blackett Laboratory, Imperial College
London, London, UK P. Barclay STFC Rutherford Appleton Laboratory, Harwell
Oxford, Didcot, UK A. de Bari Sezione INFN Pavia and Dipartimento di Fisica
Nucleare e Teorica, Pavia, Italy R. Bayes School of Physics and Astronomy,
Kelvin Building, The University of Glasgow, Glasgow, UK V. Bayliss STFC
Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK R. Bertoni
Sezione INFN Milano Bicocca, Dipartimento di Fisica G. Occhialini, Milano,
Italy V. J. Blackmore Department of Physics, University of Oxford, Denys
Wilkinson Building, Oxford, UK email: [email protected] A.
Blondel DPNC, Section de Physique, Université de Genève, Geneva, Switzerland
S. Blot Enrico Fermi Institute, University of Chicago, Chicago, IL, USA M.
Bogomilov Department of Atomic Physics, St. Kliment Ohridski University of
Sofia, Sofia, Bulgaria M. Bonesini Sezione INFN Milano Bicocca, Dipartimento
di Fisica G. Occhialini, Milano, Italy C. N. Booth Department of Physics and
Astronomy, University of Sheffield, Sheffield, UK D. Bowring Lawrence
Berkeley National Laboratory, Berkeley, CA, USA S. Boyd Department of
Physics, University of Warwick, Coventry, UK T. W. Bradshaw STFC Rutherford
Appleton Laboratory, Harwell Oxford, Didcot, UK U. Bravar University of New
Hampshire, Durham, NH, USA A. D. Bross Fermilab, Batavia, IL, USA M.
Capponi Sezione INFN Roma Tre e Dipartimento di Fisica, Roma, Italy T.
Carlisle Department of Physics, University of Oxford, Denys Wilkinson
Building, Oxford, UK G. Cecchet Sezione INFN Pavia and Dipartimento di
Fisica Nucleare e Teorica, Pavia, Italy G. Charnley The Cockcroft Institute,
Daresbury Science and Innovation Centre,Daresbury, Cheshire, UK J. H. Cobb
Department of Physics, University of Oxford, Denys Wilkinson Building, Oxford,
UK D. Colling Department of Physics, Blackett Laboratory, Imperial College
London, London, UK N. Collomb STFC Daresbury Laboratory, Daresbury,
Cheshire, UK L. Coney University of California, Riverside, CA, USA P. Cooke
Department of Physics, University of Liverpool, Liverpool, UK M. Courthold
STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK L. M.
Cremaldi University of Mississippi, Oxford, MS, USA A. DeMello Lawrence
Berkeley National Laboratory, Berkeley, CA, USA A. Dick Department of
Physics, University of Strathclyde, Glasgow, UK A. Dobbs Department of
Physics, Blackett Laboratory, Imperial College London, London, UK P. Dornan
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK S. Fayer Department of Physics, Blackett Laboratory, Imperial College
London, London, UK F. Filthaut NIKHEF, Amsterdam, The Netherlands Also at
Radboud University Nijmegen, Nijmegen, The Netherlands A. Fish Department of
Physics, Blackett Laboratory, Imperial College London, London, UK T.
Fitzpatrick Fermilab, Batavia, IL, USA R. Fletcher University of
California, Riverside, CA, USA D. Forrest School of Physics and Astronomy,
Kelvin Building, The University of Glasgow, Glasgow, UK V. Francis STFC
Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK B. Freemire
Illinois Institute of Technology, Chicago, IL, USA L. Fry STFC Rutherford
Appleton Laboratory, Harwell Oxford, Didcot, UK A. Gallagher STFC Daresbury
Laboratory, Daresbury, Cheshire, UK R. Gamet Department of Physics,
University of Liverpool, Liverpool, UK S. Gourlay Lawrence Berkeley National
Laboratory, Berkeley, CA, USA A. Grant STFC Daresbury Laboratory, Daresbury,
Cheshire, UK J. S. Graulich DPNC, Section de Physique, Université de Genève,
Geneva, Switzerland S. Griffiths The Cockcroft Institute, Daresbury Science
and Innovation Centre,Daresbury, Cheshire, UK P. Hanlet Illinois Institute
of Technology, Chicago, IL, USA O. M. Hansen CERN, Geneva, Switzerland Also
at University of Oslo, Norway G. G. Hanson University of California,
Riverside, CA, USA P. Harrison Department of Physics, University of Warwick,
Coventry, UK T. L. Hart University of Mississippi, Oxford, MS, USA T.
Hartnett STFC Daresbury Laboratory, Daresbury, Cheshire, UK T. Hayler STFC
Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK C. Heidt
University of California, Riverside, CA, USA M. Hills STFC Rutherford
Appleton Laboratory, Harwell Oxford, Didcot, UK P. Hodgson Department of
Physics and Astronomy, University of Sheffield, Sheffield, UK A. Iaciofano
Sezione INFN Roma Tre e Dipartimento di Fisica, Roma, Italy S. Ishimoto High
Energy Accelerator Research Organization (KEK), Institute of Particle and
Nuclear Studies, Tsukuba, Ibaraki, Japan G. Kafka Illinois Institute of
Technology, Chicago, IL, USA D. M. Kaplan Illinois Institute of Technology,
Chicago, IL, USA Y. Karadzhov DPNC, Section de Physique, Université de
Genève, Geneva, Switzerland Y. K. Kim Enrico Fermi Institute, University of
Chicago, Chicago, IL, USA D. Kolev Department of Atomic Physics, St. Kliment
Ohridski University of Sofia, Sofia, Bulgaria Y. Kuno Osaka University,
Graduate School of Science, Department of Physics, Toyonaka, Osaka, Japan P.
Kyberd Brunel University, Uxbridge, UK W. Lau Department of Physics,
University of Oxford, Denys Wilkinson Building, Oxford, UK J. Leaver
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK M. Leonova Fermilab, Batavia, IL, USA D. Li Lawrence Berkeley National
Laboratory, Berkeley, CA, USA A. Lintern STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK M. Littlefield Brunel University,
Uxbridge, UK K. Long Department of Physics, Blackett Laboratory, Imperial
College London, London, UK G. Lucchini Sezione INFN Milano Bicocca,
Dipartimento di Fisica G. Occhialini, Milano, Italy T. Luo University of
Mississippi, Oxford, MS, USA C. Macwaters STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK B. Martlew The Cockcroft Institute,
Daresbury Science and Innovation Centre,Daresbury, Cheshire, UK J. Martyniak
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK A. Moretti Fermilab, Batavia, IL, USA A. Moss The Cockcroft Institute,
Daresbury Science and Innovation Centre,Daresbury, Cheshire, UK A. Muir The
Cockcroft Institute, Daresbury Science and Innovation Centre,Daresbury,
Cheshire, UK I. Mullacrane The Cockcroft Institute, Daresbury Science and
Innovation Centre,Daresbury, Cheshire, UK J. J. Nebrensky Brunel University,
Uxbridge, UK D. Neuffer Fermilab, Batavia, IL, USA A. Nichols STFC
Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK R. Nicholson
Department of Physics and Astronomy, University of Sheffield, Sheffield, UK
J. C. Nugent School of Physics and Astronomy, Kelvin Building, The University
of Glasgow, Glasgow, UK Y. Onel Department of Physics and Astronomy,
University of Iowa, Iowa City, IA, USA D. Orestano Sezione INFN Roma Tre e
Dipartimento di Fisica, Roma, Italy E. Overton Department of Physics and
Astronomy, University of Sheffield, Sheffield, UK P. Owens The Cockcroft
Institute, Daresbury Science and Innovation Centre,Daresbury, Cheshire, UK V.
Palladino Sezione INFN Napoli and Dipartimento di Fisica, Università Federico
II,Complesso Universitario di Monte S. Angelo, Napoli, Italy J. Pasternak
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK F. Pastore Sezione INFN Roma Tre e Dipartimento di Fisica, Roma, Italy
C. Pidcott Department of Physics, University of Warwick, Coventry, UK M.
Popovic Fermilab, Batavia, IL, USA R. Preece STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK S. Prestemon Lawrence Berkeley
National Laboratory, Berkeley, CA, USA D. Rajaram Illinois Institute of
Technology, Chicago, IL, USA S. Ramberger CERN, Geneva, Switzerland M. A.
Rayner Department of Physics, University of Oxford, Denys Wilkinson Building,
Oxford, UK Now at DPNC, Université de Genève, Geneva, Switzerland S.
Ricciardi STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK A.
Richards Department of Physics, Blackett Laboratory, Imperial College London,
London, UK T. J. Roberts Muons, Inc., Batavia, IL, USA M. Robinson
Department of Physics and Astronomy, University of Sheffield, Sheffield, UK
C. Rogers STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK K.
Ronald Department of Physics, University of Strathclyde, Glasgow, UK P.
Rubinov Fermilab, Batavia, IL, USA R. Rucinski Fermilab, Batavia, IL, USA
I. Rusinov Department of Atomic Physics, St. Kliment Ohridski University of
Sofia, Sofia, Bulgaria H. Sakamoto Osaka University, Graduate School of
Science, Department of Physics, Toyonaka, Osaka, Japan D. A. Sanders
University of Mississippi, Oxford, MS, USA E. Santos Department of Physics,
Blackett Laboratory, Imperial College London, London, UK T. Savidge
Department of Physics, Blackett Laboratory, Imperial College London, London,
UK P. J. Smith Department of Physics and Astronomy, University of Sheffield,
Sheffield, UK P. Snopok Illinois Institute of Technology, Chicago, IL, USA
F. J. P. Soler School of Physics and Astronomy, Kelvin Building, The
University of Glasgow, Glasgow, UK T. Stanley STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK D. J. Summers University of
Mississippi, Oxford, MS, USA M. Takahashi Department of Physics, Blackett
Laboratory, Imperial College London, London, UK J. Tarrant STFC Rutherford
Appleton Laboratory, Harwell Oxford, Didcot, UK I. Taylor Department of
Physics, University of Warwick, Coventry, UK L. Tortora Sezione INFN Roma
Tre e Dipartimento di Fisica, Roma, Italy Y. Torun Illinois Institute of
Technology, Chicago, IL, USA R. Tsenov Department of Atomic Physics, St.
Kliment Ohridski University of Sofia, Sofia, Bulgaria C. D. Tunnell
Department of Physics, University of Oxford, Denys Wilkinson Building, Oxford,
UK G.Vankova Department of Atomic Physics, St. Kliment Ohridski University
of Sofia, Sofia, Bulgaria V. Verguilov DPNC, Section de Physique, Université
de Genève, Geneva, Switzerland S. Virostek Lawrence Berkeley National
Laboratory, Berkeley, CA, USA M. Vretenar CERN, Geneva, Switzerland K.
Walaron School of Physics and Astronomy, Kelvin Building, The University of
Glasgow, Glasgow, UK S. Watson STFC Rutherford Appleton Laboratory, Harwell
Oxford, Didcot, UK C. White The Cockcroft Institute, Daresbury Science and
Innovation Centre,Daresbury, Cheshire, UK C. G. Whyte Department of Physics,
University of Strathclyde, Glasgow, UK A. Wilson STFC Rutherford Appleton
Laboratory, Harwell Oxford, Didcot, UK H. Wisting DPNC, Section de Physique,
Université de Genève, Geneva, Switzerland M. Zisman Lawrence Berkeley
National Laboratory, Berkeley, CA, USA
(Received: date / Accepted: date – Entered later)
###### Abstract
A novel single-particle technique to measure emittance has been developed and
used to characterise seventeen different muon beams for the Muon Ionisation
Cooling Experiment (MICE). The muon beams, whose mean momenta vary from 171 to
281 MeV/$c$, have emittances of approximately 1.2–2.3 $\pi$ mm-rad
horizontally and 0.6–1.0 $\pi$ mm-rad vertically, a horizontal dispersion of
90–190 mm and momentum spreads of about 25 MeV/$c$. There is reasonable
agreement between the measured parameters of the beams and the results of
simulations. The beams are found to meet the requirements of MICE.
## 1 Introduction
A future high-energy Neutrino Factory or Muon Collider will require an intense
source of muons. The large volume of phase space occupied by muons at
production must be reduced before they are accelerated and stored. The short
muon lifetime prohibits the use of conventional cooling techniques; another
technique must be developed to maximise the muon flux delivered to a storage
ring.
Ionisation cooling is the only practical approach. A muon passing through a
low-$Z$ material loses energy by ionisation, reducing its transverse and
longitudinal momenta. The longitudinal momentum is restored by accelerating
cavities, with the net effect of reducing the divergence of the beam and thus
the transverse phase space the beam occupies.
The muon beams at the front-end of a Neutrino Factory or Muon Collider will be
similar. They are expected to have a very large transverse normalised
emittance of $\varepsilon_{N}\approx$ 12–20 $\pi$ mm-rad and momentum spreads
of 20 MeV/$c$ or more about a central momentum of 200 MeV/$c$. The transverse
emittance must be reduced to 2- 5 $\pi$ mm-rad (depending on the subsequent
acceleration scheme) for a Neutrino Factory [1, 2, 3, 4]. Further transverse
and longitudinal cooling is required for a Muon Collider. Emittances of
$0.4\,\pi$ mm-rad and $1\,\pi$ mm-rad are desired in the transverse and
longitudinal planes respectively, where the latter is achieved by emittance
exchange [5].
The Muon Ionisation Cooling Experiment (MICE) will be the first experiment to
demonstrate the practicality of muon ionisation cooling. This paper describes
measurements of the muon beams that will be used by MICE.
## 2 The MICE Experiment
Figure 1: MICE upstream beam line.
MICE will measure the ionisation cooling efficiency of one “Super Focus-Focus”
(SFOFO) lattice cell [6] based on the cooling-channel design of Neutrino
Factory Feasibility Study 2 [1]. A detailed description of the cooling cell is
contained in [7]. Since ionisation cooling depends on momentum (via the
dependence of energy loss and multiple scattering in materials), the MICE
experiment has been designed to measure the performance of the cell for beams
of 140 to 240 MeV/$c$ with large momentum spreads; liquid hydrogen and other
low-$Z$ absorber materials will be studied.
The expected reduction in emittance ($\approx$10% using liquid hydrogen) is
too small to be measured conventionally, where methods typically attain 10%
precision. MICE will therefore make single-particle measurements using
scintillating fibre trackers [8] inside superconducting solenoids (the
“spectrometer solenoids”) at each end of the cooling cell. Cherenkov detectors
and time-of-flight (TOF) detectors provide upstream particle identification;
the TOFs will also allow the muons to be timed with respect to the RF phase. A
pre-shower detector and a muon ranger will provide particle identification
downstream of the cooling section.
### 2.1 MICE beam requirements
For a realistic demonstration of cooling the beams used should closely
resemble those expected at the front end of a Neutrino Factory, _i.e._ , they
should have a large momentum spread and a large normalised emittance. The
emittance must be variable to allow the equilibrium emittance—which depends on
the absorber material and the optics of the channel—to be measured.
The MICE beam line has been designed to produce beams of three different
emittances at each of three central momenta. These beams are named by the
convention “$\left(\varepsilon_{N},p_{z}\right)$” according to their
normalised transverse emittance at the entrance to the cooling section and
longitudinal momentum at the centre of the first absorber. The nominal (RMS)
input emittances are $\varepsilon_{N}=3$, 6 and $10\,\pi$ mm-rad; the central
momenta are 140, 200 and 240 MeV/$c$. The baseline beam configuration is
$\left(\varepsilon_{N},p_{z}\right)$ = ($6\,\pi$ mm-rad, 200 MeV/$c$).
The beams of different emittances will be generated by means of a “diffuser”,
which allows a variable thickness of high-$Z$ material to be inserted into the
beam at the entrance to the upstream spectrometer solenoid. Scattering
increases the emittance of the beam to the desired values and, as a
consequence of the energy lost in the material, beams with a higher emittance
downstream of the diffuser must have a higher momentum upstream. An important
requirement is that the muon beam downstream of the diffuser is correctly
matched in the spectrometer solenoid.
### 2.2 MICE muon beam line
The new muon beam line for MICE (at the ISIS proton synchrotron, Rutherford
Appleton Laboratory) is shown in Figure 1 and described at length in [9]. A
titanium target [10] samples the proton beam, creating pions that are captured
by the upstream quadrupoles (Q1–3) and momentum-selected by the first dipole
(D1). The beam is transported to the Decay Solenoid, which focuses the pions
and captures the decay muons.
The second dipole (D2) can be tuned to select muons emitted backwards in the
pion rest frame to obtain a high purity muon beam. The final transport is
through two large-aperture quadrupole triplets, Q4–6 and Q7–9, that focus the
beam onto the diffuser. Each of the three quadrupole triplets is arranged to
focus-defocus-focus in the horizontal plane; the beam line can be operated in
either polarity. The optics of this section are determined by the desired
emittance of the beam in the cooling channel and the requirement of matching
into the spectrometer solenoid.
A time-of-flight station (TOF0) and two aerogel Cherenkov detectors are
located just after the Q4–6 triplet; a second time-of-flight station (TOF1) is
located after the final triplet (Q7–9). A low-mass scintillating-fibre beam-
position monitor (BPM) is located close to Q7. For the $\mu^{+}$ beams, a
variable thickness polyethylene absorber is introduced upstream of D2 to
reduce the flux of protons incident on TOF0.
The TOF detectors are described in [9, 11]. Each station consists of two
perpendicular ($x,y$) planes of 25.4 mm thick scintillator slabs. Each end of
each slab is coupled to a fast photomultiplier and subsequent electronics
[12]. The measured timing resolutions are $\sigma_{t}=55$ ps and
$\sigma_{t}=53$ ps at TOF0 and TOF1 respectively [13]. The differences in the
arrival times of light at each end of the slabs are used to obtain transverse
position measurements with resolutions of $\sigma_{x}=9.8$ mm at TOF0 and
$\sigma_{x}=11.4$ mm at TOF1 [14].
### 2.3 Beam line design
The initial design of the baseline $\left(\varepsilon_{N},p_{z}\right)$ =
($6\,\pi$ mm-rad, 200 MeV/$c$) $\mu^{+}$ beam was made using the TURTLE beam
transport code [15] assuming a 1 cm thick lead diffuser. The design was then
optimised with G4beamline [16], with matching conditions in the upstream
spectrometer solenoid of $\alpha_{x}=\alpha_{y}=0$ and
$\beta_{x}=\beta_{y}=333$ mm [9]. The baseline beam design does not compensate
for horizontal dispersion introduced at D2. The remaining
$\left(\varepsilon_{N},p_{z}\right)$ beam settings were obtained by scaling
the magnet currents of the baseline case according to the the local muon
momentum, accounting for the energy loss of muons in the beam line material,
_i.e._ scaled by a factor $p_{\mathrm{new}}/p_{\mathrm{base}}$. Hence, the
beam line will transport 18 different beams to the cooling channel with
$\varepsilon_{N}=3,6,10\,\pi$ mm-rad and $p_{z}=140,200,240$ MeV/$c$, after
the diffuser, in two beam polarities.
The “re-scaled” beam line settings will transport muons to the cooling channel
with the desired momenta but are not necessarily matched in the first
spectrometer as scattering in the diffuser changes the optical parameters.
Because the diffuser is thin, the beta function will decrease by the same
ratio as the emittance is increased and therefore the final optics and
diffuser thicknesses cannot be determined until the inherent emittances of the
input beams are known.
The beam line was commissioned in MICE “Step I” in 2010–2011. Data were taken
for eight positive and nine negative beam settings to verify the beam line
design and determine the characteristics of the beams, in particular their
momentum distributions, emittances and dispersions. The result of the
commissioning is presented below.
## 3 Characterising the MICE beams
Emittance is the area occupied by a charged particle beam, in two, four, or
six-dimensional trace-space, given by $\varepsilon=\sqrt{\det\Sigma}$ where
$\Sigma$ is the covariance matrix. In two dimensions,
$\Sigma=\left(\begin{array}[]{cc}\sigma_{xx}&\sigma_{xx^{\prime}}\\\
\sigma_{x^{\prime}x}&\sigma_{x^{\prime}x^{\prime}}\end{array}\right)\equiv\left(\begin{array}[]{cc}\varepsilon\beta&-\varepsilon\alpha\\\
-\varepsilon\alpha&\varepsilon\gamma\end{array}\right),$
where, for example, $\sigma_{xx}=\overline{x}\,\overline{x}-\overline{xx}$ and
$\overline{x}$ denotes the mean. The covariance matrix can also be expressed
in terms of the Twiss parameters $\alpha,\beta,\gamma$, and $\varepsilon$
giving a full parameterisation of the beam.
Several different methods exist for measuring the emittance of beams [17].
Commonly, beam profile monitors are used to measure the RMS beam size,
$\sigma_{x}$, at several positions. At least three profile monitors are
required to determine the three elements of the covariance matrix and hence
the emittance of the beam; the transfer matrices between the profile monitors
must be known. These methods do not require individual particles to be tracked
but are ultimately limited by the spatial resolution of the detectors and the
intensity of the beam.
By contrast, the MICE muon beam is large in spatial extent and its intensity
is low compared to conventional primary beams. The emittance and optical
parameters of such a beam can be measured if either the trace space co-
ordinates, $(x,x^{\prime}),(y,y^{\prime})$ of individual particles can be
measured at a single plane or, as in the new method described here, the
spatial co-ordinates of individual particles are measured at two detectors and
the transfer matrix between the detectors is known.
In the later Steps of MICE the beam emittance will be measured by a
scintillating fibre tracker inside a 4T solenoid. This detector was not
present during Step I and the new method was developed to characterise the
beam using only the two TOF detectors. The relative times and $(x,y)$
positions of single particles are measured in the two TOF stations and muons
are selected by broad time-of-flight cuts. Each muon is tracked through the
Q7–9 quadrupole triplet, determining the trace space angles $x^{\prime}$ and
$y^{\prime}$ at each plane. Simultaneously, the muon momenta is measured by
time-of-flight, which is important as the beam has a large momentum spread and
the transfer matrix between the two detectors depends strongly on momentum.
The covariance matrix of the beam is then obtained from a large sample of
muons so measured. The method is described briefly below; its detailed
development is given in [14].
Figure 2: The time-of-flight system and beam line section used to characterise
the beam.
Figure 2 shows the section of beam line used for the measurements. The pole
tip radius of the quadrupoles is 176 mm. The two TOF detectors have active
areas of 400 mm $\times$ 400 mm and 420 mm $\times$ 420 mm, respectively, and
were separated by 7.705 m during the 2010 commissioning; their combined time
resolution of 76 ps allows the momenta of muons to be determined with a
resolution of $\sigma_{p}=3.7$ MeV/$c$ for $p_{z}=230$ MeV/$c$.
### 3.1 Measurement technique
The measurement algorithm proceeds iteratively. An initial estimate of $p_{z}$
is made by assuming that a muon travels along the $z$-axis between the two TOF
counters. This estimate is used to determine the $x$ and $y$ transfer
matrices, $M_{x}(p_{z})$ and $M_{y}(p_{z})$, between TOF0 and TOF1. Once the
transfer matrices are known, the trace-space vectors $(x_{0},x_{0}^{\prime})$
and $(y_{0},y_{0}^{\prime})$, and $(x_{1},x_{1}^{\prime})$ and
$(y_{1},y_{1}^{\prime})$, at TOF0 and TOF1 respectively, are obtained from the
position measurements $(x_{0},y_{0})$ and $(x_{1},y_{1})$ by a rearrangement
of the transport equations:
$\left(\begin{array}[]{c}x_{1}\\\
x_{1}^{\prime}\end{array}\right)=M_{x}\left(\begin{array}[]{c}x_{0}\\\
x_{0}^{\prime}\end{array}\right),$ $\left(\begin{array}[]{c}y_{1}\\\
y_{1}^{\prime}\end{array}\right)=M_{y}\left(\begin{array}[]{c}y_{0}\\\
y_{0}^{\prime}\end{array}\right)\,.$ (1)
Explicitly
$\left(\begin{array}[]{c}x_{0}^{\prime}\\\
x_{1}^{\prime}\end{array}\right)=\frac{1}{M_{12}}\left(\begin{array}[]{cc}-M_{11}&1\\\
-1&M_{22}\end{array}\right)\left(\begin{array}[]{c}x_{0}\\\
x_{1}\end{array}\right),$ (2)
where $M_{ij}$ are the (momentum dependent) elements of $M_{x}$, and similarly
for $(y_{0}^{\prime},y_{1}^{\prime})$. The estimates of
$(x_{0},x_{0}^{\prime})$, $(y_{0},y_{0}^{\prime})$, and $p_{z}$ are used to
track the muon between TOF0 and TOF1 and obtain an improved estimate of the
trajectory and a correction, $\Delta s$, to the path length. To ensure
convergence to a stable solution, only half the predicted $\Delta s$ was
applied before recalculating the momentum from the time-of-flight; the
procedure was repeated ten times for each muon although a convergent solution
was found after typically five iterations. Finally, a small correction
($\approx$ 1.5 MeV/$c$) is applied to account for energy loss in the material,
including air, between the TOF counters.
Figure 3: Quadrupole gradients for the ($6\,\pi$ mm-rad, 200 MeV/$c$)
baseline muon beam (colour online).
In order to obtain the transfer matrices, the focusing gradients of
quadrupoles Q7–9 were determined by fitting the results of an OPERA [18] field
model of the quadrupole with two hyperbolic tangent functions [14]. Figure 3
shows the focusing gradients of the Q7–9 triplet. The quadrupoles are thick
and their fields overlap substantially. A more computationally efficient, and
sufficiently accurate, “top-hat” model of the magnets was used to obtain
$\Delta s$ [14].
Equation 2 for $x_{1}^{\prime}$, which is used to determine the horizontal
emittance at TOF1, can be expressed as
$x_{1}^{\prime}=A(p_{z})x_{0}+B(p_{z})x_{1}$
and mutatis mutandis for $y^{\prime}$. The coefficients $A(p_{z})$ and
$B(p_{z})$ for the baseline (6, 200) beam, with mean $p_{z}\approx 230$
MeV/$c$, are shown in Figure 4. Both $A$ and $B$ are strongly momentum
dependent below 200 MeV/$c$.
Figure 4: The reconstruction coefficients $A(p_{z})$ (top) and $B(p_{z})$
(bottom) for the (6, 200) baseline muon beam. The solid (blue) lines are for
$x^{\prime}$ (horizontal); the dashed (red) lines are for $y^{\prime}$
(vertical).
The procedure described above enabled the reconstruction of the trace space
vectors at both TOF counters as well as the momenta of single muons. The path
length correction, which could be as much as 15–20 mm, was necessary to avoid
a systematic underestimate of $p_{z}$ of about 4 MeV/$c$.
The momentum distributions and the $(x_{1},x_{1}^{\prime})$ and
$(y_{1},y_{1}^{\prime})$ covariance matrices, $\Sigma_{x,y}$, at the upstream
side of TOF1 for each measured beam were obtained from all the muons recorded
for that beam. The effective optical parameters and the emittances of each
beam were deduced from $\Sigma_{x,y}$ as described in Section 4.3. The
systematic uncertainty on the measurements is discussed in Section 4.4
### 3.2 Monte Carlo simulations of the MICE beam line
Monte Carlo simulations were made for six of the 18 possible beam settings to
check the beam line design software. The ($6\,\pi$ mm-rad, $p_{z}$ = 140, 200,
240 MeV/$c$) $\mu^{+}$ and $\mu^{-}$ beams were simulated in two steps.
G4beamline was used to track particles from the target as far as TOF0; the
G4MICE Monte Carlo [19] was then used to track muons between TOF0 and TOF1.
Both simulations contained descriptions of the material in, and surrounding,
the beam line and magnet models, including the apertures of the quadrupoles
Q4–9, using the optics designed for the corresponding beams. The simulations
suggest that the final emittance of the beams before the diffuser is $\approx
1\,\pi$ mm-rad, partly due to scattering in the material in the beam line but
limited by the aperture of the quadrupoles. Dispersion in the horizontal plane
due to D2 is expected.
### 3.3 Performance of the reconstruction algorithm
Figure 5: Difference between reconstructed and true momenta for a simulated
200 MeV/$c$ muon beam. Figure 6: Reconstructed trace space angles versus true
simulated values.
The performance of the reconstruction algorithm was determined by smearing the
true simulated coordinates of the muons at each TOF plane with the measured
time and position resolutions of the TOFs. The trace-space vectors were
reconstructed by the method described in Section 3.1 to obtain a set of
simulated reconstructed muons.
Figure 5 shows, for a simulated $(6,200)\,\mu^{-}$ beam, the difference
between reconstructed and true momenta. The RMS width of the distribution of
3.7 MeV/$c$ confirms that the momentum resolution is dominated by the timing
resolution of the TOF system. It is sufficiently small to measure the large
expected widths of approximately 20 MeV/$c$ of the momentum distributions.
Figure 6 shows the agreement between the true and reconstructed angles,
$x_{1}^{\prime}$ and $y_{1}^{\prime}$ for the simulated $(6,200)\,\mu^{-}$
beam. The average angular resolutions, $\sigma_{x_{1}^{\prime}}$ and
$\sigma_{y_{1}^{\prime}}$, for this beam are approximately 29 and 8 mrad
respectively. They are determined by the position resolution of the TOF
counters and multiple scattering, and depend on momentum as $x_{1}^{\prime}$
and $y_{1}^{\prime}$ are obtained from position measurements using the
momentum-dependent elements of the transfer matrix. The angular resolutions
are small but not negligible compared with the expected widths of the
$x_{1}^{\prime}$ and $y_{1}^{\prime}$ distributions.
## 4 Results of the measurements and comparison with simulations
Data were taken for eight positive and nine negative re-scaled beams that,
when used in conjunction with the diffuser, will generate the full range of
desired emittances (see Section 2.3); the polarity of the decay solenoid was
kept the same for both positive and negative beams. Muons in the data were
selected by broad time-of-flight cuts chosen for each nominal beam momentum.
The simulations used in this analysis suggest that the pion contamination at
TOF0 of the $\mu^{-}$ data is about one percent and less than five percent for
the $\mu^{+}$ sample [14]. Recent measurements [20] indicate a somewhat
smaller pion contamination.
Figure 7: Reconstructed longitudinal momentum distributions at TOF1 for six
MICE beams compared with simulations. The dotted (red), dash-dotted (blue) and
shaded distributions are simulation, reconstructed simulation and data
respectively. Distributions are normalised to contain equal numbers of events.
### 4.1 Longitudinal momentum
Table 1: Mean and RMS widths of the longitudinal momentum distributions for six beams compared with the corresponding simulations. | | Data | Simulation
---|---|---|---
Beam | Mean $p_{z}$ | RMS $p_{z}$ | Mean $p_{z}$ | RMS $p_{z}$
| | MeV/$c$ | MeV/$c$ | MeV/$c$ | MeV/$c$
| $\left(6,140\right)$ | 176.4$\pm$2.3 | 22.8$\pm$0.3 | 173.7$\pm$2.1 | 19.5$\pm$0.2
$\mu^{-}$ | $\left(6,200\right)$ | 232.2$\pm$2.5 | 23.6$\pm$0.3 | 229.3$\pm$0.8 | 21.0$\pm$0.1
| $\left(6,240\right)$ | 271.0$\pm$3.7 | 24.5$\pm$0.3 | 270.5$\pm$0.9 | 22.2$\pm$0.1
| $\left(6,140\right)$ | 176.5$\pm$2.0 | 24.4$\pm$0.3 | 176.6$\pm$3.7 | 25.5$\pm$0.5
$\mu^{+}$ | $\left(6,200\right)$ | 229.2$\pm$2.4 | 25.9$\pm$0.3 | 230.8$\pm$1.4 | 28.9$\pm$0.2
| $\left(6,240\right)$ | 267.7$\pm$2.9 | 25.8$\pm$0.3 | 269.2$\pm$4.2 | 31.3$\pm$0.5
Figure 7 shows the distributions of $p_{z}$ at TOF1 for six beams compared
with the results of the simulations. Overall the measured and simulated
distributions agree well in shape and width. The $\mu^{+}$ beams have a
slightly greater momentum spread than the $\mu^{-}$ beams, due to energy loss
fluctuations in the proton absorber. The agreement between the measured and
simulated momentum distributions is better for the $\mu^{-}$ beams than it is
for the $\mu^{+}$ beams. Since the mean momentum is dictated by D2, the
agreement between the measured and simulated mean momenta at TOF1 confirms the
beam line design. The mean momenta and the RMS widths of the measured beams
are given in Table 1. The systematic error on $p_{z}$ is mainly due to the
$\pm 35$ ps calibration uncertainty on the absolute time of flight value [14]
and is estimated to be less than 3 MeV/$c$ for all momenta below 300 MeV/$c$.
### 4.2 Transverse spatial distributions
Figure 8: Spatial distributions in the transverse plane at TOF1 for
simulation (left), reconstructed simulation (centre), and data (right) for a
(6, 200) $\mu^{-}$ beam, normalised to the same total contents. Simulated
muons in the shaded area cross uncalibrated regions of TOF1 and are excluded
from further analysis.
---
Figure 9: Root mean square beam widths, $\sigma_{x}$, $\sigma_{y}$, at TOF1
versus $p_{z}$. Solid black circles: $\mu^{-}$ data, open black circles:
$\mu^{+}$ data, solid red triangles: $\mu^{-}$ simulation, open red triangles:
$\mu^{+}$ simulation. The nominal “$p_{z}=140$” MeV/$c$ beams correspond to
momenta in the range 170–190 MeV/$c$, “$p_{z}=200$” to 220–250 MeV/$c$, and
“$p_{z}=240$” to 250–290 MeV/$c$.
Figure 8 shows a comparison of the spatial distributions in the transverse
plane at TOF1 for a simulated (6, 200) $\mu^{-}$ beam before and after
reconstruction, and data for the same beam. The effect of smearing by the
reconstruction procedure is small. Muons crossing the shaded area are excluded
from the simulation (and hence the reconstruction) as they pass through
uncalibrated regions of TOF1. Since muons must cross both a horizontal and
vertical slab of the TOF to be counted, these regions are excluded from the
data. Figure 9 shows the RMS horizontal and vertical beam sizes, $\sigma_{x}$
and $\sigma_{y}$, versus mean $p_{z}$ for all the measured beams and the six
simulated beams. The sizes of the positive and negative muon beams are very
similar both vertically and horizontally. The measured vertical beam size is
about 10–20% smaller than suggested by the simulations. The horizontal beam
size is about 10% smaller than the $\mu^{-}$ simulations but wider than the
$\mu^{+}$ simulations.
Figure 10: Horizontal ($x$) and vertical ($y$) trace space distributions at
TOF1 for simulation (top), reconstructed simulation (centre) and data (bottom)
for a (6, 200) $\mu^{-}$ beam. The ellipses correspond to $\chi^{2}=6$ (see
text). Figure 11: Distributions of $\chi^{2}$ for data (solid, shaded,
black) and reconstructed simulation (dash-dot, blue) for the (6, 200)
$\mu^{-}$ beam. Left: horizontal $\chi^{2}$; right: vertical $\chi^{2}$ at
TOF1.
Figure 10 shows the horizontal ($x,x^{\prime}$) and vertical ($y,y^{\prime}$)
trace-space distributions at TOF1 for the (6, 200) $\mu^{-}$ beam and the same
distributions from the simulations with and without smearing due to the
reconstruction. There is very good qualitative agreement between data and
reconstructed simulations in both the horizontal and vertical trace spaces.
The smearing due to the reconstruction is apparent. The distributions have a
dense core and diffuse halo. The boundaries of the distributions reflect the
apertures of the quadrupoles, principally Q9, transported to the TOF1
measurement plane downstream, and the size of TOF1. The vertical divergence of
the beam is approximately three times smaller than the horizontal divergence.
Figure 11 shows the $x$ and $y$ amplitude distributions of muons in the (6,
200) $\mu^{-}$ beam at TOF1 in terms of $\chi^{2}_{x,y}$ where
$\chi^{2}_{x}=[(x-\bar{x}),(x^{\prime}-\bar{x}^{\prime})]\Sigma_{x}^{-1}[(x-\bar{x}),(x^{\prime}-\bar{x}^{\prime})]^{T}={A_{x}}/{{\varepsilon_{x}}}\,,$
$A_{x}$ is the amplitude of a muon in trace-spaceaaaThis is sometimes referred
to as ‘single particle emittance’ [21]. and
${\varepsilon_{x}}=\sqrt{\det\Sigma_{x}}$ is the emittance of the ensemble.
The distributions of $\chi^{2}$ for the reconstructed simulation are shown for
comparison. The initial exponential behaviour of the distribution suggests
that the beam has a quasi-Gaussian core up to $\chi^{2}\approx 6$ and a non-
Gaussian tail. The high amplitude tails of the distributions are slightly
larger for the data than for the simulation.
### 4.3 Determination of emittances and effective optical parameters
The optical parameters and emittances of each beam were determined from the
covariance matrices [22] as
$\displaystyle\varepsilon_{x}$ $\displaystyle=$
$\displaystyle\sqrt{\det\Sigma_{x}},$ $\displaystyle\beta_{x}$
$\displaystyle=$ $\displaystyle\frac{\Sigma_{11}}{\varepsilon_{x}},$
$\displaystyle\alpha_{x}$ $\displaystyle=$
$\displaystyle-\frac{\Sigma_{12}}{\varepsilon_{x}},$
and similarly for $y$. Each of the beams, however, has a large momentum spread
and $\alpha$ and $\beta$ vary with momentum. The parameters determined from
the measurements are therefore effective parameters which describe the
distributions in trace-space.
The reconstructed covariance matrices at TOF1 will differ from the true
covariance matrices because of the finite spatial and angular resolution of
the reconstruction, and multiple scattering in the air between the TOFs (which
cannot be included in the simple transfer matrices used). The finite
resolution leads to a small increase in the apparent emittance of the beams;
scattering will lead to an underestimate of the emittances.
A small correction was made for the effects of resolution and scattering by
subtracting a “resolution” matrix from each measured covariance matrix. The
resolution matrices were estimated from the simulations by taking the
difference between the covariance matrices of the reconstructed and true
simulated beams. These resolution matrices were subtracted from the measured
covariance matrices to obtain corrected, measured covariance matrices, _i.e._
,
$\displaystyle\Sigma_{\rm Corrected}$ $\displaystyle=$
$\displaystyle\Sigma_{\rm Measured}-\Sigma_{\rm Resolution}$ $\displaystyle=$
$\displaystyle\Sigma_{\rm Measured}-(\Sigma_{\rm Reco-sim}-\Sigma_{\rm True-
sim})\,.$
Since simulations were made for only the six (6 $\pi$ mm-rad, $p_{z}$) beams,
the resolution matrices estimated for these beams have been used to correct
the measured covariance matrices for other beams at the same nominal momentum.
As variances are very sensitive to outliers, muons in the high amplitude tails
of the $(x,x^{\prime})$ and $(y,y^{\prime})$ distributions were excluded by
requiring $\chi^{2}_{x,y}<6$ before the corrected covariance matrices were
calculated. The ellipses on Figure 10 show the areas of the distributions
included by this cut.
Figure 12: RMS emittance ellipses in ($x,x^{\prime}$) trace-space for data
without correction for the measurement resolution (black dotted line),
corrected data (black solid line) and true simulation (red dashed line).
Figure 12 shows, for the six beams for which simulations were made, the
horizontal ($x,x^{\prime}$) RMS emittance ellipses for the uncorrected data,
the data after correction for resolution and the true simulation. The effect
of the resolution correction is to reduce the apparent emittance of the beams
and to rotate the ellipses into better alignment with the true simulation.
Figure 13: Horizontal emittance after correction for measurement resolution
and multiple scattering versus mean $p_{z}$ of the seventeen measured beams.
Solid black circles: $\mu^{-}$ data, open black circles: $\mu^{+}$ data, solid
red triangles: $\mu^{-}$ simulation, open red triangles: $\mu^{+}$ simulation.
The nominal “$p_{z}=140$” MeV/$c$ beams correspond to momenta in the range
170–190 MeV/$c$, “$p_{z}=200$” to 220–250 MeV/$c$, and “$p_{z}=240$” to
250–290 MeV/$c$. Figure 14: Vertical emittance after correction for
measurement resolution and multiple scattering versus mean $p_{z}$ of the
seventeen measured beams. Solid black circles: $\mu^{-}$ data, open black
circles: $\mu^{+}$ data, solid red triangles: $\mu^{-}$ simulation, open red
triangles: $\mu^{+}$ simulation. The nominal “$p_{z}=140$” MeV/$c$ beams
correspond to momenta in the range 170–190 MeV/$c$, “$p_{z}=200$” to 220–250
MeV/$c$, and “$p_{z}=240$” to 250–290 MeV/$c$.
The measured emittances discussed below have not been corrected upward for the
$\chi^{2}_{x,y}<6$ requirement, which has also been applied to the simulated
data, because the long non-Gaussian tail of the amplitude distribution (see
Figure 11) is not well-described by the simulations. For a pure Gaussian
distribution 5% of the muons would have $\chi^{2}>6$ and the correction would
increase the measured values of emittance by approximately 20%.
Figure 13 shows the measured horizontal emittances, after resolution
correction, of all the seventeen beams versus the mean $p_{z}$ of the beam and
the true emittances of the six simulated beams. The correction reduces the
measured emittances by $0.6\,\pi$ mm-rad on average; the largest correction is
$-0.7\,\pi$ mm-rad for the (10,140) $\mu^{+}$ beam. Figure 14 shows the
measured vertical emittances of all the seventeen beams versus the mean
$p_{z}$ of the beam and the emittances of the six simulated beams. The
correction increases the measured vertical emittances by about 10%. Clipping
occurs in the vertical plane as Q4 and Q7 are vertically defocusing. This
collimates the beam, resulting in more uniform emittances compared to the
horizontal plane.
### 4.4 Systematic uncertainties
The error bars shown on Figures 13 and 14 include both statistical and
systematic errors. Sources of systematic error fall into three broad
categories; those that affect the transverse position measurement, momentum
reconstruction, and path length corrections. The largest contribution to the
uncertainty on the emittance measurement derives from the effective speed of
light in the TOF slabs, which directly determines the measured RMS width of
the spatial and angular distributions. The various sources, summarised in
Table 2, were determined by examining the change in the reconstructed
emittance and optical parameters when the positions of the TOF detectors and
magnet currents were varied in simulation.
The TOF offsets arise from the uncertainty on their surveyed positions in the
beam line. In each instance, a simulation was produced with one TOF offset by
up to 1 cm in $x,y$ or $z$ and the muon positions recorded. These positions
were input into the reconstruction procedure, which assumes the beam line
elements are located as given by survey. The largest uncertainty occurs when
the TOFs are offset in the longitudinal ($z$) direction, which directly
affects the momentum measurement by altering the distance $\Delta L$.
The uncertainty on the quadrupole triplet position in survey was investigated
in the same manner as for the TOFs. However, since this does not affect the
distance, $\Delta L$, it has a negligible effect on the momentum calculation
and a plays a minor role in the path length correction assigned to a muon. The
currents in the quadrupoles are known to better than 1%, and the effect of
changing these currents was determined. A change in the quadrupole current by
1% has a small effect on the reconstructed path length of a muon, when
compared to the nominal currents, and is a minor source of uncertainty on the
emittance measurement. The uncertainty on $p_{z}$ has a much larger effect on
the transfer matrix used than any scaling due to an uncertainty on the
quadrupole currents (_cf._ Figure 4).
### 4.5 Results
Table 2: Contributions to the errors on the emittance measurements as percentage relative error. Source | $\delta\epsilon_{x}$ | $\delta\alpha_{x}$ | $\delta\beta_{x}$ | $\delta\eta_{x}$ | $\delta\eta_{x}^{\prime}$ | $\delta\epsilon_{y}$ | $\delta\alpha_{y}$ | $\delta\beta_{y}$ | $\delta p_{z}$
---|---|---|---|---|---|---|---|---|---
TOF1 offsets | $x$ | 0.47 | 0.74 | 0.47 | 1.39 | 0.69 | 0.014 | 0.05 | $\approx 0$ | $\approx 0$
$y$ | $\approx 0$ | 0.01 | $\approx 0$ | 0.29 | 0.17 | 0.02 | 0.06 | $\approx 0$ | 0.71
TOF0 offsets | $x$ | 0.04 | 0.07 | 0.03 | 0.13 | 0.22 | $\approx 0$ | 0.08 | 0.01 | $\approx 0$
$y$ | $\approx 0$ | $\approx 0$ | $\approx 0$ | 0.02 | $\approx 0$ | $\approx 0$ | $\approx 0$ | $\approx 0$ | $\approx 0$
$\Delta L$ | 2.10 | 0.32 | 2.11 | 1.69 | 3.30 | 2.74 | 30.17 | 2.78 | 0.71
Q789 currents | 0.051 | $\approx 0$ | 0.03 | 0.008 | 0.02 | 0.04 | 0.036 | 0.035 | 0.002
Q789 offsets | $x$ | 0.08 | 0.13 | 0.08 | 0.17 | 0.99 | $\approx 0$ | 0.08 | 0.01 | $\approx 0$
$y$ | $\approx 0$ | $\approx 0$ | $\approx 0$ | 0.01 | $\approx 0$ | $\approx 0$ | 0.01 | $\approx 0$ | $\approx 0$
Effective $c$ in scintillator | 4.87 | 0.05 | 5.23 | 2.22 | 1.59 | 4.05 | 41.27 | 4.09 | 0.11
Total (%) | 5.32 | 0.82 | 5.66 | 3.14 | 3.87 | 4.89 | 51.12 | 4.94 | 1.02
Table 3: The characterised Step I beams. | Beam | $\langle p_{z}\rangle$ (MeV/$c$) | $\sigma_{pz}$ (MeV/$c$) | $\varepsilon_{x}$ ($\pi$ mm-rad) | $\alpha_{x}$ | $\beta_{x}(m)$ | $\varepsilon_{y}$ ($\pi$ mm-rad) | $\alpha_{y}$ | $\beta_{y}$ (m)
---|---|---|---|---|---|---|---|---|---
| $\varepsilon_{N}$ | $p_{z}$ | | | | | | | |
$\mu^{-}$ | 3 | 140 | 171.58$\pm$2.39 | 22.81$\pm$ 0.32 | 2.28$\pm$0.12 | 0.50$\pm$0.01 | 1.49$\pm$0.09 | 0.95$\pm$0.05 | -0.55$\pm$0.28 | 3.62$\pm$0.18
200 | 223.24$\pm$2.72 | 24.02$\pm$ 0.29 | 1.74$\pm$0.09 | 0.49$\pm$0.01 | 1.69$\pm$0.10 | 0.78$\pm$0.04 | -0.50$\pm$0.25 | 3.71$\pm$0.19
240 | 260.55$\pm$3.24 | 24.49$\pm$ 0.30 | 1.49$\pm$0.08 | 0.49$\pm$0.01 | 1.80$\pm$0.10 | 0.75$\pm$0.04 | -0.41$\pm$0.21 | 3.65$\pm$0.18
6 | 140 | 176.43$\pm$2.27 | 22.83$\pm$ 0.29 | 2.17$\pm$0.12 | 0.52$\pm$0.01 | 1.57$\pm$0.09 | 0.96$\pm$0.05 | -0.54$\pm$0.28 | 3.64$\pm$0.18
200 | 232.22$\pm$2.51 | 23.62$\pm$ 0.26 | 1.53$\pm$0.08 | 0.55$\pm$0.01 | 1.85$\pm$0.10 | 0.78$\pm$0.04 | -0.51$\pm$0.26 | 3.80$\pm$0.19
240 | 270.96$\pm$3.65 | 24.53$\pm$ 0.33 | 1.51$\pm$0.08 | 0.48$\pm$0.01 | 1.80$\pm$0.10 | 0.73$\pm$0.04 | -0.39$\pm$0.20 | 3.51$\pm$0.18
10 | 140 | 183.46$\pm$2.35 | 22.75$\pm$ 0.29 | 2.01$\pm$0.11 | 0.53$\pm$0.01 | 1.62$\pm$0.09 | 0.92$\pm$0.05 | -0.56$\pm$-0.29 | 3.68$\pm$0.18
200 | 247.23$\pm$3.56 | 24.20$\pm$ 0.35 | 1.23$\pm$0.07 | 0.59$\pm$0.01 | 2.22$\pm$0.13 | 0.75$\pm$0.04 | -0.52$\pm$-0.27 | 3.81$\pm$0.19
240 | 281.89$\pm$3.65 | 25.28$\pm$ 0.33 | 1.65$\pm$0.09 | 0.56$\pm$0.01 | 1.82$\pm$0.10 | 0.64$\pm$0.03 | -0.39$\pm$0.20 | 3.40$\pm$0.17
$\mu^{+}$ | 3 | 200 | 222.69$\pm$2.40 | 26.49$\pm$ 0.29 | 1.98$\pm$0.11 | 0.49$\pm$0.01 | 1.58$\pm$0.09 | 0.83$\pm$0.04 | -0.40$\pm$0.20 | 3.44$\pm$0.17
240 | 257.97$\pm$2.83 | 26.37$\pm$ 0.29 | 1.59$\pm$0.08 | 0.57$\pm$0.01 | 1.87$\pm$0.11 | 0.76$\pm$0.04 | -0.31$\pm$0.16 | 3.40$\pm$0.17
6 | 140 | 176.45$\pm$1.98 | 24.36$\pm$ 0.27 | 2.32$\pm$0.12 | 0.45$\pm$0.01 | 1.50$\pm$0.09 | 0.95$\pm$0.05 | -0.48$\pm$0.25 | 3.59$\pm$0.18
200 | 229.16$\pm$2.36 | 25.87$\pm$ 0.27 | 1.91$\pm$0.10 | 0.50$\pm$0.01 | 1.61$\pm$0.09 | 0.81$\pm$0.04 | -0.38$\pm$0.19 | 3.42$\pm$0.17
240 | 267.65$\pm$2.85 | 25.79$\pm$ 0.28 | 1.69$\pm$0.09 | 0.54$\pm$0.01 | 1.76$\pm$0.10 | 0.76$\pm$0.04 | 0.26$\pm$0.14 | 3.23$\pm$0.16
10 | 140 | 182.42$\pm$2.05 | 23.87$\pm$ 0.27 | 2.16$\pm$0.12 | 0.47$\pm$0.01 | 1.56$\pm$0.09 | 0.92$\pm$0.05 | -0.48$\pm$0.24 | 3.59$\pm$0.18
200 | 243.39$\pm$2.65 | 26.77$\pm$ 0.29 | 1.66$\pm$0.09 | 0.51$\pm$0.01 | 1.78$\pm$0.10 | 0.78$\pm$0.04 | -0.38$\pm$0.19 | 3.37$\pm$0.17
240 | 274.77$\pm$2.94 | 24.79$\pm$ 0.27 | 1.78$\pm$0.09 | 0.51$\pm$0.01 | 1.65$\pm$0.09 | 0.76$\pm$0.04 | -0.22$\pm$0.11 | 3.07$\pm$0.15
The measured emittances and optical parameters are given in Table 3. The
horizontal and vertical beta functions lie in the ranges 1.49
m$<\beta_{x}<2.22$ m and 3.07 m$<\beta_{y}<3.81$ m. The values of the
horizontal and vertical $\alpha$ parameters, $0.45<\alpha_{x}<0.59$ and
$-0.56<\alpha_{y}<-0.22$, show that the beams converge to a horizontal focus
roughly 700 mm downstream of TOF1 but diverge vertically. The emittances will
be increased by scattering in TOF1.
The measured horizontal emittances and simulations agree to within 10%. Some
of the emittance of the beams can be attributed to multiple scattering in
TOF0. The emittance growth in $x$ ($y$) is expected to be
$\Delta\varepsilon_{x,y}^{2}=\sigma_{x,y}^{2}\theta_{\rm ms}^{2}$ where
$\theta_{\rm ms}^{2}=(13.6{\rm\,MeV/}c)^{2}/(p^{2}\beta^{2})\Delta X/X_{0}$
is the mean square scattering angle in the $\Delta X=0.125X_{0}$ of material
in TOF0. For 200 MeV/$c$ muons and $\sigma_{x}=70$ mm,
$\Delta\varepsilon=1.9\,\pi$ mm-rad for a beam of zero divergence, although
the effective emittance at TOF1 is limited by the aperture of the Q7–9
triplet. The fall in measured emittance with increasing $p_{z}$ seen in
Figures 13 and 14 can be attributed to scattering via the dependence of
$\theta_{\rm ms}$ on $p_{z}$.
There is some emittance growth in the $\approx 8$ m of air between TOF0 and
TOF1. Since the Q7–9 triplet focusses horizontally but is weakly defocusing
vertically, this emittance growth is less in the horizontal than the vertical
plane. For an on-axis beam, $\delta\varepsilon_{y}$ is estimated to be less
than $0.4\,\pi$ mm-rad. The resolution correction described previously
includes a small upwards correction for this emittance growth, and has the
largest effect on the measured vertical emittance. The remaining disagreement
between the measured and simulated vertical emittances can be attributed to
the difference in RMS vertical beam sizebbbThe RMS beam size in Figure 9 is
calculated and shown without the $\chi^{2}<6$ cut to demonstrate the physical
size of the beam, whereas the emittance calculation includes it. shown in
Figure 9.
Figure 15: The horizontal dispersion coefficient, $\eta$, versus mean $p_{z}$
for the seventeen beams. Solid black circles: $\mu^{-}$ data, open black
circles: $\mu^{+}$ data, Solid red triangles: $\mu^{-}$ simulation smeared
with measurement resolution. The nominal “$p_{z}=140$” MeV/$c$ beams
correspond to momenta in the range 170–190 MeV/$c$, “$p_{z}=200$” to 220–250
MeV/$c$, and “$p_{z}=240$” to 250–290 MeV/$c$.
The measured horizontal emittances shown in Figure 13 include (for both data
and simulation) the effect of dispersion. The dispersion in $x$ at the exit of
the D2 bending magnet is transformed by the optics of the beam transport into
dispersion in $x$ and $x^{\prime}$ at the TOF1 measurement plane. The
intrinsic horizontal emittances of the beams have been obtained from the
covariance matrices by subtracting the dispersion characterised by $\eta$ and
$\eta^{\prime}$ [23]:
$\displaystyle\Sigma_{11}$
$\displaystyle\rightarrow\Sigma_{11}-\eta^{2}\delta^{2}$
$\displaystyle\Sigma_{12}$
$\displaystyle\rightarrow\Sigma_{12}-\eta\eta^{\prime}\delta^{2}$
$\displaystyle\Sigma_{11}$
$\displaystyle\rightarrow\Sigma_{11}-\eta^{{}^{\prime}2}\delta^{2}$
where $\eta=\langle x\delta\rangle/\langle\delta^{2}\rangle$,
$\eta^{\prime}=\langle x^{\prime}\delta\rangle/\langle\delta^{2}\rangle$ and
$\delta=(p_{z}-\bar{p}_{z})/\bar{p}_{z}$. Figure 15 shows $\eta$ versus
$\langle p_{z}\rangle$ for all the beams and the simulations for the three
negative beams. The dispersions are similar for the $\mu^{+}$ and $\mu^{-}$
beams and are reproduced by the simulations for the negative beams. The
positive beam simulations are not shown as they did not reproduce the data
well. The reason for this is under investigation. The dispersion-corrected
intrinsic horizontal emittances and $\eta$ and $\eta^{\prime}$ are given in
Table 4. The intrinsic horizontal emittances are, on average, $0.25\,\pi$ mm-
rad smaller than the effective horizontal emittances.
Table 4: Horizontal dispersion and the intrinsic emittances of the Step I beams. | Beam | $\eta_{x}$ (mm) | $\eta^{\prime}_{x}$ (rad) | $\varepsilon_{x}$ ($\pi$ mm-rad) | $\alpha_{x}$ | $\beta_{x}$ (m)
---|---|---|---|---|---|---
| $\varepsilon_{N}$ | $p_{z}$ | | | | |
$\mu^{-}$ | 3 | 140 | 90.28 | 0.07 | 2.08$\pm$0.11 | 0.60$\pm$0.01 | 1.56$\pm$0.09
200 | 123.78 | 0.09 | 1.53$\pm$0.08 | 0.65$\pm$0.01 | 1.82$\pm$0.10
240 | 137.58 | 0.11 | 1.26$\pm$0.07 | 0.68$\pm$0.01 | 1.99$\pm$0.11
6 | 140 | 89.37 | 0.08 | 1.97$\pm$0.11 | 0.64$\pm$0.01 | 1.66$\pm$0.09
200 | 106.27 | 0.10 | 1.31$\pm$0.07 | 0.72$\pm$0.01 | 2.06$\pm$0.12
240 | 157.91 | 0.11 | 1.26$\pm$0.07 | 0.68$\pm$0.01 | 1.98$\pm$0.11
10 | 140 | 96.03 | 0.07 | 1.83$\pm$0.10 | 0.64$\pm$0.01 | 1.71$\pm$0.10
200 | 132.78 | 0.08 | 1.04$\pm$0.06 | 0.79$\pm$0.01 | 2.47$\pm$0.14
240 | 145.71 | 0.11 | 1.40$\pm$0.08 | 0.75$\pm$0.01 | 2.02$\pm$0.12
$\mu^{+}$ | 3 | 200 | 122.96 | 0.03 | 1.85$\pm$0.10 | 0.56$\pm$0.00 | 1.58$\pm$0.09
240 | 156.47 | 0.03 | 1.45$\pm$0.08 | 0.66$\pm$0.01 | 1.87$\pm$0.11
6 | 140 | 95.91 | 0.04 | 2.18$\pm$0.12 | 0.52$\pm$0.00 | 1.51$\pm$0.09
200 | 131.16 | 0.04 | 1.76$\pm$0.09 | 0.58$\pm$0.00 | 1.62$\pm$0.09
240 | 172.97 | 0.04 | 1.54$\pm$0.08 | 0.64$\pm$0.01 | 1.76$\pm$0.10
10 | 140 | 103.27 | 0.04 | 2.03$\pm$0.11 | 0.54$\pm$0.01 | 1.57$\pm$0.09
200 | 138.50 | 0.03 | 1.53$\pm$0.08 | 0.59$\pm$0.01 | 1.78$\pm$0.10
240 | 189.64 | 0.04 | 1.61$\pm$0.09 | 0.61$\pm$0.01 | 1.64$\pm$0.09
## 5 Summary
A single-particle method for measuring the properties of the muon beams to be
used by MICE has been developed. Timing measurements using two time-of-flight
counters allow the momentum of single muons to be measured with a resolution
of better than 4 MeV/$c$ and a systematic error of $<3$ MeV/$c$. The ability
to measure $p_{z}$ to this precision will complement the momentum measurements
of the solenoidal spectrometers. For low transverse amplitude particles, the
measurement of $p_{z}$ in the TOF counters is expected to have better
resolution than that of the spectrometers, which are primarily designed for
measuring $p_{t}$.
The same method allows the trace-space distributions at the entrance to MICE
to be measured to $\approx 5\%$ and hence the emittances and dispersions of
the beams. The emittances are found to be approximately 1.2–2.3 $\pi$ mm-rad
horizontally and 0.6–1.0 $\pi$ mm-rad vertically; the average horizontal
dispersion, $\eta$, is measured to be 129 mm, although it depends on the
nominal $(\varepsilon_{N},p_{z})$ beam setting. The positive and negative muon
beams are found to have very similar properties.
As a final check on the suitability of the beams for use by MICE, a set of
measured muons for the (6, 200) baseline beam was propagated from TOF1 to the
diffuser and through a simulation of the experiment. Even without further
software selection (for example, on the rather asymmetric momentum
distribution) the beam was found to be relatively well matched [24]. In
practice, some further fine-tuning of the magnet currents and diffuser
thickness should be sufficient to generate a well-matched beam suitable for
the demonstration of ionisation cooling by MICE.
### Acknowledgements
The work described here was made possible by grants from Department of Energy
and National Science Foundation (USA), the Instituto Nazionale di Fisica
Nucleare (Italy), the Science and Technology Facilities Council (UK), the
European Community under the European Commission Framework Programme 7, the
Japan Society for the Promotion of Science and the Swiss National Science
Foundation, in the framework of the SCOPES programme, whose support we
gratefully acknowledge. We are also grateful to the staff of ISIS for the
reliable operation of ISIS.
## References
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|
arxiv-papers
| 2013-06-06T19:03:14 |
2024-09-04T02:49:46.194538
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "The MICE Collaboration: D. Adams (16), D. Adey (23, b), A. Alekou (19,\n c), M. Apollonio (19, d), R. Asfandiyarov (12), J. Back (23), G. Barber (19),\n P. Barclay (16), A. de Bari (5), R. Bayes (17), V. Bayliss (16), R. Bertoni\n (3), V. J. Blackmore (20, a), A. Blondel (12), S. Blot (29), M. Bogomilov\n (1), M. Bonesini (3), C. N. Booth (21), D. Bowring (27), S. Boyd (23), T. W.\n Bradshaw (16), U. Bravar (30), A. D. Bross (25), M. Capponi (6), T. Carlisle\n (20), G. Cecchet (5), G. Charnley (14), J. H. Cobb (20), D. Colling (19), N.\n Collomb (15), L. Coney (34), P. Cooke (18), M. Courthold (16), L. M. Cremaldi\n (33), A. DeMello (27), A. Dick (22), A. Dobbs (19), P. Dornan (19), S. Fayer\n (19), F. Filthaut (10, f), A. Fish (19), T. Fitzpatrick (25), R. Fletcher\n (34), D. Forrest (17), V. Francis (16), B. Freemire (28), L. Fry (16), A.\n Gallagher (15), R. Gamet (18), S. Gourlay (27), A. Grant (15), J. S. Graulich\n (12), S. Griffiths (14), P. Hanlet (28), O. M. Hansen (11, h), G. G. Hanson\n (34), P. Harrison (23), T. L. Hart (33), T. Hartnett (15), T. Hayler (16), C.\n Heidt (34), M. Hills (16), P. Hodgson (21), A. Iaciofano (6), S. Ishimoto\n (9), G. Kafka (28), D. M. Kaplan (28), Y. Karadzhov (12), Y. K. Kim (29), D.\n Kolev (1), Y. Kuno (8), P. Kyberd (24), W. Lau (20), J. Leaver (19), M.\n Leonova (25), D. Li (27), A. Lintern (16), M. Littlefield (24), K. Long (19),\n G. Lucchini (3), T. Luo (33), C. Macwaters (16), B. Martlew (14), J.\n Martyniak (19), A. Moretti (25), A. Moss (14), A. Muir (14), I. Mullacrane\n (14), J. J. Nebrensky (24), D. Neuffer (25), A. Nichols (16), R. Nicholson\n (21), J. C. Nugent (17), Y. Onel (31), D. Orestano (6), E. Overton (21), P.\n Owens (14), V. Palladino (4), J. Pasternak (19), F. Pastore (6), C. Pidcott\n (23), M. Popovic (25), R. Preece (16), S. Prestemon (27), D. Rajaram (28), S.\n Ramberger (11), M. A. Rayner (20, j), S. Ricciardi (16), A. Richards (19), T.\n J. Roberts (26), M. Robinson (21), C. Rogers (16), K. Ronald (22), P. Rubinov\n (25), R. Rucinski (25), I. Rusinov (1), H. Sakamoto (8), D. A. Sanders (33),\n E. Santos (19), T. Savidge (19), P. J. Smith (21), P. Snopok (28), F. J. P.\n Soler (17), T. Stanley (16), D. J. Summers (33), M. Takahashi (19), J.\n Tarrant (16), I. Taylor (23), L. Tortora (6), Y. Torun (28), R. Tsenov (1),\n C. D. Tunnell (20), G.Vankova (1), V. Verguilov (12), S. Virostek (27), M.\n Vretenar (11), K. Walaron (17), S. Watson (16), C. White (14), C. G. Whyte\n (22), A. Wilson (16), H. Wisting (12), M. Zisman (27) ((1) Department of\n Atomic Physics, St. Kliment Ohridski University of Sofia, Sofia, Bulgaria (2)\n Institute for Cryogenic and Superconductivity Technology, Harbin Institute of\n Technology, Harbin, PR China (3) Sezione INFN Milano Bicocca, Dipartimento di\n Fisica G. Occhialini, Milano, Italy (4) Sezione INFN Napoli and Dipartimento\n di Fisica, Universit\\`a Federico II, Complesso Universitario di Monte S.\n Angelo, Napoli, Italy (5) Sezione INFN Pavia and Dipartimento di Fisica\n Nucleare e Teorica, Pavia, Italy (6) Sezione INFN Roma Tre e Dipartimento di\n Fisica, Roma, Italy (7) Kyoto University Research Reactor Institute, Osaka,\n Japan (8) Osaka University, Graduate School of Science, Department of\n Physics, Toyonaka, Osaka, Japan (9) High Energy Accelerator Research\n Organization (KEK), Institute of Particle and Nuclear Studies, Tsukuba,\n Ibaraki, Japan (10) NIKHEF, Amsterdam, The Netherlands (11) CERN, Geneva,\n Switzerland (12) DPNC, Section de Physique, Universit\\'e de Gen\\`eve, Geneva,\n Switzerland (13) Paul Scherrer Institut, Villigen, Switzerland (14) The\n Cockcroft Institute, Daresbury Science and Innovation Centre, Daresbury,\n Cheshire, UK (15) STFC Daresbury Laboratory, Daresbury, Cheshire, UK (16)\n STFC Rutherford Appleton Laboratory, Harwell Oxford, Didcot, UK (17) School\n of Physics and Astronomy, Kelvin Building, The University of Glasgow,\n Glasgow, UK (18) Department of Physics, University of Liverpool, Liverpool,\n UK (19) Department of Physics, Blackett Laboratory, Imperial College London,\n London, UK (20) Department of Physics, University of Oxford, Denys Wilkinson\n Building, Oxford, UK (21) Department of Physics and Astronomy, University of\n Sheffield, Sheffield, UK (22) Department of Physics, University of\n Strathclyde, Glasgow, UK (23) Department of Physics, University of Warwick,\n Coventry, UK (24) Brunel University, Uxbridge, UK (25) Fermilab, Batavia, IL,\n USA (26) Muons, Inc., Batavia, IL, USA (27) Lawrence Berkeley National\n Laboratory, Berkeley, CA, USA (28) Illinois Institute of Technology, Chicago,\n IL, USA (29) Enrico Fermi Institute, University of Chicago, Chicago, IL, USA\n (30) University of New Hampshire, Durham, NH, USA (31) Department of Physics\n and Astronomy, University of Iowa, Iowa City, IA, USA (32) Jefferson Lab,\n Newport News, VA, USA (33) University of Mississippi, Oxford, MS, USA (34)\n University of California, Riverside, CA, USA (35) Brookhaven National\n Laboratory, Upton, NY, USA (a) email: [email protected] (b) Now\n at Fermilab, Batavia, IL, USA (c) Now at CERN, Geneva, Switzerland (d) Now at\n Diamond Light Source, Harwell Science and Innovation Campus, Didcot,\n Oxfordshire, UK (f) Also at Radboud University Nijmegen, Nijmegen, The\n Netherlands (g) Permanent address Institute of Physics, Universit\\'e\n Catholique de Louvain, Louvain-la-Neuve, Belgium (h) Also at University of\n Oslo, Norway (j) Now at DPNC, Universit\\'e de Gen\\`eve, Geneva, Switzerland\n (k) Now at University of Huddersfield, UK)",
"submitter": "Victoria Blackmore",
"url": "https://arxiv.org/abs/1306.1509"
}
|
1306.1756
|
# Two-color narrowband photon pair source with high brightness
based on clustering in a monolithic waveguide resonator
Kai-Hong Luo, Harald Herrmann, Stephan Krapick, Raimund Ricken,
Viktor Quiring, Hubertus Suche, Wolfgang Sohler, and Christine Silberhorn
Integrated Quantum Optics, Applied Physics, University of Paderborn, Warburger
Str. 100, D-33098, Paderborn, Germany
###### Abstract
We report on an integrated non-degenerate narrowband photon pair source
produced at 890 nm and 1320nm via type II parametric down-conversion in a
periodically poled waveguide with high-reflective dielectric mirrors deposited
on the waveguide end faces. The conversion spectrum consists of three clusters
and only 3 to 4 longitudinal modes with about 150 MHz bandwidth in each
cluster. The high conversion efficiency in the waveguide, together with the
spectral clustering in the double resonator, leads to a high brightness of
$3\times 10^{3}~{}$pairs/(s$\cdot$mW$\cdot$MHz). The compact and rugged
monolithic design makes the source a versatile device for various applications
in quantum communication.
###### pacs:
42.50.Ex, 42.65.Wi, 42.50.Dv, 42.65.Ky
Single photon pair sources with narrow bandwidth and long coherence time play
an essential role in applications of quantum information processing; in
particular for quantum key distribution GisinRMP2002 , long distance quantum
communication DuanN2001 , and the realization of quantum networks KimbleN2008
. To overcome current limitations of long distance quantum communication due
to transmission losses, quantum repeater architectures have been proposed
BriegelPRL1998 ; SimonPRL2007 ; SangouardRMP2011 . These typically require
photon pairs with one wavelength in the telecommunication band and one
wavelength that matches the absorption line of the storage medium in a quantum
memory (QM) LvovskyNP2009 ; WalmsleyNP2010 ; LamNP2011 . Such QMs usually have
their absorption line in the visible or near infrared, i.e. far away from the
telecommunication range, and the bandwidth is typically in the range of a few
to several 100 MHz. Among the most promising materials for high-bandwidth QM’s
are solid-state atomic ensembles, specifically rare-earth ion doped crystals
or glasses TittelLPR2010 ; TittelN2011 . For example, a Nd3+-doped Y2SiO5
crystal has been successfully applied as an multi-mode atomic frequency comb
memory with an absorption bandwidth of about 120 MHz GisinN2011 .
A well-known method to generate photon pairs is parametric down conversion
(PDC) HarrisPRL1967 ; BurnhamPRL1970 . In such a process, a medium with with
$\chi^{(2)}$ nonlinearity splits a single pump photon into two photons of
lower energy, named the signal and the idler, obeying with energy conservation
and phase matching. However, the loose phase-matching condition usually leads
to a broad bandwidth typically exceeding several 100 GHz. As a result, a
strong filtering is required to match the acceptance bandwidth of the QMs with
the drawback of losing most of the generated photon pairs. To overcome this
bottle-neck, narrowband photon-pair sources are desired with an adapted
bandwidth and a high spectral brightness.
One promising approach to generate such narrowband photon pairs is to use
resonance enhancement of PDC within a cavity, also called optical parametric
oscillator (OPO) far below the threshold OuPRL1999 ; WangPRA2004 ;
KuklewiczPRL2006 ; BensonPRL2009 ; PolzikOL2009 ; PanPRL2008 ; GuoOL2008 ;
WolfgrammPRL2011 ; PomaricoNJP2009 ; PomaricoNJP2012 ; URenLP2010 ;
ChuuAPL2012 ; FortschNC2013 ; FeketePRL2013 . PDC is enhanced at the
resonances of the cavity but inhibited at non-resonant frequencies. In this
way the spectral density is redistributed in comparison to the non-resonant
case and, thus, ideally a filter-free source with actively reduced bandwidth
and without sacrificing the photon flux level can be realized OuPRL1999 .
In most of the cavity-enhanced PDC experiments reported so far, the photon
pairs were frequency degenerate or close to frequency degeneracy. Thus, within
the phase-matching bandwidth a comb of narrow lines is generated with a line
spacing corresponding to the free spectral range (FSR) of the resonator. In
the non-degenerate case, however, the material dispersion results in different
FSRs for signal and idler. As maximum enhancement is only obtained if both
signal and idler are resonant simultaneously, PDC is generated only in certain
regions of the spectrum, so called ’clusters’ EckardtJOSAB1991 . Within each
cluster PDC occurs only at a few longitudinal modes. In this year the
clustering approach has been applied for the first time to generate photon
pairs with bandwidth of 2 MHz within 3 clusters (spaced 44 GHz) and 4
longitudinal modes inside of each cluster using a bulk crystal cavity etalon
FeketePRL2013 .
Although a monolithic, resonant PDC source exploiting whispering gallery modes
has been presented recently FortschNC2013 , most of the demonstrated resonant
PDC sources used bulk crystals (either PPLN or PPKTP) as the nonlinear
material placed in an external resonator. However, it is well-known that PDC
in a waveguide is typically 2 to 3 orders of magnitude more efficient than in
bulk crystals TanzilliEP2002 ; FiorentinoOE2007 due to the strong confinement
of the optical fields along the whole interaction length. Obviously, the
resonator approach can be transferred to waveguide structures. Moreover, if
the resonator is formed by mirrors directly deposited on the waveguide end-
faces, a compact and rugged design requiring no cavity alignment can be
realized. Tuning of the resonance frequency can be accomplished by varying the
optical path length in the waveguide, for instance by temperature tuning
SchreiberSPIE2001 .
Initial experiments with a resonant PPLN waveguide source based on degenerate
type I phase-matching have been demonstrated in PomaricoNJP2009 . A detailed
theoretical study revealed that using type II phase-matching should offer a
reduced number of longitudinal modes PomaricoNJP2012 . This is due to the fact
that the large birefringence in this system provides a larger difference
between the FSR’s of the signal and the idler fields. Moreover, in this study
it was pointed out that an optimized performance of such a resonant source
must carefully take into consideration waveguide losses and resonator
outcoupling efficiency to select design parameters like waveguide length or
mirror reflectivities.
In this Letter, we present the first experimental realization of such an
integrated compact photon pair source based on a doubly resonant waveguide
exploiting type II phase-matching. The detailed structure of the integrated
waveguide chip is shown in Fig. 1. The source consists of a 14.5 mm long Ti-
indiffused waveguide in Z-cut PPLN. The poling period of $\Lambda=$4.44 $\mu$m
was chosen to provide first-order type II phase-matched PDC to generate TE-
polarized signal photons around 890 nm and TM-polarized photons around 1320 nm
when pumped at $\lambda_{p}=$532 nm in TE-polarization. This wavelength
combination was chosen to fit to the Nd-based QM GisinN2011 .
Figure 1: (color online) Integrated narrowband photon pair source composed of
a Ti indiffused waveguide in a PPLN substrate with $\Lambda=4.44~{}\mu$m
poling period and dielectric mirrors with high reflectivities at both signal
and idler wavelengths deposited on the waveguide end-faces. The upper inset is
a zoom to domain structure of the periodically poled waveguide area, and the
black dotted lines denote the inverted domains.
Waveguide losses have been measured to be $\alpha_{s}\approx 0.05~{}$dB/cm and
$\alpha_{i}\approx 0.06~{}$dB/cm for signal and idler, respectively. To
implement the resonant source dielectric layers composed of alternating layer
stacks of SiO2 and TiO2 were deposited on the end-faces of the waveguide.
Based on the design rules given in PomaricoNJP2012 the resonator was modeled
with a finesse exceeding 20 in order to provide the spectral narrowing. This
could be best realized with an asymmetric resonator with high reflectivities
for signal and idler at the front mirror and reflectivities around 90 % for
the rear mirror (outcoupling mirror). With this asymmetry of the mirror
reflectivities, the ratio of outcoupled signal (idler) photons to lost photons
before escaping this cavity is about $\eta_{s}\approx 0.41$ ($\eta_{i}\approx
0.37$), respectively. Thus, the overall photon pair escape probability is
given as ${\eta_{pp}}={\eta_{s}}{\eta_{i}}$, which means about 15% of the
generated photon pairs leave the cavity as couples at the desired output
mirror.
In practice, a stack with 17 layers deposited as front mirror provides a
reflectivity of $R\approx 99\%$ for both wavelengths and the rear mirror
consisting of 13 layers has the targeted $R\approx 90\%$. The reflectivities
of front and rear mirrors at 532 nm are 38% and 8%, respectively, to enable
efficient incoupling of pump and to prevent triple resonance effects. After
mirror deposition the resonator was characterized carefully by launching
appropriate narrowband light into the waveguide and monitoring the
transmission while varying the cavity length via temperature tuning. We
obtained a finesse of $\mathcal{F}=22$ and $\mathcal{F}=25$ for the signal and
idler wavelengths, respectively.
Figure 2: (color online) Experimental setup for the generation and
characterization of the photon pairs from the resonant waveguide, combining
PDC characterization and coincidence measurements. The detailed structure of
the waveguide resonator chip in the oven is shown in Fig. 1. The cyan dashed
circle means the optical beam paths are alternative. AOM: acousto-optical
modulator; HWP: half-wave plate; DM: dichroic mirror; SMFC: single-mode fiber
coupler; MMF: multi-mode fiber; VBG: volume Bragg grating; PZT: piezoelement;
APD: avalanche photodiode; TDC: time-to-digital converter.
The experimental setup to study the resonant source is shown in Fig. 2. The
sample is always pumped with a laser at 532 nm with a specified bandwidth of
less than 1 MHz. To avoid the effects of photorefraction, pump pulses with a
typical length of about 200 ns and a repetition rate of about 100 kHz are
extracted from the cw-source by using an acousto-optical modulator (AOM). By
using a half wave plate (HWP) together with a polarizer, the pump power is
tunable from 0.1 mW to 10 mW. The sample is heated to temperatures around 160
∘C to obtain quasi-phase-matching for the desired wavelength combination and
to prevent luminescence and deterioration due to photorefraction. During the
measurements the sample temperature is stabilized to about $\pm$1 mK.
To characterize the generated PDC in the signal wavelength range the output
from the waveguide is coupled to a spectrometer system (Andor Shamrock 303i
with iKon-M 934 CCD camera) with a resolution of about 0.15 nm. In Fig. 3
recorded PDC spectra of one waveguide prior and after mirror deposition are
shown. The spectrum of the uncoated waveguide shows a main peak with a
bandwidth (FWHM) of 0.4 nm ($\approx$ 151 GHz), as predicted for the 14.5 mm
long interaction length. The spectra of the same waveguide with dielectric
mirrors, however, shows a pronounced sub-structure. Although the resolution of
the spectrometer is too coarse to reveal details of the spectra, one can
already derive that the pair generation occurs within three clusters with a
cluster separation of about 0.2 nm ($\approx$ 75 GHz). This separation is
determined by the difference of the FSRs of signal and idler, which
theoretically are FSR${}_{s}\sim 4.4$ GHz and FSR${}_{i}\sim 4.7$ GHz
PomaricoNJP2012 ; Luo2013 . The cluster separation is half of the bandwidth of
the PDC envelope. Ideally, there is a dominant cluster in the the central of
normalized phase-matching, and another two symmetric clusters at the side
wings of the envelope with about 41% intensity. Thus, three clusters can be
observed within this envelope with a strong temperature dependence as verified
by comparing the two spectra shown in Fig. 3 with only 30 mK temperature
difference.
Figure 3: (color online) PDC spectra measured using the uncoated waveguide
(black square) and the resonant waveguide at various temperatures (blue star,
red circle). The background counts can be attributed to the dark noise of CCD
camera.
Moreover, these measurements are already a proof of resonance enhancement as
can be easily seen by comparing the spectra of the same waveguide before
coating and after coating shown in Fig. 3. For the measurements, all of the
parameters, apart from the temperature, have been kept constant. In
particular, the pump power about 1 mW in front of the incoupling lens has not
been changed. If the cavity of the resonant sample only act as a narrowband
filter, the overall level of the PDC measured with the spectrometer should
have dropped significantly, caused by the integration over 0.15 nm, which is
performed by the spectrometer due to its limited resolution. However, as the
detected power levels are almost identical for the resonant and the non-
resonant sample, we can conclude that due to the cavity there is a spectral
density redistribution resulting in an enhanced PDC generation at the cavity
resonances.
To investigate details of the spectra of the resonant source we studied the
internal structure within a single cluster. A volume Bragg grating (VBG,
OptiGrate 900) with a spectral bandwidth of 0.17 nm is inserted into the
signal beam path as shown in Fig. 2 to act as bandpass filter to select a
single cluster. The filtered light is routed via a single mode fiber to a
scanning confocal Fabry-Pérot resonator with 15 GHz FSR and a finesse of about
20. Its transmission is analyzed using a single photon detection module
(Perkin Elmer Avalanche photodiode SPCM-AQR-14) to record the signal photons
transmitted from the Fabry-Pérot which is tuned by applying a voltage ramp to
the piezo-driven mirror mount.
Figure 4: (color online) Signal spectra recorded with a confocal scanning
Fabry-Pérot with 15 GHz FSR. The two spectra vary in temperature by 6mK around
161.57 ∘C.
Corresponding measurement results are shown in Fig. 4, they reveal the modal
structure within a single cluster. The spectrum consists of longitudinal modes
with 4.5 GHz frequency separation. By finely tuning the temperature, we are
able to suppress the two ’satellite’ modes, leaving a predominantly single
mode operation – as in the upper case of Fig. 4. A slight shift of the
temperature by less than 6 mK leads to a spectrum composed of two modes with
almost equal strength surrounded by weak additional satellite modes. The
measured linewidth of each longitudinal mode of about 750 MHz is mainly
determined by the resolution of the scanning Fabry-Pérot resonator, but it is
not the natural bandwidth of the generated PDC photons.
To investigate the spectral linewidth of the photons and the correlation
between photon pairs, coincidences between signal and idler have been
characterized by measuring the arrival times of the respective photons with
the set-up shown in Fig. 2. In the left diagram of Fig. 5, the results from
such a measurement are shown. It reveals a correlation time (FWHM of the
coincidence peak) of about 2.1 ns. This is significantly broader than the
corresponding results obtained with non-resonant samples showing a width of
about 0.5 ns, which is determined by the finite resolution of our measurements
system, due to timing jitters of the detection system. The correlation time is
inversely proportional to the bandwidth of the down-conversion fields. From
the measured correlation time $\tau_{coh}$ of 2.1 ns, according to
$\tau_{coh}=1/{\pi\Delta\nu}$, where $\Delta\nu$ is the bandwidth of the down-
converted photons, a spectral bandwidth of about 150 MHz can be deduced. This
is in good accordance with the theoretically predicted width of the resonances
calculated for the given cavity parameters.
The presence of the cavity implies that the signal and idler photons may be
emitted at distinct times, corresponding to a different number of round-trips
within the cavity. Thus, the shape of the coincidence curve should be
determined by exponential functions. In the right diagram of Fig. 5 the
measurement result is re-drawn using a logarithmic scaling together with
exponential fits for the rising and falling parts. The slight asymmetry
reflects the different finesses of signal and idler resulting in slightly
different leakage times out of the resonator.
Figure 5: (color online) Measured coincidences of photon pairs as a function
of arrival time difference between the signal and idler photons using linear
scaling (left) and logarithmic scaling (right) with exponential fits. Figure
6: (color online) Measured signal-signal autocorrelation as function of
arrival time difference between two signal photons.
Such coincidence measurements can also be evaluated to determine the
efficiency of the PDC generation process Klyshko1980 . From the ratio of the
coincidence to single counts the generated photon pair rate can be determined.
We found that this generation rate is almost equal for both the non-resonant
and resonant waveguides. In the resonant case, however, the spectral
distribution is strongly confined to only a few longitudinal modes, whereas in
the non-resonant case it is distributed over the entire phase-matching
bandwidth. For our source we determined a normalized generation rate (inside
the resonator) of about $7\times 10^{6}~{}$pairs/(s$\cdot$mW). Assuming these
are distributed over three inequivalent clusters with three longitudinal modes
with different weights within each cluster as shown in Fig. 4, we can estimate
that about 39% of the generated photon pairs are within the most dominant
longitudinal mode with 150 MHz bandwidth. Taking into account the photon pair
escape probability of 15 %, the spectral brightness can be estimated to be
$B=3\times 10^{3}~{}$pairs/(s$\cdot$mW$\cdot$MHz).
An alternative method to characterize the spectral properties of the source is
the analysis of the auto-correlation Glauber function $g^{(2)}(\tau)$
Glauber1963 . The number of effective modes K can be estimated from the
normalized auto-correlation function at zero time delay according to
$g^{(2)}(0)=1+1/K$ McNeilPRA1983 ; ChristNJP2011 . We have measured the
signal-signal autocorrelation by splitting the signal radiation behind behind
the VBG (i.e. a single cluster was selected) using a 50:50 single mode fiber
coupler (SMFC) and routing the two output ports of the SMFC to individual
silicon single photon detectors.
An example of a measured $g^{(2)}(\tau)$ characteristic is shown in Fig. 6.
Around zero time delay $g^{(2)}$ strongly exceeds 1 proving the generation of
non-classical light. It is shown that from the measured data
$g^{(2)}(0)\approx 1.4$. As a result, an effective mode number of filtered
signal beam $K=2.5$ can be estimated. On the other hand, the auto-correlation
is only constant for the idler unfiltered beam. This is in reasonably good
qualitative agreement with the measured spectra shown in Fig. 4, where we
observed three modes (with different amplitudes) within a single cluster. A
refined theoretical model would be necessary to study quantitatively the
relationship between $g^{(2)}(\tau)$ and the observed spectra.
In summary, we have experimentally demonstrated a compact, bright and
narrowband photon pair source by exploiting clustering in a doubly resonant
Ti:PPLN waveguide. In the type II phase-matched, nondegenerate PDC source
photon pairs are generated with one photon matching the absorption line of a
Nd-based QM, and the wavelength of the idler photon (around 1320 nm) is
compatible for long distance transmission over standard telecom fibers.
Although the measured bandwidth of about 150 MHz is still about two orders of
magnitude larger than the bandwidth demonstrated with bulk optical versions
FeketePRL2013 , the compact and rugged design and the large brightness of
about $3\times 10^{3}~{}$pairs/(s$\cdot$mW$\cdot$MHz) makes this device a
versatile source for various quantum applications. In particular, it has a
great potential if an ultimate narrow bandwidth is not required. Beyond
further fundamental studies and work on technological improvements, future
activities can focus on the development of pure, filter-free tunable single
mode sources using pulsed light. One can exploit the use of the cluster effect
together with a double-pass pumping scheme to restrict PDC generation to only
a single cluster ChuuAPL2012 . Further improvement of the resonator quality
might enable the restriction to, ultimately, a single longitudinal mode, which
can further be tuned via a monolithicly integrated electro-optic modulator.
The authors thank Benjamin Brecht and Malte Avenhaus for helpful discussions.
We gratefully acknowledge the support by the European Union through the QuReP
project (no. 247743).
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|
arxiv-papers
| 2013-06-07T15:57:52 |
2024-09-04T02:49:46.208999
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Kai-Hong Luo, Harald Herrmann, Stephan Krapick, Raimund Ricken, Viktor\n Quiring, Hubertus Suche, Wolfgang Sohler, and Christine Silberhorn",
"submitter": "Kai-Hong Luo",
"url": "https://arxiv.org/abs/1306.1756"
}
|
1306.1781
|
# The Composition of Wage Differentials between Migrants and Natives
Panagiotis Nanos [email protected] Christian Schluter
[email protected] Economics Division, University of Southampton,
Highfield, Southampton, SO17 1BJ, UK Aix-Marseille Université (Aix-Marseille
School of Economics) , CNRS & EHESS, Centre de la Vieille Charité, 13002
Marseille, France
###### Abstract
We consider the role of unobservables, such as differences in search
frictions, reservation wages, and productivities for the explanation of wage
differentials between migrants and natives. We disentangle these by estimating
an empirical general equilibrium search model with on-the-job search due to
Bontemps et al. (1999) on segments of the labour market defined by occupation,
age, and nationality using a large scale German administrative dataset.
The native-migrant wage differential is then decomposed into several parts,
and we focus especially on the component that we label “migrant effect”, being
the difference in wage offers between natives and migrants in the same
occupation-age segment in firms of the same productivity. Counterfactual
decompositions of wage differentials allow us to identify and quantify their
drivers, thus explaining within a common framework what is often labelled the
unexplained wage gap.
###### keywords:
immigrants , decomposition of wage differentials , job search , turnover JEL
Classification: J31 , J61 , J63
## 1 Introduction
The empirical literature on the labour market experience of immigrants often
focuses on differences in observable characteristics between migrants and
natives to explain wage differentials. Less explored is the role of
unobservables, such as differences in search frictions, reservation wages, and
productivities. Yet, it is precisely these factors that modern search theory
emphasises to be important for wage dispersion. We examine and disentangle the
role of these various unobservables in explaining migrant-native wage
differentials by adapting to the migrant context the empirical general
equilibrium search model with on-the-job search due to Bontemps et al. (1999).
The estimation of this structural model on segments of the labour market
defined by occupation, age, and nationality enables us to decompose the
native-migrant wage differential into several parts. In particular, we focus
on the component that we label “migrant effect”, being the difference in wage
offers between similar native and immigrant workers in firms of the same
productivity. This effect is of interest as we thus control for firm-level
differences as measured by their productivities, which have recently been
shown using firm-level data to contribute systematically to the wage gap
(Aydemir and Skuterud (2008) in the case of Canada, de Matos (2012) for
Portugal, and Bartolucci (2013b) for the German case).111The migrant effect
corresponds to within-firm wage differentials of workers with similar
observable characteristics reported in these papers. One particular advantage
of our approach is that we do not require firm-level data (data
confidentiality promises usually deny public access), as the productivity
distribution emerges as an equilibrium relationship. We estimate the migrant
effect on internationally accessible German administrative data, the
scientific use file known as IABS which is a 2% subsample of the German
employment register. This enables us to contribute to the recent literature on
the immigrant-native wage gap as follows. While the role of observables is
well understood for explaining the wage gap, the role of unobservables is less
so. Such wage gaps arise when, for instance, migrants have systematically
lower reservation wages (whose role is examined in detail in Albrecht and
Axell (1984)), or when firms in a migrant-native segmented labour market
(which we discuss below) are less productive in the migrant segment, or when
wage-posting firms in one segment derive greater monopsony power from e.g.
greater search frictions. Our analysis focuses on the roles of differences in
the job turnover parameters, behavioural differences induced by differences in
reservation wages, and productivity differences.222 The migrant effect is not
synonymous with (taste-based) discrimination as we do not model this
explicitly (for two approaches see Bowlus and Eckstein (2002), and Flabbi
(2010)). Instead, similar to Bowlus (1997) and Bartolucci (2013a) in the
context of the gender wage gap, we have an indirect link: if market
discrimination exists and influences behavioural patterns, it will be captured
in those parameters, while other avenues of wage-impacting discrimination will
be picked up by the productivity distributions. In contrast to costly taste-
based discrimination, the new monopsony theory suggests the possibility of
profitable monopsonistic discrimination stemming e.g. from differential search
friction (Manning (2003)). For instance Barth and Dale-Olsen (2009) and Hirsch
et al. (2010) consider the gender wage gap in the light of this. We relate the
migrant effect to the Hirsch and Jahn (2012) analysis of monopsonistic
discrimination in Section 2.4. Within a common framework, we establish the
relative importance of each of these factors. Having estimated the model’s
parameters and thus the actual wage gap and migrant effect, we quantify the
roles of the various unobservables in several counterfactual experiments.
The structural model is estimated on a large German administrative panel.
Germany is a particularly interesting and relevant case since it hosts the
largest numbers of foreign nationals in Europe, and immigration is known to be
predominantly low-skilled. According to Eurostat, 7.13 million foreign
nationals resided in Germany in 2010, about 8.7% of the total population. The
size of the IABS allows us to stratify the analysis by nationality, occupation
and age. The resulting subsamples are sufficiently large to permit precise
estimation of the model’s structural parameters. Moreover, since this is
administrative data, the usual concerns about the quality of survey data in a
migrant context (sample size, measurement accuracy, and use of retrospective
information) are absent.
We briefly describe some aspects of our applications of the structural model.
In order to control for heterogeneity in observables, we follow common
estimation practice in the search-theory literature by partitioning the labour
market into many segments. These segments are defined in terms of occupation,
age, and nationality.333The term “nationality” rather than “immigrant status”
is used here for greater precision given the coding practices of the German
Statistical Office. Most German data sources record nationality and not
country of birth since German nationality was conferred by descent until the
year 2000, when Germany changed its legislation to ius soli (this change does
not affect our sample). Given the skill profile of migrants, we consider only
the low and medium skill occupations. Each segment is thus assumed to be
potentially a separate labour market, characterised by its own job turnover
parameters (the job arrival and separation rates). Turning to the
unobservables (for the econometrician), firms in each segment differ in terms
of productivity, and workers differ in terms of reservation wages. Such
reservation wage heterogeneity is plausible given the absence of a legal
minimum wage in Germany, and the fact that the location decisions of labour
migrants in Roy-style models are usually based on comparisons of expected
incomes in source and host country. Migrants might trade-off wage and non-wage
job characteristics differently to natives, given their well-known clustering.
Besides this preference component, reservation wages also feature an
institutional one, but this is less important as contributory unemployment
insurance benefits are independent of immigrant status.
The assumption of separate markets for natives and immigrants and the
associated notion of job segmentation conforms to existing international
empirical evidence. For instance, using Portuguese data, de Matos (2012) shows
that immigrants “work in different industries and occupations than natives”
(p.10), and the sorting of immigrants is also observed by Aydemir and Skuterud
(2008) for Canada. As regards Germany, D’Amuri et al. (2010) observe that
recent immigrants are significantly more likely to compete with established
immigrants rather than with natives. Velling (1995) is an early paper to
report “evidence of strong occupational segregation” (p.1) between natives and
immigrants. This finding has recently been reaffirmed by Lehmer and Ludsteck
(2011), Brücker and Jahn (2011), Bartolucci (2013b), and Glitz (2012) who
concludes that “ethnic segregation [..] is endemic in the German labour
market” (p.15).444At the same time, these papers provide complementary
perspectives on the native-immigrant wage gap in Germany: descriptive Oaxaca-
Blinder decompositions (Velling (1995), Lehmer and Ludsteck (2011)), wage
setting (Brücker and Jahn (2011)), monopsonistic discrimination (Hirsch and
Jahn (2012)), while Bartolucci (2013b) provides an interpretation in terms of
taste-based discrimination. D’Amuri et al. (2010) pursue a different concern
and estimate the wage and employment effects of recent immigration in Western
Germany (and find little evidence for adverse effects on native wages and
employment levels). This segmentation is also consistent with the evidence of
strong occupational immobility we find in our data (which has also been
observed for other countries, e.g. by de Matos (2012) for Portugal).555The
segmentation assumption has also been imposed routinely in recent search-based
structural analyses of the gender wage gap. For instance, Flabbi (2010)
considers only whites possessing a college degree, Bowlus (1997) considers two
education groups, and Bartolucci (2013a) considers four sectors and two skill
groups. Our partition is finer as we also consider three age groups in
addition to our three occupation groups (and our estimates remain unbiased
should the true partition be such that some segments be aggregated).
For each occupation-age segment, we estimate using maximum likelihood the job
turnover parameters, the parameters characterising the reservation wage
distribution, and the firms’ productivity distribution. We find substantial
differences in Germany between natives and foreigners. The segment-specific
raw average log wage gaps in our data range from .09 to .45, the overall log
wage gap being .22, which is in line with reports in the literature for
Germany (e.g. Dustmann et al. (2010) report an unconditional average log wage
gap of .23, Hirsch and Jahn (2012) report a gap of .2, while Lehmer and
Ludsteck (2011) report predicted wage gaps ranging from .08 to .44 depending
on nationality). Turning to the qualitative implications of our model
estimates, we find that migrants experience job separations more often than
natives but also find jobs more quickly. However, the net effect is such that
migrants typically experience greater search frictions. The job turnover
parameters decline in age. Across all segments and nationality, transitions
into new jobs happen more quickly than transitions into unemployment. This
finding of migrants’ higher job separation and offer rates is consistent with
differences in employment protection; in particular, Sa (2011) reports that
migrants in Germany are much more likely than natives to work on temporary
contracts. As regards the reservation wage distribution, there are some
workers in all segments with high reservation wages who turn down new job
offers when wage offers are too low. However, migrant workers are less
demanding on average than natives.
Migrants receive wage offers that are lower than those for natives controlling
for the same productivity. This migrant effect is the largest for clerks and
service workers, and small for unskilled workers. In particular, the average
migrant effect for the skilled ranges between 12% and 15% of the average wage
gap, and for clerks and service workers the range is 23% to 39%. For all
occupation groups, the migrant effect declines across age groups. These
estimates imply that the largest part of the within-group native-migrant wage
gap is explained by differences in the productivity distribution (one
explanation for such productivity differences is advanced in de Matos (2012)).
At the same time, the migrant effect is significant in many segments, and, if
expressed in terms of the average segment-specific wage of natives, it is
found to be consistent with estimates of “unexplained wage differences”
reported in the literature for Germany based on standard Oaxaca-Blinder
decompositions (for instance, Lehmer and Ludsteck (2011) report a range from 4
to 17%) or complementary approaches (Hirsch and Jahn (2012) report 6% while
Bartolucci (2013b) suggests discrimination effects ranging between 7 and 17%).
Our counterfactual decomposition approach allows us to quantify the (marginal
and joint) roles of the underlying drivers of the migrant effect in terms of
labour market turnover parameters and behavioural differences captured by the
reservation wage distribution. We find that reducing the job separation rate
for migrants to that of natives typically leads to a large reduction in the
migrant effect. This is of interest to policy makers since this parameter is
targetable by e.g. deploying measures to improve migrants’ employment
protection.
This paper is organised as follows. In Section 2, we set out the model as well
as the estimation approach. A validation exercise, reported in the Appendix,
verifies that the estimation of the structural parameters works well. Section
2.4 introduces the migrant effect, the decomposition of the actual wage
differential, and the counterfactual scenarios in the context of the simulated
data (which are later re-examined in Section 5 with the real data). Section 3
describes the data used for the analysis. The estimation results are presented
in Section 4, and the resulting decompositions in Section 5. Section 6
concludes.
## 2 The Analytical Framework
The search model with wage-posting and on-the-job search has been described
and discussed extensively before in the literature. Therefore, only its most
salient features will be outlined. We use the extension of the Burdett and
Mortensen (1998) model, and the subsequent empirical generalisation and
implementation of van den Berg and Ridder (1998), due to Bontemps et al.
(1999). This extends the basic setting by introducing productivity
heterogeneity among firms, which improves the fit of the model to wage data,
and heterogeneity among workers in terms of the unobserved opportunity cost of
employment, which improves the fit to the unemployment duration data. As
discussed above, the latter is very plausible in the migration context against
the background of Germany’s institutional rules.
The labour market is partitioned into many segments, defined in our empirical
implementation by age, occupation and nationality. Each segment is considered
as a labour market for which the following model and estimation approach
applies. The structural parameters are of course allowed to vary across
segments, but for notational simplicity we suppress a segment index. This
segmentation assumption precludes individuals moving from one segment to
another, which is consistent with the evidence of occupational immobility in
Germany presented below and the external evidence discussed in the
Introduction. If the labour market is integrated over some stipulated
segments, then the estimates of the structural parameters should be the same
statistically; the segments can then be added to improve estimation
efficiency. In line with the segmentation hypothesis we find that the
estimated structural parameters differ across occupation-age-nationality
groups. We proceed to outline the model for one labour market segment.
### 2.1 The Model of a Labour Market Segment
The labour market segment is populated by a fixed continuum of workers with
measure $M$, and a fixed continuum of firms with measure normalised to one.
Firms differ in terms of (the marginal) productivity (of labour) $p$ with
distribution $\Gamma$. Unemployed workers differ in terms of their reservation
wages $b$ with distribution $H$.
At any point in time, a worker is either unemployed or employed, and searches
for jobs both off and on the job. Individuals draw offers by sampling firms
using a uniform sampling scheme. Jobs are terminated at the exogenous rate
$\delta$, and job offers arrive at the common rate $\lambda$ irrespective of
the worker’s state. This is a restrictive assumption but necessary for
identification.666 This assumption yields, for the unemployed, a simple
solution for the opportunity cost of employment: it is simply equal to $b$. If
job offer arrival rates were to differ, Mortensen and Neumann (1988) show that
this opportunity cost would be an intractable function of all the primitives
of the model, leading to feedback to workers’ optimal strategies from wages
and firm behaviour. Let $k=\lambda/\delta$.
Job offers are, of course, unobservable to the econometrician. The job offer
distribution is denoted by $F$, whereas the observable wage or earnings
distribution (i.e. of accepted wages) is denoted by $G$. Let
$\left[\underline{w},\overline{w}\right]$ denote the support of $F$, and, for
notational convenience, $\overline{F}=\left[1-F\right]$. $F$ is related to $G$
through an equilibrium condition implied by the theoretical structure. Firms
post wages and there is no bargaining.777For an analysis of wage determination
in the presence of heterogeneity, search on-the-job, and strategic wage
bargaining, see Cahuc et al. (2006). They find no significant bargaining power
for intermediate and low skilled workers in France.
Workers are risk neutral and maximise their expected steady state discounted
future income. Their optimal strategy has the reservation wage property: an
employed individual moves to a new employer if the offered wage exceeds the
current wage (so the model does not allow for wage cuts); an unemployed
individual accepts a new job if the offer exceeds $b$, and otherwise rejects
the offer and remains unemployed. On-the-job search thus generates further ex-
post heterogeneity in reservation wages.
In steady-state equilibrium, the flows of workers into and out of the
unemployment pool are equal, which determines the unemployment rate $u$.
Consider the stock of employed workers who earn a wage less than or equal to
$w$. Two sources constitute the outflow from this stock, namely: (i) exogenous
job separations at rate $\delta$ and subsequent transits into unemployment,
and (ii) wage upgrading as employed workers move to poaching firms. The
combined outflow is thus $(1-u)G(w)(\delta+\lambda\overline{F}(w))$. The flow
into this stock consists of unemployed individuals who receive wage offers
above their reservation wage. Conditional on $b$, the probability of this
event is $u\lambda[F(w)-F(b)]$. The marginal inflow is obtained by integrating
up to $w$ over the distribution of $b$ in the stock of the unemployed.
Denoting the latter by $H_{u}$, the steady state equation for the labour
market yields the relationship between $H_{u}$ and $H$, namely
$uH_{u}(b)=\int_{-\infty}^{b}[1+k\overline{F}(x)]^{-1}dH(x)$.
Equating inflows and outflows relates the wage offer distribution $F$ to the
realised wage distribution $G$. To be precise, Bontemps et al. (1999,
Proposition 2) show that the unemployment rate $u$ and the actual wage
distribution $G$ satisfy
$u=\left[\frac{1}{1+k}H\left(\underline{w}\right)+\int_{\underline{w}}^{\overline{w}}\frac{1}{1+k\overline{F}\left(x\right)}dH\left(x\right)\right]+\left[1-H\left(\overline{w}\right)\right]$
(1)
$G\left(w\right)=\frac{H\left(w\right)-\left[1+k\overline{F}\left(w\right)\right]\left[\frac{1}{1+k}H\left(\underline{w}\right)+\int_{\underline{w}}^{w}\frac{1}{1+k\overline{F}\left(x\right)}dH\left(x\right)\right]}{\left[1+k\overline{F}\left(w\right)\right]\left(1-u\right)}.$
(2)
Risk neutral firms have constant-returns-to-scale technologies, and post wages
that maximise steady state profit flows, the profit per worker being $p-w$.
Firms do not observe the reservation wage of a potential employee. In
equilibrium, firms offer wages to workers that are smaller than their
productivity level, so firms have some monopsony power. Bontemps et al. (1999,
Proposition 9) show that in equilibrium there exists an increasing function
$K$ which maps the productivity distribution $\Gamma$ into the wage offer
distribution $F$, so that the wage offer satisfies $w=K(p)$ with
$K\left(p\right)=p-\left[\frac{\underline{p}-\underline{w}}{\left(1+k\right)^{2}}H\left(\underline{w}\right)+\int_{\underline{p}}^{p}\frac{H\left(K\left(x\right)\right)}{1+k\left[1-\Gamma\left(x\right)\right]^{2}}dx\right]\frac{\left[1+k\left[1-\Gamma\left(p\right)\right]\right]^{2}}{H\left(K\left(p\right)\right)}$
(3)
and $F\left(w\right)=\Gamma\left(K^{-1}\left(w\right)\right)$. Hence given the
frictional parameter $k$, the reservation wage distribution $H$ and the
productivity distribution $\Gamma$, equation (3) yields the wage offer
distribution $F$, which then via (1) yields the equilibrium unemployment rate
and through (2) the actual wage distribution $G$.
Our dataset does not include measures of firm productivity but, of course,
extensive wage data. Using expressions of the key quantities in terms of the
actual wage density $g$, the productivity distribution $\Gamma$ becomes
estimable. In particular, it can be shown that
$\left(1-u\right)=\frac{k}{\left(1+k\right)\int_{\underline{w}}^{\overline{w}}\frac{g\left(t\right)}{H\left(t\right)}dt},$
(4) $\displaystyle\frac{1}{\left[1+k\overline{F}\left(w\right)\right]}$
$\displaystyle=$
$\displaystyle\left(1-u\right)\int_{\underline{w}}^{w}\frac{g\left(t\right)}{H\left(t\right)}dt+\frac{1}{\left[1+k\right]}.$
(5)
Equation (4) follows from (5) with $w=\overline{w}$. The equilibrium
productivity levels are
$p=K^{-1}\left(w\right)=w+\frac{H\left(w\right)}{2\left(1-u\right)g\left(w\right)\left[1+k\overline{F}\left(w\right)\right]+h\left(w\right)}.$
(6)
### 2.2 Identification
We seek to estimate this model using data by labour market segment on
employment and unemployment durations, as well as data on wages and accepted
wage offers. These data are sufficient to identify888Eckstein and van den Berg
(2007) discuss identification issues in empirical search models more
generally. the structural parameters, once the reservation wage distribution
is parametrised. We assume that $H$ is a normal distribution with unknown
location and scale parameters, $(\mu,\sigma)\equiv\theta$. Since arrivals of
job offers and separations are assumed to follow Poisson processes, sojourn
times are exponentially distributed.
In particular, the wage data identify the wage distribution $G$, and the
minimum and the maximum of the observed wages identify the infimum
$\underline{w}$ and the supremum $\overline{w}$ of the wage offer
distribution. The steady state flow equations in form of (4) and (5) then
identify the wage offer distribution $F$ given $\lambda/\delta$ and
$H(.;\theta)$, which yield the productivity distribution $\Gamma$ via (3). The
job separation rate is identified from job durations ending in a transition to
unemployment, as these are exponential variates with parameter $\delta$, the
mean duration being $\delta^{-1}$. Job durations ending in a transition to
another job with wage $w$ are exponential with parameter $\lambda\bar{F}(w)$.
Together with unemployment durations ending in a transition to a job with wage
$w$ these identify the remaining parameters $\lambda$ and $\theta$. Since the
reservation wage is unobservable, the marginal unemployment durations are
mixtures of exponentials, $\Pr\\{T_{u}\leq t|b\leq
w\\}=1-\int_{-\infty}^{w}\exp(-\lambda\bar{F}(b)t)dH_{u}(b;\theta|b\leq w)$.
Absent such mixing, when $H$ is degenerate and all agents accept all wage
offers above the common reservation wage, transitions to a new job from each
labour market state would permit separate identification of the job offer
arrival rates, and thus would give rise to testable overidentification
restrictions. In the presence of unobservable heterogeneity captured by $H$,
overidentification restrictions only arise with additional data that would
permit, for instance, an independent estimation of the wage offer distribution
(see e.g. Christensen et al. (2005) for such an approach).
### 2.3 Maximum Likelihood Contributions for Labour Market Segments
The preceding constructive identification argument suggests that we can
estimate the structural parameters using maximum likelihood on our data on
unemployment and employment durations and wages. The likelihood contributions
we consider in detail next differ slightly from those in Bontemps et al.
(1999) since our data are flow and not stock samples. The validation exercise
reported in B verifies the good performance of our estimation procedure on
artificial data. The density of accepted wages, and thus $G$, is estimated
using kernel methods, and enters all likelihoods as a nuisance parameter.
Consider first the likelihood contributions of unemployed agents. Since the
unemployment rate is a function of the model parameters, it needs to enter the
sampling plan. In equilibrium, the probability of encountering an unemployed
individual is given by (4). Since the reservation wage $b$ is unobservable, it
needs to be integrated out. We distinguish between individuals for whom
$b\leq\underline{w}$ as they accept all job offers, a mass of
$H(\underline{w})$, and those for whom $b>\underline{w}$ as they reject offers
below $b$. Recall that $F\left(\underline{w}\right)=0$, and we assume that all
individuals included in our sample would accept at least one wage offer
$w\in[\underline{w},\overline{w}]$. This implies that the $\sup$ of $H$ is
lower than the $\sup$ of $F$, $\overline{b}\leq\overline{w}$, so this
specification does not take into account cases of permanently unemployed
individuals. Conditional on $b$, the distribution of unemployment durations in
our flow sample is exponential with parameter
$\lambda\overline{F}\left(b\right)$. The accepted wage, $w$, is a realisation
of the wage offer distribution truncated at $b$:
$f\left(w\right)/\overline{F}\left(b\right)$. The likelihood contribution of
an unemployed $L_{u}$ is thus, having substituted out $u$,
$L_{u}\left(\lambda,\delta,\theta\right)=\lambda^{\left(1-d_{r}\right)}\exp\left(-\lambda
t\right)\frac{H\left(\underline{w}\right)}{1+k}\left[f\left(w\right)\right]^{\left(1-d_{r}\right)}+$
$+\int_{\underline{w}}^{w}\left\\{\left[\lambda\overline{F}\left(b\right)\right]^{\left(1-d_{r}\right)}\exp[-\lambda\overline{F}\left(b\right)t]\left[\frac{f\left(w\right)}{\overline{F}\left(b\right)}\right]^{\left(1-d_{r}\right)}\frac{1}{\left[1+k\overline{F}\left(b\right)\right]}\right\\}dH(b),$
(7)
where $d_{r}$ is a dummy variable equal to one if the spell is right-censored
(the only relevant censoring in our data). In this case it is only known that
the unemployment duration exceeds $t$.
We turn to the likelihood contributions of employed workers, denoted by
$L_{e}$. The probability of sampling an employed individual receiving a wage
$w$ is $\left(1-u\right)g\left(w\right)$. We have further data on the job
duration and the exit state. Let $v$ be a dummy variable equal to one if the
destination of an employment spell is unemployment, and zero if the
destination is another job. We have two competing risks: Exits to unemployment
occur with probability
$\delta/\left[\delta+\lambda\overline{F}\left(w\right)\right]$ and exits to
higher paying jobs occur with probability
$\lambda\overline{F}\left(w\right)/\left[\delta+\lambda\overline{F}\left(w\right)\right]$.
Conditional on being employed with wage $w$, the job duration has an
exponential distribution with parameter
$\left[\delta+\lambda\overline{F}\left(w\right)\right]$. If a transit to
unemployment is observed at duration $t$, this implies that the duration of
the other latent risk factor exceeds $t$, the joint density factorises, and we
have $\delta\exp(-\delta t)\exp(-\lambda\overline{F}\left(w\right))$.
Therefore
$L_{e}\left(\lambda,\delta,\theta\right)=\left(1-u\right)g\left(w\right)\exp\left\\{-\left[\delta+\lambda\overline{F}\left(w\right)\right]t\right\\}\times\left\\{\delta^{v}\left[\lambda\overline{F}\left(w\right)\right]^{\left(1-v\right)}\right\\}^{\left(1-d_{r}\right)},$
(8)
where $(1-u)$ is given by equation (4). If an employment spell is right-
censored, indicated by $d_{r}$, we only know that the job duration exceeds
$t$.
### 2.4 Migrants, Natives, Wage Differentials and the Migrant Effect:
Concepts and Simulated Data
We develop an illustrative example in order to introduce our key concepts.
Consider two labour market segments, one occupied by natives (N) and the other
by immigrants (F). Workers in either segment exhibit the same observable
characteristics (in our empirical application below we consider the same skill
and age group). We calibrate the two segments (in line with the empirical
results) as follows: the job turnover parameters of migrants are assumed to be
higher than those of natives, $\delta_{F}=.016>.005=\delta_{N}$ and
$\lambda_{F}=.13>.07=\lambda_{N}$, while natives have higher mean reservation
wages, $\mu_{F}=45<60=\mu_{N}$. The productivity distribution in the segment
for natives is assumed to first order stochastically dominates that of
migrants: $\Gamma_{F}(p)=1-(\underline{p}_{F}/p)^{\alpha}$ and
$\Gamma_{N}(p)=1-(\underline{p}_{N}/p)^{\alpha}$ with $\alpha=2.1$,
$\underline{p}_{F}=40$, and $\underline{p}_{N}=50$. The validation exercise
reported in B discusses the estimation results.
Figure 1: Wage offer curves for natives and migrants, and the “migrant
effect”.
For this economy, the aggregate wage gap is substantial (equal to 32.02), but
differences in the productivity distributions are likely to play an important
role (recall the discussion in the Introduction). Figure 1 Panel A depicts the
resulting wage offers given by (3) as functions of productivity. These enable
us to consider a component of the wage gap which we label “migrant effect”,
depicted in Panel B, being the difference in wage offers between similar
native and immigrant workers in firms of the same productivity:
$w_{N}(p)-w_{F}(p)$. This effect is of interest since we thus control for
firm-level differences as measured by their productivities.
This concept of the migrant effect suggests to decompose the aggregate wage
differential999For a decomposition of wage differentials in a reduced form
setting, see Dustmann and Theodoropoulos (2010). Note that their decomposition
considers, as we do, the wage offer function, but their empirical approach
does not recover it from the data. between migrants and natives,
$\int_{A}w_{N}(p)d\Gamma_{N}(p)-\int_{A}w_{F}(p)d\Gamma_{F}(p)$, into the
aggregate migrant effect and a weighted difference between firm productivities
(where $A$ denotes the intersection of the supports of the productivity
distributions). Solving for the aggregate migrant effect, we thus have
$\displaystyle\int_{A}\left[w_{N}(p)-w_{F}(p)\right]d\Gamma_{N}(p)$
$\displaystyle=$
$\displaystyle\int_{A}w_{N}(p)d\Gamma_{N}(p)-\int_{A}w_{F}(p)d\Gamma_{F}(p)$
$\displaystyle-$
$\displaystyle\int_{A}w_{F}(p)d\left[\Gamma_{N}(p)-\Gamma_{F}(p)\right].$
We briefly comment on the relationship between the migrant effect and the
concept of monopsonistic discrimination, as examined in e.g. Hirsch and Jahn
(2012). The latter is measured by these authors indirectly from a search-model
inspired decomposition of the long run wage elasticity of labour supply using
reduced-form job separation models that are estimated separately on data for
migrants and natives. In our model, greater monopsony power of firms (measured
by the absolute or relative distance between productivity, i.e. the 45 degree
line, and wages as illustrated in Figure 1.A) in the migrant segment gives
rise to the migrant effect. Our approach enables us to go beyond measuring the
migrant effect, as we explain it within a common framework in terms of the
relative importance of differences in the job turnover parameters and
behavioural differences induced by differences in reservation wages. In
particular, a closer inspection of (3) shows that the wage offers are
complicated functions of these structural parameters,
$w_{i}(p|\underline{p}_{i},\alpha_{i},\mu_{i},\sigma_{i},\lambda_{i},\delta_{i})$
for $i\in\\{N,F\\}$.
#### 2.4.1 Counterfactual Wage Decompositions
In order to identify the principal drivers of the migrant effect, and to
conduct policy experiments, we consider next a second decomposition of the
wage gap based on counterfactuals. In particular, we ask: what would be the
migrant effect and the wage differential if one group is imputed
counterfactually parameter values of the other group? For instance, choosing
natives as the reference group and equalising counterfactually the reservation
wage distribution parameters $(\mu,\sigma)$, the counterfactual migrant effect
is, using (2.4),
$\displaystyle\int_{A}[w_{N}(p|\underline{p}_{N},\alpha_{N},\mu_{N},\sigma_{N},\lambda_{N},\delta_{N})-w_{F}(p|\underline{p}_{F},\alpha_{F},\mu_{N},\sigma_{N},\lambda_{F},\delta_{F})]d\Gamma_{N}(p)$
$\displaystyle=$
$\displaystyle\int_{A}w_{N}(p|\underline{p}_{N},\alpha_{N},\mu_{N},\sigma_{N},\lambda_{N},\delta_{N})d\Gamma_{N}(p)-\int_{A}w_{F}(p|\underline{p}_{F},\alpha_{F},\mu_{N},\sigma_{N},\lambda_{F},\delta_{F})d\Gamma_{F}(p)$
$\displaystyle-$
$\displaystyle\int_{A}w_{F}(p|\underline{p}_{F},\alpha_{F},\mu_{N},\sigma_{N},\lambda_{F},\delta_{F})d[\Gamma_{N}(p)-\Gamma_{F}(p)]$
with $\Gamma_{i}(p)=\Gamma_{i}(p|\underline{p}_{i},\alpha_{i})$ for
$i\in\\{N,F\\}$.
Table 1: Counterfactual decompositions of the wage differential using natives
as the reference group.
| Counterfactually | Remaining | Wage | Migrant
---|---|---|---|---
| equalised para. | differing para. | differential | effect
(1) | | $\underline{p},\alpha,\mu,\sigma,\lambda,\delta$ | 32.022 | 6.825
(2) | $\mu,\sigma$ | $\underline{p},\alpha,\lambda,\delta$ | 30.096 | 3.747
(3) | $\delta$ | $\underline{p},\alpha,\mu,\sigma,\lambda$ | 28.973 | 1.954
(4) | $\lambda$ | $\underline{p},\alpha,\mu,\sigma,\delta$ | 34.029 | 10.032
(5) | $\mu,\sigma,\delta$ | $\underline{p},\alpha,\lambda$ | 27.423 | -0.524
(6) | $\alpha,\mu,\lambda$ | $\underline{p},\alpha,\delta$ | 31.694 | 6.300
(7) | $\lambda,\delta$ | $\underline{p},\alpha,\mu,\sigma$ | 30.459 | 4.328
(8) | $\mu,\sigma,\lambda,\delta$ | $\underline{p},\alpha$ | 28.758 | 1.610
(9) | $\underline{p},\alpha$ | $\mu,\sigma,\lambda,\delta$ | | 4.904
(10) | $\underline{p},\alpha,\mu,\sigma$ | $\lambda,\delta$ | | 1.932
(11) | $\underline{p},\alpha,\delta$ | $\mu,\sigma,\lambda$ | | 0.750
(12) | $\underline{p},\alpha,\lambda$ | $\mu,\sigma,\delta$ | | 7.814
(13) | $\underline{p},\alpha,\mu,\sigma,\delta$ | $\lambda$ | | -1.842
(14) | $\underline{p},\alpha,\mu,\sigma,\lambda$ | $\delta$ | | 4.400
(15) | $\underline{p},\alpha,\lambda,\delta$ | $\mu,\sigma$ | | 2.741
Notes: Based on the DGP given in Appendix Table 16, and the decomposition of
equation (2.4.1). Rows 9$+$: the wage differential equals the migrant effect
because the productivity distributions are the same.
Table 1 collects the exhaustive list of possible counterfactual experiments,
and the resulting quantifications of both the counterfactual migrant effect
and wage differential (the first term on the right hand-side of (2.4.1)). The
reference group consists of natives. In column 1 we list the parameters we
counterfactually equalise, so $(\mu,\sigma)$ in row and experiment 2 is a
shorthand for $\mu_{F}=\mu_{N}$ and $\sigma_{F}=\sigma_{N}$. The residual
parameters enumerated in column 2 constitute thus the sources of the remaining
wage differences. In the first experiment, reported in row 1, no parameters
are equalised, hence the reported results are based on actual wages (i.e. we
use the actual wage decomposition (2.4)). In experiment 9 and later, we
equalise the two parameters of the productivity distribution, $\underline{p}$
and $\alpha$ (Bartolucci (2013a) labels such differences in the productivity
distribution parameters “segregation”). This nils the last term in equation
(2.4.1), so migrant effect and wage differential are equalised. In all
experiments we use simulated data based on the DGP of Appendix Table 16 but
the results reported next are in line with our data-based empirical results
for the comparative statics and policy experiments reported in Section 5.2.
The actual migrant effect of 6.8, reported in experiment 1, is substantial,
about 21% of the wage differential. At the same time this implies that the
largest contribution to the native-migrant wage gap is made by the differences
between the productivity distributions. Turning to the drivers of the migrant
effect, experiments 13-15 consider the marginal roles of $\delta$, $\lambda$,
and $(\mu,\sigma)$. Recalling that $\lambda_{F}>\lambda_{N}$ explains the
negative sign in experiment 13. Also note that $\delta_{F}>\delta_{N}$, and
$\mu_{F}<\mu_{N}$ while $\sigma_{F}=\sigma_{N}$. Experiment 14 suggests that
the difference in the separation rates plays a large quantitative role in the
determination of the migrant effect, the latter being 4.4; the complementary
insight is that, by experiment 3, equalising the job separation rates reduces
the migrant effect to 29% of its former size. The differences in mean
reservation wages, considered in experiment 15, leads to a smaller migrant
effect of 2.7. The joint effect of $\delta$ and $(\mu,\sigma)$, reported in
experiment 12, equals 7.8, and is slightly larger than the sum of the two
marginal effects. We defer discussing the policy implications of these results
to Section 5.2 as these are similar to those based on our empirical results.
## 3 The Data
The empirical analysis is based on the 2% subsample of the German employment
register provided by the Institute of Employment Research, known as IABS
(75-04 distribution). For a detailed description of the dataset, see Bender et
al. (2000). This large administrative dataset for Germany, covering the period
1975-2004 consists of mandatory notifications made by employers to social
security agencies. These notifications are made on behalf of workers,
employees, and trainees who pay social security contributions. This means that
self-employed individuals, civil servants, and workers in marginal employment
are not included. Notifications are made at the beginning and at the end of an
employment or unemployment spell. Information on individuals not experiencing
transitions during a calendar year is updated by means of an annual report.
Hence, we are able to use a flow sample in our empirical analysis.
Apart from wages, transfer payments, and spell markers, the dataset contains
some standard demographic measures, including nationality, as well as
occupation and firm markers. The education variable is not used since its
problems, particularly in the migrant context, are well-known and skills are
better measured by the occupation (see Fitzenberger et al. (2006) for a
detailed discussion; we do not use the suggested imputations since the
education variable for migrants, when observed, is likely to be of poor
quality, as discussed in Brücker and Jahn (2011, p. 296 point (ix)) and Lehmer
and Ludsteck (2011, p. 900)). Wage records in the IABS are top coded at the
social security contribution ceiling. However, this ceiling is not binding for
our population of interest, namely individuals (natives and foreigners) in low
and middle skill occupations. We use real wages in 1995 prices. The
occupational information is provided in extensive (three digit codes) but non-
standard form. We therefore map this coding into 10 major groups based on the
International Standard Classification of Occupations (ISCO-88). The Data
Appendix provides some details. Since immigration is known to be predominantly
low skilled, we select from these 10 groups 3 low and middle skilled
occupations, namely (1) unskilled blue-collar workers, (2) clerks and low-
service workers, and (3) skilled blue-collar workers.
The data allows us to distinguish between three labour market states:
employed, recipient of transfer payments (i.e. unemployment benefits,
unemployment assistance and income maintenance during participation in
training programs) and out of sample. Unfortunately, none of the two last
categories corresponds exactly to the economic concept of unemployment. This
issue is discussed in several studies, see e.g. Fitzenberger and Wilke (2010).
For example, participants in a training program are transfer payment
recipients despite being in employment (they are considered unemployed from an
administrative point of view), while individuals that are registered
unemployed but are no longer entitled to receive benefits appear to be out of
the labour force. Therefore, the dataset provides a representative sample of
those employed and covered by the social security system, but somewhat mis-
represents those in the state of unemployment. For our purposes, all
individuals who are out of sample between two different spells are classified
as unemployed, so only two labour market states are considered: unemployment
and employment. The definition of unemployment used in our analysis is
therefore somewhat broad: we assume that unemployment is proxied by non-
employment, strictly speaking non-employment is an upper-bound for
unemployment.
Nationality is included as a binary variable indicating whether an individual
is German or a foreign national. German nationality is usually conferred by
descent, and not by place of birth. The data set does not report place of
birth. Given this coding practice, some young foreign nationals might be born
and raised in Germany. At the same time, ethnic Germans who immigrated from
the former Soviet Union after the fall of the Berlin Wall will be classified
as German, although they usually speak little German and have low skills.
However, Dustmann et al. (2010) have argued that the former issue is
ignorable, and we address the second by repeating the estimation using the
subsample of individuals that were present in the data before the fall of the
Berlin Wall, see the analysis in Section 4.5.3.
### 3.1 The Sample
The data used in our empirical analysis is restricted to male full-time
workers aged 25 to 55 years old residing in West-Germany (East Germany is
excluded because of the peculiar transition processes taking place in the wake
of unification). This sample is grouped into cells by occupation, nationality,
and age. We define three age groups (25-30, 30-40, and 40-55) to proxy for
potential experience. The aim of the grouping is to arrive at cells in which
individuals are fairly homogeneous, and which are sufficiently large for the
subsequent econometric investigation.
Table 2: Occupational Immobility: Share of Stayers by Segment | Age group | Natives | Foreigners
---|---|---|---
Unskilled | | 89.52% | | 88.27% |
| Twenties | | 85.72% | | 85.45%
| Thirties | | 88.03% | | 88.56%
| Fourtyplus | | 92.54% | | 92.38%
Clerks | | 90.06% | | 88.52% |
| Twenties | | 88.03% | | 87.33%
| Thirties | | 87.44% | | 89.00%
| Fourtyplus | | 91.82% | | 91.89%
Skilled | | 92.48% | | 92.56% |
| Twenties | | 90.35% | | 91.03%
| Thirties | | 90.22% | | 92.43%
| Fourtyplus | | 94.26% | | 95.25%
The model is estimated using a flow sample of employed and unemployed
individuals, who experienced a transition from their original state within the
period 1995-2000. We consider the first such transition, and any subsequent
transitions are ignored. For all these individuals we can determine the
beginning of their original state, so that all durations are complete. The
only exception is constituted by a small number of individuals who disappear
from the dataset in the period 1995-2000, in which case their durations are
considered censored. We note that the period 1995-2000 was a period of fairly
stable growth (around 2%, with SD=.007) and unemployment (around 8%, with
SD=.007). Focussing on this stable period reduces the scope for biases arising
from asymmetric responses of natives and foreigners to the business cycle.
Foreigners in our sample are predominantly low skilled: 94% of the population
of foreigners are included in our three occupational groups, while the
corresponding number for natives is approximately 86%. The remainder
occupational category is the highly skilled, which we have excluded because of
their small share in the population of migrants (moreover, their earnings are
excessively top-coded). Table 2 considers the occupational immobility by
labour market segment. It is evident that occupational mobility is small, as
most workers remain in the same class. This gives further support to our
segmentation hypothesis, and such occupational immobility has also been found
for other countries (e.g. by de Matos (2012) for Portugal).
Table 3: Descriptives for the transition data.
| | Natives | Foreigners
---|---|---|---
Age | Transitions | Services | Unskilled | Skilled | Services | Unskilled | Skilled
25-30 | All | 8060 | 5097 | 11939 | 1887 | 2347 | 3023
from E | 6088 | 3085 | 8450 | 1438 | 1670 | 2155
E$\rightarrow$ U | 2132 | 1764 | 4418 | 718 | 997 | 1225
E$\rightarrow$ E | 3432 | 1037 | 3562 | 373 | 351 | 550
from U | 1972 | 2012 | 3489 | 449 | 677 | 868
U$\rightarrow$ E | 1879 | 1932 | 3275 | 431 | 637 | 795
$E_{censored}$ | 524 | 284 | 470 | 347 | 322 | 380
$U_{censored}$ | 93 | 80 | 214 | 18 | 40 | 73
30-40 | All | 12800 | 7748 | 15381 | 2074 | 2752 | 3681
from E | 10723 | 5506 | 12448 | 1637 | 2067 | 2830
E$\rightarrow$ U | 2988 | 2644 | 5284 | 735 | 1128 | 1451
E$\rightarrow$ E | 6717 | 2400 | 6157 | 453 | 477 | 795
from U | 2077 | 2242 | 2933 | 437 | 685 | 851
U$\rightarrow$ E | 1853 | 2055 | 2601 | 393 | 619 | 749
$E_{censored}$ | 1018 | 462 | 1007 | 449 | 462 | 584
$U_{censored}$ | 224 | 187 | 332 | 44 | 66 | 102
40-55 | All | 16900 | 12770 | 24530 | 1494 | 2938 | 5004
from E | 13912 | 9399 | 19127 | 1146 | 2090 | 3726
E$\rightarrow$ U | 4538 | 4467 | 8973 | 505 | 1101 | 2019
E$\rightarrow$ E | 6671 | 3206 | 6848 | 329 | 513 | 1024
from U | 2988 | 3371 | 5403 | 348 | 848 | 1278
U$\rightarrow$ E | 1554 | 2013 | 2130 | 244 | 540 | 582
$E_{censored}$ | 2703 | 1726 | 3306 | 312 | 476 | 683
$U_{censored}$ | 1434 | 1358 | 3273 | 104 | 308 | 696
Notes: “Censoring” refers to a drop out from the administrative register.
Table 3 summarises the labour market transitions for all nationality-age-
occupation cells observed in our flow data. For both natives and foreigners,
we observe many more transitions from employment than from unemployment.
However, for natives, the majority of transitions from employment are to
another job, whereas for the majority of foreigners the destination is
unemployment. Hence, in terms of the structural parameters, we expect higher
separation rates for foreigners, $\delta_{F}>\delta_{N}$. The duration data
for the unemployed, examined briefly in the next subsection, suggests that
foreigners exit more quickly, so that we expect $\lambda_{F}>\lambda_{N}$ at
least for this group.
Turning to the wage data, Table 4 reports for each labour market segment the
mean and standard deviation of wages (measured by daily gross wages in 1995
DM), as well as the average log wage gap,
$\Delta\log(w)\equiv\log(w_{N})-\log(w_{F})$. Natives receive substantially
higher mean wages than foreigners across all occupation groups. The segment-
specific raw average log wage gaps in our data range from .09 to .45. The
overall log wage gap of .22 is in line with reports in the literature for
Germany (e.g. Dustmann et al. (2010) report an unconditional average log wage
gap of .23, Hirsch and Jahn (2012) report a gap of .2, while Lehmer and
Ludsteck (2011) report predicted wage gaps ranging from .08 to .44). The three
occupational groups can be partially ordered in terms of mean wages: mean
wages for the skilled exceed those for the unskilled for all age groups and
across nationalities. Foreign clerks and low-service workers assume an
intermediate position, but mean wages of natives in this group can exceed
those for skilled workers.
Table 4: The average wage gap in the transition data by labour market segment.
| | Services | Unskilled | Skilled
---|---|---|---|---
Age | Wages | Native | Migrant | Native | Migrant | Native | Migrant
25-30 | mean | 122.36 | 88.94 | 107.77 | 92.54 | 124.74 | 111.07
| sd | 41.86 | 44.15 | 37.68 | 36.09 | 29.94 | 35.21
| $\Delta\log(w)$ | .32 | .15 | .11
30-40 | mean | 156.35 | 99.38 | 120.94 | 97.99 | 135.79 | 116.65
| sd | 51.22 | 55.02 | 38.24 | 36.61 | 32.04 | 36.22
| $\Delta\log(w)$ | .45 | .21 | .15
40-55 | mean | 158.17 | 112.74 | 125.05 | 107.49 | 138.29 | 126.20
| sd | 48.09 | 56.81 | 36.71 | 36.89 | 33.29 | 33.50
| $\Delta\log(w)$ | .33 | .15 | .09
Notes: $\Delta\log(w)\equiv\log(w_{N})-\log(w_{F})$. The overall log wage gap
is .22. Wage dating: for transitions from employment (E$\rightarrow$ {U,E}),
these are the last earned wages in this state, for transition out of
unemployment (U$\rightarrow$ E) these are the first wages earned in the new
job.
Figure 2: Estimates of the density of accepted wages by labour market
segments.
Notes: Natives (solid lines) v. foreigners (dashed lines).
Rather than only restricting attention to the mean wage, Figure 2 depicts the
kernel estimates of the realised wage densities (the solid lines refer to
natives). The most pronounced distributional difference exist for the semi-
skilled workers (clerks and service workers), and the differences persist
across age groups. By contrast, for all other occupations, the differences
decrease in age. The density estimates also exhibit “blips” in the far left
tails of the wage densities. This bimodality leads to problems in the
estimation of the model, manifesting themselves by the occurrence of spikes in
the estimated productivity density. We overcome this issue by truncating the
wage distributions at the 5% percentile, which is a common cut-off in the
literature (see e.g. Bowlus (1997) or Flabbi (2010)). The estimation of the
reservation wage distribution is, of coure, likely to be sensitive to the
choice of the cut-off point. We therefore explore the robustness of our
parameter estimates below in Section 4.5, and find that the frictional
parameters are fairly stable, while $\mu$ increases usually somewhat as the
truncation increases from 3% to 7%.
#### 3.1.1 Reduced Form Estimates: The Importance of Unobservable
Heterogeneity
Before embarking on the estimation of the model, we first explore
descriptively whether there is scope for unobserved heterogeneity to play a
role in explaining unemployment durations. To this end, we estimate standard
reduced-form proportional hazard (PH) and mixed proportional hazard (MPH)
models for the unemployed, controlling incrementally for duration dependence
and unobserved heterogeneity. Since the conditional unemployment durations in
the structural model are exponential with parameter $\lambda\bar{F}(b)$ and
the marginal durations are a mixture of such exponentials, we first estimate
an exponential PH model, and then allow for duration dependence by estimating
a Weibull specification. As the latter confounds dynamic sorting driven by
unobservable heterogeneity and genuine duration dependence (see e.g. van den
Berg (2001)), we then estimate MPH models using the common gamma frailty
(assumed to be independent of the covariates). Note, however, that these
reduced-form parameters do not identify the parameters of the structural model
as the former are complicated functions of the latter. In all models we
condition on interactions between age and occupational groups in order to
mirror our subsequent structural analysis of the corresponding labour market
segments.
Table 5 reports the results. Across all models the migrant dummy is positive
throughout, so that their job offer arrival rates exceed those of natives. The
Weibull PH model suggests the presence of duration dependence, but the MPH
reveals this to be caused by dynamic sorting: once unobservable heterogeneity
is controlled for, the Weibull parameter does not differ statistically from 1.
Hence Weibull and exponential MPH models yield similar coefficient estimates.
This inferred absence of duration dependence is consistent with the structural
model, as it cannot generate genuine duration dependence but does yield
dynamic sorting through unobserved heterogeneity in reservation wages.
Table 5: Reduced-form unemployment duration models
| | (1) | (2) | (3) | (4)
---|---|---|---|---|---
| | Exponential | Weibull | Weibull§ | Exponential§
Migrant | .087∗∗∗ | .069∗∗∗ | .049∗ | 0.046∗
| | (.020) | (.020) | (.027) | (.027)
Clerks $\times$ Twenties | 1.368∗∗∗ | 1.212∗∗∗ | 1.409∗∗∗ | 1.431∗∗∗
| | (.035) | (.036) | (.047) | (.047)
Clerks $\times$ Thirties | 1.059∗∗∗ | .984∗∗∗ | 1.197∗∗∗ | 1.217∗∗∗
| | (.034) | (.035) | (.047) | (.046)
Clerks $\times$ Fourtyplus | .037 | .011 | -0.005 | -0.006
| | (.037) | (.037) | (.045) | (.046)
Skilled $\times$ Twenties | 1.500∗∗∗ | 1.327∗∗∗ | 1.602∗∗∗ | 1.631∗∗∗
| | (.031) | (.031) | (.043) | (.041)
Skilled $\times$ Thirties | .914∗∗∗ | .867∗∗∗ | 1.155∗∗∗ | 1.182∗∗∗
| | (.032) | (.033) | (.045) | (.044)
Skilled$\times$ Fourtyplus | -0.429∗∗∗ | -0.451∗∗∗ | -0.531∗∗∗ | -0.539∗∗∗
| | (.034) | (.034) | (.041) | (.042)
Unskilled$\times$ Twenties | 1.297∗∗∗ | 1.153∗∗∗ | 1.392∗∗∗ | 1.417∗∗∗
| | (.035) | (.035) | (.047) | (.046)
Unskilled $\times$ Thirties | .864∗∗∗ | .828∗∗∗ | 1.062∗∗∗ | 1.083∗∗∗
| | (.035) | (.034) | (.046) | (.046)
duration | $ln(\alpha)$ | | -.222∗∗∗ | -.023∗ |
dependence | | (.006) | (.012) |
unobserved | $\theta$ | | | .702∗∗∗ | .770∗∗∗
heterogeneity | | | ( .042) | (.024)
* 1.
Notes. Standard errors in parentheses, ${}^{*}(p<0.1)$, ${}^{***}(p<0.001)$.
Reference groups: Unskilled $\times$ Fourtyplus. §Frailty is Gamma
distributed.
## 4 Estimation Results
We proceed to estimate the structural parameters of the model, i.e. the job
offer arrival rate, $\lambda$, the match destruction rate, $\delta$, and the
parameters of the distribution of workers’ reservation values, $(\mu,\sigma)$,
as well as the density of firms’ productivity in each segment. Each occupation
group is considered in turn, and we segment for each occupation the labour
market further by age and nationality. The average migrant effects and the
wage decompositions are then quantified in detail in Section 5 below.
Table 6: Structural parameter estimates: Unskilled blue collar workers
Age | Nation. | $\mu$ | $\sigma$ | $\lambda$ | $\delta$ | $k=\lambda/\delta$
---|---|---|---|---|---|---
25-30 | N | 53.76 | 11.10 | .0666 | .0257 | 2.59
| [51.74-56.02] | [9.67-14.06] | [.0487-.0891] | [.0241-.0268] |
F | 50.15 | 17.47 | .1705 | .0339 | 5.03
| [46.91-52.04] | [14.86-20.34] | [.1447-.1932] | [.0307-.0358] |
30-40 | N | 50.97 | 8.76 | .0416 | .0098 | 4.24
| [49.06-53.77] | [6.95-11.10] | [.0356-.0583] | [.0092-.0106] |
F | 49.35 | 15.86 | .1071 | .0167 | 6.41
| [46.33-50.78] | [13.14-20.2] | [.0762-.1261] | [.0162-.0178] |
40-55 | N | 54.05 | 10.10 | .0355 | .0051 | 6.96
| [51.81-55.98] | [8.56-11.87] | [.0281-.0412] | [.0048-.0056] |
F | 50.44 | 8.12 | .0353 | .0072 | 4.90
| [47.62-52.88] | [6.40-11.92] | [.0221-.0501] | [.0067-.0075] |
* 1.
Notes: In brackets: the 2.5% and 97.5% percentiles of the bootstrap
distribution.
### 4.1 Unskilled Blue Collar Workers
Table 6 reports the results. Across all three age groups, the labour turnover
parameters of migrants exceed those of natives,
$\hat{\delta}_{F}>\hat{\delta}_{N}$ and $\hat{\lambda}_{F}>\hat{\lambda}_{N}$.
Migrants experience job separations more often, but this is partially
compensated by them also finding new jobs more quickly. All job turnover
parameters fall in age. Across age groups and nationality, transitions into
new jobs happen more quickly than transitions into unemployment,
$\hat{\lambda}>\hat{\delta}$. Foreigners have slightly lower mean reservation
wages, $\hat{\mu}_{F}<\hat{\mu}_{N}$, but confidence intervals overlap. The
estimates are fairly stable across age. The estimates for the reservation wage
distribution for both groups imply that not all new job offers are accepted:
there are some workers with high reservation wages who would and do turn down
new job offers with insufficiently high wages.
In Figure 3 we consider some implications of the estimated model for the
young. Panel A plots the wage offer functions, panel B the reservation wage
density, whilst panel C plots the estimated productivity densities101010These
are obtained as follows. Given the parameter estimates and kernel estimate of
the realised wage density, the unemployment rate $u$ is estimated using
equation (4), and the wage offer distribution $F$ follows from (5); the
productivity distribution is then estimable from equation (6).. It is evident
that the productivity densities for both groups are well approximated by a
Pareto density. The slopes for sufficiently high productivities are very
similar. Turning to wage offers (panel A), for low productivities foreigners
do not do worse than natives, while for log productivities above 5 natives
receive better wage offers. Overall, the figure suggests a positive but small
migrant effect, and this is confirmed by our quantifications reported in
Section 5.
Figure 3: Unskilled blue collar workers aged 25-30.
### 4.2 Clerks and Low-Service Workers
Table 7: Structural parameter estimates: Clerks & service workers
Age | Nation. | $\mu$ | $\sigma$ | $\lambda$ | $\delta$ | $k=\lambda/\delta$
---|---|---|---|---|---|---
25-30 | N | 65.60 | 14.39 | .0984 | .0194 | 5.07
| [61.92-66.53] | [11.61-15.8] | [.0697-.0836] | [.0189-.0199] |
F | 36.09 | 13.65 | .0701 | .0272 | 2.58
| [30.88-41.69] | [8.6-17.17] | [.0624-.0886] | [.0259-.0284] |
30-40 | N | 72.66 | 9.42 | .0423 | .0073 | 5.79
| [68.41-75.12] | [7.54-10.39] | [.0355-.0530] | [.0071-.0076] |
F | 43.27 | 7.40 | .0593 | .0157 | 3.77
| [40.88-45.62] | [6.41-9.57] | [.0478-.0703] | [.0151-.0162] |
40-55 | N | 73.07 | 7.92 | .0698 | .0035 | 19.94
| [70.51-75.12] | [7.07-9.16] | [.0603-.0841] | [.0031-.0037] |
F | 49.04 | 6.86 | .0759 | .0077 | 9.86
| [46.38-51.94] | [5.22-8.41] | [.0565-.0911] | [.0072-.0081] |
* 1.
Notes: As for Table 6.
Table 7 reports the results for this occupational group, for which we observed
in Table 4 the largest average wage gap. As before, job separation rates for
foreigners exceed those of natives, decline in age, and are smaller than job
offer arrival rates. Except for the young, the transition rates of foreigners
exceed those of natives. But unlike the case of the unskilled, differences in
mean reservation wages are substantial: foreigners are substantially less
demanding, on average, than natives. These means increase in age. Figure 4
panel C suggests that productivities are again well approximated by a Pareto
form, and panel A suggests that the maximal migrant effect is substantial.
Figure 4: Clerks and service workers aged 25-30.
### 4.3 Skilled Blue-Collar Workers
Table 8: Structural parameter estimates: Skilled blue collar workers
Age | Nation. | $\mu$ | $\sigma$ | $\lambda$ | $\delta$ | $k=\lambda/\delta$
---|---|---|---|---|---|---
25-30 | N | 81.15 | 4.52 | .0801 | .0158 | 5.07
| [77.64-83.97] | [3.81-6.59] | [.0684-.0911] | [.0121-.0179] |
F | 66.38 | 14.05 | .1067 | .0225 | 4.74
| [62.88-69.04] | [11.88-17.32] | [0.955-0.1182] | [.0170-.0268] |
30-40 | N | 76.68 | 8.85 | .0698 | .0068 | 10.26
| [73.82-77.90] | [7.69-9.71] | [.0621-.0760] | [.0063-.0071] |
F | 69.30 | 8.06 | .0866 | .0124 | 6.98
| [65.57-72.05] | [7.34-9.16] | [.0681-.0946] | [.0119-.0127] |
40-55 | N | 79.71 | 6.44 | .0408 | .0035 | 11.66
| [77.18-80.94] | [5.62-7.01] | [.0343-.0478] | [.0033-.0036] |
F | 75.05 | 7.33 | .0449 | .0049 | 9.16
| [71.48-78.24] | [6.51-8.70] | [.0325-.0512] | [.0045-.0051] |
* 1.
Notes: As for Table 6.
For the skilled blue-collar workers, the by now familiar pattern emerges too,
as is evident from Table 8: both turnover parameters are higher for migrants,
and decline in age. As regards mean reservation wages, foreigners are less
demanding than natives, but the gap is not as wide as for clerks and service
workers, and it declines in age. Focussing on the young in Figure 5,
productivities are Pareto like. The migrant effect, captured in Panel A, is
modest.
Figure 5: Skilled blue collar workers aged 25-30.
### 4.4 General Discussion
Comparing the results across occupations, we observe similar patterns.
Migrants experience job separations more often than natives but also typically
find jobs more quickly, and job turnover parameters tend to decline in age.
These findings are in line with differences in employment protection observed
in Sa (2011), who reports that migrants in Germany are much more likely than
natives to work on temporary contracts. The findings are also consistent with
the other dimensions of segregation extensively documented in Glitz (2012).
Across all segments and nationality, transitions into new jobs happen more
quickly than transitions into unemployment. Overall, search frictions, as
measured by $(\lambda/\delta)^{-1}$, are of the same order of magnitude across
all occupational groups, decrease in age (except for unskilled foreigners in
which case they are stable), and are larger for foreigners than for natives
for the skilled and clerks and service workers. Thus, the higher job offer
arrival rate for foreigners cannot compensate for their higher job separation
rates. As regards the reservation wage distribution, across all segments there
are some workers with high reservation wages who turn down new job offers when
wage offers are too low.111111These results differ from estimates for
Netherlands (van den Berg and Ridder (1998)) and France (Bontemps et al.
(1999)) since both countries have a binding legal minimum wage. Similar to
these studies, however, we observe that job separation parameter $\delta$ is
approximately one order of magnitude smaller than the estimated job offer
arrival rate $\lambda$. Our results are comparable to those reported by
Bartolucci (2013a) for Germany obtained from a different empirical search
model applied to a different market segmentation. For low qualified male
workers in the manufacturing sector he reports a job separation rate of .03
and a job offer arrival rate of .3. Our estimates of $\delta$ range from .004
to .03, and the job offer arrival rate ranges from .04 to .17. Our result
regarding the mean reservation wage, the lack of a statistically significant
difference between natives and migrants for most groups, is also consistent
with external evidence reported in Bergemann et al. (2011, Table 2) who use
the IZA evaluation sample of individuals who entered unemployment in late 2007
and early 2008: the means of the self-reported reservation wages are not
statistically significantly different between natives and migrants, and their
levels are comparable to ours. Migrant workers are on average less demanding
than natives. Firm productivities are well approximated by Pareto forms.
However, migrants receive wage offers that are lower than for natives who have
the same productivity. This migrant effect is the largest for clerks and
service workers, and small for unskilled workers. The drivers of the migrant
effect are the subject of Section 5.
### 4.5 Robustness Checks
#### 4.5.1 Return Migration
A concern for our estimates in the migrant segments might be the effect of
return migration, when such returnees leave Germany out of employment. In
order to investigate the sensitivity of our estimates to this issue, we
consider restricted samples of migrants who should, in principle, be available
for work after their employment transition, by requiring foreigners to be
observable in the data 6 months after their transition. This restriction leads
to a net dropout of foreigners (relative to that of natives) across segments
between 7.7% and 15.3%.121212These rates are consistent with Gundel and Peters
(2008, Table 1) who, using GSOEP data for the period 1984-2005, suggest that
among male immigrants aged less than 60 the return rate is 10% Table 9
juxtaposes the estimates for these restricted samples (labelled noRetMig) to
our unrestricted estimates. We find that most parameter estimates remain
relatively stable (the occasional fall in $\hat{\sigma}$ reflects the extent
to which the sample restriction increases the homogeneity of the group; the
more homogeneous the sample, the smaller the estimated reservation wage
dispersion).
Table 9: Sensitivity analysis - the effect of excluding return migrants Occupation | Age | Group | $\mu$ | $\sigma$ | $\lambda$ | $\delta$
---|---|---|---|---|---|---
Unskilled | 25-30 | full | 50.15 | 17.47 | 0.1705 | 0.0339
noRetMig | 49.88 | 10.14 | 0.1207 | 0.0433
30-40 | full | 49.35 | 15.86 | 0.1071 | 0.0167
noRetMig | 50.65 | 8.53 | 0.0588 | 0.0219
40-55 | full | 50.44 | 8.12 | 0.0353 | 0.0072
noRetMig | 48.53 | 3.28 | 0.0495 | 0.0087
Skilled | 25-30 | full | 66.38 | 14.05 | 0.1067 | 0.0225
noRetMig | 65.00 | 9.05 | 0.0871 | 0.0281
30-40 | full | 69.30 | 8.06 | 0.0866 | 0.0124
noRetMig | 68.72 | 5.86 | 0.0660 | 0.0142
40-55 | full | 75.05 | 7.33 | 0.0449 | 0.0049
noRetMig | 62.74 | 7.44 | 0.0379 | 0.0058
| 25-30 | full | 36.09 | 13.65 | 0.0701 | 0.0272
| noRetMig | 37.84 | 11.37 | 0.0749 | 0.0344
Clerks | 30-40 | full | 43.27 | 7.40 | 0.0593 | 0.0157
& Services | noRetMig | 45.11 | 9.09 | 0.0689 | 0.0201
| 40-55 | full | 49.04 | 6.86 | 0.0759 | 0.0077
| noRetMig | 48.23 | 5.39 | 0.1036 | 0.0098
#### 4.5.2 The effect of truncating the wage distribution
Table 10: Sensitivity analysis - the effects of truncation Occupation | Age | Trunc. | Foreigners | Natives
---|---|---|---|---
| | | $\mu$ | $\sigma$ | $\lambda$ | $\delta$ | $\mu$ | $\sigma$ | $\lambda$ | $\delta$
Unskilled | 25-30 | 3% | 45.31 | 19.68 | .1698 | .0344 | 42.77 | 15.85 | .0603 | .0256
5% | 50.15 | 17.47 | .1705 | .0339 | 53.76 | 11.10 | .0666 | .0257
7% | 54.47 | 15.18 | .1593 | .0333 | 56.72 | 9.53 | .0783 | .0254
30-40 | 3% | 47.89 | 16.51 | .1215 | .0167 | 38.62 | 11.64 | .0321 | .0099
5% | 49.35 | 15.86 | .1071 | .0167 | 50.97 | 8.76 | .0416 | .0098
7% | 55.62 | 12.56 | .1000 | .0162 | 57.53 | 9.10 | .0306 | .0095
40-55 | 3% | 40.18 | 3.86 | .0435 | .0074 | 38.62 | 11.64 | .0321 | .0099
5% | 50.44 | 8.12 | .0353 | .0072 | 54.05 | 10.10 | .0355 | .0051
7% | 52.83 | 7.64 | .0298 | .0071 | 53.87 | 14.73 | .0276 | .0049
Skilled | 25-30 | 3% | 57.41 | 18.69 | .0915 | .0229 | 72.78 | 9.66 | .0611 | .0162
5% | 66.38 | 14.05 | .1067 | .0225 | 81.15 | 4.52 | .0801 | .0158
7% | 71.79 | 10.38 | .1061 | .0219 | 82.65 | 3.72 | .0729 | .0154
30-40 | 3% | 63.53 | 12.14 | .0695 | .0127 | 73.51 | 9.44 | .0798 | .0069
5% | 69.30 | 8.06 | .0866 | .0124 | 76.68 | 8.85 | .0698 | .0068
7% | 72.36 | 6.56 | .0579 | .0121 | 87.08 | 4.58 | .0612 | .0064
40-55 | 3% | 67.90 | 7.57 | .0557 | .0045 | 69.17 | 8.12 | .0407 | .0036
5% | 75.05 | 7.33 | .0449 | .0049 | 79.71 | 6.44 | .0408 | .0035
7% | 77.32 | 5.81 | .0392 | .0045 | 83.31 | 5.40 | .0583 | .0035
| 25-30 | 3% | 35.43 | 14.04 | .0628 | .0269 | 58.30 | 18.68 | .1113 | .0198
| 5% | 36.09 | 13.65 | .0701 | .0272 | 65.60 | 14.39 | .0984 | .0194
| 7% | 35.44 | 14.04 | .0628 | .0269 | 68.11 | 13.14 | .0953 | .0191
| 30-40 | 3% | 41.98 | 7.83 | .0608 | .0156 | 62.80 | 15.13 | .0555 | .0075
Clerks | 5% | 43.27 | 7.40 | .0593 | .0157 | 72.66 | 9.42 | .0423 | .0073
& Services | 7% | 47.34 | 5.79 | .0464 | .0154 | 78.16 | 5.92 | .0490 | .0072
| 40-55 | 3% | 47.89 | 8.88 | .0534 | .0078 | 66.87 | 10.03 | .0452 | .0035
| 5% | 49.04 | 6.86 | .0759 | .0077 | 73.07 | 7.92 | .0698 | .0035
| 7% | 48.59 | 5.09 | .0635 | .0076 | 78.09 | 6.47 | .0709 | .0034
Our samples have been truncated at 5% at the left tail of the wage
distribution, a common cut-off in the literature. Here, we examine the
sensitivity of our estimates to varying the cut-off from 3% to 7%. Table 10
reports the results. Across all segments, the frictional parameters $\delta$
and $\lambda$ are very stable. An increase in the truncation is expected to
lead to an increase in the estimated mean reservation wage. This increase,
however, turns out to be typically very modest. We conclude that our estimates
are robust.
#### 4.5.3 Ethnic German Immigrants
Table 11: Native workers: full and restricted sample results
Occupation | Age | Group | $\mu$ | $\sigma$ | $\lambda$ | $\delta$
---|---|---|---|---|---|---
Unskilled | 30-40 | all | 50.97 | 8.76 | .0416 | .0098
pre ’88 | 48.46 | 5.22 | .0362 | .0090
40-55 | all | 54.05 | 10.10 | .0355 | .0051
pre ’88 | 51.95 | 6.66 | .0203 | .0046
Skilled | 30-40 | all | 76.68 | 8.85 | .0698 | .0068
pre ’88 | 88.53 | 2.78 | .0700 | .0060
40-55 | all | 79.71 | 6.44 | .0408 | .0035
pre ’88 | 81.34 | 7.51 | .0407 | .0032
Clerks | 30-40 | all | 72.66 | 9.42 | .0423 | .0073
& Services | pre ’88 | 71.11 | 9.12 | .0496 | .0067
40-55 | all | 73.07 | 7.92 | .0698 | .0035
| pre ’88 | 72.66 | 8.93 | .0413 | .0033
* 1.
Notes: “all” refers to the full sample of native workers, “pre ’88” to the
sample of natives observed before 1988.
The inflow of foreign-born ethnic Germans in the late 1980’s and early 1990’s
changed the composition of the group of natives. While qualifying for a German
passport by descent, many did not speak German and were more similar to the
group of foreign nationals considered above. However, these ethnic German
immigrants are not directly identifiable in our data and thus latent in the
group of natives. This arguable misclassification could lead to biases in our
estimates for native workers. To check the robustness of our results to such
changes in the population of German citizens, we estimate the model using the
subsample of native workers that are also present in the data set before 1988
(labelled pre’88), the year before the inflow of ethnic Germans occurred.
Table 11 reports our estimates, and for ease of comparison, juxtaposes these
to our earlier results for the unrestricted sample (labelled all). The young
age group is excluded from this exercise since many in this group would be too
young to be employed pre 1988. The estimates are fairly similar in the full
sample and the subsample, which suggests that the presence of ethnic Germans
has only little effect on the estimates of the structural parameters for
natives.
## 5 Migrant Effects and Wage Decompositions
We proceed to examine actual and counterfactual decompositions of the wage
differential by considering the scenarios of Section 2.4.1. The discussion
there has highlighted the importance of the productivity distribution, and we
operationalise the decomposition as follows.
### 5.1 Calibration Details
Our estimation has yielded, given the (estimate of the) actual wage
distribution $G$, the estimated wage offer functions
$w_{i}^{e}(p|\hat{\lambda},\hat{\delta},\hat{\mu},\hat{\sigma})$. Given the
Pareto-like productivity distributions, we calibrate wage offer functions
$w_{i}(p|\hat{\lambda},\hat{\delta},\hat{\mu},\hat{\sigma},\underline{p},\alpha)$
based on Pareto productivity distributions by minimising the integrated
absolute deviations between $w_{i}^{e}(p|.)$ and
$w_{i}(p|.,\underline{p},\alpha)$. Table 12 reports the calibrated
parameters.131313 These are also consistent with alternative estimates based
on the shapes of Figures 3 \- 5. The approximate linearity in the productivity
plots suggests a simple (graphical) estimator of the shape parameter of the
Pareto distributions: use OLS to estimate the regression of log density on log
productivity (and add 1).
Table 12: Calibrated parameters of the Pareto productivity distribution. Age Group | Nationality | Unskilled | Skilled | Clerks
---|---|---|---|---
$\underline{p}$ | $\alpha$ | $\underline{p}$ | $\alpha$ | $\underline{p}$ | $\alpha$
25-30 | Natives | 79.789 | 2.511 | 81.282 | 3.172 | 67.677 | 2.449
Foreigners | 47.632 | 2.205 | 51.010 | 2.146 | 43.053 | 1.468
30-40 | Natives | 84.343 | 2.894 | 71.212 | 3.076 | 104.849 | 3.096
Foreigners | 57.071 | 2.611 | 61.414 | 2.661 | 41.818 | 1.463
40-55 | Natives | 70.263 | 2.842 | 69.293 | 3.045 | 72.222 | 3.197
Foreigners | 70.202 | 2.833 | 63.838 | 2.896 | 34.545 | 1.738
Figure 6 illustrates these calibrations for young workers in the three
occupations, as well as the counterfactual experiment of improving the job
turnover situation of foreigners by lowering their job separation rate to
those of natives, $\delta_{F}\equiv\delta_{N}$. The first two columns of the
figure show the close match between $w^{e}(p)$ (which we have seen before in
Figure 3) and $w(p)$. Column three depicts the calibrated wage offers
$w_{N}(p)$ (solid line) and $w_{F}(p)$ (dashed line), as well as the
counterfactual $w_{F}(p|.,\hat{\delta}_{N})$ (dotted line). The reduction in
the separation rate for foreigners from $\hat{\delta}_{F}$ to
$\hat{\delta}_{N}$ ‘rotates’ the wage offer curve up: for lower
productivities, the improvement is negligible, but for very high
productivities foreigners receive wage offers equal to or better than those
for natives. This results in the improvement in the density of accepted wages
depicted in the fourth column of the figure.
### 5.2 Results
Tables 13 to 15 report by age group the average migrant effect (row 1), as
well as the results of the counterfactual experiments which follow the
structure of Table 1. We can anticipate the qualitative results of these
experiments based on numerical comparative statics exercises which show (for
the set of parameters considered) that wage offers increase in the job offer
arrival rate $\lambda$ as search frictions decrease, and, contrariwise,
decrease in the job separation rate $\delta$ as search frictions increase. Of
course, as $k=\lambda/\delta\to\infty$ and search frictions disappear, by eq.
(3), wage offers converge to the competitive wage. Wage offers increase in the
mean reservation wage $\mu$, since by the reservation wage property of job
search only sufficiently high wage offers are accepted out of unemployment,
but the effect of $\sigma$ is ambiguous. Since we found that job separation
rates for foreigners always exceed those of natives, setting
$\delta_{F}=\delta_{N}$ increases their wage offers, which implies a reduction
both of the wage gap and the migrant effect. Similarly, we found that
$\lambda_{F}>\lambda_{N}$ (except for young clerks and service workers), so
reducing the foreigners’ job offer arrival rate to that of natives reduces
their wage offers, which implies an increase both in the wage gap and the
migrant effect. As regards reservation wages, we found that foreigners are on
average less demanding than natives, but the overall effect of the joint
experiment involving $(\mu,\sigma)$ is ambiguous given the ambiguous effect of
$\sigma$. All these qualitative effects are observed in the results tables
(experiments (1)-(4)). The principal objective of the tables is then to
quantify the impacts in order to understand the principal drivers of the
migrant effect.
We comment first on the level of the average migrant effect. Across all age
groups, the absolute migrant effect is the largest for clerks and service
workers, followed by the skilled, and is negligible for the unskilled. In
relative terms, the migrant effect of the skilled accounts for 12-15% of the
average wage gap, and for clerks and service workers for 23-39%, the average
effect being 19.6 %. Expressed in terms of the average segment-specific wage
of natives (see Table 4), the migrant effect amounts to 9.2-18.3% for clerks
and service workers, 1.6-7.7% for skilled workers, 0.7-2.6% for unskilled
workers, and 5.6% for the full sample. The latter estimates are consistent
with estimates of “unexplained wage differences” reported in the literature
for Germany based on standard Oaxaca-Blinder decompositions (for instance,
Lehmer and Ludsteck (2011) report a range from 4 to 17%) or complementary
approaches (Hirsch and Jahn (2012) report 6% while Bartolucci (2013b) suggests
discrimination effects ranging between 7 and 17%). The observed difference
between the wage differential and the migrant effect also implies that the
largest part of the native-migrant wage gap is explained by differences in the
productivity distribution, which is confirmed in experiment (9) by the drop in
the wage gap (which now equals the migrant effect by construction). Policy
interventions that seek to reduce the productivity gap will thus reduce the
wage gap. We turn to the various experiments, highlighting the role of search
frictions.
Consider first the role of the mean reservation wage $\mu$ (experiment (2)).
The gap in mean reservation wages is the largest for clerks and service
workers whilst the dispersion parameters are fairly similar. Raising then the
foreigners’ mean reservation wages shows that the substantial migrant effect
for this occupational group is reduced to between 41% and 61% of its former
level. For the skilled, we only observe a significant gap in mean reservation
wages for the young, and an equalisation of $(\mu,\sigma)$ reduces the migrant
effect to 48% of its former level. For the other age groups, and for the
unskilled, differences in $\mu$ between natives and foreigners are either
small or negligible, so equalisations have little effect. Once productivity
differences have been eliminated, a comparison between experiments (9) and
(10) shows that for clerks and service workers, the relative improvement in
the migrant effect due to the additional equalisation of $(\mu,\sigma)$ is
slightly larger (the migrant effect is now between 20% and 30% of the level
generated in experiment (9)). Turning to the policy implications, although
foreigners are on average less demanding than natives, we believe that
foreigners’ reservation wages should be less a concern for policy
interventions which are migrant-centred (as emphasised by recent policy
debates in the EU, e.g. EUCommission (2012, p. 28)) and seek to reduce the
migrant effect. Nor would any migrant-targeted benefit increase be politically
feasible in the light of the debate about welfare magnets.
By the same token, job arrival rates for foreigners typically exceed those of
natives, and thus should equally be of little policy concern. In fact, the
experiments (4) show that reducing this rate to that of natives only
substantially increases the migrant effect for the unskilled in the two first
age groups; for all other groups the induced increase in the migrant effect is
fairly small. This is also in line with the observation that $\lambda$ falls
in age for the unskilled and skilled.
We turn to the remaining frictional parameter, the job separation rate
$\delta$. Recall that foreigners’ job separation rates are larger than those
for natives, and sometimes substantially so, and that search frictions
experienced by them are typically larger than those of natives as a higher
$\lambda$ cannot compensate for the higher $\delta$. Hence there is scope for
migrant-centred policy interventions that seek to reduce their search
frictions, such as improving migrants’ employment protection. This scope,
however, decreases in age, as $\delta$ falls in age across all occupational
groups. Reducing the foreigners’ job separation rates to that of natives has
the largest absolute impact for clerks and service workers, followed by the
skilled. For the unskilled, the migrant effect is already fairly small, and an
equalisation of $\delta$ reduces the remainder further.
## 6 Conclusion
The use of the structural empirical general equilibrium search model with on-
the-job search has enabled us to disentangle the role of various unobservables
for the explanation of wage differentials between migrants and natives. In
particular, we have examined differences in search frictions, reservation
wages, and productivities in segments of the labour market defined by
occupation, age, and nationality using a large scale German administrative
dataset. The resulting decompositions of the actual and counterfactual wage
differential quantify the marginal and joint roles of the various factors.
## Acknowledgements
Financial support from the NORFACE research programme on Migration in Europe
-Social, Economic, Cultural and Policy Dynamics is gratefully acknowledged.
Thanks to Norface conference participants, especially G. Peri and C. Dustmann
for comments, as well as N. Theodoropoulos. We also wish to thank our three
referees whose detailed and very constructive comments have helped to improve
the paper.
This study uses the factually anonymous regional file of the IAB Employment
Sample (IABS) 1975-2004. Data access was provided via a Scientific Use File
supplied by the Research Data Centre (FDZ) of the German Federal Employment
Agency (BA) at the Institute for Employment Research (IAB).
Figure 6: Calibration, migrant effects, and wage densities for young workers.
Notes. Row 1: unskilled blue collar workers. Row 2: skilled blue collar
workers. Row 3: clerks and lower service workers.
Table 13: Wage differential decomposition and average migrant effects: Ages
25-30.
| | | Unskilled | Skilled | Clerks
---|---|---|---|---|---
| Counterfactually | Remaining | Wage | Migrant | Wage | Migrant | Wage | Migrant
| equalised para. | differing para. | differential | effect | differential | effect | differential | effect
(1) | | $\underline{p},\alpha,\mu,\sigma,\lambda,\delta$ | 61.717 | 2.779 | 62.463 | 9.560 | 43.426 | 16.929
(2) | $\mu,\sigma$ | $\underline{p},\alpha,\lambda,\delta$ | 61.740 | 2.793 | 60.964 | 4.609 | 38.357 | 6.995
(3) | $\delta$ | $\underline{p},\alpha,\mu,\sigma,\lambda$ | 60.806 | 0.349 | 61.260 | 7.691 | 40.637 | 14.205
(4) | $\lambda$ | $\underline{p},\alpha,\mu,\sigma,\delta$ | 64.804 | 11.272 | 63.437 | 11.146 | 40.639 | 14.206
(5) | $\mu,\sigma,\delta$ | $\underline{p},\alpha,\lambda$ | 60.828 | 0.373 | 59.900 | 3.150 | 36.253 | 5.321
(6) | $\mu,\sigma,\lambda$ | $\underline{p},\alpha,\delta$ | 64.803 | 11.134 | 61.804 | 5.798 | 36.254 | 5.322
(7) | $\lambda,\delta$ | $\underline{p},\alpha,\mu,\sigma$ | 63.917 | 8.793 | 62.237 | 9.201 | 37.861 | 11.681
(8) | $\mu,\sigma,\lambda,\delta$ | $\underline{p},\alpha$ | 63.930 | 8.729 | 60.766 | 4.334 | 34.028 | 3.660
(9) | $\underline{p},\alpha$ | $\mu,\sigma,\lambda,\delta$ | | -4.336 | | 4.142 | | 11.569
(10) | $\underline{p},\alpha,\mu,\sigma$ | $\lambda,\delta$ | | -4.962 | | 0.264 | | 3.501
(11) | $\underline{p},\alpha,\delta$ | $\mu,\sigma,\lambda$ | | -6.137 | | 2.610 | | 9.065
(12) | $\underline{p},\alpha,\lambda$ | $\mu,\sigma,\delta$ | | 2.989 | | 5.529 | | 9.066
(13) | $\underline{p},\alpha,\mu,\sigma,\delta$ | $\lambda$ | | -6.756 | | -1.088 | | 1.710
(14) | $\underline{p},\alpha,\mu,\sigma,\lambda$ | $\delta$ | | 2.330 | | 1.458 | | 1.711
(15) | $\underline{p},\alpha,\lambda,\delta$ | $\mu,\sigma$ | | 0.647 | | 3.840 | | 6.812
Notes: Based on the decomposition of equation (2.4.1). Rows 9$+$: the wage
differential equals the migrant effect because the productivity distributions
are the same. The parameter estimates are reported in Tables 6\- 8.
Table 14: Wage differential decomposition and average migrant effects: Ages
30-40.
| | | Unskilled | Skilled | Clerks
---|---|---|---|---|---
| Counterfactually | Remaining | Wage | Migrant | Wage | Migrant | Wage | Migrant
| equalised para. | differing para. | differential | effect | differential | effect | differential | effect
(1) | | $\underline{p},\alpha,\mu,\sigma,\lambda,\delta$ | 59.484 | 0.854 | 29.662 | 3.904 | 89.489 | 28.642
(2) | $\mu,\sigma$ | $\underline{p},\alpha,\lambda,\delta$ | 59.699 | 1.456 | 28.539 | 2.228 | 87.086 | 17.559
(3) | $\delta$ | $\underline{p},\alpha,\mu,\sigma,\lambda$ | 57.916 | -2.964 | 27.915 | 1.956 | 84.023 | 19.278
(4) | $\lambda$ | $\underline{p},\alpha,\mu,\sigma,\delta$ | 62.756 | 9.059 | 30.355 | 4.696 | 91.897 | 33.358
(5) | $\mu,\sigma,\delta$ | $\underline{p},\alpha,\lambda$ | 58.089 | -2.472 | 26.937 | 0.476 | 82.447 | 11.304
(6) | $\mu,\sigma,\lambda$ | $\underline{p},\alpha,\delta$ | 63.106 | 10.004 | 29.159 | 2.922 | 88.962 | 20.378
(7) | $\lambda,\delta$ | $\underline{p},\alpha,\mu,\sigma$ | 60.872 | 4.298 | 28.503 | 2.603 | 86.404 | 23.116
(8) | $\mu,\sigma,\lambda,\delta$ | $\underline{p},\alpha$ | 61.136 | 5.027 | 27.482 | 1.065 | 84.526 | 13.974
(9) | $\underline{p},\alpha$ | $\mu,\sigma,\lambda,\delta$ | | -3.091 | | 2.547 | | 14.515
(10) | $\underline{p},\alpha,\mu,\sigma$ | $\lambda,\delta$ | | -2.585 | | 1.097 | | 3.011
(11) | $\underline{p},\alpha,\delta$ | $\mu,\sigma,\lambda$ | | -5.709 | | 0.758 | | 8.259
(12) | $\underline{p},\alpha,\lambda$ | $\mu,\sigma,\delta$ | | 3.264 | | 3.286 | | 18.087
(13) | $\underline{p},\alpha,\mu,\sigma,\delta$ | $\lambda$ | | -5.249 | | -0.549 | | -2.022
(14) | $\underline{p},\alpha,\mu,\sigma,\lambda$ | $\delta$ | | 3.938 | | 1.763 | | 5.737
(15) | $\underline{p},\alpha,\lambda,\delta$ | $\mu,\sigma$ | | -0.563 | | 1.349 | | 10.719
Notes: As for Table 13.
Table 15: Wage differential decomposition and average migrant effects: Ages
40-55.
| | | Unskilled | Skilled | Clerks
---|---|---|---|---|---
| Counterfactually | Remaining | Wage | Migrant | Wage | Migrant | Wage | Migrant
| equalised para. | differing para. | differential | effect | differential | effect | differential | effect
(1) | | $\underline{p},\alpha,\mu,\sigma,\lambda,\delta$ | 2.817 | 2.723 | 19.784 | 2.327 | 64.712 | 14.591
(2) | $\mu,\sigma$ | $\underline{p},\alpha,\lambda,\delta$ | 1.728 | 1.631 | 18.883 | 1.172 | 62.721 | 6.298
(3) | $\delta$ | $\underline{p},\alpha,\mu,\sigma,\lambda$ | 1.145 | 1.056 | 18.971 | 1.404 | 62.127 | 11.248
(4) | $\lambda$ | $\underline{p},\alpha,\mu,\sigma,\delta$ | 2.788 | 2.694 | 20.030 | 2.608 | 65.002 | 15.012
(5) | $\mu,\sigma,\delta$ | $\underline{p},\alpha,\lambda$ | 0.125 | 0.032 | 18.114 | 0.304 | 60.519 | 4.125
(6) | $\mu,\sigma,\lambda$ | $\underline{p},\alpha,\delta$ | 1.700 | 1.603 | 19.113 | 1.433 | 62.954 | 6.553
(7) | $\lambda,\delta$ | $\underline{p},\alpha,\mu,\sigma$ | 1.120 | 1.030 | 19.193 | 1.655 | 62.374 | 11.533
(8) | $\mu,\sigma,\lambda,\delta$ | $\underline{p},\alpha$ | 0.100 | 0.008 | 18.325 | 0.541 | 60.737 | 4.321
(9) | $\underline{p},\alpha$ | $\mu,\sigma,\lambda,\delta$ | | 2.716 | | 1.699 | | 7.375
(10) | $\underline{p},\alpha,\mu,\sigma$ | $\lambda,\delta$ | | 1.623 | | 0.601 | | 1.676
(11) | $\underline{p},\alpha,\delta$ | $\mu,\sigma,\lambda$ | | 1.049 | | 0.828 | | 5.173
(12) | $\underline{p},\alpha,\lambda$ | $\mu,\sigma,\delta$ | | 2.687 | | 1.966 | | 7.660
(13) | $\underline{p},\alpha,\mu,\sigma,\delta$ | $\lambda$ | | 0.025 | | -0.225 | | -0.160
(14) | $\underline{p},\alpha,\mu,\sigma,\lambda$ | $\delta$ | | 1.595 | | 0.851 | | 1.905
(15) | $\underline{p},\alpha,\lambda,\delta$ | $\mu,\sigma$ | | 1.023 | | 1.064 | | 5.359
Notes: As for Table 13.
## Appendix A Data Appendix: Variable Description
Our sample only includes full-time working men aged 25-55 years old residing
in West Germany. In what follows, we describe how we construct the key
variables used in our empirical analysis.
Age: The age variable is constructed using information on the date of birth
and the year in which the spell took place. Date of birth is not available for
individuals who were under 16 years old at their first observed spell or over
65 years old at their last observed spell. In such cases, we assume that
workers were 15 years old at their first spell and 67 years old at their last
spell.
Labour Market Status: The information provided in the data set are sufficient
to distinguish between three labour market states: employed, recipient of
transfer payments, and out of sample. In our analysis, we employ the broad
definition of unemployment, as proposed by Fitzenberger and Wilke (2010), and
assume that unemployment is proxied by non-employment. Using this definition
of unemployment, we only consider two labour market states (employment and
unemployment) since being out of sample is equivalent to being unemployed.
However, this strategy may lead to the mis-classification of non-participants
as unemployed: for example, an individual that had an employment spell in her
late teens, subsequently went to university, and reappeared in the sample in
her late twenties would be classified as unemployed despite the fact that she
was not in the labour market. To correct for this problem, individuals that
are out of sample are only classified as unemployed if their out of sample
duration does not exceed the average duration of transfer payment recipients’
spells.
Spells: Due to the annual reporting system, all spells have a maximum duration
of one year. We merge all consecutive annual spells during which the
individual does not experience a change in her labour market status, i.e. she
either remains unemployed or employed with the same employer. We use firm-
identifiers included in the dataset to determine when a worker changes
employers. The new merged spells record the start date, the end date, the
duration of the spell, the employment status, the average wage under the same
employer, and the transition experienced by the individual (job-to-
unemployment, job-to-job, unemployment-to-job).
Wages: The dataset reports gross daily wages and does not provide information
on hours worked. We therefore exclude part-time employees, trainees, interns,
and at-home workers from the sample since the wage information is not
comparable for these groups. Wages are truncated at the social security
contributions threshold (DM10) and censored at the social security
contributions ceiling (DM300). For workers with wages below the social
security contribution threshold, we use wages of adjacent employment spells.
Wage censoring is not pronounced as the social security contributions ceiling
is not binding in our sample as we focus on low-wage workers who are not
likely to earn wages in excess of this upper bound.
Since the focus of our analysis is the transitions experienced by workers in
the early 1990s, all wages are reported in DM and adjusted to real 1995 prices
using the German Consumer Price Index. For all individuals who experience wage
variation during employment spells, we compute the average per period wage of
each worker under the same employer.
Occupation: The dataset includes extensive information on occupations (three-
digit codes), which is used to classify individuals to 10 major groups based
on the International Standard Classification of Occupations (ISCO-88).
Exploiting the detailed index of occupational titles of the ISCO-88, we are
able to map the code list from the Federal Employment Service of Germany
included in the IABS into the following ISCO-88 major groups: (1) Legislators,
Senior Officials, and Managers; (2) Professionals; (3) Technicians and
Associate Professionals; (4) Clerks; (5) Service Workers and Shop & Market
Sales Workers; (6) Skilled Agricultural and Fishery Workers; (7) Craft and
Related Trades Workers; (8) Plant and Machine Operators and Assemblers; (9)
Elementary Occupations; (10) Armed Forces.
We restrict attention to low- and middle-skill occupations, where the
concentration of foreigners is higher. Specifically, we consider three
occupational groups that are defined as follows: (1) Unskilled blue-collar
workers, which includes individuals classified in the ISCO-88 major groups 8
and 9; (2) Skilled blue-collar workers, which includes individuals classified
in the ISCO-88 major group 7; (3) Clerks & low-service workers, which includes
individuals classified in the ISCO-88 major groups 4 and 5.
## Appendix B Estimation: A Validation Exercise
Given the complexity of both the model and the estimating equations, it is of
interest to test their performance in a simulation exercise. In this appendix,
we carry out such a validation exercise.
The data generating process uses the parametrisations discussed above: arrival
of job offers and separations follow Poisson processes, and the reservation
wage distribution is normal. The particular calibration, given in Table 16,
distinguishes between the segments for natives (subscripted N) and immigrants
(subscripted F), and uses values similar to those encountered in our data. We
also need to stipulate either a realised wage distribution $G$, or a
productivity distribution $\Gamma$. Since we observe wages but not
productivities in our data, we specify a productivity distribution here in
order to verify that the model-implied wage distributions “look realistic”
(i.e. share the principal features of real wage distributions). Since the
empirical results suggest that productivities are Pareto-like, we assume this
explicitly here: $\Gamma_{F}(p)=1-(\underline{p}_{F}/p)^{\alpha}$ and
$\Gamma_{N}(p)=1-(\underline{p}_{N}/p)^{\alpha}$ with $\alpha=2.1$,
$\underline{p}_{F}=40$, and $\underline{p}_{N}=50$. Hence the productivity
distribution in the segment for natives first order stochastically dominates
that of migrants. We also compute the model-implied unemployment rate $u$.
Using this Data Generating Process (DGP), we draw 400 samples of 2000
observations each and estimate the model by maximum likelihood.
Table 16: Natives and immigrants: DGP and parameter estimates. | $\mu_{N}$ | $\mu_{F}$ | $\sigma_{N}$ | $\sigma_{F}$ | $\lambda_{N}$ | $\lambda_{F}$ | $\delta_{N}$ | $\delta_{F}$ | $u_{N}$ | $u_{F}$
---|---|---|---|---|---|---|---|---|---|---
True Value | 60 | 45 | 10 | 10 | .07 | .13 | .005 | .016 | .1214 | .1838
Mean | 56.23 | 40.88 | 8.61 | 10.18 | .0887 | .1181 | .0050 | .0173 | .1145 | .1822
Median | 56.33 | 40.96 | 8.43 | 10.17 | .0835 | .1136 | .0050 | .0173 | .1142 | .1819
2.5 perc. | 53.46 | 36.62 | 5.63 | 6.86 | .0566 | .0939 | .0047 | .0164 | .1053 | .1711
97.5 perc. | 59.88 | 45.21 | 12.38 | 13.62 | .1403 | .1671 | .0053 | .0181 | .1246 | .1935
Section 2.4 has considered the economic implications of the estimation
results. Here, the main focus is on the quality of the estimates. Table 16
reports the results. All structural parameters are estimated well as the true
values are included in the 95% bootstrap confidence intervals (the table
reports the 2.5 and 97.5 % confidence limits). The means of the job turnover
parameters are particularly well estimated. The mean of the reservation wage
distribution $H$ is somewhat below the true value; this underestimate is
perhaps not too surprising since the model effectively only considers the
right tail of $H$ (i.e. reservation wages $b$ that satisfy $b>\underline{w}$).
The predicted unemployment rate is also very close to the theoretical value.
Figure 1 above has depicted the implied wage offers as a function of
productivities141414The computation of the wage offer curves for the
validation exercise based on a given productivity distribution $\Gamma$ is
more involved than in our empirical analysis below. In the latter case, given
the estimates of the structural parameters and the wage density, $F(w)$
follows straightforwardly from equation (5) and the productivity values follow
from (6). In the former case,
$F\left(w\right)=\Gamma\left(K^{-1}\left(w\right)\right)$, and $K(p)$ in (3),
defined implicitly, is estimated progressively: starting from $\underline{p}$,
$p$ is incremented by a small step $\varepsilon_{p}$, and
$K(p+\varepsilon_{p})$ is found through a local search based on (3), whence
$p+2\varepsilon_{p}$ is considered. The confidence bands are computed
pointwise, and simply determined by the relevant tail quantiles of the
bootstrap distribution. while Figure 6 has depicted the skewed densities of
realised wages, which do have a shape often encountered in empirical work (see
e.g. Figure 2).
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|
arxiv-papers
| 2013-06-07T17:21:39 |
2024-09-04T02:49:46.217706
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Panagiotis Nanos and Christian Schluter",
"submitter": "Panagiotis Nanos",
"url": "https://arxiv.org/abs/1306.1781"
}
|
1306.1871
|
# TWO-DIMENSIONAL NUMERICAL SIMULATIONS OF SUPERCRITICAL ACCRETION FLOWS
REVISITED
Xiao-Hong Yang11affiliation: Shanghai Astronomical Observatory, Chinese
Academy of Sciences, 80 Nandan Road, Shanghai 200030, China;
[email protected], [email protected] 22affiliation: Department of Physics,
Chongqing University, Chongqing 400044, China , Feng Yuan11affiliation:
Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai 200030, China; [email protected], [email protected] , Ken
Ohsuga33affiliation: National Astronomical Observatory of Japan, Osawa,
Mitaka, Tokyo 181-8588, Japan , and De-Fu Bu11affiliation: Shanghai
Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road,
Shanghai 200030, China; [email protected], [email protected]
###### Abstract
We study the dynamics of super-Eddington accretion flows by performing two-
dimensional radiation-hydrodynamic simulations. Compared with previous works,
in this paper we include the $T_{\theta\phi}$ component of the viscous stress
and consider various values of the viscous parameter $\alpha$. We find that
when $T_{\theta\phi}$ is included, the rotational speed of the high-latitude
flow decreases, while the density increases and decreases at the high and low
latitudes, respectively. We calculate the radial profiles of inflow and
outflow rates. We find that the inflow rate decreases inward, following a
power law form of $\dot{M}_{\rm in}\propto r^{s}$. The value of $s$ depends on
the magnitude of $\alpha$ and is within the range of $\sim 0.4-1.0$.
Correspondingly, the radial profile of density becomes flatter compared with
the case of a constant $\dot{M}(r)$. We find that the density profile can be
described by $\rho(r)\propto r^{-p}$ and the value of $p$ is almost same for a
wide range of $\alpha$ ranging from $\alpha=0.1$ to $0.005$. The inward
decrease of inflow accretion rate is very similar to hot accretion flows,
which is attributed to the mass loss in outflows. To study the origin of
outflow, we analyze the convective stability of the slim disk. We find that
depending on the value of $\alpha$, the flow is marginally stable (when
$\alpha$ is small) or unstable (when $\alpha$ is large). This is different
from the case of hydrodynamical hot accretion flow where radiation is
dynamically unimportant and the flow is always convectively unstable. We
speculate that the reason for the difference is because radiation can
stabilize convection. The origin of outflow is thus likely because of the
joint function of convection and radiation, but further investigation is
required.
###### Subject headings:
accretion, accretion disk – black hole physics – hydrodynamics – methods:
numerical – radiative transfer
## 1\. Introduction
One milestone in black hole accretion is the standard thin disk model (Shakura
& Sunyaev 1973; Pringle 1981). This model applies below the Eddington
accretion rate defined as $\dot{M}_{\rm Edd}\equiv 10L_{\rm Edd}/c^{2}$
($L_{\rm Edd}$ is the Eddington luminosity). In this model, all the viscously
dissipated energy is immediately radiated away. When the accretion rate is
above the Eddington rate, advection begins to become important and the
accretion model is described by the “slim disk” (Abramowicz et al. 1988; see
also Begelman & Meier 1982). In a slim disk, the radiative efficiency is lower
than that of the standard thin disk because of energy advection and photons
trapping effects. The energy dissipated in the disk is advected with the
accreting matter, since the radiative diffusion timescale is longer than the
accretion timescale. Note that the slim-disk model cannot correctly treat the
photon trapping, because photon trapping is basically a multi-dimensional
effect (Ohsuga et al. 2002; 2003). The potential applications of slim-disks
include narrow-line Seyfert galaxies (Mineshige et al. 2000) and ultraluminous
X-ray sources (Watarai et al. 2001; Vierdayanti et al. 2006).
The above-mentioned pioneer works on slim disk are all one dimensional and
analytical. Multi-dimensional and time-dependent numerical simulations
obviously can reveal important additional information about the dynamics of
the accretion flow. Many radiation-hydrodynamic (RHD; Eggum et al. 1987, 1988;
Okuda 2002; Okuda et al. 2005; Ohsuga et al. 2005) and radiation
magnetohydrodynamic (MHD; Ohsuga et al. 2009; Ohsuga & Mineshige 2011)
numerical simulation works on slim disk have been performed. Among these
works, Ohsuga et al. (2005) obtain a quasi-steady structure of the
supercritical accretion flows and outflows by a two-dimensional global
simulation. Their results broadly confirm the main properties predicted by the
analytical slim-disk model. Moreover, they show that the accretion flow is
composed of the disk region around the equatorial plane and the outflow region
above and below the disk. Ohsuga & Mineshige (2007) further identify that the
supercritical accretion is feasible because a very large radiation energy
density actually produces a small radiative flux as well as a force, because
of the large optical depth and photon trapping effects.
Recently, there are some works studying the thermal stability of radiation
pressure dominated thin disk using shearing box MHD numerical simulation with
radiative transfer (Hirose et al. 2009; Jiang et al. 2013). Hirose et
al.(2009) found that the disk is thermally stable, while Jiang et al. (2013)
found that it is unstable. The reason for the discrepancy is discussed in the
latter work.
In the RHD simulation of Ohsuga et al. (2005), an anomalous shear stress is
included to mimic the angular momentum transfer. However, in reality, we
expect Maxwell stresses associated with MHD turbulence driven by the magneto-
rotational instability (MRI) to provide angular momentum transport in
accretion flows (Balbus & Hawley 1998). Since MRI is driven only by shear
associated with the orbital dynamics, when an anomalous shear stress is
adopted we should only set the two azimuthal components ($r\phi$ and
$\theta\phi$) of the stress tensor to be non-zero (Stone et al.1999). This is
confirmed by the local shearing box simulations, which indicate that the
azimuthal components of the Maxwell stress are one order of magnitude larger
than the poloidal components (e.g., Hawley et al.1995, 1996; Stone et al.
1996). In Ohsuga et al. (2005) only the $r\phi$ component of the viscous
stress is included, while the $\theta\phi$ component is neglected. So our
first aim in this paper is to examine the effect of including the $\theta\phi$
component on the dynamics of the slim disk. It should play an important role
in the angular momentum transport between different latitudes and might
suppress the Kelvin–Helmholtz instability on the boundary between the outflow
regions and the disk.
Another aim is to study the effect of the magnitude of $\alpha$. Ohsuga et
al.(2005) only consider one value of $\alpha$. Previous HD numerical
simulations of non-radiative accretion flows have shown that the structure and
dynamics of accretion flow depend on the value of $\alpha$ (Igumenshchev &
Abramowicz 1999; Stone et al. 1999; Yuan et al. 2012). The radial profile of
inflow rate becomes significantly flatter when $\alpha$ becomes larger (Yuan
et al. 2012). Different than the non-radiative accretion flow, a slim disk is
dominated by radiation. It is thus unclear whether or how the dynamics of
accretion flow depend on the value of $\alpha$.
Yet another aim of the present work, which is perhaps more interesting, is to
examine the radial profiles of inflow rate. In almost all analytical models of
accretion disks, the mass accretion rate is assumed to be constant with
radius. The validity of this assumption, however, has never been proved. For
the hot accretion flows, both HD and MHD numerical simulations have found that
the mass inflow rate (refer to Equation (4) for definition) decreases inward
(e.g., Stone et al. 1999; Stone & Pringle 2001; Yuan et al. 2012, and
references therein). This is one of the most important findings of global
simulation of accretion flows, because this result supplies an important clue
to revealing the dynamics of accretion flow. Various models have been proposed
to explain this result such as adiabatic inflow–outflow solution (Blandford &
Begelman 1999, 2004; Begelman 2012) and convection-dominated accretion flow
(CDAF; Narayan et al. 2000; Quataert & Gruzinov 2000). In the former, it is
assumed that the inward decrease of inflow rate is because of mass loss in the
outflow, while in the latter the flow is assumed to be convectively unstable,
and a convective envelope solution is constructed that can also explain the
simulations. No outflow is needed in CDAFs. Recent numerical simulations have
shown that MHD accretion flows are convectively stable (Narayan et al. 2012;
Yuan et al. 2012). Moreover, by comparing the properties of inflow and outflow
on the base of their HD and MHD numerical simulation data, Yuan et al. (2012)
argue that mass outflow should be significant. They propose that inward
decrease of accretion rate is due to outflow. Li et al. (2013) obtained a
similar conclusion. In most of the published papers, the inflow and outflow
rates (Equation(4) and (5)) are computed at each instant of time by using
instantaneous velocities, and then the time-averaged. Narayan et al. (2012;
see also Sadowski et al. 2013 for the case of spin black holes), however,
found that the outflow rate is much lower since they think the outflow rate
should be calculated by doing the time-average first (see Yuan et al. 2012 for
more discussions on the discrepancy of the two approaches of calculating the
outflow rate). While it is agreed that outflow exists, so far the strength of
outflow is still an open question. F.Yuan et al. (2013, in preparation) have
also studied the mechanism of producing outflow. It was found that in the HD
case it is buoyancy associated with the convective instability, while it is
mainly magnetic centrifugal force in the case of MHD accretion flow. It is
obviously interesting to see whether the radial profile of the inflow rate of
a slim disk behaves like a hot accretion flow,and if it does, whether the
outflow is produced by the buoyancy associated with the convective
instability.
Table 1Summary of Simulations
Models | Run | $\alpha$ | $\dot{m}_{\rm input}$ | $\theta$ | Stress Tensor | $N_{r}\times N_{\theta}$ | ts(orbits) | tf (orbits) | $\dot{m}_{\rm acc}$ | L
---|---|---|---|---|---|---|---|---|---|---
A | 1a | 0.1 | 1000 | 0$\sim$ $\pi$ | $T_{r\phi}$ | 96$\times$225 | 11.5 | 46.1 | 184.5 | 2.4
B | 1b | 0.1 | 1000 | 0$\sim$ $\pi$ | $T_{r\phi}$,$T_{\theta\phi}$ | 96$\times$225 | 11.5 | 46.1 | 160.0 | 2.2
| 2b | 0.1 | 3000 | 0$\sim$ $\pi$ | $T_{r\phi}$,$T_{\theta\phi}$ | 96$\times$225 | 11.5 | 46.1 | 374.5 | 3.2
| 3b | 0.05 | 3000 | 0$\sim$ $\pi$ | $T_{r\phi}$,$T_{\theta\phi}$ | 96$\times$225 | 11.5 | 46.1 | 382.8 | 3.6
| 4b | 0.01 | 3000 | 0$\sim$ $\pi$ | $T_{r\phi}$,$T_{\theta\phi}$ | 96$\times$225 | 63.5 | 98.1 | 240.3 | 4.2
| 5b | 0.005 | 3000 | 0$\sim$ $\pi$ | $T_{r\phi}$,$T_{\theta\phi}$ | 96$\times$225 | 150.0 | 184.6 | 125.3 | 3.6
* Note: Columns 1 and 2: the classification of our models and their number, respectively. Column 3: the viscous parameter ($\alpha$). Column 4: the mass injection rate ($\dot{m}_{\rm{input}}$) in unit of the critical mass accretion rate, $\dot{M}_{\rm crit}\equiv L_{\rm Edd}/c^{2}$. Columns 5, 6, and 7: the computational domain in the $\theta$ direction, the components of stress tensor, and the quantity of grid cells. Column 8: the approximate time $t_{s}$ (in units of the orbital time at $r$=100 $r_{s}$) after which the accretion flows become quasi-steady. Column 9: the final time $t_{f}$. Columns 10 and 11: the mass accretion rate on the BH ($\dot{m}_{\rm acc}$ ) in unit of the critical mass accretion rate, and the luminosity ($L$) in unit of $L_{\rm Edd}$, respectively.
The paper is organized as follows. We describe our models and numerical method
in Section 2. The simulation results are presented in Section 3.Section 4 is
devoted to a summary and discussions.
## 2\. Models and numerical method
The RHD equations are the same as those in Ohsuga et al. (2005). In the RHD
equations, the flux-limited diffusion (FLD) approximation developed by
Levermore & Pomraning (1981) is adopted. We neglect the self-gravity of the
disk and use the pseudo-Newtonian potential to mimic the general relativistic
effects, $\psi=-GM/(r-r_{s})$ (Paczyńsky & Wiita 1980), where $G$ is the
gravitational constant, $M$ is the mass of the black hole, $r_{s}=2GM/c^{2}$
the Schwarzschild radius, and $r$ is the radius. We assume that the gas is in
local thermodynamic equilibrium and neglect the frequency dependence of the
opacities.
As we state in Section 1, we adopt a stress tensor $\bm{T}$ to mimic the shear
stress, which is in reality should be replaced by the magnetic stress
associated with MHD turbulence driven by the MRI. In most cases, following
Stone et al. (1999), we assume that the only non-zero components of $\bm{T}$
are the azimuthal components:
$\bm{T}_{r\phi}=\eta r\frac{\partial}{\partial
r}\left(\frac{\upsilon_{\phi}}{r}\right),$ (1)
$\bm{T}_{\theta\phi}=\frac{\eta\sin(\theta)}{r}\frac{\partial}{\partial\theta}\left(\frac{\upsilon_{\phi}}{\textit{\rm
sin}(\theta)}\right).$ (2)
Here, the dynamical viscosity coefficient $\eta$ is described as a function of
the pressure $\eta=\alpha(p_{g}+\lambda E_{0})/\Omega_{K}$, where $\Omega_{k}$
is the Keplerian angular speed, $p_{g}$ is the gas pressure, $E_{0}$ is the
radiation energy density, and $\lambda$ is the flux limiter (Levermore &
Pomraning 1981). The viscous dissipative function is given by
$(\bm{T}_{r\phi}^{2}+\bm{T}_{\theta\phi}^{2})/\eta$.
All of our models are calculated in spherical coordinates
($r$,$\theta$,$\phi$). The origin is set at a central black hole of
$M=10M_{\odot}$. The size of the computational domain is $2r_{s}\leq r\leq
500r_{s}$ and $0\leq\theta\leq\pi$ or $0\leq\theta\leq\pi/2$. The inner
boundary ($r_{\rm in}$) must be smaller than the sonic point of the accretion
flow, which ensures that the inner boundary conditions do not affect the
simulation results. Abramowicz et al. (1988) show that for a slim disk the
location of the sonic point depends on the accretion rate and the viscosity
parameter $\alpha$. For small $\alpha$ and a high accretion rate, the sonic
point locates in the range of $(2-3)r_{s}$. So we set $r_{\rm in}=2r_{s}$.
The computational domain is divided into $N_{r}\times N_{\theta}$ grid cells.
A non-uniform grid is employed in the $r$ direction. The grid points in the
$r$ direction are equally distributed logarithmically, i.e., $\triangle\ln r=$
constant. In the $\theta$ direction, in order to better resolve the flow at
the equator and to not lose the resolution at the axis, we adopt the mixed
grid. Twenty grids are uniformly distributed within the $\pi/8$ from the axis,
i.e., $\triangle\theta=\pi/160$. Other grids are distributed in the angular
range of $\pi/8\leq\theta\leq 7\pi/8$ or $\pi/8\leq\theta\leq\pi/2$ in such a
way that $\triangle\cos(\theta)=3\pi/(4(N_{\theta}-20))$ or
$3\pi/(8(N_{\theta}-20))$. The outflow boundary condition is adopted at the
inner radial boundary, i.e. the values of physical variables in the ghost
zones are set to the values at the inner radial boundary. The outer radial
boundary condition is the same as that employed by Ohsuga et al. (2005), who
suggested that the matter, having a specific angular momentum corresponding to
the Keplerian angular momentum at $r=100r_{s}$, is continuously injected into
the computational domain from the outer boundary near the equator. The
injected gas distributes within 0.05$\pi$ from the equator in the Gaussian
function; their radial velocity is set according to Equation(3.61) in Kato et
al. (1998), while their poloidal velocity is set to be zero. In the angular
direction, we employ axisymmetry relative to the axis and reflection symmetry
relative to the equator (when $0<\theta<\pi/2$), respectively. The
computational domain is initially filled with a hot, rarefied, and optically
thin atmosphere. The numerical approach can be found in Ohsuga et al. (2005).
Figure 1.— Snapshots of the logarithm of density (colors), overlaid with
velocity vectors (arrows). Top: $t$ = 92.541 orbits of Run 2b ($\alpha=0.1$);
Bottom: 254.085 orbits of Run 5b ($\alpha=0.005$).
The properties of all of the simulations are listed in Table 1, where Columns
1 and 2 give the classification of our models and their number; Column 3 gives
the value of $\alpha$; Column 4 gives the mass injection rate
($\dot{m}_{\rm{input}}$) in units of the critical mass accretion rate,
$\dot{M}_{\rm{crit}}\equiv L_{\rm{Edd}}/c^{2}$; Columns 5, 6, and 7 give the
computational domain in the $\theta$ direction, the components of stress
tensor, and the quantity of grid cells, respectively; Column 8 gives the
approximate dynamical time $t_{s}$ (all times in this paper are reported in
units of the orbital time at $r$=100$r_{s}$) after which the accretion flows
become quasi-steady; Column 9 gives the final time $t_{f}$; and Columns 10 and
11 give the mass accretion rate onto the black hole ($\dot{m}_{\rm{acc}}$) in
the units of the critical mass accretion rate and the luminosity ($L$) in the
units of $L_{\rm Edd}$, respectively. According to the component of stress
tensor and the angular range of computational domain, the simulations are
divided into two groups, namely Models A and B. Models A contains only the
$r\phi$ component of stress tensor, while Model B contains both the $r\phi$
and $\theta\phi$ components.
Figure 1 displays a two-dimensional density distribution overlaid with
velocity vectors. The top and bottom panels are for $t$=92.541 orbits of Run
2b and 254.085 orbits of Run 5b, respectively.
Figure 2.— Angular profiles of a variety of time-averaged variables for Run 1a
($T_{\theta\phi}=0$; black) and Run 1b (with $T_{\theta\phi}\neq 0$; red) at
$r$=5$r_{s}$ (solid lines) and 30$r_{s}$ (dashed lines).
Figure 3.— Angular profiles of time-averaged Bernoulli parameters for Run 1a
($T_{\theta\phi}=0$; black) and Run 1b (with $T_{\theta\phi}\neq 0$; red) at
$r$=5$r_{s}$ (solid lines) and 30$r_{s}$ (dashed lines).
## 3\. Results
### 3.1. The Effects of $T_{\theta\phi}$
To investigate the effect of $T_{\theta\phi}$ on the dynamics of accretion
flow, we compare Models A and B. Figure 2 shows the angular structure of time-
averaged variables at $r=5r_{s}$ and 30 $r_{s}$ for Run 1a and Run 1b with
$\dot{m}_{\rm{input}}=1000$. In this paper, all the time-averaged quantities
are obtained by averaging 100 data files within 23 orbits after the accretion
flows have achieved the quasi-steady state. In Figure 2, the black solid
($r=5r_{s}$) and dashed ($r=30r_{s}$) lines correspond to Run 1a while the red
solid ($r=5r_{s}$) and dashed ($r=30r_{s}$) lines correspond to Run 1b. The
figure shows obvious differences of the flow structure, especially at the
high-latitude region.
The angular profiles of angular velocity ($\upsilon_{\phi}/(r\sin(\theta))$)
show that the flow in Run 1a rotates faster than that of Run 1b at the high-
latitude region. Close to the equator, the rotation of Run 1a is slightly
slower than Run 1b. Because $T_{\theta\phi}$ transports angular momentum
between different latitudes, for Run 1a the angular momentum between different
latitudes can not be transported, although the high-latitude flow rotates
faster than the low-latitude one. Therefore, it is seen that the angular
velocity increases from the equator to almost the axis (although the angular
velocity at the rotating axis is set to be zero). When $\bm{T}_{\theta\phi}$
is included in Run 1b, the angular momentum can be transported from the
quickly rotating flows at high latitude to the slowly rotating flows at low
latitude. This is why the discrepancy of the rotation velocity between the
equator and the high-latitude region is smaller in Run 1b than in Run 1a. The
discrepancy of the rotation velocity at the equator between Run 1a and Run 1b
is much smaller than in the high-latitude region, since the density at the
high-latitude region is much smaller than in the equator.
The angular profiles of radial velocity ($v_{r}$) show obvious differences
between Run 1a and Run 1b, especially at the high-latitude region. The
positive value of $v_{r}$ indicates that the flows are outflows, while the
negative value indicates the inflow nature of flows. At large radii such as
$\sim 30r_{s}$, the high-latitude flow (the flow within $30^{\circ}$ from the
axis) is mainly outflow. The speed of the high-latitude outflow of Run 1a is
higher than that of Run 1b. At small radii such as $r<5r_{s}$, the flow is
inflow at all latitudes for both models. The angular profiles of radial
velocity agree with the angular profiles of radial forces, as shown in Figure
2. The radial radiation force dominates the radial component of net force at
the high-latitude region at $\sim 30r_{s}$; therefore, it is the dominant
force changing the angular distribution of the radial velocity there. At small
radii, gravity is the dominant force.
The angular profiles of density ($\rho$) of Run 1a and Run 2b are similar,
with the maximum density located at the equator. Compared with Run 1a, the
density of the high-latitude region of Run 1b is higher, while at the equator
is slightly lower. The angular profiles of gas temperature ($T_{\rm gas}$)
show that the disk of Run 1a and Run 1b has nearly the same gas temperature at
$30r_{s}$. For Run 1b the gas temperature of high-latitude outflow is lower
than that of Run 1a. For the high-latitude outflow, the lower the gas
temperature, the higher the density.
Figure 4.— Angular profiles of a variety of time-averaged variables from Run
2b ($\alpha=0.1$; black) and Run 5b ($\alpha=0.005$; red) at $r=5r_{s}$ (solid
lines) and $30r_{s}$ (dashed lines).
Figure 5.— Angular distribution of the radial forces per unit mass for Run 5b
at $r=5r_{s}$ (top panel) and $30r_{s}$ (bottom panel). The radial forces
include gravity (red dashed line), centrifugal force (red solid line),
radiation force (blue dashed line), gas-pressure gradient force (blue solid
line), and their sum (black solid line).
Figure 6.— Angular distribution of the angular forces per unit mass for Run 5b
at $r=5r_{s}$ (top panel) and $30r_{s}$ (bottom panel). The angular forces
include the angular component of radiation force (red solid line), centrifugal
force (red dotted line), gas-pressure gradient force (red dashed line), their
sum (is shown by the black solid line.)
The angular profiles of the “radiation” temperature ($T_{\rm
rad}\equiv(E_{0}/a)^{\frac{1}{4}}$, where $a$ is the radiation constant) show
that compared with Run 1a, the radiation temperature for Run 1b is lower near
the equator and higher near the rotating axis . For the super-critical
accretion flow, the radiation force (including also the gradient of the
radiation pressure) is the dominant force driving the high-latitude outflow.
The radiation force acting on unit mass is given by
$\bm{f}=\frac{\chi}{c\rho}\bm{F}_{0}$, where the flux-mean opacity $\chi$ is
the sum of components due to absorption and scattering. The scattering in the
high-latitude outflow is the dominant factor for the opacity so that
$\chi\propto\rho$ and $\bm{f}\propto\bm{F}_{0}$. Here we employ the
approximation and have $\bm{F}_{0}=-\frac{c\lambda}{\chi}\nabla E_{0}$. In the
optically thin limit, $|\bm{F}_{0}|=cE_{0}$. In general, the high-latitude
outflow has high gas temperature and low density, so the flow is optically
thin. For Run 1a, the opacity of outflow near the rotating axis is close to
the optically thin limit, while for Run 1b the opacity of outflow is away from
the optically thin limit because of the higher density. Therefore, although
Run 1a has lower radiation energy density at the high-latitude region than Run
1b, the radiation flux of Run 1a is larger than that of Run 1b, which is shown
by the angular profiles of radial radiation flux in Figure 2. Therefore, at
the high-latitude region the radiative force of Run 1a is larger, exceeding
that of Run 1b. Compared with Run 1a, the density of high-latitude outflow for
Run 1b is higher so that the driving force of high-latitude outflow is weaker.
For the strict steady state and for inviscid hydrodynamic flow, the Bernoulli
parameter Be (Be $\equiv\upsilon^{2}/2+\gamma p/(\gamma-1)\rho-GM/(r-r_{s})$)
is conserved along the streamline. Therefore, the positive sign of Be is often
used to be the necessary condition for the outflow to escape to infinity.
However, in our case, these conditions are not satisfied, thus Be is no longer
conserved (e.g., Yuan et al. 2012). In fact, the initial result from our
ongoing work indicates that Be can increase along the trajectory of outflow
elements (F.Yuan et al. 2013, in preparation). This means that an outflow with
a negative Be can also potentially escape to infinity. Despite these
uncertainties, we still show in Figure 3 the angular distribution of the time-
averaged Bernoulli parameter Be because its value may still play some role in
determining the properties of outflow. We can see that at the region close to
the axis ($\theta<25^{\circ}$) and $r=30r_{s}$, the value of Be in Run 1b is
about one order of magnitude lower than that in Run 1a. From Figure 2, we can
see that including $T_{\theta\phi}$ for the polar outflows, rotational speed,
radial speed and temperature decrease can decrease the specific Be of outflow.
In general, the value of Be is larger close to the axis. When compared with
the hydro and MHD numerical simulations of hot accretion flow presented in
Yuan et al. (2012), it is interesting to note that the angular distribution of
Be is more similar to the MHD case (their Model D in Yuan et al. (2012))
rather than to the hydro case (their Models A, B, and C). This is perhaps
because the radiation acceleration in the $r$ direction is much stronger in
the region close to the axis than in other directions. In the case of Models
A, B, C in Yuan et al. (2012), there is no such force. But in the MHD case,
the magnetic pressure force plays a similar role with the radiation force
here.
### 3.2. The Effect of the Viscous Parameter $\alpha$
Figure 4 shows the angular distribution of the time-averaged flow for Model B
(with $\dot{m}_{\rm{input}}=3000$; i.e., Run 2b and 5b) at $r=5r_{s}$ and
$r=30r_{s}$, respectively. We can see that the viscosity parameter $\alpha$
obviously affects the angular distribution of accretion flow. The angular
profiles of angular velocity ($v_{\phi}/(r\text{sin}(\theta$))) show a large
difference close to the axis because of the different $\alpha$. When $\alpha$
is smaller, the flow rotates faster. A smaller $\alpha$ results in a smaller
$\bm{T}_{\theta\phi}$; thus, the angular momentum cannot be efficiently
transported from high to low latitudes. The angular profile of the radial
velocity ($v_{r}/c$) is different for different $\alpha$. The speed of the
high-latitude outflow in Run 15b is higher than that of Run 2b. At
$r=30r_{s}$, the high-speed outflow is constrained to be within $30^{\circ}$
from the axis. With the decrease of $\alpha$, the range of $\theta$ within
which high-latitude outflow takes place becomes larger toward the equator. At
$r=5r_{s}$, the radial velocity is negative over most $\theta$.
The plots of angular profiles of density, gas temperature, and “radiation”
temperature show that these three quantities change rapidly with $\theta$ for
Run 5b. With the decrease of $\alpha$, the density and radiation energy
density concentrate toward the equator, and the flow can be divided into two
areas with different temperature, i.e.,a low-temperature area near the equator
and a high-temperature area near the axis.
Figure 7.— Radial structure of some time-average quantities. The left panel is
for Run 2b ($\alpha=0.1$; red line) and Run 3b ($\alpha=0.05$; blue line),
while the right panel is for Run 4b ($\alpha=0.01$; red line) and Run 5b
($\alpha=0.005$; blue line). The plots from top to bottom are for density, gas
and radiation temperature, the ratio of gas pressure to radiation pressure,
specific angular momentum, and radial velocity, respectively. The dot-dashed
lines denote the profile of the one-dimensional solution. The black solid line
is to guide our eyes. In panel (D), the black solid line indicates the
Keplerian angular momentum.
Figure 8.— Time-averaged advection factor $q_{\text{adv}}$ (cf. Equation (3))
near the equator for the models of $\dot{m}_{\text{input}}=3000$. The solution
is averaged over an angle between $\theta=84^{\text{o}}$ and
$\theta=96^{\text{o}}$. Dotted lines indicate $q_{\text{adv}}$=1.
Figure 9.— Radial profiles of inflow (dashed line), outflow (dotted line), and
net rates (solid line) of Model B for various $\alpha$. This result is very
similar to the case of hot accretion flows, see text for details.
Figure 10.— Angular distribution of inflow (solid lines) and outflow (dashed
lines) rates of Model B. The black and red lines are for $r=5r_{s}$ and
$r=30r_{s}$.
In order to understand the angular profiles of radial velocity, we plot in
Figuer. 5 the angular distribution of radial forces acting on unit mass for
Run 5b at $r=5$ (top panel) and $30r_{s}$ (bottom panel), respectively. We can
see that the angular distribution of net force (solid lines) is similar to the
angular distribution of $v_{r}$ shown in Figure 4. In addition, Figure 5 shows
that within $10^{\circ}<\theta<90^{\circ}$ the radial component of centrifugal
force efficiently counteracts the gravity and even exceeds the gravity at some
degree. The top panel shows that the flows within $30^{\circ}$ from the
equator are super-Keplerian at $30r_{s}$. At $r\sim 10r_{s}$, the flows are
super-Keplerian within $40^{\circ}$ from the equator. However, within $\sim
20^{\circ}$ from the axis, the radial component of centrifugal force rapidly
decreases and cannot efficiently counteract the gravity. Hence, the equivalent
potential well is deeper near the axis than near the equator. On the other
hand, the radial component of the gas-pressure gradient force is negligible
within $60^{\circ}$ of the equator, while near the axis this component is not
negligible but is not the dominant force. The radial net force within
$60^{\circ}$ of the equator is dominated by the radial radiation force, the
gravity, and the radial centrifugal force. In the inner region, the radial net
force near the axis is dominated by the gravity, so that we see that there is
inflow near the axis. In the outer region, the radial net force near the axis
is dominated by the radial radiation force, so that we can see that there is
strong outflow near the axis. Hence, the radial radiation force plays an
important role in maintaining the radial equilibrium of flows near the equator
and driving the high-latitude outflow in the outer region.
Figure 6 shows the angular distribution of angular forces. It is seen that the
angular component of net force is nearly zero near the equator, which
indicates that flows are in force equilibrium near the equator. This is
because that the angular component of gas-pressure gradient force is also
negligible, and the radiation force balances the centrifugal force in the
$\theta$-direction. The angular motion of flows near the axis is controlled by
the centrifugal force and gas-pressure gradient force, while the angular
component of radiation force is negligible near the axis.
Figure 11.— Radial distribution of time-averaged values of flux-weighted Be
(in units of $v_{k}^{2}$), $T_{\rm gas}$ (in units of $T_{\rm vir}$), $v_{r}$
(in unit of $v_{k}$), and specific angular momentum. The solid and dashed
lines are for inflow and outflow, respectively. The black and red lines are
for Run 2b ($\alpha=0.1$) and Run 5b ($\alpha=0.005$), respectively. The
dotted line in the bottom right panel denotes the Keplerian angular momentum
at the equator.
### 3.3. Radial Structure of Accretion Flows
The radial structure of a slim disk has been solved using a vertical-
integrated one-dimensional method (e.g., Abramowicz et al. 1988). However, the
solution is one dimensional, thus the method cannot treat more viscous
components besides $T_{r\phi}$ and cannot address outflow. Here, on the basis
of the solution of two-dimensional simulation, we plot the radial structure of
the time-averaged flow near the equator in Figure 7. The solution is averaged
over the angle between $\theta=84^{\circ}$ and $\theta=96^{\circ}$. In Figure
7, the left panel is for Run 2b (red line) and Run 3b (blue line), while the
right panel is for Run 4b (red line) and Run 5b (blue line).
Panel (A) of Figure 7 shows the radial profiles of density. The dot-dashed
lines indicate the self-similar solution of the slim-disk model (Wang & Zhou
1999). We find that the density profile of different models can be well
described by a power law function, $\rho(r)\propto r^{-p}$ with $p\approx
0.55$. This result is much flatter than the self-similar solution of the slim
disk where $\rho(r)\propto r^{-1.5}$ (Wang & Zhou 1999), but it is very
similar to the case of hot accretion flows (Stone et al. 1999; Yuan et al.
2012, and references therein). The reason for the discrepancy is that in the
self-similar solution the mass accretion rate $\dot{M}(r)$ is assumed to be a
constant of radius, while as we will show in Section 3.4, the accretion rate
actually decreases inward because of the mass loss in outflows. Moreover, it
is interesting to note that the value of power law index of the density
profile, $p$, is quite “universal” for different models, although these models
have different $\alpha$ and different radial profiles of inflow rate (refer to
Section3.4). Bu et al. (2013) studied the effects of initial and boundary
conditions in simulations of accretion flow. They find a similar result,
namely the density profile is more converged compared with the diverse radial
profile of inflow rate.
Our result is apparently different from Figure 11 in Ohsuga et al. (2005), in
which they find $\rho(r)\propto r^{-1.5}$. For comparison with Ohsuga et al.
(2005), we have checked the models in Ohsuga et al. (2005) and found that only
when $\dot{m}_{\rm input}=500$ and 1000, and $\alpha=0.1$, the density profile
can be described by $\rho(r)\propto r^{-1.5}$. All other models have
$\rho(r)\propto r^{-0.55}$. Moreover, in the former case, the $\rho(r)\propto
r^{-1.5}$ density profile only holds in the range of $r\lesssim 30r_{s}$.
Beyond this radius, the profile becomes much flatter. But in the latter case,
the $\rho(r)\propto r^{-0.55}$ density profile holds until $r\sim 80r_{s}$, as
shown by Figure 6. The reason for the discrepancy is unclear.
Figure 12.— Convective stability analysis of Run 2b ($\alpha=0.1$; left) and
Run 5b ($\alpha=0.005$; right) at $t=69.233$ and $230.776$ orbits,
respectively. The dark region denotes negative $N_{\rm eff}^{2}$, i.e.,
convectively unstable region.
Panel (B) of Figure 7 shows the radial profiles of gas temperature $T_{\rm
gas}$ (red and blue dashed lines) and “radiation” temperature $T_{\rm rad}$
(red and blue solid lines). $T_{\rm rad}$ can be approximately described by a
radial power law function of $T_{{\rm rad}}\propto r^{-0.42}$, as the black
solid line shows. This is again flatter than the self-similar solution of the
slim-disk model, which has $T_{{\rm rad}}\propto r^{-5/8}$ as shown by the
dot-dashed line. We think the reason is because the compression work becomes
weaker because of the presence of outflow. In the inner region close to the
black hole, $T_{{\rm rad}}$ becomes flatter. We can find that $T_{{\rm gas}}$
and $T_{{\rm rad}}$ are nearly equal in the outer region, while the gas
temperature is higher than $T_{{\rm rad}}$ in the inner region. The radius
where the two temperatures deviate depends on the mass-injection rate and
viscous parameter $\alpha$. The higher the mass-injected rate and the smaller
the $\alpha$, the smaller the “deviation” radius. However, when $\alpha=0.01$
and 0.005, the “deviation radius” is located around the inner boundary. The
discrepancy of the two temperatures is because of the inefficient coupling
between gas and radiation. The energy transfer between gas and radiation is
controlled by the absorption opacity $\kappa_{\text{p}}$ (refer to the term
$|4\pi\kappa_{\text{p}}B-c\kappa_{\text{p}}E_{0}|$ ($B$ is the blackbody
radiation intensity) of Equations (7) and (8) in Ohsuga et al. 2005). The
absorption opacity $\kappa_{\text{p}}\propto\rho^{2}T_{\text{gas}}^{-7/2}$ is
due to free-free absorption and bound-free absorption. We find
$\kappa_{\text{p}}\propto r^{0.37}$, i.e., $\kappa_{\text{p}}$ decreases
inward. Therefore, at a small radius, the coupling between radiation and gas
is weak, and the temperature equilibrium between the gas and the radiation
field is not achieved before the gas falls onto the black hole. Smaller
$\alpha$ gives rise to smaller radial velocity, which provides more time to
transfer the energy of the gas to the radiation field. So, the “deviation
radius” moves inward with the decrease of $\alpha$.
Panel (D) in Figure 7 shows the radial distribution of the specific angular
momentum. We can see that $\alpha$ can affect the angular momentum
distribution, especially in the vicinity of the black hole ($r<$10$r_{s}$). In
the case of $\alpha$=0.01 and 0.005, the specific angular momentum becomes
super-Keplerian in the range of 3$-$6 $r_{s}$. $\dot{m}_{\rm input}$ also
affects the angular momentum distribution. The models with smaller $\alpha$
and lower $\dot{m}_{\rm input}$ have slightly flatter distribution. This
result does not agree with that of Abramowicz et al. (1988). Abramowicz et al.
(1988) identified the tendency of the specific angular momentum distribution
to become flatter with the increase of the accretion rate when the accretion
rate is less than $800\dot{M}_{\rm{crit}}$. They could not study the higher
accretion rate, because their method fails for
$\dot{M}>800\dot{M}_{\rm{crit}}$. The reason for the discrepancy is unclear.
To analyze the energetics of all the models, we define the advection factor
($q_{\text{adv}}$) of accretion flow as follows:
$q_{\rm adv}\equiv\frac{Q_{\rm adv}}{Q_{\rm vis}}=1-\frac{Q^{-}_{\rm
rad}}{Q_{\rm vis}},$ (3)
where $Q_{\rm adv}$ is the gas and radiation energy advection rate, $Q_{\rm
vis}=T^{2}/\eta$ is the viscous dissipation rate, and $Q^{-}_{\rm rad}$ is the
radiation cooling rate. Here $Q^{-}_{\rm rad}$ is defined as the $\theta$
component of $-\nabla\cdot F$ ($F$ is the radiation flux). Both
$Q^{-}_{\text{rad}}$ and $Q_{\rm{vis}}$ are obtained first by time-averaging
and then by averaging the quantity over an angle between $\theta=84^{\circ}$
and $\theta=96^{\circ}$. Figure 8 shows the time-averaged advection factor
near the equator for the models of $\dot{m}_{\rm{input}}=3000$ (i.e., Runs 2b,
3b, 4b, and 5b). We can see that $q_{\rm{adv}}$ is approximately close to a
constant in the range of $(5-70)r_{s}$. The value of $q_{\rm{adv}}$ deceases
inward when $r<10r_{s}$, especially for the models with small $\alpha$. Since
the opacity of accretion flow is dominated by scattering opacity and the half-
thickness of accretion decreases inward, the vertical opacity of accretion
flow decreases inward. Therefore, the photon trapping effect becomes weaker
and the radiation becomes stronger. $Q^{-}_{\rm{rad}}$ can even be larger than
$Q_{\rm{vis}}$, so $q_{\rm{adv}}$ is negative, as shown by Figure 8. This is
consistent with Abramowicz et al. (1988). This indicates that advection plays
a heating rather than cooling role, similar to the case of luminous hot
accretion flows (Yuan 2001).
Figure 13.— Effective temperature (left) and multi-color blackbody spectra
(right) of a slim-disk model (Run 5b with $\alpha=0.005$ and $T_{r\phi}$ and
$T_{\theta\phi}$). The solid and dashed lines correspond to the numerical
simulation and the one-dimensional global solution, respectively.
### 3.4. Inflow and Outflow: Rates and Properties
Following Stone et al. (1999), we define the mass inflow ($\dot{m}_{\rm in}$)
and outflow rates ($\dot{m}_{\rm{out}}$), in units of the critical mass
accretion rate $\dot{M}_{\rm{crit}}$, as the following time-averaged and
angle-integrated quantities:
$\dot{m}_{\rm in}(r)=-\frac{c^{2}}{L_{\rm Edd}}\int^{\pi}_{0}2\pi
r^{2}\rho\text{min}(\bm{v}_{r},0)\text{sin}\theta d\theta,$ (4) $\dot{m}_{\rm
out}(r)=\frac{c^{2}}{L_{\rm Edd}}\int^{\pi}_{0}2\pi
r^{2}\rho\text{max}(\bm{v}_{r},0)\text{sin}\theta d\theta.$ (5)
The net mass accretion rate, $\dot{m}_{\rm net}(r)=\dot{m}_{\rm
in}(r)-\dot{m}_{\rm out}(r)$, is the accretion rate that finally falls onto
the black hole. It is noted that the above rates are obtained by time-
averaging the integral rather than integrating the time-averages. According to
Equation (5), we know that the outflow rate is not a correct measurement of
real outflow, although it can provide some important information. This is
because it includes the contribution of turbulent motion.
Figure 9 shows the radial distribution of inflow, outflow, and net rates of
Model B ($\dot{m}_{\rm{input}}=3000$; i.e., Runs 2b, 3b, 4b, and 5b). We can
see that the inflow and outflow rates decrease inward, as in the case of hot
accretion flows (Stone, Pringle et al. 1999; Yuan et al. 2012, and references
therein). The radial profiles of $\dot{m}_{\text{in}}$ and
$\dot{m}_{\text{out}}$ can be described by a power law function of radius.
Using $\dot{m}_{\text{in}}(r)\propto r^{s}$ to fit the inflow rate in the
range of $(8-50)r_{s}$, we find that the value of $s$ is not sensitive to
$\dot{m}_{\rm input}$ but mainly determined by the value of $\alpha$. A large
$\alpha$ corresponds to a small $s$. We have calculated models with different
$\dot{m}_{\rm{input}}$ and derived the average values of $s$. They are $\sim
0.37,0.44,0.76$ ,and $0.98$ for models with $\alpha=0.1,0.05,0.01$, and
$0.005$, respectively. The dependence of $s$ on $\alpha$ can be approximately
described by a power law function $s\propto\alpha^{-0.33}$. This result is
again similar to the case of hot accretion flows. Yuan et al. (2012) find that
$s\sim 0.54$ and $0.65$ for $\alpha=0.01$ and $0.001$, respectively. Note that
the definition of $\alpha$ is different in the two works. We find that the
value of $\alpha$ in this paper corresponds to a smaller $\alpha$ in Yuan et
al. (2012). The quantitative comparison of the value of $\alpha$ is difficult.
As argued in Yuan et al. (2012) and F. Yuan et al. (2013, in preparation), in
the case of hot accretion flow, the inward decrease of inflow rate is due to
mass loss via outflows. We believe this is also the case for the slim disk.
The similar slope in the slim disk and hot accretion flow implies that the
strength of outflow is similar in the two cases. We will discuss the possible
origin of outflow below.
In order to analyze the angular distribution of the mass accretion rate, we
time-average the following inflow and outflow rates as a function of $\theta$:
$\dot{m}_{\rm in}(\theta)=-2\pi
r^{2}\rho\text{min}(\bm{v}_{r},0)\text{sin}\theta\Delta\theta\frac{c^{2}}{L_{\rm
Edd}},$ (6) $\dot{m}_{\rm out}(\theta)=2\pi
r^{2}\rho\text{max}(\bm{v}_{r},0)\text{sin}\theta\Delta\theta\frac{c^{2}}{L_{\rm
Edd}}.$ (7)
Similar to Equation (5), Equation (7) also does not correctly measure the
angular distribution of real outflow. The results are shown in Figure 10 for
Model B. We can see that their angular distributions are nearly symmetric to
the equator and become broader with the increase of radius and/or $\alpha$.
The distribution is different than the case of hot accretion flows (refer to
Figures 2 and 3 in Yuan et al. 2012). Firstly, in the case of slim disks, the
inflow and outflow rates are “synchronous”, i.e, they reach their maximum at
the same $\theta$ angle. But in the case of hot accretion flows, they are not
(compare Figure 10 in the present paper and Figures 2 and 3 in Yuan et al.
2012). Secondly, the outflow rate centers around the equatorial plane in the
present case, while in the case of hot accretion flows, the maximum of outflow
rate is located roughly at the surface of the disk (their Model B and Model
C). In that case, the inflow and outflow rates are not symmetric to the
equatorial plane. The reason for the discrepancy is unclear. The outflow
becomes stronger with the increase of radius. This is consistent with Figure 9
and is easy to understand. Compared with the region around the equator, the
inflow and outflow rates are almost negligible at the high-latitude region,
where the radial velocity of outflow is very high. This is because the density
of high-latitude flows is very low. The high-latitude outflow is likely driven
by radiation, but its contribution to outflow rate seems to be negligible. The
outflow rate is dominated by the low-latitude outflow. The nature of the low-
latitude outflow, namely whether they are real systematic outflow or simply
turbulence, is an important question and needs to be studied in the future.
Another question is whether the origins of outflow are real. In the case of
hot accretion flows, Yuan et al. (2012) identify the mechanism of producing
the outflow by buoyancy when magnetic field is absent. This is because a hydro
accretion flow is convectively unstable. In Section 3.5 of the present paper,
we analyze the convective stability of slim disks.
Following Yuan et al. (2012), we analyze some properties of inflow and
outflow, such as the Bernoulli parameter, gas temperature, radial velocity,
and angular momentum. The motivation is to study the mechanism of producing
outflow. We calculate the flux-weighted quantities (refer to Equations (8) and
(9) in Yuan et al. 2012) and then time-average the quantities. Figure 11 shows
the radial distribution of flux-weighted quantities ($Be$ in units of
$v_{k}^{2}$, where $v_{k}$ is the Keplerian velocity; $T_{\rm gas}$ in units
of the virial temperature $T_{\rm vir}\equiv\frac{GMm_{p}}{3kr}$, where
$m_{p}$ is the photon mass and $k$ the Boltzmann constant;and $v_{r}$ in unit
of $v_{k}$) of inflow and outflow for Run 2b and Run 5b, respectively. In both
models, Be is negative. The value of Be is in general smaller than that of hot
accretion flows (Yuan et al. 2012), which is of course because the energy loss
in slim disk is stronger.
The top right panel in Figure 11 shows that the temperature of outflow is
higher than that of inflow. This seems to suggest that the mechanism of
outflow production is because of buoyancy, like in the case of hydrodynamical
hot accretion flow. Moreover, the discrepancy of the two temperatures is
larger when $\alpha$ is larger (Run 2b). The convective stability of slim disk
is analyzed in Section 3.5. The bottom left panel in Figure 11 shows that the
radial velocity of outflow can be well described by $v_{r}/v_{k}\sim$ const.
This is similar to hot accretion flows (Yuan et al. 2012). It is interesting
to note that when $\alpha$ is smaller, the radial velocities of both inflow
and outflow are smaller. For inflow, this is easy to understand. For outflow,
the discrepancy of the radial velocity may be related to the discrepancy of
the temperature between inflow and outflow. As we can see from the top right
panel, when $\alpha$ is smaller, the discrepancy is smaller, and thus the
buoyancy may be weaker. The reason why the temperature discrepancy in the case
of small $\alpha$ is smaller is perhaps related to the magnitude of the
convective energy flux. The convective energy flux transports energy along
surfaces of constant $r\sin^{2}(\theta)$ (Quataert & Gruzinov 2000). The
magnitude of this flux is proportional to the convective diffusion coefficient
$\alpha_{c}$, and a smaller $\alpha$ corresponds to a smaller $\alpha_{c}$
(Narayan et al. 2000). Therefore, when $\alpha$ is larger, the fluids at the
surface are more strongly heated by the stronger convective energy flux. At
last, the bottom right panel shows that the angular momenta of inflow and
outflow are almost identical. This is similar to the case of hydrodynamical
hot accretion flows (Yuan et al. 2012).
### 3.5. Convective Stability
In the case of hydrodynamical hot accretion flow, it is shown that the inward
decrease of inflow rate is because of mass loss in outflow produced by the
buoyancy associated with convective instability (Yuan et al. 2012). In this
section, we analyze the convective stability of the slim disk on the basis of
our simulation data.
The energy equation of accretion flows can be written as
$Q^{-}_{\rm adv}=Q^{+}_{\rm vis}-Q^{-}_{\rm rad},$ (8)
where $Q^{-}_{\rm adv}=\Sigma\upsilon_{r}T(dS/dr)$ ($\Sigma$ is surface
density, and $S$ is the specific entropy). In general, $Q^{+}_{\rm
vis}>Q^{-}_{\rm rad}$ so that $\Sigma\upsilon_{r}T(dS/dr)>0$ and $dS/dr<0$.
The inward increase of entropy is thought of as a necessary condition of
convective instability for rotating flow. A series of simulation studies
(Igumenshchev & Abramowicz 1999; Stone et al. 1999; Igumenshchev & Abramowicz
2000; Yuan & Bu 2010) verified that hot accretion flows are convectively
unstable, confirming the prediction of Narayan & Yi (1994). Compared with hot
accretion flows, the physics of a slim disk is different. In a slim disk, the
gas particles can efficiently radiate, but the photons are trapped and hence
cannot efficiently escape from the system because of the large scattering
optical depth. In addition, a slim disk is supported by the radiation
pressure, and hence the specific entropy is dominated by the radiation
photons. Sadowski et al. (2009, 2011) found that a radiation pressure-
supported disk is convectively unstable. Gu (2012) revisited this problem,
taking into account the local energy balance between the viscous heating and
the advective and radiative cooling, and found that a slim disk is
convectively stable. Gu (2012) thought that the significant difference in the
results between their work and Sadowski et al (2009). is probably related to
the different approaches for describing the vertical structure. But in Gu
(2012), a self-similar solution of the radius is adopted, the consequence of
which is unknown.
Here we analyze the convective stability of a slim disk on the basis of our
simulation data. We assume that the Höiland criterion (e.g., Begelman & Meier
1982) is applicable to the slim disk. The condition in cylindrical coordinates
($R,\phi,z$) for convective stability in a rotating accretion flow (Tassoul
1978) is
$N^{2}_{\rm eff}=N^{2}_{R}+N^{2}_{z}+\kappa^{2}>0,$ (9)
where $N_{\rm eff}$ is defined as an effective frequency, $\kappa$ is the
epicyclic frequency which is calculated by
$\kappa^{2}=\frac{1}{R^{3}}\frac{\partial l^{2}}{\partial R}$ ($l$ is the
specific angular momentum), and $N_{R}$ and $N_{z}$ is the $R$ and $z$
component of the well-known Brunt–Väisälä frequency, respectively, which can
be calculated by
$N^{2}_{R}=-\frac{1}{\gamma_{r}\rho}\frac{\partial P}{\partial
R}\frac{\partial S}{\partial R}$ (10)
and
$N^{2}_{z}=-\frac{1}{\gamma_{r}\rho}\frac{\partial P}{\partial
z}\frac{\partial S}{\partial z},$ (11)
where $dS\propto d\text{ln}(\frac{P}{\rho^{\gamma_{r}}})$, $P$ is the total
pressure, and $\gamma_{r}$ is the adiabatic index. Here $P=p_{g}$ and
$\gamma_{r}=5/3$ are employed for the flows of low-density ($\rho<10^{-5}$
g/cm3) and high-gas temperature ($T_{\rm gas}>10^{9}$ K) (i.e., outflow or
corona region), while $P=E_{0}/3$ and $\gamma_{r}=4/3$ are employed for other
radiation-dominated flows (i.e., the disk body) where radiation and gas are
effectively coupled because of the large scattering opacity.
The results are shown in Figure 12. The left panel is for Run 2b at $t=69.233$
orbits, while the right panel for Run 5b at $t=230.777$ orbits. The dark
region denotes the unstable region, i.e., $N^{2}_{\text{eff}}<0$. We can see
that the results are somewhat subtle. For Run 2b nearly half of the region is
convectively unstable, while for Run 5b the fraction of the unstable region is
much less. Given this result, we can perhaps say that Run 2b is marginally
convectively unstable and Run 5b is stable. In both cases, compared with the
case of a hydrodynamical hot accretion flow (refer to Figure 6 in Yuan & Bu
2010), the unstable region becomes significantly smaller. We speculate that
the difference between the present work and a hydrodynamical hot accretion
flow (i.e., Figure 6 in Yuan & Bu 2010) is due to radiation. Recall that in an
magnetohydrdynamical hot accretion flow, the magnetic field plays a role of
stabilizing the convection (Balbus & Hawley 2002; Narayan et al. 2002). Here
radiation plays an effectively similar role although the underlying physics
may be different. Physically, when radiation is important as in a slim disk,
radiation can directly transport energy, thus the “need” for convection to
transport energy becomes weaker. Why is the unstable region in Run 2b larger
than that in Run 5b? The only difference between Run 2b and Run 5b is the
value of $\alpha$, which is 0.1 and 0.005, respectively. Compared with the
case of a large $\alpha$, when $\alpha$ is smaller, the convective energy flux
is smaller, thus the role of radiation to stabilize the flow becomes
relatively more effective. This explains the difference between Run 2b and Run
5b. This may also explain why the temperature discrepancy between inflow and
outflow in Run 2b is larger than in Run 5b (refer to Figure 11).
### 3.6. Multi-color Blackbody Spectra
As an initial step of the application of our numerical simulation, in this
section we compare the emitting spectra from two-dimensional numerical
simulations and one-dimensional analytical global solutions. For simplicity,
we assume a multi-color blackbody spectrum on the basis of effective
temperatures. In reality, however, Compton scattering plays an important role
in the emitted spectrum (Kawashima et al. 2012). We identify the radiation
temperature $T_{\rm rad}$ at the photosphere (where the effective optical
depth equals 1) as the effective temperature. The effective optical depth is
calculated by
$\tau_{\textbf{eff}}=\sqrt{\tau_{\text{ab}}(\tau_{\text{ab}}+\tau_{\text{sc}})}$,
where $\tau_{\text{ab}}$ and $\tau_{\text{sc}}$ are the absorption and Thomson
scattering optical depth integrated from the outer boundary along the
Z-direction. We adopt the free free absorption opacity ($\kappa_{\text{ff}}$).
Taking Run 5b as an example, we have calculated the emitted spectrum based on
the simulation data. Only the range of $3.5r_{s}<r<80r_{s}$ is considered. The
reason is that only the region of $r<80r_{s}$ has achieved steady state while
the region of $r<3.5r_{s}$ is effectively optically thin due to small
absorption optical depth caused by high gas temperatures. We also calculate
the global solutions of one-dimensional model corresponding to the parameters
of Run 5b(refer to Abramowicz et al. 1988 and Watarai et al. 2000 for the
calculation approach) and the corresponding spectrum. Figure 13 shows the
calculation results. We note that the calculated luminosity based on our two-
dimensional numerical simulation agrees well with that obtained by Kawashima
et al. (2012). Significant differences of temperature and subsequently emitted
spectrum between the two models can be found. This result can give us some
initial idea of the difference between simple one-dimensional calculation and
the more realistic two-dimensional simulation, which should be useful when
applying the slim disk theory to observations. The detailed calculation of the
spectrum can be found in Kawashima et al. (2012) and beyond the scope of the
present work.
## 4\. SUMMARY AND DISCUSSIONS
In this paper, we have performed a two-dimensional radiative hydrodynamical
numerical simulation of slim disks. The technical differences between the
present work and Ohsuga et al. (2005) are that we include an additional
component of viscous stress, i.e., $T_{\theta\phi}$, and consider various
values of the viscous parameter $\alpha$. We find that the component
$T_{\theta\phi}$ plays an important role in transporting the angular momentum
between different latitudes. As a result, compared with the case of no
$T_{\theta\phi}$ component (Ohsuga et al. 2005), the high-latitude outflow
(within $30^{\circ}$ from the axis) rotates slower, while the flow close to
the equatorial plane rotates faster. In addition, we find that the high-
latitude outflow has higher density, lower speed, and a smaller Bernoulli
parameter. For the effect of the magnitude of $\alpha$, we find that the
models with different $\alpha$ have similar radial structure but different
angular structure. The value of $\alpha$ strongly affects the radial velocity
and the value of $Be$ of outflow at a high latitude.
We have paid more attention in the present work to studying the physics of
slim disks. We have calculated the radial profiles of inflow and outflow rates
defined by Equations (4) and (5). We have found that both of them decrease
inward. Specifically, the inflow rate can be well described by a power law
form, $\dot{M}_{\rm in}\propto r^{s}$. The value of $s$ is not sensitive to
the accretion rate but is mainly dependent on the value of $\alpha$. For
$\alpha=0.1,0.05,0.01$ ,and $0.005$, $s\sim 0.37,0.44,0.76$, and $0.98$,
respectively (Figure 9). Correspondingly, the radial profile of density
becomes flatter compared with the case of a constant $\dot{M}(r)$. The density
profile can be described by $\rho(r)\propto r^{-p}$. It is interesting to note
that the value of $p$ is within a narrow range, $p\approx 0.55$ for
$\alpha\sim 0.005-0.1$ (Figure 7).
These results are very similar to a hydrodynamical hot accretion flow. In that
case, Yuan et al. (2012) show that the inward decrease of inflow rate is
because of the mass loss in outflow. We believe this is also the case for the
present slim disk. In the case of hot accretion flows, the mechanism of
producing outflow is identified to be buoyancy associated with the convective
instability of the accretion flow. To investigate the origin of outflow in the
slim disk, we first calculate and compare the properties of inflow and
outflow. We have found that the temperature of inflow is lower than that of
outflow. The discrepancy is larger when $\alpha$ is larger (Figure 11). This
suggests the existence of convective instability at some level, especially
when $\alpha$ is large. We then analyze the convective stability of the
accretion flow on the basis of our simulation data. The result is somewhat
subtle. When $\alpha=0.1$, about half of the region of the accretion flow is
convectively unstable, but when $\alpha=0.005$, less than half of the region
is unstable (Figure 12). Recall that a non-radiative accretion flow is
convectively unstable, and we speculate that radiation can stabilize the
convection. Physically this is because radiation can also take away energy,
like convection. The effectiveness of this stabilizing seems to depend on the
magnitude of $\alpha$. When $\alpha$ is smaller, it is more effective, i.e.,
the accretion flow tends to be more convectively stable (Section 3.5).
Returning to the issue of the radial profile of inflow rate, two questions
arise. The first question is what is the mechanism of producing outflow,
especially if the slim disk is roughly convectively stable when $\alpha$ is
small? We speculate that the outflow may be produced by radiation force. Or
more precisely speaking, both convection and radiation force can produce
outflow. Their relative importance may depend on $\alpha$. When $\alpha$ is
small, the convective energy flux is weaker, thus radiation force will be the
dominant mechanism of producing outflow. The force analysis presented in
Figure 5 already suggests the importance of radiation force. We will study
this problem in more detail in a future work. If this speculation is correct,
the mechanism of producing outflow in slim disks and hot accretion flows is
quite different. We note in this context that the angular distribution of the
outflow and inflow rates in our simulation are quite different from the case
of hot accretion flows (refer to Section 3.4). This may be evidence for the
different origin of outflow in slim disk and hot accretion flows.
Combing the cases of slim disks and the hydrodynamical hot accretion flow, we
find that the slope of the radial profile of inflow rate is quite similar,
although the mechanisms of producing outflow in the two cases are likely
different, as we state above. In fact, the slope is even similar to the case
of magnetohydrodynamical hot accretion flow as well. In that case, the
mechanism of producing outflow is identified to be a Lorentz force such as
magnetocentrifugal force (Yuan et al. 2012; F. Yuan et al. 2013, in
preparation). What is the reason for the same slope in spite of different
mechanisms? Begelman (2012) may provide an answer to the question of why the
hydrodynamical and magnetohydrodynamical hot accretion flows have the same
radial profile of inflow rate, see also the summary presented in Yuan et al.
(2012). Now the similarity among the three cases seems to indicate that the
analysis in Begelman (2012) also applies to radiation-dominated slim disks.
Finally, our whole investigation presented in this paper is based on the
assumption that the mass accretion rate can be super-Eddington. However,
observations of a large sample of AGNs with 407 sources show that almost all
active galactic nuclei (AGNs) are radiating below $L_{\rm Edd}$ (Kollmeier et
al. 2006). Later, Steinhardt & Elvis (2010) extended this study to a much
larger sample consisting of 62,185 quasars from the Sloan Digital Sky Survey,
and they got a similar conclusion (but see Kelly & Shen (2013) for a different
opinion). Liu et al. 2013 most recently address this sub-Eddington puzzle. The
basic idea is that because of radiative feedback, the mass accretion rate at
the inner accretion flow can be self-regulated and thus cannot be super-
Eddington. Unfortunately they find that this mechanism cannot fully solve this
problem.
We thank Wei-Min Gu and Aleksander Sadowski for helpful discussions and the
anonymous referee for constructive comments. This work was supported in part
by the Natural Science Foundation of China (grants 11103059, 11121062,
11133005, and 11003052), the National Basic Research Program of China (973
Program 2009CB824800), the CAS/SAFEA International Partnership Program for
Creative Research Teams, and the Fundamental Research Funds for the Central
Universities (No.CQDXWL-2012-019). The simulations were carried out both at
Shanghai Supercomputer Center and the Super Computing Platform of Shanghai
Astronomical Observatory.
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|
arxiv-papers
| 2013-06-08T02:47:02 |
2024-09-04T02:49:46.234117
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Xiao-Hong Yang (CQU), Feng Yuan (SHAO), Ken Ohsuga (NAOJ), and De-Fu\n Bu (SHAO)",
"submitter": "Xiao-Hong Yang",
"url": "https://arxiv.org/abs/1306.1871"
}
|
1306.1917
|
# Clifford and Euclidean translations of circles
Niels Lubbes
###### Abstract
Celestials are surfaces that admit at least two families of circles. We
classify celestials that are the Clifford translation of a circle along a
circle in the three-sphere. A Clifford torus is well known to be the Clifford
translation of a great circle along a great circle in the three-sphere. Our
contribution to this classical subject is that a celestial with four families
of circles and no real singularities is Möbius equivalent to a Clifford torus.
The main result of this paper is that a celestial of degree eight with a
family of great circles is the Clifford translation of a great circle along a
little circle. This allows us to classify such celestials up to homeomorphism.
We conclude with a classification of Euclidean translational celestials and
show that they are not Möbius equivalent to Clifford translations of circles.
###### Contents
1. 1 Introduction
2. 2 Geometry background
3. 3 Elliptic type of Clifford celestials
4. 4 Two-ruled Clifford celestials
5. 5 One-ruled Clifford celestials
6. 6 Euclidean translational celestials
7. 7 Acknowledgements
## 1 Introduction
A celestial is an irreducible embedded surface that admits at least two one-
dimensional families of circles (§2.6 and §2.7). The surfaces in Figure 1 are
examples of celestials. Such surfaces contain at least two circles through a
generic closed point. Thus a celestial can be obtained by moving a circle in
space along a closed loop in two different ways. The radius of the circle is
in general allowed to change during its motion. In this paper we investigate
the case where the movements are translations in a metric space and thus the
radius remains constant. The terminology of this introduction will be made
precise in §2 and we provide pointers to the relevant subsections.
| |
---|---|---
Figure 1a | Figure 1b | Figure 1c
Marcel Berger [2, II.7, page 100] shares some historical insights concerning
celestials. In particular he mentions a sculpture in the Strasbourg cathedral
which illustrates so called Villarceau circles as in Figure 1a. Although Yvon
Villarceau [23] published about these circles in 1848, the cathedral was built
between 1176 and 1439. Gaston Darboux [6] mentions around 1880 that celestials
carry either infinite or at most six families of circles. For modern
treatments see [1, Chapters 18-20] and [5, VII]. After 1980 this topic started
to revive again [3, 22, 12, 20]. More recently celestials have been
investigated in [19] and [18] with also in mind the applications in geometric
modeling. In [16] we classified celestials $S$ up to Euclidean type (§2.7) and
up to isomorphism of their real enhanced Picard groups ${\mathcal{P}}(S)$
(§2.5).
The three-sphere will be our model for three-dimensional Möbius-, elliptic-
and Euclidean- geometry (§2.2, §2.3 and §2.4). The elliptic transformations of
the three-sphere that are translations with respect to the elliptic metric are
called Clifford translations (§2.3).
A Clifford celestial is the Clifford translation of a circle along a circle in
the three-sphere (§2.7). The celestials in Figure 1 are stereographic
projections of Clifford celestials. When both circles are great we obtain a
Clifford torus as in Figure 1a (§2.7). In Figure 1b only one circle is great
and in Figure 1c both circles are little.
For the classification of Clifford celestials $S$ we consider an elliptic
invariant called the elliptic type (§2.7) and the algebraic structure
${\mathcal{P}}(S)$ (§2.5) which is Möbius invariant. We also consider the
singular locus of $S$ which is Möbius invariant as well.
Felix Klein [13, page 234] established that the elliptic type of a celestial
determines whether this celestial is a Clifford torus. We recall this result
in Theorem 1.a). Thus we can see from the elliptic type of a celestial whether
it is the Clifford translation of a great circle along a great circle. It
turns out that ${\mathcal{P}}(S)$ is a Möbius invariant with the same property
(Theorem 2) and as a consequence we find that a celestial with no real
singularities and four families of circles must be Möbius equivalent to a
Clifford torus (Corollary 1).
If $S$ is the Clifford translation of a great circle along a little circle
then $S$ is either a Clifford torus or a celestial of degree eight as in
Figure 1b. If $S$ is of degree eight then its elliptic type is stated in
Theorem 1.b). Conversely, the elliptic type of a celestial uniquely determines
whether this celestial is the Clifford translation of a great circle along a
little circle (Theorem 1.b)). We are able to characterize the singular locus
of $S$ (Theorem 3) and its topology (Theorem 4). It is remarkable that a
celestial of degree eight that admits a family of great circles must be a
Clifford celestial (Corollary 2). It should be noted that a celestial with two
families of great circles is not necessarily a Clifford torus (Example 1).
If a celestial is the Clifford translation of a little circle along a little
circle then its elliptic type is as in Theorem 1.c). We conjecture the
converse of Theorem 1.c), thus that the property of a celestial being the
Clifford translation of a little circle along a little circle is uniquely
determined by its elliptic type. We conclude this paper with a classification
of Euclidean translational celestials and we show that Clifford celestials are
not Euclidean translational (Theorem 5).
## 2 Geometry background
We work in the category of real algebraic varieties and we denote projective
$n$-space by ${\mathbb{P}}^{n}$. A real variety $X$ is a complex variety
together with a complex conjugation
$X\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}X$.
### 2.1 Translations
Let $(M,d)$ be a metric space and let $G$ denote its group of isometries. The
translations of $M$ are defined as
$T=\\{~{}g\in G~{}|~{}d(v,g(v))=d(w,g(w))\textrm{ for all }v,w\in M~{}\\}.$
Let $T_{\sharp}\subset T$ denote a group with the property that for all
$v,w\in M$ there exists a unique $t\in T_{\sharp}$ such that $t(v)=w$. Let
$m\in M$ be a distinguished point. If $C\subset M$ is a subset such that $m\in
C$ then we define,
$T_{\sharp}(C)=\\{~{}t\in T_{\sharp}~{}|~{}t(m)\in C~{}\\}.$
We call $C^{\prime}$ a translation axis of $T_{\sharp}(C)$ if and only if
$t(c)\in C^{\prime}$ for all $t\in T_{\sharp}(C)$ and $c\in C^{\prime}$. If
$C_{1}$ and $C_{2}$ are curves in $M$ such that $m\in C_{1}\cap C_{2}$ then
the $T_{\sharp}$-translation of $C_{1}$ along $C_{2}$ is defined as
$C_{1}\ast_{\sharp}C_{2}=\\{~{}t(C_{1})~{}|~{}t\in T_{\sharp}(C_{2})~{}\\}.$
In this article we will implicitly assume that $m\in C_{1}\cap C_{2}$.
### 2.2 Möbius geometry
The Möbius three-sphere is defined as
${\mathbb{S}}^{3}:=\\{~{}x\in{\mathbb{P}}^{4}~{}|~{}-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=0~{}\\}.$
The Möbius transformations $PO(1,4)$ are defined as the projective
isomorphisms of ${\mathbb{S}}^{3}$. The circles are defined as conics in
${\mathbb{S}}^{3}$.
### 2.3 Elliptic geometry
The elliptic absolute is defined as
$E:=\\{~{}x\in{\mathbb{S}}^{3}~{}|~{}x_{0}=0~{}\\}.$
The elliptic transformations are defined as the Möbius transformations that
preserve $E$. The central projection of ${\mathbb{S}}^{3}$ identifies the
antipodal points and is defined as
$\tau:{\mathbb{S}}^{3}\rightarrow{\mathbb{P}}^{3},\quad(x_{0}:x_{1}:x_{2}:x_{3}:x_{4})\mapsto(x_{1}:x_{2}:x_{3}:x_{4}),$
with branching locus $\tau(E)$. We call a circle $C\subset{\mathbb{S}}^{3}$
great if $\tau(C)$ is a line and little otherwise. There exists a unique great
circle $C$ through any two non-antipodal points $v,w\in{\mathbb{S}}^{3}$ with
$C\cap E=\\{a,b\\}$. The elliptic metric on the real points of
$({\mathbb{S}}^{3}\setminus E)$ is defined in terms of half the logarithm of a
cross ratio,
$d_{e}(v,w):=\frac{1}{2}\log[\tau(a),\tau(v),\tau(w),\tau(b)],$
or $d_{e}(v,w):=0$ if $v$ and $w$ are antipodal. The elliptic transformations
are the isometries with respect to this metric. The elliptic absolute $E$
admits two families of lines, called left generators and right generators
respectively. The left Clifford translations $T_{L}$ ⟦right Clifford
translations $T_{R}$⟧ are defined as the elliptic translations that preserve
the left ⟦right⟧ generators. It follows from Proposition 1.a) below that
$T_{\sharp}$ of §2.1 is equal to either $T_{L}$ or $T_{R}$. By convention we
mean with Clifford translations always the left version unless explicitly
stated otherwise.
###### Proposition 1.
(Clifford translations)
* a)
Let $S^{3}$ be an affine chart of ${\mathbb{S}}^{3}$ defined by $x_{0}\neq 0$.
If we identify $S^{3}$ with the unit quaternions $Q$ then the left ⟦right⟧
Clifford translations are the group actions $s\mapsto q\star s$ ⟦$s\mapsto
s\star q$⟧ for $s\in S^{3}$, $q\in Q$ and $\star$ the Hamiltonian product.
* b)
If $C_{1}$ and $C_{2}$ are curves in ${\mathbb{S}}^{3}$ then
$C_{1}\ast_{L}C_{2}=C_{2}\ast_{R}C_{1}$.
* c)
If a curve $C$ is the left ⟦right⟧ Clifford translation of a curve
$C^{\prime}$ then $C$ and $C^{\prime}$ intersect the same pair of left ⟦right⟧
generators.
* d)
A great circle $C$ is the left ⟦right⟧ Clifford translation of a great circle
$C^{\prime}$ if and only if $C$ and $C^{\prime}$ intersect the same pair of
left ⟦right⟧ generators.
###### Proof.
See [5, Section 7.7] for a). Assertion b) now follows from the quaternions
being associative. Assertions c) and d) follow from the left ⟦right⟧ Clifford
translations preserving the left ⟦right⟧ generators. For assertion d) note
that great circles intersect $E$ in complex conjugate points and that there is
a unique great circle through two points. ∎
### 2.4 Euclidean geometry
The Euclidean absolute in ${\mathbb{S}}^{3}$ is defined as
$A:=\\{~{}x\in{\mathbb{S}}^{3}~{}|~{}x_{0}-x_{4}=0~{}\\}.$
The stereographic projection is conformal and up to Möbius equivalence defined
as
$\pi:{\mathbb{S}}^{3}\rightarrow{\mathbb{P}}^{3},\quad(x_{0}:x_{1}:x_{2}:x_{3}:x_{4})\mapsto(x_{0}-x_{4}:x_{1}:x_{2}:x_{3}),$
with $(1:0:0:0:1)$ the center of projection [4, Section 8]. The Euclidean
metric on the real points of $({\mathbb{S}}^{3}\setminus A)$ is defined as
$d_{a}(v,w):=\sqrt{\overset{}{\underset{i\in[1,3]}{\sum}}\left(\frac{\pi(v)_{i}}{\pi(v)_{0}}-\frac{\pi(w)_{i}}{\pi(w)_{0}}\right)^{2}}.$
The Euclidean transformations are defined as the isometries with respect to
this metric. Euclidean similarities are the Möbius transformations that
preserve $A$. The unique family of lines of the Euclidean absolute are called
Euclidean generators. The Euclidean translations $T_{A}$ are the Euclidean
transformations that preserve the Euclidean generators. In Euclidean geometry
we set $T_{\sharp}$ in §2.1 equal to $T_{A}$. The Euclidean rotations are
Möbius transformations that preserve both the Euclidean- and elliptic-
absolutes.
### 2.5 Real enhanced Picard groups
Let $X$ be a complex surface together with a complex conjugation
$X\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}X$. The real enhanced
Picard group is defined as
${\mathcal{P}}(X):=(~{}\textrm{Pic}X,K,\cdot,h,\sigma_{*}),$
where $\textrm{Pic}X$ denotes the Picard group [14, Section 1.1], $K$ is the
canonical divisor class of $X$, $\cdot$ denotes the bilinear intersection
product on divisor classes, $h^{i}(C)$ assigns the $i$-th Betti number to a
divisor class $C\in\textrm{Pic}X$ with respect to sheaf cohomology [9, Section
0.3], and
$\textrm{Pic}X\stackrel{{\scriptstyle\sigma_{*}}}{{\longrightarrow}}\textrm{Pic}X$
is an involution induced by $\sigma$. We consider real enhanced Picard groups
isomorphic if and only if there exists an isomorphism of the Picard groups
that preserves $K$ and is compatible with $\cdot$, $h$ and $\sigma_{*}$. We
call ${\mathcal{P}}(X)$ of
* •
type S2 if and only if ${\mathcal{P}}(X)={\mathbb{Z}}\langle H,F\rangle$,
$K=-2(H+F)$, $H^{2}=F^{2}=0$, $HF=1$, $\sigma_{*}(H)=F$ and there exist no
class $C\in{\mathcal{P}}(X)$ such that $h^{0}(C)>0$ and $C^{2}<0$.
* •
type S4 if and only if ${\mathcal{P}}(X)={\mathbb{Z}}\langle
H,Q_{1},\ldots,Q_{5}\rangle$, $-K=3H-Q_{1}-\ldots-Q_{5}$, $H^{2}=1$,
$Q_{i}Q_{j}=-\delta_{ij}$, $HQ_{i}=0$ for all $i,j\in[1,5]$,
$\sigma_{*}:$ $(H,Q_{1},\ldots,Q_{5})$ $\mapsto$ $($ $2H-Q_{1}-Q_{2}-Q_{3}$,
$H-Q_{2}-Q_{3}$, $H-Q_{1}-Q_{3}$, $H-Q_{1}-Q_{2}$, $Q_{5}$, $Q_{4}$ $)$ and
$h^{0}(H-Q_{1})=h^{0}(H-Q_{2})=h^{0}(H-Q_{3})=h^{0}(2H-Q_{1}-Q_{2}-Q_{4}-Q_{5})=2$.
* •
type S8 if and only if ${\mathcal{P}}(X)={\mathbb{Z}}\langle H,F\rangle$,
$K=-2(H+F)$, $H^{2}=F^{2}=0$, $HF=1$, $\sigma_{*}$ is the identity,
$h^{0}(F)=h^{0}(H)=2$ and there exist no class $C\in{\mathcal{P}}(X)$ such
that $h^{0}(C)>0$ and $C^{2}<0$.
* •
type E4 if and only if ${\mathcal{P}}(X)={\mathbb{Z}}\langle H,F\rangle$,
$K=-2(H+F)$, $H^{2}=F^{2}=0$, $HF=1$ and $\sigma_{*}$ is the identity,
$h^{0}(F)=h^{0}(H)=2$ and there exist no class $C\in{\mathcal{P}}(X)$ such
that $h^{0}(C)>0$ and $C^{2}<0$.
* •
type U1 if and only if ${\mathcal{P}}(X)={\mathbb{Z}}\langle H\rangle$,
$K=-3H$, $H^{2}=1$ and $\sigma_{*}$ is the identity.
### 2.6 Classes of curves, singular points and families
We define a surface pair $(X,D)$ as a nonsingular surface $X$ together with a
divisor class $D$, such that the map
$X\stackrel{{\scriptstyle\varphi_{D}}}{{\longrightarrow}}Y\subset{\mathbb{P}}^{h^{0}(D)-1}$
associated to $D$ is a birational morphism that does not contract exceptional
curves. The polarized model $Y$ of $(X,D)$ is defined as $\varphi_{D}(X)$. In
this paper we will consider linear projections $Z$ of $Y$ such that the center
of projection does not lie on $Y$,
$X\stackrel{{\scriptstyle\varphi_{D}}}{{\longrightarrow}}Y\subset{\mathbb{P}}^{h^{0}(D)-1}\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}Z\subset{\mathbb{P}}^{n}.$
If $C\subset Z$ is a curve then we can strict transform $C$ along
$(\nu\circ\varphi_{D})$ and associate a class
$[C]\in{\mathcal{P}}(X),$
to this curve [14, Section 1.1]. The divisor class $[p]\in{\mathcal{P}}(X)$ of
an isolated singularity $p\in Z$ is defined as the class of the curves in $X$
that are contracted onto $p$ by $(\nu\circ\varphi_{D})$ [7, Section 8.2.7]. A
family of curves on a surface $Z$ parametrized by a nonsingular curve $I$ is
defined as an irreducible codimension one algebraic subset,
$F\subset Z\times I,$
such that the first projection is dominant. The divisor class
$[F]\in{\mathcal{P}}(X)$ of a family $F$ is defined as the divisor class of a
generic curve in $F$. The arithmetic genus $p_{a}(C)$ and the geometric genus
$p_{g}(C)$ of a curve $C\subset Z$ can be computed using [10, Proposition
V.1.5],
$p_{a}(C)=\frac{[C]^{2}+[C][K]}{2}+1,\qquad
p_{g}(C)=p_{a}(C)-\overset{}{\underset{p\in C}{\sum}}\delta_{p}(C),$
where $\delta_{p}(C)$ is the delta invariant of a point $p\in C$ [17, page
85].
### 2.7 Celestials and their invariants
A celestial is defined as an irreducible surface that admits at least two
families of circles. A Clifford celestial is the Clifford translation of a
circle along a circle. A Clifford torus is the Clifford translation of a great
circle along a great circle. We call $S\subset{\mathbb{S}}^{3}$ $n$-ruled if
and only if $S$ admits exactly $n$ families of great circles for
$n\in{\mathbb{Z}}_{\geq 0}\cup\\{\infty\\}$. Recall from §2.6 that families of
circles are by definition one-dimensional, thus a Clifford torus is two-ruled
and a great two-sphere in ${\mathbb{S}}^{3}$ is $\infty$-ruled.
The real enhanced Picard group ${\mathcal{P}}(S)$ of a surface
$S\subset{\mathbb{P}}^{n}$ with surface pair $(X,D)$ is defined as
${\mathcal{P}}(X)$. See [16, Theorem 9] for a classification of celestials in
${\mathbb{S}}^{3}$ up to isomorphism of their real enhanced Picard groups.
The elliptic type of a surface $S\subset{\mathbb{S}}^{3}$ is defined as the
scheme theoretic intersection $\tau(S)\cap\tau(E)$. If this intersection
consists of $2n$ complex conjugate lines for some $n>0$ then we denote the
elliptic type of $S$ as
$(d;m_{1},\ldots,m_{n}),$
such that $d=\deg\tau(S)$ and $m_{i}$ is the algebraic multiplicity in
$\tau(S)$ of a pair of complex conjugate lines indexed by $i$. See [8, Section
4.3] for the notion of algebraic multiplicity.
Suppose that the vertex $v=(1:0:0:0:1)$ of Euclidean absolute $A$ is the
center of the stereographic projection $\pi$. By abuse of notation we denote
$\pi(A\setminus v)$ as $\pi(A)$. The Euclidean type of a surface
$S\subset{\mathbb{S}}^{3}$ is defined as the scheme theoretic intersection
$\pi(S)\cap\pi(A)$. We denote the Euclidean type of $S$ as $(d,m)$ if $\pi(S)$
is of degree $d$ and $\pi(A)\subset\pi(S)$ has algebraic multiplicity $m$. See
[16, Theorem 3] for a classification of celestials in ${\mathbb{S}}^{3}$ up to
Euclidean type.
## 3 Elliptic type of Clifford celestials
In this section we will classify Clifford celestials up to elliptic type. The
elliptic type of a celestial with at least one family of great circles
determines whether its great circles are Clifford translations of each other.
Recall that $E$ denotes the elliptic absolute.
###### Lemma 1.
(intersection with elliptic absolute)
If $S\subset{\mathbb{S}}^{3}$ is the Clifford translation of circle $C_{1}$
along circle $C_{2}$ then the intersection $S\cap E$ consists of two left
generators and two right generators. Moreover, $S$ is either of degree four
and the four generators are smooth in $S$—or—$S$ is of degree eight and the
four generators are double lines in $S$.
###### Proof.
It follows from Proposition 1.b) that $S$ is a celestial. We know from [16,
Theorem 3] that $S$ is of degree either two, four or eight. The Clifford
translations of each point in $C_{1}$ trace out a non-vanishing vector field
on $S\setminus E$. It follows from the hairy ball theorem and $E$ not having
real points that $S$ is not of degree two.
Let $F_{1}=(l(C_{1}))_{l\in T_{L}(C_{2})}$ and $F_{2}=(r(C_{2}))_{r\in
T_{R}(C_{1})}$ be families of circles on $S$. Assume that both $F_{1}$ and
$F_{2}$ have no base points on $E$. It follows from Proposition 1.c) that
$C_{1}$ traces out two left generators and $C_{2}$ traces out two right
generators. If $S$ is of degree eight then we know from [16, Theorem 11] that
lines in $S$ are projections of conics in its polarized model. Thus the lines
in $E$ have algebraic multiplicity two. It follows from Bezout’s theorem that
the two left and two right generators account for all components in $S\cap E$.
The remaining case is that either $F_{1}$ or $F_{2}$ has base points on $E$.
We assume without loss of generality that $F_{1}$ has base points on $E$. We
know from [16, Theorem 9, Theorem 13] that $S$ is of degree four. Both $F_{1}$
and $F_{2}$ cover each point of $S$. Thus through each point in $S\cap E$ goes
a conic or a line component of a conic in $F_{1}$ and a conic or a line
component of a conic in $F_{2}$. Conics in $F_{1}$ and $F_{2}$ can have a
common line component but not a common conic. Let the base points of $F_{1}$
be defined by $C_{1}\cap E=\\{p,q\\}$. Let $r\in S\cap E$ such that
$r\notin\\{p,q\\}$. Now suppose that $C^{\prime}$ is the conic in $F_{1}$
through $r$. Then $C^{\prime}\cap E=\\{p,q,r\\}$ and thus by Bezout’s theorem
$C^{\prime}\subset E$. Through each point of $C^{\prime}\subset E$ goes a
conic or a line component of a conic in $F_{2}$. It follows from Proposition
1.c) that $C^{\prime}$ consist of two coplanar line components that are
contained in $E$. Since $S\cap E$ is real also the complex conjugate line
components are contained. From Bezout’s theorem it follows that these four
smooth lines account for all the components in $S\cap E$. ∎
###### Lemma 2.
(real enhanced Picard groups)
Let $S\subset{\mathbb{S}}^{3}$ be a one-ruled celestial of degree eight.
* a)
The real enhanced Picard group ${\mathcal{P}}(S)$ is of type S8 with $-K$ the
class of hyperplane sections of $S$. The class of the families of great
⟦little⟧ circles is $F$ ⟦$H$⟧.
* b)
The real enhanced Picard group ${\mathcal{P}}(\tau(S))$ is of type E4 with
$H+2F$ the class of hyperplane sections of $\tau(S)$. The class of the
families of lines ⟦conics⟧ is $F$ ⟦$H$⟧.
###### Proof.
a) From [16, Theorem 3] it follows that $S$ is a smooth Del Pezzo surface of
degree eight and contains no smooth lines. The polarized model of $S$ is the
two-uple embedding of a smooth quadric into ${\mathbb{P}}^{8}$.
b) From [15, Theorem 14] it follows that $(X,D)$ is the Hirzebruch surface
${\textbf{F}}_{0}$. This surface pairs occurs at the end of an adjoint chain
in [15, Section 10] with $2D+K=F$. ∎
Theorem 1.a) was already known to Felix Klein [13, page 234], [5, Theorem
7.94]. The assumption of being —non-quartic— in Theorem 1.c) can be omitted
after Lemma 3, Theorem 2 and Corollary 1.c).
###### Theorem 1.
(elliptic type of celestials)
* a)
A celestial $S\subset{\mathbb{S}}^{3}$ is the Clifford translation of a great
circle $C_{1}$ along a great circle $C_{2}$ if and only if $S$ is of elliptic
type $(2;1,1)$.
* b)
A one-ruled celestial $S\subset{\mathbb{S}}^{3}$ is the left or right Clifford
translation of a great circle $C_{1}$ along a little circle $C_{2}$ if and
only if $S$ is of elliptic type $(4;2,1)$.
* c)
If a—non-quartic—celestial $S\subset{\mathbb{S}}^{3}$ is the Clifford
translation of a little circle $C_{1}$ along a little circle $C_{2}$ then $S$
is of elliptic type $(8;2,2)$.
###### Proof.
a) Since $S$ is two-ruled its central projection $\tau(S)$ is a smooth quadric
with two families of lines and $S$ is of degree four. The “$\Rightarrow$”
direction now follows from Lemma 1. For “$\Leftarrow$” we observe that
$\tau(S)\cap\tau(E)$ consist of two left and two right generators. This claim
now follows from Proposition 1.d).
Let $F_{1}=(l(C_{1}))_{l\in T_{L}(C_{2})}$ and $F_{2}=(r(C_{2}))_{r\in
T_{R}(C_{1})}$ be families of curves on $S$.
“$\Rightarrow$” for b): We know from [16, Theorem 3] that $S$ is a weak Del
Pezzo surface of degree two, four or eight. Suppose by contradiction that $S$
is of degree four. Then $\tau(S)$ is a singular quadric with one family of
lines. By Proposition 1.d) the lines in the ruling intersect the same
generators of $\tau(E)$. Contradiction. Since $S$ is one-ruled it follows that
$\deg S=8$ and thus $\deg\tau(S)=4$. From Lemma 1 it follows that $S\cap E$
consist of four double lines. The central projection of the great circles in
$F_{1}$ are lines in $\tau(S)$ and intersect two left generators of $\tau(E)$.
The central projection of the little circles in $F_{2}$ are conics in
$\tau(S)$ that intersect $\tau(E)$ tangentially along two right generators. It
follows from Bezout’s theorem that the intersection multiplicity of $\tau(S)$
with $\tau(E)$ is eight. Therefore we conclude that the singular right
generators of $E$ that are double lines in $S$ are centrally projected to
smooth lines in $\tau(S)$.
“$\Leftarrow$” for b): Since $\tau(S)$ contains a singular curve in $\tau(E)$,
$S$ contains a singular curve as well. It follows from [16, Theorem 3] that
$S$ is of degree eight. Suppose that $F_{1}$ ⟦$F_{2}$⟧ is the family of great
⟦little⟧ circles on $S$. From Lemma 2.a) it follows that ${\mathcal{P}}(S)$ is
of type S8 with $[F_{1}]=F$ and $[F_{2}]=H$. The singular lines in $\tau(E)$
are central projections of singular left generators. The great circles in
$F_{1}$ are centrally projected to smooth lines and thus both singular left
generators belong to $F_{2}$. From Proposition 1.d) and $[F_{1}][F_{2}]=1$ it
follows that the great circles in $F_{1}$ are Clifford translations. It is
left to show that little circles $C_{2}$ and $C_{2}^{\prime}$ in $F_{2}$ are
Clifford translations along a generic great circle $C_{1}$ in $F_{1}$. The
central projection $\tau(S)$ is of degree four and admits a family of lines
$G_{1}$ and a family of conics $G_{2}$. From Lemma 2.b) it follows that
${\mathcal{P}}(\tau(S))$ is of type E4 with $[G_{1}]=F$, $[G_{2}]=H$ and
$[\tau(E)]=2(H+2F)$. Since $[G_{1}][G_{2}]=1$ and $[G_{1}][\tau(E)]=2$ it
follows that $\tau(C_{1})\cap\tau(C_{2})=\\{p_{1}\\}$,
$\tau(C_{1})\cap\tau(C_{2}^{\prime})=\\{p_{2}\\}$ and
$\tau(C_{1})\cap\tau(E)=\\{p_{3},p_{4}\\}$. Recall that elliptic distance
between $q_{1}\in\tau^{-1}(p_{1})$ and $q_{2}\in\tau^{-1}(p_{2})$ is defined
in terms of the cross ratio of $(p_{i})_{i\in[1,4]}$. Since $h^{0}([G_{2}])=2$
we find that the map $\varphi_{[G_{2}]}$ associated to $[G_{2}]$ is a map onto
${\mathbb{P}}^{1}$ and thus the family $G_{2}$ is defined by the fibers of
this map. The map $\varphi_{[G_{2}]}$ restricted to $\tau(C_{1})$ defines an
isomorphism $\tau(C_{1})\cong{\mathbb{P}}^{1}$. Thus
$(\varphi_{[G_{2}]}(p_{i}))_{i\in[1,4]}$ has the same cross ratio as
$(p_{i})_{i\in[1,4]}$. This cross ratio does not depend on the choice of
$\tau(C_{1})$ and thus $\tau(C_{2})$ and $\tau(C_{2}^{\prime})$ are right
Clifford translations of each other along $\tau(C_{1})$.
c) It follows from Lemma 1 that $S$ is of degree eight and $S\cap E$ consists
of four double lines. From [16, Theorem 3] we know that the conics of $S$ do
not admit base points. Moreover, $S$ has exactly two families of little
circles and no other families of circles. It follows that $S$ is zero-ruled
and thus $\tau(S)$ is of degree eight. The central projections of little
circles in $S$ are conics in $\tau(S)$ that intersect $E$ tangentially. From
Bezout’s theorem it follows that the intersection multiplicity of $\tau(S)$
with $\tau(E)$ is sixteen and thus $S$ must have elliptic type $(8;2,2)$. ∎
## 4 Two-ruled Clifford celestials
A two-ruled Clifford celestial $S\subset{\mathbb{S}}^{3}$ is a Clifford torus.
We will classify Clifford tori up to isomorphism of their real enhanced Picard
group. We find as a consequence that Clifford celestials of degree four are
Clifford tori and homeomorphic to the topological torus. Recall that $A$
denotes the Euclidean absolute.
###### Lemma 3.
(real enhanced Picard group of quartic Clifford celestial)
If $S\subset{\mathbb{S}}^{3}$ is of degree four and the Clifford translation
of a circle along a circle then ${\mathcal{P}}(S)$ is of type S4 and the
classes of the components in $S\cap E$ are depicted in Figure 3.
Figure 2
###### Proof.
Let $D$ be the divisor class of a hyperplane section of $S$. Let
$\\{~{}L_{i}\in{\mathcal{P}}(S)~{}|~{}i\in I~{}\\}$ be the divisor classes of
lines in $S$. Let $\\{~{}P_{i}\in{\mathcal{P}}(S)~{}|~{}i\in J~{}\\}$ be
divisor classes of isolated singularities of $S$. For $L$ and $L^{\prime}$ in
${\mathcal{P}}(S)$ we define $L\circledast L^{\prime}>0$ if and only if either
$LL^{\prime}>0$ or ($LP_{i}>0$ and $L^{\prime}P_{i}>0$ for some $i\in J$).
Claim 1: $D=L_{1}+L_{2}+L_{3}+L_{4}+\overset{}{\underset{i\in
J^{\prime}\subset J}{\sum}}P_{i}$ with $\sigma_{*}(L_{1})=L_{2}$,
$\sigma_{*}(L_{3})=L_{4}$ and $L_{1}\circledast L_{3}=L_{1}\circledast
L_{4}=L_{2}\circledast L_{3}=L_{2}\circledast L_{4}>0$.
From Lemma 1 it follows that $E\cap S$ consists of two pairs of complex
conjugate lines and $[S\cap E]=D$. Although the classes of lines might have
zero intersection, the lines could intersect at an isolated singularity.
Claim 2: If $F\in{\mathcal{P}}(S)$ is the divisor class of a family of
Clifford translated circles then $\overset{}{\underset{i\in
I}{\sum}}F\circledast L_{i}>0$ and $F=\overset{}{\underset{i\in
I}{\sum}}c_{i}L_{i}+\overset{}{\underset{i\in J^{\prime\prime}\subset
J}{\sum}}P_{i}$ with $c_{i}\in[0,2]$.
The first assertion of this claim follows from Proposition 1.c). From Lemma 1
we know that $F$ has nongeneric circles splitting up in two lines that might
intersect at an isolated singularity.
Claim : The assertion of this lemma is valid.
In [16, Theorem 9] we classified the real enhanced Picard groups of celestials
in ${\mathbb{S}}^{3}$ (see also [15, Proposition 2 and Proposition 4]). For
each four divisor classes of lines we check there exists $(P_{i})_{i\in
J^{\prime}}$ such that claim 1 is validated with $D=-K$. For each divisor
class of a family of circles we check whether it validates claim 2. There must
be at least two such families. It follows that ${\mathcal{P}}(S)$ is of type
S4 with in Figure 3 the only possible configuration of lines in the elliptic
absolute. ∎
###### Theorem 2.
(Clifford torus)
A celestial $S\subset{\mathbb{S}}^{3}$ is Möbius equivalent to a Clifford
torus if and only if ${\mathcal{P}}(S)$ is of type S4.
###### Proof.
Claim 1: There is up to Möbius equivalence a one-dimensional family of
Clifford tori.
The angle between two great circles is Möbius invariant so there exists at
least a one-dimensional family of Clifford tori. The central projection of a
Clifford torus is a quadric surface and its intersection with $\tau(E)$ is
prescribed in Lemma 1. There is a one-dimensional choice of complex conjugate
left generators in $\tau(E)$, and this choice uniquely determines the quadric
surface.
Claim 2: There is up to Möbius equivalence a one-dimensional family of
celestials $S\subset{\mathbb{S}}^{3}$ such that ${\mathcal{P}}(S)$ is of type
S4.
Suppose that $[p]=Q_{1}-Q_{4}$ as in Figure 3 for isolated singularity $p\in
S$. The complex stereographic projection $\pi(S)$ with center $p$ is a
singular quadric surface with vertex $\pi(q)$ such that
$[q]=H-Q_{2}-Q_{3}-Q_{5}$ and $\sigma_{*}([p])=[q]$. The hyperplane at
infinity section of $\pi(S)$ consist of a conic that is tangent to $\pi(A)$ at
$\pi(a)$ and $\pi(b)$ with $[a]=H-Q_{1}-Q_{3}-Q_{4}$, $[b]=Q_{2}-Q_{5}$ and
$\sigma_{*}([a])=[b]$. Up to Möbius equivalence there exists a one-dimensional
family of quadrics with prescribed intersection, and thus this claim holds.
Claim : The assertion of this theorem is valid.
It follows from Theorem 1.a) that a Clifford torus is of degree four. From
Lemma 3 we know that a Clifford torus has real enhanced Picard group of type
S4. By dimension counting and continuity it follows from claim 1 and claim 2
that this theorem holds. ∎
###### Corollary 1.
(Clifford torus)
* a)
A surface $S\subset{\mathbb{S}}^{3}$ admits four families of circles and no
real singularities if and only if $S$ is Möbius equivalent to a Clifford
torus.
* b)
A Clifford torus is Möbius equivalent to the inverse stereographic projection
of a ring torus.
* c)
A Clifford torus is not the Clifford translation of a little circle along a
little circle.
###### Proof.
Assertions a) and b) follow from the classification of ${\mathcal{P}}(S)$ for
celestials $S\subset{\mathbb{S}}^{3}$ in [16, Theorem 9]. If we consider a
Clifford torus as the inverse projection of a ring torus then the two families
of little circles correspond to the Euclidean circles of revolution
(horizontal red circles in Figure 1a) and the orbits of rotation (blue circles
in Figure 1a). Euclidean rotations are not Clifford translations and thus
assertion c) follows. ∎
###### Example 1.
(Clifford torus)
Suppose that $S$ is a Clifford torus. In Figure 1a we see the stereographic
projection $\pi(S)$ with the two families $F_{1}$ and $F_{2}$ of great
circles. From Theorem 2 we know that ${\mathcal{P}}(S)$ is of type S4. Since
$[F_{1}][F_{2}]=2$ we find that $[F_{1}]=H-Q_{3}$ and
$[F_{2}]=2H-Q_{1}-Q_{2}$. For the remaining families $F_{3}$ and $F_{4}$ of
little circles we conclude that $[F_{3}]=H-Q_{1}$ and $[F_{4}]=H-Q_{2}$. In
Figure 1a we see the families $F_{3}$ and $F_{4}$ of $\pi(S)$. In Figure 1b
⟦Figure 1c⟧ we see $\tau(S)$ ⟦$\pi(S)$⟧ and illustrations of families $F_{1}$
and $F_{3}$. The Clifford torus $S$ is also the Clifford translation of a
great circle along a little circle.
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Figure 3a | Figure 3b | Figure 3c
A celestial in ${\mathbb{S}}^{3}$ with six families of circles is Möbius
equivalent to a two-ruled celestial but is not the Clifford translation of a
circle along a circle. $\vartriangleleft$
## 5 One-ruled Clifford celestials
In order to classify one-ruled Clifford celestials up to elliptic and
homeomorphic equivalence, we will analyze one-ruled celestials
$S\subset{\mathbb{S}}^{3}$ of degree eight. We use ${\mathcal{P}}(\tau(S))$
and the sectional delta invariant to reduce the singular locus of the central
projection $\tau(S)$ to two possible cases. It is remarkable that this allows
us to completely characterize the singular locus of $S$ itself. As a corollary
we find that one-ruled celestials of degree eight are Clifford celestials.
Let $Z\subset{\mathbb{P}}^{n}$ be a surface and let $P$ be a generic
hyperplane section of $Z$. We define the sectional delta invariant of a curve
$C\subset Z$ as the sum of delta invariants of points in $P$ that are also in
$C$:
$\hat{\delta}(C,Z):=\overset{}{\underset{p\in C\cap P}{\sum}}\delta_{p}(P).$
###### Lemma 4.
(sectional delta invariants)
* a)
A hyperplane section of a surface $Z\subset{\mathbb{P}}^{n}$ is a curve that
is singular at the singular locus of $Z$ and a generic hyperplane section of
$Z$ is smooth outside the singular locus.
* b)
If $S\subset{\mathbb{S}}^{3}$ is a surface with singular component $C\subset
S$ such that $C\nsubseteq E$ then the sectional delta invariant of $C$ equals
twice the sectional delta invariant of $\tau(C)\subset\tau(S)$.
* c)
If $S\subset{\mathbb{S}}^{3}$ is a one-ruled celestial of degree eight then
the sectional delta invariant of the singular locus of $\tau(S)$ equals three.
* d)
If $S\subset{\mathbb{S}}^{3}$ is a one-ruled celestial of degree eight then
the sectional delta invariant of the singular locus of $S$ is either four or
eight.
###### Proof.
a) Outside the singular locus of $Z$ the linear projection $\nu$ defined as in
§2.6 is an isomorphism. The pullback of hyperplane sections of $Z$ are
hyperplane sections of $Y$. It follows from Bertini’s theorem [10, Theorem
8.18] that a generic hyperplane section of $Y$ is smooth.
b) The central projection $\tau$ is linear and two-to-one outside $E$. It
follows that locally around a singularity of a hyperplane section $\tau$ is an
analytic isomorphism. Thus the projection of this singularity has the same
delta invariant.
The proofs of claim 1 and claim 2 below are left to the reader.
Claim 1: If $(P,L)$ is the surface pair of a generic hyperplane in
${\mathbb{P}}^{3}$ then ${\mathcal{P}}(P)$ is of type U1 with $L=H$.
Claim 2: If $(Q,M)$ is the surface pair of a generic hyperplane section of
${\mathbb{S}}^{3}$ then ${\mathcal{P}}(Q)$ is of type S2 with $M=H+F$.
c) Suppose that $(X,D)$ is the surface pair of
$\tau(S)\subset{\mathbb{P}}^{3}$. It follows from Lemma 2.b) that the
arithmetic genus $p_{a}(D)=0$ and thus the geometric genus of a generic
hyperplane section of $\tau(S)$ equals zero. From claim 1 we know that the
arithmetic genus of a generic hyperplane section of $\tau(S)$ equals
$p_{a}(4L)=3$. This claim now follows from $p_{g}(4L)=0$ and assertion a).
Let $S\subset{\mathbb{S}}^{3}$ be a one-ruled celestial of degree eight and
let $C\subset S$ be a generic hyperplane section of $S$.
Claim 3: $p_{g}(C)=1$.
From Lemma 2.a) we know that $[C]=D$ with $p_{a}(D)=1$. Note that the pullback
of $C$ in the polarized model is smooth and thus $p_{a}(D)=p_{g}(C)$.
Claim 4: $p_{a}(C)\in\\{5,8,9\\}$.
From claim 2 it follows that $M[C]=8$ for $[C]=aH+bF\in{\mathcal{P}}(Q)$ and
$a,b\in{\mathbb{Z}}$. From $(aH+bF)^{2}=2ab\geq 0$, $(H+F)(aH+bF)=a+b=8$ and
$p_{a}(aH+bF)=ba-a-b+1>0$ it follows that this claim holds.
d) We assume that the center of stereographic projection $\pi$ is on the two-
sphere $Q$ such that $C\subset Q$ but outside $C$. Thus $\pi(C)\subset\pi(Q)$
is a planar curve of degree eight. From claim 1 it follows that $(P,L)$ is the
surface pair of $\pi(Q)$ and thus $p_{a}(\pi(C))=21$ with $[\pi(C)]=8L$ in
${\mathcal{P}}(P)$. The geometric genus is birational invariant and thus we
conclude from claim 3 that $p_{g}(\pi(C))=1$. From [16, Theorem 3] we know
that $\pi(S)$ is of Euclidean type $(8,4)$. Let $\Delta$ denote the sum of
delta invariants of the two singularities of $\pi(C)$ at $\pi(A)$. The
algebraic multiplicity of $\pi(A)\subset\pi(S)$ is four and thus $\Delta\geq
12$. From $p_{g}(\pi(C))=1$ it follows that $\Delta\in\\{12,14,16,18\\}$. From
assertion a) it follows that $p_{g}(C)=p_{a}(C)-(20-\Delta)=1$ with
$[C]\in{\mathcal{P}}(S)$. Assertion d) now follows from claim 4. ∎
###### Lemma 5.
(singular locus of central projection)
If $S\subset{\mathbb{S}}^{3}$ is a one-ruled celestial of degree eight then
${\mathcal{P}}(\tau(S))={\mathbb{Z}}\langle H,F\rangle$ is of type E4 and the
singular locus of $\tau(S)$ consists of components with algebraic multiplicity
two in either one of the following configurations.
1. 1.
A real line $W_{0}$ with $[W_{0}]=2F$ and two skew lines $W_{1}$ and $W_{2}$
with $[W_{1}]=[W_{2}]=H$.
2. 2.
A line $W_{0}$ with $[W_{0}]=2F$ and a line $W_{1}$ with $[W_{1}]=H$.
###### Proof.
Let $(X,D)$ denote the surface pair of $\tau(S)$. From Lemma 2.b) we know that
${\mathcal{P}}(\tau(S))$ is of type E4 with $D=H+2F$. The linear projection of
its polarized model $Y$ is denoted by
$Y\subset{\mathbb{P}}^{5}\stackrel{{\scriptstyle\nu}}{{\longrightarrow}}\tau(S)\subset{\mathbb{P}}^{3}$.
Let $W\subset\tau(S)$ denote the singular locus of $\tau(S)$ with irreducible
components $(W_{i})_{i}$.
Claim 1: Component $W_{i}$ has algebraic multiplicity two in $\tau(S)$ and
$(\deg W_{i})_{i}\in\\{$ $(1)$, $(1,1)$, $(1,1,1)$, $(2,1)$, $(3)$ $\\}$.
From [16, Theorem 13] it follows that the singular curves in $S$ have
algebraic multiplicity at most two and thus $\tau(S)$ as well. This claim now
follows from Lemma 4.a,c).
Claim 2: If $\deg W_{i}=1$ then $[W_{i}]\in\\{2F,H\\}$ and if $\deg W_{i}=2$
then $[W_{i}]=D$.
A generic point on $W_{i}$ has two preimages via $\nu$. Thus if $\deg W_{i}=1$
then the preimage of $W_{i}$ is a conic that might be reducible. From
$[W_{i}]=aH+bF$, $(aH+bF)^{2}=2ab\geq 0$ and $D(aH+bF)=2a+b=2$ for
$a,b\in{\mathbb{Z}}$ it follows that $[W_{i}]\in\\{2F,H\\}$. If $\deg W_{i}=2$
then $W_{i}$ is a hyperplane section.
The notation ${\mathcal{H}}(C)$ denotes a hyperplane section of $\tau(S)$ that
contains the curve $C\subset\tau(S)$.
Claim 3: If $C\subset\tau(S)$ is a generic conic with ${\mathcal{H}}(C)=C\cup
C^{\prime}$ then $C^{\prime}$ consists of one or two lines that intersect $C$
at both $W$ and a smooth point.
We have $[C]=H$ and thus $[C^{\prime}]=2F$. From $p_{a}(C^{\prime})\leq 0$ it
follows that $C^{\prime}$ is either a line along which ${\mathcal{H}}(C)$ is
tangent, a singular line or two lines. The polarized model $Y$ contains a line
and smooth conic through each point. This claim now follows from $HF=1$ and
Lemma 4.a).
Claim 4: If $L\subset\tau(S)$ is a generic line with ${\mathcal{H}}(L)=L\cup
Q$ then $Q$ is a cubic with a singularity at $W$. The cubic $Q$ intersects $L$
at most one time outside $W$ and the hyperplane ${\mathcal{H}}(L)$ intersects
$W$ at most one time outside $L$.
We have $[L]=F$ and thus $[Q]=H+F$. From $p_{a}(H+F)=0$ and Lemma 4.a) it
follows that $Q$ has a singularity at $W$. The remaining assertions follow
from Lemma 4.a) and $F(H+F)=1$.
In the remaining proof we make a case distinction on claim 1 and claim 2 using
claim 3 and claim 4.
Claim 5: If $(\deg W_{i})_{i}=(1)$ then $([W_{i}])_{i}\neq(2F)$.
Suppose by contradiction that $\deg W=1$ and $[W]=2F$. From claim 3 we know
that ${\mathcal{H}}(C)$ contains a line through $W$. There must be at least
three lines through a generic point on $W$ and thus $W$ has algebraic
multiplicity at least three. Contradiction.
Claim 6: If $(\deg W_{i})_{i}\in\\{(1),(1,1),(1,1,1)\\}$ then $([W_{i}])_{i}$
$\notin$ $\\{$ $(H)$, $(H,H)$, $(H,H,H)$ $\\}$.
Suppose by contradiction that $W$ consists of lines with class $H$. Then the
preimage of $W_{0}$ via $\nu$ consists of a conic. From claim 3 it follows
that a generic conic $C$ intersects $W$. It follows that $W$ has algebraic
multiplicity at least three. Contradiction.
Claim 7: If $(\deg W_{i})_{i}=(1,2)$ then $([W_{i}])_{i}\neq(H,D)$.
Suppose by contradiction that $\deg W_{0}=1$ with $[W_{0}]=H$ and $[W_{1}]=D$.
A generic line with class $F$ intersects both $W_{0}$ and $W_{1}$. From claim
3 and Lemma 4.a) it follows that a generic conic $C$ intersects a line in
${\mathcal{H}}(C)$ at both $W_{0}$ and $W_{1}$. Contradiction.
Claim 8: If $(\deg W_{i})_{i}=(1,2)$ then $([W_{i}])_{i}\neq(2F,D)$.
Suppose by contradiction that $\deg W_{0}=1$ with $[W_{0}]=2F$ and
$[W_{1}]=D$. A generic line $L$ with class $F$ does not meet $W_{0}$ because
$W_{0}$ is of algebraic multiplicity two. According to claim 4,
${\mathcal{H}}(L)$ intersects $W$ in at most one point outside $L$.
Contradiction.
Claim 9: $(\deg W_{i})_{i}\neq(3)$.
Suppose by contradiction that $W$ is an irreducible cubic curve. From claim 4
we know that ${\mathcal{H}}(L)$ intersects $W$ in at most one point outside a
generic line $L$. It follows that $L$ meets $W$ at least two times. From claim
3 it follows that $L\subset{\mathcal{H}}(C)$ meets $C$ in a smooth point. From
Lemma 4.a) it follows that $C$ intersects $L$ only at $W$. Contradiction.
Claim : The assertion of this lemma holds.
If $W_{i}$ has class $2F$ then generic ${\mathcal{H}}(W_{i})$ defines a family
of conics. Since $\tau(S)$ only admits one family of conics it follows that at
most one singular component has class $2F$. The singularity of cubic $Q$ in
claim 4 with $[Q]=H+F$ parametrizes a real line $W_{0}$. We make a case
distinction on claim 1 and claim 2. From claim 5-9 it follows that this lemma
holds. ∎
###### Lemma 6.
(singular locus of octic one-ruled celestial)
If $S\subset{\mathbb{S}}^{3}$ is a one-ruled celestial of degree eight then
${\mathcal{P}}(S)={\mathbb{Z}}\langle H,F\rangle$ is of type S8 and the
singular locus $V\subset S$ is of algebraic multiplicity two. The components
of $V$ are
* •
a great circle $V_{0}$ with $[V_{0}]=F$,
* •
left generators $V_{1}$ and $V_{2}$ with $[V_{1}]=[V_{2}]=H$, and
* •
right generators $V_{a}$ and $V_{b}$ with $[V_{a}]=[V_{b}]=F$.
The central projections $\tau(V_{1})$ and $\tau(V_{2})$ are singular lines in
$\tau(S)$. The central projections $\tau(V_{a})$ and $\tau(V_{b})$ are smooth
lines in $\tau(S)$.
###### Proof.
From Lemma 2.a) we know that ${\mathcal{P}}(S)$ is of type S8. From Lemma 5 we
know the possible linear components $(W_{i})_{i}$ of the singular locus
$W\subset\tau(S)$. Let $V_{i}$ denote the preimage of $W_{i}$ via the central
projection.
Claim 1: If $W_{i}\nsubseteq\tau(E)$ then $S$ contains a double conic
$V_{c}\subset E$.
From Lemma 4.b,c) we deduce that $V_{0}\cup V_{1}$ has sectional delta
invariant six. From Lemma 4.d) it follows that $S$ has an additional singular
component $V_{c}\subset E$ of degree at most two. Since the component must be
real it cannot be a line.
Claim 2: $W=W_{0}\cup W_{1}\cup W_{2}$ with $W_{0}\nsubseteq\tau(E)$,
$W_{1}\subset\tau(E)$ and $W_{2}\subset\tau(E)$.
Assume by contradiction that $W_{i}\nsubseteq\tau(E)$. Then it follows from
claim 1 that $V=V_{0}\cup V_{1}\cup V_{c}$. Thus $\tau(V_{c})\subset\tau(E)$
is a smooth real conic with class $H\in{\mathcal{P}}(\tau(S))$ and without
real points. A generic line in $\tau(S)$ with class
$F\in{\mathcal{P}}(\tau(S))$ intersects this conic once. Thus $\tau(S)$ does
not contain lines with real points. Contradiction. From Lemma 5 we know that
$W_{0}$ is real and thus $W_{0}\nsubseteq\tau(E)$. It follows that $W_{1}$ and
$W_{2}$ are complex conjugate in $\tau(E)$.
Claim 3: The component $V_{0}$ is a great circle.
From Lemma 5 it follows that $\tau(V_{0})=W_{0}$ is real. It follows from
claim 2 that $V_{0}$ is either a great circle or two coplanar complex
conjugate lines. Assume by contradiction the latter case. Then the
intersection of the lines is a real point in $E$. Contradiction.
Claim : The assertions of this lemma are valid.
From Lemma 5 we know that $W_{1}$ and $W_{2}$ are projections of irreducible
conics. The lines that cover $\tau(S)$ intersect skew lines $W_{1}$ and
$W_{2}$. From claim 2 it follows that $V_{1}$ and $V_{2}$ are left generators
with class $H$. From Lemma 5 and claim 3 it follows that $[V_{0}]=F$. If
$S\cap E=V_{1}\cup V_{2}\cup V^{\prime}$ then $[V^{\prime}]=2F$ since $[S\cap
E]=D$. From $p_{a}(2F)\leq 0$ it follows that $V^{\prime}$ consists either of
two components with class $F$ or one component with class $F$ and intersection
multiplicity two. We observe that $\tau(S)\cap\tau(E)=W_{1}\cup
W_{2}\cup\tau(V^{\prime})$ with $[\tau(S)\cap\tau(E)]=2D$ and
$[\tau(V^{\prime})]=4F$ in ${\mathcal{P}}(\tau(S))$. It follows that
$V^{\prime}$ consists of two right generators $V_{a}$ and $V_{b}$ with class
$F$ in ${\mathcal{P}}(S)$. Since $S$ contains no smooth lines these generators
are singular and by Bezout’s theorem of algebraic multiplicity two. The
central projections $\tau(V_{a})$ and $\tau(V_{b})$ are smooth lines in
$\tau(S)\cap\tau(E)$ along which conics intersect with multiplicity two. ∎
###### Theorem 3.
(octic one-ruled celestials)
Let $S\subset{\mathbb{S}}^{3}$ be a one-ruled celestial of degree eight.
* a)
The elliptic type of $S$ is $(4;2,1)$ and ${\mathcal{P}}(\tau(S))$ is of type
E4. The singular locus of $\tau(S)$ consists of a line $W_{0}$ with
$[W_{0}]=2F$ and two complex conjugate lines $W_{1}\subset\tau(E)$ and
$W_{2}\subset\tau(E)$ with $[W_{1}]=[W_{2}]=H$.
* b)
The singular locus $V=V_{0}\cup V_{1}\cup V_{2}\cup V_{a}\cup V_{b}$ of $S$ is
as in Lemma 6. The sectional delta invariants of the components
$(V_{0},V_{1},V_{2},V_{a},V_{b})$ are $(2,2,2,1,1)$.
###### Proof.
Assertion a) follows from Lemma 5 and Lemma 6. Assertion b) follows from Lemma
6 and Lemma 4.d). Observe that the sectional delta invariant of $S$ is eight
with $\tau(V_{a})$, $\tau(V_{b})$ smooth and $\tau(V_{1})$, $\tau(V_{2})$
singular. Thus $V_{1}$ and $V_{2}$ each account for sectional delta invariant
two. ∎
###### Corollary 2.
(octic one-ruled celestials)
* a)
If $S\subset{\mathbb{S}}^{3}$ is a one-ruled celestial then either $\tau(S)$
is a quadric cone or $S$ is of degree eight with elliptic type $(4;1,2)$.
* b)
If we Clifford translate a great circle along a little circle but not along a
great circle then exactly two translated great circles will coincide.
* c)
If a celestial in ${\mathbb{S}}^{3}$ of degree eight has a family of great
circles then this surface is a Clifford translation of a great circle along a
little circle.
###### Proof.
From [16, Theorem 3] we know that $S$ is of degree four or eight. A one-ruled
celestial of degree four is centrally projected to a quadric cone. For
assertion b) we remark that a Clifford torus is also the Clifford translation
of a great circle along a little circle. See Example 2 for the stereographic
projection of the coincidence locus $V_{0}$ of two great circles. Aside
Theorem 3 the assertions follow from Theorem 1. ∎
###### Example 2.
(octic one-ruled celestials)
Suppose that $S\subset{\mathbb{S}}^{3}$ is an octic one-ruled celestial. We
use the notation as in Theorem 3. In Figure 2 we observe that the planar
section of $\pi(S)$ that contains the singular circle $\pi(V_{0})$ in the
middle, consists of two other circles. Indeed the hyperplane sections through
$W_{0}\subset\tau(S)$ pull back to two-spheres through $V_{0}\subset S$. These
two-spheres contain aside $V_{0}$ two antipodal little circles.
---
Figure 4
We now choose the center of stereographic projection $\pi$ on $V_{0}$. In
Figure 2 we see three different examples of $\pi(S)$ where the circles
intersect the line $\pi(V_{0})$ either complex, tangentially or real. These
cases can be seen as a generalization of the ring-, horn- and spindle-torus
respectively.
| |
---|---|---
Figure 5a | Figure 5b | Figure 5c
$\vartriangleleft$
###### Theorem 4.
(topology of octic one-ruled celestials)
If $S\subset{\mathbb{S}}^{3}$ is a one-ruled celestial of degree eight then
$S$ is homeomorphic to either two exclusive tori glued together along a circle
(Figure 4a), a torus (Figure 4b), or two inclusive tori glued together along a
circle (Figure 4c).
| |
---|---|---
Figure 6a | Figure 6b | Figure 6c
###### Proof.
From Lemma 2.a) it follows that the polarized model $Y\subset{\mathbb{P}}^{8}$
of $S$ is the two-uple embedding of a smooth quadric. From Poincaré-Hopf
theorem it follows that $Y$ is a homeomorphic to a topological torus and $S$
is a linear projection of $Y$. We denote this linear projection by $\nu$. From
Theorem 3 it follows that there are two conics $V_{01}\subset Y$ and
$V_{02}\subset Y$ such that $\nu(V_{01})=\nu(V_{02})=V_{0}$ with $[V_{0}]=F$.
If $C\subset Y$ is a conic with $[C]=H$ then $HF=1$ and thus
$\\{p_{1}\\}=V_{01}\cap C$ and $\\{p_{2}\\}=V_{02}\cap C$. If
$\nu(p_{1})=\nu(p_{2})$ then we obtain up to homeomorphism Figure 4b. If
$\nu(p_{1})\neq\nu(p_{2})$ then $\nu(C)$ divides $S$ in two different
compartments. In this case either $\nu(p_{1})$ and $\nu(p_{2})$ are complex
conjugate (Figure 4a) or both real (Figure 4c). ∎
## 6 Euclidean translational celestials
###### Lemma 7.
(intersection with Euclidean absolute)
If $S\subset{\mathbb{S}}^{3}$ is the Euclidean translation of circle $C_{1}$
along circle $C_{2}$ then the intersection $S\cap A$ consists of two or four
Euclidean generators. Moreover, these Euclidean generators are either all
smooth or all double lines in $S$.
###### Proof.
As we Euclidean translate $C_{1}$ along $C_{2}$ each point in $C_{1}$ traces
out a circle and thus $S$ is a celestial. From [16, Theorem 3] it follows that
$S$ is of degree either two, four or eight. The Clifford translations of each
point in $C_{1}$ trace out a non-vanishing vector field on $S\setminus A$. It
follows from the hairy ball theorem that if $S$ is a two-sphere then $S$ meets
the vertex of $A$ and thus $\pi(S)$ is a plane.
Let $F_{1}=(t(C_{1}))_{t\in T_{A}(C_{2})}$ and $F_{2}=(t(C_{2}))_{t\in
T_{A}(C_{1})}$ be families of circles on $S$. The remaining proof is almost
word for word the same as the proof of Lemma 1 except that both “left
generator” and “right generator” are replaced by “Euclidean generator”. Also
its possible that $F_{1}$ or $F_{2}$ has a single base point at the vertex of
$A$. The details are left to the reader. ∎
###### Theorem 5.
(Euclidean translational celestials)
An Euclidean translation of a circle along a circle is the inverse
stereographic projection of either a plane (Figure 5a), a circular cylinder
(Figure 5b), an elliptic cylinder (Figure 5c) or a quartic celestial covered
by two families of parallel circles (Figure 5d).
| | |
---|---|---|---
Figure 7a | Figure 7b | Figure 7c | Figure 7d
An Euclidean translational celestial is not Möbius equivalent to a Clifford
celestial.
###### Proof.
We assume that $S\subset{\mathbb{S}}^{3}$ is the Euclidean translation of a
generic circle $C_{1}$ along a generic circle $C_{2}$. If $S$ is of Euclidean
type $(d,c)$ then it follows from Lemma 7 that $c=0$. From [16, Theorem 3] we
conclude that $S$ of Euclidean type $(1,0)$, $(2,0)$ or $(4,0)$. If $S$ is of
Euclidean type $(1,0)$ or $(2,0)$ then this classification follows from the
classification in [16, Theorem 6]. If $S$ is of Euclidean type $(4,0)$ then
$S$ is a Del Pezzo surface of degree eight. In this case it follows from [16,
Theorem 13] that the singular locus of $S$ consists of four double lines
intersecting at a point of algebraic multiplicity four and either one or two
circles. Moreover, $\pi(C_{1})$ and $\pi(C_{2})$ must be circles. It follows
from Theorem 1 and Lemma 3 that $S$ is not Möbius equivalent to a Clifford
celestial for all cases. ∎
## 7 Acknowledgements
I would like to thank Rimas Krasauskas, Helmut Pottmann, Mikhail Skopenkov and
Severinas Zube for their insights and inspiration concerning celestials as the
quaternion product of circles. I also thank Rachid Ait-Haddou, Michael Barton,
Heidi Dahl and Jean-Marie Morvan for interesting discussions and corrections.
The images of Figure 2 and Figure 2 were made with Surfex [11]. The remaining
figures were generated using Sage [21].
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#### Address of author
King Abdullah University of Science and Technology, Thuwal, Kingdom of Saudi
Arabia
email: [email protected]
|
arxiv-papers
| 2013-06-08T13:19:25 |
2024-09-04T02:49:46.246571
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Niels Lubbes",
"submitter": "Niels Lubbes",
"url": "https://arxiv.org/abs/1306.1917"
}
|
1306.2041
|
# Dynamical tides excited in rotating stars of different masses and ages and
the formation of close in orbits
S. V. Chernov 1, J. C. B. Papaloizou 2 and P.B.Ivanov1
1Astro Space Centre, P.N. Lebedev Physical Institute, 84/32 Profsoyuznaya
Street, Moscow, 117997, Russia
2 DAMTP, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA E-mail: [email protected] (SVCh)E-mail:
[email protected] (JCBP)E-mail: [email protected] (PBI)
(Accepted. Received; in original form)
###### Abstract
We study the tidal response of rotating solar mass stars, as well as more
massive rotating stars, of different ages in the context of tidal captures
leading to either giant exoplanets on close in orbits, or the formation of
binary systems in star clusters. To do this, we adopt approaches based on
normal mode and associated overlap integral evaluation, developed in a
companion paper by Ivanov et al., and direct numerical simulation, to evaluate
energy and angular momentum exchanges between the orbit and normal modes. The
two approaches are found to be in essential agreement apart from when
encounters occur near to pseudosynchronization, where the stellar angular
velocity and the orbital angular velocity at periastron are approximately
matched. We find that the strength of tidal interaction being expressed in
dimensionless natural units is significantly weaker for the more massive
stars, as compared to the solar mass stars, because of the lack of significant
convective envelopes in the former case. On the other hand the interaction is
found to be stronger for retrograde as opposed to prograde orbits in all
cases. In addition, for a given pericentre distance, tidal interactions also
strengthen for more evolved stars on account of their radial expansion. In
agreement with previous work based on simplified polytropic models, we find
that energy transferred to their central stars could play a significant role
in the early stages of the circularisation of potential ’Hot Jupiters’.
###### keywords:
hydrodynamics - celestial mechanics - planetary systems: formation, planet
-star interactions, stars: binaries: close, rotation, oscillations
††pagerange: Dynamical tides excited in rotating stars of different masses and
ages and the formation of close in orbits–References††pubyear: 2010
## 1 Introduction
Tidal interaction leads to synchronisation and orbital circularisation of
close binary stars (eg. Zahn 1977, Hut 1981). It may also result in double
star or star-planet systems that undergo close encounters in marginally
unbound orbits becoming tidally captured into highly eccentric orbits that
then begin to circularise (eg. Press & Teukolsky 1977). A process of this kind
is believed to account for giant exoplanets in close orbits with periods of a
few days (eg. Weidenschilling & Marzari 1996, Rasio & Ford 1996).
The determination of the tidal evolution requires the calculation of the
response of the tidally perturbed body. This involves energy and angular
momentum exchange between its normal modes and the orbit leading to its
evolution. In a companion paper, Ivanov et al. (2013), subsequently referred
to as Paper 1, we developed general procedures for calculating tidal energy
and angular momentum exchange rates, for bodies in periodic orbits, that are
associated with an identifiable regular spectrum of low frequency rotationally
modified gravity modes, for rotating stars with realistic structure.
These are likely to give rise to the dominant tidal response in bodies with
stratification, where the tidal forcing frequencies significantly exceed the
inverse of the convective time scale associated with any convection zone, so
that any effective turbulent viscosity is inefficient. This is also expected
to be the case for rotating stars, when the dominant tidal forcing frequencies
as viewed in the rotating frame exceed twice the rotation frequency with the
consequence that inertial modes are not efficiently excited in convective
regions.
In Paper 1 we also gave expressions from which the energy and angular momentum
transferred to stellar modes of oscillation as a result of parabolic
encounters can be calculated. A process that could lead to tidal captures and
also governs the initial phase of orbital circularisation when the orbit is
very eccentric (eg. Ivanov & Papaloizou 2004). Evaluation of the response
arising from normal modes requires calculation of mode eigenfrequencies and
corresponding overlap integrals that determine the strength of mode coupling
with the tidal potential (eg. Press & Teukolsky 1977). This procedure was
discussed in some detail in Paper 1 for the case when the traditional
approximation, appropriate for low frequency modes in stratified layers, was
adopted. We remark that, as discussed in more detail in Paper 1, tidal
phenomena such as energy and angular momentum exchange through parabolic
encounters, or orbital evolution in the regime of so-called moderately strong
viscosity (eg. Zahn 1977, Goodman $\&$ Dickson 1998), where propagating
rotationally modified gravity waves attain short wavelengths, and so are
dissipated before reaching boundaries from which they can be reflected, are
such that results are independent of the precise specification of the
dissipation process. In this regime, the wave dissipation should also occur on
a time scale that is significantly longer than the locally excited wave period
which will also be characteristic of the time for excitation due to tidal
perturbation.
In this paper we apply the formalism developed in Paper 1, where only Sun-like
stars were considered, to calculate the normal modes and their associated
overlap integrals for a range of tidal forcing frequencies for two models of a
rotating solar mass star with different ages, as well as several models of
more massive rotating stars, with different ages. The dependence of these
quantities on the existence and extent of convective regions and the
transition between convective and radiative regions is elucidated. We also
compare results obtained from the normal mode approach of Paper 1 to those
obtained from direct numerical simulations of parabolic encounters (eg.
Papaloizou & Ivanov 2010) delineating when there is good agreement between the
two approaches. Our results are then applied to the tidal capture and initial
orbital circularisation of giant exoplanets for both prograde and retrograde
orbits and also the tidal capture of stars to form binary systems in stellar
clusters (eg. Fabian, Pringle $\&$ Rees 1975, Press $\&$ Teukolsky 1977).
The plan of the paper is as follows. In section 2 we describe the stellar
models for which we calculated the quantities that enable their exchange of
energy and angular momentum under tidal gravitational perturbation due to a
companion to be calculated. These quantities are the overlap integrals and the
low frequency rotationally modified $g$ mode spectrum and they are discussed
in detail in Paper 1. We consider models in the range of $1-5M_{\odot}$ with a
variety of ages. As indicated in Paper 1, the extent of any convective
envelope and/or core plays a significant role in determining the strength of
tidal interaction as also does the detailed form of the transition between
convective and radiative regions.
In section 3 we discuss the properties of the numerically calculated mode
spectra and overlap integrals for the stellar models considered. We also
derive the rotational splitting coefficients which give the first order shifts
of mode eigenfrequencies as a result of stellar rotation. In the non rotating
case, the overlap integrals were found to be markedly larger for Sun-like
stars as compared to either a polytrope with index 3 or more massive models
with much less extensive convective regions. This is because of the convective
envelope and is expected from the theory developed in Paper 1.
The overlap integrals are also calculated for rotating models under the
neglect of centrifugal distortion and the adoption of the traditional
approximation as indicated in Paper 1. Results for angular velocities of
rotation in units of the critical rotation rate in the range $0.1-0.4$ are
presented.
We go on to apply our results to evaluate the energy and angular momentum
exchanged as a result of a parabolic encounter with a companion. These enable
the possibility of tidal capture from unbound orbits to be assessed. In
addition the time scale for the initial stages of orbital circularisation to
occur for low planetary mass companions is estimated.
We compare energy and angular momentum transfers obtained through the normal
mode/overlap integral approach to results obtained from solving the encounter
problem as an initial value problem numerically (Papaloizou & Ivanov 2010;
Ivanov & Papaloizou 2011) for the full range of rotation rates and for
pericentre distances that are not too large to make calculation intractable.
Both prograde and retrograde encounters are considered. We found that the
methods are in good agreement apart from the situation where the system is
close to pseudosynchronization. In this case the effective tidal forcing
frequencies are comparable to the rotation frequency and inertial modes, not
taken into account in the normal mode approach can play a significant role. As
the characteristic tidal forcing frequencies are expected to be significantly
larger than stellar rotation frequencies, inertial modes are unlikely to be
excited in the star during the initial stages of the formation of close in
giant planet orbits of ’Hot Jupiters’. Accordingly we do not pursue the issue
of inertial modes further in this paper.
Finally in section 5 we summarize and discuss our results. We remark that as
in our earlier work (Ivanov & Papaloizou 2011), which considered polytropic
models with index, 3, we find that tidal interaction with the central star is
significantly stronger for retrograde orbits and that it could play a
significant role in the circularisation process for giant planets, so
potentially reducing the amount of potentially destructive energy dissipation
in the planetary interior.
## 2 Formulation of the problem and details of numerical methods
### 2.1 Coordinate system and notation
The basic definitions and notation adopted in this paper are the same as in
Paper 1. We use either a spherical coordinate system $(r,\phi,\theta)$ or
associated cylindrical polar coordinate system $(\varpi,\phi,z)$ with origin
at the centre of mass of the star. When viewed in an inertial frame, the
unperturbed star rotates uniformly about the $z$ axis with angular velocity
$\Omega$. For our reference frame, we adopt the rotating frame in which the
unperturbed star appears at rest.
### 2.2 Stellar models considered
Our calculations are for rotating stars of different masses and ages. As in
paper 1 centrifugal distortion is neglected with the consequence that
equilibrium structures are not modified by rotation. Therefore standard
spherically symmetric models are used.
Model | Mass | Radius | Age | Mean density
---|---|---|---|---
1p | 1 | 1 | | 1
1a | 1 | 0.91 | $1.67\cdot 10^{8}$ | 1.33
1b | 1 | 1 | $4.41\cdot 10^{9}$ | 1
1.5a | 1.5 | 2.08 | $1.27\cdot 10^{7}$ | 0.166
1.5b | 1.5 | 1.46 | $5.96\cdot 10^{7}$ | 0.482
1.5c | 1.5 | 1.82 | $1.58\cdot 10^{9}$ | 0.249
2a | 2 | 2.68 | $6.81\cdot 10^{6}$ | 0.104
2b | 2 | 1.63 | $2.93\cdot 10^{7}$ | 0.462
2c | 2 | 2.25 | $5.93\cdot 10^{8}$ | 0.175
2d | 2 | 2.91 | $8.44\cdot 10^{8}$ | 0.0811
5a | 5 | 2.69 | $2.54\cdot 10^{6}$ | 0.257
Table 1: Radii, masses, ages and mean densities for models used in our
calculations. Radii, masses and mean densities are expressed in units
corresponding to the present day Sun, stellar ages are expressed in years.
We consider models of stars of masses $M_{*}=1,$ $1.5,$ $2,$ and $5M_{\odot}$
of different ages. Stellar masses are expressed in solar masses
($1M_{\odot}=1.9891\cdot 10^{33}g$). Radii, $R_{*},$ are expressed in solar
radii ($1R_{\odot}=6.9551\cdot 10^{10}cm$) and ages, expressed in years, are
given in Table 1. Additionally, we have calculated all quantities of interest
for a stellar model consisting of a polytrope with index $n=3.$ The mass and
radius are scaled to solar values and the adiabatic index $\Gamma=5/3.$ This
serves as reference model for our analysis and is referred to as model 1p.
All realistic stellar models apart from model 1b were kindly provided to us by
I.W. Roxburgh. The numerical code used to obtain these models is discussed in
Roxburgh (2008). Model 1b is for the present day Sun. It is discussed in
Christensen-Dalsgaard et al. (1996). Unlike models described elsewhere in a
similar context (see McMillan, McDermott $\&$ Taam 1987) our models have
metallicity appropriate for population I stars. The zero age hydrogen mass
fraction X=0.7 and the mass fraction of heavy elements Z=0.02 for all models.
Convective heat transport is described by a standard form of mixing length
theory ( Kippenhahn et al. 2013). Mixing in convective and semi-convective
zones is dealt with by incorporating diffusion into the equations governing
the evolution of the chemical abundances ( see Eggleton 1972). The diffusion
coefficient is taken to be $D_{conv}=v_{conv}{l}/6,$ where $v_{conv}$ is the
convective velocity and ${l}$ is the mixing length.
Figure 1: The density $\rho$ (in units of the mean density
$\tilde{\rho}=3M_{*}/(4\pi R_{*}^{3})$) and square of the Brunt - V$\ddot{\rm
a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency, $N^{2}$ (in units of
$GM_{*}/R_{*}^{3}$) as functions of the radius $r$ expressed in units of
$R_{*}.$ The solid curves correspond to model 1b while the dashed curves are
for the model 1a. The curves monotonically decreasing with $r$ are for the
density distributions, while the curves having maxima at some values of $r$
are for the Brunt - V$\ddot{\rm a}$is$\ddot{\rm a}$l$\ddot{\rm a}$
frequencies. Figure 2: Same as in Fig. 1 but for models with
$M_{*}=1.5M_{\odot}$. Solid, dashed and dotted curves are for models 1.5c,
1.5b and 1.5a, respectively. Note that there are regions very close to the
surface where a weak density inversion occurs in these models. However, the
values of the density where this occurs are below the minimum level plotted.
Figure 3: Same as in Fig. 1 but for models with $M_{*}=2M_{\odot}$. Solid,
dashed, dotted and dot-dashed curves are for models 2d, 2c, 2b and 2a,
respectively. Note that the dashed and dot-dashed curves for the density
almost coincide Figure 4: Same as in Fig. 1 but for model 5a with
$M_{*}=5M_{\odot}$.
As is discussed in Paper 1 when low frequency gravity modes are considered,
their important properties are found to be mainly determined by the functional
forms of the density, $\rho,$ and the Brunt - V$\ddot{\rm a}$is$\ddot{\rm
a}$l$\ddot{\rm a}$ frequency $N$. In particular, the locations of stably
stratified radiative regions, where $N^{2}>0,$ and the behaviour of $N$ in the
neighbourhood of transitions from stably stratified to convective regions, are
particularly significant. For the models described here the functional forms
in these transition regions are strongly affected by the evolutionary history
of the chemical composition profiles. Although chemical diffusion is included,
the coefficient is zero in radiative regions and large in convective regions
so that mixing is efficient there. This results in the possibility of very
rapidly varying or even discontinuous abundance profiles according to how the
boundary between a radiative and convective zone moves with evolution. Suppose
this interface moves with speed $v_{I},$ we can form the dimensionless
quantity $v_{I}l/D_{conv}=6v_{I}/v_{conv}\sim t_{conv}/t_{ev},$ the latter
ratio being the ratio of the convection time scale to the evolutionary time
scale.
On dimensionless grounds we might expect the width of the transition region,
$w_{rc},$ between a convective and radiative zone would be given by an
expression of the form $w_{rc}\sim lF(t_{conv}/t_{ev}),$ where the form of the
function $F$ is determined by the way the diffusion coefficient vanishes as
the radiative zone is entered and the evolutionary history. Simple modelling
suggests that $F$ is a monotonic function of its argument, with the variation
being more rapid for diffusion coefficients with more rapid cutoffs. For
example a linear variation is expected when the diffusion coefficient inside
the radiative region vanishes as $(r-r_{I})^{2}/l^{2}$, with $r_{I}$ being a
distance of one mixing length away from a retreating convection zone boundary.
Note that, $t_{conv}/t_{ev}\sim 10^{-(9-10)}$, is very small leading to the
possibility of very thin transition layers when the evolutionary history is
such that the composition in a convective zone differs significantly from its
immediate surroundings. At such a transition the mean molecular weight changes
while hydrostatic equilibrium enforces continuity of the pressure. There is a
then a rapid change in the density with an associated spike in the density
gradient and the square of the Brunt-V$\ddot{\rm a}$is$\ddot{\rm
a}$l$\ddot{\rm a}$ frequency. As this results from the density gradient, a
large effect can be produced with there being only small jumps in the density
or chemical profile.
Plots of the density $\rho$ and the square of the Brunt - V$\ddot{\rm
a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency, $N^{2},$ against radius, for the
models used in our calculations are given in Figs. 1-4. In Fig. 1 we
illustrate these two quantities the two models with $M_{*}=1M_{\odot}$
discussed in detail and adopted in Paper 1. Note that negative values of
$N^{2}$ that occur in convective regions are not shown. These models exhibit a
similar and rather simple structure. They have a radiative core and convective
envelope with a transition at a similar total radius fraction. The young Sun-
like star, model 1a, has a smaller radius than model 1b, which corresponds to
the present day Sun. It accordingly has a larger mean density and is
therefore, less susceptible to tidal influence from a perturbing companion
with a given orbital period.
Models 1.5a-1.5c, illustrated in Fig. 2, are for a star of mass
$M_{*}=1.5M_{\odot}$ at different ages. Their structure is more complex than
that of Sun-like stars. The youngest model 1.5a (plotted with dotted curves)
is similar to models 1a and 1b in that it also has convective envelope with a
convective core being absent. We remark that a weak density inversion occurs
very close to the surface in these models. This is a well known effect that
occurs as hydrogen is ionised in convective envelopes where the energy
transport due to convection is inefficient (see eg. Latour 1970). The
ionisation zone is very thin in such cases, and such that the reduction in the
mean molecular weight, that occurs with weak pressure variation there, causes
the density inversion. The density in these layers is very small and so they
do not affect any of the tidal response calculations presented in this paper
significantly. However, on account of the general similarity of the models, we
expect that the overlap integrals, $Q,$ characterising the strength of tidal
interactions and discussed in detail in Paper 1, will have similar properties
to those obtained for Sun-like stars. But, the density at the base of
convection zone, expressed in terms of the mean density
$\tilde{\rho}=3M_{*}/(4\pi R_{*}^{3})$, $\rho_{cb},$ is smaller than the
corresponding quantity for models 1a and 1b. We find $\rho_{cb}\sim
10^{-3}\tilde{\rho}$ and $\sim 10^{-2}\tilde{\rho}$ for model 1.5a, and either
of models 1a or 1b respectively.
The analytic WKBJ theory developed in Paper 1 predicts that the overlap
integrals, being determined by the presence of a convective envelope, are
proportional to $\sqrt{\rho}_{cb}$ in the limit of sufficiently small
eigenfrequencies (see equations (111), (113) and (115) of Paper 1).
Accordingly, we expect that for a given value of the eigenfrequency, $Q$
values for model 1.5a will be smaller than those for younger models, by a
factor corresponding to the square root of the ratio of the respective values
of $\rho_{cb}.$ Here we recall that as the mode spectrum is dense the overlap
integrals are regarded as continuous functions of frequency. Let us note that
energy and angular momentum exchanges due to tides associated with a normal
mode are, in general, proportional to the square of the appropriate overlap
integral (see Paper 1). That means that in a situation where the contribution
from the convective envelope and the region of its boundary with the inner
radiative zone determines the value of the overlap integral, energy and
angular momentum exchanges are approximately proportional to $\rho_{cb}$, see
also equation (13) of Goodman $\&$ Dickson (1998).
Our expectation is confirmed by calculation, see Fig. 8. The evolved models
1.5b and 1.5c have both convective envelopes and convective cores. However,
$\rho_{c}$ at the base of convective envelope is relatively small, being $\sim
4.6\cdot 10^{-5}$ for model 1.5c and $\sim 6\cdot 10^{-6}$ for model 1.5b.
This has the consequence that the contribution to the overlap integrals
associated with the presence of a convective envelope is strongly suppressed.
Models 1.5b and 1.5c also have convective cores. But, the transition region
between the radiative region and the convective core is extremely sharp in
these models ( see discussion in Section 2.2 above), and must be considered as
a discontinuity for eigenfunctions with characteristic wavelength larger than
the typical size of the transition zone. The width of the latter, $\Delta r,$
is of the order of or smaller than the grid size with $\Delta r/r<2.5\times
10^{-4}$, $2.5\times 10^{-5}$ for models 1.5b and 1.5c, respectively. We
emphasise that the detailed form of this transition region should be
determined from a complete treatment of convection, including overshoot, see
eg. Roxburgh (1978), Zahn (1991), and the effects of stellar rotation, etc.
This cannot be undertaken at present111Note that future advances in
astroseismology may lead to some observational constraints on details of
transitions between radiative and convective regions in the near future, see
eg. Silva Aguirre et al. (2011).. However, when eigenfunctions have typical
wavelengths larger than the size of the transition zone, one cannot assume
that the Brunt-V$\ddot{\rm a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency
increases as a power of distance from the boundary of the convective region
and perform a standard WKBJ analysis that assumes the response wavelength is
significantly shorter than the transition width. Accordingly estimates of
quantities characterising tidal interactions, such as overlap integrals or
related quantities, for example the quantity $E_{2}$ used by Zahn (1977),
based on this assumption are not valid.
Let us estimate typical periods of modes, where such calculations are
potentially formally invalid. For definiteness we consider model 1.5b. We
assume that the width of the transition zone is as small as suggested by our
numerical model. From Fig. 2 we see that for this model, the value of Brunt -
V$\ddot{\rm a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency at the maximum of the
’spike’ close to the convective core is $\approx 6.8$ and the radius of
convective core $r_{c}\approx 0.087$ in our dimensionless units.
From the WKBJ theory applied to gravity waves, the characteristic wavelength,
$\lambda$, can be estimated through $\lambda\approx r\omega/(\sqrt{6}N)$,
where $\omega$ is the eigenfrequency and it is assumed that the star is non-
rotating, see eg. Christensen-Dalsgaard (1998). On the other hand $\lambda$
should be larger than the width of the transition region, estimated above as
$\sim 2.5\times 10^{-4}r$ or smaller. Thus we obtain $\omega>4.2\cdot 10^{-3}$
in natural units for this inequality to be valid. From table 1 it follows that
this corresponds to periods $<40\,days$. This means that estimates based on a
standard WKBJ analysis may be inapplicable for all periods of interest. Let us
stress, however, that the width of the transition region may be much larger
than was assumed in order to obtain this estimate (see discussion in Section
2.2).
In Fig. 3 we show the forms of the density and the square of Brunt -
V$\ddot{\rm a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency for models with
$M_{*}=2M_{\odot}$. The youngest model, model 2a (dot-dashed curve) is fully
radiative. We expect that for this model, overlap integrals are suppressed at
low frequencies as compared to models with convective regions. The overlap
integrals associated with such models are expected to decrease with
eigenfrequency, $\omega$, faster than any power of $\omega,$ as happens for a
polytropic star represented as model 1p. Thus, tidal interactions determined
by low frequency gravity modes become rather inefficient in this case. Models
2b-2d are similar to those with mass $M_{*}=1.5M_{\odot}$ with the difference
that the convective envelope is practically absent. Therefore, contributions
to the overlap integrals coming from the envelope region should be very small.
There are also almost discontinuous transitions from radiative envelopes to
convective cores. Thus we conclude that previous estimates of the strength of
tidal interactions based on the regular behaviour of $N^{2}$ close to this
transition may need revision.
Finally, in Fig 4 we plot the density and square of the Brunt - V$\ddot{\rm
a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency for a model of a young star with
$M_{*}=5M_{\odot}$. The structure of this model is rather simple and similar
to the cases of evolved stars with $M_{*}=2M_{\odot}$. There is no convective
envelope in this model and, again, there is a quite sharp transition between
the radiative region and convective core.
## 3 Properties of stellar eigenmodes: eigenspectra, overlap integrals and
rotational splitting coefficients
In this Section we consider the quantities that determine the tidal
interactions for given orbital parameters and properties of decay of free
stellar oscillation either due to viscosity or non-linear effects. These are
the eigenfrequencies of free pulsations $\omega$, normalised overlap integrals
$\hat{Q}$, and, in case of small rotational frequency
$\Omega\ll\Omega_{*}\equiv\sqrt{{GM_{*}/R_{*}^{3}}}$, the coefficients $\beta$
which determine the splitting of eigenfrequencies due to rotation in the non-
rotating frame. We discus how these quantities depend on stellar structure
The overlap integrals are discussed in detail in Paper 1. Here we briefly
recall them for completeness. In general, $\hat{Q}$ is given by the expression
$\hat{Q}=Q/\sqrt{n},\quad
Q=\left({\mbox{\boldmath${\xi}$}}{|}\int_{0}^{2\pi}d\phi
e^{-im\phi}\nabla(r^{2}Y^{m}_{2})\right),$ (1)
where it is implied that the inner scalar product of any two complex vectors
${\mbox{\boldmath${\eta}$}_{1}}$ and ${\mbox{\boldmath${\eta}$}_{2}}$ is
determined by integration over the cylindrical coordinates $\varpi$ and $z$ as
$({\mbox{\boldmath${\eta}$}}_{1}|{\mbox{\boldmath${\eta}$}}_{2})=\int\varpi
d\varpi
dz\rho({\mbox{\boldmath${\eta}$}}_{1}^{*}\cdot{\mbox{\boldmath${\eta}$}}_{2}),$
(2)
Here $*$ denotes the complex conjugate, $Y^{m}_{2}$ is the spherical function,
${\xi}$ is the Lagrangian displacement vector corresponding to a particular
eigenmode with eigenfrequency $\omega$. It is assumed that in all expressions
the dependence of ${\xi}$ on the azimuthal angle $\phi$,
${\mbox{\boldmath${\xi}$}}\propto e^{im\phi}$, is factored out. In general the
azimuthal mode number, $m,$ is such that $|m|\leq 2.$ However, we shall
consider only $|m|=2$ below as this is the most important case (see Paper 1).
The norm $n$ is determined by the expression
$n=\pi(({\mbox{\boldmath${\xi}$}}|{\mbox{\boldmath${\xi}$}})+({\mbox{\boldmath${\xi}$}}|{\mbox{\boldmath${\cal
C}$}}{\mbox{\boldmath${\xi}$}})/\omega^{2}),$ (3)
where ${\cal C}$ is an integro-differential self-adjoint operator, which when
operating on $-\mbox{\boldmath${\xi}$}$ gives the restoring acceleration due
to the action of gravity and pressure forces.
When the star is non-rotating the overlap integrals reduce to the form given
by Press $\&$ Teukolsky (1977). A similar expression can be obtained in the
so-called traditional approximation discussed below and in Paper 1. In that
case we have
$Q=\alpha Q_{r},{\rm with}\quad
Q_{r}=\int^{R_{*}}_{0}r^{3}dr\rho(2\xi+\Lambda\xi^{S}),$ (4)
and $n=n_{st}+n_{r},$
${\rm where}\hskip
2.84526ptn_{st}=\int_{0}^{R_{*}}r^{2}dr\rho(\xi_{j}^{2}+\Lambda(\xi^{S})^{2}),{\rm
and}\hskip 2.84526ptn_{r}=\nu I_{r}I_{\theta}$ (5)
with $I_{r}=\int_{0}^{R_{*}}r^{2}dr\rho(\xi^{S})^{2}$ and
$\nu=2\Omega/\omega$. In the expressions (4) and (5) $\xi(r)$ and $\xi^{S}(r)$
give the radial dependences of the radial component of the displacement vector
and the angular components, respectively, in the traditional approximation.
The quantities $\Lambda$, $\alpha$ and $I_{\theta}$ are functions of $\nu$,
their explicit form and the dependence on $\nu$ are discussed in Paper 1. Note
that $\Lambda(\nu)$ is an eigenvalue associated with acceptable solutions of
the Laplace tidal equation.
When the star does not rotate, the Laplace tidal equation reduces to the
Legendre equation, the solution of which is the associated Legendre function.
We then have $\alpha=1$, $\Lambda=6$, $n=n_{st}$ and the expressions (4) and
(5) reduce to their standard form as given by e.g. Press $\&$ Teukolsky (1977)
and Ivanov $\&$ Papaloizou (2004).
Hereafter we express all quantities of interest in natural units. Thus,
eigenfrequencies are expressed in terms of the natural stellar frequency
$\Omega_{*},$ and the overlap integrals in terms of $\sqrt{M_{*}}R_{*}.$
### 3.1 Eigenfrequencies and overlap integrals for non-rotating stars
Let us first discuss the eigenfrequencies and overlap integrals for non-
rotating models. In this case we set $\alpha_{i}=1$ and $n_{r}=0$ in (4) and
(5), respectively. Eigenfrequencies, and eigenfunctions were obtained by a
shooting method described in Section 5.2 of Paper 1. Here we note that we
integrated the standard full set of four equations describing adiabatic
pulsations (eg. Christensen-Dalsgaard 1998) to find these quantities for
relatively large values of eigenfrequencies $\omega>0.3-0.5$ depending on the
particular model. These were then used to evaluate the overlap integrals. For
smaller eigenfrequencies, the Cowling approximation is used to find the
eigenfrequencies and the expression (78) of Paper 1 was employed to find the
overlap integrals. This expression is equivalent to the original one presented
above provided the Cowling approximation is used to find eigenmodes.
We checked that in the intermediate region for which $\omega\sim 0.3-0.5$ both
methods give practically the same results. The advantage of using the
expression (78) of Paper 1 in the low frequency limit is the fact that the
integrand in this expression is less oscillatory in comparison to the
integrand in the original expression, thus allowing us to significantly
increase accuracy of determination of $\hat{Q}$ in this limit where the
eigenfunctions contain a large number of nodes.
Models of massive stars have very rapid variations of of $N^{2}$, see Figs 2-4
in the transition regions between radiative envelopes and convective cores,.
Since the characteristic radial extent of these variations can be of the order
of the initial stellar model grid size a treatment of discontinuities may be
needed. We prefer, however, to avoid this situation in our approach. To deal
with this issue, we worked with computational grids which had a much larger
number of grid points than were originally used to represent the structure of
the stellar models. State variables were interpolated onto our more refined
grid as smooth functions. There were then no discontinuities in the
eigenfunctions on the refined grid, see Figs 5, 6.
Figure 5: The radial dependence of the radial and tangential components of the
displacement vector, $\xi$ and $\xi^{S},$ for typical eigenmodes for models 2a
and 2d. These are plotted in arbitrary units as functions of the radius $r,$
in the inner region $0<r<0.15$. Dotted and dot dashed curves (black and red in
the on-line version) and dashed and solid curves (green and blue in the on-
line version) show $\xi$ and $\xi^{S}$, respectively, for models 2a and 2d.
Fig. 2 indicates that model 2a is mostly radiative, while model 2d has a sharp
transition from the exterior radiative region to the convective core which is
situated at $r_{c}\sim 0.05$. It is seen that the eigenfunctions corresponding
to the mostly radiative model are smooth while those corresponding to the
model with a convective core demonstrate a very sharp change of behaviour in
the vicinity of $r\sim r_{c}$. Figure 6: Same as Fig. 5 but a region very
close to $r_{c}$ is shown. One can see that although the eigenfunctions
corresponding to model 2d may look discontinuous in Fig. 5, in fact they are
smooth. Figure 7: The overlap integrals $\hat{Q}$ as functions of mode
eigenfrequency $\omega$ for models 1a and 1b plotted with dashed (green in the
on-line version) and solid curves (red in the on-line version), respectively.
The black dotted curve plots the overlap integrals for a polytrope with $n=3.$
Note that for low frequencies, these are much smaller than the ones
corresponding to stellar models with realistic structure. Symbols show the
positions of eigenfrequencies, with diamonds, squares and circles representing
the results for models 1a, 1b and the polytropic model, respectively. The
smooth curves are interpolated through these. Figure 8: As for Fig. 7 but for
models with $M_{*}=1.5M_{\odot}$. Dot dashed (blue in the on-line version),
dashed (green in the on-line version) and solid (red in the on-line version)
curves are for models 1.5a, 1.5b and 1.5c, respectively. We show the results
for models 1p and 1b, plotted as black dotted and dot dot dashed curves
respectively, for comparison. Diamonds, squares and circles show positions of
particular eigenfrequencies for models 1.5a, 1.5b and 1.5c, respectively.
Figure 9: As for Fig. 7 but for models with $M_{*}=2M_{\odot}$. Dot dot dashed
(magenta in the on-line version), dot dashed (blue in the on-line version),
dashed (green in the on-line version) and solid (red in the on-line version)
curves are for models 2a, 2b, 2c and 2d, respectively. Triangles, diamonds,
squares and circles show positions of particular eigenfrequencies for models
2a, 2b, 2c and 2d, respectively. The dotted curve gives the results for model
1p. Figure 10: Same as Fig. 7 but for model 5a of a young star with
$M_{*}=5M_{\odot}.$ The solid (red in the on-line version ) curve and circles
show the results calculated for this model. The dotted curve is for model 1p.
The results of calculations of overlap integrals are shown in Figs. 7-10. In
Fig. 7 we show the results for models 1p, 1a and 1b. Since the polytropic
model 1p has been discussed extensively elsewhere (eg. Press $\&$ Teukolsky
1977, Lee $\&$ Ostriker 1986, Ivanov $\&$ Papaloizou 2004 and references
therein) and overlap integrals corresponding to models 1a and 1b are discussed
in detail in Paper 1, here we only mention that the polytropic model has much
smaller overlap integrals when $\omega<0.4$. This is due to the fact that
$\hat{Q}$ corresponding to Sun-like stars has contributions arising from the
presence of a convective envelope. These decay as a power of $\omega$ in the
limit $\omega\rightarrow 0,$ while the overlap integrals of the polytropic
model may be shown to decay faster than any power of $\omega.$ Note that the
overlap integrals for the solar model have also been calculated recently by
Weinberg et al (2012). Our calculations for the model 1b agree quite well with
their results. It is of interest to note that the overlap integrals for this
model are not monotonic at large frequencies. This effect is even more
prominent for the models with $M_{*}=1.5M_{\odot}$ and $2M_{\odot}$ discussed
below, where several peaks in the values of $\hat{Q}$ at values of frequencies
corresponding to pressure modes are observed, see Figs. 8 and 9. Since this
effect is not important for our purposes we do not discuss it here. However,
we would like to mention that it appears to be rather generic, e.g. it is
present in the dependence of $\hat{Q}$ on $\omega$ in a model of red giant
star, see Fuller et al (2012).
Fig 8 shows results for models 1.5a-1.5c. For the range of frequencies
plotted, only model 1.5a has overlap integrals larger than those of the
polytrope at small values of $\omega.$ As discussed above this is because of
the fact that this model has a rather extended convective envelope. From our
analytical theory developed in Paper 1, it follows that in the case of model
1.5a, the contribution determined by the presence of this region should be
roughly three times smaller than that arising for Sun-like stars. As seen from
8, this is confirmed by our numerical results. More evolved models 1.5b and
1.5c have rather small overlap integrals in the range of frequencies plotted.
They are even smaller than those of the polytropic star. This indicates that
tidal interactions determined by the excitation of eigenmodes in the shown
range of frequencies are relatively weak for these models222Note that the
strength of tidal interactions also depends on the ratio of the central and
stellar mean density, see below.
Results for stars of mass $M_{*}=2M_{\odot}$ are shown in Fig. 9. This case is
rather similar to the previous one. However, the more massive stars do not
have well pronounced convective envelopes and, therefore, their overlap
integrals are rather small for the range of frequencies shown, being several
times smaller than those for a polytropic star. Finally, in Fig. 10 the
overlap integrals of a young star with $M_{*}=5M_{\odot}$ are shown. They are
rather similar to, though slightly smaller than those of a polytropic star for
the range of frequencies shown. This means that in this frequency range the
contribution determined by the presence of a convective core is probably not
seen.
### 3.2 Rotational splitting coefficients
Figure 11: The rotational splitting coefficients $\beta_{r}$ as functions of
$\omega$ for models with $M_{*}=M_{\odot}$ and $M_{*}=5M_{\odot}$. Dotted,
solid, dashed and dot dashed curves are for models 1p, 1b, 1a and 5a,
respectively. Open circles show positions of numerically calculated
eigenfrequencies. Figure 12: Same as Fig. 11 but for $M_{*}=1.5M_{\odot}$.
Solid, dashed and dotted curves are for models 1.5c, 1.5b and 1.5a,
respectively. Figure 13: Same as Fig. 11 but for $M_{*}=2M_{\odot}$. Solid,
dashed, dotted and dot dashed curves are for models 2d, 2c, 2b and 2a,
respectively.
When the rotation frequency of a star, $\Omega$, is small in comparison to its
natural frequency: $\Omega\ll\Omega_{*}$ one may treat effects due rotation in
a simplified manner (eg. Lai 1997, Ivanov $\&$ Papaloizou 2011). From first
order perturbation theory, the eigenfrequencies, $\omega_{j}$, are shifted
with respect to their values for a non-rotating star, $\omega_{0,j}$, by an
amount proportional to $\Omega$. We take this shift into account but assume
that the overlap integrals are unchanged.
In the inertial frame we have (see eg. Christensen-Dalsgaard 1998)
$\omega_{j}=\omega_{0,j}+m\beta_{r}\Omega,$ (6)
where $m=0,\pm 1,\pm 2$ is the azimuthal number and $\beta_{r}$ are
dimensionless coefficients determining the magnitude of the rotational
splitting. Note that when the rotation axis is perpendicular to the orbital
plane, terms with $m=\pm 1$ do not contribute to the energy and angular
momentum transfer as a result of a periastron flyby. The rotational splitting
coefficients $\beta_{r}$ can be expressed as a ratio of two integrals
involving the components of the mode Lagrangian displacement. When
$\omega_{0,j}\gg 1$ they are close to unity and when $\omega_{0,j}\ll 1$ they
tend to $1-1/(l(l+1)={5/6}$ for spherical harmonic index, $l=2,$ (eg.
Christensen-Dalsgaard 1998). In the intermediate range of eigenfrequencies
these integrals have to be evaluated numerically. We calculate them for our
stellar models and illustrate them for models 1p, 1a, 1b and 5a in Fig. 11.
Results for models 1.5a - 1.5c are illustrated in Fig. 12, and for models
2a-2d in Fig. 13. One can see from these figures that the dependence of
$\beta_{r}$ on $\omega_{0,j}$ is not necessarily monotonic, as was also found
for of the overlap integrals.
From the results presented below, we find that use of the perturbative
description of the influence of rotation on tides described above yields
results that agree quite well with those obtained from a more accurate
approach based on the traditional approximation (Unno et al. 1989), even for
quite large stellar angular velocities $\Omega\sim 0.4.$ This is the case when
either the stellar rotation is retrograde with respect to that of the orbital
motion, or prograde with respect to the orbital motion, but with the angular
frequency being smaller in magnitude than approximately the value
corresponding to pseudosynchronization, for which there is zero angular
momentum transfer.
### 3.3 The overlap integrals in the traditional approximation
Figure 14: We show the radial contribution to the overlap integrals
$\hat{Q}_{r}$ for model 1b of the present day Sun as functions of
eigenfrequency $\omega$ for different values of $\Omega.$ The cases of
$|\Omega|=0.42$, $0.21$, $0.11$ and the non-rotating case are shown using
solid (red in the on-line version), dashed (blue in the on-line version), dot
dashed (green in the on-line version) and black dotted curves, respectively.
The curves of the same type with smaller (larger) values of $\hat{Q}_{r}$ for
a given value of $\omega$ are calculated for retrograde (prograde) directions
of rotation with respect to the pattern rotation associated with the forcing
potential. The curves corresponding to the three cases with prograde rotation
almost coincide. Symbols show the positions of eigenfrequencies with squares,
triangles, diamonds and circles corresponding to $|\Omega|=0.42$, $0.21$,
$0.11$ and $0$, respectively. Figure 15: As in Fig. 14 but for model 1.5a.
The cases $|\Omega|=0$, $0.25$ and $0.5$ are plotted using black dotted, dot
dashed (blue in the on-line version) and solid (red in the on-line version)
curves, respectively. The curves corresponding to the three cases with
prograde rotation almost coincide. Squares, triangles and circles show
positions of eigenfrequencies for $|\Omega|=0.5$, $0.25$ and $0$,
respectively.
As explained in Paper 1 and above, when the traditional approximation is used
the overlap integrals $\hat{Q}$ can be represented as products of ’angular’
and ’radial’ contributions. Thus $\hat{Q}=\alpha\hat{Q}_{r}$, where the
angular contribution $\alpha$ is given in Fig. 1 of Paper 1. The radial
contribution, $\hat{Q}_{r},$ is illustrated in Fig. 14 and 15, for models 1b
and 1.5a, respectively. Note that the curves corresponding to the largest
retrograde rotation are not monotonic, with the overlap integrals having a
pronounced minimum at some value of the eigenfrequency. This effect is
explained in Paper 1. Namely, as follows from the discussion above, the values
of the radial contributions to overlap integrals for stars with extended
convective envelopes are mainly determined by contributions coming from the
convective envelope and from the vicinity of the base of the convective zone.
It can be shown (see Paper 1, equation (111)) that these contributions are
proportional to the factor $(1-30/\Lambda(\nu))$, where we recall that
$\Lambda$ is the eigenvalue associated with acceptable solutions of the
Laplace tidal equation and $\nu=2\Omega/\omega$. It turns out that in case of
retrograde rotation, the value of $\Lambda$ corresponding to the frequencies,
where this minimum is observed, is approximately equal to thirty, and,
therefore, the main contributions to the overlap integrals are strongly
suppressed.
We use below the traditional approximation described in detail in Paper 1 to
calculate the energy and angular momentum transfer for model 1b as a result of
a flyby of a perturber on a parabolic orbit for $\Omega=\pm 0.11$, $\pm 0.21$
and $\pm 0.42$ as well as for model 1.5a for $\Omega=\pm 0.25$ and $\pm 0.5$.
These are compared with results obtained by numerically solving the flyby
problem directly as an initial value problem.
## 4 Transfers of energy and angular momentum arising from parabolic
encounters and the tidal capture problem
In this Section we discuss the well known problem of calculating the energy
and angular momentum transferred to the normal modes of a star as a result of
a parabolic encounter with a perturber treated as a point mass. We consider
realistic stellar models. In general, we assume that the stellar rotation axis
is perpendicular to the orbital plane. For a discussion of the case of a
general inclination between the stellar rotation axis and the orbital angular
momentum vector, see Ivanov $\&$ Papaloizou (2011).
The magnitude of the energy transferred, $\Delta E$, depends on whether it is
calculated in the inertial or rotating frame. In general, we have
$\Delta E_{I}=\Delta E+\Omega\Delta L,$ (7)
where $\Delta E_{I}$ and $\Delta E$ are the energy transferred as calculated
in the inertial and rotating frame, respectively, $\Omega$ is the stellar
angular velocity, and $\Delta L$ is the amount of angular momentum
transferred.
It is convenient to introduce natural units for the energy and angular
momentum transferred. Thus, we express $\Delta E_{I}$ and $\Delta E$ in units
of $E_{*}=Gm^{2}_{p}/((1+q)^{2}R_{*})$, where $G$ is the gravitational
constant, $m_{p}$ is the mass of a perturbing body, and $q=m_{p}/M_{*}.$ We
express $\Delta L$ in units of
$L_{*}={q^{2}(1+q)^{-2}}M_{*}\sqrt{GM_{*}R_{*}}$. We remark that all
quantities in equations (54) of Paper 1 can also be represented in natural
units 333The coefficients $A_{m}$ should be expressed in units of
$Gm_{p}/(R^{3}_{p}\Omega_{p})$, where $R_{p}$ is the periastron distance and a
typical periastron passage frequency
$\Omega_{p}=\sqrt{GM_{*}(1+q)/R^{3}_{p}}$. The overlap integral should be in
the units of $\sqrt{M_{*}}R_{*}$ while the eigenfrequencies and stellar
angular frequency are in the units of $\Omega_{*}$. Once the ratio
$\Omega/\Omega_{*}$ is specified, the energy and angular momentum transferred
expressed in natural units are functions of only one parameter (see eg. Press
$\&$ Teukolsky 1977, Ivanov $\&$ Papaloizou 2004, 2007)
$\eta=\sqrt{{1\over 1+q}{\left({R_{p}\over
R_{*}}\right)}^{3}}=3.05\sqrt{\bar{\rho}}P_{orb},$ (8)
where we recall that $R_{p}$ is the periastron distance. The quantity
$\bar{\rho}$ is the ratio of the mean stellar density to the solar value,
$\bar{\rho}={R_{\odot}^{3}M_{*}/(R_{*}^{3}M_{\odot})},$ and $P_{orb}$ is
orbital period of a circular orbit which has the same value of the orbital
angular momentum as the parabolic orbit under consideration, expressed in
units of one day.
Assuming that tidal evolution approximately conserves angular momentum (see,
eg. Ivanov $\&$ Papaloizou (2011) for a discussion of this approximation)
$P_{orb}$ characterises the orbital period of the binary system after the
process of tidal circularisation is complete. Note that when $q\ll 1$ as for
dynamic tides induced in a central star in exoplanetary systems, the condition
$\eta=1$ corresponds to a grazing encounter with periastron distance equal to
the stellar radius. Thus, for these systems only
$P_{orb}>P_{crit}=0.325/\sqrt{\bar{\rho}}$ are possible.
We remark that a number of authors (eg. Press $\&$ Teukolsky 1977, Lee $\&$
Ostriker 1986, Giersz 1986, McMillan, McDermott $\&$ Taam 1987) express
results in terms of another dimensionless quantity, $T{{}_{2}}(\eta)$ 444We
remark that we consider the quadrupole component of the forcing potential.,
which is related to the dimensionless energy transfer, $\Delta E$, through
$T_{2}(\eta)=\eta^{4}\Delta E.$ (9)
We use equation (9) to compare our results with whose obtained by previous
authors.
When $\Delta E_{I}$ and $\Delta E$ are expressed in natural units, their
dependence on $\eta$ may be used to compare the strength of tidal interactions
of stars with different masses and radii. However, for systems with given
orbital parameters and $m_{p}$ it also depends on the average density being
larger for stars with smaller $\bar{\rho}$, mainly through the dependence of
$\eta$ on this quantity. Therefore, in a similar way to our previous studies
(Ivanov $\&$ Papaloizou 2004, 2007, 2011) we introduce the tidal
circularisation time
$T_{ev}=15\left({M_{*}\over M_{\odot}}\right)\left({R_{*}\over
R_{\odot}}\right)\left({M_{J}\over m_{p}}\right){1\over\Delta
E_{I}}\sqrt{a_{in}},$ (10)
where $M_{J}$ is the mass of Jupiter ( see equation (104) of Ivanov &
Papaloizou 2007). From here on, it is implied that energy and angular momentum
transfers are expressed in natural units, and $T_{ev}$ is expressed in years.
Under the assumption that the energy transfers arising from consecutive
periastron passages can be simply added, $T_{ev}$ gives a characteristic time
scale for the tidal evolution of the semimajor axis of a highly eccentric
orbit with initial semi-major axis, $a_{in}$, in units of $10$ AU. We stress
that this time scale applies only to the initial stages of circularisation and
will be characteristic of the whole process, only if it can proceed
efficiently enough at small eccentricities (see Ivanov & Papaloizou 2011 for a
discussion). We set $m_{p}=M_{J}$ and $a_{in}=1$ hereafter, generalisation to
other values of $m_{p}$ simply follows from the form of equation (10).
It is convenient to use equation (10) when considering tidal interactions in
systems containing exoplanets. However, dynamic tides may be also important
for other problems, such as eg. the tidal capture of stars to form binary
systems in stellar clusters (eg. Fabian, Pringle $\&$ Rees 1975, Press $\&$
Teukolsky 1977). To characterise the strength of tidal interactions in such a
setting it is convenient to introduce ’the capture radius’ $R_{cap}$, defined
by the condition that when the periastron distance for a tidal encounter
between two initially unbound stars is equal to $R_{cap},$ the initial
relative kinetic energy of these stars, when they are very far apart, is equal
to the amount of energy transferred due to tidal interactions, i.e.
$\Delta E_{I}^{(1)}+\Delta E_{I}^{(2)}={1\over 2}{M_{*}^{(1)}M_{*}^{(2)}\over
M_{*}^{(1)}+M_{*}^{(2)}}v_{rel}^{2},$ (11)
where the upper index, $i,$ denotes quantities associated with star $i,$ and
$v_{rel}$ is the initial relative velocity of the stars with respect to each
other. Assuming that the binary consists of two identical stars, which rotate
in the same sense with respect to their orbital motion, we get
$\Delta E_{I}(\eta_{cap})=3.125\cdot 10^{-4}v_{*}^{2},\quad
v_{*}=\sqrt{({M_{\odot}\over M_{*}}{R_{*}\over R_{\odot}})}{v_{rel}\over
10km/s},$ (12)
and $\eta_{cap}$ is related to $R_{cap}$ through equation (8) with
$R_{p}=R_{cap}$.
### 4.1 Energy and angular momentum transfer as a result of a parabolic
encounter from direct numerical solution of the linear initial value problem
We have calculated the energy and angular momentum transferred to a star as a
result of an encounter with a perturber on a parabolic orbit by solving the
linear initial value problem directly. We refer to this procedure as the
direct numerical approach. The method adopted follows from that described in
Papaloizou & Ivanov (2010) and Ivanov & Papaloizou (2011). The equations
solved are (36)-(41) of Ivanov and Papaloizou (2011) with the following
modifications. In that work a polytrope of index $n=3$ and a constant
adiabatic index $\gamma=5/3$ was considered. However, here we consider
realistic stellar models for which this varies. Accordingly the quantity
$P^{1/\gamma}/\rho\nabla(P^{\prime}/P^{1/\gamma})$ in equation (36) of Ivanov
and Papaloizou (2011) was replaced by $F_{ad}/\rho\nabla(P^{\prime}/F_{ad}),$
where
$F_{ad}=\int^{P}_{P_{s}}\frac{1}{\Gamma_{1}P}dP,$ (13)
where $\Gamma_{1}=(d\ln P/d\ln\rho)_{adiabatic}$ and $P_{s}$ is the surface
boundary or photospheric pressure. This quantity is readily obtainable for the
models provided by numerical integration. Elsewhere in equations (36)-(41) of
Ivanov and Papaloizou (2011), $\gamma$ was replaced by $\Gamma_{1}.$
The presence of regions with negative $N^{2}\equiv\omega^{2}_{BV}$ gave rise
to linearly convectively unstable eigenmodes which would ultimately dominate
the solution. In order to remove such modes, as long as the density gradient
was negative, we redefined $\Gamma_{1}$ in such regions such that
$N^{2}\rightarrow 0$ there. This procedure amounts to stating that during
linear perturbation, the relationship between $P$ and $\rho$ in these regions
is maintained. Here we are adopting the common approximation that the layers
are effectively adiabatically stratified (eg. Ogilvie & Lin 2007). This is
equivalent to the condition that the convective, or frictional time scale (cf.
Zahn 1977) be significantly longer than the rotation period, which is expected
to be satisfied for the cases we consider.
In some models, there were small low density regions where the density
gradient became positive ( see discussion in Section 2.2 above). For these
regions, again $N^{2}$ was set to zero, but $\Gamma_{1}$ was not allowed to
become negative, instead being set to be the largest positive value attained
on the grid from the first procedure. In this way an incompressibility
condition is approached. However, as the values of the density and pressure
were very small, this did not lead to numerical difficulties or, as shown by
numerical tests, affect results significantly. As in our previous work, most
simulations were carried out on a $200\times 200$ numerical grid with $m=2$
which gives the dominant contribution. Resolution tests were carried out by
doubling the resolution to $400\times 400$ and as in our previous work showed
good convergence in these cases with variable $\Gamma_{1}.$
### 4.2 Non-rotating stars
In this Section we consider non rotating stars, accordingly $\Omega=0.$ In
this case $\Delta E_{I}=\Delta E$. The results of numerical calculations of
$\Delta E$ and $T_{ev}$ for the stellar models presented in table 1 are shown
in Figs. 16-23.
Figure 16: Energy transferred to the normal modes of a non-rotating star with
$M_{*}=M_{\odot}$ as the result of a parabolic flyby of a perturber of mass
$m_{p}=M_{J}$, expressed in units of $E_{*}$, as a function of the parameter
$\eta$. The solid, dashed, and dotted curves are for models 1b, 1a and 1p,
respectively. The dot dashed curve shows results calculated using the purely
analytic expressions for the mode eigenfrequencies and overlap integrals
obtained in Paper 1 for model 1b. This almost coincides with the solid line.
The dot dot dashed curve shows the result of Giersz (1986). Circles show the
energy transfer calculated for model 1b using the direct numerical approach.
Figure 17: The evolution time $T_{ev}$ as a function of the orbital period
after circularisation, $P_{orb}$, for the same models as illustrated in Fig.
16. Again, the solid, dashed, and dotted curves are for models 1b, 1a and 1p,
respectively.
In Fig 16 we show the dependence of $\Delta E$ on $\ eta$ calculated for our
models of Sun-like stars, 1a and 1b, together with the same quantities
calculated for a polytropic star with solar mass and radius. Circles show
results obtained from the direct numerical approach. One can see that the
approach based on normal mode calculation gives practically the same energy
transfer as the direct numerical approach. A small deviation for $\eta=8$ are
probably due to the effect of numerical diffusion and finite integration time.
As seen from Fig.16, the energy transfers for realistic models are
significantly larger than those for the polytropic model for large enough
values of $\eta.$ Clearly, this is due to much larger values of the overlap
integrals for the realistic models at small eigenfrequencies, see Fig. 7.
Interestingly, the energy transfers for models 1a and 1b are quite close to
each other regardless of the fact that the overlap integrals corresponding to
model 1a are larger than those of model 1b. This is explained by observation
that the number of eigenmodes contributing to the tidal interaction is larger
in the case of model 1b. This compensates for smaller values of the overlap
integrals.
The dot dashed curve shown in Fig. 16 is obtained using the formalism
described in paper 1 for model 1b. This curve is calculated by purely analytic
means. We see that there is quite good agreement between the analytic and
numerical results. In the range $2<\eta<10$ the deviation is at most about 40
per cent, when $\eta>10,$ corresponding curves practically coincide. The dot
dot dashed curve in the same figure shows the result of Giersz (1986) for a
solar model. The energy transfer found Giersz (1986) is significantly smaller
than we obtain in this paper. Since our results are obtained using three
independent methods, we believe that the Giersz (1986) result underestimates
the energy transfer, though the origin of the discrepancy is unclear. One
possible explanation is that the number of eigenmodes used in his calculation
was too small.
From Fig. 17 it follows that the evolution time scale $T_{ev}$ is smaller for
model 1b, than for model 1a, for a given value of $P_{orb}$555 Note that our
calculations for model 1a are not realistic when $T_{ev}$ exceeds its age
$\approx 1.7\cdot 10^{8}$ years. More realistic calculations of $T_{ev}$
should employ a set of overlap integrals and eigenfrequencies calculated for a
grid of stellar models of different ages.. The fact that the model of present
day Sun, model 1b, can be tidally excited more efficiently than the
corresponding polytropic model, leads to the conclusion that taking into
account realistic stellar models can significantly increase estimates of the
contribution of tides exerted on the star for the orbital evolution of Jupiter
mass exoplanets on highly eccentric orbits. The enhanced tidal interaction can
produce a significant change of the orbital semimajor axis in a time of less
than $4\cdot 10^{9}$ years when $P_{orb}<4.$ Note that this contribution can
be further amplified for stars rotating in the opposite sense to that of the
orbital motion, as discussed in Lai (1997), Ivanov $\&$ Papaloizou (2011) and
below. Furthermore, tides exerted on the star become even more efficient for
more massive planets, see eg. Ivanov $\&$ Papaloizou (2004), (2007). We also
remark that a consequence of the above is that a significant component of the
energy liberated by orbital circularisation may be dissipated in the star
rather than the planet, thus alleviating the possibility of the potential
destruction of the planet (see eg. the discussion in Ivanov & Papaloizou
2004).
Figure 18: Same as in Fig. 16 but for the models with $M_{*}=1.5M_{\odot}$.
The two dotted curves are for our ’reference’ models 1p and 1b. The polytropic
star, model 1p, has a smaller value of $\Delta E$ for a given value of $\eta.$
Solid, dashed and dot dashed curves are for models 1.5c, 1.5b and 1.5a,
respectively. As in Fig. 16, symbols indicate the amount of energy transferred
that was obtained adopting the direct numerical approach. The circle and
square are for models 1.5a and 1.5c, respectively. The dot dot dashed curve
shows the results of McMillan, McDermott $\&$ Taam 1987 for a population II
model. Figure 19: The evolution time $T_{ev}$ for models with
$M_{*}=1.5M_{\odot}$ as a function of $P_{orb}$. Curves of a given type
plotted in Fig. 18 and Fig. 19 correspond to the same models.
In Figs. 18 and 19 we show the energy transfer $\Delta E$ and the evolution
time $T_{ev}$ calculated for more massive stellar models with
$M_{*}=1.5M_{\odot}$. The dotted curves on both Figs. are for our reference
models, namely the polytropic model 1p, and the solar model 1b. As seen from
Fig. 18 the energy transfer expressed in the natural units is significantly
smaller for all the more massive models as compared to the solar model. This
is obviously because of the smaller values of the overlap integrals, see Fig.
8. However, at large values of $\eta,$ model 1.5a gives larger values for the
energy transfer than the polytropic model. As discussed above, this is due to
the presence of rather extended convective envelope in model 1.5a, which
results in larger values of the overlap integrals at small eigenfrequencies as
compared to models 1p, 1.5b, and 1.5c.
The symbols in Fig. 18 indicate the energy transfer obtained from the direct
numerical approach for models 1.5a and 1.5c. As seen from Fig.18, the
agreement between this method and the normal mode approach is excellent. The
dot dashed curve shows the results of McMillan, McDermott $\&$ Taam (1987) for
a Population II $1.5M_{\odot}$ star. At sufficiently large values of $\eta,$
the energy transfer for their model is significantly smaller than that for
model 1.5a, but larger than that for models 1.5b and 1.5c. Unfortunately,
McMillan, McDermott $\&$ Taam (1987) did not provide details of their model.
Thus it is unclear as to whether this behaviour is a consequence of the form
of the Brunt - V$\ddot{\rm a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency, as is
indicated from consideration of our models.
However, it is important to point out that, for a given $P_{orb},$ the
circularisation times, $T_{ev}$, corresponding to the more massive models are
significantly smaller than those for the Sun-like models. This is due to the
fact that the mean density of the more massive models is significantly smaller
than the mean density of the Sun-like models, see table 1, which leads to
smaller values of $\eta$ for the more massive models for a given value of
$P_{orb}$, see equation (8). In particular, model 1.5c having the largest age
$\sim 1.6\cdot 10^{9}$ years, has $T_{ev}$ of the order of or smaller than its
age, for planets having $P_{orb}<4$. The increase of the efficiency of the
tidal interaction for model 1.5c, as compared to models 1.5b and 1b, can also
be viewed as a consequence of the expansion of the star that takes place as a
result of evolution.
Figure 20: Same as in Fig. 18 but for the models with $M_{*}=2M_{\odot}$.
Solid, dashed, dot dashed and dot dot dashed curves are for models 2d, 2c, 2b
and 2c respectively, the dotted curves are the same as in Fig. 18. Figure 21:
The evolution time $T_{ev}$ for models with $M_{*}=2M_{\odot}$ as a function
of $P_{orb}$. Curves of the same type in Fig. 20 and Fig. 21 correspond to the
same models. Figure 22: The energy transfer $\Delta E$ for the model 5a with
$M=5M_{\odot}$ shown by the solid curve, the dotted curves show our reference
models 1p and 1b as in the previous Figs. Figure 23: Same as Fig. 22 but for
the evolution time $T_{ev}$.
Models with $M=2M_{\odot}$ and $M=5M_{\odot}$ are qualitatively similar to
models 1.5b and 1.5c. In all cases the energy transfer is even smaller that
for the polytropic model, while the evolution times $T_{ev}$ are smaller than
those for the reference models 1p and 1b, due to the smaller mean densities of
the massive stars. Note that the energy transfers calculated for models 2c and
2d are very close to each other. It is also interesting to note that the value
of the mean density evolves non-monotonically with time. For
$M_{*}=2M_{\odot},$ model 2d with the greatest age $\sim 10^{9}$ years, this
is smallest. This leads to possibility of significant evolution of the
semimajor axis, with $T_{ev}<10^{9}$ years, for exoplanets orbiting stars with
$M_{*}=2M_{\odot}$ and final orbital periods, assuming the circularisation
process can be completed of $P_{orb}<7$ days, solely due to tides exerted on
the star.
### 4.3 Rotating stars in the traditional approximation
In this Section we present results for rotating models 1b and 1.5a within the
framework of the traditional approximation described in detail in Paper 1. We
compare these results to those obtained from the perturbative approach, where
it is assumed that the overlap integrals are not changed by rotation, and the
mode eigenfrequencies as viewed in the inertial frame are shifted by a factor
$m\beta_{r}\Omega$, as given by equation (6). The perturbative approach is
described in more detail in Ivanov $\&$ Papaloizou (2011) and references
therein. Results are also compared to those obtained following the numerical
approach, that is by solving the encounter problem directly as an initial
value problem (see section 4.1). In all of this it is assumed that the
rotation axis is perpendicular to the orbital plane, with both prograde and
retrograde encounters with respect to the direction of the stellar rotation
being considered 666For discussion of the general case of an arbitrary
inclination of the stellar rotational axis, see Ivanov $\&$ Papaloizou
(2011)..
### 4.4 A comparison of eigenspectra obtained from normal mode calculations
with those obtained from the direct numerical approach
Figure 24: The amplitude of the time Fourier transform of the velocity
component in the $\theta$ direction evaluated at a characteristic interior
point as a function of frequency $\omega.$ The case of non-rotating model 1b
and $\eta=4\sqrt{2}$ is presented. Circles show the positions of
eigenfrequencies calculated by finding the normal modes, assuming the
traditional approximation directly (the normal mode method). Figure 25: Same
as Fig. 24 but for a retrograde tidal encounter with $\Omega=0.58$.
In order to check whether our direct numerical method and the normal modes
approach agree with each other we consider $\theta$-component of the perturbed
velocity at a characteristic position in the star as a function of time and
make a Fourier transform of the signal. The results are shown in Figs. 24 and
25 for a tidal encounter of a non-rotating star and a retrograde encounter
with $\Omega=0.58$, respectively. In the latter case positive and negative
values of the frequency $\omega$ correspond to perturbations propagating in
the direction of stellar rotation and opposite to this direction,
respectively. The value of the amplitude scaling of the Fourier transform is
arbitrary. Peaks in these Figs. indicate the approximate positions of free
normal mode pulsations. Symbols show the positions of eigenfrequencies
calculated using the normal mode approach.
As seen from Fig. 24, in the case of the non-rotating star, mainly positive
eigenfrequencies are excited in the course of tidal encounter. Positions of
peaks are in rather good agreement with the normal mode calculations for modes
having $|\omega|>1$. Smaller values of $\omega$ are not well resolved on
account of the finite time duration of the run.
Contrary to the non-rotating case, the retrograde tidal encounter mainly
excites eigenmodes with negative eigenfrequencies, see Fig. 25. Again, the
most prominent peaks approximately in the range $-3<\omega<-1$ are in a rather
good agreement with the normal mode method. The eigenfrequencies found from
the normal mode method are, however, shifted towards $\omega=0$ with respect
to the positions of the corresponding peaks, the shift being larger for
eigenmodes with larger absolute values of $\omega$. The situation is similar
for modes with positive values of $\omega$ in the range $1<\omega<3.5$, but in
this case mode eigenvalues are shifted towards larger values of $\omega$ with
respect to corresponding peak positions. This disagreement is possibly
determined by the fact that we use the Cowling and traditional approximations
to calculate the eigenspectrum in the normal mode approach. Since smaller
absolute values of the eigenfrequencies result in larger values of the
associated transfers of energy and angular momentum, we expect that the direct
numerical approach gives smaller values of these quantities in the case of
retrograde encounters and larger values for prograde encounters. This is
indeed obtained in our calculations, see the discussion below. Note too the
absence of significant peaks in the inertial range for which
$-1.16<\omega<1.16.$ This is where inertial modes associated with the
convective envelope would be expected to show up. However, when confined to a
spherical annulus inertial waves may focus on to wave attractors becoming
singular in the inviscid limit . Then corresponding discrete inviscid modes
may not exist (eg. Ogilvie & Lin 2004, Ogilvie 2013), but instead a continuous
spectrum, with the consequence that very prominent resonant spikes are not
seen in the response. Nonetheless the relatively small response in the
inertial range indicates that inertial waves are not important in this
particular case.
### 4.5 Energy and angular momentum transfer for rotating stars
Figs. 26-30 are for model 1b. In these Figs. we show results obtained from the
full numerical approach applied to Sun-like stars having $|\Omega|=0.42$,
$0.21$, $0.11$ and $0.$ These are represented by solid, dashed, dot dashed and
dotted curves, respectively. In Figs. 26-29 the retrograde encounters have a
larger value of the transferred quantity at a given value of $\eta$ then the
prograde encounters. The squares, triangles, diamonds and circles in Figs.
26-28 show the corresponding results obtained using the direct numerical
approach, respectively for $|\Omega|=0.42$, $0.21$, $0.11$ and $0$.
Figure 26: The energy transferred in the rotating frame expressed in natural
units as a function of $\eta.$ Curves of different style correspond to
different absolute values of the angular velocity $\Omega$ and symbols
indicate results obtained using the direct numerical approach. See the text
for the allocation of the curves with different styles. Figure 27:
Illustration of inertial waves in the convective envelope for the prograde
encounter of model 1b with $\eta=8$ and $\Omega=0.21.$ Contours of the product
of the Lagrangian displacement in the $\phi$ direction and $\sqrt{\rho}$ are
shown in an upper quadrant at a time after the encounter is over and the
energy transfer has been completed. The presence of rotationally modified $g$
modes of order up to 15 can be seen in the radiative core while the convective
envelope shows the presence of inertial waves that show reflections as well as
graze the boundary with the radiative core. The radial coordinate is expressed
in $cm.$ Figure 28: Same as Fig. 26, but for the absolute value of transferred
angular momentum, $|\Delta L|$. The locations where $\Delta L$ decreases
dramatically in magnitude correspond to pseudosynchronization (see text)
Figure 29: Same as Figs 26 and 28, but for the absolute value of energy in the
inertial frame, $|\Delta E_{I}|$. We recall that $\Delta E_{I}$ is related to
$\Delta E$ and $\Delta L$ through equation (7). Additionally, we show the
energy transfer calculated with the framework of the perturbative approach by
dot dot dashed lines. Different symbols on these lines correspond to different
rotation rates. Figure 30: The evolution time $T_{ev}$ defined through
equation 10 as a function of the orbital period after circularisation
$P_{orb}$ for the rotating solar models. Curves of a given style apply to the
same rotation rates as in Fig. 26. Figure 31: The capture radius $R_{cap}$
calculated according to equation (12) for model 1b as a function of a
’typical’ relative velocity $v_{*}$. Curves of a given style apply to the same
rotation rates as in Fig. 26, with larger values of $R_{cap}$ for a given
value of $v_{*}$ corresponding to retrograde encounters. Figure 32: The energy
exchange $\Delta E$ as a function of $\eta$ for a rotating model 1.5a. Solid
curves and squares, dashed curves and triangles, and finally dotted curves and
circles correspond to $|\Omega|=0.5$, $0.25$ and $0$, respectively. Figure 33:
Same as Fig. 32 but the transfer of angular momentum $\Delta L$ is shown.
Figure 34: The evolution time scale $T_{ev}$ as a function of the orbital
period $P_{orb}$ for model 1.5a. Curves of a given style represent the same
rotation rates as in Figs. 32 and 33. For a a given absolute value of
$\Omega,$ the retrograde case corresponds to a smaller value of $T_{ev}$ for a
given $P_{orb}.$ Figure 35: The radius $R_{cap}$ as a function of velocity
$v_{*}$ for rotating 1.5a model. Curves of a given style apply as for Fig. 32.
For a a given absolute value of $\Omega,$ the retrograde case corresponds to a
larger value of $R_{cap}$ for a given $P_{orb}.$
In Fig. 26 we show the transfer of energy in the inertial frame, $\Delta E$,
versus $\eta$. We see that retrograde encounters always have a larger value of
$\Delta E$ than prograde encounters, for a given $\eta$, with this effect
being more extreme for larger absolute values of the angular velocity. The
physical reason for this behaviour is discussed in detail in Lai (1997) and
Ivanov $\&$ Papaloizou (2011). It is interesting to note that prograde
encounters with non zero angular velocity produce larger values of $\Delta E$
than the case with $\Omega=0,$ when $\eta$ is large. Within the framework of
rotationally modified gravity modes under the traditional approximation, this
is explained by noting that in the limit of very large values of $\eta,$ the
eigenmodes mainly determining the value of $\Delta E$ propagate in retrograde
direction with respect to the stellar rotation in the rotating frame, but at
the same time, they propagate in the prograde direction in the inertial frame,
since the pattern speed of these modes is smaller than the angular velocity of
rotation. From the results of eg. Ivanov $\&$ Papaloizou 2011 it follows that
the energy transfer due to these modes is proportional to
$\eta^{-2}[\sum_{i}Q_{i}^{2}I^{2}_{2,-2}(y)]$, where
$y=\eta(2\Omega-|\omega_{i}|)$, the quantities $I_{l,m}(y)$ are discussed in
Press $\&$ Teukolsky (1977), see also Ivanov $\&$ Papaloizou (2007). The
function $I^{2}_{2,-2}(y)$ has a maximum at $y\approx 2$ and decreases towards
smaller and larger values of $y.$ Thus when $y=1,3$, it has values
approximately one half of its maximum value. Therefore, in order to crudely
estimate the energy transfer one may assume that only the contribution of
eigenmodes having eigenfrequencies such that $1<y<3$ need to be considered and
that all these modes have $I_{2,-2}(y)=I_{2,-2}(y=2)$. The absolute values of
the eigenfrequencies are close to $\omega_{max}=2\Omega-2/\eta,$ being defined
by the condition $y(\omega_{max})=2$. When $\eta\gg 1$ we can estimate the
number of participating modes to be $\propto 1/\eta$. On the other hand, the
typical frequency $\omega_{max}$ slightly increases when $\eta$ gets larger,
tending asymptotically to $2\Omega.$ For the rotation rates considered in this
Paper the overlap integrals corresponding to $\omega_{i}\approx 2\Omega$
increase quite sharply with increasing $\omega_{i}$, see Figs. 14 and 15. We
have checked that the fact the $Q_{i}$ increase with increasing $\eta$ and
$\omega_{max},$ approximately compensates for the decrease in the number of
modes giving a sizable contribution to $\Delta E.$ Therefore, the sum of the
contributions of these modes behaves approximately as a constant over a range
of values of $\eta.$ In this range $\Delta E\propto\eta^{-2}$ and both $\Delta
E_{I}$ and $\Delta L$ are negative. Note that this regime persists as long as
the number of terms in the sum is larger than one, which corresponds to
$\eta<\eta_{max}\approx 2/\Delta\omega$, where $\Delta\omega$ is the distance
between two neighbouring eigenfrequencies having
$\omega_{i}\approx\omega_{max}.$ When $\eta>\eta_{max},$ $\Delta E$ decreases
faster than $\eta^{-2}$. When $\Omega=0.42$ we find $\eta_{max}\approx 70$ for
the Sun-like models and we have checked that indeed this behaviour is observed
in our results.
As discussed above the values of $\Delta E$ obtained from the direct numerical
approach for the non-rotating star are in excellent agreement with the normal
mode method, with only the values corresponding to $\eta=8$ deviating by about
20%. This deviation may be explained by the influence of numerical viscosity,
which leads to relatively more dissipation over the long run times necessary
when $\eta$ is large. Such runs become prohibitive for $\eta>8.$ The case of
$\Omega=0.11$ is quite similar to the non rotating case, with only one sizable
deviation of the order of 40% associated with the retrograde encounter with
$\eta=8.$ When $\Omega=0.21$ the deviations are less than than 25% for
retrograde encounters and less than 30% for prograde encounters with $\eta\leq
4\sqrt{2}$. There is however, a large disagreement for the prograde encounter
with $\eta=8$ for which the direct numerical approach gives $\Delta E$
approximately 2.5 times larger than the normal mode method. Convergence checks
showed that this discrepancy was not due to lack of numerical convergence of
the direct numerical approach.
As seen from Fig. 28, for $\Omega=0.21$ this $\eta$ is close to the value
where $\Delta L=0$, where the star is in a state of pseudosynchronization (see
e.g. Papaloizou $\&$ Ivanov 2005, 2011, Ivanov $\&$ Papaloizou 2004, 2007 and
references therein). In a similar problem of a tidal encounter of a polytropic
rotating star discussed in Papaloizou $\&$ Ivanov (2011), an analogous
disagreement between the direct numerical and normal mode approaches was
observed.
Close to the state of pseudosynchronization the normal mode approach indicates
that when $\eta$ is fixed and $\Delta E$ and the absolute value of $\Delta L$
are considered to be functions of $\Omega,$ $\Delta E$ has a deep minimum at
$\Omega=\Omega_{ps},$ being the value for which pseudosynchronization occurs.
At that point $\Delta L=0.$ The differences between the two numerical
approaches may result in a shift in the location of this minimum, which
because of its depth, causes a large discrepancy when the methods are
compared.
In addition, our normal mode approach does not take into account the
contribution of inertial waves, which can be excited in the convective regions
and increase the amount of transferred energy as viewed in the rotating frame.
This effect would be most marked at pseudosynchronization. One would expect
that the inclusion of inertial modes would increase $\Delta E.$ To estimate
the possible magnitude of the effect, we note that Papaloizou & Ivanov (2005)
found that for a polytrope with $n=1.5,$
$\Delta E=6.5\times 10^{-3}E_{*}/\eta^{6}.$ (14)
Although only the convective envelope resembles such a polytrope, we use this
estimate. In fact there are two corrections, the first arising from the
truncation of the envelope at $r\sim 0.7R_{*},$ is expected to increase
$\Delta E$ by about an order of magnitude (Ogilvie 2013). The second, due to
the fact that the envelope is on top of a more centrally condensed model than
the polytrope and so has a lower base density, is expected to decrease $\Delta
E$ by a similar factor, thus in order to make rough estimates, we simply
assume these effects approximately cancel out. Use of (14) for $\eta=8,$ gives
$\Delta E\sim 2.5\times 10^{-8}$ which is about five times the value indicated
in Fig. 26. This indicates that inertial modes are likely to be significant
under conditions of pseudosynchronization for $\eta=8.$ Similar estimates
indicate that is is also the case for $\eta>8.$
In support the above discussion, we comment that the excitation of inertial
waves is seen in our simulation of the prograde encounter of model 1b with
$\eta=8$ and $\Omega=0.21.$ To illustrate this, contours of the product of the
Lagrangian displacement in the $\phi$ direction and $\sqrt{\rho}$ are shown in
an upper quadrant at a time after the encounter is over and the energy
transfer has been completed in Fig.27. The square of this quantity is
proportional to the kinetic energy density of the disturbance associated with
motion in the $\phi$ direction. The presence of rotationally modified $g$
modes is evident in the radiative core. Inertial waves are seen in the
convective envelope. These show some reflections and graze the boundary with
the radiative core as would be expected for critical latitude phenomena (see
Papaloizou & Ivanov 2010) .
When $\Omega=0.42$ the agreement between two approaches is less good. The
difference between $\Delta E$ is typically a factor of two for the prograde
encounters and a factor of three for retrograde encounters at larger $\eta.$
The fact that such retrograde encounters give a larger disagreement can be
explained by the shift of the eigenfrequencies of the dominant excited modes
as viewed in the rotating frame. These propagate against the sense of rotation
of the star towards larger absolute values as $\eta$ increases. In this case
both the traditional and Cowling approximations used in our normal mode
approach become less appropriate.
In Fig. 28 we plot the amount of angular momentum transferred, $\Delta L,$ as
a function of $\eta.$ For prograde encounters, in contrast to $\Delta E,$
$\Delta L$ changes sign at a value of $\eta,$ where the star rotates at the
pseudosynchronization rate. Accordingly, we plot the absolute value of $\Delta
L$ in this figure. The form of $|\Delta L,|$ for prograde encounters is non-
monotonic, having a deep minimum for $\eta\equiv\eta_{1},$ where
$\Omega=\Omega_{ps}$. When $\eta<\eta_{1},$ $\Delta L$ is positive (ie.
directed in the sense of stellar rotation), on the other hand when
$\eta>\eta_{1},$ it is negative. In the case of retrograde encounters $\Delta
L$ is always positive (ie. directed in the sense of the orbital motion). The
behaviour of the deviation between the direct numerical and normal mode
approaches is similar to that found for $\Delta E.$ We see again a better
agreement for prograde encounters, with the exception of the encounter having
$\Omega=0.21$ and $\eta=8$, where the deviation is rather large. This may be
explained as before.
Overall, our results indicate quantitative agreement between the two
approaches when the angular frequency is relatively small, say, $\Omega\leq
0.2$ except for rotation rates close to $\Omega_{ps}.$ For faster rotators the
agreement is not so good with the direct numerical approach giving values of
$\Delta E$ that are a factor of $2-3$ smaller for retrograde encounters and a
factor of $2-3$ larger for prograde encounters as long as $\eta<8.$ These
discrepancies probably arise from the neglect of inertial waves in the normal
mode treatment as well as use of the traditional approximation and the neglect
of self-gravity. Excitation of inertial waves would cause $\Delta E$ to
increase near to pseudosynchronization while the neglect of self-gravity and
the use of the traditional approximation become less appropriate for modes
excited at the high relative forcing frequencies that occurs for large
retrograde stellar rotation.
In Fig. 29 we plot the energy transfer in the inertial frame, $\Delta E_{I}$
related to $\Delta E$ and $\Delta L$ through equation (7). As for the angular
momentum transfer, it is negative for prograde encounters with
$\eta>\eta_{2}$, where $\eta_{2}\sim\eta_{1}$, and, therefore, absolute values
are plotted. The energy exchanged for prograde encounters has a sharp minimum
at $\eta=\eta_{2}$, which moves towards larger values of $\eta$ as the
magnitude of $\Omega$ decreases. Note that $\Delta E_{I}$ is always positive
for retrograde encounters. Let us recall that solid, dashed, dot dashed and
dotted curves apply to $|\Omega|=0.42$, $0.21$, $0.11$ and $0$, respectively.
Together with these results we also show the energy transfer calculated
adopting the ’perturbative’ approach where it is assumed that the overlap
integrals are not modified by rotation and the eigenfrequencies can be
calculated using (6). The respective curves are represented by dot dot dashed
lines. Symbols on these lines show different values of $|\Omega|$ with
circles, squares and diamonds corresponding to $|\Omega|=0.42$, $0.21$,
$0.11$, respectively. Remarkably, the perturbative approach agrees quite well
with the one based on the traditional approximation, especially for retrograde
encounters, even for the largest value of $|\Omega|=0.42$ adopted. In the case
of prograde encounters there is quantitative agreement when $\eta<\eta_{2}$,
and, accordingly, $\Delta E_{I}>0$. Since the perturbative approach does not
require the calculation of the overlap integrals for every given value of
$\Omega,$ the evaluation of $\Delta E_{I}$ is simplified to a great extent. It
suffices to use the overlap integrals obtained for non-rotating stars together
with the frequency splitting coefficients, $\beta_{r},$ given above for a
number of stellar models (see eg. Ivanov $\&$ Papaloizou 2011).
In Fig. 30 we plot the characteristic timescale of evolution of the semimajor
axis given by equation (10) as a function of $P_{orb}.$ The line styles are as
for Fig. 26. It is seen that for a given value of $P_{orb},$ retrograde
encounters have smaller values of $T_{ev}$ than the corresponding prograde
encounter. One can see, that for fast rotators, rotation has a significant
influence on the strength of tidal encounters ( see also Lai 1997 and Ivanov
$\&$ Papaloizou 2011). For example, when $|\Omega|=0.11,$777This corresponds
to a rotation period of the star of approximately one day. the binary system
may significantly change its semimajor axis in less than $10^{9}$yrs for
$P_{orb}<2.7$days and $<4$days for prograde and retrograde encounters,
respectively. Although stellar rotation significantly slows down in time this
effect may contribute to explaining observed exoplanetary systems containing
Hot Jupiters with a significant mismatch between directions of their orbital
angular momentum and the rotation axis of their central stars.
In Fig. 31 we show the tidal capture radius $R_{cap}$ for the rotating 1b
model. As seen from this Fig. the value of $R_{cap}$ is larger for retrograde
encounters as compared to prograde encounters as was first noted by Lai
(1997). However, this effect is prominent only when either, the rotation rate
is quite large, or the characteristic relative velocity, $v_{*}$, is small. It
is instructive to compare the dependence of $R_{cap}$ on the rotation rate
found here with results of Lai (1997), bearing in mind, however, that in that
work, a tidal encounter of a $n=1.5$ polytrope having $M_{*}=0.4M_{\odot}$ and
$R_{*}=0.5R_{\odot}$ with a point mass with $m_{p}=1.5M_{\odot}$ was
considered. When $v_{rel}=2.5km/s$ we find $R_{cap}\approx 2.7R_{*}$ and
$4.2R_{*}$ for prograde and retrograde encounters with the rather large value,
$\Omega=0.42,$ On the other hand Lai (1997) gives values of $R_{cap}\approx
4.2R_{*}$ and $6.4R_{*}$ for the same encounter, but with $\Omega=0.6$. The
ratio of the radii is approximately $0.65$ in both cases. Since the rotation
rate of our model is smaller, the relative variation of the capture radius
produced by changing from retrograde to prograde rotation is somewhat larger
in models of stars with realistic Sun-like structure as expected.
Finally, in Figs. 32-35 we plot the same quantities as in Figs. 26, 28, 30 and
31, but for the rotating model 1.5a. The absolute values of $\Omega$ are
$|\Omega|=0.5$, $0.25$ and $0.$ The results behave in a similar way to those
found for the rotating Sun-like star, with the difference that the transfers
of energy and angular momentum for a given value of $\eta,$ are smaller for
the more massive star, as is found in the non-rotating case. The evolution
timescales $T_{ev}$ are, however, larger for the Sun-like star due to its
larger average density. For example, model 1.5a with $\Omega=0.25$ gives an
evolution timescale of less than $10^{9}$years for retrograde encounters, when
the orbital period after circularisation $P_{orb}<12$ days. Of course, as in
the previous case, in order to make realistic calculations, one must take into
account the evolution of the stellar structure as it affects the mean density
of the star, as well as the braking of the stellar rotation with age.
## 5 Conclusions and Discussion
In this paper we have calculated the energy and angular momentum transferred
to a number of Population I stellar models with different masses, ages and
states of rotation through dynamical tides, as a result of an encounter with a
companion on a parabolic orbit. The results were used to estimate the initial
evolution time scale of the semimajor axis of a highly eccentric orbit.
Complementary methods based on calculation of the normal mode response and a
direct numerical approach involving the solution of the encounter problem as
an initial value problem were used. These showed quantitative agreement for
small and moderate rotation rates $|\Omega|<0.2$ as long as the distance of
closest approach was small enough that the stellar angular velocity is less
than the so-called pseudosynchronization frequency.
It was shown that when the energy and angular momentum transferred is
expressed in natural units that factor out the dependence on stellar mass and
radius so that encounters are characterised by the tidal parameter, $\eta,$
these quantities depend significantly on the stellar structure. The tidal
transfers are significantly larger for models having sufficiently extended
convective envelopes. Thus, when $\eta=8,$ other things being equal, the non-
rotating 1.5a model undergoes an energy transfer approximately $4$ times
larger than occurs for models 1.5b and 1.5c ,which is explained by the
presence of a more extended convective envelope. However, as the solar models
1a and 1b have more extended convective envelopes than models 1.5b and 1.5c
which are older than model 1.5a, the energy transferred to them is greater.
The effect increases in significance for larger values of $\eta.$ Thus when
$\eta=8,$ it is a factor of $15$ times larger.
Since the models of more massive stars with $M_{*}=2M_{\odot}$ and
$5M_{\odot}$ that we considered essentially do not have convective envelopes,
the energy transferred to them, expressed in natural units, is rather small.
Thus model 5a has a value of $\Delta E$ ten times smaller than that for model
1b when $\eta=8.$ Stellar rotation was found to play an important role, with
dynamic tides being significantly amplified for retrograde encounters, and
weakened for prograde ones, see also Lai (1997) and Ivanov $\&$ Papaloizou
(2011).
We studied the effect of rotation using the direct numerical approach, the
normal mode approach adopting the traditional approximation, and also simply
treating the effects of rotation by perturbation theory. In the latter
treatment, it was assumed that the overlap integrals are not modified by
rotation but that eigenfrequencies are shifted by an amount proportional to
the product of the splitting coefficient $\beta_{r}$ and rotation frequency
$\Omega,$ as expected from first order perturbation theory.
It was shown that the perturbative approach gives results in quantitative
agreement with the treatment based on the normal mode approach with the
traditional approximation, even for fast rotators, as long as the energy
transferred in the inertial frame $\Delta E_{I}$ was positive, being
approximately equivalent to the condition that the star rotated at less than
the pseudosynchronization rate.
As implied by our discussion in section 4.5, this is also the condition for
the forcing frequencies to be large enough that the excitation of inertial
modes is not expected to play an important role. A condition for this to apply
for prograde encounters can be approximately found by requiring that the
characteristic forcing frequency, $\Omega_{*}/\eta,$ exceed $2\Omega.$ Making
use of equation (8), we obtain the condition as
$P_{orb}<\frac{\Omega_{*}}{6.1\sqrt{\bar{\rho}}\Omega},$ (15)
Noting that for a typical rotation period of a T Tauri star of $6$ days and
solar parameters, equation (15) gives $P_{orb}<8.4$ days, the implication is
that the excitation of stellar inertial modes do not play a significant role
in the tidal capture of hot Jupiters into final prograde orbits with periods
of a few days.
This simplifies applications of the theory to particular systems since use of
the the perturbative approach requires only the calculation of the overlap
integrals, and the splitting coefficients for a given non rotating stellar
model. These are provided for all models considered in this paper. In addition
to the methods described above, we also applied the purely analytic approach
developed in Paper 1 to the solar model 1b, and showed that it gave results
differing from those obtained numerically by at most 40 per cent in the range
$2<\eta<30$. It is important to stress again that the energy and angular
momentum transfers are significantly larger for this model as compared to
models of more massive stars and a ’reference’ model of $n=3$ polytrope.
The stellar mean density plays an important role in applications to particular
astrophysical systems. This is because tides become relatively more efficient
for radially extended low-density objects. In particular, timescale $T_{ev}$
for the evolution of the semimajor axis, considered as a function of the
orbital period after the period of circularisation $P_{orb},$ becomes smaller
for more rarefied evolved massive models regardless of the fact that other
effects, such as possessing a smaller convective envelope, act in the
direction of making tidal interaction less efficient as discussed above. Thus,
$T_{ev}$ is less than $10^{9}$yrs for non-rotating models 1b and 1.5c when
$P_{orb}<3.3$days and $<4.3$days, respectively. For fast rotators this time
can be significantly reduced for retrograde encounters. Thus, in the case of a
1.5a model rotating with the angular frequency $\Omega=0.25$
$T_{ev}<10^{9}$yrs when $P_{orb}<12$days.
When the theory of dynamic tides is applied to particular astrophysical
processes, such as the tidal circularisation of exoplanet orbits starting with
a high eccentricity induced by gravitational scattering, or the process of
tidal capture of stars in stellar clusters, it is important to calculate the
overlap integrals and the coefficients $\beta_{r}$ for a grid of stellar
models of different ages. It is also important to understand the evolution of
the stellar structure, and rotation rate as a function of time. The outcome of
tidal evolution may differ significantly for stars with different masses and
different rotational history.
As discussed in Paper 1 the overlap integrals are also important for
discussing the tidal evolution of binaries with small orbital eccentricity. In
particular, for forcing frequencies large enough that inertial modes are not
expected to be excited, they fully determine the effect of tidal interactions
in the regime of ’moderately large viscosity’, see eg. Goodman $\&$ Dickson
1998 and Paper 1. The results of Zahn (1977) can only be recovered only when
the overlap integrals are $\propto\,\omega^{17/6}.$ This dependence
approximately holds for Sun-like stars in the limit of $\omega\rightarrow 0$.
It is determined by the functional form of the square of the Brunt-V$\ddot{\rm
a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency in the neighbourhood of the
transition from radiative to convective regions. In particular the
$\omega^{17/6}$ dependence of the overlap integrals requires that, in the
neighbourhood of the transition, the square of the Brunt-V$\ddot{\rm
a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency is a linear function of the
difference between a given radius and the radius of the transition. This
assumption may not be valid for a massive star, where the transition from the
radiative envelope to the convective core can be extremely sharp. In
particular, the Zahn (1977) theory does not apply to binaries with ultra-short
periods $P_{orb}\sim 10\Omega_{*}^{-1}$, where the overlap integrals decrease
much faster with $\omega,$ see the discussion of Figs. 7-10 in the text. A
theory appropriate for large orbital periods must consider the origin of, and
take into account, possible rapid variations of the the Brunt-V$\ddot{\rm
a}$is$\ddot{\rm a}$l$\ddot{\rm a}$ frequency in the neighbourhood of the
convective to radiative transition. This is left for future work.
## Acknowledgements
We are grateful to I. W. Roxburgh for providing stellar models and he and G.
I. Ogilvie for fruitful discussions. We also thank S. V. Vorontsov for useful
comments.
PBI and SVCh were supported in part by Federal programme ”Scientific
personnel” contract 8422, by RFBR grant 11-02-00244-a, by grant no. NSh
2915.2012.2 from the President of Russia and by programme 22 of the Presidium
of Russian Academy of Sciences.
Additionally, PBI was supported in part by the Dynasty Foundation and thanks
DAMTP, University of Cambridge for hospitality.
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|
arxiv-papers
| 2013-06-09T17:15:53 |
2024-09-04T02:49:46.259940
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S.V. Chernov, J. C. B. Papaloizou, P. B. Ivanov",
"submitter": "Sergey Chernov",
"url": "https://arxiv.org/abs/1306.2041"
}
|
1306.2239
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-087 LHCb-PAPER-2013-012 10 June 2013
First observation of the decay $B^{0}_{s}\rightarrow\phi\kern
4.14793pt\overline{\kern-4.14793ptK}{}^{*0}$
The LHCb collaboration†††Authors are listed on the following pages.
The first observation of the decay $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ is reported. The analysis is
based on a data sample corresponding to an integrated luminosity of 1.0 fb-1
of $pp$ collisions at $\sqrt{s}=7\,$$\mathrm{\,Te\kern-1.00006ptV}$, collected
with the LHCb detector. A yield of ${30\pm 6}$
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ decays is found in the mass
windows
${1012.5<M(K^{+}K^{-})<1026.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and
${746<M(K^{-}\pi^{+})<1046{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$. The
signal yield is found to be dominated by $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, and the corresponding
branching fraction is measured to be ${{\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\left(1.10\pm
0.24\,\mathrm{(stat)}\pm 0.14\,\mathrm{(syst)}\pm
0.08\left(f_{d}/f_{s}\right)\right)\times 10^{-6}}$, where the uncertainties
are statistical, systematic and from the ratio of fragmentation fractions
$f_{d}/f_{s}$ which accounts for the different production rate of $B^{0}$ and
$B^{0}_{s}$ mesons. The significance of $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal is 6.1 standard
deviations. The fraction of longitudinal polarization in
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decays is found to be ${f_{0}=0.51\pm 0.15\,\mathrm{(stat)}\pm
0.07\,\mathrm{(syst)}}$.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso57,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso58, V.
Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W.
Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,p, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz25, K.
Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V.
Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, A.
Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De
Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di
Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil
Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A.
Dziurda25, A. Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S.
Eidelman33, D. van Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L.
Eklund50,37, I. El Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C.
Färber11, G. Fardell49, C. Farinelli40, S. Farry12, V. Fave38, D. Ferguson49,
V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32,
M. Fiore16, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O.
Francisco2, M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E.
Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini57,
Y. Gao3, J. Garofoli57, P. Garosi53, J. Garra Tico46, L. Garrido35, C.
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Greening54, S. Gregson46, O. Grünberg59, B. Gui57, E. Gushchin32, Yu.
Guz34,37, T. Gys37, C. Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C.
Haines46, S. Hall52, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T.
Harnew45, J. Harrison53, T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A. Hicheur1, E.
Hicks51, D. Hill54, M. Hoballah5, M. Holtrop40, C. Hombach53, P. Hopchev4, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, C.
Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, M. Karacson37, T.M. Karbach37,
I.R. Kenyon44, U. Kerzel37, T. Ketel41, A. Keune38, B. Khanji20, O.
Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A.
Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P.
Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33, T.
Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A. Lai15,
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Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez
Asamar35, N. Lopez-March38, H. Lu3, D. Lucchesi21,p, J. Luisier38, H. Luo49,
F. Machefert7, I.V. Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54,
G. Manca15,d, G. Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G.
Martellotti24, A. Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez
Santos41, D. Martins Tostes2, A. Massafferri1, R. Matev37, Z. Mathe37, C.
Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R.
McNulty12, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A.
Milanes8, M.-N. Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53, P.
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Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
A. Zhokhov30, L. Zhong3, A. Zvyagin37.
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 Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, 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
## 1 Introduction
The measurement of $C\\!P$ asymmetries in flavour-changing neutral-current
processes provides a crucial test of the Standard Model (SM). In particular,
loop-mediated (penguin) decays of $B$ mesons are sensitive probes for physics
beyond the SM. Transitions between the quarks of the third and second
generation ($b\rightarrow s$) or between the quarks of the third and first
generation ($b\rightarrow d$) are complementary since SM $C\\!P$ violation is
tiny in $b\rightarrow s$ transitions and an observation of $C\\!P$ violation
would indicate physics beyond the SM. For $b\rightarrow d$ transitions the SM
branching fraction is an order of magnitude smaller than $b\rightarrow s$ due
to the relative suppression of $|V_{td}|^{2}/|V_{ts}|^{2}$. It is particularly
useful to have experimental information on pairs of channels related by
$d\leftrightarrow s$ exchange symmetry to test that the QCD contribution to
the decay is independent of the initial $B^{0}$ or $B^{0}_{s}$ meson.
The BaBar and Belle experiments have performed measurements of $b\rightarrow
sq\overline{q}$ processes, such as $B^{0}\rightarrow\phi
K^{0}_{\rm\scriptscriptstyle S}$,
$B^{0}\rightarrow\eta^{\prime}K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{0}\rightarrow f_{0}K^{0}_{\rm\scriptscriptstyle S}$ [1, 2, 3], and of
$b\rightarrow dq\overline{q}$ penguin diagrams, such as $B^{0}\rightarrow
K^{0}_{\rm\scriptscriptstyle S}K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{+}\rightarrow K^{+}K^{0}_{\rm\scriptscriptstyle S}$ [4, 5]. These modes
contain pseudo-scalar or scalar mesons in their final state whereas
$B^{0}_{(s)}\rightarrow VV^{\prime}$ decays, where $V$ and $V^{\prime}$ are
light vector mesons, provide a valuable additional source of information
because the angular distributions give insight into the physics of hadronic
$B$ meson decays and the interplay between the strong and weak interactions
they involve. From the V$-$A structure of the weak interaction and helicity
conservation in the strong interaction, the final state of these decays is
expected to be highly longitudinally polarized. This applies to both tree and
penguin decays. The BaBar and Belle experiments have confirmed that
longitudinal polarization dominates in $b\rightarrow u$ tree processes such as
$B^{0}\rightarrow\rho^{+}\rho^{-}$ [6, 7], $B^{+}\rightarrow\rho^{0}\rho^{+}$
[8, 9] and $B^{+}\rightarrow\omega\rho^{+}$ [10]. However, measurements of the
polarization in decays with both tree and penguin contributions, such as
$B^{0}\rightarrow\rho^{0}K^{*0}$ and $B^{0}\rightarrow\rho^{-}K^{*+}$ [11] and
in $b\rightarrow s$ penguin decays, $B^{0}\rightarrow\phi K^{*0}$ [12, 13],
$B^{0}_{s}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
[14] and $B^{0}_{s}\rightarrow\phi\phi$ [15, 16, *Aaij:2013qha], indicate a
low value of the longitudinal polarization fraction comparable with, or even
smaller than, the transverse fraction.
The $B^{0}_{(s)}\rightarrow VV^{\prime}$ decays can be described by models
based on perturbative QCD, or QCD factorization and SU(3) flavour symmetries.
Whilst some authors predict a longitudinal polarization fraction
$f_{0}\mathord{\sim}0.9$ for tree-dominated and $\mathord{\sim}0.75$ for
penguin decays [18, *Suzuki:2002yk, 20], other studies have proposed different
mechanisms such as penguin annihilation [21, 22] and QCD rescattering [23] to
accommodate smaller longitudinal polarization fractions $\mathord{\sim}0.5$,
although the predictions have large uncertainties. A review on the topic of
polarization in $B$ decays can be found in Ref. [24].
There are only two other $B^{0}_{(s)}\rightarrow VV^{\prime}$ penguin modes
that correspond to $b\rightarrow d$ loops. The first is the $B^{0}\rightarrow
K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. The BaBar
collaboration reported the discovery of this channel with $6\,\sigma$
significance and a measurement of its branching fraction ${\cal
B}(B^{0}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=(1.28\,{}^{+0.35}_{-0.30}\pm
0.11)\times 10^{-6}$ [25]. This is in tension with the results of the Belle
collaboration that published an upper limit of ${\cal B}(B^{0}\rightarrow
K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0})<0.8\times 10^{-6}$ at
the $90\%$ confidence level [26]. The BaBar publication also reported a
measurement of the longitudinal polarization ${f_{0}=0.80^{+0.12}_{-0.13}}$
[25], which is large compared to those from $B^{0}\rightarrow\phi K^{*0}$
($f_{0}=0.494\pm 0.036$ [13]), $B^{0}_{s}\rightarrow\phi\phi$ ($f_{0}=0.365\pm
0.025$ [16]) and $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ ($f_{0}=0.31\pm 0.13$ [14]).
The mode $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ is the other $b\rightarrow d$
penguin decay into vector mesons that has not previously been observed. This
decay is closely linked to $B^{0}\rightarrow\phi K^{*0}$, differing in the
spectator quark and the final quark in the loop, as shown in Fig. 1.111Both
the decays $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\rightarrow\phi K^{*0}$
could also have contributions from QCD singlet-penguin amplitudes [21]. From
the aforementioned relation between $b\rightarrow s$ and $b\rightarrow d$
transitions, their relative branching fractions should scale as
$|V_{td}|^{2}/|V_{ts}|^{2}$ and their polarization fractions are expected to
be very similar. Moreover, since both decays share the same final state,
except for charge conjugation, $B^{0}\rightarrow\phi K^{*0}$ is the ideal
normalization channel for the determination of the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
branching fraction. The $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay is also related to
$B^{0}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$,
since their loop diagrams only differ in the spectator quark (${s\text{
instead of }d}$), although it has been suggested that S-wave interference
effects might break the SU(3) symmetry relating two channels [27]. Finally, it
is also interesting to explore the relation of the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decay with the $B^{0}\rightarrow\rho^{0}K^{*0}$ mode since the penguin loop
diagrams of these modes are related by the $d\leftrightarrow s$ exchange. The
$B^{0}\rightarrow\rho^{0}K^{*0}$ decay also has a $b\rightarrow u$ tree
diagram, but it is expected that the penguin contribution is dominant, since
the branching fraction is comparable to that of the pure penguin
$B^{0}\rightarrow\phi K^{*0}$ decay.
The most stringent previous experimental limit on the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
branching fraction is ${{\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})<1.0\times 10^{-3}}$ at the $90\%$
confidence level [24], whereas calculations based on the QCD factorization
framework predict a value of ${(0.4\,{}^{+0.5}_{-0.3})\times 10^{-6}}$ [21]
while in perturbative QCD a value of ${(0.65\,{}^{+0.33}_{-0.23})\times
10^{-6}}$ [28] is obtained. The precise determination of the branching
fraction tests these models and provides a probe for physics beyond the SM.
The study of the angular distributions in the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ channel provides a measurement of
its polarization. In Ref. [28], a prediction of
$f_{0}=0.712\,{}^{+0.042}_{-0.048}$ is made for the longitudinal polarization
fraction, using the perturbative QCD approach, that can be compared to the
experimental result.
In this paper the first observation of the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay, with $\phi\rightarrow
K^{+}K^{-}$ and $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow
K^{-}\pi^{+}$, is reported and the determination of its branching fraction and
polarizations are presented. The study is based on data collected by the LHCb
experiment at CERN from the $\sqrt{s}=7\,$$\mathrm{\,Te\kern-1.00006ptV}$
proton-proton collisions of LHC beams. The dataset corresponds to an
integrated luminosity of 1.0 fb-1.
Figure 1: Feynman diagrams for the $B^{0}_{s}\rightarrow\phi\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ and the $B^{0}\rightarrow\phi
K^{*0}$ decays.
## 2 Detector and software
The LHCb detector [29] 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 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
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
(RICH) detectors [30]. 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 [31].
The trigger [32] 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. The software trigger used in this analysis requires
a two-, three- or four-track secondary vertex with a high sum of the $p_{\rm
T}$ of the tracks and significant displacement from the primary $pp$
interaction vertices (PVs). At least one track should have $\mbox{$p_{\rm
T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and impact parameter $\chi^{2}$
($\chi^{2}_{{\rm IP}}$) with respect to all primary interactions greater than
16. The $\chi^{2}_{{\rm IP}}$ is defined as the difference between the
$\chi^{2}$ of a PV reconstructed with and without the considered track. A
multivariate algorithm [33] is used for the identification of secondary
vertices consistent with the decay of a $b$ hadron.
In the simulation, $pp$ collisions are generated using Pythia 6.4 [34] with a
specific LHCb configuration [35]. Decays of hadronic particles are described
by EvtGen [36], in which final state radiation is generated using Photos [37].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [38, *Agostinelli:2002hh] as
described in Ref. [40].
## 3 Signal selection
Signal $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidates are formed from
$\phi\rightarrow K^{+}K^{-}$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow K^{-}\pi^{+}$
decays.222Inclusion of charge conjugated processes is implied in this work,
unless otherwise stated. The pairs of charged particles in the
$\phi\rightarrow K^{+}K^{-}$ and the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow K^{-}\pi^{+}$
candidates must combine to give invariant masses
${1012.5<M(K^{+}K^{-})<1026.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and
${746<M(K^{-}\pi^{+})<1046{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$,
consistent with the known $\phi$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ masses [24]. Each of the four
tracks is required to have $\mbox{$p_{\rm
T}$}>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{{\rm IP}}$ $>9$.
Kaons and pions are distinguished by use of a log-likelihood algorithm that
combines information from the RICH detectors and other properties of the event
[30]. The final state particles are identified by requiring that the
difference in log-likelihoods of the kaon and pion mass hypotheses is
$\mathrm{DLL}_{K\pi}$ $>2$ for each kaon candidate and $<0$ for the pion
candidate. In addition, the difference in log-likelihoods of the proton and
kaon hypotheses, $\mathrm{DLL}_{pK}$, is required to be $<0$ for the kaon from
the $\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay. This suppresses
background from $\mathchar 28931\relax^{0}_{b}$ decays. This requirement is
not necessary for the kaons from the $\phi$ candidate owing to the narrow
$K^{+}K^{-}$ invariant mass window.
The $K^{-}\pi^{+}$ pair that forms the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidate is required to
originate from a common vertex with a $\chi^{2}$ per number of degrees of
freedom ($\chi^{2}/{\rm ndf}$) $<9$, and to have a positive cosine of the
angle between its momentum and the reconstructed $B^{0}_{(s)}$ candidate
flight direction, calculated with the $B^{0}_{(s)}$ decay vertex and the best
matching primary vertex. The $K^{-}\pi^{+}$ combination is also required to
have $\mbox{$p_{\rm T}$}>900{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The same
conditions are imposed on the $\phi$ candidate.
The $B^{0}_{(s)}$ candidates are also required to fulfil some minimal
selection criteria: the $\phi$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ candidates must form a vertex
with $\chi^{2}/{\rm ndf}<15$; the distance of closest approach between their
trajectories must be less than $0.3\,\rm\,mm$; and they must combine to give
an invariant mass within
${4866<M(K^{+}K^{-}K^{-}\pi^{+})<5866{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$.
In addition, a geometrical-likelihood based selection (GL) [41, 42] is
implemented using as input variables properties of the $B^{0}_{(s)}$ meson
candidate. These are
* •
the $B^{0}_{(s)}$ candidate impact parameter (${\rm IP}$) with respect to the
closest primary vertex;
* •
the decay time of the $B^{0}_{(s)}$ candidate;
* •
the $p_{\rm T}$ of the $B^{0}_{(s)}$ candidate;
* •
the minimum $\chi^{2}_{{\rm IP}}$ of the four tracks with respect to all
primary vertices in the event; and
* •
the distance of closest approach between the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $\phi$ candidates’
trajectories reconstructed from their respective daughter tracks.
The GL is trained to optimize its discrimination power using representative
signal and background samples. For the signal a set of
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
simulated events is used. For the background a sample of events where, in
addition to the signal selections, other than those on the masses,
requirements of
$999.5<M(K^{+}K^{-})<1012.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ or
${1026.5<M(K^{+}K^{-})<1039.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ for
the $\phi$ candidate and
${M(K^{+}K^{-}K^{-}\pi^{+})>5413{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ for
the four-body mass are applied. The selection of only the high-mass
$B^{0}_{(s)}$ sideband is motivated by the nature of the background in that
region, which is purely combinatorial, whereas the low-mass sideband contains
partially reconstructed $B$ meson decays that have topological similarities to
the signal.
## 4 Suppression of background from other $b$-hadron decays
A small background from $B^{0}_{s}\rightarrow\phi\phi$ decays, where one of
the kaons from the $\phi$ is misidentified as a pion, is found to contaminate
the signal. Candidate $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays are therefore required to
be outside of the window defined by
${1012.5<M(K^{+}K^{-})<1026.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$ and
${5324<M(K^{+}K^{-}K^{+}K^{-})<5424{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$
in the $K^{+}K^{-}$ and $K^{+}K^{-}K^{+}K^{-}$ invariant masses when the mass
hypothesis for the sole pion of the decay is switched into a kaon. In
simulated events this selection removes $0.12\%$ of the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
signal decays and does not affect the $B^{0}\rightarrow\phi K^{*0}$ decay
mode. Other possible reflections, such as $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, are found to be
negligible.
In order to remove background from $B^{0}_{s}\rightarrow
D_{s}^{\mp}(\phi\pi^{\mp})K^{\pm}$ decays when the $\pi^{\mp}$ and the
$K^{\pm}$ mesons form a
$\accentset{\scalebox{0.4}{(}\raisebox{-1.7pt}{\bf{--}}\scalebox{0.4}{)}}{K}^{*0}$
candidate, events with the invariant mass of the $K^{+}K^{-}\pi^{\mp}$ system
within
${1953.5<M(K^{+}K^{-}\pi^{\mp})<1983.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}}$,
consistent with the known $D^{+}_{s}$ mass [24], are excluded.
Background from $b$-hadron decays containing a misidentified proton has also
been considered. For candidate $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, the kaon with the largest
$\mathrm{DLL}_{pK}$ is assigned the proton mass and the four-body invariant
mass recomputed. The largest potential background contribution arises from
$\kern 1.00006pt\overline{\kern-1.00006pt\mathchar
28931\relax}^{0}_{b}\rightarrow K^{+}K^{-}\overline{}p\pi^{+}$ where the
antiproton is misidentified as the kaon originating from the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ meson, and $\mathchar
28931\relax^{0}_{b}\rightarrow K^{+}K^{-}K^{-}p$, where the proton is
misidentified as the pion originating from the $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ meson. Simulation shows that
these decays produce wide four-body mass distributions which peak around
$5450{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$5500{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. This background
contribution is considered in the fit model discussed below. Other
$B^{0}_{(s)}$ decay modes containing a $\mathchar 28931\relax\rightarrow
p\pi^{-}$ decay or background from $\mathchar 28931\relax^{+}_{c}\rightarrow
pK^{-}\pi^{+}$ decays are found to be negligible.
## 5 Fit to the four-body mass spectrum
The sample of $1277$ candidates, selected as described in Sections 3 and 4,
contains many $B^{0}\rightarrow\phi K^{*0}$ decays whereas only a small
contribution from $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays is anticipated. Both
signals are parametrized with identical shapes, differing only in the mass
shift of $87.13{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ between the $B^{0}$
and $B^{0}_{s}$ mesons [24] which is fixed in the fit. The signal shapes are
described by the sum of Crystal Ball (CB) [43] and Gaussian functions that
share a common mean. The CB function, which contains most of the signal, is a
combination of a Gaussian function with a power law tail, accounting for the
intrinsic detector resolution and the radiative tail toward low masses,
respectively. The Gaussian shape describes events reconstructed with worse
mass resolution, which produce a contamination of $B^{0}\rightarrow\phi
K^{*0}$ decays in the region of the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ signal peak. The dependence
between the Gaussian and CB resolutions, $\sigma_{{\rm G}}$ and $\sigma_{{\rm
CB}}$, respectively, is found to be
$\sigma_{{\rm G}}=\sqrt{\sigma_{{\rm
CB}}^{2}+(24.74{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}})^{2}},$ (1)
from a data sample of $25\times 10^{3}$
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
decays. This channel is topologically very similar to the signal and is almost
background free. The fit to this sample also provides the power law exponent
of the CB function tail, which is subsequently fixed in the
$B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})$ and
$B^{0}\rightarrow(K^{+}K^{-})(K^{+}\pi^{-})$ mass models. The parameter that
governs the transition from the Gaussian shape to the power law function in
the CB function is unrestrained in the fit. The other unrestrained fit
parameters include: the central $B$ meson mass, the width of the CB function,
the fractional yield contained in the Gaussian function and the total signal
yield.
In addition to the $B^{0}$ and $B^{0}_{s}$ signal shapes, three more
components are included. The first accounts for partially reconstructed $B$
meson decays into $\phi$ and $K$ or $K^{*}$ excited states where a pion has
been lost. This is described by a convolution of the ARGUS shape [44] with a
Gaussian distribution. The second contribution is due to $\mathchar
28931\relax^{0}_{b}\rightarrow K^{+}K^{-}K^{-}p$ and $\kern
1.00006pt\overline{\kern-1.00006pt\mathchar 28931\relax}^{0}_{b}\rightarrow
K^{+}K^{-}\overline{}p\pi^{+}$ decays and is modelled with a histogram
obtained from simplified simulations. The third contribution is an exponential
function to account for combinatorial background.
The data passing the selection criteria are fitted using an extended unbinned
maximum likelihood fit. The invariant mass distribution of the candidates,
together with the fit contribution, is shown in Fig. 2. The yields of
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ and
${B^{0}\rightarrow(K^{+}K^{-})(K^{+}\pi^{-})}$ decays are $30\pm 6$ and
$1000\pm 32$, respectively. The fit model is validated with $10,000$ pseudo-
experiments, generated with simplified simulations, which show that the signal
yields are unbiased. Table 1 summarizes the signal and background
contributions resulting from the fit. A likelihood ratio test is employed to
assess the statistical significance of the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ signal yield. This is
performed using $\sqrt{2{\rm ln}(\mathcal{L}_{\rm s+b}/\mathcal{L}_{\rm b})}$,
where $\mathcal{L}_{\rm s+b}$ and $\mathcal{L}_{\rm b}$ are the maximum values
of the likelihoods for the signal-plus-background and background-only
hypotheses, respectively.333The applicability of this method has been verified
from the parabolic behaviour of the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ signal yield profile of
$-2\ln\mathcal{L}_{\rm s+b}$ about its minimum. This calculation results in
$6.3\,\sigma$ significance for the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ signal. The fit gives
$\sigma_{\rm CB}=15.0\pm 1.1{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for the
invariant mass resolution. Integration in a $\pm
30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ mass window yields $26.4\pm 5.7$
signal candidates and $8.2\pm 1.3$ background events, composed of $5.4\pm 0.2$
from ${B^{0}\rightarrow(K^{+}K^{-})(K^{+}\pi^{-})}$, $2.1\pm 1.3$ from
$\mathchar 28931\relax^{0}_{b}$ and $0.7\pm 0.4$ from combinatorial
contributions.
In order to explore systematic effects in the signal yield originating in the
fit model two effects were considered. First, the amount of
${B^{0}\rightarrow(K^{+}K^{-})(K^{+}\pi^{-})}$ events under the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ signal is governed by the
$24.74{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ factor in Eq. 1. Similarly,
the contamination of misidentified $\mathchar 28931\relax^{0}_{b}$ decays
under the signal is controlled by a tail that is parametrized. An extended
likelihood is built by multiplying the original likelihood function by
Gaussian distributions of these two nuissance parameters with standard
deviations of $20\%$ of their nominal values at which they are centered. The
corresponding systematic uncertainty in the signal yield is obtained by
performing a fit that maximizes this modified likelihood. The systematic
contribution is calculated subtracting the statistical uncertainty in
quadrature and found to be $\pm 1.2$ events. Including this uncertainty
results in a significance of $6.2\sigma$. Effects of other systematic
uncertainties, discussed in Sect. 9, have negiglible impact in the signal
significance.
Figure 2: Four-body $K^{+}K^{-}K^{-}\pi^{+}$ invariant mass distribution. The points show the data, the blue solid line shows the overall fit, the solid dark red shaded region is the $B^{0}_{s}\rightarrow\phi\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ signal, the light blue shaded region corresponds to the $B^{0}\rightarrow\phi K^{*0}$ signal, the grey dotted line is the combinatorial background and the green dashed line and magenta dashed-dotted lines are the partially reconstructed and misidentified $\mathchar 28931\relax^{0}_{b}$ backgrounds. Table 1: Results of the fit to the sample of selected candidates. Contribution | Yield
---|---
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ | $\>\>30\pm 6$
$B^{0}\rightarrow\phi K^{*0}$ | $1000\pm 32$
Partially reconstructed background | $\>\>218\pm 15$
$\mathchar 28931\relax^{0}_{b}$ background | $\>\>13\pm 8$
Combinatorial background | $\>\>10\pm 6$
## 6 Determination of the S-wave contribution
The $B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})$ signal is expected to be
mainly due to $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays, although there are
possible non-resonant contributions and $K^{+}K^{-}$ and $K^{-}\pi^{+}$ pairs
from other resonances. To estimate the S-wave contributions, it is assumed
that the effect is the same for $B^{0}\rightarrow\phi K^{*0}$ and
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decays, therefore allowing the larger sample of $B^{0}\rightarrow\phi K^{*0}$
decays to be used. The effect of this assumption is considered as a source of
systematic uncertainty in Sect. 8.
The $K^{+}K^{-}$ invariant mass distribution for $\phi$ candidates within a
$\pm 30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window of the known $B^{0}$
mass is described by a relativistic spin-1 Breit-Wigner distribution convolved
with a Gaussian shape to account for the effect of resolution. A linear term
is added to describe the S-wave contribution. The purity resulting from this
fit is $0.95\pm 0.01$ in a $\pm 7{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
window around the known $\phi$ mass.
The $K^{+}\pi^{-}$ pairs are parametrized by the incoherent sum of a
relativistic spin-1 Breit-Wigner amplitude and a shape that describes non-
resonant and $K^{*0}(1430)$ S-wave contributions introduced by the LASS
experiment [13, 45]. The fraction of events from $K^{*0}$ decays within a $\pm
150{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window around the $K^{*0}$ mass
results in a purity of $0.89\pm 0.02$. When combining the $K^{+}K^{-}$ and
$K^{+}\pi^{-}$ contributions, the total $\phi K^{*0}$ purity is found to be
$0.84\pm 0.02$. This purity can be translated into a p-value, quantifying the
probability that the entire ${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$
signal is due to decays other than $\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$. After combining with the
${B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})}$ significance the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ is
observed with $6.1\,\sigma$ significance.
Figure 3: Invariant mass distributions for (left) $K^{+}K^{-}$ and (right)
$K^{\mp}\pi^{\pm}$ pairs in a $\pm 30{\mathrm{\,Me\kern-0.90005ptV\\!/}c^{2}}$
window around the (top) $B^{0}_{s}$ and (bottom) $B^{0}$ mass. The solid blue
line is the overall fit, the green dashed line corresponds to $B^{0}$ cross-
feed into the $B^{0}_{s}$ mass window, the red dotted line is the S-wave
contribution and the light blue is the combinatorial background.
## 7 Determination of the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ branching fraction
The branching fraction is calculated with the $B^{0}\rightarrow\phi K^{*0}$
channel as normalization. Both decays pass the same selection and share almost
identical topologies. However, since the two decay channels can have different
polarizations, their angular distributions may differ which would cause a
difference in their detection efficiencies. A factor
$\lambda_{f_{0}}=\frac{\epsilon^{B^{0}\rightarrow\phi
K^{*0}}}{\epsilon^{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}=\frac{1-0.29f_{0}^{B^{0}\rightarrow\phi
K^{*0}}}{1-0.29f_{0}^{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}$
is calculated, where $\epsilon^{B^{0}\rightarrow\phi K^{*0}}$ and
$\epsilon^{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ are the efficiencies for the
$B^{0}\rightarrow\phi K^{*0}$ and $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decays reconstruction,
$f_{0}^{B^{0}\rightarrow\phi K^{*0}}$ and
$f_{0}^{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ their longitudinal polarization
fractions, determined in Sect. 9 for the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mode, and the factor 0.29 is
obtained from simulation.
The value of ${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})$ is computed from
${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\lambda_{f_{0}}\times\frac{f_{d}}{f_{s}}\times{\cal
B}(B^{0}\rightarrow\phi K^{*0})\times\frac{N_{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}}{N_{B^{0}\rightarrow\phi
K^{*0}}},$ (2)
where $N_{B^{0}_{s}\rightarrow\phi\kern
1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ and $N_{B^{0}\rightarrow\phi
K^{*0}}$ are the numbers of $B^{0}_{s}$ and $B^{0}$ decays, respectively, and
$f_{d}/f_{s}=3.75\pm 0.29$ [46] is the ratio of hadronization factors needed
to account for the different production rates of $B^{0}$ and $B^{0}_{s}$
mesons. With the values given in Table 2, the result,
${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=(1.10\pm 0.24)\times 10^{-6},$
is obtained, where only the statistical uncertainty is shown.
Table 2: Input values for the branching fraction computation. Parameter | Value
---|---
$\lambda_{f_{0}}$ | $1.01\pm 0.06$
$N_{B^{0}\rightarrow\phi K^{*0}}$ | $1000\pm 32\>\>\>\>$
$N_{B^{0}_{s}\rightarrow\phi\kern 1.39998pt\overline{\kern-1.39998ptK}{}^{*0}}$ | $30\pm 6\>\>$
${\cal B}(B^{0}\rightarrow\phi K^{*0})$ | $(9.8\pm 0.6)\times 10^{-6}$ [24]
As a cross-check, a different decay mode,
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$, with
$J/\psi\rightarrow\mu^{+}\mu^{-}$, has been used as a normalization channel.
Special requirements were imposed to harmonize the selection of this reference
with that for the signal. The obtained result is fully compatible with the
$B^{0}\rightarrow\phi K^{*0}$ based value.
## 8 Systematic uncertainties on the branching fraction
Four main sources of systematic effects in the determination of the branching
fraction are identified: the fit model, the dependence of the acceptance on
the longitudinal polarization, the purity of the signal and the uncertainty in
the relative efficiency of $B^{0}_{s}$ and $B^{0}$ detection.
Alternatives to the fit model discussed in Sect. 5 give an uncertainty of $\pm
1.2$ in the signal yield. This results in a relative systematic uncertainty of
$\pm 0.04$ on the branching fraction.
The systematic uncertainty in the acceptance correction factor
$\lambda_{f_{0}}$ originates from the uncertainties of the longitudinal
polarization fractions, $f_{0}$, in the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\rightarrow\phi K^{*0}$
channels and is found to be $\pm 0.06$.
As described in Sect. 6 an S-wave contribution of $0.16\pm 0.02$ was found in
the $K^{+}K^{-}$ and $K^{-}\pi^{+}$ mass windows of the $B^{0}\rightarrow\phi
K^{*0}$ candidates. The uncertainty caused by the assumption that this
fraction is the same in $B^{0}$ and $B^{0}_{s}$ decays is estimated to be
$50\%$ of the S-wave contribution. This results in a $\pm 0.08$ contribution
to the systematic uncertainty. This uncertainty also accounts for uncanceled
interference terms between the $K^{*0}$, the $\phi$ and their corresponding
S-waves. These contributions are linear in the sine or cosine of polarization
angles [13] and cancel after integration. The dependence of the acceptance on
the angles violates this cancellation contributing $\pm 0.04$ to the total
$\pm 0.08$ S-wave uncertainty.
The $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and $B^{0}\rightarrow\phi K^{*0}$
final states are very similar and a detector acceptance efficiency ratio $\sim
1$ is expected. However, small effects, such as the mass shift
$M(B^{0}_{s})-M(B^{0})$, translate into slightly different $p_{\rm T}$
distributions for the daughter particles. This results in an efficiency ratio
of $1.005$, as determined from simulation. The deviation of $\pm 0.005$ from
unity is taken as a systematic uncertainty that is propagated to the branching
fraction.
Finally, the uncertainty in the knowledge of the $B^{0}\rightarrow\phi K^{*0}$
decay branching fraction of $\pm 0.6\times 10^{-6}$ is also accounted for and
results in a relative uncertainty of 0.06 in the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
decay branching fraction.
A summary of the systematic uncertainties is shown in Table 3. The final
result for the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay branching fraction is
${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\left(1.10\pm
0.24\,\mathrm{(stat)}\pm 0.14\,\mathrm{(syst)}\pm
0.08\left(\frac{f_{d}}{f_{s}}\right)\right)\times 10^{-6},$
which corresponds to a ratio with the $B^{0}\rightarrow\phi K^{*0}$ decay
branching fraction of:
$\frac{{\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})}{{\cal B}(B^{0}\rightarrow\phi
K^{*0})}=0.113\pm 0.024\,\mathrm{(stat)}\pm 0.013\,\mathrm{(syst)}\pm
0.009\left(\frac{f_{d}}{f_{s}}\right).$
Table 3: Sources of systematic uncertainty in the branching fraction measurement. The total uncertainty is the addition in quadrature of the individual sources. Source | Relative uncertainty in $\cal B$
---|---
Fit model | $0.04\phantom{0}$
$f_{0}$ | $0.06\phantom{0}$
Purity | $0.08\phantom{0}$
Acceptance | $0.005$
$\cal B$($B^{0}\rightarrow\phi K^{*0}$) | $0.06\phantom{0}$
Total | $0.12\phantom{0}$
## 9 Polarization analysis
The $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})$
decay proceeds via two intermediate spin-1 particles. The angular distribution
of the decay is described by three transversity amplitudes $A_{0}$,
$A_{\parallel}$ and $A_{\perp}$ [47]. These can be obtained from the
distribution of the decay products in three angles $\theta_{1}$, $\theta_{2}$
and $\varphi$, defined in the helicity frame. The convention for the angles is
shown in Fig. 4. A flavour-averaged and time-integrated polarization analysis
is performed assuming that the $C\\!P$-violating phase is zero and that an
equal amount of $B^{0}_{s}$ and $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ mesons are produced. Under
these assumptions, the decay rate dependence on the polarization angles can be
written as
$\displaystyle\frac{{\rm d}^{3}\Gamma}{{\rm d}{\rm cos}\theta_{1}\,{\rm d}{\rm
cos}\theta_{2}\,{\rm d}\varphi}$ $\displaystyle\propto$
$\displaystyle|A_{0}|^{2}\cos^{2}\theta_{1}\cos^{2}\theta_{2}+|A_{\parallel}|^{2}\frac{1}{2}\sin^{2}\theta_{1}\sin^{2}\theta_{2}\cos^{2}\varphi$
$\displaystyle+$
$\displaystyle|A_{\perp}|^{2}\frac{1}{2}\sin^{2}\theta_{1}\sin^{2}\theta_{2}\sin^{2}\varphi+|A_{0}||A_{\parallel}|\cos\delta_{\parallel}\frac{1}{2\sqrt{2}}\sin
2\theta_{1}\sin 2\theta_{2}\cos\varphi.$
Additional terms accounting for the S-wave and interference contributions, as
in Ref. [13], are also considered. These terms are set to the values obtained
for the $B^{0}\rightarrow\phi K^{*0}$ sample.
The polarization fractions are defined from the amplitudes as:
$f_{j}=|A_{j}|^{2}/(|A_{0}|^{2}+|A_{\parallel}|^{2}+|A_{\perp}|^{2})$ (with
$j=0,\parallel,\perp$). In addition to the polarization fractions the cosine
of the phase difference between $A_{0}$ and $A_{\parallel}$,
$\cos\delta_{\parallel}$, is accessible in this study.
Figure 4: Definition of the angles in $B^{0}_{s}\rightarrow\phi\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ decays where $\theta_{1}$
($\theta_{2}$) is the $K^{+}$ ($K^{-}$) emission angle with respect to the
direction opposite to the $B^{0}_{s}$ meson in the $\phi$ ($\kern
1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$) rest frame and $\varphi$ is the
angle between the $\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$ and
$\phi$ decay planes in the $B^{0}_{s}$ rest frame.
The determination of the angular amplitudes depends on the spectrometer
acceptance as a function of the polarization angles $\theta_{1}$ and
$\theta_{2}$. The acceptance was found not to depend on $\varphi$. A
parametrization of the acceptance as a function of $\theta_{1}$ and
$\theta_{2}$ is calculated using simulated data and is used to correct the
differential decay rate by scaling Eq. 9. Additionally, a small correction for
discrepancies in the $p_{\rm T}$ spectrum and the trigger selection of the $B$
mesons between simulation and data is introduced.
The data in a $\pm 30{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ window around
the $B^{0}_{s}$ mass are fitted to the final angular distribution. The fit
accounts for two additional ingredients: the tail of the $B^{0}\rightarrow\phi
K^{*0}$ decays, that are polarized with a longitudinal polarization fraction
of $f_{0}=0.494$ [13], and the combinatorial background, parametrized from the
distributions of events in the high-mass $B$ sideband
$5450<M(K^{+}K^{-}K^{-}\pi^{+})<5840{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$
after relaxing the selection requirements. The latter accounts for both the
combinatorial and misidentified $\mathchar 28931\relax^{0}_{b}$ backgrounds.
The systematic uncertainties in the determination of the angular parameters
are calculated modifying the analysis and computing the difference with the
nominal result. Three elements are considered.
* •
The uncertainty in the S-wave fraction. This is computed modifying the S-wave
contribution by $50\%$ of its value. This covers within $2\,\sigma$ an S-wave
fraction from 0 to $30\%$, consistent with that typically found in decays of
$B$ mesons to final states containing a $K^{*0}$ meson.
* •
The spectrometer acceptance. This contribution is calculated comparing the
results considering or neglecting the above-mentioned $p_{\rm T}$ and trigger
corrections to the acceptance.
* •
The combinatorial background. The background model derived from the $B$ mass
sideband is replaced by a uniform angular distribution.
Figure 5: Result of the fit to the angular distribution of the
$B^{0}_{s}\rightarrow\phi\kern 1.79997pt\overline{\kern-1.79997ptK}{}^{*0}$
candidates in (left) $\cos\theta_{1}$ and (right) $\cos\theta_{2}$. The red
dotted line corresponds to the combinatorial background under the $B^{0}_{s}$
signal, the green dashed line is the $B^{0}\rightarrow\phi K^{*0}$ signal in
the $B^{0}_{s}$ region and the grey dotted-dashed line corresponds to the sum
of the S-wave and the interference terms.
The different contributions to the systematic uncertainty are given in Table 4
and the one-dimensional projections of the angular distributions are shown
Fig. 5. Other possible systematic sources, such as the uncertainty in the
polarization parameters of the $B^{0}\rightarrow\phi K^{*0}$, are found to be
negligible.
Considering all the above, the values obtained are
$\displaystyle f_{0}$ $\displaystyle=\phantom{-}0.51\pm
0.15\,\mathrm{(stat)}\pm 0.07\,\mathrm{(syst)},$ $\displaystyle f_{\parallel}$
$\displaystyle=\phantom{-}0.21\pm 0.11\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)},$ $\displaystyle\cos\delta_{\parallel}$
$\displaystyle=-0.18\pm 0.52\,\mathrm{(stat)}\pm 0.29\,\mathrm{(syst)}.$
These results for the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay are consistent with the
values measured in $B^{0}\rightarrow\phi K^{*0}$ decays of $f_{0}=0.494\pm
0.036$, $f_{\parallel}=0.212\pm 0.035$ and $\cos\delta_{\parallel}=-0.74\pm
0.10$ [13].
Table 4: Systematic uncertainties of the angular parameters. Effect | $\Delta f_{0}$ | $\Delta f_{\parallel}$ | $\Delta\cos\delta_{\parallel}$
---|---|---|---
S-wave | $0.07\phantom{0}$ | $0.02\phantom{0}$ | $0.29\phantom{0}$
Acceptance | $0.007$ | $0.005$ | $0.002$
Combinatorial background | $0.02\phantom{0}$ | $0.01\phantom{0}$ | $0.01\phantom{0}$
Total | $0.07\phantom{0}$ | $0.02\phantom{0}$ | $0.29\phantom{0}$
## 10 Summary and conclusions
A total of $30\pm 6$ $B^{0}_{s}\rightarrow(K^{+}K^{-})(K^{-}\pi^{+})$
candidates have been observed within the mass windows
$1012.5<M(K^{+}K^{-})<1026.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$746<M(K^{-}\pi^{+})<1046{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The result
translates into a significance of $6.2\,\sigma$. The analysis of the
$K^{+}K^{-}$ and the $K^{-}\pi^{+}$ mass distributions is consistent with
$(84\pm 2)\%$ of the signal originating from resonant $\phi$ and $\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ mesons. The significance of the
$B^{0}_{s}\rightarrow\phi\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$
resonant contribution is calculated to be $6.1\,\sigma$. The branching
fraction of the decay is measured to be
${\cal B}(B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0})=\left(1.10\pm
0.24\,\mathrm{(stat)}\pm 0.14\,\mathrm{(syst)}\pm
0.08\left(\frac{f_{d}}{f_{s}}\right)\right)\times 10^{-6},$
using the $B^{0}\rightarrow\phi K^{*0}$ decay as a normalization channel. This
result is roughly three times the theoretical expectation in QCD factorization
of $(0.4\,{}^{+0.5}_{-0.3})\times 10^{-6}$ [21] and larger than the
perturbative QCD value of ${(0.65\,{}^{+0.33}_{-0.23})\times 10^{-6}}$ [28],
although the values are compatible within $1\,\sigma$. The result is also
higher than the expectation of ${\cal B}(B^{0}\rightarrow\phi
K^{*0})\times|V_{td}|^{2}/|V_{ts}|^{2}$. Better precision on both the
theoretical and experimental values would allow this channel to serve as a
probe for physics beyond the SM.
An angular analysis of the $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ decay results in the polarization
fractions and phase difference
$\displaystyle f_{0}$ $\displaystyle=\phantom{-}0.51\pm
0.15\,\mathrm{(stat)}\pm 0.07\,\mathrm{(syst)},$ $\displaystyle f_{\parallel}$
$\displaystyle=\phantom{-}0.21\pm 0.11\,\mathrm{(stat)}\pm
0.02\,\mathrm{(syst)},$ $\displaystyle\cos\delta_{\parallel}$
$\displaystyle=-0.18\pm 0.52\,\mathrm{(stat)}\pm 0.29\,\mathrm{(syst)}.$
The small value obtained for the longitudinal polarization fraction follows
the trend of the $b\rightarrow s$ penguin decays $B^{0}\rightarrow\phi
K^{*0}$, $B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ and
$B^{0}_{s}\rightarrow\phi\phi$. The comparison with the decay
$B^{0}\rightarrow K^{*0}\kern 1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$,
where $f_{0}=0.80^{+0.12}_{-0.13}$ [25], shows a $2\,\sigma$ discrepancy. This
is very interesting since the loop-mediated amplitudes of each decay differ
only in the flavour of the spectator quark. The result is also compatible with
the longitudinal polarization fraction $f_{0}=0.40\pm 0.14$ measured in
$B^{0}\rightarrow\rho^{0}K^{*0}$ decays [11], the penguin amplitude of which
is related to $B^{0}_{s}\rightarrow\phi\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$ by $d\leftrightarrow s$ exchange.
Finally, the result is smaller than the prediction of perturbative QCD,
$f_{0}=0.712\,{}^{+0.042}_{-0.048}$, given in Ref. [28].
## 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); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from 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 thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-06-10T16:13:14 |
2024-09-04T02:49:46.281461
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, 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, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, O. Callot, 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. Castillo Garcia,\n M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini,\n M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M.\n Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras,\n P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau,\n G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C.\n D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, 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, D. Derkach, O. Deschamps, F. Dettori,\n A. Di Canto, H. Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil\n Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A.\n Dzyuba, S. Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S.\n Eisenhardt, U. Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch.\n Elsasser, D. Elsby, A. Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S.\n Farry, V. Fave, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M.\n Ferro-Luzzi, S. Filippov, M. Fiore, C. Fitzpatrick, M. Fontana, F.\n Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas,\n E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao,\n J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E.\n Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C.\n G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa\n G\\'andara, R. Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani,\n A. Grecu, E. Greening, S. Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu.\n Guz, T. Gys, C. Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T.\n Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T.\n Hartmann, J. He, V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E.\n van Herwijnen, A. Hicheur, E. Hicks, D. Hill, M. Hoballah, M. Holtrop, C.\n Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D.\n Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, F. Jing, M. John, D. Johnson, C.R. Jones, C.\n Joram, B. Jost, M. Kaballo, S. Kandybei, M. Karacson, T.M. Karbach, I.R.\n Kenyon, U. Kerzel, T. Ketel, A. Keune, B. Khanji, O. Kochebina, 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, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A.\n Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch,\n T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees, R. Lef\\`evre,\n A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B. Leverington, Y.\n Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I.\n Longstaff, J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S.\n Malde, G. Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G.\n Martellotti, A. Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez\n Santos, D. Martins Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi,\n E. Maurice, A. Mazurov, J. McCarthy, A. McNab, R. McNulty, B. Meadows, F.\n Meier, M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez,\n S. Monteil, D. Moran, P. Morawski, M.J. Morello, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C.\n Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A.\n Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana, M.S.\n Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C. dos\n Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky,\n A. Romero Vidal, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz Valls, G.\n Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, C. Salzmann,\n B. Sanmartin Sedes, M. Sannino, R. Santacesaria, C. Santamarina Rios, E.\n Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie, D.\n Savrina, P. Schaack, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B.\n Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia,\n A. Sciubba, M. Seco, A. Semennikov, I. Sepp, N. Serra, J. Serrano, P.\n Seyfert, M. Shapkin, I. Shapoval, P. Shatalov, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, T.\n Skwarnicki, N.A. Smith, E. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F.\n Soomro, D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F.\n Stagni, S. Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M.\n Straticiuc, U. Straumann, V.K. Subbiah, S. Swientek, V. Syropoulos, M.\n Szczekowski, P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E.\n Teodorescu, F. Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand,\n M. Tobin, S. Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S.\n Tourneur, M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M.\n Ubeda Garcia, A. Ukleja, D. Urner, U. Uwer, V. Vagnoni, G. Valenti, R.\n Vazquez Gomez, P. Vazquez Regueiro, S. Vecchi, J.J. Velthuis, M. Veltri, G.\n Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X. Vilasis-Cardona, A.\n Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V. Vorobyev, C. Vo\\ss, H.\n Voss, R. Waldi, R. Wallace, S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson,\n A.D. Webber, D. Websdale, M. Whitehead, J. Wicht, J. Wiechczynski, D.\n Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson,\n J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Antonio Romero Vidal",
"url": "https://arxiv.org/abs/1306.2239"
}
|
1306.2404
|
# An Aggregation-Based Overall Quality Measurement for Visualization
Weidong Huang CSIRO, Australia
[email protected]
###### Abstract
Aesthetics are often used to evaluate the quality of graph drawings. However,
the existing aesthetic criteria are useful in judging the extents to which a
drawing conforms to particular drawing rules. They have limitations in
evaluating overall quality. Currently the overall quality of graph drawings is
mainly evaluated based on personal judgments and user studies. Personal
judgments are not reliable, while user studies can be costly to run.
Therefore, there is a need for a direct measure of overall quality. This
measure can be used by visualization designers to quickly compare the quality
of drawings at hand at the design stage and make decisions accordingly. In an
attempt to meet this need, we propose a measure that measures overall quality
based on aggregation of individual aesthetic criteria. We present a user study
that validates this measure and demonstrates its capacity in predicting the
performance of human graph comprehension. The implications of the proposed
measure for future research are discussed.
###### Index Terms:
Visualization, graph drawing, aesthetics, overall quality, measurement
## I Introduction
In drawing graphs, one of the issues we face is how to lay them out as
empirical research has shown that layout affects how a graph is perceived
(e.g., [9]). A range of rules for laying out graphs, or aesthetics, have been
proposed in the field of graph drawing. Examples of such aesthetics include
minimum crossings and maximum symmetries. It is commonly believed that
drawings satisfying these aesthetics are more effective. As a result,
aesthetics have been widely used as quality criteria in evaluating automatic
graph drawing algorithms and interfaces of graph visualization systems (e.g.,
[5, 4]).
However, the existing aesthetic criteria are useful in judging the extents to
which a drawing conforms to specific drawing rules. They have limitations in
evaluating the overall quality. Part of the reason is that aesthetics often
conflict with each other and cannot be implemented in full at the same time.
This conflicting relationship affects current practices of graph drawing
greatly. On the one hand, many algorithms for automatic graph drawing are
designed to optimize only one or two aesthetics, and different algorithms
focus on different aesthetics [2]. This makes it difficult for an algorithm
user to choose which algorithm to use when he or she has more than one
candidate algorithm at hand. We do not know whether an algorithm that is to
minimize the number of crossings will produce better drawings than another
algorithm that is to maximize symmetries. On the other hand, it is generally
acknowledged that the best layout is the balance of aesthetics. However,
seeking a compromise between a set of aesthetics only gives us a better chance
of producing good drawings [12]. We still do not know whether a drawing
produced based on one set of aesthetics is better than another drawing that is
produced based on a different set of aesthetics.
Due to the lack of appropriate measures, the overall quality of graph
visualizations is often evaluated based on personal judgments and user
studies. However, personal judgments are subjective and are not reliable,
while user studies can be costly to run. Therefore, there is a need for a
reliable and objective measure so that we can evaluate overall quality at the
early design stage of a visualization process. This measure will help
visualization designers to quickly judge or compare the quality of the
drawings in consideration and make decisions accordingly.
In an attempt to meet this need, we propose an overall quality measure of
layout. This measure takes into account individual aesthetic criteria and
gives a single numerical value. In this paper, we explain how this measure is
computed. We present a study to validate the quality measure and to
demonstrate its capability in predicting the performance of human graph
comprehension.
## II Related work
Two main approaches have been used in evaluating graph drawings: 1) approaches
based on computational measures such as commonly used aesthetics and
specifically developed measures; and 2) approaches based on empirical measures
such as expert opinion, user preference and task performance. In this section,
selected studies are given as examples.
Didimo et al. [4] conducted a study that evaluated different graph drawing
algorithms. The quality of the resultant drawings was compared based on the
extent to which they conformed to each of a set of aesthetics. The aesthetics
used for the comparison purpose included number of crossings, crossing angle
resolution, geodesic edge tendency and vertex angles. The results indicated
that some algorithms had a better trade-off between these criteria than other
algorithms.
Brandes and Pich [3] compared distance-based graph drawing algorithms. In this
study, the drawing quality was measured as how well the Euclidean distance
between any two nodes represented their graph-theoretic distance. The results
suggested that minimization of weighted stress yielded better layout than
force-directed placement in terms of pairwise distances between nodes. Similar
quality measurements were also developed in other studies to evaluate whether
drawings have specific layout features (e.g., [18]).
While these computational approaches evaluate quality of specific visual
properties of drawings directly, empirical methods have been used to evaluate
overall quality. Huang et al. [9] conducted a study that evaluated quality of
sociograms that were drawn based on different drawing conventions and edge
crossings. Subjects were asked to perform network-related tasks and indicate
their preference for drawings conventions and crossings. Task performance
including response times and error rates were also recorded. One of their
findings was that a drawing that was the most preferred might not be the one
that induced the best performance. Other studies in this category include [1,
14, 6]. Evaluating quality based on expert opinion, Hachul and Junger [8]
conducted a study that compared algorithms for drawing general large graphs.
Rather than using a quantitative measurement, the authors evaluated the
drawing quality by commenting how well the layout displayed the graph
structure.
The above-mentioned approaches are useful at certain circumstances of a
visualization process and for specific purposes. However, a direct,
empirically validated measure is more preferable for a quick assessment of
overall quality.
## III The Proposed Quality Measure
Aesthetics used in graph drawing have either been empirically validated or
widely acknowledged for their association with human graph comprehension [16,
2]. Further, when it comes to the performance of human graph comprehension,
each aesthetic has a role to play, and it is the joint effect of these
aesthetics that is more relevant. Based on the overall quality measure
proposed by Huang et al. [13], we measure overall visual quality ($y$) as an
aggregation of aesthetics ($x$) as below:
$y=\sum\limits_{i=0}^{n}x_{i}$ (1)
For a measure to be useful, it should be objective (give the same value when
used by different assessors), reliable (give the same value when used at
different times), sensitive to changes (be able to tell the difference when
there is a change to the visualization), easy to measure (take only a few
steps to compute), be comparable (be able to give continuous numerical values,
rather than categorical), and be predictive of task performance.
In addition, although being subjective, it is our belief that for this measure
to be generically useful, aesthetics to be considered in equitation 1 should
be context or application independent, be applicable to general graphs and
reflect specific local features of layout.
Keeping these requirements in mind, we chose the following four most discussed
aesthetics as the aesthetic elements of equation 1:
1. 1.
Minimize the number of edge crossings ($cross\\#$)
2. 2.
Maximize crossing angle resolution ($crossRes$)
3. 3.
Maximize node angular resolution($angularRes$)
4. 4.
Uniformize edge lengths ($uniEdge$)
Among these aesthetics, $cross\\#$ is measured as the number of crossings in
the drawing; a smaller value is better. $CrossRes$ is measured as the minimum
size of all crossing angles; a larger value is better. $AngularRes$ is
measured as the minimum size of angles formed by any two neighboring edges; a
larger value is better. $UniEdge$ is measured as the standard deviation of all
edge lengths; a smaller value is better. The scores of these aesthetics are
measured on different scales. To be able to combine them together, they need
to be transformed to $z$ scores first. To be more specific, suppose that we
have three drawings of the same graph that we would like to compare, and they
have 2, 7, and 6 crossings, respectively. That is, the scores ($x$) of
$cross\\#$ are 2, 7 and 6. The mean of these three scores ($Mean$) is 5 and
their standard deviation ($StDev$) is 2.65. Then, the $z$ score of $cross\\#$
for each of these three drawings can be computed as below:
$z_{cross\\#}=\frac{x-Mean}{StDev}$ (2)
and it is $-1.14$, $0.76$ and $0.38$, respectively. The other aesthetics can
be standardized into $z$ scores in the same way.
In aggregating $z$ scores, the scales must be made going the same direction.
That is, higher values are always better. As such, equation 1 can be refined
and the overall quality score ($O$) can be computed as below:
$O=-z_{cross\\#}+z_{crossRes}+z_{angularRes}-z_{uniEdge}$ (3)
## IV Experiment
In this section, we describe an experiment that was designed to validate the
proposed measure and test its predictability.
### IV-A Design
To validate the measure, we generated a set of random graphs with similar
structures. For each graph, a number of drawings were generated using a
specific algorithm so that the relative quality levels between these drawings
were known beforehand. The overall quality scores of these drawings were
computed using the proposed measure. These computed scores were then tested to
see whether they were consistent with the pre-known quality.
To test the capacity of the proposed measure in predicting task performance,
we used a random graph with reasonable size and complexity. This graph was
randomly drawn a number of times. However, this time we did not know their
overall quality beforehand. Instead, we recruited users to perform a typical
graph reading task. We measured task response time, accuracy, cognitive load
and _visualization efficiency_ [11]. We also measured overall quality of each
drawing using the proposed measure. We then ran regression tests to see
whether the task performance data were significantly associated with the
measured overall quality.
In addition, as mentioned in section 2, task performance measures are often
used for quality evaluation. We would like to compare how performance measures
and the proposed measure performed in differentiating drawings based on
overall quality. We therefore included the drawings for the validity test in
our user study. As a result, the experiment included two blocks of the
drawings: one block for validity and the other for predictability. The
experiment employed a within-subject design.
### IV-B Stimuli
Figure 1: An example of the four drawing conditions of a graph for validity
For validity, we generated 20 different graphs based on the Erdos-Renyi model
of random graphs [19], with each having 30 nodes and 40 edges. These graphs
were then drawn using a force-directed algorithm. The algorithm applies forces
on the nodes and edges of a random initial layout, and moves them accordingly.
This process is repeated until an equilibrium state is reached. It is known
that each time the process is repeated, the overall layout is generally
improved. To create experimental conditions, we recorded the layout when the
process had been repeated 3000, 6000, 9000 and 12000 times for each graph. As
a result, four conditions were obtained: c3, c6, c9 and c12. Each graph had
one drawing in each condition and each condition had 20 drawings. The drawing
quality improved across the conditions from c3, c6, c9 to c12. Figure 1 shows
the four drawings of a graph.
Figure 2: Three examples of the thirty drawings of a graph for predictability
For predictability, we used a graph that had 39 nodes and 48 edges. This graph
was randomly drawn thirty times using a force-directed algorithm, resulting in
30 drawings in total. Each of these drawings was obtained by using a random
combination of a different initial layout, a different number of iterations
for convergence and a different set of parameters that were used to define the
forces. Figure 2 shows three examples of them.
To avoid possible fatigue or boredom caused by too many drawings, the twenty
graphs for validity were divided into two halves. Only one half was chosen in
an alternating order and the corresponding drawings of the chosen half were
used in each trial. As a result, each subject viewed 10 $\times$ 4
(conditions) = 40 drawings for validity and 30 drawings for predictability,
that is, 70 stimuli in total.
### IV-C Subjects
Thirty-five subjects volunteered to participate in the study. These subjects
were undergraduate students. All of them had normal or corrected-to-normal
vision.
### IV-D The task
The task was to find the shortest path between two pre-specified nodes.
Subjects were required to response by indicating the length of the path. For
each graph, the two nodes were randomly chosen with two pre-conditions. The
first condition was that there was only one shortest path between them. The
second was that the shortest path length was between 3 to 5 inclusive for
validity and 4 for predictability. Given a graph, the same path was used
across the drawings of it for a subject, while the paths could be different
between subjects. Using different paths for different subjects was to ensure
that the impact of overall layout, rather than a specific part of the drawing,
was reflected in performance data.
### IV-E The online system
A custom-built system was used to display the drawing images. The system was
designed to display the $70$ stimuli in a random order with one constraint and
to highlight the two pre-specified nodes as red. The constraint for the
randomization of the order was that no two drawings from the predictability
block were displayed in a row. It displayed one of the red nodes first. The
subject looked at the node and hit a key on the keyboard to have the whole
drawing displayed. Then the subject started looking for the answer. Once the
answer was found, the subject was required to hit a key immediately. Once the
key was hit, the time spent for the answer was recorded and an answer screen
was shown. The answer screen had six boxes on the top, with a number near each
box representing a possible answer to the task. There were also nine smaller
boxes with numbers indicating possible levels of mental effort devoted to the
task (from 1 being the lowest to 9 being the highest). The subject was
required to respond by clicking on one box of each set. After two answers had
been given, the subject hit a key to have the responses recorded and to
continue with the next drawing.
### IV-F Procedure
Before the experiment, subjects were given time to read the information and
tutorial documents, understand the graph concepts and the task, and practice
with the system. They were also free to ask questions. Subjects were
instructed to perform the task as quickly as possible without compromising
accuracy.
Once ready, subjects indicated to the experimenter and the experiment started.
At any time when the answer screen was displayed, subjects could have a break
as they wished before a key was hit. Therefore, the pace of the experiment was
controlled by subjects in order to prevent fatigue. After the online task was
completed, subjects were debriefed about the study purposes. The whole session
took about 40 minutes on average for each subject. Drinks and snacks were
provided after the study.
### IV-G Results
#### IV-G1 Validity test
Task completion time, responses to the task and mental effort were recorded.
Based on the recorded data, visualization efficiency for each drawing was
computed using a formula suggested by Huang et al. [11] to have an overall
evaluation of task performance. Overall quality of each drawing was also
computed using equation 3. In this part of the study, dependent variables were
time, accuracy, effort, efficiency and overall quality, while the independent
variable was iteration number (of the force-directed algorithm).
TABLE I: Mean Values of Dependent Variables Variable | C3 | C6 | C9 | C12
---|---|---|---|---
Time (sec.) | 9.91 | 9.51 | 7.19 | 7.11
Effort | 3.60 | 3.26 | 3.27 | 3.09
Accuracy | 0.69 | 0.75 | 0.76 | 0.76
Efficiency | -0.74 | -0.22 | 0.28 | 0.48
Overall quality | -2.14 | 0.04 | 1.02 | 1.08
We obtained 10 (drawings) $\times$ 4 (conditions) $\times$ 35 (subjects) =
1440 experimental data entries for each dependent variable. The mean values
were computed and the results are shown in Table I. First, on performance
measures, it can be seen that the subjects generally spent less time, exerted
less effort and were more accurate when iteration number increased from
condition c3, c6, c9 to c12. The performance efficiency also showed a clear
increase across the conditions. In other words, the means of the performance
measures were in good agreement with the pre-known overall quality.
TABLE II: Results of ANOVA with Post-Hoc Comparisons Variable | _F_(3,57) | _p_ | Condition Pairs Detected Different (out of 6)
---|---|---|---
Time | 2.804 | 0.048 | (c3, c9), (c3, c12)
Effort | 7.491 | 0.000 | (c3, c6), (c3, c9), (c3, c12)
Accuracy | 2.118 | 0.108 | none
Efficiency | 7.442 | 0.000 | (c3, c6), (c3, c9), (c3, c12)
Overall quality | 28.596 | 0.000 | all pairs but (c9, c12)
To test whether these trends of changes were significant at the level of 0.05,
we ran repeated ANOVA tests with post-hoc comparisons of the Least Square
Difference (LSD) method on each of the dependent variables. The results are
shown in Table II. There was a significant main effect on time, effort and
efficiency ($p<0.05$), but not on accuracy ($p>0.05$). Post-hoc comparisons
revealed that these dependent variables had different levels of capacity in
detecting condition differences. More specifically, out of the six condition
pairs in total, time data only found that two pairs of the conditions were
different; effort found three; and efficiency found three, while no difference
was shown in accuracy between any pair of the conditions.
Second, on overall quality, the mean overall quality value was -2.14 for c3,
0.04 for c6, 1.02 for c9 and 1.08 for c12 (see the bottom row of Table I).
These values showed that the measured overall quality increased while
iteration number increased across the conditions from c3 to c12. To see
whether this trend of increase was significant, we ran a repeated ANOVA with
post-hoc comparisons. The results are shown in the bottom row of Table II. The
repeated ANOVA indicated that the main effect on overall quality was
statistically significant, $F(3,57)=28.596$, $p<0.001$. Post-hoc comparisons
indicated that all conditions were statistically different from each other,
except the condition pair of c9 and c12.
#### IV-G2 Predictability test
Thirty-five subjects each viewed $30$ drawings for the shortest path task. The
time they spent and their responses to the task and mental effort were
recorded. In this part of the study, dependent variables were time, accuracy,
effort and efficiency, while the predictor variable was overall quality. The
obtained raw data were processed accordingly for each of the dependent
variables. And the overall quality of each drawing was computed using equation
3. In the end, we had $30$ data entries for each variable for data analysis.
Figure 3: Scatter diagrams between dependent variables and overall quality
We expected that the measured overall quality was negatively correlated with
time and effort, and positively correlated with accuracy and efficiency. To
test our hypotheses, we first plotted the scores of overall quality and the
scores of each dependent variable as Cartesian coordinates to generate a
scatter diagram, with overall quality on the horizontal axis and the dependent
variable on the vertical axis. This was to have a general idea about the
relationships between the two variables. The obtained diagrams are shown in
Figure 3. In these scatter diagrams, it appeared, as expected, that overall
quality had a negative relationship with time and effort, and a positive
relationship with accuracy and efficiency.
Then, we ran simple linear regression tests to see whether the observed
relationships were statistically significant. We regressed each dependent
variable on overall quality, and the results are shown in Table III. In this
table, $\beta$ is a standardized coefficient. The absolute value of it gives
the size of effect that the predictor had on a dependent variable, while the
sign implies the direction of that effect. According to common rules of thumb,
effect size is small if $\beta$ is less than 0.10, is large if $\beta$ is more
than 0.50, and is medium if $\beta$ is between 0.10 and 0.50.
TABLE III: Results of Simple Linear Regression Tests Dependent Var. | Predictor | $\beta$ | _F_(1,28) | _p_
---|---|---|---|---
Time | overall quality | -0.539 | 11.478 | 0.002
Effort | overall quality | -0.692 | 25.796 | 0.000
Accuracy | overall quality | 0.575 | 13.835 | 0.001
Efficiency | overall quality | 0.717 | 29.625 | 0.000
The overall regression test of time was significant, $F(1,28)=11.478$,
$p<0.01$. Time was negatively correlated with overall quality, $\beta=-0.539$.
The overall regression test of effort was significant, $F(1,28)=25.796$,
$p<0.001$. Effort was negatively correlated with overall quality,
$\beta=-0.692$. The overall regression test of accuracy was significant,
$F(1,28)=13.835$, $p=0.001$. Accuracy was positively correlated with overall
quality, $\beta=0.575$. The overall regression test of efficiency was
significant, $F(1,28)=29.625$, $p<0.001$. Efficiency was positively correlated
with overall quality, $\beta=0.717$.
### IV-H Discussion
Our data analysis for validity revealed that there was a significant overall
difference shown in the data of performance measures including time, effort
and efficiency, but not in the accuracy data. This indicated that the subjects
had followed the instructions closely and did not compromise accuracy for
speed in performing their tasks. The analysis also showed that the proposed
overall quality measure was able to detect the quality difference between the
drawings in the four conditions as expected.
The further post-hoc pairwise comparisons revealed that the proposed measure
was able to find more condition pairs being different than any of the task
performance and visualization efficiency measures did (see Table II). This, on
the one hand, indicated that the actual differences between conditions were
small, which is good for the validity purpose. On the other hand, it indicated
that the proposed overall quality measure was more sensitive to quality
changes than performance measures. This should not be surprising if we
consider that the proposed measure measures overall quality directly, while
performance measures measure indirectly. In addition, performance measures
require conduction of user studies, and factors associated with a user study
could negatively affect performance measures in evaluating overall quality.
More specifically, among other factors, human factors, methodological issues,
data analysis methods and the choice of tasks can each have a certain role in
the evaluation process, affecting the measurement in one way or another.
Our data analysis for predictability revealed that each of the dependent
variables had a significant correlation with the predictor variable, overall
quality. And the significant correlations came with large effect sizes as
shown in the $\beta$ values.
In summary, our tests demonstrated that given a graph, the proposed measure
was able to not only differentiate drawings based on overall quality, but also
significantly predict the performance of human graph comprehension. In other
words, the proposed measure is a valid measure of overall quality.
## V General discussion
In this paper, we proposed a measure that measures overall quality by
combining individual aesthetic criteria into a single value. We have presented
a user study that provides empirical evidence validating the proposed measure
and verifying the predictive capacity of it. It is also shown in this
particular study that the proposed measure performs better than performance
measures in evaluating overall quality. It should be noted that this
comparison was only to demonstrate the sensitivity of our new measure. And
this finding should not be interpreted as an argument of one measure being
used against another since they are essentially different kinds of measures
serving for different purposes.
The proposed measure uses the $z$ scores of its aesthetic components to
compute overall quality. Z scores have also been used for the purpose of
measurement in the field of usability engineering [17]. Usability is often
considered as a multi-dimensional construct of user experience, in which each
dimension is measured by a different metric. For example, ease of learning,
ease of use, user preference and task satisfaction. The $z$ scores of these
metrics are then combined to reflect the overall usability. Despite this, it
is important to note that there is a fundamental difference between the
overall usability and our overall layout quality. The former is empirical and
is measured based on human user experience, while the latter is computational
and is measured based on layout aesthetics.
One limitation of the proposed measure is that it assumes a linear
relationship between overall quality and its associated component aesthetics,
as implied in equation 3. This is clearly
an oversimplification and the reality can be more complex. For example, a
study of Huang et al. [10] reveals that there exists a significant quadratic
relationship between $crossRes$ and drawing quality measured by task response
time. Although more studies are needed so that the actual relationships
between them can be fully understood, the empirical evidence presented in this
paper shows that our simplified version does give valid and useful insight
into the relative overall quality between drawings.
Finally, our experiment has limitations. For example, we used only the
shortest path search task. The obtained findings could be more widely
applicable if different types of tasks were used [15]. Therefore, more studies
are needed to verify the validity of our proposed measure.
## References
* [1] D. Archambault and H. C. Purchase, The Mental Map and Memorability in Dynamic Graphs, _Proceedings of PacificVis’12_ , 89-96, 2012.
* [2] G. di Battista, P. Eades, R.Tamassia, and I. Tollis, _Graph Drawing: Algorithms for the Visualization of Graphs_ , Prentice Hall, 1998.
* [3] U. Brandes, and C. Pich, An Experimental Study on Distance-based Graph Drawing, _Proceedings of GD’08_ , 218-229, 2009.
* [4] W. Didimo, G. Liotta, and S. Romeo. Topology-driven force-directed algorithms,_Proceedings of GD’10_ , 165-176, 2010.
* [5] C. Dunne, and B. Shneiderman, Improving graph drawing readability by incorporating readability metrics, _TR No. HCIL2009-13_ , University of Maryland, 2009.
* [6] T. Dwyer, B. Lee, D. Fisher, K. I. Quinn, P. Isenberg, G. Robertson, and C. North, A Comparison of User-Generated and Automatic Graph Layouts, _IEEE Transactions on Visualization and Computer Graphics_ , 15 (6), 961-968, 2009.
* [7] P. Erdos and A. Renyi, On random graphs, _Publicationes Mathematicae_ , vol. 6, 290-297, 1959.
* [8] S. Hachul, and M. Junger, An experimental comparison of fast algorithms for drawing general large graphs, _Proceedings of GD’05_ , 235-250, 2005.
* [9] W. Huang, and S.-H. Hong, Layout effects: Comparison of sociogram drawing conventions, _Technical report no. 575_ , University of Sydney, 2005.
* [10] W. Huang, P. Eades, and S.-H. Hong, Effects of crossing angles, _Proceedings of PacificVis’08_ , 41-46, 2008.
* [11] W. Huang, P. Eades, and S.-H. Hong, Measuring effectiveness of graph visualizations: a cognitive load perspective, _Information Visualization_ , 8 (3): 139-152, 2009.
* [12] W. Huang, P. Eades, S.-H. Hong, and C.-C. Lin, Improving force-directed graph drawings by making compromises between aesthetics, _Proceedings of VLHCC’10_ , 176-183, 2010.
* [13] W. Huang, P. Eades, S.-H. Hong, and C.-C. Lin, Improving Multiple Aesthetics Produces Better Graph Drawings. _Journal of Visual Languages and Computing_ , http://dx.doi.org/10.1016/j.jvlc.2011.12.002.
* [14] C. Korner, and D. Albert, Speed of comprehension of visualized ordered sets, _Journal of Experimental Psychology: Applied_ , 8, 57-71, 2002.
* [15] B. Lee, C. Plaisant, C. S. Parr, J.-D. Fekete, and N. Henry, Task taxonomy for graph visualization, _Proceedings of BELIV’06_ , 1-5, 2006.
* [16] H. C. Purchase, R. F. Cohen, and M. James, Validating graph drawing aesthetics, _Proceedings of GD’95_ , 435-446, 1995.
* [17] T. Tullis and B. Albert, _Measuring the User Experience: Collecting, Analyzing, and Presenting Usability Metrics_ , Morgan Kaufmann Publishers, 2008.
* [18] F. Zaidi, D. Archambault, and G. Melançon, Evaluating the Quality of Clustering Algorithms Using Cluster Path Lengths, _Proceedings of ICDM’10_ , 2-56, 2010.
|
arxiv-papers
| 2013-06-11T02:01:17 |
2024-09-04T02:49:46.296531
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Weidong Huang",
"submitter": "Weidong Huang",
"url": "https://arxiv.org/abs/1306.2404"
}
|
1306.2459
|
# Fast Search for Dynamic Multi-Relational Graphs
Sutanay Choudhury
Lawrence Holder
George Chin
Pacific Northwest National Laboratory, USA [email protected]
Washington State University, USA [email protected] Pacific Northwest National
Laboratory, USA [email protected] John Feo
Pacific Northwest National Laboratory, USA [email protected]
(2013)
###### Abstract
Acting on time-critical events by processing ever growing social media or news
streams is a major technical challenge. Many of these data sources can be
modeled as multi-relational graphs. Continuous queries or techniques to search
for rare events that typically arise in monitoring applications have been
studied extensively for relational databases. This work is dedicated to answer
the question that emerges naturally: how can we efficiently execute a
continuous query on a dynamic graph? This paper presents an exact subgraph
search algorithm that exploits the temporal characteristics of representative
queries for online news or social media monitoring. The algorithm is based on
a novel data structure called the _Subgraph Join Tree (SJ-Tree)_ that
leverages the structural and semantic characteristics of the underlying multi-
relational graph. The paper concludes with extensive experimentation on
several real-world datasets that demonstrates the validity of this approach.
###### category:
H.2.4 Systems Query processing
###### keywords:
Continuous Queries; Dynamic Graph Search; Subgraph Matching
††conference: DyNetMM’13, June 23, 2013, New York, USA.
## 1 Introduction
Social networks, social media websites and mainstream news media are driving
an exponential growth in online content. This information barrage presents
both a formidable challenge and an opportunity to applications that thrive on
situational awareness. Examples of such applications include emergency
response, cyber security, intelligence and finance where the data stream is
monitored continuously for specific events. Timeliness of the detection
carries paramount importance for such applications. The applications derive
their competitive edge from fast detection as late detection may not have much
value due to incurred damage to resources. Our work is motivated by queries
that look for rare events, have a time constraint on the time to discovery and
never need a bulk retrieval of historic data due to their monitoring nature.
Continuous queries evolved in the field of relational databases to address
applications with precisely the above characteristics. A continuous query
system is defined as one where a query logically runs continuously over time
as opposed to being executed intermittently [1, 2]. Thus, continuous query
processing is data-driven or trigger oriented. Many of the prominent news or
social media streams can be represented as multi-relational data sources.
Multi-relational graphs are often an attractive representation for data
sources with sparsity. The problem of monitoring events in such data streams
can be viewed as continuously searching the dynamic graph for patterns that
represent events of interest.
Figure 1: A graph query for monitoring emergencies in social media and news
streams.
Fig. 1 shows a graph pattern that represents such an event. An operator may
substitute the "keyword" with "fire" or "accident" and register several
queries. Articles refer to articles in news as well as social media posts.
This query will capture events that are reported from the same location.
Observe that we specify the label for only one vertex in this query, the rest
of the vertices have only type specified. Therefore, we are using the labeled
vertex to anchor into a context and report when multiple events with that
context are detected in the data stream.
Graph search involves finding exact or approximate matches for a query
subgraph in a larger graph. It has been studied extensively and is formally
defined as the problem of subgraph isomorphism: given a pattern or query graph
(henceforth described as query graph) $G_{q}$ and a larger input graph
(henceforth described as the data graph) $G_{d}$, find all isomorphisms of
$G_{q}$ in $G_{d}$. Following the definition of isomorphism, the matching
involves finding a one-to-one correspondence between the vertices of a
subgraph of $G_{d}$ and vertices of $G_{q}$ such that all vertex adjacencies
are preserved. Now consider the challenges in applying traditional graph
search techniques to this problem. Unless carefully adapted, a standard search
function will search the entire data graph repeatedly and retrieve the same
search results. Also, many of the best performing graph search algorithms rely
on indexing the graph. Even with an interval as large as 5 minutes, rebuilding
the index of a massive graph repeatedly is infeasible. This motivates
exploration of incremental algorithms for continuous queries, although the
general problem of incremental subgraph isomorphism is proven to be NP-
complete as well [3].
Queries like the one shown in Fig. 1 share a number of common attributes.
First, they all involve an implicit time window to suggest the timeliness
aspect associated with the query. Clearly, the length of the time window
varies depending on the application context. The average monitoring time
window for a high volume social media stream may be in tens of minutes whereas
the equivalent period for online news may be in hours or days. Second, all
these queries aim to discover a number of temporal events that share the same
context, such as a common set of keywords and location. Lastly, a multi-
relational graph often takes the form of k-partite graphs [4, 5] where each
partite set represents a group of entities of the same type. For queries as
ones described in Fig. 1, each event that is represented by an article or a
tweet can be viewed as a k-partitite subgraph.
We exploit these three features to implement a continuous query processing
framework for multi-relational graphs. First, by utilizing a rolling time
window we continuously prune partial search results that would otherwise need
to be tracked and would contribute to the combinatorial growth in memory
utilization. Second, the temporal property of the vertices and edges
representing events suggests that it is logical to search for distinct
subgraphs where such “temporal" vertices or edges are ordered, thus
significantly reducing the search space. Finally, we take advantage of the
multi-relational structure of the data and the characteristics of temporal
events to avoid expensive joins. Given a multi-relational query graph we
decompose it in a hierarchical fashion. We design a data structure called the
_Subgraph Join Tree_ , or henceforth referred as the _SJ-Tree_ to model the
hierarchical decomposition and store matches with various subgraphs of the
query graph as represented in the tree. This paper demonstrates the validity
of this decomposition approach towards query processing. Automated
construction of SJ-Tree from an arbitrary query graph is not discussed in this
paper. We refer to the smallest units of the decomposed query graph as “search
primitives", which almost always consists of more than one edge. As new edges
arrive over time, we continuously perform (a) “local searches" to look for
matches with the search primitives and (b) use the decomposition structure to
“join" them into progressively larger matches. This represents a middle ground
between the periodic application of a graph search algorithm on the data graph
and the approach that would have been employed by a traditional stream
database. Stream databases have no alternative but to model each edge in the
query graph as a separate join operator. Our model can express this degenerate
case where an edge is represented as a search primitive in the SJ-Tree, but
the performance is extremely poor. By grouping subgraphs into search
primitives, we can simplify the query plan, significantly improve performance
by multiple orders of magnitude, and perhaps most importantly, reason about
the trade-offs involved and explore a large space of possible optimizations.
### 1.1 Contributions
Our contributions from this research are summarized below.
1\. We introduce a data structure called SJ-Tree for query graph decomposition
(section 4) and present a novel subgraph search algorithm (5) for continuous
queries on dynamic multi-relational graphs.
2\. We compare our performance with the incremental subgraph isomorphism
algorithm developed by Fan et al. [3] and show that our approach provides
improvements by multiple orders of magnitude (section 6).
3\. We present a series of experiments on representative online news (New York
Times), co-authorship networks (DBLP) and social media data sources (Tencent
Weibo) modeled as multi-relational graphs. The scale of these datasets are
orders of magnitude larger than previously reported research [6, 3] in the
literature (section 6).
4\. We present a theoretical model for complexity analysis of the search
algorithm. We also provide an extensive experimental analysis of the
algorithm’s performance as a function of the frequency distribution of vertex
labels for verification of the theoretical model.
Figure 2: More examples of monitoring queries on multi-relational graphs. The
query at the top can be used to discover events in a certain context. Set one
of the keywords to “Oil" and run the query to discover various events that
center around oil, such as price movements, discovery, accident etc. By
setting the keyword to “buyout", the bottom query can be used to detect when
news surface about a merger between two companies.
### 1.2 Problem Statement
Every edge in a dynamic graph has a timestamp associated with it and
therefore, for any subgraph $g$ of a dynamic graph we can define a time
duration $\tau(g)$ which is equal to the duration between the earliest and
latest edge belonging to $g$. Given a dynamic multi-relational graph $G_{d}$,
a query graph $G_{q}$ and a time window $t_{W}$, we report whenever a subgraph
$g_{d}$ that is isomorphic to $G_{q}$ appears in $G_{d}$ such that
$\tau(g_{d})<t_{W}$. The isomorphic subgraphs are also referred to as matches
in the subsequent discussions. If $M(G^{k}_{d})$ is the cumulative set of all
matches discovered until time step $k$ and $E_{k+1}$ is the set of edges that
arrive at time step $k+1$, we present an algorithm to compute a function
$f\left(G_{d},G_{q},E_{k+1}\right)$ which returns the incremental set of
matches that result from updating $G_{d}$ with $E_{k+1}$ and is equal to
$M(G^{k+1}_{d})-M(G^{k}_{d})$. We assume that the graph only receives edge
inserts and no deletions.
## 2 Background
### 2.1 Multi-Relational Graphs
Single relational graphs have been widely used to model systems comprised of
homogeneous elements related by a single type of relation. A social network
where vertices represent people and edges represent connections between people
is an example of a single-relational graph. A multi-relational graph becomes a
useful construct for modeling heterogeneous relations between a possibly
heterogeneous set of entities.
Definition 2.1.1 Multi-Relational Graph A multi-relational graph denoted as
$G=(V,E)$, is a graph representation of a multi-relational database. If the
database contains $K$ entity types as $\xi_{1},...\xi_{K}$, then the vertex
set $V(G)$ is partitioned into $K$ sets $V_{1},...,V_{K}$. For any vertex
$v\in V_{k}$, the label for the vertex is represented from the domain of the
entity type $\xi_{k}$. The edges of the graph are the relations between
various entities as indicated in an entity-relation model. Thus, an edge in
the graph $e\in E(G),e=\left(v_{i},v_{j}\right)$ is an instance of a relation
$R_{ij}$ between entities $\xi_{i}$ and $\xi_{j}$.
A graph representation of such a multi-relational database takes the form of a
K-partite graph [5], if there are no relations between homogeneous entities or
equivalently, if there are no edges between vertices that belong to the same
partite set. In practice, such relationships are not rare. Examples of such
linkages are citation links between articles and social ties between two
members in a network. However, we omit unary relationships from our multi-
relational model. Our omission of unary relationships is driven by usability
and a desire for simplicity. Fig. 1 and 2 show a number of examples embodying
a range of events. Consider the example in Fig. 2 that detects a series of
articles that refer to the same set of keywords; one may wish to introduce
unary relationships in the graph to indicate citation between articles and
thus, focus only on articles with high citation counts. However, such queries
can be alternatively represented by adding a query constraint to the vertices
that require them to have a minimum degree. Or, such relations could be
represented using an intermediate vertex of a different type. Thus, for the
scope of this work we define patterns of interest as query graphs that are
subgraphs of the K-partite multi-relational graph.
### 2.2 Continuous Queries
A continuous query can be described as computing a function $f$ over a stream
$S$ continuously over time and notifying the user whenever the output of $f$
satisfies a user-defined constraint [1]. They are distinguished from ad-hoc
query processing by their high selectivity (looking for unique events) and
need to detect newer updates of interest as opposed to retrieving lots of past
information. In this paradigm the primary objective is to notify a listener as
soon as the query is matched. One may view conventional databases as passive
repositories with large collections of data that work in a request-response
model whereas continuous queries are data-driven or trigger oriented. These
features coupled with real-time demands challenge many of the fundamental
assumptions for conventional databases and establish continuous query
processing on relational data streams as a major research area. The literature
on database research from the past two decades is abundant with work on
continuous query systems [7, 8]. Babcock et al. [9] provide an excellent
overview of continuous query systems and their design challenges.
### 2.3 Graph Queries
Graph querying techniques have been studied extensively in the field of
pattern recognition over nearly four decades [10]. Our work is focused on
subgraph isomorphism which is as defined as follows.
Definition 2.2.1 Subgraph Isomorphism Given the query graph $G_{q}$ and a
matching subgraph of the data graph ($G_{d}$) denoted as
$G^{{}^{\prime}}_{d}$, a matching between $G_{q}$ and $G^{{}^{\prime}}_{d}$
involves finding a bijective function $f:V(G_{q})\rightarrow
V(G^{{}^{\prime}}_{d})$ such that for any two vertices $u_{1},u_{2}\in
V(G_{q})$, $(u_{1},u_{2})\in E(G_{q})\Rightarrow(f(u_{1}),f(u_{2}))\in
E(G^{{}^{\prime}}_{d})$.
Two popular subgraph isomorphism algorithms were developed by Ullman [11] and
Cordella et al. [12]. The VF2 algorithm [12] employs a filtering and
verification strategy and outperforms the original algorithm by Ullman. Over
the past decade, the database community has focused strongly on developing
indexing and query optimization techniques to speed up the searching process.
A common theme of such approaches is to index vertices based on k-hop
neighborhood signatures derived from labels and other properties such as
degrees, spectral properties and centrality [13, 14, 15, 16]. Other major
areas of work involve join-order optimization [17] and search techniques for
alternative representations such as similarity search in a multi-dimensional
vector space [18].
## 3 Related Work
Investigation of subgraph isomorphism for dynamic graphs did not receive much
attention until recently. It introduces new algorithmic challenges because we
can-not afford to index a dynamic graph frequently enough for applications
with real-time constraints. In fact this is a problem with searches on large
static graphs as well [19]. There are two alternatives in that direction. We
can search for a pattern repeatedly or we can adopt an incremental approach.
The work by Fan et al. [3] presents incremental algorithms for graph pattern
matching. However, their solution to subgraph isomorphism is based on the
repeated search strategy. Chen et al. [6] proposed a feature structure called
the node-neighbor tree to search multiple graph streams using a vector space
approach. They relax the exact match requirement and require significant pre-
processing on the graph stream. Our work is distinguished by its focus on
temporal queries and handling of partial matches as they are tracked over time
using a novel data structure. From a data-organization perspective, the SJ-
Tree approach has similarities with the Closure-Tree [20]. However, the
closure-tree approach assumes a database of independent graphs and the
underlying data is not dynamic. There are strong parallels between our
algorithm and the very recent work by Sun et al. [19], where they implement a
query-decomposition based algorithm for searching a large static graph in a
distributed environment. Our work is distinguished by the focus on continuous
queries that involves maintenance of partial matches as driven by the query
decomposition structure, and optimizations for real-time query processing.
## 4 Incremental Query Processing
### 4.1 Naive Approach
We begin with a simplistic solution to motivate an incremental approach for
continuous query processing. For every new edge that is added to $G_{d}$, we
detect if the edge matches any edge in the query graph. This check can be
performed minimally by examining 1) if there are edges in the query graph with
the same type as the new edge and 2) if the endpoint vertices of the new edge
match with the corresponding edges in the query graph based on their types and
attributes. Once an edge is considered as a matching candidate, the next step
is to consider different combinations of matches it can participate in.
Figure 3: Illustration of a naive incremental algorithm. Assme AB matches with
PQ, and AC matches with both QR and QS.
A simple illustration of this matching process is shown in Fig. 3. While
intuitively simple, this approach falls prey to combinatorial explosion very
quickly. It finds the match with the query graph at the cost of creating many
partial matches. Assume that the $G_{d}$ receives a large number of edges that
match with the query graph edge between A and B (Fig. 3a). Let’s denote this
edge as $e_{AB}$. Therefore, a large number of partial matches will be created
with mapping information for $e_{AB}$. Subsequently, every future edge that
matches with $e_{AC}$ will need to be matched or checked against all the
existing partial matches for augmenting into a larger match. While the
subgraph matching problem has an inherent exponential nature associated with
it, a better algorithm will restrict the growth of the number of partial
matches to track and still produce the correct result.
### 4.2 Our Approach
Our objective is to introduce an approach that guides the search process to
look for specific subgraphs of the query graph and follow specific transitions
from small to larger matches. Following are the main intuitions that drive
this approach,
1. 1.
Instead of looking for a match with the entire graph or just any edge of the
query graph, partition the query graph into smaller subgraphs and search for
them.
2. 2.
Track the matches with individual subgraphs and combine them to produce
progressively larger matches.
3. 3.
Define a join order in which the individual matching subgraphs will be
combined. Do not look for every possible way to combine the matching
subgraphs.
Although the current work is completely focused on temporal queries, the graph
decomposition approach is suited for a broader class of applications and
queries. The key aspect here is to search for substructures without incurring
too much cost. Even if some subgraphs of the query graph are matched in the
data, we will not attempt to assemble the matches together without following
the join order. Thus, if there are substructures that are too frequent,
joining them and producing larger partial matches will be too expensive
without a stronger guarantee of finding a complete match. On the other hand,
if there is a substructure in the query that is rare or indicates high
selectivity, we should start assembling the partial matches together only
after that substructure is matched.
Figure 4: Illustration of query decomposition in SJ-Tree.
### 4.3 Subgraph Join Tree (SJ-Tree)
We introduce a tree structure called _Subgraph Join Tree (SJ-Tree)_ that
supports the above intuitions for implementing a join order based on
selectivity of substructures of the query graph.
Definition 4.1.1 A SJ-Tree $T$ is defined as a binary tree comprised of the
node set $N_{T}$. Each $n\in N_{T}$ corresponds to a subgraph of the query
graph $G_{q}$. Let’s assume $V_{SG}$ is the set of corresponding subgraphs and
$|V_{SG}|=|N_{T}|$. Additional properties of the SJ-Tree are defined below.
Property 1. The subgraph corresponding to the root of the SJ-Tree is
isomorphic to the query graph. Thus, for $n_{r}=root\\{T\\}$,
$V_{SG}\\{n_{r}\\}\equiv G_{q}$.
Property 2. The subgraph corresponding to any internal node of $T$ is
isomorphic to the output of the join operation between the subgraphs
corresponding to its children. If $n_{l}$ and $n_{r}$ are the left and right
child of $n$, then $V_{SG}\\{n\\}=V_{SG}\\{n_{l}\\}\Join V_{SG}\\{n_{r}\\}$.
Given two graphs $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$, the join
operation is defined as $G_{3}=G_{1}\Join G_{2}$, such that
$G_{3}=(V_{3},E_{3})$ where $V_{3}=V_{1}\cup V_{2}$ and $E_{3}=E_{1}\cup
E_{2}$.
Property 3. Each node in the SJ-Tree maintains a set of matching subgraphs. We
define a function $matches(n)$ that for any node $n\in N_{T}$, returns a set
of subgraphs of the data graph. If $M=matches(n)$, then $\forall G_{m}\in M$,
$G_{m}\equiv V_{SG}\\{n\\}$.
Property 4. Each internal node $n$ in the SJ-Tree maintains a subgraph, CUT-
SUBGRAPH($n$) that equals the intersection of the query subgraphs of its child
nodes.
For any internal node $n\in N_{T}$ such that CUT-SUBGRAPH$(n)\neq\emptyset$,
we also define a projection operator $\Pi$. Assume that $G_{1}$ and $G_{2}$
are isomorphic, $G_{1}\equiv G_{2}$. Also define $\Phi_{V}$ and $\Phi_{E}$ as
functions that define the bijective mapping between the vertices and edges of
$G_{1}$ and $G_{2}$. Consider $g_{1}$, a subgraph of $G_{1}$: $g_{1}\subseteq
G_{1}$. Then $g_{2}=\Pi(G_{2},g_{1})$ is a subgraph of $G_{2}$ such that
$V(g_{2})=\Phi_{V}(V\left(g_{1}\right))$ and
$E(g_{2})=\Phi_{E}(E\left(g_{1}\right))$.
Conceptually, the SJ-Tree is an index structure where keys track the
occurrence of matching subgraphs in the data graph. Our decision to use a
binary tree as opposed to an n-ary tree is influenced by the simplicity and
lowering the combinatorial cost of joining matches from multiple children.
Analyzing the trade-offs between different tree models (e.g. left-deep vs
bushy) is part of future work.
## 5 Continuous Query Algorithm
Figure 5: a) Example query b) Occurrence of a match in the search graph c)
Combining two partial matches to form the complete match
We present a subgraph search algorithm (Algo. 1 and 2) that utilizes the SJ-
Tree structure (referred to as $T$). The search process is illustrated in Fig.
5. The input to PROCESS-CONT-QUERY is the dynamic graph $G_{d}$, the SJ-Tree
($T$) corresponding to the query graph and the set of incoming edges. Each
leaf of the SJ-Tree represents an unique subgraph of the query graph. Lines
4-6 in Algo. 1 describe the search for each of these unique subgraphs around
every incoming edge. Any discovered match is added to the SJ-Tree (line 9).
Algorithm 1 PROCESS-CONT-QUERY($G_{d}$, T, edges)
1:$leaf$-$nodes=$GET-LEAF-NODES$(T)$
2:for all $e_{s}\in edges$ do
3: UPDATE-GRAPH($G_{d},e_{s}$)
4: for all $n\in leaf$-$nodes$ do
5: $g^{q}_{sub}=$GET-QUERY-SUBGRAPH$(T,n)$
6: $matches=$LOCAL-SEARCH($G_{d},g^{q}_{sub},e$)
7: if $matches\neq\emptyset$ then
8: for all $m\in matches$ do
9: T.UPDATE-SJ-TREE$(n,m)$
### 5.1 Local Search
The LOCAL-SEARCH function performs a subgraph isomorphism check around the
neighborhood of every incoming edge $e$. The query decomposition often reduces
the local search to performing star queries where the center of the query is
the vertex representing a temporal event. The peripheral vertices of the star
query are the other entities that represent various attributes of the event.
Further, in the context of real-time search, if the current time is $t$ and
the query specifies a time window of length $t_{W}$ then all edges that have a
timestamp older than $(t-t_{W})$ are ignored from the search. In addition to
filtering search candidates, we also periodically prune the SJ-Tree to remove
partial matches that are older than $t_{W}$ from the current time.
Algorithm 2 UPDATE-SJ-TREE($node,m)$
1:$sibling=sibling[node]$
2:$parent=parent[node]$
3:$k=$GET-JOIN-KEY(CUT-SUBGRAPH[$parent$], $m$)
4:$H_{s}$ = match-tables[$sibling$]
5:$M^{k}_{s}$ = GET($H_{s},k$)
6:for all $m_{s}\in M^{k}_{s}$ do
7: $m_{sup}$ = JOIN($m_{s}m$)
8: if parent = root then
9: PRINT(’MATCH FOUND : ’, $m_{sup}$)
10: else
11: UPDATE-SJ-TREE($parent,m_{sup}$)
12:ADD($matches[node],m$)
13:ADD($match-tables[node],k,m$)
### 5.2 Partial Match Join and Aggregation
This subsection describes the process outlined in UPDATE-SJ-TREE. The SJ-Tree
data structure maintains the sibling and parent information for every node as
distinct arrays (Algo. 2, line 1-2). Each node in the SJ-Tree maintains a hash
table that supports storing multiple objects with the same key. This
collection of tables are denoted by the match-tables property of the SJ-Tree
(Algo 2., line 4). GET() and ADD() provides lookup and update operations on
the hash tables. Whenever a new matching subgraph $g$ (available as a property
of the partial match $m$) is added to a node in the SJ-Tree, we compute a key
using its projection $(\Pi(g))$ and insert the key and the matching subgraph
into the hash table. The projection is obtained by hashing a string
representation of the subgraph.
When a new match is inserted into a leaf node we check to see if it can be
combined (referred as JOIN()) with any matches that are contained in the
collection maintained at its sibling node. A successful combination of
matching subgraphs between the leaf and its sibling node leads to the
insertion of a larger match at the parent node. This process is repeated as
long as larger matching subgraphs can be produced by moving up in the SJ-Tree.
A complete match is found when two matches belonging to the children of the
root node are combined successfully.
The JOIN operation between partial matches is critical to the overall query
processing performance. Suppose we have a query that finds a sequence of two
events with a common set of attributes. Assume that two matching events
${event}_{1}$ and ${event}_{2}$ are found with timestamps $\tau_{1}$ and
$\tau_{2}$ respectively, with $\tau_{1}<\tau_{2}$. For all practical purposes
we can report the sequence $\\{{event}_{1},{event}_{2}\\}$ and ignore the out
of order combination. Therefore, given two partial matches $M_{1}$ and $M_{2}$
with edge sets $\\{E_{i},E_{j}\\}$ and $\\{E_{m},E_{n}\\}$ respectively, the
join algorithm rejects all combinations of these two sets that do not
represent a monotonic order based on timestamps. This is accomplished by
computing a range of timestamps for each partial match. If $t_{low}[M_{i}]$
and $t_{high}[M_{i}]$ are the lowest and highest timestamp for match $M_{i}$,
then we require that $t_{high}[M_{1}]<t_{low}[M_{2}]$ for joining $M_{1}$ and
$M_{2}$.
### 5.3 Complexity Analysis
There are two primary tasks in processing every edge in the continuous query
algorithm, (1) performing a local search for a small subgraph of the query
graph and in case of a successful search, (2) updating the SJ-Tree with the
partial match. For the multi-relational queries described in this paper, the
local search reduces to performing a star query. Assuming the LOCAL-SEARCH is
cheap for star queries, we can approximate the cost of the continuous query
processing for every edge to a small constant in case of a failed local search
and to that of the UPDATE-SJ-TREE() for a successful search. If $C_{k}$ is the
expected value for the cost of insert and joins at node $k$ in the SJ-Tree,
then the complexity of updating the tree is $O((\bar{C_{k}})^{h})$, where
$\bar{C_{k}}$ denotes the expected value of $C_{k}$ over all nodes. This is
not a tight bound, and a more accurate bound can be obtained estimating
$C_{k}$ based on the frequency of the subgraphs satisfying the label
constraints in the query graph. Accurate estimation of the frequency of an
arbitrary subgraph is hard. Therefore, we resort to obtaining a loose bound in
terms of the label constraints. Assume that $v_{q}$ has the lowest degree
among all labeled vertices in the query graph and $v_{s}$ is the corresponding
vertex in the data graph. Then $n_{k}$ is bounded by ${d_{s}}\choose{d_{q}}$,
where $d_{s}$ and $d_{q}$ are the degrees of $v_{s}$ and $v_{q}$. The storage
complexity for SJ-Tree is $O(h\bar{C_{k}}|E(G_{q})|)$.
## 6 Experimental Results
This section is dedicated to answering two questions: 1) How does our
continuous graph query algorithm compare with the state of the art? To answer
this, we compare our algorithm’s performance with the IncIsoMatch algorithm
presented in [3]. 2) How does our query algorithm perform on real-world
datasets? We provide the answers from exhaustive experimentation on three
real-world datasets through systematic query selection.
Graph dataset | vertices | edges | vertex types | edge types
---|---|---|---|---
New York Times | 39,523 | 68,682 | 4 | 4
DBLP | 3.158M | 3.26M | 2 | 1
Tencent Weibo | 2.5M | 89.6M | 4 | 5
Our metric is the time to process increments of a fixed number of edges (1k or
100k) whichever is closer to 1% of the test dataset size. The times reported
only include the time spent in the query processing section. We use the query
template as shown in Fig. 6. To develop a performance model in terms of the
label distribution, we sample the degree distribution of every vertex type and
divide the range of the degree distributions into ten intervals. For each
interval, five closest candidate vertices are selected for testing purposes.
Selection of multiple vertices around each bin allows us to systematically
observe the impact of increasing the degree of the labeled vertex in the query
graph.
### 6.1 Experimental Setup
The results were obtained by using a single core on a 48-core shared memory
system comprising 2.3 GHz Opteron 6176 SE processors and 256 GB RAM. The
processor cache size is 512KB and each system node has 32 GB RAM. The code was
compiled with g++ 4.1.2-52 with -O3 optimization flag on Linux 2.6.18.
### 6.2 New York Times
Figure 6: The query template used to find temporal events. Experiments are
performed using queries with 4 event vertices and 2 feature vertices. One
feature vertex is labeled and all other vertices specify only types. Figure 7:
Results from queries finding four articles with a common keyword and location.
Legends indicate degree of query label. Figure 8: Comparison with IncIsoMatch.
MQD refers to the Multi-Relational Query Decomposition algorithm from this
paper.
We use a news dataset from New York Times collected over Aug-Oct 2011
111http://data.nytimes.com. Each article in the dataset contains a number of
facets that belong to four type of entities. Each of the articles and facets
are represented as vertices in the graph. Each edge that connects an article
with a facet carries a timestamp that is the publication time of the article.
Following the template shown in Fig. 6, we run a query that finds four
articles where all the articles have a common keyword and location. For the
location vertex we specify the labels shown in the top diagram in Fig. 7 and
observe the performance. As the figures indicate, selecting labels that
correspond to vertices with increasing degrees increases the running time of
the query.
Next, we compare our approach with the IncIsoMatch algorithm described by Fan
et al. [3]. The VF2 algorithm [12] is adapted to implement the graph search
functionality as outlined in IncIsoMatch. We specify a label on the feature
marked with $\dagger$ and select a label with one of the highest degrees for
that vertex type. The queries are as follows: 1) Find four articles with a
common keyword and a common organization$\dagger$, 2) Find four articles with
a common entity and a common keyword $\dagger$ and 3) Find four articles with
a common entity and a common location $\dagger$. Fig. 8 shows a performance
improvement from our algorithm by several orders of magnitude. The multiple
orders of improvement in performance is attributed to the strictly ordered
aggregation of partial matches in the SJ-Tree and the temporal property based
optimizations. The performance gap between the processing time of the two
algorithms increases as the graph grows larger. We attribute this to the
nature of the IncIsoMatch where it performs a search around every new edge in
the graph. The search spans all vertices around the endpoints of the new edge
as long as they are within $k$-hops, where $k$ is the diameter of the query
graph. As the data graph grows denser, even for a query graph with small or
modest size, the $k$-hop subgraph accumulates a large number of edges and the
search becomes increasingly expensive.
### 6.3 DBLP Co-Authorship Network
Figure 9: Performance results for queries on the DBLP dataset. The spikes in
the plot can be attributed to the bursty nature of scientific publishing where
authors target the same group of conferences and journals every year.
We build a multi-relational graph representation of the DBLP citation network
111dblp.uni-trier.de/xml with two types of entities: authors and articles. The
author name and the title of the article are stored as labels of respective
vertices. We run a query to find an author (author 1) who has co-authored four
papers with a specified author (author 2). Following the previously shown
query template, our query graph has four article vertices and two author
vertices. Only one author vertex is labeled. We observe the degree
distribution of the “author" vertices and select names with progressively
increasing degrees. The results are shown in Fig. 9. It can be seen that the
performance of the algorithm is quite stable for a modestly large network with
nearly 3M+ edges. Additionally, the results show that even though vertex
degree is a good indicator of the query performance, there are other factors
at play. The graph describes author-article relationship; therefore, the
degree of an author vertex provides the number of authored articles. It does
not provide the information about the number of co-authors of a person.
Searching for a person who publishes a given number of articles with fewer co-
authors will lead to more partial matches and increase per-edge query
processing time. The consistent high processing times for the query containing
"Hsinchun-Chen" despite smaller degree in the graph is a result of this
aspect.
Figure 10: Query processing time for the Tencent Weibo dataset for queries
with varying selectivity.
### 6.4 Social Media
Finally, we present our results on a data set collected from Tencent Weibo, a
Chinese microblogging social network222www.kddcup2012.org/c/kddcup2012-track1.
The data set provides a temporal history of item recommendations to registered
users of the social network. We build a graph with 4 vertex types (users,
items, keywords and categories) and 5 edge types (item-in-category, item-has-
keyword, item-reco-accept, item-reco-reject and user-profile-has-keyword).
Our test query is to detect a series of item acceptances by a group of users
described by a common keyword. Following the previously shown query template,
our query graph has four user vertices and one item and keyword vertex. We
specify a label on the item and seek to discover the keyword that characterize
the users accepting or rejecting that item. The results are shown in Fig. 10.
The figure suggests a clear trend. It shows that as the graph grows large the
query processing time eventually rises sharply. It also shows that the rise
happens earlier for low-selectivity queries where the specified label has
higher degree in the graph. This is because the number of partial matches
grows rapidly in the event of a successful search around a high degree vertex.
Every partial match from the past can potentially be merged with the latest
partial match, and the partial match collection grows combinatorially over
time.
This brings us to implementing the temporal window based pruning. We select
the query with the highest degree label (degree(item) = 299199, Fig. 10) for
which the rise in the processing time was sharpest. We set the time window
$t_{W}$ to 1 day and prune the SJ-Tree after processing every 5 million edges.
The results from the windowing enabled search is shown in Fig. 11. Observe
that the peaks in the processing time are smaller than ones observed in Fig.
10 by an order of magnitude.
Figure 11: Query processing time for the Tencent Weibo dataset with temporal
match pruning applied on every 5 million edges.
This is an extremely promising result for practical applications. Figure 11
suggests that it would take 10 seconds on average to process 100k edges for a
query with very low selectivity. This translates into a throughput of 0.01
million edges/second or 864 million edges per day. At the time of this
writing, high volume data streams such as Twitter receive nearly 300-400
million posts every day. Considering that every user action translates into
multiple edges in a graph, one may expect around billions of edges everyday.
The throughput can be expected to be much higher for a query with moderate
selectivity. Thus, we believe this level of throughput on a very low-
selectivity query gets us close to executing real-time graph queries on such
high volume data streams.
## 7 Conclusion and Future Work
We present a novel graph decomposition based approach for continuous queries
on multi-relational graphs. We introduce the SJ-Tree structure, whose nodes
represent the hierarchical decomposition of the query graph. The SJ-Tree
systematically tracks the evolving matches in the data graph as they
transition from smaller to larger matches based on the query graph
decomposition. We present experimental analysis on several real-world datasets
such as New York Times, DBLP and Tencent Weibo and show that our SJ-Tree based
algorithm coupled with temporal optimizations clearly outperforms the state of
the art [3] by multiple orders of magnitude. Our experiments demonstrate that
it is possible to execute complex multi-relational graph queries in a real-
time setting. To our knowledge, the results presented in this paper are the
best reported performance for such queries. These initial results are highly
promising in that they suggest possible ways of auto-selecting optimal values
for query processing parameters based on the data distribution. Our main
theoretical contribution is to demonstrate that for a prominent class of
multi-relational queries where the local search is cheap, we can execute graph
queries in time that is exponential to the height of the SJ-Tree.
Development of query planning algorithms to generate a SJ-Tree for any query
graph by exploiting its structural and semantic characteristics is the next
logical step. Query planning for 1) complex graph queries where a complete
temporal ordering may not be possible, 2) trade-offs between different query
decomposition strategies and 3) exploring different query classes and
determining the optimal trade-off between local search and joins in the SJ-
Tree represent areas of future work.
## Acknowledgment
The authors are grateful to Dr. Bill Howe at the University of Washington for
his suggestions that have improved this paper. Presented research is based on
work funded under the Center for Adaptive Supercomputing Software -
Multithreaded Architecture (CASS-MT) at the US Department of Energy’s Pacific
Northwest National Laboratory, which is operated by Battelle Memorial
Institute.
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* [20] H. He and A. K. Singh, “Closure-tree: An index structure for graph queries,” ser. ICDE ’06.
|
arxiv-papers
| 2013-06-11T09:21:42 |
2024-09-04T02:49:46.307508
|
{
"license": "Public Domain",
"authors": "Sutanay Choudhury, Lawrence Holder, George Chin, John Feo",
"submitter": "Sutanay Choudhury",
"url": "https://arxiv.org/abs/1306.2459"
}
|
1306.2460
|
# StreamWorks - A system for Dynamic Graph Search
Sutanay Choudhury
Lawrence Holder
George Chin
Pacific Northwest National Laboratory, USA [email protected]
Washington State University, USA [email protected] Pacific Northwest National
Laboratory, USA [email protected] Abhik Ray
Sherman Beus
John Feo
Washington State University, USA [email protected] Pacific Northwest National
Laboratory, USA [email protected] Pacific Northwest National Laboratory,
USA [email protected]
(2013)
###### Abstract
Acting on time-critical events by processing ever growing social media, news
or cyber data streams is a major technical challenge. Many of these data
sources can be modeled as multi-relational graphs. Mining and searching for
subgraph patterns in a continuous setting requires an efficient approach to
incremental graph search. The goal of our work is to enable real-time search
capabilities for graph databases. This demonstration will present a dynamic
graph query system that leverages the structural and semantic characteristics
of the underlying multi-relational graph.
###### category:
H.2.4 Systems Query processing
###### keywords:
Continuous Queries; Dynamic Graph Search; Subgraph Matching
††conference: SIGMOD’13, June 22–27, 2013, New York, New York, USA.
## 1 Introduction
Social networks, social media websites and mainstream news media are driving
an exponential growth in online content and network traffic. This information
barrage presents both a formidable challenge and an opportunity to
applications that thrive on situational awareness. Domains such as emergency
response, cyber security, intelligence and finance has many applications that
continuously monitor the data stream to look for specific events. Timeliness
of the detection carries paramount importance for such applications. The
applications derive their competitive edge from fast detection as late
detection may not have much value due to incurred damage to resources. Our
work is motivated by queries that look for rare events, have a time constraint
on the time to discovery and never need a bulk retrieval of historic data due
to their monitoring nature.
The field of relational databases studied the topic of continuous queries to
address applications with precisely the above characteristics. A continuous
query (CQ) system is defined as one where a query logically runs continuously
over time as opposed to being executed intermittently [3, 4, 2]. Many of the
prominent news, social media or cyber data streams can be represented as
multi-relational graphs. Following the sprit of CQ systems, our work can be
viewed as continuously searching a temporally evolving (henceforth referred as
dynamic) graph for graph based patterns representing various events of
interest.
Our proposed demonstration will showcase StreamWorks (Fig. 1) - an analytics
framework for dynamic graphs. With StreamWorks, a user can register graph
queries to find events as they emerge in the data graph. The novelty of
StreamWorks lies in its incremental graph search algorithm based on a query
decomposition approach. The registered queries are decomposed into sub-
patterns using a novel data structure called the SJ-Tree [6] that
systematically tracks the evolution of matches in the underlying graph. The
query decomposition is performed by utilizing statistics and summaries about
the data graph such as degree distribution, vertex and edge type distribution
and multi-retlational triad distribution.
Figure 1: Various components for graph mining and search.
### 1.1 Demonstration features
We will present an interface to compose and execute graph queries, and query
planning. Further, we will provide visualization of the evolving graph,
results from graph queries, and relevant statistics.
## 2 Background and Related Work
### 2.1 Problem Statement
Our theoretical contribution is the development of an incremental subgraph
isomorphism algorithm for dynamic graphs [6]. Given a pattern or query graph
(henceforth described as query graph) $G_{q}$ and a larger input graph
(henceforth described as the data graph) $G_{d}$, an isomorphism of $G_{q}$ in
$G_{d}$ is defined as the matching that involves a one-to-one correspondence
between the vertices of a subgraph of $G_{d}$ and vertices of $G_{q}$ such
that all vertex adjacencies are preserved.
Every edge in a dynamic graph has a timestamp associated with it and
therefore, for any subgraph $g$ of a dynamic graph we can define a time
interval $\tau(g)$ which is equal to the interval between the earliest and
latest edge belonging to $g$. Given a dynamic multi-relational graph $G_{d}$,
a query graph $G_{q}$ and a time window $t_{W}$, we report whenever a subgraph
$g_{d}$ that is isomorphic to $G_{q}$ appears in $G_{d}$ such that
$\tau(g_{d})<t_{W}$. The isomorphic subgraphs are also referred to as matches
in the subsequent discussions. If $M(G^{k}_{d})$ is the cumulative set of all
matches discovered until time step $k$ and $E_{k+1}$ is the set of edges that
arrive at time step $k+1$, we present an algorithm to compute a function
$f\left(G_{d},G_{q},E_{k+1}\right)$ which returns the incremental set of
matches that result from updating $G_{d}$ with $E_{k+1}$ and is equal to
$M(G^{k+1}_{d})-M(G^{k}_{d})$.
### 2.2 Related Work
Investigation of subgraph isomorphism for dynamic graphs did not receive much
attention until recently. It introduces new algorithmic challenges because we
can-not afford to index a dynamic graph frequently enough for applications
with real-time constraints. In fact this is a problem with searches on large
static graphs as well [8]. There are two alternatives in that direction. We
can search for a pattern repeatedly or we can adopt an incremental approach.
The work by Fan et al. [7] presents incremental algorithms for graph pattern
matching. However, their solution to subgraph isomorphism is based on the
repeated search strategy. Chen et al. [5] proposed a feature structure called
the node-neighbor tree to search multiple graph streams using a vector space
approach. They relax the exact match requirement and require significant pre-
processing on the graph stream. Our work is distinguished by its focus on
temporal queries and handling of partial matches as they are tracked over time
using a novel data structure. There are strong parallels between our algorithm
and the very recent work by Sun et al. [8], where they implement a query-
decomposition based algorithm for searching a large static graph in a
distributed environment. Our work is distinguished by the focus on continuous
queries that involves maintenance of partial matches as driven by the query
decomposition structure, and optimizations for real-time query processing.
## 3 Incremental Query Processing
### 3.1 Our Approach
A simplistic approach to solving this problem would be to check, for every
edge update, if that edge matches one in the query graph. Once an edge is
considered as a matching candidate, the next step is to consider different
combinations of matches it can participate in. While intuitively simple, this
approach falls prey to combinatorial explosion very quickly. Our objective is
to introduce an approach that guides the search process to look for specific
subgraphs of the query graph and follow specific transitions from small to
larger matches. Following are the main intuitions that drive this approach,
1\. Instead of looking for a match with the entire graph or just any edge of
the query graph, partition the query graph into smaller subgraphs and search
for them.
2\. Track the matches with individual subgraphs and combine them to produce
progressively larger matches.
3\. Define a join order in which the individual matching subgraphs will be
combined. Do not look for every possible way to combine the matching
subgraphs.
Although the current work is completely focused on temporal queries, the graph
decomposition approach is suited for a broader class of applications and
queries. The key aspect here is to search for substructures without incurring
too much cost. Even if some subgraphs of the query graph are matched in the
data, we will not attempt to assemble the matches together without following
the join order. Thus, if there are substructures that are too frequent,
joining them and producing larger partial matches will be too expensive
without a stronger guarantee of finding a complete match. On the other hand,
if there is a substructure in the query that is rare or indicates high
selectivity, we should start assembling the partial matches together only
after that substructure is matched.
Figure 2: Illustration of query decomposition in SJ-Tree. The graph shown in
the root node represents a query to find three articles or posts with a common
keyword and location.
### 3.2 Subgraph Join Tree (SJ-Tree)
We introduce a tree structure called _Subgraph Join Tree (SJ-Tree)_ that
supports the above intuitions for implementing a search and join order based
on selectivity of substructures of the query graph. Fig. 2 shows an example
decomposition of a query graph.
Definition 4.1.1 A SJ-Tree $T$ is defined as a binary tree comprised of the
node set $N_{T}$. Each $n\in N_{T}$ corresponds to a subgraph of the query
graph $G_{q}$. Let’s assume $V_{SG}$ is the set of corresponding subgraphs and
$|V_{SG}|=|N_{T}|$. Additional properties of the SJ-Tree are defined below.
Property 1. The subgraph corresponding to the root of the SJ-Tree is
isomorphic to the query graph. Thus, for $n_{r}=root\\{T\\}$,
$V_{SG}\\{n_{r}\\}\equiv G_{q}$.
Property 2. The subgraph corresponding to any internal node of $T$ is
isomorphic to the output of the join operation between the subgraphs
corresponding to its children. If $n_{l}$ and $n_{r}$ are the left and right
child of $n$, then $V_{SG}\\{n\\}=V_{SG}\\{n_{l}\\}\Join V_{SG}\\{n_{r}\\}$.
Given two graphs $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$, the join
operation is defined as $G_{3}=G_{1}\Join G_{2}$, such that
$G_{3}=(V_{3},E_{3})$ where $V_{3}=V_{1}\cup V_{2}$ and $E_{3}=E_{1}\cup
E_{2}$.
Property 3. Each node in the SJ-Tree maintains a set of matching subgraphs. We
define a function $matches(n)$ that for any node $n\in N_{T}$, returns a set
of subgraphs of the data graph. If $M=matches(n)$, then $\forall G_{m}\in M$,
$G_{m}\equiv V_{SG}\\{n\\}$.
Property 4. Each internal node $n$ in the SJ-Tree maintains a subgraph, CUT-
SUBGRAPH($n$) that equals the intersection of the query subgraphs of its child
nodes.
## 4 System Overview
### 4.1 Query Planning
With the subgraph join-tree data structure in mind, the next task is to
automatically decompose a query graph $G_{q}$ and create a subgraph join tree
based on the decomposition. Broadly our aim is to decompose the query graph
into a number of smaller graphs, which we refer to as search primitives, and
perform local searches for these primitives. We use the term local search to
refer to a subgraph search performed in the neighborhood of an edge in the
data graph for a small query subgraph. The primitives are restricted to small
and "selective" query subgraphs to keep the local search efficient. An
important goal of the decomposition process is to push the most selective
subgraph at the lowest level in the subgraph join-tree to reduce the number of
partial matches.
### 4.2 Query Execution
Our proposed subgraph matching algorithm contains two primary tasks. First,
for every incoming edge we perform a local search to detect a match with the
smallest subgraphs associated with the leaves of the SJ-Tree. When a match is
found with the subgraph corresponding to the leaf node of the SJ-Tree, we
initialize a match structure and insert it into the collection maintained at
that leaf node. Upon insertion of a match into a leaf node we check to see if
it can be combined with any matches that are contained in the collection
maintained at its sibling node. A successful combination of matching subgraphs
between the leaf or intermediate node and its sibling node leads to the
insertion of a larger match at the respective parent node. This process is
repeated as long as larger matching subgraphs can be produced by moving up in
the SJ-Tree. A complete match is found when two matches belonging to the
children of the root node are combined successfully.
### 4.3 Summarization
Summarization involves collecting summary statistics about the data graph to
use for query planning. We collect three different types of information 1)
degree distribution 2) distribution of vertex and edge types, 3) the frequency
distribution of multi-relational triad structures. Incorporation of triad
statistics into the query decomposition process is a work in progress at the
time of this writing. Continuously collecting the statistics information from
the data stream and updating the query decomposition and search strategy
remains an area for future work.
## 5 Target Applications
We focus on two major application domains: cyber-security and news/social
media monitoring. The following subsections present a quick snapshot of some
motivating queries.
### 5.1 Cyber-Security
A cyber system is naturally described as a graph with physical machines, IP
addresses, users, and software services as entities (vertices). The
relationships between these entities such as communication between machines,
association of a physical machine and an IP address, login of a user on a
machine etc are modeled as edges in the graph. From a security perspective,
updates to this dynamic graph can be constantly monitored to detect events
such as worm spread, virus attack, denial-of-service attack etc.. We construct
graph-based representation of these events (Fig. 3) and query the data graph
to detect occurrences of malicious events.
### 5.2 News and Social Media
Various online news or social media data sources can be represented as multi-
relational graphs. Entities such as articles, events, people, location,
organizations and keywords can be represented as vertices in the graph. Next,
graph based queries can be executed to detect the occurrence of various events
in the news stream. Fig. shows some example queries and Fig. 5 shows a map-
based visualization of the a series of queries executed on New York Times data
111http://data.nytimes.com.
Figure 3: Examples queries to detect cyber attacks.
## 6 Demonstration Setup
### 6.1 Setup
Dataset: We will demonstrate the query capabilities on internet traffic data
obtained from www.caida.org. The number of records in these datasets typically
varies between 50-100 million/hour.
Software/Hardware The queries will be executed on a 48-core shared memory
system running Linux 2.6.18 and comprising 2.3 GHz AMD Opteron 6176 SE
processors and 256 GB memory. Each system node has 32 GB memory attached to
it. The graph query engine is implemented in C++.
### 6.2 User Interface
There are three major focus areas for visualization and UI design.
* •
Our primary target audience includes journalists, emergency responders,
intelligence professionals who are not expected to use StreamWorks using an
API. Fig. 4 shows an experimental user interface for visual query composition.
The user interface will retrieve metadata information such as vertex and edge
types and their attributes to assist in drawing a query graph.
Figure 4: Prototype of an interface for visual graph query composition.
* •
The query graphs are a representation of events of interest; hence, we are
developing map (Fig. 5) and tabular views (Fig. 6) that show occurrence of
events in a geospatial and temporal context. The goal is to keep the
underlying graph representation transparent to the user. Query results from
any graph with location information available as a vertex attribute can be
displayed on the map view.
Figure 5: A visualization of the output from a collection of graph queries.
The queries are similar to Fig. 2. Each query graph specifies a label (such as
"politics", "accident" etc.) on the keyword vertex to indicate the event of
interest. Figure 6: Grid-based visualization showing cascading effect of a
Smurf DDoS attack across subnetworks (blue dots).
* •
Graph-based visualization of the results from subgraph queries is critical for
developers and API users. Therefore, we are adapting and applying the Gephi
graph visualization and manipulation software [1] to render snapshots of the
data graph and encode the partial and complete matches. This is also useful to
observe the choice of different query decomposition strategies. To illustrate,
Fig. 7 shows snapshots of emerging subgraph patterns in a computer network
that are identified and tracked using different SJ-Tree structures. The
percentages show the fraction of query graph being matched as measured by the
number of edges. Each SJ-Tree is shown next to its associated emerging
subgraph pattern snapshots. The colors of the subgraph patterns in the
snapshots correspond to particular partitions in the associated SJ-Tree to
indicate the level or degree of partial matching to the query graph.
Figure 7: Emerging matches for Smurf DDoS subgraph patterns in a dynamic
computer network using different query plans.
## 7 Acknowledgments
Presented research is based on work funded under the CASS-MT project at
Pacific Northwest National Laboratory, which is operated by Battelle Memorial
Institute.
## References
* [1] Gephi, an open source graph visualization and manipulation software, www.gephi.org.
* [2] D. J. Abadi, D. Carney, U. Çetintemel, M. Cherniack, C. Convey, S. Lee, M. Stonebraker, N. Tatbul, and S. Zdonik. Aurora: a new model and architecture for data stream management. The VLDB Journal, 12:120–139, August 2003.
* [3] S. Chandrasekaran, O. Cooper, A. Deshpande, M. J. Franklin, J. M. Hellerstein, W. Hong, S. Krishnamurthy, S. R. Madden, F. Reiss, and M. A. Shah. Telegraphcq: continuous dataflow processing. SIGMOD ’03.
* [4] J. Chen, D. J. DeWitt, F. Tian, and Y. Wang. Niagaracq: a scalable continuous query system for internet databases. In Proceedings of the 2000 ACM SIGMOD international conference on Management of data, SIGMOD ’00, pages 379–390, New York, USA, 2000. ACM.
* [5] L. Chen and C. Wang. Continuous subgraph pattern search over certain and uncertain graph streams. IEEE Trans. on Knowl. and Data Eng., 22(8):1093–1109, Aug. 2010\.
* [6] S. Choudhury, L. Holder, A. Ray, G. Chin, and J. Feo. Continuous queries for multi-relational graphs. Pacific Northwest National Laboratory technical report, PNNL-SA-90326, http://arxiv.org/abs/1209.2178, 2012.
* [7] W. Fan, J. Li, J. Luo, Z. Tan, X. Wang, and Y. Wu. Incremental graph pattern matching. SIGMOD ’11, 2011.
* [8] Z. Sun, H. Wang, H. Wang, B. Shao, and J. Li. Efficient subgraph matching on billion node graphs. PVLDB, 5(9), 2012.
|
arxiv-papers
| 2013-06-11T09:24:06 |
2024-09-04T02:49:46.316724
|
{
"license": "Public Domain",
"authors": "Sutanay Choudhury, Lawrence Holder, George Chin, Abhik Ray, Sherman\n Beus, John Feo",
"submitter": "Sutanay Choudhury",
"url": "https://arxiv.org/abs/1306.2460"
}
|
1306.2478
|
# Jellett-Minkowski’s formula revisited.
Isoperimetric inequalities for submanifolds in an ambient manifold with
bounded curvature
Vicent Gimeno Department of Mathematics-INIT, Universitat Jaume I, Castelló
de la Plana, Spain [email protected]
###### Abstract.
In this paper we provide an extension to the Jellett-Minkowski’s formula for
immersed submanifolds into ambient manifolds which possesses a pole and radial
curvatures bounded from above or below by the radial sectional curvatures of a
rotationally symmetric model space. Using this Jellett-Minkowski’s generalized
formula we can focus on several isoperimetric problems. More precisely, on
lower bounds for isoperimetric quotients of any precompact domain with smooth
boundary, or on the isoperimetric profile of the submanifold and its modified
volume. In the particular case of a model space with strictly decreasing
radial curvatures, an Aleksandrov type theorem is provided.
###### Key words and phrases:
Jellett-Minkowski-formula and Aleksandrov type theorem and isoperimetric
problem and Mean curvature
###### 1991 Mathematics Subject Classification:
35P15
Work partially supported by DGI grant MTM2010-21206-C02-02.
## 1\. Introduction
Given a precompact domain $\Omega\subset P$ with smooth boundary
$\partial\Omega$ in a $m-$dimensional submanifold $P^{m}$ of the Euclidean
space $\mathbb{R}^{n}$, by the Jellett formula one obtains (see [Jel53, Cha93,
Cha78])
(1.1) $m\text{V}(\Omega)+\int_{\Omega}\langle\tau,H\rangle
dV=\int_{\partial\Omega}\langle\tau,\nu\rangle dA\quad,$
where $\tau$ is the vector position in $\mathbb{R}^{n}$, $V(\Omega)$ is the
volume of $\Omega$, $\nu$ is the unit normal vector pointed outward to
$\partial\Omega$ and, $dV$ and $dA$ are the induced Riemannian volume and area
densities on $\Omega$ and $\partial\Omega$ respectively.
The so-called Minkowski formula (see for instance [MR09, Formula A in Theorem
6.11]) follows from the above formula for the particular case of a closed
$m$-dimensional submanifold $S$ immersed in $\mathbb{R}^{n}$ :
(1.2) $mV(S)+\int_{S}\langle\tau,H\rangle dV=0\quad.$
Making use of this formula, Jellett [Jel53] proved in the middle of the
nineteenth century that a star-shaped constant mean curvature surface
$S\subset\mathbb{R}^{3}$ is a round sphere (see [BS09] for an extension of
results in this direction). In 1956 A. D Aleksandrov [Ale56] improved that
result asserting that any closed, embedded hypersurface in $\mathbb{R}^{n}$
with constant mean curvature is a round sphere.
In this paper we provide an extension of the Jellett-Minkowski’s formula (1.1)
into a more general setting. Our setting will be a $m$-dimensional submanifold
$P^{m}$ immersed in a $n$-dimensional ambient manifold $(N,o)$ with a pole $o$
and radial curvatures $K_{N}$ (see §3) bounded from above or below by the
radial sectional curvatures of a rotationally symmetric model space
$M_{w}^{n}=\mathbb{R}^{+}\times\mathbb{S}_{1}^{n-1}\cup\\{o_{w}\\}$, with
center point $o_{w}$ and warped metric tensor $g_{M_{w}^{n}}$ constructed
using the positive and increasing warping function
$w:\mathbb{R}^{+}\to\mathbb{R}^{+}$ in such a way that
$g_{M_{w}^{n}}=dr^{2}+w(r)^{2}g_{\mathbb{S}_{1}^{n-1}}$ (see §2.1 for precise
definition and for the conditions that $w$ should attain in order to
$M_{w}^{n}$ have smooth metric tensor $g_{M^{n}_{w}}$ around $r=0$). By the
Jellet-Minkowski generalized formula we can obtain -among other results- an
Aleksandrov type theorem for rotationally symmetric model manifolds.
The first step to generalize the Jellett-Minkowski’s formula is to define a
_generalized $w$-vector position_ $\tau_{w}$ using the gradient $\nabla^{N}r$
of the distance function $r$ to the pole $o$ in the ambient manifold $N$
(1.3) $\tau_{w}:=\frac{w}{w^{\prime}}\nabla^{N}r\quad.$
Our second steep is to define a _weighted densities_ $d\mu_{w}$ and
$d\sigma_{w}$ on the submanifold using the warping function $w$ and the
induced Riemannian densities $dV$ and $dA$
(1.4) $\displaystyle d\mu_{w}$ $\displaystyle:=w^{\prime}(r)dV\quad,$
$\displaystyle d\sigma_{w}$ $\displaystyle:=w^{\prime}(r)dA\quad.$
Using the weighted densities $d\mu_{w}$ and $d\sigma_{w}$ for any domain
$\Omega$ with smooth boundary $\partial\Omega$ we can define the $w$-_weighted
volume_ $\mu_{w}(\Omega)$ and the $w$-_weighted area_
${\sigma_{w}}(\partial\Omega)$
(1.5) $\displaystyle\mu_{w}(\Omega)$
$\displaystyle:=\int_{\Omega}d\mu_{w}\quad,$
$\displaystyle\sigma_{w}(\partial\Omega)$
$\displaystyle:=\int_{\partial\Omega}d\sigma_{w}\quad.$
With these previous definitions we can state following Jellett-Minkowski’s
generalized formula
###### Main Theorem (Jellett-Minkowski’s generalized formula).
Let $\varphi:P^{m}\to N^{n}$ be an immersion into a $n-$dimensional ambient
manifold $N$ which possesses a pole and its radial sectional curvatures
$K_{N}$ at any point $p\in N$ are bounded by above (or below ) by the radial
curvatures $K_{w}$ of a model space $M_{w}^{n}$
$K_{N}\left(p\right)\leq
K_{M_{w}^{n}}\left(r\left(p\right)\right)=-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\quad\left(\text{respectively
}K_{N}\left(p\right)\geq-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\right)\quad.$
Suppose moreover, that $w^{\prime}>0$ . Then for any precompact domain
$\Omega$ with smooth boundary $\partial\Omega$
(1.6) $\displaystyle m\mu_{w}(\Omega)+\int_{\Omega}\langle\tau_{w},H\rangle
d\mu_{w}\leq(\geq)\int_{\partial\Omega}\langle\tau_{w},\nu\rangle
d\sigma_{w}\quad,$
where $\tau_{w}$ is the generalized $w$-vector position, $\mu_{w}(\Omega)$ is
the $w$-weighted volume of $\Omega$, $d\mu_{w}$ and $d\sigma_{w}$ are the
$w$-weighted densities, and $\nu$ is the unit normal vector to
$\partial\Omega$.
###### Remark a.
Observe since $w$ is a positive increasing function expressions (1.3), (1.4)
(1.5) are well defined. In the particular case when $w(r)=r$ one obtains
(1.7) $\displaystyle d\mu_{w}$
$\displaystyle=dV\quad\mu_{w}(\Omega)=V(\Omega)$ $\displaystyle d\sigma_{w}$
$\displaystyle=dA\quad\sigma_{w}(\partial\Omega)=A(\partial\Omega)\quad,$
and (1.6) in the above theorem becomes the same expression than in (1.1) but
instead of an equality, one obtains an inequality.
###### Remark b.
Recall that given an isometric immersion $\varphi:(P,g_{P})\to(N,g_{N})$ and
the Levi-Civita connections $\nabla^{N}$ , $\nabla^{P}$ over $N$ and $P$
respectively, the second fundamental form $B^{P}(X,Y)$ for any two vector
fields $X$ and $Y$ of $P$ is
$B^{P}(X,Y)=\nabla^{N}_{X}Y-\nabla^{P}_{X}Y\quad.$
In this paper we use the following convention to define the mean curvature
vector
$H:=\text{tr}_{g}(B^{P})\quad.$
Observe that we do not divide by the dimension of the submanifold.
###### Remark c.
In the particular case when the ambient manifold is a model space inequality
(1.6) becomes an equality (see [Bre13]) and we can rededuce the formula
[Bre13, equation (6)]
The structure of the paper is as follows.
In §2 we explore the application of the above Jellett-Minkowski’s generalized
formula obtaining:
1. (1)
In the particular case in which the ambient manifold is a model space, we
obtain an equality in the generalized Jellett-Minkowski formula. That allow us
to provide a slight new Aleksandrov type theorem (theorem 2.5) for
rotationally symmetric model spaces in the line of [Bre13] and to study its
application to the isoperimetric problem.
2. (2)
The relation of the Jellett-Minkowski formula to the $k$-isoperimetric
quotients.
3. (3)
Bounds for the isoperimetric profile of isometric immersions into a geodesic
ball in a Cartan-Hadamard in terms of the total mean curvature of the
submanifolds and its conformal type.
4. (4)
A gap result for the modified volume of complete non-compact manifolds into a
Cartan-Hadamard ambient manifold.
5. (5)
Finally, for minimal immersions, in corollary 2.18 \- that is an extension of
[CG92, theorem 4] \- , we provide $w-$modified isoperimetric inequalities.
In §3 we recall previous results due to the Hessian comparison of the
extrinsic distance function in order to in §4,§5,§6, §7, §8, §9, §10, §11, §12
prove the main theorem, the Aleksandrov type theorem (theorem 2.5) and
corollaries 2.2, 2.3, 2.7, 2.8, 2.10, 2.11, 2.15, 2.17 and 2.18.
## 2\. Applications of the Jellett-Minkowski’s generalized formula
### 2.1. Aleksandrov type theorem on $w-$model spaces
The Jellet-Minkowski generalized formula becomes an equality in the case of an
immersion into a $w-$model space. Model spaces or also called $w-$Model spaces
or rotationally symmetric model spaces, are generalized manifolds of
revolution using warped products. Let us recall here the following definition
of a model space.
###### Definition 2.1 (See [GW79, Gri99, Gri09]).
A $w-$model space $M_{w}^{n}$ is a simply connected $n$-dimensional smooth
manifold $M_{w}^{n}$ with a point $o_{w}\in M_{w}^{n}$ called the _center
point of the model space_ such that $M_{w}^{n}-\\{o_{w}\\}$ is isometric to a
smooth warped product with base $B^{1}=(\,0,\,\Lambda)\,\,\subset\,\mathbb{R}$
(where $\,0<\Lambda\leq\infty$ ), fiber $F^{n-1}=S^{n-1}_{1}$ (i.e. the unit
$(n-1)-$sphere with standard metric), and positive warping function
$w:\,[\,0,\,\Lambda\,)\to\mathbb{R}_{+}$. Namely:
(2.1)
$g_{M_{w}^{n}}=\pi^{*}\left(g_{(\,0,\,\Lambda)}\right)+(w\circ\pi)^{2}\sigma^{*}\left(g_{S^{n-1}_{1}}\right)\quad,$
being $\pi:M_{w}^{n}\to(\,0,\,\Lambda)$ and $\sigma:M_{w}^{n}\to S^{n-1}_{1}$
the projections onto the factors of the warped product.
Despite of the freedom in the choose of the $w$ function in the above
definition there exist certain restrictions around $r\to 0$. In order to
attain $M_{w}^{n}$ a smooth metric tensor around $o_{w}$ the positive warping
function $w$ should hold the following equalities (see [GW79, Pet98]) :
(2.2) $\displaystyle w(0)=0\quad,$ $\displaystyle w^{\prime}(0)=1\quad,$
$\displaystyle w^{(2k)}(0)=0\quad,$
where $w^{(2k)}(r)$ are the even derivatives of $w$.
The parameter $\Lambda$ in the above definition is called the _radius of the
model space_. If $\Lambda=\infty$, then $o_{w}$ is a pole of $M_{w}^{n}$.
The expression of the metric tensor (2.1) is often written as
$g_{M_{w}^{n}}=dr^{2}+\left(w(r)\right)^{2}d\Theta^{2}$, where $d\Theta^{2}$
denotes the standard metric on $S^{n-1}_{1}$ ( $d\Theta^{2}=g_{S^{n-1}_{1}}$).
The usual examples of model spaces are the real space forms
###### Remark d.
The simply connected space forms $\mathbb{K}^{m}(b)$ of constant curvature $b$
can be constructed as $w-$models with any given point as center point using
the warping functions
(2.3) $w(r)=w_{b}(r)=\begin{cases}\frac{1}{\sqrt{b}}\sin(\sqrt{b}\,r)&\text{if
$b>0$}\\\ \phantom{\frac{1}{\sqrt{b}}}r&\text{if $b=0$}\\\
\frac{1}{\sqrt{-b}}\sinh(\sqrt{-b}\,r)&\text{if $b<0$}\quad.\end{cases}$
Note that for $b>0$ the function $w_{b}(r)$ admits a smooth extension to
$r=\pi/\sqrt{b}$. For $\,b\leq 0\,$ any center point is a pole.
Applying now the generalized Jellett-Minkowski formula on embedded
hypersurfaces $\Sigma$ in a model space bounding a domain $\Omega$ we can
state
###### Corollary 2.2.
Let $M_{w}^{n}$ be a $w-$model space with positive warping function $w$ and
positive derivative $w^{\prime}$ on $(0,\Lambda)$ being $\Lambda$ the radius
of the model space. Then for any closed, embedded, orientable hypersurface
$\Sigma$ bounding a domain $\Omega$
(2.4)
$\frac{\sigma_{w}(\Sigma)}{\mu_{w}(\Omega)}\geq\frac{1}{n\sup_{\Sigma}(|\tau_{w}|)}\quad.$
Recall that a constant mean curvature hypersurface is a hypersurface $\Sigma$
with constant pointed inward mean curvature $h$ :
$h=-\langle H,\nu\rangle=\text{ constant on }\Sigma\quad,$
being $\nu$ the unit normal outward vector to $\Sigma$.
For constant mean curvatures hypersurfaces using the generalized Jellett-
Minkowski formula we obtain
###### Corollary 2.3.
Let $M_{w}^{n}$ be a $w-$model space with positive warping function $w$ and
positive derivative $w^{\prime}$ on $(0,\Lambda)$ being $\Lambda$ the radius
of the model space. Then for any closed, embedded, orientable hypersurface
$\Sigma$ with constant mean curvature bounding a domain $\Omega$
(2.5) $h=\frac{(n-1)}{n}\frac{\sigma_{w}(\Sigma)}{\mu_{w}(\Omega)}\quad.$
###### Remark e.
Observe that the above stated constant mean curvature hypersurfaces are mean
convex hypersurfaces ($h>0$).
Given a point $p\in M_{w}^{n}-\\{o_{w}\\}$ in a model space $M_{w}^{n}$ the
distance $r$ to the center point $o_{w}$ is given by $r(p)=\pi(p)$. A
$2$-plane $\Pi_{\text{radial}}$ in $T_{p}M_{w}^{n}$ is a _radial plane_ if it
contains the radial direction ($\nabla r\in\Pi_{\text{radial}}$) and a
$2$-plane $\Pi_{\text{fiber}}$ is a _tangent to the fiber plane_ if it is
perpendicular to the radial direction ($\Pi_{\text{fiber}}\perp\nabla r$). The
sectional curvatures of the radial or tangent to the fibers planes depends
only on the distance to the center of the model space (see proposition 3.2).
For model spaces with an accurate control of the sectional curvatures of the
radial and tangent to the fiber planes. We obtain the following Aleksandrov
type theorem
###### Theorem 2.4 (From Theorem 1.4 of [Bre13]).
Let $(M_{w}^{n},o_{w})$ be a $w-$model space with center $o_{w}$, positive
warping function $w$ and positive derivative $w^{\prime}$ on $(0,\Lambda)$
being $\Lambda$ the radius of the model space. Suppose moreover that the
sectional curvatures $K_{\Pi_{\text{fiber}}}$ of planes tangents to the fiber
and the radial sectional currvatures $K_{\Pi_{\text{rad}}}$ satisfies the
following relation
(2.6) $K_{\Pi_{\text{fiber}}}\geq K_{\Pi_{\text{rad}}}.$
Then, if the radial sectional curvature is a monotone function non increasing
on the distance to the center of the model
($\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq 0$), every closed, embedded, orientable
hypersurface $\Sigma$ with constant mean curvature bounding a domain $\Omega$
is an umbilic submanifold. In the particular case when
$K_{\Pi_{\text{fiber}}}>K_{\Pi_{\text{rad}}}$ and
$\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq 0$, every closed, embedded, orientable
hypersurface $\Sigma$ with constant mean curvature bounding a domain $\Omega$
is a sphere centered at $o_{w}$.
###### Proof.
By proposition 3.2
(2.7)
$\frac{d}{dr}K_{\Pi_{\text{rad}}}=\left(-\frac{w^{\prime\prime}}{w}\right)^{\prime}=\frac{w^{\prime}}{w}\left(-\frac{w^{\prime\prime\prime}}{w^{\prime}}+\frac{w^{\prime\prime}}{w}\right).$
Assuming the positivity of $w^{\prime}$, we obtain that inequality (2.6) and
the condition $\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq 0$ are attained if and
only if the warping function satisfies
(2.8)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w}\geq-\frac{w^{\prime\prime\prime}}{w^{\prime}}\quad.$
And that implies the condition H3’ of [Bre13, Theorem 1.4]. Finally, the
condition
(2.9) $K_{\Pi_{\text{fiber}}}>K_{\Pi_{\text{rad}}}$
is equivalent to condition H4’. ∎
Following the proof of the Theorem 1.4 of [Bre13] we can state the following
sligth modifiqued theorem
###### Theorem 2.5 (Aleksandrov type theorem).
Let $(M_{w}^{n},o_{w})$ be a $w-$model space with center $o_{w}$, positive
warping function $w$ and positive derivative $w^{\prime}$ on $(0,\Lambda)$
being $\Lambda$ the radius of the model space. Suppose moreover that the
sectional curvatures $K_{\Pi_{\text{fiber}}}$ of planes tangents to the fiber
and the radial sectional currvatures $K_{\Pi_{\text{rad}}}$ satisfies the
following relation
(2.10) $K_{\Pi_{\text{fiber}}}\geq K_{\Pi_{\text{rad}}}.$
Then:
1. (1)
If the radial sectional curvature is a monotone function non increasing on the
distance to the center of the model ($\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq
0$), every closed, embedded, orientable hypersurface $\Sigma$ with constant
mean curvature bounding a domain $\Omega$ is an umbilic submanifold. In the
particular case when $K_{\Pi_{\text{fiber}}}>K_{\Pi_{\text{rad}}}$ and
$\frac{d}{dr}K_{\Pi_{\text{rad}}}\leq 0$, every closed, embedded, orientable
hypersurface $\Sigma$ with constant mean curvature bounding a domain $\Omega$
is a sphere centered at $o_{w}$.
2. (2)
If the radial sectional curvature is a strictly decreasing function on the
distance to the center of the model ($\frac{d}{dr}K_{\Pi_{\text{rad}}}<0$),
every closed, embedded, orientable hypersurface $\Sigma$ with constant mean
curvature bounding a domain $\Omega$ is a sphere centered at $o_{w}$.
###### Remark f.
Observe that in the above theorem the new item (2)
($\frac{d}{dr}K_{\Pi_{\text{rad}}}<0$) implies (by proposition 3.2)
(2.11)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w}>-\frac{w^{\prime\prime\prime}}{w^{\prime}}\quad.$
An example of a model space satisfying the Hypotheses of the above theorem is
a generalized paraboloid
###### Example 2.6 (Generalized paraboloids).
The usual parabolid $P$ in $\mathbb{R}^{3}$
$P=\\{(x,y,z)\in\mathbb{R}^{3}\,|\,z=x^{2}+y^{2}\\}$
can be obtained as a surface of revolution with metric tensor
$g_{P}=dr^{2}+\left(\frac{1}{2}\text{arcsinh}(2r)\right)^{2}d\theta^{2}.$
In this way, we define the generalized paraboloid as a $n-$dimensional model
space $M_{w}^{n}$ with warping function
$w(r)=\frac{1}{2}\text{arcsinh}(2r).$
After a few calculation one can check that
(2.12)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w}>-\frac{w^{\prime\prime\prime}}{w^{\prime}}\quad.$
Hence, the closed, embedded, orientable and constant mean curvature
hypersurfaces in the generalized paraboloid are spheres centered at the center
of the model.
The above theorem allow us to focus on the isoperimetric problem in such model
spaces. The isoperimetric problem in a Riemannian manifold $M$ consists in
studying, among the compact hypersurfaces $\Sigma\subset M$ enclosing a region
$\Omega$ of volume $\text{V}(\Omega)=v$, those which minimize the area
$\text{A}(\Sigma)$. For a given volume $v$, an _isoperimetric region_ is a
region of volume $v$ and with minimum area on the boundary. Using [Ros05,
Theorem 1.1] we know that if the $M^{n}$ is compact the isoperimetric problem
has solution and moreover for $n\leq 7$ the isoperimetric hypersurface (the
boundary of the isoperimetric region) is smooth and with constant mean
curvature. Therefore,
###### Corollary 2.7.
Let $(M_{w}^{n},o_{w})$ be a $w-$model space with center $o_{w}$, dimension
$n\leq 7$, positive warping function $w$ and positive derivative $w^{\prime}$
on $(0,\Lambda)$ being $\Lambda$ the radius of the model space. Suppose
moreover that the sectional curvatures $K_{\Pi_{\text{fiber}}}$ of planes
tangents to the fiber and the radial sectional currvatures
$K_{\Pi_{\text{rad}}}$ satisfies the following relation
(2.13) $K_{\Pi_{\text{fiber}}}\geq K_{\Pi_{\text{rad}}}.$
Suppose moreover that the radial sectional curvature is a strictly decreasing
function on the distance to the center of the model
($\frac{d}{dr}K_{\Pi_{\text{rad}}}<0$) and $\Lambda<\infty$. Then the
isoperimetric regions are geodesic balls centered at $o_{w}$.
###### Remark g.
Let us emphasize here, that by [HHM99, theorem 9.1] the unique length-
minimizing simple closed curve enclosing a given area is a circle centered at
the origin for any plane with smooth, rotationally symmetric, complete metric
such that the Gauss curvature is a strictly decreasing function from the
origin. Hence, the above corollary is a (partial) generalization on the case
of dimension greater than $2$. See also [BM02, corollary 2.4] and [MM09,
corollary 2.3].
Given a domain $\Omega$ of volume $\text{V}(\Omega)$, denoting by
$\omega_{n-1}$ the volume of the standard sphere $S^{n-1}_{1}$ and using the
following function $\text{Rad}:\mathbb{R}^{+}\to\mathbb{R}^{+}$
(2.14) $v\to\text{Rad}(v)\quad\text{such that}\quad
v=\omega_{n-1}\int_{0}^{\text{Rad}(v)}w^{n-1}(t)dt,$
we obtain by the expression of the volume of a geodesic ball $B^{w}_{R}$
centered at $o_{w}$ (see equation (3.8))
(2.15) $\text{V}(\Omega)=\text{V}(B_{\text{Rad}(\text{V}(\Omega))}^{w}).$
Therefore, taking into account and the area of a geodesic sphere $S^{w}_{R}$
of radius $R$ centered at $o_{w}$ in a model space (see equation (3.7) ),
under the hypotheses of corollary 2.7 for any hypersurface $\Sigma$ bounding a
domain $\Omega$ of volume $V(\Omega)=v$, we have the following isoperimetric
inequality:
(2.16)
$A(\Sigma)\geq\omega_{n-1}w(\text{Rad}(v))^{n-1}=\text{A}(S^{w}_{\text{Rad}(v)}).$
Isoperimetric inequalities have several applications, among them, the above
inequality allow us to obtain an isoperimetric inequality for the first
eigenvalue of the Laplacian for the Dirichlet problem (see [Cha84] for
details).
###### Corollary 2.8.
Under the hypotheses of corollary 2.7 suppose moreover that the function
(2.17) $\mathcal{I}:\mathbb{R}^{+}\to\mathbb{R}^{+},\quad
t\to\mathcal{I}(t)=\frac{\omega_{n-1}w(\text{Rad}(t))^{n-1}}{t}$
is an non-increasing function. Then, for any precompact domain $\Omega\subset
M^{n}_{w}$ with smooth boundary $\partial\Omega$ and volume
$\text{V}(\Omega)=v$, the first eigenvalue of the Laplacian for the Dirichlet
problem $\lambda_{1}(\Omega)$ is bounded from below by
(2.18)
$\lambda_{1}(\Omega)\geq\frac{1}{4}\left[\mathcal{I}\left(v\right)\right]^{2}.$
### 2.2. Isoperimetric inequalities and the Jellett-Minkowski formula
In the setting of a controlled radial curvature ambient manifold with a pole,
and certain restrictions on the mean curvature of the submanifold, the
generalized Jellett-Minkowski formula is enough to get isoperimetric
inequalities (lower bounds to the isoperimetric profile) or to characterize
the $k$-isoperimetric quotient of a domain in the submanifold. The
$k-$isoperimetric quotient is defined as follows
###### Definition 2.9 (See also [CF91, Cha93]).
Given a domain $\Omega$ with smooth boundary $\partial\Omega$, the _$w$
-weighted $k$-isoperimetric quotient_ $\mathcal{I}^{w}_{k}(\Omega)$ is given
by
(2.19)
$\mathcal{I}^{w}_{k}(\Omega):=\frac{\sigma_{w}(\partial\Omega)}{\mu_{w}(\Omega)^{\frac{k-1}{k}}}\quad,$
where $\mu_{w}(\Omega)$ and $\sigma_{w}(\partial\Omega)$ are the $w$-weighted
volume and $w$-weighted area respectively.
To recall the definition of the isoperimetric profile in a Riemannian manifold
$M$, let us denote by $\mathcal{O}_{M}$ the set of relatively compact open
subsets of $M$ with smooth boundary. The isoperimetric profile $I_{M}$ is a
function $I_{M}:[0,\operatorname{Vol}(M)]\to\mathbb{R}^{+}$ such that
(2.20) $I_{M}(v):=\begin{cases}0\text{ if }v=0\\\
\inf_{\Omega\in\mathcal{O}_{M}}\\{\text{A}(\partial\Omega)\,:\,\text{V}(\Omega)=v\\}\end{cases}$
In this subsection we will examine under appropriate settings the relation
between the Jellett-Minkowski’s generalized formula and the $k-$isoperimetric
quotients, the isoperimetric profile, and the modified volume of a
submanifold.
#### 2.2.1. Lower bounds to the $k$-isoperimetric quotient of a domain
The next application of the generalized Jellett-Minkowski formula will be to
obtain lower bounds to the $k$-isoperimetric profile of a precompact domain
with smooth boundary. Hence, as a direct consequence of theorem Main Theorem
and the Hölder’s inequality we can state
###### Corollary 2.10.
Let $\varphi:P^{m}\to N^{n}$ be an immersion into a $n-$dimensional ambient
manifold $N$ which possesses a pole and its radial sectional curvatures
$K_{N}$ at any point $p\in N$ are bounded by above by the radial curvatures
$K_{w}$ of a model space $M_{w}^{n}$
$K_{N}\left(p\right)\leq
K_{w}\left(r\left(p\right)\right)=-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\quad.$
Suppose moreover, that $w^{\prime}>0$ . Then for any precompact domain
$\Omega$ with smooth boundary $\partial\Omega$
(2.21)
$\mathcal{I}^{w}_{k}(\Omega)\geq\frac{m}{\sup_{\Omega}|\tau_{w}|}\mu_{w}(\Omega)^{1/k}-\left(\int_{\Omega}|H|^{k}d\mu_{w}\right)^{1/k}\quad.$
#### 2.2.2. Immersions into a geodesic ball. Isoperimetric profile, total
mean curvature and Parabolicity
Applying corollary 2.10 to the special setting of an immersion into a geodesic
ball in a Cartan-Hadamard ambient manifold we obtain the following corollary
for the isoperimetric profile.
###### Corollary 2.11.
Let $\varphi:P^{m}\to B_{R}^{N}(o)$ be an complete immersion into a geodesic
ball $B_{R}^{N}(o)$ of radius $R$ centered in $o\in N$ in a Cartan-Hadamart
ambient manifold $N$ with sectional curvatures $K_{N}\leq 0$ bounded from
above. Suppose that the submanifold has finite $k$-norm of the mean curvature,
namely, $\int_{P}|H|^{k}dV<\infty$, for some $k>1$. Then the isoperimetric
profile is bounded from below by
(2.22)
$I_{P}(v)\geq\left(\frac{mv^{1/k}}{R}-\left(\int_{P}|H|^{k}dV\right)^{1/k}\right)v^{\frac{k-1}{k}}\quad,$
where $dV$ is the Riemannian density in $P$.
###### Remark h.
The existence of isometric and complete immersions into a ball in in a Cartan-
Hadamard is out of any doubt. In fact, by the celebrated Nash’s imbedding
theorem [Nas56], any Riemannian $n$-manifold with a $C^{\infty}$ positive
metric has a $C^{\infty}$ isometric imbedding in
$\frac{1}{2}(n+1)(3n+1)$-dimensional Euclidean space, and in fact, in any
small portion of this space (as for instance a geodesic ball of this space).
The existence of immersions with finite integral of the norm of the mean
curvature is also well known (at least in the limit case $H=0$). In [MM05,
MM06] F. Martín and S. Morales in -according to S.T Yau- highly non-trivial
refinement of a Nadirashvili’s method [Nad96] constructed a complete minimal
immersion of a $2$-dimensional disc into every open convex set of
$\mathbb{R}^{3}$.
###### Remark i.
From corollary 2.10 applying as a warping function $w(r)=r$ we can deduce that
if we have a submanifold $P$ immersed in a ball $B_{R}^{N}(o)$ of radius $R$
centered at $o\in N$ in a Cartan-Hadamard ambient manifold $N$ with norm of
the mean curvature vector $|H|$ bounded from above by
(2.23) $|H|\leq\frac{m\,\epsilon}{R}\quad,$
for $0<\epsilon<1$, the isoperimetric quotients are bounded from below by
(2.24) $I_{k}(\Omega)\geq\frac{m(1-\epsilon)}{R}\text{V}(\Omega)^{1/k}\quad,$
for any $k>1$. But that implies
(2.25)
$\frac{\text{A}(\partial\Omega)}{\text{V}(\Omega)}\geq\frac{m(1-\epsilon)}{R}\quad,$
and therefore $P$ has positive Cheeger constant, positive fundamental tone and
$P$ is Hyperbolic (see [Gri99] for the relation of positive fundamental tone
and non-parabolicity).
Hence, the mean curvature of an isometric immersion into a geodesic ball of a
Cartan-Hadamard manifold is closely related to the conformal type problem.
The geodesic balls are the isoperimetric domains in the Euclidean space.
Therefore, in the $n$-dimensional Euclidean space, the isoperimetric profile
is given by
(2.26) $I_{\mathbb{R}^{n}}(v)=C_{n}v^{\frac{n-1}{n}}\quad.$
Applying corollary 2.11, proving by contradiction, we can easily state that
for any immersion of $\mathbb{R}^{n}$ into a ball $B_{R}^{\mathbb{R}^{n+k}}$
of $\mathbb{R}^{n+k}$, $\int_{\mathbb{R}^{n}}|H|^{n}dV=\infty$. In general
###### Corollary 2.12.
Let $\varphi:P^{m}\to B_{R}^{N}(o)$ be an complete immersion into a geodesic
ball $B_{R}^{N}(o)$ of radius $R$ centered in $o\in N$ in a Cartan-Hadamart
ambient manifold $N$ with sectional curvatures $K_{N}\leq 0$ bounded from
above.
Suppose that for some $k>1$ there exist a constant $C_{k}$ such that the
isoperimetric profile is bounded from above by
(2.27) $I_{P}(v)\leq C_{k}v^{\frac{k-1}{k}}\quad.$
Then, either
(2.28) $\int_{P}|H|^{k}dV=\infty\quad.$
or
(2.29)
$\text{V}(P)\leq\left(\frac{C_{k}+\left(\int_{P}|H|^{k}d\text{V}\right)^{\frac{1}{k}}}{m}\right)^{k}R^{k}\quad.$
If moreover the submanifold has the non-shrinking property (see definition
below) we can state further geometric and analytic properties
###### Definition 2.13 (Non-shrinking property).
A manifold $M$ of infinite volume has the _non-shrinking property_ if for any
$v>0$ there exist $R_{v}$ such that
(2.30) $\inf_{x\in M}\text{V}(B_{R_{v}}^{M}(x))\geq v\quad.$
Let us emphasize here, that there exist well known manifolds with that
property
###### Example 2.14 (Non-shrinking manifolds).
The most elemental example of a manifold with non-shrinking property is
$\mathbb{R}^{n}$. Since given a point $x\in\mathbb{R}^{n}$ the volume
$V(B_{R}(x))$ of a geodesic ball of radius $R$ centered at $x$ is an
increasing function of $R$, for any $v>0$ one can easily found $R_{v}$ such
that $V(B_{R_{v}}(x))>v$, and by the homogeneity of $\mathbb{R}^{n}$ we obtain
the non-shrinking property. The same is true for the hyperbolic space
$\mathbb{H}^{n}$, and in general for any Cartan-Hadamard manifold.
Using that non-shrinking property we can state that
###### Corollary 2.15.
Under the assumptions of corollary 2.11, suppose moreover that $P$ has non-
shrinking property. Then
1. (1)
For any point $x\in P$ the geodesic balls of $P$, $B^{P}_{R}(x)$, of radius
$R$ centered at $x$ satisfies
(2.31) $\liminf_{R\to\infty}R^{-k}\text{V}(B^{P}_{R}(x))>0,$
2. (2)
If moreover $k>2$, $P$ possesses a positive Green’s function (or equivalently,
it has transient Brownian motion).
As a reverse of the above corollary we can state
###### Corollary 2.16.
Let $\varphi:P^{m}\to B^{N}_{R}(0)$ be an isometric immersion of a parabolic
submanifold $P$ into the ball $B_{R}^{N}(o)$ of radius $R$ centered at $o\in
N$ in a Cartan-Hadamard manifold $N$. Then:
1. (1)
or, for any $k>2$
$\int_{P}|H|^{k}dV=\infty\quad.$
2. (2)
Or, $P$ has not the non-shrinking property.
By the above corollary any immersion of a parabolic manifold with non-positive
sectional curvature into a geodesic ball of a Cartan-Hadamard manifold has
infinite norm of its mean curvature vector.
#### 2.2.3. The volume of complete non-compact submanifolds into a Cartan-
Hadamard ambient space
In [CMV12] unifying results from [CL98, DCWX10, Fre96, FX10] is proved that
given an isometric immersion $\varphi:P\to N$ of a complete non-compact
manifold $P$ in a manifold $N$ with bounded geometry (i.e., $N$ has sectional
curvature bounded from above and injectivity radius bounded from below by a
positive constant), if any end $E$ of $P$ has finite $L^{p}$-norm of the mean
curvature vector of $\varphi$, $\|H\|_{L^{p}(E)}<\infty$, for some $m\leq
p\leq\infty$ then $E$ must have infinite volume.
Note that $p\geq m$ is not a removable condition. Indeed in Example 4.3 of
[CMV12] is shown a complete non-compact hypersurface $P^{m}$ in
$\mathbb{R}^{m+1}$ , with $m\geq 3$, of finite volume and mean curvature
vector with finite $L^{p}$-norm, for any $0\leq p<m-1$.
In the particular case of a $m$-dimensional submanifold in a Cartan-Hadamard
ambient manifold with sectional curvatures bounded from above by a negative
constant $b$, for any $p\geq 2$, and any dimension $m$ of the submanifold , we
can obtain lower bounds for the $w_{b}$-weighted volume of the submanifold in
terms of the $w_{b}$-weighted $L^{p}$-norm of the norm of the mean curvature
vector, being $w_{b}$ the warping function given in remark d, i.e.,
$w_{b}=\frac{1}{\sqrt{-b}}\sinh(\sqrt{-b}\,r)\quad.$
In such a setting we can state
###### Corollary 2.17.
Let $\varphi:P^{m}\to N$ be an immersion into a Cartan-Hadamard manifold $N$
with sectional curvatures $K_{N}$ bounded from above by a negative constant
$K_{N}\leq b<0$. Then for any $p\geq 2$,
1. (1)
either
$\mu_{w_{b}}(P)^{\frac{1}{p}}\leq\frac{\|H\|_{L^{p}_{w_{b}}(P)}}{m\sqrt{-b}}\quad,$
2. (2)
or
$\mu_{w_{b}}(P)=\infty\quad.$
where $\mu_{w_{b}}(P)$ is the $w_{b}$-weighted volume of $P$, and
$\|H\|_{L^{p}_{w_{b}}(P)}$ is the $w_{b}$-weighted $L^{p}$-norm of the norm of
the mean curvature vector, namely
(2.32)
$\|H\|_{L^{p}_{w_{b}}(P)}=\left(\int_{P}|H|^{p}d\mu_{w_{b}}\right)^{\frac{1}{p}}\quad.$
#### 2.2.4. Isoperimetric inequalities on minimal submanifolds
Given a simple close curve $C$ in the flat plane, bounding a domain $D$.
Denoting by $L$ and $A$ the length of $C$ and the area of $D$ respectively,
the classical isoperimetric inequality states that
$4\pi A\leq L^{2}\quad.$
For minimal submanifolds $P$ with smooth boundary $\partial P$ lying on a
geodesic ball we can state a similar inequality using the $w$-weighted volume
###### Corollary 2.18.
Let $P^{m}$ be a $m-$dimensional compact manifold with smooth boundary
$\partial P$. Let $\varphi:P^{m}\to N^{n}$ a minimal immersion into a
$n-$dimensional ambient manifold $N$ which possesses a pole $o\in N$ and its
radial sectional curvatures $K_{N}$ at any point $p\in N$ are bounded by above
by the radial curvatures $K_{w}$ of a model space $M_{w}^{n}$
$K_{N}\left(p\right)\leq
K_{M_{w}^{n}}\left(r\left(p\right)\right)=-\frac{w^{\prime\prime}}{w}\left(r\left(p\right)\right)\quad.$
Suppose moreover, that $w^{\prime}\geq c>0$. $\varphi^{-1}(o)\in P$ and
$\varphi(\partial P)$ lies in a geodesic sphere centered at the pole $o$. Then
(2.33) $\displaystyle cm^{m}V_{m}\,\mu_{w}(P)^{m-1}\leq A(\partial
P)^{m}\quad,$
where $\mu_{w}(P)$ is the $w$-weighted volume of $P$, $A(\partial P)$ is the
Riemannian area of $\partial P$ and $V_{m}$ is the volume of the unit ball in
$\mathbb{R}^{m}$.
###### Remark j.
Applying the above corollary but using $w=w_{b}$ given in remark d we get, as
a particular case, the isoperimetric inequality of theorem 4 of [CG92]. See
also [Pal99] for an other sort of isoperimetric inequalities on minimal
submanifolds properly immersed into a Cartan-Hadamard ambient manifold.
## 3\. Preliminaries
### 3.1. Manifold with a pole and extrinsic distance function
We assume throughout the most part of the paper that
$\varphi:P^{m}\longrightarrow N^{n}$ is an isometric immersion of a complete
non-compact Riemannian $m$-manifold $P^{m}$ into a complete Riemannian
manifold $N^{n}$ with a pole $o\in N$. Recall that a pole is a point $o$ such
that the exponential map
$\exp_{o}\colon T_{o}N^{n}\to N^{n}$
is a diffeomorphism. For every $x\in N^{n}-\\{o\\}$ we define
$r(x)=r_{o}(x)=\operatorname{dist}_{N}(o,x)$, and this distance is realized by
the length of a unique geodesic from $o$ to $x$, which is the radial geodesic
from $o$. We also denote by $r|_{P}$ or by $r$ the composition
$r\circ\varphi:P\to\mathbb{R}_{+}\cup\\{0\\}$. This composition is called the
extrinsic distance function from $o$ in $P^{m}$. The gradients of $r$ in $N$
and $r|_{P}$ in $P$ are denoted by $\nabla^{N}r$ and $\nabla^{P}r$,
respectively. Then we have the following basic relation, by virtue of the
identification, given any point $x\in P$, between the tangent vector fields
$X\in T_{x}P$ and $\varphi_{*_{x}}(X)\in T_{\varphi(x)}N$
(3.1) $\nabla^{N}r=\nabla^{P}r+(\nabla^{N}r)^{\bot},$
where $(\nabla^{N}r)^{\bot}(\varphi(x))=\nabla^{\bot}r(\varphi(x))$ is
perpendicular to $T_{x}P$ for all $x\in P$.
Since the manifold with a pole has a well defined radial vector field, we cab
define the radial sectional curvatures.
###### Definition 3.1.
Let $o$ be a point in a Riemannian manifold $N$ and let $x\in N-\\{o\\}$. The
sectional curvature $K_{N}(\sigma_{x})$ of the two-plane $\sigma_{x}\in
T_{x}N$ is then called a $o$-radial sectional curvature of $N$ at $x$ if
$\sigma_{x}$ contains the tangent vector to a minimal geodesic from $o$ to
$x$. We denote these curvatures by $K_{o,N}(\sigma_{x})$.
### 3.2. $w-$model spaces
The model spaces has two different roles in this paper, the first of them is
the role as an ambient manifold and the second one is as a controller of the
curvature restrictions. The sectional curvatures of a model space can be
explicitly obtained using the warped function $w$.
###### Proposition 3.2 (See [GW79, Gri99, O’N83]).
Let $M_{w}^{m}$ be a $w-$model with center point $o_{w}$. Then the
$o_{w}$-radial sectional curvatures of $M_{w}^{m}$ at every $x\in\pi^{-1}(r)$
(for $\,r\,>\,0\,$) are all identical and determined by the radial function
$\,K_{w}(r)\,$ defined as follows:
(3.2)
$K_{p_{w},M_{w}}(\sigma_{x})\,=\,K_{w}(r)\,=\,-\frac{w^{\prime\prime}(r)}{w(r)}\quad.$
And the sectional curvatures $K(\Pi_{S^{w}_{r}})$ of the $2-$planes
$\Pi_{S^{w}_{r}}$ tangents to $S_{r}^{w}=\pi^{-1}(r)$ are equal to
(3.3)
$K(\Pi_{S^{w}_{r}})=\frac{1-\left(w^{\prime}\left(r\right)\right)^{2}}{w(r)}\quad.$
We can also explicitly calculate the mean curvature of the geodesic spheres.
###### Proposition 3.3 (See [O’N83] p. 206).
Let $M_{w}^{n}$ be a $w-$model with warping function $w(r)$ and center
$o_{w}$. The distance sphere of radius $r$ and center $o_{w}$ in $M_{w}^{n}$,
denoted as $S^{w}_{r}$, is the fiber $\pi^{-1}(r)$. This distance sphere has
the following constant mean curvature vector in $M_{w}^{n}$
(3.4)
$H_{{\pi^{-1}}(r)}=-n\,\eta_{w}(r)\,\nabla^{M}\pi=-n\,\eta_{w}(r)\,\nabla^{M}r\quad,$
where the mean curvature function $\eta_{w}(r)$ is defined by
(3.5) $\eta_{w}(r)=\frac{w^{\prime}(r)}{w(r)}=\frac{d}{dr}\ln(w(r))\quad.$
In particular we have for the constant curvature space forms
$\mathbb{K}^{m}(b)$:
(3.6) $\eta_{w_{b}}(r)=\begin{cases}\sqrt{b}\cot(\sqrt{b}\,r)&\text{if
$b>0$}\\\ \phantom{\sqrt{b}}1/r&\text{if $b=0$}\\\
\sqrt{-b}\coth(\sqrt{-b}\,r)&\text{if $b<0$}\quad.\end{cases}$
The area of the geodesic sphere $S^{w}_{R}(o_{w})$ of radius $R$ centered at
$o_{w}$ is completely determined via $w$ by the volume of the fiber
(3.7) $A(S^{w}_{R}(o_{w}))=\omega_{n-1}w^{n-1}(R),$
And the volume of the corresponding ball $B_{R}^{w}(o_{w})$, for which the
fiber is the boundary
(3.8) $V(B^{w}_{R}(o_{w}))=\omega_{n-1}\int_{0}^{R}w^{n-1}(t)dt,$
being $\omega_{n-1}$ in (3.7) and (3.8) the volume of the standard sphere
$S^{n-1}_{1}$.
### 3.3. Hessian and Laplacian comparison
The 2.nd order analysis of the restricted distance function $r_{|_{P}}$
defined on manifolds with a pole is governed by the Hessian comparison (see
[GW79, Theorem A]).
The Hessian of a restricted function in a submanifold and the Hessian of the
function in the ambient space are related by the following proposition
###### Proposition 3.4.
Given an isometric immersion $\varphi:P^{m}\to N^{n}$, and given a smooth
function $f:N\to\mathbb{R}$, then:
(3.9)
$\operatorname{Hess}^{P}(f\circ\varphi)(X,Y)=\operatorname{Hess}^{N}f(X,Y)+\langle
B^{P}(X,Y),\nabla^{N}f\rangle.$
In the case of radial functions of a model space
###### Proposition 3.5.
let $M_{w}^{m}$ denote a $w-$model with center $o_{w}$. Let
$r:M_{w}^{n}\to\mathbb{R}^{+}$ denote the distance function to the center
$o_{w}$. Then for any smooth function $F:\mathbb{R}\to\mathbb{R}$,
(3.10) $\displaystyle\operatorname{Hess}^{M_{w}^{n}}F\circ r(X,Y)=$
$\displaystyle\left(F^{\prime\prime}\circ r-\left(F^{\prime}\circ
r\right)\left(\eta_{w}\circ r\right)\right)\langle X,\nabla r\rangle\langle
Y,\nabla r\rangle$ $\displaystyle+\left(F^{\prime}\circ
r\right)\left(\eta_{w}\circ r\right)\left(\langle X,Y\rangle\right).$
Now, we can state a comparison theorem when one of the spaces is a model space
$M^{m}_{w}$ using [GW79]:
###### Theorem 3.6 (See [GW79], Theorem A).
Let $N^{n}$ be a manifold with a pole $p$, let $M_{w}^{m}$ denote a $w-$model
with center $p_{w}$. Suppose that $m\leq n$ and that every $p$-radial
sectional curvature at $x\in N-\\{p\\}$ is bounded from above (or below) by
the $p_{w}$-radial sectional curvatures in $M_{w}^{m}$ as follows:
(3.11)
$K_{p,N}(\sigma_{x})\leq-\frac{w^{\prime\prime}(r)}{w(r)}\quad\left(\text{respectively
}K_{p,N}(\sigma_{x})\geq-\frac{w^{\prime\prime}(r)}{w(r)}\right)$
for every radial two-plane $\sigma_{x}\in T_{x}N$ at distance
$r=r(x)=\operatorname{dist}_{N}(p,x)$ from $p$ in $N$. Then the Hessian of the
distance function in $N$ satisfies
(3.12) $\displaystyle\operatorname{Hess}^{N}(r(x))(X,X)$
$\displaystyle\geq(\leq)\operatorname{Hess}^{M_{w}^{m}}(r(y))(Y,Y)$
$\displaystyle=\eta_{w}(r)\left(1-\langle\nabla^{M}r(y),Y\rangle_{M}^{2}\right)$
$\displaystyle=\eta_{w}(r)\left(1-\langle\nabla^{N}r(x),X\rangle_{N}^{2}\right)$
for every unit vector $X$ in $T_{x}N$ and for every unit vector $Y$ in
$T_{y}M$ with $\,r(y)=r(x)=r\,$ and
$\,\langle\nabla^{M}r(y),Y\rangle_{M}=\langle\nabla^{N}r(x),X\rangle_{N}\,$.
Hence, from proposition 3.4, proposition 3.5, and theorem 3.6 after few
calculations one obtains
###### Corollary 3.7.
Given an isometric immersion $\varphi:P^{m}\to N^{n}$. Suppose again that the
assumptions of Theorem 3.6 are satisfied. Then, for every smooth function
$f(r)$ with $f^{\prime}(r)\geq 0\,\,\textrm{for all}\,\,\,r$ :
(3.13) $\displaystyle\Delta^{P}(f\circ r)\,\geq(\leq)$
$\displaystyle\left(\,f^{\prime\prime}(r)-f^{\prime}(r)\eta_{w}(r)\,\right)\|\nabla^{P}r\|^{2}$
$\displaystyle+mf^{\prime}(r)\left(\,\eta_{w}(r)+\frac{1}{m}\langle\,\nabla^{N}r,\,H_{P}\,\rangle\,\right)\quad,$
where $H_{P}$ denotes the mean curvature vector of $P$ in $N$.
## 4\. Proof of the Jellett-Minkowski’s generalized formula (main theorem)
In order to prove the Jellett-Minkowski’s generalized formula we only have to
apply the divergence theorem to an appropriate function. Let us define the
following function $F:\mathbb{R}^{+}\to\mathbb{R}^{+}$ given by
(4.1) $F(t):=\int_{0}^{t}w(s)ds\quad.$
Using corollary 3.7
(4.2) $\displaystyle\Delta^{P}F$
$\displaystyle\geq(\leq)mw(r)\left(\eta_{w}(r)+\frac{1}{m}\langle\nabla^{N}r,H\rangle\right)$
$\displaystyle=$
$\displaystyle\left(m+\langle\tau_{w},H\rangle\right)w^{\prime}(r)\quad.$
Applying the divergence theorem to the domain $\Omega\subset P$
(4.3) $\displaystyle\int_{\partial\Omega}\langle\nabla^{P}F,\nu\rangle
dA\geq(\leq)m\mu_{w}(\Omega)+\int_{\Omega}\langle\tau_{w},H\rangle
d\mu_{w}\quad.$
Therefore
(4.4) $\displaystyle\int_{\partial\Omega}\langle\tau_{w},\nu\rangle
d\sigma_{w}\geq(\leq)m\mu_{w}(\Omega)+\int_{\Omega}\langle\tau_{w},H\rangle
d\mu_{w}\quad.$
And the theorem follows.
## 5\. Proof of corollary 2.2 and corollary 2.3
Since $\Sigma=\partial\Omega$ ($\partial\Sigma=\emptyset$), and $\Omega$ is
totally geodesic submanifold by the generalized Jellett-Minkowsi formula
(5.1) $\displaystyle n\mu_{w}(\Omega)$
$\displaystyle=\int_{\Sigma}\langle\tau_{w},\nu\rangle d\sigma_{w}\quad,$
$\displaystyle(n-1)\sigma_{w}(\Sigma)$
$\displaystyle=-\int_{\Sigma}\langle\tau_{w},H\rangle d\sigma_{w}\quad.$
From the first line of the above equations
(5.2)
$n\mu_{w}(\Omega)\leq\max_{\Sigma}|\tau_{w}|\int_{\Sigma}d\sigma_{w}=\max_{\Sigma}|\tau_{w}|\sigma_{w}(\Sigma)\quad,$
And the corollary 2.2 follows. On the other hand, by the second equality of
(5.1) if $\Sigma$ is a constant mean curvature hypersurface
(5.3) $\displaystyle(n-1)\sigma_{w}(\Sigma)$
$\displaystyle=h\int_{\Sigma}\langle\tau_{w},\nu\rangle
d\sigma_{w}=h\,n\,\mu_{w}(\Omega)\quad.$
And corollary 2.3 is proven.
## 6\. Proof of the theorem 2.5
This proof follows the proof done in [Bre13]. First of all, we need the
following Heintze-Karcher [HK78] inequality
###### Theorem 6.1 (Heintze-Karcher inequality).
Let $M_{w}^{n}$ be a $w-$model space with center $o_{w}$, positive warping
function $w$ and positive derivative $w^{\prime}$ on $(0,\Lambda)$ being
$\Lambda$ the radius of the model space. Suppose moreover that the warping
function satisfies the following inequalities:
(6.1)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w}\geq-\frac{w^{\prime\prime\prime}}{w^{\prime}}\quad,$
then in every closed, embedded, orientable and convex mean curvature
hypersurface $\Sigma$ bounding a domain $\Omega$ the following inequality
holds
(6.2) $\left(n-1\right)\int_{\Sigma}\frac{1}{h}d\sigma_{w}\geq
n\mu_{w}(\Omega)\quad.$
With equality in 6.2 if $\Sigma$ is umbilic, and in the particular case when
$\frac{w^{\prime\prime}}{w}\neq\frac{w^{\prime\prime\prime}}{w^{\prime}}$,
equality in 6.2 implies that $\Sigma$ is a sphere centered at $o_{w}$.
From corollary 2.3 it is clear that if the hypersurface is a constant mean
hypersurface bounding a domain, the surface is mean convex and attains
equality in inequality (6.2). Hence, the only thing to do in order to prove
theorem 2.5 is to prove the above theorem.
###### Remark k.
The proof of the above theorem follows from the proof of Theorem 3.5 of
[Bre13]. The only new piece is the condition
$\frac{w^{\prime\prime}}{w}\neq\frac{w^{\prime\prime\prime}}{w^{\prime}}$ and
its rigidity consequences arising from remark l. For completeness in order to
attain remark l we have to repeat part of the proof done in section 3 of
[Bre13].
###### Proof.
Since $w^{\prime}>0$ we can use the following conformally modified metric
$g_{c}=\frac{1}{w^{\prime}}g_{M_{w}^{n}}$ being $g_{M_{w}^{n}}$ the metric
tensor in $M_{w}^{n}$. For each point $p\in\overline{\Omega}$, we denote by
$u(p)=d_{g_{c}}(p,\Sigma)$ the distance to $p$ from $\Sigma$ with respect to
the metric $g_{c}$, and we denote by
$\Phi:\Sigma\times[0,\infty)\to\overline{\Omega}$ the normal exponential map
with respect to $g_{c}$. Namely, for each point $x\in\Sigma$, the curve
$t\to\Phi(x,t)$ is a geodesic with respect to $g_{c}$, and we have
(6.3) $\Phi(x,0)=x,\quad\left.\frac{\partial}{\partial
t}\Phi(x,t)\right|_{t=0}=-w^{\prime}(r(x))\nu(x).$
Let us define
(6.4) $\displaystyle A$
$\displaystyle:=\left\\{(x,t\in\Sigma\times[0,\infty)|\,u(\Phi(x,t))=t\right\\}$
$\displaystyle A^{*}$
$\displaystyle:=\left\\{(x,t\in\Sigma\times[0,\infty)|\,(x,t+\delta)\in
A\text{ for some }\delta>0\right\\}$ $\displaystyle\Sigma_{t}^{*}$
$\displaystyle=\Phi(A^{*}\cap(\Sigma\times\\{t\\}).$
We also denote by $h$ and $B^{\Sigma_{t}^{*}}$ the mean curvature and the
second fundamental form of $\Sigma^{*}_{t}$ with respect to the metric
$g_{M_{w}^{n}}$. Hence,
###### Proposition 6.2 (See proposition 3.2 of [Bre13]).
The mean curvature of $\Sigma_{t}^{*}$ is positive and satisfies the
differential inequality
$\frac{\partial}{\partial
t}\left(\frac{w^{\prime}}{h}\right)\leq-\frac{1}{n-1}\left(w^{\prime}\right)^{2}.$
###### Proof.
Since $\nu=-\frac{\nabla u}{|\nabla u|}$ is the outward-pointing unit normal
vector to $\Sigma_{t}^{*}$ with respect to the metric $g_{M_{w}^{n}}$, by the
variation formulas, the mean curvature of $\Sigma_{t}^{*}$ satisfies the
following equation
(6.5) $\frac{\partial}{\partial
t}h=\Delta_{\Sigma^{*}_{t}}w^{\prime}+\left(\text{Ricc}^{M_{w}^{n}}(\nu,\nu)+\|B^{\Sigma^{*}_{t}}\|^{2}\right)w^{\prime}.$
Using proposition 3.4 for any orthonormal basis $\\{e_{i}\\}_{i=1}^{n-1}$ of
$\Sigma_{t}^{*}$
(6.6) $\displaystyle\Delta_{\Sigma_{t}^{*}}w^{\prime}$
$\displaystyle=\sum_{i=1}^{n-1}\operatorname{Hess}^{\Sigma_{t}^{*}}w^{\prime}(e_{i},e_{i})$
$\displaystyle=\sum_{i=1}^{n-1}\operatorname{Hess}^{M_{w}^{n}}w^{\prime}(e_{i},e_{i})+\sum_{i=1}^{n-1}\langle
B^{\Sigma_{t}^{*}}(e_{i},e_{i}),\nabla^{M_{w}^{n}}w^{\prime}\rangle$
$\displaystyle=\Delta^{M_{w}^{n}}w^{\prime}-\operatorname{Hess}^{M_{w}^{n}}w^{\prime}(\nu,\nu)+\langle
H,\nabla^{M_{w}^{n}}w^{\prime}\rangle,$
applying proposition 3.5
(6.7) $\displaystyle\Delta_{\Sigma_{t}^{*}}w^{\prime}=$ $\displaystyle
w^{\prime\prime\prime}+(n-1)w^{\prime\prime}\eta_{w}-\operatorname{Hess}^{M_{w}^{n}}w^{\prime}(\nu,\nu)+\langle
H,\nabla^{M_{w}^{n}}w^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
w^{\prime}\left[\left(\frac{w^{\prime\prime\prime}}{w^{\prime}}-\frac{w^{\prime\prime}}{w}\right)\left(1-\langle\nabla^{M_{w}^{n}}r,\nu\rangle^{2}\right)+\frac{w^{\prime\prime}}{w}\left(n-1\right)\right]$
$\displaystyle+\langle H,\nabla^{M_{w}^{n}}w^{\prime}\rangle.$
Taking into account that $1-\langle\nabla^{M_{w}^{n}}r,\nu\rangle^{2}\geq 0$
and $\frac{w^{\prime\prime\prime}}{w^{\prime}}\geq\frac{w^{\prime\prime}}{w}$
(6.8) $\displaystyle\Delta_{\Sigma_{t}^{*}}w^{\prime}\geq$ $\displaystyle
w^{\prime}\frac{w^{\prime\prime}}{w}(n-1)+\langle
H,\nabla^{M_{w}^{n}}w^{\prime}\rangle$ $\displaystyle=$ $\displaystyle
w^{\prime}\frac{w^{\prime\prime}}{w}(n-1)-h\langle\nu,\nabla^{M_{w}^{n}}w^{\prime}\rangle.$
###### Remark l.
If we have equality in inequality (6.8) then
(6.9)
$\left(\frac{w^{\prime\prime\prime}}{w^{\prime}}-\frac{w^{\prime\prime}}{w}\right)\left(1-\langle\nabla^{M_{w}^{n}}r,\nu\rangle^{2}\right)=0.$
In the particular case when
$\frac{w^{\prime\prime\prime}}{w^{\prime}}\neq\frac{w^{\prime\prime}}{w}$ we
would get
(6.10) $\langle\nabla^{M_{w}^{n}}r,\nu\rangle^{2}=1$
In order to make use of equality (6.5) we need estimate
$\text{Ricc}^{M_{w}^{n}}(\nu,\nu)$ and $\|B^{\Sigma_{t}^{+}}\|^{2}$. Those
estimates are taken care of in the next two lemmas
###### Lemma 6.3 (From Proposition 3.2).
Let $M_{w}^{n}$ be a $w-$model space. Suppose that
(6.11)
$\frac{1}{w^{2}}-\left(\frac{w^{\prime}}{w}\right)^{2}\geq-\frac{w^{\prime\prime}}{w},$
then for any unit vector $\nu$,
(6.12) $\text{Ricc}^{M_{w}^{n}}(\nu,\nu)\geq-(n-1)\frac{w^{\prime\prime}}{w}.$
###### Lemma 6.4.
For any hypersurface $S$,
(6.13) $h^{2}\leq(n-1)\|B^{S}\|^{2}$
with equality if and only if $S$ is umbilic.
Applying inequalities (6.8),(6.12) and (6.13) to equality (6.5) we get
(6.14) $\frac{\partial h}{\partial
t}\geq-h\langle\nu,\nabla^{M_{w}^{n}}w^{\prime}\rangle+w^{\prime}\frac{h^{2}}{n-1}$
Therefore
(6.15) $\displaystyle\frac{\partial}{\partial
t}\left(\frac{h}{w^{\prime}}\right)=$
$\displaystyle\frac{1}{w^{\prime}}\frac{\partial h}{\partial
t}-\frac{h}{\left(w^{\prime}\right)^{2}}\frac{\partial w^{\prime}}{\partial
t}$ $\displaystyle\geq$
$\displaystyle\frac{-h}{w^{\prime}}\langle\nabla^{M_{w}^{n}}w^{\prime},\nu\rangle+\frac{h^{2}}{n-1}-\frac{h}{\left(w^{\prime}\right)^{2}}\frac{\partial
w^{\prime}}{\partial t}.$
By the evolution equation (6.3)
(6.16) $\frac{\partial w^{\prime}}{\partial
t}=-w^{\prime}\nu(w^{\prime})=-w^{\prime}\langle\nabla^{M_{w}^{n}}w^{\prime},\nu\rangle.$
Hence,
(6.17) $\displaystyle\frac{\partial}{\partial
t}\left(\frac{h}{w^{\prime}}\right)\geq$ $\displaystyle\frac{h^{2}}{n-1}.$
And the proposition follows. ∎
Following the proof of [Bre13, Theorem 3.5] consider the quantity
(6.18) $Q(t)=(n-1)\int_{\Sigma^{*}_{t}}\frac{w^{\prime}}{h}dA.$
Therefore we can use a similar proposition to proposition 3.4 in [Bre13]
###### Proposition 6.5.
(6.19) $Q(0)-Q(\tau)\geq n\int_{u\leq\tau}w^{\prime}dV.$
Letting $\tau\to\infty$ in the above proposition,
(6.20) $\displaystyle(n-1)\int_{\Sigma}\frac{1}{h}d\sigma_{w}$
$\displaystyle=(n-1)\int_{\Sigma^{*}_{t}}\frac{w^{\prime}}{h}dA$
$\displaystyle=Q(0)\geq
n\int_{u<\infty}w^{\prime}dV=n\int_{\Omega}w^{\prime}dV$
$\displaystyle=n\mu_{w}(\Omega).$
And the inequality (6.2) follows.
\begin{picture}(2640.0,2638.0)(9931.0,-1830.0)\end{picture}
Figure 1. By equality (6.10) $\Sigma$ is a sphere or a finite union of
spheres, hence $\Omega$ is either a geodesic ball centered at $o_{w}$ ( A ),
or a geodesic annulus centered at $o_{w}$ ( B ) or a finite union of annuli
(B+C ) or A geodesic ball centered at $o_{w}$ with a finite union of geodesic
annuli centered at $o_{w}$ (A + B + C ).
$\nu$$\nu$
Figure 2. The product $h=-\langle\nu,H\rangle$ is not constant in the boundary
of an annulus centered at the center of the model.
Observe that equality in (6.2) implies equality in (6.13) and hence umbilic
submanifold. If moreover we assume
$\frac{w^{\prime\prime\prime}}{w^{\prime}}\neq\frac{w^{\prime\prime}}{w}$
equality in (6.2) implies equality (6.10). If $\Sigma$ has $k>0$ connected
components $\Sigma_{1},\cdots,\Sigma_{k}$, each one is a sphere centered at
$o_{w}$. Hence, $\Omega$ is of one of the following types (see figure 1):
1. (1)
A geodesic ball centered at $o_{w}$.
2. (2)
A finite union of geodesic annuli centered at $o_{w}$.
3. (3)
A geodesic ball centered at $o_{w}$ with a finite union of geodesic annuli
centered at $o_{w}$.
But we can prove that $\Omega$ does not contain annuli. Because, if $\Omega$
contains an annulus since $\eta_{w}>0$, $H$ is always pointing to $o_{w}$ (see
proposition 3.3) and the unit normal $\nu$ is always pointing outward to
$\Omega$, therefore the product $h=-\langle\nu,H\rangle$ has to be not
constant (see figure 2) in contradiction to the assumption of constant $h$.
Thus finally, $\Omega$ is a ball and $\Sigma$ is a sphere centered at $o_{w}$.
∎
## 7\. Proof of corollary 2.8
By the Cheeger inequality (see [Cha84, theorem 3, chapter IV])
(7.1)
$\lambda_{1}(\Omega)\geq\frac{1}{4}\left(\inf_{\mathcal{O}\subset\Omega}\frac{\text{A}(\partial\mathcal{O})}{\text{V}(\mathcal{O})}\right)^{2},$
where $\mathcal{O}$ ranges on the open subdomains of $\Omega$. By the
isoperimetric inequality (2.16)
(7.2) $\displaystyle\lambda_{1}(\Omega)$
$\displaystyle\geq\frac{1}{4}\left(\inf_{\mathcal{O}\subset\Omega}\frac{\omega_{n-1}w(\text{Rad}(\text{V}(\mathcal{O}))^{n-1}}{\text{V}(\mathcal{O})}\right)^{2}$
$\displaystyle=\frac{1}{4}\left(\inf_{\mathcal{O}\subset\Omega}\mathcal{I}(\text{V}(\mathcal{O}))\right)^{2}.$
Taking into account that $\mathcal{I}$ is non-increasing and
$\text{V}(\mathcal{O})\leq\text{V}(\Omega)$, the corollary follows.
## 8\. Proof of corollary 2.10
For any $p>0$, we use the density $d\mu_{w}$ to define the
$L^{p}_{w}(\Omega)$-space, by declaring a measurable function $f$ to be an
element of $L^{p}_{w}(\Omega)$ if the integral
(8.1) $\left(\int_{\Omega}|f|^{p}d\mu_{w}\right)^{1/p}$
is finite. The $L^{p}_{w}(\Omega)$-norm of $f$, $\|f\|_{L^{p}_{w}(\Omega)}$,
is given by the above expression. Hölder’s inequality states that for $p,q>1$
satisfying
(8.2) $1/p+1/q=1\quad,$
one has for $\phi\in\text{L}^{p}_{w}(\Omega)$,
$\psi\in\text{L}^{q}_{w}(\Omega)$ ,
(8.3)
$\int_{\Omega}|\phi\psi|d\mu_{w}\leq\|\phi\|_{L^{p}_{w}(\Omega)}\|\psi\|_{L^{q}_{w}(\Omega)}\quad.$
Now, from the Jellett-Minkowski’s generalized formula we get
(8.4) $\sup_{\Omega}|\tau_{w}|\sigma_{w}(\partial\Omega)\geq
m\mu_{w}(\Omega)-\sup_{\Omega}|\tau_{w}|\int_{\Omega}|H|d\mu_{w}.$
Applying the Hölder inequality to the last integral in the above inequality
(8.5) $\sup_{\Omega}|\tau_{w}|\sigma_{w}(\partial\Omega)\geq
m\mu_{w}(\Omega)-\sup_{\Omega}|\tau_{w}|\left(\int_{\Omega}|H|^{k}d\mu_{w}\right)^{1/k}\mu_{w}(\Omega)^{\frac{k-1}{k}}\quad,$
and the corollary follows.
## 9\. Proof of corollary 2.11
Applying corollary 2.10 to the setting of corollary 2.11 (namely $w(t)=t$,
$d\mu=dV$) we obtain
(9.1)
$\mathcal{I}_{k}(\Omega)\geq\frac{m\text{V}(\Omega)^{1/k}}{R}-\left(\int_{P}|H|^{k}dV\right)^{1/k}$
Taking into account the definition of $\mathcal{I}_{k}$ and the isoperimetric
profile, the corollary follows.
## 10\. Proof of corollary 2.15
Since $P$ has non-shrinking property, there exist $\rho$ such that for any
$o\in P$
(10.1)
$V(B_{\rho}(o))>\left(\frac{R}{m}\right)^{k}\int_{P}|H|^{k}dV+\epsilon\quad,$
for some $\epsilon>0$.
Therefore, by inequality (9.1)
(10.2) $\mathcal{I}_{k,\rho}(P)>0\quad.$
Applying now [CF91, Theorem 5 and inequality (15)] the corollary is proven.
## 11\. Proof of corollary 2.18
The first thing to do in order to prove the corollary 2.18 is to study the
behavior of the volume of the extrinsic balls. Recall that an extrinsic ball
$D_{R}(o)$ centered to the pole $o\in N$ and with radius $R$ is the sublevel
set of the extrinsic distance function. Namely,
(11.1) $D_{R}(o)=\varphi^{-1}\left(B_{R}^{N}\left(o\right)\right)\quad,$
being $B_{R}^{N}$ the geodesic ball of $N$ of radius $R$ centered at the pole
$o\in N$. Note that we can construct the order-preserving bijection
$F:\mathbb{R}^{+}\to\mathbb{R}^{+}\quad t\to F(t)$
given by equation (4.1).
Since $\varphi:P\to N$ is a minimal immersion into a manifold with a pole $N$,
applying equation (4.2) we have
(11.2) $\Delta^{P}F\circ r\geq mw^{\prime}\circ r\quad.$
Taking into account that $w^{\prime}>0$, by the maximum principle there exist
$R_{T}$ such that
(11.3) $P=D_{R_{T}}(o)\quad.$
Now we need the following monotonicity formula
###### Proposition 11.1.
Under the assumptions of corollary 2.18, the function
$f:\mathbb{R}^{+}\to\mathbb{R}^{+}$ given by
(11.4) $f(R):=\frac{\mu_{w}(D_{R})}{w(R)^{m}}\quad,$
is a nondecreasing function of $R$, and
(11.5) $f(R)\geq cV_{m}\quad.$
###### Proof.
Using theorem Main Theorem taking into account that $P$ is minimal and
$\partial D_{R}$ lies in a geodesic sphere of $N$ of radius $R$, we obtain
(11.6) $m\mu_{w}(D_{R})\leq\frac{1}{\eta_{w}(R)}\sigma_{w}(\partial
D_{R})\quad.$
By the coarea formula we get
(11.7) $\sigma_{w}(\partial D_{R})\leq\frac{d}{dR}\mu_{w}(D_{R})\quad.$
Therefore, using inequalities (11.6) and (11.7) together
(11.8)
$\frac{d}{dR}\ln\left(\mu_{w}\left(D_{R}\right)\right)\geq\frac{d}{dR}\ln\left(w\left(R\right)^{m}\right)\quad.$
Hence, we obtain the desired monotonicity formula. Observe also that
(11.9) $\lim_{R\to 0}\frac{\mu_{w}(D_{R})}{w(R)^{m}}\geq\lim_{R\to
0}c\frac{V(D_{R})}{w(R)^{m}}\geq cV_{m}\quad.$
And the proposition follows.∎
On the other hand, from inequality (11.6)
(11.10) $\sigma_{w}(\partial D_{R})\geq m\mu_{w}(D_{R})\eta_{w}(R)\quad.$
But taking into account the definition of the extrinsic ball and $w$-weighted
area, and using the above proposition
(11.11) $\displaystyle A(\partial D_{R})\geq$
$\displaystyle\frac{m}{w(R)}\mu_{w}(D_{R})=m\left(\frac{\mu_{w}(D_{R})}{w(R)^{m}}\right)^{\frac{1}{m}}\mu_{w}(D_{R})^{1-1/m}$
$\displaystyle\geq$ $\displaystyle
m\left(cV_{m}\right)^{\frac{1}{m}}\mu_{w}(D_{R})^{1-1/m}\quad.$
Hence, finally the corollary follows changing $R$ by $R_{T}$ in the above
inequality.
## 12\. Proof of corollary 2.17
Applying inequality (8.5) to the extrinsic ball $D_{R}$ taking into account
that $\sup_{D_{R}}|\tau_{w}|\leq\frac{1}{\sqrt{-b}}$ we get
(12.1) $m\mu_{w}(D_{R})\leq\frac{\sigma_{w}(\partial
D_{R})}{\sqrt{-b}}+\frac{1}{\sqrt{-b}}\left(\int_{D_{R}}|H|^{p}d\mu_{w}\right)^{1/p}\mu_{w}(D_{R})^{\frac{p-1}{p}}\quad,$
by inequality (11.7),
(12.2)
$m\sqrt{-b}-\|H\|_{L_{w}^{p}(P)}\mu_{w}(D_{R})^{\frac{-1}{p}}\leq\frac{d}{dR}\ln\mu_{w}(D_{R})\quad,$
Given $R_{0}>0$, for any $R\geq R_{0}$
(12.3)
$m\sqrt{-b}-\|H\|_{L_{w}^{p}(P)}\mu_{w}(D_{R_{0}})^{\frac{-1}{p}}\leq\frac{d}{dR}\ln\mu_{w}(D_{R})\quad,$
If the submanifold has finite volume, there exist a divergent sequence
$\\{R_{i}\\}_{i=1}^{\infty}$ such that
(12.4)
$m\sqrt{-b}-\|H\|_{L_{w}^{p}(P)}\mu_{w}(D_{R_{0}})^{\frac{-1}{p}}\leq\limsup\frac{d}{dR}\ln\mu_{w}(D_{R_{i}})=0.$
And therefore the corollary follows letting $R_{0}$ tend to infinity.
## Acknowledgments
The author is very grateful to Simon Brendle for his useful discussion about
his paper [Bre13] and to Antonio Cañete for show me the paper [MM09].
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|
arxiv-papers
| 2013-06-11T10:18:33 |
2024-09-04T02:49:46.324094
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Vicent Gimeno",
"submitter": "Vicent Gimeno",
"url": "https://arxiv.org/abs/1306.2478"
}
|
1306.2507
|
# Eigenvalue Spectra of Modular Networks
Tiago P. Peixoto [email protected] Institut für Theoretische Physik,
Universität Bremen, Hochschulring 18, D-28359 Bremen, Germany
###### Abstract
A large variety of dynamical processes that take place on networks can be
expressed in terms of the spectral properties of some linear operator which
reflects how the dynamical rules depend on the network topology. Often such
spectral features are theoretically obtained by considering only local node
properties, such as degree distributions. Many networks, however, possess
large-scale modular structures that can drastically influence their spectral
characteristics, and which are neglected in such simplified descriptions. Here
we obtain in a unified fashion the spectrum of a large family of operators,
including the adjacency, Laplacian and normalized Laplacian matrices, for
networks with generic modular structure, in the limit of large degrees. We
focus on the conditions necessary for the merging of the isolated eigenvalues
with the continuous band of the spectrum, after which the planted modular
structure can no longer be easily detected by spectral methods. This is a
crucial transition point which determines when a modular structure is strong
enough to affect a given dynamical process. We show that this transition
happens in general at different points for the different matrices, and hence
the detectability threshold can vary significantly depending on the operator
chosen. Equivalently, the sensitivity to the modular structure of the
different dynamical processes associated with each matrix will be different,
given the same large-scale structure present in the network. Furthermore, we
show that, with the exception of the Laplacian matrix, the different
transitions coalesce into the same point for the special case where the
modules are homogeneous, but separate otherwise.
###### pacs:
89.75.Hc, 02.70.Hm, 05.10.-a, 64.60.aq
Networks form the substrate of a dominating class of interacting complex
systems, on which various dynamical processes take place. Many of the most
important types of dynamics such as random walks Noh and Rieger (2004);
Samukhin _et al._ (2008), diffusion, synchronization Barahona and Pecora
(2002); Arenas _et al._ (2008); Almendral and Díaz-Guilera (2007) and
epidemic spreading Wang _et al._ (2003); Castellano and Pastor-Satorras
(2010); Goltsev _et al._ (2012) have central properties which are directly
expressed via the spectral features of matrices associated with the network
topology Dorogovtsev _et al._ (2003); Chung _et al._ (2003); Kim and Motter
(2007), such as the mixing time of random walks, epidemic thresholds and the
synchronization speed of oscillators, to name a few. Virtually all of these
processes will be affected by large-scale modular structures present in the
network Newman (2011), which is reflected in its spectral properties Ergün and
Kühn (2009); Kühn and van Mourik (2011); Chauhan _et al._ (2009); Nadakuditi
and Newman (2012). Since such large-scale modularity is a ubiquitous property
in real networks Newman (2011), describing the spectral features resulting
from this is a crucial step in understanding how these systems function.
Additionally, the information encoded in the eigenvectors of these matrices
are central to the nontrivial task of detecting large-scale features in
empirical networks Fortunato (2010); Newman (2006); Fiedler (1973); Pothen
_et al._ (1990); Nadakuditi and Newman (2012), and from it is possible to
derive general bounds on the detectability of existing community structure
Nadakuditi and Newman (2012).
In this work, we formulate an unified framework to obtain the eigenvalue
spectrum associated with arbitrary modular structures, parameterized as
stochastic block models Holland _et al._ (1983); Fienberg _et al._ (1985);
Faust and Wasserman (1992); Karrer and Newman (2011). The framework allows the
straightforward calculation of a large class of matrices which include the
adjacency, Laplacian and normalized Laplacian matrices, and is exact in the
limit of large degrees. It contrasts with previous work Kühn and van Mourik
(2011) which is exact in the limit of small degrees, but depends on the
solution of a number of self-consistency equations which are solved
stochastically. Here we show that if the block structure is sufficiently well
pronounced, it will trigger the appearance of isolated eigenvalues, with
associated eigenvectors strongly correlated with the block partition. If the
block structure becomes too weak (but nonvanishing), the isolated eigenvalues
merge with the continuous band, and the eigenvectors are no longer correlated
with the block partition. This has important consequences to the detectability
of modular structure in networks Nadakuditi and Newman (2012) but also to a
large class of dynamical processes since after this transition takes place one
should not expect the modular structure to play a significant role. We show
that in general the different matrices have different sensitivities to the
imposed block structure, and exhibit these transitions for different
modularity strengths.
_Unified framework. —_ Any given undirected network can be encoded via its
adjacency matrix $\bm{A}$, which has entries $A_{ij}=1$ if node $i$ is
adjacent to $i$, or $A_{ij}=0$ otherwise. The Laplacian matrix is defined as
$\bm{L}=\bm{D}-\bm{A}$, where $\bm{D}$ is a diagonal matrix containing the
vertex degrees, $D_{ij}=\delta_{ij}k_{i}$. Finally, the normalized Laplacian
is defined as $\bm{\mathcal{L}}=\bm{I}-\bm{D}^{-1/2}\bm{A}\bm{D}^{-1/2}$. Here
we use a general parametrization which contains these matrices as special
cases, via the matrix $\bm{W}=\bm{C}+\bm{M}$, where $\bm{C}$ is a random
diagonal matrix, and $\bm{M}$ is a random symmetric matrix. Simply by choosing
$\\{\bm{C}=0,\bm{M}=\bm{A}\\}$, $\\{\bm{C}=D,\bm{M}=-\bm{A}\\}$ and
$\\{\bm{C}=\bm{I},\bm{M}=-\bm{D}^{-1/2}\bm{A}\bm{D}^{-1/2}\\}$, we recover
$\bm{A}$, $\bm{L}$ and $\bm{\mathcal{L}}$, respectively. We may write
$\bm{W}=\bm{C}+\bm{\mathcal{M}}+{\left<\bm{M}\right>}=\bm{\mathcal{X}}+{\left<\bm{M}\right>}$,
such that the matrix $\bm{\mathcal{X}}=\bm{C}+\bm{\mathcal{M}}$, with
$\bm{\mathcal{M}}=\bm{M}-{\left<\bm{M}\right>}$, has off-diagonal entries with
zero mean. The spectrum of $\bm{\mathcal{X}}$ can be obtained via its average
resolvent ${\left<(z\bm{I}-\bm{\mathcal{X}})^{-1}\right>}$, using the
Stieltjes transform
$\rho(z)=-\frac{1}{N\pi}\operatorname{Im}\operatorname{Tr}{\left<(z\bf{I}-\bm{\mathcal{X}})^{-1}\right>}$,
with $z$ approaching the real line from above. Given an arbitrary random
matrix $\bm{X}$ with zero-mean off-diagonal entries, if the variance of the
entries is sufficiently large, we can use the approximation Nadakuditi and
Newman (2013),
${\left<[\bm{X}^{-1}]_{ii}\right>}\simeq\sum_{X_{ii}}\frac{P^{i}(X_{ii})}{X_{ii}-\sum_{j}{\left<[\bm{X}^{-1}]_{jj}\right>}{\left<a^{2}_{j}\right>}},$
(1)
and ${\left<[\bm{X}^{-1}]_{ij}\right>}=0$ for $i\neq j$, where $\bm{a}$ is the
$i$th column of $\bm{X}$, with the diagonal element removed, and it is assumed
that the diagonal elements $X_{ii}$ can only take discrete values, distributed
according to $P^{i}(X_{ii})$. We use Eq. 1 to compute the average resolvent of
the matrix $\bm{\mathcal{X}}$. We consider random graphs parameterized as
stochastic block models Holland _et al._ (1983); Fienberg _et al._ (1985);
Faust and Wasserman (1992) where $N$ nodes are divided into $B$ distinct
blocks, where each block $r$ has $n_{r}$ nodes, and the matrix entry $e_{rs}$
specifies the number of edges between blocks $r$ and $s$, which are otherwise
randomly placed. Hence, in the considered cases, the expected value of
$\bm{M}$ is simply a function of the block memberships, i.e.
${\left<C_{ii}\right>}=[\bm{C}_{B}]_{b_{i},b_{i}}=c_{b_{i}}$ and
${\left<M_{ij}\right>}=[\bm{M}_{B}]_{b_{i},b_{j}}$, with $\bm{C}_{B}$ and
$\bm{M}_{B}$ being matrices of size $B\times B$, and the vector $\bm{b}$ of
size $N$ and entries in the range $[1,B]$ specifies the block memberships.
When applying this to Eq. 1 with $\bm{X}=z\bm{I}-\bm{\mathcal{X}}$, we may use
the fact the averages on both sides of Eq. 1 can only depend on the block
membership of the respective nodes. Thus, using the shorthand
$t_{r}(z)\equiv{\left<[(z\bm{I}-\bm{\mathcal{X}})^{-1}]_{ii}\right>}$ for
$i\in r$, we obtain,
$t_{r}(z)=\sum_{c}\frac{p^{r}_{c}}{z-c-\sum_{s}\sigma^{2}_{rs}n_{s}t_{s}(z)},$
(2)
where $p^{r}_{c}$ is probability distribution of the diagonal elements $c$ for
block $r$, and $\sigma^{2}_{rs}$ is the variance of the elements of
$\bm{\mathcal{M}}$, labeled according to block membership, which is identical
to the variance of $\bm{M}$. The spectrum of $\bm{\mathcal{X}}$ may be finally
obtained via
$\rho(z)=-\frac{1}{N\pi}\sum_{r}n_{r}\operatorname{Im}t_{r}(z).$ (3)
In order to obtain the spectrum of $\bm{W}$, we employ an argument developed
in Ref. Benaych-Georges and Nadakuditi (2011), and note that in order for $z$
to be an eigenvalue of $\bm{W}=\bm{\mathcal{X}}+\bm{M}$, we must have
$\det(z\bm{I}-(\bm{\mathcal{X}}+{\left<\bm{M}\right>}))=0$, which can be
rewritten as
$\det(z\bm{I}-\bm{\mathcal{X}})\det(\bm{I}-(z\bm{I}-\bm{\mathcal{X}})^{-1}{\left<\bm{M}\right>})=0$.
Thus, if the second determinant is zero for a given $z$, it will be an
eigenvalue of $\bm{W}$ but not of $\bm{\mathcal{X}}$. These additional
eigenvalues may be obtained via the ensemble average
$\det(\bm{I}-{\left<(z\bm{I}-\bm{\mathcal{X}})^{-1}\right>}{\left<\bm{M}\right>})=0$,
which will hold if the matrix
${\left<(z\bm{I}-\bm{\mathcal{X}})^{-1}\right>}{\left<\bm{M}\right>}$ has an
eigenvalue equal to one. Since this matrix has a maximum rank equal to $B$,
its nonzero eigenvalues will be identical to the $B\times B$ matrix
$\bm{T}(z)\bm{M}_{B}\bm{N}$, where $\bm{T}(z)$ and $\bm{N}$ are diagonal
$B\times B$ matrices containing the values of $t_{r}(z)$ and $n_{r}$,
respectively. Hence, the existence of additional eigenvalues of $\bm{W}$ may
obtained by solving,
$\det(\bm{I}_{B}-\bm{T}(z)\bm{M}_{B}\bm{N})=0,$ (4)
simultaneously with $\rho(z)=0$. Eqs. 2, 3 and 4 provide a complete recipe for
obtaining the desired spectrum, provided we know the $B\times B$ matrices
$\sigma^{2}_{rs}$ and $\bm{M}_{B}$ as well as the diagonal entry distribution
$p^{r}_{c}$. For the three matrices of interest they are easily computed as
$\\{p^{r}_{c}=\delta_{0,c};\;\sigma^{2}_{rs}=[\bm{M}_{B}]_{rs}=e_{rs}/n_{r}n_{s}\\}$
for $\bm{A}$,
$\\{p^{r}_{c}=P(c,e_{r}/n_{r});\;[\bm{M}_{B}]_{rs}=-e_{rs}/n_{r}n_{s};\;\sigma^{2}_{rs}=e_{rs}/n_{r}n_{s}\\}$
for $\bm{L}$, with $P(c,\lambda)$ being a Poisson distribution on $c$ with
average $\lambda$, and
$\\{p^{r}_{c}=\delta_{1,c};\;[\bm{M}_{B}]_{rs}=-e_{rs}/\sqrt{n_{r}e_{r}n_{s}e_{s}};\;\sigma^{2}_{rs}\simeq
e_{rs}/e_{r}e_{s}\\}$ for $\bm{\mathcal{L}}$. We emphasize that, since the
approximation in Eq. 1 was used, the obtained spectrum should be correct only
in the limit of sufficiently large degrees. If this holds, the theory
reproduces in very good detail the spectrum of empirical networks, as can be
seen in Fig. 1. The spectrum is composed of a continuous band, as well as a
number of isolated eigenvalues, which correspond very well to the solutions of
Eqs. 3 and 4, respectively. The same is true for the spectrum of the matrices
$\bm{\mathcal{L}}$ and $\bm{L}$ (Fig. 2). The spectrum of $\bm{L}$ is special,
since it contains an elaborate fine structure, with many fringes, and an
interleaving of the continuous band (Eq. 3) with the isolated eigenvalues (Eq.
4). The continuous band has no well-defined edge, with fringes which extend
through the whole spectrum, but with decaying amplitudes. Despite such
detailed structure, the theory captures these features very well, as can be
seen in Fig. 2 (see also the Supplemental Material).
\begin{overpic}[unit=1cm,width=216.81pt,trim=0.0pt 0.0pt 0.0pt
0.0pt,clip]{example_adj_N20000_ak300_nalt25-band.pdf}
\put(18.0,32.0){\includegraphics[width=62.87344pt]{example_graph.pdf}}
\put(70.0,25.0){\includegraphics[width=62.87344pt]{example_graph_matrix.pdf}}
\end{overpic}
Figure 1: _Top:_ Continuous band of the matrix $\bm{A}$ for the block
structure in the inset (right: $e_{rs}$ matrix and block sizes $n_{r}$, left:
graphical representation). The solid line corresponds to Eq. 3, and the grey
histogram is averaged over $25$ network realizations with $N=2\times 10^{4}$,
and ${\left<k\right>}=300$._Bottom:_ The same, but with the isolated
eigenvalues added. The grey vertical lines are average empirical values,
whereas the solid (orange) curve corresponds to the determinant of Eq. 4. The
vertical (green) line segments mark the eigenvalues of the matrix
$\bm{C}_{B}+\bm{M}_{B}\bm{N}$.
Figure 2: Eigenvalue spectrum of the normalized Laplacian matrix
$\bm{\mathcal{L}}$ (top) and Laplacian matrix $\bm{L}$ (bottom) for the block
structure of Fig. 1.
For isolated eigenvalues which are sufficiently detached from the spectral
band, Eq. 2 may be approximated by $t_{r}\approx 1/(z-c_{r})$, in which case
Eq. 4 amounts to $\det(z\bm{I}_{B}-(\bm{C}_{B}+\bm{M}_{B}\bm{N}))=0$, where
$\bm{C}_{B}$ is a diagonal matrix with the $c_{r}$ values. If this holds, the
detached eigenvalues will correspond to the spectrum of the matrix
$\bm{C}_{B}+\bm{M}_{B}\bm{N}$.
At the edges of the continuous band the purely real solution to Eq. 2 becomes
unstable, and the largest eigenvalue of the Jacobian
$J_{rs}(z)\equiv\partial\hat{t}_{r}/\partial
t_{s}=\sum_{c}p^{r}_{c}\sigma^{2}_{rs}n_{s}/(z-c-\sum_{s}\sigma^{2}_{rt}n_{t}t_{t}(z))^{2}$,
where $\hat{t}_{r}$ is the right-hand side of Eq. 2, becomes equal to one.
Hence, one may find the edges of the continuous band by solving
$\det(\bm{I}_{B}-\bm{J}(z))=0$, simultaneously with $\rho(z)=0$.
Figure 3: _Left:_ Extremal eigenvalues of $\bm{A}$ (top) and
$\bm{\mathcal{L}}$ (bottom), for the block structure of Fig. 1, as a function
of the parameter $c$ defined in the text. The solid lines are solutions of Eq.
4, and the data points are empirical values for $N=2\times 10^{4}$. The dotted
vertical line marks the detachment transition. _Right, top (bottom):_
Eigenvector values for second and third largest (smallest) eigenvalues of
$\bm{A}$ ($\bm{\mathcal{L}}$), for different values of $c$. The circles
(stars) correspond to the empirical (theoretical) average values for each
block.
_Eigenvectors. —_ The eigenvector equation
$(\bm{\mathcal{X}}+\bm{M})\bm{v}=z\bm{v}$ can be rewritten as
$(z\bm{I}-\mathcal{X})^{-1}\bm{M}\bm{v}=\bm{v}$. Taking the ensemble average,
we get
${\left<(z\bm{I}-\mathcal{X})^{-1}\right>}\bm{M}{\left<\bm{v}\right>}={\left<\bm{v}\right>}$.
Since the average values of $\bm{v}$ can only depend on the block memberships,
and ${\left<(z\bm{I}-\mathcal{X})^{-1}\right>}$ is diagonal we get
$\bm{T}(z)\bm{M}_{B}\bm{N}\bm{v}_{B}=\bm{v}_{B},$ (5)
where $\bm{v}_{B}$ contain the average values of $v$ for each block.
If the block structure is made sufficiently tenuous, all but the most extremal
detached eigenvalues will approach progressively the continuous band. At some
point, before the graph becomes fully random, they will merge with the
continuous band, and the associated eigenvectors will no longer convey any
information on the existing block structure. An example is shown in Fig. 3,
which shows the full spectrum of the block structure given by
$e_{rs}=ce_{rs}^{0}+(1-c)e^{0}_{r}e^{0}_{s}/2E$, with $e_{rs}^{0}$ being the
same block structure shown in Fig. 1, and $e^{0}_{r}=\sum_{s}e_{rs}^{0}$. The
parameter $c$ interpolates between a random graph ($c=0$) and the original
block structure ($c=1$), while preserving the same degree distribution. As
show in Fig. 3, for a specific value of $c=c^{*}>0$ all but the most extremal
eigenvalue merge with the continuous band, and for $c<c^{*}$ the eigenvector
values are no longer discernibly correlated with the planted block structure.
It is important to notice that the transition point $c^{*}$ is different for
the matrices $\bm{A}$ and $\bm{\mathcal{L}}$, and thus the different spectra
will have different sensitivities to the planted block structure. This can be
seen in more detail by considering a simpler two-block system with
$n_{1}/N=w$, $n_{2}/N=1-w$ and $e_{rs}=E[c\delta_{rs}+(1-c)/2]$, which is a
diagonal block structure with the parameter $c$ controlling the block
segregation and $w$ the degree asymmetry 111Note that the parameter $c$ does
not change the degree distribution.. In Fig. 4 is shown the extremal
eigenvalues for the three matrices as a function of $c$, compared with
empirical values. For the normalized Laplacian matrix $\bm{\mathcal{L}}$, the
extremal eigenvalue is very insensitive to the parameter $w$ 222The curves
_do_ change, however only very subtly.. The matrix $\bm{A}$ displays, on the
other hand, different transition points, depending on $w$, with larger values
of $c^{*}$ for larger degree asymmetries. The spectral band for the matrix
$\bm{L}$ has no well-defined edge; hence, the transition point on a finite
network will depend on the system size. The observable edge of the band is
obtained by computing the extremal statistics of $\rho(z)$ (see the
Supplemental Material), and matches well the observed values, as can be seen
in Fig. 4. A comparison of the transition points can be seen in the lower
right of Fig. 4, where it is also included the values for the modularity
matrix $\bm{B}=\bm{A}-\bm{k}\bm{k}^{T}/2E$, where $\bm{k}$ is a vector with
node degrees, often used for community detection Newman (2006), which can also
be calculated with the presented method in an entirely analogous fashion.
Since for this specific block structure it has systematically the lowest
threshold $c^{*}$ among the others, this seems to corroborate the hypothesis
in Refs. Nadakuditi and Newman (2012); Radicchi (2013) that $\bm{B}$ may
posses optimal characteristics in some scenarios. On the other hand, the
comparatively worst behavior of the Laplacian $\bm{L}$ raises issues with its
use for this purpose (as in e.g. Ref. Newman (2013)).
Figure 4: Top, left (right): Second largest (smallest) eigenvalue of $\bm{A}$
($\bm{\mathcal{L}}$), for the asymmetric two-block structure described in the
text. The dashed curves are the theoretical values, and the data points are
obtained from network realizations with $N=2\times 10^{4}$ and
${\left<k\right>}=300$. Bottom, left: Second smallest eigenvalue of $L$. The
dashed curves are the expected values for $N=2\times 10^{4}$ (see Supplemental
Material). Bottom, right: Transition point $c^{*}$ as a function of $w$ for
the matrices $\bm{A}$, $\bm{\mathcal{L}}$ and the modularity matrix $\bm{B}$.
_Homogeneous blocks.—_ Further analytical progress can be made by assuming
that the blocks are homogeneous, such that the right-hand side of Eq. 2 is the
same for all blocks. This means that they must all share the same properties
such as size $n_{r}$ and average degree $e_{r}/n_{r}$. The solution in case
$p^{r}_{c}=\delta_{d,c_{r}}$ (i.e. for both $\bm{A}$ and $\bm{\mathcal{L}}$)
will then be simply $t(z)=(z-d\pm\sqrt{(d-z)^{2}-4a})/2a$ with
$a=a_{r}=N\sum_{s}\sigma_{rs}^{2}/B$, which will result in the usual
semicircle distribution $\rho(z)=\sqrt{4a-(z-d)^{2}}/2a\pi$ for
$|z-d|<2\sqrt{a}$; otherwise, $\rho(z)=0$. The detached eigenvalues will be
given by the solution of $\det(\bm{I}-t(z)N\bm{M}_{B}/B)=0$. Hence there will
be a one-to-one correspondence between the nonzero eigenvalues $\lambda_{i}$
of $\bm{M}_{B}$ and the detached eigenvalues $z_{i}=d+at_{i}+1/t_{i}$, where
$t_{i}=B/N\lambda_{i}$, as long as $|z_{i}-d|>2\sqrt{a}$; otherwise, they will
merge with the continuous band. By making $|z_{i}-d|=2\sqrt{a}$, one obtains
that this transition happens at $\lambda_{i}=\pm\sqrt{a}B/N$. Both for
$\bm{A}$ and $\bm{\mathcal{L}}$ one can see that this transition occurs at the
same point: If one writes the block matrix as
$e_{rs}=N{\left<k\right>}m_{rs}$, such that $\sum_{rs}m_{rs}=1$, this
transition translates to
$\lambda_{m}^{2}=\frac{1}{{\left<k\right>}B^{2}},$ (6)
where $\lambda_{m}$ is an eigenvalue of the $m_{rs}$ matrix. The fact that the
detachment transition is identical for both $\bm{A}$ and $\bm{\mathcal{L}}$ is
a special property of the homogeneous block structure, and does not hold in
general, as we have shown previously 333It can also be shown that Eq. 6 also
holds for the modularity matrix $\bm{B}$..
$l=1$
$l=2$
$l=3$
$l=4$
$l=5$
Figure 5: Top: Detachment transitions for the nested partition model
described in the text with $B_{1}=2$ as a function of the mixing parameter
$c$, and for different nesting depths $l$, for $\bm{A}$ and
$\bm{\mathcal{L}}$. The data points correspond to network realizations with
$N=2\times 10^{4}$ and ${\left<k\right>}=300$, and the solid lines are
theoretical values. Bottom: Example of $e_{rs}$ matrices with $B_{1}=2$ for
different values of $l$.
As a concrete example of an homogeneous structure, we consider a nested
version of the usual planted partition model Condon and Karp (2001), inspired
by similar constructions done in Refs. Leskovec _et al._ (2008); Palla _et
al._ (2010). We define a seed structure with $B_{1}$ blocks and
$[\bm{m}_{1}]_{rs}=\delta_{rs}c/B_{1}+(1-\delta_{rs})(1-c)/B_{1}(B_{1}-1)$,
and construct a nested matrix of depth $l$ via
$\bm{m}_{l}=\bm{m}_{l-1}\otimes\bm{m}_{l-1}$ where $\otimes$ denotes the
Kronecker product. The eigenvalues of the matrix $\bm{m}_{l}$ are given by
$\lambda^{i}_{m_{l}}=((cB_{1}-1)/(B_{1}(B_{1}-1)))^{l-i}/{B_{1}^{i}}$, for
$i\in[0,l]$. Thus, from Eq. 6 one obtains a series of transitions, where a
deeper level of the nested structure “fades away,” and the spectrum is
indistinguishable from that of a $l-1$ structure (see Fig. 5). The transition
of the shallowest level happens at ${\left<k\right>}=((B-1)/(cB-1))^{2}$,
which is the same as the regular planted partition model Nadakuditi and Newman
(2012). This transition marks the point at which more general inference
methods should also fail to detect the imposed partition Decelle _et al._
(2011).
In summary, we presented an unified framework to obtain the full spectrum of
random networks with modular structure, in the limit of large degrees. We
showed that the detachment transition of the isolated eigenvalues is a general
feature which determines how strongly the existing modular structure affects
the different spectra. The different matrices react differently to the imposed
modular structure and have different transition points. Only when the blocks
are homogeneous do some of these transitions collapse together. Hence, in
general, the detectability threshold of the imposed block structure may depend
strongly on the actual spectrum which is observed.
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See pages - of sup_mat.pdf
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arxiv-papers
| 2013-06-11T12:42:05 |
2024-09-04T02:49:46.340812
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tiago P. Peixoto",
"submitter": "Tiago Peixoto",
"url": "https://arxiv.org/abs/1306.2507"
}
|
1306.2577
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-094 LHCb-PAPER-2013-025 11 June 2013
Measurement of the differential branching fraction of the decay $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$
The LHCb collaboration†††Authors are listed on the following pages.
The differential branching fraction of the decay $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ is
measured as a function of the square of the dimuon invariant mass, $q^{2}$. A
yield of $78\pm 12$ $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ decays is observed using data, corresponding to an
integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, collected by the LHCb
experiment at a centre-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$. A
significant signal is found in the $q^{2}$ region above the square of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass, while at lower-$q^{2}$
values upper limits are set on the differential branching fraction.
Integrating the differential branching fraction over $q^{2}$, while excluding
the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ regions,
gives a branching fraction of ${\cal B}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-})=(0.96\pm
0.16\mathrm{\,(stat)}\pm 0.13\mathrm{\,(syst)}\pm 0.21(\mathrm{norm}))\times
10^{-6}$, where the uncertainties are statistical, systematic and due to the
normalisation mode, $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$, respectively.
Submitted to Physics Letters B
© CERN on behalf of the LHCb collaboration, license
http://creativecommons.org/licenses/by/3.0/CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F.
Dettori41, A. Di Canto11, F. Di Ruscio23,k, H. Dijkstra37, M. Dogaru28, S.
Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F.
Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S.
Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S.
Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch.
Elsasser39, A. Falabella14,e, C. Färber11, G. Fardell49, C. Farinelli40, S.
Farry51, V. Fave38, D. Ferguson49, V. Fernandez Albor36, F. Ferreira
Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16, C. Fitzpatrick37, M.
Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C.
Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas Torreira36, D.
Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53,
J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M.
Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V.
Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P.
Griffith44, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou58, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, B.
Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45,
J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51,
P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, A. Hicheur1, E.
Hicks51, D. Hill54, M. Hoballah5, M. Holtrop40, C. Hombach53, P. Hopchev4, W.
Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50,
V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E.
Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R.
Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M.
Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, A. Keune38, B.
Khanji20, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M.
Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G.
Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,j, V. Kudryavtsev33,
T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A.
Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C.
Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40,
J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T.
Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C.
Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-
March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, J. Maratas5, U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11,
G. Martellotti24, A. Martens8, A. Martín Sánchez7, M. Martinelli40, D.
Martinez Santos41, D. Martins Tostes2, A. Massafferri1, R. Matev37, Z.
Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,32,37,e, B. Mc Skelly51, J.
McCarthy44, A. McNab53, R. McNulty12, B. Meadows56,54, F. Meier9, M.
Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S.
Monteil5, D. Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R.
Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B.
Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49,
S. Neubert37, N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o,
M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54,
A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S.
Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora
Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-
Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, G.
Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan58, E. Picatoste Olloqui35,
B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F.
Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10,
B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51,
C. Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
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Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, D.A. Roberts57, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37,
V. Romanovsky34, A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H.
Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N.
Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, C. Salzmann39,
B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C. Santamarina
Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18,l, C. Satriano24,m, A.
Satta23, M. Savrie16,e, D. Savrina30,31, P. Schaack52, M. Schiller41, H.
Schindler37, M. Schlupp9, M. Schmelling10, B. Schmidt37, O. Schneider38, A.
Schopper37, M.-H. Schune7, R. Schwemmer37, B. Sciascia18, A. Sciubba24, M.
Seco36, A. Semennikov30, I. Sepp52, N. Serra39, J. Serrano6, P. Seyfert11, M.
Shapkin34, I. Shapoval16,42, P. Shatalov30, Y. Shcheglov29, T. Shears51,37, L.
Shekhtman33, O. Shevchenko42, V. Shevchenko30, A. Shires52, R. Silva
Coutinho47, M. Sirendi46, T. Skwarnicki58, N.A. Smith51, E. Smith54,48, J.
Smith46, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro18, D. Souza45,
B. Souza De Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F. Stagni37, S.
Stahl11, O. Steinkamp39, S. Stoica28, S. Stone58, B. Storaci39, M.
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M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D. Tonelli37, S. Topp-
Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38, M.T. Tran38, M.
Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37,
A. Ukleja27, D. Urner53, A. Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G.
Valenti14, A. Vallier7, M. Van Dijk45, R. Vazquez Gomez18, P. Vazquez
Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g,
G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V.
Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S.
Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D.
Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L.
Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48,
J. Wimberley57, J. Wishahi9, M. Witek25, S.A. Wotton46, S. Wright46, S. Wu3,
K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, R. Young49, X. Yuan3, O.
Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C.
Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
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 Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The decay $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ is a rare ($b\\!\rightarrow s$) flavour-changing
neutral current process that in the Standard Model proceeds through
electroweak loop (penguin and $W^{\pm}$ box) diagrams. Since non-Standard
Model particles may also participate in these loop diagrams, measurements of
this and similar decays can be used to search for physics beyond the Standard
Model. In the past, more emphasis has been placed on the study of rare decays
of mesons than of baryons, in part due to the theoretical complexity of the
latter [1]. In the particular system studied in this Letter, the decay
products include only a single hadron, simplifying the theoretical modelling
of hadronic physics in the final state.
The study of $\mathchar 28931\relax^{0}_{b}$ baryon decays is of considerable
interest for two reasons. Firstly, as the $\mathchar 28931\relax^{0}_{b}$
baryon has non-zero spin, there is the potential to improve the limited
understanding of the helicity structure of the underlying Hamiltonian, which
cannot be extracted from mesonic decays [2, 1]. Secondly, as the composition
of the $\mathchar 28931\relax^{0}_{b}$ baryon may be considered as the
combination of a heavy quark with a light diquark system, the hadronic physics
differs significantly from that of the $B$ meson decay. This may allow this
aspect of the theory to be tested, which may lead to improvements in
understanding of $B$ mesons.
Theoretical aspects of the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ decay
have been considered both in the SM and in various scenarios of physics beyond
the Standard Model [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Although
based on the same effective Hamiltonian as that for the corresponding mesonic
transitions, the hadronic form factors for the $\mathchar 28931\relax^{0}_{b}$
baryon case are less well-known due to the smaller number of experimental
constraints. This leads to a large spread in the predicted branching
fractions. The differential branching fraction as a function of the square of
the dimuon invariant mass, $q^{2}\equiv m_{\mu^{+}\mu^{-}}^{2}$, is of
particular interest. The approaches taken by the theoretical calculations
depend on the $q^{2}$ region. By comparing predictions with data as a function
of $q^{2}$, these different methods of treating form factors are tested.
The first observation of the decay $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ by the
CDF collaboration [16] had a signal yield of $24\pm 5$ events, corresponding
to an absolute branching fraction ${\cal B}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-})=(1.73\pm
0.42\mathrm{\,(stat)}\pm 0.55\mathrm{\,(syst)})\times 10^{-6}$, with evidence
for signal at $q^{2}$ above the square of the mass of the $\psi{(2S)}$
resonance.
Following previous measurements of rare decays involving dimuon final states
[17, 18], a first measurement by LHCb of the differential and total branching
fractions for the rare decay $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ is
reported. The inclusion of charge conjugate modes is implicit throughout. The
rates are normalised with respect to the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decay, with
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$.
This analysis uses a $pp$ collision data sample, corresponding to an
integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, collected during 2011 at a
centre-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$.
## 2 Detector and software
The LHCb detector [19] 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 (VELO) 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 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
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum.
Charged hadrons are identified using two ring-imaging Cherenkov detectors
[20]. 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 [21].
The trigger [22] 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. Candidate events are first required to pass a
hardware trigger which selects muons with a transverse momentum,
$\mbox{$p_{\rm T}$}>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 an impact parameter greater
than 100$\,\upmu\rm m$ with respect to all of the primary $pp$ interaction
vertices (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 in the event.
A candidate $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ or $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decay that is directly responsible for triggering
both the hardware and software triggers is denoted as “trigger on signal”. An
event in which a $\mathchar 28931\relax^{0}_{b}$ baryon is reconstructed in
either of these modes but none of the daughter particles are necessary for the
trigger decision is referred to as “trigger independent of signal”. As these
two categories of event are not mutually exclusive, the overlap may be used to
estimate the efficiency of the trigger selection directly from data.
In the simulation, $pp$ collisions are generated using Pythia 6.4 [23] with a
specific LHCb configuration [24]. Decays of hadronic particles are described
by EvtGen [25] in which final state radiation is generated using Photos [26].
The interaction of the generated particles with the detector and its response
are implemented using the Geant4 toolkit [27, *Agostinelli:2002hh] as
described in Ref. [29].
## 3 Candidate selection
Candidate $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ (signal mode) and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ (normalisation mode) decays are reconstructed
from muon, $\mathchar 28931\relax$ baryon and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates. The
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates are reconstructed
via their dimuon decays and therefore the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decay is an ideal normalisation process. The
dimuon candidates are formed from two oppositely-charged particles identified
as muons [21, 20]. Good track quality is ensured by requiring
$\chi^{2}/\mathrm{ndf}$ ($\chi^{2}$ per degree of freedom) $<4$ for a track
fit. The candidates must also have $\chi^{2}_{\rm IP}$ with respect to any
primary interaction greater than 16, where $\chi^{2}_{\rm IP}$ is defined as
the difference in $\chi^{2}$ of a given PV reconstructed with and without the
considered track. These $\mu^{+}\mu^{-}$ pairs are required to have an
invariant mass of less than 5050${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
to be consistent with originating from a common vertex ($\chi^{2}_{\rm
vtx}/\mathrm{ndf}<9$).
Candidate $\mathchar 28931\relax$ decays are reconstructed in the $\mathchar
28931\relax\\!\rightarrow p\pi^{-}$ mode from two oppositely-charged particles
that either both originate within the acceptance of the VELO (“long $\mathchar
28931\relax$” candidates), or both originate outside the acceptance of the
VELO (“downstream $\mathchar 28931\relax$” candidates). Tracks are required to
have $\mbox{$p_{\rm T}$}>0.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and
$\mathchar 28931\relax$ candidates must have $\chi^{2}_{\rm
vtx}/\mathrm{ndf}<30$ ($<25$ for downstream $\mathchar 28931\relax$
candidates), a decay time of at least 2${\rm\,ps}$, and a reconstructed
invariant mass within 30${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
world average value [30]. Due to the distinct kinematics and topology of the
$\mathchar 28931\relax$ decay, it is not necessary to impose particle
identification requirements on the decay products of the $\mathchar
28931\relax$ candidate.
Candidate $\mathchar 28931\relax^{0}_{b}$ decays are formed by combining
$\mathchar 28931\relax$ and dimuon candidates that originate from a common
vertex ($\chi^{2}_{\rm vtx}/\mathrm{ndf}<8$), have $\chi^{2}_{\rm IP}<9$,
$\chi^{2}_{\rm VS}>100$ and an invariant mass in the interval
4.9–7.0${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The $\chi^{2}_{\rm VS}$ is
defined as the difference in $\chi^{2}$ between fits in which the $\mathchar
28931\relax^{0}_{b}$ decay vertex is assumed to coincide with the PV and
allowing the decay vertex to be distinct from the PV. Candidates must also
point to the associated PV by requiring the angle between the $\mathchar
28931\relax^{0}_{b}$ momentum vector and the vector between the PV and the
$\mathchar 28931\relax^{0}_{b}$ decay vertex is less than 8$\rm\,mrad$. The
associated PV is the one relative to which the $\mathchar 28931\relax^{0}_{b}$
candidate has the lowest $\chi^{2}_{\rm IP}$ value.
The final selection is based on a neural network classifier [31, 32] with 15
variables as input. The single most important variable is the $\chi^{2}$ from
a kinematic fit [33] that constrains the decay products of the $\mathchar
28931\relax^{0}_{b}$, the $\mathchar 28931\relax$ and the dimuon systems to
originate from their respective vertices. Other variables that contribute
significantly are the momentum and transverse momentum of the $\mathchar
28931\relax^{0}_{b}$ candidate, the $\chi^{2}_{\rm IP}$ and track
$\chi^{2}/\mathrm{ndf}$ for both muons, the $\chi^{2}_{\rm IP}$ of the
$\mathchar 28931\relax^{0}_{b}$ candidate, and the separation of the
$\mathchar 28931\relax$ and $\mathchar 28931\relax^{0}_{b}$ vertices.
Downstream and long $\mathchar 28931\relax$ decays have separate inputs to the
neural network for $\chi^{2}_{\rm IP}$ and $\chi^{2}_{\rm VS}$ because of the
differing track resolution and kinematics. In the final selection of
$\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ candidates, the $\mu^{+}\mu^{-}$ invariant mass
is required to be in the interval
3030–3150${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The signal sample used to
train the neural network consists of simulated $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ events,
while background is taken from data in the upper sideband of the $\mathchar
28931\relax^{0}_{b}$ candidate mass spectrum, between 6.0 and
7.0${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, which is dominated by
candidates with dimuon mass in the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}$ region. The requirement on the output of the neural network is chosen
to maximise $N_{\mathrm{S}}/\sqrt{N_{\mathrm{S}}+N_{\mathrm{B}}}$, where
$N_{\mathrm{S}}$ and $N_{\mathrm{B}}$ are the expected numbers of signal and
background events, respectively. To ensure an appropriate normalisation of
$N_{\mathrm{S}}$, the number of $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ candidates after the preselection is scaled by
the measured ratio of branching fractions between the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ and
$\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays [16], and the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
branching fraction [30]. The value of $N_{\mathrm{B}}$ is derived from the
background training sample normalised to the number of candidates in the
signal region after preselection. The $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ signal
candidates exclude the $q^{2}$ regions of
8.68–10.09${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ and
12.86–14.18${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$, which are dominated
by contributions from the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and
$\psi{(2S)}$ resonances, respectively. The effect of finite $q^{2}$ resolution
is negligible. Relative to the preselected event sample, the neural network
retains $(76.0\pm 0.3)\,\%$ of the rare decay signal while rejecting $(95.9\pm
0.2)\,\%$ of the background.
## 4 Peaking backgrounds
Backgrounds are studied using simulated samples of $b$ hadrons in which the
final state includes two muons. For the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ channel, the only significant contribution found
is from $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays, with
$K^{0}_{\rm\scriptscriptstyle S}\\!\rightarrow\pi^{+}\pi^{-}$, which has the
same topology as the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ mode. This contribution leads to a broad shape
that peaks below the $\mathchar 28931\relax^{0}_{b}$ mass region and is
accommodated in the mass fit described later.
For the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ channel, sources of peaking background are
considered in the $q^{2}$ ranges of interest. The contributions identified are
$\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays in which an energetic photon is radiated
from either of the muons, and $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ decays, where
$K^{0}_{\rm\scriptscriptstyle S}\\!\rightarrow\pi^{+}\pi^{-}$ and a pion is
misreconstructed as a proton. The $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays contribute in the $q^{2}$ region just
below $m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}^{2}$, and populate a
mass region significantly below the $\mathchar 28931\relax^{0}_{b}$ mass. The
contribution from the $B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\mu^{+}\mu^{-}$ decays is estimated by taking the number of
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ events found in the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ fit, and scaling this by the ratio of world
average branching fractions between the decay processes $B^{0}\\!\rightarrow
K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ and
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ (including the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$
branching fraction) [30]. This gives fewer than 10 events integrated over
$q^{2}$, which is small relative to the expected total background levels.
## 5 Yields
### 5.1 Fit description
The yields of signal and background events in the data are determined in the
mass range 5.35–5.85${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ using unbinned,
extended maximum likelihood fits, for the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ and the
$\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ modes. The likelihood function has the form
$\mathcal{L}=e^{-(N_{\mathrm{S}}+N_{\mathrm{B}}+N_{\mathrm{P}})}\times\prod_{i=1}^{N}[N_{\mathrm{S}}P_{\mathrm{S}}(m_{i})+N_{\mathrm{B}}P_{\mathrm{B}}(m_{i})+N_{\mathrm{P}}P_{\mathrm{P}}(m_{i})]\;,$
(1)
where $N_{\mathrm{S}}$, $N_{\mathrm{B}}$ and $N_{\mathrm{P}}$ are number of
signal, combinatorial and peaking background events, respectively, and
$P_{j}(m_{i})$ are the corresponding probability density functions (PDFs). The
mass of the $\mathchar 28931\relax^{0}_{b}$ candidate, $m_{i}$, is determined
by a kinematic fit of the full decay chain in which the proton and pion are
constrained such that the $p\pi^{-}$ invariant mass corresponds to the Łbaryon
mass [30].
The signal shape, in both $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ and
$\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ modes, is described by the sum of two Gaussian
functions that share a common mean but have independent widths. The
combinatorial background is parametrised by a first-order polynomial, while
the background due to
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays is modelled by an exponential
function (with a cut-off) convolved with a Gaussian function.
For the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ mode, the widths and common mean in the signal
parametrisation are free parameters. The contribution of the narrower Gaussian
function is fixed to be 86 % of the total yield based on studies with
simulated data. The parameters describing the shape of the peaking background
are fixed to those derived from simulated
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ decays.
For the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ decay, the signal shape parameters are fixed
according to the result of the fit to $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ data. Studies with simulated data show that the
signal shape parameters in both decay modes are consistent with one another,
the only deviations being in the tails of the mass distribution. These are due
to small differences in the momentum spectra of the muons and energy loss from
radiative effects, and are negligible given the uncertainties inherent in the
size of the current data sample. The peaking background is found to be
negligible in the $q^{2}$ regions considered and is therefore excluded from
the fit.
### 5.2 Fit results
Figure 1: Invariant mass distribution of the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ candidates. The histogram shows data, the solid
red line is the overall fit function, the dotted blue line represents the sum
of the combinatorial and peaking backgrounds and the dash-dotted green line
the combinatorial background component.
The invariant mass distributions of the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ candidates is shown in Fig. 1. The fitted
function provides a good description of the data, with a $\chi^{2}$/ndf
corresponding to a probability of 47 %. The numbers of signal, combinatorial
background and peaking background events are found to be $2680\pm 64$,
$1294\pm 83$ and $1501\pm 85$, respectively, and the widths of the Gaussian
functions are $16.0\pm 0.4$ and $33\pm
5$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, compatible with simulation.
The invariant mass distribution for the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ process,
integrated over $q^{2}$ and in six $q^{2}$ intervals, are shown in Figs. 2 and
3, respectively. The yields, both integrated and differential in $q^{2}$, are
summarised in Table 1. The same $q^{2}$ intervals as in Ref. [16] are used to
facilitate comparison with the CDF measurements. The statistical significance
of the observed signal yields in Table 1 are evaluated as
$\sqrt{2\Delta\ln{\mathcal{L}}}$, where $\Delta\ln{\mathcal{L}}$ is the change
in the logarithm of the likelihood function when the signal component is
excluded from the fit, relative to the nominal fit in which it is present.
Significant signal yields are only apparent for
$q^{2}>m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}^{2}$. Yields at
lower-$q^{2}$ values are compatible with zero, consistent with previous
observations [16].
Figure 2: Invariant mass distribution of the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ candidates, integrated over all $q^{2}$ values, together with the fit function described in the text. The histogram shows data, the solid red line is the overall fit function and the dotted blue line represents the background component. Figure 3: Invariant mass distributions for the rare decay $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ candidates, in six $q^{2}$ intervals, together with the fit function described in the text. The histogram shows data, the solid red line is the overall fit function and the dotted blue line represents the background component. Table 1: Signal ($N_{\mathrm{S}}$) and background ($N_{\mathrm{B}}$) decay yields obtained from the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ mass fit in each $q^{2}$ interval. The integrated yield is the result of a fit without separation of the data into distinct $q^{2}$ regions. The statistical significance is calculated as described in the text. $q^{2}$ interval [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ] | $N_{\mathrm{S}}$ | $N_{\mathrm{B}}$ | Significance
---|---|---|---
0.00 – 2.00 | $2\pm 3$ | $34\pm 6$ | 0.8
2.00 – 4.30 | $4\pm 3$ | $42\pm 7$ | 1.4
4.30 – 8.68 | $4\pm 5$ | $134\pm 12$ | 1.0
10.09 – 12.86 | $13\pm 5$ | $52\pm 8$ | 3.4
14.18 – 16.00 | $14\pm 4$ | $20\pm 5$ | 4.9
16.00 – 20.30 | $44\pm 7$ | $24\pm 6$ | 9.8
Integrated yield | $\,78\pm 12$ | $310\pm 19$ | 8.9
## 6 Efficiency
The measurement of the differential branching fraction of $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ relative
to $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ benefits from the cancellation of several
potential sources of systematic uncertainty in the ratio of efficiencies,
$\varepsilon_{\rm rel}={\varepsilon_{\rm tot}}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-})/{\varepsilon_{\rm tot}}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax)$. The efficiency for each of the decays is
calculated according to
${\varepsilon_{\rm
tot}}=\varepsilon(\mathrm{geometry})\;\varepsilon(\mathrm{selection}|\mathrm{geometry})\;\varepsilon(\mathrm{trigger}|\mathrm{selection})\;,$
(2)
where the first term represents the efficiency for the final state particles
to be within the LHCb angular acceptance, the second term the combined
efficiency for candidate detection, reconstruction and selection, and the
rightmost term the efficiency for an event to satisfy the trigger requirements
if it is reconstructed and selected. All efficiencies are evaluated using
simulated data. A phase space model is used for $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays. The model used for $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ decays
includes $q^{2}$ and angular dependence as described in Ref. [34], together
with Wilson coefficients based on Refs. [35, 36]. Interference effects from
charmonium contributions are not included.
With these models, the geometric acceptance is found to be 16 % for $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays and in the range 16–20 % ($q^{2}$
dependent) for the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ channel. The overall efficiency to reconstruct and
select the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ decays varies from 1.3 % in the lowest $q^{2}$
interval to values around 2.5 % in the higher-$q^{2}$ regions. The $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decay has a similar efficiency to the
larger-$q^{2}$ regions of the rare decay. The trigger efficiency is calculated
using an emulation of the hardware trigger, combined with the same software
stage of the trigger that was used for data. The trigger efficiency increases
from approximately 50 % to 80 % for the lowest to highest $q^{2}$ regions,
respectively. An independent cross-check of the trigger efficiency is
performed using $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ data by calculating the ratio of yields that are
both classified as trigger on signal and trigger independent of signal
relative to those that are only classified as trigger independent of signal.
This data-driven method gives an efficiency of $(75\pm 7)$ %, which is
consistent with that of $(70.5\pm 0.3)$ % computed from simulation.
The relative efficiency for the ratio of branching fractions in each $q^{2}$
interval, calculated from the absolute efficiencies described above, are given
in Table 2. The rise in relative efficiency as a function of increasing
$q^{2}$ is dominated by two effects. Firstly, at low $q^{2}$ the muons have
lower momenta and therefore have a lower probability of satisfying the trigger
requirements. Secondly, at low $q^{2}$ the Łbaryon has a larger fraction of
the $\mathchar 28931\relax^{0}_{b}$ momentum and is more likely to decay
outside of the acceptance. The uncertainties combine both statistical and
systematic contributions (with the latter dominating) and include a small
correlated uncertainty due to the use of a single sample of $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays as the normalisation channel for all
$q^{2}$ intervals. The systematic uncertainties are described in more detail
in Sect. 7.
Table 2: Total relative efficiency, $\varepsilon_{\mathrm{rel}}$, between $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ and $\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax$ decays. The uncertainties are the combination of both statistical and systematic components, and are dominated by the latter. $q^{2}$ interval [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ] | $\varepsilon_{\mathrm{rel}}$
---|---
0.00–2.00 | $0.48\pm 0.07$
2.00–4.30 | $0.74\pm 0.08$
4.30–8.68 | $0.88\pm 0.09$
10.09–12.86 | $1.19\pm 0.12$
14.18–16.00 | $1.36\pm 0.14$
16.00–20.30 | $1.28\pm 0.15$
## 7 Systematic uncertainties
### 7.1 Yields
Three separate sources of systematic uncertainty on the measured yields are
considered for both the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ and $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ decay
modes: the definition of the signal PDF, the definition of the background PDF
and the choice of the fixed parameters used in the fits to data.
For the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, the default signal PDF is replaced by a
single Gaussian function. A 2.0 % change in signal yield relative to the
default fit is observed and assigned as the systematic uncertainty. The shape
of the combinatorial background function is changed from the default first-
order polynomial to a second-order polynomial. The 1.8 % change in the signal
yield is assigned as the systematic uncertainty. To estimate the sensitivity
of the background process
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ to differences between data and
simulation, the shape of this background is varied in the fit. A relative
uncertainty of 4.7 % is assigned. For $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ decays,
as the parameter values of the signal PDF are from fits to the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ data, the uncertainty in the signal shape is
accounted for by using the signal shape parameters and covariance matrix
obtained from the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ mass fit. The dependence on the shape of the
signal PDF is investigated by fitting data using the parameters determined
from the single-Gaussian function treatment of the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ data described above. The combinatorial
background modelling is studied in the same way as for the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays. The systematic uncertainties on the yield
in each $q^{2}$ interval are summarised in Table 3, where the total is the sum
in quadrature of the three individual components.
Table 3: Absolute systematic uncertainties on the yields for the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ decay. | $q^{2}$ interval [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ]
---|---
Source | 0.00– | 2.00– | 4.30– | 10.09– | 14.18– | 16.00–
| 2.00 | 4.30 | 8.68 | 12.86 | 16.00 | 20.30
Signal PDF | 0.08 | 0.08 | 0.16 | 0.4 | 0.08 | 2.3
Combinatorial background | 2.7 | 0.7 | 0.21 | 3.5 | 2.2 | 2.5
Signal shape parameters | 0.04 | 0.08 | 0.09 | 0.4 | 0.17 | 1.1
Total | 2.7 | 0.7 | 0.28 | 3.5 | 2.2 | 3.5
No additional uncertainty is assigned to account for finite peaking
background, as constraining it to the prediction from simulated
$B^{0}\\!\rightarrow K^{0}_{\rm\scriptscriptstyle S}\mu^{+}\mu^{-}$ decays has
a negligible effect.
### 7.2 Relative efficiencies
In measuring the $q^{2}$ dependence of the differential branching fraction,
three types of correlation are taken into account: those between the
normalisation and signal decays; those between the different $q^{2}$ regions;
and those between the geometric, selection and trigger efficiencies. For
simplicity, correlations among $q^{2}$ intervals are taken into account where
a systematic uncertainty is significant and neglected where a given
uncertainty is small compared to the dominant sources. Overall, the dominant
systematic effect identified is that related to the current knowledge of the
angular structure of the decays and $q^{2}$ dependence of the decay channels.
The uncertainty due to the finite size of simulated samples used is comparable
to that from other sources considered, and is summarised together with all
other contributions to the relative efficiency in Table 4, where the total is
the sum in quadrature of the individual components.
Table 4: Absolute systematic uncertainties on the total relative efficiency, $\varepsilon_{\mathrm{rel}}$. | $q^{2}$ interval [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ]
---|---
Source | 0.00– | 2.00– | 4.30– | 10.09– | 14.18– | 16.00–
| 2.00 | 4.30 | 8.68 | 12.86 | 16.00 | 20.30
Simulated sample size | 0.014 | 0.015 | 0.015 | 0.025 | 0.04 | 0.032
Decay structure | 0.05 | 0.07 | 0.08 | 0.11 | 0.13 | 0.12
Polarisation | 0.007 | 0.007 | 0.011 | 0.014 | 0.015 | 0.05
$\mathchar 28931\relax$ reconstruction efficiency | 0.027 | 0.009 | 0.003 | $<$0.001 | 0.003 | 0.004
Production kinematics | 0.023 | 0.005 | 0.007 | 0.026 | 0.014 | 0.05
Neural network | 0.021 | 0.027 | 0.032 | 0.021 | 0.002 | 0.04
Total | 0.07 | 0.08 | 0.09 | 0.12 | 0.14 | 0.15
#### 7.2.1 Decay structure and production polarisation
The main factors that affect the detection efficiencies are the angular
structure of the decays and the production polarisation. Although these arise
from different parts of the process, the efficiencies are linked and therefore
are treated together.
For the $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ decay, the impact of the limited knowledge of the
production polarisation, $P_{b}$, is estimated by comparing the default
efficiency with that in either of the fully polarised scenarios, $P_{b}=\pm
1$, taking the larger difference as the associated uncertainty. To assess the
systematic uncertainty due to the decay structure, the efficiency from the
default model [34, 35, 36] is compared with that from the phase space decay,
taking the larger of this difference or the statistical precision as the
systematic uncertainty.
For the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ mode, the default phase space decay is compared
with the efficiency derived using the model from Ref. [37], which depends on
the polarisation parameter $P_{b}$ and four complex amplitudes. While fixing
$P_{b}=0$, a scan of the four complex amplitudes is made and the distribution
of the change in efficiency relative to the default is constructed. The sum in
quadrature of the mean and r.m.s. of this distribution is assigned as the
systematic uncertainty due to the decay structure.
To assess the importance of the production polarisation, this exercise is
repeated while setting $P_{b}=\pm 1$. The sum in quadrature of the mean and
r.m.s. of the distribution of deviations from the default gives the combined
effect of decay structure and production polarisation. The systematic
uncertainty due to production polarisation alone is determined by subtracting
in quadrature the systematic uncertainty due to the decay structure.
The impact of $P_{b}$ on the efficiencies is found to be small using the fully
polarised scenarios, which are a conservative variation relative to the recent
measurement of Ref. [38].
#### 7.2.2 Lifetime of $\mathchar 28931\relax^{0}_{b}$ baryon
The $\mathchar 28931\relax^{0}_{b}$ baryon lifetime used throughout is
1.425${\rm\,ps}$ [30] and the systematic uncertainty associated with this
assumption is investigated by varying the lifetime by one standard deviation
(0.032${\rm\,ps}$). No significant effect is found.
#### 7.2.3 Reconstruction efficiency for $\mathchar 28931\relax$ baryon
The $\mathchar 28931\relax$ baryon is reconstructed from either long or
downstream tracks, and their relative proportions differ between data and
simulation. For simulated $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, $(21.1\pm 0.2)$ % of $\mathchar
28931\relax$ baryon candidates are reconstructed from long tracks, compared to
$(26.4\pm 0.7)$ % in data. For the phase space decay distribution of simulated
$\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ decays, $(21.5\pm 0.1)$ % (integrated over $q^{2}$)
are long tracks, indicating that both decay modes have a similar behaviour. To
account for a potential effect due to the different fractions of long and
downstream tracks observed in data and simulation, the efficiencies are first
determined separately for $\mathchar 28931\relax$ baryon candidates formed
exclusively from long and from downstream tracks. A new relative efficiency is
then determined, setting the fraction of downstream tracks to 27 % for
simulated $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays, and increasing it by 5 % in each $q^{2}$
interval for simulated $\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-}$ decays. The systematic uncertainty from this source
is assigned as the difference between this reweighted efficiency and the
default case.
#### 7.2.4 Production kinematics
There is a small difference between data and simulation in the momentum and
transverse momentum distributions of the Łbaryon produced in the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decays. Simulated data are reweighted to
reproduce these distributions in data, and the differences in the relative
efficiencies with respect to the default are assigned as the systematic
uncertainty due to production kinematics.
#### 7.2.5 Modelling of neural network observables
A discrepancy is observed between data and simulation in the neural network
response for $\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ decay candidates. This is due to differences
between $\chi^{2}$ distributions in data and simulation. A systematic
uncertainty is assigned as the change relative to the default efficiency after
all efficiencies are recalculated using reweighted neural network input
variables.
## 8 Results and conclusion
The relative differential branching fraction is measured in each $q^{2}$
interval as
$\frac{1}{{\cal B}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax)}\frac{\mathrm{d}{\cal B}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-})}{\mathrm{d}q^{2}}=\frac{N_{\mathrm{S}}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-})}{N_{\mathrm{S}}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax)}\frac{1}{\varepsilon_{\mathrm{rel}}}{\cal
B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-})\frac{1}{\Delta q^{2}}\;,$ (3)
where $\Delta q^{2}$ represents the width of the given $q^{2}$ interval.
For $q^{2}$ regions in which no statistically significant signal is observed,
an upper limit on $\mathrm{d}{\cal B}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-})/\mathrm{d}q^{2}$ is calculated using the following
Bayesian approach. The signal PDF for $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ decays
is reparametrised in terms of the relative differential rate of Eq. 3,
$N_{\mathrm{S}}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax)$, $\varepsilon_{\mathrm{rel}}$ and $\cal
B$(${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\\!\rightarrow\mu^{+}\mu^{-}$). The known uncertainties on the
$\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$ yield and $\varepsilon_{\mathrm{rel}}$ are
included in the fit with Gaussian constraints and the profile likelihood over
the relative branching fraction is then obtained. An upper limit is set at the
value where the posterior likelihood corresponds to 90 % (95 %). A uniform
prior between zero and $3\times 10^{-3}$ is used. The limits on the absolute
differential branching fractions are given by the product of the relative
limit and $\cal B$($\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$) and include the uncertainty on $\cal
B$($\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax$) from Ref. [30].
Table 5: Measured relative differential branching fraction, $(1/{\cal B}(\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax))\;\mathrm{d}{\cal B}(\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-})/\mathrm{d}q^{2}$. The first uncertainty is statistical and the second is systematic. The systematic uncertainty includes the small, correlated component due to ${\cal B}({J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\\!\rightarrow\mu^{+}\mu^{-})=(5.93\pm 0.06)\times 10^{-2}$ [30]. The rightmost column gives the 90 % (95 %) confidence level upper limit (UL) on the relative branching fraction in $q^{2}$ intervals where no significant signal is observed. $q^{2}$ interval [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ] | $\dfrac{1}{{\cal B}(\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax)}\dfrac{\mathrm{d}{\cal B}}{\mathrm{d}q^{2}}$ $[10^{-4}({\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}})^{-1}]$ | UL $[10^{-4}({\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}})^{-1}]$
---|---|---
0.00 – 2.00 | $0.45\pm 0.62\pm 0.64$ | 1.7 (2.1)
2.00 – 4.30 | $0.50\pm 0.41\pm 0.11$ | 1.3 (1.5)
4.30 – 8.68 | $0.25\pm 0.27\pm 0.03$ | 0.7 (0.9)
10.09 – 12.86 | $0.90\pm 0.34\pm 0.26$ | –
14.18 – 16.00 | $1.26\pm 0.38\pm 0.25$ | –
16.00 – 20.30 | $1.76\pm 0.29\pm 0.27$ | –
Table 6: Measured differential branching fraction, $\mathrm{d}{\cal B}(\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-})/\mathrm{d}q^{2}$, for $\cal B$($\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax$)$=(6.2\pm 1.4)\times 10^{-4}$ [30], where the first uncertainty is statistical, the second systematic and the third from the uncertainty in $\cal B$($\mathchar 28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\mathchar 28931\relax$). The rightmost column gives the 90 % (95 %) confidence level upper limit (UL) on the branching fraction in $q^{2}$ intervals where no significant signal is observed. $q^{2}$ interval [${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ ] | $\mathrm{d}{\cal B}/\mathrm{d}q^{2}$ $[10^{-7}({\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}})^{-1}]$ | UL $[10^{-7}({\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}})^{-1}]$
---|---|---
0.00 – 2.00 | $0.28\pm 0.38\pm 0.40\pm 0.06$ | 1.2 (1.5)
2.00 – 4.30 | $0.31\pm 0.26\pm 0.07\pm 0.07$ | 0.9 (1.1)
4.30 – 8.68 | $0.15\pm 0.17\pm 0.02\pm 0.03$ | 0.5 (0.6)
10.09 – 12.86 | $0.56\pm 0.21\pm 0.16\pm 0.12$ | –
14.18 – 16.00 | $0.79\pm 0.24\pm 0.15\pm 0.17$ | –
16.00 – 20.30 | $1.10\pm 0.18\pm 0.17\pm 0.24$ | –
Figure 4: Measured differential branching fraction for the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ decay.
In regions without a significant signal, the 90 % confidence level upper
limits are also shown. The uncertainties due to components that are fully
correlated across all $q^{2}$ bins, e.g. the branching fraction of the
normalisation channel from Ref. [30], are not included in this figure. The
dashed red line with the filled area shows the theoretical prediction from
Ref. [14].
The measured relative differential branching fraction is presented in Table 5,
while the absolute differential branching fraction is given in Table 6 and
shown in Fig. 4. The integrated relative branching fraction is obtained as the
sum of the differential rates in six $q^{2}$ intervals (weighted by $\Delta
q^{2}$). This gives the integral over the full phase space, with the exception
of the $q^{2}$ regions corresponding to the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and $\psi{(2S)}$ resonances. In
this integration the statistical uncertainties are added in quadrature.
Systematic uncertainties on the $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ yield
and the relative efficiency are treated as uncorrelated. The remaining
systematic uncertainties, including the statistical and systematic
uncertainties in the normalisation mode yield from Ref. [30], are treated as
fully correlated. This leads to the relative branching fraction of
$\frac{{\cal B}(\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-})}{{\cal B}(\mathchar
28931\relax^{0}_{b}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\mathchar 28931\relax)}=(1.54\pm 0.30\mathrm{\,(stat)}\pm
0.20\mathrm{\,(syst)}\pm 0.02\,(\mathrm{norm}))\times 10^{-3}\;,$
which corresponds to the absolute branching fraction
${\cal B}(\mathchar 28931\relax^{0}_{b}\\!\rightarrow\mathchar
28931\relax\mu^{+}\mu^{-})=(0.96\pm 0.16\mathrm{\,(stat)}\pm
0.13\mathrm{\,(syst)}\pm 0.21\,(\mathrm{norm}))\times 10^{-6}\;,$
where the last uncertainty accounts for the branching fraction of the
normalisation mode [30].
These new measurements of the branching fraction and differential branching
fraction for the rare decay $\mathchar
28931\relax^{0}_{b}\\!\rightarrow\mathchar 28931\relax\mu^{+}\mu^{-}$ are
based on a yield of $78\pm 12$ signal decays obtained from data, corresponding
to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, collected at a centre-
of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$. Evidence for this process
is found for $q^{2}>m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}^{2}$ and
is compatible with previous measurements by the CDF collaboration [16]. Within
the precision of measurements presented in this Letter, the Standard Model
predictions of Ref. [14] provide a good description of the data.
## 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); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from 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 thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-06-11T16:42:07 |
2024-09-04T02:49:46.352982
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, 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, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R.\n Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M.\n Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M.\n Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M.\n Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra\n Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, P. Griffith, 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, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, M. Holtrop, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten,\n R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M.\n Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, 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, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O.\n Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C.\n Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi,\n P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A.\n Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, B. Mc\n Skelly, J. McCarthy, A. McNab, R. McNulty, B. Meadows, F. Meier, M. Meissner,\n M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D.\n Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D. Nguyen, T.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem,\n A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch,\n A. Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, 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. Salzmann, B. Sanmartin Sedes, M. Sannino,\n R. Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n I. Sepp, N. Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, P.\n Shatalov, Y. Shcheglov, T. Shears, L. Shekhtman, O. Shevchenko, V.\n Shevchenko, A. Shires, R. Silva Coutinho, M. Sirendi, T. Skwarnicki, N.A.\n Smith, E. Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro,\n D. Souza, B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S.\n Stahl, O. Steinkamp, S. Stoica, S. Stone, B. Storaci, M. Straticiuc, U.\n Straumann, V.K. Subbiah, L. Sun, S. Swientek, V. Syropoulos, M. Szczekowski,\n P. Szczypka, T. Szumlak, S. T'Jampens, M. Teklishyn, E. Teodorescu, F.\n Teubert, C. Thomas, E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S.\n Tolk, D. Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur,\n M.T. Tran, M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda\n Garcia, A. Ukleja, D. Urner, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti,\n A. Vallier, M. Van Dijk, R. Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez\n Sierra, S. 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"submitter": "Michal Kreps",
"url": "https://arxiv.org/abs/1306.2577"
}
|
1306.2740
|
# A Partial Hamiltonian Approach for Current Value Hamiltonian Systems
R. Naza, F. M. Mahomedb and Azam Chaudhryc
a Centre for Mathematics and Statistical Sciences, Lahore School of Economics,
Lahore, 53200, Pakistan
b Differential Equations, Continuum Mechanics and Applications, School of
Computational and Applied Mathematics, University of the Witwatersrand, Wits
2050, South Africa
c Department of Economics, Lahore School of Economics, Lahore, 53200, Pakistan
Abstract
We develop a partial Hamiltonian framework to obtain reductions and closed-
form solutions via first integrals of current value Hamiltonian systems of
ordinary differential equations (ODEs). The approach is algorithmic and
applies to many state and costate variables of the current value Hamiltonian.
However, we apply the method to models with one control, one state and one
costate variable to illustrate its effectiveness. The current value
Hamiltonian systems arise in economic growth theory and other economic models.
We explain our approach with the help of a simple illustrative example and
then apply it to two widely used economic growth models: the Ramsey model with
a constant relative risk aversion (CRRA) utility function and Cobb Douglas
technology and a one-sector AK model of endogenous growth are considered. We
show that our newly developed systematic approach can be used to deduce
results given in the literature and also to find new solutions.
keyword: Current value Hamiltonian, partial Hamiltonian approach, economic
growth models
## 1 Introduction
There has been extensive use of dynamic optimization in economic modeling and
many of these models use the current value Hamiltonian whenever the integrand
function contains a discount factor. These models range from those used for
neoclassical economic growth ([1], [2]) to optimal firm-level investment [3]
and human capital and earnings [4]. Pontrygin’s maximum principle provides a
set of necessary conditions for the solution of the continuous time optimal
control problem involving a current value Hamiltonian and a dynamical system
of ODEs is obtained for control, state and costate variables. Beginning with
[5] there have been various approaches, both qualitative and quantitative (see
[6] for a good account of these), to deal with dynamic economic models arising
from current value Hamiltonian system and most of these models were solved
using numerical approaches (like [7]) or linear approximations around steady
states ([8]). The critical problem is that for the underlying nonlinear
dynamical system in economics there is a lack of a general analytical solution
procedure not only for higher order systems but even for systems with one
state and costate variable.
It is true to say that nonlinear dynamical systems evade closed-form solutions
in general. However, the lack of a general procedure inhibits the search for
reductions and solutions of such type of nonlinear equations even when
solutions do exist. Having said that, there are some well-known closed-form
solutions that appear in the literature (see, e.g. [9, 10, 11, 12, 13, 14]).
These solutions have been obtained by seemingly disparate approaches.
Independent of the knowledge of explicit solutions, dynamic local stability of
certain systems (see [15, 16, 17]) have been characterized by qualitative or
numerical approaches.
Several important contributions have been made in the analysis of nonlinear
dynamical systems of economic models. Here we focus on a new approach which
yields first reductions and closed-form solutions for such systems of ODEs. We
develop a Hamiltonian framework for several control, state and costate
variables. Therefore, the method we develop is applicable to an arbitrary
system of ODEs. However, we apply it to a system of two ODEs in order to show
its effectiveness. In the case of higher order systems of ODEs, the approach
may require the use of algebraic computing.
The layout of the paper is as follows. The partial Hamiltonian approach is
developed in Section 2. In Section 3 we provide a simple illustrative example
to show how our approach works. The Ramsey model with a constant relative risk
aversion (CRRA) utility function with Cobb Douglas technology and the one-
sector AK model of endogenous growth are studied in Section 4 and known
solutions are deduced via our partial Hamiltonian approach. Conclusions are
finally presented in Section 5.
## 2 A Hamiltonian version of the Noether-type theorem
Herein we develop a partial Hamiltonian approach for current value
Hamiltonians which do not satisfy the canonical Hamilton equations. This is
done for several control, state and costate variables.
Let $t$ be the independent variable and
$(q,p)=(q^{1},...,q^{n},p_{1},...,p_{n})$ the phase space coordinates. The
derivatives of $q^{i}$, $p_{i}$ with respect to $t$ are
$\dot{p}_{i}=D(p_{i}),\;\dot{q}_{i}=D(q_{i}),\;i=1,2,\cdots,n,$ (1)
where
$D=\frac{\partial}{\partial t}+\dot{q}_{i}\frac{\partial}{\partial
q_{i}}+\dot{p}_{i}\frac{\partial}{\partial p_{i}}+\cdots$ (2)
is the total derivative operator with respect to $t$. The summation convention
is utilized for repeated indices. The variables $t,q,p$ are independent and
connected only by the differential relations (1).
There are some well-known operators which are defined in the space of the
variables $(t,q,p)$ and its prolongations. We introduce them.
In addition to the Euler operator
$\frac{\delta}{\delta q^{i}}=\frac{\partial}{\partial
q^{i}}-D\frac{\partial}{\partial\dot{q}^{i}},i=1,2,\cdots,n,$ (3)
one also has the variational operator
$\frac{\delta}{\delta p_{i}}=\frac{\partial}{\partial
p_{i}}-D\frac{\partial}{\partial\dot{p}_{i}},i=1,2,\cdots,n.$ (4)
The action of the operators (3) and (4) on
$L(t,q,\dot{q})=p_{i}\dot{q}^{i}-H(t,q,p)$ (5)
equated to zero yields the canonical Hamilton equations
$\displaystyle\dot{q}^{i}=\frac{\partial H}{\partial p_{i}},$ (6)
$\displaystyle\dot{p}_{i}=-\frac{\partial H}{\partial q^{i}},i=1,\ldots,n.$
That is ${\delta L}/{\delta q^{i}}=0$ and ${\delta L}/{\delta p_{i}}=0$
results in (6). Equation (5) is the well-known Legendre transformation which
relates the Hamiltonian and Lagrangian, where $p_{i}=\partial
L/\partial\dot{q}^{i}$ and $\dot{q}^{i}=\partial H/\partial\dot{p}_{i}$.
Generators of point symmetries in the space $(t,q,p)$ are operators of the
form
$X=\xi(t,q,p)\frac{\partial}{\partial
t}+\eta^{i}(t,q,p)\frac{\partial}{\partial
q^{i}}+\zeta_{i}(t,q,p)\frac{\partial}{\partial p_{i}}.$ (7)
The operator in (7) is a generator of a point symmetry of the canonical
Hamiltonian system (6) if ([18])
$\displaystyle\dot{\eta}^{i}-\dot{q}^{i}\dot{\xi}-X(\frac{\partial H}{\partial
p_{i}})=0,$
$\displaystyle\dot{\zeta}_{i}-\dot{p}_{i}\dot{\xi}+X(\frac{\partial
H}{\partial q^{i}})=0,\;i=1,\ldots,n$ (8)
on the system (6).
Hamiltonian symmetries in evolutionary or canonical form have been considered
([18]). Furthermore, symmetry properties of the Hamiltonian action have been
investigated in the space $(t,q,p)$ by [19] and [20]. In the latter, the
authors considered the general form of the symmetries (7) and provided a
Hamiltonian version of Noether’s theorem.
The following important results which are analogs of Noether symmetries and
the Noether theorem (see [18, 21, 22, 20] for a discussion) were established.
Theorem 1 (Hamilton action symmetries): A Hamiltonian action
$p_{i}dq^{i}-Hdt$ (9)
is invariant up to gauge $B(t,q,p)$ with respect to a group generated by (7)
if and only if the condition
$\zeta_{i}\frac{\partial H}{\partial
p_{i}}+p_{i}D(\eta^{i})-X(H)-HD(\xi)-D(B)=0,$ (10)
holds.
Theorem 2 (Hamiltonian version of Noether’s theorem): The canonical Hamilton
system (6) which is invariant has the first integral
$I=p_{i}\eta^{i}-\xi H-B$ (11)
for some gauge function $B=B(t,q,p)$ if and only if the Hamiltonian action is
invariant up to divergence with respect to the operator $X$ given in (7) on
the solutions to equations (6).
We now focus our attention on systems of equations which are not in the
canonical form (6). Therefore the Theorems 1 and 2 do not apply for these
systems. We need an extension of the existing results which we carry out
below.
Since the current value Hamiltonian (see, e.g. [23]) satisfies
$\displaystyle\dot{q}^{i}=\frac{\partial H}{\partial p_{i}},$ (12)
$\displaystyle\dot{p}^{i}=-\frac{\partial H}{\partial
q^{i}}+\Gamma_{i},\;i=1,2,\cdots,n,$
where $\Gamma_{i}$ is a nonzero function, we seek an extension of the results
relating to the canonical Hamiltonian system to the system (12) so that we can
obtain first integrals of system (12) in an algorithmic manner. We also refer
to an $H$ that satisfies (12) as a partial Hamiltonian.
It is opportune to remark that $X$ as in (7) is a generator of point symmetry
of the current value Hamiltonian system (12) if
$\displaystyle\dot{\eta}^{i}-\dot{q}^{i}\dot{\xi}-X(\frac{\partial H}{\partial
p_{i}})=0,$
$\displaystyle\dot{\zeta}_{i}-\dot{p}_{i}\dot{\xi}+X(\frac{\partial
H}{\partial q^{i}}-\Gamma_{i})=0,\;i=1,\ldots,n$ (13)
on the system (12). Note that (13) is evidently different from (8) due to the
nonzero term $\Gamma_{i}$.
We introduce the definition of what we call the partial Hamiltonian operator
below. This is motivated by the analogous definition of the partial Noether
operator given in [24, 25].
Definition 1: An operator $X$ of the form (7) is a partial Hamiltonian
operator corresponding to a current value Hamiltonian as in (12), if there
exists a function $B(t,q,p)$ such that
$\zeta_{i}\frac{\partial H}{\partial
p_{i}}+p_{i}D(\eta^{i})-X(H)-HD(\xi)=D(B)+(\eta^{i}-\xi\frac{\partial
H}{\partial p_{i}})(-\Gamma_{i})$ (14)
holds.
Note that if $H$ is a present value Hamiltonian, then equation (14) becomes
the usual determining equation for symmetries of the Hamiltonian action since
$\Gamma_{i}=0$ in this case.
Also one can immediately see from (14) that if $X$ and $Y$ are partial
Hamiltonian operators, then so is a linear combination of these Hamiltonian
operators.
We now have the following important theorem on how one constructs first
integrals for the system (12). That is we present the partial Hamiltonian
approach for current value Hamiltonians. This is achieved for several control,
state and costate variables.
Theorem 3 (partial Hamiltonian version pf the partial Noether theorem): An
operator $X$ of the form (7) is a partial Hamiltonian operator of the current
value Hamiltonian $H$ corresponding to system (12) if and only if (11) is its
first integral.
Proof: The result follows by straightforward differentiation of the first
integral formula (11) on the solutions of system (12). However, one has to
remember that the terms involving $\dot{p}_{i}$ need to be replaced by the
right hand side of the second equation of system (12) which has a non-zero
function $\Gamma_{i}$. Another way to show this to be the case is to utilize
$D(H)|_{\dot{q}^{i}=\frac{\partial H}{\partial
p_{i}},\dot{p}^{i}=-\frac{\partial H}{\partial
q^{i}}+\Gamma_{i}}=H_{t}+\Gamma_{i}\frac{\partial H}{\partial p_{i}}$ (15)
as well as the identity
$\displaystyle\zeta_{i}\dot{q}^{i}+p_{i}D(\eta^{i})-X(H)-HD(\xi)-D(B)-(\eta^{i}-\xi\frac{\partial
H}{\partial p_{i}})(-\Gamma_{i})$
$\displaystyle=\xi(D(H)-H_{t}-\Gamma_{i}\frac{\partial H}{\partial
p_{i}})-\eta^{i}(\dot{p}^{i}+\frac{\partial H}{\partial
q^{i}}-\Gamma_{i})+\zeta_{i}(\dot{q}^{i}-\frac{\partial H}{\partial p_{i}})$
$\displaystyle+D(p_{i}\eta^{i}-\xi H-B)$ (16)
which holds for any smooth functions $H(t,q,p)$ and suitable functions
$B(t,q,p)$ and $\Gamma_{i}$. This identity follows from direct computations.
Remark. An approach in proving Theorem 3 is by invoking the Legendre
transformation (5) on the partial Noether operators and partial Noether
theorem given in [24].
## 3 A Simple illustrative example
Consider the following mathematical example:
Maximize
$\int_{0}^{\infty}[\alpha q-\beta q^{2}-\alpha u^{2}-\gamma u]e^{-rt}dt$ (17)
subject to
$\dot{q}=u,$ (18)
where $\alpha,\beta,\gamma$ are all positive, $r$ is a discount factor, $q(t)$
is the state variable and $u(t)$ is the control variable.
Hamiltonian function and maximum principle:
The current value Hamiltonian function is defined as
$H(t,q,p,u)=\alpha q-\beta q^{2}-\alpha u^{2}-\gamma u+pu$ (19)
where $p(t)$ is called the costate variable. The necessary first order
conditions for optimal control are [23]:
$\frac{\partial H}{\partial u}=0$ (20) $\dot{q}=\frac{\partial H}{\partial p}$
(21) $\dot{p}=-\frac{\partial H}{\partial q}+rp$ (22)
Equation (20)-(22) with $H$ given by (19) yields
$p=2\alpha u+\gamma$ (23) $\dot{q}=u$ (24) $\dot{p}=2\beta q-\alpha+pr$ (25)
Equations (23)-(25) need to be solved for $p(t),q(t),u(t)$. Of course, the
direct way to solve this problem is to eliminate $p$, $u$ by utilizing
(23)-(25) in order to obtain a scalar linear second order ordinary
differential equation in $q$, which is amenable to straightforward
integration. We explain here how we can find the solution by using the partial
Hamiltonian approach introduced above.
Determination of Partial Hamiltonian operators:
The partial Hamiltonian operator determining equation is given in (14)
Expansion of equation (14) yields
$\displaystyle p(\eta_{t}+\dot{q}\eta_{q})-\eta(\alpha-2\beta q)-(\alpha
q-\beta q^{2}-\alpha u^{2}-\gamma u+pu)(\xi_{t}+\dot{q}\xi_{q})$ (26)
$\displaystyle=B_{t}+\dot{q}B_{q}+(\eta-\xi u)(-rp),$
in which we assume that $\xi=\xi(t,q)$, $\eta=\eta(t,q)$, $B=B(t,q)$.
Note that one can also assume these functions to be dependent on $p$. We have
chosen $(t,q)$ dependence to simplify the calculations here and in the models
considered in Section 4 with the purpose of deriving solutions. This
assumption leads to at least one partial Hamiltonian operator. More general
assumptions are used in the event that one does not obtain an operator.
With the help of (23)-(24), Equation (26) can be written as
$\displaystyle(2\alpha u+\gamma)(\eta_{t}+u\eta_{q})-(\xi_{t}+u\xi_{q})(\alpha
q-\beta q^{2}-\alpha u^{2}-\gamma u+2\alpha u^{2}+\gamma u)$ (27)
$\displaystyle-\eta(\alpha-2\beta q)=B_{t}+uB_{q}+(\eta-\xi u)(-2r\alpha
u-\gamma r).$
Separating equation (27) with respect to powers of $u$ as $\xi,\eta,B$ do not
contain $u$, we have
$\displaystyle u^{3}:-\alpha\xi_{q}=0,$ (28) $\displaystyle
u^{2}:2\alpha\eta_{q}-\alpha\xi_{t}=2\alpha r\xi,$ (29) $\displaystyle
u:2\alpha\eta_{t}+\gamma\eta_{q}=B_{q}-2\alpha r\eta+\gamma r\xi,$ (30)
$\displaystyle u^{0}:\gamma\eta_{t}-\eta(\alpha-2\beta q)-\xi_{t}(\alpha
q-\beta q^{2})=B_{t}-r\gamma\eta.$ (31)
System (28)-(30) yields
$\displaystyle\xi=a(t),\;\eta=(\frac{1}{2}\dot{a}+ra)q+b(t),$ (32)
$\displaystyle B=\alpha(\frac{1}{2}\ddot{a}+r\dot{a})q^{2}+\alpha
r(\frac{1}{2}\dot{a}+ra)q^{2}+2\alpha\dot{b}q+2\alpha
rbq+\frac{1}{2}\gamma\dot{a}q+d(t).$
Substituting $\xi,\eta,B$ from (32) in (31) and then separating w.r.t powers
of $q$ we have
$\displaystyle q^{2}:\frac{1}{2}\alpha\dddot{a}+\frac{3}{2}\alpha
r\ddot{a}+(\alpha r-2\beta)\dot{a}-2\beta ra=0,\qquad$ (33) $\displaystyle
q:\frac{3}{2}(r\gamma-\alpha)\dot{a}+r(r\gamma-\alpha)a+2b\beta=2\alpha\ddot{b}+2\alpha
r\dot{b},$ (34) $\displaystyle q^{0}:\gamma\dot{b}-\alpha b+\gamma
rb=\dot{d}.\qquad\qquad\qquad\qquad\qquad\qquad$ (35)
The solution of equations (33)-(35) for $a,b,d$ with general
$\alpha,\beta,\gamma,r$ is purely formal and depend on the roots of the
characteristic equation. Clearly there are three lengthy solutions for $a$ and
two for $b$. To be transparent, we have selected values. Therefore we seek a
solution of equations (33)-(35) for $a,b,d$ with specific values
$\alpha=\gamma=r=1$ and $\beta=2$ and we arrive at
$\displaystyle a(t)=c_{1}e^{-t}+c_{2}e^{2t}+c_{3}e^{-4t},$ $\displaystyle
b(t)=c_{4}e^{t}+c_{5}e^{-2t},$ (36) $\displaystyle
d(t)=c_{4}e^{t}+c_{5}e^{-2t}+c_{6},$
where $c_{1},\cdots,c_{6}$ are arbitrary constants. Finally, we obtain the
following $\xi,\eta,B$ after substituting $a,b,d$ from (36) into (32)
$\displaystyle\xi=c_{1}e^{-t}+c_{2}e^{2t}+c_{3}e^{-4t},$
$\displaystyle\eta=(\frac{1}{2}c_{1}e^{-t}+2c_{2}e^{2t}-c_{3}e^{-4t})q+c_{4}e^{t}+c_{5}e^{-2t},$
(37) $\displaystyle
B(t)=(6c_{2}e^{2t}+3c_{3}e^{-4t})q^{2}+(-\frac{1}{2}c_{1}e^{-t}+c_{2}e^{2t}-2c_{3}e^{-4t}$
$\displaystyle+4c_{4}e^{t}-2c_{5}e^{-2t})q+c_{4}e^{t}+c_{5}e^{-2t}+c_{6}.$
The generators $X_{i}$ form a vector space. By choosing one of the constants
as one and the rest as zero in turn we have the following five operators and
gauge terms:
$\displaystyle X_{1}=e^{-t}\frac{\partial}{\partial
t}+\frac{1}{2}qe^{-t}\frac{\partial}{\partial q},\;B_{1}=-\frac{1}{2}e^{-t}q$
$\displaystyle X_{2}=e^{2t}\frac{\partial}{\partial
t}+2qe^{2t}\frac{\partial}{\partial q},\;B_{2}=6q^{2}e^{2t}+qe^{2t}$
$\displaystyle X_{3}=e^{-4t}\frac{\partial}{\partial
t}-qe^{-4t}\frac{\partial}{\partial q},\;B_{3}=3q^{2}e^{-4t}-2qe^{-4t}$ (38)
$\displaystyle X_{4}=e^{t}\frac{\partial}{\partial q},B_{4}=4qe^{t}+e^{t}$
$\displaystyle X_{5}=e^{-2t}\frac{\partial}{\partial
q},B_{5}=-2qe^{-2t}+e^{-2t}.$
In general the $X_{i}$’s are not symmetries of the system
$\displaystyle\dot{q}=\frac{1}{2}p-\frac{1}{2},$
$\displaystyle\dot{p}=4q+p-1,$ (39)
which considered as a non-canonical Hamiltonian system (23)-(25) admits the
operators (38). For example in the case of $X_{4}$ we have that the first of
equations (13) gives $\zeta=2e^{t}$. However, the second equation of (13) is
not satisfied as easily can be verified.
Construction of first integrals from partial Hamiltonian operators and gauge
terms:
Now, first integrals satisfying $DI=0$, on the solutions, corresponding to
operators and gauge terms given in (38) can be computed from (11) and the
following integrals result.
$\displaystyle I_{1}=[\frac{1}{2}pq-(q-2q^{2}-u^{2}-u+pu)+\frac{1}{2}]e^{-t},$
$\displaystyle I_{2}=[2pq-(q-2q^{2}-u^{2}-u+pu)-6q^{2}-q]e^{2t},$ (40)
$\displaystyle I_{3}=[-pq-(q-2q^{2}-u^{2}-u+pu)-3q^{2}+2q]e^{-4t},$
$\displaystyle I_{4}=[p-4q-1]e^{t},$ $\displaystyle I_{5}=[p+2q-1]e^{-2t}.$
There are five first integrals, two of which are functionally independent.
Optimal solution via first integrals:
Equations (23)-(25) need to be solved for $p(t),q(t),u(t)$ with
$\alpha=\gamma=r=1,\beta=2$. We demonstrate here how one can find a solution
by using first integrals. We derive the solution associated with the first
integral $I_{4}$. As $DI=0$, on the solutions, and thus $I=constant$, we have
$[p-4q-1]e^{t}=A_{1},$ (41)
where $A_{1}$ is an arbitrary constant and this gives
$p(t)=4q+1+A_{1}e^{-t}.$ (42)
From (23), $u=\frac{p-1}{2}$ and after using $p$ from (42), we have
$u(t)=\frac{4q+A_{1}e^{-t}}{2}.$ (43)
Thus if $q(t)$ is known we can get the optimal path $p(t)$ and $u(t)$ from
(42) and (43). Equation (24) with $u$ from (43) yields
$\dot{q}=\frac{4q+A_{1}e^{-t}}{2},$ (44)
and this is a first order linear equation in $q(t)$. The solution of equation
(44) is
$q(t)=\frac{A_{1}}{2}e^{-t}+A_{2}e^{2t},$ (45)
where $A_{1}$ and $A_{2}$ are arbitrary constants which we can specify if we
have given initial and terminal conditions. One can use any one of the first
integrals (40) to obtain the general solution to this linear problem.
Generally a first integral provides a reduction of order of the system by one.
One can also achieve reduction to quadratures of the system by invoking first
integrals as we see in the examples that follow.
## 4 Optimal path of some economic models
### 4.1 Ramsey neoclassical model with CRRA utility function
We consider the following Ramsey neoclassical growth model [8], [26], where
the representative consumer’s utility maximization problem is defined as
${Max}\quad\int_{0}^{\infty}e^{-rt}c^{1-\sigma}dt,\;\sigma\not=0,1$ (46)
subject to the capital accumulation equation and parameter restriction
$\dot{k}(t)=k^{\beta}-\delta k-c,\;k(0)=k_{0},\;0<\beta<1,$ (47)
where $c(t)$ is the consumption per person, $k(t)$ is the capital labor ratio,
$\beta,\delta,r$ are the capital share, depreciation rate, rate of time
preferences respectively. The intertemporal elasticity of substitution is
given by $1/{\sigma}$ and $k_{0}$ is the initial capital stock.
The current value Hamiltonian function for this model is defined as
$H(t,c,k,\lambda)=c^{1-\sigma}+\lambda(k^{\beta}-\delta k-c),$ (48)
where $\lambda(t)$ is the costate variable. The necessary first order
conditions for optimal control are
$\lambda=(1-\sigma)c^{-\sigma}$ (49) $\dot{k}=k^{\beta}-\delta k-c$ (50)
$\dot{\lambda}=-\lambda(\beta k^{\beta-1}-\delta)+\lambda r$ (51)
and the transversality condition is
$\lim_{t\to\infty}e^{-rt}\lambda(t)k(t)=0.$ (52)
From (49) and (51), the growth rate of consumption is given by
$\frac{\dot{c}}{c}=\frac{\beta}{\sigma}k^{\beta-1}-\frac{1}{\sigma}(\delta+r).$
(53)
We seek a solution $\lambda(t),k(t),c(t)$ of equations (49)-(51) by utilizing
the Hamiltonian approach. The partial Hamiltonian determining equation (14)
for the Hamiltonian (48) yields
$\displaystyle\lambda(\eta_{t}+\dot{k}\eta_{k})-\eta\lambda(\beta
k^{\beta-1}-\delta)-[c^{1-\sigma}+\lambda(k^{\beta}-\delta
k-c)](\xi_{t}+\dot{k}\xi_{k})$ (54)
$\displaystyle=B_{t}+\dot{k}B_{k}+(\eta-\xi{\partial
H\over\partial\lambda})(-r\lambda),$
in which we assume that $\xi=\xi(t,k)$, $\eta=\eta(t,k)$, $B=B(t,k)$. The same
reason applies here as for the illustrative example. Equation (54) with the
help of (49)-(51) can be written as
$\displaystyle(1-\sigma)c^{-\sigma}[\eta_{t}+(k^{\beta}-\delta
k-c)\eta_{k}]-\eta(1-\sigma)c^{-\sigma}(\beta k^{\beta-1}-\delta)$ (55)
$\displaystyle-[c^{1-\sigma}+(1-\sigma)c^{-\sigma}(k^{\beta}-\delta
k-c)][\xi_{t}+(k^{\beta}-\delta k-c)\xi_{k}]$
$\displaystyle=B_{t}+(k^{\beta}-\delta
k-c)B_{k}-r(1-\sigma)c^{-\sigma}[\eta-\xi(k^{\beta}-\delta k-c)].$
Separating equation (55) with respect to powers of the control variable $c$,
we have
$\displaystyle c^{2-\sigma}:-\sigma\xi_{k}=0,$ (56) $\displaystyle
c^{1-\sigma}:-\eta_{k}(1-\sigma)-\sigma\xi_{t}+r(1-\sigma)\xi=0,$ (57)
$\displaystyle c^{-\sigma}:\eta_{t}+(k^{\beta}-\delta k)\eta_{k}-\eta(\beta
k^{\beta-1}-\delta)$ $\displaystyle-(k^{\beta}-\delta
k)\xi_{t}+r\eta-r\xi(k^{\beta}-\delta k)=0,$ (58) $\displaystyle
c,\;c^{0}:B_{k}=0,\;B_{t}=0.$ (59)
Equations (56), (57) and (59) result in
$\xi=a_{1}(t),\;\eta=(-\frac{\sigma}{1-\sigma}\dot{a}_{1}+ra_{1})k+a_{2}(t),\;B=0.$
(60)
Equation (58) with $\xi,\eta,B$ from (60) gives $a_{2}=0$ and then reduces to
$\displaystyle k^{\beta}:\dot{a}_{1}-\frac{\beta
r(1-\sigma)}{\beta\sigma-1}a_{1}=0,\;\beta\sigma\not=1,$ (61) $\displaystyle
k:-\sigma\ddot{a}_{1}+[r(1-2\sigma)+\delta(1-\sigma)]\dot{a}_{1}+r(1-\sigma)(r+\delta)a_{1}=0.$
(62)
Equation (61) is valid if $\sigma\beta\not=1$, for the case where the
capital’s share is not equal to the intertemporal elasticity of substitution.
Equations (61) and (62) yield
$a_{1}(t)=c_{1}e^{\delta\beta(1-\sigma)t}$ (63)
with
$\sigma=\frac{r+\delta}{\beta\delta}.$ (64)
The restriction on the parameters (64) is the same as the one given in [8, 26]
and our approach yields this during the solution process. Now $\xi,\eta$ and
$B$ are given by
$\xi=c_{1}e^{\delta\beta(1-\sigma)t},\;\eta=-c_{1}\delta
e^{\delta\beta(1-\sigma)t}k,B=0,$ (65)
and the only partial Hamiltonian operator is
$X=e^{\delta\beta(1-\sigma)t}\frac{\partial}{\partial t}-\delta
e^{\delta\beta(1-\sigma)t}k\frac{\partial}{\partial k},\;B=0.$ (66)
The following first integral corresponding to the partial Hamiltonian operator
and gauge terms given in (66) can be computed from (11):
$I=e^{\delta\beta(1-\sigma)t}[-\sigma
c^{1-\sigma}+(\sigma-1)c^{-\sigma}k^{\beta}].$ (67)
We write (67) as a constant, i.e.
$-\sigma
c^{1-\sigma}+(\sigma-1)c^{-\sigma}k^{\beta}=A_{1}e^{\delta\beta(\sigma-1)t}.$
(68)
From equation (68), we have
$k=[\frac{A_{1}}{\sigma-1}c^{\sigma}e^{\delta\beta(\sigma-1)t}+\frac{\sigma}{\sigma-1}c]^{\frac{1}{\beta}}.$
(69)
Our next goal is to get either $c$ or $k$. If $A_{1}=0$ we arrive at the well-
known solution given in [8, 26]. Equation (69) for $A_{1}=0$ yields
$c(t)=(1-\frac{\beta\delta}{r+\delta})k^{\beta}$ (70)
where $\frac{\sigma-1}{\sigma}=1-\frac{\beta\delta}{r+\delta}$ by (64).
Substituting $c$ from equation (70) in Equation (50) results in
$\dot{k}+\delta k=(\frac{\beta\delta}{r+\delta})k^{\beta}.$ (71)
The solution of equation (71) subject to the initial condition $k(0)=k_{0}$ is
given by
$k(t)=[\frac{\beta}{r+\delta}+(k_{0}^{1-\beta}-\frac{\beta}{r+\delta})e^{-(1-\beta)\delta
t}]^{\frac{1}{1-\beta}}.$ (72)
The solutions (70) and (72) are the same as the ones derived in [8, 26] and
satisfy the transversality condition given by (52). This guarantees that our
approach works. For $A_{1}\not=0$, we can get more solutions. Now we
substitute (69) into equation (53) determining $c$, viz.
$\frac{d}{dt}(ce^{\beta\delta t})=\frac{\beta}{\sigma}ce^{\delta\beta
t}[\frac{A_{1}}{\sigma-1}c^{\sigma}e^{\delta\beta(\sigma-1)t}+\frac{\sigma}{\sigma-1}c]^{1-\frac{1}{\beta}}.$
(73)
Introducing
$S=ce^{\beta\delta t},$ (74)
equation (73) directly results in
$\frac{\beta}{\sigma}e^{\delta(1-\beta)t}dt=\frac{dS}{S[\frac{A_{1}}{\sigma-1}S^{\sigma}+\frac{\sigma}{\sigma-1}S]^{1-\frac{1}{\beta}}},$
(75)
which provides the general solution. Here, one operator and thus one first
integral was sufficient to work out the solution. In general, one requires
two, as we have a system of two first order equations, which we wish to solve.
### 4.2 One-Sector Model of Endogenous growth: The AK model
We consider the following one-sector model of endogenous growth presented in
[8] where the representative consumer’s utility maximization problem is
${Max}\quad\int_{0}^{\infty}e^{-(\rho-n)t}\frac{c^{1-\theta}-1}{1-\theta}dt,\;\theta>0,\;\theta\not=1$
(76)
subject to
$\dot{a}(t)=(r-n)a+w-c,\;c(0)=c_{0},$ (77)
where $c(t)$ is the consumption per person, $a(t)$ is the assets per person,
$r(t)$ is the interest rate, $w(t)$ is the wage rate, and $n$ is the growth
rate of population. Suppose firms have the linear production function
$y=f(k)=Ak$ (78)
where $A>0$. The marginal product of capital is not diminishing, i.e.
$f^{\prime\prime}=0$ and this property makes it different from neoclassical
production function. The marginal product of capital is the constant $A$ and
the marginal product of labor is zero. Thus
$r=A-\delta,\ w=0$ (79)
where $\delta\geq 0$ is the depreciation rate. It is assumed that the economy
is closed and $a(t)=k(t)$ holds. If we take $a=k$, $r=A-\delta$ and $w=0$ then
our optimal control problem is to maximize (76) subject to
$\dot{k}=(A-\delta-n)k-c,\;c(0)=c_{0}.$ (80)
The current value Hamiltonian function is defined as
$H(t,c,A,\lambda)=\frac{c^{1-\theta}-1}{1-\theta}+\lambda[(A-\delta-n)k-c],$
(81)
where $c(t)$ is control variable, $k(t)$ is the state variable and
$\lambda(t)$ is the costate variable. The necessary first order conditions for
optimal control are
$\lambda=c^{-\theta}$ (82) $\dot{k}=(A-\delta-n)k-c$ (83)
$\dot{\lambda}+(A-\delta-\rho)\lambda=0.$ (84)
The transversality condition is
$\lim_{t\to\infty}e^{-(\rho-n)t}\lambda(t)k(t)=0$ (85)
and from (82) and (84), the growth rate of consumption is given by
$\frac{\dot{c}}{c}=\frac{1}{\theta}(A-\delta-\rho).$ (86)
Now we solve this model by utilizing our partial Hamiltonian approach. The
partial Hamiltonian operator determining equation (14) for Hamiltonian (81)
with $\xi=\xi(t,k)$, $\eta=\eta(t,k)$, $B=B(t,k)$ can be written as
$\displaystyle c^{-\theta}[\eta_{t}+((A-\delta-n)k-c)\eta_{k}]-\eta
c^{-\theta}(A-\delta-n)$
$\displaystyle-[\frac{c^{1-\theta}-1}{1-\theta}+c^{-\theta}((A-\delta-n)k-c)][\xi_{t}+((A-\delta-n)k-c)\xi_{k}]$
(87)
$\displaystyle=B_{t}+B_{k}((A-\delta-n)k-c)-c^{-\theta}(\rho-n)[\eta-\xi((A-\delta-n)k-c)],$
where we have used equations (82)-(84). By following the same procedure for
equation (87) as described in the previous two examples, we finally have
$\xi=a_{1}(t),\;\eta=[-\frac{\theta}{1-\theta}\dot{a}_{1}+(\rho-n)a_{1}]k+a_{2}(t),\;B=\frac{1}{1-\theta}a_{1}(t),$
(88)
$-\frac{\theta}{1-\theta}\ddot{a}_{1}+[\rho-A+\delta-\frac{(\rho-n)\theta}{1-\theta}]\dot{a}_{1}+(\rho-n)(\rho-A+\delta)a_{1}=0,$
(89) $\dot{a}_{2}+(\rho-A+\delta)a_{2}=0.$ (90)
Solving equation (89) for $a_{1}(t)$, we have
$a_{1}(t)=c_{1}e^{-(\rho-n)t}+c_{2}e^{\frac{(\rho-A+\delta)(1-\theta)}{\theta}t},$
(91)
and equation (90) yields
$a_{2}(t)=c_{3}e^{(A-\delta-\rho)t}.$ (92)
Thus $\xi,\eta,B$ are given by
$\displaystyle\xi=c_{1}e^{-(\rho-n)t}+c_{2}e^{\frac{(\rho-A+\delta)(1-\theta)}{\theta}t},$
(93)
$\displaystyle\eta=\frac{\rho-n}{1-\theta}c_{1}ke^{-(\rho-n)t}-(n-A+\delta)c_{2}ke^{\frac{(\rho-A+\delta)(1-\theta)}{\theta}t}+c_{3}e^{(A-\delta-\rho)t},$
$\displaystyle
B=\frac{1}{1-\theta}[c_{1}e^{-(\rho-n)t}+c_{2}e^{\frac{(\rho-A+\delta)(1-\theta)}{\theta}t}],$
The following first integrals corresponding to operators and gauge terms given
in (93) are computed from (11):
$\displaystyle
I_{1}=e^{-(\rho-n)t}c^{-\theta}k[\frac{\rho-n\theta+(\theta-1)(A-\delta)}{(1-\theta)}]-\frac{\theta}{1-\theta}c^{1-\theta}e^{-(\rho-n)t}$
(94) $\displaystyle
I_{2}=\frac{\theta}{\theta-1}c^{1-\theta}e^{\frac{(\rho-A+\delta)(1-\theta)}{\theta}t},\;\;I_{3}=c^{-\theta}e^{-(\rho-A+\delta)t}.$
Now we explain how to solve the AK model using the first integrals
$I_{1},I_{3}$. Setting $I_{1}=a_{1}$ and $I_{3}=a_{2}$ after some
simplifications and using the initial condition $c(0)=c_{0}$, we have
$k(t)=\frac{1-\theta}{\phi\theta}a_{1}e^{(\rho-n)t}c^{\theta}(t)+\frac{1}{\phi}c(t),$
(95)
where
$\phi=\frac{1}{\theta}[\rho-n\theta+(\theta-1)(A-\delta)]$ (96)
and
$c(t)=c_{0}e^{(\frac{A-\delta-\rho}{\theta})t},c_{0}=a_{2}^{-\frac{1}{\theta}}.$
(97)
Note that $(A-\delta-\rho)>0$ the consumption $c(t)$ given in (97) increases
with time. Substituting the value of consumption $c(t)$ in (97) into (95),
capital is given by
$k(t)=\frac{1-\theta}{\phi\theta}a_{1}c_{0}^{\theta}e^{(A-\delta-n)t}+\frac{1}{\phi}c_{0}e^{(\frac{A-\delta-\rho}{\theta})t},$
(98)
where the constant $a_{1}$ can be determined from the transversality
condition. The solutions (97) and (98) are the same as given in [8] and here
we deduced these by utilizing our partial Hamiltonian approach. The
transversality condition can be rewritten as
$\lim_{t\to\infty}e^{-(\rho-n)t}c^{-\theta}k=0$ (99)
and we need to show $\lim_{t\to\infty}e^{-(\rho-n)t}c^{-\theta}k$ is zero.
Using (97) and (98) we have
$\lim_{t\to\infty}\frac{1-\theta}{\phi\theta}a_{1}+\frac{1}{\phi}c_{0}^{1-\theta}e^{-\phi
t}$ (100)
and it tends to zero only if we choose constant $a_{1}=0$ and assume that
$\phi=\frac{1}{\theta}[\rho-n\theta+(\theta-1)(A-\delta)]>0$. This further
results in the following restriction on the parameters
$\rho+\delta>(1-\theta)(A-\delta)+n\theta+\delta$ (101)
and (95) reduces to
$k(t)=\frac{1}{\phi}c(t).$ (102)
The detailed interpretations of the solutions obtained here are given in [8].
## 5 Concluding remarks
A systematic way to obtain reductions and closed-form solutions via first
integrals of Hamiltonian systems commonly arising in economic growth theory
and other economic models is developed. This is an algorithmic approach and
can be applied to many state and costate variables of the current value
Hamiltonian. However, we applied our method to systems with one control, one
state and one costate variable. The approach was explained with the help of
one simple illustrative example. We first studied two economic growth models,
the Ramsey model with a constant relative risk aversion (CRRA) utility
function and Cobb Douglas technology and the one-sector AK model of endogenous
growth. For the Ramsey model, the solutions derived from our methodology were
the same as those derived by [8, 26]. The restriction on the parameters was
obtained in a systematic way during the solution process unlike in other
models where it was assumed. The solutions were valid for $\sigma\beta\not=1$,
the case where the capital’s share is not equal to the intertemporal
elasticity of substitution. The first integrals and closed-form solutions for
the one-sector AK model of endogenous growth were also re-derived by our
partial Hamiltonian approach.
We have shown that our systematic approach can be used to deduce results given
in the literature and we also found new solutions for a variety of models.
## Acknowledgments
FMM is grateful to the staff of the Lahore School of Economics, Pakistan for
their warm hospitality during which time this work was commenced and
completed. In particular he thanks Dr. Shahid Amjad Chaudhry for his
friendship and constant support. FMM also thanks the NRF of South Africa for
research support through a grant.
## References
* [1] Ramsey, F., 1928. A Mathematical theory of saving. Economic Journal 38, 543-559.
* [2] Lucas, R., 1988. On the Mechanics of Economic Development. Journal of Monetary Economics 22, 3-42.
* [3] Eisner, R., Strotz, R., 1963. The determinants of Business investment. Impacts of monetary policy, Englewood cliffs.
* [4] Ben-Porath Yoram., 1967. The production of Human Capital and the life cycle of Earnings. The Journal of Political Economy 75,352-365.
* [5] Cass, D., Shell, K., 1976. Introduction to Hamiltonian dynamics in Economics. Journal of Economic theory 12, 1-10.
* [6] Ruiz-Tamarit, J.R., Ventura-Marco, M., 2011. Solution to nonlinear MHDS arising from optimal growth problems. Mathematical Social Sciences 61, 86-96.
* [7] Mulligan, C. B., Sala-i-Martin, X., 1993\. Transitional dynamics in two-sector models of endogenous growth. The Quaterly Journal of Economics 108, 739 773
* [8] Barro, R. J., Sala-i-Martin, X., 2004. Economic growth. The MIT press, Cambridge.
* [9] Ruiz-Tamarit, J.R., 2008. The closed-form solution for a family of four-dimension nonlinear MHDS. Journal of Economic Dynamics & Control 32, 1000-1014.
* [10] Chilarescu, C., 2008. An analytical solutions for a model of endogenous growth. Economic Modelling 25, 1175-1182.
* [11] Chilarescu. C., 2009. A closed-form solution to the transitional dynamics of the Lucas Uzawa model. Economic Modelling 26, 135-138.
* [12] Hiraguchi, R., 2009. A note on the closed-form solution to the Lucas-Uzawa model with externality. Journal of Economic Dynamics & Control 33, 1757-1760.
* [13] Guerrini, L., 2010. A closed-form solution to the Ramsey model with logistic population growth. Economic Modelling 27, 1178-1182.
* [14] Diele, F., Marangi, C., Ragni, S., 2011. Exponential Lawson integration for nearly Hamiltonian systems arising in optimal control. Mathematics and Computers in Simulation 81, 1057-1067.
* [15] Rodriguez, A., 2004. On the local stability of the solution to optimal control problems. Journal of Economic Dynamics & Control 28, 2475-2484.
* [16] Brock, W. A., Scheinkman, J., 1976. Global asymptotic stability of optimal control systems with applications to the theory of economic growth. Journal of Economic Theory 12, 164-190.
* [17] Rodriguez, A., 1996. On the local stability of the stationary solution to variational problems. Journal of Economic Dynamics & Control 20, 415-431.
* [18] Olver, P. J., 1993. Application of Lie Groups to Differential Equations. Second Edition, Springer, New York.
* [19] Prince, G. E., Leach, P. G. L., 1980. The Lie theory of extended group in Hamiltonian mechanics oscillator. The Journal of the Australian Mathematical Society Series B Applied Mathematics 3, 941-961.
* [20] Dorodnitsyn, V., Kozlov,R., 2010. Invariance and first integrals of continuous and discrete Hamitonian equations. Journal of Engineering Mathematics 66, 253-270.
* [21] Ibragimov, N. H.,(Editor) 1994-1996. CRC Handbook of Lie Group Analysis of Differential Equations, Vols. 1-3 (Boca Raton, FL: Chemical Rubber Company).
* [22] Ibragimov, N. H., Kara, A. H., Mahomed, F. M., 1998. Lie-Bäcklund and Noether Symmetries with Applications. Nonlinear Dynamics 15, 115-136.
* [23] Chiang, A. C., 1992. Elements of dynamic optimization. McGraw Hill.
* [24] Kara, A. H., Mahomed, F. M., Naeem, I., Soh, C. W., 2007. Partial Noether operators and first integrals via partial Lagrangians. Mathematical Methods in the Applied Sciences 30, 2079-2089.
* [25] Kara, A. H., Mahomed, F. M., 2006. Noether-type symmetries and conservation laws via partial Lagragians. Nonlinear Dynamics 45, 367-383.
* [26] Ragni, S., Diele, F., Marangi, C., 2010. Steady-state invariance in high-order Runge-Kutta discretization of optimal growth models. Journal of Economic Dynamics & Control, 34, 1248-1259.
|
arxiv-papers
| 2013-06-12T08:07:44 |
2024-09-04T02:49:46.370612
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "R. Naz, F. M. Mahomed and Azam Chaudhry",
"submitter": "Rehana Naz Prof",
"url": "https://arxiv.org/abs/1306.2740"
}
|
1306.2750
|
# A $\gamma$ Doradus Candidate In Eclipsing Binary BD And?
E. Sipahi [email protected] H. A. Dal Ege University, Science
Faculty, Department of Astronomy and Space Sciences, 35100 Bornova, İzmir,
Turkey
###### Abstract
The BVR photometric light curves of the eclipsing binary BD And were obtained
in 2008 and 2009. We estimated the mass ratio of the system as 0.97 and the
photometric solutions were derived. The results show that BD And is a detached
binary system, whose components have a little temperature difference of about
40 K. By analyzing photometric available light minimum times, we also derived
an update ephemeris and found for the first time a possible periodic
oscillation with an amplitude of 0.011 days and a period of 9.6 years. The
results indicate that the periodic oscillation could be caused by a third
component physically attached to the eclipsing binary. After removing the
light variations due to the eclipses and proximity effects, the light-curve
distortions are further explained by the pulsation of the primary component
with a dominant period of $\sim$1 day. In accordance with the position of the
primary component on the Hertzsprung-Russell diagram and its pulsation period,
the primary component of BD And could be an excellent $\gamma$ Doradus
candidate. It is rarely phenomenon that a component of the eclipsing binary
system is a $\gamma$ Doradus variable.
###### keywords:
techniques: photometric — (stars:) binaries (including multiple): close —
(stars:) binaries: eclipsing — (stars: variables:) $\gamma$ Doradus — stars:
individual: (BD And)
††journal: New Astronomy
## 1 Introduction
According to the catalogue of parameters for eclipsing binaries (Brancewicz &
Dworak, 1980), BD And (GSC 03635-01320) is a $\beta$ Lyr type eclipsing
binary, which has a spectral type of F8, with a period of $0^{d}.462902$. The
binary system BD And was mainly observed to derive minimum times (Kim et al.,
2006; Hubscher et al., 2006; Hubscher & Walter, 2007; Hubscher et al., 2009a,
b). Shaw (1994) included BD And into the catalog of near-contact binary
systems, while both Giuricin et al. (1983) and Malkov et al. (2006) listed the
system in their tables of Algol systems. BD And has a blue magnitude
$B=11^{m}.6$ as reported in the SIMBAD database and an interesting light curve
variation. Its properties are relatively poorly known compared to those of
other short-period binaries. There is no spectroscopic study for BD And in the
literature.
Although BD And is a well observed system and many minima times have been
derived, there is no completed light curve in the literature. Therefore, this
binary was included in our observing plan for understanding the properties of
the light variation and studying the period change. In this paper, the
photometric light curves of BD And in the BVR-bands were observed in 2008 and
2009. Those observations were used to determine the photometric solution. At
the same time, orbital period changes were analyzed with all photometric light
minimum times. This is the first detailed study of this interesting binary.
## 2 Observations
Observations were acquired with a thermoelectrically cooled ALTA $U+47$
$1024\times 1024$ pixel CCD camera attached to a 35 cm - Schmidt - Cassegrains
type MEADE telescope at Ege University Observatory. The observations made in
BVR bands were continued in August 20, 23, 24, 25, September 04, 15 and
October 12, 13, 20 in the season of 2008, while we continued the observations
in August 03, 04, 10, 13, 18, 25, 26, 29, 30, September 28 and October 07, 08
in the season of 2009. Some basic parameters of the program stars are listed
in Table 1. In the table, GSC 3635 1816 and GSC 3635 838 were observed as
comparison stars. Although BD And and the comparison stars are very close in
the sky, differential atmospheric extinction corrections were applied. The
atmospheric extinction coefficients were obtained from observations of the
comparison stars on each night. Heliocentric corrections were also applied to
the times of the observations. The mean averages of the standard deviations
are $0^{m}.021$, $0^{m}.013$, and $0^{m}.019$ for observations acquired in the
BVR bands, respectively. To compute the standard deviations of observations,
we used the standard deviations of the reduced differential magnitudes in the
sense comparisons minus check stars for each night.
All the data of BD And were phased by using the minimum time and period taken
from Kreiner (2004). Then, the V-band light curve shown in Figure 1 was
derived. The V-band light curve phased with the light-elements given in the
literature demonstrated some clues about the properties of the system, as
following: (1) There is only one minimum. However, (2) This minimum seems to
be two groups of minima, a shallow one inside a deeper one. (3) There is also
light variability at maxima phases.
## 3 Light Curve Analysis
### 3.1 A Model with The First Approach
We took JHK brightness of the system ($J=9^{m}.504$, $H=9^{m}.164$,
$K=9^{m}.055$) from the NOMAD Catalogue (Zacharias et al., 2004). Using these
brightness, we derived dereddened colours as a $(J-H)_{0}=0^{m}.185$ and
$(H-K)_{0}=0^{m}.040$ for the system. Using the calibrations given by Tokunaga
(2000), we derived the temperature of the primary component as 7000 K
depending on these dereddened colours, which indicate that its spectral type
is F2V. Photometric analysis of BD And was carried out using the PHOEBE
V.0.31a software (Prša & Zwitter, 2005), which depends on the method used in
the version 2003 of the Wilson-Devinney Code (Wilson & Devinney, 1971; Wilson,
1990).
The BVR light curves, which were phased with the light-elements given by
Kreiner (2004), were analysed simultaneously with the ”detached system”,
”semi-detached system with the primary component filling its Roche-Lobe” and
”semi-detached system with the secondary component filling its Roche-Lobe”
modes. In the process of the computation, we initially adopted the following
fixed parameters: the mean temperature of the primary component ($T_{1}$), the
gravity-darkening exponents of $g_{1}=g_{2}=1.0$ (Lucy, 1967) and the
bolometric albedo coefficients of $A_{1}=A_{2}=1.0$ (Rucinski, 1969). The
commonly adjustable parameters employed are the orbital inclination ($i$), the
mean temperature of the secondary component ($T_{2}$), the potentials of the
components ($\Omega_{1}$ and $\Omega_{2}$) and the monochromatic luminosity of
the primary component ($L_{1}$).
The mass ratio $q$ can be estimated using the empirical relation (Kjurkchieva
& Ivanov, 2006);
$q~{}=~{}\frac{M_{2}}{M_{1}}~{}\approx~{}(\frac{T_{2}}{T_{1}})^{1.7}$ (1)
However, the secondary minimum does not appear in the light curves constructed
with the light elements given in the literature. Because of this, using
different $q$ values from 0.1 to 2, we tried to find an acceptable solution.
During the analyses, the first attempts indicated that the secondary component
of the system should be a faint-late type star. So, we searched the solution
in detail for $q$ values in range of 0.1 and 0.6. On the other hand, we could
not reach a unique solution for none of the different $q$ values from 0.1 to
2. None of the obtained solutions can not be statistically acceptable in the
astrophysical sense. This result demonstrated that there is a problem in
phasing of the data due to the wrong minimum time or especially the wrong
period. Considering the absence of the secondary minimum in the light curve
(Fig. 1), we decided that the wrong parameter must be the period.
### 3.2 The Solution with Real Period
Considering minima separation into two groups in the primary minima of the
light curve phased with the light-elements given in the literature, we
examined all consecutive minima one by one. We saw that the depths of minima
are regularly changing from one cycle to the next one. It is clear that the
orbital period should be two times of those given in the literature. The most
remarkable amplitude variation should be seen at the secondary minimum. This
is because the pulsating component is the primary one and this pulsating
primary component passes in front of the secondary one during the secondary
minimum. The effect of the pulsation is not obviously seen around the primary
minimum because the non-pulsating secondary component partly covers the
pulsating component. Using $0^{d}.925804$ value for the period, all the data
were re-phased. Then, we re-derived the light curve, which is shown in the
upper panel of Figure 2. The shape of the light curve agrees with the Algol
type as was published by Malkov et al. (2006).
In this case, as seen in Figure 2, there are two minima with a little
amplitude difference in the light curve, one of them lies at the phase of 1.0,
while the second one lies at the phase of 1.5. In addition, the low amplitude
variation seen at out-of-eclipses in Figure 1 exhibits itself better than the
previous. Moreover, the low amplitude is seen around the second minimum from
the phase of 1.2 to 1.7. In this light curve, although there is not any level
difference among the primary minima, the levels of the secondary minima are
still varying from one cycle to the next one. This variation should be a part
of the low amplitude variation seen around the secondary minimum, because they
all are following themselves with the same low amplitude. The part of this
variation in the secondary minimum is shown in the bottom panel of Figure 2.
The light variability seen both at the maxima and in the secondary minimum
should be originated due to the primary star.
In this configuration, the preliminary analysis indicates that the system is a
detached binary. Therefore, ”detached mode” was applied to our analysis. The
Wilson-Devinney code is based on Roche geometry which is sensitive to the mass
ratio. To find a photometric mass ratio, the solutions were obtained for a
series of fixed values of the mass ratio from $q=0.70$ to 1.4 in increments of
0.05. The sum of the squared residuals, ($\Sigma res^{2}$), for the
corresponding mass ratios are plotted in Figure 3, where the lowest value of
($\Sigma res^{2}$) was found at $q=0.97$. Therefore, it indicates that the
most likely mass ratio appears to be approximately 0.97. The photometric
solutions for 0.97 are listed in Table 2 and corresponding light curves are
plotted with black lines in Figure 4. The solution parameters obtained from
the BVR light curves analyses seem to be acceptable from the astrophysical
point of view.
## 4 Discussion and Conclusions
### 4.1 Estimated Absolute Parameters and Evalutionary Status
Although there is not any radial velocity curve of the system, we tried to
estimate the absolute parameters of the components. Considering its spectral
type, we computed the mass of the primary component, using the calibrations
given by Tokunaga (2000), and the mass of the secondary component was
calculated from the estimated mass ratio of the system. Using Kepler’s third
law, we first calculated the semi-major axis ($a=5.79$ $R_{\odot}$), and then
the mean radii of the components. All the estimated absolute parameters are
listed in Table 3. In Figure 5, we plotted the distribution of the radii
versus the masses for the components of the system. In the figure, the open
circle represents the secondary component, while the filled circle represents
the primary component. The lines represent the ZAMS and TAMS theoretical model
developed by Pols et al. (1998).
### 4.2 The Light Variations Out-of-Eclipses
Intrinsic light variations are clearly seen at all the phases without the
primary minimum. This indicates that the primary component of the system is
responsible for this variation. The light variation of BD And consists of
proximity effects, eclipses, and intrinsic variations due from the primary
component. The eclipses and the proximity effects in the light curves can be
calculated from light curve analysis. The intrinsic light variations less
affect these quantities. After extracting the synthetic light curves from the
BVR light curves as seen in Figure 4, we looked for frequencies, which could
arise from the outside light variation. The frequency analysis was performed
using PERIOD04 (Lenz & Breger, 2005). The peaks were searched for a range from
0 to 167 cd-1. No relevant features have been detected at frequencies higher
than $\sim$10 cd-1, up to the Nyquist limit of $\sim$167 cd-1. The prominent
features are situated in the frequency interval 0.7-3.2 cd-1. In the analysis,
we found just one dominant frequency in BVR-bands and these frequencies are
listed in Table 4, in which the amplitudes, phases and their corresponding
errors are given. The amplitude spectrum of the dominant frequency of BD And
data in the BVR-bands are shown in Figure 6. Unfortunately, the BVR data are
not adequate for searching the existence of other possible frequencies.
The dominant period of the low amplitude light variations caused by the
primary component is about 0d.9988. This case indicates that the primary
component is a candidate for $\gamma$ Dor type pulsating stars. According to
Kaye et al. (1999), $\gamma$ Dor type pulsation causes a sinusoidal light
curves with amplitudes of a few $mmag$ to a few per cent in a time-scales of
0d.4 - 3d.0. These type stars generally exhibit multi-periodic pulsations.
Although the $\gamma$ Doradus stars are located on the main-sequence together
with $\delta$ Scuti star, they are generally located especially around the
cool border of $\delta$ Scuti star instability strip. They also have a
different pulsation mechanism compared to $\delta$ Scuti stars (Handler &
Shobbrook, 2002). According to Henry et al. (2005), there are three points to
identify a star as a $\gamma$ Doradus variable: (1) the stars spectral type
should be a late A or early F, (2) luminosity class should be IV or V, (3) the
star should have periodic photometric variability in the $\gamma$ Doradus
period range that is attributable to pulsation.
Our results obtained in this study demonstrate that the primary component
exhibits all of them. The relationship between the $\delta$ Scuti and $\gamma$
Doradus pulsators is a debating subject (Handler & Shobbrook, 2002). Handler
and Shobbrook (2002) indicated that the pulsation constants ($Q$) is a
parameter to classify a star as $\delta$ Scuti or $\gamma$ Doradus stars.
Using the equation (2) taken from Handler and Shobbrook (2002), we found a
pulsation constant for the primary component of BD And as 0d.74. The case also
indicates that the primary component is a $\gamma$ Doradus type pulsator. The
primary component of BD And is plotted as an asteriks together with known
$\gamma$ Doradus stars, whose data were taken from Henry et al. (2005), in
Figure 7. The pulsating primary component of the BD And lies just at the
middle of the instability strip of $\gamma$ Doradus type pulsators. Our
results reveal that the primary component of BD And could be a $\gamma$
Doradus type variable. Its physical characteristics are the same with the
nature of VZ CVn (İbanoǧlu et al., 2007). VZ CVn is the only known system of
this type.
$\log Q_{\rm i}~{}=~{}C~{}+~{}0.5\log g~{}+~{}0.1M_{\rm bol}~{}+~{}\log T_{\rm
eff}~{}+~{}\log P_{\rm i}$ (2)
### 4.3 The O-C Variation
In the literature, 116 minima times in total have been obtained for the
system. 73 minima times were obtained from photoelectric and CCD observations,
while the rest are photographic and visual measurements. The minima times
obtained photographically have very large standard deviation. Using the SPSS
V17.0 software (Green et al., 1999) and GrahpPad Prism V5.02 software (Dawson
& Trapp, 2004; Motulsky, 2007), we statistically analysed all the minima times
to search whether there is any regular variation, or not. In the first step,
we analysed all the data together, but the obtained correlation coefficients
of the fits were found to be very low. In addition, $p-values$ were found to
be higher than the threshold level of significance ($\alpha-value$) of 0.005.
As it is well known that both $p-~{}and~{}\alpha-values$ are used to test
whether the obtained fit is statistically acceptable, or not (Wall & Jenkins,
2003). In the second step, we separated the data into the groups, as
photoelectric, CCD, visual and photographical observations. Then, both groups
of the data were analysed separately. According to $p-value$, although an
acceptable fit was obtained just from the photoelectric and CCD observations,
no acceptable fit was found from the photographic and visual observations. As
a results, we used just photoelectric and CCD observations in the analyses.
All minima were taken from the literature and from the more recent studies
listed in Table 5. We obtained fourteen new epochs of minimum during the
observations and seven of them have already been published by Sipahi et al.
(2009). We used these minima times listed in Table 5 to determine the light
elements by using the least squares method. We have found the new light
elements as following:
$JD~{}(Hel.)~{}=~{}24~{}55057.4656(5)~{}+~{}0^{d}.9258051(1)~{}\times~{}E.$
(3)
We plot the residuals of all the eclipse timings as $(O-C)$ in the upper panel
of Figure 8.
There is a sinusoidal oscillation in the O-C residuals for BD And. The
oscillating characteristic may be caused by the light-time effect due to the
existence of the third body orbiting the eclipsing binary or a possible result
of magnetic activity cycles of the system. Since no spectroscopic solutions
are available in the literature of BD And, its absolute parameters can not be
directly determined. We estimated the primary mass as 1.56 $M_{\odot}$, and
the radius as 1.42 $R_{\odot}$ by assuming the primary component to be a
normal and main-sequence F2 star. Based on our photometric solutions, the mass
of the secondary component was found to be $M_{2}=1.51$ $M_{\odot}$, while the
separation ($a$) between the two components was calculated as 5.79
$R_{\odot}$. Both the observed colours and the light curve solution of the
system show that the components are not late type stars. Therefore, the
magnetic activity cycle is not a possible mechanism to cause the period
variation. However, the BVR light curve analysis gives a physically acceptable
values for a third light contribution (see Table 2). Therefore, the period
oscillation may be caused due to the light-time effect. A weighted least-
squares solution for two parameters ($T_{0}$ and P) of the linear ephemeris of
BD And and five parameters ($a_{12}sini^{\prime}$, $e^{\prime}$, $w^{\prime}$,
$T^{\prime}$ and $P^{\prime}$) of the light-time effect are presented in Table
6. The observational points and theoretical best fit curve are plotted against
epoch number in Figure 8. The parameters we derived for the light-time effect
enable calculations of the mass function, using the Equation (4) taken from
Kopal (1978):
$f(m)~{}=~{}\frac{(a_{12}sini^{\prime})^{3}}{(P^{\prime})^{2}}=\frac{(M_{3}~{}sini^{\prime})^{3}}{(M_{1}+M_{2}+M_{3})^{2}}$
(4)
where $M_{1}$, $M_{2}$ and $M_{3}$ are the masses of the binary, and the third
body, respectively. We obtained the mass function as $f(m)=0.0089$ $M_{\odot}$
for the third body. The mass of the third component can be estimated with
Equation (4), depending on the orbital inclination. We calculated the third
body masses at different inclination, $i^{\prime}$ values, which are shown in
Table 6. Here, the total mass of the eclipsing system was taken as 3.07
$M_{\odot}$. The result parameters given in Table 6 suggest that BD And has an
eccentric orbit around the mass center of the third-body system with a period
of $\sim$9.6 years.
Furthermore, a significant contribution of the third light was found in the
light curve analysis. Thus, it confirms that third body may produce the
sinusoidal variation of the period. It is more likely that this third light,
used in our light curve solutions, was caused by a third body in the system.
Although BD And was included into the list of near-contact binary by Shaw
(1994), our results showed that BD And is a detached binary system with a
little temperature difference of about 40 K between the two components. BD And
seems to be an analogue of those detached systems with a possible third body,
such as V2080 Cyg (İbanoǧlu et al., 2008).
The aim of the present paper is to draw attention to eclipsing binary BD And
which shows interesting features photometrically. Of course, our solutions are
based on photometric observations only. For better understanding of the
properties and the evolutionary state of BD And, spectroscopic observations
are needed.
## Acknowledgments
The authors wish to thank Prof. Dr. Ömer Lütfi Deǧirmenci for his help with
the $O-C$ analysis. The authors acknowledge generous allotments of observing
time at the Ege University Observatory. We also thank the referee for useful
comments that have contributed to the improvement of the paper.
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Figure 1: Upper and lower panels show the differential magnitudes versus time
(in HJD) and phase. Here, the data were phased by using the light-elements
given by Kreiner (2004). Figure 2: The V-band light curve is shown in the
upper panel. Here, the data were phased by using twice the period given by
Kreiner (2004). In the bottom panel, the secondary minima are plotted for a
closer look to the minima for better visibility of light variations. In the
figure, the colours represent observations obtained in different nights.
Figure 3: The variation of the sum of weighted squared residuals versus mass
ratio in the ”q search”. Figure 4: BD And’s light curves (dimmer - filled
circle) observed in BVR bands and the synthetic curves (line) derived from the
light curve solutions in each band are shown in the left side panels. The
prewhitening curves (dark filled circle) obtained from the residual data are
shown in the right side panels. Figure 5: The places of the components of BD
And in the Mass-Radius distribution. The black-line and dashed line represent
the ZAMS and TAMS theoretical models developed by Pols et al. (1998). Figure
6: The result of the frequency analysis are shown from the upper to the bottom
for the pulsating component of BD And in the BVR filter, respectively. High
frequency spectrum of the residuals were obtained after removing the binary
model. Figure 7: The place of the primary component of BD And among $\gamma$
Doradus type stars in the HR diagram. In the figure, the small filled circles
represent $\gamma$ Doradus type stars listed in Henry et al. (2005). The
asterisk represents the primary component of the system. The dash dotted lines
(red) represent the borders of the area, in which $\gamma$ Doradus type stars
take place. In addition we plotted the hot (HB) and cold (CB) borders of the
$\delta$ Scuti stars for comparison. In the figure, the purpel open circles
represent some semi- and un-detached binaries taken from Soydugan et al.
(2006) and references therein. The ZAMS and TAMS were taken from Girardi et
al. (2000), while the borders of the Instability Strip were computed from
Rolland et al. (2002). Figure 8: The O-C diagram depending only on
observations of BD And. In the upper panel, all available O-C data are shown,
while the O-C data used in the analyses are shown in the bottom panel. In both
panels, the filled circles represent the CCD observations, while the asterisk
represents the photoelectric observations. In bottom panel, the dashed curve
represents the predicted light-time effect. The open circles represent the
photographic observations, while the plus represent the visual observations.
Table 1: Basic parameters for the observed stars. Star | Alpha (J2000) | Delta (J2000) | V
---|---|---|---
| (h m s) | (∘ ′ ′′) | (mag)
BD And | 23 07 05.162 | +50 57 30.95 | 10.84
GSC 3635 1816 | 23 07 09.720 | +50 57 35.80 | 12.20
GSC 3635 838 | 23 07 11.290 | +50 55 23.40 | 11.70
Table 2: The parameters obtained from the light curve analysis. Parameter | Value
---|---
$q$ | 0.97
$i$ (∘) | 85.88$\pm$0.07
$T_{1}$ (K) | 7000
$T_{2}$ (K) | 6962$\pm$10
$\Omega_{1}$ | 5.108$\pm$0.012
$\Omega_{2}$ | 5.585$\pm$0.019
$L_{1}/L_{T}(B)$ | 0.570$\pm$0.048
$L_{1}/L_{T}(V)$ | 0.567$\pm$0.046
$L_{1}/L_{T}(R)$ | 0.565$\pm$0.045
$L_{2}/L_{T}(B)$ | 0.422$\pm$0.048
$L_{2}/L_{T}(V)$ | 0.424$\pm$0.046
$L_{2}/L_{T}(R)$ | 0.424$\pm$0.045
$L_{3}/L_{T}(B)$ | 0.008$\pm$0.037
$L_{3}/L_{T}(V)$ | 0.009$\pm$0.038
$L_{3}/L_{T}(R)$ | 0.011$\pm$0.039
$g_{1}$, $g_{2}$ | 1.0, 1.0
$A_{1}$, $A_{2}$ | 1.0, 1.0
$x_{1,bol}$, $x_{2,bol}$ | 0.471, 0.473
$x_{1,B}$, $x_{2,B}$ | 0.599, 0.609
$x_{1,V}$, $x_{2,V}$ | 0.495, 0.501
$x_{1,R}$, $x_{2,R}$ | 0.399, 0.404
$<r_{1}>$ | 0.246$\pm$0.001
$<r_{2}>$ | 0.215$\pm$0.001
Table 3: The estimated absolute parameters derived for BD And system. Parameter | Primary | Secondary
---|---|---
Mass ($M_{\odot}$) | 1.56 | 1.51
Radius ($R_{\odot}$) | 1.42 | 1.24
Luminosity ($L_{\odot}$) | 4.34 | 3.24
$M_{bol}$ (mag) | 3.15 | 3.46
$log~{}g$ | 4.33 | 4.43
Table 4: The results of the frequency analysis. Filter | | Frequency (c/d) | Amplitude (mag) | Phase | SNR
---|---|---|---|---|---
B | $f_{1}$ | 1.00138$\pm$0.00007 | 0.0210$\pm$0.0006 | 0.63$\pm$0.01 | 9.03
V | $f_{1}$ | 1.00125$\pm$0.00008 | 0.0171$\pm$0.0003 | 0.25$\pm$0.01 | 9.16
R | $f_{1}$ | 1.00129$\pm$0.00007 | 0.0127$\pm$0.0006 | 0.03$\pm$0.02 | 8.32
Table 5: Times of minimum light for BD And. HJD (+24 00000) | Cycle | $(O-C)$ | Method | Ref.
---|---|---|---|---
27784.1790 | -29459.0 | 0.0142 | PG | Kreiner (2004)
28081.3540 | -29138.0 | 0.0056 | PG | Kreiner (2004)
28749.3280 | -28416.5 | 0.0110 | PG | Kreiner (2004)
29476.0790 | -27631.5 | 0.0048 | PG | Kreiner (2004)
34600.4190 | -22096.5 | 0.0119 | PG | Azarnova (1956)
34677.2480 | -22013.5 | -0.0009 | PG | Azarnova (1956)
34930.4620 | -21740.0 | 0.0053 | PG | Azarnova (1956)
34962.4030 | -21705.5 | 0.0060 | PG | Azarnova (1956)
34987.3960 | -21678.5 | 0.0023 | PG | Azarnova (1956)
34993.4150 | -21672.0 | 0.0035 | PG | Azarnova (1956)
35012.3890 | -21651.5 | -0.0015 | PG | Azarnova (1956)
39035.4680 | -17306.0 | -0.0098 | PG | Kreiner (2004)
39061.4030 | -17278.0 | 0.0026 | PG | Kreiner (2004)
39443.2930 | -16865.5 | -0.0021 | PG | Kreiner (2004)
41599.4940 | -14536.5 | -0.0019 | PG | Kreiner (2004)
41602.2740 | -14533.5 | 0.0007 | PG | Kreiner (2004)
42525.2990 | -13536.5 | -0.0023 | PG | Kreiner (2004)
43050.6920 | -12969.0 | -0.0039 | VIS | Baldwin & Samolyk (1996)
43073.8260 | -12944.0 | -0.0150 | VIS | Baldwin & Samolyk (1996)
43380.7500 | -12612.5 | 0.0045 | VIS | Baldwin & Samolyk (1996)
43429.8160 | -12559.5 | 0.0028 | VIS | Baldwin & Samolyk (1996)
43755.6940 | -12207.5 | -0.0027 | VIS | Baldwin & Samolyk (1996)
44022.7920 | -11919.0 | 0.0005 | VIS | Baldwin & Samolyk (1996)
44485.6760 | -11419.0 | -0.0182 | VIS | Baldwin & Samolyk (1996)
Table 5: Continued. HJD (+24 00000) | Cycle | $(O-C)$ | Method | Ref.
---|---|---|---|---
44593.5460 | -11302.5 | -0.0046 | VIS | Baldwin & Samolyk (1996)
44885.6470 | -10987.0 | 0.0048 | VIS | Baldwin & Samolyk (1996)
45291.6090 | -10548.5 | 0.0012 | VIS | Baldwin & Samolyk (1996)
45622.5880 | -10191.0 | 0.0047 | VIS | Baldwin & Samolyk (1996)
46682.6260 | -9046.0 | -0.0045 | VIS | Baldwin & Samolyk (1996)
46713.6530 | -9012.5 | 0.0081 | VIS | Baldwin & Samolyk (1996)
46756.6850 | -8966.0 | -0.0099 | VIS | Baldwin & Samolyk (1996)
46769.6480 | -8952.0 | -0.0082 | VIS | Baldwin & Samolyk (1996)
47062.6560 | -8635.5 | -0.0176 | VIS | Baldwin & Samolyk (1996)
47861.6450 | -7772.5 | 0.0014 | VIS | Baldwin & Samolyk (1996)
48897.6240 | -6653.5 | 0.0041 | VIS | Baldwin & Samolyk (1996)
48954.5560 | -6592.0 | -0.0009 | VIS | Baldwin & Samolyk (1996)
49278.5940 | -6242.0 | 0.0052 | VIS | Baldwin & Samolyk (1996)
49554.4764 | -5944.0 | -0.0024 | PE | Agerer & Hubscher (1995)
49567.4378 | -5930.0 | -0.0023 | PE | Agerer & Hubscher (1995)
49578.5492 | -5918.0 | -0.0005 | PE | Agerer & Hubscher (1995)
49585.4923 | -5910.5 | -0.0010 | PE | Agerer & Hubscher (1995)
49646.5962 | -5844.5 | -0.0002 | CCD | Hubscher et al. (2006)
49688.2583 | -5799.5 | 0.0006 | CCD | Hubscher et al. (2006)
49899.8190 | -5571.0 | 0.0148 | VIS | Baldwin & Samolyk (1996)
49948.4109 | -5518.5 | 0.0019 | CCD | Hubscher et al. (2006)
49954.4298 | -5512.0 | 0.0031 | CCD | Hubscher et al. (2006)
49983.5990 | -5480.5 | 0.0094 | VIS | Baldwin & Samolyk (1996)
49989.6080 | -5474.0 | 0.0007 | VIS | Baldwin & Samolyk (1996)
Table 5: Continued. HJD (+24 00000) | Cycle | $(O-C)$ | Method | Ref.
---|---|---|---|---
50020.6260 | -5440.5 | 0.0042 | CCD | Baldwin & Samolyk (1996)
50044.7000 | -5414.5 | 0.0072 | VIS | Baldwin & Samolyk (1996)
50081.2650 | -5375.0 | 0.0029 | CCD | Hubscher et al. (2006)
50346.5092 | -5088.5 | 0.0039 | CCD | Hubscher et al. (2006)
50380.3009 | -5052.0 | 0.0037 | CCD | Blättler et al. (1996)
50380.3016 | -5052.0 | 0.0044 | CCD | Hubscher et al. (2006)
50391.4150 | -5040.0 | 0.0081 | CCD | Acerbi et al. (1997)
51306.5670 | -4051.5 | 0.0015 | CCD | Kreiner (2004)
51509.3195 | -3832.5 | 0.0026 | CCD | Blättler et al. (2000)
52195.3368 | -3091.5 | -0.0019 | CCD | Blättler et al. (2001)
52955.4219 | -2270.5 | -0.0030 | PE | Hubscher et al. (2006)
53268.3486 | -1932.5 | 0.0014 | CCD | Hubscher et al. (2006)
53277.6059 | -1922.5 | 0.0007 | CCD | Baldwin (2007)
53340.5611 | -1854.5 | 0.0011 | CCD | Baldwin (2007)
53345.1917 | -1849.5 | 0.0027 | CCD | Chun-Hwey et al. (2006)
53347.0419 | -1847.5 | 0.0013 | CCD | Chun-Hwey et al. (2006)
53347.9676 | -1846.5 | 0.0012 | CCD | Chun-Hwey et al. (2006)
53352.1361 | -1842.0 | 0.0035 | CCD | Chun-Hwey et al. (2006)
53406.2942 | -1783.5 | 0.0020 | CCD | Hubscher et al. (2006)
53632.6500 | -1539.0 | -0.0016 | CCD | Baldwin (2007)
53638.2096 | -1533.0 | 0.0032 | CCD | Chun-Hwey et al. (2006)
53675.2413 | -1493.0 | 0.0027 | CCD | Chun-Hwey et al. (2006)
53690.0535 | -1477.0 | 0.0020 | CCD | Chun-Hwey et al. (2006)
53690.0547 | -1477.0 | 0.0032 | PE | Kreiner (2004)
Table 5: Continued. HJD (+24 00000) | Cycle | $(O-C)$ | Method | Ref.
---|---|---|---|---
53696.0723 | -1470.5 | 0.0030 | CCD | Chun-Hwey et al. (2006)
53696.9980 | -1469.5 | 0.0029 | CCD | Chun-Hwey et al. (2006)
53994.6444 | -1148.0 | 0.0029 | CCD | Baldwin (2007)
54024.2705 | -1116.0 | 0.0032 | PE | Hubscher & Walter (2007)
54295.5302 | -823.0 | 0.0019 | CCD | Hubscher et al. (2009a)
54310.8077 | -806.5 | 0.0037 | CCD | Baldwin (2007)
54379.3165 | -732.5 | 0.0029 | CCD | Hubscher et al. (2009b)
54390.4269 | -720.5 | 0.0036 | PE | Hubscher & Walter (2007)
54596.8802 | -497.5 | 0.0023 | CCD | Kreiner (2004)
54676.4992 | -411.5 | 0.0020 | VIS | Hubscher et al. (2010a)
54680.6656 | -407.0 | 0.0023 | CCD | Kreiner (2004)
54702.4229 | -383.5 | 0.0032 | CCD | Sipahi et al. (2009)
54703.3488 | -382.5 | 0.0033 | CCD | Sipahi et al. (2009)
54707.5158 | -378.0 | 0.0041 | CCD | Sipahi et al. (2009)
54709.3668 | -376.0 | 0.0035 | CCD | Sipahi et al. (2009)
54714.4586 | -370.5 | 0.0034 | CCD | Sipahi et al. (2009)
54728.3450 | -355.5 | 0.0027 | VIS | Hubscher et al. (2010a)
54752.4148 | -329.5 | 0.0016 | CCD | Sipahi et al. (2009)
54753.3401 | -328.5 | 0.0011 | CCD | Sipahi et al. (2009)
54798.2421 | -280.0 | 0.0015 | PE | Hubscher et al. (2009b)
54834.3488 | -241.0 | 0.0018 | PE | Hubscher et al. (2010a)
55039.4130 | -19.5 | 0.0001 | PE | Hubscher et al. (2010b)
55057.4661 | 0.0 | 0.0000 | CCD | This Study
55058.8557 | 1.5 | 0.0009 | CCD | Kreiner (2004)
Table 5: Continued.
HJD (+24 00000) | Cycle | $(O-C)$ | Method | Ref.
---|---|---|---|---
55062.5586 | 5.5 | 0.0006 | CCD | This Study
55069.5015 | 13.0 | -0.0001 | PE | This Study
55070.4273 | 14.0 | -0.0001 | PE | This Study
55071.3532 | 15.0 | 0.0000 | CCD | Hubscher et al. (2010b)
55103.2916 | 49.5 | -0.0019 | CCD | This Study
55113.4756 | 60.5 | -0.0017 | CCD | This Study
55135.6958 | 84.5 | -0.0009 | PE | Diethelm (2010)
55481.4814 | 458.0 | -0.0036 | PE | Hubscher (2011)
55483.3337 | 460.0 | -0.0029 | CCD | Hubscher (2011)
55497.6850 | 475.5 | -0.0016 | PE | Diethelm (2011)
55514.3479 | 493.5 | -0.0032 | PE | Paschke & Brát (2006)
55547.2144 | 529.0 | -0.0028 | PE | Paschke & Brát (2006)
55547.2150 | 529.0 | -0.0022 | PE | Paschke & Brát (2006)
55480.5562 | 457.0 | -0.0030 | PE | Hubscher et al. (2012a)
55832.3609 | 837.0 | -0.0043 | PE | Hubscher & Lehmann (2012b)
55839.3033 | 844.5 | -0.0055 | PE | Hubscher & Lehmann (2012b)
55873.5574 | 881.5 | -0.0062 | PE | Hubscher & Lehmann (2012b)
55877.2615 | 885.5 | -0.0053 | PE | Hubscher & Lehmann (2012b)
55890.2201 | 899.5 | -0.0080 | PE | Diethelm (2013)
56233.6979 | 1270.5 | -0.0040 | PE | This Study
* $Note:$
In the fourth column, PG refers to photographic observation, VIS refers to
visual observations, PE refers to photoelectric observation, CCD refers to CCD
observation.
Table 6: The parameters derived from $O-C$ analysis of BD And. Parameter | Value
---|---
$T_{o}$ | 55057.4656(5)
$P$ (day) | 0.9258051(1)
$T^{\prime}$ | 53300(90)
$P^{\prime}$ (year) | 9.56(2)
$e^{\prime}$ | 0.35(8)
$w^{\prime}$ (∘) | 2.76(8)
$a_{12}sini^{\prime}$ (AU) | 0.93(9)
$A$ (day) | 0.0054(2)
$f(m)$ | 0.0089(6) $M_{\odot}$
$M_{3}$($i^{\prime}$=$30^{\circ}$) | 0.99 $M_{\odot}$
$M_{3}$($i^{\prime}$=$45^{\circ}$) | 0.70 $M_{\odot}$
$M_{3}$($i^{\prime}$=$60^{\circ}$) | 0.58 $M_{\odot}$
$M_{3}$($i^{\prime}$=$75^{\circ}$) | 0.50 $M_{\odot}$
$M_{3}$($i^{\prime}$=$90^{\circ}$) | 0.47 $M_{\odot}$
|
arxiv-papers
| 2013-06-12T08:40:50 |
2024-09-04T02:49:46.379485
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "E. Sipahi, H. A. Dal",
"submitter": "Hasan Ali Dal",
"url": "https://arxiv.org/abs/1306.2750"
}
|
1306.2806
|
11institutetext: Uppsala University, Sweden 22institutetext: University of
Edinburgh, UK 33institutetext: LIAFA, Univ Paris Diderot, Sorbonne Paris Cité,
CNRS, France 44institutetext: University of Turin, Italy
# Solving Parity Games on Integer Vectors ††thanks: Technical Report EDI-INF-
RR-1417 of the School of Informatics at the University of Edinburgh, UK.
(http://www.inf.ed.ac.uk/publications/report/). Full version (including
proofs) of material presented at CONCUR 2013 (Buenos Aires, Argentina).
arXiv.org - CC BY 3.0.
Parosh Aziz Abdulla Supported by Uppsala Programming for Multicore
Architectures Research Center (UpMarc). 11 Richard Mayr Supported by Royal
Society grant IE110996.22 Arnaud Sangnier 33 Jeremy Sproston Supported by the
project AMALFI (University of Turin/Compagnia di San Paolo) and the MIUR-PRIN
project CINA.44
###### Abstract
We consider parity games on infinite graphs where configurations are
represented by control-states and integer vectors. This framework subsumes two
classic game problems: parity games on vector addition systems with states
(vass) and multidimensional energy parity games. We show that the
multidimensional energy parity game problem is inter-reducible with a subclass
of single-sided parity games on vass where just one player can modify the
integer counters and the opponent can only change control-states. Our main
result is that the minimal elements of the upward-closed winning set of these
single-sided parity games on vass are computable. This implies that the Pareto
frontier of the minimal initial credit needed to win multidimensional energy
parity games is also computable, solving an open question from the literature.
Moreover, our main result implies the decidability of weak simulation
preorder/equivalence between finite-state systems and vass, and the
decidability of model checking vass with a large fragment of the modal
$\mu$-calculus.
## 1 Introduction
In this paper, we consider integer games: two-player turn-based games where a
color (natural number) is associated to each state, and where the transitions
allow incrementing and decrementing the values of a finite set of integer-
valued counters by constants. We refer to the players as Player $0$ and Player
$1$.
We consider the classical parity condition, together with two different
semantics for integer games: the energy semantics and the vass semantics. The
former corresponds to multidimensional energy parity games [7], and the latter
to parity games on vass (a model essentially equivalent to Petri nets [8]). In
energy parity games, the winning objective for Player $0$ combines a
qualitative property, the classical parity condition, with a quantitative
property, namely the energy condition. The latter means that the values of all
counters stay above a finite threshold during the entire run of the game. In
vass parity games, the counter values are restricted to natural numbers, and
in particular any transition that may decrease the value of a counter below
zero is disabled (unlike in energy games where such a transition would be
immediately winning for Player 1). So for vass games, the objective consists
only of a parity condition, since the energy condition is trivially satisfied.
We formulate and solve our problems using a generalized notion of game
configurations, namely partial configurations, in which only a subset $C$ of
the counters may be defined. A partial configuration $\gamma$ denotes a
(possibly infinite) set of concrete configurations that are called
instantiations of $\gamma$. A configuration $\gamma^{\prime}$ is an
instantiation of $\gamma$ if $\gamma^{\prime}$ agrees with $\gamma$ on the
values of the counters in $C$ while the values of counters outside $C$ can be
chosen freely in $\gamma^{\prime}$. We declare a partial configuration to be
winning (for Player $0$) if it has an instantiation that is winning. For each
decision problem and each set of counters $C$, we will consider the
$C$-version of the problem where we reason about configurations in which the
counters in $C$ are defined.
#### Previous Work.
Two special cases of the general $C$-version are the abstract version in which
no counters are defined, and the concrete version in which all counters are
defined. In the energy semantics, the abstract version corresponds to the
unknown initial credit problem for multidimensional energy parity games, which
is coNP-complete [6, 7]. The concrete version corresponds to the fixed initial
credit problem. For energy games without the parity condition, the fixed
initial credit problem was solved in [4] (although it does not explicitly
mention energy games but instead formulates the problem as a zero-reachability
objective for Player $1$). It follows from [4] that the fixed initial credit
problem for $d$-dimensional energy games can be solved in $d$-EXPTIME (resp.
$(d-1)$-EXPTIME for offsets encoded in unary) and even the upward-closed
winning sets can be computed. An EXPSPACE lower bound is derived by a
reduction from Petri net coverability. The subcase of one-dimensional energy
parity games was considered in [5], where both the unknown and fixed initial
credit problems are decidable, and the winning sets (i.e., the minimal
required initial energy) can be computed. The assumption of having just one
dimension is an important restriction that significantly simplifies the
problem. This case is solved using an algorithm which is a generalization of
the classical algorithms of McNaughton [13] and Zielonka [16].
However, for general multidimensional energy parity games, computing the
winning sets was an open problem, mentioned, e.g., in [6].
In contrast, under the vass semantics, all these integer game problems are
shown to be undecidable for dimensions $\geq 2$ in [2], even for simple
safety/coverability objectives. (The one-dimensional case is a special case of
parity games on one-counter machines, which is PSPACE-complete). A special
subcase are single-sided vass games, where just Player $0$ can modify counters
while Player $1$ can only change control-states. This restriction makes the
winning set for Player $0$ upward-closed, unlike in general vass games. The
paper [14] shows decidability of coverability objectives for single-sided vass
games, using a standard backward fixpoint computation.
#### Our Contribution.
First we show how instances of the single-sided vass parity game can be
reduced to the multidimensional energy parity game, and vice-versa. I.e.,
energy games correspond to the single-sided subcase of vass games. Notice
that, since parity conditions are closed under complement, it is merely a
convention that Player $0$ (and not Player $1$) is the one that can change the
counters.
Our main result is the decidability of single-sided vass parity games for
general partial configurations, and thus in particular for the concrete and
abstract versions described above. The winning set for Player $0$ is upward-
closed (wrt. the natural multiset ordering on configurations), and it can be
computed (i.e., its finitely many minimal elements). Our algorithm uses the
Valk-Jantzen construction [15] and a technique similar to Karp-Miller graphs,
and finally reduces the problem to instances of the abstract parity problem
under the energy semantics, i.e., to the unknown initial credit problem in
multidimensional energy parity games, which is decidable by [7].
From the above connection between single-sided vass parity games and
multidimensional energy parity games, it follows that the winning sets of
multidimensional energy parity games are also computable. I.e., one can
compute the Pareto frontier of the minimal initial energy credit vectors
required to win the energy parity game. This solves the problem left open in
[6, 7].
Our results imply further decidability results in the following two areas:
semantic equivalence checking and model-checking. Weak simulation preorder
between a finite-state system and a general vass can be reduced to a parity
game on a single-sided vass, and is therefore decidable. Combined with the
previously known decidability of the reverse direction [3], this implies
decidability of weak simulation equivalence. This contrasts with the
undecidability of weak bisimulation equivalence between vass and finite-state
systems [11]. The model-checking problem for vass is decidable for many
linear-time temporal logics [10], but undecidable even for very restricted
branching-time logics [8]. We show the decidability of model-checking for a
restricted class of vass with a large fragment of the modal $\mu$-calculus.
Namely we consider vass where some states do not perform any updates on the
counters, and these states are used to guard the for-all-successors modal
operators in this fragment of the $\mu$-calculus, allowing us to reduce the
model-checking problem to a parity game on single-sided vass.
## 2 Integer Games
#### Preliminaries.
We use ${\mathbb{N}}$ and ${\mathbb{Z}}$ to denote the sets of natural numbers
(including $0$) and integers respectively. For a set $A$, we define $|{A}|$ to
be the cardinality of $A$. For a function $f:A\mapsto B$ from a set $A$ to a
set $B$, we use $f[a\leftarrow b]$ to denote the function $f^{\prime}$ such
that $f(a)=b$ and $f^{\prime}(a^{\prime})=f(a^{\prime})$ if $a^{\prime}\neq
a$. If $f$ is partial, then $f(a)=\bot$ means that $f$ is undefined for $a$.
In particular $f[a\leftarrow\bot]$ makes the value of $a$ undefined. We define
${\it dom}\left({f}\right):=\left\\{{a}|\;{f(a)\neq\bot}\right\\}$.
#### Model.
We assume a finite set ${\mathcal{C}}$ of counters. An integer game is a tuple
${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$ where $Q$ is a finite
set of states, $T$ is a finite set of transitions, and
$\kappa:Q\mapsto\left\\{0,1,2,\ldots,k\right\\}$ is a coloring function that
assigns to each $q\in Q$ a natural number in the interval $[0..k]$ for some
pre-defined $k$. The set $Q$ is partitioned into two sets $Q_{0}$ (states of
Player $0$) and $Q_{1}$ (states of Player $1$). A transition $t\in T$ is a
triple $\left\langle{q_{1},{\it op},q_{2}}\right\rangle$ where $q_{1},q_{2}\in
Q$ are states and ${\it op}$ is an operation of one of the following three
forms (where $c\in{\mathcal{C}}$ is a counter): (i) ${c}\mbox{\small+ +}$
increments the value of $c$ by one; (ii) ${c}\mbox{\small- -}$ decrements the
value of $c$ by one; (iii) ${\it nop}$ does not change the value of any
counter. We define ${\tt source}\left(t\right)=q_{1}$, ${\tt
target}\left(t\right)=q_{2}$, and ${\tt op}\left(t\right)={\it op}$. We say
that ${\mathcal{G}}$ is single-sided in case ${\it op}={\it nop}$ for all
transitions $t\in T$ with ${\tt source}\left(t\right)\in Q_{1}$. In other
words, in a single-sided game, Player $1$ is not allowed to changes the values
of the counters, but only the state.
#### Partial Configurations.
A partial counter valuation $\vartheta:{\mathcal{C}}\mapsto{\mathbb{Z}}$ is a
partial function from the set of counters to ${\mathbb{Z}}$. We also write
$\vartheta(c)=\bot$ if $c\notin{\it dom}\left({\vartheta}\right)$. A partial
configuration $\gamma$ is a pair $\left\langle{q,\vartheta}\right\rangle$
where $q\in Q$ is a state and $\vartheta$ is a partial counter valuation. We
will also consider nonnegative partial configurations, where the partial
counter valuation takes values in ${\mathbb{N}}$ instead of ${\mathbb{Z}}$. We
define ${\tt state}\left({\gamma}\right):=q$, ${\tt
val}\left({\gamma}\right):=\vartheta$, and
$\kappa\left({\gamma}\right):=\kappa\left({{\tt
state}\left({\gamma}\right)}\right)$. We generalize assignments from counter
valuations to configurations by defining
$\left\langle{q,\vartheta}\right\rangle[c\leftarrow
x]=\left\langle{q,\vartheta[c\leftarrow x]}\right\rangle$. Similarly, for a
configuration $\gamma$ and $c\in{\mathcal{C}}$ we let $\gamma(c):={\tt
val}\left({\gamma}\right)(c)$, ${\it dom}\left({\gamma}\right):={\it
dom}\left({{\tt val}\left({\gamma}\right)}\right)$ and $|{\gamma}|:=|{{\it
dom}\left({\gamma}\right)}|$. For a set of counters $C\subseteq{\mathcal{C}}$,
we define $\Theta^{C}:=\left\\{{\gamma}|\;{{\it
dom}\left({\gamma}\right)=C}\right\\}$, i.e., it is the set of configurations
in which the defined counters are exactly those in $C$. We use $\Gamma^{C}$ to
denote the restriction of $\Theta^{C}$ to nonnegative partial configurations.
We partition $\Theta^{C}$ into two sets $\Theta^{C}_{0}$ (configurations
belonging to Player $0$) and $\Theta^{C}_{1}$ (configurations belonging to
Player $1$), such that $\gamma\in\Theta^{C}_{i}$ iff ${\it
dom}\left({\gamma}\right)=C$ and ${\tt state}\left({\gamma}\right)\in Q_{i}$
for $i\in\left\\{0,1\right\\}$. A configuration is concrete if ${\it
dom}\left({\gamma}\right)={\mathcal{C}}$, i.e.,
$\gamma\in\Theta^{{\mathcal{C}}}$ (the counter valuation ${\tt
val}\left({\gamma}\right)$ is defined for all counters); and it is abstract if
${\it dom}\left({\gamma}\right)=\emptyset$, i.e.,
$\gamma\in\Theta^{\emptyset}$ (the counter valuation ${\tt
val}\left({\gamma}\right)$ is not defined for any counter). In the sequel, we
occasionally write $\Theta$ instead of $\Theta^{{\mathcal{C}}}$, and
$\Theta_{i}$ instead of $\Theta^{{\mathcal{C}}}_{i}$ for
$i\in\left\\{0,1\right\\}$. The same notations are defined over nonnegative
partial configurations with $\Gamma$, and $\Gamma^{C}_{i}$ and $\Gamma_{i}$
for $i\in\left\\{0,1\right\\}$. For a nonnegative partial configuration
$\gamma=\left\langle{q,\vartheta}\right\rangle\in\Gamma$, and set of counters
$C\subseteq{\mathcal{C}}$ we define the restriction of $\gamma$ to $C$ by
$\gamma^{\prime}={\gamma}|{C}=\left\langle{q^{\prime},\vartheta^{\prime}}\right\rangle$
where $q^{\prime}=q$ and $\vartheta^{\prime}(c)=\vartheta(c)$ if $c\in C$ and
$\vartheta^{\prime}(c)=\bot$ otherwise.
#### Energy Semantics.
Under the energy semantics, an integer game induces a transition relation
$\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{E}}$ on the set of
partial configurations as follows. For partial configurations
$\gamma_{1}=\left\langle{q_{1},\vartheta_{1}}\right\rangle$,
$\gamma_{2}=\left\langle{q_{2},\vartheta_{2}}\right\rangle$, and a transition
$t=\left\langle{q_{1},{\it op},q_{2}}\right\rangle\in T$, we have
$\gamma_{1}\stackrel{{\scriptstyle
t}}{{\longrightarrow}}_{\mathcal{E}}\gamma_{2}$ if one of the following three
cases is satisfied: (i) ${\it op}={c}\mbox{\small+ +}$ and either both
$\vartheta_{1}(c)=\bot$ and $\vartheta_{2}(c)=\bot$ or
$\vartheta_{1}(c)\neq\bot$, $\vartheta_{2}(c)\neq\bot$ and
$\vartheta_{2}=\vartheta_{1}[c\leftarrow\vartheta_{1}(c)+1]$; (ii) ${\it
op}={c}\mbox{\small- -}$, and either both $\vartheta_{1}(c)=\bot$ and
$\vartheta_{2}(c)=\bot$ or $\vartheta_{1}(c)\neq\bot$,
$\vartheta_{2}(c)\neq\bot$ and
$\vartheta_{2}=\vartheta_{1}[c\leftarrow\vartheta_{1}(c)-1]$; (iii) ${\it
op}={\it nop}$ and $\vartheta_{2}=\vartheta_{1}$. Hence we apply the operation
of the transition only if the relevant counter value is defined (otherwise,
the counter remains undefined). Notice that, for a partial configuration
$\gamma_{1}$ and a transition $t$, there is at most one $\gamma_{2}$ with
$\gamma_{1}\stackrel{{\scriptstyle
t}}{{\longrightarrow}}_{\mathcal{E}}\gamma_{2}$. If such a $\gamma_{2}$
exists, we define $t(\gamma_{1}):=\gamma_{2}$; otherwise we define
$t(\gamma_{1}):=\bot$. We say that $t$ is enabled at $\gamma$ if
$t(\gamma)\neq\bot$. We observe that, in the case of energy semantics, $t$ is
not enabled only if ${\tt state}\left({\gamma}\right)\neq{\tt
source}\left(t\right)$.
#### VASS Semantics.
The difference between the energy and vass semantics is that counters in the
case of vass range over the natural numbers (rather than the integers), i.e.
the vass semantics will be interpreted over nonnegative partial
configurations. Thus, the transition relation
$\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{V}}$ induced by an
integer game ${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$ under the
vass semantics differs from the one induced by the energy semantics in the
sense that counters are not allowed to assume negative values. Hence
$\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{V}}$ is the
restriction of $\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{E}}$ to
nonnegative partial configurations. Here, a transition
$t=\left\langle{q_{1},{c}\mbox{\small- -},q_{2}}\right\rangle\in T$ is enabled
from $\gamma_{1}=\left\langle{q_{1},\vartheta_{1}}\right\rangle$ only if
$\vartheta_{1}(c)>0$ or $\vartheta_{1}(c)=\bot$. We assume without restriction
that at least one transition is enabled from each partial configuration (i.e.,
there are no deadlocks) in the vass semantics (and hence also in the energy
semantics). Below, we use ${\tt
sem}\in\left\\{{\mathcal{E}},{\mathcal{V}}\right\\}$ to distinguish the energy
and vass semantics.
#### Runs.
A run $\rho$ in semantics ${\tt sem}$ is an infinite sequence
$\gamma_{0}\stackrel{{\scriptstyle t_{1}}}{{\longrightarrow}}_{\tt
sem}\gamma_{1}\stackrel{{\scriptstyle t_{2}}}{{\longrightarrow}}_{\tt
sem}\cdots$ of transitions between concrete configurations. A path $\pi$ in
${\tt sem}$ is a finite sequence $\gamma_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\tt sem}\gamma_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\tt sem}\cdots\gamma_{n}$ of transitions between
concrete configurations. We say that $\rho$ (resp. $\pi$) is a $\gamma$-run
(resp. $\gamma$-path) if $\gamma_{0}=\gamma$. We define $\rho(i):=\gamma_{i}$
and $\pi(i):=\gamma_{i}$. We assume familiarity with the logic LTL. For an LTL
formula $\phi$ we write $\rho\models_{\mathcal{G}}\phi$ to denote that the run
$\rho$ in ${\mathcal{G}}$ satisfies $\phi$. For instance, given a set $\beta$
of concrete configurations, we write $\rho\models_{\mathcal{G}}\Diamond\beta$
to denote that there is an $i$ with $\gamma_{i}\in\beta$ (i.e., a member of
$\beta$ eventually occurs along $\rho$); and write
$\rho\models_{\mathcal{G}}\Box\Diamond\beta$ to denote that there are
infinitely many $i$ with $\gamma_{i}\in\beta$ (i.e., members of $\beta$ occur
infinitely often along $\rho$).
#### Strategies.
A strategy of Player $i\in\left\\{0,1\right\\}$ in ${\tt sem}$ (or simply an
$i$-strategy in ${\tt sem}$) $\sigma_{i}$ is a mapping that assigns to each
path $\pi=\gamma_{0}\stackrel{{\scriptstyle t_{1}}}{{\longrightarrow}}_{\tt
sem}\gamma_{1}\stackrel{{\scriptstyle t_{2}}}{{\longrightarrow}}_{\tt
sem}\cdots\gamma_{n}$ with ${\tt state}\left({\gamma_{n}}\right)\in Q_{i}$, a
transition $t=\sigma_{i}(\pi)$ with $t(\gamma_{n})\neq\bot$ in ${\tt sem}$. We
use $\Sigma_{i}^{\tt sem}$ to denote the sets of $i$-strategies in ${\tt
sem}$. Given a concrete configuration $\gamma$, $\sigma_{0}\in\Sigma_{0}^{\tt
sem}$, and $\sigma_{1}\in\Sigma_{1}^{\tt sem}$, we define ${\tt
run}\left(\gamma,\sigma_{0},\sigma_{1}\right)$ to be the unique run
$\gamma_{0}\stackrel{{\scriptstyle t_{1}}}{{\longrightarrow}}_{\tt
sem}\gamma_{1}\stackrel{{\scriptstyle t_{2}}}{{\longrightarrow}}_{\tt
sem}\cdots$ such that (i) $\gamma_{0}=\gamma$, (ii)
$t_{i+1}=\sigma_{0}(\gamma_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\tt sem}\gamma_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\tt sem}\cdots\gamma_{i})$ if ${\tt
state}\left({\gamma_{i}}\right)\in Q_{0}$, and (iii)
$t_{i+1}=\sigma_{1}(\gamma_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\tt sem}\gamma_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\tt sem}\cdots\gamma_{i})$ if ${\tt
state}\left({\gamma_{i}}\right)\in Q_{1}$. For $\sigma_{i}\in\Sigma_{i}^{\tt
sem}$, we write $[i,\sigma_{i},{\tt sem}]:\gamma\models_{\mathcal{G}}\phi$ to
denote that ${\tt
run}\left(\gamma,\sigma_{i},\sigma_{1-i}\right)\models_{\mathcal{G}}\phi$ for
all $\sigma_{1-i}\in\Sigma_{1-i}^{\tt sem}$. In other words, Player $i$ has a
winning strategy, namely $\sigma_{i}$, which ensures that $\phi$ will be
satisfied regardless of the strategy chosen by Player $1-i$. We write $[i,{\tt
sem}]:\gamma\models_{\mathcal{G}}\phi$ to denote that $[i,\sigma_{i},{\tt
sem}]:\gamma\models_{\mathcal{G}}\phi$ for some $\sigma_{i}\in\Sigma_{i}^{\tt
sem}$.
#### Instantiations.
Two nonnegative partial configurations $\gamma_{1},\gamma_{2}$ are said to be
disjoint if (i) ${\tt state}\left({\gamma_{1}}\right)={\tt
state}\left({\gamma_{2}}\right)$, and (ii) ${\it
dom}\left({\gamma_{1}}\right)\cap{\it dom}\left({\gamma_{2}}\right)=\emptyset$
(notice that we require the states to be equal). For a set of counters
$C\subseteq{\mathcal{C}}$, and disjoint partial configurations
$\gamma_{1},\gamma_{2}$, we say that $\gamma_{2}$ is a $C$-complement of
$\gamma_{1}$ if ${\it dom}\left({\gamma_{1}}\right)\cup{\it
dom}\left({\gamma_{2}}\right)=C$, i.e., ${\it dom}\left({\gamma_{1}}\right)$
and ${\it dom}\left({\gamma_{2}}\right)$ form a partitioning of the set $C$.
If $\gamma_{1}$ and $\gamma_{2}$ are disjoint then we define
$\gamma_{1}\oplus\gamma_{2}$ to be the nonnegative partial configuration
$\gamma:=\left\langle{q,\vartheta}\right\rangle$ such that $q:={\tt
state}\left({\gamma_{1}}\right)={\tt state}\left({\gamma_{2}}\right)$,
$\vartheta(c):={\tt val}\left({\gamma_{1}}\right)(c)$ if ${\tt
val}\left({\gamma_{1}}\right)(c)\neq\bot$, $\vartheta(c):={\tt
val}\left({\gamma_{2}}\right)(c)$ if ${\tt
val}\left({\gamma_{2}}\right)(c)\neq\bot$, and $\vartheta(c):=\bot$ if both
${\tt val}\left({\gamma_{1}}\right)(c)=\bot$ and ${\tt
val}\left({\gamma_{2}}\right)(c)=\bot$. In such a case, we say that $\gamma$
is a $C$-instantiation of $\gamma_{1}$. For a nonnegative partial
configuration $\gamma$ we write $\left\llbracket{\gamma}\right\rrbracket_{C}$
to denote the set of $C$-instantiations of $\gamma$. We will consider the
special case where $C={\mathcal{C}}$. In particular, we say that $\gamma_{2}$
is a complement of $\gamma_{1}$ if $\gamma_{2}$ is a
${\mathcal{C}}$-complement of $\gamma_{1}$, i.e., ${\tt
state}\left({\gamma_{2}}\right)={\tt state}\left({\gamma_{1}}\right)$ and
${\it dom}\left({\gamma_{1}}\right)={\mathcal{C}}-{\it
dom}\left({\gamma_{2}}\right)$. We use $\overline{\gamma}$ to denote the set
of complements of $\gamma$. If $\gamma_{2}\in\overline{\gamma_{1}}$, we say
that $\gamma=\gamma_{1}\oplus\gamma_{2}$ is an instantiation of $\gamma_{1}$.
Notice that $\gamma$ in such a case is concrete. For a nonnegative partial
configuration $\gamma$ we write $\left\llbracket{\gamma}\right\rrbracket$ to
denote the set of instantiations of $\gamma$. We observe that
$\left\llbracket{\gamma}\right\rrbracket=\left\llbracket{\gamma}\right\rrbracket_{{\mathcal{C}}}$
and that $\left\llbracket{\gamma}\right\rrbracket=\left\\{\gamma\right\\}$ for
any concrete nonnegative configuration $\gamma$.
#### Ordering.
For nonnegative partial configurations $\gamma_{1},\gamma_{2}$, we write
$\gamma_{1}\sim\gamma_{2}$ if ${\tt state}\left({\gamma_{1}}\right)={\tt
state}\left({\gamma_{2}}\right)$ and ${\it dom}\left({\gamma_{1}}\right)={\it
dom}\left({\gamma_{2}}\right)$. We write $\gamma_{1}\sqsubseteq\gamma_{2}$ if
${\tt state}\left({\gamma_{1}}\right)={\tt state}\left({\gamma_{2}}\right)$
and ${\it dom}\left({\gamma_{1}}\right)\subseteq{\it
dom}\left({\gamma_{2}}\right)$. For nonnegative partial configurations
$\gamma_{1}\sim\gamma_{2}$, we write $\gamma_{1}\preceq\gamma_{2}$ to denote
that ${\tt state}\left({\gamma_{1}}\right)={\tt
state}\left({\gamma_{2}}\right)$ and ${\tt
val}\left({\gamma_{1}}\right)(c)\leq{\tt val}\left({\gamma_{2}}\right)(c)$ for
all $c\in{\it dom}\left({\gamma_{1}}\right)={\it
dom}\left({\gamma_{2}}\right)$. For a nonnegative partial configuration
$\gamma$, we define
${\gamma}\\!\uparrow:=\left\\{{\gamma^{\prime}}|\;{\gamma\preceq\gamma^{\prime}}\right\\}$
to be the upward closure of $\gamma$, and define
${\gamma}\\!\downarrow:=\left\\{{\gamma^{\prime}}|\;{\gamma^{\prime}\preceq\gamma}\right\\}$
to be the downward closure of $\gamma$. Notice that
${\gamma}\\!\uparrow={\gamma}\\!\downarrow=\left\\{\gamma\right\\}$ for any
abstract configuration $\gamma$. For a set $\beta\subseteq\Gamma^{C}$ of
nonnegative partial configurations, let
${\beta}\\!\uparrow:=\cup_{\gamma\in\beta}{\gamma}\\!\uparrow$. We say that
$\beta$ is upward-closed if ${\beta}\\!\uparrow=\beta$. For an upward-closed
set $\beta\subseteq\Gamma^{C}$, we use ${\it min}\left(\beta\right)$ to denote
the (by Dickson’s Lemma unique and finite) set of minimal elements of $\beta$.
#### Winning Sets of Partial Configurations.
For a nonnegative partial configuration $\gamma$, we write $[i,{\tt
sem}]:\gamma\models_{\mathcal{G}}\phi$ to denote that
$\exists\gamma^{\prime}\in\left\llbracket{\gamma}\right\rrbracket.[i,{\tt
sem}]:\gamma^{\prime}\models_{\mathcal{G}}\phi$, i.e., Player $i$ is winning
from some instantiation $\gamma^{\prime}$ of $\gamma$. For a set
$C\subseteq{\mathcal{C}}$ of counters, we define
${\mathcal{W}}[{{\mathcal{G}},{\tt
sem},i,C}]({\phi}):=\left\\{{\gamma\in\Gamma^{C}}|\;{[{\tt
sem},i]:\gamma\models_{\mathcal{G}}\phi}\right\\}$. If
${\mathcal{W}}[{{\mathcal{G}},{\tt sem},i,C}]({\phi})$ is upward-closed, we
define the Pareto frontier as ${\tt Pareto}[{{\mathcal{G}},{\tt
sem},i,C}]({\phi}):={\it min}\left({\mathcal{W}}[{{\mathcal{G}},{\tt
sem},i,C}]({\phi})\right)$.
#### Properties.
We show some useful properties of the ordering on nonnegative partial
configurations. Note that for nonnegative partial configurations, we will not
make distinctions between the energy semantics and the vass semantics; this is
due to the fact that in nonnegative partial configurations and in their
instantiations we only consider positive values for the counters. For the
energy semantics, as we shall see, this will not be a problem since we will
consider winning runs where the counter never goes below $0$. We now show
monotonicity and (under some conditions) “reverse monotonicity” of the
transition relation wrt. $\preceq$. We write
$\gamma_{1}\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\tt sem}\gamma_{2}$
if there exists $t$ such that $\gamma_{1}\stackrel{{\scriptstyle
t}}{{\longrightarrow}}_{\tt sem}\gamma_{2}$.
###### Lemma 1
Let $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$ be nonnegative partial
configurations. If (i)
$\gamma_{1}\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{2}$,
and (ii) $\gamma_{1}\preceq\gamma_{3}$, then there is a $\gamma_{4}$ such that
$\gamma_{3}\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{4}$
and $\gamma_{2}\preceq\gamma_{4}$. Furthermore, if (i)
$\gamma_{1}\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{2}$,
and (ii) $\gamma_{3}\preceq\gamma_{1}$, and (iii) ${\mathcal{G}}$ is single-
sided and (iv) $\gamma_{1}\in\Gamma_{1}$, then there is a $\gamma_{4}$ such
that
$\gamma_{3}\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{4}$
and $\gamma_{4}\preceq\gamma_{2}$.
We consider a version of the Valk-Jantzen lemma [15], expressed in our
terminology.
###### Lemma 2
[15] Let $C\subseteq{\mathcal{C}}$ and let $U\subseteq\Gamma^{C}$ be upward-
closed. Then, ${\it min}\left(U\right)$ is computable if and only if, for any
nonnegative partial configuration $\gamma$ with ${\it
dom}\left({\gamma}\right)\subseteq C$, we can decide whether
$\left\llbracket{\gamma}\right\rrbracket_{C}\cap U\neq\emptyset$.
## 3 Game Problems
Abstract Energy decidable [7]$C$-version Single-Sided vassdecidable, Corollary
2Concrete Single-Sided vassdecidable$C$-version Energydecidable, Corollary
3Concrete EnergydecidableParetoSingle-Sided vasscomputable, Theorem 3.3Pareto
Energycomputable, Theorem 3.4Algorithm 1Lemma 5Lemma 4TrivialTrivialSection
4Section 4Lemma 5 Figure 1: Problems considered in the paper and their
relations. For each property, we state the lemma that show its
decidability/computability. The arrows show the reductions of problem
instances that we show in the paper.
Here we consider the parity winning condition for the integer games defined in
the previous section. First we establish a correspondence between the vass
semantics when the underlying integer game is single-sided, and the energy
semantics in the general case. We will show how instances of the single-sided
vass parity game can be reduced to the energy parity game, and vice-versa.
Figure 1 depicts a summary of our results. For either semantics, an instance
of the problem consists of an integer game ${\mathcal{G}}$ and a partial
configuration $\gamma$. For a given set of counters $C\subseteq{\mathcal{C}}$,
we will consider the $C$-version of the problem where we assume that ${\it
dom}\left({\gamma}\right)=C$. In particular, we will consider two special
cases: (i) the abstract version in which we assume that $\gamma$ is abstract
(i.e., ${\it dom}\left({\gamma}\right)=\emptyset$), and (ii) the concrete
version in which we assume that $\gamma$ is concrete (i.e., ${\it
dom}\left({\gamma}\right)={\mathcal{C}}$). The abstract version of a problem
corresponds to the unknown initial credit problem [6, 7], while the concrete
one corresponds to deciding if a given initial credit is sufficient or, more
generally, computing the Pareto frontier (left open in [6, 7]).
#### Winning Conditions.
Assume an integer game ${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$
where $\kappa:Q\mapsto\left\\{0,1,2,\ldots,k\right\\}$. For a partial
configuration $\gamma$ and $i:0\leq i\leq k$, the relation
$\gamma\models_{\mathcal{G}}({\tt color}=i)$ holds if $\kappa\left({{\tt
state}\left({\gamma}\right)}\right)=i$. The formula simply checks the color of
the state of $\gamma$. The formula $\gamma\models_{\mathcal{G}}\overline{\tt
neg}$ holds if ${\tt val}\left({\gamma}\right)(c)\geq 0$ for all $c\in{\it
dom}\left({\gamma}\right)$. The formula states that the values of all counters
are nonnegative in $\gamma$. For $i:0\leq i\leq k$, the predicate ${\it
even}(i)$ holds if $i$ is even. Define the path formula ${\tt
Parity}:=\bigvee_{(0\leq i\leq k)\wedge{\it
even}(i)}\left(\left(\Box\Diamond({\tt
color}=i)\right)\wedge\left(\bigwedge_{i<j\leq k}\Diamond\Box\neg({\tt
color}=j)\right)\right)$. The formula states that the highest color that
appears infinitely often along the path is even.
#### Energy Parity.
Given an integer game ${\mathcal{G}}$ and a partial configuration $\gamma$, we
ask whether $[0,{\mathcal{E}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}\wedge(\Box\overline{\tt neg})$, i.e., whether Player $0$ can force a
run in the energy semantics where the parity condition is satisfied and at the
same time the counters remain nonnegative. The abstract version of this
problem is equivalent to the unknown initial credit problem in classical
energy parity games [6, 7], since it amounts to asking for the existence of a
threshold for the initial counter values from which Player $0$ can win. The
nonnegativity objective $(\Box\overline{\tt neg})$ justifies our restriction
to nonnegative partial configurations in our definition of the instantiations
and hence of the winning sets.
###### Theorem 3.1
[7] The abstract energy parity problem is decidable.
The winning set ${\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$ is upward-closed for
$C\subseteq{\mathcal{C}}$. Intuitively, if Player $0$ can win the game with a
certain value for the counters, then any higher value for these counters also
allows him to win the game with the same strategy. This is because both the
possible moves of Player $1$ and the colors of configurations depend only on
the control-states.
###### Lemma 3
For any $C\subseteq{\mathcal{C}}$, the set
${\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$ is upward-closed.
Since this winning set is upward-closed, it follows from Dickson’s Lemma that
it has finitely many minimal elements. These minimal elements describe the
Pareto frontier of the minimal initial credit needed to win the game. In the
sequel we will show how to compute this set ${\tt
Pareto}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg})}):={\it
min}\left({\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})\right)$; cf. Theorem 3.4.
#### VASS Parity.
Given an integer game ${\mathcal{G}}$ and a nonnegative partial configuration
$\gamma$, we ask whether $[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}$, i.e., whether Player $0$ can force a run in the vass semantics where
the parity condition is satisfied. (The condition $\Box\overline{\tt neg}$ is
always trivially satisfied in vass.) In general, this problem is undecidable
as shown in [2], even for simple coverability objectives instead of parity
objectives.
###### Theorem 3.2
[2] The VASS Parity Problem is undecidable.
We will show that decidability of the vass parity problem is regained under
the assumption that ${\mathcal{G}}$ is single-sided. In [14] it was already
shown that, for a single-sided vass game with reachability objectives, it is
possible to compute the set of winning configurations. However, the proof for
parity objectives is much more involved.
#### Correspondence of Single-Sided vass Games and Energy Games.
We show that single-sided vass parity games can be reduced to energy parity
games, and vice-versa. The following lemma shows the direction from vass to
energy.
###### Lemma 4
Let ${\mathcal{G}}$ be a single-sided integer game and let $\gamma$ be a
nonnegative partial configuration. Then
$[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$ iff
$[0,{\mathcal{E}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$.
Hence for a single-sided ${\mathcal{G}}$ and any set
$C\subseteq{\mathcal{C}}$, we have
${\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})={\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$. Consequently, using Lemma 3 and
Theorem 3.1, we obtain the following corollary.
###### Corollary 1
Let ${\mathcal{G}}$ be single-sided and $C\subseteq{\mathcal{C}}$.
1. 1.
${\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt Parity}})$ is upward-
closed.
2. 2.
The $C$-version single-sided vass parity problem is reducible to the
$C$-version energy parity problem.
3. 3.
The abstract single-sided vass parity problem (i.e., where $C=\emptyset$) is
decidable.
The following lemma shows the reverse reduction from energy parity games to
single-sided vass parity games.
###### Lemma 5
Given an integer game ${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$,
one can construct a single-sided integer game
${\mathcal{G}}^{\prime}=\left\langle{Q^{\prime},T^{\prime},\kappa^{\prime}}\right\rangle$
with $Q\subseteq Q^{\prime}$ such that
$[0,{\mathcal{E}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$ iff
$[0,{\mathcal{V}}]:\gamma\models_{{\mathcal{G}}^{\prime}}{\tt Parity}$ for
every nonnegative partial configuration $\gamma$ of ${\mathcal{G}}$.
Proof sketch. Since ${\mathcal{G}}^{\prime}$ needs to be single-sided, Player
$1$ cannot change the counters. Thus the construction forces Player $0$ to
simulate the moves of Player $1$. Whenever a counter drops below zero in
${\mathcal{G}}$ (and thus Player $0$ loses), Player $0$ cannot perform this
simulation in ${\mathcal{G}}^{\prime}$ and is forced to go to a losing state
instead. ∎
#### Computability Results.
The following theorem (shown in Section 4) states our main computability
result. For single-sided vass parity games, the minimal elements of the
winning set ${\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt Parity}})$
(i.e., the Pareto frontier) are computable.
###### Theorem 3.3
If ${\mathcal{G}}$ is single-sided then ${\tt
Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt Parity}})$ is computable.
In particular, this implies decidability.
###### Corollary 2
For any set of counters $C\subseteq{\mathcal{C}}$, the $C$-version single-
sided vass parity problem is decidable.
From Theorem 3.3 and Lemma 5 we obtain the computability of the Pareto
frontier of the minimal initial credit needed to win general energy parity
games.
###### Theorem 3.4
${\tt Pareto}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$ is computable for any game
${\mathcal{G}}$.
###### Corollary 3
The $C$-version energy parity problem is decidable.
## 4 Solving Single-Sided VASS Parity Games (Proof of Theorem 3.3)
Consider a single-sided integer game
${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$ and a set
$C\subseteq{\mathcal{C}}$ of counters. We will show how to compute the set
${\tt Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt Parity}})$. We reduce
the problem of computing the Pareto frontier in the single-sided vass parity
game to solving the abstract energy parity game problem, which is decidable by
Theorem 3.1.
We use induction on $k=|{C}|$. As we shall see, the base case is
straightforward. We perform the induction step in two phases. First we show
that, under the induction hypothesis, we can reduce the problem of computing
the Pareto frontier to the problem of solving the $C$-version single-sided
vass parity problem (i.e., we need only to consider individual nonnegative
partial configurations in $\Gamma^{C}$). In the second phase, we introduce an
algorithm that translates the latter problem to the abstract energy parity
problem.
#### Base Case.
Assume that $C=\emptyset$. In this case we are considering the abstract
single-sided vass parity problem. Recall that
${\gamma}\\!\uparrow=\left\\{\gamma\right\\}$ for any $\gamma$ with ${\it
dom}\left({\gamma}\right)=\emptyset$. Since $C=\emptyset$, it follows that
${\tt Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})=\left\\{{\gamma}|\;{({\it
dom}\left({\gamma}\right)=\emptyset)\wedge\left([0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}\right)}\right\\}$. In other words, computing the Pareto frontier in
this case reduces to solving the abstract single-sided vass parity problem,
which is decidable by Corollary 1.
#### From Pareto Sets to vass Parity.
Assuming the induction hypothesis, we reduce the problem of computing the set
${\tt Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt Parity}})$ to the
$C$-version single-sided vass parity problem, i.e., the problem of checking
whether $[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$ for some
$\gamma\in\Gamma^{C}$ when the underlying integer game is single-sided. To do
that, we will instantiate the Valk-Jantzen lemma as follows. We instantiate
$U\subseteq\Gamma^{C}$ in Lemma 2 to be
${\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt Parity}})$ (this set
is upward-closed by Corollary 1 since ${\mathcal{G}}$ is single-sided). Take
any nonnegative partial configuration $\gamma$ with ${\it
dom}\left({\gamma}\right)\subseteq C$. We consider two cases. First, if ${\it
dom}\left({\gamma}\right)=C$, then we are dealing with the $C$-version single-
sided vass parity game which will show how to solve in the sequel. Second,
consider the case where ${\it dom}\left({\gamma}\right)=C^{\prime}\subset C$.
By the induction hypothesis, we can compute the (finite) set ${\tt
Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt Parity}})={\it
min}\left({\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt
Parity}})\right)$. Then to solve this case, we use the following lemma.
###### Lemma 6
For all nonnegative partial configurations $\gamma$ such that ${\it
dom}\left({\gamma}\right)=C^{\prime}\subset C$, we have
$\left\llbracket{\gamma}\right\rrbracket_{C}\cap{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})\neq\emptyset$ iff
$\gamma\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt
Parity}})$.
Hence checking
$\left\llbracket{\gamma}\right\rrbracket_{C}\cap{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})\neq\emptyset$ amounts to simply comparing $\gamma$ with the elements
of the finite set ${\tt
Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt Parity}})$, because
${\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt Parity}})$ is
upward-closed by Corollary 1.
#### From vass Parity to Abstract Energy Parity.
We introduce an algorithm that uses the induction hypothesis to translate an
instance of the $C$-version single-sided vass parity problem to an equivalent
instance of the abstract energy parity problem.
The following definition and lemma formalize some consequences of the
induction hypothesis. First we define a relation that allows us to directly
classify some nonnegative partial configurations as winning for Player $1$
(resp. Player $0$).
###### Definition 1
Consider a nonnegative partial configuration $\gamma$ and a set of nonnegative
partial configurations $\beta$. We write $\beta\lhd\gamma$ if: (i) for each
$\hat{\gamma}\in\beta$, ${\it dom}\left({\hat{\gamma}}\right)\subseteq C$ and
$|{\gamma}|=|{\hat{\gamma}}|+1$, and (ii) for each $c\in{\it
dom}\left({\gamma}\right)$ there is a $\hat{\gamma}\in\beta$ such that
$\hat{\gamma}\preceq\gamma[c\leftarrow\bot]$.
###### Lemma 7
Let $\beta=\bigcup_{C^{\prime}\subseteq C,|C^{\prime}|=|C|-1}{\tt
Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt Parity}})$ be the
Pareto frontier of minimal Player $0$ winning nonnegative partial
configurations with one counter in $C$ undefined. Let
$\\{c_{i},\dots,c_{j}\\}={\mathcal{C}}-C$ be the counters outside $C$.
1. 1.
For every $\hat{\gamma}\in\beta$ with $\\{c\\}=C-{\it
dom}\left({\hat{\gamma}}\right)$ there exists a minimal finite number
$v(\hat{\gamma})$ s.t. $\left\llbracket{\hat{\gamma}[c\leftarrow
v(\hat{\gamma})]}\right\rrbracket\cap{{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})}\neq\emptyset$.
2. 2.
For every $\hat{\gamma}\in\beta$ there is a number $u(\hat{\gamma})$ s.t.
$\hat{\gamma}[c\leftarrow v(\hat{\gamma})][c_{i}\leftarrow
u(\hat{\gamma}),\dots,c_{j}\leftarrow
u(\hat{\gamma})]\in{{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})}$, i.e., assigning value $u(\hat{\gamma})$ to counters outside $C$
is sufficient to make the nonnegative configuration winning for Player $0$.
3. 3.
If $\gamma\in\Gamma^{C}$ is a Player $0$ winning nonnegative partial
configuration, i.e.,
$\left\llbracket{\gamma}\right\rrbracket\cap{{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})}\neq\emptyset$, then $\beta\lhd\gamma$.
The third part of this lemma implies that if $\neg(\beta\lhd\gamma)$ then we
can directly conclude that $\gamma$ is not winning for Player $0$ (and thus
winning for Player $1$) in the parity game.
Now we are ready to present the algorithm (Algorithm 1).
#### Input and output of the algorithm.
The algorithm inputs a single-sided integer game
${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$, and a nonnegative
partial configuration $\gamma$ where ${\it dom}\left({\gamma}\right)=C$. To
check whether $[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$, it
constructs an instance of the abstract energy parity problem. This instance is
defined by a new integer game ${\mathcal{G}}^{\it out}=\left\langle{Q_{\it
out},T_{\it out},\kappa_{\it out}}\right\rangle$ with counters in
${\mathcal{C}}-C$, and a nonnegative partial configuration $\gamma^{\it out}$.
Since we are considering the abstract version of the problem, the
configuration $\gamma^{\it out}$ is of the form $\gamma^{\it
out}=\left\langle{q^{\it out},\vartheta_{\it out}}\right\rangle$ where ${\it
dom}\left({\vartheta_{\it out}}\right)=\emptyset$. The latter property means
that $\gamma^{\it out}$ is uniquely determined by the state $q^{\it out}$ (all
counter values are undefined). Lemma 9 relates ${\mathcal{G}}$ with the newly
constructed ${\mathcal{G}}^{\it out}$.
Input: ${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$: Single-Sided
Integer Game; $\gamma\in\Gamma^{C}$ with $|{C}|=k>0$.
Output: ${\mathcal{G}}^{\it out}=\left\langle{Q^{\it out},T^{\it
out},\kappa^{\it out}}\right\rangle$: integer game;
$q^{\it out}\in Q^{\it out}$; $\gamma^{\it out}=\left\langle{q^{\it
out},\vartheta_{\it out}}\right\rangle$ where ${\it dom}\left({\vartheta_{\it
out}}\right)=\emptyset$; $\lambda:Q_{\it out}\cup T_{\it
out}\mapsto\;\Gamma^{C}\cup T$
1
2$\beta\leftarrow\bigcup_{(C^{\prime}\subseteq
C)\wedge|{C^{\prime}}|=|{C}|-1}{\tt
Pareto}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}^{\prime}}]({{\tt
Parity}})$ ;
3
4$\;T^{\it out}\leftarrow\emptyset$; ${\tt new}\left({q^{\it out}}\right)$;
$\kappa\left({q^{\it out}}\right)\leftarrow\kappa\left({\gamma}\right)$;
$\lambda\left({q^{\it out}}\right)\leftarrow\gamma$; $Q_{\it
out}\leftarrow\left\\{q^{\it out}\right\\}$;
5 if _$\lambda\left({q^{\it out}}\right)\in\Gamma_{0}$_ then $Q_{0}^{\it
out}\leftarrow\left\\{q^{\it out}\right\\}$; $Q_{1}^{\it
out}\leftarrow\emptyset$;
6 else $Q_{1}^{\it out}\leftarrow\left\\{q^{\it out}\right\\}$; $Q_{0}^{\it
out}\leftarrow\emptyset$;
7 ;
8 ${\tt ToExplore}\leftarrow\left\\{q^{\it out}\right\\}$ ;
9
10while _${\tt ToExplore}\neq\emptyset$_ do
11 Pick and remove a $q\in{\tt ToExplore}$;
12 if _$\neg(\beta\lhd\lambda\left({q}\right))$_ then
13 $\kappa^{\it out}\left({q}\right)\leftarrow 1$; $T^{\it out}\leftarrow
T^{\it out}\cup\left\\{\left\langle{q,{\it nop},q}\right\rangle\right\\}$
14 else if _$\exists q^{\prime}.\left(q^{\prime},q\right)\in\left(T^{\it
out}\right)^{*}\wedge\left(\lambda(q^{\prime})\prec\lambda\left({q}\right)\right)$
_ then
15 $\kappa^{\it out}\left({q}\right)\leftarrow 0$; $T^{\it out}\leftarrow
T^{\it out}\cup\left\\{\left\langle{q,{\it nop},q}\right\rangle\right\\}$
16 else for _each $t\in T$ with $t(\lambda\left({q}\right))\neq\bot$_ do
17 if _$\exists q^{\prime}.\left(q^{\prime},q\right)\in\left(T^{\it
out}\right)^{*}.\lambda\left({q^{\prime}}\right)=t(\lambda\left({q}\right))$_
then
18 $T^{\it out}\leftarrow T^{\it out}\cup\left\\{\left\langle{q,{\tt
op}\left(t\right),q^{\prime}}\right\rangle\right\\}$;
$\lambda\left({\left\langle{q,{\tt
op}\left(t\right),q^{\prime}}\right\rangle}\right)\leftarrow t$
19 else
20 ${\tt new}\left({q^{\prime}}\right)$;
$\kappa\left({q^{\prime}}\right)\leftarrow\kappa\left({t(\lambda\left({q}\right))}\right)$;
$\lambda(q^{\prime})\leftarrow t(\lambda\left({q}\right))$;
21 if _$\lambda\left({q^{\prime}}\right)\in\Gamma_{0}$_ then $Q_{0}^{\it
out}\leftarrow Q_{0}^{\it out}\cup\left\\{q^{\prime}\right\\}$;
22 else $Q_{1}^{\it out}\leftarrow Q_{1}^{\it
out}\cup\left\\{q^{\prime}\right\\}$;
23 ;
24 $T^{\it out}\leftarrow T^{\it out}\cup\left\\{\left\langle{q,{\tt
op}\left(t\right),q^{\prime}}\right\rangle\right\\}$;
$\lambda(\left\langle{q,{\tt
op}\left(t\right),q^{\prime}}\right\rangle)\leftarrow t$;
25 ${\tt ToExplore}\leftarrow{\tt ToExplore}\cup\left\\{q^{\prime}\right\\}$;
26
27 ;
28 ;
29
30
Algorithm 1 Building an instance of the abstract energy parity problem.
#### Operation of the algorithm.
The algorithm performs a forward analysis similar to the classical Karp-Miller
algorithm for Petri nets. We start with a given nonnegative partial
configuration, explore its successors, create loops when previously visited
configurations are repeated and define a special operation for the case when
configurations strictly increase. The algorithm builds the graph of the game
${\mathcal{G}}^{\it out}$ successively (i.e., the set of states $Q^{\it out}$,
the set of transitions $T^{\it out}$, and the coloring of states $\kappa$).
Additionally, for bookkeeping purposes inside the algorithm and for reasoning
about the correctness of the algorithm, we define a labeling function
$\lambda$ on the set of states and transitions in ${\mathcal{G}}^{\it out}$
such that each state in ${\mathcal{G}}^{\it out}$ is labeled by a nonnegative
partial configuration in $\Gamma^{C}$, and each transition in
${\mathcal{G}}^{\it out}$ is labeled by a transition in ${\mathcal{G}}$.
The algorithm first computes the Pareto frontier ${\tt
Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt Parity}})$ for all
counter sets $C^{\prime}\subseteq{\mathcal{C}}$ with $|{C^{\prime}}|=|{C}|-1$.
This is possible by the induction hypothesis. It stores the union of all these
sets in $\beta$ (line 1). At line 1, the algorithms initializes the set of
transitions $T^{\it out}$ to be empty, creates the first state $q^{\it out}$,
defines its coloring to be the same as that of the state of the input
nonnegative partial configuration $\gamma$, labels it by $\gamma$, and then
adds it to the set of states $Q^{\it out}$. At line 1 it adds $q^{\it out}$ to
the set of states of Player $0$ or Player $1$ (depending on where $\gamma$
belongs), and at line 1 it adds $q^{\it out}$ to the set ${\tt ToExplore}$.
The latter contains the set of states that have been created but not yet
analyzed by the algorithm.
After the initialization phase, the algorithm starts iterating the while-loop
starting at line 1. During each iteration, it picks and removes a new state
$q$ from the set ${\tt ToExplore}$ (line 1). First, it checks two special
conditions under which the game is made immediately losing (resp. winning) for
Player $0$.
Condition 1: If $\neg(\beta\lhd\lambda\left({q}\right))$ (line 1), then we
know by Lemma 7 (item 3) that the nonnegative partial configuration
$\lambda\left({q}\right)$ is not winning for Player $0$ in ${\mathcal{G}}$.
Therefore, we make the state $q$ losing for Player $0$ in ${\mathcal{G}}^{\it
out}$. To do that, we change the color of $q$ to $1$ (any odd color will do),
and add a self-loop to $q$. Any continuation of a run from $q$ is then losing
for Player $0$ in ${\mathcal{G}}^{\it out}$.
Condition 2: If Condition 1 did not hold then the algorithm checks (at line 1)
whether there is a predecessor $q^{\prime}$ of $q$ in ${\mathcal{G}}^{\it
out}$ with a label $\lambda\left({q^{\prime}}\right)$ that is strictly smaller
than the label $\lambda\left({q}\right)$ of $q$, i.e.,
$\lambda\left({q^{\prime}}\right)\prec\lambda\left({q}\right)$. (Note that we
are not comparing $q$ to arbitrary other states in ${\mathcal{G}}^{\it out}$,
but only to predecessors.) If that is the case, then the state $q$ is made
winning for Player $0$ in ${\mathcal{G}}^{\it out}$. To do that, we change the
color of $q$ to $0$ (any even color will do), and add a self-loop to $q$. The
intuition for making $q$ winning for Player $0$ is as follows. Since
$\lambda\left({q^{\prime}}\right)\prec\lambda\left({q}\right)$, the path from
$\lambda\left({q^{\prime}}\right)$ to $\lambda\left({q}\right)$ increases the
value of at least one of the defined counters (those in $C$), and will not
decrease the other counters in $C$ (though it might have a negative effect on
the undefined counters in ${\mathcal{C}}-C$). Thus, if a run in
${\mathcal{G}}$ iterates this path sufficiently many times, the value of at
least one counter in $C$ will be pumped and becomes sufficiently high to allow
Player $0$ to win the parity game on ${\mathcal{G}}$, provided that the
counters in ${\mathcal{C}}-C$ are initially instantiated with sufficiently
high values. This follows from the property $\beta\lhd\lambda\left({q}\right)$
and Lemma 7 (items 1 and 2).
If none of the tests for Condition1/Condition2 at lines 1 and 1 succeeds, the
algorithm continues expanding the graph of ${\mathcal{G}}^{\it out}$ from $q$.
It generates all successors of $q$ by applying each transition $t\in T$ in
${\mathcal{G}}$ to the label $\lambda\left({q}\right)$ of $q$ (line 1). If the
result $t(\lambda\left({q}\right))$ is defined then there are two possible
cases. The first case occurs if we have previously encountered (and added to
$Q^{\it out}$) a state $q^{\prime}$ whose label equals
$t(\lambda\left({q}\right))$ (line 1). Then we add a transition from $q$ back
to $q^{\prime}$ in ${\mathcal{G}}^{\it out}$, where the operation of the new
transition is the same operation as that of $t$, and define the label of the
new transition to be $t$. Otherwise (line 1), we create a new state
$q^{\prime}$, label it with the nonnegative configuration
$t(\lambda\left({q}\right))$ and assign it the same color as
$t(\lambda\left({q}\right))$. At line 1 $q^{\it out}$ is added to the set of
states of Player $0$ or Player $1$ (depending on where $\gamma$ belongs). We
add a new transition between $q$ and $q^{\prime}$ with the same operation as
$t$. The new transition is labeled with $t$. Finally, we add the new state
$q^{\prime}$ to the set of states to be explored.
###### Lemma 8
Algorithm 1 will always terminate.
Lemma 8 implies that the integer game ${\mathcal{G}}^{\it out}$ is finite (and
hence well-defined). The following lemma shows the relation between the input
and output games ${\mathcal{G}},{\mathcal{G}}^{\it out}$.
###### Lemma 9
$[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$ iff
$[0,{\mathcal{E}}]:\gamma^{\it out}\models_{{\mathcal{G}}^{\it out}}{\tt
Parity}\wedge\Box\overline{\tt neg}$ .
Proof sketch. The left to right implication is easy. Given a Player $0$
winning strategy in ${\mathcal{G}}$, one can construct a winning strategy in
${\mathcal{G}}^{\it out}$ that uses the same transitions, modulo the labeling
function $\lambda\left({}\right)$. The condition $\Box\overline{\tt neg}$ in
${\mathcal{G}}^{\it out}$ is satisfied since the configurations in
${\mathcal{G}}$ are always nonnegative and the parity condition is satisfied
since the colors seen in corresponding plays in ${\mathcal{G}}^{\it out}$ and
${\mathcal{G}}$ are the same.
For the right to left implication we consider a Player $0$ winning strategy
$\sigma_{0}$ in ${\mathcal{G}}^{\it out}$ and construct a winning strategy
$\sigma_{0}^{\prime}$ in ${\mathcal{G}}$. The idea is that a play $\pi$ in
${\mathcal{G}}$ induces a play $\pi^{\prime}$ in ${\mathcal{G}}^{\it out}$ by
using the same sequence of transitions, but removing all so-called pumping
sequences, which are subsequences that end in Condition 2. Then
$\sigma_{0}^{\prime}$ acts on history $\pi$ like $\sigma_{0}$ on history
$\pi^{\prime}$. For a play according to $\sigma_{0}^{\prime}$ there are two
cases. Either it will eventually reach a configuration that is sufficiently
large (relative to $\beta$) such that a winning strategy is known by induction
hypothesis. Otherwise it contains only finitely many pumping sequences and an
infinite suffix of it coincides with an infinite suffix of a play according to
$\sigma_{0}$ in ${\mathcal{G}}^{\it out}$. Thus it sees the same colors and
satisfies ${\tt Parity}$. ∎
Since $\gamma^{\it out}$ is abstract and the abstract energy parity problem is
decidable (Theorem 3.1) we obtain Theorem 3.3.
The termination proof in Lemma 8 relies on Dickson’s Lemma, and thus there is
no elementary upper bound on the complexity of Algorithm 1 or on the size of
the constructed energy game ${\mathcal{G}}^{\it out}$. The algorithm in [4]
for the fixed initial credit problem in pure energy games without the parity
condition runs in $d$-exponential time (resp. $(d-1)$-exponential time for
offsets encoded in unary) for dimension $d$, and is thus not elementary
either. As noted in [4], the best known lower bound is EXPSPACE hardness,
easily obtained via a reduction from the control-state reachability (i.e.,
coverability) problem for Petri nets.
## 5 Applications to Other Problems
### 5.1 Weak simulation preorder between VASS and finite-state systems
Weak simulation preorder [9] is a semantic preorder on the states of labeled
transition graphs, which can be characterized by weak simulation games. A
configuration of the game is given by a pair of states $(q_{1},q_{0})$. In
every round of the game, Player $1$ chooses a labeled step
$q_{1}\stackrel{{\scriptstyle a}}{{\longrightarrow}}q_{1}^{\prime}$ for some
label $a$. Then Player $0$ must respond by a move which is either of the form
$q_{0}\stackrel{{\scriptstyle\tau^{*}a\tau^{*}}}{{\longrightarrow}}q_{0}^{\prime}$
if $a\neq\tau$, or of the form
$q_{0}\stackrel{{\scriptstyle\tau^{*}}}{{\longrightarrow}}q_{0}^{\prime}$ if
$a=\tau$ (the special label $\tau$ is used to model internal transitions). The
game continues from configuration $(q_{1}^{\prime},q_{0}^{\prime})$. A player
wins if the other player cannot move and Player $0$ wins every infinite play.
One says that $q_{0}$ weakly simulates $q_{1}$ iff Player $0$ has a winning
strategy in the weak simulation game from $(q_{1},q_{0})$. States in different
transition systems can be compared by putting them side-by-side and
considering them as a single transition system.
We use $\left\langle{Q,T,\Sigma,\lambda}\right\rangle$ to denote a labeled
vass where the states and transitions are defined as in Section 2, $\Sigma$ is
a finite set of labels and $\lambda:T\mapsto\Sigma$ assigns labels to
transitions.
It was shown in [3] that it is decidable whether a finite-state labeled
transition system weakly simulates a labeled vass. However, the decidability
of the reverse direction was open. (The problem is that the weak
$\stackrel{{\scriptstyle\tau^{*}a\tau^{*}}}{{\longrightarrow}}$ moves in the
vass make the weak simulation game infinitely branching.) We now show that it
is also decidable whether a labeled vass weakly simulates a finite-state
labeled transition system. In particular this implies that weak simulation
equivalence between a labeled vass and a finite-state labeled transition
system is decidable. This is in contrast to the undecidability of weak
bisimulation equivalence between vass and finite-state systems [11].
###### Theorem 5.1
It is decidable whether a labeled vass weakly simulates a finite-state labeled
transition system.
Proof sketch. Given a labeled vass and a finite-state labeled transition
system, one constructs a single-sided vass parity game s.t. the vass weakly
simulates the finite system iff Player $0$ wins the parity game. The idea is
to take a controlled product of the finite system and the vass s.t. every
round of the weak simulation game is encoded by a single move of Player $1$
followed by an arbitrarily long sequence of moves by Player $0$. The move of
Player $1$ does not change the counters, since it encodes a move in the finite
system, and thus the game is single-sided. Moreover, one enforces that every
sequence of consecutive moves by Player $0$ is finite (though it can be
arbitrarily long), by assigning an odd color to Player $0$ states and a higher
even color to Player $1$ states.
### 5.2 $\mu$-Calculus model checking VASS
While model checking vass with linear-time temporal logics (like LTL and
linear-time $\mu$-calculus) is decidable [8, 10], model checking vass with
most branching-time logics (like EF, EG, CTL and the modal $\mu$-calculus) is
undecidable [8]. However, we show that Theorem 3.3 yields the decidability of
model checking single-sided vass with a guarded fragment of the modal
$\mu$-calculus. We consider a vass $\left\langle{Q,T}\right\rangle$ where the
states, transitions and semantics are defined as in Section 2, and reuse the
notion of partial configurations and the transition relation defined for the
vass semantics on integer games. We specify properties on such vass in the
positive $\mu$-calculus $L^{\textit{pos}}_{\mu}$ whose atomic propositions $q$
refer to control-states $q\in Q$ of the input vass.
The syntax of the positive $\mu$-calculus $L^{\textit{pos}}_{\mu}$ is given by
the following grammar:
$\phi::=q~{}\mid~{}X~{}\mid~{}\phi\wedge\phi~{}\mid~{}\phi\vee\phi~{}\mid~{}\Diamond\phi~{}\mid~{}\Box\phi~{}\mid~{}\mu
X.\phi~{}\mid~{}\nu X.\phi$ where $q\in Q$ and $X$ belongs to a countable set
of variables $\mathcal{X}$. The semantics of $L^{\textit{pos}}_{\mu}$ is
defined as usual (see appendix). To each closed formula $\phi$ in
$L^{\textit{pos}}_{\mu}$ (i.e., without free variables) it assigns a subset of
concrete configurations $\llbracket\phi\rrbracket$.
The model-checking problem of vass with $L^{\textit{pos}}_{\mu}$ can then be
defined as follows. Given a vass $\mathcal{S}=\left\langle{Q,T}\right\rangle$,
a closed formula $\phi$ of $L^{\textit{pos}}_{\mu}$ and an initial
configuration $\gamma_{0}$ of $\mathcal{S}$, do we have
$\gamma_{0}\in\llbracket\phi\rrbracket$? If the answer is yes, we will write
$\mathcal{S},\gamma_{0}\models\phi$. The more general global model-checking
problem is to compute the set $\llbracket\phi\rrbracket$ of configurations
that satisfy the formula. The general unrestricted version of this problem is
undecidable.
###### Theorem 5.2
[8] The model-checking problem of vass with $L^{\textit{pos}}_{\mu}$ is
undecidable.
One way to solve the $\mu$-calculus model-checking problem for a given Kripke
structure is to encode the problem into a parity game [12]. The idea is to
construct a parity game whose states are pairs, where the first component is a
state of the structure and the second component is a subformula of the given
$\mu$-calculus formula. States of the form
$\left\langle{q,\Box\phi}\right\rangle$ or
$\left\langle{q,\phi\wedge\psi}\right\rangle$ belong to Player $1$ and the
remainder belong to Player $0$. The colors are assigned to reflect the nesting
of least and greatest fixpoints. We can adapt this construction to our context
by building an integer game from a formula in $L^{\textit{pos}}_{\mu}$ and a
vass $\mathcal{S}$, as stated by the next lemma.
###### Lemma 10
Let $\mathcal{S}$ be a vass, $\gamma_{0}$ a concrete configuration of
$\mathcal{S}$ and $\phi$ a closed formula in $L^{\textit{pos}}_{\mu}$. One can
construct an integer game ${\mathcal{G}}(\mathcal{S},\phi)$ and an initial
concrete configuration $\gamma^{\prime}_{0}$ such that
$[0,{\mathcal{V}}]:\gamma^{\prime}_{0}\models_{{\mathcal{G}}(\mathcal{S},\phi)}{\tt
Parity}$ if and only if $\mathcal{S},\gamma_{0}\models\phi$.
Now we show that, under certain restrictions on the considered vass and on the
formula from $L^{\textit{pos}}_{\mu}$, the constructed integer game
${\mathcal{G}}(\mathcal{S},\phi)$ is single-sided, and hence we obtain the
decidability of the model-checking problem from Theorem 3.3. First, we reuse
the notion of single-sided games from Section 2 in the context of vass, by
saying that a vass $\mathcal{S}=\left\langle{Q,T}\right\rangle$ is single-
sided iff there is a partition of the set of states $Q$ into two sets $Q_{0}$
and $Q_{1}$ such that ${\it op}={\it nop}$ for all transitions $t\in T$ with
${\tt source}\left(t\right)\in Q_{1}$. The guarded fragment
$L^{\textit{sv}}_{\mu}$ of $L^{\textit{pos}}_{\mu}$ for single-sided vass is
then defined by guarding the $\Box$ operator with a predicate that enforces
control-states in $Q_{1}$. Formally, the syntax of $L^{\textit{sv}}_{\mu}$ is
given by the following grammar:
$\phi::=q~{}\mid~{}X~{}\mid~{}\phi\wedge\phi~{}\mid~{}\phi\vee\phi~{}\mid~{}\Diamond\phi~{}\mid~{}Q_{1}\wedge\Box\phi~{}\mid~{}\mu
X.\phi~{}\mid~{}\nu X.\phi$, where $Q_{1}$ stands for the formula
$\bigvee_{q\in Q_{1}}q$. By analyzing the construction of Lemma 10 in this
restricted case, we obtain the following lemma.
###### Lemma 11
If $\mathcal{S}$ is a single-sided vass and $\phi\in L^{\textit{sv}}_{\mu}$
then the game ${\mathcal{G}}(\mathcal{S},\phi)$ is equivalent to a single-
sided game.
By combining the results of the last two lemmas with Corollary 1, Theorem 3.3
and Corollary 2, we get the following result on model checking single-sided
vass.
###### Theorem 5.3
1. 1.
Model checking $L^{\textit{sv}}_{\mu}$ over single-sided vass is decidable.
2. 2.
If $\mathcal{S}$ is a single-sided vass and $\phi$ is a formula of
$L^{\textit{sv}}_{\mu}$ then $\llbracket\phi\rrbracket$ is upward-closed and
its set of minimal elements is computable.
## 6 Conclusion and Outlook
We have established a connection between multidimensional energy games and
single-sided vass games. Thus our algorithm to compute winning sets in vass
parity games can also be used to compute the minimal initial credit needed to
win multidimensional energy parity games, i.e., the Pareto frontier.
It is possible to extend our results to integer parity games with a mixed
semantics, where a subset of the counters follow the energy semantics and the
rest follow the vass semantics. If such a mixed parity game is single-sided
w.r.t. the vass counters (but not necessarily w.r.t. the energy counters) then
it can be reduced to a single-sided vass parity game by our construction in
Section 3. The winning set of the derived single-sided vass parity game can
then be computed with the algorithm in Section 4.
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## Appendix
### Proof of Lemma 1
Let $\gamma_{1}=\left\langle{q_{1},\vartheta_{1}}\right\rangle$,
$\gamma_{2}=\left\langle{q_{2},\vartheta_{2}}\right\rangle$, and
$\gamma_{3}=\left\langle{q_{3},\vartheta_{3}}\right\rangle$ be nonnegative
partial configurations and let $t=\left\langle{q^{\prime}_{1},{\it
op},q^{\prime}_{2}}\right\rangle$ in $T$.
Assume that $\gamma_{1}\stackrel{{\scriptstyle
t}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{2}$ and that
$\gamma_{1}\preceq\gamma_{3}$. From this we know that
$q_{1}=q^{\prime}_{1}=q_{3}$, that $q_{2}=q^{\prime}_{2}$ and also that ${\it
dom}\left({\gamma_{1}}\right)={\it dom}\left({\gamma_{2}}\right)={\it
dom}\left({\gamma_{3}}\right)$. There are several cases for a transition $t$
that can be taken from $\gamma_{1}$. If ${\it op}$ is an increment or a ${\it
nop}$ operation then only the control-state matters for taking the transition.
If ${\it op}$ is a decrement transition then the initial value of the
decremented counter has to be either $\bot$ or $\geq 1$. Since this is the
case for $\vartheta_{1}$ and since $\gamma_{1}\preceq\gamma_{3}$, we deduce
that this also holds for $\vartheta_{3}$. Then we obtain the nonnegative
partial configuration
$\gamma_{4}=\left\langle{q_{4},\vartheta_{4}}\right\rangle$ from $\gamma_{3}$
by following the rule of the transition relation $\stackrel{{\scriptstyle
t}}{{\longrightarrow}}_{\mathcal{V}}$. Moreover, we can deduce that
$\gamma_{2}\preceq\gamma_{4}$, because any operation on the undefined counters
leaves the counters undefined, and for the other counters one can easily prove
that for all $c^{\prime}\in{\it dom}\left({\gamma_{1}}\right)$,
$\vartheta_{4}(c^{\prime})=\vartheta_{2}(c^{\prime})+(\vartheta_{3}(c^{\prime})-\vartheta_{1}(c^{\prime}))$.
Now suppose that $\gamma_{1}\stackrel{{\scriptstyle
t}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{2}$, that
$\gamma_{3}\preceq\gamma_{1}$, that ${\mathcal{G}}$ is single-sided and that
$\gamma_{1}\in\Gamma_{1}$. It follows that $q_{1}=q^{\prime}_{1}=q_{3}$, that
$q_{2}=q^{\prime}_{2}$ and also that ${\it dom}\left({\gamma_{1}}\right)={\it
dom}\left({\gamma_{2}}\right)={\it dom}\left({\gamma_{3}}\right)$.
Furthermore, since $\gamma_{1}\in\Gamma_{1}$, we deduce that $q_{1}\in Q_{1}$
and, since ${\mathcal{G}}$ is single-sided, we have that ${\it op}={\it nop}$.
Hence, by definition of the transition relation $\stackrel{{\scriptstyle
t}}{{\longrightarrow}}_{\mathcal{V}}$, we obtain $\vartheta_{1}=\vartheta_{2}$
and so by choosing $\gamma_{4}=\left\langle{q_{2},\vartheta_{3}}\right\rangle$
we obtain that $\gamma_{3}\stackrel{{\scriptstyle
t}}{{\longrightarrow}}\gamma_{4}$. Since $\gamma_{3}\preceq\gamma_{1}$, we
have $\vartheta_{3}(c)\leq\vartheta_{1}(c)$ for all $c\in{\it
dom}\left({\gamma_{1}}\right)$ and hence $\gamma_{4}\preceq\gamma_{2}$.
### Proof of Lemma 2
Usually the Valk and Jantzen Lemma, which allows the computation of the
minimal elements of an upward-closed set of vectors of naturals, is stated a
bit differently by using vectors of naturals with $\omega$ at some indexes to
represent any integer values (see for instance in [1]). In our context, the
$\omega$ are replaced by undefined values for the counters in the considered
nonnegative partial configurations, but the idea is the same. The usual way to
express the Valk and Jantzen Lemma is as follows: For
$C\subseteq{\mathcal{C}}$ and an upward-closed set $U\subseteq\Gamma^{C}$,
${\it min}\left(U\right)$ is computable if and only if for any nonnegative
partial configuration $\gamma$ with ${\it dom}\left({\gamma}\right)\subseteq
C$, one can decide whether
${\left\llbracket{\gamma}\right\rrbracket_{C}}\\!\downarrow\cap
U\neq\emptyset$. Now we show that this way of formalizing the Valk and Jantzen
Lemma is equivalent to the statement of Lemma 2.
First, if we assume that ${\it min}\left(U\right)$ is computable, then it is
obvious that for any nonnegative partial configuration $\gamma$ with ${\it
dom}\left({\gamma}\right)\subseteq C$, we can decide whether
$\left\llbracket{\gamma}\right\rrbracket_{C}\cap U\neq\emptyset$. In fact, it
suffices to check whether there exists a $\gamma_{1}\in{\it
min}\left(U\right)$ such that for all $c\in{\it dom}\left({\gamma}\right)$, we
have $\gamma(c)\geq\gamma_{1}(c)$ (since $U$ is upward-closed). Since ${\it
min}\left(U\right)$ is finite, it is possible check this condition for all
nonnegative partial configurations $\gamma_{1}$ in ${\it min}\left(U\right)$.
Now assume that for any nonnegative partial configuration $\gamma$ with ${\it
dom}\left({\gamma}\right)\subseteq C$, we can decide whether
$\left\llbracket{\gamma}\right\rrbracket_{C}\cap U\neq\emptyset$. Consider a
configuration $\gamma_{1}$ with ${\it dom}\left({\gamma_{1}}\right)\subseteq
C$. First note that ${\gamma_{1}}\\!\downarrow$ is a finite set and also that
${\left\llbracket{\gamma_{1}}\right\rrbracket_{C}}\\!\downarrow=\bigcup_{\gamma_{2}\in{\gamma_{1}}\\!\downarrow}\left\llbracket{\gamma_{2}}\right\rrbracket_{C}$.
But since ${\gamma_{1}}\\!\downarrow$ is finite, and since we can decide
whether $\left\llbracket{\gamma_{2}}\right\rrbracket_{C}\cap U\neq\emptyset$
for each $\gamma_{2}\in{\gamma_{1}}\\!\downarrow$, we can decide whether
${\left\llbracket{\gamma_{1}}\right\rrbracket_{C}}\\!\downarrow\cap
U\neq\emptyset$. By the Valk and Jantzen Lemma, ${\it min}\left(U\right)$ is
computable.
### Proof of Lemma 3
We will show that the set
${\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$ is upward-closed. Let
$\gamma_{1},\gamma_{2}\in\Gamma^{C}$ (with
$\gamma_{1}=\left\langle{q_{1},\vartheta_{1}}\right\rangle$ and
$\gamma_{2}=\left\langle{q_{1},\vartheta_{2}}\right\rangle$) such that
$\gamma_{1}\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$ and $\gamma_{1}\preceq\gamma_{2}$. In
order to prove that ${\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$ is upward-closed, we need to show that
$\gamma_{2}\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$.
Since $\gamma_{1}\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$, there exists
$\gamma^{\prime}_{1}\in\left\llbracket{\gamma_{1}}\right\rrbracket$ such that
$[0,{\mathcal{E}}]:\gamma^{\prime}_{1}\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$, i.e., there exists
$\gamma^{\prime}_{1}=\left\langle{q_{1},\vartheta^{\prime}_{1}}\right\rangle\in\left\llbracket{\gamma_{1}}\right\rrbracket$
and $\sigma_{0}\in\Sigma_{0}^{\mathcal{E}}$ such that ${\tt
run}\left(\gamma^{\prime}_{1},\sigma_{0},\sigma_{1}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$ for all
$\sigma_{1}\in\Sigma_{1}^{\mathcal{E}}$. Let us first define the following
concrete configuration
$\gamma^{\prime}_{2}=\left\langle{q_{1},\vartheta^{\prime}_{2}}\right\rangle$
with:
$\vartheta^{\prime}_{2}(c)=\left\\{\begin{array}[]{ll}\vartheta_{2}(c)&\mbox{
if }c\in{\it dom}\left({\gamma_{2}}\right)\\\ \vartheta^{\prime}_{1}(c)&\mbox{
if }c\notin{\it dom}\left({\gamma_{2}}\right)\\\ \end{array}\right.$
By definition we have
$\gamma^{\prime}_{2}\in\left\llbracket{\gamma_{2}}\right\rrbracket$ and since
$\gamma_{1}\preceq\gamma_{2}$, we also have
$\gamma^{\prime}_{1}\preceq\gamma^{\prime}_{2}$ (i.e.
$\vartheta^{\prime}_{1}(c)\leq\vartheta^{\prime}_{2}(c)$ for all
$c\in{\mathcal{C}}$). We want to show that
$[0,{\mathcal{E}}]:\gamma^{\prime}_{2}\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$, i.e., that player $0$ has a winning
strategy from the concrete configuration $\gamma^{\prime}_{2}$.
We now show how to build a winning strategy
$\sigma_{0}^{\prime}\in\Sigma_{0}^{\mathcal{E}}$ for player $0$ from the
configuration $\gamma^{\prime}_{2}$. First to any $\gamma^{\prime}_{2}$-path
$\pi=\gamma^{\prime\prime}_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\mathcal{E}}\gamma^{\prime\prime}_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\mathcal{E}}\cdots\gamma^{\prime\prime}_{n}$ we
associate the $\gamma^{\prime}_{1}$-path
$\alpha(\pi)=\gamma^{\prime\prime\prime}_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\mathcal{E}}\gamma^{\prime\prime\prime}_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\mathcal{E}}\cdots\gamma^{\prime\prime\prime}_{n}$
where for all $j\in\left\\{0,\ldots,n\right\\}$, if
$\gamma^{\prime\prime}_{j}=\left\langle{q^{\prime\prime}_{j},\vartheta^{\prime\prime}_{j}}\right\rangle$
then
$\gamma^{\prime\prime\prime}_{j}=\left\langle{q^{\prime\prime}_{j},\vartheta^{\prime\prime\prime}_{j}}\right\rangle$
with
$\vartheta^{\prime\prime\prime}_{j}(c)=\vartheta^{\prime\prime}_{j}(c)-(\vartheta^{\prime}_{2}(c)-\vartheta^{\prime}_{1}(c))$
for all $c\in{\mathcal{C}}$ (i.e. to obtain $\alpha(\pi)$ from $\pi$, we
decrement each counter valuation by the difference between
$\vartheta^{\prime}_{2}(c)-\vartheta^{\prime}_{1}(c)$). Note that
$\alpha(\pi)$ is a valid path since we are considering the energy semantics
where the counters can take negative values. Now we define the strategy
$\sigma_{0}^{\prime}\in\Sigma_{0}^{\mathcal{E}}$ for player $0$ as
$\sigma_{0}^{\prime}(\pi)=\sigma_{0}(\alpha(\pi))$ for each
$\gamma^{\prime}_{2}$-path $\pi$. Here again the strategy is well defined
since in energy games the enabledness of a transition depends only on the
control-state and not on the counter valuation.
We will now prove that for all strategies
$\sigma_{1}^{\prime}\in\Sigma_{1}^{\mathcal{E}}$, we have ${\tt
run}\left(\gamma^{\prime}_{2},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. Let
$\sigma_{1}^{\prime}\in\Sigma_{1}^{\mathcal{E}}$. Using $\sigma_{1}^{\prime}$,
we will construct another strategy $\sigma_{1}\in\Sigma_{1}^{\mathcal{E}}$ and
prove that if ${\tt
run}\left(\gamma^{\prime}_{1},\sigma_{0},\sigma_{1}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$ then ${\tt
run}\left(\gamma^{\prime}_{2},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. Before we give the definition of
$\sigma_{1}$, we introduce another notation. To any $\gamma^{\prime}_{1}$-path
$\pi=\gamma^{\prime\prime\prime}_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\mathcal{E}}\gamma^{\prime\prime\prime}_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\mathcal{E}}\cdots\gamma^{\prime\prime\prime}_{n}$
we associate the $\gamma^{\prime}_{2}$-path
$\overline{\alpha}(\pi)=\gamma^{\prime\prime}_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\mathcal{E}}\gamma^{\prime\prime}_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\mathcal{E}}\cdots\gamma^{\prime\prime}_{n}$ where
for all $j\in\left\\{0,\ldots,n\right\\}$, if
$\gamma^{\prime\prime\prime}_{j}=\left\langle{q^{\prime\prime}_{j},\vartheta^{\prime\prime\prime}_{j}}\right\rangle$
then
$\gamma^{\prime\prime}_{j}=\left\langle{q^{\prime\prime}_{j},\vartheta^{\prime\prime}_{j}}\right\rangle$
with
$\vartheta^{\prime\prime}_{j}(c)=\vartheta^{\prime\prime\prime}_{j}(c)+(\vartheta^{\prime}_{2}(c)-\vartheta^{\prime}_{1}(c))$
for all $c\in{\mathcal{C}}$ (i.e., to obtain $\overline{\alpha}(\pi)$ from
$\pi$, we increment each counter valuation by the difference between
$\vartheta^{\prime}_{2}(c)-\vartheta^{\prime}_{1}(c)$). Note that
$\overline{\alpha}(\pi)$ is a valid path. Now we define the strategy
$\sigma_{1}\in\Sigma_{1}^{\mathcal{E}}$ for Player $1$ as
$\sigma_{1}(\pi)=\sigma_{1}^{\prime}(\overline{\alpha}(\pi))$ for each
$\gamma^{\prime}_{1}$-path $\pi$.
We extend in the obvious way the function $\alpha()$ [resp.
$\overline{\alpha}()$] to $\gamma^{\prime}_{2}$-run [resp. to
$\gamma^{\prime}_{1}$-run]. Then one can easily check that we have
$\alpha({\tt
run}\left(\gamma^{\prime}_{2},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right))={\tt
run}\left(\gamma^{\prime}_{1},\sigma_{0},\sigma_{1}\right)$ and that ${\tt
run}\left(\gamma^{\prime}_{2},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)=\overline{\alpha}({\tt
run}\left(\gamma^{\prime}_{1},\sigma_{0},\sigma_{1}\right))$ by construction
of the strategy $\sigma_{0}^{\prime}$ and $\sigma_{1}$. First, remember that
$\sigma_{0}$ is a winning strategy for Player $0$ from the configuration
$\gamma^{\prime}_{1}$. Thus we have ${\tt
run}\left(\gamma^{\prime}_{1},\sigma_{0},\sigma_{1}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. Since in $\overline{\alpha}({\tt
run}\left(\gamma^{\prime}_{1},\sigma_{0},\sigma_{1}\right))$ the sequence of
control-states are the same and all the counter valuations along the path are
greater or equal to the ones seen in ${\tt
run}\left(\gamma^{\prime}_{1},\sigma_{0},\sigma_{1}\right)$ (remember that we
add to each configuration, to each counter $c$ the quantity
$\vartheta^{\prime}_{2}(c)-\vartheta^{\prime}_{1}(c)\geq 0$), this allows us
to deduce that $\overline{\alpha}({\tt
run}\left(\gamma^{\prime}_{1},\sigma_{0},\sigma_{1}\right))\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. Hence we have ${\tt
run}\left(\gamma^{\prime}_{2},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$.
Finally we have proved that there exist
$\gamma^{\prime}_{2}\in\left\llbracket{\gamma_{2}}\right\rrbracket$ and
$\sigma_{0}^{\prime}\in\Sigma_{0}^{\mathcal{E}}$ such that ${\tt
run}\left(\gamma^{\prime}_{2},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$ for all
$\sigma_{1}^{\prime}\in\Sigma_{1}^{\mathcal{E}}$. So Player $0$ has a winning
strategy from an instantiation of the configuration $\gamma_{2}$. Hence
$\gamma_{2}\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$.
### Proof of Lemma 4
Let ${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$ be a single-sided
integer game and $\gamma\in\Gamma^{C}$ a nonnegative partial configuration.
First we will assume that $[0,{\mathcal{E}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. This means that there exists
$\gamma^{\prime}=\left\langle{q,\vartheta}\right\rangle\in\left\llbracket{\gamma}\right\rrbracket$
and $\sigma_{0}\in\Sigma_{0}^{\mathcal{E}}$ such that ${\tt
run}\left(\gamma^{\prime},\sigma_{0},\sigma_{1}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$ for all
$\sigma_{1}\in\Sigma_{1}^{\mathcal{E}}$. The idea we will use here is that
since the strategy $\sigma_{0}$ keeps the value of the counters positive, then
the same strategy can be followed under the vass semantics, and furthermore
this strategy will be a winning strategy for the vass parity game. Let us
formalize this idea. We build the strategy
$\sigma_{0}^{\prime}\in\Sigma_{0}^{\mathcal{V}}$ as follows: for any path
$\pi=\gamma_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\mathcal{V}}\cdots\gamma_{n}$, we have
$\sigma_{0}^{\prime}(\pi)=\sigma_{0}(\pi)$ if
$\sigma_{0}(\pi)(\gamma_{n})\neq\bot$ (under the vass semantics) and otherwise
$\sigma_{0}^{\prime}(\pi)$ equals any enabled transition. Note that this
definition is valid since any path in the vass semantics is also a path in the
energy semantics. We consider now a strategy
$\sigma_{1}^{\prime}\in\Sigma_{1}^{\mathcal{V}}$. This strategy can be easily
extended to a strategy $\sigma_{1}\in\Sigma_{1}^{\mathcal{E}}$ for the energy
game by playing any transition when the input path is not a path valid under
the vass semantics. First note that since $\sigma_{0}$ is a winning strategy
in the energy parity game we have ${\tt
run}\left(\gamma^{\prime},\sigma_{0},\sigma_{1}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. From the way we build the strategies, we
deduce that ${\tt
run}\left(\gamma^{\prime},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)={\tt
run}\left(\gamma^{\prime},\sigma_{0},\sigma_{1}\right)$. Since the colors seen
along a run depend only of the control-state, we deduce that ${\tt
run}\left(\gamma^{\prime},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)\models_{\mathcal{G}}{\tt
Parity}$. Hence we have proven that
$[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$.
We now assume that $[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}$. This means that there exists
$\gamma^{\prime}=\left\langle{q,\vartheta}\right\rangle\in\left\llbracket{\gamma}\right\rrbracket$
and $\sigma_{0}\in\Sigma_{0}^{\mathcal{V}}$ such that ${\tt
run}\left(\gamma^{\prime},\sigma_{0},\sigma_{1}\right)\models_{\mathcal{G}}{\tt
Parity}$ for all $\sigma_{1}\in\Sigma_{1}^{\mathcal{V}}$. We build a strategy
$\sigma_{0}^{\prime}\in\Sigma_{0}^{\mathcal{E}}$ as follows: for any path
$\pi$ in the vass semantics $\sigma_{0}^{\prime}(\pi)=\sigma_{0}(\pi)$;
otherwise, if $\pi$ is not a valid path under the vass semantics,
$\sigma_{0}^{\prime}(\pi)$ is equal to any transition enabled in the last
configuration of the path. Take now a strategy
$\sigma_{1}^{\prime}\in\Sigma_{0}^{\mathcal{E}}$ for Player 1 in the energy
parity game. From $\sigma_{1}^{\prime}$, we define a strategy
$\sigma_{1}\in\Sigma_{0}^{\mathcal{V}}$ as follows: for any path $\pi$ in the
vass semantics, let $\sigma_{1}(\pi)=\sigma_{1}^{\prime}(\pi)$. Note that
since the game is single-sided this strategy is well defined; in fact, in a
single-sided game, in the states of Player 1, all the outgoing transitions are
enabled in the energy and in the vass semantics (because in single-sided
games, Player 1 does not change the counter values). But then we have ${\tt
run}\left(\gamma^{\prime},\sigma_{0},\sigma_{1}\right)={\tt
run}\left(\gamma^{\prime},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)$ and
since ${\tt
run}\left(\gamma^{\prime},\sigma_{0},\sigma_{1}\right)\models_{\mathcal{G}}{\tt
Parity}$ and since it is a valid run under the vass semantics, we deduce that
the values of the counters always remain positive. Consequently we have ${\tt
run}\left(\gamma^{\prime},\sigma_{0}^{\prime},\sigma_{1}^{\prime}\right)\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. We conclude that
$[0,{\mathcal{E}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$.
### Proof of Lemma 5
Let ${\mathcal{G}}=\left\langle{Q,T,\kappa}\right\rangle$ be an integer game.
From it we build a single-sided integer game
${\mathcal{G}}^{\prime}=\left\langle{Q^{\prime},T^{\prime},\kappa^{\prime}}\right\rangle$
as follows:
* •
$Q^{\prime}=Q\uplus\left\\{q_{t}\mid t\in
T\right\\}\uplus\left\\{q_{\ell}\right\\}$ (where $\uplus$ denotes the
disjoint union operator), with $Q_{0}^{\prime}=Q_{0}\uplus\left\\{q_{t}\mid
t\in T\right\\}\uplus\left\\{q_{\ell}\right\\}$ and $Q_{1}^{\prime}=Q_{1}$;
* •
$T^{\prime}$ is the smallest set of transitions such that, for each transition
$t=\left\langle{q_{1},{\it op},q_{2}}\right\rangle$ in $T$, the following
conditions are respected:
* –
$\left\langle{q_{1},{\it nop},q_{t}}\right\rangle\in T^{\prime}$;
* –
$\left\langle{q_{t},{\it op},q_{2}}\right\rangle\in T^{\prime}$;
* –
$\left\langle{q_{t},{\it nop},q_{\ell}}\right\rangle\in T^{\prime}$;
* –
$\left\langle{q_{\ell},{\it nop},q_{\ell}}\right\rangle\in T^{\prime}$;
* •
$\kappa^{\prime}$ is defined as follows:
* –
for all $q\in Q$, $\kappa^{\prime}(q)=\kappa(q)$;
* –
for all $t\in T$, $\kappa^{\prime}(q_{t})=0$;
* –
$\kappa^{\prime}(q_{\ell})=1$.
By construction ${\mathcal{G}}^{\prime}$ is single-sided. Also note that once
the system enters the losing state $q_{\ell}$, Player 0 loses the game since
the only possible infinite run from this state remains in $q_{\ell}$ and the
color associated to this state is odd (it is equal to $1$). Figure 2 depicts
the encoding of transitions of the form $\left\langle{q_{1},{c}\mbox{\small-
-},q_{2}}\right\rangle$.
$q_{1}$$q_{t}$$q_{2}$$q_{\ell}$${\it nop}$${c}\mbox{\small- -}$${\it
nop}$${\it nop}$ Figure 2: Translating a transition
$\left\langle{q_{1},{c}\mbox{\small- -},q_{2}}\right\rangle$ from an energy
game to a single-sided vass game. Note that $\kappa\left({q_{\ell}}\right)$ is
odd.
We will now prove that ${\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt
neg}})={\mathcal{W}}[{{\mathcal{G}}^{\prime},{\mathcal{V}},0,C}]({{\tt
Parity}})\cap\left\\{\gamma\mid{\tt state}\left({\gamma}\right)\in
Q\right\\}$. First let
$\gamma\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$. This means that there exists
$\gamma^{\prime}\in\left\llbracket{\gamma}\right\rrbracket$ and
$\sigma_{0}\in\Sigma_{0}^{\mathcal{E}}$ such that
$[0,\sigma_{0},{\mathcal{E}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. From $\sigma_{0}$, we will build a
winning strategy $\sigma_{0}^{\prime}\in\Sigma_{0}^{\mathcal{V}}$ for player 0
in ${\mathcal{G}}^{\prime}$. Let us first introduce some notation. To a path
in ${\mathcal{G}}^{\prime}$, $\pi=\gamma_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{t_{1}}\stackrel{{\scriptstyle
t^{\prime}_{1}}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{t_{2}}\stackrel{{\scriptstyle
t^{\prime}_{2}}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{2}\cdots\gamma_{n}$
with ${\tt state}\left({\gamma_{n}}\right)\in Q$, we associate the path
$\beta(\pi)=\gamma_{0}\stackrel{{\scriptstyle
t1}}{{\longrightarrow}}_{\mathcal{E}}\gamma_{1}\stackrel{{\scriptstyle
t2}}{{\longrightarrow}}_{\mathcal{E}}\cdots\gamma_{n}$ in ${\mathcal{G}}$ (by
construction of ${\mathcal{G}}^{\prime}$ such a path exists). The strategy
$\sigma_{0}^{\prime}$ is then defined as follows. For all paths
$\pi=\gamma_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{2}\cdots\gamma_{n}$ in
${\mathcal{G}}^{\prime}$:
* •
if ${\tt state}\left({\gamma_{n}}\right)\in Q$, then
$\sigma_{0}^{\prime}(\pi)=\left\langle{{\tt
state}\left({\gamma_{n}}\right),{\it nop},q_{t}}\right\rangle$ with
$t=\sigma_{0}(\beta(\pi))$;
* •
if ${\tt state}\left({\gamma_{n}}\right)=q_{t}$ for some transition
$t=\left\langle{q_{1},{\it op},q_{2}}\right\rangle\in T$, then if
$\left\langle{q_{t},{\it op},q_{2}}\right\rangle$ is enabled in $\gamma_{n}$,
$\sigma_{0}^{\prime}(\pi)=\left\langle{q_{t},{\it op},q_{2}}\right\rangle$,
otherwise $\sigma_{0}^{\prime}=\left\langle{q_{t},{\it
nop},q_{\ell}}\right\rangle$;
* •
if ${\tt state}\left({\gamma_{n}}\right)=q_{\ell}$, then
$\sigma_{0}^{\prime}(\pi)=\left\langle{q_{\ell},{\it
nop},q_{\ell}}\right\rangle$.
One can then easily verify using the definition of ${\mathcal{G}}^{\prime}$
and of the strategy $\sigma_{0}^{\prime}$ that since
$[0,\sigma_{0},{\mathcal{E}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}\wedge\Box\overline{\tt neg}$, we have
$[0,\sigma_{0}^{\prime},{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt
Parity}$ and hence that
$\gamma\in{\mathcal{W}}[{{\mathcal{G}}^{\prime},{\mathcal{V}},0,C}]({{\tt
Parity}})\cap\left\\{\gamma\mid{\tt state}\left({\gamma}\right)\in
Q\right\\}$.
The proof that if we take
$\gamma\in{\mathcal{W}}[{{\mathcal{G}}^{\prime},{\mathcal{V}},0,C}]({{\tt
Parity}})\cap\left\\{\gamma\mid{\tt state}\left({\gamma}\right)\in Q\right\\}$
then $\gamma$ belongs also to
${\mathcal{W}}[{{\mathcal{G}},{\mathcal{E}},0,C}]({{\tt
Parity}\wedge\Box\overline{\tt neg}})$ is done similarly.
### Proof of Lemma 6
Let $\gamma$ be a nonnegative partial configuration such that ${\it
dom}\left({\gamma}\right)=C^{\prime}\subset C$. Suppose that
$\left\llbracket{\gamma}\right\rrbracket_{C}\cap{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})\neq\emptyset$, i.e., there is a
$\gamma_{1}\in\left\llbracket{\gamma}\right\rrbracket_{C}$ where
$\gamma_{1}\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})$. Since
$\gamma_{1}\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})$ there is a
$\gamma_{2}\in\left\llbracket{\gamma_{1}}\right\rrbracket$ with
$[0,{\mathcal{V}}]:\gamma_{2}\models_{\mathcal{G}}{\tt Parity}$. Notice that
$\gamma_{2}\in\left\llbracket{\gamma}\right\rrbracket$. It follows that
$\gamma\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt
Parity}})$.
Now, suppose that
$\gamma\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt
Parity}})$. By definition there is a
$\gamma_{1}\in\left\llbracket{\gamma}\right\rrbracket$ such that
$[0,{\mathcal{V}}]:\gamma_{1}\models_{\mathcal{G}}{\tt Parity}$. Define
$\gamma_{2}$ by $\gamma_{2}(c):=\gamma_{1}(c)$ for all $c\in C$ and
$\gamma_{2}(c):=\bot$ for all $c\notin C$. Then
$\gamma_{2}\in\left\llbracket{\gamma}\right\rrbracket_{C}$ and
$\gamma_{2}\in{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})$, hence
$\left\llbracket{\gamma}\right\rrbracket_{C}\cap{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,C}]({{\tt
Parity}})\neq\emptyset$.
### Proof of Lemma 7
1. 1.
Consider a partial nonnegative configuration $\hat{\gamma}\in\beta$ where
$c\in C-{\it dom}\left({\hat{\gamma}}\right)$.
Since
$\left\llbracket{\hat{\gamma}}\right\rrbracket\cap{{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})}\neq\emptyset$, there exists a minimal finite number
$v(\hat{\gamma})$ s.t.
${\mathcal{W}}(\hat{\gamma}):=\left\llbracket{\hat{\gamma}[c\leftarrow
v(\hat{\gamma})]}\right\rrbracket\cap{{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})}\neq\emptyset$.
2. 2.
In particular, ${\mathcal{W}}(\hat{\gamma})$ is upward-closed w.r.t. the
counters in ${\mathcal{C}}-C$ and ${\it
min}\left({\mathcal{W}}(\hat{\gamma})\right)$ is finite. Let $u(\hat{\gamma})$
be the maximal constant appearing in ${\it
min}\left({\mathcal{W}}(\hat{\gamma})\right)$. Thus, an instantiation of
$\hat{\gamma}[c\leftarrow v(\hat{\gamma})]$ where the counters in
${\mathcal{C}}-C$ have values $\geq u(\hat{\gamma})$ is certainly winning for
Player $0$, i.e., in
${{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})}$.
3. 3.
The first condition of Def. 1 is satisfied by the definition of $\beta$.
Moreover, since $\gamma\in\Gamma^{C}$ and
$\left\llbracket{\gamma}\right\rrbracket\cap{{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})}\neq\emptyset$, for every $c\in C$ we have
$\left\llbracket{\gamma[c\leftarrow\bot]}\right\rrbracket\cap{{\mathcal{W}}[{{\mathcal{G}},{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})}\neq\emptyset$. Since $\beta$ are by definition the minimal
nonnegative configurations (with a domain which is exactly one element smaller
than $C$) that have this property, there must exist some element
$\hat{\gamma}\in\beta$ s.t. $\hat{\gamma}\preceq\gamma$. Therefore, also the
second condition of Def. 1 is satisfied and we get $\beta\lhd\gamma$. ∎
### Proof of Lemma 8
We assume the contrary and derive a contradiction. If Algorithm 1 does not
terminate then, in the graph of the game ${\mathcal{G}}^{\it out}$, it will
build an infinite sequence of states $q_{0},\ldots,q_{k},\ldots$ such that,
for all $i,j\in{\mathbb{N}}$, the following properties hold: $i<j$ implies
(a)
$(q_{i},q_{j})\in\left(T^{\it out}\right)^{*}$, and,
(b)
$\lambda\left({q_{i}}\right)\neq\lambda\left({q_{j}}\right)$, and,
(c)
$\lambda\left({q_{i}}\right)\not\prec\lambda\left({q_{j}}\right)$.
The property (a) comes from the way we build the transition relation when
adding new state to the set to ${\tt ToExplore}$ at Line 17 of the algorithm
(and from the fact that the vass is finitely branching and hence so is the
graph of the game ${\mathcal{G}}^{\it out}$). The property (b) is deduced
thanks to the test at Line 12 that necessarily fails infinitely often,
otherwise the algorithm would terminate. The property (c) is obtained thanks
to the test at Line 9 which must also fail infinitely often if the algorithm
does not terminate. Since the number of counters is fixed, the set
$(\Gamma^{C},\preceq)$ is well-quasi-ordered by Dickson’s Lemma. Hence in the
infinite sequence of states $q_{0},\ldots,q_{k},\ldots$ there must appear two
states $q_{i}$ and $q_{j}$ with $i<j$ such that
$\lambda\left({q_{i}}\right)\preceq\lambda\left({q_{j}}\right)$, which is a
contradiction to the conjunction of (b) and (c). This allows us to conclude
that the Algorithm 1 necessarily terminates.
### Proof of Lemma 9
We show both directions of the equivalence.
#### Left to right implication.
If $[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$ then there
exists a concrete nonnegative configuration
$\gamma_{0}\in\left\llbracket{\gamma}\right\rrbracket$ with
$\gamma_{0}=\gamma\oplus\gamma^{\prime}$ s.t.
$[0,{\mathcal{V}}]:\gamma_{0}\models_{\mathcal{G}}{\tt Parity}$, i.e.,
$\gamma^{\prime}$ assigns values to the counters in ${\mathcal{C}}-C$.
Moreover, we have $\gamma^{\it out}=\left\langle{q^{\it out},\vartheta_{\it
out}}\right\rangle$ where $\lambda\left({q^{\it out}}\right)=\gamma$ and ${\it
dom}\left({\vartheta_{\it out}}\right)=\emptyset$.
Using the winning strategy $\sigma_{0}\in\Sigma_{0}^{\mathcal{V}}$ of Player
$0$ in ${\mathcal{G}}$ from $\gamma_{0}$, we will construct a winning strategy
$\sigma_{0}^{\prime}\in\Sigma_{0}^{\mathcal{E}}$ of Player $0$ from a concrete
configuration $\gamma^{\prime}_{0}\in\left\llbracket{\gamma^{\it
out}}\right\rrbracket$ in ${\mathcal{G}}^{\it out}$, where
$\gamma^{\prime}_{0}=\left\langle{q^{\it out},{\tt
val}\left({\gamma^{\prime}}\right)}\right\rangle$. We do this by maintaining a
correspondence between nonnegative configurations in both games and between
the used sequences of transitions. Let $\pi=\gamma_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}_{\mathcal{V}}\gamma_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}_{\mathcal{V}}\dots\gamma_{n}$ a partial play in
${\mathcal{G}}$, and $\pi^{\prime}=\gamma_{0}^{\prime}\stackrel{{\scriptstyle
t_{1}^{\prime}}}{{\longrightarrow}}_{\mathcal{E}}\gamma_{1}^{\prime}\stackrel{{\scriptstyle
t_{2}^{\prime}}}{{\longrightarrow}}_{\mathcal{E}}\dots\gamma_{n}^{\prime}$ a
partial play in ${\mathcal{G}}^{\it out}$.
We will define $\sigma_{0}^{\prime}$ to ensure that either the following
invariant holds for all $i\geq 0$ or Condition 2 holds for some
$\gamma_{n}^{\prime}$ and the invariant holds for all $i\leq n$.
1. 1.
$\lambda\left({t_{i}^{\prime}}\right)=t_{i}$
2. 2.
$\lambda\left({{\tt
state}\left({\gamma_{i}^{\prime}}\right)}\right)=\gamma_{i}|C$
3. 3.
${\tt val}\left({\gamma_{i}^{\prime}}\right)=\gamma_{i}|({\mathcal{C}}-C)$
4. 4.
$\kappa\left({\gamma_{i}^{\prime}}\right)=\kappa\left({\gamma_{i}}\right)$
These conditions are satisfied for the initial states at $i=0$, since
$\lambda\left({{\tt
state}\left({\gamma_{0}^{\prime}}\right)}\right)=\lambda\left({q^{\it
out}}\right)=\gamma=\gamma_{0}|C$, ${\tt
val}\left({\gamma_{0}^{\prime}}\right)={\tt
val}\left({\gamma^{\prime}}\right)=\gamma_{0}|({\mathcal{C}}-C)$ and
$\kappa\left({\gamma_{0}^{\prime}}\right)=\kappa\left({q^{\it
out}}\right)=\kappa\left({\gamma}\right)=\kappa\left({\gamma_{0}}\right)$.
For the step we choose $\sigma_{0}^{\prime}(\pi^{\prime}):=t_{n+1}^{\prime}$
s.t. $\lambda\left({t_{n+1}^{\prime}}\right)=t_{n+1}=\sigma_{0}(\pi)$ which
maintains the invariant.
It cannot happen that Condition 1 holds in $\pi^{\prime}$. All visited
nonnegative configurations $\gamma_{i}$ in the winning play $\pi$ are also
winning for Player $0$. By Lemma 7 (item 3), we have $\beta\lhd\gamma_{i}|C$
and thus $\beta\lhd\gamma_{i}|C=\lambda\left({{\tt
state}\left({\gamma_{i}^{\prime}}\right)}\right)$ so that Condition 1 is false
at $\gamma_{i}^{\prime}$.
Since ${\mathcal{G}}$ is a vass-game, we have $\gamma_{i}\geq 0$ for all
$i\geq 0$. Therefore $\lambda\left({{\tt
state}\left({\gamma_{i}^{\prime}}\right)}\right)=\gamma_{i}|C\geq 0$ and ${\tt
val}\left({\gamma_{i}^{\prime}}\right)=\gamma_{i}|({\mathcal{C}}-C)\geq 0$.
Thus the same transitions are possible in ${\mathcal{G}}^{\it out}$ as in
${\mathcal{G}}$.
In the case where Condition 2 eventually holds in ${\mathcal{G}}^{\it out}$,
Player $0$ trivially wins the game in ${\mathcal{G}}^{\it out}$. Otherwise we
have ${\tt
val}\left({\gamma_{i}^{\prime}}\right)=\gamma_{i}|({\mathcal{C}}-C)\geq 0$ for
all $i\geq 0$ and thus the nonnegativity condition $\Box\overline{\tt neg}$ of
${\mathcal{G}}^{\it out}$ is satisfied by $\pi^{\prime}$.
Finally, since the parity condition is satisfied by $\pi$ and
$\kappa\left({\gamma_{i}}\right)=\kappa\left({\gamma_{i}^{\prime}}\right)$,
the parity condition is also satisfied by $\pi^{\prime}$. Therefore
$\sigma_{0}^{\prime}$ is winning for Player $0$ in ${\mathcal{G}}^{\it out}$
from $\gamma^{\prime}_{0}\in\left\llbracket{\gamma^{\it out}}\right\rrbracket$
and thus we obtain $[0,{\mathcal{E}}]:\gamma^{\it
out}\models_{{\mathcal{G}}^{\it out}}{\tt Parity}\wedge\Box\overline{\tt neg}$
as required.
#### Right to left implication.
If $[0,{\mathcal{E}}]:\gamma^{\it out}\models_{{\mathcal{G}}^{\it out}}{\tt
Parity}\wedge\Box\overline{\tt neg}$ then there exists a concrete nonnegative
configuration $\gamma^{\prime}\in\left\llbracket{\gamma^{\it
out}}\right\rrbracket$ s.t.
$[0,{\mathcal{E}}]:\gamma^{\prime}\models_{{\mathcal{G}}^{\it out}}{\tt
Parity}\wedge\Box\overline{\tt neg}$. Due to the concreteness of
$\gamma^{\prime}$ and the $\Box\overline{\tt neg}$ property, we also have
$[0,{\mathcal{V}}]:\gamma^{\prime}\models_{{\mathcal{G}}^{\it out}}{\tt
Parity}$. Thus Player $0$ has a winning strategy $\sigma_{0}$ in the vass
parity game on ${\mathcal{G}}^{\it out}$ from the concrete nonnegative
configuration $\gamma^{\prime}$.
Using $\sigma_{0}$, we will construct a winning strategy $\sigma_{0}^{\prime}$
for Player $0$ in the vass parity game on ${\mathcal{G}}$ from some
nonnegative configuration
$\gamma_{0}\in\left\llbracket{\gamma}\right\rrbracket$. Let
$\gamma_{0}=\gamma\oplus\gamma^{\prime\prime}$, where $\gamma^{\prime\prime}$
is some yet to be constructed function assigning sufficiently high values to
counters in ${\mathcal{C}}-C$. We only prove the sufficient condition that a
winning strategy $\sigma_{0}^{\prime}$ exists, but do not construct a Turing
machine that implements it. This is because $\sigma_{0}^{\prime}$ uses the
numbers $v(\hat{\gamma})$ and $u(\hat{\gamma})$ from Lemma 7 that are not
computed here.
In order to construct $\sigma_{0}^{\prime}$ and $\gamma^{\prime\prime}$, we
need some definitions. Consider a sequence of transitions in
${\mathcal{G}}^{\it out}$ that leads from $q$ to $q^{\prime}$ ending with
Condition 2 at line 1 in the algorithm. We call this sequence a pumping
sequence. Its effect is nonnegative on all counters in $C$ and strictly
increasing in at least one of them, although its effect may be negative on
counters in ${\mathcal{C}}-C$. Due to the finiteness of ${\mathcal{G}}^{\it
out}$ (by Lemma 8), the number of different pumping sequences is bounded by
some number $p$ and their maximal length is bonded by some number $l$. For the
given finite $\beta=\bigcup_{C^{\prime}\subseteq C,|C^{\prime}|=|C|-1}{\tt
Pareto}[{{\mathcal{G}},{\mathcal{V}},0,C^{\prime}}]({{\tt Parity}})$ we use
the constants from Lemma 7 to define the following finite upper bounds
$v:={\it max}\left(\\{v(\hat{\gamma})\ |\ \hat{\gamma}\in\beta\\}\right)$ and
$u:={\it max}\left(\\{u(\hat{\gamma})\ |\ \hat{\gamma}\in\beta\\}\right)$.
Now we define $\sigma_{0}^{\prime}$. The intuition is as follows. Either the
current nonnegative configuration is already known to be winning for Player
$0$ by induction hypothesis (if the current nonnegative configuration is
sufficiently large compared to nonnegative configurations in
$\left\llbracket{\beta}\right\rrbracket$) in which case he plays according to
his known winning strategy from the induction hypothesis. Otherwise, for a
given history $\pi$ in ${\mathcal{G}}$, Player $0$ plays like for a history
$\pi^{\prime}$ in ${\mathcal{G}}^{\it out}$, where $\pi^{\prime}$ is derived
from $\pi$ as follows. For $\pi^{\prime}$ we first use a sequence of
transitions in ${\mathcal{G}}^{\it out}$ whose labels (see line 13 of the
algorithm) correspond to the sequence of transitions in $\pi$, but then we
remove all subsequences from $\pi^{\prime}$ which are pumping sequences in
${\mathcal{G}}^{\it out}$. Thus Player $0$ plays from nonnegative
configurations in ${\mathcal{G}}$ that are possibly larger than the
corresponding (labels of) nonnegative configurations in ${\mathcal{G}}^{\it
out}$ on the counters in $C$. The other counters in ${\mathcal{C}}-C$ might
differ between the games and will have to be chosen sufficiently high by the
initial $\gamma^{\prime\prime}$ to stay positive during the game (see below).
We show that the history of the winning game in ${\mathcal{G}}$ will contain
only finitely many such pumping sequences, and thus finite initial values
(encoded in $\gamma^{\prime\prime}$) for the counters in ${\mathcal{C}}-C$
will suffice to win the game.
Let $\pi=\gamma_{0}\stackrel{{\scriptstyle
t_{1}}}{{\longrightarrow}}\gamma_{1}\stackrel{{\scriptstyle
t_{2}}}{{\longrightarrow}}\cdots\gamma_{n}$ be a path in ${\mathcal{G}}$,
where Player $0$ played according to strategy $\sigma_{0}^{\prime}$. Our
strategy $\sigma_{0}^{\prime}$ will maintain the invariant that $\pi$ induces
a sequence of states $\hat{\pi}=q_{0},q_{1},\dots,q_{n}$ in
${\mathcal{G}}^{\it out}$. The sequence $\hat{\pi}$ is almost like a path in
${\mathcal{G}}^{\it out}$ with transitions whose label is the same as the
transitions in $\pi$, except that it contains back-jumps to previously visited
states whenever a pumping sequence is completed.
Let $q_{0}=q^{\it out}$. For the step from $q_{i}$ to $q_{i+1}$ there are two
cases. For a given transition $t_{i}$ in ${\mathcal{G}}$ appearing in $\pi$
there is a unique transition $t_{i}^{\prime}$ in ${\mathcal{G}}^{\it out}$
with $\lambda\left({t_{i}^{\prime}}\right)=t_{i}$. As an auxiliary
construction we define the state $q_{i+1}^{\prime}$, which is characterized
uniquely by
$\lambda\left({q_{i+1}^{\prime}}\right)=t_{i}^{\prime}(\lambda\left({q_{i}}\right))$.
If there is a $j\leq i$ s.t. the sequence from $q_{j}$ to $q_{i+1}^{\prime}$
is a pumping sequence and $q_{j}$ is not part of a previously identified
pumping sequence (the construction ensures that there can be at most one such
$j$), then let $q_{i+1}:=q_{j}$, i.e., we jump back to the beginning of the
pumping sequence. Otherwise, if no pumping sequence is completed at
$q_{i+1}^{\prime}$, then let $q_{i+1}=q_{i+1}^{\prime}$, so that we have
$q_{i}\stackrel{{\scriptstyle t_{i}^{\prime}}}{{\longrightarrow}}q_{i+1}$.
From the sequence $\hat{\pi}$ we obtain a genuine path $\pi^{\prime}$ in
${\mathcal{G}}^{\it out}$ by deleting all pumping sequences from $\hat{\pi}$.
In the case where $\gamma_{n}$ belongs to Player $0$ we define
$\sigma_{0}^{\prime}(\pi)$ by case distinction.
1. 1.
We let $\sigma_{0}^{\prime}(\pi):=t_{i}$ where
$\lambda\left({t_{i}}\right)=\sigma_{0}(\pi^{\prime})$, except when the
condition of the following case 2 holds.
2. 2.
By $\lambda\left({q_{0}}\right)=\gamma$ and
$\gamma_{0}=\gamma\oplus\gamma^{\prime\prime}$ we have
${\lambda\left({q_{0}}\right)}\preceq{\gamma_{0}}|{C}$. Since the effects of
the sequences of transitions in $\pi$ and $\hat{\pi}$ are the same, and
pumping sequences have a nondecreasing effect on the counters in $C$, we
obtain ${\lambda\left({q_{i}}\right)}\preceq{\gamma_{i}}|{C}$ for all $i\geq
0$.
Since $\sigma_{0}$ is winning in ${\mathcal{G}}^{\it out}$ we have
$\beta\lhd\lambda\left({q_{i}}\right)$ and thus $\beta\lhd{\gamma_{i}}|{C}$.
By Def. 1, there exists some $\hat{\gamma}\in\beta$ and counter $c\notin{\it
dom}\left({\hat{\gamma}}\right)$ s.t.
$\hat{\gamma}\preceq{\gamma_{i}}|{C}[c\leftarrow\bot]$.
Condition for case 2: If $\gamma_{i}(c)\geq v(\hat{\gamma})$ and
$\gamma_{i}(c^{\prime})\geq u(\hat{\gamma})$ for every counter
$c^{\prime}\in{\mathcal{C}}-C$ then, by Lemma 7 (items 1 and 2) and
monotonicity (Lemma 1), Player $0$ has a winning strategy
$\sigma_{0}^{\prime\prime}$ from $\gamma_{i}$. In this case
$\sigma_{0}^{\prime}$ henceforth follows this winning strategy
$\sigma_{0}^{\prime\prime}$.
Now we show that $\sigma_{0}^{\prime}$ is winning for Player $0$ in
${\mathcal{G}}$ from the initial nonnegative configuration
$\gamma_{0}=\gamma\oplus\gamma^{\prime\prime}$ for some sufficiently large but
finite $\gamma^{\prime\prime}$. We distinguish two cases, depending on whether
case 2 above is reached or not.
If Case 2 is reached: Consider the case where condition 2 above holds at some
reached game nonnegative configuration $\gamma_{n}$. Every pumping sequence
$\alpha$ has nondecreasing effect on all counters in $C$ and strictly
increases at least some counter $c_{\alpha}\in C$. Thus if $\hat{\pi}$
contains the pumping sequence $\alpha$ at least $v$ times, then
$\gamma_{n}(c_{\alpha})-\lambda\left({q_{n}}\right)(c_{\alpha})\geq v$ and in
particular $\gamma_{n}(c_{\alpha})\geq v$. If additionally, $\gamma_{n}$ is
sufficiently large on the counters outside $C$, i.e.,
$\gamma_{n}(c^{\prime})\geq u$ for every counter
$c^{\prime}\in{\mathcal{C}}-C$, then case 2 above applies and the winning
strategy $\sigma_{0}^{\prime\prime}$ takes over.
The path $\pi$ (resp. $\hat{\pi}$) can contain at most $v*p$ pumping sequences
of a combined length that is bounded by $v*p*l$ before the first condition
$\gamma_{n}(c_{\alpha})\geq v$ becomes true for some pumping sequence
$\alpha$. In this case it is sufficient for $\sigma_{0}^{\prime\prime}$ to win
if the values in the counters in ${\mathcal{C}}-C$ are $\geq u$ at nonnegative
configuration $\gamma_{n}$. How large does a counter
$c^{\prime}\in{\mathcal{C}}-C$ need to be at the (part of the) initial
nonnegative configuration $\gamma^{\prime\prime}$ in order to satisfy this
additional condition later at $\gamma_{n}$? Since $\sigma_{0}$ is winning in
the vass game from $\gamma^{\prime}$ in ${\mathcal{G}}^{\it out}$, an initial
value $\gamma^{\prime}(c^{\prime})$ is sufficient to keep the counter
$c^{\prime}$ above $0$ in the game on ${\mathcal{G}}^{\it out}$. Thus an
initial value of $\gamma^{\prime}(c^{\prime})+u$ is sufficient to keep the
counter $c^{\prime}$ above $u$ in the game on ${\mathcal{G}}^{\it out}$.
Moreover, the game played according to $\sigma_{0}^{\prime}$ in
${\mathcal{G}}$ contains the same transitions (modulo the labeling
$\lambda\left({\dots}\right)$) as the game played according to $\sigma_{0}$ on
${\mathcal{G}}^{\it out}$, except for the $\leq v*p*l$ extra transitions in
pumping sequences. Since a single transition can decrease a counter by at most
one, an initial counter value of
$\gamma^{\prime\prime}(c^{\prime})=\gamma^{\prime}(c^{\prime})+u+v*p*l$ is
sufficient in order to have $c^{\prime}\geq u$ whenever case 2 applies and
then $\sigma_{0}^{\prime\prime}$ (and thus $\sigma_{0}^{\prime}$) is winning
for Player $0$. The counters in $C$ are always large enough by construction,
since ${\lambda\left({q_{i}}\right)}\preceq{\gamma_{i}}|{C}$ for all $n\geq
i\geq 0$. The parity objective is satisfied by $\sigma_{0}^{\prime}$, since it
is satisfied by $\sigma_{0}^{\prime\prime}$ on the infinite suffix of the
game.
If Case 2 is not reached: Otherwise, if case 2 is not reached, then the vass
game on ${\mathcal{G}}$ played according to $\sigma_{0}^{\prime}$ is like the
vass game on ${\mathcal{G}}^{\it out}$ played according to $\sigma_{0}$,
except for the finitely many interludes of pumping sequences, of which there
are at most $p*v$ (with a combined length $\leq v*p*l$). Since $\sigma_{0}$ is
winning the vass game on ${\mathcal{G}}^{\it out}$ from $\gamma^{\prime}$,
this keeps the counters nonnegative. At most $v*p*l$ extra transitions happen
in ${\mathcal{G}}$ (in the pumping sequences) and a single transition can
decrement a counter by at most one. Thus it is sufficient for staying
nonnegative in ${\mathcal{G}}$ if
$\gamma^{\prime\prime}(c^{\prime})\geq\gamma^{\prime}(c^{\prime})+v*p*l$ for
all $c^{\prime}\in{\mathcal{C}}-C$. The counters in $c\in C$ trivially stay
nonnegative, since ${\lambda\left({q_{i}}\right)}\preceq{\gamma_{i}}|{C}$ for
all $i\geq 0$. The parity objective is satisfied, since the colors of the
nonnegative configurations $\gamma_{i}$ and $q_{i}$ in $\pi$ and $\hat{\pi}$
coincide, the colors of an infinite suffix of $\hat{\pi}$ coincide with the
colors of an infinite suffix of $\pi^{\prime}$ and $\pi^{\prime}$ satisfies
the parity objective as $\sigma_{0}$ is winning in ${\mathcal{G}}^{\it out}$.
Combination of the cases. While $\sigma_{0}^{\prime}$ might not be able to
enforce either of the two cases described above, one of them will certainly
hold in any play. We define the (part of the) initial nonnegative
configuration $\gamma^{\prime\prime}$ to be sufficiently high to win in either
case, by taking the maximum of the requirements for the cases.
We let $\gamma^{\prime\prime}(c^{\prime}):=\gamma^{\prime}(c)+u+v*p*l$ for all
$c^{\prime}\in{\mathcal{C}}-C$ and obtain that $\sigma_{0}^{\prime}$ is a
winning strategy for Player $0$ in the parity game on ${\mathcal{G}}$ from the
initial nonnegative configuration
$\gamma_{0}=\gamma\oplus\gamma^{\prime\prime}\in\left\llbracket{\gamma}\right\rrbracket$.
Thus $[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$, as required.
∎
### Proof of Theorem 5.1
Given a labeled finite-state system $\left\langle{S,\stackrel{{\scriptstyle
a}}{{\longrightarrow}},\Sigma}\right\rangle$ and a labeled vass
$\left\langle{Q,T,\Sigma,\lambda}\right\rangle$ with initial states $s_{0}$
and $\left\langle{q_{0},\vartheta}\right\rangle$, respectively, we construct a
single-sided integer game ${\mathcal{G}}=\left\langle{Q_{0}\uplus
Q_{1},T^{\prime},\kappa}\right\rangle$ with initial configuration
$\gamma=\left\langle{\left\langle{s_{0},q_{0},1}\right\rangle,\vartheta}\right\rangle$
s.t. $\left\langle{q_{0},\vartheta}\right\rangle$ weakly simulates $s_{0}$ if
and only if $[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$. Then
decidability follows from Theorem 3.3.
Let $Q_{1}=\\{\left\langle{s,q,1}\right\rangle\ |\ s\in S,q\in Q\\}\cup\\{{\it
win}_{0}\\}$ and $Q_{0}=\\{\left\langle{s,q,0}\right\rangle\ |\ s\in S,q\in
Q\\}\cup\\{\left\langle{s,q^{a},0}\right\rangle\ |\ s\in S,q\in
Q,a\in\Sigma\\}\cup\\{{\it lose}_{0}\\}$. Let $\kappa\left({Q_{1}}\right)=2$
and $\kappa\left({Q_{0}}\right)=1$, i.e., Player $0$ wins the parity game iff
states belonging to Player $1$ are visited infinitely often.
Now we define $T^{\prime}$. For every finite-state system transition
$s\stackrel{{\scriptstyle a}}{{\longrightarrow}}s^{\prime}$ and every $q\in
Q$, we add a transition $\left\langle{\left\langle{s,q,1}\right\rangle,{\it
nop},\left\langle{s^{\prime},q^{a},0}\right\rangle}\right\rangle$. Here the
state $q^{a}$ encodes the choice of the symbol $a$ by Player $1$, which
restricts the future moves of Player $0$. For every vass transition
$t=\left\langle{q_{1},{\it op},q_{2}}\right\rangle\in T$ with label
$\lambda\left({t}\right)=\tau$ and every $s\in S,a\in\Sigma$ we add a
transition $\left\langle{\left\langle{s,q_{1}^{a},0}\right\rangle,{\it
op},\left\langle{s,q_{2}^{a},0}\right\rangle}\right\rangle$. This encodes the
first arbitrarily long sequence of $\tau$-moves in the Player $0$ response of
the form $\tau^{*}a\tau^{*}$. For every vass transition
$t=\left\langle{q_{1},{\it op},q_{2}}\right\rangle\in T$ with label
$\lambda\left({t}\right)=a\neq\tau$ and $s\in S$ we add a transition
$\left\langle{\left\langle{s,q_{1}^{a},0}\right\rangle,{\it
op},\left\langle{s,q_{2},0}\right\rangle}\right\rangle$. This encodes the
$a$-step in in the Player $0$ response of the form $\tau^{*}a\tau^{*}$.
Moreover, we add transitions
$\left\langle{\left\langle{s,q^{\tau},0}\right\rangle,{\it
nop},\left\langle{s,q,0}\right\rangle}\right\rangle$ for all $s\in S,q\in Q$
(since a $\tau$-move in the weak simulation game does not strictly require a
response step). For every vass transition $t=\left\langle{q_{1},{\it
op},q_{2}}\right\rangle\in T$ with label $\lambda\left({t}\right)=\tau$ and
$s\in S$ we add a transition
$\left\langle{\left\langle{s,q_{1},0}\right\rangle,{\it
op},\left\langle{s,q_{2},0}\right\rangle}\right\rangle$. This encodes the
second arbitrarily long sequence of $\tau$-moves in the Player $0$ response of
the form $\tau^{*}a\tau^{*}$. Finally, for all $s\in S,q\in Q$ we add
transitions $\left\langle{\left\langle{s,q,0}\right\rangle,{\it
nop},\left\langle{s,q,1}\right\rangle}\right\rangle$. Here Player $0$ switches
the control back to Player $1$. He cannot win by delaying this switch
indefinitely, because the color of the states in $Q_{0}$ is odd.
The following transitions encode the property of the simulation game that a
player loses if he gets stuck. For every state in $q\in Q_{1}$ with no
outgoing transitions we add a transition $\left\langle{q,{\it nop},{\it
win}_{0}}\right\rangle$. In particular this creates a loop at state ${\it
win}_{0}$. Since the color of ${\it win}_{0}$ is even, this state is winning
for Player $0$. For every state in $q\in Q_{0}$ with no outgoing transitions
we add a transition $\left\langle{q,{\it nop},{\it lose}_{0}}\right\rangle$.
In particular this creates a loop at state ${\it lose}_{0}$. Since the color
of ${\it lose}_{0}$ is odd, this state is losing for Player $0$.
This construction yields a single-sided integer game, since all transitions
from states in $Q_{1}$ have operation ${\it nop}$.
A round of the weak simulation game is encoded by the moves of the players
between successive visits to a state in $Q_{1}$. A winning strategy for Player
$0$ in the weak simulation game directly induces a winning strategy for Player
$0$ in the parity game ${\mathcal{G}}$, since the highest color that is
infinitely often visited is $2$, and thus
$[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$. Conversely,
$[0,{\mathcal{V}}]:\gamma\models_{\mathcal{G}}{\tt Parity}$ implies a winning
strategy for Player $0$ in the parity game on ${\mathcal{G}}$ which ensures
that color $2$ is seen infinitely often. Therefore, states in $Q_{1}$ are
visited infinitely often. Thus, either infinitely many rounds of the weak
simulation game are simulated or state ${\it win}_{0}$ is reached in
${\mathcal{G}}$ and Player $1$ gets stuck in the weak simulation game. In
either case, Player $0$ wins the weak simulation game and
$\left\langle{q_{0},\vartheta}\right\rangle$ weakly simulates $s_{0}$. ∎
### Semantics of $L^{\textit{pos}}_{\mu}$
The syntax of the positive $\mu$-calculus $L^{\textit{pos}}_{\mu}$ is given by
the following grammar:
$\phi::=q~{}\mid~{}X~{}\mid~{}\phi\wedge\phi~{}\mid~{}\phi\vee\phi~{}\mid~{}\Diamond\phi~{}\mid~{}\Box\phi~{}\mid~{}\mu
X.\phi~{}\mid~{}\nu X.\phi$ where $q\in Q$ and $X$ belongs to a countable set
of variables $\mathcal{X}$.
Free and bound occurrences of variables are defined as usual. We assume that
no variable has both bound and free occurrences in some $\phi$, and that no
two fixpoint subterms bind the same variable (this can always be ensured by
renaming a bound variable). A formula is closed if it has no free variables.
Without restriction, we do not use any negation in our syntax. Negation can be
pushed inward by the usual dualities of fixpoints, and the negation of an
atomic proposition referring to a control-state can be expressed by a
disjunction of propositions referring to all the other control-states.
We now give the interpretation over the vass $\left\langle{Q,T}\right\rangle$
of a formula of $L^{\textit{pos}}_{\mu}$ according to an environment
$\rho:\mathcal{X}\rightarrow 2^{\Gamma}$ which associates to each variable a
subset of concrete configurations. Given $\rho$, a formula $\phi\in
L^{\textit{pos}}_{\mu}$ represents a subset of concrete configurations,
denoted by $\llbracket\phi\rrbracket_{\rho}$ and defined inductively as
follows.
$\begin{array}[]{lcl}\llbracket
q\rrbracket_{\rho}&=&\left\\{\gamma\in\Gamma\mid{\tt
state}\left({\gamma}\right)=q\right\\}\\\ \llbracket
X\rrbracket_{\rho}&=&\rho(X)\\\
\llbracket\phi\wedge\psi\rrbracket_{\rho}&=&\llbracket\phi\rrbracket_{\rho}\cap\llbracket\psi\rrbracket_{\rho}\\\
\llbracket\phi\vee\psi\rrbracket_{\rho}&=&\llbracket\phi\rrbracket_{\rho}\cup\llbracket\psi\rrbracket_{\rho}\\\
\llbracket\Diamond\phi\rrbracket_{\rho}&=&\left\\{\gamma\in\Gamma\mid\exists\gamma^{\prime}\in\llbracket\phi\rrbracket_{\rho}\mbox{~{}s.t.~{}}\gamma\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{V}}\gamma^{\prime}\right\\}\\\
\llbracket\Box\phi\rrbracket_{\rho}&=&\left\\{\gamma\in\Gamma\mid\forall\gamma^{\prime}\in\Gamma,\gamma\stackrel{{\scriptstyle}}{{\longrightarrow}}_{\mathcal{V}}\gamma^{\prime}\mbox{
implies }\gamma^{\prime}\in\llbracket\phi\rrbracket_{\rho}\right\\}\\\
\llbracket\mu
X.\phi\rrbracket_{\rho}&=&\bigcap\left\\{\Gamma^{\prime}\subseteq\Gamma\mid\llbracket\phi\rrbracket_{\rho[X\leftarrow\Gamma^{\prime}]}\subseteq\Gamma^{\prime}\right\\}\\\
\llbracket\nu
X.\phi\rrbracket_{\rho}&=&\bigcup\left\\{\Gamma^{\prime}\subseteq\Gamma\mid\Gamma^{\prime}\subseteq\llbracket\phi\rrbracket_{\rho[X\leftarrow\Gamma^{\prime}]}\right\\}\\\
\end{array}$
where the notation $\rho[X\leftarrow\Gamma^{\prime}]$ is used to define an
environment equal to $\rho$ on every variable except on $X$ where it returns
$\Gamma^{\prime}$. We recall that $(2^{\Gamma},\subseteq)$ is a complete
lattice and that, for every $\phi\in L^{\textit{pos}}_{\mu}$ and every
environment $\rho$, the function $G:2^{\Gamma}\mapsto 2^{\Gamma}$, which
associates to $\Gamma^{\prime}\subseteq\Gamma$ the set
$G(\Gamma^{\prime})=\llbracket\phi\rrbracket_{\rho[X\leftarrow\Gamma^{\prime}]}$,
is monotonic. Hence, by the Knaster-Tarski Theorem, the set $\llbracket\mu
X.\phi\rrbracket_{\rho}$ (resp. $\llbracket\nu X.\phi\rrbracket_{\rho}$) is
the least fixpoint (resp. greatest fixpoint) of $G$, and it is well-defined.
Finally we denote by $\llbracket\phi\rrbracket$ the subset of configurations
$\llbracket\phi\rrbracket_{\rho_{0}}$ where $\rho_{0}$ is the environment
which assigns the empty set to each variable.
### Proof of Lemma 10
We consider a vass $\mathcal{S}=\left\langle{Q,T}\right\rangle$ and $\phi$ a
formula in $L^{\textit{pos}}_{\mu}$. We will use in this proof the set of
subformulae of $\phi$, denoted by $\mathit{sub}(\phi)$. For formulae in
$L^{\textit{pos}}_{\mu}$ we assume that no variable is bounded by the same
fixpoint. Hence given a formula $\phi$ and a bounded variable
$X\in\mathcal{X}$, we can determine uniquely the subformula of $\phi$ that
bounds the variable $X$; such a formula will be denoted by $\phi_{X}$. We also
denote by $\mathit{free}(\phi)$ the set of free variables in $\phi$. The
integer game
${\mathcal{G}}(\mathcal{S},\phi)=\left\langle{Q^{\prime},T^{\prime},\kappa}\right\rangle$
is built as follows:
* •
$Q^{\prime}=Q\times\mathit{sub}(\phi)$
* •
The transition relation $T^{\prime}$ is the smallest set respecting the
following conditions for all the formulae $\psi\in\mathit{sub}(\phi)$:
* –
If $\psi=q$ with $q\in Q$, then
$\left\langle{\left\langle{q^{\prime},\psi}\right\rangle,{\it
nop},\left\langle{q^{\prime},\psi}\right\rangle}\right\rangle$ belongs to
$T^{\prime}$ for all states $q^{\prime}$ in $Q$;
* –
If $\psi=X$ with $X\in\mathcal{X}$ and $X\notin\mathit{free}(\phi)$, then
$\left\langle{\left\langle{q,\psi}\right\rangle,{\it
nop},\left\langle{q,\phi_{X}}\right\rangle}\right\rangle$ belongs to
$T^{\prime}$ for all states $q$ in $Q$;
* –
If $\psi=X$ with $X\in\mathcal{X}$ and $X\in\mathit{free}(\phi)$, then
$\left\langle{\left\langle{q,\psi}\right\rangle,{\it
nop},\left\langle{q,\psi}\right\rangle}\right\rangle$ belongs to $T^{\prime}$
for all states $q$ in $Q$;
* –
If $\psi=\psi^{\prime}\wedge\psi^{\prime\prime}$ or
$\psi=\psi^{\prime}\vee\psi^{\prime\prime}$ then
$\left\langle{\left\langle{q,\psi}\right\rangle,{\it
nop},\left\langle{q,\psi^{\prime}}\right\rangle}\right\rangle$ and
$\left\langle{\left\langle{q,\psi}\right\rangle,{\it
nop},\left\langle{q,\psi^{\prime\prime}}\right\rangle}\right\rangle$ belong to
$T^{\prime}$ for all states $q$ in $Q$;
* –
If $\psi=\Diamond\psi^{\prime}$ or $\psi=\Box\psi^{\prime}$ then for all
states $q\in Q$ and for all transitions $\left\langle{q,{\it
op},q^{\prime}}\right\rangle\in T$, we have
$\left\langle{\left\langle{q,\psi}\right\rangle,{\it
op},\left\langle{q^{\prime},\psi^{\prime}}\right\rangle}\right\rangle$ in
$T^{\prime}$;
* –
If $\psi=\mu X.\psi^{\prime}$ or $\psi=\nu X.\psi^{\prime}$, then
$\left\langle{\left\langle{q,\psi}\right\rangle,{\it
nop},\left\langle{q,\psi^{\prime}}\right\rangle}\right\rangle$ for all states
$q\in Q$.
* •
A state $\left\langle{q,\psi}\right\rangle$ belongs to $Q_{0}^{\prime}$ if and
only if:
* –
$\psi=q^{\prime}$ with $q^{\prime}\in Q$, or,
* –
$\psi=X$ with $X\in\mathcal{X}$, or,
* –
$\psi=\psi^{\prime}\vee\psi^{\prime\prime}$, or,
* –
$\psi=\Diamond\psi^{\prime}$, or,
* –
$\psi=\mu X.\psi^{\prime\prime}$, or,
* –
$\psi=\nu X.\psi^{\prime\prime}$.
* •
A state $\left\langle{q,\psi}\right\rangle$ belongs to $Q_{1}^{\prime}$ if and
only if:
* –
$\psi=\psi^{\prime}\wedge\psi^{\prime\prime}$, or,
* –
$\psi=\Box\psi^{\prime}$.
* •
The coloring function $\kappa$ is then defined as follows:
* –
for all $q,q^{\prime}\in Q$, if $q^{\prime}=q$ then
$\kappa{\left\langle{q,q^{\prime}}\right\rangle}=0$ and if $q^{\prime}\neq q$
then $\kappa{\left\langle{q,q^{\prime}}\right\rangle}=1$;
* –
for all $q\in Q$ and all $X\in\mathit{free}(\phi)$,
$\kappa{\left\langle{q,X}\right\rangle}=1$;
* –
for all $q\in Q$, for all subformulae $\psi\in\mathit{sub}(\phi)$ if
$\psi\neq\mu X.\psi^{\prime\prime}$ and $\psi\neq\nu X.\psi^{\prime\prime}$
and $\psi\neq q^{\prime}$ with $q^{\prime}\in Q$ and $\psi\neq X$ with
$X\in\mathit{free}(\phi)$ , then
$\kappa{\left\langle{q,\psi}\right\rangle}=0$;
* –
for all $q\in Q$, for all subformulae $\psi\in\mathit{sub}(\phi)$ such that
$\psi\neq\mu X.\psi^{\prime\prime}$,
$\kappa{\left\langle{q,\psi}\right\rangle}=m$ where $m$ is the smallest odd
number greater or equal to the alternation depth of $\psi$;
* –
for all $q\in Q$, for all subformulae $\psi\in\mathit{sub}(\phi)$ such that
$\psi\neq\mu X.\psi^{\prime\prime}$,
$\kappa{\left\langle{q,\psi}\right\rangle}=m$ where $m$ is the smallest even
number greater or equal to the alternation depth of $\psi$;
Before providing the main property of the game
${\mathcal{G}}(\mathcal{S},\phi)$, we introduce a new winning condition which
will be useful in the sequel of the proof. This winning condition uses an
environment $\rho:\mathcal{X}\rightarrow 2^{\Gamma}$ and is given by the
formula ${\tt
Parity}\vee\bigvee_{X\in\mathit{free}(\phi)}\Diamond(X\wedge\rho(X))$ where
$X\wedge\rho(X)$ holds in the configurations of the form
$\left\langle{\left\langle{q,X}\right\rangle,\vartheta}\right\rangle$ such
that $\left\langle{q,\vartheta}\right\rangle\in\rho(X)$. It states that a run
is winning if it respects the parity condition or if at some point it
encounters a configuration of the form
$\left\langle{\left\langle{q,X}\right\rangle,\vartheta}\right\rangle$ with
$X\in\mathit{free}(\phi)$ and
$\left\langle{q,\vartheta}\right\rangle\in\rho(X)$. We denote by ${\tt
Cond}(\phi,\rho)$ the formula
$\bigvee_{X\in\mathit{free}(\phi)}\Diamond(X\wedge\rho(X))$.
We will now prove the following property: for all formulae $\phi$ in
$L^{\textit{pos}}_{\mu}$, for all concrete configurations
$\gamma=\left\langle{q,\vartheta}\right\rangle$ of $\mathcal{S}$ and all
environments $\rho:\mathcal{X}\rightarrow 2^{\Gamma}$, we have
$\gamma\in\llbracket\phi\rrbracket_{\rho}$ iff
$[0,{\mathcal{V}}]:\left\langle{\left\langle{q,\phi}\right\rangle,\vartheta}\right\rangle\models_{{\mathcal{G}}(\mathcal{S},\phi)}{\tt
Parity}\vee{\tt Cond}(\phi,\rho)$, i.e., iff
$\left\langle{\left\langle{q,\phi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}(\mathcal{S},\phi),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}\vee{\tt Cond}(\phi,\rho)})$.
We reason by induction on the length of $\phi$. For the base case with
$\phi=q$ with $q\in Q$ or $\phi=X$ with $X\in\mathcal{X}$ the property
trivially holds. We then proceed with the induction reasoning. It is easy to
prove that the property holds for formulae of the form
$\phi^{\prime}\wedge\phi^{\prime\prime}$ or
$\phi^{\prime}\vee\phi^{\prime\prime}$ if the property holds for
$\phi^{\prime}$ and $\phi^{\prime\prime}$ and the same for formulae of the
form $\Diamond\phi^{\prime}$ and $\Box\phi^{\prime}$. We consider now a
formula $\phi$ of the form $\mu X.\psi$ and assume that the property holds for
the formula $\psi$. Let $G:2^{\Gamma}\mapsto 2^{\Gamma}$ be the function which
associates to any subset of configurations $\Gamma^{\prime}$ the set
$G(\Gamma^{\prime})=\llbracket\psi\rrbracket_{\rho[X\leftarrow\Gamma^{\prime}]}$.
By induction hypothesis we have $\left\langle{q,\vartheta}\right\rangle\in
G(\Gamma^{\prime})$ iff
$\left\langle{\left\langle{q,\psi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}(\mathcal{S},\psi),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}\vee{\tt Cond}(\psi,\rho[X\leftarrow\Gamma^{\prime}])})$. We denote by
$\mu G$ the least fixpoint of $G$. We want to prove that
$\left\langle{q,\vartheta}\right\rangle\in\mu G$ iff
$\left\langle{\left\langle{q,\mu
X.\psi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}(\mathcal{S},\mu
X.\psi),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt Parity}\vee{\tt Cond}(\mu
X.\psi,\rho)})$. We define the following set of configurations
$\Gamma_{\mu}=\left\\{\left\langle{q,\vartheta}\right\rangle\in\Gamma\mid\left\langle{\left\langle{q,\mu
X.\psi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}(\mathcal{S},\mu
X.\psi),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt Parity}\vee{\tt Cond}(\mu
X.\psi,\rho)})\right\\}$. So finally what we want to prove is that $\mu
G=\Gamma_{\mu}$.
* •
We begin by proving that $\mu G\subseteq\Gamma_{\mu}$. By definition $\mu
G=\bigcap\left\\{\Gamma^{\prime}\subseteq\Gamma\mid
G(\Gamma^{\prime})\subseteq\Gamma^{\prime}\right\\}$. Hence it is enough to
prove that $G(\Gamma_{\mu})\subseteq\Gamma_{\mu}$. Let
$\left\langle{q,\vartheta}\right\rangle\in G(\Gamma_{\mu})$. This means that
$\left\langle{\left\langle{q,\psi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}(\mathcal{S},\psi),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}\vee{\tt Cond}(\psi,\rho[X\leftarrow\Gamma_{\mu}])})$ by definition of
$G$. We want to prove that $\left\langle{\left\langle{q,\mu
X.\psi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}(\mathcal{S},\psi),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}\vee{\tt Cond}(\mu X.\psi,\rho)})$. First note that the configuration
$\left\langle{\left\langle{q,\mu X.\psi}\right\rangle,\vartheta}\right\rangle$
belongs to Player 0, and from this configuration, Player 0 has a unique choice
which is to go to the state
$\left\langle{\left\langle{q,\psi}\right\rangle,\vartheta}\right\rangle$. Then
from $\left\langle{\left\langle{q,\psi}\right\rangle,\vartheta}\right\rangle$,
if Player $0$ plays as in the game ${\mathcal{G}}(\mathcal{S},\psi)$ where it
has a winning strategy, there are two options:
1. 1.
a control-state of the form $\left\langle{q^{\prime},X}\right\rangle$ is never
encountered and in that case Player 0 wins because it was winning in
${\mathcal{G}}(\mathcal{S},\psi)$ and the run performed is the same;
2. 2.
a control-state of the form $\left\langle{q^{\prime},X}\right\rangle$ is
encountered, but in that case, Player 0 is necessarily in a configuration
$\left\langle{\left\langle{q^{\prime},X}\right\rangle,\vartheta}\right\rangle$
with
$\left\langle{q^{\prime},\vartheta}\right\rangle\in\rho[X\leftarrow\Gamma_{\mu}](X)$
(by definition of the winning condition in ${\mathcal{G}}(\mathcal{S},\psi)$),
ie with $\left\langle{q^{\prime},\vartheta}\right\rangle\in\Gamma_{\mu}$. But
this means that from this configuration, Player 0 has a winning strategy for
the game ${\mathcal{G}}(\mathcal{S},\phi)$.
Hence we have shown that
$\left\langle{q,\vartheta}\right\rangle\in\Gamma_{\mu}$ and consequently
$G(\Gamma_{\mu})\subseteq\Gamma_{\mu}$. This allows us to deduce that $\mu
G\subseteq\Gamma_{\mu}$.
* •
We will now prove that $\Gamma_{\mu}\subseteq\mu G$. For this we will prove
that for all $\Gamma^{\prime}\subseteq\Gamma$ such that
$G(\Gamma^{\prime})=\Gamma^{\prime}$, we have
$\Gamma_{\mu}\subseteq\Gamma^{\prime}$. This will in fact imply that
$\Gamma_{\mu}\subseteq\mu G$, since $\mu G$ is the least fixpoint of the
function $G$. Let $\Gamma^{\prime}\subseteq\Gamma$ such that
$G(\Gamma^{\prime})=\Gamma^{\prime}$ and let
$\left\langle{q,\vartheta}\right\rangle\in\Gamma_{\mu}$. We reason _by
contradiction_ and assume that
$\left\langle{q,\vartheta}\right\rangle\notin\Gamma^{\prime}$. Since
$\left\langle{q,\vartheta}\right\rangle\in\Gamma_{\mu}$, this means that
Player 0 has a winning strategy to win in the game
${\mathcal{G}}(\mathcal{S},\mu X.\psi))$ from the configuration
$\left\langle{\left\langle{q,\mu X.\psi}\right\rangle,\vartheta}\right\rangle$
with the objective ${\tt Parity}\vee{\tt Cond}(\mu X.\psi,\rho)$. Since
$\left\langle{q,\vartheta}\right\rangle\notin\Gamma^{\prime}=G(\Gamma^{\prime})$,
this means that there is no winning strategy for Player 0 in the game
${\mathcal{G}}(\mathcal{S},\psi)$ from configuration
$\left\langle{\left\langle{q,\psi}\right\rangle,\vartheta}\right\rangle$ with
the objective ${\tt Parity}\vee{\tt
Cond}(\psi,\rho[X\leftarrow\Gamma^{\prime}])$. Since Player 0 has a winning
strategy to win in the game ${\mathcal{G}}(\mathcal{S},\mu X.\psi)$, we can
adapt this strategy to the game ${\mathcal{G}}(\mathcal{S},\psi)$ (by
restricting it to the path possible in this game and beginning one step
later). But since this strategy is not winning in the game
${\mathcal{G}}(\mathcal{S},\psi)$ with the objective ${\tt Parity}\vee{\tt
Cond}(\psi,\rho[X\leftarrow\Gamma^{\prime}])$, it means that there is a path
$\pi_{0}$ in ${\mathcal{G}}(\mathcal{S},\psi)$ that respects this strategy and
this path necessarily terminates in a state of the form
$\left\langle{\left\langle{q_{1},X}\right\rangle,\vartheta_{1}}\right\rangle$
with
$\left\langle{q_{1},\vartheta_{1}}\right\rangle\notin\rho[X\leftarrow\Gamma^{\prime}](X)$,
i.e., with
$\left\langle{q_{1},\vartheta_{1}}\right\rangle\notin\Gamma^{\prime}$
(otherwise this strategy which is winning in ${\mathcal{G}}(\mathcal{S},\mu
X.\psi)$ would also be winning in ${\mathcal{G}}(\mathcal{S},\psi)$). On the
other hand, in ${\mathcal{G}}(\mathcal{S},\mu X.\psi)$,
$\left\langle{\left\langle{q_{1},X}\right\rangle,\vartheta_{1}}\right\rangle$
has a unique successor which is $\left\langle{\left\langle{q_{1},\mu
X.\psi}\right\rangle,\vartheta_{1}}\right\rangle$ and from which Player 0 has
a winning strategy since we have followed a winning strategy in the game
${\mathcal{G}}(\mathcal{S},\mu X.\psi)$ that has lead us to that
configuration. Hence we have
$\left\langle{q_{1},\vartheta_{1}}\right\rangle\notin\Gamma^{\prime}$ and
$\left\langle{q_{1},\vartheta_{1}}\right\rangle\in\Gamma_{\mu}$. So from
$\left\langle{q_{1},\vartheta_{1}}\right\rangle$ we can perform a similar
reasoning following the winning strategy in ${\mathcal{G}}(\mathcal{S},\mu
X.\psi)$ to reach a configuration
$\left\langle{\left\langle{q_{2},X}\right\rangle,\vartheta_{2}}\right\rangle$
such that
$\left\langle{q_{2},\vartheta_{2}}\right\rangle\notin\Gamma^{\prime}$ and
$\left\langle{q_{2},\vartheta_{2}}\right\rangle\in\Gamma_{\mu}$. Finally, by
performing the same reasoning we succeed in building an infinite play in
${\mathcal{G}}(\mathcal{S},\mu X.\psi)$ which follows a winning strategy and
such that the sequence of the visited configurations is of the form:
$\left\langle{\left\langle{q,\mu
X.\psi}\right\rangle,\vartheta}\right\rangle\ldots\left\langle{\left\langle{q_{1},\mu
X.\psi}\right\rangle,\vartheta_{1}}\right\rangle\ldots\left\langle{\left\langle{q_{2},\mu
X.\psi}\right\rangle,\vartheta_{2}}\right\rangle\ldots$
Note that for all $i\geq 1$, $\kappa(\left\langle{\left\langle{q_{i},\mu
X.\psi}\right\rangle,\vartheta_{i}}\right\rangle)$ is the maximal priority in
the game ${\mathcal{G}}(\mathcal{S},\mu X.\phi)$ and it is odd by definition
of the game. This means that the path we obtain following a winning strategy
for Player 0 is losing, which is a contradiction. Hence we have
$\left\langle{q,\vartheta}\right\rangle\in\Gamma^{\prime}$. From this we
deduce that $\Gamma_{\mu}\subseteq\mu G$.
If we consider a formula $\phi$ of the form $\nu X.\psi$, a reasoning similar
to the previous one can be performed in order to show that the property holds.
Thanks to the previous proof, for all formulae $\phi$ in
$L^{\textit{pos}}_{\mu}$, for all concrete configurations
$\gamma=\left\langle{q,\vartheta}\right\rangle$ of $\mathcal{S}$, we have
$\gamma\in\llbracket\phi\rrbracket_{\rho_{0}}$ iff
$\left\langle{\left\langle{q,\phi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}(\mathcal{S},\phi)),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}\vee{\tt Cond}(\phi,\rho_{0})})$ where $\rho_{0}$ is the environment
which assigns to each variable the empty set. This means that for all formulae
$\phi$ in $L^{\textit{pos}}_{\mu}$, for all concrete configurations
$\gamma=\left\langle{q,\vartheta}\right\rangle$ of $\mathcal{S}$, we have
$\gamma\in\llbracket\phi\rrbracket_{\rho_{0}}$ iff
$\left\langle{\left\langle{q,\phi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}(\mathcal{S},\phi)),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})$ because ${\tt Cond}(\phi,\rho_{0})$ is equivalent to the formula
which is always false. By denoting
$\gamma^{\prime}=\left\langle{\left\langle{q,\phi}\right\rangle,\vartheta}\right\rangle$,
we have hence that
$[0,{\mathcal{V}}]:\gamma^{\prime}\models_{{\mathcal{G}}(\mathcal{S},\phi)}{\tt
Parity}$ if and only if $\mathcal{S},\gamma\models\phi$.
### Proof of Lemma 11
Let $\mathcal{S}=\left\langle{Q,T}\right\rangle$ be a single-sided vass and
$\phi\in L^{\textit{sv}}_{\mu}$. Strictly speaking the construction of the
game ${\mathcal{G}}(\mathcal{S},\phi)$ proposed in the proof of Lemma 10 does
not build a single-sided game. However we can adapt this construction in order
to build an equivalent single-sided game. In this manner we adapt the
construction to the case of $\phi\in L^{\textit{sv}}_{\mu}$ by changing the
rules for the outgoing transitions for states in the game of the form
$\left\langle{q,Q_{1}\wedge\Box\psi}\right\rangle$. To achieve this we build a
game
${\mathcal{G}}^{\prime}(\mathcal{S},\phi)=\left\langle{Q^{\prime},T^{\prime},\kappa}\right\rangle$
the same way as ${\mathcal{G}}(\mathcal{S},\phi)$ except that we perform the
following change in the definition of transition relation for states of the
form $\left\langle{q,Q_{1}\wedge\Box\psi}\right\rangle$:
* •
If $\psi=Q_{1}\wedge\Box\psi^{\prime}$ then for all states $q\in Q_{1}$, for
all transitions $\left\langle{q,{\it nop},q^{\prime}}\right\rangle\in T$, we
have $\left\langle{\left\langle{q,\psi}\right\rangle,{\it
nop},\left\langle{q^{\prime},\psi^{\prime}}\right\rangle}\right\rangle$ in
$T^{\prime}$, and, for all states $q\in Q_{0}$, we have
$\left\langle{\left\langle{q,\psi}\right\rangle,{\it
nop},\left\langle{q,\psi}\right\rangle}\right\rangle$ in $T^{\prime}$.
Then the states of the form $\left\langle{q,Q_{1}\wedge\Box\psi}\right\rangle$
will belong to Player 1 and the coloring of such states will be defined as
follows:
* •
for all $q\in Q$, for all subformulae $\psi\in\mathit{sub}(\phi)$, if
$\psi=Q\wedge\Box\psi^{\prime}$ then if $q\in Q_{1}$,
$\kappa(\left\langle{q,\psi}\right\rangle)=0$ else
$\kappa(\left\langle{q,\psi}\right\rangle)=1$.
Apart from these changes the definition of the game
${\mathcal{G}}^{\prime}(\mathcal{S},\phi)$ is equivalent to the one of
${\mathcal{G}}(\mathcal{S},\phi)$. By construction, since $\mathcal{S}$ is
single-sided and by definition of $L^{\textit{sv}}_{\mu}$, we have that such
an integer game ${\mathcal{G}}^{\prime}(\mathcal{S},\phi)$ is single-sided.
Furthermore, for any concrete configuration
$\gamma=\left\langle{\left\langle{q,\psi}\right\rangle,\vartheta}\right\rangle$,
one can easily show that
$[0,{\mathcal{V}}]:\gamma\models_{{\mathcal{G}}(\mathcal{S},\phi)}{\tt
Parity}$ iff
$[0,{\mathcal{V}}]:\gamma\models_{{\mathcal{G}}^{\prime}(\mathcal{S},\phi)}{\tt
Parity}$.
### Proof of Theorem 5.3
Let $\mathcal{S}=\left\langle{Q,T}\right\rangle$ be a single-sided vass,
$\phi$ be closed formula of $L^{\textit{sv}}_{\mu}$ and $\gamma_{0}$ be an
initial configuration of $\left\langle{Q,T}\right\rangle$. Using Lemma 10 and
Lemma 11, we have that
$[0,{\mathcal{V}}]:\gamma^{\prime}_{0}\models_{{\mathcal{G}}^{\prime}(\mathcal{S},\phi)}{\tt
Parity}$ if and only if $\mathcal{S},\gamma_{0}\models\phi$ where
${\mathcal{G}}^{\prime}(\mathcal{S},\phi)$ is a single-sided integer game.
Hence, thanks to Corollary 2, we can deduce that the model-checking problem of
$L^{\textit{sv}}_{\mu}$ over single-sided vass is decidable. Furthermore, by
using the result of these two lemmas we have that
$\left\langle{q,\vartheta}\right\rangle\in\llbracket\phi\rrbracket_{\rho_{0}}$
iff
$\left\langle{\left\langle{q,\phi}\right\rangle,\vartheta}\right\rangle\in{\mathcal{W}}[{{\mathcal{G}}^{\prime}(\mathcal{S},\phi),{\mathcal{V}},0,C}]({{\tt
Parity}})$. Hence by Corollary 1, we deduce that
$\llbracket\phi\rrbracket_{\rho_{0}}$ is upward-closed and by Theorem 3.3 that
we can compute its set of minimal elements which is equal to
$\left\\{\left\langle{q,\vartheta}\right\rangle\mid\left\langle{\left\langle{q,\phi}\right\rangle,\vartheta}\right\rangle\in{\tt
Pareto}[{{\mathcal{G}}^{\prime}(\mathcal{S},\phi),{\mathcal{V}},0,{\mathcal{C}}}]({{\tt
Parity}})\right\\}$.
|
arxiv-papers
| 2013-06-12T12:52:01 |
2024-09-04T02:49:46.391881
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Parosh Aziz Abdulla, Richard Mayr, Arnaud Sangnier, Jeremy Sproston",
"submitter": "Richard Mayr",
"url": "https://arxiv.org/abs/1306.2806"
}
|
1306.2826
|
# Modified binary encounter Bethe model for electron-impact ionization
M. Guerra1 [email protected] F. Parente1 [email protected] P.
Indelicato2 [email protected] J. P. Santos1 [email protected] 1
Centro de Física Atómica, CFA, Departamento de Física, Faculdade de Ciências e
Tecnologia, FCT, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal 2
Laboratoire Kastler Brossel, École Normale Supérieure, CNRS, Université P. et
M. Curie – Paris 6, Case 74; 4, place Jussieu, 75252 Paris CEDEX 05, France
###### Abstract
Theoretical expressions for ionization cross sections by electron impact based
on the binary encounter Bethe (BEB) model, valid from ionization threshold up
to relativistic energies, are proposed.
The new modified BEB (MBEB) and its relativistic counterpart (MRBEB)
expressions are simpler than the BEB (nonrelativistic and relativistic)
expressions because they require only one atomic parameter, namely the binding
energy of the electrons to be ionized, and use only one scaling term for the
ionization of all sub-shells.
The new models are used to calculate the K-, L- and M-shell ionization cross
sections by electron impact for several atoms with $Z$ from 6 to 83.
Comparisons with all, to the best of our knowledge, available experimental
data show that this model is as good or better than other models, with less
complexity.
###### keywords:
Electron Impact , Cross sections , K-shell , L-shell , M-shell
††journal: International Journal of Mass Spectrometry
## 1 Introduction
Knowledge of ionization and excitation cross sections is of fundamental
importance for understanding collision-dynamics and electron-atom
interactions, as well as in several applied fields such as radiation science,
plasma physics, astrophysics and also elemental analysis using X-ray
fluorescence (XRF), Auger electron spectroscopy (AES), electron energy loss
spectroscopy (EELS) and electron probe microanalysis (EPMA). These areas of
study need enormous and continuous quantities of data, within a certain
accuracy level, for different targets over a wide range of energy values.
Electron impact ionization and excitation have been actively studied by many
research groups since the 1920’s. Most of the work produced was based on
classical collision theory, and several first principle theories were
developed [14, 55, 69, 3, 4, 75]. The most important work in the field of
electron-atom collision was made by Bethe (1930) who derived the correct form
of the ionization cross section shape for high-energy collisions [3] using the
plane-wave Born approximation (PWBA). Since then, several empirical and semi-
empirical models have been proposed to describe electron impact ionization of
atoms and molecules [21, 71, 72, 73, 43, 45], and several reviews on them were
published [56, 32]. However, each of these models works only on a limited
range of target atoms and/or electron energy values and accuracies are in most
cases very low. With the advance of quantum mechanical computational methods,
some very accurate ab initio calculations were performed. Nevertheless, these
calculations are very time-consuming, limiting the domain of applicability of
such models [62, 53, 6, 7].
In the last years, many analytical formulas have been developed to overcome
these difficulties, some of them empirical [50, 5, 23] and others derived from
first principles [39, 40, 18, 70].
The binary-encounter-Bethe (BEB) model proposed by Kim and Rudd [39]
successfully combines the binary-encounter theory with the dipole interaction
of the Bethe theory for fast incident electrons [3], and meets the above
mentioned requirements. The BEB method, using an analytic formula that
requires only the incident particle energy ($T$), the target particle’s
binding energy ($B$) and the target particle’s kinetic energy ($U$), generates
direct ionization cross section curves for neutral atoms, which are reliable
in intensity ($\pm$ 20%) and shape from the ionization threshold to a few keV
in the incident energy [38, 41], or to thousands keV [57] if we consider its
relativistic version (RBEB) [40].
The factor $1/(T+U+B)$ was the only ad-hoc term considered in the BEB model
(cf. Eq. (57) in Ref. [39] ), accounting for the projectile’s kinetic energy
change upon entering the atomic cloud.
Although this type of scaling has been inserted in several theories such as
the PWBA [37], its success remains to be explained, even though it is an
practical way to account for the electron exchange, distortion and
polarization effects that are absent in the first-order PWBA.
Kim and Rudd [39] noticed that they had to modify the scaling of the BEB/RBEB
models. Comparisons to experimental data [57] suggested that a simple average
of the BEB cross sections with the $1/(T+U+B)$ and $1/T$ terms reproduces the
experimental K-shell ionization cross section data at low to intermediate $Z$
values, and the results obtained with the classical term $1/T$ follow closely
the experimental data for L-Shell ionization. Thus, in order to take advantage
of the success of the BEB/RBEB models, it is necessary to choose one of the
terms $1/T$, $1/(T+U+B)$ and $1/2[1/T+1/(T+U+B)]$ according to the sub-shell
to be ionized.
In this work, we use a different scaling for the BEB/RBEB models, in which,
instead of using several scaling terms depending the ionization sub-shell, we
adopt a $1/(T+C)$ term for all sub-shells, where $C$ is a constant for each
$Z$. This constant is related with the energy change of the incident electron
in the field of the nucleus and the bound electrons of the target atom.
This article is organized as follows. A brief outline of the underlying theory
is presented in Sec. 2. The results are compared with available experimental
and theoretical data in Sec. 3. The conclusions are presented in Sec. 4.
## 2 Theory
The relativistic theory of the BEB and RBEB models is given in detail in Refs.
[39, 40]. Below, therefore, we restrict ourselves to a rather brief account of
the basic expressions, just enough for discussing the role of the scaling
denominator in the ionization cross sections computation.
The term $1/T$ in the Bethe cross section was included originally to normalize
the cross section to the incoming electron flux per unit area perpendicular to
the incident beam direction.
This term was modified by Burguess [10, 9], and later by Vriens [9, 10, 74,
22], who replaced it by $1/(T+U+B)$, with the argument that the effective
kinetic energy of the incident electron seen by the target is $T$ plus the
energy of the bound electron. This denominator can be seen as the scaling
factor to represent the correlation between the two colliding electrons.
Although the BEB and RBEB models have been very successful in reproducing the
ionization cross sections, as mentioned previously, the scaling factor may be
adapted in order to take into account where ionization takes place.
In the model presented in this article we replace all the used scaling factors
in the BEB/RBEB models by the $1/(T+C)$, where $C$ is a factor that depends
only on $Z$.
Considering that the $C(Z)$ function in the term $1/[T+C(Z)]$ is related to
the shielding of the nuclear charge by the bound electrons of the target atom,
and that the binding energy of the K-shell electrons in neutral atoms (in
a.u.) scales as $0.4240Z^{2.1822}$ (Casnati et al. [11]), we may assume that
$C(Z)$ should have an almost quadratic form. Therefore, as a first
approximation, we adopt $C(Z)$ to be equal to the hydrogenic energy levels
expression, i.e., $C(Z)=Z_{\textrm{eff}}^{2}/(2n^{2})$, where $n$ the
principal quantum number, and $Z_{\textrm{eff}}$ is the effective nuclear
charge that accounts for the electronic shielding and electronic correlation.
Moreover, in order to emulate the energy change of the incident electron when
it penetrates the electronic cloud, we assume a linear combination of the
corresponding sub-shell hydrogen-like energy levels for the function $C(Z)$,
which, in atomic units, can be written as
$C_{n\ell j}(Z)=a\frac{Z_{\textrm{eff,}n\ell
j}^{2}}{2n^{2}}+b\frac{Z_{\textrm{eff,}n^{\prime}\ell^{\prime}j^{\prime}}^{2}}{2n^{\prime
2}},$ (1)
where $a$ and $b$ are constants. An analysis of the experimental results
across the whole $Z$ spectra leads to the use of $a=0.3$ and $b=0.7$.
From the data published by Clementi et al. [12, 13], we have obtained
$Z_{\textrm{eff},1s}=0.9834Z-0.1947$ and $Z_{\textrm{eff},2s}=0.7558Z-1.1724$.
Replacing these functions in Eq. (1), we get for K-shell ionization
$C_{1s1/2}(Z)=0.126-0.213\ Z+0.195\ Z^{2}.$ (2)
In the cases where the $Z_{\textrm{eff}}$ is not known, we may use the well-
known approximation that considers the effective nuclear charge to be given by
the atomic number minus the inner electrons up to the sub-shell being ionized.
### 2.1 Modified binary encounter Bethe model
The modified binary encounter Bethe model (MBEB) total ionization cross
section, in reduced units, is written as
$\sigma_{\textrm{MBEB}}=\frac{S}{t+c}\left[\frac{1}{2}\left(1-\frac{1}{t^{2}}\right)\ln
t+\left(1-\frac{1}{t}\right)-\frac{\ln t}{t+1}\right],$ (3)
where the reduced units are expressed as
$\displaystyle t$ $\displaystyle=$ $\displaystyle T/B,$ $\displaystyle c$
$\displaystyle=$ $\displaystyle(C/B)2R,$ $\displaystyle S$ $\displaystyle=$
$\displaystyle 4\pi a_{0}^{2}N(R/B)^{2}.$ (4)
In Eq. (4), $C$ is the scaling constant given by Eq. (1), $N$ is the
occupation number, $a_{0}$ is the Bohr’s radius ($5.29\times 10^{-11}$ m), and
$R$ is the Rydberg energy ($13.6$ eV).
The relativistic counterpart of the modified binary encounter Bethe model
(MRBEB) reads
$\displaystyle\sigma_{\textrm{MRBEB}}$ $\displaystyle=$
$\displaystyle\frac{4\pi
a_{0}^{2}\alpha^{4}N}{\left(\beta_{t}^{2}+c\beta_{b}^{2}\right)2b^{\prime}}\left\\{\frac{1}{2}\left[\ln\left(\frac{\beta_{t}^{2}}{1-\beta_{t}^{2}}\right)-\beta_{t}^{2}-\ln\left(2b^{\prime}\right)\right]\left(1-\frac{1}{t^{2}}\right)\right.$
(5) $\displaystyle\left.+1-\frac{1}{t}-\frac{\ln
t}{t+1}\frac{1+2t^{\prime}}{\left(1+t^{\prime}/2\right)^{2}}+\frac{b^{\prime
2}}{\left(1+t^{\prime}/2\right)^{2}}\frac{t-1}{2}\right\\},$
where
$\displaystyle\beta_{t}^{2}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{\left(1+t^{\prime}\right)^{2}},\ \ \ \ \ \ \ \ \ \ \ \ \
t^{\prime}=T/mc^{2},$ $\displaystyle\beta_{b}^{2}$ $\displaystyle=$
$\displaystyle 1-\frac{1}{\left(1+b^{\prime}\right)^{2}},\ \ \ \ \ \ \ \ \ \ \
\ \ b^{\prime}=B/mc^{2},$ (6)
and $\alpha$ is the fine structure constant, $c$ is the speed of light in
vacuum, and $m$ is the electron mass.
## 3 Results
The present MBEB/MRBEB models produce reliable cross sections between the
threshold and the peak without using any experiment-dependent parameters.
As an illustration, we apply the nonrelativistic MBEB and relativistic MRBEB
expressions to the K-shell ionization of C, Ne, Si, Sc, Ti, V, Cr, Fe, Zn, Co,
Sr, and Ag, to the L-shell ionization of Se, Kr, Ag, Sb, Xe, and Ba, and to
the M-shell ionization of Pb and Bi.
Contrary to the BEB/RBEB models, which require two input parameters ($B$ and
$U$), the MBEB/MRBEB models require only the knowledge of one parameter, the
binding energy $B$. For the binding energies of inner-shell electrons, one can
use experimental values [17] to match experimental thresholds precisely, or
theoretical binding energies from Dirac-Fock wave functions that are reliable
to 1% or better in general. The values of $B$ of the elements studied in this
work are listed in Table 1. For the carbon atom the K-shell binding energy was
taken from Ref. [8], while the remaining elements K-shell binding energies
were obtained from Ref. [17]. The L- and M-shell binding energies were
evaluated using the MDFGME code developed by J. P. Desclaux and P. Indelicato
[16, 31].
The electron occupation number was set to $N=2$ for $s_{1/2}$ and $p_{1/2}$
orbitals, $N=4$ for $p_{3/2}$ and $d_{3/2}$ orbitals and $N=6$ for $d_{5/2}$
orbitals.
### 3.1 K-shell ionization
On Fig. 1 (for C, Ne, Si, Sc, Ti, and V) and Fig. 2 ( for Cr, Fe, Zn, Co, Sr
and Ag), we compare the present MBEB [Eq. (3)] and MRBEB cross sections [Eq.
(5)] to all available experimental data, to the empirical cross sections by
Hombourger et al. [29], Haque et al. [23], and to the analytical model by Bote
et al. [5], which results from a fit to a database of cross sections
calculated using the plane-wave (PWBA) and distorted-wave (DWBA) Born
approximations. For overvoltages ($t=T/B$) lower than 16, the fit was done to
the DWBA database, and for $t>16$ the PWBA database was used, since, for high-
energies, the difference between the DWBA and PWBA cross sections is
negligible. The DWBA/PWBA model, labeled as DWBA for simplicity, provides
ionization cross section values that agree with those in the DWBA/PWBA
database to within about 1%, except for projectiles with near-threshold
energies. Since both the Hombourger et al. model and the XCVTS model of Haque
et al. are empirical, the range of validity of such models is limited by the
availability of experimental data. Furthermore, the XCVTS model uses a scaling
term with different coefficients for different shells as in the unmodified
BEB/RBEB expressions.
In the analysis of Figs. 1 and 2, as discussed previously by Santos et al.
[57], caution is warranted when comparing the experimental and theoretical
data represented. Experimental data are mainly obtained through the detection
of X-rays or Auger electrons emitted when bound electrons fill the K-shell
vacancies created by electron impact. However, K-shell vacancies can be
created not only by direct ionization but also by excitations of K electrons
to unoccupied bound states. Since most theories, including the MBEB/MRBEB
models, are designed for only direct ionization by electron impact,
experimental data may exceed the theoretical data by the amount due to
excitations of K electrons to bound levels. Therefore, unless experimental
data have explicitly excluded the K-shell vacancies created by excitation,
comparisons of theories and experiments may have an inherent ambiguity of
$\sim$10%.
Below we discus the cases that we analyzed. In order to compare the
experimental values to the different theoretical results, we used the reduced
$\chi^{2}$, $Q$, defined by $Q=\chi^{2}/\nu$, where $\nu$ is the number of
experimental data points:
* 1.
Carbon: The relativistic and nonrelativistic cross sections are almost
identical for $T<1$ keV. The present MRBEB cross section, the DWBA and the
XCTVS results are in good agreement with the experimental data by Egerton et
al. [19], Tawara et al. [68], and Isaacson et al. [33] (with the reduced
$\chi^{2}$, $Q$, equal to 0.91, 0.65 and 0.73, respectively), while the
experimental data by Hink et al. [25] display an increasing trend toward lower
$T$ not seen in any other theory or experiment.
* 2.
Neon: The relativistic and nonrelativistic cross sections are almost identical
for $T<$ 100 keV. The theoretical cross sections are in fairly good agreement
with experimental data by Tawara et al. [68], Glupe et al. [20], and Platten
et al. [51].
* 3.
Silicon: We see the beginning of the relativistic rise at $T>$ 100 keV, which
is not followed by the nonrelativistic MBEB. In this high $T$ region, all
theoretical relativistic data agree with the experimental data by Ishii et al.
[34] and Shchagin et al. [64], with $Q$ values from 0.4 (Hombourger) to 0.7
(DWBA).
* 4.
Scandium: The experimental results by An et al. [2] are not in agreement with
any of the theories presented here, so new experimental data are required to
better understand this case.
* 5.
Titanium: The experiments are divided into two groups. The experimental cross
sections by Jessenberger et al. [35] lie above all theoretical data in the
peak region, while the experimental cross sections by He et al. [24] are lower
than all theoretical data.
* 6.
Vanadium: The MRBEB cross section values for vanadium are in good agreement
with the experimental data by An et al. [2], having the lowest $Q$ value of
all theoretical models, which ranges from 20.9 to 130.5.
* 7.
Chromium: We notice that all experimental data except the one from He et al.
[24] for chromium agree with the represented theoretical models, confirming
the trend of the experimental data by He et al. observed in Ti.
* 8.
Iron: Although there is a general agreement between the theoretical data and
the experimental results, the MRBEB model underestimates slightly the
ionization cross sections in the peak region.
* 9.
Zinc: The MRBEB cross sections are in good agreement with the experimental
data by Tang et al. [67] at low $T$, and with the only experimental value at
high $T$ from Ishii et al. [34]. This is confirmed by the low $Q$ value of 1.1
that we find, to be compared to the high value of 18.3 for the XCVTS model.
Nevertheless, the MRBEB values become larger than the other three theoretical
cross section values beyond $T=1$ MeV. There is thus a strong need of new
experiment for $T>1$ MeV is desirable to distinguish different predictions
from different theories.
* 10.
Cobalt: The experimental data by An et al. [1] agree very well with the MRBEB
model, from threshold to the ionization peak, which produces the lowest $Q$
value of all models in a range from 0.2 to 10.6.
* 11.
Strontium: The theoretical data disagree among them and with the experimental
data. However, we observe that the MRBEB model ($Q$=3.2) follows more closely
the experimental data by Shevelko et al. [65] at low $T$, while the DWBA
($Q$=6.3) and the Hombourger ($Q$=10.0) models follow more closely the
experimental data by Middleman et al. [49] at high $T$.
* 12.
Silver: Ten sets of experimental data are compared with the MRBEB cross
sections and other theories. Again, experiments are divided into groups near
the peak. The experimental data by Davis et al. [15] agree well with the
Hombourger cross sections. The experimental data by Schneider et al. [59],
Kiss et al. [42], Hoffman et al. [28] agree with the MRBEB cross sections. The
data by el Nasr et al. [63] and Hubner et al. [30] disagree with all the
presented theoretical cross sections. Although all theoretical cross sections
agree in the vicinity of $T=500$ keV, the difference between the present MRBEB
cross section values and the other theoretical relativistic cross section
values is widening at $T=1$ MeV, amplifying the trend observed in Zn and Sr.
The silver atom is another example for which definitive measurements would
help to distinguish different theories.
### 3.2 L- and M-shell ionization
In order to investigate the range of applicability of the approach presented
in this work besides the K-shell ionization, we have also applied the MBEB and
MRBEB models to the L-shell ionization of Se, Kr, Ag, Sb, Xe and Ba, and to
M-shell ionization of Pb and Bi.
On Fig. 3 and Fig. 4 the MBEB and MRBEB cross sections for the L- shell (for
Se, Kr, Ag, Sb, Xe, and Ba) and M-shell (for Pb and Bi), respectively, are
displayed as well as the theoretical results obtained with the DWBA, XCVTS and
Lotz [45, 46] models, and by Scofield [61], and the experimental available
data for the analyzed elements. The Lotz empirical expression, proposed more
than 30 years ago, is one of the most successful formulas for calculating
total direct ionization of any given state.
Concerning the L-Shell ionization, we notice that the MRBEB cross sections are
in good agreement with the experimental data for the analyzed elements, except
Xe, having the lowest $Q$ value for Se, Kr, Sb and Ba (1.2, 0.6, 0.5, 0.6,
respectively), and the second lowest for Ag (1.4). The theoretical data
disagree among them, namely in the peak region; the DWBA values produce the
highest peak, followed in equal ground by the XCVTS and the Lotz curves, and
finally by the MBEB and MRBEB curves. It should be pointed out that the
experimental data by Hippler et al. [26] for Xe exhibits the greater
uncertainty (about 30%) among the studied cases. This uncertainty is less than
17% for the other elements. The experimental data for the M-shell ionization
is scarce and exist only for high incident electron energies, in the
relativistic regime ($T>10^{4}$ keV). In this high region, all theoretical
relativistic data agree with the experimental data by Ishii et al. [34] and
Hoffman et al. [28], with $Q$ values equal to 0.4 (XCVTs), 0.7 (MRBEB), and
1.7 (DWBA) for Pb, and 0.3 (XCVTs), 0.9 (MRBEB), and 2.0 (DWBA) for Bi. The
comparison among the theoretical data have the same outcome obtained for the
L-Shell.
## 4 Conclusions
The new MBEB and MRBEB models presented in this work require only one atomic
parameter, namely the binding energy of the electrons to be ionized, and,
contrary to the BEB/RBEB models, use only one scaling term (1/(T + C)) for the
ionization of all sub-shells.
The MBEB and MRBEB expressions were used to obtain the K-, L-, and M shell
ionization cross sections by electron impact for several atoms with Z from 6
to 83.
We pointed out that the comparison of the MRBEB cross sections to experimental
values contains inherent ambiguities, because the MRBEB model predicts cross
sections for the direct ionization of electrons of a definite sub-shell, while
most experimental data are based on all sub-shell vacancies created by direct
ionization as well as excitations to bound levels.
As show on Figs. 1, 2, 3, and 4, relativistic effects become increasingly
important as the binding energies of the elements increase. Hence,
relativistic theory must be used for treating both atomic structure and
collision dynamics for medium to heavy atoms.
The presented comparisons show that the MRBEB model produces reliable K-, L-
and M-shell ionization cross sections between the threshold and several MeV
with an accuracy of $\sim$20%, or better, without using empirical parameters.
The simple relativistic MRBEB expression presented in this article provides a
continuous coverage of K-, L- and M-shell ionization cross sections by
electron impact from the threshold to relativistic incident energies, making
this expression ideally suited for modeling systems where ionization cross
sections for a wide range of incident energies are required, such as fusion
plasmas.
## Acknowledgements
This research was supported in part by FCT (PEst-OE/FIS/UI0303/2011, Centro de
F sica Atómica), by the French-Portuguese collaboration (PESSOA Program,
contract no 441.00), by the Acções Integradas Luso-Francesas (contract no
F-11/09), and by the Programme Hubert Curien. M.G. acknowledges the support of
the FCT, under Contract No. SFRH/BD/38691/2007. Laboratoire Kastler Brossel is
“Unité Mixte de Recherche du CNRS, de l’ ENS et de l’UPMC n∘ 8552”.
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Table 1: Binding energy $B$ values for the K-, L- and M-shells. The $B$ value
for C is from Ref. [8]. The remaining K-shell and L-shell $B$ values are from
Ref. [17]. The M-shell binding energies were evaluated using the MDFGME code
[16, 31]
Element | | | | $B$(eV) | | |
---|---|---|---|---|---|---|---
| K-Shell | L-shell | M-shell
| | L1 | L2 | L3 | M1 | M2 | M3 | M4 | M5
C | 296.07 | | | | | | | |
Ne | 866.90 | | | | | | | |
Si | 1840.05 | | | | | | | |
Sc | 4489.37 | | | | | | | |
Ti | 4964.58 | | | | | | | |
V | 5463.76 | | | | | | | |
Cr | 5989.02 | | | | | | | |
Fe | 7110.75 | | | | | | | |
Co | 7708.75 | | | | | | | |
Zn | 9660.76 | | | | | | | |
Se | | 1652.44 | 1474.72 | 1433.98 | | | | |
Kr | | 1916.30 | 1729.66 | 1677.25 | | | | |
Sr | 16107.20 | | | | | | | |
Ag | 25515.59 | 3807.34 | 3525.83 | 3350.96 | | | | |
Sb | | 4698.44 | 4381.90 | 4132.33 | | | | |
Xe | | 5452.89 | 5103.83 | 4782.16 | | | | |
Ba | | 5995.90 | 5623.29 | 5247.04 | | | | |
Pb | | | | | 3905.53 | 3601.14 | 3110.21 | 2628.17 | 2525.49
Bi | | | | | 4056.25 | 3744.91 | 3223.11 | 2731.84 | 2623.08
Figure 1: Electron impact K-shell ionization cross sections for (a) C, (b) Ne,
(c) Si, (d) Sc, (e) Ti, (f) V. Thick solid curve, present MRBEB cross section
Eq. (5); dash-dash curve, MBEB cross section Eq. (3) dot-dot curve, DWBA by
Bote et al. [5]; dot-dash curve, relativistic empirical formula by Hombourger
[29]; short dot-dash curve, XCVTS semiempirical formula by Haque et al. [23];
Experimental data by Egerton et al. [19], Tawara et al. [68], Hink et al.
[25], Isaacson et al. [33], Glupe et al. [20], Platten et al. [51], Ishii et
al. [34], Kamiya et al. [36], Hoffman et al. [28], Shchagin et al. [64], He et
al. [24], Jessenberger et al. [35], and An et al. [2]. Figure 2: Electron
impact K-shell ionization cross sections for (a) Cr, (b) Fe, (c) Zn, (d) Co,
(e) Sr and (f) Ag. Thick solid curve, present MRBEB cross section Eq. (5);
dash-dash curve, MBEB cross section Eq. (3) dot-dot curve, DWBA by Bote et al.
[5]; dot-dash curve, relativistic empirical formula by Hombourger [29]; short
dot-dash curve, XCVTS semiempirical formula by Haque et al. [23]; Experimental
data by Llovet et al. [44], He et al. [24], Luo et al. [47] (Cr), Luo et al.
[48] (Fe), Scholz et al. [60], Ishii et al. [34], Tang et al. [67], An et al.
[1], Shevelko et al. [65], Middleman et al. [49], Davis et al. [15] Schneider
et al. [59], Shima et al. [66], Rester et al. [52], Kiss et al. [42], Schlenk
et al. [58], El Nasr et al. [63], Hubner et al. [30], Ricz et al. [54], and
Hoffman et al. [28]. Figure 3: Electron impact L-shell ionization cross
sections for (a) Se, (b) Kr, (c) Ag, (d) Sb, (e) Xe and (f) Ba. Thick solid
curve, present MRBEB cross section Eq. (5); dash-dash curve, MBEB cross
section Eq. (3) dot-dash curve, relativistic empirical formula by Lotz [45,
46]; dot-dot curve, DWBA by Bote et al. [5]; short dot-dash curve, XCVTS
semiempirical formula by Haque et al. [23]; $\times$, DWBA values by Scofield
et al. [61]; Experimental data by Ishii et al. [34], Kiss et al. [42], Hippler
et al. [26], and Hoffman et al. [27]. Figure 4: Electron impact M-shell
ionization cross sections for (a) Pb and (f) Bi. Thick solid curve, present
MRBEB cross section Eq. (5); dash-dash curve, MBEB cross section Eq. (3) dot-
dash curve, relativistic empirical formula by Lotz [45, 46]; dot-dot curve,
DWBA by Bote et al. [5]; short dot-dash curve, XCVTS semiempirical formula by
Haque et al. [23]; Experimental data by Ishii et al. [34], and Hoffman et al.
[27] .
|
arxiv-papers
| 2013-06-12T13:42:40 |
2024-09-04T02:49:46.408827
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Guerra, F. Parente, P. Indelicato, J.P. Santos",
"submitter": "Mauro Guerra",
"url": "https://arxiv.org/abs/1306.2826"
}
|
1306.2860
|
# Neat-flat modules
Engin Büyükaşık Izmir Institute of Technology, Department of Mathematics,
35430, Urla, Izmir, Turkey [email protected] and Yılmaz Durğun
Izmir Institute of Technology, Department of Mathematics,
Gülbahçeköyü, 35430, Urla, Izmir, Turkey. [email protected]
###### Abstract.
Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called neat-flat if
any short exact sequence of the form $0\to K\to N\to M\to 0$ is neat-exact
i.e. any homomorphism from a simple right $R$-module $S$ to $M$ can be lifted
to $N$. We prove that, a module is neat-flat if and only if it is simple-
projective. Neat-flat right $R$-modules are projective if and only if $R$ is a
right $\sum$-$CS$ ring. Every finitely generated neat-flat right $R$-module is
projective if and only if $R$ is a right $C$-ring and every finitely generated
free right $R$-module is extending. Every cyclic neat-flat right $R$-module is
projective if and only if $R$ is right $CS$ and right $C$-ring. Some
characterizations of neat-flat modules are obtained over the rings whose
simple right $R$-modules are finitely presented.
###### Key words and phrases:
closed submodule, neat submodule, extending modules, $m$-injective module,
$C$-ring, $CS$-ring
###### 2000 Mathematics Subject Classification:
16D10, 16D40, 16D70, 16E30
## 1\. Introduction
Throughout, $R$ is an associative ring with identity and all modules are
unitary right $R$-modules. For an R-module $M$, $M^{+}$, $E(M)$,
$\operatorname{Soc}(M)$ will denote the character module, injective hull, the
socle of $M$, respectively. A subgroup $A$ of an abelian group $B$ is called
_neat_ in $B$ if $pA=A\cap pB$ for each prime integer $p$. The notion of neat
subgroup generalized to modules by Renault (see, [renault:neat]). Namely, a
submodule $N$ of $R$-module $M$ is called _neat_ in $M$, if for every simple
$R$-module $S$, every homomorphism $f:S\to M/N$ can be lifted to a
homomorphism $g:S\to M$. Equivalently, $N$ is neat in $M$ if and only if
$\operatorname{Hom}(S,g):\,\operatorname{Hom}(S,M)\to\operatorname{Hom}(S,M/N)$
is an epimorphism for every simple $R$-module $S$. Neat submodules have been
studied extensively by many authors (see, [Wisbauer-et.al:t-complementedandt-
supplementedmodules], [Fuchs:NeatSubmodulesOverIntegralDomain],
[Mermut:Ph.D.tezi], [Stenstrom:highsubmodulesandpurity],
[Stenstrom:Puresubmodules]). An $R$-module $M$ is called _m-injective_ if for
any maximal right ideal I of $R$, any homomorphism $f:I\to M$ can be extended
to a homomorphism $g:R\to M$ (see, [septimi:minjective], [Mermut:Ph.D.tezi],
[Ozdemir:Ph.D.tezi], [smith:Injectivemodulesandprimeideals],
[Wang:onmaximalinjective], [Xiang:maxflatmaxinjective]). Note that,
$m$-injective modules are called max-injective in [Wang:onmaximalinjective].
It turns out that, a module $M$ is $m$-injective if and only if
$\operatorname{Ext}^{1}_{R}(R/\emph{I},M)=0$ for any maximal right ideal I of
$R$ if and only if $M$ is a neat submodule in every module containing it i.e.
any short exact sequence of the form $0\rightarrow M\rightarrow N\rightarrow
L\rightarrow 0$ is neat-exact (see, [septimi:minjective, Theorem 2]). A ring
$R$ is a right $C$-ring if for every proper essential right ideal $I$ of $R$,
the module $R/I$ has a simple module, (see, [Renault:Cring]). Any right
semiartinian ring is a $C$-ring, and a domain is a $C$-ring if and only if
every torsion $R$-module contains a simple module. By
[smith:Injectivemodulesandprimeideals, Lemma 4], $R$ is a right $C$-ring if
and only if every $m$-injective module is injective.
Motivated by the relation between $m$-injective modules and neat submodules,
we investigate the modules $M$, for which any short exact sequence ending with
$M$ is neat-exact. Namely, we call $M$ _neat-flat_ if for any epimorphism
$f:N\rightarrow M$, the induced map
$\operatorname{Hom}(S,\,N)\rightarrow\operatorname{Hom}(S,M)$ is surjective
for any simple right $R$-module $S$.
In [mao:whendoeseverysimplehaveaprojectiveenvelope], a right $R$-module $M$ is
called _simple-projective_ if for any simple right $R$-module $N$, every
homomorphism $f:N\rightarrow M$ factors through a finitely generated free
right $R$-module $F$, that is, there exist homomorphisms $g:N\rightarrow F$
and $h:F\rightarrow M$ such that $f=hg.$ Simple-projective modules and a
generalization of these modules have been studied in
[mao:whendoeseverysimplehaveaprojectiveenvelope] and [rada], respectively. By
using simple-projective modules, the authors, characterize the rings whose
simple (resp. finitely generated) right modules have projective (pre)envelope
in the sense of [xu:flatcoverofmodules]. Clearly, projective modules and
modules with $\operatorname{Soc}(M)=0$ are simple-projective. Also, a simple
right $R$-module is simple-projective if and only if it is projective. Hence,
$R$ is a semisimple Artinian ring if and only if every right $R$-module is
simple-projective (see, [mao:whendoeseverysimplehaveaprojectiveenvelope,
Remark 2.2.]).
The paper is organized as follows.
In section 3, we prove that, a right $R$-module $M$ is neat-flat if and only
if $M$ is simple-projective (Theorem 3.2). The right socle of $R$ is zero if
and only if neat-flat modules coincide with the modules that have zero socle
(Proposition 3.3). We also investigate the rings over which neat-flat modules
are projective. Namely, we prove that, (1) every neat-flat module is
projective if and only if $R$ is a right $\sum$-$CS$ ring (Theorem 3.5); (2)
every finitely generated neat-flat module is projective if and only if $R$ is
a right $C$-ring and every finitely generated free right $R$-module is
extending (Theorem 3.6); (3) every cyclic right $R$-module is projective if
and only if $R$ is right $CS$ and right $C$-ring (Corollary 3.7).
In section 4, we consider neat-flat modules over the rings whose simple right
modules are finitely presented. In this case, the Auslander-Bridger tranpose
of any simple right $R$-module is a finitely presented left $R$-module. This
fact is used to obtain several characterization of neat-flat modules. Also, we
examine the relation between the flat, absolutely pure and neat-flat modules
over such rings.
For the unexplained concepts and results we refer the reader to [AF],
[extendingmodules], [Wisbauer:Foundationsofmoduleandringtheory] and
[xu:flatcoverofmodules].
## 2\. preliminaries
The class of neat-exact sequences form a proper class in the sense of
[Buchsbaum]. This fact leads to an important characterization of neat-flat
modules (see, Lemma 3.1). This characterization become crucial in the proof of
the results in the present paper. In this section, we give some definitions
and results which are used in the sequel.
Let $R$ be an associative ring with identity and $\mathcal{P}$ be a class of
short exact sequences of right $R$-modules and $R$-module homomorphisms. If a
short exact sequence $\mathbb{E}:0\to A\overset{f}{\to}B\overset{g}{\to}C\to
0$ belongs to $\mathcal{P}$, then $f$ is said to be a
$\mathcal{P}$-_monomorphism_ and $g$ is said to be a
$\mathcal{P}$-_epimorphism_. A short exact sequence $\mathbb{E}$ is determined
by each of the monomorphisms $f$ and the epimorphisms $g$ uniquely up to
isomorphism.
###### Definition 2.1.
The class $\mathcal{P}$ is said to be _proper_ (in the sense of Buchsbaum) if
it satisfies the following conditions [maclane:homology]:
1. P-1)
If a short exact sequence $E$ is in $\mathcal{P}$, then $\mathcal{P}$ contains
every short exact sequence isomorphic to $E$ .
2. P-2)
$\mathcal{P}$ contains all splitting short exact sequences.
3. P-3)
The composite of two $\mathcal{P}$-monomorphisms is a
$\mathcal{P}$-monomorphism if this composite is defined.
4. P-4)
The composite of two $\mathcal{P}$-epimorphisms is a $\mathcal{P}$-epimorphism
if this composite is defined.
5. P-5)
If $g$ and $f$ are monomorphisms, and $g\,\circ f$ is a
$\mathcal{P}$-monomorphism, then $f$ is a $\mathcal{P}$-monomorphism.
6. P-6)
If $g$ and $f$ are epimorphisms, and $g\,\circ f$ is a
$\mathcal{P}$-epimorphism. then $g$ is a $\mathcal{P}$-epimorphism.
From now on, $\mathcal{P}$ will denote a proper class. A module $M$ is called
$\mathcal{P}$-flat if every short exact sequence of the form $0\to A\to B\to
M\to 0$ is in $\mathcal{P}.$
For a class $\mathcal{M}$ of right $R$-modules, let
$\tau^{-1}(\mathcal{M})=\\{\mathbb{E}\mid M\otimes\mathbb{E}\,\text{ exact for
each}M\in\mathcal{M}\\}$, and
$\pi^{-1}(\mathcal{M})=\\{\mathbb{E}\mid\operatorname{Hom}(M,\,\mathbb{E})\,\,\text{is
exact for each}M\in\mathcal{M}\\}.$ Then the $\tau^{-1}(\mathcal{M})$ and
$\pi^{-1}(\mathcal{M})$ are proper classes (see,
[Sklyarenko:RelativeHomologicalAlgebra]). The classes $\tau^{-1}(\mathcal{M})$
and $\pi^{-1}(\mathcal{M})$ are called flatly generated and projectively
generated by $\mathcal{M}$, respectively.
###### Theorem 2.2.
[Sklyarenko:RelativeHomologicalAlgebra, Theorem 8.1] Let $\mathcal{M}$ be a
class of modules and $\mathbb{E}$ be a short exact sequence. Then
$\mathbb{E}\in\tau^{-1}(\mathcal{M})$ if and only if
$\mathbb{E}^{+}\in\pi^{-1}(\mathcal{M}).$
Let $M$ be a finitely presented right $R$-module. Then there is an exact
sequence $\gamma:P_{0}\overset{f}{\to}P_{1}\overset{g}{\to}M$ where $P_{0}$
and $P_{1}$ are finitely generated projective right $R$-modules. By applying
the functor $(-)^{*}=\operatorname{Hom}_{R}(-,R)$ to this sequence, we get:
$0\to\operatorname{Hom}_{R}(M,R)\overset{g^{*}}{\to}Hom_{R}(P_{0},R)\overset{f^{*}}{\to}\operatorname{Hom}_{R}(P_{1},R).$
If the right side of this sequence of left $R$-modules filled by the module
$Tr_{\gamma}(M):=Coker(f^{*})=P_{1}^{*}/\operatorname{Im}(f^{*})$ then we
obtain the exact sequence
$\gamma^{*}:P_{0}^{*}\overset{f^{*}}{\to}P_{1}^{*}\overset{\sigma}{\to}Tr_{\gamma}(M)\to
0$ where $\sigma$ is the canonical epimorphism. For a finitely generated
projective $R$-module $P$, its dual $P^{*}=\operatorname{Hom}_{R}(P,R)$ is a
finitely generated projective right $R$-module. So $P_{0}^{*}$ and $P_{1}^{*}$
are finitely generated projective modules, hence the exact sequence
$\gamma^{*}$ is a presentation for the finitely presented right $R$-module
$Tr_{\gamma}(M)$ which is called the Auslander-Bridger tranpose of the
finitely presented $R$-module $M$, (see [auslenderbridge:stablemoduletheory]).
###### Proposition 2.3.
[Sklyarenko:RelativeHomologicalAlgebra, Corollary 5.1] For any finitely
presented right $R$-module $M$ and any short exact sequence $\mathbb{E}$ of
right $R$-modules, the sequence $\operatorname{Hom}(M,\mathbb{E})$ is exact if
and only if the sequence $\mathbb{E}\otimes Tr(M)$ is exact.
###### Theorem 2.4.
[Sklyarenko:RelativeHomologicalAlgebra, Theorem 8.3] Let $\mathcal{M}$ be a
set of finitely presented left $R$-modules. Let
$Tr(\mathcal{M})=\left\\{Tr(M)|M\in\mathcal{M}\right\\}$. Then we have
$\pi^{-1}(\mathcal{M})=\tau^{-1}(Tr(\mathcal{M}))$ and
$\tau^{-1}(\mathcal{M})=\pi^{-1}(Tr(\mathcal{M})).$
## 3\. Neat-flat modules
By definition, the class of neat-exact sequences is projectively generated by
the class of simple right $R$-modules. Hence neat-exact sequences form a
proper class. For the following lemma we refer to
[mishina:abeliangroupsandmodules, Proposition 1.12-1.13]. Its proof is
included for completeness.
###### Lemma 3.1.
The following are equivalent for a right $R$-module $M$.
1. (1)
$M$ is neat-flat.
2. (2)
Every exact sequence $0\to A\to B\to M\to 0$ is neat exact.
3. (3)
There exists a neat exact sequence $0\to K\to F\to M\to 0$ with $F$
projective.
4. (4)
There exists a neat exact sequence $0\to K\to F\to M\to 0$ with $F$ neat-flat.
###### Proof.
$(1)\Rightarrow(2)\Rightarrow(3)\Rightarrow(4)$ are clear.
$(4)\Rightarrow(1)$ Let $0\to A\to B\overset{g}{\to}M\to 0$ be any short exact
sequence. We claim that $g$ is a neat epimorphism. By (4), there exists a neat
exact sequence $0\to K\to F\overset{s}{\to}M\to 0$ with $F$ neat-flat. We
obtain a commutative diagram with exact rows
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{t}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$
in which the right square is a pullback diagram. Since $F$ is neat-flat, $t$
is a neat epimorphism. Then $gu=st$ is a neat epimorphism by 2.1 P-4), and so
$f$ is a neat epimorphism by 2.1 P-6). This completes the proof. ∎
###### Theorem 3.2.
Let $R$ be a ring and $M$ be an $R$-module. Then $M$ is simple-projective if
and only if $M$ is neat-flat.
###### Proof.
Suppose $M$ is simple-projective and $s:R^{(I)}\rightarrow M$ be an
epimorphism. Let $S$ be simple right $R$-module and $f:S\rightarrow M$ be a
homomorphism. As $M$ is simple-projective $f$ factors through a finitely
generated free module i.e. there are homomorphisms $h:S\rightarrow R^{n}$ and
$g:R^{n}\rightarrow M$ such that $f=gh.$ Since $R^{n}$ is projective, there is
a homomorphism $t:R^{n}\rightarrow R^{(I)}$ such that $g=st.$ We get the
following diagram
$\textstyle{R^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\scriptstyle{t}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{h}$$\textstyle{R^{(I)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{M}$
Then $f=gh=sth$, and so the induced map
$\operatorname{Hom}(S,\,R^{(I)})\rightarrow\operatorname{Hom}(S,\,M)\rightarrow
0$ is surjective. Therefore the sequence $0\to\operatorname{Ker}s\to
R^{(I)}\overset{s}{\to}M\to 0$ is neat exact. Hence $M$ is neat-flat by Lemma
3.1(3).
Conversely, let $M$ be a neat-flat module. Then there is a neat exact sequence
$0\to K\to F\overset{g}{\to}M\to 0$ with $F$ free by Lemma 3.1. Let $S$ be a
simple module and $f:S\rightarrow M$ be any homomorphism. Then there is a
homomorphism $h:S\rightarrow F$ such that $f=gh.$ As $S$ is finitely
generated, $h(S)\subseteq H$ for some finitely generated free submodule of
$F$. Then we get $f=gh=(gi)h^{\prime}$ where $i:H\rightarrow F$ is the
inclusion and $h^{\prime}:S\rightarrow H$ is the homomorphism defined as
$h^{\prime}(x)=h(x)$ for each $x\in S.$ Therefore $f$ factors through $H$, and
so $M$ is simple projective. ∎
Let $M$ be a right $R$-module with $\operatorname{Soc}(M)=0.$ Then
$\operatorname{Hom}(S,\,M)=0$ for any simple right $R$-module $S$, and so $M$
is neat-flat.
###### Proposition 3.3.
Let $R$ be a ring and $M$ be any $R$-module. The following are equivalent:
1. (1)
$\operatorname{Soc}(R_{R})=0$.
2. (2)
$M$ is neat-flat right $R$-module if and only if $\operatorname{Soc}(M)=0$.
###### Proof.
$(1)\Rightarrow(2)$ Suppose $M$ is a neat-flat right $R$-module. Then there is
a neat exact sequence $0\to K\to P\to\ M\to 0$ with $P$ projective by Lemma
3.1. Then the sequence $\operatorname{Hom}_{R}(S,P)\to\
\operatorname{Hom}_{R}(S,M)\to 0$ is exact for any simple right $R$-module
$S$. We have $\operatorname{Soc}(P)=0$ by (1). Then
$\operatorname{Hom}_{R}(S,P)=0$, and so $\operatorname{Soc}(M)=0$. The
converse is clear.
$(2)\Rightarrow(1)$ Since every projective module is neat-flat,
$\operatorname{Soc}(R_{R})=0$ by (2). ∎
###### Proposition 3.4.
[mao:whendoeseverysimplehaveaprojectiveenvelope, Proposition 2.4]The class of
simple-projective right $R$-modules is closed under extensions, direct sums,
pure submodules, and direct summands.
Recall that, a submodule $N$ of a module $M$ is called _closed (or a
complement) in $M$_, if $N$ has no proper essential extension in $M$, i.e.
$N\unlhd K\leq M$ implies $N=K.$ A module $M$ is said to be an extending
module or a $CS$-module if every closed submodule of $M$ is a direct summand
of $M$. $R$ is a right $CS$ ring if $R_{R}$ is $CS$. $M$ is called (countably)
$\sum$-$CS$ module if every direct sum of (countably many) copies of $M$ is
$CS$, (see, for example, [extendingmodules]). The $\sum$-$CS$ rings were first
introduced and termed as co-$H$-rings in [oshiro]. Closed submodules are neat
by [Stenstrom:highsubmodulesandpurity, Proposition 5]. By [generalov, Theorem
5], every neat submodule is closed if and only if $R$ is a right $C$-ring.
###### Theorem 3.5.
Let $R$ be a ring. The following are equivalent.
1. (1)
Every neat-flat right $R$-module is projective.
2. (2)
$R$ is a right $\sum$-$CS$ ring.
###### Proof.
$(1)\Rightarrow(2)$ Let $P$ be a projective $R$-module and $N$ be a closed
submodule of $P$. Then $N$ is a neat submodule of $P$. So that $P/N$ is neat-
flat by Lemma 3.1 and so $P/N$ is projective by (1). Therefore the sequence
$0\rightarrow N\rightarrow P\rightarrow P/N\rightarrow 0$ splits, and so $N$
is a direct summand of $P$. Hence $R$ is a $\sum$-$CS$ ring.
$(2)\Rightarrow(1)$ Every right $\sum$-$CS$ ring is both right and left
perfect by [oshiro, Theorem 3.18]. Hence, $R$ is a right $C$-ring by [AF,
Theorem 28.4]. Let $M$ be a neat-flat right $R$-module. Then there is a neat
exact sequence $\mathbb{E}:\,\,0\to K\hookrightarrow P\to\ M\to 0$ with $P$
projective by Lemma 3.1. Since $R$ is right $C$-ring, $K$ is closed in $P$ by
[generalov, Theorem 5]. Hence the sequence $\mathbb{E}$ splits by (2), and so
$M$ is projective. ∎
###### Theorem 3.6.
Let $R$ be a ring. The following are equivalent.
1. (1)
Every finitely generated neat-flat right $R$-module is projective.
2. (2)
$R$ is a right $C$-ring and every finitely generated free right $R$-module is
extending.
###### Proof.
$(1)\Rightarrow(2)$ Let $I$ be an essential right ideal of $R$ with
$\operatorname{Soc}(R/I)=0$. Then $\operatorname{Hom}(S,R/I)=0$ for each
simple right $R$-module $S$ and hence $I$ is neat ideal of $R$. So $R/I$ is
neat-flat by Lemma 3.1. But it is projective by (1), and so $I$ is direct
summand of $R$. This is contradict with essentiality of $I$ in $R$. So that
$R$ is a right $C$-ring.
Let $F$ be a finitely generated free right $R$-module and $K$ a closed
submodule of $F$. Since every closed submodule is neat, $F/K$ is neat-flat by
Lemma 3.1. Then $F/K$ is projective by (1), and so $K$ is a direct summand of
$F$.
$(2)\Rightarrow(1)$ Let $M$ be a finitely generated neat-flat right
$R$-module. Then there is an exact sequence
$0\to\operatorname{Ker}(f)\hookrightarrow F\to M\to 0$ with $F$ finitely
generated free right $R$-module. By Lemma 3.1 $\operatorname{Ker}(f)$ is neat
submodule of $F$. Since $R$ is $C$-ring, $\operatorname{Ker}(f)$ is closed
submodule of $F$ by [generalov, Theorem 5]. Then
$0\to\operatorname{Ker}(f)\hookrightarrow F\to M\to 0$ is a split exact
sequence. So $M$ is projective. ∎
Following the proof of Theorem 3.6, we obtain the following corollary.
###### Corollary 3.7.
Every cyclic neat-flat right $R$-module is projective if and only if $R$ is
both right $CS$ and right $C$-ring.
###### Remark 3.8.
Let $M$ be a right $R$-module. Then the socle series $\\{S_{\alpha}\\}$ of $M$
is defined as: $S_{1}=\operatorname{Soc}(M)$,
$S_{\alpha}/S_{\alpha-1}=\operatorname{Soc}(M/S_{\alpha-1}),$ and for a limit
ordinal $\alpha$, $S_{\alpha}=\cup_{\beta<\alpha}S_{\beta}.$ Put
$S=\cup\\{S_{\alpha}\\}$. Then, by construction $M/S$ has zero socle. $M$ is
semiartinian (i.e. every proper factor of $M$ has a simple module) if and only
if $S=M$ (see, for example, [extendingmodules]).
From the proof of Theorem 3.5, we see that the condition that, every free
right $R$-module is extending implies $R$ is a right $C$-ring. In the
following example we show that, if every finitely generated free right
$R$-module is extending, then $R$ need not be a right $C$-ring. Hence the
right $C$-ring condition in 3.6 is necessary.
###### Example 3.9.
Let $R$ be the ring of all linear transformations (written on the left) of an
infinite dimensional vector space over a division ring. Then $R$ is prime,
regular, right self-injective and $\operatorname{Soc}(R_{R})\neq 0$ by
[Goodearl:vonneumannregularrings, Theorem 9.12]. As $R$ is a prime ring,
$\operatorname{Soc}(R_{R})$ is an essential ideal of $R_{R}.$ Let $S$ be as in
Remark 3.8, for $M=R$. Then $S\neq R$, by [Clark-
Smith:OnSemiartinianandinjectivityconditions, Lemma 1(2)]. Since $R/S$ has
zero socle, $S$ is a neat submodule of $R_{R}$. On the other hand, $S$ is not
a closed submodule of $R$, otherwise $S$ would be a direct summand of $R$
because $R$ is right self injective (i.e. extending). Therefore $R$ is not a
right $C$-ring. Also, as $R$ is right self injective $R^{n}$ is injective, and
so extending for every $n\geq 1.$
## 4\. Rings whose simple Right modules are finitely presented
In this section, we consider neat-flat modules over the rings whose simple
right modules are finitely presented. The reason for considering these rings
is that, the Auslander-Bridger tranpose of simple right $R$-modules is a
finitely presented left $R$-module over such rings.
###### Definition 4.1.
Let $R$ be a ring and $n$ a nonnegative integer. A right $R$-module $M$ is
called $n$-presented if it has a finite $n$-presentation, i.e., there is an
exact sequence $F_{n}\to F_{n-1}\to\ldots F_{1}\to F_{0}\to M\to 0$ in which
every $F_{i}$, is a finitely generated free right $R$-module
[ding:onncoherentrings].
###### Lemma 4.2.
[ding:onncoherentrings, Lemma 2.7] Let $R$ and $S$ be rings, and $n$ a fixed
positive integer. Consider the situation $(_{R}A,_{R}B_{S},C_{S})$ with
${}_{R}A$ $n$-presented and $C_{S}$ injective. Then there is an isomorphism
$\operatorname{Tor}^{R}_{n-1}(\operatorname{Hom}_{{}_{S}}(B,C),A)\cong\operatorname{Hom}_{S}(\operatorname{Ext}^{n-1}_{R}(A,B),C)$
.
###### Proposition 4.3.
[Relativehomologicalalgebra, Proof of Proposition 5.3.9.] Every R-module $M$
is a pure submodule of a pure injective R-module $M^{++}$.
Let $M$ be a right $R$-module. $M$ is called absolutely pure (or FP-injective)
if $\operatorname{Ext}^{1}(N,M)=0$ for any finitely presented right $R$-module
$N$, i.e. $M$ is a pure submodule of its injective hull $E(M)$. For any right
$R$-module $M$, the character module $M^{+}$ is a pure injective right
$R$-module, (see, [Relativehomologicalalgebra, Proposition 5.3.7]).
###### Remark 4.4.
Note that, if every simple right $R$-module is finitely presented, then every
pure submodule is neat. So that, in this case, any right flat $R$-module is
neat-flat.
Using, Theorem 2.4, we obtain the following characterization of neat-flat
modules.
###### Theorem 4.5.
Let $R$ be a ring such that every simple right $R$-module is finitely
presented. Then $M$ is a neat-flat right $R$-module if and only if
$\operatorname{Tor}_{1}(M,\,Tr(S))=0$ for each simple right $R$-module $S$.
###### Proof.
Let $M$ be an $R$-module and $\mathbb{E}:0\to K\overset{f}{\to}F\to M\to 0$ be
a short exact sequence with $F$ projective. Let $S$ be simple right
$R$-module. Tensoring $\mathbb{E}$ by $Tr(S)$ we get the exact sequence
$0=\operatorname{Tor}_{1}(F,Tr(S))\to\operatorname{Tor}_{1}(M,Tr(S))\to
K\otimes Tr(S)\overset{f\otimes 1_{Tr(S)}}{\longrightarrow}F\otimes Tr(S).$
Now, suppose $M$ is neat-flat. Then $\mathbb{E}$ is neat-exact by Lemma 3.1.
So that $f\otimes 1_{Tr(S)}$ is monic, by Theorem 2.4. Hence
$\operatorname{Tor}_{1}(M,Tr(S))=0$.
Conversely, suppose $\operatorname{Tor}_{1}(M,Tr(S))=0$ for each simple right
$R$-module $S$. Then the sequence $0\to K\otimes Tr(S)\to F\otimes Tr(S)$ is
exact, and so the sequence $0\to K\to F\to M\to 0$ is neat-exact by Theorem
2.4. Then $M$ is neat-flat by Lemma 3.1.
∎
###### Corollary 4.6.
Let $R$ be a ring such that every simple right $R$-module is finitely
presented and $M$ be an arbitrary $R$-module. If $M$ is absolutely pure, then
$M^{+}$ is neat-flat.
###### Proof.
Let $S$ be a simple right $R$-module. By our assumption $S$ is finitely
presented, and so $Tr(S)$ is finitely presented $R$-module. Then
$\operatorname{Ext}^{1}(Tr(S),M)=0$, because $M$ is absolutely pure. We have,
$0=\operatorname{Ext}^{1}(Tr(S),M)^{+}\cong\operatorname{Tor}_{1}(M^{+},Tr(S))$
by Lemma 4.2. Hence $\operatorname{Tor}_{1}(M^{+},Tr(S))=0$, and so $M^{+}$ is
neat-flat by Theorem 4.5. ∎
###### Corollary 4.7.
Let $R$ be a ring such that every simple $R$-module is finitely presented and
$M$ be a right $R$-module. If $M$ is injective, then $M^{+}$ is neat-flat.
###### Lemma 4.8.
Let $R$ be a ring such that every simple $R$-module is finitely presented and
$M$ be a right $R$-module. Then $M$ is neat-flat if and only if $M^{++}$ is
neat-flat.
###### Proof.
Let $\mathcal{M}$ be the set of all representatives of simple right
$R$-modules. Suppose $M$ is a neat-flat $R$-module. Then there exists a neat-
exact sequence $\mathbb{E}:0\to K\to F\to M\to 0$ with $F$ projective by Lemma
3.1. By Theorem 2.4, $\mathbb{E}\in\tau^{-1}(Tr(\mathcal{M}))$. Then
$\mathbb{E}^{+}\in\pi^{-1}(Tr(\mathcal{M}))$ by Theorem 2.2, and so
$\mathbb{E}^{+}\in\tau^{-1}(\mathcal{M})$ by Theorem 2.4. Again by Theorem 2.4
and Theorem 2.2 we have $\mathbb{E}^{++}:0\to K^{++}\to F^{++}\to M^{++}\to
0\in\pi^{-1}(\mathcal{M})=\tau^{-1}(Tr(\mathcal{M})).$
Since $F$ is projective, $F^{+}$ is injective by [Rotman:HomologicalAlgebra,
Theorem 3.52]. Thus $F^{++}$ is neat-flat by Corollary 4.7. Then $M^{++}$ is
neat-flat, since $E^{++}$ is neat exact, and neat-flat modules closed under
neat quotient by Lemma 3.1.
Conversely, suppose $M^{++}$ is neat-flat. Since $M$ is a pure submodule of
$M^{++}$ by Proposition 4.3, $M$ is neat-flat by Theorem 3.2 and Proposition
3.4. ∎
###### Definition 4.9.
A right $R$-module $M$ is called _max-flat_ if
$\operatorname{Tor}_{R}^{1}(M,R/I)=0$ for every maximal left ideal $I$ of $R$
(see, [Xiang:maxflatmaxinjective]).
Note that a right $R$-module $M$ is max-flat if and only if $M^{+}$ is
$m$-injective by the standard isomorphism
$\operatorname{Ext}^{1}(S,M^{+})\cong\operatorname{Tor}_{1}(M,S)^{+}$ for all
simple left $R$-module $S$.
Using the similar arguments of [Xiang:maxflatmaxinjective, Theorem 4.5], we
can prove the following. The proof is omitted.
###### Theorem 4.10.
Let $R$ be a ring such that every simple right $R$-module is finitely
presented and $M$ be a right $R$-module. Then the followings are hold.
1. (1)
$M$ is an $m$-injective right $R$-module if and only if $M^{+}$ is max-flat.
2. (2)
$M$ is an $m$-injective right $R$-module if and only if $M^{++}$ is
$m$-injective.
3. (3)
$M$ is a max-flat right $R$-module if and only if $M^{++}$ is max-flat.
###### Proposition 4.11.
[septimi:minjective, Theorem 3]The following are equivalent for a right
$R$-module $M$:
1. (1)
$M$ is an $m$-injective $R$-module.
2. (2)
$\operatorname{Soc}(E(M)/M)=0$.
###### Proposition 4.12.
Assume that every neat-flat right $R$-module is flat. Then the following are
hold.
1. (1)
Every $m$-injective right $R$-module is absolutely pure.
2. (2)
For every right $R$-module $M$, $M$ is max-flat if and only if $M$ is flat.
###### Proof.
(1) Let $M$ be an $m$-injective right $R$-module. By Proposition 4.11,
$\operatorname{Soc}(E(M)/M)=0$, and so $E(M)/M$ is neat-flat. Then $E(M)/M$ is
flat by our hypothesis. Hence $M$ is a pure submodule of $E(M)$, and so $M$ is
an absolutely pure module.
(2) Assume $M$ is a max-flat right $R$-module. Then $M^{+}$ is $m$-injective,
and so it is absolutely pure by (1). But $M^{+}$ pure injective by
[Relativehomologicalalgebra, Proposition 5.3.7], so $M^{+}$ is injective. Then
$M$ is flat by [Rotman:HomologicalAlgebra, Theorem 3.52]. The converse
statement is clear.
∎
###### Theorem 4.13.
[Flatandprojectivecharactermodules, Theorem 1] The following statements are
equivalent:
1. (1)
$R$ is a right coherent ring.
2. (2)
$M_{R}$ is absolutely pure if and only if $M^{+}$ is a flat module.
3. (3)
$M_{R}$ is absolutely pure if and only if $M^{++}$ is an injective left
R-module.
4. (4)
${}_{R}M$ is flat if and only if $M^{++}$ is a flat left R-module.
###### Proposition 4.14.
Consider the following statements.
1. (1)
Every neat-flat right $R$-module is flat, and every simple right $R$-module is
finitely presented.
2. (2)
$M$ is an $m$-injective right $R$-module if and only if $M^{+}$ is a flat left
$R$-module.
3. (3)
$R$ is a right coherent ring, and $M$ is an $m$-injective right $R$-module if
and only if $M$ is an absolutely pure right $R$-module.
Then $(1)\Rightarrow(2)\Leftrightarrow(3).$
###### Proof.
$(1)\Rightarrow(3)$ By Proposition 4.12(1), every $m$-injective right
$R$-module is absolutely pure. On the other hand, every absolutely pure right
$R$-module is $m$-injective since every simple right $R$-module is finitely
presented by (1). Then, for every right $R$-module $M$, $M$ is absolutely pure
if and only if $M$ is $m$-injective, if and only if $M^{+}$ is max-flat by
Theorem 4.10(2), if and only if $M^{+}$ is a flat module by Proposition
4.12(2). Hence $R$ is a right coherent ring by
[Flatandprojectivecharactermodules, Theorem 1]. This proves (3).
$(2)\Rightarrow(3)$ Let $M$ be a left $R$-module. We claim that, $M$ is a flat
$R$-module if and only if $M^{++}$ is a flat module. If $M$ is flat, then
$M^{+}$ is injective by [Rotman:HomologicalAlgebra, Theorem 3.52], and so
$M^{++}$ is flat left $R$-module by (2). Conversely, if $M^{++}$ is a flat
module, then $M$ is flat since $M$ is a pure submodule of $M^{++}$ by
Proposition 4.3 and flat modules are closed under pure submodules (see,
[Lam:lecturesonmodulesandrings, Corollary 4.86]). So $R$ is a right coherent
ring by Theorem 4.13. The last part of (3) follows by (2) and Theorem 4.13
again.
$(3)\Rightarrow(2)$ By Theorem 4.13. ∎
###### Proposition 4.15.
Let $R$ be a ring such that every simple right $R$-module is finitely
presented. The following statements are equivalent:
1. (1)
$M$ is an absolutely pure left $R$-module if and only if
$\operatorname{Ext}_{R}^{1}(Tr(S),M)=0$ for each simple right $R$-module $S$.
2. (2)
$M$ is a flat right $R$-module if and only if $M$ is a neat-flat $R$-module.
###### Proof.
$(1)\Rightarrow(2)$ Let $M$ be a neat-flat right $R$-module. Then
$\operatorname{Tor}_{1}(M,\,Tr(S))=0$ for each simple right $R$-module $S$ by
Theorem 4.5. By the standard adjoint isomorphism we have,
$\operatorname{Ext}^{1}(Tr(S),M^{+})\cong\operatorname{Tor}_{1}(M,Tr(S))^{+}=0$.
Then $M^{+}$ is absolutely pure left $R$-module by (1). But $M^{+}$ pure
injective, so $M^{+}$ is injective. Then $M$ is flat by
[Rotman:HomologicalAlgebra, Theorem 3.52]. The converse is clear.
$(2)\Rightarrow(1)$ Let $M$ be a $R$-module such that
$\operatorname{Ext}^{1}(Tr(S),M)=0$ for each simple $R$-module $S$. Then, by
Lemma 4.2,
$0=\operatorname{Ext}^{1}(Tr(S),M)^{+}=\operatorname{Tor}_{1}(M^{+},Tr(S))$.
So, $M^{+}$ is neat-flat by Theorem 4.5, and it is flat by (2). But $R$ is
right coherent by Proposition 4.14, so $M$ is absolutely pure by Theorem 4.13.
The converse is clear.
∎
Acknowledgments
Some part of this paper was written while the second author was visiting Padua
University, Italy. He wishes to thank the members of the Department of
Mathematics for their kind hospitality and the Scientific and Technical
Research Council of Turkey (TÜBİTAK) for their financial support.
## References
|
arxiv-papers
| 2013-06-12T15:14:17 |
2024-09-04T02:49:46.419261
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Engin B\\\"uy\\\"uka\\c{s}{\\i}k and Y{\\i}lmaz Dur\\u{g}un",
"submitter": "Yilmaz Dur\\u{g}un",
"url": "https://arxiv.org/abs/1306.2860"
}
|
1306.2861
|
# Bayesian Inference and Learning in Gaussian Process
State-Space Models with Particle MCMC
###### Abstract
State-space models are successfully used in many areas of science, engineering
and economics to model time series and dynamical systems. We present a fully
Bayesian approach to inference _and learning_ (i.e. state estimation and
system identification) in nonlinear nonparametric state-space models. We place
a Gaussian process prior over the state transition dynamics, resulting in a
flexible model able to capture complex dynamical phenomena. To enable
efficient inference, we marginalize over the transition dynamics function and
infer directly the joint smoothing distribution using specially tailored
Particle Markov Chain Monte Carlo samplers. Once a sample from the smoothing
distribution is computed, the state transition predictive distribution can be
formulated analytically. Our approach preserves the full nonparametric
expressivity of the model and can make use of sparse Gaussian processes to
greatly reduce computational complexity.
Roger Frigola1, Fredrik Lindsten2, Thomas B. Schön2,3 and Carl E. Rasmussen1
1\. Dept. of Engineering, University of Cambridge, UK, {rf342,cer54}@cam.ac.uk
2\. Div. of Automatic Control, Linköping University, Sweden,
[email protected]
3\. Dept. of Information Technology, Uppsala University, Sweden,
[email protected]
## 1 Introduction
State-space models (SSMs) constitute a popular and general class of models in
the context of time series and dynamical systems. Their main feature is the
presence of a latent variable, the _state_
$\mathbf{x}_{t}\in\mathsf{X}\triangleq\mathbb{R}^{n_{x}}$, which condenses all
aspects of the system that can have an impact on its future. A discrete-time
SSM with nonlinear dynamics can be represented as
$\displaystyle\mathbf{x}_{t+1}$
$\displaystyle=f(\mathbf{x}_{t},\mathbf{u}_{t})+\mathbf{v}_{t},$ (1a)
$\displaystyle\mathbf{y}_{t}$
$\displaystyle=g(\mathbf{x}_{t},\mathbf{u}_{t})+\mathbf{e}_{t},$ (1b)
where $\mathbf{u}_{t}$ denotes a known external input, $\mathbf{y}_{t}$
denotes the measurements, $\mathbf{v}_{t}$ and $\mathbf{e}_{t}$ denote i.i.d.
noises acting on the dynamics and the measurements, respectively. The function
$f$ encodes the dynamics and $g$ describes the relationship between the
observation and the unobserved states.
We are primarily concerned with the problem of learning general nonlinear
SSMs. The aim is to find a model that can adaptively increase its complexity
when more data is available. To this effect, we employ a Bayesian
nonparametric model for the dynamics (1a). This provides a flexible model that
is not constrained by any limiting assumptions caused by postulating a
particular functional form. More specifically, we place a Gaussian process
(GP) prior [1] over the unknown function $f$. The resulting model is a
generalization of the standard parametric SSM. The functional form of the
observation model $g$ is assumed to be known, possibly parameterized by a
finite dimensional parameter. This is often a natural assumption, for instance
in engineering applications where $g$ corresponds to a sensor model – we
typically know what the sensors are measuring, at least up to some unknown
parameters. Furthermore, using too flexible models for both $f$ and $g$ can
result in problems of non-identifiability.
We adopt a fully Bayesian approach whereby we find a posterior distribution
over all the latent entities of interest, namely the state transition function
$f$, the hidden state trajectory
$\mathbf{x}_{0:T}\triangleq\\{\mathbf{x}_{i}\\}_{i=0}^{T}$ and any hyper-
parameter $\boldsymbol{\theta}$ of the model. This is in contrast with
existing approaches for using GPs to model SSMs, which tend to model the GP
using a finite set of target points, in effect making the model parametric
[2]. Inferring the distribution over the state trajectory
$p(\mathbf{x}_{0:T}\mid\mathbf{y}_{0:T},\mathbf{u}_{0:T})$ is an important
problem in itself known as _smoothing_. We use a tailored particle Markov
Chain Monte Carlo (PMCMC) algorithm [3] to efficiently sample from the
smoothing distribution whilst marginalizing over the state transition
function. This contrasts with conventional approaches to smoothing which
require a fixed model of the transition dynamics. Once we have obtained an
approximation of the smoothing distribution, with the dynamics of the model
marginalized out, learning the function $f$ is straightforward since its
posterior is available in closed form given the state trajectory. Our only
approximation is that of the sampling algorithm. We report very good mixing
enabled by the use of recently developed PMCMC samplers [4] and the exact
marginalization of the transition dynamics.
There is by now a rich literature on GP-based SSMs. For instance, Deisenroth
et al. [5, 6] presented refined approximation methods for filtering and
smoothing for already learned GP dynamics and measurement functions. In fact,
the method proposed in the present paper provides a vital component needed for
these inference methods, namely that of learning the GP model in the first
place. Turner et al. [2] applied the EM algorithm to obtain a maximum
likelihood estimate of parametric models which had the form of GPs where both
inputs and outputs were parameters to be optimized. This type of approach can
be traced back to [7] where Ghahramani and Roweis applied EM to learn models
based on radial basis functions. Wang et al. [8] learn a SSM with GPs by
finding a MAP estimate of the latent variables and hyper-parameters. They
apply the learning in cases where the dimension of the observation vector is
much higher than that of the latent state in what becomes a form of dynamic
dimensionality reduction. This procedure would have the risk of overfitting in
the common situation where the state-space is high-dimensional and there is
significant uncertainty in the smoothing distribution.
## 2 Gaussian Process State-Space Model
(a) Standard GP regression
(b) GP-SSM
Figure 1: Graphical models for standard GP regression and the GP-SSM model.
The thick horizontal bars represent sets of fully connected nodes.
We describe the generative probabilistic model of the Gaussian process SSM
(GP-SSM) represented in Figure 1 by
$\displaystyle f(\mathbf{x}_{t})$
$\displaystyle\sim\mathcal{GP}\big{(}m_{\boldsymbol{\theta}_{\mathbf{x}}}(\mathbf{x}_{t}),k_{\boldsymbol{\theta}_{\mathbf{x}}}(\mathbf{x}_{t},\mathbf{x}_{t}^{\prime})\big{)},$
(2a) $\displaystyle\mathbf{x}_{t+1}\mid\mathbf{f}_{t}$
$\displaystyle\sim\mathcal{N}(\mathbf{x}_{t+1}\mid\mathbf{f}_{t},\mathbf{Q}),$
(2b) $\displaystyle\mathbf{y}_{t}\mid\mathbf{x}_{t}$ $\displaystyle\sim
p(\mathbf{y}_{t}\mid\mathbf{x}_{t},\boldsymbol{\theta}_{\mathbf{y}}),$ (2c)
and $\mathbf{x}_{0}\sim p(\mathbf{x}_{0})$, where we avoid notational clutter
by omitting the conditioning on the known inputs $\mathbf{u}_{t}$. In
addition, we put a prior $p(\boldsymbol{\theta})$ over the various hyper-
parameters
$\boldsymbol{\theta}=\\{\boldsymbol{\theta}_{\mathbf{x}},\boldsymbol{\theta}_{\mathbf{y}},\mathbf{Q}\\}$.
Also, note that the measurement model (2c) and the prior on $\mathbf{x}_{0}$
can take any form since we do not rely on their properties for efficient
inference.
The GP is fully described by its mean function and its covariance function. An
interesting property of the GP-SSM is that any _a priori_ insight into the
dynamics of the system can be readily encoded in the mean function. This is
useful, since it is often possible to capture the main properties of the
dynamics, e.g. by using a simple parametric model or a model based on first
principles. Such simple models may be insufficient on their own, but useful
together with the GP-SSM, as the GP is flexible enough to model complex
departures from the mean function. If no specific prior model is available,
the linear mean function $m({\mathbf{x}}_{t})=\mathbf{x}_{t}$ is a good
generic choice. Interestingly, the prior information encoded in this model
will normally be more vague than the prior information encoded in parametric
models. The measurement model (2c) implicitly contains the observation
function $g$ and the distribution of the i.i.d. measurement noise
$\mathbf{e}_{t}$.
## 3 Inference over States and Hyper-parameters
Direct learning of the function $f$ in (2a) from input/output data
$\\{\mathbf{u}_{0:T-1},\mathbf{y}_{0:T}\\}$ is challenging since the states
$\mathbf{x}_{0:T}$ are not observed. Most (if not all) previous approaches
attack this problem by reverting to a parametric representation of $f$ which
is learned alongside the states. We address this problem in a fundamentally
different way by marginalizing out $f$, allowing us to respect the
nonparametric nature of the model. A challenge with this approach is that
marginalization of $f$ will introduce dependencies across time for the state
variables that lead to the loss of the Markovian structure of the state-
process. However, recently developed inference methods, combining sequential
Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) allow us to tackle this
problem. We discuss marginalization of $f$ in Section 3.1 and present the
inference algorithms in Sections 3.2 and 3.3.
### 3.1 Marginalizing out the State Transition Function
Targeting the joint posterior distribution of the hyper-parameters, the latent
states _and_ the latent function $f$ is problematic due to the strong
dependencies between $\mathbf{x}_{0:T}$ and $f$. We therefore marginalize the
dynamical function from the model, and instead target the distribution
$p(\boldsymbol{\theta},\mathbf{x}_{0:T}\mid\mathbf{y}_{1:T})$ (recall that
conditioning on $\mathbf{u}_{0:T-1}$ is implicit). In the MCMC literature,
this is referred to as collapsing [9]. Hence, we first need to find an
expression for the marginal prior
$p(\boldsymbol{\theta},\mathbf{x}_{0:T})=p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta})p(\boldsymbol{\theta})$.
Focusing on $p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta})$ we note that,
although this distribution is not Gaussian, it can be represented as a product
of Gaussians. Omitting the dependence on $\boldsymbol{\theta}$ in the
notation, we obtain
$p(\mathbf{x}_{1:T}\mid\boldsymbol{\theta},\mathbf{x}_{0})=\
\prod_{t=1}^{T}p(\mathbf{x}_{t}\mid\boldsymbol{\theta},\mathbf{x}_{0:t-1})=\prod_{t=1}^{T}\mathcal{N}\big{(}\mathbf{x}_{t}\mid\boldsymbol{\mu}_{t}(\mathbf{x}_{0:t-1}),\mathbf{\Sigma}_{t}(\mathbf{x}_{0:t-1})\big{)},$
(3a) with $\displaystyle\boldsymbol{\mu}_{t}(\mathbf{x}_{0:t-1})$
$\displaystyle=\mathbf{m}_{t-1}+\mathbf{K}_{t-1,0:t-2}\widetilde{\mathbf{K}}^{-1}_{0:t-2}\
(\mathbf{x}_{1:t-1}-\mathbf{m}_{0:t-2}),$ (3b)
$\displaystyle\mathbf{\Sigma}_{t}(\mathbf{x}_{0:t-1})$
$\displaystyle=\widetilde{\mathbf{K}}_{t-1}-\mathbf{K}_{t-1,0:t-2}\widetilde{\mathbf{K}}^{-1}_{0:t-2}\mathbf{K}_{t-1,0:t-2}^{\top}$
(3c)
for $t\geq 2$ and $\boldsymbol{\mu}_{1}(\mathbf{x}_{0})=\mathbf{m}_{0}$,
$\mathbf{\Sigma}_{1}(\mathbf{x}_{0})=\widetilde{\mathbf{K}}_{0}$. Equation (3)
follows from the fact that, once conditioned on $\mathbf{x}_{0:t-1}$, a one-
step prediction for the state variable is a standard GP prediction. Here, we
have defined the mean vector
$\mathbf{m}_{0:t-1}\triangleq\begin{bmatrix}m({\mathbf{x}}_{0})^{\top}&\dots&m({\mathbf{x}}_{t-1})^{\top}\end{bmatrix}^{\top}$
and the $(n_{x}t)\times(n_{x}t)$ positive definite matrix $\mathbf{K}_{0:t-1}$
with block entries
$[\mathbf{K}_{0:t-1}]_{i,j}=k({\mathbf{x}}_{i-1},{\mathbf{x}}_{j-1})$. We use
two sets of indices, as in $\mathbf{K}_{t-1,0:t-2}$, to refer to the off-
diagonal blocks of $\mathbf{K}_{0:t-1}$. We also define
$\widetilde{\mathbf{K}}_{0:t-1}=\mathbf{K}_{0:t-1}+\mathbf{I}_{t}\otimes\mathbf{Q}$.
We can also express (3a) more succinctly as,
$p(\mathbf{x}_{1:t}\mid\boldsymbol{\theta},\mathbf{x}_{0})=|(2\pi)^{n_{x}t}\widetilde{\mathbf{K}}_{0:t-1}|^{-\frac{1}{2}}\exp(-\frac{1}{2}(\mathbf{x}_{1:t}-\mathbf{m}_{0:t-1})^{\top}\widetilde{\mathbf{K}}_{0:t-1}^{-1}(\mathbf{x}_{1:t}-\mathbf{m}_{0:t-1})).$
(4)
This expression looks very much like a multivariate Gaussian density function.
However, we emphasize that this is not the case since both
$\mathbf{m}_{0:t-1}$ and $\widetilde{\mathbf{K}}_{0:t-1}$ depend (nonlinearly)
on the argument $\mathbf{x}_{1:t}$. In fact, (4) will typically be very far
from Gaussian.
### 3.2 Sequential Monte Carlo
With the prior (4) in place, we now turn to posterior inference and we start
by considering the joint smoothing distribution
$p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta},\mathbf{y}_{0:T})$. The sequential
nature of the proposed model suggests the use of SMC. Though most well known
for filtering in Markovian SSMs – see [10, 11] for an introduction – SMC is
applicable also for non-Markovian latent variable models. We seek to
approximate the sequence of distributions
$p(\mathbf{x}_{0:t}\mid\boldsymbol{\theta},\mathbf{y}_{0:t})$, for
$t=0,\,\dots,\,T$. Let
$\\{\mathbf{x}_{0:t-1}^{i},\mathbf{w}_{t-1}^{i}\\}_{i=1}^{N}$ be a collection
of weighted particles approximating
$p(\mathbf{x}_{0:t-1}\mid\boldsymbol{\theta},\mathbf{y}_{0:t-1})$ by the
empirical distribution,
$\widehat{p}(\mathbf{x}_{0:t-1}\mid\boldsymbol{\theta},\mathbf{y}_{0:t-1})\triangleq\sum_{i=1}^{N}\mathbf{w}_{t-1}^{i}\delta_{\mathbf{x}_{0:t-1}^{i}}(\mathbf{x}_{0:t-1})$.
Here, $\delta_{\mathbf{z}}(\mathbf{x})$ is a point-mass located at
$\mathbf{z}$. To propagate this sample to time $t$, we introduce the auxiliary
variables $\\{\mathbf{a}_{t}^{i}\\}_{i=1}^{N}$, referred to as _ancestor
indices_. The variable $\mathbf{a}_{t}^{i}$ is the index of the ancestor
particle at time $t-1$, of particle $\mathbf{x}_{t}^{i}$. Hence,
$\mathbf{x}_{t}^{i}$ is generated by first sampling $\mathbf{a}_{t}^{i}$ with
$\mathbb{P}(\mathbf{a}_{t}^{i}=j)=\mathbf{w}_{t-1}^{j}$. Then,
$\mathbf{x}_{t}^{i}$ is generated as,
$\displaystyle\mathbf{x}_{t}^{i}\sim
p(\mathbf{x}_{t}\mid\boldsymbol{\theta},\mathbf{x}_{0:t-1}^{\mathbf{a}_{t}^{i}},\mathbf{y}_{0:t}),$
(5)
for $i=1,\,\dots,\,N$. The particle trajectories are then augmented according
to
$\mathbf{x}_{0:t}^{i}=\\{\mathbf{x}_{0:t-1}^{\mathbf{a}_{t}^{i}},\mathbf{x}_{t}^{i}\\}$.
Sampling from the one-step predictive density is a simple (and sensible)
choice, but we may also consider other proposal distributions. In the above
formulation the resampling step is implicit and corresponds to sampling the
ancestor indices (cf. the auxiliary particle filter, [12]). Finally, the
particles are weighted according to the measurement model,
$\mathbf{w}_{t}^{i}\propto
p(\mathbf{y}_{t}\mid\boldsymbol{\theta},\mathbf{x}_{t}^{i})$ for
$i=1,\,\dots,\,N$, where the weights are normalized to sum to 1.
### 3.3 Particle Markov Chain Monte Carlo
There are two shortcomings of SMC: (i) it does not handle inference over
hyper-parameters; (ii) despite the fact that the sampler targets the joint
smoothing distribution, it does in general not provide an accurate
approximation of the full joint distribution due to _path degeneracy_. That
is, the successive resampling steps cause the particle diversity to be very
low for time points $t$ far from the final time instant $T$.
To address these issues, we propose to use a particle Markov chain Monte Carlo
(PMCMC, [3, 13]) sampler. PMCMC relies on SMC to generate samples of the
highly correlated state trajectory within an MCMC sampler. We employ a
specific PMCMC sampler referred to as particle Gibbs with ancestor sampling
(PGAS, [4]), given in Algorithm 1. PGAS uses Gibbs-like steps for the state
trajectory $\mathbf{x}_{0:T}$ and the hyper-parameters $\boldsymbol{\theta}$,
respectively. That is, we sample first $\mathbf{x}_{0:T}$ given
$\boldsymbol{\theta}$, then $\boldsymbol{\theta}$ given $\mathbf{x}_{0:T}$,
etc. However, the full conditionals are not explicitly available. Instead, we
draw samples from specially tailored Markov kernels, leaving these
conditionals invariant. We address these steps in the subsequent sections.
Algorithm 1 Particle Gibbs with ancestor sampling (PGAS)
1. 1.
Set $\boldsymbol{\theta}[0]$ and ${\mathbf{x}}_{1:T}[0]$ arbitrarily.
2. 2.
For $\ell\geq 1$ do
1. (a)
Draw $\boldsymbol{\theta}[\ell]$ conditionally on ${\mathbf{x}}_{0:T}[\ell-1]$
and $\mathbf{y}_{0:T}$ as discussed in Section 3.3.2.
2. (b)
Run CPF-AS (see [4]) targeting
$p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta}[\ell],\mathbf{y}_{0:T})$,
conditionally on ${\mathbf{x}}_{0:T}[\ell-1]$.
3. (c)
Sample $k$ with $\mathbb{P}(k=i)=w_{T}^{i}$ and set
${\mathbf{x}}_{1:T}[\ell]={\mathbf{x}}_{1:T}^{k}$.
3. 3.
end
#### 3.3.1 Sampling the State Trajectories
To sample the state trajectory, PGAS makes use of an SMC-like procedure
referred to as a conditional particle filter with ancestor sampling (CPF-AS).
This approach is particularly suitable for non-Markovian latent variable
models, as it relies only on a forward recursion (see [4]). The difference
between a standard particle filter (PF) and the CPF-AS is that, for the
latter, one particle at each time step is specified _a priori_. Let these
particles be denoted
$\widetilde{\mathbf{x}}_{0:T}=\\{\widetilde{\mathbf{x}}_{0},\,\dots,\,\widetilde{\mathbf{x}}_{T}\\}$.
We then sample according to (5) only for $i=1,\,\dots,\,N-1$. The $N$th
particle is set deterministically:
$\mathbf{x}_{t}^{N}=\widetilde{\mathbf{x}}_{t}$. To be able to construct the
$N$th particle trajectory, $\mathbf{x}_{t}^{N}$ has to be associated with an
ancestor particle at time $t-1$. This is done by sampling a value for the
corresponding ancestor index $\mathbf{a}_{t}^{N}$. Following [4], the ancestor
sampling probabilities are computed as
$\displaystyle\widetilde{\mathbf{w}}_{t-1\mid
T}^{i}\propto\mathbf{w}_{t-1}^{i}\frac{p(\\{\mathbf{x}_{0:t-1}^{i},\widetilde{\mathbf{x}}_{t:T}\\},\mathbf{y}_{0:T})}{p(\mathbf{x}_{0:t-1}^{i},\mathbf{y}_{0:t-1})}\propto\mathbf{w}_{t-1}^{i}\frac{p(\\{\mathbf{x}_{0:t-1}^{i},\widetilde{\mathbf{x}}_{t:T}\\})}{p(\mathbf{x}_{0:t-1}^{i})}=\mathbf{w}_{t-1}^{i}p(\widetilde{\mathbf{x}}_{t:T}\mid\mathbf{x}_{0:t-1}^{i}).$
(6)
where the ratio is between the unnormalized target densities up to time $T$
and up to time $t-1$, respectively. The second proportionality follows from
the mutual conditional independence of the observations, given the states.
Here, $\\{\mathbf{x}_{0:t-1}^{i},\widetilde{\mathbf{x}}_{t:T}\\}$ refers to a
path in $\mathsf{X}^{T+1}$ formed by concatenating the two partial
trajectories. The above expression can be computed by using the prior over
state trajectories given by (4). The ancestor sampling weights
$\\{\widetilde{\mathbf{w}}_{t-1\mid T}^{i}\\}_{i=1}^{N}$ are then normalized
to sum to 1 and the ancestor index $\mathbf{a}_{t}^{N}$ is sampled with
$\mathbb{P}(\mathbf{a}_{t}^{N}=j)=\mathbf{w}_{t-1\mid t}^{j}$.
The conditioning on a prespecified collection of particles implies an
invariance property in CPF-AS, which is key to our development. More
precisely, given $\widetilde{\mathbf{x}}_{0:T}$ let
$\widetilde{\mathbf{x}}_{0:T}^{\prime}$ be generated as follows:
1. 1.
Run CPF-AS from time $t=0$ to time $t=T$, conditionally on
$\widetilde{\mathbf{x}}_{0:T}$.
2. 2.
Set $\widetilde{\mathbf{x}}_{0:T}^{\prime}$ to one of the resulting particle
trajectories according to
$\mathbb{P}(\widetilde{\mathbf{x}}_{0:T}^{\prime}=\mathbf{x}_{0:T}^{i})=\mathbf{w}_{T}^{i}$.
For any $N\geq 2$, this procedure defines an ergodic Markov kernel
$M_{\boldsymbol{\theta}}^{N}(\widetilde{\mathbf{x}}_{0:T}^{\prime}\mid\widetilde{\mathbf{x}}_{0:T})$
on $\mathsf{X}^{T+1}$, leaving the _exact_ smoothing distribution
$p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta},\mathbf{y}_{0:T})$ invariant [4].
Note that this invariance holds for any $N\geq 2$, i.e. the number of
particles that are used only affect the mixing rate of the kernel
$M_{\boldsymbol{\theta}}^{N}$. However, it has been experienced in practice
that the autocorrelation drops sharply as $N$ increases [4, 14], and for many
models a moderate $N$ is enough to obtain a rapidly mixing kernel.
#### 3.3.2 Sampling the Hyper-parameters
Next, we consider sampling the hyper-parameters given a state trajectory and
sequence of observations, i.e. from
$p(\boldsymbol{\theta}\mid{\mathbf{x}}_{0:T},\mathbf{y}_{0:T})$. In the
following, we consider the common situation where there are distinct hyper-
parameters for the likelihood
$p(\mathbf{y}_{0:T}\mid\mathbf{x}_{0:T},\boldsymbol{\theta}_{\mathbf{y}})$ and
for the prior over trajectories
$p(\mathbf{x}_{0:T}\mid\boldsymbol{\theta}_{\mathbf{x}})$. If the prior over
the hyper-parameters factorizes between those two groups we obtain
$p(\boldsymbol{\theta}\mid\mathbf{x}_{0:T},\mathbf{y}_{0:T})\propto
p(\boldsymbol{\theta}_{\mathbf{y}}\mid\mathbf{x}_{0:T},\mathbf{y}_{0:T})\
p(\boldsymbol{\theta}_{\mathbf{x}}\mid\mathbf{x}_{0:T})$. We can thus proceed
to sample the two groups of hyper-parameters independently. Sampling
$\boldsymbol{\theta}_{\mathbf{y}}$ will be straightforward in most cases, in
particular if conjugate priors for the likelihood are used. Sampling
$\boldsymbol{\theta}_{\mathbf{x}}$ will, nevertheless, be harder since the
covariance function hyper-parameters enter the expression in a non-trivial
way. However, we note that once the state trajectory is fixed, we are left
with a problem analogous to Gaussian process regression where
$\mathbf{x}_{0:T-1}$ are the inputs, $\mathbf{x}_{1:T}$ are the outputs and
$\mathbf{Q}$ is the likelihood covariance matrix. Given that the latent
dynamics can be marginalized out analytically, sampling the hyper-parameters
with slice sampling is straightforward [15].
## 4 A Sparse GP-SSM Construction and Implementation Details
A naive implementation of the CPF-AS algorithm will give rise to
$\mathcal{O}(T^{4})$ computational complexity, since at each time step
$t=1,\,\dots,\,T$, a matrix of size $T\times T$ needs to be factorized.
However, it is possible to update and reuse the factors from the previous time
step, bringing the total computational complexity down to the familiar
$\mathcal{O}(T^{3})$. Furthermore, by introducing a sparse GP model, we can
reduce the complexity to $\mathcal{O}(M^{2}T)$ where $M\ll T$. In Section 4.1
we introduce the sparse GP model and in Section 4.2 we provide insight into
the efficient implementation of both the vanilla GP and the sparse GP.
### 4.1 FIC Prior over the State Trajectory
An important alternative to GP-SSM is given by exchanging the vanilla GP prior
over $f$ for a sparse counterpart. We do not consider the resulting model to
be an approximation to GP-SSM, it is still a GP-SSM, but _with a different
prior over functions_. As a result we expect it to sometimes outperform its
non-sparse version in the same way as it happens with their regression
siblings [16].
Most sparse GP methods can be formulated in terms of a set of so called
inducing variables [17]. These variables live in the space of the latent
function and have a set $\mathcal{I}$ of corresponding inducing inputs. The
assumption is that, conditionally on the inducing variables, the latent
function values are mutually independent. Although the inducing variables are
marginalized analytically – this is key for the model to remain nonparametric
– the inducing inputs have to be chosen in such a way that they, informally
speaking, cover the same region of the input space covered by the data.
Crucially, in order to achieve computational gains, the number $M$ of inducing
variables is selected to be smaller than the original number of data points.
In the following, we will use the fully independent conditional (FIC) sparse
GP prior as defined in [17] due to its very good empirical performance [16].
As shown in [17], the FIC prior can be obtained by replacing the covariance
function $k(\cdot,\cdot)$ by,
$k^{\rm
FIC}(\mathbf{x}_{i},\mathbf{x}_{j})=s(\mathbf{x}_{i},\mathbf{x}_{j})+\delta_{ij}\big{(}k(\mathbf{x}_{i},\mathbf{x}_{j})-s(\mathbf{x}_{i},\mathbf{x}_{j})\big{)},$
(7)
where $s(\mathbf{x}_{i},\mathbf{x}_{j})\triangleq
k(\mathbf{x}_{i},\mathcal{I})k(\mathcal{I},\mathcal{I})^{-1}k(\mathcal{I},\mathbf{x}_{j})$,
$\delta_{ij}$ is Kronecker’s delta and we use the convention whereby when $k$
takes a set as one of its arguments it generates a matrix of covariances.
Using the Woodbury matrix identity, we can express the one-step predictive
density as in (3), with
$\displaystyle\boldsymbol{\mu}_{t}^{\rm FIC}(\mathbf{x}_{0:t-1})$
$\displaystyle=\mathbf{m}_{t-1}+\mathbf{K}_{t-1,\mathcal{I}}\mathbf{P}\mathbf{K}_{\mathcal{I},0:t-2}\boldsymbol{\Lambda}^{-1}_{0:t-2}\
(\mathbf{x}_{1:t-1}-\mathbf{m}_{0:t-2}),$ (8a)
$\displaystyle\mathbf{\Sigma}^{\rm FIC}_{t}(\mathbf{x}_{0:t-1})$
$\displaystyle=\widetilde{\mathbf{K}}_{t-1}-\mathbf{S}_{t-1}+\mathbf{K}_{t-1,\mathcal{I}}\mathbf{P}\mathbf{K}_{\mathcal{I},t-1},$
(8b)
where
$\mathbf{P}\triangleq(\mathbf{K}_{\mathcal{I},\mathcal{I}}+\mathbf{K}_{\mathcal{I},0:t-2}\boldsymbol{\Lambda}^{-1}_{0:t-2}\mathbf{K}_{0:t-2,\mathcal{I}})^{-1}$,
$\boldsymbol{\Lambda}_{0:t-2}\triangleq{\rm
diag}[\widetilde{\mathbf{K}}_{0:t-2}-\mathbf{S}_{0:t-2}]$ and
$\mathbf{S}_{\mathcal{A},\mathcal{B}}\triangleq\mathbf{K}_{\mathcal{A},\mathcal{I}}\mathbf{K}_{\mathcal{I},\mathcal{I}}^{-1}\mathbf{K}_{\mathcal{I},\mathcal{B}}$.
Despite its apparent cumbersomeness, the computational complexity involved in
computing the above mean and covariance is $\mathcal{O}(M^{2}t)$, as opposed
to $\mathcal{O}(t^{3})$ for (3). The same idea can be used to express (4) in a
form which allows for efficient computation. Here $\operatorname*{diag}$
refers to a block diagonalization if $\mathbf{Q}$ is not diagonal.
We do not address the problem of choosing the inducing inputs, but note that
one option is to use greedy methods (e.g. [18]). The fast forward selection
algorithm is appealing due to its very low computational complexity [18].
Moreover, its potential drawback of interference between hyper-parameter
learning and active set selection is not an issue in our case since hyper-
parameters will be fixed for a given run of the particle filter.
### 4.2 Implementation Details
As pointed out above, it is crucial to reuse computations across time to
attain the $\mathcal{O}(T^{3})$ or $\mathcal{O}(M^{2}T)$ computational
complexity for the vanilla GP and the FIC prior, respectively. We start by
discussing the vanilla GP and then briefly comment on the implementation
aspects of FIC.
There are two costly operations of the CPF-AS algorithm: (i) sampling from the
prior (5), requiring the computation of (3b) and (3c) and (ii) evaluating the
ancestor sampling probabilities according to (6). Both of these operations can
be carried out efficiently by keeping track of a Cholesky factorization of the
matrix
$\widetilde{\mathbf{K}}(\\{\mathbf{x}_{0:t-1}^{i},\widetilde{\mathbf{x}}_{t:T-1}\\})=\mathbf{L}_{t}^{i}\mathbf{L}_{t}^{i\top}$,
for each particle $i=1,\,\dots,\,N$. Here,
$\widetilde{\mathbf{K}}(\\{\mathbf{x}_{0:t-1}^{i},\widetilde{\mathbf{x}}_{t:T-1}\\})$
is a matrix defined analogously to $\widetilde{\mathbf{K}}_{0:T-1}$, but where
the covariance function is evaluated for the concatenated state trajectory
$\\{\mathbf{x}_{0:t-1}^{i},\widetilde{\mathbf{x}}_{t:T-1}\\}$. From
$\mathbf{L}_{t}^{i}$, it is possible to identify sub-matrices corresponding to
the Cholesky factors for the covariance matrix
$\mathbf{\Sigma}_{t}(\mathbf{x}_{0:t-1}^{i})$ as well as for the matrices
needed to efficiently evaluate the ancestor sampling probabilities (6).
It remains to find an efficient update of the Cholesky factor to obtain
$\mathbf{L}_{t+1}^{i}$. As we move from time $t$ to $t+1$ in the algorithm,
$\widetilde{\mathbf{x}}_{t}$ will be replaced by $\mathbf{x}_{t}^{i}$ in the
concatenated trajectory. Hence, the matrix
$\widetilde{\mathbf{K}}(\\{\mathbf{x}_{0:t}^{i},\widetilde{\mathbf{x}}_{t+1:T-1}\\})$
can be obtained from
$\widetilde{\mathbf{K}}(\\{\mathbf{x}_{0:t-1}^{i},\widetilde{\mathbf{x}}_{t:T-1}\\})$
by replacing $n_{x}$ rows and columns, corresponding to a rank $2n_{x}$
update. It follows that we can compute $\mathbf{L}_{t+1}^{i}$ by making
$n_{x}$ successive rank one updates and downdates on $\mathbf{L}_{t}^{i}$. In
summary, all the operations at a specific time step can be done in
$\mathcal{O}(T^{2})$ computations, leading to a total computational complexity
of $\mathcal{O}(T^{3})$.
For the FIC prior, a naive implementation will give rise to
$\mathcal{O}(M^{2}T^{2})$ computational complexity. This can be reduced to
$\mathcal{O}(M^{2}T)$ by keeping track of a factorization for the matrix
$\mathbf{P}$. However, to reach the $\mathcal{O}(M^{2}T)$ cost all
intermediate operations scaling with $T$ has to be avoided, requiring us to
reuse not only the matrix factorizations, but also intermediate matrix-vector
multiplications.
## 5 Learning the Dynamics
Algorithm 1 gives us a tool to compute
$p(\mathbf{x}_{0:T},\boldsymbol{\theta}\mid\mathbf{y}_{1:T})$. We now discuss
how this can be used to find an explicit model for $f$. The goal of learning
the state transition dynamics is equivalent to that of obtaining a predictive
distribution over $\mathbf{f}^{*}=f(\mathbf{x}^{*})$, evaluated at an
arbitrary test point $\mathbf{x}^{*}$,
$p(\mathbf{f}^{*}\mid\mathbf{x}^{*},\mathbf{y}_{1:T})=\int
p(\mathbf{f}^{*}\mid\mathbf{x}^{*},\mathbf{x}_{0:T},\boldsymbol{\theta})\
p(\mathbf{x}_{0:T},\boldsymbol{\theta}\mid\mathbf{y}_{1:T})\
\text{d}\mathbf{x}_{0:T}\,\text{d}\mathbf{\boldsymbol{\theta}}.$ (9)
Using a sample-based approximation of
$p(\mathbf{x}_{0:T},\boldsymbol{\theta}\mid\mathbf{y}_{1:T})$, this integral
can be approximated by
$p(\mathbf{f}^{*}\mid\mathbf{x}^{*},\mathbf{y}_{1:T})\approx\frac{1}{L}\sum_{\ell=1}^{L}p(\mathbf{f}^{*}\mid\mathbf{x}^{*},\mathbf{x}_{0:T}[\ell],\boldsymbol{\theta}[\ell])=\frac{1}{L}\sum_{\ell=1}^{L}\mathcal{N}(\mathbf{f}^{*}\mid\boldsymbol{\mu}^{\ell}(\mathbf{x}^{*}),\mathbf{\Sigma}^{\ell}(\mathbf{x}^{*})),$
(10)
where $L$ is the number of samples and
$\boldsymbol{\mu}^{\ell}(\mathbf{x}^{*})$ and
$\mathbf{\Sigma}^{\ell}(\mathbf{x}^{*})$ follow the expressions for the
predictive distribution in standard GP regression if
$\mathbf{x}_{0:T-1}[\ell]$ are treated as inputs, $\mathbf{x}_{1:T}[\ell]$ are
treated as outputs and $\mathbf{Q}$ is the likelihood covariance matrix. This
mixture of Gaussians is an expressive representation of the predictive density
which can, for instance, correctly take into account multimodality arising
from ambiguity in the measurements. Although factorized covariance matrices
can be pre-computed, the overall computational cost will increase linearly
with $L$.The computational cost can be reduced by thinning the Markov chain
using e.g. random sub-sampling or kernel herding [19].
In some situations it could be useful to obtain an approximation from the
mixture of Gaussians consisting in a single GP representation. This is the
case in applications such as control or real time filtering where the cost of
evaluating the mixture of Gaussians can be prohibitive. In those cases one
could opt for a pragmatic approach and learn the mapping
$\mathbf{x}^{*}\mapsto\mathbf{f}^{*}$ from a cloud of points
$\\{\mathbf{x}_{0:T}[\ell],\mathbf{f}_{0:T}[\ell]\\}_{\ell=1}^{L}$ using
sparse GP regression. The latent function values $\mathbf{f}_{0:T}[\ell]$ can
be easily sampled from the normally distributed
$p(\mathbf{f}_{0:T}[\ell]\mid\mathbf{x}_{0:T}[\ell],\boldsymbol{\theta}[\ell])$.
## 6 Experiments
### 6.1 Learning a Nonlinear System Benchmark
Consider a system with dynamics given by
$x_{t+1}=ax_{t}+bx_{t}/(1+x_{t}^{2})+cu_{t}+v_{t},v_{t}\sim\mathcal{N}(0,q)$
and observations given by $y_{t}=dx_{t}^{2}+e_{t},e_{t}\sim\mathcal{N}(0,r)$,
with parameters $(a,b,c,d,q,r)=(0.5,25,8,0.05,10,1)$ and a known input
$u_{t}=\cos(1.2(t+1))$. One of the difficulties of this system is that the
smoothing density $p(\mathbf{x}_{0:T}\mid\mathbf{y}_{0:T})$ is multimodal
since no information about the sign of $x_{t}$ is available in the
observations. The system is simulated for $T=200$ time steps, using log-normal
priors for the hyper-parameters, and the PGAS sampler is then run for $50$
iterations using $N=20$ particles. To illustrate the capability of the GP-SSM
to make use of a parametric model as baseline, we use a mean function with the
same parametric form as the true system, but parameters $(a,b,c)=(0.3,7.5,0)$.
This function, denoted _model B_ , is manifestly different to the actual state
transition (green vs. black surfaces in Figure 2), also demonstrating the
flexibility of the GP-SSM.
Figure 2 (left) shows the samples of $\mathbf{x}_{0:T}$ (red). It is apparent
that the distribution covers two alternative state trajectories at particular
times (e.g. $t=10$). In fact, it is always the case that this bi-modal
distribution covers the two states of opposite signs that could have led to
the same observation (cyan). In Figure 2 (right) we plot samples from the
smoothing distribution, where each circle corresponds to
$(\mathbf{x}_{t},\mathbf{u}_{t},\mathbb{E}[\mathbf{f}_{t}])$. Although the
parametric model used in the mean function of the GP (green) is clearly not
representative of the true dynamics (black), the samples from the smoothing
distribution accurately portray the underlying system. The smoothness prior
embodied by the GP allows for accurate sampling from the smoothing
distribution even when the parametric model of the dynamics fails to capture
important features.
To measure the predictive capability of the learned transition dynamics, we
generate a new dataset consisting of $10\thinspace 000$ time steps and present
the RMSE between the predicted value of $f(\mathbf{x}_{t},\mathbf{u}_{t})$ and
the actual one. We compare the results from GP-SSM with the predictions
obtained from two parametric models (one with the true model structure and one
linear model) and two known models (the ground truth model and model B). We
also report results for the sparse GP-SSM using an FIC prior with 40 inducing
points. Table 1 summarizes the results, averaged over 10 independent training
and test datasets. We also report the RMSE from the joint smoothing sample to
the ground truth trajectory.
Figure 2: Left: Smoothing distribution. Right: State transition function (black: actual transition function, green: mean function (model B) and red: smoothing samples). Table 1: RMSE to ground truth values over 10 independent runs. RMSE | prediction of $\mathbf{f}^{*}|\mathbf{x}_{t}^{*},\mathbf{u}_{t}^{*},\mathrm{data}$ | smoothing $\mathbf{x}_{0:T}|\mathrm{data}$
---|---|---
Ground truth model (known parameters) | – | $2.7\pm 0.5$
GP-SSM (proposed, model B mean function) | $1.7\pm 0.2$ | $3.2\pm 0.5$
Sparse GP-SSM (proposed, model B mean function) | $1.8\pm 0.2$ | $2.7\pm 0.4$
Model B (fixed parameters) | $7.1\pm 0.0$ | $13.6\pm 1.1$
Ground truth model, learned parameters | $0.5\pm 0.2$ | $3.0\pm 0.4$
Linear model, learned parameters | $5.5\pm 0.1$ | $6.0\pm 0.5$
### 6.2 Learning a Cart and Pole System
We apply our approach to learn a model of a cart and pole system used in
reinforcement learning. The system consists of a cart, with a free-spinning
pendulum, rolling on a horizontal track. An external force is applied to the
cart. The system’s dynamics can be described by four states and a set of
nonlinear ordinary differential equations [20]. We learn a GP-SSM based on 100
observations of the state corrupted with Gaussian noise. Although the training
set only explores a small region of the 4-dimensional state space, we can
learn a model of the dynamics which can produce one step ahead predictions
such the ones in Figure 3. We obtain a predictive distribution in the form of
a mixture of Gaussians from which we display the first and second moments.
Crucially, the learned model reports different amounts of uncertainty in
different regions of the state-space. For instance, note the narrower error-
bars on some states between $t=320$ and $t=350$. This is due to the model
being more confident in its predictions in areas that are closer to the
training data.
Figure 3: One step ahead predictive distribution for each of the states of the
cart and pole system. Black: ground truth. Colored band: one standard
deviation from the mixture of Gaussians predictive.
## 7 Conclusions
We have shown an efficient way to perform fully Bayesian inference and
learning in the GP-SSM. A key contribution is that our approach retains the
full nonparametric expressivity of the model. This is made possible by
marginalizing out the state transition function, which results in a non-
trivial inference problem that we solve using a tailored PGAS sampler.
A particular characteristic of our approach is that the latent states can be
sampled from the smoothing distribution even when the state transition
function is unknown. Assumptions about smoothness and parsimony of this
function embodied by the GP prior suffice to obtain high-quality smoothing
distributions. Once samples from the smoothing distribution are available,
they can be used to describe a posterior over the state transition function.
This contrasts with the conventional approach to inference in dynamical
systems where smoothing is performed conditioned on a model of the state
transition dynamics.
#### References
## References
* [1] C. E. Rasmussen and C. K. I. Williams, _Gaussian Processes for Machine Learning_. MIT Press, 2006.
* [2] R. Turner, M. P. Deisenroth, and C. E. Rasmussen, “State-space inference and learning with Gaussian processes,” in _13th International Conference on Artificial Intelligence and Statistics_ , ser. W&CP, Y. W. Teh and M. Titterington, Eds., vol. 9, Chia Laguna, Sardinia, Italy, May 13–15 2010, pp. 868–875.
* [3] C. Andrieu, A. Doucet, and R. Holenstein, “Particle Markov chain Monte Carlo methods,” _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , vol. 72, no. 3, pp. 269–342, 2010.
* [4] F. Lindsten, M. Jordan, and T. B. Schön, “Ancestor sampling for particle Gibbs,” in _Advances in Neural Information Processing Systems 25_ , P. Bartlett, F. Pereira, C. Burges, L. Bottou, and K. Weinberger, Eds., 2012, pp. 2600–2608.
* [5] M. Deisenroth, R. Turner, M. Huber, U. Hanebeck, and C. Rasmussen, “Robust filtering and smoothing with Gaussian processes,” _IEEE Transactions on Automatic Control_ , vol. 57, no. 7, pp. 1865 –1871, july 2012.
* [6] M. Deisenroth and S. Mohamed, “Expectation Propagation in Gaussian process dynamical systems,” in _Advances in Neural Information Processing Systems 25_ , P. Bartlett, F. Pereira, C. Burges, L. Bottou, and K. Weinberger, Eds., 2012, pp. 2618–2626.
* [7] Z. Ghahramani and S. Roweis, “Learning nonlinear dynamical systems using an EM algorithm,” in _Advances in Neural Information Processing Systems 11_ , M. J. Kearns, S. A. Solla, and D. A. Cohn, Eds. MIT Press, 1999.
* [8] J. Wang, D. Fleet, and A. Hertzmann, “Gaussian process dynamical models,” in _Advances in Neural Information Processing Systems 18_ , Y. Weiss, B. Schölkopf, and J. Platt, Eds. Cambridge, MA: MIT Press, 2006, pp. 1441–1448.
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* [12] M. K. Pitt and N. Shephard, “Filtering via simulation: Auxiliary particle filters,” _Journal of the American Statistical Association_ , vol. 94, no. 446, pp. 590–599, 1999.
* [13] F. Lindsten and T. B. Schön, “Backward simulation methods for Monte Carlo statistical inference,” _Foundations and Trends in Machine Learning_ , vol. 6, no. 1, pp. 1–143, 2013.
* [14] F. Lindsten and T. B. Schön, “On the use of backward simulation in the particle Gibbs sampler,” in _Proceedings of the 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , Kyoto, Japan, Mar. 2012.
* [15] D. K. Agarwal and A. E. Gelfand, “Slice sampling for simulation based fitting of spatial data models,” _Statistics and Computing_ , vol. 15, no. 1, pp. 61–69, 2005.
* [16] E. Snelson and Z. Ghahramani, “Sparse Gaussian processes using pseudo-inputs,” in _Advances in Neural Information Processing Systems (NIPS)_ , Y. Weiss, B. Schölkopf, and J. Platt, Eds., Cambridge, MA, 2006, pp. 1257–1264.
* [17] J. Quiñonero-Candela and C. E. Rasmussen, “A unifying view of sparse approximate Gaussian process regression,” _Journal of Machine Learning Research_ , vol. 6, pp. 1939–1959, 2005.
* [18] M. Seeger, C. Williams, and N. Lawrence, “Fast Forward Selection to Speed Up Sparse Gaussian Process Regression,” in _Artificial Intelligence and Statistics 9_ , 2003.
* [19] Y. Chen, M. Welling, and A. Smola, “Super-samples from kernel herding,” in _Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence (UAI 2010)_ , P. Grünwald and P. Spirtes, Eds. AUAI Press, 2010.
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|
arxiv-papers
| 2013-06-12T15:20:28 |
2024-09-04T02:49:46.426030
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Roger Frigola, Fredrik Lindsten, Thomas B. Sch\\\"on, Carl E. Rasmussen",
"submitter": "Roger Frigola",
"url": "https://arxiv.org/abs/1306.2861"
}
|
1306.2908
|
A Heavy Ion Fireball freeze-out Dipion Cocktail for Au-Au Collisions at
$\sqrt{s_{NN}}$=200 GeV (Part 1).
R.S. Longacrea
aBrookhaven National Laboratory, Upton, NY 11973, USA
###### Abstract
In this paper we develop all the ingredients that come into play in the
freeze-out of the heavy ion fireball for Au-Au collisions at
$\sqrt{s_{NN}}$=200 GeV into dipions. The resonance production of particles
that decay into dipions plus minijets that also decay into dipions are
explored. The final state re-scattering of the pions from minijets play an
important role in the mass spectrum of the cocktail. Mass shifts due to
minijet interference which depend on the volume of the re-scattering pions is
presented. The effective mass balance function is explained. The dipion mass
spectrum within a $p_{t}$ range(intermediate $p_{t}$) is fitted using thermal
and minijet amplitudes.
## 1 Introduction
The ultra-relativistic heavy ion collision starts out as a state of high
density nuclear matter called the Quark Gluon Plasma(QGP) and expands rapidly
to freeze-out. During the freeze-out phase quarks and gluons form a system of
strongly interacting hadrons. These hadrons continue to expand in a thermal
manner until no further scattering is possible because the system becomes to
dilute. However this transition from quarks and gluons(partons) into hadrons
is not a smooth affair. The expansion is very rapid and some faster or hard
scattered partons fragment directly into hadron through a minijet[1] process.
Thus we have thermal and minijet hadrons present in the last scattering of the
hadrons. This mixture of sources is considered in this paper and applied to
the dipion mass spectrum of the heavy ion fireball formed in Au-Au collisions
at $\sqrt{s_{NN}}$=200.
The paper is organized in the following manner:
Sec. 1 is the introduction to the cross sections for $\pi$ $\pi$ scattering.
Sec. 2 develops a two component model with direct production and decay plus a
background $\pi$ $\pi$ component which must re-scatter in order to satisfy
unitarity. Sec. 3 consider two channel unitary scattering $\pi$ $\pi$ and $K$
$\overline{K}$ for $J^{PC}$ = $0^{++}$ and shows what re-scattering would look
like in the $\pi$ $\pi$ channel. Sec. 4 introduces the balance function for
dipion data and applies it to the two component model. Sec. 5 applies the two
component model to dipion data within a $p_{t}$ range. Sec. 6 presents the
summary and discussion. Finally there are two appendices. Appendix A show the
mathematical details of how the two component model is worked out. Appendix B
determines the maximum value of the $\alpha$ parameter using photo-production
data reported in Ref.[2].
### 1.1 $\pi$ $\pi$ scattering cross section
For the first part of this story we will define what a scattering cross
section is. We will first only consider elastic scattering of pions. Two pions
can scatter at a certain energy which we will call $M_{\pi\pi}$. The
differential cross section $\sigma$ at a given $M_{\pi\pi}$ is
$\frac{d\sigma}{d\phi
d\theta}=\frac{1}{K^{2}}\left|\sum_{\ell}(2\ell+1)T_{\ell}P_{\ell}(cos\theta)\right|^{2}$
(1)
where $\phi$ and $\theta$ are the azimuthal and scattering angles,
respectively. $T_{\ell}$ is a complex scattering amplitude and $\ell$ is the
angular momentum. $P_{\ell}$ is the Legendre polynomial, which is a function
of $cos\theta$. $K$ is the flux factor equal to the pion momentum in the
center of mass. The $T_{\ell}$ elastic scattering amplitudes are complex
amplitudes described by one real number which is in units of angles. The form
of the amplitude is
$T_{\ell}=e^{i\delta_{\ell}}sin\delta_{\ell}$ (2)
We note that $\delta$ depends on the value of $\ell$ and $M_{\pi\pi}$. We will
use the phase shifts given in Ref[3].
## 2 The two component model
Let us consider two pions scattering in the final state of the heavy ion
collision. The scattering will be in some $\ell$ partial wave. The
$M_{\pi\pi}$ of the scattering dipion system will depend on the probability of
the phase space of the overlapping pions. The pions emerge from a close
encounter in a defined quantum state with a random phase. We will call this
amplitude $A$ and note that the absolute value squared of the amplitude is
proportional to the phase space overlap. The emerging pions can re-scatter
through the quantum state of the pions, which is a partial wave or a phase
shift. We have amplitude $A$ plus $A$ times the re-scattering of pions through
the phase shift consistent quantum state of $A$. The correct unitary way to
describe this process is given by Ref[4] equation(4.5)
$T=\frac{V_{1}U_{1}}{D_{1}}+\frac{\left(V_{2}+\frac{D_{12}V_{1}}{D_{1}}\right)\left(U_{2}+\frac{D_{12}U_{1}}{D_{1}}\right)}{D_{2}-\frac{D_{12}^{2}}{D_{1}}}$
(3)
In the above equation we have two terms, 1 and 2. The first term denoted by 1
is the $\pi\pi$ scattering through $\ell$-wave which will become the amplitude
$A$ mentioned above, where $V$ is the incoming and $U$ is the outgoing
$\pi\pi$ system. The second term denoted by 2 is the direct production of the
$\pi\pi$ system in the $\ell$-wave with $V$ being the production, the
propagation being $D$ and the decay being $U$. We see that there are terms
$D_{12}$ which involves a loop of pions between scattering pions and the
formation of a resonance($\ell$ =1 would be a $\rho$) by the pions.
### 2.1 Final equation for the two component model
The complete derivation is in Appendix A. From the appendix we get two terms,
one being the direct production of the resonance or $\pi\pi$ $\ell$-wave phase
shift and the second being the resonance from re-scattering. The final
equation 6 has two important factors, one is two-body phase space and the
other is a coefficient $\alpha$. This coefficient is related to the real part
of the $\pi\pi$ re-scattering loop and is given by equation 4. When the pions
re-scatter or interact at a close distance or a point the real part of
$\alpha$ has its maximum value of $\alpha_{0}$. While if the pions re-scatter
or interact at a distance determined by the diffractive limit the value of
$\alpha$ is zero.
In the equation 6 $|T|^{2}$ is the cross section for $\ell$ partial wave
produced, where $D$ is the direct production amplitude and $A$ is the
amplitude introduced above for the re-scattering pions into the $\ell$-wave
with $\delta_{\ell}$ the $\pi\pi$ phase shift[3]. The $q$ is the $\pi\pi$
center of mass momentum. At a given $p_{t}$ and $y$ bin, $D$ will have a
thermal factor as a function of $M_{\pi\pi}$. The $\alpha$ which is the real
part of the re-scattering factor has a simple form given by
$\alpha=(1.0-\frac{r^{2}}{r_{0}^{2}})\alpha_{0}$ (4)
where $r$ is the radius of re-scattering in fm’s and $r_{0}$ is 1.0 fm or the
limiting range of the strong interaction ranging to $r$ = 0.0 for point like
interactions.
The dependence of $A$ is calculated by the phase space overlap of dipions
added as four vectors and corrected for proper time, with the sum having the
correct $p_{t}$ , $y$ and phase space weighting for $\ell^{th}$ partial wave
for a given $M_{\pi\pi}$.
Finally we must use the correct two body phase space. For a two body system of
pions, phase space goes to an constant as $M_{\pi\pi}$ goes to infinity. Let
us choose this constant to be unity. Phase space which is denoted by PS is
equal to
$PS=\frac{2qB_{\ell}(q/q_{s})}{M_{\pi\pi}}$ (5)
where $B_{\ell}$ is a Blatt-Weisskopf-barrier factor[5] for $\ell$ angular
momentum quantum number. The $q_{s}$ is the momentum related to the range of
interaction of the $\pi\pi$ scattering. 1 fm is the usual interaction distance
which implies that $q_{s}$ is .200 GeV/c. For the $\rho$ meson $\ell$=1 the
barrier factor is $B_{1}$ = $\frac{(q/q_{s})^{2}}{(1+(q/q_{s})^{2})}$. The
phase space factor PS as a function of $q$ near the $\pi\pi$ threshold is
given by $q^{2\ell+1}$. Thus in the appendix we use $q^{2\ell+1}$ for the
factor PS except for equation 6 which is the final equation.
$|T|^{2}=|D|^{2}\frac{sin^{2}\delta_{\ell}}{PS}+\frac{|A|^{2}}{PS}\left|\alpha
sin\delta_{\ell}+PScos\delta_{\ell}\right|^{2}$ (6)
## 3 Re-scattering through the Swave $\pi\pi$ is a two channel problem
In Sec. 2 we derived equation 6 considering only elastic scattering of the
$\pi\pi$ system. If we consider the Dwave it couples to the $f_{2}(1270)$
($J^{PC}$ = $2^{++}$) with 85% of the cross section in the $\pi\pi$ channel.
The Pwave couples to the $\rho(770)$ ($J^{PC}$ = $1^{--}$) where 100% is in
the $\pi\pi$ channel. The Swave $\pi\pi$ ($J^{PC}$ = $0^{++}$) couples to two
resonances the $\sigma$ and the $f_{0}(980)$. The $\sigma$ is purely elastic
while the $f_{0}(980)$ is split between the $\pi$ $\pi$ and $K$ $\overline{K}$
channels. These two channels plus two resonances gives an additional
complexity to the re-scattering problem.
In order to handle the $\pi\pi$ and $K$ $\overline{K}$ channels we will use
the K-matrix approach. When we are below the $K$ $\overline{K}$ threshold the
system is only a one channel problem and the K-matrix is only a single term of
the matrix
$K_{11}=tan\delta_{0}.$ (7)
We see that when $\delta_{0}$ = $90^{\circ}$ that there will be a pole in the
K-matrix. It is standard to expand the K-matrix as a sum of poles.
$K_{11}=\sum_{i}{\frac{2\gamma_{i}^{2}q_{\pi\pi}}{M_{\pi\pi}}\over{(M_{i}^{2}-M_{\pi\pi}^{2})}}$
(8)
Where $\gamma_{i}$ is the coupling of the pole($i^{th}$) to the $\pi\pi$
channel, $q_{\pi\pi}$ is the center of mass momentum of the $\pi\pi$ channel
and $M_{i}$ is the mass of the pole($i^{th}$). The T-matrix is given by
$T_{11}=e^{i\delta_{0}}sin\delta_{0}={K_{11}\over{(1-iK_{11})}}=\,(1-iK)^{-1}K$
(9)
When both channels are open $j=1$ $\pi$ $\pi$ and $j=2$ $K$ $\overline{K}$,
the K-matrix is given by
$K_{11}=\sum_{i}{\frac{2\gamma_{i1}^{2}q_{1}}{M_{1}}\over{(M_{i}^{2}-M_{1}^{2})}},K_{21}=\sum_{i}{\frac{2\gamma_{i2}\sqrt{q_{2}}\gamma_{i1}\sqrt{q_{1}}}{M_{1}}\over{(M_{i}^{2}-M_{1}^{2})}},K_{12}=\sum_{i}{\frac{2\gamma_{i1}\sqrt{q_{1}}\gamma_{i2}\sqrt{q_{2}}}{M_{1}}\over{(M_{i}^{2}-M_{1}^{2})}},K_{22}=\sum_{i}{\frac{2\gamma_{i2}^{2}q_{2}}{M_{1}}\over{(M_{i}^{2}-M_{1}^{2})}}.$
(10)
The T-matrix is given by
$T=\,(\delta-iK)^{-1}K.$ (11)
We fit the Swave ($J^{PC}$ = $0^{++}$) $\pi\pi$ of Ref[3] using three poles
for the $\sigma$, $f_{0}$ and some background from higher mass poles. The
$T_{11}$ amplitude is shown in Figure 1.
Figure 1: The $T_{11}$ amplitude for the Swave ($J^{PC}$ = $0^{++}$) $\pi\pi$
comes from a fit to of Ref[3] using three K-matrix poles for the $\sigma$,
$f_{0}$ and some background from higher mass poles.
Two pions scattering in the final state of the heavy ion collision in a Swave
will be our amplitude $A$, where the emerging pions can re-scatter through the
Swave. We have amplitude $A$ plus $A$ times the re-scattering of pions through
the Swave phase shift.
The term of equation 6 $PSe^{i\delta_{0}}cos\delta_{0}$ is equal to
$PS(1+ie^{i\delta_{0}}sin\delta_{0})$. $T_{11}$ which is equal to
$e^{i\delta_{0}}sin\delta_{0}$ for the one channel case becomes $T_{11}$ =
$\eta e^{i\delta_{0}}sin\delta_{0}$ for the two channel case. Thus the re-
scattering term becomes $PS(1+i\eta e^{i\delta_{0}}sin\delta_{0})$ or
$PS(1+iT_{11})$. Using our K-matrix fit to the Swave one obtain the re-
scattering term plotted in Figure 2. In the lower mass we see a shift of the
spectrum to a lower mass. In the next section will see the same effect for
$\rho(770)$ resonance that re-scattering will shift its mass to lower values.
This shift will be caused by a direct production plus the re-scattering adding
together creating a shifted $\rho(770)$[6]. We see that the $f_{0}$ is a
narrow resonance. The $f_{0}$ resonates at the $K$ $\overline{K}$ threshold.
Direct production of the $f_{0}$ gives a bump at the $K$ $\overline{K}$
threshold and the re-scattering of $\pi\pi$ also gives such a bump at the $K$
$\overline{K}$ threshold(Figure 2). Therefore we will only consider the
$f_{0}$ as a resonance being directly produced and decaying into $\pi\pi$ near
the $K$ $\overline{K}$ threshold.
## 4 The balance function for dipion effective mass
Up to this point in the paper we did not specify the charge of the pions
considered. With the idea of the balance function we look at the creation of
pairs of opposite charge pions. The QGP fireball begins mostly neutral without
a large excess of charge. Pairs of quarks and anti-quarks are created finally
forming hadrons mainly pions. Thus for every $\pi^{+}$ there is a $\pi^{-}$
which balance out the charge. The balance function measures the kinematic
variable between the balancing charge[7]. In our case we want to measure
effective mass of the $\pi^{+}\pi^{-}$ pairs. The dipion effective mass
balance function is defined by
$B(M_{\pi\pi})=\frac{1}{N}\sum_{events}\sum_{pairs}\frac{1}{2}\biggl{\\{}\frac{(M_{\pi^{+}\pi^{-}}-M_{\pi^{+}\pi^{+}})}{N_{+}}+\frac{(M_{\pi^{+}\pi^{-}}-M_{\pi^{-}\pi^{-}})}{N_{-}}\biggr{\\}},$
(12)
where $N$ is the number of events, $N_{+}$ is the number of positive pions per
event, and $N_{-}$ is the number of negative pions per event.
Let us consider the effective mass dipion balance function for p-p collisions
$\sqrt{s_{NN}}$=200 GeV[8]. The effective mass shows a large bump at the
$K^{0}$ mass from $K^{0}_{S}\rightarrow\pi^{+}\pi^{-}$. Above this bump there
is another mass bump which could be the $\rho(770)$. At this mass Pwave should
be the most important partial wave so let us assume this is the case. The
balance function drops off with mass so we assume an exponential fall off with
$M_{\pi^{+}\pi^{-}}$. The $\rho(770)$ should be directly produced and decay
through the Pwave phase shift. We will use equation 6 to fit the p-p balance
function. Figure 3 shows the fit to the STAR data, where the solid line is the
fit and dashed line is the $\rho(770)$. The exponential is amplitude A and the
dotted line is $A$ plus $A$ times the re-scattering of pions through the Pwave
phase shift. The value of $\alpha$ is 0.37 which means the radius of re-
scattering of the pions is 0.9 fm using equation 4 with the value of
$\alpha_{0}$ equal to 2. The solid line which is a good fit to the data peaks
at .73 GeV/c. We see that the shift of this peak is caused by the dotted line
added to the dashed line.
Figure 2: We plot for the Swave ($J^{PC}$ = $0^{++}$) $\pi\pi$ re-scattering
term $\left|PS(1+iT_{11})\right|^{2}$. The $T_{11}$ amplitude comes from a fit
to of Ref[3] using three K-matrix poles for the $\sigma$, $f_{0}$ and some
background from higher mass poles.
Figure 3: The fit to the p-p balance function STAR data[8], where the solid
line is the fit and dashed line is the $\rho(770)$. The exponential is
amplitude A and the dotted line is $A$ plus $A$ times the re-scattering of
pions through the Pwave phase shift.
We turn to the central Au-Au balance function measured by STAR[8]. Again using
the same functions that we used above we fit the data. The solid line is the
fit and dashed line is the $\rho(770)$. The exponential function is amplitude
A and the dotted line is $A$ plus $A$ times the re-scattering of pions through
the Pwave phase shift. The value of $\alpha$ is 0.44 which means the radius of
re-scattering of the pions is 0.88 fm using equation 4 with the value of
$\alpha_{0}$ equal to 2. We see it is a little smaller volume where pions can
re-scatter.
## 5 The dipion effective mass cocktail over a $p_{t}$ range in Au-Au
In order to form this cocktail we need to consider three important ingredients
that come into play. First is the thermal production of resonances that decay
into $\pi^{+}\pi^{-}$ as a function of dipion $p_{t}$. Second is the minijet
production of $\pi^{+}\pi^{-}$ that is not through a resonance decay and
determine its dipion effective mass spectrum as a function of dipion $p_{t}$.
Also we need the break up of the spectrum into partial waves. Third is
rewriting equation 6 in a form that uses resonance or Breit-Wigner parameters
(mass, widths) instead of phase shifts plus modify the equation to use the
derived minijet amplitudes.
### 5.1 Thermal production of resonances
Thermal resonance production will have a Boltzmann weighting of the dipion
effective mass spectrum. Since we are projecting in $p_{t}$ this weighting
will be an exponential function of the transverse mass divided by the
temperature[6,9-13].
$Weight(M_{\pi\pi})=\frac{M_{\pi\pi}}{\sqrt{M_{\pi\pi}^{2}+p^{2}_{t}}}exp\frac{-\sqrt{M_{\pi\pi}^{2}+p^{2}_{t}}}{T}$
(13)
This weight times the Breit-Wigner line shape is the thermal production of the
resonance which decays into the dipion system. The Breit-Wigner line shape is
given by
$BW(M_{\pi\pi})=\frac{M_{\pi\pi}M_{0}\Gamma}{(M^{2}_{0}-M^{2}_{\pi\pi})^{2}+M^{2}_{0}\Gamma^{2}}.$
(14)
Where $\Gamma$ is the $M_{\pi\pi}$ dependent total width
$\Gamma=\Gamma_{0}{\frac{qB_{\ell}(q/q_{s})}{M_{\pi\pi}}\over{\frac{q_{0}B_{\ell}(q_{0}/q_{s})}{M_{0}}}}$
(15)
with $\Gamma_{0}$ being the total width at resonance, $B_{\ell}$ is the Blatt-
Weisskopf-barrier factor[5] for the $\ell$ of the resonance, $q$ is the
$\pi\pi$ center mass momentum, $q_{0}$ is $q$ at resonance, $M_{0}$ is the
mass of the resonance, and $q_{s}$ is center mass momentum related to the
size(1.0 fm is used $q_{s}$ = .200 GeV/c).
Figure 4: The fit to the Au-Au balance function STAR data[8], where the solid
line is the fit and dashed line is the $\rho(770)$. The exponential is
amplitude A and the dotted line is $A$ plus $A$ times the re-scattering of
pions through the Pwave phase shift.
### 5.2 Minijet production of dipions
Partons that under go a hard scattering fragment into hadrons[1, 14]. These
hadrons become part of the outward flow of hadrons with the thermal hadrons.
Hadrons that have long life times will decay outside the freeze-out volume.
For these long lived resonances the dipion spectrum will be given by the
Breit-Wigner line shape of the last subsection. Thus the source of these
resonances either thermal or minijet fragmentation will not be apparent.
Resonances like the $\eta$($c\tau$ = 154000 fm), the $\omega$($c\tau$ = 24
fm), the $\eta^{\prime}$($c\tau$ = 100 fm), the $K^{*}$($c\tau$ = 4 fm), and
the $\phi$($c\tau$ = 50 fm) are decaying outside the freeze-out volume. All
other resonances decay inside this volume like the $\sigma$($c\tau$ = 1/3 fm),
the $\rho(770)$($c\tau$ = 1.3 fm), and the $f_{2}(1270)$($c\tau$ = 1 fm).
Pions from these decays become a source for re-scattering with pion directly
produced or the ones that arise from decays.
We will use the minijet fragmentation code of PYTHIA[14] in order to estimate
the dipion effective mass spectrum. We set up correct kinetics by using
minijets that are predicted by the program HIJING[15] for Au-Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV. We cycle through the minijets produced by HIJING
forming dipion pair($\pi^{+}\pi^{-}$) combinations within a given minijet
fragmentation. We remove the long lived resonance listed above because they
can be handled through the direct thermal production. All pairs that come
directly from the second class short lived resonances are also not considered.
They also can be handled through the direct thermal production. However
combinations of their decay pions with other pions are considered as a source
of minijet dipions. We see from Figure 5 that the hadrons that fragment from
the minijets are moving in the same direction and will interact with each
other.
Figure 5: A minijet parton shower.
The dipions from this minijet source can be selected for its dipion $p_{t}$
range and decomposed into each partial wave $\ell$($\ell$wave) obtaining an
amplitude A for equation 6. We can separate out this dipion spectrum by using
statistical and kinematic weighting. At the highest dipion mass the S, P, D,
and F waves have a $2\ell+1$ statistical weighting. S is 1, P is 3, D is 5,
and F is 7 making 16 units of probability. The kinematic weighting is given by
the Blatt-Weisskopf-barrier factors. At each dipion mass we have a dipion
spectrum from the minijets with a selection on dipion $p_{t}$ which we call
$jet(M_{\pi^{+}\pi^{-}})$. Let us define Z as the ratio of the center mass
dipion momentum $q$ divided $q_{s}$ which is .200 GeV/c(size of 1.0 fm) all
squared
$Z=\frac{q^{2}}{q_{s}^{2}}.$ (16)
The Fwave minijet dipion weight is given by
$F=\frac{7}{16}\frac{Z^{3}}{(Z^{3}+6Z^{2}+45Z+225)}.$ (17)
The Dwave minijet dipion weight is given by
$D=\frac{5}{9}(1.0-F)\frac{Z^{2}}{(Z^{2}+3Z+9)}.$ (18)
The Pwave minijet dipion weight is given by
$P=\frac{3}{4}(1.0-D-F)\frac{Z}{(Z+1)}.$ (19)
The Swave minijet dipion weight is given by
$S=(1.0-P-D-F).$ (20)
Thus $F(M_{\pi^{+}\pi^{-}})=Fjet(M_{\pi^{+}\pi^{-}})$,
$D(M_{\pi^{+}\pi^{-}})=Djet(M_{\pi^{+}\pi^{-}})$,
$P(M_{\pi^{+}\pi^{-}})=Pjet(M_{\pi^{+}\pi^{-}})$,and
$S(M_{\pi^{+}\pi^{-}})=Sjet(M_{\pi^{+}\pi^{-}})$.
Let us choose a dipion $p_{t}$ range and plot the above minijet spectrum. We
choose 1.6 GeV/c $<$ $p_{t}$ $<$ 1.8 GeV/c and plot in Figure 6 the Swave,
Pwave, Dwave, and Fwave from minijets coming from HIJING for Au-Au collisions
at $\sqrt{s_{NN}}=$ 200 GeV.
Figure 6: The dipion $p_{t}$ range 1.6 GeV/c $<$ $p_{t}$ $<$ 1.8 GeV/c showing
the Swave, Pwave, Dwave, and Fwave from minijets coming from HIJING for Au-Au
collisions at $\sqrt{s_{NN}}=$ 200 GeV.
### 5.3 Equation 6 for Breit-Wigner parameters
In this subsection we will alter equation 6 so it can use Breit-Wigner
parameters (mass, width) instead of phase shifts. We will also need to modify
the re-scattering part of the equation in order to have the correct threshold
behavior we have just introduced into the minijet partial waves above. The
phase shift can be written for the $\ell^{th}$ wave as
$cot\delta_{\ell}=\frac{(M_{\ell}^{2}-M_{\pi\pi}^{2})}{M_{\ell}\Gamma_{\ell}},$
(21)
where $M_{\ell}$ is the mass of the resonance in the $\ell$wave and
$\Gamma_{\ell}$ is its total width.
$\Gamma_{\ell}=\Gamma_{0\ell}{\frac{qB_{\ell}(q/q_{s})}{M_{\pi\pi}}\over{\frac{q_{\ell}B_{\ell}(q_{\ell}/q_{s})}{M_{\ell}}}}$
(22)
with $\Gamma_{0\ell}$ the total width at resonance, $B_{\ell}$ is the Blatt-
Weisskopf-barrier factor for the $\ell$ of the resonance, $q$ is the $\pi\pi$
center mass momentum, $q_{\ell}$ is $q$ at resonance, $M_{\ell}$ is the mass
of the resonance, and $q_{s}$ is center mass momentum related to the size(1.0
fm is used $q_{s}$ = .200 GeV/c).
Using equation 21 we rewrite equation 6 as
$|T_{\ell}|^{2}=|D_{\ell}|^{2}\frac{sin^{2}\delta_{\ell}}{PS_{\ell}}+\frac{|A_{\ell}|^{2}sin^{2}\delta_{\ell}}{PS_{\ell}}\left|\alpha+PS_{\ell}cot\delta_{\ell}\right|^{2}$
(23)
The $D_{\ell}$ is the thermal production term and is constant except for the
Boltzmann weight. The expected threshold behavior $q^{2\ell+1}$ comes from the
$sin\delta_{\ell}$ term. Since there is $sin^{2}\delta_{\ell}$ one of the
$q^{2\ell+1}$ is killed off by dividing by $PS_{\ell}$. In Figure 6 we have
put into our minijet $A_{\ell}$ the correct threshold $q^{2\ell+1}$ so we need
kill off the $q^{2\ell+1}$ of the other $sin\delta_{\ell}$ term. Therefore
equation 6 for our minijet $A_{\ell}$ we will use
$|T_{\ell}|^{2}=|D_{\ell}|^{2}\frac{sin^{2}\delta_{\ell}}{PS_{\ell}}+\frac{|A_{\ell}|^{2}sin^{2}\delta_{\ell}}{PS_{\ell}^{2}}\left|\alpha+PS_{\ell}cot\delta_{\ell}\right|^{2}$
(24)
Rewriting equation 6 for each partial wave with Breit-Wigner parameters the
first term becomes
$|T_{\ell}|_{1}^{2}=|D_{\ell}|^{2}\frac{M_{\pi\pi}^{2}}{\sqrt{M_{\pi\pi}^{2}+p^{2}_{t}}}exp\frac{-\sqrt{M_{\pi\pi}^{2}+p^{2}_{t}}}{T}\frac{M_{\ell}\Gamma_{\ell}}{(M_{\ell}^{2}-M_{\pi\pi}^{2})^{2}+M_{\ell}^{2}\Gamma_{\ell}^{2}},$
(25)
while the second term
$|T_{\ell}|_{2}^{2}=|A_{\ell}|^{2}\frac{M_{\ell}^{2}\Gamma_{\ell}^{2}}{(M_{\ell}^{2}-M_{\pi\pi}^{2})^{2}+M_{\ell}^{2}\Gamma_{\ell}^{2}}\left|\alpha+\frac{2qB_{\ell}(\frac{q}{q_{s}})(M_{\ell}^{2}-M_{\pi\pi}^{2})}{M_{\pi\pi}M_{\ell}\Gamma_{\ell}}\right|^{2}\left(\frac{M_{\pi\pi}^{2}}{4q^{2}B_{\ell}^{2}(\frac{q}{q_{s}})}\right).$
(26)
$|T|^{2}=\sum_{\ell}|T_{\ell}|^{2}$ (27)
where
$|T_{\ell}|^{2}=|T_{\ell}|_{1}^{2}+|T_{\ell}|_{2}^{2}$ (28)
and
$|A_{0}|^{2}=S(M_{\pi^{+}\pi^{-}})$,$|A_{1}|^{2}=P(M_{\pi^{+}\pi^{-}})$,$|A_{2}|^{2}=D(M_{\pi^{+}\pi^{-}})$,
and $|A_{3}|^{2}=F(M_{\pi^{+}\pi^{-}})$.
### 5.4 STAR data dipion $p_{t}$ range (1.6 GeV/c $<$ $p_{t}$ $<$ 1.8 GeV/c)
We have fitted a dipion $p_{t}$ range using equation 27 above for the STAR
data Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to 80% centrality. We
included minijets up to $\ell$ = 3 and resonances $\sigma$ $\ell$ = 0,
$\rho(770)$ $\ell$ = 1, and $f_{2}(1270)$ $\ell$ = 2. Using the arguments of
Sec. 3 we added the $f_{0}$ as a direct thermal term ($|T_{0}|_{1}^{2}$) and
only the $\sigma$ interfered with $\ell$ = 0 minijet background. Two other
thermal terms are present in the cocktail, the $K^{0}_{S}$ and the
$\omega_{0}$.
Finally the threshold effective mass region .280 GeV to .430 GeV is dominated
by the Swave and receives contributions from minijet fragmentation, $\pi\pi$
Swave phase shift, $\eta$ decay, HBT adding to the like sign $\pi\pi$
distribution that has been subtracted away from the unlike sign $\pi\pi$ and
the coulomb correction between the charged pions. The minijet fragmentation is
the least known of the effects since we relied on PYTHIA, however there are
large uncertainty in all the other effects. So for this fit we let the minijet
fragmentation be free to fit the data and let the Breit-Wigner parameters for
the $\sigma$ determine the Swave phase shifts plus leaving out all other
effects. The results of this fit is shown in Figure 7. Table I shows the
Breit-Wigner parameters used in the fit.
Table I. The Bret-Wigner Parameters of the fit.
Table I
---
resonance | mass(GeV) | width(GeV)
$\sigma$ | 1.011 | 1.015
$f_{0}$ | 0.973 | 0.041
$\rho$ | 0.748 | 0.147
$f_{2}$ | 1.275 | 0.185
Figure 7: Fit to STAR dipion effective mass distribution (1.6 GeV/c $<$
$p_{t}$ $<$ 1.8 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 27. See text for complete information.
The value of $\alpha$ for this $p_{t}$ range is .806. For coherent photo-
production of the $\rho(770)$ from STAR data[2] the value of $\alpha$ is 2.0
or $\alpha_{0}$ because the two pions emerge from a point source(see Appendix
B). Thus the radius of the size associated with re-scattering is .773 fm for
this $p_{t}$ range from equation 4. Since the thermal term for resonances
could also come from minijet fragmentation, we need additional studies to
determine how much production comes from minijets and how much comes from the
fireball directly[16].
## 6 Summary and Discussion
In this article we start with the basic definition of elastic $\pi\pi$
scattering. Next we show how re-scattering of pions depends on the unitary
condition that interactions present in the phase shift of an orbital state
must interact all the time. The process of parton fragmentation into dipion
states through unitarity leads to a equation of production and re-scattering
in a given orbital quantum number. This equation (equation 6) has two
components in each orbital state: one being the thermal production of
resonances in a dipion orbital state, the other is the re-scattering of
dipions coming from parton or minijet fragmentation into the dipion orbital
state which do not come directly from the resonance. Unitarity requires that
there most be re-scatter through the resonance of the phase shift. This
equation is used over and over again with the details presented in the
appendix.
Equation 6 considers only elastic scattering of the $\pi\pi$ system. We
considered the Dwave which couples to the $f_{2}(1270)$ ($J^{PC}$ = $2^{++}$)
with 85% of the cross section in the $\pi\pi$ channel. The Pwave which coupled
to the $\rho(770)$ ($J^{PC}$ = $1^{--}$) where 100% is in the $\pi\pi$
channel. The Swave $\pi\pi$ ($J^{PC}$ = $0^{++}$) couples to two resonances
the $\sigma$ and the $f_{0}(980)$. The $\sigma$ is purely elastic while the
$f_{0}(980)$ is split between the $\pi$ $\pi$ and $K$ $\overline{K}$ channels.
We saw that the $f_{0}$ was a narrow resonance. The $f_{0}$ resonates at the
$K$ $\overline{K}$ threshold. Direct production of the $f_{0}$ gives a bump at
the $K$ $\overline{K}$ threshold and the re-scattering of $\pi\pi$ also gives
such the same bump at the $K$ $\overline{K}$ threshold Sec. 3 (Figure 2).
Therefore we considered the $f_{0}$ as a resonance being directly produced and
decaying into $\pi\pi$ near the $K$ $\overline{K}$ threshold.
We used equation 6 to fit the dipion effective mass balance function. We
assumed the Pwave was the most important partial wave in the dipion effective
mass range of the fit. The balance function drops off with mass so we assumed
an exponential fall off with $M_{\pi^{+}\pi^{-}}$. The two parts of equation 6
where the first part being the $\rho(770)$ directly produced and decaying
through the Pwave phase shift with the second part being the exponential
function plus re-scattering through the $\rho(770)$. With this simple model we
were able to fit the p-p balance (Figure 3) and the central Au-Au
balance(Figure 4). The observed mass shifts were due to the re-scattering
effect, being $\sim$40 MeV.
We chose a dipion $p_{t}$ range to do a cocktail fit to the effective dipion
mass spectrum. We needed three important ingredients in order to do this fit.
First is the thermal production of resonances that decay into $\pi^{+}\pi^{-}$
as a function of dipion $p_{t}$. Second is the dipion effective mass spectrum
as a function of dipion $p_{t}$ coming from minijet production not through
resonance decay. Third we needed to rewrite equation 6 in a form that uses
resonance or Breit-Wigner parameters (mass, widths) instead of phase shifts.
Once these three ingredients were developed we were successful in doing a
cocktail fit (Figure 7). Since the thermal term for resonances could also come
from minijet fragmentation, we need additional studies to determine how much
production comes from minijets and how much comes from the fireball
directly[16]. Other fits to more $p_{t}$ ranges are considered in Ref.[17].
## 7 Acknowledgments
This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886. The author thanks William Love for the STAR analysis of
the angular correlation data from Run 4. Also for his assistance in the
production of figures. It is sad that he is gone.
## Appendix A Appendix
Starting with equation (4.5) from Ref[4]
$T=\frac{V_{1}U^{\prime}_{1}}{D_{1}}+\frac{\left(V_{2}+\frac{D_{12}V_{1}}{D_{1}}\right)\left(U^{\prime}_{2}+\frac{D_{12}U^{\prime}_{1}}{D_{1}}\right)}{D_{2}-\frac{D_{12}^{2}}{D_{1}}}$
(29)
In order to have the correct threshold kinematics, we define
$U^{\prime}_{1}=U_{1}\sqrt{q^{2\ell+1}}$ (30)
$U^{\prime}_{2}=U_{2}\sqrt{q^{2\ell+1}}$ (31)
where $q$ is the $\pi\pi$ center of mass momentum and $\ell$ is the value of
the angular momentum. The amplitude $A$ of the text is given by
$\frac{V_{1}U_{1}}{D_{1}}=A$ (32)
Thus we have
$\frac{V_{1}U^{\prime}_{1}}{D_{1}}=A\sqrt{q^{2\ell+1}}$ (33)
The phase shift for the $\ell^{th}$ partial wave will be given by
$\delta_{\ell}$, where
$\frac{U^{\prime}_{2}U^{\prime}_{2}}{D_{2}}=e^{i\delta_{\ell}}sin\delta_{\ell}$
(34)
The above equality is true if the $D_{1}$ mode plays no role in the $\pi\pi$
scattering in the $\ell^{th}$ partial wave. But in the initial state there is
a large production of $D_{1}$. The $U^{\prime}$s are the basic coupling of the
$D^{\prime}$s to the $\pi\pi$ system. In order to decouple $D_{1}$ from the
$\pi\pi$ system $U_{1}$ must go to zero. We can maintain a finite production
of $D_{1}$ if we define
$V_{1}=\frac{1}{U_{1}}$ (35)
Thus the first term in the equation becomes
$\frac{V_{1}U^{\prime}_{1}}{D_{1}}=\frac{\frac{1}{U_{1}}U_{1}\sqrt{q^{2\ell+1}}}{D_{1}}=\frac{\sqrt{q^{2\ell+1}}}{D_{1}}=A\sqrt{q^{2\ell+1}}$
(36)
The form of $D_{12}$ is given by
$D_{12}=\alpha U_{1}U_{2}+iq^{2\ell+1}U_{1}U_{2}$ (37)
$D_{12}$ is the real and imaginary part of the two pion loop from state 1 to
state 2. The $U^{\prime}$s are the $\pi\pi$ couplings and the imaginary part
goes to zero at the $\pi\pi$ threshold. The $\alpha$ factor is $\alpha_{0}$
for re-scattering coming from a point source, but goes to zero when re-
scattering is diffractive. A simple form for $\alpha$ is given by
$\alpha=(1.0-\frac{r^{2}}{r_{0}^{2}})\alpha_{0}$ (38)
where $r$ is the radius of re-scattering in fm’s and $r_{0}$ is 1.0 fm or the
limiting range of the strong interaction. The second term of the first
equation is
$\frac{\left(V_{2}+\frac{D_{12}V_{1}}{D_{1}}\right)\left(U^{\prime}_{2}+\frac{D_{12}U^{\prime}_{1}}{D_{1}}\right)}{D_{2}-\frac{D_{12}^{2}}{D_{1}}}$
(39)
Rewriting
$\frac{\left(V_{2}+\frac{\alpha
U_{1}U_{2}V_{1}}{D_{1}}+iq^{2\ell+1}\frac{U_{1}U_{2}V_{1}}{D_{1}}\right)\left(U^{\prime}_{2}+\frac{D_{12}U^{\prime}_{1}}{D_{1}}\right)}{D_{2}-\frac{D_{12}^{2}}{D_{1}}}$
(40)
Let us make substitutions
$V_{1}=\frac{1}{U_{1}},U_{2}=\frac{U^{\prime}_{2}}{\sqrt{q^{2\ell+1}}},\frac{1}{D_{1}}=A,D_{12}=0$
(41)
The second term becomes
$\frac{\left(V_{2}+\frac{A\alpha
U^{\prime}_{2}}{\sqrt{q^{2\ell+1}}}+i\sqrt{q^{2\ell+1}}AU^{\prime}_{2}\right)U^{\prime}_{2}}{D_{2}}$
(42)
The first term is
$\frac{V_{1}U^{\prime}_{1}}{D_{1}}=A\sqrt{q^{2\ell+1}}$ (43)
Adding the first and the second terms and substituting the phase shift,
$T=\frac{V_{2}}{U_{2}}\frac{e^{i\delta_{\ell}}sin\delta_{\ell}}{\sqrt{q^{2\ell+1}}}+A\left(\frac{e^{i\delta_{\ell}}\alpha
sin\delta_{\ell}}{\sqrt{q^{2\ell+1}}}+\sqrt{q^{2\ell+1}}e^{i\delta_{\ell}}cos\delta_{\ell}\right)$
(44)
The term with the factor $\frac{V_{2}}{U_{2}}$ is the direct production of the
dipion system. We shall call this amplitude $D$. The re-scattered amplitude is
$A$ and is modified by the dipion phase shift. These two amplitudes have some
random phase and are not coherent. Thus the cross section is
$|T|^{2}=|D|^{2}\frac{sin^{2}\delta_{\ell}}{PS}+\frac{|A|^{2}}{PS}\left|\alpha
sin\delta_{\ell}+PScos\delta_{\ell}\right|^{2}$ (45)
## Appendix B Appendix
In this appendix we determine the value $\alpha$ coming from a point
source(thus $\alpha_{0}$ see equation 4 and 38) using photo-production data
reported in Ref.[2]. When a photon interacts with a strong field a $\rho$ or a
$\pi^{+}$ $\pi^{-}$ pair is formed in a Pwave, both states are coherent with
each other and must be added together(see equation 3 of Ref.[2]). We write
equation 3 using the phase shift of Pwave $\pi^{+}$ $\pi^{-}$ scattering and
the width($\Gamma_{0}$) of the $\rho$ resonance. Threshold behavior is added
using the phase space factor (PS) for Pwave production (see equation 5 of Sec.
2.1).
$\frac{dN}{dM_{\pi\pi}}=\left|A_{\rho}\frac{sin\delta
e^{i\delta}}{\Gamma_{0}^{1/2}(\frac{PS}{PS_{0}})^{1/2}}+(\frac{PS}{PS_{0}})^{1/2}B_{\pi\pi}\right|^{2}$
(46)
The Pwave phase factor is
$PS=\frac{\frac{2q(q/q_{s})^{2}}{(1+(q/q_{s})^{2})}}{M_{\pi\pi}},$ (47)
where $q$ is the center mass momentum of the $\pi$ $\pi$ and $q_{s}$ is
related to the range of interaction of the $\pi\pi$ scattering. 1 fm is the
usual interaction distance which implies that $q_{s}$ is .200 GeV/c.
$M_{\pi\pi}$ is the effective mass of the system. The Pwave phase factor at
the $\rho$ resonance is
$PS_{0}=\frac{\frac{2q(q_{0}/q_{s})^{2}}{(1+(q_{0}/q_{s})^{2})}}{M_{\rho}},$
(48)
where $q_{0}$ is the center mass momentum of the $\pi$ $\pi$ at the $\rho$
mass.
From equation 46 we pull out a common phase space ratio from both terms,
$\frac{dN}{dM_{\pi\pi}}=\frac{1}{(\frac{PS}{PS_{0}})}\left|A_{\rho}\frac{sin\delta
e^{i\delta}}{\Gamma_{0}^{1/2}}+(\frac{PS}{PS_{0}})B_{\pi\pi}\right|^{2}.$ (49)
The ratio of the $\pi$ $\pi$ production amplitude over the $\rho$ production
amplitude as averaged over world data is
$\frac{B_{\pi\pi}}{A_{\rho}}=0.85.$ (50)
Using equation 50 and substituting this relation for $A_{\rho}$ into equation
49 we obtain
$\frac{dN}{dM_{\pi\pi}}=\frac{B_{\pi\pi}^{2}}{(\frac{PS}{PS_{0}})}\left|\frac{sin\delta
e^{i\delta}}{0.85\Gamma_{0}^{1/2}}+(\frac{PS}{PS_{0}})\right|^{2}.$ (51)
The second term of equation 51 can be expanded because 1 =
$e^{-i\delta}e^{i\delta}$ becoming
$\frac{dN}{dM_{\pi\pi}}=\frac{B_{\pi\pi}^{2}}{(\frac{PS}{PS_{0}})}\left|\frac{sin\delta
e^{i\delta}}{0.85\Gamma_{0}^{1/2}}+(\frac{PS}{PS_{0}})cos\delta
e^{i\delta}-i(\frac{PS}{PS_{0}})sin\delta e^{i\delta}\right|^{2}.$ (52)
There is a common phase factor($e^{i\delta}$) on all three terms which has a
magnitude of 1. The magnitude for all three terms can be written as
$\frac{dN}{dM_{\pi\pi}}=(\frac{PS}{PS_{0}})B_{\pi\pi}^{2}sin^{2}\delta+\frac{B_{\pi\pi}^{2}}{PS(PS_{0})}\left|\frac{PS_{0}sin\delta}{0.85\Gamma_{0}^{1/2}}+PScos\delta\right|^{2}.$
(53)
The second term has an amplitude of the form of equation 6 with
$\alpha sin\delta+PScos\delta.$ (54)
This implies that the maximum value of $\alpha$ is $\alpha_{0}$ and given by
$\alpha_{0}=\frac{PS_{0}}{0.85\Gamma_{0}^{1/2}}\simeq 2.0.$ (55)
## References
* [1] T. Trainor, Phys. Rev. C 80 (2009) 044901.
* [2] B.I. Abelev et al., Phys. Rev. C 77 (2008) 034910.
* [3] G. Grayner et. al., Nucl. Phys. B 75 (1974) 189.
* [4] R. Aaron and R.S. Longacre, Phys. Rev. D 24 (1981) 1207\.
* [5] F. von Hippel and C. Quigg, Phys. Rev. 5 (1972) 624.
* [6] P. Fachini et al., J.Phys.G G34 (2007) 431.
* [7] S.A. Bass, P. Danielewicz, and S. Pratt, Phys. Rev. Lett. 85 (2000) 2689.
* [8] M.M. Aggarwal et al., Phys. Rev. C 82 (2010) 024905.
* [9] E.V. Shuryak and G.E. Brown, Nucl. Phys. A 717 (2003) 322.
* [10] H.W. Barz et al., Phys. Lett. B265 (1991) 219.
* [11] R. Rapp, Nucl. Phys. A 725 (2003) 254.
* [12] W. Broniowski et al., Phys. Rev. C 68 (2003) 034911.
* [13] W. Bauer and S. Pratt, Phys. Rev. C 68 (2003) 064905.
* [14] T. Sjostrand, M. van Zijil, Phys. Rev. D 36 (1987) 2019.
* [15] X.N. Wang and M. Gyulassy, Phys. Rev. D 44 (1991) 3501.
* [16] R.S. Longacre, arXiv:1306.3493v1[hep-ph].
* [17] R.S. Longacre, arXiv:1306.3493v2[hep-ph].
|
arxiv-papers
| 2013-06-12T17:49:48 |
2024-09-04T02:49:46.435967
|
{
"license": "Public Domain",
"authors": "Ron S. Longacre",
"submitter": "Ron S. Longacre",
"url": "https://arxiv.org/abs/1306.2908"
}
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1306.2967
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arxiv-papers
| 2013-06-12T20:23:07 |
2024-09-04T02:49:46.444696
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{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Mohsen Joneidi, Mostafa Sadeghi",
"submitter": "Mohsen Joneidi",
"url": "https://arxiv.org/abs/1306.2967"
}
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1306.3049
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# On graded Gorenstein injective dimension
Afsaneh Esmaeelnezhad and Parviz Sahandi Department of Mathematics,
University of Kharazmi, Tehran, Iran [email protected] Department of
Mathematics, University of Tabriz, Tabriz, Iran, and School of Mathematics,
Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746,
Tehran Iran. [email protected], [email protected]
###### Abstract.
There are nice relations between graded homological dimensions and ordinary
homological dimensions. We study the Gorenstein injective dimension of a
complex of graded modules denoted by ${}^{*}\operatorname{Gid}$, and derive
its properties. In particular we prove the Chouinard’s like formula for
${}^{*}\operatorname{Gid}$, and compare it with the usual Gorenstein injective
dimension.
###### Key words and phrases:
graded rings, graded modules, Chouinard’s formula, injective dimension,
Gorenstein injective dimension
###### 2010 Mathematics Subject Classification:
13D05,13D02,13A02
P. Sahandi was supported in part by a grant from IPM (No. 91130030).
## 1\. Introduction
Let $R$ be a Noetherian $\mathbb{Z}$-graded ring. In [9] and [10], Fossum and
Fossum-Foxby have studied the graded homological dimension of graded modules
and compare them with classical homological dimensions. They shown that for a
graded $R$-module $M$, one has
$\,{}^{*}\operatorname{id}_{R}M\leq\operatorname{id}_{R}M\leq\,^{*}\operatorname{id}_{R}M+1$
where $\operatorname{id}_{R}M$ (resp. $\,{}^{*}\operatorname{id}_{R}M$)
denotes for the injective dimension of $M$ in the category of $R$-modules
(resp. category of graded $R$-modules). It is natural to ask how these
inequalities hold for the Gorenstein injective dimension
$\operatorname{Gid}_{R}M$. In this paper we give an answer this question.
Section 2 of this paper is devoted to review some hyper-homological algebra
for the derived category of the graded ring $R$. In Section 3 we define the
∗injective dimension of complexes of graded modules and homogeneous
homomorphisms, and derived its properties. In the final section we define the
graded Gorenstein injective dimension of complexes of graded modules and
homogeneous homomorphisms denoted by $\,{}^{*}\operatorname{Gid}$. Among other
results we show that if the graded ring $R$ admits a $\\!{}^{*}$dualizing
complex, or is a non-negatively graded ring, then
$\,{}^{*}\operatorname{Gid}_{R}X\leq\operatorname{Gid}_{R}X\leq\,^{*}\operatorname{Gid}_{R}X+1$,
where $\operatorname{Gid}_{R}X$ is the Gorenstein injective dimension of $X$
over $R$ (see Corollary 4.19). Also in this case we prove a Chouinard’s like
formula for $\,{}^{*}\operatorname{Gid}_{R}X$ (see Theorem 4.18). Our source
of graded rings and modules are [3] and [10].
Throughout this paper $R$ is a commutative Noetherian $\mathbb{Z}$-graded
ring.
## 2\. Derived category of complexes of graded modules
We use the notation from the appendix of [5]. Let $X$ be a complex of
$R$-modules and $R$-homomorphisms. The _supremum_ and the _infimum_ of a
complex $X$, denoted by $\sup(X)$ and $\inf(X)$ are defined by the supremum
and infimum of $\\{i\in\mathbb{Z}|\mbox{H}_{i}(X)\neq 0\\}$. If $m$ is an
integer and $X$ is a complex, then $\Sigma^{m}X$ denotes the complex $X$
_shifted_ $m$ degrees to the left; it is given by
$(\Sigma^{m}X)_{\ell}=X_{\ell-m}\text{ and
}\partial^{\Sigma^{m}X}_{\ell}=(-1)^{m}\partial^{X}_{\ell-m}$
for $\ell\in\mathbb{Z}$.
The symbol ${\mathcal{D}}(R)$ denotes the _derived category_ of $R$-complexes.
The full subcategories ${\mathcal{D}}_{\sqsubset}(R)$,
${\mathcal{D}}_{\sqsupset}(R)$, ${\mathcal{D}}_{\square}(R)$ and
${\mathcal{D}}_{0}(R)$ of ${\mathcal{D}}(R)$ consist of $R$-complexes $X$
while $\mbox{H}_{\ell}(X)=0$, for respectively $\ell\gg 0$, $\ell\ll 0$,
$|\ell|\gg 0$ and $\ell\neq 0$. Homology isomorphisms are marked by the sign
$\simeq$. The right derived functor of the homomorphism functor of
$R$-complexes and the left derived functor of the tensor product of
$R$-complexes are denoted by ${\mathbf{R}}\operatorname{Hom}_{R}(-,-)$ and
$-\otimes^{\mathbf{L}}_{R}-$, respectively.
Let $M=\oplus_{n\in\mathbb{Z}}M_{n}$ and $N=\oplus_{n\in\mathbb{Z}}N_{n}$ be
two graded $R$-modules. The ${}^{*}\operatorname{Hom}$ functor is defined by
${}^{*}\operatorname{Hom}_{R}(M,N)=\bigoplus_{i\in\mathbb{Z}}\operatorname{Hom}_{i}(M,N)$,
such that $\operatorname{Hom}_{i}(M,N)$ is a $\mathbb{Z}$-submodule of
$\operatorname{Hom}_{R}(M,N)$ consisting of all $\varphi:M\to N$ such that
$\varphi(M_{n})\subseteq N_{n+i}$ for all $n\in\mathbb{Z}$. In general
${}^{*}\operatorname{Hom}_{R}(M,N)\neq\operatorname{Hom}_{R}(M,N)$ but
equality holds if M is finitely generated, see [3, Exercise 1.5.19]. Also the
tensor product $M\otimes_{R}N$ of $M$ and $N$ is a graded module with
$(M\otimes_{R}N)_{n}$ is generated (as a $\mathbb{Z}$-module) by elements
$m\otimes n$ with $m\in M_{i}$ and $n\in N_{j}$ where $i+j=n$.
Let $\\{M_{\alpha}\\}_{\alpha\in I}$ be a family of graded $R$-modules. Then
$\bigoplus_{\alpha}M_{\alpha}$ becomes a graded $R$-module with
$(\bigoplus_{\alpha}M_{\alpha})_{n}=\bigoplus_{\alpha}(M_{\alpha})_{n}$ for
all $n\in\mathbb{Z}$, see [10, Page 289]. Also recall that the direct product
exists in the category of graded modules. Then the direct product is denoted
by ${}^{*}\prod_{\alpha}M_{\alpha}$ and
$(^{*}\prod_{\alpha}M_{\alpha})_{n}=\prod_{\alpha}(M_{\alpha})_{n}$ for all
$n\in\mathbb{Z}$, see [10, Page 289]. In this case there are the following
bijections [10, Page 289]
${}^{*}\operatorname{Hom}_{R}(\bigoplus_{\alpha}M_{\alpha},-)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\\!^{*}\prod_{\alpha}\\!^{*}\operatorname{Hom}_{R}(M_{\alpha},-),$
${}^{*}\operatorname{Hom}_{R}(-,\\!^{*}\prod_{\alpha}M_{\alpha})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\\!^{*}\prod_{\alpha}\\!^{*}\operatorname{Hom}_{R}(-,M_{\alpha}).$
Likewise direct and inverse limits are exists in the category of graded
modules with
$(\\!^{*}\lim_{\longrightarrow}M_{\alpha})_{n}=\lim_{\longrightarrow}(M_{\alpha})_{n},$
$(\\!^{*}\lim_{\longleftarrow}M_{\alpha})_{n}=\lim_{\longleftarrow}(M_{\alpha})_{n},$
see [10, Page 289]. Let $(R,\mathfrak{m})$ be a ∗local (Noetherian) ring $R$,
that is, a graded ring with a unique homogeneous maximal ideal $\mathfrak{m}$.
The $\mathfrak{m}$-∗adic completion of $R$ is
$\\!{}^{*}\widehat{R}=\\!^{*}\lim_{\longleftarrow}R/\mathfrak{m}^{n},$
which is a Noetherian graded ring by [9, Corollary VIII.2]. It is known that
the $\mathfrak{m}$-∗adic completion $\,{}^{*}\widehat{R}$ is a flat $R$-module
[10, Corollary 3.3], and that if
$E:=\\!^{*}\operatorname{E}_{R}(R/\mathfrak{m})$ is the ∗ injective envelope
of $R/\mathfrak{m}$ over $R$, then
$\,{}^{*}\operatorname{Hom}_{R}(E,E)\cong\\!^{*}\widehat{R}$ [9, Theorem
VIII.3].
The symbol ${}^{*}{\mathcal{C}}(R)$ denotes the category of complexes of
graded $R$-modules and homogeneous differentials. Note that the category of
graded modules is an abelian category, hence ${}^{*}{\mathcal{C}}(R)$ has a
derived category, (see [15]), which will be denoted by
$\,{}^{*}{\mathcal{D}}(R)$. Analogously we have
$\,{}^{*}{\mathcal{C}}_{\sqsubset}(R)$,
$\,{}^{*}{\mathcal{C}}_{\sqsupset}(R)$, $\,{}^{*}{\mathcal{C}}_{\square}(R)$
and $\,{}^{*}{\mathcal{C}}_{0}(R)$ (resp.
$\,{}^{*}{\mathcal{D}}_{\sqsubset}(R)$,
$\,{}^{*}{\mathcal{D}}_{\sqsupset}(R)$, $\,{}^{*}{\mathcal{D}}_{\square}(R)$
and $\,{}^{*}{\mathcal{D}}_{0}(R)$) which are the full subcategories of
$\,{}^{*}{\mathcal{C}}(R)$ (resp. $\,{}^{*}{\mathcal{D}}(R)$). If we use the
notation $X\in^{*}\mathcal{C}_{(\sharp)}(R)$, we mean
$\mbox{H}(X)\in^{*}\mathcal{C}_{\sharp}(R)$.
For $R$-complexes $X$ and $Y$ of graded modules, with homogeneous
differentials $\partial^{X}$ and $\partial^{Y}$, we define the _homomorphism
complex_ ${}^{*}\operatorname{Hom}_{R}(X,Y)$ as follows:
${}^{*}\operatorname{Hom}_{R}(X,Y)_{\ell}=\\!^{*}\prod_{p\in\mathbb{Z}}\\!^{*}\operatorname{Hom}_{R}(X_{p},Y_{p+\ell})$
and when $\psi=(\psi_{p})_{p\in\mathbb{Z}}$ belongs to
${}^{*}\operatorname{Hom}_{R}(X,Y)_{\ell}$ the family
$\partial_{\ell}^{{}^{*}\operatorname{Hom}_{R}(X,Y)}(\psi)$ in
${}^{*}\operatorname{Hom}_{R}(X,Y)_{\ell-1}$ has $p$-th component
$\partial_{\ell}^{{}^{*}\operatorname{Hom}_{R}(X,Y)}(\psi)_{p}=\partial^{Y}_{p+\ell}\psi_{p}-(-1)^{\ell}\psi_{p-1}\partial^{X}_{p}.$
When $X\in\,^{*}{\mathcal{C}}_{\sqsupset}^{f}(R)$ and
$Y\in\,^{*}{\mathcal{C}}_{\sqsubset}(R)$ all the products
$\\!{}^{*}\prod_{p\in\mathbb{Z}}\\!^{*}\operatorname{Hom}_{R}(X_{p},Y_{p+\ell})$
are finite. Thus using [3, Exercise 1.5.19], we have
$\\!{}^{*}\prod_{p\in\mathbb{Z}}\\!^{*}\operatorname{Hom}_{R}(X_{p},Y_{p+\ell})=\bigoplus_{p\in\mathbb{Z}}\operatorname{Hom}_{R}(X_{p},Y_{p+\ell}),$
for every $\ell\in\mathbb{Z}$. Therefore
${}^{*}\operatorname{Hom}_{R}(X,Y)=\operatorname{Hom}_{R}(X,Y)$.
We also define the _tensor product complex_ $X\otimes_{R}Y$ as follows:
$(X\otimes_{R}Y)_{\ell}=\bigoplus_{p\in\mathbb{Z}}(X_{p}\otimes_{R}Y_{\ell-p})$
and the $\ell$-th differential $\partial_{\ell}^{X\otimes_{R}Y}$ is given on a
generator $x_{p}\otimes y_{\ell-p}$ in $(X\otimes_{R}Y)_{\ell}$, where $x_{p}$
and $y_{\ell-p}$ are homogeneous elements, by
$\partial_{\ell}^{X\otimes_{R}Y}(x_{p}\otimes
y_{\ell-p})=\partial^{X}_{p}(x_{p})\otimes
y_{\ell-p}+(-1)^{p}x_{p}\otimes\partial^{Y}_{\ell-p}(y_{\ell-p}).$
If $X$ and $Y$ are $R$-complexes of graded modules, then
$\,{}^{*}\operatorname{Hom}_{R}(X,-)$, $\,{}^{*}\operatorname{Hom}_{R}(-,Y)$,
and $X\otimes_{R}-$ are functors on $\,{}^{*}{\mathcal{C}}(R)$.
Note that any object of $\,{}^{*}{\mathcal{C}}_{\sqsubset}(R)$ has an
∗injective resolution by [15, Page 47], and any object of
$\,{}^{*}{\mathcal{C}}_{\sqsupset}(R)$ has an ∗projective resolution by [15,
Page 48]. The right derived functor of the $\\!{}^{*}\operatorname{Hom}$
functor in the category of graded complexes is denoted by
$\mathbf{R}\,^{*}\operatorname{Hom}_{R}(-,-)$ and set
$\,{}^{*}\operatorname{Ext}^{i}_{R}(-,-)=\mbox{H}_{-i}(\mathbf{R}\,^{*}\operatorname{Hom}_{R}(-,-))$.
It is easily seen that if $R$ is a Noetherian $\mathbb{Z}$-graded ring and $X$
a homologically finite complex of graded modules and
$Y\in\\!^{*}\mathcal{C}(R)$ then
$\mathbf{R}\,^{*}\operatorname{Hom}_{R}(X,Y)={\mathbf{R}}\operatorname{Hom}_{R}(X,Y)$.
Also the left derived functor of $-\otimes_{R}-$ in the category of graded
complexes is denoted by $-\otimes_{R}^{\mathbf{L}^{*}}-$. Since ∗projective
graded $R$-modules coincide with projective $R$-modules by [10, Proposition
3.1] we easily see that $-\otimes_{R}^{\mathbf{L}^{*}}-$ coincides with the
ordinary left derived functor of $-\otimes_{R}-$ in the category of complexes.
So we use $-\otimes^{\mathbf{L}}_{R}-$ instead of
$-\otimes_{R}^{\mathbf{L}^{*}}-$.
For the homomorphism and the tensor product functors we have the following
useful proposition, see [5, A.2.8, A.2.10, and A.2.11] for the ungraded case.
The proof is the same as the ungraded case so we omit it.
###### Proposition 2.1.
Let $S$ be a graded ring which is an $R$-algebra.
∗Adjointness. Let $Z,Y\in\,^{*}{\mathcal{C}}(S)$ and
$X\in\,^{*}{\mathcal{C}}(R)$. Then $Z\otimes_{S}Y\in\,^{*}{\mathcal{C}}(S)$
and $\\!{}^{*}\operatorname{Hom}_{R}(Y,X)\in\,^{*}{\mathcal{C}}(S)$, and there
is an isomorphism of $S$-complexes
$\rho_{ZYX}:^{*}\operatorname{Hom}_{R}(Z\otimes_{S}Y,X)\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\\!^{*}\operatorname{Hom}_{S}(Z,\\!^{*}\operatorname{Hom}_{R}(Y,X)),$
which is natural in $Z,Y$ and $X$.
∗Tensor-Evaluation. Let $Z,Y\in\,^{*}{\mathcal{C}}(S)$ and
$X\in\,^{*}{\mathcal{C}}(R)$. Then
${}^{*}\operatorname{Hom}_{S}(Z,Y)\in\,^{*}{\mathcal{C}}(R)$ and
$Y\otimes_{R}X\in\,^{*}{\mathcal{C}}(S)$, and there is a natural morphism of
$S$-complexes
$\omega_{ZYX}:^{*}\operatorname{Hom}_{S}(Z,Y)\otimes_{R}X\longrightarrow\\!^{*}\operatorname{Hom}_{S}(Z,Y\otimes_{R}X).$
The morphism is invertible under each of the next two extra conditions
* (i)
$Z\in\\!^{*}\mathcal{C}_{\square}^{fp}(S)$,
$Y\in\\!^{*}\mathcal{C}_{\sqsupset}(S)$, and
$X\in\\!^{*}\mathcal{C}_{\sqsupset}(R)$; or
* (ii)
$Z\in\\!^{*}\mathcal{C}_{\square}^{fp}(S)$,
$Y\in\\!^{*}\mathcal{C}_{\sqsubset}(S)$, and
$X\in\\!^{*}\mathcal{C}_{\square}(R)$.
∗Hom-Evaluation. Let $Z,Y\in\,^{*}{\mathcal{C}}(S)$ and
$X\in\,^{*}{\mathcal{C}}(R)$. Then
${}^{*}\operatorname{Hom}_{S}(Z,Y)\in\,^{*}{\mathcal{C}}(R)$ and
$\,{}^{*}\operatorname{Hom}_{R}(Y,X)\in\,^{*}{\mathcal{C}}(S)$, and there is a
natural morphism of $S$-complexes
$\theta_{ZYX}:Z\otimes_{S}\\!^{*}\operatorname{Hom}_{R}(Y,X)\longrightarrow\\!^{*}\operatorname{Hom}_{R}(\\!^{*}\operatorname{Hom}_{S}(Z,Y),X).$
The morphism is invertible under each of the next two extra conditions
* (i)
$Z\in\\!^{*}\mathcal{C}_{\square}^{fp}(S)$,
$Y\in\\!^{*}\mathcal{C}_{\sqsupset}(S)$, and
$X\in\\!^{*}\mathcal{C}_{\sqsubset}(R)$; or
* (ii)
$Z\in\\!^{*}\mathcal{C}_{\sqsupset}^{fp}(S)$,
$Y\in\\!^{*}\mathcal{C}_{\sqsubset}(S)$, and
$X\in\\!^{*}\mathcal{C}_{\square}(R)$.
By $Z\in\\!^{*}\mathcal{C}^{fp}(S)$ we mean that $Z$ consists of finitely
generated projective $S$-modules.
We recall the definition of the _depth_ and _width_ of complexes. Let
$\mathfrak{a}$ be an ideal in a ring $R$ and $X$ a complex of graded
$R$-modules. The $\mathfrak{a}$-$0pt$ and $\mathfrak{a}$-$0pt$ of $X$ over $R$
are defined respectively by
$\displaystyle 0pt(\mathfrak{a},X):=$
$\displaystyle-\sup{\mathbf{R}}\operatorname{Hom}_{R}(R/{\mathfrak{a}},X),$
$\displaystyle 0pt(\mathfrak{a},X):=$
$\displaystyle\inf(R/{\mathfrak{a}}\otimes^{\mathbf{L}}_{R}X).$
If $(R,\mathfrak{m})$ is a local ring then set $0pt_{R}X:=0pt(\mathfrak{m},X)$
and $0pt_{R}X:=0pt(\mathfrak{m},X)$.
Now let $(R,\mathfrak{m})$ be a $\,{}^{*}$local graded ring and $X$ be a
complex of graded $R$-modules. By [3, Proposition 1.5.15(c)],
$-\otimes_{R}R_{\mathfrak{m}}$ is a faithfully exact functor on the category
of graded $R$-modules. Then we have
$\displaystyle 0pt(\mathfrak{m},X)=$
$\displaystyle\inf\\{i|\mbox{H}_{i}(R/\mathfrak{m}\otimes^{\mathbf{L}}_{R}X)\neq
0\\}$ $\displaystyle=$
$\displaystyle\inf\\{i|\mbox{H}_{i}(R/\mathfrak{m}\otimes^{\mathbf{L}}_{R}X)\otimes_{R}R_{\mathfrak{m}}\neq
0\\}$ $\displaystyle=$
$\displaystyle\inf\\{i|\mbox{H}_{i}(R_{\mathfrak{m}}/\mathfrak{m}R_{\mathfrak{m}}\otimes^{\mathbf{L}}_{R_{\mathfrak{m}}}X_{\mathfrak{m}})\neq
0\\}$ $\displaystyle=$ $\displaystyle
0pt(\mathfrak{m}R_{\mathfrak{m}},X_{\mathfrak{m}})=0pt_{R_{\mathfrak{m}}}X_{\mathfrak{m}}.$
Likewise we have $0pt(\mathfrak{m},X)=0pt_{R_{\mathfrak{m}}}X_{\mathfrak{m}}$.
###### Proposition 2.2.
Let $(R,\mathfrak{m},k)$ be a ∗local ring and $X\in\mathcal{C}_{\sqsupset}(R)$
and set $E:=\\!^{*}\operatorname{E}_{R}(k)$. Then
$0pt(\mathfrak{m},X)=0pt(\mathfrak{m},\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(X,E)).$
###### Proof.
We have the following computations:
$\displaystyle 0pt(\mathfrak{m},X)=$
$\displaystyle\inf(k\otimes^{\mathbf{L}}_{R}X)=-\sup\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(k\otimes^{\mathbf{L}}_{R}X,E)$
$\displaystyle=$
$\displaystyle-\sup\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(k,\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(X,E))$
$\displaystyle=$ $\displaystyle
0pt(\mathfrak{m},\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(X,E)).$
The second equality hods since $E$ is faithfully ∗injective and the third one
uses the ∗adjointness isomorphism. ∎
The following lemma is the graded version of one of Foxby’s accounting
principles [5, Lemma A.7.9].
###### Lemma 2.3.
Let $(R,\mathfrak{m},k)$ be a ∗local ring. Then
* (a)
If $X\in\,^{*}\mathcal{C}(k)$, then there is a quasiisomorphism
$\mbox{H}(X)\rightarrow X$.
* (b)
If $X\,^{*}\mathcal{C}_{(\sqsubset)}(R)$ and $W\in\,^{*}\mathcal{C}(k)$ then
$\inf\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(W,X)=\inf\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(k,X)-\sup
W.$
###### Proof.
Part $(a)$ is easy since very graded $k$-module is free by [3, Exercise
1.5.20]. For part $(b)$ note that $W=k\otimes^{\mathbf{L}}_{k}W$. Then using
the ∗adjointness isomorphism we have
$\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(W,X)=\mathbf{R}\\!^{*}\operatorname{Hom}_{k}(W,\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(k,X))=\\!^{*}\operatorname{Hom}_{k}(W,\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(k,X)).$
Thus using part $(a)$ we have
$\displaystyle\inf\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(W,X)=$
$\displaystyle\inf\\{\ell|\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(W,X)_{\ell}\neq
0\\}$ $\displaystyle=$
$\displaystyle\inf\\{\ell|\,^{*}\prod_{i+j=\ell}\\!^{*}\operatorname{Hom}_{k}(W_{-j},\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(k,X)_{i})\neq
0\\}$ $\displaystyle=$
$\displaystyle\inf\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(k,X)-\sup W.$
This completes the proof. ∎
The following equality is the graded version of [20, Lemma 2.6].
###### Proposition 2.4.
Let $(R,\mathfrak{m},k)$ be a ∗local ring
$X,Y\in\,^{*}\mathcal{C}_{\square}(R)$, then
$0pt(\mathfrak{m},\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(X,Y))=0pt(\mathfrak{m},X)+0pt(\mathfrak{m},Y)-0pt(\mathfrak{m},R).$
###### Proof.
Use Lemma 2.3 and the same argument as [20, Lemma 2.6]. ∎
Let $a\in R$ be homogeneous and set $\alpha=\deg(a)$. Then the complex $0\to
R(-\alpha)\stackrel{{\scriptstyle a}}{{\to}}R\to 0$ concentrated in degrees 1
and 0 is called the _Koszul complex_ of $a$, and denoted by $K(a)$, where
$R(-\alpha)$ denotes the graded $R$-module with grading given by
$R(-\alpha)_{n}=R_{n-\alpha}$. Note that $K(a)\in^{*}\mathcal{C}(R)$. Now let
$\mathfrak{a}$ be a homogeneous ideal of $R$ and $a_{1},\cdots,a_{n}$ be a set
of generators of $\mathfrak{a}$ by homogeneous elements. The Koszul complex of
$\mathfrak{a}$, denoted by $K:=K(\mathfrak{a})$, and define as
$K(a_{1})\otimes_{R}\cdots\otimes_{R}K(a_{n})$. It is shown in [12] that
$0pt(\mathfrak{a},X)=\inf(K\otimes_{R}X)$.
Let $\alpha:X\to Y$ be a homogeneous morphism between $R$-complexes of graded
$R$-modules. The mapping cone of $\alpha$ is a complex of graded $R$-modules
given by
$\mathcal{M}(\alpha)_{\ell}:Y_{\ell}\oplus X_{\ell-1}$
and
$\delta_{\ell}^{\mathcal{M}(\alpha)}(y_{\ell},x_{\ell-1})=(\delta_{\ell}^{Y}(y_{\ell})+\alpha_{\ell-1}(x_{\ell-1}),-\delta_{\ell-1}^{X}(x_{\ell-1})).$
Note that $\delta_{\ell}^{\mathcal{M}(\alpha)}$ is a homogeneous
differentiation from $Y_{\ell}\oplus X_{\ell-1}$ to $Y_{\ell-1}\oplus
X_{\ell-2}$. It is easy to see that the morphism $\alpha:X\to Y$ between
complexes of graded $R$-modules is quasi-isomorphism if and only if the
mapping cone of $\alpha$ is homologically trivial (see [5, Lemma A.1.19]).
Also for the covariant and contravariant functors
${}^{*}\operatorname{Hom}_{R}(V,-)$ and ${}^{*}\operatorname{Hom}_{R}(-,W)$ we
have the following:
$\mathcal{M}(\,^{*}\operatorname{Hom}_{R}(V,\alpha))=\,^{*}\operatorname{Hom}_{R}(V,\mathcal{M}(\alpha)),$
$\mathcal{M}(\,^{*}\operatorname{Hom}_{R}(\alpha,W))=\Sigma^{1}\,{}^{*}\operatorname{Hom}_{R}(\mathcal{M}(\alpha),W).$
See [5, A2.1.2 and A.2.1.4].
The ungraded version of the following result contained in [7, Proposition
2.7].
###### Proposition 2.5.
Let $\mathfrak{B}$ be a class of graded $R$-modules, and $\alpha:X\to Y$ be a
quasiisomorphism between complexes of graded $R$-modules such that
${}^{*}\operatorname{Hom}_{R}(\alpha,V):\,^{*}\operatorname{Hom}_{R}(Y,V)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\,^{*}\operatorname{Hom}_{R}(X,V)$
is quasiisomorphism for every module $V\in\mathfrak{B}$. Let
$\widetilde{V}\in\,^{*}\mathcal{C}(R)$ be a complex consisting of modules from
$\mathfrak{B}$. Then the induced morphism,
${}^{*}\operatorname{Hom}_{R}(\alpha,\widetilde{V}):\,^{*}\operatorname{Hom}_{R}(Y,\widetilde{V})\longrightarrow\,^{*}\operatorname{Hom}_{R}(X,\widetilde{V}),$
is a quasiisomorphism, provided that either
* (a)
$\widetilde{V}\in\,^{*}\mathcal{C}_{\sqsubset}(R)$, or
* (b)
$X,Y\in\,^{*}\mathcal{C}_{\sqsubset}(R)$.
###### Proof.
In either cases it is enough to show that
${}^{*}\operatorname{Hom}_{R}(\mathcal{M}(\alpha),\widetilde{V})$ is
homologically trivial. On the other hand by graded version of [7, Lemma 2.5]
the result holds if
${}^{*}\operatorname{Hom}_{R}(\mathcal{M}(\alpha),\widetilde{V}_{\ell})$ is
homologically trivial for each $\ell\in\mathbb{Z}$ which this is our
assumption. ∎
## 3\. ∗injective dimension
The injective dimension of a complex $X$ is defined and studied in [1],
denoted by $\operatorname{id}_{R}X$. A graded module $J$ is called ∗injective
if it is an injective object in the category of graded modules. The injective
dimension of a graded module $M$ in the category of graded modules, is denoted
by ${}^{*}\operatorname{id}_{R}M$ (cf. [10, 15, 3]). The ∗injective dimension
of a complex of graded modules $X$ is studied in [15, Page 83]. Let
$n\in\mathbb{Z}$. A homologically left bounded complex of graded modules $X$,
is said to have ∗injective dimension at most $n$, denoted by
${}^{*}\operatorname{id}_{R}X\leq n$, if there exists an ∗injective resolution
$X\to I$, such that $I_{i}=0$ for $i<-n$. If ${}^{*}\operatorname{id}_{R}X\leq
n$ holds, but ${}^{*}\operatorname{id}_{R}X\leq n-1$ does not, we write
${}^{*}\operatorname{id}_{R}X=n$. If ${}^{*}\operatorname{id}_{R}X\leq n$ for
all $n\in\mathbb{Z}$ we write ${}^{*}\operatorname{id}_{R}X=-\infty$. If
${}^{*}\operatorname{id}_{R}X\leq n$ for no $n\in\mathbb{Z}$ we write
${}^{*}\operatorname{id}_{R}X=\infty$. The following theorem inspired by [1,
Theorem 2.4.I and Corollary 2.5.I].
###### Theorem 3.1.
For $X\in$ ${}^{*}\mathcal{D}_{\sqsubset}(R)$ and $n\in\mathbb{Z}$ the
following are equivalent:
* (1)
${}^{*}\operatorname{id}_{R}X\leq n.$
* (2)
$n\geq-\sup U-\inf(\mathbf{R}^{*}\operatorname{Hom}_{R}(U,X))$ for all $U\in$
${}^{*}\mathcal{D}_{\square}(R)$ and $\mbox{H}(U)\neq 0$.
* (3)
$n\geq-\inf X$ and ${}^{*}\operatorname{Ext}^{n+1}_{R}(R/J,X)=0$ for every
homogeneous ideal $J$ of $R$.
* (4)
$n\geq-\inf X$ and for any (resp. some) ∗injective resolution $I$ of $X$, the
graded $R$-module $\operatorname{Ker}(\partial_{-n}:I_{-n}\to I_{-n-1})$ is
∗injective.
Moreover the following hold:
${}^{*}\operatorname{id}_{R}X=$
$\displaystyle\sup\\{j\in\mathbb{Z}|^{*}\operatorname{Ext}^{j}_{R}(R/J,X)\neq
0\text{ for some homogeneous ideal }J\\}$ $\displaystyle=$
$\displaystyle\sup\\{-\sup(U)-\inf(\mathbf{R}^{*}\operatorname{Hom}_{R}(U,X))|U\ncong
0\text{ in }^{*}\mathcal{D}_{\square}(R)\\}.$
###### Proof.
$(1)\Rightarrow(2)$ Let $t:=\sup U$ and $I$ be an ∗injective resolution of
$X$, such that, for all $i<-n$, $I_{i}=0$. Then we have
${}^{*}\operatorname{Ext}^{i}_{R}(U,X)\cong\mbox{H}_{-i}(^{*}\operatorname{Hom}_{R}(U,I)).$
Since ${}^{*}\operatorname{Hom}_{R}(U,I)_{-i}=0$ for $-i<-n-t$, the assertion
follows.
$(2)\Rightarrow(3)$ It is trivial that
${}^{*}\operatorname{Ext}^{n+1}_{R}(R/J,X)=0$ for every homogeneous ideal $J$
of $R$. For the second assertion let $U=R$ in (2). So that
$\operatorname{Ext}^{i}_{R}(R,X)=\\!^{*}\operatorname{Ext}^{i}_{R}(R,X)=0$ for
$i>n$. Now by [1, Lemma 1.9(b)], we have $\mbox{H}_{-i}(X)=0$ for $-i<-n$.
This means that $n\geq-\inf X$.
$(3)\Rightarrow(4)$ By hypothesis of (4) $\mbox{H}_{i}(I)=0$ for $i<-n$. Thus
the complex
$\cdots\to 0\to 0\to I_{-n}\to I_{-n-1}\to\cdots\to I_{i}\to
I_{i-1}\to\cdots,$
gives an ∗injective resolution of $\operatorname{Ker}\partial_{-n}$. In
particular
$\\!{}^{*}\operatorname{Ext}^{1}_{R}(R/J,\operatorname{Ker}\partial_{-n})=\mbox{H}_{-n-1}\\!^{*}\operatorname{Hom}_{R}(R/J,I)=\\!^{*}\operatorname{Ext}^{n+1}_{R}(R/J,X)=0$
for every homogeneous ideal $J$ of $R$. Thus $\operatorname{Ker}\partial_{-n}$
is ∗injective by [10, Corollary 4.3].
$(4)\Rightarrow(1)$ Let $I$ be any ∗injective resolution of $X$. By (5) we
have $\operatorname{Ker}\partial_{-n}$ is ∗injective. Thus
$\\!{}^{*}\operatorname{id}_{R}X<-n$ by definition.
The next two equalities are trivial. ∎
For a local ring $(R,\mathfrak{m},k)$ and for an $R$-complex $X$ and
$i\in\mathbb{Z}$ the $i$th Bass number and Betti number of $X$ are defined
respectively by
$\mu^{i}_{R}(X):=\operatorname{dim}_{k}\mbox{H}_{-i}({\mathbf{R}}\operatorname{Hom}_{R}(k,X))$
and
$\beta_{i}^{R}(X):=\operatorname{dim}_{k}\mbox{H}_{i}(k\otimes_{R}^{\mathbf{L}}X).$
It is well-known that for $X\in{\mathcal{D}}_{\sqsubset}(R)$ one has (cf. [1,
Proposition 5.3.I])
$\operatorname{id}_{R}X=\sup\\{m\in\mathbb{Z}|\exists\mathfrak{p}\in\operatorname{Spec}(R);\mu^{m}_{R_{\mathfrak{p}}}(X_{\mathfrak{p}})\neq
0\\}.$
As a graded analogue we have:
###### Proposition 3.2.
For $X\in$ ${}^{*}\mathcal{D}_{\sqsubset}(R)$ we have the following equality
${}^{*}\operatorname{id}_{R}X=\sup\\{m\in\mathbb{Z}|\exists\mathfrak{p}\in^{*}\operatorname{Spec}(R);\mu^{m}_{R_{\mathfrak{p}}}(X_{\mathfrak{p}})\neq
0\\}.$
###### Proof.
The argument is the same as proof of [1, Proposition 5.3.I] with some changes.
Denote the supremum by $i$. By Theorem 3.1, we have
$\\!{}^{*}\operatorname{id}_{R}X\geq i$. Hence the equality holds if
$i=\infty$. Thus assume that $i$ is finite. By Theorem 3.1 we have to show
that if $\\!{}^{*}\operatorname{Ext}^{j}_{R}(M,X)\neq 0$ for some finitely
generated graded $R$-module $M$, then $j\leq i$; this implies that
$\\!{}^{*}\operatorname{id}_{R}X\leq i$. The elements of $Ass(M)$ are
homogeneous prime ideals. Thus we have a filtration $0=M_{0}\subset
M_{1}\subset\cdots\subset M_{t}=M$ of graded submodules of $M$ such that for
each $i$ we have $M_{i}/M_{i-1}\cong R/\mathfrak{p}_{i}$ with
$\mathfrak{p}_{i}\in\operatorname{Supp}M$ and is homogeneous. From the long
exact sequence of $\\!{}^{*}\operatorname{Ext}^{j}_{R}(-,X)\neq 0$ we have the
set
$\\{\mathfrak{q}\in\\!\operatorname{Spec}(R)|\text{ there is an }h\geq j\text{
such that }\\!^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{q},X)\neq 0\\},$
is not empty. Let $\mathfrak{p}$ maximal in this set, and for a homogeneous
$x\in R\backslash\mathfrak{p}$ consider the exact sequence
$0\to R/\mathfrak{p}\stackrel{{\scriptstyle x}}{{\to}}R/\mathfrak{p}\to
R/(\mathfrak{p}+Rx)\to 0.$
It induces an exact sequence
$\\!{}^{*}\operatorname{Ext}^{h}_{R}(R/(\mathfrak{p}+Rx),X)\to\\!^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{p},X)\stackrel{{\scriptstyle
x}}{{\to}}\\!^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{p},X)\to\\!^{*}\operatorname{Ext}^{h+1}_{R}(R/(\mathfrak{p}+Rx),X)$
in which the left-hand term is trivial because of the maximality of
$\mathfrak{p}$. Thus
$\\!{}^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{p},X)\stackrel{{\scriptstyle
x}}{{\to}}\\!^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{p},X)$ is injective
for all homogeneous elements $x\in R\backslash\mathfrak{p}$, hence so is the
homogeneous localization homomorphism
$\\!{}^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{p},X)\to\\!^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{p},X)_{(\mathfrak{p})}$.
Thus the free $R_{(\mathfrak{p})}/\mathfrak{p}R_{(\mathfrak{p})}$-module
$\\!{}^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{p},X)_{(\mathfrak{p})}$ is
nonzero. Consequently
$(\\!^{*}\operatorname{Ext}^{h}_{R}(R/\mathfrak{p},X)_{(\mathfrak{p})})_{\mathfrak{p}R_{(\mathfrak{p})}}\cong\\!^{*}\operatorname{Ext}^{h}_{R_{\mathfrak{p}}}(R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}},X_{\mathfrak{p}})$
is nonzero. This implies that $j\leq h\leq i$. ∎
###### Remark 3.3.
(1) A graded module is called ∗projective if it is a projective object in the
category of graded modules. By [10, Proposition 3.1] the ∗projective graded
$R$-modules coincide with projective $R$-modules. The projective dimension of
a graded module $M$ in the category of graded modules, is denoted by
${}^{*}\operatorname{pd}_{R}M$ (cf. [10]). Let $n\in\mathbb{Z}$. A
homologically right bounded complex of graded modules $X$, is said to have
∗projective dimension at most $n$, denoted by
${}^{*}\operatorname{pd}_{R}X\leq n$, if there exists a ∗projective resolution
$P\to X$, such that $P_{i}=0$ for $i>n$. If ${}^{*}\operatorname{pd}_{R}X\leq
n$ holds, but ${}^{*}\operatorname{pd}_{R}X\leq n-1$ does not, we write
${}^{*}\operatorname{pd}_{R}X=n$. If ${}^{*}\operatorname{pd}_{R}X\leq n$ for
all $n\in\mathbb{Z}$ we write ${}^{*}\operatorname{pd}_{R}X=-\infty$. If
${}^{*}\operatorname{pd}_{R}X\leq n$ for no $n\in\mathbb{Z}$ we write
${}^{*}\operatorname{pd}_{R}X=\infty$.
(2) For $X\in\\!^{*}\mathcal{D}_{\sqsupset}(R)$ by the same method as in [1,
Theorem 2.4.P and Corollary 2.5.P] we have
${}^{*}\operatorname{pd}_{R}X=$
$\displaystyle\sup\\{j\in\mathbb{Z}|^{*}\operatorname{Ext}^{j}_{R}(X,N)\neq
0\text{ for some graded }R\text{-module }N\\}$ $\displaystyle=$
$\displaystyle\sup\\{\inf(U)-\inf(\mathbf{R}^{*}\operatorname{Hom}_{R}(X,U))|U\ncong
0\text{ in }^{*}\mathcal{D}_{\square}(R)\\}.$
(3) It is easy to see that for $X\in\\!^{*}\mathcal{D}_{\sqsupset}(R)$, we
have $\\!{}^{*}\operatorname{pd}_{R}X\leq\operatorname{pd}_{R}X$.
The proof of the following proposition is easy so we omit it (see [3, Theorem
1.5.9]). Let $J$ be an ideal of the graded ring $R$. Then the graded ideal
$J^{*}$ is denoted to the ideal generated by all homogeneous elements of $J$.
It is well-known that if $\mathfrak{p}$ is a prime ideal of $R$, then
$\mathfrak{p}^{*}$ is a homogeneous prime ideal of $R$ by [3, Lemma 1.5.6].
###### Proposition 3.4.
Assume that $X\in\,^{*}{\mathcal{D}}_{\square}(R)$ and $\mathfrak{p}$ is a non
homogeneous prime ideal in $R$. Then
$\mu^{i+1}_{R_{\mathfrak{p}}}(X_{\mathfrak{p}})=\mu^{i}_{R_{\mathfrak{p}^{*}}}(X_{\mathfrak{p}^{*}})$
and
$\beta_{i}^{R_{\mathfrak{p}}}(X_{\mathfrak{p}})=\beta_{i}^{R_{\mathfrak{p}^{*}}}(X_{\mathfrak{p}^{*}})$for
any integer $i\geq 0$.
###### Corollary 3.5.
Let $X\in\,^{*}{\mathcal{D}}_{\square}(R)$ and $\mathfrak{p}$ be a non-
homogeneous prime ideal in $R$. Then
$0ptX_{\mathfrak{p}}=0ptX_{\mathfrak{p}^{*}}+1.$
###### Proof.
Using Proposition 3.4, we can assume that both $0ptX_{\mathfrak{p}}$ and
$0ptX_{\mathfrak{p}^{*}}$ are finite. So the equality follows from the fact
that over a local ring $(R,\mathfrak{m},k)$ we have
$0pt_{R}X=\inf\\{i\in\mathbb{Z}|\mu^{i}_{R}(X)\neq 0\\}$. ∎
Foxby in [11] defined the _small support_ of a homologically right bounded
complex $X$ over a Noetherian ring $R$, denoted by $\mbox{supp}_{R}X$, as
$\operatorname{supp}_{R}X=\\{\mathfrak{p}\in\operatorname{Spec}R|\exists
m\in\mathbb{Z}:\beta_{m}^{R_{\mathfrak{p}}}(X_{\mathfrak{p}})\neq 0\\}.$
Let $\,{}^{*}\operatorname{supp}_{R}X$ be a subset of
$\operatorname{supp}_{R}X$ consisting of homogeneous prime ideals of
$\operatorname{supp}_{R}X$. Then from Proposition 3.4 we see that
$\mathfrak{p}\in\operatorname{supp}_{R}X$ if and only if
$\mathfrak{p}^{*}\in\,^{*}\operatorname{supp}_{R}X$. Also using [11,
Proposition 2.8] and Corollary 3.5 (or directly from Proposition 3.4) we have
$0pt_{R_{\mathfrak{p}}}X_{\mathfrak{p}}<\infty\Leftrightarrow
0pt_{R_{\mathfrak{p}^{*}}}X_{\mathfrak{p}^{*}}<\infty.$
###### Proposition 3.6.
Assume that $X\in\,^{*}{\mathcal{D}}_{\square}(R)$, and $\mathfrak{p}$ is a
non homogeneous prime ideal in $R$. Then
$0pt_{R_{\mathfrak{p}}}X_{\mathfrak{p}}=0pt_{R_{\mathfrak{p}^{*}}}X_{\mathfrak{p}^{*}}$
###### Proof.
We can assume that both $0ptX_{\mathfrak{p}}$ and $0ptX_{\mathfrak{p}^{*}}$
are finite numbers. And the argument is dual to the proof of [3, Theorem
1.5.9]. ∎
The ungraded version of the following theorem was proved for modules by
Chouinard [4, Corollary 3.1] and extended to complexes by Yassemi [20, Theorem
2.10].
###### Theorem 3.7.
Let $X\in$ $\,{}^{*}\mathcal{D}_{\square}(R)$. If
$\,{}^{*}\operatorname{id}_{R}X<\infty$ then
${}^{*}\operatorname{id}_{R}X=\sup\\{0ptR_{\mathfrak{p}}-0ptX_{\mathfrak{p}}|\mathfrak{p}\in\,^{*}\\!\operatorname{Spec}(R)\\}.$
###### Proof.
We have the following computations
${}^{*}\operatorname{id}_{R}X=$
$\displaystyle\sup\\{m\in\mathbb{Z}|\exists\mathfrak{p}\in\,^{*}\operatorname{Spec}(R):\mu^{m}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\neq
0\\}$ $\displaystyle=$
$\displaystyle\sup\\{m\in\mathbb{Z}|\exists\mathfrak{p}\in\,^{*}\\!\operatorname{Spec}(R):\mbox{H}_{m}({\mathbf{R}}\operatorname{Hom}_{R_{\mathfrak{p}}}(\kappa(p),M_{\mathfrak{p}}))\neq
0\\}$ $\displaystyle=$
$\displaystyle\sup\\{-\inf{\mathbf{R}}\operatorname{Hom}_{R_{\mathfrak{p}}}(\kappa(\mathfrak{p}),M_{\mathfrak{p}})|\mathfrak{p}\in\,^{*}\\!\operatorname{Spec}(R)\\}$
$\displaystyle=$
$\displaystyle\sup\\{0ptR_{\mathfrak{p}}-0pt_{R_{\mathfrak{p}}}M_{\mathfrak{p}}|\mathfrak{p}\in\,^{*}\\!\operatorname{Spec}(R)\\}.$
The first equality is by Proposition 3.2 and the last one is by [20, Lemma
2.6(a)]. ∎
The following corollary was already known for graded modules in [10, Corollary
4.12].
###### Corollary 3.8.
For every $X\in\,^{*}\mathcal{D}_{\square}(R)$, we have
$\,{}^{*}\operatorname{id}_{R}X\leq\operatorname{id}_{R}X\leq\,^{*}\operatorname{id}_{R}X+1.$
###### Proof.
First of all note that by proposition 3.4, $\operatorname{id}_{R}X<\infty$ if
and only if $\,{}^{*}\operatorname{id}_{R}X<\infty$. The first inequality is
clear by Theorem 3.7 and [20, Theorem 2.10]. For the second one let
$\mathfrak{p}\in\operatorname{Spec}R$ be such that
$\operatorname{id}_{R}X=0ptR_{\mathfrak{p}}-0pt_{R_{\mathfrak{p}}}M_{\mathfrak{p}}$
by [20, Theorem 2.10]. By Corollary 3.5 and Proposition 3.6 we have
$0ptR_{\mathfrak{p}}-0pt_{R_{\mathfrak{p}}}M_{\mathfrak{p}}\leq
0ptR_{\mathfrak{p}^{*}}-0pt_{R_{\mathfrak{p}^{*}}}M_{\mathfrak{p}^{*}}+1\leq\,^{*}\operatorname{id}_{R}X+1,$
where the second inequality holds by Theorem 3.7. ∎
Here we define the $\\!{}^{*}$dualizing complex for a graded ring and prove
some related results that we need in the next section.
###### Definition 3.9.
A ∗dualizing complex for a graded ring $R$ is a homologically finite and
bounded complex $D$, of graded $R$-modules, such that
$\,{}^{*}\operatorname{id}_{R}D<\infty$ and the homothety morphism
$\psi:R\to\textbf{R}\,^{*}\operatorname{Hom}_{R}(D,D)$ is invertible in
$\,{}^{*}\mathcal{D}(R)$.
###### Corollary 3.10.
Any ∗dualizing complex for $R$ is a dualizing complex for $R$.
The proof of the following lemma is the same as [15, Chapter V, Proposition
3.4].
###### Lemma 3.11.
Let $(R,\mathfrak{m},k)$ be a ∗local ring and that $D$ is a ∗dualizing complex
for $R$. Then there exists an integer $t$ such that
$\mbox{H}^{t}(\textbf{R}\,^{*}\operatorname{Hom}_{R}(k,D))\cong k$ and
$\mbox{H}^{i}(\textbf{R}\,^{*}\operatorname{Hom}_{R}(k,D))=0$ for $i\neq t$.
Assume that $(R,\mathfrak{m})$ is a ∗local ring. A ∗dualizing complex $D$ is
said to be _normalized ∗dualizing complex_, if $t=0$ in the lemma. It is easy
to see that a suitable shift of any ∗dualizing complex is a normalized one.
Also using [15, Chapter V, Proposition 3.4] we see that if $D$ is a normalized
∗dualizing complex for $(R,\mathfrak{m})$, then $D_{\mathfrak{m}}$ is a
normalized dualizing complex for $R_{\mathfrak{m}}$.
###### Lemma 3.12.
Let $(R,\mathfrak{m},k)$ be a ∗local ring and that $D$ is a normalized
∗dualizing complex for $R$. Then there exists a natural functorial isomorphism
on the category of graded modules of finite length to itself
$\phi:\mbox{H}^{0}(\textbf{R}\,^{*}\operatorname{Hom}_{R}(-,D))\to\,^{*}\operatorname{Hom}_{R}(-,\\!^{*}\operatorname{E}_{R}(k)),$
where $\\!{}^{*}E_{R}(k)$ is the ∗injective envelope of $k$ over $R$.
###### Proof.
Since $D$ is a normalized ∗dualizing complex for $R$,
$T:=\mbox{H}^{0}(\textbf{R}\,^{*}\operatorname{Hom}_{R}(-,D))$ is an additive
contravariant exact functor from the category of graded modules of finite
length to itself. Let $M$ be a graded $R$-module and $m\in M$ is homogeneous
element of degree $\alpha$. Then $\epsilon_{m}:R(-\alpha)\to M$ is a
homogeneous morphism which sends $1$ into $m$. Thus we have a homogeneous
morphism $\phi(M):T(M)\to\\!^{*}\operatorname{Hom}_{R}(M,T(R))$ which sends a
homogeneous element $x\in T(M)$ to a morphism
$f_{x}\in^{*}\operatorname{Hom}_{R}(M,T(R))$ such that
$f_{x}(m)=T(\epsilon_{m})(x)$ for every homogeneous element $m\in M$. It is
easy to see that it is functorial on $M$. Thus we showed that there is a
natural functorial morphism $\phi:T\to\\!^{*}\operatorname{Hom}_{R}(-,T(R))$.
Therefore by the same method of [14, Lemma 4.4 and Propositions 4.5], there is
a functorial isomorphism
$\phi:\mbox{H}^{0}(\textbf{R}\,^{*}\operatorname{Hom}_{R}(-,D))\to\,^{*}\operatorname{Hom}_{R}(-,\,^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n})),$
from the category of graded modules of finite length to itself. Using the
technique of proof of [14, Proposition 4.7] in conjunction with [10, Corollary
4.3], we see that
$\,{}^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n})$ is an
∗injective $R$-module. Since $D$ is a normalized ∗dualizing complex for $R$ we
have
$\,{}^{*}\operatorname{Hom}_{R}(k,\,^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n}))\cong\mbox{H}^{0}(\textbf{R}\,^{*}\operatorname{Hom}_{R}(k,D))\cong
k.$
Thus in particular we can embed $k$ to
$\,{}^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n})$. To show
that $\,{}^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n})$ is an
∗essential extension of $k$, let $Q$ be a graded submodule of
$\,{}^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n})$ such that
$k\cap Q=0$. Then ${}^{*}\operatorname{Hom}_{R}(k,Q)$ can be embed in
${}^{*}\operatorname{Hom}_{R}(k,\,^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n}))\cong
k.$
Therefore ${}^{*}\operatorname{Hom}_{R}(k,Q)=0$. On the other hand
$\operatorname{Ass}(T(R/\mathfrak{m}^{n}))$ is in $V(\mathfrak{m})$ for each
$n\in\mathbb{N}$. Now by [19, Proposition 2.1], the fact that each prime ideal
of
$\operatorname{Ass}(\,^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n}))$
is the annihilator of a homogeneous element [3, Lemma 1.5.6], and the
definition of $\,{}^{*}\displaystyle\lim_{\longrightarrow}$, we have
$\operatorname{Ass}(\,^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n}))\subseteq\bigcup_{n\in\mathbb{N}}\operatorname{Ass}(T(R/\mathfrak{m}^{n}))\subseteq
V(\mathfrak{m}).$
Consequently $Q$ has support in $V(\mathfrak{m})$, so that $Q=0$. Therefore
$\,{}^{*}\displaystyle\lim_{\longrightarrow}T(R/\mathfrak{m}^{n})\cong\\!^{*}\operatorname{E}_{R}(k)$.
∎
Let $\mathfrak{a}$ be an ideal of $R$. The right derived _local cohomology
functor_ with support in $\mathfrak{a}$ is denoted by
$\mathbf{R}\Gamma_{\mathfrak{a}}(-)$. Its right adjoint,
$\mathbf{L}\Lambda^{\mathfrak{a}}(-)$, is the left derived _local homology
functor_ with support in $\mathfrak{a}$ (see [13] for detail).
Now we have the following proposition, which its proof use Lemma 3.12, and the
argument is the same as [15, Chapter V, Proposition 6.1].
###### Proposition 3.13.
Let $(R,\mathfrak{m},k)$ be a ∗local ring and that $D$ be a normalized
∗dualizing complex for $R$. Then
${\mathbf{R}}\Gamma_{\mathfrak{m}}(D)\simeq\\!^{*}\operatorname{E}_{R}(k)$.
## 4\. ∗Gorenstein injective dimension
In this section we introduce the concept of $\\!{}^{*}$Gorenstein injective
dimension of complexes and we derive its main properties. In particular we
prove a Chouinard’s like formula for this dimension, and compare it with the
usual Gorenstein injective dimension.
###### Definition 4.1.
A graded $R$-module $N$ is called $\,{}^{*}$Gorenstein injective, if there
exists an acyclic complex I of $\,{}^{*}$injective $R$-modules and homogeneous
homomorphisms such that $M\cong\operatorname{Ker}(I^{0}\to I^{1})$ and for
every $\,{}^{*}$injective module $E$, the complex
$\,{}^{*}\operatorname{Hom}_{R}(E,\textbf{I})$ is exact.
It is clear that every ∗injective $R$-module is $\,{}^{*}$Gorenstein
injective. So that every $Y\in\,^{*}\mathcal{D}_{\sqsubset}(R)$ has a
$\,{}^{*}$Gorenstein injective resolution. The ∗Gorenstein injective dimension
of $Y\in\,^{*}\mathcal{D}_{\sqsubset}(R)$ denoted by
$\,{}^{*}\operatorname{Gid}_{R}Y$, is define as:
$\,{}^{*}\operatorname{Gid}_{R}Y:=\inf\left\\{\sup\\{\ell\in Z|B_{-\ell}\neq
0\\}\bigg{|}\begin{array}[]{l}B_{\ell}\text{ is }\,^{*}\text{Gorenstein
injective and}\\\ B\in\,^{*}\mathcal{D}_{\sqsubset}(R)\text{ is isomorphic to
}Y\end{array}\right\\}.$
By a careful revision of the proof of dual version of [16, Theorems 2.5 and
2.20] we have the following two results.
###### Theorem 4.2.
The class of ∗Gorenstein injective $R$-modules is ∗injectively resolving, that
is for any short exact sequence $0\to X^{\prime}\to X\to X^{\prime\prime}\to
0$ with $X^{\prime}$ ∗Gorenstein injective $R$-module, $X^{\prime\prime}$ is
∗Gorenstein injective if and only if $X$ is ∗Gorenstein injective.
###### Proposition 4.3.
Let $N$ be a graded $R$-module with finite ∗Gorenstein injective dimension,
and let $n$ be an integer. Then the following conditions are equivalent:
* (1)
${}^{*}\\!\operatorname{Gid}_{R}N\leq n$.
* (2)
${}^{*}\\!\operatorname{Ext}^{i}_{R}(L,N)=0$ for all $i>n$, and all
$R$-modules $L$ with finite ${}^{*}\operatorname{id}_{R}L$.
* (3)
${}^{*}\\!\operatorname{Ext}^{i}_{R}(I,N)=0$ for all $i>n$, and all ∗injective
$R$-modules $I$.
* (4)
For every exact sequence $0\rightarrow N\rightarrow
H^{0}\rightarrow\cdots\rightarrow H^{n-1}\rightarrow C^{n}\rightarrow 0$ where
$H^{0},\cdots,H^{n-1}$ are ∗Gorenstein injectives, then $C^{n}$ is also
∗Gorenstein injective.
Consequently, the ∗Gorenstein injective dimension of $M$ is determined by the
formulas:
${}^{*}\operatorname{Gid}_{R}N=$
$\displaystyle\sup\\{i\in\mathbb{N}_{0}|\,^{*}\operatorname{Ext}^{i}_{R}(L,N)\neq
0\text{ for some graded }R\text{-module }L\text{ with
finite}\,^{*}\operatorname{id}_{R}L\\}$ $\displaystyle=$
$\displaystyle\sup\\{i\in\mathbb{N}_{0}|\,^{*}\operatorname{Ext}^{i}_{R}(I,N)\neq
0\text{ for some graded }^{*}\text{injective module }I\\}.$
The ungraded version of the following theorem is in [7, Theorem 2.8]. The
proof uses Proposition 2.5 and the same technique of proof of [7, Theorem
2.8].
###### Theorem 4.4.
Let $V\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}W$ be a
quasiisomorphism between complexes of graded $R$-modules, where each module in
$V$ and $W$ has finite ∗projective dimension or finite ∗injective dimension.
If $B\in\,^{*}\mathcal{C}_{\sqsubset}(R)$ is a complex of ∗Gorenstein
injective modules, then the induced morphism
${}^{*}\operatorname{Hom}_{R}(W,B)\to\,^{*}\operatorname{Hom}_{R}(V,B)$
is a quasiisomorphism under each of the next two condition:
* (a)
$V,W\in\,^{*}\mathcal{C}_{\sqsupset}(R)$, or
* (b)
$V,W\in\,^{*}\mathcal{C}_{\sqsubset}(R)$.
###### Corollary 4.5.
Assume that $Y\simeq B$ where $B\in$ ${}^{*}\mathcal{C}_{\sqsubset}(R)$ is a
complex of ∗Gorenstein injective modules. If $U\simeq V$, where $V\in$
${}^{*}\mathcal{C}_{\sqsupset}(R)$ is a complex in which each module has
finite ∗projective dimension or finite ∗injective dimension, then
$\mathbf{R}^{*}\\!\operatorname{Hom}_{R}(U,Y)\simeq\,^{*}\operatorname{Hom}_{R}(V,B).$
###### Proof.
Is the same as [7, Corollary 2.10] using Theorem 4.4(a). ∎
The ungraded version of the following theorem is contained in [7, Theorem
3.3], and its proof is dual of [7, Theorem 3.1]. We present the proof of
(3)$\Rightarrow$(4) for later use.
###### Theorem 4.6.
Let $Y\in$ ${}^{*}\mathcal{D}_{\sqsubset}(R)$ be a complex of finite
∗Gorenstein injective dimension. For $n\in\mathbb{Z}$ the following are
equivalent:
* (1)
${}^{*}\operatorname{Gid}_{R}Y\leq n.$
* (2)
$n\geq-\sup U-\inf\mathbf{R}^{*}\\!\operatorname{Hom}_{R}(U,Y)$ for all $U\in$
${}^{*}\mathcal{D}_{\square}(R)$ of finite ∗projective or finite ∗injective
dimension with $H(U)\neq 0$.
* (3)
$n\geq-\inf\mathbf{R}^{*}\\!\operatorname{Hom}_{R}(J,Y)$ for all ∗injective
$R$-modules $J$.
* (4)
$n\geq-\inf Y$ and for any left-bounded complex $B\simeq Y$ of ∗Gorenstein
injective modules, the $\operatorname{Ker}(B_{-n}\rightarrow B_{-(n+1)})$ is a
∗Gorenstein injective module.
Moreover the following hold:
${}^{*}\operatorname{Gid}_{R}Y=$
$\displaystyle\sup\\{-\sup(U)-\inf\mathbf{R}^{*}\operatorname{Hom}_{R}(U,Y)|^{*}\operatorname{id}_{R}(U)<\infty\text{
and }H(U)\neq 0\\}$ $\displaystyle=$
$\displaystyle\sup\\{-\inf\mathbf{R}^{*}\\!\operatorname{Hom}_{R}(J,Y)|J\text{
is }^{*}\text{injective}\\}.$
###### Proof.
(2)$\Rightarrow$(3) and (4)$\Rightarrow$(1) are clear. $(1)\Rightarrow(2)$ is
dual of $(1)\Rightarrow(2)$ in [7, Theorem 3.1].
(3)$\Rightarrow$(4): To establish $n\geq-\inf Y$, it is sufficient to show
that
$\sup\\{-\inf\mathbf{R}^{*}\\!\operatorname{Hom}_{R}(J,Y)|J\text{ is
}^{*}\text{injective}\\}\geq-\inf Y.\quad(*)$
By assumption $g:=\\!^{*}\operatorname{Gid}_{R}(Y)$ is finite, so $Y\simeq B$
for some complex of ∗Gorenstein injective modules:
$B=0\to B_{s}\to B_{s-1}\to\cdots\to B_{-g+1}\to B_{-g}\to 0.$
Now it is clear that $-g\leq\inf Y$. By Lemma 4.5, for any ∗injective module
$J$, the complex $\\!{}^{*}\operatorname{Hom}_{R}(J,B)$ is isomorphic to
$\mathbf{R}^{*}\operatorname{Hom}_{R}(J,Y)$ in ${}^{*}\mathcal{D}(R)$. If
$g=-\inf Y$ then the differential $\partial_{-g+1}:B_{-g+1}\to B_{-g}$ is not
surjective. Now by the definition of ∗Gorenstein injective modules there
exists an ∗injective module $J$ such that $J\to B_{-g}$ is surjective. Notice
that the differential ${}^{*}\operatorname{Hom}_{R}(J,\partial_{-g+1})$ in the
complex ${}^{*}\operatorname{Hom}_{R}(J,B)$ is not surjective, for otherwise,
for any $\phi\in^{*}\operatorname{Hom}_{R}(J,B_{-g})$ there exists a
$\psi\in^{*}\operatorname{Hom}_{R}(J,B_{-g+1})$ such that
$\phi=\partial_{-g+1}\psi$. This implies that $\partial_{-g+1}$ is surjective
which is a contradiction. Therefore ${}^{*}\operatorname{Hom}_{R}(J,B)$ has
nonzero homology in degree $-g=\inf Y$. This gives $(*)$. Next assume that
$g>-\inf Y=-t$ and consider the exact sequence
$B:0\to Z_{t}^{B}\to B_{t}\to B_{t-1}\to\cdots\to B_{-g+1}\to B_{-g}\to
0.\quad(**)$
It shows that ${}^{*}\operatorname{Gid}_{R}Z^{B}_{t}\leq t+g$ and it is not
difficult to see that the equality must hold. For otherwise
${}^{*}\operatorname{Gid}_{R}Y<g$ and by Proposition 4.3,
$\mbox{H}_{-g}(\mathbf{R}^{*}\operatorname{Hom}_{R}(J,Y))\cong\\!^{*}\operatorname{Ext}^{t+g}(J,Z^{B}_{t})\neq
0$ for some ∗injective module $J$. Therefore $(*)$ follows, which gives the
inequality $n\geq-\inf Y$.
To prove the second part of (4) let $B$ be a left-bounded complex of
∗Gorenstein injective modules such that $B\simeq Y$. By assumption
${}^{*}\operatorname{Gid}_{R}Y$ is finite, so there exists a bounded complex
$\tilde{B}$ of ∗Gorenstein injective modules such that $\tilde{B}\simeq Y$.
Since $n\geq-\inf Y=-\inf\tilde{B}$, the kernel $Z^{\tilde{B}}_{n}$ fits in an
exact sequence
$0\to
Z^{\tilde{B}}_{-n}\to\tilde{B}_{-n}\to\tilde{B}_{-n-1}\to\cdots\to\tilde{B}_{t}\to
0.$
By Proposition 4.3 and the isomorphism
${}^{*}\operatorname{Ext}^{i}(J,Z^{\tilde{B}}_{-n})\simeq\mbox{H}_{-(n+i)}(\mathbf{R}^{*}\operatorname{Hom}_{R}(J,Y))=0$
for $i>0$ we get that $Z^{\tilde{B}}_{-n}$ is ∗Gorenstein injective.
Now it is enough to prove that if $I$ and $B$ are left bounded complexes of
respectively ∗injective and ∗Gorenstein injective modules and $B\simeq Y\simeq
I$ then the kernel $Z^{I}_{-n}$ is ∗Gorenstein injective if and only if
$Z^{B}_{-n}$ is so. Let $B$ and $I$ be such complexes. As $I$ consists of
∗injectives by [15, Chapter I, Lemma 4.5] there is a quasi isomorphism
$\pi:B\rightarrow I$ which induces a quasi isomorphism between the complexes
$\pi\supset_{n}:B\supset_{n}\rightarrow I\supset_{n}$. The mapping cone
$\mathcal{M}(\pi\supset_{n})=\cdots\to B_{n-1}\oplus I_{n}\to B_{n}\oplus
I_{n+1}\to B_{n+1}\oplus Z^{B}_{n}\to Z^{I}_{n}\to 0$
is bounded exact such that all modules but the two right-most ones are
∗Gorenstein injective modules. It follows by Theorem 4.2 that $Z^{I}_{n}$ is
∗Gorenstein injective if and only if $B_{n+1}\oplus Z^{B}_{n}$ is so, which is
tantamount to $Z^{B}_{n}$ being ∗Gorenstein injective.
The two equalities are immediate consequences of the equivalence of (1)-(4). ∎
###### Corollary 4.7.
Let $Y\in$ ${}^{*}\mathcal{D}_{\sqsubset}(R)$. Then
${}^{*}\operatorname{Gid}_{R}Y\leq^{*}\operatorname{id}_{R}Y,$
with equality if ${}^{*}\operatorname{id}_{R}Y$ is finite.
###### Proof.
Is the same as [5, Proposition 6.2.6] using Theorem 4.6 and Corollary 3.8. ∎
Recall the _finitistic injective dimension_ of $R$ which defined as
$\operatorname{FID}(R):=\sup\\{\operatorname{id}_{R}M|M\text{ is an
}R\text{-module with }\operatorname{id}_{R}M<\infty\\}.$
The _finitistic Gorenstein injective dimension_ $\mbox{FGID}(R)$, _finitistic
∗injective dimension_ ${}^{*}\operatorname{FID}(R)$ and _finitistic
∗Gorenstein injective dimension_ ${}^{*}\mbox{FGID}(R)$ are define similarly.
In [16, Theorem 2.29], Holm proved that
$\mbox{FGID}(R)=\operatorname{FID}(R)$. Also it is known that over a
commutative Noetherian ring $R$ we have
$\operatorname{FID}(R)\leq\operatorname{dim}R$ by [2, Corollary 5.5] and [17,
II. Theorem 3.2.6].
The following theorem is the graded version of Holm’s result [16, Theorem
2.29]. Its proof is dual of [16, Theorem 2.28]. We give the sketch of proof.
###### Theorem 4.8.
If $R$ is any graded ring
${}^{*}\operatorname{FGID}(R)=^{*}\operatorname{FID}(R)$.
###### Proof.
Using Corollary 4.7 we have
${}^{*}\operatorname{FID}(R)\leq\\!^{*}\operatorname{FGID}(R)$. If $N$ is a
graded $R$-module with $0<\,^{*}\operatorname{Gid}_{R}N<\infty$, then the
graded version of [16, Theorem 2.15] gives a graded $R$-module $C$ such that
${}^{*}\operatorname{id}_{R}C=\,^{*}\operatorname{Gid}_{R}N-1$. Therefore
${}^{*}\operatorname{FGID}(R)\leq\,^{*}\operatorname{FID}(R)+1$. Now for the
reverse inequality
$\\!{}^{*}\operatorname{FGID}(R)\leq\,^{*}\operatorname{FID}(R)$, assume that
$0<\,^{*}\operatorname{FGID}(R)=m<\infty$. Pick a module $N$ with
${}^{*}\operatorname{Gid}_{R}N=m$. Hence the same technique of [18, Lemma
2.2], gives a graded module $T$ such that ${}^{*}\operatorname{id}_{R}T=m$. ∎
###### Corollary 4.9.
Let $Y\in$ ${}^{*}\mathcal{D}_{\sqsubset}(R)$ be a complex of finite
∗Gorenstein injective dimension. Then
${}^{*}\operatorname{Gid}_{R}Y\leq\operatorname{FID}(R)-\inf Y.$
###### Proof.
Note that by $(**)$ in the proof of Theorem 4.6 above we have
$\\!{}^{*}\operatorname{Gid}_{R}Y=\\!^{*}\operatorname{Gid}_{R}M-\inf Y$ for
some graded $R$-module $M$. Thus
$\\!{}^{*}\operatorname{Gid}_{R}Y\leq\\!^{*}\mbox{FGID}(R)-\inf
Y=\\!^{*}\operatorname{FID}(R)-\inf Y\leq\operatorname{FID}(R)-\inf Y$. ∎
Let $D$ be a $\\!{}^{*}$dualizing complex of $R$. Then $D$ is a dualizing
complex of $R$, so we have the Bass category $B(R)$ with respect to $D$ (cf.
[7, Page 237]). It is known that for $Y\in\mathcal{D}_{\sqsubset}(R)$, we have
$Y\in$ $B(R)$ if and only if $\operatorname{Gid}_{R}Y<\infty$, [7, Theorem
4.4].
###### Definition 4.10.
Let $D$ be a $\\!{}^{*}$dualizing complex of $R$. The ∗Bass category
${}^{*}B(R)$ with respect to $D$ is define as:
${}^{*}B(R):=\left\\{Y\in^{*}\mathcal{D}_{\square}(R)\bigg{|}\begin{array}[]{l}\epsilon_{Y}:D\otimes_{R}^{\mathbf{L}}\mathbf{R}^{*}\\!\operatorname{Hom}_{R}(D,Y)\to
Y\text{ is an iso-}\\\ \text{morphism and
}\mathbf{R}^{*}\operatorname{Hom}_{R}(D,Y)\in\\!^{*}\mathcal{D}_{\square}(R)\end{array}\right\\}.$
It is easily seen that a complex $Y\in^{*}\mathcal{D}_{\square}(R)$ is in
$B(R)$ if and only if is in $\\!{}^{*}B(R)$.
###### Theorem 4.11.
Assume that $R$ admits a $\\!{}^{*}$dualizing complex $D$. Then For $Y\in$
${}^{*}\mathcal{D}_{\sqsubset}(R)$ the following are equivalent:
* (1)
$Y\in\\!^{*}B(R)$.
* (2)
${}^{*}\operatorname{Gid}_{R}Y<\infty$.
###### Proof.
It is dual to the proof of [7, Theorem 4.1]. In fact the proof uses that if
$N$ is a graded $R$-module satisfying both $N\in$ ${}^{*}B(R)$ and
${}^{*}\operatorname{Ext}^{m}_{R}(J,N)=0$ for all integer $m>0$ and all
∗injective $R$-module $J$, then $N$ is ∗Gorenstein injective, which its proof
is dual to [7, Lemma 4.6]. ∎
###### Theorem 4.12.
Let $(R,\mathfrak{m},k)$ be a ∗local ring that admits a ∗dualizing complex
$D$. For a complex $Y\in$ ${}^{*}\mathcal{D}_{\square}(R)$ of finite
∗Gorenstein injective dimension, we have
$0pt(\mathfrak{m},Y)=0pt(\mathfrak{m},R)+\inf\textbf{R}^{*}\operatorname{Hom}_{R}(\\!^{*}\operatorname{E}_{R}(k),Y).$
###### Proof.
We follow the method of [7, Theorem 6.5]. By Theorem 4.11, $Y\in\\!^{*}B(R)$;
in particular $Y\simeq
D\otimes^{\mathbf{L}}_{R}\textbf{R}^{*}\operatorname{Hom}_{R}(D,Y)$.
Furthermore, we can assume that $D$ is a normalized ∗dualizing complex, so
that by Proposition 3.13 we have
${\mathbf{R}}\Gamma_{\mathfrak{m}}(D)\cong\\!^{*}\operatorname{E}_{R}(k)$ and
that $D_{\mathfrak{m}}$ is a normalized dualizing complex for
$R_{\mathfrak{m}}$. We compute as follows:
$\displaystyle 0pt(\mathfrak{m},Y)=$ $\displaystyle
0pt(\mathfrak{m},D\otimes^{\mathbf{L}}_{R}\textbf{R}^{*}\operatorname{Hom}_{R}(D,Y))$
$\displaystyle=$ $\displaystyle
0pt(\mathfrak{m},D)+0pt(\mathfrak{m},\textbf{R}^{*}\operatorname{Hom}_{R}(D,Y))$
$\displaystyle=$ $\displaystyle\inf
D_{\mathfrak{m}}+\inf\textbf{L}\Lambda^{\mathfrak{m}}\mathbf{R}\\!^{*}\operatorname{Hom}_{R}(D,Y)$
$\displaystyle=$ $\displaystyle
0ptR_{\mathfrak{m}}+\inf\textbf{R}^{*}\operatorname{Hom}_{R}({\mathbf{R}}\Gamma_{\mathfrak{m}}(D),Y)$
$\displaystyle=$ $\displaystyle
0pt(\mathfrak{m},R)+\inf\textbf{R}^{*}\operatorname{Hom}_{R}(\\!^{*}\operatorname{E}_{R}(k),Y).$
The second equality is by [20, Theorem 2.4(b)], the third one by the fact that
$D_{\mathfrak{m}}$ is homologically finite and by [13, Theorem 2.11], and the
forth one by [13, 2.6], and the fact that
$\textbf{R}^{*}\operatorname{Hom}_{R}(D,Y)={\mathbf{R}}\operatorname{Hom}_{R}(D,Y)$.
∎
The ungraded version of the following result is in [7, Proposition 5.5].
###### Proposition 4.13.
Assume that $R$ admits a $\\!{}^{*}$dualizing complex and let $Y\in$
${}^{*}\mathcal{D}_{\square}(R)$. Then for any homogeneous prime ideal
$\mathfrak{p}\in R$ there is an inequality
${}^{*}\operatorname{Gid}_{R_{(\mathfrak{p})}}Y_{(\mathfrak{p})}\leq^{*}\operatorname{Gid}_{R}Y.$
###### Proof.
It is enough to show that if $N$ is a $\\!{}^{*}$Gorenstein injective, then
$N_{(\mathfrak{p})}$ is $\\!{}^{*}$Gorenstein injective over
$R_{(\mathfrak{p})}$. This is similar to the proof of [5, Theorem 6.2.13]
using Corollary 4.9. ∎
The following proposition is the graded version of [8, Lemma 2.1]. For part
$(b)$ we follow the technique of [8, Lemma 2.1]. We present the proof to give
some hints for the graded analogues. Before doing that we need a lemma.
###### Lemma 4.14.
Let $(R,\mathfrak{m})$ be a ∗local non-negatively graded ring. Then the
$\mathfrak{m}$-∗adic completion ${}^{*}\widehat{R}$ of $R$, is a ∗faithfully
flat $R$-module, that is ${}^{*}\widehat{R}$ is $R$-flat and for any graded
$R$-module $M$, $M=0$ if and only if $M\otimes_{R}\,^{*}\widehat{R}=0$.
###### Proof.
It is well known that ${}^{*}\widehat{R}$ is a flat $R$-module by [10,
Corollary 3.3]. Now let $M$ be a graded $R$-module such that
$M\otimes_{R}\,^{*}\widehat{R}=0$. Note that
$\mathfrak{m}=\mathfrak{m}_{0}\oplus R_{1}\oplus R_{2}\oplus\cdots$, where
$\mathfrak{m}_{0}$ is the unique maximal ideal of $R_{0}$. So that
$(^{*}\widehat{R})_{0}=\displaystyle\lim_{\longleftarrow}(R/\mathfrak{m}^{n})_{0}=\displaystyle\lim_{\longleftarrow}R_{0}/\mathfrak{m}_{0}^{n}=\widehat{R_{0}}$,
where $\widehat{R_{0}}$ is the $\mathfrak{m}_{0}$-adic completion of $R_{0}$.
Let $t$ be an integer and set $M_{(t)}:=M_{t}\oplus M_{t+1}\oplus
M_{t+2}\oplus\cdots$, which is a graded submodule of $M$. Hence
$M_{(t)}\otimes_{R}\,^{*}\widehat{R}=0$. Therefore
$(M_{(t)}\otimes_{R}\,^{*}\widehat{R})_{t}=0$. Since
$(M_{(t)}\otimes_{R}\,^{*}\widehat{R})_{t}$ is generated as a
$\mathbb{Z}$-module by $x\otimes r$ for $x\in M_{t}$ and
$r\in(^{*}\widehat{R})_{0}=\widehat{R_{0}}$ and $t$ is arbitrary, we see that
$M\otimes_{R}\widehat{R_{0}}=0$. On the other hand we have
$R_{0}\otimes_{R}\widehat{R_{0}}=R_{0}\otimes_{R}(R_{0}\otimes_{R_{0}}\widehat{R_{0}})=(R_{0}\otimes_{R}R_{0})\otimes_{R_{0}}\widehat{R_{0}}=(R_{0}\otimes_{R_{0}}R_{0})\otimes_{R_{0}}\widehat{R_{0}}=R_{0}\otimes_{R_{0}}\widehat{R_{0}}=\widehat{R_{0}}$.
Hence
$0=M\otimes_{R}\widehat{R_{0}}=(M\otimes_{R_{0}}R_{0})\otimes_{R}\widehat{R_{0}}=M\otimes_{R_{0}}(R_{0}\otimes_{R}\widehat{R_{0}})=M\otimes_{R_{0}}\widehat{R_{0}}$.
Since $\widehat{R_{0}}$ is a faithfully flat $R_{0}$-module, we get that
$M=0$. This completes the proof. ∎
###### Proposition 4.15.
Let $N$ be a ∗Gorenstein injective $R$-module. Then under each of the
following conditions
* (a)
$R$ admits a ∗dualizing complex; or
* (b)
$R$ is a non-negatively graded ring,
one has
$0ptR_{\mathfrak{p}}\leqslant 0pt_{R_{\mathfrak{p}}}N_{\mathfrak{p}}$
for every $\mathfrak{p}$ in ${}^{*}\operatorname{Spec}R$, and equality holds
if $\mathfrak{p}$ is a maximal element in ${}^{*}\operatorname{supp}_{R}N$.
###### Proof.
For part $(a)$ let $\mathfrak{p}$ be a homogenous prime ideal. Consider the
∗local ring
$(S,\mathfrak{n}):=(R_{(\mathfrak{p})},\mathfrak{p}R_{(\mathfrak{p})})$. By
Proposition 4.13, $B:=N_{(\mathfrak{p})}$ is a ∗Gorenstein injective
$S$-module. Thus
${}^{*}\operatorname{Ext}^{i}_{S}(\\!^{*}\operatorname{E}_{S}(S/\mathfrak{n}),B)=0$
for all $i>0$, that is
$\inf\textbf{R}^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{E}_{S}(S/\mathfrak{n}),B)\geq
0$. On the other hand since $B_{\mathfrak{n}}\cong N_{\mathfrak{p}}$ and
$S_{\mathfrak{n}}\cong R_{\mathfrak{p}}$, by Theorem 4.12 we have
$0ptN_{\mathfrak{p}}=0ptR_{\mathfrak{p}}+\inf\textbf{R}^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{E}_{S}(S/\mathfrak{n}),B).$
Thus $0ptR_{\mathfrak{p}}\leqslant 0pt_{R_{\mathfrak{p}}}N_{\mathfrak{p}}$.
Now if $0pt_{R_{\mathfrak{p}}}N_{\mathfrak{p}}$ is finite, we have
$\inf\textbf{R}^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{E}_{S}(S/\mathfrak{n}),B)<\infty$.
Using Lemma 4.5 we have
$\textbf{R}^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{E}_{S}(S/\mathfrak{n}),B)\simeq^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{E}_{S}(S/\mathfrak{n}),B).$
Therefore the infimum must be zero. This proves the second statement.
For $(b)$ assume that $\mathfrak{p}$ is a homogeneous prime ideal and $T$ is a
graded $R_{(\mathfrak{p})}$-module with
$\,{}^{*}\operatorname{pd}_{R_{(\mathfrak{p})}}T<\infty$. A standard dimension
shifting argument shows that
$\,{}^{*}\operatorname{Ext}^{i}_{R_{(\mathfrak{p})}}(T,N_{(\mathfrak{p})})=0$
for all $i>0$. Set $d=0pt(\mathfrak{p}R_{(\mathfrak{p})},R_{(\mathfrak{p})})$
and choose a homogeneous maximal $R_{(\mathfrak{p})}$-regular sequence ${\bf
x}$ in $\mathfrak{p}R_{(\mathfrak{p})}$ by [3, Proposition 1.5.11]. Since the
∗projective dimension of $R_{(\mathfrak{p})}/({\bf x})$ is finite we have
$\displaystyle 0\leq$
$\displaystyle\inf\mathbf{R}\\!^{*}\operatorname{Hom}_{R_{(\mathfrak{p})}}(R_{(\mathfrak{p})}/({\bf
x}),N_{(\mathfrak{p})})$ $\displaystyle\leq$ $\displaystyle
0pt(\mathfrak{p}R_{(\mathfrak{p})},\mathbf{R}\\!^{*}\operatorname{Hom}_{R_{(\mathfrak{p})}}(R_{(\mathfrak{p})}/({\bf
x}),N_{(\mathfrak{p})}))$ $\displaystyle=$ $\displaystyle
0pt(\mathfrak{p}R_{(\mathfrak{p})},N_{(\mathfrak{p})})-d=0pt_{R_{\mathfrak{p}}}N_{\mathfrak{p}}-d,$
where the second inequality holds by [6, 4.3], and the first equality follows
from Proposition 2.4.
Now let $\mathfrak{p}$ be a maximal element in
${}^{*}\operatorname{supp}_{R}N$. Set $S=R_{(\mathfrak{p})}$ which is a ∗local
ring with depth $d$, homogeneous maximal ideal
$\mathfrak{n}=\mathfrak{p}R_{(\mathfrak{p})}$, $B=N_{(\mathfrak{p})}$, and
$l=R_{(\mathfrak{p})}/\mathfrak{p}R_{(\mathfrak{p})}$. One has
$\mathfrak{n}\in\operatorname{supp}_{S}B$ and by exactly the same method of
proof of [8, Lemma 2.1] we have
$\,{}^{*}\operatorname{Ext}^{i}_{S}(T,B)=0=\,^{*}\operatorname{Ext}^{i}_{S}(E,B)$
for all $i>0$ and every graded $S$-module $T$ with
$\,{}^{*}\operatorname{pd}_{S}T$ finite and
$E:=\\!^{*}\operatorname{E}_{S}(l)$.
Let $K$ denote the Koszul complex on a homogeneous system of generators for
$\mathfrak{n}$. Since $\mbox{H}_{i}(K\otimes_{S}E)$ are Artinian, and by [3,
Corollary 1.6.13], we have $\mathfrak{n}\mbox{H}_{i}(K\otimes_{S}E)=0$, we see
that $\mbox{H}_{i}(K\otimes_{S}E)$ are finitely generated. So there is a
resolution $L\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}K\otimes_{S}E$
by finitely generated free S-modules. Using the ∗Hom-evaluation we have
$K\otimes_{S}(E\otimes^{\mathbf{L}}_{S}\\!{}^{*}\operatorname{Hom}_{S}(E,B))\simeq
L\otimes_{S}\\!^{*}\operatorname{Hom}_{S}(E,B)\cong\\!^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{Hom}_{S}(L,E),B).$
The free resolution above induces a quasiisomorphism $\alpha$ from
$\,{}^{*}\operatorname{Hom}_{S}(K\otimes_{S}E,E)\cong\\!^{*}\operatorname{Hom}_{S}(K,\\!^{*}\widehat{S})$
to $\,{}^{*}\operatorname{Hom}_{S}(L,E)$. The mapping cone
$C=\mathcal{M}(\alpha)$ is a bounded complex of direct sums of
${}^{*}\widehat{S}$ and $E$. Thus
$\,{}^{*}\operatorname{Ext}^{i}_{S}(C_{j},B)=0$ for all $i>0$ and all $j$.
Hence, an application of $\,{}^{*}\operatorname{Hom}_{S}(-,B)$ yields a
quasiisomorphism
$\,{}^{*}\operatorname{Hom}_{S}(\alpha,B):\\!^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{Hom}_{S}(L,E),B)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\\!^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{Hom}_{S}(K,\\!^{*}\widehat{S}),B).$
The modules in the complex $\,{}^{*}\operatorname{Hom}_{S}(K,^{*}\widehat{S})$
are Ext-orthogonal to the modules in the mapping cone of an injective
resolution $B\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}H$. Therefore,
one has
$\\!{}^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{Hom}_{S}(K,\\!^{*}\widehat{S}),B)\simeq\\!^{*}\operatorname{Hom}_{S}(\\!^{*}\operatorname{Hom}_{S}(K,\\!^{*}\widehat{S}),H)$
by the graded analogue of [7, Lemma 2.4]. Now piece together the last three
quasiisomorphisms, and use ∗Hom-evaluation to obtain
$K\otimes_{S}(E\otimes^{\mathbf{L}}_{S}\\!{}^{*}\operatorname{Hom}_{S}(E,B))\simeq
K\otimes_{S}\mathbf{R}^{*}\operatorname{Hom}_{S}(\\!^{*}\widehat{S},B).$
Therefore by [6, (4.2) and (4.11)], the complexes
$E\otimes^{\mathbf{L}}_{S}\\!{}^{*}\operatorname{Hom}_{S}(E,B)$ and
$\mathbf{R}^{*}\operatorname{Hom}_{S}(\\!^{*}\widehat{S},B)$ have the same
width. From [20, Theorem 2.4(b)] and Proposition 2.4 we have
$0pt(\mathfrak{n},E)+0pt(\mathfrak{n},\,^{*}\operatorname{Hom}_{S}(E,B))=0pt(\mathfrak{n},B).$
Now we can see that $0pt(\mathfrak{n},\,^{*}\operatorname{Hom}_{S}(E,B))=0$
(see proof of [8, Lemma 2.1]). Consequently
$0pt_{R_{\mathfrak{p}}}N_{\mathfrak{p}}=0pt(\mathfrak{n},B)=0pt(\mathfrak{n},E)=0pt(\mathfrak{n},\\!^{*}\widehat{S})$
using Proposition 2.2. Now since $\\!{}^{*}\widehat{S}$ is a flat $S$-module
we have
$\operatorname{Ext}^{i}_{S}(S/\mathfrak{n},\\!^{*}\widehat{S})\cong\operatorname{Ext}^{i}_{S}(S/\mathfrak{n},S)\otimes_{S}\\!^{*}\widehat{S}.$
Keep in mind that
$\operatorname{Ext}^{i}_{S}(S/\mathfrak{n},S)=\,^{*}\operatorname{Ext}^{i}_{S}(S/\mathfrak{n},S)$
is a graded $S$-module. Hence using Lemma 4.14 we have
$\operatorname{Ext}^{i}_{S}(S/\mathfrak{n},\\!^{*}\widehat{S})=0$ if and only
if $\operatorname{Ext}^{i}_{S}(S/\mathfrak{n},S)=0$. Therefore
$0pt(\mathfrak{n},\\!^{*}\widehat{S})=0pt(\mathfrak{n},S)=d$. ∎
###### Lemma 4.16.
Let $X$ be a complex of graded $R$-modules. For any homogeneous surjective
homomorphism $i:Y_{n}\longrightarrow X_{n}$ of graded $R$-modules there is a
commutative diagram
$\textstyle{Y=\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{=}$$\textstyle{Y_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{i^{\prime}}$$\textstyle{Y_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{i}$$\textstyle{X_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\cdots}$$\textstyle{X=\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{n+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{\prime}}$$\textstyle{X_{n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{\prime}}$$\textstyle{X_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{\prime}}$$\textstyle{X_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$
such that $Y$ is a complex of graded $R$-modules, that $i^{\prime}$ is
surjective and $\operatorname{Ker}i\cong\operatorname{Ker}i^{\prime}$ and the
induced map $\mbox{H}(Y)\longrightarrow\mbox{H}(X)$ is an isomorphism.
###### Proof.
Let $\alpha=\alpha^{\prime}i$ and $\beta:Y_{n+1}\longrightarrow Y_{n}$ be the
pullback of $\beta^{\prime}$ along $i$ thus
$Y_{n+1}=\\{(x,y)|x\in X_{n+1},y\in Y_{n}\text{ and
}\beta^{\prime}(x)=i(y)\\}.$
It is clear that $Y_{n+1}$ is a graded $R$-module. Let $i^{\prime}:Y_{n+1}\to
X_{n+1}$ be a map defined by $i^{\prime}(x,y)=x$ which is seen to be
surjective. Define $\gamma:X_{n+2}\to Y_{n+1}$ by
$\gamma(x)=(\gamma^{\prime}(x),0)$ for every $x\in X_{n+2}$. It is clear that
$Y$ is a complex and $\operatorname{Ker}i\cong\operatorname{Ker}i^{\prime}$.
Therefore the induced map $\mbox{H}(Y)\longrightarrow\mbox{H}(X)$ is an
isomorphism. ∎
###### Corollary 4.17.
Let $Y\in$ ${}^{*}\mathcal{D}_{\square}(R)$ such that
${}^{*}\operatorname{Gid}_{R}Y=g>0$. Then there is an exact triangle $G\to
I\to Y\to\Sigma G$ such that $G$ is a ∗Gorenstein injective module and $I$ is
a complex of graded $R$-modules such that ${}^{*}\operatorname{id}_{R}I=g$.
###### Proof.
Without loss of generality we can assume that $Y$ has the form
$0\to I_{0}\to I_{-1}\to\cdots\to I_{-g+1}\to B_{-g}\to 0,$
where the $I_{j}$s are ∗injective and $B_{-g}$ is ∗Gorenstein injective. By
definition $B_{-g}$ is homomorphic image of some $\\!{}^{*}$injective module
$J_{0}$. Thus by Lemma 4.16 we have a commutative diagram
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{-g+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{-g+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{J_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{-g+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I_{-g+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{-g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$$\textstyle{0}$$\textstyle{0}$
such that $\operatorname{Ker}i\cong\operatorname{Ker}i^{\prime}$. Since
$\operatorname{Ker}i$ is a ∗Gorenstein injective module, using Theorem 4.2,
$G_{-g+1}$ is also ∗Gorenstein injective. By repeating this argument $g$ times
we get the desired exact triangle $G\to I\to Y\to\Sigma G$. ∎
The following equality extends Chouinard’s formula [4] and [8, Theorem 2.2] to
the ∗Gorenstein injective dimension.
###### Theorem 4.18.
Assume that $R$ admits a $\\!{}^{*}$dualizing complex or $R$ is a non-
negatively graded ring. Let $Y\in$ ${}^{*}\mathcal{D}_{\square}(R)$ of finite
∗Gorenstein injective dimension. Then there is an equality
${}^{*}\operatorname{Gid}_{R}Y=\sup\\{0ptR_{\mathfrak{p}}-0pt_{R_{\mathfrak{p}}}Y_{\mathfrak{p}}|\mathfrak{p}\in^{*}\operatorname{Spec}R\\}.$
###### Proof.
The argument is similar to [8, Theorem 2.2], just use Proposition 4.15 and
Corollary 4.17. ∎
In the following corollary we compare the ∗Gorenstein injective dimension with
the usual Gorenstein injective dimension.
###### Corollary 4.19.
Let $Y\in$ ${}^{*}\mathcal{D}_{\square}(R)$.Then under each of the following
conditions
* (a)
$R$ admits a ∗dualizing complex; or
* (b)
$R$ is a non-negatively graded ring, and ${}^{*}\operatorname{Gid}_{R}Y$ and
$\operatorname{Gid}_{R}Y$ are finite,
one has
${}^{*}\operatorname{Gid}_{R}Y\leq\operatorname{Gid}_{R}Y\leq^{*}\operatorname{Gid}_{R}Y+1.$
###### Proof.
If $R$ admits a ∗dualizing complex, then Theorem 4.11, implies that
${}^{*}\operatorname{Gid}_{R}Y<\infty$, if and only if
$\operatorname{Gid}_{R}Y<\infty$. In each both cases the inequalities follow
from Theorem 4.18 and [8, Theorem 2.2]. ∎
## References
* [1] L. L. Avramov and H. B. Foxby, Homological dimensions of unbounded complexes, J. Pure Appl. Algebra. 71, (1991), 129–155.
* [2] H. Bass, Injective dimension in Noetherian rings, Trans. Amer. Math. Soc., 102, (1962), 18–29.
* [3] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics. 39, Cambridge University Press, Cambridge, 1998.
* [4] Leo G. Chouinard II, On finite weak and injective dimension, Proc. Amer. Math. Soc. 60, (1976), 57–60.
* [5] L. W. Christensen, Gorenstein dimensions, Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin, 2000.
* [6] L. W. Christensen, H. B. Foxby, and A. Frankild, Restricted homological dimensions and Cohen-Macaulayness, J. Algebra, 251, (2002), no. 1, 479–502.
* [7] L.W. Christensen, A. Frankild, and H. Holm On Gorenestein Projective, Injective and Flat dimensions - A Functorial description with applications, J. Algebra, 302, (2006), 231-279.
* [8] L. W. Christensen and S. Sather-Wagstaff, Transfer of Gorenstein dimensions along ring homomorphisms, J. Pure Appl. Algebra. 214, (2010), 982–989.
* [9] R. M. Fossum, The structure of indecomposable injective modules, Math. Scand. 36, (1975), 291–312.
* [10] R. M. Fossume and H. B. Foxby, The Category of Graded Modules, Math. Scand. 35, (1974), 288–300.
* [11] H. B. Foxby, Bounded complexes of flat modules, J. Pure Appl. Algebra, 15, (1979), 149–172.
* [12] H. B. Foxby and S. Iyengar, Depth and amplitude for unbounded complexes, Commutative algebra (Grenoble/Lyon, 2001), 119–137, Contemp. Math. 331, Amer. Math. Soc. Providence, RI, 2003.
* [13] A. Frankild, Vanishing of local homology modules, Math. Z., 244, (3), (2003), 615–630.
* [14] A. Grothendieck, Local cohomology, Lecture notes in Math. 41 Springer Verlag, 1967.
* [15] R. Hartshorne, Residues and Duality, Lecture Notes in math. 20, Springer-Verlag, Heidelberg, 1966.
* [16] H. Holm, Gorenstein homological dimensions, J. Pure Appl. Algebra, 189, (2004), 167-193.
* [17] M. Raynaud and L. Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13, (1971), 1–89.
* [18] P. Sahandi and T. Sharif, Dual of the Auslander-Bridger formula and GF-perfectness, Math. Scand., 101, (2007), 5–18.
* [19] A. Singh and I. Swanson, Associated primes of local cohomology modules and of Frobenius powers, Int. Math. Res. Not. 33, (2004), 1703–1733.
* [20] S. Yassemi, Width of complexes of modules, Acta Math. Vietnam. 23(1), (1998) 161–169.
|
arxiv-papers
| 2013-06-13T07:58:34 |
2024-09-04T02:49:46.451728
|
{
"license": "Public Domain",
"authors": "Afsaneh Esmaeelnezhad and Parviz Sahandi",
"submitter": "Parviz Sahandi Dr.",
"url": "https://arxiv.org/abs/1306.3049"
}
|
1306.3051
|
# A convenient implementation of the overlap between arbitrary Hartree-Fock-
Bogoliubov vacua for projection
Zao-Chun Gao [email protected] Qing-Li Hu Y. S. Chen China Institute of
Atomic Energy, P.O. Box 275 (10), Beijing 102413, P.R. China
###### Abstract
Overlap between Hartree-Fock-Bogoliubov(HFB) vacua is very important in the
beyond mean-field calculations. However, in the HFB transformation, the $U,V$
matrices are sometimes singular due to the exact emptiness ($v_{i}=0$) or full
occupation ($u_{i}=0$) of some single-particle orbits. This singularity may
cause some problem in evaluating the overlap between HFB vacua through
Pfaffian. We found that this problem can be well avoided by setting those zero
occupation numbers $u_{i},v_{i}$ to some tiny values denoted by
$\varepsilon(>0)$, which numerically satisfies $1+\varepsilon^{2}=1$ (e.g.,
$\varepsilon=10^{-8}$ when using the double precision data type). This
treatment does not change the HFB vacuum state because
$u_{i}^{2},v_{i}^{2}=\varepsilon^{2}$ are numerically zero relative to 1.
Therefore, for arbitrary HFB transformation, we say that the $U,V$ matrices
can always be nonsingular. From this standpoint, we present a new convenient
Pfaffian formula for the overlap between arbitrary HFB vacua, which is
especially suitable for symmetry restoration. Testing calculations have been
performed for this new formula. It turns out that our method is reliable and
accurate in evaluating the overlap between arbitrary HFB vacua.
###### keywords:
Hartree-Fock-Bogoliubov method, beyond mean-field method, Pfaffian
††journal: Physics Letter B
## 1 Introduction
The Hartree-Fock-Bogoliubov (HFB) approximation has been a great success in
understanding interacting many-body quantum systems in all fields of physics.
However, the beyond mean-field effects (e.g., the nuclear vibration and
rotation) are missing in the HFB calculations. Methods that go beyond mean-
field, such as the Generator Coordinate Method(GCM) and the projection method,
are expected to take those missing effects into consideration and present
better description of the many-body quantum system. In the beyond mean-field
calculations, operator matrix elements and overlaps between multi-
quasiparticle HFB states are basic blocks. These matrix elements and overlaps
can be evaluated using the generalized Wick’s theorem (GWT)[1, 2], or
equivalently using Pfaffian [3, 4, 5, 6, 7], or using the compact formula in
Ref.[8]. However, in the efficient calculations (e.g., see [5]), all of the
matrix elements and overlaps require the value of the overlap between HFB
vacua.
Thus, the reliable and accurate evaluation of the overlap between HFB vacua is
very important for the stability and the efficiency of the beyond mean-field
calculations. Especially in cases near to the Egido pole [9], the overlap
between HFB vacua is very tiny, and a small error could lead to a large
uncertainty of the matrix elements. In the past, numerical calculations of the
overlap were performed with the Onishi formula [10]. Unfortunately, the Onishi
formula leaves the sign of the overlap undefined due to the square root of a
determinant. Several efforts have been made to overcome this sign problem [11,
12, 13, 14, 15, 16]. In 2009, Robledo proposed a different overlap formula
with the Pfaffian rather than the determinant [17]. This formula completely
solves the sign problem but requires the inversion of the matrix $U$ in the
Bogoliubov transformation. To avoid the singularity of $U$, the formula for
the limit when several orbits are fully occupied is given in Ref. [18].
Simultaneously, the limit when some orbits are exact empty was also considered
to reduce the computational cost. Meanwhile, various Pfaffian formulae for the
overlap between HFB vacua have been proposed by several authors [3, 6, 7]. In
Ref. [7], the overlap formula does not require the inversion of $U$, but the
empty orbits in the Fock space should be omitted.
In practical calculation, one should first identify the singularity of the
matrices $U$ and $V$ in the Bogoliubov transformation. This can be easily
tested with the Bloch-Messiah theorem (see details in Ref. [19]). The matrices
$U$ and $V$ can be decomposed as $U=D\bar{U}C$ and $V=D^{*}\bar{V}C$. Here,
$D$ and $C$ are unitary matrices. $\bar{U}$ and $\bar{V}$ refer to the BCS-
transformation and are constructed from the occupation numbers $u_{i},v_{i}$
with $0\leq u_{i},v_{i}\leq 1$ and $u_{i}^{2}+v_{i}^{2}=1$ (see Eq. (7.9) and
Eq. (7.12) in Ref.[19]). The limits of fully occupied ($u_{i}=0,v_{i}=1$) and
fully empty ($u_{i}=1,v_{i}=0$) levels have been carefully treated in Refs.
[6, 7, 18] to avoid the collapse of the overlap computation.
However, we note that in most realistic cases the $v_{i}$’s can be extremely
close to 0 or 1 but not exact 0 or 1. Strictly speaking, these levels with
such extreme emptiness or occupation should be considered but may lead to
exotic values ( extremely huge or extremely tiny) of the Pfaffian in the
proposed formulae. What is worse, the Pfaffian values are easily out of the
scope of the double precision data type and cause the computation collapsed.
Careful treatment must be made to avoid such data overflow. In this paper, we
implement an accurate and reliable calculation for the overlap between
arbitrary HFB vacua in a unified way. For the cases of ($u_{i}=1,v_{i}=0$) and
($u_{i}=0,v_{i}=1$), we treat them as the cases of
($u_{i}=1,v_{i}=\varepsilon$) and ($u_{i}=\varepsilon,v_{i}=1$), respectively.
The tiny quantity $\varepsilon>0$ is chosen such that $\varepsilon^{2}$ should
be numerical zero relative to 1 in the practical calculation. In other words,
$\varepsilon$ should numerically satisfy $1+\varepsilon^{2}=1$. Under this
condition, $\varepsilon$ may be chosen as large as possible so that the
calculated Pfaffian values are not necessarily too huge or too tiny. For
instance, one can choose $\varepsilon=10^{-8}$ when using double precision.
Because $v_{i}^{2}(u_{i}^{2})=\varepsilon^{2}$ is actually zero relative to
$u_{i}^{2}(v_{i}^{2})=1$ in practical calculations, this treatment does not
change the HFB vacuum at all. Therefore, without losing the generality, we
assume that all levels in the Fock space are partly occupied, but some of
their $u_{i},v_{i}$ values are allowed to be extremely close to 0 or 1.
Ideally, $U,V$ are nonsingular in our assumption, and we can derive a new
formula for the overlap between the HFB vacua based on the work of Bertsch and
Robledo [7]. This formula is especially convenient for the symmetry
restoration. Numerical calculations have been carried out for heavy nuclear
system to test the precision of the new formula by comparing with the Onishi
formula.
In section 2, the formalism of the new overlap formula is given. Section 3
provides an example of numerical calculation. A summary is given in section 4.
## 2 The overlap between the HFB vacua
We denote $\hat{c}^{\dagger}_{i}$ and $\hat{c}_{i}$ as the creation and
annihilation operators defined in an $M$-dimensional Fock-space. The Hartree-
Fock-Bogoliubov(HFB) transformation is
$\left(\begin{array}[]{c}\hat{\beta}\\\
\hat{\beta}^{\dagger}\end{array}\right)=\left(\begin{array}[]{cc}U^{\dagger}&V^{\dagger}\\\
V^{T}&U^{T}\end{array}\right)\left(\begin{array}[]{c}\hat{c}\\\
\hat{c}^{\dagger}\end{array}\right).$ (1)
Here, we assume $U$ and $V$ are nonsingular matrices, and their shapes are
$M\times M$. The HFB vacuum (unnormalized) can be written as
$\displaystyle|\phi\rangle=\hat{\beta}_{1}\hat{\beta}_{2}...\hat{\beta}_{M}|-\rangle,$
(2)
where $|-\rangle$ is the true vacuum. By definition, one has
$\displaystyle\hat{\beta}_{i}|\phi\rangle=0\quad\mathrm{for}\quad 1\leq i\leq
M.$ (3)
The second HFB vacuum $|\phi^{\prime}\rangle$ is defined in the same way, but
the prime, ‘′’, is attached to the corresponding symbols to show difference.
The overlap between $|\phi\rangle$ and$|\phi^{\prime}\rangle$ is given by
$\displaystyle\langle\phi|\phi^{\prime}\rangle$ $\displaystyle=$
$\displaystyle\langle-|\hat{\beta}^{\dagger}_{M}\hat{\beta}^{\dagger}_{M-1}...\hat{\beta}^{\dagger}_{1}\hat{\beta}^{\prime}_{1}\hat{\beta}^{\prime}_{2}...\hat{\beta}^{\prime}_{M}|-\rangle$
(4) $\displaystyle=$ $\displaystyle
s_{M}\langle-|\hat{\beta}^{\dagger}_{1}\hat{\beta}^{\dagger}_{2}...\hat{\beta}^{\dagger}_{M}\hat{\beta}^{\prime}_{1}\hat{\beta}^{\prime}_{2}...\hat{\beta}^{\prime}_{M}|-\rangle,$
where, $s_{M}=(-1)^{[M(M-1)/2]}$. If $M$ is even, $s_{M}=(-1)^{M/2}$.
Following the technique of Bertsch and Robledo [7], one can obtain
$\displaystyle\langle\phi|\phi^{\prime}\rangle$ $\displaystyle=$
$\displaystyle
s_{M}\mathrm{pf}\left(\begin{array}[]{cc}V^{T}U&V^{T}V^{\prime*}\\\
-V^{\prime\dagger}V&U^{\prime\dagger}V^{\prime*}\end{array}\right).$ (7)
The shape of the matrix in Eq. (7) is $2M\times 2M$, and no empty levels are
omitted. For the norm overlap $\langle\phi|\phi\rangle$, it is real and
positive. From Eq.(7) and the Bloch-Messiah theorem, one can get
$\displaystyle\langle\phi|\phi\rangle=s_{M}\mathrm{pf}\left(\begin{array}[]{cc}V^{T}U&V^{T}V^{*}\\\
-V^{\dagger}V&U^{\dagger}V^{*}\end{array}\right)=\prod_{i=1}^{M/2}v_{i}^{2}.$
(10)
Denoting $\prod_{i=1}^{M/2}v_{i}$ by $\mathfrak{N}$, the normalized quasi-
particle vacuum, $|\psi\rangle$, can be written as
$\displaystyle|\psi\rangle=\frac{|\phi\rangle}{\mathfrak{N}}.$ (11)
Then, one finds that
$\displaystyle\langle\psi|\psi^{\prime}\rangle=\frac{s_{M}}{\mathfrak{NN^{\prime}}}\mathrm{pf}\left(\begin{array}[]{cc}V^{T}U&V^{T}V^{\prime*}\\\
-V^{\prime\dagger}V&U^{\prime\dagger}V^{\prime*}\end{array}\right).$ (14)
In the symmetry restoration, the general rotational operator, involving the
spin and particle number projection, may be written as
$\displaystyle\hat{\mathbb{R}}(\Xi)=\hat{R}(\Omega)e^{-i\hat{N}\phi_{n}}e^{-i\hat{Z}\phi_{p}},$
(15)
where $\hat{R}(\Omega)$ is the rotation operator, and $\Omega$ refers to the
three Euler angles $\alpha,\beta,\gamma$. $e^{-i\hat{N}\phi_{n}}$ and
$e^{-i\hat{Z}\phi_{p}}$ are ‘gauge’ rotational operators induced by the
neutron and proton number projection. $\hat{N}$ and $\hat{Z}$ are neutron and
proton number operators, respectively. $\phi_{n}$ and $\phi_{p}$ are ”gauge”
angles for neutron and proton, respectively. $\Xi$ refers to
$(\Omega,\phi_{n},\phi_{p})$. The matrix element
$\langle\psi|\hat{\mathbb{R}}(\Xi)|\psi^{\prime}\rangle$ needs to be
calculated. Let’s define the general rotation transformation for symmetry
restoration,
$\displaystyle\hat{\mathbb{R}}(\Xi)\left(\begin{array}[]{cc}\hat{c}\\\
\hat{c}^{\dagger}\end{array}\right)\hat{\mathbb{R}}^{\dagger}(\Xi)=\left(\begin{array}[]{cc}\mathbb{\mathbb{D}}^{\dagger}(\Xi)&0\\\
0&\mathbb{\mathbb{D}}^{T}(\Xi)\end{array}\right)\left(\begin{array}[]{cc}\hat{c}\\\
\hat{c}^{\dagger}\end{array}\right),$ (22)
where $\mathbb{D}_{ij}(\Xi)=\langle i|\hat{\mathbb{R}}(\Xi)|j\rangle$, and
$|i(j)\rangle=\hat{c}^{\dagger}_{i(j)}|-\rangle$. The $\mathbb{D}(\Xi)$ matrix
has the dimension $M\times M$. One can get
$\displaystyle\hat{\mathbb{R}}(\Xi)\left(\begin{array}[]{cc}\hat{\beta}^{\prime}\\\
\hat{\beta}^{\prime\dagger}\end{array}\right)\hat{\mathbb{R}}^{\dagger}(\Xi)=\mathcal{D}(\Xi)\left(\begin{array}[]{cc}\hat{c}\\\
\hat{c}^{\dagger}\end{array}\right),$ (27)
where
$\displaystyle\mathcal{D}(\Xi)=\left(\begin{array}[]{cc}[\mathbb{D}(\Xi)U^{\prime}]^{\dagger}&[\mathbb{D}^{*}(\Xi)V^{\prime}]^{\dagger}\\\
{[\mathbb{D}^{*}(\Xi)V^{\prime}]^{T}}&{[\mathbb{D}(\Xi)U^{\prime}]^{T}}\end{array}\right).$
(30)
By comparing Eq.(27) with Eq.(1), one can obtain the rotated overlap by
replacing $U^{\prime}$ and $V^{\prime}$ in Eq.(14) with
$\mathbb{D}(\Xi)U^{\prime}$ and $\mathbb{D}^{*}(\Xi)V^{\prime}$, respectively.
Thus
$\displaystyle\mathbb{N}_{\mathrm{pf}}(\Xi)=\langle\psi|\hat{\mathbb{R}}(\Xi)|\psi^{\prime}\rangle=\frac{s_{M}}{\mathfrak{NN^{\prime}}}\mathrm{pf}[\mathcal{M}(\Xi)],$
(31)
where
$\mathcal{M}(\Xi)=\left(\begin{array}[]{cc}V^{T}U&V^{T}\mathbb{D}(\Xi)V^{\prime*}\\\
-V^{\prime\dagger}\mathbb{D}^{T}(\Xi)V&U^{\prime\dagger}V^{\prime*}\end{array}\right).$
(32)
This formula is essentially the same as the one proposed by Bertsch and
Robledo [7], but we will transform it into a new form. Supposing that there is
a $\Xi_{0}$ satisfying $\mathbb{N}_{\mathrm{pf}}(\Xi_{0})\neq 0$, we have
$\displaystyle\frac{\mathbb{N}_{\mathrm{pf}}(\Xi)}{\mathbb{N}_{\mathrm{pf}}(\Xi_{0})}$
$\displaystyle=$
$\displaystyle\frac{\mathrm{pf}[\mathcal{M}(\Xi)]}{\mathrm{pf}[\mathcal{M}(\Xi_{0})]}=\frac{\mathrm{pf}\left[P\mathcal{M}(\Xi)P^{T}\right]}{\mathrm{pf}\left[P\mathcal{M}(\Xi_{0})P^{T}\right]}$
(33) $\displaystyle=$
$\displaystyle\frac{\mathrm{pf}[\mathcal{W}(\Xi)]}{\mathrm{pf}[\mathcal{W}(\Xi_{0})]},$
where
$\displaystyle\mathcal{W}(\Xi)=\left(\begin{array}[]{cc}[U^{\prime}V^{\prime-1}]^{\dagger}&-\mathbb{D}^{T}(\Xi)\\\
\mathbb{D}(\Xi)&UV^{-1}\end{array}\right),$ (36)
and $P$ is
$\displaystyle P=\left(\begin{array}[]{cc}0&(V^{\prime\dagger})^{-1}\\\
(V^{T})^{-1}&0\end{array}\right).$ (39)
Therefore, one can get
$\displaystyle{\mathbb{N}_{\mathrm{pf}}(\Xi)}=\mathcal{C}{\mathrm{pf}[\mathcal{W}(\Xi)]},$
(40)
where, the coefficient $\mathcal{C}$ is actually independent of $\Xi_{0}$, and
can be written as
$\displaystyle\mathcal{C}=\frac{\mathbb{N}_{\mathrm{pf}}(\Xi_{0})}{\mathrm{pf}[\mathcal{W}(\Xi_{0})]}=\frac{s_{M}}{\mathfrak{NN^{\prime}}\mathrm{det}P}={s_{M}\Delta\mathfrak{NN^{\prime}}}.$
(41)
Here, $\Delta$ is a phase determined by
$\displaystyle\Delta=\mathrm{det}D^{*}\mathrm{det}D^{\prime}\mathrm{det}C\mathrm{det}C^{\prime*}.$
(42)
In Eq.(40), we have used the Bloch-Messiah theorem and the following equation
$\displaystyle\mathrm{det}P=\mathrm{det}[(V^{\prime\dagger})^{-1}]\mathrm{det}[(V^{T})^{-1}].$
(43)
Eq.(40) looks more convenient to be implemented and may save some computing
time in contrast to Eq.(31), where extra evaluation of
$V^{T}\mathbb{D}(\Xi)V^{\prime*}$ is required for each mesh point in the
integral of projection.
For comparison, let us present a brief introduction of the overlap of the
Onishi formula [10]. The unitary transformation of the quasi-particles under
rotation $\hat{\mathbb{R}}(\Xi)$ can be written as
$\displaystyle\hat{\mathbb{R}}(\Xi)\left(\begin{array}[]{cc}\hat{\beta}^{\prime}\\\
\hat{\beta}^{\prime\dagger}\end{array}\right)\hat{\mathbb{R}}^{\dagger}(\Xi)=\left(\begin{array}[]{cc}\mathbb{X}(\Xi)&\mathbb{Y}(\Xi)\\\
\mathbb{Y}^{*}(\Xi)&\mathbb{X}^{*}(\Xi)\end{array}\right)\left(\begin{array}[]{cc}\hat{\beta}\\\
\hat{\beta}^{\dagger}\end{array}\right),$ (50)
where
$\displaystyle\mathbb{X}(\Xi)$ $\displaystyle=$ $\displaystyle
U^{\prime\dagger}\mathbb{D}^{\dagger}(\Xi)U+V^{\prime\dagger}\mathbb{D}^{T}(\Xi)V,$
$\displaystyle\mathbb{Y}(\Xi)$ $\displaystyle=$ $\displaystyle
U^{\prime\dagger}\mathbb{D}^{\dagger}(\Xi)V^{*}+V^{\prime\dagger}\mathbb{D}^{T}(\Xi)U^{*}.$
(51)
The Onishi formula is then expressed as (see Ref.[20]),
$\displaystyle\mathbb{N}_{\mathrm{Onishi}}(\Xi)=\langle\psi|\hat{\mathbb{R}}(\Xi)|\psi^{\prime}\rangle$
(52) $\displaystyle=$
$\displaystyle(\pm)\sqrt{\mathrm{det}[\mathbb{X}(\Xi)]}e^{-i(M_{n}\phi_{n}+M_{p}\phi_{p})/2},$
where $M_{n}$ and $M_{p}$ are the numbers of neutron and proton orbits in the
Fock space, respectively. The value of $\mathrm{det}[\mathbb{X}(\Xi)]$ is a
complex number, and the sign of the square root is left undefined. Extra
efforts must be made to determine the sign before the application of the
Onishi formula. For instance, in the Projected Shell Model [21] without
particle number projection, the overlap between the BCS vacua is real and
positive, thus there is no sign ambiguity and the Onishi formula works.
## 3 Numerical test of the overlap formulae
Although the sign problem is solved in Eq.(31) and Eq.(40), one can imagine
that $\mathfrak{N}$, $\mathfrak{N}^{\prime}$ are extremely tiny numbers by
definition. Thus $\mathrm{pf}[\mathcal{M}(\Xi)]$ is also very tiny, but
$\mathrm{pf}[\mathcal{W}(\Xi)]$ should be huge. Numerical accuracy of Eqs.
(31) and (40) needs to be carefully tested. It is believed that the Onishi
formula is accurate except for its undetermined sign. So, it is helpful to
compare the numerical values of the overlaps using Eq.(31), Eq.(40) and
Eq.(52).
To demonstrate the accuracy and the reliability of the Eqs. (31) and (40),
numerical calculations are performed for the typical example of the deformed
heavy nucleus 226Th. For projection, we should take
$|\psi\rangle=|\psi^{\prime}\rangle$, and then $\Delta=1$.
The $U,V$ matrices are obtained from the Nilsson+BCS method. The single
particle levels are generated from the Nilsson Hamiltonian with the standard
parameters [22]. The single-particle model space contains 5 neutron major
shells with $N=$4-8 and 5 proton major shells with $N=$3-7, i.e., the Fock
space has 145 neutron levels ($M_{n}=290$) and 110 proton levels
($M_{p}=220$). The numbers of the active neutrons and protons are 96 and 70,
respectively. The quadrupole deformation is taken to be $\epsilon_{2}=0.2$.
Here, we only consider the axial symmetry for simplicity.
In the no pairing case, the BCS vacuum becomes a pure slater determinant,
which is a challenge for Eq.(40) because all $v_{i}$’s above the Fermi surface
are zero. Consequently, $\mathfrak{N}=0$ and $\mathcal{W}(\Xi)$ is meaningless
due to the singularity of $V$. Here, we use the double precision data type and
set $v_{i}=\varepsilon=10^{-8}$ for those $v_{i}=0$ orbits to avoid the
collapse of calculation. Therefore we have
$\displaystyle\langle\phi_{n}|\phi_{n}\rangle=(10^{-16})^{\frac{290-96}{2}}=10^{-1552},$
$\displaystyle\langle\phi_{p}|\phi_{p}\rangle=(10^{-16})^{\frac{220-70}{2}}=10^{-1200},$
where, $|\phi_{n}\rangle$ and $|\phi_{p}\rangle$ are BCS vacua for neutrons
and protons, respectively, and
$|\phi\rangle=|\phi_{n}\rangle|\phi_{p}\rangle$. The tiny numbers $10^{-1552}$
and $10^{-1200}$ are too far out of the scope of the double precision data
($\sim 10^{\pm 307}$). To avoid the data overflow, we multiply the tiny
variable by $10^{200}$ several times until the scaled absolute value falls
into the interval $[10^{-200},10^{200}]$. In other words, we use a number $y$
and an integer number $k$ to express a tiny number $x$ through
$x=y\times(10^{-200})^{k}$. If $x$ is a huge number, then $k$ is negative.
Figure 1: (Color online) Overlaps of the ground state neutron slater
determinant for 226Th as functions of $\phi_{n}$ with Euler angles
$\alpha=\gamma=0^{\circ},\beta=10^{\circ}$, calculated with present formula
[Eq.(40)] and the Onishi formula [Eq.(52) with ‘+’ sign].
Re$[\mathbb{N}(\phi_{n})]$ and Im$[\mathbb{N}(\phi_{n})]$ are the real and
imaginary parts of the overlap.
However, for the Onishi formula of Eq.(52), we do not need to change $v_{i}=0$
to $v_{i}=\varepsilon$. The overlaps for the neutron part, calculated with
Eq.(40) and Eq.(52), are compared in Fig.1. The curves of Eq.(40) are
continuous, but the sign uncertainty of Eq.(52) causes the discontinuity.
However, if one copies the sign of Eq.(40) to Eq.(52), one can compare
numerical difference between Eq.(40) and Eq.(52) using the following quantity,
$R$,
$\displaystyle
R=\left|\frac{\mathbb{N}_{\mathrm{Onishi}}(\phi_{n})}{\mathbb{N}_{\mathrm{pf}}(\phi_{n})}-1\right|.$
(53)
In all calculations, we found that $R<10^{-12}$ with double precision. This
confirms that a small change of $v_{i}$ from zero to $\varepsilon$ almost does
not affect numerical accuracy. However, it is crucial to keep Eq.(40) valid.
Yet notice that $\mathbb{N}_{\mathrm{pf}}(\Xi)$ in Eq.(40) is obtained from a
product of tiny and giant numbers. The same calculations have also been done
with Eq.(31), and we also get $R<10^{-12}$. Thus we have presented an
alternative way of using Eq.(31), where we set $v_{i}=\varepsilon$ for those
empty orbits rather than omitting them[7].
Once the overlap is available, it is straightforward to perform the symmetry
restoration. The deformed BCS vacuum of 226Th has been projected onto good
particle number and spin. Therefore, one can test how precise the numerical
calculations with Eq.(40) satisfy
$\displaystyle\sum_{N,Z,I}\langle\psi|\hat{P}^{N}\hat{P}^{Z}\hat{P}^{I}_{00}|\psi\rangle=1,$
(54)
where $\hat{P}^{N}$, $\hat{P}^{Z}$, and $\hat{P}^{I}_{MK}$ are neutron-number,
proton-number, and spin projection operators, respectively. For the above
vacuum state without pairing (i.e. the ground state slater determinant), the
particle numbers of both neutrons and protons are good. Indeed, our particle
number projection (using 16 mesh points in the integral) shows that
$\langle\psi_{n}|\hat{P}^{N}|\psi_{n}\rangle=1$ ($N=96$), or 0 ($N\neq 96$)
with numerical errors less than $10^{-13}$. Calculations for the protons also
have the same accuracy. This again shows the reliability of Eq. (40). Angular
momentum projection is also performed on the same state in addition to the
particle number projection. The amplitude of
$\langle\psi|\hat{P}^{N}\hat{P}^{Z}\hat{P}^{I}_{00}|\psi\rangle$ with
($N=96,Z=70$) is plotted as a function of spin $I$ in Fig.2. In the integral
of the spin projection, 100 mesh points are taken, and the range of spin is
$0\leq I\leq 70$, and we indeed reproduced Eq.(54) with numerical error around
$10^{-12}$.
Figure 2: The amplitude of projection,
$\langle\psi|\hat{P}^{N}\hat{P}^{Z}\hat{P}^{I}_{00}|\psi\rangle$, as a
function of spin $I$ at $N=96$ and $Z=70$ using Eq.(40). $|\psi\rangle$ is the
axially deformed BCS vacuum but without pairing.
We also have tested Eq.(40) in the projection of the triaxially deformed
vacuum with normal pairing, which seems more convenient to use Eq.(40). With
the present method, similar accuracy has also been achieved.
## 4 Summary
Following the strategy of Bertsch and Robledo [7], we have proposed a new
formula of the overlap between HFB vacua by using the Pfaffian identity and
assuming that the inverse of the $V$ matrix exists. This formula is especially
convenient and efficient in the symmetry restoration, and has the same high
accuracy as the Onishi formula as well as the correct sign. The reliability of
the present formula has been tested by carrying out the calculations of the
overlap and the quantum number projection for the heavy nucleus 226Th. In the
testing calculations, one has to be faced with two numerical problems: (1) The
extreme (huge or tiny) quantities are certainly encountered, and we have
properly treated this situation to avoid data overflow (see the text). (2) For
those empty orbits with $v_{i}=0$, which make Eq.(40) invalid, one can change
$v_{i}$ to a small quantity $\varepsilon(>0)$ to avoid the singularity of $V$
matrix. It turns out that such treatments work very well. Testing calculations
have confirmed that the present formula is even applicable to the pure slater
determinant without losing the numerical accuracy. Thus it is promising that
Eq.(40) may be applicable in evaluating the overlap between arbitrary HFB
vacua.
Acknowledgements Z. G. thanks Prof. Y. Sun and Dr. F. Q. Chen for the
stimulating and fruitful discussions. The authors acknowledge support from the
National Natural Science Foundation of China under Contract Nos. 11175258,
11021504 and 11275068.
## References
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* [6] B. Avez, M. Bender, Phys. Rev. C. 85 (2012) 034325.
* [7] G.F. Bertsch and L.M. Robledo, Phys. Rev. Lett. 108 (2012) 042505.
* [8] S. Perez-Martin and L.M. Robledo, Phys. Rev. C 76 (2007) 064314.
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* [15] K. Hara, A. Hayashi and P. Ring, Nucl. Phys. A 385 (1982) 14.
* [16] M. Oi and N. Tajima, Phys. Lett. B 606 (2005) 43.
* [17] L.M. Robledo, Phys. Rev. C 79 (2009) 021302(R).
* [18] L.M. Robledo, Phys. Rev. C 84 (2011) 014307.
* [19] P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer-Verlag, 1980.
* [20] K.W. Schmid, Prog. Part. Nucl. Phys. 52 (2004) 565.
* [21] K. Hara and Y. Sun, Int. J. Mod. Phys. E 4 (1995) 637.
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|
arxiv-papers
| 2013-06-13T08:11:13 |
2024-09-04T02:49:46.462058
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zao-Chun Gao, Qing-Li Hu and Y. S. Chen",
"submitter": "Zao-Chun Gao",
"url": "https://arxiv.org/abs/1306.3051"
}
|
1306.3127
|
# Behavior in a Shared Resource Game with Cooperative, Greedy, and Vigilante
Players
Christopher Griffin111 Dr. Griffin is a faculty member at the Applied Research
Laboratory and Department of Mathematics, Penn State University, University
Park, PA 16802, E-mail: [email protected] and George Kesidis222Dr. Kesidis
is a faculty member in the Departments of Electrical Engineering and Computer
Science and Engineering, Penn State University, University Park, PA 16802,
E-mail: [email protected]
###### Abstract
We study a problem of trust in a distributed system in which a common resource
is shared by multiple parties. In such naturally information-limited settings,
parties abide by a behavioral protocol that leads to fair sharing of the
resource. However, greedy players may defect from a cooperative protocol and
achieve a greater than fair share of resources, often without significant
adverse consequences to themselves. In this paper, we study the role of a few
vigilante players who also defect from a cooperative resource-sharing protocol
but only in response to perceived greedy behavior. For a simple model of
engagement, we demonstrate surprisingly complex dynamics among greedy and
vigilante players. We show that the best response function for the greedy-
player under our formulation has a jump discontinuity, which leads to
conditions under which there is no Nash equilibrium. To study this property,
we formulate an exact representation for the greedy player best response
function in the case when there is one greedy player, one vigilante player and
$N-2$ cooperative players. We use this formulation to show conditions under
which a Nash equilibrium exists. We also illustrate that in the case when
there is no Nash equilibrium, then the discrete dynamic system generated from
fictitious play will not converge, but will oscillate indefinitely as a result
of the jump discontinuity. The case of multiple vigilante and greedy players
is studied numerically. Finally, we explore the relationship between
fictitious play and the better response dynamics (gradient descent) and
illustrate that this dynamical system can have a fixed point even when the
discrete dynamical system arising from fictitious play does not.
## I Introduction
In this paper, we study the problem of trust in a distributed system in which
a common resource is shared by many parties or players. In such distributed
systems, cooperation and trust are required for the fair and efficient use of
a common resource by a plurality of parties/players. Often in such naturally
information-limited settings, the players abide by a behavioral protocol that
leads to fair sharing of resource. However, a greedy player may defect from a
cooperative protocol and achieve a greater than fair share of resources, often
without significant adverse consequences if any. This problem has a long
history, e.g., [1, 2, 3, 4], and a broad range of applications - e.g., in [5],
the problem of efficient cooperation of two processes that a share resource is
studied from a control-theoretic perspective. The more general problem of
trust and cooperation remains an active area of research in multiple
disciplines [6, 7, 8]. A principle challenge is attribution, and perhaps even
detection, of deviation from cooperative behavior by some greedy players.
Upon detection of greedy behavior (essentially, detection of a breech of
trust), all players may defect from cooperative behavior leading to a less
efficient uncooperative (anarchistic) equilibrium or possibly deadlock and a
“tragedy of the commons” [9]. In this paper, we consider a much more measured
response by only a small number of “vigilante” players that also defect from
cooperative play but only after greedy behavior has been detected. The
intention of such vigilante play is to entice greedy players back to
cooperative play by creating a near deadlock situation in which all players
suffer. For an “objective based” model of engagement, we show surprisingly
complex behavior among greedy and vigilante players.
Specifically, we assume a shared resource can be accessed by any of $N$ users
at any time, but two users cannot access the resource at the same time. Each
user $i$ chooses a probability $q_{i}$ of accessing the resource at any given
time. Thus, the probability that user $i$ can access the resource is:
$T_{i}(q_{1},\dots,q_{N})=q_{i}\prod_{j\neq i}(1-q_{j})$ (1)
An example of this model is a synchronous, random-access ALOHA local-area
communications network [10]. In this system, users transmit at random and
simultaneous communications cause collision, which results in failed
communication. Cooperative use of a resource is common in communications
systems in which all users assume that most, if not all, other users adhere to
agreed upon protocols of behavior, e.g., Internet protocols like TCP
congestion control, even if cooperation is not in their immediate best
interest. Various distributed mechanisms have been implemented to
cooperatively desynchronize demand (e.g., TCP, ALOHA, CSMA). Typically, when
congestion is detected, all end-devices are expected to slow down their
transmission rates and then slowly increase again hoping to find a fair and
efficient equilibrium. However, if some users employ alternative
implementations of the prescribed (“by rule”) protocols, e.g., ones that slow
down less than they should, or even increase their transmission rate in the
presence of congestion, the result could be an unfair allocation or even
congestion collapse, see, e.g., [11, 12]. There is a steadily growing
literature on communications that analyzes the equilibria of different
distributed network resource allocation games, e.g., [13, 14, 15, 16, 17, 18,
19, 20, 21, 22, 23]; these results are relevant to more general resource
sharing problems. The experience with TCP in particular, e.g., [24], has shown
that developers do create versions of the protocol that depart from the
standard, cooperative (by-rule) congestion-avoidance algorithm, like Turbo
TCP, but that the great majority of end-hosts employ the standard cooperative
protocol.
Our objective in this paper is to formulate a model that combines the
objective functions of greedy players, vigilante players and cooperative
players. Cooperative players follow a prescribed (fair) protocol and are not
selfish utility maximizers. Greedy players are selfish utility maximizers
whose objective is to take-over the resource. A vigilante player prefers to
follow a fair resource sharing protocol, but will increase her transmission
rate to punish perceived greediness. As a part of this work, we show that the
cyclic behavior induced in [25] through fixed rules can result from a
discontinuity in the best-response function.
The remainder of this paper is organized as follows: In Section II we lay out
the preliminary formulae used in the remainder of this paper. In Section III
we provide details on our model, including greedy and altruistic player
utility functions. We analyze a two-player system in Section IV we explicitly
study a simplified two player shared channel model and characterize the jump
discontinuity in the best response function of the greedy player and its
effect on Nash Equilibria. In Section V we study multi-player systems
numerically when there are multiple greedy players or multiple vigilante
players and compare our results to the results of better-response dynamics.
Finally we provide conclusions and future directions in Section VI.
## II Mathematical Preliminaries
Let $q\in[0,1]$ be the transmission probability for a cooperative player in
our distributed resource game. In a game with $N$ players, $q=1/N$, the fair
allocation of the resource to a cooperative player. Let $g\in[0,1]$ be the
resource access probability of the greedy player. Presumably, $g\geq q$ for
any fixed $N$. Finally, let $a\in[0,1]$ be the resource access probability of
the vigilante player. Presumably, $a\geq q$ for any $N$. The expected resource
access probability for a greedy player is:
$\Theta(g,a):=g(1-a)\left(1-\frac{1}{N}\right)^{N-2},$ (2)
with the corresponding expected resource access probability for the vigilante
player is:
$\Phi(g,a):=a(1-g)\left(1-\frac{1}{N}\right)^{N-2}=\Theta(a,g).$ (3)
All other players access the resource with probability:
$\frac{1}{N}(1-a)(1-g)\left(1-\frac{1}{N}\right)^{N-3}.$
In the absence of knowledge of the vigilante, the greedy player expects
$a=1/N$ and thus would like to maximize $\Theta\left(g,\tfrac{1}{N}\right)$,
which can be accomplished by setting $g=1$ to obtain a resource access
probability of:
$\Theta_{0}:=\left(1-\frac{1}{N}\right)^{N-1}.$ (4)
In the absence of knowledge of the greedy player, the vigilante player expects
$g=1/N$ and expects a resource access probability of:
$\Phi_{0}:=\frac{1}{N}\left(1-\frac{1}{N}\right)^{N}$ (5)
## III Mathematical Model
Suppose now the vigilante player expects a (single) greedy player. Using an
estimate of her resource access probability $\hat{\Phi}$, an estimate can be
obtained for $g$ as:
$\hat{g}:=\left[\frac{a\left({\frac{N-1}{N}}\right)^{N-2}-\hat{\Phi}}{a\left({\frac{N-1}{N}}\right)^{N-2}}\right]_{0}^{1}.$
(6)
The vigilante player now wishes to enforce fairness unilaterally, by modifying
her access probability to punish greedy players. However, it is possible the
vigilante player is sensitive to her impact on the community e.g., in the case
when the greedy player is only a little greedy. In this case, the objective
function of the vigilante player to be minimized can be written as:
$U_{a}(g,a;\rho):=\left(\Theta(g,a)-\Phi_{0}\right)^{2}+\rho\left(a-\frac{1}{N}\right)^{2}.$
(7)
Here $\rho$ is a control parameter that adjusts the extent to which the
vigilante is willing to sacrifice her principles of good behavior to punish a
greedy player. As we will see, this parameter can have a substantial impact on
existence of the underlying system equilibria.
Conversely, the greedy player wishes to maximize his resource access
probability and is willing to violate the communal policy of fairness (e.g.,
$g=1/N$) to do so. However, the greedy player realizes there may be a
vigilante who will punish him for bad behavior and hence may modulate his
behavior back toward the communal norm if he detects his expected resource
access probability $\hat{\Theta}$ is well below his desired value
$\Theta_{0}$. The greedy player’s objective function to be minimized can be
formulated as:
$U_{g}(g,a;\lambda):=\left(\Theta(g,a)-\Theta_{0}\right)^{2}\cdot\left(1+\lambda\left(g-\frac{1}{N}\right)^{2}\right).$
(8)
Note that as $\Theta(g,a)-\Theta_{0}$ approaches zero, then for any fixed
value $\lambda$, $\lambda\left(g-\tfrac{1}{N}\right)^{2}$ also approaches zero
and the effect of $\left(g-\tfrac{1}{N}\right)^{2}$ diminishes. Thus a
successful greedy player ignores the fact he is not playing fairly, while an
unsuccessful greedy player will throttle back his greediness to try to find a
better outcome. We note that the function $U_{g}$ has three (first order)
critical points given by:
$C_{g}=\left\\{\frac{N-1}{N(1-a)},\frac{N-3a+2}{(1-a)N}\pm\sqrt{-\frac{8}{\lambda}+{\frac{\left(a-2+N\right)^{2}}{{N}^{2}\left(a-1\right)^{2}}}}\right\\},$
while the function $U_{a}$ has a single critical point given by:
$\frac{1}{4}\,{\frac{4\,{g}^{2}-g+2\,\rho}{{g}^{2}+\rho}}.$ (9)
Throughout the remainder of this paper, we will study the game in which both
the greedy and vigilante players are utility minimizers whose decisions affect
each other. In the sequel, we refer to this game as
$\mathcal{G}(U_{g},U_{a})$.
## IV Analysis of $N$ Player System
The fact that the objective functions are quartic in $g$ and quadratic in $a$
leads to a complex analytical problem for arbitrary $N\geq 2$. We show that
the best response function of the Greedy player may have a jump discontinuity
and characterize it completely when it does.
Given a value $a\in[0,1]$, the best response function for the greedy player,
denoted by $\beta_{g}(a;\lambda)$ is the set of values of $g$ that minimize
$U_{g}$ for the given value of $a$. We note that when this point-to-set map is
a function, then it may be discontinuous, as shown in Figure 1. This
discontinuity is caused by the non-convexity of $U_{g}$ in $g$. An interesting
result of this phenomenon is the fact that the game $\mathcal{G}(U_{g},U_{a})$
may not have any Nash equilibrium (NE), leading to interesting discrete time
dynamic behavior.
Let $\beta_{a}(g;\rho)$ be the best response function for the vigilante player
(defined analogously for $\beta_{g}(a;\lambda)$). Recall from [26] (Chapter 1)
that a pair $(g^{*},a^{*})$ is a NE if and only if
$g^{*}\in\beta_{g}(a^{*};\lambda)$ and $a^{*}\in\beta_{a}(g^{*};\rho)$.
Suppose that $\beta_{a}(g;\rho)$ and $\beta_{g}(a;\lambda)$ are functions
(rather than point-to-set maps). A pair $(g^{*},a^{*})$ is a NE if and only if
$g^{*}=\beta_{g}(a^{*};\lambda)$ and $g^{*}\in\beta_{a}^{-1}(a^{*};\rho)$ (or
likewise $a^{*}=\beta_{a}(g^{*};\rho)$ and
$a^{*}\in\beta_{g}^{-1}(g^{*};\lambda)$). Here $\beta^{-1}_{a}$ and
$\beta^{-1}_{g}$ are the usual inverse relations.
(a) Nash Equilibrium (b) No Nash Equilibrium
Figure 1: In the first figure, the NE is located at the intersection of the
two curves, in this case $\beta_{g}(a;\lambda)$ and $\beta_{a}^{-1}(g,\rho)$.
In the second figure, no such intersection occurs.
We now illustrate two cases for the game where $N=10$; that is there is one
vigilante player and one greedy player and eight cooperative players. In one
case, a NE exists and in the other no NE exists. Fix $\lambda=10$. For
$\rho=0.001$, a (unique) NE exists while for $\rho=0.01$ there is no NE. The
two cases are illustrated in Figure 1.
We can solve precisely for the point of discontinuity in the best response
function and obtain a complete characterization of the discontinuous best-
response curve $\beta_{g}(a;\lambda)$. We have already established that there
are three critical points that may come into play in finding (local) minima of
the function $U_{g}$. The discontinuity is caused by the best response moving
among two of these three points as well as the boundary value $g=1$.
We can prove easily that
$r_{1}:=\frac{N-1}{N(1-a)}$ (10)
is a global minima. To see this, note $U_{g}(r_{1})=0$ and $U_{g}$ itself is
strictly non-negative and thus $r_{1}$ must be a global minima since $U_{g}$
attains $0$ at this value.
We can also see that when
$r_{3}:=\frac{N-3a+2}{(1-a)N}-\sqrt{-\frac{8}{\lambda}+{\frac{\left(a-2+N\right)^{2}}{{N}^{2}\left(a-1\right)^{2}}}},$
is real and distinct from:
$r_{2}:=\frac{N-3a+2}{(1-a)N}+\sqrt{-\frac{8}{\lambda}+{\frac{\left(a-2+N\right)^{2}}{{N}^{2}\left(a-1\right)^{2}}}},$
then it is a local minima. To see this, note that evaluating the second
derivative of $U_{g}$ at $r_{3}$ yields:
$-\frac{1}{2}\,{N}^{2}\left({\frac{N-1}{N}}\right)^{2\,N}\left(s_{{1}}\cdot\gamma+s_{{2}}\right)\left(N-1\right)^{-4}$
where:
$s_{{1}}=3\,N\lambda\,\left(-1+a\right)\left(a-2+N\right)$
$s_{{2}}=8\,{N}^{2}{a}^{2}-16\,{N}^{2}a-{N}^{2}\lambda-2\,Na\lambda-{a}^{2}\lambda+8\,{N}^{2}+\\\
4\,N\lambda+4\,a\lambda-4$
and
$\gamma=\sqrt{-{\frac{s_{{2}}}{{N}^{2}\left(1-a\right)^{2}\lambda}}}$
Our assumption that $r_{3}$ is real implies that $s_{2}<0$. Further, our
assumption that $a\in(0,1)$ implies that $s_{1}<0$. Clearly, $\gamma>0$ (using
the customary positive branch of the square root function). It follows that
$s_{1}\gamma+s_{2}<0$. Thus, $U_{g}^{\prime}(r_{3})>0$ and $r_{3}$ is a local
minima.
As a corollary to the previous result, we note that when it exists and is
distinct from $r_{3}$, the critical point $r_{2}$ is a local maximum. To see
this, we observe that $U_{g}(g,a;\lambda)$ is a fourth order polynomial in $g$
with a positive coefficient for $g^{4}$ when we assume $a>0$ and $\lambda>0$.
The corollary follows from the previous results and this fact.
We now observe that the first critical point $r_{1}$ is strictly less than 1
when $a<1/N$. For $a\geq 1/N$, $r_{1}\geq 1$. Thus we have proved that for
$a\in[0,\tfrac{1}{N}]$, the behavior of $\beta_{g}(a;\lambda)$ on the left-
side of the discontinuity is defined by the function:
$\beta_{g}^{-}(a;\lambda):=\min\left\\{1,\frac{N-1}{N(1-a)}\right\\}=\min\\{1,r_{1}\\}.$
(11)
Let $a^{+}$ be the point of discontinuity. We have already shown that
$a^{+}\geq 1/N$. Clearly now to the right of $a^{+}$, the value of
$\beta_{g}(a;\lambda)$ is controlled by the third critical point in $C_{g}$.
Thus we have:
$\beta_{g}^{+}(a;\lambda):=\frac{N-3a+2}{(1-a)N}-\sqrt{-\frac{8}{\lambda}+{\frac{\left(a-2+N\right)^{2}}{{N}^{2}\left(a-1\right)^{2}}}}$
(12)
For $a\in[1/N,a^{+}]$, $\beta_{g}(a;\lambda)$ takes on its boundary value
$g^{*}=1$. In reality, the best response is a $g^{*}>1$, but this is not
possible. It now suffices to compute $a^{+}$. This can be done by solving for
the value of $a$ so that:
$U_{g}(1,a;\lambda)=U_{g}\left(r_{3},a;\lambda\right)$ (13)
Assuming $a^{+}$ is the (unique) root on $[1/N,1]$ of Equation 13 we now may
write:
$\beta_{g}(a;\lambda):=\begin{cases}\beta_{g}^{-}(a;\lambda)&\text{if
$a<a^{+}$}\\\ \beta_{g}^{+}(a;\lambda)&\text{otherwise}\end{cases}$ (14)
Multiple (non-extraneous) roots for Equation 13, simply indicate the presence
of additional jump discontinuities as the best response moves back and forth
between the boundary value $g=1$ and $g=r_{3}$. In practice we have not
observed additional jump discontinuities and we conjecture that for any
$\lambda$ there is a unique $a^{+}\in[1/N,1]$ that completely characterizes
the discontinuity point.
Suppose the Vigilante and Greedy players engage in iterated play and that each
player can estimate his/her throughput and hence the other player’s strategy.
From this information, each player can compute his/her best response using
$\beta_{g}(a;\lambda)$ and $\beta_{a}(g;\rho)$. The player’s strategy at time
$t\geq 0$ can then be updated according to the rule:
$\displaystyle g^{t+1}=(\beta_{g}(a^{t};\lambda)-g^{t})\epsilon_{g}+g^{t}$
(15) $\displaystyle
a^{t+1}=(\beta_{a}(g^{t};\lambda)-a^{t})\epsilon_{a}+a^{t}$ (16)
Here $\epsilon_{g}$ and $\epsilon_{a}$ are parameters that control the extent
of the player’s jump. In the case when there is no Nash equilibria, we observe
oscillatory behavior caused by the jump discontinuity in $\beta_{g}$. The
oscillation size is directly related to the size of $\epsilon_{g}$ and
$\epsilon_{a}$. This is illustrated in Figure 2.
Figure 2: The oscillation of the two players strategies in a discrete step
iterated game is caused by the jump discontinuity of $\beta_{g}$.
By contrast, when there is a Nash equilibrium, the system converges to it (as
would be expected). This is illustrated in Figure 3.
Figure 3: The existence of a NE ensures that iterated play converges to a
system equilibrium.
## V Numerical Analysis of Multi-Player Systems
We now consider two scenarios: (i) We show that the better response behavior
given by Jacobi iteration can have convergent behavior, even in the case when
there is no Nash equilibrium, illustrating the differences in convergence
between better and best response play. (ii) We show that the presence of an
additional greedy player yields non-trivial behavioral changes on the part of
the greedy and vigilante strategies as a result of the computation of
$\hat{g}$ (see Expression 6).
### V-A Comparison to Differential Play
In convex game-theoretic analysis, it is not uncommon to investigate the
system of differential equations generated by Jacobi iteration (see e.g.,
[27]). For us, these are defined by:
$\left\\{\begin{aligned} \dot{a}=-\frac{\partial U_{a}(g,a;\rho)}{\partial
a}\\\ \dot{g}=-\frac{\partial U_{g}(g,a;\lambda)}{\partial
g}\end{aligned}\right.$ (17)
This model is meant to suggest that the players, rather than computing their
best response to (an estimate) of the other player’s strategy will follow an
(infinitesimal) gradient descent. If a point $(g^{*},a^{*})$ is an interior NE
(that is, it is not on the boundary) then necessarily, ${\partial
U_{a}(g^{*},a^{*};\rho)}/{\partial a}={\partial
U_{g}(g^{*},a^{*};\lambda)}/{\partial g}=0$; i.e., each interior NE is
necessarily a fixed point of the system in Expression 17. We note that this is
a necessary condition for an interior NE, not a sufficient condition in the
case of non-convex player objective functions.
We have already observed that when $\lambda=10$ and $\rho=0.01$, there is no
NE. However, there is an interior fixed point for System 17. Identifying a
solution for System 17 requires identifying the roots of a complex set of
polynomial equations. These can be solved in closed form (no polynomial has a
degree higher than 4) but the closed form solutions do not yield any intuition
into the properties of the underlying model. What is interesting, is that
there exist real-valued fixed points of the differential equation system for
which the system is stable, even when the fixed point is not a NE. In
particular, when $\rho=0.01$, then the point of stability is: $g\approx
0.203$, $a\approx 0.297$, while for $\rho=0.001$, the point of stability is
$g\approx 0.175$, $a\approx 0.429$, where the second fixed point is the same
as the Nash equilibrium. The intersection of the best response curves occurs
when $\beta_{g}(a;\lambda)=r_{3}$ while $\beta_{a}(g;\rho)$ is (always)
computed as:
$\left[\arg_{a}\left(\frac{\partial U_{a}(g,a;\lambda)}{\partial
a}=0\right)\right]^{1}_{0}$ (18)
Thus the intersection of $\beta_{g}(a;\lambda)$ and $\beta_{a}(g;\rho)$ must
occur at a stability point for System 17. We can show that in both cases these
points are globally stable by analyzing the eigenvalues of the Jacobian matrix
of the linearized system. One can verify that when $\rho=0.01$, the
eigenvalues of the Jacobian matrix are approximated by $\\{-1.501,-0.053\\}$,
while for $\rho=0.001$ the eigenvalues of the Jacobian matrix are approximated
by $\\{-1.981,-0.021\\}$. Thus by Theorem 3.1 of [28], the fixed points of the
nonlinear systems are stable, even if these points do not correspond to a NE.
This is illustrated in Figure 4.
Figure 4: The phase portrait of the corresponding Jacobi Iteration with
$N=10$, $\lambda=10$, $\rho=0.01$. Note this attracting fixed is not a NE.
It is also worth noting that this fixed point is not globally attracting.
There are initial conditions for which the system moves toward deadlock, which
$g=1.0$. These dynamics will only be realized if the players follow a gradient
descent strategy, rather than using their best response strategies.
### V-B Additional Greedy and Vigilante Players
An interesting property of this model is its behavior in the presence of
multiple greedy or vigilante players. In these cases, it may be impossible for
a vigilante player to know the number of greedy players. Consequently, she may
choose to assume there is always (exactly) one greedy player and use
Expression (6) to estimate $g$ for use in $\beta_{a}(g;\rho)$. In the case
when there is more than one greedy player, this will lead the vigilante to
overestimate the individual strategies of the greedy players, but this
assumption is consistent with what a vigilante could actually communicate.
Under this assumption, the vigilante uses the formula:
$\hat{\Theta}(\hat{g},a):=\hat{g}(1-a)\left(1-\frac{1}{N}\right)^{N-2}.$ (19)
Then the vigilante will attempt to minimize:
$U_{a}(\hat{g},a;\rho)=\left(\hat{\Theta}(\hat{g},a)-\Phi_{0}\right)^{2}+\rho\left(a-\frac{1}{N}\right)^{2}.$
(20)
Meanwhile, for $M$ greedy players we have:
$\Theta_{i}(g_{i},g_{-i},a):=g_{i}(1-a)\left(1-\frac{1}{N}\right)^{N-M-1}\prod_{k\neq
i}(1-g_{k}).$ (21)
The functions $U_{g_{i}}(g_{i},a;\lambda_{i})$ are defined analogously. Notice
that greedy player $i$ does not need to know about the existence of greedy
player $j$ for these objective functions to make sense.
In the case when there are additional vigilante players, then we modify
Expression (20) slightly to:
$U_{a_{i}}(\hat{g},a_{i};\rho_{i})=\left(\hat{\Theta}(\hat{g},a_{i})-\Phi_{0}\right)^{2}+\rho_{i}\left(a_{i}-\frac{1}{N}\right)^{2}$
Additional vigilante players will simply see vigilante activity as the result
of a greedy play. Some interesting behaviors occur in both the case when there
are additional greedy or vigilante players. In the case when
$\lambda_{1}=\lambda_{2}=10$ and $\rho=0.01$, we obtain convergence to a NE,
unlike when there was only a single greedy player with $\lambda=10$ and
$\rho=0.01$. This is illustrated in Figure 5.
Figure 5: Convergence in the case when $N=10$, $\lambda_{1}=\lambda_{2}=10$
and $\rho=0.01$ and there are two greedy players.
In this case, the two greedy player converge to the same value at equilibrium.
There are still parameters (as before) for which the system does not converge,
but it is interesting to note that the introduction of additional greedy
players causes convergence for parameters that were non-convergent in the
single greedy-player case.
Finally, we consider the case with two vigilante players and one greedy
player. As one would expect, the two vigilante players overestimate the greedy
player’s move and the system converges to a near deadlock state, with the two
vigilante players unable to recover from the fact that they don’t know about
each other [25]. This is illustrated in Figure 6.
Figure 6: Convergence in the case when $N=10$, $\lambda=10$ and
$\rho_{1}=\rho_{2}=0.001$ and there are two vigilante players.
On the other hand, if the vigilantes adjust their $\rho_{i}$ $(i=1,2)$ upward
to be more sensitive to their play, then the system does not converge, but
oscillates as in the case with one greedy player and one vigilante player. In
this case, however, the oscillation is about access rates $g$ that are almost
fair. This is illustrated in Figure 7.
Figure 7: Convergence in the case when $N=10$, $\lambda=10$ and
$\rho_{1}=\rho_{2}=0.1$ and there are two vigilante players.
## VI Conclusions and Future Directions
In this paper, we formulated a multiplayer distributed resource access game in
which some players have a greedy objective function and other players behave
as vigilantes modifying their access probabilities to punish perceived
greediness. Greedy players will back-off from a pure greedy strategy if the
greedy strategy leads to poor payoff. We showed that the best response
function for the greedy player under our formulation has a jump discontinuity,
which leads to conditions under which there is no Nash equilibrium in the
game. To understand this property, we formulated an exact representation for
the greedy player’s best response function in the case when there was one
greedy player and one vigilante player. We used this formulation to show
conditions under which a Nash equilibrium exists. We also illustrated that in
the case when there is no Nash Equilibrium, then the discrete dynamic system
generated from fictitious play does not converge, but oscillates indefinitely
as a result of the jump discontinuity. Finally, we discussed the cases when
there was more than one greedy player and more than one vigilante.
In the future, we will investigate theoretical results on this model when
there are a (small) number of vigilante and greedy players. It is clear from
Figure 2 that the oscillations caused by the jump discontinuity have a
somewhat complex periodic behavior. It would be interesting to understand how
this periodicity is related to $\epsilon_{g}$ and $\epsilon_{a}$. In addition
to this, we will study and compare in detail the discrete dynamical system
arising from fictitious play to the continuous dynamics that arise from
better-response dynamics (gradient descent or Jacobi iteration). Finally,
there is a unique control theoretic problem embedded in this model. In the
case where there were multiple vigilante’s, we saw that it was easy for the
vigilante’s to overreact to each other. However, by modifying their respective
$\rho_{i}$, the system was brought to a better point of (dynamic) stability
(see Figures 6 and 7). Dynamically controlling $\rho_{i}$ to improve system
performance in the case of multiple greedy and vigilante players is of
interest.
## Acknowledgments
Portions of Dr. Kesidis’ work were supported by the National Science
Foundation.
## References
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|
arxiv-papers
| 2013-06-13T14:50:42 |
2024-09-04T02:49:46.470471
|
{
"license": "Public Domain",
"authors": "Christopher Griffin and George Kesidis",
"submitter": "Christopher Griffin",
"url": "https://arxiv.org/abs/1306.3127"
}
|
1306.3160
|
# Modeling and Control of Rare Segments in BitTorrent with Epidemic
Dynamics††thanks: A shorter version of this paper that did not include the
$N$-segment lumped model was presented in May 2011 at IEEE ICC, Kyoto.
C. Griffin111C. Griffin is the with Applied Research Laboratory, Penn State
University, University Park, PA 16802, E-mail: [email protected] G.
Kesidis222G. Kesidis is the with Depts. of Electrical Engineering and Computer
Science and Engineering, Penn State University, University Park, PA 16802,
E-mail: [email protected] P. Antoniadis333P. Antoniadis is with the Communications
Systems Group, ETH Zurich, E-mail: [email protected] and S. Fdida444S.
Fdida are with the Computer Science Dept., Univ. Pierre & Marie Curie (LIP6),
[email protected]
###### Abstract
Despite its existing incentives for leecher cooperation, BitTorrent file
sharing fundamentally relies on the presence of seeder peers. Seeder peers
essentially operate outside the BitTorrent incentives, with two caveats: slow
downlinks lead to increased numbers of “temporary” seeders (who left their
console, but will terminate their seeder role when they return), and the
copyright liability boon that file segmentation offers for permanent seeders.
Using a simple epidemic model for a two-segment BitTorrent swarm, we focus on
the BitTorrent rule to disseminate the (locally) rarest segments first. With
our model, we show that the rarest-segment first rule minimizes transition
time to seeder (complete file acquisition) and equalizes the segment
populations in steady-state. We discuss how alternative dissemination rules
may beneficially increase file acquisition times causing leechers to remain in
the system longer (particularly as temporary seeders). The result is that
leechers are further enticed to cooperate. This eliminates the threat of
extinction of rare segments which is prevented by the needed presence of
permanent seeders. Our model allows us to study the corresponding trade-offs
between performance improvement, load on permanent seeders, and content
availability, which we leave for future work. Finally, interpreting the two-
segment model as one involving a rare segment and a “lumped” segment
representing the rest, we study a model that jointly considers control of rare
segments and different uplinks causing “choking,” where high-uplink peers will
not engage in certain transactions with low-uplink peers.
## 1 Introduction
There are several different incentives in the BitTorrent protocol: the
segmentation of the data object (file) into pieces555Alternatively called
chunks, segments or blocks in the BitTorrent literature and herein. to promote
swapping of pieces among peers in a swarm, the dissemination strategy of the
file pieces (rarest-first), the uplink reciprocity (choking) strategy when
swapping pieces, and the optimistic unchoking strategy. The configuration of
these rules can significantly affect performance under different scenarios and
assumptions (e.g., the size of a swarm and of individual neighborhoods of
interacting peers [1], the amount of asymmetry between, and distribution of,
uplink capacities, etc.).
There is a significant literature on modeling the properties of the existing
BitTorrent algorithm, e.g., [2, 3, 4, 5, 6], some with an aim to improve its
performance. Our model is different from those explored in [7, 8, 4] for
BitTorrent, and we compute different quantities of interest. In [9], the
authors propose a “fluid” model of a single torrent/swarm (as we do in the
following) and fit it to (transient) data drawn from aggregate swarms. In
[10], they consider a similar model with normalized terms the effect of which
is a nonlinear time-dilation of the transient dynamics. The connection to
branching process models [8, 11] is simply that ours only tracks the number of
active peers who possess or demand the file under consideration, i.e., a
single swarm. In [12] a strategy called BitMax is proposed that can fully use
upload capacities of “resourceful” peers and thus improve performance without
the reciprocity strategy implemented today in BitTorrent. So, there is a clear
tension between maximizing global performance and fairness in BitTorrent [13].
This means that if certain peers are required to share more resources than
they will need to consume themselves, they might choose not to join the system
or try to prematurely defect. Studies of incentives primarily focus on
reciprocity mechanisms in terms of upload bandwidth, see e.g., [14, 15] with
the objective of fairness.
Although this is a theoretically interesting question, in practice there are
many users that are typically understood as not behaving rationally on
BitTorrent, including the significant number of seeders both for popular and
unpopular content [16]. The presence of “permanent” seeders is enabled by the
typically flat-rate pricing (without quotas) for residential Internet access
[17], and the file segmentation itself provides some limitation of liability
for illegal dissemination of copyrighted material666Swarm discovery via third
parties, e.g., search and downloading torrent files from certain web sites,
offers additional limitation of liability for permanent seeders.. Also,
segment extinction is precluded by the presence of permanent seeders.
It is also well understood that peers spending additional time on-line will
improve overall content availability, since while participating in a swarm
downloading content, peers do disseminate file pieces belonging to other
swarms up to and including the point at which they acquire the entire file and
become seeders.
With the presence of permanent seeders, the extinction of rare pieces is not a
threat. The presence of temporary seeders is desirably increased by extended
download times, as the leecher peers may leave their console while waiting and
become seeders while they are absent [16].
A delaying strategy may be also implemented by seeders to limit their upload
capacity in an ad hoc fashion; a simulation [18] studied a seeder strategy to
reduce its upload throughput particularly for non-popular items. A general
public good model was proposed in [19] focusing on content availability in
which the main contribution of peers is their time on-line instead of upload
capacity.
A main objective of this paper is to study strategies of file-segment
dissemination, based on segment rarity, to explore the trade-off between
improving download performance and enticing cooperative activity through
longer downloading times for leecher peers. That is, deviations from rarest
first segment distribution will have the beneficial effect of additionally
delaying the leechers, all or just some of them, with segment extinction
precluded by the presence of permanent seeders. As such, we are interested in
the transient behavior of a given swarm, rather than a generic transaction
process among a fixed group of peers. The model we use will reflect this
emphasis.
A shorter version of this paper was presented in May 2011 at IEEE ICC, Kyoto.
In that paper, we studied only a two-segment model. In this paper, we extend
the results of the original paper by considering an $N$ segment model with an
intentionally rare final segment as well as two peer classes – one with a high
throughput rate and the other with a low throughput rate. We derive the
differential equation model for this case and show how this variation affects
sojourn time from leecher to seeder in both classes.
The remainder of this paper is organized as follows. In Section 2 and 3, we
describe a two-piece deterministic epidemic model of a swarm, similar to one
previously studied in [20] as a special case of a deterministic limit of a
stochastic sequential transactional model; but here we consider a control
parameter governing which piece is disseminated given that there is choice. In
Section 4, we argue that the “bang-bang” globally rarest first is optimal in
terms of overall download time. In Section 5, we describe the equilibria under
continuous globally (and locally) rarest first control. In Section 6, we
discuss how the rarity of file segments can be deliberately controlled by the
seeders. In Section 7, we interpret a two-segment model as one modeling a rare
segment and a lumping of the rest to study rare-segment control under choking
due to differences in uplink bandwidths of the peers. Finally, we conclude in
Section 8 with a summary.
## 2 Epidemic model
Let $\lambda$ be the total peer arrival rate to a specific swarm, where newly
arrived peers possess no part of the data object $F$ being disseminated in the
swarm. The quantity $\delta$ is the death rate for seeders–individuals that
possess the entire file and seed the population with its segments. Let $\beta$
be the download rate parameter of client-server transactions, which depend on
the size of the file being transmitted and the associated willingness of the
server peer to participate in the transaction (for nothing in return from the
client peer).
### 2.1 One Segment Model
In the absence of BitTorrent incentives, we have the single-segment case
$\dot{x}_{l}=-\beta_{0}x_{l}x_{s}+\lambda_{l}\quad\dot{x}_{s}=\beta_{0}x_{l}x_{s}-\delta
x_{s}+\lambda_{s},$
where $x_{l}$ are the leechers (and do not possess any parts of the file),
$x_{s}$ are the seeders (who possess the complete file), and $\lambda$ their
exogenous arrival rates to the swarm. The successful transaction rate is
proportional to the contact rate between a member of the seeder and leecher
populations, which we assume to be proportional to the product of their sizes
[21]. We chose this model for concreteness; other types of models, some
similar to the above, have also been extensively studied, e.g., urn,
replicator, Volterra-Lotka, and coupon-collector [22].
Therefore there are two types of peers in the seeder state: those that arrive
as seeders and tend to remain longer and those that arrive formerly as
leechers and tend to remain briefly. Rather than using a fixed population of
“permanent” seeders for the former category, we model them by a small external
arrival process $\lambda_{s}$, giving an average population of
$\lambda_{s}/\delta_{s}$ by Little’s formula [23], with
$\frac{1}{\delta_{s}}\gg\frac{1}{\delta_{l}}$. The mean lifetime $1/\delta$ of
a typical seeder is therefore the weighted average:
$\frac{1}{\delta}=\frac{\lambda_{l}/\delta_{l}}{\lambda_{l}/\delta_{l}+\lambda_{s}/\delta_{s}}\cdot\frac{1}{\delta_{l}}+\frac{\lambda_{s}/\delta_{s}}{\lambda_{l}/\delta_{l}+\lambda_{s}/\delta_{s}}\cdot\frac{1}{\delta_{s}}$
(1)
The globally attracting stable equilibrium is given by
$\mathbf{x}^{*}=(x_{l}^{*},x_{s}^{*})=\left(\frac{\delta\lambda_{l}}{\beta(\lambda_{s}+\lambda_{l})},~{}\frac{\lambda_{l}+\lambda_{s}}{\delta}\right),$
### 2.2 Two-Segment Model
Consider splitting file $F$ into two segments, $a$ and $b$. In this case, the
model for this system becomes:
$\left\\{\begin{aligned} \dot{x}_{l}&=\lambda_{l}-\beta
x_{l}(x_{a}+x_{b}+x_{s})\\\ \dot{x}_{a}&=-x_{a}(\beta x_{s}+\gamma
x_{b})+\beta x_{l}(x_{a}+u(x_{a},x_{b})x_{s})\\\ \dot{x}_{b}&=-x_{b}(\beta
x_{s}+\gamma x_{a})+\beta x_{l}(x_{b}+[1-u(x_{a},x_{b})]x_{s})\\\
\dot{x}_{s}&=\lambda_{s}+\beta(x_{a}+x_{b})x_{s}+2\gamma x_{a}x_{b}-\delta
x_{s}\end{aligned}\right.$ (2)
Here, $\gamma$ represents is rate parameter for swap/trade transactions
($\beta$ and $\gamma$ may be decreasing functions $N$). The “control” function
$u\in[0,1]$ represents how the seeder may distribute the segments based on its
knowledge of their relative prevalence; whereas in BitTorrent, the rarity of
the segment is locally determined among peers that are directly transacting,
i.e., BitTorrent uses locally rarest first policy to determine which segments
to disseminate. In [20], $u\equiv 1/2$ was assumed.
We assume that $\gamma\geq\beta>\beta_{0}$, where the former inequality is
owing to stronger “server” incentives in a swap transaction compared to a
client server transaction. Also, the latter inequality is owing to $a$ and $b$
being smaller than $F$ and peers would generally be more reluctant to transmit
the entire file $F$ in one shot out of concerns of liability for copyright
violation. In [20], for $u\equiv 1/2$ (constant control), we showed
convergence of a scaled stochastic discrete transactional process to the above
epidemic dynamics and compared pure client-server to the two-segment system in
terms of time to transition from leecher to seeder.
It is worth noting that BitTorrent does not function precisely in this way.
Permanent BitTorrent seeders do not truly have control over the pieces they
chose to transmit to members of the swarm, as swarms use pull (as opposed to
push) request frames. This model assumes that seeders (through some mechanism)
will have control over the fragments they push to seeders through the
$u(x_{a},x_{b})$ parameter.
## 3 Two-Segment Selection Control
For simplicity in the following, we focus on the two-segment swarm (2). When
$u\equiv 1/2$, then half the successful transactions between $x_{l}$ and
$x_{s}$ result in an arrival to the $x_{a}$ population (that possess only the
first segment of $F$), and the other half an arrival to the $x_{b}$ population
(in both cases, a departure from the $x_{a}$ population, of course). When
$u\equiv 1/2$, there is always at least one equilibrium solution given by
[20]:
$\displaystyle x_{l}=$
$\displaystyle\frac{\lambda_{l}}{\beta}\left(\sigma_{0}+\frac{\lambda_{l}+\lambda_{s}}{\delta}\right)^{-1}$
$\displaystyle\quad x_{a}=$ $\displaystyle\frac{\sigma_{0}}{2},$ (3)
$\displaystyle x_{b}=$ $\displaystyle\frac{\sigma_{0}}{2},$
$\displaystyle\quad x_{s}=$
$\displaystyle\frac{\lambda_{l}+\lambda_{s}}{\delta}$
where $\sigma_{0}$ is the unique positive root of the quadratic equation:
$\sigma_{0}^{2}+2\kappa_{0}\sigma_{0}-{2\lambda_{l}}/{\gamma}$ and
$\kappa_{0}:=\frac{\beta(\lambda_{l}+\lambda_{s})}{\gamma\delta}$.
The case of a constant control (allowing for a constant $u\in[0,1]$) is
complex and leads to the analysis of a quartic equation without providing much
insight into the system. We consider this case in Section 7 through a
numerical study. In general, the control will vary as a function of state
$(x_{l},x_{a},x_{b},x_{s})$. The continuous globally rarest first control is
simply
$u(x_{a},x_{b})=\frac{x_{b}}{x_{a}+x_{b}}$ (4)
The presumption here is that the seeders have an estimate of the ratio of
population sizes $x_{a}/x_{b}$. A non-continuous, “bang-bang” version of this
rule, requiring less information for the seeders, is
$u(x_{a},x_{b})=\begin{cases}1&\text{if $x_{a}<x_{b}$}\\\ \frac{1}{2}&\text{if
$x_{a}=x_{b}$}\\\ 0&\text{if $x_{a}>x_{b}$}\end{cases}$ (5)
Again, BitTorrent used a locally rarest first control consistent with (5).
Note that both of these controls admit the equilibrium for $u\equiv 1/2$ of
the previous section.
## 4 Minimizing Traversal Time
Consider the Mayer optimal control problem, with control $u(\mathbf{x},t)$:
$\left\\{\begin{aligned}
\min_{u}\;\;&x_{l}(T)+x_{a}(T)+x_{b}(T)~{}\text{subject to:}\\\ &\text{the
system model (\ref{eqn:SystemModel}),}\\\ &\mathbf{x}(0)=\mathbf{x}^{0},\\\
&u\in[0,1],\,\mathbf{x}(t)\geq 0\;\;\forall t\in[0,T].\end{aligned}\right.$
(6)
Here we assume that $T$ is a finite ending time that may be arbitrarily large.
Naturally we assume that $\mathbf{x},u\in\mathcal{L}^{2}[0,t]$, the space of
bounded square integrable functions and $T$ is some arbitrary large but finite
end of time.
The objective is motivated by Little’s formula [23] which states that, for a
stationary regime, the sojourn time from arrival to a swarm as leecher to the
transition to seeder is
$\frac{x_{l}^{*}+x^{*}_{a}+x^{*}_{b}}{\lambda_{l}}.$ (7)
Our assertion that $T$ is finite comes from the qualitative analysis of the
differential equations given in System (2). We argue that Expression (5) is
the control that minimizes the objective subject to these epidemic dynamics,
beginning from an arbitrary initial point $\mathbf{x}(0)$. Expression (5) is
the fully discrete form of Expression (4).
There is always at least one attracting equilibrium point for the epidemic
dynamics (2) when $u\equiv 1$ (this is the value of Expression (5) when
$x_{a}<x_{b}$). This equilibrium occurs at:
$x^{*}_{a}=\frac{\lambda\delta}{(\lambda_{s}+\lambda)\beta},\quad
x^{*}_{b}=0.$
If we assume that $u$ is defined by Expression (5) and that
$x_{a}(0)<x_{b}(0)$, we note that before the foregoing equilibrium is reached,
we obtain $x_{a}(t)=x_{b}(t)$ for some $t$, and we return to the dynamics in
the case when $u\equiv 1/2$ (see Figure 1). In this case, we will maintain
$x_{a}(t)=x_{b}(t)$ and move to the equilibrium point already identified in
Expression (3). A similar argument holds when $x_{b}(0)<x_{a}(0)$ in which
case $u\equiv 0$. Again we will return to the dynamics when $u\equiv 1/2$
before $x_{a}$ reaches $0$, which is the equilibrium in this case.
Figure 1: Phase plot in $(x_{a},x_{b})$ space showing the bang-bang controller
pushing $x_{a}$ to equal $x_{b}$ and then proceeding to a globally attracting
stationary point. In this example, $\beta=2$, $\gamma=3$, $\lambda_{s}=1$,
$\lambda_{l}=4$ and $\delta=2$.
Observe that the Hamiltonian of the control problem (6) will be linear in $u$.
Thus, the bang-bang controller of Expression (5) is optimal [24]. The bang-
bang controller will switch its state depending on the adjoint dynamics of the
system and may be singular for certain adjoint dynamics. The only reasonable
singular control in this case is $u\equiv 1/2$ (used whenever $x_{a}=x_{b}$),
while a reasonable proxy for the adjoint conditions is given in Expression
(5).
We illustrate the optimality of the discontinuous globally rarest first
controller through a numerical example. Figure 2 at left shows the optimal
controller, which was computed by discretization, pushing $x_{a}=x_{b}$ and
then maintaining this state.
(a) Bang-Bang Control (b) Continuous Control
Figure 2: Left: The optimal controller is shown on the bottom, while
$x_{a}(t)$ and $x_{b}(t)$ are shown above. Note the optimal controller driving
$x_{a}=x_{b}$. This controller was computed using discretization method.
Right: The continuous globally rarest first function drives $x_{a}$ to to
$x_{b}$ however, the convergence rate is slower than for the true optimal
control function.
We can contrast this to the continuous approximation of the globally rarest
first controller given in Expression (4). In Figure 2 at right we illustrate
the effect the continuous globally rarest first controller has on the values
of $x_{a}$ and $x_{b}$. Note that the two converge much more slowly than in
the optimal case.
Note that since Problem (6) is of the Mayer type, the specific form of the
controller will matter most when $T$ is small, i.e., when $T$ is much less
than the time required to reach an attracting equilibrium. This is precisely
the behavior we see in Figures 2. The optimal (bang-bang) controller attempts
to drive the system to its equilibrium point as quickly as possible since it
is here that steady-state component of Problem (6), given by (7), is
minimized. It does so by pushing the system to a spot on the diagonal (where
$x_{a}=x_{b}$); since thereafter $u\equiv 1/2$, the systems with constant
control ($u\equiv 1/2$ always), discrete globally rarest first control, or
continuous globally rarest first control, will share equilibrium points (this
is illustrated in the next section). For large values of $T$, the dominant
component of the objective will be the the equilibrium point to which the
system tends, not the transient component which may be sensitive to the choice
among such controllers.
## 5 Equilibria under Continuous Globally Rarest First Control
More interesting equilibria are possible under the continuous form of globally
rarest first controller (4). Equilibrium analysis focusing on the values of
$x_{a}^{*}$ and $x_{b}^{*}$ (by first solving for and substituting out
$x_{s}^{*}$ and $x_{l}^{*}$) yields:
$x_{l}^{*}=\lambda_{l}{\beta}^{-1}\left(x^{*}_{{a}}+x^{*}_{{b}}-{\frac{\lambda_{s}+2\,\gamma\,x^{*}_{{a}}x^{*}_{{b}}}{\beta\,x^{*}_{{a}}+\beta\,x^{*}_{{b}}-\delta}}\right)^{-1}.$
(8)
$x_{s}=-{\frac{\lambda_{s}+2\,\gamma\,x_{{a}}x_{{b}}}{\beta\,x_{{a}}+\beta\,x_{{b}}-\delta}}$
(9)
In the following, suppress the superscript “*” for notational simplicity.
We can solve the simpler simultaneous nonlinear equations
$\dot{x}_{a}+\dot{x}_{b}=0$ and $\dot{x}_{a}-\dot{x}_{b}=0$ to obtain
equilibrium solutions for $x_{a}$ and $x_{b}$ respectively. From the former,
we get:
$x_{a}=-{\frac{x_{{b}}\lambda_{l}\,\beta+\beta\,\lambda_{s}\,x_{{b}}-\lambda_{l}\,\delta}{2\,\gamma\,\delta\,x_{{b}}+\lambda_{l}\,\beta+\beta\,\lambda_{s}}}$
(10)
We can then substitute this expression into $\dot{x}_{a}-\dot{x}_{b}=0$ to
obtain a ratio of polynomials in $x_{b}$ whose numerator is:
$-2\left(2\gamma\delta
x_{b}^{2}+2\beta(\lambda_{l}+\lambda_{s})x_{b}-\lambda_{l}\delta\right)\left[a_{0}+a_{1}x_{b}+a_{2}x_{b}^{2}+a_{3}x_{b}^{3}+a_{4}x_{b}^{4}\right]$
(11)
where:
$\left\\{\begin{aligned}
&a_{0}={\lambda_{l}}^{4}{\beta}^{2}+{\lambda_{s}}^{3}{\beta}^{2}\lambda_{l}+3\,\lambda_{s}\,{\beta}^{2}{\lambda_{l}}^{3}+3\,{\lambda_{s}}^{2}{\beta}^{2}{\lambda_{l}}^{2}\\\
&a_{1}=3\,{\lambda_{s}}^{2}\lambda_{l}\,\delta\,\gamma\,\beta-{\lambda_{l}}^{2}\gamma\,{\delta}^{3}+6\,\lambda_{s}\,{\lambda_{l}}^{2}\delta\,\gamma\,\beta+3\,\delta\,\gamma\,\beta\,{\lambda_{l}}^{3}\\\
&a_{2}=3\,{\beta}^{2}\lambda_{s}\,{\lambda_{l}}^{2}\gamma+3\,{\beta}^{2}{\lambda_{s}}^{2}\lambda_{l}\,\gamma+\gamma\,{\delta}^{2}\lambda_{l}\,\beta\,\lambda_{s}+2\,{\gamma}^{2}\lambda_{l}\,{\delta}^{2}\lambda_{s}\\\
&\hskip
30.00005pt+2\,{\gamma}^{2}{\lambda_{l}}^{2}{\delta}^{2}+\gamma\,\beta\,{\delta}^{2}{\lambda_{l}}^{2}+{\beta}^{2}\gamma\,{\lambda_{s}}^{3}+{\beta}^{2}\gamma\,{\lambda_{l}}^{3}\\\
&a_{3}=4\,\lambda_{s}\,\lambda_{l}\,\delta\,{\gamma}^{2}\beta+2\,{\lambda_{s}}^{2}\delta\,{\gamma}^{2}\beta+2\,{\lambda_{l}}^{2}\delta\,{\gamma}^{2}\beta-2\,{\gamma}^{2}{\delta}^{3}\lambda_{l}\\\
&a_{4}=2\,{\gamma}^{2}{\delta}^{2}\lambda_{l}\,\beta+2\,{\gamma}^{2}{\delta}^{2}\beta\,\lambda_{s}\end{aligned}\right.$
The roots of $\dot{x}_{a}-\dot{x}_{b}$ are governed by the roots of the
quadratic polynomial:
$2\gamma\delta
x_{b}^{2}+2\beta(\lambda_{l}+\lambda_{s})x_{b}-\lambda_{l}\delta$ (12)
and a quartic polynomial:
$a_{0}+a_{1}x_{b}+a_{2}x_{b}^{2}+a_{3}x_{b}^{3}+a_{4}x_{b}^{4}$ (13)
### 5.1 Quadratic Equation Case
###### Proposition 5.1.
A unique strictly positive, real solution with $x_{a}=x_{b}$ exists for (10)
and (12).
###### Proof.
The roots of (12) are given by:
$\frac{-1}{4\gamma\delta}\left(2\beta(\lambda+\rho)\pm
2\sqrt{\beta^{2}(\lambda+\rho)^{2}+2\lambda\gamma\delta^{2}}\right)$ (14)
The fact that all parameters are positive yields the expression:
$2\beta(\lambda+\rho)=2\lvert\sqrt{\beta^{2}(\lambda+\rho)^{2}}\rvert<2\lvert\sqrt{\beta^{2}(\lambda+\rho)^{2}+2\lambda\gamma\delta^{2}}\rvert$
(15)
Thus,
$2\beta(\lambda+\rho)-2\sqrt{\beta^{2}(\lambda+\rho)^{2}+2\lambda\gamma\delta^{2}}<0$
(16)
while
$2\beta(\lambda+\rho)+2\sqrt{\beta^{2}(\lambda+\rho)^{2}+2\lambda\gamma\delta^{2}}>0$
(17)
The factor of $-1/4\gamma\delta$ leads us to conclude that there is one
positive root and one negative root always for (12). Thus:
$x_{b}^{*}=\frac{-1}{4\gamma\delta}\left(2\beta(\lambda+\rho)-2\sqrt{\beta^{2}(\lambda+\rho)^{2}+2\lambda\gamma\delta^{2}}\right)$
(18)
is a non-extraneous equilibrium solution. Substituting this value in the
expression for $x_{a}$ and simplifying algebraically yields:
$x_{a}^{*}=\frac{-1}{4\gamma\delta}\left(2\beta(\lambda+\rho)-2\sqrt{\beta^{2}(\lambda+\rho)^{2}+2\lambda\gamma\delta^{2}}\right)$
(19)
as well. That is, the two roots are equal and positive. This completes the
proof. ∎
### 5.2 Quartic Equation Case
The roots of the quadratic equation (12) just considered are not necessarily
those of the quartic equation (13). We make use of the following known result
(see, e.g., Theorem 1 of [25]):
###### Theorem 5.2.
For the quartic equation,
$f(x)=c_{0}+4c_{1}x+6c_{2}x^{2}+4c_{3}x^{3}+c_{4}x^{4}$, define the following
terms: $G=c_{4}^{2}c_{1}-3c_{4}c_{3}c_{2}+2c_{3}^{3}$,
$H=c_{4}c_{2}-c_{3}^{2}$, $I=c_{4}c_{0}-4c_{3}c_{1}+3c_{2}^{2}$,
$J=\left\lvert\begin{array}[]{ccc}c_{4}&c_{3}&c_{2}\\\ c_{3}&c_{2}&c_{1}\\\
c_{2}&c_{3}&c_{4}\end{array}\right\rvert,$
and the discriminant $\Delta=I^{3}-27J^{2}$. Then $f(x)=0$ has no real roots
if and only if:
1. 1.
$\Delta=0$, $G=0$, $12H^{2}-c_{4}I=0$ and $H>0$; or
2. 2.
$\Delta>0$ and (a) $H\geq 0$; or (b) $H<0$ and $12H^{2}-c_{4}^{2}I<0$
For (13) we can evaluate the discriminant and determine conditions on
$\lambda_{s}$ when $\Delta<0$, which will create a real-root for (13). The
sign of the discriminant is governed by a quadratic expression on
$\lambda_{s}$. We identify terms:
$\lambda_{0,1}=\frac{(\eta\xi+1)^{2}}{4\gamma\,{\xi}^{3}\eta}\left[10\,\beta\,\gamma+{\gamma}^{2}~{}\pm\sqrt{68\,{\beta}^{2}{\gamma}^{2}+20\,\beta\,{\gamma}^{3}+{\gamma}^{4}+64\,\gamma\,{\beta}^{3}}\right]$
(20)
Given the fact that the parameters are always positive, the square root is
always real and thus for any set of parameters, we can identify a condition
under which $\Delta=0$. Let $\lambda_{0}$ and $\lambda_{1}$ denote the two
roots defined above. Evaluating a point directly between the two roots yields:
$\lambda_{s}^{+}={\frac{\left(\eta\,\xi+1\right)^{2}\left(\gamma+10\,\beta\right)}{4{\xi}^{3}\eta}}$
(21)
Evaluating $\Delta$ at $\lambda_{s}^{+}$ yields the simple expression:
$\Delta(\lambda_{s}^{+})=\frac{1}{8}\left(\eta\,\xi+1\right)^{4}\left(\gamma+16\,\beta\right)\left(\gamma+2\,\beta\right)^{2}$
(22)
Since all parameters are positive, we can see that when
$\lambda_{s}\in(\lambda_{0},\lambda_{1})$, then $\Delta>0$. If we choose
$\lambda_{s}$ outside of this range and fix $\beta$, $\gamma$, $\eta$ and
$\xi$ we can identify the example off-diagonal equilibrium solutions
illustrated in the main text.
For the case:
$\lambda_{l}\geq\delta\geq\lambda_{s},$
let $\eta=\lambda_{l}/\delta$ and $\xi=\delta/\lambda_{s}$ with $\eta,\xi\geq
1$. The quartic discriminant $\Delta$ will depend on a quadratic form in
$\lambda_{s}$ with roots at $\lambda_{0}$ and $\lambda_{1}$ and
$\lambda_{0}<\lambda_{1}$. Between these values, the quartic equation has no
real roots. Outside these values, the quartic has at least two real roots
which correspond to equilibrium points for $x_{a}$ and $x_{b}$ that are off-
diagonal (i.e., $x_{a}\neq x_{b}$).
### 5.3 Off-Diagonal Equilibria
Real solutions to the quartic equation are interesting because they lead to
off-diagonal equilibrium solutions in cases where the values of the parameters
are widely skewed. For the case $\beta=2$, $\gamma=3$, $\eta=1.1$, $\xi=1.1$
we obtain:
$\lambda_{0}=-0.759\text{ and }\lambda_{1}=39.121.$ (23)
Choosing $\lambda_{s}=40>\lambda_{1}$, so that $\delta=44$ and
$\lambda_{l}=48.4$. The resulting (real) off-diagonal equilibrium points
$(x_{a}^{*},x_{b}^{*})$ derived with these parameters in the quartic are:
$(5.237,~{}0.772)\text{ and }(0.772,~{}5.237).$
This example illustrates the existence of off-diagonal equilibrium points. A
field plot of this case is shown in Figure 3 (Left). The blue lines are the
trajectories of the dynamical system with representative starting points. Off-
diagonal equilibrium points are shown as black diamonds.
It is interesting to note that the off-diagonal equilibrium points partition
the phase plane into a central region of stability flanked by two regions of
instability [26]. Thus, when parameter values are highly skewed, the on-
diagonal equilibrium point is not a global attractor as it appears to be when
it is the unique non-extraneous equilibrium (see Figure 3 on Left). The
trajectories illustrate a component of this region of attraction. It is clear
that these equilibrium points should not occur in the bang-bang control case –
evaluating Little’s formula for the on-diagonal equilibrium point in this case
shows that the on-diagonal equilibrium correctly minimizes the objective
function of the control problem (6). This is discussed in the next section.
The results shown on the equilibrium points of the continuous locally rarest
first control are summarized in the following theorem.
###### Theorem 5.3.
For the dynamics given in Expression (2) under (4), there is alway at least
one point of equilibrium occurring at:
$\displaystyle x_{s}^{*}=\frac{\lambda_{l}+\lambda_{s}}{\delta}$
$\displaystyle
x_{l}^{*}=\frac{\lambda_{l}\gamma\delta}{\beta\left((\lambda_{l}+\lambda_{s})(\gamma-\beta)+\sqrt{\beta^{2}(\lambda_{l}+\lambda_{s})^{2}+2\lambda_{l}\gamma\delta^{2}}\right)}$
$\displaystyle
x_{a}^{*}=x_{b}^{*}=\frac{-1}{4\gamma\delta}\left(2\beta(\lambda_{l}+\lambda_{s})-2\sqrt{\beta^{2}(\lambda_{l}+\lambda_{s})^{2}+2\lambda_{l}\gamma\delta^{2}}\right)$
Furthermore, if $\lambda_{s}\not\in[\lambda_{0},\lambda_{1}]$ and
$\lambda_{s}>0$, with $\lambda_{0}$ and $\lambda_{1}$ defined in Expression
(20), then the system may admit at least one other equilibrium point; in this
case, $x_{l}^{*}$ and $x_{s}^{*}$ remain as defined, but $x_{a}^{*}$ and
$x_{b}^{*}$ may take on non-equal values.
We illustrate the on diagonal equilibrium that always exists in Figure 3 (Left
and Right) for the case when $\beta=2$, $\gamma=3$, $\lambda_{s}=1$,
$\lambda_{l}=4$ and $\delta=2$. The black line shows a representative sample
path for this dynamical system.
(a) Off-Diagonal Equilibrium (b) On Diagonal Equilibrium
Figure 3: (Left) Phase plot in $(x_{a},x_{b})=(x,y)$ space showing regions of
stability and instability with off-diagonal equilibrium points. (Right) Phase
plot in $(x_{a},x_{b})=(x,y)$ space showing the stability of the on-diagonal
equilibrium point. The black line shows a representative sample path for this
dynamical system.
## 6 Discussion: Controlling segment rarity
Controlled rarity is the action of keeping a collectible object from a set
intentionally rare. Controlled rarity is used to sell packs of trading cards,
i.e., collectors buy additional packs seeking the rare card to complete their
set. In BitTorrent, the rarity of certain segments can be deliberately
controlled to increase swarm sojourn time and encourage additional cooperative
(uploading) behavior by leechers. This can serve to stabilize the swarm and
prevent collapse.
As a simple example in the two-segment case, the seeders can use
$u(x_{a},x_{b})=x_{a}/(x_{a}+kx_{b})$ for some constant $k$. Clearly, as $k>1$
increases, the seeders will increasingly tend to disseminate segment $b$ even
when segment $a$ is rarer in the swarm. Based on the results of the previous
section, the sojourn time from leecher to seeder is larger under any such rule
with compared to the globally rarest first rule (5). At an extreme, seeders
could substitute $1-u$ for $u$ given in (5) to extend leecher sojourn times.
Heretofore, we have described three control policies, (4), (5), and $u\equiv
1/2$. We note that these three control policies share equilibrium points with
$x_{a}^{*}=x_{b}^{*}$. But for (4), we computed off-diagonal equilibria with
$x_{a}^{*}+x_{b}^{*}=5.237+0.772=6.009$. However, for this example, the on-
diagonal equilibria are $x_{a}^{*}=x_{b}^{*}=2.248$, i.e.,
$x_{a}^{*}+x_{b}^{*}=4.496$ which is less than that of the off-diagonal
equilibria. So, in this way, we see that, even in a stationary regime, control
(4) may still lead to longer leecher sojourn times (through Little’s Formula)
than the optimal bang-bang control (5).
Our model allows us to devise beneficial delaying strategies in terms of
reduced load for the permanent seeders, overall content availability, and
performance–at least for a percentage of the leechers, since clearly those
that will be delayed will not have any gain for the specific swarm. We leave
this for future work.
## 7 Jointly modeling uplink-based choking and rare segments with a lumped
non-rare segment model
In this section, we reinterpret the two-segment model as representing a
general BitTorrent swarm with one segment intentionally rare. That is, all
other (non-rare) segments are lumped together by assuming that their
collective acquisition occurs at the same time-scale as that of the single
rare segment. Thus, the four types of players based on the segments they
possess are: leechers, seeders, players with all but the rarest segment (i.e.,
$N-1$ segments), and players with the rare segment.
Let $\beta_{k}$ ($k\in\\{r,N-1\\}$) be the probability of successful
transaction, again assuming contact/attempts are proportional to the
probability of the populations. The probability $\beta_{r}$ for transactions
involving the rare segment are such that $\beta_{r}\leq\beta_{N-1}$, where
$\beta_{N-1}$ is the lumped parameter corresponding to peers with all $N-1$
other (non-rare) segments.
As a result, we get the following epidemic dynamics as a variation of those
considered above assuming all uplinks are the same:
$\displaystyle\dot{x}_{l}=$
$\displaystyle\lambda_{l}-(\beta_{r}x_{r}+\beta_{N-1}x_{N-1}+[u\beta_{r}+(1-u)\beta_{N-1}]x_{s})x_{l}$
(24) $\displaystyle\dot{x}_{r}=$
$\displaystyle\beta_{r}(ux_{s}+x_{r})x_{l}-(\beta_{N-1}x_{s}+\beta_{r}x_{N-1})x_{r}$
(25) $\displaystyle\dot{x}_{N-1}=$
$\displaystyle\beta_{N-1}((1-u)x_{s}+x_{N-1})x_{l}-\beta_{r}(x_{s}+x_{r})x_{N-1}$
(26) $\displaystyle\dot{x}_{s}=$
$\displaystyle\lambda_{s}+\beta_{N-1}x_{s}x_{r}+\beta_{r}(2x_{r}+x_{s})x_{N-1}-\delta
x_{s}$ (27)
where we require $\lambda_{s},\lambda_{l}>0$ to prevent extinction. Also,
consider how the seeder can control the system by varying $u$ governing seeder
contact with leechers (whether a rare segment is shared). For a simple
example, take $\beta_{r}=1=\beta_{N-1}$, $\lambda_{l}=1$, $\lambda_{s}=0.01$
and $\delta=0.1$. In Figure 4, we plot as a function of $u$ the mean (i.e.,
equilibrium, $\dot{\underline{x}}=0$) delay from leecher to seeder, which is
by Little’s theorem [23]
$\displaystyle\frac{x_{l}+x_{r}+x_{N-1}}{\lambda_{l}}.$
The equilibrium point for these “symmetric” dynamics, involving the roots of
(12), can be computed as follows: For the symmetric case where
$\beta_{r}=1=\beta_{N-1}$ and $u=0.5$, the diagonal equilibrium solution is
$\displaystyle x_{l}^{*}=$
$\displaystyle\frac{x_{N-1}^{*}\lambda_{l}\delta}{\lambda_{l}\delta-
x_{N-1}^{*}(\lambda_{s}+\lambda_{l})},$ (28) $\displaystyle
x_{r}^{*},x_{N-1}^{*}=$
$\displaystyle\frac{-(\lambda_{s}+\lambda_{l})+\sqrt{(\lambda_{s}+\lambda_{l})^{2}+2\delta^{2}\lambda_{l}}}{2\delta},$
(29) $\displaystyle x_{s}^{*}=$
$\displaystyle\frac{\lambda_{s}+\lambda_{l}}{\delta},$ (30)
where $x_{N-1}^{*}=x_{r}^{*}$ is the positive root (12).
Other solutions (when $u\neq 1/2$) are computed as the stationary points of
the differential system.
Figure 4: Mean delay from leecher to seeder vs $u$
To consider both a high and low uplink bandwidth cases, we use superscripts
$({\sf hi})$ and $({\sf lo})$ on the parameters above, and let
$x_{k}:=x^{({\sf hi})}_{k}+x^{({\sf lo})}_{k}$ for $k\in\\{l,r,N-1,s\\}$. We
assume all peers know which segment is the rare one. We also assume uplink
bandwidth discrepancies only eliminate the swap terms $x_{r}^{({\sf
hi})}x_{N-1}^{({\sf lo})}$ through choking, i.e., a high-uplink peer will
trade with a low-uplink peer if that low-uplink peer is providing the rare
segment. Since all transactions are one-way when leechers are involved, we
have the dynamics:
$\displaystyle\dot{x}_{l}^{({\sf lo})}=\lambda_{l}^{({\sf
lo})}-[\beta_{r}x_{r}+\beta_{N-1}x_{N-1}+(u\beta_{r}+(1-u)\beta_{N-1})x_{s}]x_{l}^{({\sf
lo})}$ (31) $\displaystyle\dot{x}_{l}^{({\sf hi})}=\lambda_{l}^{({\sf
hi})}-[\beta_{r}x_{r}+\beta_{N-1}x_{N-1}+(u\beta_{r}+(1-u)\beta_{N-1})x_{s}]x_{l}^{({\sf
hi})},$ (32)
Leecher dyanmics without uplink considerations above are obtained by adding
these two equations with $\lambda_{k}:=\lambda^{({\sf hi})}_{k}+\lambda^{({\sf
lo})}_{k}$ for $k\in\\{l,s\\}$. The complete set of differential equations
describing system behavior are:
$\displaystyle\dot{x}_{r}^{({\sf lo})}=$
$\displaystyle\beta_{r}(ux_{s}+x_{r})x_{l}^{({\sf
lo})}-(\beta_{N-1}x_{s}+\beta_{r}x_{N-1})x_{r}^{({\sf lo})}$ (33)
$\displaystyle\dot{x}_{r}^{({\sf hi})}=$
$\displaystyle\beta_{r}(ux_{s}+x_{r})x_{l}^{({\sf
hi})}-(\beta_{N-1}x_{s}+\beta_{r}x_{N-1}^{({\sf hi})})x_{r}^{({\sf hi})}$ (34)
$\displaystyle\dot{x}_{N-1}^{({\sf lo})}=$
$\displaystyle\beta_{N-1}((1-u)x_{s}+x_{N-1})x_{l}^{({\sf
lo})}-\beta_{r}(x_{s}+x_{r}^{({\sf lo})})x_{N-1}^{({\sf lo})}$ (35)
$\displaystyle\dot{x}_{N-1}^{({\sf hi})}=$
$\displaystyle\beta_{N-1}((1-u)x_{s}+x_{N-1})x_{l}^{({\sf
hi})}-\beta_{r}(x_{s}+x_{r})x_{N-1}^{({\sf hi})}$ (36)
$\displaystyle\dot{x}_{s}^{({\sf lo})}=$ $\displaystyle\lambda_{s}^{({\sf
lo})}+(\beta_{N-1}x_{s}+\beta_{r}x_{N-1})x_{r}^{({\sf
lo})}+\beta_{r}(x_{s}+x_{r}^{({\sf lo})})x_{N-1}^{({\sf lo})}-\delta^{({\sf
lo})}x_{s}^{({\sf lo})}$ (37) $\displaystyle\dot{x}_{s}^{({\sf hi})}=$
$\displaystyle\lambda_{s}^{({\sf
hi})}+(\beta_{N-1}x_{s}+\beta_{r}x_{N-1}^{({\sf hi})})x_{r}^{({\sf
hi})}+\beta_{r}(x_{s}+x_{r})x_{N-1}^{({\sf hi})}-\delta^{({\sf
hi})}x_{s}^{({\sf hi})}$ (38)
Note how the final terms involving population products (i.e., Hamer terms
[21]) on the right-hand-side differ for the high and low uplink population
dynamics to account for choking.
For these coupled dynamics involving both high-uplink and low-uplink peers, we
considered a special case with the number of high-uplink peers much smaller
than those of low-uplink peers, specifically with: $\lambda_{l}^{({\sf
lo})}=10$, $\lambda^{({\sf hi})}_{s}=0.1$, $\lambda_{l}^{({\sf hi})}=1$,
$\lambda_{s}^{({\sf hi})}=0.01$, $\delta^{({\sf lo})}=9$, $\delta^{({\sf
hi})}=0.9$, $\beta_{r}=\beta_{N}-1=1$. The sensitivity of the delay from
leecher to seeder for the low-uplink peers was illustrated in Figure 4. The
same quantity for the two-type system is plotted in Figure 5, where we also
plot the continuous globally-rarest-first optimal-control sojourn times.
Figure 5: Mean delay from leecher to seeder for high-uplink peers vs $u$ with
$\delta^{({\sf lo})}=9$ and $\delta^{({\sf hi})}=0.9$
Note the shape of the curve describing the sojourn time as a function of $u$
is highly affected by the parameter choices. For example, choosing smaller
values $\delta^{({\sf lo})}=1$ and $\delta^{({\sf hi})}=0.1$ produces Figure
6.
Figure 6: Mean delay from leecher to seeder for high-uplink peers vs $u$ with
$\delta^{({\sf lo})}=1$ and $\delta^{({\sf hi})}=0.1$
The continuous globally rarest first control produces almost identical
behavior to the case when $u\equiv 1/2$. This is because both the bang-bang
controller and its continuous variation drive the system toward the case when
the number of rare-segments holders is identical to the number of “$N-1$
segments” holders. When this occurs, the values of the three controllers are
identical. In computing these plots, the number of rare segment holders was
initialized lower than the number of $N-1$ segment holders. This suggests that
there is little to be gained from the more complex rarest first controllers
but a substantial amount to be gained in terms of swarm stability by setting
$u\neq 1/2$.
## 8 Summary
Ostensibly under BitTorrent incentives, the locally-rarest-first rule’s
objective is to prevent extinction of certain segments. The file segmentation
system itself is intended to extend a peer’s time in a swarm and thereby
increase their cooperation (swapping activity). Choking and optimistic
unchoking mechanisms result in a clustering of peers according to their
allocated uplinks [1], i.e., peers will naturally tend to swap with others
with similar uplink rates. Unchoking is intended to give peers with low
uplinks a chance to increase (rehabilitate) their uplinks. Note how these
incentives may be undermined by a large number of seeder peers. When these
incentives do not work because many leecher peers are simply unwilling or
unable to increase their uplinks and when the present seeders are congested,
unchoking may allow peers to access needed segments even if it means acquiring
them at a rate significantly slower than their own allocated uplinks for file-
sharing.
Typically, there are a persistent number of “permanent” seeders which
exclusively perform server transactions in the swarm that operate “outside” of
these incentives to distribute segments and prevent segment extinction [16].
These permanent seeders are fostered by: fixed-rate pricing frameworks for
Internet access, and limited liability for copyright infringement afforded by
the file segmentation framework itself as well as by third-party swarm
discovery (e.g., downloading torrents via certain web sites).
Given a positive departure rate for seeders, the presence of permanent seeders
is here modeled by the assumption that $\lambda_{s}>0$, i.e., a small but
persistent arrival rate of seeders.
We discussed the optimality properties of the locally-rarest-first segment
distribution policy and how to employ different policies that create
relatively rare segments. The leechers are thereby enticed to stay somewhat
longer in the swarm and, consequently, cooperate more.
In the local swarm, delays will increase the availability of segments that can
be exchanged between leechers. Globally, they would cause leechers to become
temporary seeders for more time in other swarms or even under certain
assumptions in the local one [16].
We note that delays will result in larger average total leecher populations
$x_{l}$ that will consequently create a lesser burden on the permanent seeder
population $x_{s}$. To reflect the limited uplink capacity of the permanent
seeders, we might want to reduce the parameter $\beta$ as $x_{l}$ increases so
that the $\beta x_{l}$ factor is fixed, i.e., so that the segment transfer
rate $\beta x_{l}x_{s}$ reflects these limits.
Finally, we also studied the system interpreting the two-segment model as a
rare segment and a aggregation of the rest, naturally leading to different
associated uplink parameters. We numerically showed how leecher and seeder
sojourn times were affected by the (lumped) model parameters, particularly the
proportion of the population that are seeders, and argued that taking the
control $u\neq 1/2$ leads to good stability properties.
## 9 Acknowledgements
G. Kesidis’ work was funded in part by NSF CNS NeTS Grant No. 0915928.
## References
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* [2] B. Cohen, “BitTorrent protocol specification,” http://www.bittorrent.com/protocol.html.
* [3] ——, “Incentives build robustness in bittorrent,” in _Workshop on Economics of Peer-to-Peer Systems_ , Berkeley, CA, May 2003.
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* [5] P. Antoniadis, C. Courcoubetis, and R. Mason, “Comparing economic incentives in peer-to-peer networks,” _Compute Networks_ , vol. 46, no. 1, pp. 133–146, 2004.
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* [9] L. Guo, S. Chen, Z. Xiao, E. Tan, X. Ding, and X. Zhang, “Measurements, modeling and analysis of bittorrent-like systems,” in _Measurements, modeling and analysis of BitTorrent-like systems_ , October 2005.
* [10] I. Norros, H. Reittu, and T. Eirola., “On the stability of two-chunk file-sharing systems.” _Queueing Systems: Theory and Applications_ , vol. 67, no. 3, 2011.
* [11] Z. Ge, D. R. Figueiredo, S. Jaiswal, J. Kurose, and D. Towsley, “Modeling peer-to-peer file sharing systems,” in _Proc. IEEE INFOCOM_ , San Francisco, CA, April 2004.
* [12] N. Laoutaris, D. Carra, and P. Michiardi, “Uplink allocation beyond choke/unchoke or how to divide and conquer best.” in _Proceedings of the 2008 ACM CoNEXT Conference_ , 2008.
* [13] B. Fan, J. Lui, and D.-M. Chiu, “The design trade-offs of BitTorrent-like file sharing protocols,” _IEEE/ACM Transactions on Networking (TON)_ , vol. 17, no. 2, pp. 365–376, 2009.
* [14] R. Ma, S. Lee, J. Lui, and D. Yau., “A game theoretic approach to provide incentive and service differentiation in P2P networks,” in _Proc. of the joint international conference on Measurement and modeling of computer systems_ , New York, NY, 2004, pp. 189–198.
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* [16] J. Bieber, M. Kenney, N. Torre, and L. Cox., “An empirical study of seeders in BitTorrent,” Duke University, Computer Science Dept., Tech. Rep. CS-2006-08, 2006.
* [17] A. Tangpong and G. Kesidis, “A simple file-sharing game relating uplink choking, flat-rate pricing and seeding,” in _Proc. CISS_ , Princeton, NJ, 2010.
* [18] E. Adar, “Drawing crowds and bit welfare,” _ACM SIGecom Exchanges_ , vol. 5, no. 4, pp. 31–40, 2005.
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* [20] T. Konstantopoulos, G. Kesidis, and P. Sousi, “A stochastic epidemiological model and a deterministic limit for BitTorrent-like peer-to-peer file-sharing networks,” in _Workshop on Network Control and Optimization (NET-COOP)_ , vol. Springer LNCS 5425, September 2008.
* [21] D. Daley and J. Gani., _Epidemic Modeling: An Introduction_. Cambridge University Press, 1999.
* [22] M. Vojnovic and L. Massoulie, “Coupon replication systems,” _IEEE/ACM Transactions on Networking (TON)_ , vol. 16, no. 3, Jube 2008.
* [23] R. M. Wolff, _Stochastic Modeling and the Theory of Queues_. Englewood Cliffs, NJ: Prentice-Hall, 1989.
* [24] D. Kirk, _Optimal Control Theory: An Introduction_. Dover Press, 2004.
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|
arxiv-papers
| 2013-06-13T16:33:03 |
2024-09-04T02:49:46.478673
|
{
"license": "Public Domain",
"authors": "Christopher Griffin and George Kesidis and Panayotis Antoniadis and\n Serge Fdida",
"submitter": "Christopher Griffin",
"url": "https://arxiv.org/abs/1306.3160"
}
|
1306.3383
|
# Autonomous Demand Side Management based on Energy Consumption Scheduling and
Instantaneous Load Billing: An Aggregative Game Approach ††thanks: This work
of He (Henry) Chen was supported by International Postgraduate Research
Scholarship (IPRS), Australian Postgraduate Award (APA), and Norman I Price
Supplementary scholarship. ††thanks: H. Chen, Y. Li, and B. Vucetic are with
School of Electrical and Information Engineering, The University of Sydney,
Sydney, NSW 2006, Australia (email: [email protected],
[email protected], [email protected]). ††thanks: R. Louie is
with the Electronic and Computer Engineering, Hong Kong University of Science
and Technology, Clear Water Bay, Hong Kong (email:[email protected]).
He (Henry) Chen, Yonghui Li,
Raymond H. Y. Louie, and Branka Vucetic
###### Abstract
In this paper, we investigate a practical demand side management scenario
where the selfish consumers compete to minimize their individual energy cost
through scheduling their future energy consumption profiles. We adopt an
instantaneous load billing scheme to effectively convince the consumers to
shift their peak-time consumption and to fairly charge the consumers for their
energy consumption. For the considered DSM scenario, an aggregative game is
first formulated to model the strategic behaviors of the selfish consumers. By
resorting to the variational inequality theory, we analyze the conditions for
the existence and uniqueness of the Nash equilibrium (NE) of the formulated
game. Subsequently, for the scenario where there is a central unit calculating
and sending the real-time aggregated load to all consumers, we develop a one
timescale distributed iterative proximal-point algorithm with provable
convergence to achieve the NE of the formulated game. Finally, considering the
alternative situation where the central unit does not exist, but the consumers
are connected and they would like to share their estimated information with
others, we present a distributed synchronous agreement-based algorithm and a
distributed asynchronous gossip-based algorithm, by which the consumers can
achieve the NE of the formulated game through exchanging information with
their immediate neighbors.
###### Index Terms:
Smart grid, demand side management, aggregative game, Nash equilibrium,
distributed iterative proximal-point method, distributed agreement (consensus)
method, distributed gossip-based algorithm.
## I Introduction
Recently, demand side management (DSM) has emerged as one of the key
techniques to transform today’s aging power grid into a more efficiently and
more reliably operated smart grid [1, 2]. Thanks to the two-way communication
capabilities of smart grid, real-time pricing [3] has been regarded as a
promising technique to implement DSM due to its ability to effectively
convince consumers to shift their peak-time energy consumption to non-peak
time. In real-time pricing schemes, the energy price for a certain operation
period is normally designed to be proportional to the aggregated load of all
consumers during the considered period [3, 4, 5, 6]. As a result, the
consumers would prefer to consume more energy during non-peak times rather
than peak times in order to decrease their energy cost. This can improve the
operation efficiency of the whole grid since its demand is flattened.
In a real-time pricing based DSM framework, the billing mechanism (i.e., how
to charge the consumers for their energy usage) is of great importance since
it may significantly affect the consumers’ motivation to participate in the
DSM program. However, there has only been limited work investigating this
important billing issue. [4] proposed a simple billing approach, where the
consumers were charged in proportional to their total energy consumption for
the next operation period. This total load billing method can minimize the
whole grid energy cost. However, the consumers are charged the same amount if
they consume the same total amount of electricity, regardless in peak or off-
peak times, which leads to unfair charging for the consumers who use less
electricity in peak times [5]. To address this problem, [5, 6] proposed a new
billing approach, where each consumer is charged based on his/her
instantaneous load in each time slot during the next operation period. As a
result, the consumers will be charged more if they consume more during peak
times and this can effectively improve the fairness of charging between
different consumers [5]. In this paper, the billing approach proposed in [5,
6] is termed as instantaneous load billing, in contrast to the total load
billing in [4]. Based on the proposed billing approach, [6] also developed a
classical non-cooperative game for the DSM scenario where the traditional
consumers as well as consumers owing distributed energy sources and/or energy
storage compete to reduce their energy bills. However, the main analysis and
results in [5, 6] are only valid when the energy price is a _linear_ function
of the total load of all consumers in each time slot. Very recently, [7]
extended [6] to the scenario with a _general_ energy price function. Based on
the proximal decomposition method [8], synchronous and asynchronous algorithms
were respectively developed in [6] and [7] for the consumers to achieve their
optimal strategies in a distributed manner.
In this paper, we develop three novel distributed algorithms for autonomous
DSM scenario, which enable the selfish consumers to optimize their own energy
payment through scheduling their future energy consumption. The key
contributions of this paper, with a particular emphasis on the differences
with [6, 7], are summarized as follows:
(1) Inspired by [5, 6, 7], we adopt the instantaneous load billing scheme to
effectively convince the consumers to shift their peak-time energy consumption
and fairly charge the consumers. In this paper, we are interested in a
practical _polynomial_ energy price model instead of the general energy price
model considered in [7], since the polynomial model has been widely adopted in
power systems (e.g., spot market price model [9, 10]). By exploring the
aggregative property of the instantaneous load billing scheme that the energy
cost of each consumer only depends on its own and all consumers’ aggregated
energy consumption profiles [6, 7], we develop a novel aggregative game111An
aggregative game is a special kind of the non-cooperative game where each
player’s payoff is parameterized by its own action and the aggregative of the
actions taken by all players [11, 12][13, Ch. 4]. to model the strategic
behaviors of the selfish consumers. Additionally, we perform new theoretical
analysis for the Nash equilibrium (NE) of the formulated game. This analysis
will be facilitated by using advanced variational inequality theory [14]. As
shown in this paper, the formulation of the aggregative game can facilitate
the game analysis, the algorithm design and the convergence proof for the
proposed algorithms. In our previous work [15], a distributed and parallel
gradient projection algorithm was proposed for the considered DSM framework.
(2) For the algorithm design, we first consider the same setup as in [6, 7]
where a central unit exists and broadcasts the real-time aggregated energy
consumption profile to all consumers. In this case, the synchronous and
asynchronous proximal decomposition algorithms proposed in [6, 7] can be
directly applied to compute the NE of the formulated game. It should be noted
that the algorithms in [6, 7] are _two timescale_ , which is due to the nature
of the problem (i.e., the mapping function associated with distributed
generation and storage is monotone) in [6, 7]. However, as shown later, the
formulated problem in this paper can be guaranteed to possess strictly
monotone mapping. Thus, we may not need to apply the two timescale algorithms,
which are generally harder to implement in online settings than the _one
timescale_ algorithms [16]. Motivated by this, we develop a distributed
iterative proximal-point algorithm to achieve the NE of the formulated game.
This new algorithm is a parallel and one timescale algorithm and the choice of
algorithm parameters does not depend on the system arguments.
(3) Considering the alternative situation without a central unit but where the
consumers are connected and they exchange their estimated information with
others, we develop a distributed agreement (consensus)-based algorithm, by
which the consumers can achieve the NE of the formulated game through
exchanging information with their immediate neighbors. Although information
exchanges are required between the consumers in this algorithm, no private
information (e.g., the exact energy consumption profile of each consumer) is
shared between the consumers, thus effectively protecting the consumers’
privacy. Moreover, the parameters of this algorithm can also be chosen without
knowing the system’s arguments a priori.
(4) Although the central unit is not necessary for the aforementioned
agreement-based algorithm, synchronization between the consumers and
coordination in terms of algorithm step sizes are still required, which are
challenging in very large networks. Motivated by this, we develop a
distributed asynchronous gossip-based algorithm for computing the NE of the
formulated game without the need of a central unit. In this developed
algorithm, synchronization is not required between the consumers. Besides the
asynchronous updates, the consumers are allowed to use uncoordinated step
sizes that are based on the frequency of the consumer update. Note that
although the distributed consensus and gossip algorithms are well-known
techniques, their application to achieve the NE of the formulated game is not
straightforward at all and is not feasible without the formulation of the
aggregative game in this paper.
_Notations_ : All the vectors, except as specially stated, are column vectors.
${\bf{x}}^{T}$ and
${\left\|{\bf{x}}\right\|_{2}}=\sqrt{{{\bf{x}}^{T}}{\bf{x}}}$ denote the
transpose and Euclidean norm of a vector $\bf{x}$, respectively. $A\times B$
is the cartesian product of sets $A$ and $B$.
${\bf{x}}=\left({{{\bf{x}}_{n}}}\right)_{n=1}^{N}$ denotes the operation of
concatenating all vectors ${{{\bf{x}}_{1}},\ldots,{{\bf{x}}_{N}}}$ into a
single column vector, i.e.,
${\bf{x}}={\left({{\bf{x}}_{1}^{T},\ldots,{\bf{x}}_{N}^{T}}\right)^{T}}$. To
emphasize the $n$-th element within ${\bf{x}}$, we sometimes write $\left({\bf
x}_{n},{\bf x}_{-n}\right)$ instead of $\bf x$ with ${\bf
x}_{-n}=\left({{{\bf{x}}_{m}}}\right)_{m=1,m\neq n}^{N}$. We use
${\left[{\;\cdot\;}\right]_{\mathcal{K}}}$ to denote the Euclidean projection
operator onto a set $\mathcal{K}$. ${\nabla_{\bf{x}}}f\left({\bf{x}}\right)$
and $\nabla_{\bf{x}}^{2}f\left({\bf{x}}\right)$ respectively denote the
gradient vector and Hessian matrix of a scalar function
$f\left({\bf{x}}\right)$, while ${\bf{J}}_{\bf
x}{\bf{F}}\left({\bf{x}}\right)$ denotes the Jacobian matrix of a vector
function ${\bf{F}}\left({\bf{x}}\right)$.
The rest of this paper is organized as follows. The system model and the
instantaneous load billing scheme are described in Section II. Section III
formulates the aggregated game and analyzes the existence and uniqueness for
the NE of the formulated game. The three new distributed algorithms are
proposed in Section IV-VI, respectively. In section VII, numerical results are
presented to illustrate and validate the theoretical analysis. Finally,
Section VIII concludes this paper.
## II System Model
We consider an electricity network comprised of $N$ consumers, which are
served by a common energy provider. We denote the set of these consumers as
$\mathcal{N}=\\{1,\ldots,N\\}$. Each consumer is equipped with an energy
management controller unit, which has full responsibility for scheduling the
consumer’s energy consumption. In addition, there exists a two-way
communications network connecting each consumer to the energy provider.
Similar to [4, 17], we assume that the energy requirement of each consumer is
determined in advance for $H$ future time slots. Each time slot can represent
different timing horizons, e.g., one hour of a day.
### II-A Energy Consumption Model
We consider an energy consumption model as in [17], where the $n$th
($n\in{\mathcal{N}}$) consumer’s energy consumption profile can be formulated
as
${{\bf{q}}_{n}}={\left({q_{n}^{1},\ldots,q_{n}^{H}}\right)^{T}},$ (1)
where ${q_{n}^{h}}$ is the energy consumption of consumer $n$ in the $h$th
time slot and it is subject to the following constraints:
$q_{n}^{h,\min}\leq q_{n}^{h}\leq
q_{n}^{h,\max}\;{\rm{and}}\;\sum\nolimits_{h=1}^{H}{q_{n}^{h}=}{E_{n}},$ (2)
where $q_{n}^{h,\min}$ and $q_{n}^{h,\max}$ denote consumer $n$’s minimum and
maximum energy levels222Note that the minimum and maximum energy levels can be
estimated in practice by sophisticated predictive techniques, such as machine
learning and stochastic signal processing [17]. Moreover, the approaches
presented in this work can be easily extended to the appliance-level energy
consumption model [4]. in time slot $h$, respectively, and $E_{n}$ is the
total energy requirement of consumer $n$ over all time slots. Therefore, the
individual feasible energy consumption set of consumer $n$ can be expressed as
$\begin{split}{\mathcal{Q}_{n}}=&\left\\{{{\bf{q}}_{n}}:\sum\nolimits_{h=1}^{H}{q_{n}^{h}=}{E_{n}},\;{\rm
and}\right.\\\ &\;\;\;\left.q_{n}^{h,\min}\leq q_{n}^{h}\leq
q_{n}^{h,\max},\;\forall h\in{\mathcal{H}}\right\\},\end{split}$ (3)
where ${\mathcal{H}}=\left\\{{1,\ldots,H}\right\\}$ is the set of all future
$H$ time slots. The feasible energy consumption set of all consumers can thus
be expressed as
${\cal Q}={{\cal Q}_{1}}\times\ldots\times{{\cal Q}_{N}}.$ (4)
### II-B Instantaneous Load Billing
To effectively convince the consumers to shift their peak-time energy
consumption and fairly charge the consumers for their energy consumption, we
adopt the instantaneous load billing scheme [5, 6, 7], where the energy price
(the cost of one unit energy) of a certain time slot is set as an increasing
and smooth function of the total demand in that time slot, and the consumers
are charged based on the instantaneous energy price as well as the energy
amount they consume in each time slot. Instead of the general price model[7],
we focus on a practical and specific polynomial energy price model in this
paper, which has been widely adopted in power systems (e.g., the spot market
price model [9, 10]). Specifically, the energy price of the $h$th time slot is
given by:
${p_{h}}\left({{L_{h}}}\right)=a_{h}{\left(L_{h}\right)^{b_{h}}}+c_{h},$ (5)
where $a_{h},~{}b_{h},~{}c_{h}$ are time slot-specific parameters with
$a_{h}>0$, $b_{h}\geq 1$, $c_{h}\geq 0$, and $L_{h}$ is the total energy
consumed by all consumers in time slot $h$. It is should note that the price
function in (5) can readily account for the important characteristics of
energy prices that are needed for DSM in smart grid. For example, the
increasing and convex price function ensures that the energy price will grow
more rapidly as the aggregated load increases. This can effectively convince
the consumers to shift their peak-time consumption to non-peak hours, thereby
flattening the overall demand curve and reducing the need for carbon-intensive
and expensive peaking power plants. Therefore, the considered energy price
model can improve the efficiency of the energy provider, and motivate and
engage the energy provider to enforce such price model.
Follow the adopted energy price model, the total energy cost for consumer $n$
over all future $H$ time slots can thus be given by:
${\mathcal{B}}_{n}\left({{{\bf{q}}_{n}},{{\bf{q}}_{-n}}}\right)=\sum\nolimits_{h=1}^{H}{\left[{{p_{h}}\left({\sum\nolimits_{m=1}^{N}{q_{m}^{h}}}\right)q_{n}^{h}}\right]},$
(6)
where ${{\bf{q}}_{-n}}=\left({{{\bf{q}}_{m}}}\right)_{m=1,m\neq n}^{N}$
denotes the $(N-1)H\times 1$ vector of all consumers’ energy consumption
profiles, except the $n$th one. This is in contrast to the total load billing
method in [4], where the energy payment of the $n$th consumer is calculated by
${{\cal
B}_{n}^{{\rm{TLB}}}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{-n}}}\right)=\frac{{{E_{n}}}}{{\sum\nolimits_{m=1}^{N}{{E_{m}}}}}\sum\limits_{h=1}^{H}{\left[{{p_{h}}\left({\sum\limits_{m=1}^{N}{q_{m}^{h}}}\right)\sum\limits_{m=1}^{N}{q_{m}^{h}}}\right]}.$
(7)
It has been shown in [5] that the adopted billing method in (6) is fairer than
the total load billing method given in (7). This will also be validated by the
simulation results in this paper.
Note that (6) can be further rewritten as
$\displaystyle{{\cal
B}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)=\sum\nolimits_{h=1}^{H}{\left[{p_{h}\left({q_{\Sigma}^{h}}\right)q_{n}^{h}}\right]},$
(8)
where ${{\bf{q}}_{\Sigma}}=\sum\nolimits_{m=1}^{N}{{{\bf{q}}_{m}}}$ denotes
the aggregated energy consumption profile of all consumers over future $H$
time slots and $q_{\Sigma}^{h}=\sum\nolimits_{m=1}^{N}{q_{m}^{h}}$ is the
$h$th element of ${{\bf{q}}_{\Sigma}}$. From (8), we can see that the
calculation of the total energy cost of each consumer only requires the
knowledge of the aggregated energy consumption profile of all consumers
(${{\bf{q}}_{\Sigma}}$), and that the individual consumption profile of each
consumer (${{\bf{q}}_{-n}}$) is not required any more.
## III Game Formulation and Analysis
In this section, we formulate an aggregative game for the considered DSM
scenario. By employing variational inequality theory, we then analyze the
existence and uniqueness of the NE for the formulated aggregative game.
### III-A Aggregative Game Formulation
We consider the scenario where all consumers are selfish. In particular, each
consumer aims to minimize his/her total cost through energy consumption
scheduling. Mathematically, this will involve the $n$th consumer
($n\in{\mathcal{N}}$) solving the following optimization problem:
$\begin{array}[]{*{20}{c}}~{}{\mathop{\min}\limits_{{{\bf{q}}_{n}}}\;{{\mathcal{B}}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)}\\\
{{\rm{s}}{\rm{.}}\;{\rm{t}}{\rm{.}}\;{{\bf{q}}_{n}}\in\mathcal{Q}_{n}}\\\
\end{array}.$ (9)
We can observe from (9) that the consumers solve optimization problems which
are coupled with the aggregated energy consumption of all consumers. Hence,
this energy consumption control scenario can be modeled by the following
aggregative game [11, 12][13, Ch. 4]:
* •
_Players_ : The $N$ consumers.
* •
_Actions_ : Each consumer selects its energy consumption ${\bf
q}_{n}\in{\mathcal{Q}}_{n}$ to minimize his/her total energy cost.
* •
_Payoffs_ : The total energy cost
${\mathcal{B}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)$ defined
in (8).
For convenience, we denote this Nash equilibrium (NE) problem as
$\mathcal{G}=\left\langle{\mathcal{N},\left\\{\mathcal{Q}_{n}\right\\},\left\\{{\mathcal{B}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)\right\\}}\right\rangle$.
In the following subsection, we will employ variational inequality theory [14]
to analyze the formulated game.
### III-B NE Analysis
Before proceeding, it is convenient to first present the following lemma
regarding the properties of the formulated game’s action sets and payoff
functions:
###### Lemma 1
For each $n=1,\ldots,N$, the set ${\mathcal{Q}}_{n}\in\mathbb{R}^{H}$ is
convex and compact, and each function
${\mathcal{B}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)$ is
continuously differentiable in ${{\bf{q}}_{n}}$. For each $n\in{\mathcal{N}}$
and each fixed tuple ${{\bf{q}}_{-n}}$, the function ${{\cal
B}_{n}}\left({\cdot\;,\;\cdot\;+\sum\nolimits_{m=1,m\neq
n}^{N}{{{\bf{q}}_{m}}}}\right)$ is convex in ${\bf q}_{n}$ over the set
${\mathcal{Q}}_{n}$.
###### Proof:
See Appendix -A. ∎
Under Lemma 1 and according to [14, Prop. 1.4.2], we have the following lemma:
###### Lemma 2
The NE of the formulated game $\mathcal{G}$ is equivalent to the solution of
the variational inequality (VI) problem333Given a subset ${\mathcal{K}}$ of
the Euclidean $N$-dimensional space $\mathbb{R}^{N}$ and a mapping $\bf F$:
${\mathcal{K}}\rightarrow\mathbb{R}^{N}$, the variational inequality problem,
denoted VI$\left({\mathcal{K}},{\bf F}\right)$, is to find a vector ${\bf
x}^{\ast}\in{\mathcal{K}}$ such that
${\left({{\bf{y}}-{{\bf{x}}^{*}}}\right)^{T}}{\bf{F}}\left({{{\bf{x}}^{*}}}\right)\geq
0,\;\forall{\bf{y}}\in{\mathcal{K}}$. denoted by VI$\left({\mathcal{Q}},{\bf
F}\right)$ where ${\cal Q}={{\cal Q}_{1}}\times\ldots\times{{\cal Q}_{N}}$ and
$\displaystyle{\bf{F}}\left({\bf{q}}\right)=\left({{{\bf{F}}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)}\right)_{n=1}^{N},$
(10)
with
${{\bf{F}}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)={\nabla_{{{\bf{q}}_{n}}}}{{\cal
B}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right).$ (11)
By investigating the monotonicity property of the mapping
${\bf{F}}\left({\bf{q}}\right)$, we can derive the following proposition:
###### Proposition 1
If the price parameter ${b_{h}}$ satisfies ${b_{h}}<3+4/\left({N-1}\right)$
for any $h\in{\mathcal{H}}$, then the formulated aggregative game admits a
unique NE.
###### Proof:
See Appendix -B. ∎
###### Remark 1
As can be seen from Proposition 1, only a specific relationship between the
exponential factor of the polynomial price function and the number of
consumers is required to guarantee the uniqueness of the NE. Specifically, the
exponential factor of the price function is subject to an upper bound, which
is inversely proportional to the number of consumers $N$.
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\square$
One could consider to solve the aforementioned game in a centralized manner,
where a central unit adopts the algorithms proposed in [14, Ch. 12] to solve
the associated VI problem. However, such an approach requires each consumer to
release detailed information about their energy consumption feasible set,
which may lead to consumers’ privacy and security concerns. To overcome this
issue, in the following sections, we will develop three different distributed
algorithms to achieve the NE of the formulated aggregative game for the
scenarios with and without a central unit, which calculates the aggregated
load and broadcasts it to consumers in each iteration of the algorithm.
## IV Distributed Iterative Proximal-point Algorithm with A Central Unit
In this section, we consider the same setting as in [6, 7] where there is a
central unit, which can provide the consumers with the latest information of
the aggregated energy consumption profile after all consumers update their
individual ones. In this case, we develop a distributed iterative Proximal-
point algorithm to achieve the NE of the formulated aggregative game.
Before presenting our algorithm, it is worth mentioning that the formulated
game can also be solved by the synchronous and asynchronous proximal
decomposition algorithms proposed in [6] and [7], which were guaranteed to
converge under some conditions on the algorithm parameters. The distributed
algorithms in [6, 7] were proposed based on the proximal decomposition method
[8] and solved a sequence of regulated versions of the original problem, each
of which may need a distributed iterative process in itself. This is actually
a two timescale approach (i.e., the proximal method updates at a slower
timescale while solutions of the regularized problems change at a faster
timescale) and is generally harder to implement in online settings [16].
Additionally, the regulation parameter of such kind of algorithms has to be
chosen centrally since it is normally dependent on the system arguments. It
should be noted that the two-timescale property of the algorithms in[6, 7] is
due to the nature of the problem (i.e., the mapping function associated with
distributed generation and storage is monotone). However, as shown in Section
III, the formulated problem in this paper can be guaranteed to possess
strictly monotone mapping. Thus, we may not need to apply the two timescale
algorithms. Motivated by this, we present a single timescale distributed
algorithm based on the iterative regulation technique [16, 18], which requires
only one projection step in each iteration. This algorithm is formally
described in Algorithm 1.
1: Set ${t}=1$ and each consumer $n\in{\mathcal{N}}$ chooses a random ${\bf
q}_{n}{\left(1\right)}$ from their feasible set ${\mathcal{Q}}_{n}$ and sends
it to the central unit. The central unit calculates
${\bf{q}}_{\Sigma}{\left({{1}}\right)}=\sum\nolimits_{n=1}^{N}{{\bf{q}}_{n}{\left({{1}}\right)}}$
and broadcasts it to the consumers. Given the values of the step-size
$\gamma(t)$ and the parameter $\theta>0$.
2: If a suitable termination criterion is satisfied: $\rm{STOP}$.
3: For each consumer $n\in{\mathcal{N}}$: 3.1: Receive
${\bf{q}}_{\Sigma}{\left({{t}}\right)}$ from the central unit. 3.2: Update the
energy consumption profile by
$\begin{split}{{\bf{q}}_{n}}\left({t+1}\right)=&\left[{{{\bf{q}}_{n}}\left(t\right)-\gamma\left(t\right)\left({{{\bf
F}_{n}}\left({{{\bf{q}}_{n}}\left(t\right),{\bf{q}}_{\Sigma}{\left({{t}}\right)})}\right)+}\right.}\right.\\\
&{\left.{\left.{\theta\left({{{\bf{q}}_{n}}\left(t\right)-{{\bf{q}}_{n}}\left({t-1}\right)}\right)}\right)}\right]_{{{\cal
Q}_{n}}}}.\end{split}$ 3.3: Send the update
${\bf{q}}_{n}{\left({{t}+1}\right)}$ to the central unit.
4: $t\leftarrow t+1$; go to $\rm{STEP}$ 2\.
Algorithm 1 : Distributed Iterative Proximal-point Algorithm
The convergence property of Algorithm 1 is summarized in the following
proposition:
###### Proposition 2
Assume that the condition in Proposition 1 holds. Then, the sequence of the
energy consumption profile $\\{{\bf{q}}(t)\\}$ generated by Algorithm 1
converges to the unique NE of the game ${\mathcal{G}}$ if the step-size
$\gamma(t)$ satisfies the following:
$\sum\nolimits_{t=1}^{\infty}{\gamma\left(t\right)=\infty}\;{\rm{and}}\;\sum\nolimits_{t=1}^{\infty}{{\gamma^{2}}\left(t\right)<\infty}.$
(12)
###### Proof:
See Appendix -C. ∎
As shown above, Algorithm 1 can converge to the NE of the formulated game when
there is a central unit that calculates the aggregated energy consumption
profile ${\bf q}_{\Sigma}$ and broadcasts it to all consumers in each
iteration. However, the developed Algorithm 1 and the algorithms in [6, 7]
cannot be directly implemented for situations where the central unit does not
exist, in which case the consumers thus do not have ready access to the
aggregated energy consumption profile. Motivated by this issue, we will
develop a distributed synchronous agreement-based algorithm and a distributed
asynchronous gossip-based algorithm to achieve the NE of the formulated game
in the following sections.
## V Distributed Synchronous Agreement-based Algorithm without A Central Unit
In this section, we consider an alternative scenario where the central unit
does not exist, but the consumers are connected in some manner and they are
willing to share their estimated information through local communication. For
this setting, we develop a distributed agrement-based algorithm, through which
the consumers can achieve the NE of the game $\mathcal{G}$ via exchanging
information with their immediate neighbors. In the developed algorithm, the
connection topology of the consumers is modeled as an undirected (not
necessarily complete) static graph. In practice, such a connection can be
established through either wired or wireless communication techniques.
Specifically, the connection can be implemented by employing the power line
communication technique or using the resources of cellular networks to
establish a virtual private network. As these techniques are widely deployed,
the connection of a large number of consumers in large areas is feasible.
Since only immediate connected consumers exchange information, the amount of
data to be exchanged at each iteration of the developed algorithm is
proportional to the numbers of connections between the consumers.
Recall that, in the formulated aggregative game, each consumer’s payoff is
only determined by his/her own energy consumption profile and the aggregated
energy consumption profile of all consumers. Hence, the unique NE of the
formulated game is achieved when the consumers reach an agreement (consensus)
on the aggregated energy consumption profile. Following this equivalence and
inspired by [13, Ch. 4], we develop a distributed agreement-based algorithm to
achieve the unique NE of the considered aggregative game. In each iteration of
this algorithm, each consumer $n\in{\mathcal{N}}$ executes the following three
steps:
* •
_Step 1_ : Estimate the average energy consumption of all consumers through a
weighted combination of his/her own estimation and the estimation of the
immediate neighbors in the last iteration. We use ${\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(t\right)$ to denote the average energy consumption of all consumers
estimated by the consumer $n$ in the $t$th iteration. Then, the aggregated
load of the whole network estimated by consumer $n$ is
$N{\bf{\hat{q}}}_{n}^{{}_{\rm A}}\left(t\right)$.
* •
_Step 2_ : Update his/her energy consumption profile based on the estimated
aggregated load through executing a Euclidean projection operation.
* •
_Step 3_ : Update his/her own estimation of the average energy consumption.
1: Set $t=1$. Choose any feasible starting point
${\bf{q}}\left(1\right)=\left({{{\bf{q}}_{n}}\left(1\right)}\right)_{n=1}^{N}$
and set ${\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(1\right)={{{\bf{q}}_{n}}\left(1\right)}$ for every
$n\in{\mathcal{N}}$. Given the weight parameters $w_{n,k}$ and the step-size
$\alpha\left(t\right)$.
2: If a suitable termination criterion is satisfied: $\rm{STOP}$.
3: Each consumer $n\in{\mathcal{N}}$ updates his/her energy consumption
profile and the estimated average energy consumption of all consumers via
executing
$\displaystyle\footnotesize\begin{split}&~{}~{}~{}~{}~{}~{}~{}{\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(t\right)={w_{n,n}}{\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(t\right)+\sum\nolimits_{k\in{{\cal
D}_{n}}}w_{n,k}{\bf{\hat{q}}}_{k}^{{}_{\rm A}}\left(t\right),\\\
&~{}~{}~{}~{}~{}~{}~{}{{\bf{q}}_{n}}\left({t+1}\right)={\left[{{{\bf{q}}_{n}}\left(t\right)-\alpha\left(t\right){{\bf{F}}_{n}}\left({{{\bf{q}}_{n}}\left(t\right),N{\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(t\right)}\right)}\right]_{{{\mathcal{Q}}_{n}}}},\\\
&~{}~{}~{}~{}~{}~{}~{}{\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left({t+1}\right)={\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(t\right)+{{\bf{q}}_{n}}\left({t+1}\right)-{{\bf{q}}_{n}}\left(t\right).\end{split}$
4: $t\leftarrow t+1$; go to $\rm{STEP}$ 2\.
Algorithm 2 : Synchronous Agreement-based Algorithm
To proceed, it is convenient to first model the connection topology between
consumers. For simplicity but without loss of generality, we model the
connection topology of the consumers as an undirected static graph
${\mathcal{M}}\left({\mathcal{N}},{\mathcal{E}}\right)$ with ${\mathcal{N}}$
being the set of all consumers and ${\mathcal{E}}$ being the set of undirected
edges among the consumers. The notation $\left\\{n,k\right\\}\in{\mathcal{E}}$
means that consumer $n$ and consumer $k$ are immediate neighbors, and
${\mathcal{D}}_{n}$ denotes the set of consumer $n$’s neighbors, i.e., ${{\cal
D}_{n}}=\left\\{{\left.{k\in{\cal
N}}\right|\left\\{{k,n}\right\\}\in{\mathcal{E}}}\right\\}$. Now we are ready
to present the distributed agreement-based algorithm, which is formally
described in Algorithm 2, where the notation $w_{n,k}$ denotes the nonnegative
weight that consumer $n$ assigns to the estimate of consumer $k$, which is set
to zero if $k\notin{{\mathcal{D}}_{n}}$ and $n\neq k$.
In terms of the convergence of Algorithm 2, we have the following proposition:
###### Proposition 3
Assume that the undirected graph
${\mathcal{M}}\left({\mathcal{N}},{\mathcal{E}}\right)$ is connected, the
step-size $\\{\alpha(t)\\}$ is monotonically decreasing with $t$ and satisfies
the following:
$\sum\nolimits_{t=0}^{\infty}{\alpha\left(t\right)}=\infty,\;{\rm{and}}\;{\sum\nolimits_{t=0}^{\infty}{\left[{\alpha\left(t\right)}\right]}^{2}}<\infty,$
(13)
and the weights adopted by the consumers meets the following444Note that the
summations in the following equations are actually equivalent to that over the
set ${\mathcal{D}}_{n}$. This is because that ${w_{n,k}}=0$ if consumer $k$ is
not a neighbor of consumer $n$.:
$\sum\nolimits_{k=1}^{N}{{w_{n,k}}=1},\forall
n,\;{\rm{and}}\;\sum\nolimits_{n=1}^{N}{{w_{n,k}}=1},\forall k,$ (14)
and the condition in Proposition 1 holds. Then the sequence $\left\\{{\bf
q}(t)\right\\}$ generated by Algorithm 2 converges to the unique NE of the
formulated game $\mathcal{G}$.
###### Proof:
See Appendix -D. ∎
###### Remark 2
In this paper, we use the following formula for the weights [13, Ch. 4]:
${w_{n,k}}=\left\\{\begin{array}[]{l}{\tau\mathord{\left/{\vphantom{\tau{\left[{{{\max}_{n}}\left|{{{\cal
D}_{n}}}\right|}\right]}}}\right.\kern-1.2pt}{\left[{{{\max}_{n}}\left|{{{\cal
D}_{n}}}\right|}\right]}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;n\neq k\\\
1-\left|{{{\cal
D}_{n}}}\right|{\tau\mathord{\left/{\vphantom{\tau{\left[{{{\max}_{n}}\left|{{{\cal
D}_{n}}}\right|}\right]}}}\right.\kern-1.2pt}{\left[{{{\max}_{n}}\left|{{{\cal
D}_{n}}}\right|}\right]}}\;\;\;{\rm{if}}\;n=k\\\ \end{array}\right.,$ (15)
where $\left|{{{\cal D}_{n}}}\right|$ denotes the cardinality of the set
${{{\cal D}_{n}}}$, and $0<\tau<1$ is used to measure the relative proportion
of the neighbors’ estimates in each consumer’s estimation of the average
energy consumption. It is straightforward to validate that the weights in (15)
satisfy the conditions in (14). Other choices of the weights can be found in
[19].
Although information exchanges are required between the consumers in Algorithm
2, the consumers only need to share their estimations of the average energy
consumption of all consumers instead of their exact energy consumption
profiles with their immediate neighbors. Thus, the developed algorithm can
avoid the consumers’ security and privacy concerns. $~{}~{}\square$
Note that although the central unit is not necessary for the developed
Algorithm 2, synchronization between the consumers and coordination in terms
of algorithm step sizes are still required, which are challenging in very
large networks. Motivated by this, we will develop a distributed asynchronous
algorithm in next section.
## VI Distributed Asynchronous Gossip-based Algorithm without A Central Unit
In this section, we develop a distributed asynchronous gossip-based algorithm
for computing the NE of the formulated game without the need of a central
unit. The consumers perform their estimations and updates in the same way as
in the Algorithm 2, but the updates occur asynchronously instead of
synchronously. The developed algorithm allows the consumers to use
uncoordinated step size values. More specifically, the consumers can choose
the step size based on their own information-update frequency. The graph model
for the connection topology of the consumers in Section V is also applicable
in this section.
1: Set $t=1$. Choose any feasible starting point
${\bf{q}}\left(1\right)=\left({{{\bf{q}}_{n}}\left(1\right)}\right)_{n=1}^{N}$
and set ${\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(1\right)={{{\bf{q}}_{n}}\left(1\right)}$ for every
$n\in{\mathcal{N}}$.
2: If a suitable termination criterion is satisfied: $\rm{STOP}$.
3: Each consumer $n\in\left\\{I^{t},J^{t}\right\\}$ counts the number of
updates that he/she has executed up to time $t$ inclusively (denoted by
$\epsilon_{n}(t)$), sets the step size as $\alpha_{n}(t)=1/\epsilon_{n}(t)$,
and updates his/her energy consumption profile and the estimated average
energy consumption of all consumers via executing
$\displaystyle\footnotesize\begin{split}&~{}~{}~{}~{}~{}~{}~{}{\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(t\right)=\frac{{\bf{\hat{q}}}_{I^{t}}^{{}_{\rm
A}}\left(t\right)+{\bf{\hat{q}}}_{J^{t}}^{{}_{\rm A}}\left(t\right)}{2},\\\
&~{}~{}~{}~{}~{}~{}~{}{{\bf{q}}_{n}}\left({t+1}\right)={\left[{{{\bf{q}}_{n}}\left(t\right)-\alpha_{n}\left(t\right){{\bf{F}}_{n}}\left({{{\bf{q}}_{n}}\left(t\right),N{\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(t\right)}\right)}\right]_{{{\mathcal{Q}}_{n}}}},\\\
&~{}~{}~{}~{}~{}~{}~{}{\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left({t+1}\right)={\bf{\hat{q}}}_{n}^{{}_{\rm
A}}\left(t\right)+{{\bf{q}}_{n}}\left({t+1}\right)-{{\bf{q}}_{n}}\left(t\right).\end{split}$
4: $t\leftarrow t+1$; go to $\rm{STEP}$ 2\.
Algorithm 3 : Asynchronous Gossip-based Algorithm
To allow for asynchronous updates, we adopt the gossip protocol [20] to model
the consumers’ exchange of their estimations for the average energy
consumption of all consumers. In this protocol, each consumer is assumed to
have a clock which ticks according to a Poisson process with rate 1. At a tick
of his/her clock, consumer $n$ contacts a randomly selected555Here, we
consider that each neighbor has an equal chance of being selected. neighbor
$k\in{\mathcal{D}}_{n}$ to exchange information. With reference to [20], the
consumers’ clocks processes can be equivalently modeled as a single virtual
clock that ticks according to a Poisson process with rate $N$. We assume that
only one consumer communicates with its neighbor at each tick of the virtual
clock and we use $Z^{t}$ to denote $t$th tick of the virtual Poisson process.
We discretize time so that the instant $t$ corresponds to the time-slot
$\left[Z^{t-1},Z^{t}\right)$. At each time $t$, every consumer $n$ has his/her
consumption profile ${\bf q}_{n}(t)$ and estimation of the average energy
consumption of all consumers ${\bf{\hat{q}}}_{n}^{{}_{\rm A}}\left(t\right)$.
Let $I^{t}\in{\mathcal{N}}$ denote the consumer whose local clock ticked at
time $t$. Note that $I^{t}$ is uniformly distributed in the set $\mathcal{N}$
since the Poisson clocks at each consumer are independent. Moreover, the
memoryless property of the Poisson arrival process ensures that the process
$\left\\{I^{t}\right\\}$ is independent and identically distributed. We use
$J^{t}$ to denote the consumer randomly contacted by the consumer $I^{t}$,
where $J^{t}$ is a neighbor of the consumer $I^{t}$, i.e.,
$J^{t}\in{\mathcal{D}}_{I^{t}}$. Then, these two consumers will exchange their
estimations of average energy consumption and update their own energy
profiles. The developed asynchronous gossip-based algorithm is formally
described in Algorithm 3. As can be seen from Algorithm 3, the consumers
perform the same updates as in the synchronous Algorithm 2, but only two
randomly selected consumers update their estimations of average energy
consumption and their own energy profiles at each iteration, while the other
consumers do not update.
For the convergence of Algorithm 3, we have the following proposition and the
proof follows from Appendixes -B--D, the adopted step sizes and [13, Ch. 4,
Prop. 12]:
###### Proposition 4
Assume that the condition in Proposition 1 holds and the undirected graph
${\mathcal{M}}\left({\mathcal{N}},{\mathcal{E}}\right)$ is connected. Then,
the sequence of the energy consumption profile $\\{{\bf{q}}(t)\\}$ generated
by Algorithm 3 converges to the unique NE of the game ${\mathcal{G}}$ almost
surely.
###### Remark 3
In this developed algorithm, synchronization is not required between the
consumers. Besides the asynchronous updates, the consumers are allowed to use
uncoordinated step sizes that are based on the frequency of the consumers’
updates. Specifically, consumer $n$ uses the step size
$\alpha_{n}(t)=\frac{1}{\epsilon_{n}(t)},~{}n\in\left\\{I^{t},J^{t}\right\\},$
(16)
at the $t$th iteration, where ${\epsilon_{n}(t)}$ denotes the numbers of
updates that consumer $n$ has performed up to time $t$ inclusively. In
addition, analogous to Algorithm 2, no private information is required to
exchange between the consumers in Algorithm 3.
It is worth mentioning that the pairwise gossip protocol (i.e, only a random
pair of consumers is chosen to update at each iteration) is adopted for
simplicity in this paper. The developed algorithm can be extended to the
general setup where a random subset of consumers (more than one pair) exchange
their estimations and update their energy profiles at each iteration. This
will be considered in our future work. $\square$
## VII Numerical Results
In this section, we present some numerical results to validate the above
theoretical analysis and illustrate the performance of the developed
algorithms.
In the following simulation results, we consider the residential scenario
consisting of $N=50$ consumers, where the consumers determine their energy
consumption for the following whole day, which starts from 8 AM. Each time
slot is set as one hour, i.e., $H=24$ and the first time slot corresponds to
the hour between 8 AM and 9 AM. In Fig. 1, we provide a typical energy
consumption interval of a residential consumer [21, Figs. 2.5-2.7],[22].
Considering that different consumers may have different energy consumption
interval in practice, the ‘Low limit’ and ‘Upper limit’ of each consumer in
each time slot are formed by respectively adding a random real number to the
corresponding value in Fig. 1. Then, the initial energy consumption of a
certain consumer in each time slot, $q_{n}^{h}(1)$, is uniformly chosen
between his/her corresponding ‘Low limit’ and ‘Upper limit’. The numerical
results show that the selected consumption parameters yield the total energy
consumption of every consumer in the order of 10 kWh to 30 kWh, which is
representative of a residential consumer [22].
Figure 1: The typical energy consumption interval of a residential consumer.
Figure 2: The connection topology for the consumers used in Algorithm 2 and 3.
The consumers are denoted by the circles and their connections are represented
by the solid lines.
According to Fig. 1, we classify the whole time horizon into three segments:
off-peak hours (12 AM to 7 AM), mid-peak hours (7 AM to 4 PM and 10 PM to 12
AM), and on-peak hours (4 PM to 10 PM). We also set $a_{h}$ equal to 0.003,
0.004 and 0.005 for the off-peak, mid-peak and on-peak hours, respectively,
and parameters $b_{h}$ and $c_{h}$ are set equal to 1.2 and 0 for $\forall
h\in\mathcal{H}$, respectively. In the considered DSM scenario, the value
$q_{n}^{h,\min}$ for the $n$th consumer is set to his/her ‘Low limit’ of the
$h$th time slot. In addition, the values of $q_{n}^{h,\max}$ for mid-peak and
on-peak hours are set to his/her maximum value of the ‘Upper limit’, while the
values of $q_{n}^{h,\max}$ for the off-peak hours are uniformly chosen from
the interval $[0.4,0.6]$. The values of $E_{n}$ are chosen to be equal to the
sum of the consumers’ initial energy consumption profiles before applying the
DSM program. Moreover, the parameters for the algorithms are chosen666We refer
the readers to [18] for more discussion on the choice of algorithm parameters.
as follows: $\gamma\left(t\right)={t^{-0.51}}$ and $\theta=0.2$ for Algorithm
1, and $\alpha\left(t\right)={t^{-0.51}}$ and $\tau=0.5$ for Algorithm 2.
Finally, a randomly generated connection structure of the consumers for
Algorithm 2 and 3 are given in Fig. 2, where two consumers are directly linked
means that they are immediate neighbors, who can exchange information in the
iterations of the algorithms.
(a) Convergence of Algorithms 1 and 2
(b) Convergence of Algorithm 3
Figure 3: The convergence of the developed algorithms in terms of the
consumers’ total energy cost.
Figure 4: The aggregated energy consumption profiles of all consumers before
and after the DSM program.
Fig. 3 plots the total energy cost for three different consumers versus the
number of iterations of the developed algorithms. It can be observed from Fig.
3 (a) that both Algorithm 1 and Algorithm 2 converge to the NE of the
formulated game very quickly. Specifically, the energy cost of each consumer
has already achieved a relatively stable state after the first 10 iterations,
which verifies the validness of both Proposition 2 and Proposition 3, as well
as displaying the high efficiency of the developed algorithms. Fig. 3 (b) is
plotted to illustrate the convergence performance of the Algorithm 3. As can
be observed from Fig. 3 (b) that the total energy cost of different consumers
approach to coincide with that obtained by Algorithm 1 and 2 after $200$
iterations. This validates the results given in Proposition 4. Note that due
to space limitations, we only show results in Fig. 3 for three randomly
selected consumers, although it can be shown that similar results also hold
for the other consumers and a wide range of settings with different
parameters.
In Fig 4, we compare the aggregated energy consumption profiles of all
consumers corresponding to the situations before and after DSM program. We
clearly observe from Fig. 4 that the proposed DSM scheme effectively
encourages the consumers to shift their energy consumption from peak to non-
peak hours. We also investigate the peak-to-average ratio (PAR) of the
aggregated load defined as
${\rm{PAR}}=\frac{{H{{\max}_{h}}q_{\Sigma}^{h}}}{{\sum\nolimits_{h=1}^{H}{q_{\Sigma}^{h}}}}.$
(17)
The simulation results show that the PAR decreases from $2.3189$ to $1.6161$
(i.e., $30.31$% less) before and after the DSM program. This will result in a
generally flattened demand profile, which will not only reduce the consumers’
energy cost but also benefit the efficiency of the whole power grid.
To show that the adopted billing method can fairly charge the consumers, we
plot the energy consumption profiles of consumer $43$ and consumer $50$ after
applying the proposed DSM program in Fig. 5. Their total daily energy
requirements are $E_{43}=20.63$ (kWh) and $E_{50}=19.99$ (kWh), respectively.
If the total load billing method [4] was used, consumer $43$ would be charged
more than consumer $50$ since $E_{43}>E_{50}$. However, as can be observed
from Fig. 5, the on-peak energy usage of consumer $50$ is larger than that of
consumer $43$. This can also be reflected by the PAR values of these two
consumers, i.e., ${\rm{PAR}}_{50}=1.9438$ and ${\rm{PAR}}_{43}=1.7863$. Thus,
it may be not fair to charge consumer $43$ more than consumer $50$ simply
because he/she consumes more energy totally. In contrast, our numerical
results show that consumer $43$ and consumer $50$ will finally be charged
${\mathcal{B}}_{43}=10.56$ and ${\mathcal{B}}_{50}=10.66$ (i.e.,
${\mathcal{B}}_{43}<{\mathcal{B}}_{50}$) after the proposed DSM program. This
result is understandable since the adopted billing method considers not only
how much the consumers consume the energy totally but also when the consumers
use the energy. By this example, we show that the adopted billing approach can
charge the consumers more fairly, thereby motivating the consumers to
participate in the DSM program.
Fig. 6 compares the total energy cost of all consumers for three different
cases with different number of consumers. As expected, it can be observed from
Fig. 6 that the total energy cost is significantly reduced after the proposed
DSM program. We also compare the performance of the proposed game-theoretical
DSM program with the optimal one obtained by solving the following social
welfare optimization problem:
$\begin{array}[]{l}\mathop{\min}\limits_{\left\\{{{{\bf{q}}_{1}},\ldots,{{\bf{q}}_{N}}}\right\\}}\;\sum\nolimits_{n=1}^{N}{{{\mathcal{B}}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{-n}}}\right)}\\\
\;\;\;\;\;\;\;{\rm{s}}{\rm{.t}}{\rm{.}}\;\;{{\bf{q}}_{n}}\in{{\mathcal{Q}}_{n}},\;\forall
n\in{\mathcal{N}}\\\ \end{array}.$ (18)
From Fig. 6, we can observe that the total energy cost achieved by the
proposed DSM program is almost the same with the optimal one777The theoretical
analysis of this observation (i.e., the price of anarchy analysis for the
formulated game) is out of the scope of this paper and will be considered in
future work.. Thus, we can claim that the proposed DSM framework qualifies as
a practically appealing candidate for the DSM of future smart grid.
Figure 5: The energy consumption profiles of consumer 43 and consumer 50 after
the DSM program.
Figure 6: Comparison of total energy cost of all consumers before the DSM
program, after the DSM program, and obtained by social welfare optimization.
## VIII Conclusions
In this paper, we formulated an aggregative game for the demand side
management program based on energy consumption scheduling and instantaneous
load billing, where the consumers are selfish and compete to minimize their
individual energy cost. The sufficient condition for the existence and
uniqueness of Nash equilibrium (NE) of the formulated game was subsequently
given and proved. Based on the formulation of the aggregative game, we
developed three distributed algorithms to achieve the NE of the formulated
game, corresponding to the two scenarios where the consumers can or cannot
access the real-time information of the aggregated load. In these algorithms,
the choice for the algorithm parameters do not depend on the arguments of the
system and no private information is required to exchange between consumers.
Numerical results showed that the developed algorithms can quickly converge to
the NE of the formulated game and efficiently convince the consumers to shift
their on-peak consumption, which are beneficial to both the consumers and the
whole grid.
### -A Proof of Lemma 1
It is evident that the statements in the first part of Lemma 1 holds. Hence,
we only need to prove the convexity of ${{\cal
B}_{n}}\left({\cdot\;,\;\cdot\;+\sum\nolimits_{m=1,m\neq
n}^{N}{{{\bf{q}}_{m}}}}\right)$ in ${{\bf{q}}_{n}}$ for every fixed
${{\bf{q}}_{-n}}$. This can be achieved by proving that the Hessian of ${{\cal
B}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)$ is positive
semidefinite [23]. After some algebraic manipulation, we have
$\begin{split}\nabla_{{{\bf{q}}_{n}}}^{2}{{\cal
B}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)={\rm{diag}}\left\\{{\left[{q_{n}^{h}{p_{h}}^{\prime\prime}\left({{q_{\Sigma}^{h}}}\right)+2{p_{h}}^{\prime}\left({{q_{\Sigma}^{h}}}\right)}\right]_{h=1}^{H}}\right\\}.\end{split}$
(19)
Since (19) is a diagonal matrix with all diagonal elements being positive, the
Hessian matrix of
${{\mathcal{B}}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{-n}}}\right)$ is positive
semidefinite. This completes the proof.
### -B Proof of Proposition 1
Based on Lemma 2, the proof of this proposition follows if we can show that
the formulated VI$\left({\mathcal{Q}},{\bf F}\right)$ in Lemma 2 only
possesses one solution. With reference to [8, Thm. 4.1][24, Thm. 2.3], the
VI$\left({\mathcal{Q}},{\bf F}\right)$ admits a unique solution if the mapping
${\bf{F}}\left({\bf{q}}\right)$ is strictly monotone888A mapping $\bf
F\left({\bf{x}}\right)$: ${\mathcal{K}}\ni{\bf{x}}\rightarrow\mathbb{R}^{N}$
is said to be _strictly monotone_ on ${\mathcal{K}}$ if
$\left({{\bf{x}}-{\bf{y}}}\right)^{T}{\left({{\bf{F}}\left({\bf{x}}\right)-{\bf{F}}\left({\bf{y}}\right)}\right)}>0,~{}\forall\bf
x,\bf y\in{\mathcal{K}}\;\rm{and}\;\bf x\neq\bf y$. over $\mathcal{Q}$ since
the feasible set $\mathcal{Q}$ is compact and convex.
To prove the strict monotonicity of the mapping
${\bf{F}}\left({\bf{q}}\right)$, it suffices to show that
$\begin{split}&\sum\nolimits_{h=1}^{H}{\sum\nolimits_{n=1}^{N}{\left[{\left({q_{n}^{h}-s_{n}^{h}}\right)\left({{\nabla_{q_{n}^{h}}}{{\mathcal{B}}_{n}}\left({\bf{q}}\right)-{\nabla_{s_{n}^{h}}}{{\mathcal{B}}_{n}}\left({\bf{s}}\right)}\right)}\right]}}>0,\\\
\end{split}$ (20)
for any
${\bf{q}}=\left({{{\bf{q}}_{n}}}\right)_{n=1}^{N},\;{\bf{s}}=\left({{{\bf{s}}_{n}}}\right)_{n=1}^{N}\in{\cal
Q}$.
Let ${{\bf{l}}^{h}}=\left({q_{1}^{h},\ldots,q_{N}^{h}}\right)^{T}$ and
${{\bf{j}}^{h}}=\left({s_{1}^{h},\ldots,s_{N}^{h}}\right)^{T}$, then we can be
re-write (20) as
$\begin{split}&\sum\nolimits_{h=1}^{H}{\left[{\left({{{\bf{l}}^{h}}-{{\bf{j}}^{h}}}\right)^{T}\left({{\nabla_{{{\bf{l}}^{h}}}}{\mathcal{B}}_{n}^{h}\left({{{\bf{l}}^{h}}}\right)-{\nabla_{{{\bf{j}}^{h}}}}{\mathcal{B}}_{n}^{h}\left({{{\bf{j}}^{h}}}\right)}\right)}\right]}>0,\end{split}$
(21)
where
${\mathcal{B}}_{n}^{h}\left({{{\bf{l}}^{h}}}\right)={p_{h}}\left({q_{\Sigma}^{h}}\right)q_{n}^{h}$,
and ${\nabla_{{{\bf{l}}^{h}}}}{\cal
B}_{n}^{h}\left({{{\bf{l}}^{h}}}\right)={\left({{\nabla_{q_{1}^{h}}}{\cal
B}_{1}^{h}\left({{{\bf{l}}^{h}}}\right),{\nabla_{q_{2}^{h}}}{\cal
B}_{2}^{h}\left({{{\bf{l}}^{h}}}\right),\ldots,{\nabla_{q_{N}^{h}}}{\cal
B}_{n}^{h}\left({{{\bf{l}}^{h}}}\right)}\right)^{T}}$.
We observe that a sufficient condition for (21) to hold is if
$\left({{{\bf{l}}^{h}}-{{\bf{j}}^{h}}}\right)^{T}\left[{{\bf
g}_{h}\left({{{\bf{l}}^{h}}}\right)-{\bf
g}_{h}\left({{{\bf{j}}^{h}}}\right)}\right]>0,\;\forall h\in{\mathcal{H}},$
(22)
where
${{\bf{g}}_{h}}\left({{{\bf{l}}^{h}}}\right)={\nabla_{{{\bf{l}}^{h}}}}{\cal
B}_{n}^{h}\left({{{\bf{l}}^{h}}}\right)$, which is defined for the sake of
notation.
Recall the definition of a strictly monotone mapping, we can obtain that (22)
holds if the mapping ${{\bf{g}}_{h}}\left({{{\bf{l}}^{h}}}\right)$ is strictly
monotone. With reference to [8, Eq. (4.8)], the condition in (22) can be shown
to be equivalent to proving the Jacobian matrix of
${{\bf{g}}_{h}}\left({{{\bf{l}}^{h}}}\right)$ is positive definite. Since the
transpose operation does not change the definite property of a given matrix,
what we only to prove is that the transpose of the Jacobian matrix of
${{\bf{g}}_{h}}\left({{{\bf{l}}^{h}}}\right)$, denoted by
${{\bf{G}}_{h}}\left({{{\bf{l}}^{h}}}\right)={\nabla_{{{\bf{l}}^{h}}}}{{\bf{g}}_{h}}\left({{{\bf{l}}^{h}}}\right)$,
is positive definite.
To proceed, we have the $(n,m)$th entry of
${{\bf{G}}_{h}}\left({{{\bf{l}}^{h}}}\right)$ after some algebraic
manipulation given by
${\left[{{{\bf{G}}_{h}}\left({{{\bf{l}}^{h}}}\right)}\right]_{n,m}}=\left\\{\begin{array}[]{l}\sigma_{h}\left[{2{q_{\Sigma}^{h}}+\left({{b_{h}}-1}\right)q_{n}^{h}}\right],\;{\mathop{\rm
if}\nolimits}\;n=m\\\
\sigma_{h}\left[{{q_{\Sigma}^{h}}+\left({{b_{h}}-1}\right)q_{n}^{h}}\right],\;\;\;{\mathop{\rm
if}\nolimits}\;n\neq m\end{array}\right.$ (23)
where
$\sigma_{h}={a_{h}}{b_{h}}{\left({{q_{\Sigma}^{h}}}\right)^{{b_{h}}-2}}$.
Since the matrix ${{\bf{G}}_{h}}\left({{{\bf{l}}^{h}}}\right)$ may not be
symmetric, we can prove its positive definiteness by showing that the
symmetric matrix
${{\bf{G}}_{h}}\left({{{\bf{l}}^{h}}}\right)+{{\bf{G}}_{h}}{\left({{{\bf{l}}^{h}}}\right)^{T}}=\sigma_{h}\left({\underbrace{{{\bf{z}}^{h}}{\bf{1}}^{T}+{{\bf{1}}}\left({{\bf{z}}^{h}}\right)^{T}}_{{{\mathcal{T}}_{b}}}+2{q_{\Sigma}^{h}}{\bf{I}}}\right),$
(24)
is positive definite [25], where
${{\bf{z}}^{h}}={q_{\Sigma}^{h}}{\bf{1}}+\left({b_{h}-1}\right){{\bf{l}}^{h}}$,
${\bf{1}}$ is a $N\times 1$ vector with every element of 1. This is equivalent
to showing that the smallest eigenvalue of this matrix is positive.
After some appropriate calculations [25], the two non-zero eigenvalues of the
matrix ${{\mathcal{T}}_{b}}$ in (24) are given by
$\begin{split}&\eta_{{{\mathcal{T}}_{b}}}^{1}=\left({N+1{\rm{+}}{b_{h}}}\right){q_{\Sigma}^{h}}+\sqrt{N\left({{\bf{z}}^{h}}\right)^{T}{{\bf{z}}^{h}}}-2{q_{\Sigma}^{h}},\\\
&\eta_{{{\mathcal{T}}_{b}}}^{2}=\left({N+1{\rm{+}}{b_{h}}}\right){q_{\Sigma}^{h}}-\sqrt{N\left({{\bf{z}}^{h}}\right)^{T}{{\bf{z}}^{h}}}-2{q_{\Sigma}^{h}}.\end{split}$
(25)
Note that $N\geq 2$ is implicit here.
Since $\eta_{{{\mathcal{T}}_{b}}}^{1}\geq\eta_{{{\mathcal{T}}_{b}}}^{2}$, the
smallest eigenvalue of the matrix
${{\bf{G}}_{h}}\left({{{\bf{l}}^{h}}}\right)+{{\bf{G}}_{h}}{\left({{{\bf{l}}^{h}}}\right)^{T}}$
can be expressed as
${\eta_{\min}}=\sigma_{h}\min\left({\left({N+1{\rm{+}}{b_{h}}}\right){q_{\Sigma}^{h}}-\sqrt{N\left({{\bf{z}}^{h}}\right)^{T}{{\bf{z}}^{h}}},2{q_{\Sigma}^{h}}}\right),$
(26)
where the second $2{q_{\Sigma}^{h}}$ term in the $\min$ function arises
because there are $N-2$ zero eigenvalues in the matrix ${{\mathcal{T}}_{b}}$.
To further simplify (26), we have
$\begin{split}\left({{\bf{z}}^{h}}\right)^{T}{{\bf{z}}^{h}}&={\sum\nolimits_{n=1}^{N}{\left({{q_{\Sigma}^{h}}+\left({{b_{h}}-1}\right)q_{n}^{h}}\right)}^{2}}\\\
&\leq\left({N-1+{{\left({{b_{h}}}\right)}^{2}}}\right){\left({{q_{\Sigma}^{h}}}\right)^{2}},\end{split}$
(27)
Substituting (27) into (26), we obtain
${\eta_{\min}}\geq\sigma_{h}\min\left({\kappa_{h}{q_{\Sigma}^{h}},2{q_{\Sigma}^{h}}}\right),$
(28)
where
$\kappa_{h}=\left({N+1{\rm{+}}{b_{h}}}\right)-\sqrt{N\left({N-1+{{\left({{b_{h}}}\right)}^{2}}}\right)}$.
Since $\sigma_{h}>0$, we observe from the right hand side of (28) that
${\eta_{\min}}>0$ if $\kappa_{h}>0$, or equivalently
${\left({N+1{\rm{+}}{b_{h}}}\right)^{2}}>N\left({N-1+{{\left({{b_{h}}}\right)}^{2}}}\right)$,
which we can re-written as
$\left({1{\rm{+}}{b_{h}}}\right)\left({\left({N-1}\right){b_{h}}-\left({3N+1}\right)}\right)<0$.
Thus, a sufficient condition for $\kappa_{h}>0$ is
${b_{h}}<3+4/\left({N-1}\right).$ (29)
This completes the proof.
### -C Proof of Proposition 2
The proof for the convergence of a general iterative proximal-point algorithm
and the corresponding necessary conditions were presented in [18, Sec. 3].
Therefore, we only need to prove that the formulated NEP
${\mathcal{G}}_{\boldsymbol{\lambda}}$ meets all the required conditions
listed in [18, Assumption (A3)].
Firstly, it is evident that the set $\mathcal{Q}$ is compact, and that
$\left\|{\bf{q}}\right\|_{2}$ and $\left\|{\bf F}\left(\bf
q\right)\right\|_{2}$ are both bounded for $\forall{\bf{q}}\in\mathcal{Q}$.
Secondly, as we have proved in Appendix -B, the mapping ${\bf F}\left(\bf
q\right)$ is strictly monotone on $\bf q$ when the price parameter ${b_{h}}$
satisfies ${b_{h}}<3+4/\left({N-1}\right)$ for $\forall h\in{\mathcal{H}}$.
Therefore, we only need to prove that ${\bf F}\left(\bf q\right)$ is Lipschitz
continuous over ${\mathcal{Q}}$, i.e., show that there exists a real constant
${c_{1}^{\rm{lip}}}>0$ such that, for all $\bf{q,~{}s}\in{\mathcal{Q}}$,
${\left\|{{\bf{F}}\left({\bf{q}}\right)-{\bf{F}}\left({\bf{s}}\right)}\right\|_{2}}\leq{c_{1}^{\rm{lip}}}{\left\|{{\bf{q}}-{\bf{s}}}\right\|_{2}}\;.$
(30)
According to the definition of Euclidean norm, (30) can be shown to hold if
each element of the function ${\bf{F}}\left({\bf{q}}\right)$, denoted by
$f_{n}^{h}\left({\bf{q}}\right)=d{\mathcal{B}}_{n}\left({\bf{q}}\right)/dq_{n}^{h}$,
is Lipschitz continuous in $\bf q$, i.e., for any $n\in{\mathcal{N}}$ and any
$h\in{\mathcal{H}}$, there exists a real constant ${c_{n,h}^{\rm{lip}}}>0$
such that, for all $\bf{q,~{}s}\in{\mathcal{Q}}$,
$\left|{f_{n}^{h}\left({\bf{q}}\right)-f_{n}^{h}\left({\bf{s}}\right)}\right|\leq
c_{n,h}^{{\rm{lip}}}{\left\|{{\bf{q}}-{\bf{s}}}\right\|_{2}}.$ (31)
We now proceed to prove (31) holds. After some algebraic manipulation, we have
$f_{n}^{h}\left({\bf{q}}\right)={p_{h}}^{\prime}\left({q_{\Sigma}^{h}}\right)q_{n}^{h}+{p_{h}}\left({q_{\Sigma}^{h}}\right)=f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right).$
(32)
Substituting (32) into the left-hand side of (31), we have
$\begin{split}\left|{f_{n}^{h}\left({\bf{q}}\right)-f_{n}^{h}\left({\bf{s}}\right)}\right|=&\left|{f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right)-f_{n}^{h}\left({s_{n}^{h},s_{\Sigma}^{h}}\right)}\right|\\\
\leq&\left|{f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right)-f_{n}^{h}\left({s_{n}^{h},q_{\Sigma}^{h}}\right)}\right|+\\\
&\left|{f_{n}^{h}\left({s_{n}^{h},q_{\Sigma}^{h}}\right)-f_{n}^{h}\left({s_{n}^{h},s_{\Sigma}^{h}}\right)}\right|,\end{split}$
(33)
where $s_{\Sigma}^{h}=\sum\nolimits_{n=1}^{N}{s_{n}^{h}}$ and the inequality
follows according to the triangular inequality.
Now by recalling that
${p_{h}}\left({{q_{\Sigma}^{h}}}\right)=a_{h}{\left(q_{\Sigma}^{h}\right)^{b_{h}}}+c_{h}$,
we can rewrite the function
${f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right)}$ as
$\begin{split}f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right)={a_{h}}{\left({q_{\Sigma}^{h}}\right)^{{b_{h}}-1}}\left({{b_{h}}q_{n}^{h}+q_{\Sigma}^{h}}\right)+{c_{h}}.\end{split}$
(34)
With reference to [26, Ch. 12], it is straightforward to deduce that the
function ${f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right)}$ in (34) is
Lipschitz continuous in $q_{n}^{h}$ for a fixed $q_{\Sigma}^{h}$ and is
Lipschitz continuous in $q_{\Sigma}^{h}$ for a fixed $q_{n}^{h}$. That is
there exists two real constant $c_{n,h,1}^{\rm{lip}},c_{n,h,2}^{\rm{lip}}>0$
such that for any $q_{n}^{h}$ and $s_{n}^{h}$,
$\begin{split}&\left|{f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right)-f_{n}^{h}\left({s_{n}^{h},q_{\Sigma}^{h}}\right)}\right|\leq
c_{n,h,1}^{\rm{lip}}\left|{q_{n}^{h}-s_{n}^{h}}\right|\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=c_{n,h,1}^{\rm{lip}}\sqrt{{{\left({q_{n}^{h}-s_{n}^{h}}\right)}^{2}}}\leq
c_{n,h,1}^{\rm{lip}}{\left\|{{\bf{q}}-{\bf{s}}}\right\|_{2}},\end{split}$ (35)
and for any $q_{\Sigma}^{h}$ and $s_{\Sigma}^{h}$,
$\begin{split}&\left|{f_{n}^{h}\left({s_{n}^{h},q_{\Sigma}^{h}}\right)-f_{n}^{h}\left({s_{n}^{h},s_{\Sigma}^{h}}\right)}\right|\leq
c_{n,h,2}^{\rm{lip}}\left|{q_{\Sigma}^{h}-s_{\Sigma}^{h}}\right|\\\ &\leq
c_{n,h,2}^{\rm{lip}}\sqrt{\sum\nolimits_{n=1}^{N}{{{\left({q_{n}^{h}-s_{n}^{h}}\right)}^{2}}}}\leq
c_{n,h,2}^{\rm{lip}}{\left\|{{\bf{q}}-{\bf{s}}}\right\|_{2}}.\end{split}$ (36)
By substituting (35) and (36) into (33), we deduce that it can always find a
real constant ${c_{n,h}^{\rm{lip}}}>0$ such that (31) holds for all
$\bf{q,~{}s}\in{\mathcal{Q}}$. This completes the proof.
### -D Proof of Proposition 3
The sufficient conditions (i.e., [13, Ch. 4, Assumptions 8-13]) and the
rigorous proofs for the convergence of a general distributed agreement-based
algorithm have been provided in [13, Ch. 4.1]. Hence, the proof of the
Proposition 3 follows if the formulated aggregative game meets all the
required conditions listed in [13, Ch. 4, Assumptions 8-13]. Based on the
analysis in Lemma 1 and Proposition 1, the adopted structure of the weights
and the assumption on the step-size, we can claim that the considered
aggregative game has already satisfied all the conditions except the one
stated in [13, Ch. 4, Assumptions 10]. Therefore, the remaining task is to
prove that the formulated game also meets the condition in [13, Ch. 4,
Assumptions 10]. More specifically, we need to show that each mapping
${{\bf{F}}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)$ is Lipschitz
continuous in ${{\bf{q}}_{\Sigma}}$ for every fixed ${\bf
q}_{n}\in{\mathcal{Q}}_{n}$.
Analogous to the analysis in Appendix -C,
${{\bf{F}}_{n}}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)$ is Lipschitz
continuous in ${{\bf{q}}_{\Sigma}}$ if each element of this function,
$f_{n}^{h}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)$, is Lipschitz
continuous in ${{\bf{q}}_{\Sigma}}$. The validity for the Lipschitz continuity
of $f_{n}^{h}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)$ follows since
$f_{n}^{h}\left({{{\bf{q}}_{n}},{{\bf{q}}_{\Sigma}}}\right)=f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right)$
and ${f_{n}^{h}\left({q_{n}^{h},q_{\Sigma}^{h}}\right)}$ is Lipschitz
continuous in $q_{\Sigma}^{h}$ for a fixed $q_{n}^{h}$ (cf. Appendix -C). This
completes the proof.
## Appendix A Acknowledgement
The authors would like to thank the anonymous reviewers for their valuable
comments and suggestions, which improved the quality of the paper. The authors
also thank Dr. Gregor Verbic, Prof. David Hill, Dr. Archie Chapman and Dr.
Peng Wang for their helpful discussion.
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* [23] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge University Press, 2004.
* [24] R. Cominetti, F. Facchinei, and J. Lasserre, “Computation of generalized nash equilibria: Recent advancements,” in _Modern Optimization Modelling Techniques_ , ser. Advanced Courses in Mathematics - CRM Barcelona. Springer Basel, 2012, pp. 131–204.
* [25] E. Altman, T. Basar, T. Jimenez, and N. Shimkin, “Competitive routing in networks with polynomial costs,” _IEEE Trans. Automatic Control_ , vol. 47, no. 1, pp. 92–96, 2002.
* [26] K. Eriksson, D. Estep, and C. Johnson, _Applied Mathematics: Body and Soul (Volume I and III)_. New York: Springer-Verlag, 2004.
| He (Henry) Chen (S’10) received his B.E. degree in Communication
Engineering and M.E. degree (research) in Communication and Information System
both from Shandong University, China, in 2008 and 2011, respectively. He was
awarded the Outstanding Bachelor Thesis of Shandong University in 2008 and the
Outstanding Master Thesis of Shandong Province in 2012. He is currently
working towards the Ph.D. degree in electrical engineering at the University
of Sydney, Sydney, Australia. His current research interests include demand
side management of smart grid, wireless communications powered by wireless
energy transfer and the applications of game theory, optimization theory, as
well as varational inequality theory in these areas. His research is supported
by International Postgraduate Research Scholarship (IPRS), Australian
Postgraduate Award (APA), and Norman I Price Supplementary Scholarship.
---|---
| Yonghui Li (M’04-SM’09) received his Ph.D. degree in November 2002 from
Beijing University of Aeronautics and Astronautics. From 1999-2003, he was
affiliated with Linkair Communication Inc, where he held a position of project
manager with responsibility for the design of physical layer solutions for the
LAS-CDMA system. Since 2003, he has been with the Centre of Excellence in
Telecommunications, the University of Sydney, Australia. He is now an
Associate Professor in School of Electrical and Information Engineering,
University of Sydney. He was the Australian Queen Elizabeth II Fellow and is
currently the Australian Future Fellow. His current research interests are in
the area of wireless communications, with a particular focus on machine-to-
machine communications, cooperative communications, coding techniques and
wireless sensor networks. He holds a number of patents granted and pending in
these fields. He is an executive editor for European Transactions on
Telecommunications (ETT). He has also been involved in the technical committee
of several international conferences, such as ICC, Globecom, etc.
---|---
| Raymond H. Y. Louie (S’06-M’10) received the combined B.E. degree in
electrical engineering and B.Sc. degree in computer science from the
University of New South Wales, Sydney, Australia, in 2006 and the Ph.D. degree
in electrical engineering from the University of Sydney, Australia, in 2010.
He was then an ARC Australian Postdoctoral Fellow at the University of Sydney,
and is now a Visiting Assistant Professor at the Hong Kong University of
Science and Technology. His research interests include biomedical engineering,
cognitive radio, ad hoc networks, MIMO systems, cooperative communications,
network coding, and multivariate statistical theory. Dr. Louie was awarded a
Best Paper Award at IEEE Globecom 2010.
---|---
| Branka Vucetic (SM’00-F’03) currently holds the Peter Nicol Russel Chair of
Telecommunications Engineering at the University of Sydney and serves the
Director of Centre of Excellence in Telecommunications. She is an
internationally recognized expert in wireless communications and coding. She
has published more than three hundred research papers and co-authored four
books in telecommunications and coding theory. Prof. Vucetic is an IEEE
Fellow. Her most significant research contributions have been in the field of
channel coding and its applications in wireless communications. The research
of Prof. Vucetic has involved collaborations with industry and government
organisations in Australia and several other countries.
---|---
|
arxiv-papers
| 2013-06-14T12:45:36 |
2024-09-04T02:49:46.496307
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "He Chen, Yonghui Li, Raymond H. Y. Louie, and Branka Vucetic",
"submitter": "He Chen",
"url": "https://arxiv.org/abs/1306.3383"
}
|
1306.3397
|
# Asymptotic formula for the tail of the maximum of smooth Gaussian fields on
non locally convex sets
Jean-Marc Azaïs and Viet-Hung Pham
Institut de Mathématiques de Toulouse
Université Paul Sabatier (Toulouse III), France
###### Abstract
In this paper we consider the distribution of the maximum of a Gaussian field
defined on non locally convex sets. Adler and Taylor or Azaïs and Wschebor
give the expansions in the locally convex case. The present paper generalizes
their results to the non locally convex case by giving a full expansion in
dimension 2 and some generalizations in higher dimension. For a given class of
sets, a Steiner formula is established and the correspondence between this
formula and the tail of the maximum is proved. The main tool is a recent
result of Azaïs and Wschebor that shows that under some conditions the
excursion set is close to a ball with a random radius. Examples are given in
dimension 2 and higher.
Key-words: Stochastic processes, Gaussian fields, Rice formula, distribution
of the maximum, non locally convex indexed set.
Classifications: 60G15, 60G60, 60G70.
## 1 Introduction
Let $\mathcal{X}=\\{X(t):t\in S\subset{\mathbb{R}}^{n}\\}$ be a random field
with real values and let $M_{S}$ be its maximum (or supremum) on $S$.
Computing the distribution of the maximum is a very important issue from the
theoretical point of view and also has a great impact on applications,
especially in spatial statistics. This problem has therefore received a great
deal of attention from many authors.
However an exact result is known only in very few cases, (see Azaïs and
Wschebor [5]). In other cases, the only available results are asymptotic
expansions or bounds mainly in the case of stationary Gaussian random fields.
One of the most well-known and quite general methods is the "double-sum
method", first proposed by Pickands [11] and extended by Piterbarg [12], [13].
The main idea of this method is to use the inclusion-exclusion principle and
the Bonferroni inequality after dividing the parameter set into suitable
smaller subsets. It was first proposed in dimension 1 : $n=1$ (in this case we
use the classical terminology of "random processes" instead of "random
field"). More precisely, for some particular processes, i.e., the "$\alpha$
processes", Pickands proposed an equivalent for the tail of the maximum.
However, the result depends on some unknown constants,referred to as Pickands’
constants and just gives an equivalent.
Another method is the "tube method" proposed by Sun [16]. She observed that if
the Karhunen-Loève expansion of the field is finite in the sense that there
exist a finite number of random variables
$\xi_{1},\ldots,\xi_{k}\overset{i.i.d}{\sim}\mathcal{N}(0,1)$ such that at
every point $t$ in the parameter set, the value of the field at this point
$X(t)$ can be expressed as
$X(t)=a_{t}^{1}\xi_{1}+\ldots+a_{t}^{k}\xi_{k},$
where the vector $(a_{t}^{1},\ldots,a_{t}^{k})$ has unit norm since
$\textrm{Var}(X(t))=1$, then the original parameter set can be transformed
into a subset of the unit sphere $\mathcal{S}^{k-1}$. She then used Weyl’s
tube formula to compute the polynomial expansion of the volume of the tube
around a subset of the unit sphere and derived the asymptotic formula of the
tail of the distribution of the maximum from this expansion. She is the first
one who realizes the strong connection between the geometric functionals of
the parameter set (the coefficients of the polynomial expansion) and the tail
of the distribution. When the Karhunen-Loève expansion is not finite, she uses
a truncation argument to derive an asymptotic formula with two terms. Later
on, this method was extended by Takemura and Akimichi [17], [18].
In the 1940s, in his pioneering work, Rice [14] considered a stationary
process $\mathcal{X}$ with $\mathcal{C}^{1}$ paths defined on the compact
interval $[0,T]$. He observed that for every level $u$:
$\begin{array}[]{rl}\displaystyle\mathbb{P}\left(\underset{t\in[0,T]}{\max}X(t)\geq
u\right)&\displaystyle\leq\mathbb{P}(X(0)\geq u)+\mathbb{P}\left(\exists
t\in[0,T]:\,X(t)=u,X^{\prime}(t)\geq 0\right)\\\
&\displaystyle\leq\mathbb{P}(X(0)\geq
u)+\mathbb{E}\left(\textrm{card}\left\\{t\in[0,T]:\,X(t)=u,X^{\prime}(t)\geq
0\right\\}\right),\\\ \end{array}$
where the last expectation can be evaluated by the famous Rice-Kac formula.
This upper bound was later proved to be sharp by Piterbarg [15]. This Rice-Kac
formula is the starting point of the following methods dealing with the random
fields: the "Rice method" by Azaïs and Delmas [3], [8], the "direct method" by
Azaïs and Wschebor [5] and the "Euler characteristic method" by Adler and
Taylor [1]. These methods use a multidimensional Rice-Kac formula :
Generalized Rice formula (Azaïs and Wschebor) or Metatheorem (Adler and
Taylor).
In the direct method, Azaïs and Wschebor used some results from the random
matrix theory to compute the expectation of the absolute value of the
determinant of the Hessian that appears in the Rice formula. They obtained an
upper bound for the tail of the distribution depending on some geometric
functionals of the parameter set. This upper bound is also sharp.
Adler and Taylor combined differential and integral geometry to find the
"Euler characteristic method" that gives one of most frequently used results
in this area. They considered stratified sets, i.e. locally convex Whitney
stratified manifolds. First, they used the Metatheorem to compute the
expectation of the Euler characteristic of the excursion set (see Theorem
12.4.1) and, second, they proved that the difference between the above
expectation and the excursion probability (the tail of the distribution) is
super exponentially smaller (see Theorem 14.3.3). Note that the geometric
functionals of the parameter set appear in the expectation of the Euler
characteristic under the name of Lipschitz-Killing curvatures.
We recall an important example when the parameter set $S$ is a convex body in
${\mathbb{R}}^{2}$ (compact, convex, with non-empty interior) and
$\mathcal{X}$ is an isotropic centered Gaussian field defined on some
neighborhood of $S$ and satisfying ${\textnormal{V}ar}(X(t)=1$ and
$\textrm{Var}(X^{\prime}(t))=I_{n}$, where $I_{n}$ is the identity matrix of
size $n$ . Let us denote:
$M_{S}=\max_{t\in S}X(t).$
Then, under some regularity conditions, the Euler characteristic method gives:
$\mathbb{P}(M_{S}\geq u)=\overline{\Phi}(u)+\frac{\sigma_{1}(\partial
S)}{2\sqrt{2\pi}}\varphi(u)+\frac{\sigma_{2}(S)}{2\pi}u\varphi(u)+o\left(\varphi\left((1+\alpha)u\right)\right),$
(1)
for some $\alpha>0$, where $\overline{\Phi}(u)$ and $\varphi(u)$ are the tail
distribution and the density of a standard normal variable, $\sigma_{2}(S)$ is
the area of $S$ and $\sigma_{1}(S)$ is the perimeter of $S$ . Note that the
coefficient $1$ of the term $\overline{\Phi}(u)$ can be interpreted as the
Euler characteristic of $S$.
Adler and Taylor use the local convexity that can be defined as the fact that
for every point $t\in S$, the support cone $C_{t}$ generated by the set of
directions
$\Biggl{\\{}\lambda\in{\mathbb{R}}^{2}:\;\|\lambda\|=1,\,\exists s_{n}\in
S\;\textnormal{such\; that}\;s_{n}\rightarrow
t\;\textnormal{and}\;\frac{s_{n}-t}{\|s_{n}-t\|}\rightarrow\lambda\Biggr{\\}},$
(2)
is convex, plus some regularity conditions (see, for example [1, Section 8.2])
($\|.\|$ is the Euclidean norm). Similarly, Azaïs and Wschebor [6, p. 231] use
the condition:
$\kappa(S)=\underset{t\in S}{\sup}\underset{s\in S,\;s\neq
t}{\sup}\frac{\textnormal{dist}(s-t,C_{t})}{\|s-t\|^{2}}<\infty$ (3)
where dist is the Euclidean distance.
However none of these methods is able to provide a full expansion for the
asymptotic formula in the non-locally convex cases, even the very simple case
of $S$ being "the angle" that is the union of two segments with the angle
$\beta\in(0,\pi)$, see Figure 1,
$S_{1}$$S_{2}$$\beta$ Figure 1: The angle, an example of non-local convexity.
which is presented in [1, Section 14.4.4]. By a full expansion, we mean a
formula of the type (1) with three terms in dimension 2 and n+1 in the general
case.
We are therefore interested in the following question:
"Can we find some full expansions for the tail of the maximum in some non-
locally convex cases in dimension $2$ and higher?"
In a previous article [4], we gave an upper bound for the tail of the
distribution for quite general parameter sets $S$. More precisely, if $S$ is
the Hausdorff limit of connected polygons $S_{n}$, if $\mathcal{X}$ is a
stationary centered Gaussian field with variance $1$ and
$\textrm{Var}(X^{\prime}(t))=I_{n}$ defined on a neighborhood of $S$ then for
every level $u$:
$\mathbb{P}\\{M_{S}\geq
u\\}\leq\overline{\Phi}(u)+\frac{\liminf_{n}\sigma_{1}(S_{n})\varphi(u)}{2\sqrt{2\pi}}+\frac{\sigma_{2}(S)}{2\pi}\left[c\varphi(u\textrm{/}c)+u\Phi(u\textrm{/c})\right]\varphi(u),$
(4)
where $c=\sqrt{\textnormal{Var}(X^{\prime\prime}_{11}(t))-1}$,
$X^{\prime\prime}_{11}(t)=\frac{\partial^{2}X(t)}{\partial t^{2}_{1}}$
$\sigma_{2}$ is the area and $\sigma_{1}$ is the perimeter.
Note that (4) can be applied to polygons taking the simpler form:
$\mathbb{P}\\{M_{S}\geq
u\\}\leq\overline{\Phi}(u)+\frac{\sigma_{1}(S)\varphi(u)}{2\sqrt{2\pi}}+\frac{\sigma_{2}(S)}{2\pi}\left[c\varphi(u\textrm{/}c)+u\Phi(u\textrm{/c})\right]\varphi(u),$
(5)
When the polygon is convex, we can check that (5) is sharp by comparing (1)
and (5). However we do not have such information in the non-convex case.
Recently Azaïs and Wschebor [7] proposed a new method, still based on the
generalized Rice formula, to derive the asymptotic formula when the parameter
set $S$ is fractal. They also gave an asymptotic expansion with two terms in
the case of a parameter set with a finite perimeter (defined as an outer
Minkowsky content). However, for example in dimension 2, this result does not
give the coefficient of $\overline{\Phi}(u)$, which is the third term by order
of importance.
Section 2 is devoted to dimension 2. We define a quite general class of
parameter sets in ${\mathbb{R}}^{2}$ (see Definition 1) and derive the
asymptotic formula for the tail of the maximum of the random fields defined on
these parameter sets. This is our main result (Theorem 1). It shows that the
coefficient corresponding to $\overline{\Phi}(u)$ is not always equal to the
Euler characteristic of the parameter set and, in fact, it is derived from the
Steiner formula that gives the volume (area) of the tube around $S$. Here
again, we emphasize the strong connection between the tube formula of the
parameter set and the tail of the maximum.
In Section 3, we examine this connection by considering some examples. We use
elementary geometry to compute the tube formula, obtain the geometric
functionals, and then immediately obtain the asymptotic expansion of the tail
distribution. All the examples correspond to new results. In particular, the
examples in Subsection 3.5 and 3.6 could shed new light on this problem. We
also conjecture that the strong connection still occurs in dimensions higher
than $2$ and 3, and even in fractal dimension.
### Hypotheses and notation
We will use the following assumption on the random field $\mathcal{X}$
througout this paper:
Assumption $A$: $\mathcal{X}$ is a random field defined on a ball
$B\subset{\mathbb{R}}^{n}$ satisfying:
* i.
$\mathcal{X}$ is a stationary centered Gaussian field.
* ii.
Almost surely the paths of $X(t)$ are of class $\mathcal{C}^{3}$.
* iii.
$\textnormal{Var}(X(t))=1$ and $\textnormal{Var}(X^{\prime}(t))$ is the
identity matrix.
* iv.
For all $s\neq t\in B$, the distribution of
$(X(s),X(t),X^{\prime}(s),X^{\prime}(t))$ does not degenerate.
* v.
For all $t\in B,\;\gamma\in\mathcal{S}^{n-1}$, the distribution of
$(X(t),X^{\prime}(t),X^{\prime\prime}(t)\gamma)$ does not degenerate.
We use the following additional notation and hypotheses.
* •
$\mathcal{S}^{n-1}$ is the unit sphere in ${\mathbb{R}}^{n}$.
* •
$S$ is a compact subset of $B$ at a positive distance from the boundary
$\partial B$ and satisfies some regularity properties (see Definition 1).
* •
$B(t,r)$ is the ball of radius $r$ centered at $t$.
* •
$M_{Z}$ is the maximum of $X(t)$ on the set $Z\subset{\mathbb{R}}^{n}$.
* •
$S^{+\epsilon}$ is the tube around $S$ defined as:
$S^{+\epsilon}=\left\\{t\in{\mathbb{R}}^{n}:\;\textnormal{dist}(t,S)\leq\epsilon\right\\}.$
## 2 Main results
Firstly, we define, the class of parameter sets $S$ that will be considered in
dimension 2.
###### Definition 1 (Two dimensional sets with piecewise-$\mathcal{C}^{2}$
boundary).
We assume that the compact set $S$ consists of a finite number of connected
components of the same nature. We describe in detail the case where $S$ has
only one connected component. $S$ contains three parts:
* •
The interior $\overset{\circ}{S}$.
* •
The non isolated curves: they form the boundary of $\overset{\circ}{S}$, and
consist of a finite number of closed continuous piecewise-$\mathcal{C}^{2}$
curves. One of these curves is the exterior boundary and the others are the
boundaries of the holes inside $S$. We assume that these curves are simple and
do not intersect each other. Each of these curves is parameterized by its arc
length using the positive orientation.
* •
The isolated curves: they consist of a finite number of disjoint self-avoiding
continuous and piecewise-$\mathcal{C}^{2}$ open curves. If
$\overset{\circ}{S}$ is not empty, each curve has exactly one extremity that
belongs to $\overline{\overset{\circ}{S}}$. If $\overset{\circ}{S}$ is empty,
then $S$ consists of only one isolated curve.
Irregular points are the points where the parametrization of the curve is no
longer $\mathcal{C}^{2}$. They divide the curves above into a finite number of
edges. An edge of a non-isolated curve will be referred to as non-isolated
edge, and similarly for an isolated edge. To limit the number of
configurations, we assume that an irregular point belongs to one of four
following categories:
* •
Convex binary points: the intersection of two non-isolated edges and the
support cone defined by (2) is convex.
* •
Concave binary points: as above but the support cone is not convex. Denote
$\beta\in[0,\pi)$ as the discontinuity of the angle of the tangent.
* •
Angle points: they are the intersection of two edges belonging to the same
isolated curve. Denote $\beta\in[0,\pi)$ by the discontinuity of the angle
(the orientation does not matter) as in Figure 1.
* •
Concave ternary points: the intersection of two non isolated edges
$E_{1},E_{2}$ and one isolated one $E_{3}$. In the main result, these points
will be considered with multiplicity two. We associate two concave angles to
each of these points:
\- $\beta_{1}$: the discontinuity of the angle of the tangent when we pass
from $E_{1}$ to $E_{3}$.
\- $\beta_{2}$: the discontinuity of the angle of the tangent when we pass
from $E_{3}$ to $E_{2}$.
To obtain a rather simple result, we only consider the concave ternary points
such that $\beta_{1}+\beta_{2}\leq\pi$, and we exclude more complicated
situations such as point of order four or the existence of handles, for
example.
Finally, the $\beta$’s described above will be referred to as concave angles.
$\beta$$E_{1}$$E_{2}$$E_{3}$$\beta_{1}$$\beta_{2}$ Figure 2: Convex, concave
binary and concave ternary points, respectively.
Remark. It should be observed that the sets with piecewise-$\mathcal{C}^{2}$
boundary considered here are Whitney stratified manifolds in the sense of [1,
Section 8.1] with some additional restrictions. We refer readers to this book
for more details.
Our proof will be hereditary proof. We will start from a set without concave
points and use the result recalled in the Appendix to establish an expansion
for such a set. We will then proceed by union to extend the result to the
general class of sets of Definition 1. To do so, we need a definition of the
property that will be extended by union. This is the object of the following
definition.
###### Definition 2 (Steiner formula heuristic property).
A compact subset $S$ of ${\mathbb{R}}^{2}$ is said to satisfy the Steiner
formula heuristic (SFH) if it satisfies the following conditions:
* •
There exist two non-negative constants $L_{1}(S)$ and $L_{0}(S)$ such that, as
$\epsilon$ tends to $0$,
$\sigma_{2}(S^{+\epsilon})=\sigma_{2}(S)+\epsilon
L_{1}(S)+\pi\epsilon^{2}L_{0}(S)+o(\epsilon^{2}).$ (6)
* •
For all processes $X(t)$ satisfying Assumption $A$,
$\mathbb{P}(M_{S}\geq
u)=L_{0}(S)\overline{\Phi}(u)+L_{1}(S)\frac{\varphi(u)}{2\sqrt{2\pi}}+\sigma_{2}(S)\frac{u\varphi(u)}{2\pi}+o\left(u^{-1}\varphi(u)\right),$
(7)
as $u\to\infty$.
Remarks.
* 1.
There exist some generalizations of the Steiner formula that hold true for
every closed set, see [10]. The present form is more restrictive.
* 2.
If $S$ is a convex body, then (6) will take the form : for all $\epsilon>0$
$\sigma_{2}(S^{+\epsilon})=\sigma_{2}(S)+\epsilon
L_{1}(S)+\pi\epsilon^{2}L_{0}(S).$ (8)
$L_{1}(S)$ is just the perimeter $\sigma_{1}(S)$ and $L_{0}(S)$ is the Euler
characteristic of $S$ which is equal to $1$.
If, in addition, the number of irregular points of $S$ (points where the
support cone is not a half space) is finite, then on the basis of the result
of Adler and Taylor, (7) follows. Thus a convex body with a finite number of
irregular points satisfies the SFH property.
* 3.
If $S$ has a positive reach in the sense that there exists a positive constant
$r$ such that for all $t\in S^{+r}$, $t$ has only one projection on $S$, then
(8) is true for all $\epsilon<r$ (see [2], [9]). Moreover, if $S$ is a set
with a piecewise-$\mathcal{C}^{2}$ boundary in the sense of Definition 1 and
satisfies $\kappa(S)<\infty$ where $\kappa(S)$ is defined in (3), then (7)
still holds true (see Appendix).
* 4.
In the most general cases, the constant $L_{1}(S)$ is the outer Minkowski
content of $S$ ($\textnormal{OMC}(S)$), which is defined, when it exists, by:
$\sigma_{2}(S^{+\epsilon})=\sigma_{2}(S)+\epsilon\textnormal{OMC}(S)+o(\epsilon).$
For more details, see [2]. It corresponds to the definition of the perimeter
of a curve in convex geometry. It can differ from the length of the boundary
of $S$, for example in the case of "the square with whiskers" (see Figure 3).
Figure 3: The square with whiskers.
In this case, the length of the boundary is equal to the perimeter of the
square plus the length of the whiskers, while $\textnormal{OMC}(S)$ is equal
to the perimeter of the square plus two times the length of the whiskers. In
addition it should be noticed that $L_{0}(S)$ is not always equal to the Euler
characteristic (see Subsection 3.4).
We are now able to state our main result.
###### Theorem 1.
Let $S$ be a compact subset in ${\mathbb{R}}^{2}$ with a
piecewise-$\mathcal{C}^{2}$ boundary and with concave angles
$\beta_{1},\ldots,\beta_{k}$ as defined in Definition 1. Let $\mathcal{X}$ be
a random field satisfying Assumption $A$. Let $M_{S}$ be the maximum of $X(t)$
on $S$.
Then $S$ satisfies the SFH and:
$\mathbb{P}(M_{S}\geq
u)=\left[\chi(S)+\frac{1}{\pi}\sum_{i=1}^{k}\left(\frac{\beta_{i}}{2}-\tan\frac{\beta_{i}}{2}\right)\right]\overline{\Phi}(u)+\frac{\textnormal{OMC}(S)}{2\sqrt{2\pi}}\varphi(u)+\frac{\sigma_{2}(S)}{2\pi}u\varphi(u)+o\left(u^{-1}\varphi(u)\right),$
(9)
where $\chi(S)$ is the Euler characteristic of $S$ that is equal to the number
of connected components minus the number of holes.
In addition, the outer Minkowski content $\textnormal{OMC}(S)$ is equal to the
length of the non-isolated edges plus twice the length of the isolated edges.
For an illustration of this theorem, see the examples in Section 3. In order
to prove the main theorem, we need some auxiliary lemmas. Firstly, we recall a
well-known result on Gaussian processes [6].
###### Lemma 1 (Borel-Sudakov-Tsirelson inequality).
Let $\mathcal{X}$ be a centered Gaussian field defined on a parameter $Z$ such
that it is almost surely bounded. Then $\mathbb{E}(M_{Z})<\infty$, and, for
all $u>0$,
$\mathbb{P}\left(M_{Z}-\mathbb{E}(M_{Z})\geq
u\right)\leq\exp(-u^{2}/(2\sigma^{2}_{Z})),$
where $\displaystyle\sigma^{2}_{Z}=\underset{t\in
Z}{\sup}\mathbb{E}(X^{2}(t))$.
An easy consequence of the BST inequality is that, for each $\epsilon>0$,
there exists a constant $C_{\epsilon}>0$ such that for all $u>0$:
$\mathbb{P}(M_{Z}\geq u)\leq
C_{\epsilon}\exp\left(\frac{-u^{2}}{2(\sigma^{2}_{Z}+\epsilon)}\right).$ (10)
We will use the above observation to prove the following lemma.
###### Lemma 2.
Let $\mathcal{X}$ be a random field satisfying Assumption $A$. Let
$Z_{1},\ldots,Z_{k}$ be some compact subsets of $B$ such that:
$Z_{1}\cap\ldots\cap Z_{k}=\emptyset.$
Then there exist two constants $\theta>1$ and $C$ such that for all $u>0$,
$\mathbb{P}\left(M_{Z_{1}}\geq u,\,\ldots,\,M_{Z_{k}}\geq u\right)\leq
C.\exp(-\theta u^{2}/2).$
###### Proof.
On the set $Z:=Z_{1}\times\ldots\times Z_{k}$, we consider the Gaussian field
$Y$ defined by:
$Y(t_{1},\ldots,t_{k})=X(t_{1})+\ldots+X(t_{k}).$
Then
$\mathbb{P}\left(M_{Z_{1}}\geq u,\,\ldots,\,M_{Z_{k}}\geq
u\right)\leq\mathbb{P}\left(\underset{t\in Z}{\sup}Y(t)\geq k.u\right).$
Applying (10) to the Gaussian field $Y$, we see that for each $\epsilon>0$,
there exists a constant $C_{\epsilon}>0$ such that for all $u>0$:
$\mathbb{P}(\underset{t\in Z}{\sup}Y(t)\geq k.u)\leq
C_{\epsilon}\exp\left(\frac{-k^{2}u^{2}}{2(\sigma^{2}_{Z}+\epsilon)}\right),$
where
$\sigma^{2}_{Z}=\underset{t\in Z}{\sup}\mathbb{E}(Y^{2}(t))=\underset{t\in
Z}{\sup}\mathbb{E}\left[(X(t_{1})+\ldots+X(t_{k}))^{2}\right].$
Since $\mathbb{E}(X^{2}(t_{i}))=1$,
$\mathbb{E}\left(X(t_{i})X(t_{j})\right)<1$ if $t_{i}\neq t_{j}$ and $Z$ is
compact, we have $\sigma^{2}_{Z}<k^{2}$. By choosing $\epsilon>0$ such that
$k^{2}>\sigma^{2}_{Z}+\epsilon$, the result follows. ∎
Since we look at a result of the type (9), every event with proba
$o\left(u^{-1}\varphi(u)\right)$ can be neglected and will be called
"negligible". Lemma 2 shows that the event $(M_{Z_{1}}\geq
u,\,\ldots,\,M_{Z_{k}}\geq u)$ is negligible as $u\to+\infty$.
The following lemma is a recent result of Azaïs and Wschebor [7].
###### Lemma 3.
Let $X$ be a random field satisfying Assumption $A$ and $\alpha$ be a given
real number $0<\alpha<1$. Then the following events are negligible:
* $A_{1}=\left\\{\exists\;\mbox{a local maximum in}\;B\mbox{ with value}\;\geq u+1\right\\}$.
* $A_{2}=\left\\{\exists\;\mbox{two or more local maxima in}\;\overset{\circ}{B}\mbox{ with value}\;\geq u\right\\}$.
* $\begin{array}[]{rl}A_{3}=&\left\\{\exists\,\mbox{a local maximum}\;t\in\overset{\circ}{B}\right.\\\ &\left.\displaystyle\mbox{such that}\;u<X(t)<u+1,\;\min\left\\{\gamma^{T}X^{\prime\prime}(s)\gamma:\,s\in B(t,u^{-\beta}),\,\gamma=\frac{s-t}{\|s-t\|}\right\\}\leq-X(t)-u^{\alpha}\right\\},\end{array}$
where $\alpha$ and $\beta$ are some positive constants in $(0,1)$, satisfying
$\beta>(1-\alpha)/2$.
* $\begin{array}[]{rl}A_{4}=&\bigg{\\{}\exists\,\mbox{a local maximum}\;t\in B\\\ &\left.\displaystyle\mbox{such that}\;u<X(t)<u+1,\;\max\left\\{\gamma^{T}X^{\prime\prime}(s)\gamma:\,s\in B(t,u^{-\beta}),\,\gamma=\frac{s-t}{\|s-t\|}\right\\}\geq-X(t)+u^{\alpha}\right\\}.\end{array}$
Let us comment on Lemma 3. Consider the event $\\{M>u\\}\cap
A_{1}^{c}\cap\cdots\cap A_{4}^{c}$ that differs from the event of interest
$\\{M>u\\}$ by a negligible probability.
Because we are in $A_{2}^{c}$, there exists at most in $B$ one local maximum
with a value larger than $u$. This implies that the excursion set:
$K_{u}:=\\{s\in B:\,X(s)\geq u\\}$
consists of one connected component. Moreover , because we are in
$A_{1}^{c}\cap A_{4}^{c}$, and thanks to a Taylor expansion:
$X(s)=X(t)+\frac{1}{2}\|s-t\|^{2}\gamma^{T}X^{\prime\prime}(\eta)\gamma,$
this component is included in:
$B(t,\overline{r})\ \ ;\mbox{ with }\ \
\overline{r}=\sqrt{2\frac{X(t)-u}{u-u^{\alpha}}}$ (11)
and where $t$ is the location of the local maximum.
This implies that, for $u$ large enough, $t$ lies in $\overset{\circ}{B}$.
Using the fact that we are in $A_{3}^{c}$ , we obtain, in the same manner,
$B(t,r)\subset K_{u}$
with
$\underline{r}=\sqrt{2\frac{X(t)-u}{X(t)+u^{\alpha}}}$
Eventually, we obtain:
$\mathbb{P}(\exists\,\mbox{a local maxima}\;t\in\overset{\circ}{B}:\,t\in
S^{+\underline{r}})+o(u^{-1}\varphi(u))\leq\mathbb{P}(M_{S}\geq
u)\leq\mathbb{P}(\exists\,\mbox{a local maxima}\;t\in\overset{\circ}{B}:\,t\in
S^{+\overline{r}})+o(u^{-1}\varphi(u)).$
On the basis of this observation, Azaïs and Wschebor derived an asymptotic
formula for the excursion distribution $\mathbb{P}(M_{S}\geq u)$. For more
details see [7].
Our starting point in this paper is the following lemma that extend the ideas
of Azaïs and Wschebor by considering several sets.
###### Lemma 4.
Let $\mathcal{X}$ be a random field satisfying Assumption $A$ and
$S_{1},\ldots,S_{m}$ be $m$ subsets of $B$ at a positive distance from
$\partial B$. Assume that there exist two constants $C>0$ and $0\leq d<n$ such
that:
$\lambda_{n}\left(S_{1}^{+\epsilon}\cap\ldots\cap
S_{m}^{+\epsilon}\right)=\left(C+o(1)\right)\epsilon^{n-d}\;as\;\epsilon\rightarrow
0.$ (12)
Then, as $u\rightarrow+\infty$,
$\mathbb{P}\left(\forall i=1\ldots m:\;M_{S_{i}}\geq
u\right)=u^{d-1}\varphi(u)\left(\frac{C}{2^{d/2}\pi^{n/2}}\Gamma\left(1+(n-d)/2\right)+o(1)\right),$
(13)
where $\Gamma$ is the Gamma function.
###### Proof.
Using Lemma 3 we have the upper-bound
$\begin{array}[]{rl}&\displaystyle\mathbb{P}\left(\forall i=1\ldots
m:\;M_{S_{i}}\geq u\right)\\\
\leq&o\left(u^{-1}\varphi(u)\right)+\mathbb{P}\left(\exists
t\in\overset{\circ}{B}:\;X(.)\textnormal{ \;has\; a\; local\; maximum\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}}\right)\\\
\leq&o\left(u^{-1}\varphi(u)\right)+\mathbb{E}\left(\textnormal{card}\left\\{t\in\overset{\circ}{B}:\;X(.)\textnormal{
\;has\; a\; local\; maximum\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}}\right\\}\right).\end{array}$
Applying the Rice formula (see [6, Chapter 6]),
$\displaystyle E:=$
$\displaystyle\mathbb{E}\left(\textnormal{card}\left\\{t\in\overset{\circ}{B}:\;X(.)\textnormal{
\;has\; a\; local\; maxima\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}}\right\\}\right)$
$\displaystyle=$
$\displaystyle\int_{u}^{+\infty}dx\int_{\overset{\circ}{B}}\mathbb{E}\left(|\det(X^{\prime\prime}(t))|\mathbb{I}_{\\{X^{\prime\prime}(t)\preceq
0\\}}\mathbb{I}_{\\{t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}}\\}}\mid
X(t)=x,\;X^{\prime}(t)=0\right)p_{X(t),X^{\prime}(t)}(x,0)\sigma_{n}(dt)$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{n/2}}\int_{u}^{+\infty}\sigma_{n}(\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\overline{r}^{*}})\mathbb{E}\left(|\det(X^{\prime\prime}(0))|\mathbb{I}_{\\{X^{\prime\prime}(0)\preceq
0\\}}\mid X(0)=x,X^{\prime}(0)=0\right)\varphi(x)\,dx,$
where $X^{\prime\prime}(0)\preceq 0$ means that the matrix
$X^{\prime\prime}(0)$ is semi definite negative, $p_{X(t),X^{\prime}(t)}(x,0)$
is the value of the joint density function of the random vector
$(X(t),X^{\prime}(t))$ at the point $(x,0)$, and $\overline{r}^{*}$ is the
value of $\overline{r}$ given by (11) when $X(t)=x$. We use the stationary
property of the field here and the fact that $X(t)$ and $X^{\prime}(t)$ are
two independent Gaussian vectors.
Using the following result (see Azaïs and Delmas [3]):
$\mathbb{E}\left(|\det(X^{\prime\prime}(0))|\mathbb{I}_{\\{X^{\prime\prime}(0)\preceq
0\\}}\mid
X(0)=x,X^{\prime}(0)=0\right)=x^{n}+O\left(x^{n-2}\right)\;\textnormal{as}\;x\rightarrow\infty,$
and hypothesis (12), we have, since we are in $A_{1}^{c}$:
$\begin{array}[]{rl}E=&\displaystyle\frac{1}{(2\pi)^{n/2}}\int_{u}^{u+1}x^{n}\varphi(x)C\left[2\frac{x-u}{u-u^{\alpha}}\right]^{(n-d)/2}dx+o\left(u^{d-1}\varphi(u)\right)\\\
=&\displaystyle\frac{Cu^{(n+d)/2}}{2^{d/2}\pi^{n/2}}\int_{u}^{u+1}\varphi(x)(x-u)^{(n-d)/2}dx+o\left(u^{d-1}\varphi(u)\right).\\\
\end{array}$
By the change of variable $x=u+y/u$,
$\begin{array}[]{rl}E=&\displaystyle\frac{C}{2^{d/2}\pi^{n/2}}u^{d-1}\varphi(u)\int_{0}^{u}\exp\left(-y-\frac{y^{2}}{2u^{2}}\right)y^{(n-d)/2}dy+o\left(u^{d-1}\varphi(u)\right)\\\
=&\displaystyle
u^{d-1}\varphi(u)\left(\frac{C}{2^{d/2}\pi^{n/2}}\Gamma(1+(n-d)/2)+o(1)\right).\end{array}$
We then obtain the upper bound as above.
For the lower bound, we see that
$\begin{array}[]{rl}&\displaystyle\mathbb{P}\left(\forall i=1\ldots
m:\;M_{S_{i}}\geq u\right)\\\
\geq&o\left(u^{-1}\varphi(u)\right)+\mathbb{P}\left(\exists
t\in\overset{\circ}{B}:\;X(.)\textnormal{ \;has\; a\; local\; maximum\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\underline{r}}\right).\\\
\end{array}$
Set
$\mathcal{M}^{\underline{r}}=\textnormal{card}\left\\{t\in\overset{\circ}{B}:\;X(.)\textnormal{
\;has\; a\; local\; maximum\;
at}\;t,\;X(t)>u,\;t\in\underset{i=1}{\overset{m}{\cap}}S_{i}^{+\underline{r}}\right\\}.$
It is proven in [13] or [7] that
$0\leq\mathbb{E}(\mathcal{M}^{\underline{r}})-\mathbb{P}(\mathcal{M}^{\underline{r}}\geq
1)\leq\mathbb{E}(\mathcal{M}^{\underline{r}}(\mathcal{M}^{\underline{r}}-1))/2\leq\mathbb{E}(\mathcal{M}_{u}(\mathcal{M}_{u}-1))/2=o\left(u^{-1}\varphi(u)\right),$
where
$\mathcal{M}_{u}=\textnormal{card}\left\\{t\in\overset{\circ}{B}:\;X(.)\textnormal{
\;has\; a\; local\; maximum\; at}\;t,\;X(t)>u\right\\}.$
Then
$\mathbb{P}\left(\min_{i}\\{M_{S_{i}}\\}\geq u\right)\geq
o\left(u^{-1}\varphi(u)\right)+\mathbb{E}(\mathcal{M}^{\underline{r}}).$
Here, using the Rice formula again and by the same arguments, we obtain the
same equivalent formula for both the upper and lower bounds. The result then
follows. ∎
The main idea is to use the inclusion-exclusion principle to compute the
probability of the union of events $\\{M_{S_{i}}>u\\}$ through Lemma 4 that
gives probability of the intersection of some of them. Let us give a simple
introductory example. Suppose that $S=S_{1}\cup S_{2}$ with $S_{1}$ and
$S_{2}$ satisfy the SFH as in Definition 2. Suppose, in addition, that the
condition (12) is met, i.e.,
$\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon})=\left(C+o(1)\right)\epsilon^{2}.$
Then, using Lemma 4, we have an expansion of $\mathbb{P}\left(M_{S_{1}}\geq
u,\,M_{S_{2}}\geq u\right)$ and, by consequence, an expansion of
$\mathbb{P}(M_{S}\geq u)$ with an error of $o(u^{-1}\varphi(u))$.
However, in general, we need to decompose $S$ into three or even four sets.
The next lemma is the basis of our method. It shows that the Steiner formula
heuristic property (SFH) is heredity in the sense that: if we start from some
subsets in ${\mathbb{R}}^{2}$ satisfying the SFH, then under some conditions,
the union of these subsets also satisfies the SFH. Therefore, to prove the
main theorem, we just prove that the considered parameter set can be expressed
as the union of the subsets satisfying the SFH.
###### Lemma 5.
Let $S_{1},\;S_{2},\;S_{3}$ and $S_{4}$ be four compact subsets in
${\mathbb{R}}^{2}$ such that:
* 1.)
For every $i=1,2,3,4$, $S_{i}$ satisfies the SFH .
* 2.)
$S_{1}\cup S_{2},\;S_{2}\cup S_{3},\;S_{3}\cup S_{4},\;$ and $S_{4}\cup S_{1}$
satisfy the SFH.
* 3.)
$S_{2}\cap S_{4}=\emptyset$ and $S_{1}\cap S_{3}\cap S_{4}=\emptyset$.
* 4.)
As $\epsilon$ tends to $0$, there exist two positive constants $C_{13}$ and
$C_{123}$ such that
$\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{3}^{+\epsilon})=\left(C_{13}+o(1)\right)\epsilon^{2}\;\;\textnormal{and}\;\;\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon}\cap
S_{3}^{+\epsilon})=\left(C_{123}+o(1)\right)\epsilon^{2}.$ (14)
Then $S=S_{1}\cup S_{2}\cup S_{3}\cup S_{4}$ also satisfies the SFH and:
$\begin{array}[]{rl}-\indent L_{1}(S)=&\displaystyle L_{1}(S_{1}\cup
S_{2})+L_{1}(S_{2}\cup S_{3})+L_{1}(S_{3}\cup S_{4})+L_{1}(S_{4}\cup
S_{1})-\sum_{i=1}^{4}L_{1}(S_{i}),\\\ -\indent L_{0}(S)=&\displaystyle
L_{0}(S_{1}\cup S_{2})+L_{0}(S_{2}\cup S_{3})+L_{0}(S_{3}\cup
S_{4})+L_{0}(S_{4}\cup
S_{1})-\sum_{i=1}^{4}L_{0}(S_{i})+\frac{C_{123}-C_{13}}{\pi}.\\\ \end{array}$
Note that in many cases, Lemma 5 will be used with $S_{4}=\emptyset$ and will
take a simpler form.
###### Proof.
In order to prove that $S$ satisfies the SFH, we need to shows the
correspondence between the tube formula of $S$ as in (6) and the asymptotic
expansion for the tail of the distribution as in (7).
$\bullet$ First, we consider the tube formula of $S$. We prove the following
equality for the volume of $S^{+\epsilon}$ for a sufficiently small
$\epsilon$:
$\displaystyle A:=$
$\displaystyle\sigma_{2}(S^{+\epsilon})=B:=\sigma_{2}((S_{1}\cup
S_{2})^{+\epsilon})+\sigma_{2}((S_{2}\cup
S_{3})^{+\epsilon})+\sigma_{2}((S_{3}\cup
S_{4})^{+\epsilon})+\sigma_{2}((S_{4}\cup S_{1})^{+\epsilon})$
$\displaystyle-\sigma_{2}(S_{1}^{+\epsilon})-\sigma_{2}(S_{2}^{+\epsilon})-\sigma_{2}(S_{3}^{+\epsilon})-\sigma_{2}(S_{4}^{+\epsilon})+\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon}\cap S_{3}^{+\epsilon})-\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{3}^{+\epsilon}).$ (15)
Concerning $A$, we can observe that $A=\sigma_{2}(S_{1}^{+\epsilon}\cup
S_{2}^{+\epsilon}\cup S_{3}^{+\epsilon}\cup S_{4}^{+\epsilon})$ and use the
inclusion-exclusion principle to obtain a full expansion. Doing the same on
$B$ we see that the following quantity is missing:
$-\\{2,4\\}+\\{1,2,4\\}+\\{2,3,4\\}+\\{1,3,4\\}-\\{1,2,3,4\\},$
where, for example, $\\{2,4\\}=\sigma_{2}(S_{2}^{+\epsilon}\cap
S_{4}^{+\epsilon})$. Our hypotheses shows that these quantities vanish as soon
as $\epsilon$ is small enough. This proves (15) and implies that:
$\sigma_{2}(S^{+\epsilon})=\sigma_{2}(S)+\epsilon
L_{1}(S)+\pi\epsilon^{2}L_{0}(S)+o(\epsilon^{2}),$
where
$\begin{array}[]{rl}-\indent L_{1}(S)=&\displaystyle L_{1}(S_{1}\cup
S_{2})+L_{1}(S_{2}\cup S_{3})+L_{1}(S_{3}\cup S_{4})+L_{1}(S_{4}\cup
S_{1})-\sum_{i=1}^{4}L_{1}(S_{i}),\\\ -\indent L_{0}(S)=&\displaystyle
L_{0}(S_{1}\cup S_{2})+L_{0}(S_{2}\cup S_{3})+L_{0}(S_{3}\cup
S_{4})+L_{0}(S_{4}\cup
S_{1})-\sum_{i=1}^{4}L_{0}(S_{i})+\frac{C_{123}-C_{13}}{\pi}.\\\ \end{array}$
$\bullet$ For the excursion probability on $S$, using the inclusion-exclusion
principle once again,
$\begin{array}[]{rl}\mathbb{P}(M_{S}\geq u)=&\mathbb{P}(M_{S_{1}\cup S_{2}\cup
S_{3}\cup S_{4}}\geq u)\\\
=&\displaystyle\sum_{i=1}^{4}\mathbb{P}(M_{S_{i}}\geq u)-\sum_{1\leq i<j\leq
4}\mathbb{P}(M_{S_{i}}\geq u,\,M_{S_{j}}\geq u)\\\ &\displaystyle+\sum_{1\leq
i<j<k\leq 4}\mathbb{P}(M_{S_{i}}\geq u,\,M_{S_{j}}\geq u,\,M_{S_{k}}\geq
u)-\mathbb{P}(M_{S_{i}}\geq u,\,\forall i=1,2,3,4).\end{array}$
On the basis of Lemma 2, it is easy to see that the events $\\{M_{S_{2}}\geq
u,\,M_{S_{4}}\geq u\\}$ and $\\{M_{S_{1}}\geq u,\,M_{S_{3}}\geq
u,\,M_{S_{4}}\geq u\\}$ have negligible probabilities ($o(u^{-1}\varphi(u))$),
yielding:
$\displaystyle\mathbb{P}(M_{S}\geq u)=$
$\displaystyle\sum_{i=1}^{4}\mathbb{P}(M_{S_{i}}\geq
u)-\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u)-\mathbb{P}(M_{S_{2}}\geq
u,\,M_{S_{3}}\geq u)$ $\displaystyle\displaystyle-\mathbb{P}(M_{S_{3}}\geq
u,\,M_{S_{4}}\geq u)-\mathbb{P}(M_{S_{4}}\geq u,\,M_{S_{1}}\geq
u)-\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{3}}\geq u)$
$\displaystyle\displaystyle+\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq
u,\,M_{S_{3}}\geq u)+o\left(u^{-1}\varphi(u)\right)$ $\displaystyle=$
$\displaystyle\displaystyle\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq
u)+\mathbb{P}(M_{S_{2}}\geq u,\,M_{S_{3}}\geq u)+\mathbb{P}(M_{S_{3}}\geq
u,\,M_{S_{4}}\geq u)$ $\displaystyle+\mathbb{P}(M_{S_{4}}\geq
u,\,M_{S_{1}}\geq u)-\displaystyle\sum_{i=1}^{4}\mathbb{P}(M_{S_{i}}\geq
u)-\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{3}}\geq u)$
$\displaystyle\displaystyle+\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq
u,\,M_{S_{3}}\geq u)+o\left(u^{-1}\varphi(u)\right).$
Now, using the SFH property in the first and second conditions and applying
Lemma 4 for two probabilities $\mathbb{P}(M_{S_{1}}\geq u,\;M_{S_{3}}\geq u)$
and $\mathbb{P}(M_{S_{1}}\geq u,\;M_{S_{2}}\geq u,\;M_{S_{3}}\geq u)$, we can
deduce that:
$\mathbb{P}(M_{S}\geq
u)=L_{0}(S)\overline{\Phi}(u)+L_{1}(S)\frac{\varphi(u)}{2\sqrt{2\pi}}+\sigma_{2}(S)\frac{u\varphi(u)}{2\pi}+o\left(u^{-1}\varphi(u)\right),$
where the constants $L_{0}(S)$ and $L_{1}(S)$ are defined as in the statement.
Since a correspondence exists between the two formulas obtained, we have
proved the SFH property of $S$. ∎
An introductory example to understand the method
To introduce our method, we consider the case of the simplest non-convex
polygon shown in Figure 4. Note that in this case, we have exactly one concave
binary point with concave angle $\beta$.
$S_{1}$$S_{2}$$S_{3}$ Figure 4: Non-convex polygon with concave binary
irregular point.
$S$ is decomposed into three polygons $S_{1},S_{2}$ and $S_{3}$ with a zero
measure intersection, as indicated in Figure 4. These polygons are convex so
they satisfy the SFH as well as $S_{1}\cup S_{2}$ and $S_{2}\cup S_{3}$.
To apply Lemma 5, with $S_{4}=\emptyset$, it remains to compute the areas of
$(S_{1}^{+\epsilon}\cap S_{3}^{+\epsilon})$ and $(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon}\cap S_{3}^{+\epsilon})$. Elementary geometry shows that
$(S_{1}^{+\epsilon}\cap S_{3}^{+\epsilon})$ consists of two sections of a disc
with angle $(\pi-\beta)$ and two quadrilaterals of area
$\epsilon^{2}\tan(\beta/2)$ each, whereas in $(S_{1}^{+\epsilon}\cap
S_{2}^{+\epsilon}\cap S_{3}^{+\epsilon})$, one quadrilateral is replaced by a
section of a disc of angle $\beta$ (see Figure 5).
$S_{1}$$S_{2}$$S_{3}$$\beta$ Figure 5: Intersection of $\epsilon$-neighborhood
sets.
Thus,
$\sigma_{2}(S_{1}^{+\epsilon}\cap
S_{3}^{+\epsilon})=\left[(\pi-\beta)+2\tan\frac{\beta}{2}\right]\epsilon^{2},$
$\sigma_{2}(S_{1}^{+\epsilon}\cap S_{2}^{+\epsilon}\cap
S_{3}^{+\epsilon})=\left[(\pi-\beta)+\frac{\beta}{2}+\tan\frac{\beta}{2}\right]\epsilon^{2}.$
Then using (14), we can define the constants $C_{123}$ and $C_{13}$, and
compute that:
$C_{123}-C_{13}=\frac{\beta}{2}-\tan\frac{\beta}{2}.$
This quantity measures the non convexity of the concave binary point. Since
the $L_{0}-$ constants of $S_{2},\,S_{1}\cup S_{2}$ and $S_{2}\cup S_{3}$ are
both equal to $1$ in this case, an application of Lemma 5 shows that the
coefficient of $\overline{\Phi}(u)$ in the expansion of the tail of $M_{S}$ is
now $\displaystyle 1+\frac{\beta/2-\tan(\beta/2)}{\pi}$.
### Proof of the main theorem
Using the above lemmas, we are able now to prove the main theorem. If the
parameter set $S$ consists of several disjoint connected components, then by
using Lemma 2 the tail distribution of the maxima defined on these components
can be added with an error of $o\left(u^{-1}\varphi(u)\right)$, and the right-
hand side of (9) is also additive, we can assume in the sequel that $S$ is
connected.
Our proof is based on induction on the number of concave points of $S$. It
should be recalled here that there are three types of concave points: binary,
angle and ternary (see Definition 1).
$\bullet$ Suppose that $S$ has no concave point. Two cases will be considered:
depending on whether $\overset{\circ}{S}$ is empty or not.
If $\overset{\circ}{S}$ is empty, then $S$ consists of only one isolated edge.
Using the parametrization of the unique edge, we see that $M_{S}$ is just the
maximum of a smooth random process (with parameter of dimension 1). By using
the Rice method for the number of up-crossings, Piterbarg [12] or Rychlik [15]
showed that $S$ satisfies the SFH.
If $\overset{\circ}{S}$ is not empty, then $S$ cannot have isolated edges and
$S$ has a positive reach in the sense of Federer [9] because the curvature on
the compact edges is bounded. Therefore,
$\sigma_{2}(S^{+\epsilon})=\chi(S)\pi\epsilon^{2}+\textnormal{OMC}(S)\epsilon+\sigma_{2}(S),$
(16)
for small enough $\epsilon$. On the other hand, on the basis of Theorem 8.12
of Azaïs and Wschebor [6], it can deduced that the SFH applies (see Appendix
for details).
$\bullet$ Suppose $S$ has at least one concave point. We will decompose $S$
into the subsets whose number of concave points is strictly smaller than that
of $S$. The induction hypothesis then ensures that these subsets will satisfy
the SFH. We can therefore use Lemma 5 to "glue" them together and to show that
the union $S$ also satisfies the SFH. In fact, our method is based on the
"destruction" of concave points as in the introductory example. More
precisely, there are four possibilities regarding $P$:
* 1.
Concave binary point on the exterior boundary of $S$. We decompose $S$ into
three compact subsets $S_{1},\,S_{2}$ and $S_{3}$ as in Figure 4. By
decomposition we mean an essential partition, i.e. that $S=S_{1}\cup S_{2}\cup
S_{3}$ and that $S_{1}$, $S_{2}$ and $S_{3}$ have disjoint interiors. The
decomposition is as follows: at $P$ we prolong the two tangents inward and
construct two $\mathcal{C}^{2}$ paths that avoid all the holes and end at one
regular point on the exterior boundary such that these two paths have no
intersection other than point $P$. This is always possible because the
connected open set $\overset{\circ}{S}$ is path connected. We then define
$S_{1}$, $S_{2}$ and $S_{3}$ as in Figure 6. To apply Lemma 5 with
$S_{4}=\emptyset$, we need to verify all the required conditions.
To compute $\sigma_{2}(S_{1}^{+\epsilon}\cap S_{3}^{+\epsilon})$ and
$\sigma_{2}(S_{1}^{+\epsilon}\cap S_{2}^{+\epsilon}\cap S_{3}^{+\epsilon})$,
we can locally replace the edges starting from $P$ by their tangents with an
error of $O(\epsilon^{3})$ and thus $o(\epsilon^{2})$. In that case the
computation of these areas is exactly the same as in the introductory example.
$S_{1}$$S_{3}$$S_{2}$$P$ Figure 6: Decomposition of $S$ at a binary concave
point on the exterior boundary.
On the other hand, let us consider the number of concave points of
$S_{1},\,S_{2},\,S_{3},\,S_{1}\cup S_{2}$ and $S_{2}\cup S_{3}$. Due to the
way we have constructed these sets, we have destroyed the concavity of $P$ in
the sense that with respect to these subsets, $P$ becomes a convex binary
point or a regular one. We can see that an irregular point of these subsets is
also an irregular set of $S$ unless it is $P$ or one of the other endpoints on
the exterior boundary of the two prolonged paths. However, we have proved that
$P$ is no longer a concave point with respect to these subsets. Moreover,
since the other endpoint is chosen to be a regular point on the exterior
boundary of $S$ and since the support cones at this point with respect to
these subsets (as defined in (2)) are included in the one with respect to $S$,
these support cones are convex; then this endpoint is therefore not a concave
point. Hence, a concave point of the constructed subsets is also a concave
point of $S$ and $P$ is a concave point only with respect to $S$. These
subsets therefore have a number of concave points equal at most to the one of
$S$ minus 1. They therefore satisfy the SFH by induction.
Since all the required conditions are met, on the basis of Lemma 5, $S$
satisfies the SFH with the desired constants.
* 2.
Concave binary point on the boundary of a hole inside $S$. Drawing two
prolonged paths as above, we simply decompose $S$ into two subsets $S_{2}$ and
$S^{\prime}$. We then divide $S^{\prime}$ into three subsets as follows: we
also choose two regular points on the boundary of the hole and two
corresponding regular points on the exterior boundary of $S$ and construct two
smooth curves so that they connect one regular point on the boundary of the
hole with the corresponding one on the exterior boundary, and do not intersect
themselves or two curves from the irregular point or additional holes. Again,
this is possible because $\overset{\circ}{S}$ is path-connected.
$S_{1},S_{2},S_{3},S_{4}$ are then constructed as in Figure 7.
$S_{1}$$S_{3}$$S_{2}$$S_{4}$$P$ Figure 7: Decomposition at a concave point on
the interior boundary.
We will use Lemma 5 to prove that $S$ satisfies the SFH. Indeed, the
computation of the areas $\sigma_{2}(S_{1}^{+\epsilon}\cap S_{3}^{+\epsilon})$
and $\sigma_{2}(S_{1}^{+\epsilon}\cap S_{2}^{+\epsilon}\cap
S_{3}^{+\epsilon})$ and the arguments to show that the subsets in the first
and second conditions of Lemma 5 satisfy the SFH remain the same as in the
case above. The third condition about the empty intersections is easily
verified from the construction. We can therefore deduce the SFH property of
$S$.
* 3.
Concave ternary point. We define $S_{1}$ as the isolated edge containing $P$,
$S_{2}=\\{P\\}$ and $S_{3}$ as the closure of the complement of $S_{1}$ (see
Figure 8).
$S_{3}$$S_{2}$$S_{1}$ Figure 8: Decomposition at a concave ternary point.
By the same arguments as in the above cases, we can check that all the
required conditions in Lemma 5 are met. $S$ then satisfies the SFH.
* 4.
Angle point. We do the same as in the concave ternary point case (see Figure
9).
$S_{1}$$S_{3}$$S_{2}$ Figure 9: At an angle point.
We have proved that $S$ satisfies the SFH. In order to establish (9), we need
to compute the constant $L_{0}(S)$.
Firstly, we have seen that when $S$ contains no concave points:
$L_{0}(S)=\chi(S)$
Secondly, Lemma 5 shows that when we "glue" $S_{1},S_{2},S_{3}$ and $S_{4}$
together, each concave points causes a distortion to the additivity which is
equal to:
$\frac{\beta_{i}/2-\tan(\beta_{i}/2)}{\pi}.$
Therefore we eventually have:
$L_{0}(S)=\chi(S)+\sum_{i=1}^{k}\frac{\beta_{i}/2-\tan(\beta_{i}/2)}{\pi}$
and we are done.
## 3 Examples
In this section, we give some examples that are direct applications or direct
generalizations of Theorem 1. All these results are new and rather unexpected.
In most two-dimensional cases, the parameter set $S$ has the
piecewise-$\mathcal{C}^{2}$ boundary as in Definition 1, satisfying the SFH.
Therefore, in order to derive the asymptotic formula for the tail of the
maximum, we just use elementary geometry to compute the area of the tube and
consider the corresponding coefficients.
### 3.1 The angle
Let $S$ be the angle as in Figure 1. On the basis of Theorem 1, $S$ satisfies
the SFH. Using elementary geometry, we can compute that for small enough
$\epsilon$,
$\lambda_{2}(S^{+\epsilon})=2\left(\lambda_{1}(S_{1})+\lambda_{1}(S_{2})\right)\epsilon+\left(\pi+\beta/2-\tan(\beta/2)\right)\epsilon^{2},$
where $\lambda_{1}(.)$ is simply the length of the segment. We then have:
$\mathbb{P}(M_{S}\geq
u)=\left(1+\frac{\beta/2-\tan(\beta/2)}{\pi}\right)\overline{\Phi}(u)+\frac{\lambda_{1}(S_{1})+\lambda_{1}(S_{2})}{\sqrt{2\pi}}\varphi(u)+o\left(u^{-1}\varphi(u)\right).$
### 3.2 The multi-angle
This is an extension of the angle case. Let $S$ be a self-avoiding continuous
curve that is the union of $k+1$ curves with concave angles
$\\{\beta_{1},\ldots,\beta_{k}\\}$. In this case, the induction process in the
proof of the main theorem can be seen as the induction on the number of
segments, i.e., we add one more segment into the union each time. Here again
using elementary geometry, for small enough $\epsilon$:
$\lambda_{2}(S^{+\epsilon})=2\lambda_{1}(S)\epsilon+\left(\pi+\underset{i=1}{\overset{k}{\sum}}\left(\beta_{i}/2-\tan(\beta_{i}/2)\right)\right)\epsilon^{2},$
where $\lambda_{1}(S)$ is the length of the curve that is equal to the sum of
the lengths of the segments. Hence we immediately have the asymptotic formula:
$\mathbb{P}(M_{S}\geq
u)=\left(1+\frac{\underset{i=1}{\overset{k}{\sum}}\left(\beta_{i}/2-\tan(\beta_{i}/2)\right)}{\pi}\right)\overline{\Phi}(u)+\frac{\sigma_{1}(S)}{\sqrt{2\pi}}\varphi(u)+o\left(u^{-1}\varphi(u)\right).$
### 3.3 The empty square
Let $S$ be the empty square, i.e. the boundary of a square in
${\mathbb{R}}^{2}$. This case is very similar to the multi-angle case, but the
curves are no longer self-avoiding. In this case, the induction process on the
number of segments still works. We can therefore deduce that $S$ satisfies the
SFH. The elementary geometry shows that for small enough $\epsilon$:
$\lambda_{2}(S^{+\epsilon})=2\lambda_{1}(S)\epsilon+(\pi-4)\epsilon^{2};$
then, as a consequence,
$\mathbb{P}(M_{S}\geq
u)=\frac{\pi-4}{\pi}\overline{\Phi}(u)+\frac{\sigma_{1}(S)}{\sqrt{2\pi}}\varphi(u)+o\left(u^{-1}\varphi(u)\right).$
### 3.4 The full square with whiskers
We consider "the square with whiskers" as in Figure 3. In this case, $S$ has
two concave ternary points. From the main theorem, we know that $S$ satisfies
the SFH. Therefore, since we can compute the area of the tube as
$\lambda_{2}(S^{+\epsilon})=\lambda_{2}(S)+\textnormal{OMC}(S)\epsilon+(2\pi-4)\epsilon^{2},$
for small enough $\epsilon$ ; we have the expansion:
$\mathbb{P}(M_{S}\geq
u)=\frac{2\pi-4}{\pi}\overline{\Phi}(u)+\frac{\textnormal{OMC}(S)}{2\sqrt{2\pi}}\varphi(u)+\frac{\sigma_{2}(S)}{2\pi}u\varphi(u)+o\left(u^{-1}\varphi(u)\right).$
### 3.5 An irregular locally convex set
In this subsection, we consider a strange and interesting example. We consider
$S$ as the union of two tangent curves as in Figure 10.
Figure 10: Two tangent edges.
Suppose that $S_{1}$ is a section of a circle of radius $R$ and $S_{2}$ is a
segment tangent to that circle. For small enough $\epsilon$ , the area of the
intersection between two tubes is:
$\frac{\pi}{2}\epsilon^{2}+\frac{(R+\epsilon)^{2}}{2}\arcsin\frac{2\sqrt{R\epsilon}}{R+\epsilon}-(R-\epsilon)\sqrt{R\epsilon}=\frac{\pi}{2}\epsilon^{2}+\frac{8}{3}\sqrt{R}\epsilon^{3/2}+O(\epsilon^{5/2}).$
In the above equation, we used the fact that for small enough $x$ ,
$\arcsin x=x+\frac{1}{2}\frac{x^{3}}{3}+\frac{1\cdot 3}{2\cdot
4}\frac{x^{5}}{5}+\ldots\;.$
It is clear that the order of the area of the intersection is not of $2$ as in
Condition (14), so we cannot apply Lemma 5 directly. In this example, the area
of the intersection contains two order: $2$ and $3/2$.
The asymptotic formulas for the tail of the maximum of the random fields
defined on $S_{1}$ and $S_{2}$ are well understood since they are one-
dimensional cases. Then by the inclusion-exclusion principle,
$\mathbb{P}(M_{S}\geq u)=\mathbb{P}(M_{S_{1}}\geq u)+\mathbb{P}(M_{S_{2}}\geq
u)-\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u\right).$
To compute $\mathbb{P}(M_{S}\geq u)$, we need to derive an expansion for
$\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u\right)$.
By carefully examining in the proof of Lemma 4, we can choose $\alpha$ such
that the difference between the upper and the lower bounds of the probability
$\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u)$ is negligible. Indeed, as in
Lemma 4, after substituting the area of the intersection of the tubes into the
expectation, we obtain the upper bound:
$\begin{array}[]{rl}&\displaystyle\frac{1}{2\pi}\int_{u}^{u+1}\left(x^{2}+O(1)\right)\varphi(x)\left[\frac{\pi}{2}\frac{2(x-u)}{u-u^{\alpha}}+\frac{8}{3}\sqrt{R}\left(2\frac{x-u}{u-u^{\alpha}}\right)^{3/4}+O\left(\left(2\frac{x-u}{u-u^{\alpha}}\right)^{5/4}\right)\right]dx+o(u^{-1}\varphi(u))\\\
=&\displaystyle\frac{1}{2\pi}\int_{u}^{u+1}x^{2}\varphi(x)\left[\frac{\pi}{2}\frac{2(x-u)}{u-u^{\alpha}}+\frac{8}{3}\sqrt{R}\left(2\frac{x-u}{u-u^{\alpha}}\right)^{3/4}\right]dx+o(u^{-1}\varphi(u)),\end{array}$
and, similarly, the lower one:
$\frac{1}{2\pi}\int_{u}^{u+1}x^{2}\varphi(x)\left[\frac{\pi}{2}\frac{2(x-u)}{x+u^{\alpha}}+\frac{8}{3}\sqrt{R}\left(2\frac{x-u}{x+u^{\alpha}}\right)^{3/4}\right]dx+o(u^{-1}\varphi(u)).$
To control the difference between them, we firstly consider the term:
$\begin{array}[]{rl}D_{1}=&\displaystyle\int_{u}^{u+1}x^{2}\varphi(x)\left[\left(\frac{x-u}{u-u^{\alpha}}\right)^{3/4}-\left(\frac{x-u}{x+u^{\alpha}}\right)^{3/4}\right]dx\\\
=&\displaystyle\int_{u}^{u+1}x^{2}\varphi(x)(x-u)^{3/4}\frac{(x+u^{\alpha})^{3/4}-(u-u^{\alpha})^{3/4}}{(x+u^{\alpha})^{3/4}(u-u^{alpha})^{3/4}}dx.\end{array}$
Since
$a^{3/4}-b^{3/4}=\frac{a^{3}-b^{3}}{\left(a^{3/4}+b^{3/4}\right)\left(a^{3/2}+b^{3/2}\right)}=\frac{(a-b)(a^{2}+ab+b^{2})}{\left(a^{3/4}+b^{3/4}\right)\left(a^{3/2}+b^{3/2}\right)}$
for $a=x+u^{\alpha}$ and $b=u-u^{\alpha}$, and we can replace $x,a,b$ by $u$,
then:
$\begin{array}[]{rl}D_{1}\leq&\displaystyle(const)\int_{u}^{u+1}u^{2}\varphi(x)(x-u)^{3/4}\frac{\left(x-u+2u^{\alpha}\right)u^{2}}{u^{3+3/4}}dx\\\
\leq&\displaystyle(const)u^{\alpha+1/4}\int_{u}^{u+1}\varphi(x)(x-u)^{3/4}dx.\end{array}$
Here using the change of variable $x=u+y/u$ once again,
$D_{1}\leq(const)\frac{u^{\alpha+1/4}}{u^{1+3/4}}\varphi(u)\int_{0}^{u}\exp\left(-y-\frac{y^{2}}{2u^{2}}\right)y^{3/4}dy.$
Therefore, if we choose $\alpha<1/2$ then $D_{1}=o(u^{-1}\varphi(u))$. For the
second term :
$\int_{u}^{u+1}x^{2}\varphi(x)\left[\frac{x-u}{u-u^{\alpha}}-\frac{x-u}{x+u^{\alpha}}\right]dx,$
we can use the same arguments. Note that this case is simpler.
In conclusion, we have proved that if $\alpha<1/2$, then the difference
between the upper and the lower bounds of the probability
$\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u)$ is negligible. As in Lemma 4,
we have the following expansion:
$\mathbb{P}(M_{S_{1}}\geq u,\,M_{S_{2}}\geq
u)=\frac{8\sqrt{R}}{2^{1/4}3\pi}\Gamma(7/4)u^{-1/2}\varphi(u)+\frac{\overline{\Phi}(u)}{2}+o(u^{-1}\varphi(u)).$
Thus, we have:
###### Proposition 1.
Using the above notation
$\mathbb{P}(M_{S_{1}\cup S_{2}}\geq
u)=\frac{3\overline{\Phi}(u)}{2}-\frac{8\sqrt{R}}{2^{1/4}3\pi}\Gamma(7/4)u^{-1/2}\varphi(u)+\frac{\sigma_{1}(S_{1})+\sigma_{1}(S_{2})}{\sqrt{2\pi}}\varphi(u)+o\left(u^{-1}\varphi(u)\right).$
(17)
This example is an apparent counter-example to the results of Adler and
Taylor. More precisely, $S$ is clearly a piecewise smooth locally convex
manifold: it is easy to check that at the intersection of the circle and the
straight line, the support cone is limited to one direction and is thus
convex. Thus if the random field $X$ is sufficiently smooth, it seems that
Theorem 14.3.3 of [1] implies the validity of the Euler characteristic
heuristic and Theorem 12.4.2 of [1] gives an expansion of the Euler
characteristic function that should apply. This would be clearly in
contradiction with the term $u^{-1/2}\varphi(u)$ in (17).
In fact, there is no contradiction: Theorem 14.3.3 also demands the manifold
to be regular in the sense of Definition 9.22 of [1] and the present set is
not a cone space in the sense of Definition 8.3.1 of [1]. This shows that the
local convexity itself is not sufficient.
It is surprising to see that in (17), the asymptotic formula contains three
terms corresponding to the powers: $-1$ (in $\overline{\Phi}(u)$), $-1/2$ and
$0$. This is the first time we can see such a combination; in all the well-
known cases before, we only saw a combination of integer powers. We emphasize
that this strange combination comes from the tube formula of the parameter
set.
### 3.6 Some examples in dimension 3
Lemmas 4 and 5 can be applied in higher dimensions. However, in dimension 3,
for example, they do not make it possible to obtain a full Taylor expansion
that would contain, in general, four terms.
In fact, the coefficient of $\overline{\Phi}(u)$ cannot be determined for non-
locally convex sets. We give some examples below.
* •
$S$ is a dihedral that is the union of two non-coplanar rectangles $S_{1}$ and
$S_{2}$, with a common edge such that the angle of the dihedral is $\alpha$,
see Figure 11.
$\alpha$$S_{1}$$S_{2}$ Figure 11: Example of a dihedral.
Using the inclusion-exclusion principle, we are just concerned with the
probability $\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u\right)$. Using
Lemma 4 in the case where $n=3$ and $d=1$, we obtain the expansion of this
probability with only one term and an error of$o\left(\varphi(u)\right)$.
Then,
$\begin{array}[]{rl}\mathbb{P}(M_{S}\geq
u)=&\quad\displaystyle\frac{\sigma_{1}(\partial S_{1})+\sigma_{1}(\partial
S_{2})-\sigma_{1}(S_{1}\cap
S_{2})((\pi+\alpha)/2+\cot(\alpha/2))/\pi}{2\sqrt{2\pi}}\varphi(u)\\\
&\displaystyle+\;\frac{\sigma_{2}(S_{1})+\sigma_{2}(S_{2})}{2\pi}u\varphi(u)+o\left(\varphi(u)\right).\end{array}$
* •
$S$ has the $L-$shape, as in Figure 12.
Figure 12: L-shape.
Then, by decomposing $S$ into three hyper-rectangles $S_{1},\;S_{2}$ and
$S_{3}$ that are indicated by the dotted lines with $S_{3}$ between the two
others, we can apply Lemma 5 with a slight modification that since, in this
cas, $n=3$ and $d=1$, then the asymptotic formulas for
$\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u\right)$ and
$\mathbb{P}\left(M_{S_{1}}\geq u,\,M_{S_{2}}\geq u,\,M_{S_{3}}\geq u\right)$
are of order $\varphi(u)$ and the error is $o\left(\varphi(u)\right)$ (from
Lemma 4). We then have an expansion with three terms as follows:
$\mathbb{P}(M_{S}\geq
u)=\frac{\varphi(u)L_{1}(S)}{2\sqrt{2\pi}}+\frac{L_{2}(S)u\varphi(u)}{2\pi}+\frac{L_{3}(S)(u^{2}-1)\varphi(u)}{(2\pi)^{3/2}}+o\left(\varphi(u)\right),$
(18)
where the coefficients $\\{L_{i}(S),\;i=1,\ldots,3\\}$ are given by the
Steiner formula and will be defined at the end of this section.
* •
In a more complicated case, i.e., non-convex trihedral (see Figure 13).
Figure 13: Example of a non-convex trihedral.
In this case, we have three concave edges in the sense that the angles inside
the trihedral at these edges are strictly greater than $\pi$. We will destroy
this concavity by extending the planes (faces) containing these edges so that
they decompose $S$ into smaller convex subsets $\\{S_{i}\\}$ with disjoint
interiors (see the dotted lines in the figure). Observe that the intersection
between two subsets is of one of four types: empty set, a single point, an
edge or a face. If it is a face, then the union of these two subsets is also
convex. An intersection between three or more subsets is one of three types:
empty set, a single point or an edge. Using the inclusion-exclusion principle,
we need to find the expansion of the probability of the intersection of the
events $M_{S_{i_{k}}}\geq u$ for some $k$. Concerning the intersection of the
$\\{S_{i_{k}}\\}$, we have the following cases:
* 1.
Empty set. On the basis of Lemma 2, the probability of the intersection of the
events $M_{S_{i_{k}}}\geq u$ is $o(u^{-1}\varphi(u))$.
* 2.
A single point. By applying Lemma 4 in the case $d=0$, the probability
considered is also $o(u^{-1}\varphi(u))$.
* 3.
An edge. By applying Lemma 4 in the case $n=3$ and $d=0$, the expansion for
the probability considered is of the order $\varphi(u)$ with the error
$o(\varphi(u))$.
* 4.
A face. This case just happens when we consider the intersection between two
subsets. Since both of these subsets and their union are convex, the expansion
for the tail distribution of the maxima defined on them is well-known. We can
therefore compute the expansion for the probability considered by the
inclusion-exclusion principle.
We therefore obtain an asymptotic formula for $\mathbb{P}(M_{S}\geq u)$, as in
(18).
In general, by the same arguments and using induction, when $S$ is a polytope,
$\mathbb{P}(M_{S}\geq
u)=\frac{\varphi(u)L_{1}(S)}{2\sqrt{2\pi}}+\frac{L_{2}(S)u\varphi(u)}{2\pi}+\frac{L_{3}(S)(u^{2}-1)\varphi(u)}{(2\pi)^{3/2}}+o\left(\varphi(u)\right),$
where
* -
$L_{3}(S)$ is the volume of $S$.
* -
$L_{2}(S)$ is one half of the surface area.
* -
To compute $L_{1}(S)$, we consider two types of edge: convex and concave. An
internal dihedral angle is associated to each edge $i$. If this angle is less
than or equal to $\pi$, the edge is considered to be convex and the angle is
denoted by $\alpha_{i}$. Let $h$ be the number of such edges. If the angle is
larger than $\pi$ the edge is considered to be "concave" and the angle is
denoted by $\beta_{i}$. Let $k$ be the number of such angles, then:
$L_{1}(S)=\sum_{i=1}^{h}\frac{(\pi-\alpha_{i})}{2\pi}l_{i}+\sum_{i=1}^{k}\frac{\cot(\beta_{i}/2)}{\pi}l_{i},$
where $l_{i}$ is the length of edge $i$.
## Conclusion
In all the examples considered, the Steiner formula for the tube governs the
expansion of the tail of the maximum as if the excursion set was precisely a
unique ball with a random radius. We have not found any counter-example to
that principle and we therefore conjecture that the result is true for a much
wider class of sets than those considered in this paper.
## 4 Appendix
In this appendix, we prove that a compact connected set in ${\mathbb{R}}^{2}$
with piecewise-$\mathcal{C}^{2}$ boundary and without concave irregular points
will satisfy the SFH. This is very similar to the general result of Adler and
Taylor, see Theorem 14.3.3 in [1]. However, these authors just clarified and
specified this theorem in the convex case and we think that there is a need to
provide the following proof.
Firstly, the Steiner formula (16) has already been established. We now
consider the excursion probability. We recall the following definitions
* •
Let $S_{2}$ be the interior of $S$; $S_{1}$ be the union of the
$\mathcal{C}^{2}$ edges and $S_{0}$ be the union of the convex irregular
points.
* •
For $t\in S_{j}$, $X^{\prime}_{j}(t)$ and $X^{\prime\prime}_{j}(t)$ are the
first and second derivatives of $X$ along $S_{j}$ respectively;
$X^{\prime}_{j,N}(t)$ denotes the outward normal derivative.
In our case, it is easy to see that:
$\kappa(S)=\underset{t\in S}{\sup}\underset{s\in S,\;s\neq
t}{\sup}\frac{\textnormal{dist}(s-t,C_{t})}{\|s-t\|^{2}}<\infty.$
In order to apply Theorem 8.12 and Corollary 8.13 of Azaïs and Wschebor [6],
we have to check the conditions (A1) to (A5) (see [6, p. 185]). The first
three are regularity conditions that are included in Assumption A. Note that
since the edges are of dimension 1, a direct proof of the Rice formula can be
performed without assuming that they are of class $\mathcal{C}^{3}$ as in
(A1).
* •
The condition (A4) states that the maximum is attained at a single point. It
can be deduced from the Bulinskaya lemma (Proposition 6.11 in [6]) since for
$s\neq t$, $(X(s),X(t),X^{\prime}(s),X^{\prime}(t))$ has a non-degenerate
distribution.
* •
The condition (A5) states that there is almost surely no point $t\in S$ such
that $X^{\prime}(t)=0$ and $\det\left(X^{\prime\prime}(t)\right)=0$. It can be
deduced from Proposition 6.5 in [6] applied to the process, $X^{\prime}(t)$,
which is $\mathcal{C}^{2}$.
Since all the required conditions are met, we have:
$\underset{u\rightarrow+\infty}{\liminf}-2u^{-2}\log\big{[}\int_{u}^{\infty}p^{E}(x)dx-\mathbb{P}\\{M_{S}\geq
u\\}\big{]}\geq 1+\underset{t\in
S}{\inf}\frac{1}{\sigma_{t}^{2}+\kappa_{t}^{2}}>1,$ (19)
where
* •
$p^{E}(x)$ is the approximation of the density of the maximum given by the
Euler characteristic method. More precisely,
$\begin{array}[]{rl}p^{E}(x)=&\displaystyle\sum_{t\in
S_{0}}\mathbb{E}\left(\mathbb{I}_{X^{\prime}_{0}(t)\in\widehat{C}_{t,0}}\mid
X(t)=x\right)\varphi(x)\\\
&+\displaystyle\sum_{j=1}^{2}(-1)^{j}\int_{S_{j}}\mathbb{E}\left(\det\left(X^{\prime\prime}_{j}(t)\right)\mathbb{I}_{X^{\prime}_{j,N}(t)\in\widehat{C}_{t,j}}\mid
X(t)=x,\,X^{\prime}_{j}(t)=0\right)\frac{\varphi(x)}{(2\pi)^{j/2}}dt,\\\
\end{array}$ (20)
where $\widehat{C}_{t,j}$ is the dual cone of the support cone $C_{t}$,
$\widehat{C}_{t,j}=\\{z\in{\mathbb{R}}^{2}:\;\langle z,x\rangle\geq
0,\;\forall\;x\in C_{t}\\}.$
* •
$\displaystyle\sigma_{t}^{2}=\underset{s\in
S\setminus\\{t\\}}{\sup}\frac{\textnormal{Var}\left(X(s)\mid
X(t),X^{\prime}(t)\right)}{(1-\textnormal{Cov}(X(s),X(t)))^{2}}.$
* •
$\displaystyle\kappa_{t}=\underset{s\in
S\setminus\\{t\\}}{\sup}\frac{\textnormal{dist}\left(\frac{\partial}{\partial
t}\textnormal{Cov}(X(s),X(t)),C_{t}\right)}{1-\textnormal{Cov}(X(s),X(t))}.$
We compute $p^{E}(x)$ as follows:
* •
When $j=2$, there is no normal space and $X^{\prime}_{2,N}(t)$ makes no sense.
It is easy to see that (see, for example, Azaïs and Wschebor [6, p. 244])
$\int_{S_{2}}\mathbb{E}\left(\det\left(X^{\prime\prime}_{2}(t)\right)\mid
X(t)=x,\,X^{\prime}_{2}(t)=0\right)dt=\sigma_{2}(S)(x^{2}-1).$
* •
When $j=0$, $X^{\prime}_{0,N}(t)\\-=X^{\prime}(t)$ and:
$\mathbb{E}\left(\mathbb{I}_{X^{\prime}(t)\in\widehat{C}_{t,0}}\mid
X(t)=x\right)=\frac{\mathcal{A}(\widehat{C}_{t,0})}{2\pi};$
where $\mathcal{A}(\widehat{C}_{t,0})$ is the angle of the cone that is equal
to the discontinuity of the angle of the tangent at the irregular point $t$.
* •
When $j=1$, we consider a point $t$ on an edge $L$ of the exterior boundary.
Note that in this case, the support cone $C_{t}$ is just a half-plane, so the
event $\\{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}\\}$ can be viewed as
$\\{X^{\prime}_{1,N}(t)\geq 0\\}$.
At the point $t$, the second derivative along the curve can be expressed as:
$X^{\prime\prime}_{1}(t)=X^{\prime\prime}_{T}(t)+C(t)X^{\prime}_{1,N}(t),$
where $X^{\prime\prime}_{T}$ is the second derivative in the tangent direction
and $C(t)$ is the signed curvature at $t$.
It is easy to check that the covariance function of the vector
$(X^{\prime\prime}_{T},X^{\prime}_{1,N},X,X_{1}^{\prime})$ is:
$\left(\begin{array}[]{cccc}\textnormal{Var}(X^{\prime\prime}_{T})&0&-1&0\\\
0&1&0&0\\\ -1&0&1&0\\\ 0&0&0&1\\\ \end{array}\right).$
Therefore, for such an edge $L$,
$\mathbb{E}\left(X^{\prime\prime}_{1}(t)\mathbb{I}_{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}}\mid
X(t)=x,\,X^{\prime}_{1}(t)=0\right)=\mathbb{E}\left(\left(-x+C(t)X^{\prime}_{1,N}(t)\right)\mathbb{I}_{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}}\right)=\frac{-x}{2}+\frac{C(t)}{\sqrt{2\pi}}$
and
$-\int_{L}\mathbb{E}\left(X^{\prime\prime}_{1}(t)\mathbb{I}_{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}}\mid
X(t)=x,\,X_{1}^{\prime}(t)=0\right)\frac{\varphi(x)}{\sqrt{2\pi}}dt=\frac{\sigma_{1}(L)x}{2\sqrt{2\pi}}\varphi(x)-\frac{\varphi(x)}{2\pi}\int_{L}C(t)dt.$
The quantity $-\int_{L}C(t)dt$ can be viewed as the variation of the angle of
the tangent from the beginning to the end of this edge.
Since we complete a whole turn in the positive orientation:
$\sum_{\textnormal{ irregular points of the ext.
boundary}}\mathcal{A}(\widehat{C}_{t})\quad+\sum_{\textnormal{edges of the
ext. boundary}}-\int_{L_{i}}C(t)dt=2\pi.$
For a point $t$ on an edge $L_{i}$ of the interior boundary (holes), the
interpretation of the second derivative changes into:
$X^{\prime\prime}_{1}(t)=X^{\prime\prime}_{T}(t)-C(t)X^{\prime}_{1,N}(t).$
Therefore,
$-\int_{L_{i}}\mathbb{E}\left(X^{\prime\prime}_{1}(t)\mathbb{I}_{X^{\prime}_{1,N}(t)\in\widehat{C}_{t,1}}\mid
X(t)=x,\,X_{1}^{\prime}(t)=0\right)\frac{\varphi(x)}{\sqrt{2\pi}}dt=\frac{\sigma_{1}(L_{i})x}{2\sqrt{2\pi}}\varphi(x)+\frac{\varphi(x)}{2\pi}\int_{L_{i}}C(t)dt.$
For the boundary of a hole inside $S$,
$\sum_{\textnormal{irregular
points}}\mathcal{A}(\widehat{C}_{t})+\sum_{\textnormal{edges}}\int_{L_{i}}C(t)dt=-2\pi.$
In conclusion, substituting into (20),
$p^{E}(x)=\chi(S)\varphi(x)+\frac{\sigma_{1}(\partial
S)}{2\sqrt{2\pi}}x\varphi(x)+\frac{\sigma_{2}(S)}{2\pi}(x^{2}-1)\varphi(x),$
since the Euler characteristic $\chi(S)$ is equal to $1$ (the number of
connected components) minus the number of the holes.
Integrating $p^{E}(x)$, we obtain the asymptotic expansion:
$\mathbb{P}(M_{S}\geq u)=\chi(S)\overline{\Phi}(u)+\frac{\sigma_{1}(\partial
S)}{2\sqrt{2\pi}}\varphi(u)+\frac{\sigma_{2}(S)}{2\pi}u\varphi(u)+Rest,$
where $Rest$ is super-exponentially smaller in the sense of (19). This implies
a correspondence between the asymptotic expansion and the Steiner formula.
## References
* [1] R.J. Adler and J. Taylor, Randoms fields and Geometry, Springer, New York, 2007.
* [2] L. Ambrosio, A. Colesanti and E. Villa, Outer Minkowski content for some classes of closed sets, Math. Ann. 342 (2008), no. 4, 727-748.
* [3] J.M. Azaïs and C. Delmas, Asymptotic expansions for the distribution of the maximum of Gaussian random fields, Extremes 5 (2002), no. 2, 181-212.
* [4] J.M. Azaïs and V.H. Pham, The record method for two and three dimensional parameters random fields, ALEA, Lat. Am. J. Probab. Math. Stat. 11 (2014), no. 1, 161-183.
* [5] J.M. Azaïs and M. Wschebor, A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail, Stochastic Process. Appl. 118 (2008), no. 7, 1190-1218.
* [6] J.M. Azaïs and M. Wschebor, Level sets and extrema of random processes and fields, John Wiley and Sons, 2009.
* [7] J.M. Azaïs and M. Wschebor, The tail of the maximum of smooth Gaussian fields on fractal sets, Journal of Theoretical Probability (2012).
* [8] C. Delmas, Distribution du maximum d’un champ aléatoire et application statistiques, Ph.D. thesis, Université Paul Sabatier (2001).
* [9] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-481.
* [10] D. Hug, G. Last and W. Weil, A local Steiner-type formula for general closed sets and applications, Math. Z. 246 (2004), no. 1-2, 237-272.
* [11] J. Pickands III,Asymptotic properties of the maximum in a stationary Gaussian process, Trans. Amer. Math. Soc. 145 (1969), 75-86.
* [12] V.I. Piterbarg,Asymptotic methods in the theory of Gaussian processes and fields, Translated from the Russian by V. V. Piterbarg. Revised by the author. Translations of Mathematical Monographs, 148. American Mathematical Society, Providence, RI, 1996.
* [13] V.I. Piterbarg, Rice’s method for Gaussian random fields, Fundamental and Applied Mathematics, 1996, vol.2, No 1, 20 pp. (in Russian).
* [14] S.O. Rice, The Distribution of the Maxima of a Random Curve,. Amer. J. Math. 61 (1939), no. 2, 409-416.
* [15] I. Rychlik, New bounds for the first passage, wave-length and amplitude densities, Stochastic Process. Appl. 34 (1990), no. 2, 313-339.
* [16] J. Sun, Tail probabilities of the maxima of Gaussian random fields, Ann. Probab. 21 (1993), 34-71.
* [17] A. Takemura and S. Kuriki, On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains, Ann. Appl. Probab. 12 (2002), 768-796.
* [18] A. Takemura and S. Kuriki, Tail probability via the tube formula when the critical radius is zero, Bernoulli 9 (2003), no. 3, 535-558.
* [19] J. Taylor, A. Takemura and R.J. Adler, Validity of the expected Euler characteristic heuristic, Ann. Probab. 33 (2005), no. 4, 1362-1396.
[email protected]
[email protected]
|
arxiv-papers
| 2013-06-14T13:50:53 |
2024-09-04T02:49:46.508211
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/",
"authors": "Jean-Marc Aza\\\"is, Viet-Hung Pham",
"submitter": "Pham Viet-Hung",
"url": "https://arxiv.org/abs/1306.3397"
}
|
1306.3422
|
# Spontaneous centralization of control in a network of company ownerships
Sebastian M. Krause [email protected] Tiago P. Peixoto
[email protected] Stefan Bornholdt [email protected] Institut
für Theoretische Physik, Universität Bremen, D-28359 Bremen, Germany
###### Abstract
We introduce a model for the adaptive evolution of a network of company
ownerships. In a recent work it has been shown that the empirical global
network of corporate control is marked by a central, tightly connected “core”
made of a small number of large companies which control a significant part of
the global economy. Here we show how a simple, adaptive “rich get richer”
dynamics can account for this characteristic, which incorporates the increased
buying power of more influential companies, and in turn results in even higher
control. We conclude that this kind of centralized structure can emerge
without it being an explicit goal of these companies, or as a result of a
well-organized strategy.
## I Introduction
The worldwide network of company ownership provides crucial information for
the systemic analysis of the world economy Schweitzer _et al._ (2009); Farmer
_et al._ (2012). A complete understanding of its properties and how they are
formed has a wide range of potential applications, including assessment and
evasion of systemic risk Battiston _et al._ (2012), collusion and antitrust
regulation Gulati _et al._ (2000); Gilo _et al._ (2006), market monitoring
Diamond (1984); Chirinko and Elston (2006), and strategic investment Teece
(1992). Recently, Vitali et al Vitali _et al._ (2011) inferred the network
structure of global corporate control, using the Orbis 2007 marketing database
111http://www.bvdinfo.com/products/company-information/international/orbis.
Analyzing its structure, they found a tightly connected “core” made of a small
number of large companies (mostly financial institutions) which control a
significant part of the global economy. A central question which arises is
what is the dominant mechanism behind this centralization of control. The
answer is not obvious, since the decision of firms to buy other firms can be
driven by diverse goals: Banks act as financial intermediaries doing
monitoring for uninformed investors Diamond (1984); Chirinko and Elston
(2006), managers can improve their power by buying other firms instead of
paying dividends Jensen (1986), speculation on stock prices as well as
dividend earnings can be a significant source of revenue Modigliani and Miller
(1958); La Porta _et al._ (2000); Jensen (1986), and companies can have
strategic advantages, e.g. due to knowledge sharing Teece (1992); Hamel
(1991); Dyer and Singh (1998). Another possible hypothesis for control
centralization is that managers collude to form influential alliances: Indeed,
agents (e.g. board members) often work for different firms in central
positions Battiston and Catanzaro (2004). Although all these factors are
likely to play a role, we here investigate a different hypothesis, namely that
a centralized structure may arise spontaneously, as a result of a simple
“richt-get-richer” dynamics Simon (1955), without any explicit underlying
strategy from the part of the companies. We consider a simple adaptive
feedback mechanism Gross and Blasius (2008), which incorporates the indirect
control that companies have on other companies they own, which in turn
increases their buying power. The higher buying power can then be used to buy
portions of more important companies, or a larger number of less important
ones, which further increases their relative control, and progressively
marginalizes smaller companies. We show that this simple dynamical ingredient
suffices to reproduce many of the qualitative features observed in the real
data Vitali _et al._ (2011), including the emergence of a core-periphery
structure and the relative portion of control exerted by the dominating core.
Although this does not preclude the possibility that companies may take
advantage and further consolidate their privileged positions in the network,
it does suggest that deliberate strategizing may not be the dominating factor
which leads to global centralization.
## II Model description
We consider a network of $N$ companies, where a directed edge between two
nodes $j\to i$ means company $j$ owns a portion of company $i$. The relative
amount of $i$ which $j$ owns is given by the matrix $w_{ij}$ (i.e. the
ownership shares), such that $\sum_{j}w_{ij}=1$. We note that it is possible
for self-loops to exist, i.e. a company can in principle buy its own shares.
In the following, we describe a model with two main mechanisms: 1. The
evolution of the relative control of companies, given a static network; 2\.
The evolution of the network topology via adaptive rewiring of the edges.
### II.1 Evolution of control
Here we assume that if $j$ owns $i$, it exerts some influence on $i$ in a
manner which is proportional to $w_{ij}$. If we let $v_{j}$ describe the
relative amount of control a company $j$ has on other companies, we can write
$v_{j}=1-\alpha+\alpha\sum_{i}A_{ij}w_{ij}v_{i},$ (1)
where $A_{ij}$ is the adjacency matrix, the parameter $\alpha$ determines the
propagation of control and $1-\alpha$ is an intrinsic amount of independence
between companies 222Eq. 1 can be seen as a weighted version of the Katz
centrality index Katz (1953), which is one of many ways of measuring the
relative centrality of nodes in a directed network, such as PageRank Page _et
al._ (1999) and HITS Kleinberg (1999). It converges for $0\leq\alpha<1$ and we
enforce normalization with $\sum_{i}v_{i}=N$.. We further assume that the
control value $v_{j}$ directly affects other features such as profit margins,
and overall market influence, such that the buying power of companies with
large $v_{j}$ is also increased. This means that the ownership of a company
$i$ is distributed among the owners $j$, proportionally to their control
$v_{j}$, i.e.
$w_{ij}=\frac{A_{ij}v_{j}}{\sum_{l}A_{il}v_{l}},$ (2)
(see Fig. 1). These equations are assumed to evolve in a faster time scale,
such that equilibrium is reached before the topology changes, as described in
the next section.
Figure 1: Illustration of the control of firms including indirect control
(left) and the ownership being proportional to the control (right), as
described in the text.
### II.2 Evolution of the network topology
Companies may decide to buy or sell shares of a given company at a given time.
The actual mechanisms regulating these decisions are in general complicated
and largely unknown, since they may involve speculation, actual market value,
and other factors, which we do not attempt to model in detail here. Instead,
we describe these changes probabilistically, where an edge may be deleted or
inserted randomly in the network, and such moves may be accepted or rejected
depending on how much it changes the control of the nodes involved. For
simplicity, we force the total amount of edges in the network to be kept
constant, such that a random edge deletion is always accompanied by a random
edge insertion. Such “moves” may be rejected or accepted, based on the change
they bring to the $v_{j}$ values of the companies involved. If we let $m$ be
the company which buys new shares of company $l$, and $j$ which sells shares
of company $i$, the probability that the move is accepted is
$p=\min\left(1,e^{\beta(\tilde{w}_{lm}v_{l}-w_{ij}v_{i})}\right),$ (3)
where $w_{ij}$ is computed before the move and $\tilde{w}_{lm}$ afterwards,
and the parameter $\beta$ determines the capacity companies have to foresee
the advantage of the move, such that for $\beta=0$ all random moves are
accepted, and for $\beta\to\infty$ they are only accepted if the net gain is
positive (see Fig. 2). Note that in Eq. 3 it is implied that companies with
larger control will tend to buy more than companies with smaller control,
which is well justified by our assumption that control is correlated with
profit and wealth.
Figure 2: Illustration of the adaptive process, before the rewiring (left) and
afterwards (right), as described in the text.
The overall dynamics is composed by performing many rewiring steps as
described above, until an equilibrium is reached, i.e. the observed network
properties do not change any longer. In order to preserve a separation of time
scales between the control and rewiring dynamics, we performed a sufficiently
large number of iterations of Eqs. 1 and 2 before each attempted edge move.
## III Centralization of control
A typical outcome of the dynamics can be seen in Fig. 3 for a network with
$N=3\times 10^{4}$ nodes and average degree $\left<k\right>=2$, after an
equilibration time of about $6\times 10^{9}$ steps. In contrast to the case
with $\beta=0$, which results in a fully random graph, for a sufficiently high
value of $\beta$ the distribution of firm ownerships (i.e. the out-degree of
the nodes) becomes very skewed, with a bimodal form. We can divide the most
powerful companies into a broad range which owns shares from $10$ to about
$150$ other companies, and a separate group with $k_{\text{out}}>150$. The
correlation matrix of this network shows that these high-degree nodes are
connected strongly among themselves, and own a large portion of the remaining
companies (see Fig. 3). This corresponds to a highly connected “core” of about
45 nodes with $\left<k_{\text{sub}}\right>\approx 39.8$, which is highlighted
in red in Fig. 3c and can be seen separately in Fig. 3d. The distribution of
in-degree (not shown) is bimodal as well with highest values for the inner
core. With values up to $k_{\rm in}=50$, the highest in-degree (number of
owners) is considerably below the highest out-degree (number of firms owned at
once).
(a)
(b)
(c)
(d)
Figure 3: (a) Degree distribution of the resulting network for
$\left<k\right>=2$, a control propagation value of $\alpha=0.5$, $N=30000$ and
different values of prior knowledge $\beta$; (b) Degree correlation matrix for
$\beta=10$, showing the resulting core-periphery structure; (c) Graph layout
of the whole network, with red nodes representing a chosen fraction of the
most highly connected core, and blue ones the periphery; (d) Subgraph of the
most powerful companies with $v_{i}>20$ (about 100). The node colors and sizes
correspond to the $v_{i}$ values.
Figure 4: Left: Distribution of inherited control $v_{i}-(1-\alpha)$ for
$\alpha=0.5$ and different values of $\beta$; Right: Relative fraction of
control as a function of fraction of most powerful companies.
Similarly to the out-degree, the distribution of control values $v_{i}$ is
also bimodal for larger values of $\beta$, as can be seen in Fig. 4, and is
strongly correlated with the out-degree values. The total fraction of
companies controlled by the most powerful ones is very large, as shown on the
right panel of Fig. 4. For instance, we see that a fraction of around $0.15\%$
of the central core controls about $57\%$ of all companies. The companies with
intermediary values of control (and out-degree) also possess a significant
part of the global control, e.g. around $.85\%$ of the most powerful control
an additional $25\%$ of the network. It is important to emphasize the
difference between these two classes of companies for two reasons: Firstly the
inner core inherits control from intermediate companies without the need to
gather up all the minor companies. In fact the ownership links going out from
the inner core (about $10^{4}$) is enough to cover the direct control of only
a third of all companies, while the effective control is more than a half.
Secondly, the fraction of intermediary companies increases for larger
networks. For a network with $N=3\times 10^{5}$, the inner core includes a
fraction of only $0.04\%$, controlling an effective $41\%$ of the total
companies. Nonetheless, all the most powerful companies together account for
around $1\%$ of the network and $82\%$ of the total control; values which do
not change considerably with system size.
Let us compare the results presented so far with empirical data presented in
Vitali _et al._ (2011). For different reasons, this comparison can only be
qualitative. First of all, the empirical data includes economic agents with
different functions (shareholders, transnational companies and participated
companies) out of different sectors (eg. financial and real economy), while we
consider identical agents. Secondly, we force every company to be owned 100%,
while the empirical data neglects restrained shares and diversified holdings.
Thirdly, the control analysis in Vitali _et al._ (2011) is done somewhat
differently: All the $600,508$ economic agents were considered for the
topological characterization, while many companies (80% of all agents there)
were neglected for the control analysis. In the empirical data, a strongly
connected component of $1,318$ companies controls more than a half of all
companies arranged in the out component. This concentration is compatible with
the core-periphery structure presented in Fig. 3, however the empirical data
does not show a distinct bimodal structure. Nonetheless, there are highly
connected substructures in the core, e.g. a structure with 22 highly connected
financial companies ($\left<k_{\rm sub}\right>\approx 12$) was highlighted in
Battiston _et al._ (2012). The control concentration in the empirical data
was reported as a fraction of $0.5\%$ which controls $80\%$ of the network.
This is similar to the results of our model (see Fig. 4 on the right). There
are, however, features that our model does not reproduce, the most important
of which being the out-degree distribution of the network, which in Vitali
_et al._ (2011) is very broad, and displays no discernible scales, where in
our case it is either bimodal or Poisson-like. One possible explanation for
this discrepancy is that we have focused on equilibrium steady-state
configurations of the dynamics, whereas the real economy is surely far away
from such an equilibrium. A more precise model would need to incorporate such
transient dynamics in a more realistic way. Nevertheless, the general tendency
of the control to be concentrated on relatively few companies is evident in
such equilibrium states, and features very prominently in the empirical data
as well.
### III.1 Transition to centralization
To investigate the transition from homogeneous no centralized networks with
increasing $\beta$, we measured the inverse participation ratio
$I=\left[\frac{1}{TN}\sum_{ti}v_{i}(t)^{2}\right]^{-1}$ with the time $t$
summing over a sufficiently long time window of length $T$ after
equilibration. Since $\frac{1}{N}\leq I\leq 1$, we expect $I=1$ in the
perfectly homogeneous case where $v_{i}=1$ for all nodes, and $I=\frac{1}{N}$
if only one node has $v_{i}>0$, and the control is maximally concentrated. As
can be seen in Fig. 5, we observe a smooth transition from very homogeneous
companies connected in fully random manner for $\beta=0$, to a pronounced
concentration of control for increased $\beta$, for which the aforementioned
core-periphery is observed. The transition becomes more abrupt when either the
average degree $\left<k\right>$ is increased or the parameter $\alpha$ (which
determines the fraction of inherited control) is decreased.
Figure 5: Inverse participation ratio
$I=\left[\frac{1}{TN}\sum_{ti}v_{i}(t)^{2}\right]^{-1}$ as a function of
$\beta$, for a network with $N=10^{4}$, and for (left) $\left<k\right>=2$ and
different values of $\alpha$ and (right) $\alpha=0.5$ and different values of
$\left<k\right>$.
Figure 6: Distribution of out degrees (left) and inherited control
$v_{i}-(1-\alpha)$ (right) for $\beta=10$, $\left<k\right>=2$ and $N=30000$ as
in Fig. 3 and 4, but for different values of $\alpha$.
Centralization of control can emerge in different ways depending on the
parameters $\alpha$ and $\beta$. In Fig. 6, it is shown that different values
of $\alpha$ for a high value of $\beta=10$ can lead to a detached controlling
core ($\alpha=0.2$) or to broadly distributed control values ($\alpha=0.8$).
With smaller values of $\alpha$, indirect control is suppressed and companies
can gain power only by owning large numbers of marginal companies. E.g.: for
$\alpha=0.2$, this leads to a highly connected core of $41$ companies having
$\left<k_{\rm sub}\right>\approx 18.2$, the rest of the companies have very
little influence. For larger values of $\alpha$, indirect control has a larger
effect, which leads to a hierarchical network where companies with small
numbers of owned firms $k_{\rm out}$ may nevertheless inherit large control
values $v_{i}$. The case with $\alpha=0.5$ and $\beta=10$ shown in Figs. 3 and
4 exhibits a mixture of these two scenarios. The transition to a centralized
core also occurs when increasing $\beta$ and keeping $\alpha$ constant (see
right panel in Fig. 5).
Figure 7: Left: Graph layout of a $10\times 10$ lattice with $\alpha=0.9$. The
vertex sizes and colors correspond to the $v_{i}$ values, and the edge
thickness to the $w_{ij}$ values. Right: Distribution of inherited control
$v_{i}-(1-\alpha)$ for static poisson graphs having $\left<k\right>=2$ and
$N=30\,000$, with different values of $\alpha$ (for $\alpha=0.5$ and
$\alpha=0.8$ shifted). The dashed line is a power law with exponent $-1$.
One interesting aspect of the centralization of control as we have formulated
is that it is not entirely dependent on the adaptive dynamics, and occurs also
to some extent on graphs which are static. Simply solving Eqs. 1 and 2 will
lead to a non-trivial distribution of control values $v_{i}$ which depend on
the (in this case fixed) network topology and the control inheritance
parameter $\alpha$. In Fig. 7 is shown on the left the control values obtained
for a square $2D$ lattice with periodic boundary conditions, and bidirectional
edges. What is observed is a spontaneous symmetry breaking, where despite the
topological equivalence shared between all nodes, a hierarchy of control is
formed, which is not unique and will vary between each realization of the
dynamics. A similar behavior is also observed for fully random graphs, as
shown on the right of Fig. 7, where the distribution of control values becomes
increasingly broader for larger values of $\alpha$, asymptotically approaching
a power-law $\rho(v)\sim v^{-1}$ for $\alpha\to 1$. This behavior is similar
to a phase transition at $\alpha=1$, where at this point Eq. 1 no longer
converges to a solution.
## IV Conclusion
We have tested the hypothesis that a rich-get-richer process using a simple,
adaptive dynamics is capable of explaining the phenomenon of concentration of
control observed in the empirical network of company ownership Vitali _et
al._ (2011). The process we proposed incorporates the indirect control that
companies have on other companies they own, which increases their buying power
in a feedback fashion, and allows them to gain even more control. In our
model, the system spontaneously organizes into a steady-state comprised of a
well-defined core-periphery structure, which reproduces many qualitative
observations in the real data presented in Vitali _et al._ (2011), such as
the relative portion of control exerted by the dominating companies. Our model
shows that this kind of centralized structure can emerge without it being an
explicit goal of the companies involved. Instead, it can emerge simply as a
result of individual decisions based on local knowledge only, with the effect
that powerful companies can increase their relative advantage even further.
It is interesting to compare our model to other agent based models featuring
agents competing for centrality. The emergence of hierarchical, centralized
states with interesting patterns of global order was reported for agents
creating links according to game theory Holme and Ghoshal (2006); Lee _et
al._ (2011); Do _et al._ (2010) as well as for very simple effective rules of
rewiring according to measured centrality König and Tessone (2011); Bardoscia
_et al._ (2013). The latter is combined with phase transitions according to
the noise in the rewiring process. The stylized model of a society studied in
Bardoscia _et al._ (2013) shows a hierarchical structure, if the individuals
have a preference for social status. The intuitive emergence of hierarchy is
associated with shrinking mobility of single agents within the hierarchy. This
effect is present in our model as well and deserves further investigation.
Our results may shed light on certain antitrust regulation strategies. As we
found that a simple mechanism without collusion suffices for control
centralization, any regulation which is targeted to diminish such activities
may prove fruitless. Instead, targeting the self-organizing features which
lead to such concentration, such as e.g. limitations on the indirect control
of shareholders representing other companies, may appear more promising.
## References
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* Gulati _et al._ (2000) R. Gulati, N. Nohria, and A. Zaheer, Strategic management journal 21, 203 (2000).
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* Vitali _et al._ (2011) S. Vitali, J. B. Glattfelder, and S. Battiston, PLoS ONE 6, e25995 (2011).
* Note (1) http://www.bvdinfo.com/products/company-information/international/orbis.
* Jensen (1986) M. Jensen, Corporate Finance, and Takeovers. American Economic Review 76 (1986).
* Modigliani and Miller (1958) F. Modigliani and M. H. Miller, The American Economic Review 48, 261 (1958).
* La Porta _et al._ (2000) R. La Porta, F. Lopez-de Silanes, A. Shleifer, and R. W. Vishny, The Journal of Finance 55, 1 (2000).
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* Dyer and Singh (1998) J. H. Dyer and H. Singh, Academy of Management Review 23, 660 (1998).
* Battiston and Catanzaro (2004) S. Battiston and M. Catanzaro, The European Physical Journal B - Condensed Matter and Complex Systems 38, 345 (2004).
* Simon (1955) H. A. Simon, Biometrika 42, 425 (1955).
* Gross and Blasius (2008) T. Gross and B. Blasius, Journal of The Royal Society Interface 5, 259 (2008).
* Note (2) Eq. 1 can be seen as a weighted version of the Katz centrality index Katz (1953), which is one of many ways of measuring the relative centrality of nodes in a directed network, such as PageRank Page _et al._ (1999) and HITS Kleinberg (1999). It converges for $0\leq\alpha<1$ and we enforce normalization with $\sum@\slimits@_{i}v_{i}=N$.
* Holme and Ghoshal (2006) P. Holme and G. Ghoshal, Physical Review Letters 96, 098701 (2006).
* Lee _et al._ (2011) S. Lee, P. Holme, and Z.-X. Wu, Physical Review Letters 106, 028702 (2011).
* Do _et al._ (2010) A.-L. Do, L. Rudolf, and T. Gross, New Journal of Physics 12, 063023 (2010).
* König and Tessone (2011) M. D. König and C. J. Tessone, Physical Review E 84, 056108 (2011).
* Bardoscia _et al._ (2013) M. Bardoscia, G. De Luca, G. Livan, M. Marsili, and C. J. Tessone, Journal of Statistical Physics 151, 440 (2013).
* Katz (1953) L. Katz, Psychometrika 18, 39 (1953).
* Page _et al._ (1999) L. Page, S. Brin, R. Motwani, and T. Winograd, Stanford Infolab , 17 (1999).
* Kleinberg (1999) J. M. Kleinberg, J. ACM 46, 604–632 (1999).
|
arxiv-papers
| 2013-06-14T15:12:15 |
2024-09-04T02:49:46.519740
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Sebastian M. Krause, Tiago P. Peixoto, Stefan Bornholdt",
"submitter": "Tiago Peixoto",
"url": "https://arxiv.org/abs/1306.3422"
}
|
1306.3483
|
# On the realization problem of plane real algebraic curves as Hessian curves
Angelito Camacho Calderón and Adriana Ortiz Rodríguez Work supported by
CONACyT Work partially supported by DGAPA-UNAM grant PAPIIT-IN108112 and
N103010
###### Abstract
The Hessian Topology is a subject having interesting relations with several
areas, for instance, differential geometry, implicit differential equations,
analysis and singularity theory. In this article we study the problem of
realization of a real plane curve as the Hessian curve of a smooth function.
The plane curves we consider are constituted either by only outer ovals or
inner ovals. We prove that some of such curves are realizable as Hessian
curves.
Keywords: Hessian curves, Hessian Topology, Real algebraic curves
MS classification: 53A15; 53A05.
## Introduction
It is well known that the points of a generic smooth surface in
$\mathbb{R}^{3}$ ($\mathbb{R}$P3) are classified as elliptic, parabolic and
hyperbolic. The geometric structure of such a surface is described by the
behavior of the sets conformed by these points, that is, the set of parabolic
points is a smooth curve called parabolic curve whose complement is a disjoint
union of elliptic and hyperbolic domains. A problem that has been studied for
a long time is the description of this geometric structure, in particular the
topology of the parabolic curve [13], [9], [5], [2], [3], among others. When
the surface is expressed locally as the graph $\,z=f(x,y)\,$ of a smooth
function $f$, the parabolic curve is the image under $f$, of the Hessian curve
of $f$, which is the set of zeros of the Hessian polynomial of $f$, Hess
$f:=f_{xx}f_{yy}-f_{xy}^{2}$. Moreover, if the surface is the graph of a
smooth function defined on the whole plane, then this local description is
global, and so, to understand the behavior of the parabolic curve in this
case, it is enough to study the Hessian curve.
Some topics concerning the Hessian Topology [1], [4] (problems 2000-1, 2000-2,
2001-1, 2002-1) are based in the study of the following problems. How many
compact connected components can have the (smooth) parabolic curve of the
graph of a real polynomial of degree $n$ in two variables? and, how are they
distributed? [10], [11], [6], [7], [8]. Another such a problem is to determine
the number of connected components of the parabolic curve of a smooth
algebraic surface of fixed degree $n$ in the real projective space [14], [10],
[6]. The realization problem of a smooth plane curve as the Hessian curve of a
smooth function is another such a problem of this subject, which is treated
here. The meaning of all these problems is to find out if the property of
being Hessian curve imposes conditions on the topology and geometry of curves
and surfaces.
A connected component of a compact smooth plane curve is called outer oval if
it is contained in the outside component of the complement of each of the
connected components of the curve. In this paper we give a family of real
polynomials $\\{f_{k}\\}_{k\geq 2}\,$ such that the Hessian curve of $f_{k}$
consists exactly of $k-2$ outer ovals. Jointly, in Theorem 2, we give a
detailed description of the geometric structure of the graph of each one of
these polynomials.
In Theorem 10, for each even natural number $k$, we present real polynomials
of degree $\,k\,$ such that its Hessian curve is formed exactly by $k-2$
concentric circles. We also describe the geometric structure of the graph of
these polynomials. In an analogue form, we contribute to the odd case with a
family of real functions $\\{f_{r}\\}_{r\geq 1}\,$ such that the Hessian curve
of $f_{r}$ are constituted by $2r-1$ concentric circles, Theorem 11.
To conclude, we show in Proposition 12, that if a smooth plane curve $\,C\,$
is equivalent by an affine transformation on the plane to any of the previous
Hessian curves, then $C$ is also a Hessian curve.
## Preliminaries
Points on a smooth surface (not necessarily generic) in $\mathbb{R}^{3}$
($\mathbb{R}$P3) can be classified in terms of the maximal order of contact of
the tangent lines to the surface at each point. A point $p$ of a surface is
elliptic if all lines tangent to the surface at $p$ have a contact of order
$2$ with the surface at that point. The tangent lines having an order of
contact at least three with the surface are known as asymptotic lines. A point
$p$ is hyperbolic if it has exactly two asymptotic lines. We say that an
hyperbolic point is an inflexion point if it has at least one asymptotic line
with order of contact greater than $3$.
The points on the surface that are not elliptic or hyperbolic will be called
parabolic points and the set constituted by them will be referred to as
parabolic curve. This curve is formed by the following types of points (some
of them may not appear). A generic parabolic point is a point which has
exactly one asymptotic line and the order of contact of this line with the
surface is $3$. A special parabolic point (other authors call it Gaussian cusp
or godron) is a point at which the parabolic curve is smooth locally, the
4-jet (without constant neither linear terms) of the function defining the
surface at this point is not a perfect square and moreover, it has exactly one
asymptotic line whose order of contact is at least 4. Finally, a degenerate
parabolic point is a parabolic point that is not generic or special. It is
important to mention that all these type of points are invariant under the
action of the affine group on 3-space.
The directions determined on the $xy$-plane of the asymptotic lines under the
projection $\pi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{2},\
(x,y,z)\mapsto(x,y)$, are the solutions of the quadratic differential
equation:
$f_{xx}(x,y)dx^{2}+2f_{xy}(x,y)dxdy+f_{yy}(x,y)dy^{2}=0,$
where the quadratic differential form on the left will be referred to as the
second fundamental form of $f$. Its discriminant defined as
$\Delta_{II_{f}}=f_{xy}^{2}-f_{xx}f_{yy},$
allows us to characterize the type of point in the graph of $f$. That is,
$(p,f(p))$ is elliptic, parabolic or hyperbolic if $\Delta_{II_{f}}(p)$ is
negative, zero or positive, respectively.
## Results and Proofs
## 1 Outer ovals
We recall that a connected component of a compact smooth plane curve is called
outer oval if it is contained in the outside component of the complement of
each of the connected components of the curve.
###### Definition 1
A set of polynomials $\,l_{1},\ldots,l_{n}\in\mathbb{R}[x,y]\,$ of degree one
are in good position if for each $\,i=1,\ldots,n,\,$ the straight line
$\,l_{i}(x,y)$= $0$ contains no critical point of the function $\,\prod_{j\neq
i}l_{j}$.
###### Theorem 2
Let $\,m,n\,$ two natural numbers. Consider the polynomial
$\,f(x,y)=\prod_{i=0}^{m}l_{i}(x,y)\prod_{j=0}^{n}{g_{j}}(x,y)$
of degree $\,m+n+2\,$ given as the product of polynomials of degree one,
$l_{0}(x,y)=y-ax,\,{g_{0}}(x,y)=y-bx,\,l_{i}(x,y)=x-a_{i},\,{g_{j}}(x,y)=x-b_{j}$,
with $\,a,b,a_{i},b_{j}\in\mathbb{R},\,i=1,\ldots,m,j=1,\ldots,n,\,a\neq b\,$
and $\,a_{m}<\cdots a_{1}<0<b_{1}<\cdots<b_{n}$. If the set of polynomials
$\,\\{l_{i}(x,y),\,g_{j}(x,y):i=0,\ldots,m,j=0,\ldots,n\\}\,$ are in good
position, then the Hessian curve of $f$ consists exactly of $\,m+n\,$ outer
ovals. Moreover, the graph of $f$ has $\,3(m+n)\,$ special parabolic points
and its inflexion curve is the set $\,\\{f(x,y)=0\\}$.
Proof. In order to prove that the inflexion curve is the set
$\,\\{f(x,y)=0\\}$, we state the following Remark which is a consequence of
the definition of $f(x,y)$.
###### Remark 3
The restriction of $f$ to any straight line $m\in\mathbb{R}^{2}$ is a one
variable real polynomial such that all its inflexion points are non
degenerated.
###### Lemma 4
The inflexion curve of the graph of $f$ is the set $\,\\{f(x,y)=0\\}$.
Proof. Because the order of contact of each straight line $\,l_{i}(x,y)=0,$
$\,i=1,\ldots,n,\,$ with the graph of $f$ is infinite then these lines are
contained in the inflexion curve. Inversely, suppose that $p$ is a point of
the inflexion curve and does not belong to the set $\,L:=\\{f(x,y)=0\\}$. So,
at least one asymptotic direction of $p$ has order of contact equal to or
greater than 4 with the graph of $f$. This implies that the restriction of $f$
to the line, which is the projection to the $xy$-plane of this asymptotic
line, has a degenerated inflexion point. This contradicts Remark 3.
###### Lemma 5
A point is in the intersection of the parabolic curve with the inflexion curve
if and only if this point is a special parabolic point or a degenerated
parabolic point.
Proof. If a point belongs to both curves, parabolic and inflexion, it has an
asymptotic line with order of contact at least 4 with the surface. Then, the
point is special parabolic or degenerated. Inversely, because a special
parabolic point satisfies that its asymptotic line has order of contact at
least 4 with the surface, it lies in the inflexion curve. It only remains to
prove that any degenerated point lies in the inflexion curve. Note that a
degenerated point has one or an infinite number of asymptotic lines. For the
first case, if the Hessian curve is singular at this point, the asymptotic
line has order of contact at least 4 with the surface, so it lies in the
inflexion curve. Remaining in the first case, suppose that the Hessian curve
is smooth at the point and the 4-jet (without constant neither linear terms)
of the function is a perfect square. This implies that the point lies in the
inflexion curve. For the case in which every tangent line at the point is an
asymptotic line, we have that the second fundamental form of $f$ at the point
is identically zero, that is, each tangent line has order of contact at least
4 with the surface.
###### Lemma 6
The intersection of the parabolic curve with the inflexion curve consists of
$\,3(m+n)$ points.
Proof. After some straightforward calculus we have that the restriction of the
Hessian of $f$ to the straight line $\,x=a_{i},\,$ for $i=1,,\ldots,m,\,$ is
$\displaystyle\mathrm{Hess}f_{|_{x=a_{i}}}(x,y)$ $\displaystyle=$
$\displaystyle-(2y-(a+b)a_{i})^{2}\Big{(}\prod_{j=1}^{n}(a_{i}-b_{j})\sum_{k=1}^{m}\prod_{s\neq
k}^{m}(a_{i}-a_{s})\Big{)}^{2}$ $\displaystyle=$
$\displaystyle-(2y-(a+b)a_{i})^{2}\,C_{i}^{2},$
where $\,C_{i}=\prod_{j=1}^{n}(a_{i}-b_{j})\sum_{k=1}^{m}\prod_{s\neq
k}^{m}(a_{i}-a_{s})\,$ is a nonzero real number for each $\,i=1,\ldots,m.$
Then $\mathrm{Hess}f_{|_{x=a_{i}}}(x,y)=0\,$ if and only if
$\,y=\frac{(a+b)a_{i}}{2},\,i=1,\ldots,m.$ The points
$\left(a_{i},\frac{(a+b)a_{i}}{2}\right),$ $i=1,\ldots,m,$ are between the
line $l_{0}(x,y)=0$ and the line $g_{0}(x,y)=0$ because
$\,ba_{i}<\frac{(a+b)a_{i}}{2}<aa_{i}$. Moreover, the other points on the
lines $l_{i}=0,\,i=1,\ldots,m,$ are hyperbolic points.
By a similar analysis for the lines $\,x=b_{j}\,$ for $j=1,,\ldots,n$ we have
that the points $\left(b_{j},\frac{(a+b)b_{j}}{2}\right),$ $j=1,\ldots,n,$ are
parabolic and they are in the interior of the compact segment
$[ab_{j},bb_{j}]$ of the line $\,x=b_{j}\,$.
Now, let us analyze the intersection of the parabolic curve with the straight
lines $l_{0}(x,y)=0$ and $g_{0}(x,y)=0$. Replacing $\,y=ax$ in the Hessian
polynomial of $f$ we obtain
$\displaystyle\mathrm{Hess}f(x,y)|_{y=ax}=-(a-b)^{2}\left[\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\right.\qquad\qquad\qquad$
$\displaystyle\left.+\,x\left(\sum_{l=1}^{m}\prod_{i\neq
l}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})\right)\right]^{2}.$
Note that the expression being inside of the square brackets is the first
derivative of the one variable polynomial $\displaystyle
g(x)=x\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})$ of degree $m+n+1$
which has $m+n+1$ simple zeros and $m+n$ simple critical points. This implies
that $\mathrm{Hess}f(x,y)|_{y=ax}$ has $m+n$ simple real roots.
By an analogous analysis for the straight line $\,y=bx$ we obtain that
$\mathrm{Hess}f(x,y)|_{y=bx}$ has $m+n$ simple real roots.
###### Lemma 7
If the set of polynomials
$\,\\{l_{i}(x,y),\,g_{j}(x,y):i=0,\ldots,m,j=0,\ldots,n\\}\,$ are in good
position, then $\,\\{f(x,y)=0\\}$ contains no degenerated parabolic point.
Before starting with the proof of this Lemma we state the following Remark
which is obtained by a suitable grouping of terms of the Hessian polynomial of
$f$.
###### Remark 8
The Hessian polynomial of $f$ can be written as
$\mathrm{Hess}f(x,y)=\beta(x)\left(y-\frac{(a+b)x}{2}\right)^{2}+\alpha(x),$
where
$\displaystyle\alpha(x)=-(a-b)^{2}\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})\left[\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})\right.\qquad\qquad\quad$
$\displaystyle+2x\left(\sum_{l=1}^{m}\prod_{i\neq
l}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})\right)\qquad\quad\,\,\,$
$\displaystyle+\frac{x^{2}}{2}\left(\sum_{l=1}^{m}\sum_{s\neq
l}^{m}\prod_{i\neq
l,s}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\sum_{k=1}^{n}\sum_{t\neq
k}^{n}\prod_{j\neq k,t}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})\right.$
$\displaystyle\left.\left.+2\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\sum_{l=1}^{m}\prod_{i\neq l}^{m}(x-a_{i})\right)\right],\hskip
150.79968pt$
$\displaystyle\beta(x)=2\left(y-\frac{(a+b)x}{2}\right)^{2}\left[\prod_{i=1}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})\bigg{(}\sum_{l=1}^{m}\sum_{s\neq
l}^{m}\prod_{i\neq l,s}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})\right.$
$\displaystyle+\sum_{k=1}^{n}\sum_{t\neq k}^{n}\prod_{j\neq
k,t}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})+2\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\sum_{l=1}^{m}\prod_{i\neq l}^{m}(x-a_{i})\bigg{)}\qquad\quad$
$\displaystyle\left.-2\left(\sum_{l=1}^{m}\prod_{i\neq
l}^{m}(x-a_{i})\prod_{j=1}^{n}(x-b_{j})+\sum_{k=1}^{n}\prod_{j\neq
k}^{n}(x-b_{j})\prod_{i=1}^{m}(x-a_{i})\right)\right].\qquad\qquad$
Proof. (Lemma 7). We shall prove that the points lying in both curves,
parabolic and inflexion, are all special points.
The next calculus shows that the second fundamental form of $f$ at each point
$q\in\\{f(x,y)=0\\}\,$ is nonzero. Let suppose that $\,q\in\\{l_{i}(x,y)=0\\}$
for some $i\in\\{0,\ldots,m\\}$ (the following arguments are valid for the
case that $q$ belongs to some $l_{i}(x,y)=0$). So, $\,f(x,y)=l(x,y)l_{i}(x,y)$
where $l(x,y)=\prod_{k=0,k\neq i}^{m}l_{k}(x,y)\prod_{j=0}^{n}g_{j}(x,y).$
Then,
$\displaystyle f_{xx}(q)=2\frac{\partial l_{i}}{\partial x}(q)l_{x}(q),\qquad
f_{yy}(q)=2\frac{\partial l_{i}}{\partial y}(q)l_{y}(q),$ $\displaystyle
f_{xy}(q)=\frac{\partial l_{i}}{\partial x}(q)l_{y}(q)+\frac{\partial
l_{i}}{\partial y}l_{x}(q).\qquad\qquad$
The hypothesis of good position implies that $q$ can not be critical point of
$l$. So, $l_{x}(q)\neq 0$ or $l_{y}(q)\neq 0$ and the second fundamental form
of $f$ at $q$ is non degenerated. That is, each point has exactly one
asymptotic line.
Now, let $p$ be a point in the intersection of the parabolic curve with the
inflexion curve. We shall prove that the Hessian curve at $p$ is non singular.
The proof will be in two cases.
* •
The point $p$ lies in any of the lines $\,l_{0}(x,y)=0\,$ or $\,g_{0}(x,y)=0$.
After an appropriate affine transformation of the $xy$-plane, we can suppose
that such a line is the line $y=0$. Denote by $\,x_{1},\ldots,x_{m+n+1}$ the
intersection points of the line $y=0$ with all other lines. Since
$f(x,y)=y\prod_{k=1}^{m+n+1}r_{k}(x,y)$, where $\,r_{k}(x,y)$ are real
polynomials of degree one, then,
$f_{y}(x,y)|_{y=0}=\prod_{k=1}^{m+n+1}r_{k}(x,y)|_{y=0}.$
Define $\,f_{y}(x,y)|_{y=0}=F(x)$. The polynomial $F(x)$ is of degree $m+n+1$
and its roots are real and simple because they are the points
$\,x_{1},\ldots,x_{m+n+1}$. Moreover, it has $m+n$ non degenerated critical
points, that is, the function $\,F^{\prime}(x)=f_{xy}(x,y)|_{y=0}\,$ has $m+n$
simple real zeroes: exactly one in the interior of each segment
$\,[x_{i},x_{i+1}],\,i=1,\ldots,m+n$ and $\,f_{xxy}(x,y)|_{y=0}\neq 0\,$ at
these roots. Note that
$(\mathrm{Hess}f(x,y))|_{y=0}=-(f_{xy}(x,y))^{2}|_{y=0}=-(F^{\prime}(x))^{2}.$
So, the point $p$, after applying the affine transformation, has the form
$\tilde{p}=(\tilde{x},0)$, where $\tilde{x}$ is one of the critical points of
$F$. Moreover, $\,f_{xy}(x,y),\,f_{xx}(x,y)$ are zero at this point. Finally,
since the second fundamental form of $f$ at $\tilde{p}$ is non zero and
$\,f_{xxy}(\tilde{p})\neq 0$, then
$\displaystyle\left(\frac{\partial}{\partial
y}\mathrm{Hess}f\right)(\tilde{p})=(f_{xx}f_{yyy}+f_{xxy}f_{yy}-2f_{xy}f_{xyy})|_{\tilde{p}}$
$\displaystyle=f_{xxy}(\tilde{p})f_{yy}(\tilde{p})\neq 0.\hskip 68.00206pt$
* •
The point $p$ lies in one of the lines $\,l_{1}(x,y)=0,\ldots,l_{m}(x,y)=0,$
$g_{1}(x,y)=0,\ldots,g_{n}(x,y)=0.$ In the proof of Lemma 6 it is shown that
$p$ has the form $\left(x_{0},\frac{(a+b)x_{0}}{2}\right)\,$ where
$\,x_{0}\in\\{a_{1},\ldots,a_{m},b_{1},\ldots,b_{n}\\}$. So, by Remark 8 we
have $\,\frac{\partial}{\partial
y}\mathrm{Hess}f(p)=\alpha^{\prime}(x_{0})\neq 0$.
Note that the asymptotic line at each one of these points is a line of the set
$\,\\{l_{i}(x,y)=0,\,g_{j}(x,y)=0:i=0,\ldots,m,j=0,\ldots,n\\}$ because the
parabolic curve is tangent to this set.
It only remains to prove that the 4-jet (without constant neither linear
terms) of the function at these points is free square. For the points
$\left(a_{i},\frac{(a+b)a_{i}}{2}\right),$ this assertion follows by noting
that the coefficient of the $xy^{2}$ term is nonzero while the coefficient of
$y^{4}$ is zero.
In order to prove the compacity of the parabolic curve consider the bounded
connected components of the complement of $\,\\{f(x,y)=0\\}\,$ in the
$xy$-plane. Take its closure of each one and the union of them. Let us denote
by $W$ the complement of this union.
###### Lemma 9
The points belonging to the set $W$ are hyperbolic.
Proof. We partition the set $W$ in 4 regions which are represented in Figure
1.
Figure 1: Partition of the set $W$ in 4 regions
Consider Region I. It was shown in the proof of Lemma 6 that the points lying
in some of the straight lines $\,l_{i}(x,y)=0,\,{g_{j}}(x,y)=0$, with
$\,i=0,\ldots,m,\,j=0,\ldots,n\,$ of Region I (or II) are hyperbolic. So, it
only remains to show that the points of Region I and belonging to the
complement of $\,\\{f(x,y)=0\\}\,$ are hyperbolic. Let us denote by $q$ such a
point. Consider the straight line $m$ on the $xy$-plane parallel to the
$y$-axis and passing through $q$. The restriction $\,f|_{m}\,$ of $\,f$ to
this line is a one variable polynomial of degree 2 with two real roots (the
intersections with the straight lines $\,l_{0}(x,y)=0,\,g_{0}(x,y)=0$). So,
$f(q)$ and the derivative of $\,f|_{m}\,$ at $q$ are positive or negative.
That is $\,f|_{m}\,$ is convex or concave at $q$, respectively.
Suppose that $f(q)$ is positive. Denote by $t$ the straight line tangent to
the level curve $\,f(x,y)=f(q)$ at $q$. Because the derivative of $\,f|_{t}\,$
is zero, the line $t$ is different from the line $m$. The function
$\,f|_{t}\,$ has a non degenerated critical point at $q$ which is a maximum.
So, the function $\,f|_{t}\,$ is concave at $q$. The two previous conclusions
imply that $(q,f(q))$ is a hyperbolic point. The case $\,f(q)<0\,$ is
analogous.
The analysis for Region II is similar.
Now, we consider Region III which is the unbounded connected component
determined by the lines $\,x=b_{n},\,y=ax\,$ and $\,y=bx$. Let $q$ be a point
in such a set. Consider the straight lines $\,m,l\,$ parallels to the lines
$g_{0}(x,y)=0,\,l_{0}(x,y)=0$, respectively, and passing through $q$. The
lines $\,m,\,l$ divide the $xy$-plane in four sectors, one of which, denoted
by $A$, is totally contained in Region III. Take a straight line $t$ such that
it passes through $q$, it divides the sector $A$ in two unbounded sectors and
it is not parallel to any line of $L$. So, the roots of $\,f|_{t}\,$ are in
the left side of $q$.
If $\,f(q)>0$, then the derivative of $\,f|_{t}\,$ at $q$ is positive. So
$\,f|_{t}\,$ is convex at $q$. On the other hand, consider the straight line
passing through $q$ which is parallel to the $y$-axis. Denote it by $r$. So,
$\,f|_{r}\,$ is concave at $q$ because it is a one variable polynomial of
degree two and $\,f(q)>0$. Then, the point $(q,f(q))$ is hyperbolic. The case
for Region IV is similar.
Since there is one special parabolic point at each compact segment of the
straight lines $g_{0}(x,y)=0,\,l_{0}(x,y)=0$, there is at least one connected
component of the Hessian curve on each compact polygone delimited by the lines
$g_{j}(x,y)=0,\,l_{i}(x,y)=0,\,i=0,\ldots,m,\,j=0,\ldots,n.$ Because there are
$m+n$ such as compact polygons, then there are at least $m+n$ connected
components of the Hessian curve. Now, the goal is to prove that the Hessian
curve has exactly $m+n$ connected components. In fact, we shall show that the
Hessian curve has at most $2(m+n)$ vertical tangent lines since this implies
that the Hessian curve has at most $m+n$ connected components.
Let us prove it. Consider the expression of the Hessian polynomial of $f$
given in Remark 8. The parabolic points having a vertical tangent line to the
parabolic curve are given by
$\Big{\\{}\frac{\partial}{\partial
y}\mbox{Hess}f(x,y)=0\Big{\\}}\cap\Big{\\{}\mbox{Hess}f(x,y)=0\Big{\\}}.$
Note that if $x_{0}$ is a real root of the polynomial $\beta(x)$ then
$\alpha(x_{0})\neq 0$ because in another case the points of the line $x=x_{0}$
would be parabolic and it contradicts Lemma 9. So, the parabolic points having
a vertical tangent line are the points
$\,\left(\tilde{x_{0}},\frac{(a+b)\tilde{x_{0}}}{2}\right)\,$ such that
$\tilde{x_{0}}$ is a real root of $\alpha(x)$. The proof concludes by noting
that $\alpha(x)$ is a one variable real polynomial of degree $2(m+n)$.
## 2 Concentric circles. Even case
###### Theorem 10
Consider the real polynomial of degree $\,2n$
$\,f(x,y)=\prod_{i=1}^{n}(x^{2}+y^{2}-m_{i}^{2})\,$
with $\,0<m_{1}<m_{2}<\cdots<m_{n},\,$ $\,m_{i}\in\mathbb{R},i=1,\ldots,n$.
Then, its Hessian curve is non singular and it is formed by $2(n-1)$
concentric circles. Moreover, the unbounded connected component of the
complement of the Hessian curve on the $xy$-plane is elliptic.
Proof. After a straightforward calculus the Hessian polynomial of $f$ is
$\mathrm{Hess}(f)=4\,s(x,y)\,t(x,y),\quad\mbox{where}$ $\displaystyle
s(x,y)=\sum_{j=1}^{n}\prod_{i\neq j}^{n}(x^{2}+y^{2}-m_{i}^{2})\,\quad\mbox{
and}\hskip 82.51282pt$ $\displaystyle t(x,y)=\sum_{j=1}^{n}\prod_{i\neq
j}^{n}(x^{2}+y^{2}-m_{i}^{2})+2(x^{2}+y^{2})\sum_{j=1}^{n}\sum_{l\neq
j}^{n}\prod_{i\neq j,l}^{n}(x^{2}+y^{2}-m_{i}^{2}).$
Note that the graph of the Hessian of $f$, $\,z-\mathrm{Hess}f(x,y)=0,$ is the
rotation about the $z$-axis of the graph of the function
$\,4\tilde{s}(x)\tilde{t}(x),\,$ where
$\widetilde{s}(x)=\sum_{j=1}^{n}\prod_{i\neq j}^{n}(x^{2}-m_{i}^{2})\,\mbox{
and}\qquad$ $\widetilde{t}(x)=\sum_{j=1}^{n}\prod_{i\neq
j}^{n}(x^{2}-m_{i}^{2})+2(x^{2})\sum_{j=1}^{n}\sum_{l\neq j}^{n}\prod_{i\neq
j,l}^{n}(x^{2}-m_{i}^{2}).$
So, the solution curves of $\,s(x,y)=0$ $(\,t(x,y)=0)\,$ correspond to the non
negative real roots of $\widetilde{s}(x)$ $(\,\widetilde{t}(x))$. We shall
prove now that $\widetilde{s}(x)$ and $\widetilde{t}(x)$ have $n-1$ simple
positive real roots, each one. Define the one variable real polynomial
$\widetilde{f}(x):=\prod_{i=1}^{n}(x^{2}-m_{i}^{2}).$
The degree of this polynomial is $\,2n$, it has $2n$ non zero simple roots,
$2n-1$ non degenerated critical points and $2n-2$ simple inflexion points.
Because
$\widetilde{f}_{x}(x)=2x\sum_{j=1}^{n}\prod_{i\neq
j}^{n}(x^{2}-m_{i}^{2})=2x\widetilde{s}(x),$
then $\,\widetilde{s}(x)$ has $2n-2$ non zero simple real roots from which,
$n-1$ are positive by symmetry. Moreover, since
$\widetilde{f}_{xx}(x)=2\sum_{j=1}^{n}\prod_{i\neq
j}^{n}(x^{2}-m_{i}^{2})+4x^{2}\sum_{j=1}^{n}\sum_{l\neq j}^{n}\prod_{i\neq
j,l}^{n}(x^{2}-m_{i}^{2})=\widetilde{t}(x),$
then $\widetilde{t}(x)$ has $2n-2$ simple real roots (different from the roots
of $\widetilde{s}(x)$). Because $\widetilde{f}$ is symmetric with respect to
the origin, then $\widetilde{t}(x)$ has $n-1$ positive roots. Moreover, the
unbounded connected component of the complement of the Hessian curve on the
$xy$-plane is elliptic because the functions $\tilde{s}$ and $\tilde{t}$ are
positive if $\,x>m_{n}.$
## 3 Concentric circles. Odd case
###### Theorem 11
Consider the real valued differentiable function
$\,f(x,y)=\prod_{k=1}^{n}\frac{x^{2}+y^{2}-k^{2}}{x^{2}+y^{2}+1}.$
Its Hessian curve is non singular and it is formed by $\,2n-1\,$ concentric
circles. Moreover, the unbounded connected component of the complement of the
Hessian curve on the $xy$-plane is hyperbolic.
Proof. After a straightforward computation the Hessian polynomial of $f$ is
$\mathrm{Hess}f(x,y)=\frac{4\,s(x,y)\,t(x,y)}{(x^{2}+y^{2}+1)^{2n+3}},\,\,\mbox{
where}$ $\displaystyle s(x,y):=\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{i\neq
j}^{n}(x^{2}+y^{2}-i^{2})\bigg{)},\hskip 110.96556pt$ $\displaystyle
t(x,y):=(1-3x^{2}-3y^{2})\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{i\neq
j}^{n}(x^{2}+y^{2}-i^{2})\bigg{)}+\hskip 28.45274pt$
$\displaystyle\qquad\qquad+2(x^{2}+y^{2})\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\sum_{l\neq
j}^{n}\bigg{(}(l^{2}+1)\prod_{k\neq
j,l}^{n}(x^{2}+y^{2}-k^{2})\bigg{)}\bigg{)}.$
Note that the graph of the Hessian polynomial of $f$, is the rotation about
the $z$-axis of the graph of the function
$\,\frac{4\tilde{s}(x)\tilde{t}(x)}{(x^{2}+1)^{2n+3}}$, where
$\displaystyle\tilde{s}(x)=\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{k\neq
j}^{n}(x^{2}-k^{2})\bigg{)},\,\,\quad\quad\tilde{t}(x)=(1-3x^{2})\sum_{j=1}^{n}\bigg{(}(j^{2}+$
$\displaystyle+1)\prod_{k\neq
j}^{n}(x^{2}-k^{2})\bigg{)}+2x^{2}\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\sum_{l\neq
j}^{n}\bigg{(}(l^{2}+1)\prod_{k\neq j,l}^{n}(x^{2}-k^{2})\bigg{)}\bigg{)}.$
We analyze now the zeros of $\,\tilde{s}(x)\,$ and $\,\tilde{t}(x)$. Consider
the differentiable function $\,\widetilde{f}:\mathbb{R}\rightarrow\mathbb{R}$
$\widetilde{f}(x)=\prod_{k=1}^{n}\frac{x^{2}-k^{2}}{x^{2}+1}.$
This function has exactly $2n$ simple zeros from which $n$ are positive.
Moreover,
$\displaystyle\widetilde{f}_{x}(x)=\frac{2x}{(x^{2}+1)^{n+1}}\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{i\neq
j}^{n}(x^{2}-i^{2})\bigg{)}=\frac{2x\,\widetilde{s}(x)}{(x^{2}+1)^{n+1}},\quad\qquad$
$\displaystyle\widetilde{f}_{xx}(x)=\frac{2}{(x^{2}+1)^{n+2}}\Bigg{[}(1-3x^{2})\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\prod_{i\neq
j}^{n}(x^{2}-i^{2})\bigg{)}+\qquad\qquad$
$\displaystyle+2x^{2}\sum_{j=1}^{n}\bigg{(}(j^{2}+1)\sum_{l\neq
j}^{n}\bigg{(}(l^{2}+1)\prod_{k\neq
j,l}^{n}(x^{2}-k^{2})\bigg{)}\bigg{)}\Bigg{]}=\frac{2\,\widetilde{t}(x)}{(x^{2}+1)^{n+2}}.$
Since $\widetilde{f}(x)$ has at least $2n-1$ critical points and the degree of
$\tilde{s}(x)$ is $2n-2$ then, the polynomial $\tilde{s}(x)$ has exactly
$2n-2$ simple roots different from zero ($n-1$ are positive). So, the set
$\,s(x,y)=0\,$ is a disjoint union of $n-1$ concentric circles.
Because the critical points of $\widetilde{f}(x)$ are non degenerated and it
is symmetric respect to the origin, then $\widetilde{f}(x)$ has at least $n-1$
positive inflexion points in the interval $(-n,n),$ exactly one point between
two consecutive critical points. That is, $\widetilde{t}(x)$ has $n-1$
positive real roots in the interval $[0,n)$.
Denote by $\,x_{0}$ the critical point of $\,\tilde{f}(x)$ lying in the
interval $(n-1,n).$ Note that the sign of
$\,\tilde{s}(n-1)=\left((n-1)^{2}+1\right)\prod_{k\neq
n-1}^{n}\left((n-1)^{2}-k^{2}\right)\,$ is negative while the sign of
$\,\tilde{s}(n)=(n^{2}+1)\prod_{i=1}^{n-1}(n^{2}-i^{2})\,$ is positive. From
these remarks and because the function $\,\widetilde{f}$ has exactly one
critical point at each interval $\,(i,i+1),\,i=1,\ldots,n-1$, we have that
$\,x_{0}$ is a minimum. Moreover, $\tilde{t}(x_{0})$ is positive. After a
straightforward calculation, it can be seen that $\,\tilde{t}(n^{2})\,$ is
negative. So, by the main value Theorem, there exists a real number
$\,c\in(x_{0},n^{2})\,$ such that $\tilde{t}(c)=0.$ It implies that
$\tilde{t}(x)$ has $n$ positive real roots. That is, the set $\,t(x,y)=0\,$ is
a disjoint union of $n$ concentric circles different from the circles
described by the equation $\,s(x,y)=0.$ We conclude that the Hessian curve of
$\,f\,$ consists of $2n-1$ concentric circles.
###### Proposition 12
If a smooth plane curve $\,C\,$ is equivalent by an affine transformation of
the plane, to a Hessian curve, then $\,C\,$ is also a Hessian curve.
Proof. Suppose that the curve $C$ is defined by $\,g(x,y)=0,\,$ where $g$ is a
smooth real valued function defined on the plane. By hypothesis, the curve $C$
is equivalent by an affine transformation to a Hessian curve, that is,
$\,g(x,y)=(\mbox{Hess }f)\circ T(x,y),$ where
$T:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is an affine transformation of the
plane whose determinant (of the linear part) is denoted by $J$. After a
straightforward calculation, we have that
$\mbox{Hess }\left(\frac{f\circ T(x,y)}{J}\right)=(\mbox{Hess }f)\circ
T(x,y).$
So, the curve $C$ is also a Hessian curve.
## References
* [1] Arnold V.I., Astroidal geometry of hypocycloides and the Hessian topology of hyperbolic polynomials, Russ. Math. Surv., (2001), 56, no. 6, 1019-1083.
* [2] Arnold V. I., Remarks on Parabolic Curves on Surfaces and the Higher-Dimensional Mobius-Sturm Theory, Funct. Anal. Appl. 31 No. 4 (1997), 227-239.
* [3] Arnold V. I., On the problem of realization of a given Gaussian curvature function, Topol. Methods Nonlinear Anal. 11 (1998), no. 2, 199-206.
* [4] Arnold V. I., Arnold’s Problems, Springer-Verlag, 2004.
* [5] Banchoff T., Gaffney T. and Mc.Crory C., Cusps of Gauss Mappings, Res. Notes Math., Pitman 1982.
* [6] Bertrand B. $\&$ Brugallé E., On the number of connected components of the parabolic curve, C. R. Math. Acad. Sci. Paris 348 (2010), no. 5-6, 287–289.
* [7] Hernández Martínez L. I., Ortiz-Rodríguez A. $\&$ Sánchez-Bringas F., On the affine geometry of the graph of a real polynomial, J. Dyn. Control Syst., 18, No.4 (2012), 455-465.
* [8] Hernández Martínez L. I., Ortiz Rodríguez A. $\&$ Sánchez Bringas F., On the Hessian geometry of a real polynomial hyperbolic near infinity. Adv. Geom. 13, No.2 (2013), 277-292.
* [9] Kergosien, Y.L.; Thom, R. Sur les points paraboliques des surfaces, C. R. Acad. Sci. Paris, Série A, t. 290 (1980), 705-710.
* [10] Ortiz Rodríguez A., Quelques aspects sur la géométrie des surfaces algébriques réelles, Bull. Sci. Math., 127, No.2, (2003), 149 - 177.
* [11] Ortiz Rodríguez A., Sottile F., Real Hessian Curves, Bol. Soc. Mat. Mexicana (3), 13, no.1 (2007), 157-166.
* [12] 18, No.4 (2012), 455-465 Platonova O. A., Singularities of the mutual of a surface and a line, Russ. Math. Surv. 36 (1981), 248-249.
* [13] Salmon G., A treatise in Analytic Geometry of Three Dimensions, Chelsea Publ., 1927.
* [14] Segre B., THE NON-SINGULAR CUBIC SURFACES: a New Method of Investigation with Special Reference to Questions of Reality, Clarendon Press, 1942.
Adriana Ortiz Rodríguez, Instituto de Matemáticas, Universidad Nacional
Autónoma de México, Area de la Inv. Cient., Circuito Exterior, C. U., México
D.F 04510, México.
e-mail: [email protected]
Angelito Camacho Calderón, Instituto de Matemáticas, unidad Cuernavaca,
Universidad Nacional Autónoma de México, Av. Universidad s/n. Col. Lomas de
Chamilpa, c.p.62210, Cuernavaca, Morelos.
e-mail: [email protected]
|
arxiv-papers
| 2013-06-14T18:52:39 |
2024-09-04T02:49:46.527485
|
{
"license": "Public Domain",
"authors": "Angelito Camacho Calder\\'on and Adriana Ortiz Rodr\\'i guez",
"submitter": "Adriana Ortiz-Rodriguez",
"url": "https://arxiv.org/abs/1306.3483"
}
|
1306.3493
|
A Heavy Ion Fireball freeze-out Dipion Cocktail for Au-Au Collisions at
$\sqrt{s_{NN}}$=200 GeV $p_{t}$ dependence(Part 2).
R.S. Longacrea
aBrookhaven National Laboratory, Upton, NY 11973, USA
###### Abstract
In this paper we use methods developed in Part 1 of The Dipion Cocktail, to
fit the $p_{t}$ dependence of dipions for mid-central Au-Au collisions at
$\sqrt{s_{NN}}$=200 GeV. For the minijet fragmentation part we use PYTHIA
fragmentation as described in Part 1. For the thermal resonance production we
use an exponential growth behavior. The interference between the direct
production of dipion pairs from non-resonance minijet fragmentation and re-
scattering through resonance states gives a measure of the size of the re-
scattering region. This size is contain in the $\alpha$ parameter of Part1. We
assumed a relationship between the $\alpha$ parameter and the mass shift of
the $\rho$ resonance. The data used for the fits comes from the RHIC collider
as measured in the STAR experiment.
## 1 Introduction
The ultra-relativistic heavy ion collision starts out as a state of high
density nuclear matter called the Quark Gluon Plasma(QGP) and expands rapidly
to freeze-out. During the freeze-out phase quarks and gluons form a system of
strongly interacting hadrons. These hadrons continue to expand in a thermal
manner until no further scattering is possible because the system becomes to
dilute. However this transition from quarks and gluons(partons) into hadrons
is not a smooth affair. The expansion is very rapid and some faster or hard
scattered partons fragment directly into hadron through a minijet[1] process.
Thus we have thermal and minijet hadrons present in the last scattering of the
hadrons. The Dipion Cocktail Part 1 considered this mixture of sources and
applied it to the dipion mass spectrum of the heavy ion fireball formed in Au-
Au collisions at $\sqrt{s_{NN}}$=200. Part 1 showed that both thermal or soft
production of hadrons and the minijet fragmented hadrons can be described
through a set of unified formal equations. Part 2(this paper) applies this
formalism to the $p_{t}$ dependence of dipions for Au-Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV and 40% to 80% centrality.
The paper is organized in the following manner:
Sec. 1 Introduction. Sec. 2 Review of the two component model which we use to
fit the dipion data within a set of $p_{t}$ ranges. Sec. 3 Discussion of the
relationship between the $\alpha$ parameter and the mass and widths of
resonances. Sec. 4 we present a fit to 19 $p_{t}$ ranges for Au-Au collisions
at $\sqrt{s_{NN}}=$ 200 GeV and 40% to 80% centrality. Sec. 5 Summary and
Discussion.
## 2 Two component model with Breit-Wigner parameters
In this section we will alter equation 6 of Part 1 so it can use Breit-Wigner
parameters (mass, width) instead of phase shifts. We will also need to modify
the re-scattering part of the equation in order to have the correct threshold
behavior we have introduced in Part 1 for the minijet partial waves. The phase
shift can be written for the $\ell^{th}$ wave as
$cot\delta_{\ell}=\frac{(M_{\ell}^{2}-M_{\pi\pi}^{2})}{M_{\ell}\Gamma_{\ell}},$
(1)
where $M_{\ell}$ is the mass of the resonance in the $\ell$wave and
$\Gamma_{\ell}$ is its total width.
$\Gamma_{\ell}=\Gamma_{0\ell}{\frac{qB_{\ell}(q/q_{s})}{M_{\pi\pi}}\over{\frac{q_{\ell}B_{\ell}(q_{\ell}/q_{s})}{M_{\ell}}}}$
(2)
with $\Gamma_{0\ell}$ the total width at resonance, $B_{\ell}$ is the Blatt-
Weisskopf-barrier factor[2] for the $\ell$ of the resonance, $q$ is the
$\pi\pi$ center mass momentum, $q_{\ell}$ is $q$ at resonance, $M_{\ell}$ is
the mass of the resonance, and $q_{s}$ is center mass momentum related to the
size(1.0 fm is used $q_{s}$ = .200 GeV/c).
Using equation 1 we rewrite equation 6 of Part 1 as
$|T_{\ell}|^{2}=|D_{\ell}|^{2}\frac{sin^{2}\delta_{\ell}}{PS_{\ell}}+\frac{|A_{\ell}|^{2}sin^{2}\delta_{\ell}}{PS_{\ell}}\left|\alpha+PS_{\ell}cot\delta_{\ell}\right|^{2}$
(3)
The $D_{\ell}$ is the thermal production term and is constant except for the
Boltzmann weight(see equation 13 in Part 1). The expected threshold behavior
$q^{2\ell+1}$ comes from the $sin\delta_{\ell}$ term. Since there is
$sin^{2}\delta_{\ell}$ one of the $q^{2\ell+1}$ is killed off by dividing by
$PS_{\ell}$. In Figure 6 of Part 1 we see we have put into our minijet
$A_{\ell}$ the correct threshold $q^{2\ell+1}$ so we need to kill off the
$q^{2\ell+1}$ of the other $sin\delta_{\ell}$ term. Therefore the above
equation for our minijet $A_{\ell}$ we will use
$|T_{\ell}|^{2}=|D_{\ell}|^{2}\frac{sin^{2}\delta_{\ell}}{PS_{\ell}}+\frac{|A_{\ell}|^{2}sin^{2}\delta_{\ell}}{PS_{\ell}^{2}}\left|\alpha+PS_{\ell}cot\delta_{\ell}\right|^{2}$
(4)
Rewriting equation 6 of Part 1 for each partial wave with Breit-Wigner
parameters the first term becomes
$|T_{\ell}|_{1}^{2}=|D_{\ell}|^{2}\frac{M_{\pi\pi}^{2}}{\sqrt{M_{\pi\pi}^{2}+p^{2}_{t}}}exp\frac{-\sqrt{M_{\pi\pi}^{2}+p^{2}_{t}}}{T}\frac{M_{\ell}\Gamma_{\ell}}{(M_{\ell}^{2}-M_{\pi\pi}^{2})^{2}+M_{\ell}^{2}\Gamma_{\ell}^{2}},$
(5)
while the second term
$|T_{\ell}|_{2}^{2}=|A_{\ell}|^{2}\frac{M_{\ell}^{2}\Gamma_{\ell}^{2}}{(M_{\ell}^{2}-M_{\pi\pi}^{2})^{2}+M_{\ell}^{2}\Gamma_{\ell}^{2}}\left|\alpha+\frac{2qB_{\ell}(\frac{q}{q_{s}})(M_{\ell}^{2}-M_{\pi\pi}^{2})}{M_{\pi\pi}M_{\ell}\Gamma_{\ell}}\right|^{2}\left(\frac{M_{\pi\pi}^{2}}{4q^{2}B_{\ell}^{2}(\frac{q}{q_{s}})}\right).$
(6)
$|T|^{2}=\sum_{\ell}|T_{\ell}|^{2}$ (7)
where
$|T_{\ell}|^{2}=|T_{\ell}|_{1}^{2}+|T_{\ell}|_{2}^{2}$ (8)
and
$|A_{0}|^{2}=S(M_{\pi^{+}\pi^{-}})$,$|A_{1}|^{2}=P(M_{\pi^{+}\pi^{-}})$,$|A_{2}|^{2}=D(M_{\pi^{+}\pi^{-}})$,
and $|A_{3}|^{2}=F(M_{\pi^{+}\pi^{-}})$. S, P, D and F comes from subsection
5.2 of Part 1.
## 3 A Relationship between the $\alpha$ parameter and the mass and widths of
resonances.
Equation 6 of Part 1 has an important factor the coefficient $\alpha$. This
coefficient is related to the real part of the $\pi\pi$ re-scattering loop and
is given by equation 9. When the pions re-scatter or interact at a close
distance or a point the real part of the loop $\alpha$ has its maximum value
of $\alpha_{0}$. While if the pions re-scatter or interact at a distance
determined by the diffractive limit the value of $\alpha$ is zero. The
$\alpha$ which is the real part of the re-scattering factor has a simple form
given by
$\alpha=(1.0-\frac{r^{2}}{r_{0}^{2}})\alpha_{0}$ (9)
where $r$ is the radius of re-scattering in fm’s and $r_{0}$ is 1.0 fm or the
limiting range of the strong interaction ranging to $r$ = 0.0 for point like
interactions.
When $\pi\pi$ pairs interact at the diffractive limit their phase shift should
be the same as the phase shift of the vacuum. The same statement is true for
$\pi\pi$ interacting at a point since asymptotic freedom demands that the
strong interaction should have no effect. However values of $\alpha$ in
between zero and $\alpha_{0}$ represent a confined volume where strongly
interacting gluons, quarks and virtual mesons may influence the phase shift of
the $\pi\pi$ system. Phase shifts under a Breit-Wigner parameters assumption
depend on the mass and width of the resonance parameter. In the next section
we use fits to data to determine the relationship between Breit-Wigner
parameters and the value of $\alpha$.
## 4 STAR data dipion $p_{t}$ range (0.2 GeV/c $<$ $p_{t}$ $<$ 4.0 GeV/c)
We have fitted 19 dipion $p_{t}$ ranges(see Table I) using equation 7 above
for the STAR Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to 80%
centrality data. We included minijets up to $\ell$ = 3 and resonances $\sigma$
$\ell$ = 0, $\rho(770)$ $\ell$ = 1, and $f_{2}(1270)$ $\ell$ = 2. Using the
arguments of Sec. 3 of Part 1, we added the $f_{0}$ as a direct thermal term
($|T_{0}|_{1}^{2}$) and only the $\sigma$ interfered with $\ell$ = 0 minijet
background. Two other thermal terms are present in the cocktail, the
$k^{0}_{s}$ and the $\omega_{0}$. All the thermal terms have an exponential
behavior with dipion $p_{t}$. The spectrum of the minijet partial waves is
obtained from PYTHIA[3](see Sec. 5.2 of Part 1). We let the data determine
which minijet partial wave to add. We find only Swave minijet background is
important until $p_{t}$ equal to 1.1 GeV/c. Above 1.5 GeV/c all four minijet
partial waves are used up to Fwave. It should be noted Dwave and Fwave are
small effects. We used PDG[4] for the $f_{2}(1270)$ mass = 1.275 GeV and width
= .185 GeV. The $f_{0}$ was fitted obtaining mass = 0.9727 $\pm$ .0039 GeV and
width = 0.04512 $\pm$ 0.01128 GeV. The $\sigma$ mass and width used was fixed
because it was ill determined. The mass used was mass = 1.011 GeV and width =
1.015 GeV. The $\rho$ mass and width is explained below.
Finally the threshold effective mass region .280 GeV to .430 GeV is dominated
by the Swave and receives contributions from minijet fragmentation, $\pi\pi$
Swave phase shift, $\eta$ decay, HBT adding to the like sign $\pi\pi$
distribution that has been subtracted away from the unlike sign $\pi\pi$ and
the coulomb correction between the charged pions. The minijet fragmentation is
the least known of the effects since we relied on PYTHIA, however there are
large uncertainty in all the other effects. So for these fits we let the
minijet fragmentation be free to fit the data and let the Breit-Wigner
parameters for the $\sigma$ determine the Swave phase shifts plus leaving out
all other effects.
For the $\alpha$ parameter in $p_{t}$ bins up to 1.1 GeV/c the minijet Swave
interference is the determining factor. Above 1.1 GeV/c the Pwave interference
becomes most important. The values of $\alpha$ which gives a reasonable fit
are shown in Table II.
We have determined that the $\sigma$ pole or Breit-Wigner parameters is so far
away from the real axis thus it is too short lived to be influenced by
hadronic interactions. The $\rho$ phase shift being of a life time comparable
to hadronic interaction taking place becomes most sensitive. We have found as
a function of $\alpha$ the best of $\rho$ width is always 0.147 GeV with an
error of $\pm$ .007 GeV. The mass however decreases as $\alpha$ grows,
reaching a minimum of 0.738 GeV at an $\alpha$ of 0.907. This is a mass shift
of 37 MeV. An $\alpha$ of 0.504 is the smallest $\alpha$ we find in our fits.
A mass of 0.775 GeV is the best fit when the value of $\alpha$ is at 0.504.
Using equation 9 in Table II we determine the radius of $\pi\pi$ re-scattering
for each $p_{t}$ bin. The value of $\alpha_{0}$ used in Table II is equal to
2.0 as determined in Appendix B of Part 1. Table II shows an interesting
density effect around dipion $p_{t}$ of 0.6 to 1.0 GeV/c. If one consider that
$p_{t}$ maybe related to fireball size through the idea of hubble flow, then
pions with a $p_{t}$ of around 0.4 GeV/c maybe coming from a less dense region
in the central part of the fireball. This could be a density wave effect.
Table I. The $p_{t}$ ranges bins we fit.
Table I
---
$p_{t}$ bin number | lower edge(GeV/c) | upper edge(GeV/c)
1 | 0.2 | 0.4
2 | 0.4 | 0.6
3 | 0.6 | 0.8
4 | 0.8 | 1.0
5 | 1.0 | 1.2
6 | 1.2 | 1.4
7 | 1.4 | 1.6
8 | 1.6 | 1.8
9 | 1.8 | 2.0
10 | 2.0 | 2.2
11 | 2.2 | 2.4
12 | 2.4 | 2.6
13 | 2.6 | 2.8
14 | 2.8 | 3.0
15 | 3.0 | 3.2
16 | 3.2 | 3.4
17 | 3.4 | 3.6
18 | 3.6 | 3.8
19 | 3.8 | 4.0
Table II. The $\alpha$ value, $\rho$ mass value and radius in each $p_{t}$
range.
Table II
---
$p_{t}$(GeV/c) | $\alpha$ | $\rho$ mass(GeV) | radius(f)
0.3 | 0.907 $\pm$ .028 | 0.738 $\pm$ .004 | .739 $\pm$ .010
0.5 | 0.806 $\pm$ .025 | 0.748 $\pm$ .003 | .773 $\pm$ .008
0.7 | 0.706 $\pm$ .022 | 0.755 $\pm$ .002 | .804 $\pm$ .007
0.9 | 0.706 $\pm$ .022 | 0.755 $\pm$ .002 | .804 $\pm$ .007
1.1 | 0.806 $\pm$ .025 | 0.748 $\pm$ .003 | .773 $\pm$ .008
1.3 | 0.806 $\pm$ .025 | 0.748 $\pm$ .003 | .773 $\pm$ .008
1.5 | 0.806 $\pm$ .025 | 0.748 $\pm$ .003 | .773 $\pm$ .008
1.7 | 0.806 $\pm$ .025 | 0.748 $\pm$ .003 | .773 $\pm$ .008
1.9 | 0.806 $\pm$ .025 | 0.748 $\pm$ .003 | .773 $\pm$ .008
2.1 | 0.605 $\pm$ .019 | 0.759 $\pm$ .002 | .835 $\pm$ .006
2.3 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
2.5 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
2.7 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
2.9 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
3.1 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
3.3 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
3.5 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
3.7 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
3.9 | 0.504 $\pm$ .016 | 0.775 $\pm$ .001 | .865 $\pm$ .004
The 19 dipion spectrum for the $p_{t}$ bins are shown in Figure 1 through
Figure 19.
Figure 1: Fit to STAR dipion effective mass distribution (0.2 GeV/c $<$
$p_{t}$ $<$ 0.4 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 2: Fit to STAR dipion effective mass distribution (0.4 GeV/c $<$
$p_{t}$ $<$ 0.6 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 3: Fit to STAR dipion effective mass distribution (0.6 GeV/c $<$
$p_{t}$ $<$ 0.8 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 4: Fit to STAR dipion effective mass distribution (0.8 GeV/c $<$
$p_{t}$ $<$ 1.0 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 5: Fit to STAR dipion effective mass distribution (1.0 GeV/c $<$
$p_{t}$ $<$ 1.2 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 6: Fit to STAR dipion effective mass distribution (1.2 GeV/c $<$
$p_{t}$ $<$ 1.4 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 7: Fit to STAR dipion effective mass distribution (1.4 GeV/c $<$
$p_{t}$ $<$ 1.6 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 8: Fit to STAR dipion effective mass distribution (1.6 GeV/c $<$
$p_{t}$ $<$ 1.8 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 9: Fit to STAR dipion effective mass distribution (1.8 GeV/c $<$
$p_{t}$ $<$ 2.0 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 10: Fit to STAR dipion effective mass distribution (2.0 GeV/c $<$
$p_{t}$ $<$ 2.2 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 11: Fit to STAR dipion effective mass distribution (2.2 GeV/c $<$
$p_{t}$ $<$ 2.4 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 12: Fit to STAR dipion effective mass distribution (2.4 GeV/c $<$
$p_{t}$ $<$ 2.6 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 13: Fit to STAR dipion effective mass distribution (2.6 GeV/c $<$
$p_{t}$ $<$ 2.8 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 14: Fit to STAR dipion effective mass distribution (2.8 GeV/c $<$
$p_{t}$ $<$ 3.0 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 15: Fit to STAR dipion effective mass distribution (3.0 GeV/c $<$
$p_{t}$ $<$ 3.2 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 16: Fit to STAR dipion effective mass distribution (3.2 GeV/c $<$
$p_{t}$ $<$ 3.4 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 17: Fit to STAR dipion effective mass distribution (3.4 GeV/c $<$
$p_{t}$ $<$ 3.6 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 18: Fit to STAR dipion effective mass distribution (3.6 GeV/c $<$
$p_{t}$ $<$ 3.8 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
Figure 19: Fit to STAR dipion effective mass distribution (3.8 GeV/c $<$
$p_{t}$ $<$ 4.0 GeV/c) for Au-Au collisions at $\sqrt{s_{NN}}=$ 200 GeV 40% to
80% centrality using equation 7. See text for complete information.
The direct cross sectional yield of the thermally produced states into
$\pi^{+}$ $\pi^{-}$ is shown in Figure 20. The states are $k_{s}$, $\rho$,
$f_{0}$ and $f_{2}$ and the yields come from the exponential fits to the
direct thermal component of equation 7.
Figure 20: The direct thermal $p_{t}$ yeild into the $\pi^{+}\pi^{-}$ channel
as seen in the STAR dipion effective mass fit for the $k_{s}$, $\rho$, $f_{0}$
and $f_{2}$ using equation 7. See text for complete information.
## 5 Summary and Discussion
The Dipion Cocktail Part 2 applies a formalism derived in The Dipion Cocktail
Part 1 which considered a mixture of sources present in the dipion mass
spectrum of the heavy ion fireball. Part 1 showed that both thermal or soft
production of hadrons and the minijet fragmented hadrons can be described
through a set of unified formal equations. Part 2(this paper) applies this
Part 1 formalism to the $p_{t}$ dependence of dipions for Au-Au collisions at
$\sqrt{s_{NN}}=$ 200 GeV and 40% to 80% centrality.
Part 1 started with the basic definition of elastic $\pi\pi$ scattering. Next
showed how re-scattering of pions depends on the unitary condition that
interactions present in the phase shift of an orbital state must interact all
the time. The process of parton fragmentation into dipion states through
unitarity leads to a equation of production and re-scattering in a given
orbital quantum number. This equation(equation 7) has two components in each
orbital state: one being the thermal production of resonances in a dipion
orbital state, the other is the re-scattering of dipions coming from parton or
minijet fragmentation into the dipion orbital state which do not come directly
from the resonance. Unitarity requires that there most be re-scatter through
resonance phase shifts which we defined through Breit-Wigner parameters (mass,
width).
We have fitted 19 dipion $p_{t}$ ranges(see Table I) using equation 7. We
included minijets up to $\ell$ = 3 and resonances $\sigma$ $\ell$ = 0,
$\rho(770)$ $\ell$ = 1, and $f_{2}(1270)$ $\ell$ = 2. Using the arguments of
Sec. 3 of Part 1, we added the $f_{0}$ as a direct thermal term
($|T_{0}|_{1}^{2}$) and only the $\sigma$ interfered with $\ell$ = 0 minijet
background. Two other thermal terms are present in the cocktail, the
$k^{0}_{s}$ and the $\omega_{0}$. All the thermal terms have an exponential
behavior with dipion $p_{t}$ and are shown Figure 20. The spectrum of the
minijet partial waves is obtained from PYTHIA[3](see Sec. 5.2 of Part 1). We
let the data determine which minijet partial wave to add. We find only Swave
minijet background is important until $p_{t}$ equal to 1.1 GeV/c. Above 1.5
GeV/c all four minijet partial waves are used up to Fwave. It should be noted
Dwave and Fwave are small effects. We used PDG[4] for the $f_{2}(1270)$ mass =
1.275 GeV and width = .185 GeV. The $f_{0}$ was fitted obtaining mass = 0.9727
$\pm$ .0039 GeV and width = 0.04512 $\pm$ 0.01128 GeV. The $\sigma$ mass and
width used was fixed because it was ill determined. The mass used was mass =
1.011 GeV and width = 1.015 GeV.
For the $\alpha$ parameter in $p_{t}$ bins up to 1.1 GeV/c the minijet Swave
interference is the determining factor. Above 1.1 GeV/c the Pwave interference
becomes most important. The values of $\alpha$ which gives a reasonable fit
are shown in Table II.
We have determined that the $\sigma$ pole or Breit-Wigner parameters is so far
away from the real axis thus it is too short lived to be influenced by
hadronic interactions. The $\rho$ phase shift being of a life time comparable
to hadronic interaction taking place becomes most sensitive. We have found as
a function of $\alpha$ the best of $\rho$ width is always 0.147 GeV with an
error of $\pm$ .007 GeV. The mass however decreases as $\alpha$ grows,
reaching a minimum of 0.738 GeV at an $\alpha$ of 0.907. This is a mass shift
of 37 MeV. An $\alpha$ of 0.504 is the smallest $\alpha$ we find in our fits.
A mass of 0.775 GeV is the best fit when the value of $\alpha$ is at 0.504.
Using equation 9 in Table II we determine the radius of $\pi\pi$ re-scattering
for each $p_{t}$ range. Table II shows an interesting density effect around
dipion $p_{t}$ of 0.6 to 1.0 GeV/c. If one consider that $p_{t}$ maybe related
to fireball size through the idea of hubble flow, then pions with a $p_{t}$ of
around 0.4 GeV/c maybe coming from a less dense region in the central part of
the fireball. This could be a density wave effect.
## 6 Acknowledgments
This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886. The author thanks William Love for the STAR analysis of
the angular correlation data from Run 4. Also for his assistance in the
production of figures. It is sad that he is gone.
## References
* [1] T. Trainor, Phys. Rev. C 80 (2009) 044901.
* [2] F. von Hippel and C. Quigg, Phys. Rev. 5 (1972) 624.
* [3] T. Sjostrand, M. van Zijil, Phys. Rev. D 36 (1987) 2019.
* [4] J. Beringer et al. (Particle Data Group), J. Phys. D86 (2012) 010001\.
|
arxiv-papers
| 2013-06-14T19:35:54 |
2024-09-04T02:49:46.535964
|
{
"license": "Public Domain",
"authors": "Ron S. Longacre",
"submitter": "Ron S. Longacre",
"url": "https://arxiv.org/abs/1306.3493"
}
|
1306.3498
|
# Coincidence and Common Fixed Point Results for Generalized $\alpha$-$\psi$
Contractive Type Mappings with Applications
Priya Shahi∗, Jatinderdeep Kaur, S. S. Bhatia
###### Abstract.
A new, simple and unified approach in the theory of contractive mappings was
recently given by Samet _et al._ (Nonlinear Anal. 75, 2012, 2154-2165) by
using the concepts of $\alpha$-$\psi$-contractive type mappings and
$\alpha$-admissible mappings in metric spaces. The purpose of this paper is to
present a new class of contractive pair of mappings called generalized
$\alpha$-$\psi$ contractive pair of mappings and study various fixed point
theorems for such mappings in complete metric spaces. For this, we introduce a
new notion of $\alpha$-admissible w.r.t $g$ mapping which in turn generalizes
the concept of $g$-monotone mapping recently introduced by
$\acute{C}$iri$\acute{c}$ et al. (Fixed Point Theory Appl. 2008(2008), Article
ID 131294, 11 pages). As an application of our main results, we further
establish common fixed point theorems for metric spaces endowed with a partial
order as well as in respect of cyclic contractive mappings. The presented
theorems extend and subsumes various known comparable results from the current
literature. Some illustrative examples are provided to demonstrate the main
results and to show the genuineness of our results.
Keywords: Common fixed point; Contractive type mapping; Partial order; Cyclic
mappings.
Mathematics Subject Classification (2000): 54H25, 47H10, 54E50.
∗Corresponding author: Priya Shahi, School of Mathematics and Computer
Applications, Thapar University, Patiala 147004, Punjab, India. Email address:
[email protected]
School of Mathematics and Computer Applications,
Thapar University, Patiala-147004, Punjab, India.
Email addresses: [email protected]∗, [email protected],
[email protected]
## 1\. Introduction
Fixed point theory has fascinated many researchers since 1922 with the
celebrated Banach fixed point theorem. There exists a vast literature on the
topic and this is a very active field of research at present. Fixed point
theorems are very important tools for proving the existence and uniqueness of
the solutions to various mathematical models (integral and partial
differential equations, variational inequalities etc). It is well known that
the contractive-type conditions are very indispensable in the study of fixed
point theory. The first important result on fixed points for contractive-type
mappings was the well-known Banach-Caccioppoli theorem which was published in
1922 in [1] and it also appears in [2]. Later in 1968, Kannan [3] studied a
new type of contractive mapping. Since then, there have been many results
related to mappings satisfying various types of contractive inequality, we
refer to ([4], [5], [6], [7], [8] etc) and references therein.
Recently, Samet _et al._ [9] introduced a new category of contractive type
mappings known as $\alpha$-$\psi$ contractive type mapping. The results
obtained by Samet _et al._ [9] extended and generalized the existing fixed
point results in the literature, in particular the Banach contraction
principle. Further, Karapinar and Samet [10] generalized the
$\alpha$-$\psi$-contractive type mappings and obtained various fixed point
theorems for this generalized class of contractive mappings.
The study related to common fixed points of mappings satisfying certain
contractive conditions has been at the center of vigorous research activity.
In this paper, some coincidence and common fixed point theorems are obtained
for the generalized $\alpha$-$\psi$ contractive pair of mappings. Our results
unify and generalize the results derived by Karapinar and Samet [10], Samet
_et al._ [9], $\acute{C}$iri$\acute{c}$ et al. [11] and various other related
results in the literature. Moreover, from our main results, we will derive
various common fixed point results for metric spaces endowed with a partial
order and that for cyclic contractive mappings. The presented results extend
and generalize numerous related results in the literature.
## 2\. Preliminaries
First we introduce some notations and definitions that will be used
subsequently.
###### Definition 2.1.
(See [9]). Let $\Psi$ be the family of functions
$\psi:[0,\infty)\rightarrow[0,\infty)$ satisfying the following conditions:
(i) $\psi$ is nondecreasing.
(ii) $\displaystyle\sum_{n=1}^{+\infty}\psi^{n}(t)<\infty$ for all $t>0$,
where $\psi^{n}$ is the $n^{th}$ iterate of $\psi$.
These functions are known as (c)-comparison functions in the literature. It
can be easily verified that if $\psi$ is a (c)-comparison function, then
$\psi(t)<t$ for any $t>0$.
Recently, Samet _et al._ [9] introduced the following new notions of
$\alpha$-$\psi$-contractive type mappings and $\alpha$-admissible mappings:
###### Definition 2.2.
Let $(X,d)$ be a metric space and $T:X\rightarrow X$ be a given self mapping.
$T$ is said to be an $\alpha$-$\psi$-contractive mapping if there exists two
functions $\alpha:X\times X\rightarrow[0,+\infty)$ and $\psi\in\Psi$ such that
$\alpha(x,y)d(Tx,Ty)\leq\psi(d(x,y))$
for all $x,y\in X$.
###### Definition 2.3.
Let $T:X\rightarrow X$ and $\alpha:X\times X\rightarrow[0,+\infty)$. $T$ is
said to be $\alpha$-admissible if
$x,y\in X$, $\alpha(x,y)\geq 1\Rightarrow\alpha(Tx,Ty)\geq 1$.
The following fixed point theorems are the main results in [9]:
###### Theorem 2.1.
Let $(X,d)$ be a complete metric space and $T:X\rightarrow X$ be an
$\alpha$-$\psi$-contractive mapping satisfying the following conditions:
(i) $T$ is $\alpha$-admissible;
(ii) there exists $x_{0}\in X$ such that $\alpha(x_{0},Tx_{0})\geq 1$;
(iii) $T$ is continuous.
Then, $T$ has a fixed point, that is, there exists $x^{*}\in X$ such that
$Tx^{*}=x^{*}$.
###### Theorem 2.2.
Let $(X,d)$ be a complete metric space and $T:X\rightarrow X$ be an
$\alpha$-$\psi$-contractive mapping satisfying the following conditions:
(i) $T$ is $\alpha$-admissible;
(ii) there exists $x_{0}\in X$ such that $\alpha(x_{0},Tx_{0})\geq 1$;
(iii) if $\\{x_{n}\\}$ is a sequence in $X$ such that
$\alpha(x_{n},x_{n+1})\geq 1$ for all $n$ and $x_{n}\rightarrow x\in X$ as
$n\rightarrow+\infty$, then $\alpha(x_{n},x)\geq 1$ for all $n$.
Then, $T$ has a fixed point.
Samet _et al._ [9] added the following condition to the hypotheses of Theorem
2.1 and Theorem 2.2 to assure the uniqueness of the fixed point:
(C): For all $x,y\in X$, there exists $z\in X$ such that $\alpha(x,z)\geq 1$
and $\alpha(y,z)\geq 1$.
Recently, Karapinar and Samet [10] introduced the following concept of
generalized $\alpha$-$\psi$-contractive type mappings:
###### Definition 2.4.
Let $(X,d)$ be a metric space and $T:X\rightarrow X$ be a given mapping. We
say that $T$ is a generalized $\alpha$-$\psi$-contractive type mapping if
there exists two functions $\alpha:X\times X\rightarrow[0,\infty)$ and
$\psi\in\Psi$ such that for all $x,y\in X$, we have
$\displaystyle\displaystyle\alpha(x,y)d(Tx,Ty)\leq\psi(M(x,y)),$
where $\displaystyle
M(x,y)=\max\left\\{d(x,y),\frac{d(x,Tx)+d(y,Ty)}{2},\frac{d(x,Ty)+d(y,Tx)}{2}\right\\}$.
Further, Karapinar and Samet [10] established fixed point theorems for this
new class of contractive mappings. Also, they obtained fixed point theorems on
metric spaces endowed with a partial order and fixed point theorems for cyclic
contractive mappings.
###### Definition 2.5.
[12] Let $X$ be a non-empty set, $N$ is a natural number such that $N\geq 2$
and $T_{1},T_{2},...,T_{N}:X\rightarrow X$ are given self-mappings on $X$. If
$w=T_{1}x=T_{2}x=...=T_{N}x$ for some $x\in X$, then $x$ is called a
coincidence point of $T_{1},T_{2},...,T_{N-1}$ and $T_{N}$, and $w$ is called
a point of coincidence of $T_{1},T_{2},...,T_{N-1}$ and $T_{N}$. If $w=x$,
then $x$ is called a common fixed point of $T_{1},T_{2},...,T_{N-1}$ and
$T_{N}$.
Let $f,g:X\rightarrow X$ be two mappings. We denote by $C(g,f)$ the set of
coincidence points of $g$ and $f$; that is,
$\displaystyle C(g,f)=\\{z\in X:gz=fz\\}$
## 3\. Main results
We start the main section by introducing the new consepts of
$\alpha$-admissible w.r.t $g$ mapping and generalized $\alpha$-$\psi$
contractive pair of mappings.
###### Definition 3.1.
Let $f,g:X\rightarrow X$ and $\alpha:X\times X\rightarrow[0,\infty)$. We say
that $f$ is $\alpha$-admissible w.r.t $g$ if for all $x,y\in X$, we have
$\alpha(gx,gy)\geq 1\Rightarrow\alpha(fx,fy)\geq 1$.
###### Remark 3.1.
Clearly, every $\alpha$-admissible mapping is $\alpha$-admissible w.r.t $g$
mapping when $g=I$.
The following example shows that a mapping which is $\alpha$-admissible w.r.t
$g$ may not be $\alpha$-admissible.
###### Example 3.2.
Let $X=[1,\infty)$. Define the mapping $\alpha:X\times X\rightarrow[0,\infty)$
by
$\alpha(x,y)=\left\\{\begin{array}[]{ll}2&\mbox{if }x>y\\\
\displaystyle\frac{1}{3}&otherwise\end{array}\right.$
Also, define the mappings $f,g:X\rightarrow X$ by
$f(x)=\displaystyle\frac{1}{x}$ and $g(x)=e^{-x}$ for all $x\in X$.
Suppose that $\alpha(x,y)\geq 1$. This implies from the definition of $\alpha$
that $x>y$ which further implies that $\displaystyle\frac{1}{x}<\frac{1}{y}$.
Thus, $\alpha(fx,fy)\ngeq 1$, that is, $f$ is not $\alpha$-admissible.
Now, we prove that $f$ is $\alpha$-admissible w.r.t $g$. Let us suppose that
$\alpha(gx,gy)\geq 1$. So,
$\alpha(gx,gy)\geq 1\Rightarrow gx>gy\Rightarrow
e^{-x}>e^{-y}\Rightarrow\displaystyle\frac{1}{x}>\frac{1}{y}\Rightarrow\alpha(fx,fy)\geq
1$
Therefore, $f$ is $\alpha$-admissible w.r.t $g$.
In what follows, we present examples of $\alpha$-admissible w.r.t $g$
mappings.
###### Example 3.3.
Let $X$ be the set of all non-negative real numbers. Let us define the mapping
$\alpha:X\times X\rightarrow[0,+\infty)$ by
$\alpha(x,y)=\left\\{\begin{array}[]{c l}1&if\hskip 5.69046ptx\geq y,\\\
0&if\hskip 5.69046ptx<y.\\\ \end{array}\right.$
and define the mappings $f,g:X\rightarrow X$ by $f(x)=e^{x}$ and $g(x)=x^{2}$
for all $x\in X$. Thus, the mapping $f$ is $\alpha$-admissible w.r.t $g$.
###### Example 3.4.
Let $X=[1,\infty)$. Let us define the mapping $\alpha:X\times
X\rightarrow[0,+\infty)$ by
$\alpha(x,y)=\left\\{\begin{array}[]{c l}3&if\hskip 5.69046ptx,y\in[0,1],\\\
\frac{1}{2}&\hskip 5.69046ptotherwise.\\\ \end{array}\right.$
and define the mappings $f,g:X\rightarrow X$ by
$f(x)=ln\left(1+\displaystyle\frac{x}{3}\right)$ and $g(x)=\sqrt{x}$ for all
$x\in X$. Thus, the mapping $f$ is $\alpha$-admissible w.r.t $g$.
Next, we present the new notion of generalized $\alpha$-$\psi$ contractive
pair of mappings as follows:
###### Definition 3.5.
Let $(X,d)$ be a metric space and $f,g:X\rightarrow X$ be given mappings. We
say that the pair $(f,g)$ is a generalized $\alpha$-$\psi$ contractive pair of
mappings if there exists two functions $\alpha:X\times
X\rightarrow[0,+\infty)$ and $\psi\in\Psi$ such that for all $x,y\in X$, we
have
(1) $\displaystyle\alpha(gx,gy)d(fx,fy)\leq\psi(M(gx,gy)),$
where $\displaystyle
M(gx,gy)=\max\left\\{d(gx,gy),\frac{d(gx,fx)+d(gy,fy)}{2},\frac{d(gx,fy)+d(gy,fx)}{2}\right\\}$.
Our first result is the following coincidence point theorem.
###### Theorem 3.1.
Let $(X,d)$ be a complete metric space and $f,g:X\rightarrow X$ be such that
$f(X)\subseteq g(X)$. Assume that the pair $(f,g)$ is a generalized
$\alpha$-$\psi$ contractive pair of mappings and the following conditions
hold:
(i) $f$ is $\alpha$-admissible w.r.t. $g$;
(ii) there exists $x_{0}\in X$ such that $\alpha(gx_{0},fx_{0})\geq 1$;
(iii) If $\\{gx_{n}\\}$ is a sequence in $X$ such that
$\alpha(gx_{n},gx_{n+1})\geq 1$ for all $n$ and $gx_{n}\rightarrow gz\in g(X)$
as $n\rightarrow\infty$, then there exists a subsequence $\\{gx_{n(k)}\\}$ of
$\\{gx_{n}\\}$ such that $\alpha(gx_{n(k)},gz)\geq 1$ for all $k$.
Also suppose $g(X)$ is closed. Then, $f$ and $g$ have a coincidence point.
###### Proof.
In view of condition (ii), let $x_{0}\in X$ be such that
$\alpha(gx_{0},fx_{0})\geq 1$. Since $f(X)\subseteq g(X)$, we can choose a
point $x_{1}\in X$ such that $fx_{0}=gx_{1}$. Continuing this process having
chosen $x_{1},x_{2},...,x_{n}$, we choose $x_{n+1}$ in $X$ such that
(2) $\displaystyle fx_{n}=gx_{n+1},n=0,1,2,...$
Since $f$ is $\alpha$-admissible w.r.t $g$, we have
$\alpha(gx_{0},fx_{0})=\alpha(gx_{0},gx_{1})\geq
1\Rightarrow\alpha(fx_{0},fx_{1})=\alpha(gx_{1},gx_{2})\geq 1$
Using mathematical induction, we get
(3) $\displaystyle\alpha(gx_{n},gx_{n+1})\geq 1,\forall\hskip
2.84544ptn=0,1,2,...$
If $fx_{n+1}=fx_{n}$ for some $n$, then by (2),
$\displaystyle fx_{n+1}=gx_{n+1},n=0,1,2,...$
that is, $f$ and $g$ have a coincidence point at $x=x_{n+1}$, and so we have
finished the proof. For this, we suppose that $d(fx_{n},fx_{n+1})>0$ for all
$n$. Applying the inequality (1) and using (3), we obtain
(4) $\displaystyle d(fx_{n},fx_{n+1})$ $\displaystyle\leq$
$\displaystyle\alpha(gx_{n},gx_{n+1})d(fx_{n},fx_{n+1})$ $\displaystyle\leq$
$\displaystyle\psi(M(gx_{n},gx_{n+1}))$
On the other hand, we have
$\displaystyle M(gx_{n},gx_{n+1})$ $\displaystyle=$
$\displaystyle\max\left\\{d(gx_{n},gx_{n+1}),\displaystyle\frac{d(gx_{n},fx_{n})+d(gx_{n+1},fx_{n+1})}{2},\displaystyle\frac{d(gx_{n},fx_{n+1})+d(gx_{n+1},fx_{n})}{2}\right\\}$
$\displaystyle\leq$
$\displaystyle\max\\{d(fx_{n-1},fx_{n}),d(fx_{n},fx_{n+1})\\}$
Owing to monotonicity of the function $\psi$ and using the inequalities (2)
and (4), we have for all $n\geq 1$
(5) $\displaystyle d(fx_{n},fx_{n+1})$ $\displaystyle\leq$
$\displaystyle\psi(\max\left\\{d(fx_{n-1},fx_{n}),d(fx_{n},fx_{n+1})\right\\}$
If for some $n\geq 1$, we have $d(fx_{n-1},fx_{n})\leq d(fx_{n},fx_{n+1})$,
from (5), we obtain that
$\displaystyle\hskip 56.9055ptd(fx_{n},fx_{n+1})$ $\displaystyle\leq$
$\displaystyle\psi(d(fx_{n},fx_{n+1}))<d(fx_{n},fx_{n+1}),$
a contradiction. Thus, for all $n\geq 1$, we have
(6)
$\displaystyle\max\left\\{d(fx_{n-1},fx_{n}),d(fx_{n},fx_{n+1})\right\\}=d(fx_{n-1},fx_{n})$
Notice that in view of (5) and (6), we get for all $n\geq 1$ that
(7) $\displaystyle d(fx_{n},fx_{n+1})$ $\displaystyle\leq$
$\displaystyle\psi(d(fx_{n-1},fx_{n})).$
Continuing this process inductively, we obtain
(8) $\displaystyle d(fx_{n},fx_{n+1})$ $\displaystyle\leq$
$\displaystyle\psi^{n}(d(fx_{0},fx_{1})),\hskip 14.22636pt\forall n\geq 1.$
From (8) and using the triangular inequality, for all $k\geq 1$, we have
(9) $\displaystyle d(fx_{n},fx_{n+k})$ $\displaystyle\leq$ $\displaystyle
d(fx_{n},fx_{n+1})+...+d(fx_{n+k-1},fx_{n+k})$ $\displaystyle\leq$
$\displaystyle\sum_{p=n}^{n+k-1}\psi^{p}(d(fx_{1},fx_{0}))$
$\displaystyle\leq$
$\displaystyle\sum_{p=n}^{+\infty}\psi^{p}(d(fx_{1},fx_{0}))$
Letting $p\rightarrow\infty$ in (9), we obtain that $\\{fx_{n}\\}$ is a Cauchy
sequence in $(X,d)$. Since by (2) we have
$\\{fx_{n}\\}=\\{gx_{n+1}\\}\subseteq g(X)$ and $g(X)$ is closed, there exists
$z\in X$ such that
(10) $\displaystyle\displaystyle\lim_{n\rightarrow\infty}gx_{n}=gz.$
Now, we show that $z$ is a coincidence point of $f$ and $g$. On contrary,
assume that $d(fz,gz)>0$. Since by condition (iii) and (10), we have
$\alpha(gx_{n(k)},gz)\geq 1$ for all $k$, then by the use of triangle
inequality and (1) we obtain
(11) $\displaystyle d(gz,fz)$ $\displaystyle\leq$ $\displaystyle
d(gz,fx_{n(k)})+d(fx_{n(k)},fz)$ $\displaystyle\leq$ $\displaystyle
d(gz,fx_{n(k)})+\alpha(gx_{n(k)},gz)d(fx_{n(k)},fz)$ $\displaystyle\leq$
$\displaystyle d(gz,fx_{n(k)})+\psi(M(gx_{n(k)},gz)$
On the other hand, we have
$\displaystyle M(gx_{n(k)},gz)$ $\displaystyle=$
$\displaystyle\max\left\\{d(gx_{n(k)},gz),\frac{d(gx_{n(k)},fx_{n(k)})+d(gz,fz)}{2},\frac{d(gx_{n(k)},fz)+d(gz,fx_{n(k)})}{2}\right\\}$
Owing to above equality, we get from (11),
$\displaystyle d(gz,fz)$ $\displaystyle\leq$ $\displaystyle
d(gz,fx_{n(k)})+\psi(M(gx_{n(k)},gz)$ $\displaystyle\leq$ $\displaystyle
d(gz,fx_{n(k)})+$
$\displaystyle\psi\left(\max\left\\{d(gx_{n(k)},gz),\frac{d(gx_{n(k)},fx_{n(k)})+d(gz,fz)}{2},\frac{d(gx_{n(k)},fz)+d(gz,fx_{n(k)})}{2}\right\\}\right)$
Letting $k\rightarrow\infty$ in the above inequality yields $\displaystyle
d(gz,fz)\leq\psi\left(\frac{d(fz,gz)}{2}\right)<\frac{d(fz,gz)}{2}$, which is
a contradiction. Hence, our supposition is wrong and $d(fz,gz)=0$, that is,
$fz=gz$. This shows that $f$ and $g$ have a coincidence point. ∎
The next theorem shows that under additional hypotheses we can deduce the
existence and uniqueness of a common fixed point.
###### Theorem 3.2.
In addition to the hypotheses of Theorem 3.1, suppose that for all $u,v\in
C(g,f)$, there exists $w\in X$ such that $\alpha(gu,gw)\geq 1$ and
$\alpha(gv,gw)\geq 1$ and $f,g$ commute at their coincidence points. Then $f$
and $g$ have a unique common fixed point.
###### Proof.
We need to consider three steps:
Step 1. We claim that if $u,v\in C(g,f)$, then $gu=gv$. By hypotheses, there
exists $w\in X$ such that
(12) $\displaystyle\alpha(gu,gw)\geq 1,\alpha(gv,gw)\geq 1$
Due to the fact that $f(X)\subseteq g(X)$, let us define the sequence
$\\{w_{n}\\}$ in $X$ by $gw_{n+1}=fw_{n}$ for all $n\geq 0$ and $w_{0}=w$.
Since $f$ is $\alpha$-admissible w.r.t $g$, we have from (12) that
(13) $\displaystyle\alpha(gu,gw_{n})\geq 1,\alpha(gv,gw_{n})\geq 1$
for all $n\geq 0$. Applying inequality (1) and using (13), we obtain
(14) $\displaystyle d(gu,gw_{n+1})$ $\displaystyle=$ $\displaystyle
d(fu,fw_{n})$ $\displaystyle\leq$ $\displaystyle\alpha(gu,gw_{n})d(fu,fw_{n})$
$\displaystyle\leq$ $\displaystyle\psi(M(gu,gw_{n}))$
On the other hand, we have
(15) $\displaystyle M(gu,gw_{n})$ $\displaystyle=$
$\displaystyle\max\left\\{d(gu,gw_{n}),\frac{d(gu,fu)+d(gw_{n},fw_{n})}{2},\frac{d(gu,fw_{n})+d(gw_{n},fu)}{2}\right\\}$
$\displaystyle\leq$
$\displaystyle\max\left\\{d(gu,gw_{n}),d(gu,gw_{n+1})\right\\}$
Using the above inequality, (14) and owing to the monotone property of $\psi$,
we get that
(16) $\displaystyle d(gu,gw_{n+1})$ $\displaystyle\leq$
$\displaystyle\psi(\max\left\\{d(gu,gw_{n}),d(gu,gw_{n+1})\right\\})$
for all $n$. Without restriction to the generality, we can suppose that
$d(gu,gw_{n})>0$ for all $n$. If
$\max\\{d(gu,gw_{n}),d(gu,gw_{n+1})\\}=d(gu,gw_{n+1})$, we have from (16) that
(17) $\displaystyle d(gu,gw_{n+1})\leq\psi(d(gu,gw_{n+1}))<d(gu,gw_{n+1})$
which is a contradiction. Thus, we have
$\max\\{d(gu,gw_{n}),d(gu,gw_{n+1})\\}=d(gu,gw_{n})$, and
$d(gu,gw_{n+1})\leq\psi(d(gu,gw_{n}))$ for all $n$. This implies that
(18) $\displaystyle d(gu,gw_{n})\leq\psi^{n}(d(gu,gw_{0})),\hskip
2.84544pt\forall n\geq 1$
Letting $n\rightarrow\infty$ in the above inequality, we infer that
(19) $\displaystyle\lim_{n\rightarrow\infty}d(gu,gw_{n})=0$
Similarly, we can prove that
(20) $\displaystyle\lim_{n\rightarrow\infty}d(gv,gw_{n})=0$
It follows from (19) and (20) that $gu=gv$.
Step 2. Existence of a common fixed point: Let $u\in C(g,f)$, that is,
$gu=fu$. Owing to the commutativity of $f$ and $g$ at their coincidence
points, we get
(21) $\displaystyle g^{2}u=gfu=fgu$
Let us denote $gu=z$, then from (21), $gz=fz$. Thus, $z$ is a coincidence
point of $f$ and $g$. Now, from Step 1, we have $gu=gz=z=fz$. Then, $z$ is a
common fixed point of $f$ and $g$.
Step 3. Uniqueness: Assume that $z^{*}$ is another common fixed point of $f$
and $g$. Then $z^{*}\in C(g,f)$. By step 1, we have $z^{*}=gz^{*}=gz=z$. This
completes the proof. ∎
In what follows, we furnish an illustrative example wherein one demonstrates
Theorem 3.2 on the existence and uniqueness of a common fixed point.
###### Example 3.6.
Consider $X=[0,+\infty)$ equipped with the usual metric $d(x,y)=|x-y|$ for all
$x,y\in X$. Define the mappings $f:X\rightarrow X$ and $g:X\rightarrow X$ by
$f(x)=\left\\{\begin{array}[]{c l}\displaystyle
2x-\displaystyle\frac{3}{2}&if\hskip 2.84544ptx>2,\\\
\displaystyle\frac{x}{3}&if\hskip 2.84544pt0\leq x\leq 2.\\\
\end{array}\right.$
and
$g(x)=\displaystyle\frac{x}{2}\hskip 2.84544pt\forall\hskip 0.28436ptx\in X$.
Now, we define the mapping $\alpha:X\times X\rightarrow[0,+\infty)$ by
$\alpha(x,y)=\left\\{\begin{array}[]{c l}1&if\hskip 2.84544pt(x,y)\in[0,1],\\\
0&otherwise.\\\ \end{array}\right.$
Clearly, the pair $(f,g)$ is a generalized $\alpha$-$\psi$ contractive pair of
mappings with $\psi(t)=\displaystyle\frac{4}{5}t$ for all $t\geq 0$. In fact,
for all $x,y\in X$, we have
$\displaystyle\alpha(gx,gy).d(fx,fy)=1.\left|\displaystyle\frac{x}{3}-\displaystyle\frac{y}{3}\right|$
$\displaystyle\leq$
$\displaystyle\frac{4}{5}\left|\displaystyle\frac{x}{2}-\displaystyle\frac{y}{2}\right|$
$\displaystyle=$ $\displaystyle\displaystyle\frac{4}{5}d(gx,gy)$
$\displaystyle\leq$
$\displaystyle\displaystyle\frac{4}{5}M(gx,gy)=\psi(M(gx,gy))$
Moreover, there exists $x_{0}\in X$ such that $\alpha(gx_{0},fx_{0})\geq 1$.
Infact, for $x_{0}=1$, we have
$\alpha\left(\displaystyle\frac{1}{2},\displaystyle\frac{1}{3}\right)=1$.
Now, it remains to show that $f$ is $\alpha$-admissible w.r.t $g$. In so
doing, let $x,y\in X$ such that $\alpha(gx,gy)\geq 1$. This implies that
$gx,gy\in[0,1]$ and by the definition of $g$, we have $x,y\in[0,2]$.
Therefore, by the definition of $f$ and $\alpha$, we have
$f(x)=\displaystyle\frac{x}{3}\in[0,1]$,
$f(y)=\displaystyle\frac{y}{3}\in[0,1]$ and $\alpha(fx,fy)=1$.
Thus, $f$ is $\alpha$-admissible w.r.t $g$. Clearly, $f(X)\subseteq g(X)$ and
$g(X)$ is closed.
Finally, let $\\{gx_{n}\\}$ be a sequence in $X$ such that
$\alpha(gx_{n},gx_{n+1})\geq 1$ for all $n$ and $gx_{n}\rightarrow gz\in g(X)$
as $n\rightarrow+\infty$. Since $\alpha(gx_{n},gx_{n+1})\geq 1$ for all $n$,
by the definition of $\alpha$, we have $gx_{n}\in[0,1]$ for all $n$ and
$gz\in[0,1]$. Then, $\alpha(gx_{n},gz)\geq 1$. Now, all the hypotheses of
Theorem 3.1 are satisfied. Consequently, $f$ and $g$ have a coincidence point.
Here, 0 is a coincidence point of $f$ and $g$. Also, clearly all the
hypotheses of Theorem 3.2 are satisfied. In this example, 0 is the unique
common fixed point of $f$ and $g$.
## 4\. Consequences
In this section, we will show that many existing results in the literature can
be obtained easily from our Theorem 3.2.
### 4.1. Standard Fixed Point Theorems
By taking $\alpha(x,y)=1$ for all $x,y\in X$ in Theorem 3.2, we obtain
immediately the following fixed point theorem.
###### Corollary 4.1.
Let $(X,d)$ be a complete metric space and $f,g:X\rightarrow X$ be such that
$f(X)\subseteq g(X)$. Suppose that there exists a function $\psi\in\Psi$ such
that
(22) $\displaystyle d(fx,fy)\leq\psi(M(gx,gy)),$
for all $x,y\in X$. Also, suppose $g(X)$ is closed. Then, $f$ and $g$ have a
coincidence point. Further, if $f$, $g$ commute at their coincidence points,
then $f$ and $g$ have a common fixed point.
By taking $g=I$ in Corollary 4.1, we obtain immediately the following fixed
point theorem.
###### Corollary 4.2.
(see Karapinar and Samet [10]). Let $(X,d)$ be a complete metric space and
$f:X\rightarrow X$. Suppose that there exists a function $\psi\in\Psi$ such
that
(23) $\displaystyle d(fx,fy)\leq\psi(M(x,y)),$
for all $x,y\in X$. Then $f$ has a unique fixed point.
The following fixed point theorems can be easily obtained from Corollaries 4.1
and 4.2.
###### Corollary 4.3.
Let $(X,d)$ be a complete metric space and $f,g:X\rightarrow X$ be such that
$f(X)\subseteq g(X)$. Suppose that there exists a function $\psi\in\Psi$ such
that
(24) $\displaystyle d(fx,fy)\leq\psi(d(gx,gy)),$
for all $x,y\in X$. Also, suppose $g(X)$ is closed. Then, $f$ and $g$ have a
coincidence point. Further, if $f$, $g$ commute at their coincidence points,
then $f$ and $g$ have a common fixed point.
###### Corollary 4.4.
(Berinde [13]). Let $(X,d)$ be a complete metric space and $f:X\rightarrow X$.
Suppose that there exists a function $\psi\in\Psi$ such that
(25) $\displaystyle d(fx,fy)\leq\psi(d(x,y)),$
for all $x,y\in X$. Then $f$ has a unique fixed point.
###### Corollary 4.5.
($\acute{C}$iri$\acute{c}$ [14]). Let $(X,d)$ be a complete metric space and
$T:X\rightarrow X$ be a given mapping. Suppose that there exists a constant
$\lambda\in(0,1)$ such that
$\displaystyle
d(fx,fy)\leq\lambda\max\left\\{d(x,y),\frac{d(x,fx)+d(y,fy)}{2},\frac{d(x,fy)+d(y,fx)}{2}\right\\}$
for all $x,y\in X$. Then $T$ has a unique fixed point.
###### Corollary 4.6.
(Hardy and Rogers [15]) Let $(X,d)$ be a complete metric space and
$T:X\rightarrow X$ be a given mapping. Suppose that there exists constants
$A,B,C\geq 0$ with $(A+2B+2C)\in(0,1)$ such that
$\displaystyle d(fx,fy)\leq Ad(x,y)+B[d(x,fx)+d(y,fy)]+C[d(x,fy)+d(y,fx)],$
for all $x,y\in X$. Then $T$ has a unique fixed point.
###### Corollary 4.7.
(Banach Contraction Principle [1]) Let $(X,d)$ be a complete metric space and
$T:X\rightarrow X$ be a given mapping. Suppose that there exists a constant
$\lambda\in(0,1)$ such that
$\displaystyle d(fx,fy)\leq\lambda d(x,y)$
for all $x,y\in X$. Then $T$ has a unique fixed point.
###### Corollary 4.8.
(Kannan [3]) Let $(X,d)$ be a complete metric space and $T:X\rightarrow X$ be
a given mapping. Suppose that there exists a constant $\lambda\in(0,1/2)$ such
that
$\displaystyle d(fx,fy)\leq\lambda[d(x,fx)+d(y,fy)],$
for all $x,y\in X$. Then $T$ has a unique fixed point.
###### Corollary 4.9.
(Chatterjee [16]) Let $(X,d)$ be a complete metric space and $T:X\rightarrow
X$ be a given mapping. Suppose that there exists a constant
$\lambda\in(0,1/2)$ such that
$\displaystyle d(fx,fy)\leq\lambda[d(x,fy)+d(y,fx)],$
for all $x,y\in X$. Then $T$ has a unique fixed point.
### 4.2. Fixed Point Theorems on Metric Spaces Endowed with a Partial Order
Recently, there have been enormous developments in the study of fixed point
problems of contractive mappings in metric spaces endowed with a partial
order. The first result in this direction was given by Turinici [17], where he
extended the Banach contraction principle in partially ordered sets. Some
applications of Turinici’s theorem to matrix equations were presented by Ran
and Reurings [18]. Later, many useful results have been obtained regarding the
existence of a fixed point for contraction type mappings in partially ordered
metric spaces by Bhaskar and Lakshmikantham [4], Nieto and Lopez [7, 19],
Agarwal et al. [20], Lakshmikantham and $\acute{C}$iri$\acute{c}$ [6] and
Samet [21] etc. In this section, we will derive various fixed point results on
a metric space endowed with a partial order. For this, we require the
following concepts:
###### Definition 4.1.
[10] Let $(X,\preceq)$ be a partially ordered set and $T:X\rightarrow X$ be a
given mapping. We say that $T$ is nondecreasing with respect to $\preceq$ if
$x,y\in X,x\preceq y\Rightarrow Tx\preceq Ty$.
###### Definition 4.2.
[10] Let $(X,\preceq)$ be a partially ordered set. A sequence
$\\{x_{n}\\}\subset X$ is said to be nondecreasing with respect to $\preceq$
if $x_{n}\preceq x_{n+1}$ for all $n$.
###### Definition 4.3.
[10] Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$.
We say that $(X,\preceq,d)$ is regular if for every nondecreasing sequence
$\\{x_{n}\\}\subset X$ such that $x_{n}\rightarrow x\in X$ as
$n\rightarrow\infty$, there exists a subsequence $\\{x_{n(k)}\\}$ of
$\\{x_{n}\\}$ such that $x_{n(k)}\preceq x$ for all $k$.
###### Definition 4.4.
[11] Suppose $(X,\preceq)$ is a partially ordered set and $F,g:X\rightarrow X$
are mappings of $X$ into itself. One says $F$ is $g$-non-decreasing if for
$x,y\in X$,
(26) $\displaystyle g(x)\preceq g(y)\hskip 14.22636ptimplies\hskip
14.22636ptF(x)\preceq F(y).$
###### Definition 4.5.
Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$. We
say that $(X,\preceq,d)$ is $g$-regular where $g:X\rightarrow X$ if for every
nondecreasing sequence $\\{gx_{n}\\}\subset X$ such that $gx_{n}\rightarrow
gz\in X$ as $n\rightarrow\infty$, there exists a subsequence $\\{gx_{n(k)}\\}$
of $\\{gx_{n}\\}$ such that $gx_{n(k)}\preceq gz$ for all $k$.
We have the following result.
###### Corollary 4.10.
Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$ such
that $(X,d)$ is complete. Assume that $f,g:X\rightarrow X$ be such that
$f(X)\subseteq g(X)$ and $f$ be a $g$-non-decreasing mapping w.r.t $\preceq$.
Suppose that there exists a function $\psi\in\Psi$ such that
(27) $\displaystyle d(fx,fy)\leq\psi(M(gx,gy)),$
for all $x,y\in X$ with $gx\preceq gy$. Suppose also that the following
conditions hold:
(i) there exists $x_{0}\in X$ s.t $gx_{0}\preceq fx_{0}$;
(ii) $(X,\preceq,d)$ is $g$-regular.
Also suppose $g(X)$ is closed. Then, $f$ and $g$ have a coincidence point.
Moreover, if for every pair $(x,y)\in C(g,f)\times C(g,f)$ there exists $z\in
X$ such that $gx\preceq gz$ and $gy\preceq gz$, and if $f$ and $g$ commute at
their coincidence points, then we obtain uniqueness of the common fixed point.
###### Proof.
Define the mapping $\alpha:X\times X\rightarrow[0,\infty)$ by
(30) $\displaystyle\alpha(x,y)=\left\\{\begin{array}[]{ll}1&\mbox{if }x\preceq
y\hskip 5.69046ptor\hskip 5.69046ptx\succeq y\\\
0&otherwise\end{array}\right.$
Clearly, the pair $(f,g)$ is a generalized $\alpha$-$\psi$ contractive pair of
mappings, that is,
$\displaystyle\alpha(gx,gy)d(fx,fy)\leq\psi(M(gx,gy)),$
for all $x,y\in X$. Notice that in view of condition (i), we have
$\alpha(gx_{0},fx_{0})\geq 1$. Moreover, for all $x,y\in X$, from the
$g$-monotone property of $f$, we have
(31) $\displaystyle\alpha(gx,gy)\geq 1\Rightarrow gx\hskip
2.84544pt\preceq\hskip 2.84544ptgy\hskip 2.84544ptor\hskip 2.84544ptgx\succeq
gy\Rightarrow fx\preceq fy\hskip 2.84544ptor\hskip 2.84544ptfx\succeq
fy\Rightarrow\alpha(fx,fy)\geq 1.$
which amounts to say that $f$ is $\alpha$-admissible w.r.t $g$. Now, let
$\\{gx_{n}\\}$ be a sequence in $X$ such that $\alpha(gx_{n},gx_{n+1})\geq 1$
for all $n$ and $gx_{n}\rightarrow gz\in X$ as $n\rightarrow\infty$. From the
$g$-regularity hypothesis, there exists a subsequence $\\{gx_{n(k)}\\}$ of
$\\{gx_{n}\\}$ such that $gx_{n(k)}\preceq gz$ for all $k$. So, by the
definition of $\alpha$, we obtain that $\alpha(gx_{n(k)},gz)\geq 1$. Now, all
the hypotheses of Theorem 3.1 are satisfied. Hence, we deduce that $f$ and $g$
have a coincidence point $z$, that is, $fz=gz$.
Now, we need to show the existence and uniqueness of common fixed point. For
this, let $x,y\in X$. By hypotheses, there exists $z\in X$ such that
$gx\preceq gz$ and $gy\preceq gz$, which implies from the definition of
$\alpha$ that $\alpha(gx,gz)\geq 1$ and $\alpha(gy,gz)\geq 1$. Thus, we deduce
the existence and uniqueness of the common fixed point by Theorem 3.2. ∎
The following results are immediate consequences of Corollary 4.10.
###### Corollary 4.11.
Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$ such
that $(X,d)$ is complete. Assume that $f,g:X\rightarrow X$ and $f$ be a
$g$-non-decreasing mapping w.r.t $\preceq$. Suppose that there exists a
function $\psi\in\Psi$ such that
(32) $\displaystyle d(fx,fy)\leq\psi(d(gx,gy)),$
for all $x,y\in X$ with $gx\preceq gy$. Suppose also that the following
conditions hold:
(i) there exists $x_{0}\in X$ s.t $gx_{0}\preceq fx_{0}$;
(ii) $(X,\preceq,d)$ is $g$-regular.
Also suppose $g(X)$ is closed. Then, $f$ and $g$ have a coincidence point.
Moreover, if for every pair $(x,y)\in C(g,f)\times C(g,f)$ there exists $z\in
X$ such that $gx\preceq gz$ and $gy\preceq gz$, and if $f$ and $g$ commute at
their coincidence points, then we obtain uniqueness of the common fixed point.
###### Corollary 4.12.
Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$ such
that $(X,d)$ is complete. Assume that $f,g:X\rightarrow X$ and $f$ be a
$g$-non-decreasing mapping w.r.t $\preceq$. Suppose that there exists a
constant $\lambda\in(0,1)$ such that
(33) $\displaystyle
d(fx,fy)\leq\lambda\max\left\\{d(gx,gy),\frac{d(gx,fx)+d(gy,fy)}{2},\frac{d(gx,fy)+d(gy,fx)}{2}\right\\},$
for all $x,y\in X$ with $gx\preceq gy$. Suppose also that the following
conditions hold:
(i) there exists $x_{0}\in X$ s.t $gx_{0}\preceq fx_{0}$;
(ii) $(X,\preceq,d)$ is $g$-regular.
Also suppose $g(X)$ is closed. Then, $f$ and $g$ have a coincidence point.
Moreover, if for every pair $(x,y)\in C(g,f)\times C(g,f)$ there exists $z\in
X$ such that $gx\preceq gz$ and $gy\preceq gz$, and if $f$ and $g$ commute at
their coincidence points, then we obtain uniqueness of the common fixed point.
###### Corollary 4.13.
Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$ such
that $(X,d)$ is complete. Assume that $f,g:X\rightarrow X$ and $f$ be a
$g$-non-decreasing mapping w.r.t $\preceq$. Suppose that there exists
constants $A,B,C\geq 0$ with $(A+2B+2C)\in(0,1)$ such that
(34) $\displaystyle d(fx,fy)\leq
Ad(gx,gy)+B[d(gx,fx)+d(gy,fy)]+C[d(gx,fy)+d(gy,fx)],$
for all $x,y\in X$ with $gx\preceq gy$. Suppose also that the following
conditions hold:
(i) there exists $x_{0}\in X$ s.t $gx_{0}\preceq fx_{0}$;
(ii) $(X,\preceq,d)$ is $g$-regular.
Also suppose $g(X)$ is closed. Then, $f$ and $g$ have a coincidence point.
Moreover, if for every pair $(x,y)\in C(g,f)\times C(g,f)$ there exists $z\in
X$ such that $gx\preceq gz$ and $gy\preceq gz$, and if $f$ and $g$ commute at
their coincidence points, then we obtain uniqueness of the common fixed point.
###### Corollary 4.14.
Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$ such
that $(X,d)$ is complete. Assume that $f,g:X\rightarrow X$ and $f$ be a
$g$-non-decreasing mapping w.r.t $\preceq$. Suppose that there exists a
constant $\lambda\in(0,1)$ such that
(35) $\displaystyle d(fx,fy)\leq\lambda(d(gx,gy)),$
for all $x,y\in X$ with $gx\preceq gy$. Suppose also that the following
conditions hold:
(i) there exists $x_{0}\in X$ s.t $gx_{0}\preceq fx_{0}$;
(ii) $(X,\preceq,d)$ is $g$-regular.
Also suppose $g(X)$ is closed. Then, $f$ and $g$ have a coincidence point.
Moreover, if for every pair $(x,y)\in C(g,f)\times C(g,f)$ there exists $z\in
X$ such that $gx\preceq gz$ and $gy\preceq gz$, and if $f$ and $g$ commute at
their coincidence points, then we obtain uniqueness of the common fixed point.
###### Corollary 4.15.
Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$ such
that $(X,d)$ is complete. Assume that $f,g:X\rightarrow X$ and $f$ be a
$g$-non-decreasing mapping w.r.t $\preceq$. Suppose that there exists
constants $A,B,C\geq 0$ with $(A+2B+2C)\in(0,1)$ such that
(36) $\displaystyle d(fx,fy)\leq\lambda[d(gx,fx)+d(gy,fy)],$
for all $x,y\in X$ with $gx\preceq gy$. Suppose also that the following
conditions hold:
(i) there exists $x_{0}\in X$ s.t $gx_{0}\preceq fx_{0}$;
(ii) $(X,\preceq,d)$ is $g$-regular.
Also suppose $g(X)$ is closed. Then, $f$ and $g$ have a coincidence point.
Moreover, if for every pair $(x,y)\in C(g,f)\times C(g,f)$ there exists $z\in
X$ such that $gx\preceq gz$ and $gy\preceq gz$, and if $f$ and $g$ commute at
their coincidence points, then we obtain uniqueness of the common fixed point.
###### Corollary 4.16.
Let $(X,\preceq)$ be a partially ordered set and $d$ be a metric on $X$ such
that $(X,d)$ is complete. Assume that $f,g:X\rightarrow X$ and $f$ be a
$g$-non-decreasing mapping w.r.t $\preceq$. Suppose that there exists
constants $A,B,C\geq 0$ with $(A+2B+2C)\in(0,1)$ such that
(37) $\displaystyle d(fx,fy)\leq\lambda[d(gx,fy)+d(gy,fx)],$
for all $x,y\in X$ with $gx\preceq gy$. Suppose also that the following
conditions hold:
(i) there exists $x_{0}\in X$ s.t $gx_{0}\preceq fx_{0}$;
(ii) $(X,\preceq,d)$ is $g$-regular.
Also suppose $g(X)$ is closed. Then, $f$ and $g$ have a coincidence point.
Moreover, if for every pair $(x,y)\in C(g,f)\times C(g,f)$ there exists $z\in
X$ such that $gx\preceq gz$ and $gy\preceq gz$, and if $f$ and $g$ commute at
their coincidence points, then we obtain uniqueness of the common fixed point.
Remarks
* •
Letting $g=I_{X}$ in Corollary 4.11 we obtain Corollary 3.12 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.12 we obtain Corollary 3.13 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.13 we obtain Corollary 3.14 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.14 we obtain Corollary 3.15 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.15 we obtain Corollary 3.16 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.16 we obtain Corollary 3.17 in [10].
### 4.3. Fixed Point Theorems for Cyclic Contractive Mappings
As a generalization of the Banach contraction mapping principle, Kirk _et al._
[22] in 2003 introduced cyclic representations and cyclic contractions. A
mapping $T:A\cup B\rightarrow A\cup B$ is called cyclic if $T(A)\subseteq B$
and $T(B)\subseteq A$, where $A,B$ are nonempty subsets of a metric space
$(X,d)$. Moreover, $T$ is called a cyclic contraction if there exists
$k\in(0,1)$ such that $d(Tx,Ty)\leq kd(x,y)$ for all $x\in A$ and $y\in B$.
Notice that although a contraction is continuous, cyclic contractions need not
be. This is one of the important gains of this theorem. In the last decade,
several authors have used the cyclic representations and cyclic contractions
to obtain various fixed point results. see for example ([23, 24, 25, 26, 27,
28]).
###### Corollary 4.17.
Let $(X,d)$ be a complete metric space, $A_{1}$ and $A_{2}$ are two nonempty
closed subsets of $X$ and $f,g:Y\rightarrow Y$ be two mappings, where
$Y=A_{1}\cup A_{2}$. Suppose that the following conditions hold:
(i) $g(A_{1})$ and $g(A_{2})$ are closed;
(ii) $f(A_{1})\subseteq g(A_{2})$ and $f(A_{2})\subseteq g(A_{1})$;
(iii) $g$ is one-to-one;
(iv) there exists a function $\psi\in\Psi$ such that
(38) $\displaystyle d(fx,fy)\leq\psi(M(gx,gy)),\hskip 2.84544pt\forall(x,y)\in
A_{1}\times A_{2}.$
Then, $f$ and $g$ have a coincidence point $z\in A_{1}\cap A_{2}$. Further, if
$f$, $g$ commute at their coincidence points, then $f$ and $g$ have a unique
common fixed point that belongs to $A_{1}\cap A_{2}$.
###### Proof.
Due to the fact that $g$ is one-to-one, condition (iv) is equivalent to
(39) $\displaystyle d(fx,fy)\leq\psi(M(gx,gy)),\hskip
2.84544pt\forall(gx,gy)\in g(A_{1})\times g(A_{2}).$
Now, since $A_{1}$ and $A_{2}$ are closed subsets of the complete metric space
$(X,d)$, then $(Y,d)$ is complete.
Define the mapping $\alpha:Y\times Y\rightarrow[0,\infty)$ by
(42) $\displaystyle\alpha(x,y)=\left\\{\begin{array}[]{ll}1&if\hskip
2.84544pt(x,y)\in(g(A_{1})\times g(A_{2}))\cup(g(A_{2})\times g(A_{1}))\\\
0&otherwise\end{array}\right.$
Notice that in view of definition of $\alpha$ and condition (iv), we can write
(43) $\displaystyle\alpha(gx,gy)d(fx,fy)\leq\psi(M(gx,gy))$
for all $gx\in g(A_{1})$ and $gy\in g(A_{2})$. Thus, the pair $(f,g)$ is a
generalized $\alpha$-$\psi$ contractive pair of mappings.
By using condition (ii), we can show that $f(Y)\subseteq g(Y)$. Moreover,
$g(Y)$ is closed.
Next, we proceed to show that $f$ is $\alpha$-admissible w.r.t $g$. Let
$(gx,gy)\in Y\times Y$ such that $\alpha(gx,gy)\geq 1$; that is,
(44) $\displaystyle(gx,gy)\in(g(A_{1})\times g(A_{2}))\cup(g(A_{2})\times
g(A_{1}))$
Since $g$ is one-to-one, this implies that
(45) $\displaystyle(x,y)\in(A_{1}\times A_{2})\cup(A_{2}\times A_{1})$
So, from condition (ii), we infer that
(46) $\displaystyle(fx,fy)\in(g(A_{2})\times g(A_{1}))\cup(g(A_{1})\times
g(A_{2}))$
that is, $\alpha(fx,fy)\geq 1$. This implies that $f$ is $\alpha$-admissible
w.r.t $g$.
Now, let $\\{gx_{n}\\}$ be a sequence in $X$ such that
$\alpha(gx_{n},gx_{n+1})\geq 1$ for all $n$ and $gx_{n}\rightarrow gz\in g(X)$
as $n\rightarrow\infty$. From the definition of $\alpha$, we infer that
(47) $\displaystyle(gx_{n},gx_{n+1})\in(gA_{1}\times gA_{2})\cup(gA_{2}\times
gA_{1})$
Since $(gA_{1}\times gA_{2})\cup(gA_{2}\times gA_{1})$ is a closed set with
respect to the Euclidean metric, we get that
(48) $\displaystyle(gz,gz)\in(gA_{1}\times gA_{2})\cup(gA_{2}\times gA_{1}),$
thereby implying that $gz\in g(A_{1})\cap g(A_{2})$. Therefore, we obtain
immediately from the definition of $\alpha$ that $\alpha(gx_{n},gz)\geq 1$ for
all $n$.
Now, let $a$ be an arbitrary point in $A_{1}$. We need to show that
$\alpha(ga,fa)\geq 1$. Indeed, from condition (ii), we have $fa\in g(A_{2})$.
Since $ga\in g(A_{1})$, we get $(ga,fa)\in g(A_{1})\times g(A_{2})$, which
implies that $\alpha(ga,fa)\geq 1$.
Now, all the hypotheses of Theorem 3.1 are satisfied. Hence, we deduce that
$f$ and $g$ have a coincidence point $z\in A_{1}\cup A_{2}$, that is, $fz=gz$.
If $z\in A_{1}$, from (ii), $fz\in g(A_{2})$. On the other hand, $fz=gz\in
g(A_{1})$. Then, we have $gz\in g(A_{1})\cap g(A_{2})$, which implies from the
one-to-one property of $g$ that $z\in A_{1}\cap A_{2}$. Similarly, if $z\in
A_{2}$, we obtain that $z\in A_{1}\cap A_{2}$.
Notice that if $x$ is a coincidence point of $f$ and $g$, then $x\in A_{1}\cap
A_{2}$. Finally, let $x,y\in C(g,f)$, that is, $x,y\in A_{1}\cap A_{2}$,
$gx=fx$ and $gy=fy$. Now, from above observation, we have $w=x\in A_{1}\cap
A_{2}$, which implies that $gw\in g(A_{1}\cap A_{2})=g(A_{1})\cap g(A_{2})$
due to the fact that $g$ is one-to-one. Then, we get that $\alpha(gx,gw)\geq
1$ and $\alpha(gy,gw)\geq 1$. Then our claim holds.
Now, all the hypotheses of Theorem 3.2 are satisfied. So, we deduce that
$z=A_{1}\cap A_{2}$ is the unique common fixed point of $f$ and $g$. This
completes the proof. ∎
The following results are immediate consequences of Corollary 4.17.
###### Corollary 4.18.
Let $(X,d)$ be a complete metric space, $A_{1}$ and $A_{2}$ are two nonempty
closed subsets of $X$ and $f,g:Y\rightarrow Y$ be two mappings, where
$Y=A_{1}\cup A_{2}$. Suppose that the following conditions hold:
(i) $g(A_{1})$ and $g(A_{2})$ are closed;
(ii) $f(A_{1})\subseteq g(A_{2})$ and $f(A_{2})\subseteq g(A_{1})$;
(iii) $g$ is one-to-one;
(iv) there exists a function $\psi\in\Psi$ such that
$\displaystyle d(fx,fy)\leq\psi(d(gx,gy)),\hskip 2.84544pt\forall(x,y)\in
A_{1}\times A_{2}.$
Then, $f$ and $g$ have a coincidence point $z\in A_{1}\cap A_{2}$. Further, if
$f$, $g$ commute at their coincidence points, then $f$ and $g$ have a unique
common fixed point that belongs to $A_{1}\cap A_{2}$.
###### Corollary 4.19.
Let $(X,d)$ be a complete metric space, $A_{1}$ and $A_{2}$ are two nonempty
closed subsets of $X$ and $f,g:Y\rightarrow Y$ be two mappings, where
$Y=A_{1}\cup A_{2}$. Suppose that the following conditions hold:
(i) $g(A_{1})$ and $g(A_{2})$ are closed;
(ii) $f(A_{1})\subseteq g(A_{2})$ and $f(A_{2})\subseteq g(A_{1})$;
(iii) $g$ is one-to-one;
(iv) there exists a constant $\lambda\in(0,1)$ such that
$\displaystyle
d(fx,fy)\leq\lambda\max\left\\{d(gx,gy),\frac{d(gx,fx)+d(gy,fy)}{2},\frac{d(gx,fy)+d(gy,fx)}{2}\right\\}\hskip
2.84544pt\forall(x,y)\in A_{1}\times A_{2}.$
Then, $f$ and $g$ have a coincidence point $z\in A_{1}\cap A_{2}$. Further, if
$f$, $g$ commute at their coincidence points, then $f$ and $g$ have a unique
common fixed point that belongs to $A_{1}\cap A_{2}$.
###### Corollary 4.20.
Let $(X,d)$ be a complete metric space, $A_{1}$ and $A_{2}$ are two nonempty
closed subsets of $X$ and $f,g:Y\rightarrow Y$ be two mappings, where
$Y=A_{1}\cup A_{2}$. Suppose that the following conditions hold:
(i) $g(A_{1})$ and $g(A_{2})$ are closed;
(ii) $f(A_{1})\subseteq g(A_{2})$ and $f(A_{2})\subseteq g(A_{1})$;
(iii) $g$ is one-to-one;
(iv) there exists a constant $\lambda\in(0,1)$ such that
$\displaystyle d(fx,fy)\leq
Ad(gx,gy)+B[d(gx,fx)+d(gy,fy)]+C[d(gx,fy)+d(gy,fx)],\hskip
2.84544pt\forall(x,y)\in A_{1}\times A_{2}.$
Then, $f$ and $g$ have a coincidence point $z\in A_{1}\cap A_{2}$. Further, if
$f$, $g$ commute at their coincidence points, then $f$ and $g$ have a unique
common fixed point that belongs to $A_{1}\cap A_{2}$.
###### Corollary 4.21.
Let $(X,d)$ be a complete metric space, $A_{1}$ and $A_{2}$ are two nonempty
closed subsets of $X$ and $f,g:Y\rightarrow Y$ two mappings, where
$Y=A_{1}\cup A_{2}$. Suppose that the following conditions hold:
(i) $g(A_{1})$ and $g(A_{2})$ are closed;
(ii) $f(A_{1})\subseteq g(A_{2})$ and $f(A_{2})\subseteq g(A_{1})$;
(iii) $g$ is one-to-one;
(iv) there exists a constant $\lambda\in(0,1)$ such that
$\displaystyle d(fx,fy)\leq\lambda(d(gx,gy)),\hskip 2.84544pt\forall(x,y)\in
A_{1}\times A_{2}.$
Then, $f$ and $g$ have a coincidence point $z\in A_{1}\cap A_{2}$. Further, if
$f$, $g$ commute at their coincidence points, then $f$ and $g$ have a unique
common fixed point that belongs to $A_{1}\cap A_{2}$.
###### Corollary 4.22.
Let $(X,d)$ be a complete metric space, $A_{1}$ and $A_{2}$ are two nonempty
closed subsets of $X$ and $f,g:Y\rightarrow Y$ be two mappings, where
$Y=A_{1}\cup A_{2}$. Suppose that the following conditions hold:
(i) $g(A_{1})$ and $g(A_{2})$ are closed;
(ii) $f(A_{1})\subseteq g(A_{2})$ and $f(A_{2})\subseteq g(A_{1})$;
(iii) $g$ is one-to-one;
(iv) there exists a constant $\lambda\in(0,1)$ such that
$\displaystyle d(fx,fy)\leq\lambda[d(gx,fx)+d(gy,fy)],\hskip
2.84544pt\forall(x,y)\in A_{1}\times A_{2}.$
Then, $f$ and $g$ have a coincidence point $z\in A_{1}\cap A_{2}$. Further, if
$f$, $g$ commute at their coincidence points, then $f$ and $g$ have a unique
common fixed point that belongs to $A_{1}\cap A_{2}$.
###### Corollary 4.23.
Let $(X,d)$ be a complete metric space, $A_{1}$ and $A_{2}$ are two nonempty
closed subsets of $X$ and $f,g:Y\rightarrow Y$ be two mappings, where
$Y=A_{1}\cup A_{2}$. Suppose that the following conditions hold:
(i) $g(A_{1})$ and $g(A_{2})$ are closed;
(ii) $f(A_{1})\subseteq g(A_{2})$ and $f(A_{2})\subseteq g(A_{1})$;
(iii) $g$ is one-to-one;
(iv) there exists a constant $\lambda\in(0,1)$ such that
$\displaystyle d(fx,fy)\leq\lambda[d(gx,fy)+d(gy,fx)],\hskip
2.84544pt\forall(x,y)\in A_{1}\times A_{2}.$
Then, $f$ and $g$ have a coincidence point $z\in A_{1}\cap A_{2}$. Further, if
$f$, $g$ commute at their coincidence points, then $f$ and $g$ have a unique
common fixed point that belongs to $A_{1}\cap A_{2}$.
Remarks
* •
Letting $g=I_{X}$ in Corollary 4.18 we obtain Corollary 3.19 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.19 we obtain Corollary 3.20 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.20 we obtain Corollary 3.21 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.21 we obtain Corollary 3.22 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.22 we obtain Corollary 3.23 in [10].
* •
Letting $g=I_{X}$ in Corollary 4.23 we obtain Corollary 3.24 in [10].
## 5\. Acknowledgement
The first author gratefully acknowledges the University Grants Commission,
Government of India for financial support during the preparation of this
manuscript.
## References
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* [21] Samet, B.: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. TMA (2010) doi:10.1016/j.na.2010.02.026
* [22] Kirk, W. A., Srinivasan, P. S., Veeramani, P.: Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4, 79 -89 (2003).
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* [26] Pacurar, M., Rus I. A.:Fixed point theory for cyclic $\varphi$-contractions, Nonlinear Anal. 72, 1181-1187 (2010).
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|
arxiv-papers
| 2013-06-14T09:51:28 |
2024-09-04T02:49:46.542126
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Priya Shahi, Jatinderdeep Kaur, S. S. Bhatia",
"submitter": "Priya Shahi",
"url": "https://arxiv.org/abs/1306.3498"
}
|
1306.3533
|
arxiv-papers
| 2013-06-15T00:11:43 |
2024-09-04T02:49:46.551787
|
{
"license": "Public Domain",
"authors": "Satya Mandal and Sarang Sane",
"submitter": "Satya Mandal",
"url": "https://arxiv.org/abs/1306.3533"
}
|
|
1306.3604
|
# OFDM Synthetic Aperture Radar Imaging with Sufficient Cyclic Prefix
Tian-Xian Zhang and Xiang-Gen Xia, Fellow, IEEE Tian-Xian Zhang is with the
School of Electronic Engineering, University of Electronic Science and
Technology of China, Chengdu, Sichuan, P.R. China, 611731\. Fax:
+86-028-61830064, Tel: +86-028-61830768, E-mail: [email protected]. His
research was supported by the Fundamental Research Funds for the Central
Universities under Grant ZYGX2012YB008 and by the China Scholarship Council
(CSC) and was done when he was visiting the University of Delaware, Newark, DE
19716, USA. Xiang-Gen Xia is with the Department of Electrical and Computer
Engineering, University of Delaware, Newark, DE 19716, USA. Email:
[email protected]. Xia’s research was partially supported by the Air Force
Office of Scientific Research (AFOSR) under Grant FA9550-12-1-0055.
###### Abstract
The existing linear frequency modulated (LFM) (or step frequency) and random
noise synthetic aperture radar (SAR) systems may correspond to the frequency
hopping (FH) and direct sequence (DS) spread spectrum systems in the past
second and third generation wireless communications. Similar to the current
and future wireless communications generations, in this paper, we propose OFDM
SAR imaging, where a sufficient cyclic prefix (CP) is added to each OFDM
pulse. The sufficient CP insertion converts an inter-symbol interference (ISI)
channel from multipaths into multiple ISI-free subchannels as the key in a
wireless communications system, and analogously, it provides an inter-range-
cell interference (IRCI) free (high range resolution) SAR image in a SAR
system. The sufficient CP insertion along with our newly proposed SAR imaging
algorithm particularly for the OFDM signals also differentiates this paper
from all the existing studies in the literature on OFDM radar signal
processing. Simulation results are presented to illustrate the high range
resolution performance of our proposed CP based OFDM SAR imaging algorithm.
###### Index Terms:
Cyclic prefix (CP), inter-range-cell interference (IRCI), orthogonal
frequency-division multiplexing (OFDM), synthetic aperture radar (SAR)
imaging, swath width matched pulse (SWMP), zero sidelobes.
## I Introduction
Synthetic aperture radar (SAR) can perform well to image under almost all
weather conditions [1], which, in the past decades, has received considerable
attention. Several types of SAR systems using different transmitted signals
have been well developed and analyzed, such as the linear frequency modulated
(LFM) chirp radar [2], linear/random step frequency radar [1, 3], and random
noise radar [4, 5, 6].
Recently, orthogonal frequency-division multiplexing (OFDM) signals have been
used in radar applications, which may provide opportunities to achieve
ultrawideband (UWB) radar. OFDM radar signal processing was first presented in
[7] and was also studied in [8, 9, 10, 11, 12, 13]. Adaptive OFDM radar was
investigated for moving target detection and low-grazing angle target tracking
in [14, 15, 16]. Using OFDM signals for SAR applications was proposed in [17,
18, 19, 20, 21, 22, 23]. In [17, 18, 19], adaptive OFDM signal design was
studied for range ambiguity suppression in SAR imaging. The reconstruction of
the cross-range profiles is studied in [22, 23]. Signal processing of a
passive OFDM radar using digital audio broadcast (DAB), digital video
broadcast (DVB), Wireless Fidelity (WiFi) or worldwide inoperability for
microwave access (WiMAX) signals for target detection and SAR imaging was
investigated in [24, 25, 26, 27, 28, 29, 30]. However, all the existing OFDM
radar (including SAR) signal processing is on radar waveform designs with
ambiguity function analyses to mitigate the interferences between range/cross-
range cells using multicarrier signals similar to the conventional waveform
designs and the radar receivers, such as SAR imaging algorithms, are basically
not changed. The most important feature of OFDM signals in communications
systems, namely, converting an intersymbol-interference (ISI) channel to
multiple ISI-free subchannels, when a sufficient cyclic prefix (CP) is
inserted, has not been utilized so far in the literature. In this paper, we
will fully take this feature of the OFDM signals into account to propose OFDM
SAR imaging where a sufficient CP is added to each OFDM pulse, as the next
generation high range resolution SAR imaging. In our proposed SAR imaging
algorithm, not only the transmission side but also the receive side are
different from the existing SAR imaging methods. To further explain it, let us
briefly overview some of the key signalings in SAR imaging.
To achieve long distance imaging, a pulse with long enough time duration is
used to carry enough transmit energy [31]. The received pulses from different
scatterers are overlapped with each other and cause energy interferences
between these scatterers. To mitigate the impact of the energy interferences
and achieve high resolution, the transmitted pulse is coded using frequency or
phase modulation (i.e., LFM signal and step frequency signal) or random noise
type signals in random noise radar to achieve a bandwidth $B$ which is large
compared to that of an uncoded pulse with the same time duration [31]. This is
similar to the spread spectrum technique in communications systems. Then,
pulse compression techniques are applied at the receiver to yield a narrow
compressed pulse response. Thus, the reflected energies from different range
cells can be distinguished [31]. However, the energy interferences between
different range cells, that we regard as inter-range-cell interference (IRCI),
still exist because of the sidelobes of the ambiguity function of the
transmitted signal, whose sidelobe magnitude is roughly $\sqrt{N}$ if the
mainlobe of the modulated signal pulse is $N$ [32]. This IRCI is much
significant in SAR imaging [2]. Despite of the IRCI, the existing well-
known/used SAR are LFM (or step frequency) SAR [2, 3] and random noise SAR [4,
5, 6].
We are adopting the OFDM technique [33] that is the key technology in the
latest wireless communications standards, such as long term evolution (LTE)
[34] and WiFi. If we think about only one user in a wireless communications
system, there are similarity and difference between wireless communications
systems and SAR imaging systems. The similarity is that both systems are
transmitting and receiving signals reflected from various scatterers and the
numbers of multipaths depend on the transmitted signal bandwidths. The
difference is that in communications systems, the receiver cares about the
transmitted signal, while in SAR imaging systems, the receiver cares about the
scatterers within different range cells (with different time delays) that
reflect and cause multipath signals. The multipaths cause ISI in
communications, while the multipaths cause the IRCI in SAR imaging systems.
The higher the bandwidth is, the more multipaths there are in communications
systems, and the more range cells there are for a fixed imaging scene (or
swath width), i.e., the higher the range resolution is, in SAR imaging
systems. The existing well-known/used LFM (or step frequency) and random noise
SAR systems, in fact, correspond to the two spread spectrum systems, i.e.,
frequency hopping (FH) and direct sequence (DS) in CDMA systems [35], which
work well in non-high bandwidth wireless communications systems, such as the
second and the third generations of cellular communications, of less than $10$
MHz (roughly) bandwidth. They, however, may not work well for a system of a
much higher signal bandwidth, such as $20$ MHz in LTE, due to the severe ISI
caused by too many multipaths. In contrast, since OFDM with a sufficient CP
can convert an ISI channel to multiple ISI-free subchannels, as mentioned
earlier, it is used in the latest LTE and WiFi standards. As an analogy, one
expects that LFM and random noise SAR may not work too well for high bandwidth
radar systems where there are too many range cells in one cross range, which
cause severe IRCI due to the significant sidelobes of the ambiguity functions
of the LFM and random signals. On the other hand, in order to have a high
range resolution, a high bandwidth is necessary. Therefore, borrowing from
wireless communications, in this paper we propose to use OFDM signals with
sufficient CP to deal with the IRCI problem as a next generation SAR imaging
to produce a high range resolution SAR image. We show that, in our proposed
OFDM SAR imaging with a sufficient CP, the sidelobes are ideally zero, and for
any range cell, there will be no IRCI from other range cells in one cross
range.
Another difference for OFDM communications systems and CP based OFDM SAR
systems is as follows. It is known in communications that, for an OFDM system,
a Doppler frequency shift is not desired, while the azimuth domain (or cross
range direction) in a SAR imaging system is, however, generated from the
relative Doppler frequency shifts between the radar platform and the
scatterers. One might ask how the OFDM signals are used to form a SAR image.
This question is not difficult to answer. The range distance between radar
platform and image scene is known and the radar platform moving velocity is
known too. Thus, the Doppler shifts are also known, which can be used to
generate the cross ranges similar to other SAR imaging techniques, and can be
also used to compensate the Doppler shift inside one cross range and correct
the range migration.
This paper is organized as follows. In Section II, we propose our CP based
OFDM SAR imaging algorithm. In Section III, we present some simulations to
illustrate the high range resolution property of the proposed CP based OFDM
SAR imaging and also the necessity of a sufficient CP insertion in an OFDM
signal. In Section IV, we conclude this paper and point out some future
research problems.
## II System Model and CP Based OFDM SAR Imaging
In this section, we first describe the OFDM SAR signal model, and then propose
the corresponding SAR imaging algorithm.
### II-A OFDM SAR signal model
In this paper, we consider the monostatic broadside stripmap SAR geometry as
shown in Fig. 1. The radar platform is moving parallelly to the $y$-axis with
an instantaneous coordinate $\left(0,y_{p}(\eta),H_{p}\right)$, $H_{p}$ is the
altitude of the radar platform, $\eta$ is the relative azimuth time referenced
to the time of zero Doppler, $T_{a}$ is the synthetic aperture time defined by
the azimuth time extent the target stays in the antenna beam. For convenience,
let us choose the azimuth time origin $\eta=0$ to be the zero Doppler sample.
Consider an OFDM signal with $N$ subcarriers, a bandwidth of $B$ Hz, and let
${\mbox{\boldmath{$S$}}}=\left[S_{0},S_{1},\ldots,S_{N-1}\right]^{T}$
represent the complex weights transmitted over the subcarriers, and
$\sum\limits_{k=0}^{N-1}\left|S_{k}\right|^{2}=N$. Then, a discrete time OFDM
signal is the inverse fast Fourier transform (IFFT) of the vector $S$ and the
OFDM pulse is
$s\left(t\right)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}S_{k}\textrm{exp}\left\\{j2\pi
k\Delta ft\right\\},\ t\in\left[0,T+T_{GI}\right],$ (1)
where $\Delta f=\frac{B}{N}=\frac{1}{T}$ is the subcarrier spacing.
$\left[0,T_{GI}\right)$ is the time duration of the guard interval that
corresponds to the CP in the discrete time domain as we shall see later in
more details and its length $T_{GI}$ will be specified later too, $T$ is the
length of the OFDM signal excluding CP. Due to the periodicity of the
exponential function $\textrm{exp}(\cdot)$ in (1), the tail part of
$s\left(t\right)$ for $t$ in $\left(T,T+T_{GI}\right]$ is the same as the head
part of $s\left(t\right)$ for $t$ in $\left[0,T_{GI}\right)$. Let $f_{c}$ be
the carrier frequency of operation, the transmitted signal is given by
$s_{1}\left(t\right)=\textrm{Re}\left\\{\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}S_{k}\textrm{exp}\left\\{j2\pi
f_{k}t\right\\}\right\\},\ t\in\left[0,T+T_{GI}\right],$ (2)
where $f_{k}=f_{c}+k\Delta f$ is the $k$th subcarrier frequency.
Figure 1: Monostatic stripmap SAR geometry.
After the demodulation to baseband, the complex envelope of the received
signal from a static point target in the $m$th range cell can be written in
terms of fast time $t$ and slow time $\eta$
$\begin{array}[]{ll}u_{m}\left(t,\eta\right)&={g_{m}}\varepsilon_{a}(\eta)\textrm{exp}\left\\{-j4\pi
f_{c}\frac{R_{m}(\eta)}{c}\right\\}\\\
&\times\frac{1}{\sqrt{N}}\sum\limits_{k=0}^{N-1}S_{k}\textrm{exp}\left\\{\frac{j2\pi
k}{T}\left[t-\frac{2R_{m}(\eta)}{c}\right]\right\\}+w(t,\eta),\
t\in\left[\frac{2R_{m}(\eta)}{c},\frac{2R_{m}(\eta)}{c}+T+T_{GI}\right],\\\
\end{array}$ (3)
where $\varepsilon_{a}(\eta)=p_{a}^{2}\left(\theta(\eta)\right)$ is the
azimuth envelope,
$p_{a}(\theta)\approx\textrm{sinc}\left(\frac{0.886\theta}{\beta_{bw}}\right)$
[2], $\textrm{sinc}(x)=\frac{\textrm{sin}(x)}{x}$ is the sinc function,
$\theta$ is the angle measured from boresight in the slant range plane,
$\beta_{bw}=\frac{0.866\lambda}{L_{a}}$ is the azimuth beamwidth, $L_{a}$ is
the effective length of the antenna, $g_{m}$ is the radar cross section (RCS)
coefficient caused from the scatterers in the $m$th range cell within the
radar beam footprint, and $c$ is the speed of light. $w(t,\eta)$ represents
the noise. $R_{m}(\eta)$ is the instantaneous slant range between the radar
and the $m$th range cell with the coordinate $(x_{m},y_{m},0)$ and it can be
written as
$R_{m}(\eta)=\sqrt{\bar{R}_{m}^{2}+\left(y_{m}-y_{p}\left(\eta\right)\right)^{2}}=\sqrt{\bar{R}_{m}^{2}+v_{p}^{2}\eta^{2}},$
(4)
where $\bar{R}_{m}=\sqrt{x_{m}^{2}+H_{p}^{2}}$ is the slant range when the
radar platform and the target in the $m$th range cell are the closest
approach, and $v_{p}$ is the effective velocity of the radar platform.
Figure 2: Illustration diagram of one range line.
Then, the complex envelope of the received signal from all the range cells in
a swath can be written as
$u(t,\eta)=\sum_{m}u_{m}(t,\eta).$ (5)
At the receiver, with the A/D converter, the received signal is sampled with
sampling interval $T_{s}=\frac{1}{B}$ and the range resolution is
$\rho_{r}=\frac{c}{2B}$. Assume that the swath width for the radar is $R_{w}$.
Let $M=\frac{R_{w}}{\rho_{r}}$ that is determined by the radar system. Then, a
range profile can be divided into $M$ range cells as shown in Fig. 2. As we
mentioned earlier, the main reason why OFDM has been successfully adopted in
both recent wireline and wireless communications systems is its ability to
deal with multipaths (they cause ISI in communications) that become more
severe when the signal bandwidth is larger. In our radar applications here,
the response of each range cell, formed by the summation of the responses of
all scatterers within this range cell, contains its own delay and phase. Thus,
to a transmitted pulse, each range cell can be regarded as one path of
communications. $M$ range cells correspond to $M$ paths. Excluding one main
path (i.e., the nearest range cell), there will be $M-1$ multipaths. To
convert the ISI caused from the multipaths to the ISI free case in
communications, a guard interval (or CP) needs to be added to each OFDM block
and the CP length can not be smaller than the number of multipaths that is
$M-1$ in this paper. Although in the radar application here, ISI is not the
concern, the $M$ range cell paths are superposed (or interfered) together in
the radar return signal, which is the same as the ISI in communications. So,
in order to convert these interfered $M$ range cells to individual range cells
without any IRCI, similar to OFDM systems in communications, the CP length
should be at least $M-1$. For convenience, we use CP length $M-1$ in this
paper, i.e., a CP of length $M-1$ is added at the beginning of an OFDM pulse,
and then the guard interval length $T_{GI}$ in the analog transmission signal
is $T_{GI}=(M-1)T_{s}$. Notice that $T=NT_{s}$, so the time duration of an
OFDM pulse is $T_{o}=T+T_{GI}=(N+M-1)T_{s}$. In this paper, we assume $N\geq
M$, i.e., the number of subcarriers of the OFDM signal is at least the number
of range cells in a swath (or a cross range), which is similar to the
application in communications [33, 35]. When $N<M$, the IRCI occurs and the
detailed reason will be seen later.
Figure 3: Transmission comparison of OFDM signals in (a) communications
systems and (b) SAR systems.
In communications applications, to achieve a high transmission throughput, the
OFDM pulses are transmitted consecutively as shown in Fig. 3(a). However, in
SAR imaging applications, for monostatic case, the transmitter and receiver
share the same antenna, which can not both transmit and receive signals at the
same time and transmission throughput is not a concern. Thus, transmitted
signals and radar return signals are usually separated in time. This implies
that a reasonable receive window is needed between two consecutive pulses as
shown in Fig. 3(b). For convenience, similar to what is commonly done in SAR
systems, in this paper, we assume that the pulse repetition interval (PRI) is
long enough so that all of the range cells in a swath fall within the receive
window. Therefore, the PRI length $T_{\textrm{PRI}}$ should be
$T_{\textrm{PRI}}=\frac{1}{\textrm{PRF}}>\left(\frac{2R_{w}}{c}+T_{o}\right),$
(6)
where $R_{w}=M\rho_{r}$ is the swath width. We want to emphasize here that in
our common SAR imaging applications, the pulse repetition frequency (PRF) may
not be too high [2] and there is sufficient time duration to add a CP (a guard
interval) for an OFDM pulse.
For Fig. 2, we notice that $R_{m}(\eta)=R_{0}(\eta)+m\rho_{r}$. Thus,
$t-\frac{2R_{m}(\eta)}{c}$ in (3) can be written as
$\begin{array}[]{ll}t-\frac{2R_{m}(\eta)}{c}&=t-\frac{2\left(R_{0}(\eta)+m\rho_{r}\right)}{c}\\\
&=t-t_{0}(\eta)-mT_{s},\\\ \end{array}$ (7)
where for each $\eta$, the constant time delay
$t_{0}(\eta)=\frac{2R_{0}(\eta)}{c}$ is independent of $m$. Let the sampling
be aligned with the start of the received pulse after $t_{0}(\eta)$ seconds
for the first arriving version of the transmitted pulse. Combining with (3),
(7) and (5), $u(t,\eta)$ can be converted to the discrete time linear
convolution of the transmitted sequence with the weighting RCS coefficients
$d_{m}$, and the received sequence can be written as
$u_{i}=\sum_{m=0}^{M-1}d_{m}s_{i-m}+w_{i},\ i=0,1,\ldots,N+2M-3,$ (8)
where
$d_{m}=g_{m}\varepsilon_{a}(\eta)\textrm{exp}\left\\{-j4\pi
f_{c}\frac{R_{m}(\eta)}{c}\right\\},$ (9)
in which $4\pi f_{c}\frac{R_{m}(\eta)}{c}$ in the exponential is the azimuth
phase, and $s_{i}$ is the complex envelope of the OFDM pulse in (1) with time
duration $t\in\left[0,T+T_{GI}\right]$ for $T=NT_{s}$ and $T_{GI}=(M-1)T_{s}$.
After sampling at $t=iT_{s}$, (1) can be recast as:
$s_{i}=s\left(iT_{s}\right)=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}S_{k}\textrm{exp}\left\\{\frac{j2\pi
ki}{N}\right\\},\ i=0,1,\ldots,N+M-2,$ (10)
and $s_{i}=0$ if $i<0$ or $i>N+M-2$.
Notice that the transmitted sequence with CP is
$\tilde{{\mbox{\boldmath{$s$}}}}=\left[s_{0},s_{1},\ldots,s_{N+M-2}\right]^{T}$,
where
$\left[s_{0},\ldots,s_{M-2}\right]^{T}=\left[s_{N},\ldots,s_{N+M-2}\right]^{T}$.
The vector
${\mbox{\boldmath{$s$}}}=\left[s_{0},s_{1},\ldots,s_{N-1}\right]^{T}$ is
indeed the IFFT of the vector
${\mbox{\boldmath{$S$}}}=\left[S_{0},S_{1},\ldots,S_{N-1}\right]^{T}$.
### II-B Range compression
When the signal in (8) is received, the first and the last $M-1$ samples111The
reason to remove both the head and the tail $M-1$ samples is because the total
number of received signal samples in (8) is $N+2(M-1)$. Because of the receive
window between the OFDM pulses as shown in Fig. 3(b), the tail $M-1$ samples
in (8) are not affected by the follow-up OFDM pulses. However, they do not
have the full $M$ RCS coefficients from all the $M$ range cells. If there is
no receive window between the OFDM pulses, the transmission is shown in Fig.
3(a) as in communications, we only remove the head $M-1$ samples from the
received signal sequence $u_{n}$ and use the next $N$ samples of $u_{n}$
starting from $n=M-1$. are removed, and then, we obtain
$u_{n}=\sum_{m=0}^{M-1}d_{m}s_{n-m}+w_{n},\ n=M-1,M,\ldots,N+M-2.\\\ $ (11)
Then, the received signal
${\mbox{\boldmath{$u$}}}=\left[u_{M-1},u_{M},\ldots,u_{N+M-2}\right]^{T}$ is
$\begin{bmatrix}u_{M-1}\\\ u_{M}\\\ \vdots\\\
u_{N+M-2}\end{bmatrix}=\begin{bmatrix}d_{M-1}&\cdots&d_{0}&\cdots&\cdots&0&\cdots&0\\\
0&d_{M-1}&\cdots&d_{0}&\cdots&0&\cdots&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
0&\cdots&0&\cdots&\cdots&d_{M-1}&\cdots&d_{0}\end{bmatrix}\begin{bmatrix}s_{0}\\\
\vdots\\\ s_{M-1}\\\ s_{M}\\\ \vdots\\\
s_{N+M-2}\end{bmatrix}+\begin{bmatrix}w_{M-1}\\\ w_{M}\\\ \vdots\\\
w_{N+M-2}\end{bmatrix}.$ (12)
Since
$\left[s_{0},\ldots,s_{M-2}\right]^{T}=\left[s_{N},\ldots,s_{N+M-2}\right]^{T}$,
it is not hard to see that the vector
$\tilde{{\mbox{\boldmath{$s$}}}}=\left[s_{0},s_{1},\ldots,\right.$
$\\!\\!\\!\left.s_{N+M-2}\right]^{T}$ in (12) can be replaced by its tail
part222This part is slightly different from what appears in communications
applications [33, Ch. 5.2], [35, Ch. 12.4] where the vector
${\mbox{\boldmath{$s$}}}^{\prime}$ in (13) is replaced by the head part, $s$,
of the vector $\tilde{{\mbox{\boldmath{$s$}}}}$.
${\mbox{\boldmath{$s$}}}^{\prime}$, which can be also seen, in, for example,
[35, Ch. 12.4], then, the matrix representation (12) is equivalent to the
following representation:
${\mbox{\boldmath{$u$}}}={\mbox{\boldmath{$H$}}}{\mbox{\boldmath{$s$}}}^{\prime}+{\mbox{\boldmath{$w$}}},$
(13)
where
${\mbox{\boldmath{$s$}}}^{\prime}=\left[s_{M-1},s_{M},\cdots,s_{N+M-2}\right]^{T}=\left[s_{M-1},\cdots,s_{N-1},s_{0},\cdots,s_{M-2}\right]^{T}$,
${\mbox{\boldmath{$w$}}}=\left[w_{M-1},w_{M},\ldots,\right.$
$\\!\\!\\!\left.w_{N+M-2}\right]^{T}$ and $H$ is built by superposing the
first $M-1$ columns of the weighting RCS coefficient matrix in (12) to its
last $M-1$ columns. And $H$ can be given by the following $N$ by $N$ matrix:
${\mbox{\boldmath{$H$}}}=\begin{bmatrix}d_{0}&0&\cdots&0&d_{M-1}&\cdots&d_{1}\\\
\vdots&\ddots&\ddots&\vdots&\ddots&\ddots&\vdots\\\
d_{M-2}&\cdots&d_{0}&0&\cdots&0&d_{M-1}\\\
d_{M-1}&d_{M-2}&\cdots&d_{0}&0&\cdots&0\\\
0&\ddots&\ddots&\vdots&\ddots&\ddots&\vdots\\\
\vdots&\ddots&d_{M-1}&d_{M-2}&\cdots&d_{0}&0\\\
0&\cdots&0&d_{M-1}&d_{M-2}&\cdots&d_{0}\end{bmatrix}.$ (14)
One can see that the matrix $H$ in (14) is a circulant matrix that can be
diagonalized by the discrete Fourier transform (DFT) matrix of the same size.
The OFDM demodulator then performs a fast Fourier transform (FFT) on the
vector $u$:
$U_{k}=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}u_{n+M-1}\textrm{exp}\left\\{\frac{-j2\pi
kn}{N}\right\\},\ k=0,1,\ldots,N-1.$ (15)
From (13)-(14), the above $U_{k}$ can be expressed as:
$U_{k}=D_{k}S_{k}^{\prime}+W_{k},\ k=0,1,\ldots,N-1,$ (16)
where $\left[S_{0}^{\prime},S_{1}^{\prime},\cdots,S_{N-1}^{\prime}\right]^{T}$
is the FFT of the vector ${\mbox{\boldmath{$s$}}}^{\prime}$, a cyclic shift of
the vector $s$ of amount $M-1$, i.e.,
$S_{k}^{\prime}=S_{k}\textrm{exp}\left\\{\frac{j2\pi k(M-1)}{N}\right\\}$,
$W_{k}$ is the FFT of the noise, and
$D_{k}=\sum_{m=0}^{M-1}d_{m}\textrm{exp}\left\\{\frac{-j2\pi mk}{N}\right\\}.$
(17)
Then, the estimate of $D_{k}$ is
$\hat{D}_{k}=\frac{U_{k}}{S_{k}^{\prime}}=\frac{U_{k}}{S_{k}\textrm{exp}\left\\{\frac{j2\pi
k(M-1)}{N}\right\\}}=D_{k}+\frac{W_{k}}{S_{k}}\textrm{exp}\left\\{\frac{-j2\pi
k(M-1)}{N}\right\\},\ k=0,1,\ldots,N-1.$ (18)
Notice that if $S_{k}$ is small, the noise is enhanced. Thus, for the
constraint condition $\sum\limits_{k=0}^{N-1}\left|S_{k}\right|^{2}=N$, from
(18) the optimal signal $S_{k}$ should have constant module for all $k$.
The vector
${\mbox{\boldmath{$D$}}}=\left[D_{0},D_{1},\ldots,D_{N-1}\right]^{T}$ is
indeed the $N$-point FFT of $\sqrt{N}{\mbox{\boldmath{$\gamma$}}}$, where
$\gamma$ is the weighting RCS coefficient vector:
${\mbox{\boldmath{$\gamma$}}}=\left[d_{0},d_{1},\cdots,d_{M-1},\underbrace{0,\cdots,0}_{N-M}\right]^{T}.$
(19)
So, the estimate of $d_{m}$ can be achieved by the $N$-point IFFT on the
vector
$\hat{{\mbox{\boldmath{$D$}}}}=\left[\hat{D}_{0},\hat{D}_{1},\ldots,\hat{D}_{N-1}\right]^{T}$:
$\hat{d}_{m}=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}\hat{D}_{k}\textrm{exp}\left\\{\frac{j2\pi
mk}{N}\right\\},\ m=0,\ldots,M-1.\\\ $ (20)
Then, we obtain the following estimates of the $M$ range cell weighting RCS
coefficients:
$\hat{d}_{m}={\sqrt{N}}d_{m}+\tilde{w}_{m}^{\prime},\ m=0,\ldots,M-1,$ (21)
where $\tilde{w}_{m}^{\prime}$ is from the noise and its variance is the same
as that in (18) since the IFFT implemented in (20) is a unitary transform. In
$(\ref{estimated})$, $d_{m}$ can be completely recovered without any IRCI from
other range cells. From (9), when $d_{m}$ are determined, the RCS coefficients
$g_{m}$ are determined, and vice versa.
Figure 4: Transmitted and received timing of OFDM SAR system for $M=N$.
For $N>M$, there are some zeros in the vector $\gamma$ in (19). Considering
(19) and (20), we notice that part of the transmitted OFDM sequence is used to
estimate the unreal weighting RCS coefficients, i.e., the zeros in $\gamma$.
When $N=M$, there is no zeros included in the vector $\gamma$ in (19). We name
this special case as swath width matched pulse (SWMP). The OFDM pulse length
(excluding CP) $T=NT_{s}$ and the swath width $M\rho_{r}$ follow the
relationship of $M\rho_{r}=\frac{cT}{2}$, i.e., the range resolution of a
pulse with time duration $T$ is just the swath width $M\rho_{r}$ without range
compression processing. And $\frac{cT}{2}$ is the maximal swath width that we
can obtain without IRCI. Thus, the optimal time duration of the OFDM pulse is
$T_{o}=(2N-1)T_{s}$ with CP length $N-1$, which is the maximal possible CP
length for an OFDM sequence of length $N$.
Since the first and the last $M-1$ samples of the received sequence $u_{i}$ in
(8) are removed, the receiver only needs to sample the received signal from
$t=T_{0}$ to $t=T_{0}+T$, even when the radar return signal starts to arrive
at time $t=T$ as shown in Fig. 4. Therefore, the minimum range of the OFDM
radar is $\frac{cT}{2}$, the same as that of the traditional pulse radar of
transmitted pulse length $T$. Notice that the minimum range is just the same
as the maximal swath width. Thus, if we want to increase the swath width to,
e.g., $10$ km, the transmitted pulse duration $T_{o}$ should be increased to
about $133.3\ \mu s$ and the minimum range is increased333The pulse length
here is much longer than the traditional radar pulse and the number $N$ of
subcarriers in OFDM signals is large too. While long pulses of LFM signals
need high frequency linearity and stability and thus are not easy to generate,
although long LFM pulses are not necessary, long OFDM pulses (i.e., large $N$)
are not difficult to generate since OFDM signals can be easily generated by
the IFFT operation as in (1). Moreover, the conventional multiple channel SAR
technology [36] for wide imaged swath width can be used to reduce the OFDM
pulse length and the minimum range. to $10$ km. Also, the minimum receive
window is just the OFDM pulse length $T$ (excluding CP) as Fig. 4, and the PRI
length follows $T_{\textrm{PRI}}>T_{o}+T$. As a remark, different from the
applications in communications where the CP is an overhead and may reduce the
transmission data rate, in the SWMP case in SAR applications here, the longer
the CP length is, the less the IRCI is, which leads to a better (high
resolution) SAR image.
When $N<M$, according to (10) the signal vector
$\left[s_{0},\cdots,s_{M-1},s_{M},\cdots,s_{N+M-2}\right]^{T}$ in (12) is
$\left[s_{0},\ldots,s_{N-1},s_{0},\ldots,s_{N-1},\ldots,s_{(N+M-2)_{N}}\right]^{T}$,
where $(n)_{N}$ is the residue of $n$ modulo $N$. Thus, the $N$ by $N$ matrix
$H$ in (13) and (14) becomes
${\mbox{\boldmath{$H$}}}=\begin{bmatrix}\tilde{d}_{0}&\tilde{d}_{N-1}&\cdots&\tilde{d}_{1}\\\
\tilde{d}_{1}&\tilde{d}_{0}&\cdots&\tilde{d}_{2}\\\
\vdots&\vdots&\ddots&\vdots\\\
\tilde{d}_{N-1}&\tilde{d}_{N-2}&\cdots&\tilde{d}_{0}\end{bmatrix},$ (22)
where $\tilde{d}_{n}=\sum\limits_{i:\ 0\leq iN+n\leq M-1}d_{iN+n},\
n=0,\ldots,N-1$. One can see, $\tilde{d}_{n}$ is a summation of weighting RCS
coefficients from several range cells, i.e., each $\tilde{d}_{n}$ has IRCI.
Then, following the OFDM approach (15)-(21), what we can solve is the
superposed weighting RCS coefficients $\tilde{d}_{n}$, i.e., IRCI occurs. This
is the reason why we require $N\geq M$ in this paper.
After the range compression, combining the equations (3), (8), (9) and (21),
the range compressed signal can be written as
$u_{ra}(t,\eta)=\sqrt{N}\sum\limits_{m=0}^{M-1}\hat{g}_{m}\delta\left(t-\frac{2R_{m}(\eta)}{c}\right)\varepsilon_{a}(\eta)\textrm{exp}\left\\{-j4\pi
f_{c}\frac{R_{m}(\eta)}{c}\right\\}+w_{ra}(t,\eta),$ (23)
where $\delta\left(t-\frac{2R_{m}(\eta)}{c}\right)$ is the delta function with
non-zero value at $t=\frac{2R_{m}(\eta)}{c}$, which indicates that the
estimates $\hat{g}_{m}$ of the RCS coefficient values ${g}_{m}$ are not
affected by any IRCI from other range cells after the range compression.
$\hat{g}_{m}$ can be obtained via (9) using the estimate $\hat{d}_{m}$ in
(20). In the delta function, the target range migration is incorporated via
the azimuth varying parameter $\frac{2R_{m}(\eta)}{c}$. Also, the azimuth
phase in the exponential is unaffected by the range compression.
Comparing with (3), we notice that the range compression gain in (23) is equal
to $\sqrt{N}$, and the noise powers are the same in (23) and (3) when $S_{k}$
have constant module. Thus, the signal-to-noise ratio (SNR) gain after the
range compression is $N$. For an LFM signal pulse with time duration
$T_{L}=T_{o}=(N+M-1)T_{s}$ and the same transmitted signal energy as in (1),
it is well known that the SNR gain after range compression is $N+M-1$, which
is equal to the time-bandwidth product (TBP) of the LFM signal pulse [2].
Clearly, $N<N+M-1<2N$. This implies that the LFM range compression SNR gain is
larger (but not too much larger) than that of the OFDM pulse. However, the
IRCI exists because of the sidelobes of the ambiguity function, resulting in a
significant imaging performance degradation. In fact, the sidelobe magnitude
is roughly in the order of $\sqrt{N}$ in this case and all the sidelobes from
scatterers in all other $M-1$ range cells will be added to the $m$th range
cell for an arbitrary $m,\ 0\leq m\leq M-1$. One can see that when $M$ is
roughly more than $\sqrt{N}$, the scatterers in the $m$th range cell will be
possibly buried by the sidelobes of the scatterers in other range cells and
therefore can not be well detected and imaged. This is similarly true for a
random noise radar. In contrast, since the sidelobes are ideally $0$ in the
OFDM signal here, all $M$ scatterers can be ideally detected and imaged
without any IRCI as long as $N\geq M$, which may provide a high range
resolution image.
### II-C Discussion on the design of weights $S_{k}$
As one has seen from (15)-(21), in order to estimate the weighting RCS
coefficients, the noise needs to be divided by $S_{k}$, which may be
significantly enhanced if $S_{k}$ is small. As mentioned earlier, in this
regard, the optimal weights $S_{k}$ should have constant module.
A special case of constant modular weights is that all $S_{k}$ are the same,
i.e., a constant. In this special case, the signal sequence $s_{i}$ to
transmit is the delta sequence, i.e., $s_{0}=\sqrt{N}$, and $s_{i}=0$ if
$0<i\leq N-1$, which is equivalent to the case of short rectangular pulse of
pulse length $\left[0,\ \frac{T}{N}\right]$. When a high range resolution is
required, a large bandwidth $B$ is needed and then there will be a large
number $M$ of range cells in a swath. This will require a large $N$. In this
case, such a short pulse with length $\left[0,\ \frac{T}{N}\right]$ and power
$N$ may not be easily implemented [31]. This implies that constant weights
$S_{k}$ may not be a good choice for the proposed OFDM signals.
Another case is when all the weights $S_{k}$ are completely random, i.e., they
are independently and identically distributed (i.i.d.). In this case, the mean
power of the transmitted signal $s_{i}$ is constant for every $i$. This gives
us the interesting property for an OFDM signal, namely, although its bandwidth
is as large as the short pulse of length $\left[0,\ \frac{T}{N}\right]$, its
mean energy is evenly spread over much longer ($N$ times longer) pulse
duration, which makes it much easier to generate and implement in a practical
system than the short pulse case.
In terms of the peak-to-average power ratio (PAPR) of transmitted signals, the
former case corresponds to the worst case, i.e., the highest PAPR case that is
$N$, while the later case corresponds to the best case in the mean sense,
i.e., the lowest PAPR case that is $1$. After saying so, the above i.i.d.
weight $S_{k}$ case is only in the statistical sense. In practice, a
deterministic weight sequence $S_{k}$ is used, which can be only a pseudo-
random noise (PN) sequence and therefore, its $N$-point inverse discrete
Fourier transform $s_{i}$ (and/or its analog waveform $s(t)$) may not have a
constant power and in fact, its PAPR may be high (although may not be the
highest) compared with the LFM radar or the random noise radar. This will be
an interesting future research problem on how to deal with the high PAPR
problem of OFDM signals for radar applications. Note that there have been many
studies for the PAPR reduction in communications community, see for example
[33, 35]. If we only consider the finite time domain signal values, i.e., the
IDFT, $s_{i}$, of the weights $S_{k}$ in (10), we can use a Zadoff-Chu
sequence as $S_{k}$ that is, in fact, a discrete LFM signal, and then its
IDFT, $s_{i}$, has constant module as well [37, 38]. In this case, both the
weights $S_{k}$ and the discrete time domain signal values $s_{i}$ have
constant module, i.e., the discrete PAPR (the peak power over the mean power
of $s_{i}$) is $1$.
As a remark, if one only considers the discrete transmitted signal sequence
$s_{i}$, it can be, in fact, from any radar signal, such as LFM or random
noise radar signal as follows.
Let $s^{\prime}\left(t\right)$ be any radar transmitted signal and set
$s_{i}=s^{\prime}\left(\frac{iT}{N}\right),\ 0\leq i\leq N-1$. Then, we can
always find the corresponding weights $S_{k},\ 0\leq k\leq N-1$, that are just
the $N$-point FFT of $s_{i},\ 0\leq i\leq N-1$. Then, the analog OFDM waveform
$s\left(t\right)$ in (1) can be thought of as an interpolation of this
discrete time sequence $s_{i}$. Thus, as $N$ goes large, the analog OFDM
waveform $s\left(t\right)$ can approach the given radar waveform
$s^{\prime}\left(t\right)$.
### II-D Insufficient CP case
Let us consider the CP length to be $\bar{M}$ and $\bar{M}<M-1$. In this case,
the length of CP is insufficient. When an insufficient CP is used, if OFDM
pulses are transmitted consecutively without any waiting interval as in
communications systems, the OFDM blocks will interfer each other due to the
multipaths, which is called inter-block-interference (IBI). However, this will
not occur in our radar application in this paper, since the second OFDM pulse
needs to wait for receiving all the radar return signals of the first
transmitted OFDM pulse as we have explained in Section II.A earlier. Although
there is no IBI, the insufficient CP will cause the inter-carrier-interference
(ICI) that leads to the IRCI as shown below. In this case, (13) can be recast
as
$\begin{array}[]{ll}\Delta{\mbox{\boldmath{$u$}}}&={\mbox{\boldmath{$u$}}}-\bar{{\mbox{\boldmath{$u$}}}}\\\
&={\mbox{\boldmath{$H$}}}{\mbox{\boldmath{$s$}}}^{\prime}-\bar{{\mbox{\boldmath{$H$}}}}{\mbox{\boldmath{$s$}}}^{\prime}+{\mbox{\boldmath{$w$}}},\end{array}$
(24)
where
$\bar{{\mbox{\boldmath{$H$}}}}=\begin{bmatrix}0&\cdots&d_{M-1}&d_{M-2}&\cdots&d_{\bar{M}+1}&0&\cdots&0\\\
0&\cdots&0&d_{M-1}&\cdots&d_{\bar{M}+2}&0&\cdots&0\\\
\vdots&\cdots&\vdots&\ddots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&\cdots&0&0&\ddots&d_{M-1}&0&\cdots&0\\\
\vdots&\cdots&\vdots&\ddots&\ddots&\vdots&\vdots&\cdots&\vdots\\\
0&\cdots&0&0&\cdots&0&0&\cdots&0\\\ \end{bmatrix}.$ (25)
The $n$th element of $\Delta{\mbox{\boldmath{$u$}}}$ can be expressed as
$\Delta u_{n}=u_{n}-\bar{u}_{n},\ n=M-1,M,\ldots,N+M-2,$ (26)
where $u_{n}$ is the same as (11), i.e., from
${\mbox{\boldmath{$H$}}}{\mbox{\boldmath{$s$}}}^{\prime}$ and noise $w$, and
$\bar{u}_{n}=\sum_{\tilde{m}=\bar{M}-M+2+n}^{M-1}d_{\tilde{m}}s_{n-\tilde{m}}.\\\
$ (27)
Notice that $\bar{u}_{n}=0$ for $n=2M-\bar{M}-2,\cdots,N+M-2$. Then, the
$N$-point FFT is performed on the vector $\Delta{\mbox{\boldmath{$u$}}}$
$U_{k}=D_{k}S_{k}-\bar{U}_{k}+W_{k},\ k=0,1,\ldots,N-1,$ (28)
where
$\bar{U}_{k}=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}\bar{u}_{n+M-1}\textrm{exp}\left\\{\frac{-j2\pi
kn}{N}\right\\}.$ (29)
After the $N$-point IFFT is performed on the vector
$\left[\frac{\bar{U}_{0}}{S_{0}^{\prime}},\frac{\bar{U}_{1}}{S_{1}^{\prime}},\cdots,\frac{\bar{U}_{N-1}}{S_{N-1}^{\prime}}\right]^{T}$,
we can obtain
$\hat{d}_{m}=\sqrt{N}d_{m}-\xi_{m}+\tilde{w}_{m},\ m=0,1,\ldots,M-1,$ (30)
where
$\xi_{m}=\frac{1}{N}\sum_{k=0}^{N-1}\frac{1}{S_{k}}\sum_{n=0}^{M-\bar{M}-2}\sum_{\tilde{m}=\bar{M}-M+2+n}^{M-1}d_{\tilde{m}}s_{n+M-1-\tilde{m}}\textrm{exp}\left\\{\frac{-j2\pi
k(n-m)}{N}\right\\}.$ (31)
We remark that $\xi_{m}$ is the IRCI resulted from the insufficient CP, and
$\xi_{m}$ is related to the reflectivities of the neighboring range cells and
the transmitted signal. From the second and third summation signs in $\xi_{m}$
in (31) and the equation (27), a smaller $\bar{M}$ leads to more range cells
involved in the interference, resulting in a stronger IRCI. The performance
degradation with different insufficient CP lengths of $\bar{M}$ will be shown
in the simulations in Section III later.
## III Simulations and Performance Discussions
This section is to present some simulations and discussions for our proposed
CP based OFDM SAR imaging. The simulation stripmap SAR geometry is shown in
Fig. 1. The azimuth processing is similar to the conventional stripmap SAR
imaging [2] as shown in Fig. 5(a). For computational efficiency, a fixed value
of $R_{c}$ located at the center of the range swath is set as the reference
range cell as [2]. Then, the range cell migration correction (RCMC) and the
azimuth compression are implemented in the whole range swath using $R_{c}$.
For convenience, we do not consider the noise in this section as what is
commonly done in SAR image simulations. For comparison, we also consider the
range Doppler algorithm (RDA) using LFM and random noise signals as shown in
the block diagram of Fig. 5. Since the performance of a step frequency signal
SAR is similar to that of an LFM signal SAR, here we only consider LFM signal
SAR in our comparisons. In Fig. 5 (b), the secondary range compression (SRC)
is implemented in the range and azimuth frequency domain, the same as the
Option 2 in [2, Ch. 6.2]. In Fig. 5 (c), the range compression of the random
noise signal and the conventional OFDM signal are achieved by the correlation
between the transmitted signals and the range time domain data. Notice that
the difference of these three imaging methods in Fig. 5 is the range
compression, while the RCMC and azimuth compression are identical.
Figure 5: Block diagram of SAR imaging processing: (a) CP based OFDM SAR; (b)
LFM SAR; (c) Random noise SAR and conventional OFDM SAR.
The simulation experiments are performed with the following parameters as a
typical SAR system: PRF = $800$ Hz, the bandwidth is $B=150$ MHz, the antenna
length is $L_{a}=1$ m, the carrier frequency $f_{c}=9$ GHz, the synthetic
aperture time is $T_{a}=1$ sec, the effective radar platform velocity is
$v_{p}=150$ m/sec, the platform height of the antenna is $H_{p}=5$ km, the
slant range swath center is $R_{c}=5\sqrt{2}$ km, the sampling frequency
$f_{s}=150$ MHz, the number of range cells is $M=96$ with the center at
$R_{c}$. For the convenience of FFT/IFFT computation, we set
$T=\frac{512}{150}\ \mu s\approx 3.41\ \mu s$, then the number of subcarriers
for the OFDM signal is $N=512$. The CP length is $95$ that is sufficient and
the CP time duration is $T_{GI}=\frac{95}{150}\ \mu s\approx 0.63\ \mu s$.
Thus, the time duration of an OFDM pulse is $T_{o}=\frac{607}{150}\ \mu
s\approx 4.05\ \mu s$. The complex weight vectors over the subcarriers of the
CP based OFDM signal and the conventional OFDM signal are set to be vectors of
binary PN sequence of values $-1$ and $1$. Meanwhile, for the transmission
energies of the three SAR imaging methods to be the same, the time durations
of an LFM pulse, a conventional OFDM pulse and a random noise pulse are also
$4.05\ \mu s$.
A point target is assumed to be located at the range swath center. Without
considering the additive noise, the normalized range profiles of the point
spread function are shown in Fig. 6 and the details around the mainlobe area
are shown in its zoom-in image. It can be seen that the sidelobes are much
lower for the CP based OFDM signal than those of the other three signals,
while the $3$ dB mainlobe widths of the four signals are all the same.
Figure 6: The range profiles of the point spread function. Figure 7: The
azimuth profiles of the point spread function.
The normalized azimuth profiles of the point spread function of the three
methods are shown in Fig. 7. The results show that the azimuth profiles of the
point spread function are similar for all the four signals of LFM, CP based
OFDM, conventional OFDM and random noise.
We then consider an extended object with the shape of a tank constructed by a
few single point scatterers, and the original reflectivity profile is shown as
Fig. 8. The results indicate that the imaging performance using the CP based
OFDM SAR is better than the other three signals. Specifically, the boundaries
of the extended object are observed less blurred by using the CP based OFDM
SAR imaging. In Fig. 8, we also consider the imaging of the tank with our
proposed method when the CP length is zero and the transmitted OFDM pulse is
the same as the conventional OFDM pulse. As a remark, comparing Fig. 8(e) and
Fig. 8(f), one may see that the SAR imaging performance degradation is
significant when CP length is zero. This is because our proposed range
reconstruction method in the receiver, as mentioned at (15)-(21), is for CP
based OFDM SAR imaging and different with the traditional matched filter SAR
imaging method. Thus, sufficient CP should be included in the transmitted OFDM
pulse to achieve IRCI free range reconstruction.
We next consider the importance of adding a sufficient CP in our proposed CP
based OFDM SAR imaging. We consider a single range line (a cross range) with
$M=96$ range cells, and targets are included in $18$ range cells, the
amplitudes are randomly generated and shown as the red circles in Fig. 9, and
the RCS coefficients of the other range cells are set to be zero. The
normalized imaging results are shown as the blue asterisks. The results
indicate that the imaging is precise when the length of CP is $95$, i.e.,
sufficient CP length, in Fig. 9(a), the amplitudes of the range cells without
targets are lower than $-300$ dB, which are due to the computer numerical
errors. With the decrease of the CP length the imaging performance is degraded
and the IRCI is increased. Specifically, the zero amplitude range cells become
non-zero anymore and some targets are even submerged by the IRCI as shown in
Fig. 9(b) and Fig. 9(c). We also show the imaging results with the
conventional OFDM SAR image as in Fig. 9(d). The curves in Fig. 9(d) indicate
that some targets are submerged by the IRCI from other range cells. In Fig. 9,
we notice that, when the CP lengthes are $95$ and $80$ (as in Fig. 9(a) and
Fig. 9(b), respectively), the imaging performances of our proposed method
outperform the conventional OFDM SAR image in Fig. 9(d). However, the imaging
performance with zero length CP is worse than the conventional OFDM SAR image,
although they have the same transmitted OFDM waveform, which is again because
our proposed range reconstruction method at the receiver is different from the
conventional method. It also further indicates that a sufficient CP is
important for our proposed CP based OFDM SAR imaging.
Figure 8: Imaging results of simulated reflectivity profile for a tank: (a)
Original tank; (b) LFM SAR; (c) Random noise SAR; (d) OFDM SAR with sufficient
CP; (e) Conventional OFDM signal SAR; (f) CP based OFDM SAR with CP length
$=0$.
We also consider a single range line (a cross range) with $M$ range cells, in
which the RCS coefficients are set $g_{m}=1,\ m=0,\ldots,M-1$. After the CP
based OFDM SAR imaging with different lengths of CP, we calculate the mean
square errors (MSE) between the energy normalized imaging results and the
original RCS coefficients $g_{m}$. The results are achieved from the average
of $1000$ independent Monte Carlo simulations and are shown in Fig. 10. The
curves suggest that the performance degradation occurs when the length of CP
is less than $M-1$, i.e., insufficient. The MSE is supposed to be zero when
the CP length is $M-1$ that is sufficient. However, one can still observe some
errors in Fig. 10, which is because errors may occur by using a fixed
reference range cell $R_{c}$ in the imaging processing (i.e., RCMC and azimuth
compression), and the wider swath (or a larger $M$) causes the larger imaging
error. Thus, the MSE is slightly larger when $M$ is larger.
Figure 9: A range line imaging with different CP lengths. Red circles denote
the real target amplitudes, blue asterisks denote imaging results. Figure 10:
The mean square errors for insufficient CP lengths.
## IV Conclusion
In this paper, by using the most important feature of OFDM signals in
communications systems, namely converting an ISI channel to multiple ISI-free
subchannels, we have proposed a novel method for SAR imaging using OFDM
signals with sufficient CP. The sufficient CP insertion provides an IRCI free
(high range resolution) SAR image. We first established the CP based OFDM SAR
imaging system model and then derived the CP based OFDM SAR imaging algorithm
with sufficient CP and showed that this algorithm has zero IRCI (or IRCI-free)
for each cross range. We also analyzed the influence when the CP length is
insufficient. By comparing with the LFM SAR and the random noise SAR imaging
methods, we then finally provided some simulations to illustrate the high
range resolution property of the proposed CP based OFDM SAR imaging and also
the necessity of a sufficient CP insertion in an OFDM signal. The main
features of the proposed SAR imaging are highlighted below.
* •
The sufficient CP length $M-1$ is determined by the number of range cells $M$
within a swath, which is directly related to the range resolution of the SAR
system.
* •
The optimal time duration of the OFDM pulse is $T_{o}=(2N-1)T_{s}$ with CP
length $N-1$. The minimum range of the proposed CP based OFDM SAR is the same
as the maximal swath width.
* •
The range sidelobes are ideally zero for the proposed CP based OFDM SAR
imaging, which can provide high range resolution potential for SAR systems.
From our simulations, we see that the imaging performance of the CP based OFDM
SAR is better than those of the LFM SAR and the random noise SAR, which may be
more significant in MIMO radar applications.
* •
The imaging performance of the CP based OFDM SAR is degraded and the IRCI is
increased when the CP length is insufficient.
Some future researches may be needed for our proposed CP based OFDM SAR
imaging systems. One of them is the high PAPR problem of the OFDM signals as
what has been pointed out in Section II-C. Another problem is that OFDM
signals are sensitive to an unknown Doppler shift, such as, that induced from
an known moving target, which may damage the orthogonality of OFDM subcarriers
and result in the potential ICI. Also, how does our proposed OFDM SAR imaging
method work for a distributed target model [39]? Some of these problems are
under our current investigations.
## Acknowledgment
The authors would like to thank the editor and the anonymous reviewers for
their useful comments and suggestions that have improved the presentation of
this paper.
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|
arxiv-papers
| 2013-06-15T20:29:10 |
2024-09-04T02:49:46.557563
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tian-Xian Zhang and Xiang-Gen Xia",
"submitter": "Tian-Xian Zhang",
"url": "https://arxiv.org/abs/1306.3604"
}
|
1306.3663
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-095 LHCb-PAPER-2013-004 Oct. 1, 2013
Measurement of $B$ meson production cross-sections in proton-proton collisions
at $\sqrt{s}=$ 7 TeV
The LHCb collaboration†††Authors are listed on the following pages.
The production cross-sections of $B$ mesons are measured in $pp$ collisions at
a centre-of-mass energy of $7\mathrm{\,Te\kern-1.00006ptV}$, using data
collected with the LHCb detector corresponding to a integrated luminosity of
$0.36\mbox{\,fb}^{-1}$. The $B^{\rm+}$, $B^{\rm 0}$ and $B^{\rm 0}_{s}$ mesons
are reconstructed in the exclusive decays
$B^{\rm+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{\rm 0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{\rm 0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$,
with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$,
$K^{*0}\rightarrow K^{+}\pi^{-}$ and $\phi\rightarrow K^{+}K^{-}$. The
differential cross-sections are measured as functions of $B$ meson transverse
momentum $p_{\rm T}$ and rapidity $y$, in the range $0<p_{\rm
T}<40\;{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$. The integrated
cross-sections in the same $p_{\rm T}$ and $y$ ranges, including charge-
conjugate states, are measured to be
$\begin{array}[]{lcl}\sigma(pp\rightarrow B^{+}+X)&=&38.9\pm 0.3\,({\rm
stat.})\pm 2.5\,({\rm syst.})\,\pm 1.3\,({\rm norm.})\,{\rm\rm\,\upmu b},\\\
\sigma(pp\rightarrow B^{0}+X)&=&38.1\pm 0.6\,({\rm stat.})\pm 3.7\,({\rm
syst.})\,\pm 4.7\,({\rm norm.})\,{\rm\rm\,\upmu b},\\\ \sigma(pp\rightarrow
B^{0}_{s}+X)&=&10.5\pm 0.2\,({\rm stat.})\pm 0.8\,({\rm syst.})\,\pm
1.0\,({\rm norm.})\,{\rm\rm\,\upmu b},\end{array}$
where the third uncertainty arises from the pre-existing branching fraction
measurements.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, C. Abellan Beteta35,n, B. Adeva36, M. Adinolfi45, C. Adrover6, A.
Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S.
Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S.
Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, R.B.
Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso57,
E. Aslanides6, G. Auriemma24,m, S. Bachmann11, J.J. Back47, C. Baesso58, V.
Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W.
Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S.
Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, M. Benayoun8, G.
Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O.
Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk57,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia57, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton57, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,p, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, M.
Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M. Chrzaszcz25, K.
Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V.
Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P.
Collins37, A. Comerma-Montells35, A. Contu15, A. Cook45, M. Coombes45, S.
Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, S.
Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y. David40, I. De
Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39, J.M. De Miranda1, L. De
Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del
Buono8, D. Derkach14, O. Deschamps5, F. Dettori41, A. Di Canto11, H.
Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D.
Dossett47, A. Dovbnya42, F. Dupertuis38, R. Dzhelyadin34, A. Dziurda25, A.
Dzyuba29, S. Easo48,37, U. Egede52, V. Egorychev30, S. Eidelman33, D. van
Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El
Rifai5, Ch. Elsasser39, D. Elsby44, A. Falabella14,e, C. Färber11, G.
Fardell49, C. Farinelli40, S. Farry12, V. Fave38, D. Ferguson49, V. Fernandez
Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16,
C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2,
M. Frank37, C. Frei37, M. Frosini17,f, S. Furcas20, E. Furfaro23,k, A. Gallas
Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini57, Y. Gao3, J. Garofoli57,
P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E.
Gersabeck11, M. Gersabeck53, T. Gershon47,37, Ph. Ghez4, V. Gibson46, V.V.
Gligorov37, C. Göbel58, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H.
Gordon54, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37,
E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, O.
Grünberg59, B. Gui57, E. Gushchin32, Yu. Guz34,37, T. Gys37, C.
Hadjivasiliou57, G. Haefeli38, C. Haen37, S.C. Haines46, S. Hall52, T.
Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53,
T. Hartmann59, J. He37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando
Morata36, E. van Herwijnen37, A. Hicheur1, E. Hicks51, D. Hill54, M.
Hoballah5, M. Holtrop40, C. Hombach53, P. Hopchev4, W. Hulsbergen40, P.
Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50, V. Iakovenko43,
M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, F.
Jing3, M. John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M.
Kaballo9, S. Kandybei42, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, U.
Kerzel37, T. Ketel41, A. Keune38, B. Khanji20, O. Kochebina7, I. Komarov38,
R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32,
K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M.
Kucharczyk20,25,j, V. Kudryavtsev33, T. Kvaratskheliya30,37, V.N. La Thi38, D.
Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E.
Lanciotti37, G. Lanfranchi18,37, C. Langenbruch37, T. Latham47, C.
Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A.
Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11,
Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37,
S. Lohn37, I. Longstaff50, J.H. Lopes2, E. Lopez Asamar35, N. Lopez-March38,
H. Lu3, D. Lucchesi21,p, J. Luisier38, H. Luo49, F. Machefert7, I.V.
Machikhiliyan4,30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G.
Mancinelli6, U. Marconi14, R. Märki38, J. Marks11, G. Martellotti24, A.
Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41, D.
Martins Tostes2, A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E.
Maurice6, A. Mazurov16,32,37,e, J. McCarthy44, A. McNab53, R. McNulty12, B.
Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N.
Minard4, J. Molina Rodriguez58, S. Monteil5, D. Moran53, P. Morawski25, M.J.
Morello22,r, R. Mountain57, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28,
B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1,
M. Needham49, N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o,
M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54,
A. Novoselov34, A. Oblakowska-Mucha26, V. Obraztsov34, S. Oggero40, S.
Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, M. Orlandea28, J.M. Otalora
Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal57, A. Palano13,b, M.
Palutan18, J. Panman37, A. Papanestis48, M. Pappagallo50, C. Parkes53, C.J.
Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, G.N. Patrick48, C.
Patrignani19,i, C. Pavel-Nicorescu28, A. Pazos Alvarez36, A. Pellegrino40, G.
Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, D.L. Perego20,j, E. Perez
Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, G.
Pessina20, K. Petridis52, A. Petrolini19,i, A. Phan57, E. Picatoste Olloqui35,
B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F.
Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, D. Popov10, B.
Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C.
Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,q, W. Qian4, J.H.
Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N.
Rauschmayr37, G. Raven41, S. Redford54, M.M. Reid47, A.C. dos Reis1, S.
Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5,
P. Robbe7, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34,
A. Romero Vidal36, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz
Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B.
Saitta15,d, C. Salzmann39, B. Sanmartin Sedes36, M. Sannino19,i, R.
Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A.
Sarti18,l, C. Satriano24,m, A. Satta23, M. Savrie16,e, D. Savrina30,31, P.
Schaack52, M. Schiller41, H. Schindler37, M. Schlupp9, M. Schmelling10, B.
Schmidt37, O. Schneider38, A. Schopper37, M.-H. Schune7, R. Schwemmer37, B.
Sciascia18, A. Sciubba24, M. Seco36, A. Semennikov30, I. Sepp52, N. Serra39,
J. Serrano6, P. Seyfert11, M. Shapkin34, I. Shapoval16,42, P. Shatalov30, Y.
Shcheglov29, T. Shears51,37, L. Shekhtman33, O. Shevchenko42, V. Shevchenko30,
A. Shires52, R. Silva Coutinho47, T. Skwarnicki57, N.A. Smith51, E.
Smith54,48, M. Smith53, M.D. Sokoloff56, F.J.P. Soler50, F. Soomro18, D.
Souza45, B. Souza De Paula2, B. Spaan9, A. Sparkes49, P. Spradlin50, F.
Stagni37, S. Stahl11, O. Steinkamp39, S. Stoica28, S. Stone57, B. Storaci39,
M. Straticiuc28, U. Straumann39, V.K. Subbiah37, S. Swientek9, V.
Syropoulos41, M. Szczekowski27, P. Szczypka38,37, T. Szumlak26, S. T’Jampens4,
M. Teklishyn7, E. Teodorescu28, F. Teubert37, C. Thomas54, E. Thomas37, J. van
Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, D. Tonelli37, S. Topp-
Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38, M.T. Tran38, M.
Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40, M. Ubeda Garcia37,
A. Ukleja27, D. Urner53, U. Uwer11, V. Vagnoni14, G. Valenti14, R. Vazquez
Gomez35, P. Vazquez Regueiro36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g,
G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-
Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V.
Vorobyev33, C. Voß59, H. Voss10, R. Waldi59, R. Wallace12, S. Wandernoth11, J.
Wang57, D.R. Ward46, N.K. Watson44, A.D. Webber53, D. Websdale52, M.
Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L. Wiggers40, G.
Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48, J. Wishahi9, M.
Witek25, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z.
Xing57, Z. Yang3, R. Young49, X. Yuan3, O. Yushchenko34, M. Zangoli14, M.
Zavertyaev10,a, F. Zhang3, L. Zhang57, W.C. Zhang12, Y. Zhang3, A. Zhelezov11,
A. Zhokhov30, L. Zhong3, A. Zvyagin37.
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 Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57Syracuse University, Syracuse, NY, United States
58Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
59Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, 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
## 1 Introduction
Measurements of beauty production in multi-TeV proton-proton ($pp$) collisions
at the LHC provide important tests of quantum chromodynamics. State of the art
theoretical predictions are given by the fixed-order plus next-to-leading
logarithm (FONLL) approach [1, 2, 3]. In these calculations, the dominant
uncertainties arise from the choice of the renormalisation and factorisation
scales, and the assumed b-quark mass [4]. The primary products of
$b\overline{}b$ hadronisation are $B^{\rm+}$, $B^{\rm 0}$, $B^{\rm 0}_{s}$ and
their charge-conjugate states (throughout the paper referred to as $B$ mesons)
formed by one $\bar{b}$ quark bound to one of the three light quarks ($u$, $d$
and $s$). Accurate measurements of the cross-sections probe the validity of
the production models. At the LHC, $b\overline{}b$ production has been studied
in inclusive $b\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X$
decays [5, 6] and semileptonic [7, 8] decays. Other measurements, using fully
reconstructed $B$ mesons, have also been performed by the LHCb and CMS
collaborations [9, 10, 11, 12].
In this paper, a measurement of the production cross-sections of $B$ mesons
(including their charge-conjugate states) is presented. This study is
performed in the transverse momentum range $0<p_{\rm
T}<40\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and rapidity range $2.0<y<4.5$
using data, corresponding to a integrated luminosity of
$0.36\mbox{\,fb}^{-1}$, collected in $pp$ collisions at centre-of-mass energy
of $7\mathrm{\,Te\kern-1.00006ptV}$ by the LHCb experiment. The $B$ mesons are
reconstructed in the exclusive decays
$B^{\rm+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$,
$B^{\rm 0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ and
$B^{\rm 0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$,
with ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$,
$K^{*0}\rightarrow K^{+}\pi^{-}$ and $\phi\rightarrow K^{+}K^{-}$.
The LHCb detector [13] 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 placed downstream. The
combined tracking system has momentum resolution $\Delta p/p$ 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
better than 20$\,\upmu\rm m$ for transverse momentum higher than 3
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Charged hadrons are identified using
two ring-imaging Cherenkov (RICH) detectors. 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.
The events used in this analysis are selected by a two-stage trigger system
[14]. The first stage is hardware based whilst the second stage is software
based. At the hardware stage events containing either a single muon or a pair
of muon candidates, with high transverse momentum, are selected. In the
subsequent software trigger the decision of the single-muon or dimuon hardware
trigger is confirmed and a muon pair with an invariant mass consistent with
the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [15] is
required. To reject high-multiplicity events with a large number of $pp$
interactions, global event cuts on the hit multiplicities of subdetectors are
applied.
## 2 Candidate selection
The selection of $B$ meson candidates starts by forming
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay
candidates. These are formed from pairs of oppositely-charged particles that
are identified as muons and have $p_{\rm
T}>0.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Good quality of the reconstructed
tracks is ensured by requiring the $\chi^{2}/{\rm ndf}$ of the track fit to be
less than 4, where ndf is the number of degrees of freedom of the fit. The
muon candidates are required to originate from a common vertex and the
$\chi^{2}/{\rm ndf}$ of the vertex fit is required to be less than 9. The mass
of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate is required to
be around the known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [15],
between $3.04$ and $3.14\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.
Kaons used to form $B^{\rm+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$ candidates are required to have $p_{\rm T}$ larger than 0.5
${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. Information from the RICH detector
system is not used in the selection since the
$B^{\rm+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ decay
is Cabibbo suppressed. Candidates for $K^{*0}\rightarrow K^{+}\pi^{-}$ and
$\phi\rightarrow K^{+}K^{-}$ decays are formed from pairs of oppositely-
charged hadron candidates. Since the background levels of these two channels
are higher than for $B^{\rm+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}$decay, the hadron identification information provided by the RICH
detectors is used. Kaons used to form $K^{*0}$ candidates in the $B^{\rm
0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ channel and
$\phi$ candidates in the $B^{\rm
0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ channel are
selected by cutting on the difference between the log-likelihoods of the kaon
and pion hypotheses provided by the RICH detectors ($\mathrm{DLL}_{K\pi}>0$).
The pions used to form $K^{*0}$ candidates are required to be inconsistent
with the kaon hypothesis (${\rm DLL}_{\pi K}>-5$). The same track quality cuts
used for muons are applied to kaons and pions. The $K^{*0}$ and $\phi$ meson
candidates are constructed requiring a good vertex quality ($\chi^{2}/{\rm
ndf}<16$) and $p_{\rm T}>1.0\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. The
masses of the $K^{*0}$ and $\phi$ candidates are required to be consistent
with their known masses [15], in the intervals
$0.826-0.966\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ and
$1.008-1.032\,{\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$, respectively.
The ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidate is combined with
a $K^{+}$, $K^{*0}$ or $\phi$ candidate to form a $B^{\rm+}$, $B^{\rm 0}$ or
$B^{\rm 0}_{s}$ meson, respectively. A vertex fit [16] is performed that
constrains the daughter particles to originate from a common point and the
mass of the muon pair to match the known
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass [15]. The $\chi^{2}/{\rm
ndf}$ returned by this fit is required to be less than 9. To further reduce
the combinatorial background due to particles produced in the primary $pp$
interaction, only $B$ candidates with a decay time larger than 0.3 ${\rm ps}$,
which corresponds to about 6 times the decay time resolution, are kept. In the
$B^{\rm 0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$
samples, duplicate candidates are found that share the same
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ particle but have pion tracks
that are reconstructed several times from one track. In these cases only one
of the candidates is randomly retained. Duplicate candidates of other sources
in the other decay modes occur at a much lower rate and are retained. Finally,
the fiducial requirements $0<p_{\rm
T}<40\;{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$ are applied to
the $B$ meson candidates.
## 3 Cross-section determination
The differential production cross-section for each $B$ meson species is
calculated as
$\displaystyle\frac{{\rm d^{2}}\sigma(B)}{{\rm d}p_{\rm T}\;{\rm d}y}$
$\displaystyle=$ $\displaystyle\frac{N_{B}(p_{\rm T},y)}{\epsilon_{\rm
tot}(p_{\rm T},y)\;{\cal L}_{\rm int}\;{\cal
B}(B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)\;\Delta p_{\rm
T}\;\Delta y},$
where $N_{B}(p_{\rm T},y)$ is the number of reconstructed signal candidates in
a given $(p_{\rm T},y)$ bin, $\epsilon_{\rm tot}(p_{\rm T},y)$ is the total
efficiency in a given $(p_{\rm T},y)$ bin, ${\cal L}_{\rm int}$ is the
integrated luminosity, ${\cal
B}(B\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}X)$ is the product
of the branching fractions of the decays in the complete decay chain, and
$\Delta p_{\rm T}$ and $\Delta y$ are the widths of the bin. The width of each
$y$ bin is fixed to 0.5 while the widths of the $p_{\rm T}$ bins vary to allow
for sufficient number of candidates in each bin.
The signal yield in each bin of $p_{\rm T}$ and $y$ is determined using an
extended unbinned maximum likelihood fit to the invariant mass distribution of
the reconstructed $B$ candidates. The fit model includes two components: a
double-sided Crystal Ball function to model the signal and an exponential
function to model the combinatorial background. The former is an extension of
the Crystal Ball function [17] that has tails on both the low- and the high-
mass side of the peak described by separate parameters, which are determined
from simulation. For the $B^{\rm+}$ channel, the $K$-$\pi$ misidentified
$B^{\rm+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$ decay
is modelled by a shape that is found to fit the distribution of simulated
events. The invariant mass distributions of the selected $B$ candidates and
the fit results in one $p_{\rm T}$ and $y$ bin are shown in Fig. 1.
Figure 1: Invariant mass distributions of the selected candidates for (top
left) $B^{\rm+}$ and (top right) $B^{\rm 0}$ decays, both in the range
$4.5<p_{\rm T}<5.0\;{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$, $3.0<y<3.5$, and
(bottom) $B^{\rm 0}_{s}$ decay in the range $4.0<p_{\rm
T}<5.0\;{\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$, $3.0<y<3.5$. The results of the
fit to the model described in the text are superimposed. The Cabibbo-
suppressed background is barely visible in the top left plot.
For the $B^{\rm 0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{*0}$ and $B^{\rm
0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$ decay
channels, an additional non-resonant S-wave component (where the
$K^{+}\pi^{-}$ and the $K^{+}K^{-}$ originate directly from $B^{\rm 0}$ and
$B^{\rm 0}_{s}$ decays, and not via $K^{*0}$ or $\phi$ resonances) is also
present. The amount of this component present in each case is determined from
an independent fit to the $K^{+}\pi^{-}$ or $K^{+}K^{-}$ mass distribution,
respectively, integrating the $p_{\rm T}$ and $y$ range. The signal component
is described by a relativistic Breit-Wigner function, and the S-wave
background by a phase space function. From the fit results, the S-wave
fractions are determined to be $\sim 6\%$ for $B^{\rm
0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\pi^{-}$ and
$\sim 3\%$ for $B^{\rm 0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}K^{-}$ decays. The yields of $B$ mesons are then corrected
according to the S-wave fractions.
The geometrical acceptance as well as the reconstruction and selection
efficiencies, except for the hadron identification efficiencies, are
determined using simulated signal events. The $pp$ collisions are generated
using Pythia 6.4 [18] with a specific LHCb configuration [19]. Decays of
hadronic particles are described by EvtGen [20], in which final state
radiation is generated using Photos [21]. The interaction of the generated
particles with the detector and its response are implemented using the Geant4
toolkit [22, *Agostinelli:2002hh] as described in Ref. [24]. The hadron
identification efficiencies are measured using tracks from the decay
$D^{*+}\rightarrow D^{0}\pi^{+}$ with $D^{0}\rightarrow K^{-}\pi^{+}$,
selected without using information from the RICH detectors [25].
Only candidates where the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ is
responsible for the trigger decision are used. The trigger efficiency is
measured in data using the tag and probe method described in Ref. [14].
The luminosity is measured using Van der Meer scans and a beam-gas imaging
method [26]. The integrated luminosity of the data sample used in this
analysis is determined to be $362\pm 13\mbox{\,pb}^{-1}$.
The branching fraction ${\cal B}(B^{\rm
0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0})$ measured by
the Belle collaboration [27] is used in the determination of the $B^{\rm 0}$
cross-section, since it includes the effect of the S-wave interference, while
other measurements do not. The measurement of ${\cal B}(B^{\rm
0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)$ given in
Ref. [28] is used in the determination of the $B^{\rm 0}_{s}$ cross-section.
Since this branching fraction measurement used the average ratio of
fragmentation fractions $f_{s}/f_{d}$ from Ref. [29, 30], the result in this
paper cannot be taken as an independent measurement of $f_{s}/f_{d}$. The
other branching fractions are obtained from Ref. [15].
## 4 Systematic uncertainties
The measurements are affected by systematic uncertainties in the determination
of the signal yields, efficiencies, branching fractions and luminosity, as
summarised in Table 1. The total systematic uncertainty is obtained from the
sum in quadrature of all contributions.
Table 1: Relative systematic uncertainties (in %), given as single values or
as ranges, when they depend on the ($p_{\rm T}$, y) bin.
Source | $B^{\rm+}$ | $B^{\rm 0}$ | $B^{\rm 0}_{s}$
---|---|---|---
Signal fit model | 2.5 | 1.3 | 1.2
Fit range | 0.1 | 1.0 | 0.4
Non-resonant background | - | 2.2 | 1.9
Combinatorial background | 0.6 | 0.8 | 0.3
Bin size | $0.1-10.9$ | $0.1-19.3$ | $0.1-13.2$
Duplicate candidates | - | 3.1 | -
Trigger efficiency | $2.4-7.9$ | $2.6-7.9$ | $2.6-6.5$
Tracking efficiency | $2.4-7.4$ | $4.4-8.3$ | $4.4-8.5$
Vertex quality cut | 1.0 | 0.9 | 0.2
Muon identification | $0.7-4.9$ | $0.8-5.0$ | $0.8-5.8$
Hadron identification | - | 1.0 | 0.8
Angular distribution | - | $0.1-0.3$ | $0.1-4.7$
$p_{\rm T}$ distribution | - | $0-24.4$ | -
Branching fractions | 3.3 | 12.3 | 10.0
Luminosity | 3.5 | 3.5 | 3.5
Uncertainties on the signal yields arise from imperfect knowledge of the
signal shape, non-resonant background and finite size of the bins. The
uncertainty from the signal shape is estimated by comparing the fitted and
generated signal yields in simulation. The non-resonant background ratios
determined in this analysis are compared with those from measurements with
angular fits [28] and the differences are assigned as systematic
uncertainties. By varying the $p_{\rm T}$ or $y$ binning, the uncertainty for
changing the binning in $p_{\rm T}$ is found to be small while that for $y$ is
non-negligible in the low $y$ bin. An uncertainty is assigned due to the
procedure of removal of duplicate candidates in $B^{\rm
0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0}$ events. For
the other modes this effect is found to be negligible. The uncertainties from
the background shape, misidentified
$B^{\rm+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}$
background and mass fit range are small.
Uncertainties on the efficiencies arise from the trigger, tracking, particle
identification, angular distribution, $p_{\rm T}$ spectrum and vertex fit
quality cut. The systematic uncertainty from the trigger efficiency is
evaluated by comparing the efficiency measured using a trigger-unbiased sample
of simulated ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ events with that
determined from the simulation. The effect of the global event cuts in the
trigger is found to be negligible. The tracking efficiencies are estimated
with a tag and probe method [31] using ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\rightarrow\mu^{+}\mu^{-}$ events in both data and simulation. The
simulated efficiencies, used to determine the cross-section, are corrected
according to the differences between data and simulation. The tracking
uncertainty includes two components: the first is from the data-simulation
difference correction; the second is due to the uncertainty on the hadronic
interaction length of the detector used in the simulation. Possible systematic
biases in the determination of the hadron identification efficiency are
estimated using simulated events and comparing the true efficiency with that
obtained by applying the same procedure as for the data. The muon
identification uncertainty is estimated by comparing the efficiency in
simulation with that measured, on data, using a tag and probe method. The
systematic uncertainties due to the uncertainties on the angular distributions
of $B^{\rm 0}$ and $B^{\rm 0}_{s}$ decays [15, 32] are taken into account by
simulating the effect of varying the central values of the polarization
amplitudes by $\pm 1$ sigma. In the first $p_{\rm T}$ bin of the $B^{\rm 0}$
sample, the agreement of the $p_{\rm T}$ distributions between data and
simulation is not as good as in the other bins. The discrepancy is assigned as
an additional uncertainty for that bin. The vertex fit quality cut uncertainty
is estimated from the data to simulation comparison. By calculating the signal
yields and efficiencies separately for data taken with two magnet polarities,
the results are found to be stable.
The systematic uncertainties from the branching fractions are calculated with
their correlations taken into account. Since the ${\cal B}(B^{\rm
0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{*0})$ and ${\cal
B}(B^{\rm 0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi)$
have been measured with sizeable uncertainty, the corresponding uncertainties
are listed separately in the cross-section results. The absolute luminosity
scale is measured with $3.5\%$ uncertainty, which is dominated by the beam
current uncertainty [26].
## 5 Results and conclusion
The measured differential production cross-sections of $B$ mesons in bins of
$p_{\rm T}$ and $y$ are shown in Fig. 2. These results are integrated
separately over $y$ and $p_{\rm T}$, and compared with the FONLL predictions
[3], as shown in Fig. 3 and Fig. 4, respectively. The hadronisation fractions
$f_{u}=f_{d}=(33.7\pm 2.2)$% and $f_{s}=(9.0\pm 0.9)$% from Ref. [29] are used
to fix the overall scale of FONLL. The uncertainty of the FONLL computation
includes the uncertainties on the $b$-quark mass, renormalisation and
factorisation scales, and CTEQ 6.6 [33] parton distribution functions. Good
agreement is seen between the FONLL predictions and measured data.
The integrated cross-sections of the $B$ mesons with $0<p_{\rm
T}<40\;{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $2.0<y<4.5$ are
$\begin{array}[]{rcl}\sigma(pp\rightarrow B^{+}\,X)&=&38.9\pm 0.3\,({\rm
stat.})\pm 2.5\,({\rm syst.})\,\pm 1.3\,({\rm norm.})\,{\rm\rm\,\upmu b},\\\
\sigma(pp\rightarrow B^{0}\,X)&=&38.1\pm 0.6\,({\rm stat.})\pm 3.7\,({\rm
syst.})\,\pm 4.7\,({\rm norm.})\,{\rm\rm\,\upmu b},\\\ \sigma(pp\rightarrow
B^{0}_{s}\,X)&=&10.5\pm 0.2\,({\rm stat.})\pm 0.8\,({\rm syst.})\,\pm
1.0\,({\rm norm.})\,{\rm\rm\,\upmu b},\end{array}$
where the third uncertainties arise from the uncertainties of the branching
fractions used for normalisation. The $B^{\rm+}$ result is in good agreement
with a previous measurement by LHCb [9]. These represent the first
measurements of $B^{\rm 0}$ and $B^{\rm 0}_{s}$ meson production cross-
sections in $pp$ collisions in the forward region at centre-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$.
Figure 2: Differential production cross-sections for (top) $B^{\rm+}$,
(middle) $B^{\rm 0}$ and (bottom) $B^{\rm 0}_{s}$ mesons, as functions of
$p_{\rm T}$ for each $y$ interval.
Figure 3: Differential production cross-sections for (top) $B^{\rm+}$,
(middle) $B^{\rm 0}$ and (bottom) $B^{\rm 0}_{s}$ mesons, as functions of
$p_{\rm T}$ integrated over the whole $y$ range. The open circles with error
bars are the measurements (not including uncertainties from normalisation
channel branching fractions and luminosity) and the blue shaded areas are the
uncertainties from the branching fractions. The red dashed lines are the upper
and lower uncertainty limits of the FONLL computation [3].
Figure 4: Differential production cross-sections for (top) $B^{\rm+}$,
(middle) $B^{\rm 0}$ and (bottom) $B^{\rm 0}_{s}$ mesons, as functions of $y$
integrated over the whole $p_{\rm T}$ range. The black open circles with error
bars are the measurements (not including uncertainties from normalisation
channel branching fractions and luminosity) and the blue shaded areas are the
uncertainties from the branching fractions. The red dashed lines are the upper
and lower uncertainty limits of the FONLL computation [3].
## Acknowledgements
We thank M. Cacciari for providing the FONLL predictions for the $B$ meson
production cross-sections. 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); ANCS/IFA (Romania); MinES, Rosatom,
RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT
(Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United
Kingdom); NSF (USA). We also acknowledge the support received from 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 thankful for the computing resources put at
our disposal by Yandex LLC (Russia), as well as to the communities behind the
multiple open source software packages that we depend on.
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|
arxiv-papers
| 2013-06-16T14:17:52 |
2024-09-04T02:49:46.569172
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, C. Abellan Beteta, B. Adeva, M. Adinolfi,\n C. Adrover, A. Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander,\n S. Ali, G. Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio,\n Y. Amhis, L. Anderlini, J. Anderson, R. Andreassen, R.B. Appleby, O. Aquines\n Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G. Auriemma,\n S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W. Baldini, R.J. Barlow, C.\n Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I.\n Bediaga, S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, M. Benayoun, G.\n Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M.\n van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T.\n Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W.\n Bonivento, S. Borghi, A. Borgia, 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, I. Burducea, A. Bursche, G. Busetto, J. Buytaert, S.\n Cadeddu, O. Callot, 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. Castillo Garcia,\n M. Cattaneo, Ch. Cauet, M. Charles, Ph. Charpentier, P. Chen, N. Chiapolini,\n M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M.\n Clemencic, H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras,\n P. Collins, A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau,\n G. Corti, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R. Currie, C.\n D'Ambrosio, P. David, P.N.Y. David, I. De Bonis, K. De Bruyn, S. De Capua, M.\n De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P. De Simone, D. Decamp,\n M. Deckenhoff, L. Del Buono, D. Derkach, O. Deschamps, F. Dettori, A. Di\n Canto, H. Dijkstra, M. Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D.\n Dossett, A. Dovbnya, F. Dupertuis, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S.\n Easo, U. Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, D. Elsby, A.\n Falabella, C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D.\n Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S.\n Filippov, M. Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O.\n Francisco, M. Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas\n Torreira, D. Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi,\n J. Garra Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck,\n T. Gershon, Ph. Ghez, V. Gibson, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A.\n Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A.\n Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S.\n Gregson, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, S.C. Haines, S. Hall, T. Hampson, S.\n Hansmann-Menzemer, N. Harnew, S.T. Harnew, J. Harrison, T. Hartmann, J. He,\n V. Heijne, K. Hennessy, P. Henrard, J.A. Hernando Morata, E. van Herwijnen,\n A. Hicheur, E. Hicks, D. Hill, M. Hoballah, M. Holtrop, C. Hombach, P.\n Hopchev, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain, D. Hutchcroft, D.\n Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A. Jaeger, E. Jans, P.\n Jaton, F. Jing, M. John, D. Johnson, C.R. Jones, C. Joram, B. Jost, M.\n Kaballo, S. Kandybei, M. Karacson, T.M. Karbach, I.R. Kenyon, U. Kerzel, T.\n Ketel, A. Keune, B. Khanji, O. Kochebina, I. Komarov, R.F. Koopman, P.\n Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K. Kreplin, M. Kreps, G.\n Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V. Kudryavtsev, T.\n Kvaratskheliya, V.N. La Thi, D. Lacarrere, G. Lafferty, A. Lai, D. Lambert,\n R.W. Lambert, E. Lanciotti, G. Lanfranchi, C. Langenbruch, 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, L. Li\n Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff,\n J.H. Lopes, E. Lopez Asamar, N. Lopez-March, H. Lu, D. Lucchesi, J. Luisier,\n H. Luo, F. Machefert, I.V. Machikhiliyan, F. Maciuc, O. Maev, S. Malde, G.\n Manca, G. Mancinelli, U. Marconi, R. M\\\"arki, J. Marks, G. Martellotti, A.\n Martens, A. Mart\\'in S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins\n Tostes, A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A.\n Mazurov, J. McCarthy, A. McNab, R. McNulty, B. Meadows, F. Meier, M.\n Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, M.J. Morello, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, N. Neufeld, A.D. Nguyen, T.D. Nguyen, C.\n Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, D.L.\n Perego, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, D. Popov, B. Popovici, C.\n Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch, A.\n Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. 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Wicht, J. Wiechczynski, D.\n Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams, M. Williams, F.F. Wilson,\n J. Wishahi, M. Witek, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z.\n Xing, Z. Yang, R. Young, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev,\n F. Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong,\n A. Zvyagin",
"submitter": "Bo Liu",
"url": "https://arxiv.org/abs/1306.3663"
}
|
1306.3672
|
# Can planetesimals form by collisional fusion?
Dhrubaditya Mitra1, J.S. Wettlaufer1,2, and Axel Brandenburg1,3 1Nordita, KTH
Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23,
SE-10691 Stockholm, Sweden
2Yale University, New Haven, CT, USA
3Department of Astronomy, AlbaNova University Center, Stockholm University,
SE-10691 Stockholm, Sweden
(, Revision: 1.61 )
###### Abstract
As a test bed for the growth of protoplanetary bodies in a turbulent
circumstellar disk we examine the fate of a boulder using direct numerical
simulations of particle seeded gas flowing around it. We provide an accurate
description of the flow by imposing no-slip and non-penetrating boundary
conditions on the boulder surface using the immersed boundary method pioneered
by Peskin (2002). Advected by the turbulent disk flow, the dust grains collide
with the boulder and we compute the probability density function (PDF) of the
normal component of the collisional velocity. Through this examination of the
statistics of collisional velocities we test the recently developed concept of
collisional fusion which provides a physical basis for a range of collisional
velocities exhibiting perfect sticking. A boulder can then grow sufficiently
rapidly to settle into a Keplerian orbit on disk evolution time scales.
###### Subject headings:
accretion, accretion disks – planets and satellites: formation –
protoplanetary disks – turbulence
## 1\. Introduction
### 1.1. Accretion Disks and Protoplanets
Planet formation is hypothesized to occur through the growth of protoplanetary
bodies formed from gas, dust and ice grains in an accretion disk around a
central star (Armitage, 2010). The complex scenario of the planet formation
process involves the following four stages. Firstly, the initial collapse of
interstellar gas to create the central protostar ($\sim 0.1$ My); secondly,
the slow accretion of mass onto the star and the formation of primary
planetesimals within the evolving accretion disk ($\sim$ My); thirdly, a phase
($\sim$ My) of reduced accretion rate allowing the photoevaporative wind to
divide the disk into an inner and an outer region at a radius determined by
the ratio of the stellar accretion rate to the mass loss rate due to
photoevaporation; finally, there is a clearing phase ($\sim 0.1$ My) during
which the inner disk accretes onto the star while the lightest elements of the
outer disk are removed due to direct exposure to photoevaporative UV flux.
Recent cosmochemical evidence reveals that the long held view of a $\sim$ My
age difference between Ca-Al-rich inclusions (CAIs) within carbonaceous
chondrite meteorites and chondrules within chondrites can be refuted (Connelly
et al., 2012). To the extent that these data demonstrate commensurability over
disk lifetime scales of CAI and chondrule formation, the detailed transient
development of matter within circumstellar disks becomes all the more
compelling for studies that can isolate essential physical processes. Here we
focus on fundamental aspects of the second stage above. This stage is crucial
for understanding how the material that forms the building blocks of planets
can organize into bodies that thwart the radiative pressure effects in the
subsequent stages that sweep the disk of small particles and gas.111A
different hypothesis originally due to Safronov and Goldreich and Ward (see
e.g. Goldreich et al., 2004; Armitage, 2010; Youdin, 2010, for a review) leads
to planetesimals by the gravitational collapse of the disk material. We do not
consider this here.
The accretion disk is treated as a two phase system defined by a fluid phase
(‘gas’) and solid particles (‘dust’) advected by the fluid. Ubiquitous
attractive long range van der Waals and electrostatic interactions facilitate
the agglomeration and growth of small (micron or smaller) dust grains that are
brought into proximity by the turbulent flow of the gas. However, depending on
the material and the mechanical and thermodynamic conditions of a particle-
particle collision, sticking (through a number of mechanisms), fragmentation,
or bouncing will determine the fate and the size distribution of accreting
matter from the small scales upward (Blum & Wurm, 2008; Wettlaufer, 2010; Zsom
et al., 2011).
Because the central star creates a radially decaying pressure gradient, the
gas moves at a slightly sub-Keplerian speed. Thus, depending on the position
in the disk, there are a range of particle sizes that experience a strong
“headwind” and so lose angular momentum, thereby driving them into the central
star on time scales as rapidly as a century (Armitage, 2010; Youdin, 2010). We
are concerned with the long standing problem of how, when objects grow and
begin to experience the local headwind, they can accumulate matter
sufficiently quickly to slow their drift inward. To focus the question, we
examine in some detail how a meter sized object grows by accretion of small
particles mediated by turbulent flows of the gas.
### 1.2. Hydrodynamic Preliminaries
The typical value of the “disk Mach number” $\mathcal{M}_{d}$ is based on the
Keplerian velocity $v_{\rm kepler}$, which in the thin disk approximation is
$\mathcal{M}_{d}=\frac{v_{\rm kepler}}{c_{\rm s}}\approx\frac{r}{h},$ (1)
where $r$ is the radial position in the disk and $h$ is its vertical scale
height. At $1$ AU, $h/r\approx 0.02$ and hence $\mathcal{M}_{d}\approx 50$
(see, e.g., Armitage, 2010, p. 40). Now, as noted above, because the central
star creates a radially decreasing pressure gradient, the gas moves at a sub-
Keplerian speed $v_{\rm wind}=\eta v_{\rm kepler}$ where $\eta$ can be as
small as $10^{-3}$ depending on the position in the disk.
To understand the effects of the interaction between the dust and the gas, we
begin by considering a solid body of spherical shape with radius $R_{\rm SB}$,
moving through a gas with kinematic viscosity $\nu$ and speed $v_{\rm wind}$.
We estimate its Reynolds number as,
$\mbox{Re}_{\rm SB}=\frac{v_{\rm wind}R_{\rm SB}}{\nu}=\frac{v_{\rm
wind}}{c_{\rm s}}\,\frac{R_{\rm SB}}{\lambda}\,\frac{\lambda c_{\rm
s}}{\nu}\approx\mathcal{M}\,\frac{R_{\rm SB}}{\lambda},$ (2)
where $\mathcal{M}\equiv v_{\rm wind}/c_{\rm s}$ is the Mach number of the
headwind and $\lambda$ the mean-free-path of the gas molecules. Importantly,
for this estimate we have used the well-known expression for the viscosity of
gases $\nu\sim c_{\rm s}\lambda$ (see e.g., Lifshitz & Pitaevskii, 1981,
section 8). Now, because $\mathcal{M}=\eta\mathcal{M}_{d}$, we can have
$\mathcal{M}\approx 0.05$, and hence, so long as $R_{\rm
SB}<\mathcal{M}^{-1}\lambda\approx 20\,\lambda$, the local Reynolds number of
the solid body is less than unity. For $R_{\rm SB}\sim\lambda$ the size of the
solid body is well below the smallest hydrodynamic length scale in the gas and
its motion is then described by the simple drag law
$\frac{d{\bm{v}}_{\rm SB}}{dt}=\frac{1}{\tau_{\rm SB}}\left({\bm{v}}_{\rm
SB}-{\bm{U}}\right),$ (3)
where ${\bm{v}}_{\rm SB}$ is the velocity of the particle, ${\bm{U}}$ is the
local velocity of the gas, and $\tau_{\rm SB}$ is the so-called stopping time
describing the deceleration of particle motion relative to the gas. When a
particle is smaller than the typical hydrodynamic length scale in the problem,
$\tau_{\rm SB}$ is given by the Epstein drag law,
$\tau_{\rm SB}^{\rm Ep}=\frac{\rho_{\rm SB}}{\rho_{\rm g}}\frac{R_{\rm
SB}}{c_{\rm s}},$ (4)
where $\rho_{\rm SB}$ is the material density of the solid particles and
$\rho_{\rm g}$ is the gas density. When $\mathcal{M}^{-1}\lambda>R_{\rm
SB}>\lambda$, the relevant drag law is that of Stokes and $\tau_{\rm SB}$ is
given by
$\tau_{\rm SB}^{\rm St}=\frac{2}{9}\frac{\rho_{\rm SB}}{\rho_{\rm
g}}\frac{R_{\rm SB}^{2}}{\nu}.$ (5)
Despite the fact that when $R_{\rm SB}>\mathcal{M}^{-1}\lambda$, the simple
drag law (3) no longer describes the motion of the dust particles, most
numerical approaches to these problems (see, e.g., Johansen et al., 2007;
Armitage, 2010; Nelson & Gressel, 2010; Carballido et al., 2010, 2011)
continue to use it because a more accurate description is computationally
prohibitive. Here we will call bodies of approximately this size “boulders”.
The mean-free-path $\lambda$ in an accretion disk varies with radius; e.g.,
according to the minimum mass solar nebula model $\lambda$ ranges from
$\approx 10{\rm cm}$ at approximately $1.5$ AU to $\approx 10{\rm m}$ at $10$
AU. Hence $R_{\rm boulder}$ ranges from $\sim 2\,{\rm m}$ in inner disk
regions to $\sim 200\,{\rm m}$ at about $10$ AU. A more accurate approximation
of the motion of such particles is given by the Maxey-Riley equation (Maxey &
Riley, 1983), which assumes a spherical geometry. While the Maxey-Riley
approach is appealing on fundamental grounds, it has yet to be used in
simulations of fully developed turbulence.
### 1.3. Bouncing, Sticking, Fusing
A crucial and often-used assumption is that all collisions have a sticking
probability of unity. Indeed, under such an assumption planetesimal growth
under a wide range of disk conditions is sufficiently rapid that there is no
loss to the central star. Clearly, however, the probability of sticking
depends, among other things, on the collisional velocity, the material
properties of the colliding bodies, the ambient temperature, and the relative
particle size. It is a commonly accepted picture that for collisional
velocities $V_{c}$ above a certain threshold value, $V_{\rm th}\sim 0.1$–10 cm
s-1, particle agglomeration is not possible; and elastic rebound overcomes
attractive surface and intermolecular forces (e.g., Chokshi et al., 1993).
However, for bodies covered with ice, experimental (Blum & Wurm, 2008) and
theoretical (Wettlaufer, 2010) studies of collisions between dust grains and
meter-sized objects have elucidated the range of collisional velocities (which
depends on the relative particle size) over which perfect sticking occurs.
This latter work considers the basic role of the phase behavior of matter
(phase diagrams, amorphs and polymorphs) in leading to so-called collisional
fusion. In this fusion process, a physical basis for efficient sticking is
provided through collisional melting/amphorphization/polymorphization and
subsequent fusion/annealing to extend the collisional velocity range of
sticking to $\Delta V_{c}\sim$ 1–100 m s${}^{-1}\gg V_{\rm th}$, which
encompasses both typical turbulent rms (root-mean-square) speeds and the
velocity differences between boulders and small grains $\sim 1$–50 m s-1.
Moreover, bodies of high melting temperature and multicomponent materials,
such as silicon and olivine, can fuse in this manner depending on the details
of their phase diagrams. Hence, in principle, the approach provides a
framework for sticking from the inner to the outer nebula. Here, we explore
the influence of such a range, $\Delta V_{c}$, on the growth of a boulder in a
simulated disk.
### 1.4. Summary of Approach
The fate of the boulder is studied from a reference frame fixed to it, while
the gas flows around it. We provide an accurate description of the flow by
performing a direct numerical simulation (DNS) with no-slip and non-
penetrating boundary conditions on the boulder surface using a numerical
technique called Immersed Boundary Method (Peskin, 2002). Hence, there is no
ad hoc approximation involved in describing the mutual interaction between the
boulder and the gas flow. However, at present, it is computationally
prohibitive to solve for more than one boulder using this DNS scheme.
Consequently we focus our study on the flow mediated collisions between one
boulder and many “effectively” point sized dust grains whose sizes are much
smaller than $R_{\rm boulder}$. Our principal approximations in treating the
motion of the dust grains are (a) to use Equation (3) and (b) to ignore the
back-reaction of the dust grains onto the flow. Advected by the turbulent disk
flow, the dust grains collide with the boulder and we compute the PDF of the
normal component of the collisional velocity.
## 2\. Model
The mechanism of formation of planetesimals from dust grains is modeled by the
same tools that are used to study, for example, hydrometeor growth in the
terrestrial atmosphere, namely the coagulation/fragmentation equations of
Smoluchowski (1916); see, e.g., Armitage (2010), for a recent review. The
Smoluchowski equations are integro-differential equations that require two
crucial ingredients: the probability distribution function of relative
collisional velocities of the bodies in question and their sticking
efficiency. The former, particularly for the inner disk region, is strongly
influenced by turbulence. Recently, there has been significant progress in
calculating the statistical properties of individual particle velocities
(Carballido et al., 2011; Nelson & Gressel, 2010) and, perhaps more
importantly, pairwise relative velocities (Carballido et al., 2010) from
direct numerical simulations. Similar results have also been obtained from
both phenomenological (Ormel & Cuzzi, 2007; Cuzzi & Hogan, 2003) and shell
(Hubbard, 2012) models of turbulence. While these approaches provide key
insights and intuition, they also leave open aspects with which the strategy
we take is not burdened, such as (a) the use of the simple drag law (3) to
describe the motion of boulders, (b) the ability to obtain only the root-mean-
square collision velocity, rather than the PDF of collision velocities
(Carballido et al., 2010; Hubbard, 2012, are exceptions), (c) not modeling
actual collisions, so that collisional velocities are inferred from looking at
relative velocities at small distances. To calculate the PDF of collisional
velocities between a boulder and small dust grains, such approximations may be
too simplistic because of the presence of a boundary layer around the larger
object. Indeed, Garaud et al. (2013) have recently pointed out the importance
of using the PDF of collisional velocities instead of simply the root-mean-
square value. However, taking this into account in a global (or even local)
simulation of a disk is computationally prohibitive. Therefore, we take an
initial modest step to try and understand such collisions by solving the
equations of motion for weakly compressible fluids in two dimensions with a
circular object–the boulder–inside. We ignore two classes of collisions, (a)
between dust grains themselves, and (b) between two or more boulders.
### 2.1. Numerical method
${\bm{g}}$
Figure 1.— A sketch of our computational domain. The domain is divided into
two halves. The left half contains the “boulder” sketched by the blue circle.
In the right half the fluid is acted upon by an external white-in-time force
which is non-zero only in the part of the domain limited by the two dashed
lines. The turbulence thus generated is moved toward the “boulder” by the
action of weak body force ${\bm{g}}$ along the arrow shown in the figure. The
body force does not act directly on the particles, which are introduced
continuously in a small area in the right half of the domain. Initial
positions of a few particles are shown as red dots.
Our computational domain is a rectangular box divided into two equal parts
(Fig. 1). In the right half, fluid turbulence is generated by external forcing
that is non-zero between the two dashed lines shown in Fig. 1. The turbulence
thus generated is moved toward the “boulder” by the action of a body force
${\bm{g}}$ in the direction of the arrow shown in the figure. This body force
is responsible for generating a mean flow, which models the head-wind faced by
a boulder–the circular object at the left half of the domain. The boundary
layer around the boulder is fully resolved by imposing non-penetrating and no-
slip boundary conditions using the immersed boundary method. After the flow
has reached a stationary state, we introduce $N_{\rm p}=2\times 10^{4}$
particles into the right half of the domain as depicted in Fig. 1. The motion
of these particles obeys the simple drag law,
$\frac{d{\bm{v}}_{\rm p}}{dt}=\frac{1}{\tau_{\rm p}}\left({\bm{v}}_{\rm
p}-{\bm{U}}\right),$ (6)
with the characteristic drag time of the “dust particles” $\tau_{\rm p}$. As
noted before and as is clear from context, no such assumption need be made for
the boulder. The back-reaction from the dust grains to the gas is ignored.
When a dust grain collides with the boulder it is removed from the simulation
and a new dust grain is introduced in the right half of the domain. We use the
Pencil Code222http://pencil-code.googlecode.com/ in which the immersed
boundary method was first implemented by Haugen & Kragset (2010).
### 2.2. Parameters
The characteristic large-scale velocity is the root-mean-square velocity in
the streamwise direction, $v_{\rm wind}\equiv\langle v^{2}_{y}\rangle^{1/2}$.
We always use the Reynolds number corresponding to the central solid body,
defined by
$\mbox{Re}_{\rm SB}\equiv v_{\rm wind}R_{\rm SB}/\nu.$ (7)
And the Stokes number of the “dust particles” is defined by
$\mbox{St}\equiv\tau_{\rm p}/\tau_{\rm L},$ (8)
where $\tau_{\rm L}=L_{y}/v_{\rm wind}$ with $L_{y}$ being the length of our
domain along the streamwise direction; from right to left in Fig. 1. By virtue
of limiting our simulations to two dimensions we can access a larger range of
particle Reynolds numbers $\mbox{Re}_{\rm SB}$, from $30$ to $1000$ with
resolutions ranging from ${\tt 128}\times{\tt 512}$ to ${\tt 512}\times{\tt
2048}$ grid points. The surface of the boulder is resolved with ${\tt 100}$ to
${\tt 400}$ grid points.
## 3\. Results
$v_{\rm wind}$$y$$x$
Figure 2.— Contour plot of vorticity in the upper half of our domain. The
black circle at the center of the domain is the circular object. The arrow
shows the time and space averaged direction of $v_{\rm wind}$.
$v_{\rm wind}$
Figure 3.— Plot showing how the boulder would grow if all collisions were
perfectly sticky. The arrow shows the direction of $v_{\rm wind}$. The growth
for two different runs (a) $\mbox{Re}_{\rm SB}\approx 29$, $\mbox{St}\approx
0.5$ ($\ast$), and (b) $\mbox{Re}_{\rm SB}\approx 1000$, $\mbox{St}\approx
0.6$ ($\square$), for the same total time duration are shown. The inner semi-
circle shows the initial surface of the boulder.
A representative snapshot of the vorticity field is given in Fig. 2. A movie,
available online at: http://www.youtube.com/watch?v=-Fr5Q2Kp0wo, shows that,
although over a spatiotemporal average there is a streamwise mean flow on the
boulder, there are large fluctuations. At a particular instant the direction
of the gas velocity at the boulder surface can deviate significantly from the
streamwise direction. Furthermore we observe that most of the collisions do
not occur at the front face of the boulder but there is a significant number
of collisions that deviate from centrality; see Fig. 3. Note, however, that
there are almost no collisions on the backside of the boulder. A clear
implication of this is that, for perfect sticking of all collisions, an
initially spherical boulder evolves into a non spheroidal body and hence may
begin to tumble in the disk.
### 3.1. PDF of collisional velocities
(a)
(b)
(c)
Figure 4.— PDF of collisional velocities for $\mbox{Re}_{\rm SB}\approx
1000$,$\mbox{St}\approx 0.6$. (a) Log-log (base $10$) plot for small $v_{\rm
n}$; $P(v_{\rm n})\sim(v_{\rm n}/v_{\rm wind})^{2}$, the straight line has a
slope of $2$. (b) Semi-log (base $10$) at large $v_{\rm n}$. The straight
line, which is a fit to the points denoted by the symbol $\ast$, has slope
$0.96$ (c) The PDF with the two approximations at small and larger $v_{\rm n}$
plotted together. Figure 5.— Probability distribution function, $P(v_{n})$,
of the normal component of collisional velocity versus $(v_{n}/v_{\rm
wind})^{2}$ for four different runs: (a) $\mbox{Re}_{\rm SB}\approx 29$,
$\mbox{St}\approx 0.5$ ($\ast$), (b) $\mbox{Re}_{\rm SB}\approx 69$,
$\mbox{St}\approx 1$ ($\square$), (c) $\mbox{Re}_{\rm SB}\approx
516$,$\mbox{St}\approx 0.7$ ($\vartriangle$), (d) $\mbox{Re}_{\rm SB}\approx
1000$,$\mbox{St}\approx 0.6$ ($\lozenge$). Figure 6.— Probability
distribution function, $P(v_{n})$, of normal component of collisional velocity
versus $[v_{n}/v_{\rm wind}]^{2}$ for two different Stokes numbers;
$\mbox{St}\approx 0.5$ ($\ast$), and $\mbox{St}\approx 0.1$ ($\vartriangle$),
for $\mbox{Re}_{\rm SB}\approx 29$.
The criterion for a collision is that the distance between a dust grain and
the boulder becomes less than a grid point. After this collision we remove the
dust grain from the simulation. For $\mbox{Re}_{\rm SB}\approx 1000$ in Fig.
4; we plot the PDF, $P(v_{\rm n})$, of the component of the velocity of the
dust grain normal to the surface of the boulder, $v_{\rm n}$. At small $v_{\rm
n}$, $P(v_{\rm n})\sim v_{\rm n}^{2}$ (Fig. 4a) and at large $v_{\rm n}$ the
fall off is $\sim\exp[-(v_{\rm n}/v_{\rm 0})^{2}]$ (Fig. 4b). However, as
shown in Fig. 4(c), over the whole range it is difficult to fit the PDF with a
Maxwellian distribution.
Now we consider how the PDF changes as the Stokes and Reynolds numbers of the
flow change. We vary the Reynolds number by changing the viscosity of the
flow. Hence, a change in Reynolds number also changes $v_{\rm wind}$, and this
leads to a change in the Stokes number333As we change viscosity holding all
other variables, including the body force ${\bm{g}}$, constant, $v_{\rm wind}$
also changes. This changes $\tau_{\rm L}=L_{y}/v_{\rm wind}$ which
consequently changes St through (8). . Therefore, in our approach to the
numerical treatment of the flow, it is not possible to perform a systematic
study of the Reynolds number dependence of the PDF at fixed Stokes number.
However, in order to produce an effective treatment of such a circumstance, we
present in Fig. 5 the PDFs for different Reynolds numbers wherein the Stokes
numbers are not too different from each other. We see that for small
$\mbox{Re}_{\rm SB}$ the peak of the PDF lies very close to $v_{\rm wind}$,
but as $\mbox{Re}_{\rm SB}$ increases the peak moves to smaller velocities by
only a very small amount. Although $\mbox{Re}_{\rm SB}$ changes by almost a
factor of $20$ the position of the peak (normalized by $v_{\rm wind}$) only
changes from $0.6$ to $0.3$. A more dramatic change is observed in the PDF as
the Stokes number is changed from $0.5$ to $0.1$ when $\mbox{Re}\approx 29$ is
held fixed, as shown in Fig. 6. In particular, the tail of the PDF at high
$v_{\rm n}$ is severely cut off as the Stokes number is decreased by a factor
of $5$. One can understand this as follows; when the Stokes number decreases,
the dust grains begin to follow streamlines and hence never collide with the
boulder. The implication of this is clearly that a smaller Stokes number
implies a smaller number of high-impact collisions. Nevertheless, the most
striking result for the problem at hand is the insensitivity of the PDF to
$\mbox{Re}_{\rm SB}$ and St.
### 3.2. From DNS to disk astrophysics
Our simulations take place in the reference frame of the boulder. Although the
boulder is also comoving with the local gas with velocity $v_{\rm kepler}$,
the head wind corresponds to $v_{\rm wind}$ in our simulations, thereby
setting the velocity scale. The radius of the boulder is taken to be $\approx
10\,{\rm m}$. The magnitude of the headwind in the disk is estimated to be
$v_{\rm wind}\approx 10^{-3}v_{\rm kepler}\approx 3\times 10^{3}\,{\rm
cm\,s}^{-1}$ (see e.g. Armitage, 2010, page 130). In the astrophysical
literature it is common to non-dimensionalize $\tau_{\rm p}$ with $\Omega_{\rm
kepler}$, the Keplerian frequency, to define the orbital Stokes number,
$\mbox{St}_{\rm kepler}$. Here, we use the largest eddy time scale $\tau_{\rm
L}=L_{y}/v_{\rm wind}$, to obtain St. These two Stokes numbers are related by
$\mbox{St}_{\rm kepler}=\mbox{St}\frac{\tau_{L}}{\tau_{\rm orb}}$ (9)
where $\tau_{\rm orb}$ is the characteristic time scale of the Keplerian orbit
is defined by
$\tau_{\rm orb}=\frac{R_{\rm orb}}{v_{\rm kepler}}$ (10)
where $R_{\rm orb}$ is the orbital radius. Using the definition of the two
time scales $\tau_{\rm L}$ and $\tau_{\rm orb}$, we obtain the ratio of the
two Stokes numbers to be
$\frac{\mbox{St}_{\rm kepler}}{\mbox{St}}=\frac{L_{y}}{R_{\rm
orb}}\frac{v_{\rm kepler}}{v_{\rm wind}}\approx 10^{-5}$ (11)
where we have used $R_{\rm orb}=1{\rm AU}$, $L_{y}=50R_{\rm SB}\approx 500m$,
and $v_{\rm wind}=\eta v_{\rm kepler}$ with $\eta=10^{-3}$. We have used
Stokes number ranging from $\mbox{St}=0.1$ to $2$ which in turn gives
$\mbox{St}_{\rm kepler}\approx 10^{-6}$ to $2\times 10^{-5}$. We use the same
conventions used in the Supplementary Information of Johansen et al. (2007) to
convert the value of $\mbox{St}_{\rm kepler}$ to a radius of the dust grain;
this implies that our “dust particles” are of the size of tenth of millimeters
or smaller. Clearly the “dust particles” are smaller than hydrodynamic scales,
and hence it is justified to consider them as point objects whose motion are
described by the Epstein drag law.
### 3.3. Collisional fusion
The PDFs of collisional velocities show that, irrespective of the Reynolds
number and the Stokes number within the range considered by us, most
collisions occur at velocities rather near to $v_{\rm wind}$. To illustrate
this in Fig. 5 we have drawn two vertical dashed lines at $(v_{\rm n}/v_{\rm
wind})^{2}=0.2$ and $(v_{\rm n}/v_{\rm wind})^{2}=1.2$. The area under the PDF
between the two lines includes approximately $95\%$ of the total number of
collisions. Translated to parameters in the disk, this implies that, if there
is a mechanism by which dust grains with velocities ranging from $0.2\,v_{\rm
wind}$ to $1.2\,v_{\rm wind}$ would stick to a boulder, then we could consider
$95\%$ of collisions to have a perfect sticking probability.
Roughly speaking, this implies a range of velocities $6-36\,{\rm m\,s}^{-1}$.
These collisional velocities are far too high for the bodies to fuse by
attractive intermolecular forces. An alternative scenario by which the
colliding bodies can fuse at high speed has been suggested by Wettlaufer
(2010). As discussed in section 1.3, the very high local pressures that occur
during a collision can lead to phase change. If, when the pressure begins to
relax during rebound the momentarily liquified (or disordered) interfacial
material re-freezes (or anneals) before particle separation, then fusion can
occur. The idea was demonstrated when the colliding bodies are covered by ice,
but the theory is generally applicable to all materials whose phase diagram is
known in detail. An example of the process in a high melting temperature
material (silicon) was noted in Wettlaufer (2010). Hence, whether the range of
collisional velocities over which such process can occur in a material such as
olivine matches with the range we find here is a topic of ongoing research.
Note that here the particle Reynolds number $\mbox{Re}_{\rm SB}$ varies
linearly with the particle radius but the range over which most of the
collisions occur does not depend sensitively on $\mbox{Re}_{\rm SB}$, and
hence not on the particle radius. Thus, runaway growth of the boulder through
the accretion of dust grains is a viable mechanism in areas of the disk where
collisional fusion can operate in the range we obtain.
## 4\. Discussion and Conclusion
To describe the motion of micron sized dust grains in a protoplanetary disk
the simple drag law of Equation (6) is sufficient. Theoretical estimates (see
e.g., Armitage, 2010, p. 120) suggests that micron sized dust particles in the
inner disk (about $5$ AU) can grow up to a size of $10$’s of centimeters if we
assume that the presence of turbulence increases the number of collisions and
that almost all collisions result in coagulation by long-ranged intermolecular
forces. But the process that allows them to continue to grow to the size of
planetesimals is not well understood. As the dust grains grow, at some stage
they become boulders and their local Reynolds number exceeds unity. At this
stage we need a more accurate description of their interaction with the gas
than the one provided by (3). Here we provide such a description of a boulder
colliding with dust grains by using the immersed boundary method of Peskin
(2002). Remarkably, we find that the PDF of collisional velocities depends
weakly on $\mbox{Re}_{\rm SB}$ and St. In particular, we find that, if
collisional fusion between dust grains is possible in the range of collisional
velocities $\Delta V_{c}$ between $0.2$ to $1.2\,v_{\rm wind}$, then
approximately $95\%$ of the collisions exhibit perfect sticking and runaway
growth of a boulder to a planetesimal is possible. Whether collisional fusion
can occur in this range is a problem of material science under extreme
conditions and is the subject of ongoing research and a future paper.
Recent studies (Garaud et al., 2013; Windmark et al., 2012) have pointed out
that the PDF of collisional velocities is a crucial ingredient to the
coagulation-fragmentation models. In particular, Windmark et al. (2012) have
assumed the PDF of collisional velocities to be Maxwellian, and have concluded
that, by virtue of considering a PDF that is continuous at small values of its
argument, growth by sticking is possible even if the sticking efficiency is
determined by long-ranged intermolecular forces (sticking with efficiency
unity if the relative velocity of collisions is less than 5 cm s-1). Here, we
determine numerically the PDFs for the classes of collisions between boulders
and dust grains and find that it cannot be simply described by a Maxwellian
distribution - although it does have an exponential tail. It is well known
that in turbulent flows the PDF of the velocities of a tracer particle is
Gaussian. We do not know of any study of the PDF of velocity of inertial
particles (particles that obey (6)) in turbulent flows, but it is reasonable
to assume that it would also be Gaussian. If such an assumption holds, then we
expect the PDF of collisional velocities to have an exponential tail, so long
as the size of particles is not comparable. Were the colliding particles to be
of roughly the same size, the PDF may indeed have a power-law tail by virtue
of intermittency.
In an earlier paper, Sekiya & Takeda (2003) found that dust monomers advected
by a steady laminar flow do not collide with a spherical solid body with of
radius much larger than the hydrodynamic length scale. The crucial limitation
in their work was to assume the flow to be laminar. Here, we have considered
turbulent flow and have obtained a different result, i.e., a significant
percentage of the dust particles do hit the solid body with the PDF of
collisional velocities peaking around the speed of the head wind.
There exists an alternative scenario of planetesimal formation (Johansen et
al., 2007) in which the boulders are described by the simple drag law (3) but
their back-reaction on the gas is accounted for. This is predicted to give
rise to “streaming instabilities” which form boulder clusters around high
pressure regions. Such clusters are then expected to coagulate by mutual
gravitational interaction. In the light of the arguments presented in the
present paper, this streaming instability scenario requires further
investigation. This is because basic physical principles tell us that the
description of the motion of the boulder is inadequately described by (3).
While the immersed boundary method can potentially solve this problem we need
to have massive computational resources to examine the fate of many boulders.
We conclude by pointing out the limitations of our study. Firstly, here we
confine ourselves to two dimensions. On the one hand, this has the virtue of
permitting a larger range of $\mbox{Re}_{\rm SB}$ that can be easily accessed
numerically. On the other hand we cannot capture the richness of particle
fusion in the remaining dimension. However, we believe that this may imply
that the growth of the particle we have studied to be a lower bound. Secondly,
when collisional fusion starts operating the initial spherical object we study
will not remain spherical. This may quantitatively affect further growth in a
manner that depends on how the boulder tumbles through the disk. Thirdly, the
turbulence in our flow is generated by external forcing. It would be
appropriate to use shearing-box simulations in three dimensions where the flow
is driven by magneto-rotational instability. We believe that these rather
clear limitations do not detract from the robust results obtained in this
study, which clarify the microphysical questions for a range of colliding
materials and the computational fluid dynamics issues that will advance a
sober assessment of planetesimal formation processes.
## Acknowledgments
Financial support from the European Research Council under the AstroDyn
Research Project 227952, and the Swedish Research Council grant 2011-5423 is
gratefully acknowledged. J.S.W. thanks the Wenner-Gren and John Simon
Guggenheim Foundations, and the Swedish Research Council. We also thank the
anonymous referee for his/her useful suggestions.
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|
arxiv-papers
| 2013-06-16T15:59:52 |
2024-09-04T02:49:46.578466
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dhrubaditya Mitra (1), J.S. Wettlaufer (1,2), and Axel Brandenburg\n (1,3) ((1) NORDITA (2) Yale University (3) Stockholm University)",
"submitter": "Dhrubaditya Mitra",
"url": "https://arxiv.org/abs/1306.3672"
}
|
1306.3744
|
# GRAVITATIONAL BINARY-LENS EVENTS WITH PROMINENT EFFECTS OF LENS ORBITAL
MOTION
H. Park1,34, A. Udalski2,33, C. Han1,34,37, A. Gould3,34, J.-P. Beaulieu4,35,
Y. Tsapras5,6,36,
and
M. K. Szymański2, M. Kubiak2, I. Soszyński2, G. Pietrzyński2,7, R. Poleski2,3,
K. Ulaczyk2, P. Pietrukowicz2,
S. Kozłowski2, J. Skowron2, Ł. Wyrzykowski2,8
(The OGLE Collaboration),
J.-Y. Choi1, D. L. Depoy9, Subo Dong10, B. S. Gaudi3, K.-H. Hwang1, Y. K.
Jung1, A. Kavka3, C.-U. Lee11,
L. A. G. Monard12, B,-G. Park11, R. W. Pogge3, I. Porritt13, I.-G. Shin1, J.
C. Yee3
(The $\mu$FUN Collaboration),
M. D. Albrow14, D. P. Bennett15, J. A. R. Caldwell16, A. Cassan4, C.
Coutures4, D. Dominis17, J. Donatowicz18,
P. Fouqué19, J. Greenhill20, M. Huber21, U. G. Jørgensen22, S. Kane23, D.
Kubas4, J. -B. Marquette4, J. Menzies24,
C. Pitrou4, K. R. Pollard14, K. C. Sahu25, J. Wambsganss26, A. Williams27, M.
Zub26
(The PLANET Collaboration),
A. Allan28, D. M. Bramich29, P. Browne30, M. Dominik30, K. Horne30, M.
Hundertmark30, N. Kains29, C. Snodgrass31,
I. A. Steele32, R. A. Street5
(The RoboNet Collaboration) 1Department of Physics, Institute for
Astrophysics, Chungbuk National University, Cheongju 371-763, Korea 2Warsaw
University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland
3Department of Astronomy, The Ohio State University, 140 West 18th Avenue,
Columbus, OH 43210, USA 4Institut d’Astrophysique de Paris, UMR 7095 CNRS -
Université Pierre & Marie Curie, 98bis Bd Arago, 75014 Paris, France 5Las
Cumbres Observatory Global Telescope Network, 6740 Cortona Drive, Suite 102,
Goleta, CA 93117, USA 6School of Mathematical Sciences, Queen Mary,
University of London, Mile End Road, London E1 4NS, UK 7Universidad de
Concepción, Departamento de Astronomia, Casilla 160-C, Concepción, Chile
8Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK 9Department of Physics and Astronomy, Texas A&M University,
College Station, TX 77843, USA 10Institute for Advanced Study, Einstein
Drive, Princeton, NJ 08540, USA 11Korea Astronomy and Space Science
Institute, 776 Daedukdae-ro, Yuseong-gu, Daejeon 305-348, Korea 12Klein Karoo
Observatory, Calitzdorp, and Bronberg Observatory, Pretoria, South Africa
13Turitea Observatory, Palmerston North, New Zealand 14Department of Physics
and Astronomy, University of Canterbury, Private Bag 4800, Christchurch 8020,
New Zealand 15Department of Physics, University of Notre Dame, 225 Nieuwland
Science Hall, Notre Dame, IN 46556-5670, USA 16McDonald Observatory,
University of Texas, Fort Davis, TX 79734, USA 17Astrophysikalisches Institut
Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany 18Department of
Computing, Technical University of Vienna, Wiedner Hauptstrasse 10, A-1040
Vienna, Austria 19Observatoire Midi-Pyrénées, Laboratoire d’Astrophysique,
UMR 5572, Université Paul Sabatier - Toulouse 3, 14 avenue Edouard Belin,
31400 Toulouse, France 20School of Mathematics and Physics, University of
Tasmania, Private Bag 37, Hobart, TAS 7001, Australia 21Institute for
Astronomy, University of Hawaii, 2680 Woodlawn Drive Honolulu, HI 96822-1839,
USA 22Niels Bohr Institute, Astronomical Observatory, Juliane Maries vej 30,
2100 Copenhagen, Denmark 23NASA Exoplanet Science Institute, Caltech, MS
100-22, 770 South Wilson Avenue, Pasadena, CA 91125, USA 24South African
Astronomical Observatory, PO Box 9 Observatory 7935, South Africa 25Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
26Astronomisches Rechen-Institut (ARI), Zentrum für Astronomie der Universität
Heidelberg (ZAH), Mönchhofstr. 12-14, 69120, Heidelberg, Germany 27Perth
Observatory, Walnut Road, Bickley, Perth 6076, Australia 28School of Physics,
University of Exeter Stocker Road, Exeter, Devon, EX4 4QL, UK 29European
Southern Observatory, Karl-Schwarzschild-Straße 2, 85748 Garching bei München,
Germany 30SUPA, University of St. Andrews, School of Physics and Astronomy,
North Haugh, St. Andrews, KY16 9SS, UK 31Max Planck Institute for Solar
System Research, Max-Planck-Str. 2, 37191 Katlenburg-Lindau, Germany
32Astrophysics Research Institute, Liverpool John Moores University, Twelve
Quays House, Egerton Wharf, Birkenhead, Wirral., CH41 1LD, UK 33The OGLE
Collaboration 34The $\mu$FUN Collaboration 35The PLANET Collaboration 36The
RoboNet Collaboration 37Corresponding author
###### Abstract
Gravitational microlensing events produced by lenses composed of binary masses
are important because they provide a major channel to determine physical
parameters of lenses. In this work, we analyze the light curves of two binary-
lens events OGLE-2006-BLG-277 and OGLE-2012-BLG-0031 for which the light
curves exhibit strong deviations from standard models. From modeling
considering various second-order effects, we find that the deviations are
mostly explained by the effect of the lens orbital motion. We also find that
lens parallax effects can mimic orbital effects to some extent. This implies
that modeling light curves of binary-lens events not considering orbital
effects can result in lens parallaxes that are substantially different from
actual values and thus wrong determinations of physical lens parameters. This
demonstrates the importance of routine consideration of orbital effects in
interpreting light curves of binary-lens events. It is found that the lens of
OGLE-2006-BLG-277 is a binary composed of a low-mass star and a brown dwarf
companion.
###### Subject headings:
gravitational lensing: micro – orbital motion – binaries: general
## 1\. INTRODUCTION
Progress in gravitational microlensing experiments for the last two decades
has enabled a great increase in the number of event detections from tens of
events per year at the early stage to several thousands per year in current
experiments. Among discovered lensing events, an important portion are
produced by lenses composed of two masses (Mao & Paczyński, 1991).
One reason why binary-lens events are important is that these events provide a
major channel to determine the physical parameters of lenses. For the
determination of the lens parameters from observed lensing light curves, it is
required to simultaneously measure the lens parallax $\pi_{\rm E}$ and the
angular Einstein radius $\theta_{\rm E}$. The lens parallax is measured from
long-term deviations in lensing light curves caused by the positional change
of an observer induced by the orbital motion of the Earth around the Sun:
parallax effect (Gould, 1992). On the other hand, the Einstein radius is
measured from deviations in lensing light curves affected by the finite size
of a source star: finite-source effect (Gould, 1994; Witt & Mao, 1994). With
the measured values of $\pi_{\rm E}$ and $\theta_{\rm E}$, the mass and
distance to the lens are determined respectively by $M_{\rm tot}=\theta_{\rm
E}/(\kappa\pi_{\rm E})$ and $D_{\rm L}={\rm AU}/(\pi_{\rm E}\theta_{\rm
E}+\pi_{\rm S})$, where $\kappa=4G/(c^{2}{\rm AU})$, ${\rm AU}$ is an
Astronomical Unit, $\pi_{\rm S}={\rm AU}/D_{\rm S}$, and $D_{\rm S}$ is the
distance to the lensed star (Gould, 1992; Gould et al., 2006). For single-lens
events, the chance to measure $\theta_{\rm E}$ is very low because finite-
source effects occur only for very rare events with extremely high
magnifications in which the lens passes over the surface of the source star,
e.g., Choi et al. (2012). By contrast, the chance to measure $\theta_{\rm E}$
is high for binary-lens events because most of these events involve source
stars’ caustic crossings or approaches during which finite-source effects are
important. As a result, the majority of gravitational lenses with measured
physical parameters are binaries.
It is known that changes of lens positions caused by the orbital motion of a
binary lens can induce long-term deviations in lensing light curves, similar
to deviations induced by parallax effects. Since this was first detected for
the event MACHO-97-BLG-41 (Bennett et al., 1999; Albrow et al., 2000; Jung et
al., 2013), orbital effects have been considered for more binary-lens events,
e.g., An et al. (2002); Jaroszyński et al. (2005); Skowron et al. (2011); Shin
et al. (2011). However, analyses have been carried out only for a limited
number of events. An important obstacle of orbital analyses is the heavy
computation required to consider the time variation of the caustic morphology
caused by the orbit-induced changes of the binary separation and orientation.
As a result, routine orbital analyses for general binary-lens events became
possible very recently after being able to utilize efficient modeling software
and powerful computing resources.
Considering orbital effects is important for accurate determinations of
physical lens parameters. Since orbital and parallax effects induce similar
long-term deviations, it might be that orbital effects can be mimicked by
parallax effects. Then, if only parallax effects are considered for events
affected by orbital effects, the determined physical parameters would be
different from their true values. In this work, we demonstrate the importance
of considering orbital effects by presenting analyses of two binary-lens
events.
Table 1Telescopes
Event | Telescopes
---|---
OGLE-2006-BLG-277 | OGLE, 1.3 m Warsaw, LCO, Chile
| $\mu$FUN, 1.3 m SMARTS, CTIO, Chile
| PLANET, 1.5 m Boyden, South Africa
| PLANET, 1.0 m Canopus, Australia
| PLANET, 0.6 m Perth, Australia
| PLANET, 1.54 m Danish, Chile
| RoboNet, 2.0 m LT, La Palma, Spain
OGLE-2012-BLG-0031 | OGLE, 1.3 m Warsaw, LCO, Chile
| $\mu$FUN, 1.3 m SMARTS, CTIO, Chile
| $\mu$FUN, 0.36 m Turitea, New Zealand
| $\mu$FUN, 0.36 m KKO, South Africa
| RoboNet, 2.0 m FTS, Australia
| Robonet, 2.0 m LT, La Palma, Spain
Note. — LCO: Las Campanas Observatory, CTIO: Cerro Tololo Inter-American
Observatory, KKO: Klein Karoo Observatory, LT: Liverpool Telescope, FTS:
Faulkes Telescope South
## 2\. Observation
The events analyzed in this work are OGLE-2006-BLG-277 and OGLE-2012-BLG-0031.
Both events occurred on stars toward the Galactic bulge field with equatorial
coordinates $(\alpha,\delta)_{\rm J2000}=(18^{\rm h}01^{\rm m}14^{\rm
s}\hskip-2.0pt.84,-27^{\circ}48^{\prime}36^{\prime\prime}\hskip-2.0pt.2)$,
corresponding to the Galactic coordinates
$(l,b)=(2.71^{\circ},-2.39^{\circ})$, for OGLE-2006-BLG-277 and
$(\alpha,\delta)_{\rm J2000}=(17^{\rm h}50^{\rm m}50^{\rm
s}\hskip-2.0pt.53,-29^{\circ}10^{\prime}48^{\prime\prime}\hskip-2.0pt.8)$,
corresponding to $(l,b)=(0.38^{\circ},-1.10^{\circ})$, for OGLE-2012-BLG-0031.
The events were discovered from survey observations conducted by the Optical
Gravitational Lensing Experiment (OGLE: Udalski, 2003). In addition to the
survey observation, the events were additionally observed by follow-up groups
including the Probing Lensing Anomalies NETwork (PLANET: Albrow et al., 1998),
the Microlensing Follow-Up Network ($\mu$FUN: Gould et al., 2006), and the
RoboNet (Tsapras et al., 2009) groups. In Table 1, we list survey and follow-
up groups who participated in observations of the individual events along with
the telescopes they employed as well as their locations. We note that the
event OGLE-2006-BLG-277 was previously analyzed by Jaroszyński et al. (2010),
but the analysis was based on only OGLE data. We, therefore, reanalyze the
event based on all combined data considering higher-order effects.
Reductions of data were done using photometry codes developed by the
individual groups, mostly based on difference image analysis (Alard & Lupton,
1998; Woźniak et al., 2001; Bramich, 2008; Albrow et al., 2009). In order to
use data sets acquired from different observatories, we readjust the error
bars. For this, we first add a quadratic error term so that the cumulative
distribution of $\chi^{2}$ ordered by magnifications is approximately linear
in data counts, and then rescale errors so that $\chi^{2}$ per degree of
freedom $(\chi^{2}/{\rm dof})$ becomes unity.
Figure 1.— Light curve of OGLE-2006-BLG-277. In the legends indicating
observatories, the subscript of each observatory denotes the passband. The
subscript “$N$” denotes that no filter is used. The insets in the upper panel
show the enlargement of the caustic-crossing parts of the light curve. The
lower four panels show the residuals of data from four different models.
In Figures 1 and 2, we present the light curves of the individual events. Both
light curves exhibit sharp spikes that are characteristic features of caustic-
crossing binary-lens events. The spikes occur in pairs because the caustic
forms a closed curve. Usually, the inner region between two spikes has a
“U”-shape trough as is in OGLE-2006-BLG-277. For OGLE-2012-BLG-0031, the inner
region exhibits a complex pattern. Such a pattern can be produced when the
source trajectory runs approximately tangent to the fold of a caustic.
## 3\. Modeling
### 3.1. Standard Model
Figure 2.— Light curve of OGLE-2012-BLG-0031. Notations are the same as those
in Fig. 1.
Knowing that the events were produced by binary lenses, we conduct modeling of
the observed light curves. Basic description of a binary-lens event requires 7
lensing parameters. Among them, the first three parameters describe the lens-
source approach. These parameters include the time of the closest approach of
the source to a reference position383838 For a binary lens with a projected
separation less than the Einstein radius, $s<1$, we set the reference position
of the lens as the center of mass of the binary lens. For a binary with a
separation greater than the Einstein radius, $s>1$, on the other hand, we set
the reference as the photocenter that is located at a position with an offset
$q/[s(1+q)]$ from the middle position between the two lens components. of the
binary lens, $t_{0}$, the lens-source separation at $t_{0}$ in units of the
Einstein radius, $u_{0}$, and the time required for the source to cross the
Einstein radius, $t_{\rm E}$ (Einstein time scale). The Einstein ring
represents the source image for an exact lens-source alignment and its radius
$\theta_{\rm E}$ is usually used as the length scale of lensing phenomena. The
Einstein radius is related to the physical lens parameters by $\theta_{\rm
E}=(\kappa{M}\pi_{\rm rel})^{1/2}$, where $M$ is the mass of the lens and
$\pi_{\rm rel}={\rm AU}(D^{-1}_{\rm L}-D^{-1}_{\rm S})$ is the relative lens-
source parallax. Another three lensing parameters describe the binary lens.
These parameters include the projected separation, $s$ (in units of
$\theta_{\rm E}$), the mass ratio between the binary lens components, $q$, and
the angle between the source trajectory and the binary axis, $\alpha$ (source
trajectory angle). The last parameter is the normalized source radius
$\rho_{\ast}=\theta_{\ast}/\theta_{\rm E}$, where $\theta_{\ast}$ is the
angular source radius. This parameter is needed to describe the parts of light
curves affected by finite-source effects, which are important when a source
star crosses over or approaches close to caustics formed by a binary lens.
In modeling the light curves based on the standard lensing parameters
(standard model), searches for best-fit solutions have been done in two steps.
In the first step, we identify local solutions by inspecting $\chi^{2}$
distributions in the parameter space. For this, we use both grid search and
downhill approach. We choose ($s,q,\alpha$) as grid parameters because lensing
magnifications can vary dramatically with small changes of these parameters.
By contrast, lensing magnifications vary smoothly with changes of the other
parameters, and thus we search for the solutions of these parameters by
minimizing $\chi^{2}$ using a downhill approach. We use the Markov Chain Monte
Carlo (MCMC) method for the $\chi^{2}$ minimization. In the second step, we
refine the lensing parameters for the individual local solutions by allowing
all parameters to vary. Then, the best-fit solution is obtained by comparing
$\chi^{2}$ values of the individual local solutions. We estimate the
uncertainties of the lensing parameters based on the distributions of the
parameters obtained from the MCMC chain of solutions.
Table 2Model Parameters
Model | $\chi^{2}/{\rm dof}$ | $t_{0}$ | $u_{0}$ | $t_{\rm E}$ | $s$ | $q$ | $\alpha$ | $\rho_{\ast}$ | $\pi_{{\rm E},N}$ | $\pi_{{\rm E},E}$ | $ds/dt$ | $d\alpha/dt$
---|---|---|---|---|---|---|---|---|---|---|---|---
| | (HJD’) | | (days) | | | | ($10^{-3}$) | | | (${\rm yr}^{-1}$) | (${\rm yr}^{-1}$)
OGLE-2006-BLG-277 | | | | | | | | | | | |
Standard | 2652.6 | 3941.620 | 0.157 | 39.13 | 1.374 | 2.600 | 1.477 | 5.83 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
| /1499 | $\pm$0.020 | $\pm$0.002 | $\pm$0.08 | $\pm$0.001 | $\pm$0.037 | $\pm$0.003 | $\pm$0.02
Parallax only | 1811.0 | 3941.723 | 0.169 | 39.30 | 1.371 | 2.512 | 1.485 | 5.90 | 0.45 | 0.54 | $\cdots$ | $\cdots$
| /1497 | $\pm$0.025 | $\pm$0.003 | $\pm$0.08 | $\pm$0.001 | $\pm$0.035 | $\pm$0.003 | $\pm$0.02 | $\pm$0.07 | $\pm$0.01
Orbit only | 1528.2 | 3943.066 | 0.170 | 38.78 | 1.347 | 2.033 | -1.485 | 5.98 | $\cdots$ | $\cdots$ | 0.73 | -0.33
| /1497 | $\pm$0.031 | $\pm$0.005 | $\pm$0.07 | $\pm$0.001 | $\pm$0.030 | $\pm$0.005 | $\pm$0.02 | $\pm$0.02 | $\pm$0.11
Orbit+Parallax | 1511.9 | 3943.071 | -0.168 | 37.90 | 1.348 | 1.981 | 1.457 | 6.03 | 1.13 | -0.05 | 0.69 | 1.21
| /1495 | $\pm$0.031 | $\pm$0.005 | $\pm$0.13 | $\pm$0.001 | $\pm$0.030 | $\pm$0.006 | $\pm$0.02 | $\pm$0.16 | $\pm$0.04 | $\pm$0.03 | $\pm$0.22
OGLE-2012-BLG-0031 | | | | | | | | | | | |
Standard | 2580.5 | 6022.532 | 0.046 | 59.17 | 0.477 | 0.294 | 0.800 | 5.48 | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$
| /2403 | $\pm$0.042 | $\pm$0.001 | $\pm$0.59 | $\pm$0.003 | $\pm$0.010 | $\pm$0.009 | $\pm$0.11
Parallax only | 2430.4 | 6022.233 | -0.047 | 56.47 | 0.510 | 0.223 | -0.739 | 5.63 | -0.29 | 0.10 | $\cdots$ | $\cdots$
| /2405 | $\pm$0.043 | $\pm$0.001 | $\pm$0.66 | $\pm$0.003 | $\pm$0.008 | $\pm$0.009 | $\pm$0.11 | $\pm$0.08 | $\pm$0.02
Orbit only | 2422.1 | 6022.364 | 0.051 | 54.88 | 0.511 | 0.264 | 0.774 | 6.80 | $\cdots$ | $\cdots$ | 0.43 | 3.63
| /2405 | $\pm$0.042 | $\pm$0.001 | $\pm$0.68 | $\pm$0.003 | $\pm$0.011 | $\pm$0.009 | $\pm$0.19 | $\pm$0.08 | $\pm$0.20
Orbit+Parallax | 2419.6 | 6022.350 | -0.051 | 54.58 | 0.511 | 0.268 | -0.773 | 6.81 | -0.09 | 0.03 | 0.47 | -2.98
| /2407 | $\pm$0.042 | $\pm$0.001 | $\pm$0.77 | $\pm$0.003 | $\pm$0.010 | $\pm$0.009 | $\pm$0.21 | $\pm$0.13 | $\pm$0.02 | $\pm$0.07 | $\pm$0.39
Note. — HJD’=HJD-2,450,000
For magnification computations affected by finite-source effects, we use the
“map-making method” developed by Dong et al. (2006). In this method, a map of
rays for a given binary lens with a separation $s$ and a mass ratio $q$ is
constructed by using the inverse ray-shooting technique (Schneider & Weiss,
1986; Kayser et al., 1986; Wambsganss, 1997). In this technique, rays are
uniformly shot from the image plane, bent according to the lens equation, and
land on the source plane. The lens equation for a binary lens is represented
by
$\zeta=z-{m_{1}\over\overline{z}-\overline{z}_{\rm
L,1}}-{m_{2}\over\overline{z}-\overline{z}_{\rm L,2}},$ (1)
where $m_{1}={1}/(1+q)$ and $m_{2}=qm_{1}$ are the mass fractions of the
individual binary lens components, $\zeta=\xi+i\eta$, $z=x+iy$, and $z_{{\rm
L},i}=x_{{\rm L},i}+iy_{{\rm L},i}$ denote the positions of the source, image,
and lens expressed in complex notions, respectively, and $\overline{z}$
denotes the complex conjugate of $z$. With the constructed map, the finite-
source magnification for a given position of a source with a normalized radius
$\rho_{\ast}$ is computed as the ratio of the number density of rays within
the source to that on the image plane. This method saves computation time by
enabling to produce many light curves resulting from various source
trajectories based on a single map. In addition, the method enables to speed
up computation by allotting computation into multiple CPUs. We further
accelerate computation by using semi-analytic hexadecapole approximation
(Pejcha & Heyrovský, 2009; Gould, 2008) for finite magnification computations.
In our finite-source computations, we consider the limb-darkening effect of
the source star by modeling the surface brightness profile as
$S_{\lambda}\propto 1-\Gamma_{\lambda}(1-3\cos\phi/2)$, where
$\Gamma_{\lambda}$ is the linear limb-darkening coefficient, $\lambda$ is the
passband, and $\phi$ denotes the angle between the line of sight toward the
source star and the normal to the source surface. The limb-darkening
coefficients are adopted from Claret (2000) considering the source type that
is determined based on the source locations in the color-magnitude diagrams.
We find that the source star of OGLE-2006-BLG-277 is a K-type giant star. For
OGLE-2012-BLG-0031, the lensed star is located in a very reddened region,
causing difficulties in precisely characterizing the star based on its color
and brightness. Nevertheless, it is found that the source is a giant. The
adopted coefficients are $\Gamma_{V}=0.74$, $\Gamma_{R}=0.64$ and
$\Gamma_{I}=0.53$ for both events. For data sets obtained without any filter,
we choose a mean value of the $R$ and $I$ band coefficients, i.e.,
$\Gamma_{N}=(\Gamma_{R}+\Gamma_{I})/2$, where the subscript “$N$” denotes that
no filter is used.
In Table 2, we list the best-fit solutions of the lensing parameters obtained
from standard modeling for the individual events. In Figures 1 and 2, we also
present the residuals from the fits. It is found that even though the fits
basically describe the main features of the observed light curves, there exist
important residuals that last throughout both events.
### 3.2. Higher-order Effects
Long-term residuals from the standard models suggest that one needs to
consider higher-order effects in order to better describe the lensing light
curves. Since it is known that such long-term residuals are caused by the
parallax and/or lens orbital effects, we conduct additional modeling
considering both higher-order effects.
To describe parallax effects, it is necessary to include two parameters
$\pi_{{\rm E},N}$ and $\pi_{{\rm E},E}$, which represent the two components of
the lens parallax vector $\mbox{\boldmath$\pi$}_{\rm E}$ projected onto the
sky along the north and east equatorial coordinates, respectively. The
magnitude of the parallax vector, ${\pi}_{\rm E}=(\pi_{{\rm
E},N}^{2}+\pi_{{\rm E},E}^{2})^{1/2}$, corresponds to the relative lens-source
parallax scaled to the Einstein radius of the lens, i.e. $\pi_{\rm E}=\pi_{\rm
rel}/\theta_{\rm E}$ (Gould, 2004). The direction of the parallax vector
corresponds to the relative lens-source motion in the frame of the Earth at a
reference time of the event. In our modeling, we use $t_{0}$ as the reference
time. Parallax effects cause the source motion relative to the lens to deviate
from rectilinear.
To first order approximation, the lens orbital motion is described by two
parameters $ds/dt$ and $d\alpha/dt$, which represent the change rates of the
normalized binary separation and the source trajectory angle, respectively
(Albrow et al., 2000; An et al., 2002). In addition to causing the relative
lens-source motion to deviate from rectilinear, the orbital effect causes
further deviation in lensing light curves by deforming the caustic over the
course of the event due to the change of the binary separation.
Figure 3.— Geometry of the best-fit models for OGLE-2006-BLG-277 (upper
panel) and OGLE-2012-BLG-0031 (lower panel). Small dots and closed solid
curves represent the lens positions and caustics at two different times
$t_{1}$ and $t_{2}$. Black solid curves with arrows represent source
trajectories. The size of the small empty circle at the tip of the arrow of
each source trajectory represents the source size. The abscissa and ordinate
are parallel with and perpendicular to the binary axis, respectively. All
lengths are normalized by the Einstein ring radius.
In Table 2, we list the results of modeling considering the higher-order
effects. For each event, we conduct 3 sets of additional modeling in which the
parallax effect and orbital effect are considered separately (“parallax only”
and “orbital only”) and both effects are simultaneously considered
(“orbit+parallax”). In the lower panels of Figures 1 and 2, we present the
residuals of the individual models. In Figure 3, we present the geometry of
the lens systems of the best-fit solutions, where the source trajectory with
respect to the lens components and the resulting caustics are shown. We note
that the relative lens positions and caustics vary in time due to the orbital
motion of the lens and thus we mark the positions at two different moments.
For both events, we find that the dominant second-order effect is the lens
orbital motion. The dominance of the orbital effect is evidenced by the fact
that the models considering only the orbital effect result in fits as good as
those considering both the parallax and orbital effects. It is found that the
consideration of orbital effects improves the fits by $\Delta\chi^{2}=1124.4$
and $158.4$ compared to the standard models of OGLE-2006-BLG-277 and
OGLE-2012-BLG-0031, respectively. On the other hand, the improvements by
additionally considering the parallax effect are merely $\Delta\chi^{2}=16.3$
and $2.5$ for the individual events.
Figure 4.— Distributions of $\chi^{2}$ in the space of the parallax
parameters $\pi_{{\rm E},E}$ and $\pi_{{\rm E},N}$ for OGLE-2006-BLG-277
(upper panels) and OGLE-2012-BLG-0031 (lower panels). For each event, the
distribution in the left panel is obtained from modeling considering only the
parallax effect, while the distribution in the right panel is constructed by
considering both the orbital and parallax effects. Different contours
correspond to $\Delta\chi^{2}<1$ (red), $4$ (yellow), $9$ (green), $16$ (sky
blue), $25$ (blue), and $36$ (purple), respectively.
To be noted is that parallax effects can mimic orbital effects to some extent
for both events. We find that the improvements of the fits by the parallax
effect are $\Delta\chi^{2}=841.6$ (cf. $\Delta\chi^{2}=1124.4$ improvement by
the orbital effect) and $150.1$ (cf. $\Delta\chi^{2}=158.4$ by the orbital
effect) for OGLE-2006-BLG-277 and OGLE-2012-BLG-0031, respectively. In
addition, the values of the lens parallax determined without considering
orbital effects substantially differ from those determined by considering
orbital effects. This can be seen in Figure 4 where we present $\chi^{2}$
distributions in the space of the parallax parameters. For OGLE-2006-BLG-277,
the measured lens parallax is $\pi_{\rm E}=1.13\pm 0.16$ when both parallax
and orbital effects are considered, while $\pi_{\rm E}=0.70\pm 0.05$ when only
the parallax effect is considered. For OGLE-2012-BLG-0031, modeling
considering only parallax effects results in a lens parallax $\pi_{\rm
E}=0.31\pm 0.08$, while the lens parallax is consistent with zero in 3$\sigma$
level in the model considering additional orbital effects. These facts imply
that orbital effects can masquerade as parallax effects and thus lens parallax
values measured based on modeling not considering orbital effects can result
in wrong values. This leads to wrong determinations of physical lens
parameters because masses and distances to lenses are determined from measured
values of the lens parallax.
It was pointed out by Batista et al. (2011) and Skowron et al. (2011) that the
parallax component perpendicular to the relative lens-source motion,
$\pi_{{\rm E},\bot}$, is strongly correlated with the orbital parameter
$d\alpha/dt$, causing a degeneracy between $\pi_{{\rm E},\bot}$ and
$d\alpha/dt$. They argued that this degeneracy occurs because the lens-source
motion in the direction perpendicular to the Sun-Earth axis induces deviations
in lensing light curves similar to those induced by the rotation of the
binary-lens axis. For both events OGLE-2006-BLG-277 and OGLE-2012-BLG-0031,
the direction of the relative lens-source motion is similar to east-west
direction, and thus $\pi_{{\rm E},\bot}\sim\pi_{{\rm E},N}$. According to this
degeneracy, the lens parallax vectors estimated by the “parallax only” and the
“orbit+parallax” models should result in similar values of $\pi_{{\rm E},E}$,
while values of $\pi_{{\rm E},N}$ can be widely different. However, both the
events analyzed in this work do not conform to the previous prediction. This
implies that the parallax-orbit degeneracy is much more complex than
previously thought, and thus it is essential to study the degeneracy in all
cases where higher-order effects are detected.
We determine the physical lens parameters based on the best-fit solutions
(orbit+parallax models). For this, we first determine the Einstein radius. The
Einstein radius is determined by $\theta_{\rm E}=\theta_{\ast}/\rho_{\ast}$,
where the normalized source radius $\rho_{\ast}$ is measured from the modeling
and the angular stellar radius is determined based on the source type. The
measured Einstein radius of the lens of OGLE-2006-BLG-277 is is $\theta_{\rm
E}=1.35\pm 0.11$ ${\rm mas}$. This corresponds to the relative lens-source
proper motion $\mu=\theta_{\rm E}/t_{\rm E}=13.0\pm 1.0$ ${\rm mas}$ ${\rm
yr^{-1}}$. With the measured mass ratio between the lens components, then the
masses of the individual lens components are $M_{1}=M_{\rm tot}/(1+q)=0.049\pm
0.014$ $M_{\odot}$ and $M_{2}=qM_{\rm tot}/(1+q)=0.097\pm 0.027$ $M_{\odot}$,
respectively. Therefore, the lens is composed of a low-mass star and a brown
dwarf. The distance to the lens is $D_{\rm L}=0.60\pm 0.14$ ${\rm kpc}$. The
close distance explains the relatively high proper motion ($13.0\pm 1.0$ ${\rm
mas}$ ${\rm yr^{-1}}$). With the physical parameters combined with orbital
parameters, we evaluate the ratio of tranverse kinetic to potential energy
$\left({\rm KE\over\rm PE}\right)_{\bot}={(r_{\bot}/{\rm AU})^{2}\over
8{\pi}^{2}(M_{\rm tot}/M_{\odot})}\left[\left({1\over s}{ds\over
dt}\right)^{2}+\left({d\alpha\over dt}\right)^{2}\right],$ (2)
where $r_{\bot}$ denotes the projected binary separation (Dong et al., 2009).
The ratio should obey $\rm(KE/PE)_{\bot}\leq KE/PE<1$ for kinetically stable
binary orbit. We find $\rm(KE/PE)_{\bot}=0.20\pm 0.04$. For
OGLE-2012-BLG-0031, it is difficult to determine the physical lens parameters
not only because the source type is uncertain but also because the lens
parallax is consistent with zero.
## 4\. Summary and Conclusion
We analyzed two binary-lens events OGLE-2006-BLG-277 and OGLE-2012-BLG-0031
for which the light curves exhibit significant residuals from standard binary-
lens models. From modeling considering higher-order effects, we found that the
residuals were greatly removed by considering the effect of the lens orbital
motion. We also found that parallax effects could mimic orbital effects to
some extent and the parallax values measured not considering the orbital
effect could result in dramatically different value from true values, and thus
wrong determinations of physical lens parameters. We also found that the lens
of OGLE-2006-BLG-277 was a binary composed of a low-mass star and a brown
dwarf companion.
Work by CH was supported by Creative Research Initiative Program
(2009-0081561) of National Research Foundation of Korea. AG was supported by
NSF grant AST 1103471. The OGLE project has received funding from the European
Research Council under the European Community’s Seventh Framework Programme
(FP7/2007-2013) / ERC grant agreement no. 246678 to AU. S.D. was supported
through a Ralgh E. and Doris M. Hansmann Membership at the IAS and NSF grant
AST-0807444. DMB, MD, MH, RAS and YT would like to thank the Qatar Foundation
for support from QNRF grant NPRP-09-476-1-078. Dr. David Warren gave financial
support to Mt. Canopus Observatory.
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|
arxiv-papers
| 2013-06-17T06:27:41 |
2024-09-04T02:49:46.587827
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "H. Park, A. Udalski, C. Han, A. Gould, J.-P. Beaulieu, Y. Tsapras, M.\n K. Szyma\\'nski, M. Kubiak, I. Soszy\\'nski, G. Pietrzy\\'nski, R. Poleski, K.\n Ulaczyk, P. Pietrukowicz, S. Koz{\\l}owski, J. Skowron, {\\L}. Wyrzykowski,\n J.-Y. Choi, D. L. Depoy, Subo Dong, B. S. Gaudi, K.-H. Hwang, Y. K. Jung, A.\n Kavka, C.-U. Lee, L. A. G. Monard, B.-G. Park, R. W. Pogge, I. Porritt, I.-G.\n Shin, J. C. Yee, M. D. Albrow, D. P. Bennett, J. A. R. Caldwell, A. Cassan,\n C. Coutures, D. Dominis, J. Donatowicz, P. Fouqu\\'e, J. Greenhill, M. Huber,\n U. G. J{\\o}rgensen, S. Kane, D. Kubas, J. -B. Marquette, J. Menzies, C.\n Pitrou, K. R. Pollard, K. C. Sahu, J. Wambsganss, A. Williams, M. Zub, A.\n Allan, D. M. Bramich, P. Browne, M. Dominik, K. Horne, M. Hundertmark, N.\n Kains, C. Snodgrass, I. A. Steele and R. A. Street",
"submitter": "Cheongho Han",
"url": "https://arxiv.org/abs/1306.3744"
}
|
1306.3811
|
# Suppression of the Rayleigh-Plateau instability on a vertical fibre coated
with wormlike micelle solutions111Accepted in Soft Matter.
DOI:10.1039/C3SM27940E
F. Boulogne1,2, M.A. Fardin3,4, S. Lerouge3,4,
L. Pauchard1, F. Giorgiutti-Dauphiné1
###### Abstract
We report on the Rayleigh-Plateau instability in films of giant micelles
solutions coating a vertical fibre. We observe that the dynamics of thin films
coating the fibre could be very different from the Newtonian or standard Non-
Newtonian cases. By varying the concentration of the components of the
solutions and depending on the film thickness, we show for the first time that
the Rayleigh-Plateau instability can be stabilized using surfactant solutions.
Using global rheology and optical visualisations, we show that the development
of shear-induced structures is required to stabilize the micellar film along
the fibre. Assuming that the viscoelastic properties of the shear-induced
state can be described by a simple model, we suggest that, in addition to the
presence of shear-induced structures, the latter must have an elastic modulus
greater than a critical value evaluated from a linear stability analysis.
Finally, our analysis provides a way of estimating the bulk elasticity of the
shear-induced state.
_1 UPMC Univ Paris 06, Univ Paris-Sud, CNRS, F-91405. Lab FAST, Bat 502,
Campus Univ, Orsay, F-91405, France.
2 [email protected]
3 Laboratoire Matière et Systèmes Complexes, CNRS UMR 7057, Université Paris
Diderot, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cédex 13, France.
4 The Academy of Bradylogists _
## 1 Introduction
When a liquid film is coating a fibre, it undergoes spatial thickness
variations. The mechanism of this instability has been understood by Lord
Rayleigh in the 19th century[33]. Due to the energetic cost of free surfaces
(surface tension), the liquid tends to minimize its surface area by breaking a
cylinder into a serie of regularly spaced droplets. This instability can be
observed on free falling jets and it underlies many natural phenomena, like
the dew drops on coweb [7]. The coating of fibres has been extensively studied
by Quéré et al. in the situation where the fibre is drawn out of a bath [31].
They compared their measurements with theoretical predictions from Landau,
Levich and Derjaguin on the film thickness [23, 12].
In the case of a Newtonian fluid flowing down a vertical fibre, flow regimes
have been depicted as a function of physical parameters. For high flow rates
and/or fibre radii, the flow is dominated by inertia and the nature of the
instability is convective [14]. On the contrary, when the fibre size is
smaller than the capillary length and for creeping flow (low Reynolds number),
the instability is absolute and the physics is dominated by surface tension
[14]. In such absolute regimes, we recently studied the flow of polymer
solutions [6]. We investigated experimentally the influence of two non-
Newtonian properties, shear-thinning effect and first normal stress
difference, on the growth rate of the instability and on the morphology of the
drops. The pattern is globally the same as in the case of Newtonian fluids:
the first normal stress difference just tends to slightly decrease the growth
rate and to smooth the drop shape.
Beside polymers, a broad variety of molecules assemble to form non-Newtonian
fluids. For instance, it is well-known that above a Critical Micellar
Concentration (CMC), surfactant molecules can self-assemble to form aggregates
called micelles. The size and the shape of the micelles depend on the
structure of the surfactant molecule, on the surfactant concentration and on
the presence of additives like simple or organic salts [21]. In some range of
parameters, the micelles are giant worms often called ”living polymers”
because thay can entangle like polymers but they can also break and fuse
continuously. Solutions of worm-like micelles present remarkable rheological
properties such as shear-thickening or shear-banding effects, extensively
studied by theoreticians and experimentalists [2, 8]. These non-linear
properties are often associated with the development of out of equilibrium
structures induced by the shear flow. These shear-induced structures present
strong viscoelastic properties leading to specific behaviors like the
oscillation of a falling sphere [22] or the incomplete retraction of a
filament after a pinch-off [36]. These examples highlight the consequence of
the viscoelastic properties of the flow-induced phases.
In this paper, we investigate the flow, at low Reynolds number, along a
vertical fibre of semi-dilute giant micelles solutions. By varying the
concentration of the chemical compounds of the solutions and the thickness of
the film along the fibre, we observe different morphologies for the micellar
film. In some conditions, the film exhibits the expected Rayleigh-Plateau
instability, as it has been observed so far, in Newtonian fluids or
viscoelastic polymer solutions while for other conditions, remarkably, the
micellar film remains stable. Using global rheology and optical
visualisations, we demonstrate that the stabilisation process along the fibre
is connected with the development, above a characteristic shear stress, of
shear-induced structures. We also show that the development of shear-induced
structures is necessary but not sufficient to stabilize the micellar film.
Using a simple viscoelastic model (Kelvin-Voigt) to describe the viscoelastic
properties of the shear-induced state, we suggest that the elastic modulus
characterizing the shear-induced structures has to be greater than a critical
value provided by the linear stability analysis.
The paper is organized as follows. In Section 2, we present the experimental
setup and the chemical system. In Section 3, we report different flow regimes
depending on surfactant and/or salt concentrations and film thicknesses. In
Section 4, we present the rheological behavior of our micellar systems and we
visualize the material under shear to confirm the presence of shear-induced
structures. The next two Sections are devoted to the conditions required to
stabilize the micellar film on the fibre. Section 5 focus on the stress
applied to the film by gravity and its relation with the rheological behaviour
of the solution and Section 6 studies the role of the shear-induced film
elasticity on the Rayleigh-Plateau instability.
## 2 Experimental details
### 2.1 Experimental setups
Figure 1: (a) Notations and schematic view of the vertical fibre and the
injection device. (b-d) Pictures illustrating the different morphologies of
the film flowing along the fibre: (b) Unstable morphology which corresponds to
drops sliding on a quasi static liquid film. (c) Stable film. (d) Film
morphology corresponding to a gel-like block moving down the fibre. The
diameter of the fibre is $0.56$ mm.
The experimental setup used for the study of the Rayleigh-Plateau instability
is depicted in Figure 1(a). An upper tank, of diameter $14$ cm, is connected
to a valve. This valve is composed of two axisymmetric cones. The adjustment
of the gap between these cones controls the flow rate i.e. the film thickness
on the fibre. Guided by a nozzle, the liquid flows along a vertical nylon
fibre. The fibre is about $60$ cm long and has a radius $R=0.28$ mm. Vertical
position and centering of the nozzle are crucial to ensure an axisymmetric
coating. This position is adjusted using an optical support with a precision
of 2.4 arc sec (Model U200-A2K, Newport). Observations are done with a
telecentric lens and a high speed camera (about $500$ images/s) delivering a
resolution of $0.024$ mm/pixel (Fig. 1(b-c)).
The rheological properties of the samples have been characterized using a
stress-controlled rheometer (Anton Paar, MCR 501) at $20.00\pm 0.01^{\circ}$C.
The temperature was adjusted with a Peltier plate. We used a cone and plate
geometry (radius $49.988$mm, angle $0.484^{\circ}$). A solvent trap was used
to limit evaporation during the measurements. Rheological protocol consists in
applying stress sweep, the stress value being imposed during 1000 s at each
step. The flow curve is obtained by averaging the shear rate response over the
last 100 s of each step.
The global rheological measurements have been complemented by rheo-optical
observations of the flow performed in a Taylor-Couette (TC) flow geometry
(gap: $1.13$ mm, length: $4$ cm), adapted to the shaft of a stress-controlled
rheometer (Anton Paar MCR 301). The inner cylinder is rotating while the outer
transparent cylinder is fixed in the laboratory frame. Direct visualizations
were made in the plane velocity gradient/vorticity (i.e. the plane made by the
radial direction and the cylinders axis), the gap of the TC cell being
illuminated with white light. Images of the gap were collected using a CCD
camera. The rheological signal was recorded simultaneously, allowing direct
correlation with optical visualizations [25, 16]. Note that similar rheograms
were obtained with TC and cone and plate geometries.
### 2.2 Materials
We focus on aqueous micellar solutions made of cetyltrimethylammonium bromide
(CTAB) and sodium salicylate (NaSal) purchased from Sigma-Aldrich and used
without further purification. NaSal is an organic salt, sometimes designated
as a co-surfactant since it participates to the micellar structure by taking
place in-between the polar head groups of surfactant molecules [32]. The
addition of this salt also screens repulsive electrostatic interactions
between CTA+ groups and it reduces the spontaneous curvature of the system
[26]. As a result, the micellar uniaxial growth is promoted because of the
large energetic cost of hemispherical endcaps. All the solutions investigated
range in the semi-dilute regime. For the range of concentration chosen here,
the micelles are locally cylindrical and slightly entangled [2].
To prepare chemical solutions, we mixed in pure water (milliQ quality,
resistivity: $18$ M$\Omega$.cm) the weight percentages of NaSal and CTAB until
a complete dissolution. Samples were stored at rest at $23^{\circ}$C in
darkness to avoid any degradation. The solutions were allowed to reach
equilibrium for at least three days in these conditions before experiments
were performed. We carefully checked the homogeneity of solutions before any
measurement. All measurements along the fibre were carried out at $T=20.0\pm
0.5^{\circ}$C.
The equilibrium surface tension of the solutions studied in this paper is
$36.0\pm 0.5$ mN/m (Wilhelmy plate method with a Krüss tensiometer), a value
consistent with measurements reported in the literature for solutions above
the critical micellar concentration [11].
## 3 Morphologies of the liquid film along a vertical fibre
We focused on low Reynolds number flows ($Re\lesssim 1$) dominated by surface
tension ($R<l_{c}=\sqrt{\frac{\gamma}{\rho g}}=1.9$ mm), for which the flow is
absolutely unstable [14]. We studied the flow along the vertical fibre of a
large set of wormlike micelles solutions. Depending at once on the surfactant
concentration [CTAB], the salt concentration [NaSal] and the film thickness
$h$, we observe three possible morphologies of the film along the fibre as
displayed in Fig. 1(b-d):
* •
The film can be unstable with respect to the Rayleigh-Plateau instability. The
associated morphology corresponds to drops sliding on a quasi-static liquid
film (Fig. 1(b)).
* •
The film can be stable and in this case, the associated morphology corresponds
to a flat film (Fig. 1(c)).
* •
The film can be subjected to some kind of fracture leading to separate blocks
moving down the fiber (Fig. 1(d)). The morphology is essentially flat but the
film is discontinuous.
In order to illustrate the effect of the various parameters on the flow along
the fibre, we construct a flow-phase diagram (see Fig. 2a) in the plane
([CTAB], [NaSal]), allowing to identify different domains.
At low NaSal concentrations, the apparent viscosity is low ($\lesssim 50$
mPa.s) which corresponds to inertial regimes [14]. At high NaSal
concentrations, due to a high zero shear viscosity ($>100$ Pa.s), the fluid
hardly or could not flow though our experimental device (”No flow” domain).
These two domains are excluded from the present study which, focuses on the
intermediate range of NaSal concentrations, typically between $0.2$ and $0.5$
wt.%. In this intermediate range, three domains, denoted by $\cal{A}$,
$\cal{B}$ and $\cal{C}$, can be distinguished according to the sequence of
morphologies observed as a function of the film thickness.
Figure 2: (a): Flow-phase diagram showing domains depending on surfactant
(CTAB) and cosurfactant (NaSal) concentrations. Five domains are identified:
Inertial (green $+$), ”No flow” (magenta $\times$), $\cal{A}$ (blue
$\triangle$), $\cal{B}$ (red $\Box$) and $\cal{C}$ (orange $\circ$). (b):
Schematic representations of the different sequences of film morphologies as a
function of the film thickness associated with the different domains
$\cal{A}$, $\cal{B}$ and $\cal{C}$ in the flow-phase diagram.
* •
Domain $\cal{A}$
In this domain, the micellar solutions behave as usual Newtonian fluids [14]
or polymer solutions [6] for all the film thicknesses accessible with our
experiment: the film is first flat and then, the Rayleigh-Plateau instability
develops (Fig. 1(b)).
* •
Domain $\cal{B}$
In this domain, regardless of the film thickness, the micellar solutions do
not exhibit the Rayleigh-Plateau instability. For sufficiently small
thicknesses (i.e. low flow rates), the film remains flat and continuous over
the whole fibre extent (about $60$ cm). From this stable morphology, an
increase of the film thickness leads to a breaking of the liquid thread into
blocks of a few centimeters long, sliding along the fibre (Fig. 1 (d)). Note
that, visually the blocks seem to have a gel-like character with a large
elasticity, in contrast to the solutions at rest. Similar viscoelastic threads
were observed in the case of pinch-off and filament retraction in extensional
flow of wormlike micelles [36, 3].
* •
Domain $\cal{C}$
In between the two domains $\cal{A}$ and $\cal{B}$, the domain $\cal{C}$ is
characterized by the existence of the three morphologies (see Fig. 1(b-d)).
For small thicknesses, the films are destabilized by the Rayleigh-Plateau
instability with patterns similar to those described in domain $\cal{A}$. As
its thickness is increased, the film adopts a stable morphology corresponding
to a flat continuous liquid thread. Finally, for sufficiently large
thicknesses, the flat and continuous liquid thread breaks into gel-like blocks
as described in domain $\cal{B}$.
Figure 3: From a vertical fibre where a stable film is flowing down, we tilt
the fibre horizontaly leading to the development of the Rayleigh-Plateau
instability.
The different sequences of morphologies described above and observed
respectively in the domains $\cal{A}$, $\cal{B}$ and $\cal{C}$ are schematized
in Fig. 2(b).
When a stable morphology is observed, we have noticed that if we tilt the
fibre horizontally, after a few seconds, drops can appear (figure 3). This
observation suggests that the flow under the action of gravity is necessary to
stabilize the film. Particularly, this shows that the flow properties of the
micellar solutions are involved.
Note that this effect is different from the stabilizing process depicted by
Quéré [30] with Newtonian fluids. In his experiments, stable films flowing
down a vertical fibre were obtained by saturating the Rayleigh-Plateau
instability thanks to the advection of the flows. This saturation (i.e. flat
film) occurs if the film thickness is smaller than
$h_{c}=\frac{R^{3}}{l_{c}^{2}}$. In our experiment, $h_{c}=6\mu$m while
$h>200\mu$m, and the flow is absolutely unstable .
## 4 Rheo-optical properties under simple shear flow
In contrast to the free jets dominated by extensionnal flow [9, 15, 4], the
flow along the fibre is dominated by shear. We thus performed rheological
experiments under simple shear flow in order to characterize the rheological
properties of the micellar solutions used in this study. The measurement of
the flow properties is combined with direct optical visualisations of the
samples illuminated with white light in order to detect possible changes in
the structure of the micellar fluid. In the different domains $\cal{A}$,
$\cal{B}$ and $\cal{C}$ identified in the flow-phase diagram, we observe the
same type of flow behaviour. Figure 4(a) displays a representative flow curve
gathered from a sample belonging to domain $\cal{C}$ and obtained in stress-
controlled mode. The choice of this mode has been motivated by the fact that
the maximum stress in the liquid flowing down the fibre can be easily
estimated (See Section 5). The response of the samples to simple shear flow is
highly nonlinear, with successive transitions, from shear-thickening to
different degrees of shear-thinning. Such complex evolution of the shear
stress as a function of the shear rate has already been observed for dilute
micellar solutions, well-known to exhibit a shear-thickening transition [20,
19]. Following Refs. [20, 19], we divide the flow curve into four distinct
regimes, defined by “critical” shear stresses, denoted $\sigma_{c}$,
$\sigma_{s}$ and $\sigma_{f}$.
* •
Regime I ($\sigma<\sigma_{c}$)
This regime corresponds to the primary response of the micellar solution to
exceedingly low shear stresses. For dilute systems, we expect a linear
increase of the shear stress as a function of the shear rate, associated with
a Newtonian behaviour while for semi-dilute systems, a slight decrease of the
viscosity as a function of the shear rate is usually observed, due to partial
alignment of the micelles by the flow [18, 20]. Unfortunately, reaching this
regime requires shear stresses that cannot be imposed by the rheometer, what
leads to the lack of experimental data points in Fig. 4. Note that in these
conditions, $\sigma_{c}$ is only roughly defined.
* •
Regime II ($\sigma_{c}<\sigma<\sigma_{s}$)
This regime is characterized by a shear-thickening transition, the apparent
viscosity of the material increasing with the shear rate (see Fig. 4). This
transition is associated with a re-entrant behavior of the flow curve since
the shear rate first decreases and then increases with the shear stress in
this regime. Due to the re-entrant character of the transition, this regime
can only be observed at imposed stress. The upper boundary of this regime is
noted $\sigma_{s}$ and corresponds to the shear stress for which the apparent
viscosity reaches a maximum. Images of the sample in the shear-thickening
regime do not exihibit any changes compared to the situation at rest. The gap
of the TC cell appears homogeneous suggesting that the changes in the
structure of the fluid associated with the shear-thickening transition, occur
at sub-micronic scale.
* •
Regime III ($\sigma_{s}<\sigma<\sigma_{f}$)
In this range of applied stresses, the flow becomes shear-thinning and direct
visualisations show that the gap of the TC cell remains homogeneous (see Fig.
6 (a-c)).
* •
Regime IV ($\sigma>\sigma_{f}$)
This regime is characterized by a strong degree of shear-thinning associated
with the existence of a stress plateau, the onset of which is denoted by a
“critical” shear stress $\sigma_{f}$. The stress plateau generates a jump in
the shear rate resulting in a sharp decrease of the apparent viscosity. The
stress $\sigma_{f}$ is determined by increasing the shear stress step by step
with an appropriate sampling as illustrated in Fig. 5. The plateau is
characterized by a large jump in the shear rate at imposed stress, or
equivalently a strong increase of the shear rate versus time at fixed stress
(see inset in Fig. 5) and a large increase of the first normal stress
difference $N_{1}$, indicative of nonlinear viscolelastic properties of the
material under shear. Note that below or above $\sigma_{f}$, variations of the
shear rate are lower than $10$% over the duration of the measure ($1000$ s).
For the sample in Fig. 5, we estimate $\sigma_{f}=4.1\pm 0.3$ Pa. In this
regime, direct visualisations in the TC geometry reveal a drastic change in
the structure of the fluid (see Fig.6 (d-e)). A fraction of the sample becomes
slightly turbid leading to variations in the refraction index observable with
visible light. These observations suggest that, for $\sigma>\sigma_{f}$, a new
phase characterized by a length scale in the order of a few microns nucleates
in the gap of the TC device [35]. We observed that the nucleation occurs from
the edges of the cylinders and progressively extends along the inner cylinder,
the growth being favoured at the bottom of the cell which has a conic shape.
At fixed stress, the turbidity fluctuations observed on the pictures evolve in
time with a complex dynamics. The proportion of the induced phase increases
with the imposed shear stress, and for sufficiently high shear stresses, the
induced phase invades the whole gap (see Fig.6 (f-g)). This scenario is
reminiscent of the shear-banding transition even if there is no clear evidence
for an organisation into two bands separated by a well-defined interface [24].
By monitoring the shear-rate as a function of time, we noticed that the
appearance of the induced phase is correlated with the presence of a kink in
the $\dot{\gamma}(t)$ curve which appears sooner as the stress increases. This
kink is also observed in the cone-plate geometry for $\sigma>\sigma_{f}$ (See
inset in Figure 5) and is then the signature for the appearance of the induced
phase.
Figure 4: (a): Shear stress $\sigma$ and (b): apparent viscosity $\eta$ as a
function of the shear rate $\dot{\gamma}$ for a solution of CTAB ($2.55$ wt%)
and NaSal ($0.32$wt.%). The temperature is T=$20.00\pm 0.01^{\circ}$C. The
rheological protocol consists in stress sweep experiment: at each step, the
shear stress is kept fixed during $1000$ s. The resulting shear rate
$\dot{\gamma}(t)$ reaches a stationnary value after a time period that depends
on the applied shear stress. The flow curve is obtained by averaging the
measured shear rate over the last $100$ s of each step. $\sigma_{c}$,
$\sigma_{s}$ and $\sigma_{f}$ denotes characteristic shear stresses that
delimit different regimes in the flow curve. Figure 5: Determination of the
stress plateau value ($\sigma_{f}$). Stress $\sigma$ (red $+$) and first
normal stress difference $N_{1}$ (green $\times$) versus shear rate for
$2.55$wt.% CTAB and $0.32$wt.% NaSal (T=$20.00\pm 0.01^{\circ}$C). At each
step, the shear stress is imposed during $1000$ s and the measured shear rate
results from the average over the last $100$ s of each step. Inset: Time
evolution of the shear rate $\dot{\gamma}$ for a fresh sample at $\sigma=4.5$
Pa. The black arrow indicates the signature linked to the apparition of SIS.
Beyond this purely phenomenological description, our observations deserve
further comments. As mentioned above, similar complex succession of
rheological transitions in the flow curve have already been observed in dilute
micellar solutions of TTAA/NaSal [20, 19]. However, the direct observations
reported by the authors largely differ from the present study: Indeed, in the
shear-thickening regime (regime II), the fluid phase of regime I was found to
coexist with a viscous phase of gel-like shear-induced structures while in
regime III, the induced gel-like phase invaded the whole gap of the flow
geometry. Furthermore, the rheological response in regime III was also
different, the apparent viscosity being constant, while in our case the system
is shear-thinning. This behaviour was ascribed to the existence of a plug flow
with slip at the walls. Finally, above $\sigma_{f}$, fracture of the gel-like
phase followed by flow instabilities were observed.
Here we do not observe any phase coexistence in regime II. However, this does
not rule out the fact that the shear-thickening transition is associated with
a change of the structure of the system but rather suggests that this change
of the structure occurs at sub-micronic scale which prevents direct
visualisation. Our situation could correspond to the minimal scenario for the
shear-thickening transition in micellar systems, namely the uniaxial growth of
the micelles [1, 29, 27, 24]. On the other hand, we clearly observe a phase
coexistence in regime IV, above $\sigma_{f}$, the induced phase being
characterized by a large first-normal stress difference. Note that, at
equilibrium, the samples under investigation are only weakly entangled: their
concentration is associated with the very beginning of the semi-dilute regime,
in a “transition” range where the rheological behaviour is between pure shear-
thickening of dilute rod-like micellar systems, and pure shear-banding of
strongly entangled wormlike micelles [17, 24]. In this context, the shear-
banding-like behaviour observed in the present study is likely to correspond
to a transition from the state induced during the shear-thickening regime
(regime II) towards another induced “phase” structured at larger scale (regime
IV).
Note that in the literature on wormlike micelles, the shear-induced state
associated with the shear-thickening transition was called SIS for shear-
induced structure [28], or SIP for shear-induced phase [5] interchangeably.
Recent studies [37] have also used FISP to denote irreversible “flow-induced
structured phase” obtained in extensional flow. However, in the present paper,
we make no distinction between the expressions “shear-induced structures” and
“shear-induced phase” and in the following we will refer to SIS as the
reversible structures (or “phase”) induced at larger scale in regime IV.
Figure 6: Views of the gap of a Couette cell in the plane velocity
gradient/vorticity (noted $(\vec{\nabla}\vec{v},\vec{\nabla}\times\vec{v})$)
for different applied shear stresses. Each image is taken after a time $t\sim
1000$ s. The solution is composed of $2.55$wt.% CTAB and $0.32$wt.% NaSal. The
top and bottom sides correspond to the outer and inner wall respectively. For
$\sigma>\sigma_{f}=4$ Pa, structures appear near the inner wall. Arrows
indicate some turbidity fluctuations indicative of SIS with micronic
characteristic length scale.
## 5 Stress induced by gravity and development of shear-induced structures
As described in section 3, the film morphology depends on the film thickness
$h$. Our working hypothesis is that the transition between unstable to stable
morphologies along the fibre might be linked to the presence of the SIS. To
clarify and to go further on that point, we will focus on solutions in domain
$\cal{C}$. This choice is motivated by the fact that solutions which belong to
this domain present all three morphologies. We first study the influence of
the CTAB concentration. We use a set of chemical solutions for which CTAB
concentration is in the range $[1.8,3.2]$wt.% keeping
$[\textrm{NaSal}]=0.32$wt.% constant. For each solution, we keep increasing
the flow rate. The results are reported in figure 7 where we observe
successively unstable, stable and block morphologies as the film thickness
increases.
Figure 7: Film morphologies for different film thicknesses and various
surfactant concentrations ($[\textrm{NaSal}]=0.32$wt.%). The black line is a
fit of film thicknesses $h_{f}=\sigma_{f}/(\rho g)$ ($\square$) related to
stress plateau values. It separates unstable and stable flow regimes.
To explain this transition between stable and unstable morphologies, and since
the rheology is controlled by the applied shear stress, we compare the stress
plateau $\sigma_{f}$ extracted from the rheological curves to the
gravitational stress $\sigma_{g}$ applied on the film. This stress
$\sigma_{g}=\rho gh$ ranges between 2 and 8 Pa in our experiments and is then
comparable to the value of the stress plateau $\sigma_{f}$. We define the film
thickness $h_{f}$ related to the stress plateau $\sigma_{f}$ as
$h_{f}=\sigma_{f}/(\rho g)$.
Figure 7 illustrates the succession of film morphologies as a function of the
film thickness $h$ for different surfactant concentrations and a fixed salt
concentration. We also report, for each sample, the value
$h_{f}=\sigma_{f}/(\rho g)$ gathered from the rheological experiments.
Remarkably, the unstable-stable transition is found to occur as $h\simeq
h_{f}$. Thus, we can conclude that the condition $\sigma_{g}>\sigma_{f}$ is
necessary to stabilize the film. This suggests that the stabilization of the
film is connected to the presence of the SIS as $\sigma_{f}$ is the
characteristic stress value for the development of these structures.
Consequently, the SIS are able to be formed, before the Rayleigh-Plateau
instability acts, if $\sigma>\sigma_{f}$. This fixes the first condition to
stabilize the film. As we will see in the next section, this condition is
necessary but not sufficient.
## 6 Effect of the SIS elasticity on the instability growth rate
In this section, we study a second set of chemical solutions in order to
complete our understanding. The question we attempt to answer is: why do films
become suddenly always unstable if the concentrations are slightly changed
near the border between domains $\cal{A}$ and $\cal{C}$? To cross these two
domains, we work with a constant CTAB concentration
($[\textrm{CTAB}]=2.55$wt.%) while NaSal concentrations vary in the range
$[0.25,0.4]$wt.%. Figure 8 displays the film morphology as a function of the
thickness for different salt concentrations covering domains $\cal{A}$ and
$\cal{C}$ at fixed surfactant concentration. Based on rheological data, the
thickness of the film $\sigma_{f}/(\rho g)$ beyond which the SIS are likely to
develop is also reported. A good agreement is again observed in domain
$\cal{C}$ between the expected thickness computed from rheological results and
the effective thickness above which the films are stable. However, there is a
discrepancy for solutions in domain $\cal{A}$ since films thicker than $h_{f}$
are unstable. According to our analysis, the SIS still develop but they do not
stabilize the film.
Figure 8: Film morphologies for different film thicknesses and various salt
concentrations ($[\textrm{CTAB}]=2.55$wt.%). The black line is a fit of film
thicknesses $h_{f}=\sigma_{f}/(\rho g)$ related to stress plateau values.
To understand these observations, we focus on two effects: the surface tension
which is responsible for the instability and the bulk elasticity which tends
to slow down the film deformations.
It is worth noting that the constitutive relation describing the SIS is still
unknown. Since SIS present strong elastic properties, we attempt a description
of the SIS by a simple Kelvin-Voigt model allowing analytic calculations [34,
10]. The Kelvin-Voigt model takes into account the viscous dissipation at
short time scales and elasticity at long time scales. The stress tensor
$\sigma_{ij}$ is function of two parameters: viscosity $\eta_{0}$ and elastic
modulus $G_{0}$:
$\sigma_{ij}(t)=G_{0}\epsilon_{ij}+\eta_{0}\dot{\epsilon}_{ij}$ (1)
where $\epsilon_{ij}$ is the strain and the dot represents the time
derivative.
To obtain the dispersion relation for this constitutive equation, we perform a
linear stability calculation. The Fourier transform of the expression (1)
reduces the equation to
$\sigma_{ij}(\omega)=\eta(\omega)\dot{\epsilon}_{ij}(\omega)$ (2)
where $\omega$ is a complex frequency and with
$\eta(\omega)=\frac{iG_{0}}{\omega}+\eta_{0}$.
Experimentally, we observe that the film thicknesses $h(z)$ and the fibre
radius $R$ have the same order of magnitude. Under this consideration, we can
calculate the dispersion relation in our geometry[13].
In the cylindrical reference frame $(r,\theta,z)$ (See figure 1(a)), the
$z$-component of the momentum balance, taking into account the lubrication
approximation, becomes:
$\rho g-\partial_{z}p+\frac{\eta}{r}\partial_{r}(r\partial_{r}u)=0$ (3)
where $u$ is the fluid velocity along the fibre ($z$ direction). The boundary
conditions are: no-slip on the fibre ($u(r=R)=0$) and zero tangential stress
at the liquid-air interface ($\partial_{r}u(r=R+h)=0$). The pressure gradient
is caused by surface curvatures and it is expressed from Laplace’s law:
$\partial_{z}p=-\gamma\left(\frac{\partial_{z}h}{(R+h)^{2}}+\partial_{zzz}h\right)$
(assuming $\partial_{z}h\ll 1$). Solving this equation, the fluid velocity
profile is $u(r)=\frac{\partial_{z}p-\rho
g}{4\eta}\left[(r^{2}-R^{2})-2(R+h)^{2}\ln\left(\frac{r}{R}\right)\right]$.
Incompressibility leads to:
$\frac{\partial h}{\partial t}+\frac{\partial q}{\partial z}=0$ (4)
where the flow rate per unit length is
$q=\frac{1}{2\pi(R+h)}\int_{R}^{R+h}u(r)2\pi rdr$.
To conduct a linear stability analysis, we develop the film thickness as
$h=h_{0}+h_{1}e^{i(kz-\omega t)}$ where $k$ is the real wave number. Thus, we
derive the following dispersion relation
$\omega=k\frac{\rho gR^{2}\psi(\alpha)}{16\eta(\omega)}-i\frac{\gamma
Rh_{0}^{3}\phi(\alpha)}{3\eta(\omega)(R+h_{0})}\left(k^{4}-\frac{k^{2}}{(R+h_{0})^{2}}\right)$
(5)
with $\alpha=h_{0}/R$. Two dimensionless functions reflect the geometry:
$\psi(x)=\frac{-x(2+x)(6+5x(2+x))+12(1+x)^{4}\ln(1+x)}{(1+x)^{2}}$ and
$\phi(x)=\frac{3(4(x+1)^{4}\ln(x+1)-x(x+2)(3x(x+2)+2))}{16x^{3}}$
The imaginary part of $\omega$ corresponds to the instability growth rate
noted $\Omega(k)$:
$\Omega(k)=\frac{-\gamma
Rh_{0}^{3}\phi(\alpha)}{3\eta_{0}(R+h_{0})}\left(k^{4}-\frac{k^{2}}{(R+h_{0})^{2}}\right)-\frac{G_{0}}{\eta_{0}}$
(6)
For $\Omega(k)<0$, the system is stable, whereas for $\Omega(k)>0$ the system
is unstable. If $G_{0}=0$, the solution for Newtonian fluids is recovered. The
evolution of the growth rate with $k$ is plotted in figure 9.
We observe that the bulk elasticity ($G_{0}$) plays a stabilizing effect on
the film as illustrated in figure 9: the marginal stability curve is shifted
to the negative $\Omega$. If $\textrm{max}(\Omega)=\Omega(k_{max})<0$ (blue
curve in figure 9), all modes are damped by the elasticity resulting in a
stable film.
Figure 9: Growth rate of the instability $\Omega$ vs the wave number $k$ from
the Kelvin-Voigt model illustrating the stabilizing effect of the bulk
elasticity $G_{0}$. The case $G_{0}=0$ (in green) is the marginal stability
curve for Newtonian fluids (viscosity $\eta_{0}$). As $G_{0}$ increases, the
marginal curves (in blue) is shifted to the negative growth rate domain.
Note that, in this model, the wave number for the maximum of the growth rate
is $k_{max}=\frac{1}{\sqrt{2}(R+h_{0})}$ and is independent of the elastic
modulus.
Solving the condition $\Omega(k_{max})=0$ defines a critical elastic modulus
$G_{0}^{c}$:
$G_{0}^{c}(h)=\frac{\gamma h^{3}R\phi(h/R)}{12(R+h)^{5}}$ (7)
Experimentally, we can estimate $G_{0}^{c}$ from the border between domains
$\cal{A}$ and $\cal{C}$. From the data presented in figure 8, we can evaluate
for $[\textrm{CTAB}]\simeq 2.55\textrm{wt.}\%\simeq 70$mM,
$[\textrm{NaSal}]\simeq 0.33\textrm{wt.}\%\simeq 20$mM: $G_{0}\simeq
G_{0}^{c}(h_{f})=1.0$ Pa (with $h=h_{f}=0.4$ mm).
Thus the bulk elasticity of the SIS has to be sufficient to shift the curve of
the growth rate to the negative values, in order to stabilize the film. To
summarize, we have shown that the elastic shear induced structures are
developed in our flow if $\rho gh>\sigma_{f}$. If this condition is fulfilled,
we expect a stable film only if the bulk elasticity is greater than
$G_{0}^{c}$. As the result, we can explain the apparent discrepancy of the
unstable region in figure 8. From the transient zone to the unstable one (i.e.
increasing salt concentration), the bulk elasticity decreases such as
$G_{0}<G_{0}^{c}$.
The value $G_{0}^{c}=1$ Pa can be compared to oscillatory shear flow tests. As
suggested by the flow curve in figure 4, the linear regime could not be
reached with our rheometer. The first test consists in the measurement of the
elastic modulus versus time for a shear stress amplitude of $5\times 10^{-4}$
Pa at a frequency $f=1$Hz. The elastic modulus increases in time from $\sim
10^{-2}$ Pa to a few Pascal ($[\textrm{CTAB}]=2.55$ wt.% and
$[\textrm{NaSal}]=0.32$ wt.%). In a second test, a constant shear stress
$\sigma>\sigma_{f}$ is applied to the sample followed by an oscillating stress
of small amplitude. Even if the SIS has time to partially relax, such
experiments can provide a reasonnable order of magnitude of the moduli
($G^{\prime}$ and $G"$) of the SIS at high angular velocities. We found a
value for the elastic modulus around $1.5$ Pa, consistent with the value
$G_{0}^{c}=1$ Pa gathered from the flow along the fibre and considering that
the viscoelastic properties of the SIS can be modelled by a Kelvin-Voigt
model. Our results suggest that the elastic modulus of the solutions under
shear is larger than the one for the solution at rest and that the value
required to stabilize the flow could be deduced from the Kelvin-Voigt model.
## 7 Conclusion
In this paper, we studied the flow of CTAB/NaSal solutions on a vertical
fibre. Structure modifications occuring from the chemical composition and the
shear flow provides a large variety of flowing regimes. The flow on the fiber
can actually be unstable and similar to the flow of newtonian fluids: the
capillary driven Rayleigh-Plateau instability produces drops sliding on the
fibre. For another set of solutions, the film can stay uniform along the fiber
provided the film thickness satisfies $h>h_{f}=\sigma_{f}/(\rho g)$. From
rheo-optical measurements, we found that the condition for stabilizing the
flow is that the characteristic stress on the film should be larger than a
critical stress $\sigma_{f}$. This critical stress has been identified as the
onset of emergence of micronic structures induced by the shear-flow, i.e.,
shear-induced structures (SIS). A last flowing regime is observed for high
film thicknesses and in a given range of concentrations: in this last regime,
the film breaks in a series of gel-like blocks sliding along the fibre.
Our analysis suggests that the bulk elasticity of the SIS is responsible for
the decrease of the instability growth rate until inhibited the Rayleigh-
Plateau instability for a negative growth rate. Two conditions are required to
prevent the destabilization of the film: the presence of the SIS and a
sufficient elasticity of theses structures to inhibit the instability driven
by the surface tension. A linear analysis of stability assuming a Kelvin-Voigt
model for SIS, evidences a stable flow provided the elastic modulus of the SIS
is higher than a critical value. By comparing with experimental results on
systems which can go through an unstable to stable regime, we can estimate the
value of the elastic modulus value of these SIS. In the future, the inspection
of the micronic SIS can provide a better understanding of the micellar
structure inducing the bulk elasticity.
## 8 Acknowledgment
The authors thank Triangle de la Physique for the rheometer (Anton Paar, MCR
501) and Fédération Paris VI for the high-speed camera. Also, thanks to Jérôme
Delacotte, Christophe Clanet and Marina Moreno Luna for discussions.
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|
arxiv-papers
| 2013-06-17T11:05:01 |
2024-09-04T02:49:46.601926
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Fran\\c{c}ois Boulogne, Marc-Antoine Fardin, Sandra Lerouge, Ludovic\n Pauchard, Fr\\'ed\\'erique Giorgiutti-Dauphin\\'e",
"submitter": "Fran\\c{c}ois Boulogne",
"url": "https://arxiv.org/abs/1306.3811"
}
|
1306.4128
|
# Constant Modulus Algorithms Using Hyperbolic Givens Rotation
A. Ikhlef [email protected] R. Iferroujene [email protected] A.
Boudjellal [email protected]. K. Abed-Meraim
[email protected] A. Belouchrani [email protected].
ECE Dep., Univ. of British Columbia, 2356 Main Mall, Vancouver, V6T 1Z4,
Canada. EE Dep., Ecole Nationale Polytechnique, BP 182 EL Harrach, 16200
Algiers, Algeria. Polytech’Orleans, PRISME Laboratory, 12 Rue de Blois, 45067
Orleans, France.
###### Abstract
We propose two new algorithms to minimize the constant modulus (CM) criterion
in the context of blind source separation. The first algorithm, referred to as
Givens CMA (G-CMA) uses unitary Givens rotations and proceeds in two stages:
prewhitening step, which reduces the channel matrix to a unitary one followed
by a separation step where the resulting unitary matrix is computed using
Givens rotations by minimizing the CM criterion. However, for small sample
sizes, the prewhitening does not make the channel matrix close enough to
unitary and hence applying Givens rotations alone does not provide
satisfactory performance. To remediate to this problem, we propose to use non-
unitary Shear (Hyperbolic) rotations in conjunction with Givens rotations.
This second algorithm referred to as Hyperbolic G-CMA (HG-CMA) is shown to
outperform the G-CMA as well as the Analytical CMA (ACMA) in terms of
separation quality. The last part of this paper is dedicated to an efficient
adaptive implementation of the HG-CMA and to performance assessment through
numerical experiments.
###### keywords:
Blind Source Separation, Constant Modulus Algorithm, Adaptive CMA, Sliding
Window, Hyperbolic Rotations, Givens Rotations.
††journal: Elsevier Signal Processing Journal
## 1 Introduction
During the last two decades, Blind Source Separation (BSS) has attracted an
important interest. The main idea of BSS consists of finding the transmitted
signals without using pilot sequences or a priori knowledge on the propagation
channel. Using BSS in communication systems has the main advantage of
eliminating training sequences, which can be expensive or impossible in some
practical situations, leading to an increased spectral efficiency. Several BSS
criteria have been proposed in the literature e.g. [31, 32]. The CM criterion
is probably the best known and most studied higher order statistics based
criterion in blind equalization [3, 4, 12, 13] and signal separation [5, 7, 8,
9] areas. It exploits the fact that certain communication signals have the
constant modulus property, as for example phase modulated signals. The
Constant Modulus Algorithm (CMA) was developed independently by [1, 2] and was
initially designed for PSK signals. The CMA principle consists of preventing
the deviation of the squared modulus of the outputs at the receiver from a
constant. The main advantages of CMA, among others, are its simplicity,
robustness, and the fact that it can be applied even for non-constant modulus
communication signals.
Many solutions to the minimization of the CM criterion have been proposed (see
[9] and references therein). The CM criterion was first minimized via adaptive
Stochastic Gradient Algorithm (SGA) [2] and later on many variants have been
devised. It is known, in adaptive filtering, that the convergence rate of the
SGA is slow. To improve the latter, the authors in [19] proposed an
implementation of the CM criterion via the Recursive Least Squares (RLS)
algorithm. The author in [6] proposed to rewrite the CM criterion as a least
squares problem, which is solved using an iterative algorithm named Least
Squares CMA (LS-CMA). In [10], the authors proposed an algebraic solution for
the minimization of the CM criterion. The proposed algorithm is named
Analytical CMA (ACMA) and consists of computing all the separators, at one
time, through solving a generalized eigenvalue problem. The main advantage of
ACMA is that, in the noise free case, it provides the exact solution, using
only few samples (the number of samples must be greater than or equal to
$M^{2}$, where $M$ is the number of transmitting antennas). Moreover, the
performance study of ACMA showed that it converges asymptotically to the
Wiener receiver [11]. However, the main drawback of ACMA is its numerical
complexity especially for a large number of transmitting antennas. An adaptive
version of ACMA was also developed in [9]. More generally, an abundant
literature on the CM-like criteria and the different algorithms used to
minimize them exists including references [9, 4, 27, 28, 29].
In this paper, we propose two algorithms to minimize the CM criterion. The
first one, referred to as Givens CMA (G-CMA), performs prewhitening in order
to make the channel matrix unitary then, it applies successive Givens
rotations to find the resulting matrix through minimization of the CM
criterion. For large number of samples, prewhitening is effective and the
transformed channel matrix is very close to unitary, however, for small sample
sizes, it is not, and hence results in significant performance loss. In order
to compensate the effect of the ineffective prewhitening stage, we propose to
use Shear rotations [25, 17]. Shear rotations are non-unitary hyperbolic
transformations which allow to reduce departure from normality. We note that
the authors in [25, 17, 16, 26] used Givens and Shear rotations in the context
of joint diagonalization of matrices. We thus propose a second algorithm,
referred to as Hyperbolic G-CMA (HG-CMA), that uses unitary Givens rotations
in conjunction with non-unitary Shear rotations. The optimal parameters of
both complex Shear and Givens rotations are computed via minimization of the
CM criterion. The proposed algorithms have a lower computational complexity as
compared to the ACMA. Moreover, unlike the ACMA which requires a number of
samples greater than the square of the number of transmitting antennas, G-CMA
and HG-CMA do not impose such a condition. Finally, we propose an adaptive
implementation of the HG-CMA using sliding window which has the advantages of
fast convergence and good separation quality for a moderate computational cost
comparable to that of the methods in [6, 8, 9].
The remainder of the paper is organized as follows. Section 2 introduces the
problem formulation and assumptions. In Sections 3 and 4, we introduce the
G-CMA and HG-CMA, respectively. Section 5 is dedicated to the adaptive
implementation of the HG-CMA. Some numerical results and discussion are
provided in Section 6, and conclusions are drawn in Section 7.
## 2 Problem Formulation
Consider the following multiple-input multiple-output (MIMO) memoryless system
model with $M$ transmit and $N$ receive antennas:
$\mathbf{y}(n)=\mathbf{x}(n)+\mathbf{b}(n)=\mathbf{A}\mathbf{s}(n)+\mathbf{b}(n)$
(1)
where $\mathbf{s}(n)=[s_{1}(n),s_{2}(n),\ldots,s_{M}(n)]^{T}$ is the $M\times
1$ source vector, $\mathbf{b}(n)=[b_{1}(n),b_{2}(n),\ldots,b_{N}(n)]^{T}$ is
the $N\times 1$ additive noise vector, $\mathbf{A}$ represents the $N\times M$
MIMO channel matrix, and
$\mathbf{y}(n)=[y_{1}(n),y_{2}(n),\ldots,y_{N}(n)]^{T}$ is the $N\times 1$
received vector.
In the sequel, we assume that the channel matrix $\mathbf{A}$ is full column
rank (and hence $N\geq M$), the source signals are discrete valued (i.e.,
generated from a finite alphabet), zero-mean, independent and identically
distributed (i.i.d.), mutually independent random processes, and the noise is
additive white independent from the source signals. Note that these
assumptions are quite mild and generally satisfied in communication
applications.
Our main goal is to recover the source signals blindly, i.e., using only the
received data. For this purpose, we need to compute an $M\times N$ separation
(receiver) matrix $\mathbf{W}$ such that $\mathbf{W}\mathbf{y}(n)$ results in
the source signals, i.e.
$\mathbf{z}(n)=\mathbf{W}\mathbf{y}(n)=\mathbf{W}\mathbf{A}\mathbf{s}(n)+\bar{\mathbf{b}}(n)=\mathbf{G}\mathbf{s}(n)+\bar{\mathbf{b}}(n)$
(2)
where $\mathbf{z}(n)=[z_{1}(n),z_{2}(n),\ldots,z_{M}(n)]^{T}$ is the $M\times
1$ vector of the estimated source signals, $\mathbf{G}=\mathbf{W}\mathbf{A}$
is the $M\times M$ global system matrix and
$\bar{\mathbf{b}}(n)=\mathbf{W}\mathbf{b}(n)$ is the filtered noise at the
receiver output. Ideally, in BSS, matrix $\mathbf{W}$ separates the source
signals except for a possible permutation and up to scalar factors111To remove
these ambiguities, when necessary, side information or a short training
sequence is always required., i.e.
$\mathbf{W}\mathbf{x}(n)=\mathbf{P}\mathbf{\Lambda}\mathbf{s}(n)$ (3)
where $\mathbf{P}$ is a permutation matrix and $\mathbf{\Lambda}$ is a non-
singular diagonal matrix.
In the sequel, we propose to use the well known CMA to achieve the desired
BSS. In other words, we propose to estimate the separation matrix by
minimizing the CM criterion:
$\mathcal{J}(\mathbf{W})=\sum_{j=1}^{K}\sum_{i=1}^{M}\left(|z_{ij}|^{2}-1\right)^{2}$
(4)
where $z_{ij}$ is the $(i,j)$th entry of $\mathbf{Z}=\mathbf{W}\mathbf{Y}$,
with $\mathbf{Y}=[\mathbf{y}(1),\mathbf{y}(2),\ldots,\mathbf{y}(K)]$ ($K$
being the sample size). This CM criterion has been used by many authors and
has been shown to lead to the desired source separation for CM signals222In
fact, the CMA can be used for sub-Gaussian sources (not necessary of constant
modulus) as proved in [18]. and large sample sizes as stated below.
###### Theorem 1
If $K$ is large enough such that columns of matrix
$\mathbf{S}=[\mathbf{s}(1),\mathbf{s}(2),\\\ \ldots,\mathbf{s}(K)]$ include
all possible combinations of source vectors333Note that this is a sufficient
condition only. $\mathbf{s}(n)$, then the criterion $\mathcal{J}(\mathbf{W})$
(where $\mathbf{W}$ is such that $\mathbf{WA}$ is non singular) is minimized
if and only if $\mathbf{W}$ satisfies:
$\mathbf{W}\mathbf{A}=\mathbf{P}\mathbf{\Lambda}$ (5)
or, in the absence of noise:
$\mathbf{W}\mathbf{Y}=\mathbf{P}\mathbf{\Lambda}\mathbf{S}$ (6)
where $\mathbf{P}$ is an $M\times M$ permutation matrix and $\mathbf{\Lambda}$
is an $M\times M$ diagonal non-singular matrix.
###### Proof 1
The proof can easily be derived from that of Theorem $3.2$ in [24].
## 3 Givens CMA (G-CMA)
In this section, we propose a new algorithm, referred to as G-CMA, based on
Givens rotations, for the minimization of the CM criterion444Part of this
section’s work has been presented in [14].. It is made up of two stages:
1. 1.
Prewhitening: the prewhitening stage allows to convert the arbitrary channel
matrix into a unitary one. Hence, this reduces finding an arbitrary separation
matrix to finding a unitary one [32]. Moreover, prewhitening has the advantage
of reducing vector size (data compression) in the case where $N>M$ and
avoiding trivial undesired solutions.
2. 2.
Givens rotations: After prewhitening, the new channel matrix is unitary and
can therefore be computed via successive Givens rotations. Here, we propose to
compute the optimal parameters of these rotations through minimizing the CM
criterion.
The prewhitening matrix $\mathbf{B}$ can be computed by using the classical
eigendecomposition of the covariance matrix of the received signal
$\mathbf{Y}$ (often, it is computed as the inverse square root of the data
covariance matrix, $\frac{1}{K}\mathbf{Y}\mathbf{Y}^{H}$ [32]). The whitened
signal can then be written as:
$\bar{\mathbf{Y}}=\mathbf{B}\mathbf{Y}$ (7)
Therefore, assuming the noise free case and that the prewhitening matrix
$\mathbf{B}$ is computed using the exact covariance matrix, we have:
$\bar{\mathbf{Y}}=\mathbf{B}\mathbf{A}\mathbf{S}=\mathbf{V}^{H}\mathbf{S}$ (8)
where $\mathbf{V}=\mathbf{A}^{H}\mathbf{B}^{H}$ is an $M\times M$ unitary
matrix. From (8), it is clear that, in order to find the source signals, it is
sufficient to find the unitary matrix $\mathbf{V}$ and hence the separator can
simply be expressed as: $\mathbf{W}=\mathbf{V}\mathbf{B}$, which, in the
absence of noise, results in
$\mathbf{Z}=\mathbf{W}\mathbf{Y}=\mathbf{V}\mathbf{B}\mathbf{Y}=\mathbf{V}\bar{\mathbf{Y}}=\mathbf{V}\mathbf{V}^{H}\mathbf{S}=\mathbf{S}$.
Now, to minimize the CM criterion in (4) w.r.t. to matrix $\mathbf{V}$, we
propose an iterative algorithm where $\mathbf{V}$ is rewritten using Givens
rotations. Indeed, in Jacobi-like algorithms [30], the unitary matrix
$\mathbf{V}$ can be decomposed into product of elementary complex Givens
rotations $\mathbf{\Psi}_{pq}$ such that:
$\mathbf{V}=\prod_{N_{Sweeps}}~{}\prod_{1\leq p<q\leq M}\mathbf{\Psi}_{pq}$
(9)
where $N_{Sweeps}$ refers to the number of sweeps (iterations555In this paper
we will use the terms iteration and sweep interchangeably.) and the Givens
rotation matrix $\mathbf{\Psi}_{pq}$ is a unitary matrix where all diagonal
elements are one except for two elements $\psi_{pp}$ and $\psi_{qq}$.
Likewise, all off-diagonal elements of $\mathbf{\Psi}_{pq}$ are zero except
for two elements $\psi_{pq}$ and $\psi_{qp}$. Elements
$\psi_{pp},\psi_{pq},\psi_{qp}$, and $\psi_{qq}$ are given by:
$\displaystyle\left[\begin{array}[]{cc}\psi_{pp}&\psi_{pq}\\\
\psi_{qp}&\psi_{qq}\end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\cos(\theta)&e^{\jmath\alpha}\sin(\theta)\\\
-e^{-\jmath\alpha}\sin(\theta)&\cos(\theta)\end{array}\right]$ (14)
To compute $\mathbf{\Psi}_{pq}$, we need to find only the rotation angles
$(\theta,\alpha)$. The idea here is to choose the rotation angles
$(\theta,\alpha)$ such that the CM criterion $\mathcal{J}(\mathbf{V})$ is
minimized. For this purpose, let us consider the unitary transformation666For
simplicity, we keep using notation $\bar{\mathbf{Y}}$ even though the latter
matrix is transformed at each iteration of the proposed algorithm.
$\breve{\mathbf{Y}}=\mathbf{\Psi}_{pq}\bar{\mathbf{Y}}$. Given the structure
of $\mathbf{\Psi}_{pq}$, this unitary transformation changes only the elements
in rows $p$ and $q$ of $\bar{\mathbf{Y}}$ according to:
$\breve{y}_{pj}=\cos(\theta)\bar{y}_{pj}+e^{\jmath\alpha}\sin(\theta)\bar{y}_{qj}\mbox{
and
}\breve{y}_{qj}=-e^{-\jmath\alpha}\sin(\theta)\bar{y}_{pj}+\cos(\theta)\bar{y}_{qj}$
(15)
where $\bar{y}_{ij}$ refers to the $(i,j)$th entry of $\bar{\mathbf{Y}}$.
The algorithm consists of minimizing iteratively the criterion in (4) by
applying successive Givens rotations, with initialization of
$\mathbf{V}=\mathbf{I}$. $\mathbf{\Psi}_{pq}$ are computed such that
$\mathcal{J}(\mathbf{\Psi}_{pq})$ is minimized at each iteration. In order to
minimize $\mathcal{J}(\mathbf{\Psi}_{pq})$, we propose to express it as a
function of $(\theta,\alpha)$. Since the application of Givens rotation matrix
$\mathbf{\Psi}_{pq}$ to $\bar{\mathbf{Y}}$ modifies only the two rows $p$ and
$q$, the terms that depend on $(\theta,\alpha)$ are those corresponding to
$i=p$ or $i=q$ in (4). Considering (14) and (15), we have:
$\displaystyle\begin{array}[]{l}\mathcal{J}(\mathbf{\Psi}_{pq})=\sum_{j=1}^{K}\left[\big{(}|\breve{y}_{pj}|^{2}-1\big{)}^{2}+\big{(}|\breve{y}_{qj}|^{2}-1\big{)}^{2}\right]+\sum_{j=1}^{K}\sum_{i=1,i\neq
p,q}^{M}\big{(}|\bar{y}_{ij}|^{2}-1\big{)}^{2}\end{array}$ (17)
On the other hand, by considering (15) and the following equalities:
$\displaystyle\begin{array}[]{l}\cos^{2}(\theta)=\frac{1}{2}(1+\cos(2\theta)),\sin^{2}(\theta)=\frac{1}{2}(1-\cos(2\theta)),\sin(2\theta)=2\sin(\theta)\cos(\theta)\end{array}$
(19)
and after some manipulations, we obtain:
$\displaystyle\begin{split}|\breve{y}_{pj}|^{2}=\mathbf{t}_{j}^{T}\mathbf{v}+\frac{1}{2}\big{(}|\bar{y}_{pj}|^{2}+|\bar{y}_{qj}|^{2}\big{)}\mbox{
and
}|\breve{y}_{qj}|^{2}=-\mathbf{t}_{j}^{T}\mathbf{v}+\frac{1}{2}\big{(}|\bar{y}_{pj}|^{2}+|\bar{y}_{qj}|^{2}\big{)}\end{split}$
(20)
with:
$\displaystyle\mathbf{v}=[\cos(2\theta),~{}\sin(2\theta)\cos(\alpha),~{}\sin(2\theta)\sin(\alpha)]^{T}$
(21)
$\displaystyle\mathbf{t}_{j}=\Big{[}\frac{1}{2}\big{(}|\bar{y}_{pj}|^{2}-|\bar{y}_{qj}|^{2}\big{)},~{}\Re(\bar{y}_{pj}\bar{y}_{qj}^{*}),~{}\Im(\bar{y}_{pj}\bar{y}_{qj}^{*})\Big{]}^{T}$
(22)
where $\Re(a)$ and $\Im(a)$ denote real and imaginary parts of $a$,
respectively. Using (20), we get:
$\displaystyle\big{(}|\breve{y}_{pj}|^{2}-1\big{)}^{2}$
$\displaystyle+\big{(}|\breve{y}_{qj}|^{2}-1\big{)}^{2}=2\mathbf{v}^{T}\mathbf{t}_{j}\mathbf{t}_{j}^{T}\mathbf{v}+2\left(\frac{|\bar{y}_{pj}|^{2}+\bar{y}_{qj}|^{2}}{2}-1\right)^{2}$
(23)
Then, plugging (23) into (17) yields:
$\displaystyle\mathcal{J}(\mathbf{\Psi}_{pq})$ $\displaystyle=$ $\displaystyle
2\sum_{j=1}^{K}\mathbf{v}^{T}\mathbf{t}_{j}\mathbf{t}_{j}^{T}\mathbf{v}+2\sum_{j=1}^{K}\left(\frac{|\bar{y}_{pj}|^{2}+|\bar{y}_{qj}|^{2}}{2}-1\right)^{2}$
(24) $\displaystyle+$ $\displaystyle\sum_{j=1}^{K}\sum_{i=1\atop i\neq
p,q}^{M}\big{(}|\bar{y}_{ij}|^{2}-1\big{)}^{2}$
Given that the second and third summations in (24) do not depend on
$(\theta,\alpha)$, the minimization problem is equivalent to the minimization
of:
$\mathcal{F}(\mathbf{\Psi}_{pq})=\mathbf{v}^{T}\mathbf{T}\mathbf{v}$ (25)
where $\mathbf{T}=\sum_{j=1}^{K}\mathbf{t}_{j}\mathbf{t}_{j}^{T}$ and
$\|\mathbf{v}\|=1$. Finally, the solution $\mathbf{v}$ that minimizes (25) is
given by the unit norm eigenvector of $\mathbf{T}$ corresponding to the
smallest eigenvalue777This is a $3\times 3$ eigenvalue problem that can be
solved explicitly.. Given $\mathbf{v}=[v_{1},v_{2},v_{3}]^{T}$ we have:
$\displaystyle\begin{split}\cos(\theta)=\sqrt{\frac{1+v_{1}}{2}}\mbox{ and
}e^{\jmath\alpha}\sin(\theta)=\frac{v_{2}+\jmath
v_{3}}{\sqrt{2(1+v_{1})}}\end{split}$ (26)
Using (26), the computation of $\mathbf{\Psi}_{pq}$ follows directly from
(14). The G-CMA algorithm is summarized in Table 1 (for simplicity, we use the
same notation for the data and its transformed version).
Initialization: $\mathbf{V}=\mathbf{I}$
---
1\. Prewhitening: $\bar{\mathbf{Y}}=\mathbf{B}\mathbf{Y}$, where $\mathbf{B}$
is the prewhitening matrix.
2\. Complex Givens rotations:
for $i=1:N_{Sweeps}$
for $p=1:M-1$
for $q=p+1:M$
Compute $\mathbf{\Psi}_{pq}$ using (26)
$\bar{\mathbf{Y}}=\mathbf{\Psi}_{pq}\bar{\mathbf{Y}}$
$\mathbf{V}=\mathbf{\Psi}_{pq}\mathbf{V}$
end for
end for
end for
3\. After convergence, computation of the separation matrix:
$\mathbf{W}=\mathbf{V}\mathbf{B}$
4\. Separation: $\hat{\mathbf{S}}=\mathbf{W}\mathbf{Y}=\bar{\mathbf{Y}}$.
Table 1: The Givens CMA (G-CMA) algorithm.
The G-CMA algorithm described above requires that the number of samples
available at the receiver is large enough so that the prewhitening step
results in an equivalent channel matrix close to unitary, for which the use of
Givens rotations is effective. However, for small numbers of samples,
prewhitening may result in an equivalent channel matrix not close to unitary,
in which case, applying G-CMA alone is ineffective. Next, we propose to solve
this problem by introducing the Hyperbolic Givens rotations.
## 4 Hyperbolic Givens CMA (HG-CMA)
As stated in the previous section, the use of Givens rotations in the case of
small numbers of samples is not effective. To overcome this limitation, we
introduce here the use of Hyperbolic Givens rotations. The latter consist of
applying Shear rotations and Givens rotations alternatively. Matrix
$\mathbf{W}$ can be decomposed into product of elementary complex Shear
rotations, Givens rotations and normalization transformation as follows:
$\mathbf{W}=\prod_{N_{Sweeps}}~{}~{}\prod_{1\leq p<q\leq
M}\mathbf{D}_{pq}~{}\mathbf{\Psi}_{pq}~{}\mathbf{H}_{pq}$ (27)
where $\mathbf{D}_{pq}$, $\mathbf{\Psi}_{pq}$ and $\mathbf{H}_{pq}$ denote
normalization, unitary Givens and non-unitary Shear transformations,
respectively. The unitary matrix $\mathbf{\Psi}_{pq}$ is defined in (14).
Similar to $\mathbf{\Psi}_{pq}$, $\mathbf{H}_{pq}$ is equal to the identity
matrix except for the elements $h_{pp},h_{pq},h_{qp}$ and $h_{qq}$ that are
given by:
$\displaystyle\left[\begin{array}[]{cc}h_{pp}&h_{pq}\\\
h_{qp}&h_{qq}\end{array}\right]$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}\cosh(\gamma)&e^{\jmath\beta}\sinh(\gamma)\\\
e^{-\jmath\beta}\sinh(\gamma)&\cosh(\gamma)\end{array}\right]$ (32)
where $\gamma\in\mathbb{R}$ is the hyperbolic transformation parameter and
$\beta\in[-\frac{\pi}{2},\frac{\pi}{2}]$ is an angle parameter (equal to zero
in the real case). The normalization transformation
$\mathbf{D}_{pq}=\mathbf{D}_{pq}(\lambda_{p},\lambda_{q})$ is a diagonal
matrix with diagonal elements equal to one except for the two elements
$d_{pp}=\lambda_{p}$, and $d_{qq}=\lambda_{q}$.
In the following derivation, we consider the square case where $N=M$ (if
$N>M$, one can use signal subspace projection as in [9]).
### 4.1 Non-Unitary Shear Rotations
By applying $\mathbf{H}_{pq}$ to the received signal, we get:
$\tilde{\mathbf{Y}}=\mathbf{H}_{pq}~{}\mathbf{Y}$ (33)
From (32), only the $p$th and $q$th rows of $\mathbf{Y}$ are affected
according to:
$\displaystyle\begin{split}\tilde{y}_{pj}&=&\cosh(\gamma){y_{pj}}+e^{\jmath\beta}\sinh(\gamma){y_{qj}}\mbox{
and
}\tilde{y}_{qj}&=&e^{-\jmath\beta}\sinh(\gamma){y_{pj}}+\cosh(\gamma){y_{qj}}\end{split}$
(34)
In order to compute $\mathbf{H}_{pq}$, we propose to minimize the CM cost
function in (4) w.r.t. $\mathbf{H}_{pq}$:
$\displaystyle\mathcal{J}(\mathbf{H}_{pq})=\sum_{j=1}^{K}(|\tilde{y}_{pj}|^{2}-1)^{2}+(|\tilde{y}_{qj}|^{2}-1)^{2}+\sum_{j=1}^{K}\sum_{i=1\atop
i\neq p,q}^{M}({|\bar{y}_{ij}|^{2}}-1)^{2}$ (35)
By considering (34) and the following equalities:
$\displaystyle\begin{split}\sinh(2\gamma)=2\sinh(\gamma)\cosh(\gamma),\cosh^{2}(\gamma)=\frac{1}{2}(\cosh(2\gamma)+1),\sinh^{2}(\gamma)=\frac{1}{2}(\cosh(2\gamma)-1)\end{split}$
(36)
and after some straightforward derivations, we obtain:
$\displaystyle|\tilde{y}_{pj}|^{2}=\mathbf{r}_{j}^{T}\mathbf{u}+\frac{1}{2}(|y_{pj}|^{2}-|{y_{qj}|^{2}})\mbox{
and
}|\tilde{y}_{qj}|^{2}=\mathbf{r}_{j}^{T}\mathbf{u}-\frac{1}{2}(|y_{pj}|^{2}-|{y_{qj}|^{2}})$
(37)
with:
$\displaystyle\mathbf{u}=\left[\cosh(2\gamma),\;\;\cos(\beta)\;\sinh(2\gamma),\;\;\sin(\beta)\;\sinh(2\gamma)\right]^{T}$
(38)
$\displaystyle\mathbf{r}_{j}=\left[\frac{1}{2}\left(|y_{pj}|^{2}+|{y_{qj}|^{2}}\right),\;\;\Re\left(y_{pj}y_{qj}^{*}\right),\;\;\Im\left(y_{pj}y_{qj}^{*}\right)\right]^{T}$
(39)
Using the results in (37), we can rewrite the first two terms in (35) as:
$\displaystyle\left(|\tilde{y}_{pj}|^{2}-1\right)^{2}+\left(|\tilde{y}_{qj}|^{2}-1\right)^{2}$
$\displaystyle=$ $\displaystyle
2\mathbf{u}^{T}\mathbf{r}_{j}\mathbf{r}_{j}^{T}\mathbf{u}-4\mathbf{u}^{T}\mathbf{r}_{j}$
(40) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(|\bar{y}_{pj}|^{2}-|{\bar{y}_{qj}|^{2}}\right)^{2}+2$
Then, by substituting (40) into (35), we obtain:
$\displaystyle\mathcal{J}(\mathbf{u})=2\left(\sum_{j=1}^{K}\mathbf{u}^{T}\mathbf{r}_{j}\mathbf{r}_{j}^{T}\mathbf{u}-2\mathbf{u}^{T}\mathbf{r}_{j}\right)$
$\displaystyle+$ $\displaystyle
2\sum_{j=1}^{K}\big{[}\frac{1}{4}({|\bar{y}_{pj}|^{2}}-{|\bar{y}_{qj}|^{2}})^{2}+1\big{]}$
(41) $\displaystyle+$ $\displaystyle\sum_{j=1}^{K}\sum_{i=1\atop i\neq
p,q}^{M}({|\bar{y}_{ij}|^{2}}-1)^{2}$
We note that only the first term on the right hand side of the equality (41)
depends on $(\gamma,\beta)$, and hence the minimization of (41) is equivalent
to the minimization of:
$\mathcal{F}(\mathbf{u})=\sum_{j=1}^{K}\mathbf{u}^{T}\mathbf{r}_{j}\mathbf{r}_{j}^{T}\mathbf{u}-2\mathbf{u}^{T}\mathbf{r}_{j}$
(42)
This optimisation problem can be achieved in three different ways: by
computing the exact solution, by taking linear approximation to zero, and with
semi linear approximation.
#### 4.1.1 Exact Solution
In this approach, we compute the optimum solution using the Lagrange
multiplier method. The optimization problem can be expressed as:
$\displaystyle\min_{\mathbf{u}}~{}~{}\mathcal{F}(\mathbf{u})~{}~{}~{}\textrm{s.t.}~{}~{}~{}\mathbf{u}^{T}\mathbf{J}_{3}\mathbf{u}=1$
(43)
where $\mathbf{J}_{3}=\mbox{diag}\left(\left[1,-1,~{}-1\right]\right)$ so that
constraint is equivalent to $\cosh^{2}(2\gamma)-\sinh^{2}(2\gamma)=1$. The
Lagrangian of the optimization problem in (43) can be written as:
$\mathcal{L}(\mathbf{u},\lambda)=\mathbf{u}^{T}\mathbf{R}\mathbf{u}-2\mathbf{r}^{T}\mathbf{u}+\lambda(\mathbf{u}^{T}\mathbf{J}_{3}\mathbf{u}-1)$
(44)
where $\mathbf{R}=\sum_{j=1}^{K}\mathbf{r}_{j}\mathbf{r}_{j}^{T}$ is a
$(3\times 3)$ symmetric matrix, $\mathbf{r}=\sum_{j=1}^{K}\mathbf{r}_{j}$,
$\mathbf{u}$ and $\mathbf{r}_{j}$ are defined in (38) and (39), respectively.
The solution that minimizes the Lagrangian in (44) can be expressed as:
$\mathbf{u}=(\mathbf{R}+\lambda\mathbf{J}_{3})^{-1}\mathbf{r}$ (45)
where $\lambda$ is the solution of:
$\mathbf{u}^{T}\mathbf{J}_{3}\mathbf{u}=1\Longleftrightarrow\mathbf{r}^{T}(\mathbf{R}+\lambda\mathbf{J}_{3})^{-1}\mathbf{J}_{3}(\mathbf{R}+\lambda\mathbf{J}_{3})^{-1}\mathbf{r}=1$
(46)
which is a $6$-th order polynomial equation (see appendix A) of the form:
$P_{6}(\lambda)=c_{0}\lambda^{6}+c_{1}\lambda^{5}+c_{2}\lambda^{4}+c_{3}\lambda^{3}+c_{4}\lambda^{2}+c_{5}\lambda+c_{6}=0$.
The desired solution $\lambda$ is the real-valued root of the above polynomial
that corresponds to the minimum value of (44). Finally, given the solution
$\mathbf{u}=[u_{1}\;u_{2}\;u_{3}]^{T}$ in (45), the Shear transformation
entries are computed as:
$h_{pp}=h_{qq}=\sqrt{\frac{u_{1}+1}{2}}\mbox{ and
}h_{pq}=h_{qp}^{*}=\frac{(u_{2}+\jmath u_{3})}{2h_{pp}}$ (47)
Note that, for the computation of each Shear rotation matrix, we need to
perform a $3\times 3$ matrix inversion and solve a $6$-th order polynomial
equation. Hence, as the number of sweeps and transmit antennas increases, the
complexity increases. In the following, we present two suboptimal solutions
that have less complexity and close performance compared to the exact one.
#### 4.1.2 Semi-Exact Solution
We denote this approach by semi-exact solution, since for computing $\beta$ we
take the approximation in (48), while for the angle rotation $\gamma$ we
compute an exact solution using the Lagrange multiplier method. By considering
the first order approximation around zero of $\sinh$ and $\cosh$, we have:
$\displaystyle\sinh(2\gamma)\approx 2\sinh(\gamma)\approx 2\gamma\mbox{ and
}\cosh(2\gamma)\approx\cosh(\gamma)\approx 1$ (48)
Using (48) in (38), equation (42) can be expressed as:
$\displaystyle\mathcal{F}(\gamma,\beta)=\sum_{j=1}^{K}r_{j}^{(1)}\left(r_{j}^{(1)}-2\right)$
$\displaystyle+$ $\displaystyle
4\gamma\left[\cos(\beta)r_{j}^{(2)}\left(r_{j}^{(1)}-1\right)+\sin(\beta)r_{j}^{(3)}\left(r_{j}^{(1)}-1\right)\right]$
(49) $\displaystyle+$ $\displaystyle
4\gamma^{2}\left(\cos(\beta)r_{j}^{(2)}+\sin(\beta)r_{j}^{(3)}\right)^{2}$
where $r_{j}^{(i)}$ is the $i$th element of $\mathbf{r}_{j}$. The linear
approximation of (49) for $\gamma$ close to zero (which corresponds to simply
neglecting the terms involving $\gamma^{n}$ for $n\geq 2$) can be obtained by
discarding the last term of (49):
$\begin{array}[]{l}\mathcal{F}(\gamma,\beta)\approx\sum_{j=1}^{K}r_{j}^{(1)}\left(r_{j}^{(1)}-2\right)+4\gamma\left[\cos(\beta)r_{j}^{(2)}\left(r_{j}^{(1)}-1\right)+\sin(\beta)r_{j}^{(3)}\left(r_{j}^{(1)}-1\right)\right]\end{array}$
(50)
The minimization of (50) obtained by zeroing its derivative) leads to:
$\begin{array}[]{l}\beta=\mathrm{arctan}\left(\frac{\sum_{j=1}^{K}r_{j}^{(3)}\;\left(r_{j}^{(1)}-1\right)}{\sum_{j=1}^{K}r_{j}^{(2)}\;\left(r_{j}^{(1)}-1\right)}\right)\end{array}$
(51)
Once we have $\beta$, let us define:
$\displaystyle\tilde{\mathbf{u}}=[\cosh(2\gamma),\;\sinh(2\gamma)]^{T}$ (52)
$\displaystyle\tilde{\mathbf{r}}_{j}=\left[\frac{1}{2}\left(|y_{pj}|^{2}+|{y_{qj}|^{2}}\right),~{}\cos(\beta)\Re(y_{pj}y_{qj}^{*})+\sin(\beta)\Im(y_{pj}y_{qj}^{*})\right]^{T}$
(53)
and hence, finding $\gamma$ which minimizes (42) implies solving the following
optimization problem:
$\min_{\tilde{\mathbf{u}}}~{}~{}\mathcal{K}(\tilde{\mathbf{u}})~{}~{}~{}\textrm{s.t.}~{}~{}~{}\tilde{\mathbf{u}}^{T}\mathbf{J}_{2}\tilde{\mathbf{u}}=1$
(54)
where $\mathbf{J}_{2}=\mbox{diag}\left([1,~{}-1]\right)$ and:
$\displaystyle\begin{array}[]{l}\mathcal{K}(\tilde{\mathbf{u}})=\sum_{j=1}^{K}\tilde{\mathbf{u}}^{T}\tilde{\mathbf{r}}_{j}\tilde{\mathbf{r}}_{j}^{T}\tilde{\mathbf{u}}-2\tilde{\mathbf{u}}^{T}\tilde{\mathbf{r}}_{j}\end{array}$
(56)
By defining
$\tilde{\mathbf{R}}=\sum_{j=1}^{K}\tilde{\mathbf{r}}_{j}\tilde{\mathbf{r}}_{j}^{T}$
and $\tilde{\mathbf{r}}=\sum_{j=1}^{K}\tilde{\mathbf{r}}_{j}$, the
optimization of (54) using Lagrange multiplier leads to:
$\tilde{\mathbf{u}}=(\tilde{\mathbf{R}}+\lambda\mathbf{J}_{2})^{-1}\tilde{\mathbf{r}}$
(57)
where $\lambda$ is the solution of:
$\tilde{\mathbf{u}}^{T}\mathbf{J}_{2}\tilde{\mathbf{u}}=1\Longleftrightarrow\tilde{\mathbf{r}}^{T}(\tilde{\mathbf{R}}+\lambda\mathbf{J}_{2})^{-1}\mathbf{J}_{2}(\tilde{\mathbf{R}}+\lambda\mathbf{J})^{-1}\tilde{\mathbf{r}}=1$
(58)
This is a $4$-th order polynomial equation (see appendix A) of the form:
$P_{4}(\lambda)=c_{0}\lambda^{4}+c_{1}\lambda^{3}+c_{2}\lambda^{2}+c_{3}\lambda+c_{4}=0$.
The desired solution $\lambda$ is the real-valued root of the above polynomial
that corresponds to the minimum value of (56). Finally, given the solution
$\tilde{\mathbf{u}}=[\tilde{u}_{1}\;\tilde{u}_{2}]^{T}$ in (57) and $\beta$ in
(51), the Shear transformation entries can be obtained as:
$\displaystyle h_{pp}$
$\displaystyle=h_{qq}=\sqrt{\frac{1}{2}(\tilde{u}_{1}+1)}\mbox{ and }h_{pq}$
$\displaystyle=h_{qp}^{*}=e^{\jmath\beta}\frac{\tilde{u}_{2}}{2h_{pp}}$ (59)
We note that in this solution, for the computation of each Shear rotation
matrix, we need to solve a $4$-th order polynomial equation. Hence, the
complexity of this solution is clearly less than that of the exact one.
#### 4.1.3 Solution with Linear Approximation to Zero
In this approach, we compute $\beta$ as in (51) and then we compute $\gamma$
which minimizes (56) by considering the approximation in (48). We define:
$\tilde{\mathbf{R}}=\left[\begin{array}[]{cc}\tilde{r}_{11}&\tilde{r}_{12}\\\
\tilde{r}_{21}&\tilde{r}_{22}\end{array}\right]~{}~{}\mathrm{and}~{}~{}\tilde{\mathbf{r}}=\left[\begin{array}[]{cc}\tilde{r}_{1}\\\
\tilde{r}_{2}\end{array}\right]$ (60)
and using (34), (56) can be written as:
$\mathcal{K}(\gamma)=\frac{1}{2}(\tilde{r}_{11}+\tilde{r}_{22})\cosh(4\gamma)+\tilde{r}_{12}\sinh(4\gamma)-2\tilde{r}_{1}\cosh(2\gamma)-2\tilde{r}_{2}\sinh(2\gamma)$
(61)
By taking the first derivative of (61) with respect to $\gamma$, using (48),
and setting the result equal to zero, we obtain:
$\sinh(2\gamma)(\tilde{r}_{11}+\tilde{r}_{22}-\tilde{r}_{1})+\cosh(2\gamma)(\tilde{r}_{12}-\tilde{r}_{2})=0$
(62)
Which solution is:
$\displaystyle\begin{array}[]{l}\gamma=\frac{1}{2}\mathrm{arctanh}\left(\frac{\sum_{j=1}^{K}\left[\left(\cos(\beta)r_{j}^{(2)}+\sin(\beta)r_{j}^{(3)}\right)\left(1-r_{j}^{(1)}\right)\right]}{\sum_{j=1}^{K}\left[\left((r_{j}^{(1)})^{2}-r_{j}^{(1)}\right)+\left(\cos(\beta)r_{j}^{(2)}+\sin(\beta)r_{j}^{(3)}\right)^{2}\right]}\right)\end{array}$
(64)
Given $\beta$ in (51) and $\gamma$ in (64), the computation of
$\mathbf{H}_{pq}$ follows directly. This solution has the lowest complexity
among the three considered ones.
### 4.2 Unitary Givens Rotation
After the Shear transformation, we now apply the Givens transformation to the
result of the Shear rotation as:
$\mathbf{\underline{Y}}=\mathbf{\Psi}_{pq}\tilde{\mathbf{Y}}$ (65)
The unitary matrix $\mathbf{\Psi}_{pq}$ is computed in the same way as in
Section 3.
### 4.3 Normalization Rotations
The last algorithm’s transform is a normalization step. In our CM criterion in
(4), we have set the constant equal to one while in the original CM criterion
it is chosen equal to
$C_{i}=E[\left|s_{i}\right|^{4}]/E[\left|s_{i}\right|^{2}]$. Somehow, this
normalization step is introduced to compensate for this constant choice (the
value of $C_{i}$ is supposed unknown in a blind context).
It has been shown in the two previous subsection that both Givens and
hyperbolic transformations affect only the rows of indices $p$ and $q$ of the
data bloc $\mathbf{\underline{Y}}$ which means that only these two rows need
to be normalized:
$\mathbf{Z}=\mathbf{D}_{(pq)}(\lambda_{p},\lambda_{q})~{}\mathbf{\underline{Y}}$
(66)
The optimal parameters $(\lambda_{p},\lambda_{q})$ are calculated so that they
minimize the CM criterion in (4) w.r.t.
$\mathbf{D}_{(pq)}(\lambda_{p},\lambda_{q})$. The CM criterion is expressed in
this case as (constant terms are omitted):
$\mathcal{J}_{D}(\lambda_{p},\lambda_{q})=\sum_{j=1}^{K}(\lambda_{p}^{4}\left|\underline{y}_{pj}\right|^{4}-2\lambda_{p}^{2}\left|\underline{y}_{pj}\right|^{2})+\sum_{j=1}^{K}(\lambda_{q}^{4}\left|\underline{y}_{qj}\right|^{4}-2\lambda_{q}^{2}\left|\underline{y}_{qj}\right|^{2})$
(67)
Optimal normalization parameters can be obtained at the zeros of the
derivatives of (67) with respect to these two parameters as follows:
$\displaystyle\begin{array}[]{lll}\lambda_{p}=\sqrt{\sum_{j=1}^{K}\left|\underline{y}_{pj}\right|^{2}/\sum_{j=1}^{K}\left|\underline{y}_{pj}\right|^{4}}~{}~{}\mbox{
and
}\lambda_{q}=\sqrt{\sum_{j=1}^{K}\left|\underline{y}_{qj}\right|^{2}/\sum_{j=1}^{K}\left|\underline{y}_{qj}\right|^{4}}\end{array}$
(69)
The HG-CMA algorithm is summarized in Table 2.
Initialization: $\mathbf{W}=\mathbf{I}$
---
Signal subspace projection if $N>M$
for $i=1:N_{Sweeps}$
for $p=1:M-1$
for $q=p+1:M$
Compute $\mathbf{H}_{pq}$:
\- using (47) for exact solution
\- using (51) and (59) for semi exact solution
\- using (51) and (64) for linear approximation to zero (preferred)
$\mathbf{Y}=\mathbf{H}_{pq}\mathbf{Y}$
$\mathbf{W}=\mathbf{H}_{pq}\mathbf{W}$
Compute $\mathbf{\Psi}_{pq}$ using (26)
$\mathbf{Y}=\mathbf{\Psi}_{pq}\mathbf{Y}$
$\mathbf{W}=\mathbf{\Psi}_{pq}\mathbf{W}$
Compute $\mathbf{D}_{pq}$ using (69)
$\mathbf{Y}=\mathbf{D}_{pq}\mathbf{Y}$
$\mathbf{W}=\mathbf{D}_{pq}\mathbf{W}$
end for
end for
end for
Separation: $\hat{\mathbf{S}}=\mathbf{W}\mathbf{Y}=\mathbf{Y}$.
Table 2: The Hyperbolic Givens CMA (HG-CMA) algorithm.
## 5 Adaptive HG-CMA
To make an adaptive version of the HG-CMA algorithm, let us consider a sliding
bloc of size $K$,
$\mathbf{Y}^{(t-1)}=\left[\mathbf{y}(t-K),...,\mathbf{y}(t-2),\mathbf{y}(t-1)\right]$
which is updated at each new acquisition of a new sample $\mathbf{y}(t)$ (at
time instant $t$). The main idea of the adaptive HG-CMA is to apply only one
sweep of complex rotations on the sliding window at each time instant and
update the separation matrix $\mathbf{W}$ by this sweep of rotations.
The numerical cost of the HG-CMA is of order $O(KM^{2})$ (assuming $K>M$) but
can be reduced to $O(KM)$ flops per iteration if we use only one or two
rotations per time instant. In the simulation experiments, we compare the
performance of the algorithm in the 3 following cases:
* 1.
When we use one complete sweep (i.e. $M(M-1)/2$ rotations)
* 2.
When we use one single rotation which indices are chosen according to an
automatic selection (i.e. automatic incrementation) throughout the iterations
in such a way all search directions are visited periodically.
* 3.
When we use two rotations per iteration (time instant): one pair of indices is
selected according to the maximum deviation criterion:
$\begin{array}[]{l}(p,q)=arg\max\sum_{k=1}^{K}(|y_{pk}|^{2}-1)^{2}+(|y_{qk}|^{2}-1)^{2}\end{array}$
(70)
the other rotation indices are selected automatically.
Comparatively, the adaptive ACMA [9] costs approximately $O(M^{3})$ flops per
iteration and the LS-CMA888We consider here an adaptive version of the LS-CMA
using the same sliding window as for our algorithm. costs $O(KM^{2}+M^{3})$.
Interestingly, as shown in section 6, the sliding window length $K$ can be
chosen of the same order as the number of sources $M$ without affecting much
the algorithm’s performance. In that case, the numerical cost of HG-CMA
becomes similar to that of the adaptive ACMA. The adaptive HG-CMA algorithm is
summarized in Table 3. Note that, the normalization step is done outside the
sweep loop which reduces slightly the numerical cost.
Initialization: $\mathbf{W}^{(K)}=\mathbf{I}_{M}$
---
For $t=K+1,K+2,...$ do
$\mathbf{y}(t)=\mathbf{W}^{(t-1)}~{}\mathbf{y}(t)$
$\mathbf{Y}^{(t)}=\left[\mathbf{y}(t-K),...,\mathbf{y}(t-1),\mathbf{y}(t)\right]$
$\mathbf{W}^{(t)}=\mathbf{W}^{(t-1)}$
For all $1\leq p<q\leq M$ do
Compute $\mathbf{H}_{(pq)}$ using (51) and (64)
Compute $\mathbf{\Psi}_{(pq)}$ using (26)
Update
$\mathbf{W}^{(t)}=\mathbf{\Psi}_{(pq)}~{}\mathbf{H}_{(pq)}~{}\mathbf{W}^{(t)}$
Update
$\mathbf{Y}^{(t)}=\mathbf{\Psi}_{(pq)}~{}\mathbf{H}_{(pq)}~{}\mathbf{Y}^{(t)}$
end For
For $1\leq p\leq M$, compute $\lambda_{p}$ using (69), end For
Compute $\mathbf{D}=\mbox{diag}([\lambda_{1},\cdots,\lambda_{M}])$
Update $\mathbf{W}^{(t)}=\mathbf{D}~{}\mathbf{W}^{(t)}$ and
$\mathbf{Y}^{(t)}=\mathbf{D}~{}\mathbf{Y}^{(t)}$
end For
Table 3: Adaptive HG-CMA Algorithm.
## 6 Numerical Results
Some numerical results are now presented in order to assess the performance of
the proposed algorithms. For comparison we use ACMA [10] and LS-CMA [9] as a
benchmark. As performance measure, we use the signal to interference and noise
ratio (SINR) defined as:
$\displaystyle\begin{array}[]{l}\textrm{SINR}=\frac{1}{M}\sum_{k=1}^{M}\textrm{SINR}_{k}\mbox{
with }~{}~{}\textrm{SINR}_{k}=\frac{|g_{kk}|^{2}}{\sum\limits_{\ell,\ell\neq
k}|g_{k\ell}|^{2}+\mathbf{w}_{k}\mathbf{R}_{b}\mathbf{w}_{k}^{H}}\end{array}$
(72)
where $\textrm{SINR}_{k}$ is the signal to interference and noise ratio at the
$k$th output $g_{ij}=\mathbf{w}_{i}\mathbf{a}_{j}$, where $\mathbf{w}_{i}$ and
$\mathbf{a}_{j}$ are the $i$th row vector and $j$th column vector of matrices
$\mathbf{W}$ and $\mathbf{A}$, respectively.
$\mathbf{R}_{b}=E[\mathbf{b}\mathbf{b}^{H}]=\sigma_{b}^{2}\mathbf{I}_{N}$ is
the noise covariance matrix. The source signals are assumed to be of unit
variance.
We use the data model in (1); The system inputs are independent, uniformly
distributed and drawn from 8-PSK, or 16-QAM constellations. The channel
matrices $\mathbf{A}$ are generated randomly at each run but with controlled
conditioning (their entries are generated as i.i.d. Gaussian variables).
Unless otherwise specified, we consider $M=5$ transmit and $N=7$ receive
antennas. The noise variance is determined according to the desired signal to
noise ratio (SNR). In all figures the results are averaged over 1000
independent realizations (Monte Carlo runs).
Fig. 1 depicts the SINR of HG-CMA vs. the SNR. We compare the three solutions,
i.e., linear approximation to zero, semi-exact and exact solutions for Shear
rotations in HG-CMA for 8-PSK and 16-QAM constellations. The sample size is
$K=100$ and the number of sweeps is set equal to 10. We observe that the three
solutions have almost the same performance for both 8-PSK and 16-QAM
constellations. Therefore, in the following simulations, in HG-CMA, we will
consider the linear approximation to zero solution.
In Fig. 2, we investigate the effect of the number of sweeps on the
performance of G-CMA and HG-CMA. The figure shows the SINR vs. the SNR for
different numbers of sweeps. In this simulation, we assumed 8-PSK
constellation and $K=100$ samples. We observe that, as expected, the
performance is improved by increasing the number of sweeps and from 5 sweeps
upwards, the performance remains unchangeable. In the rest of this section we
consider $10$ sweeps in G-CMA and HG-CMA. Moreover, we can see that for small
number of iterations HG-CMA is much better than G-CMA and the gap between them
decreases as the number of iterations increases.
Fig. 3 compares the proposed HG-CMA and G-CMA algorithms with ACMA in terms of
SINR vs. SNR for 8-PSK constellation and various numbers of samples. We
observe that, as expected, the larger the number of samples, the better the
performance for all algorithms. For small number of samples, i.e. $K=20$, we
observe that HG-CMA significantly outperforms ACMA and G-CMA. We also observe
that G-CMA performs better than ACMA for low to moderate SNR while for
$\mathrm{SNR}>23~{}\mathrm{dB}$, ACMA becomes better. The reason that ACMA
performs worse than HG-CMA is that the number of samples $K=20$ is less than
the number of transmit antennas squared $M^{2}$, i.e., $K=20<M^{2}=25$ and as
we stated above for ACMA to achieve good performance in the case of PSK
constellations the number of samples $K$ must be at least greater than $M^{2}$
[11]. For $K=100$, HG-CMA still provides the best performance while the
performance of ACMA becomes very close to that of HG-CMA and better than that
of G-CMA. We can say that for small or moderate number of samples the proposed
algorithms are more suitable as compared to ACMA even for PSK constellations.
In Fig. 4, we consider the case of 16-QAM constellation. We notice that the
proposed HG-CMA and G-CMA algorithms provide better performance as compared to
ACMA. We also observe that, unlike the 8-PSK case in Fig. 3, the performance
of HG-CMA and G-CMA are close in the case of 16-QAM. Moreover, we can see that
the gap between the performance of the proposed algorithms and ACMA gets
smaller as the number of samples $K$ increases. We can say that the proposed
HG-CMA and G-CMA algorithms are more suitable as compared to ACMA for non-
constant modulus constellations, since they provide better performance for a
lower computational cost.
In Figs. 5 and 6, we plot the SINR of HG-CMA, G-CMA and ACMA vs. the number of
samples $K$ for 8-PSK and 16-QAM constellations, respectively. We compare the
performance of the proposed algorithms HG-CMA and G-CMA with ACMA for
different antenna configurations and SNR=30 dB. In both figures we observe
that, the larger the number of samples, the better the performance. In Fig. 5,
in the case of 8-PSK constellation, we observe that HG-CMA provides the best
performance. For small number of samples, G-CMA outperforms ACMA. However, for
large number of samples ACMA performs better. In Fig. 6 for 16-QAM, HG-CMA and
G-CMA outperform ACMA and the gap is larger for small number of samples and
decreases as the number of samples increases.
In Figs. 7 and 8 we plot the symbol error rate (SER) of HG-CMA, G-CMA and ACMA
vs. SNR for different number of samples $K$ for 8-PSK and 16-QAM
constellations, respectively. We considered $M=5$ and $N=7$. In Fig. 7, for
8-PSK case, we notice that the proposed HG-CMA provides the best performance.
We also observe that G-CMA outperforms ACMA for small number of samples, here
$K=20$. However, for large number of samples ACMA performs better than G-CMA
for all SNRs. Note that for very large SNR and $K\geq M^{2}$ it is expected
that ACMA outperforms HG-CMA since ACMA in this case provides the optimal
(exact in the noiseless case) solution. In the case of 16-QAM in Fig. 8, we
observe that the proposed HG-CMA and G-CMA algorithms always outperform ACMA,
even for large number of samples. Therefore, we can conclude that the proposed
HG-CMA and G-CMA are preferable to ACMA in the case of non-constant modulus
constellations, i.e. 16-QAM, for any number of samples. In the case of
constant modulus constellations, e.g. PSK, HG-CMA and G-CMA are better than
ACMA for small number of samples. However, for large number of samples and the
range of interest of SNR from $0-30$ dB, HG-CMA and ACMA have close
performance and ACMA is better than G-CMA.
To assess the performance of the adaptive HG-CMA, we consider here, unless
stated otherwise, a $5\times 5$ MIMO system (i.e. $M=5$), an i.i.d. 8-PSK
modulated sequences as input sources, and the processing window size is set
equal to $K=2M$. In Fig. 9, we compare the convergence rates and separation
quality of adaptive HG-CMA (with different number of rotations per time
instant), LS-CMA and adaptive ACMA. One can observe that adaptive HG-CMA
outperforms the two other algorithms in this simulation context. Even with
only two rotations per time instant, our algorithm leads to high separation
quality with fast convergence rate (typically, few tens of iterations are
sufficient to reach the steady state level).
In Fig. 10, the plots represent the steady state SINR (obtained after 1000
iterations) versus the SNR. One can see that the adaptive HG-CMA has no floor
effect (as for the LS-CMA and adaptive ACMA) and its SINR increases almost
linearly with the SNR in dB.
In Fig. 11, the SNR is set equal to $20dB$ and the plots represent again the
steady state SINR versus the number of sources $M$. Severe performance
degradation is observed (when the number of sources increases) for the LS-CMA
and adaptive ACMA while the adaptive HG-CMA performance seems to be
unaffected. In Fig. 12, the plots illustrate the algorithms performance versus
the chosen processing window size999This concerns only LS-CMA and adaptive HG-
CMA as the adaptive ACMA in [9] uses an exponential window with parameter
$\beta=0.995$. $K$. Surprisingly, HG-CMA algorithm reaches its optimal
performance with relatively short window sizes ($K$ can be chosen of the same
order as $M$).
In the last experiment (Fig. 13), we consider 16-QAM sources (with non CM
property). In that case, all algorithms performance are degraded but adaptive
HG-CMA still outperforms the two other algorithms. To improve the performance
in the case of non constant modulus signals, one needs to increase the
processing window size as illustrated by this simulation result but more
importantly, one needs to use more elaborated cost functions which combines
the CM criterion with alphabet matching criteria e.g. [12, 13].
Figure 1: Average SINR of HG-CMA vs. SNR. $M=5$, $N=7$, $K=100$, 8-PSK,
16-QAM, and the number of sweeps is 10. Figure 2: Average SINR of HG-CMA and
G-CMA vs. SNR. The effect of the number of sweeps on the performance of G-CMA.
$M=5$, $N=7$, $K=100$, and 8-PSK. Figure 3: Average SINR of HG-CMA, G-CMA, and
ACMA vs. SNR for different numbers of samples $K$. 8-PSK case, $M=5$, $N=7$,
and 10 sweeps. Figure 4: Average SINR of HG-CMA, G-CMA and ACMA vs. SNR for
different numbers of samples $K$. 16-QAM case, $M=5$, $N=7$, and 10 sweeps.
Figure 5: Average SINR of HG-CMA, G-CMA and ACMA vs. the number of samples $K$
for different antenna configurations. 8-PSK case, SNR=30 dB, and 10 sweeps.
Figure 6: Average SINR of HG-CMA, G-CMA and ACMA vs. the number of samples $K$
for different antenna configurations. 16-QAM case, SNR=30 dB, and 10 sweeps.
Figure 7: Average symbol error rate of HG-CMA, G-CMA and ACMA vs. SNR for
different numbers of samples $K$. 8-PSK case, $M=5$, $N=7$, and 10 sweeps.
Figure 8: Average symbol error rate of HG-CMA, G-CMA and ACMA vs. SNR for
different numbers of samples $K$. 16-QAM case, $M=5$, $N=7$, and 10 sweeps.
Figure 9: SINR vs. Time Index: $SNR=20dB$, $M=N=5$, $K=10$, 8-PSK. Figure 10:
SINR vs. SNR: $M=N=5$, $K=10$, 8-PSK. Figure 11: SINR vs. Source Number:
$SNR=20dB$, $K=2M$, 8-PSK. Figure 12: SINR vs. Bloc Size K: $M=N=5$, 8-PSK.
Figure 13: SINR vs. SNR: $M=N=5$, 16-QAM.
## 7 Conclusion
We proposed two algorithms, G-CMA and HG-CMA, for BSS in the context of MIMO
communication systems based on the CM criterion. In G-CMA we combined
prewhitening and Givens rotations and in HG-CMA we combined Shear rotations
and Givens rotations. G-CMA is appropriate for large number of samples since
in this case prewhitening is accurate. However, in the case of small number of
samples HG-CMA is preferred since Shear rotations allow to compensate for the
prewhitening stage, i.e., reduce the departure from normality. For PSK
constellations and small number of samples, we showed that the proposed HG-CMA
and G-CMA algorithms are better than the conventional ACMA. However for large
number of samples HG-CMA and ACMA have close performance and ACMA outperforms
G-CMA. In the case of 16-QAM constellation, HG-CMA and G-CMA outperform
largely the conventional ACMA for small number of samples.
Also, for the HG-CMA, a moderate complexity adaptive implementation is
considered with the advantages of fast convergence rate and high separation
quality. The simulation results illustrate its effectiveness as compared to
the adaptive implementations of ACMA and LS-CMA. They show that the sliding
window size can be chosen as small as twice the number of sources without
significant performance loss. Also, they illustrate the trade off between the
convergence rate and the algorithm’s numerical cost as a function of the
number of used rotations per iteration. As a perspective, the proposed
technique can be adapted for the optimization of more elaborated cost
functions which combine the CM criteria with alphabet matching criteria.
## 8 Appendix A
It has been shown in subsection 4.1.1 that the optimal solution in the sense
of minimizing the CM criterion in (35) is given by (see equation (45)):
$\mathbf{u}=(\mathbf{R}+\lambda\mathbf{J}_{3})^{-1}\mathbf{r}$ (73)
where $\lambda$ is the solution of:
$\mathbf{u}^{T}\mathbf{J}_{3}\mathbf{u}=1\Longleftrightarrow\mathbf{r}^{T}(\mathbf{R}+\lambda\mathbf{J}_{3})^{-1}\mathbf{J}_{3}(\mathbf{R}+\lambda\mathbf{J}_{3})^{-1}\mathbf{r}=1$
(74)
In the following, we will show that (74) is a $6$-th order polynomial
equation. Let the $3\times 3$ matrices $\mathbf{U}$ and
$\mathbf{\Lambda}=\mbox{diag}\left[\lambda_{1}~{}\lambda_{2}~{}\lambda_{3}\right]$
be the generalized eigenvectors and eigenvalues matrices of the matrix pair
($\mathbf{R}$, $\mathbf{J}_{3}$), i.e.
$\mathbf{R}=\mathbf{J}_{3}~{}\mathbf{U}~{}\mathbf{\Lambda}~{}\mathbf{U}^{-1}$
(75)
and hence:
$\left(\mathbf{R}+\lambda\mathbf{J}_{3}\right)^{-1}=\mathbf{U}\left(\mathbf{\Lambda}+\lambda\mathbf{I}_{3}\right)^{-1}\mathbf{U}^{-1}\mathbf{J}_{3}$
(76)
replacing (76) in (74) leads to:
$\mathbf{r}^{T}\mathbf{U}\left(\mathbf{\Lambda}+\lambda\mathbf{I}_{3}\right)^{-2}\mathbf{U}^{-1}\mathbf{J}_{3}\mathbf{r}=\mathbf{a}^{T}\left(\mathbf{\Lambda}+\lambda\mathbf{I}_{3}\right)^{-2}\mathbf{b}=1$
(77)
where
$\mathbf{a}^{T}=\mathbf{r}^{T}\mathbf{U}=\left[a_{1}~{}a_{2}~{}a_{3}\right]$
and
$\mathbf{b}=\mathbf{U}^{-1}\mathbf{J}_{3}\mathbf{r}=\left[b_{1}~{}b_{2}~{}b_{3}\right]^{T}$.
Knowing that
$\left(\mathbf{\Lambda}+\lambda\mathbf{I}_{3}\right)^{-2}=\mbox{diag}\left[(\lambda+\lambda_{1})^{-2},~{}(\lambda+\lambda_{2})^{-2},~{}(\lambda+\lambda_{3})^{-2}\right]$,
(77) is rewritten as:
$\sum_{i=1}^{3}\frac{a_{i}b_{i}}{\left(\mathbf{\lambda}+\lambda_{i}\right)^{2}}=1$
(78)
which is equivalent to:
$\prod_{i=1}^{3}\left(\mathbf{\lambda}+\lambda_{i}\right)^{2}-\sum_{i=1}^{3}a_{i}b_{i}\prod_{j=1,j\neq
i}^{3}\left(\mathbf{\lambda}+\lambda_{j}\right)^{2}=0$ (79)
Which is a $6$-th order polynomial equation of the form
$P_{6}(\lambda)=c_{0}\lambda^{6}+c_{1}\lambda^{5}+c_{2}\lambda^{4}+c_{3}\lambda^{3}+c_{4}\lambda^{2}+c_{5}\lambda+c_{6}=0$
with:
$\displaystyle\begin{array}[]{lll}c_{0}=1,~{}~{}c_{1}=2\sum_{i=1}^{3}\lambda_{i},~{}~{}c_{2}=\sum_{i=1}^{3}\left(\lambda_{i}^{2}+4\prod_{j=1,j\neq
i}^{3}\lambda_{j}\right)-\mathbf{a}^{T}\mathbf{b}\\\
c_{3}=2\sum_{i=1}^{3}\left(\left(\lambda_{i}^{2}-a_{i}b_{i}\right)\sum_{j=1,j\neq
i}^{3}\lambda_{j}\right),~{}~{}c_{6}=\prod_{i=1}^{3}\lambda_{i}^{2}-\sum_{i=1}^{3}a_{i}b_{i}\prod_{j=1,j\neq
i}^{3}\lambda_{j}^{2}\\\
c_{4}=\lambda_{1}^{2}\lambda_{2}^{2}\left(1+\lambda_{3}^{2}\right)+4\prod_{i=1}^{3}\lambda_{i}\sum_{i=1}^{3}\lambda_{i}-\sum_{i=1}^{3}a_{i}b_{i}\left(\sum_{j=1,j\neq
i}^{3}\lambda_{j}^{2}+4\prod_{j=1,j\neq i}^{3}\lambda_{j}\right)\\\
c_{5}=2\left(\prod_{i=1}^{3}\lambda_{i}\right)\left(\sum_{i=1}^{3}\prod_{j=1,j\neq
i}^{3}\lambda_{j}\right)-\sum_{i=1}^{3}a_{i}b_{i}\left(\sum_{j=1,j\neq
i}^{3}\lambda_{j}\right)\left(\prod_{j=1,j\neq i}^{3}\lambda_{j}\right)\\\
\end{array}$
Using the same reasoning, we can find the coefficients of the $4$-th order
polynomial equation in (58);
$P_{4}(\lambda)=c_{0}\lambda^{4}+c_{1}\lambda^{3}+c_{2}\lambda^{2}+c_{3}\lambda^{1}+c_{4}=0$.
$\displaystyle\begin{array}[]{lll}c_{0}=1,~{}~{}c_{1}=2\sum_{i=1}^{2}\tilde{\lambda_{i}},~{}~{}c_{2}=\sum_{i=1}^{2}\tilde{\lambda_{i}}^{2}+4\prod_{j=1,j\neq
i}^{2}\tilde{\lambda_{j}}-\tilde{\mathbf{a}}^{T}\tilde{\mathbf{b}}\\\
c_{3}=2\sum_{i=1}^{2}\left(\tilde{\lambda_{i}}^{2}-\tilde{a}_{i}\tilde{b}_{i}\right)\sum_{j=1,j\neq
i}^{2}\tilde{\lambda_{j}},~{}~{}c_{4}=\prod_{i=1}^{2}\tilde{\lambda_{i}}^{2}-\sum_{i=1}^{2}\tilde{a}_{i}\tilde{b}_{i}\prod_{j=1,j\neq
i}^{2}\tilde{\lambda_{j}}^{2}\end{array}$
with
$\tilde{\mathbf{a}}^{T}=\tilde{\mathbf{r}}^{T}\tilde{\mathbf{U}}=\left[\tilde{a}_{1}~{}\tilde{a}_{2}\right]$
and
$\tilde{\mathbf{b}}=\tilde{\mathbf{U}}^{-1}\mathbf{J}_{2}\tilde{\mathbf{r}}=\left[\tilde{b}_{1}~{}\tilde{b}_{2}\right]^{T}$.
Where the $2\times 2$ matrices $\tilde{\mathbf{U}}$ and
$\tilde{\mathbf{\Lambda}}=\mbox{diag}\left[\tilde{\lambda}_{1}~{}\tilde{\lambda}_{2}\right]$
represent the generalized eigendecomposition of the matrix pair
($\tilde{\mathbf{R}}$, $\mathbf{J}_{2}$).
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|
arxiv-papers
| 2013-06-18T10:17:56 |
2024-09-04T02:49:46.622136
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Aissa Ikhlef, Redha Iferroujene, Abdelouahab Boudjellal, Karim\n Abed-Meraim, Adel Belouchrani",
"submitter": "Abdelouahab Boudjellal boudjwahab",
"url": "https://arxiv.org/abs/1306.4128"
}
|
1306.4131
|
# On characterization of toric varieties
Ilya Karzhemanov
###### Abstract.
We study the conjecture due to V. V. Shokurov on characterization of toric
varieties. We also consider one generalization of this conjecture. It is shown
that none of the characterizations holds true in dimension $\geq 3$. Some
weaker versions of the conjecture(s) are verified.
MS 2010 classification: 14M25, 14E30
Key words: toric variety, Picard number, log pair, vector bundle
## 1\. Introduction
### 1.1.
There is an abundance of results on characterization of algebraic varieties
with a transitive group action. Some of those we are familiar with are [28],
[19], [41], [9] (for projective spaces and hyperquadrics) and [16], [42] (for
Grassmannians and other Hermitian symmetric spaces). At the same time, not
very much is known, from the point of view of characterization, for varieties
with other, less transitive, group actions. Amongst the first that come into
ones mind are toric varieties and (more generally) reductive varieties (see
[1], [2] (and also [5], [31]) for foundations of the reductive (resp.
spherical) theory). In the same spirit, there is a related Hirzebruch problem
to describe compactifications of $\mathbb{C}^{n}$, the topic studied in
numerous papers (see [32], [10], [11] and [36] for instance). Postponing the
discussion of all these matters until another paper, lets focus on the toric
case.
To begin with, we mention recent paper [21] (which elaborates on [18]), where
an arbitrary (smooth complete) toric variety $T$ is characterized by the
property that a certain sheaf
$\mathcal{R}_{T}\in\text{Ext}^{1}(\mathcal{O}_{T}^{\oplus
h^{1,1}(X,\mathbb{C})},\Omega^{1}_{T})$ (called _potential_) splits into a
direct sum of line bundles $\mathcal{O}_{T}(-D_{\alpha})$. Unfortunately, this
is not quite an effective criterion, and we are up for a “numerical” one.
### 1.2.
Let $X\longrightarrow Z\ni o$ be an algebraic variety1)1)1)All varieties, if
not specified, are assumed to be normal, projective and defined over
$\mathbb{C}$. over a scheme (germ) $Z\ni o$. Put $n:=\dim X$ and let $n\geq 2$
in what follows.
Consider a $\mathbb{Q}$-boundary2)2)2)We will be using freely notions and
facts from the minimal model theory (see [4], [23], [22]). $D:=\sum
d_{i}D_{i}$, where $D_{i}$ are (not necessarily prime) Weil divisors on $X$
and $0\leq d_{i}\leq 1$ for all $i$. Through the rest of the paper $X$ and $D$
will be subject to the following constraints:
* •
the pair $(X,D)$ is log canonical (lc for short);
* •
the divisor $-(K_{X}+D)$ is nef;
* •
singularities of $X$ are $\mathbb{Q}$-factorial (or _$X$ is
$\mathbb{Q}$-factorial_).
Actually, the last condition is redundant (and the first one is too general)
for the forthcoming considerations. Indeed, one may always apply a dlt
modification $h:\tilde{X}\longrightarrow X$, with $\mathbb{Q}$-factorial
$\tilde{X}$, dlt pair $(\tilde{X},h_{*}^{-1}(D))$ and
$K_{\tilde{X}}+h_{*}^{-1}(D)\equiv h^{*}(K_{X}+D)$.3)3)3)$\equiv$ stands for
the numerical equivalence. Replacing the pair $(X,D)$ by
$(\tilde{X},h_{*}^{-1}(D))$ doesn’t effect the arguments of the present paper.
Let $N^{1}(X)$ be the Néron-Severi group of $X$. One defines the number
$r(X,D)$ as the dimension of $\mathbb{Q}$-vector subspace in
$N^{1}(X)\otimes\mathbb{Q}$ generated by all the $D_{i}$’s. Alternatively, if
one drops the $\mathbb{Q}$-factoriality assumption, $r(X,D)$ is defined as the
rank of the group generated by all the $D_{i}$’s modulo algebraic equivalence.
Clearly, $r(X,D)\leq\rho(X)=$ the Picard number of $X$. A finer relation
between these gadgets is provided by the following:
###### Conjecture 1.3 (V. V. Shokurov).
For $(X/Z\ni o,D)$ as above, the estimate $\sum d_{i}\leq r(X,D)+\dim X$
holds. Moreover, the equality is achieved iff the pair $(X,\llcorner
D\lrcorner)$ is (formally) toric, i.e. $X$ is toric and $\llcorner D\lrcorner$
is its boundary (all over $Z\ni o$).
Notice that when $X$ is a genuine toric variety and $D$ is its boundary, with
$D_{i}$ irreducible and $d_{i}=1$ for all $i$, then obviously $K_{X}+D\sim 0$
and $\sum d_{i}=r(X,D)+\dim X$, thus motivating the last statement of the
above Conjecture. Lets give another
###### Example 1.4 (cf. [34, Example 1.3]).
Take $Z:=X$, so that $(X/X\ni o,D)$ is a singularity germ. Let
$X:=(xy+zt=0)\subset\mathbb{C}^{4}$ and $D:=(xy=0)\cap X$. Then we have
$r(X,D)=1$ and $K_{X}=0$. Hence $\sum d_{i}=r(X,D)+\dim X$ and $X\ni o$ is
obviously toric. In fact this is not a coincidence, as the local version (i.e.
with $Z=X$) of Conjecture 1.3 was proved in [22, 18.22 – 18.23]. Namely, given
$(X/X\ni o,D)$ such that each $D_{i}$ is $\mathbb{Q}$-Cartier, the estimate
$\sum d_{i}\leq\dim X$ holds. Moreover, in the case of equality one has
$X\simeq(\mathbb{C}^{n}\ni 0)/\mathfrak{A}$, where $\mathfrak{A}$ is a finite
abelian group acting diagonally on $\mathbb{C}^{n}$ and $D_{i}$ correspond to
$\mathfrak{A}$-invariant hyperplanes $(x_{i}=0)\subset\mathbb{C}^{n}$ with
respect to the factorization morphism $\mathbb{C}^{n}\longrightarrow X$. In
particular, the pair $(X,\llcorner D\lrcorner)$ is toric.
###### Remark 1.5.
In view of Example 1.4, it is tempting to ask whether $X$ (and $D$) in
Conjecture 1.3 is by any means related to a toric variety, say whether
$(X,\llcorner D\lrcorner)$ is _formally_ (not necessarily regularly or even
analytically) isomorphic to a toric pair? A $\text{char}~{}p>0$ version of
Conjecture 1.3 might also be of some interest: for instance, when the ground
field is $\overline{\mathbb{F}}_{p}$, is $(X,\llcorner D\lrcorner)$ a toric
pair (up to Frobenius twist)? Finally, one may consider a weaker version of
Conjecture 1.3, with “$X$ is toric” replaced by “$X$ admits a torification”
(compare with e.g. [24]).
### 1.6.
In order to attack Conjecture 1.3, one may assume that $Z\neq X$ (see Example
1.4). Then, replacing $Z$ by a formal neighborhood of $o$ we are led to the
case $Z=\text{Spec}~{}\mathbb{C}[[t]]$. Moreover, taking an embedding
$\mathbb{C}[[t]]\hookrightarrow\mathbb{C}$ one may assume that $Z=o$.
The first proof of Conjecture 1.3, for $\dim X=2$, had appeared in [39,
Theorem 6.4]. The case when $\dim X=3$, the pair $(X,D)$ is plt and
$K_{X}+D\equiv 0$ had been treated in [38], and Conjecture 1.3 was proved in
full there under the stated conditions.4)4)4)Note that under the extremal
condition $\sum d_{i}=r(X,D)+\dim X$ all (potentially toric) $3$-folds in [38]
turned out to be actually toric and smooth (see [38, Theorem 1.2]). Lets also
point out a birational (“rough”) version of Conjecture 1.3 (cf. Theorem 1.15
below), proved in [34] assuming the _Weak Adjunction Conjecture_. Finally,
general strategy towards the proof of Conjecture 1.3, and more, was developed
in [27] and illustrated there (in passing) at some crucial points. We now
recall some of the matters from [27].
First of all, one generalizes Conjecture 1.3 as follows. Put
$c(X,D):=r(X,D)+\dim X-\sum d_{i}$
(the quantity $c(X,D)$ was called in [27] _complexity_ of the pair $(X,D)$).
Similarly, one defines
$ac(X,D):=\rho(X)+\dim X-\sum d_{i}$
(the _absolute complexity_), and for consistency lets also introduce
$\displaystyle c(X):=\inf\\{c(X,D)\ |\ D\ \text{is a boundary on}\ X\
\text{such that}$ $\displaystyle\text{the divisor}\ -(K_{X}+D)\ \text{is nef
and the pair}\ (X,D)\ \text{is lc}\\}$
and similarly for $ac(X)$, with $c(X,D)$ replaced by $ac(X,D)$ (evidently,
$c(X)\leq ac(X)$). Then one observes that for $d_{i}$’s all integer, the
condition $c(X,D)=0$ (so that the pair $(X,D)$ is toric by Conjecture 1.3) is
equivalent to $c(X,D)<1$. This had led the author of [27] to make his
###### Conjecture 1.7 (J. McKernan).
For $(X,D)$ as above, the following holds:
1. 1)
$c(X)\geq 0$,
2. 2)
If $ac(X,D)<2$, then $X$ is a rational variety,
3. 3)
If $c(X,D)<1$, then there is a divisor $D^{\prime}$ such that the pair
$(X,D^{\prime})$ is toric. Moreover, $\llcorner D\lrcorner\subseteq
D^{\prime}$ and $D^{\prime}-S$ is linearly equivalent to a divisor with
support in $D$, where $S$ is either empty or an irreducible divisor.
Obviously, Conjecture 1.7 $\Longrightarrow$ Conjecture 1.3.
###### Remark 1.8.
Note that one can not loose the assumptions in Conjectures 1.3, 1.7. Indeed,
for $X:=\mathbb{P}^{1}\times\mathbb{P}^{1}\times E$ and
$D:=0\times\mathbb{P}^{1}\times E+\infty\times\mathbb{P}^{1}\times
E+\mathbb{P}^{1}\times 0\times E+\mathbb{P}^{1}\times\infty\times E$, $E$ is
an elliptic curve, we have $c(X,D)=c(X)=1$, but $X$ is not even rationally
connected (cf. Conjecture 1.7, 2)). Also, taking $X:=\mathbb{F}_{m}$ and
$D:=2E_{\infty}+\displaystyle\sum_{i=1}^{m+2}F_{i}$, where $E_{\infty}$ is the
negative section and $F_{i}$ are fibers of the natural projection
$\mathbb{F}_{m}\longrightarrow\mathbb{P}^{1}$ (so that $K_{X}+D=0$), we get
$c(X,D)\leq 0$, but the pair $(X,D)$ is non-toric (cf. Conjecture 1.7, 3)). At
the same time, as one immediately verifies, slightly more general versions of
Conjectures 1.3, 1.7, studied in [39], [34], [34] and [27], are equivalent to
those stated above.
### 1.9.
The aim of the present paper is to show that Conjecture 1.7 is not true as
stated when $\dim X\geq 3$ (cf. Remark 4.26). Before spelling out the details
lets give a
###### Definition 1.10.
(one of many possible) Call an algebraic variety $X$ _fake toric_ (or
_$\mathfrak{f}$ -toric_ for short) if there exists a $\mathbb{Q}$-boundary $D$
on $X$, _quasi-toric boundary_ (or _qt-boundary_), such that the pair $(X,D)$
satisfies all the conditions in 1.2 (up to passing to a
$\mathbb{Q}$-factorialization), the pair $(X,\llcorner D\lrcorner)$ is non-
toric, but $c(X,D)<1$. Denote by $\mathfrak{T}^{\mathfrak{f},n}$ the class of
all $\mathfrak{f}$-toric varieties of dimension $n$.
Here is our main result:
###### Theorem 1.11.
For every $n\geq 3$, the class $\mathfrak{T}^{\mathfrak{f},n}\neq\emptyset$.
More precisely, there is $X\in\mathfrak{T}^{\mathfrak{f},n}$ with a qt-
boundary $D$ such that $K_{X}+D\equiv 0$ and $ac(X,D)=\frac{1}{2}$.
The proof of Theorem 1.11 occupies Section 4 below. We indicate that our
construction of varieties from $\mathfrak{T}^{\mathfrak{f},n}$ is very much
motivated by the exposition in [27]. In fact, the arguments of [27]
essentially rely on two assertions, namely that an algebraic variety
isomorphic to a toric variety in codimension $1$ is also toric (see [27, §3]),
and that the divisor $K_{X}+D$ has an _integral_ $\mathbb{Q}$-complement (see
[37], [39]), i.e. one can always find an integral Weil divisor $D^{\prime}\geq
D$ (for $D$ considered up to the linear equivalence) such that the pair
$(X,D^{\prime})$ is lc and $K_{X}+D^{\prime}\equiv 0$ (see [27, §4]). Using
these together with a $\dim X$-inductive argument (cf. 1.13 below) eventually
proves Conjecture 1.7.
The outlined strategy applies well when $\dim X=2$ and justifies Conjecture
1.3 in this case (cf. [39], [38, Proposition 2.1], [37, Theorem 8.5.1]). At
the same time, rational scrolls (a. k. a. $\mathbb{P}^{N}$-bundles over
$\mathbb{P}^{1}$) are all toric, so that it is reasonable to test Conjecture
1.7 on $X$ having the form of a projectivization $\mathbb{P}(\mathcal{E})$ for
some indecomposable vector bundle $\mathcal{E}$ over $\mathbb{P}^{2}$. And it
is this $X$ for which the method of [27], together with Conjecture 1.7,
breaks. Essentially, when $\dim X=3$, we obtain $X$ as the blowup of the
smooth quadric $Q\subset\mathbb{P}^{4}$ in a line $L\subset Q$, followed by
factorizing via two $\linebreak\mathbb{Z}/_{\displaystyle
2\mathbb{Z}}$-actions. Both $\mathbb{Z}/_{\displaystyle 2\mathbb{Z}}$ act on
$Q$ preserving $L$ and are constructed as follows. One
$\mathbb{Z}/_{\displaystyle 2\mathbb{Z}}$ corresponds to the Galois action for
the $2$-to-$1$ projection $Q\longrightarrow\mathbb{P}^{3}$ and another
$\mathbb{Z}/_{\displaystyle 2\mathbb{Z}}$ is a lift to $Q$ of the Galois
involution on $\mathbb{P}^{3}$ corresponding to the quotient morphism
$\mathbb{P}^{3}\longrightarrow\mathbb{P}(1,1,1,2)$ (one can easily see such a
lifting does exist). It is then an exercise to find $D$ as in Theorem 1.11 and
simple fan considerations show that $X$ is non-toric (see Section 4 for
further details).5)5)5)Our arguments to prove Theorem 1.11 are so designed in
order to allow one describe the class $\mathfrak{T}^{\mathfrak{f},n}$ in full
for every $n$ (cf. 4.28, A) below). We plan to advance on this way elsewhere.
Finally, observe at this point that so obtained $X$ is _singular_ , while for
smooth $X$’s Conjecture 1.3 has been proved recently in [43].
###### Remark 1.12.
Lets collect some heuristics which provides an indication for that the above
mentioned “toric-in-codimension-$1$” and “$\mathbb{Q}$-complementary”
assertions do not hold in general. Firstly, in the former case given two
varieties $X_{1},X_{2}$ (we assume both $X_{i}$ to be $\mathbb{Q}$-factorial),
with $X_{1}$ toric and $\theta:X_{1}\dashrightarrow X_{2}$ an isomorphism in
codimension $1$, it is easy to see that the indeterminacy locus of $\theta$ is
torus-invariant (see e.g. [23, Lemma 6.39]). Furthermore, as long as $X_{1}$
is a Mori dream space (see [15]), $\theta$ can be factored into a sequence of
torus-invariant $\Delta$-flips w. r. t. a movable divisor $\Delta$ on $X_{1}$,
so that $\theta$ and $X_{2}$ are also toric. This may work for instance when
both $X_{i}$ admit integral boundaries $D_{i}$ and birational contractions
$f_{i}:X_{i}\longrightarrow Y$ such that $c(X_{i},D_{i})=0$ and $Y$ (hence
$(Y,f_{i*}(D_{i}))$) is toric (see [27], the discussion before$/$after
Definition-Lemma 3.4).6)6)6)Note however that there is a substantial gap in
[27] at this point. Namely, let $E_{i}$ be the $f_{i}$-exceptional divisor,
inducing a discrete valuation $v_{i}$ on the field $\mathbb{C}(Y)$. Then it is
claimed in [27] that (for $(X_{i},D_{i})$, etc. as given, but without the
assumption on $c(X_{i},D_{i})$) there is a sequence of toric blowups
extracting $v_{i}$ (starting with the blowup of $f_{i}(E_{i})$). But this
doesn’t occur in general (globally at least) because the scheme $f_{i}(E_{i})$
may not be reduced (this is a popular spot of erroneous usage of [23, Lemma
2.45] – replacing the initial scheme $f_{i}(E_{i})$ by $f_{i}(E_{i})_{red}$).
For example, consider $Y:=\mathbb{C}^{2}$ with (toric) boundary
$\Delta:=(xy=0)$, and let $f_{1}:X_{1}\longrightarrow Y$ be the blowup of the
scheme $Z:=((x+y^{2})^{2}=y^{3}=0)$ (supported on $(0,0)$). In other words,
$f_{1}$ is the weighted blowup with weights $(3,2)$, so that
$K_{X}+f_{1*}^{-1}\Delta=f_{1}^{*}(K_{Y}+\Delta)-E_{1}$ and the pair
$(X_{1},D_{1}:=f_{1*}^{-1}\Delta+E_{1})$ is lc. However, the map $f_{1}$
doesn’t extend to a toric morphism
$\widetilde{f_{1}}:\widetilde{X_{1}}\longrightarrow\mathbb{P}^{2}$ for a toric
surface $\widetilde{X_{1}}$ compactifying $X_{1}$ (resp. $\mathbb{P}^{2}$
compactifying $Y$) because $((x+y^{2})^{2},y^{3})$, the defining ideal of $Z$,
does not coincide with $(x^{2},y^{3})$. One can find non-toric
$\widetilde{X_{1}},\widetilde{f_{1}}$ though (cf. $(X,D)$ in 4.21 below). But
for general $\mathbb{Q}$-boundaries $D_{i}$, $f_{i*}(D_{i})$ need not be
supported on the toric boundary of $Y$, and this brings us to the
“$\mathbb{Q}$-complementary” part. In the latter case, one might arrive at a
variety $X$ acted by finite groups $\displaystyle\prod_{i=1}^{k}G$, any $k\geq
1$, which yields a sequence $X^{i}$ of quotients –
$X^{0}:=X,~{}X^{1}:=X/G,~{}X^{2}:=X^{1}/G,\ldots$ – admitting a Gromov-
Hausdorff limit $X^{\infty}$. Pretending for a moment that all $X^{i}$ are
smooth, $X^{\infty}$ turns out to be proper algebraic (see [7]), carrying an
$\infty$-degree map $X\longrightarrow X^{\infty}$, which is absurd. However,
we propose the _exceptional_ complements do not occur for the
$\mathfrak{f}$-toric pairs, as well as for the toric ones (cf. [39], [35],
[17], [25], [26]).
### 1.13.
We conclude by stating positive versions of Conjecture 1.7 we were able to
find:
###### Proposition 1.14 (Yu. G. Prokhorov).
Let $(X,D)$ be as in 1.2. Assume also that $K_{X}+D\equiv 0$ and $d_{i}=1$ for
all $i$. Then
1. 1)
$c(X,D)\geq 0$.
2. 2)
If $c(X,D)=0$, then $X$ is rationally connected. Furthermore, if $X$ has only
terminal singularities, then $D$ has a rationally connected component.
(Proposition 1.14 is proved in Section 2 below. The arguments are entirely due
to Yu. G. Prokhorov.)
###### Theorem 1.15.
For $(X,D)$ as in Proposition 1.14, if $ac(X,D)=0$, then $X$ is rational.
Furthermore, if $ac(X,D)=1$ and $X$ is rationally connected, then $X$ is
rational as well.
###### Remark 1.16.
In Theorem 1.15, it might be possible to replace $ac(X,D)$ (resp. $D$ being
integral) by $c(X,D)$ (resp. $\llcorner D\lrcorner\neq 0$), but we couldn’t
extend our arguments to this setting.
It is in the proof of Theorem 1.15, where we use a nice inductive trick
suggested in [27]. Namely, since $\text{Supp}\,D$ consists of $\geq\rho(X)+1$
(irreducible) divisors, we can find a map (a “Morse function”)
$f:X\longrightarrow\mathbb{P}^{1}$ whose fibers satisfy the hypotheses of
Theorem 1.15 (see 3.7 below), and one may then argue by induction (see Section
3 for further details).
## 2\. Proof of Proposition 1.14
### 2.1.
As $K_{X}\equiv-D\neq 0$, we can run the $K_{X}$-MMP (see [4, Corollary
1.3.2]). Furthermore, since $K_{X}+D\equiv 0$, the components of $D$ are not
contracted on each step (see e.g. [23, Lemma 3.38]). Note also that the
quantity $c(X,D)$ doesn’t increase on each step. In particular, we may assume
there is an extremal contraction $\phi:X\longrightarrow Y$ which is a Mori
fiber space. Finally, as one can easily see by restricting $D$ to generic
fiber $F$ of $\phi$, there is a horizontal component $D_{0}\subset D$, i.e.
$\phi(D_{0})=Y$. We may take $D_{0}$ irreducible.
###### Lemma 2.2 (cf. [38, Corollary 3.7.1]).
We have either $D_{0}\cap D_{i}\neq\emptyset$ for all $i$, or there is exactly
one $j$ such that $D_{0}\cap D_{j}=\emptyset$. In the latter case,
$\phi:X\longrightarrow Y$ is a conic bundle with two sections $D_{0}$ and
$D_{j}$.
###### Proof.
Suppose first that $\dim X/Y>1$. Then, since $D_{0}\big{|}_{F}$ is ample, one
gets $D_{0}\cap D_{i}\neq\emptyset$ for all $i$. Further, if $\dim X/Y=1$ and
$D_{0}$ doesn’t intersect at least two of the $D_{i}$, say $D_{0}\cap
D_{1}=D_{0}\cap D_{2}=\emptyset$, then $D_{0},D_{1},D_{2}$ are multisections
of $\phi$ and we obtain
$0\equiv\big{(}K_{X}+D\big{)}\big{|}_{F}=K_{F}+D\big{|}_{F}>0$
on $F=\mathbb{P}^{1}$, a contradiction. Thus there is exactly one $D_{j}$ with
$D_{j}\cap D_{0}=\emptyset$. Then, since
$K_{F}=\mathcal{O}_{\mathbb{P}^{1}}(-2)$, we have $D_{0}\cdot F=D_{j}\cdot
F=1$. Hence both $D_{0},D_{j}$ are sections of $\phi$. ∎
Take the divisor
$B:=\sum_{i\neq 0,\ D_{i}\cap D_{0}\neq\emptyset}D_{i},$
normalization $\nu:D_{0}^{\nu}\longrightarrow D_{0}$ and let
$B^{\nu}:=\text{Diff}_{\scriptscriptstyle D_{0}}(B)$ be the different of $B$
on $D_{0}^{\nu}$ (see e.g. [40, §3]). Note that
$K_{D_{0}^{\nu}}+B^{\nu}=\nu^{*}\big{(}\big{(}K_{X}+D\big{)}\big{|}_{D_{0}}\big{)}\equiv
0$
in our case.
Further, cutting with hyperplane sections we obtain that $B^{\nu}=N+\nu^{*}B$,
where $N$ is the preimage of the non-normal locus on $D_{0}$ (cf. [40, §3]).
Also, by Inversion of Adjunction (see [20]), the pair $(D_{0}^{\nu},B^{\nu})$
is lc (because $(X,D)$ is). So, by the discussion at the beginning of 1.2 we
may replace the pair $(X,D)$ with $(D_{0}^{\nu},B^{\nu})$. Moreover, we have
$B^{\nu}=N+\nu^{*}B=N+\sum_{i\neq 0}\nu^{*}D_{i},$
which easily yields $c(D_{0}^{\nu},B^{\nu})\leq c(X,D)$ (see Lemma 2.2), and
Proposition 1.14, 1) follows by induction on the dimension (notice that
$B^{\nu}$ is integral by construction).
Suppose now that $c(X,D)=0$. Then $c(D_{0}^{\nu},B^{\nu})=0$ and $D_{0}$ is
rationally connected by induction. Furthermore, as $F$ is a Fano variety with
only log terminal singularities, it is rationally connected (see [14,
Corollary 1.3]). This implies that $X$ is rationally connected (see [13,
Corollary 1.3]). Finally, if $X$ has only terminal singularities, then its dlt
modification $\tilde{X}\longrightarrow X$ (cf. 1.2) is a small birational
morphism, and Proposition 1.14, 2) follows.
###### Corollary 2.3.
For $X$ as in Proposition 1.14 (in fact for any rationally connected $X$) and
any $\mathbb{Q}$-Cartier divisors
$L_{1},L_{2}\in\text{Pic}(X)\otimes\mathbb{Q}$, we have $L_{1}\equiv L_{2}\iff
L_{1}\sim_{\mathbb{Q}}L_{2}$.
###### Proof.
We may replace $X$ by its resolution $X^{\prime}$. Let
$g:X^{\prime}\longrightarrow X$ be some birational contraction. Lets also
replace each $L_{i}$ by $g^{*}(L_{i})$. The assertion now follows from
$h^{1}(\mathcal{O}_{X^{\prime}})=0$ ($=\dim\text{Alb}_{X^{\prime}}+1$).
Indeed, for $L_{1},L_{2}$ both Cartier and effective the condition $L_{1}\sim
L_{2}$ (equivalent to $L_{1}\equiv L_{2}$, as $\text{Alb}_{X^{\prime}}=0$)
simply asserts that there is a rational map ($=$ a function
$\in\mathbb{C}(X)$) $f:X\dashrightarrow\mathbb{P}^{1}$, with
$f^{-1}(0)=L_{1},~{}f^{-1}(\infty)=L_{2}$. The latter property doesn’t depend
on the resolution $g$. The argument for arbitrary $\mathbb{Q}$-Cartier $L_{i}$
is similar. ∎
## 3\. Proof of Theorem 1.15
### 3.1.
In what follows, we assume that $ac(X,D)=c(X,D)\leq 1$. The proof of Theorem
1.15 will go essentially by induction on both $\dim X$ and $\rho(X)$. In the
former case, $\dim X=2$ is the base of induction (cf. the beginning of 1.6),
while in the latter case we have the following:
###### Proposition 3.2.
The assertion of Theorem 1.15 holds when $\rho(X)=1$.
###### Proof.
After taking cyclic coverings of $X$ w. r. t. various $D_{i}$’s (see e.g. [40,
§2]), the arguments in the proof of [34, Corollary 2.8] apply and show that
$X\simeq\mathbb{P}^{n}/\mathfrak{A}\qquad\text{or}\qquad Q/\mathfrak{B},$
where $Q\subset\mathbb{P}^{n+1}$ is a smooth quadric,
$\mathfrak{A}\subset\text{Aut}(\mathbb{P}^{n}),\mathfrak{B}\subset\text{Aut}(Q)$
are finite abelian groups. Note that the case $X=\mathbb{P}^{n}/\mathfrak{A}$
has been treated in [34, Corollary 2.8] (and in fact the pair $(X,D)$ turns
out to be toric).
Suppose now that $X=Q/\mathfrak{B}$. Let $Q^{\mathfrak{B}}$ be the locus of
$\mathfrak{B}$-fixed points on $Q$.
###### Lemma 3.3.
$Q^{\mathfrak{B}}\neq\emptyset$.
###### Proof.
Notice that the $\mathfrak{B}$-action on $Q$ is induced by some (projective)
action on $\mathbb{P}^{n+1}$, since $\rho(Q)=1$ and $K_{Q}$ is
$\mathfrak{B}$-invariant. In particular, there is a $\mathfrak{B}$-invariant
line $l\subset\mathbb{P}^{n+1}$, so that the locus $l\cap Q$ is also
$\mathfrak{B}$-invariant. Now, if $l\cap Q$ is a finite set, then $|l\cap
Q|\leq 2$. Put $l\cap Q=\\{P_{1},P_{2}\\}$ for some $P_{i}\in Q$. Then it is
evident that $P_{i}\in Q^{\mathfrak{B}}$ (after possibly replacing $l$ by
another $\mathfrak{B}$-invariant line). Finally, the case when $|l\cap
Q|=\infty$, i.e. $l\subset Q$, is obvious. ∎
Take any $P\in Q^{\mathfrak{B}}$ and consider the ($\mathfrak{B}$-equivariant
birational) linear projection $Q\dashrightarrow\mathbb{P}^{n}$ from $P$. Then
the $\mathfrak{B}$-action descends to $\mathbb{P}^{n}$ and we have
$Q/\mathfrak{B}$ and $\mathbb{P}^{n}/\mathfrak{B}$ birationally isomorphic,
with rational $\mathbb{P}^{n}/\mathfrak{B}$. Proposition 3.2 is proved. ∎
###### Remark 3.4.
In general, if $X$ and each of $D_{i}$ are defined over a field k
($\text{char}~{}\textbf{k}=0$), one can easily adjust the arguments in the
proof of Proposition 3.2 to show that $X$ is k-rational, once given that
$\rho\big{(}X\otimes_{\textbf{k}}\bar{\textbf{k}}\big{)}=1$ over the algebraic
closure $\bar{\textbf{k}}$.
### 3.5.
Let $\phi:X\longrightarrow Y$ be an extremal contraction. Put
$D_{Y}:=\phi_{*}(D),D_{Y,i}:=\phi_{*}(D_{i})$ and suppose that $\phi$ is
divisorial. Note that $\phi_{*}(K_{X})=K_{Y}$ because singularities of $Y$ are
rational. Hence $K_{Y}+D_{Y}=\phi_{*}(K_{X}+D)\equiv 0$ and $c(Y,D_{Y})$ is
defined.
###### Lemma 3.6.
In the above setting, the assertion of Theorem 1.15 holds.
###### Proof.
It is clear that $0\leq c(Y,D_{Y})\leq c(X,D)$, and the result follows from
Proposition 3.2 and induction on $\rho(X)$. ∎
### 3.7.
We now turn to the construction of a pencil on $X$, as indicated in 1.9.
Firstly, we may assume that
$D_{1}\sim_{\mathbb{Q}}\sum\delta_{i}D_{i}=:D^{\prime}_{1}$ for some
$\delta_{i}\in\mathbb{Q}$, where $\delta_{2}\neq 0$ say (see Corollary 2.3).
Secondly, we may assume w. l. o. g. that $\delta_{i}\geq 0$ for all $i$
(replacing, if necessary, $D_{1}$ by $D_{1}+\sum_{j}\delta^{\prime}_{j}D_{j}$
for some $\delta_{j}\in\mathbb{Q}_{\geq 0}$), which gives a map
$f:X\dashrightarrow\mathbb{P}^{1}$ with
$kD_{1}=f^{-1}(0),kD^{\prime}_{1}=f^{-1}(\infty)$ for some $k\in\mathbb{N}$.
Finally, since the indeterminacies of $f$ are located on some of the
$D_{i}$’s, we will assume (for the sake of simplicity) that $f$ is undefined
precisely on $D_{1}\cap D_{2}$ (in general, one has to consider merely a
bigger number of $D_{j}$’s, satisfying $D_{1}\cap\bigcup D_{j}=$ the
indeterminacy locus of $f$).
Further, there is a local analytic isomorphism
$\big{(}X,D_{1}+D_{2},D_{1}\cap
D_{2}\big{)}\simeq\big{(}\mathbb{C}^{n},\\{x_{1}x_{2}=0\\},\\{x_{1}=x_{2}=0\\}\big{)}/\mathbb{Z}_{m}(1,q,0,\ldots,0)$
at the generic point of $D_{1}\cap D_{2}$, where $m,q\in\mathbb{N},(m,q)=1$
(see e.g. [40, Proposition 3.9]). In particular, if
$\phi:\tilde{X}\longrightarrow X$ is the blowup of $D_{1}\cap D_{2}$, then we
have $K_{\tilde{X}}+\phi_{*}^{-1}(D)+E\equiv\phi^{*}(K_{X}+D)$ for the
$\phi$-exceptional divisor $E$. Thus, after taking a number of subsequent
blowups of $D_{1}\cap D_{i}$ we may assume that
$f:X\longrightarrow\mathbb{P}^{1}$ is _regular_.
### 3.8.
Let $F$ be a fiber of $f$. Notice that $K_{X}\cdot Z=-D\cdot Z\leq 0$ for
generic $F$ and all curves $Z\subset F$.
###### Lemma 3.9.
$D\cdot Z>0$ for some $Z,F$ as above.
###### Proof.
Suppose the contrary. Then $D_{i}\cdot Z=0$ for all $i$ and $Z\subset F$.
Hence $f(D_{i})=\text{pt}$ for all $i$. But the latter is impossible because
$D_{i}$ generate $\text{Pic}(X)\otimes\mathbb{Q}$. ∎
Among those $Z\subset F$ with $K_{X}\cdot Z<0$ (see Lemma 3.9) there is an
extremal ray of the cone $\overline{NE}(X)$, denoted as $R$ say. The curve
representing $R$ sits in an irreducible component of $F^{\prime}\subseteq F$,
so that once $F$ is reducible, $F\setminus F^{\prime}\neq\emptyset$, we get an
extremal birational contraction $\phi:X\longrightarrow Y$ over
$\mathbb{P}^{1}$. Moreover, we eliminate the case when $\phi$ is flipping by
applying the $K_{X}$-flip (and termination of MMP with scaling). On the other
hand, if $\phi$ is divisorial (in which case it contracts $F^{\prime}$), then
we are done by Lemma 3.6.
Thus we may assume that $\rho(X)=2$ (i.e. $f:X\longrightarrow\mathbb{P}^{1}$
is a Mori fibration).
###### Lemma 3.10.
$D^{\prime}_{1}=D_{2}$ and $D_{1},D_{2}$ are the only components of $D$
contracted by $f$. Moreover, $F$ is $\mathbb{Q}$-factorial and $\rho(F)=1$.
###### Proof.
This is proved in [34, Proposition 3.6, Lemma 2.10]. ∎
Now exactly as in [34, Lemma 2.10, (iii)], Lemma 3.10, Remark 3.4 and
induction on $\dim$ imply rationality of $X$. Theorem 1.15 is completely
proved.
## 4\. Proof of Theorem 1.11
### 4.1.
Fix a smooth projective surface $S$. By $\mathcal{E}$ (resp. $\mathcal{F}$) we
will denote a vector bundle (resp. a coherent sheaf) on $S$. Lets recall some
notions and facts about $\mathcal{E}$ and $\mathcal{F}$ (see e.g. [6], [8],
[30]).
First of all, one defines the Chern classes $c_{i}:=c_{i}(\mathcal{E})\in
H^{2i}(S,\mathbb{Z})$ for $\mathcal{E}$ (and similarly for $\mathcal{F}$,
using a locally free resolution), with $c_{1}=\det\mathcal{E}$. In fact,
letting $W:=\mathbb{P}(\mathcal{E})$ there is a natural inclusion
$H^{*}(S,\mathbb{Z})\hookrightarrow H^{*}(W,\mathbb{Z})$ induced by the
projection $\pi:W\longrightarrow S$, and the following identity (called the
Hirsch formula) holds:
$H^{2}+H\cdot c_{1}+c_{2}=0.$
Here $H:=\mathcal{O}(1)\in H^{2}(W,\mathbb{Z})$ is the Serre line bundle on
$W$ and $c_{i}$ are identified with $\pi^{*}(c_{i})$ (cf. Remark 4.2). In
particular, for $r:=\text{rk}~{}\mathcal{E}=2$ we have
$H^{3}=-c_{1}^{2}-c_{2},$
which together with the Euler’s formula
$K_{W}=-rH+\pi^{*}(K_{S}+c_{1})$
gives
$-K_{W}^{3}=6K_{S}^{2}+10c_{1}^{2}+24K_{S}\cdot c_{1}-8c_{2}.$
###### Remark 4.2.
If the structure group of $\mathcal{E}$ is not $SU(2)$ (or
$H^{1}(S,\mathbb{Z})\neq 0$), the previous formulae (except for the Euler’s
one and with suitably corrected $-K_{W}^{3}$) should be read with all the
$c_{i}$ up to $\pm$. In fact, when $\mathcal{E}\simeq\mathcal{E}^{*}$ (the
dual of $\mathcal{E}$) and $\simeq$ is non-canonical, there is no preference
in choosing $\pi^{*}(c_{i})$ or $-\pi^{*}(c_{i})$ as one may change the
orientation in the fibers of $\mathcal{E}$. In particular, the Hirsch formula
turns into $H^{2}\pm H\cdot c_{1}\pm c_{2}=0$ (with both $c_{i}$ having
definite sign at once).
Further, let $Z\subset S$ be a $0$-dimensional subscheme supported at a finite
number of points $p_{1},\ldots,p_{m}$. Put
$\ell_{i}:=\dim~{}\mathcal{O}_{S,p_{i}}/I_{Z,p_{i}}$ and
$\ell(Z):=\sum_{i}\ell_{i}$ (the _length_ of $Z$). One can show that
$c_{2}(j_{*}Z)=-\ell(Z)\in H^{4}(S,\mathbb{Z})$, where $j:Z\longrightarrow X$
is an embedding. In particular, if $\mathcal{E}$ admits a splitting
(4.3) $0\to\mathcal{L}\to\mathcal{E}\to\mathcal{L}^{\prime}\otimes I_{Z}\to 0$
for some line bundles $\mathcal{L},\mathcal{L}^{\prime}\in\text{Pic}(S)$, we
obtain (using the Whitney formula)
(4.4) $c_{1}=c_{1}(\mathcal{L})+c_{1}(\mathcal{L}^{\prime}),\qquad
c_{2}=c_{1}(\mathcal{L})\cdot c_{1}(\mathcal{L}^{\prime})+\ell(Z).$
###### Example 4.5.
Let $s\in H^{0}(S,\mathcal{E})$ be a section. Assume for simplicity that the
zero locus $(s)_{0}\subset S$ of $s$ has codimension $\geq 2$. Then one gets
an exact sequence of sheaves $0\to\mathcal{O}_{S}\stackrel{{\scriptstyle
s}}{{\longrightarrow}}\mathcal{E}\stackrel{{\scriptstyle\wedge
s}}{{\longrightarrow}}\bigwedge^{2}\mathcal{E}\otimes I_{(s)_{0}}\to 0$ (cf.
(4.3)).
### 4.6.
Assume now that $r=2$. Let $\mathcal{L}\subset\mathcal{E}$ be a line subbundle
such that the sheaf $\mathcal{E}/\mathcal{L}$ is torsion free. Then by working
locally it is easy to obtain an exact sequence (4.3). All such sequences
(_extensions_ of $\mathcal{E}$) are classified by the group
$\text{Ext}^{1}(\mathcal{L}^{\prime}\otimes I_{Z},\mathcal{L})$, which in the
case when $Z$ is a locally complete intersection coincides with
$\mathcal{O}_{Z}$. More precisely, since $\mathcal{L},\mathcal{L}^{\prime}$,
etc. are defined up to a $\mathbb{C}^{*}$-action, the classifying space for
the extensions is the projectivization
$\mathbb{P}(\text{Ext}^{1}(\mathcal{L}^{\prime}\otimes
I_{Z},\mathcal{L}))=\mathbb{P}(\mathcal{O}_{Z})$.
One may apply the preceding considerations, together with the formulae (4.4)
and Serre’s criterion,7)7)7)Implying in particular that (4.3) provides a
locally free extension when $H^{2}(S,L^{\prime-1}\otimes L)=0$. to prove the
following:
###### Proposition 4.7 (R. L. E. Schwarzenberger).
In the previous notations, for $S=\mathbb{P}^{2}$ and any
$(a_{1},a_{2})\in\mathbb{Z}^{2}$ there exists $\mathcal{E}$
($\text{rk}~{}\mathcal{E}=2$) as above, with $c_{i}=a_{i}$.
###### Example 4.8.
In (4.3), take $\mathcal{L}=\mathcal{L}^{\prime}=\mathcal{O}_{S}$ and
$Z=\text{pt}$. Then the corresponding $\mathcal{E}$ has $c_{1}=0$ and
$c_{2}=1$. In particular, $\mathcal{E}$ is indecomposable. Furthermore, the
discussion in 4.6 shows that such $\mathcal{E}$ is unique up to isomorphism.
### 4.9.
From now on $S=\mathbb{P}^{2}$ and $\mathcal{E}$ is as in Example 4.8. In this
case, since $h^{0}(S,\mathcal{E})=h^{0}(W,H)\geq 1$, we have
$H=\Sigma+\alpha L,$
where $\Sigma$ is a section of $\pi$, $L:=\pi^{*}(l)$, $l\subset
S=\mathbb{P}^{2}$ is a line and $\alpha\in\mathbb{Z}_{\geq 0}$. Recall also
that
$K_{W}=-2H-3L.$
###### Remark 4.10.
Working locally, it is easy to see that $\Sigma\simeq\mathbb{F}_{1}$. In
particular, the divisor $-K_{\Sigma}$ is ample.
###### Proposition 4.11.
The divisor $-(K_{W}+\delta_{1}\Sigma+\delta_{2}L)$ is ample on $W$ for some
$0\leq\delta_{1},\delta_{2}\ll 1$.
###### Proof.
We have $\overline{NE}(W)=\mathbb{R}_{\scriptscriptstyle\geq 0}\cdot
C\oplus\mathbb{R}_{\scriptscriptstyle\geq 0}\cdot F$ for a fiber $F$ of $\pi$
and some numerical class $C\in N_{1}(W)\otimes\mathbb{R}$. Note that $L\cdot
C>0$, since otherwise $L\equiv 0$.
###### Lemma 4.12.
$\Sigma\cdot C<0$.
###### Proof.
Suppose that $\Sigma\cdot C\geq 0$. Then $H$ and $\Sigma$ are nef and
$-K_{W}+H$ is ample. In particular, the linear system $|mH|$ is basepoint-
free, $m\gg 1$. Further, using the formulae in 4.1 we find $H^{2}\cdot
L=0,~{}\Sigma^{2}\cdot L=-2\alpha$. The latter gives $\alpha=0$ for
$\Sigma\cdot L$ is an effective $1$-cycle. Hence $H\sim\Sigma$. But then,
since $H\cdot\Sigma\cdot L=0$, we obtain that $H^{2}\equiv 0$. On the other
hand, we have $H^{3}=\pm 1$ (cf. 4.1 and Remark 4.2), a contradiction. ∎
One may write $C=\displaystyle\lim_{i}C_{i}$ for some $C_{i}\in
N^{1}(W)\otimes\mathbb{Q}$. Lemma 4.12 gives $\Sigma\cdot C_{i}<0$ for $i\gg
1$, i.e. $C_{i}\subset\Sigma$ for all such $i$. Then, since
$-(K_{W}+\Sigma)\big{|}_{\Sigma}=-K_{\Sigma}$ is ample (see Remark 4.10), we
get
$-(K_{W}+\Sigma+\delta_{2}L)\cdot C_{i}>0,$
which implies that $-(K_{W}+\Sigma+\delta_{2}L)\cdot C\geq 0$.
###### Lemma 4.13.
$-(K_{W}+\Sigma+\delta_{2}L)\cdot C>0$.
###### Proof.
Indeed, once $(K_{W}+\Sigma+\delta_{2}L)\cdot C=0$ for all $\delta_{2}$, we
have $L\equiv 0$, a contradiction. ∎
Further, since
$-(K_{W}+\Sigma+\delta_{2}L)\cdot F=1,$
Lemma 4.13 shows that the divisor $-(K_{W}+\Sigma+\delta_{2}L)$ is ample. The
latter also holds for $-(K_{W}+\delta_{1}\Sigma+\delta_{2}L)$. Proposition
4.11 is proved. ∎
According to Proposition 4.11 the pair $(W,\delta_{1}\Sigma+\delta_{2}L)$ is a
log Fano. In particular we have
$\overline{NE}(W)=\mathbb{R}_{\scriptscriptstyle\geq 0}\cdot
C\oplus\mathbb{R}_{\scriptscriptstyle\geq 0}\cdot F$ for a fiber $F$ of $\pi$
and some curve $C\in N^{1}(W)$. Furthermore, the extremal contraction
$\phi:W\longrightarrow Y$ of $C$ is birational for $\Sigma\cdot C<0$.
###### Lemma 4.14.
$\phi(\Sigma)$ is a curve.
###### Proof.
Indeed, if $\phi(\Sigma)$ is a point, then from $H^{2}\cdot
L=0,\Sigma^{2}\cdot L=-2\alpha$ (cf. the proof of Lemma 4.12) we obtain that
$\phi$ is given by the linear system $|m(\Sigma+2\alpha L)|,m\gg 1$.
Furthermore, $H^{3}=\pm 1$ yields $\Sigma^{3}=3\alpha^{2}\pm 1$, and from
$(\Sigma+2\alpha L)^{2}\cdot\Sigma=0$ we find that $\alpha=\pm 1$ (with
$\Sigma^{3}=4$).
Write
$K_{W}\equiv\phi^{*}(K_{Y})+\delta\Sigma$
for some rational $\delta>-1$. Intersecting with $\Sigma\cdot L$, we get
$\delta=(3-2\alpha)/2\alpha\in\\{1/2,-5/2\\}$, i.e. $\delta=1/2$ (and
$\alpha=1$). But then $\Sigma\simeq\mathbb{P}^{2}$ (see [29]), which
contradicts Remark 4.10. ∎
### 4.15.
Lemma 4.14 implies that
$\phi\big{|}_{\Sigma}:\mathbb{F}_{1}\longrightarrow\mathbb{P}^{1}$ is the
natural projection. Note also that $L\cdot\Sigma\equiv F+C$ and
$\Sigma\cdot C=-2\alpha-1.$
Hence $\phi$ is given by the linear system $|m(\Sigma+(2\alpha+1)L)|,m\gg 1$.
Then, using $H^{3}=\pm 1$ and $\Sigma^{3}=3\alpha^{2}\pm 1$ (cf. the proof of
Lemma 4.14), we obtain
$0=(\Sigma+(2\alpha+1)L)^{2}\cdot\Sigma=3\alpha^{2}\pm
1-4\alpha(2\alpha+1)+(2\alpha+1)^{2}\in\\{-\alpha^{2},-\alpha^{2}+2\\},$
which forces $\alpha=0$ (and $\Sigma^{3}=-1$).
Further, the divisor $-K_{W}=2(H+L)+L$ (for $H\sim\Sigma$) is ample, so that
$Y$ is smooth (near $\phi(\Sigma)$) and $\phi:W\longrightarrow Y$ is the
blowup of $\phi(\Sigma)$ (see [29]). Moreover, as $-K_{W}^{3}=46$ (see 4.1),
$Y$ must be a quadric in $\mathbb{P}^{4}$ with $\phi(\Sigma)\subset Y$ being a
line (see e.g. [3]). Then we can write
$-K_{W}=\Sigma+\Delta+L^{\prime}+L^{\prime\prime},$
where $\Delta\in|H+L|$ is a general surface (isomorphic to the del Pezzo
surface of degree $7$), $L^{\prime},L^{\prime\prime}\sim L$ are generic
(isomorphic to $\mathbb{P}^{1}\times\mathbb{P}^{1}$), and the pair
$(W,B:=\Sigma+\Delta+L^{\prime}+L^{\prime\prime})$ is lc. In particular, we
have $ac(W)\leq 1$ (cf. 1.6).
### 4.16.
Let $[x_{0}:x_{1}:x_{2}]$ be projective coordinates on $\mathbb{P}^{2}$.
Choose two cyclic groups $G_{1},G_{2}\simeq\mathbb{Z}/_{\displaystyle
2\mathbb{Z}}$ with generators $g_{1},g_{2}$, respectively, acting on
$\mathbb{P}^{2}$ as follows:
(4.17)
$g_{1}:\left[x_{0}:x_{1}:x_{2}\right]\mapsto\left[-x_{0}:x_{1}:x_{2}\right],\
g_{2}:\left[x_{0}:x_{1}:x_{2}\right]\mapsto\left[-x_{0}:-x_{1}:x_{2}\right].$
One may also choose the point $Z$ corresponding to $\mathcal{E}$ to be
$G_{i}$-invariant. Then we get $g_{i}^{*}\mathcal{E}\simeq\mathcal{E}$ for
both $i$ (see Example 4.8). Hence the $G_{i}$-actions lift from
$\mathbb{P}^{2}$ to (biregular) actions on $W$. Note however that $G_{i}$ may
no longer commute when acting on $W$ – lifting from $\mathbb{P}^{2}$ depends
on the choice of a fiber coordinate (cf. Remark 4.2). Such an “ambiguity” is
the cornerstone in the forthcoming constructions of an $\mathfrak{f}$-toric
variety (cf. Remark 1.12).
Consider the divisor $R_{1}:=\pi^{*}(x_{0}=0)$. We may assume
$R_{1}\cap\Sigma\not\supset F$ (cf. the beginning of 4.15). In this case,
group $G_{1}$ acts trivially on $R_{1}$. Indeed, contraction
$\phi:W\longrightarrow Y$ is $G_{1}$-equivariant and $Y\subset\mathbb{P}^{4}$
is a smooth quadric whose equation can be chosen to be
$x_{0}^{2}=x_{1}x+x_{2}y$. Here $x_{i}$ are as earlier, $x,y$ are
complementary projective coordinates on $\mathbb{P}^{4}$ and $G_{1}$ acts just
on $x_{0}$ via $\pm$. This gives $R_{1}$ as stated.
Further, taking the equation of $Y$ in the form
$x_{0}^{2}=x_{0}x+x_{1}^{2}+x_{2}y$, the group $G_{2}$ will act via $\pm$ on
$x$ (in addition to $x_{0},x_{1}$), for $G_{2}$-equivariant morphism
$\phi:W\longrightarrow Y$. One can see that this delivers two _non-commuting_
actions of $G_{1},G_{2}$ on $W$, which descend to those (4.17) on
$\mathbb{P}^{2}$. Then, identifying the space $H^{0}(W,\Delta)$ with
$\mathbb{C}^{5}$ we may set $G_{i}$ to act like this:
(4.18) $g_{1}:(x_{0},x_{1},x_{2},x,y)\mapsto(-x_{0},x_{1},x_{2},x,y)$
and
(4.19) $g_{2}:(x_{0},x_{1},x_{2},x,y)\mapsto(-x_{0},-x_{1},x_{2},x+x_{0},y).$
Now lets form a $\mathbb{C}$-algebra
$\mathfrak{R}:=\displaystyle\bigoplus_{k\geq 0}H^{0}(W,k\Delta)\oplus
H^{0}(W,k\Sigma)$
and determine its $G_{1}$-, $G_{2}$-invariant subalgebra
$\mathfrak{R}_{G_{1},G_{2}}:=\mathfrak{R}^{G_{1}}\cap\mathfrak{R}^{G_{2}}$.
Namely, the latter is isomorphic to the $G_{2}$-invariant part of the
$\mathbb{C}$-algebra generated by $x_{1},x_{2},x,y$ and the section defining
$\Sigma$, where $G_{2}$ acts just on $x_{1}$ via $\pm$. This yields an
algebraic variety $X:=\mathrm{Proj}~{}\mathfrak{R}_{G_{1},G_{2}}$ together
with the natural morphism $p:W\longrightarrow X$.
###### Remark 4.20.
Forgetting about $W$ for a moment, the preceding construction of $X$ is a
reminiscence of the following. One starts with the quadric
$Y\subset\mathbb{P}H^{0}(W,\Delta)$ and the $G_{i}$-actions on it given by
(4.18), (4.19). Taking $\phi_{*}(\Delta)$ and dropping $\Sigma$, one defines
the ring $\mathfrak{R}^{Y}$ for $Y$ exactly as $\mathfrak{R}$ above, together
with the invariant part $\mathfrak{R}_{G_{1},G_{2}}^{Y}$. Then the induced
morphism $Y\longrightarrow\mathrm{Proj}~{}\mathfrak{R}_{G_{1},G_{2}}^{Y}$ is
composed of two Galois $\mathbb{Z}/_{\displaystyle 2\mathbb{Z}}$-covers, the
first being the linear projection $Y\longrightarrow\mathbb{P}^{3}$ from the
point $[1:0:\ldots:0]$ and the second one
$\mathbb{P}^{3}\longrightarrow\mathbb{P}(1,1,1,2)$ being the usual
$\mathbb{Z}/_{\displaystyle 2\mathbb{Z}}$-quotient, where the corresponding
action on $\mathbb{P}^{3}$ is obtained from (4.19) by setting $x_{0}:=0$.
### 4.21.
It follows from the discussion in Remark 4.20 that the “quotient” morphism
$p:W\longrightarrow X$ at the end of 4.16 is _finite_ (and so $X$ is a
$3$-fold). Furthermore, $p$ is also composed of two subsequent Galois
$\mathbb{Z}/_{\displaystyle 2\mathbb{Z}}$-covers, with $R_{1}$ being the
ramification divisor for the first cover $W\longrightarrow W_{1}$ and
similarly the image of $R_{2}:=\pi^{*}(x_{1}=0)$ on $W_{1}$ for the second one
$W_{1}\longrightarrow X$ (cf. (4.18), (4.19)). Then from the Hurwitz formula
(applied twice) we get
$K_{W}\equiv
p^{*}\big{(}K_{X}+\frac{1}{2}~{}\widetilde{R_{1}}+\frac{1}{2}~{}\widetilde{R_{2}}\big{)},$
where $\widetilde{R_{i}}:=p(R_{i})$. Here we have crucially used the fact that
$G_{i}$-actions on $W$ don’t commute (this results in (4.18), (4.19)).
Otherwise, with commuting $G_{1},G_{2}$, there’ll be just one ramification
divisor $R_{1}$, which yields $ac(X,D)\geq 1$ for the pair $(X,D)$ below (and
hence no $\mathfrak{f}$-toric variety can appear on this way).
In the pair $(W,B)$ from 4.15, the divisor $\Sigma$ is $G_{i}$-invariant for
both $i$, and we take $L^{\prime}:=\pi^{*}(x_{2}=0)$ to be $G_{i}$-invariant
as well. There is also $G_{i}$-invariant divisor $\Delta:=(y=0)\sim H+L$ such
that $(W,B)$ remains lc. Given these $L^{\prime}$ and $\Delta$, we obtain
(4.22)
$-4(\widetilde{\Sigma}+\widetilde{\Delta}+\widetilde{L^{\prime}})+\widetilde{L^{\prime\prime}}=p_{*}(K_{W})\equiv
4(K_{X}+\frac{1}{2}~{}\widetilde{R_{1}}+\frac{1}{2}~{}\widetilde{R_{2}})$
for $\widetilde{\Sigma}:=p(\Sigma)$ with $p_{*}(\Sigma)=4\widetilde{\Sigma}$,
and similarly for $\widetilde{L^{\prime}}:=p(L^{\prime})$, etc. Also notice
that
(4.23)
$\widetilde{L^{\prime}}\equiv\widetilde{\Delta}-\widetilde{\Sigma},\qquad\widetilde{L^{\prime\prime}}\equiv
4\widetilde{L^{\prime}},\qquad\widetilde{R_{1}}\equiv
2\widetilde{L^{\prime}}\equiv\widetilde{R_{2}}.$
The following assertions are straightforward (by reducing to
$Y\subset\mathbb{P}^{4}$ as in Remark 4.20):
###### Lemma 4.24.
The linear system $|2\widetilde{\Delta}|$ is basepoint-free and the resulting
map $\alpha:X\longrightarrow\mathbb{P}(1,1,1,2)$ is an extremal birational
contraction (of $\widetilde{\Sigma}$).
###### Lemma 4.25.
The linear system $|\widetilde{R_{1}}|$ is basepoint-free and the resulting
map $X\longrightarrow\mathbb{P}^{2}$ is an extremal contraction.
Applying (4.22), (4.23) and Lemma 4.25 we find
$K_{X}+\widetilde{\Sigma}+\widetilde{\Delta}+\widetilde{L^{\prime}}+\frac{1}{2}~{}\big{(}L_{1}+L_{2}+L_{3}\big{)}\equiv
0$
for some generic $L_{i}\in|\widetilde{R_{1}}|$. In particular, the pair
$\big{(}X,D:=\widetilde{\Sigma}+\widetilde{\Delta}+\widetilde{L^{\prime}}+\frac{1}{2}\big{(}L_{1}+L_{2}+L_{3}\big{)}\big{)}$
is lc, having $ac(X,D)=\frac{1}{2}$.
###### Remark 4.26.
One can easily modify the preceding pair $(X,D)$ and derive that $ac(X)=0$
(thus disproving Conjecture 1.3 as well). Namely, since the linear system
$|\widetilde{R_{1}}|$ is basepoint-free, we may replace
$\frac{1}{2}\big{(}L_{1}+L_{2}\big{)}$ simply by $L_{1}$. Now it follows from
[37, Proposition 4.3.2] (applied to the morphism
$\alpha:X\longrightarrow\mathbb{P}(1,1,1,2)$ in Lemma 4.24) that the modified
pair $(X,D)$ is $1$-complementary. Alternatively, one may observe that
$L_{i}\equiv\alpha^{*}(\mathcal{O}(2)),\widetilde{\Delta}\equiv\alpha^{*}(\mathcal{O}(1))$
for the generator $\mathcal{O}(1)$ of the class group of
$\mathbb{P}(1,1,1,2)$, which easily leads to an integral boundary $D$ with
$ac(X,D)=0$.
Theorem 1.11 ($n=3$ case) now follows from the next
###### Proposition 4.27.
Variety $X$ is non-toric.8)8)8)Compare with [38, Lemma 3.4].
###### Proof.
Suppose the contrary. The singular locus of $X$ contains the curve
$\widetilde{F}:=p(\Sigma\cap L^{\prime})$ such that locally at the general
point on $\widetilde{F}$ singularities of $X$ are of type
$\mathbb{C}^{*}\times\displaystyle\frac{1}{2}(1,1)$. There are also four
$\displaystyle\frac{1}{2}(1,1,1)$-points $A_{i}\in X$, with
$A_{1},A_{2}\in\widetilde{F}$ say, and $A_{3},A_{4}\not\in\widetilde{F}$.
Let $N\simeq\mathbb{Z}^{3}$ and $\Lambda\subset
N\otimes_{\mathbb{Z}}\mathbb{R}$ be the lattice and the fan, respectively,
associated with $X$. Fix a cone $\sigma\subset\Lambda$ (with the corresponding
affine toric neighborhood
$U_{\sigma}:=\text{Spec}~{}\mathbb{C}\left[\sigma^{\vee}\cap M\right]$ on $X$,
where $M:=\text{Hom}(N,\mathbb{Z})$) so that $\widetilde{F}$ is identified
with a $2$-stratum on $\sigma$. Let also
$\alpha:X\longrightarrow\mathbb{P}(1,1,1,2)=:\mathbb{P}$ be as in Lemma 4.24.
By assumption, morphism $\alpha$ is toric,
$\alpha(\widetilde{F})\subset\mathbb{P}$ is a “line” and $\Lambda$ looks like
as in Figure 1.
Figure 1.
Here $\text{Supp}\,\Lambda$ coincides with the support of the fan
$\Lambda_{\mathbb{P}}$ of $\mathbb{P}$ (i.e. $\Lambda$ subdivides
$\Lambda_{\mathbb{P}}$). Note also that vector $OC$ corresponds to the surface
$\widetilde{\Sigma}$ and the face of $\sigma$ containing $OC$ corresponds to
$\alpha(\widetilde{F})$.
Now, since $A_{3},A_{4}\not\in\widetilde{F}$, we must also have
$A_{3},A_{4}\not\in U_{\sigma}$ and $A_{3},A_{4}\not\in\widetilde{\Sigma}$.
But the latter is impossible for $\mathbb{P}$ containing only one singular
point. Proposition 4.27 is proved. ∎
In order to prove Theorem 1.11 for any $n\geq 4$ it suffices to take
$\mathfrak{X}:=X\times(\mathbb{P}^{1})^{n-3}$. It is easy to see (modulo
Proposition 4.27) that $\mathfrak{X}$ is non-toric. At the same time we have
$ac(\mathfrak{X},\mathfrak{D})=\frac{1}{2}$, where $\mathfrak{D}$ is the
pullback to $\mathfrak{X}$ of the boundary $D$ on $X$ plus the sum of generic
divisors = pullbacks w. r. t. the projection
$\mathfrak{X}\longrightarrow(\mathbb{P}^{1})^{n-3}$, so that
$K_{\mathfrak{X}}+\mathfrak{D}\equiv 0$.9)9)9)Of course, one could
alternatively _start_ with the quadric $Y$ in the previous arguments, blow up
a line on it and arrive at $W$. All such $W$’s are (obviously) isomorphic
because the group $\text{Aut}(Y)$ acts transitively on the set of lines
contained in $Y$. Then considerations of 4.16, 4.21 apply literally. But
still, again, we think the present vector bundle point of view is more natural
and suitable for generalizations (cf. 4.28, A)).
### 4.28.
Fix some $X\in\mathfrak{T}^{f,n}$. We conclude by asking the following
questions:
1. A)
Is $X$ always obtained as (an extremal contraction$/$a blowup of) the quotient
$\mathbb{P}(\mathcal{E})/G$ for some toric variety $T$, (semistable) vector
bundle $\mathcal{E}$ on $T$ and a finite group
$G\circlearrowright\mathbb{P}(\mathcal{E})$? (Notice that $\mathcal{E}$ from
Example 4.8 is semistable.)
2. B)
Does $X$ admit a regular $(\mathbb{C}^{*})^{k}$-action for some $k\geq 1$ (cf.
[33])?
3. C)
Is $X$ a compactification of the torus $(\mathbb{C}^{*})^{n}$ (cf. 1.1 and
Remark 1.5)?
4. D)
Is $X$ a Mori dream space$/$an FT variety$/\ldots$ (see [15], [12])?
5. E)
Does the set $\mathcal{C}:=\left\\{ac(X)\right\\}_{X\in\mathfrak{T}^{f,n}}$
satisfy the A. C. C. condition?
Acknowledgments. I am grateful to Yu. G. Prokhorov for introducing me to the
subject treated in this paper. Also thanks to J. McKernan and D. Stepanov for
challenging emails. Finally, the work owes much to hospitality of the Courant
Institute, University of Edinburgh and Max Planck Institute for Mathematics.
Partial financial support was provided by CRM fellowship and NSERC grant.
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|
arxiv-papers
| 2013-06-18T10:28:14 |
2024-09-04T02:49:46.632268
|
{
"license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/",
"authors": "Ilya Karzhemanov",
"submitter": "Ilya Karzhemanov",
"url": "https://arxiv.org/abs/1306.4131"
}
|
1306.4415
|
# Cosmological Tests Using GRBs, the Star Formation Rate
and Possible Abundance Evolution
Jun-Jie Wei1,2, Xue-Feng Wu1,3,4, Fulvio Melia5, Da-Ming Wei1,6, and Long-Long
Feng1,3
1Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008,
China
2Graduate University of Chinese Academy of Sciences, Beijing 100049, China
3Chinese Center for Antarctic Astronomy, Nanjing 210008, China
4Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing University-
Purple Mountain Observatory, Nanjing 210008, China
5Department of Physics, The Applied Math Program, and Department of Astronomy,
The University of Arizona, AZ 85721, USA
6Key Laboratory of Dark Matter and Space Astronomy, Chinese Academy of
Sciences, Nanjing, 210008, China Email:[email protected] Woodruff Simpson
Fellow
###### Abstract
The principal goal of this paper is to use attempts at reconciling the Swift
long gamma-ray bursts (LGRBs) with the star formation history (SFH) to compare
the predictions of $\Lambda$CDM with those in the $R_{\rm h}=ct$ Universe. In
the context of the former, we confirm that the latest _Swift_ sample of GRBs
reveals an increasing evolution in the GRB rate relative to the star formation
rate (SFR) at high redshifts. The observed discrepancy between the GRB rate
and the SFR may be eliminated by assuming a modest evolution parameterized as
$(1+z)^{0.8}$—perhaps indicating a cosmic evolution in metallicity. However,
we find a higher metallicity cut of $Z=0.52Z_{\odot}$ than was seen in
previous studies, which suggested that LGRBs occur preferentially in metal
poor environments, i.e., $Z\sim 0.1-0.3Z_{\odot}$. We use a simple power-law
approximation to the high-_z_ ($\ga 3.8$) SFH, i.e., $R_{\rm
SF}\propto[(1+z)/4.8]^{\alpha}$, to examine how the high-_z_ SFR may be
impacted by a possible abundance evolution in the _Swift_ GRB sample. For an
expansion history consistent with $\Lambda$CDM, we find that the Swift
redshift and luminosity distributions can be reproduced with reasonable
accuracy if $\alpha=-2.41_{-2.09}^{+1.87}$. For the $R_{\rm h}=ct$ Universe,
the GRB rate is slightly different from that in $\Lambda$CDM, but also
requires an extra evolutionary effect, with a metallicity cut of
$Z=0.44Z_{\odot}$. Assuming that the SFR and GRB rate are related via an
evolving metallicity, we find that the GRB data constrain the slope of the
high-_z_ SFR in $R_{\rm h}=ct$ to be $\alpha=-3.60_{-2.45}^{+2.45}$. Both
cosmologies fit the GRB/SFR data rather well. However, in a one-on-one
comparison using the Aikake Information Criterion, the best-fit $R_{\rm h}=ct$
model is statistically preferred over the best-fit $\Lambda$CDM model with a
relative probability of $\sim 70\;\%$ versus $\sim 30\;\%$.
###### keywords:
Gamma-ray bursts: general–Methods: statistical–Stars: formation–Cosmology:
theory, observations
## 1 Introduction
Our understanding of the star formation history (SFH) in the Universe
continues to be refined with improving measurement techniques and a broader
coverage in redshift—now extending out to $z\ga 6$. However, direct star
formation rate (SFR) measurements are quite challenging at these high
redshifts, particularly towards the faint end of the galaxy luminosity
function. Using ultraviolet and far-infrared observations, Hopkins & Beacom
(2006) constrained the cosmic SFH out to $z\approx 6$, and found that the SFR
rapidly increases at $z\la 1$, remains almost constant in the redshift range
$1\la z\la 4$, and then shows a steep decline with slope $\sim-8$ at $z\ga 4$.
The sharp drop at $z\ga 4$ may be due to significant dust extinction at such
high redshifts. Li (2008) derived the SFR out to $z=7.4$ by adding new
ultraviolet measurements and obtained a shallower decay ($\sim-4.46$) in this
range. The high-_z_ SFR has also been constrained using observations of color-
selected Lyman break galaxies (LBG; Bouwens et al. 2008; Mannucci et al. 2007;
Verma et al. 2007) and Ly$\alpha$ Emitters (LAE; Ota et al. 2008). Several of
the more prominent SFR determinations are summarized in Figure 1 below. One
can see from this plot that, due to the inherent difficulty of making and
interpreting these measurements, the various determinations can disagree with
each other even after taking the uncertainties into account.
Gamma-ray bursts (GRBs) are the most luminous transient events in the cosmos.
Owing to their high luminosity, GRBs can be detected out to the edge of the
visible Universe, constituting a powerful tool for probing the cosmic star
formation rate from a different perspective, i.e., by studying the death rate
of massive stars rather than observing them directly during their lives. Since
the successful launch of the _Swift_ satellite, the number of measured GRB
redshifts has increased rapidly, and thus a reliable statistical analysis is
now possible. The statistical analysis on the GRB redshift distributions have
been well investigated (e.g. Shao et al. 2011; Robertson & Ellis 2012; Dado &
Dar 2013). It is believed that long bursts (LGRBs) with durations $T_{\rm
90}>2$ s (where $T_{\rm 90}$ is the time over which $90\%$ of the prompt
emission was observed; Kouveliotou et al. 1993) are powered by the core
collapse of massive stars (e.g., Woosley 1993a; Paczynski 1998; Woosley &
Bloom 2006), an idea given strong support by several confirmed associations
between LGRBs and supernovae (Stanek et al. 2003; Hjorth et al. 2003; Chornock
et al. 2010).
Figure 1: The cosmic star formation rate as a function of redshift. The high-
_z_ SFR (shaded band) is constrained by the _Swift_ GRB data, and is
characterized by a power-law index $-5.07<\alpha<-1.05$ (see § 4.2). Some
observationally-determined SFRs are also shown for comparison.
This scenario—known as the collapsar model—suggests that the cosmic GRB rate
should in principle trace the cosmic star formation rate (Totani 1997; Wijers
et al. 1998; Blain & Natarajan 2000; Lamb & Reichart 2000; Porciani & Madau
2001; Piran 2004; Zhang & Mészáros 2004; Zhang 2007). However, observations
seem to indicate that the rate of LGRBs does not strictly follow the SFR, but
instead increases with cosmic redshift faster than the SFR, especially at
high-z (Daigne et al. 2006; Le & Dermer 2007; Yüksel & Kistler et al. 2007;
Salvaterra & Chincarini 2007; Guetta & Piran 2007; Li 2008; Kistler et al.
2008; Salvaterra et al. 2009, 2012). This has led to the introduction of
several possible mechanisms that could produce such an observed enhancement to
the GRB rate (Daigne et al. 2006; Guetta & Piran 2007; Le & Dermer 2007;
Salvaterra & Chincarini 2007; Kistler et al. 2008, 2009; Li 2008; Salvaterra
et al. 2009, 2012; Campisi et al. 2010; Qin et al. 2010; Wanderman & Piran
2010; Cao et al. 2011; Virgili et al. 2011; Robertson & Ellis 2012; Elliott et
al. 2012). The idea that appears to have gained some traction is the
possibility that the difference between the GRB rate and the SFR is due to an
enhanced evolution parameterized as $(1+z)^{\delta}$ (Kistler et al. 2008),
which may encompass the effects of cosmic metallicity evolution (Langer &
Norman 2006; Li 2008), an evolution in the stellar initial mass function (Xu &
Wei 2009; Wang & Dai 2011), and possible selection effects (see, e.g., Coward
et al. 2008, 2013; Lu et al. 2012).
Of course, if we knew the mechanism responsible for the difference between the
GRB rate and the SFR, we could constrain the high-_z_ SFR very accurately
using the GRB data alone. This limitation notwithstanding, GRBs have indeed
already been used to estimate the SFR in several instances, including the
following representative cases: Chary et al. (2007) estimated a lower limit to
the SFR of $0.12\pm 0.09$ and $0.09\pm 0.05$ $M_{\bigodot}$ $\rm yr^{-1}$ $\rm
Mpc^{-3}$ at $z=4.5$ and $6$, respectively, using deep observations of three
$z\sim 5$ GRBs with the _Spitzer_ Space Telescope; Yüksel et al. (2008) used
_Swift_ GRB data to constrain the SFR in the range $z=4-7$ and found that no
steep drop exists in the SFR up to at least $z\sim 6.5$; Kistler et al. (2009)
constrained the SFR using four years of _Swift_ observations and found that
the SFR to $z\ga 8$ was consistent with LBG-based measurements; Wang & Dai
(2009) studied the high-_z_ SFR up to $z\sim 8.3$, but found that the SFR at
$z\ga 4$ showed a steep decay with a slope of $\sim-5.0$; and Ishida et al.
(2011) used the principal component analysis method to measure the high-_z_
SFR from the GRB data and found that the level of star formation activity at
$z\approx 4$ could have been already as high as the present-day one ($\approx
0.01$ $M_{\bigodot}$ $\rm yr^{-1}$ $\rm Mpc^{-3}$).
The question of how the GRB redshift distribution is related to the SFH is
clearly still not completely answered, but there is an additional important
ingredient that has hitherto been ignored in this ongoing discussion—the
impact on this relationship from the assumed cosmological expansion itself.
Our principal goal in this paper is to update and enlarge the GRB sample using
the latest catalog of 254 Swift LGRBs in order to carry out a comparative
analysis between $\Lambda$CDM and the $R_{\rm h}=ct$ Universe. We wish to
examine the influence on these results due to the background cosmology, and
see to what extent the implied abundance evolution depends on the expansion
scenario. We will assemble our sample in § 2, and discuss our method of
analysis in § 3. A possible mechanism of evolution and the implied high-_z_
SFR are investigated in § 4, together with a direct comparison between the two
cosmologies. Our discussion and conclusions are presented in § 5.
## 2 The _Swift_ GRB observations
_Swift_ has enabled observers to greatly extend the reach of GRB measurements
relative to the pre-_Swift_ era, resulting in the creation of a rich data set.
To obtain reliable statistics, we consider long bursts detected by Swift up to
July, 2013, with accurate redshift measurements and durations exceeding
$T_{\rm 90}>2$ s. We calculate the isotropic-equivalent luminosity of a GRB
using
$L_{\rm iso}={E_{\rm iso}(1+z)\over T_{\rm 90}}\;,$ (1)
where $E_{\rm iso}$ is the rest-frame isotropic equivalent $1-10^{4}$ keV
gamma-ray energy. The low-luminosity ($L_{\rm iso}<10^{49}$ erg $\rm s^{-1}$)
GRBs are not included in our sample because they may belong to a distinct
population (Soderberg et al. 2004; Cobb et al. 2006; Liang et al. 2007;
Chapman et al. 2007).
With these criteria, we combine the samples presented in Butler et al. (2007,
2010), Perley et al. (2009), Sakamoto et al. (2011), Greiner et al. (2011),
Kr$\ddot{\rm u}$hler et al. (2011), Hjorth et al. (2012), and Perley & Perley
(2013). For GRBs where the samples disagree, we choose the most recently
measured redshifts. The combined catalog containts 258 GRBs with known
redshifts and redshift upper limits, but four GRBs (051002, 051022, 060505,
and 071112C) have incomplete fluence or burst duration measurements and are
discarded. The remaining 254 long duration GRBs with redshifts or redshift
limits serve as our base GRB catalog. Our final sample is listed in Table 1,
which includes the following information for each GRB: (1) its name; (2) the
redshift $z$; (3) the burst duration $T_{\rm 90}$; and (4) the isotropic-
equivalent energy $E_{\rm iso}$. The quantities $T_{\rm 90}$ and $E_{\rm iso}$
of 231 GRBs are directly taken from the
catalog111http://butler.lab.asu.edu/Swift/index.html of Butler et al. (2007,
2010) and those of 14 others (050412, 050607, 050713A, 060110, 060805A,
060923A, 070521, 071011, 080319A, 080320, 080516, 081109, 081228, and 090904B)
are from Robertson & Ellis (2012). The duration $T_{\rm 90}$ of the nine
remaining GRBs (050406, 050502B, 051016B, 060602A, 070419B, 080325, 090404,
090417B, and 090709A) are taken from Sakamoto et al. (2011), while their
isotropic energy $E_{\rm iso}$ is calculated from the 15 to 150 keV fluences
reported by Sakamoto et al. (2011); we correct the observed fluence in a given
bandpass to the cosmological rest frame ($1-10^{4}$ keV in this analysis).
Table 1: GRB Catalog. GRB | _z_ | $T_{90}$ | log $E_{\rm iso}^{\Lambda{\rm CDM}}$ | log ${E}_{\rm iso}^{R_{\rm h}=ct}$ | GRB | _z_ | $T_{90}$ | log $E_{\rm iso}^{\Lambda{\rm CDM}}$ | log ${E}_{\rm iso}^{R_{\rm h}=ct}$
---|---|---|---|---|---|---|---|---|---
| | (s) | (erg) | (erg) | | | (s) | (erg) | (erg)
130701A | 1.155 | 4.62 $\pm$ 0.09 | $52.32^{+0.07}_{-0.03}$ | $52.23\par^{+0.07}_{-0.03}$ | 080520 | 1.545 | 2.97 $\pm$ 0.24 | $51.05^{+\par 5.78}_{-0.16}$ | $50.96^{+5.78}_{-0.16}$
130612A | 2.006 | 6.64 $\pm$ 1.06 | $51.70^{+0.31}_{-0.09}$ | $51.62\par^{+0.31}_{-0.09}$ | 080516 | $3.6^{a}$ | $5.75^{b}$ | $53.08^{+\par 0.22}_{-0.17}$c | $53.04^{+0.22}_{-0.17}$
130610A | 2.092 | 48.45 $\pm$ 2.35 | $52.71^{+0.44}_{-0.10}$ | $52.63\par^{+0.44}_{-0.10}$ | 080430 | 0.767 | 16.20 $\pm$ 0.78 | $51.60^{+\par 0.34}_{-0.09}$ | $51.51^{+0.34}_{-0.09}$
130606A | 5.913 | 278.52 $\pm$ 3.54 | $53.39^{+0.36}_{-0.08}$ | $53.39\par^{+0.36}_{-0.08}$ | 080413B | 1.1 | 7.04 $\pm$ 0.43 | $52.20^{+\par 0.06}_{-0.05}$ | $52.10^{+0.06}_{-0.05}$
130604A | 1.06 | 78.07 $\pm$ 9.81 | $51.90^{+0.50}_{-0.09}$ | $51.81\par^{+0.50}_{-0.09}$ | 080413A | 2.433 | 46.62 $\pm$ 0.13 | $52.97^{+\par 0.30}_{-0.08}$ | $52.90^{+0.30}_{-0.08}$
130603B | 0.3564 | 2.20 $\pm$ 0.01 | $50.89^{+0.66}_{-0.15}$ | $50.83\par^{+0.66}_{-0.15}$ | 080411 | 1.03 | 58.29 $\pm$ 0.46 | $53.38^{+\par 0.17}_{-0.08}$ | $53.28^{+0.17}_{-0.08}$
130514A | 3.6 | 220.32 $\pm$ 5.60 | $53.60^{+0.12}_{-0.05}$ | $53.55\par^{+0.12}_{-0.05}$ | 080330 | 1.51 | 66.10 $\pm$ 0.98 | $51.63^{+\par 0.99}_{-0.06}$ | $51.54^{+0.99}_{-0.06}$
130511A | 1.3033 | 4.95 $\pm$ 0.82 | $51.24^{+0.70}_{-0.14}$ | $51.14\par^{+0.70}_{-0.14}$ | 080325 | $1.78^{d}$ | $162.82^{e}$ | $53.12^{+\par 0.04}_{-0.04}$f | $53.03^{+0.04}_{-0.04}$
130505A | 2.27 | 292.81 $\pm$ 33.84 | $54.31^{+0.45}_{-0.23}$ | $54.23\par^{+0.45}_{-0.23}$ | 080320 | $7^{g}$ | $13.80^{b}$ | $53.53^{+\par 0.58}_{-0.07}$c | $53.56^{+0.58}_{-0.07}$
130427B | 2.78 | 7.04 $\pm$ 0.26 | $52.50^{+0.39}_{-0.09}$ | $52.44\par^{+0.39}_{-0.09}$ | 080319C | 1.95 | 32.88 $\pm$ 3.27 | $52.80^{+\par 0.37}_{-0.09}$ | $52.72^{+0.37}_{-0.09}$
130427A | 0.3399 | 324.70 $\pm$ 2.50 | $53.66^{+0.19}_{-0.11}$ | $53.60\par^{+0.19}_{-0.11}$ | 080319B | 0.937 | 147.32 $\pm$ 2.50 | $54.58^{+\par 0.26}_{-0.17}$ | $54.49^{+0.26}_{-0.17}$
130420A | 1.297 | 114.84 $\pm$ 4.84 | $52.72^{+0.07}_{-0.05}$ | $52.63\par^{+0.07}_{-0.05}$ | 080319A | $2.2^{g}$ | $43.60^{b}$ | $53.47^{+\par 0.38}_{-0.06}$c | $53.39^{+0.38}_{-0.06}$
130418A | 1.217 | 97.92 $\pm$ 2.26 | $51.77^{+0.14}_{-0.08}$ | $51.68\par^{+0.14}_{-0.08}$ | 080310 | 2.4266 | 361.92 $\pm$ 3.75 | $52.78^{+\par 0.78}_{-0.07}$ | $52.71^{+0.78}_{-0.07}$
130408A | 3.758 | 5.64 $\pm$ 0.31 | $53.08^{+0.55}_{-0.13}$ | $53.04\par^{+0.55}_{-0.13}$ | 080210 | 2.641 | 43.89 $\pm$ 4.36 | $52.72^{+\par 0.39}_{-0.08}$ | $52.65^{+0.39}_{-0.08}$
130215A | 0.597 | 89.05 $\pm$ 8.39 | $51.89^{+0.31}_{-0.07}$ | $51.81\par^{+0.31}_{-0.07}$ | 080207 | 2.0858 | 310.98 $\pm$ 9.34 | $53.05^{+\par 0.23}_{-0.07}$ | $52.96^{+0.23}_{-0.07}$
130131B | 2.539 | 4.74 $\pm$ 0.21 | $52.23^{+0.03}_{-0.03}$ | $52.16\par^{+0.03}_{-0.03}$ | 080129 | 4.349 | 45.60 $\pm$ 3.00 | $52.90^{+\par 0.42}_{-0.20}$ | $52.87^{+0.42}_{-0.19}$
121229A | 2.707 | 26.64 $\pm$ 2.15 | $51.85^{+0.95}_{-0.10}$ | $51.79\par^{+0.95}_{-0.10}$ | 071227 | 0.383 | 2.20 $\pm$ 0.16 | $50.45^{+\par 0.60}_{-0.22}$ | $50.39^{+0.60}_{-0.22}$
121211A | 1.023 | 184.14 $\pm$ 2.31 | $51.80^{+0.65}_{-0.09}$ | $51.70\par^{+0.65}_{-0.09}$ | 071122 | 1.14 | 79.20 $\pm$ 4.88 | $51.55^{+\par 0.64}_{-0.14}$ | $51.46^{+0.64}_{-0.14}$
121201A | 3.385 | 39.04 $\pm$ 2.93 | $52.39^{+0.38}_{-0.08}$ | $52.34\par^{+0.38}_{-0.08}$ | 071117 | 1.331 | 6.48 $\pm$ 0.76 | $52.29^{+\par 0.18}_{-0.07}$ | $52.20^{+0.18}_{-0.07}$
121128A | 2.2 | 25.65 $\pm$ 5.47 | $52.98^{+0.10}_{-0.07}$ | $52.91\par^{+0.10}_{-0.07}$ | 071031 | 2.692 | 187.18 $\pm$ 7.12 | $52.61^{+\par 0.45}_{-0.07}$ | $52.54^{+0.45}_{-0.07}$
121027A | 1.77 | 69.30 $\pm$ 1.90 | $52.39^{+0.11}_{-0.09}$ | $52.31\par^{+0.11}_{-0.09}$ | 071021 | 2.145 | 204.96 $\pm$ 17.95 | $53.00^{+\par 0.43}_{-0.14}$ | $52.92^{+0.43}_{-0.14}$
121024A | 2.298 | 12.46 $\pm$ 0.39 | $52.40^{+0.38}_{-0.16}$ | $52.32\par^{+0.38}_{-0.16}$ | 071020 | 2.145 | 4.40 $\pm$ 0.27 | $53.00^{+\par 0.43}_{-0.14}$ | $52.92^{+0.43}_{-0.14}$
120922A | 3.1 | 179.54 $\pm$ 6.27 | $53.28^{+0.21}_{-0.04}$ | $53.22\par^{+0.21}_{-0.04}$ | 071011 | $5^{g}$ | $80.90^{b}$ | $54.37^{+\par 0.34}_{-0.19}$c | $54.36^{+0.34}_{-0.19}$
120909A | 3.93 | 617.70 $\pm$ 30.95 | $53.68^{+0.48}_{-0.09}$ | $53.64\par^{+0.48}_{-0.09}$ | 071010B | 0.947 | 34.68 $\pm$ 1.02 | $52.26^{+\par 0.09}_{-0.03}$ | $52.16^{+0.09}_{-0.03}$
120907A | 0.97 | 6.27 $\pm$ 0.28 | $51.29^{+0.40}_{-0.05}$ | $51.20\par^{+0.40}_{-0.05}$ | 071010A | 0.98 | 22.40 $\pm$ 1.70 | $51.13^{+\par 0.81}_{-0.07}$ | $51.04^{+0.81}_{-0.07}$
120815A | 2.358 | 9.68 $\pm$ 1.21 | $52.01^{+0.90}_{-0.09}$ | $51.94\par^{+0.90}_{-0.09}$ | 071003 | 1.605 | 148.32 $\pm$ 0.68 | $53.27^{+\par 0.35}_{-0.15}$ | $53.17^{+0.35}_{-0.15}$
120811C | 2.671 | 25.20 $\pm$ 1.26 | $52.88^{+0.02}_{-0.10}$ | $52.81\par^{+0.02}_{-0.10}$ | 070810A | 2.17 | 7.68 $\pm$ 0.41 | $51.97^{+\par 0.13}_{-0.05}$ | $51.89^{+0.13}_{-0.05}$
120802A | 3.796 | 50.16 $\pm$ 1.52 | $52.83^{+0.09}_{-0.07}$ | $52.79\par^{+0.09}_{-0.07}$ | 070802 | 2.45 | 14.72 $\pm$ 0.61 | $51.71^{+\par 0.46}_{-0.08}$ | $51.63^{+0.46}_{-0.08}$
120729A | 0.8 | 78.65 $\pm$ 6.50 | $51.86^{+0.40}_{-0.08}$ | $51.77\par^{+0.40}_{-0.08}$ | 070721B | 3.626 | 330.66 $\pm$ 6.28 | $53.51^{+\par 0.32}_{-0.19}$ | $53.47^{+0.32}_{-0.19}$
120724A | 1.48 | 49.17 $\pm$ 4.33 | $51.78^{+0.65}_{-0.10}$ | $51.68\par^{+0.65}_{-0.10}$ | 070714B | 0.92 | 64.18 $\pm$ 1.60 | $51.50^{+\par 0.60}_{-0.15}$ | $51.41^{+0.60}_{-0.15}$
120722A | 0.9586 | 37.31 $\pm$ 2.46 | $51.68^{+0.71}_{-0.03}$ | $51.59\par^{+0.71}_{-0.03}$ | 070612A | 0.617 | 254.74 $\pm$ 3.63 | $52.30^{+\par 0.40}_{-0.09}$ | $52.22^{+0.40}_{-0.09}$
120712A | 4.1745 | 18.46 $\pm$ 1.08 | $52.97^{+0.19}_{-0.07}$ | $52.94\par^{+0.19}_{-0.07}$ | 070611 | 2.04 | 11.31 $\pm$ 0.45 | $51.72^{+\par 0.30}_{-0.10}$ | $51.64^{+0.30}_{-0.10}$
120404A | 2.876 | 40.50 $\pm$ 1.49 | $52.65^{+0.30}_{-0.08}$ | $52.58\par^{+0.30}_{-0.08}$ | 070529 | 2.4996 | 112.21 $\pm$ 2.94 | $52.98^{+\par 0.40}_{-0.16}$ | $52.91^{+0.40}_{-0.16}$
120327A | 2.813 | 71.20 $\pm$ 2.33 | $53.00^{+0.17}_{-0.05}$ | $52.93\par^{+0.17}_{-0.05}$ | 070521 | $1.35^{h}$ | $38.60^{b}$ | $53.40^{+\par 0.38}_{-0.15}$c | $53.31^{+0.38}_{-0.15}$
120326A | 1.798 | 72.72 $\pm$ 3.08 | $52.49^{+0.07}_{-0.03}$ | $52.40\par^{+0.07}_{-0.03}$ | 070518 | 1.16 | 5.34 $\pm$ 0.19 | $50.94^{+\par 0.75}_{-0.06}$ | $50.85^{+0.75}_{-0.06}$
120119A | 1.728 | 70.40 $\pm$ 4.32 | $53.33^{+0.08}_{-0.04}$ | $53.24\par^{+0.08}_{-0.04}$ | 070508 | 0.82 | 21.20 $\pm$ 0.25 | $52.90^{+\par 0.09}_{-0.06}$ | $52.81^{+0.09}_{-0.06}$
120118B | 2.943 | 30.78 $\pm$ 2.85 | $52.81^{+0.55}_{-0.04}$ | $52.75\par^{+0.55}_{-0.04}$ | 070506 | 2.31 | 3.55 $\pm$ 0.17 | $51.42^{+\par 0.28}_{-0.09}$ | $51.35^{+0.28}_{-0.09}$
111229A | 1.3805 | 2.79 $\pm$ 0.25 | $50.97^{+0.72}_{-0.06}$ | $50.88\par^{+0.72}_{-0.06}$ | 070419B | $1.9591^{i}$ | $238.14^{e}$ | $53.38^{+\par 0.01}_{-0.01}$f | $53.29^{+0.01}_{-0.01}$
111228A | 0.714 | 101.40 $\pm$ 1.31 | $52.56^{+0.11}_{-0.10}$ | $52.48\par^{+0.11}_{-0.10}$ | 070419A | 0.97 | 161.25 $\pm$ 8.87 | $51.39^{+\par 0.42}_{-0.09}$ | $51.29^{+0.42}_{-0.09}$
111123A | 3.1516 | 235.20 $\pm$ 6.58 | $53.39^{+0.15}_{-0.07}$ | $53.34\par^{+0.15}_{-0.07}$ | 070411 | 2.954 | 108.56 $\pm$ 3.62 | $53.02^{+\par 0.34}_{-0.08}$ | $52.96^{+0.34}_{-0.08}$
111209A | 0.677 | 4.64 $\pm$ 0.33 | $51.18^{+0.77}_{-0.17}$ | $51.09\par^{+0.77}_{-0.17}$ | 070318 | 0.836 | 51.00 $\pm$ 2.32 | $51.98^{+\par 0.41}_{-0.10}$ | $51.89^{+0.41}_{-0.10}$
111107A | 2.893 | 31.59 $\pm$ 2.44 | $52.52^{+0.44}_{-0.11}$ | $52.46\par^{+0.44}_{-0.11}$ | 070306 | 1.497 | 261.36 $\pm$ 6.65 | $52.80^{+\par 0.39}_{-0.08}$ | $52.71^{+0.39}_{-0.08}$
111008A | 4.9898 | 75.66 $\pm$ 2.25 | $53.69^{+0.34}_{-0.06}$ | $53.68\par^{+0.34}_{-0.06}$ | 070208 | 1.165 | 52.48 $\pm$ 0.85 | $51.47^{+\par 0.34}_{-0.13}$ | $51.37^{+0.34}_{-0.13}$
110818A | 3.36 | 77.28 $\pm$ 5.61 | $53.16^{+0.40}_{-0.07}$ | $53.11\par^{+0.40}_{-0.07}$ | 070129 | 2.3384 | 92.15 $\pm$ 2.24 | $52.49^{+\par 0.11}_{-0.09}$ | $52.41^{+0.11}_{-0.09}$
110808A | 1.348 | 39.38 $\pm$ 3.44 | $51.45^{+0.91}_{-0.09}$ | $51.36\par^{+0.91}_{-0.09}$ | 070110 | 2.352 | 47.70 $\pm$ 1.54 | $52.45^{+\par 0.30}_{-0.08}$ | $52.38^{+0.30}_{-0.08}$
110801A | 1.858 | 400.40 $\pm$ 1.99 | $52.80^{+0.19}_{-0.09}$ | $52.72\par^{+0.19}_{-0.09}$ | 070103 | 2.6208 | 10.92 $\pm$ 0.14 | $51.70^{+\par 0.47}_{-0.09}$ | $51.63^{+0.47}_{-0.09}$
110731A | 2.83 | 46.56 $\pm$ 7.14 | $53.56^{+0.32}_{-0.14}$ | $53.50\par^{+0.32}_{-0.14}$ | 061222B | 3.355 | 42.00 $\pm$ 2.15 | $52.92^{+\par 0.39}_{-0.08}$ | $52.87^{+0.39}_{-0.08}$
110715A | 0.82 | 13.15 $\pm$ 1.40 | $52.48^{+0.04}_{-0.03}$ | $52.39\par^{+0.04}_{-0.03}$ | 061222A | 2.088 | 81.65 $\pm$ 4.24 | $53.32^{+\par 0.25}_{-0.07}$ | $53.24^{+0.25}_{-0.07}$
110503A | 1.613 | 9.31 $\pm$ 0.64 | $53.07^{+0.16}_{-0.08}$ | $52.98\par^{+0.16}_{-0.08}$ | 061126 | 1.159 | 26.78 $\pm$ 0.46 | $52.89^{+\par 0.39}_{-0.14}$ | $52.80^{+0.39}_{-0.14}$
110422A | 1.77 | 26.73 $\pm$ 0.29 | $53.65^{+0.03}_{-0.02}$ | $53.57\par^{+0.03}_{-0.02}$ | 061121 | 1.314 | 83.00 $\pm$ 12.50 | $53.30^{+\par 0.24}_{-0.11}$ | $53.20^{+0.24}_{-0.11}$
110213A | 1.46 | 43.12 $\pm$ 3.47 | $52.72^{+0.26}_{-0.08}$ | $52.62\par^{+0.26}_{-0.08}$ | 061110B | 3.44 | 32.39 $\pm$ 0.45 | $53.12^{+\par 0.37}_{-0.26}$ | $53.07^{+0.37}_{-0.26}$
110205A | 2.22 | 277.02 $\pm$ 4.67 | $53.48^{+0.10}_{-0.04}$ | $53.41\par^{+0.10}_{-0.04}$ | 061110A | 0.757 | 47.04 $\pm$ 1.80 | $51.46^{+\par 0.43}_{-0.09}$ | $51.38^{+0.43}_{-0.09}$
110128A | 2.339 | 17.10 $\pm$ 0.70 | $52.36^{+0.49}_{-0.22}$ | $52.28\par^{+0.49}_{-0.22}$ | 061021 | 0.3463 | 12.06 $\pm$ 0.32 | $51.40^{+\par 0.38}_{-0.15}$ | $51.34^{+0.38}_{-0.15}$
101225A | 0.847 | 63.00 $\pm$ 6.97 | $51.43^{+0.64}_{-0.33}$ | $51.34\par^{+0.64}_{-0.33}$ | 061007 | 1.261 | 74.90 $\pm$ 0.51 | $54.17^{+\par 0.33}_{-0.17}$ | $54.08^{+0.33}_{-0.17}$
101219B | 0.55 | 41.80 $\pm$ 1.45 | $51.47^{+0.52}_{-0.08}$ | $51.39\par^{+0.52}_{-0.08}$ | 060927 | 5.4636 | 23.03 $\pm$ 0.26 | $52.95^{+\par 0.10}_{-0.06}$ | $52.95^{+0.10}_{-0.06}$
101213A | 0.414 | 175.68 $\pm$ 15.30 | $51.85^{+0.32}_{-0.17}$ | $51.78\par^{+0.32}_{-0.17}$ | 060926 | 3.2 | 7.05 $\pm$ 0.39 | $51.95^{+\par 1.13}_{-0.08}$ | $51.90^{+1.13}_{-0.08}$
100906A | 1.727 | 116.85 $\pm$ 0.69 | $53.14^{+0.21}_{-0.07}$ | $53.05\par^{+0.21}_{-0.07}$ | 060923A | $4^{g}$ | $51.50^{b}$ | $53.30^{+\par 0.20}_{-0.10}$c | $53.27^{+0.20}_{-0.10}$
100901A | 1.408 | 459.19 $\pm$ 10.66 | $52.26^{+0.57}_{-0.12}$ | $52.17\par^{+0.57}_{-0.12}$ | 060912A | 0.937 | 5.92 $\pm$ 0.35 | $51.92^{+\par 0.26}_{-0.12}$ | $51.83^{+0.26}_{-0.12}$
100816A | 0.8049 | 2.50 $\pm$ 0.22 | $51.75^{+0.15}_{-0.06}$ | $51.66\par^{+0.15}_{-0.06}$ | 060908 | 1.8836 | 18.48 $\pm$ 0.17 | $52.61^{+\par 0.18}_{-0.07}$ | $52.53^{+0.18}_{-0.07}$
100814A | 1.44 | 176.96 $\pm$ 3.61 | $52.79^{+0.16}_{-0.05}$ | $52.70\par^{+0.16}_{-0.05}$ | 060906 | 3.686 | 72.96 $\pm$ 9.41 | $53.11^{+\par 0.43}_{-0.04}$ | $53.07^{+0.43}_{-0.04}$
100728B | 2.106 | 11.52 $\pm$ 0.78 | $52.39^{+0.33}_{-0.07}$ | $52.31\par^{+0.33}_{-0.07}$ | 060904B | 0.703 | 171.04 $\pm$ 2.29 | $51.49^{+\par 0.28}_{-0.09}$ | $51.40^{+0.28}_{-0.09}$
100728A | 1.567 | 222.00 $\pm$ 6.89 | $53.82^{+0.14}_{-0.08}$ | $53.73\par^{+0.14}_{-0.08}$ | 060814 | 0.84 | 159.16 $\pm$ 4.08 | $52.95^{+\par 0.03}_{-0.18}$ | $52.86^{+0.03}_{-0.18}$
100621A | 0.542 | 66.33 $\pm$ 1.27 | $52.46^{+0.05}_{-0.03}$ | $52.38\par^{+0.05}_{-0.03}$ | 060805A | $3.8^{g}$ | $4.93^{b}$ | $52.26^{+\par 0.65}_{-0.12}$c | $52.22^{+0.65}_{-0.12}$
100615A | 1.398 | 43.46 $\pm$ 1.30 | $52.62^{+0.08}_{-0.05}$ | $52.53\par^{+0.08}_{-0.05}$ | 060729 | 0.54 | 119.14 $\pm$ 1.40 | $51.49^{+\par 0.33}_{-0.08}$ | $51.41^{+0.33}_{-0.08}$
100513A | 4.772 | 65.10 $\pm$ 4.39 | $52.92^{+0.37}_{-0.08}$ | $52.90\par^{+0.37}_{-0.08}$ | 060719 | 1.532 | 57.00 $\pm$ 0.84 | $52.16^{+\par 0.55}_{-0.03}$ | $52.07^{+0.55}_{-0.03}$
100425A | 1.755 | 43.56 $\pm$ 1.03 | $51.81^{+0.73}_{-0.12}$ | $51.72\par^{+0.73}_{-0.12}$ | 060714 | 2.711 | 118.72 $\pm$ 1.87 | $52.90^{+\par 0.42}_{-0.05}$ | $52.83^{+0.42}_{-0.05}$
100424A | 2.465 | 110.25 $\pm$ 5.30 | $52.50^{+0.30}_{-0.08}$ | $52.42\par^{+0.30}_{-0.08}$ | 060708 | 1.92 | 7.50 $\pm$ 0.45 | $51.78^{+\par 0.20}_{-0.07}$ | $51.70^{+0.20}_{-0.07}$
100418A | 0.624 | 9.63 $\pm$ 0.81 | $50.73^{+0.77}_{-0.04}$ | $50.65\par^{+0.77}_{-0.04}$ | 060707 | 3.425 | 75.14 $\pm$ 2.46 | $52.80^{+\par 0.14}_{-0.07}$ | $52.75^{+0.14}_{-0.07}$
100316B | 1.18 | 4.30 $\pm$ 0.34 | $51.08^{+0.86}_{-0.03}$ | $50.99\par^{+0.86}_{-0.03}$ | 060614 | 0.125 | 108.80 $\pm$ 0.86 | $51.40^{+\par 0.07}_{-0.08}$ | $51.37^{+0.07}_{-0.08}$
100302A | 4.813 | 31.72 $\pm$ 3.11 | $52.36^{+0.72}_{-0.04}$ | $52.35\par^{+0.72}_{-0.04}$ | 060607A | 3.082 | 102.55 $\pm$ 3.35 | $52.97^{+\par 0.32}_{-0.08}$ | $52.91^{+0.32}_{-0.08}$
100219A | 4.667 | 31.05 $\pm$ 2.84 | $52.46^{+0.55}_{-0.13}$ | $52.44\par^{+0.55}_{-0.13}$ | 060605 | 3.78 | 18.54 $\pm$ 1.16 | $52.34^{+\par 0.53}_{-0.10}$ | $52.30^{+0.53}_{-0.10}$
091208B | 1.063 | 15.21 $\pm$ 1.31 | $52.16^{+0.17}_{-0.07}$ | $52.06\par^{+0.17}_{-0.07}$ | 060604 | 2.1357 | 39.90 $\pm$ 0.70 | $51.73^{+\par 0.96}_{-0.10}$ | $51.65^{+0.96}_{-0.10}$
091127 | 0.49 | 9.57 $\pm$ 0.56 | $52.16^{+0.31}_{-0.02}$ | $52.09\par^{+0.31}_{-0.02}$ | 060602A | $0.787^{i}$ | $74.68^{e}$ | $51.98^{+\par 0.04}_{-0.04}$f | $51.89^{+0.04}_{-0.04}$
091109A | 3.076 | 49.68 $\pm$ 4.60 | $53.13^{+0.31}_{-0.22}$ | $53.08\par^{+0.31}_{-0.22}$ | 060526 | 3.221 | 295.55 $\pm$ 4.01 | $52.73^{+\par 0.47}_{-0.03}$ | $52.68^{+0.47}_{-0.03}$
091029 | 2.752 | 39.96 $\pm$ 1.28 | $52.91^{+0.06}_{-0.07}$ | $52.85\par^{+0.06}_{-0.07}$ | 060522 | 5.11 | 74.10 $\pm$ 2.30 | $52.87^{+\par 0.40}_{-0.08}$ | $52.86^{+0.40}_{-0.08}$
091024 | 1.092 | 114.73 $\pm$ 4.95 | $52.80^{+0.37}_{-0.15}$ | $52.70\par^{+0.37}_{-0.15}$ | 060512 | 0.4428 | 8.37 $\pm$ 0.36 | $50.31^{+\par 0.65}_{-0.09}$ | $50.24^{+0.65}_{-0.09}$
091020 | 1.71 | 39.00 $\pm$ 1.07 | $52.67^{+0.30}_{-0.08}$ | $52.58\par^{+0.30}_{-0.08}$ | 060510B | 4.9 | 229.89 $\pm$ 2.77 | $53.37^{+\par 0.19}_{-0.08}$ | $53.36^{+0.19}_{-0.08}$
091018 | 0.971 | 4.44 $\pm$ 0.15 | $51.82^{+0.10}_{-0.05}$ | $51.72\par^{+0.10}_{-0.05}$ | 060502A | 1.51 | 30.24 $\pm$ 4.18 | $52.47^{+\par 0.39}_{-0.10}$ | $52.38^{+0.39}_{-0.10}$
090927 | 1.37 | 18.36 $\pm$ 1.33 | $51.35^{+0.71}_{-0.07}$ | $51.26\par^{+0.71}_{-0.07}$ | 060428B | 0.348 | 20.46 $\pm$ 0.62 | $50.31^{+\par 0.28}_{-0.10}$ | $50.25^{+0.28}_{-0.10}$
090926B | 1.24 | 126.36 $\pm$ 5.21 | $52.56^{+0.06}_{-0.03}$ | $52.47\par^{+0.06}_{-0.03}$ | 060418 | 1.489 | 103.24 $\pm$ 10.33 | $52.93^{+\par 0.28}_{-0.06}$ | $52.84^{+0.28}_{-0.06}$
090904B | $5^{j}$ | $64.00^{b}$ | $53.54^{+0.18}_{-0.18}$c | $53.53\par^{+0.18}_{-0.18}$ | 060306 | 3.5 | 60.96 $\pm$ 0.80 | $52.88^{+\par 0.15}_{-0.06}$ | $52.84^{+0.15}_{-0.06}$
090814A | 0.696 | 113.16 $\pm$ 12.99 | $51.39^{+0.24}_{-0.08}$ | $51.30\par^{+0.24}_{-0.08}$ | 060223A | 4.41 | 8.40 $\pm$ 0.28 | $52.50^{+\par 0.17}_{-0.07}$ | $52.48^{+0.17}_{-0.07}$
090812 | 2.452 | 99.76 $\pm$ 15.30 | $53.32^{+0.38}_{-0.12}$ | $53.25\par^{+0.38}_{-0.12}$ | 060210 | 3.91 | 369.94 $\pm$ 20.65 | $53.63^{+\par 0.36}_{-0.08}$ | $53.59^{+0.36}_{-0.08}$
090809 | 2.737 | 192.92 $\pm$ 5.24 | $52.16^{+0.74}_{-0.13}$ | $52.09\par^{+0.74}_{-0.13}$ | 060206 | 4.045 | 6.06 $\pm$ 0.16 | $52.63^{+\par 0.12}_{-0.07}$ | $52.60^{+0.12}_{-0.07}$
090726 | 2.71 | 51.03 $\pm$ 0.97 | $52.27^{+0.49}_{-0.10}$ | $52.21\par^{+0.49}_{-0.10}$ | 060202 | 0.783 | 205.92 $\pm$ 2.52 | $51.83^{+\par 0.41}_{-0.07}$ | $51.74^{+0.41}_{-0.07}$
090715B | 3 | 267.54 $\pm$ 4.54 | $53.39^{+0.28}_{-0.09}$ | $53.33\par^{+0.28}_{-0.09}$ | 060124 | 2.296 | 8.16 $\pm$ 0.19 | $51.84^{+\par 0.44}_{-0.10}$ | $51.76^{+0.44}_{-0.10}$
090709A | $1.8^{d}$ | $88.73^{e}$ | $52.61^{+0.05}_{-0.05}$f | $52.52\par^{+0.05}_{-0.05}$ | 060116 | 6.6 | 36.00 $\pm$ 1.21 | $53.30^{+\par 0.38}_{-0.12}$ | $53.32^{+0.38}_{-0.12}$
090618 | 0.54 | 115.20 $\pm$ 0.43 | $53.17^{+0.04}_{-0.03}$ | $53.10\par^{+0.04}_{-0.03}$ | 060115 | 3.53 | 109.89 $\pm$ 1.14 | $52.79^{+\par 0.17}_{-0.07}$ | $52.75^{+0.17}_{-0.07}$
090529 | 2.625 | 79.79 $\pm$ 3.52 | $52.41^{+0.24}_{-0.09}$ | $52.34\par^{+0.24}_{-0.09}$ | 060110 | $5^{g}$ | $21.10^{b}$ | $53.92^{+\par 0.35}_{-0.08}$c | $53.91^{+0.35}_{-0.08}$
090519 | 3.85 | 81.77 $\pm$ 6.00 | $53.18^{+0.38}_{-0.24}$ | $53.14\par^{+0.38}_{-0.24}$ | 060108 | 2.03 | 15.28 $\pm$ 1.10 | $51.78^{+\par 0.62}_{-0.06}$ | $51.70^{+0.62}_{-0.06}$
090516 | 4.109 | 228.48 $\pm$ 9.45 | $53.73^{+0.38}_{-0.10}$ | $53.69\par^{+0.38}_{-0.10}$ | 051227 | 0.714 | 4.30 $\pm$ 0.19 | $50.90^{+\par 0.57}_{-0.23}$ | $50.81^{+0.57}_{-0.23}$
090429B | 9.4 | 5.80 $\pm$ 0.29 | $52.74^{+0.13}_{-0.07}$ | $52.81\par^{+0.13}_{-0.07}$ | 051117B | 0.481 | 10.45 $\pm$ 0.25 | $50.23^{+\par 0.56}_{-0.11}$ | $50.16^{+0.56}_{-0.11}$
090424 | 0.544 | 50.28 $\pm$ 0.53 | $52.43^{+0.06}_{-0.05}$ | $52.36\par^{+0.06}_{-0.05}$ | 051111 | 1.55 | 50.96 $\pm$ 2.45 | $52.70^{+\par 0.33}_{-0.09}$ | $52.61^{+0.33}_{-0.09}$
090423 | 8.26 | 12.36 $\pm$ 0.59 | $52.93^{+0.09}_{-0.07}$ | $52.98\par^{+0.09}_{-0.07}$ | 051109A | 2.346 | 4.90 $\pm$ 0.30 | $52.35^{+\par 0.49}_{-0.08}$ | $52.28^{+0.49}_{-0.08}$
090418 | 1.608 | 57.97 $\pm$ 0.85 | $52.95^{+0.31}_{-0.15}$ | $52.86\par^{+0.31}_{-0.15}$ | 051016B | $0.9364^{i}$ | $4.02^{e}$ | $51.15^{+\par 0.06}_{-0.06}$f | $51.06^{+0.06}_{-0.06}$
090417B | $0.345^{d}$ | $282.49^{e}$ | $51.41^{+0.03}_{-0.03}$f | $51.35\par^{+0.03}_{-0.03}$ | 051006 | 1.059 | 26.46 $\pm$ 0.53 | $52.02^{+\par 0.34}_{-0.20}$ | $51.93^{+0.34}_{-0.20}$
090407 | 1.4485 | 147.52 $\pm$ 1.02 | $51.71^{+0.74}_{-0.14}$ | $51.62\par^{+0.74}_{-0.14}$ | 051001 | 2.4296 | 55.90 $\pm$ 1.63 | $52.38^{+\par 0.07}_{-0.11}$ | $52.31^{+0.07}_{-0.11}$
090404 | $3^{d}$ | $82.01^{e}$ | $53.30^{+0.02}_{-0.02}$f | $53.24\par^{+0.02}_{-0.02}$ | 050922C | 2.198 | 4.56 $\pm$ 0.12 | $52.60^{+\par 0.30}_{-0.08}$ | $52.52^{+0.30}_{-0.08}$
090313 | 3.375 | 90.24 $\pm$ 6.75 | $52.67^{+0.67}_{-0.05}$ | $52.62\par^{+0.67}_{-0.05}$ | 050915A | 2.5273 | 21.39 $\pm$ 0.59 | $52.26^{+\par 0.52}_{-0.12}$ | $52.19^{+0.52}_{-0.12}$
090205 | 4.6497 | 10.68 $\pm$ 0.69 | $52.09^{+0.59}_{-0.09}$ | $52.07\par^{+0.59}_{-0.09}$ | 050908 | 3.35 | 10.80 $\pm$ 0.64 | $52.11^{+\par 0.26}_{-0.09}$ | $52.06^{+0.26}_{-0.09}$
090113 | 1.7493 | 8.80 $\pm$ 0.13 | $52.01^{+0.48}_{-0.08}$ | $51.92\par^{+0.48}_{-0.08}$ | 050904 | 6.29 | 197.20 $\pm$ 2.26 | $54.13^{+\par 0.22}_{-0.13}$ | $54.15^{+0.22}_{-0.13}$
090102 | 1.547 | 30.69 $\pm$ 1.21 | $53.15^{+0.31}_{-0.17}$ | $53.06\par^{+0.31}_{-0.17}$ | 050826 | 0.297 | 34.44 $\pm$ 1.87 | $50.53^{+\par 0.52}_{-0.24}$ | $50.48^{+0.52}_{-0.24}$
081228 | $3.4^{a}$ | $3.00^{b}$ | $52.57^{+0.19}_{-0.15}$c | $52.52\par^{+0.19}_{-0.15}$ | 050824 | 0.83 | 37.95 $\pm$ 4.02 | $51.19^{+\par 2.47}_{-0.12}$ | $51.10^{+2.47}_{-0.12}$
081222 | 2.77 | 33.48 $\pm$ 1.44 | $53.18^{+0.10}_{-0.05}$ | $53.12\par^{+0.10}_{-0.05}$ | 050822 | 1.434 | 104.88 $\pm$ 2.63 | $52.37^{+\par 0.64}_{-0.03}$ | $52.28^{+0.64}_{-0.03}$
081221 | 2.26 | 34.23 $\pm$ 0.64 | $53.53^{+0.04}_{-0.03}$ | $53.45\par^{+0.04}_{-0.03}$ | 050820A | 2.6147 | 239.68 $\pm$ 0.37 | $53.40^{+\par 0.34}_{-0.20}$ | $53.33^{+0.34}_{-0.20}$
081203A | 2.1 | 254.28 $\pm$ 26.94 | $53.24^{+0.34}_{-0.10}$ | $53.16\par^{+0.34}_{-0.10}$ | 050819 | 2.5043 | 46.80 $\pm$ 4.85 | $52.00^{+\par 0.92}_{-0.11}$ | $51.93^{+0.92}_{-0.11}$
081121 | 2.512 | 19.38 $\pm$ 0.96 | $53.21^{+0.40}_{-0.11}$ | $53.14\par^{+0.40}_{-0.11}$ | 050814 | 5.3 | 27.54 $\pm$ 1.71 | $52.73^{+\par 0.21}_{-0.09}$ | $52.72^{+0.21}_{-0.09}$
081118 | 2.58 | 66.55 $\pm$ 5.08 | $52.46^{+0.68}_{-0.06}$ | $52.39\par^{+0.68}_{-0.06}$ | 050803 | 0.422 | 88.20 $\pm$ 1.35 | $51.40^{+\par 0.44}_{-0.15}$ | $51.33^{+0.44}_{-0.15}$
081109 | $0.98^{k}$ | $221.00^{b}$ | $52.61^{+0.28}_{-0.23}$c | $52.52\par^{+0.28}_{-0.23}$ | 050802 | 1.71 | 14.25 $\pm$ 0.60 | $52.27^{+\par 0.35}_{-0.08}$ | $52.18^{+0.35}_{-0.08}$
081029 | 3.8479 | 169.10 $\pm$ 8.55 | $53.17^{+0.25}_{-0.20}$ | $53.14\par^{+0.25}_{-0.20}$ | 050801 | 1.56 | 5.88 $\pm$ 0.20 | $51.31^{+\par 0.63}_{-0.06}$ | $51.22^{+0.63}_{-0.06}$
081028 | 3.038 | 275.59 $\pm$ 9.68 | $53.07^{+0.12}_{-0.08}$ | $53.01\par^{+0.12}_{-0.08}$ | 050730 | 3.969 | 60.48 $\pm$ 2.26 | $52.92^{+\par 0.42}_{-0.12}$ | $52.88^{+0.42}_{-0.12}$
081008 | 1.9685 | 199.32 $\pm$ 11.52 | $52.82^{+0.21}_{-0.08}$ | $52.74\par^{+0.21}_{-0.08}$ | 050724 | 0.258 | 2.50 $\pm$ 0.04 | $49.96^{+\par 0.49}_{-0.08}$ | $49.92^{+0.49}_{-0.08}$
081007 | 0.5295 | 5.55 $\pm$ 0.26 | $50.87^{+0.28}_{-0.09}$ | $50.79\par^{+0.28}_{-0.09}$ | 050713A | $3.6^{g}$ | $94.90^{b}$ | $54.19^{+\par 0.37}_{-0.13}$c | $54.15^{+0.37}_{-0.13}$
080928 | 1.692 | 284.90 $\pm$ 12.16 | $52.46^{+0.38}_{-0.08}$ | $52.37\par^{+0.38}_{-0.08}$ | 050607 | $4^{g}$ | $48.00^{b}$ | $53.09^{+\par 0.38}_{-0.05}$c | $53.06^{+0.38}_{-0.05}$
080916A | 0.689 | 62.53 $\pm$ 3.24 | $51.92^{+0.11}_{-0.05}$ | $51.84\par^{+0.11}_{-0.05}$ | 050603 | 2.821 | 9.80 $\pm$ 0.39 | $53.63^{+\par 0.40}_{-0.15}$ | $53.56^{+0.40}_{-0.15}$
080913 | 6.7 | 8.19 $\pm$ 0.26 | $52.85^{+0.41}_{-0.09}$ | $52.87\par^{+0.41}_{-0.09}$ | 050525 | 0.606 | 9.10 $\pm$ 0.04 | $52.32^{+\par 0.02}_{-0.02}$ | $52.24^{+0.02}_{-0.02}$
080905B | 2.374 | 103.97 $\pm$ 4.68 | $52.55^{+0.39}_{-0.08}$ | $52.47\par^{+0.39}_{-0.08}$ | 050505 | 4.27 | 60.20 $\pm$ 1.35 | $53.21^{+\par 0.38}_{-0.10}$ | $53.18^{+0.38}_{-0.10}$
080810 | 3.35 | 453.15 $\pm$ 5.09 | $53.56^{+0.27}_{-0.19}$ | $53.50\par^{+0.27}_{-0.19}$ | 050502B | $5.2^{i}$ | $16.62^{e}$ | $52.82^{+\par 0.04}_{-0.04}$f | $52.81^{+0.04}_{-0.04}$
080805 | 1.505 | 111.84 $\pm$ 9.11 | $52.62^{+0.22}_{-0.17}$ | $52.53\par^{+0.22}_{-0.17}$ | 050416A | 0.6535 | 2.91 $\pm$ 0.18 | $51.00^{+\par 0.19}_{-0.09}$ | $50.92^{+0.19}_{-0.09}$
080804 | 2.2 | 61.74 $\pm$ 8.81 | $53.21^{+0.45}_{-0.18}$ | $53.13\par^{+0.45}_{-0.18}$ | 050412 | $4.5^{g}$ | $26.50^{b}$ | $54.00^{+\par 0.79}_{-0.26}$c | $53.98^{+0.79}_{-0.26}$
080721 | 2.602 | 29.92 $\pm$ 2.29 | $54.06^{+0.42}_{-0.20}$ | $53.99\par^{+0.42}_{-0.20}$ | 050406 | $2.7^{i}$ | $4.79^{e}$ | $51.56^{+\par 0.09}_{-0.09}$f | $51.49^{+0.09}_{-0.09}$
080710 | 0.845 | 139.05 $\pm$ 10.01 | $51.91^{+0.46}_{-0.23}$ | $51.82\par^{+0.46}_{-0.23}$ | 050401 | 2.9 | 34.41 $\pm$ 0.34 | $53.52^{+\par 0.35}_{-0.09}$ | $53.46^{+0.35}_{-0.09}$
080707 | 1.23 | 30.25 $\pm$ 0.43 | $51.55^{+0.52}_{-0.07}$ | $51.45\par^{+0.52}_{-0.07}$ | 050319 | 3.24 | 153.55 $\pm$ 2.20 | $52.67^{+\par 0.62}_{-0.05}$ | $52.62^{+0.62}_{-0.05}$
080607 | 3.036 | 83.66 $\pm$ 0.83 | $54.46^{+0.20}_{-0.14}$ | $54.40\par^{+0.20}_{-0.14}$ | 050318 | 1.44 | 30.96 $\pm$ 0.09 | $52.08^{+\par 0.08}_{-0.09}$ | $51.98^{+0.08}_{-0.09}$
080605 | 1.6398 | 19.57 $\pm$ 0.32 | $53.33^{+0.19}_{-0.08}$ | $53.24\par^{+0.19}_{-0.08}$ | 050315 | 1.949 | 94.60 $\pm$ 1.66 | $52.77^{+\par 0.48}_{-0.01}$ | $52.68^{+0.48}_{-0.01}$
080604 | 1.416 | 125.28 $\pm$ 5.37 | $51.86^{+0.46}_{-0.09}$ | $51.77\par^{+0.46}_{-0.09}$ | 050223 | 0.5915 | 17.38 $\pm$ 0.60 | $50.87^{+\par 0.29}_{-0.08}$ | $50.79^{+0.29}_{-0.08}$
080603B | 2.69 | 59.50 $\pm$ 0.51 | $52.80^{+0.07}_{-0.07}$ | $52.74\par^{+0.07}_{-0.07}$ | 050126 | 1.29 | 28.71 $\pm$ 1.91 | $51.90^{+\par 0.58}_{-0.12}$ | $51.81^{+0.58}_{-0.12}$
aRedshift from Greiner et al. (2011). b$T_{90}$ taken from Robertson & Ellis
(2012). c$E_{\rm iso}$ taken from Robertson & Ellis (2012). dRedshift from
Perley & Perley (2013). e$T_{90}$ taken from Sakamoto et al. (2011). f$E_{\rm
iso}$ calculated from the fluence provided by Sakamoto et al. (2011). gDark
GRB redshift limit from Perley et al. (2009). hRedshift from Perley et al.
(2009). iRedshift from Hjorth et al. (2012). jDark GRB redshift limit from
Greiner et al. (2011). kRedshift from Kr$\ddot{\rm u}$hler et al. (2011).
Since we will use the cumulative redshift distribution $N(<z)$ of this sample
as the basis for our analysis, it is important to consider its uncertainties.
Redshift measurements are strongly biased towards optically bright afterglows,
and are more easily made when the afterglow is not obscured by dust (see,
e.g., Greiner et al. 2011). The phenomenon of so-called dark GRBs with
suppressed optical counterparts could influence whether the observed $N(<z)$
is representative of that for all long-duration GRBs. Perley et al. (2009)
have considered this important issue by attempting to constrain the redshift
distribution of dark GRBs through deep searches that successfully located
faint optical and near-infrared counterparts. The Perley et al. (2009) work
provides us with one redshift and nine redshift upper limits for a subsample
of dark GRBs in our catalog. Greiner et al. (2011) and Kr$\ddot{\rm u}$hler et
al. (2011) have pursued this effort in parallel and have provided three
additional redshifts and one redshift upper limit for dark GRBs in our
catalog. Via host galaxy measurements, Hjorth et al. (2012) and Perley &
Perley (2013) have also provided nine additional redshifts for dark GRBs that
we have added to our catalog. We assume that the subsamples of dark GRBs with
redshift upper limits presented by Perley et al. (2009), Greiner et al.
(2011), and Kr$\ddot{\rm u}$hler et al. (2011) are representative of that
class, and therefore optionally incorporate those limits to characterize the
effects of possible incompleteness of the Swift sample with firm redshift
determinations.
Figure 2: The luminosity-redshift distribution of 254 _Swift_ GRBs in
$\Lambda$CDM (left panel) and $R_{\rm h}=ct$ (right panel). The blue dots
represent the bursts with redshift upper limits. The shaded regions represent
the luminosity threshold adopted in our calculations (see text and equation
18).
Our final sample includes 254 GRBs, whose luminosity-redshift distribution is
shown in Figure 2. A determination of $E_{\rm iso}$ requires the assumption of
a particular cosmological model. In this figure, we show the resulting
distributions for both $\Lambda$CDM (left panel) and $R_{\rm h}=ct$ (right
panel). As presented in the various sources used to compile our catalog,
quantities such as $E_{\rm iso}$ are estimated assuming a $\Lambda$CDM
cosmology. Here, we must therefore recalibrate them for use in $R_{\rm h}=ct$.
The differences between these two models222See also Melia (2012b) for a more
pedagogical description of the $R_{\rm h}=ct$ Universe. are summarized in
Melia (2012a,2013a,2013b), Melia & Shevchuk (2012), Melia & Maier (2013), and
Wei et al. (2013). The luminosity distance in $\Lambda$CDM is given by the
expression
$D_{L}^{\Lambda{\rm CDM}}(z)={c\over
H_{0}}{(1+z)\over\sqrt{\mid\Omega_{k}\mid}}\;sinn\left\\{\mid\Omega_{k}\mid^{1/2}\times\int_{0}^{z}{dz\over\sqrt{(1+z)^{2}(1+\Omega_{m}z)-z(2+z)\Omega_{\Lambda}}}\right\\}\;,$
(2)
where $c$ is the speed of light, and $H_{0}$ is the Hubble constant at the
present time. In this equation, $\Omega_{m}\equiv\rho_{m}/\rho_{c}$ is the
energy density of matter written in terms of today’s critical density,
$\rho_{c}\equiv 3c^{2}H_{0}^{2}/8\pi G$. Also, $\Omega_{\Lambda}$ is the
similarly defined density of dark energy, and $\Omega_{k}$ represents the
spatial curvature of the Universe—appearing as a term proportional to the
spatial curvature constant $k$ in the Friedmann equation. In addition, $sinn$
is $\sinh$ when $\Omega_{k}>0$ and $\sin$ when $\Omega_{k}<0$. For a flat
Universe with $\Omega_{k}=0$, this equation simplifies to the form
$(1+z)c/H_{0}$ times the integral.
In the $R_{\rm h}=ct$ Universe, the luminosity distance is given by the much
simpler expression
$D_{L}^{R_{\rm h}=ct}=\frac{c}{H_{0}}(1+z)\ln(1+z)\;.$ (3)
The factor $c/H_{0}$ is in fact the gravitational horizon $R_{\rm h}(t_{0})$
at the present time, so we may also write the luminosity distance as
$D_{L}^{R_{\rm h}=ct}=R_{\rm h}(t_{0})(1+z)\ln(1+z)\;.$ (4)
We have found the equivalent isotropic energy in the $R_{\rm h}=ct$ Universe
using the expression
${E}_{\rm iso}^{R_{\rm h}=ct}=E_{\rm iso}^{\Lambda{\rm
CDM}}\times\left({D^{R_{\rm h}=ct}_{L}\over D^{\Lambda{\rm
CDM}}_{L}}\right)^{2}\;,$ (5)
where $E_{\rm iso}^{\Lambda{\rm CDM}}$ is the previously published value.
## 3 The Model
The observed rate of GRBs per unit time at redshifts $\in(z,z+dz)$ with
luminosity $\in(L,L+dL)$ is given by
$\frac{dN}{dt\,dz\,dL}=\frac{\dot{\rho}_{\rm
GRB}(z)}{1+z}\frac{\Delta\Omega}{4\pi}\frac{dV_{\rm com}(z)}{dz}\,\Phi(L)\;,$
(6)
where $\dot{\rho}_{\rm GRB}(z)$ is the co-moving rate density of GRBs,
$\Phi(L)$ is the beaming-convolved luminosity function (LF), the factor
$(1+z)^{-1}$ accounts for the cosmological time dilation and
$\Delta\Omega=1.4$ sr is the solid angle covered on the sky by _Swift_
(Salvaterra & Chincarini 2007). The co-moving volume is calculated using
$\frac{dV_{\rm com}}{dz}=4\pi D_{\rm com}^{2}\frac{dD_{\rm com}}{dz}.$ (7)
In the standard ($\Lambda$CDM) model, the co-moving luminosity distance is
given as
$D_{\rm com}^{\rm\Lambda
CDM}(z)\equiv\frac{c}{H_{0}}\int_{0}^{z}\frac{dz^{\prime}}{\sqrt{\Omega_{\rm
m}(1+z^{\prime})^{3}+\Omega_{\Lambda}}}\;,$ (8)
where we now adopt concordance values of the cosmological parameters:
$H_{0}=70$ km $\rm s^{-1}$ $\rm Mpc^{-1}$, $\Omega_{\rm m}=0.3$, and
$\Omega_{\Lambda}=0.7$, and assume a spatially flat expansion. In the $R_{\rm
h}=ct$ Universe, the co-moving luminosity distance is given by the much
simpler expression
$D_{\rm com}^{R_{\rm h}=ct}(z)=\frac{c}{H_{0}}\ln(1+z)$ (9)
which, as we have noted previously, has only one free parameter—the Hubble
constant $H_{0}$. For the sake of consistency, we will adopt the standard
$H_{0}=70$ km $\rm s^{-1}$ $\rm Mpc^{-1}$ throughout this analysis.
As discussed above, we assume that the GRB rate density is related to the
cosmic SFR density $R_{\rm SF}(z)$ and a possible evolution effect $f(z)$,
given as
$\dot{\rho}_{\rm GRB}(z)=k_{\rm GRB}R_{\rm SF}(z)f(z)\;,$ (10)
where $k_{\rm GRB}$ is the GRB formation efficiency to be determined from the
observations.
Because of the faintness of sub-luminous galaxies at high redshifts, as well
as the uncertainty of the dust extinction (in terms of the amount of dust as
well as the dust attenuation law), it is difficult to observe LBG’s at high
redshifts. Consequently, the LBG samples are incomplete, and the star
formation history at $z\ga 4$ is not well constrained by the data. For
relatively low redshifts ($z\la 4$), the star formation rate density $R_{\rm
SF}$ has been fitted with a piecewise power law (Hopkins & Beacom 2006; Li
2008), which in $\Lambda$CDM (with the concordance, WMAP parameters) may be
written
$\log_{10}R_{\rm SF}^{\rm\Lambda CDM}(z)=a+b\log_{10}(1+z)\;,$ (11)
where
$(a,b)=\left\\{\begin{array}[]{ll}(-1.70,3.30),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z<0.993\\\
(-0.727,0.0549),~{}~{}~{}~{}~{}0.993<z<3.8,\\\ \end{array}\right.$ (12)
and $R_{\rm SF}$ is in units of $\rm M_{\bigodot}$ $\rm yr^{-1}$ $\rm
Mpc^{-3}$. To convert from one cosmology to another, our procedure is as
follows: the co-moving volume is proportional to the co-moving distance cubed,
$V_{\rm com}\propto D_{\rm com}^{3}$, and the co-moving volume between
redshifts $z-\Delta z$ and $z+\Delta z$ is $V_{\rm com}(z,\Delta z)\propto
D_{\rm com}^{3}(z+\Delta z)-D_{\rm com}^{3}(z-\Delta z)$. Since the luminosity
is proportional to the co-moving distance squared, $L\propto D_{\rm com}^{2}$,
the SFR density for a given redshift range is (Hopkins 2004)
$R_{\rm SF}(z)\propto\frac{L(z)}{V_{\rm com}(z,\Delta z)}\propto\frac{D_{\rm
com}^{2}(z)}{D_{\rm com}^{3}(z+\Delta z)-D_{\rm com}^{3}(z-\Delta z)}.$ (13)
Thus, the SFR in the redshift range $z=0-3.8$ for the $R_{\rm h}=ct$ Universe
becomes
$\log R_{\rm SF}^{R_{\rm h}=ct}(z)=a+b\log(1+z)\;,$ (14)
where
$(a,b)=\left\\{\begin{array}[]{ll}(-1.70,3.52),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}z<0.993\\\
(-0.507,-0.46),~{}~{}~{}~{}~{}0.993<z<3.8.\\\ \end{array}\right.$ (15)
For the GRB luminosity function (LF) $\Phi(L)$, several models have been
adopted in the literature: a single power law with an exponential cut-off at
low luminosity (exponential LF), a broken power law, and a Schechter function.
Here, we use the exponential LF
$\Phi(L)\propto\left(\frac{L}{L_{\star}}\right)^{-a_{\rm
L}}\exp\left(-\frac{L_{\star}}{L}\right)\;,$ (16)
where $a_{\rm L}$ is the power-law index and $L_{\star}$ is the cutoff
luminosity. The normalization constant of the LF is calculated assuming a
minimum luminosity $L_{\rm min}=10^{49}$ erg s-1. The LF will be taken to be
non-evolving in this paper.
Finally, when considering an instrument having a flux threshold, the expected
number of GRBs with luminosity $L_{\rm iso}>L_{\rm lim}$ and redshift
$z\in(z_{1},z_{2})$ during an observational period T should be
$\begin{split}N=\frac{\Delta\Omega\;T}{4\pi}\int_{z_{1}}^{z_{2}}\frac{\dot{\rho}_{\rm
GRB}(z)}{1+z}\frac{dV_{\rm com}(z)}{dz}\;dz\int_{\rm max[L_{\rm min},L_{\rm
lim}(z)]}^{\infty}\Phi(L)\;dL\;.\end{split}$ (17)
The luminosity threshold appearing in equation (17) may be approximated using
a bolometric energy flux limit $F_{\rm lim}=1.2\times 10^{-8}$ erg $\rm
cm^{-2}$ $\rm s^{-1}$ (Li 2008), for which
$L_{\rm lim}=4\pi D_{\rm L}^{2}F_{\rm lim}\;,$ (18)
where $D_{\rm L}$ is the luminosity distance to the burst (either $D_{\rm
L}^{\Lambda{\rm CDM}}$ or $D_{\rm L}^{R_{\rm h}=ct}$, as the case may be).
## 4 A Comparative Analysis of $\Lambda$CDM and The $R_{\rm h}=ct$ Universe
### 4.1 A possible evolutionary effect
The _Swift_ /BAT trigger is quite complex. Its algorithm has two modes: the
count rate trigger and the image trigger (Fenimore et al. 2003; Sakamoto et
al. 2008, 2011). Rate triggers are measured on different timescales (4 ms to
64 s), with a single or several backgrounds. Image triggers are found by
summing images over various timescales and searching for uncataloged sources.
So the sensitivity of the BAT is very difficult to parametrize exactly (Band
2006). Moreover, although the rate density $R_{\rm SF}(z)$ is now reasonably
well measured from $z=0$ to $4$, it is not well constrained at $z\ga 4$. To
avoid the complications that would arise from the use of a detailed treatment
of the _Swift_ threshold and the star formation rate at high-_z_ , we will
adopt a model-independent approach by selecting only GRBs with $L_{\rm
iso}\geq L_{\rm lim}$ and $z<4$, as Kistler et al. (2008) did in their
treatment. The cut in luminosity333Note that although the luminosity distances
are formulated differently in the two cosmologies we are examining here,
distance measures in the optimized $\Lambda$CDM model are very close to those
in $R_{\rm h}=ct$, so this cutoff does not bias either model. is chosen to be
equal to the threshold at the highest redshift of the sample, i.e., $L_{\rm
lim}\approx 1.8\times 10^{51}$ erg $\rm s^{-1}$. The cuts in luminosity and
redshift minimize selection effects in the GRB data. With these conditions,
our final tally of GRBs is 118 for $\Lambda$CDM and 111 in $R_{\rm h}=ct$.
These data are delimited by the red dashed lines in Figure 2.
Now, since $L_{\rm lim}$ is constant, the integral of the LF in equation (17)
can be treated as a constant coefficient, no matter what the specific form of
$\Phi(L)$ happens to be. That is, we may write
$\begin{split}N(<z)\propto\int_{0}^{z}R_{\rm
SF}(z)\frac{f(z)}{1+z}\frac{dV_{\rm com}}{dz}\;dz\;.\end{split}$ (19)
Figure 3 shows the cumulative redshift distribution of observed GRBs (steps),
normalized over the redshift range $0<z<4$. The gray-shaded region shows how
the distribution shifts in the limiting cases of all dark GRBs occurring at
$z=0$ or the upper redshift limits determined by Perley et al. (2009), Greiner
et al. (2011), and Kr$\ddot{\rm u}$hler et al. (2011). In the left panel of
this figure, we assume the $\Lambda$CDM cosmology, and compare the observed
GRB cumulative redshift distribution with three types of redshift evolution,
characterized through the function $f(z)$. If this function is constant
(dotted red line), the expectation from the SFR alone (i.e., the non-evolution
case) is incompatible with the observations. If we parameterize the possible
evolution effect as $f(z)\propto(1+z)^{\delta}$, we find that the $\chi^{2}$
statistic is minimized for $\delta=0.8$, which is consistent with that of
Robertson & Ellis (2012) and Wang (2013). The weak redshift evolution
($\delta=0.8$) can reproduce the observed cumulative GRB rate density best
(dashed green line). At the $2\sigma$ confidence level, the value of $\delta$
lies in the range $0.07<\delta<1.53$. In the limiting case where all the dark
GRBs are local, the power-law index is constrained to be $-0.56<\delta<0.98$
at the $2\sigma$ confidence level. The peak probability occurs for
$\delta=0.21$. Instead, if all the dark GRBs are at their maximum possible
redshift, the power-law index moves to $0.19<\delta<1.67$ ($2\sigma$) with a
peak near $\delta\approx 0.93$. Clearly, the additional uncertainty arising
from the inclusion of dark GRBs is an important consideration. If dark GRBs
occur at their maximum allowed redshifts, the distribution is more heavily
weighted towards higher values of $z$ and would therefore indicate a stronger
redshift dependence of the relationship between the GRB rate and the SFR. We
will discuss the third type of evolution shortly.
Figure 3: Left: Cumulative distribution of 118 _Swift_ GRBs with $z<4$ and
$L_{\rm iso}>1.8\times 10^{51}$ erg $s^{-1}$, assuming the standard
($\Lambda$CDM) model. The black steps and the gray area indicate the
cumulative distribution of GRBs with firm reshifts and the uncertainty owing
to dark GRBs. Three fitting results using equation (19) are also shown: the
red dotted line corresponds to the SFR on its own (i.e., $f(z)$ is constant),
the pink dashed line corresponds to $f(z)\propto(1+z)^{\delta}$, with
$\delta=0.80$, and the blue solid line corresponds to
$f(z)\propto\Theta(\epsilon,z)$, with $\epsilon=0.52$. Right: Same as the left
panel, but for 111 _Swift_ GRBs and with $\delta=1.03$ and $\epsilon=0.44$ in
the $R_{h}=ct$ Universe.
The collapsar model predicts that long bursts should occur preferentially in
metal poor environments. From a theoretical standpoint, this is not surprising
since lower metallicity leads to weaker stellar winds and hence less angular
momentum loss, resulting in the retention of rapidly rotating cores in stars
at the time of their explosion, as implied by simulations of the collapsar
model for GRBs (e.g., Woosley 1993b; MacFayden & Woosley 1999; Yoon & Langer
2005). It has therefore been suggested that the observationally required
evolution may be due mainly to the cosmic evolution in metallicity.
According to Langer & Norman (2006), the fractional mass density belonging to
metallicity below $Z=\epsilon Z_{\odot}$ (where $Z_{\odot}$ is the solar metal
abundance, and $\epsilon$ is determined by the metallicity threshold for the
production of GRBs) at a given redshift _z_ can be calculated using
$\Theta(\epsilon,z)=\hat{\Gamma}(\kappa+2,\epsilon^{\beta}10^{0.15\beta
z})/\Gamma(\kappa+2)$, where $\kappa=-1.16$ is the power-law index in the
Schechter distribution function of galaxy stellar masses (Panter et al. 2004),
$\beta=2$ is the slope of the linear bisector fit to the galaxy stellar mass-
metallicity relation (Savaglio 2006), and $\hat{\Gamma}(a,x)$ and $\Gamma(x)$
are the incomplete and complete Gamma functions, respectively. To test this
interpretation of the anomalous evolution, we parameterize the evolution
function as $f(z)\propto\Theta(\epsilon,z)$, and show the result of an
evolving metallicity as a blue line in the left panel of Figure 3. This
theoretical curve agrees very well with the observations. The best fit to the
observations yields $\epsilon=0.52$. At the $2\sigma$ confidence level, the
value of $\epsilon$ lies in the range $0.19<\epsilon<0.85$. A comparison
between this curve and that obtained with $f(z)=(1+z)^{0.8}$ shows that the
differences between these two fits is not very significant. Therefore, we
confirm that the anomalous evolution in $\Lambda$CDM may be due to an evolving
metallicity. However, in contrast to previous studies that suggest a
metallicity cut of $Z_{\rm th}\la 0.3Z_{\odot}$ (Woosley & Heger 2006; Langer
& Norman 2006; Salvaterra & Chincarini 2007; Li 2008; Campisi et al. 2010;
Salvaterra et al. 2012), we find that only the higher metallicity cut $Z_{\rm
th}=0.52Z_{\odot}$ is consistent with the data, in agreement with the
conclusions of Hao & Yuan (2013). It is worth mentioning that the higher
metallicity cut is also more consistent with recent studies of the long GRB
host galaxies (Graham et al. 2009; Levesque et al. 2010a,b; Michalowski et al.
2012).
The right panel of Figure 3 shows the cumulative redshift distribution of 111
_Swift_ GRBs with $L_{\rm lim}=1.8\times 10^{51}$ erg $\rm s^{-1}$ and $z<4$
in the $R_{\rm h}=ct$ Universe. The result of our fitting from the SFR alone
(i.e., with a constant $f[z]$) is shown as a dotted red line, which again is
incompatible with the observations. An additional evolutionary effect,
parametrized as $f(z)\propto(1+z)^{1.03}$ is required (dashed green line). At
the $2\sigma$ confidence level, the value of $\delta$ lies in the range
$0.32<\delta<1.71$. If the dark GRB sample with redshift limits is assumed to
be local ($z\approx 0$), the $2\sigma$ interval is $-0.38<\delta<1.12$ with a
peak near $\delta=0.37$. Instead, if all dark GRBs are at their maximum
possible redshift, the power-law index moves to $0.43<\delta<1.87$ ($2\sigma$)
with a peak near $\delta\approx 1.15$. Clearly, if dark GRBs occur at their
maximum allowed redshifts, the distribution is more heavily weighted toward
higher redshifts and the extra redshift evolution effect still exists in the
$R_{\rm h}=ct$ Universe. If we instead designate the evolutionary effect as
$f(z)\propto\Theta(\epsilon,z)$, the evolving metallicity agrees very well
with the observations (blue line). The best fit to the observations yields
$\epsilon=0.44\pm 0.28(2\sigma)$. Clearly, the evolutionary effect in both the
$\Lambda$CDM and the $R_{\rm h}=ct$ cosmologies can be accounted for with a
metallicity cutoff at $Z_{\rm th}$ ($0.52Z_{\odot}$ for the former and
$0.44Z_{\odot}$ for the latter).
In the next section, we will consider the implications of these findings for
the star-formation history, assuming that GRBs trace both star formation and a
possible evolutionary effect. We will adopt the best fitting values
$\delta=0.80$ or $\epsilon=0.52$ for a reasonable description of the
evolutionary effect in $\Lambda$CDM, and $\delta=1.03$ or $\epsilon=0.44$ in
the $R_{\rm h}=ct$ Universe.
### 4.2 Constraints on the high-_z_ star formation history in $\Lambda$CDM
and the $R_{\rm h}=ct$ Universe
The SFR is well measured at low-_z_ now. For high-_z_ ($z\ga 4$), a decrease
to the SFR was seemingly implied by the work of Hopkins & Beacom (2006), which
was confirmed by observations of LBGs and GRBs. Nonetheless, given the poor
coverage of these remote regions, the SFR trends towards high-_z_ are still
rather ambiguous. For this reason, previous studies have included all
possibilities: one in which the star-formation history continues to plateau,
or in which it drops off, or even increases with increasing redshift (see,
e.g., Daigne et al. 2006). In our analysis, we will introduce a free parameter
$\alpha$ to parameterize the high-_z_ history as a power law at redshifts
$z\geq 3.8$:
$R_{\rm
SF}(z)=\left\\{\begin{array}[]{ll}0.20\left(\frac{1+z}{4.8}\right)^{\alpha},~{}~{}~{}~{}~{}{\rm
for~{}~{}~{}\Lambda CDM}\\\
0.15\left(\frac{1+z}{4.8}\right)^{\alpha},~{}~{}~{}~{}~{}{\rm
for~{}~{}~{}R_{\rm h}=ct},\\\ \end{array}\right.$ (20)
and we will attempt to constrain this index $\alpha$ using the GRB
observations. The normalization constant in this expression is set by the
requirement that $R_{\rm SF}$ be continuous across $z=3.8$.
We optimize the values of each model’s free parameters, including the index
$\alpha$ of high-_z_ SFR, the GRB formation efficiency $k_{\rm GRB}$, and the
GRB LF, by minimizing the $\chi^{2}$ statistic jointly fitting the observed
redshift distribution and luminosity distribution of bursts in our sample with
firm measurements of their redshift. The observed number of GRBs in each
redshift bin $z\in(z_{1},z_{2})$ is given by equation (17), while, the
observed number of events in each luminosity bin $L_{\rm iso}\in(L_{\rm
1},L_{\rm 2})$ is given by
$\begin{split}N_{(L_{1},L_{2})}=\frac{\Delta\Omega\;T}{4\pi}\int_{L_{1}}^{L_{2}}\Phi(L)\;dL\int_{0}^{z_{\rm
max}(L)}\frac{\dot{\rho}_{\rm GRB}(z)}{1+z}\frac{dV_{\rm
com}(z)}{dz}\,dz\;,\end{split}$ (21)
where $T\sim 8.6$ yr is the observational period, and $z_{\rm max}(L_{\rm
iso})$ is the maximum redshift out to which a GRB of luminosity $L_{\rm iso}$
can be detected; this is obtained by solving the equation $L_{\rm
lim}(z)=L_{\rm iso}$ for each assumed cosmology.
Table 2: Best-fitting Results in Different Cosmological Models.
Model | $\alpha$ | $k_{\rm GRB}$ | $L_{\star}$ | $a_{L}$ | $\chi^{2}$ | AIC
---|---|---|---|---|---|---
| | ($10^{-9}$ $M_{\odot}^{-1}$) | ($10^{49}$ erg $\rm s^{-1}$) | | |
| | $\Lambda$CDM | | | |
No evol | $-2.48_{-1.46}^{+1.45}$ | $6.21_{-0.88}^{+0.36}$ | $1.03_{-0.39}^{+0.39}$ | $1.41_{-0.04}^{+0.03}$ | 66.5 | 74.5
Density evol ($\delta=0.80$) | $-3.06_{-2.01}^{+2.01}$ | $4.39_{-1.12}^{+0.67}$ | $0.46_{-0.48}^{+0.48}$ | $1.51_{-0.07}^{+0.08}$ | 55.4 | 63.4
Metallicity evol ($\epsilon=0.52$) | $-2.41_{-2.09}^{+1.87}$ | $13.3_{-2.6}^{+3.1}$ | $1.19_{-0.29}^{+0.29}$ | $1.51_{-0.05}^{+0.09}$ | 56.0 | 64.0
| | $R_{\rm h}=ct$ | | | |
No evol | $-3.27_{-1.39}^{+1.44}$ | $7.22_{-1.05}^{+0.33}$ | $1.11_{-0.68}^{+0.68}$ | $1.40_{-0.04}^{+0.03}$ | 67.4 | 75.4
Density evol ($\delta=1.03$) | $-4.47_{-2.34}^{+2.30}$ | $3.77_{-1.01}^{+0.49}$ | $1.06_{-0.66}^{+0.66}$ | $1.54_{-0.05}^{+0.11}$ | 58.6 | 66.6
Metallicity evol ($\epsilon=0.44$) | $-3.60_{-2.45}^{+2.45}$ | $19.5_{-4.4}^{+4.2}$ | $1.11_{-0.26}^{+0.34}$ | $1.50_{-0.02}^{+0.14}$ | 54.3 | 62.3
Notes.The total number of data points in the fit is 42, including 33 points
for the redshift distribution and 9 points for the luminosity distribution.
We report the best-fit parameters together with their $1\sigma$ confidence
level for different models in Table 2. In the last two columns, we give the
total $\chi^{2}$ value (i.e., the sum of the $\chi^{2}$ values obtained from
the fit of the redshift and luminosity distributions) and the Akaike
information criterion (AIC) score, respectively. For each fitted model, the
AIC is given by ${\rm AIC}=\chi^{2}+2k$, where $k$ is the number of free
parameters. If there are three models for the data, $\mathcal{M}_{1}$,
$\mathcal{M}_{2}$, and $\mathcal{M}_{3}$, and they have been separately
fitted, the one with the least resulting AIC is the one favored by this
criterion. A more quantitative ranking of models can be computed as follows.
If ${\rm AIC}_{\alpha}$ comes from model $\mathcal{M}_{\alpha}$, the
unnormalized confidence in $\mathcal{M}_{\alpha}$ is given by the “Akaike
weight” $\exp(-{\rm AIC}_{\alpha}/2)$. Informally, in a three-way comparison,
the relative probability that $\mathcal{M}_{\alpha}$ is statistically
preferred is
${\cal L}(\mathcal{M}_{\alpha})=\frac{\exp(-{\rm AIC}_{\alpha}/2)}{\exp(-{\rm
AIC}_{1}/2)+\exp(-{\rm AIC}_{2}/2)+\exp(-{\rm AIC}_{3}/2)}\;.$ (22)
Figure 4: Distributions in $z$ and $L$ of 244 _Swift_ GRBs with firm redshift
measurements in the $\Lambda$CDM cosmology (the solid points and steps, with
the number of GRBs in each bin indicated by a dark point with Poisson error
bars). The dotted lines (red) show the expected distribution for the case of
no evolution. The results of density and metallicity evolution models are
shown with green dashed lines and blue solid lines, respectively. Figure 5:
Same as Figure 4, but for the $R_{\rm h}=ct$ Universe.
#### 4.2.1 No-Evolution Model
This model is for the GRB rate that purely follows the SFR. Figure 4 shows the
$z$ and $L$ distributions of 244 _Swift_ GRBS in the $\Lambda$CDM cosmology.
If the function $f(z)$ is constant (dotted red line), the expectation from the
SFR alone (i.e., the non-evolution case) does not provide a good
representation of the observed $z$ and $L$ distributions of our sample. In
particular, the rate of GRBs at high-_z_ is under-predicted and the fit of the
$L$ distribution is not as good as those of the density evolution model or
metallicity threshold model, more fully described below. This is confirmed by
a more detailed statistical analysis. Indeed, on the basis of the AIC model
selection criterion, we can discard this model as having a likelihood of only
$\sim 0.2\;\%$ of being correct compared to the other two $\Lambda$CDM models.
Figure 5 shows the redshift and luminosity distributions of 244 _Swift_ GRBS
in the $R_{\rm h}=ct$ Universe. The results of our fitting from the SFR alone
(i.e., with a constant $f[z]$) are indicated with dotted red lines, which
again are incompatible with the observations. On the basis of the AIC model
selection criterion, we can discard the no-evolution model as having a
likelihood of only $\sim 0.1\;\%$ of being correct compared to the other two
$R_{\rm h}=ct$ models.
#### 4.2.2 Density Evolution Model
This model assumes that the GRB rate follows the SFR in conjunction with an
additional evolution characterized by $(1+z)^{\delta}$. In $\Lambda$CDM, we
find that $\delta=0.80$ reproduces the observed $z$ and $L$ distributions
(green dashed lines in Figure 4) quite well. In this model, the slope of the
high-_z_ SFR is characterized by an index $\alpha=-3.06_{-2.01}^{+2.01}$. The
range of high-_z_ SFH’s with $\alpha\in(-5.07,-1.05)$ is marked with a shaded
band in figure 1, in comparison with the available data. It is interesting to
note that Wang (2013) derived a similar slope ($\alpha=-3.0$) for the high-_z_
SFR. Wu et al. (2012) showed that the GRB formation rate in $\Lambda$CDM
decreases with a power index of $\sim-3.8$ for $z\ga 4$, in good agreement
with the SFR we derive here at the $1\sigma$ confidence level. Using the AIC
model selection criterion, we find that among the $\Lambda$CDM models, this
one is statistically preferred with a relative probability $\sim 57.3\;\%$.
In the $R_{\rm h}=ct$ Universe, we simultaneously fit the observed $z$ and $L$
distributions of _Swift_ GRBs using $\dot{\rho}_{\rm GRB}(z)=k_{\rm GRB}R_{\rm
SFR}(z)(1+z)^{\delta}$, together with the piecewise smooth $R_{\rm SFR}(z)$
concatenated from equations (14) and (20); the best fit corresponds to
$\delta=1.03$. We find that a high-_z_ SFR with slope
$\alpha=-4.47_{-2.34}^{+2.30}$ is required to reproduce both the observed $z$
and $L$ distributions (green dashed lines in figure 5). Again using the AIC
model selection criterion, we find that this model is somewhat disfavored
statistically compared to the other two $R_{\rm h}=ct$ models, with a relative
probability of $\sim 10.4\;\%$.
#### 4.2.3 Metallicity Evolution Model
This model assumes that the GRB rate is proportional to the star formation
history with an additional evolution in cosmic metallicity (i.e.,
$f[z]\propto\Theta[\epsilon,z]$). For $\Lambda$CDM, we find that a high-$z$
SFR with index $\alpha=-2.41_{-2.09}^{+1.87}$ and a metallicity evolution
parameter $\epsilon=0.52$ fits the data best (blue solid lines in Figure 4).
The $\chi_{\rm dof}^{2}$ for this fit is $56.0/42=1.33$. In general, fitting
the observations with this model produces better consistency than the non-
evolution model. According to the AIC, the metallicity evolution model in
$\Lambda$CDM is slightly disfavored compared to the more general density
evolution model, but the differences are statistically insignificant ($\sim
42.5\;\%$ for the former versus $\sim 57.3\;\%$ for the latter). We conclude
that in the context of $\Lambda$CDM, the required density evolution may be due
to an evolving metallicity.
In the context of the $R_{\rm h}=ct$ Universe, the best fit is produced with a
high-$z$ SFR with index $\alpha=-3.60_{-2.45}^{+2.45}$ and a metallicity
evolution with $\epsilon=0.44$. The $\chi_{\rm dof}^{2}$ for this fit is
$54.3/42=1.29$. This model is represented by the blue solid lines in figure 5.
The AIC shows that the likelihood of this model being correct is $\sim
89.5\;\%$ compared to the other two $R_{\rm h}=ct$ models examined above.
Unlike the situation with $\Lambda$CDM, here there is a clear indication that
abundance evolution is required to account for the SFR/GRB data.
## 5 Discussion and Conclusions
We have used the cumulative redshift distribution of the latest sample of
_Swift_ GRBs above a fixed luminosity limit, together with the star formation
history over the interval $z\in(0,4)$, to compare the predictions of
$\Lambda$CDM and the $R{\rm h}=ct$ Universe. With $\Lambda$CDM as the
background spacetime, earlier work had already demonstrated that in this
cosmology the SFR underproduces the GRB rate density at high redshifts. It has
been suggested that this effect can be understood if a modest evolution,
parameterized as $f(z)=(1+z)^{0.80}$, is included; we have confirmed in both
$\Lambda$CDM and $R_{\rm h}=ct$ that this factor may be readily explained as
an evolution in metallicity. However, we have also found that a comparison
with the observational data shows that a relatively high metallicity cut
($Z=0.52Z_{\odot}$ in $\Lambda$CDM and $Z=0.44Z_{\odot}$ in $R_{\rm h}=ct$) is
required, in contrast to previous work that suggested LGRBs occur
preferentially in low metallicity environments, i.e., $Z\sim
0.1-0.3Z_{\odot}$.
For both cosmologies, we have shown that if these results are correct, then by
assuming that such trends continue beyond $z\simeq 4$, the adoption of a
simple power-law approximation for the high-_z_ ($\ga 3.8$) SFR , i.e.,
$R_{\rm SF}\propto[(1+z)/4.8]^{\alpha}$, we may also constrain the slope
$\alpha$ using the GRB data. We have found for $\Lambda$CDM that the SFR at
$z\ga 3.8$ shows a decay with slope $\alpha=-2.41_{-2.09}^{+1.87}$. And using
a simple relationship between the GRB rate density and the SFR, including an
evolution in metallicity, we have demonstrated that the $z$ and $L$
distributions of 244 _Swift_ GRBs can be well fitted by our updated SFH, using
a threshold in the metallicity for GRB production.
The best fit for the redshift distribution of the _Swift_ GRBs in the $R_{\rm
h}=ct$ Universe requires a slightly different rate than that in $\Lambda$CDM,
though still with an additional evolutionary effect, which could be a high
metallicity cut of $Z=0.44Z_{\odot}$. Assuming that the GRB rate is related to
the SFR with this evolving metallicity, we have found that in the $R_{\rm
h}=ct$ Universe the slope of the high-_z_ SFR would be
$\alpha=-3.60_{-2.45}^{+2.45}$.
The principal goal of this work has been to directly compare the predictions
of $\Lambda$CDM and $R_{\rm h}=ct$ and their ability to account for the
GRB/SFR observations. Aside from the issue of whether or not the GRB-redshift
distribution is consistent with the SFR in either model, we have also examined
which of these two cosmologies fits the data better, and is therefore
statistically preferred by the Aikake Information Criterion in a one-on-one
comparison.
To keep the complexity of this problem manageable, we have chosen to find the
best fits to the data by optimizing four free parameters ($\alpha$, $k_{\rm
GRB}$, $L_{\star}$ and $a_{L}$), though the models themselves were held fixed
by the concordance values of $H_{0}$, $\Omega_{m}$, $\Omega_{\Lambda}$ and the
dark-energy equation-of-state in the case of $\Lambda$CDM, and the same value
of $H_{0}$ for $R_{\rm h}=ct$. The two models produce very similar profiles in
the distance-redshift relationship (Melia 2012a; Wei et al. 2013), so it is
not very surprising to see that both can account quite well for the observed
SFR-GRB rate correlation.
However, the AIC does not favor these models equally. From Table 2, we find
that a direct comparison between the best $\Lambda$CDM fit (entry 2 in this
table) and the best $R_{\rm h}=ct$ fit (entry 6) favors the latter with a
relative probability $\sim 63.4\;\%$ versus $\sim 36.6\;\%$ for the standard
model. If we further assume that the required evolutionary effect is indeed
due to changes in metallicity, so that we now compare entries 3 and 6 in Table
2, then the AIC favors $R_{\rm h}=ct$ with a relative probability $\sim
70.0\;\%$ versus $\sim 30.0\;\%$ for $\Lambda$CDM. However, if the required
evolutinary effect is simply due to density and not changes in metallicity
(entries 2 and 5), the AIC favors $\Lambda$CDM with a relative probability of
$\sim 83.2\;\%$ versus $\sim 16.8\;\%$ for $R_{\rm h}=ct$.
The statistical significance of these likelihoods has been investigated
theoretically, e.g., by Yanagihara & Ohmoto (2005). Its variability has also
been studied empirically; for example, by repeatedly comparing $\Lambda$CDM to
other cosmological models on the basis of data sets generated by a bootstrap
method (Tan & Biswas 2012). It is known that the AIC is increasingly accurate
when the number of data points is large, but it is felt that in all cases, the
magnitude of the difference $\Delta=\allowbreak{\rm AIC}_{2}-\nobreak{\rm
AIC}_{1}$ should provide a numerical assessment of the evidence that model 1
is to be preferred over model 2. A rule of thumb that has been used in the
literature is that if $\Delta\la 3$, it is mildly strong; and if $\Delta\ga
5$, it is quite strong.
Therefore, our conclusion from the comparative study we have reported here is
that—based on the currently available GRB/SFR observations—the $R_{\rm h}=ct$
Universe is mildly favored over $\Lambda$CDM in a one-on-one comparison if the
required evolution is due to changes in metallicity (for which $\Delta\approx
1.7$). However, $\Lambda$CDM is mildly favored over $R_{\rm h}=ct$ (with
$\Delta\approx 3.2$) if instead the evolution is with density.
The prevailing view at the moment seems to be that changes in metallicity are
responsible for the required evolution so, in this context, the GRB/SFR data
tend to be more consistent with the predictions of $R_{\rm h}=ct$ than those
of the concordance model. Note that the likelihood estimates we have made here
were based on the use of priors for $\Lambda$CDM. Were we to optimize $H_{0}$
along with the other four parameters (for both models), and $\Omega_{m}$,
$\Omega_{\Lambda}$ and the dark-energy equation of state for $\Lambda$CDM, we
could certainly lower their $\chi^{2}$ for the best fits, but the AIC strongly
penalizes models with many free parameters. The $\chi^{2}$ values listed for
$\Lambda$CDM in Table 2 would need to decrease by at least 6 in order to
compensate for the increase due to the factor $2k$ in the expression ${\rm
AIC}=\chi^{2}+2k$. This seems unlikely since the fits using the concordance
model are already rather good.
Refinements in future measurements of the GRB rate and SFR may show that the
currently believed explanation for their differences (i.e., an evolution in
metallicity) is incorrect. In that case, a reassessment of these comparisons
may produce different results. As of now, however, it appears that the SFR
underproduces the observed GRB rate unless some additional evolution were
present to broaden their disparity with increasing redshift. We have found
that such an evolution is consistent with a relatively high metallicity cutoff
for the LGRBs.
## Acknowledgments
We thank X. H. Cui, X. Kang, E. W. Liang, and F. Y. Wang for helpful
discussions. This work is partially supported by the National Basic Research
Program (“973” Program) of China (Grants 2014CB845800 and 2013CB834900), the
National Natural Science Foundation of China (grants Nos. 10921063, 11273063,
11322328, and 11373068), the One-Hundred-Talents Program and the Youth
Innovation Promotion Association of the Chinese Academy of Sciences, and the
Natural Science Foundation of Jiangsu Province. F.M. is grateful to Amherst
College for its support through a John Woodruff Simpson Lectureship, and to
Purple Mountain Observatory in Nanjing, China, for its hospitality while this
work was being carried out. This work was partially supported by grant
2012T1J0011 from The Chinese Academy of Sciences Visiting Professorships for
Senior International Scientists, and grant GDJ20120491013 from the Chinese
State Administration of Foreign Experts Affairs. We also thank the anonymous
referee for providing many comments and suggestions that have led to a
significant improvement in the presentation of the material in this paper.
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* Yoon & Langer (2005) Yoon S.-C., Langer N., 2005, A&A, 443, 643
* Zhang (2007) Zhang B., 2007, ChJAA, 7, 1
* Zhang & Mészáros (2004) Zhang B., Mészáros P., 2004, IJMPA, 19, 2385
|
arxiv-papers
| 2013-06-19T03:14:47 |
2024-09-04T02:49:46.650861
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jun-Jie Wei, Xue-Feng Wu, Fulvio Melia, Da-Ming Wei, and Long-Long\n Feng",
"submitter": "Jun-Jie Wei",
"url": "https://arxiv.org/abs/1306.4415"
}
|
1306.4435
|
# Profile for a simultaneously blowing up solution to a complex valued
semilinear heat equation
Nejla Nouaili,
[email protected]
CEREMADE, Université Paris Dauphine, Paris Sciences et Lettres.
Hatem Zaag111This author is supported by the ERC Advanced Grant no. 291214,
BLOWDISOL and by ANR Project ANAÉ ref. ANR-13-BS01-0010-03.,
[email protected]
Université Paris 13, Sorbonne Paris Cité,
LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France
###### Abstract
We construct a solution to a complex nonlinear heat equation which blows up in
finite time $T$ only at one blow-up point. We also give a sharp description of
its blow-up profile. The proof relies on the reduction of the problem to a
finite dimensional one and the use of index theory to conclude. We note that
the real and imaginary parts of the constructed solution blow up
simultaneously, with the imaginary part dominated by the real.
Mathematical Subject classification: 35K57, 35K40, 35B44.
Keywords: Simultaneous blow-up, Complex heat equation.
## 1 Introduction
This paper is concerned with blow-up solutions of the complex heat equation
$\partial_{t}u=\Delta u+u^{2},$ (1)
where $u(t):x\in\mathbb{R}^{N}\to\mathbb{C}$ and $\Delta$ denotes the
Laplacian.
If we write $u(x,t)=v(x,t)+i\tilde{v}(x,t)$, where $v$ and
$\tilde{v}\in\mathbb{R}$, we obtain the following reaction-diffusion system:
$\begin{array}[]{lll}\partial_{t}v&=&\Delta v+v^{2}-\tilde{v}^{2},\\\
\partial_{t}\tilde{v}&=&\Delta\tilde{v}+2v\tilde{v},\end{array}$ (2)
where $(x,t)\in\mathbb{R}^{N}\times(0,T)\mbox{, }v(0,x)=v_{0}(x)$ and
$\tilde{v}(0,x)=\tilde{v}_{0}(x)$.
Equation (1) has a strong relation with the viscous Constantin-Lax-Majda
equation, which is a one dimensional model for the vorticity equation. For
more details see Okamoto, Sakajo and Wunsch [OSW08], Sakajo [Sak03a] and
[Sak03b] and Guo, Ninomiya, Shimojo and Yanagida in [GNSY13].
The Cauchy problem for system (2) can be solved in
$(L^{\infty}(\mathbb{R}^{N}))^{2}$, locally in time. We say that
$u(t)=v(t)+i\tilde{v}(t)$ blows up in finite time $T<\infty$, if $u(t)$ exists
for all $t\in[0,T)$ and $\lim_{t\to
T}\|v(t)\|_{L^{\infty}}+\|\tilde{v}(t)\|_{L^{\infty}}=+\infty.$ In that case,
$T$ is called the blow-up time of the solution. A point
$x_{0}\in\mathbb{R}^{N}$ is said to be a blow-up point if there is a sequence
$\\{(x_{j},t_{j})\\}$, such that $x_{j}\to x_{0}$, $t_{j}\to T$ and
$|v(x_{j},t_{j})|+|\tilde{v}(x_{j},t_{j})|\to\infty$ as $j\to\infty$. The set
of all blow-up points is called the blow-up set.
When $u$ is real (i.e., $\tilde{v}\equiv 0$), then system (2) is reduced to
the scalar equation
$\partial_{t}u=\Delta u+u^{p}\mbox{, where }p=2.$ (3)
The blow-up question for equation (3), with $p>1$, has been studied
intensively by many authors and no list can be exhaustive.
When it comes to deriving the blow-up profile, the situation is completely
understood in one space dimension (however, less is understood in higher
dimensions, see Velázquez [Vel92, Vel93a, Vel93b] and Zaag [Zaa02a, Zaa02b,
Zaa02c] for partial results). In one space dimension, given $a$ a blow-up
point, this is the situation:
* •
either
$\sup_{|x-a|\leq
K\sqrt{(T-t)\log(T-t)}}\left|(T-t)u(x,t)-f\left(\frac{x-a}{\sqrt{(T-t)\log(T-t)}}\right)\right|\to
0,$ (4)
* •
or for some $m\in\mathbb{N}$, $m\geq 2$, and $C_{m}>0$
$\sup_{|x-a|<K(T-t)^{1/2m}}\left|(T-t)u(x,t)-f_{m}\left(\frac{C_{m}(x-a)}{(T-t)^{1/2m}}\right)\right|\to
0,$ (5)
as $t\to T$, for any $K>0$, where
$f(z)=\frac{8}{8+|z|^{2}}\mbox{ and }f_{m}(z)=\frac{1}{1+|z|^{2m}}\mbox{, for
all }z\in\mathbb{R}^{N}.$ (6)
From Bricmont and Kupiainen [BK94] and Herrero and Velázquez [HV93], we have
examples of initial data leading to each of the aboved-mentioned scenarios.
Note that (4) corresponds to the fundamental mode of the harmonic oscillator
in the leading order, whereas (5) corresponds to higher modes. Moreover,
Herrero and Velázquez proved the genericity of the behavior (4) in one space
dimension in [HV92] and [HV94], and only announced the result in the higher
dimensional case (the result has never been published). Note also that the
stability of such a profile with respect to initial data has been proved by
Fermanian Kammerer, Merle and Zaag in [FKZ00] and [FKMZ00]. For more results
on equation (3), see [Bal77], [GK85], [GK87], [GK89], [HV93], [HV94], [MM04],
[MM09], [MZ98], [MZ00], [Miz07] and [QS07].
As we inferred above, equation (1) appears as a complex generalization of the
real valued equation (3). Note that there is another complex generalization of
(3). Indeed, Filippas and Merle consider in [FM95] the following equation:
$\partial_{t}u=\Delta u+|u|^{p-1}u\mbox{ with }u\in\mathbb{C}\mbox{ and }p>1,$
(7)
and generalize to this equation the results first proved in the real case by
Giga and Kohn [GK85, GK87, GK89]. Our equation (1) appears then as a “twin“ of
equation (7). However there is a fundamental difference between the two.
Indeed, equation (7) has a variational structure, which allows to use various
energy techniques, unlike equation (1), where such techniques certainly fail.
Therefore, considering equation (1) appears as a highly challenging task.
Considering equation (1) with non-real solutions, we have the following blow-
up results from [GNSY13]:
(A) A non-simultaneous blow-up criterion, see Theorem 1.5 in [GNSY13]:
Assume that
$v_{0},\;\tilde{v}_{0}\in C^{1}(\mathbb{R}^{N}),\;0\leq v_{0}\leq
M,\;v_{0}\not\equiv M,\;0<\tilde{v}_{0}\leq L,$ (8)
$\lim_{|x|\to\infty}v_{0}(x)=M,\;\;\lim_{|x|\to\infty}\tilde{v}_{0}(x)=0,$ (9)
for some constants $L>0$ and $M>0$. Then, the solution of (2) blows up at time
$t=T(M)$ with $\tilde{v}\not\equiv 0$. Moreover, the component $v$ blows up
only at space infinity and $\tilde{v}$ remains bounded.
(B) A Fourier-based blow-up criterion, see Theorem 1.2 in [GNSY13]:
If the Fourier transform of initial data of (1) is real and positive, then the
solution blows up.
(C) A simultaneous blow-up criterion, see Theorem 1.3 in [GNSY13]:
If $v_{0}$ is even, $\tilde{v}_{0}$ is odd with $\tilde{v}_{0}(x)>0$ for
$x>0$, then the fact that the blow-up set is compact implies that $v$ and
$\tilde{v}$ blow up simultaneously.
Following the description in (4) and (5) together with the work of [GNSY13],
we see that the blow-up profile derivation remains open, in the non-real case.
Indeed, either we have the description (4) or (5), with a zero imaginary part,
or we have blow-up solutions from [GNSY13], with a non-trivial imaginary part,
and no profile description.
In this paper, we give the first example of a complex-valued blow-up solution
of equation (1), with a non-trivial imaginary part, and a full description of
its blow-up profile, obeying behavior (4) (note that our method extends with
no difficulty to the construction of an analogous solution obeying the
behavior (5); however, the proof should be even more technical). Let us note
that the blow-up behavior we give here is not predicted by [GNSY13] (see
details in the remarks following our result). More precisely, this is our
result:
###### Theorem 1
(Existence of a blow-up solution for equation (1) with the description of its
profile) There exists $T>0$ such that equation (1) has a solution
$u(x,t)=v(x,t)+i\tilde{v}(x,t)$ in $\mathbb{R}^{N}\times[0,T)$ such that:
(i) the solution $u$ blows up in finite time $T$ only at the origin;
(ii) there holds that
$\left\|(T-t)u(.,t)-f\left(\frac{.}{\sqrt{(T-t)|\log(T-t)|}}\right)\right\|_{L^{\infty}}\leq\frac{C}{\sqrt{|\log(T-t)|}},$
(10)
where $f$ is defined by (6).
(iii) For all $R>0$,
$\sup_{|x|\leq
R\sqrt{T-t}}\left|(T-t)\tilde{v}(x,t)-\frac{\sum_{i=1}^{N}C_{i}}{|\log(T-t)|^{2}}\left(\frac{x_{i}^{2}}{T-t}-2\right)\right|\leq\frac{C}{|\log(T-t)|^{\alpha}},$
(11)
where $(C_{1},C_{2},..,C_{N})\neq(0,0,..,0)$ and $2<\alpha\leq 2+\eta$ for
some small $\eta>0$.
(iv) For all $x\neq 0$, $u(x,t)\to u^{*}(x)$ uniformly on compacts sets of
$\mathbb{R}^{N}\backslash\\{0\\}$, and
$u^{*}(x)\sim\frac{16|\log|x||}{|x|^{2}}\mbox{ as }x\to 0.$ (12)
Remarks:
1) Note that the real and imaginary parts of $u$ blow up simultaneously at
$x=0$. However the real part dominates the imaginary part in the sense that
$v(0,t)\sim\frac{1}{T-t}>>\frac{2\left|\sum_{i=1}^{N}C_{i}\right|}{(T-t)|\log(T-t)|^{2}}\sim|\tilde{v}(0,t)|\mbox{
as }t\to\infty.$
2) As announced right before the statement of our theorem, the solution we
construct is new and doesn’t obey the criteria given in [GNSY13]: this is
clear from (24) below. The proof relies on the reduction of the problem to a
$2(N+1)-$dimensional problem (a $4-$dimensional one if $N=1$; see Lemma 3.4
below). In the real case treated by Merle and Zaag in [MZ97], the problem was
of dimension $N+1$. Since that number is equal to the dimension of the blow-up
parameters ($1$ for the blow-up time and $N$ for the blow-up point), the
authors of [MZ97] were able to show the stability of the behavior (10) with
respect to initial data, of course in the real case. Here, in the complex
case, since the dimension of our problem $2(N+1)$ exceeds that of the blow-up
parameters $(N+1)$, we suspect our solution to be unstable with respect to
perturbations in initial data.
Our proof uses some ideas developed by Bricmont and Kupiainen [BK94] and Merle
and Zaag [MZ97] to construct a blow-up solution for the semilinear heat
equation (3) obeying the behavior (4). In [EZ11], Ebde and Zaag use the same
ideas to show the persistence of the profile (4) under perturbations of
equation (1) in the real case by lower order terms involving $u$ and $\nabla
u$. In [MZ08], Masmoudi and Zaag adapted that method to the case of the
following complex Ginzburg-Landau equation, where no gradient structure
exists:
$\partial_{t}u=(1+i\beta)\Delta u+(1+i\delta)|u|^{p-1}u\mbox{, with $\beta$
and $\delta$ are reals},$
(note that the case $\beta=0$ and $\delta$ small was first studied by Zaag in
[Zaa98]).
More precisely, the proof relies on the understanding of the dynamics of the
selfsimilar version of (2) (see system (17) below) around the profile (4).
Moreover, we proceed in two steps:
* •
First, we reduce the question to a finite-dimensional problem: we show that it
is enough to control a $(N+1)$-dimensional variable in order to control the
solution (which is infinite dimensional) near the profile.
* •
Second, we proceed by contradiction to solve the finite-dimensional problem
and conclude using index theory.
Surprisingly enough, we would like to mention that this kind of methods has
proved to be successful in various situations including hyperbolic and
parabolic PDE, in particular with energy-critical exponents. This was the case
for the construction of multi-solitons for the semilinear wave equation in one
space dimension by Côte and Zaag [CZ13], the wave maps by Raphaël and
Rodnianski [RR12], the Schrödinger maps by Merle, Raphaël and Rodnianski
[MRR11], the critical harmonic heat flow by Schweyer [Sch12] and the two-
dimensional Keller-Segel equation by Raphaël and Schweyer [RS13].
We proceed in 3 sections to prove Theorem 1. We first give in Section 2 an
equivalent formulation of the problem in the scale of the well-known
similarity variables. Section 3 is devoted to the proof of the similary
variables formulation (this is a central part in our argument). Finally, we
conclude the proof of Theorem 1 in Section 4.
Acknowledgement. We would like to thank the referee for his valuable remarks
which undoubtedly improved the presentation of our paper.
## 2 Formulation of the problem
For simplicity, we give the proof in one dimension. The adaptation to higher
dimensions is straightforward. We would like to find initial data
$u_{0}=v_{0}+i\tilde{v}_{0}$ such that the solution $u=v+i\tilde{v}$ of
equation (2) blows up in time $T$ with
$\displaystyle\lim_{t\to
T}\left\|(T-t)u(x,t)-f\left(\frac{x}{\sqrt{(T-t)|\log(T-t)|}}\right)\right\|_{L^{\infty}}=0,$
(13)
where $f$ is defined in (6).
This is the main estimate and the other results of Theorem 1 will appear as
by-products of the proof (see Section 4 for the proof of all the estimates of
Theorem 1).
Introducing the following self-similar transformation of problem (2):
$\begin{array}[]{c}w(y,s)=(T-t)v(x,t)\mbox{,
}\tilde{w}(y,s)=(T-t)\tilde{v}(x,t),\\\ y=\frac{x-a}{\sqrt{T-t}}\mbox{,
}s=-\log(T-t),\end{array}$ (14)
we see that (13) is equivalent to finding $s_{0}>0$ and initial data at
$s_{0}$, $W_{0}(y,s_{0})=w_{0}(y,s_{0})+i\tilde{w}_{0}(y,s_{0})$, such that
$W(y,s)=w(y,s)+i\tilde{w}(y,s)$ satisfies
$\displaystyle\lim_{s\to\infty}\left\|W(y,s)-f\left(\frac{y}{\sqrt{s}}\right)\right\|_{L^{\infty}}=0.$
(15)
Introducing
$w=\varphi+q\mbox{ and }\tilde{w}=\tilde{q}\mbox{ where
}\varphi(y,s)=f\left(\frac{y}{\sqrt{s}}\right)+\frac{1}{4s},$ (16)
the problem is then reduced to constructing a function $Q=q+i\tilde{q}$ such
that
$\lim_{s\to\infty}\|Q(y,s)\|_{L^{\infty}}=0,$
and $(q,\tilde{q})$ is a solution of the following equation for all
$(y,s)\in\mathbb{R}\times[s_{0},\infty)$,
$\begin{array}[]{lll}\displaystyle\partial_{s}q&=&\displaystyle({\cal{L}}+V)q+b(y,s)+R(y,s),\\\
\displaystyle\partial_{s}\tilde{q}&=&\displaystyle({\cal{L}}+V)\tilde{q}+\tilde{b}(y,s),\end{array}$
(17)
where
$\displaystyle{\cal{L}}=\partial_{y}^{2}-\frac{1}{2}y\partial_{y}+1\mbox{,
}V(y,s)=2\left(\varphi(y,s)-1\right),$ (18) $b(y,s)=q^{2}-\tilde{q}^{2}\mbox{,
}\tilde{b}(y,s)=2q\tilde{q},$ (19)
and
$\displaystyle
R(y,s)=\partial_{y}^{2}\varphi-\frac{1}{2}y\partial_{y}\varphi-\varphi+\varphi^{2}-\partial_{s}\varphi.$
(20)
The control of $q$ and $\tilde{q}$ near $0$ obeys two facts:
* •
Localization: the fact that our profile $\varphi(y,s)$ dramatically changes
its value from $1+\frac{1}{4s}$ in the region near $0$ to $\frac{1}{4s}$ in
the region near infinity, according to a free boundary moving at the speed
$\sqrt{s}$. This will require different treatments in the regions
$|y|<2K_{0}\sqrt{s}$ and $|y|>2K_{0}\sqrt{s}$ for some $K_{0}$ to be chosen.
* •
Spectral information: the fact that the operator ${\cal L}$ is selfadjoint,
$b$ and $\tilde{b}$ are quadratic in $(q,\tilde{q})$ and that
$\|R(s)\|_{L^{\infty}}+\|V(s)\|_{L^{2}_{\rho}}\to 0\mbox{ as }s\to\infty$
from (16) and (18), which shows that the dynamics of equation (17) near $0$
are driven by the spectral properties of ${\cal L}$. This will require a
decomposition of the solution according to the spectrum of ${\cal L}$. Note
that the operator ${\cal L}$ is self-adjoint in the Hilbert space
$L^{2}_{\rho}=\\{g\in L^{2}_{loc}(\mathbb{R},\mathbb{C})\mbox{,
}\|g\|_{L^{2}_{\rho}}^{2}\equiv\int_{\mathbb{R}}|g|^{2}e^{-\frac{|y|^{2}}{4}}dy<+\infty\\}\mbox{
where }\rho(y)=\displaystyle\frac{e^{-\frac{|y|^{2}}{4}}}{(4\pi)^{1/2}}.$
The spectrum of ${\cal L}$ is explicitly given by
$spec({\cal L})=\\{1-\frac{m}{2}\mbox{, }m\in\mathbb{N}\\}.$
All the eigenvalues are simple, the eigenfunctions are dilations of Hermite’s
polynomial and given by
$h_{m}(y)=\sum_{n=0}^{[\frac{m}{2}]}\frac{m!}{n!(m-2n)!}(-1)^{n}y^{m-2n}.$
(21)
Note that ${\cal L}$ has two positive (or expanding) directions ($\lambda=1$
and $\lambda=\frac{1}{2}$), and a zero direction ($\lambda=0$). Complying with
the localization and spectral information facts, we will decompose $q$ and
$\tilde{q}$ accordingly as stated above:
* •
first, we consider a non-increasing cut-off function $\chi_{0}\in
C^{\infty}_{0}(\mathbb{R}^{+},[0,1])$ such that $supp(\chi_{0})\subset[0,2]$,
$\chi_{0}(\xi)=1$ for $\xi<1$ and $\chi_{0}(\xi)=0$ for $\xi>2$, then
introduce
$\chi(y,s)=\chi_{0}\left(\frac{|y|}{K_{0}\sqrt{s}}\right),$
where $K_{0}\geq 1$ will be chosen large enough so that various technical
estimates hold. Then, we write $q=q_{b}+q_{e}$ and
$\tilde{q}=\tilde{q}_{b}+\tilde{q}_{e}$, where the inner parts and the outer
parts are given by
$q_{b}=q\chi\mbox{, }\tilde{q}_{b}=\tilde{q}\chi\mbox{, }q_{e}=q(1-\chi)\mbox{
and }\tilde{q}_{e}=\tilde{q}(1-\chi).$
Let us remark that
$supp(q_{b}(s))\subset B(0,2K_{0}\sqrt{s})\mbox{ and
}supp(q_{e}(s))\subset\mathbb{R}\setminus B(0,K_{0}\sqrt{s}),$
and the same holds for $\tilde{q}_{b}$ and $\tilde{q}_{e}$.
* •
Second, we study $q_{b}$ and $\tilde{q}_{b}$ using the structure of ${\cal
L}$, isolating the nonnegative directions. More precisely we decompose $q_{b}$
and $\tilde{q}_{b}$ as follows
$\begin{array}[]{lll}q_{b}(y,s)&=&\sum_{0}^{2}q_{m}(s)h_{m}(y)+q_{-}(y,s),\\\
\tilde{q}_{b}(y,s)&=&\sum_{0}^{2}\tilde{q}_{m}(s)h_{m}(y)+\tilde{q}_{-}(y,s),\\\
\end{array}$ (22)
where $q_{m}$ (respectively $\tilde{q}_{m}$) is the projection of $q_{b}$
(respectively $\tilde{q}_{b}$) on $h_{m}$, $q_{-}(y,s)=P_{-}(q_{b})$
(respectively $\tilde{q}_{-}(y,s)=P_{-}(\tilde{q}_{b})$) and $P_{-}$ is the
projection on $\\{h_{i},\;i\geq 3\\}$ the negative subspace of the ${\cal L}$.
In summary, we can decompose $q$ (respectively $\tilde{q}$) in 5 components as
follows:
$\begin{array}[]{lll}q(y,s)&=&\sum_{m=0}^{2}q_{m}(s)h_{m}(y)+q_{-}(y,s)+q_{e}(y,s),\\\
\tilde{q}(y,s)&=&\sum_{m=0}^{2}\tilde{q}_{m}(s)h_{m}(y)+\tilde{q}_{-}(y,s)+\tilde{q}_{e}(y,s).\end{array}$
(23)
Here and throughout the paper, we call $q_{-}(y,s)$ (respectively
$\tilde{q}_{-}$) the negative part of $q$ (respectively $\tilde{q}$), $q_{2}$
(respectively $\tilde{q}_{2}$), the null mode of $q$ (respectively
$\tilde{q}$).
## 3 The construction method in selfsimilar variables
This section is devoted to the proof of the existence of a solution
$(q,\tilde{q})$ of system (17) satisfying
$\|q(s)\|_{L^{\infty}}+\|\tilde{q}(s)\|_{L^{\infty}}\to 0$. This is a central
argument in our proof. In Section 4, we use this solution and give the proof
of Theorem 1. Though we refer to the earlier work by Merle and Zaag [MZ97] for
purely technical details, we insist on the fact that we can completely split
from that paper as long as ideas and arguments are considered. We hope that
the explanation of the strategy we give in this section will be more reader
friendly.
We proceed in 3 subsections:
\- In the first subsection, we give all the arguments of the proof without the
details, which are left for the following subsection (readers not interested
in technical details may stop here).
\- In the second subsection, we give various estimates concerning initial
data.
\- In the third subsection, we give the dynamics of system (17) near the zero
solution, in accordance with the decomposition (23).
### 3.1 The proof without technical details
Given $s_{0}>0$, we consider initial data for equation (17) of the following
form:
$\begin{array}[]{lll}\displaystyle
q_{d_{0},d_{1}}(y,s_{0})&=&\displaystyle\frac{A}{s_{0}^{2}}(d_{0}+d_{1}y)\chi(2y,s_{0}),\\\
\displaystyle\tilde{q}_{\tilde{d}_{0},\tilde{d}_{1},\tilde{d}_{2}}(y,s_{0})&=&\displaystyle\left[\frac{\tilde{A}}{s_{0}^{\alpha}}(\tilde{d}_{0}+\tilde{d}_{1}y)+\frac{\tilde{B}}{s_{0}^{2}}h_{2}(y)\right]\chi(2y,s_{0}),\end{array}$
(24)
for some constants $A$, $\tilde{A}$ and $\tilde{B}$ will be fixed later and
the parameters $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in[-2,2]^{4}$. The
solution of equation (17) with initial data (24) will be denoted by
$(q,\tilde{q})(s_{0},A,\tilde{A},\tilde{B},(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1}),y,s)$,
or, when there is no ambiguity by
$(q,\tilde{q})(s_{0},d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1},y,s)$ or even
$(q,\tilde{q})(y,s)$. We will show that given $\tilde{B}\in\mathbb{R}$, if $A$
and $\tilde{A}$ are fixed large enough, then $s_{0}$ is fixed large enough
depending on $A$, $\tilde{A}$ and $\tilde{B}$, we can also fixe the parameters
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in[-2,2]^{4}$, so that the solution
$(q,\tilde{q})\left(s_{0},A\,\tilde{A},\tilde{B},d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1},y,s\right)$
will converge to $0$ as $s\to\infty$. Thanks to the decomposition given in
(23), in order to control $(q,\tilde{q})(s)$ near $(0,0)$, it is enough to
control it in some shrinking set defined as follows:
###### Definition 3.1
(Definition of a shrinking set for the components of $(q,\tilde{q})$) For all
$A\geq 1$, $\tilde{A}\geq 1$, $0<\eta<\frac{1}{2}$, $2<\alpha<2+\eta$ and
$s\geq e$, we define $V_{A}(s)$ (respectively $\tilde{V}_{\tilde{A}}(s)$) as
the set of all functions $r$ (respectively $\tilde{r}$) in $L^{\infty}$ such
that:
$|r_{m}(s)|\leq As^{-2}\;m=0,1,\;\;|r_{2}(s)|\leq A^{2}(\log s)s^{-2},$
$\forall y\in\mathbb{R},\;|r_{-}(y,s)|\leq
A(1+|y|^{3})s^{-2},\;\|r_{e}(s)\|_{L^{\infty}}\leq A^{2}s^{-\frac{1}{2}},$
(respectively
$|\tilde{r}_{m}(s)|\leq\tilde{A}s^{-\alpha}\;m=0,1,\;\;|\tilde{r}_{2}(s)|\leq\tilde{A}^{2}s^{-2+\eta},$
$\forall
y\in\mathbb{R},\;|\tilde{r}_{-}(y,s)|\leq\tilde{A}(1+|y|^{3})s^{-\alpha},\;\|\tilde{r}_{e}(s)\|_{L^{\infty}}\leq\tilde{A}^{2}s^{-\alpha+3/2}),$
where $r_{-}$, $r_{e}$ and $r_{m}$ (respectively $\tilde{r}_{-}$,
$\tilde{r}_{e}$ and $\tilde{r}_{m}$) are defined in Section 2.
As a matter of fact, if $s\geq e$ and $(r,\tilde{r})\in
V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$, then one easily derives that
$\|r(s)\|_{L^{\infty}}\leq C\frac{A^{2}}{\sqrt{s}}\mbox{ and
}\|\tilde{r}(s)\|_{L^{\infty}}\leq C\frac{\tilde{A}^{2}}{s^{\alpha-3/2}},$
(25)
(see Proposition 3.7 page 157 in [MZ97] for details). Thus, our aim become the
following:
###### Proposition 3.2
(Existence of a solution of (17) trapped in
$V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$) There exists ${A_{1}}$ such that for
all $A\geq{A_{1}}$ and $\tilde{A}\geq{A_{1}}$, $0<\eta<\frac{1}{10}$ and
$2<\alpha<2+\eta$, there exists ${s_{0,1}}(A,\tilde{A})$ such that for all
$s_{0}\geq{s_{0,1}}$ and $|\tilde{B}|\leq 1$, there exists
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})$, such that,
if $(q,\tilde{q})$ is a solution of (17) with initial data at $s_{0}$ given by
(24), then
$\forall s\geq s_{0},\;\;q(s)\in V_{A}(s)\mbox{ and
}\tilde{q}(s)\in\tilde{V}_{\tilde{A}}(s).$
The aim of this section is to prove this proposition.
In the following lemma, we find a set
$D_{A,\tilde{A},\tilde{B},\eta,\alpha,s_{0}}=D_{s_{0}}$ such that
$(q,\tilde{q})(s_{0})\in V_{A}(s_{0})\times\tilde{V}_{\tilde{A}}(s_{0})$,
whenever $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$. More
precisely, we claim the following:
###### Lemma 3.3
(Choice of parameters $d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1}$ to have
initial data in $V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$ at $s=s_{0}$) For
each $|\tilde{B}|\leq 1$, $A\geq 1$, $\tilde{A}\geq 1$, $0<\eta<\frac{1}{10}$
and $2<\alpha<2+\eta$, there exists $s_{0,2}(A,\tilde{A},\tilde{B})\geq e$
such that for all $s_{0}\geq s_{0,2}$:
If initial data for equation (17) are given by (24): then, there exists a
cuboid
$D_{A,\tilde{A},\tilde{B},\eta,\alpha,s_{0}}=D_{s_{0}}\subset[-2,2]^{4},$ (26)
such that, for all $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$,
we have
$(q,\tilde{q})(s_{0},A,\tilde{A},\tilde{B},d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in
V_{A}(s_{0})\times\tilde{V}_{\tilde{A}}(s_{0}).$
Proof: The proof is purely technical and follows as the analogous step in
[MZ97], for that reason we refer the reader to Lemma 3.5 page 156 and Lemma
3.9 page 160 in [MZ97]. $\blacksquare$
Let us consider $|\tilde{B}|\leq 1$, $A\geq 1$, $\tilde{A}\geq 1$,
$0<\eta<1/10$, $2<\alpha<2+\eta$,
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$ and $s_{0}\geq
s_{0,1}$ defined in Lemma 3.3. From the local Cauchy theory, we define a
maximal solution $(q,\tilde{q})$ to equation (17) with initial data (24), and
a maximal time
$s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in[s_{0},+\infty]$ such that,
for all $s\in[s_{0},s_{*})$, $(q,\tilde{q})(s)\in
V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$ and:
* •
either $s_{*}=\infty$,
* •
or $s_{*}<\infty$ and from continuity,
$(q,\tilde{q})(s_{*})\in\partial\left(V_{A}(s_{*})\times\tilde{V}_{\tilde{A}}(s_{*})\right)$,
in the sense that when $s=s^{*}$, one ’$\leq$’ symbol in the definition of
$V_{A}(s_{*})$ and $\tilde{V}_{\tilde{A}}(s_{*})$ is replaced by the symbol
’$=$’.
Our aim is to show that for all $|\tilde{B}|\leq 1$, for $A$, $\tilde{A}$ and
$s_{0}$ large enough, one can find a parameter
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})$ in $D_{s_{0}}$ such that
$s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})=\infty.$ (27)
We argue by contradiction, and assume that for all
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$,
$s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})<+\infty$. As we have just
stated, one of the symbols ’$\leq$’ in the definition of $V_{A}(s)$ and
$\tilde{V}_{\tilde{A}}(s)$ should be replaced by ’$=$’ symbols when $s=s_{*}$.
In fact, this is possible only with the ’$\leq$’ symbols concerning the
components $q_{0}$, $q_{1}$, $\tilde{q}_{0}$ or $\tilde{q}_{1}$, as one sees
in the following:
###### Lemma 3.4
(Reduction to a finite dimensional problem) There exists $A_{3}>0$ such that
for each $A\geq A_{3}$ and $\tilde{A}\geq A_{3}$ there exists
$s_{0,3}(A,\tilde{A})\geq s_{0,2}(A,\tilde{A})$ such that if $s_{0}\geq
s_{0,3}$, then
$(q_{0}(s_{*}),q_{1}(s_{*}),\tilde{q}_{0}(s_{*}),\tilde{q}_{1}(s_{*}))\in\partial\left(\left[-\frac{A}{s_{*}^{2}},\frac{A}{s_{*}^{2}}\right]^{2}\times\left[-\frac{\tilde{A}}{s_{*}^{\alpha}},\frac{\tilde{A}}{s_{*}^{\alpha}}\right]^{2}\right)$.
Proof: This is a direct consequence of the dynamics of equation (17), as we
will show in Subsection 3.3 below.
Just to give a flavor of the argument, we invite the reader to look at
Proposition 3.8 below, where we project system (17) on the different
components of $q$ and $\tilde{q}$ introduced in (23). There, one can see that
the components $q_{2}$, $q_{-}$ and $q_{e}$ (respectively $\tilde{q}_{2}$,
$\tilde{q}_{-}$ and $\tilde{q}_{e}$) correspond to decreasing directions of
the flow and since they are “small“ at $s=s_{0}$ (see Lemma 3.7 below), they
remain small up for $s\in[s_{0},s_{*}]$, and can not touch their bounds. Thus,
only $q_{0}$, $q_{1}$, $\tilde{q}_{0}$ or $\tilde{q}_{1}$ may touch their
boundary at $s=s_{*}$.
For more details on the arguments, see Subsection 3.3 below. This ends the
proof of Lemma 3.4.$\blacksquare$
From Lemma 3.4, we may define the rescaled flow $\Phi$ at $s=s_{*}$ for the
four expanding directions, namely $q_{0}$, $q_{1}$, $\tilde{q}_{0}$ and
$\tilde{q}_{1}$, as follows:
$\begin{array}[]{lccl}\Phi:&D_{s_{0}}&\to&\partial([-1,1]^{4})\\\
&(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})&\to&\displaystyle\left(\frac{s_{*}^{2}q_{0}}{A},\frac{s_{*}^{2}q_{1}}{A},\frac{s_{*}^{\alpha}\tilde{q}_{0}}{\tilde{A}},\frac{s_{*}^{\alpha}\tilde{q}_{1}}{\tilde{A}}\right)_{d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1}}(s_{*}).\end{array}$
(28)
In particular,
$\mbox{either }\omega q_{m}(s_{*})=\frac{A}{s_{*}^{2}}\mbox{ or
}\tilde{\omega}q_{\tilde{m}}(s_{*})=\frac{\tilde{A}}{s_{*}^{\alpha}},$ (29)
for some $m$, $\tilde{m}\in\\{0,1\\}$ and $\omega$,
$\tilde{\omega}\in\\{-1,1\\}$, both depending on
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})$. In the following lemma, we show
that $q_{m}$ (or $\tilde{q}_{m}$) actually crosses its boundary at $s=s_{*}$,
resulting in the continuity of $s_{*}$ and $\Phi$. More precisely, we have the
following:
###### Lemma 3.5 (Transverse crossing)
There exists $A_{4}>0$ such that for all $A\geq A_{4}$ and $\tilde{A}\geq
A_{4}$, there exists $s_{0,4}(A,\tilde{A})\geq s_{0,3}(A,\tilde{A})$ such that
if $s_{0}\geq s_{0,4}$ and (29) holds for some $s_{*}\geq s_{0,4}$, then
$\mbox{either }\omega\frac{dq}{ds}(s_{*})>0\mbox{ or
}\tilde{\omega}\frac{d\tilde{q}_{\tilde{m}}}{ds}(s_{*})>0.$ (30)
Clearly, from the transverse crossing, we see that
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\mapsto
s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\mbox{ is continuous,}$
hence by definition (28), $\Phi$ is continuous. In order to find a
contradiction and conclude, we crucially use the particular form we choose for
initial data in (24). More precisely, we have the following:
###### Lemma 3.6 (Degree $1$ on the boundary)
There exists $A_{5}>0$ such that for each $A\geq A_{5}$ and $\tilde{A}\geq
A_{5}$, there exists $s_{0,5}(A,\tilde{A})\geq s_{0,4}(A,\tilde{A})$ such that
if $s_{0}\geq s_{0,4}$, then the mapping
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\to(q_{0}(s_{0}),q_{1}(s_{0}),\tilde{q}_{0}(s_{0}),\tilde{q}_{1}(s_{0}))$
maps $\partial D_{s_{0}}$ into
$\partial\left([-\frac{A}{s_{0}^{2}},\frac{A}{s_{0}^{2}}]^{2}\times[-\frac{\tilde{A}}{s_{0}^{\alpha}},\frac{\tilde{A}}{s_{0}^{\alpha}}]^{2}\right)$,
and has degree one on the boundary.
Indeed, from this lemma and the transverse crossing property of Lemma 3.5, we
see that if $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in\partial D_{s_{0}}$,
then $s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})=s_{0}$,
$\Phi(s_{*}(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1}),d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})=\left(\frac{s^{2}_{*}q_{0}}{A},\frac{s^{2}_{*}q_{1}}{A},\frac{s^{\alpha}_{*}\tilde{q}_{0}}{A},\frac{s^{\alpha}_{*}\tilde{q}_{1}}{A}\right)(s_{0})$
and $\Phi$ defined in (28) is a continuous function from the cuboid
$D_{s_{0}}\subset\mathbb{R}^{4}$ to $\partial[-1,1]^{4}$, whose restriction to
$\partial D_{s_{0}}$ is of degree 1. This is a contradiction. Thus,
Proposition 3.2 is proved, and from identity (25), we have constructed a
solution $(q,\tilde{q})$ to system (17), such that
$\|q(s)\|_{L^{\infty}}+\|\tilde{q}(s)\|_{L^{\infty}}\to 0\mbox{ as
}s\to\infty.$
In the following subsections, we give the proofs of the technical steps of the
current subsection (namely Lemmas 3.3, 3.4, 3.5 and 3.6), referring to earlier
work when the proof is the same.
### 3.2 Preparation of initial data
In this subsection, we study initial data given by (24). More precisely, we
state a lemma which directly implies Lemmas 3.3 and 3.6. It also shows the
(relative) smallness of the components $q_{2}$, $q_{-}$, $q_{e}$,
$\tilde{q}_{2}$, $\tilde{q}_{-}$ and $\tilde{q}_{e}$, an information which
will be useful for the next subsection, dedicated to the dynamics of equation
(17), crucial for the proofs of the reduction to a finite dimensional problem
(Lemma 3.4) and the transverse crossing property (Lemma 3.5). More precisely,
we claim the following:
###### Lemma 3.7
(Decomposition of initial data in different components) For each
$|\tilde{B}|\leq 1$, $A\geq 1$, $\tilde{A}\geq 1$, $0<\eta<\frac{1}{10}$ and
$2<\alpha<2+\eta$, there exists $s_{0,6}(A,\tilde{A},\tilde{B})\geq e$ such
that for all $s_{0}\geq s_{0,6}$:
(i) there exists a cuboid
$D_{s_{0}}\subset[-2,2]^{4},$ (31)
such that the mapping
$(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\to(q_{0}(s_{0}),q_{1}(s_{0}),\tilde{q}_{0}(s_{0}),\tilde{q}_{1}(s_{0}))$
is linear and one to one from $D_{s_{0}}$ onto
$[-\frac{A}{s_{0}^{2-\eta}},\frac{A}{s_{0}^{2-\eta}}]^{2}\times[-\frac{\tilde{A}}{s_{0}^{\alpha}},\frac{\tilde{A}}{s_{0}^{\alpha}}]^{2}$
and maps the boundary $\partial D_{s_{0}}$ into the boundary
$\partial\left([-\frac{A}{s_{0}^{2}},\frac{A}{s_{0}^{2}}]^{2}\times[-\frac{\tilde{A}}{s_{0}^{\alpha}},\frac{\tilde{A}}{s_{0}^{\alpha}}]^{2}\right)$.
Moreover, it is of degree one on the boundary.
(ii) For all $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})\in D_{s_{0}}$, we have
$\begin{array}[]{l}|q_{2}(s_{0})|\leq CAe^{-\gamma s_{0}}\mbox{, for some
}\gamma>0,\;\;|q_{-}(y,s_{0})|\leq\frac{c}{s_{0}^{2}}(1+|y|^{3})\mbox{ and
}q_{e}(y,s_{0})=0,\\\ |d_{0}|+|d_{1}|\leq 1,\end{array}$ (32)
and
$\begin{array}[]{l}|\tilde{q}_{2}(s_{0})-\frac{\tilde{B}}{s_{0}^{2}}|\leq
C\tilde{A}e^{-\gamma s_{0}}\mbox{, for some
}\gamma>0,\;\;|\tilde{q}_{-}(y,s_{0})|\leq\frac{c}{s_{0}^{\alpha}}(1+|y|^{3})\mbox{
and }\tilde{q}_{e}(y,s_{0})=0,\\\ |\tilde{d}_{0}|+|\tilde{d}_{1}|\leq
1.\end{array}$ (33)
Proof: Since we have almost the same definition of the set $V_{A}$, and almost
the same expression of initial data as in [MZ97], we refer the reader to Lemma
3.5 page 156 and Lemma 3.9 page 160 from [MZ97]. $\blacksquare$
### 3.3 Reduction to a finite-dimensional problem
This subsection is dedicated to the proof of Lemmas 3.4 and 3.5. They both
follow from the understanding of the flow of equation (17) in the set
$V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$. Accordingly, this crucially relies
on the projection of equation (17) with respect to the decomposition given in
(23). More precisely, we claim the following:
###### Proposition 3.8 (Dynamics of equation (17))
There exists $A_{7}\geq 1$ such that for all $A\geq A_{7}$, $\tilde{A}\geq
A_{7}$, $0<\eta<\frac{1}{10}$, $2<\alpha<2+\eta$ and $\theta\geq 0$, there
exists $s_{0,7}(A,\tilde{A},\theta)$ such that the following holds for all
$s_{0}\geq s_{0,7}$:
Assume that for some $\tau\geq s_{0}$ and for all $s\in[\tau,\tau+\theta]$,
$(q(s),\tilde{q}(s))\in V_{A}(s)\times\tilde{V}_{A}(s).$
Then, the following holds for all $s\in[\tau,\tau+\theta]$:
(i)(Differential inequalities satisfied by the expanding and null modes) For
$m=0$ and $1$, we have:
$\left|q^{\prime}_{m}(s)-(1-\frac{m}{2})q_{m}(s)\right|\leq\frac{C}{s^{2}},$
$\left|\tilde{q}^{\prime}_{m}(s)-(1-\frac{m}{2})\tilde{q}_{m}(s)\right|\leq\frac{C\tilde{A}^{2}}{s^{3-\eta}},$
$\left|\tilde{q}^{{}^{\prime}}_{2}(s)+\frac{2}{s}\tilde{q}_{2}(s)\right|\leq\frac{C\tilde{A}}{s^{\alpha+1}}.$
(ii)(Control of the null and negative modes) Moreover, we have:
$\displaystyle|q_{2}(s)|$
$\displaystyle\leq\frac{\tau^{2}}{s^{2}}|q_{2}(\tau)|+\frac{CA(s-\tau)}{s^{3}},$
$\displaystyle|\tilde{q}_{2}(s)|$
$\displaystyle\leq\frac{\tau^{2}}{s^{2}}|\tilde{q}_{2}(\tau)|+\frac{C\tilde{A}(s-\tau)}{s^{\alpha+1}},$
$\displaystyle\left\|\frac{q_{-}(s)}{1+|y|^{3}}\right\|_{L^{\infty}}$
$\displaystyle\leq
Ce^{-\frac{(s-\tau)}{2}}\left\|\frac{q_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+C\frac{e^{-(s-\tau)^{2}}\|q_{e}(\tau)\|_{L^{\infty}}}{s^{3/2}}+\frac{C(1+s-\tau)}{s^{2}},$
$\displaystyle\left\|\frac{\tilde{q}_{-}(s)}{1+|y|^{3}}\right\|_{L^{\infty}}$
$\displaystyle\leq
Ce^{-\frac{(s-\tau)}{2}}\left\|\frac{\tilde{q}_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+C\frac{e^{-(s-\tau)^{2}}\|\tilde{q}_{e}(\tau)\|_{L^{\infty}}}{s^{3/2}}+\frac{C(1+s-\tau)}{s^{\alpha}},$
$\displaystyle\|q_{e}(s)\|_{L^{\infty}}$ $\displaystyle\leq
Ce^{-\frac{(s-\tau)}{2}}\|q_{e}(\tau)\|_{L^{\infty}}+C\frac{e^{s-\tau}}{s^{3/2}}\left\|\frac{q_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+\frac{C(1+s-\tau)}{s^{1/2}},$
$\displaystyle\|\tilde{q}_{e}(s)\|_{L^{\infty}}$ $\displaystyle\leq
Ce^{-\frac{(s-\tau)}{2}}\|\tilde{q}_{e}(\tau)\|_{L^{\infty}}+C\frac{e^{s-\tau}}{s^{3/2}}\left\|\frac{\tilde{q}_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+\frac{C(1+s-\tau)}{s^{\alpha-3/2}}.$
Let us first insist on the fact that the derivation of Lemmas 3.4 and 3.5
follows from Proposition 3.8, exactly as in the real case treated in [MZ97]
(see pages 163 to 166 and 158 to 159 in [MZ97] ). For that reason, we only
focus in the following on the proof of Proposition 3.8.
Proof of Proposition 3.8:
The proof of Proposition 3.8 consists in the projection of the two equations
of system (17) on the different components of $q$ and $\tilde{q}$ defined in
(23).
When $\tilde{q}\equiv 0$, the proof is already available from Lemma 3.13 pages
167 and Lemma 3.8 page 158 from [MZ97].
When $\tilde{q}\not\equiv 0$, since the equation satisfied by $\tilde{q}$ in
(17) shares the same linear part as the equation in $q$, the proof is similar
to the argument in [MZ97]. For that reason, we only give the ideas here, and
kindly ask the interested reader to look at Lemma 3.13 page 167 and Lemma 3.8
page 158 in [MZ97] for the technical details.
(i) Multiplying the two equations in (17) by $\chi(y,s)k_{m}(y)\rho(y)$, for
$m=0,1,2$ and integrating in $y\in\mathbb{R}$, we proceed as in pages 158-159
from [MZ97] and we get the differential inequalities given in (i) with no
difficulties.
(ii) For convenience, we separate the contribution of $q$ and $\tilde{q}$ in
the quadratic term $b(y,s)$ defined in (19) by writing $b(y,s)=B(y,s)-N(y,s)$
with
$B(y,s)=q^{2}\mbox{, and }N(y,s)=\tilde{q}^{2}.$ (34)
Let us first recall equations of $(q,\tilde{q})$ in their Duhamel formulation,
$\begin{array}[]{lll}q(s)&=&K(s,\tau)q(\tau)+\int_{\tau}^{s}d\sigma
K(s,\sigma)B(q(\sigma))+\int_{\tau}^{s}d\sigma
K(s,\sigma)R(\sigma)-\int_{\tau}^{s}d\sigma K(s,\sigma)N(\sigma),\\\
\tilde{q}(s)&=&K(s,\tau)\tilde{q}(\tau)+\int_{\tau}^{s}d\sigma
K(s,\sigma)\tilde{b}(\sigma),\end{array}$ (35)
where $K$ is the fundamental solution of the operator ${\cal L}+V$. We write
$q=\alpha+\beta+\gamma+\delta$ and $\tilde{q}=\tilde{\alpha}+\tilde{\beta}$,
where
$\begin{array}[]{l}\alpha(s)=K(s,\tau)q(\tau),\;\;\beta(s)=\int_{\tau}^{s}d\sigma
K(s,\sigma)B(q(\sigma)),\\\ \gamma(s)=\int_{\tau}^{s}d\sigma
K(s,\sigma)R(\sigma),\;\;\delta(s)=-\int_{\tau}^{s}d\sigma
K(s,\sigma)N(\sigma).\end{array}$ (36)
$\tilde{\alpha}(s)=K(s,\tau)\tilde{q}(\tau),\;\;\tilde{\beta}(s)=\int_{\tau}^{s}d\sigma
K(s,\sigma)\tilde{b}(\sigma).\\\ $ (37)
We assume that $(q(s),\tilde{q}(s))\in V_{A}(s)\times\tilde{V}_{A}(s)$ for
each $s\in[\tau,\tau+\theta]$. Clearly, proceeding as the derivation of Lemma
3.13 page 167 in [MZ97], (ii) of Proposition 3.8 follows from the following:
###### Lemma 3.9 (Projection of the Duhamel formulation)
There exists $A_{8}\geq 1$ such that for all $A\geq A_{8}$, $\tilde{A}\geq
A_{8}$ and $\theta>0$ there exists $s_{0,8}(A,\tilde{A},\theta)\geq
s_{0,7}(A)$, such that for all $s_{0}\geq s_{0,8}(A,\tilde{A},\theta)$, if we
assume that for some $\tau\geq s_{0}$ and for all $s\in[\tau,\tau+\theta]$,
$q(s)\in V_{A}(s)$ and $\tilde{q}(s)\in\tilde{V}_{\tilde{A}}(s)$, then
(i) (Linear terms)
$\begin{array}[]{lll}|\alpha_{2}(s)|&\leq&\frac{\tau^{2}}{s^{2}}|q_{2}(\tau)|+\frac{CA(s-\tau)}{s^{3}},\\\
\left\|\frac{\alpha_{-}(s)}{1+|y|^{3}}\right\|_{L^{\infty}}&\leq&Ce^{-\frac{(s-\tau)}{2}}\left\|\frac{q_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+C\frac{e^{-(s-\tau)^{2}}\|q_{e}(\tau)\|_{L^{\infty}}}{s^{3/2}}+\frac{C}{s^{2}},\\\
\|\alpha_{e}(s)\|_{L^{\infty}}&\leq&Ce^{-\frac{(s-\tau)}{2}}\|q_{e}(\tau)\|_{L^{\infty}}+Ce^{s-\tau}s^{3/2}\left\|\frac{q_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+\frac{C}{\sqrt{s}},\end{array}$
(38)
and
$\begin{array}[]{lll}|\tilde{\alpha}_{2}(s)|&\leq&\frac{\tau^{2}}{s^{2}}|\tilde{q}_{2}(\tau)|+\frac{CA(s-\tau)}{s^{\alpha+1}},\\\
\left\|\frac{\tilde{\alpha}_{-}(s)}{1+|y|^{3}}\right\|_{L^{\infty}}&\leq&Ce^{-\frac{(s-\tau)}{2}}\left\|\frac{\tilde{q}_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+C\frac{e^{-(s-\tau)^{2}}\|\tilde{q}_{e}(\tau)\|_{L^{\infty}}}{s^{3/2}}+\frac{C}{s^{\alpha}},\\\
\|\tilde{\alpha}_{e}(s)\|_{L^{\infty}}&\leq&Ce^{-\frac{(s-\tau)}{2}}\|\tilde{q}_{e}(\tau)\|_{L^{\infty}}+Ce^{s-\tau}s^{3/2}\left\|\frac{\tilde{q}_{-}(\tau)}{1+|y|^{3}}\right\|_{L^{\infty}}+\frac{C}{s^{\alpha-3/2}}.\end{array}$
(39)
(ii) (Nonlinear terms)
$\displaystyle|\beta_{2}(s)|$ $\displaystyle\leq\frac{(s-\tau)}{s^{3}},$
$\displaystyle|\beta_{-}(y,s)|$
$\displaystyle\leq\frac{(s-\tau)}{s^{2}}(1+|y|^{3}),$
$\displaystyle\|\beta_{e}(s)\|_{L^{\infty}}$
$\displaystyle\leq\frac{(s-\tau)}{\sqrt{s}},$ $\displaystyle|\delta_{2}(s)|$
$\displaystyle\leq C\frac{(s-\tau)}{s^{3}},$ $\displaystyle|\delta_{-}(y,s)|$
$\displaystyle\leq C\frac{(s-\tau)}{s^{2}}(1+|y|^{3}),$
$\displaystyle\|\delta_{e}(s)\|_{L^{\infty}}$ $\displaystyle\leq
C\frac{(s-\tau)}{\sqrt{s}},$ $\displaystyle|\tilde{\beta}_{2}(s)|$
$\displaystyle\leq\frac{(s-\tau)}{s^{\alpha+1}},$
$\displaystyle|\tilde{\beta}_{-}(y,s)|$
$\displaystyle\leq\frac{(s-\tau)}{s^{\alpha}}(1+|y|^{3}),$
$\displaystyle\|\tilde{\beta}_{e}(s)\|_{L^{\infty}}$
$\displaystyle\leq\frac{(s-\tau)}{s^{\alpha-3/2}}.$
(iii) (Source term)
$|\gamma_{2}(s)|\leq C(s-\tau)s^{-3},\;|\gamma_{-}(y,s)|\leq
C(s-\tau)(1+|y|^{3})s^{-2},\;\|\gamma_{e}(s)\|_{L^{\infty}}\leq(s-\tau)s^{-1/2}.$
Proof: We consider, $A\geq 1$, $\tilde{A}\geq 1$, $\theta>0$, and
$s_{0}\geq\theta$. The terms $\alpha$, $\beta$ and $\gamma$ are already
present in the case of the real-valued semilinear heat equation, so we refer
to Lemma 3.13 page 167 in [MZ97] for the estimates involving them. As for
$\tilde{\alpha}$, since the definition of $\tilde{V}_{\tilde{A}}(s)$ is
different from the definition of $V_{A}(s)$, the reader will have absolutely
no difficulty to adapt Lemma 3.13 of [MZ97] to the new situation. Thus, we
only focus on the new terms $\delta(y,s)$ and $\tilde{\beta}(y,s)$. Note that
since $s_{0}\geq\theta$, if we take $\tau\geq s_{0}$, then $\tau+\theta\leq
2\tau$ and if $\tau\leq\sigma\leq s\leq\tau+\theta$, then
$\frac{1}{2\tau}\leq\frac{1}{s}\leq\frac{1}{\sigma}\leq\frac{1}{\tau}.$
Let us first derive the following bounds when $(q(s),\tilde{q}(s))\in
V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$:
###### Proposition 3.10 (Bounds for $(q(s),\tilde{q}(s))\in
V_{A}(s)\times\tilde{V}_{\tilde{A}}(s)$)
For all $s\geq e$, we consider $r\in V_{A}(s)$ and
$\tilde{r}\in\tilde{V}_{\tilde{A}}(s)$, where the shrinking sets $V_{A}(s)$
and $\tilde{V}_{\tilde{A}}(s)$ are given in Definition 3.1. Then, we have:
$\begin{array}[]{l}(i)\;for\;all\;y\in\mathbb{R},\;\;|r(y,s)|\leq
CA^{2}\frac{\log s}{s^{2}}(1+|y|^{3}),\\\ (ii)\;\|r(s)\|_{L^{\infty}}\leq
C\frac{A^{2}}{\sqrt{s}},\\\
(iii)\;for\;all\;y\in\mathbb{R},\;\;|\tilde{r}(y,s)|\leq
C\frac{\tilde{A}^{2}}{s^{2-\eta}}(1+|y|^{3}),\;\;|\tilde{r}_{b}(y,s)|\leq\frac{C\tilde{A}}{s^{\alpha-3/2}},\\\
(iv)\;\|\tilde{r}(s)\|_{L^{\infty}}\leq
C\frac{\tilde{A}^{2}}{s^{\alpha-3/2}}.\end{array}$
Proof: The proof is omitted since it is the same as the corresponding part in
[MZ97]. See Proposition 3.7 page 157 in [MZ97] for details. $\blacksquare$
Then, we recall from Bricmont and Kupiainen [BK94] the following estimates on
$K(s,\sigma)$, the semigroup generated by ${\cal L}+V$:
###### Lemma 3.11
(Properties of $K(s,\sigma)$):
(i) For all $s\geq\sigma>1$ and $y$, $x\in\mathbb{R}$, we have
$|K(s,\sigma,y,x)|\leq Ce^{(s-\sigma){\cal L}}(y,x),$
where $e^{\psi{\cal L}}$ is given by
$e^{\psi{\cal L}}(y,x)=\frac{e^{\psi}}{\sqrt{4\pi(1-e^{-\psi})}}{\rm
exp}\left[-\frac{(ye^{-\psi/2}-x)^{2}}{4(1-e^{-\psi})}\right].$
(ii)We have for all $s\geq\tau\geq 1$, with $s\leq 2\tau$,
$\displaystyle\left|\int K(s,\tau,y,x)(1+|x|^{m})dx\right|\leq C\int
e^{(s-\tau){\cal L}}(y,x)(1+|x|^{m})dx\leq e^{s-\tau}(1+|y|^{m}).$ (40)
Proof:
(i) See page 181 in [MZ97]
(ii) See Corollary 3.14 page 168 in [MZ97]. $\blacksquare$
Estimates on $\delta$ defined in (36):
Consider $s\in[\tau,\tau+\theta]$. Since $\tilde{q}(s)\in\tilde{V}_{A}(s)$ by
assumption, using (iii) and (iv) of Lemma 3.10, we see that
$\forall
y\in\mathbb{R},\;|\tilde{q}(y,s)|\leq\min\left(\frac{C\tilde{A}^{2}}{s^{2-\eta}}(1+|y|^{3}),\frac{C\tilde{A}^{2}}{s^{\alpha-3/2}}\right),$
(41)
hence by definition (34) of $N$, we obtain
$\forall y\in\mathbb{R},\;|N(y,s)|\leq
C\tilde{A}^{4}\min\left(\frac{(1+|y|^{3})}{s^{\alpha+\frac{1}{2}-\eta}},\frac{1}{s^{2\alpha-3}},\frac{1+|y|^{6}}{s^{4-2\eta}}\right).$
(42)
Using Lemma 3.11 and the definition (36) of $\delta$, we write
$\begin{array}[]{lll}|\delta(y,s)|&\leq&\displaystyle\int_{\tau}^{s}d\sigma\int_{\mathbb{R}}\left|K(s,\sigma,y,x)N(x,\sigma)\right|dx\\\
&\leq&\displaystyle\int_{\tau}^{s}d\sigma\int_{\mathbb{R}}e^{(s-\sigma){\cal
L}}(y,x)\frac{C\tilde{A}^{4}(1+|x|^{3})}{s^{\alpha+1/2-\eta}}dx\\\
&\leq&\displaystyle\frac{C\tilde{A}^{4}(s-\tau)}{s^{\alpha+1/2-\eta}}e^{s-\tau}(1+|y|^{3})\leq\frac{(s-\tau)}{s^{2}}(1+|y|^{3}),\\\
\end{array}$ (43)
for $s_{0}$ large enough, since $\eta<1/2$.
Using the following bounds in (42) and proceeding similarly, we see that
$\forall
y\in\mathbb{R},\;|\delta(y,s)|\leq(s-\tau)\min\left(\frac{1+|y|^{3}}{s^{2}},\frac{1}{\sqrt{s}},\frac{1+|y|^{6}}{s^{3}}\right),$
since $\alpha>2$, $\eta<1/2$, and provided that $s_{0}$ is large enough.
By definition of $q_{m}$, $q_{-}$ and $q_{e}$ for $m\leq 2$, we write
$\begin{array}[]{lll}|\delta_{m}(s)|&\leq&\left|\int_{\mathbb{R}}\chi(y,s)\delta(y,s)k_{m}(y)\rho(y)dy\right|\leq
C\int_{\mathbb{R}}|\delta(y,s)|(1+|y|^{2})\rho(y)dy\leq\frac{C(s-\tau)}{s^{3}},\\\
|\delta_{-}(y,s)|&=&\left|\chi(y,s)\delta(y,s)-\sum_{i=0}^{2}\delta_{i}(s)k_{i}(y)\right|\leq(s-\tau)(1+|y|^{3})\frac{C}{s^{2}}.\\\
|\delta_{e}(y,s)|&=&\left|(1-\chi(y,s))\delta(y,s)\right\|\leq(s-\tau)\frac{C}{\sqrt{s}}.\end{array}$
(44)
Estimates on $\tilde{\beta}$ defined in (37):
Consider $s\in[\tau,\tau+\theta]$. Since $q(s)\in V_{A}(s)$ by assumption,
using (i) and (ii) of Lemma 3.10, we see that
$\forall y\in\mathbb{R},\;|q(y,s)|\leq CA^{2}\min\left(\frac{\log
s}{s^{2}}(1+|y|^{3}),\frac{1}{\sqrt{s}}\right).$
Using (41) and the definition (19) of $\tilde{b}$, we see that
$\forall y\in\mathbb{R},\;|\tilde{b}(y,s)|\leq
CA^{2}\tilde{A}^{2}\min\left(\frac{\log
s}{s^{\alpha+1/2}}(1+|y|^{3}),\frac{1}{s^{\alpha-1}},\frac{1+|y|^{6}}{s^{4-\eta}}\right).$
Using the definition (37) of $\tilde{\beta}$ and arguing as for estimate
(LABEL:esdelta), we see that
$\forall
y\in\mathbb{R},\;|\tilde{\beta}(y,s)|\leq(s-\tau)\min\left(\frac{1+|y|^{3}}{s^{\alpha}},\frac{1}{s^{\alpha-3/2}},\frac{1+|y|^{6}}{s^{\alpha+1}}\right),$
provided that $s_{0}$ is large enough, since $\eta<\frac{1}{2}$ and
$\alpha<2+\eta<3-\eta$. Arguing as for (44), we get the desired estimates.
This concludes the proof of Lemma 3.9. $\blacksquare$
Since item (ii) of Proposition 3.8 follows from Lemma 3.9, exactly as for
Lemma 3.13 page 167 in [MZ97], this also ends the proof of Proposition 3.8 .
$\blacksquare$
Since Lemmas 3.4 and 3.5 follow from Proposition 3.8 exactly as in [MZ97] (see
pages 163 to 166 and 158 to 159 in that paper), this is also the conclusion of
the proof of Lemmas 3.4 and 3.5. Recalling that we have already justified that
Lemmas 3.3 and 3.6 hold (see Lemma 3.7 above), and given that Proposition 3.2
is the consequence of Lemmas 3.3, 3.4, 3.5 and 3.6, this is also the
conclusion of Proposition 3.2. $\blacksquare$
## 4 Asymptotic behavior of $u(t)$
We prove Theorem 1 in this section. We will first derive (ii) and (iii) from
Section 3, then we will prove (i) and (iv).
Consider $0<|\tilde{B}|\leq 1$. Using Proposition 3.2, Lemma 3.7 and
Proposition 3.8, we see that if $A=\tilde{A}=\max(1,{A_{1}},A_{7})$,
$0<\eta<\frac{1}{10}$, $2<\alpha<2+\eta$ and $T\leq T_{9}(\tilde{B})$ for some
$T_{9}(\tilde{B})\leq\min(T_{1},T_{6},T_{7})$ where $T_{i}=-\log s_{0,i}$,
then there exists a parameter $(d_{0},d_{1},\tilde{d}_{0},\tilde{d}_{1})$ such
that if $(q(s_{0}),\tilde{q}(s_{0}))$ is given by (24), where $s_{0}=-\log T$,
then
$\begin{array}[]{ll}\forall s\geq-\log T,\;q(s)\in
V_{A}(s),\;\tilde{q}(s)\in\tilde{V}_{\tilde{A}}(s),&\;\left|\tilde{q}_{2}^{{}^{\prime}}(s)+\frac{2}{s}\tilde{q}_{2}(s)\right|\leq
C\frac{\tilde{A}}{s^{\alpha+1}}\leq\frac{\mu_{0}}{s^{\alpha+1}},\\\ \mbox{with
}\mu_{0}=\frac{\alpha-2}{4}|\tilde{B}|s_{0}^{\alpha-2},&\end{array}$ (45)
and
$\left|\tilde{q}_{2}(s_{0})-\frac{\tilde{B}}{s_{0}^{2}}\right|\leq
C\tilde{A}e^{-\gamma s_{0}}\leq\frac{|\tilde{B}|}{4s_{0}^{2}}.$
As announced earlier, we use this property to derive (ii) and (iii) of Theorem
1, then we will prove (i) and (iv).
(ii) This directly follows from (45), (25) and the selfsimilar transformation
(14).
(iii) From (45), we see that
$\forall s\geq-\log
T,\;|\left(s^{2}\tilde{q}_{2}\right)^{{}^{\prime}}|\leq\frac{\mu_{0}}{s^{\alpha-1}},$
(46)
which means that $s^{2}\tilde{q}_{2}(s)$ has some limit $l$ as $s\to\infty$.
Integrating this inequality between $s$ and $+\infty$, we obtain
$|s^{2}\tilde{q}_{2}(s)-l|\leq\frac{\mu_{0}}{(2-\alpha)s^{\alpha-2}}.$ (47)
Putting $s=s_{0}$ in this identity, then using (45), we see that
$|s_{0}^{2}\tilde{q}_{2}(s_{0})-l|\leq\frac{|\tilde{B}|}{4}\mbox{ and
}|s_{0}^{2}\tilde{q}_{2}(s_{0})-\tilde{B}|\leq\frac{|\tilde{B}|}{4},$
Thus, it follows that
$|l-\tilde{B}|\leq\frac{|\tilde{B}|}{2}\mbox{, hence
}|l|\geq\frac{|\tilde{B}|}{2}>0\mbox{ and }l\not\equiv 0.$
We then write from the decomposition (23) that for all $s\geq-\log T$, $R>0$
and $|y|\leq R$, $\tilde{q}_{e}(y,s)=0$, hence,
$\tilde{q}(y,s)-\frac{l}{s^{2}}h_{2}(y)=\sum_{i=0}^{1}\tilde{q}_{i}(s)h_{i}(y)+(\tilde{q}_{2}(s)-\frac{l}{s^{2}})h_{2}(y)+\tilde{q}_{-}(y,s).$
Using the fact that for all $s\geq-\log T$,
$\tilde{q}(s)\in\tilde{V}_{\tilde{A}}(s)$ (see (45) above), Definition 3.1 for
$\tilde{V}_{\tilde{A}}(s)$, together with (47), we see that for all
$s\geq-\log T,\;R>0\mbox{ and }|y|\leq R$
$\left|\tilde{q}(y,s)-\frac{l}{s^{2}}h_{2}(y)\right|\leq\frac{C(\tilde{A},R)}{s^{\alpha}}.$
Using the definition (16) of $\tilde{q}$ and (14) of $\tilde{w}$, we get the
desired conclusion.
(i) If $x_{0}=0$, then we see from (10) and (11) that $|v(0,t)|\sim(T-t)^{-1}$
as $t\to T$. Hence $u$ blows up at time $T$ at $x_{0}=0$. It remains to prove
that any $a\neq 0$ is not a blow-up point. The following result from Giga and
Kohn [GK89] allows us to conclude:
###### Proposition 4.1
(Giga and Kohn - No blow-up under the ODE threshold) For all $C_{0}>0$, there
is $\eta_{0}>0$ such that if $v(\xi,\tau)$ solves
$\left|v_{t}-\Delta v\right|\leq C_{0}(1+|v|^{p})$
and satisfies
$|v(\xi,\tau)|\leq\eta_{0}(T-t)^{-1}$
for all $(\xi,\tau)\in B(a,r)\times[T-r^{2},T)$ for some $a\in\mathbb{R}$ and
$r>0$, then $v$ does not blow up at (a,T).
Proof: See Theorem 2.1 page 850 in [GK89]. Note that the proof of Giga and
Kohn is valid also when $u$ is complex valued. $\blacksquare$
Indeed, since we see from (10) that
$\sup_{|x-x_{0}|\leq|x_{0}|/2}(T-t)^{-1}|u(x,t)|\leq\left|\varphi\left(\frac{|x_{0}|/2}{\sqrt{(T-t)|\log(T-t)|}}\right)\right|+\frac{C}{\sqrt{|\log(T-t)|}}\to
0$
as $t\to T$, $x_{0}$ is not a blow-up point of $u$ from Proposition 4.1. This
concludes the proof of (i) of Theorem 1.
(iv) Arguing as Merle did in [Mer92], we derive the existence of a blow-up
profile $u^{*}\in C^{2}(\mathbb{R}^{*})$ such that $u(x,t)\to u^{*}(x)$ as
$t\to T$, uniformly on compact sets of $\mathbb{R}^{*}$. The profile
$u^{*}(x)$ is not defined at the origin. In the following, we would like to
find its equivalent as $x\to 0$ and show that it is in 444 singular at the
origin. We argue as in Masmoudi and Zaag [MZ08]. Consider $K_{0}>0$ to be
fixed large enough later. If $x_{0}\neq 0$ is small enough, we introduce for
all $(\xi,\tau)\in\mathbb{R}\times[-\frac{t_{0}(x_{0})}{T-t_{0}(x_{0})},1)$,
$\displaystyle V(x_{0},\xi,\tau)$ $\displaystyle=(T-t_{0}(x_{0}))v(x,t),$ (48)
$\displaystyle\tilde{V}(x_{0},\xi,\tau)$
$\displaystyle=(T-t_{0}(x_{0}))\tilde{v}(x,t),$ (49)
$\displaystyle\mbox{where, }x$
$\displaystyle=x_{0}+\xi\sqrt{T-t_{0}(x_{0})},\;t=t_{0}(x_{0})+\tau(T-t_{0}(x_{0})),$
(50)
and $t_{0}(x_{0})$ is uniquely determined by
$|x_{0}|=K_{0}\sqrt{(T-t_{0}(x_{0}))|\log(T-t_{0}(x_{0}))|}.$ (51)
From the invariance of problem (2) under dilation,
$(V(x_{0},\xi,\tau),\tilde{V}(x_{0},\xi,\tau))$ is also a solution of (2) on
its domain. From (50), (51), (11) and (10), we have
$\sup_{|\xi|<2|\log(T-t_{0}(x_{0}))|^{1/4}}\left|V(x_{0},\xi,0)-f(K_{0})\right|\leq\frac{C}{|\log(T-t_{0}(x_{0}))|^{1/4}}\to
0\mbox{ as }x_{0}\to 0$
and
$\sup_{|\xi|<2|\log(T-t_{0}(x_{0}))|^{1/4}}\left|\tilde{V}(x_{0},\xi,0)\right|\leq\frac{C}{|\log(T-t_{0}(x_{0}))|^{1/4}}\to
0\mbox{ as }x_{0}\to 0.$
Using the continuity with respect to initial data for problem (2) associated
to a space-localization in the ball $B(0,|\xi|<|\log(T-t_{0}(x_{0}))|^{1/4})$,
we show as in Section 4 of [Zaa98] that
$\begin{array}[]{l}\sup_{|\xi|\leq|\log(T-t_{0}(x_{0}))|^{1/4},\;0\leq\tau<1}|V(x_{0},\xi,\tau)-U_{K_{0}}(\tau)|\leq\epsilon(x_{0})\mbox{
as }x_{0}\to 0\\\
\sup_{|\xi|\leq|\log(T-t_{0}(x_{0}))|^{1/4},\;0\leq\tau<1}|\tilde{V}(x_{0},\xi,\tau)|\leq\epsilon(x_{0})\mbox{
as }x_{0}\to 0,\end{array}$
where $U_{K_{0}}(\tau)=((1-\tau)+\frac{K_{0}^{2}}{8})^{-1}$ is the solution of
the PDE (2) with constant initial data $\varphi(K_{0})$. Making $\tau\to 1$
and using (50), we see that
$\begin{array}[]{lll}v^{*}(x_{0})=\lim_{t\to
T}v(x,t)&=&(T-t_{0}(x_{0}))^{-1}\lim_{\tau\to 1}V(x_{0},0,\tau)\\\
&\sim&(T-t_{0}(x_{0}))^{-1}U_{K_{0}}(1)\end{array}$
as $x_{0}\to 0$. We note also that
$|\tilde{v}^{*}(x_{0})|\leq\epsilon(x_{0})(T-t_{0}(x_{0}))^{-1}.$
Since we have from (51)
$\log(T-t_{0}(x_{0}))\sim 2\log|x_{0}|\mbox{ and
}T-t_{0}(x_{0})\sim\frac{|x_{0}|^{2}}{2K_{0}^{2}|\log|x_{0}||},$
as $x_{0}\to 0$, this yields (iv) of Theorem 1 and concludes the proof of
Theorem 1. $\blacksquare$
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|
arxiv-papers
| 2013-06-19T06:38:25 |
2024-09-04T02:49:46.664590
|
{
"license": "Public Domain",
"authors": "Nejla Nouaili and Hatem Zaag",
"submitter": "Nejla Nouaili",
"url": "https://arxiv.org/abs/1306.4435"
}
|
1306.4489
|
EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)
CERN-PH-EP-2013-099 LHCb-PAPER-2013-029 July 25, 2013
Searches for $B^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ decays
The LHCb collaboration†††Authors are listed on the following pages.
The results of searches for
$B^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ decays are reported. The analysis is based on a
data sample, corresponding to an integrated luminosity of
$1.0\mbox{\,fb}^{-1}$ of $pp$ collisions, collected with the LHCb detector. An
excess with 2.8 $\sigma$ significance is seen for the decay
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and an upper limit on the branching fraction is set at
the 90 % confidence level: ${\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)<4.8\times 10^{-6}$, which is the first such limit. No
significant signals are seen for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p$
and $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ decays, for which the corresponding limits are
set: ${\cal B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)<5.2\times 10^{-7}$, which significantly improves the
existing limit; and ${\cal
B}(B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+})<5.0\times 10^{-7}$, which is the first limit on
this branching fraction.
Submitted to JHEP
© CERN on behalf of the LHCb collaboration, license CC-BY-3.0.
LHCb collaboration
R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z.
Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G.
Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24,37, S. Amato2, S. Amerio21,
Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, J.E. Andrews57,
R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M.
Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J.
Back47, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C.
Barschel37, S. Barsuk7, W. Barter46, Th. Bauer40, A. Bay38, J. Beddow50, F.
Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-
Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39,
M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A.
Bizzeti17,h, P.M. Bjørnstad53, T. Blake37, F. Blanc38, J. Blouw11, S. Blusk58,
V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A.
Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den
Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45,
H. Brown51, I. Burducea28, A. Bursche39, G. Busetto21,q, J. Buytaert37, S.
Cadeddu15, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P.
Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R.
Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho
Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R.
Cenci57, M. Charles54, Ph. Charpentier37, P. Chen3,38, N. Chiapolini39, M.
Chrzaszcz25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M.
Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E.
Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M.
Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C.
Craik47, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, P. David8, P.N.Y.
David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian39,
J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M.
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Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, G.
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B. Sanmartin Sedes36, M. Sannino19,i, R. Santacesaria24, C. Santamarina
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Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37.
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 Padova, Padova, Italy
22Sezione INFN di Pisa, Pisa, Italy
23Sezione INFN di Roma Tor Vergata, Roma, Italy
24Sezione INFN di Roma La Sapienza, Roma, Italy
25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of
Sciences, Kraków, Poland
26AGH - University of Science and Technology, Faculty of Physics and Applied
Computer Science, Kraków, Poland
27National Center for Nuclear Research (NCBJ), Warsaw, Poland
28Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest-Magurele, Romania
29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia
30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia
31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow,
Russia
32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN),
Moscow, Russia
33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State
University, Novosibirsk, Russia
34Institute for High Energy Physics (IHEP), Protvino, Russia
35Universitat de Barcelona, Barcelona, Spain
36Universidad de Santiago de Compostela, Santiago de Compostela, Spain
37European Organization for Nuclear Research (CERN), Geneva, Switzerland
38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
39Physik-Institut, Universität Zürich, Zürich, Switzerland
40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands
41Nikhef National Institute for Subatomic Physics and VU University Amsterdam,
Amsterdam, The Netherlands
42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine
43Institute for Nuclear Research of the National Academy of Sciences (KINR),
Kyiv, Ukraine
44University of Birmingham, Birmingham, United Kingdom
45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United
Kingdom
46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom
47Department of Physics, University of Warwick, Coventry, United Kingdom
48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom
49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United
Kingdom
50School of Physics and Astronomy, University of Glasgow, Glasgow, United
Kingdom
51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom
52Imperial College London, London, United Kingdom
53School of Physics and Astronomy, University of Manchester, Manchester,
United Kingdom
54Department of Physics, University of Oxford, Oxford, United Kingdom
55Massachusetts Institute of Technology, Cambridge, MA, United States
56University of Cincinnati, Cincinnati, OH, United States
57University of Maryland, College Park, MD, United States
58Syracuse University, Syracuse, NY, United States
59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de
Janeiro, Brazil, associated to 2
60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11
aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS),
Moscow, Russia
bUniversità di Bari, Bari, Italy
cUniversità di Bologna, Bologna, Italy
dUniversità di Cagliari, Cagliari, Italy
eUniversità di Ferrara, Ferrara, Italy
fUniversità di Firenze, Firenze, Italy
gUniversità di Urbino, Urbino, Italy
hUniversità di Modena e Reggio Emilia, Modena, Italy
iUniversità di Genova, Genova, Italy
jUniversità di Milano Bicocca, Milano, Italy
kUniversità di Roma Tor Vergata, Roma, Italy
lUniversità di Roma La Sapienza, Roma, Italy
mUniversità della Basilicata, Potenza, Italy
nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain
oHanoi University of Science, Hanoi, Viet Nam
pInstitute of Physics and Technology, Moscow, Russia
qUniversità di Padova, Padova, Italy
rUniversità di Pisa, Pisa, Italy
sScuola Normale Superiore, Pisa, Italy
## 1 Introduction
The production of baryon-antibaryon pairs in $B$ meson decays is of
significant experimental and theoretical interest. For example, in the case of
$p\overline{}p$ pair production, the observed decays $B^{0}\rightarrow\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{(*)0}p\overline{}p$ [1, 2],
$B^{+}\rightarrow K^{(*)+}p\overline{}p$ [3, 4, 5, 6, 7], $B^{0}\rightarrow
K^{(*)0}p\overline{}p$ [4, 6] and $B^{+}\rightarrow\pi^{+}p\overline{}p$ [4,
5] all have an enhancement near the $p\overline{}p$ threshold.111Throughout
this paper, the inclusion of charge-conjugate processes is implied. Possible
explanations for this behaviour include the existence of an intermediate state
in the $p\overline{}p$ system [8] and short-range correlations between $p$ and
$\overline{}p$ in their fragmentation [9, 10, 11]. Moreover, for each of these
decays, the branching fraction is approximately $10\,\%$ that of the
corresponding decay with $p\overline{}p$ replaced by $\pi^{+}\pi^{-}$ [12]. In
contrast, the decay $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ has not yet been observed; the most restrictive upper
limit being ${\cal B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)<8.3\times 10^{-7}$ at 90 % confidence level [13],
approximately fifty times lower than the branching fraction for
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$
decays [14]. This result is in tension with the theoretical prediction of
${\cal B}(B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)=(1.2\pm 0.2)\times 10^{-6}$ [15]. Improved experimental
information on the $B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ decay would help to understand the process of dibaryon
production.
In this paper, the results of a search for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p$
and $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ decays are presented. No prediction or experimental limit
exists for the branching fraction ${\cal
B}(B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)$, but it is of interest to measure the suppression
relative to $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ [16]. In addition, a search for the decay
$B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ is performed, for which no published measurement
exists. All branching fractions are measured relative to that of the decay
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$, which is well suited for this purpose due to its
similar topology to the signal decays. Additionally, the lower background
level and its more precisely measured branching fraction make it a more
suitable normalisation channel than the companion $B^{0}$ mode.
## 2 Detector and dataset
The LHCb detector [17] 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 placed downstream. The
combined tracking system provides 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
(IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum
($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov
detectors [18]. 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 [19]. The trigger [20] 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.
The analysis uses a data sample, corresponding to an integrated luminosity of
$1.0\mbox{\,fb}^{-1}$ of $pp$ collision data at a centre-of-mass energy of
$7\mathrm{\,Te\kern-1.00006ptV}$, collected with the LHCb detector during
2011. Samples of simulated events are also used to determine the signal
selection efficiency, to model signal event distributions and to investigate
possible background contributions. In the simulation, $pp$ collisions are
generated using Pythia 6.4 [21] with a specific LHCb configuration [22].
Decays of hadronic particles are described by EvtGen [23], in which final
state radiation is generated using Photos [24]. The interaction of the
generated particles with the detector and its response are implemented using
the Geant4 toolkit [25, *Agostinelli:2002hh] as described in Ref. [27].
## 3 Trigger and selection requirements
The trigger requirements for this analysis exploit the signature of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decay,
and hence are the same for the signal and the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ control channel. At the hardware stage either one or two
identified muon candidates are required. In the case of single muon triggers,
the transverse momentum of the candidate is required to be larger than
$1.5{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. For dimuon candidates a requirement
on the product of the $p_{\rm T}$ of the muon candidates is applied,
$\sqrt{\mbox{$p_{\rm T}$}_{1}\mbox{$p_{\rm
T}$}_{2}}>1.3{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$. In the subsequent software
trigger, at least one of the final state muons is required to have both
$\mbox{$p_{\rm T}$}>1.0{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and ${\rm
IP}>100\,\upmu\rm m$. Finally, the muon tracks are required to form a vertex
that is significantly displaced from the primary vertices (PVs) and to have
invariant mass within $120{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of the
known ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mass,
$m_{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ [12].
The selection uses a multivariate algorithm (hereafter referred to as MVA) to
reject background. A neural network is trained on data using the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ control channel as a proxy for the signal decays.
Preselection criteria are applied in order to obtain a clean sample of the
control channel decays. The muons from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay must be well identified
and have $\mbox{$p_{\rm T}$}>500{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. They
should also form a vertex with $\chi^{2}_{\rm vtx}<12$ and have invariant mass
within the range
$-48<m_{\mu^{+}\mu^{-}}-m_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}}<43{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The separation of the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ vertex from all PVs must be
greater than $3\rm\,mm$. The pion candidates must be inconsistent with the
muon hypothesis, have $\mbox{$p_{\rm
T}$}>200{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$ and have minimum $\chi^{2}_{\rm
IP}$ with respect to any of the PVs greater than 9, where the $\chi^{2}_{\rm
IP}$ is defined as the difference in $\chi^{2}$ of a given PV reconstructed
with and without the considered track. In addition, the scalar sum of their
transverse momenta must be greater than
$600{\mathrm{\,Me\kern-1.00006ptV\\!/}c}$. The $B$ candidate formed from the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ and two oppositely charged
hadron candidates should have $\chi^{2}_{\rm vtx}<20$ and a minimum
$\chi^{2}_{\rm IP}$ with respect to any of the PVs less than 30. In addition,
the cosine of the angle between the $B$ candidate momentum vector and the line
joining the associated PV and the $B$ decay vertex ($B$ pointing angle) should
be greater than 0.99994.
The mass distribution of candidate
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ decays remaining after the preselection is then fitted
in order to obtain signal and background distributions of the variables that
enter the MVA training, using the sPlot technique [28]. The fit model is
described in Sec. 4. The variables that enter the MVA training are chosen to
minimise any difference in the selection between the signal and control
channels. Different selection algorithms are trained for the
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ mode and for the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ mode, with slightly different sets of variables.
The variables in common between the selections are the minimum $\chi^{2}_{\rm
IP}$ of the $B$ candidate; the cosine of the $B$ pointing angle; the
$\chi^{2}$ of the $B$ and ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
candidate vertex fits; the $\chi^{2}$ per degree of freedom of the track fit
of the charged hadrons; and the minimum IP of the muon candidates. For the
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ selection the following additional variables are
included: the $p_{\rm T}$ of the charged hadron and
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ candidates; the $p_{\rm T}$ of
the $B$ candidate; and the flight distance and flight distance significance
squared of the $B$ candidate from its associated PV. For the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ selection only the momentum and $p_{\rm T}$ of the
muon candidates are included as additional variables.
The MVAs are trained using the NeuroBayes package [29]. Two different figures
of merit are considered to find the optimal MVA requirement. The first is that
suggested in Ref. [30]
${\cal{Q}}_{1}=\frac{\epsilon_{\rm MVA}}{{a/2}+\sqrt{B_{\rm MVA}}}\,,$ (1)
where $a=3$ and quantifies the target level of significance, $\epsilon_{\rm
MVA}$ is the efficiency of the selection of the signal candidates, which is
determined from simulated signal samples, and $B_{\rm MVA}$ is the expected
number of background events in the signal region; which is estimated by
performing a fit to the invariant mass distribution of the data sidebands. The
second figure of merit is an estimate of the expected 90 % confidence level
upper limit on the branching fraction in the case that no signal is observed
${\cal{Q}}_{2}=\frac{1.64\,\sigma_{N_{\rm sig}}}{\epsilon_{\rm MVA}}\,,$ (2)
where $\sigma_{N_{\rm sig}}$ is the expected uncertainty on the signal yield,
which is estimated from pseudo-experiments generated with the background-only
hypothesis. The maximum of the first and the minimum of the second figure of
merit are found to occur at very similar values. For the
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$
($B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$) decay, requirements are chosen such that
approximately 50 % (99 %) of the signal is retained while reducing the
background to 20 % (70 %) of its level prior to the cut. The background level
for the $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ decay is very low due to its proximity to
threshold, and only a loose MVA requirement is necessary.
The particle identification (PID) selection for the signal modes is optimised
in a similar way using Eq. (1). It is found that, for the signal channels,
placing a tight requirement on the proton with a higher value for the
logarithm of the likelihood ratio of the proton and pion hypotheses [18] and a
looser requirement on the other proton results in much better performance than
applying the same requirement on both protons. No PID requirements are made on
the pion track in the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ mode.
The acceptance and selection efficiencies are determined from simulated signal
samples, except for those of the PID requirements, which are determined from
data control samples to avoid biases due to known discrepancies between data
and simulation. High-purity control samples of $\mathchar
28931\relax\rightarrow p\pi^{-}$ ($D^{0}\rightarrow K^{-}\pi^{+}$) decays with
no PID selection requirements applied are used to tabulate efficiencies for
protons (pions) as a function of their momentum and $p_{\rm T}$. The
kinematics of the simulated signal events are then used to determine an
average efficiency. Possible variations of the efficiencies over the multibody
phase space are considered. The efficiencies are determined in bins of the
Dalitz plot, $m^{2}_{{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}h^{+}}$ vs.
$m^{2}_{h^{+}h^{-}}$, where $h=\pi,p$; the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay angle (defined as the
angle between the $\mu^{+}$ and the $p\overline{}p$ system in the
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ rest frame); and the angle
between the decay planes of the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
and the $h^{+}h^{-}$ system. The variation with the Dalitz plot variables is
the most significant. For the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ control sample, the distribution of the signal in the
phase space variables is determined using the sPlot technique and these
distributions are used to find a weighted average efficiency.
A number of possible background modes, such as cross-feed from
$B^{0}_{(s)}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}h^{+}h^{\prime-}$ final states (where $h^{(\prime)}=\pi,K$), have been
studied using simulation. None of these are found to give a significant
peaking contribution to the $B$ candidate invariant mass distribution once all
the selection criteria had been applied. Therefore, all backgrounds in the
fits to the mass distributions of
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ candidates are considered as being combinatorial
in nature. For the fits to the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ control channel, some particular backgrounds are taken
into account, as described in the following section.
After all selection requirements are applied, 854 and 404 candidates are found
in the invariant mass ranges
$[5167,5478]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$[5129,5429]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ decays, respectively. The efficiency ratios, with
respect to the $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ normalisation channel, including contributions from
detector acceptance, trigger and selection criteria (but not from PID) are
$0.92\pm 0.16$, $0.85\pm 0.12$ and $0.17\pm 0.04$ for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$, respectively. In addition, the relative PID
efficiencies are found to be $0.78\pm 0.02$, $0.79\pm 0.02$ and $1.00\pm 0.03$
for $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$, respectively. The systematic uncertainties
arising from these values are discussed in Sec. 5.
## 4 Fit model and results
Signal and background event yields are estimated by performing unbinned
extended maximum likelihood fits to the invariant mass distributions of the
$B$ candidates. The signal probability density functions (PDFs) are
parametrised as the sum of two Crystal Ball (CB) functions [31], where the
power law tails are on opposite sides of the peak. This form is appropriate to
describe the asymmetric tails that result from a combination of the effects of
final state radiation and stochastic tracking imperfections. The two CB
functions are constrained to have the same peak position, equal to the value
fitted in the simulation. The resolution parameters are allowed to vary within
a Gaussian constraint, with the central value taken from the simulation and
scaled by the ratio of the values found in the control channel data and
corresponding simulation. The proximity to threshold of the signal decays
provides a mass resolution of 1–3${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$,
whereas for the normalisation channel it is
6–9${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. The tail parameters and the
relative normalisation of the two CB functions are taken from the simulated
distributions and fixed for the fits to data.
A second-order polynomial function is used to describe the combinatorial
background component in the
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ spectrum while an exponential function is used for the
same component in the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ and
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ channels. The parameters of these functions are allowed
to vary in the fits. There are several specific backgrounds that contribute to
the $B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ invariant mass spectrum [14], which need to be
explicitly modelled. In particular, the decay
${B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{+}\pi^{-}}$, where a kaon is misidentified as a pion, is modelled by
an exponential function. The yield of this contribution is allowed to vary in
order to enable a better modelling of the background in the low mass region.
Two additional sources of peaking background are considered: partially
reconstructed decays, such as
${B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}(\rho\gamma)}$; and decays where an additional low momentum
pion is included from the rest of the event, such as
${B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}}$. Both
distributions are fitted with a non-parametric kernel estimation, with shapes
fixed from simulation. The yields of these components are also fixed to values
estimated from the known branching fractions and selection efficiencies
evaluated from simulation.
In order to validate the stability of the fit, a series of pseudo-experiments
have been generated using the PDFs described above. The experiments are
conducted for a wide range of generated signal yields. No significant bias is
observed in any of the simulation ensembles; any residual bias being accounted
for as a source of systematic uncertainty.
The fits to the data are shown in Figs. 1 and 2. The signal yields are
$N(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)=5.9\,^{\,+5.9}_{\,-5.1}\pm 2.5$,
$N(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)=21.3\,^{\,+8.6}_{\,-7.8}\pm 2.6$ and
$N(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+})=0.7\,^{\,+3.2}_{\,-2.5}\pm 0.7$, where the first
uncertainties are statistical and the second are systematic and are described
in the next section. The numbers of events in the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ normalisation channel are found to be $2120\pm 50$ and
$4021\pm 76$ (statistical uncertainties only) when applying the selection
requirements for the
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ measurements, respectively.
The statistical significances of the signal yields are computed from the
change in the fit likelihood when omitting the corresponding component,
according to $\sqrt{2\ln(L_{\rm sig}/L_{0})}$, where $L_{\rm sig}$ and $L_{0}$
are the likelihoods from the nominal fit and from the fit omitting the signal
component, respectively. The statistical significances are found to be
$1.2\,\sigma$, $3.0\,\sigma$ and $0.2\,\sigma$ for the decays
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$, respectively. The statistical likelihood curve is
convolved with a Gaussian function of width given by the systematic
uncertainty. The resulting negative log likelihood profiles are shown in Fig.
3. The total significances of each signal are found to be $1.0\,\sigma$,
$2.8\,\sigma$ and $0.2\,\sigma$ for the modes
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$, respectively.
Figure 1: Invariant mass distribution of (a)
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and (b)
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ candidates after the full selection. Each
component of the fit model is displayed on the plot: the signal PDFs are
represented by the dot-dashed violet and dashed green line; the combinatorial
background by the dotted red line; and the overall fit is given by the solid
blue line. The fit pulls are also shown, with the red lines corresponding to
$2\,\sigma$. The $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ yield is multiplied by five in order to make the
signal position visible.
Figure 2: Invariant mass distribution of
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ candidates after the full selection for the (a)
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and (c)
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ searches. The corresponding logarithmic plots are
shown in (b) and (d). Each component of the fit is represented on the plot:
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ signal (green dashed),
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ signal (violet dot-dashed),
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}\pi^{-}$
background (black falling hashed),
${B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\eta^{\prime}}$ background (cyan rising hashed), and combinatorial
background (red dotted). The overall fit is represented by the solid blue
line.
Figure 3: Negative log-likelihood profiles for the (a)
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$, (b)
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$, and (c)
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ signal yields. The red dashed line corresponds to
the statistical-only profile while the blue line includes all the systematic
uncertainties.
## 5 Systematic uncertainties
Many potential sources of systematic uncertainty are reduced by the choice of
the normalisation channel. Nonetheless, some factors remain that could still
affect the measurements of the branching fractions. The sources and their
values are summarised in Table 1.
Precise knowledge of the selection efficiencies for the modes is limited both
by the simulation sample size and by the variation of the efficiency over the
multi-body phase space, combined with the unknown distribution of the signal
over the phase space. The simulation sample size contributes an uncertainty of
approximately $1\,\%$ in each of the channels, and the effect of efficiency
variation across the phase space, determined from the spread of values
obtained in bins of the relevant variables, is evaluated to be $17\,\%$,
$14\,\%$ and $23\,\%$ for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ decays, respectively. The large systematic
uncertainties reflect the unknown distribution of signal events across the
phase space. In contrast, the uncertainty for the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ normalisation channel is estimated by varying the
binning scheme in the phase space variables and is found to be only $1\%$ for
both the $B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ MVA selections. Possible biases due to training
the MVA using the control channel were investigated and found to be
negligible.
The proton PID efficiency is measured using a high-purity data sample of
${\mathchar 28931\relax\\!\rightarrow p\pi^{-}}$ decays. By repeating the
method with a simulated control sample, and considering the difference with
the simulated signal sample, the associated systematic uncertainties are found
to be $3\,\%$, $3\,\%$ and $2\,\%$ for the modes
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$, respectively. Furthermore, the limited sample
sizes give an additional $1\,\%$ uncertainty. In the
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ channel there is an additional source of
uncertainty due to the different reconstruction efficiencies for the extra
pion track in data and simulation, which is determined to be less than
$2\,\%$.
The effect of approximations made in the fit model is investigated by
considering alternative functional forms for the various signal and background
PDFs. The nominal signal shapes are replaced with a bifurcated Gaussian
function with asymmetric exponential tails. The background is modelled with an
exponential function for
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ decays, whereas a second-order polynomial function is
used for $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ and the normalisation channel. Combined in
quadrature, these sources change the fitted yields by $2.5$, $2.6$ and $0.7$
events, which correspond to $42\,\%$, $12\,\%$ and $92\,\%$ for the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ modes, respectively. The bias on the determination
of the fitted yield is studied with pseudo-experiments. No significant bias is
found, and the associated systematic uncertainty is $0.2$, $0.3$ and $0.2$
events ($4\,\%$, $1\,\%$ and $26\,\%$) for the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ modes, respectively.
Since a $B^{0}_{s}$ meson decay is used for the normalisation, the results for
${\cal B}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)$ and ${\cal
B}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+})$ rely on the knowledge of the ratio of the
fragmentation fractions, measured to be $f_{s}/f_{d}=0.256\pm 0.020$ [32],
introducing a relative uncertainty of $8\,\%$. It is assumed that
$f_{u}=f_{d}$. The uncertainty on the measurement of the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}$ branching fraction includes a contribution from this
source. Hence, to avoid double counting, it is omitted when evaluating the
systematic uncertainties on the absolute branching fractions.
A series of cross-checks are performed to test the stability of the fit
result. The PID and MVA requirements are tightened and loosened. The fit range
is restricted to $[5229,5416]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and
$[5129,5379]{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ for
$B^{0}_{(s)}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ decays, respectively. No significant change in the
results is observed in any of the cross-checks.
Table 1: Systematic uncertainties on the branching fraction ratios of the decays $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p$, $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p$ and $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p\pi^{+}$ measured relative to $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\pi^{+}\pi^{-}$. The total is obtained from the sum in quadrature of all contributions. Source | Uncertainty on the branching fraction ratio (%)
---|---
| $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p$ | $B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p$ | $B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p\pi^{+}$
Event selection | $\phantom{1}{1}$ | $\phantom{1}{1}$ | $\phantom{1}{1}$
Efficiency variation | $17$ | $14$ | $23$
PID simulation sample size | $\phantom{1}{1}$ | $\phantom{1}{1}$ | $\phantom{1}{1}$
PID calibration method | $\phantom{1}{3}$ | $\phantom{1}{3}$ | $\phantom{1}{2}$
Tracking efficiency | — | — | $\phantom{1}{2}$
Fit model | $42$ | $12$ | $92$
Fit bias | $\phantom{1}{4}$ | $\phantom{1}{1}$ | $26$
Fragmentation fractions | $\phantom{1}{8}$ | — | $\phantom{1}{8}$
Total | $46$ | $19$ | $98$
## 6 Results and conclusions
The relative branching fractions are determined according to
$\frac{{\cal{B}}({B_{q}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p(\pi^{+})})}{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}=\frac{\epsilon^{\rm
sel}_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}}}{\epsilon^{\rm
sel}_{{B_{q}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p(\pi^{+})}}}\times\phantom{=}\frac{\epsilon^{\rm
PID}_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}}}{\epsilon^{\rm
PID}_{{B_{q}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p(\pi^{+})}}}\times\frac{N_{{B_{q}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p(\pi^{+})}}}{N_{B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-}}}\times\frac{f_{s}}{f_{{q}}},$ (3)
where $\epsilon^{\rm sel}$ is the selection efficiency, $\epsilon^{\rm PID}$
is the particle identification efficiency, and $N$ is the signal yield. The
results obtained are
$\displaystyle\frac{{\cal{B}}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)}{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}$ $\displaystyle\,=\hskip 5.69046pt$ $\displaystyle(1.0$
$\,{}^{+1.0}_{-0.9}\,$ $\displaystyle\pm\,$ $\displaystyle 0.5)\times
10^{-3}\,,$
$\displaystyle\frac{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)}{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}$ $\displaystyle\,=\hskip 5.69046pt$ $\displaystyle(1.5$
$\,{}^{+0.6}_{-0.5}\,$ $\displaystyle\pm\,$ $\displaystyle 0.3)\times
10^{-2}\,,$
$\displaystyle\frac{{\cal{B}}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+})}{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}$ $\displaystyle\,=\hskip 5.69046pt$
$\displaystyle(0.27$ $\,{}^{+1.23}_{-0.95}\,$ $\displaystyle\pm\,$
$\displaystyle 0.26)\times 10^{-3}\,,$
where the first uncertainty is statistical and the second is systematic. The
absolute branching fractions are calculated using the measured branching
fraction of the normalisation channel ${\cal
B}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})=(1.98\pm 0.20)\times 10^{-4}$ [16]
$\displaystyle{\cal{B}}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)$ $\displaystyle\,=\hskip 5.69046pt$ $\displaystyle(2.0$
$\,{}^{+1.9}_{-1.7}\mathrm{\,[stat]}$ $\displaystyle\pm\,$ $\displaystyle
0.9\mathrm{\,[syst]}$ $\displaystyle\pm\,$ $\displaystyle
0.1\,[\rm{norm}])\times 10^{-7},$
$\displaystyle{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)$ $\displaystyle\,=\hskip 5.69046pt$ $\displaystyle(3.0$
$\,{}^{+1.2}_{-1.1}\mathrm{\,[stat]}$ $\displaystyle\pm\,$ $\displaystyle
0.6\mathrm{\,[syst]}$ $\displaystyle\pm\,$ $\displaystyle
0.3\,[\rm{norm}])\times 10^{-6},$
$\displaystyle{\cal{B}}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+})$ $\displaystyle\,=\hskip 5.69046pt$
$\displaystyle(0.54$ $\,{}^{+2.43}_{-1.89}\mathrm{\,[stat]}$
$\displaystyle\pm\,$ $\displaystyle 0.52\mathrm{\,[syst]}$
$\displaystyle\pm\,$ $\displaystyle 0.03\,[\rm{norm}])\times 10^{-7},$
where the third uncertainty originates from the control channel branching
fraction measurement. The dominant uncertainties are statistical, while the
most significant systematic come from the fit model and from the variation of
the efficiency over the phase space.
Since the significances of the signals are below $3\,\sigma$, upper limits at
both $90\,\%$ and $95\,\%$ confidence levels (CL) are determined using a
Bayesian approach, with a prior that is uniform in the region with positive
branching fraction. Integrating the likelihood (including all systematic
uncertainties), the upper limits are found to be
$\displaystyle\frac{{\cal{B}}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)}{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}$ $\displaystyle<$ $\displaystyle 2.6~{}(3.0)\times
10^{-3}~{}~{}~{}{\rm at}~{}~{}90\,\%\ (95\,\%)~{}{\rm CL}\,,$
$\displaystyle\frac{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)}{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}$ $\displaystyle<$ $\displaystyle 2.4~{}(2.7)\times
10^{-2}~{}~{}~{}{\rm at}~{}~{}90\,\%\ (95\,\%)~{}{\rm CL}\,,$
$\displaystyle\frac{{\cal{B}}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+})}{{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}\pi^{+}\pi^{-})}$ $\displaystyle<$ $\displaystyle 2.5~{}(3.1)\times
10^{-3}~{}~{}~{}{\rm at}~{}~{}90\,\%\ (95\,\%)~{}{\rm CL}\,,$
and the absolute limits are
$\displaystyle{\cal{B}}(B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)$ $\displaystyle<$ $\displaystyle 5.2~{}(6.0)\times
10^{-7}~{}~{}~{}{\rm at}~{}~{}90\,\%\ (95\,\%)~{}{\rm CL}\,,$
$\displaystyle{\cal{B}}(B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p)$ $\displaystyle<$ $\displaystyle 4.8~{}(5.3)\times
10^{-6}~{}~{}~{}{\rm at}~{}~{}90\,\%\ (95\,\%)~{}{\rm CL}\,,$
$\displaystyle{\cal{B}}(B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+})$ $\displaystyle<$ $\displaystyle 5.0~{}(6.1)\times
10^{-7}~{}~{}~{}{\rm at}~{}~{}90\,\%\ (95\,\%)~{}{\rm CL}\,.$
In summary, using the data sample collected in 2011 by the LHCb experiment
corresponding to an integrated luminosity of $1.0\mbox{\,fb}^{-1}$ of $pp$
collisions at $\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, searches for the
decay modes $B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$,
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ are performed. No significant signals are seen,
and upper limits on the branching fractions are set. A significant improvement
in the existing limit for
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p$
decays is achieved and first limits on the branching fractions of
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ and
$B^{+}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p\pi^{+}$ decays are established. The limit on the
$B^{0}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}p\overline{}p$
branching fraction is in tension with the theoretical prediction [15]. The
significance of the
$B^{0}_{s}\\!\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}p\overline{}p$ signal is $2.8\,\sigma$, which motivates new theoretical
calculations of this process as well as improved experimental searches using
larger datasets.
## 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); ANCS/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov
Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER
(Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We
also acknowledge the support received from 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 thankful for the computing resources put at our disposal by
Yandex LLC (Russia), as well as to the communities behind the multiple open
source software packages that we depend on.
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|
arxiv-papers
| 2013-06-19T11:07:12 |
2024-09-04T02:49:46.676865
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "LHCb collaboration: R. Aaij, B. Adeva, M. Adinolfi, C. Adrover, A.\n Affolder, Z. Ajaltouni, J. Albrecht, F. Alessio, M. Alexander, S. Ali, G.\n Alkhazov, P. Alvarez Cartelle, A.A. Alves Jr, S. Amato, S. Amerio, Y. Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, J.E. Andrews, R.B. Appleby, O.\n Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E. Aslanides, G.\n Auriemma, M. Baalouch, S. Bachmann, J.J. Back, C. Baesso, V. Balagura, W.\n Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W. Barter, Th. Bauer, A. Bay,\n J. Beddow, F. Bedeschi, I. Bediaga, S. Belogurov, K. Belous, I. Belyaev, E.\n Ben-Haim, G. Bencivenni, S. Benson, J. Benton, A. Berezhnoy, R. Bernet, M.-O.\n Bettler, M. van Beuzekom, A. Bien, S. Bifani, T. Bird, A. Bizzeti, P.M.\n Bj{\\o}rnstad, T. Blake, F. Blanc, J. Blouw, S. Blusk, V. Bocci, A. Bondar, N.\n Bondar, W. Bonivento, S. Borghi, A. Borgia, T.J.V. Bowcock, E. Bowen, C.\n Bozzi, T. Brambach, J. van den Brand, J. Bressieux, D. Brett, M. Britsch, T.\n Britton, N.H. Brook, H. Brown, I. Burducea, A. Bursche, G. Busetto, J.\n Buytaert, S. Cadeddu, O. Callot, M. Calvi, M. Calvo Gomez, A. Camboni, P.\n Campana, D. Campora Perez, A. Carbone, G. Carboni, R. Cardinale, A. Cardini,\n H. Carranza-Mejia, L. Carson, K. Carvalho Akiba, G. Casse, L. Castillo\n Garcia, M. Cattaneo, Ch. Cauet, R. Cenci, M. Charles, Ph. Charpentier, P.\n Chen, N. Chiapolini, M. Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L.\n Clarke, 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, B. Couturier, G.A. Cowan, D.C. Craik, S. Cunliffe, R.\n Currie, C. D'Ambrosio, P. David, P.N.Y. David, A. Davis, I. De Bonis, K. De\n Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva, P.\n De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, F. Di Ruscio, H. Dijkstra, M.\n Dogaru, S. Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella,\n C. F\\\"arber, G. Fardell, C. Farinelli, S. Farry, V. Fave, D. Ferguson, V.\n Fernandez Albor, F. Ferreira Rodrigues, M. Ferro-Luzzi, S. Filippov, M.\n Fiore, C. Fitzpatrick, M. Fontana, F. Fontanelli, R. Forty, O. Francisco, M.\n Frank, C. Frei, M. Frosini, S. Furcas, E. Furfaro, A. Gallas Torreira, D.\n Galli, M. Gandelman, P. Gandini, Y. Gao, J. Garofoli, P. Garosi, J. Garra\n Tico, L. Garrido, C. Gaspar, R. Gauld, E. Gersabeck, M. Gersabeck, T.\n Gershon, Ph. Ghez, V. Gibson, L. Giubega, V.V. Gligorov, C. G\\\"obel, D.\n Golubkov, A. Golutvin, A. Gomes, H. Gordon, M. Grabalosa G\\'andara, R.\n Graciani Diaz, L.A. Granado Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E.\n Greening, S. Gregson, P. Griffith, 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, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, A. Hicheur, E. Hicks, D. Hill, M.\n Hoballah, M. Holtrop, C. Hombach, P. Hopchev, W. Hulsbergen, P. Hunt, T.\n Huse, N. Hussain, D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten,\n R. Jacobsson, A. Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D.\n Johnson, C.R. Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M.\n Karacson, T.M. Karbach, I.R. Kenyon, T. Ketel, A. Keune, B. Khanji, O.\n Kochebina, 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, T. Kvaratskheliya, V.N. La Thi, D.\n Lacarrere, G. Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G.\n Lanfranchi, C. Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van\n Leerdam, J.-P. Lees, R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O.\n Leroy, T. Lesiak, B. Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C.\n Linn, B. Liu, G. Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H.\n Lu, D. Lucchesi, J. Luisier, H. Luo, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi,\n P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A.\n Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, B. Mc\n Skelly, J. McCarthy, A. McNab, R. McNulty, B. Meadows, F. Meier, M. Meissner,\n M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S. Monteil, D.\n Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I. Mous, F.\n Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T. Nakada, R.\n Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D. Nguyen, T.D.\n Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin, T. Nikodem,\n A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, M. Orlandea, J.M. Otalora Goicochea, P.\n Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman, A.\n Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson, G. Passaleva, G.D.\n Patel, M. Patel, G.N. Patrick, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini, E. Perez\n Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, G. Pessina, K. Petridis, A. Petrolini, A. Phan, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch,\n A. Puig Navarro, G. Punzi, W. Qian, J.H. Rademacker, B. Rakotomiaramanana,\n M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S. Redford, M.M. Reid, A.C.\n dos Reis, S. Ricciardi, A. Richards, K. Rinnert, V. Rives Molina, D.A. Roa\n Romero, P. Robbe, D.A. Roberts, E. Rodrigues, P. Rodriguez Perez, S. Roiser,\n V. Romanovsky, A. Romero Vidal, 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. Salzmann, B. Sanmartin Sedes, M. Sannino,\n R. Santacesaria, C. Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C.\n Satriano, A. Satta, M. Savrie, D. Savrina, P. Schaack, M. Schiller, H.\n Schindler, M. Schlupp, M. Schmelling, B. Schmidt, O. Schneider, A. Schopper,\n M.-H. Schune, R. Schwemmer, B. Sciascia, A. Sciubba, M. Seco, A. Semennikov,\n I. Sepp, N. 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"submitter": "Rafael Coutinho",
"url": "https://arxiv.org/abs/1306.4489"
}
|
1306.4540
|
# M dwarf catalog of LAMOST pilot survey
Z.P. Yi 11affiliation: National Astronomical Observatories(NAOC), Chinese
Academy of Sciences, Beijing 100012, China. 22affiliation: Shandong University
at Weihai, 180 Wenhuaxi Road, Weihai, 264209, China. 33affiliation: Key
Laboratory of Optical Astronomy, NAOC, Chinese Academy of Sciences, Beijing
100012, China. 44affiliation: University of Chinese Academy of Sciences, 19A
Yuquan Road, Shijingshan District, Beijing, 100049, China , A.L. Luo
11affiliation: National Astronomical Observatories(NAOC), Chinese Academy of
Sciences, Beijing 100012, China. 33affiliation: Key Laboratory of Optical
Astronomy, NAOC, Chinese Academy of Sciences, Beijing 100012, China.
88affiliation: Correspondence Author , Y.H. Song 11affiliation: National
Astronomical Observatories(NAOC), Chinese Academy of Sciences, Beijing 100012,
China. 33affiliation: Key Laboratory of Optical Astronomy, NAOC, Chinese
Academy of Sciences, Beijing 100012, China. , J.K. Zhao11affiliation: National
Astronomical Observatories(NAOC), Chinese Academy of Sciences, Beijing 100012,
China. 33affiliation: Key Laboratory of Optical Astronomy, NAOC, Chinese
Academy of Sciences, Beijing 100012, China. , Z.X. Shi 11affiliation: National
Astronomical Observatories(NAOC), Chinese Academy of Sciences, Beijing 100012,
China. 44affiliation: University of Chinese Academy of Sciences, 19A Yuquan
Road, Shijingshan District, Beijing, 100049, China , P. Wei11affiliation:
National Astronomical Observatories(NAOC), Chinese Academy of Sciences,
Beijing 100012, China. 44affiliation: University of Chinese Academy of
Sciences, 19A Yuquan Road, Shijingshan District, Beijing, 100049, China , J.J.
Ren 11affiliation: National Astronomical Observatories(NAOC), Chinese Academy
of Sciences, Beijing 100012, China. 44affiliation: University of Chinese
Academy of Sciences, 19A Yuquan Road, Shijingshan District, Beijing, 100049,
China , F.F. Wang11affiliation: National Astronomical Observatories(NAOC),
Chinese Academy of Sciences, Beijing 100012, China. 33affiliation: Key
Laboratory of Optical Astronomy, NAOC, Chinese Academy of Sciences, Beijing
100012, China. , X. Kong 11affiliation: National Astronomical
Observatories(NAOC), Chinese Academy of Sciences, Beijing 100012, China.
33affiliation: Key Laboratory of Optical Astronomy, NAOC, Chinese Academy of
Sciences, Beijing 100012, China. ,Y.B. Li 11affiliation: National Astronomical
Observatories(NAOC), Chinese Academy of Sciences, Beijing 100012, China.
33affiliation: Key Laboratory of Optical Astronomy, NAOC, Chinese Academy of
Sciences, Beijing 100012, China. , P. Du,11affiliation: National Astronomical
Observatories(NAOC), Chinese Academy of Sciences, Beijing 100012, China.
22affiliation: Shandong University at Weihai, 180 Wenhuaxi Road, Weihai,
264209, China. 33affiliation: Key Laboratory of Optical Astronomy, NAOC,
Chinese Academy of Sciences, Beijing 100012, China. , W. Hou 11affiliation:
National Astronomical Observatories(NAOC), Chinese Academy of Sciences,
Beijing 100012, China. 44affiliation: University of Chinese Academy of
Sciences, 19A Yuquan Road, Shijingshan District, Beijing, 100049, China , Y.X.
Guo 11affiliation: National Astronomical Observatories(NAOC), Chinese Academy
of Sciences, Beijing 100012, China. 33affiliation: Key Laboratory of Optical
Astronomy, NAOC, Chinese Academy of Sciences, Beijing 100012, China. , S.
Zhang 11affiliation: National Astronomical Observatories(NAOC), Chinese
Academy of Sciences, Beijing 100012, China. 44affiliation: University of
Chinese Academy of Sciences, 19A Yuquan Road, Shijingshan District, Beijing,
100049, China , Y.H. Zhao11affiliation: National Astronomical
Observatories(NAOC), Chinese Academy of Sciences, Beijing 100012, China.
33affiliation: Key Laboratory of Optical Astronomy, NAOC, Chinese Academy of
Sciences, Beijing 100012, China. , S.W. Sun 11affiliation: National
Astronomical Observatories(NAOC), Chinese Academy of Sciences, Beijing 100012,
China. 33affiliation: Key Laboratory of Optical Astronomy, NAOC, Chinese
Academy of Sciences, Beijing 100012, China. National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100012, China
[email protected], [email protected] J.C. Pan22affiliation: Shandong University at
Weihai, 180 Wenhuaxi Road, Weihai, 264209, China. Shandong University at
Weihai, 180 Wenhuaxi Road, Weihai, 264209, China L.Y. Zhang55affiliation:
College of Science/Department of Physics, Guizhou University, Guiyang 550025,
China College of ScienceDepartment of Physics, Guizhou University, Guiyang,
550025, China; A. A. West66affiliation: Department of Astronomy, Boston
University, 725 Commonwealth Ave, Boston, MA 02215, USA Department of
Astronomy, Boston University, 725 Commonwealth Ave, Boston,MA 02215, USA H.B.
Yuan77affiliation: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing, 100871 China Kavli Institute for Astronomy and
Astrophysics, Peking University, Beijing, 100871 China
###### Abstract
We present a spectroscopic catalog of 67082 M dwarfs from the LAMOST pilot
survey. For each spectrum of the catalog, spectral subtype, radial velocity,
equivalent width of H${\alpha}$, a number of prominent molecular band indices
and the metal sensitive parameter $\zeta$ are provided . Spectral subtype have
been estimated by a remedied Hammer program (Original Hammer: Covey et al.
2007), in which indices are reselected to obtain more accurate auto-classified
spectral subtypes. All spectra in this catalog have been visually inspected to
confirm the spectral subtypes. Radial velocities have been well measured by
our developed program which uses cross-correlation method and estimates
uncertainty of radial velocity as well. We also examine the magnetic activity
properties of M dwarfs traced by H${\alpha}$ emission line. The molecular band
indices included in this catalog are temperature or metallicity sensitive and
can be used for future analysis of the physical properties of M dwarfs. The
catalog is available on the website http://sciwiki.lamost.org/MCatalogPilot/.
star: late type — method: analysis — data: catalog
## 1 Introduction
M dwarfs are the most common stars in the Galaxy (Bochanski et al. 2010).
Their main sequence lifetime is even longer than the present age of the
universe (Laughlin et al. 1997). Therefore, M dwarfs can be used to trace the
structure and the evolution of the Milky Way. Besides investigating the
properties of the Galaxy, M dwarfs are important for identifying potentially
habitable extrasolar planets (Charbonneau et al. 2009). Many previous studies
have been completed using catalog of M dwarfs, for instance, tracing the
Galactic disk kinematics (Hawley et al. 1996; Gizis et al. 2002; Lépine et al.
2003; Bochanski et al. 2005, 2007a, 2010), studying the structure of the
Galaxy (Reid et al.1997; Kerber et al. 2001; Woolf et al. 2012) and computing
the stellar initial mass function (Covey et al. 2008b; Bochanski et al. 2010).
To research these scientific topics, some fundamental and preliminary analysis
need to be performed in advance, including spectral type classification
(Kirkpatrick et al. 1991; Reid et al.1995; Kirkpatrick et al. 1999; Martin et
al. 1999; Cruz et al. 2002), radial velocities measurement (Bochanski et al.
2007b), metallicity estimation (Gizis 1997; Lépine et al. 2003; Woolf et al.
2006; Lépine et al. 2007) and an analysis of magnetic activity (Reid et al.
1995; Hawley et al. 1996; Gizis et al. 2000; West et al. 2004; west et al.
2011).
Because of the difficulty of obtaining spectra from these faint objects,
studies on M dwarfs of a decade ago were limited by the number of M dwarf
spectra (e.g. Delfosse et al. 1998, 1999, with 118 M stars). However, with the
development of modern astronomical facilities, the number of M dwarf spectra
increases dramatically. Reid et al. (1995) obtained a spectroscopy catalog of
1746 stars, containing primarily M dwarf spectra and a small number of $A\sim
K$ spectra. SDSS later sharply expanded the number of M dwarfs spectroscopic
sample. West et al. (2008) presented a spectroscopic catalog of more than
44,000 M dwarfs from the Sloan Digital Sky Survey (SDSS) Data Release 5 and
then as a part of the SEGUE (the Sloan Extension for Galactic Understanding
and Exploration) survey (Yanny et al. 2009), over 50,000 additional M dwarf
candidates provided a new sight for probing the structure, kinematics, and
evolution of the Milky Way. West et al. (2011) presented the latest
spectroscopic catalog including 70841 M dwarfs spectra from SDSS Data Release
7 (Abazajian, K. et al. 2009), providing fundamental parameters of M dwarfs
for future use of the sample to probe galactic chemical evolution.
The Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST, also
called Guo Shou Jing Telescope) is a National Major Scientific Project
undertaken by the Chinese Academy of Science (Wang et al. 1996, Cui et al.
2012). The survey contains two main parts: the LAMOST ExtraGAlactic Survey
(LEGAS), and the LAMOST Experiment for Galactic Understanding and Exploration
(LEGUE) survey of Milky Way stellar structure. LAMOST has a field of view as
large as 20 square degrees, and at the same time a large effective aperture
that varies from 3.6 to 4.9 meters in diameter. A limiting magnitude that
LAMOST can reach is as faint as r = 19 at resolution R = 1800. This telescope
therefore has a great potential to efficiently survey a large volume of space
for stars and galaxies. From October 2011 to May 2012, LAMOST has completed
the pilot survey and released 319000 spectra (Luo et al. 2012). From the fall
of 2012, LAMOST has begun the general survey. Within 4-5 years, LAMOST will
observe at least 2.5 million stars in a contiguous area in the Galactic halo,
and more than 7.5 million stars in the low galactic latitude areas around the
plane. The spectra collected for such a huge sample of stars will provide a
legacy that allow us to learn detailed information on stellar kinematics,
chemical compositions well beyond SDSS/SEGUE. There is a detailed description
of LAMOST spectra survey in Zhao, et al (2012) and the survey science plan of
LEGUE in Deng, et al (2012).
Among the spectra that LAMOST pilot survey obtained, M type spectra account
for near ten percent of all stellar spectra (LAMOST 1D pipeline recognized as
stellar spectra, regardless of S/N). In terms of this proportion, the total
number of M dwarfs spectra in the entire survey will be near one million. Such
a big sample will enable a number of research topics in exploring the
evolution and the structure of the Milky Way. Because countless molecules
bands leads to peculiar morphology of M-type stellar spectra, during the
process of LAMOST data analysis, M type spectra have to be treated separately
to derive fundamental parameters, while other type stellar spectra are input
to the ULySS program (Wu et al. 2011) for parameters of temperature, gravity
and metal abundance. Our attention is to drive accurate fundamental parameters
from M dwarfs spectra obtained in LAMOST pilot survey. This will lay the
foundation for further research on the Galactic structure and kinematics.
In this paper we describe the M dwarfs spectra from LAMOST pilot survey and
the methods adopted to derive the fundamental parameters including spectral
subtype, radial velocity, equivalent width of H${\alpha}$, a number of
prominent molecular band indices, the metal sensitive parameter $\zeta$ , and
their uncertainties. In section 2 we describe observational spectra from
LAMOST and the spectral quality of M dwarfs. In section 3 we discuss how to
determine the spectral types for M dwarfs. The radial velocity and its
uncertainty of M dwarfs are discussed in section 4. Magnetic activity,
molecular band indices and metal sensitive parameter are discussed
respectively in section 5. We summarize the results of this paper in section
6.
## 2 LAMOST pilot data and Observations
Now the resolution of LAMOST spectra is R=1500 over a wavelength range of
$3700\AA\sim 9000\AA$. Two arms of each spectrograph cover the entire
wavelength range with 200 $\AA$ of overlap. The spectral coverage of blue is
$3700\AA\sim 5900\AA$ while that of red is $5700\AA\sim 9000\AA$. The raw data
have been reduced with LAMOST 2D and 1D pipeline (Luo et al. 2004), including
bias subtraction, cosmic-ray removal, spectral trace and extraction, flat-
fielding, wavelength calibration, and sky subtraction. The red side has higher
throughput than the blue. Therefore, it is easy to obtain high quality spectra
of M dwarfs than other types of stars because most of the light of M dwarfs is
in the red band. Although the latest version of LAMOST pipeline can accurately
process the spectra in steps of wavelength calibration and flat-fielding,
there is still some uncertainty in the continuum level because of a lack of a
high-precision companion photometric survey telescope to aid in absolute flux
calibration. In LAMOST sample, most high quality M dwarf spectra matched well
with the corresponding templates, but a small fraction of spectra showed
different slope than they should, which may caused by interstellar reddening.
We need to consider about these factors which could impact the accuracy of our
results. To estimate the effect of continuum uncertainties, we compared 348
objects that both LAMOST and SDSS have their spectra. Generally, the S/N of
the spectra of these objects in LAMOST are lower than in SDSS. Most of the 348
samples show flux difference smaller than 20%. Figure 1 shows an example of
the comparison. The blue band is from SDSS and the red one is from LAMOST,
both are from observing the same object. The two spectra morphologically agree
well and the flux difference of two spectra is not more than 10%.
To separate M dwarfs from other stars, we selected the spectra that were
classified as M or K7 by the LAMOST 1D pipeline. LAMOST 1D pipeline carries
out chi-squared fits of the observed spectra to the templates. The templates
were constructed by linear combinations of eigen spectra (from decomposition
of a set of SDSS spectra) and low-order polynomials (Luo et al. 2012). Most
M-type stellar spectra were correctly recognized by the 1D pipeline, but near
one fifth M-type stellar spectra (mainly early-type M) were misclassified as
K7 dwarfs. Therefore, all K7 type spectra need to be inspected to search for
M-type spectra. The total number of the candidates is 98,887. After visually
inspection using the Hammer spectral typing facility (Covey et al. 2007), we
excluded K7 type spectra, double stars and poor quality spectra. We ignored
giants contamination because giant contamination rate is very low (Covey et
al. 2008b). Our final LAMOST catalog contains 67,082 M dwarfs.
It is important to note that not all spectra of this M dwarf sample are in the
released stellar spectra from LAMOST pilot survey. The released stellar
spectra need meet S/N larger than 10 (in the g and r bands). But for many
M-type stellar spectra, even with a lower S/N (in r band), they can be easily
identified and assigned with an appropriate spectral type. Thus, we collected
all M-type spectra into this M dwarf sample irrespective of S/N.
## 3 SPECTRAL TYPES
Spectral subtype of an M dwarf is one of the most important fundamental
parameters, which relates to the temperature and the mass of the M dwarf.
There are two primary approaches to classify a M dwarf according its spectrum.
The one uses the overall slope of the spectrum, which requires accurate
spectrophotometric calibration over the full optical wavelength. The other
matches the relative strength of atomic and molecular features in the spectrum
which has been normalized by dividing by an estimated continuum. Considering
the flux uncertainty of LAMOST M dwarf spectra, we choose the second method to
derive the spectral types of the LAMOST M dwarfs.
The Hammer (Covey et al. 2007) is an IDL-based code that uses the relative
strength of features. It has been widely used for classifying stellar spectra
(Lee et al. 2008a; West et al. 2011; Woolf et al. 2012; Dhital et al. 2012),
especially for M dwarf spectra. For late-type stars classification, the Hammer
computes 16 molecular band head indices of each spectrum, including indices of
CaI, MgI, CaH3, TiO5, VO, NaI, Cs, CrH, CaII, etc. The wavelength of the
indices covers from $4000\AA$ to $9100\AA$. The Hammer matches these indices
with the indices computed from templates, and the spectral type of the closest
template is selected as the spectral type of the observed spectra. However, as
described in the previous findings, the automatic Hammer tends to classify
some later-than-M5 spectra as an earlier subtype (West et al. 2011).
To try to remedy the Hammer of the late M classification problem, we first
tested the Hammer code with known type M dwarf spectra. Bochanski et al
(2007b, here after Bochanski2007b) derived low-mass M0$\sim$L0 template
spectra that were computed from over 4000 SDSS spectra, and used them for
medium-resolution radial velocity standards. We adopted Bochanski2007b
template to test the Hammer automatic spectral typing. We used the Hammer
automatically classify the Bochanski2007b templates and the results are shown
in the second column of Table 2. In contrast of the first column of the table,
M1, M5, M6, M7 and M9 template are allocated with an earlier subtype.
We inspected each mismatch case and found that the spectral region between
6000Å and 7000Å and the one beyond 8000Å are not well matched between the
Bochanski2007b template and the Hammer template. It means that the original
Hammer indices are not adequate to discriminate all of the subtypes. We run an
ensemble learning method Random Forest (Breiman, 2001) in search of the most
important features for classifying M dwarf spectra.
Random forest is an ensemble classifier, which consists of many decision trees
and aggregates their results. The method injects randomness to guarantee trees
in the forest are different. This somewhat counterintuitive strategy turns out
to perform very well comparing to many other classifiers, including
discriminant analysis, support vector and neural networks (Liaw et al. 2002).
Random forests have become increasingly popular in many scientific fields
(C.Strobl et al. 2008). And variable importance measures of random forest have
been receiving increased attention as a method of features selection in many
classification tasks in bioinformatics and related scientific fields (Díaz-
Uriarte R. et al.2006, C.Strobl et al. 2007 ).
Before we classified Bochanski2007b M dwarf template using random forest, we
divided each spectrum from 6000Å to 9000Å into 600 regions, each region covers
5Å. The adopted five pseudo continuum regions are: 6130Å$\sim$ 6134Å ,
6545Å$\sim$ 6549Å , 7042Å$\sim$ 7046Å , 7560Å$\sim$ 7564Å , 8125Å$\sim$ 8129Å
. Mean flux of each region from 600 regions was computed and then was divided
by the mean flux of each pseudo continuum. Finally, 3000 indices were obtained
for each spectrum. These indices and corresponding subtypes were input to
construct the forest. Knowledge or patterns were learnt from the inputs during
construction so that the constructed forest can classify a spectrum according
to what it learnt. Simultaneously the variable importance measurements were
provided by the forest.
The important features (the numerators of important indices) supplied by
Random Forest method are shown in Figure 2. As a large amount of LAMOST
spectra do not have high quality in the blue part, and most of obvious
features of M dwarfs that can help to distinguish the spectral subtypes locate
in 6000$\sim$ 9000Å , the spectral range we care about is limited to
6000Å$\sim$ 9000Å . According to the features importance list, we adjusted the
indices of the Hammer by adding three new indices and keeping the original
Hammer indices that are in 6000$\sim$9000Å . The three indices and their
corresponding wavelength ranges are shown in Table 1. CaH6385, TiO8250 are
sensitive to temperature. The range of index 6545 is near H${\alpha}$ and is
often used as a part of continuum to compute the equivalent width of
H${\alpha}$. So we take the index 6545 as a pseudo continuum index, and named
it Color6545. Table 1 shows the wavelength ranges of each index. The original
Hammer indice CaI, MgI and Color1 are removed because they are out of the
wavelength of 6000$\sim$9000Å . After adjusting indices of the Hammer, there
are still 16 indices. The 16 indices values of Hammer M dwarf template are
calculated again. Figure 3 showed the values of recomputed indices and three
extra indices. In Figure 3, the points in the same color are indices values in
the same subtype. Lines from red to blue are corresponding to the subtypes of
M0 to M9. This figure indicates which indices are good in distinguishing
subtypes of M dwarf spectra.
We then classified the Bochanski2007b template using the modified Hammer. The
results are showed in the third column of Table 2. The classification result
of each template is right now. In order to verify the performance of Hammer
after adjusting indices, we further tested the code with 70,841 spectra from
the SDSS DR7 M dwarf catalog (West et al 2011). All spectral subtypes in the
catalog were derived by visual inspection. The results are showed in Figure 4.
The left panel of the figure is the differences distribution of all spectral
subtypes for total 70,841 spectra. It shows a number of spectra that were
classified a later subtype by the original Hammer, are partially corrected by
the modified Hammer. The accuracy of the modified Hammer is higher than the
original Hammer. The right panel shows the differences distribution of
subtypes for later than M5 spectra and indicates the original Hammer
classifies a larger fraction late-type M dwarfs as an earlier subtype (as
indicated by West et al. 2011). This is greatly remedied by the improved
Hammer. According to the statistical results of all the data, the mean subtype
difference is 1 subtype before adjusting indices while the mean subtype
difference is 0.6 subtypes for modified Hammer.
We used the amended Hammer to classify the 67,082 M dwarf spectra from LAMOST
pilot survey. Figure 5 shows the signal to noise (S/N) distribution of M
dwarfs from LAMOST pilot survey. The S/N was computed in the range of
6900Å$\sim$ 8160Å. Figure 6 shows the spectral subtypes distribution of M
dwarfs in this catalog. From this distribution, early-type M dwarfs account
for a large proportion, and the number of late-type M dwarfs from M6 to M9 is
small (only 724). This is likely due to target selection effects and the
resolution capacity limitation of LAMOST telescope.
## 4 Radial Velocity
The radial velocity (RV) of each M dwarf was measured by the cross-correlation
method. Each observed spectrum was cross-correlated with the Bochanski2007b M
dwarf template of best matched subtype. In order to decrease the impact of
inaccuracy of flux calibration, we used a cubic polynomial to rectify the
observed spectra to best fitted template spectra. We constrained the range of
radial velocities to $\pm$500km/s, then moved the observed spectra from
-500km/s to 500km/s in a 2km/s step. After each move, the observed spectra was
multiplied an optimal cubic polynomial to cross-correlate the observed
spectrum, with the corresponding template fitting all correlation values to
produce a Gaussian peak. The corresponding radial velocity of the Gaussian
peak was chosen as our final radial velocity. A bootstrap estimate was
conducted to access the internal error of radial velocity estimation.
We used spectra from the SDSS DR7 M dwarf catalog to test our RV measure
method. Figure 7 shows the results. The left panel shows the RV comparison to
the West2011 values. The right panel shows the distribution of RV differences,
in which 67843 RVs differences between -100km/s and +100km/s are shown. Figure
7 indicates that the RVs we measured generally agree with the RVs West et al.
measured. The mean of two RVs difference is 0.17km/s while the standard
deviation is 6.4km/s, which is less than the reported uncertainties of
West2011. The larger scatter of RV around the center of the figure is due to
low S/N of spectra, which can be seen in Figure 8. It is intrinsically
difficult to derive accurate RVs from these spectra with low S/N. There are a
group of points in the bottom of the left panel of Figure 7. We visually
inspected the Na doublet and found the RVs of these spectra measured by
West2011 have larger uncertainties. A Na doublet at 8183Å and 8195Å fitting
example is showed in the Figure 9, in which template spectrum was plotted in
red and observed spectrum was plotted in blue. The observed spectrum was
corrected to zero radial velocity by RV values respectively from west2011 (top
panel) and from our method (bottom panel). We further selected a subsample to
check the performance of our method. In this subsample, the S/Ns are between
10 and 20 and the differences of two RVs are between 50 and 200. Of total 479
spectra from this sample, for about 180 spectra, our RV is better than West’s,
and about 104 worse than West’s and for the rest 195 spectra, the accuracy of
two RVs is the same.
There are many factors which may cause radial velocity uncertainties. The
uncertainty of using cross-correlation method is due to some reasons such as
the resolution of the spectra, signal noise ratio, the accuracy of the
wavelength calibration and flux calibration, and matching the spectral type of
the template to the observed spectra. We used the best matched spectral type
template and cross-correlated it with the observed spectra, which minimizes
the error introduced by spectral type mismatch. For flux calibration problem,
LAMOST has relative flux calibration instead of absolute calibration, which
may lead to inaccuracy of spectral flux, and for the wavelength calibration,
the RV error caused by inaccurate wavelength calibration of LAMOST spectra is
less than 10km/s (Luo2012).
We computed RV and error for all spectra of our catalog and got a mean
internal error 11.5 km/s. All spectra of this catalog then were corrected
using our radial velocity value for further measurements of H${\alpha}$
emission line and molecular band indices.
## 5 MAGNETIC ACTIVITY & MOLECULAR BAND INDICES
H${\alpha}$ emission line is the best indicator of chromospheric magnetic
activity in M dwarfs due to their red colors. We estimated the magnetic
activity of M dwarfs according to the methods of West et al. (2004, 2011). We
used a total 14$\AA$ wavelength region for calculation of equivalent width of
H${\alpha}$. The central wavelength is 6564.66Å in vacuum with 7Å on either
side. The continua regions are 6555.0Å$\sim$ 6560.0Å and 6570.0Å$\sim$6575.0Å
. Our magnetic activity criteria are similar to the west2011 criteria. As our
sample contained all spectra of M dwarf from the pivot survey irrespective of
S/N, we add an additional S/N criterion (5) to obtain a more clean activity
sample. Our criteria is (1) The S/N of continuum near H${\alpha}$ is larger
than 3, (2) the EW of H${\alpha}$ must larger than 1, (3) EW is larger than
three times of error, (4) the height of the emission line is larger than three
times of noise in the adjacent continuum, (5) the S/N of 6500$\sim$6550Å and
6575Å$\sim$6625Å is larger than 10. A star is classified as inactive, it
should meet the criteria of (1) and (5) mentioned before to insure the
spectrum has higher S/N, besides the spectrum has no detectable emission.
Using the criteria, from 67082 M dwarf spectra, 2312 of them are H${\alpha}$
active while 26074 are H${\alpha}$ inactive, the H${\alpha}$ activity fraction
of M0 to M5 listed in Table 3. We confirm that later subtypes have higher
active fractions and the trend of the active fraction from M0 to M5 is in
agreement with West2011. But in this sample, the number of late-type M dwarfs
from M6 to M9 is 724, and only 174 spectra with S/N$>$10\. This is likely due
to target selection effects and the resolution capacity limitation of LAMOST
telescope. The number of late-type M dwarfs is too small to produce a
reasonable activity fraction if using the same magnetic activity criteria.
Therefore the activity fraction of M6$\sim$ M9 was not provided here.
We have also computed important molecular band features TiO1$\sim$TiO5,
CaH1$\sim$CaH3, and CaOH according to the wavelength ranges defined by Reid et
al (1995a). The errors of these indices are given as well.
A rough indicator of metallicity $\zeta$ was computed, which was defined by
Lépine et al. (2007). This $\zeta$ based on the strength of TiO5, CaH2 and
CaH3 molecular bands. According to the indicator M spectra can be divided into
different metallicity classes: dwarf, subdwarf, extreme dwarf and ultra
subdwarfs. However, Lépine et al. (2012) found $\zeta$ are not sensitive
enough to diagnose metallicity variations in dwarfs of subtypes M2 and
earlier. Dhital et al. (2012) refined the indice $\zeta$ and the new $\zeta$
can better fit their observed M sample, in which most are M0$\sim$M3 dwarfs
spectra. We measured the parameter $\zeta$ and its error according to the
refined definition of $\zeta$ by Dhital et al.. Note that Mann et al. (2013)
further tested the $\zeta$ parameter with their sample and found $\zeta$
correlates with [Fe/H] for super-solar metallicities, but $\zeta$ does not
always correctly identify metal-poor M dwarfs.
## 6 SUMMARY
We present an M dwarf spectral catalog from LAMOST pilot survey, which
consists of 67082 M dwarfs. In this catalog, spectral subtypes, radial
velocities, magnetic activity, equivalent widths of H${\alpha}$ and various
molecular band indices are provided. Spectral subtypes have been derived by
the remedied Hammer program and then confirmed by visually inspection. The
amended Hammer results have an average offset of 0.6 subtype compared to the
visually inspection results. The radial velocities of the sample have been
measured with cross-correlation method, with an average 11.5km/s internal
error. We also estimate the magnetic activity of M dwarfs through measuring
the equivalent width and strength of H${\alpha}$.
The M dwarf catalog of LAMOST pilot survey provides the first glance of LAMOST
M-type stellar spectra and it is the first step of our research. Our
subsequent work will focus on obtaining more information about M dwarfs, such
as kinematics, metallicity, distance and mass through cross-matching with
other catalogs. The formal LAMOST survey will enlarge the sample of M dwarfs
rapidly and even get the largest spectroscopic sample of M dwarfs. Large
samples can enable more and scientific studies of M dwarfs and provide more
statistically significant results to explore the structure and evolution of
the Milky Way.
We would like to thank Dr. Wei Du for helpful discussions toward the
implementation and development of data process software and Ms. Min-Yi, Lin
for continuing help with astronomical research methods. This research had made
use of LAMOST data and SDSS data. This research is supported by the National
Natural Science Foundation of China(Grant Nos
10973021,11078013,10978010,11078019 and 11263001).
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Figure 1: Comparison between LAMOST and SDSS spectra, top panel shows two
spectra with SDSS spectra in blue and LAMOST spectra in red; bottom panel is
the difference of the two spectra.
Figure 2: The Hammer/Random Forest features. The two spectra are the same M4
dwarf spectrum from Bochanski2007b template. Top: The blue crosses identify
regions that are used by the Hammer ,18 total regions. Bottom: The 18 most
important features, which represent the numerators of the important indices
that the random forest recommends, are highlighted in red stars.
Figure 3: The modified Hammer indices, including 13 Original Hammer indices in
$6000\sim 9000$Åand newly added three indices. The new-added three indices:
CaH6385, TiO8250,Color6545. Each line corresponds to a different subtype.
Lines from red to blue standards for subtypes from M0 to M9. If certain index
value increases or reduces monofonically with the type from M0 M9, it
indicates that the index is a good spectral type tracer and it is suitable for
spectrum type classification. For example, VO7912, TiOB, TiO8250 and Color6545
are good spectral type tracers.
Figure 4: Distribution of differences between Hammer/improved Hammer and
visually inspected spectral subtype for M dwarfs. The top panel is the
distribution of differences for total 70841 spectra(West et al.2011). The
bottom panel is the distribution of differences for later than M5 spectra.
Figure 5: S/N distribution of M dwarfs in the LAMOST pilot survey Figure 6:
Spectral subtype distribution of M dwarfs in the LAMOST pilot survey
Figure 7: RV comparison. The left panel shows the comparison between RVs
measured using our method and the RVs measured by West. The right is the
distribution of RVs differences. Figure 8: The relation of differences of RVs
as a function of S/N. When S/N is less than 20, there is more scatter in the
differences in the RV values.
Figure 9: Example of Na doublet fitting at 8183 Åand 8195 Å. The template
spectra are plotted in red and the observed spectra are plotted in blue. The
observed spectra was corrected to zero radial velocity respectively according
to the RV values measured by west2011 (top panel) and by our method (bottom
panel)
Table 1: New-added Three Indices Index | $\lambda_{w}$[Å] | $\lambda_{c}$ont[Å]
---|---|---
CaH6385 | 6385.0 | 6389.0 | 6545.0 | 6549.0
TiO8250 | 8250.0 | 8254.0 | 7560.0 | 7564.0
Color6545 | 6545.0 | 6549.0 | 7560.0 | 7564.0
Table 2: Original Hammer and improved Hammer classification of Bochanski2007 M dwarf templates Bochanski2007b | Original Hammer | Improved Hammer | Eyecheck
---|---|---|---
Subtype | Autotype | Autotype | Subtype
M0 | M0 | M0 | M0
M1 | M0 | M1 | M1
M2 | M2 | M2 | M2
M3 | M3 | M3 | M3
M4 | M4 | M4 | M4
M5 | M4 | M5 | M5
M6 | M5 | M6 | M6
M7 | M6 | M7 | M7
M8 | M8 | M8 | M8
M9 | M8 | M9 | M9
Table 3: H${\alpha}$ activity fraction on each subtype SPT | Total | Active | Inactive | Active Fraction
---|---|---|---|---
M0 | 15657 | 267 | 10408 | 2.50
M1 | 18594 | 471 | 7805 | 5.69
M2 | 17941 | 650 | 5231 | 11.05
M3 | 9500 | 600 | 1970 | 23.35
M4 | 3619 | 258 | 479 | 35.01
M5 | 1047 | 53 | 82 | 39.26
|
arxiv-papers
| 2013-06-19T13:26:47 |
2024-09-04T02:49:46.687763
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Z.P. Yi, A.L. Luo, Y.H. Song, J.K. Zhao, Z.X. Shi, P. Wei, J.J. Ren,\n F.F. Wang, X. Kong, Y.B. Li, P. Du, W. Hou, Y.X. Guo, S. Zhang, Y.H. Zhao,\n S.W. Sun, J.C. Pan, L.Y. Zhang, A. A. Wes, H.B. Yuan",
"submitter": "Xiaoyan Chen",
"url": "https://arxiv.org/abs/1306.4540"
}
|
1306.4588
|
11institutetext: Tim Gershon 22institutetext: Department of Physics,
University of Warwick, Coventry, United Kingdom and European Organization for
Nuclear Research (CERN), Geneva, Switzerland, 22email:
[email protected]
# Flavour Physics in the LHC Era
Tim Gershon
###### Abstract
These lectures give a topical review of heavy flavour physics, in particular
$C\\!P$ violation and rare decays, from an experimental point of view. They
describe the ongoing motivation to study heavy flavour physics in the LHC era,
the current status of the field emphasising key results from previous
experiments, some selected topics in which new results are expected in the
near future, and a brief look at future projects.
## 1 Introduction
The concept of “flavour physics” was introduced in the 1970s Browder:2008em
> The term flavor was first used in particle physics in the context of the
> quark model of hadrons. It was coined in 1971 by Murray Gell-Mann and his
> student at the time, Harald Fritzsch, at a Baskin-Robbins ice-cream store in
> Pasadena. Just as ice cream has both color and flavor so do quarks.
Leptons also come in different flavours, and flavour physics covers the
properties of both sets of fermions. Counting the fundamental parameters of
the Standard Model (SM), the 3 lepton masses, 6 quark masses and 4 quark
mixing (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv parameters are related
to flavour physics. In case neutrino masses are introduced, the new parameters
(at least 3 more masses and 4 more mixing parameters) are also related to
flavour physics. This large number of free parameters is behind several of the
mysteries of the SM:
* •
Why are there so many different fermions?
* •
What is responsible for their organisation into generations / families?
* •
Why are there 3 generations / families each of quarks and leptons?
* •
Why are there flavour symmetries?
* •
What breaks the flavour symmetries?
* •
What causes matter – antimatter asymmetry?
Unfortunately these mysteries will not be answered in these lectures – they
are mentioned here simply because it is important to bear in mind their
existence. Instead the focus will be on specific topics in the flavour-
changing interactions of the charm and beauty quarks,111 It is one of the
peculiarities of our field that “heavy flavour physics” does not include
discussion of the heaviest flavoured particle, the top quark. with occasional
digressions on related topics.
While our main interest is in the properties of the charm and beauty quarks,
due to the strong interaction, experimental studies must be performed using
one or more of the many different charmed or beautiful hadrons. These can
decay to an even larger multitude of different final states, making learning
the names of all the hadrons a big challenge for flavour physicists. Moreover,
hadronic effects can often obscure the underlying dynamics. Nevertheless, it
is the hadronisation that results in the very rich phenomenology that will be
discussed, so one should bear in mind that Bigi:2005av
> The strong interaction can be seen either as the “unsung hero” or the
> “villain” in the story of quark flavour physics.
## 2 Motivation to study heavy flavour physics in the LHC era
There are two main motivations for ongoing experimental investigations into
heavy flavour physics: (i) $C\\!P$ violation and its connection to the matter-
antimatter asymmetry of the Universe; (ii) discovery potential far beyond the
energy frontier via searches for rare or SM forbidden processes. These will be
discussed in turn below.
First let us consider one of the mysteries listed above (What breaks the
flavour symmetries?) to see how it is connected to these motivations. In the
SM, the vacuum expectation value of the Higgs field breaks the electroweak
symmetry. Fermion masses arise from the Yukawa couplings of the quarks and
charged leptons to the Higgs field, and the CKM matrix arises from the
relative misalignment of the Yukawa matrices for the up- and down-type quarks.
Consequently, the only flavour-changing interactions are the charged current
weak interactions. This means that there are no flavour-changing neutral
currents (the GIM mechanism Glashow:1970gm ), a feature of the SM which is not
generically true in most extended theories. Flavour-changing processes provide
sensitive tests of this prediction; as an example, many new physics (NP)
models induce contributions to the $\mu\rightarrow e\gamma$ transition at
levels close to (or even above!) the current experimental limit, recently made
more restrictive by the MEG experiment Adam:2013mnn , ${\cal
B}(\mu^{+}\rightarrow e^{+}\gamma)<5.7\times 10^{-13}$ at 90% confidence level
(CL). Improved experimental reach in this and related charged lepton flavour
violation searches therefore provides interesting and unique NP discovery
potential (for a review, see, e.g., Ref. Marciano:2008zz ).
### 2.1 $C\\!P$ violation
As mentioned above, the CKM matrix arises from the relative misalignment of
the Yukawa matrices for the up- and down-type quarks:
$V_{CKM}=U_{u}U^{\dagger}_{d}\,,$ (1)
where $U_{u}$ and $U_{d}$ diagonalise the up- and down-type quark mass
matrices respectively. Hence, $V_{CKM}$ is a $3\times 3$ complex unitary
matrix. Such a matrix is in general described by 9 (real) parameters, but 5
can be absorbed as unobservable phase differences between the quark fields.
This leaves 4 parameters, of which 3 can be expressed as Euler mixing angles,
but the fourth makes the CKM matrix complex – and hence the weak interaction
couplings differ for quarks and antiquarks, i.e. $C\\!P$ violation arises.
The expression “$C\\!P$ violation” refers to the violation of the symmetry of
the combined $C$ and $P$ operators, which replace particle with antiparticle
(charge conjugation) and invert all spatial co-ordinates (parity)
respectively. Therefore $C\\!P$ violation provides absolute discrimination
between particle and antiparticle: one cannot simply swap the definition of
which is called “particle” with a simultaneous redefinition of left and
right.222 The importance of $C\\!P$ violation in this regard was noted by
Landau Landau:1957tp following the observation of parity violation Lee:1956qn
; Wu:1957my . There is a third discrete symmetry, time reversal ($T$), and it
is important to note that there is a theorem that states that $C\\!PT$ must be
conserved in any locally Lorentz invariant quantum field theory Lueders:1992dq
. Therefore, under rather reasonable assumptions, an observation of $C\\!P$
violation corresponds to an observation of $T$ violation, and vice versa.
Nonetheless, it remains of interest to establish $T$ violation without
assumptions regarding other symmetries Angelopoulos:1998dv ; Lees:2012uka .
The four parameters of the CKM matrix can be expressed in many different ways,
but two popular choices are the Chau-Keung (PDG) parametrisation –
$(\theta_{12},\theta_{13},\theta_{23},\delta)$ Chau:1984fp – and the
Wolfenstein parametrisation – $(\lambda,A,\rho,\eta)$ Wolfenstein:1983yz . In
both cases a single parameter ($\delta$ or $\eta$) is responsible for all
$C\\!P$ violation. This encapsulates the predictivity that makes the CKM
theory such a remarkable success: it describes a vast range of phenomena at
many different energy scales, from nuclear beta transitions to single top
quark production, all by only four parameters (plus hadronic effects).
Let us digress a little into history. In 1964, $C\\!P$ violation was
discovered in the kaon system Christenson:1964fg , but it was not until 1973
that Kobayashi and Maskawa proposed that the effect originated from the
existence of three quark families Kobayashi:1973fv . On a shorter time-scale,
in 1967 Sakharov noted that $C\\!P$ violation was one of three conditions
necessary for the evolution of a matter-dominated universe, from a symmetric
initial state Sakharov:1967dj :
1. 1.
baryon number violation,
2. 2.
$C$ and $C\\!P$ violation,
3. 3.
thermal inequilibrium.
This observation evokes the prescient concluding words of Dirac’s 1933 Nobel
lecture, discussing his successful prediction of the existence of antimatter,
in the form of the positron Dirac :
> If we accept the view of complete symmetry between positive and negative
> electric charge so far as concerns the fundamental laws of Nature, we must
> regard it rather as an accident that the Earth (and presumably the whole
> solar system), contains a preponderance of negative electrons and positive
> protons. It is quite possible that for some of the stars it is the other way
> about, these stars being built up mainly of positrons and negative protons.
> In fact, there may be half the stars of each kind. The two kinds of stars
> would both show exactly the same spectra, and there would be no way of
> distinguishing them by present astronomical methods.
Dirac was not aware of the existence of $C\\!P$ violation, that breaks the
complete symmetry of the laws of Nature. Moreover, modern astronomical methods
do allow to search for antimatter dominated regions of the Universe, and none
have been observed (though searches, for example by the PAMELA and AMS
experiments, are ongoing). Therefore, $C\\!P$ violation appears to play a
crucial role in the early Universe.
We can illustrate this with a simple exercise. Suppose we start with equal
amounts of matter ($X$) and antimatter ($\bar{X}$). The matter $X$ decays to
final state $A$ (with baryon number $N_{A}$) with probability $p$ and to final
state $B$ (baryon number $N_{B}$) with probability $(1-p)$. The antimatter,
$\bar{X}$, decays to final state $\bar{A}$ (with baryon number $-N_{A}$) with
probability $\bar{p}$ and final state $\bar{B}$ (baryon number $-N_{B}$) with
probability $(1-\bar{p})$. The resulting baryon asymmetry is
$\Delta N_{\rm
tot}=N_{A}p+N_{B}(1-p)-N_{A}\bar{p}-N_{B}(1-\bar{p})=(p-\bar{p})(N_{A}-N_{B})\,.$
So clearly $\Delta N_{\rm tot}\neq 0$ requires both $p\neq\bar{p}$ and
$N_{A}\neq N_{B}$, i.e. both $C\\!P$ violation and baryon number violation.
It is natural to next ask whether the magnitude of the baryon asymmetry of the
Universe could be caused by the $C\\!P$ violation in the CKM matrix. The
baryon asymmetry can be quantified relative to the number of photons in the
Universe,
$\Delta N_{B}/N_{\gamma}=(N({\rm baryon})-N({\rm antibaryon}))/N_{\gamma}\sim
10^{-10}\,.$
This can be compared to a dimensionless and parametrisation invariant measure
of the amount of $C\\!P$ violation in the SM, $J\times P_{u}\times
P_{d}/M^{12}$, where
* •
$J=\cos(\theta_{12})\cos(\theta_{23})\cos^{2}(\theta_{13})\sin(\theta_{12})\sin(\theta_{23})\sin(\theta_{13})\sin(\delta)$
,
* •
$P_{u}=(m_{t}^{2}-m_{c}^{2})(m_{t}^{2}-m_{u}^{2})(m_{c}^{2}-m_{u}^{2})$,
* •
$P_{d}=(m_{b}^{2}-m_{s}^{2})(m_{b}^{2}-m_{d}^{2})(m_{s}^{2}-m_{d}^{2})$,
* •
and $M$ is the relevant scale, which can be taken to be the electroweak scale,
${\cal O}(100\mathrm{\,Ge\kern-1.00006ptV})$.
The parameter $J$ is known as the Jarlskog parameter Jarlskog:1985ht , and is
expressed above in terms of the Chau-Keung parameters. Putting all the numbers
in, we find a value for the asymmetry of $\sim 10^{-17}$, much below the
observed $10^{-10}$. This is the origin of the widely accepted statement that
the SM $C\\!P$ violation is insufficient to explain the observed baryon
asymmetry of the Universe. Note that this occurs primarily not because $J$ is
small, but rather because the electroweak mass scale is far above the mass of
most of the quarks. Therefore, to explain the baryon asymmetry of the
Universe, there must be additional sources of $C\\!P$ violation that occur at
high energy scales. There is, however, no guarantee that these are connected
to the $C\\!P$ violation that we know about. The new sources may show up in
the quark sector via discrepancies with CKM predictions (as will be discussed
below), but could equally appear in the lepton sector as $C\\!P$ violation in
neutrino oscillations. Or, for that matter, new sources could be flavour-
conserving and be found in measurements of electric dipole moments, or could
be connected to the Higgs sector, or the gauge sector, or to extra dimensions,
or to other NP. In any case, precision measurements of flavour observables are
generically sensitive to additions to the SM, and hence are well-motivated.
In this context, it is worth noting the enticing possibility of
“leptogenesis”, where the baryon asymmetry is created via a lepton asymmetry
(see, e.g., Ref. Davidson:2008bu for a review). In the case that neutrinos
are Majorana particles – i.e. they are their own antiparticles – the right-
handed neutrinos may be very massive, which provides an immediate connection
with the needed high energy scale. Experimental investigation of this concept
requires the determination of the lepton mixing (PMNS) Pontecorvo:1957cp ;
Maki:1962mu matrix, and proof whether or not neutrinos are Majorana
particles. The recent determination of the neutrino mixing angle $\theta_{13}$
An:2012eh ; Ahn:2012nd provides an important step forward; the next
challenges are to establish $C\\!P$ violation in neutrino oscillations and to
observe (or limit) neutrinoless double beta decay processes.
### 2.2 Rare processes
We have already digressed into history, and we should avoid doing so too much,
but it is striking how often NP has shown up at the precision frontier before
“direct” discoveries at the energy frontier. Examples include: the GIM
mechanism being established before the discovery of charm; $C\\!P$ violation
being discovered and the CKM theory developed before the discovery of the
bottom and top quarks; the observation of weak neutral currents before the
discovery of the $Z$ boson. In particular, loop processes are highly sensitive
to potential NP contributions, since SM contributions are suppressed or
absent.
As a specific example of this we can consider the loop processes involved in
oscillations of neutral flavoured mesons. (Rare decay processes will be
discussed in more detail below.) There are four such pseudoscalar particles in
nature ($K^{0}$, $D^{0}$, $B^{0}$ and $B_{s}^{0}$) which can oscillate into
their antiparticles via both short-distance (dispersive) and long-distance
(absorptive) processes, as illustrated in Fig. 1. Representing such a meson
generically by $M^{0}$, the evolution of the particle-antiparticle system is
given by the time-dependent Schrödinger equation,
$i\frac{\partial}{\partial t}\left(\begin{array}[]{c}M^{0}\\\
\bar{M}^{0}\end{array}\right)=\left(M-\frac{i}{2}\Gamma\right)\left(\begin{array}[]{c}M^{0}\\\
\bar{M}^{0}\end{array}\right)\,,$ (2)
where the effective333 The complete Hamiltonian would include all possible
final states of decays of $M^{0}$ and $\bar{M}^{0}$. Hamiltonian
$H=M-\frac{i}{2}\Gamma$ is written in terms of $2\times 2$ Hermitian matrices
$M$ and $\Gamma$. Note that the $C\\!PT$ theorem requires that $M_{11}=M_{22}$
and $\Gamma_{11}=\Gamma_{22}$, i.e. that particle and antiparticle have
identical masses and lifetimes.
Figure 1: Illustrative diagrams of (left) short-distance (dispersive)
processes in $B^{0}_{s}$ mixing; (right) long-distance (absorptive) processes
in $K^{0}$ mixing.
The physical states are eigenstates of the effective Hamiltonian, and are
written
$M_{\rm L,H}=pM^{0}\pm q\bar{M}^{0}\,,$ (3)
where $p$ and $q$ are complex coefficients that satisfy
$\left|p\right|^{2}+\left|q\right|^{2}=1$. Here the subscript labels L and H
distinguish the eigenstates by their nature of being lighter or heavier; in
some systems the labels S and L are instead used for short-lived and long-
lived respectively (the choice depends on the values of the mass and width
differences; the labels 1 and 2 are also sometimes used, usually to denote the
$C\\!P$ eigenstates). $C\\!P$ is conserved (in mixing) if the physical states
correspond to the $C\\!P$ eigenstates, i.e. if $\left|q/p\right|=1$. Solving
the Schrödinger equation gives
$\left(\frac{q}{p}\right)^{2}=\frac{M_{12}^{*}-\frac{i}{2}\Gamma_{12}^{*}}{M_{12}-\frac{i}{2}\Gamma_{12}}\,,$
(4)
with eigenvalues given by $\lambda_{\rm L,H}=m_{\rm
L,H}-\frac{i}{2}\Gamma_{\rm
L,H}=(M_{11}-\frac{i}{2}\Gamma_{11})\pm(q/p)(M_{12}-\frac{i}{2}\Gamma_{12})$,
corresponding to mass and width differences $\Delta m=m_{\rm H}-m_{\rm L}$ and
$\Delta\Gamma=\Gamma_{\rm H}-\Gamma_{\rm L}$ given by
$\displaystyle(\Delta m)^{2}-\frac{1}{4}(\Delta\Gamma)^{2}$ $\displaystyle=$
$\displaystyle
4(\left|M_{12}\right|^{2}+\frac{1}{4}\left|\Gamma_{12}\right|^{2})\,,$ (5)
$\displaystyle\Delta m\Delta\Gamma$ $\displaystyle=$ $\displaystyle 4\,{\rm
Re}(M_{12}\Gamma_{12}^{*})\,.$ (6)
Note that with this notation, which is the same as that of Ref. CPVreview ,
$\Delta m$ is positive by definition while $\Delta\Gamma$ can have either
sign.444 With the definition given, $\Delta\Gamma$ is predicted to be negative
for $B^{0}$ and $B^{0}_{s}$ mesons in the SM, and hence the sign-flipped
definition is often encountered in the literature, e.g. in Ref. Bmixreview .
Rather than going into the details of the formalism (which can be found in,
e.g., Ref. Nierste:2009wg ) let us instead take a simplistic picture.
* •
The value of $\Delta m$ depends on the rate of the mixing diagram of Fig.
1(left). This depends on CKM matrix elements, together with various other
factors that are either known or (in the case of decay constants and bag
parameters) can be calculated using lattice QCD. Moreover for the $B$ mesons,
these other factors can be made to cancel in the $\Delta m_{d}/\Delta m_{s}$
ratio, such that the measured value of this quantity gives a theoretically
clean determination of $\left|V_{td}/V_{ts}\right|^{2}$.
* •
The value of $\Delta\Gamma$, on the other hand, depends on the widths of
decays of the meson and antimeson into common final states (such as
$C\\!P$-eigenstates). Therefore, $\Delta\Gamma$ is large for the $K^{0}$
system, where the two pion decay dominates, small for $D^{0}$ and $B^{0}$
mesons, where the most favoured decays are to flavour-specific or quasi-
flavour-specific final states, and intermediate in the $B^{0}_{s}$ system.
* •
Finally $C\\!P$ violation in mixing tends to zero (i.e. $q/p\approx 1$) if
${\rm arg}(\Gamma_{12}/M_{12})=0$, $M_{12}\ll\Gamma_{12}$ or
$M_{12}\gg\Gamma_{12}$.
This simplistic picture is sufficient to explain qualitatively the
experimental values of the mixing parameters given in Table 2.2. It should be
noted that $\Delta\Gamma(B^{0}_{s})$ has become well-measured only very
recently (as discussed below), and that the experimental sensitivity for the
$C\\!P$ violation parameters in all of the $D^{0}$, $B^{0}$ and $B^{0}_{s}$
systems is still far from that of the SM prediction, making improved
measurements very well motivated.
Table 1: Qualitative expectations and measured values for the neutral meson
mixing parameters. Experimental results are taken from Refs. Beringer:1900zz ;
KLreview ; Amhis:2012bh . The definition of $a_{\rm sl}$ is given in footnote
6.
| $\Delta m$ | $\Delta\Gamma$ | $\left|q/p\right|$
---|---|---|---
| ($x=\Delta m/\Gamma$) | ($y=\Delta\Gamma/(2\Gamma))$) | ($a_{\rm sl}\approx 1-\left|q/p\right|^{2}$)
$K^{0}$ | large | $\sim$ maximal | small
| $\sim 500$ | $\sim 1$ | $(3.32\pm 0.06)\times 10^{-3}$
$D^{0}$ | small | small | small
| $(0.63\pm 0.19)\%$ | $(0.75\pm 0.12)\%$ | $0.52\,^{+0.19}_{-0.24}$
$B^{0}$ | medium | small | small
| $0.770\pm 0.008$ | $0.008\pm 0.009$ | $-0.0003\pm 0.0021$
$B^{0}_{s}$ | large | medium | small
| $26.49\pm 0.29$ | $0.075\pm 0.010$ | $-0.0109\pm 0.0040$
Thus, neutral meson oscillations are rare processes described by parameters
that can be both predicted in the SM and measured experimentally. All
measurements to date are consistent with the SM predictions (though see
below). These results can then be used to put limits on non-SM contributions.
This can be done within particular models, but the model-independent approach,
described in, e.g., Ref. Isidori:2010kg is illustrative. The NP contribution
is expressed as a perturbation to the SM Lagrangian,
${\cal L}_{\rm eff}={\cal L}_{\rm
SM}+\Sigma\frac{c_{i}^{(d)}}{\Lambda^{d-4}}{\cal O}_{i}^{d}({\rm SM\
fields})\,,$ (7)
where the dimension $d$ of higher than 4 has an associated scale $\Lambda$ and
couplings $c_{i}$.555 In Eq. (7) it is assumed that the NP modifies the SM
operators; more generally extensions to the operator basis are also possible.
Given the observables in a given neutral meson system, NP contributions
described effectively as four-quark operators ($d=6$) can be constrained,
either by putting bounds on $\Lambda$ for a fixed value of $c_{i}$ (typically
1), or by putting bounds on $c_{i}$ for a fixed value of $\Lambda$ (typically
$1\mathrm{\,Te\kern-1.00006ptV}$). In the former case bounds of ${\cal
O}(100\mathrm{\,Te\kern-1.00006ptV})$ are obtained; in the latter case the
bounds can be ${\cal O}(10^{-9})$ or below Isidori:2010kg , with the strongest
(weakest) bounds being in the $K^{0}$ ($B^{0}_{s}$) sectors. A similar
analysis, but with more up-to-date inputs has been performed in Ref.
Lenz:2012az , with results illustrated in Fig. 2. The mixing amplitude,
normalised to its SM value, is denoted by $\Delta$, and experimental
constraints give $({\rm Re}\Delta,{\rm Im}\Delta)$ consistent with $(1,0)$
(i.e. with the SM) for both $B^{0}$ and $B^{0}_{s}$ systems.
Figure 2: Constraints on NP contributions in (left) $B^{0}$ and (right)
$B^{0}_{s}$ mixing Lenz:2012az .
This is a very puzzling situation. Limits on the NP scale give values of at
least $100\mathrm{\,Te\kern-1.00006ptV}$ for generic couplings. But, as
discussed elsewhere, we expect NP to appear at the
$\mathrm{\,Te\kern-1.00006ptV}$ scale to solve the hierarchy problem (and to
provide a dark matter candidate, etc.) If NP is indeed at this scale, NP
flavour-changing couplings must be small. But why? This is the so-called “new
physics flavour problem”.
A theoretically attractive solution to this problem, known as minimal flavour
violation (MFV) D'Ambrosio:2002ex , exploits the fact that the SM flavour-
changing couplings are also small. Therefore, if there is a perfect alignment
of the flavour violation in a NP model with that in the SM, the tension is
reduced. The MFV paradigm is highly predictive, stating that there are no new
sources of $C\\!P$ violation and also that the correlations between certain
observables share their SM pattern (the ratio of branching fractions of
$B^{0}\rightarrow\mu^{+}\mu^{-}$ and $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$
being a good example). Therefore, once physics beyond the SM is discovered, it
will be an important goal to establish whether or not it is minimally flavour
violating. This further underlines that the flavour observables carry
information about physics at very high scales.
Nonetheless, it must be reiterated that there are several important
observables that are not yet well measured, and that could rule out MFV. For
example, the bounds on NP scales obtained above (from Ref. Isidori:2010kg ) do
not include results on $C\\!P$ violation in mixing in the $B^{0}$ and
$B^{0}_{s}$ sectors. In fact, the D0 collaboration has reported a measurement
of an anomalous effect Abazov:2011yk of the inclusive same-sign dimuon
asymmetry, which is $3.9\sigma$ away from the SM prediction (of very close to
zero Lenz:2011ti ). This measurement is sensitive to an approximately equal
combination of the parameters of $C\\!P$ violation in $B^{0}$ and $B^{0}_{s}$
mixing, $a_{\rm sl}^{d}$ and $a_{\rm sl}^{s}$,666 The $a_{\rm sl}$ parameters,
so named because the asymmetries are measured using semileptonic decays, are
related to the $p$ and $q$ parameters by $a_{\rm
sl}=(1-\left|q/p\right|^{4})/(1+\left|q/p\right|^{4})$. however some
sensitivity to the source of the asymmetry can be obtained by applying
additional constraints on the impact parameter to obtain a sample enriched in
either oscillated $B^{0}$ or $B^{0}_{s}$ candidates. In addition, $a_{\rm
sl}^{d}$ and $a_{\rm sl}^{s}$ can be measured individually. The latest world
average, shown in Fig. 3, gives $a_{\rm sl}^{d}=-0.0003\pm 0.0021$, $a_{\rm
sl}^{s}=-0.0109\pm 0.0040$ Amhis:2012bh . Improved measurements are needed to
resolve the situation.
Figure 3: World average of constraints on the parameters describing $C\\!P$
violation in $B^{0}$ and $B^{0}_{s}$ mixing, $a_{\rm sl}^{d}$ and $a_{\rm
sl}^{s}$. The green ellipse comes from the D0 inclusive same-sign dimuon
analysis Abazov:2011yk ; the blue shaded bands give the world average
constraints on $a_{\rm sl}^{d}$ and $a_{\rm sl}^{s}$ individually; the red
ellipse is the world average including all constraints Amhis:2012bh .
## 3 Current experimental status of heavy quark flavour physics
### 3.1 The CKM matrix and the Unitarity Triangle
Much of the experimental programme in heavy quark flavour physics is devoted
to measurements of the parameters of the CKM matrix. As discussed above, the
CKM matrix can be written in terms of the Wolfenstein parameters, which
exploit the observed hierarchy in the mixing angles:
$\displaystyle V_{\rm CKM}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\\\ \end{array}\right)$ (11)
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}1-\lambda^{2}/2&\lambda&A\lambda^{3}(\rho-i\eta)\\\
-\lambda&1-\lambda^{2}/2&A\lambda^{2}\\\
A\lambda^{3}(1-\rho-i\eta)&-A\lambda^{2}&1\\\ \end{array}\right)+{\cal
O}\left(\lambda^{4}\right)\,,$ (15)
where the expansion parameter $\lambda$ is the sine of the Cabibbo angle
($\lambda=\sin\theta_{\rm C}\approx V_{us}$). It should be noted that although
the hierarchy is highly suggestive, there is no underlying reason known for
this pattern; moreover, the pattern in the lepton sector is completely
different. Note also that at ${\cal O}(\lambda^{3})$ in the Wolfenstein
parametrisation, the complex phase in the CKM matrix enters only in the
$V_{ub}$ and $V_{td}$ (top right and bottom left) elements, but this is purely
a matter of convention – only relative phases are observable.
The unitarity of the CKM matrix, $V^{\dagger}_{\rm CKM}V_{\rm CKM}=V_{\rm
CKM}V^{\dagger}_{\rm CKM}=1$, puts a number of constraints on the magnitudes
and relative phases of the elements. Among these relations, one which has been
precisely tested is
$\left|V_{ud}\right|^{2}+\left|V_{us}\right|^{2}+\left|V_{ub}\right|^{2}=1\,,$
(16)
where the measurements of $\left|V_{ud}\right|^{2}$ from, e.g., super-allowed
$\beta$ decays and $\left|V_{us}\right|^{2}$ from leptonic and semileptonic
kaon decays are indeed consistent with the prediction to within one part in
$10^{3}$ thetaCreview .777 The contribution from $\left|V_{ub}\right|^{2}$ is
at the level of $10^{-5}$ and therefore negligible for this test at current
precision.
The unitarity condition also results in six constraints,
$\Sigma_{i}V_{u_{i}d_{j}}V_{u_{i}d_{k}}^{*}=\Sigma_{i}V_{u_{j}d_{i}}V_{u_{k}d_{i}}^{*}=0$
($u_{i,j,k}\in(u,c,t),\,d_{i,j,k}\in(d,s,b),\,j\neq k$), for example
$V_{ud}V_{ub}^{*}+V_{cd}V_{cb}^{*}+V_{td}V_{tb}^{*}=0\,,$ (17)
which correspond to three complex numbers summing to zero, and hence can be
represented as triangles in the complex plane. The triangles have very
different shapes, but all of them have the same area, which is given by half
of the Jarlskog parameter Jarlskog:1985ht . The specific triangle relation
given in Eq. (17) is particularly useful to visualise the constraints from
various different measurements, as shown in the iconic images from the
CKMfitter Charles:2004jd and UTfit Bona:2005vz collaborations, reproduced in
Fig. 4. Conventionally, this “Unitary Triangle” (UT) is rescaled by
$V_{cd}V_{cb}^{*}$ so that by definition the position of the apex is
$\bar{\rho}+i\bar{\eta}\equiv-\frac{V_{ud}V_{ub}^{*}}{V_{cd}V_{cb}^{*}}\,,$
(18)
where $\left(\overline{\rho},\overline{\eta}\right)$ Buras:1994ec are related
to the Wolfenstein parameters by
$\rho+i\eta\;=\;\frac{\sqrt{1-A^{2}\lambda^{4}}(\bar{\rho}+i\bar{\eta})}{\sqrt{1-\lambda^{2}}\left[1-A^{2}\lambda^{4}(\bar{\rho}+i\bar{\eta})\right]}\,.$
(19)
Figure 4: Constraints on the Unitarity Triangle as compiled by (left)
CKMfitter Charles:2004jd , (right) UTfit Bona:2005vz .
Two popular naming conventions for the UT angles exist in the literature:
$\alpha\equiv\phi_{2}=\arg\left[-\frac{V_{td}V_{tb}^{*}}{V_{ud}V_{ub}^{*}}\right],\hskip
5.69046pt\beta\equiv\phi_{1}=\arg\left[-\frac{V_{cd}V_{cb}^{*}}{V_{td}V_{tb}^{*}}\right],\hskip
5.69046pt\gamma\equiv\phi_{3}=\arg\left[-\frac{V_{ud}V_{ub}^{*}}{V_{cd}V_{cb}^{*}}\right].$
(20)
The $\left(\alpha,\beta,\gamma\right)$ set is used in these lectures. The
lengths of the sides $R_{u}$ and $R_{t}$ of the UT are given by
$R_{u}=\left|\frac{V_{ud}V_{ub}^{*}}{V_{cd}V_{cb}^{*}}\right|=\sqrt{\overline{\rho}^{2}+\overline{\eta}^{2}}\,,\hskip
14.22636ptR_{t}=\left|\frac{V_{td}V_{tb}^{*}}{V_{cd}V_{cb}^{*}}\right|=\sqrt{(1-\overline{\rho})^{2}+\overline{\eta}^{2}}\,.$
(21)
A major achievement of the past decade or so has been to significantly improve
the precision of the parameters of the UT. In particular, the primary purpose
of the so-called “$B$ factory” experiments, BaBar and Belle, was the
determination of $\sin 2\beta$ using
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ (and related modes). This was carried
out using completely new experimental techniques to probe $C\\!P$ violation in
a very different way to previous experiments in the kaon system. In
particular, if we denote the amplitude for a $B^{0}$ meson to decay to a
particular final state $f$ as $A_{f}$, and that for the charge conjugate
process as $\bar{A}_{\bar{f}}$, then using the parameters $p$ and $q$ from Eq.
(3), we define the parameter
$\lambda_{f}=\frac{q}{p}\frac{\bar{A}_{\bar{f}}}{A_{f}}$ and the following
categories of $C\\!P$ violation in hadronic systems:888 Considering the
possibility that $C\\!P$ violation may be observed in the lepton sector as
differences of oscillation parameters between neutrinos and antineutrinos (in
appearance experiments), it is worth noting that this would be another
different category.
1. 1.
$C\\!P$ violation in mixing ($\left|q/p\right|\neq 1$),
2. 2.
$C\\!P$ violation in decay ($\left|\bar{A}_{\bar{f}}/A_{f}\right|\neq 1$),
3. 3.
$C\\!P$ violation in interference between mixing and decay (${\rm
Im}\left(\lambda_{f}\right)\neq 0$).
Additionally, in the literature the concepts of indirect and direct $C\\!P$
violation are often encountered: the former is where the effect is consistent
with originating from a single phase in the mixing amplitude, while the latter
cannot be accounted for in such a way. Following this categorisation, $C\\!P$
violation in decay (the only category available to baryons or charged mesons)
is direct, while $C\\!P$ violation in mixing and interference can be indirect
so long as only one measurement is considered – but if two such measurements
give different values, this also establishes direct $C\\!P$ violation.
### 3.2 Determination of $\sin(2\beta)$
The determination of $\sin(2\beta)$ from
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$ Carter:1980tk ; Bigi:1981qs , exploits
the fact that some measurements of $C\\!P$ violation in interference between
mixing and decay can be cleanly interpreted theoretically, since hadronic
factors do not contribute. The full derivation of the decay-time-dependent
decay rate of $B^{0}$ mesons to a $C\\!P$ eigenstate $f$ is a worthwhile
exercise for the reader, and can be found in, e.g., Refs. Branco:1999fs ;
Bigi:2000yz . The result, for mesons that are known to be either $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ or $B^{0}$ at time $t=0$, is
$\displaystyle\Gamma(\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{\rm
phys}\rightarrow f(t))$ $\displaystyle=$
$\displaystyle\frac{e^{-t/\tau(B^{0})}}{2\tau(B^{0})}\left[1+S_{f}\sin(\Delta
mt)-C_{f}\cos(\Delta mt)\right],$ $\displaystyle\Gamma(B^{0}_{\rm
phys}\rightarrow f(t))$ $\displaystyle=$
$\displaystyle\frac{e^{-t/\tau(B^{0})}}{2\tau(B^{0})}\left[1-S_{f}\sin(\Delta
mt)+C_{f}\cos(\Delta mt)\right],$ (22)
where
$S_{f}=\frac{2\,{\rm Im}(\lambda_{f})}{1+|\lambda_{f}|^{2}}\,,\ {\rm and}\
C_{f}=\frac{1-|\lambda_{f}|^{2}}{1+|\lambda_{f}|^{2}}\,.$ (23)
In these expressions $\Delta\Gamma$ has been assumed to be negligible, as
appropriate for the $B^{0}$ system. Assuming further $\left|q/p\right|=1$,
then for decays dominated by a single amplitude, $C_{f}=0$ and
$S_{f}=\sin({\rm arg}(\lambda_{f}))$, and so for
$B^{0}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip
2.0mu}K^{0}_{\rm\scriptscriptstyle S}$, $S=\sin(2\beta)$, to a very good
approximation.
The experimental challenge for the measurement of $\sin(2\beta)$ then lies in
the ability to measure the coefficient of the sinusoidal oscillation of the
decay-time-asymmetry. Until recently, the most copious sources of cleanly
reconstructed $B$ mesons came from accelerators colliding electrons with
positrons at the $\Upsilon(4S)$ resonance (a $b\bar{b}$ bound state just above
the threshold for decay into pairs of $B$ mesons). For symmetric colliders,
the $B$ mesons are produced at rest, and therefore lifetime measurements are
not possible. A boost is necessary, which can be advantageously achieved by
making the $e^{+}e^{-}$ collisions asymmetric.999 Boosted $b$ hadrons can also
be obtained in hadron colliders, as will be discussed below. One strong
feature of this approach is that the quantum correlations of the $B$ mesons
produced in $\Upsilon(4S)$ decay are retained, so that the decay of one into a
final state that tags its flavour ($B^{0}$ or $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}$) can be used to set the clock to
$t=0$ and specify the flavour of the other at that time.
The concept of the asymmetric $B$ factory was such a good one that two were
built: PEP-II at SLAC, colliding $9.0\mathrm{\,Ge\kern-1.00006ptV}$ $e^{-}$
with $3.1\mathrm{\,Ge\kern-1.00006ptV}$ $e^{+}$, and KEKB at KEK
($8.0\mathrm{\,Ge\kern-1.00006ptV}$ $e^{-}$ on
$3.5\mathrm{\,Ge\kern-1.00006ptV}$ $e^{+}$). These have achieved world record
luminosities, with peak instantaneous luminosities above
$10^{34}\rm\,cm^{-2}\rm{\,s}^{-1}$, and a combined data sample of over
$1\mbox{\,ab}^{-1}$, corresponding to over $10^{9}$ $B\bar{B}$ pairs. The
detectors (BaBar Aubert:2001tu ; BABAR:2013jta and Belle Abashian:2000cg
respectively) share many common features, such as silicon vertex detectors,
gas based drift chambers, electromagnetic calorimeters based on Tl-doped CsI
crystals, and $1.5\,{\rm T}$ solenoidal magnetic fields. The main difference
is in the technology used to separate kaons from pions: a system based on the
detection of internally reflected Cherenkov light for BaBar, and a combination
of aerogel Cherenkov counters and a time-of-flight system for Belle.
Through the measurement of $\sin 2\beta$, BaBar Aubert:2001nu and Belle
Abe:2001xe were able to make the first observations of $C\\!P$ violation
outside the kaon sector, thus validating the Kobayashi-Maskawa mechanism. The
latest (and, excluding upgrades, most likely final) results from BaBar
Aubert:2009aw and Belle Adachi:2012et shown in Fig. 5 give a clear visual
confirmation of the large $C\\!P$ violation effect. The world average value,
using determinations based on $b\rightarrow c\bar{c}s$ transitions, is
Amhis:2012bh
$\sin 2\beta=0.682\pm 0.019\ {\rm which\ gives}\
\beta=(21.5\,^{+0.8}_{-0.7})^{\circ}\,.$ (24)
Figure 5: Results from (left) BaBar Aubert:2009aw and (right) Belle
Adachi:2012et on the determination of $\sin 2\beta$.
### 3.3 Determination of $\alpha$
Additional measurements are needed to over-constrain the UT and thereby test
the Standard Model. The angle $\alpha$ can, in principle, be determined in a
similar way as $\beta$, but using a decay mediated by the $b\rightarrow
u\bar{u}d$ tree-diagram which carries the relative weak phase $\gamma$ (since
$\pi-\left(\beta+\gamma\right)=\alpha$ by definition). However, in any such
decay a contribution from the $b\rightarrow d$ loop (“penguin”) amplitude,
which carries a different weak phase, is also possible. This complicates the
interpretation of the observables, since $S\neq\sin(2\alpha)$; on the other
hand direct $C\\!P$ violation becomes observable, if the relative strong phase
is non-zero. Constraints on $\alpha$ can still be obtained using a channel in
which the penguin contribution either can be shown to be small, or can be
corrected for using an isospin analysis Gronau:1990ka . The world average,
$\alpha=\left(89.0\,^{+4.4}_{-4.2}\right)^{\circ}$, is dominated by
constraints from the $B^{0}\rightarrow\rho^{+}\rho^{-}$ decay Aubert:2007nua ;
Abe:2007ez , which is consistent with having negligible penguin contribution.
### 3.4 The sides of the Unitarity Triangle
The lengths of the sides of the UT have also been constrained by various
observables. The value of $R_{t}$ depends on $\left|V_{td}\right|$, and can be
determined from $b\rightarrow d$ transitions such as the rate of $B^{0}$
oscillations, i.e. $\Delta m_{d}$, or the branching fraction
$B\rightarrow\rho\gamma$. In both cases, theoretical uncertainties are reduced
if the measurement is performed relative to that for the corresponding
$b\rightarrow s$ transition. The most precise constraint to date comes from
the ratio of $\Delta m_{d}$ Amhis:2012bh ; Abe:2004mz ; LHCb-PAPER-2012-032
and $\Delta m_{s}$ Amhis:2012bh ; Abulencia:2006ze ; LHCb-PAPER-2013-006 and
gives $\left|\frac{V_{td}}{V_{ts}}\right|=0.211\pm 0.001\pm 0.005$, where the
first uncertainty is experimental and the second theoretical (originating from
lattice QCD calculations).
The value of $R_{u}$ depends on $\left|V_{ub}\right|$ and can be determined
from $b\rightarrow u$ tree-level transitions. Semileptonic decays allow
relatively clean theoretical interpretation,101010 Fully leptonic decays are
even cleaner theoretically, but are experimentally scarce. Such modes will be
discussed below. but still require QCD calculations to go from the parton
level transition to the observed (semi-hadronic) final state (for a recent
review, see Ref. Luth:2011zz ). Two approaches have been pursued: exclusive
decays, such as $B^{0}\rightarrow\pi^{-}e^{+}\nu$, and inclusive decays,
$B\rightarrow X_{u}e^{+}\nu$. The theory of inclusive decays is based on the
operator product expansion (discussed in Sec. 5.7) and would be extremely
clean, were it not for the fact that experimentally cuts are needed to remove
the more prevalent $b\rightarrow c$ transition. Exclusive decays tend to have
less background from $b\rightarrow c$ processes. The differential branching
fractions can be translated in constraints on $\left|V_{ub}\right|$ using
knowledge of form-factors at the kinematic limit obtained from lattice QCD
calculations, together with phenomenological models that extrapolate over the
whole phase space. The most precise results use $B\rightarrow\pi\ell^{+}\nu$
decays ($\ell=e,\mu$) delAmoSanchez:2010zd ; delAmoSanchez:2010af ; Ha:2010rf
, and give an “exclusive” determination of $\left|V_{ub}\right|$ that is,
however, in tension with the “inclusive” value Amhis:2012bh :
$\displaystyle\left|V_{ub}\right|_{\rm excl.}$ $\displaystyle=$
$\displaystyle\left[3.23\times\left(1.00\pm 0.05\pm 0.08\right)\right]\times
10^{-3}\,,$ $\displaystyle\left|V_{ub}\right|_{\rm incl.}$ $\displaystyle=$
$\displaystyle\left[4.42\times\left(1.000\pm 0.045\pm
0.034\right)\right]\times 10^{-3}\,.$
where the first uncertainties are experimental and the second theoretical.
Since the origin of the discrepancy, which is also seen in determinations of
$\left|V_{cb}\right|$ from $b\rightarrow c\ell\nu$ transitions, is not
understood, the uncertainty is scaled to give
$\displaystyle\left|V_{ub}\right|_{\rm avg.}$ $\displaystyle=$
$\displaystyle\left[3.95\times\left(1.000\pm 0.096\pm
0.099\right)\right]\times 10^{-3}\,.$
The results on $\beta$, $\alpha$, $R_{t}$ and $R_{u}$ are the most
constraining inputs to the CKM fits shown in Fig. 4 Charles:2004jd ;
Bona:2005vz . While the results are all consistent with the Standard Model
prediction of a single source of $C\\!P$ violation, there are some tensions
that deserve further investigation. Moreover, there are still certain
important observables where large NP contributions are possible, as will be
discussed in more detail below.
## 4 Flavour physics at hadron colliders
Results from the $B$ factory experiments provided enormous progress in the
understanding of heavy flavour physics (only a very brief selection has been
discussed above). Nonetheless, many results remain statistically limited, and
the $B^{0}_{s}$ sector is relatively unexplored. To progress further, it is
necessary to have a copious source of production of all flavours of $b$
hadron. As shown in Table 4, high energy hadron colliders satisfy these
criteria, but present significant experimental challenges: to be able to
identify the decays of interest from the high multiplicity environment, and to
reject the even more copious rate of minimum bias events.111111 Experiments at
$e^{+}e^{-}$ machines also have to reject effectively backgrounds from QED
processes, but this can be done at trigger level with simple requirements.
Table 2: Summary of some relevant properties for $b$ physics in different
experimental environments. Adapted from Ref. GibsonHCPSS .
| $e^{+}e^{-}\rightarrow\Upsilon(4S)\rightarrow B\bar{B}$ | $p\bar{p}\rightarrow b\bar{b}X$ | $pp\rightarrow b\bar{b}X$
---|---|---|---
| | ($\sqrt{s}=2\mathrm{\,Te\kern-1.00006ptV}$) | ($\sqrt{s}=14\mathrm{\,Te\kern-1.00006ptV}$)
| PEP-II, KEKB | Tevatron | LHC
Production cross-section | $1\rm\,nb$ | $\sim 100\rm\,\upmu b$ | $\sim 500\rm\,\upmu b$
Typical $b\bar{b}$ rate | $10{\rm\,Hz}$ | $\sim 100{\rm\,kHz}$ | $\sim 500{\rm\,kHz}$
Pile-up | 0 | 1.7 | 0.5–20
$b$ hadron mixture | $B^{+}B^{-}$ (50%), $B^{0}\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ (50%) | $B^{+}$ (40%), $B^{0}$ (40%), $B^{0}_{s}$ (10%),
| | $\mathchar 28931\relax^{0}_{b}$ (10%), others ($<1\%$)
$b$ hadron boost | small ($\beta\gamma\sim 0.5$) | large ($\beta\gamma\sim 100$)
Underlying event | $B\bar{B}$ pair alone | Many additional particles
Production vertex | Not reconstructed | Reconstructed from many tracks
$B^{0}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}$ pair production | Coherent (from $\Upsilon(4S)$ decay) | Incoherent
Flavour tagging power | $\epsilon D^{2}~{}\sim 30\%$ | $\epsilon D^{2}~{}\sim 5\%$
Figure 6: The LHCb detector Alves:2008zz .
The LHCb detector Alves:2008zz , shown in Fig. 6, has been designed to meet
these challenges. It is in essence a forward spectrometer (covering the
acceptance region that optimises its flavour physics capability), with a
dipole magnet, a precision silicon vertex detector and strong particle
identification capability. Tracks can be identified as different hadron
species using information from ring-imaging Cherenkov detectors, while
calorimeters and muon detectors enable charged leptons to be distinguished and
also provide trigger signals. The trigger system LHCb-DP-2012-004 uses these
hardware level signals to reduce the rate from the maximum LHC bunch-crossing
rate of $40{\rm\,MHz}$ to the $1{\rm\,MHz}$ rate at which the detector can be
read out. A software trigger then searches for inclusive signatures of
$b$-hadron decays such as high-$p_{\rm T}$ signals and displaced vertices, and
also performs reconstruction of several exclusive $b$ and $c$ decay channels,
in order to further reduce the rate to a level that can be written to offline
data storage ($3{\rm\,kHz}$ in 2011, $5{\rm\,kHz}$ in 2012).
During the LHC run, the detector operated with data taking efficiency above
$90\%$, with instantaneous luminosity around $3\,(4)\times
10^{32}\rm\,cm^{-2}\rm{\,s}^{-1}$ recording data samples of
$1\,(2)\mbox{\,fb}^{-1}$ at $\sqrt{s}=7\,(8)\mathrm{\,Te\kern-1.00006ptV}$ in
2011 (2012).121212 Note that these values already exceed the LHCb design
luminosity of $2\times 10^{32}\rm\,cm^{-2}\rm{\,s}^{-1}$. The luminosity is
less than that delivered to ATLAS and CMS, since the experimental design
requires low pile-up, i.e. a low number of $pp$ collisions per bunch-crossing.
However, this allows the luminosity to be “levelled” and remain at a constant
value throughout the LHC fill, providing very stable data taking-
conditions.131313 Similar stability was achieved at $e^{+}e^{-}$ colliders by
a completely different method referred to as trickle (or continuous)
injection. In addition, the polarity of the dipole magnet is reversed
regularly, which enables cancellation of detector asymmetries in various
measurements.
In addition to LHCb, it must be noted that the “general purpose detectors”
ATLAS and CMS at the LHC, and CDF and D0 at the Tevatron, have capability to
study flavour physics. For most of these experiments, their programme is,
however, restricted to decay modes triggered by high $p_{\rm T}$ muons, but
CDF benefited from a two-track trigger Ristori:2010zz that enabled a broader
range of measurements to be performed.
### 4.1 Heavy flavour production and spectroscopy
The capabilities of the different experiments can be demonstrated from the
measurements of production cross-sections that have been performed by each.
Most have studied ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ production
(e.g. Refs. Acosta:2004yw ; Aad:2011sp ; Chatrchyan:2011kc ; LHCb-
PAPER-2011-003 ; LHCb-PAPER-2012-039 ; LHCb-PAPER-2013-016 ) as well as $b$
hadron production using decay modes involving muons or
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ mesons Aaltonen:2007zza ;
Aaltonen:2009xn ; Abachi:1994kj ; Aad:2012jga ; Chatrchyan:2012hw ; LHCb-
PAPER-2010-002 . However, only CDF and LHCb have been able to study prompt
charm production Acosta:2003ax ; LHCb-PAPER-2012-041 .141414 Measurements of
charm production and other processes by ALICE are not included in this
discussion. Although ALICE can study production at low luminosity, it cannot
perform competitive studies of flavour changing processes. The cross-sections
measured confirm the theoretical predictions, and enable the values for
integrated luminosity to be translated into more easily comprehensible terms.
For example, with $1\mbox{\,fb}^{-1}$ recorded at
$\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}$, and the measured $b\bar{b}$
production cross-section LHCb-PAPER-2010-002 ; LHCb-PAPER-2011-018 , it is
easily shown that over $10^{11}$ $b\bar{b}$ quark pairs have been produced in
the LHCb acceptance. This compares to the combined BaBar and Belle data sample
of $\sim 10^{9}$ $B\bar{B}$ meson pairs. Consequently, for any channel where
the efficiency, including effects from reconstruction, trigger and offline
selection, is not too small, LHCb has the world’s largest data sample. This
further emphasises the need for an excellent trigger in order to perform
flavour physics at hadron colliders.
Production measurements such as those mentioned above test QCD models, and are
important (and highly-cited) results. However, since they are not within the
remit of flavour-changing interactions of the charm and beauty quarks, they
will not be discussed further here. Nonetheless, a brief digression into
studies of another aspect of QCD, that of spectroscopy, will be worthwhile.
This covers the study of properties of hadronic states such as lifetimes,
masses, decay channels and quantum numbers, and also the discoveries of new
states. Indeed, some of the most highly-cited papers from recent flavour
physics experiments relate to such topics, including the discovery of the
$X(3872)$ particle by Belle Choi:2003ue and of the $D_{sJ}$ states by BaBar
Aubert:2003fg and CLEO Besson:2003cp . The first new particles discovered at
the LHC, prior to the Higgs boson, were hadronic states Aad:2011ih ;
Chatrchyan:2012ni ; LHCB-PAPER-2012-012 . More recently, significant progress
has been made in understanding the nature of the $X(3872)$ LHCb-PAPER-2013-001
. New results are eagerly anticipated in several related areas, for example to
clarify the situation regarding the existence of charged charmonium-like
states, claimed by Belle Choi:2007wga ; Mizuk:2009da ; Mizuk:2008me but not
confirmed by BaBar Aubert:2008aa ; Lees:2011ik , which would be smoking gun
signatures for an exotic hadronic nature.151515 New claims of charged
charmonium-like states have recently been made Ablikim:2013mio ; Liu:2013dau .
Recent claims of charged bottomonium-like states by Belle Belle:2011aa ;
Adachi:2012cx seem to strengthen the case that such exotics can exist in
nature, but one should note that history teaches us that not all claimed
states turn out to be real Hom:1976cv .
The topic of spectroscopy also provides a useful illustration of the
importance of triggering for flavour physics experiments at hadron colliders.
In 2008, the BaBar experiment discovered the $\eta_{b}$ meson using the
process $e^{+}e^{-}\rightarrow\Upsilon(3S)\rightarrow\eta_{b}\gamma$, where
only the photon is reconstructed and the signal is inferred from a peak in the
photon energy spectrum Aubert:2008ba . The $\eta_{b}$ meson is the
pseudoscalar $b\bar{b}$ ground state. It is the lightest bottomonium state, so
why did it take more than 30 years after the discovery of the $\Upsilon(1S)$
meson Herb:1977ek (the lightest vector $b\bar{b}$ state) to see it in
experiments? In particular, since hadron collisions produce particles with all
possible quantum numbers, why was it not observed at, e.g., the Tevatron? The
answer lies in the fact that the vector state decays to dimuons, which have a
distinctive trigger signature. The dominant decay channels of the $\eta_{b}$
are expected to be multibody hadronic final states, which make its observation
in a hadronic environment extremely challenging.
## 5 Key observables in the LHC era
### 5.1 Direct $C\\!P$ violation
As mentioned above, a condition for direct $C\\!P$ violation is
$\left|\bar{A}_{\bar{f}}/A_{f}\right|\neq 1$. In order for this to be realised
we need the amplitude to be composed of at least two parts with different weak
and strong phases. This is often realised by tree ($T$) and penguin ($P$)
amplitudes (example diagrams are shown in Fig. 7), so that
$A_{f}=\left|T\right|e^{i(\delta_{T}-\phi_{T})}+\left|P\right|e^{i(\delta_{P}-\phi_{P})}\
\ {\rm and}\ \
\bar{A}_{\bar{f}}=\left|T\right|e^{i(\delta_{T}+\phi_{T})}+\left|P\right|e^{i(\delta_{P}+\phi_{P})}\,,$
(25)
where the strong (weak) phases $\delta_{T,P}$ ($\phi_{T,P}$) keep the same
(change) sign under the $C\\!P$ transformation by definition. The $C\\!P$
asymmetry is defined from the rate difference between the particle involving
the quark ($D$ or $\bar{B}$) and that containing the antiquark ($\bar{D}$ or
$B$). Using the definition for $B$ decays, we trivially find
$A_{C\\!P}=\frac{\left|\bar{A}_{\bar{f}}\right|^{2}-\left|A_{f}\right|^{2}}{\left|\bar{A}_{\bar{f}}\right|^{2}+\left|A_{f}\right|^{2}}=\frac{2\left|T\right|\left|P\right|\sin(\delta_{T}-\delta_{P})\sin(\phi_{T}-\phi_{P})}{\left|T\right|^{2}+\left|P\right|^{2}+2\left|T\right|\left|P\right|\cos(\delta_{T}-\delta_{P})\cos(\phi_{T}-\phi_{P})}\,.$
(26)
Therefore, for large direct $C\\!P$ violation effects to occur, we need
$\left|P/T\right|$, $\sin(\delta_{T}-\delta_{P})$ and
$\sin(\phi_{T}-\phi_{P})$ to all be ${\cal O}(1)$.
Figure 7: SM (left) tree and (right) penguin diagrams for the decays
$B^{0}\rightarrow K^{+}\pi^{-}$.
Charmless $B$ decays, i.e. decays of $B$ mesons to final states that do not
contain charm quarks, provide good possibilities for the observation of direct
$C\\!P$ violation, since many decays have both tree and penguin contributions
with similar levels of CKM suppression. These are of interest to search for
NP, since the penguin loop diagrams are sensitive to potential contributions
from new particles. An excellent example is $B^{0}\rightarrow K^{+}\pi^{-}$,
which provided the first observation of direct $C\\!P$ violation outside the
kaon sector, and has a world average value of $A_{C\\!P}(B^{0}\rightarrow
K^{+}\pi^{-})=-0.086\pm 0.007$ Lees:2012kx ; Duh:2012ie ; LHCb-PAPER-2012-029
; CDFnote10726 ; Amhis:2012bh . Curiously, the $C\\!P$ violation effect
observed in $B^{+}\rightarrow K^{+}\pi^{0}$ decays is rather different:
$A_{C\\!P}(B^{+}\rightarrow K^{+}\pi^{0})=0.040\pm 0.021$ Lees:2012kx ;
Duh:2012ie ; Amhis:2012bh , although naïvely changing the spectator quark in
Fig. 7 suggests that similar values should be expected. This is referred to as
the “$K\pi$ puzzle”, and could in principle be a hint for NP, though the more
mundane explanation of larger than expected QCD corrections cannot be ruled
out at present. Several methods are available to test the QCD explanations,
which motivate improved measurements of other $K\pi$ modes (in particular, of
$A_{C\\!P}(B^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{0})$), of
similar decay modes with three-body final states ($K\rho,K^{*}\pi$), and of
charmless two-body $B^{0}_{s}$ decays. On this last topic, following
pioneering work by CDF Aaltonen:2011jv ; CDFnote10726 , LHCb has recently
reported both the first decay time-dependent analysis of $B^{0}_{s}\rightarrow
K^{+}K^{-}$ LHCb-CONF-2012-007 and the first observation of $C\\!P$ violation
in $B^{0}_{s}\rightarrow K^{-}\pi^{+}$ decays LHCb-PAPER-2013-018 , which
demonstrate good prospects for progress in the coming years.
With regard to three-body decays, it is worth noting that despite hundreds of
measurents by the $B$ factories, the significance of the world average in any
other charmless $B^{+}$ or $B^{0}$ decay mode does not exceed $5\,\sigma$,
though channels such as $B^{+}\rightarrow\eta K^{+}$ and
$B^{+}\rightarrow\rho^{0}K^{+}$ approach this level. However, very recently,
LHCb has demonstrated that large $C\\!P$ violation effects occur in specific
regions of the phase space of three-body charmless decays such as
$B^{+}\rightarrow K^{+}\pi^{+}\pi^{-}$ LHCb-CONF-2012-018 ; LHCb-CONF-2012-028
; LHCb-PAPER-2013-027 . Further study is necessary to quantify the effect and
identify its origin.
### 5.2 The UT angle $\gamma$ from $B\rightarrow DK$ decays
The angle $\gamma$ of the CKM Unitarity Triangle is unique in that it is the
only $C\\!P$-violating parameter that can be measured using only tree-level
decays. This makes it a benchmark Standard Model reference point. Improving
the precision with which $\gamma$ is known is one of the primary goals of
contemporary flavour physics, and this will only become more important after
NP is discovered, since it will be essential to disentangle SM and NP
contributions to $C\\!P$-violating observables.
The phase $\gamma$ can be determined exploiting the fact that in decays of the
type $B\rightarrow DK$, the $b\rightarrow c\bar{u}s$ and $b\rightarrow
u\bar{c}s$ amplitudes can interfere if the neutral charmed meson is
reconstructed in a final state that is accessible to both $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays. There are many possible
such final states, with various experimental advantages and disadvantages.
These include $C\\!P$ eigenstates, doubly- or singly-Cabibbo-suppressed decays
and multibody final states. Moreover, decays of different $b$ hadrons can all
be used to provide constraints on $\gamma$. Two particularly interesting
approaches are to study decay time-dependent asymmetries of
$B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}$ decays Aleksan:1991nh and to study
the Dalitz plot (i.e. phase-space) dependent asymmetries in $B^{0}\rightarrow
DK^{+}\pi^{-}$ decays Gershon:2008pe ; Gershon:2009qc . First results from
LHCb show promising potential for these decays LHCb-CONF-2012-029 ; LHCb-
PAPER-2013-022 . All such measurements will help to improve the overall
precision in a combined fit.
Figure 8: Illustration of the concept behind the determination of $\gamma$
using $B^{\pm}\rightarrow DK^{\pm}$ decays. For $B^{-}$ decays the amplitudes
add with relative phase $\delta-\gamma$, while for $B^{+}$ the relative phase
is $\delta+\gamma$. Here the simplest case with $D$ decays to $C\\!P$
eigenstates (such as $K^{+}K^{-}$) is shown, but the method can be extended to
any final state accessible to both $D^{0}$ and $\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}$ decays.
The basic concept behind the method is illustrated in Fig. 8 for
$B^{-}\rightarrow D_{C\\!P}K^{-}$ decays. It must be emphasised that due to
the absence of loop contributions to the decay it is extremely clean
theoretically Brod:2013sga . This, and the abundance of different final states
accessible, means that all parameters can be determined from data. The
relevant parameters are the weak phase $\gamma$, an associated strong ($C\\!P$
conserving) phase difference between the $b\rightarrow c\bar{u}s$ and
$b\rightarrow u\bar{c}s$ decay amplitudes, labelled $\delta_{B}$, and the
ratio of their magnitudes, $r_{B}$. The small value of $r_{B}(B^{-}\rightarrow
DK^{-})\sim 10\,\%$ means that large event samples are necessary to obtain
good constraints on $\gamma$, and only recently has the first $5\sigma$
observation of $C\\!P$ violation in $B\rightarrow DK$ decays been achieved
LHCb-PAPER-2012-001 . Larger values of $r_{B}$ are expected in
$B^{0}\rightarrow DK^{*0}$ and $B^{0}_{s}\rightarrow D^{\mp}_{s}K^{\pm}$
decays, but until now the samples available in these channels have not been
been sufficient to give meaningful constraints on $\gamma$. The available
measurements use $B^{(*)-}\rightarrow D^{(*)}K^{(*)-}$ decays, with the latest
combinations from each experiment giving (BaBar)
$\gamma=(69\,^{+17}_{-16})^{\circ}$ Lees:2013zd , (Belle)
$\gamma=(68\,^{+15}_{-14})^{\circ}$ Trabelsi:2013uj and (LHCb)
$\gamma=(71\,^{+15}_{-16})^{\circ}$ LHCb-PAPER-2013-020 . Significant progress
in this area is anticipated from LHCb in the coming years.161616 Updates using
more data have already started to appear from LHCb LHCb-CONF-2013-004 ; LHCb-
CONF-2013-006 .
### 5.3 Mixing and $C\\!P$ violation in the $B^{0}_{s}$ system
A complete analysis of the time-dependent decay rates of neutral $B$ mesons
must also take into account the lifetime difference between the eigenstates of
the effective Hamiltonian, denoted by $\Delta\Gamma$. This is particularly
important in the $B^{0}_{s}$ system, since the value of $\Delta\Gamma_{s}$ is
non-negligible. Neglecting $C\\!P$ violation in mixing, the relevant
replacements for Eq. (22) are Dunietz:2000cr
$\begin{array}[]{lcr}\lx@intercol\Gamma(\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\,{}_{\rm phys}\rightarrow
f(t))={\cal
N}\frac{e^{-t/\tau(B^{0}_{s})}}{4\tau(B^{0}_{s})}\Big{[}\cosh(\frac{\Delta\Gamma
t}{2})+\hfil\lx@intercol&\\\ &\lx@intercol\hfil S_{f}\sin(\Delta
mt)-C_{f}\cos(\Delta mt)+A^{\Delta\Gamma}_{f}\sinh(\frac{\Delta\Gamma
t}{2})\Big{]},\lx@intercol\\\ \lx@intercol\Gamma(B^{0}_{s}\,{}_{\rm
phys}\rightarrow f(t))={\cal
N}\frac{e^{-t/\tau(B^{0}_{s})}}{4\tau(B^{0}_{s})}\Big{[}\cosh(\frac{\Delta\Gamma
t}{2})-\hfil\lx@intercol&\\\ &\lx@intercol\hfil S_{f}\sin(\Delta
mt)+C_{f}\cos(\Delta mt)+A^{\Delta\Gamma}_{f}\sinh(\frac{\Delta\Gamma
t}{2})\Big{]}.\lx@intercol\\\ \end{array}$ (27)
where ${\cal N}$ is a normalisation factor and
$A^{\Delta\Gamma}_{f}=-\frac{2\,{\rm
Re}(\lambda_{f})}{1+|\lambda_{f}|^{2}}\,.$ (28)
Note that, by definition,
$\left(S_{f}\right)^{2}+\left(C_{f}\right)^{2}+\left(A^{\Delta\Gamma}_{f}\right)^{2}=1\,.$
(29)
Also $A^{\Delta\Gamma}_{f}$ is a $C\\!P$-conserving parameter, unlike $S_{f}$
and $C_{f}$ (since it appears with the same sign in equations for both $\kern
1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$ and $B^{0}_{s}$ states).
Nonetheless, it provides sensitivity to ${\rm arg}(\lambda_{f})$, which means
that interesting results can be obtained from untagged time-dependent analyses
(a.k.a. effective lifetime measurements Fleischer:2011cw ).
The formalism of Eq. (27) is usually invoked for the determination of the
$C\\!P$ violation phase in $B^{0}_{s}$ oscillations, $\phi_{s}=-2\beta_{s}$,
using $B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\phi$
decays. However, in that case things are complicated even further by the fact
that the final state, containing two vector particles, is an admixture of
$C\\!P$-even and $C\\!P$-odd which must be disentangled by angular
analysis.171717 A somewhat more straightforward analysis can be done with the
$B^{0}_{s}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}f_{0}(980)$
decay LHCb-PAPER-2012-006 . Moreover, there is a potential contribution from
S-wave $K^{+}K^{-}$ pairs within the $\phi$ mass window used in the analysis.
However, all of these features can be turned to the benefit of the analysis,
providing better sensitivity and allowing to resolve an ambiguity in the
results LHCb-PAPER-2011-028 . A compilation of the latest results is shown in
Fig. 9.181818 Very recent LHCb LHCb-PAPER-2013-002 and ATLAS ATLAS:2013nla
updates are not included. Although great progress has been made over the last
few years, the experimental precision does not yet challenge the theoretical
uncertainty, and so further updates are of great interest.
Figure 9: Compilation of the latest results Amhis:2012bh in the $\phi_{s}$
vs. $\Delta\Gamma_{s}$ plane from LHCb LHCb-CONF-2012-002 , CDF
Aaltonen:2012ie , D0 Abazov:2011ry and ATLAS Aad:2012kba .
### 5.4 Mixing-induced $C\\!P$ violation in hadronic $b\rightarrow s$ penguin
decay modes
As discussed in Sec. 5.1, decay modes mediated by penguin diagrams are
potentially sensitive to NP effects, although it is a considerable challenge
to disentangle QCD effects. One useful approach is to study mixing-induced
$C\\!P$ violation effects in channels that are dominated by the penguin
transition, so that little or no tree (or other) contribution is expected.
Such channels include $B^{0}\rightarrow\phi K^{0}_{\rm\scriptscriptstyle S}$,
$B^{0}\rightarrow\eta^{\prime}K^{0}_{\rm\scriptscriptstyle S}$,
$B^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}K^{0}_{\rm\scriptscriptstyle
S}K^{0}_{\rm\scriptscriptstyle S}$, $B^{0}_{s}\rightarrow\phi\phi$ and
$B^{0}_{s}\rightarrow K^{*0}\kern
1.99997pt\overline{\kern-1.99997ptK}{}^{*0}$.191919 The decay
$B^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle S}\pi^{0}$ is also of great
interest since the tree contribution can be controlled using isospin relations
to other $B\rightarrow K\pi$ decays. For the $B^{0}$ decays, the formalism is
the same as given in Eq. (22), and the parameters are expected in the SM to be
given, to good approximations, by $C_{f}\approx 0$, $S_{f}\approx\sin(2\beta)$
(up to a sign, given by the $C\\!P$ eigenvalue of the final state). These
channels have been studied extensively by BaBar and Belle: early results
provided hints for discrepancies with the SM predictions, but the significance
of the deviation diminshed with improved results Lees:2012kxa ;
Nakahama:2010nj ; Aubert:2008ad ; Chen:2006nk ; Lees:2011nf . For the
$B^{0}_{s}$ decays, the formalism is as given in Eq. (27) (though with
modifications due to the vector-vector final states), and the SM expectation
is that $C\\!P$ violation effects vanish, to a good approximation, since the
very small phase in the $b\rightarrow s$ decay cancels that in the
$B^{0}_{s}$–$\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}$
oscillations. First results have been reported by LHCb LHCb-PAPER-2011-012 ;
LHCb-PAPER-2012-004 ; LHCb-PAPER-2013-007 , and will reach a very interesting
level of sensitivity as more data is accumulated.
### 5.5 Charm mixing and $C\\!P$ violation
In the charm system the mixing parameters $x=\Delta m/\Gamma$ and
$y=\Delta\Gamma/(2\Gamma)$ are both small, $x,y\ll 1$. Therefore, a Taylor
expansion can be performed on the generic expression of Eq. (27) to give
$\begin{array}[]{lcr}\Gamma(\kern
1.99997pt\overline{\kern-1.99997ptD}{}^{0}\,_{\rm phys}\rightarrow
f(t))&=&{\cal
N}\frac{e^{-t/\tau(D^{0})}}{4\tau(D^{0})}\Big{[}1-C_{f}+\left(S_{f}x+A^{\Delta\Gamma}_{f}y\right)\Gamma
t\Big{]},\\\ \Gamma(D^{0}\,_{\rm phys}\rightarrow f(t))&=&{\cal
N}\frac{e^{-t/\tau(D^{0})}}{4\tau(D^{0})}\Big{[}1+C_{f}-\left(S_{f}x-A^{\Delta\Gamma}_{f}y\right)\Gamma
t\Big{]}.\end{array}$ (30)
Hence an untagged analysis of $D^{0}\rightarrow K^{+}K^{-}$ can measure
$A^{\Delta\Gamma}_{f}y$ (also known as $y_{C\\!P}$), while a tagged analysis
can additionally probe $S_{f}x$. Since the mixing parameters are small, the
focus until now has been to establish definitively oscillation effects, but in
the coming years the main objective will be to observe or limit $C\\!P$
violation in the charm system, which is expected to be very small in the SM.
Note that in case the source of $D^{0}$ mesons is either from $D^{*+}$ decays
or semileptonic $b$-hadron decays, the flavour tagging is very effectively
achieved from the charge of the associated pion or lepton, respectively. Many
other final states can be used to gain additional sensitivity to charm mixing
and $C\\!P$ violation parameters, a recent example being the observation of
charm mixing at LHCb using $D^{0}\rightarrow K^{+}\pi^{-}$ decays LHCb-
PAPER-2012-038 . The result of this analysis, and the world average
constraints on the $x$ and $y$ parameters in the $D^{0}$ system,202020 Note
that in Fig. 10, the definition of the $x$ and $y$ parameters in the charm
system is different from that in Sec. 2.2 – in this definition the $C\\!P$
violating phase in $D^{0}$ oscillations is assumed to be small, and $x$ can be
either positive or negative. are shown in Fig. 10.
Figure 10: (Left) Decay-time evolution of the ratio, $R$, of
$D^{0}\rightarrow K^{+}\pi^{-}$ to $D^{0}\rightarrow K^{-}\pi^{+}$ yields
(points) with the projection of the mixing allowed (solid line) and no-mixing
(dashed line) fits overlaid, from Ref. LHCb-PAPER-2012-038 ; (Right) World
average constraints on the $x$ and $y$ parameters in the $D^{0}$ system
Amhis:2012bh .
Direct $C\\!P$ violation in the charm system can also be used to test the SM.
One interesting recent result has been the measurement of $\Delta A_{C\\!P}$,
which is the difference between the direct $C\\!P$ violation parameters of
$D^{0}\rightarrow K^{+}K^{-}$ and $D^{0}\rightarrow\pi^{+}\pi^{-}$ decays. By
measuring the difference, a cancellation of production and detection
asymmetries can be exploited, while the physical $C\\!P$ asymmetry may be
enhanced.212121 The $C\\!P$ asymmetries in $D^{0}\rightarrow K^{+}K^{-}$ and
$D^{0}\rightarrow\pi^{+}\pi^{-}$ decays are expected to be of opposite sign
due to U-spin symmetry. This method was first used by LHCb LHCb-
PAPER-2011-023 and then by CDF Aaltonen:2012qw and Belle Ko:ICHEP , all
indicating a larger than expected effect. This prompted a great deal of
theoretical activity, summarised in Ref. LHCb-PAPER-2012-031 , with the
conclusion that a SM origin of the $C\\!P$ violation, although unlikely, was
not ruled out. Many further studies were proposed to test both SM and NP
hypotheses, and these remain of great interest and will be pursued. However,
the most recent results by LHCb LHCb-PAPER-2013-003 ; LHCb-CONF-2013-003
suggest that the central value is smaller than previous thought, and therefore
the SM explanation becomes harder to rule out.
### 5.6 Photon polarisation in radiative $B$ decays
The $b\rightarrow s\gamma$ transition is an archetypal flavour-changing
neutral-current (FCNC) transition, and has been considered a sensitive probe
for NP since the first measurements of its rate Ammar:1993sh ; Alam:1994aw .
The latest results for the inclusive branching fraction Amhis:2012bh are
consistent with the SM prediction Misiak:2006zs
$\displaystyle{\cal B}\left(B\rightarrow X_{s}\gamma\right)^{\rm
exp}_{E_{\gamma}>1.7\mathrm{\,Ge\kern-0.70004ptV}}$ $\displaystyle=$
$\displaystyle(3.43\pm 0.21\pm 0.07)\times 10^{-4}\,,$ (31)
$\displaystyle{\cal B}\left(B\rightarrow X_{s}\gamma\right)^{\rm
th}_{E_{\gamma}>1.7\mathrm{\,Ge\kern-0.70004ptV}}$ $\displaystyle=$
$\displaystyle(3.15\pm 0.23)\times 10^{-4}\,.$ (32)
However, additional observables, such as $C\\!P$ and isospin asymmetries
provide complementary sensitivity and still have experimental uncertainties
much larger than those of the theoretical predictions of their values in the
SM.
One particularly interesting observable is the polarisation of the emitted
photon in $b\rightarrow s\gamma$ decays, since the $V-A$ structure of the SM
weak interaction results in a high degree of polarisation, that is not
necessarily reproduced in extended models. Until now, the most promising
approach to probe the polarisation has been from time-dependent asymmetry
measurements of $B^{0}\rightarrow K^{0}_{\rm\scriptscriptstyle
S}\pi^{0}\gamma$ Atwood:1997zr ; Atwood:2004jj but the most recent
measurements Aubert:2008gy ; Ushiroda:2006fi still have large uncertainties.
LHCb can use several different methods to study photon polarisation in
$b\rightarrow s\gamma$ transitions, such as measuring the effective lifetime
in $B^{0}_{s}\rightarrow\phi\gamma$ decays Muheim:2008vu . Although all such
measurements are highly challenging, the observed yields in
$B^{0}_{s}\rightarrow\phi\gamma$ LHCb-PAPER-2012-019 and other related
channels such as $B^{0}\rightarrow K^{*0}e^{+}e^{-}$ LHCb-PAPER-2013-005
suggest there are good prospects for significant progress in the coming years.
### 5.7 Angular observables in $B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays
The $b\rightarrow sl^{+}l^{-}$ FCNC transitions offer similar, but
complementary, sensitivity to NP as $b\rightarrow s\gamma$, but are
experimentally more convenient to study, in particular when the lepton pair is
muonic, i.e. $l^{+}l^{-}=\mu^{+}\mu^{-}$. The multi-body final state provides
access to a wide range of kinematic observables, several of which have clean
theoretical predictions (especially at low values of the dilepton invariant
mass squared, $q^{2}$). This makes these decays a superb laboratory for NP
tests.
The theoretical framework for these (and other) processes is the operator
product expansion. This is an effective theory, applicable for $b$ physics,
which describes the weak interactions of the SM by integrating out the heavier
($W$, $Z$, $t$) fields. As such it can be considered a modern version of the
Fermi theory of beta decay. Conceptually, it can be expressed as
${\cal L}_{(\rm full\ EW\times QCD)}\longrightarrow{\cal L}_{\rm eff}={\cal
L}_{\rm QED\times QCD}{\small\left(\begin{array}[]{c}{\rm quarks\neq t}\\\
{\rm leptons}\end{array}\right)}+\Sigma_{n}C_{n}{\cal O}_{n}\,,$ (33)
where ${\cal O}_{n}$ represent the local interaction terms, and $C_{n}$ are
coupling constants that are referred to as Wilson coefficients.222222 As
written here the $C_{n}$ include the Fermi coupling and the CKM matrix
elements, but usually these terms are factored out. The Wilson coefficients
encode information on the weak scale, and are calculable and known in the SM
(at least to leading order). Moreover, they are affected by NP, so comparing
the measured values with their expectations provides tests of the SM. A more
detailed description of the operator product expansion can be found in, e.g.
Ref. Buras:2005xt .
For the purposes of discussing $b\rightarrow sl^{+}l^{-}$ decays, the Wilson
coefficients of interest are $C_{7}$ (which also affects $b\rightarrow
s\gamma$), $C_{9}$ and $C_{10}$. The differential decay distribution, for the
inclusive process, can be written Lee:2006gs
$\frac{d^{2}\Gamma}{dq^{2}\,d\cos\theta_{l}}=\frac{3}{8}\left[(1+\cos^{2}\theta_{l})H_{T}(q^{2})+2\cos\theta_{l}H_{A}(q^{2})+2(1-\cos^{2}\theta_{l})H_{L}(q^{2})\right]$
(34)
where $\theta_{l}$ is the angle between the momentum vectors of the positively
charged lepton and the opposite of the decaying $b$ hadron in the dilepton
rest frame.232323 The full decay distribution for $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ and other $B\rightarrow Vl^{+}l^{-}$
($V=\rho,\omega,K^{*},\phi$) decays includes two other angles: the decay angle
of the vector meson (usually denoted $\theta_{V}$) and the angle between the
two decay planes (usually denoted $\phi$). The coefficients are given by
$\displaystyle H_{T}(q^{2})$ $\displaystyle\propto$ $\displaystyle
2q^{2}\left[\left(C_{9}+2C_{7}\frac{m_{b}^{2}}{q^{2}}\right)^{2}+C_{10}^{2}\right]$
$\displaystyle H_{A}(q^{2})$ $\displaystyle\propto$
$\displaystyle-4q^{2}C_{10}\left(C_{9}+2C_{7}\frac{m_{b}^{2}}{q^{2}}\right)$
(35) $\displaystyle H_{L}(q^{2})$ $\displaystyle\propto$
$\displaystyle\left[\left(C_{9}+2C_{7}\right)^{2}+C_{10}^{2}\right]\,.$
Note that the term involving $H_{A}$ depends linearly on $\cos\theta_{l}$ and
hence gives rise to a $q^{2}$-dependent forward backward asymmetry, $A_{\rm
FB}$. The shape of $A_{\rm FB}$, in particular the value of $q^{2}$ at which
it crosses zero, can be predicted with low uncertainty in the SM. The
expressions for exclusive processes, such as $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$, are conceptually similar to those of Eqs. (34) and
(5.7), but are more complicated as they also involve hadronic form factors. On
the other hand, exclusive channnels also provide additional observables that
can be studied (such as the longitudinal polarisation of the $K^{*0}$ meson,
$F_{\rm L}$), some of which can be precisely predicted in the SM, and are
sensitive to NP contributions.
Figure 11: (Left) Differential branching fraction and (right) $A_{\rm FB}$ of
$B^{0}\rightarrow K^{*0}\mu^{+}\mu^{-}$ decays in bins of $q^{2}$ as measured
by LHCb LHCb-PAPER-2013-019 .
The decay rates and angular distributions of $B^{0}\rightarrow
K^{*0}\mu^{+}\mu^{-}$ decays have been studied by many experiments, with the
most precise results to date, from LHCb LHCb-PAPER-2013-019 , shown in Fig.
11. This analysis provides the first measurement of the $A_{\rm FB}$ zero
crossing point, $q_{0}^{2}=4.9\pm 0.9\,{\rm GeV}^{2}/c^{4}$, consistent with
the SM prediction. Significant progress, including improved measurements of
other NP-sensitive angular observables, can be expected in the coming years.
### 5.8 The very rare decay $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$
The “killer app.” for flavour physics as a tool to probe for (and potentially
discover) NP is the very rare decay $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$. The
branching fraction is highly suppressed in the SM due to a combination of
three factors, none of which are necessarily reproduced in extended models:
(i) the absence of tree-level FCNC transitions; (ii) the $V-A$ structure of
the weak interaction that results in helicity suppression of purely leptonic
decays of flavoured pseudoscalar mesons; (iii) the hierarchy of the CKM matrix
elements. In particular, in the minimally supersymmetric extension of the SM,
the presence of a pseudoscalar Higgs particle alleviates the helicity
suppression and enhances the branching fraction by a factor proportional to
$\tan^{6}\beta/M_{A_{0}}^{4}$, where $\tan\beta$ is the ratio of Higgs’ vacuum
expectation values, and $M_{A_{0}}$ is the pseudoscalar Higgs mass. Therefore,
in the region of phase-space where $\tan\beta$ is not too small, and
$M_{A_{0}}$ is not too large, the decay rate can be significantly enhanced
above its SM expectation Buras:2012ru ,242424 Note that, due to the non-zero
value of the decay width difference in the $B^{0}_{s}$ system, this value
needs to be corrected upwards by $\sim 9\%$ to obtain the experimentally
measured (i.e., decay time integrated) quantity DeBruyn:2012wk .
${\cal B}\left(B^{0}_{s}\rightarrow\mu^{+}\mu^{-}\right)^{\rm SM}=\left(3.2\pm
0.3\right)\times 10^{-9}\,.$ (36)
Due to the very clean signature of this decay, it has been studied by
essentially all high-energy hadron collider experiments. The crucial
components to obtain good sensitivity are high luminosity, a large $B$
production cross-section within the acceptance, and good vertex and mass
resolution to reject the background. Although ATLAS Aad:2012pn and CMS
Chatrchyan:2012rga have collected more luminosity, at present the strengths
of the LHCb detector have allowed it to obtain the most precise results for
this mode. Following a series of increasingly restrictive upper limits LHCb-
PAPER-2011-004 ; LHCb-PAPER-2011-025 ; LHCb-PAPER-2012-007 , LHCb has recently
obtained the first evidence, with $3.5\sigma$ significance, for the decay
LHCb-PAPER-2012-043 , as shown in Fig. 12. The branching fraction is measured
to be
${\cal
B}\left(B^{0}_{s}\rightarrow\mu^{+}\mu^{-}\right)=\left(3.2\,^{+1.4}_{-1.2}({\rm
stat})\,^{+0.5}_{-0.3}({\rm syst})\right)\times 10^{-9}\,,$ (37)
in agreement with the SM prediction.
Figure 12: Invariant mass distribution of selected
$B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ candidates, with fit result overlaid
LHCb-PAPER-2012-043 .
Further updates of this measurement are keenly anticipated, and are likely to
appear at regular intervals throughout the lifetime of the LHC. It is worth
noting that even in case the $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$ branching
fraction remains consistent with the SM, the decay provides an additional
handle on NP through its effective lifetime Buras:2013uqa . Moreover, it will
be important to study also the even more suppressed
$B^{0}\rightarrow\mu^{+}\mu^{-}$ decay, since the ratio of the $B^{0}$ and
$B^{0}_{s}$ branching fractions is a benchmark test of MFV.
## 6 Future flavour physics experiments
As stressed in the previous sections, the first results from the LHC have
already provided dramatic advances in flavour physics, and significant further
progress is anticipated in the coming years. However, the instantaneous
luminosity of LHCb is limited due to the fact that its subdetectors are read
out at $1{\rm\,MHz}$. As shown in Fig. 13 (left), increasing the luminosity
requires tightening of the hardware trigger thresholds in order not to exceed
this limit. This then results in lower efficiencies, especially for decay
channels triggered by the calorimeter (i.e., channels without muons in the
final state), so that there is no net gain in yield. Therefore, after several
years of operation at the optimal instantaneous luminosity at $\sqrt{s}=13\
{\rm or}\ 14\mathrm{\,Te\kern-1.00006ptV}$,252525 Note that heavy flavour
cross-sections increase with $\sqrt{s}$, so a significant boost in yields is
expected when moving to higher energies. the time required to double the
accumulated statistics becomes excessively long.
Figure 13: (Left) Scaling of yields with instantaneous in certain decay
channels at LHCb CERN-LHCC-2012-007 , showing the limitation caused by the
$1{\rm\,MHz}$ readout. Note that during 2012 LHCb operated at an instantaneous
luminosity of $4\times 10^{32}\rm\,cm^{-2}\rm{\,s}^{-1}$. (Right) Illustration
of the key components of the LHCb subdetector upgrades.
As should be clear from the discussions above, it remains of great importance
to pursue a wide range of flavour physics measurements and improve their
precision to the level of the theoretical uncertainty, and therefore it is of
clear interest to get past the $1{\rm\,MHz}$ readout limitation. The concept
of the LHCb upgrade CERN-LHCC-2011-001 ; CERN-LHCC-2012-007 is to read out
the full detector at $40{\rm\,MHz}$ (which corresponds to the maximum bunch
crossing rate, with $25{\rm\,ns}$ spacing) and implement the trigger fully in
software. This will allow the experiment to run at higher luminosities, up to
$1\ {\rm or}\ 2\times 10^{33}\rm\,cm^{-2}\rm{\,s}^{-1}$, and will also
significantly improve the efficiency for modes currently triggered by
calorimeter signals at the hardware level. The accumulated samples in most key
modes will increase by around two orders of magnitude compared to what was
collected in 2011. Moreover, with a flexible trigger scheme, the capability to
search for other signatures of NP will be enhanced, so that the upgraded
experiment can be considered a general purpose detector in the forward region.
The LHCb upgrade is planned to occur during the second long shutdown of the
LHC, in 2018. Since its target luminosity is still below that which can be
delivered by the LHC, it does not depend (though it is consistent with) the
HL-LHC machine upgrade.
There are several other flavour physics experiments that will be coming online
on a similar same timescale. The KEKB accelerator and Belle experiment are
being upgraded Abe:2010sj , in order to allow luminosities almost two orders
of magnitude larger than has previously been achieved. Compared to the LHCb
upgrade, the $e^{+}e^{-}$ environment is advantageous for decay modes with
missing energy and for inclusive measurements. Some of the key channels for
Belle2 are lepton flavour violating decays of $\tau$ leptons, mixing-induced
$C\\!P$ asymmetries in decays such as $B^{0}\rightarrow\phi
K^{0}_{\rm\scriptscriptstyle S}$ and
$B^{0}\rightarrow\eta^{\prime}K^{0}_{\rm\scriptscriptstyle S}$, and the
leptonic decay $B^{+}\rightarrow\tau^{+}\nu$ (which can be considered a
counterpart of $B^{0}_{s}\rightarrow\mu^{+}\mu^{-}$, and is sensitive to the
exchange of charged Higgs particles) Browder:2008em ; Aushev:2010bq .
In addition, the NA62 NA62-10-07 and K0T0 Yamanaka:2012yma experiments will
search for the $K^{+}\rightarrow\pi^{+}\nu\overline{\nu}$ and
$K^{0}_{\rm\scriptscriptstyle L}\rightarrow\pi^{0}\nu\overline{\nu}$ decays,
respectively. Long considered the “holy grail” of kaon physics these decays
are highly suppressed in the SM and have clean theoretical predictions. The
new generation of experiments should be able to observe these channels for the
first time, if they occur at around the SM rate.
## 7 Conclusion
Flavour physics continues to present many mysteries, and these demand
continued experimental and theoretical investigation. Heavy flavour physics is
complementary to other sectors of the global particle physics programme such
as the high-$p_{\rm T}$ experiments at the LHC, and neutrino oscillation and
low energy precision experiments. The prospects are good for significant
progress in the coming few years and, with upgraded experiments planned to
come online in the second half of this decade, beyond.
###### Acknowledgements.
These lectures were delivered, in variously modified forms, at the Hadron
Collider Physics Summer School 2010 in Fermilab, USA, the 2012 Spring School
“Bruno Touschek” of the Frascati National Laboratories in Frascati, Italy and
the $69^{\rm th}$ Scottish Universities Summer School in Physics in St.
Andrews, UK, 2012. The author is grateful to the organisers of these meetings
for invitations and support, and to the participants for stimulating
discussions and questions. This work is supported by the Science and
Technology Facilities Council (United Kingdom), CERN, and by the European
Research Council under FP7.
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|
arxiv-papers
| 2013-06-19T15:50:07 |
2024-09-04T02:49:46.697183
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Tim Gershon",
"submitter": "Tim Gershon",
"url": "https://arxiv.org/abs/1306.4588"
}
|
1306.4615
|
# $K$-Adaptive Partitioning for Survival Data, with an Application to Cancer
Staging
Soo-Heang Eo111Department of Statistics, Korea University, Seoul, Korea Hyo
Jeong Kang222Department of Pathology, Asan Medical Center, Seoul, Korea Seung-
Mo Hong333Corresponding author, Department of Pathology, University of Ulsan
College of Medicine, Asan Medical Center, 88 Olympic-ro 43-gil, Songpa-gu,
Seoul, Korea, 138-736, [email protected] HyungJun Cho444Corresponding author,
Department of Statistics, Korea University, Anam-dong, Seongbuk-gu, Seoul,
Korea, 136-701, [email protected]
###### Abstract
In medical research, it is often needed to obtain subgroups with heterogeneous
survivals, which have been predicted from a prognostic factor. For this
purpose, a binary split has often been used once or recursively; however,
binary partitioning may not provide an optimal set of well separated
subgroups. We propose a multi-way partitioning algorithm, which divides the
data into $K$ heterogeneous subgroups based on the information from a
prognostic factor. The resulting subgroups show significant differences in
survival. Such a multi-way partition is found by maximizing the minimum of the
subgroup pairwise test statistics. An optimal number of subgroups is
determined by a permutation test. Our developed algorithm is compared with two
binary recursive partitioning algorithms. In addition, its usefulness is
demonstrated with a real data of colorectal cancer cases from the Surveillance
Epidemiology and End Results program. We have implemented our algorithm into
an R package kaps, which is freely available in the Comprehensive R Archive
Network (CRAN).
Keywords: Multi-way split, Change point detection, Recursive partitioning,
Staging system, SEER database
## 1 Introduction
Clinicians are interested in obtaining a handful of subgroups or stages with
heterogeneous survivals by partitioning a prognostic factor for prognostic
diagnosis [1]. The tumor node metastasis (TNM) staging system is the most
widely used cancer staging system, and provides critical information about
prognosis and about estimation for responsiveness to specific treatment for
cancer patients [2]. The TNM staging system is composed of three
classifications: T classification based on the extent or size of the primary
tumor, N classification determined by the involvement of the regional lymph
nodes (LNs), and M classification by distant metastasis. Each T, N, or M
classification is decided by grouping cases with similar prognosis. When T
classification, based solely on the size of the primary tumor such as breast
cancer, or N classification, in several gastrointestinal tract cancers, was
determined, an increased tumor size or increased number of metastatic regional
LNs is linked with a worse prognosis of cancer patients.
Figure 1: Tree diagram from the log-rank survival tree for the colorectal
cancer data. Each oval including a split rule depicts an intermediate node and
each rectangle with the node number (Node), the number of observations (n),
and the median survival time (med) describes a terminal node. An observation
goes to the left subnode if and only if the condition is satisfied.
Information related to the intermediate node is presented on the right side of
the ovals.
Such a staging system can be constructed by various partitioning techniques.
Several studies have been conducted for partitioning a prognostic factor [3,
4, 5, 6, 7, 8]. Various test statistics have been employed to obtain subgroups
with different survivals or cancer stages; however, these approaches only
revealed two subgroups such as low- and high-risk patients [9, 10]. Tree-
structured or recursive partitioning methods were also utilized to find an
optimal set of cutpoints [11, 12], so as to obtain several heterogeneous
subgroups. Binary recursive partitioning selects the best point at the first
split, but its subsequent split points may not be optimal in combination. Some
subgroups differ substantially in survival, but others may differ barely or
insignificantly.
For illustration, we consider the data regarding colorectal cancer [2] from
the Surveillance Epidemiology and End Results (SEER), which can be obtained
from the SEER website (http://seer.cancer.gov). The number of metastatic LNs
acts as a prognostic factor to obtain several heterogeneous subgroups with
different levels of survival. For analysis, we selected 65,186 cases with more
than 12 examined LNs because the examination of more than 12 LNs is accepted
for proper evaluation of the prognosis of patients with colorectal cancers
[13]. Figure 1 shows a tree-diagram for the colorectal cancer data by the
tree-based method used in [12]. The Kaplan-Meier survival curves for the
resulting subgroups are also displayed in Figure 2. This indicates that
survivals of some subgroups differ insignificantly or their differences are
not equal-spaced.
We propose an algorithm for overcoming these limitations and introduce a
convenient software program in this paper. Our algorithm evaluates multi-way
split points simultaneously and finds an optimal set of cutpoints for a
prognostic factor. In addition, an optimal number of cutpoints is selected by
a permutation test. The algorithm was implemented into an R package kaps,
which can be used conveniently and freely via the Comprehensive R Archive
Network (CRAN, http://cran.r-project.org/package=kaps).
Figure 2: Kaplan-Meier survival curves of the subgroups selected by the log-
rank survival tree for the colorectal cancer data.
The rest of the paper is organized as follows. In Section 2, we propose novel
staging algorithm called $K$-adaptive partitioning algorithm. In Section 3,
porposed algorithm is compared with two recursive partitioning techniques
through a simulation study. In Section 4, the algorithm is applied to the
colorectal cancer data from the SEER database. In Section 5, our concluding
remarks are provided. In the Appendix, an R package kaps [14] is described and
illustrated with a simple example.
## 2 Proposed method
In this section, we describe and summarize our proposed algorithm for finding
the best split set of cutpoints on a prognostic factor and for selecting the
optimal number of subgroups or cutpoints in survival data. We call the
algorithm $K$-Adaptive Partitioning for Survival data, or KAPS for short.
### 2.1 Finding the best split set
Let $T_{i}$ be a survival time, $C_{i}$ a censoring status, and $X_{i}$ be an
ordered covariate for the $i^{th}$ observation. We observe the triples
$(Y_{i},\delta_{i},X_{i})$ and define
$Y_{i}=\min(T_{i},C_{i})\;\mbox{ and }\;\delta_{i}=I(T_{i}\leq C_{i}),$
which represent the observed response variable and the censoring indicator,
respectively.
Our aim is to divide the whole data $\mathfrak{D}$ into $K$ heterogeneous
subgroups $\mathfrak{D}_{1},\mathfrak{D}_{2},\ldots,\mathfrak{D}_{K}$ based on
the information of $X$. All the heterogeneous subgroups should differ
significantly in survival. Rather than having both extremely poor and well
separated subgroups, it is more useful to have only fairly well separated
subgoups. In other words, all the subgroups need to show greater pairwise
differences than a certain criterion. To achieve this purpose, our algorithm
is constructed in the following manner. Suppose $X$ consists of many unique
values for possible splitting. A split set (denoted by $s$) consisting of one
or more cutpoints on $X$ divides the data $\mathfrak{D}$ into two or more
subgroups. That is, a split set with $(K-1)$ cutpoints generates $K$ disjoint
subgroups. There exists a number of possible split sets (denoted by $S$)
because there are a number of combinations of different cutpoints on $X$.
To compare the subgroups in terms of survival, we can utilize $\chi^{2}$
statistics as test statistics from the log-rank or Gehan-Wilcoxon tests [15].
Let $\chi_{1}^{2}$ be the $\chi^{2}$ statistic with one degree of freedom (df)
for comparing the $g^{th}$ and $h^{th}$ of $K$ subgroups created by a split
set $s_{K}$ when $K$ is given. For a split set $s_{K}$ of $\mathfrak{D}$ into
$\mathfrak{D}_{1},\mathfrak{D}_{2},\ldots,\mathfrak{D}_{K}$, the test
statistic for a measure of deviance can be defined as
$T_{1}(s_{K})=\min_{1\leq g<h\leq K}\chi_{1}^{2}\;\mbox{for}\;s_{K}\in S_{K},$
(1)
where $S_{K}$ is a collection of split sets $s_{K}$ generating $K$ disjoint
subgroups. By this, we find the worst pair with the smallest test statistic
out of the ($K-1$) adjacent pairs of $K$ subgroups constructed by $s_{K}$.
Then, take $s_{K}^{*}$ as the best split set such that
$T_{1}^{*}(s_{K}^{*})=\max_{s_{K}\in S_{K}}T_{1}(s_{K}).$ (2)
The best split $s_{K}^{*}$ is a set of $(K-1)$ cutpoints which clearly
separate the data $\mathfrak{D}$ into $K$ disjoint subsets of the data:
$\mathfrak{D}_{1},\mathfrak{D}_{2},\ldots,\mathfrak{D}_{K}$. The worst pair of
the $K$ subsets should show significant differences in survival. The overall
performance can be evaluated by the overall test statistic
$T_{K-1}^{*}(s_{K}^{*})=\chi_{K-1}^{2}$ statistic for comparing all $K$
subgroups from $s_{K}^{*}$. When $K=2$, the overall test statistic is the same
as (2). When two or more split sets have the maximum of the minimum pairwise
statistics, they can be compared by their overall test statistics. The
algorithm is summarized as follows.
Algorithm 1. Finding the best split set for given $K$
* Step
1: Compute chi-squared test statistics $\chi_{1}^{2}$ for all possible pairs,
$g$ and $h$, of $K$ subgroups by $s_{K}$, where $1\leq g<h\leq K$ and $s_{K}$
is a split set of ($K-1$) cutpoints generating $K$ disjoint subgroups.
* Step
2: Obtain the minimum pairwise statistic $T_{1}(s_{K})$ by minimizing
$\chi_{1}^{2}$ for all possible pairs, $i.e.$, $T_{1}(s_{K})=\min_{1\leq
g<h\leq K}\chi_{1}^{2}\;\mbox{for}\;s_{K}\in S_{K},$where $S_{K}$ is a
collection of split sets $s_{K}$ generating $K$ disjoint subsets of the data.
* Step
3: Repeat Steps 1 and 2 for all possible split sets $S_{K}$.
* Step
4: Take the best split set $s_{K}^{*}$ such that
$T_{1}^{*}(s_{K}^{*})=\max_{s_{K}\in S_{K}}T_{1}(s_{K})$. When two or more
split sets have the maximum $T_{1}^{*}$ of the minimum pairwise statistics,
choose the best split set with the largest overall statistic $T_{K-1}^{*}$.
### 2.2 Selecting the optimal number of subgroups
One of the important issues is to determine a reasonable number of subgroups,
i.e. the selection of an optimal $K$. The binary tree-based approaches [11,
12, 7, 8] find optimal binary splits recursively, and then determine their
tree sizes using certain criteria. As described in Section 2.1, we find an
optimal multi-way split at a time for the given number of subgroups. In
addition, we need to choose only one of a possible number of subgroups. Prior
information in each field may be useful. For a data-driven objective choice,
we here suggest a statistical procedure to choose an optimal number of
subgroups.
Let $s_{K}^{*}$ and $T_{1}^{*}(s_{K}^{*})$ be the best split set and the
minimum pairwise statistic using the raw data for each $K$. The data can be
reconstructed by matching their labels after permuting the labels of $X$ with
retaining the labels of $(Y,\delta)$. The survival time $Y$ is independent of
the covariate $X$ in the reconstructed data, which is called the permuted
data. When the permuted data are allocated into each subgroup by $s_{K}^{*}$,
there should be no significant differences in survival among the subgroups.
The repetition of this procedure generates the null distribution of the test
statistics. If we repeat this procedure many times ($R$ times), and then we
obtain the permutation $p$-value $p_{K}$ for each $K$. This is the ratio where
the minimum pairwise statistics of the permuted data are greater than or equal
to that of the raw data, $i.e.$,
$p_{K}=\sum_{r=1}^{R}I(T_{1}^{(r)}(s_{K}^{*})\geq
T_{1}^{*}(s_{K}^{*}))/R,\;K=2,3,\ldots,$
where $T_{1}^{(r)}(s_{K}^{*})$ is the $r^{th}$ repeated minimum pairwise
statistic for the permuted data. In addition, we correct the $p$-values for
multiple comparison because there are ($K-1$) comparisons between two adjacent
subgroups when there are $K$ subgroups. For example, the corrected $p$-value
can be obtained using Bonferroni correction, $i.e.$,
$p_{K}^{c}={p_{K}}/{(K-1)},K=2,3,\ldots$. Lastly, we choose the largest number
to discover as many significantly different subgroups as possible, given that
the corrected $p$-values are smaller than or equal to a pre-determined
significance level, $e.g.$ $\alpha=0.05$. Formally,
$\hat{K}=\max\\{K|\;p_{K}^{c}\leq\alpha,K=2,3,\ldots\\}.$ (3)
The algorithm is summarized as follows.
Algorithm 2. Selecting the optimal number of subgroups ($K$)
* Step
1: Find $s_{K}^{*}$ and $T_{1}(s_{K}^{*})$ with the raw data for each $K$
using Algorithm 1.
* Step
2: Construct the permuted data by permuting the labels of $X$ whilst retaining
the labels of $(Y,\delta)$.
* Step
3: Allocate the permuted data into each subgroup by $s_{K}^{*}$.
* Step
4: Obtain the minimum pairwise statistic $T_{1}^{(r)}(s_{K}^{*})$ for the
permuted data.
* Step
5: Repeat steps 2 to 4 $R$ times, and then obtain
$T_{1}^{(1)}(s_{K}^{*}),T_{1}^{(2)}(s_{K}^{*}),\ldots,T_{1}^{(R)}(s_{K}^{*})$.
* Step
6: Compute the permutation $p$-value $p_{K}$ for each $K$, $i.e.$,
$p_{K}=\sum_{r=1}^{R}I(T_{1}^{(r)}(s_{K}^{*})\geq
T_{1}(s_{K}^{*}))/R,\;K=2,3,\ldots.$
* Step
7: Correct the permutation $p$-value $p_{K}$ by correcting for multiple
comparisons, $e.g.$, corrected $p$-value
$p_{K}^{c}={p_{K}}/{(K-1)},K=2,3,\ldots$.
* Step
8: Select the largest $K$ when the corrected $p$-values are less than or equal
to $\alpha$, $i.e.$, $\hat{K}=\max\\{K|\;p_{K}^{c}\leq\alpha,K=2,3,\ldots\\}.$
## 3 Simulation study
In this section, we investigate the performance of our proposed method (kaps)
with simulated data. For comparison, we employ two recursive partitioning
algorithms: survival CART [16] and conditional inference tree [17]. The former
is descended from the traditional CART for survival data and the latter is
based on maximally selected rank statistics [5]. They were implemented in the
R packages rpart [18] and party [17], respectively. Our algorithm was
implemented in the R package kaps [14].
### 3.1 Simulation setting
To generate simulated data, we assume that survival time $T_{i}$ is generated
from exponential distribution with a parameter $\lambda_{i}$ and and censoring
times $C_{i}$ is generated from Uniform distribution with appropriate
parameters. Then we observe $Y_{i}=\min(T_{i},C_{i})$ and
$\delta_{i}=I(T_{i}\leq C_{i})$, where $i=1,2,\ldots,n$. In addition, a
prognostic factor $X_{i}$ is generated from a discrete uniform distribution
with a range of 1 and 20, i.e., DU$(1,20)$. We first consider the following
stepwise model (SM) defining parameter $\lambda_{i}$ as follows.
$\lambda_{i}=\begin{cases}0.02,&\;\;\;\;\;\;\;\;\;X_{i}\leq 7,\\\
0.04,&\;\;7<X_{i}\leq 14,\\\ 0.08,&14<X_{i},\end{cases}$
This model has three different hazard rates that are distinguished by two
cutpoints 7 and 14. In addition, we consider the following linear model (LM)
defining parameter $\lambda_{i}$ as follows.
$\lambda_{i}=0.1X_{i}.$
In this model, $\lambda_{i}$ depends on $X_{i}$ linearly. It follows that
$Y_{i}$ depends on $X_{i}$ nonlinearly. This model has a number of different
hazard rates. For each model, we generate a simulated data set of 200
observations with average censoring rates of 15% or 30%. For testing, we
independently generate a test data set of sample size 200 observations. All
the simulation experiments are repeated 100 times independently.
### 3.2 Simulation results
Table 1: Overall and minimum pairwise log-rank statisitics (standard errors) by each of rpart, ctree, and kaps for the stepwise model (SM) and linear model (LM) with average censoring rates (CR) of 15% or 30%. The minimum pairwise statistic is the smallest one among all the pairs. For SM, the log-rank statistics are provided for reference (ref) when the true cutpoints are used. Model | K | Method | CR = 15% | CR = 30%
---|---|---|---|---
| | | Overall | Pairwise | Overall | Pairwise
| | ref | 48.68 (1.17) | 9.06 (0.42) | 39.84 (1.29) | 7.13 (0.37)
| | rpart | 35.39 (1.29) | 3.88 (0.43) | 27.55 (1.24) | 2.56 (0.30)
SM | 3 | ctree | 38.97 (1.28) | 5.21 (0.46) | 30.90 (1.21) | 3.94 (0.38)
| | kaps | 39.69 (1.35) | 7.11 (0.47) | 31.42 (1.27) | 5.04 (0.38)
| | rpart | 43.57 (1.17) | 43.57 (1.17) | 38.73 (1.03) | 38.73 (1.03)
| 2 | ctree | 38.94 (1.16) | 38.94 (1.16) | 36.29 (1.01) | 36.29 (1.01)
| | kaps | 44.52 (1.15) | 44.52 (1.15) | 38.74 (1.11) | 38.74 (1.11)
| | rpart | 48.07 (1.47) | 6.87 (0.50) | 43.21 (1.11) | 6.32 (0.44)
LM | 3 | ctree | 55.64 (1.39) | 12.94 (1.14) | 47.99 (1.40) | 8.33 (0.64)
| | kaps | 54.96 (1.39) | 13.83 (0.63) | 47.95 (1.36) | 11.33 (0.59)
| | rpart | 59.60 (1.59) | 2.59 (0.26) | 53.01 (1.33) | 2.30 (0.22)
| 4 | ctree | 59.82 (1.35) | 3.17 (0.37) | 52.05 (1.34) | 2.10 (0.21)
| | kaps | 61.27 (1.39) | 3.22 (0.24) | 53.48 (1.34) | 2.70 (0.23)
(a) rpart
(b) rpart
(c) ctree
(d) ctree
(e) kaps
(f) kaps
Figure 3: Cutpoint selection for the stepwise model (SM) with average
censoring rate 30%. The histograms and the scatterplots of two selected
cutpoints, $C_{1}$ and $C_{2}$, for each method are displayed in the left and
right columns, respectively. The 95% confidence ellipse is superimposed on
each scatterplot.
(a) SM model with CR = 15%
(b) SM model with CR = 30%
(c) LM model with CR = 15%
(d) LM model with CR = 30%
Figure 4: Proportions of the selected numbers of subgroups by rpart, ctree,
and kaps for the stepwise model (SM) and linear model (LM) with averaged
censoring rate (CR) 15% or 30%
We first study whether the selection of cutpoints is correct when the number
of subgroups $K$ is specified. In addition, we investigate whether the
cutpoint selection affects the partition performance, which is measured by the
overall log-rank statistic and the minimum pairwise log-rank statistic. The
overall log-rank statistic is for testing the differences of all the subgroups
and the minimum pairwise log-rank statistic is the smallest from all the
pairs.
For the SM, it is reasonable to select cutpoints 7 and 14 because the hazard
rates are distinguished by these two cutpoints. On the other hand, it is not
clear which points should be selected for the LM. Thus, we investigate the
frequencies of selected cutpoints by each of rpart, ctree, and kaps for the
SM. As seen in the histograms in Figure 3, kaps often selects points around 7
and 14, while rpart and ctree often tend to select the points between 7 and
14. The scatterplots in Figure 3 show the distributions of the selected
cutpoints in two-dimensional space where each axis indicates each cutpoint.
The cutpoints of kaps are mostly distributed in a smaller ellipse (almost
circular), while those of rpart and ctree are distributed in larger ellipses.
Therefore, we can say that kaps selects the true cutpoints better than rpart
and ctree.
For SM, the two true cutpoints 7 and 14 are known. Thus, when the true
cutpoints are used for partitioning, the overall statistics are 48.68 and
39.84 and the minimum pairwise statistics are 9.06 and 7.13 when CR are 15%
and 30%, respectively. It can be shown that kaps has the largest overall and
minimum pairwise statistics from the three methods, while rpart has the
smallest value of these statistics. Therefore, the correct selection of
cutpoints leads to an improved performance in partitioning. For LM, the true
cutpoints are unknown. Moreover, it is not known how many subgroups will be
best. Thus, we assume 2, 3, and 4 subgroups ($K=2,3,4$) because these would be
useful in practice. When $K=2$, the overall statistics are the same as the
pairwise statistics because there are only two subgroups. The results show
that kaps is slightly better and ctree is slightly worse than the others,
although all the methods perform well. When $K=3$, all the methods lead to
significant differences between all pairs of subgroups. However, kaps performs
the best, while rpart is the worst. When $K=4$, none of the methods find a
significant difference for the worst pair although kaps is slightly better.
This implies that it is reasonable to have three subgroups ($K=3$) in this
case.
We next explore how many subgroups are selected by each method. For each
method, the default option was used and the minimum sample size in each
subgroup was 10% of the data. Figure 4 displays the histograms of the
subgroups selected by each method for SM and LM with CR of 15% or 30%. For SM,
rpart tends to identify two subgroups and ctree identifies two or three
subgroups. In contrast, kaps most often identifies three subgroups. For LM,
rpart tends to identify two subgroups and ctree identifies three or four
subgroups. On the other hand, kaps identifies three or four, but mostly four
subgroups. This implies that rpart identifies a smaller number of subgroups
while kaps does a larger number.
## 4 Example
Table 2: Numbers of metastasis lymph nodes for the staging systems by AJCC, kaps, ctree, and rpart. Minimum pairwise statistics and their corresponding pairs of subgroups are given at the bottom. Subgroup | AJCC | rpart | ctree | kaps
---|---|---|---|---
Subgroup 1 | 0 | 0 | 0 | 0
Subgroup 2 | 1 | 1,2,3 | 1 | 1
Subgroup 3 | 2,3 | 4 $\sim$ 10 | 2,3 | 2,3
Subgroup 4 | 4,5,6 | $\geq$ 11 | 4,5 | 4,5,6
Subgroup 5 | $\geq$7 | — | 6,7,8 | 7,8,9,10
Subgroup 6 | — | — | $\geq$9 | $\geq$ 11
Min. pairwise statistic | 131.23 | 932.30 | 78.35 | 131.23
Corresponding pair | (2, 3) | (3, 4) | (3, 4) | (2, 3)
(a) AJCC
(b) rpart
(c) ctree
(d) kaps
Figure 5: Kaplan-Meier survival curves for the subgroups identified by AJCC,
rpart, ctree, and kaps
We apply our proposed method to the colorectal cancer data from the
Surveillance Epidemiology and End Results (SEER) database
(http://seer.cancer.gov). The SEER data includes information about a variety
of cancers and has been collected from various locations and sources in the US
since 1973 and it is continually expanded to cover more areas and
demographics. It includes incidence and population data associated with age,
gender, race, year of diagnosis, and geographic areas. We here utilized the
data consisting of patients with colorectal cancer, which were used to develop
a new cancer staging system. We use the number of metastatic lymph nodes (LNs)
as an ordered prognostic factor, which was used for the N classification of
the current TNM staging system of the American Joint Committee of Cancer
(AJCC). For analysis, 65,186 cases were selected with 12 or more examined LNs
because this many LNs need to be examined for evaluating the prognosis of
colorectal cancer patients [13].
Table 2 shows the numbers of metastatic LNs for the stages discovered by
rpart, ctree, and kaps, including the N classification of the current TNM
staging system of American Joint Committee of Cancer (AJCC). The minimum
pairwise log-rank statistics and their corresponding pairs of subgroups are
given at the bottom of this table. AJCC consists of 5 subgroups and the worst
pair of subgroups is (2, 3) with a minimum pairwise log-rank statistic of
131.23. That is, two subgroups 2 and 3 are {LNs = 1} and {LNs = 2 or 3}. rpart
has the largest minimum pairwise statistic, but it discovers only 4 subgroups.
In contrast, ctree and kaps identify one more subgroup than AJCC. Our kaps has
a smaller minimum pairwise statistic than ctree. Thus, we can say that kaps
performs better. The survival curves for the subgroups are shown in Figure 5.
rpart shows well-separated curves, but they are only 4 subgroups. Our kaps
consists of 6 subgroups, all of which are shown to be fairly well-separated in
Figure 5.
## 5 Conclusion
In this paper, we have proposed a multi-way partitioning algorithm for
censored survival data. It divides the data into $K$ heterogeneous subgroups
based on the information of a prognostic factor. The resulting subgroups show
significant differences in survival. Rather than a mixture of extremely poorly
and well-separated subgroups, our developed algorithm aims to generate only
fairly well-separated subgroups even though there is no extremely well-
separated subgroup. For this purpose, we identify a multi-way partition which
maximizes the minimum of the pairwise test statistics among subgroups. The
partition consists of two or more cutpoints, whose number is determined by a
permutation test.
Our developed algorithm is compared with two binary recursive partitioning
algorithms, which are widely used in R. The simulation study implies that our
algorithm outperforms the others. In addition, its usefulness was demonstrated
using a real colorectal cancer data set from the SEER database. We have
implemented our algorithm in an R package kaps, which is convenient to use and
freely available in R via the Comprehensive R Archive Network (CRAN,
http://cran.r-project.org/package=kaps).
## Acknowledgement
This research was supported by the Basic Science Research Program through the
National Research Foundation of Korea (NRF) funded by the Ministry of
Education, Science and Technology (2010-0007936). It was also supported by the
Asan Institute for Life Sciences, Seoul, Korea (2013-554).
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## Appendix
The algorithm described in this paper was implemented into an R package kaps
[14] which is available at the Comprehensive R Archive Network (CRAN,
http://cran.r-project.org/package=kaps). In the Appendix, we illustrate the
use of the algorithm with a simple example.
### Overview
A package kaps was written in R language [19] which allows clean interface
implementation and great extension. The package depends on methods, survival
[20], Formula [21] and coin [22] packages. The package Formula is utilized to
handle multiple parts on the right-hand side of the formula object for
convenient use. The package coin is used for the permutation test for the
selection of optimal number of subgroups. In addition, the packages locfit
[23], foreach [24] and doMC [25] are suggested to give fancy visualization and
minimize computational cost, respectively.
### Main Function
The $K$-adaptive partitioning algorithm can be conducted by a function kaps().
Usage and input arguments for kaps() are as follows. The type of the arguments
is given in brackets.
kaps(formula, data, K = 2:4, mindat, type = c("perm", "NULL"), ...)
* •
formula [S4 class Formula]: a Formula object with a response variable on the
left hand side of the $\sim$ operator and covariate terms on the right side.
The response has to be a survival object with survival time and censoring
status in the Surv function.
* •
data [data.frame]: a data frame with variables used in the formula. It needs
at least three variables including survival time, censoring status, and a
covariate.
* •
K [vector]: the number of subgroups. The default value is 2:4.
* •
mindat [scalar]: the minimum number of observations at each subgroup. The
default value is 5% of data.
* •
type [character]: a type of optimal subgroup selection algorithm. At this
stage, we offer two options. The option ”perm” utilizes permutation test,
while ”NULL” passes a selection algorithm.
* •
$\ldots$ [S4 class kapsOptions]: a list of minor parameters.
The primary arguments used for analysis are formula and data. All of the
information created by kaps() is stored into an object from the kaps S4 class.
The output structure is given in Table 3. In addition, five generic functions
are available for the class: show-method, print-method, plot-method, predict-
method and summary-method.
Table 3: The main slots for the kaps S4 class. Slot | Type | Description
---|---|---
call | language | evaluated function call
formula | Formula | formula to be used
data | data.frame | data to be used in the model fitting
groupID | vector | subgroup classified
index | vector | an index for the selected K
split.pt | vector | cut-off points selected
results | list | results for each $K$
Options | kapsOptions | minor parameters to be used
X | scalar | test statistic with the worst pair of subgroups
Z | scalar | overall test statistic
pair | numeric | selected pair of subgroups
### Illustrative example
To illustrate the function kaps with various options, we use a simple
artificial data, toy, which consists of 150 artificial observations of the
survival time (time), its censoring status (status) and the number of
metastasis lymph nodes (LNs) (meta) as a covariate. The data can be called up
from the package kaps:
R> library("kaps")R> data(toy)R> head(toy)
meta status time1 1 0 02 4 1 263 0 1 224 9 1 155 0 1 706 1 0 96
Here we utilize just 3 variables: meta, status and time. The number of
metastasis LNs, meta, is used as an ordered prognostic factor for finding
heterogeneous subgroups. The available data have the following structure:
R> str(toy)
’data.frame’:150 obs. of 3 variables: $ meta : int 1 4 0 9 0 1 0 5 0 0 ... $
status: num 0 1 1 1 1 0 0 0 1 0 ... $ time : num 0 26 22 15 70 96 97 10 32 127
...
#### Selecting a set of cut-off points for given $K$
Suppose we specify the number of subgroups in advance. For instance, $K=3$. To
select an optimal set of two cut-off points when $K=3$, the function kaps is
called via the following statements
R> fit1 <\- kaps(Surv(time, status) ~ meta, data = toy, K = 3)R> fit1
Call:kaps(formula = Surv(time, status) ~ meta, data = toy, K = 3)K-Adaptive
Partitioning for Survival DataSamples= 150 Optimal K=3Selecting a set of cut-
off points: Xk df Pr(>|Xk|) X1 df Pr(>|X1|) adj.Pr(|X1|) cut-off pointsK=3
36.8 2 0 7.2 1 0.0073 0.014701 0, 10 *\---Signif. codes: 0 ‘***’ 0.001 ‘**’
0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1P-values of pairwise comparisons 0<=meta<=0
0<meta<=100<meta<=10 1e-04 -10<meta<=38 <.0000 0.0073 On the R command, we
first create an object fit1 by the function kaps() with the three input
arguments formula, data, and K. The object fit1 has the S4 class kaps. The
function show returns the outputs of the object, consisting of three parts:
Call, Selecting a set of cut-off points, and P-values of pairwise comparisons.
The first part, Call, displays the model formula with a dataset and a number
for $K$. In this example, the prognostic factor, $meta$, is used to find three
heterogeneous subgroups since $K=3$. Next, the information regarding the
selection of an optimal set of cut-off points is provided for given $K$ in the
table. In this part, the Xk ($T_{K-1}^{2}$) and X1 ($T_{1}^{2}(s_{K}^{*})$)
mean the overall and minimum pairwise test statistics, and the Pr(>|Xk|) and
Pr(>|X1|) denote their corresponding $p$-values. The adj.Pr(|X1|) indicates a
permuted $p$-value for the worst-pair with the smallest test statistic.
When $K=3$, an optimal set of two cut-off points selected by the algorithm is
$s_{K}^{*}=\\{0,10\\}$. The two cut-off points are used to partition the data
into three groups: $meta=0$, $0<meta\leq 10$, and $10<meta\leq 38$. For the
three subgroups, the overall test statistic $T_{K-1}^{2}$ (Xk), the degree of
freedom (df), and the $p$-value (Pr(|Xk|)) are given. Note that if $K$ is not
significant, the output part is changed from "Optimal K=3" to "Optimal K<3".
It means the value of the argument $K$ may be less than the present input
value. Lastly, the $p$-values of pairwise comparisons among all the pairs of
subgroups are provided.
The $p$-values can be adjusted for multiple comparison, as shown below.
R> fit2 <\- kaps(Surv(time, status) ~ meta, data = toy, K=3,\+
p.adjust.methods = "holm")R> fit2
Call:kaps(formula = Surv(time, status) ~ meta, data = toy, K = 3,
p.adjust.methods = "holm")K-Adaptive Partitioning for Survival DataSamples=
150 Optimal K=3Selecting a set of cut-off points: Xk df Pr(>|Xk|) X1 df
Pr(>|X1|) adj.Pr(|X1|) cut-off pointsK=3 36.8 2 0 7.2 1 0.0073 0.012101 0, 10
*\---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1P-values of
pairwise comparisons 0<=meta<=0 0<meta<=100<meta<=10 2e-04 -10<meta<=38 <.0000
0.0073 It is based on the internal function p.adjust. The default value of
p.adjust.methods is ”none”. The only difference between the objects fit1 and
fit2 is the $p$-values of pairwise comparisons. For more information, refer to
the help page of the function p.adjust. The Kaplan-Meier survival curves can
be obtained by
R> plot(fit1)
Figure 6: Kaplan-Meier survival curves for the toy dataset with three
subgroups: G1= {$meta=0$}, G2= {$0<meta\leqq 10$}, and G3= {$10<meta\leqq
38$}.
It provides Kaplan-Meier survival curves for the selected subgroups as seen in
Figure 6. The method summary shows the tabloid information for the subgroups.
It consists of the number of observations (N), the survival median time (Med),
and the 1-year (yrs.1), 3-year (yrs.3), and 5-year (yrs.5) survival times. The
rows mean orderly for all the data (All) and each subgroup.
R> summary(fit1)
N Med yrs.1 yrs.3 yrs.5All 150 57 0.813 0.609 0.488Group=1 76 107 0.946 0.803
0.655Group=2 60 35 0.749 0.456 0.368Group=3 14 11 0.357 0.143 0.000
#### Finding an optimal $K$
The number ($K$) of subgroups is usually unknown and may not therefore be
specified in advance. Rather, an optimal $K$ can be selected by the algorithm
for a given range of $K$ as follows:
R> fit3 <\- kaps(Surv(time, status) ~ meta, data = toy, K = 2:4)R> fit3
Call:kaps(formula = Surv(time, status) ~ meta, data = toy, K = 2:4)K-Adaptive
Partitioning for Survival DataSamples= 150 Optimal K=3Selecting a set of cut-
off points: Xk df Pr(>|Xk|) X1 df Pr(>|X1|) adj.Pr(|X1|) cut-off pointsK=2
26.4 1 0 26.37 1 0.0000 0.00000 8 ***K=3 36.8 2 0 7.20 1 0.0073 0.01240 0, 10
*K=4 38.0 3 0 1.89 1 0.1692 0.16752 0, 3, 6\---Signif. codes: 0 ‘***’ 0.001
‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1P-values of pairwise comparisons 0<=meta<=0
0<meta<=100<meta<=10 1e-04 -10<meta<=38 <.0000 0.0073
Optimal sets of cut-off points are selected for each $K$, as seen in the
output with the title ”Selecting a set of cut-off points”. The explanation for
the output is the same as that of the previous subsection. Then an optimal $K$
is selected by the algorithm with permutation test as described in Section
2.2, respectively. In the output, Xk and X1 indicate the overall and worst-
pair test statistics. Their degrees of freedom and $p$-values are followed in
the output. The "adj.Pr(|X1|)" is the Bonferroni corrected permuted $p$ value
for the worst pair by which we make a decision for the optimal $K$. In this
example, an optimal $K$ is 3 because the worst pairs of comparisons were
significant with significance level $\alpha=0.05$ when $K=2\mbox{ and }3$, and
the worst-pair $p$-value for $K=4$ is rapidly increased.
The test statistic for determining an optimal $K$ can be displayed by
R> plot(fit3) It generates the four plots shown in Figure 7. The top left
panel is the scatterplot of survival times against the prognostic factor
$meta$ with the line fitted by local censored regression [26]. The top right
panel is the Kaplan-Meier survival curves for the subgroups selected with the
optimal $K$. At the bottom are displayed the plots of the overall and worst-
pair $p$ values against $K$. The dotted lines indicate thresholds for
significance ($\alpha=0.05$).
Figure 7: The top left panel is the scatter plot of survival times against the
prognostic factor with the line fitted by local censored regression. The top
right panel is the Kaplan-Meier survival curves for the selected subgroups.
The panels at the bottom are the plots of the overall and worst-pair
$p$-values against $K$ with significance level $\alpha=0.05$.
The outputs for $K$s can also be printed out. For instance, when $K$ is 4, the
output is printed out as follows.
R> print(fit3, K= 4)
P-values of pairwise comparisons when K = 4 0<=meta<=0 0<meta<=3
3<meta<=60<meta<=3 1e-04 - -3<meta<=6 0.2812 0.1687 -6<meta<=38 <.0000 0.1151
0.0092 It gives information about pairwise comparisons for a specific $K$.
### System requirements, availability and installation
kaps is an R package developed by employing the following R packages: methods,
survival, Formula and coin. It requires R ($>$3.0.0) and runs under Windows
and Unix like operating systems. The source code of development version and
detailed installation guide for kaps are freely available under the terms of
GNU license from https://sites.google.com/site/sooheangstat/. The stable
version of kaps is also available at the Comprehensive R Archive Network
(http://cran.r-project.org/package=kaps).
Project name | $K$-Adaptive Partitioning for Survival Data
---|---
Operating system(s) | Platform independent
Other requirements | None
Programming language | R ($\geq$3.0.0)
License | GNU GPL version 3
|
arxiv-papers
| 2013-06-19T16:57:06 |
2024-09-04T02:49:46.711643
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Soo-Heang Eo, Hyo Jeong Kang, Seung-Mo Hong, HyungJun Cho",
"submitter": "Soo-Heang Eo",
"url": "https://arxiv.org/abs/1306.4615"
}
|
1306.4640
|
††thanks: Official contribution of the National Institute of Standards and
Technology; not subject to copyright in the United States. Certain commercial
products may be identified in order to adequately specify or describe the
subject matter of this work. In no case does such identification imply
recommendation or endorsement by the National Institute of Standards and
Technology, nor does it imply that the equipment identified is necessarily the
best available for the purpose.
# Stratified Sampling for the Ising Model: A Graph-Theoretic Approach
Amanda Streib [email protected] National Institute of Standards and
Technology Noah Streib [email protected] National Institute of Standards
and Technology Isabel Beichl [email protected] National Institute of
Standards and Technology Francis Sullivan [email protected] Center for
Computing Sciences
###### Abstract
We present a new approach to a classical problem in statistical physics:
estimating the partition function and other thermodynamic quantities of the
ferromagnetic Ising model. Markov chain Monte Carlo methods for this problem
have been well-studied, although an algorithm that is truly practical remains
elusive. Our approach takes advantage of the fact that, for a fixed bond
strength, studying the ferromagnetic Ising model is a question of counting
particular subgraphs of a given graph. We combine graph theory and heuristic
sampling to determine coefficients that are independent of temperature and
that, once obtained, can be used to determine the partition function and to
compute physical quantities such as mean energy, mean magnetic moment,
specific heat, and magnetic susceptibility.
Ising model, partition function, graph theory, heuristic sampling
###### pacs:
02.70.-c, 02.10.Ox
## I Introduction
Computing thermodynamic quantities of the ferromagnetic Ising model has been a
fundamental problem in statistical physics since the early 20th century Ising
(1925), where the demonstration of the model’s phase transition served as the
first rigorous proof that small changes at an atomic scale can lead to large,
observable changes Peierls (1936). Singularities in the thermodynamic
quantities indicate the critical temperature at which the phase transition
occurs. The partition function $Z$ of the Ising model and its partial
derivatives determine these quantities. While $Z$ has been found exactly in
special cases Onsager (1944); Yang (1952), there is unlikely to exist an
efficient method of finding $Z$ in general Jerrum and Sinclair (1993).
Therefore, the task of estimating $Z$ has drawn significant effort from the
physics and computer science communities Cipra (1987). However, an algorithm
that is truly practical has yet to be found. In this paper, we present a new
heuristic sampling approach with the goal of solving real-world instances
quickly.
The classical approach to this problem is to sample from the Gibbs
distribution using a Markov chain Metropolis _et al._ (1953); Swendsen and
Wang (1987); Jerrum and Sinclair (1993). Ideally, the algorithm will require
only a polynomial number of samples to estimate $Z$ at a particular
temperature, but even then this process must be repeated for each temperature
of interest. In contrast, each run of our heuristic sampling algorithm,
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$, estimates certain
coefficients that are independent of temperature. Once obtained, these
coefficients can be used to compute $Z$, mean energy, mean magnetization,
specific heat, and magnetic susceptibility at all temperatures by simply
evaluating polynomials with these coefficients.
For a fixed bond strength, computing $Z$ is equivalent to counting subgraphs
of a graph $G$. Let $x_{k,e}$ denote the number of subgraphs of $G$ with $2k$
odd vertices and $e$ edges. Using the high-temperature expansion, we can write
$Z$ and its derivatives as polynomials whose coefficients come from the set of
$x_{k,e}$. For each $k$, $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$
generates a search tree whose leaves are the set of subgraphs with $2k$ odd
vertices, and then implements the stratified sampling method of Chen Chen
(1992) to estimate the $x_{k,e}$. In the absence of an applied field, the
problem of estimating $Z$ reduces to estimating $x_{0,e}$ for all $e$. As will
become clear, it is simple to restrict
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ to subgraphs with no odd-
degree vertices, which significantly reduces the complexity of the algorithm
in this special case.
## II Definitions and Terminology
In this section, we introduce important notions from statistical physics and
graph theory.
### II.1 Ising Model
Given a graph $G=(V,E)$ with $|V|=n$ and $|E|=m$, a _spin configuration_
$\sigma=\sigma(G)$ is an assignment of spins in $\\{+1,-1\\}$ to the elements
of $V$. The energy of $\sigma$ is given by the Hamiltonian
$H(\sigma)=-J\sum_{(x,y)\in E}\sigma_{x}\sigma_{y}-B\sum_{x\in V}\sigma_{x},$
where $J$ is the interaction energy (bond strength) and $B$ is the external
magnetic field. In this paper we restrict the ferromagnetic case, fixing
$J=1$.To model the physical reality of a ferromagnet, the probability assigned
to state $\sigma$ is given by the Gibbs distribution, defined as $e^{-\beta
H(\sigma)}/Z$, where $\beta=(k_{\operatorname{\mathcal{B}}}T)^{-1}$ is
proportional to inverse temperature and $k_{\operatorname{\mathcal{B}}}$ is
Boltzmann’s constant. The normalizing constant $Z=\sum_{\sigma}\exp(-\beta
H(\sigma))$ is also called the partition function.
Following the notation of Jerrum and Sinclair (1993), let $\lambda=\tanh(\beta
J)$ and $\mu=\tanh(\beta B)$. The high-temperature expansion is defined by
$Z=AZ^{\prime}$, where $A=(2\cosh(\beta B))^{n}\cosh(\beta J)^{m}$ is an
easily computed constant, and
$Z^{\prime}=\sum_{X\subseteq
E}\lambda^{|E(X)|}\mu^{|\operatorname{o}(X)|}~{},$
where the sum is taken over all subsets $X$ of the edges of $G$. In a slight
abuse of notation, we let $X$ also refer to the graph with vertex-set $V$ and
edge-set $X$. In this manner, $E(X)$ is the edge-set of $X$,
$\operatorname{o}(X)$ is the the set of odd-degree vertices in $X$, and all
subgraphs in this paper are spanning and labeled.
Since all graphs have an even number of vertices of odd degree, Jerrum and
Sinclair Jerrum and Sinclair (1993) write $Z^{\prime}$ as a polynomial in
$\mu^{2}$: $Z^{\prime}=\sum_{k=0}^{\lfloor n/2\rfloor}c_{k}\mu^{2k},$ where
$c_{k}=\sum_{X~{}:~{}|\operatorname{o}(X)|=2k}\lambda^{|E(X)|}~{}.$ Notice
that we can compute $Z^{\prime}$ for any choice of $\mu$ given the values of
the $c_{k}$, making the $c_{k}$ independent of the magnetic field. However, we
wish to have full temperature-independence, so we write
$c_{k}=\sum_{e=0}^{m}x_{k,e}\lambda^{e}\quad\text{and}\quad
Z^{\prime}=\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{e=0}^{m}x_{k,e}\lambda^{e}\mu^{2k},$ (1)
where $x_{k,e}$ is as defined in the introduction. As we shall see,
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ is designed to estimate the
$x_{k,e}$. Thus, $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ yields an
estimate of $Z^{\prime}$, and hence $Z$ as well, at all temperatures
simultaneously.
While $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ is defined for all
graphs $G$, the graphs with the most physical significance are the square
lattices (grids) with periodic boundary conditions in two and three
dimensions. Therefore, all of the computations provided in this paper utilize
such graphs, and we shall refer to the $s\times s$ square lattice with
periodic boundary conditions simply as the $s\times s$ grid.
### II.2 Cycle Bases
We now introduce some elementary algebraic graph theory which
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ uses (for more on this topic,
see Diestel (2006)). The _symmetric difference_ of two subgraphs $X_{1}$ and
$X_{2}$ of $G$, written $X_{1}\oplus X_{2}$, is the subgraph of $G$ that
contains precisely those edges in exactly one of $X_{1}$ and $X_{2}$. One may
consider this operation as addition of subgraphs over the field
$\mathbb{F}_{2}=\\{0,1\\}$. Notice that an edge $e$ is in
$\bigoplus_{i=1}^{t}X_{i}$ if and only if $e$ appears in an odd number of
these subgraphs.
Let $\mathcal{E}_{0}$ be the set of _even subgraphs_ , those subgraphs with no
vertices of odd degree. Since the symmetric difference of two even subgraphs
is again an even subgraph, we may view $\mathcal{E}_{0}$ as a vector space
over $\mathbb{F}_{2}$, called the _cycle space_ of $G$. The dimension of the
cycle space is $m-n+1$. Hence, every set of $m-n+1$ linearly independent even
subgraphs forms a _cycle basis_ $\mathcal{C}$ of $G$. Further, every even
subgraph has a _unique_ representation using the elements of $\mathcal{C}$,
and $|\mathcal{E}_{0}|=2^{m-n+1}$.
When $X\in\mathcal{E}_{0}$, the parity of each vertex in $X\oplus Y$ is the
same in $Y$. Now consider a subgraph $P$ of $G$ with
$\operatorname{o}(P)=\\{v_{1},v_{2},\ldots,v_{2k}\\}$. The set
$\mathcal{E}_{0}\oplus P:=\\{X\oplus P:X\in\mathcal{E}_{0}\\}$
is exactly the $2^{m-n+1}$ subgraphs whose odd vertices are
$\operatorname{o}(P)$. Therefore, the set of subgraphs with $2k$ odd vertices,
$\mathcal{E}_{k}$, is $\bigcup_{S}\mathcal{E}_{0}\oplus P_{S}$, where the
union is over all $S\subseteq V$ of size $2k$ and $P_{S}$ is _any_ subgraph
with $\operatorname{o}(P)=S$.
Cycle bases have a long history in combinatorics Mac Lane (1937), and are used
both in theory and applications Kavitha _et al._ (2009). A _fundamental cycle
basis_ is defined as the cycles in $T+e$ for each $e\in E(G)-E(T)$, for a
spanning tree $T$ of $G$. Since spanning trees can be found quickly (see e.g.
Cormen _et al._ (2009)), so can fundamental bases. _Minimum cycle bases_ ,
which are bases with the fewest total edges, have proven helpful in practice
and can also be found in polynomial time Berger _et al._ (2004).
## III Algorithms
Our main data structure is a search-tree; a rooted tree in which each node
represents a subgraph of $G$. For each $k$, we shall define a search-tree
$\tau_{k}$ whose leaves are precisely $\mathcal{E}_{k}$. Our goal is to
estimate $x_{k,e}$, the number of leaves of $\tau_{k}$ that have $e$ edges.
Tree search algorithms have a lengthy history in computer science Pearl
(1984). A classical example of such is an algorithm of Knuth Knuth (1975) for
estimating properties of a backtrack tree. To estimate the number of leaves,
for example, Knuth’s algorithm explores a random path down the tree from the
root, choosing a child uniformly at random at each step. It then returns the
product of the number of children of each node seen along the path. It is easy
to see that this estimator is unbiased; i.e. the expected value is the number
of leaves.
For our application, we want the number of leaves of $\tau_{k}$ of a certain
type (with $e$ edges). We achieve this via Chen’s generalization of Knuth’s
algorithm, which was originally introduced to reduce the variance of the
estimator. Since Chen’s work lies at the heart of our approach, we take the
next section to explain it in further detail. In Section III.2, we describe
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$. In SM , we present an
alternative to $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$, which is
related to Jerrum and Sinclair (1993). This approach, which we call
$\operatorname{\mathcal{A}_{\text{\sc{edge}}}}$, may be more appropriate in
the presence of an external field, but is outperformed by
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ when $B=0$.
### III.1 Stratified Sampling
We describe in Algorithm 1 a simplified version of the stratified sampling
algorithm introduced by Chen Chen (1992). Let $\tau$ be a search tree and
choose a _stratifier_ for $\tau$ — a way of partitioning the nodes into sets
called _strata_.111In general, the stratifier must satisfy a few technical
conditions. However, as long as we require each strata to contain nodes from a
single level of $\tau$, we are guaranteed that these conditions are met. For
each stratum $\alpha$, Algorithm 1 produces a representative
$s_{\alpha}\in\alpha$ and a weight $w_{\alpha}$, which is an unbiased estimate
of the number of nodes in $\alpha$.
For Algorithm 1, let $Q_{\operatorname{1}}$ and $Q_{\operatorname{2}}$ be
queues. Each node $s$ of $\tau$ has a weight $w$, and we write $(s,w)$ to
represent this pair. The input is the root $r$ of $\tau$, a method for
determining the children of a node in $\tau$, and the stratifier. The output
is the set of $(s_{\alpha},w_{\alpha})$. If the algorithm never encounters an
element of $\alpha$, it returns $(\emptyset,0)$ for $\alpha$.
List of Algorithms 1 Chen’s Algorithm
_initialize_ : $Q_{\operatorname{1}}=\\{(r,1)\\}$,
$Q_{\operatorname{2}}=\\{\\}$, $i=0$.
while $i<\text{number of levels in $\tau$}$ do
while $Q_{\operatorname{1}}\not=\emptyset$ do
output the first element $(s,w)$ of $Q_{\operatorname{1}}$
for each child $t$ of $s$ in $\tau$ do
if $Q_{\operatorname{2}}$ contains an element $(u,w_{u})$ in the same stratum
as $t$ then
update $w_{u}=w+w_{u}$
w. prob. $w/w_{u}$ replace $(u,w_{u})$ with $(t,w_{u})$ in
$Q_{\operatorname{2}}$
else
add $(t,w)$ to $Q_{\operatorname{2}}$
pop $(s,w)$ off of $Q_{\operatorname{1}}$
set $Q_{\operatorname{1}}=Q_{\operatorname{2}}$ and reset
$Q_{\operatorname{2}}=\emptyset$
$i$++
### III.2 Cycle-Addition Algorithm
Let $S\subseteq V$ of size $2k$ and recall that $P_{S}$ is any subgraph of $G$
with $\operatorname{o}(P)=S$. Let
$\mathcal{C}=\\{C_{1},C_{2},\ldots,C_{m-n+1}\\}$ be a cycle basis of $G$.
Define $\tau(\mathcal{C},P_{S})$ as the search-tree determined by the
following rules:
* 1.
$P_{S}$ is the root of $\tau(\mathcal{C},P_{S})$, and
* 2.
each node $X$ at level $0\leq i<m-n+1$ has two children:
$X\oplus\mathcal{C}_{i-1}$ and $X$.
Now $\tau_{k}$ is the tree with artificial root node $R$ whose $\binom{n}{2k}$
children correspond to the roots of $\tau(\mathcal{C},P_{S})$, one for each
distinct subset of size $2k$.
\begin{overpic}[scale={.5}]{k4_4odds.eps} \put(8.0,75.0){$G$}
\put(40.5,75.0){$C_{1}$} \put(64.5,75.0){$C_{2}$} \put(88.0,75.0){$C_{3}$}
\put(38.5,66.0){$P_{S}$} \put(28.0,55.0){$\oplus C_{1}$}
\put(9.0,35.0){$\oplus C_{2}$} \put(59.0,35.0){$\oplus C_{2}$}
\put(-0.5,14.5){$\oplus C_{3}$} \put(24.75,14.5){$\oplus C_{3}$}
\put(49.75,14.5){$\oplus C_{3}$} \put(75.25,14.5){$\oplus C_{3}$}
\put(53.5,69.0){${\bf 1}$} \put(28.25,49.5){${\bf 1}$} \put(79.0,49.5){${\bf
1}$} \put(17.0,29.0){${\bf 2}$} \put(67.5,29.0){${\bf 2}$}
\put(22.0,11.0){${\bf 2}$} \put(60.0,11.0){${\bf 2}$} \put(72.0,11.0){${\bf
4}$} \end{overpic} Figure 1:
In order to implement Algorithm 1, we define the stratifier for each
$\tau(\mathcal{C},P_{S})$ by: the nodes $X$ and $Y$ in
$\tau(\mathcal{C},P_{S})$ belong to the same stratum if and only if $X$ and
$Y$ are in the same level of $\tau(\mathcal{C},P_{S})$, and $|E(X)|=|E(Y)|$.
The inputs to $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ are a graph
$G$, an integer $k$ in $[0,n/2]$, and an integer $N$. The output of each of
the $N$ runs of Algorithm 1, as a subroutine of
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$, is a set of
$(s_{\alpha},w_{\alpha})$ pairs. Consider a representative $s_{\alpha}$ that
is a leaf node in the tree $\tau(\mathcal{C},P_{S})$, and suppose $s_{\alpha}$
has $e$ edges. Then $\binom{n}{2k}w_{\alpha}$ is our estimate of $x_{k,e}$,
since each sample represents all $\binom{n}{2k}$ choices of $S$.
List of Algorithms 2 $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$
Choose a cycle basis $\mathcal{C}$ of $G$
for $j\in[1,N]$ do
Choose $S\subseteq V$ with $|S|=2k$
Find $P_{S}$
Run Algorithm 1 on $\tau(\mathcal{C},P_{S})$
for $e\in[0,m]$ do
Let $\alpha$ be the stratum corresponding to the bottom level of
$\tau(\mathcal{C},P_{S})$ and $e$ edges, and output $\binom{n}{2k}$ times the
average of the $N$ estimates of $w_{\alpha}$ as $x_{k,e}$
Figure 1 shows an example of $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$
with $k=2$, $S=V(G)$, $N=1$, and $G$, $P_{S}$,
$\mathcal{C}=\\{C_{1},C_{2},C_{3}\\}$, and $\tau(\mathcal{C},P_{S})$ as
depicted. The graphs bounded by solid circles are the strata representatives,
and their weights are in bold just above. The solid edges of
$\tau(\mathcal{C},P_{S})$ connect the nodes seen by
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$. The output is $x_{2,2}=2$,
$x_{2,3}=4$, $x_{2,6}=2$, and $x_{2,e}=0$ for $e\in\\{0,1,4,5\\}$.
In Figure 2a, we show the output of many runs of
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ on a $4\times 4$ grid for
$k\in[0,4]$, and use this output (and that for $k\in[5,8]$) with Equation 1 to
get Figure 2a, using four values of $\lambda$. While the $c_{k}$ are log-
concave Jerrum and Sinclair (1993), the $x_{k,e}$ may not be.
\begin{overpic}[scale={.19}]{xkes_4x4x1.eps} \put(55.0,32.0){\tiny{$k=0$}}
\put(55.0,43.0){\tiny{$k=1$}} \put(55.0,52.0){\tiny{$k=2$}}
\put(55.0,57.0){\tiny{$k=3$}} \put(55.0,62.0){\tiny{$k=4$}}
\put(2.0,20.0){\begin{sideways}\parbox{42.67912pt}{\footnotesize{$\log_{2}(x_{k,e})$}}
\end{sideways}} \put(47.0,-1.0){\small{$e$}} \end{overpic} (a)
\begin{overpic}[scale={.19}]{ck_all_4x4x1_4lam.eps}
\put(47.0,59.0){\tiny{$\lambda=1$}} \put(47.0,51.0){\tiny{$\lambda=0.7$}}
\put(47.0,43.0){\tiny{$\lambda=0.414$}} \put(47.0,29.0){\tiny{$\lambda=0.1$}}
\put(2.0,20.0){\begin{sideways}\parbox{42.67912pt}{\footnotesize{$\log_{2}(c_{k})$}}
\end{sideways}} \put(47.0,-1.0){\small{$k$}} \end{overpic} (b)
Figure 2: (Color online)
### III.3 No external field
In the absence of an external field, we only need to run
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ for $k=0$. This represents a
huge time savings in comparison to the case $B\not=0$, as then we need to run
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ for all $k\in[0,n/2]$.
Furthermore, we must choose $S=\emptyset$, which eliminates this step from the
algorithm.
### III.4 Details
The algorithm $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ is really a
class of algorithms, each corresponding to the choice of cycle basis, the
order of the subgraphs in the basis, the subsets $S$, and the roots $P_{S}$.
We briefly discuss these choices here and elaborate further in SM .
The choice of cycle basis is central to the performance of
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$. Experimentally, minimum
cycle bases have outperformed fundamental and random cycle bases in terms of
overall speed and variance. However, it remains an interesting open problem to
determine the optimal basis for
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$.
As for the choice of $S$, we know that $k=0$ implies $S=\emptyset$. But for
$k>0$, we must choose $S$.222Except if $G$ itself is even, in which case there
is no choice for $k=n/2$ either. We would like every subset of $V(G)$ of size
$2k$ to appear as $S$ at least once. However, when $k$ is near $n/4$,
$\binom{n}{2k}$ is exponentially large in $n$.333Ideally, we would partition
the subsets of $V(G)$ into isomorphism classes $\\{V_{i}\\}_{i=1}^{t}$ and
choose a representative for class $V_{i}$ to act as $S$ in
$|V_{i}|N/\binom{n}{2k}$ instances. However, the number of such classes can
also be exponentially large. So instead we are forced to select a reasonable
number of such subsets that work well in
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$.
Once $S=\\{v_{1},v_{2},\dots,v_{2k}\\}$ is chosen, we must find $P_{S}$. One
such method is to use a spanning tree $T$ of $G$ to create
$P_{S}=\bigoplus_{i=1}^{k}\mathcal{P}_{{2i-1},{2i}}$, where
$\mathcal{P}_{{2i-1},{2i}}$ is the path from $v_{2i-1}$ to $v_{2i}$ in $T$.
## IV Performance
### IV.1 Convergence
In Section V, we show how to get unbiased estimates of $Z^{\prime}$ from our
unbiased estimates of the $x_{k,e}$. To evaluate the efficiency of the
algorithm, we need to know how many samples ($N$) we need to be reasonably
confident about our estimate of $Z^{\prime}$. The answer depends on the
relative variance of our estimate of $Z^{\prime}$.444By the Central Limit
Theorem, we need
$N=\frac{z_{\delta/2}^{2}}{\epsilon^{2}}\left(\frac{E[(Z^{\prime})^{2}]}{E^{2}[Z^{\prime}]}-1\right)$
to be within $\epsilon$ with probability $1-\delta$, where $z_{\delta/2}$
comes from the normal distribution, and
$\frac{E[(Z^{\prime})^{2}]}{E^{2}[Z^{\prime}]}-1$ is precisely relative
variance. As heuristic sampling methods are relatively new, there are not many
tools for computing the variance of these algorithms. Experimentally, such
methods have been shown to work well in practice, but a robust theoretical
foundation is lacking Bezáková _et al._ (2012); Pearl (1984). Therefore,
analyzing the variance for this problem remains an important open question
which deserves further study.
\begin{overpic}[scale={.19}]{rel_var_c0_4x4_10000000.eps}
\put(2.0,-22.0){\begin{sideways}\parbox{147.95433pt}{\small{Rel. Var. of
$Z^{\prime}$}} \end{sideways}} \put(47.0,0.0){$\beta^{-1}$} \end{overpic} (a)
\begin{overpic}[scale={.19}]{even_only_times_c_code_hours_cubic_growth.eps}
\put(3.0,8.0){\begin{sideways}\parbox{71.13188pt}{Running Time}\end{sideways}}
\put(49.0,0.0){$n$} \end{overpic} (b)
Figure 3: (Color online)
In our simulations, we find that although
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ is temperature independent,
the variance is not. Figure 3a shows the relative sample variance of our
estimate of $Z^{\prime}$ as a function of temperature, for a $4\times 4$ grid
with no external field. The highest sample variance occurs at the critical
temperature, $\beta^{-1}\approx 2.269$. However, even at the critical
temperature, our estimate of $Z^{\prime}$ converges quickly. Figure 4 presents
six separate runs of $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ with
$B=0$, and shows the convergence to $Z^{\prime}$ for each run as a function of
the number of samples. The exact value of $Z^{\prime}$ is displayed as the
straight black line.
\begin{overpic}[scale={.45}]{convergence_c0_250000_2-26918_nologs.eps}
\put(1.0,30.0){$Z^{\prime}$} \put(35.0,1.0){Number of Samples} \end{overpic}
Figure 4: (Color online)
### IV.2 Running time
The number of operations of a single run of Algorithm 1 as a subroutine of
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ is a function of the number
of strata used in $\tau(\mathcal{C},P_{S})$ and the number of operations
performed to process each node in $Q_{\operatorname{1}}$. Recall that our
stratifier partitions nodes according to their level in
$\tau(\mathcal{C},P_{S})$ and number of edges. Clearly, each level has at most
$m+1$ strata. Further, there are $m-n+2$ levels. Hence, the number of strata
used is at most $(m-n+2)(m+1)$. For each node in $Q_{\operatorname{1}}$,
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ examines its two children, so
the total number of nodes of $\tau(\mathcal{C},P_{S})$ used by the subroutine
is at most $2(m-n+2)m=O(m^{2}).$ For each of these nodes, we take the
symmetric difference of two subgraphs and count the number of edges remaining,
each of which is an $O(m)$ operation. Thus, each run of Algorithm 1 as a
subroutine of $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ terminates
after $O(m^{3})$ operations. For square lattices in dimension $d$, the number
of operations is $O(d^{3}n^{3})$, as $m=dn$.
### IV.3 Implementation
We implemented $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ in C, using
GMP to deal with the large weights generated by the algorithm. In Figure 3b,
we plot our experimental running times for $\sqrt{n}\times\sqrt{n}$ grids
against the curve $f(n)=1.25\cdot 10^{-14}n^{3}$, which matches up well with
our bound of $O(n^{3})$.
Typically, one stores graphs as matrices or lists. However, we greatly improve
the running time of $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ by
storing each subgraph $X$ as an integer whose bitstring $b_{X}$ has length
$m$; $b_{X}(e)=1$ if and only if $e\in E(X)$. Here, $b_{X\oplus
Y}=b_{X}~{}\text{\sc{xor}}~{}b_{Y}$, so taking symmetric differences is quite
fast. Further, $|E(X)|$ is simply the number of ones in the bitstring, which
can also be computed quickly.
One may achieve another increase in speed if machine-level instructions for
the operation XOR are used for large integers. Most modern micro-processors
have such capabilities, as they are used in scientific computing Intel
Corporation (2013).
## V Physical Quantities
In this section, we show how to use the estimates of the $x_{k,e}$ to
calculate physical quantities. Let $f(X)=f(|\operatorname{o}(X)|,|E(X)|)$ be
any function on subgraphs $X$ which depends only on the number of odd vertices
and the number of edges of $X$. We can calculate the expected value of $f$
with respect to the distribution
$\pi^{\prime}(X)=\lambda^{|E(X)|}\mu^{|\operatorname{o}(X)|}/Z^{\prime}$ from
our estimates of the $x_{k,e}$ by
$\mathbb{E}[f]=\frac{1}{Z^{\prime}}\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{e=0}^{m}f(k,e)x_{k,e}\lambda^{e}\mu^{2k}.$ (2)
Notice if $f$ is identically $1$, $Z^{\prime}\mathbb{E}[f]=Z^{\prime}$, and so
we can approximate $Z^{\prime}$, and hence $Z$, by simply looking at the
double sum. In Theorem 1, we show that important physical quantities can also
be expressed as $\mathbb{E}[f]$ for suitable choices of $f$. The proof of
Theorem 1 involves taking partial derivatives of $\ln Z$ with respect to
$\beta$ and $B$ following the method of Jerrum and Sinclair (1993). As these
calculations are tedious but easy, we leave the details to SM .
###### Theorem 1.
The mean magnetic moment, mean energy, magnetic susceptibility, and specific
heat can each be written as sums of expectations of random variables over the
distribution $\pi^{\prime}$.
In Figure 5, we show estimates of mean energy and specific heat from
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ with $N=50,000,000$ on a
$16\times 16$ grid as a function of $\beta^{-1}$. These figures match those of
(Gould and Tobochnik, 2010, p. 252) nicely.
\begin{overpic}[scale={.19}]{mean_energy_16x16_50000000_c0_cfiles_no_binoms.eps}
\put(0.0,35.0){\large{$\frac{{\bf\varepsilon}}{n}$}}
\put(45.0,0.0){\scriptsize{$\beta^{-1}$}} \end{overpic} (a)
\begin{overpic}[scale={.19}]{specific_heat_16x16_50000000_c0_cfiles.eps}
\put(-5.5,35.0){$\frac{\mathcal{C}}{nk_{\operatorname{\mathcal{B}}}}$}
\put(45.0,0.0){\scriptsize{$\beta^{-1}$}} \end{overpic} (b)
Figure 5: (Color online)
## VI Conclusions
The algorithm $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ is a completely
new approach to the problem of estimating $Z$. To our knowledge it is the
first heuristic sampling method for this problem. For this reason, it is
difficult to compare the running time of
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ with the current best-known
algorithms, which are all Markov chain Monte Carlo methods. What is clear is
that $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ gives us an estimate of
$Z$ at _all temperatures simultaneously_ in only $O(m^{3})$ operations, where
the constant hidden by the big-O notation is small. Bounding the variance of
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ is an important open problem
which is necessary to give a real understanding of its efficiency. However, if
the goal is to get _some_ estimate as fast as possible,
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ is an excellent choice.
Besides analyzing the variance of
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$, there are several other
directions for future work. For example, there are many choices made in
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ which could be optimized,
such as the choice of cycle basis. These choices could affect the variance
significantly. One might consider other tree-search algorithms and compare
their performance with that of
$\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$ and
$\operatorname{\mathcal{A}_{\text{\sc{edge}}}}$. We also plan to investigate
more extensively the connections between our heuristic method and MCMC
methods.
###### Acknowledgements.
We wish to thank Professor Ted Einstein at the University of Maryland for
discussing our ideas and for reminding us of the difference between physics
and mathematics.
## References
* Ising (1925) E. Ising, Zeitschrift für Physik 31, 253 (1925).
* Peierls (1936) R. Peierls, Proc. Camb. Phil. Soc. 32, 477 (1936).
* Onsager (1944) L. Onsager, Phys. Rev. 65, 117 (1944).
* Yang (1952) C. N. Yang, Phys. Rev. 85, 808 (1952).
* Jerrum and Sinclair (1993) M. Jerrum and A. Sinclair, SIAM Journal on Computing 22, 1087 (1993).
* Cipra (1987) B. A. Cipra, Amer. Math. Monthly 94, 937 (1987).
* Metropolis _et al._ (1953) N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Journal of Chemical Physics 21, 1087 (1953).
* Swendsen and Wang (1987) R. H. Swendsen and J.-S. Wang, Phys. Rev. Lett. 58, 86 (1987).
* Chen (1992) P. C. Chen, SIAM Journal on Computing 21, 295 (1992).
* Diestel (2006) R. Diestel, _Graph theory_, Graduate Texts in Mathematics Series (Springer London, Limited, 2006).
* Mac Lane (1937) S. Mac Lane, Fundamenta Mathematicae 28, 22 (1937).
* Kavitha _et al._ (2009) T. Kavitha, C. Liebchen, K. Mehlhorn, D. Michail, R. Rizzi, T. Ueckerdt, and K. A. Zweig, Computer Science Review 3, 199 (2009).
* Cormen _et al._ (2009) T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, _Introduction to Algorithms, Third Edition_ (MIT Press and McGraw-Hill, 2009).
* Berger _et al._ (2004) F. Berger, P. Gritzmann, and S. de Vries, Algorithmica 40, 51 (2004).
* Pearl (1984) J. Pearl, _Heuristics: Intelligent Search Strategies for Computer Problem Solving_ (Addison-Wesley Pub. Co., Inc., Boston, MA, 1984).
* Knuth (1975) D. E. Knuth, Math. Comp. 29, 121 (1975).
* (17) See Supplemental Material at [URL will be inserted by publisher] for an alternative stratified sampling algorithm $\operatorname{\mathcal{A}_{\text{\sc{edge}}}}$, for further discussion of issues related to the choices made in $\operatorname{\mathcal{A}_{\text{\sc{cycle}}}}$, and for the proof of Theorem 1.
* Bezáková _et al._ (2012) I. Bezáková, A. Sinclair, D. Štefankovič, and E. Vigoda, Algorithmica 64, 606 (2012).
* Intel Corporation (2013) Intel Corporation, _Intel ® 64 and IA-32 Architectures Software Developer’s Manuals_ (2013 (accessed May 7, 2013)).
* Gould and Tobochnik (2010) H. Gould and J. Tobochnik, _Statistical and Thermal Physics: With Computer Applications_ (Princeton University Press, 2010).
|
arxiv-papers
| 2013-06-19T18:22:11 |
2024-09-04T02:49:46.719708
|
{
"license": "Public Domain",
"authors": "Amanda Streib, Noah Streib, Isabel Beichl, Francis Sullivan",
"submitter": "Noah Streib",
"url": "https://arxiv.org/abs/1306.4640"
}
|
1306.4853
|
CHAPTER: DECLARATION OF AUTHORSHIP
I, Dominic Hosler, declare that the work presented in this thesis, except where otherwise stated, is based on my own research and has not been submitted previously for a degree in this or any other university.
Parts of the work reported in this thesis have been published as follows:
§ PUBLICATIONS
* Dominic Hosler, Carsten van de Bruck and Pieter Kok. “Information gap for classical and quantum communication in a Schwarzschild spacetime.” In: Physical Review A 85.4 (2012-04-24), page 042312.
* Eduardo Martín-Martínez, Dominic Hosler and Miguel Montero. “Fundamental limitations to information transfer in accelerated frames.” In: Physical Review A 86.6 (2012-12-07), page 062307.
* Dominic Hosler and Pieter Kok. “Parameter estimation over a relativistic quantum channel.” (2013-06-13) arXiv: 1306.3144. Submitted to: Physical Review A.
CHAPTER: ABSTRACT
In this Ph.D. thesis, I investigate the communication abilities of non-inertial observers and the precision to which they can measure parametrized states.
I introduce relativistic quantum field theory with field quantisation, and the definition and transformations of mode functions in Minkowski, Schwarzschild and Rindler spaces.
I introduce information theory by discussing the nature of information, defining the entropic information measures, and highlighting the differences between classical and quantum information.
I review the field of relativistic quantum information.
We investigate the communication abilities of an inertial observer to a relativistic observer hovering above a Schwarzschild black hole, using the Rindler approximation.
We compare both classical communication and quantum entanglement generation of the state merging protocol, for both the single and dual rail encodings.
We find that while classical communication remains finite right up to the horizon, the quantum entanglement generation tends to zero.
We investigate the observers' abilities to precisely measure the parameter of a state that is communicated between Alice and Rob.
This parameter was encoded to either the amplitudes of a single excitation state or the phase of a NOON state.
With NOON states the dual rail encoding provided greater precision, which is different to the results for the other situations.
The precision was maximum for a particular number of excitations in the NOON state.
We calculated the bipartite communication for Alice-Rob and Alice-AntiRob beyond the single mode approximation.
Rob and AntiRob are causally disconnected counter-accelerating observers.
We found that Alice must choose in advance with whom, Rob or AntiRob she wants to create entanglement using a particular setup.
She could communicate classically to both.
CHAPTER: COPYRIGHT NOTICE
Dominic Hosler (2013)
This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/ or send a letter to Creative Commons, 444 Castro Street, Suite 900, Mountain View, California, 94041, USA.
The author, Dominic Hosler, reserves his moral right to be identified as the author of this work.
This is reflected in the attribution clause of the applied license.
CHAPTER: ACKNOWLEDGEMENTS
Firstly I would like to thank my supervisor, Pieter Kok for all the support and direction he has given me throughout my studies.
I have really enjoyed working with someone who has such an enthusiasm for physics.
We have shared many interesting discussions.
It is safe to say I would not have achieved all that I have without him.
I have been fortunate enough during my studies towards my Ph.D. to work with some really talented physicists.
Carsten van de Bruck, with whom I collaborated when working on communication near a black hole.
During my trip to Madrid, I was hosted by Juan León with whom I worked on interesting photon localization problems.
The same trip started a collaboration with Eduardo Martín-Martínez and Miguel Montero, I am grateful to be able to work with both of them.
My thanks also goes to my friends and co-workers in the office, who have all supported me.
These were, Carlos Pérez-Delgado, Marcin Zwierz, Samantha Walker, Mark Pearce, Sam Coveney, Michael Woodhouse and Matt Hewitt.
We have shared many in-depth discussions about related and non-related subjects.
They also provided plenty of welcome relief from study.
I would like to thank my family.
My wife Sascha for her continual support while I was studying, her understanding for the nature of my chosen career path and her encouragement when I hit stumbling blocks.
She is always there for me, to listen and help with anything she can.
My daughter Caoimhe for being a source of inspiration and motivation.
I would like to thank my uncle, Akin Kumoluyi for making the possibility of getting a Ph.D. seem more achievable.
I remember his Ph.D. graduation ceremony when I was about 7, it was then I decided that I too could get a Ph.D. if I chose.
I would like to extend gratitude to my grandparents, Mary and Charles Hosler for loaning me their spare room to use as an office whilst writing up.
Finally, I wish to express sincere gratitude to my parents Adam and Sandra Hosler, who dedicated a lot of their time to home educating me and making me believe I could achieve anything.
Without them I would not be where I am today.
For the publication “Information gap for classical and quantum communication in a Schwarzschild spacetime”, Pieter Kok, Carsten van de Bruck and I would like to thank M. Wilde for valuable comments on the manuscript, and J. Dunningham and V. Palge for stimulating discussions.
For the publication “Parameter estimation using NOON states over a relativistic quantum channel”, Pieter Kok, and I would like to thank Michael Skotiniotis for helpful discussions.
For the publication “Fundamental limitations to information transfer in accelerated frames”, Eduardo Martín-Martínez, Miguel Montero and I would like to thank Pieter Kok for useful discussions and Mark Wilde for helpful suggestions.
During the research and preparation of this manuscript, I visited Madrid and I would like to thank, IFF, CSIC, Madrid for their hospitality.
I also acknowledge funding support for the trip from a Santander Research Mobility Award.
The stick people used in a number of my figures are inspired by characters from the web-comic xkcd[http://xkcd.com].
Missing 'biblatex' package
The bibliography requires the 'biblatex' package.
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IOP Publishing
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titleUnruh effect in quantum information beyond the single-mode approximation
Addison Wesley
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labeltitleScalar production in Schwarzschild and Rindler metrics
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titleScalar production in Schwarzschild and Rindler metrics
The Royal Society
labeltitleRapid Solution of Problems by Quantum Computation
journaltitleProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
titleRapid Solution of Problems by Quantum Computation
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labeltitleA Father Protocol for Quantum Broadcast Channels
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University of Vienna
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howpublishedDiploma Thesis, University of Vienna
titleRelativistic Effects in Quantum Entanglement
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titleEntanglement generation in relativistic quantum fields
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seriesSTOC '96
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Springer Berlin / Heidelberg
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|
arxiv-papers
| 2013-06-20T12:41:02 |
2024-09-04T02:49:46.733606
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Dominic Hosler",
"submitter": "Dominic Hosler",
"url": "https://arxiv.org/abs/1306.4853"
}
|
1306.4876
|
Comments: 11 pages, 8 figures Subjects: Mesoscale and Nanoscale Physics (cond-
mat.mes-hall)
# Exciton condensation in quantum wells. Excitonic hydrodynamics. Autosolitons
V.I. Sugakov Institute for Nuclear Research,National Academy of Sciences
Kyiv,Ukraine) [email protected]
###### Abstract
The hydrodynamics equations for indirect excitons in the double quantum wells
are obtained taking into account 1) a possibility of an exciton condensed
phase formation, 2) the presence of a pumping, 3) the finite value of the
exciton lifetime, 4) the exciton scattering by defects. New types of solutions
in the form of bright and dark autsolitons are obtained for the exciton
density. The role of localized and free exciton states is analyzed in a
formation of the emission spectra.
double quantum wells; exciton condensation; solitons
###### pacs:
71.35.Lk, 73.21.Fg
## I Introduction
While seeking the exciton Bose-Einstain condensation in double quantum wells,
the several non-trivial effects were observed. A very large exciton lifetime
is a special feature of excitons in double quantum well in the presence of an
electric field directed in parallel to the normal of quantum well plain 1 .
The effect occurs due to the separation of electrons and holes into the
different wells, which causes a very weak overlapping their wave function and
the damping of the mutual recombination. The large lifetime allows one to
create a high concentration of the excitons at small pumping and to study a
manifestation of the effects of exciton-exciton interaction. Excitons with
electrons and holes localized in different wells are called ’indirect
excitons’. The nonzero dipole moment of the indirect excitons should cause
their mutual repulsion that complicates the creation of the exciton condensed
phase. These properties and the facts, that excitons have the integer spin and
the small mass, stimulated the search of the exciton Bose-Einstein
condensation in double quantum wells. These investigations gave a number of
new results. So, the narrow band with unusual properties was observed and
studied by Timofeev’s group2 ; 3 ; 4 in the emission spectra of the indirect
excitons in $AlGaAs$ system. It was shown that this band appears at the some
threshold pumping, also the peculiarities of the temperature and pumping
dependencies were revealed. The authors 3 ; 4 builded a phase diagram
”threshold pumping - temperature”.
The nontrivial results were found in a spatial distribution of exciton
emission from quantum well. In the papers 5 , an appearance of a ring outside
the laser spot was observed in the emission from double quantum well. The ring
radius exceeded significantly the exciton diffusion length. The explanation of
the appearance of the ring was given in the papers 7 ; 8 under the
assumption, that holes are captured by the well more effectively than
electrons, and, in addition, there are donors in the crystal, which create
some concentration of free electrons. As a result a region rich by holes
arises in a vicinity of the laser spot. Outside of this region the quantum
well is enriched by free electrons. On the boundary of the region the
processes of a recombination take place, which causes the a creation of
excitons on the ring and the appearance of a spatial distribution of the
emission in the form of the ring.
Intriguing facts appear under an investigation of the spatial distribution of
the exciton density. Different spatial nonhomogeneous structures were observed
in the emission of the indirect excitons at the pumping greater a critical
value. Thus, in the paper 5 a division of the emission ring was observed on
separate fragments periodically localized along the ring. In the paper 9 , in
which the excitation of the quantum well carried out through a window in
metallic electrode, the authors found in the luminescence spectrum a
periodical structure of the islands situated along the ring under perimeter of
the window. The number of the islands growths with increasing of the radius of
the window. The similar results were obtained not only in double quantum
wells, but also in a wide quantum well in the presence of a strong electric
field, which divides electrons and holes between different sides of the
quantum well. As the result the excitons with charges, strong separated in the
space, and with large lifetime are created 10 . On such dipole excitons in the
wide quantum well, the effects similar to those on excitons in double quantum
wells were observed,namely the appearance of periodical structures of the
islands under the window in a electrode11 and on the ring outside the laser
spot All . Recently Timofeev and coauthors 12 ; Gorb presented examples of
the structures in the emission spectra at the different forms of windows in
the electrode: the rectangle, two circles and others. In the paper 13 the
authors, choosing the form of the electrode, created additional periodical
potential for excitons. It was found, that besides the periodical structure
imposed by external conditions, the partition of the emission into fragments
was observed in the direction, in which the potential is almost uniform.
The phenomenon of the symmetry loss and the creation of structures in the
emission spectra of indirect excitons stimulated a series of theoretical
investigations 14 ; 15 ; 16 ; 17 ; 18 ; 18a ; 18b ; 18d . The authors of the
work 14 considered the instability, which arises under kinetics of level
occupations by the particles with the Bose-Einstein statistics. Namely, the
growth of the occupation of the level with zero moment should stimulate the
transitions of excitons to this level. But the density of excitons was found
greater, and the temperature was found lower than these values observed on the
experiments. Some authors explain the appearance of the periodicity by Bose
condensation of excitons 15 ; 16 . There is a suggestion to describe the
system by a nonlinear Schrodinger equation 17 ; 18 . Also a possibility of the
Mott transition in considered systems was studied 18a . The periodicity
appearance in the exciton system was investigated in 18b using Bogolyubov’s
equations under some approximations of inter-exciton interaction. Let note the
paper 18d ,in which the observed in 5 appearance of the islands was explained
from classical model diffusion equation with Coulomb interaction between
electrons,holes, and excitons. In the listed above works the main efforts were
applied for the ascertainment of the principal possibility of the appearance
of the periodicity of the exciton density distribution. A specific application
of the results for an explanation of the numerous experiments of different
kind (at different pumping and temperature, nonhomogeneous external fields)
was not employed.
An appearance of an instability of the uniform distribution of the exciton
density and a formation of the periodical distribution were shown in the work
18e from a position of self-organization processes in non-equilibrium systems
for excitons with attractive interaction. The phenomenon has the threshold
behavior with respect to a pumping. After the successful observation of
periodical structures in the system of indirect excitons in double quantum
well on the base of AlGaAs crystal by Timofeev’s and Butov’s groups, the
several works were fulfilled 19 ; 20 ; 21 ; 22 ; 23 ; 23a ; 23b in this model
devoted to explanation of the experiments. The theoretical approaches of the
works 19 ; 20 ; 21 ; 22 ; 23 ; 23a ; 23b are based on the following
assumptions.
1.There is an exciton condensed phase caused by the attractive interaction
between excitons. As was mentioned, there are the dipole-dipole repulsion
interaction between excitons. But the simple calculations show, that the
exchange and van der Waals interactions exceed the dipole-dipole repulsion at
certain distances between excitons, if the distance between quantum wells is
not very far and the exciton dipole moment is not too large. An existence of
attractive interaction between excitons is confirmed by the calculations of
biexcitons 24 ; 25 ; 26 ; 26a , and under investigation of many-exciton system
27 .
2.The finite value of the exciton lifetime plays an important role in the
formation of a spatial distribution of exciton condensed phases. As usually,
the exciton lifetime exceeds significantly the duration of the establishment
of a local equilibrium. By this reason, the lifetime of excitons is suggested
to be equal infinity under the solutions of many exciton problems. But, the
taking into account the finiteness of the exciton lifetime is necessary in the
study of the spatial distribution phases in two-phase systems, because the
exciton lifetime is less than the time of the establishment of the equilibrium
between phases. The last time is determined by slow diffusion processes and is
great. Just the finite exciton lifetime restricts the maximal size of the
exciton condensed phase and it causes the existence of a correlation in
positions of separate regions of the condensed phase. As the result the
spatial structures of the condensed phase appear in the shape of separate
islands (in two dimensional case), parameters of which and the mutual position
depend on the exciton lifetime. Thus, the created spatial structures are non-
equilibrium and they are a consequence of a self-organization in non-
equilibrium systems.
The theory, developed in the works 19 ; 20 ; 21 ; 22 ; 23 ; 23a ; 23b , has
explained many features of the indirect exciton manifestation in the double
quantum wells on the base $AlGaAs$ crystal. So, the observed in 4 behavior of
narrow band as a function of the pumping and the temperature is presented in
20 . It is follow from these calculations in accordance with the experiment 4
, that the intensity of the emission of the narrow band decreases linearly
with temperature at fixed pumping and rises ultralinearly with the growth of
the pumping at fixed temperature . The theory 20 has explained the
experimental ”threshold pumping \- temperature” phase diagram obtained in 3 ;
5 . The works 19 ; 21 ; 22 ; 23 were dedicated to the interpretation of the
appearance of the periodically situated islands in the emission of spectra
from the both the ring outside the laser spot, founded in 5 , and from the
region of the double quantum well under perimeter of the window in electrode,
founded in 9 . The theory described the sizes of the islands, the distances
between them, their appearance and vanishing depending on pumping and
temperature. The calculated behavior of the condensed phase islands around two
laser spots observed in 8 is presented in 23b . At approaching the centers of
lasers spots, the rings of the emission around two spots transform from two
rings around two centers to the deformed single ring with two laser spots
inside the ring in according with experiment 8 . The theory 20 explained an
appearance of a spike observed in But in emission of the indirect excitons
after the shutdown of a pumping by the increasing of the exciton lifetime in a
consequence of the removal of an Auger processes. In the paper22 the
fragmentation of the inner ring in the laser spot was predicted. These
fragmentation was observed recently in the experiment Rem .
Several theoretical investigations were made for new systems, which were not
studied by experiments so far. It was shown, that a periodical structure of
the condensed phase islands arise at the light irradiation in the quantum well
under electrode with a slot Sug . The chain of the islands moves in the
presence of a linear potential along the slot. The process reminds the Gunn
effect in semiconductors and can be called ”excitonic Gunn effect”. The
pumping dependence study of exciton density distribution in the well under the
electrode in the shape of a disk presents all stages of the phase transitions:
from the islands of the condensed phase in a gas phase till the islands of the
gas phase in an environment of the condensed phase 23c . The investigation of
excitonic pulse moving in external fields 23d shown that maximum exciton
density remains constant during the exciton lifetime and there is a
possibility of a control of pulse moving by an another laser if the formation
of the pulse occurs by excitons in the condensed phase.
While developing the theory, two approaches of the theory of phase transitions
were used: the model of nucleation (Lifshits-Slyozov) and the model of
spinodal decomposition ( Cahn-Hillart). These models were generalized on the
particles with the finite lifetime, that is important for interpretation of
the experimental results. The involvement of Bose-Einstein condensation for
excitons was not required for the explanation of the experiments, the
considered condensation is the condensation in real space. Among many
experiments, explained by the theory, there are two ones, presented below,
which did not considered jet in the framework of the presented theory.
A. It was shown in the work 30 , that maximum of the frequency dependence of
the emission from the region between the islands is lower, than maximum of the
emission frequency from the islands, so from the region, where the exciton
density is large. The difference of the maxima is small, it is less than the
width of the emission band. But on the base of this date the authors came to
the conclusion, that there is the repulsion interaction between excitons only.
This result contradicts the main assumptions of the model of the works 19 ; 20
; 21 ; 22 ; 23 ; 23a about the presence of attractive interaction between
excitons, that exist at some distance between excitons and causes the creation
of a condensed phase. In the Appendix we present a possible explanation of
such effect in the case of the attractive interaction between excitons SugUJP
. The explanation is based on the presence in the well of localized exciton
states, levels of which are situated lower than the exciton levels and become
saturated with increasing pumping. The emission band is determined by the free
and localized exciton states. The exciton states form the upper part of the
band. With increasing pumping the number of excitons and the blue part of the
band emission growth. In the case of an attractive interaction, a lowering of
the exciton levels caused by the exciton condensation is small in comparison
with the whole width of the band. As the result the maximum of the emission
shifts to higher side with increasing the pumping, if the emissions from the
exciton condensed phase and from the localized states are not separated.
B. In the works 12 ; 31 ; 32 a coherence was observed in the emission spectra
from island 32 , or even from different islands 12 ; 31 . The coherence was
revealed in the interference of the emission from the different spatial
points.
The effect $B$ is not considered in the presented paper. For its study, the
microscopic model of the condensed phase is needed. But a qualitative
explanation may be given. The interference of the wave functions does not
observed on the experiment directly, the interference of electromagnetic waves
is shown on the experiment. Because the electromagnetic field and scattered
field are coherent, the interference of the emission from two islands may
arise as a result of an imposition of electromagnetic field emitted by some
island and the scattered field by other island. It was shown in the papers 33
; 34 ; 35 , that the strong correlation between exciton densities at different
points takes place in the case when the exciton condensed phase exists. Also
there is a sharp maximum of the Fourier transformation of the two point
correlation function. It is reason of a mutual connection of the wave emitted
from some point with the wave scattered by other region. But the quantitative
calculations require the date of microscopic model of the exciton condensed
phase, particularly, the numerical value of the polarizability is needed.
In the presented paper the hydrodynamics equation for excitons is obtained for
the case, if excitons are in condensed phase. The equations allow to describe
the moving of the complicated system composed of two phases: gas and condensed
ones. We have studied the spatial distribution of both the exciton density and
exciton flux in the case of condensation at steady-state pumping. From
analysis of different solution of the equation for exciton density it was
shown the existence of exciton autosolitons at some parameters of the system.
Also the possible explanation of the effect $A$ is given taking into account
the presence of localized states, which become totaly occupied with increasing
pumping.
It is necessary to emphasize that besides the system, we investigated, there
is an another one, in which the exciton condensation is studied. It is the
exciton condensation in bilayer quantum Hall system. In this system the two
layers are filled by electrons in the presence of strong magnetic field with
total (for two wells) Landau level filling factor $\nu_{T}=1$ and with total
density $n_{T}=n_{1}+n_{2}$, where $n_{i}$ is the density in the separate well
($i=1,2$). An electron in the lowest Landau level of one layer bounded to hole
in the lowest Landau level of the other layer is considered as exciton. The
review of collective effects in such system is presented inEis . The bilayer
quantum Hall system differs from the system that we consider. The electrons
are created in the bilayer quantum Hall system by donors and the system is
equilibrium. We studied the system in which the excitons are created by the
light,the system is non-equilibrium. The excitons have the finite value of the
lifetime and this fact influences significantly on the behavior of the
collective states, particularly, on the formation of the spatial structures.
## II Hydrodynamics of exciton condensed phase
The hydrodynamic equations of excitons were obtained and analyzed in the work
36 . In comparison with this paper, we obtain SugUJP the hydrodynamic
equations of excitons generalizing the Navier-Stokes equations taking into
account the finite exciton lifetime, the pumping of exciton and the existence
of an exciton condensed phase caused by interaction between excitons. The
system is described by the exciton density $n\equiv n(\vec{r},t)$ and by the
velocity of the exciton liquid $\vec{u}\equiv\vec{u}(\vec{r},t)$. The equation
of the continuity of the exciton density is rewritten in the form
$\frac{\partial n}{\partial t}+div(n\vec{u})=G-\frac{n}{\tau_{ex}},$ (1)
where $G$ is the pumping (the number of excitons created for unit time in unit
area of the quantum well), $\tau_{ex}$ is the exciton lifetime. In the
comparison with the typical equation for a liquid, the presented equation for
excitons contains the terms, that describe the pumping and the finite lifetime
of the excitons.
The equation for the movement of the unit volume of an exciton liquid is
rewritten in the form
$\frac{\partial mnu_{i}}{\partial t}=-\frac{\partial\Pi_{ik}}{\partial
x_{k}}-\frac{mnu_{i}}{\tau_{sc}},$ (2)
where $m$ is the exciton mass, $\Pi_{ik}$ is the tensor of the density of the
exciton flux.
$\Pi_{ik}=P_{ik}+mnu_{i}u_{k}-{\sigma}^{\prime}_{ik},$ (3)
where $P_{ik}$ is the pressure tensor, ${\sigma}^{\prime}_{ik}$ is the
viscosity tensor of a tension.
In comparison with the typical Navier-Stokes equation a braking of exciton
liquid by phonons and by defects is introduced in Eq. (2). In the equation (2)
we neglected by the momentum change caused by the creation and the
annihilation of excitons. Indeed, the momentum change in the unit time and in
the unit volume owing to an disappearance of the excitons has the order
$mnu/\tau_{ex}$. Since $\tau_{ex}>>\tau_{sc}$ this value is much less the last
term in the formula (2). The momentum change due to the addition of new
excitons by pumping is small too, because mean exciton moment, created by
external light, is close to zero.
Introducing coefficients of the viscosity and using Eq. (1), Eq.(2) may be
rewritten in the form
$\displaystyle\rho\left({\frac{\partial u_{i}}{\partial
t}+\left({u_{k}\frac{\partial}{\partial
x_{k}}}\right)u_{i}}\right)=-\frac{\partial P_{ik}}{\partial x_{k}}+\eta\Delta
u_{i}$ $\displaystyle+(\varsigma+\eta/3)\left({\frac{\partial}{\partial
x_{i}}}\right)div\vec{u}-\frac{\rho u_{i}}{\tau_{sc}},$ (4)
$\rho=mn$ is the mass of excitons in unit volume.
Let us consider the tensor of pressure. To find the connection between the
tensor and others parameters it is needed to use the equation of the state. We
suggest that the state of the local equilibrium is realized and the state of
the system may be described by a free energy, which depends on spatial
coordinate. Let us present the functional of the free energy in the form
$F=\int{d\vec{r}\left({\frac{K}{2}(\nabla n)^{2}+f(n)}\right)}.$ (5)
The first term in the integrand (5) describes the energy of non-homogeneity.
The pressure tensor will be obtained from equation of the state of the system.
At the given presentation of the free energy, the pressure tensor is
determined by the formula 37
$P_{\alpha\beta}=\left({p-\frac{K}{2}(\vec{\nabla}n)^{2}-Kn\Delta
n}\right)\delta_{\alpha\beta}+K\frac{\partial n}{\partial
x_{\alpha}}\frac{\partial n}{\partial x_{\beta}},$ (6)
where $p=n{f}^{\prime}(n)-f(n)$ is the equation of the state, $p$ is the
isotropic pressure.
Taking into account (6), we rewrite the equation (II) finally in the form
SugUJP
$\displaystyle\frac{\partial u_{i}}{\partial t}+u_{k}\frac{\partial
u_{i}}{\partial x_{k}}+\frac{1}{m}\frac{\partial}{\partial
x_{i}}\left({-K\Delta n+\frac{\partial f}{\partial n}}\right)+\nu\Delta u_{i}$
$\displaystyle+(\varsigma/m+\nu/3)\left({\frac{\partial}{\partial
x_{i}}}\right)div\vec{u}+\frac{u_{i}}{\tau_{sc}}=0.$ (7)
Eqs.(1,II) are the equations of hydrodynamics for an exciton system. These
equations differ from the hydrodynamic equations, investigated in 36 , by the
presence of the terms of the right side in Eq.(1), which take into account the
lifetime and pumping, and by the presence of the third term in Eq.(II), which
describes a condensed phase. It follows from the estimations, made in the work
36 , that the terms with the viscosity coefficients are small and we shall
neglect them.
In the case of a steady state irradiation of the system, the Eqs.(1) and (II)
have the solution $n=G\tau$, $u=0$. In order to investigate the stability of
this solution we consider the behavior of a small fluctuation of the exciton
density and the velocity from these values: $n\rightarrow n+\delta
n\exp(i\vec{k}\cdot\vec{r}+\lambda t)$, $u=\delta
u\exp(i\vec{k}\cdot\vec{r}+\lambda(\vec{k})t)$. After substitution these
expressions in Eqs.(1, II), we obtain in the linear approximation with respect
to the fluctuations the following expression
$\displaystyle\lambda_{\pm}(\vec{k})=\frac{1}{2}(-(1/\tau_{sc}+1/\tau_{ex})$
$\displaystyle\pm\sqrt{(1/\tau_{sc}-1/\tau_{ex})^{2}-\frac{4k^{2}n}{m}(k^{2}K+\frac{\partial^{2}f}{\partial
n^{2}})}),$ (8)
It is follow from (II), that both parameters $\lambda_{\pm}(\vec{k})$ have a
negative real part at small and large values of vector $\vec{k}$ and,
therefore, the uniform solution of hydrodynamics equation is stable. But the
value $\lambda_{+}(\vec{k})$ may be positive in some interval of vector
$\vec{k}$ , when $\frac{\partial^{2}f}{\partial n^{2}}$ becomes negative. In
these case the uniform distribution of the exciton density is unstable with
respect to a formation of nonhomogeneous structures. The instability arises at
some threshold value of exciton density and at some critical value of the wave
vector. Analysis of the equation (II) gives the following expression for the
critical values of the wave vector $k_{c}$ and the exciton density $n_{c}$
$k_{c}^{4}=\frac{m}{Kn_{c}\tau_{sc}\tau_{ex}},$ (9)
$\frac{k_{c}^{2}n_{c}}{m}\left(k_{c}^{2}K+\frac{\partial f(n_{c})}{\partial
n_{c}^{2}}\right)+\frac{1}{\tau_{sc}\tau_{ex}}=0.$ (10)
For stable particles ($\tau_{ex}\rightarrow\infty$) the equations (9,10) give
the condition $\frac{\partial f}{\partial n^{2}}=0$, that is condition for
spinodal decomposition for a system in the equilibrium case.
Depending on parameters the Eqs.(1,II) describe the ballistic and diffusion
movement of the exciton system. The relaxation time $\tau_{sc}$ plays the
important role in a formation of the exciton moving. Due to arising of the
nonhomogeneous structures, the exciton currents in the system exist
($\vec{j}=n\vec{u}\neq 0$) even under the uniform steady-state pumping.
Excitons are moving from regions with the small exciton density to the regions
with the high density. In the presented paper we shall consider the spatial
distribution of exciton density and exciton current in the double quantum well
under steady-state pumping. In this case the exciton carrent is small and we
suggest the existence of the next conditions
$\frac{\partial u_{i}}{\partial t}<<u_{i}/\tau_{sc}.$ (11)
$u_{k}\frac{\partial u_{i}}{\partial x_{k}}<<u_{i}/\tau_{sc}.$ (12)
Particularly, the Eq.(11) holds under the study of the steady-state exciton
distribution. The fulfilment of Eq.(12) will be shown later after some
numerical calculations.
Using the conditions (11) and (12) we obtain from Eq.(II) the value of the
velocity $\vec{u}$
$\vec{u}=-\frac{\tau_{sc}}{m}\vec{\nabla}\left({-K\Delta n+\frac{\partial
f}{\partial n}}\right),$ (13)
As the result the equation for the exciton density current may be presented in
the form
$\vec{j}=n\vec{u}=-M\nabla\mu,$ (14)
where $\mu=\delta F/\delta n$ is the chemical potential of the system,
$M=nD/\kappa T$ is the mobility, $D=\kappa T\tau_{sc}/m$ is the diffusion
coefficient of excitons.
Therefore, the equation for the exciton density (1) equals
$\displaystyle\frac{\partial n}{\partial t}=\frac{D}{\kappa
T}(-Kn\Delta^{2}n-K\vec{\nabla}n\cdot\vec{\nabla}\Delta n)$
$\displaystyle+\frac{D}{\kappa
T}\vec{\nabla}\cdot\left(n\frac{\partial^{2}f}{\partial
n^{2}}\vec{\nabla}n\right)+G-\frac{n}{\tau_{ex}}.$ (15)
Just in the form of (II) we investigated a spatial distribution of the exciton
density at exciton condensation at different dependencies $f$ on $n$ [22, 23,
33, 38, 39]. So, our previous consideration of the problem corresponds to the
diffusion movement of hydrodynamics equations (1,II). At some conditions,
applied to the functional $F$ the uniform solution is unstable, and the
spatial structure arises in the system. For the system under study the
condensed phase appears, if the function $f(n)$ describes a phase transition.
In the papers mentioned above the examples of such dependencies were given.
Here, we analyze an other dependence $f(n)$, which often is also used in the
theory of phase transitions. We shall approximate the density of the free
energy in the form
$f=\kappa
Tn(\ln(n/n_{a})-1)+a\frac{n^{2}}{2}+b\frac{n^{4}}{4}+c\frac{n^{6}}{6},$ (16)
where $a$, $b$, $c$ are the constant values. Three last terms in the formula
(16) are the main terms, they arise due to an exciton-exciton interaction and
describe the phase transition. The first term was introduced in order to
describe the system in a space, where the exciton concentration is small(if
such region exists in the system). With increasing the exciton density the
term $a\frac{n^{2}}{2}$ must manifests itself firstly. It gives the
contribution the $an$ value to chemical potential. The origin of this term is
connected in our system with the dipole-dipole interaction that should become
apparent at the beginning due to its long-range nature. For estimations of $a$
we may use for the dipole-dipole exciton interaction in double quantum well
the plate capacitor formula $an=4\pi e^{2}dn/\epsilon$, where $d$ is the
distance between wells, $\epsilon$ is the dielectric constant. This formula is
used usually for a determination of the exciton density from the the
experimental meaning of the blue shift of the frequency of the exciton
emission with the rise of the density. It is follow from the formula that
$a=4\pi e^{2}d/\epsilon$. This expression is approximate because it does not
take into account the exciton-exciton correlations 25 ; 37b . When the exciton
density growths the last two terms in (14) begin to play a role. An existence
of the condensed phase requires that the value $b$ was negative($b<0$). For
stability of the system at large $n$ the parameter $c$ should be positive
($c>0$). It is suggested in the model, that the condensed phase arises due to
the exchange and van der Waals interactions. For the system with the large
distance between wells the dipole-dipole interaction exceeds attractive
interaction and the condensed phase does not arise. The disappearance of
chemical potential minimum as a function of $n$ with increasing of the
parameter $a$ (with increasing the distance between the quantum wells) is
presented in Fig1.
Figure 1: Qualitative dependence of the chemical potential on $n$. The
parameter $a$ for the thin line is greater than for the thick line. The
parameters $b$ and $c$ are the same for both curves.
Let us introduce the dimensionless parameters: $\tilde{n}=n/n_{o},$ where
$n_{o}=\left({a/c}\right)^{1/4}$, $\tilde{b}=b/(ac)^{1/2}$,
$\tilde{\vec{r}}=\vec{r}/\xi$, where $\xi=\left({K/a}\right)^{1/2}$ is the
coherence length , $\tilde{t}=t/t_{0}$, where $t_{0}=\frac{\kappa
TK}{Dn_{o}a^{2}}$, $D_{1}=\frac{\kappa T}{an_{o}}$, $\tilde{G}=Gt_{0}/n_{0}$,
$\tilde{\tau}_{ex}=\tau/t_{0}$. As the result the equation (II) is reduced to
the form (hereinafter the symbol $\sim$ will be omitted in the equation)
$\displaystyle\frac{\partial n}{\partial t}=D_{1}\Delta n-n\Delta^{2}n+n\Delta
n(1+3bn^{2}+5n^{4})$ $\displaystyle-\vec{\nabla}n\cdot\vec{\nabla}\Delta
n+(\vec{\nabla}n)^{2}(1+9bn^{2}+25n^{4})+G-\frac{n}{\tau_{ex}}.$ (17)
The solutions of the equation (II) are presented in Fig.2 in the one-
dimensional case ($n(\vec{r},t)\equiv n(z,t))$) for three values of the
steady-state uniform pumping.
Figure 2: The spatial dependence of the exciton density at the different
value of the pumping: for the continues line $G=0.0055$, for the periodical
line $G=0.008$, for the dashed line $G=0.0092$. $D_{1}=0.03$, $b=-1.9$
The solutions are obtained at the initial conditions $n(z,0)=0$ and the
boundary conditions
$n^{\prime}(0,t)=n^{\prime}(L,t)=n^{\prime\prime}(0,t)=n^{\prime\prime}(L,t)=0$,
where $L$ is the size of the system. The periodical solution exists in the
some interval of the pumping $G_{c1}<G<G_{c2}$. At the great size of the
system the structure of the solution (the period and the amplitude of the
lattice) does not depend on the boundary conditions. At the given parameters
the periodical solution exists at $0.0055<G<0.0092$. Outside this region, the
solution describes a uniform system: the gas phase at the pumping less the
lower boundary value and the condensed phase at the pumping greater upper
boundary value. Upper part of the periodical distribution corresponds to
condensed phase, the lower one corresponds to the gas phase. The size of the
condensed phase increases with the change of the pumping from $G_{c1}$ to
$G_{c2}$. At $G>G_{c2}$ the state with uniform distribution of condensed phase
emerges.
Figure 3: The spatial dependence of the exciton current at $G=0.008$,
$D_{1}=0.03$, $b=-1.9$.
Fig.3 shows the spatial dependence of the exciton current calculated by the
formula (14). The current equals zero in the centers of the condensed and gas
phases and it has a maximum in the region of a transition from the condensed
phase to the gas phase.
Let us do some estimations. It is seen from Fig.2, that the exciton density in
the condensed phase redouble approximately in comparison with the density in
the gas phase. The results for the currents in Fig.3 is presented in
dimensionless units: $\tilde{j}=j/j_{o}$, where $j_{o}=n_{o}u_{o}$,
$u_{o}=(\tau_{sc}n_{o}a)/(m\xi)$ is the unit of the velocity. The exciton
density presented in Fig.2 in dimensionless units ($\tilde{n}=n/n_{o}$). It is
seen from Fig.2 that$\tilde{n}\sim 1$, and the magnitude of $n$ has an order
on $n_{o}$. So, for estimations we may suppose that $n_{o}a$ corresponds to
the shift of the luminescence line with increasing the exciton density, also
the magnitude of $\xi$ has the order of the size of the condensed phase. For
the following magnitudes of parameters
$\tau_{sc}=10^{-11}\textmd{s},n_{o}a=2\cdot 10^{-3}\textmd{eV},m=2\cdot
10^{-28}\textmd{g},\xi=2\cdot 10^{-4}\textmd{cm}$, we obtain $u_{o}\sim
10^{6}\textmd{cm/s}$. According to calculations (see Fig.3) the magnitudes of
the current and the velocity are in two orders of the values less than their
unit. So, at chosen parameters, the maximal amplitude of the velocity of the
exciton density flux in the dissipative structure in the double quantum well
equals $10^{4}\textmd{cm/s}$. In order to verify the fulfilment of the
condition (12), let us suppose that ($\partial u_{i})/(\partial x_{k}\sim
u/l$), where $l$ is the period of the structure. It follows from experiments 5
; 9 that $l\sim(5\div 10)\mu\textmd{m})$. Using these date we obtain that the
condition (12) is satisfied very good. This condition is violated at
$\tau_{sc}\geq 10^{-9}\textmd{s}$. It is the very large value. The calculation
using uncertainty principle from the band width of the narrow line ($2\cdot
10^{-4}\textmd{eV}$)2 ; 3 gives more less the magnitude $3\cdot
10^{-12}\textmd{s}$. Therefore, the formation of nonuniform exciton
dissipative structures in the double quantum well occurs by the diffusion
movement of excitons. For a proof of the main equation (II) the last term in
Eq. (2), which describes the loss of the momentum due to scattering on defects
and phonons, is of importance. Just this term describes the processes, which
cause a decay of the exciton flux. From the viewpoint of the possibility of
the appearance of superfluidity, the situation for excitons is more
complicated than that for the liquid helium and for the atoms of alkali metals
at ultralow temperatures. In the last systems, the phonons (moving of
particles) are intrinsic compound part of the system spectrum, the interaction
between phonons (particles) are the interaction between of atoms of the system
and does not cause the change of the complete momentum of the system and its
moving as whole. Phonons and defects for excitons are external subsystems,
which brake the exciton moving. Therefore, for the creation of exciton
superfluidity it is needed, that the value of $\tau_{sc}$ growths
significantly. It is possible for exciton polaritons, which weekly interact
with phonons; and there is a certain experimental evidence on an observation
of the polariton condensation 40 . For the indirect excitons the critical
temperature of the superfluid transition is strongly lowered by
inhomogeneities 40a ; 40b . So, the question about the possibility of the
superfluidity existence for the indirect excitons on the base AlGaAs system is
open.
Thus, the peculiarities, that are observed at large densities of the indirect
excitons, may be explained by the phase transitions in the system of the
particles with attractive interactions and the finite value of the lifetime
without an involvement of the Bose-Einstein condensation.
## III Excitonic autosolitons
As it was shown, at $n<n_{c1}(G<G_{c1})$ the uniform solution of Eq. (II) is
stable. But, at some limits of a pumping at $G<G_{c1}$ there exists a
stationary solution for the exciton density distribution localized in a space.
For example, with the parameters used for calculations of the exciton
distribution in Fig.3, the threshold value of the pumping equals
$G_{c1}=$0.0055; but, at a steady-state pumping there is the spatial
nonuniform solution of the equation (II) at $G<G_{c1}$ in the form of an
isolated spike. It may be obtained solving Eq.(II) at the pumping, which
consist of a constant value $G_{0}$ and an additional pulse $dG$ with the
maxima in the some point of the space and in the time moment
$dG=s\,\exp[-w(z-z_{0})^{2}]\,\exp[-p(t-t_{0})^{2}]$ (18)
where $s,\,w,\,p$ are parameters. The formula (18) describes a pulse of the
pumping, which acts during some time interval with the maximum in the point
$z_{0}$.
The solution of Eq.(II) obtained under an application of the addition pulse
(18) in the region $z_{0}=L/2$ has at $t\to\infty$ the form presented in
Fig.4.
Figure 4: The spatial dependence of the exciton density of the excitonic
autosoliton at the pumping $G=0.005<G_{c1}$; $D_{1}=0.03$, $b=-1.9$.
The solution exists at $t\to\infty$, i.e. at the times, when the action of the
addition pulse is absent already. The shape of the spike $n(z)$ does not
depend on parameters $s,\;w,\;p$, except cases, when at least one of these
parameters tends to zero and becomes less some value. In addition, the
solution in the form, presented in Fig.4, arises also, if the additional pulse
is absent, but there is some distribution of the exciton density in the
initial time $t=0$:
$n(z,0)=s_{0}\exp(-w(z-z_{0})^{2}).$ (19)
Fig.5 shows the distribution of exciton current in the vicinity of the
localized solution. The current changes the sign in the center of the
localized state of the density.
Figure 5: The spatial dependence of the exciton current in the localized
state at the pumping $G=0.005<G_{c1}$; $D_{1}=0.03$, $b=-1.9$.
We have verified by direct calculations, that the solution presented in Fig.4
in the form of a localized distribution of the density is stable. We call the
state, that describes this solution localized in space, by excitonic
autosoliton. The spatial dependence of the exciton density will be designated
by $n_{as}(z)$. The autosolitons exist in the some regions of the pumping
$G_{cas}<G<G_{c1}$. At chosen parameters in Fig.4 the autosolitons arise under
conditions $0.003<G<0.0055$. The solutions exist in the form of the
autosoliton side by side with the uniform solutions. The lower boundary
$G_{cas}$ depends on exciton parameters, particulary, on the exciton lifetime.
Excitonic autosolitons correspond to solitary solutions of nonlinear equations
for the excitons (II). The name“auto” is introduced according to 42 ; it
underlines, that the solitary solutions arise in the dissipative system in a
contrast to ”solitons”, which appear in conservative systems. The obtained
here autosolitons correspond to the ”static autosolitons” according to the
classification 42 . The solutions in the form of the autosolitons are
degenerated: if there is a solitary solution $n_{as}(z)$,then
$n_{as}(z-z_{0})$ will be also solution at the arbitrary $z_{0}$ (in the
infinite medium). But, if there is an external field in a system, which
creates a spatial dependent additional potential for excitons, the solitary
excitation moves. Thus, at linear spatial dependence of the additional
potential energy in the formula of the free energy (16) the term $\delta
V=-dz$ should be added. In this case the equation (II) has the solution in the
form of autowaves $n_{as}(z-vt)$, where $v$ is the velocity of the autowave.
In the region, in which the periodic solution of the exciton density takes
place ($G(c1)<G<G(c2)$), such autowaves were investigated in the work Sug .
Localized solutions exist also in the some region at the pumping greater the
value, at which the periodical structure arises ($G>G(c2)$). The dependence of
the exciton density may be obtained from Eq.(II) choosing an additional
pumping pulse in the form (18), but at $a<0$. An example of such solution is
presented in Fig. 6. These structures appear in the form of a dip, and can be
called ”dark autosolitons” by analogy with the soliton’s terminology.
Figure 6: The spatial dependence of the exciton density of the dark
autosoliton at $G=0.0095>G_{c2}$; $D_{1}=0.03$, $b=-1.9$ .
To explain the appearance of the autosoliton-type solution we recall that the
phase transition are investigated in the paper. As is known, there exists the
region between spinodal and binodal, in which the creation of a nucleus of new
phase is needed for phase transition. The size of the nucleus should exceed
some critical value. The obtained critical value of $n_{c}$ (Eqs.(9,10)) is
based under the consideration of the small fluctuations and corresponds really
to the boundary of the spinodal region corrected by non-equilibrium state of
the system. The autosoliton arises at pulses larger some critical value. So,
the appearance of autosolitons corresponds to the appearance of the nucleus
outside the spinodal boundary for phase transitions of the stable particles. A
size of the new phase of the stable particles increases with time, while the
distribution density of the unstable particles (excitons) does not depend on
the time (at a steady-state pumping). It explains the existence of the
localized states of exciton distribution outside the region of the instability
of an uniform distribution. It should be noted, that in the phase transition
approach of nucleation-growth with an generation on unstable particles 19 ; 20
; 21 the existence of the condensed phase islands outside the spinodal region
is taking into account automatically, because this approach consider
fluctuations, which do not presented in the equation (II).
The model of the nucleation-growth, which we used under the investigation of
the exciton condensation in 19 ; 20 ; 21 , is close to the methods, used in 43
; 44 ; 45 under the investigation of electron-hole droplets in germanium and
silicon. But in contrast to the works 43 ; 44 ; 45 we took into account the
correlation in the spatial positions of the droplets (the islands in the two-
dimensional case), that allows to describe the different structures, mutual
positions of of islands, their concentration, the formation of the periodicity
and other properties, which are observed in quantum wells on the base of the
AlGaAs system.
## IV Conclusions
In the work, the several problems of the theory of indirect exciton
condensation in quantum well on the base AlGaAs crystal are considered.
The hydrodynamic equations for excitons were build with taking into account a
possibility of the condensed phase formation, the finite lifetime of excitons,
the scattering of excitons by defects. The analysis of the equations was
fulfilled for the diffusive exciton flow. The equations explain many spatial
excitonic structures obtained under experimental investigations of indirect
excitons in quantum wells on the base AlGaAs without an involvement of the
Bose-Einstein condensation of excitons.
Also, solutions of the equations in a form of solitary localized states (the
bright and dark exciton autosolitons) were found. They exist side by side with
the steady-state exciton density state and may be excited by the additional
pulse greater some threshold value. In the model of the nucleation-growth
these solutions correspond to the nuclei of the condensed phase in a gas phase
(the bright autosolitons) and to the nuclei of the gas phase in the condensed
phase (the dark autosolitons).
It was shown, that the results of the experiment 30 , in which has been
obtained, that the emission spectrum from the condensed phase region is
shifted to shortwave side in a comparison of the emission spectra from the
region of the gas phase, do not contradict to the model with an attractive
interaction between excitons. For the explanation it was taking into account,
that a formation of the emission spectrum occurs by both free and trapped
excitons.
## Appendix A Distribution of excitons between localized and delocalized
states
According to 30 the frequency of the emission from the islands on the ring
around the laser spot is higher, than the frequency of emission from the
region between islands. The authors made the conclusion 30 , that interaction
between excitons is repulsive, and, therefore, the formation of the condensed
phase by attractive interaction is impossible. It contradicts the main
assumption of the works 19 ; 20 ; 21 ; 22 ; 23 , though these works explain
many experiments. Now we remove this contradiction, taking into account the
presence of localized excitons.
Residual donors and acceptors, defects, inhomogeneous thickness of the wells
create an accidental fluctuating potential, which may be the reason of
appearance of the localized levels. Till now the explanation of the creation
of the localized states is not determined definitely, but their existence is
confirmed by the presence of an emission in the region of the frequencies less
the frequency of the exciton band emission and broadening of exciton lines. At
the low temperature and at the small pumping the main part of the band
consists of the emission from defect centers, the part of the exciton emission
growths with increasing pumping. Let us consider the relation between the
contribution to the emission band intensity from free excitons and the
excitons localized on defects. Since the defect structure of the samples
depends on their preparation, a solution of this problem can not be solved in
general. We shall use some approximations.
We shall consider the energy distribution of electron-hole pairs at steady-
state irradiation. Such pair may be the delocalized exciton in the exciton
band and an electron and hole localized near a defect or in the region with a
modified thickness of the well. For small density of the excitation (the
electron-hole pairs) the interaction between them may be neglected. Due to
long-range character the dipole-dipole interaction, which appear with
increasing the excitation density, gives an identical shift of delocalized and
localized levels. It means that this interaction do not influence on mutual
distribution of the localized and delocalized states. We shall suggest that
the localized states are saturable, namely, every center may capture a
restricted number of excitations. In our calculations we shall assume that
only single excitation may be localized on the defect. Another excitations are
or absent or have very low binding energy and are unstable. The dependence of
a density of localized states on energy was chosen in the exponential form,
namely $\rho(E)=\alpha N_{l}exp(\alpha E)$, where $N_{l}$ is the density of
the defect centers, $E$ is the depth of the trap level. The exciton states
(free and localized) are distributed on levels after a creation of electrons
and holes by an external irradiation and their subsequent recombination and
relaxation. Because the time of the relaxation is much less than the exciton
lifetime, the distribution of excitation between free and localized states
corresponds to to a state of thermodynamical equilibrium. In considered model
we should obtain the distribution of electron-hole pairs, a population of
which on a single level may be changed from zero to infinity for $E>0$ (the
free exciton states) and from zero to one for $E<0$ (the localized states).
Formally, in considered system the free excitons have Bose-Einstein statistic
and localized excitations obey Fermi-Dirac statistic. At small exciton density
Bose-Einstein and Boltsmann statistics give the similar results for the free
excitons, but the application of Fermi-Dirac statistic for localized states
with single level for one trap is important. The equation for the energy
distribution may be find from minima of large canonical distribution
$w(n_{k},n_{i})=\exp\left(\frac{\Omega+N\mu-E}{\kappa T}\right),$ (20)
where $N=\Sigma_{i}n_{i}+\Sigma_{k}n_{k}$,
$E=\Sigma_{i}n_{i}E_{i}+\Sigma_{k,l}n_{k}E_{k}$, $n_{i}=0,1$,
$n_{k}=0,1,...\infty$, $k$ is the wave vector of the exciton, $l$ designates
the singular levels. $\mu$ is the exciton chemical potential.
The distribution of excitons over free and localized level is determined from
minimum of the functional (20). As the result we obtain the following
conditions for the mean values of the free exciton density $n$ and the density
of the localized states $n_{L}$
$n_{ex}=\frac{g\nu}{4\pi
E_{ex}a_{ex}^{2}}\int_{0}^{\infty}\frac{dE}{\exp(\frac{E-\mu}{\kappa T})-1},$
(21) $n_{L}=\alpha N_{l}\int_{-\infty}^{0}\frac{\exp(\alpha
E)dE}{\exp(\frac{E-\mu}{\kappa T})+1},$ (22)
where $a_{ex}=(\hbar^{2}\varepsilon)/(\mu_{ex}e^{2})$ and
$E_{ex}=(\mu_{ex}e^{4})/(2\varepsilon^{2}\hbar^{2})$ are the radius and the
energy of the exciton in the ground state in bulk material, $g=4$, $\mu_{ex}$
is the reduced mass of the exciton, $\nu$ is the ratio of the reduced and the
total mass of the exciton. The chemical potential $\mu$ is determined from
condition
$n_{L}+n=G\tau_{ex},$ (23)
where $G\tau_{ex}$ is the whole number of excitation (free and localized) per
unit surface.
The dependence of the distribution of the free and the localized exciton on
pumping is presented in Fig.7 as function of whole number of the excitation
presented in units of $1/a_{ex}^{2}$
Figure 7: The dependence of the density of the free (thick line) and trapped
(thin line) excitons on the pumping. The parameters of the system:
$T=2K,N_{l}=0.001/a_{ex}^{2},\alpha=300(\textmd{eV})^{-1}$
Let us the exciton radius equals $10\textmd{nm}$. Then the concentration of
the traps and the width of the distribution of trap levels, chosen under
calculations of Fig.7, have the order of $10^{9}\textmd{cm}^{-2}$ and
$0.003\textmd{eV}$, correspondingly.
As it is seen from Fig.7 that the number of the localized excitations exceeds
at small pumping the number of free excitons and the emission band should be
determined by the emission from the traps. With increasing pumping the
occupation of the trap levels become saturated. For chosen parameters the
concentration of excitations at the saturation is a value of the order of
$10^{9}\textmd{cm}^{-2}$. Simultaneously with the saturation of the localized
levels the exciton density growths. As the result, the shortwave part of the
emission band should be increased with increasing pumping. When the exciton
density becomes great, the collective exciton effects begin to manifest
themselves. The equations (21,22) do not take into account the interactions
between the excitations, and special models and theories are needed for
descriptions of collective effects. The appearance of a narrow line was
observed in 2 with increasing pumping on the shortwave part on the exciton
emission band. Simultaneously, the patterns arise in the emission spectra. The
narrow line appeared after the localized states becomes occupied. According to
2 this line is connected with the exciton Bose-Einstain condensation.
According to our model 19 , the islands of condensed phase arise, if the
exciton density become higher some threshold value, and the narrow band
corresponds to the condensed phase, caused by the attractive interaction
between excitons.
Figure 8: The distribution of excitations in the traps and in the states of
exciton band. The thick line in fig.8a corresponds to the energy per single
exciton in the condensed phase. On the right (fig.8b) the upper line describes
the whole emission from the island (the emission of both the condemned phase
and trapped excitons), the low line describes the emission of the trapped
excitons.
The energy per single exciton in the condensed phase is less than the energy
of free excitons (the thick line in Fig.8), but the gain of energy under
condensation is less than the whole bandwidth, which are formed by the
localized and delocalized states. The gain of energy is significantly less
than the binding energy of the exciton to an electron -hole drop in silicon
and germanium. According to 3 ; 4 the narrow band is shifted to the red side
with increasing pumping in the value less than 0.5 meV, while the whole
bandwidth has the order of 2 meV. So, the energy of photons emitted from the
islands of condensed phase is higher than the energy of photons emitted by
traps. The excitons can not leave the condensed phase (the islands) and move
to the traps (to the states with lower energy) since the levels of the traps
are occupied already. Thus, the emission frequency of the condensed phase is
larger, than the emission frequency of the gas phase, even at the attractive
interaction between excitons. It may be reason of obtained in 30 results in
which the maximum of the frequency of the emission from the islands is higher
than the maximum frequency from the regions between the islands.
The qualitative results coincide with the results obtained from the solution
of kinetics equations in SugUJP for level distributions. The similar behavior
of the distribution of free and trapped excitons takes place for an other
energy dependence of the density of localized states. We confirmed the results
for the gaussian density distribution.
The results may be applied for explanation of intensity and temperature
dependence of the exciton emission of dipolar excitons in InGaAs coupled
double quantum wells Schin . The authors observed with increasing pumping the
growth of the shortwave side of the emission band, the narrowing of the band.
They obtained that the shortwave side of the band is very sharp. These results
may be explained by suggestion, that the lower part of the band is formed by
localized states. After the saturation of the localized states with increasing
pumping, the excitations begin to occupy the levels of free excitons. Since
the density of exciton state is much greater than the density of defect states
the shortwave edge of the band is sharp. The exciton condensed phase may not
arise in the system investigated in Schin as the distance between wells
($17\textmd{nm}$) is greater than the distance in the system investigated by
the Timofeev’s group 2 ($13\textmd{nm}$), so the repulsive dipole-dipole
interaction between excitons in Schin may be larger than the attractive
interaction.
It should be noted, that in the work 5 the band of the emission from the
condensed phase is wider than the narrow band that was observed in the work 4
, where the another method of the creation of excitons was applied. Maybe, it
is related to the fact, that in the work 5 , excitons were created in the
region of the p-n transition: the electrons approach this region from one
side, and the holes move to the region from other side. The excitons are
situated in a space between the regions rich by electrons from one side, and
by holes from other side. To combine into excitons, electrons and holes should
surely go through the region of the condensed phase. Therefore, this region
contains the charges, which create an electric field and cause the broadening
of the emission band.
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|
arxiv-papers
| 2013-06-20T13:42:31 |
2024-09-04T02:49:46.743817
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "V. I. Sugakov",
"submitter": "Volodymyr Sugakov Iosiphovich",
"url": "https://arxiv.org/abs/1306.4876"
}
|
1306.5074
|
A simultaneous decomposition of five real quaternion matrices with
applications
Zhuo-Heng He, Qing-Wen Wang
Department of Mathematics, Shanghai University, Shanghai 200444. P.R. China
E-mail: [email protected]
> Abstract: In this paper, we construct a simultaneous decomposition of five
> real quaternion matrices in which three of them have the same column
> numbers, meanwhile three of them have the same row numbers. Using the
> simultaneous matrix decomposition, we derive the maximal and minimal ranks
> of some real quaternion matrices expressions. We also show how to choose the
> variable real quaternion matrices such that the real quaternion matrix
> expressions achieve their maximal and minimal ranks. As an application, we
> give a solvability condition and the general solution to the real quaternion
> matrix equation $BXD+CYE=A$. Moreover, we give a simultaneous decomposition
> of seven real quaternion matrices.
> Keywords: Matrix decomposition; Matrix equation; Quaternion; Rank
> 2010 AMS Subject Classifications: 15A24, 15A09, 15A03
>
## 1\. Introduction
Let $\mathbb{R}$ be the real number fields. Let $\mathbb{H}^{m\times n}$ be
the set of all $m\times n$ matrices over the real quaternion algebra
$\mathbb{H}=\big{\\{}a_{0}+a_{1}i+a_{2}j+a_{3}k\big{|}~{}i^{2}=j^{2}=k^{2}=ijk=-1,a_{0},a_{1},a_{2},a_{3}\in\mathbb{R}\big{\\}}.$
For a quaternion matrix $A,$ we denote the conjugate transpose, the column
right space, the row left space of $A$ by
$A^{\ast},\mathcal{R}\left(A\right)$, $\mathcal{N}\left(A\right),$
respectively, the dimension of $\mathcal{R}\left(A\right)$ by
$\dim\mathcal{R}\left(A\right).$ By [12], $\dim\mathcal{R}\left(A\right)=\dim$
$\mathcal{N}\left(A\right),$ which is called the rank of the quaternion matrix
$A$ and denoted by $r(A).$ It is easy to see that for any nonsingular matrices
$P$ and $Q$ of appropriate sizes, $A$ and $PAQ$ have the same rank [34].
Moreover, for $A\in\mathbb{H}^{m\times n},$ by [12], there exist invertible
matrices $P$ and $Q$ such that
$PAQ=\left(\begin{array}[]{ll}I_{r}&0\\\ 0&0\end{array}\right)$
where $r=r(A),$ $I_{r}$ is the $r\times r$ identity matrix.
There is no need to emphasize the importance of matrix decomposition. Matrix
decomposition is not only a basic approach in matrix theory, but also an
applicable tool in other areas in mathematics. Many papers have presented
different matrix decompositions for different purposes (e.g. [2], [4],
[13]-[18], [24], [25], [31], [33]). For matrix factorization, there are many
kinds of decompositions, for instance, the generalized singular decompose
([8], [18]), the restricted singular value decomposition of matrix triplets
[32], the decomposition of triple matrices which two of them have the same row
numbers meanwhile two of them have the same column numbers ([4], [11]), the
decomposition of the matrix triplet $(A,B,C)$ ([13], [15], [26]), the
decomposition of a matrix quaternity in which two of them have the same column
numbers, meanwhile three of them have the same row numbers [27].
The research on maximal and minimal ranks of partial matrices started in later
1980s. Some optimization problems on ranks of matrix expressions attract much
attention from both theoretical and practical points of view. Minimal and
maximal ranks can be used in control theory (e.g. [5], [6]). Cohen et al. [10]
and H.J. Woerdeman [28]-[30] considered the maximal and minimal ranks of
$3\times 3$ partial block matrix
$\displaystyle\begin{bmatrix}A_{11}&A_{12}&X\\\ A_{21}&A_{22}&A_{23}\\\
Y&A_{32}&A_{33}\end{bmatrix}$
over $\mathbb{C}$. Cohen and Dancis [9] gave the minimal rank and extremal
inertias of the Hermitian matrix
$\displaystyle\begin{bmatrix}A&B&X\\\ B^{*}&C&D\\\
X^{*}&D^{*}&E\end{bmatrix}.$
Liu [16] obtained the maximal and minimal ranks of $A-BXC$ using the
restricted singular value decomposition (RSVD) of the matrix triplet
$(C,A,B)$. Liu and Tian [14] derived the maximal and minimal ranks of $A-BX-
CY$ using the QQ-SVD of the matrix triplet $(C,A,B)$. Tian ([19], [20])
studied the maximal and minimal ranks of the matrix expression
$\displaystyle p(X,Y)=A-BXD-CYE$ (1.1)
using generalized inverses of matrices. Chu, Hung and Woerdeman [7] also
considered the maximal and minimal ranks of (1.1).
The aim of this paper is to revisit the maximal and minimal ranks of the real
quaternion matrix expression (1.1) through a simultaneous decomposition of the
real quaternion matrix array
$\displaystyle\begin{matrix}A&B&C\\\ D&&\\\ E&&\end{matrix}.$ (1.2)
Our method is different from the methods mentioned in [7], [19] and [20]. We
can derive the maximal and minimal ranks of the real quaternion matrix
expression
$\displaystyle
f_{2}(X_{1},X_{2},X_{3},X_{4})=A-B_{1}X_{1}-X_{2}C_{2}-B_{3}X_{3}C_{3}-B_{4}X_{4}C_{4}.$
(1.3)
We also consider the solvability condition and minimal rank of the general
solution to the real quaternion matrix equation
$\displaystyle BXD+CYE=A.$ (1.4)
Moreover, we also give a simultaneous decomposition of the real quaternion
matrix array
$\displaystyle\begin{matrix}A&B&C&D\\\ E&&&\\\ F&&&\\\ G&&&\end{matrix},$
which plays an important role in investigating the extreme ranks of the matrix
expression $A-BXE-CYF-DZG$.
## 2\. A simultaneous decomposition of the real quaternion matrix array (5.1)
Now we give the main theorem of this section.
###### Theorem 2.1.
Let $A\in\mathbb{H}^{m\times n},B\in\mathbb{H}^{m\times
p_{1}},C\in\mathbb{H}^{m\times p_{2}},D\in\mathbb{H}^{q_{1}\times n}$ and
$E\in\mathbb{H}^{q_{2}\times n}$ be given. Then there exist nonsingular
matrices $P\in\mathbb{H}^{m\times m}$, $Q\in\mathbb{H}^{n\times
n},T_{1}\in\mathbb{H}^{p_{1}\times p_{1}},T_{2}\in\mathbb{H}^{p_{2}\times
p_{2}},V_{1}\in\mathbb{H}^{q_{1}\times q_{1}},V_{2}\in\mathbb{H}^{q_{2}\times
q_{2}}$ such that
$\displaystyle
A=PS_{A}Q,B=PS_{B}T_{1},C=PS_{C}T_{2},D=V_{1}S_{D}Q,E=V_{2}S_{E}Q,$ (2.1)
where
$S_{A}=\left[\begin{array}[c]{ccccccc}0&0&0&0&0&I_{m_{1}}&0\\\
0&A_{1}&A_{2}&A_{3}&0&0&0\\\ 0&A_{4}&A_{5}&A_{6}&0&0&0\\\
0&A_{7}&A_{8}&A_{9}&0&0&0\\\ 0&0&0&0&I_{m_{5}}&0&0\\\ I_{m_{6}}&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{array}\right],S_{B}=\left[\begin{array}[c]{ccc}0&0&B_{1}\\\
I_{m_{2}}&0&0\\\ 0&I_{m_{3}}&0\\\ 0&0&0\\\ 0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right],S_{C}=\left[\begin{array}[c]{ccc}C_{1}&C_{2}&0\\\
0&0&0\\\ 0&I_{m_{3}}&0\\\ 0&0&I_{m_{4}}\\\ 0&0&0\\\ 0&0&0\\\
0&0&0\end{array}\right],$
$S_{D}=\left[\begin{array}[c]{ccccccc}0&I_{n_{2}}&0&0&0&0&0\\\
0&0&I_{n_{3}}&0&0&0&0\\\
D_{1}&0&0&0&0&0&0\end{array}\right],S_{E}=\left[\begin{array}[c]{ccccccc}E_{1}&0&0&0&0&0&0\\\
E_{2}&0&I_{n_{3}}&0&0&0&0\\\ 0&0&0&I_{n_{4}}&0&0&0\end{array}\right],$
$\displaystyle
m_{1}+m_{5}+m_{6}=r\begin{bmatrix}A&B&C\end{bmatrix}+r\begin{bmatrix}A\\\ D\\\
E\end{bmatrix}-r\begin{bmatrix}A&B&C\\\ D&0&0\\\ E&0&0\end{bmatrix},$ (2.2)
$\displaystyle m_{2}=r\begin{bmatrix}A&B&C\\\ D&0&0\\\
E&0&0\end{bmatrix}-r\begin{bmatrix}A&C\\\ D&0\\\
E&0\end{bmatrix},m_{4}=r\begin{bmatrix}A&B&C\\\ D&0&0\\\
E&0&0\end{bmatrix}-r\begin{bmatrix}A&B\\\ D&0\\\ E&0\end{bmatrix},$ (2.3)
$\displaystyle m_{3}=r\begin{bmatrix}A&B\\\ D&0\\\
E&0\end{bmatrix}+r\begin{bmatrix}A&C\\\ D&0\\\
E&0\end{bmatrix}-r\begin{bmatrix}A&B&C\\\ D&0&0\\\
E&0&0\end{bmatrix}-r\begin{bmatrix}A\\\ D\\\ E\end{bmatrix},$ (2.4)
$\displaystyle n_{2}=r\begin{bmatrix}A&B&C\\\ D&0&0\\\
E&0&0\end{bmatrix}-r\begin{bmatrix}A&B&C\\\
E&0&0\end{bmatrix},n_{4}=r\begin{bmatrix}A&B&C\\\ D&0&0\\\
E&0&0\end{bmatrix}-r\begin{bmatrix}A&B&C\\\ D&0&0\end{bmatrix},$ (2.5)
$\displaystyle n_{3}=r\begin{bmatrix}A&B&C\\\
D&0&0\end{bmatrix}+r\begin{bmatrix}A&B&C\\\
E&0&0\end{bmatrix}-r\begin{bmatrix}A&B&C\\\ D&0&0\\\
E&0&0\end{bmatrix}-r\begin{bmatrix}A&B&C\end{bmatrix}.$ (2.6)
###### Proof.
The proof is inspired by [15] and [26]. The whole procedure is constructive
and consists of seven steps.
Step 1. We can find two nonsingular matrices $P_{1}$ and $Q_{1}$ such that
$\displaystyle
P_{1}\begin{bmatrix}B&C\end{bmatrix}=\begin{bmatrix}B^{(1)}&C^{(1)}\\\
0&0\end{bmatrix},\begin{bmatrix}D\\\
E\end{bmatrix}Q_{1}=\begin{bmatrix}D^{(1)}&0\\\ E^{(1)}&0\end{bmatrix},$
where $\begin{bmatrix}B^{(1)}&C^{(1)}\end{bmatrix}$ has full row rank, and
$\begin{bmatrix}D^{(1)}\\\ E^{(1)}\end{bmatrix}$ has full column rank. Denote
$\displaystyle P_{1}AQ_{1}=\begin{bmatrix}A_{1}^{(1)}&A_{2}^{(1)}\\\
A_{3}^{(1)}&A_{4}^{(1)}\end{bmatrix}.$
Step 2. There exist two nonsingular matrices $P_{2}$ and $Q_{2}$ such that
$\displaystyle P_{2}A_{4}^{(1)}Q_{2}=\begin{bmatrix}I_{m_{5}}&0\\\
0&0\end{bmatrix},~{}m_{5}=r(A_{4}^{(1)}).$
Then
$\displaystyle diag(I,P_{2})\begin{bmatrix}A_{1}^{(1)}&A_{2}^{(1)}\\\
A_{3}^{(1)}&A_{4}^{(1)}\end{bmatrix}diag(I,Q_{2}):=\begin{bmatrix}A_{1}^{(2)}&A_{2}^{(2)}&A_{3}^{(2)}\\\
A_{4}^{(2)}&I_{m_{5}}&0\\\ A_{5}^{(2)}&0&0\end{bmatrix},$ $\displaystyle
diag(I,P_{2})\begin{bmatrix}B^{(1)}&C^{(1)}\\\
0&0\end{bmatrix}:=\begin{bmatrix}B^{(2)}&C^{(2)}\\\ 0&0\\\
0&0\end{bmatrix},\begin{bmatrix}D^{(1)}&0\\\
E^{(1)}&0\end{bmatrix}diag(I,Q_{2}):=\begin{bmatrix}D^{(2)}&0&0\\\
E^{(2)}&0&0\end{bmatrix},$
where $\begin{bmatrix}B^{(2)}&C^{(2)}\end{bmatrix}$ has full row rank, and
$\begin{bmatrix}D^{(2)}\\\ E^{(2)}\end{bmatrix}$ has full column rank.
Step 3. Set
$\displaystyle P_{3}=\begin{bmatrix}I&-A_{2}^{(2)}&0\\\ 0&I&0\\\
0&0&I\end{bmatrix},Q_{3}=\begin{bmatrix}I&0&0\\\ -A_{4}^{(2)}&I&0\\\
0&0&I\end{bmatrix}.$
Then we have
$\displaystyle P_{3}\begin{bmatrix}A_{1}^{(2)}&A_{2}^{(2)}&A_{3}^{(2)}\\\
A_{4}^{(2)}&I_{m_{5}}&0\\\
A_{5}^{(2)}&0&0\end{bmatrix}Q_{3}:=\begin{bmatrix}A_{1}^{(3)}&0&A_{2}^{(3)}\\\
0&I_{m_{5}}&0\\\ A_{3}^{(3)}&0&0\end{bmatrix},$ $\displaystyle
P_{3}\begin{bmatrix}B^{(2)}&C^{(2)}\\\ 0&0\\\
0&0\end{bmatrix}:=\begin{bmatrix}B^{(3)}&C^{(3)}\\\ 0&0\\\
0&0\end{bmatrix},\begin{bmatrix}D^{(2)}&0&0\\\
E^{(2)}&0&0\end{bmatrix}Q_{3}:=\begin{bmatrix}D^{(3)}&0&0\\\
E^{(3)}&0&0\end{bmatrix}.$
Step 4. We can choose nonsingular matrices $P_{4},Q_{4},P_{5}$ and $Q_{5}$
such that such that
$\displaystyle P_{4}A_{2}^{(3)}Q_{4}=\begin{bmatrix}I_{m_{1}}&0\\\
0&0\end{bmatrix},~{}P_{5}A_{3}^{(3)}Q_{5}=\begin{bmatrix}I_{m_{6}}&0\\\
0&0\end{bmatrix},~{}r(A_{2}^{(3)})=m_{1},~{}r(A_{3}^{(3)})=m_{6}.$
Then
$\displaystyle diag(P_{4},I,P_{5})\begin{bmatrix}A_{1}^{(3)}&0&A_{2}^{(3)}\\\
0&I_{m_{5}}&0\\\
A_{3}^{(3)}&0&0\end{bmatrix}diag(Q_{4},I,Q_{5}):=\begin{bmatrix}A_{1}^{(4)}&A_{2}^{(4)}&0&I_{m_{1}}&0\\\
A_{3}^{(4)}&A_{4}^{(4)}&0&0&0\\\ 0&0&I_{m_{5}}&0&0\\\ I_{m_{6}}&0&0&0&0\\\
0&0&0&0&0\end{bmatrix},$ $\displaystyle
diag(P_{4},I,P_{5})\begin{bmatrix}B^{(3)}&C^{(3)}\\\ 0&0\\\
0&0\end{bmatrix}:=\begin{bmatrix}B_{1}^{(4)}&C_{1}^{(4)}\\\
B_{2}^{(4)}&C_{2}^{(4)}\\\ 0&0\\\ 0&0\\\ 0&0\end{bmatrix},$
$\displaystyle\begin{bmatrix}D^{(3)}&0&0\\\
E^{(3)}&0&0\end{bmatrix}diag(Q_{4},I,Q_{5}):=\begin{bmatrix}D_{1}^{(4)}&D_{2}^{(4)}&0&0&0\\\
E_{1}^{(4)}&E_{2}^{(4)}&0&0&0\end{bmatrix}.$
Especially, we have
$\displaystyle
r\begin{bmatrix}B&C\end{bmatrix}=r\begin{bmatrix}B^{(3)}&C^{(3)}\end{bmatrix}=r\begin{bmatrix}B_{1}^{(4)}&C_{1}^{(4)}\\\
B_{2}^{(4)}&C_{2}^{(4)}\end{bmatrix},r\begin{bmatrix}D&E\end{bmatrix}=r\begin{bmatrix}D^{(3)}\\\
E^{(3)}\end{bmatrix}=r\begin{bmatrix}D_{1}^{(4)}&D_{2}^{(4)}\\\
E_{1}^{(4)}&E_{2}^{(4)}\end{bmatrix},$
i.e., $\begin{bmatrix}B_{2}^{(4)}&C_{2}^{(4)}\end{bmatrix}$ has full row rank,
and $\begin{bmatrix}D_{2}^{(4)}\\\ E_{2}^{(4)}\end{bmatrix}$ has full column
rank.
Step 5. Set
$\displaystyle P_{6}=\begin{bmatrix}I&0&0&-A_{1}^{(4)}&0\\\
0&I&0&-A_{3}^{(4)}&0\\\ 0&0&I&0&0\\\ 0&0&0&I&0\\\
0&0&0&0&I\end{bmatrix},Q_{6}=\begin{bmatrix}I&0&0&0&0\\\ 0&I&0&0&0\\\
0&0&I&0&0\\\ 0&-A_{2}^{(4)}&0&I&0\\\ 0&0&0&0&I\end{bmatrix}.$
Then we have
$\displaystyle P_{6}\begin{bmatrix}A_{1}^{(4)}&A_{2}^{(4)}&0&I_{m_{1}}&0\\\
A_{3}^{(4)}&A_{4}^{(4)}&0&0&0\\\ 0&0&I_{m_{5}}&0&0\\\ I_{m_{6}}&0&0&0&0\\\
0&0&0&0&0\end{bmatrix}Q_{6}:=\begin{bmatrix}0&0&0&I_{m_{1}}&0\\\
0&A_{1}^{(5)}&0&0&0\\\ 0&0&I_{m_{5}}&0&0\\\ I_{m_{6}}&0&0&0&0\\\
0&0&0&0&0\end{bmatrix},$ $\displaystyle
P_{6}\begin{bmatrix}B_{1}^{(4)}&C_{1}^{(4)}\\\ B_{2}^{(4)}&C_{2}^{(4)}\\\
0&0\\\ 0&0\\\ 0&0\end{bmatrix}:=\begin{bmatrix}B_{1}^{(5)}&C_{1}^{(5)}\\\
B_{2}^{(5)}&C_{2}^{(5)}\\\ 0&0\\\ 0&0\\\
0&0\end{bmatrix},\begin{bmatrix}D_{1}^{(4)}&D_{2}^{(4)}&0&0&0\\\
E_{1}^{(4)}&E_{2}^{(4)}&0&0&0\end{bmatrix}Q_{6}:=\begin{bmatrix}D_{1}^{(5)}&D_{2}^{(5)}&0&0&0\\\
E_{1}^{(5)}&E_{2}^{(5)}&0&0&0\end{bmatrix},$
where $\begin{bmatrix}B_{2}^{(5)}&C_{2}^{(5)}\end{bmatrix}$ has full row rank,
and $\begin{bmatrix}D_{2}^{(5)}\\\ E_{2}^{(5)}\end{bmatrix}$ has full column
rank.
Step 6. We can find six nonsingular matrices
$P_{7},Q_{7},W_{B},W_{C},W_{D},W_{E}$ such that
$\displaystyle
P_{7}\begin{bmatrix}B_{2}^{(5)}&C_{2}^{(5)}\end{bmatrix}\begin{bmatrix}W_{B}&0\\\
0&W_{C}\end{bmatrix}=\begin{bmatrix}I_{m_{2}}&0&0&0&0&0\\\
0&I_{m_{3}}&0&0&I_{m_{3}}&0\\\ 0&0&0&0&0&I_{m_{4}}\end{bmatrix},$
$\displaystyle\begin{bmatrix}W_{D}&0\\\
0&W_{E}\end{bmatrix}\begin{bmatrix}D_{2}^{(5)}\\\
E_{2}^{(5)}\end{bmatrix}Q_{7}=\begin{bmatrix}I_{n_{2}}&0&0\\\ 0&I_{n_{3}}&0\\\
0&0&0\\\ 0&0&0\\\ 0&I_{n_{3}}&0\\\ 0&0&I_{n_{4}}\end{bmatrix}.$
Then
$\displaystyle diag(I,P_{7},I,I,I)\begin{bmatrix}0&0&0&I_{m_{1}}&0\\\
0&A_{1}^{(5)}&0&0&0\\\ 0&0&I_{m_{5}}&0&0\\\ I_{m_{6}}&0&0&0&0\\\
0&0&0&0&0\end{bmatrix}diag(I,Q_{7},I,I,I):=$
$\displaystyle\left[\begin{array}[c]{ccccccc}0&0&0&0&0&I_{m_{1}}&0\\\
0&A_{1}^{(6)}&A_{2}^{(6)}&A_{3}^{(6)}&0&0&0\\\
0&A_{4}^{(6)}&A_{5}^{(6)}&A_{6}^{(6)}&0&0&0\\\
0&A_{7}^{(6)}&A_{8}^{(6)}&A_{9}^{(6)}&0&0&0\\\ 0&0&0&0&I_{m_{5}}&0&0\\\
I_{m_{6}}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\end{array}\right],$ $\displaystyle
diag(I,P_{7},I,I,I)\begin{bmatrix}B_{1}^{(5)}&C_{1}^{(5)}\\\
B_{2}^{(5)}&C_{2}^{(5)}\\\ 0&0\\\ 0&0\\\
0&0\end{bmatrix}\begin{bmatrix}W_{B}&0\\\
0&W_{C}\end{bmatrix}:=\begin{bmatrix}B_{1}^{(6)}&B_{2}^{(6)}&B_{3}^{(6)}&C_{1}^{(6)}&C_{2}^{(6)}&C_{3}^{(6)}\\\
I_{m_{2}}&0&0&0&0&0\\\ 0&I_{m_{3}}&0&0&I_{m_{3}}&0\\\ 0&0&0&0&0&I_{m_{4}}\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\end{bmatrix},$
$\displaystyle\begin{bmatrix}W_{D}&0\\\
0&W_{E}\end{bmatrix}\begin{bmatrix}D_{1}^{(5)}&D_{2}^{(5)}&0&0&0\\\
E_{1}^{(5)}&E_{2}^{(5)}&0&0&0\end{bmatrix}diag(I,Q_{7},I,I,I):=\begin{bmatrix}D_{1}^{(6)}&I_{n_{2}}&0&0&0&0&0\\\
D_{2}^{(6)}&0&I_{n_{3}}&0&0&0&0\\\ D_{3}^{(6)}&0&0&0&0&0&0\\\
E_{1}^{(6)}&0&0&0&0&0&0\\\ E_{2}^{(6)}&0&I_{n_{3}}&0&0&0&0\\\
E_{3}^{(6)}&0&0&I_{n_{4}}&0&0&0\end{bmatrix}.$
Step 7. Set
$\displaystyle
P_{8}=\begin{bmatrix}I&-B_{1}^{(6)}&-B_{2}^{(6)}&-C_{3}^{(6)}&0&0&0\\\
0&I&0&0&0&0&0\\\ 0&0&I&0&0&0&0\\\ 0&0&0&I&0&0&0\\\ 0&0&0&0&I&0&0\\\
0&0&0&0&0&I&0\\\
0&0&0&0&0&0&I\end{bmatrix},~{}Q_{8}=\begin{bmatrix}I&0&0&0&0&0&0\\\
-D_{1}^{(6)}&I&0&0&0&0&0\\\ -D_{2}^{(6)}&0&I&0&0&0&0\\\
-E_{3}^{(6)}&0&0&I&0&0&0\\\ 0&0&0&0&I&0&0\\\ 0&0&0&0&0&I&0\\\
0&0&0&0&0&0&I\end{bmatrix}.$
Then we have
$\displaystyle
P_{8}\begin{bmatrix}B_{1}^{(6)}&B_{2}^{(6)}&B_{3}^{(6)}&C_{1}^{(6)}&C_{2}^{(6)}&C_{3}^{(6)}\\\
I_{m_{2}}&0&0&0&0&0\\\ 0&I_{m_{3}}&0&0&I_{m_{3}}&0\\\ 0&0&0&0&0&I_{m_{4}}\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\end{bmatrix}:=\begin{bmatrix}0&0&B_{1}&C_{1}&C_{2}&0\\\
I_{m_{2}}&0&0&0&0&0\\\ 0&I_{m_{3}}&0&0&I_{m_{3}}&0\\\ 0&0&0&0&0&I_{m_{4}}\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\end{bmatrix},$
$\displaystyle\begin{bmatrix}D_{1}^{(6)}&I_{n_{2}}&0&0&0&0&0\\\
D_{2}^{(6)}&0&I_{n_{3}}&0&0&0&0\\\ D_{3}^{(6)}&0&0&0&0&0&0\\\
E_{1}^{(6)}&0&0&0&0&0&0\\\ E_{2}^{(6)}&0&I_{n_{3}}&0&0&0&0\\\
E_{3}^{(6)}&0&0&I_{n_{4}}&0&0&0\end{bmatrix}Q_{8}:=\begin{bmatrix}0&I_{n_{2}}&0&0&0&0&0\\\
0&0&I_{n_{3}}&0&0&0&0\\\ D_{1}&0&0&0&0&0&0\\\ E_{1}&0&0&0&0&0&0\\\
E_{2}&0&I_{n_{3}}&0&0&0&0\\\ 0&0&0&I_{n_{4}}&0&0&0\end{bmatrix},$
$\displaystyle P_{8}\left[\begin{array}[c]{ccccccc}0&0&0&0&0&I_{m_{1}}&0\\\
0&A_{1}^{(6)}&A_{2}^{(6)}&A_{3}^{(6)}&0&0&0\\\
0&A_{4}^{(6)}&A_{5}^{(6)}&A_{6}^{(6)}&0&0&0\\\
0&A_{7}^{(6)}&A_{8}^{(6)}&A_{9}^{(6)}&0&0&0\\\ 0&0&0&0&I_{m_{5}}&0&0\\\
I_{m_{6}}&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{array}\right]Q_{8}:=\left[\begin{array}[c]{ccccccc}\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&0&I_{m_{1}}&0\\\
\phi_{5}&A_{1}^{(6)}&A_{2}^{(6)}&A_{3}^{(6)}&0&0&0\\\
\phi_{6}&A_{4}^{(6)}&A_{5}^{(6)}&A_{6}^{(6)}&0&0&0\\\
\phi_{7}&A_{7}^{(6)}&A_{8}^{(6)}&A_{9}^{(6)}&0&0&0\\\ 0&0&0&0&I_{m_{5}}&0&0\\\
I_{m_{6}}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\end{array}\right],$
where
$\displaystyle\phi_{1}=-\phi_{2}D_{1}^{(6)}-\phi_{3}D_{2}^{(6)}-\phi_{4}E_{3}^{(6)},$
$\displaystyle\phi_{2}=-B_{1}^{(6)}A_{1}^{(6)}-B_{2}^{(6)}A_{4}^{(6)}-C_{3}^{(6)}A_{7}^{(6)},~{}\phi_{3}=-B_{1}^{(6)}A_{2}^{(6)}-B_{2}^{(6)}A_{5}^{(6)}-C_{3}^{(6)}A_{8}^{(6)},$
$\displaystyle\phi_{4}=-B_{1}^{(6)}A_{3}^{(6)}-B_{2}^{(6)}A_{6}^{(6)}-C_{3}^{(6)}A_{9}^{(6)},~{}\phi_{5}=-A_{1}^{(6)}D_{1}^{(6)}-A_{2}^{(6)}D_{2}^{(6)}-A_{3}^{(6)}E_{3}^{(6)},$
$\displaystyle\phi_{6}=-A_{4}^{(6)}D_{1}^{(6)}-A_{5}^{(6)}D_{2}^{(6)}-A_{6}^{(6)}E_{3}^{(6)},~{}\phi_{7}=-A_{7}^{(6)}D_{1}^{(6)}-A_{8}^{(6)}D_{2}^{(6)}-A_{9}^{(6)}E_{3}^{(6)}.$
Using $I_{m_{1}},I_{m_{6}}$ as the pivots to eliminate the blocks
$\phi_{1},\phi_{2},\phi_{3},\phi_{4},\phi_{5},\phi_{6},\phi_{7}$,
respectively, we obtain
$\displaystyle\begin{bmatrix}I&0&0&0&0&-\phi_{1}&0\\\ 0&I&0&0&0&-\phi_{5}&0\\\
0&0&I&0&0&-\phi_{6}&0\\\ 0&0&0&I&0&-\phi_{7}&0\\\ 0&0&0&0&I&0&0\\\
0&0&0&0&0&I&0\\\
0&0&0&0&0&0&I\end{bmatrix}\left[\begin{array}[c]{ccccccc}\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&0&I_{m_{1}}&0\\\
\phi_{5}&A_{1}^{(6)}&A_{2}^{(6)}&A_{3}^{(6)}&0&0&0\\\
\phi_{6}&A_{4}^{(6)}&A_{5}^{(6)}&A_{6}^{(6)}&0&0&0\\\
\phi_{7}&A_{7}^{(6)}&A_{8}^{(6)}&A_{9}^{(6)}&0&0&0\\\ 0&0&0&0&I_{m_{5}}&0&0\\\
I_{m_{6}}&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{array}\right]\begin{bmatrix}I&0&0&0&0&0&0\\\
0&I&0&0&0&0&0\\\ 0&0&I&0&0&0&0\\\ 0&0&0&I&0&0&0\\\ 0&0&0&0&I&0&0\\\
0&-\phi_{2}&-\phi_{3}&-\phi_{4}&0&I&0\\\ 0&0&0&0&0&0&I\end{bmatrix}$
$\displaystyle:=\left[\begin{array}[c]{ccccccc}0&0&0&0&0&I_{m_{1}}&0\\\
0&A_{1}&A_{2}&A_{3}&0&0&0\\\ 0&A_{4}&A_{5}&A_{6}&0&0&0\\\
0&A_{7}&A_{8}&A_{9}&0&0&0\\\ 0&0&0&0&I_{m_{5}}&0&0\\\ I_{m_{6}}&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{array}\right].$
After Step 7, we have established the decomposition.
∎
## 3\. Maximal and minimal ranks of (1.1) and (1.3)
As applications of Theorem 2.1, we in this section consider the maximal and
minimal ranks of (1.1) and (1.3) over $\mathbb{H}$. We also show how to choose
the variable real quaternion matrices such that the real quaternion matrix
expressions achieve their maximal and minimal ranks.
###### Theorem 3.1.
Let $A,B,C,D,E$ be given quaternion matrices, and $p(X,Y)$ be as given in
(1.1). Then
$\displaystyle\mathop{\max}\limits_{X,Y}r\left[{p\left({X,Y}\right)}\right]=\min\left\\{r\left[{\begin{array}[]{*{20}{c}}A\\\
D\\\
E\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&B&C\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&B\\\
E&0\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&C\\\
D&0\end{array}}\right]\right\\},$
$\displaystyle\mathop{\min}\limits_{X,Y}r\left[{p\left({X,Y}\right)}\right]$
$\displaystyle=r\left[{\begin{array}[]{*{20}{c}}A\\\ D\\\
E\end{array}}\right]+r\left[{\begin{array}[]{*{20}{c}}A&B&C\end{array}}\right]+\max\Bigg{\\{}r\left[{\begin{array}[]{*{20}{c}}A&B\\\
E&0\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&B&C\\\
E&0&0\end{array}}\right]$
$\displaystyle-r\left[{\begin{array}[]{*{20}{c}}A&B\\\ D&0\\\
E&0\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&C\\\
D&0\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&B&C\\\
D&0&0\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&C\\\ D&0\\\
E&0\end{array}}\right]\Bigg{\\}}.$
###### Proof.
From Theorem 2.1, we can rewrite the expression $p(X,Y)$ in (1.1) in the
following canonical form
$\displaystyle p(X,Y)=P(S_{A}-S_{B}T_{1}XV_{1}S_{D}-S_{C}T_{2}YV_{2}S_{E})Q.$
Because $P,Q$ are nonsingular, the rank of $p(X,Y)$ can be written as
$\displaystyle
r[p(X,Y)]=r(S_{A}-S_{B}T_{1}XV_{1}S_{D}-S_{C}T_{2}YV_{2}S_{E}).$
Put $\widehat{X}=T_{1}XV_{1},\widehat{Y}=T_{2}YV_{2}$. Partition the matrices
$\widehat{X}$ and $\widehat{Y}$
$\displaystyle\widehat{X}=\begin{bmatrix}X_{1}&X_{2}&X_{3}\\\
X_{4}&X_{5}&X_{6}\\\
X_{7}&X_{8}&X_{9}\end{bmatrix},\widehat{Y}=\begin{bmatrix}Y_{1}&Y_{2}&Y_{3}\\\
Y_{4}&Y_{5}&Y_{6}\\\ Y_{7}&Y_{8}&Y_{9}\end{bmatrix}.$
Hence,
$\displaystyle
S_{B}\widehat{X}S_{D}=\left[\begin{array}[c]{ccccccc}*&*&*&0&0&0&0\\\
&X_{1}&X_{2}&0&0&0&0\\\ &X_{4}&X_{5}&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0\end{array}\right],S_{C}\widehat{Y}S_{E}=\left[\begin{array}[c]{ccccccc}*&0&*&*&0&0&0\\\
0&0&0&0&0&0&0\\\ &0&Y_{5}&Y_{6}&0&0&0\\\ &0&Y_{8}&Y_{9}&0&0&0\\\
0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\end{array}\right].$
Through the new notation, the matrix expression $A-BXD-CYE$ can be rewritten
as
$\displaystyle
S_{A}-S_{B}\widehat{X}S_{D}-S_{C}\widehat{Y}S_{E}=\left[\begin{array}[c]{ccccccc}*&*&*&*&0&I_{m_{1}}&0\\\
&A_{1}-X_{1}&A_{2}-X_{2}&A_{3}&0&0&0\\\
&A_{4}-X_{4}&A_{5}-X_{5}-Y_{5}&A_{6}-Y_{6}&0&0&0\\\
&A_{7}&A_{8}-Y_{8}&A_{9}-Y_{9}&0&0&0\\\ 0&0&0&0&I_{m_{5}}&0&0\\\
I_{m_{6}}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0\end{array}\right].$
Obviously,
$\displaystyle
r(S_{A}-S_{B}\widehat{X}S_{D}-S_{C}\widehat{Y}S_{E})=m_{1}+m_{5}+m_{6}+r(\Omega),$
where
$\displaystyle\Omega=\left[\begin{array}[c]{ccc}A_{1}-X_{1}&A_{2}-X_{2}&A_{3}\\\
A_{4}-X_{4}&A_{5}-X_{5}-Y_{5}&A_{6}-Y_{6}\\\
A_{7}&A_{8}-Y_{8}&A_{9}-Y_{9}\end{array}\right]:=\left[\begin{array}[c]{ccc}Z_{1}&Z_{2}&A_{3}\\\
Z_{3}&Z_{4}&Z_{5}\\\ A_{7}&Z_{6}&Z_{7}\end{array}\right],$ (3.7)
where $Z_{1},\cdots,Z_{7}$ are arbitrary real quaternion matrices. It is
easily seen from (3.7) that
$\displaystyle\mathop{\max}\left\\{r(A_{3}),r(A_{7})\right\\}\leq
r(\Omega)\leq$
$\displaystyle\mathop{\min}\left\\{m_{2}+m_{3}+m_{4},n_{2}+n_{3}+n_{4},r(A_{3})+n_{2}+n_{3}+m_{3}+m_{4},r(A_{7})+n_{3}+n_{4}+m_{2}+m_{3}\right\\}.$
(3.8)
Now, we choose $Z_{1},\cdots,Z_{7}$ such that $\Omega$ reach the upper and
lower bounds in (3). There exist nonsingular matrices
$P_{1},Q_{1},P_{2},Q_{2}$ such that
$\displaystyle P_{1}A_{3}Q_{1}=\begin{bmatrix}I_{a}&0\\\
0&0\end{bmatrix},~{}P_{2}A_{7}Q_{2}=\begin{bmatrix}I_{b}&0\\\
0&0\end{bmatrix}.$
Case 1. Assume that $b=r(A_{7})\leq r(A_{3})=a.$ Set
$\displaystyle Z_{2}=0,Z_{3}=0,Z_{4}=0,Z_{5}=0,Z_{6}=0.$
Then
$\displaystyle\Omega=\left[\begin{array}[c]{cc}Z_{1}&A_{3}\\\
A_{7}&Z_{7}\end{array}\right].$
Denote
$\displaystyle P_{1}Z_{1}Q_{2}=\begin{bmatrix}W_{1}&W_{2}\\\ W_{3}&W_{4}\\\
W_{5}&W_{6}\end{bmatrix},~{}P_{2}Z_{7}Q_{1}=\begin{bmatrix}U_{1}&U_{2}&U_{3}\\\
U_{4}&U_{5}&U_{6}\end{bmatrix},$
where $W_{1},\cdots,W_{4},U_{1},\cdots,U_{4}$ are arbitrary real quaternion
matrices. Then we have
$\displaystyle r(\Omega)=r\begin{bmatrix}W_{1}&W_{2}&I_{b}&0&0\\\
W_{3}&W_{4}&0&I_{a-b}&0\\\ W_{5}&W_{6}&0&0&0\\\ I_{b}&0&U_{1}&U_{2}&U_{3}\\\
0&0&U_{4}&U_{5}&U_{6}\end{bmatrix}.$
Set
$\displaystyle W_{1}=I_{b},W_{2}=0,W_{3}=0,W_{4}=0,W_{5}=0,W_{6}=0,$
$\displaystyle U_{1}=I_{b},U_{2}=0,U_{3}=0,U_{4}=0,U_{5}=0,U_{6}=0.$
Then $r(\Omega)=a=r(A_{3})$. The case $r(A_{7})\geq r(A_{3})$ can be shown
similarly. Hence, $\mathop{\max}\left\\{r(A_{3}),r(A_{7})\right\\}\leq
r(\Omega).$
Case 2. Assume that
$\displaystyle
m_{2}+m_{3}+m_{4}\leq\mathop{\min}\left\\{n_{2}+n_{3}+n_{4},r(A_{3})+n_{2}+n_{3}+m_{3}+m_{4},r(A_{7})+n_{3}+n_{4}+m_{2}+m_{3}\right\\}.$
Then
$\displaystyle\Omega=\left[\begin{array}[c]{ccccc}W_{1}&W_{2}&W_{3}&I_{a}&0\\\
W_{4}&W_{5}&W_{6}&0&0\\\ W_{7}&W_{8}&W_{9}&W_{10}&W_{11}\\\
I_{b}&0&W_{12}&W_{13}&W_{14}\\\ 0&0&W_{15}&W_{16}&W_{17}\end{array}\right].$
where $W_{1},\cdots,W_{17}$ are arbitrary real quaternion matrices. Set
$\displaystyle\begin{bmatrix}W_{4}&W_{5}&W_{6}\end{bmatrix}=\begin{bmatrix}I_{m_{4}-b}&0\end{bmatrix},\begin{bmatrix}W_{11}\\\
W_{14}\\\
W_{17}\end{bmatrix}=\begin{bmatrix}I_{m_{3}+b+m_{2}-a}&0\end{bmatrix},$
$\displaystyle W_{i}=0,i=1,2,3,7,8,9,10,12,13,15,16.$
Then we have $r(\Omega)=m_{2}+m_{3}+m_{4}.$ The case
$\displaystyle
n_{2}+n_{3}+n_{4}\leq\mathop{\min}\left\\{m_{2}+m_{3}+m_{4},r(A_{3})+n_{2}+n_{3}+m_{3}+m_{4},r(A_{7})+n_{3}+n_{4}+m_{2}+m_{3}\right\\}.$
can be shown similarly.
Case 3. Assume that
$\displaystyle
r(A_{3})+n_{2}+n_{3}+m_{3}+m_{4}\leq\mathop{\min}\left\\{m_{2}+m_{3}+m_{4},n_{2}+n_{3}+n_{4},r(A_{7})+n_{3}+n_{4}+m_{2}+m_{3}\right\\}.$
Then we have $m_{2}-a\geq n_{2}+n_{3}$ and $n_{4}-a\geq m_{3}+m_{4}$. Set
$\displaystyle\begin{bmatrix}W_{4}&W_{5}&W_{6}\end{bmatrix}=\begin{bmatrix}I_{n_{2}+n_{3}}\\\
0\end{bmatrix},\begin{bmatrix}W_{11}\\\ W_{14}\\\
W_{17}\end{bmatrix}=\begin{bmatrix}I_{m_{3}+m_{4}}&0\end{bmatrix},W_{i}=0,i=1,2,3,7,8,9,10,12,13,15,16.$
Then we have
$\displaystyle
r(\Omega)=a+n_{2}+n_{3}+m_{3}+m_{4}=r(A_{3})+n_{2}+n_{3}+m_{3}+m_{4}.$
The case
$\displaystyle
r(A_{7})+n_{3}+n_{4}+m_{2}+m_{3}\leq\mathop{\min}\left\\{m_{2}+m_{3}+m_{4},n_{2}+n_{3}+n_{4},r(A_{3})+n_{2}+n_{3}+m_{3}+m_{4}\right\\}.$
can be shown similarly.
Hence,
$\displaystyle
m_{1}+m_{5}+m_{6}+\mathop{\max}\left\\{r(A_{3}),r(A_{7})\right\\}\leq r(A-BXD-
CYE)\leq m_{1}+m_{5}+m_{6}+$
$\displaystyle\mathop{\min}\left\\{m_{2}+m_{3}+m_{4},n_{2}+n_{3}+n_{4},r(A_{3})+n_{2}+n_{3}+m_{3}+m_{4},r(A_{7})+n_{3}+n_{4}+m_{2}+m_{3}\right\\}.$
It follows from Theorem 2.1 that
$\displaystyle
m_{1}+m_{2}+m_{3}+r(A_{7})+m_{5}+m_{6}+n_{3}+n_{4}=r\begin{bmatrix}A&B\\\
E&0\end{bmatrix},$ (3.9) $\displaystyle
m_{1}+n_{2}+n_{3}+m_{3}+m_{4}+r(A_{3})+m_{5}+m_{6}=r\begin{bmatrix}A&C\\\
D&0\end{bmatrix}.$ (3.10)
Combining (2.2)-(2.6), (3.9) and (3.10), we can obtain the results.
∎
###### Corollary 3.2.
Let $A,B,C,D,E$ be given, and $p(X,Y)$ be as given in (1.1). Assume that
$\displaystyle\mathcal{R}(B)\subseteq\mathcal{R}(C),~{}\mathcal{R}(E^{*})\subseteq\mathcal{R}(D^{*}).$
Then
$\displaystyle\mathop{\max}\limits_{X,Y}r\left[{p\left({X,Y}\right)}\right]=\min\left\\{r\left[{\begin{array}[]{*{20}{c}}A\\\
D\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&C\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&B\\\
E&0\end{array}}\right]\right\\},$
$\displaystyle\mathop{\min}\limits_{X,Y}r\left[{p\left({X,Y}\right)}\right]$
$\displaystyle=r\left[{\begin{array}[]{*{20}{c}}A\\\
D\end{array}}\right]+r\left[{\begin{array}[]{*{20}{c}}A&C\end{array}}\right]+r\left[{\begin{array}[]{*{20}{c}}A&B\\\
E&0\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&C\\\
E&0\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&B\\\
D&0\end{array}}\right].$
By the method mentioned in the proof of Theorem 3.1, We can choose variable
real quaternion matrices $X$ and $Y$ such that the real quaternion matrix
expression attains its maximal and minimal ranks.
###### Corollary 3.3.
Let $A\in\mathbb{H}^{m\times n},B\in\mathbb{H}^{m\times
k},C\in\mathbb{H}^{l\times n}$ be given, and
$\displaystyle f_{1}(X,Y)=A-BX-YC.$ (3.11)
Then
$\displaystyle\mathop{\max}\limits_{X,Y}r\left[{f_{1}\left({X,Y}\right)}\right]=\min\left\\{m,n,r\left[{\begin{array}[]{*{20}{c}}A&B\\\
C&0\end{array}}\right]\right\\},\mathop{\min}\limits_{X,Y}r\left[{f_{1}\left({X,Y}\right)}\right]=r\left[{\begin{array}[]{*{20}{c}}A&B\\\
C&0\end{array}}\right]-r(B)-r(C).$
By the method mentioned in the proof of Theorem 3.1, We can choose variable
real quaternion matrices $X$ and $Y$ such that the real quaternion matrix
expression (3.11) attains its maximal and minimal ranks.
###### Theorem 3.4.
Let $A\in\mathbb{H}^{m\times n},B_{1}\in\mathbb{H}^{m\times
k},C_{2}\in\mathbb{H}^{l\times n},B_{3}\in\mathbb{H}^{m\times
k_{3}},C_{3}\in\mathbb{H}^{l_{3}\times n},B_{4}\in\mathbb{H}^{m\times
k_{4}},C_{4}\in\mathbb{H}^{l_{4}\times n}$ be given, and
$f_{2}(X_{1},X_{2},X_{3},X_{4})$ be as given in (1.3). Then
$\displaystyle\mathop{\max}\limits_{\left\\{{{X_{i}}}\right\\}}r\left[{f_{2}\left({{X_{1}},{X_{2}},{X_{3}},{X_{4}}}\right)}\right]=$
$\displaystyle\min\Bigg{\\{}m,n,r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}\\\
{{C_{2}}}&0\\\ {{C_{3}}}&0\\\ {{C_{4}}}&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ \end{array}}\right],$ $\displaystyle
r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\ {{C_{2}}}&0&0\\\
{{C_{4}}}&0&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ \end{array}}\right]\Bigg{\\}},$
$\displaystyle\mathop{\min}\limits_{\left\\{{{X_{i}}}\right\\}}r\left[{f_{2}\left({{X_{1}},{X_{2}},{X_{3}},{X_{4}}}\right)}\right]$
$\displaystyle=r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}\\\ {{C_{2}}}&0\\\
{{C_{3}}}&0\\\ {{C_{4}}}&0\\\
\end{array}}\right]+r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ \end{array}}\right]-r({B_{1}})-r({C_{2}})$
$\displaystyle+\max\left\\{r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ {{C_{4}}}&0&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right],\right.$
$\displaystyle\left.r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ {{C_{3}}}&0&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right]\right\\}.$
###### Proof.
Our proof is just similar to the proof in [23]. Note that
$\displaystyle\begin{bmatrix}A-B_{3}X_{3}C_{3}-B_{4}X_{4}C_{4}&B_{1}\\\
C_{2}&0\end{bmatrix}=\begin{bmatrix}A&B_{1}\\\
C_{2}&0\end{bmatrix}-\begin{bmatrix}B_{3}\\\
0\end{bmatrix}X_{3}\begin{bmatrix}C_{3}&0\end{bmatrix}-\begin{bmatrix}B_{4}\\\
0\end{bmatrix}X_{4}\begin{bmatrix}C_{4}&0\end{bmatrix}.$ (3.12)
It follows from Theorem 3.1 that there exist $\widehat{X_{3}}$,
$\widehat{X_{4}}$ $\widetilde{X_{3}}$ and $\widetilde{X_{4}}$ such that the
real quaternion matrix (3.12) attains its maximal and minimal ranks, i.e.
$\displaystyle
r\begin{bmatrix}A-B_{3}\widehat{X_{3}}C_{3}-B_{4}\widehat{X_{4}}C_{4}&B_{1}\\\
C_{2}&0\end{bmatrix}=\mathop{\max}\limits_{\left\\{{{X_{3}},X_{4}}\right\\}}r\begin{bmatrix}A-B_{3}X_{3}C_{3}-B_{4}X_{4}C_{4}&B_{1}\\\
C_{2}&0\end{bmatrix}$
$\displaystyle=\min\Bigg{\\{}r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}\\\
{{C_{2}}}&0\\\ {{C_{3}}}&0\\\ {{C_{4}}}&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ \end{array}}\right]\Bigg{\\}},$
$\displaystyle
r\begin{bmatrix}A-B_{3}\widetilde{X_{3}}C_{3}-B_{4}\widetilde{X_{4}}C_{4}&B_{1}\\\
C_{2}&0\end{bmatrix}=\mathop{\min}\limits_{\left\\{{{X_{3}},X_{4}}\right\\}}r\begin{bmatrix}A-B_{3}X_{3}C_{3}-B_{4}X_{4}C_{4}&B_{1}\\\
C_{2}&0\end{bmatrix}$
$\displaystyle=r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}\\\ {{C_{2}}}&0\\\
{{C_{3}}}&0\\\ {{C_{4}}}&0\\\
\end{array}}\right]+r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ \end{array}}\right]$
$\displaystyle+\max\left\\{r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ {{C_{4}}}&0&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right],\right.$
$\displaystyle\left.r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ {{C_{3}}}&0&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right]\right\\}.$
By Corollary 3.3, we can find $\widehat{X_{1}}$, $\widehat{X_{2}}$,
$\widetilde{X_{1}}$ and $\widetilde{X_{2}}$ such that
$\displaystyle
r\left[{f_{1}\left({\widehat{X_{1}},\widehat{X_{2}},\widehat{X_{3}},\widehat{X_{4}}}\right)}\right]=\min\left\\{m,n,r\left[{\begin{array}[]{*{20}{c}}A-B_{3}\widehat{X_{3}}C_{3}-B_{4}\widehat{X_{4}}C_{4}&B_{1}\\\
C_{2}&0\end{array}}\right]\right\\}$
$\displaystyle=\min\left\\{m,n,r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}\\\
{{C_{2}}}&0\\\ {{C_{3}}}&0\\\ {{C_{4}}}&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ \end{array}}\right]\right\\},$
$\displaystyle
r\left[{f_{1}\left({\widetilde{X_{1}},\widetilde{X_{2}},\widetilde{X_{3}},\widetilde{X_{4}}}\right)}\right]=r\left[{\begin{array}[]{*{20}{c}}A-B_{3}\widetilde{X_{3}}C_{3}-B_{4}\widetilde{X_{4}}C_{4}&B_{1}\\\
C_{2}&0\end{array}}\right]-r(B_{1})-r(C_{2})$
$\displaystyle=r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}\\\ {{C_{2}}}&0\\\
{{C_{3}}}&0\\\ {{C_{4}}}&0\\\
\end{array}}\right]+r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ \end{array}}\right]-r({B_{1}})-r({C_{2}})$
$\displaystyle+\max\left\\{r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ {{C_{4}}}&0&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right],\right.$
$\displaystyle\left.r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ {{C_{3}}}&0&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right]\right\\}.$
∎
###### Theorem 3.5.
Let
$\displaystyle
f_{3}(X_{1},X_{2},X_{3},X_{4})=A-B_{1}X_{1}C_{1}-B_{2}X_{2}C_{2}-B_{3}X_{3}C_{3}-B_{4}X_{4}C_{4}$
(3.13)
be a linear matrix expression with four two-sided terms over $\mathbb{H}$,
where
$\displaystyle\mathcal{R}(B_{i})\subseteq\mathcal{R}(B_{2}),~{}\mathcal{R}(C_{j}^{*})\subseteq\mathcal{R}(C_{1}^{*}),~{}i=1,3,4,~{}j=2,3,4.$
Then,
$\displaystyle\mathop{\max}\limits_{\left\\{{{X_{i}}}\right\\}}r\left[{f_{3}\left({{X_{1}},{X_{2}},{X_{3}},{X_{4}}}\right)}\right]=$
$\displaystyle\min\Bigg{\\{}r\begin{bmatrix}A&B_{2}\end{bmatrix},r\begin{bmatrix}A\\\
C_{1}\end{bmatrix},r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}\\\
{{C_{2}}}&0\\\ {{C_{3}}}&0\\\ {{C_{4}}}&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ \end{array}}\right],$ $\displaystyle
r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\ {{C_{2}}}&0&0\\\
{{C_{4}}}&0&0\\\
\end{array}}\right],r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ \end{array}}\right]\Bigg{\\}},$
$\displaystyle\mathop{\min}\limits_{\left\\{{{X_{i}}}\right\\}}r\left[{f_{3}\left({{X_{1}},{X_{2}},{X_{3}},{X_{4}}}\right)}\right]$
$\displaystyle=r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}\\\ {{C_{2}}}&0\\\
{{C_{3}}}&0\\\ {{C_{4}}}&0\\\
\end{array}}\right]+r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ \end{array}}\right]+r\begin{bmatrix}A\\\
C_{1}\end{bmatrix}+r\begin{bmatrix}A&B_{2}\end{bmatrix}-r\begin{bmatrix}A&B_{1}\\\
C_{1}&0\end{bmatrix}-r\begin{bmatrix}A&B_{2}\\\ C_{2}&0\end{bmatrix}$
$\displaystyle+\max\left\\{r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ {{C_{4}}}&0&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right],\right.$
$\displaystyle\left.r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{3}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0&0\\\ {{C_{3}}}&0&0&0\\\
\end{array}}\right]-r\left[{\begin{array}[]{*{20}{c}}A&{{B_{1}}}&{{B_{4}}}\\\
{{C_{2}}}&0&0\\\ {{C_{3}}}&0&0\\\ {{C_{4}}}&0&0\\\
\end{array}}\right]\right\\}.$
###### Proof.
Applying the method mentioned in Theorem 3.4 and Corollary 3.2, we can get the
results. ∎
###### Remark 3.1.
Tian [22] derived the maximal and minimal ranks of (1.1), (1.3), (3.11) and
(3.13) over $\mathbb{C}$.
## 4\. Minimal rank of the general solution to real quaternion matrix
equation (1.4)
We in this section consider a solvability condition and the general solution
to real quaternion matrix equation (1.4). Moreover, we study the minimal rank
of the general solution to real quaternion matrix equation (1.4).
###### Theorem 4.1.
Let $A,B,C,D,E$ be given. Then the real matrix equation (1.4) is consistent if
and only if
$\displaystyle
r\begin{bmatrix}A&C&B\end{bmatrix}=r\begin{bmatrix}C&B\end{bmatrix},r\begin{bmatrix}A\\\
D\\\ E\end{bmatrix}=r\begin{bmatrix}D\\\ E\end{bmatrix},r\begin{bmatrix}A&B\\\
E&0\end{bmatrix}=r\begin{bmatrix}0&B\\\
E&0\end{bmatrix},r\begin{bmatrix}A&C\\\
D&0\end{bmatrix}=r\begin{bmatrix}0&C\\\ D&0\end{bmatrix}.$ (4.1)
In this case, the general solution to (1.4) can be expressed as
$\displaystyle X=T_{1}^{-1}\begin{bmatrix}A_{1}&A_{2}&X_{3}\\\
A_{4}&X_{5}&X_{6}\\\
X_{7}&X_{8}&X_{9}\end{bmatrix}V_{1}^{-1},Y=T_{2}^{-1}\begin{bmatrix}Y_{1}&Y_{2}&Y_{3}\\\
Y_{4}&A_{5}-X_{5}&A_{6}\\\ Y_{7}&A_{8}&A_{9}\end{bmatrix}V_{2}^{-1},$
where $A_{1},A_{2},A_{4},A_{5},A_{6},A_{8},A_{9},T_{1},V_{1},T_{2},V_{2}$ are
defined as in Theorem 2.1, and $X_{3},X_{5},X_{6},X_{7},$
$X_{8},X_{9},Y_{1},Y_{2},Y_{3},Y_{4},Y_{7}$ are arbitrary real quaternion
matrices.
###### Remark 4.1.
J.K. Baksalary and R. Kala [1], A.B. Özgüler [17] derived the solvability
condition (4.1).
###### Theorem 4.2.
Let $A\in\mathbb{H}^{m\times n},B\in\mathbb{H}^{m\times
p_{1}},C\in\mathbb{H}^{m\times p_{2}},D\in\mathbb{H}^{q_{1}\times n}$ and
$E\in\mathbb{H}^{q_{2}\times n}$ be given. Suppose that real quaternion matrix
equation (1.4) is consistent. Then
$\displaystyle\mathop{\min}\limits_{BXD+CYE=A}r\left({X}\right)=r\begin{bmatrix}A&C\end{bmatrix}+r\begin{bmatrix}A\\\
E\end{bmatrix}-r\begin{bmatrix}A&C\\\ E&0\end{bmatrix},$
$\displaystyle\mathop{\min}\limits_{BXD+CYE=A}r\left({Y}\right)=r\begin{bmatrix}A&B\end{bmatrix}+r\begin{bmatrix}A\\\
D\end{bmatrix}-r\begin{bmatrix}A&B\\\ D&0\end{bmatrix}.$
###### Proof.
It follows from Theorem 4.1 that the solution $X$ in (1.4) can be expressed as
$\displaystyle X=T_{1}^{-1}\begin{bmatrix}A_{1}&A_{2}&X_{3}\\\
A_{4}&X_{5}&X_{6}\\\ X_{7}&X_{8}&X_{9}\end{bmatrix}V_{1}^{-1},$
where $A_{1},A_{2},A_{4},T_{1},V_{1}$ are defined as in Theorem 2.1, and
$X_{3},X_{5},X_{6},X_{7},X_{8},X_{9}$ are arbitrary real quaternion matrices.
Since real quaternion matrix equation (1.4) is consistent, i.e., the equations
in (4.1) hold, we get $A_{3}=0,A_{7}=0,B_{1}=0,D_{1}=0,$ where
$A_{3},A_{7},B_{1}$ and $D_{1}$ are defined as in Theorem 2.1. Note that
$\displaystyle\begin{bmatrix}A_{1}&A_{2}&X_{3}\\\ A_{4}&X_{5}&X_{6}\\\
X_{7}&X_{8}&X_{9}\end{bmatrix}=\begin{bmatrix}A_{1}&A_{2}&X_{3}\\\
A_{4}&0&0\\\ X_{7}&0&0\end{bmatrix}+\begin{bmatrix}0&0\\\ I&0\\\
0&I\end{bmatrix}\begin{bmatrix}X_{5}&X_{6}\\\
X_{8}&X_{9}\end{bmatrix}\begin{bmatrix}0&I&0\\\ 0&0&I\end{bmatrix}.$ (4.2)
First, we consider the minimal rank of $X$. Applying Theorem 3.1 to (4.2)
yields
$\displaystyle\mathop{\min}\limits_{BXD+CYE=A}r\left({X}\right)=$
$\displaystyle\mathop{\min}\limits_{X_{3},X_{5},X_{6},X_{7},X_{8},X_{9}}r\begin{bmatrix}A_{1}&A_{2}&X_{3}\\\
A_{4}&X_{5}&X_{6}\\\ X_{7}&X_{8}&X_{9}\end{bmatrix}$ $\displaystyle=$
$\displaystyle\mathop{\min}\limits_{X_{3},X_{7}}\left(r\begin{bmatrix}A_{1}&A_{2}&X_{3}\end{bmatrix}+r\begin{bmatrix}A_{1}\\\
A_{4}\\\ X_{7}\end{bmatrix}-r(A_{1})\right)$ $\displaystyle=$ $\displaystyle
r\begin{bmatrix}A_{1}&A_{2}\end{bmatrix}+r\begin{bmatrix}A_{1}\\\
A_{4}\end{bmatrix}-r(A_{1}).(SetX_{3}=0,X_{7}=0)$
Applying (2.2)-(2.6) to $r\begin{bmatrix}A&C\end{bmatrix}+r\begin{bmatrix}A\\\
E\end{bmatrix}-r\begin{bmatrix}A&C\\\ E&0\end{bmatrix}$ yields
$\displaystyle r\begin{bmatrix}A&C\end{bmatrix}+r\begin{bmatrix}A\\\
E\end{bmatrix}-r\begin{bmatrix}A&C\\\
E&0\end{bmatrix}=r\begin{bmatrix}A_{1}&A_{2}\end{bmatrix}+r\begin{bmatrix}A_{1}\\\
A_{4}\end{bmatrix}-r(A_{1})=\mathop{\min}\limits_{BXD+CYE=A}r\left({X}\right).$
∎
###### Remark 4.2.
Tian [21] gave the minimal rank of the general solution to the matrix equation
(1.4). Our method is different from this in [21].
## 5\. A simultaneous decomposition of seven real quaternion matrices
In this section, we give a simultaneous decomposition of the real quaternion
matrix array
$\displaystyle\begin{matrix}A&B&C&D\\\ E&&&\\\ F&&&\\\ G&&&\end{matrix}.$
(5.1)
The proof will be given in another paper.
###### Theorem 5.1.
Let $A\in\mathbb{H}^{m\times n},B\in\mathbb{H}^{m\times
p_{1}},C\in\mathbb{H}^{m\times p_{2}},D\in\mathbb{H}^{m\times
p_{3}},E\in\mathbb{H}^{q_{1}\times n},F\in\mathbb{H}^{q_{2}\times n}$ and
$G\in\mathbb{H}^{q_{3}\times n}$be given. Then there exist nonsingular
matrices $P\in\mathbb{H}^{m\times m}$, $Q\in\mathbb{H}^{n\times
n},T_{1}\in\mathbb{H}^{p_{1}\times p_{1}},T_{2}\in\mathbb{H}^{p_{2}\times
p_{2}},T_{3}\in\mathbb{H}^{p_{3}\times p_{3}},V_{1}\in\mathbb{H}^{q_{1}\times
q_{1}},V_{2}\in\mathbb{H}^{q_{2}\times q_{2}},V_{3}\in\mathbb{H}^{q_{3}\times
q_{3}}$ such that
$\displaystyle
A=PS_{A}Q,B=PS_{B}T_{1},C=PS_{C}T_{2},D=PS_{D}T_{3},E=V_{1}S_{E}Q,F=V_{2}S_{F}Q,G=V_{3}S_{G}Q,$
(5.2)
where
$S_{A}=\left[\begin{array}[c]{ccccccccccccc}0&0&0&0&0&0&0&0&0&0&0&I_{p_{1}}&0\\\
0&A_{11}&A_{12}&A_{13}&A_{14}&A_{15}&A_{16}&A_{17}&A_{18}&A_{19}&0&0&0\\\
0&A_{21}&A_{22}&A_{23}&A_{24}&A_{25}&A_{26}&A_{27}&A_{28}&A_{29}&0&0&0\\\
0&A_{31}&A_{32}&A_{33}&A_{34}&A_{35}&A_{36}&A_{37}&A_{38}&A_{39}&0&0&0\\\
0&A_{41}&A_{42}&A_{43}&A_{44}&A_{45}&A_{46}&A_{47}&A_{48}&A_{49}&0&0&0\\\
0&A_{51}&A_{52}&A_{53}&A_{54}&A_{55}&A_{56}&A_{57}&A_{58}&A_{59}&0&0&0\\\
0&A_{61}&A_{62}&A_{63}&A_{64}&A_{65}&A_{66}&A_{67}&A_{68}&A_{69}&0&0&0\\\
0&A_{71}&A_{72}&A_{73}&A_{74}&A_{75}&A_{76}&A_{77}&A_{78}&A_{79}&0&0&0\\\
0&A_{81}&A_{82}&A_{83}&A_{84}&A_{85}&A_{86}&A_{87}&A_{88}&A_{89}&0&0&0\\\
0&A_{91}&A_{92}&A_{93}&A_{94}&A_{95}&A_{96}&A_{97}&A_{98}&A_{99}&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&I_{p_{2}}&0&0\\\ I_{p_{3}}&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0\end{array}\right],$
$S_{B}=\left[\begin{array}[c]{cccccc}0&0&0&0&0&B_{1}\\\ I_{m_{1}}&0&0&0&0&0\\\
0&I_{m_{2}}&0&0&0&0\\\ 0&0&I_{m_{3}}&0&0&0\\\ 0&0&0&I_{m_{4}}&0&0\\\
0&0&0&0&I_{m_{5}}&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\end{array}\right],S_{C}=\left[\begin{array}[c]{cccccc}0&0&0&C_{1}&C_{2}&C_{3}\\\
0&0&0&I_{m_{1}}&0&0\\\ 0&0&0&0&I_{m_{2}}&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ I_{m_{4}}&0&0&0&0&0\\\ 0&I_{m_{6}}&0&0&0&0\\\
0&0&I_{m_{7}}&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\end{array}\right],$
$S_{D}=\left[\begin{array}[c]{cccccc}0&D_{1}&D_{2}&D_{3}&D_{4}&D_{5}\\\
0&0&0&0&I_{m_{1}}&0\\\ 0&0&0&0&0&0\\\ 0&0&0&I_{m_{3}}&0&0\\\
0&I_{m_{4}}&0&0&0&0\\\ 0&0&0&0&0&0\\\ 0&I_{m_{4}}&0&0&0&0\\\
0&0&I_{m_{6}}&0&0&0\\\ 0&0&0&0&0&0\\\ I_{m_{8}}&0&0&0&0&0\\\ 0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 0&0&0&0&0&0\end{array}\right],$
$S_{E}=\left[\begin{array}[c]{ccccccccccccc}0&I_{n_{1}}&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&I_{n_{2}}&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&I_{n_{3}}&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&I_{n_{4}}&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&I_{n_{5}}&0&0&0&0&0&0&0\\\
E_{1}&0&0&0&0&0&0&0&0&0&0&0&0\end{array}\right],$
$S_{F}=\left[\begin{array}[c]{ccccccccccccc}0&0&0&0&0&0&I_{n_{4}}&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&I_{n_{6}}&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&I_{n_{7}}&0&0&0&0\\\
F_{1}&I_{n_{1}}&0&0&0&0&0&0&0&0&0&0&0\\\
F_{2}&0&I_{n_{2}}&0&0&0&0&0&0&0&0&0&0\\\
F_{3}&0&0&0&0&0&0&0&0&0&0&0&0\end{array}\right],$
$S_{G}=\left[\begin{array}[c]{ccccccccccccc}0&0&0&0&0&0&0&0&0&I_{n_{8}}&0&0&0\\\
G_{1}&0&0&0&I_{n_{4}}&0&I_{n_{4}}&0&0&0&0&0&0\\\
G_{2}&0&0&0&0&0&0&I_{n_{6}}&0&0&0&0&0\\\
G_{3}&0&0&I_{n_{3}}&0&0&0&0&0&0&0&0&0\\\
G_{4}&I_{n_{1}}&0&0&0&0&0&0&0&0&0&0&0\\\
G_{5}&0&0&0&0&0&0&0&0&0&0&0&0\end{array}\right].$
It is easy to derive the numbers of
$p_{1},\cdots,p_{3},m_{1},\cdots,m_{8},n_{1},\cdots,n_{8}$.
###### Remark 5.1.
Tian [21] derived the maximal rank of the matrix expression
$\displaystyle k(X,Y,Z)=A-BXE-CYF-DZG.$ (5.3)
However, Tian did not give the proof. Applying Theorem 5.1, we can find the
maximal and minimal ranks of the matrix expression (5.3). Moreover, we can
give a solvability condition and the general solution to the matrix equation
$\displaystyle BXE+CYF+DZG=A.$ (5.4)
The corresponding results and their applications will be given in another
paper.
###### Remark 5.2.
Similarly, we can establish a simultaneous decomposition of $[A,B,C,D]$, where
$A=A^{*}$. Using the results on the simultaneous decomposition of $[A,B,C,D]$,
we can derive the maximal rank and extremal inertias of the Hermitian matrix
expression
$\displaystyle
h(X,Y,Z)=A-BXB^{*}-CYC^{*}-DZD^{*},~{}X=X^{*},~{}Y=Y^{*},~{}Z=Z^{*}.$ (5.5)
In addition, we can give a solvability condition and the general solution to
the matrix equation
$\displaystyle BXB^{*}+CYC^{*}+DZD^{*}=A,~{}X=X^{*},~{}Y=Y^{*},~{}Z=Z^{*}.$
(5.6)
The corresponding results and their applications will be given in another
paper.
###### Remark 5.3.
Tian [23] derived the maximal and minimal inertias of the Hermitian matrix
expression
$\displaystyle f_{k}(X_{1},X_{2},\cdots,X_{k})=A-B_{1}X_{1}B_{1}^{*}-\cdots-
B_{k}X_{k}B_{k}^{*},X_{i}=X_{i}^{*}$ (5.7)
using generalized inverses of matrices. Chen, He and Wang [3] derived the
maximal rank of the Hermitian matrix expression (5.7) through a simultaneous
decomposition of the matrix $[A,B_{1},\cdots,B_{k}]$, where $A$ is Hermitian.
The approach in this paper is different with this in [3].
## 6\. Conclusions
We have established a simultaneous decomposition of five real quaternion
matrices in which three of them have the same column numbers, meanwhile three
of them have the same row numbers. Using the simultaneous matrix
decomposition, we have presented the maximal and minimal ranks of the real
quaternion matrix expressions (1.1) and (1.3). We have given a necessary and
sufficient condition for the existence of the general solution to the real
quaternion matrix equation (1.4). The expression of the general solution to
(1.4) has also been given when it is solvable. Moreover, we have derived the
minimal rank of the general solution to the real quaternion matrix equation
(1.4). The results of this paper may be generalized to an arbitrary division
ring (with an involutional anti-automorphism).
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|
arxiv-papers
| 2013-06-21T08:32:28 |
2024-09-04T02:49:46.761794
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Zhuoheng He, Qingwen Wang",
"submitter": "Fayou Zhao",
"url": "https://arxiv.org/abs/1306.5074"
}
|
1306.5079
|
conjtheorem cortheorem lemmatheorem facttheorem claimtheorem proptheorem
definitiontheorem conj cor lemma fact claim prop definition exampletheorem
example rmktheorem rmk
# Comparison Theorems for Manifold with Mean Convex Boundary
Jian Ge
###### Abstract.
Let $M^{n}$ be an $n$-dimensional Riemannian manifold with boundary $\partial
M$. Assume that Ricci curvature is bounded from below by $(n-1)k$, for
$k\in\mathds{R}$, we give a sharp estimate of the upper bound of
$\rho(x)=\operatorname{d}(x,\partial M)$, in terms of the mean curvature bound
of the boundary. When $\partial M$ is compact, the upper bound is achieved if
and only if $M$ is isometric to a disk in space form. A Kähler version of
estimation is also proved. Moreover we prove a Laplace comparison theorem for
distance function to the boundary of Kähler manifold and also estimate the
first eigenvalue of the real Laplacian.
## 0\. Introduction
Let $M^{n}$ be an $n$-dimensional Riemannian manifold with boundary $\partial
M$, $\rho(x)=\operatorname{d}(x,\partial M)$ be the distance function to the
boundary. In a recent paper [Li12], the author proved $\max_{x\in
M}\rho(x)\leq 1/k$ under the assumptions that the Ricci curvature
$\operatorname{Ric}\geq 0$ and the mean curvature $H$ of the boundary
satisfies $H\geq(n-1)k$ for $k>0$, the equality holds if and only if $M$
isometric to the Euclidean ball of radius $1/k$. The argument is essentially
the well known Jacobi field estimates. The idea can be traced back to [HK78],
although estimations of volume is main topic there instead of the distance
functions, also they treated compact manifold only. Under a stronger
assumption that the sectional curvatures bounded from below, Dekster estimated
$\rho$ and lenght of more general curves in [Dek77]. Alexander and Bishop
studied more general Alexandrov spaces with certain convexity condition on the
boundary in [AB10], among many other things, one estimate of the upper bound
of $\rho$ has also been derived for such spaces. In this note we generalize
the above mentioned theorem to Riemannian manifold with lower Ricci curvature
bound $k\in\mathds{R}$ and give an unified proof for all $k$. i.e. we prove
the following
###### Theorem 0.1.
Let $M^{n}$ be a complete $n$-dimensional Riemannian manifold with lower Ricci
curvature bound $(n-1)k$ and boundary $\partial M$. Assume the mean curvature
$H$ of the boundary satisfies $H\geq(n-1)h$. For the case $k\leq 0$ we assumer
further that $h>\sqrt{-k}$. Let $\rho(x)=\operatorname{d}(x,\partial M)$ be
the distance to the boundary. Then
$\rho(x)\leq\left\\{\begin{aligned}
\frac{1}{\sqrt{|k|}}&\coth^{-1}\Big{(}\frac{h}{\sqrt{|k|}}\Big{)}\quad&{\rm
for}\quad k<0\\\ &\frac{1}{h}\quad&{\rm for}\quad k=0\\\
\frac{1}{\sqrt{k}}&\cot^{-1}\Big{(}\frac{h}{\sqrt{k}}\Big{)}\quad&{\rm
for}\quad k>0\\\ \end{aligned}\right.$ (0.1)
If we assume further that $\partial M$ is bounded, then the upper bound in
(0.1) implies that $M$ is compact. Note also that the condition $h>\sqrt{-k}$
for $k\leq 0$ is also sharp. Since the distance function $\rho(x)$ defined on
the warp product $[0,\infty)\times_{e^{-t}}\SS^{n-1}$ is unbounded and
satisfies the mean convex condition for $h=\sqrt{-k}=1$. Note one can also
give a Laplace comparison of $\rho$ for manifold with lower Ricci curvature
bound and mean convex boundary, which is implicitly proved in [HK78]. Hence we
will only list theorem here and omitted the proof. However, the proof is
similar as the Kähler version comparison proved in section 2. Define
$\textsf{H}_{k}(r)=(n-1)\textbf{{c}}_{k}(r)/\textbf{{s}}_{k}(r),$
which is the mean curvature of the geodesic sphere of radius $r$ in simply
connected space form with constant sectional curvature $k$, for
$r\in(0,\pi/\sqrt{k})$ if $k>0$. See (1.2) and (1.3) for the definition of
$\textbf{{s}}_{k}$ and $\textbf{{c}}_{k}$. Hence we have
###### Theorem 0.2 (Implicitly given in [HK78]).
Let $M^{n}$ be a complete $n$-dimensional Riemannian manifold with lower Ricci
curvature bound $(n-1)k$ and boundary $\partial M$. Assume the mean curvature
$H$ of the boundary satisfies $H\geq\textsf{H}_{k}(h)>0$. Let
$\rho(x)=\operatorname{d}(x,\partial M)$ be the distance to the boundary. Then
$\Delta\rho(x)\leq-\textsf{H}_{k}(h-\rho(x)).$
Let $\SS^{n}_{k}$ be the simply connected $n$-dimensional space form of
constant sectional curvature $k$. Let $D_{X}(p,r)$ denote the close disk of
radius $r$ centered at point $p\in X$. i.e. $D_{X}(p,r)=\\{x\in
X|\operatorname{d}(x,p)\leq r\\}$. The proof of the rigidity part in [Li12]
for $\operatorname{Ric}\geq 0$ does not extend to general lower curvature
bound, hence we use Theorem 0.2 and the idea in Cheeger-Gromoll’s proof of
splitting theorem [CG71], one can prove the following rigidity theorem for
Theorem 0.1.
###### Theorem 0.3.
Let $M^{n}$ be a complete $n$-dimensional Riemannian manifold with lower Ricci
curvature bound $(n-1)k$ and boundary $\partial M$. Assume the mean curvature
$H$ of the boundary satisfies $H\geq\textsf{H}_{k}(h)$. Let
$\rho(x)=\operatorname{d}(x,\partial M)$ be the distance to the boundary. Let
$\ell=\max_{x\in M}\rho(x)$, then $\ell=h$ if and only if $M$ is isometric to
$D_{\SS^{n}_{k}}(h)$.
It is well known that results related to Riemannian curvatures are sometimes
also hold for Kähler manifold when hypothesis are suitably phrased in terms of
bi-sectional or holomorphic sectional curvature. In fact the idea of the proof
of Theorem 0.1 can be used to prove the following estimate for Kähler
manifold. It suits our purposes well in this note to avoid complex vector
spaces. In fact we will treat Kähler manifold as a Riemannian manifold with
metric $g$ admitting a parallel skew-symmetric linear transformation $J$ on
the tangent bundle such that $J^{2}=-I$, where $I$ is the identity
transformation of $TM$. Let $R$ be the Riemannian curvature tensor of $g$ with
the convention
$R(X,Y,Z,W)=\langle-\nabla_{X}\nabla_{Y}Z+\nabla_{Y}\nabla_{X}Z+\nabla_{[X,Y]}Z,W\rangle.$
Hence the sectional curvature of a plane $\sigma$ is $R(X,Y,X,Y)$, where $X,Y$
are an orthonormal basis of $\sigma$. Let $\sigma_{1}$ and $\sigma_{2}$ are
two planes in $T_{x}M$ each invariant under $J$. Let $X$ and $Y$ be unit
vectors in $\sigma_{1}$ and $\sigma_{2}$ respectively. Then the holomorphic
bisectional curvature of $\sigma_{1}$ and $\sigma_{2}$ is defined by:
${\rm bisec}(\sigma_{1},\sigma_{2})=R(X,JX,Y,JY).$
By the Bianchi identity we have
${\rm bisec}(\sigma_{1},\sigma_{2})=R(X,Y,X,Y)+R(X,JY,X,JY)$
Notice that our definition of bi-sectional curvature differs by factor $2$ to
the one gave in [LW05]. By writing the inequality ${\rm bisec}\geq 2k$ we mean
$R(X,Y,X,Y)+R(X,JY,X,JY)\geq 2k(|X|^{2}|Y|^{2}+\langle X,Y\rangle^{2}+\langle
X,JY\rangle^{2}),$
for any $X,Y\in T_{x}M$ and any $x\in M$. Note that the complex projective
space $\mathds{CP}^{n}$ with the Fubini-study metric is normalized to have
constant holomorphic sectional curvature $4$. The condition ${\rm bisec}\geq
2k$ are stronger than the condition $\operatorname{Ric}\geq(2n+2)k$, however
the conclusions one can get are also stronger. Define
$\textsf{KH}_{k}(r)=\textbf{{c}}_{4k}(r)/\textbf{{s}}_{4k}(r)+(2n-2)\textbf{{c}}_{k}(r)/\textbf{{s}}_{k}(r),$
which is the mean curvature of the geodesic sphere of radius $r$ in simply
connected complex space form with constant holomorphic sectional curvature
$4k$. One easily verify that $\textsf{KH}_{k}(r)$ is monotone decreasing in
it’s natural domain of definition. We have the following Kähler version of
Theorem 0.1
###### Theorem 0.4.
Let $M^{n}$ be a Kähler manifold of complex dimension $n$ with bisectional
curvature ${\rm bisec}\geq 2k$, for $k\in\mathds{R}$. If $\partial M$ is mean
convex with mean curvature $H\geq\textsf{KH}_{k}(h)$, if $k=1$ we assume
$0<h<\pi/2$. Let $\rho(x)=\operatorname{d}(x,\partial M)$ be the distance to
the boundary. Then for all $x\in M$,
$\rho(x)\leq h.$
In fact if we put a stronger assumption on the convexity of $\partial M$, then
Theorem 0.4 can also be viewed as a consequence of the following Laplacian
comparison theorem for $\rho(x)$:
###### Theorem 0.5.
Let $M^{n}$ be a Kähler manifold of complex dimension $n$ with bisectional
curvature ${\rm bisec}\geq 2k$, for $k\in\mathds{R}$. If the second
fundamental form of $\partial M$ with respect to inner normal direction $\nu$
satisfy
$\operatorname{II}(J\nu,J\nu)\geq\textbf{{c}}_{4k}(h)/\textbf{{s}}_{4k}(h)$
and $\operatorname{II}(V,V)+\operatorname{II}(JV,JV)\geq
2\textbf{{c}}_{k}(h)/\textbf{{s}}_{k}(h)$ for any $V\perp J\nu$, if $k=1$ we
assume $0<h<\pi/2$. Let $\rho(x)=\operatorname{d}(x,\partial M)$ be the
distance to the boundary. Then
$\Delta\rho(x)\leq-\textsf{KH}_{k}(h-\rho(x))\leq-\textsf{KH}_{k}(h).$
The Laplace comparison for distance function to a point with arbitrary
holomorphic bisectional curvature lower bound is given by Li-Wang [LW05].
###### Theorem 0.6 (Li-Wang’s Laplace Comparison, [LW05]).
Let $M^{n}$ be a Kähler manifold of complex dimension $n$ with holomorphic
bisectional curvature ${\rm bisec}\geq 2k$, for $k\in\mathds{R}$. Let
$r(x)=\operatorname{d}(x,o)$, where $o\in M$ is a fixed point in $M$. Then
$\Delta r(x)\leq\textsf{KH}_{k}(r(x)).$
We also note that a complex Hessian comparison theorem for the Busemann
function on complete Kähler manifold with non-negative holomorphic bisectional
curvature was proved by Greene-Wu [GW78]. Cao-Ni [CN05] proved the complex
Hessian comparison theorem for the distance function on Kähler manifold with
non-negative holomorphic bisectional curvature. Our techniques used in the
proofs of Theorem 0.4 and Theorem 0.5 are inspired by [GW78] and [CS05]. The
Laplacian comparison of distance function to the boundary is a natural
complement of the comparison theorems given in [LW05]. As an application of
the Laplace comparison, we estimate the lower bound of the first eigenvalue of
the Laplacian for Kähler manifold with boundary using the idea of Song-Ying Li
and Xiaodong Wang [LW12]. They gave an estimation in the similar settings with
a fixed second fundamental form $2n$.
###### Theorem 0.7.
Assume $M^{n}$ satisfies the condition in Theorem 0.5 with
$\textsf{KH}_{k}(h)\geq 0$ and also assume $M$ is compact. Denote the first
Dirichlet eigenvalue of the Laplacian by $\lambda_{1}$. Then
$\lambda_{1}\geq\Big{(}\frac{\textsf{KH}_{k}(h)}{2}\Big{)}^{2}.\\\ $
It is my pleasure to thank Werner Ballmann for useful comments.
## 1\. Proof of the Theorem 0.1 and Theorem 0.3
In this note, all proofs are given in an uniform flavor, i.e. independent of
the sign of $k$. Hence let’s recall some definitions.
###### Definition .
Given a real constant $k$, we let $\textbf{{s}}_{k}$ denote the solution to
the ordinary differential equation
$\left\\{\begin{aligned} \phi^{\prime\prime}+k\phi&=0,\\\
\phi(0)=0,\quad&\phi^{\prime}(0)=1\end{aligned}\right.$ (1.1)
Setting $\textbf{{c}}_{k}(t)=\textbf{{s}}^{\prime}_{k}(t)$, we clearly get
$\textbf{{c}}^{\prime}_{k}(t)=-k\textbf{{s}}_{k}(t)$ and $\textbf{{c}}_{k}$
satisfies $\textbf{{c}}_{k}^{\prime\prime}+k\textbf{{c}}_{k}=0$ , with initial
condition $\textbf{{c}}_{k}(0)=1,\textbf{{c}}^{\prime}_{k}(0)=0$.
The explicit expressions are given by
$\textbf{{s}}_{k}(t)=\left\\{\begin{aligned}
&\frac{\sinh{(\sqrt{|k|}t)}}{\sqrt{|k|}}\quad&{\rm for}\quad k<0,\\\
&t\quad\quad&{\rm for}\quad k=0,\\\
&\frac{\sin{(\sqrt{k}t)}}{\sqrt{k}}\quad&{\rm for}\quad k>0,\\\
\end{aligned}\right.$ (1.2) $\textbf{{c}}_{k}(t)=\left\\{\begin{aligned}
&\cosh{(\sqrt{|k|}t)}\quad&{\rm for}\quad k<0\\\ &1\quad\quad&{\rm for}\quad
k=0\\\ &\cos{(\sqrt{k}t)}\quad&{\rm for}\quad k>0\\\ \end{aligned}\right.$
(1.3)
We sum up several basic formulas of $\textbf{{s}}_{k}$ and $\textbf{{c}}_{k}$,
the proof is straightforward calculations.
###### Proposition .
$\displaystyle\textbf{{s}}_{k}$ $\displaystyle{}^{\prime}=\textbf{{c}}_{k};$
(1.4) $\displaystyle\textbf{{c}}_{k}^{\prime}$
$\displaystyle=-k\textbf{{s}}_{k};$ $\displaystyle\textbf{{c}}_{k}(A+B)$
$\displaystyle=\textbf{{c}}_{k}(A)\textbf{{c}}_{k}(B)-k\textbf{{s}}_{k}(A)\textbf{{s}}_{k}(B);$
$\displaystyle\textbf{{s}}_{k}(A+B)$
$\displaystyle=\textbf{{s}}_{k}(A)\textbf{{c}}_{k}(B)+\textbf{{c}}_{k}(A)\textbf{{s}}_{k}(B).$
###### Proof of the Theorem 0.1.
For any fixed $\ell>0$ (if $k\geq 0$ we also assume $\ell<\pi/\sqrt{k}$), we
consider the differential equation
$\left\\{\begin{aligned} f^{\prime\prime}(s)+kf(s)&=0\\\ f(0)&=0\\\
f(\ell)&=1\\\ \end{aligned}\right.$ (1.5)
One easily verified
$f(s)=\frac{\textbf{{s}}_{k}(s)}{\textbf{{s}}_{k}(\ell)}$ (1.6)
is the solution. For any point $x\in M$, let $p\in\partial M$ such that
$\operatorname{d}(p,x)=\rho(x)$. Let $\gamma:[0,\ell]\to M$ be a shortest
geodesic joining $x$ to $p$, i.e. $\gamma(0)=x,\gamma(\ell)=p$ and
$\ell=\rho(x)$. By the first variation formula, $\gamma^{\prime}(\ell)\perp
T_{p}(\partial M)$. For and vector field $V$ normal to $\gamma$, the second
variational formula of arc-length gives
$\displaystyle L^{\prime\prime}(V,V)$
$\displaystyle=\int_{0}^{\ell}\Big{(}|V^{\prime}(s)|^{2}-|V(s)|^{2}K(\gamma^{\prime}(s),V(s))\Big{)}ds$
(1.7)
$\displaystyle\quad-\langle\nabla_{V(\ell)}(\gamma^{\prime}(\ell)),V(\ell)\rangle$
$\displaystyle\geq 0,$
where $K(v,w)$ denotes the sectional curvature of the plane spanned by $v$ and
$w$. Choose an orthonormal basis $E_{1},\cdots,E_{n-1}$ of $T_{p}(\partial
M)$. Let $E_{i}(s)$ be the parallel transportation of $E_{i}$ along $\gamma$.
Define
$V_{i}(s)=f(s)E_{i}(s)$
Let $V=V_{i}$ in (1.7), sum over $i$ for $1$ to $(n-1)$ and apply the boundary
condition of (1.5), one get
$\int_{0}^{\ell}\Big{(}(n-1)f^{\prime
2}(s)-f^{2}(s)\operatorname{Ric}(\gamma^{\prime}(s))\Big{)}ds-H_{p}\geq 0.$
where $\operatorname{Ric}(\gamma^{\prime}(s))$ denotes the Ricci curvature
along the direction $\gamma^{\prime}(s)$ and $H_{p}$ denotes the mean
curvature of $\partial M$ at $p$ w.r.t. the inner normal direction
$-\gamma^{\prime}(\ell)$:
$H_{p}=\sum_{i=1}^{n-1}\langle\nabla_{V_{i}(\ell)}\gamma^{\prime}(\ell),V_{i}(\ell)\rangle.$
Integration by parts for the term $f^{\prime 2}$ and make use of the
differential equation (1.5), one get:
$(n-1)[ff^{\prime}]|_{0}^{\ell}-H_{p}\geq 0.$
Since $H\geq(n-1)h>0$, we have:
$f^{\prime}(\ell)\geq h.$ (1.8)
That is $\textbf{{c}}_{k}(\ell)/\textbf{{s}}_{k}(\ell)\geq h$. Hence the
theorem follows from the explicit expressions (1.2) and (1.3). ∎
###### Remark .
It can be seen from the proof that the condition $h>\sqrt{|k|}$ is essential
to get an upper bound when solving (1.8). For positive $k$, the estimates also
holds for mean curvature bounded by a negative constant.
In [CG71], Cheeger and Gromoll proved the celebrate splitting theorem for open
manifold with non-negative Ricci curvature, the key step is that the sum two
Busemann functions constructed out of a line is super-harmonic and achieves a
minimum along this line, hence the minimal principle of super-harmonic
function implies it must be constant. Our proof of the Rigidity Theorem 0.3 is
similar to Cheeger-Gromoll’s proof. Note Li’s proof of the rigidity for $k=0$
given in [Li12] is along a slightly different line, where a rigidity theorem
for isoperemetric inequality plays an important role.
###### Proof of Rigidity Theorem 0.3.
Since $\partial M$ is compact, Theorem 0.1 implies that $M$ itself is compact.
Hence there exists $x_{0}\in M$ such that $\rho(x_{0})=h=\max_{M}(\rho(x))$.
Define
$r(x)=\operatorname{d}(x,x_{0}).$
The classical Laplacian comparison theorem implies that
$\Delta r(x)\leq\textsf{H}_{k}(r(x)).$ (1.9)
Theorem 0.2 implies
$\Delta\rho(x)\leq-\textsf{H}_{k}(h-\rho(x)).$ (1.10)
Follow the idea of Cheeger-Gromoll mentioned above, one define:
$F(x)=r(x)+\rho(x)-h.$
By triangle inequality, we clearly have $F(x)\geq 0$ for $x\in M$. Let
$\gamma:[0,h]\to M$ be a length minimizing geodesic joining $x_{0}$ and
$p\in\partial M$. Hence $F(\gamma(t))\equiv 0$. One calculate
$\displaystyle\Delta F(x)$ $\displaystyle=\Delta r(x)+\Delta\rho(x)$ (1.11)
$\displaystyle\leq\textsf{H}_{k}(r(x))-\textsf{H}_{k}(h-\rho(x))$
$\displaystyle\leq\textsf{H}_{k}(r(x))-\textsf{H}_{k}(r(x))$
$\displaystyle\leq 0$
where we used (1.9) and (1.10) for the first inequality. The second inequality
follows by the triangle inequality $h-\rho(x)\leq r(x)$ and the fact that
$\textsf{H}_{k}(t)$ is monotone decreasing. Therefore $F$ is a nonnegative
super-harmonic function on $M$ and achieves the minimal $0$ at some interior
point $\gamma(s)$ for $s\in(0,h)$. Hence $F$ must be identically $0$. This
shows the smoothness of the $r$ and $\rho$ in $M-\\{x_{0},\partial M\\}$. In
fact if $x\in M-\\{x_{0},\partial M\\}$, then $x$ can be jointed to both
$x_{0}$ and $\partial M$ by geodesic segments $\sigma_{1}$ and $\sigma_{2}$.
If we put this two segment together then it has length
$h=\operatorname{d}(x_{0},\partial M)$, such a segment must be smooth. Hence
$r$ is smooth and $\rho=h-r$ is also smooth. Also note that the geodesic never
bifurcate, which implies the uniqueness of such geodesic connecting $x_{0}$ to
$\partial M$ passing through $x$. Hence $M=D_{M}(x_{0},h)$, $\partial
M=\partial D_{M}(x_{0},h)$ and moreover
$\exp:B_{T_{x_{0}}M}(o,h)\to M$
is a diffeomorphism. The equality $\Delta F=0$ also implies
$\Delta r=H_{k}(r)=(n-1)\frac{\textbf{{c}}_{k}(r)}{\textbf{{s}}_{k}(r)},$
Taking derivative of $\textsf{H}_{k}(r)$ with respect to the direction
$\partial_{r}:=\partial/\partial r$ one get
$\displaystyle-(n-1)k$ $\displaystyle=\partial_{r}(\Delta r)+\frac{(\Delta
r)^{2}}{n-1}$ (1.12) $\displaystyle\leq\partial_{r}(\Delta
r)+|\operatorname{Hess}r|^{2}$
$\displaystyle=-\operatorname{Ric}(\partial_{r},\partial_{r})$
$\displaystyle\leq-(n-1)k,$
where the first inequality is Cauchy-Schwartz and the second equality is
Weitzenböck formula applied for distance function $r$. Hence all inequalities
in (1.12) are equalities. In particular equality in Cauchy-Schwartz implies
that Hessian is diagonal:
$\operatorname{Hess}r=\textbf{{c}}_{k}(r)/\textbf{{s}}_{k}(r)g_{r}$ for
$0<r<h$. Therefore the metric can be written in polar coordinate as
$g=dr^{2}+\textbf{{s}}_{k}^{2}(r)ds_{n-1}^{2}$
where $ds_{n-1}^{2}$ is the standard metric on $(n-1)$-dimensional sphere of
sectional curvature 1. Hence $M$ is isometric to the disk of radius $h$ in
space form $\SS^{n}_{k}$. ∎
## 2\. Comparison Theorems for Kähler Manifolds
We first give a proof similar to the proof of Theorem 0.1.
###### Proof of Theorem 0.4.
Fix $x\in M$. Let $p\in\partial M$ be a point such that
$\operatorname{d}(x,p)=\rho(x)$. It is know that under the assumption ${\rm
bisec}\geq 2$ the diameter of $M$ is less than $\pi/2$ see [LW05]. Hence we
can assume $\ell\leq\pi/2$ for the case $k=1$. Let $\gamma:[0,\ell]\to M$ be a
geodesic such that $\gamma(0)=x,\gamma(\ell)=p$ and $\ell=\rho(x)$. Choose an
orthonormal basis of $T_{p}M$:
$\\{E_{1},E_{2},\cdots,E_{2n-1},E_{2n}\\},$
satisfying $E_{2k}=JE_{2k-1}$ for $k=1,\cdots,(n-1)$ and
$E_{2n}=\gamma^{\prime}(\ell)$. Denote by $E_{i}(s)$ the parallel translation
of $E_{i}$ along $\gamma$. Define:
$f(s)=\frac{\textbf{{s}}_{k}(s)}{\textbf{{s}}_{k}(\ell)},\quad
g(s)=\frac{\textbf{{s}}_{4k}(s)}{\textbf{{s}}_{4k}(\ell)}.$
and
$V_{i}(s)=f(s)E_{i}(s),\ \text{for }i=1,\cdots,2n-2,\quad
V_{2n-1}(s)=g(s)E_{2n-1}(s).$
By the second variational formula one have
$\begin{split}0&\leq\sum_{i=1}^{2n-1}\delta^{2}(V_{i},V_{i})\\\
&=\int_{0}^{\ell}\Big{(}(2n-2)f^{\prime
2}(s)-\sum_{i=1}^{2n-2}f^{2}(s)K(\gamma^{\prime},E_{i})\\\
&\quad\quad+g^{\prime
2}(s)-g^{2}(s)K(\gamma^{\prime},J\gamma^{\prime})\Big{)}ds-H_{p}\\\
&=(2n-2)\textbf{{c}}_{k}(\ell)/\textbf{{s}}_{k}(\ell)+\textbf{{c}}_{4k}(\ell)/\textbf{{s}}_{4k}(\ell)-H_{p}\\\
\end{split}$ (2.1)
i.e.
$\textsf{KH}_{k}(\ell)\geq\textsf{KH}_{k}(h).$
By the monotonicity of $\textsf{KH}_{k}$, we have $\ell\leq h$. ∎
The distance estimate can also be derived from the following Hessian
comparison theorem. But let us recall first some basic definitions, which can
be found in [BC64]. Let $N\in M$ a submanifold, then the $N$-Jacobi field
along a geodesic $\gamma:[0,b]\to M$ with $\gamma(0)=p\in N$ and
$\gamma^{\prime}(0)\perp T_{p}(N)$ is the unique Jacobi field $J$ satisfying:
$J(0)\in T_{p}(N),\nabla_{\gamma^{\prime}(0)}J(0)-S_{\xi}J(0)\in
T_{p}^{\perp}N,$
where $S_{\xi}$ denotes the shape operator of $N$ with respect to the normal
vector $\xi$, i.e. $\langle S_{\xi}V,W\rangle=\langle\nabla_{V}\xi,W\rangle$
and $T_{p}^{\perp}N$ the normal bundle of $N$ in $M$. Recall also that the
index form $I_{\gamma}(,)$ associated with $\gamma$, for any vector field
$V\perp\gamma^{\prime}$ along $\gamma$, it is defined by:
$I_{\gamma}(V,V)=\langle
S_{\gamma^{\prime}(0)}V(0),V(0)\rangle+\int_{0}^{\ell}\Big{|}V^{\prime}|^{2}+|V|^{2}K(\gamma^{\prime},V)\Big{)}ds.$
Note that the second fundamental form with respect to the inner normal
direction of $\partial M$ is defined by $\operatorname{II}(V(0),V(0))=-\langle
S_{\gamma^{\prime}(0)}V(0),V(0)\rangle$. We have the following
###### Proposition (The basic inequality. cf. [BC64]).
Suppose there is no focal point of $N$ on $\gamma(0,\ell]$. For all vector
field $V$ along $\gamma$ with $V(0)\in T_{p}N$, there is an unique $N$-Jacobi
field $J$ such that $J(\ell)=V(\ell)$. Moreover $I_{\gamma}(V)\geq
I_{\gamma}(J)$ and equality occurs if and only if $V=J$.
Theorem 0.5 is a consequence of the following complex Hessian comparison
theorem.
###### Theorem 2.1.
Let $M^{n}$ be a Kähler manifold of complex dimension $n$ with holomorphic
bisectional curvature ${\rm bisec}\geq 2k$, where $k=1,0$ or $-1$. If the
second fundamental form of $\partial M$ with respect to inner normal direction
$\nu$ satisfy
$\operatorname{II}(J\nu,J\nu)\geq\textbf{{c}}_{4k}(h)/\textbf{{s}}_{4k}(h)$
and $\operatorname{II}(V,V)+\operatorname{II}(JV,JV)\geq
2\textbf{{c}}_{k}(h)/\textbf{{s}}_{k}(h)$ for any $V\perp J\nu$, if $k=1$ we
assume $0<h<\pi/2$. Let $\rho(x)=\operatorname{d}(x,\partial M):=\ell$ be the
distance to the boundary. Then for all $\ell<h$, we have
$\displaystyle\operatorname{Hess}\rho(J\nu,J\nu)$
$\displaystyle\leq-\frac{\textbf{{c}}_{4k}(h-\ell)}{\textbf{{s}}_{4k}(h-\ell)}$
(2.2)
$\displaystyle\operatorname{Hess}\rho(V,V)+\operatorname{Hess}\rho(JV,JV)$
$\displaystyle\leq-2\frac{\textbf{{c}}_{k}(h-\ell)}{\textbf{{s}}_{k}(h-\ell)}$
In particular,
$\Delta(\rho)(x)\leq-\textsf{KH}_{k}(h-\rho(x))\leq-\textsf{KH}_{k}(h)$ (2.3)
###### Proof.
For any $x\in M$ let $p\in\partial M$ be a point such that
$\rho(x)=\operatorname{d}(x,\partial M)=:\ell$. Let $\gamma:[0,\ell]\to M$ be
a geodesic from $p$ to $x$. By Theorem 0.4, $\ell\leq h$. It suffices to
estimate the Laplacian for non cut point $x$. Let $f$ be the solution of the
differential equation:
$\left\\{\begin{aligned} f^{\prime\prime}+4kf&=0,\\\
f(0)=1,\quad&f^{\prime}(0)=-h_{1},\end{aligned}\right.$ (2.4)
where $h_{1}=\textbf{{c}}_{4k}(h)/\textbf{{s}}_{4k}(h)$. The explicit solution
is
$f(s)=\textbf{{c}}_{4k}(s)-h_{1}\textbf{{s}}_{4k}(s).$
Also let $g$ be the solution of
$\left\\{\begin{aligned} g^{\prime\prime}+kg&=0,\\\
g(0)=1,\quad&g^{\prime}(0)=-h_{2},\end{aligned}\right.$ (2.5)
where $h_{2}=\textbf{{c}}_{k}(h)/\textbf{{s}}_{k}(h)$. The explicit solution
is
$g(s)=\textbf{{c}}_{k}(s)-h_{2}\textbf{{s}}_{k}(s).$
Since $\ell<h$, one easily verify that $f(\ell)$ and $g(\ell)$ are non-zero,
hence we can define
$\tilde{f}(s)=f(s)/f(\ell),\quad\tilde{g}(s)=g(s)/g(\ell).$
and let $\\{E_{1},\cdots,E_{2n}\\}$ be an orthonormal basis of $T_{p}M$ such
that $JE_{2i-1}=E_{2i}$ for $i=1,\cdots,(n-1)$ and
$E_{2n}=\gamma^{\prime}(0)$. Parallel translate the $E_{i}$ along $\gamma$,
one get $E_{i}(s)$ and define
$V_{i}(s)=\tilde{g}(s)E_{i},\ \text{for }\ i=1,\cdots,2n-1,\ \text{and }\
V_{2n-1}(s)=\tilde{f}(s)E_{2n-1}.$
Let $J_{i}$ be the unique $\partial M$-Jacobi field with
$J_{i}(\ell)=E_{i}(\ell)$. Hence
$\nabla^{2}\rho(E_{i}(\ell),E_{i}(\ell))=\nabla^{2}\rho(J_{i}(\ell),J_{i}(\ell))=\langle\nabla_{\gamma^{\prime}(\ell)}J_{i},J_{i}\rangle=I_{\gamma}(J_{i},J_{i}),$
By Section 2, one have $\nabla^{2}\rho(E_{i}(\ell),E_{i}(\ell))\leq
I(V_{i},V_{i})$. Hence it suffices to estimate $I(V_{i},V_{i})$ from above. In
fact we have
$\begin{split}I_{\gamma}(V_{2n-1},V_{2n-1})&=-\tilde{f}^{2}(0)\operatorname{II}(E_{2n-1},E_{2n-1})\\\
&\quad+\int_{0}^{\ell}\Big{(}(\tilde{f}^{\prime})^{2}-(\tilde{f})^{2}K(\gamma^{\prime},E_{2n-1})\Big{)}ds\\\
&\leq-\tilde{f}^{2}(0)h_{1}+\int_{0}^{\ell}\Big{(}(\tilde{f}^{\prime})^{2}-4k(\tilde{f})^{2}\Big{)}ds\\\
&=-\tilde{f}^{2}(0)h_{1}+\tilde{f}^{\prime}\tilde{f}|_{0}^{\ell}\\\
&=f^{\prime}(\ell)/f(\ell)\\\
&=\frac{-4k\textbf{{s}}_{4k}(\ell)-\textbf{{c}}_{4k}(\ell)\textbf{{c}}_{4k}(h)/\textbf{{s}}_{4k}(h)}{\textbf{{c}}_{4k}(\ell)-\textbf{{s}}_{4k}(\ell)\textbf{{c}}_{4k}(h)/\textbf{{s}}_{4k}(h)}\\\
&=-\frac{\textbf{{c}}_{4k}(h-\ell)}{\textbf{{s}}_{4k}(h-\ell)}\end{split}$
(2.6)
Similar calculation shows that
$I_{\gamma}(V_{2i-1},V_{2i-1})+I_{\gamma}(V_{2i},V_{2i})\leq
2g^{\prime}(\ell)/g(\ell)=-2\frac{\textbf{{c}}_{k}(h-\ell)}{\textbf{{s}}_{k}(h-\ell)}$
(2.7)
Sum up for $i=1$ to $2n-1$, one get
$\begin{split}\Delta\rho&\leq
I_{\gamma}(V_{2n-1},V_{2n-1})+\sum_{i=1}^{n-1}I_{\gamma}(V_{2i-1},V_{2i-1})+I_{\gamma}(V_{2i},V_{2i})\\\
&\leq f^{\prime}(\ell)/f(\ell)+2(n-1)g^{\prime}(\ell)/g(\ell)\\\
&\leq-\textsf{KH}_{k}(h-\ell).\end{split}$ (2.8)
The second inequality in (2.3) follows from the Theorem 0.4 and monotonicity
of $\textsf{KH}_{k}$. ∎
Using the same idea of [LW12], we can estimate $\lambda_{0}$ as follows.
###### Proof of Theorem 0.7.
Let $f$ be the eigenfunction correspond to $\lambda_{0}$. i.e. $f$ satisfies
$\left\\{\begin{aligned} \Delta f+\lambda_{0}f&=0,\quad&\text{on }M,\\\
f&=0,\quad&\text{on }\partial M.\end{aligned}\right.$ (2.9)
We can assume $f>0$ on $M$. For any $C^{2}$ function $g:M\to\mathds{R}$, we
let
$F(x)=f(x)e^{g(x)}.$
Clearly $F\geq 0$ on $M$ and $F|_{\partial M}=0$. Hence by compactness of $M$,
$F$ reaches its maximum at an interior point, say $p_{0}\in M$. Hence at
$p_{0}$ we have
$\nabla F(p_{0})=(\nabla f\cdot e^{g}+fe^{g}\nabla g)|_{p_{0}}=0;$
where $\nabla f$ denotes the gradient vector of $f$. Hence at $p_{0}$:
$\nabla f=-f\nabla g.$
One calculates at $p_{0}$:
$\displaystyle 0\geq\Delta F$ $\displaystyle=\Delta(fe^{g})$ (2.10)
$\displaystyle=e^{g}(\Delta f+2\langle\nabla f,\nabla g\rangle+f\Delta
g+f\langle\nabla g,\nabla g\rangle)$
$\displaystyle=e^{g}(-\lambda_{0}f+2\langle-f\nabla g,\nabla g\rangle+f\Delta
g+f\langle\nabla g,\nabla g\rangle)$
$\displaystyle=e^{g}f(-\lambda_{0}-|\nabla g|^{2}+\Delta g).$
Therefore
$\lambda_{0}\geq(\Delta g-|\nabla g|^{2})|_{p_{0}}.$ (2.11)
for and $g\in C^{2}(M)$ and $p_{0}$ depends on $g$. Hence to get a lower bound
for $\lambda_{0}$ one let $g(x)=-c\rho(x)$, where
$\rho(x)=\operatorname{d}(x,\partial M)$ and $c$ is some positive constant to
be determined. Hence by Theorem 0.5, the following inequalities hold in
barrier sense
$\Delta g\geq c\textsf{KH}_{k}(h),\quad|\nabla g|\leq c$
Hence by (2.11), the following inequality holds for all $c\geq 0$
$\lambda_{0}\geq c(\textsf{KH}_{k}(h)-c).$ (2.12)
One easily see the maximum of the right hand side of (2.12) is
$\Big{(}\frac{\textsf{KH}_{k}(h)}{2}\Big{)}^{2},$
which is achieved when $c=\textsf{KH}_{k}(h)/2\geq 0$. ∎
## References
* [AB10] Stephanie Alexander and Richard Bishop, Extrinsic curvature of semiconvex subspaces in Alexandrov geometry, Ann. Global Anal. Geom. 37 (2010), no. 3, 241-262.
* [BC64] Richard L. Bishop and Richard J Crittenden, Geometry of manifolds. Pure and Applied Mathematics, Vol. XV Academic Press, New York-London 1964 ix+273 pp.
* [CN05] H.D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kählermanifolds, Math. Ann. 331 (2005), no. 4, 795-807.
* [CS05] Jianguo Cao and Mei-Chi Shaw, A new proof of the Takeuchi theorem. Lecture notes of Seminario Interdisciplinare di Matematica. Vol. IV, 65-72, Lect. Notes Semin. Interdiscip. Mat., IV, S.I.M. Dep. Mat. Univ. Basilicata, Potenza, 2005.
* [CG71] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff. Geom. 6 (1971/72), 119-128.
* [Dek77] B. Dekster, Estimates of the length of a curve. J. Differ. Geom. 12, 101 - 118 (1977)
* [GW78] R.E. Greene and H. Wu, On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs, Abh. Math. Sem. Univ. Hamburg 47 (1978) 171-185.
* [HK78] Ernst Heintze and Hermann Karcher, A general comparison theorem with applications to volume estimates for submanifolds. Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 451-470.
* [Li12] Martin Li A Sharp Comparison Theorem for Compact Manifolds with Mean Convex Boundary. The Journal of Geometric Analysis, Published online. DOI 10.1007/s12220-012-9381-6
* [LW05] Peter Li and Jiaping Wang, Comparison theorem for Kähler manifolds and positivity of spectrum. J. Differential Geom. 69 (2005), no. 1, 43 C74.
* [LW12] Song-Ying Li and Xiaodong Wang, Bottom of the spectrum of Kahler manifolds with strongly pseudo-convex boundary, preprint.
Jian Ge
Max Planck Institute for Mathematics
Vivatsgasse 7
53111 Bonn, Germany
[email protected]
|
arxiv-papers
| 2013-06-21T09:20:56 |
2024-09-04T02:49:46.770965
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Jian Ge",
"submitter": "Jian Ge",
"url": "https://arxiv.org/abs/1306.5079"
}
|
1306.5105
|
11institutetext: Indian Institute of Astrophysics, II Block, Koramangala,
Bangalore 560 034, India, 11email: [email protected] 22institutetext:
Laboratoire d’astrophysique, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Observatoire de Sauverny, 1290 Versoix, Switzerland 33institutetext: Ulugh Beg
Astronomical Institute, Uzbek Academy of Sciences, Astronomicheskaya 33,
Tashkent, 100052, Uzbekistan 44institutetext: Institut d’Astrophysique et de
Géophysique, Université de Liège, Allée du 6 Août, 17, 4000 Sart Tilman, Liège
1, Belgium 55institutetext: Instituut voor Sterrenkunde, Katholieke
Universiteit Leuven, Celestijnenlaan 200B, 3001 Heverlee, Belgium
# COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org.
XIV. Time delay of the doubly lensed quasar SDSS J1001$+$5027
S. Rathna Kumar COSMOGRAIL: the COSmological MOnitoring of GRAvItational
Lenses††thanks: Based on observations made with the 2.0-m Himalayan Chandra
Telescope (Hanle, India), the 1.5-m AZT-22 telescope (Maidanak Observatory,
Uzbekistan), and the 1.2-m Mercator Telescope. Mercator is operated on the
island of La Palma by the Flemish Community, at the Spanish Observatorio del
Roque de los Muchachos of the Instituto de Astrofísica de Canarias.,††thanks:
Light curves are available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44,
and on http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. M. Tewes COSMOGRAIL:
the COSmological MOnitoring of GRAvItational Lenses††thanks: Based on
observations made with the 2.0-m Himalayan Chandra Telescope (Hanle, India),
the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the 1.2-m
Mercator Telescope. Mercator is operated on the island of La Palma by the
Flemish Community, at the Spanish Observatorio del Roque de los Muchachos of
the Instituto de Astrofísica de Canarias.,††thanks: Light curves are available
at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. C. S. Stalin
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. F. Courbin
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. I. Asfandiyarov
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. G. Meylan
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. E. Eulaers
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. T. P. Prabhu
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. P. Magain
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. H. Van Winckel
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org. Sh. Ehgamberdiev
COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses††thanks:
Based on observations made with the 2.0-m Himalayan Chandra Telescope (Hanle,
India), the 1.5-m AZT-22 telescope (Maidanak Observatory, Uzbekistan), and the
1.2-m Mercator Telescope. Mercator is operated on the island of La Palma by
the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos
of the Instituto de Astrofísica de Canarias.,††thanks: Light curves are
available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/557/A44, and on
http://www.cosmograil.org.COSMOGRAIL: the COSmological MOnitoring of
GRAvItational Lenses††thanks: Based on observations made with the 2.0-m
Himalayan Chandra Telescope (Hanle, India), the 1.5-m AZT-22 telescope
(Maidanak Observatory, Uzbekistan), and the 1.2-m Mercator Telescope. Mercator
is operated on the island of La Palma by the Flemish Community, at the Spanish
Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de
Canarias.,††thanks: Light curves are available at the CDS via anonymous ftp to
cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-
bin/qcat?J/A+A/557/A44, and on http://www.cosmograil.org.
(Received 21 June 2013 / Accepted 8 July 2013)
This paper presents optical R-band light curves and the time delay of the
doubly imaged gravitationally lensed quasar SDSS J1001$+$5027 at a redshift of
1.838. We have observed this target for more than six years, between March
2005 and July 2011, using the 1.2-m Mercator Telescope, the 1.5-m telescope of
the Maidanak Observatory, and the 2-m Himalayan Chandra Telescope. Our
resulting light curves are composed of 443 independent epochs, and show strong
intrinsic quasar variability, with an amplitude of the order of 0.2
magnitudes. From this data, we measure the time delay using five different
methods, all relying on distinct approaches. One of these techniques is a new
development presented in this paper. All our time-delay measurements are
perfectly compatible. By combining them, we conclude that image A is leading B
by $119.3\pm 3.3$ days ($1\sigma$, 2.8% uncertainty), including systematic
errors. It has been shown recently that such accurate time-delay measurements
offer a highly complementary probe of dark energy and spatial curvature, as
they independently constrain the Hubble constant. The next mandatory step
towards using SDSS J1001$+$5027 in this context will be the measurement of the
velocity dispersion of the lensing galaxy, in combination with deep Hubble
Space Telescope imaging.
###### Key Words.:
gravitational lensing: strong – cosmological parameters – quasar: individual
(SDSS J1001$+$5027)
## 1 Introduction
In the current cosmological paradigm, only a handful of parameters seem
necessary to describe the Universe on the largest scales and its evolution
over time. Testing this cosmological model requires a range of experiments,
characterized by different sensitivities to these parameters. These
experiments, or cosmological probes, are all affected by statistical and
systematic errors and none of them on its own can uniquely constrain the
cosmological models. This is due to the degeneracies inherent in each specific
probe, implying that the probes become truly effective in constraining
cosmology only when used in combination.
The latest cosmology results by the Planck consortium beautifully illustrate
this (Planck Collaboration 2013). In particular, the constraints obtained by
Planck on the Hubble parameter $H_{0}$, on the curvature $\Omega_{k}$, and on
the dark energy equation of state parameter $w$ rely mostly on the combination
of the baryonic acoustic oscillations measurements (BAO) with the Cosmic
microwave background (CMB) observations.
Strong gravitational lensing offers a valuable yet inexpensive complement to
independently constrain some of the cosmological parameters, through the
measurement of the so-called time delays in quasars strongly lensed by a
foreground galaxy (Refsdal 1964). The principle of the method is the
following. The travel times of photons along the distinct optical paths
forming the multiple images are not identical. These travel-time differences,
called the time delays, depend on the geometrical differences between the
optical paths, which contain the cosmological information, and on the
potential well of the lensing galaxy(ies). In practice, time delays can be
measured from photometric light curves of the multiple images of lensed
quasar: if the quasar shows photometric variations, these are seen in the
individual light curves at epochs separated by the time delay.
A precise and accurate measurement of such a time delay, in combination with a
well-constrained model for the lensing galaxy, can be used to constrain
cosmology in a way which is very complementary to other cosmological probes
(see, e.g., Linder 2011). A recent and remarkable implementation of this
approach can be found in Suyu et al. (2013a) that uses the time-delay
measurements from Tewes et al. (2013b). We note, however, that to obtain a
robust cosmological inference from this time-delay technique, particular
attention must be paid to any possible lens model degeneracies (Schneider &
Sluse 2013a; Suyu et al. 2013b; Schneider & Sluse 2013b).
So far, only a few quasar time delays have been measured convincingly, from
long and well-sampled light curves. The international
COSMOGRAIL111http://www.cosmograil.org/ (COSmological MOnitoring of
GRAvItational Lenses) collaboration is changing this situation by measuring
accurate time delays for a large number of gravitationally lensed quasars. The
goal of COSMOGRAIL is to reach an accuracy of less than 3%, including
systematics, for most of its targets.
In this paper, we present the time-delay measurement for the two-image
gravitationally lensed quasar SDSS J1001$+$5027 ($\alpha_{2000}$ =
10:01:28.61, $\delta_{2000}$ = +50:27:56.90) at $z=1.838$ (Oguri et al. 2005).
The image separation of $\Delta\theta=2.86$″(Oguri et al. 2005) and the high
declination of the target make it a relatively easy prey for medium-sized
northern telescopes and average seeing conditions. The redshift of the lensing
galaxy $z_{l}=0.415$ has been measured spectroscopically (Inada et al. 2012).
Our paper is structured as follows. Section 2 describes our monitoring, the
data reduction, and the resulting light curves. In Sect. 3 we present a new
time-delay point estimator. We add this technique to a pool of four other
existing algorithms, to measure the time delay in Sect. 4. Finally, we
summarize our results and conclude in Sect. 5.
## 2 Observations, data reduction, and light curves
Table 1: Summary of COSMOGRAIL observations of SDSS J1001$+$5027.
Telescope | Camera | FoV | Pixel scale | Monitoring period | Epochs | Exp. timeaa$a$The exposure time is given by the number of dithered exposures per epoch and their individual exposure times. | Samplingbb$b$The sampling is given as the mean (median) number of days between two consecutive epochs, excluding the seasonal gaps.
---|---|---|---|---|---|---|---
Mercator 1.2 m | MEROPE | $6.5\arcmin\times 6.5\arcmin$ | $0\aas@@fstack{\prime\prime}190$ | 2005 Mar – 2008 Dec | 239 | 5 $\times$ 360 s | 3.8 (2.0) d
HCT 2.0 m | HFOSC | $10\arcmin\times 10\arcmin$ | $0\aas@@fstack{\prime\prime}296$ | 2005 Oct – 2011 Jul | 143 | 4 $\times$ 300 s | 9.5 (6.1) d
Maidanak 1.5 m | SITE | $8.9\arcmin\times 3.5\arcmin$ | $0\aas@@fstack{\prime\prime}266$ | 2005 Dec – 2008 Jul | 41 | 7 $\times$ 180 s | 5.9 (4.1) d
Maidanak 1.5 m | SI | $18.1\arcmin\times 18.1\arcmin$ | $0\aas@@fstack{\prime\prime}266$ | 2006 Nov – 2008 Oct | 20 | 6 $\times$ 600 s | 12.6 (9.5) d
Combined | | | | 2005 Mar – 2011 Jul | 443 | 201.5 h | 3.8 (1.9) d
222
### 2.1 Observations
Figure 1: Distribution of the average observed FWHM and elongation $\epsilon$
of field stars in the images used to build the light curves of SDSS
J1001$+$5027.
We monitored SDSS J1001$+$5027 in the R band for more than six years, from
March 2005 to July 2011, with three different telescopes: the 1.2-m Mercator
Telescope located at the Roque de los Muchachos Observatory on La Palma
(Spain), the 1.5-m telescope of the Maidanak Observatory in Pamir Alai
(Uzbekistan), and the 2-m Himalayan Chandra Telescope (HCT) located at the
Indian Astronomical Observatory in Hanle (India). Table 1 details our
monitoring observations. In total we obtained photometric measurements for 443
independent epochs, with a mean sampling interval below four days. Each epoch
consists of at least three, but mostly four or more, dithered exposures.
Figure 1 summarizes the image quality of our data. The COSMOGRAIL
collaboration has now ceased the monitoring of this target, to focus on other
systems.
### 2.2 Deconvolution photometry
Figure 2: R-band image centered on SDSS J1001$+$5027\. The image is the
combination of the 210 best exposures from the Mercator telescope, for a total
exposure time of 21 hours. We use the stars labeled 1, 2, and 3 to model the
PSF and to cross-calibrate the photometry of each exposure. The position of
the two lensing galaxies G1 and G2 are indicated in the zoomed image in the
upper left. They are most clearly seen in the deconvolved images presented in
Fig. 3.
Figure 3: Two ways of modeling the light distribution for extended objects
during the deconvolution process. On the left is shown a single 360-second
exposure of SDSS J1001$+$5027 obtained with the Mercator telescope in typical
atmospheric conditions. The other panels show the parametric (top row) and
pixelized light models (bottom row) for the lens galaxies as described in the
text. The residual image for the single exposure is also shown in each case,
as well as the average residuals over the 120 best exposures. The residual
maps are normalized by the shot noise amplitude. The dark areas indicate
excess flux in the data with respect to the model. Gray scales are linear.
The image reduction and photometry closely follows the procedure described in
Tewes et al. (2013b). We performed the flat-field correction and bias
subtraction for each exposure using custom software pipelines, which address
the particularities of the different telescopes and instruments.
Figure 2 shows part of the field around SDSS J1001$+$5027, obtained by
stacking the best monitoring exposures from the Mercator telescope to reach an
integrated exposure time of 21 hours. The relative flux measurements of the
quasar images and reference stars for each individual epoch were obtained
through our COSMOGRAIL photometry pipeline, which is based on the simultaneous
MCS deconvolution algorithm (Magain et al. 1998). The stars labeled 1, 2, and
3 in Fig. 2 are used to characterize the point spread function (PSF) and
relative magnitude zero-point of each exposure.
The two quasar images A and B of SDSS J1001$+$5027 are separated by
$2.86\arcsec$, which is significantly larger than the typical separation in
strongly lensed quasars. In principle, this makes SDSS J1001$+$5027 a
relatively easy target to monitor, as the quasar images are only slightly
blended in most of our images. However, image B lies close to the primary
lensing galaxy G1. Minimizing the additive contamination by G1 to the flux
measurements of B therefore requires a model for the light distribution of G1.
In Fig. 3, we show two different ways of modeling these galaxies. Our standard
approach, shown in the bottom panels, consists in representing all extended
objects, such as the lens galaxies, by a regularized pixel grid. The values of
these pixels get iteratively updated during the deconvolution photometry
procedure. Because of obvious degeneracies, this approach may fail when a
relatively small extended object (lens galaxy) is strongly blended with a
bright point source (quasar), leading to unphysical light distributions. To
explore the sensitivity of our results to a possible bias of this kind, we
have adopted an alternative approach of representing G1 and G2 by two simply
parametrized elliptical Sersic profiles, as shown in the top row of Fig. 3.
For both cases, the residuals from single exposures are convincingly
homogeneous. Only when averaging the residuals of many exposures to decrease
the noise can the simply parametrized models be seen to yield a less good
overall fit to the data, since they cannot represent additional background
sources nor compensate for small systematic errors in the shape of the PSF.
We find that the difference between these approaches in terms of the resulting
quasar flux photometry is marginal; it is insignificant regarding the
measurement of the time delay. In all the following we will use the quasar
photometry obtained using the parametrized model (top row of Fig. 3) which is
likely to be closer to reality than our pixelized model in the immediate
surroundings of image B.
### 2.3 Light curves
Figure 4: R-band light curves of the quasars images A and B in SDSS
J1001$+$5027 from March 2005 to July 2011. The 1$\sigma$ photometric error
bars are also shown. For display purpose, the curve of quasar image B is shown
shifted in time by the measured time delay (see text). The light curves are
available in tabular form from the CDS and the COSMOGRAIL website.
Following Tewes et al. (2013b), we empirically corrected for small magnitude
and flux shifts between the light curve contributions from different
telescopes/cameras to obtain minimal dispersion in each of the combined light
curves. In the present case we chose the photometry from the Mercator
telescope as a reference, and for the data from the Maidanak and HCT
telescopes, we optimized a common magnitude shift and individual flux shifts
for A and B.
Figure 4 shows the combined 6.5-season long light curves, from which we
measure a time delay of $\Delta t_{\mathrm{AB}}=-119.3$ days (see Sect. 4). In
this figure, light curve B has been shifted by this time delay to highlight
the correspondence and temporal overlap of the data. We observe strong
intrinsic quasar variability, common to images A and B. In the period 2006 to
2007, the variability in image A is as large as 0.25 magnitudes over a single
year. In addition to this large scale correspondence, several small and short
scale intrinsic variability features are common to both curves, for instance
around December 2005 and January 2010. Our data unambiguously reveal, already
to the eye, an approximate time delay of $\Delta t_{\mathrm{AB}}\approx-120$
days, with A leading B.
### 2.4 An apparent mismatch between the light curves of the quasar images
The apparent flux ratio between the quasar images, as inferred from the time-
shifted light curves shown in Fig. 4, stays roughly in the range from 0.40 to
0.44 mag over the length of our monitoring. Strong gravitational lens models
readily explain different magnifications of the quasar images, yielding
stationary flux ratios or magnitude shifts between the light curves. Figure 4
hints, however, at a moderate correlation between some variable flux ratio and
the intrinsic quasar variability. In particular, the amplitude of the quasar
variability, in units of magnitudes, appears to be smaller in B than in A.
Potential reasons for this mismatch include the effects of microlensing by
stars of the lens galaxy, or a contamination of the photometry of B by some
additive external flux. We find that one has to subtract from curve B about
20% of its median flux to obtain an almost stationary magnitude shift of about
0.66 mag between the light curves. As this contamination would be several
times larger than the entire flux of galaxy G1, we conclude that plausible
errors of our light models for G1 cannot be responsible for the observed
discrepancy between the light curves.
## 3 A new time-delay estimator
Although an unambiguous approximation of the time delay of SDSS J1001$+$5027
can be made by eye, accurately measuring its value is not trivial, and is made
more difficult by the extrinsic variability between the light curves. Even
more obvious features of the data, such as the sampling gaps due to non-
visibility periods of the targets, could easily bias the results from a time-
delay measurement technique. The impact of these effects on the quality of the
time-delay inference clearly differs for each individual quasar lensing system
and dataset. To check for potential systematic errors, we feel that a wise
approach is to employ several numerical methods based on different fundamental
principles.
In the present section we introduce a new time-delay estimation method, based
on minimizing residuals of a high-pass filtered difference light curve between
the quasar images.
### 3.1 The difference-smoothing technique
Figure 5: Difference light curves of SDSS J1001$+$5027 as obtained by the new
difference-smoothing technique introduced in this paper. The curves are shown
for the best time-delay estimate found with this technique (top panel, $\Delta
t_{\mathrm{AB}}=-118.6$ days), and for a wrong time-delay value (bottom panel,
$\Delta t_{\mathrm{AB}}=-100.0$ days). The difference light curves $d_{i}$ are
shown as colored points. They are smoothed using a kernel of width $s=100$
days to compute the $f_{i}$ (black points). The error bars on the black points
show the uncertainty coefficients $\sigma_{f_{i}}$. The points in the
difference light curve $d_{i}$ are color-coded according to the absolute
factors of their uncertainties $\sigma_{d_{i}}$ by which they deviate from
$f_{i}$. In both panels, light curve A is used as reference, and light curve B
is shifted in flux by the same amount.
This technique is a point estimator that determines both an optimal time delay
and an optimal shift in _flux_ between two light curves, while also allowing
for smooth extrinsic variability. The correction for a flux shift between the
light curves explicitly addresses the mismatch described in Sect. 2.4,
whatever its physical explanation. This flux shift may be due to a
contamination of light curve B by residual light from the lensing galaxy, from
the lensed quasar host galaxy, or by microlensing resolving the quasar
structure.
We consider two light curves A and B sampled at epochs $t_{i}$, where Ai and
Bi are the observed magnitudes at epochs $t_{i}$, ($i=1,2,3,...,N$). We select
A as the reference curve. Light curve B is shifted in time with respect to A
by some amount $\tau$, and in _flux_ by some amount $\Delta f$. Formally, this
shifted version B′ of B is given by
$\displaystyle\mathrm{B}_{i}^{\prime}$ $\displaystyle=$
$\displaystyle-2.5\,{\rm log}\left(10^{-0.4\,\mathrm{B}_{i}}+\Delta f\right),$
(1) $\displaystyle t_{i}^{\prime}$ $\displaystyle=$ $\displaystyle
t_{i}+\tau.$ (2)
For any estimate of the time delay $\tau$ and of the flux shift $\Delta f$, we
form a _difference light curve_ , with points $d_{i}$ at epochs $t_{i}$,
$d_{i}(\tau,\Delta
t)=\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, as in Pelt et al. (1996),
and $\sigma_{\mathrm{B}_{j}}$ denotes the photometric error of the magnitude
Bj. This decorrelation length should typically be of the order of the sampling
period, small enough to not smooth out any intrinsic quasar variability
features from the light curve B. The uncertainties on each $d_{i}$ are then
calculated 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. To summarize, at this point we have a
discrete difference light curve, sampled at the epochs of curve A, built by
subtracting from light curve A a smoothed and shifted version of B. We now
smooth this difference curve $d_{i}$, again 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 a second free parameter of this method. Its
value must be chosen to be significantly larger than $\delta$. For each
$f_{i}$, we compute an uncertainty coefficient
$\sigma_{f_{i}}=\sqrt{\frac{1}{\sum_{j=1}^{N}\nu_{ij}}}.$ (8)
The idea of the present method is now to optimize the time-delay estimate
$\tau$ and flux shift $\Delta f$ to minimize residuals between the difference
curve $d_{i}$ and the much smoother $f_{i}$. Any incorrect value for $\tau$
introduces relatively fast structures that originate from the quasar
variability into $d_{i}$, and these structures will not be well represented by
$f_{i}$. Figure 5 illustrates this phenomenon in the case of SDSS J1001$+$5027
by showing $d_{i}$ and $f_{i}$ for an optimal and an arbitrarily chosen wrong
time-delay estimate. In both panels of Fig. 5, the largest deviations between
$d_{i}$ and $f_{i}$ are due to poorly constrained points with very high
$\sigma_{d_{i}}$, and are therefore not significant. However, for the
incorrect time-delay estimate, a larger number of well-constrained points of
$d_{i}$ significantly deviate from $f_{i}$ (yellow and red points). To
quantify this match between $d_{i}$ and $f_{i}$ we define a cost function in
the form of 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,\Delta f)$ using a global
optimization.
In the above description, light curves A and B are not interchangeable, thus
introducing an asymmetry into the time-delay measurement process. To avoid
this arbitrary choice of the reference curve, we systematically perform all
computations for both permutations of A and B, and minimize the sum of the two
resulting values of $\overline{\chi}^{2}$.
### 3.2 The uncertainty estimation procedure
As a point estimator, the technique described above does not provide
information on the uncertainty of its result. We stress that simple
statistical techniques such as variants of bootstrapping or resampling cannot
be used to quantify the uncertainty of such highly non-linear time-delay
estimators (Tewes et al. 2013a). These approaches are not able to discredit
“lethargic” estimators, which favor a particular solution (or a small set of
solutions) while being relatively insensitive to the actual shape of the light
curves. Furthermore, they are not sensitive to plain systematic biases of the
techniques.
Consequently, to quantify the random and systematic errors of this estimator,
for each dataset to be analyzed and as a function of its free parameters, we
follow the Monte Carlo analysis described in Tewes et al. (2013a). It consists
in applying the point estimator to a large number of fully synthetic light
curves, which closely mimic the properties of the observed data, but have
known true time delays. It is particularly important that these synthetic
curves cover a range of true time delays around a plausible solution, instead
of all having the same true time delay. Only this feature enables the method
to adequately penalize estimators with lethargic tendencies.
### 3.3 Application to SDSS J1001$+$5027
The decorrelation length $\delta$ and the width of the smoothing kernel $s$
are the two free parameters of the described technique. In this work, we
choose $\delta$ to be equal to the mean sampling of the light curves
($\delta=5.2$ days) and $s=100$ days, yielding a point estimate of $\Delta
t_{\mathrm{AB}}=-118.6$ days for the time delay. The corresponding $d_{i}$ and
$f_{i}$ difference light curves are shown in the top panel of Fig. 5. Results
of the uncertainty analysis will be presented in the next section, together
with the performance of other point estimators.
We have explored a range of alternative values for the free parameters
($s=50,100,150,200$ and $\delta=2.6,5.2,10.4$ days), and find that neither the
time-delay point estimate from the observed data, nor the error analysis is
significantly affected. The time-delay estimates resulting from these
experiments stay within 1.2 days around the reference value obtained for
$\delta=5.2$ and $s=100$ days. Regarding the uncertainty analysis, we observe
that increasing the smoothing length scale $s$ beyond 100 days decreases the
random error, but at the cost of an increasing bias, which is not surprising.
## 4 Time-delay measurement of SDSS J1001$+$5027
Figure 6: Error analysis of the four time-delay measurement techniques, based
on delay estimations on 1000 synthetic curves that mimic our SDSS J1001$+$5027
data. The horizontal axis corresponds to the value of the true time delay used
in these synthetic light curves. The gray vertical lines delimit bins of true
time delay. In each of these bins, the colored rods and 1$\sigma$ error bars
show the systematic biases and random errors, respectively, as committed by
the different techniques.
Figure 7: Time-delay measurements of SDSS J1001$+$5027, following five different methods. The total error bar shown here includes systematic and random errors. Table 2: Time-delay measurements for SDSS J1001$+$5027\. The total $1\sigma$ error bars are given. Whenever possible, we give in parenthesis the breakdown of the error budget: (random, systematic). Method | $\Delta t_{\mathrm{AB}}$ [day]
---|---
Dispersion-like technique | -120.5 +/- 6.2 (3.6, 5.0)
Difference-smoothing technique | -118.6 +/- 3.7 (3.4, 1.4)
Regression difference technique | -121.1 +/- 3.8 (3.7, 1.0)
Free-knot spline technique | -119.7 +/- 2.6 (2.4, 0.8)
GP by Hojjati et al. (2013) | -117.8 +/- 3.2
Combined estimate (see text) | -119.3 +/- 3.3
In this work, we use five different methods to measure the time delay of SDSS
J1001$+$5027 from the data shown in Fig. 4. All these methods have been
developed to address light curves affected by extrinsic variability, resulting
from microlensing or flux contamination. Three of the techniques, namely the
dispersion-like technique, the regression difference technique, and the free-
knot spline technique are described in length in Tewes et al. (2013a) and were
used to measure the time delays in the four-image quasar RX J1131$-$123 (Tewes
et al. 2013b).
In the the previous section, we presented our fourth method, the difference-
smoothing technique. These first four methods are point estimators: they
provide best estimates, without information on the uncertainty of their
results. We proceed by quantifying the accuracy and precision of these
estimators by applying them to a set of 1000 fully synthetic light curves,
produced and adjusted following Tewes et al. (2013a). These simulations
include the intrinsic variations of the quasar source, mimicking the observed
variability of SDSS J1001$+$5027, as well as extrinsic variability on a range
of time scales from a few days to several years. They share the same sampling
and scatter properties as the real observations.
Figure 6 shows the results of this analysis, depicting the delay measurement
error as a function of the true delay used to generate the synthetic light
curves. As always, this analysis naturally takes into account the intrinsic
variances of the techniques, that are due to the limited ability of the
employed global optimizers to find the absolute minima of the cost functions.
As can be seen in Fig. 6, the dispersion-like technique is strongly biased for
this particular dataset. This could be a consequence of the simplistic
polynomial correction for extrinsic variability linked to this technique. For
the other techniques, the bias remains smaller than the random error, and no
strong dependence on the true time delay is detected.
The final systematic error bar for each of these four techniques is taken as
the worst measured systematic error on the simulated light curves (biggest
colored rod in Fig. 6). The final random error is taken as the largest random
error across the range of tested time delays. Finally, the total error bar for
each technique is obtained by summing the systematic and random components in
quadrature.
In the writing process of this paper, Hojjati et al. (2013) proposed a new
independent method to measure time delays that is also able to address
extrinsic variability. Their method is based on Gaussian process modeling, and
does not rely on point estimation. It provides its own standalone estimate of
the total uncertainty. We have provided these authors with the COSMOGRAIL data
of SDSS J1001$+$5027, without letting them know our measured values. They find
$\Delta t_{\mathrm{AB}}=-117.8\pm 3.2$ days.
We include this measurement by Hojjati et al. (2013) as a fifth measurement in
our result summary, presented in Table 2 and in a more graphical form in Fig.
7. Not only do their time-delay values agree with our four estimates, but also
their error bars agree well with ours, in spite of the totally different way
of estimating them.
We have five time-delay estimates from five very different methods, and all
these estimates are compatible with each other. We now need to combine these
results. In doing this, we exclude the delay from the dispersion-like
technique that, as we show, is dominated by systematic errors. While the
estimates from the four remaining techniques are obtained with very different
methods, they are still not independent, as they all make use of the same
data. We therefore simply average the four time-delay measurements to obtain
our combined estimate, and we use the average of the total uncertainties as
the corresponding uncertainty. This leads to $\Delta t_{\mathrm{AB}}=-119.3\pm
3.3$ days, shown in black in Fig. 7.
## 5 Conclusion
In this paper, we present the full COSMOGRAIL light curves for the two images
of the gravitationally lensed quasar SDSS J1001$+$5027\. The final data, all
taken in the R band, totalize 443 observing epochs, with a mean temporal
sampling of 3.8 days, from the end of 2004 to mid-2011. The COSMOGRAIL
monitoring campaign for SDSS J1001$+$5027 is no longer in progress. It
involved three different telescopes with diameters from 1.2 m to 2 m, hence
illustrating the effectiveness of small telescopes in conducting long-term
projects with potentially high impact on cosmology.
We analyzed our light curves with five different numerical techniques,
including the three methods described in Tewes et al. (2013a). In addition, we
introduced and described a new additional method, based on representing the
extrinsic variability by a smoothed version of the difference light curve
between the quasar images. Finally, we also presented results obtained via the
technique of Hojjati et al. (2013), based on modeling of the quasar and
microlensing variations using Gaussian processes. The technique was _blindly_
applied to the data by the authors of Hojjati et al. (2013), without any prior
knowledge of the results obtained with the other four methods.
Aside from the dispersion-like technique, dominated by systematic errors, we
find that the four other methods yield similar time-delay values and similar
random and systematic error bars. Our final estimate of the time delay is
taken as the mean of these four best results, together with the mean of their
uncertainties: $\Delta t_{\mathrm{AB}}=-119.3\pm 3.3$ days, with image A
leading image B. This is a relative uncertainty of 2.8%, including systematic
errors.
The present time-delay measurement can be used in combination with lens models
to constrain cosmological parameters, in particular the Hubble parameter H0
and the curvature $\Omega_{k}$ (e.g., Suyu et al. 2013a). The accuracy reached
on cosmology with SDSS J1001$+$5027 alone or in combination with other lenses,
will rely on the availability of follow-up observations to measure: (1) the
lens velocity dispersion, (2) the mass contribution of intervening objects
along the line of sight, and (3) the detailed structure of the lensed host
galaxy of the quasar. This translates in practice into one single night of an
8m-class telescope, plus about four orbits of the Hubble Space Telescope.
###### Acknowledgements.
We thank the numerous observers who contributed to the data from the Mercator
and Maidanak telescopes, and we are grateful for the support provided by the
staff at the Indian Astronomical Observatory, Hanle and CREST, Hoskote. We
also thank A. Hojjati, A. Kim, and E. Linder for running their curve shifting
algorithm on our data. S. Rathna Kumar and C. S. Stalin acknowledge support
from the Indo-Swiss Personnel Exchange Programme
INT/SWISS/ISJRP/PEP/P-01/2012. COSMOGRAIL is financially supported by the
Swiss National Science Foundation (SNSF). We thank the referee Peter Schneider
for his timely comments that helped to improve this paper.
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|
arxiv-papers
| 2013-06-21T11:30:22 |
2024-09-04T02:49:46.778202
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Rathna Kumar, M. Tewes, C. S. Stalin, F. Courbin, I. Asfandiyarov,\n G. Meylan, E. Eulaers, T. P. Prabhu, P. Magain, H. Van Winckel, and Sh.\n Ehgamberdiev",
"submitter": "S Rathna Kumar",
"url": "https://arxiv.org/abs/1306.5105"
}
|
1306.5124
|
# Evidence for Intra-Unit-Cell magnetic order in the pseudo-gap state of
high-$\rm T_{c}$ cuprates
Y. Sidis and P. Bourges Laboratoire Léon Brillouin, CEA-CNRS, CE-Saclay,
91191 Gif sur Yvette, France [email protected]
###### Abstract
The existence of the mysterious pseudo-gap state in the phase diagram of
copper oxide superconductors and its interplay with unconventional d-wave
superconductivity has been a long standing issue for more than a decade. There
is now a growing number of experimental indications that the pseudo-gap phase
actually corresponds to a symmetry breaking state. In his theory for cuprates,
C. M. Varma proposes that the pseudo-gap is a new state of matter associated
with the spontaneous appearance of circulating current loops within $\rm
CuO_{2}$ unit cell. This intra-unit-cell order breaks time reversal symmetry,
but preserves lattice translation invariance. Polarized elastic neutron
scattering measurements provide evidence for an intra-unit-cell magnetic order
inside the pseudo-gap state. This order could be produced by the orbital-like
magnetic moments induced by the circulating current loops. The magnetic order
displays the same characteristic features in $\rm HgBa_{2}CuO_{4+\delta}$,
$\rm YBa_{2}Cu_{3}O_{6+x}$ and $\rm Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}$
demonstrating that this genuine phase is ubiquitous of the pseudo-gap of high
temperature copper oxide materials. We review the main properties
characterizing this intra-unit-cell magnetic order and discuss its interplay
or competition with other spin and charge instabilities.
## 1 Pseudo-gap state and circulating current loop theory
In the phase diagram of high temperature superconducting copper oxides,
various anomalous electronic properties have been reported in addition to the
occurrence of unconventional d-wave superconductivity. In normal state, these
materials exhibit non-Fermi liquid properties and enter a mysterious pseudo-
gap regime, characterized by the loss of density of states on certain portions
of the Fermi surface below a temperature $\rm T^{\star}$. A large variety of
theoretical models has been designed to account for these exotic electronic
properties and to shed light on their interplay with non conventional
superconductivity. Among them, the pseudo-gap theory of C. M. Varma [1, 2, 3]
proposes that the pseudo-gap state is a long range ordered state and that
fluctuations of its order parameter control not only the non Fermi liquid
properties, but also are likely to mediate the d-wave superconducting pairing
[4]. Derived from the 3-band Hubbard model, this mean-field theory postulated
that strong electronic interaction involving the Coulomb repulsion between
copper and neighboring planar oxygens give rise to the spontaneous appearance
of circulating current (CC) loops within $\rm CuO_{2}$ unit cell. The CC-loops
appear by pairs, turning clockwise and anti-clockwise. This Intra-Unit-Cell
(IUC) order breaks time reversal symmetry, but preserves lattice translation
invariance. Beyond the mean-field theory proposed by C. M. Varma, the
existence of a CC-loop order is challenged by other theoretical studies. The
numerical analysis carried by Thomale and Greiter [5, 6] does not confirm
their existence. The quantum variational Monte Carlo simulation of Weber and
co-workers [7] questions the stability of the CC-loops suggested by Varma’s
model, but suggests that a CC-loop order could become much more stable in a
5-band Hubbard model, where apical oxygen orbitals are also included. In
addition, the existence of CC-loops in ladder copper oxide materials is not
fully excluded [8, 9, 10]. Beyond pure theoretical considerations, several
experimental observations provide strong encouragement for models based CC-
loop order in copper oxide materials.
## 2 Evidence for an Intra-Unit-Cell magnetic order
Polarized neutron scattering studies carried in bilayer $\rm
YBa_{2}Cu_{3}O_{6+x}$ (Y123) [11, 12, 13, 14] and monolayer $\rm
HgBa_{2}CuO_{4+\delta}$ (Hg1201) [15, 16] have reported experimental evidence
of a long range 3D magnetic order hidden in the pseudo-gap state. The
existence of such a long range ordered magnetic state has been confirmed
recently in bilayer $\rm Bi_{2}Sr_{2}CaCuO_{8+\delta}$ (Bi2212) [17]. In
monolayer $\rm La_{2-x}Sr_{x}CuO_{4}$ (La214), a similar magnetic order has
also been observed [18], but in this system it remains 2D and short ranged.
Figure 1 shows the variation of the temperature $\rm T_{mag}$ at which the
magnetic order settles in as a function of the hole doping level (p).
Figure 1: (color online) Variation of the temperature $\rm T_{mag}$ at which
the IUC magnetic order appears as a function the hole doping level for four
cuprate families: $\rm YBa_{2}Cu_{3}O_{6+x}$ (Y123 -
$\color[rgb]{0,0,0}\fullcircle$)) [11, 13, 14], $\rm HgBa_{2}CuO_{4+\delta}$
(Hg1201- $\color[rgb]{1,0,0}\fullsquare$) [15, 16], $\rm
Bi_{2}Sr_{2}CaCuO_{8+\delta}$ (Bi2212 - $\blacklozenge$) [17], $\rm
La_{2-x}Sr_{x}CuO_{4}$ (La214 - $\blacktriangle$) [18]. The open symbols stand
for the value of the superconducting critical temperature $\rm T_{c}$ for each
sample. Note that the value of $\rm T_{c}$ is a function not only of p but
also of disorder, possibly leading to systematic differences in carrier
concentration estimates. p for Hg1201 is estimated using the $\rm T_{c}(p)$
relationship for Y123 [20].
In all the three compounds with a superconducting critical temperature $\rm
T_{c}$ as high as $\sim$90 K at optimal doping, the magnetic order develops
concomitantly with the pseudo-gap state, at least for a hole doping level
larger than p=0.09. In Hg1201 and Y123, the magnetic ordering temperature $\rm
T_{mag}$ matches the pseudo-gap temperature $\rm T^{\star}$ determined by
resistivity measurements [11, 15, 19]. In Bi2212 [17], the hole doping
dependence and the magnitude of $\rm T_{mag}$ and $\rm T^{\star}$ determined
by various techniques [21, 22, 23, 24] are quite consistent. The fact that
$\rm T^{\star}$ does not decrease linearly with increasing hole doping and
likely diplays a shoulder around optimal doping in Bi2212 [25] is well
reproduced by the hole doping dependence of $\rm T_{mag}$ (Fig. 1). In
addition, the comparison of the polarized neutron scattering measurements and
tunneling spectroscopy measurements in Bi2212 [24] suggest that the variation
of the ordered magnetic moment upon increasing hole doping seems to reproduce
the one of the pseudo-gap energy $\rm\Delta_{PG}$.
This novel 3D magnetic order does not seem to depend strongly on the
structural properties of the materials. In contrast with Y123 or Hg1201,
Bi2212 has a body centered structure. The magnetic order seems to remain 3D
independently from the nature of the stacking of Cu sites along the $\rm{\bf
c}$ axis. Furthermore, Hg1201 is a tetragonal system, Y123 is orthorhombic
along the Cu-O bonds and Bi2212 is orthorhombic along diagonals. The influence
of orthorhombicity on details of the magnetic structure is a relevant issue,
but dedicated neutron scattering studies are still under progress.
Whatever the system where it has been observed, this novel magnetic state
preserves the lattice translation invariant, but, at variance with
ferromagnets, does not give rise to a uniform magnetization [19]. These
observations imply the existence of an IUC (antiferro-)magnetic order. The
overall symmetry of this order is consistent with the so-called
CC-$\rm\theta_{II}$ phase proposed by C. M. Varma [3]. Polarized neutron
scattering studies may have detected the distribution of static magnetic
fields generated by the CC-loops. The possible detection of these magnetic
fields by local probes such as NMR, NQR or $\mu$SR is still debated [26, 27,
28, 29, 30], but should help to get a deeper understanding of the intrinsic
nature of the IUC magnetic order.
Beyond polarized neutron scattering experiments, other measurements carried
out in Y123 provide direct or indirect indications that a symmetry breaking
state develops below the pseudo-gap temperature. Indeed, resonant ultrasound
spectroscopy measurements have recently reported the first thermodynamic
evidence of a true phase transition taking place when entering the pseudo-gap
[31]. In addition, the observation of anomalies in the second derivative of
the magnetization below $\rm T_{\chi}$ [32] can be understood in term of a
coupling of the CC-loop order parameter to the uniform susceptibility [33].
Furthermore, the spontaneous appearance of an a-b anisotropy in the Nernst
coefficient at $\rm T_{\nu}$ [34] could be interpreted as a violation of the
Onsager Reciprocity principle in the CC-loop state which breaks time reversal
symmetry [35]. Finally, a polar Kerr effect has been observed within the
pseudo-gap state below $\rm T_{K}$ [36]. This effect has been initially
interpreted as a signature of a time reversal symmetry broken state [36], but
actually ascribed to a gyrotopic order [37]. As shown in Fig.2, the hole
doping dependencies of the different temperatures $\rm T_{mag}$, $\rm
T_{\chi}$, $\rm T_{K}$ and $\rm T_{\nu}$ display a certain similarity,
suggesting that the related phenomena could share a common origin.
While the appearance of an IUC magnetic order can be ascribed to CC-loops, to
date there is no direct observation of these CC-loops in superconducting
cuprates. Alternatively, a resonant X-ray diffraction study in non
superconducting CuO has claimed the first direct observation of CC-loops in
$\rm CuO_{2}$ unit cell, i.e the basic building block of superconducting
copper oxide materials [38], but has been challenged theoretically [39].
Nevertheless, such a kind of experiments in the superconducting cuprates would
be very illuminating in resolving the question of whether a CC-loop
description is appropriate for the observed neutron scattering signal in the
pseudo-gap phase [39].
Figure 2: (color online) $\rm YBa_{2}Cu_{3}O_{6+x}$ phase diagram as a
function of the hole doping p. In addition to the antiferromagnetic (AF) state
and the d-wave superconducting (SC) state, several phases can be detected. In
the lighly doped regime ($\rm 0.05<p<0.085$), incommensurate spin fluctuations
develop a quasi-1D anisotropy spontaneously below $\rm T_{nem}$ [59]. The $\rm
C_{4}$ rotational symmetry is first broken, but the translation invariance is
preserved. At lower temperature, below $\rm T_{SDW}$, the spin correlations
freeze and a spin density (SDW) state settles in [59]. In the underdoped
regime ($\rm p>0.085$), the pseudo-gap state develops below $\rm T^{\star}$
that can be determined by the departure of the electrical resistivity from a
linear T-dependence at $\rm T_{\rho}$ [34]. The IUC-magnetic order appears at
$\rm T_{mag}\sim T^{\star}$ [14]. Around that temperature an anomaly can be
observed in the second derivative of the magnetization at $\rm T_{\chi}$ [32]
and the Nernst coefficient exhibits a net a-b anisotropy at $\rm T_{\nu}$
[34]. Below $\rm T_{mag}$, a Kerr effect is observed at $\rm T_{K}$ [36].
Around the specific hole doping $\rm p\sim 1/8$, a charge density wave (CDW)
instability develops below a temperature $\rm T_{CDW}$ [68], close to $\rm
T_{K}$.
## 3 Time reversal symmetry breaking
Since the discovery of the pseudo-gap regime, by pioneering NMR measurements
in the early 90’s [40], the most accurate information concerning the
electronic properties of this phase have been provided by tunneling
spectroscopy [41] and angle resolved photo-emission (ARPES) [42] in bilayer
Bi2212. In the pseudo-gap state, left-circularly polarized photon give a
different photocurrent from right-circularly photons in ARPES measurements
[43]. This circular dichroism would imply that time reversal symmetry is
spontaneously broken in the pseudo-gap state, a phenomenon in good agreement
with a CC-loop order [44].
The recent observation of an IUC magnetic order in Bi2212 further confirms
that time reversal symmetry is broken in the pseudo-gap state. The temperature
dependencies of polarized neutron measurement in samples UD-85 and OD-70 [17]
and of the dichroic effects in samples UD-85 and OD-65 [43] match with each
other (Fig. 3). Keeping in mind that ARPES measurements are performed on thin
films and polarized neutron measurement on large single crystals, the onset of
time reversal symmetry breaking found by both types of measurements for
samples UD-85 is in a good agreement (Fig. 3). Likewise, the observations of
dichroic effect at antinodal wave vectors and of a magnetic signal on Bragg
reflections (1,0,L) indicate that the phase responsible for both effects
possess the same symmetry as the CC-$\rm\theta_{II}$ phase proposed in the CC-
loop theory for the pseudo-gap [44].
Figure 3: (color online) Comparison between polarized neutron scattering
measurements performed on Bi2212 single crystals [17] and circular dischroism
obtained from ARPES measurements on Bi2212 thin films [43]. a) The full
magnetic intensity ($\rm I_{M}$) normalized by the nuclear intensity ($\rm
I_{N}$) measured at the Bragg reflection (1,0,1). This normalization procedure
is useful for comparing different samples. ($\rm I_{M}$) is about 3 orders of
magnitude smaller than ($\rm I_{N}$). Raw polarized neutron scattering data
and a detailed description of the method used to extract such a weak magnetic
intensity have been recently given in [17, 19]. b) Circular dischroism D(T) at
the anti-nodal wave vectors, reproduced from [43]. The data associated with
underdoped samples (UD- $\rm T_{c}$) are shown with symbols
$\color[rgb]{1,0,0}\fullcircle$ and those associated with overdoped samples
(OD- $\rm T_{c}$) with symbols $\color[rgb]{0,0,1}\fullsquare$.
It is worth pointing out that the reported dichroic effect in ARPES and its
interpretation are still controversial and the subject of a long running
debate [45, 46, 47, 48, 49], since it has been argued that structural effects
could also account for the dichroic signal reported in ARPES. Whenever correct
for the analysis of ARPES measurements, these criticisms do not hold for
polarized neutron diffraction measurements, since structural effects cannot
produce a spin flip signal varying with the neutron spin polarization as
observed experimentally.
Beyond circularly polarized ARPES measurements and polarized neutron
diffraction, the observations of a polar Kerr effect at a temperature $\rm
T_{K}$ in bilayer Y123 [36] and in monolayer $\rm
Pb_{0.55}Bi_{1.5}Sr_{1.6}La_{0.4}Cu_{6+\delta}$ (Pb-Bi2201) [50] have been
first presented as a direct evidence for time reversal symmetry breaking
within the pseudo-gap state. This interpretation is now challenged by the fact
the sign of the rotation of the polarization angle is the same on opposite
surfaces of the sample, while in the usual Kerr effect the rotation angle must
reverse. While $\rm T_{K}$ coincides with $\rm T^{\star}$ in Pb-Bi2201, it
appears deep inside the pseudo-gap state in Y123 (Fig. 2). Starting from a CC-
loop order breaking time reversal symmetry at $\rm T^{\star}$, distinct
theoretical models propose different routes to account for the observed polar
Kerr once additional chiral properties are considered [51, 52].
All experimental reports available in monolayer systems (Hg1201, Pb-Bi2201)
and bilayer systems (Y123, Bi2212) provide compelling evidence that the
pseudo-gap phase is actually an ordered state, whose order parameter and
broken symmetries remains to be clearly identified.
## 4 Static and dynamical magnetic properties
Neutron scattering technique can probe simultaneously the nuclear and magnetic
correlations. Spin density wave (SDW) -like instabilities break translation
invariance and produces a magnetic response at wave vectors distinct from the
nuclear Bragg reflections. In cuprates, static or fluctuating SDWs are usually
located around the antiferromagnetic (AF) wave vector and characterized by the
wave vectors $\rm{\bf q}_{SDW}$=$\rm{\bf q}_{AF}\pm(\delta,0)$ (and/or
$\rm\pm(0,\delta)$) [53], where $\rm{\bf q}_{AF}$=(0.5,0.5) stands for the
planar AF wave vector, given in reduced lattice units (tetragonal notation)
(Fig. 4.a). At variance, the IUC magnetic order implies the existence of
staggered magnetic moments within the unit cell and corresponds to a q=0
magnetic instability. Since the IUC magnetic order preserves the lattice
translation invariance, the weak magnetic signal that it produces in a neutron
scattering experiment is superimposed with the much stronger nuclear
scattering. The use of spin polarized neutron scattering technique is then
essential to disentangle the nuclear and magnetic neutron scattered
intensities. A full polarization analysis allows a determination of the
orientation of the ordered magnetic moments.
The search for a long range 3D magnetic order in the pseudo-gap phase has been
performed on Bragg reflections (1,0,L)-(0,1,L) with integer L values in
monolayer Hg1201 [15, 16] and in bilayers Y123 [11, 12, 13, 14] and Bi2212
[17]. In the case of monolayer La214 [18], the IUC order is quasi-2D and can
be observed for any L value. In Ref. [17], the magnetic intensities along the
$\rm{\bf c}$ axis have been compared for these 4 cuprates families.
Interestingly, these intensities exhibit for all systems a similar decay with
$\rm\frac{2\pi}{c}L$ (the component of the wave vector perpendicular to the
$\rm CuO_{2}$ plane), suggesting a common magnetic origin. For all cuprate
families, the polarization analysis demonstrates that the magnetic moments
cannot be strictly perpendicular to the $\rm CuO_{2}$ planes. A conservative
estimate of $\rm 45\pm 20^{o}$ can be given for the tilt angle of these
moments with respect to the $\rm\bf{c}$ axis. While Polar Kerr effect is
usually associated with the appearance of ferromagnetic moments, it can also
be mediated by magneto-electric coupling for an antiferromagnetic order. The
tilt of magnetic moments could be an important feature which lowers the
symmetry of the IUC magnetic structure and allows the magnetic structure to be
active to produce a Kerr effet [54].
The origin of the static magnetic order reported by polarized neutron
diffraction is still an open issue. The IUC-cell order could be induced by CC-
loops in the CC-$\rm\theta_{II}$ phase. This phase is characterized by two CC-
loops per $\rm CuO_{2}$ unit cell and the loop pattern is fourfold
degenerated. However, the CC-loops are confined with the $\rm CuO_{2}$ planes,
generating orbital magnetic moments perpendicular to the planes. This is at
variance with the experimental observation. Recently, two Ising magnetic mode
has been discovered in the pseudo-gap state of Hg1201 [55, 56]. These new
modes could be ascribed to magnetic excitations associated with a change of
configuration of the CC-loop pattern. The two excitations corresponding to a
$\pm$90o rotation of a loop pattern are indeed active in the spin flip channel
in inelastic polarized neutron scattering. The quantitative analysis of the
characteristic mode energies and their weak dispersion leads to the conclusion
that the ground state cannot reduce to one of the four states of the
CC-$\rm\theta_{II}$ phase, but should be made of their quantum superposition
[57, 58]. This produces a quantum interference phenomenon in neutron
diffraction measurement: the tilt angle of the magnetic moment reported in
polarized neutron scattering measurements would then highlight the degree of
admixture of the four orthogonal states of the CC-$\rm\theta_{II}$ phase.
Alternatively, it has been proposed that CC-loops could be delocalized on the
$\rm CuO_{6}$ octahedra for a monolayer like Hg1201 [7] or on $\rm CuO_{5}$
pyramids in bilayer systems like Y123 or Bi2212 [30]. This model should
produce a magnetic signal on Bragg reflection (00L), but no signal has been
observed at Bragg position (002) neither in Y123 [11] nor in Hg1201 [15].
Furthermore, this model fails to account for the polarization analysis carried
out on the Bragg reflection (100) [19].
## 5 Competing spin instabilities
In La214 [18] for a hole doping level p=0.085, the quasi-2D and short range
IUC magnetic order appears at a rather low temperature $\rm T_{mag}\sim$120 K.
Below this temperature, the dynamical spin response at the wave vector
$\rm{\bf q}_{SDW}$ exhibits two marked modifications: (i) an enhancement of
the low energy spin fluctuations intensity and (ii) an order parameter-like
increase of the incommensurability parameter $\delta$. Both effects indicate a
deep interplay between the q=0 magnetic order and the incommensurate spin
fluctuations. In lightly doped Y123 [14] (p $\sim$ 0.08), the 3D long range
IUC order develops at $\rm T_{mag}\sim$170 K (Fig. 1), i.e very close to the
temperature $\rm T_{nem}\sim$ 150 K (Fig.2). At this characteristic
temperature, the low energy incommensurate spin fluctuations start exhibiting
a net a-b anisotropy [59]. $\rm T_{nem}$ is usually interpreted as the onset
temperature at which the $\rm C_{4}$ rotational symmetry is spontaneously
broken by strong electronic correlations pushing the system into an electronic
nematic state. In lightly doped Y123, the orthorhombic lattice distortion
serves as weak orientational field and is essential for the observation of the
a-b anisotropy. Since this state also preserves the translation invariance, it
corresponds to another competing q=0 electronic instability. At much lower
temperature, spin correlations freeze at $\rm T_{SDW}$ (Fig. 2), giving rise
to a spin density wave (SDW) state that can further coexist with the
superconducting phase [59]. In underdoped Y123 [14] ($\rm p>0.09$), the 3D IUC
magnetic order is well developed and settles in at $\rm T_{mag}\simeq
T^{\star}$. In parallel, low energy spin fluctuations around the AF wave
vector are gapped. Through the substitution of a few percents of non magnetic
Zn impurities (known to strongly weaken the superducting order), neither $\rm
T_{mag}$ [14] nor $\rm T^{\star}$ [61] are affected. On the contrary, the
average magnetic moment associated with the IUC magnetic order drops down.
This indicates that the IUC order is likely to be destroyed around Zn
impurities, but remains unaffected far away. Simultaneously, quasi-1D
incommensurate spin fluctuations are restored at low energy, suggesting that a
competing SDW order could develop where the IUC magnetic order is suppressed.
All these studies suggest a competition between the IUC magnetic state and an
incipient SDW state.
Figure 4: (color online) a) Location in the reciprocal space of the different
kinds of spin, charge and orbital-like magnetic signals. The IUC-magnetic
order is observed at the planar waves vector q =(1,0) and (0,1)
($\color[rgb]{1,0,0}\fullcircle$), but should be observable at q =(1,1)
($\color[rgb]{1,0,0}\opencircle$). A fluctuating or static charge density wave
(CDW) is characterized by the planar wave vectors $\rm{\bf
q}_{CDW}$=$\rm\pm(\epsilon,0)$ (and/or $\rm\pm(0,\epsilon)$
($\color[rgb]{0,1,1}\fullsquare$). A fluctuating or static spin density wave
(SDW) is associated with the planar wave vectors $\rm{\bf q}_{SDW}$=$\rm{\bf
q}_{AF}\pm(\delta,0)$ (and/or $\rm\pm(0,\delta)$) ($\blacklozenge$), where
$\rm{\bf q}_{AF}$=(0.5,0.5). b) Hole doping dependencies of the
incommensurability parameters $\rm\epsilon$ and $\rm\delta$ for 3 cuprate
families: (i) $\rm La_{2-x}Sr_{x}CuO_{4}$ [72], $\rm
La_{1.6-x}Nd_{0.4}Sr_{x}CuO_{4}$ [73, 74, 75, 53, 76, 77], $\rm
La_{2-x}Ba_{x}CuO_{4}$ [78] (Light green symbols) ,(ii) $\rm
YBa_{2}Cu_{3}O_{6+x}$ [59, 60, 68, 69](black symbols), (iii) $\rm
Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}$ [64] (red symbols). The parameter
$\rm\delta$ is measured using neutron scattering technique only, while
$\rm\epsilon$ can be determined using either by X-ray scattering or neutron
scattering technique. For $\rm Bi_{2}Sr_{2}caCu_{2}O_{8+\delta}$,
$\rm\epsilon$ is estimated from STM measurements.
## 6 Competing charge instabilities
The existence of an IUC order associated with the pseudo-gap state in Bi2212
system is not only supported by polarized neutron scattering measurements. In
Bi2212, the analysis of STM images at $\rm\Delta_{PG}$, the energy usually
associated with the pseudo-gap, points towards an IUC order in the pseudo-gap
state [62]. At $\rm\Delta_{PG}$, the unbalance of the electronic densities
measured at planar wave vector $\rm{\bf q}_{x}$=$\rm(1,0)$ and $\rm{\bf
q}_{y}$=$\rm(0,1)$ suggests that $\rm C_{4}$ rotational symmetry is broken in
the pseudo-gap state. The analysis of spectroscopic imaging STM is consistent
with an IUC electronic nematic order, yielding distinct electronic density on
O sites along $\rm{\bf a}$ and $\rm{\bf b}$ directions [62]. In principle, the
IUC electronic nematicity is different from the IUC loop order. However, the
mean-field analysis of the different IUC-ordering possibilities in the 3-band
Emery model indicates that the electronic nematic order and the CC-loop order
could actually coexist [63].
The IUC electronic nematicity inferred from spectroscopic imaging STM studies
corresponds to a long range order which coexists with a (short range) charge
density wave (CDW) order [62]. This order break the $\rm C_{4}$ rotational
symmetry and the lattice translation invariance. The CDW correlations are
characterized by the planar wave vectors $\rm{\bf
q}_{CDW}$=$\rm\pm(\epsilon,0)$ (and/or $\rm\pm(0,\epsilon)$) (Fig. 4.a).
According to STM measurements in Bi2212, the incommensurability parameter
$\rm\epsilon$ is typically of the order of 0.25 r.l.u and decreases linearly
with increasing hole doping [64] (Fig. 4.b). Furthermore, the fluctuating CDW
order develops below the onset of the pseudo-gap state and can be viewed as a
consequence of the pseudo-gap rather than its cause [65]. Likewise, a local
CDW order has been recently associated with topological defects in the
electronic nematic state [66].
In Y123, the observation of an a-b anisotropy in the Nernst coefficient [34]
close to $\rm T^{\star}$ (Fig 2) is often interpreted as an indication that
the $\rm C_{4}$ rotation symmetry is spontaneously broken within the pseudo-
gap state and associated with an electronic nematic state [67]. At lower
temperature, hard X-ray [68] and soft X-ray resonant scattering [69] studies
reported evidence for a charge modulation along wave vectors $\rm{\bf
q}_{CDW}$=$\rm\pm(\epsilon,0)$ and $\rm\pm(0,\epsilon)$, with $\epsilon\sim$
0.3 r.l.u (Fig.4.b). Considering X-ray measurements, one cannot yet
unambiguously distinguish between an equal distribution of domains of two
uniaxial CDWs with mutually perpendicular propagating wave vectors or a single
CDW with biaxial charge modulation. However recent sound velocity measurements
strongly support a single CDW with biaxial charge modulation [71], but is at
variance with $\rm C_{4}$ rotation symmetry breaking inferred from Nernst
coefficient. The intensity of the CDW signal grows on cooling down to $\rm
T_{c}$ below which it is suppressed. Under an external magnetic field, the CDW
is strengthened while superconductivity is depressed. NMR measurements [70]
further indicate that a static CDW shows up only under magnetic field, but
should be absent at zero field. As a consequence the CDW reported in X-ray
measurements should be still fluctuating. In Y123, the evidence of a CDW
instability is well documented in a narrow hole doping range around p $\sim$
1/8 ($\rm 0.09<p<0.13$) (Fig. 4.b, solid line). The CDW instability develops
at $\rm T_{CDW}$ well below the pseudo-gap temperature $\rm T^{\star}$ or the
temperature associated with the IUC magnetic order $\rm T_{mag}$ (Fig. 2).
In La214, there is no direct observation of a CDW order neither by neutron
scattering nor by X-ray scattering. At variance, once the system is co-doped
with Nd ($\rm La_{1.6}Nd_{0.04}Sr_{x}CuO_{4}$) [73, 74, 75, 76, 77] or Sr
replaced by Ba ($\rm La_{2-x}Ba_{x}CuO_{4}$) [78], a CDW order can be observed
below the LTO-LTT structural transition and followed by a SDW order. In that
specific cuprate family, $\epsilon$ is about twice $\delta$, suggesting a
strong interplay between charge and spin modulations. These combined charge
and spin modulations are usually interpreted within stripe models where doped
hole segregate into charge stripes separating hole poor AF domains ([53] and
references therein). The stripe order develops around the characteristic hole
doping $\rm p\sim 1/8$, where stripes could be pinned down onto the lattice.
In La214 (p=0.085) [18], where there is neither indication for a CDW order nor
for a SDW order, the IUC magnetic order remains quasi-2D and at short range.
The planar magnetic correlation lengths are consistent with a picture where
CC-loops could be confined within 2-leg copper ladders. In this scenario, the
CC-Loop order would be frustrated by the competing stripe instability, which
seems much stronger in that cuprate family than in any other superconducting
cuprate materials.
More theoretical and experimental work is needed to shed light on the
interplay between the pseudo-gap phenomenon, the IUC orders (i.e q=0
instabilities) and the spin and/or charge density wave instabilities. To this
respect, one can notice that for decades the role of oxygen in cuprates has
been overlooked, whereas an IUC magnetic order could be bound to CC-loops [1,
2, 3] or orbital oxygen moments [79]. In addition, the observed charge
modulations could be produced by a valence bond state or quadrupolar density
wave state [80], related to a non uniform electronic densities on oxygen
sites. Thus, the internal degrees of freedom of $\rm CuO_{2}$ plaquettes could
become relevant to shed light on the physics of cuprates.
## 7 Superconducting state
While polarized neutron scattering measurements show the existence of an IUC
magnetic order in the normal state below $\rm T^{\star}$, the evolution of
this order upon entering the superconducting state is still an open and
interesting issue. Indeed time reversal symmetry is violated in this state,
whereas it is preserved in a pure d-wave superconducting state [81]. However
the search for indications of the competition or coexistence between these
distinct electronic states using polarized neutron scattering technique has
been limited by a technical problem. To date, the elastic polarized neutron
scattering measurements have been performed using a standard longitudinal
polarization set-up [19], for which the neutron polarization is maintained on
the sample using a tiny magnetic guide field ($\sim$ 10 Gauss). Since a
depolarization of the neutron beam can take place when cooling down below $\rm
T_{c}$, most of polarized neutron scattering studies have been restricted to
the normal state. In early studies carried out in Y123 [11], a few data in the
superconducting state were reported, but no definitive conclusion can be drawn
from these data, since the issue of depolarization of the neutron beam was not
addressed. To overcome such a technical problem, measurements in the
superconducting state require to place the sample in a zero magnetic field
chamber, already available in a more sophisticate spherical polarization set-
up (such Cryo-PAD or Mu-PAD). Using this device, elastic polarized neutron
measurements will be safely extended into the superconducting state in a close
future.
## 8 Conclusion
Polarized elastic neutron scattering measurements indicate that an IUC
magnetic order develops in the pseudo-gap state. This novel magnetic order
displays the same characteristic features in monolayer Hg1201 and bilayer Y123
and Bi2212, demonstrating that this genuine phase is ubiquitous in the pseudo-
gap of high temperature copper oxide materials. The polarized neutron
scattering and the circularly polarized photoemission out over various cuprate
families all point towards a breaking of time reversal symmetry up on entering
the pseudo-gap state. The intrinsic nature of the IUC magnetic order remains
to be understood, but, to date, the few information we have concerning the
magnetic pattern within the $\rm CuO_{2}$ unit cell are consistent with the
CC-loop model (phase CC-$\rm\theta_{II}$) proposed by C. M. Varma [1, 2, 3].
Nevertheless, the stability of the CC-loop state is still an open theoretical
issue, in addition to the ability of such a q=0 instability to produce a gap
in the charge excitation spectrum. New theoretical and experimental
developments are needed to shed light on the interplay between the observed
IUC magnetic order and other electronic states, such as an IUC electronic
nematicity, incipient CDW/SDW orders and d-wave superconductivity.
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|
arxiv-papers
| 2013-06-21T13:17:17 |
2024-09-04T02:49:46.786894
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Y. Sidis and P. Bourges",
"submitter": "Yvan Sidis",
"url": "https://arxiv.org/abs/1306.5124"
}
|
1306.5202
|
# Nonparametric Bayesian grouping methods for spatial time-series data
Edward B. Baskerville Department of Ecology & Evolutionary Biology,
University of Michigan Trevor Bedford Institute of Evolutionary Biology,
University of Edinburgh Robert C. Reiner University of California, Davis;
Fogarty International Center, National Institutes of Health Mercedes Pascual
Department of Ecology & Evolutionary Biology, University of Michigan Howard
Hughes Medical Institute
(19 June 2013)
###### Abstract
We describe an approach for identifying groups of dynamically similar
locations in spatial time-series data based on a simple Markov transition
model. We give maximum-likelihood, empirical Bayes, and fully Bayesian
formulations of the model, and describe exhaustive, greedy, and MCMC-based
inference methods. The approach has been employed successfully in several
studies to reveal meaningful relationships between environmental patterns and
disease dynamics.
## 1 Introduction
Spatial time series resulting from complex dynamical processes, for example
environmentally driven disease dynamics, can present a formidable modeling
challenge. The purpose of this document is to describe a conceptually simple
approach for drawing insights from such data by identifying groups of spatial
locations with similar dynamics. Dynamics are assumed to follow a simple
nonparametric Markov transition model, thus eliminating the difficulty of
formulating a mechanistic model.
Aspects of this approach have been employed in several studies of spatial
disease dynamics. The basic Markov transition model is the core of a more
complex model that was used to identify relationships between cholera and
environmental variables in Dhaka, Bangladesh [1], but with groups imposed _a
priori_ rather than inferred. The empirical Bayes formulation described here
has been used to infer good groupings of regions for a study on malaria and
irrigation in northeast Gujarat [2], as well as for several other projects in
progress. This document does not detail any results, but instead serves as an
extended description of the methods used in those studies as well as some
extensions. The fully Bayesian approach is akin to one used for studying
trophic food webs [3].
In section 2, we describe the structure of the data. In section 3, we describe
the Markov transition model and give maximum-likelihood, empirical-Bayes, and
fully Bayesian formulations. In section 4, we describe exhaustive, greedy, and
Markov-chain Monte Carlo methods for inferring good groupings under the model.
## 2 Data representation
For these methods, we assume that the data set consists of time series for a
discrete set of spatial regions. The model is based on a discretization of the
data into a finite number of levels. How the discretization is performed
implies particular assumptions about the system. For case data of an
infectious disease, we suggest the following procedure as a sensible default:
1\. If the data set consists of absolute counts, first normalize by the
population over time within each spatial region in order to yield a prevalence
time series (cases per capita) for each region. 2\. Assign all zeros in the
data to level $\ell=0$. 3\. Choose a desired number of levels $L$, and divide
the remaining nonzero data into $L-1$ equal quantiles. For example, if there
are 10 nonzero data points and $L=3$, the smallest five data points will have
$\ell=1$, and the largest five will have $\ell=2$.
The methods that follow are based entirely on this discrete representation in
both space and time.
## 3 Nonparametric Markov transition model
The model used here arises from two simple assumptions: (1) that the behavior
of a time series can be characterized using transition probabilities between
discrete levels of disease prevalence; and (2) that groups of spatial
locations will obey the same transition probabilities.
### 3.1 Transition probabilities
With $L$ different levels, there are $L^{2}$ different possible transitions
from a state $i$ to a state $j$. Each of these transitions is assigned a
different probability in each group of locations,
$\displaystyle p_{ij}^{(g)}$ (1)
where $g$ is the group identifier. With $K$ total groups, there are $KL^{2}$
different transition probabilities. Furthermore, all the transition
probabilities from a particular level to other levels must sum to 1:
$\displaystyle\sum_{j=1}^{L}p_{ij}^{(g)}=1\,,$ (2)
for every starting level $i$ and every group $g$. The transition probabilities
for a particular group $g$ can be represented as a matrix, $\mathbf{P}_{g}$,
which has entries $p_{ij}^{(g)}$, and whose rows sum to 1.
### 3.2 Grouping of spatial locations
The $R$ different regions may be assigned to any number of groups $K\leq R$.
The assignment into groups can be represented as a vector $\mathbf{g}$:
$\displaystyle\mathbf{g}=\begin{pmatrix}g_{1}\\\ \vdots\\\ g_{R}\end{pmatrix}$
(3)
where $g_{r}$ is the group number that location $r$ is assigned to, $1\leq
g_{r}\leq K$. Because the group numbers are arbitrary, assignment vectors are
often written in a canonical form known as a restricted growth function, which
requires new groups to appear sequentially from left to right: $g_{r}\leq
g_{i}+1$ for all $i<r$. Mathematically, each arrangement into groups is known
as a _set partition_ , which is uniquely defined by a single restricted growth
function, but from here we refer to it simply as a _grouping_.
### 3.3 Model formulation
Within a single location $r$, we have a series of discrete levels over time,
$\displaystyle\boldsymbol{\ell}_{r}=\left\\{\ell_{r}^{(0)},\ldots,\ell_{r}^{(T)}\right\\}\,.$
(4)
for $T+1$ time points. If location $r$ is in group $g_{r}$, then the
probability of a particular transition in the data,
$\ell_{r}^{(t-1)}\rightarrow\ell_{r}^{(t)}$, is governed by the transition
probability for group $g_{r}$, $p_{\ell_{r}^{(t-1)}\ell_{r}^{(t)}}^{(g_{r})}$.
Therefore, the probability of the entire time series, given initial condition
$\ell_{0}$, group assignment $g_{r}$, and transition probabilities for the
group, $\mathbf{P}_{g_{r}}$, is
$\displaystyle\mathrm{Prob}\left[\boldsymbol{\ell}_{r}|\ell_{0},g_{r},\mathbf{P}_{g}\right]$
$\displaystyle=\prod_{t=1}^{T}p_{\ell_{r}^{(t-1)}\ell_{r}^{(t)}}^{(g_{r})}\,.$
(5)
This can be rewritten in terms of the number of transitions, $n_{ij}^{(r)}$,
between each pair of levels $i$ and $j$ in location $r$:
$\displaystyle\mathrm{Prob}\left[\mathbf{N}_{r}|g_{r},\mathbf{P}_{g_{r}}\right]$
$\displaystyle=\prod_{i=0}^{L-1}\prod_{i=0}^{L-1}\left[p_{ij}^{(g_{r})}\right]^{n_{ij}^{(r)}}\,,$
(6)
where $\mathbf{N}$ is a matrix of all transition counts $n_{ij}^{(r)}$.
Since locations in the same group have the same transition probabilities, we
can calculate the probability of the data associated with all locations in a
single group according to $\mathbf{N}_{g}$, a matrix of the total transition
counts for an entire group $g$:
$\displaystyle\mathrm{Prob}\left[\mathbf{N}_{g}|\mathbf{g},\mathbf{P}_{g}\right]$
$\displaystyle=\prod_{i=0}^{L-1}\prod_{i=0}^{L-1}\left[p_{ij}^{(g)}\right]^{n_{ij}^{(g)}}\,.$
(7)
Note that $\mathbf{N}_{g}$ is dependent on $\mathbf{g}$, since it includes the
counts for all locations in group $g$.
The probability of observing all time series in the data set is thus
$\displaystyle\mathrm{Prob}\left[\left\\{\boldsymbol{\ell}_{1},\ldots,\boldsymbol{\ell}_{m}\right\\}|\mathbf{g},\left\\{\mathbf{P}_{1}\ldots\mathbf{P}_{K}\right\\}\right]$
$\displaystyle=\prod_{g=1}^{K}\prod_{i=1}^{L}\prod_{i=1}^{L}\left[p_{ij}^{(g)}\right]^{n_{ij}^{(g)}}\,.$
(8)
### 3.4 Maximum-likelihood inference
The probability of the data, given model parameters, can be rewritten as a
function of the model parameters, given the data, a quantity known as the
likelihood:
$\displaystyle\mathcal{L}\left[\mathbf{g},\left\\{\mathbf{P}_{1}\ldots\mathbf{P}_{K}\right\\}|\left\\{\boldsymbol{\ell}_{1},\ldots,\boldsymbol{\ell}_{m}\right\\}\right]$
$\displaystyle=\mathrm{Prob}\left[\left\\{\boldsymbol{\ell}_{1},\ldots,\boldsymbol{\ell}_{m}\right\\}|\mathbf{g},\left\\{\mathbf{P}_{1}\ldots\mathbf{P}_{K}\right\\}\right]$
(9)
Under maximum likelihood, the best estimate of the parameters is where the
likelihood—the probability of observing the data—is maximized:
$\displaystyle\hat{\mathbf{g}},\left\\{\hat{\mathbf{P}}_{1}\ldots\hat{\mathbf{P}}_{K}\right\\}$
$\displaystyle=\mathrm{argmax}\,\mathcal{L}\left[\mathbf{g},\left\\{\mathbf{P}_{1}\ldots\mathbf{P}_{K}\right\\}|\left\\{\boldsymbol{\ell}_{1},\ldots,\boldsymbol{\ell}_{m}\right\\}\right]\,.$
(10)
In other words, under maximum likelihood, assuming a fixed number of groups
$K$ and a fixed discretization of the data, the goal is to find the
combination of grouping and transition probabilities that makes the data most
probable.
With the grouping $\mathbf{g}$ specified, the maximum-likelihood estimate of
each $p_{ij}^{(g)}$ is simply the empirical fraction of transitions for group
$g$ that starting at level $i$ that ended up at level $j$:
$\displaystyle\hat{p}_{ij}^{(g)}$
$\displaystyle=\frac{n_{ij}^{(g)}}{\sum_{j^{\prime}}n_{ij^{\prime}}^{(g)}}\,.$
(11)
The best grouping, similarly, is the one that maximizes the likelihood under
the maximum-likelihood estimates of the corresponding transition matrices.
Methods for identifying the best grouping are discussed in section 4.
### 3.5 Bayesian formulation
In a Bayesian model, the object to be inferred is a probability distribution
over model parameters, rather than a single point estimate. Parameters that
are deemed more likely, given the data, are given more weight in this
distribution, called the _posterior distribution_ because it is the
distribution of parameters given the data—that is, after the data has been
observed. This requires assuming a _prior distribution_ over model
parameters—the distribution before the data has been observed.
A fully Bayesian treatment can be built incrementally: first by introducing
priors for transition probabilities, but identifying only a single best
grouping; and then by introducing priors for the assignment into groups.
#### 3.5.1 Priors for transition probabilities
Each row $\mathbf{n}_{i}^{(g)}$ of the transition count matrix
$\mathbf{N}_{i}^{(g)}$ for group $g$ is governed by a categorical random
variable with parameters from the corresponding row of the probability matrix,
$\mathbf{p}_{i}^{(g)}$.
$\displaystyle\mathrm{Prob}\left[\mathbf{n}_{i}^{(g)}|\mathbf{g},\mathbf{p}_{i}^{(g)}\right]$
$\displaystyle=\prod_{j=0}^{L-1}\left[p_{ij}^{(g)}\right]^{n_{ij}^{(g)}}\,.$
(12)
The conjugate prior for the transition probabilities is the Dirichlet
distribution, which has density
$\displaystyle f(\mathbf{p}_{i}|\boldsymbol{\alpha}_{i})$
$\displaystyle=\frac{\Gamma\left(\sum_{j=0}^{L-1}\alpha_{ij}\right)}{\prod_{j=0}^{L-1}\Gamma(\alpha_{ij})}\prod_{j=0}^{L-1}\left[p_{ij}^{(g)}\right]^{(\alpha_{ij}-1)}$
(13)
where $\boldsymbol{\alpha}_{i}$ is a vector of “concentration parameters” that
governs the relative weight and evenness of the transition probabilities, and
$\Gamma(\centerdot)$ is the gamma function.
This prior distribution is _conjugate_ because the posterior distribution is
also Dirichlet, so that the parameters $\mathbf{p}_{i}^{(g)}$ are governed by
posterior concentration parameters $\alpha_{ij}^{(g)}$ equal to
$\displaystyle\alpha_{ij}^{(g)}$ $\displaystyle=\alpha_{ij}+n_{ij}^{(g)}\,.$
(14)
#### 3.5.2 Identification of prior concentration parameters
The concentration parameters $\boldsymbol{\alpha}_{i}$ for each starting level
$i$ must be chosen in order to fully specify the model. Prior information
about the system can be used to inform this choice, but if prior information
is absent, one natural choice is the non-informative Jeffreys prior, with all
$\alpha_{ij}=\frac{1}{2}$ [4].
Another approach is to apply the maximum-likelihood principle to the
concentration parameters, while allowing transition probabilities to vary, an
approach known as _maximum marginal likelihood_ or _empirical Bayes_ [5]. If
the grouping $\mathbf{g}$ is fixed, this requires finding values of each
$\alpha_{i}$ that maximize the probability of observing the data, integrated
over all transition probabilities:
$\displaystyle\hat{\boldsymbol{\alpha}}_{i}$
$\displaystyle=\mathrm{argmax}_{\boldsymbol{\alpha}_{i}}\prod_{g=1}^{K}\mathrm{Prob}\left[\mathbf{n}_{i}^{(g)}|\boldsymbol{\alpha}_{i}\right]\,,$
(15)
where
$\displaystyle\mathrm{Prob}\left[\mathbf{n}_{i}^{(g)}|\boldsymbol{\alpha}_{i}\right]$
$\displaystyle=\int_{\mathbf{p}_{i}^{(g)}}f(\mathbf{p}_{i}^{(g)}|\boldsymbol{\alpha}_{i})\mathrm{Prob}\left[\mathbf{n}_{i}^{(g)}|\mathbf{p}_{i}^{(g)}\right]\,d\mathbf{p}_{i}^{(g)}$
(16)
is analytically tractable for the conjugate prior, and equal to
$\displaystyle\mathrm{Prob}\left[\mathbf{n}_{i}^{(g)}|\boldsymbol{\alpha}_{i}\right]$
$\displaystyle=\frac{\Gamma\left(\sum_{j=0}^{L-1}\alpha_{ij}\right)}{\Gamma\left(\sum_{j=0}^{L-1}n_{ij}^{(g)}+\alpha_{ij}\right)}\prod_{j=0}^{L-1}\frac{\Gamma\left(n_{ij}^{(g)}+\alpha_{ij}\right)}{\Gamma\left(\alpha_{ij}\right)}\,.$
(17)
This can be simplified by requiring that all $\boldsymbol{\alpha}_{i}$ vectors
have the same components, or further that all $\alpha_{ij}$ parameters, take
on the same value.
#### 3.5.3 Identification of a single best grouping
Rather than inferring a full posterior distribution over both transition
probabilities and groupings, the empirical-Bayes approach can be used to
identify a single best grouping while allowing for a distribution over
transition probabilities. This can be done in conjunction with empirical-Bayes
identification of the concentration parameters, but for simplicity we assume
here that the concentration parameters $\boldsymbol{\alpha}_{i}$ are fixed.
The probability of observing the transition counts in all groups, given the
grouping $\mathbf{g}$, is:
$\displaystyle\mathrm{Prob}\left[\left\\{\mathbf{N}_{1}\ldots\mathbf{N}_{K}\right\\}|\mathbf{g},\left\\{\boldsymbol{\alpha}_{0}\ldots\boldsymbol{\alpha}_{L-1}\right\\}\right]$
$\displaystyle=\prod_{g=1}^{K}\prod_{i=0}^{L-1}\mathrm{Prob}\left[\mathbf{n}_{i}^{(g)}|\mathbf{g},\left\\{\boldsymbol{\alpha}_{0}\ldots\boldsymbol{\alpha}_{L-1}\right\\}\right]\,,$
(18)
where the inner probability of a single row of transitions in a single group
is as given in equation (17).
Via maximum likelihood, the best grouping is as given in equation (10).
In section 4 we describe three different methods for searching through the
space of possible groupings.
#### 3.5.4 Priors for groupings
The simplest approach in choosing a prior for groupings is to use a uniform
distribution, so that all groupings are _a priori_ equally likely. This choice
results in a posterior distribution over groupings that is simply proportional
to equation (18). However, such a prior implies a distribution that is
strongly biased toward an intermediate number of groupings.
Another approach is to use a combined Dirichlet-categorical distribution for
groupings. Under this model, each region has some probability of being a
member of each group $g$. The vector of group probabilities can in turn be
modeled as being drawn from a Dirichlet distribution with a shared
concentration parameter, since all groups are _a priori_ equivalent. As seen
above when marginalizing over transition probabilities, the Dirichlet and
categorical distributions can be analytically collapsed, yielding the
following probability mass function dependent on shared concentration
parameter $\delta$:
$\displaystyle\mathrm{Prob}\left[\mathbf{g}|\delta\right]$
$\displaystyle=\frac{K!}{(K-d)!}\frac{\Gamma\left(K\delta\right)}{\Gamma\left(K\delta+\sum_{g=1}^{K}n_{g}\right)}\prod_{g=1}^{K}\frac{\Gamma\left(n_{g}+\delta\right)}{\Gamma\left(\delta\right)}\,,$
(19)
where $d$ is the number of nonempty groups and $n_{g}$ is the number of
regions assigned to group $g$. The leading factor $\frac{K!}{(K-d)!}$ is
required because empty groups are indistinguishable.
Another possible prior is the Dirichlet process, which can be seen as a
limiting case of the Dirichlet-categorical model [6] where the number of
groups $K\rightarrow\infty$ and the concentration parameter $\delta\rightarrow
0$ such that $K\delta\rightarrow\alpha>0$. The probability mass function of
the Dirichlet process is
$\displaystyle\mathrm{Prob}\left[\mathbf{g}|\alpha\right]$
$\displaystyle=\frac{\alpha^{d}\Gamma(\alpha)\prod_{g=1}^{d}(n_{g}-1)!}{\Gamma(\alpha+R)}\,,$
(20)
where $R$ is the total number of regions being grouped, $d$ is the number of
nonempty groups, and $n_{g}$ is the number of regions in group $g$.
## 4 Inference and search methods for groupings
Depending on the nature of the data and the modeling approach, several methods
are available for searching through the space of possible groupings. If the
number of possible groupings of regions is small, then it may be possible to
exhaustively enumerate all of them. When the combinatorics become intractable,
approximation methods are required. When using maximum likelihood or empirical
Bayes for group arrangements, a greedy search algorithm should be able to
identify good groupings, but will fail to exhaustively explore the space. If a
fully Bayesian treatment is desired, a Markov-chain Monte Carlo approach is
necessary. Under maximum likelihood or empirical Bayes, stochastic
optimization algorithms can be used as a faster alternative.
Identification of the maximum-likelihood grouping (equation (10)) makes sense
as long as the number of parameters stays the same, which means that the
number of nonempty groups must remain fixed. If different numbers of groups
are being considered, then the comparison between groupings must penalize for
the change in degrees of freedom, for example using the Akaike information
criterion [7]. However, unlike the Bayesian approach, which directly
integrates over the uncertainty in parameter values, AIC penalizes all
parameters equally based on assumptions of independence and asymptotic
distributions. Therefore, the Bayesian approach is likely to be more robust.
### 4.1 Exhaustive enumeration
The number of different possible assignments of regions to groups is equal to
the Bell number $B_{R}$ for $R$ regions:
$\displaystyle B_{R}$
$\displaystyle=\sum_{r=0}^{R}\frac{1}{r!}\sum_{j=0}^{r}(-1)^{r-j}\binom{r}{j}j^{R}\,.$
(21)
If this number is small, then good group arrangements can be found by
enumerating every single arrangement, which might be reasonable for as many as
15 regions ($B_{15}\approx 1.4\times 10^{9}$), but quickly grows out of reach
($B_{30}\approx 8.5\times 10^{23}\approx 1.4\mathrm{mol}$). Restricting the
number of groups also increases the practical ceiling for exhaustive search.
For example, when splitting into only two groups, 30 regions can be split only
$2^{29}\approx 5.4\times 10^{8}$ different ways, which is quite tractable on
modern computers.
Note that exhaustive enumeration can be used not only to identify the best
grouping under maximum likelihood or empirical Bayes, but can also be used to
calculate the full posterior distribution over groupings given a prior
(section 3.5.4).
Because the likelihood and marginal likelihood calculations involve finding a
sum of transition counts over all members of the group, enumeration can be
vastly sped up by ensuring that each successive grouping is different from the
previous one by a single change in group membership. That way, the transition
counts can be updated by subtracting one integer per matrix entry from the old
group, and adding one integer per matrix entry to the new group. This
optimization can be achieved through the use of a _Gray code_ , which was
originally invented to eliminate the need for synchronization of multiple
electromechanical switches [8]. For example, a Gray code for five regions
being organized into two groups is:
00000, 00001, 00011, 00010, 00110, 00111, 00101, 00100,
01100, 01101, 01111, 01110, 01010, 01011, 01001, 01000
where the group numbers are 0 and 1. Note that the first group assignment is
always 0, so that these groupings are in restricted growth function form.
### 4.2 Greedy clustering
A simple greedy heuristic is agglomerative clustering, which starts by putting
every region in its own group. At each step, all possible combinations of two
groups are tested, and the one that gives the greatest increase in marginal
likelihood or AIC is selected. The process terminates either when all regions
are in the same group or no increase in marginal likelihood is possible. This
method is very fast, but does not search all possible clusterings and may
arrive at one that is suboptimal.
### 4.3 Markov-chain Monte Carlo using Gibbs sampling and MCMCMC
When exhaustive enumeration is computationally intractable, a stochastic
search method must be used. Markov-chain Monte Carlo (MCMC) approaches,
originally invented for sampling from distributions of the energy state of
interacting molecules [9], have the advantage of converging to a desired
target distribution, and can thus give samples from the full Bayesian
posterior over groupings and model parameters.
#### 4.3.1 Gibbs sampling of groupings
The most straightforward approach for this model is to use Gibbs sampling to
move regions from group to group. Under Gibbs sampling, a new value for a
parameter is drawn from the distribution conditional on all other parameters ;
the sequence of samples will converge to the target posterior distribution
[10]. Specifically, for this model, the algorithm repeats the following steps
until convergence:
1. 1.
Randomly sample initial group assignments.
2. 2.
Repeat for each region $r$:
1. (a)
Calculate the marginal likelihood of the model with $r$ assigned to each of
the possible groups, keeping all other regions’ group assignments the same.
2. (b)
Choose a new group for $r$ by sampling from the discrete probability
distribution weighted by the marginal likelihoods.
#### 4.3.2 MCMCMC
A technique known as Metropolis-coupled Markov chain Monte Carlo (MCMCMC, or
(MC)3)—can be used to avoid getting stuck at local maxima in the search space
[11]. Multiple chains are run at different “heats,” where hotter chains
explore configurations more freely, and colder chains are more likely to move
toward better solutions. This enables good solutions to be refined while other
alternatives may still be explored.
A chain at heat level $i$ explores the distribution
$\displaystyle f_{i}(\theta|D)$ $\displaystyle\propto
f(\theta)\left[f\left(D|\theta\right)\right]^{\tau_{i}}$ (22)
where $\theta$ represents parameters (e.g., groupings); $D$ is the data, and
$0\leq\tau_{i}\leq 1$.
Swaps are proposed between adjacent chains $i,j$, and the probability of
accepting a swap between chains is equal to
$\displaystyle\mathrm{Prob}[\theta_{i}\rightarrow\theta_{j},\theta_{j}\rightarrow\theta_{i}]$
$\displaystyle=\min\left\\{1,\frac{f_{i}(\theta_{j}|D)f_{j}(\theta_{i}|D)}{f_{i}(\theta_{i}|D)f_{j}(\theta_{j}|D)}\right\\}$
(23)
$\displaystyle=\min\left\\{1,\left[\frac{f(D|\theta_{j})}{f(D|\theta_{i})}\right]^{(\tau_{i}-\tau_{j})}\right\\}$
(24)
where $\theta_{i}$ is the initial configuration of chain $i$ and $\theta_{j}$
is the initial configuration of chain $j$. The cold chain, with $\tau_{0}=0$,
will converge to the target distribution $f(\theta|D)$.
## References
* [1] Reiner RC, King AA, Emch M, Yunus M, Faruque A, et al. (2012) Highly localized sensitivity to climate forcing drives endemic cholera in a megacity. PNAS 109: 2033–2036.
* [2] Baeza A, Bouma MJ, Dhiman RC, Baskerville EB, Ceccato P, et al. (2013) Long-lasting transition towards sustainable elimination of desert malaria under irrigation development. PNAS (in revision).
* [3] Baskerville EB, Dobson AP, Bedford T, Allesina S, Anderson TM, et al. (2011) Spatial guilds in the Serengeti food web revealed by a Bayesian group model. PLoS Computational Biology 7: e1002321.
* [4] Jeffreys H (1946) An invariant form for the prior probability in estimation problems. Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences 186: 453–461.
* [5] Robbins H (1956) An empirical Bayes approach to statistics. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability: Contributions to the Theory of Statistics. University of California Press, volume 1, p. 157.
* [6] Green PJ, Richardson S (2001) Modelling heterogeneity with and without the Dirichlet process. Scandinavian Journal of Statistics 28: 355–375.
* [7] Akaike H (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control 19: 716–723.
* [8] Gray F (1953). Pulse code communication. US Patent 2,632,058.
* [9] Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953) Equation of state calculations by fast computing machines. Journal of Chemical Physics 21: 1087.
* [10] Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence 6: 721–741.
* [11] Geyer C (1991) Markov chain Monte Carlo maximum likelihood. In: Keramidas E, editor, Proceedings of the 23rd Symposium on the Interface. Computing Science and Statistics, pp. 156–163.
|
arxiv-papers
| 2013-06-21T17:49:27 |
2024-09-04T02:49:46.799633
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Edward B. Baskerville, Trevor Bedford, Robert C. Reiner, and Mercedes\n Pascual",
"submitter": "Edward Baskerville",
"url": "https://arxiv.org/abs/1306.5202"
}
|
1306.5258
|
# Fuzzy games with a countable space of actions and applications to systems
of generalized quasi-variational inequalities
Monica Patriche University of Bucharest Faculty of Mathematics and Computer
Science 14 Academiei Street 010014 Bucharest, Romania
[email protected]
###### Abstract
In this paper, we introduce an abstract fuzzy economy (generalized fuzzy game)
model with a countable space of actions and we study the existence of the
fuzzy equilibrium. As applications, two types of results are obtained. The
first ones concern the existence of the solutions for systems of generalized
quasi-variational inequalities with random fuzzy mappings which we define. The
last ones are new random fixed point theorems for correspondences with values
in complete countable metric spaces.
###### keywords:
Abstract fuzzy economy, fuzzy equilibrium, incomplete information, random
fixed point, random quasi-variational inequalities, random fuzzy mapping.
## 1 INTRODUCTION
In equilibrium theory, there are two directions for the study of the models
with differential information. The first one is due to Harsanyi [9], whose
first approach refers to Nash type models, and the second one is due to Radner
[23], who introduced differential information into the Arrow-Debreu model [2].
Since then, an entire literature has developed on this topic.
A basic problem concerning the noncooperative games with differential
information was the equilibrium existence. The classical deterministic models
of abstract economies due to Borglin and Keiding [3], Shafer and Sonnenschein
[24], or Yannelis and Prabhakar [28] have had several generalizations. We can
refer the reader to Yannelis’s results, who introduced new information
concepts based on measurability requirements or Patriche’s models [17], [19],
[21], [22].
A new approach of an abstract economy with private information and a countable
set of actions [22] follows the ideas of Yu and Zhang in [29], who worked with
the distributions of the atomless correspondences.
This paper has two aims. Firstly, we extend the study of the model defined in
[22] in the fuzzy framework, by taking into account those situations in which
agents may have partial control over the actions they choose. Since we are
interested in a fuzzy setting, special attention is paid to the fuzzy
equilibrium existence of the fuzzy games emphasizing that this approach can be
used to treat more complicated economic systems. The uncertainties derived by
the individual character of agents in election situations can be interpreted
using fuzzy random mappings, which can be used for the purpose of modelling
and analyzing the economic systems. We continue the way opened by Kim and Lee
[11], who proved that the theory of fuzzy sets, initiated by Zadeh [31], has
became a good framework for obtaining results concerning fuzzy equlibrium
existence for abstract fuzzy economies. For other results in this domain, the
reader is referred to [4], [10], [12], [13], [25], [27].
Finally, we search for applications of our results concerning the existence of
the fuzzy equilibrium for the fuzzy games with countable action sets, in
connection with the systems of generalized quasi-equilibrium inequalities with
random fuzzy mappings. The variational inequalities were introduced in 1960s
by Fichera and Stampacchia, who studied equilibrium problems arising from
mechanics. Since then, this domain has been extensively developed and has been
found very useful in many diverse fields of pure and applied sciences, such as
mechanics, physics, optimization and control theory, operations research and
several branches of engineering sciences. Recently, the random variational
inequalities have been introduced and studied in [7], [8], [15], [26], [30].
We prove the existence of the solutions for systems of random generalized
quasi-variational inequalities with random fuzzy mappings which we define. As
consequences, we obtain new random fixed point theorems for correspondences
with values in complete countable metric spaces.
The paper is organized as follows. In the next section, some notational and
terminological conventions are given. In Section 3, the model of an abstract
fuzzy economy with private information and a countable space of actions is
introduced and the main result is stated. Section 4 contains existence
theorems for solutions of random quasi-variational inequalities with random
fuzzy mappings and random fixed point theorems. In the last section, classical
results are obtained as consequences.
## 2 NOTATION AND DEFINITION
Throughout this paper, we shall use the following notation:
$\mathbb{R}_{++}$ denotes the set of strictly positive reals. co$D$ denotes
the convex hull of the set $D$. $\overline{co}D$ denotes the closed convex
hull of the set $D$. $2^{D}$ denotes the set of all non-empty subsets of the
set $D$. If $D\subset Y$, where $Y$ is a topological space, cl$D$ denotes the
closure of $D$.
For the reader’s convenience, we review a few basic definitions and results
concerning continuity and measurability of correspondences.
Let $X$, $Y$ be topological spaces and $F:X\rightarrow 2^{Y}$ be a
correspondence. $F$ is said to be upper semicontinuous if for each $x\in X$
and each open set $V$ in $Y$ with $F(x)\subset V$, there exists an open
neighbourhood $U$ of $x$ in $X$ such that $F(y)\subset V$ for each $y\in U$.
$F$ is said to be lower semicontinuous (l.s.c) if for each $x\in X$ and each
open set $V$ in $Y$ with $F(x)\cap V\neq\emptyset$, there exists an open
neighbourhood $U$ of $x$ in $X$ such that $F(y)\cap V\neq\emptyset$ for each
$y\in U$.
Each correspondence $F:$ $X\rightarrow 2^{Y}$ has two natural inverses. The
upper inverse $F^{u}$ (also called the strong inverse) of a subset $A$ of $Y$
is defined by $F^{u}(A)=\left\\{x\in A:F(x)\subset A\right\\}.$ The lower
inverse $F^{l}$ (also called the weak inverse) of a subset $A$ of $Y$ is
defined by $F^{l}(A)=\left\\{x\in A:F(x)\cap A\not=\emptyset\right\\}.\vskip
6.0pt plus 2.0pt minus 2.0pt$
An important result concerning upper (lower) semicontinuous correspondences is
the following one.
###### Lemma 1
[30] Let $X$ and $Y$ be two topological spaces and let $A$ be a closed (resp.
open) subset of $X.$ Suppose $F_{1}:X\rightarrow 2^{Y}$, $F_{2}:X\rightarrow
2^{Y}$ are lower semicontinuous (resp. upper semicontinuous) such that
$F_{2}(x)\subset F_{1}(x)$ for all $x\in A.$ Then the correspondence
$F:X\rightarrow 2^{Y}$ defined by
$\mathit{F(x)=}\left\\{\begin{array}[]{c}F_{1}(x)\text{, \ \ \ \ \ \ \ if
}x\notin A\text{, }\\\ F_{2}(x)\text{, \ \ \ \ \ \ \ \ \ if }x\in
A\end{array}\right.$
is also lower semicontinuous (resp. upper semicontinuous).
Let $(T$, $\mathcal{T})$ be a measurable space, $Y$ a topological space and
$F:T\rightarrow 2^{Y}$ a corespondence. $F$ is weakly measurable if
$F^{l}(A)\in\mathcal{T}$ for each open subset $A$ of $Y.$ $F$ is measurable if
$F^{l}(A)\in\mathcal{T}$ for each closed subset $A$ of $Y.$ If $(T$,
$\mathcal{T})$ is a measurable space, $Y$ a countable set and $F:T\rightarrow
2^{Y}$ is a corespondence, then $F$ is measurable if for each $y\in Y,$
$F^{-1}(y)=\left\\{t\in T:y\in F(t)\right\\}$ is $\mathcal{T-}$measurable.
###### Lemma 2
[1]. For a correspondence $F:T\rightarrow 2^{Y}$ from a measurable space into
a metrizable space we have the following:
1. 1.
If $F$ is measurable, then it is also weakly measurable;
2. 2.
If $F$ is compact valued and weakly measurable, it is measurable.
The following properties are essential tools used to prove the existence of
equilibria for abstract economies in Section 3. We follow Yu and Zhang [29].
Let $Y$ be a countable complete metric space, $(T$, $\mathcal{T}$, $\lambda)$
an atomless probability space and $F:T\rightarrow 2^{Y}$ a measurable
corespondence. The function $f:T\rightarrow Y$ is said to be a selection of
$F$ if $f(t)\in F(t)$ for $\lambda-$almost $t\in T.$ Let us denote
$\mathcal{D}_{F}=\left\\{\lambda f^{-1}:f\text{ is a measurable selection of
}F\right\\}.\vskip 6.0pt plus 2.0pt minus 2.0pt$
We will present some regular properties of $\mathcal{D}_{F},$ also obtained by
Yu and Zhang [29].
The next lemma states the convexity of $\mathcal{D}_{F}$ for any
correspondence $F$.
###### Lemma 3
[29]. Let $Y$ be a countable complete metric space, $(T$, $\mathcal{T}$,
$\lambda)$ an atomless probability space and $F:T\rightarrow 2^{Y}$ a
measurable corespondence. Then, $\mathcal{D}_{F}$ is non-empty and convex in
the space $\mathcal{M}(Y)$ \- the space of probability measure on $Y$,
equipped with the topology of weak convergence.
The compactness of $\mathcal{D}_{F}$ is stated in the next result.
###### Lemma 4
[29]. Let $Y$ be a countable complete metric space, and $(T$, $\mathcal{T}$,
$\lambda)$ be an atomless probability space and $F:T\rightarrow 2^{Y}$ be a
measurable corespondence. If $F$ is compact valued, then, $\mathcal{D}_{F}$ is
compact in $\mathcal{M}(Y).\vskip 6.0pt plus 2.0pt minus 2.0pt$
Lemma 5 concerns the upper semicontinuity property of $\mathcal{D}_{F_{x}}.$
###### Lemma 5
[29]. Let $X$ be a metric space, $(T$, $\mathcal{T}$, $\lambda)$ an atomless
probability space, $Y$ a countable complete metric space and $F:T\times
X\rightarrow 2^{Y}$ a correspondence. Let us assume that, for any fixed $x$ in
$X$, $F(\cdot,x)$ (also denoted by $F_{x})$ is a compact-valued measurable
correspondence, and for each fixed $t\in T,$ $F(t,\cdot)$ is upper
semicontinuous on $X$. Also, let us assume that there exists a compact valued
corespondence $H:T\times X\rightarrow 2^{Y}$ such that $F(t,x)\subset H(t)$
for all $t$ and $x$. Then $\mathcal{D}_{F_{x}}$ is upper semicontinuous on
$X.$
We will also need Kuratowski-Ryll-Nardzewski Selection Theorem in order to
prove our main results in the next section.
###### Theorem 6
(Kuratowski-Ryll-Nardzewski Selection Theorem) [1]. A weakly measurable
correspondence with non-empty closed values from a measurable space into a
Polish space admits a measurable selector.
The fuzzy mappings are presented below.
Let $\tciFourier(Y)$ be a collection of all fuzzy sets over $Y.$ A mapping
$F:\Omega\rightarrow\tciFourier(Y)$ is called a fuzzy mapping. If $F$ is a
fuzzy mapping from $\Omega,$ $F(\omega)$ is a fuzzy set on $Y$ and
$F(\omega)(y)$ is the membership function of $y$ in $F(\omega).$
Let $A\in\tciFourier(Y),$ $a\in[0,1],$ then the set $(A)_{a}=\\{y\in
Y:A(y)\geq a\\}$ is called an $a-$cut set of the fuzzy set $A.$
A fuzzy mapping $F:\Omega\rightarrow\tciFourier(Y)$ is said to be measurable
if, for any given $a\in[0,1],$ $(F(\cdot))_{a}:\Omega\rightarrow 2^{Y}$ is a
measurable correspondence. A fuzzy mapping $F:\Omega\rightarrow\tciFourier(Y)$
is said to have a measurable graph if, for any given $a\in[0,1],$ the
correspondence $(F(\cdot))_{a}:\Omega\rightarrow 2^{Y}$ has a measurable
graph. A fuzzy mapping $F:\Omega\times X\rightarrow\tciFourier(Y)$ is called a
random fuzzy mapping if, for any given $x\in X,$
$F(\cdot,x):\Omega\rightarrow\tciFourier(Y)$ is a measurable fuzzy mapping.
## 3 FUZZY EQUILIBRIUM EXISTENCE FOR ABSTRACT FUZZY ECONOMIES WITH PRIVATE
INFORMATION AND A COUNTABLE SET OF ACTIONS
### 3.1 THE MODEL OF AN ABSTRACT FUZZY ECONOMY WITH PRIVATE INFORMATION
In this section we define a model of abstract fuzzy economy with private
information and a countable set of actions. This model is a generalization of
the one defined in [22]
Let $I$ be a non-empty finite set (the set of agents). For each $i\in I$, the
space of actions, $S_{i}$ is a countable complete metric space and
$(\Omega_{i},\mathcal{Z}_{i})$ is a measurable space. Let
$(\Omega,\mathcal{F})$ be the product measurable space $\underset{i\in
I}{(\prod}\Omega_{i},\underset{i\in
I}{\mathop{\textstyle\bigotimes}}\mathcal{Z}_{i})$, and let $\mu$ be a
probability measure on $(\Omega,\mathcal{F}).$ For a point
$\omega=(\omega_{1},...,\omega_{n})\in\Omega,$ define the coordinate
projections $\tau_{i}(\omega)=\omega_{i}.$ The random mapping
$\tau_{i}(\omega)$ is interpreted as player i’s private information related to
his action.
We also denote, for each $i\in I,$ Meas($\Omega_{i},S_{i})$ the set of
measurable fuzzy mappings $f$ from $(\Omega_{i},\mathcal{Z}_{i})$ to $S_{i}.$
An element $g_{i}$ of Meas($\Omega_{i},S_{i})$ is called a pure strategy for
player $i.$ A pure strategy profile $g$ is an n-vector function
$(g_{1},g_{2},...,g_{n})$ that specifies a pure strategy for each player.
We suppose that there exists a fuzzy mapping
$X_{i}:\Omega_{i}\rightarrow\mathcal{F(}S_{i})$ such that each agent $i$ can
choose an action from $(X_{i}(\omega_{i}))_{z_{i}}\subset S_{i}$ for each
$\omega_{i}\in\Omega_{i}.$
Let $D_{(X_{i}(\cdot))_{z_{i}}}$ be the set
$\left\\{(\mu\tau_{i}^{-1})g_{i}^{-1}:g_{i}\text{ is a measurable selection of
}(X_{i}(\cdot))_{z_{i}}\right\\}$ and
$\mathcal{D}_{(X(\cdot))_{z}}:=\underset{i\in
I}{\prod}\mathcal{D}_{(X_{i}(\cdot))_{z_{i}}}.$
For each $i\in I,$ let us denote $h_{g_{i}}=(\mu\tau_{i}^{-1})g_{i}^{-1},$
where $g_{i}$ is a measurable selection of $(X_{i}(\cdot))_{z_{i}}$ and
$h_{g}=(h_{g_{1}},h_{g_{2}},...,h_{g_{n}}).$
###### Definition 1
An abstract fuzzy economy (or a generalized fuzzy game) with private
information and a countable space of actions is defined as
$\Gamma=(I,((\Omega_{i},\mathcal{Z}_{i}),(S_{i},A_{i},P_{i},a_{i},b_{i},z_{i})))_{i\in
I},\mu)),$ where:
(a) $X_{i}:\Omega_{i}\rightarrow\mathcal{F}(S_{i})$ is the action (strategy)
fuzzy mapping of agent $i$,
(b) for each $\omega_{i}\in\Omega_{i},$
$A_{i}(\omega_{i},$·$):\mathcal{D}_{(X(\cdot))_{z}}\rightarrow\mathcal{F}(S_{i})$
is the random fuzzy constraint mapping of agent $i;$
(c) for each $\omega_{i}\in\Omega_{i},$
$P_{i}(\omega_{i},$·$):\mathcal{D}_{(X(\cdot))_{z}}\rightarrow\mathcal{F}(S_{i})$
is the random fuzzy preference mapping of agent $i;$
(d) $a_{i}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow(0,1]$ is a random fuzzy
constraint function and $p_{i}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow(0,1]$
is a random fuzzy preference function;
(e) $z_{i}\in(0,1]$ is such that for all
$(\omega_{i},h_{g})\in\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}},$
$(A_{i}(\omega_{i},h_{g}))_{a_{i}(h_{g})}\subset(X_{i}(\omega_{i}))_{z_{i}}$
and $(P_{i}(\omega,h_{g}))_{p_{i}(h_{g})}\subset(X_{i}(\omega))_{z_{i}}.\vskip
6.0pt plus 2.0pt minus 2.0pt$
###### Definition 2
A fuzzy equilibrium for $\Gamma$ is defined as a strategy profile
$g^{\ast}=(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})\in\mathop{\textstyle\prod}\limits_{i\in
I}$Meas$(\Omega_{i},S_{i})$ such that for each $i\in I:$
1)
$g_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},h_{g^{\ast}})_{a_{i}(h_{g^{\ast}})}$
for each $\omega_{i}\in\Omega_{i};$
2)
$(A_{i}(\omega_{i},h_{g^{\ast}}))_{a_{i}(h_{g^{\ast}})}\cap(P_{i}(\omega_{i},h_{g^{\ast}}))_{p_{i}(h_{g^{\ast}})}=\phi$
for each $\omega_{i}\in\Omega_{i}.$
### 3.2 EXISTENCE OF THE FUZZY EQUILIBRIUM FOR ABSTRACT FUZZY ECONOMIES WITH
A COUNTABLE SET OF ACTIONS
In this subsection we prove the existence of fuzzy equilibrium of abstract
fuzzy economies.
Theorem 7 is our main result.
Let us denote
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})\in\mathcal{D}_{(X(\cdot))_{z}}.$
###### Theorem 7
Let
$\Gamma=(I,((\Omega_{i},\mathcal{Z}_{i}),(S_{i},A_{i},P_{i},a_{i},b_{i},z_{i}))_{i\in
I},\mu)$ be an abstract economy with private information and a countable space
of actions, where $I$ is a finite index set such that for each $i\in I:$
a) $S_{i}$ is a countable complete metric space and
$(\Omega_{i},\mathcal{Z}_{i})$ is a measurable space; $(\Omega,\mathcal{Z})$
is the product measurable space $\underset{i\in
I}{(\prod}(\Omega_{i},\mathcal{Z}_{i}))$ and $\mu$ an atomless probability
measure on $(\Omega,Z);$
b) the fuzzy mapping $X_{i}:\Omega_{i}\rightarrow\mathcal{F}(S_{i})$ is
measurable and $(X_{i}(\cdot))_{z_{i}}:\Omega_{i}\rightarrow 2^{S_{i}}$ has
compact values;
c) the fuzzy mapping $A_{i}$ is such that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$
$\mathit{(}A_{i}(\cdot,\lambda)_{a_{i}(\lambda)}:\Omega_{i}\rightarrow
2^{S_{i}}$ is measurable and, for all $\omega_{i}\in\Omega_{i},$
$(A_{i}(\omega_{i},\cdot))_{a_{i}(\cdot)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S_{i}}$ is upper semicontinuous with non-empty compact values;
d) the fuzzy mapping $P_{i}$ is such that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$
$\mathit{(}P_{i}(\cdot,\lambda)_{p_{i}(\lambda)}:\Omega_{i}\rightarrow
2^{S_{i}}$ is measurable and, for all $\omega_{i}\in\Omega_{i},$
$(P_{i}(\omega_{i},\cdot))_{p_{i}(\cdot)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S_{i}}$ is upper semicontinuous with non-empty compact values;
e) for each $\omega_{i}\in\Omega_{i}$ and each
$g\in\mathop{\textstyle\prod}\limits_{i\in I}$Meas$(\Omega_{i},S_{i}),$
$g_{i}(\omega_{i})\not\in(P_{i}(\omega_{i},h_{g})_{p_{i}((h_{g})};$
f) the set
$U_{i}^{\omega_{i}}:=\left\\{\lambda\in\mathcal{D}_{(X(\cdot))_{z}}:(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda)}\cap
P_{i}((\omega_{i},\lambda))_{p_{i}(\lambda)}=\emptyset\right\\}$ is open in
$\mathcal{D}_{(X(\cdot))_{z}}$ in for each $\omega_{i}\in\Omega_{i}$.
Then, there exists $g^{\ast}\in\mathop{\textstyle\prod}\limits_{i\in
I}$Meas$(\Omega_{i},S_{i})$ an equilibrium for $\Gamma.$
Proof. According to Lemma 3, $D_{(X_{i}(\cdot))_{z_{i}}}$ is non-empty and
convex. According to Lemma 4, $D_{(X_{i}(\cdot))_{z_{i}}}$ is compact. For
each $i\in I$ let us denote
$U_{i}:=\left\\{(\omega_{i},\lambda)\in\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}:(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda)}\cap
P_{i}((\omega_{i},\lambda))_{p_{i}(\lambda)}=\emptyset\right\\}$ and
$U_{i}^{\omega_{i}}:=\left\\{(\omega_{i},\lambda)\in\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}:(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda)}\cap
P_{i}((\omega_{i},\lambda))_{p_{i}(\lambda)}=\emptyset\right\\}.$
According to the assumption f), the set $U_{i}^{\omega_{i}}$ is open in
$\mathcal{D}_{(X(\cdot))_{z}}.$
Let us define $F_{i}:\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S_{i}}$ by
$F_{i}(\omega_{i},\lambda)=\left\\{\begin{array}[]{c}(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda)}\cap(P_{i}(\omega_{i},\lambda))_{p_{i}(\lambda)}\text{
if }(\omega_{i},\lambda)\not\in U_{i},\\\
(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda)}\text{ if }(\omega_{i},\lambda)\in
U_{i}.\end{array}\right.$
Then, the correspondence $F_{i}$ has non-empty compact values and is
measurable with respect to $\Omega_{i}$ and upper semicontinuous with respect
to $\lambda\in\mathcal{D}_{(X(\cdot))_{z}}.$
We denote $\mathcal{D}_{F_{i}}(\lambda)=$
=$\\{h_{g_{i}}=(\mu\tau_{i}^{-1})g_{i}^{-1}:g_{i}$ is a measurable selection
of $F_{i}(\cdot,\lambda)\\}.$ Then:
i) $\mathcal{D}_{F_{i}}(\lambda)$ is non-empty because there exists a
measurable selection from the correspondence $F_{i}$ according to Kuratowski-
Ryll-Nardewski Selection Theorem.
ii) $\ \mathcal{D}_{F_{i}}(\lambda)$ is convex and compact according to Lemma
3 and Lemma 4.
We define $\Phi:$ $\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{\mathcal{D}_{(X(\cdot))_{z}}},$
$\Phi(\lambda)=\mathop{\textstyle\prod}\limits_{i\in
I}\mathcal{D}_{F_{i}}(\lambda).$
The set $\mathcal{D}_{(X(\cdot))_{z}}$ is non-empty compact and convex.
According to Lemma 5, the correspondence $\mathcal{D}_{F_{i}}$ is upper
semicontinuous. Then, the correspondence $\Phi$ is upper semicontinuous and
has non-empty compact and convex values. According to Ky Fan fixed point
Theorem [6], there exists a fixed point
$\lambda^{\ast}\in\Phi(\lambda^{\ast}).$ In particular, for each player $i,$
$\lambda_{i}^{\ast}\in\mathcal{D}_{F_{i}}(\lambda^{\ast}).$ Therefore, for
each player $i,$ there exists $g_{i}^{\ast}\in$Meas$(\Omega_{i},S_{i})$ such
that $g_{i}^{\ast}$ is a selection of $F_{i}(\cdot,\lambda^{\ast})$ and
$h_{g_{i}^{\ast}}=(\mu\tau_{i}^{-1})(g_{i}^{\ast})^{-1}=\lambda_{i}^{\ast}.$
Let us denote $h_{g^{\ast}}=(h_{g_{1}^{\ast}},...,h_{g_{n}^{\ast}}).$
We prove that $g^{\ast}$ is an equilibrium for $\Gamma.$ For each $i\in I,$
because $g_{i}^{\ast}$ is a selection of
$F_{i}(\cdot,h_{g_{1}^{\ast}},...,h_{g_{n}^{\ast}})$, it follows that
$g_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},h_{g^{\ast}}))_{a_{i}(h_{g^{\ast}})}\cap(P_{i}(\omega_{i},h_{g^{\ast}}))_{p_{i}(h_{g^{\ast}})}$
if $(\omega_{i},h_{g^{\ast}})\not\in U_{i}$ or
$g_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},h_{g^{\ast}}))_{a_{i}(h_{g^{\ast}})}$
if $(\omega_{i},h_{g^{\ast}})\in U_{i}.$
According to the assumption d), it follows that
$g_{i}^{\ast}(\omega_{i})\not\in(P_{i}(\omega_{i},h_{g^{\ast}}))_{p_{i}(h_{g^{\ast}})}$
for each $\omega_{i}\in\Omega_{i}.$ Then
$g_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},h_{g^{\ast}}))_{a_{i}(h_{g^{\ast}})}$
and $(\omega_{i},h_{g^{\ast}})\in U_{i}.$ This is equivalent with the fact
that
$g_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},h_{g^{\ast}})_{a_{i}(h_{g^{\ast}})}$
and $(A_{i}(\omega_{i},h_{g^{\ast}})_{a_{i}(h_{g^{\ast}})}\cap
P_{i}(\omega_{i},h_{g^{\ast}})_{p_{i}(h_{g^{\ast}})}=\emptyset$ for each
$\omega_{i}\in\Omega_{i}.$ Consequently,
$g^{\ast}=(g_{1}^{\ast},g_{2}^{\ast},...,g_{n}^{\ast})$ is an equilibrium for
$\Gamma.\vskip 6.0pt plus 2.0pt minus 2.0pt$
## 4 RANDOM QUASI-VARIATIONAL INEQUALITIES WITH RANDOM FUZZY MAPPINGS
### 4.1 New types of systems of generalized quasi-variational inequalities
Noor and Elsanousi [15] introduced the notion of a random variational
inequality. We also propose the next systems of generalized quasi-variational
inequalities.
Let $I$ be a non-empty and finite set. For each $i\in I$, let $S_{i}$ be a
countable complete metric space and $(\Omega_{i},\mathcal{Z}_{i})$ a
measurable space. Let $(\Omega,\mathcal{F})$ be the product measurable space
$\underset{i\in I}{(\prod}\Omega_{i},\underset{i\in
I}{\mathop{\textstyle\bigotimes}}\mathcal{Z}_{i})$ and $\mu$ a probability
measure on $(\Omega,\mathcal{F}).$ For a point
$\omega=(\omega_{1},...,\omega_{n})\in\Omega,$ define the coordinate
projections $\tau_{i}(\omega)=\omega_{i}.$We also denote, for each $i\in I,$
Meas($\Omega_{i},S_{i})$ the set of measurable fuzzy mappings $f$ from
$(\Omega_{i},\mathcal{Z}_{i})$ to $S_{i}.$ We suppose that there exists a
correspondence $X_{i}:\Omega_{i}\rightarrow S_{i}$ and let
$\mathcal{D}_{X_{i}}$ be the set {$(\mu\tau_{i}^{-1})g_{i}^{-1}:g_{i}$ is a
measurable selection of $X_{i}\\}$ and $\mathcal{D}_{X}:=\underset{i\in
I}{\prod}\mathcal{D}_{X_{i}}.$
Let $A_{i}:\Omega_{i}\times\mathcal{D}_{X}\rightarrow 2^{S_{i}}$ be a
correspondence and $\psi_{i}:\Omega_{i}\times\mathcal{D}_{X}\times
S_{i}\rightarrow\mathbb{R}\cup\\{-\infty,+\infty\\}$.
We associate the next generalized quasi-variational problem with $A_{i}$ and
$\psi_{i}$:
(1) Find $\lambda^{\ast}\in\mathcal{D}_{X}$ such that:
i) $\lambda_{i}^{\ast}(\omega_{i})\in A_{i}(\omega_{i},\lambda^{\ast});$
ii) $\sup_{y_{i}\in
A_{i}(\omega_{i},\lambda^{\ast})}\psi_{i}(\omega_{i},\lambda^{\ast},y_{i})\leq
0$ for all $\omega_{i}\in\Omega_{i}.$
If $S_{i}$ is a countable completely metrizable topological vector space, we
introduce the following definition.
Let $S_{i}^{\prime}$ be the dual space of $S_{i}$ and $G_{i}:\Omega_{i}\times
S_{i}\rightarrow 2^{S_{i}^{\prime}},$
$A_{i}:\Omega_{i}\times\mathcal{D}_{X}\rightarrow 2^{S_{i}}$ be
correspondences.
(2) Find $\lambda^{\ast}\in\mathcal{D}_{X}$ such that:
i) $\lambda^{\ast}(\omega_{i})\in A_{i}(\omega_{i},\lambda^{\ast})$
ii) $\sup_{y_{i}\in A_{i}(\omega_{i},\lambda^{\ast})}\sup_{v\in
G_{i}(\omega_{i},y_{i})}\mathop{\mathrm{R}e}\langle
v,\lambda_{i}^{\ast}(\omega_{i})-y_{i}\rangle\leq 0$ for all
$\omega_{i}\in\Omega_{i},$
where the real part of pairing between $S_{i}^{\prime}$ and $S_{i}$ is denoted
by $\mathop{\mathrm{R}e}\langle v,x\rangle$ for each $v\in S_{i}^{\prime}$ and
$x\in S_{i}.$
We will work on the next fuzzy model:
For each $i\in I$, let $S_{i}$ be a countable complete metric space and
$(\Omega_{i},\mathcal{Z}_{i})$ is a measurable space. Let
$A_{i}:\Omega_{i}\times\mathcal{D}_{X}\rightarrow\mathcal{F}(S_{i})$ be a
fuzzy mapping and let $a_{i}:\mathcal{D}_{X}\rightarrow(0,1]$ be a fuzzy
function. Let $\psi_{i}:\Omega_{i}\times\mathcal{D}_{X}\times
S_{i}\rightarrow\mathbb{R}\cup\\{-\infty,+\infty\\}$.
Now, we are introducing the next type of variational inequality:
(3) Find $\lambda^{\ast}\in\mathcal{D}_{X}$ such that for every $i\in I$,
i)
$\lambda_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})};$
ii)
$\sup_{y_{i}\in(A_{i}((\omega_{i},\lambda^{\ast})))_{a_{i}(\lambda^{\ast})}}\psi_{i}(\omega_{i},\lambda^{\ast},y_{i})\leq
0$ for all $\omega_{i}\in\Omega_{i}.$
where $(A_{i_{x^{\ast}}})_{a_{i}(x^{\ast})}=\\{z\in
Y_{i}:A_{i_{x^{\ast}}}(z)\geq a_{i}(x^{\ast})\\}.$
If $A_{i}:X\rightarrow 2^{S_{i}}$ is a classical correspondence, then we get
the variational inequality defined in (1).
Finally, we introduce the following system of generalized quasi-variational
inequalities, in case that, for each $i\in I,$ $S_{i}$ is a countable
completely metrizable topological vector space, $(\Omega_{i},\mathcal{Z}_{i})$
is measurable space,
$A_{i}:\Omega_{i}\times\mathcal{D}_{X}\rightarrow\mathcal{F}(S_{i})$ and
$G_{i}:\Omega_{i}\times S_{i}\rightarrow\mathcal{F}(S_{i}^{\prime})$ are fuzzy
mappings and $a_{i}:\mathcal{D}_{X}\rightarrow(0,1]$,
$g_{i}:\mathcal{S}_{i}\rightarrow(0,1]$ are fuzzy functions.
(4) Find $\lambda^{\ast}\in\mathcal{D}_{X}$ such that for every $i\in I$,
$\left\\{\begin{array}[]{c}\lambda_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})};\\\
\sup_{y_{i}\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})}}\sup_{v\in(G_{i}(\omega_{i},y_{i}))_{g_{i}(y_{i})}}\mathop{\mathrm{R}e}\langle
v,\lambda_{i}^{\ast}(\omega_{i})-y_{i}\rangle\leq 0\text{ }\end{array}\right.$
for all $\omega_{i}\in\Omega_{i}.$
### 4.2 The existence of the solutions of the systems of generalized quasi-
variational inequalities with random fuzzy mappings
In this section, we are establishing new results concerning the existence of
the systems of generalized random quasi-variational inequalities with random
fuzzy mappings and we also are stating random fixed point theorems. The proofs
rely on the theorem of fuzzy equilibrium existence for the abstract fuzzy
economy.
This is our first theorem.
###### Theorem 8
Let $I$ be a non-empty and finite set. For each $i\in I$, $S_{i}$ is a
countable complete metric space and $(\Omega_{i},\mathcal{Z}_{i})$ is a
measurable space. Let $(\Omega,\mathcal{Z})$ be the product measurable space
$\underset{i\in I}{(\prod}\Omega_{i},\underset{i\in
I}{\mathop{\textstyle\bigotimes}}\mathcal{Z}_{i})$, and let $\mu$ be a
probability measure on $(\Omega,\mathcal{Z}).$ Suppose that the following
conditions are satisfied for each $i\in I:$
a) the correspondence $X_{i}:\Omega_{i}\rightarrow\mathcal{F}(S_{i})$ is
measurable such that $(X_{i}(\cdot))_{z_{i}}:\Omega_{i}\rightarrow 2^{S_{i}}$
has compact values;
b) the fuzzy mapping $A_{i}$ is such that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$
$\mathit{(}A_{i}(\cdot,\lambda)_{a_{i}(\lambda)}:\Omega_{i}\rightarrow
2^{S_{i}}$ is measurable and, for all $\omega_{i}\in\Omega_{i},$
$(A_{i}(\omega_{i},\cdot))_{a_{i}(\cdot)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S_{i}}$ is upper semicontinuous with non-empty compact values;
Let us assume that the mapping
$\psi_{i}:\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}\times
S_{i}\rightarrow\mathbb{R}\cup\\{-\infty,+\infty\\}$ is such that:
(c) $\lambda\rightarrow\\{y\in
S_{i}:\psi_{i}(\omega,\lambda,y)>0\\}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S_{i}}$ is upper semicontinuous with compact values on
$\mathcal{D}_{(X(\cdot))_{z}}$ for each fixed $\omega_{i}\in\Omega_{i};$
(d) $\lambda_{i}(\omega_{i})\notin\\{y\in
S_{i}:\psi_{i}(\omega_{i},\lambda,y)>0\\}$ for each fixed
$(\omega_{i},\lambda)\in\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}};$
(e) for each $\omega_{i}\in\Omega_{i},$
$\\{\lambda\in\mathcal{D}_{(X(\cdot))_{z}}:\alpha_{i}(\omega_{i},\lambda)>0\\}$
is weakly open in $\mathcal{D}_{(X(\cdot))_{z}},$ where
$\alpha_{i}:\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}\rightarrow\mathbb{R}$
is defined by
$\alpha_{i}(\omega_{i},\lambda)=\sup_{y\in(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda))}}\psi_{i}(\omega_{i},\lambda,y)$
for each
$(\omega_{i},\lambda)\in\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}};$
(f) $\\{\omega_{i}:\alpha_{i}(\omega_{i},\lambda)>0\\}\in\mathcal{Z}_{i}$ for
each $\lambda\in\mathcal{D}_{(X(\cdot))_{z}}$.
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such that
for every $i\in I$,
i)
$\lambda_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})};$
ii)
sup${}_{y\in(A_{i}(\omega,\lambda^{\ast})_{a_{i}(\lambda^{\ast})}}\psi_{i}(\omega_{i},\lambda^{\ast},y)\leq
0$.
Proof. For every $i\in I,$ let
$P_{i}:\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}\rightarrow\mathcal{F}(S_{i})$
and $p_{i}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow(0,1]$ such that
$(P_{i}(\omega,\lambda))_{p_{i}(\lambda)}=\\{y\in
S_{i}:\psi_{i}(\omega_{i},\lambda,y)>0\\}$ for each
$(\omega_{i},\lambda)\in\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}.$
We shall show that the abstract economy
$\Gamma=(I,((\Omega_{i},\mathcal{F}_{i}),(S_{i},A_{i},P_{i},a_{i},b_{i},$
$z_{i})))_{i\in I},\mu))$ satisfies all the hypotheses of Theorem 7.
Suppose $\omega_{i}\in\Omega_{i}.$
According to c), we have that
$\lambda\rightarrow(P_{i}(\omega_{i},\lambda))_{p_{i}((\lambda)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S_{i}}$ is upper semicontinuous with non-empty values and according to d),
$\lambda_{i}(\omega_{i})\not\in(P_{i}(\omega_{i},\lambda))_{p_{i}((\lambda)}$
for each $\lambda\in\mathcal{D}_{(X(\cdot))_{z}}.$
According to the definition of $\alpha_{i},$ we note that, for each
$\omega_{i}\in\Omega_{i},$ $\
\\{\lambda\in\mathcal{D}_{(X(\cdot))_{z}}:(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda))}\cap(P_{i}(\omega_{i},\lambda))_{p_{i}(\lambda)}\neq\emptyset\\}=\\{\lambda\in\mathcal{D}_{(X(\cdot))_{z}}:\alpha_{i}(\omega,\lambda)>0\\}$
so that
$\\{\lambda\in\mathcal{D}_{(X(\cdot))_{z}}:(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda)}\cap(P_{i}(\omega_{i},\lambda))_{p_{i}(\lambda)}\neq\emptyset\\}$
is weakly open in $\mathcal{D}_{(X(\cdot))_{z}}$ by e).
According to b) and f), it follows that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$ the correspondences
$(A_{i}(\cdot,\lambda))_{a_{i}(\lambda)}:\Omega_{i}\rightarrow 2^{S_{i}}$ and
$(P_{i}(\omega_{i},\lambda))_{p_{i}(\lambda)}:\Omega_{i}\rightarrow 2^{S_{i}}$
are measurable.
Thus, the abstract fuzzy economy
$\Gamma=(I,((\Omega_{i},\mathcal{Z}_{i}),(S_{i},A_{i},P_{i},a_{i},b_{i},z_{i})))_{i\in
I},$
$\mu))$ satisfies all the hypotheses of Theorem 7. Therefore, there exists
$\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such that for every $i\in I:$
$\lambda_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})}$
and
$(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})}\cap(P_{i}(\omega_{i},\lambda^{\ast}))_{p_{i}(\lambda^{\ast})}=\phi$;
that is, there exists $\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such
that for every $i\in I:$
i)
$\lambda_{i}^{\ast}(\omega)\in(A_{i}(\omega,\lambda^{\ast}))_{a_{i}\lambda^{\ast})};$
ii)
sup${}_{y\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})}}\psi_{i}(\omega_{i},\lambda^{\ast},y)\leq
0$.
If $|$I$|$=1, we obtain the following corollary.
###### Corollary 9
Let $S$ be a countable complete metric space,
$(\Omega,\mathcal{Z},\mathcal{\mu})$ be a measure space. Suppose that the
following conditions are satisfied$:$
a) the fuzzy mapping $X:\Omega\rightarrow\mathcal{F}(S)$ is measurable such
that $(X(\cdot))_{z}:\Omega_{i}\rightarrow 2^{S}$ has compact values;
b) the fuzzy mapping $A$ is such that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$
$\mathit{(}A(\cdot,\lambda))_{a(\lambda)}:\Omega\rightarrow 2^{S}$ is
measurable and, for all $\omega\in\Omega,$
$(A(\omega,\cdot))_{a(\cdot)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow 2^{S}$
is upper semicontinuous with non-empty compact values;
The mapping $\psi:\Omega\times\mathcal{D}_{(X(\cdot))_{z}}\times
S\rightarrow\mathbb{R}\cup\\{-\infty,+\infty\\}$ is such that:
(c) $\lambda\rightarrow\\{y\in
Y:\psi(\omega,\lambda,y)>0\\}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow 2^{S}$
is upper semicontinuous with compact values on $\mathcal{D}_{(X(\cdot))_{z}}$
for each fixed $\omega\in\Omega;$
(d) $\lambda(\omega)\notin\\{y\in S:\psi(\omega,\lambda,y)>0\\}$ for each
fixed $(\omega,\lambda)\in\Omega\times\mathcal{D}_{(X(\cdot))_{z}};$
(e) for each $\omega\in\Omega,$
$\\{\lambda\in\mathcal{D}_{(X(\cdot))_{z}}:\alpha(\omega,\lambda)>0\\}$ is
weakly open in $\mathcal{D}_{(X(\cdot))_{z}},$ where
$\alpha:Z\times\mathcal{D}_{(X(\cdot))_{z}}\rightarrow\mathbb{R}$ is defined
by
$\alpha(\omega,\lambda)=\sup_{y\in(A(\omega,\lambda))_{a(\lambda))}}\psi(\omega,\lambda),y)$
for each $(\omega,\lambda)\in Z\times\mathcal{D}_{(X(\cdot))_{z}};$
(f) $\\{\omega:\alpha(\omega,\lambda)>0\\}\in\mathcal{Z}$ for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}}.$
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such that,
i)
$\lambda^{\ast}(\omega)\in(A(\omega,\lambda^{\ast}))_{a_{i}(\lambda^{\ast})};$
ii)
sup${}_{y\in(A(\omega,\lambda^{\ast})_{a(\lambda^{\ast})}}\psi(\omega,\lambda^{\ast},y)\leq
0$.
As a consequence of Theorem 8, we prove the following Tan and Yuan’s type
(1995) random quasi-variational inequality with random fuzzy mappings.
###### Theorem 10
Let $I$ be a non-empty and finite set. For each $i\in I$, $S_{i}$ is a
countable completely metrizable topological vector space and
$(\Omega_{i},\mathcal{Z}_{i})$ is a measurable space. Let
$(\Omega,\mathcal{Z})$ be the product measurable space $\underset{i\in
I}{(\prod}\Omega_{i},\underset{i\in
I}{\mathop{\textstyle\bigotimes}}\mathcal{Z}_{i})$, and $\mu$ a probability
measure on $(\Omega,\mathcal{Z}).$ Suppose that the following conditions are
satisfied for each $i\in I:$
a) the correspondence $X_{i}:\Omega_{i}\rightarrow\mathcal{F}(S_{i})$ is
measurable such that $(X_{i}(\cdot))_{z_{i}}:\Omega_{i}\rightarrow 2^{S_{i}}$
has compact values;
b) the mapping $A_{i}$ is such that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$
$\mathit{(}A_{i}(\cdot,\lambda)_{a_{i}(\lambda)}:\Omega_{i}\rightarrow
2^{S_{i}}$ is measurable and, for all $\omega_{i}\in\Omega_{i},$
$(A_{i}(\omega_{i},\cdot))_{a_{i}(\cdot)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S_{i}}$ is upper semicontinuous with non-empty compact values;
$G_{i}:\Omega_{i}\times S\rightarrow\mathcal{F}(S^{\prime})$ and
$g_{i}:S\rightarrow(0,1]$ are such that:
(c) For each fixed $(\omega_{i},y)\in\Omega_{i}\times S,$
$\lambda\rightarrow\\{y\in
S:\sup_{u\in(G_{i}(\omega_{i},y))_{g_{i}(y)}}$Re$\langle
u,\lambda_{i}(\omega_{i})-y\rangle>0\\}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S}$ is upper semicontinuous with compact values;
(d) for each fixed $\omega_{i}\in\Omega_{i},$ the set
$\\{\lambda\in\mathcal{D}_{(X(\cdot))_{z}}:\sup_{y\in(A_{i}(\omega_{i},\lambda))_{a_{i}(\lambda)})}\sup_{u\in(G_{i}(\omega_{i},y))_{g_{i}(y)}}$Re$\langle
u,\lambda_{i}(\omega_{i})-y\rangle>0\\}$ is weakly open in
$\mathcal{D}_{(X(\cdot))_{z}}$
(e)
$\\{\omega_{i}\in\Omega_{i}:\sup_{u\in(G_{i}(\omega,y))_{g_{i}(y)}}\mathit{Re}\langle
u,\lambda_{i}(\omega_{i})-y\rangle>0\\}\in\mathcal{Z}_{i}$ for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}}$.
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such that
for every $i\in I$:
i)
$\lambda_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})};$
ii) sup${}_{u\in(G_{i}(\omega_{i},y))_{g_{i}(y)}}Re\langle
u,\lambda_{i}^{\ast}(\omega_{i})-y\rangle\leq 0$ for all
$y\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})}$.
Proof. Let us define
$\psi_{i}:\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}\times
S\rightarrow\mathbb{R}\cup\\{-\infty,+\infty\\}$ by
$\psi_{i}(\omega_{i},\lambda,y)=\sup_{u\in(G_{i}(\omega_{i},y))_{g_{i}(y)}}$Re$\langle
u,\lambda_{i}(\omega_{i})-y\rangle$ for each
$(\omega_{i},\lambda,y)\in\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}\times
S.$
We have that $\lambda_{i}(\omega_{i})\notin\\{y\in
S:\psi_{i}(\omega_{i},\lambda,y)>0\\}$ for each fixed
$(\omega_{i},\lambda)\in\Omega_{i}\times\mathcal{D}_{(X(\cdot))_{z}}.$
All the hypotheses of Theorem 8 are satisfied. According to Theorem 8, there
exists $\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such that
$\lambda_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})}$
for every $i\in I$ and
sup${}_{y\in
A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})}}\sup_{u\in(G_{i}(\omega_{i},y))_{g_{i}(y)}}\mathit{Re}\langle
u,\lambda_{i}^{\ast}(\omega_{i})-y\rangle\leq 0$ for every $i\in I.\vskip
6.0pt plus 2.0pt minus 2.0pt$
If $|I|=1$, we obtain the following corollary.
###### Corollary 11
Let $S$ be a countable completely metrizable topological vector space and let
$(\Omega,\mathcal{Z},\mathcal{\mu})$ be a measure space. Suppose that the
following conditions are satisfied$:$
a) the fuzzy mapping $X:\Omega\rightarrow\mathcal{F}(S)$ is measurable such
that $(X(\cdot))_{z}:\Omega\rightarrow 2^{S}$ has compact values;
b) the fuzzy mapping $A$ is such that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$
$\mathit{(}A(\cdot,\lambda)_{a(\lambda)}:\Omega\rightarrow 2^{S}$ is
measurable and, for all $\omega\in\Omega,$
$(A(\omega,\cdot))_{a(\cdot)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow 2^{S}$
is upper semicontinuous with non-empty compact values;
$G:\Omega\times S\rightarrow\mathcal{F}(S^{\prime})$ and $g:S\rightarrow(0,1]$
are such that:
(c) For each fixed $(\omega,y)\in\Omega\times S,$ $\lambda\rightarrow\\{y\in
S:\sup_{u\in(G(\omega,y))_{g(y)}}$Re$\langle
u,\lambda(\omega)-y\rangle>0\\}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow 2^{S}$
is upper semicontinuous with compact values;
(d) for each fixed $\omega\in\Omega,$ the set
$\\{\lambda\in\mathcal{D}_{(X(\cdot))_{z}}:\sup_{y\in(A(\omega,\lambda))_{a(\lambda)})}\sup_{u\in(G(\omega,y))_{g(y)}}$Re$\langle
u,\lambda(\omega)-y\rangle>0\\}$ is weakly open in
$\mathcal{D}_{(X(\cdot))_{z}}$
(e) $\\{(\omega,\lambda):\sup_{u\in(G(\omega,y))_{g(y)}}\mathit{Re}\langle
u,\lambda(\omega)-y\rangle>0\\}\in\mathcal{Z}$ for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}}$.
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such that:
i) $\lambda^{\ast}(\omega)\in(A(\omega,\lambda^{\ast}))_{a(\lambda^{\ast})};$
ii) sup${}_{u\in(G(\omega,y))_{g(y)}}Re\langle
u,\lambda^{\ast}(\omega)-y\rangle\leq 0$ for all
$y\in(A(\omega,\lambda^{\ast}))_{a(\lambda^{\ast})}$.
We obtain the following random fixed point theorem as a particular case of
Theorem 8.
###### Theorem 12
Let $I$ be a non-empty finite set. For each $i\in I$, $S_{i}$ is a countable
complete metric space and $(\Omega_{i},\mathcal{Z}_{i})$ is a measurable
space. Let $(\Omega,\mathcal{Z})$ be the product measurable space
$\underset{i\in I}{(\prod}\Omega_{i},\underset{i\in
I}{\mathop{\textstyle\bigotimes}}\mathcal{Z}_{i})$, and $\mu$ a probability
measure on $(\Omega,\mathcal{Z}).$ Suppose that the following conditions are
satisfied for each $i\in I:$
a) the fuzzy mapping $X_{i}:\Omega_{i}\rightarrow\mathcal{F}(S_{i})$ is
measurable and $(X_{i}(\cdot))_{z_{i}}:\Omega_{i}\rightarrow 2^{S_{i}}$ has
compact values;
b) the fuzzy mapping $A_{i}$ is such that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$
$\mathit{(}A_{i}(\cdot,\lambda)_{a_{i}(\lambda)}:\Omega_{i}\rightarrow
2^{S_{i}}$ is measurable and, for all $\omega_{i}\in\Omega_{i},$
$(A_{i}(\omega_{i},\cdot))_{a_{i}(\cdot)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow
2^{S_{i}}$ is upper semicontinuous with non-empty compact values.
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such that
for every $i\in I$,
$\lambda_{i}^{\ast}(\omega_{i})\in(A_{i}(\omega_{i},\lambda^{\ast}))_{a_{i}(\lambda^{\ast})}$.
If $|I|=1$, we obtain the following result.
###### Theorem 13
Let $S$ be a countable complete metric space and let
$(\Omega,\mathcal{Z},\mathcal{\mu})$ be a measure space. Suppose that the
following conditions are satisfied$:$
a) the fuzzy mapping $X:\Omega\rightarrow\mathcal{F}(S)$ is measurable and
$(X(\cdot))_{z}:\Omega\rightarrow 2^{S}$ has compact values;
b) the mapping $A$ is such that for each
$\lambda\in\mathcal{D}_{(X(\cdot))_{z}},$
$\mathit{(}A(\cdot,\lambda)_{a(\lambda)}:\Omega\rightarrow 2^{S}$ is
measurable and, for all $\omega\in\Omega,$
$(A(\omega,\cdot))_{a(\cdot)}:\mathcal{D}_{(X(\cdot))_{z}}\rightarrow 2^{S}$
is upper semicontinuous with non-empty compact values.
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{(X(\cdot))_{z}}$ such that
$\lambda^{\ast}(\omega)\in(A(\omega,\lambda^{\ast}))_{a(\lambda^{\ast})}$.
## 5 Classical systems of generalized quasi-variational inequalities and
random fixed point theorems
If we consider classical corespondences in the last section, we obtain several
results concerning systems of generalized random quasi-variational
inequalities and random fixed points, which are new in the literature. We
present Theorem 14 as a consequence of Theorem 8.
###### Theorem 14
Let $I$ be a non-empty finite set. For each $i\in I$, $S_{i}$ is a countable
complete metric space and $(\Omega_{i},\mathcal{Z}_{i})$ is a measurable
space. Let $(\Omega,\mathcal{Z})$ be the product measurable space
$\underset{i\in I}{(\prod}\Omega_{i},\underset{i\in
I}{\mathop{\textstyle\bigotimes}}\mathcal{Z}_{i})$, and let $\mu$ be a
probability measure on $(\Omega,\mathcal{Z}).$ Suppose that the following
conditions are satisfied for each $i\in I:$
a) the correspondence $X_{i}:\Omega_{i}\rightarrow 2^{S_{i}}$ is measurable
and $(X_{i}(\cdot))_{z_{i}}:\Omega_{i}\rightarrow 2^{S_{i}}$ has compact
values;
b) the correspondence $A_{i}$ is such that for each
$\lambda\in\mathcal{D}_{X},$ $A_{i}(\cdot,\lambda):\Omega_{i}\rightarrow
2^{S_{i}}$ is measurable and, for all $\omega_{i}\in\Omega_{i},$
$A_{i}(\omega_{i},\cdot):\mathcal{D}_{X}\rightarrow 2^{S_{i}}$ is upper
semicontinuous with non-empty compact values;
Let us assume that the mapping $\psi_{i}:\Omega_{i}\times\mathcal{D}_{X}\times
S\rightarrow\mathbb{R}\cup\\{-\infty,+\infty\\}$ is such that:
(c) $\lambda\rightarrow\\{y\in
Y:\psi_{i}(\omega,\lambda,y)>0\\}:\mathcal{D}_{X}\rightarrow 2^{S}$ is upper
semicontinuous with compact values on $\mathcal{D}_{X}$ for each fixed
$\omega_{i}\in\Omega_{i};$
(d) $\lambda_{i}(\omega_{i})\notin\\{y\in
S:\psi_{i}(\omega_{i},\lambda,y)>0\\}$ for each fixed
$(\omega_{i},\lambda)\in\Omega_{i}\times\mathcal{D}_{X};$
(e) for each $\omega_{i}\in\Omega_{i},$
$\\{\lambda\in\mathcal{D}_{X}:\alpha_{i}(\omega_{i},\lambda)>0\\}$ is weakly
open in $\mathcal{D}_{X},$ where
$\alpha_{i}:\Omega_{i}\times\mathcal{D}_{X}\rightarrow\mathbb{R}$ is defined
by $\alpha_{i}(\omega_{i},\lambda)=\sup_{y\in
A_{i}(\omega_{i},\lambda)}\psi_{i}(\omega_{i},\lambda,y)$ for each
$(\omega_{i},\lambda)\in\Omega_{i}\times\mathcal{D}_{X};$
(f) $\\{\omega_{i}:\alpha_{i}(\omega_{i},\lambda)>0\\}\in\mathcal{Z}_{i}$ for
each $\lambda\in\mathcal{D}_{X}$.
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{X}$ such that for every
$i\in I$,
i) $\lambda_{i}^{\ast}(\omega_{i})\in A_{i}(\omega_{i},\lambda^{\ast});$
ii) sup${}_{y\in
A_{i}(\omega,\lambda^{\ast})}\psi_{i}(\omega_{i},\lambda^{\ast},y)\leq 0$.
The next theorems concern the existence of the random fixed point for
correspondences with values in complete countable metric spaces.
###### Theorem 15
Let $I$ be a non-empty finite set. For each $i\in I$, $S_{i}$ is a countable
complete metric space and $(\Omega_{i},\mathcal{Z}_{i})$ is a measurable
space. Let $(\Omega,\mathcal{Z})$ be the product measurable space
$\underset{i\in I}{(\prod}\Omega_{i},\underset{i\in
I}{\mathop{\textstyle\bigotimes}}\mathcal{Z}_{i})$, and let $\mu$ be a
probability measure on $(\Omega,\mathcal{Z}).$ Suppose that the following
conditions are satisfied for each $i\in I:$
a) the correspondence $X_{i}:\Omega_{i}\rightarrow 2^{S_{i}}$ is measurable
with compact values;
b) for each $\lambda\in\mathcal{D}_{X},$
$A_{i}(\cdot,\lambda):\Omega_{i}\rightarrow 2^{S_{i}}$ is measurable and, for
all $\omega_{i}\in\Omega_{i},$
$A_{i}(\omega_{i},\cdot):\mathcal{D}_{X}\rightarrow 2^{S_{i}}$ is upper
semicontinuous with non-empty compact values.
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{X}$ such that for every
$i\in I$, $\lambda_{i}^{\ast}(\omega_{i})\in
A_{i}(\omega_{i},\lambda^{\ast})$.
If $|I|=1$, we obtain the following result.
###### Theorem 16
Let $S$ be a countable complete metric space and let
$(\Omega,\mathcal{Z},\mathcal{\mu})$ be a measure space. Suppose that the
following conditions are satisfied$:$
a) the correspondence $X:\Omega\rightarrow 2^{S}$ is measurable with compact
values;
b) for each $\lambda\in\mathcal{D}_{X},$ $A(\cdot,\lambda):\Omega\rightarrow
2^{S}$ is measurable and, for all $\omega\in\Omega,$
$A(\omega,\cdot):\mathcal{D}_{X}\rightarrow 2^{S}$ is upper semicontinuous
with non-empty compact values.
Then, there exists $\lambda^{\ast}\in\mathcal{D}_{X}$ such that
$\lambda^{\ast}(\omega)\in A(\omega,\lambda^{\ast})$.
## References
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* [11] Kim W. K. and Lee K. H. (1998), Fuzzy fixed point and existence of equilibria of fuzzy games, J. Fuzzy Math. 6, 193-202.
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|
arxiv-papers
| 2013-06-21T21:31:43 |
2024-09-04T02:49:46.807530
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Monica Patriche",
"submitter": "Monica Patriche",
"url": "https://arxiv.org/abs/1306.5258"
}
|
1306.5268
|
# Static and Dynamic Aspects of
Scientific Collaboration Networks
Christian Staudt [email protected] Andrea Schumm
[email protected] Henning Meyerhenke [email protected] Robert Görke
[email protected] Dorothea Wagner [email protected] Institute of
Theoretical Informatics, Karlsruhe Institute of Technology (KIT), Am
Fasanengarten 5, 76131 Karlsruhe, Germany
###### Abstract
Collaboration networks arise when we map the connections between scientists
which are formed through joint publications. These networks thus display the
social structure of academia, and also allow conclusions about the structure
of scientific knowledge. Using the computer science publication database DBLP,
we compile relations between authors and publications as graphs and proceed
with examining and quantifying collaborative relations with graph-based
methods. We review standard properties of the network and rank authors and
publications by centrality. Additionally, we detect communities with
_modularity_ -based clustering and compare the resulting clusters to a ground-
truth based on conferences and thus topical similarity. In a second part, we
are the first to combine DBLP network data with data from the _Dagstuhl
Seminars_ : We investigate whether seminars of this kind, as social and
academic events designed to connect researchers, leave a visible track in the
structure of the collaboration network. Our results suggest that such single
events are not influential enough to change the network structure
significantly. However, the network structure seems to influence a
participant’s decision to accept or decline an invitation.
## I Introduction
In scientometrics, the quantitative study of science, network analysis has
become a prominent tool. _Coauthorship networks_ have attracted interest both
as _social networks_ and as _knowledge networks_ : They display the social
structure of academia, while their bibliographic aspect allows conclusions
about the structure of scientific knowledge. Accordingly, networks of this
kind are the objects of ongoing research: Newman ([1, 2, 3, 4]), for example,
studies properties of coauthorship networks in the realm of physics (Los
Alamos e-Print Archive, SPIRES), mathematics (Mathematical Reviews),
biomedical research (Medline) and computer science (NCSTRL), summarizing many
statistical properties of coauthorship networks. Aspects like _connectedness_
, _distance_ , _degree distribution_ , _centrality_ and _community structure_
are recurring themes in such studies. Where we follow up on these topics, we
cite relevant related work in the respective sections of this paper.
Based on the extensive publication database _DBLP_ [5], we model relations
between authors and publications as graphs, mapping almost the entire field of
computer science. This allows us to examine and quantify the collaborative
relations between researchers using graph-based methods. We compile a graph in
which edges link coauthors, as well as a bipartite author-paper graph. In the
first part, we review standard properties of the network, rank authors and
publications by centrality, and detect communities with _modularity_ -based
clustering. In the second part, we combine the network with seminar data
provided by the _Schloss Dagstuhl_ [6] conference center: The _Dagstuhl
Seminars_ assemble researchers with the goal of fostering (collaborative) work
in cutting-edge areas of computer science. We examine whether such events
leave a track in the structure of the collaboration network. For this purpose,
we apply appropriate measures to a time-resolved version of the _authorship
graph_.
We are the first to perform a joint analysis of the Dagstuhl and DBLP
datasets, which allows us to study the impact of social/academic events on the
time-evolution of the network structure. Our results suggest that a
participant’s decision to accept or decline an invitation can be predicted
from the network data to some extent. While our analysis of the DBLP data
mostly confirms properties of similar networks, the distribution of the number
of coauthors differs from data reported in [4]. We also describe an approach
to finding central researchers based on _eigenvector centrality_ in the
bipartite authorship graph, a combination that to our knowledge has not been
used before. Additionally, we apply _modularity_ clustering and compare the
detected communities to a ground-truth defined by conferences, from which we
infer distinct areas of research.
## II Preliminaries
### II-A Collaboration Network Model
As of 2011, DBLP covers about 1.5 million publications by 0.8 million authors.
The earliest work dates from 1936, and we include all works up to 2009 in our
analysis. We describe briefly how a coauthorship network is extracted from the
publication database and represented as different types of graphs. The
database associates publications and authors and thus provides two main
relations, _authorship_ and _coauthorship_ , formalized as follows:
###### Def. 1.
Given the sets of authors $\mathbf{A}$ and publications $\mathbf{P}$, the
_authorship_ relation is defined as
$\forall\\{a,p\\}\in\mathbf{A}\times\mathbf{P}:\quad a\smile p\iff\text{$a$ is
author of $p$}$
The _coauthorship_ relation between two authors from $\mathbf{A}$ is defined
as
$\forall\\{a,b\\}\in\mathbf{A}\times\mathbf{A}:a\frown b\iff\exists
p\in\mathbf{P}:a\smile p\wedge b\smile p$
From these, two graph representations of the network follow: A bipartite
_authorship graph_ (or author-paper graph) $G_{\mathbf{PA}}$, in which each
publication is connected by edges to its authors; and a _coauthorship graph_
$G_{\mathbf{A}}$, in which two authors are connected by an edge if they are
coauthors of a joint publication.
###### Def. 2.
The _authorship graph_ is a mapping from the sets of publications $\mathbf{P}$
and authors $\mathbf{A}$ to the node sets $V_{\mathbf{P}}$ and
$V_{\mathbf{A}}$, resulting in a bipartite graph
$G_{\mathbf{PA}}=(V_{\mathbf{A}},V_{\mathbf{P}},E)$, where
$\\{v_{a},v_{p}\\}\in E\iff a\smile p$
###### Def. 3.
The _coauthorship graph_ is a mapping from the set of authors $\mathbf{A}$ to
the node set $V_{\mathbf{A}}$, resulting in the graph
$G_{\mathbf{A}}=(V_{\mathbf{A}},E)$, where
$\\{v_{a},v_{b}\\}\in E\iff a\frown b$
While $G_{\mathbf{A}}$ is sufficient when focusing only on the social network
of coauthors, $G_{\mathbf{PA}}$ preserves the publications as the cause of
relations, as well as single-author publications. Table I shows the size of
the graphs constructed from the full publication data set.
graph | $n$ | $m$
---|---|---
$G_{\mathbf{PA}}$ | 2 296 586 | 3 775 881
$G_{\mathbf{A}}$ | 852 250 | 2 785 037
TABLE I: Size of resulting graphs
In order to determine whether events have effects detectable in terms of the
network (Section IV), we also track groups of authors over the course of time,
using a sequence of graphs in which each graph represents a current snapshot
of the authorship relations. This _time-resolved_ version of $G_{\mathbf{PA}}$
enables us to study the dynamics of the network: Let $t(p)$ denote the
publication date of publication $p$. Then the publications from a time segment
$[y,z],z>y$, are
$\mathbf{P}_{[y,z]}:=\\{p\in\mathbf{P}:y\leq t(p)\leq z\\}$
The respective authors of these publications are
$\mathbf{A}_{[y,z]}:=\\{a\in\mathbf{A}:\exists p\in\mathbf{P}_{[y,z]}:a\smile
p\\}$
The graph sequence is constructed on the basis of a sliding time segment, with
parameters width $w$ and increment $s$:
###### Def. 4.
The _time-resolved authorship graph_ is a sequence of graphs
$\mathcal{G}_{\mathbf{PA}}^{w,s}$ where each graph in the sequence is
constructed from the publications in $\mathbf{P}_{[y,y+w]}$ and the authors up
to $\mathbf{A}_{[y,y+w]}$ using a sliding time segment with width $w$ and
increment $s$.
Author nodes are aggregated over time, while publications are deleted for each
step in the sequence. A time segment and increment of 1 year was chosen for
the study in Section IV, the finest time resolution possible with DBLP data.
## III Network Properties and Community Structure
### III-A General Network Properties
We briefly review some general properties of the collaboration network:
#### Connectedness
$G_{\mathbf{A}}$ features a _giant connected component_ containing about 80%
of all authors. (Giant components connecting up to 90% of all authors have
previously been detected across scientific fields [3]). Aside from the 6% of
the authors without collaborations, about 14 % of author nodes are distributed
over a multitude of small components with few publications. We conclude that,
in general, authors who have worked on several publications and were part of
more than one collaborative team join the large connected component. In terms
of average distances between researchers (6.58 for a sample), we confirm the
previously reported _small world_ property for the field of computer science
[4] and DBLP in particular [7].
#### $k$-Core Structure
Figure 1: Histogram of _core numbers_ in $G_{\mathbf{A}}$ (x-axis: _core
number_ , logarithmic y-axis: frequency)
A $k$-core is a maximal subgraph in which each node is adjacent to at least
$k$ other nodes. $k$-cores refine the concept of connected components (which
form the 1-core); _$k$ -core decomposition_ reveals nested, successively more
cohesive layers of the graph. We assign each node a _core number_ , the
highest $k$ for which there is a $k$-core containing the node. Figure 1 shows
a histogram of the resulting _core numbers_ in $G_{\mathbf{A}}$ with two
logarithmic axes. The rather uniform sequence indicates uniform density and
cohesiveness of the graph, showing that the network does not have strongly
cohesive groups of authors embedded in shells of weakly connected authors [8].
A more extensive $k$-core analysis of a DBLP-based coauthorship network is
presented in [9].
#### Degree Distribution
Figure 2: Degree distribution in $G_{A}$ (logarithmic x-axis: degree,
logarithmic y-axis: frequency)
Node degree in $G_{\mathbf{A}}$ corresponds to the number of coauthors of each
author. The degree distribution is highly skewed. It indicates a _scale-free
network_ , in which the frequency $P(k)$ of nodes with degree $k$ follows a
power law, i.e. $P(k)\sim k^{-\gamma}$, with coefficient $\gamma=2.889$.
Newman [3] reports a differing power-law degree distribution in the number of
coauthors with $\gamma=3.41$ for computer science, based on NCSTRL.
#### Summary
General properties indicate that the network of collaborations in computer
science is in many respects a typical social network: It shows participation
inequality (visible as a power-law degree distribution), with a few highly
prolific authors and many smaller contributions. It also features a high
degree of connectedness, a giant component, and mostly short paths between
arbitrary pairs of nodes. Our observations are in agreement with the results
of related studies (except for the degree power-law exponent), indicating that
these properties are universal features of scientific collaboration networks.
### III-B Centrality
_Centrality measures_ were formulated to identify nodes which are structurally
prominent or influential, due to their position in the center of a network.
_Betweenness_ and _closeness centrality_ have previously been applied to
coauthorship graphs with the goal of identifying influential scientists in
their respective fields ([4, 10]). Elmacioglu et al. report a ranking of
prominent scholars by _closeness_ and _betweenness_ centrality [7]. As a
rationale, it has been stated that authors with high _betweenness_ are
important intermediates for interactions or information flows, as it allows
them to control such flows; high closeness is assumed to be an advantage for
accessing or disseminating information [7]. However, it is not clear why
academic influence should be understood mainly as the ability to mediate
interactions. Furthermore, the network of information flow in academia and the
network of coauthorship relations may be quite distinct. We therefore follow a
different approach based on _eigenvector centrality_ [11] in the bipartite
authorship graph: It assumes that an author’s influence is first of all
proportional to the amount of publications. Additionally, the contribution of
a paper to an author’s centrality should be weighted depending on the
centrality of the coauthors.
###### Def. 5.
_Eigenvector centrality_ : Given a graph $G$ with adjacency matrix $A$, we
require a centrality score $x_{i}$ of node $v_{i}$ to be proportional to the
scores of its neighbors:
$x_{i}=c\sum_{j=1}^{n}A(i,j)\ x_{j}\qquad c\neq 0$
By the Perron-Frobenius theorem, there exists a nonnegative eigenvector $x$ of
$A$ (satisfying $Ax=\frac{1}{c}x=\lambda x$) which corresponds to the largest
eigenvalue $\lambda$. An entry $x_{i}$ constitutes the desired centrality
score for vertex $v_{i}$.
Modeling the collaboration network as the bipartite graph $G_{\mathbf{PA}}$
has the benefit that it allows us to assign a centrality score to a
publication as a node, rather than just account for a publication as an edge
attribute or weight in $G_{\mathbf{A}}$ [12]. Thus, our centrality scores
express the concept that authors are central in the collaboration network to
the extent that they have collaborated on central publications with other
central authors. In this respect, the approach is similar to ranking webpages
with the PageRank algorithm, where hyperlinks are treated as votes to the
relevance of the target page and are weighted by the relevance of the source
page.
$\begin{array}[]{|c|c|}\hline\cr\text{centrality}\cdot
10^{-5}&\text{author}\\\ \hline\cr 9.76232&\text{Diane Crawford}\\\
9.45441&\text{Robert L. Glass}\\\ 9.08697&\text{Chin-Chen Chang}\\\
8.30777&\text{Edwin R. Hancock}\\\ 7.91401&\text{Grzegorz Rozenberg}\\\
7.82901&\text{Joseph Y. Halpern}\\\ 7.75409&\text{Sudhakar M. Reddy}\\\
7.69387&\text{Philip S. Yu}\\\ 7.50894&\text{Moshe Y. Vardi}\\\
7.47370&\text{Ronald R. Yager}\\\ \hline\cr\end{array}$ TABLE II: Top segment
of author ranking by centrality
(a) authors
(b) publications
Figure 3: Centrality scores (logarithmic y-axis), sorted
Figure 3 shows the distributions of centrality scores for authors and
publications. Extreme values are less frequent, and the distribution does not
exhibit a power law. Table II contains the top segment of an author ranking by
our approach to centrality. (See [13] for a comparison to a purely
productivity-based ranking of DBLP authors.) The respective ranking of
publications places papers with unusually high author counts at the top, e.g.
work on large supercomputing and database projects, and further study would be
needed to interpret publication centrality properly. With respect to the
evaluation in Section IV, it should also be noted that Dagstuhl seminar
invitees have a significantly higher median _eigenvector centrality_ score
than other authors ($3.8\cdot 10^{-6}$ versus $2.4\cdot 10^{-7}$). We
therefore propose that _eigenvector centrality_ in bipartite author-paper
networks is a promising approach for studying the role and impact of
collaborating individuals in science, and might serve as an objective measure
of influence in scientific publishing.
### III-C Modularity-driven Clustering
_Graph clustering_ comprises a variety of methods for detecting natural
communities in networks. Formally, it is concerned with partitioning the node
set into disjoint subsets (clusters), the result of which is called a
_clustering_. The notion of a cluster is usually based on the _intra-cluster
density versus inter-cluster sparsity_ paradigm, according to which a
clustering should identify groups of nodes which are internally densely
connected, while only sparse connections exist between the groups. One of the
primary measures of clustering quality based on this paradigm is _modularity_
[14].
###### Def. 6.
For a graph $G=(V,E)$ and a clustering $\zeta=\\{C_{1},\dots,C_{k}\\}$ of $G$,
_modularity_ is defined as
$\displaystyle\mathit{mod}(G,\zeta):=\sum_{C\in\zeta}\frac{|E(C)|}{|E|}-\sum_{C\in\zeta}\frac{\left(\sum_{v\in
C}deg(v)\right)^{2}}{(2\cdot|E|)^{2}}$
The measure considers the clustering’s _coverage_ (the fraction of edges
placed within a cluster) on the actual graph and subtracts the _coverage_ it
would achieve on a randomly connected version of the graph (preserving degree
distribution). _Modularity_ -based clusterings often agree with human
intuition, although criticism has emerged recently [15]. Since maximizing
_modularity_ is an $\mathcal{N}\mathcal{P}$-hard problem [16], we use a
heuristic based on _local greedy agglomeration_. The base algorithm, commonly
referred to as the Louvain Method [17], starts with a singleton clustering,
considers nodes in turn, moves them to the best neighboring cluster and
contracts the graph for the next iteration. This yields a hierarchy of graphs
with increasing coarseness where the clustering in the coarsest level induces
the resulting clustering in the original graph. Rotta et al. [18] enhance
this algorithm by a refinement phase that iteratively projects this clustering
to lower levels of the hierarchy and further improves modularity by local node
moves. We use this modified algorithm.
It is a common approach to apply a clustering method to a real world network
and then compare it to a ground-truth partition of the node set in order to
interpret the result. For example, Rodriguez et al. [19] study sensor
networks research groups and apply clustering techniques like _leading
eigenvector_ , but not _modularity_ maximization; these network-structural
communities are then compared to communities defined by socio-academic
similarities.
Figure 4: Size distribution for the 300 largest clusters (x-axis: cluster
size, y-axis: frequency)
We therefore proceed as follows: Applying local greedy agglomeration to
$G_{\mathbf{PA}}$ yields a clustering with 86761 clusters, achieving a
_modularity_ of 0.896896. The majority of clusters contain only a handful of
nodes, and likely correspond to the many tiny components of the graph, while
the dominant connected component is divided into several large clusters (see
Figure 4). With a clustering of the _authorship graph_ at hand, we attempt to
interpret such a _modularity_ -driven clustering in the context of
collaboration networks. The partition found by maximizing _modularity_ locally
identifies groups of authors who are densely connected through collaborative
ties. Our hypothesis is that we can infer a topical similarity from these
connections. More precisely, we conjecture that researchers form collaborative
ties around distinct areas of research, which is reflected in the clustering
structure of the graph. To put this hypothesis to the test, we compare the
_modularity clustering_ of $G_{\mathbf{PA}}$ to a ground-truth subdivision of
the author set based on conferences: Assuming that distinct areas of computer
science generally have dedicated conferences, we assign all authors who have
published at a particular conference to an author-cluster. (Unlike the
_modularity clustering_ , this does not yield a proper, complete and disjoint
partition of the author set, but is nonetheless informative.) Thereby we
arrive at _topical clusters_ of authors, which are suited as a ground-truth to
compare the _modularity clustering_ to.
| random | topical
---|---|---
$O$ | 0.04404 | 0.22832
$J$ | 0.00372 | 0.01390
TABLE III: Mean maximum overlap for modularity clustering and random vs
topical clustering
In order to evaluate the similarity between the two community structures, one
being the _modularity clustering_ , the other the _topical clustering_ defined
by conferences, we apply overlap measures to each pair of clusters: The
_Jaccard index_ $J(A,B):={{|A\cap B|}\over{|A\cup B|}}$ favors exact match of
the two sets; the _overlap coefficient_ $O(A,B):={{|A\cap
B|}\over{\min(|A|,|B|)}}$ treats containment of one set in the other set as a
strong match, which is more equitable when dealing with clusters of uneven
sizes. Applying these measures yields matrices of overlap values between
_modularity clusters_ and _topical clusters_. Additionally, we arrive at a
baseline for the overlap values by calculating the overlap matrix of
_modularity_ clustering and a random clustering. The random clustering is
constructed by copying the size distribution of the 250 largest modularity
clusters, but randomly assigning authors to the clusters.
In these overlap matrices, we are interested in the maximum entry for each
row, pointing to pairs of clusters that are most similar. Table III shows the
means of these maximum overlap values. It is evident that the maximum $J$ and
$O$ overlap is significantly better for _modularity clusters_ than for random
clusters. This shows that a more than coincidental relation between
_modularity clusters_ and _topical clusters_ exists. However, the values are
not close to 1.0 and indicate that the correspondence is not very strong.
Thus, factors in addition to joint conferences are influential in shaping the
community structure of the network. In the following section, we take an in-
depth look at one possible factor of this kind, namely participation in
research seminars.
## IV Impact of Seminars on Network Evolution
After describing static aspects of the network in the previous section, this
section is concerned with its dynamics: We examine whether the Dagstuhl
Seminars, as academic and social events, leave a track in the structure of the
network, preferably in the form of increased collaboration between the
participants. In the authors’ subjective experience, the seminars present
valuable opportunities for networking. Our approach to this question can be
summarized as follows: Track groups of researchers (seminar participants and
others selected as reference groups) in the time-resolved graph
$\mathcal{G}_{\mathbf{PA}}^{1,1}$ and observe their publication output as well
as their collaborative links; take into account the date of a seminar in order
to observe immediate or long-term effects. The preparations necessary for this
approach are described in the following:
### IV-A Preparations
#### Aligning Data Sets
Our data sets record a total of 11 625 seminar guests in the Dagstuhl database
and 852 250 authors in DBLP. All seminars took place in the 2000s. We align
the tests by author name, whereby some false (mis)matches cannot be avoided.
Still, a matching author in the publication database was found for 72 percent
of the seminar invitees.
#### Area Launchers
In order to detect increased collaboration which can be clearly attributed to
the seminars, we first try to identify _area launchers_. These are seminars
intended to bring together a group of researchers who have not collaborated
much before. A stated goal of the Dagstuhl Seminars is that some of them are
intended to “launch” new areas of research by fostering collaboration between
previously unaffiliated researchers, thereby contributing to emerging fields.
_Area launchers_ are relevant to us due to the following argument: If
participants develop collaborative ties in the aftermath of an _area launcher_
seminar, it is possible to attribute this more clearly to the seminar rather
than existing relationships, developed, for instance, in the course of a
common conference.
We classify a set of seminars as _area launchers_ without special knowledge
about the intent or content of the seminar, but solely from participation
data: It is assumed that well-established areas of research generally spawn
their own dedicated conference, and that the participants of such a conference
represent the researchers active in this area. By this logic, a seminar
corresponds to an established area of research if the invitees have a strong
overlap with the participants of the respective conference. Furthermore, if
researchers attend the same conference, it is likely that they are already
familiar with each other as well as each other’s work. We therefore reason
that a seminar is an _area launcher_ if its invitees do not overlap strongly
and clearly with the participants of any particular conference. From this
calculated set of seminars, 10 seminars are selected by hand and classified as
_area launchers_.
#### Measures
We quantify the publication output and intensity of collaboration among
researchers using several measures which map sets of authors to real numbers.
For example, Figure 5 shows a small number of authors (light nodes) and their
publications (dark nodes) in the _authorship graph_. Authors belonging to $A$
are colored blue. Blue lines show existing (dashed line) and nonexisting
(dotted line) coauthorship relations between pairs of authors in $A$. This
illustrates the measure $\mathit{cad}(A)$, which is the fraction of actually
existing coauthorship relations within an author set.
Figure 5: Illustrating collaboration measure $\mathit{cad}$:
$\mathit{cad}(A)=2/3$
Before introducing all measures, it is helpful to define sets of
(co)publications, copublications internal to a group, and coauthors first:
Given a set of authors $A\subseteq\mathbf{A}$, the set of their publications
$P(A)$ is equal to
$P(A):=\bigcup_{a\in A}P(a)=\bigcup_{a\in A}\\{p\in\mathbf{P}:a\smile p\\}$
The set of _copublications_ for an author $a$ consists of publications which
were written as collaborations with another author:
$C\\!P(a):=\\{p\in P(a):\exists b\in\mathbf{A}:b\smile p\\}$
For an author set $A\subseteq\mathbf{A}$, the _aggregated copublications_ are
$C\\!P(A):=\bigcup_{a\in A}C\\!P(a)$
The set of _intra-copublications_ of a set of authors is defined as
$C\\!P_{\text{intra}}(A):=\left\\{p\in C\\!P(A):\exists{a,b}\in A:a\smile
p,b\smile p\right\\}$
The set of coauthors for a given author $a\in\mathbf{A}$ are those authors
with whom $a$ has authored a collaboration.
$C\\!A(a):=\\{b\in\mathbf{A}:b\frown a\\}$
This can be generalized for a set of authors $A$:
$C\\!A(A):=\bigcup_{a\in A}C\\!A(a)$
Based on these sets, we formulate five measures, listed and defined in Table
IV. These measures are intended to answer the following questions:
* •
$\mathit{ap}(A)$: What is the general productivity of an average author from
the group?
* •
$\mathit{acp}(A)$: What is the productivity of such an author in terms of
collaborations?
* •
$\mathit{aca}(A)$: With how many other authors does an average author from the
group collaborate?
* •
$\mathit{cpr}_{\text{intra}}(A)$: Do the authors collaborate more often within
the group or outside of the group?
* •
$\mathit{cad}(A)$: How close is the group to a collaborative clique, i.e. a
group in which all authors have collaborated with each other?
measure | definition
---|---
$a\\!p(A)$ | $\frac{|P(A)|}{|A|}$
$acp(A)$ | $\frac{|C\\!P(A)|}{|A|}$
$\mathit{aca}(A)$ | $\frac{|C\\!A(A)|}{|A|}$
$cpr_{\text{intra}}(A)$ | $\frac{|C\\!P_{\text{intra}}(A)|}{|C\\!P(A)|}$
$cad(A)$ | $|\\{\\{a,b\\}\in\textstyle{A\choose 2}:a\frown b\\}|/|\textstyle{A\choose 2}|$
TABLE IV: Overview of collaboration measures and their definitions
#### Author Classes
The classes of author groups which we track are the seminar participants on
the one hand and several reference classes on the other:
* •
_seminar attendees_ ($A\\!t_{s}$): For each seminar $s$, the set of
researchers who attended the seminar.
* •
_seminar absentees_ ($A\\!b_{s}$): For each seminar $s$, the set of
researchers who were invited to the seminar but did not attend. (For some
seminars, the set was empty or very small, so these are only included if they
have a sufficient size.)
* •
_random samples_ ($R\\!S_{i}$) Contains randomly assembled sets of authors
with the size of a typical seminar.
* •
_connected samples_ ($C\\!S_{i}$) Contains sets of authors found by collecting
nodes from $G_{\mathbf{PA}}$ in a breadth-first search from a random initial
node until the typical size of a seminar is reached.
* •
_all authors_ ($\mathbf{A}$) A single set containing all authors.
### IV-B Evaluation and Results
We speculate that joint participation in a seminar leads to increased
collaboration between the participants. This would be measurable as higher
values for the collaboration measures ($\mathit{cad}$,
$\mathit{cpr}_{\text{intra}}$) on the respective subgraph. Additionally, we
measure whether seminar participation leads to a higher publication output for
the participants ($\mathit{ap}$, $\mathit{acp}$, $\mathit{aca}$). In order to
test this, seminar-related groups as well as reference groups are tracked
within the graph $\mathcal{G}^{1,1}_{\mathbf{PA}}$: For any author set $A$, a
subset $A^{\prime}\subseteq A$ has corresponding nodes $V_{A^{\prime}}$ in the
graph $G_{y}$. For all measures $M$, we evaluate $M(A^{\prime})$, yielding a
sequence of values for each group. The evaluation yields one value sequence
per author group, and thus several data points per year. All seminar-related
sequences are aligned according to the time of the seminar, in order to
compare values before and after seminar participation. We present these data
points in boxplot form (e.g. Figure 6), with the horizontal axis denoting time
relative to the seminar date and the vertical axis values of the respective
measure. By following the plotted median and quantiles along the time axis,
one can identify trends for the author class as a whole. The point in time
where a seminar occurs is marked by an arrow.
In the following section, we describe a selection of notable observations:
#### Average publication output remains rather constant
For the authors as a whole ($\mathbf{A}$), average publication output and
number of coauthors remain stable over time, even as the graph grows at an
increasing rate and author nodes accumulate.
#### Randomly grouped authors as a baseline for publication output
As a reference class, we evaluate the randomly compiled author groups $R\\!S$.
Both $\mathit{ap}$ and $\mathit{aca}$ are, on average, in the range of
0.6-0.8, showing that there are typically inactive authors in any given time
frame. As expected, there is no collaboration between authors in the random
samples.
#### Connected Sample Groups
Authors from the $C\\!S$ have a significantly higher productivity than
randomly selected authors, since breadth-first search finds high-degree nodes
with a higher probability. There is also an upward trend over time for all
measures. A possible explanation for this is that nodes gain connections over
time according to degree, if there is an underlying _preferential-attachment_
process at work (as suggested by the power-law degree distribution). Overall
$\mathit{cpr}_{\text{intra}}$ remains clearly below 0.5, showing that these
sample groups are just sections from greater collaborative clusters.
#### Attendees and absentees are equally productive
(a) $\mathit{aca}$: $A\\!t$
(b) $\mathit{aca}$: $A\\!b$
Figure 6: $\mathit{aca}$ (y-axis) for seminar attendees and absentees (x-axis:
time relative to seminar, arrow: seminar date)
The effect of seminar participation is best judged by contrasting attendees
with absentees. With respect to productivity, measured by the number of
coauthors and the number of publications, attendees and absentees are quite
similar, with some outliers among the absentees surpassing the attendees (see
Figure 6). For the productivity measures, an upward trend before the seminar
continues for a few years but then tends to reverse.
#### Attendees form a more cohesive group
(a) $\mathit{cpr}_{\text{intra}}$: $A\\!t$
(b) $\mathit{cpr}_{\text{intra}}$: $A\\!b$
Figure 7: $\mathit{cpr}_{\text{intra}}$ (y-axis) for seminar attendees and
absentees (x-axis: time relative to seminar, arrow: seminar date)
For seminar attendees, a larger fraction of their collaborations are internal
to the seminar group, both before and after the seminar (Figure 7). This
indicates that attendees already come from a more cohesive group. Values for
$\mathit{cad}$ agree with this interpretation: Clearly, those who choose to
attend the seminar form a denser subgraph in the collaboration network. There
seems to be no lasting increase in collaboration after the seminar, but a
downward trend for both attendees and absentees.
#### Area launchers are not exceptional
For the subset of seminars classified as _area launchers_ , we expect
comparatively less collaboration before the seminar, and a stronger increase
after. This effect would be most clearly captured by the measures
$\mathit{cpr}_{\text{intra}}$ (Figure 8) and $\mathit{cad}$. The plots in
Figure 8 support our reasoning about area launchers, namely that the authors
invited have a comparatively low probability of collaboration in the time
prior to the seminar: Values for $\mathit{cpr}_{\text{intra}}$ are generally
in the lower range compared to all seminars. Still, a visible change after the
time of the seminar is missing. The influence of an _area launcher_ seminar
does not seem to differ from the other seminars.
(a) $\mathit{cpr}_{\text{intra}}$: $A\\!t$
(b) $\mathit{cpr}_{\text{intra}}$: $A\\!b$
Figure 8: $\mathit{cpr}_{\text{intra}}$ (y-axis) for attendees and absentees
of _area launchers_ (x-axis: time relative to seminar, arrow: seminar date)
#### Subdivision by career stage
Suspecting that seminar participation affects researchers in early stages of
their career more strongly, we repeat a part of the evaluation with the
authors classified by career length ($\leq 5$, $\leq 15$, $>15$ years of
publication history). However, the results do not modify our conclusions: A
seminar effect for academic newcomers is no more observable than for all other
authors.
#### Summary and Interpretation
Seminar invitees are more productive and more collaborative than randomly
selected authors. Yet there is little difference between attendees and
absentees in terms of their productivity. Invited researchers are already
actively publishing, with an upward trend, prior to the time of the seminar.
For $\mathit{cpr}_{\text{intra}}$ and $\mathit{cad}$, attendees are
consistently better than absentees. This indicates that those who attend are
already a tightly connected collaborative group before the seminar, possibly
influencing their decision to participate. The general trend over time is an
increase up to the seminar and a slight decrease afterwards for both classes
of researchers. A possible explanation for the increase and decrease over time
is that invitations are biased towards researchers who are currently most
active: Invitations to seminars occur at a period of peak activity. There is,
however, no significant change of structure connected to seminars (either
significant short-term increase in collaboration directly after the seminar or
long-term increase). Most importantly, attendees and absentees do not differ
in this respect. While the focus on _area launcher_ seminars supports our
assumption that the invited researchers had collaborated less, a significant
structural change after the seminar is not visible. These results suggest that
a single event like a seminar is not influential enough to alter the network
structure of collaboration for the group of participants in ways observable
with our measures. Clearly, other factors have additional and apparently more
influence on the structure. Rather in the opposite direction, the network
structure might be employed to predict who will attend the seminar and who
will decline, since the participants evidently come from a more cohesive
group.
## V Conclusion
This paper ties in with the existing work on scientific collaboration networks
and explores several new variations of network analysis methods. The
coauthorship graph in the field of computer science constitutes in many
respects a typical social network, as observed before in similar studies: We
encounter properties such as low average distances between researchers, a
_giant connected component_ , a power-law distribution with regard to
publications and coauthors (making it a _scale-free network_), and a regular
_$k$ -core_ structure. We detect dense communities of researchers through
_modularity_ maximization, and compare the resulting clustering to ground-
truth communities defined by conferences, from which topical similarity is
inferred. The overlap between the two partitions is clearly not coincidental,
although other factors seem to be at work in shaping the community structure.
In order to identify influential researchers by their network centrality, we
test a novel combination of bipartite author-paper graph and _eigenvector
centrality_. We are the first to incorporate data on participants of the
Schloss Dagstuhl research seminars and use it to evaluate the impact of such
seminars on the evolution of collaborative ties. Since the seminars are
designed to foster collaboration on cutting-edge research topics, and many
participants experience the seminars as a valuable opportunity for networking,
we investigate whether such effects can be observed as structural changes in
the collaboration network. Seminar invitees are more productive, more
collaborative and structurally prominent compared to the average researcher.
However, our methods suggest that seminar participation does not directly
affect the structure of the collaboration network. An interesting finding of
this analysis was that researchers who choose to attend the seminar form a
distinctly more cohesive subgraph than those who decline.
## Acknowledgment
We thank Ulrik Brandes for helpful discussions during the preparation of this
work. We also thank the Schloss Dagstuhl conference center for providing us
with the necessary data on their seminars.
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|
arxiv-papers
| 2013-06-21T22:31:47 |
2024-09-04T02:49:46.816453
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Christian Staudt, Andrea Schumm, Henning Meyerhenke, Robert G\\\"orke,\n Dorothea Wagner",
"submitter": "Christian Lorenz Staudt",
"url": "https://arxiv.org/abs/1306.5268"
}
|
1306.5297
|
Drosophila embryogenesis scales uniformly across temperature in
developmentally diverse species
Steven G. Kuntz1,2∗, Michael B. Eisen1,2,3,4
1 QB3 Institute for Quantitative Biosciences, University of California,
Berkeley, California, United States of America
2 Department of Molecular and Cell Biology, University of California,
Berkeley, California, United States of America
3 Howard Hughes Medical Institute, University of California, Berkeley,
California, United States of America
4 Department of Integrative Biology, University of California, Berkeley,
California, United States of America
$\ast$ E-mail: [email protected]
## Abstract
Temperature affects both the timing and outcome of animal development, but the
detailed effects of temperature on the progress of early development have been
poorly characterized. To determine the impact of temperature on the order and
timing of events during Drosophila melanogaster embryogenesis, we used time-
lapse imaging to track the progress of embryos from shortly after egg laying
through hatching at seven precisely maintained temperatures between 17.5∘C and
32.5∘C. We employed a combination of automated and manual annotation to
determine when 36 milestones occurred in each embryo. D. melanogaster
embryogenesis takes $\sim$33 hours at 17.5∘C, and accelerates with increasing
temperature to a low of 16 hours at 27.5∘C, above which embryogenesis slows
slightly. Remarkably, while the total time of embryogenesis varies over two
fold, the relative timing of events from cellularization through hatching is
constant across temperatures. To further explore the relationship between
temperature and embryogenesis, we expanded our analysis to cover ten
additional Drosophila species of varying climatic origins. Six of these
species, like D. melanogaster, are of tropical origin, and embryogenesis time
at different temperatures was similar for them all. D. mojavensis, a sub-
tropical fly, develops slower than the tropical species at lower temperatures,
while D. virilis, a temperate fly, exhibits slower development at all
temperatures. The alpine sister species D. persimilis and D. pseudoobscura
develop as rapidly as tropical flies at cooler temperatures, but exhibit
diminished acceleration above 22.5∘C and have drastically slowed development
by 30∘C. Despite ranging from 13 hours for D. erecta at 30∘C to 46 hours for
D. virilis at 17.5∘C, the relative timing of events from cellularization
through hatching is constant across all species and temperatures examined
here, suggesting the existence of a previously unrecognized timer controlling
the progress of embryogenesis that has been tuned by natural selection as each
species diverges.
## Author Summary
Temperature profoundly impacts the rate of development of “cold-blooded”
animals, which proceeds far faster when it is warm. There is, however, no
universal relationship. Closely related species can develop at markedly
different speeds at the same temperature. This creates a major challenge when
comparing development among species, as it is unclear whether they should be
compared at the same temperature or under different conditions to maintain the
same developmental rate. Facing this challenge while working with flies
(Drosophila species), we found there was little data to inform this decision.
So, using time-lapse imaging, precise temperature-control, and computational
and manual video-analysis, we tracked the complex process of embryogenesis in
11 species at seven different temperatures. There was over a three-fold
difference in developmental rate between the fastest species at its fastest
temperature and the slowest species at its slowest temperature. However, our
finding that the timing of events within development all scaled uniformly
across species and temperatures astonished us. This is good news for
developmental biologists, since we can induce species to develop nearly
identically by growing them at different temperatures. But it also means flies
must possess some unknown clock-like molecular mechanism driving embryogenesis
forward.
## Introduction
It has long been known that Drosophila, like most poikilotherms, develops
faster at higher temperatures, with embryonic [1], larval [1, 2], and pupal
stages [3, 4], as well as total lifespan [5, 6] showing similar logarithmic
trends. While genetics, ecology, and evolution of this trait have been
investigated for over a century [2, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17],
the effects of temperature on the order and relative timing of developmental
events, especially within embryogenesis, are poorly understood.
We became interested in the relationship between species, temperature, and the
cadence of embryogenesis for practical reasons. Several years ago, we
initiated experiments looking at the genome-wide binding of transcription
factors in the embryos of divergent Drosophila species: D. melanogaster, D.
pseudoobscura, and D. virilis. With transcription factor binding a highly
dynamic process, we tried to match both the conditions (especially
temperature, which we believed would affect transcription factor binding) in
which embryos were collected and the developmental stages we analyzed.
However, our initial attempts to collect D. pseudoobscura embryos at 25∘C —the
temperature at which we collect D. melanogaster —were unsuccessful, with large
numbers of embryos failing to develop, likely a consequence of D.
pseudoobscura’s alpine origin. While D. virilis lays readily at 25∘C, we found
that their embryos develop more slowly than D. melanogaster, complicating the
collection of developmental stage-matched samples.
Having encountered such challenges with just three species, and planning to
expand to many more, we were faced with several important questions. Given
that embryogenesis occurs at different rates in different species [8, 18], how
should we time collections to get the same mix of stages we get from our
standard 2.5 –3.5 hour collections in D. melanogaster, or any other stage we
study in the future? Is it better to compare embryos collected at the same
temperature even if it is not optimal for, or even excludes, some species; or,
should we collect embryos from each species at their optimal temperature, if
such a thing exists? Should we select a temperature for each species so that
they all develop with a similar velocity? Or should we find a set of species
that develop at the same speed at a common temperature? And even if we could
match the overall rate of development, would heterochronic effects mean that
we could not get an identical mix of stages?
We found a woeful lack in the kind of data needed to answer these questions.
Powsner precisely measured the effect of temperature on the total duration of
embryogenesis in D. melanogaster [1], and Markow made similar measurements for
other Drosophila species at a fixed temperature (24∘C) [18], but the precise
timing of events within embryogenesis had been described only for D.
melanogaster at 25∘C [19, 20].
The work described here was born to address this deficiency. We used a
combination of precise temperature control, time-lapse imaging, and careful
annotation to catalog the effects of a wide range of temperatures on embryonic
development in 11 Drosophila species from diverse climates. We focused on
species with published genome sequences [21] (Table 1), as these are now
preferentially used for comparative and evolutionary studies. Of the species
we studied D. melanogaster, D. ananassae, D. erecta, D. sechellia, D.
simulans, D. willistoni, and D. yakuba are all native to the tropics, though
D. melanogaster, D. ananassae, and D. simulans have spread recently to become
increasingly cosmopolitan [17]. D. mojavensis is a sub-tropical species, while
D. virilis is a temperate species that has become holarctic and D. persimilis
and D. pseudoobscura are alpine species (Figure 1A).
## Results
### Time-lapse imaging tracks major morphological events
We used automated, time-lapse imaging to track the development of embryos held
at a constant and precise temperature from early embryogenesis (pre-
cellularization) to hatching. We maintained the temperature at $\pm$0.1∘C
using thermoelectric Peltier heat pumps. Different sets of embryos were
analyzed at temperatures ranging from 17.5∘C to 32.5∘C, in 2.5∘C increments.
Images were taken every one to five minutes, depending on the total time of
development. A minimum of four embryos from each species were imaged for each
temperature, for a total of 77 conditions. In total, time-lapse image series
were collected and analyzed from over 1000 individual embryos.
We encountered, and solved, several challenges in designing the experimental
setup, including providing the embryos with sufficient oxygen [22, 23] and
humidity. We found that glass slides were problematic due to a lack of
oxygenation and led to a $\sim$28% increase in developmental time, so we
instead employed an oxygen-permeable tissue culture membrane, mounted on a
copper plate to maintain thermal conduction. At higher temperatures, we found
that the embryos dehydrated, so humidifiers were used to increase ambient
humidity. Detailed photos of the apparatus and descriptions can be found in
Figure S1.
We used a series of simple computational transformations (implemented in
Matlab) to orient each embryo, correct for shifting focus, and adjust the
brightness and contrast of the images, creating a time-lapse movie for each
embryo. We manually examined images from 60 time-lapse series in D.
melanogaster and identified 36 distinct developmental stages [19, 20] that
could be recognized in our movies (Table 2,
http://www.youtube.com/watch?v=dYSrXK3o86I and
http://www.youtube.com/watch?v=QKVmRy3dDR0 or “D. melanogaster with labelled
stages” and “D. melanogaster with labelled stages at reduced framerate” in
DOI:10.5061/dryad.s0p50”). Due to the volume of images collected, we
implemented a semi-automated system to annotate our entire movie collection.
Briefly, images from matching stages in manually annotated D. melanogaster
movies were averaged to generate composite reference images for each stage
(Figure 2). We then used a Matlab script to find the image-matrix correlation
between each of these composite reference images to the images in each time-
lapse to estimate the timing of each morphological stage via the local
correlation maximum (Figure S2A).
Of the 36 events, the eight most unambiguous events (Figure S3), identifiable
regardless of embryo orientation, were selected for refinement and further
analysis (pole bud appears, membrane reaches yolk, pole cell invagination,
amnioproctodeal invagination, amnioserosa exposed, clypeolabrum retracts,
heart-shaped midgut, and trachea fill) (Figure S2B,C). Using a Python-scripted
graphical user interface, each of the eight events in every movie was manually
examined and the algorithm prediction adjusted when necessary. Timing of
hatching was excluded from these nine primary events because it was highly
variable, likely due to the assay conditions following dechorionation, and
suitable only as an indication of successful development, not as a reliable
and reproducible time point. The “membrane reaches yolk stage” was used
throughout as a zero point due to the precision with which the stage could be
identified in all species and from all orientations.
Links to representative time-lapse videos are provided in Table 3.
### D. melanogaster embryogenesis scales uniformly with temperature
As expected, the total time of embryogenesis of D. melanogaster had a very
strong dependence on temperature (Figure 3,
http://www.youtube.com/watch?v=-yrs4DcFFF0 or “D. melanogaster at 7
temperatures” in DOI:10.5061/dryad.s0p50). From 17.5∘C to 27.5∘C, there was a
two-fold acceleration in developmental rate, matching the previously observed
doubling of total lifespan with a 10∘C change in temperature [6]. The velocity
of embryogenesis at 30∘C is roughly the same as at 27.5∘C, and is appreciably
slower at 32.5∘C, likely due to heat stress. At 35∘C, successful development
becomes extremely rare.
To examine how these temperature-induced shifts in the total time of
embryogenesis were reflected in the relative timing of individual events, we
rescaled the time series data for each embryo so that the time from our most
reliable early landmark (the end of cellularization) to our most reliable late
landmark (trachea filling) was identical, and examined where each of the
remaining landmarks fell (Figure 3C). We were surprised to find that D.
melanogaster exhibited no major changes in its proportional developmental time
under any of the non-stressful temperature conditions tested. Therefore, at
least as far as most visually evident morphological features go, embryogenesis
scales uniformly across a two-fold range of total time.
When the embryos were under heat stress ($>$30∘C), we observed a very slight
contraction in the proportion of time between early development (pole bud
appears) to the end of cellularization (membrane reaches yolk), and a slight
contraction between the end of cellularization and mid-germ band retraction
(amnioserosa exposure).
### Embryogenesis scales uniformly across species despite significant
differences in temperature dependent developmental rate
In each of the ten additional Drosophila species we examined we observed all
of the 36 developmental landmarks we identified in D. melanogaster in the same
temporal order (Figure 4A). However, there was marked interspecies variation
in both the total time of embryogenesis at a given temperature (Figure 4B-E,
Table 3) and the way embryogenesis time varied with temperature (Figure 5).
When we examined the 10 remaining species, we found not only that the relative
timing of events was constant across temperature within a species, as observed
in D. melanogaster, but that landmarks occurred at the same relative time
between species at all non-stressful temperatures (Figures 6, Table 4).
### Developmental time is exponentially related to $\alpha/T$
Between 17.5∘C and 27.5∘C the total developmental time for all species can be
approximated relatively accurately by an exponential regression ($R^{2}>0.9$).
For all species we find that temperature T can be related to developmental
time $t_{dev}$, agreeing with a long history of temperature-dependent rate
modeling [24]:
$t_{dev}\approx e^{\frac{\alpha}{T}}$
and developmental rate v:
$ln(v)\approx-\frac{\alpha}{T}$
The parameters of these relations for each species, which includes two
independent coefficients, are included in Table 5. Also included in Table 5 is
the $Q_{10}$, an empirical description of biological rate change from a 10∘C
temperature change, for the 17.5∘C to 27.5∘C interval. At higher temperatures,
heat stress appears to counter the logarithmic trend and lengthens
developmental time. Since the temperature responses are highly reproducible,
the developmental time for each species can be modeled and predictions made
for future experiments (Figure S4).
### Effect of temperature on developmental rate is coupled to climatic origin
Seven of the eleven species we examined were of tropical origin, with only two
alpine, one subtropical and one temperature species. At mid-range temperatures
(22.5∘C - 27.5∘C), the tropical species developed the fastest, followed by the
subtropical D. mojavensis, the alpine D. pseudoobscura and D. persimilis, and
the temperate D. virilis (Figure 5), in accord with [18].
Some tropical species have expanded into temperature zones and a variety of
wild strains have been collected from a variety of climates. We examined nine
additional strains of D. melanogaster collected along the eastern United
States [25, 26]. Though collected along a tropical to temperate cline and
there was some variation between strains, no trends were seen (Figure S5A,B).
The tropical species all showed highly similar responses to temperature, even
though they originate from different continents (Africa, Asia and South
America) and are not closely related (five of the species are in the
melanogaster subgroup, but D. ananassae and D. willistoni are highly diverged
from both D. melanogaster and each other). Though they possess similar
temperature-responses, these species possess significantly different and
independent temperature response curves ($p<0.05$) and the differences are
large enough to be relevant for precise developmental experiments. These
cross-species differences tend to be, but are not necessarily, larger than
those seen between D. melanogaster strains (Figure S5C). The embryogenesis
rate for these species increases rapidly with temperature ($Q_{10}\sim 2.2$)
before slowing down at and above 30∘C (Figure S6A-F,
http://www.youtube.com/watch?v=vy6L4fmWkso or “D. ananassae at 7 temperatures”
in DOI:10.5061/dryad.s0p50). The two closely related alpine species (D.
pseudoobscura and D. persimilis) match the embryogenesis rate of the tropical
species at 17.5∘C, but accelerate far less rapidly with increasing temperature
($Q_{10}\sim 1.6$), especially at 25∘C and above (Figure S6I,J,
http://www.youtube.com/watch?v=sYi-FUXpv4Q or “D. pseudoobscura at 6
temperatures” in DOI:10.5061/dryad.s0p50). These species also show a sharp
increase in embryogenesis rate and low viability above 27.5∘C, consistent with
their cooler habitat. The subtropical D. mojavensis (Figure S6H,
http://www.youtube.com/watch?v=XWMs4oUx_mU or “D. mojavensis at 6
temperatures” in DOI:10.5061/dryad.s0p50) and temperate D. virilis (Figure
S6G, http://www.youtube.com/watch?v=eyr4ckDb0kM or “D. virilis at 6
temperatures” in DOI:10.5061/dryad.s0p50) both develop very slowly at low
temperature, but accelerate rapidly as temperature increases ($Q_{10}$ of
$\sim 2.5$ and $\sim 2.2$ respectively). D. virilis remains the slowest
species up to 30∘C, while D. mojavensis is as fast as the tropical species at
high temperatures. These species are both members of the virilis-repleta
radiation and it remains to be seen if this growth response is characteristic
of the group as a whole, independent of climate.
### Effects of heat stress
Under heat-stress, the proportionality of development is disrupted in some
embryos (Figure S7A). The effect is not uniform, as some embryos developed
proportionally under heat-stress and others exhibited significant aberrations,
largely focused in post-germband shortening stages. This can be most clearly
seen in individuals of D. ananassae, D. mojavensis, D. persimilis, and D.
pseudoobscura. We did not identify any particular stage as causing this delay,
but rather it appears to reflect a uniform slowing of development.
Early heat shock significantly disrupts development enough to noticeably
affect morphology in yolk contraction, cellularization, and gastrulation
(Figure S7B). Syncytial animals are the most sensitive to heat-shock (Figure
S7C). In D. melanogaster and several other species we observed a slight
contraction of proportional developmental time between early development (pole
bud appears) and the end of cellularization (membrane reaches yolk) under
heat-stress ($>$ 30∘C, Figure S7D). While all later stages following
cellularization maintain their proportionality even at very high temperatures,
the pre-cellularization stages take proportionally less and less time. This
indicates that at higher temperatures, some pre-cellularization kinetics scale
independently of later stages, possibly leading to mortality as the
temperature becomes more extreme.
## Discussion
We have addressed the lack of good data on the progress of embryogenesis in
different species and at different temperatures with a carefully collected and
annoted series of time-lapse movies in 11 species at seven temperatures that
span most of the viable range for Drosophila species. From a practical
standpoint, the predictable response of each species to temperature, and the
uniform scaling of events between species and temperature, provides a
relatively simple answer to the question that motivated this study - to
determine how to obtain matched samples for genomic studies: simply choose the
range of stages to collect in one strain or species, and scale the collection
and aging times appropriately. The fact that development scales uniformly over
non-extreme temperatures would seem to give some leeway in the choice of
temperature, so long as heat-stress is avoided, though it remains unclear how
molecular processes are affected by temperature.
### Uniform scaling and the timing of embryogenesis
In carrying out this survey, we were surprised to find that the relative
timing of landmark events in Drosophila embryogenesis is constant across
greater than three-fold changes in total time, spanning 15∘C and over 100
million years of independent evolution. And the fact that the same holds true
for 34 developmental landmarks at two temperatures in the zebrafish Danio
rerio [27], (the only other species for which we were able to locate similar
data), suggests that this phenomenon may have some generality. But why is this
so?
Drosophila development involves a diverse set of cellular processes including
proliferation, growth, apoptosis, migration, polarization, differentiation,
and tissue formation. One might expect (we certainly did) these different
processes to scale independently with temperature, much as different chemical
reactions do, and as a result, different stages of embryogenesis or parts of
the developing embryo would scale differentially with temperature. But this is
not the case. The simplest explanation for this observation is that a single
shared mechanism controls timing across embryogenesis throughout the genus
Drosophila. But what could such a mechanism be? One possibility is that there
is an actual clock —some molecule or set of molecules whose abundance or
activity progresses in a clocklike manner across embryogenesis and is read out
to trigger the myriad different processes that occur in the transition from a
fertilized egg to a larvae. However there is no direct evidence that such a
clock exists (although we note that there is a pulse of ecdysone during
embryogenesis with possible morphological functions [28, 29]). A more likely
explanation is that there is a common rate limiting process throughout
embryogenesis. Our data are largely silent on what this could be, but we know
from other experiments that it is cell, or at least locally, autonomous [30,
31, 32] and would have to limit processes like migration that do not require
cell division (we also note that cell division has been excluded as a
possibility in zebrafish [32]). However, energy production, yolk utilization,
transcription or protein synthesis are reasonable possibilities.
Although there are very few comparisons of the relative timing of events
during development, it has long been noted that various measurements of
developmental timing scale exponentially with $\alpha/T$ [5, 24, 6, 1, 33],
but no good explanation for this phenomenon has been uncovered. Perhaps
development is more generally limited by something that scales exponentially
with $\alpha/T$, like metabolic rate, which, we note, has been implicated
numerous times in lifespan, which is, in some ways, a measure of developmental
timing.
Gillooly and co-workers, noting the there was a relationship between metabolic
rate, temperature and animal size, have proposed a model that incorporates
mass into the Arrhenius equation to explain the relationship between these
factors in species from across the tree of life [34, 35]. We, however, do not
find that mass can explain the differences in temperature-dependence between
species. Even closely-related species, with nearly 2-fold differences in their
mass (e.g. D. melanogaster, D. simulans, D. sechellia, D. yakuba, and D.
erecta), have significant divergence in their proportionality coefficients
that do not converge at all when correcting for differences in mass through
the one quarter power scaling proposed by Gillooly, et al. This suggests that
some other factor is responsible for the differences, as has been argued by
other groups [36, 37, 18]. The relationship between climate and temperature
response raises the possibility that whatever this factor is has been subject
to selection to tune the temperature response to each species’ climate.
However, without additional data this is purely a hypothesis.
Although a common rate-limiting step is simplest explanation for uniform
scaling, it is certainly not the only one. It is possible that different rate
limiting steps or other processes control developmental velocity at different
times or in different parts of the embryo, and that they scale identically
with temperature either coincidentally, or as the result of selection (it is
important to remember that, as per Arrhenius, one does not expect different
reactions to scale identically with temperature). If this is the result of
selection, what is the selection pressure? Evolutionary developmental
biologists, perhaps most notably Stephen J. Gould, have long written about how
changes in either the absolute or relative timing of different events during
development have had significant effects on morphology throughout animal
evolution [38, 39, 40, 41]. Perhaps this is also true for fly embryogenesis,
but that any such changes in morphology are selectively disadvantageous and
have been strongly selected against. It is also likely that many developing
fly embryos experience significant changes in temperature while developing, so
there may be strong selection to maintain uniform development across
temperature to ensure normal progression while the temperature is changing.
Finally, we note that there are limits to this uniformity. At extreme
temperatures, especially high ones, things no longer scale uniformly, likely
reflecting the differential negative effects of high temperature at different
stages of embryogenesis as well as the differential ability of the embryo to
compensate for them. There are also clearly checkpoints in place that, while
not triggered during normal embryogenesis, are important in extreme or unusual
circumstances. Most strikingly, when Lucchetta et al. and Niemuth et al.
examined embryos developing in chambers that allowed for independent
temperature control of the anterior and posterior portions of the embryo, the
two parts of the embryo developed at different velocities for much of
embryogenesis [30, 31]. They found that embryos are robust to asynchrony in
timing across the embryo, though there are critical periods that, once passed,
do not permit re-synchronization of development [30], hinting at some specific
checkpoints or feedback.
### Climate and the rate of embryogenesis
The clustering of developmental timing and its temperature response with
climate —especially amongst tropical species from different continents and
parts of the Drosophila tree —suggests that this is an adaptive, or in some
cases permissive, phenotype, although with only 11 species and poor coverage
of non-tropical species this has to remain highly speculative. There are
necessarily additional components to the temperature response, as significant
variation exists within the tropical species and between D. melanogaster
strains. The virilis-repleta radiation, which includes both D. virilis and D.
mojavensis may have a climate-independent adaptation that leads to slowed
development at cooler temperatures, a feature that is hard to rationalize. The
poor response of the alpine D. pseudoobscura and D. persimilis to high
temperature is consistent with their cool climate. Nevertheless, little is
known about when and where most of these species lay their eggs and their
natural microclimates.
The clustering of developmental responses in species by their native climates
rather than their climates of collection suggests that if climate adaptation
is a contributing factor, the response arises slowly or rarely. The tested
strains of D. melanogaster were collected in temperate, subtropical, and
tropical climates and the D. simulans strain was collected in a sub-tropical
climate. Nevertheless, both species performed qualitatively like other
tropical species and unlike native species collected nearby. This suggests
that temperature responses are neither rapidly evolving (with D. melanogaster
being present in the temperate United States for over 130 years [42]) nor
primed for change in tropical species.
## Materials and Methods
### Rearing of Drosophila
Drosophila strains were reared and maintained on standard fly media at 25∘C,
except for D. persimilis and D. pseudoobscura which were reared and maintained
at 22∘C. D. melanogaster lines were raised at 18∘C and 22∘C for several years
and their temperature response profiles were observed, verifying that
transferring embryos from the ambient growth temperature for a line to the
experimental temperature did not lead to heat-shock responses and had
relatively little impact on the temperature response (Figure S8A,B). Egg-lays
were performed in medium cages on 10 cm molasses plates for 1 hour at 25∘C
after pre-clearing for all species except D. persimilis, which layed at 22∘C.
Comparisons to D. melanogaster raised and laying at 22∘C confirmed that growth
at lower temperatures does not account for all of the differences between the
tropical and alpine species (Figure S8C).To encourage egg-lay, cornmeal food
media was added to plates for D. sechellia and pickled cactus was added to
plates for D. mojavensis. Embryos were collected and dechorionated with fresh
50% bleach solution (3% hypochlorite final) for 45 to 90 seconds (based on the
species) in preparation for imaging. Dechorionation timing was selected as the
time it took for 90% of the eggs to be successfully dechorionated. This
prevented excess bleaching, as many species, such as D. mojavensis, are more
sensitive than D. melanogaster. Strains used were D. melanogaster, OreR, DGRP
R303, DGRP R324, DGRP R379, DGRP R380, DGRP R437, DGRP R705, Schmidt Ln6-3,
Schmidt 12BME10-24, and Schmidt 13FSP11-5; D. pseudoobscura, 14011-0121.94,
MV2-25; D. virilis, 15010-1051.87, McAllister V46; D. yakuba, 14021-0261.01,
Begun Tai18E2; D. persimilis, 14011-0111.49,(Machado) MSH3; D. simulans,
14021-0251.195, (Begun) simw501; D. erecta, 14021-0224.01, (TSC); D.
mojavensis wrigleyi, 15081-1352.22, (Reed) CI 12 IB-4 g8; D. sechellia,
14021-0248.25, (Jones) Robertson 3C; D. willistoni, 14030-0811.24, Powell
Gd-H4-1; D. ananassae, 14024-0371.13, Matsuda (AABBg1).
### Time-lapse Imaging
Embryos were placed on oxygen-permeable film (lumox, Greiner Bio-one), affixed
with dried heptane glue and then covered with Halocarbon 700 oil (Sigma) [43].
The lumox film was suspended on a copper plate that was temperature-regulated
with two peltier plates controlled by an H-bridge temperature controller
(McShane Inc., 5R7-570) with a thermistor feedback, accurate to $\pm$0.1∘C.
Time-lapse imaging with bright field transmitted light was performed on a
Leica M205 FA dissecting microscope with a Leica DFC310 FX camera using the
Leica Advanced Imaging Software (LAS AF) platform. Greyscale images were saved
from pre-cellularization to hatch. Images were saved every one to five
minutes, depending on the temperature. A humidifier was used to mitigate
fluctuations in ambient humidity, though fluctuations did not affect
developmental rate. Due to fluctuations in ambient temperature and humidity,
the focal plane through the halocarbon oil varied significantly. Therefore,
z-stacks were generated for each time-lapse and the most in-focus plane at
each time was computationally determined for each image using an algorithm
(implemented in Matlab) through image autocorrelation [44, 45]. Time-lapse
videos available from Dryad Digital Repository: doi:10.5061/dryad.s0p50
### Event estimation
A subset of time-lapses in D. melanogaster were analyzed to obtain a series of
representative images for each of the 36 morphological events, selected as all
events defined by [19, 46] that were reproducibly identifiable under our
conditions, described. These images were sorted based on embryo orientation
and superimposed to generate composite reference images. Images from each
time-lapse to be analyzed were manually screened to determine the time when
the membrane reaches the yolk, the time of trachea filling, and the
orientation of the embryo (Figure S3. This information was fed into a Matlab
script, along with the time-lapse images and the set of 34 composite reference
images, to estimate the time of 34 morphological events during embryogenesis
via image correlation. The same D. melanogaster reference images were used for
all species for consistency. A correlation score was generated for each frame
of the time-lapse. The running score was then smoothed (Savitzky-Golay
smoothing filter) and the expected time window was analyzed for local maxima.
The error in event calling for the computer is very large (greater than what
we see for the overall spread across individuals of a single species at a
given temperature), necessitating manual verification or correction of events.
Many of these errors are due to aberrations in the image that confuse the
computer but would not confuse a person. This results in a few bad images
having a very negative effect of the overall accuracy of the computer
analysis, but permits a significant improvement with just a little user input.
The error in manual calls is very small compared to the variation between
individuals. Computer-aided estimates were individually verified or corrected
using a python GUI for all included data.
### Statistical analysis
Statistical significance of event timing was determined by t-test with
Bonferonni multiple testing corrections. Median correction to remove outliers
was used in determining the mean and standard deviation of each developmental
event. Least-squares fitting was used to determine the linear approximation of
log-corrected developmental time for each species. Python and Matlab scripts
used in the data analysis are available at
github.com/sgkuntz/TimeLapseCode.git.
## Acknowledgments
We obtained flies from the Bloomington and UCSD stock centers. We thank Paul
Schmidt for the clinal fly lines. We would also like to thank T. Kaplan and P.
Combs for their advice and assistance on data analysis and programming, C.
Bergman for comments, Ng Wei Tian for his work on event verification, and BCK
and EDKK for their support.
## References
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## Figures
Figure 1: Geographic and climatic origin and phylogeny of analyzed Drosophila
species (A) Ancestral ranges are shown for each species [47, 17, 48]. While D.
melanogaster and D. simulans are now cosmopolitan and D. ananassae is
expanding in the tropics (green), their presumed ancestral ranges are shown.
D. virilis is holarctic (gray) and restricted from the tropics, with a poor
understanding of its ancestral range. Other species are more or less found in
their native ranges, covering a variety of climates. Sites of collection are
noted by arrows. (B) The phylogeny of the sequenced Drosophila species. Many
of the tropical species are closely related, though D. willistoni serves as a
tropical out-group compared to the melanogaster and obscura groups. Branch
lengths are based on evolutionary divergence times [49]. (C) Range sizes vary
considerably between the species. Figure 2: Developmental landmarks used in
study Many images of each stage (examples on the left) were averaged to
generate composite images (lateral view on the right) for each of the
developmental stages, of which 29 are shown. Figure 3: Developmental time of
D. melanogaster varies with temperature (A) Images of developing D.
melanogaster embryos at each temperature are shown for a selection of stages
to highlight the overall similarity of development. (B) The time individual
animals reached the various time-points are shown, with each event being a
different color. Time 0 is defined as the end of cellularization, when the
membrane invagination reaches the yolk. Between 17.5∘C and 27.5∘C the total
time of embryogenesis, $t_{dev}$ measured as the mean time between
cellularization and trachea fill, has a logarithmic relationship to
temperature described by $t_{dev}=4.02e^{37.31/T}$ where T is temperature in
∘C ($R^{2}=0.963$). (C) The developmental rate in D. melanogaster changes
uniformly with temperature, not preferentially affecting any stage. Timing
here is normalized between the end of cellularization and the filling of the
trachea. Figure 4: Drosophila species develop at different rates and respond
to temperature in distinct ways (A) Images of developing embryos of each
species are shown to scale. All species go through the same stages in the same
order at all viable temperatures. (B) At 17.5∘C all species show uniformly
long developmental times, with D. virilis and D. mojavensis being
significantly longer than other species. (C) At 22.5∘C and (D) 27.5∘C there is
considerably more variation between species. While developmental times
decrease with increasing temperature across all species, the effect is muted
in the alpine species. (E) At 30∘C, developmental rate has stopped
accelerating and the alpine species are seeing considerable slow-down in
development time. Figure 5: Temperature dependent developmental rates are
climate specific The time between the end of cellularization and trachea fill
are shown for all species at a range of temperatures. The climatic groups –
tropical (warm colors), alpine (blues), temperate (purple), and sub-tropical
(green) – clearly stand out from one another to form four general trends.
Figure 6: Proportionality of developmental stages is not affected by non-heat-
stress temperatures (A) Across species, development maintains the same
proportionality. D. pseudoobscura stands out as not being co-linear at higher
temperatures. Instead, the later part of its development is slowed and takes
up a disproportionally long time. (B) Plotting proportionality across all
species and all temperatures reveals the approximately normally distributed
proportionality of all morphological stages.
## Tables
Table 1: Drosophila species and strains Species | Stock number | Strain | Collection site
---|---|---|---
D. melanogaster | | OreR | Oregon, USA
D. pseudoobscura | 14011-0121.94 | MV2-25 | Mesa Verde, Colorado, USA
D. virilis | 15010-1051.87 | McAllister V46 | unknown, possibly Asia
D. yakuba | 14021-0261.01 | Begun Tai18E2 | Liberia
D. persimilis | 14011-0111.49 | Machado MSH3 | Mt. St. Helena, California, USA
D. simulans | 14021-0251.195 | Begun simw501 | Mexico City, Mexico
D. erecta | 14021-0224.01 | (TSC) | unknown, probably Africa
D. mojavensis wrigleyi | 15081-1352.22 | Reed CI 12 IB-4 g8 | Catalina Island, California, USA
D. sechellia | 14021-0248.25 | (Jones) Robertson 3C | Cousin Island, Seychelles
D. willistoni | 14030-0811.24 | Powell Gd-H4-1 | Guadeloupe Island, France
D. ananassae | 14024-0371.13 | Matsuda (AABBg1) | Hawaii, USA
Table 2: Major morphological events in Drosophila development Event | Stage [19, 46] | Notes
---|---|---
Posterior gap appears | 2 | Gap between yolk and vitelline membrane
Pole bud appears | 3 | Cells migrate into the posterior gap
Nuclei at periphery | 4 | Cells migrate to edges
Pole cells form | 4 | Replication of the pole cells
Yolk contraction | 4 | Light edge of embryo expands
Cellularization begins | 5 | Cell cycle 14
Membrane reaches the yolk | 5 | This is regarded as the zero time-point
Pole cells migrate | 6 | Pole cells begin anterior movement
Cephalic furrow forms | 6 | Dorsal and ventral furrows form
Pole cells invaginate | 7 | Pole cells enter dorsal furrow
Transversal fold formation | 7 | Dorsal furrows between amnioproctodeum and cephalic furrow
Cephalic furrow reclines | 8 | Dorsal furrow moves posteriorly
Amnioproctodeal invagination | 8 | Invagination approaches cephalic fold
Anterior midgut primordial | 8 | Tissue thickens at anterior ventral edge
Stomodeal plate forms | 9 | Ventral gap anterior to cephalic fold
Stomodeum invagination | 10 | Ventral furrow anterior to cephalic fold
Clypeolabral lobe forms | 10 | Dorsal, ventral furrows both present
Germ band maxima | 11 | Maximum extension of germband
Clypeolabrum rotates | 11 | Clypeolabrum shifts dorsally
Posterior gap | 11 | Gap forms before germband shortening
Gnathal bud appears | 12 | Ventral tissue between the clypeolabrum and cephalic folds moves anteriorly
Germband retraction begins | 12 | Movement begins mid-germband
Amnioserosa exposed | 12 | Germband retracted to the posterior 30% of the embryo
Germband retracted | 13 | Germband fully retracted
Dorsal divot | 14 | Dorsal gap between head and amnioserosa
Clypeolabrum retracts | 14 | Clypeolabrum pulls away from anterior vitelline membrane
Anal plate forms | 14 | Posterior depression forms
Midgut unified | 14 | Dark circle forms at embryo’s center
Heart-shaped midgut | 15 | Triangular midgut
Clypeolabrum even with ventral lobes | 16 | Ventral lobes move anteriorly to be even with clypeolabrum
Gnathal lobes pinch | 16 | Gnathal lobes meet
Convoluted gut | 16 | Separation between sections of the midgut
Head involution done | 17 | Head lobes complete anterior migration
Muscle contractions | 17 | Head begins twitching
Trachea fills | 17 | Developmental end point
Hatch | 17 | Highly variable
Table 3: Drosophila development videos
Subject | Link
---|---
D. melanogaster with labelled stages | http://www.youtube.com/watch?v=dYSrXK3o86I
D. melanogaster with labelled stages at reduced framerate | http://www.youtube.com/watch?v=QKVmRy3dDR0
D. melanogaster at 7 temperatures | http://www.youtube.com/watch?v=-yrs4DcFFF0
11 species at 17.5∘C | http://www.youtube.com/watch?v=HId_Idz-GhQ
11 species at 22.5∘C | http://www.youtube.com/watch?v=jO6JfgwMaH4
11 species at 27.5∘C | http://www.youtube.com/watch?v=vlYeuFqKQhI
D. ananassae at 7 temperatures | http://www.youtube.com/watch?v=vy6L4fmWkso
D. mojavensis at 6 temperatures | http://www.youtube.com/watch?v=XWMs4oUx_mU
D. virilis at 6 temperatures | http://www.youtube.com/watch?v=eyr4ckDb0kM
D. pseudoobscura at 6 temperatures | http://www.youtube.com/watch?v=sYi-FUXpv4Q
* All videos available at DOI:10.5061/dryad.s0p50
Table 4: The timing of specific developmental events can be predicted as a function of total developmental time Stage | Event Timing (hours post cellularization) | Percent Error
---|---|---
Pole bud appears | $t_{pba}\approx-0.093t_{dev}$ | 8%
Pole cells invaginate | $t_{pci}\approx 0.018t_{dev}$ | 40%
Amnioproctodeal invagination | $t_{api}\approx 0.035t_{dev}$ | 18%
Amnioserosa exposed | $t_{ase}\approx 0.35t_{dev}$ | 6%
Clypeolabrum retracts | $t_{clr}\approx 0.49t_{dev}$ | 4%
Heart-shaped midgut | $t_{hsm}\approx 0.57t_{dev}$ | 12%
Table 5: The developmental time of embryos between 17.5∘C and 27.5∘C is a
species-specific function of temperature
Species | Developmental Time* | $R^{2\dagger}$ | 95% Confidence Prediction Interval for Future Observations | $Q_{10}^{\ddagger}$ (27.5:17.5)
---|---|---|---|---
D. virilis# | $t_{Dvir}=5.64e^{37.08/T}$ | 0.989 | $t_{Dvir}\pm 31.937\sqrt{1.00+(\frac{1}{T}-0.04)^{2}}$ | 2.2
D. mojavensis# | $t_{Dmoj}=3.67e^{43.81/T}$ | 0.983 | $t_{Dmoj}\pm 54.263\sqrt{1.00+(\frac{1}{T}-0.05)^{2}}$ | 2.5
D. willistoni | $t_{Dwil}=3.63e^{40.50/T}$ | 0.944 | $t_{Dwil}\pm 3.122\sqrt{1.00+(\frac{1}{T}-0.04)^{2}}$ | 2.3
D. pseudoobscura | $t_{Dpse}=7.61e^{25.95/T}$ | 0.903 | $t_{Dpse}\pm 39.257\sqrt{1.00+(\frac{1}{T}-0.05)^{2}}$ | 1.7
D. persimilis | $t_{Dper}=9.31e^{21.20/T}$ | 0.961 | $t_{Dper}\pm 22.598\sqrt{1.00+(\frac{1}{T}-0.05)^{2}}$ | 1.6
D. ananassae | $t_{Dana}=2.94e^{42.68/T}$ | 0.979 | $t_{Dana}\pm 1.440\sqrt{1.00+(\frac{1}{T}-0.05)^{2}}$ | 2.4
D. yakuba | $t_{Dyak}=4.67e^{33.08/T}$ | 0.943 | $t_{Dyak}\pm 2.203\sqrt{1.00+(\frac{1}{T}-0.05)^{2}}$ | 2.0
D. erecta | $t_{Dere}=5.21e^{32.97/T}$ | 0.937 | $t_{Dere}\pm 2.689\sqrt{1.00+(\frac{1}{T}-0.04)^{2}}$ | 2.0
D. melanogaster | $t_{Dmel}=4.02e^{37.31/T}$ | 0.963 | $t_{Dmel}\pm 1.281\sqrt{1.00+(\frac{1}{T}-0.05)^{2}}$ | 2.2
D. sechellia | $t_{Dsec}=4.47e^{34.67/T}$ | 0.957 | $t_{Dsec}\pm 2.386\sqrt{1.00+(\frac{1}{T}-0.04)^{2}}$ | 2.1
D. simulans | $t_{Dsim}=3.50e^{39.14/T}$ | 0.960 | $t_{Dsim}\pm 1.883\sqrt{1.00+(\frac{1}{T}-0.05)^{2}}$ | 2.3
* End of cellularization to trachea fill in hours, where T is in ∘C
$\dagger R^{2}$, the Pearson Product-Moment’s Correlation Coefficient of
determination, is calculated following a least-squares regression across all
data points to a curve of the form ln(developmental time) = b($1\over{T}$)+a.
$\ddagger$ $Q_{10}$ is the ratio between developmental times across a 10
degree interval, in this case between 27.5∘C and 17.5∘C. A value of 2.2 would
indicate that development takes 2.2 times as long at 17.5∘C than at 27.5∘C.
$\\#$ Curve fit through 30∘C
## Supplementary Figures
Figure S1: Microscopy imaging setup (A) The imaging setup, showing the
dissecting microscope with temperature control apparatus on the automated
stage. (B) A close-up view of the temperature controlled platform flanked by
heat-sinks (blue) that sit atop the Peltier thermoelectric controllers. In the
center is a copper plate, with a thermister at the bottom to monitor plate
temperature. The holes in the green masking tape line up with holes drilled
through the copper plate and lined with a gas-permeable membrane. The masking
tape helps retain the halocarbon oil. (C) A closer view of the setup. (D) A
schematic of the setup demonstrates the temperature control and imaging
apparatus in cross-section. Figure S2: Events were predicted by computational
analysis before manual verification (A) For every time-lapse, each frame was
correlated to each of the 34 composite images. (B) The running scores for 6
different events, with their maxima (black arrows) highlighted to reflect the
estimated event time. (C) The time of amnioserosa exposure is estimated by the
strong correlation at about 450 frames into the time-lapse. Figure S3:
Identifying morphological stages (A) ‘Pole bud appears’ stage is identified by
the first appearance of cells migrating into the posterior gap of the embryo
(black arrow). (B) ‘Membrane reaches yolk’ stage is identified by the
converging of the leading edge of the invaginating cytoplasmic membrane on the
dark yolk. (C) ‘Pole cell invagination’ is identified by the completion of the
fold (black arrow) that encapsulates the pole cells (yellow arow). (D)
‘Amnioproctodeal invagination’ is identified by the point when the leading
edge of the posterior invagination (black arrow) has covered $\sim$80% of the
distance to the leading edge of the cephalic furrow (vertical yellow line) and
the pole cells have reached the interior of the embryo. (E) ‘Amnioserosa
exposed’ is identified by the point when the trailing edge of the germ band
has retracted to the posterior 30% of the embryo. (F) ‘Clypeolabrum retracts’
is identified by the withdrawal of the ventral edge of the clypeolabrum (black
arrow) from the gnathal buds and vitelline membrane to create a gap (black
arrowhead). (G) ‘Heart-shaped midgut’ is identified by the posterior
elongation of the formerly spherical developing midgut and residual yolk (dark
mass in the center of the embryo) to form a contiguous dark teardrop or heart-
shaped mass (delimited with yellow lines). (H) ‘Trachea fill’ is identified by
the rapid darkening of the trachea as they fill. The primary branches of the
trachea run along the both the left and right dorsal sides, originating at the
posterior of the embryo. Figure S4: Prediction of future observations of
development at different temperatures The behavior of developing embryos can
be predicted. The mean line (green) generated from least-squares curve-fitting
(Table 5) and the 95% confidence prediction interval for future observations
(dashed orange line) are shown for each species. Figure S5: Different D.
melanogaster wild isolate strains exhibit a limited range of temperature
responses (A) Lines (R303, R324, R379, R380, R437, and R705) collected near
Raleigh, North Carolina [25] exhibit a range of temperature responses. (B)
Clinal lines from Florida (DmelFL), Pennsylvania (DmelPA), and Maine (DmelME)
[26] exhibit a range of responses similar to those of the Raleigh lines.
Despite their clinal distribution, no trends are seen, with flies from Florida
and Maine being virtually indistinguishable. This is possibly due to their
relatively recent introduction across the cline. (C) Despite the differences
between the D. melanogaster lines above, they all (seen here grouped together
as light blue points) lie within the response range seen for the melanogaster
species subgroup, mainly falling between the responses of D. melanogaster
Ore-R and D. erecta. Like Ore-R, their growth is significantly slower than D.
yakuba, D. ananassae, D. simulans, and D. sechellia, but obeys the same
general trend. Figure S6: Ten species of Drosophila exhibit dynamic response
to temperature changes (A-F) There is some variation species to species, but
all tropical Drosophila exhibit a similar temperature response-curve to D.
ananassae. (G) Temperate D. virilis also has a steep response, though
intermediate to the previous two groups. (H) Sub-tropical D. mojavensis has a
steeper temperature response, though a similar high temperature developmental
time. (I,J) Alpine D. pseudoobscura and D. persimilis have a cold response
like the tropical species, but longer developmental times at warmer
temperatures. Figure S7: Heat-stress affects syncytial developmental
proportionality and morphology (A) At heat-stress temperatures, the
proportionality of developmental stages is affected in some, but not all,
embryos. (B) Heat stress in D. melanogaster at 32.5∘C affects morphology
during yolk contraction and gastrulation. Embryos may exhibit asynchronous
yolk-contraction (first image), uneven nuclear distribution during
cellularization (second image), or disrupted morphology during gastrulation
(third image). (C) Heat shock at 37.5∘C for 30 minutes reveals embryos
sensitivity prior to the completion of cellularization. Most animals that had
completed cellularization survived heat-shock and continued to develop
properly (blue diamonds), while no animals that had not completed
cellularization prior to heat-shock survived. All embryos that died (orange
stars) exhibited severe morphological disruptions. (D) Linear regression of
stages across different temperatures reveals that, despite significant
variance in later stages (shown in colored bars), only the pre-cellularization
time point is affected by heat-stress enough to exhibit a significantly
different slope between higher temperatures (27.5∘C and above, yellow bar) and
lower temperatures (25∘C and below, red bar). Figure S8: Temperature
conditioning of adult flies leads to some heat tolerance (A) D. melanogaster
raised for many generations at 25 ∘C, 22 ∘C, and 18 ∘C produce embryos that
show similar temperature responses, though there is some accelerated growth
when acclimatized to higher temperatures. There is no indication of severe
heat shock as embryos are moved from the acclimatized temperature to the
experimental temperature. (B) D. mojavensis and D. virilis exhibit a similar
trend of only minor differences between strains acclimatized at 25 ∘C and 22
∘C. (C) The difference between D. melanogaster raised at 22 ∘C and D.
persimilis also raised at 22 ∘C remains significant, indicating that the heat-
stress response of D. persimilis is not due simply to its being raised at 22
∘C.
|
arxiv-papers
| 2013-06-22T07:01:31 |
2024-09-04T02:49:46.828016
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Steven G. Kuntz and Michael B. Eisen",
"submitter": "Steven Kuntz",
"url": "https://arxiv.org/abs/1306.5297"
}
|
1306.5543
|
Control and manipulation of electromagentically induced transparency in a nonlinear optomechanical system with two movable mirrors
S.Shahidani ^1
M. H. Naderi^1,2
M. Soltanolkotabi^1,2
$^{1}$ Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran
$^{2}$ Quantum Optics Group, Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran
We consider an optomechanical cavity made by two moving mirrors which contains a Kerr-down conversion nonlinear crystal. We show that the coherent oscillations of the two mechanical oscillators can lead to splitting in the electromagnetically induced transparency (EIT) resonance, and appearance of an absorption peak within the transparency window. In this configuration the coherent induced splitting of EIT is similar to driving a hyperfine transition in an atomic Lambda-type three-level system by a radio-frequency or microwave field. Also, we show that the presence of nonlinearity provides an additional flexibility for adjusting the width of the transparency windows. The combination of an additional mechanical mode and the nonlinear crystal suggests new possibilities for adjusting the resonance frequency, the width and the spectral positions of the EIT windows as well as the enhancement of the absorption peak within the transparency window.
37.30.+i, 03.67.Bg, 42.50.Wk, 42.50.Pq
§ INTRODUCTION
The coherent interaction of laser radiation with multi-level atoms can induce interesting phenomena such as electromagnetically induced transparency (EIT) and electromagnetically induced absorption (EIA).
EIT is a technique for turning an opaque medium into a transparent one and EIA is a technique for enhancement of absorption of light around resonance. These techniques have been used widely to manipulate the group velocity of light[1, 2, 3], for storage of quantum information [4, 5, 6], and for enhancement of nonlinear processes[7, 8, 9, 10].
Theoretical studies and technological advances in nanofabrication, laser cooling and trapping <cit.> have made it possible to reach a considerable control over the light-matter interaction in an optomechanical system. Optomechanically induced transparency (OMIT) and absorption (OMIA) are notable examples of light beam control in optomechanical systems. In OMIT which has been predicted theoretically<cit.> and demonstrated experimentally<cit.>, the anti-Stokes scattering of an intense red-detuned optical "control" field brings about a modification in the optical response of the optomechanical cavity making it transparent in a narrow bandwidth around the cavity resonance for a probe beam.
In analogy to the atomic EIT, the happening of OMIT is accompanied by a sharp negative derivative of the dispersion profile of the cavity near the resonance and subluminal group velocity for the probe field [12, 13].
In the atomic EIT the possibility of modification of the probe laser absorption, splitting and reshaping of the EIT peak and reduction of the EIT linewidth have been studied widely[14, 15, 16, 17, 18, 19, 20].
In this work we are interested in the engineering and control of the probe response, specially OMIT resonance, in the presence of an additional mechanical mode and the Kerr-down conversion nonlinearity. To this end, we consider a cavity with two moving mirrors, driven by a strong coupling and a weak probe field which contains a nonlinear crystal consisting of a Kerr medium and a degenerate optical parametric amplifier (OPA). This exploration is motivated by the following reasons.
First, in a recent theoretical work [21] it has been shown that the coherent coupling between the two cavity modes and the mechanical mode of a moving mirror in a double cavity configuration of optomechanical system leads to the appearance of an absorption peak within the transparency window. In this configuration by changing the power of the electromagnetic field the switching between EIT and EIA is possible. This model is quite general and a variety of systems, which can be modeled by three coupled oscillators, can make the same response. A driven Fabry-Perot cavity with two vibrating mirrors can be effectively described by three coupled oscillators whenever a slightly difference in the mechanical frequencies leads to the center-of-mass-relative-motion coupling [22]. In this configuration the two mechanical oscillators are coupled to a single cavity mode.
Second, an OPA inside a cavity can considerably improve the optomechanical coupling, the normal mode splitting (NMS), and the cooling of the mechanical mirror<cit.>. This kind of cooling process which is accompanied by the enhancement of the effective damping rate of the mirror can be used to increase the width of the transparency window and reduce the group velocity of a propagating probe pulse.
Third, it has been predicted<cit.> that by tuning the Kerr nonlinearity in an optomechanical cavity one can use the cavity energy shift to reduce the photon number fluctuation and provide a coherently-controlled dynamics for the mirror.
Based on these reasons, we first investigate the effect of the additional mechanical oscillator on the OMIT resonance. We will show that if the coupling field oscillates close to the mechanical resonance frequencies there are two occasions of two-photon resonance for the probe and coupling lasers. Consequently, the coherent oscillations of the two mechanical oscillators give rise to splitting of the OMIT resonance, and appearance of an absorption peak within the transparency window. The coherent induced splitting of OMIT resonance in this configuration is similar to driving a hyperfine transition in an atomic $\Lambda$-type three-level system by a radio-frequency or microwave field.
Then we explore how EIT and EIA resonances respond to the presence of a Kerr-down conversion nonlinearity in the cavity. We will show that in the presence of Kerr-down conversion nonlinearity one can effectively control the width of the transparency window. Also, we demonstrate that to achieve a desirable control over the OMIT resonance the presence of both nonlinearities is needed.
In addition, for the three-mode nonlinear optomechanical system we show that the coherent oscillation of the center-of-mass mode which is responsible for the absorption peak and splitting in the transparency window, increases. This results in the increment of the central peak absorption.
Briefly, the combination of an additional mechanical mode and the nonlinear crystal suggests new possibilities for "engineering" the OMIT resonance.
§ THE PHYSICAL MODEL
The model we consider is an optomechanical cavity with two vibrating mirrors which contains a Kerr-down conversion nonlinear crystal (Fig.<ref>). The vibrating mirrors are treated as two independent quantum mechanical harmonic oscillator with resonance frequency $\Omega_{k}$, effective mass $m_{k}$, and energy decay rate $\gamma_{k}$ $(k=1,2)$, coupled to a common cavity mode having the resonance frequency $\omega_{0}$. The nonlinear crystal is composed of a degenerate OPA and a nonlinear Kerr medium. The cavity mode is coherently driven by a strong input coupling laser field with frequency $\omega_{c}$ and amplitude $\varepsilon_{c}$ as well as a weak probe field with frequency $\omega_{p}$ and amplitude $\varepsilon_{p}$ through the left mirror. Furthermore, the system is pumped by a coupling beam to produce parametric oscillation and induce the Kerr nonlinearity in the cavity. When the detection bandwidth is chosen such that it includes only a single, isolated, mechanical resonance and mode-mode coupling is negligible we can restrict to a single mechanical mode for each mirror so that the mechanical Hamiltonian of the mirrors is given by
\begin{equation} \label{Hm}
H_{m}= \sum_{k=1}^{2}( \frac{ p_{k}^2} {2m_{k}} +\frac{1} {2} m_{k} \Omega_{k}^2 q_{k}^2).
\end{equation}
Furthermore, in the adiabatic limit, in which the mirror frequencies are much smaller than the cavity free spectral range $c/2L$ ($c$ is the speed of light in vacuum and $L$ is the cavity length in the absence of the intracavity field) the photon scattering into the other modes can be neglected and we can restrict the model to the case of single-cavity mode[23, 24]. We also assume that the induced resonance frequency shift of the cavity and the nonlinear parameter of the Kerr medium are much smaller than the longitudinal-mode spacing in the cavity.
It should be noted that in the adiabatic limit, the number of photons generated by the Casimir, retardation, and Doppler effects is negligible [25, 26, 27].
(Color online) Schematic of the setup studied in the text. The cavity that consists of two movable mirrors contains a Kerr-down conversion system which is pumped by a coupling beam to produce parametric oscillation and induce Kerr nonlinearity in the cavity. The cavity mode is coherently driven by a strong input coupling laser field and a weak probe field through the left mirror.
Under this condition, the total Hamiltonian of the system can be written as
\begin{equation} \label{H}
\end{equation}
\begin{eqnarray}
H_{0}&=&\hbar\omega_{0} a^{\dagger} a+H_{m}
+\hbar g_{m} a^{\dagger} a (q_{1}-q_{2})\nonumber\\
&&+i\hbar (s_{in}(t) a^{\dagger}- s_{in}^{*}(t)a),\\
H_{1}&=&i \hbar G( e^ { i\theta} a^{\dagger 2} -e^ {-i\theta} a^2 )+\hbar\eta a^{\dagger 2} a^{2} .
\end{eqnarray}
The first term in $H_{0} $ is the free Hamiltonian of the cavity field with the annihilation (creation) operator $a(a^{\dagger})$, frequency $\omega_0$ and decay rate $\kappa$, $H_m$ is the free Hamiltonian of the mirrors given by Eq.(<ref>), the third term describes the optomechanical coupling between the cavity field and the mechanical oscillators due to the radiation pressure force, and the last term in $H_{0} $ describes the driving of the intracavity mode with the input laser amplitude $s_{in}(t)$. Also, the two terms in $H_{1} $ describe, respectively, the coupling of the intracavity field with the OPA and the Kerr medium; $G$ is the nonlinear gain of the OPA which is proportional to the pump power driving amplitude, $\theta$ is the phase of the field driving the OPA, and $\eta$ is the anharmonicity parameter proportional to the third order nonlinear susceptibility $\chi^{(3)}$ of the Kerr medium. We will solve this problem for the total driving field $s_{in}(t)=(\varepsilon_{c}+\varepsilon_{p}e^{-i(\omega_{p}-\omega_{c})t})e^{-i\omega_{c}t}$, where $\varepsilon_{c} =\sqrt{2\kappa P_{c}/\hbar \omega_{c}}$ $(\varepsilon_{p}=\sqrt{2\kappa P_{p}/\hbar \omega_{p}})$ and $P_{c}$ ($P_{p}$) are, respectively, the amplitude and power of the input coupling (probe) field.
The dynamics of the system is described by a set of nonlinear Langevin equations. Since we are interested in the mean response of the system to the probe field we write the Langevin equations for the mean values. In a frame rotating at the coupling laser frequency $\omega_{c}$, neglecting quantum and thermal noises we obtain
\begin{eqnarray}
\langle\dot{a}\rangle\; &=&-[ i (\omega_{0}-\omega_{c})+\kappa] \langle a\rangle- i g_{m}\langle a\rangle ( \langle q_{1}\rangle- \langle q_{2}\rangle ) \nonumber\\
2 i \eta \langle a^{\dagger}\rangle\langle a\rangle ^{2}+2G e^{i\theta}\langle a^{\dagger}\rangle +s_{in}(t),\\
\langle\dot{ q_{k}}\rangle &=& \langle p_{k}\rangle}/{m_{k}, \;(k=1,2),\\
\dot {\langle p_{k}\rangle}&=&-m_{k} \Omega_{k}^{2} \langle q_{k}\rangle +(-1)^{k}\hbar g_{m}\langle a^{\dagger}\rangle\langle a\rangle\nonumber\\
\gamma_{k}\langle p_{k}\rangle, \;(k=1,2).
\end{eqnarray}
\end{subequations}
Under the assumption that the input coupling laser field is much stronger than the probe field($\varepsilon_c\gg \varepsilon_p$), we obtain the steady-state mean values of $p$, $q$ and $a$ as
\begin{subequations}\label{qsas}
\begin{eqnarray}
p_{k}^{s}&=&0, \;(k=1,2),\\ q_{k}^{s}&=&(-1)^k\frac{\hbar g_{m}}{m_{k} \Omega_{k}^{2}}|a_{s}|^{2}\;(k=1,2), \\
a_{s}&=&\frac{\varepsilon_{c}}{\sqrt{(\Delta - 2G \sin (\theta) )^{2}+(\kappa -2G \cos (\theta)) ^{2}}},
\end{eqnarray}
\end{subequations}
where $q_{k}^{s}$ denotes the new equilibrium position of the movable mirrors and $\Delta=\omega_{0}-\omega_{c}+g_{m}( q_{1}^{s}-q_{2}^{s}) +2\eta |a_{s}|^{2}=\Delta_0+2\eta |a_{s}|^{2} $ is the effective detuning of the cavity which includes both the radiation pressure and the Kerr medium effects. It is obvious that the optical path and hence the cavity detuning are modified in an intensity-dependent way. Since the effective detuning $\Delta $ satisfies a fifth-order equation, it can have five real solutions and hence the system may exhibit multistability for a certain range of parameters. In our work we choose the parameters such that only one solution exists and the system has no bistability.
Now we consider the perturbation made by the probe field. The quantum Langevin equations for the fluctuations are given by
\begin{subequations}\label{langevin}
\begin{eqnarray}
\delta \dot a &=&-( i \Delta_{1}+\kappa) \delta a- i g_{m}a_{s} ( \delta q_{1}- \delta q_{2} ) \nonumber\\
(2G e^{i\theta}-
2 i \eta a_{s}^{2}) \delta a^{\dagger} +s_{in}(t),\\
\delta\dot{ q_{k}} &=& \delta p_{k}}/{m_{k}, \;(k=1,2),\\
\dot {\delta p_{k}}&=&-m_{k} \Omega_{k}^{2} \delta q_{k}+
(-1)^k\hbar g_{m} a_{s}( \delta a^{\dagger}+\delta a) \nonumber\\
&&-\gamma_{k}\delta p_{k},\,(k=1,2),
\end{eqnarray}
\end{subequations}
where $\Delta_{1}=\Delta_0+4\eta a_{s}^2$.
It is evident that the cavity mode is coupled only to the relative motion of the two mirrors, and it is therefore convenient to rewrite the above equations in terms of the fluctuations of the relative and center-of-mass coordinates:
\begin{eqnarray}
\delta Q&=&\frac{m_{1}}{M}\delta q_{1}+\frac{m_{2}}{M}\delta q_{2},\;\delta P=\delta p_{1}+\delta p_{2},\\
\delta q&=&\delta q_{2}-\delta q_{1},\; \frac{\delta p}{\mu}=\frac{\delta p_{2}}{m_{2}}-\frac{\delta p_{1}}{m_{1}},
\end{eqnarray}
where $M=m_{1}+m_{2}$ and $\mu =m_{1}m_{2}/M$ are the effective masses of the relative and center-of-mass modes, respectively. The linearized quantum Langevin equations for the fluctuation operators of these coordinates take the forms
\begin{subequations}\label{langevin2}
\begin{eqnarray}
\langle\delta\dot{ a}\rangle &=&-( i \Delta_{1}+\kappa) \langle\delta a\rangle+ i g_{m}a_{s} \langle\delta q\rangle \nonumber\\
(2G e^{i\theta}-2 i \eta a_{s}^{2})\langle \delta a^{\dagger}\rangle +\varepsilon_{p} e^{-i(\omega_{p}-\omega_{c})t},\\
\langle\delta \dot{q}\rangle &=& \langle{\delta p}\rangle/{\mu},\\
\langle\delta\dot{ p}\rangle&=&-\mu \Omega_{r}^{2} \langle\delta q\rangle-\gamma_{r}\langle\delta p\rangle-\mu (\Omega_{2}^{2}-\Omega_{1}^{2})\langle\delta Q\rangle\nonumber\\
\frac{\mu}{M}(\gamma_{2}-\gamma_{1})\langle\delta P\rangle+
\hbar g_{m} a_{s}(\langle \delta a^{\dagger}\rangle+\langle\delta a\rangle),\\
\langle\delta \dot{Q}\rangle &=& \langle{\delta P}\rangle/{M},\\
\langle\delta\dot{P}\rangle&=&-M\Omega_{cm}^{2}\langle \delta Q\rangle-\gamma_{cm}\langle\delta P\rangle-\mu (\Omega_{2}^{2}-\Omega_{1}^{2})\langle\delta q\rangle\nonumber\\
(\gamma_{2}-\gamma_{1})\langle\delta p\rangle,
\end{eqnarray}
\end{subequations}
where we have defined the relative motion frequency $\Omega_{r}^2=(m_{2}\Omega_{1}^2+m_{1}\Omega_{2}^2)/M$, damping rate $\gamma_{r}=(m_{2}\gamma_{1}+m_{1}\gamma_{2})/M$ and also the center-of-mass frequency $\Omega_{cm}^2=(m_{1}\Omega_{1}^2+m_{2}\Omega_{2}^2)/M$ and damping rate $\gamma_{cm}=(m_{1}\gamma_{1}+m_{2}\gamma_{2})/M$. The above equations show that even though the cavity mode interacts only with the relative motion mode, there is a coupling between the center-of-mass and relative motion modes when $\Omega_{1}\neq\Omega_{2}$ or $\gamma_{1}\neq\gamma_{2}$. We will show that the presence of this coupling makes the switching from EIT to EIA possible. Now we use a fairly standard procedure for the investigation of the probe response. Defining $\delta=\omega_{p}-\omega_{c}$, we use the following ansatz
\begin{subequations}\label{ansatz}
\begin{eqnarray}
\langle\delta a\rangle &=&A_{-}e^{-i\delta t}+A_{+}e^{i\delta t},\\
\langle\delta a^{\dagger}\rangle &=&A_{-}^{*}e^{-i\delta t}+A_{+}^{*}e^{i\delta t},\\
\langle\delta q\rangle &=&qe^{-i\delta t}+q^{*}e^{i\delta t},\\
\langle\delta Q\rangle &=& Q e^{-i\delta t}+Q^{*}e^{i\delta t}.
\end{eqnarray}
\end{subequations}
In the original frame $A_{-}$ and $A_{+}$ oscillate at $\omega_p$ and $2\omega_c-\omega_p$, respectively.
Using the input-output relation\cite{in-out}, we obtain
\begin{equation}
\varepsilon_{out}+\varepsilon_{c}e^{-i\omega_c t}+\varepsilon_p e^{-i\omega_p t}=2\kappa ( a_s+\delta a)e^{-i\omega_c t}.
\end{equation}
Substituting Eq.(\ref{ansatz}) into Eq.(\ref{langevin2}) we obtain the following equations
\begin{subequations}\label{eqs}
\begin{eqnarray}
(\Theta+i\delta)A_{-}+\Gamma( A_{+})^* +i g_m a_s q+\varepsilon_p&=&0,\\
\Gamma^{*}A_{-}+(\Theta^{*}+i\delta)( A_{+})^*-i g_m a_s q&=&0,\\
\hbar g a_s(A_{-}+(A_{+})^*)+\mu\chi_{r}(\delta)q+\Lambda Q&=&0,\\
M \chi_{cm}(\delta) Q+\Lambda q&=&0,
\end{eqnarray}
\end{subequations}
where we have defined
\begin{subequations}\label{lambda-define}
\begin{eqnarray}
\Theta &=&-(\kappa+i\Delta_1),\\
\Gamma &=&2G e^{i\theta}-2i \eta a_s^2,\\
\Lambda &=&\mu(\Omega_{1}^2-\Omega_{2}^2 +i\delta (\gamma_2-\gamma_{1})),\\
%\Lambda_{2}&=&\mu(\Omega_{1}^2-\Omega_{2}^2) +i\delta (\gamma_2-\gamma_{1})\\
\chi_{r}(\delta)\;&=&\delta^2-\Omega_{r}^2+i\delta \gamma_{r},\\
\chi_{cm}(\delta)&=&\delta^2-\Omega_{cm}^2+i\delta \gamma_{cm}.
\end{eqnarray}
\end{subequations}
From the Eq.(\ref{lambda-define}c) we find that when $\Omega_1=\Omega_2$ and $\gamma_1=\gamma_2$, $\Lambda=0$ and thus the center-of-mass motion is fully decoupled from the cavity mode and the relative motion. While whenever $\Lambda\neq0$ the three modes are all coupled.
The total output field $\varepsilon_t$, at the probe frequency is given by
\begin{equation}\label{quad}
\varepsilon_t=2\kappa A_-/\varepsilon_p=\dfrac{2\kappa}{d(\delta)}\lbrace\kappa-i(\Delta_1+ \delta) -i f(\delta)\rbrace,
\end{equation}
\begin{subequations}\label{definf}
\begin{eqnarray}
f(\delta)&=&\hbar g_m^2 a_s^2 /\chi(\delta),
\\
\chi(\delta)&=&\mu\chi_r(\delta)-\dfrac{\Lambda^2}{M \chi_{cm}(\delta)},\\
d(\delta)&=&(\kappa-i \delta)^2+\Delta_1^2-\vert\Gamma\vert^2 \nonumber\\&&+
2 (\Delta_1+Im(\Gamma)) f(\delta).
\end{eqnarray}
\end{subequations}
The real part ($\varepsilon_R$) and imaginary part ($\varepsilon_I$) of the field amplitude $\varepsilon_t$, respectively, show the absorptive and dispersive behavior of the output field at the probe frequency.
These quantities can be measured by homodyne technique [28].
The structure of the output field has some main characteristics, arising from the nonlinearity of the system and the freedom in choosing equal or unequal mechanical frequencies and damping rates.
To understand the coupling-field-induced modification of the probe response and its structure we present the results and numerical calculations in the next section.
\section{RESULTS AND DISCUSSIONS} \label{sec3}
In this section, we first consider the bare cavity optomechanical system and investigate the condition in which the coherent coupling between the mechanical and optical modes leads to OMIT and OMIA. Then we examine the effects of the Kerr-down conversion nonlinearity on these phenomena.
\subsection{Bare cavity}
To simplify our treatment for the bare cavity we can use the reasonable rotating wave approximation (RWA) to neglect the far off-resonance lower sideband ($A^{+}\simeq 0$) in the resolved sideband regime ($\kappa\ll\Omega_k,k=1,2$)\cite{rae}. In the resolved sideband regime the normal mode splitting occurs \cite{NMS1,NMS2,NMS3}. In this approximation $\varepsilon_t$ is simplified to the following form
\begin{equation}
\varepsilon_t\simeq\dfrac{2\kappa}{\kappa+i(\Delta_0-\delta)+i\dfrac{\hbar g_m^2 a_s^2}{\chi(\delta)}}.
\end{equation}
In what follows we investigate the two cases of equal and different mechanical frequencies separately.
%We work in the in the resolved sideband regime ($\kappa\ll\Omega_k,k=1,2$). In this regime the occurrence of normal mode splitting \cite{NMS1,NMS2,NMS3} guarantees the splitting of the OMIT and OMIA to be distinguishable.
% We choose $\omega_p$ to be close to the cavity frequency $\omega_p\sim\omega_0$. The probe field absorption profiles are shown in Fig. \ref{} for the following situations: (i) when the frequencies and damping rates of mechanical oscillators are the same, i.e., $\Omega_1=\Omega_2=\omega_m$ and $\gamma_1=\gamma_2=\gamma_m$ and the frequency of the coupling laser is detuned by $\omega_m$ from the cavity mode, (ii) when the mechanical frequencies are different and their damping rates are equal, i.e., $\Omega_1\neq\Omega_2$ and $\gamma_1=\gamma_2=\gamma_m$ and the frequency of the coupling laser is detuned by $\omega_m=(\Omega_1+\Omega_2)/2$ from the cavity mode.
%To simplify the investigations we can use the reasonable rotating wave approximation (RWA) for neglecting the far off-resonance lower sideband, ($A^{+}\simeq 0$) in the resolved sideband regime ($\kappa\ll\Omega_k,k=1,2$)\cite{rae}. In the resolved sideband regime normal mode splitting occurs \cite{NMS1,NMS2,NMS3}. In this approximation $\varepsilon_t$ simplifies to the following form
%\varepsilon_t\simeq\dfrac{2\kappa}{\kappa+i(\Delta_0-\delta)+i\dfrac{\hbar g_m^2 a_s^2}{\chi(\delta)}},
%This logical approximation simplifies the investigation of the probe field response.
%\beta &=&\hbar g_m^2 a_s^2,\\
\subsubsection{Equal mechanical frequencies and damping rates ($\Lambda=0$)}
First, we consider the case in which the frequencies and damping rates of the two mechanical oscillators are the same, i.e., $\Omega_1=\Omega_2=\omega_m$ and $\gamma_1=\gamma_2=\gamma_m$. As mentioned before, in this condition the radiation pressure is only coupled to the relative position of the two mirrors and the center-of-mass becomes an isolated quantum oscillator. Therefore $\chi(\delta)=\mu \chi_r(\delta)$.
When $\omega_p$ is close to the cavity frequency ($\omega_p\sim\omega_0$) and the coupling field $\omega_c$ drives the cavity on its red sideband ($\Delta_0\thicksim \omega_m$) the structure of the resonance response of the output field $\varepsilon_t$ is simplified to that of a cavity with one movable mirror and effective mass $2\mu$ :
\begin{equation}
\varepsilon_t\simeq\dfrac{2\kappa}{\kappa -i x+\lbrace\beta /(\gamma_m/2-i x)\rbrace},
\end{equation}
where $\beta =\hbar g_m^2 a_s^2/2\mu$ and $x=\delta-\omega_m$ is the detuning from the line center. Therefore the denominator of the response function is quadratic in $x$.
\subsubsection{ Different mechanical frequencies and equal damping rates ($\Lambda\neq0$)}
Now we consider the case in which the frequency of the mechanical oscillators is different $\Omega_1\neq\Omega_2$ but their damping rates are equal $\gamma_1=\gamma_2=\gamma_m$. The new aspect of this condition is the coupling between the center-of-mass and the relative motion modes which results in the anomalous EIA in the optomechanical cavity.
When $\omega_p$ is close to the cavity frequency ($\omega_p\sim\omega_0$) and the coupling field $\omega_c$ is red tuned by an amount $\omega
_m=(\Omega_1+\Omega_2)/2$ the response of the system is simplified to the following form
\begin{equation}\label{Et-OMIA}
\varepsilon_t\simeq\dfrac{2\kappa}{\kappa -i x+\dfrac{2\beta}{\delta_1 x+b_1 -\dfrac{\Lambda^2/\mu M}{\delta_2 x+b_2}}},
\end{equation}
where $\delta_1=\omega_m+\Omega_r$, $b_1=\omega_m^2-\Omega_r^2+i \omega_m\gamma_m$ and $\delta_2=\omega_m+\Omega_{cm}$, $b_2=\omega_m^2-\Omega_{cm}^2+i \omega_m\gamma_m$. Therefore the denominator of the response function is cubic in $x$.
To illustrate the numerical results we show the probe field absorption and dispersion profiles for the bare cavity in Fig.\ref{fig:re&im1} for the two cases of equal and different mechanical frequencies. We use the following set of experimentally realizable parameters \cite{teufel}: $P_c=6 $ mW,$\lambda=2\pi c/\omega_c=1064 $ nm,$\Omega_1=2\pi\times10^7$Hz, $m_1=m_2=12$ ng, $\kappa/\Omega_1=0.02 $, $\gamma_1/2\pi=\gamma_2/2\pi=200$ Hz, and $L=6$ mm.
The figure clearly shows the splitting of the transparency window due to an additional coherency in the system.
%\chi_r(\delta)&\simeq &(x+\delta_1)(\delta^{'}_{1})+i\omega_m\gamma_m,\\
%\chi_{cm}(\delta)&\simeq &(x+\delta_2)(\omega_m+\Omega_r)+i\omega_m\gamma_m.
\begin{figure}[ht]
\centering
\includegraphics[width=3.5in]{pic1.eps}
\includegraphics[width=3.5in]{pic2.eps}
\caption{
(Color online)(a)The real and (b) the imaginary parts
of the field amplitude $\varepsilon_t$ versus the normalized frequency $x/\omega_{m}$ for the bare cavity with equal mechanical frequencies $\Omega_1=\Omega_2=2\pi\times10^7$Hz(red solid line) and with different mechanical frequencies $\Omega_1=2\pi\times10^7$Hz,$\Omega_2=1.03\Omega_1$ (blue dashed line).The coupling field $\omega_c$ is red detuned by an amount $\omega_m=(\Omega_1+\Omega_2)/2$ and the two mechanical damping rates $\gamma_1$ and$\gamma_2$ are equal. }
\label{fig:re&im1}
\end{figure}
Physically, in the two mode optomechanical system ($\Lambda=0$) when the coupling field $\omega_c$ is red detuned by an amount $\omega_m$ ($\Delta_0\thicksim \omega_m$) and $\omega_p$ is close to the cavity frequency the optomechanical system behaves like a three-level Lambda medium for the probe field as shown in Fig.(\ref{fig:level}). The intense coupling laser field "dresses" the mechanical mode. In this view , the OMIT can be seen as a level splitting like an Autler-Towns doublet [29], as shown in Fig.(\ref{fig:level}).The coherent cancellation of the two resonances in the middle of the doublet, at the two-photon resonance, provides the system transmittive in a narrowband around the cavity resonance for the probe field.
\begin{figure}[ht]
\centering
\includegraphics[width=3in]{level.eps}
\caption{
Level diagram structure for the OMIT. The $\vert1\rangle\leftrightarrow \vert3\rangle$ transition is the excitation at cavity frequency and the $\vert2\rangle\leftrightarrow \vert3\rangle$ transition is the excitation of the mechanical oscillator. Coherent coupling of the mechanical and optical modes generates the destructive interference of excitation pathways in the middle of the doublet of dressed states $ \vert1_d\rangle$ and $\vert2_d\rangle$ for the probe beam.}
\label{fig:level}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=3in]{level2.eps}
\caption{
Level diagram structure for the OMIA. The $\vert1\rangle\leftrightarrow \vert3\rangle$ transition is the excitation at cavity frequency; The coupling laser is red tuned by an amount $\omega_m=(\Omega_1+\Omega_2)/2$ to induce EIT.The splitting is due to the fact that there are two occasions of two-photon resonance for the probe and coupling lasers.}
\label{fig:level2}
\end{figure}
Similarly, for the three-mode system ($\Lambda\neq0$) we can describe the happening of anomalous EIA based on a level diagram structure. In Fig.\ref{fig:level2} the $\vert1\rangle\leftrightarrow \vert3\rangle$ transition is the excitation at cavity frequency and the coupling laser is red tuned by an amount $\omega_m=(\Omega_1+\Omega_2)/2$ ($\Delta_0\sim\omega_m$) forming a $\Lambda$-type three-level system producing OMIT. But the radiation pressure induces an additional coherency between the mechanical modes giving rise to OMIT splitting. The coherent induced splitting of OMIT due to the radiation pressure is similar to driving a hyperfine transition in an atomic $\Lambda$-type three-level system by a radio-frequency or microwave field[30, 31, 32, 33, 34]. Figure \ref{fig:splitting-g} shows how the OMIT splitting varies linearly as a function of the strength of radiation pressure coupling $g_m$. The splitting in OMIT is due to the fact that there are two occasions of two-photon resonance for the probe and coupling lasers at $\delta=\Omega_1$ and $\delta=\Omega_2$.
\begin{figure}[ht]
\centering
\includegraphics[width=3in]{pic3.eps}
\caption{
(Color online)The probe field absorption profile versus the normalized frequency $x/\omega_{m}$ showing a linear OMIT splitting as a function of the normalized radiation pressure coupling $g_m/g_1$ where $g_1=\omega_c/L$. The mechanical frequencies are $\Omega_1=2\pi\times 10^7$Hz and $\Omega_2=1.05\Omega_1$. Other parameters are the same as those in Fig.\ref{fig:re&im1}.}
\label{fig:splitting-g}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=3in]{eit-control.eps}
\caption{(Color online)The absorption profile of the probe field versus the normalized frequency $x/\omega_{m}$ for a bare cavity ($G=\eta=0$) (green line) and a nonlinear cavity with $G=4\times 10^6$Hz, $\eta=0.03$Hz, $\theta=3\pi/2$(blue dashed line) and with $G=4\times 10^6$Hz, $\eta=0.04$Hz, $\theta=\pi/2$(red solid line) . The parameters are $P_c=8$mW, $m_1=m_2=15$ ng, $\Omega_1=\Omega_2=2\pi\times 10^7$Hz, $\lambda=512$ nm, $L=2$mm, and $\kappa=0.01\Omega_1$. Other parameters are the same as those in Fig.\ref{fig:re&im1}.
\label{fig:EIT-control}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=3in]{sideband.eps}
\caption{
(Color online)The parameter $2\kappa \vert A_+\vert/\varepsilon_p$ versus the normalized frequency $x/\omega_{m}$ for a bare cavity ($G=\eta=0$) (red solid line) and a nonlinear cavity ($G=1.5\kappa$, $\eta=0.03$Hz,$\theta=\pi/2$)(blue dashed line). The mechanical frequencies are $\Omega_1=\Omega_2=2\pi\times 10^7$Hz. Other parameters are the same as those in Fig.\ref{fig:re&im1}.}
\label{fig:sideband}
\end{figure}
\subsection{Nonlinear cavity}
Now we investigate the effect of the Kerr-down conversion nonlinearity on the total output field amplitude $\varepsilon_t$. Although the nonlinearity does not alter the level diagram structure of the OMIT, it manifests itself in the steady-state response of the system (Eq.(\ref{qsas})), in the optomechanical coupling rate $g_m a_s$ (Eq.(\ref{langevin})), and in the parameter $\Gamma_+$ which is responsible for a direct coupling between $A_{-}$ and $A_+$ (Eq.(\ref{eqs}a)).
In the OMIT condition the optomechanical coupling rate $g_m a_s$ is equivalent to the Rabi frequency in the atomic EIT\cite{weis}. The dependence of $ a_s$ on the nonlinearity can be used to control the width of the transparency window which is related to the effective mechanical damping rate $\gamma_{eff}$.
This parameter is approximately given by\cite{weis,safavi,agarwalrouter}
\begin{equation}
\gamma_{eff}=\gamma_m(1+C),
\end{equation}
where $C=2 \hbar( g_m a_s)^2/m \omega_m\kappa \gamma_m$ denotes the optomechanical cooperativity of the cavity\cite{weis,rae,hill}. In Fig. \ref{fig:EIT-control} we have plotted the absorption profile for different values of $G$, $\theta$ and $\eta$. It shows that by controlling these parameters the width of the transparency window can be increased or decreased in comparison with that of a bare cavity.
It should be noted that in the presence of only one of the two nonlinearities we cannot control the transparency window desirably. This can be explained by the fact that according to Eq.(\ref{quad}), in the absence of optomechanical coupling ($g_m=0$) there would be an absorption peak near the modified resonance condition of the cavity $\delta=\sqrt{\Delta_1^2-\vert\Gamma\vert^2}$. Therefore the nonlinear parameters should be choosen such that $\sqrt{\Delta_1^2-\vert\Gamma\vert^2}\simeq\Delta$, otherwise the control and probe fields induce a radiation-pressure force oscillating at the frequency $\delta$, which is not close enough to the resonance frequency of the moving mirrors to induce coherent oscillations in them. This feature leads to disappearance of OMIT in the output probe field.
\begin{figure}[ht]
\centering
\includegraphics[width=3in]{q1.eps}
\includegraphics[width=3in]{q2.eps}
\includegraphics[width=3in]{re2.eps}
\caption{(Color online)The real parts of (a) the normalized parameter $g_{m}a_{s} q/\varepsilon_p $, (b) the normalized parameter $g_{m}a_{s} Q/\varepsilon_p$ and (c) the field amplitude $\varepsilon_t$ versus the normalized frequency $x/\omega_{m}$ for a bare cavity ($G=\eta=0$) (red line) and a nonlinear cavity with $G=10^7$Hz, $\eta=0.09$Hz, $\theta=\pi/2$(blue dashed line).The mechanical frequencies are $\Omega_1=2\pi\times 10^7$Hz and $\Omega_2=1.06\Omega_1$. Other parameters are the same as those in Fig.\ref{fig:EIT-control}.
\label{fig:q1}
\end{figure}
Also, according to Eq.(\ref{eqs}a), in the presence of nonlinearity there is a direct coupling between $A_{-}$ and $A_+$ because of the factor $\Gamma$. Therefore it seems that in contrast to the bare optomechanical cavity the Stokes scattering of the light from the strong intracavity coupling field is no longer negligible. In Fig.\ref{fig:sideband} we have plotted the parameter $2\kappa \vert A_+\vert/\varepsilon_p$ as a function of the normalized frequency $x/\omega_{m}$ for a bare cavity and a cavity with Kerr-down conversion nonlinearity. As is seen, in the dip of the transparency window $2\kappa \vert A_+\vert/\varepsilon_p$ reaches its local minimum for a nonlinear cavity and its local maximum for a bare cavity. Hence even though in the presence of nonlinearity outside the OMIT window the lower sideband can also be tuned by the strong coupling field, but the contribution of the Stokes scattering around the cavity resonance is more negligible for a nonlinear cavity.
Now we consider the probe response in the presence of Kerr-down conversion nonlinearity for the second case ($\Omega_1\neq \Omega_2$ ). As stated before, the OMIT splitting and appearance of the central absorption peak are due to an additional coherent oscillation in the system which is provided by the fluctuations in the center-of-mass mode, i.e., $\langle \delta Q\rangle$. Figures \ref{fig:q1}(a) and \ref{fig:q1}(b) illustrate the effect of the nonlinearity on $q$ and $Q$, respectively. They show a shift in the coherent oscillations of $q$ which leads to the broadening of the width of the transparency windows and an increase in the coherent oscillations of the center-of-mass mode $Q$ which results in the enhancement of central peak absorption (Fig.\ref{fig:q1}(c)).
In conclusion, we have studied theoretically the effect of an additional mechanical mode and a Kerr-down conversion nonlinear crystal on the EIT resonance in an optomechanical system with two movable mirrors. We have shown that the coherent oscillations of the two mechanical oscillators can lead to splitting in the EIT resonance, and appearance of an absorption peak within the transparency window. This configuration is similar to driving a hyperfine transition in an atomic $\Lambda$-type three-level system by a radio-frequency or microwave field. Also, we have shown that in the presence of Kerr-down conversion nonlinearity by controlling the nonlinear parameters $G$, $\eta$ and $\theta$ the width of transparency can be adjusted to be greater or smaller than that of a bare cavity. The combination of an additional mechanical mode and nonlinear crystal suggests new possibilities for manipulating and controlling the EIT resonance in the optomechanical systems.
\section*{Acknowledgement}
The authors wish to thank The Office of Graduate Studies of The University of Isfahan for their support.
\bibliographystyle{apsrev4-1}
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\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
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\def\bibfnamefont#1{#1}\fi
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\def\citenamefont#1{#1}\fi
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\def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi
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\end{document}
%\end{figure} %
% the cavity mode in the rotating frame acquires the effective frequency √2 − 4G2 which is dependent on the parametric
% The nonlinear gain medium increases the optomechanical cooperativity $C$ (hence $\gamma_{eff}$) and the splitting by increasing the cavity photon number\cite{agarwal-gain}.
% the mechanical displacement amplitude $q$ Outside the EIT window, around the Autler-Townes doublet 2 ∼ ±c, the probe is strongly absorbed,
% According to Eq.(\ref{eqs}a) the effect of the nonlinearity becomes manifest not only in the steady-state response of the system (Eq.(\ref{qsas})), but also in the $\Gamma_+$ which is responsible for a direct coupling between $A_{-}$ and $A_+$. Therefore it seems that in contrast to the bare optomechanical cavity the Stokes scattering of the light from the strong intracavity coupling field is no longer negligible. In Fig.\ref{fig:sideband} we have plotted the parameter $2\kappa \vert A_+\vert/\varepsilon_p$ as a function of the normalized frequency $x/\omega_{m}$ for a bare cavity and a cavity with Kerr-down conversion nonlinearity. As is seen, in the dip of the transparency window $2\kappa \vert A_+\vert/\varepsilon_p$ reaches its local minimum for a nonlinear cavity and its local maximum for a bare cavity. Hence even though in the presence of nonlinearity the lower sideband at frequency $2\omega_c-\omega_p$ can also be tuned by the strong coupling field, but the share of the Stokes scattering around the cavity resonance is more negligible for a nonlinear cavity. This feature leads to the broadening of the transparency window as shown in Fig.\ref{fig:EIT-G,eta}.
% It is notable that since the radiation pressure effects are dependent on the intracavity photon number and the photon number is dependent on the nonlinear parameters $G$, $\theta$ and $\eta$, the two other nonlinear parameters $\theta$ and $\eta$ can also provide more flexibility for controlling the width of the transparency window. In Fig. \ref{fig:EIT-control} we have plotted the absorption profile for different values of $G$, $\theta$ and $\eta$. It shows that by controlling these parameters the width of the transparency window can be increased or decreased in comparison with that of a bare cavity.
% It should be noted that in the presence of only one of the two nonlinearities we cannot control the transparency window desirably. This can be explained by the fact that according to Eq.(\ref{quad}), in the presence of the Kerr-down conversion nonlinearity OMIT occurs whenever the nonlinear parameters $G$, $\eta$ and $\theta$ is chosen such that $\sqrt{\Delta_1^2-\vert\Gamma\vert^2}\simeq\Delta\sim\omega_m$. The frequency shift of the cavity may destroy the effective scattering of the coupling laser field towards the resonance frequency of the cavity which is not degenerate with the probe field any more. This feature can lead to normal absorption of the probe field which is not worthy.
|
arxiv-papers
| 2013-06-24T09:03:26 |
2024-09-04T02:49:46.843326
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "S. Shahidani, M. H. Naderi, M. Soltanolkotabi",
"submitter": "Sareh Shahidani",
"url": "https://arxiv.org/abs/1306.5543"
}
|
1306.5583
|
§ INTRODUCTION
Sequential Monte Carlo () methods are a class of sampling algorithms that
combine importance sampling and resampling. They have been primarily used as
“particle filters” to solve optimal filtering problems; see, for example,
Cappé et al., 2007 and Doucet and Johansen, 2011 for recent reviews. They are
also used in a static setting where a target distribution is of interest, for
example, for the purpose of Bayesian modeling. This was proposed by
Del Moral et al., 2006 and developed by Peters, 2005 and
Del Moral et al., 2006. This framework involves the construction of a
sequence of artificial distributions on spaces of increasing dimensions which
admit the distributions of interest as particular marginals.
algorithms are perceived as being difficult to implement while general
tools were not available until the development by Johansen, 2009, which
provided a general framework for implementing algorithms. algorithms
admit natural and scalable parallelization. However, there are only parallel
implementations of algorithms for many problem specific applications,
usually associated with specific related researches. Lee et al., 2010
studied the parallelization of algorithms on s with some
generality. There are few general tools to implement algorithms on
parallel hardware though multicore s are very common today and computing
on specialized hardware such as s are more and more popular.
The purpose of the current work is to provide a general framework for
implementing algorithms on both sequential and parallel hardware. There
are two main goals of the presented framework. The first is reusability. It
will be demonstrated that the same implementation source code can be used to
build a serialized sampler, or using different programming models (for
example, and ) to build parallelized samplers for multicore
s. They can be scaled for clusters using with few modifications.
And with a little effort they can also be used to build parallelized samplers
on specialized massive parallel hardware such as s using . The
second is extensibility. It is possible to write a backend for to use
new parallel programming models while reusing existing implementations. It is
also possible to enhance the library to improve performance for specific
applications. Almost all components of the library can be reimplemented by
users and thus if the default implementation is not suitable for a specific
application, they can be replaced while being integrated with other components
§ SEQUENTIAL MONTE CARLO
§.§ Sequential importance sampling and resampling
Importance sampling is a technique which allows the calculation of the
expectation of a function $\varphi$ with respect to a distribution $\pi$ using
samples from some other distribution $\eta$ with respect to which $\pi$ is
absolutely continuous, based on the identity,
\begin{equation}
\Exp_{\pi}[\varphi(X)]
= \int\varphi(x)\pi(x)\intd x
= \int\frac{\varphi(x)\pi(x)}{\eta(x)}\eta(x)\intd x
= \Exp_{\eta}\Bigl[\frac{\varphi(X)\pi(X)}{\eta(X)}\Bigr]
\end{equation}
And thus, let $\{X^{(i)}\}_{i=1}^N$ be samples from $\eta$, then
$\Exp_{\pi}[\varphi(X)]$ can be approximated by
\begin{equation}
\hat\varphi_1 =
\frac{1}{N}\sum_{i=1}^N\frac{\varphi(X^{(i)})\pi(X^{(i)})}{\eta(X^{(i)})}
\end{equation}
In practice $\pi$ and $\eta$ are often only known up to some normalizing
constants, which can be estimated using the same samples. Let $w^{(i)} =
\pi(X^{(i)})/\eta(X^{(i)})$, then we have
\begin{equation}
\hat\varphi_2 =
\frac{\sum_{i=1}^Nw^{(i)}\varphi(X^{(i)})}{\sum_{i=1}^Nw^{(i)}}
\end{equation}
\begin{equation}
\hat\varphi_3 = \sum_{i=1}^NW^{(i)}\varphi(X^{(i)})
\end{equation}
where $W^{(i)}\propto w^{(i)}$ and are normalized such that
$\sum_{i=1}^NW^{(i)} = 1$.
Sequential importance sampling () generalizes the importance sampling
technique for a sequence of distributions $\{\pi_t\}_{t\ge0}$ defined on
spaces $\{\prod_{k=0}^tE_k\}_{t\ge0}$. At time $t = 0$, sample
$\{X_0^{(i)}\}_{i=1}^N$ from $\eta_0$ and compute the weights $W_0^{(i)}
\propto \pi_0(X_0^{(i)})/\eta_0(X_0^{(i)})$. At time $t\ge1$, each sample
$X_{0:t-1}^{(i)}$, usually termed particles in the literature, is
extended to $X_{0:t}^{(i)}$ by a proposal distribution
$q_t(\cdot|X_{0:t-1}^{(i)})$. And the weights are recalculated by $W_t^{(i)}
\propto \pi_t(X_{0:t}^{(i)})/\eta_t(X_{0:t}^{(i)})$ where
\begin{equation}
\eta_t(X_{0:t}^{(i)}) =
\eta_{t-1}(X_{0:t-1}^{(i)})q_t(X_{0:t}^{(i)}|X_{0:t-1}^{(i)})
\end{equation}
and thus
\begin{align}
W_t^{(i)} \propto \frac{\pi_t(X_{0:t}^{(i)})}{\eta_t(X_{0:t}^{(i)})}
&= \frac{\pi_t(X_{0:t}^{(i)})\pi_{t-1}(X_{0:t-1}^{(i)})}
\pi_{t-1}(X_{0:t-1}^{(i)})} \notag\\
&= \frac{\pi_t(X_{0:t}^{(i)})}
\label{eq:si}
\end{align}
and importance sampling estimate of $\Exp_{\pi_t}[\varphi_t(X_{0:t})]$ can be
obtained using $\{W_t^{(i)},X_{0:t}^{(i)}\}_{i=1}^N$.
However this approach fails as $t$ becomes large. The weights tend to become
concentrated on a few particles as the discrepancy between $\eta_t$ and
$\pi_t$ becomes larger. Resampling techniques are applied such that, a new
particle system $\{\bar{W}_t^{(i)},\bar{X}_{0:t}^{(i)}\}_{i=1}^M$ is obtained
with the property,
\begin{equation}
\Exp\Bigl[\sum_{i=1}^M\bar{W}_t^{(i)}\varphi_t(\bar{X}_{0:t}^{(i)})\Bigr] =
\Exp\Bigl[\sum_{i=1}^NW_t^{(i)}\varphi_t(X_{0:t}^{(i)})\Bigr]
\label{eq:resample}
\end{equation}
In practice, the resampling algorithm is usually chosen such that $M = N$ and
$\bar{W}^{(i)} = 1/N$ for $i=1,\dots,N$. Resampling can be performed at each time
$t$ or adaptively based on some criteria of the discrepancy. One popular
quantity used to monitor the discrepancy is effective sample size
(), introduced by Liu and Chen, 1998, defined as
\begin{equation}
\ess_t = \frac{1}{\sum_{i=1}^N (W_t^{(i)})^2}
\end{equation}
where $\{W_t^{(i)}\}_{i=1}^N$ are the normalized weights. And resampling can
be performed when $\ess\le \alpha N$ where $\alpha\in[0,1]$.
The common practice of resampling is to replicate particles with large weights
and discard those with small weights. In other words, instead of generating a
random sample $\{\bar{X}_{0:t}^{(i)}\}_{i=1}^N$ directly, a random sample of
integers $\{R^{(i)}\}_{i=1}^N$ is generated, such that $R^{(i)} \ge 0$ for $i
= 1,\dots,N$ and $\sum_{i=1}^N R^{(i)} = N$. And each particle value
$X_{0:t}^{(i)}$ is replicated for $R^{(i)}$ times in the new particle system.
The distribution of $\{R^{(i)}\}_{i=1}^N$ shall fulfill the requirement of
Equation <ref>. One such distribution is a multinomial
distribution of size $N$ and weights $(W_t^{(i)},\dots,W_t^{(N)})$. See
Douc et al., 2005 for some commonly used resampling algorithms.
§.§ samplers
samplers allow us to obtain, iteratively, collections of weighted samples
from a sequence of distributions $\{\pi_t\}_{t\ge0}$ over essentially any
random variables on some spaces $\{E_t\}_{t\ge0}$, by constructing a sequence
of auxiliary distributions $\{\tilde\pi_t\}_{t\ge0}$ on spaces of increasing
dimensions, $\tilde\pi_t(x_{0:t})=\pi_t (x_t) \prod_{s=0}^{t-1}
L_s(x_{s+1},x_s)$, where the sequence of Markov kernels $\{L_s\}_{s=0}^{t-1}$,
termed backward kernels, is formally arbitrary but critically influences the
estimator variance. See Del Moral et al., 2006 for further details and
guidance on the selection of these kernels.
Standard sequential importance sampling and resampling algorithms can then be
applied to the sequence of synthetic distributions, $\{\tilde\pi_t\}_{t\ge0}$.
At time $t - 1$, assume that a set of weighted particles
$\{W_{t-1}^{(i)},X_{0:t-1}^{(i)}\}_{i=1}^N$ approximating $\tilde\pi_{t-1}$ is
available, then at time $t$, the path of each particle is extended with a
Markov kernel say, $K_t(x_{t-1}, x_t)$ and the set of particles
$\{X_{0:t}^{(i)}\}_{i=1}^N$ reach the distribution $\eta_t(X_{0:t}^{(i)}) =
\eta_0(X_0^{(i)})\prod_{k=1}^tK_t(X_{t-1}^{(i)}, X_t^{(i)})$, where $\eta_0$
is the initial distribution of the particles. To correct the discrepancy
between $\eta_t$ and $\tilde\pi_t$, Equation <ref> is applied and in
this case,
\begin{equation}
W_t^{(i)} \propto \frac{\tilde\pi_t(X_{0:t}^{(i)})}{\eta_t(X_{0:t}^{(i)})}
= \frac{\pi_t(X_t^{(i)})\prod_{s=0}^{t-1}L_s(X_{s+1}^{(i)}, X_s^{(i)})}
\propto \tilde{w}_t(X_{t-1}^{(i)}, X_t^{(i)})W_{t-1}^{(i)}
\end{equation}
where $\tilde{w}_t$, termed the incremental weights, are calculated as,
\begin{equation}
\tilde{w}_t(X_{t-1}^{(i)},X_t^{(i)}) =
\frac{\pi_t(X_t^{(i)})L_{t-1}(X_t^{(i)}, X_{t-1}^{(i)})}
{\pi_{t-1}(X_{t-1}^{(i)})K_t(X_{t-1}^{(i)}, X_t^{(i)})}
\end{equation}
If $\pi_t$ is only known up to a normalizing constant, say $\pi_t(x_t) =
\gamma_t(x_t)/Z_t$, then we can use the unnormalized incremental
\begin{equation}
w_t(X_{t-1}^{(i)},X_t^{(i)}) =
\frac{\gamma_t(X_t^{(i)})L_{t-1}(X_t^{(i)}, X_{t-1}^{(i)})}
{\gamma_{t-1}(X_{t-1}^{(i)})K_t(X_{t-1}^{(i)}, X_t^{(i)})}
\end{equation}
for importance sampling. Further, with the previously normalized
weights $\{W_{t-1}^{(i)}\}_{i=1}^N$, we can estimate the ratio of normalizing
constant $Z_t/Z_{t-1}$ by
\begin{equation}
\frac{\hat{Z}_t}{Z_{t-1}} =
\sum_{i=1}^N W_{t-1}^{(i)}w_t(X_{t-1}^{(i)},X_t^{(i)})
\end{equation}
Sequentially, the normalizing constant between initial distribution $\pi_0$
and some target $\pi_T$, $T\ge1$ can be estimated. See Del Moral et al., 2006
for details on calculating the incremental weights. In practice, when $K_t$ is
invariant to $\pi_t$, and an approximated suboptimal backward kernel
\begin{equation}
L_{t-1}(x_t, x_{t-1}) = \frac{\pi(x_{t-1})K_t(x_{t-1}, x_t)}{\pi_t(x_t)}
\end{equation}
is used, the unnormalized incremental weights will be
\begin{equation}
w_t(X_{t-1}^{(i)},X_t^{(i)}) =
\frac{\gamma_t(X_{t-1}^{(i)})}{\gamma_{t-1}(X_{t-1}^{(i)})}.
\label{eq:inc_weight_mcmc}
\end{equation}
§.§ Other sequential Monte Carlo algorithms
Some other commonly used sequential Monte Carlo algorithms can be viewed as
special cases of algorithms introduced above. The annealed importance sampling
(; Neal, 2001) can be viewed as samplers without resampling.
Particle filters as seen in the physics and signal processing literature, can
also be interpreted as the sequential importance sampling and resampling
algorithms. See Doucet and Johansen, 2011 for a review of this topic. A simple
particle filter example is used in Section <ref>
to demonstrate basic features of the library.
§ USING THE VSMC LIBRARY
§.§ Overview
The library makes use of 's template generic programming to
implement general algorithms. This library is formed by a few major
modules listed below. Some features not included in these modules are
introduced later in context.
Core The highest level of abstraction of samplers. Users
interact with classes defined within this module to create and manipulate
general samplers. Classes in this module include Sampler,
Particle and others. These classes use user defined callback
functions or callable objects, such as functors, to perform problem
specific operations, such as updating particle values and weights. This
module is documented in Section <ref>.
Symmetric Multiprocessing () This is the form of computing most
people use everyday, including multiprocessor workstations, multicore
desktops and laptops. Classes within this module make it possible to write
generic operations which manipulate a single particle that can be applied
either sequentially or in parallel through various parallel programming
models. A method defined through classes of this module can be used by
Sampler as callback objects. This module is documented in
Section <ref>.
Message Passing Interface is the de facto standard
for parallel programming on distributed memory architectures. This module
enables users to adapt implementations of algorithms written for the module such that the same sampler can be parallelized using . In
addition, when used with the module, it allows easy implementation of
hybrid parallelization such as /. In Section <ref>, an example is shown how to extend
existing parallelized samplers using this module.
OpenCL This module is similar to the two above except it eases the
parallelization through , such as for the purpose of General
Purpose Programming (). is a framework for writing
programs that can be execute across heterogeneous platforms. programs can run on either s or s. It is beyond the scope of
this paper to give a proper introduction to , and their use
in . However, later we will demonstrate the relative performance of
this programming model on both s and s in
Section <ref>.
§.§ Obtaining and installation
is a header only library. There is practically no installation step. The
library can be downloaded from <http://zhouyan.github.io/vSMC/vSMC.zip>.
After downloading and unpacking, one can start using by ensuring that
the compiler can find the headers inside the include directory. To
permanently install the library in a system directory, one can simply copy the
contents of the include directory to an appropriate place.
Alternatively, one can use the (2.8 or later) configuration script and
obtain the source by . On a Unix-like system (such as Mac OS X, BSD,
Linux and others),
git clone git://github.com/zhouyan/vSMC.git
cd vSMC
git submodule init
git submodule update
mkdir build
cd build
cmake .. -DCMAKE_BUILD_TYPE=Release
make install
For Unix-like systems, there is also a shell script build.sh that
builds all examples in this paper and produces the results and benchmarks as
in Section <ref>. See documentations in that script for details of
how to change settings for the users' platform.
§.§ Documentation
To build the reference manual, one need , version 1.8.3 or later.
Continuing the last step (still in the build directory), invoking
make docs will create a doc directory inside build, which
contains the references. Alternatively the reference manual can also be
found on <http://zhouyan.github.io/vSMC/doc/html/index.html>. It is beyond
the scope of this paper to document every feature of the library. In
many places we will refer to this reference manual for further information.
§.§ Third-party dependencies
uses counter-based for random number generating. For an
sampler with $N$ particles, constructs $N$ (statistically)
independent streams. It is possible to use millions of such streams
without a huge memory footprint or other performance penalties. Since each
particle has its own independent stream, it frees users from many
thread-safety and statistical independence considerations. It is highly
recommended that the users install this library. Within , these streams are wrapped under engines, and can be replaced by other
compatible engines seamlessly. Users only need to be familiar with
classes defined in <random> or their equivalents to
use these streams. See the documentation of the corresponding libraries
for details, as well as examples in Section <ref>.
The other third-party dependency is the library. Version 1.49 or later
is required. However, this is actually optional provided that proper features are available in the standard library, for example using with
. The headers of concern are <functional> and
<random>. To instruct to use the standard library headers instead
of falling back to the library, one needs to define the configuration
macros before including any headers. For example,
clang++ -std=c++11 -stdlib=libc++ -DVSMC_HAS_CXX11LIB_FUNCTIONAL=1 -DVSMC_HAS_CXX11LIB_RANDOM=1 -o prog prog.cpp
tells the library to use <functional> and <random>.
The availability of these headers are also checked by the configuration
§.§ Compiler support
has been tested with recent versions of , , and
. can optionally use some features to improve performance
and usability. In particular, as mentioned before, can use standard libraries instead of the library. At the time of writing,
with has the most comprehensive support of with respect
to standard compliant and feature completion. 4.8 , 2012 and 2013 also have very good support. Note that, provided the library is available, all language and library features are optional.
can be used with any conforming compilers.
§ THE VSMC LIBRARY
§.§ Core module
The core module abstracts general samplers. samplers can be viewed
as formed by a few concepts regardless of the specific problems. The following
is a list of the most commonly seen components of samplers and their
corresponding abstractions.
* A collection of all particle state values, namely
$\{X_t^{(i)}\}_{i=1}^N$. In , users need to define a class, say
T, to abstract this concept. We refer to this as the value
collection. We will slightly abuse the generic name T in this
paper. Whenever a template parameter is mentioned with the name T,
it always refers to such a value collection type unless stated otherwise.
* A collection of all particle state values and their associated
weights. This is abstracted by a Particle<T> object. We refer to
this as the particle collection. A Particle<T> object has
three primary sub-objects. One is the above type T value collection
object. Another is an object that abstracts weights
$\{W_t^{(i)}\}_{i=1}^N$. By default this is a WeightSet object. The
last is a collection of streams, one for each particle. By default
this is an RngSet object.
* Operations that perform tasks common to all samplers to these
particles, such as resampling. These are the member functions of
* A sampler that updates the particles (state values and weights) using
user defined callbacks. This is a Sampler<T> object.
* Monitors that record the importance sampling estimates of certain
functions defined for the values when the sampler iterates. These are
Monitor<T> objects. A monitor for the estimates of $E[h(X_t)]$
computes $h(X_t^{(i)})$ for each $i = 1,\dots,N$. The function value
$h(X_t)$ is allowed to be a vector.
Note that within the core module, all operations are applied to
Particle<T> objects, that is $\{W_t^{(i)},X_t^{(i)}\}_{i=1}^N$, instead
of a single particle. Later we will see how to write operations that can be
applied to individual particles and can be parallelized easily.
§.§.§ Program structures
A program usually consists of the following tasks.
* Define a value collection type T.
* Constructing a Sampler<T> object.
* Configure the behavior of initialization and updating by adding
callable objects to the sampler object.
* Optionally add monitors.
* Initialize and iterate the sampler.
* Retrieve the outputs, estimates and other informations.
In this section we document how to implement each of these tasks. Within the
library, all public classes, namespaces and free functions, are declared
in the namespace vsmc.
§.§.§ The value collection
The template parameter T is a user defined type that abstracts the
value collection. does not restrict how the values shall be actually
stored. They can be stored in memory, spread among nodes of a cluster, in memory or even in a database. However this kind of flexibility comes with a
small price. The value collection does need to fulfill two requirements. We
will see later that for most common usage, provides readily usable
implementations, on top of which users can create problem specific classes.
First, the value collection class T has to provide a constructor of the
T (SizeType N)
where SizeType is some integer type. Since allows one to allocate
the states in any way suitable, one needs to provide this constructor which
Particle<T> can use to allocate them.
Second, the class has to provide a member function named copy that
copies each particle according to replication numbers given by a resampling
algorithm. For the same reason as above has no way to know how it can
extract and copy a single particle when it is doing the resampling. The
signature of this member function may look like
template <typename SizeType>
void copy (std::size_t N, const SizeType *copy_from);
The pointer copy_from points to an array that has $N$ elements,
where $N$ is the number of particles. After calling this member function, the
value of particle i shall be copied from the particle
j = copy_from[i]. In other words, particle i is a child of
particle copy_from[i], or copy_from[i] is the parent of particle
i. If a particle j shall remain itself, then
copy_from[j] == j. How the values are actually copied is user defined.
Note that, the member function can take other forms, as usual when using template generic programming.
The actual type of the pointer copy_from, SizeType, is
Particle<T>::size_type, which depends on the type T. For
example, define the member function as the following is also allowed,
void copy (int N, const std::size_t *copy_from);
provided that Particle<T>::size_type is indeed std::size_t.
However, writing it as a member function template releases the users from
finding the actual type of pointer copy_from and the sometimes
troubling forward declaration issues. Will not elaborate such more technical
issues further.
§.§.§ Constructing a sampler
Once the value collection class T is defined. One can start
constructing samplers. For example, the following line creates a sampler
with $N$ particles
Sampler<T> sampler(N);
The number of particles is the only mandatory argument of the constructor.
There are two optional parameters. The complete signature of the constructor
explicit Sampler<T>::Sampler (Sampler<T>::size_type N,
ResampleScheme scheme = Stratified, double threshold = 0.5);
The scheme parameter is self-explanatory. provides a few built-in
resampling schemes; see the reference manual for a list of them. User defined
resampling algorithms can also be used. See the reference manual for details.
The threshold is the threshold of $\ess/N$ below which a resampling
will be performed. It is obvious that if $\text{\code{threshold}}\ge1$ then
resampling will be performed at each iteration. If
$\text{\code{threshold}}\le0$ then resampling will never be performed. Both
parameters can be changed later. However the size of the sampler can never be
changed after the construction.
§.§.§ Initialization and updating
All the callable objects that initialize, move and weight the particle
collection can be added to a sampler through a set of member functions. All
these objects operate on the Particle<T> object. Because allows one to manipulate the particle collection as a whole, in principle many
kinds of parallelization are possible.
To set an initialization method, one need to implement a function with the
following signature,
std::size_t init_func (Particle<T> particle, void *param);
or a class with operator() properly defined, such as
struct init_class
std::size_t operator() (Particle<T> particle, void *param); ;
They can be added to the sampler through
respectively. std::function or its equivalent
boost::function can also be used. For example,
std::function<std::size_t (Particle<T> , void *)> init_obj(init_func);
The addition of updating methods is more flexible. There are two kinds of
updating methods. One is simply called move in , and is performed
before the possible resampling at each iteration. These moves usually perform
the updating of the weights among other tasks. The other is called
mcmc, and is performed after the possible resampling. They are often
type moves. Multiple move's or mcmc's are also allowed. In
fact a sampler consists of a queue of move's and a queue of
mcmc's. The move's in the queue can be changed through
Sampler<T> move (const Sampler<T>::move_type new_move, bool append);
If append == true then new_move is appended to the existing
(possibly empty) queue. Otherwise, the existing queue is cleared and
new_move is added. The member function returns a reference to the
updated sampler. For example, the following move,
std::size_t move_func (std::size_t iter, Particle<T> particle);
can be added to a sampler by
sampler.move(move_func, false);
This will clear the (possibly empty) existing queue of move's and set a
new one. To add multiple moves into the queue,
sampler.move(move1, true).move(move2, true).move(move3, true);
Objects of class type with operator() properly defined can also be
used, similarly to the initialization method. The queue of mcmc's can
be used similarly. See the reference manual for other methods that can be used
to manipulate these two queues.
In principle, one can combine all moves into a single move. However, sometimes
it is more natural to think of a queue of moves. For instance, if a
multi-block Metropolis random walk consists of kernels $K_1$ and $K_2$, then
one can implement each of them as functions, say mcmc_k1 and
mcmc_k2, and add them to the sampler sequentially,
sampler.mcmc(mcmc_k1, true).mcmc(mcmc_k2, true);
Then at each iteration, they will be applied to the particle collection
sequentially in the order in which they are added.
§.§.§ Running the algorithm, monitoring and outputs
Having set all the operations, one can initialize and iterate the sampler by
sampler.initialize((void *)param);
The param argument to initialize is optional, with NULL
as its default. This parameter is passed to the user defined init_func.
The iter_num argument to iterate is also optional; the default
is $1$.
Before initializing the sampler or after a certain time point, one can add
monitors to the sampler. The concept is similar to 's monitor
statement, except it does not monitor the individual values but rather the
importance sampling estimates. Consider approximating $\Exp[h(X_t)]$, where
$h(X_t) = (h_1(X_t),\dots,h_m(X_t))$ is an $m$-vector function. The importance
sampling estimate can be obtained by $AW$ where $A$ is an $N$ by $m$ matrix
where $A(i,j) = h_j(X_t^{(i)})$ and $W = (W_t^{(i)},\dots,W_t^{(N)})^T$ is the
$N$-vector of normalized weights. To compute this importance sampling
estimate, one need to define the following evaluation function (or a class
with operator() properly defined),
void monitor_eval (std::size_t iter, std::size_t m,
const Particle<T> particle, double *res)
and add it to the sampler by calling,
sampler.monitor("some.user.chosen.variable.name", m, monitor_eval);
When the function monitor_eval is called, iter is the iteration
number of the sampler, m is the same value as the one the user passed
to Sampler<T>::monitor; and thus one does not need global variable or
other similar techniques to access this value. The output pointer res
points to an $N \times m$ output array of row major order. That is, after the
calling of the function, res[i * dim + j] shall be $h_j(X_t^{(i)})$.
After each iteration of the sampler, the importance sampling estimate will be
computed automatically. See the reference manual for various ways to retrieve
the results. Usually it is sufficient to output the sampler by
std::ofstream output("file.name");
output << sampler << std::endl;
where the output file will contain the importance sampling estimates among
other things. Alternatively, one can use the Monitor<T>::record member
function to access specific historical results. See the reference manual for
details of various overloaded version of this member function.
A reference to the value collection T object can be retrieved through
the Particle<T>::value member function. Particle<T> objects
manage the weights through a weight set object, which by default is of type
WeightSet. The Particle<T>::weight_set member function returns a
reference to this weigh set object. A user defined weight set class that
abstracts $\{W_t^{(i)}\}_{i=1}^N$ can also be used. The details involve some
more advanced template techniques and are documented in the reference
manual. One possible reason for replacing the default is to provide special
memory management of the weight set. For example, the module provides a
special weight set class that manages weights across multiple nodes and
perform proper normalization, computation of , and other tasks.
The default WeightSet object provides some ways to retrieve weights.
Here we document some of the most commonly used. See the reference manual for
details of others. The weights can be accessed one by one, for example,
Particle<T> particle = sampler.particle();
double w_i = particle.weight_set().weight(i);
double log_w_i = particle.weight_set().log_weight(i);
One can also read all weights into a container, for example,
std::vector<double> w(particle.size());
double *lw = new double[particle.size()];
Note that these member functions accept general output iterators.
§.§.§ Implementing initialization and updating
So far we have only discussed how to add initialization and updating objects
to a sampler. To actually implement them, one writes callable objects that
operate on the Particle<T> object. For example, a move can be
implemented through the following function as mentioned before,
std::size_t move_func (std::size_t iter, Particle<T> particle);
Inside the body of this function, one can change the value by manipulating the
object through the reference returned by particle.value().
When using the default weight set class, the weights can be updated through a
set of member functions of WeightSet. For example,
std::vector<double> weight(particle.size());
The set_equal_weight member function sets all weights to be equal. The
set_weight and set_log_weight member functions set the values of
weights and logarithm weights, respectively. The mul_weight and
add_log_weight member functions multiply the weights or add to the
logarithm weights by the given values, respectively. All these member functions
accept general input iterators as their arguments.
One important thing to note is that, whenever one of these member functions is
called, both the weights and logarithm weights will be re-calculated,
normalized, and the will be updated. The reason for not allowing changing
a single particle's weight is that, in a multi-threading environment, it is
possible for one to change one weight in one thread, and obtain another in
another thread without proper normalizing. Conceptually, changing one weight
actually changes all weights.
§.§.§ Generating random numbers
The Particle<T> object has a sub-object, a collection of engines
that can be used with <random> or distributions. For
each particle i, one can obtain an engine that produces an stream
independent of others by
To generate distribution random variates, one can use the
<random> library. For example,
std::normal_distribution<double> rnorm(mean, sd);
double r = rnorm(particle.rng(i));
or use the library,
boost::random::normal_distribution<double> rnorm(mean, sd);
double r = rnorm(particle.rng(i));
itself also makes use of <random> or depending
on the value of the configuration macro VSMC_HAS_CXX11LIB_RANDOM.
Though the user is free to choose which one to use in their own code, there is
a convenient alternative. For each class defined in <random>,
it is imported to the vsmc::cxx11 namespace. Therefore one can use
cxx11::normal_distribution<double> rnorm(mean, sd);
while the underlying implementation of normal_distribution can be
either standard library or . The benefit is that if one needs to
develop on multiple platforms, and only some of them support and some
of them have the library, then only the configure macro
VSMC_HAS_CXX11LIB_RANDOM needs to be changed. This can be configured
through and other build systems. Of course, one can also use an
entirely different system than those provided by .
§.§ SMP module
§.§.§ The value collection
Many typical problems' value collections can be viewed as a matrix of certain
type. For example, a simple particle filter whose state is a vector of length
Dim and type double can be viewed as an $N$ by Dim matrix
where $N$ is the number of particles. A trans-dimensional problem can use an
$N$ by $1$ matrix whose type is a user defined class, say StateType.
For this kind of problems, provide a class template
template <MatrixOrder Order, std::size_t Dim, typename StateType>
class StateMatrix;
which provides the constructor and the copy member function required by
the core module interface, as well as methods for accessing individual values.
The first template parameter (possible value RowMajor or
ColMajor) specifies how the values are ordered in memory. Usually one
shall choose RowMajor to optimize data access. The second template
parameter is the number of variables, an integer value no less than $1$ or the
special value Dynamic, in which case StateMatrix provides a
member function resize_dim such that the number of variables can be
changed at runtime. The third template parameter is the type of the state
Each particle's state is thus a vector of length Dim, indexed from
0 to Dim - 1. To obtain the value at position pos of the
vector of particle i, one can use one of the following member
StateBase<RowMajor, Dim, StateType> value(N);
StateType val = value.state(i, pos);
StateType val = value.state(i, Position<Pos>());
StateType val = value.state<Pos>(i);
where Pos is a compile time constant expression whose value is the same
as pos, assuming the position is known at compile time. One can also
read all values. To read the variable at position pos,
std::vector<StateType> vec(value.size());
value.read_state(pos, vec.begin());
Or one can read all values through an iterator,
std::vector<StateType> mat(Dim * value.size());
value.read_state_matrix(ColMajor, mat.first());
Alternatively, one can also read all values through an iterator which points
to iterators,
std::vector<std::vector<StateType> > mat(Dim);
for (std::size_t i = 0; i != Dim; ++i)
std::vector<std::vector<StateType>::iterator> iter(Dim);
for (std::size_t i = 0; i != Dim; ++i)
iter[i] = mat[i].begin();
If the compiler support <tuple>, vSMC also provides a
StateTuple class template, which is similar to StateMatrix
except that the types of values do not have to be the same for each variable.
This is similar to 's data.frame. For example, suppose each
particle's state is formed by two double's, an int and a user
defined type StateType, then the following constructs a value
collection using StateTuple,
StateTuple<ColMajor, double, double, int, StateType> value(N);
And there are a few ways to access the state values, similar to
double x0 = value.state(i, Position<0>());
int x2 = value.state<2>(i);
std::vector<StateType> vec(value.size());
state.read_state(Position<3>(), vec.begin());
See the reference manual for details.
§.§.§ A single particle
For a Particle<T> object, one can construct a SingleParticle<T>
object that abstracts one of the particle from the collection. For example,
Particle<T> particle(N);
SingleParticle<T> sp(i, particle);
create a SingleParticle<T> object corresponding to the particle at
position i. There are a few member functions of
SingleParticle<T> that makes access to individual particles easier than
through the interface of Particle<T>. Firt sp.id() returns the
value of the argument i in the above code that created this
SingleParticle<T> object. In addition, sp.rng() is equivalent to
particle.rng(i). Also sp.particle() returns a constant reference
to the Particle<T> object. And sp.particle_ptr() returns a
pointer to such a constant Particle<T> object. Note that, one cannot
get write access to a Particle<T> object through interface of
SingleParticle<T>. Instead, one can only get write access to a single
particle. For example, If T is a StateMatrix instantiation or
its derived class, then sp.state(pos) is equivalent to
particle.value().state(i, pos) and the reference it returns is mutable.
See the reference manual for more informations on the interface of the
SingleParticle class template. SingleParticle<T> objects are
usually not constructed by users, but rather by the libraries' other classes
in the module, and passed to user defined functions, as we will see very
§.§.§ Sequential and parallel implementations
Once we have the SingleParticle<T> concept, we are ready to introduce
how to write implementations of algorithms that manipulate a single
particle and can be applied to all particles in parallel or sequentially. For
sequential implementations, this can be done through five base classes,
template <typename BaseState> class StateSEQ;
template <typename T, typename D = Virtual> class InitializeSEQ;
template <typename T, typename D = Virtual> class MoveSEQ;
template <typename T, typename D = Virtual> class MonitorEvalSEQ;
template <typename T, typename D = Virtual> class PathEvalSEQ;
The template parameter BaseState needs to satisfy the general value
collection requirements in addition to a copy_particle member function,
for example, StateMatrix. Other base classes expect T to satisfy
general value collection requirements. The details of all these class
templates can be found in the reference manual. Here we use the
MoveSEQ<T> class as an example to illustrate their usage. Recall that
Sampler<T> expect a callable object which has the following signature
as a move,
std::size_t move_func (std::size_t iter, Particle<T> particle);
For the purpose of illustration, the type T is defined as,
typedef StateMatrix<RowMajor, 1, double> T;
Here is a typical example of implementation of such a function,
std::size_t move_func (std::size_t iter, Particle<T> particle)
std::vector<double> inc_weight(particle.size());
for (Particle<T>::size_type i = 0; i != particle.size(); ++i)
cxx11::normal_distribution<double> rnorm(0, 1);
particle.value().state(i, 0) = rnorm(particle.rng(i));
inc_weight[i] = cal_inc_weight(particle.value().state(i, 0);
where cal_inc_weight is some function that calculates the logarithm
incremental weights. As we see, there are three main parts of a typical move.
First, we allocate a vector inc_weight. Second, we generate normal
random variates for each particle and calculate the incremental weights. This
is done through a for loop. Third, we add the logarithm incremental
weights. The first and the third are global operations while the second
is local. The first and the third are often optional and absent. The
local operation is also usually the most computational intensive part of algorithms and can benefit the most from parallelizations.
With MoveSEQ<T>, one can derive from this class, which defines
the operator() required by the core module interface,
std::size_t operator() (std::size_t iter, Particle<T> particle);
and customize what this operator does by defining one or more of the following
three member functions, corresponding to the three parts, respectively,
void pre_processor (std::size_t iter, Particle<T> particle);
std::size_t move_state (std::size_t iter, SingleParticle<T> sp);
void post_processor (std::size_t iter, Particle<T> particle);
For example,
#include <vsmc/smp/backend_seq.hpp>
class move : public MoveSEQ<T>
public :
void pre_processor (std::size_t iter, Particle<T> particle)
std::size_t move_state (std::size_t iter, SingleParticle<T> sp)
cxx11::normal_distribution<double> rnorm(0, 1);
sp.state(0) = rnorm(sp.rng());
inc_weight_[sp.id()] = cal_inc_weight(sp.state(0));
void post_processor (std::size_t iter, Particle<T> particle)
private :
std::vector<double> inc_weight_;
The operator() of MoveSEQ<T> is equivalent to the single
function implementation as shown before.
In the simplest case, MoveSEQ only takes away the loop around Part 2.
However if one implement the move in such a way, and then replace
MoveSEQ with MoveOMP, the changing of the base class name causes
to use to parallelize the loop. For example, one can declare the
class as
#include <vsmc/smp/backend_omp.hpp>
class move : public MoveOMP<T>;
and use exactly the same implementation as before. Now when
move::operator() is called, it will be parallelized by . Other
backends are available in case is not available. Among them there are
and . In addition to these well known parallelization programming
models, also has its own implementation using <thread>.
There are other backends documented in the reference manual. To use any of
these parallelization, all one need to do is to change a few base class names.
In practice, one can use conditional compilation, for example, to use a
sequential implementation or a parallelized one, we can write,
#ifdef USE_SEQ
#include <vsmc/smp/backend_seq.hpp>
#define BASE_MOVE MoveSEQ
#ifdef USE_OMP
#include <vsmc/smp/backend_omp.hpp>
#define BASE_MOVE MoveOMP
class move : public BASE_MOVE<T>;
And we can compile the same source into different samplers with
Makefile rules such as the following,
prog-seq : prog.cpp
$(CXX) $(CXXFLAGS) -DUSE_SEQ -o prog-seq prog.cpp
prog-omp : prog.cpp
$(CXX) $(CXXFLAGS) -DUSE_omp -o prog-omp prog.cpp
Or one can configure the source file through a build system such as ,
which can also determine which programming model is available on the system.
§.§.§ Adapter
Sometimes, the cumbersome task of writing a class to implement a move and
other operations can out weight the power we gain through object oriented
programming. For example, a simple move_state-like function is all that
we need to get the work done. In this case, one can create a move through the
MoveAdapter. For example, say we have implemented
std::size_t move_state (std::size_t iter, SingleParticle<T> sp);
as a function. Then one can create a callable object acceptable to
Sampler<T>::move through
MoveAdapter<T, MoveSEQ> move_obj(move_state);
MoveAdapter<T, MoveSTD> move_obj(move_state);
MoveAdapter<T, MoveTBB> move_obj(move_state);
MoveAdapter<T, MoveCILK> move_obj(move_state);
MoveAdapter<T, MoveOMP> move_obj(move_state);
sampler.move(move_obj, false);
These are respectively, sequential, <thread>, , ,
and implementations. The first template parameter is the type of value
collection and the second is the name of the base class template. Actually,
the MoveAdapter's constructor accepts two more optional arguments, the
pre_processor and the post_processor, corresponding to the other
two aforementioned member functions. Similar adapters for the other three base
classes also exist.
Another scenario where an adapter is desired is that which backend to use
needs be decided at runtime. The above simple adapters can already be used for
this purpose. In addition, another form of the adapter is as the following,
class move;
MoveAdapter<T, MoveTBB, move> move_obj;
sampler.move(move_obj, false);
where the class move has the same definition as before but it no longer
derives from any base class. The class move is required to have a
default constructor, a copy constructor and an assignment operator.
§.§ Thread-safety and scalability considerations
When implementing parallelized algorithms, issues such as thread-safety
cannot be avoided even though the library hides most parallel constructs
from the user.
Classes in the library usually guarantee that their member functions are
thread-safe in the sense that calling the same member function on different
objects at the same time from different threads is safe. However, calling
mutable member functions on the same object from different threads is usually
not safe. Calling immutable member functions is generally safe. There are a
few exceptions,
* The constructors of Particle and Sampler are not
thread-safe. Therefore if one need to construct multiple Sampler
from different threads, a mutex protection is needed. However, subsequent
member function calls on the constructed objects are thread-safe according
to the above rules.
* Member functions that concern a single particle are generally
thread-safe in the sense that one can call them on the same object from
different threads as long as they are called for different particles. For
example Particle::rng and StateMatrix::state are
In general, one can safely manipulate different individual particles from
different threads, which is the minimal requirement for scalable
parallelization. But one cannot manipulate the whole particle collection from
different threads, for example WeightSet::set_log_weight.
User defined callbacks shall generally follow the same rules. For example, for
a MoveOMP subclass, pre_processor and post_processor does
not have be thread-safe, but move_state needs to be. In general, avoid
write access to memory locations shared by all particles from
move_state and other similar member functions. One needs to take some
extra care when using third-party libraries. For example, in our example
implementation of the move class, the rnorm object, which is
used to generate Normal random variates, is defined within move_state
instead of being a class member data even though it is created with the same
parameters for each particle. This is because the call rnorm(sp.rng())
is not thread-safe in some implementations, for example, when using the library.
For scalable performance, certain practices should be avoided when
implementing member functions such as move_state. For example, dynamic
memory allocation is usually lock-based and thus should be avoided.
Alternatively one can use a scalable memory allocator such as the one provided
by . In general, in functions such as move_state, one should avoid
using locks to guarantee thread-safety, which can be a bottleneck to parallel
§ EXAMPLE
§.§ A simple particle filter
§.§.§ The model and algorithm
This is an example used in . Through this example, we will show how to
re-implement a simple particle filter in . It shall walk one through the
basic features of the library. is the first general framework for
implementing algorithms in and serves as one of the most important
inspirations for the library. It is widely used by many researchers. One
of the goals of the current work is that users familiar with shall find
little difficulty in using the new library.
The state space model, known as the almost constant velocity model in the
tracking literature, provides a simple scenario. The state vector $X_t$
contains the position and velocity of an object moving in a plane. That is,
$X_t = (\xpos^t, \ypos^t, \xvel^t, \yvel^t)^T$. Imperfect observations $Y_t =
(\xobs^t, \yobs^t)^T$ of the positions are possible at each time instance. The
state and observation equations are linear with additive noises,
\begin{align*}
X_t &= AX_{t-1} + V_t \\
Y_t &= BX_t + \alpha W_t
\end{align*}
\begin{equation*}
A = \begin{pmatrix}
1 & \Delta & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix} \qquad
B = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
\end{pmatrix} \qquad
\alpha = 0.1
\end{equation*}
and we assume that the elements of the noise vector $V_t$ are independent
Normal with variance $0.02$ and $0.001$ for position and velocity,
respectively. The observation noise, $W_t$ comprises independent, identically
distributed $t$-distributed random variables with degree of freedom $\nu =
10$. The prior at time $0$ corresponds to an axis-aligned Gaussian with
variance $4$ for the position coordinates and $1$ for the velocity
coordinates. The particle filter algorithm is shown in Algorithm <ref>.
Set $t\leftarrow0$.
$\xpos^{(0,i)},\ypos^{(0,i)}\sim\rnorm(0,4)$ and
Weight $W_0^{(i)} \propto L(X_0^{(i)}|Y_0)$ where $L$ is
the likelihood function.
Set $t\leftarrow t + 1$.
\begin{align*}
\xpos^{(t,i)}&\sim\rnorm(\xpos^{(t-1,i)} + \Delta\xvel^{(t-1,i)}, 0.02) &
\xvel^{(t,i)}&\sim\rnorm(\xvel^{(t-1,i)}, 0.001) \\
\ypos^{(t,i)}&\sim\rnorm(\ypos^{(t-1,i)} + \Delta\yvel^{(t-1,i)}, 0.02) &
\yvel^{(t,i)}&\sim\rnorm(\yvel^{(t-1,i)}, 0.001)
\end{align*}
Weight $W_t^{(i)} \propto W_{t-1}^{(i)}L(X_t^{(i)}|Y_t)$.
Repeat the Iteration step until all data are processed.
Particle filter algorithm for the almost constant velocity model.
§.§.§ Implementations
We first introduce the body of the main function, showing the typical
work flow of a program.
#include <vsmc/core/sampler.hpp>
#include <vsmc/smp/adapter.hpp>
#include <vsmc/smp/state_matrix.hpp>
#include <vsmc/smp/backend_seq.hpp>
const std::size_t DataNum = 100;
const std::size_t ParticleNum = 1000;
const std::size_t Dim = 4;
int main ()
Sampler<cv> sampler(ParticleNum);
sampler.move(cv_move(), false);
MonitorEvalAdapter<cv, MonitorEvalSEQ> cv_est(cv_monitor_state);
sampler.monitor("pos", 2, cv_est);
sampler.initialize((void *)"pf.data");
sampler.iterate(DataNum - 1);
std::ofstream est("pf.est");
est << sampler << std::endl;
return 0;
In the main function, we constructed a sampler with ParticleNum
particles. And then we added the initialization function cv_init and
the move object of type cv_move. And then we added a monitor that will
record the importance sampling estimates of the two position parameters. Next,
we initialized the sampler with data file pf.data and iterate the
sampler until all data are processed. The last step is that we output the
results into a file called pf.est.
The class cv will be our value collection which is a subclass of
StateMatrix<RowMajor, Dim, double>. To illustrate both the core module
and the module, the initialization cv_init will be implemented
as a standalone function. The move cv_move will be implemented as a
derived class of MoveSEQ<cv>. To monitor the importance sampling
estimates of the two position parameters, we will implement a simple function
cv_monitor_state and use the adapter MonitorEvalAdapter<cv,
The value collection is an $N$ by Dim (in this case $\text{\code{Dim}}
= 4$) matrix of type double. We can simply use
StateMatrix<RowMajor, Dim, double> as our value collection. However, we
would like to enhance its functionality through inheritance. First, since the
data is shared by all particles, it is natural to bind it with the value
collection. Second, both the initialization and move will need to calculate
the log-likelihood. We can implement it as a standalone function, but since
the log-likelihood function will need to access the data, it is convenient to
implement it as a member function of the value collection cv. Here is
the full implementation of this simple value collection class
class cv : public StateMatrix<RowMajor, Dim, double>
public :
cv (size_type N) :
StateMatrix<RowMajor, Dim, double>(N),
x_obs_(DataNum), y_obs_(DataNum)
double log_likelihood (std::size_t iter, size_type id) const
const double scale = 10;
const double nu = 10;
double llh_x = scale * (state(id, 0) - x_obs_[iter]);
double llh_y = scale * (state(id, 1) - y_obs_[iter]);
llh_x = std::log(1 + llh_x * llh_x / nu);
llh_y = std::log(1 + llh_y * llh_y / nu);
return -0.5 * (nu + 1) * (llh_x + llh_y);
void read_data (const char *filename)
std::ifstream data(filename);
for (std::size_t i = 0; i != DataNum; ++i)
data >> x_obs_[i] >> y_obs_[i];
private :
std::vector<double> x_obs_;
std::vector<double> y_obs_;
The log_likelihood member function accepts the iteration number and the
particle's id as arguments. It returns the log-likelihood of the
id'th particle at iteration iter. The read_data member
function simply read the data from a file.
The initialization is implemented through the cv_init function,
std::size_t cv_init (Particle<cv> particle, void *filename);
such that, it first checks if filename is NULL. If it is not,
then we use it to read the data. So the first initialization may look like
sampler.initialize((void *)filename);
And after that, if we want to re-initialize the sampler, we can simply call,
This will reset the sampler and initialize it again but without reading the
data. If the data set is large, repeated can be very expensive. After
reading the data, we will initialize each particle's value by Normal random
variates, and calculate its log-likelihood. The last step is to set the
logarithm weights of the particle collection. Since this is not a
accept-reject type algorithm, the returned acceptance count bares no meaning.
Here is the complete implementation,
std::size_t cv_init (Particle<cv> particle, void *filename)
if (filename)
particle.value().read_data(static_cast<const char *>(filename));
const double sd_pos0 = 2;
const double sd_vel0 = 1;
cxx11::normal_distribution<double> norm_pos(0, sd_pos0);
cxx11::normal_distribution<double> norm_vel(0, sd_vel0);
std::vector<double> log_weight(particle.size());
for (Particle<cv>::size_type i = 0; i != particle.size(); ++i)
particle.value().state(i, 0) = norm_pos(particle.rng(i));
particle.value().state(i, 1) = norm_pos(particle.rng(i));
particle.value().state(i, 2) = norm_vel(particle.rng(i));
particle.value().state(i, 3) = norm_vel(particle.rng(i));
log_weight[i] = particle.value().log_likelihood(0,i);
return 0;
In this example, we read all data from a single file for simplicity. In a
realistic application, the data is often processed online – the filter is
applied when new data becomes available. In this case, the user can use the
optional argument of Sampler<T>::initialize to pass necessary
information to open a data connection instead of a file name.
In the above implementation we iterated over all particles. There are other
ways to iterate over a particle collection. First we can use
SingleParticle<cv> objects,
for (Particle<cv>::size_type i = 0; i != particle.size(); ++i)
SingleParticle<cv> sp(i, particle);
sp.state(0) = norm_pos(sp.rng());
Second, provides style iterators, Particle<T>::iterator,
whose value_type is SingleParticle<T>. Therefore one can write
the following loop
for (Particle<cv>::iterator iter = particle.begin();
iter != particle.end(); ++iter)
iter->state(0) = norm_pos(iter->rng());
Third, if one has a compiler supporting auto and range-based
for, the following is also supported,
for (auto sp : particle) sp.state(0) = norm_pos(sp.rng());
There are little or no performance difference among all these forms of loops.
However, one can choose an appropriate form to work with the interfaces of
other libraries. For example, the iterator support allows to be used
with <algorithm> and other similar libraries.
The updating method cv_move is similar to cv_init. It will
update the particle's value by adding Normal random variates. However, as we
see above, each call to cv_init causes a log_weight vector being
allocated. Its size does not change between iterations. So it can be viewed as
some resource of cv_move and it is natural to use a class object to
manage it. Here is the implementation of cv_move,
class cv_move : public MoveSEQ<cv>
public :
void pre_processor (std::size_t iter, Particle<cv> particle)
std::size_t move_state (std::size_t iter, SingleParticle<cv> sp)
const double sd_pos = std::sqrt(0.02);
const double sd_vel = std::sqrt(0.001);
const double delta = 0.1;
cxx11::normal_distribution<double> norm_pos(0, sd_pos);
cxx11::normal_distribution<double> norm_vel(0, sd_vel);
sp.state(0) += norm_pos(sp.rng()) + delta * sp.state(2);
sp.state(1) += norm_pos(sp.rng()) + delta * sp.state(3);
sp.state(2) += norm_vel(sp.rng());
sp.state(3) += norm_vel(sp.rng());
incw_[sp.id()] = sp.particle().value().log_likelihood(iter, sp.id());
return 0;
void post_processor (std::size_t iter, Particle<cv> particle)
particle.weight_set().add_log_weight( incw_[0]);
private :
std::vector<double> incw_;
First before calling any move_state, the pre_processor will be
called, as described in Section <ref>. At this step, we will
resize the vector used for storing incremental weights. After the first
resizing, subsequent calls to resize will only cause reallocation if
the size changed. In our example, the size of the particle system is fixed, so
we don't need to worry about excessive dynamic memory allocations. The
move_state member function moves each particle according to our model.
And after move_state is called for each particle, the
post_processor will be called and we simply add the logarithm
incremental weights.
For each particle, we want to monitor the $\xpos$ and $\ypos$ parameters and
get the importance sampling estimates. To extract the two values from a
particle, we can implement the following function
void cv_monitor_state (std::size_t iter, std::size_t dim,
ConstSingleParticle<cv> csp, double *res)
assert(dim <= Dim);
for (std::size_t d = 0; d != dim; ++d)
res[d] = csp.state(d);
and in the main function we construct a monitor by
MonitorEvalAdapter<cv, MonitorEvalSEQ> cv_est(cv_monitor_state);
and add it to the sampler through
sampler.monitor("pos", 2, cv_est);
If later we decided to monitor all states, we only need to change the 2
in the above line to Dim.
After we implemented all the above, compiled and ran the program, a file
called pf.est was written by the following statement in the main
est << sampler << std::endl;
The output file contains the , resampling, importance sampling estimates
and other informations in a table. This file can be read by most statistical
softwares for further analysis. For instance, we can process this file with
, and get the plot of the estimates of positions against the
observations, as shown in Figure <ref>.
Observations and estimates of a simple particle filter.
§.§ Bayesian Modeling of Gaussian mixture model
§.§.§ The model and algorithm
Since Richardson and Green, 1997, the Gaussian mixture model () has
provided a canonical example of a model-order-determination problem. We use
the same model as in Del Moral et al., 2006 to illustrate the implementation
of this classical example in Monte Carlo literature. This model is also used
in Zhou et al., 2013 for demonstration of the use of in the context of
Bayesian model comparison, which provides more details of the following
setting. The model is as follows; data $\data = (y_1,\dots,y_n)$ are
independently and identically distributed as
\begin{equation*}
y_i|\theta_r \sim \sum_{j=1}^r \omega_j\rnorm(\mu_j,\lambda_j^{-1})
\end{equation*}
where $\rnorm(\mu_j,\lambda_j^{-1})$ denotes the Normal distribution with mean
$\mu_j$ and precision $\lambda_j$; $\theta_r =
(\mu_{1:r},\lambda_{1:r},\omega_{1:r})$ and $r$ is the number of components in
each model. The parameter space is thus $\Real^r\times\Real^{+r}\times S_r$
where $S_r = \{\omega_{1:r}:0\le\omega_j\le1; \sum_{j=1}^r\omega_j=1\}$ is the
standard $(r-1)$-simplex. The priors which are the same for each component are
taken to be $\mu_j\sim\rnorm(\xi,\kappa^{-1})$,
$\lambda_j\sim\rgamma(\nu,\chi)$ and $\omega_{1:r}\sim\rdir(\rho)$ where
$\rdir(\rho)$ is the symmetric Dirichlet distribution with parameter $\rho$
and $\rgamma(\nu,\chi)$ is the Gamma distribution with shape $\nu$ and scale
$\chi$. The prior parameters are set in the same manner as in
Richardson and Green, 1997; also see Zhou et al., 2013 for details. The data
is simulated from a four components model with $\mu_{1:4} = (-3, 0,3, 6)$, and
$\lambda_j =2$, $\omega_j = 0.25$, $j = 1,\dots,4$. Our interest is to
simulate the posterior distribution of models with $r$ components, denoted by
$M_r$ and obtaining the normalizing constant for the purpose of Bayesian model
comparison <cit.>.
Numerous strategies are possible to construct a sequence of distributions for
the purpose of sampling. One option is to use for each model $M_r$,
$r\in\{1,2,\dots\}$, the sequence $\{\pi_t\}_{t=0}^{T_r}$, defined by
\begin{equation}
\pi_t(\theta_r^t) \propto
\pi(\theta_r^t|M_r)p(\data|\theta_r^t,M_r)^{\alpha(t/T_r)}.
\label{eq:geometry}
\end{equation}
where the number of distribution, $T_r$, and the annealing schedule,
$\alpha:[0,1]\to[0,1]$ may be different for each model. This leads to
Algorithm <ref>.
For each model $M_r\in\calM$ perform the following algorithm.
Set $t\leftarrow0$.
Sample $\theta_r^{(i,t)}\sim\pi(\theta_r^{(i,t)}|M_r)$.
Weight $W_0^{(i)} \propto 1$.
Set $t\leftarrow t + 1$.
Weight $W_t^{(i)} \propto W_{t-1}^{(i)}
p(\data|\theta_r^{(i,t-1)},M_r)^{\alpha(t/T_r) - \alpha([t-1]/T_r)}$.
Apply resampling if necessary.
Sample $\theta_r^{(i,t)} \sim
K_t(\cdot|\theta_r^{(i,t-1)})$, a $\pi_t$-invariant kernel.
Repeat the Iteration step up to $t = T_r$.
algorithm for Bayesian modeling of Gaussian mixture
The kernel $K_t$ in Algorithm <ref> is constructed as a
three-blocks Metropolis random walk,
* Update $\mu_{1:r}$ through a Normal random walk.
* Update $\lambda_{1:r}$ through a Normal random walk on logarithm
scale, that is, on $\log\lambda_{j}$, $j = 1, \dots, r$.
* Update $\omega_{1:r}$ through a Normal random walk on logit scale,
that is, on $\omega_{j}/\omega_r$, $j = 1,\dots,r-1$.
The standard direct estimate of the normalizing constants
[Del Moral et al., 2006] can be obtained from the output of this algorithm,
\begin{equation}
\hat\lambda_{\text{\textsc{ds}}}^{T_r,N} =
\sum_{i=1}^N \frac{\pi(\theta_r^{(i,0)}|M_r)}{\nu(\theta_0^{(i,0)})} \times
\prod_{t=1}^{T_r} \sum_{i=1}^N W_{t-1}^{(i)}
p(\data|\theta_r^{(i,t)}M_r)^{\alpha(t/T_r) - \alpha([t-1]/T_r)}
\label{eq:smc-ds}
\end{equation}
where $W_{t-1}^{(i)}$ is the importance weight of sample $\theta_{t-1}^{(i)}$.
§.§.§ Path sampling for estimation of normalizing constants
As shown in Zhou et al., 2013 the estimation of the normalizing constant
associated with our sequence of distributions can also be achieved by a Monte
Carlo approximation to the path sampling formulation given by
Gelman and Meng, 1998, also known as thermodynamic integration or Ogata's
method. Given a parameter $\alpha$ which defines a family of distributions,
$\{p_{\alpha} = q_{\alpha} / Z_\alpha\}_{\alpha \in [0,1]}$ that move smoothly
from $p_0 = q_0 / Z_0$ to $p_1 = q_1 / Z_1$ as $\alpha$ increases from zero to
one, one can estimate the logarithm of the ratio of their normalizing
constants via a simple integral relationship,
\begin{equation}
\log\biggl( \frac{Z_1}{Z_0} \biggr) =
\int_{0}^{1} \Exp_\alpha \biggl[ \rnd{\log q_{\alpha}(\cdot)}{\alpha}
\biggr] \intd\alpha, \label{eq:path_identity}
\end{equation}
where $\Exp_\alpha$ denotes expectation under $p_\alpha$. The sequence of
distributions in the algorithm for this example can be interpreted as
belonging to such a family of distributions, with $\alpha = \alpha(t/T_r)$.
The sampler provides us with a set of weighted samples obtained from a
sequence of distributions suitable for approximating this integral. At each
time $t$ we can obtain an estimate of the expectation within the integral via
the usual importance sampling estimator, and this integral can then be
approximated via a trapezoidal integration. In summary, the path sampling
estimator of the ratio of normalizing constants $\lambda^{T_r} = \log(Z_1/Z_0)$
can be approximated by
\begin{equation}
\hat\lambda_{\text{\textsc{ps}}}^{T_r,N} = \sum_{t=1}^{T_r}
\frac{1}{2}(\alpha_t - \alpha_{t - 1})(U_t^N + U_{t-1}^N)
\label{eq:path_est}
\end{equation}
\begin{equation}
U_t^N = \sum_{i=1}^N
W_t^{(i)} \rnd{\log q_{\alpha}(X_t^{(i)})}{\alpha}\Bigm|_{\alpha = \alpha_t}
\label{eq:path_import}
\end{equation}
§.§.§ Implementations
In this example we will implement the following classes.
* gmm_param is a class that abstracts the parameters of the
model, $\theta_r = (\mu_{1:r},\lambda_{1:r},\omega_{1:r})$.
* gmm_state is the value collection class.
* gmm_init is a class that implements operations used to
initialize the sampler.
* gmm_move_smc is a class that implements operations used to
update the weights as well as selecting the random walk proposal scales
and distribution parameter $\alpha(t/T_r)$.
* gmm_move_mu, gmm_move_lambda and gmm_move_weight
are classes that implement the random walks, each for one of the three
* gmm_path is a class that implements monitors for the path
sampling estimator. This class is similar to the importance sampling
monitor introduced before. It is to be used with
Sampler<gmm_state>::path_sampling. Its interface requirement will
be documented later.
* gmm_alpha_linear and gmm_alpha_prior are classes that
implement two of the many possible annealing schemes, $\alpha(t/T_r) =
t/T_r$ (linear) and $\alpha(t/T_r) = (t/T_r)^p$, $p > 1$ (prior).
* And last, the main function, which configure, initialize and
iterate the samplers.
This example is considerably more complicated than the last one. Instead of
documenting all the implementation details, for many classes we will only
show the interfaces. In most cases, the implementations are straightforward as
they are either data member accessors or straight translation of mathematical
formulations. For member functions with more complex structures, detailed
explanation will be given. Interested readers can see the source code for more
Later we will build both sequential and parallelized samplers. A few
configuration macros will be defined at compile time. For example, the
sequential sampler is compiled with the following header and macros,
#include <vsmc/smp/backend_seq.hpp>
#define BASE_SATE StateSEQ
#define BASE_INIT InitializeSEQ
#define BASE_MOVE MoveSEQ
#define BASE_PATH PathEvalSEQ
The definitions of these macros will be changed at compile time to build
parallelized samplers. For example, when using parallelization, the
header backend_omp.hpp will be used instead of backend_seq.hpp;
and StateSEQ will be changed to StateOMP along with similar
changes to the other macros. In the distributed source code, this is
configured by the build system.
Again, we first introduce the main function. The required headers are
the same as the last particle filter example in addition to the backend
headers as described above. The following variables used in the main
function will be set by the user input.
int ParticleNum;
int AnnealingScheme;
int PriorPower;
int CompNum;
std::string DataFile;
In the main function, we will create objects that set the distribution
parameter $\alpha(t/T_r)$ at each iteration according to the user input of
AnnealingScheme. Below is the main function. Note that some
source code of operations which set the parameters above are omitted.
int main ()
gmm_move_smc::alpha_setter_type alpha_setter;
if (AnnealingScheme == 1)
alpha_setter = gmm_alpha_linear(IterNum);
if (AnnealingScheme == 2)
alpha_setter = gmm_alpha_prior(IterNum, PriorPower);
Sampler<gmm_state> sampler(ParticleNum);
.move(gmm_move_smc(alpha_setter), false);
.mcmc(gmm_move_mu(), false);
.mcmc(gmm_move_lambda(), true);
.mcmc(gmm_move_weight(), true);
.initialize((void *) DataFile.c_str());
double ds = sampler.particle().value().nc().log_zconst();
double ps = sampler.path().log_zconst();
std::cout << "Standard estimate : " << ds << std::endl;
std::cout << "Path sampling estimate : " << ps << std::endl;
return 0;
The sampler first sets the number of components and allocate memory through
member function comp_num of gmm_state. Then it sets the
initialization and updating methods. Before possible resampling, a
gmm_move_smc object is added. After that, three Metropolis random walks
are appended. In addition, we add a gmm_path object to calculate the
path sampling integration. Then we initialize and iterate the sampler and get
the normalizing constant estimates.
It is obvious that the parameter class gmm_param need to store the
parameters $(\mu_{1:r},\lambda_{1:r},\omega_{1:r})$. We also associate with
each particle its log-likelihood and log-prior. Here is the definition of the
gmm_param class. We omitted definitions of some data access member
class gmm_param
public :
void comp_num (std::size_t num);
void save_old ();
double log_prior () const return log_prior_;
double log_prior () return log_prior_;
double log_likelihood () const return log_likelihood_;
double log_likelihood () return log_likelihood_;
int mh_reject_mu (double p, double u);
int mh_reject_lambda (double p, double u);
int mh_reject_weight (double p, double u);
int mh_reject_common (double p, double u);
double log_lambda_diff () const;
double logit_weight_diff () const;
void update_log_lambda ();
private :
std::size_t comp_num_;
double log_prior_, log_prior_old_, log_likelihood_, log_likelihood_old_;
std::vector<double> mu_, mu_old_;
std::vector<double> lambda_, lambda_old_;
std::vector<double> weight_, weight_old_;
std::vector<double> log_lambda_;
The comp_num member function allocate the memory for a given number of
components. The save_old member function save the current particle
states. It is used before the states are updated with the random walk
proposals, as we will see later when we implement the gmm_move_mu. The
mh_reject_mu member function accept the Metropolis acceptance
probability $p$ and a uniform $(0,1]$ random variate, say $u$; it rejects the
proposed change if $p < u$, and restore the particle state of the parameters
$\mu_{1:r}$ by those values saved by save_old. The member functions
mh_reject_lambda and mh_reject_weight do the same for the other
two set of parameters. All these three also call the mh_reject_common
which restore the stored log-likelihood and log-prior values. The use of these
member functions will be seen in the implementation of gmm_move_mu, in
the context of which their own implementation become obvious. Other member
functions provide some useful computations such as the logarithm of the
$\lambda_{1:r}$. They are used when compute the log-likelihood.
The class gmm_state contains some properties common to all particles,
such as the data and the distribution parameter $\alpha(t/T_r)$. Also, we will
have it to record the logarithm of the ratio of normalizing constants, using
the NormalizingConstant class. We will see how to update this variable
at each iteration in the implementation of gmm_move_smc. The prior
parameters are also stored in the value collection. Here is the definition of
this value collection class. Again, we omitted some data access member
class gmm_state : public BASE_STATE<StateMatrix<RowMajor, 1, gmm_param> >
public :
NormalizingConstant nc () return nc_;
const NormalizingConstant nc () const return nc_;
void alpha (double a)
a = a < 1 ? a : 1;
a = a > 0 ? a : 0;
if (a == 0)
alpha_inc_ = 0;
alpha_ = 0;
alpha_inc_ = a - alpha_;
alpha_ = a;
void comp_num (std::size_t num)
comp_num_ = num;
for (size_type i = 0; i != this->size(); ++i)
this->state(i, 0).comp_num(num);
double update_log_prior (gmm_param param) const;
double update_log_likelihood (gmm_param param) const
static const double log2pi = 1.8378770664093455; // log(2pi)
double ll = -0.5 * obs_.size() * log2pi;
for (std::size_t k = 0; k != obs_.size(); ++k)
double lli = 0;
for (std::size_t i = 0; i != param.comp_num(); ++i)
double resid = obs_[k] - param.mu(i);
lli += param.weight(i) * std::exp(
0.5 * param.log_lambda(i) -
0.5 * param.lambda(i) * resid * resid);
ll += std::log(lli);
return param.log_likelihood() = ll;
void read_data (const char *filename);
private :
NormalizingConstant nc_;
std::size_t comp_num_;
double alpha_, alpha_inc_;
double mu0_, sd0_, shape0_, scale0_;
double mu_sd_, lambda_sd_, weight_sd_;
std::vector<double> obs_;
The variable alpha_inc_ is $\Delta\alpha(t/T_r) = \alpha(t/T_r) -
\alpha((t-1)/T_r)$, which will be used when we update the weights. The
variable nc_ of type NormalizingConstant will be updated when
the weights are changed by gmm_move_smc and it will compute the
standard normalizing constant estimate
$\hat\lambda_{\text{\textsc{ds}}}^{T_r,N}$. The variables mu0_ and
sd0_ are the prior parameters of the means $\mu_{1:r}$. The variables
shape0_ and scale0_ are the prior parameters of the precisions
$\lambda_{1:r}$. The variables mu_sd_, lambda_sd_, and
weight_sd_ are the proposal scales of the three random walks,
respectively. The data access member functions of these variables are omitted
in the above source code snippet.
In the update_log_likelihood member function, the calculation is a
straightforward translation of the mathematical formulation. The
gmm_param::update_log_lambda member function is used before the loop,
which simply calculates $\log\lambda_j$ for $j = 1,\dots,r$, and stores their
values. The purpose is to avoid repeated computation of these quantities
inside the loop. When the function returns, it uses the mutable version of the
gmm_param::log_likelihood member function to update the log-likelihood
stored in the param object. This is the reason that the function is
named with a update prefix. As we will see later, whenever the
parameter values are updated, it will be followed by a call to
update_log_likelihood and update_log_prior, which is implemented
in a similar fashion. Therefore the value we get by calling
gmm_param::log_likelihood will always be “up-to-date” while no
repeated computation is involved. Surely there are other and possibly better
design choices. However, for this simple example, this design serves our
purpose well.
The initialization is implemented using the gmm_init class,
class gmm_init : public BASE_INIT<gmm_state, gmm_init>
public :
std::size_t initialize_state (vsmc::SingleParticle<gmm_state> sp)
const gmm_state state = sp.particle().value();
gmm_param param = sp.state(0);
vsmc::cxx11::normal_distribution<> rmu(
state.mu0(), state.sd0());
vsmc::cxx11::gamma_distribution<> rlambda(
state.shape0(), state.scale0());
vsmc::cxx11::gamma_distribution<> rweight(1, 1);
double sum = 0;
for (std::size_t i = 0; i != param.comp_num(); ++i)
param.mu(i) = rmu(sp.rng());
param.lambda(i) = rlambda(sp.rng());
param.weight(i) = rweight(sp.rng());
sum += param.weight(i);
for (std::size_t i = 0; i != param.comp_num(); ++i)
param.weight(i) /= sum;
return 1;
void initialize_param (vsmc::Particle<gmm_state> particle,
void *filename)
if (filename)
particle.value().read_data(static_cast<const char *>(filename));
The initialize_param member function is called before the
pre_processor, which is absent in this case and have the default
implementation which does nothing. And it processes the optional parameter of
Sampler::initialize, the file name of the data. The
initialize_state member function initialize the state values according
to the prior and update the log-prior and log-likelihood.
After initialization, at each iteration, gmm_move_smc class will
implement the updating of weights as well as selecting of the proposal scales
and the distribution parameter. For example, when using the linear annealing
scheme, we can implement a gmm_alpha_linear class as the following,
class gmm_alpha_linear
public :
gmm_alpha_linear (const std::size_t iter_num) : iter_num_(iter_num)
void operator() (std::size_t iter, Particle<gmm_state> particle)
particle.value().alpha(static_cast<double>(iter) / iter_num_);
private :
std::size_t iter_num_;
It accepts the total number of iterations $T_r$ as an argument to its
constructor. And it implements an operator() that update the
distribution parameter $\alpha(t/T_r)$. The prior annealing scheme can be
implemented similarly. For simplicity and demonstration purpose, we only allow
gmm_move_smc to be configured with different annealing schemes, and
hard code the proposal scales. An industry strength design may make this class
a template with annealing scheme and proposal scales as policy template
class gmm_move_smc
public :
typedef cxx11::function<void (std::size_t, Particle<gmm_state> )>
gmm_move_smc (const alpha_setter_type alpha_setter) :
std::size_t operator() (std::size_t iter, Particle<gmm_state> particle)
alpha_setter_(iter, particle);
double alpha = particle.value().alpha();
alpha = alpha < 0.02 ? 0.02 : alpha;
particle.value().mu_sd(0.15 / alpha);
particle.value().lambda_sd((1 + std::sqrt(1 / alpha)) * 0.15);
particle.value().weight_sd((1 + std::sqrt(1 / alpha)) * 0.2);
double coeff = particle.value().alpha_inc();
for (vsmc::Particle<gmm_state>::size_type i = 0;
i != particle.size(); ++i)
incw_[i] =
coeff * particle.value().state(i, 0).log_likelihood();
particle.value().nc().add_log_weight( incw_[0],
particle.weight_set().add_log_weight( incw_[0]);
return 0;
private :
alpha_setter_type alpha_setter_;
std::vector<double> incw_;
std::vector<double> weight_;
Note that, cxx11::function is an alias to either std::function
or boost::function, depending on the value of the configuration macro
VSMC_HAS_CXX11LIB_FUNCTIONAL. Objects of this class type can be added
to a sampler as a move. The operator() satisfies the interface
requirement of the core module. First it uses alpha_setter_ to set the
distribution parameter $\alpha(t/T_r)$. Second, it sets the proposal scales
for the three Metropolis random walks according to the current value of
$\alpha$. Then it computes the unnormalized incremental weights. The
NormalizingConstnat class has member function add_log_weight,
which is not unlike the one with the same name in WeightSet. It accepts
the logarithm of the incremental weights and a WeightSet object. The
standard normalizing constant estimates will be computed using these values.
The last, we also modify the WeightSet type object itself by adding the
logarithm of the incremental weights.
At each iteration, ramdom walks are also perfoemd. The implementations of the
random walks are straightforward. Below is the implementation of the random
walk on the mean parameters. The random walks on the other parameters are
class gmm_move_mu : public BASE_MOVE<gmm_state, gmm_move_mu>
public :
std::size_t move_state (std::size_t iter, SingleParticle<gmm_state> sp)
const gmm_state state = sp.particle().value();
gmm_param param = sp.state(0);
cxx11::normal_distribution<> rmu(0, state.mu_sd());
cxx11::uniform_real_distribution<> runif(0, 1);
double p =
param.log_prior() + state.alpha() * param.log_likelihood();
for (std::size_t i = 0; i != param.comp_num(); ++i)
param.mu(i) += rmu(sp.rng());
p = state.update_log_prior(param) +
state.alpha() * state.update_log_likelihood(param) - p;
double u = std::log(runif(sp.rng()));
return param.mh_reject_mu(p, u);
First we save the logarithm of the value of target density computed using the
old values in p, the acceptance probability. And then we call
gmm_param::save_old to save the old values. Next we update each
parameter with a proposed Normal random variates and compute the new log-prior
and the log-likelihood as well as the new value of the target density. Then we
reject it according to the Metropolis algorithm, as implemented in
gmm_param, which manages both the current states as well as the backup.
Last we need to monitor certain quantities for interference purpose. Recall
that, in the main function we used
sampler.path_sampling(gmm_path()) to set the monitoring of path
sampling integrands. The path_sampling member function requires a
callable objects with the following signature,
double path_eval (std::size_t iter, const Particle<T> , double *res);
The input parameter iter is the iteration number, the value of $t$ in
Equation <ref>. The return value shall be the value of $\alpha_t$.
The output parameter res shall store the array of values of
\begin{equation*}
\rnd{\log q_{\alpha}(X_t^{(i)})}{\alpha}\Bigm|_{\alpha = \alpha_t}.
\end{equation*}
Our implementation of gmm_path is a sub-class of an module base
class, which provides an operator() that satisfies the above interface
requirement. Its usage is similar to the MoveSEQ template introduced in
Section <ref>.
The path sampling integrands under this geometry annealing scheme are simply
the log-likelihood. Therefore the implementation of gmm_path class is
rather simple,
class gmm_path : public BASE_PATH<gmm_state, gmm_path>
public :
double path_state (std::size_t, ConstSingleParticle<gmm_state> sp)
return sp.state(0).log_likelihood();
double path_grid (std::size_t, const Particle<gmm_state> particle)
return particle.value().alpha();
§.§.§ Results
After compiling and running the algorithm, the results were consistent with
those reported in Del Moral et al., 2006. For a more in depth analyze of the
methodologies, extensions and the results see Zhou et al., 2013.
§.§.§ Extending the
implementation using
's module assumes that identical samplers are constructed on each
node, with possible different number of particles to accommodate the
difference in capacities among nodes. To extend the above implementation
for use with , first at the beginning of the main function, we add
the following,
MPIEnvironment env(argc, argv);
to initialize the environment. When the object env is destroyed at
the exit of the main function, the environment is finalized.
Second, we need to replace base value collection class template with
StateMPI. So now gmm_state is declared as the following,
class gmm_state :
public StateMPI<BASE_STATE<StateMatrix<RowMajor, 1, gmm_param> > >;
The implementation is exactly the same as before. Third, the gmm_param
class now needs to be transferable using . Unlike the situations, a
simple copy constructor is not enough. uses library, and
thus one only needs to write a serialize member function for
gmm_param such that the data can be serialized into bytes. See
documents of and serialization libraries for details. In
summary, the following member function accept an Archive object as
input, and it can perform a store or a load operation based on the
Archive type. In a load operation, the Archive object is like an
input stream and in a store operation, it is like an output stream.
template <typename Archive>
void serialize (Archive ar, const unsigned)
int num = comp_num_;
ar num;
ar log_prior_;
ar log_likelihood_;
for (std::size_t i = 0; i != comp_num_; ++i)
ar mu_[i];
ar lambda_[i];
ar weight_[i];
ar log_lambda_[i];
Fourth, after user input of the sampler parameters, we need to sync them with
all nodes. For example, for the ParticleNum parameter,
boost::mpi::communicator World;
boost::mpi::broadcast(World, ParticleNum, 0);
Last, any importance sampling estimates that are computed on each node, need
to be combined into final results. For example, the path sampling results are
now obtained through adding the results from each node together,
double ps_sum = 0;
boost::mpi::reduce(World, ps, ps_sum, std::plus<double>(), 0);
ps = ps_sum;
For the standard normalizing constant ratio estimator, we will replace
NormalizingConstant with NormalizingConstantMPI, which will
perform such and other tasks.
After these few lines of change, the sampler is now parallelized using and can be deployed to clusters and other distributed memory architecture. On
each node, the selected parallelization is used to perform
multi-threading parallelization locally. 's module will take care of
normalizing weights and other tasks.
§.§.§ Parallelization performance
One of the main motivation behind the creation of is to ease the
parallelization with different programming models. The same implementation can
be used to built different samplers based on what kind of parallel programming
model is supported on the users' platforms. In this section we compare the
performance of various parallel programming models and parallelization.
We consider five different implementations supported by 2013:
sequential, , , and <thread>. The samplers
are compiled with
CXX=icpc -std=c++11 -gcc-name=gcc-4.7 -gxx-name=g++-4.7
CXXFLAGS=-O3 -xHost -fp-model precise -DVSMC_HAS_CXX11LIB_FUNCTIONAL=1 -DVSMC_HAS_CXX11LIB_RANDOM=1
on a Ubuntu 12.10 workstation with an Xeon W3550 (3.06GHz, 4 cores, 8 hardware
threads through hyper-threading) . A four components model and $100$
iterations with a prior annealing scheme is used for all implementations. A
range of numbers of particles are tested, from $2^3$ to $2^{17}$.
For different number of particles, the wall clock time and speedup are shown
in Figure <ref>. For $10^4$ or more particles, the
differences are minimal among all the programming models. They all have
roughly 550% speedup. With smaller number of particles, 's parallelization is less efficient than other industry strength programming
models. However, with $1000$ or more particles, which is less than typical
applications, the difference is not very significant.
Performance of implementations of Bayesian modeling for
Gaussian mixture model (Linux; Xeon W3550, 3.06GHz, 4 cores, 8 threads).
implementations are also compared on the same workstation, which also
has an NVIDIA Quadro 2000 graphic card. programs can be compiled to
run on both s and s. For implementation, there are and
platforms. We use the implementation as a baseline for comparison.
The same implementation are used for all the and runtimes.
Therefore they are not particularly optimized for any of them. For the implementation, in addition to double precision, we also tested a single
precision configuration. Unlike modern s, which have the same
performance for double and single precision floating point operations (unless
instructions are used, which can have at most a speedup by a factor of
2), s penalize double precision performance heavily.
For different number of particles, the wall clock time and speed up are
plotted in Figure <ref>. With smaller number of particles,
the implementations have a high overhead when compared to the implementation. With a large number of particles, has a similar
performance as the implementation. is about 40% faster than the
implementation. This is due to more efficient vectorization and compiler
optimizations. The double precision performance of the NVIDIA has a 220%
speedup and the single precision performance has near 1600% speedup. As a
rough reference for the expected performance gain, the has a theoretical
peak performance of 24.48 GFLOPS. The has a theoretical peak performance
of 60 GFLOPS in double precision and 480 GFLOPS in single precision. This
represents 245% and 1960% speedup compared to the , respectively.
Performance of implementations of Bayesian modeling for
Gaussian mixture model (Linux; Xeon W3550 , 3.06GHz, 4 cores, 8
threads; NVIDIA Quadro 2000).
It is widely believed that programming is tedious and hard. However,
provides facilities to manage platforms and devices as well as
common operations. Limited by the scope of this paper, the implementation (distributed with the source) is not documented in this
paper. Overall the implementation has about 800 lines including both
host and device code. It is not an enormous increase in effort when compared
to the 500 lines implementation. Less than doubling the code base but
gaining more than 15 times performance speedup, we consider the programming
effort is relatively small.
§ DISCUSSION
This paper introduced a template library intended for implementing
general algorithms and constructing parallel samplers with different
programming models. While it is possible to implement many realistic
application with the presented framework, some technical proficiency is still
required to implement some problem specific part of the algorithms. Some basic
knowledge of in general and how to use a template library are also
It is shown that with the presented framework it is possible to implement
parallelized, scalable samplers in an efficient and reusable way. The
performance of some common parallel programming models are compared using an
Some future work may worth the effort to ease the implementation of algorithms further. However, there is a balance between performance,
flexibility and the ease of use. aims to be developer-friendly and to
provide users as much control as possible for all performance related aspects.
For a -like interface, users may be interested in other software such as
. In addition provides a user friendly and high performance
alternative with a focus on state-space models. Compared with these recent
developments, is less accessible to those with little or no knowledge of
. However, for researchers with expertise in and template generic
programming in particular, provides a framework within which
potential superior performance can be obtained and greater flexibility and
extensibility are possible.
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|
arxiv-papers
| 2013-06-24T11:44:30 |
2024-09-04T02:49:46.853386
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "Yan Zhou",
"submitter": "Yan Zhou Yan Zhou",
"url": "https://arxiv.org/abs/1306.5583"
}
|
1306.5879
|
# Stable intersection of middle-$\alpha$ Cantor sets
M. Pourbarat111E-mail: [email protected]
Department of Mathematics, Faculty of Mathematical Sciences,
Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran
###### Abstract
In the present paper, We introduce a pair of middle Cantor sets namely
$(C_{\alpha},C_{\beta})$ having stable intersection, while the product of
their thickness is smaller than one. Furthermore, the arithmetic difference
$C_{\alpha}-\lambda C_{\beta}$ contains at least one interval for each nonzero
number $\lambda$.
Keywords: Middle-$\alpha$ Cantor sets, stable intersection, thickness, Palis
conjecture, arithmetic differences.
AMS Classification: 37C45, 28A80, 37G25.
## 1 Introduction
Regular Cantor sets appear in dynamical systems when hyperbolic sets intersect
stable and unstable manifolds of its points. The study of metrical and
topological properties of intersection of regular Cantor sets emerges
naturally in the theory of homoclinic bifurcations. Meanwhile, stable
intersections between Cantor sets which come from stable and unstable
foliations of a horseshoe, provide examples of open sets of nonhyperbolic
diffeomorphisms after unfolding a homoclinic tangency. In these cases, the
open set of diffeomorphisms presenting persistent tangencies between the
stable and unstable foliations of the horseshoe stably has positive lower
density at the initial parameter of the bifurcation, in parametrized families
(see [M1] and [PT]).
Regular Cantor sets appear in number theory too, related to diophantine
approximations. Many Cantor sets given by combinatorial conditions on the
continued fraction of real numbers emerge in this situation. In the study of
the classical Markov and Lagrange spectra related to them, we usually deal
with the arithmetic difference of two regular Cantor sets (see [M2]). On the
other hand, the intersection of regular Cantor sets can be interpreted by
their arithmetic difference as
$K-K^{\prime}:=\big{\\{}x-y\mid~{}~{}x\in K~{},~{}y\in
K^{\prime}~{}\big{\\}}=\big{\\{}t\in\mathbb{R}\mid~{}~{}K\cap(K^{\prime}+t)\neq\emptyset~{}\big{\\}}.$
The arithmetic difference of regular Cantor sets can be so complicate even for
the simplest possible family of pairs of Cantor sets (see [MO] and [P3]). For
instance, conjecture of Palis remains still open in the affine case (see [P1],
[P2] and [PT]). Also, there exist regular Cantor sets $K$ and $K^{\prime}$,
such that $K-K^{\prime}$ has positive Lebesgue measure, but does not contain
any interval (see [S]). Having stable intersection of regular Cantor sets,
clearly implies the existence of an interval contained in their arithmetic
difference.
Before stating our main result, we pose some notations. A Cantor set $K$ is
regular or dynamically defined if:
i) there are disjoint compact intervals $K_{1},K_{2},\ldots,K_{r}$ such that
$K\subset K_{1}\cup\cdots\cup K_{r}$ and the boundary of each $K_{i}$ is
contained in $K$,
ii) there is a $C^{1+\epsilon}$ expanding map $\psi$ defined in a neighborhood
of set $K_{1}\cup K_{2}\cup\cdots\cup K_{r}$ such that $\psi(K_{i})$ is the
convex hull of a finite union of some intervals $K_{j}$ satisfying:
ii.1 For each $i$, $1\leq i\leq r$ and $n$ sufficiently big, $\psi^{n}(K\cap
K_{i})=K,$
ii.2 $K=\bigcap_{n=0}^{\infty}\psi^{-n}(K_{1}\cup K_{2}\cup\cdots\cup K_{r})$.
The set $\\{K_{1},K_{2},\cdots,K_{r}\\}$ is, by definition, a Markov partition
for $K$, and the set $D:=\bigcup_{i=1}^{r}K_{i}$ is the Markov domain of $K$.
A regular Cantor set is affine if $D\psi$ be constant on every interval
$K_{i}$. The simplest kind of affine Cantor sets are middle-$\alpha$ Cantor
sets that generalizes in the most natural way, the usual ternary Cantor set
which corresponds to p=3 in below definition. Definition. Let $p>2$ and
$\alpha:=1-\frac{2}{p}$. Then _middle_ -$\alpha$ Cantor set can be written as
$C_{\alpha}:=~{}\Big{\\{}~{}x\in\mathbb{R}~{}\mid~{}~{}x=(1-\frac{1}{p})\sum_{i=0}^{\infty}~{}\frac{a_{i}}{p^{i}}~{}~{}~{},~{}~{}~{}a_{i}\in\\{0,1\\}~{}\Big{\\}},$
that is a regular Cantor set with the Markov partition $\\{K_{1},K_{2}\\}$ and
expanding map
$\phi(x):=~{}\left\\{\begin{array}[c]{lcr}~{}~{}px~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x\in
K_{1}:=~{}[0,~{}\frac{1}{p}]\\\ px-p+1~{}~{}~{}~{}~{}~{}x\in
K_{2}:=~{}[1-\frac{1}{p},~{}1]\end{array}.\right.$
We say that the Cantor set $K$ is close on the topology $C^{1+\epsilon}$ to a
Cantor set $\widetilde{K}$ with the Markov partition
$\\{\widetilde{K_{1}},\widetilde{K_{2}},\cdots,\widetilde{K_{s}}\\}$ defined
by expanding map $\widetilde{\psi}$ if and only if $r=s$, the extremes of
$K_{i}$ are near the corresponding extremes of
$\widetilde{K_{i}},~{}i=1,2,...,r$ and supposing $\psi\in C^{1+\epsilon}$ with
Holder constant $C$, we must have $\widetilde{\psi}\in
C^{1+\widetilde{\epsilon}}$ with Holder constant $\widetilde{C}$ such that
$(\widetilde{C},\widetilde{\epsilon})$ is near $(C,\epsilon)$ and
$\widetilde{\psi}$ is close to $\psi$ in the $C^{1}$ topology.
Definition. Regular Cantor sets $K$ and $K^{\prime}$ have _stable
intersection_ if for any pair of regular Cantor sets
$(\widetilde{K},\widetilde{K^{\prime}})$ near $(K,K^{\prime})$, we have
$\widetilde{K}\cap\widetilde{K^{\prime}}\neq\emptyset$.
Besides the Hausdorff dimension, there is another fractal invariant namely
thickness introduced by Newhouse, that plays a relevant role in determining
stable intersection of regular Cantor sets (see [N]). Such thickness condition
was generalized by Moreira in [M1] as follows:
Definition. Take $U$ be a bounded gap of Cantor set $K$ and $L_{U}$, $R_{U}$
be the intervals at its left and its right, respectively, that separate it
from the closest larger gaps.
$K:$
$|$$($$)$$($$)$$($$)$$|$
$L_{U}$$U$$R_{U}$ Let $\tau_{R}(U):=~{}\frac{|R_{U}|}{|U|}$ and
$~{}\tau_{L}(U):=~{}\frac{|L_{U}|}{|U|}$. The _right thickness_ $\tau_{R}$ and
_left thickness_ $\tau_{L}$ are
$\tau_{R}(K):=~{}\inf_{U}\tau_{R}(U)~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}~{}~{}\tau_{L}(K):=~{}\inf_{U}\tau_{L}(U)$
and the Newhouse thickness $\tau(K)$ is the minimum of lateral thicknesses
$\tau_{R}(K)$ and $\tau_{L}(K)$.
One of the special characteristics of above definition is the existence of an
open and dense subset in $C^{1+\epsilon}$ topology of regular Cantor sets
whose elements have these lateral thicknesses varying continuously. But,
Newhouse thickness is continuous in $C^{1+\epsilon}$ topology of all regular
Cantor sets.
Now we state some fundamental results on stable intersection of regular Cantor
sets:
I) If $HD(K)+HD(K^{\prime})<1$, therefore no translations of $K$ and
$K^{\prime}$ have stable intersection (see [PT]),
II) If $\tau(K)\cdot\tau(K^{\prime})>1$ and $K$ is linked to $K^{\prime}$,
then $(K,K^{\prime})$ have stable intersection (see [N] and [PT]),
III) If $HD(K)+HD(K^{\prime})>1$ and $K$ is linked to $K^{\prime}$, then
$(K,K^{\prime})$ have generically stable intersection (see [MY]).
Note that, $K$ is linked to $K^{\prime}$ means closure of each gap of $K$ does
not contain $K^{\prime}$.
Moreira introduced affine Cantor sets with more than 2 expanding maps and
small lateral thicknesses that having stable intersection. He showed that
lateral thicknesses change continuously at affine Cantor sets defined by two
expanding maps. Moreover, for affine Cantor sets defined by two affine
expanding maps;
$\bullet~{}$ If $\tau_{R}(K)\cdot\tau_{L}(K^{\prime})>1$,
$\tau_{L}(K)\cdot\tau_{R}(K^{\prime})>1$ and $K$ is linked to $K^{\prime}$,
then $(K,K^{\prime})$ have stable intersection (see [M1]).
In [HMP], we constructed a pair of affine Cantor sets with the simplest
possible combinatorics which have stable intersection, while
$\tau_{R}(K)\cdot\tau_{L}(K^{\prime})<1$. These considerations motivate the
following problem, that will be discussed in this work:
###### Problem 1.
Does there exist a non empty open set in the space of affine Cantor sets
defined by two expanding maps contained in the region
$\big{\\{}(K,K^{\prime})~{}|~{}~{}~{}\tau_{R}(K)\cdot\tau_{L}(K^{\prime})<1~{}~{}and~{}~{}\tau_{L}(K)\cdot\tau_{R}(K^{\prime})<1~{}\big{\\}}$
such that their elements have stable intersection?
Indeed, we consider a pair of special middle-$\alpha$ Cantor sets that gives
an affirmative solution to this problem. The main challenge is to construct a
recurrent compact set of relative configurations, since, by the proposition in
Subsection 2.3 of [MY], any relative configuration contained in a recurrent
compact set is a configuration of stable intersection. The rest of this paper
is outlined as follows:
In Section 2, we present the necessary definitions and state the recurrence
condition on relative configurations of [MY] which implies stable intersection
of pairs of regular Cantor sets.
In Section 3, we translate this condition to the setting of affine Cantor
sets, which gives a recurrent condition on a simpler space of relative
configurations.
Constructing a recurrent compact set of the relative configurations of
middle-$\alpha$ Cantor sets, under special conditions, is the main subject of
Section 4.
## 2 Basic definitions
We will use notations similar to those of [MY] which are restated here.
Regular Cantor sets can also be defined as follows:
Let $A$ be a finite alphabet, $\mathcal{B}$ a subset of $A^{2}$, and $\Sigma$
the subshift of finite type $A^{\mathbb{Z}}$ with allowed transitions
$\mathcal{B}$. We will always assume that $\Sigma$ is topologically mixing and
every letter in $A$ occurs in $\Sigma$.
An expanding map of type $\Sigma$ is a map $g$ with the following properties:
i) the domain of $g$ is a disjoint union $\bigcup_{\mathcal{B}}I(a,b)$, where
for each $(a,b)$, $I(a,b)$ is a compact subinterval of
$I(a):=[0,~{}1]\times\\{a\\}$,
ii) for each $(a,b)\in\mathcal{B}$, the restriction of $g$ to $I(a,b)$ is a
smooth diffeomorphism onto $I(b)$ satisfying $|Dg(t)|>1$ for all $t$.
The regular Cantor set associated to $g$ is the maximal invariant set
$K:=\bigcap_{n\geq 0}g^{-n}\big{(}\bigcup_{\mathcal{B}}I(a,b)\big{)}.$
These two definitions are equivalent. On one hand, we may, in the first
definition, take $I(i):=I_{i}$ for each $i\leq r$, and, for each pair $i,j$
such that $\psi(I_{i})\supset I_{j}$, take $I(i,j)=I_{i}\cap\psi^{-1}(I_{j})$.
Conversely, in the second definition, we can consider an abstract line
containing all intervals $I(a)$ as subintervals, and
$\\{I(a,b)~{}|~{}(a,b)\in\mathcal{B}\\}$ as the Markov partition.
Also, a regular Cantor set $K$ is affine if $Dg$ be constant on every
$I(a,b)$.
Let $\Sigma^{-}:=\big{\\{}(\theta_{n})_{n\leq
0}\mid~{}(\theta_{i},\theta_{i+1})\in\mathcal{B}\ \emph{for}\
i<0~{}\big{\\}}$. We equip $\Sigma^{-}$ with the following ultrametric
distance: for
$\underline{\theta}\not=\underline{\tilde{\theta}}\in\Sigma^{-}$, set
$d(\underline{\theta},\underline{\tilde{\theta}}):=\left\\{\begin{array}[c]{lcr}1&&\theta_{0}\neq\tilde{\theta}_{0}\\\
|I(\underline{\theta}\wedge\underline{\tilde{\theta}})|&&\emph{otherwise}\end{array},\right.$
where
$\underline{\theta}\wedge\underline{\tilde{\theta}}:=(\theta_{-n},\cdots,\theta_{0})$
if $\tilde{\theta}_{-j}=\theta_{-j}$ for $0\leq j\leq n$ and
$\tilde{\theta}_{-n-1}\not=\theta_{-n-1}$.
Suppose that $\underline{\theta}\in\Sigma^{-}$ and $n\in\mathbb{N}$, let
$\underline{\theta}^{n}:=(\theta_{-n},\cdots,\theta_{0})$ and
$B(\underline{\theta}^{n})$ be the affine map from $I(\underline{\theta}^{n})$
onto $I({\theta}_{0})$ which the diffeomorphism
$k\frac{\theta}{n}:=B(\underline{\theta}^{n})\circ f_{\underline{\theta}^{n}}$
is orientation preserving. Then, for each $\underline{\theta}\in\Sigma^{-}$,
there is a smooth diffeomorphism $k^{\underline{\theta}}$ such that
$k\frac{\theta}{n}$ converges to $k^{\underline{\theta}}$ in
Diff${}^{r}_{+}\big{(}I(\theta_{0}\big{)}$, for any $r\in(1,+\infty)$,
uniformly in $\underline{\theta}$.
Next, we define renormalization operators. For $(a,b)\in\mathcal{B}$, let
$f_{a,b}:=~{}[g|_{I(a,b)}]^{-1},$
this is a contracting diffeomorphism from $I(b)$ onto $I(a,b)$. If
$\underline{a}:=~{}(a_{0},a_{1},\ldots,a_{n})$ is a word of $\Sigma$, then we
put
$f_{\underline{a}}:=~{}f_{a_{0},a_{1}}\circ\cdots f_{a_{n-1},a_{n}},$
this is a contracting diffeomorphism from $I(a_{n})$ onto a subinterval of
$I(a_{0})$, that we denote by $I(\underline{a})$. Also let
$F^{\underline{\theta}}$ be the affine map from $I(\theta_{0})$ onto
$I(\theta_{-1},\theta_{0})$ with the same orientation of
$f_{\theta_{-1},\theta_{0}}$.
Consider
$\mathcal{A}:=\big{\\{}(\underline{\theta},A)\mid~{}\underline{\theta}\in\Sigma^{-}$
and $A:~{}I(\theta_{0})~{}\longrightarrow~{}\mathbb{R}$ is an affine embedding
map}. The renormalization operators
$T_{\theta_{1},\theta_{0}}:~{}\mathcal{A}~{}\longrightarrow~{}\mathcal{A}$ are
defined by
$T_{\theta_{1},\theta_{0}}(\underline{\theta},A):=~{}(\underline{\theta}\theta_{1},A\circ
F^{\underline{\theta}\theta_{1}})~{}~{}~{}~{},~{}~{}~{}~{}(\theta_{0},\theta_{1})\in\mathcal{B}.$
For two sets of data $(A,\mathcal{B},\Sigma,g)$ and
$(A^{\prime},\mathcal{B}^{\prime},\Sigma^{\prime},g^{\prime})$ defining
regular Cantor sets $K$ and $K^{\prime}$, denote $\mathcal{C}$ be the quotient
of $\mathcal{A}\times\mathcal{A}$ by the diagonal action on the left of affine
group.
A non empty compact set $\mathcal{L}$ in $\mathcal{C}$ is recurrent if for
every $u\in\mathcal{L}$ and $\ell,\ell^{\prime}\geq 0$ with
$\ell+\ell^{\prime}>0$, when $(\underline{\theta},A)$ and
$(\underline{\theta^{\prime}},A^{\prime})$ represents $u$, then there exist
words $\underline{a}=(a_{0},\cdots,a_{\ell})$ and
$\underline{a^{\prime}}=(a_{0}^{\prime},\cdots,a_{\ell^{\prime}}^{\prime})$ in
$\Sigma$ and $\Sigma^{\prime}$, respectively, with $a_{0}=\theta_{0}$ and
$a_{0}^{\prime}=\theta_{0}^{\prime}$, such that
$\big{(}T_{\underline{a}}(\underline{\theta},A),T_{\underline{a^{\prime}}}^{\prime}(\underline{\theta^{\prime}},A^{\prime})\big{)}=v$
belongs to $\mathcal{L}^{\circ}:=\emph{int}\mathcal{L}$. The following
proposition has been proved in [MY].
###### Proposition 1.
Any relative configuration (of limit geometries) contained in a recurrent
compact set is stably intersecting.
In the end of this section, we suppose that
$\mathcal{S}:=\Sigma^{-}\times\Sigma^{{}^{\prime}-}\times\mathbb{R^{\ast}}$,
where $\mathbb{R^{\ast}}=\mathbb{R}\backslash\\{0\\}$. One can see that the
fibers of the quotient map $\mathcal{C}\longrightarrow\mathcal{S}$ are one-
dimensional and have a canonical affine structure. Moreover, this bundle map
is trivializable. We choose an explicit trivialization
$\mathcal{C}\cong\mathcal{S}\times\mathbb{R}$ in order to have a coordinate in
each fiber.
## 3 Transfer of renormalization operators on the space
$\mathcal{S}\times\mathbb{R}$
Assume that $K$ is an affine Cantor set together with the Markov partition
$\\{I(n,m)\\}_{{}_{m,n\in A}}$ and an expanding map
$\Phi\mid_{{}_{I(n,m)}}(x):=~{}p_{{}_{(n,m)}}\cdot x+q_{{}_{(n,m)}},$
where $A:=\\{1,2,3,\cdots,N\\}$. Let $I(n):=[a_{n}^{1},~{}a_{n}^{2}]$ and
$I(n,m):=[a_{n,m}^{1},~{}a_{n,m}^{2}]$, then for each $(n,m)\in\mathcal{B}$,
we have:
$\Phi\big{(}I(n,m)\big{)}=I(m).$
In the case $P_{(\theta_{0},\theta_{1})}>0$, map
$F^{\underline{\theta}\theta_{1}}$ is
$F^{\underline{\theta}\theta_{1}}:I(\theta_{1})\longrightarrow
I(\theta_{0},\theta_{1}),$
$F^{\underline{\theta}\theta_{1}}(x)=\frac{a_{\theta_{0},\theta_{1}}^{2}-a_{\theta_{0},\theta_{1}}^{1}}{a_{\theta_{1}}^{2}-a_{\theta_{1}}^{1}}(x-a_{\theta_{1}}^{1})+a_{\theta_{0},\theta_{1}}^{1}.$
Also in the opposite orientation, map $F^{\underline{\theta}\theta_{1}}$ is
$F^{\underline{\theta}\theta_{1}}(x)=\frac{a_{\theta_{0},\theta_{1}}^{2}-a_{\theta_{0},\theta_{1}}^{1}}{a_{\theta_{1}}^{2}-a_{\theta_{1}}^{1}}(x-a_{\theta_{1}}^{1})+a_{\theta_{0},\theta_{1}}^{2}.$
Therefore, in both cases, we obtain
$F^{\underline{\theta}\theta_{1}}(x)=\frac{1}{p_{(\theta_{0},\theta_{1})}}x-\frac{q_{(\theta_{0},\theta_{1})}}{p_{(\theta_{0},\theta_{1})}}.$
To continue, we construct the homeomorphism between
$\mathcal{S}\times\mathbb{R}$ and $\mathcal{C}$, then we transfer all
renormalization operators to $\mathcal{S}\times\mathbb{R}$.
###### Theorem 1.
The map
$\displaystyle
L:~{}\mathcal{C}~{}\longrightarrow~{}\mathcal{S}\times\mathbb{R}$
$\displaystyle\left[(\underline{\theta},ax+b),(\underline{\theta}^{\prime},a^{\prime}x+b^{\prime})\right]$
$\displaystyle\mapsto(\underline{\theta},\underline{\theta}^{\prime},\frac{a^{\prime}}{a},\frac{b^{\prime}-b}{a})$
is a homeomorphism between the space of relative configurations $\mathcal{C}$
and $\mathcal{S}\times\mathbb{R}$.
Proof. $L$ is well defined:
Let
$[(\underline{\theta},ax+b),(\underline{\theta}^{\prime},a^{\prime}x+b^{\prime})]=[(\underline{\theta},a_{1}x+b_{1}),(\underline{\theta}^{\prime},a_{1}^{\prime}x+b_{1}^{\prime})].$
Then there exist $c,d\in\mathbb{R}$ such that
$a_{1}x+b_{1}=c(ax+b)+d\ \ ,\ \
a_{1}^{\prime}x+b_{1}^{\prime}=c(a^{\prime}x+b^{\prime})+d,$
therefore,
$\frac{a^{\prime}}{a}=\frac{a_{1}^{\prime}}{a_{1}}\ \ \ ,\ \
\frac{b^{\prime}-b}{a}=\frac{b_{1}^{\prime}-b_{1}}{a_{1}}.$
$L$ is onto:
$\forall~{}(\underline{\theta},\underline{\theta}^{\prime},s,t)\in\mathcal{S}\times\mathbb{R},\hskip
28.45274ptL\Big{(}[(\underline{\theta},x),(\underline{\theta}^{\prime},sx+t)]\Big{)}=(\underline{\theta},\underline{\theta}^{\prime},s,t).$
Also, $L$ is one to one:
Suppose that
$L\Big{(}[(\underline{\theta},x),(\underline{\theta}^{\prime},sx+t)]\Big{)}=L\Big{(}[(\underline{\theta},x),(\underline{\theta}^{\prime},s^{\prime}x+t^{\prime})]\Big{)},$
then $s=s^{\prime}$ and $t=t^{\prime}$.
From the structure of $L$, we see that $L$ and $L^{-1}$ are continuous. $\Box$
Now we transfer the renormalization operators of relative configurations
$\mathcal{C}$ to the space $\mathcal{S}\times\mathbb{R}$. To do this, let
$\big{(}(\theta_{1},\theta_{0}),(\theta_{1}^{\prime},\theta_{0}^{\prime})\big{)}\in\mathcal{B}\times\mathcal{B}^{\prime}$,
then we obtain
$\displaystyle(\underline{\theta},\underline{\theta}^{\prime},s,t)~{}\xrightarrow{{L^{-1}}}\
\left[(\underline{\theta},x)~{},~{}(\underline{\theta}^{\prime},sx+t)\right]~{}\xrightarrow{{\big{(}T_{{}_{\theta_{1},\theta_{0}}},T_{{}_{\theta_{1}^{\prime},\theta_{0}^{\prime}}}^{\prime}\big{)}}}$
$\displaystyle\left[(\underline{\theta}~{}\theta_{1},\frac{1}{p_{(\theta_{0},\theta_{1})}}x-\frac{q_{(\theta_{0},\theta_{1})}}{p_{(\theta_{0},\theta_{1})}})\text{
},\text{
}(\underline{\theta}^{\prime}~{}\theta_{1}^{\prime},\frac{s}{p_{(\theta_{0}^{\prime},\theta_{1}^{\prime})}}x-\frac{q_{(\theta_{0}^{\prime},\theta_{1}^{\prime})}}{p_{(\theta_{0}^{\prime},\theta_{1}^{\prime})}}s+t)\newline
\right]\overset{L}{\longrightarrow}$
$\displaystyle\left(\underline{\theta}~{}\theta_{1},\underline{\theta}^{\prime}~{}\theta_{1}^{\prime},\frac{p_{(\theta_{0},\theta_{1})}}{p_{(\theta_{0}^{\prime},\theta_{1}^{\prime})}^{\prime}}~{}s~{},~{}p_{(\theta_{0},\theta_{1})}t-\frac{q_{(\theta_{0}^{\prime},\theta_{1}^{\prime})}^{\prime}}{p_{(\theta_{0}^{\prime},\theta_{1}^{\prime})}^{\prime}}p_{(\theta_{0},\theta_{1})}~{}s+q_{(\theta_{0},\theta_{1})}\right).$
We denote
$L\circ\big{(}T_{{}_{\theta_{1},\theta_{0}}},T_{{}_{\theta_{1}^{\prime},\theta_{0}^{\prime}}}^{\prime}\big{)}\circ
L^{-1}$ by
$T_{\big{(}(\theta_{1},\theta_{0}),(\theta_{1}^{\prime},\theta_{0}^{\prime})\big{)}}$
and for the sake of comfort we select those which are in these kinds:
$T_{\big{(}(\theta_{1},\theta_{0}),id\big{)}}:~{}(\underline{\theta},\underline{\theta}^{\prime},s,t)~{}\longrightarrow~{}(\underline{\theta}\theta_{1},~{}\underline{\theta}^{\prime},~{}p_{{}_{\theta_{1},\theta_{0}}}s~{},~{}p_{{}_{\theta_{1},\theta_{0}}}~{}t+q_{{}_{\theta_{1},\theta_{0}}}),$
$T_{\big{(}id,(\theta_{1}^{\prime},\theta_{0}^{\prime})\big{)}}:~{}(\underline{\theta},\underline{\theta}^{\prime},s,t)~{}\longrightarrow~{}(\underline{\theta},~{}\underline{\theta}^{\prime}\theta_{1}^{\prime},~{}\frac{s}{p_{{}_{\theta_{1}^{\prime},\theta_{0}^{\prime}}}^{\prime}}~{},~{}t-\frac{q_{{}_{\theta_{1},\theta_{0}}}^{\prime}}{p_{{}_{\theta_{1},\theta_{0}}}^{\prime}}s).~{}~{}~{}~{}~{}~{}$
## 4 Construction of recurrent sets
In this section, we introduce a pair of middle-$\alpha$ Cantor sets that have
a recurrent compact set in the relative configurations, while
$\tau(K)\cdot\tau(K^{\prime})<1$.
###### Theorem 2.
Suppose that $K$ and $K^{\prime}$ are two homogenous Cantor sets with the
convex hull $[0,~{}1]$ and expanding maps $\phi$ and $\phi^{\prime}$ as
$K:$$\frac{1}{p}$$K^{\prime}:$$\frac{1}{p}$$\frac{1}{q}$$\frac{1}{q}$
$\phi(x):=\left\\{\begin{array}[c]{lcr}~{}~{}~{}px~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x\in[0,~{}\frac{1}{p}]\\\
px-p+1~{}~{}~{}~{}~{}x\in[1-\frac{1}{p},~{}1]\end{array}\right.~{}~{}\phi^{\prime}(x):=\left\\{\begin{array}[c]{lcr}~{}~{}~{}qx~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}x\in[0,~{}\frac{1}{q}]\\\
qx-q+1~{}~{}~{}~{}~{}x\in[1-\frac{1}{q},~{}1]\end{array},\right.$
where
$p:=\gamma^{40}:=(1.0321)^{40}=3.538923071...,$
$q:=\gamma^{31}:=(1.0321)^{31}=2.663024240....$
Then pair $(K,K^{\prime})$ have stable intersection, while their thickness
product is smaller than one.
Proof. At first, we see that
$\tau(K)\cdot\tau(K^{\prime})=\frac{\frac{1}{p}}{1-\frac{2}{p}}\cdot\frac{\frac{1}{q}}{1-\frac{2}{q}}=\frac{1}{(p-2)(q-2)}=0.980062299...$
As explained at the end of Section 3, transferred renormalization operators
can be considered
$\mathbb{R^{\ast}\times\mathbb{R}~{}\longrightarrow~{}\mathbb{R}^{\ast}\times\mathbb{R}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$(s,t)\overset{T_{0}}{\longmapsto}(ps~{},~{}pt)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}*~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(s,t)\overset{T_{1}}{\longmapsto}(ps~{},~{}pt-p+1)$
$(s,t)\overset{T^{\prime}_{0}}{\longmapsto}(\frac{s}{q}~{},~{}t)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(s,t)\overset{T^{\prime}_{1}}{\longmapsto}(\frac{s}{q}~{},~{}t+\frac{q-1}{q}s)$
Let
$~{}s_{1}:=~{}\frac{\frac{1}{p}}{1-\frac{2}{q}+2(1-\frac{1}{q})\frac{1}{q^{39}}}$
,
$~{}s_{2}:=~{}\frac{1-\frac{2}{p}}{\frac{1}{q}-2(1-\frac{1}{q})\frac{1}{q^{39}}}~{}$
and
$\Delta:=~{}\big{\\{}(s,t)~{}|~{}~{}\gamma^{-1}s_{2}\leq s\leq\gamma
s_{1}~{}~{},~{}~{}-s+(1-\frac{1}{q})\frac{s}{q^{39}}\leq t\leq
1-(1-\frac{1}{q})\frac{s}{q^{39}}~{}\big{\\}}.$
Note that, since $~{}\tau_{1}\cdot\tau_{2}<1~{}$and$~{}(p-2)(q-2)<\gamma$, we
have $~{}\gamma^{-1}s_{2}<s_{1}<s_{2}<\gamma s_{1}$.
Observe that the numerical approximations are
$~{}~{}~{}~{}~{}\gamma^{-1}s_{2}\cong 1.1220161,~{}~{}s_{1}\cong
1.1349444,~{}~{}s_{2}\cong 1.1580329,~{}~{}\gamma s_{1}\cong 1.1713761.~{}$
Take
$\mathcal{L}:=~{}\Delta\setminus\Delta_{1}\cup\Delta_{2},$
with
$~{}~{}\Delta_{1}:=~{}\big{\\{}(s,t)~{}|~{}~{}L^{1}:~{}t+(1-\frac{1}{q})(1+\frac{1}{q^{39}})s>1~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}L^{2}:~{}t+(1-\frac{1}{q})\frac{s}{q^{39}}>\frac{1}{p}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}L^{3}:~{}t+(\frac{1}{q}-(1-\frac{1}{q})\frac{1}{q^{39}})s<1-\frac{1}{p}~{}\big{\\}},$
$~{}~{}\Delta_{2}:=~{}\big{\\{}(s,t)~{}|~{}~{}L^{4}:~{}t+(1-(1-\frac{1}{q})\frac{1}{q^{39}})s<1-\frac{1}{p}~{}~{}~{},~{}~{}~{}L^{5}:~{}t+(\frac{1}{q}-(1-\frac{1}{q})\frac{1}{q^{39}})s<0\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{},~{}~{}~{}L^{6}:~{}t+(1-\frac{1}{q})(1+\frac{1}{q^{39}})s>\frac{1}{p}~{}\big{\\}}.$
Figure 1:
We are going to show that compact set $\mathcal{L}$, as displayed in Figure 1,
is a recurrent set for the operators $(*)$.
Every vertical lines $s=s_{0}$ pass over itself with suitable compositions of
the operators $(*)$ since $p^{31}=q^{40}$. Therefore we can transfer the
operators $(*)$ on these lines by
###### Proposition 2.
Let $s\in\mathbb{R}^{\ast}$ and $\\{a_{k}\\}^{30}_{k=0}$ ,
$\\{b_{k}\\}^{39}_{k=0}$ be two finite sequences of numbers 0 and 1, then the
maps
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T_{s}(t):=~{}p^{31}t+a_{s},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}**~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{s}:=-(p-1)p^{30}\big{(}\sum^{30}_{k=0}\frac{a_{k}}{p^{k}}-\frac{p(q-1)}{q(p-1)}s\sum^{39}_{k=0}\frac{b_{k}}{q^{k}}\big{)}~{}~{}~{}~{}$
are return maps to the vertical line $s$.
Proof. Suppose that $\\{b_{k}\\}^{\infty}_{k=0}$ and
$\\{a_{k}\\}^{\infty}_{k=0}$ are two arbitrary sequences of numbers 0 and 1.
For every $a_{k}$ and $b_{k}$, we can consider the operators of $(*)$ in below
form
$T_{a_{k}}(s,t):=~{}(ps~{},~{}pt-(p-1)a_{k})~{}~{}~{}~{},~{}~{}~{}~{}T_{b_{k}}(s,t):=~{}(\frac{s}{q}~{},~{}t+(\frac{q-1}{q})b_{k}s)$
Let $m,n\in\mathbb{N}$, then we obtain
$~{}~{}~{}i)~{}T_{a_{m-1}}\circ....\circ
T_{a_{0}}(s,t)=\big{(}p^{m}s~{},~{}p^{m}t-(p-1)\sum^{m-1}_{k=0}a_{k}p^{m-1-k}\big{)},$
$~{}~{}~{}ii)~{}T_{b_{n-1}}\circ....\circ
T_{b_{0}}(s,t)=\big{(}\frac{s}{q^{n}}~{},~{}t+\frac{s}{q^{n}}(q-1)\sum^{n-1}_{k=0}b_{k}q^{n-1-k}\big{)}.$
To prove the relations (i) and (ii), we use induction. Case $m=n=1$ is valid.
Suppose that assertion satisfies for cases $i$ and $j$, then we have:
$~{}~{}~{}~{}T_{a_{i}}\circ T_{a_{i-1}}\circ....\circ
T_{a_{0}}(s,t)=\big{(}p^{i+1}s~{},~{}p^{i+1}t-(p-1)\sum^{i-1}_{k=0}a_{k}p^{i-k}-(p-1)a_{i}\big{)}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\big{(}p^{i+1}s~{},~{}p^{i+1}t-(p-1)\sum^{i}_{k=0}a_{k}p^{i-k}\big{)},$
$~{}~{}~{}~{}T_{b_{j}}\circ T_{b_{j-1}}\circ....\circ
T_{b_{0}}(s,t)=\big{(}\frac{s}{q^{j+1}}~{},~{}t+\frac{s}{q^{j}}(q-1)\sum^{j-1}_{k=0}b_{k}q^{j-1-k}+b_{j}\frac{(q-1)}{q}\cdot\frac{s}{q^{j}}\big{)}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\big{(}\frac{s}{q^{j+1}}~{},~{}t+\frac{s}{q^{j+1}}(q-1)\sum^{j}_{k=0}b_{k}q^{j-k}\big{)}$
and we see that the relations (i) and (ii) hold for cases $i+1$ and $j+1$.
Replace $m=31$ and $n=40$ in the relations (i) and (ii), then we obtain
$~{}~{}~{}~{}~{}~{}~{}~{}~{}T_{b_{39}}\circ....\circ T_{b_{0}}\circ
T_{a_{30}}\circ....\circ
T_{a_{0}}(s,t)=\big{(}s~{},~{}p^{31}t-(p-1)\sum^{30}_{k=0}a_{k}p^{30-k}+\\\
\frac{p^{31}s}{q^{40}}(q-1)\sum^{39}_{k=0}b_{k}q^{39-k}\big{)}=\big{(}s~{},~{}p^{31}t-(p-1)p^{30}(\sum^{30}_{k=0}\frac{a_{k}}{p^{k}}-\frac{p(q-1)}{q(p-1)}s\sum^{39}_{k=0}\frac{b_{k}}{q^{k}})\big{)}$.
This completes the proof of proposition. $\Box$
If $~{}\\{a_{ik}\\}^{30}_{k=0}~{}$ and $~{}\\{b_{ik}\\}^{39}_{k=0}~{}$ be two
arbitrary finite sequences of numbers $0,1$ and
$s^{*}:=~{}\frac{p(q-1)}{q(p-1)}s$, then all of the return maps (or operators)
are
$~{}~{}~{}~{}~{}~{}~{}~{}T_{s}(t)=p^{31}t+a_{s},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{s}:=-(p-1)p^{30}\big{(}\sum^{30}_{k=0}\frac{a_{ik}}{p^{k}}-s^{*}\sum^{39}_{k=0}\frac{b_{ik}}{q^{k}}\big{)}~{}~{}~{}~{}~{}~{}~{}a_{ik},b_{ik}=0~{}or~{}1.$
Here we deal with $2^{31+40}$ squares of length
$p^{-31}=\gamma^{-1240}=0.0{\underbrace{...}_{17}}0966...$, which project
under angle $\theta:=\cot^{-1}s$ and determine this position, see Figure 2.
Figure 2:
Definition. The set $R\subset\mathbb{R}$ is a _recurrent_ if for every element
of $R$ there exist suitable composites of maps $(**)$, so transfers that
element to $R^{\circ}$.
Let
$I_{s}^{\pm}:=~{}[a_{1},~{}b_{1}]\bigcup~{}[a^{\pm}_{2},~{}b^{\pm}_{2}]\bigcup~{}[a_{3},~{}b_{3}]$
with
$~{}~{}~{}~{}~{}~{}a_{1}:=-\frac{q(p-1)}{p(q-1)}s^{*}+(1-\frac{1}{p})\frac{s^{*}}{q^{39}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}b_{1}:=\frac{1}{p}-(1-\frac{1}{p})(1+\frac{1}{q^{39}})s^{*},$
$~{}~{}~{}~{}~{}~{}a_{2}^{-}:=-\frac{p-1}{p(q-1)}s^{*}+(1-\frac{1}{p})\frac{s^{*}}{q^{39}},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}b_{2}^{-}:=1-(1-\frac{1}{p})(1+\frac{1}{q^{39}})s^{*},$
$~{}~{}~{}~{}~{}~{}a_{2}^{+}:=1-\frac{1}{p}-\frac{q(p-1)}{p(q-1)}s^{*}+(1-\frac{1}{p})\frac{s^{*}}{q^{39}},~{}~{}~{}~{}~{}~{}~{}~{}b_{2}^{+}:=\frac{1}{p}-(1-\frac{1}{p})\frac{s^{*}}{q^{39}},$
$~{}~{}~{}~{}~{}~{}a_{3}:=1-\frac{1}{p}-\frac{p-1}{p(q-1)}s^{*}+(1-\frac{1}{p})\frac{s^{*}}{q^{39}},~{}~{}~{}~{}~{}~{}~{}~{}~{}b_{3}:=1-(1-\frac{1}{p})\frac{s^{*}}{q^{39}}.$
$I_{s}^{\pm}:~{}$$a_{1}$$b_{1}$$a_{2}^{\pm}$$b_{2}^{\pm}$$a_{3}$$b_{3}$
###### Lemma 1.
For every $s_{1}\leq s\leq\frac{q(p-1)}{p(q-1)}~{}$, the set $I_{s}^{-}$ and
for every $\frac{q(p-1)}{p(q-1)}<s\leq s_{2},~{}$ the set $I_{s}^{+}$ is a
recurrent set for maps $(**)$.
Proof. If $s_{1}\leq s\leq s_{2}$, then
$~{}~{}~{}~{}~{}~{}~{}\delta_{1}:=0.987915116...=\frac{q-1}{(p-1)(q-2)+\frac{2(p-1)(q-1)}{q^{39}}}<s^{*}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}<\frac{(p-2)(q-1)}{(p-1)-\frac{2(p-1)(q-1)}{q^{39}}}=1.00801257...=:\delta_{2}.$
Fix $s=cotg~{}\theta$ and relinquish of notation $\pm$ on our calculations.
At first, we show that for every point in interval
$J_{s}:=~{}\big{[}0-\frac{q(p-1)}{p(q-1)}(1-\frac{1}{q}+\frac{1}{q^{2}})\frac{s^{*}}{q^{38}}~{},~{}~{}\frac{1}{p^{29}}-\frac{q(p-1)}{p(q-1)}(1-\frac{1}{q})\frac{s^{*}}{q^{39}}\big{]},$
there exist suitable maps of $(**)$ that send that point to $I_{s}^{\circ}$.
We remind that $J_{s}$ is projection of the union of these 14 squares
$C_{ij}:=~{}[0,~{}\frac{1}{p^{31}}]\times[0,~{}\frac{1}{q^{40}}]+{(c_{i},c_{j})}~{}~{},~{}~{}1\leq
i,j\leq 4$
with
$(c_{i},c_{j})\in\\{c_{i}\\}^{4}_{i=1}\times\\{c_{j}\\}^{4}_{j=1}:=~{}\big{\\{}0~{},~{}(1-\frac{1}{p})\frac{1}{p^{30}}~{},~{}(1-\frac{1}{p})\frac{1}{p^{29}}~{},~{}(1-\frac{1}{p})(\frac{1}{p^{29}}+\frac{1}{p^{30}})\big{\\}}\times$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\big{\\{}0~{},~{}(1-\frac{1}{q})\frac{1}{q^{39}}~{},~{}(1-\frac{1}{q})\frac{1}{q^{38}}~{},~{}(1-\frac{1}{q})(\frac{1}{q^{38}}+\frac{1}{q^{39}})\big{\\}},$
except the squares $C_{41}$ and $C_{14}.$ Let C be one of these squares, $T$
be its correspondent operator and $\Pi_{\theta}$ be the projection map on
$\mathbb{R}^{2}$. Set $K(C):=~{}\Pi_{\theta}(C)-T^{-1}(I_{s}^{\circ})$
contains 4 components if $s\in(s_{1},~{}s_{2})$ and 2 components if $s=s_{1}$
or $s=s_{2}$.
We show that intervals $\Pi_{\theta}(C)$ overlap each other and each point of
$K(C)$ goes to $I_{s}^{\circ}$ under other operators of $(**)$. Suppose that
$T_{ij}(t)=p^{31}t+a_{ij}$ and
$T_{i^{\prime}j^{\prime}}(t)=p^{31}t+a_{i^{\prime}j^{\prime}}$ are the
corresponding operators to the squares $C_{ij}$ and
$C_{i^{\prime}j^{\prime}}$, respectively. Therefore we have
$T_{i^{\prime}j^{\prime}}(t)=T_{ij}(t)-(a_{ij}-a_{i^{\prime}j^{\prime}})$ and
we see that on special conditions point t goes to $I^{\circ}$ under the
operator $T_{i^{\prime}j^{\prime}}$, in fact,
$~{}~{}~{}~{}i)$ If
$~{}~{}~{}~{}~{}max~{}~{}\big{\\{}-\frac{1}{p}+\frac{(p-1)(q-2)}{p(q-1)}s^{*}+2(1-\frac{1}{p})\frac{s^{*}}{q^{39}}~{}~{},~{}~{}1-\frac{2}{p}-\frac{(p-1)}{p(q-1)}s^{*}+2(1-\frac{1}{p})\frac{s^{*}}{q^{39}}\big{\\}}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=max~{}~{}a_{2}^{\pm}-b_{1}<a_{ij}-a_{i^{\prime}j^{\prime}}<\frac{1}{p}+\frac{p-1}{p(q-1)}s^{*}-2(1-\frac{1}{p})\frac{s^{*}}{q^{39}}=b_{1}-a_{1},~{}~{}~{}$
then all points of last 3 components of $K(C_{ij})$ will be sent to
$I_{s}^{\circ}$ by the operator $T_{i^{\prime}j^{\prime}}$ and right side of
$\Pi_{\theta}(C_{ij})$ intersects left side of
$\Pi_{\theta}(C_{i^{\prime}j^{\prime}}),$
$~{}~{}~{}~{}ii)$ If
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}1-b_{1}=1-\frac{1}{p}+(1-\frac{1}{p})(1+\frac{1}{q^{39}})s^{*}<a_{ij}-a_{i^{\prime}j^{\prime}}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}<1+\frac{q(p-1)}{p(q-1)}s^{*}-2(1-\frac{1}{p})\frac{s^{*}}{q^{39}}=b_{3}-a_{1},$
then the last component of $K(C_{ij})$ will be sent to $I_{s}^{\circ}$ by the
operator $T_{i^{\prime}j^{\prime}}$ and right side of $\Pi_{\theta}(C_{ij})$
intersects left side of $\Pi_{\theta}(C_{i^{\prime}j^{\prime}}).$
Similar result is valid for negative cases.
We need numeral values of right and left sides of above inequality, indeed,
$~{}~{}~{}max~{}~{}a_{2}^{\pm}-b_{1}=0.008670035...,$
$~{}~{}~{}~{}~{}~{}~{}min~{}~{}b_{1}-a_{1}=0.708758125...,$
$~{}~{}~{}max~{}~{}~{}~{}1-b_{1}=1.440604766...,$
$~{}~{}~{}~{}~{}~{}~{}min~{}~{}b_{3}-a_{1}=2.134944413....$
We split interval $J_{s}$ in such cases:
Case 1. $~{}t\in K(C_{31})$, we consider the operators $T_{41}$ and $T_{42}$
as
$~{}~{}~{}~{}~{}T_{41}(t)=p^{31}t-(p-1)p^{30}(-\frac{s^{*}}{q^{38}}-\frac{s^{*}}{q{39}})=p^{31}t+\frac{p-1}{p}(q^{2}+q)s^{*},$
$~{}~{}~{}~{}~{}T_{42}(t)=p^{31}t-(p-1)p^{30}(\frac{1}{p^{30}}-\frac{s^{*}}{q^{38}}-\frac{s^{*}}{q{39}})=p^{31}t+\frac{p-1}{p}(-p+(q^{2}+q)s^{*}).$
On the other hand, the corresponding operator $C_{31}$ is
$~{}~{}~{}~{}~{}T_{31}(t)=p^{31}t-(p-1)p^{30}(-\frac{s^{*}}{q^{38}})=p^{31}t+\frac{p-1}{p}(q^{2})s^{*}$
and we obtain
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-1.925836829<a_{31}-a_{41}=\frac{p-1}{p}(-q)s^{*}<-1.887440068,$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+0.613086242<a_{31}-a_{42}=\frac{p-1}{p}(p-qs^{*})<+0.651483003,$
therefore the element t will be sent to $I_{s}^{\circ}$ by using the operators
$T_{41}$ or $T_{42}$, and interval $\Pi_{\theta}(C_{31})$ connects
$\Pi_{\theta}(C_{41})$ to $\Pi_{\theta}(C_{42}).$
Case 2. $~{}t\in K(C_{42})$, here we need the operator
$~{}~{}~{}~{}~{}T_{32}(t)=p^{31}t-(p-1)p^{30}(\frac{1}{p^{30}}-\frac{s^{*}}{q^{38}})=p^{31}t+\frac{p-1}{p}(-p+q^{2}s^{*})$
and we obtain
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{42}-a_{32}=\frac{p-1}{p}(q)s^{*}=a_{41}-a_{31},$
therefore first 3 components of $K(C_{42})$ map to $I_{s}^{\circ}$ by the
operator $T_{31}$ since
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-0.708758125...<a_{42}-a_{31}<-0.008670035...$
and the last component of $K(C_{42})$ maps to $I_{s}^{\circ}$ by using the
operator $T_{32}$ and interval $\Pi_{\theta}(C_{42})$ connects
$\Pi_{\theta}(C_{31})$ to $\Pi_{\theta}(C_{32}).$
Case 3. $~{}t\in K(C_{32})$, in this case, we consider the operator
$~{}~{}~{}~{}~{}T_{21}(t)=p^{31}t-(p-1)p^{30}(-\frac{s^{*}}{q^{39}})=p^{31}t+\frac{p-1}{p}(q)s^{*}$
and we have:
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+0.599935514<a_{32}-a_{21}=\frac{p-1}{p}(-p+(q^{2}-q)s^{*})<+0.663790257,$
therefore the operators $T_{42}$ or $T_{21}$ send the point t to
$I_{s}^{\circ}$ and interval $\Pi_{\theta}(C_{32})$ connects $\Pi(C_{42})$ to
$\Pi_{\theta}(C_{21}).$
Case 4. $~{}t\in K(C_{21})\bigcup K(C_{22})\bigcup K(C_{11})$, when $t\in
K(C_{21})$, we have $T_{11}(t)=p^{31}t$ and we see that numbers
$a_{21}-a_{32}$ and $a_{21}-a_{11}$ satisfy $(i)$ and $(ii)$ conditions above,
this says that point t goes to $I_{s}^{\circ}$ under the operators $T_{32}$ or
$T_{11}$ and interval $\Pi_{\theta}(C_{21})$ connects $\Pi_{\theta}(C_{32})$
to $\Pi(C_{11}).$ Elements $K(C_{22})$ and $K(C_{11})$ behave like (2) and (3)
cases above.
Case 5. $~{}t\in K(C_{12})\bigcup K(C_{43})$, in this case, we obtain
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{22}=\frac{p-1}{p}(-p+qs^{*}),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{12}=\frac{p-1}{p}(-p)$,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{43}=\frac{p-1}{p}(-p^{2}+(q^{2}+q)s^{*}),~{}~{}~{}~{}~{}~{}~{}a_{33}=\frac{p-1}{p}(-p^{2}+q^{2}s^{*}),$
therefore,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}a_{12}-a_{22}=\frac{p-1}{p}(-q)s^{*}=a_{33}-a_{43},$
$~{}~{}~{}-0.608256623<a_{12}-a_{43}=\frac{p-1}{p}(p^{2}-p-(q^{2}+q)s^{*})<-0.467608362,$
as above discussion, the point t goes to set $I_{s}^{\circ}$ and the
projections cover together.
Case 6. Other cases, here we use relation
$T_{ij}(t)=T_{i(j-2)}\big{(}t-(\frac{p-1}{p})\frac{1}{p^{29}}\big{)}$
that sends the point $t$ to $I_{s}^{\circ}$ since point
$t-(\frac{p-1}{p})\frac{1}{p^{29}}$ comes back to above cases.
Till now, we have shown that points of interval $J_{s}$ will be sent to
$I_{s}^{\circ}.$
Let $D$ be the union of these 14 squares and $D_{1}$ be all the squares in
$[0,~{}\frac{1}{p^{28}}]\times[0,~{}\frac{1}{q^{37}}]$, except two corner
squares
$[0,~{}\frac{1}{p^{31}}]\times[\frac{1}{q^{37}}-\frac{1}{q^{40}},~{}\frac{1}{q^{37}}]$
and
$[\frac{1}{p^{28}}-\frac{1}{p^{31}},~{}\frac{1}{p^{28}}]\times[0,~{}\frac{1}{q^{40}}]$.
We are going to pave all the squares in
$[0,~{}\frac{1}{p}]\times[0,~{}\frac{1}{q}]$ by set $D_{1}$, of course, except
two corner squares
$[0,~{}\frac{1}{p^{31}}]\times[\frac{1}{q}-\frac{1}{q^{40}},~{}\frac{1}{q}]~{}~{}~{},~{}~{}~{}[\frac{1}{p}-\frac{1}{p^{31}},~{}\frac{1}{p}]\times[0,~{}\frac{1}{q^{40}}].$
We represent this set to $\mathfrak{D}$ and we construct it by induction.
Let $D_{n}$ be the set of all the squares in
$[0,~{}\frac{1}{p^{i+1}}]\times[0,~{}\frac{1}{q^{j+1}}]$, except two corner
squares
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[0,~{}\frac{1}{p^{31}}]\times[\frac{1}{q^{j+1}}-\frac{1}{q^{40}},~{}\frac{1}{q^{j+1}}]~{}~{}~{},~{}~{}~{}[\frac{1}{p^{i+1}}-\frac{1}{p^{31}},~{}\frac{1}{p^{i+1}}]\times[0,~{}\frac{1}{q^{40}}],$
for a pair $(i,j)$ with condition $~{}1\leq i\leq 28$, $1\leq j\leq 37$ and
also $D_{n}$ supports that every line with slope $\theta=\theta_{s}$, that
passes among itself, has a conflict with a copy of $D_{1}$. Before defining
set $D_{n+1}$, as shown in Figure 3, put
$E_{n}:=~{}D_{n}+\big{(}(1-\frac{1}{p})\frac{1}{p^{i}},{0}\big{)}~{}~{}~{}~{}~{},~{}~{}~{}~{}~{}F_{n}:=~{}D_{n}+\big{(}0,(1-\frac{1}{q})\frac{1}{q^{j}}\big{)}$
Figure 3:
I) If $\Pi_{\theta}(D_{n})\bigcup\Pi_{\theta}(E_{n})$ be connected, then we
let $D_{n+1}$ be the set of all the squares in
$[0,~{}\frac{1}{p^{i}}]\times[0,~{}\frac{1}{q^{j+1}}]$, except
$[0,~{}\frac{1}{p^{31}}]\times[\frac{1}{q^{j+1}}-\frac{1}{q^{40}},~{}\frac{1}{q^{j+1}}]$
and
$[\frac{1}{p^{i}}-\frac{1}{p^{31}},~{}\frac{1}{p^{i}}]\times[0,~{}\frac{1}{q^{40}}]$
otherwise,
II) If $\Pi_{\theta}(D_{n})\bigcup\Pi_{\theta}(F_{n})$ be connected, then we
let $D_{n+1}$ be the set of all the squares in
$[0,~{}\frac{1}{p^{i+1}}]\times[0,~{}\frac{1}{q^{j}}]$, except
$[0,~{}\frac{1}{p^{31}}]\times[\frac{1}{q^{j}}-\frac{1}{q^{40}},~{}\frac{1}{q^{j}}]$
and
$[\frac{1}{p^{i+1}}-\frac{1}{p^{31}},~{}\frac{1}{p^{i+1}}]\times[0,~{}\frac{1}{q^{40}}]$.
Condition (I) is equivalent with the relation
$(1-\frac{1}{p})\frac{1}{p^{i}}-\Big{(}\frac{1}{q^{j+1}}-(1-\frac{1}{q})\frac{1}{q^{39}}\Big{)}\frac{q(p-1)}{p(q-1)}s^{*}<\frac{1}{p^{i+1}}-(1-\frac{1}{q})\frac{1}{q^{39}}\cdot\frac{q(p-1)}{p(q-1)}s^{*}\\\
\xRightarrow{\times\frac{p^{i+1}}{s^{*}}}\frac{p-2}{s^{*}}<\frac{p^{i}}{q^{j}}(\frac{p-1}{q-1})-\frac{2(p-1)p^{i}}{q^{39}}\Longrightarrow\frac{(p-2)(q-1)}{(p-1)s^{*}}+\frac{2(q-1)p^{i}}{q^{39}}<\frac{p^{i}}{q^{j}}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1)$
Condition (II) is equivalent with the relation
$0-\Big{(}\frac{1}{q^{j+1}}-(1-\frac{1}{q})\frac{1}{q^{39}}\Big{)}\frac{q(p-1)}{p(q-1)}s^{*}<\frac{1}{p^{i+1}}-(1-\frac{1}{q})(\frac{1}{q^{j}}+\frac{1}{q^{39}})\frac{q(p-1)}{p(q-1)}s^{*}\\\
\xRightarrow{\times\frac{q-1}{(p-1)s^{*}}p^{i+1}}\frac{p^{i}}{q^{j}}<\frac{q-1}{(p-1)(q-2)s^{*}}-\frac{2(q-1)p^{i}}{(q-2)q^{39}}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(2)$
At first, we show that assertion is valid for $n=1$. We can put $i=28$ and
$j=37$ in the relation (2) and then we obtain that interval
$\Pi_{\theta}\big{(}D+(0,(1-\frac{1}{q})\frac{1}{q^{37}})\big{)}$ intersects
interval $\Pi_{\theta}(D)$. This matter is similar for sets
$\Pi_{\theta}\big{(}D+((1-\frac{1}{p})\frac{1}{p^{28}},0)\big{)}$ and
$\Pi_{\theta}\big{(}D+((1-\frac{1}{p})\frac{1}{p^{28}},(1-\frac{1}{q})\frac{1}{q^{37}})\big{)}$.
The relation (1) is not valid here, and we had to use their operators. Two
squares that occur on points
$(c_{1},c_{4})=\big{(}(1-\frac{1}{p})(\frac{1}{p^{29}}+\frac{1}{p^{30}}),0\big{)}~{},~{}(c_{8},c_{5}):=~{}\big{(}(1-\frac{1}{p})\frac{1}{p^{28}},(1-\frac{1}{q})(\frac{1}{q^{37}}+\frac{1}{q^{38}}+\frac{1}{q^{39}})\big{)}$
have the following operators
$~{}~{}~{}~{}~{}T_{14}(t)=p^{31}t+\frac{p-1}{p}(-p^{2}-p),$
$~{}~{}~{}~{}~{}T_{85}(t)=p^{31}t+\frac{p-1}{p}\big{(}-p^{3}+(q^{3}+q^{2}+q)s^{*}\big{)}$
and we get
$~{}~{}~{}~{}~{}+0.025457501<a_{85}-a_{14}=\frac{p-1}{p}\big{(}-p^{3}+p^{2}+p+(q^{3}+q^{2}+q)s^{*}\big{)}<0.438403979.$
Therefore the projection of two squares $C_{84}$ and $C_{14}$ fill each
other’s middle gaps, also the square $C_{84}$ set upper than $C_{14}$ under
projection $\Pi_{\theta}$, this expresses that boundary points fall in set
$J_{s}$ or
$J_{s}+\Pi_{\theta}\big{(}(1-\frac{1}{p})\frac{1}{p^{28}},(1-\frac{1}{q})\frac{1}{q^{37}})\big{)}.$
Suppose that assertion is valid for $n$ and we check property of $D_{n+1}$.
Since $(p-2)(q-2)>1$, the relations (1) and (2) do not happen together,
therefore the process stops when
$\frac{q-1}{(p-1)(q-2)s^{*}}-\frac{2(q-1)p^{i}}{(q-2)q^{39}}-1<\frac{p^{i}}{q^{j}}-1<\frac{(p-2)(q-1)}{(p-1)s^{*}}+\frac{2(q-1)p^{i}}{q^{39}}-1.$
In fact, we show that always
$~{}~{}~{}~{}~{}~{}~{}|\frac{p^{i}}{q^{j}}-1|>max\Big{\\{}\frac{(p-2)(q-1)}{(p-1)\delta_{1}}+\frac{2(q-1)p^{i}}{q^{39}}-1~{},~{}-\big{(}\frac{q-1}{(p-1)(q-2)\delta_{2}}-\frac{2(q-1)p^{i}}{(q-2)q^{39}}-1\big{)}\Big{\\}}\\\
~{}~{}~{}~{}~{}~{}~{}=max\big{\\{}0.020343299...+\frac{2(q-1)p^{i}}{q^{39}}~{},~{}0.019937701...+\frac{2(q-1)p^{i}}{(q-2)q^{39}}\big{\\}}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(3)$
To prove the relation (3), we divide $1\leq i\leq 27$ in the following cases:
1) If $i=27$, then $\frac{p^{i}}{q^{39}}=\frac{q}{p^{4}}$ and we obtain
$~{}~{}~{}~{}~{}~{}~{}min\big{\\{}|\frac{p^{27}}{q^{j}}-1|~{}\mid~{}~{}1\leq
j\leq
37~{}\big{\\}}=~{}min\big{\\{}|\frac{p^{27}}{q^{34}}-1|~{},~{}|\frac{p^{27}}{q^{35}}-1|\big{\\}}\\\
~{}~{}~{}~{}~{}~{}~{}=~{}min\\{|\gamma^{26}-1|~{},~{}|\gamma^{-5}-1|\\}=0.146131269....$
On the other hand,
$~{}~{}~{}~{}~{}~{}\frac{2(q-1)p^{27}}{q^{39}}=\frac{2(q-1)q}{p^{4}}=0.056470184...~{}~{}~{},~{}~{}~{}\frac{2(q-1)p^{27}}{(q-2)q^{39}}=\frac{2(q-1)q}{(q-2)p^{4}}=0.085170618...,$
therefore the relation (3) happens anytime.
2) If $i=26$, then $\frac{p^{26}}{q{39}}=\frac{q}{p^{5}}$ and by
$~{}~{}~{}~{}~{}~{}~{}~{}min\big{\\{}|\frac{p^{26}}{q^{j}}-1|~{}|~{}~{}1\leq
j\leq
37~{}\big{\\}}=min\big{\\{}|\frac{p^{26}}{q^{33}}-1|~{},~{}|\frac{p^{26}}{q^{34}}-1|\big{\\}}\\\
~{}~{}~{}~{}~{}~{}=~{}min\\{|\gamma^{17}-1|~{},~{}|\gamma^{-14}-1|\\}=0.357467488...$
the relation(3) is valid.
3) If $i\leq 25$, then maximum value in the right side of the relation (3)
happens when $i=25$, thus,
$\frac{2(q-1)p^{i}}{q^{39}}=0.004508966...~{}~{}~{},~{}~{}~{}\frac{2(q-1)p^{i}}{(q-2)q^{39}}=0.006800605....$
We show that $|\frac{p^{i}}{q^{j}}-1|>0.028738306$, where $1\leq j\leq 37$.
Since $\frac{p^{i}}{q^{j}}=\gamma^{40i-31j}$ and $40i-31j\neq 0$, value
$min\big{\\{}|\frac{p^{i}}{q^{j}}-1|~{}|~{}~{}1\leq j\leq 37~{}~{},~{}~{}1\leq
i\leq 25~{}~{}\big{\\}}$
happens when $40i-31j=\pm 1$, (for example $i=7$ and $j=9$). Also, we have:
$min\\{|\gamma-1|~{},~{}|\gamma^{-1}-1|\\}=0.031101637....$
Therefore $D_{n+1}$ always exists since sets $E_{n}$ and $F_{n}$ are copies of
$D_{n}$ and the relation (3) is valid. This process continues until we
construct $\mathfrak{D}$, here is $i=1$ and $j=1$ since $(p-2)(q-2)>1$.
Therefore, every line $L_{\theta}$ that passes among the convex hull
$\mathfrak{D}$, had to meet at least one copy of set $D_{1}$ and always
projected point will be sent to $I_{s}^{\circ}$ by using one of the operators
$(**)$, like case (6) above. Also, sets $\mathfrak{D}+(1-\frac{1}{p},0),$
$\mathfrak{D}+(0,1-\frac{1}{q})$ and
$\mathfrak{D}+(1-\frac{1}{p},1-\frac{1}{q})$ behave like set $\mathfrak{D}$,
as case (6) above.
For $s=s_{1}$, $I_{s}$ is connected interval $(-s,~{}1)$, but our images are
not the points which have been obtained from the intersection of lines
$L^{3},L^{1}$ and $L^{6},L^{5}$ yet. The same situation holds for $s=s_{2}.$
Here the proof is finished. $\Box$
Lemma 2 . If $\gamma^{-1}s_{2}\leq s\leq s_{1}$ or $s_{2}\leq s\leq\gamma
s_{1}$, then interval $I_{s}=(-s,~{}1)$ is a recurrent set for the maps
$(**)$.
Proof. It is enough to show that every line with slope $\theta=\cot^{-1}s$
conflicts with one of the squares in $[0,~{}1]\times[0,~{}1]$. Like the above
lemma, we do this by induction.
Let $D_{1}:=C_{11}$ and suppose that $D_{n}$ be the set of all the squares in
$[0,~{}\frac{1}{p^{i+1}}]\times[0,~{}\frac{1}{q^{j+1}}]$, for a pair of $i,j$
and every line $L_{\theta}$ intersects one of the squares $D_{n}$. Let
$E_{n},F_{n}$ be like Lemma 1 and
I) Let $D_{n+1}$ be the set of all the squares in
$[0,~{}\frac{1}{p^{i}}]\times[0,~{}\frac{1}{q^{j+1}}]$ if
$\Pi_{\theta}(D_{n})\bigcup\Pi_{\theta}(E_{n})$ be connected otherwise,
II) Let $D_{n+1}$ be the set of all the squares in
$[0,~{}\frac{1}{p^{i+1}}]\times[0,~{}\frac{1}{q^{j}}]$ if
$\Pi_{\theta}(D_{n})\bigcup\Pi_{\theta}(F_{n})$ be connected.
If we take small sentences in the relations (1) and (2) of previous lemma and
replace $s^{*}:=~{}\frac{p(q-1)}{q(p-1)}s$, then the process stops when
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{q}{p(q-2)s}<\frac{p^{i}}{q^{j}}<\frac{q}{ps}(p-2).$
We need to show that the relation $s_{1}<\frac{p^{i}}{q^{j}}s<s_{2}$ does not
happen any time:
1) If $\gamma^{-1}s_{2}\leq s\leq s_{1}$, then
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p^{i}>q^{j}~{}\Longrightarrow~{}1<\gamma<\frac{p^{i}}{q^{j}}~{}\xRightarrow{\gamma^{-1}s_{2}<s}~{}s_{2}<\frac{p^{i}}{q^{j}}s,$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p^{i}<q^{j}~{}\Longrightarrow~{}\frac{p^{i}}{q^{j}}<\gamma^{-1}<1~{}\xRightarrow{s<s_{1}}\frac{p^{i}}{q^{j}}~{}s<\gamma^{-1}s_{1}<s_{1}.$
2) If $s_{2}\leq s\leq\gamma s_{1}$, then
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p^{i}>q^{j}~{}\Longrightarrow~{}1<\gamma<\frac{p^{i}}{q^{j}}~{}\xRightarrow{s_{2}<s}~{}s_{2}<\gamma
s_{2}<\frac{p^{i}}{q^{j}}s,$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}p^{i}<q^{j}~{}\Longrightarrow~{}\frac{p^{i}}{q^{j}}<\gamma^{-1}<1~{}\xRightarrow{s<\gamma
s_{1}}~{}\frac{p^{i}}{q^{j}}s<s_{1}.$
Thus, construction continues till $i=j=0$. This completes the proof of the
lemma. $\Box$
Now we show that $\mathcal{L}$ is a recurrent set for the operators $(*)$.
Case1. If $~{}s_{1}\leq s\leq s_{2}$, then we use Lemma 1.
Case2. If $~{}\gamma^{-1}s_{2}<s<s_{1}$ or $s_{2}<s<\gamma s_{1}$, then we use
Lemma 2 and the operators
$~{}~{}~{}~{}~{}~{}~{}~{}\pi_{2}\circ T_{0}^{\prime 40}\circ
T_{1}^{31}(t)=p^{31}t-p^{31}+1~{}~{}~{}~{},~{}~{}~{}~{}\pi_{2}\circ
T_{1}^{\prime 40}\circ T_{0}^{31}(t)=p^{31}t+(p^{31}-1)s$
of course, if it is needed.
Case3. If $~{}s=\gamma^{-1}s_{2}~{}$ or $~{}s=\gamma s_{1}$, then
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}min\big{\\{}\frac{p^{i}}{q^{j}}~{}\mid~{}~{}p^{i}>q^{j}~{}\big{\\}}=\frac{p^{7}}{q^{9}}=\gamma~{}~{}~{},~{}~{}~{}max\big{\\{}\frac{p^{i}}{q^{j}}~{}\mid~{}~{}p^{i}<q^{j}~{}\big{\\}}=\frac{p^{24}}{q^{31}}=\gamma^{-1},$
therefore point $(\gamma^{-1}s_{2},t)\in\mathcal{L}$ passes on line
$s=\frac{p^{7}}{q^{9}}\cdot\gamma^{-1}s_{2}=s_{2}$ and it had to fall in set
${s_{2}}\times(-s_{2},~{}1)$ since satisfies condition of the Lemma 2. By
using the operators $T_{0}^{\prime 40}\circ T_{1}^{31}$ and $T_{1}^{\prime
40}\circ T_{0}^{31}$ and Lemma 1, we transfer point $(s,t)$ to
$\mathcal{L}^{\circ}$.
The same situation is valid for point $(\gamma s_{1},t)\in\mathcal{L}$.
This completes the proof of the theorem. $\Box$
Replacing $K$ and $K^{\prime}$ to the forms of $C_{\alpha}$ and $C_{\beta}$,
respectively and also appropriate selection of the recurrent set in Theorem 2
yields below proposition.
###### Proposition 3.
Set $C_{\alpha}-\lambda C_{\beta}$ contains at least one interval for each
$\lambda\in\mathbb{R}^{*}$.
We close this paper by posing a question which inspired by Theorem 2 and Palis
conjecture.
###### Open Problem 1.
Does there exist an open and dense subset in the mysterious region
$\Omega:=\Big{\\{}(C_{\alpha},C_{\beta})~{}\Big{|}~{}~{}HD(C_{\alpha})+HD(C_{\beta})>1~{}~{}and~{}~{}\tau(C_{\alpha})\cdot\tau(C_{\beta})<1~{}\Big{\\}},$
such that their elements have stable intersection?
###### Acknowledgment 1.
I offer my sincerest gratitude to C. G Moreira for posing to me the problem
and his valuable comments and suggestions from the very early stage of this
research in IMPA.
## References
[HMP] B. Honary, C. G. Moreira, M. Pourbarat, Stable intersections of affine
Cantor sets, Bull Braz Math Soc, 36 (2005), no. 3, 363–378.
[M1] C. G. Moreira, Stable intersections of Cantor sets and homoclinic
bifurcations, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 6,
741–781.
[M2] C. G. Moreira, Sums of regular Cantor sets, dynamics and applications to
number theory, Periodica Mathematica Hungarica 37 (1998), no. 1-3, 55-63.
[MY] C. G. Moreira and J.-C. Yoccoz, Stable intersections of regular Cantor
sets with large Hausdorff dimension, Ann. of Math. 154 (2001), no. 1, 45–96.
[MO] P. Mendes and F. Oliveira, On the topological structure of the arithmetic
sum of two Cantor sets, Nonlinearity 7 (1994),no. 2, 329–343.
[N] S. Newhouse, Non density of Axiom A(a) on $S^{2}$, Proc. A.M.S. Symp. Pure
Math 14 (1970), 191-202.
[P1] J. Palis, A global view of dynamics and conjecture on denseness of
finitude of attractors, Astérisque No. 261 (2000), xiii–xiv, 335–347.
[P2] J. Palis, A global perspective for non-conservative dynamics, Ann. 1.H.
Poincaré AN 22 (2005), , 485-507.
[P3] M. Pourbarat, On the arithmetic difference of middle Cantor sets,
Submitted.
[PT] J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at
homoclinic bifurcations, Cambridge Univ. Press, Cambridge, 1993.
[S] A. Sannami, An example of a regular Cantor set whose difference set is a
Cantor set with positive measure, Hokkaido Math. J. 21 (1992), no. 1, 7–24.
|
arxiv-papers
| 2013-06-25T08:46:13 |
2024-09-04T02:49:46.875895
|
{
"license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/",
"authors": "M. Pourbarat",
"submitter": "Mehdi Pourbarat Dr",
"url": "https://arxiv.org/abs/1306.5879"
}
|
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